
// Make newform 825.6.a.p in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_825_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_825_6_a_p();".
// 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_825_6_a_p();".
// 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 := [-33248, 149572, -118840, -15851, 9055, 272, -176, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0], [-44, 1, 1, 0, 0, 0, 0, 0], [-44, -70, 2, 1, 0, 0, 0, 0], [55536, -36334, -7927, 1600, 164, -14, -1, 0], [99104, -85572, -13084, 7407, 176, -160, 0, 1], [89856, -146720, -15018, 10287, 344, -188, -2, 1], [-129568, 314100, 6664, -24861, 208, 504, -4, -3]];
    Rf_basisdens := [1, 1, 1, 1, 160, 320, 320, 320];
    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_825_a();"
function MakeCharacter_825_a()
    N := 825;
    order := 1;
    char_gens := [551, 727, 376];
    v := [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_825_a_Hecke(Kf)
    return MakeCharacter_825_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+1)]];
    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 := 6;
    raw_aps := [[-1, -1, 0, 0, 0, 0, 0, 0], [9, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [9, -3, -1, 0, 0, 0, -1, 0], [-121, 0, 0, 0, 0, 0, 0, 0], [-52, 42, -3, 2, 0, 1, -1, 1], [-35, -6, -5, -1, -1, -1, 10, 0], [-125, 34, 2, 3, -3, 4, -7, -3], [-739, -1, -6, 7, 3, -9, 1, -6], [126, 101, -15, 5, 7, -22, -6, -1], [-598, 44, -37, -22, 0, 9, 21, 9], [2049, 426, 141, 20, 10, 41, 45, 1], [-1613, -116, -93, 15, 3, 21, 58, -26], [-2806, -56, -121, 0, -38, -35, -5, 21], [-1013, 490, 325, -88, -2, -23, 1, 17], [-872, 438, 128, 60, 48, 106, -4, -38], [-9673, -620, -21, -68, 22, 77, 147, -27], [1624, 303, -796, 17, -23, -115, 9, 64], [-2952, 1531, 472, -199, 1, -203, -67, 40], [-2527, -1199, -362, -163, 65, 73, -165, -58], [-8060, -720, -202, -242, 22, 122, -68, -8], [-2535, -3048, 475, 175, -37, -111, 198, 126], [-12368, 129, -339, 179, 49, 86, -184, -165], [-2630, -7661, 929, -235, 11, 16, 102, 131], [-18577, 4322, 452, -30, 18, 298, 446, -208], [-5653, 3331, -644, 58, 128, -515, -226, -43], [20702, -637, 525, -433, -191, -304, 128, 235], [-43108, 6155, -1309, -557, -79, -386, -138, 107], [-29896, -3250, 1141, 644, 254, 477, -359, -349], [22936, 2632, 574, 454, 118, -558, 36, 148], [-36853, 10680, 1805, 153, -211, 215, 962, -158], [-11372, -5867, 3925, -601, -395, 266, -124, 43], [-56504, 31957, -293, 323, 133, 396, -354, -23], [-43278, -7517, 4343, -565, -511, -158, 626, 623], [-62991, -16092, 1025, 1135, -169, -89, -436, -60], [-96308, 22097, -1222, 13, -427, -67, -2645, -160], [-78498, 17843, 2028, -249, -373, -527, -577, -198], [137404, -477, 5350, 373, -39, 215, -273, -1202], [-47424, 18402, -3002, 468, 844, 1408, -814, -448], [85435, -36387, -2810, -1132, 86, 617, 1498, 539], [-163673, -700, 1241, -152, -170, 807, 4421, 59], [-53921, 4435, 2447, -647, 903, 1230, -352, -311], [-123159, 20449, -3754, 1925, -503, -735, -1101, 622], [192820, 9579, -1040, -677, -825, 613, 2855, 1554], [-59657, 29879, 5436, -754, -700, 1145, 2842, 1013], [-142232, -488, 219, 312, 298, -2907, -293, -1029], [-261288, 13089, -4494, -987, -35, 2133, 707, 1984], [400440, -20359, 5614, 2515, 1047, 457, -883, -498], [-252740, -937, -5611, 1, 223, -2232, -3834, 279], [-51329, -35965, 7079, -2723, -541, 1166, 1700, 3033], [-174861, -54464, -19167, 343, -1105, -2669, -2024, 520], [-272600, -16346, 9588, -438, 738, -2386, 2238, -696], [-344522, 50865, -6830, 5039, 483, -1843, 733, -894], [-420516, -65514, 18252, 1010, 1074, 2342, -4530, -816], [-614264, -26239, 5683, -587, -1965, -168, 