
// Make newform 74.7.d.a in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_74_d();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_74_d_Hecke();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_74_7_d_a();".
// 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_74_7_d_a();".
// 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 := [657611720884512028944, 0, 1596692663909213634249, 0, 117348928221103362406, 0, 2140498199687229457, 0, 13599755691108102, 0, 36580026955844, 0, 45282701078, 0, 28007049, 0, 8470, 0, 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, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-4921257925665401457031200541511071743997455234288, 0, -3081517034507571101947622490876940133441468954663, 0, -94105867536987384846618673287945534887203487651, 0, -732621727526351353644660282046547033780474844, 0, -2172571750214881426241678230870620738383902, 0, -2813208893823245560689659957694343457278, 0, -1782904871642358507069246048037609740, 0, -547367128967772941778063219864587, 0, -65285699492967695899218651215, 0], [1633125795135048414465100431098510319482200469917980671068114279686521808, 0, -52095813071052844428549392336737810652476674992618056715599916427924219, 0, -1498386339266296726160364454351585038002217681497706300933601120330087, 0, -11047964021213478227148339225851227979366777652303872385210323758508, 0, -31681051148233252201892024276766053612293602088401704703208409734, 0, -40334124140132505852837162445428856288037349989710700290435366, 0, -25311713961335296499371751303345949176877786338892197064860, 0, -7720668101189028822494174764546492601539366142586495247, 0, -916568700438725324687524559031340391652541305590307, 0], [-1289088224412522983852486429573357593898066889459823459328056829403740688, 0, -151905390365747048264821725723263438706401852435742856790714689554233481, 0, -2824226146610586774172062249909434583332632151607874802047985298027693, 0, -18074663145749591052719549522194009873543513702817235446749118755172, 0, -48943735246617930054231013143695657176802046120045577274484977986, 0, -60819167721182684903796119640959097283938041617535674989072674, 0, -37700592569476560958876781076059701467890772442457797026740, 0, -11417793439955877119926139372888073094256998135925107813, 0, -1349338994906724280647116335674595675839716743079873, 0], [-148048623754745344110623587708522058212926469113000661366844220286688304, 0, -139282987232275641816474248133233601501231207248737106211400583345935323, 0, 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50187143570894154247111112844191380292641698351736373391384844054033746, 0, 30984673826790253670209918925285857047800877145793788435018735383380, 0, 9360497668394074044984475855666258611227162903688081608659348237, 0, 1104367495117601163513208720365954094540198758015594326278297], [7480056852284856735876454166497833751683935578742437152141770555246188455266500176, -11261893910518330260560553111152793950491435483887109546804304306757758047104475728, 835392419674039218762029109732238129423087176635639691041167594445457575861308077, -779059399758016997804110478977815948637320777096266774231701356051767385825792141, 