
// Make newform 1160.4.a.i in Magma, downloaded from the LMFDB on 16 April 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_1160_a();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_1160_4_a_i();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_1160_4_a_i();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function ConvertToHeckeField(input: pass_field := false, Kf := [])
    if not pass_field then
        poly := [1840185, -20060541, -532152, 5124616, 230401, -416127, -17905, 13963, 418, -200, -3, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-37, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0], [-3792515400646610976, 7298747727956879979, 4082360136756612487, -1272544530253328797, -403564994907665241, 62078310152995694, 12226502021462064, -1071095438073645, -144908114222349, 5786728461593, 567988984139], [-119544934142747431305, 263683337444972311476, -45059585262624735280, -41936984901556914664, 8326742851478087739, 2036179988585218594, -397141374500116405, -37356360644386168, 6877689955408130, 229498016835954, -38628969806215], [-48498554095053070374, -161555865747495860433, 49044396786081212818, 29327980587779777089, -5610766382421086106, -1517706436781868898, 243272920427842492, 28384736743606933, -4173510398218292, -173488238985243, 23758935140794], [-104783901947205452433, 121792650398419872381, -5962499813414814791, -23936978421376104743, 3505978493334170676, 1356355625395046636, -215427386086048781, -27020531546420759, 4212095156696455, 173262108408501, -25409973951134], [7981512084816317838, -90544869860814400371, 45488259082798623632, 18366351373154122067, -6429892824151346340, -1104061701264784010, 284188059434638088, 22603623443201051, -4778015342953498, -146827469111697, 26511482308880], [25326747850990562829, -248867677595383326417, 43679344739524321021, 40853065825385776327, -6228166398292885626, -2078964934732975420, 278272647998490325, 38746804007056351, -4761037500241661, -234444356809305, 27142944470896], [-178996314039777714105, -439634828321980229874, 146234574255666949228, 72238621335437176342, -18104651930414627973, -3860375481171711994, 787432234055059447, 75561269602869790, -13330616060279534, -479702534973792, 74958689178601], [14599884230495779959, 112666215915505742418, -45168200121666336752, -21320407640937047026, 5871260628059851971, 1163845358001791018, -251282707415657289, -22665992985644454, 4164523563374922, 142710821940128, -23010207823135]];
    Rf_basisdens := [1, 1, 1, 393499598782267044, 2360997592693602264, 1180498796346801132, 1180498796346801132, 1180498796346801132, 1180498796346801132, 2360997592693602264, 393499598782267044];
    Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf);
    Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]];
    inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf);
    return Eltseq(inp_vec);
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_1160_a();"
function MakeCharacter_1160_a()
    N := 1160;
    order := 1;
    char_gens := [871, 581, 697, 321];
    v := [1, 1, 1, 1];
    // chi(gens[i]) = zeta^v[i]
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    F := CyclotomicField(order);
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),[F|F.1^e:e in v]);
    return MinimalBaseRingCharacter(chi);
end function;

function MakeCharacter_1160_a_Hecke( : Kf := Kf)
    return MakeCharacter_1160_a();
end function;


function ExtendMultiplicatively(weight, aps, character)
    prec := NextPrime(NthPrime(#aps)) - 1; // we will able to figure out a_0 ... a_prec
    primes := PrimesUpTo(prec);
    prime_powers := primes;
    assert #primes eq #aps;
    log_prec := Floor(Log(prec)/Log(2)); // prec < 2^(log_prec+1)
    F := Universe(aps);
    FXY<X, Y> := PolynomialRing(F, 2);
    // 1/(1 - a_p T + p^(weight - 1) * char(p) T^2) = 1 + a_p T + a_{p^2} T^2 + ...