2624, -1085], [-65334, -108118, -4530, -870, 814, 4368, -1184, 1786], [-139090, 84734, -790, 4582, -502, -2816, -4856, 2486], [-702363, -1648, -8893, -25, -941, -879, -306, -634], [120516, -81379, 1116, 1983, -241, -6441, -3257, 1828], [-278797, -147888, -12955, -237, -1325, 4167, 1992, 1316], [-82045, -83243, -9037, -1780, -884, -4428, -5253, -3188], [-372763, -90836, -14643, 5619, 1195, 643, -2636, -3308], [-404312, -145551, 10514, -1771, -3107, -763, 6531, -1088], [-216028, -107339, 8507, 4615, 1953, -1920, -10762, -1317], [115085, 41815, 15015, 11571, 1289, -1882, 5114, 635], [-660792, -36882, 16272, 4568, 3900, 7538, 1192, -4922], [-619292, 138986, -17198, 346, 1054, -1772, 10804, -3042], [-326488, -269569, -6410, -1211, -487, 1963, 6991, -1030], [-659464, -58088, 12676, -3718, -538, 2580, -14650, 4946], [-272322, 149955, 10147, 343, 1613, -678, -5682, 2737], [-173570, -126181, -6227, 11777, 711, 1052, -4114, -1293], [-361310, -68950, 10184, -8056, -5924, -6618, 7296, 1610], [158358, -172582, -2289, -8132, 1610, 3159, -5837, 2197], [-958146, -244045, -22138, -15925, 2035, 6463, -5395, 2384], [-1131686, 99989, -5872, -5191, -7891, -2901, -1059, -1494], [207396, -117113, 77541, -1303, -817, 8516, -5046, 345], [-205422, -53072, 648, -11450, -790, 4336, -11850, 1022], [468230, 80599, 472, 12127, 7395, -4479, 18563, -5166], [-483666, -134197, 23405, -10775, -6153, 8828, 23158, -2397], [-1051904, -1435, 6336, 6775, 3543, -7629, 13275, -4984], [943785, -163556, -11479, -1128, 9646, 7375, -14667, 3067], [860417, 236172, 52351, -9606, 1204, 4053, -9483, -1225], [-894790, 87284, -10888, 3230, 626, -7212, -4618, 7294], [352856, -127798, -45249, -2606, 364, -6557, 10781, 3665], [-467179, -97407, 10175, 1440, 5720, -7996, -39137, -184], [174639, -316452, 34757, 10372, -2950, -15349, 10725, 759], [625322, -434069, 49339, 9519, -2587, -3402, 10378, -3801], [987622, -34790, -14906, 14384, -64, 13768, -13442, 3200], [-115578, 27940, -56284, 15190, 5242, 7744, 6714, 4750], [-24610, 102165, -63785, 10471, 937, 19132, -16454, -6933], [-537604, 42942, -19336, 2438, -5738, 1194, -14378, -7868], [158224, -310084, 18604, 24502, 4842, 7568, 5042, -3718], [1681594, -58014, -163961, -6038, -1740, -8113, -10311, -2065], [-3845860, -48823, -27815, 1483, 601, -4110, 4940, -3241], [-861172, -249917, -83270, 127, -5625, 10915, 27301, 8536], [263704, -169922, 67512, 8454, -4146, 386, -8658, 12888], [3042504, 157254, 4394, 11996, -5740, -4544, 18326, 6108], [-1255912, 124003, 1043, -1331, -2193, -9834, -6552, -5431], [-1265861, 203500, -106448, 8157, 6803, -236, 8937, 251], [-1192808, 454970, -169159, -4252, 1406, 7485, -1763, -10821], [4154335, 59475, -74286, 240, 2438, 16179, 2362, 2103], [1339602, 77503, 184441, -4937, 13473, 13608, 25244, -4781], [-2846574, -216858, -58314, -6166, 4374, 22812, 6736, 994], [1714561, -483420, -71499, -3771, -1295, -10613, -45262, -7214], [1158, -158594, -216733, -14434, 3080, 12075, 2109, -339], [4465983, -641, -62891, 7691, -7259, 2808, 8904, -2643], [3085383, 231632, 96297, -16916, -470, -34693, 29609, -10205], [-6578842, -406418, -73266, -8212, 9772, -24484, -19662, 656], [2313543, 518090, -97971, 10844, 10666, 13305, -39955, 1085], [277808, -148515, 13254, 24635, -4425, 2933, -65499, 474], [6491680, 320918, 67174, -19546, -4190, 3684, -19156, -1366], [-2016708, -464768, 159114, -38476, -4440, 24838, 11726, -13738], [3163880, 620323, 19477, -18121, -10879, 5788, 3876, 11197], [-351494, -344759, -14906, -35981, -865, 26953, 6173, -17666], [-3102046, -146107, -121420, -42357, -8541, -7145, -5805, -10868], [-1330926, 