/*
Generated by Allan Steel in Magma, 2010.

Field defining polynomial: x^4 - x^3 + 4*x^2 + 3*x + 9.

Traces:    [ -3, 0 ]
Density:   [ 0.0052910053, 0.97917192 ]
Numer max: [ 9, 243 ]
Numer av:  [ 0.011512836, 10.930062 ]
Denom max: [ 12, 648 ], [ [ <2, 2>, <3, 1> ], [ <2, 3>, <3, 4> ] ]
Denom LCM: [ 12, 648 ], [ [ <2, 2>, <3, 1> ], [ <2, 3>, <3, 4> ] ]

Character:
( 189, -3, -189*zeta(39)_3 - 189, 189*zeta(39)_3, 0, 0, 0, 0, 0, -3, -3, 
    -3*zeta(39)_3, 3*zeta(39)_3 + 3, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 
    -3*zeta(39)_3, 3*zeta(39)_3 + 3, -3*zeta(39)_3, 3*zeta(39)_3 + 3, 0, 0, 0, 
    0, 0, 0, zeta(39)_13^11 + zeta(39)_13^8 + zeta(39)_13^7 + zeta(39)_13^6 + 
    zeta(39)_13^5 + zeta(39)_13^2 + 1, -zeta(39)_13^11 - zeta(39)_13^8 - 
    zeta(39)_13^7 - zeta(39)_13^6 - zeta(39)_13^5 - zeta(39)_13^2, 0, 0, 
    zeta(39)_3, zeta(39)_3, -zeta(39)_3 - 1, -zeta(39)_3 - 1, 
    -zeta(39)_3*zeta(39)_13^11 - zeta(39)_3*zeta(39)_13^8 - 
    zeta(39)_3*zeta(39)_13^7 - zeta(39)_3*zeta(39)_13^6 - 
    zeta(39)_3*zeta(39)_13^5 - zeta(39)_3*zeta(39)_13^2, 
    zeta(39)_3*zeta(39)_13^11 + zeta(39)_3*zeta(39)_13^8 + 
    zeta(39)_3*zeta(39)_13^7 + zeta(39)_3*zeta(39)_13^6 + 
    zeta(39)_3*zeta(39)_13^5 + zeta(39)_3*zeta(39)_13^2 + zeta(39)_3, 
    -zeta(39)_3*zeta(39)_13^11 - zeta(39)_3*zeta(39)_13^8 - 
    zeta(39)_3*zeta(39)_13^7 - zeta(39)_3*zeta(39)_13^6 - 
    zeta(39)_3*zeta(39)_13^5 - zeta(39)_3*zeta(39)_13^2 - zeta(39)_3 - 
    zeta(39)_13^11 - zeta(39)_13^8 - zeta(39)_13^7 - zeta(39)_13^6 - 
    zeta(39)_13^5 - zeta(39)_13^2 - 1, zeta(39)_3*zeta(39)_13^11 + 
    zeta(39)_3*zeta(39)_13^8 + zeta(39)_3*zeta(39)_13^7 + 
    zeta(39)_3*zeta(39)_13^6 + zeta(39)_3*zeta(39)_13^5 + 
    zeta(39)_3*zeta(39)_13^2 + zeta(39)_13^11 + zeta(39)_13^8 + zeta(39)_13^7 + 
    zeta(39)_13^6 + zeta(39)_13^5 + zeta(39)_13^2 )
*/

G := PermutationGroup(ATLASGroup("3G23"));
M :=
GModule(G, [
    Matrix(SparseMatrix(K, 189, 189, [
    <1, 23, 1/12*(-w^3 + 4*w^2 - 4*w + 9)>, 
    <2, 24, 1/12*(w^3 - 4*w^2 + 4*w + 3)>, 
    <3, 9, -1>, 
    <4, 16, 1/12*(w^3 - 4*w^2 + 4*w + 3)>, 
    <5, 13, 1/12*(-w^3 + 4*w^2 - 4*w + 9)>, 
    <6, 15, -1>, 
    <7, 17, -1>, 
    <8, 8, -1>, 
    <9, 3, -1>, 
