
// Make newform 41.12.a.a in Magma, downloaded from the LMFDB on 30 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_41_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_41_12_a_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_41_12_a_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 := [1505823571529486523548772, 2730444604038756827754428, -423772965407378010138968, -9491487500136134990959, 2086916929767680919453, 14611041561982637102, -3418226388308715926, -13246337520368409, 2605976341612395, 6344973355152, -1040129054956, -1503772881, 221738291, 166886, -23750, -7, 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, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-374024205359946637577821540155377461240121376672019910322265293617230239441052063939174486602148142256, 4251594411806348347365076674400534529314020697819031978325612907742022233917669918449440787202732972, 634222645053309645394592112925817956998811975532148957055957891094942855705021429992044544852594252, -28122150614404718171162442985186584667528166824772538191126626924274105469534941543115250118420040, -564591487999516188174435331121463824682862278360426113035128767035687896559510903552981678753849, 55665524936924452336129903708343017784113145258479708867578108967073998400839434600327333951913, 66704453281870573410110785970463577097144451843907833204985952120875102123616092998860831845, -45634315958778122618425143725903544715536423239523320860773047163447534413088252120376548333, 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1298132831782370698215279516361830913790220972908443442473949298064412356626600419884, 957492516254535438473901696126031560210047730236611505733674010769655489487710046, -56456537951975568680241261671872576600598731790755893347175725287655618768027437], [314387172978682841937549543924654368020430295852109748727399011381943277941740698651535112122148645732, -354051481010185366693409970924918369062560602169495395567900234887880003272557021034713220872969006944, 9771839834369474594491660348861066477312123180093855082305925995488608292964320936647335969373790664, 2385471852068822744506857348558315120165438983685864836374989812807490325482193269044629754843655577, -12693885407505721884191785189516863474961668400146023993583320631819867561639303068620785184653162, -4637649186566636962164951836812030506954372477347825939035659155913019908071590306061909148022108, 750379700104364450125853122220867524890839078228271326992830422895126960828708867225415055662, 3934708202611847208967434823896145514301839717690355436578647197589563780326361368470535386485, 3351060721002582305033591681613511983815354661100729403959630404189129469310291823549656352, -1686705768160604181780670232284257741395344072625850892307616014069818667002759666730158096, -1250332156536050220137654608122769331980652921751079895903471114997512025565532088774204, 377214627337883381929074882154315866105561884165029923915715859927869412932873361269747, 162265136807296966751100943662300364868438988867816629748325049954851927663998333702, -41729478423425211600611194201024331023047364197072544449598371468111535625726197940, -7222214149415014480568932498313237276251407716869855879197619363963508717336794, 1796250125907959095348210255237842051582846349559123246605853689333630559217279], [4966944035738147591965426808071505226448856092720734451800284433846823043058978422647443694470929301012, -466492007883436366462402910928915365206441681672466618628178184954485961630747599555326649842013022924, -130175644030570732642665336124226977630699483054720508162024870827404723191363771492394526385965647844, 5551626319250184186822375052240704676041597188798821050089886684044596741390767354992905648884212385, 471436932481472987509651301255816069118886035004855409865347569159253167674895207212839382488235583, -13752174851893533519694750946755021715654845981635733773659399266109679239604818564956180113002301, -565176009249270160299253974069824291818101507003739184415035602853093226501072083562686573711415, 13885667228494001292355012797267670024148605475733771877860038311204767904742055409968414757386, 299269898935503182315928133785314827732007236484052722037522575028810464615528505220130477434, -6786144719900617629465830315870124678383130463763945699168999163297945676238277390892721782, -75878490508168901569714923635988798413112611798985370771612549233265739962651125793095394, 1661305028523458619532813693574607241444754693166961377657355684425784827020535551879893, 9033118219392374305961931827067010168695149369117901549705716905634248039881678767043, -195213941366792407452687435146870430533380558501481634379756457340778513293086278257, -406403534093918179056645660769684245808912050458593111397848959129515558708051603, 8752590203048041055981472315397349257586711654192780181278575976188180761871216]];