    R<T> := PowerSeriesRing(FXY : Precision := log_prec + 1);
    recursion := Coefficients(1/(1 - X*T + Y*T^2));
    coeffs := [F!0: i in [1..prec]];
    coeffs[1] := 1; //a_1
    for i := 1 to #primes do
        p := primes[i];
        coeffs[p] := aps[i];
        b := p^(weight - 1) * F!character(p);
        r := 2;
        p_power := p * p;
        //deals with powers of p
        while p_power le prec do
            Append(~prime_powers, p_power);
            coeffs[p_power] := Evaluate(recursion[r + 1], [aps[i], b]);
            p_power *:= p;
            r +:= 1;
        end while;    
    end for;
    Sort(~prime_powers);
    for pp in prime_powers do
        for k := 1 to Floor(prec/pp) do
            if GCD(k, pp) eq 1 then
                coeffs[pp*k] := coeffs[pp]*coeffs[k];
            end if;
        end for;
    end for;
    return coeffs;
end function;


function qexpCoeffs()
    // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
    weight := 4;
    raw_aps := [[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [4, -1, 0, -1, 0, 0, 0, 0, 0, 0, 0], [10, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0], [3, -2, 0, -1, 0, 0, 1, 0, 0, 0, 0], [11, -4, -1, 0, 1, 0, -1, -1, 1, 0, 0], [32, 0, 0, -1, 0, 0, 0, 1, 0, 1, 0], [22, 3, 1, 0, -1, -1, 1, 0, 1, -1, 1], [-29, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [11, 4, 4, 4, -3, 3, 1, -1, 0, 0, 0], [14, -11, -3, 0, 0, 1, 0, -1, 0, -3, 1], [35, -2, -1, -2, -1, -3, -3, 2, 3, 1, -1], [55, -5, -1, 0, -6, -3, 1, 1, 1, 2, -1], [37, 3, -2, 1, 2, -3, -2, -1, 3, -4, -2], [-31, -5, -2, 12, -5, 3, 0, -2, 8, -3, 5], [83, 12, -7, 18, 4, -2, -2, 0, 5, -4, 1], [42, -22, -3, -7, -11, -3, 3, 4, 0, 3, 3], [65, 19, -4, 11, -8, 5, -3, -1, 6, 6, 8], [65, -4, -5, 3, 0, 1, 2, -1, -4, 4, -1], [-74, -24, 6, 12, 4, 5, 8, -7, 2, -5, -3], [46, 16, -14, 9, 7, -1, 4, -1, -5, 2, -2], [89, 7, 8, 20, 5, 7, 1, -8, -4, -7, 4], [135, -20, 1, 4, 17, -9, 2, -6, -10, 5, -3], [100, 37, 4, -5, -11, 0, 17, 9, 0, -6, 6], [125, -6, 26, 15, -6, 6, -4, -8, -3, -6, 2], [281, -19, 2, -25, -14, 7, -3, 13, -2, 8, -6], [374, 49, 5, -8, 12, -10, 3, -2, 0, -16, -3], [98, 10, 8, -8, -14, -8, 9, 0, -3, 10, 0], [291, -11, -3, 35, -16, 3, -3, 17, 6, 9, 15], [400, 67, -9, -11, 1, 10, -8, -11, 5, -7, 15], [295, 106, 5, -23, 13, 5, 2, -9, -2, 2, 1], [20, 65, -6, 19, 5, 18, -10, -5, 7, 14, -10], [22, 62, -17, 35, 12, -4, -19, -32, 22, -2, -21], [487, 22, 3, -14, 3, 17, -16, -2, 0, -3, -9], [-114, 124, -34, -13, -27, -2, 12, -11, 24, 5, 2], [128, 173, 25, -54, -12, -5, 9, 9, 15, -17, -7], [437, -43, -24, -79, -14, -21, 2, 17, -9, 2, 0], [727, 93, -9, 34, -6, -3, 4, -15, 10, 0, -5], [284, -28, -9, 35, 45, 5, -39, -12, 8, -21, -13], [562, 134, -28, 15, -16, -1, 12, 37, 9, 6, 4], [-387, 40, -17, -8, 1, 17, -5, -30, -7, 3, -9], [152, 330, 3, 65, 23, 10, 15, -36, 38, -30, 11], [411, -107, -11, 13, 22, -21, 10, 23, -13, -9, 9], [387, 227, -62, -18, 27, -23, 11, 8, -13, 7, 19], [451, -46, -3, -13, 18, 31, 2, -11, -56, -4, -7], [1082, 184, -8, -86, 25, -40, 36, 38, -36, -18, 0], [1241, -137, -26, 4, 20, 16, -59, -30, 45, -4, -6], [1535, -77, 76, 15, 30, -11, -11, 21, -2, 