16114492051504777269701731027668684674993154953938031093097480853561724165085601, -13825916921024805765429100999597202401031483936518725808091246260683828199974273, 105088771211707650965525286945183589273826490226821136899454272393482199420404, -86868631972227027952438469202522817184241310748594957226921156527343086418292, 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    Rf_basisdens := [1, 1, 5565579766171016398223675302671571569699942349824, 2135585458765051194563149359271119191497844212508014208038325120424960, 2135585458765051194563149359271119191497844212508014208038325120424960, 533896364691262798640787339817779797874461053127003552009581280106240, 35680850083596562713583977672071521969258970524668948636672, 480393227699763359241647159567754804528207782546403082717504800272986795591680, 480393227699763359241647159567754804528207782546403082717504800272986795591680, 2281867831573875956397824007946835321508986967095414642908147801296687279060480, 2281867831573875956397824007946835321508986967095414642908147801296687279060480, 9127471326295503825591296031787341286035947868381658571632591205186749116241920, 9127471326295503825591296031787341286035947868381658571632591205186749116241920, 12009830692494083981041178989193870113205194563660077067937620006824669889792, 12009830692494083981041178989193870113205194563660077067937620006824669889792, 2281867831573875956397824007946835321508986967095414642908147801296687279060480, 2281867831573875956397824007946835321508986967095414642908147801296687279060480, 4563735663147751912795648015893670643017973934190829285816295602593374558120960];
    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_74_d();"
function MakeCharacter_74_d()
    N := 74;
    order := 4;
    char_gens := [39];
    v := [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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_74_d_Hecke();"
function MakeCharacter_74_d_Hecke(Kf)
    N := 74;
    order := 4;
    char_gens := [39];
    char_values := [[0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := true, Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
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 := 7;
    raw_aps := [[4, 0, 0, 0, 0, 0, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [16, 0, 0, 0, 0, 0, 16, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-7, 0, 3, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 124, -1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0], [-375, -2, -2, 0, -1, 0, -375, -2, 0, 0, -1, -1, 0, 0, -2, 1, 0, 0], [-519, 23, 23, 1, -3, 0, -519, 1, 0, 0, -3, -8, 0, 0, 1, 1, 0, 1], [429, -31, -31, 0, -3, -2, 429, 17, 0, 2, -3, -4, 0, 0, 0, -1, 0, 0], [-2782, -40, -40, 3, -3, 2, -2782, -19, 0, -2, -3, -7, 0, 0, -11, -12, 0, 3], [-2277, 272, -272, 5, 16, -2, 2277, 0, 41, -2, -16, 0, 23, -3, 0, 0, 4, -5], [-1019, -82, 82, 4, -1, 6, 1019, 0, 66, 6, 1, 0, -8, 3, 0, 0, 9, -4], [-13366, 58, 170, 21, -34, -9, -3347, 104, 42, -20, 18, 21, -52, 8, -11, 0, 0, 15], [0, 1152, 0, 0, 0, 0, -2521, -60, 60, 5, 44, -7, 7, 2, 2, 20, 20, -9], [-3635, 137, 137, 9, -14, -11, -3635, -129, 0, 11, -14, 52, 0, 0, -18, -41, 0, 9], [12623, 0, 275, -8, 143, 24, 0, 232, 232, 