    Rf_basisdens := [1, 1, 101631363625339417040971353601304632945307022366188442587292417017800656452447891369055666548244480, 101631363625339417040971353601304632945307022366188442587292417017800656452447891369055666548244480, 813050909002715336327770828810437063562456178929507540698339336142405251619583130952445332385955840, 25407840906334854260242838400326158236326755591547110646823104254450164113111972842263916637061120, 101631363625339417040971353601304632945307022366188442587292417017800656452447891369055666548244480, 5807506492876538116626934491503121882588972706639339576416709543874323225854165221088895231328256, 271016969667571778775923609603479021187485392976502513566113112047468417206527710317481777461985280, 203262727250678834081942707202609265890614044732376885174584834035601312904895782738111333096488960, 813050909002715336327770828810437063562456178929507540698339336142405251619583130952445332385955840, 813050909002715336327770828810437063562456178929507540698339336142405251619583130952445332385955840, 406525454501357668163885414405218531781228089464753770349169668071202625809791565476222666192977920, 813050909002715336327770828810437063562456178929507540698339336142405251619583130952445332385955840, 23230025971506152466507737966012487530355890826557358305666838175497292903416660884355580925313024, 101631363625339417040971353601304632945307022366188442587292417017800656452447891369055666548244480];
    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_41_a();"
function MakeCharacter_41_a()
    N := 41;
    order := 1;
    char_gens := [6];
    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;

function MakeCharacter_41_a_Hecke(Kf)
    return MakeCharacter_41_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 := 12;
    raw_aps := [[-1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-29, -1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-170, -4, 1, 2, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-6251, -130, -6, -1, -1, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-82765, -629, -71, 252, -4, 21, -2, -4, -3, 0, -6, 6, 3, -3, -3, -1], [-102423, -3829, -48, 569, 23, -22, -23, 2, 3, 0, 15, -1, -18, 16, -10, -1], [-14871, -31439, -258, 3587, 2, 12, 3, 32, 63, 16, 1, -48, 18, 38, 116, -53], [-2018527, -54065, 24, 4038, -42, 131, 114, 99, -21, -115, 38, -159, 79, -23, 25, -111], [-3888838, -178496, 648, -11073, -80, 1211, -412, -109, 185, -467, -12, -141, 279, 433, 89, 531], [-11964976, -211280, 1336, -36404, 1871, 2504, -3092, 502, 1878, 520, 520, -243, -1368, 402, -286, 758], [-39599658, 291038, -16648, -78189, -284, -861, -2074, 1003, -1459, -315, -1928, 1365, 4583, -483, -479, -149], [-106948515, 1217841, 4685, 201111, -5076, 16027, -325, 2650, 165, 430, -2807, 612, 5176, -2912, 466, -3939], [115856201, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-213398378, 584132, 102854, 404945, 304, 22133, -9518, 5603, -9027, -2739, 1202, 5879, -25299, -2641, 1087, 2627], [-488862710, 8761605, 166597, 1161415, -10631, 94463, -4600, -438, -11310, 27530, 13977, -164, -7993, -8213, 4921, 6252], [-663674837, -10974219, 75208, 250269, 17311, 12538, 41379, 13170, 46077, -1578, 35633, -14769, -23106, -18792, 14906, -64919], [407496366, 4267782, 201810, 297057, -67172, -61995, 22306, -65329, -124713, 45325, -48764, 39409, 112261, -63005, -99929, 60585], [-398510021, 883113, -143840, 2532059, 19073, 108450, -101235, 63512, 63323, 90564, -47525, -7801, -32000, 43004, 43218, -3025], [1778301705, 26798271, 273709, 5256736, 172274, -351773, -95596, -138998, -43701, -95358, 14958, 37528, 55425, 38903, 26339, 19969], [644154448, 45827859, 714095, 9676196, -97727, 382196, 36280, 32549, -366005, 269931, 127683, 98353, -455460, 132910, 130852, 150111], [2134966097, 102015323, -2184082, 6709879, -119002, -856943, 38613, 21287, -140850, 173146, 36825, -184169, -115667, -119117, -55063, 195376], [-6475741954, 82794003, -4532665, -6387847, 98547, 32933, -128408, 73978, 372538, -579546, 410463, -150576, 297517, -142591, 222159, -471224], [4548759264, 121147180, -5066050, -18388658, -15828, 1545810, -188968, -167024, -49226, 230980, -66936, -118432, -654862, 166834, -5526, -6422], [-2291799179, 95901065, -5408952, -34857527, 