12, -40], [344, 103, -45, -66, -60, -4, -11, 38, -3, 40, -4], [-405, -126, 17, 14, -13, -1, 6, 24, 0, -7, 9], [1002, -110, -38, 97, -17, 6, -36, 13, -20, 1, 6], [13, 120, 2, -15, 52, 10, -6, 18, 13, 14, -34], [343, 104, 53, 90, -34, 29, 2, -22, -18, 13, -23], [-271, 108, 39, -42, -51, -31, -9, 40, -11, 17, -13], [441, -271, 40, -3, -28, 9, 52, 27, -5, -24, 16], [841, 115, 60, -86, 61, -7, -52, -42, -2, 11, -45], [376, -32, -64, -94, 95, -12, -38, -34, -34, -24, -40], [-485, 368, 37, 80, 17, 17, -63, -44, 27, -29, 19], [-794, -3, -107, 16, -66, 38, -4, 12, 16, 46, 62], [1003, -431, -52, 71, 20, 17, -25, -45, 16, -6, 8], [-819, -27, 36, -154, -39, 18, 3, 7, -45, 30, -6], [1390, -1, 21, 15, -71, -26, 10, 17, 113, -9, 53], [-771, 100, -65, -18, -35, -2, 12, 2, -35, 110, -5], [-2638, -508, -26, 76, 36, 52, -25, -40, -5, 38, -24], [-2060, 355, 31, -126, 12, -11, 84, 23, -10, -23, 9], [448, 262, 24, 61, -62, 46, -70, -55, 32, 27, 72], [181, 302, 101, 28, 35, 4, -10, 65, -126, 6, -6], [340, -115, 11, 84, -46, 48, 41, 90, 38, -30, 19], [-1304, -42, 38, -82, 24, 60, -10, -92, -56, 4, 0], [-3033, -330, -7, 12, 9, -33, 33, 38, -31, -81, -35], [1673, -210, 25, 114, -65, -26, -36, 42, 51, -42, -51], [335, 3, 56, -47, -186, 5, 100, 55, -41, 68, 40], [-2068, 449, 87, -120, 62, -96, -53, 48, 61, -4, -20], [2445, -94, 11, -254, 110, 37, 108, 36, -54, -33, -41], [789, -863, -15, -203, 77, 3, 60, -16, -66, -55, -23], [-353, 480, -11, 104, -15, 25, -81, -104, 45, -9, 87], [-2337, 925, -44, -2, 31, -15, -27, 40, 49, -21, -41], [-1164, -30, 187, -145, -61, -25, 83, 46, -40, 43, -7], [-339, 360, -99, 0, 107, 27, -51, -112, 53, -5, -83], [3161, 80, -24, 238, -122, -17, 13, -55, 132, 18, 112], [-2142, 560, -118, 26, 218, -92, -5, -54, -13, -64, -64], [-2640, 338, 22, 54, 86, 10, -114, -122, 38, -32, 22], [-4804, -203, -17, 116, -90, 99, -2, -5, 76, 83, 49], [4146, -190, 30, -31, -151, 6, 32, 145, -60, 113, 98], [-772, 347, 57, 196, -85, 31, 36, 54, 50, -77, 145], [1505, -72, -97, -6, -171, -51, 39, 124, -47, 97, -5], [-1611, 680, 87, 34, -143, -101, 97, 186, 19, -9, -3], [-2928, -351, 29, -6, 80, 38, -65, -48, -125, -10, -138], [1516, 33, 139, -137, 115, -96, 31, 23, -108, 33, -41], [-90, 177, -114, -110, -97, 2, -77, -11, 17, 61, 46], [-215, 612, 4, 17, 10, 125, 63, -2, 84, -67, 104], [-1501, 75, -160, 115, 201, -68, -115, 5, -161, 35, -40], [-1591, 1086, 45, 115, 33, -155, -148, -1, 140, -46, 1], [4817, -290, -55, -197, -103, -96, 98, 71, 63, -13, 17], [-1229, -223, 154, -295, -198, -33, 34, 115, 25, 50, 10], [-942, 402, 16, -320, -14, -56, 71, 68, -189, 94, -96], [-692, -675, -89, 64, 68, -40, -102, -104, 22, 76, 44], [-2832, 445, -305, 101, 65, -68, -7, 3, 110, 7, -89], [-775, 203, 178, 364, -181, 49, 107, -52, 5, 107, 155], [-1163, 469, -102, 22, 15, -65, -129, -190, 160, 35, -18], [-2168, -12, -79, -419, 271, 109, -38, -18, -83, -41, -49], [-1855, 175, 116, -46, 192, 52, -152, -178, -132, -34, -180], [1648, 1330, -140, 290, 68, -128, -59, 32, 215, -82, 0], [739, 1628, 77, 169, -49, 82, 192, 97, -121, 79, 93], [-2278, 192, -204, -20, -120, 43, -75, -45, 29, 27, 61], [-4368, -297, -337, -105, -147, -96, -63, 