0, 0, 17, 17, -32, 32, 1, -1, 0], [3240, 0, -1236, -25, 51, 47, 0, 114, 114, 0, 0, 63, 63, -48, 48, 47, -47, 0], [-9978, -339, -339, 58, 57, 96, -9978, 132, 0, -96, 57, 104, 0, 0, -118, 10, 0, 58], [28457, 7, -7, -43, -216, 46, -28457, 0, -374, 46, 216, 0, -102, -13, 0, 0, 75, 43], [0, 1214, 0, 0, 0, 0, -79749, -204, 204, -71, 193, 84, -84, 123, 123, -44, -44, 262], [-10431, 0, -173, 146, -15, 70, 0, -1052, -1052, 0, 0, -20, -20, -156, 156, -19, 19, 0], [0, -1188, 0, 0, 0, 0, 1027, 1190, -1190, -88, 102, 436, -436, -14, -14, 67, 67, -343], [95299, 3977, 3977, -217, 47, 115, 95299, 1394, 0, -115, 47, 91, 0, 0, 193, -167, 0, -217], [129491, 0, 7245, -618, -845, -162, 0, -1484, -1484, 0, 0, 124, 124, -66, 66, 229, -229, 0], [63521, 854, -854, -192, 44, 77, -63521, 0, 1847, 77, -44, 0, -772, -219, 0, 0, -27, 192], [-91889, 397, 397, 298, -430, -267, -91889, -3616, 0, 267, -430, 94, 0, 0, 126, 236, 0, 298], [0, -2616, 0, 0, 0, 0, 30536, 64, -64, 43, 553, -348, 348, 270, 270, -119, -119, 425], [-114104, -8119, 8119, 438, 551, 77, 114104, 0, 1063, 77, -551, 0, 872, -249, 0, 0, 692, -438], [-357099, 0, -2712, 300, 867, -136, 0, 779, 779, 0, 0, 72, 72, 707, -707, -350, 350, 0], [-336645, 8868, 8868, -54, -874, 326, -336645, -1065, 0, -326, -874, -1400, 0, 0, -493, -453, 0, -54], [270925, -3731, 3731, 687, -1787, -442, -270925, 0, 3440, -442, 1787, 0, -608, 272, 0, 0, -24, -687], [-398381, 0, 11351, 2064, -21, -104, 0, 2881, 2881, 0, 0, -277, -277, -405, 405, -334, 334, 0], [-383427, 6032, -6032, 1179, 1566, 740, 383427, 0, 6103, 740, -1566, 0, -1340, 1122, 0, 0, 195, -1179], [-115355, 0, -15316, -644, -5259, -931, 0, -7099, -7099, 0, 0, -1728, -1728, -729, 729, 10, -10, 0], [0, -47926, 0, 0, 0, 0, 82121, 7523, -7523, -50, -735, -72, 72, -415, -415, -150, -150, 1368], [793192, 0, 10212, -1317, 4081, -181, 0, -8347, -8347, 0, 0, -1688, -1688, 371, -371, -117, 117, 0], [0, -65176, 0, 0, 0, 0, -408448, 4793, -4793, 18, -576, 1817, -1817, 469, 469, -317, -317, 1434], [927262, 0, -28568, -3443, 2029, 801, 0, -4547, -4547, 0, 0, 1016, 1016, -99, 99, -963, 963, 0], [156160, 46536, -46536, -200, -1820, 215, -156160, 0, -12979, 215, 1820, 0, -2314, 680, 0, 0, -1431, 200], [856630, -43081, -43081, -527, -7668, -338, 856630, 11205, 0, 338, -7668, -784, 0, 0, 743, -1512, 0, -527], [0, -13670, 0, 0, 0, 0, 1299700, -2321, 2321, -1153, -7459, 659, -659, 1392, 1392, 663, 663, 1505], [-612767, 72976, -72976, 470, -1244, -697, 612767, 0, -14332, -697, 1244, 0, 2848, -1676, 0, 0, 1692, -470], [644044, 0, 21328, -1095, 9339, -1243, 0, -6595, -6595, 0, 0, 2426, 2426, -4791, 4791, 1186, -1186, 0], [-1381098, -17653, -17653, -139, 9755, 475, -1381098, 32540, 0, -475, 9755, 2266, 0, 0, -3713, -1101, 0, -139], [-1309467, 32258, 32258, 689, 2673, 3869, -1309467, -6079, 0, -3869, 2673, 4714, 0, 0, -1795, -1441, 0, 689], [-2525482, 0, -144492, 1, 1613, 3303, 0, 17604, 17604, 0, 0, -1635, -1635, 1277, -1277, 825, -825, 0], [3119239, 82814, -82814, -1066, 682, 1927, -3119239, 0, 42842, 1927, -682, 0, 3268, 1974, 0, 0, 1662, 1066], [-1493324, 0, -11769, -680, -9182, 3535, 0, -15643, -15643, 0, 0, 2086, 2086, 1506, -1506, 388, -388, 0], [1266225, 0, -15455, 1346, 