502370, 2516868, 383847, 398054, -484259, 46646, 646433, -545386, -1549596, 357776, 574942, -645315], [11733879754, 247467172, 6707822, 8100714, -101108, 889530, -288740, -1080248, 968732, 509102, -863288, 1325304, 1734880, -1488284, -60740, -415396], [-11270345966, -204325946, 9441098, -10481178, -445473, -1689850, 1140754, -271102, -385080, -1346276, -133398, -560299, -595688, 227274, -768466, 469984], [1278734500, -846164, -1484158, -53868006, 268126, -4992850, 2028542, -33276, 122916, -614144, 168458, 834130, -787890, 1407930, 685306, 1604564], [-33389685868, -253291160, -5548454, -15872052, -1649324, 1373012, 1357196, -802398, -1696528, 74770, -686648, -1729942, -1471712, 12884, -1303208, 1470032], [-59885423850, 321299490, -21107122, -4826850, -629633, -3703674, 1858230, 461258, 76348, 142860, -1467442, -567771, -876436, 1829122, 1456590, 495132], [-4188109781, -121900771, 6221426, -16128887, 1862646, -3524233, -769849, 879677, 5139858, -1000582, -2096445, -554391, 4779731, -2266367, -1213677, -2519616], [-94881290128, 1236381254, -10897050, -179871913, 5658916, 15526841, 3766250, 589215, 7798365, -170863, 4500354, -6966901, -2304191, -1497285, 1459843, -8089541], [-104556345854, 1008936166, 88642236, -35190744, -3728940, 4389694, 3838884, -1453926, 306898, 2861882, 2237216, 3026022, 6334450, -747670, 4044222, 10886], [-142194181241, -1947778765, -12665512, -1459069, -5135248, -23983858, -6248587, 72544, -3736499, 8363314, 542059, -8450802, 454970, -821966, 384268, -5116471], [13389573574, 3546712086, 43226204, 112361661, 2824608, -541955, 5974394, -11002511, 990635, -1844957, -7266768, 1656611, -2892431, -6931777, 782407, 6022429], [-44820196826, -7952982058, -71881852, -196925878, -542047, -50210908, -11604394, 6133538, -3615468, -6526768, -2560994, 9401519, 11352704, 1893882, -5278090, 1224340], [-27371649500, -2970742189, -93933885, 63778496, -3159471, 43686344, 4639304, -750439, -2384189, 13779459, 12668463, 1807453, -13359960, -1243518, 9567212, 4356223], [-149708833874, -1683160430, 35234758, 861730994, 10474143, 22118866, -18731054, -14318146, 13381832, -5522500, 15112434, -11085599, 6697500, 8629898, -2244946, 2892600], [87419694074, -4183490684, -104173520, -793794066, -7553148, 35534708, -3725196, 32032280, 2201554, -6088584, -25520014, 8723850, 30357084, -3427388, -3847320, -8071594], [166782528183, -8514293980, 116056732, -379376340, 19930197, -6916445, -7465833, 4455547, -12811533, 1473221, -15065016, 28138463, 19673125, -24343133, -26794497, -13144007], [296699873860, -16813829250, 53344892, 132516306, -9818834, -95282988, 22954790, 4970420, 8510786, -3254868, 6257634, -18778546, -6905960, -18560096, -4542228, 9539746], [40941177841, -1470759025, -117862037, -457777273, 21614520, 67149928, 11003032, -10369339, 21323336, 30218083, 6120900, -13537921, 21529854, -29888398, 9899114, -14819628], [-332553253909, 961695853, 822754, 766988991, -43072871, 66979876, 11431279, -31477766, 26055165, -2498268, 29330489, -48004027, 20114070, -10432960, 9948202, 22512793], [-1427995815715, -8195916676, -213983140, -946577750, 26203843, -30765125, 61384401, -13347269, -17619129, -64835315, 5413792, 29138247, -35030455, 14625295, 3667363, -25555003], [-887561114522, 24425114004, 373451334, 2029400574, -15496980, 71401200, -84130552, 5672878, 21425758, -31601526, -63484464, 24403798, 127707910, -23623046, -53184862, -12747686], [-406405887059, -1769717697, -331983740, 1372455973, -6897627, -58798970, -20044813, -378344, -57069719, 54458552, -16601107, 38975615, -56864944, 6552936, 22819202, 80575085], [-1796423752555, 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    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_41_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_41_12_a_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_41_12_a_a(:prec:=16)
    chi := MakeCharacter_41_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 12));
    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_41_12_a_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_41_12_a_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_41_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,12,sign)));
    Vf := Kernel([<2,R![3805101821770612944666624, 1862824352462280868233216, -439631294333590325166080, -1066391172386867642368, 2108418118936815493120, -6030018345505327104, -3437669352367717760, 7704756343230016, 2616137075693280, -4090170526448, -1042011886808, 1161453532, 221745194, -165789, -23735, 9, 1]>],Snew);
    return Vf;
end function;