191, 106, 75, 71], [-4276, 290, 96, 250, 20, -6, -183, -198, 187, -140, 62], [-1123, 176, 378, 103, 216, 33, -75, -70, 44, -167, -126], [3115, -852, -45, -72, 257, -125, 1, -246, 85, -269, -175], [-5659, 111, 62, -359, -38, 125, 50, 19, -57, -136, -74], [-8456, 247, -387, 104, 214, -46, -153, -122, 93, -222, -62], [-3371, -106, -35, -284, 315, 104, 162, 9, -86, -54, 6], [4949, -522, -1, -50, -86, 127, 24, -88, 16, -103, 67], [-3839, 522, -1, 98, 101, -139, 10, 146, 104, 145, -73], [-135, -582, -179, -530, -199, -129, 9, 174, 159, -45, 33], [-991, 429, -60, 423, 2, 93, -31, -35, -78, 56, 104], [-9754, -203, -95, 100, 296, -138, 19, -184, 48, -124, 1], [-9684, 1159, 253, -368, -42, 159, 43, -57, 15, -101, -3], [6410, 216, 276, 323, 38, -20, -106, 95, -246, -89, 4], [2882, 676, -164, -42, -76, -94, 37, 292, 151, 204, 102], [-329, -378, -87, -234, 199, -127, -296, -194, 174, 99, -93], [4765, -879, -62, -64, 39, 116, 75, -209, -111, 184, 12], [-748, 1341, 23, 23, 39, -144, -119, 233, 42, -119, -57], [4240, -1262, 293, -142, 335, -4, -81, 209, -162, -83, -71], [1202, -676, -8, 230, 200, -82, -59, 192, -117, -20, -22], [6822, 400, -28, 202, -200, -26, 131, 128, -203, 108, -50], [-291, 1434, -115, 421, -166, 95, -128, 245, 34, 136, 25], [-1348, 759, -11, -37, 75, -166, -4, -27, 215, -123, 61], [-683, 1273, 380, 78, -219, 22, 284, 359, -132, -22, -4], [8708, -820, -128, 435, 90, 266, 108, -51, 90, -49, 24], [-7444, 889, 153, 289, -83, -50, 104, -19, 275, -11, 113], [-4043, 478, 329, -9, -309, 239, 326, 13, -120, 32, 153], [7625, 255, -54, 38, -27, 132, -61, -335, 15, -48, 144], [-2466, -1337, -151, -82, -46, -26, 201, 110, 17, -94, 102], [10552, 1410, -158, 14, 60, -146, 36, 190, -178, 102, 6], [-5995, 537, -138, -76, 123, 46, -24, 1, 98, -154, 56], [4335, -1071, 288, 194, -203, 53, 121, 42, 350, -243, 132], [8243, -1217, 186, -20, 69, -228, -241, 65, 293, -156, -44], [3697, 718, 41, 552, 63, 165, -12, -10, 216, -193, 31], [5589, -332, -40, -43, 195, -55, 151, -62, 44, 63, 16], [-5549, -610, -197, 530, -173, 137, 146, 186, -234, 233, 83], [-4084, 1, -45, 1081, 257, 268, -148, -357, -61, -39, -101], [4143, 1129, 619, -100, 112, 265, -153, -89, -209, 12, -209], [9405, 79, -72, 410, -43, -207, -184, 166, 418, -33, 43], [-2071, -1818, -203, -86, 64, -141, -386, 64, -80, 67, -127], [3199, -243, 532, -944, -291, -202, 228, 283, -266, 206, -156], [-1673, 1650, -123, 232, 397, -59, 149, -128, -19, 41, -173], [14561, -2046, -339, -620, -62, -13, -74, -80, -36, 129, -47], [-2941, -1379, 494, -321, 317, 148, -151, -119, -163, -201, -218], [6610, -1586, 159, -609, -161, 161, 233, 234, -452, 207, 7], [6196, -40, 124, 128, -6, -60, 165, 326, -167, -174, 52], [10439, -325, 258, -98, 137, 37, -157, -318, 46, -31, -266], [-3287, -1457, -42, -203, -64, -203, -66, 23, 139, -252, 86], [5701, -1615, -27, 422, 108, 179, -73, -299, -217, 376, -23], [-11098, 1572, 2, 402, -454, 201, 140, 184, 397, 17, 186], [12474, 118, -340, -101, 213, -204, -62, 41, -198, 319, 56], [7773, 2441, 48, 618, -23, -93, 15, 36, 201, -29, -255], [-2236, 460, -208, 784, -146, -92, -369, -70, -125, -106, -60], [2892, 1346, -242, -126, 