    <10, 25, 1/12*(w^3 - 4*w^2 + 4*w + 3)>, 
    <11, 20, -1>, 
    <12, 12, -1>, 
    <13, 5, 1/12*(w^3 - 4*w^2 + 4*w + 3)>, 
    <14, 18, 1/12*(-w^3 + 4*w^2 - 4*w + 9)>, 
    <15, 6, -1>, 
    <16, 4, 1/12*(-w^3 + 4*w^2 - 4*w + 9)>, 
    <17, 7, -1>, 
    <18, 14, 1/12*(w^3 - 4*w^2 + 4*w + 3)>, 
    <19, 22, 1/12*(w^3 - 4*w^2 + 4*w + 3)>, 
    <20, 11, -1>, 
    <21, 26, 1/12*(-w^3 + 4*w^2 - 4*w + 9)>, 
    <22, 19, 1/12*(-w^3 + 4*w^2 - 4*w + 9)>, 
    <23, 1, 1/12*(w^3 - 4*w^2 + 4*w + 3)>, 
    <24, 2, 1/12*(-w^3 + 4*w^2 - 4*w + 9)>, 
    <25, 10, 1/12*(-w^3 + 4*w^2 - 4*w + 9)>, 
    <26, 21, 1/12*(w^3 - 4*w^2 + 4*w + 3)>, 
    <27, 27, -1>, 
    <28, 50, 1/12*(-w^3 + 4*w^2 - 4*w + 9)>, 
    <29, 51, 1/12*(w^3 - 4*w^2 + 4*w + 3)>, 
    <30, 36, -1>, 
    <31, 43, 1/12*(w^3 - 4*w^2 + 4*w + 3)>, 
    <32, 40, 1/12*(-w^3 + 4*w^2 - 4*w + 9)>, 
    <33, 42, -1>, 
    <34, 44, -1>, 
    <35, 35, -1>, 
    <36, 30, -1>, 
    <37, 52, 1/12*(w^3 - 4*w^2 + 4*w + 3)>, 
    <38, 47, -1>, 
    <39, 39, -1>, 
    <40, 32, 1/12*(w^3 - 4*w^2 + 4*w + 3)>, 
    <41, 45, 1/12*(-w^3 + 4*w^2 - 4*w + 9)>, 
    <42, 33, -1>, 
    <43, 31, 1/12*(-w^3 + 4*w^2 - 4*w + 9)>, 
    <44, 34, -1>, 
    <45, 41, 1/12*(w^3 - 4*w^2 + 4*w + 3)>, 
    <46, 49, 1/12*(w^3 - 4*w^2 + 4*w + 3)>, 
    <47, 38, -1>, 
    <48, 53, 1/12*(-w^3 + 4*w^2 - 4*w + 9)>, 
    <49, 46, 1/12*(-w^3 + 4*w^2 - 4*w + 9)>, 
    <50, 28, 1/12*(w^3 - 4*w^2 + 4*w + 3)>, 
    <51, 29, 1/12*(-w^3 + 4*w^2 - 4*w + 9)>, 
    <52, 37, 1/12*(-w^3 + 4*w^2 - 4*w + 9)>, 
    <53, 48, 1/12*(w^3 - 4*w^2 + 4*w + 3)>, 
    <54, 54, -1>, 
    <55, 80, 1/12*(-w^3 + 4*w^2 - 4*w - 3)>, 
    <56, 77, 1/12*(-w^3 + 4*w^2 - 4*w - 3)>, 
    <57, 66, 1/12*(-w^3 + 4*w^2 - 4*w - 3)>, 
    <58, 58, 1>, 
    <59, 70, 1/12*(-w^3 + 4*w^2 - 4*w - 3)>, 
    <60, 71, 1>, 
    <61, 63, 1>, 
    <62, 79, 1/12*(-w^3 + 4*w^2 - 4*w - 3)>, 
    <63, 61, 1>, 
    <64, 75, 1>, 
    <65, 65, 1>, 
    <66, 57, 1/12*(w^3 - 4*w^2 + 4*w - 9)>, 
    <67, 72, 1>, 
    <68, 73, 1/12*(w^3 - 4*w^2 + 4*w - 9)>, 
    <69, 76, 1/12*(w^3 - 4*w^2 + 4*w - 9)>, 
    <70, 59, 1/12*(w^3 - 4*w^2 + 4*w - 9)>, 
    <71, 60, 1>, 
    <72, 67, 1>, 
    <73, 68, 1/12*(-w^3 + 4*w^2 - 4*w - 3)>, 
    <74, 78, 1/12*(w^3 - 4*w^2 + 4*w - 9)>, 