24777, 3222, 0, -3553, -3553, 0, 0, 4455, 4455, -2470, 2470, -1667, 1667, 0], [2167599, 8490, 8490, 2619, -8759, -3239, 2167599, 9722, 0, 3239, -8759, -14066, 0, 0, -3776, 4702, 0, 2619], [2714380, 0, 22240, -2289, -869, -1901, 0, -69453, -69453, 0, 0, 2241, 2241, 2225, -2225, -2333, 2333, 0], [0, -81972, 0, 0, 0, 0, 79581, -38330, 38330, 1818, -15565, -475, 475, 1966, 1966, 228, 228, -3116], [-1145977, -134992, -134992, 1189, -5198, -3729, -1145977, -1534, 0, 3729, -5198, -8256, 0, 0, -1007, 2603, 0, 1189], [2832689, -135337, -135337, -4335, 5687, 5874, 2832689, 34222, 0, -5874, 5687, 174, 0, 0, -1684, 3514, 0, -4335], [-1092348, -22310, 22310, -38, 17536, 2644, 1092348, 0, 102201, 2644, -17536, 0, 17620, -6290, 0, 0, -891, 38], [-2190365, 49023, 49023, 11074, 8488, 897, -2190365, 33852, 0, -897, 8488, -4014, 0, 0, -3616, -9414, 0, 11074], [0, 37299, 0, 0, 0, 0, -9664873, -21657, 21657, 3676, -13769, -6076, 6076, 2334, 2334, 457, 457, -6598], [2256338, 0, 150478, 16672, 19850, -3022, 0, -1639, -1639, 0, 0, -1166, -1166, -2416, 2416, -5084, 5084, 0], [-2984765, 0, 334821, -3422, -12117, 7442, 0, -21954, -21954, 0, 0, 14553, 14553, 6619, -6619, -5667, 5667, 0], [6767160, 48178, -48178, 2412, 458, -1897, -6767160, 0, -66154, -1897, -458, 0, -26734, -3949, 0, 0, 2319, -2412], [-1796241, 448198, 448198, -6423, 563, -1746, -1796241, 71194, 0, 1746, 563, 2038, 0, 0, -3230, 7498, 0, -6423], [-7047587, 335123, -335123, -3102, 12162, 2390, 7047587, 0, -69258, 2390, -12162, 0, -5264, -6018, 0, 0, -312, 3102], [-225970, 0, -438592, 1752, -45882, 3434, 0, 1661, 1661, 0, 0, -14227, -14227, 3319, -3319, -4164, 4164, 0], [0, -601981, 0, 0, 0, 0, 3117588, -8231, 8231, -5572, -25520, 10532, -10532, -5599, -5599, 2221, 2221, -4540], [-6108074, 120814, -120814, 4774, -579, 568, 6108074, 0, 5026, 568, 579, 0, 10831, -3401, 0, 0, -19439, -4774], [4656521, -44329, -44329, -5848, -1676, 9963, 4656521, -13745, 0, -9963, -1676, 20908, 0, 0, 651, 1423, 0, -5848], [0, 243752, 0, 0, 0, 0, 668364, -51118, 51118, 5764, -25544, -12814, 12814, 15024, 15024, 13071, 13071, 5836], [-2858207, 185839, 185839, 5937, 41843, 361, -2858207, -45542, 0, -361, 41843, 32062, 0, 0, -8158, 18816, 0, 5937], [0, 553032, 0, 0, 0, 0, -21093957, -44771, 44771, -4844, 13212, 37290, -37290, -3298, -3298, -6516, -6516, 4237], [12569157, -76726, 76726, 14446, -56460, 5970, -12569157, 0, 5536, 5970, 56460, 0, -34936, 28326, 0, 0, 11720, -14446], [-1375640, 0, -9300, -9412, -69916, -13200, 0, 11193, 11193, 0, 0, -15537, -15537, -13145, 13145, 18914, -18914, 0], [-4667916, -312393, 312393, 12547, -23816, 747, 4667916, 0, -29991, 747, 23816, 0, -24016, -5621, 0, 0, -8819, -12547], [7130695, 0, 361117, 12714, -17211, -13368, 0, 121122, 121122, 0, 0, -30727, -30727, -914, 914, 4449, -4449, 0], [5207892, 0, -1074342, -24564, -31550, -15380, 0, 26096, 26096, 0, 0, 28020, 28020, -20558, 20558, 16302, -16302, 0], [0, 251368, 0, 0, 0, 0, 14369676, 156330, -156330, 8255, 99201, 12712, -12712, 9039, 9039, 14052, 14052, 5075], [-6972630, 0, 920031, 2300, 46112, 5566, 0, 52618, 52618, 0, 0, -10508, -10508, 11372, -11372, -8206, 8206, 0], [11039199, -584190, -584190, 7569, -5969, -17708, 