24, -184, -42, 524, -152, 320, 104], [5126, -1975, 461, -836, 79, -281, 124, 298, -474, 383, -131], [-10452, 1248, 278, 380, -654, 122, 186, -116, 134, 318, 146], [4060, -713, -419, -599, -301, -34, -68, 339, 235, 3, 45], [2822, -1742, -196, -157, 592, -46, -184, -127, -214, -73, -268], [-987, 1494, -401, 188, -297, -369, 65, -40, 89, 99, -63], [964, -467, -277, -909, 443, -16, 214, 141, 93, 95, -93], [-3426, -1820, 108, -753, -15, -236, -178, 483, 38, -3, -16], [8766, 2567, 167, -384, -604, 113, 231, 535, -57, 305, 143], [18820, 873, 529, 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-592, -456, -518], [5988, 4355, -1520, -2073, -1791, 404, 1820, 553, -919, 1240, 692], [-17627, -9908, 481, -3586, -1571, -97, 116, 1578, 242, 385, -271], [-1814, -4364, 774, -2561, 1325, 113, -124, -1295, -2715, 1442, -1518], [24024, 6823, -199, -3607, 1985, -612, 111, 737, 718, 679, 645], [-28984, -2318, -1450, -3316, 2378, -87, -2102, -1729, 1856, -81, -639], [-32211, 2273, 666, 1775, -2366, -135, 1852, 683, -1699, 248, 210], [-7426, -11082, 320, 2540, 286, -1016, -595, -24, 413, 1774, 244], [-14120, 14149, 743, 2408, -1754, 1423, 121, -745, -599, 635, 621], [-23906, 12255, 589, 307, -411, 1210, 1664, -1957, -885, 35, 237], [-13404, -3974, 3924, -1304, -1171, 236, 2272, 1624, 632, -676, 1244], [15666, -7634, 2492, -1230, -1490, 970, 1608, -2572, 230, -498, -594], [-64695, 5130, 54, -937, -1082, 632, -346, -2, 1723, -208, 1096], [-25354, 15303, -1693, 2594, 2578, -326, -3361, -488, 710, -1342, -405], [-67319, 18524, 873, -1156, 1423, -809, 193, 1668, -2275, 1015, -1167], [-29280, 5964, 180, 3369, -1065, 1374, 1442, -725, -920, 759, 700], [-20069, 6406, 1958, 5154, -865, -205, -1011, -1275, -222, 1250, 1206]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1160_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1160_4_a_i();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
function MakeNewformModFrm_1160_4_a_i(:prec:=11)
    chi := MakeCharacter_1160_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 4));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(2999) - 1;
    while true do
        trunc_vec := Vector(Kf, [0] cat [f_vec[i]: i in [1..prec]]);
        B := Basis(S, prec + 1);
        S_basismat := Matrix([AbsEltseq(g): g in B]);
        if Rank(S_basismat) eq Min(NumberOfRows(S_basismat), NumberOfColumns(S_basismat)) then
            S_basismat := ChangeRing(S_basismat,Kf);
            f_lincom := Solution(S_basismat,trunc_vec);
            f := &+[f_lincom[i]*Basis(S)[i] : i in [1..#Basis(S)]];
            return f;
        end if;
        error if prec eq maxprec, "Unable to distinguish newform within newspace";
        prec := Min(Ceiling(1.25 * prec), maxprec);
    end while;
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_1160_4_a_i();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function MakeNewformModSym_1160_4_a_i( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1160_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<3,R![13817344, -6918192, -12091904, 2021968, 1626344, -232420, -74572, 10077, 1352, -175, -8, 1]>],Snew);
    return Vf;
end function;