    <75, 64, 1>, 
    <76, 69, 1/12*(-w^3 + 4*w^2 - 4*w - 3)>, 
    <77, 56, 1/12*(w^3 - 4*w^2 + 4*w - 9)>, 
    <78, 74, 1/12*(-w^3 + 4*w^2 - 4*w - 3)>, 
    <79, 62, 1/12*(w^3 - 4*w^2 + 4*w - 9)>, 
    <80, 55, 1/12*(w^3 - 4*w^2 + 4*w - 9)>, 
    <81, 81, 1>, 
    <82, 97, -1>, 
    <83, 101, 1/12*(w^3 - 4*w^2 + 4*w + 3)>, 
    <84, 102, 1/12*(w^3 - 4*w^2 + 4*w + 3)>, 
    <85, 90, -1>, 
    <86, 107, 1/12*(w^3 - 4*w^2 + 4*w + 3)>, 
    <87, 87, -1>, 
    <88, 92, -1>, 
    <89, 106, -1>, 
    <90, 85, -1>, 
    <91, 100, 1/12*(w^3 - 4*w^2 + 4*w + 3)>, 
    <92, 88, -1>, 
    <93, 96, 1/12*(-w^3 + 4*w^2 - 4*w + 9)>, 
    <94, 94, -1>, 
    <95, 108, 1/12*(-w^3 + 4*w^2 - 4*w + 9)>, 
    <96, 93, 1/12*(w^3 - 4*w^2 + 4*w + 3)>, 
    <97, 82, -1>, 
    <98, 98, -1>, 
    <99, 104, 1/12*(-w^3 + 4*w^2 - 4*w + 9)>, 
    <100, 91, 1/12*(-w^3 + 4*w^2 - 4*w + 9)>, 
    <101, 83, 1/12*(-w^3 + 4*w^2 - 4*w + 9)>, 
    <102, 84, 1/12*(-w^3 + 4*w^2 - 4*w + 9)>, 
    <103, 105, 1>, 
    <104, 99, 1/12*(w^3 - 4*w^2 + 4*w + 3)>, 
    <105, 103, 1>, 
    <106, 89, -1>, 
    <107, 86, 1/12*(-w^3 + 4*w^2 - 4*w + 9)>, 
    <108, 95, 1/12*(w^3 - 4*w^2 + 4*w + 3)>, 
    <109, 124, -1>, 
    <110, 128, 1/12*(w^3 - 4*w^2 + 4*w + 3)>, 
    <111, 129, 1/12*(w^3 - 4*w^2 + 4*w + 3)>, 
    <112, 117, -1>, 
    <113, 134, 1/12*(w^3 - 4*w^2 + 4*w + 3)>, 
    <114, 114, -1>, 
    <115, 119, -1>, 
    <116, 133, -1>, 
    <117, 112, -1>, 
    <118, 127, 1/12*(w^3 - 4*w^2 + 4*w + 3)>, 
    <119, 115, -1>, 
    <120, 123, 1/12*(-w^3 + 4*w^2 - 4*w + 9)>, 
    <121, 121, -1>, 
    <122, 135, 1/12*(-w^3 + 4*w^2 - 4*w + 9)>, 
    <123, 120, 1/12*(w^3 - 4*w^2 + 4*w + 3)>, 
    <124, 109, -1>, 
    <125, 125, -1>, 
    <126, 131, 1/12*(-w^3 + 4*w^2 - 4*w + 9)>, 
    <127, 118, 1/12*(-w^3 + 4*w^2 - 4*w + 9)>, 
    <128, 110, 1/12*(-w^3 + 4*w^2 - 4*w + 9)>, 
    <129, 111, 1/12*(-w^3 + 4*w^2 - 4*w + 9)>, 
    <130, 132, 1>, 
    <131, 126, 1/12*(w^3 - 4*w^2 + 4*w + 3)>, 
    <132, 130, 1>, 
    <133, 116, -1>, 
    <134, 113, 1/12*(-w^3 + 4*w^2 - 4*w + 9)>, 
    <135, 122, 1/12*(w^3 - 4*w^2 + 4*w + 3)>, 
    <136, 150, 1>, 
    <137, 137, 1>, 
    <138, 153, 1/12*(-w^3 + 4*w^2 - 4*w - 3)>, 
    <139, 158, 1/12*(-w^3 + 4*w^2 - 4*w - 3)>, 