11039199, -67894, 0, 17708, -5969, 1966, 0, 0, -12206, 4096, 0, 7569], [-10347422, -638401, -638401, 1913, -30729, -15396, -10347422, -378035, 0, 15396, -30729, -72403, 0, 0, -18471, -16864, 0, 1913], [0, 1316556, 0, 0, 0, 0, -9564728, 59229, -59229, 429, -40853, -24211, 24211, 445, 445, -19885, -19885, 28477], [-1761034, 500967, -500967, 15928, -211, 2625, 1761034, 0, 360441, 2625, 211, 0, 30952, -48761, 0, 0, 1939, -15928], [7216791, -336185, 336185, 27234, 87824, 11274, -7216791, 0, 127346, 11274, -87824, 0, 60208, 44326, 0, 0, -1298, -27234], [-34097678, 0, 1455359, 7116, 9282, -23144, 0, -163300, -163300, 0, 0, -15182, -15182, -447, 447, -2944, 2944, 0], [19696280, 770699, -770699, 21630, 17324, 8896, -19696280, 0, -385942, 8896, -17324, 0, 67299, -11018, 0, 0, -9559, -21630], [-5546364, 1538557, -1538557, -22689, 30317, 4481, 5546364, 0, 45, 4481, -30317, 0, 10980, 1202, 0, 0, -7753, 22689], [23027349, 0, 1690038, -11421, -104814, 30559, 0, 4481, 4481, 0, 0, -23497, -23497, 27226, -27226, 1633, -1633, 0], [-13075552, -567704, 567704, -50486, 88710, 3353, 13075552, 0, 67435, 3353, -88710, 0, 68249, -17847, 0, 0, -40528, 50486], [0, -249765, 0, 0, 0, 0, -16058924, -74453, 74453, -55113, 32698, 24212, -24212, 5107, 5107, -8572, -8572, -42848], [11295557, -797848, -797848, -49866, 37046, -12316, 11295557, 102007, 0, 12316, 37046, 58634, 0, 0, -5777, 45139, 0, -49866], [3377298, 153858, 153858, 30375, 79732, 31055, 3377298, 231355, 0, -31055, 79732, 35100, 0, 0, -27739, -19579, 0, 30375], [10496114, 896833, 896833, -13284, -44353, 12102, 10496114, -150531, 0, -12102, -44353, -49202, 0, 0, 53179, 47593, 0, -13284], [-14830071, -430163, -430163, 7278, 5319, 22647, -14830071, -542561, 0, -22647, 5319, 51877, 0, 0, 48823, 23316, 0, 7278], [2896657, -533981, -533981, 1199, -59661, 10632, 2896657, -168278, 0, -10632, -59661, 38254, 0, 0, 26918, 36550, 0, 1199], [20242877, 820133, 820133, -61720, -14909, 32967, 20242877, -221101, 0, -32967, -14909, 19237, 0, 0, -25347, 2896, 0, -61720], [173343, -1059648, -1059648, 4069, 20348, 40521, 173343, 72863, 0, -40521, 20348, 25626, 0, 0, 35336, -20341, 0, 4069], [56957413, 0, 1590926, -25926, -22043, 26640, 0, -329051, -329051, 0, 0, -19292, -19292, -4603, 4603, -26308, 26308, 0], [-14996077, 719962, -719962, -17091, -81845, 5794, 14996077, 0, 256754, 5794, 81845, 0, -21858, 64206, 0, 0, 25900, 17091], [15054184, 959449, 959449, 1105, 88116, -9534, 15054184, 6123, 0, 9534, 88116, 103494, 0, 0, 78801, -39326, 0, 1105], [0, -938604, 0, 0, 0, 0, -59546728, 287527, -287527, -469, 27917, -29848, 29848, 60007, 60007, -7965, -7965, -25795], [0, -231294, 0, 0, 0, 0, -8005684, 131799, -131799, 11285, -247873, -89468, 89468, -53646, -53646, -29199, -29199, 43369], [-34409344, -1893566, -1893566, -42860, 116084, 490, -34409344, -220519, 0, -490, 116084, 4910, 0, 0, -19754, -22485, 0, -42860], [-25848900, -1317154, -1317154, 44684, 29426, -17039, -25848900, 1250174, 0, 17039, 29426, -94294, 0, 0, 2557, -36603, 0, 44684], [-103781835, 955946, -955946, 9045, 10727, 9933, 103781835, 0, 448016, 9933, -10727, 0, 6368, -56890, 0, 0, -43346, -9045], [-8340220, -378710, 378710, 61021, -132551, 36226, 