666553, -38453, 6369, -6897, -104, -39308, 18383], [-3863206, -763838, 113028, -25204, -19192, -19622, 48480, 20734], [1128639, 579046, -217483, 3332, 1114, 2357, 37197, 34145], [-5193730, -636614, -76532, 31342, -20698, 1618, 11746, -1588], [1134150, 347965, -108615, 7683, -13879, -8226, 81620, -4797], [-1154633, -96576, -137695, 30386, 1768, -16281, 51503, 5637], [-1992128, -158282, -31472, 2166, -12082, -26822, 78686, 2308], [433114, -224709, 26783, 11303, -27315, -16250, -16990, -7139], [-1106307, -444470, -157783, -26190, 20536, -11015, -36529, -17529], [7143636, -332500, 206872, -3148, 7380, 36868, 38964, -7628], [741565, -194551, 154964, -109142, -18068, -42745, 33430, 22803], [343251, 313477, 309193, 3759, 15385, 36050, 38728, -11261], [508056, 832158, -150960, 95170, 11922, -2662, -22886, -24432], [12038504, 624820, -139075, 20606, 8368, -22769, -111933, -18741], [5050652, -67499, 185851, 44861, 29547, -26208, 32956, -8483], [1734135, -432192, 125217, 46183, 10519, -22397, 44840, 8672], [-7854764, -104902, 186356, 27240, -4964, 36506, 32772, -14582], [2601048, -1090975, 42321, 22407, 15269, -12714, 43068, -20355], [5275646, 881653, -302120, 9669, 10161, -47829, 36729, 13114], [2433482, 971946, -66642, 20188, 36668, 18016, -32798, -30844], [-3152572, 586297, -31050, -31079, -23263, -7735, -13829, 59704], [-3292208, 916, -71742, -62620, -19872, -32678, 5062, 34442], [9077349, -910175, 180249, -15286, 7578, 61514, 60175, 4188], [-1880278, -2962150, -224628, 41680, 18420, 4858, -34140, 5786], [-8262391, 663022, 42893, -42245, 5847, 75657, 88336, -27918], [3711401, -537447, 306733, 9150, 3358, -15746, -40729, -14036], [-3810258, -1048409, 179289, -12577, -20167, -33248, 69700, 23627], [3576516, 573229, 64898, 34403, -34537, 33713, -99071, -28698], [1802406, -50675, 396411, -26747, 38171, 34580, 105048, -30129], [-7892186, -633088, 268660, -67044, 20284, -13548, -87064, 14500], [1797128, 679310, -348192, -31650, 5094, -51078, 4390, 26628], [-8803038, -1094473, -148394, 7021, 6801, 11827, -37137, -5126], [5079673, -1086436, 426769, 50093, 2625, -3713, -46458, -19542], [-909168, -896378, 105462, 113356, 20988, 130260, 12402, -48948], [-3641004, -567423, -48887, -9545, 8893, -43290, -185916, 14373], [2458818, 1098486, -181019, 96914, -25792, -82435, 40367, 35385], [-9031198, 2439021, 146315, 41285, 30451, 108804, 81928, -16929], [7070674, -100659, -8908, -83857, -22433, -5001, 88439, 16756], [-9429556, -2126996, -419870, 69134, -26098, -56206, -82696, 34780], [3862850, -2221190, -159708, -57296, 14276, -42342, -30324, -23810], [-5155195, 3139364, 217867, -37158, -64196, -49559, -15191, 21343], [-12577031, 71065, 130269, -165628, 3448, -28338, -229303, -16446], [-13357554, 1530517, -194747, -128663, -26253, -45414, -139522, 33865], [7621038, 2412687, 54722, 5943, -15873, 105487, 118309, -24036], [-12883173, 1738470, -268553, -3489, 15391, 60359, 174906, -4452], [6953527, 717854, 44551, -31074, -37056, 48287, -272167, 285], [9415517, -1403212, -155659, -27595, 22721, 50315, 106170, 44762], [20014068, -1345428, 589312, 95184, 1160, -53972, -62816, 17496], [-15790539, 1584312, -205317, -40896, -17166, -68119, -62073, -32955], [-624270, 4186524, -384740, -53086, -4274, -33384, -270890, -7970], [8993467, 1415777, -208044, -62899, 52541, -50813, 14753, 17056], [-15874140, -17823, -241622, -16293, -42793, -87963, -88351, 50058], [-2228234, -1884208, -227356, -29300, 18788, 65244, -8520, 41164], [-23087688, 2104805, 211312, 190389, 9465, -60597, -6679, 5374], [6129956, 1208244, -387778, 43416, -72044, 26702, -154522, -7830], [-361228, 4882301, 