    <140, 156, 1/12*(-w^3 + 4*w^2 - 4*w - 3)>, 
    <141, 144, 1>, 
    <142, 146, 1>, 
    <143, 157, 1>, 
    <144, 141, 1>, 
    <145, 161, 1/12*(-w^3 + 4*w^2 - 4*w - 3)>, 
    <146, 142, 1>, 
    <147, 155, 1/12*(w^3 - 4*w^2 + 4*w - 9)>, 
    <148, 148, 1>, 
    <149, 151, 1/12*(w^3 - 4*w^2 + 4*w - 9)>, 
    <150, 136, 1>, 
    <151, 149, 1/12*(-w^3 + 4*w^2 - 4*w - 3)>, 
    <152, 159, -1>, 
    <153, 138, 1/12*(w^3 - 4*w^2 + 4*w - 9)>, 
    <154, 162, 1/12*(w^3 - 4*w^2 + 4*w - 9)>, 
    <155, 147, 1/12*(-w^3 + 4*w^2 - 4*w - 3)>, 
    <156, 140, 1/12*(w^3 - 4*w^2 + 4*w - 9)>, 
    <157, 143, 1>, 
    <158, 139, 1/12*(w^3 - 4*w^2 + 4*w - 9)>, 
    <159, 152, -1>, 
    <160, 160, 1>, 
    <161, 145, 1/12*(w^3 - 4*w^2 + 4*w - 9)>, 
    <162, 154, 1/12*(-w^3 + 4*w^2 - 4*w - 3)>, 
    <163, 177, 1>, 
    <164, 164, 1>, 
    <165, 180, 1/12*(-w^3 + 4*w^2 - 4*w - 3)>, 
    <166, 185, 1/12*(-w^3 + 4*w^2 - 4*w - 3)>, 
    <167, 183, 1/12*(-w^3 + 4*w^2 - 4*w - 3)>, 
    <168, 171, 1>, 
    <169, 173, 1>, 
    <170, 184, 1>, 
    <171, 168, 1>, 
    <172, 188, 1/12*(-w^3 + 4*w^2 - 4*w - 3)>, 
    <173, 169, 1>, 
    <174, 182, 1/12*(w^3 - 4*w^2 + 4*w - 9)>, 
    <175, 175, 1>, 
    <176, 178, 1/12*(w^3 - 4*w^2 + 4*w - 9)>, 
    <177, 163, 1>, 
    <178, 176, 1/12*(-w^3 + 4*w^2 - 4*w - 3)>, 
    <179, 186, -1>, 
    <180, 165, 1/12*(w^3 - 4*w^2 + 4*w - 9)>, 
    <181, 189, 1/12*(w^3 - 4*w^2 + 4*w - 9)>, 
    <182, 174, 1/12*(-w^3 + 4*w^2 - 4*w - 3)>, 
    <183, 167, 1/12*(w^3 - 4*w^2 + 4*w - 9)>, 
    <184, 170, 1>, 
    <185, 166, 1/12*(w^3 - 4*w^2 + 4*w - 9)>, 
    <186, 179, -1>, 
    <187, 187, 1>, 
    <188, 172, 1/12*(w^3 - 4*w^2 + 4*w - 9)>, 
    <189, 181, 1/12*(-w^3 + 4*w^2 - 4*w - 3)>
    ])),
    Matrix(K, 189, 189,
        [ [ 49/216, -2/81, 2/81, 5/648 ], [ -7/72, 1/54, -1/54, 1/216 ], [ 1/27,
        1/18, -1/54, 1/54 ], [ 1/72, -19/162, 5/81, -13/648 ], [ -31/216, -1/18,
        1/27, -5/216 ], [ 4/27, 1/27, 1/54, 1/54 ], [ -5/72, -7/162, 1/162, 
        -7/648 ], [ -11/54, 5/162, -5/162, -1/162 ], [ -5/54, -11/162, -1/162, 
        -1/81 ], [ -1/54, -1/27, 1/54, -1/54 ], [ -1/72, -5/81, 7/162, -7/648 ],
        [ -7/108, 11/162, -1/81, 1/324 ], [ -5/72, -1/162, -1/81, -7/648 ], [ 