8340220, 0, -274064, 36226, 132551, 0, -53556, 25501, 0, 0, 18099, -61021], [-55761035, -1263001, -1263001, -64549, 139582, -44812, -55761035, 126449, 0, 44812, 139582, -99948, 0, 0, 12368, -36209, 0, -64549], [62452108, 1687019, -1687019, 65978, -59597, 4140, -62452108, 0, 544018, 4140, 59597, 0, -37586, 29854, 0, 0, 41314, -65978], [-24627416, 0, -1590709, -11484, 273680, -78334, 0, 183079, 183079, 0, 0, 15950, 15950, 24010, -24010, 62339, -62339, 0], [-47088411, 2141698, 2141698, 5188, -90110, -51357, -47088411, 115589, 0, 51357, -90110, -137892, 0, 0, -101145, 35977, 0, 5188], [-35768217, 351821, -351821, 61730, -123841, -82472, 35768217, 0, 176873, -82472, 123841, 0, -145098, 25044, 0, 0, 36859, -61730], [-27918945, 0, 1316318, -4395, 332138, 1728, 0, 113097, 113097, 0, 0, -80626, -80626, -820, 820, -44852, 44852, 0], [-104524447, 0, -2313763, -105250, 260203, -41244, 0, -7350, -7350, 0, 0, -7613, -7613, -53664, 53664, -10111, 10111, 0], [1210965, 0, -1442358, 174326, -359481, -13141, 0, 600366, 600366, 0, 0, -4222, -4222, 33234, -33234, -51429, 51429, 0], [-8565034, -1352510, 1352510, 60564, -35137, -18722, 8565034, 0, -184148, -18722, 35137, 0, 81495, 32301, 0, 0, 76813, -60564], [0, 2276192, 0, 0, 0, 0, 28072992, -662796, 662796, 8991, -108477, -113814, 113814, 65185, 65185, 28241, 28241, 46689], [0, -2184690, 0, 0, 0, 0, 53959157, 495652, -495652, -31101, 40738, -64663, 64663, 8200, 8200, 70302, 70302, -74873], [0, -1508365, 0, 0, 0, 0, -9386652, -508558, 508558, -55955, 75476, -162592, 162592, -74631, -74631, -48187, -48187, -452], [33209794, 118106, -118106, -57604, 80783, 40280, -33209794, 0, -475916, 40280, -80783, 0, 119367, 55925, 0, 0, -77047, 57604], [-44012313, 0, -3484530, -63539, -19006, 93989, 0, 660320, 660320, 0, 0, -66057, -66057, -53141, 53141, 31772, -31772, 0], [946604, 838552, -838552, 66978, 129535, -9668, -946604, 0, 210784, -9668, -129535, 0, -231976, 138390, 0, 0, 8508, -66978], [-22619911, 139319, -139319, -123417, 27113, 98229, 22619911, 0, -28608, 98229, -27113, 0, 76049, 179651, 0, 0, -61527, 123417], [-58912290, 1774193, -1774193, 63869, 29231, -114856, 58912290, 0, 116953, -114856, -29231, 0, 143921, -70057, 0, 0, 107582, -63869], [0, -3510905, 0, 0, 0, 0, 191335538, -341686, 341686, -62838, -71184, 63774, -63774, -28536, -28536, -24220, -24220, -24038], [-45986614, 3884767, -3884767, 150694, -147754, 83380, 45986614, 0, -640448, 83380, 147754, 0, 110929, -76550, 0, 0, 3737, -150694], [-169564451, 0, -4958764, -80171, -252980, -6949, 0, -246303, -246303, 0, 0, -153671, -153671, -22422, 22422, -35943, 35943, 0], [0, 5110488, 0, 0, 0, 0, 160210524, -397306, 397306, 58977, 475561, 73740, -73740, 103349, 103349, -61745, -61745, 81707], [22590155, -4818369, -4818369, 66183, -47210, 13809, 22590155, -1076093, 0, -13809, -47210, -343118, 0, 0, 840, 85223, 0, 66183], [0, -2653260, 0, 0, 0, 0, -16964154, -485202, 485202, 57084, -407914, -131718, 131718, -50939, -50939, 51248, 51248, -20438], [-31583075, 3228435, 3228435, 154167, -100602, 33666, -31583075, 306012, 0, -33666, -100602, 205260, 0, 0, -119667, 23087, 0, 154167], [56592803, 188901, 188901, 142444, -170659, -89662, 56592803, 576949, 