116577, -29413, 38097, 64838, 67732, 21659], [-2609442, 2249945, 27154, -112119, 13137, -140251, -146385, 83224], [-13416098, 2939176, -371878, 50438, 58958, 130446, 124880, 41148], [-3813470, 1685667, 35276, 69131, 34015, -142287, -15293, 14934], [-17291010, -806862, 202471, -65870, -84148, -73185, 202305, 37679], [-11190834, 1314625, 370111, 196449, -42681, 21132, -57576, 5459], [-6834589, 255528, -276212, 111825, -4241, 16076, -318815, -14361], [1813878, -2134360, 39750, -4502, -39590, -66294, -192344, -52872], [18563367, 190317, -922307, 174898, -44830, 76954, -155089, -21640], [-4488462, -568090, 228673, 19068, -34506, -74775, 37797, 78947], [9251438, 5964652, -85889, -138080, 31050, 51453, 77471, 9999], [-13454318, 5009060, 913838, -72562, 16462, -13130, 22796, -59840], [12582929, 4253407, -271279, -25447, -25061, -38992, 14450, 48967], [-11541738, -502944, 278502, -188694, 1266, -11502, -63392, -16008], [2617608, 1217760, 169368, -25660, 29052, 74464, 19964, -114036], [-7559034, -133378, -20686, 90672, 24720, -94168, 277874, -30328], [-10316014, 1840392, -110699, -112418, -1112, -101197, -135165, -12125], [21850500, 1370577, -880072, 5853, 79137, -11273, -122807, -5006], [-22773282, -835269, 304092, -160841, 51043, 63785, 148623, -62294], [-35434938, 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[132222073, -19839518, -4625805, -1079170, 707112, -755241, 788749, -157247], [136955666, 28340363, -4255002, 2708121, 130237, 867431, 829931, 259390], [143490337, 30744893, -1450898, -401022, 113568, -1658967, 735088, -433943], [-28207539, 37491680, -6019909, 612992, -207942, -94079, 340239, -712287], [-193555899, -21037954, -8496120, 1419521, 731867, -1212166, -648275, 136221], [263336098, -18454819, -1974202, -286827, 170237, -2216891, 1758311, 275832], [31493186, 30187980, -10464780, 1071292, 85652, -2186568, -375264, 668604], [-56666572, -43793804, 1179302, 1324010, -400422, 297230, 541328, 38040], [-31228794, 47651515, 945196, -1436487, -438647, -153479, -811599, 89600], [166481996, -20107625, 6146093, -681219, 66963, 3015488, -116822, 101667], [-163659130, 7440588, 4488996, 900004, 1053332, -182616, -1386872, 1252], [35042618, -42586746, -9577454, 758270, -379518, -322080, -1543768, 437110], [58828938, 30425004, 4485847, 500322, 892568, -618675, -1848871, 5149], [61794259, 2809151, -822040, -849625, 242287, 629281, -379977, 1222044], [-182516208, 5498513, 748731, -837855, 612655, -823088, 4078052, -514065], [90479139, -44935055, 6109468, -328635, -470399, 215731, 1342389, -153834], [-188211388, -11207142, 1308276, -405056, 418556, -645686, -2956916, -441398], [-68320459, -37132886, -1507171, 286538, 978024, 1474069, -426685, -278049], [376372339, -32432454, -10171985, -847035, 3761, -867085, -1755868, -289158], [26974938, -29442583, -1299136, -2543669, 124531, 607607, 1666439, -226524], [-70305978, -24981698, 5425380, -711490, -305578, -1313082, 1291898, 715832]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_825_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_825_6_a_p();".
// 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_825_6_a_p(:prec:=8)
    chi := MakeCharacter_825_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 6));
    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_825_6_a_p();".
// 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_825_6_a_p( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_825_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,6,sign)));
    Vf := Kernel([<2,R![-277200, -305880, -22268, 45922, 5160, -1251, -141, 9, 1]>],Snew);
    return Vf;
end function;