        2/27, -7/162, 1/162, -1/162 ], [ -1/216, 0, -1/54, 1/216 ], [ -11/216, 
        8/81, -7/162, 13/648 ], [ 1/72, -8/81, 2/81, -13/648 ], [ -19/108, 
        -1/162, 1/162, -5/324 ], [ 7/72, -1/162, 1/162, 5/648 ], [ -5/216, 
        7/162, -2/81, 7/648 ], [ -7/72, 17/162, -11/162, 11/648 ], [ 1/36, 1/81,
        1/162, 1/324 ], [ 1/216, -1/162, 2/81, -7/648 ], [ 7/36, 1/18, 0, 1/36 
        ], [ 7/108, -1/27, 0, -1/108 ], [ 1/18, -7/162, 2/81, -1/162 ], [ -1/8, 
        1/54, 0, -1/216 ], [ 0, -13/81, 11/162, -5/162 ], [ 17/216, 5/162, 
        1/162, 5/648 ], [ -5/216, -1/81, 1/81, -5/648 ], [ -1/6, -2/81, 2/81, 
        -2/81 ], [ -7/54, 1/6, -2/27, 1/54 ], [ -1/216, -1/18, 1/54, -1/72 ], [ 
        0, 4/81, -5/162, 1/81 ], [ 1/72, 11/81, -5/81, 19/648 ], [ -5/54, 5/81, 
        -2/81, 1/162 ], [ 1/108, 1/27, -1/54, 1/108 ], [ -17/108, 1/54, 0, 
        -1/108 ], [ -1/72, 2/81, -1/162, 1/648 ], [ 17/108, -1/81, -1/162, 5/324
        ], [ -13/216, -1/162, -1/81, -1/648 ], [ 19/216, 1/162, -1/162, 7/648 ],
        [ 1/9, 10/81, -4/81, 5/162 ], [ -23/216, -5/162, 1/81, -11/648 ], [ 
        -5/36, 11/81, -5/81, 5/324 ], [ 1/108, -1/18, 1/27, -1/108 ], [ 5/216, 
        -1/27, 1/54, -1/216 ], [ 35/216, 1/27, -1/54, 5/216 ], [ 0, -1/162, 
        1/162, -1/162 ], [ -7/108, 0, 0, -1/108 ], [ 23/216, -11/81, 5/81, 
        -13/648 ], [ -1/27, -7/81, 5/162, -1/81 ], [ -7/108, 1/162, -1/162, 
        -1/324 ], [ -1/72, 2/81, -1/162, 1/648 ], [ -1/54, -1/18, 2/81, -1/81 ],
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        -5/162, 1/54, -1/162 ], [ -1/216, 1/81, -1/162, 1/216 ], [ -1/54, 1/81, 
        -1/162, 0 ], [ 1/108, 0, 1/162, -1/324 ], [ 1/27, -2/81, 1/81, 0 ], [ 
        13/216, -1/54, 2/81, 1/648 ], [ -1/54, -1/162, -1/162, 0 ], [ 1/27, 
        -5/162, 1/54, -1/162 ], [ 1/108, 5/162, -1/162, 1/108 ], [ -5/72, 1/81, 
        -1/162, -1/216 ], [ -1/72, 1/54, -1/81, 1/648 ], [ -7/108, -4/81, 1/54, 
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        -1/216, 1/81, -1/162, 1/216 ], [ 1/216, -1/162, 1/81, -1/216 ], [ 1/216,
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        1/81, -1/108 ], [ 1/27, -4/81, 1/54, -1/162 ], [ 1/12, 1/54, -1/81, 