0, 89662, -170659, 323252, 0, 0, 57658, 48140, 0, 142444], [-14846141, 0, -4391895, -128702, -77225, 199196, 0, 453217, 453217, 0, 0, 139547, 139547, -42896, 42896, 2830, -2830, 0], [-80943293, -4010888, 4010888, -108434, -112098, 134361, 80943293, 0, -388091, 134361, 112098, 0, 141806, 4847, 0, 0, -40632, 108434], [0, -5838570, 0, 0, 0, 0, 255312946, -789491, 789491, -79127, 209909, 161746, -161746, 96167, 96167, 47629, 47629, -29237], [0, 2964039, 0, 0, 0, 0, 12294962, -437062, 437062, -17769, 44796, -125320, 125320, -5943, -5943, 3551, 3551, 275886], [0, 853753, 0, 0, 0, 0, 78823609, 156380, -156380, -208066, -537563, -59368, 59368, 115606, 115606, 56438, 56438, -203940], [0, 4921103, 0, 0, 0, 0, 87739881, 559479, -559479, -110918, 277473, 11477, -11477, -171471, -171471, 40152, 40152, 39886], [55180933, -2558219, -2558219, -112082, -692567, 28649, 55180933, 336772, 0, -28649, -692567, -57784, 0, 0, 1577, 121537, 0, -112082], [0, 3137254, 0, 0, 0, 0, -20175413, -123335, 123335, -26307, -643226, -320091, 320091, -33098, -33098, -217087, -217087, 261131], [-198183588, -1964169, 1964169, -176751, -398876, 26158, 198183588, 0, -362228, 26158, 398876, 0, -421318, 220808, 0, 0, -48946, 176751], [135235350, 0, -2404448, 193289, 126899, -187153, 0, -813129, -813129, 0, 0, 9054, 9054, 83758, -83758, -134942, 134942, 0], [78778793, 0, -3209461, 465110, -441009, 81504, 0, 518005, 518005, 0, 0, -392972, -392972, 140286, -140286, -115444, 115444, 0], [262145261, 3382282, -3382282, 55258, -557469, 89983, -262145261, 0, 2107919, 89983, 557469, 0, 44643, 166351, 0, 0, 210558, -55258], [97781587, 5733254, -5733254, -53292, -231190, -41861, -97781587, 0, -1062310, -41861, 231190, 0, -104016, -139160, 0, 0, -153598, 53292], [-105516050, 0, 5578079, 217302, 308648, -4279, 0, 371408, 371408, 0, 0, 273470, 273470, 470950, -470950, -11397, 11397, 0], [-406348482, 0, -8446340, -76953, -773547, 42293, 0, 338054, 338054, 0, 0, 167645, 167645, 99954, -99954, 29244, -29244, 0], [-66588634, 0, -5123642, -345546, -455820, -239976, 0, -1267038, -1267038, 0, 0, 63062, 63062, -183731, 183731, 30782, -30782, 0], [-61058363, 4225633, 4225633, 150789, -286187, -113775, -61058363, -3248710, 0, 113775, -286187, 11660, 0, 0, 67894, -74008, 0, 150789], [70576123, -3125876, 3125876, -146430, 342923, 47039, -70576123, 0, 2319545, 47039, -342923, 0, 235573, -221167, 0, 0, -80000, 146430], [0, -3021468, 0, 0, 0, 0, 4882794, -120264, 120264, 161946, -148406, -62263, 62263, -132564, -132564, -105945, -105945, 198606], [164918544, 4204264, -4204264, 186523, 532665, -218600, -164918544, 0, 2757378, -218600, -532665, 0, -239286, -58049, 0, 0, -269207, -186523], [77946644, 6447637, 6447637, 70083, -648496, 43265, 77946644, 421051, 0, -43265, -648496, -256362, 0, 0, -545727, -227189, 0, 70083], [-175882345, -149084, -149084, -293535, -1115064, 101597, -175882345, -786659, 0, -101597, -1115064, -317564, 0, 0, -121176, 18911, 0, -293535], [-65393272, 0, -10561948, 202034, -343932, 70258, 0, 1430540, 1430540, 0, 0, 123994, 123994, 100432, -100432, 6423, -6423, 0], [282380440, 0, 4985148, -688670, 293804, 43316, 0, -1395197, -1395197, 0, 0, 135071, 135071, -214478, 214478, 84426, -84426, 0], [0, -22767778, 0, 0, 0, 0, 293091405, 408757, -408757, -7433, -7661, -286372, 286372, -85749, -85749, 53618, 53618, 172348], [236237645, 2976947, -2976947, -301320, -234050, -168212, -236237645, 0, 1025152, -168212, 234050, 0, -661438, 209236, 0, 0, -142552, 301320], [0, -6562255, 0, 0, 0, 0, 263973831, 2518911, -2518911, -181864, 961435, 246976, -246976, -117812, -117812, -65648, -65648, 105178], [-136537952, 3369171, 3369171, 150044, 534445, 44498, -136537952, 2007758, 0, -44498, 534445, -222842, 0, 0, -215022, -499034, 0, 150044], [-44210376, 305211, 305211, 288631, -155311, 283581, -44210376, -2882537, 0, -283581, -155311, 45370, 0, 0, -123586, -489103, 0, 288631], [-155125957, -8069760, 8069760, -16648, 1372315, -10690, 155125957, 0, -3764397, -10690, -1372315, 0, 346230, -95904, 0, 0, -204295, 16648], [0, -85068, 0, 0, 0, 0, -211573313, -896282, 896282, -57609, -598337, 242790, -242790, 395539, 395539, 95233, 95233, -22448], [-190975093, 0, -6795268, 42811, 843402, 139825, 0, -157685, -157685, 0, 0, -284529, -284529, 44057, -44057, 44419, -44419, 0], [-275984406, 0, -1409850, 170992, -1125184, -406146, 0, -138260, -138260, 0, 0, 52286, 52286, -214144, 214144, 249141, -249141, 0], [395069047, -4982179, -4982179, 276818, 446535, 94850, 395069047, -2402803, 0, -94850, 446535, -21896, 0, 0, -117406, -58873, 0, 276818], [0, 19588662, 0, 0, 0, 0, -280158161, -1405193, 1405193, -34816, 187492, 671916, -671916, 88739, 88739, 344738, 344738, -219213], [-112993269, -3692398, -3692398, -125037, -758936, 181299, -112993269, -1844004, 0, -181299, -758936, 1162892, 0, 0, -383125, 88285, 0, -125037], [-348096133, 0, 9841280, -808124, -589893, 295085, 0, -65306, -65306, 0, 0, -171060, -171060, 370512, -370512, 18418, -18418, 0], [9090390, -10222002, 10222002, 204005, 21702, -115923, -9090390, 0, -5096436, -115923, -21702, 0, -322646, 77570, 0, 0, -85916, -204005], [0, 14682429, 0, 0, 0, 0, 448302427, -889513, 889513, -249730, -504561, -678430, 678430, -426997, -426997, 5102, 5102, 68256], [-310952199, 12514531, -12514531, 37660, 600688, 242442, 310952199, 0, 3039826, 242442, -600688, 0, 838221, -312085, 0, 0, -3563, -37660], [-605673672, 10275231, 10275231, -76293, -584390, -62398, -605673672, 928572, 0, 62398, -584390, -214588, 0, 0, 833186, 183110, 0, -76293]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_74_d_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_74_7_d_a();".
// 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_74_7_d_a(:prec:=18)
    chi := MakeCharacter_74_d();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 7));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(997) - 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_74_7_d_a();".
// 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_74_7_d_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_74_d();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,7,sign)));
    Vf := Kernel([<3,R![472663916932825857600, 0, 3004304740442828454249, 0, 194535437264309616102, 0, 2979392161707128817, 0, 17465487582207222, 0, 42593147514900, 0, 49247171526, 0, 29142265, 0, 8582, 0, 1]>],Snew);
    return Vf;
end function;