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        7/162, -2/81 ], [ -1/6, -1/27, 0, -1/54 ], [ -5/36, 1/18, -1/54, 1/108 
        ], [ -1/24, 1/54, -1/54, 1/216 ], [ -1/27, -2/81, -1/81, -1/162 ], [ 
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        2/81, -2/81, 1/162 ], [ 5/216, 1/162, 1/81, -11/648 ], [ 7/108, -5/162, 
        1/81, -1/324 ], [ -1/6, 1/54, -1/27, 0 ], [ 13/216, -5/81, 2/81, -7/648 
        ], [ -23/216, -7/162, 1/162, -7/648 ], [ -11/108, -17/162, 5/162, -7/324
        ], [ -1/24, -1/27, 0, -1/72 ], [ -13/216, 13/162, -7/162, 7/648 ], [ 
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        -7/162, 7/648 ], [ -1/24, 7/54, -1/18, 5/216 ], [ -1/36, 5/54, -1/27, 
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        1/54, 1/216 ], [ 1/9, -2/27, 1/54, 0 ], [ -13/108, -2/81, 1/162, -5/324 
        ], [ 1/9, 2/27, -1/54, 1/54 ], [ -7/72, -1/27, 1/54, -1/72 ], [ -1/8, 
        1/27, -1/54, -1/216 ], [ -35/216, 1/81, 1/162, -7/648 ], [ 1/72, 0, 0, 
        1/216 ], [ 25/216, -1/162, 1/162, 5/648 ], [ 41/216, 1/162, -1/162, 
        13/648 ], [ 1/9, -7/54, 1/18, -1/54 ], [ 1/216, -7/162, 2/81, -7/648 ], 
        [ -2/27, -1/81, 1/81, -1/81 ], [ 1/9, -2/27, 1/27, 0 ], [ -5/72, -5/54, 
        1/18, -5/216 ], [ -1/12, -1/108, 1/108, -1/108 ], [ 0, -1/27, 1/54, 0 ],
        [ 11/108, 7/324, -1/324, 1/324 ], [ -2/27, -19/324, 5/162, -7/324 ], [ 
        -1/108, 1/324, 2/81, -1/162 ], [ 5/108, 31/324, -5/162, 1/81 ], [ -2/27,
        2/81, -2/81, 1/162 ], [ 1/216, -11/324, -1/324, -7/648 ], [ -11/108, 
        2/81, -5/324, -1/81 ], [ 1/18, 0, 0, 0 ], [ -5/216, 1/81, -7/324, 5/648 
        ], [ 5/216, -4/81, 5/162, -11/648 ], [ 1/36, -1/36, 5/108, -1/108 ], [ 
        -2/27, -1/324, -1/162, -1/324 ], [ 0, 1/27, -1/108, 1/108 ], [ 5/108, 
        -7/162, 11/324, -1/162 ], [ -1/216, 7/162, -11/324, 13/648 ], [ 1/108, 
        5/81, -5/324, 1/162 ], [ 1/24, 1/54, 1/108, 1/216 ], [ 1/216, -2/81, 
        5/324, -1/648 ], [ -1/54, 1/162, 1/324, -1/324 ], [ 7/108, -11/162, 
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]: Check := false) where w := K.1 where K := ext<K|Polynomial(K, [9, 3, 4, -1, 1])> where K is 
RationalField();
K<w> := BaseRing(M);
K;
M;
