
// Make newform 605.6.a.p in Magma, downloaded from the LMFDB on 27 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_605_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_605_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_605_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 := [-32708279373824, 157475390795776, -51220983567360, -163614980036608, 88321308161536, 22119016261376, -15019635572992, -1362443421632, 1089062915552, 48021607312, -42703408468, -1043264932, 987264735, 14112735, -13821739, -115347, 115018, 521, -523, -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, 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, 0, 0], [-52, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-27904681467357382189917281768733782054615726626342912, -54735689425843099495480781186290052307425555110882304, 19081540397162116113054162202275283458353664783855616, 16982233011928465838759771267442317489801537921248256, 5167784138420388806249670395679846150193777889741056, -545964736045877019430335132452724184968677870973952, -1573469765333922259778481962022398792510617004285568, -152666207105483502378322000102590864373309851920416, 129446903097554843983650056599031438502605339971680, 15152185440179740640511798018529666927872289645880, -4964574633877214553761497678028972214876519903102, -597441011872194547341256145049087967738679972081, 101227109939667464398547492609536644623468521675, 12174317158136907735219680046889819799868955273, -1126654498609800403992876600992778265484913781, 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5382235612664922000640173480072771777986071482810381216096002, -271983529280189022639399398250524455355137968694215338935185, -30167420970910002081734111396581373970431943631649257268069, 621453585031665418902061399993891973520119798754729462365, 68580161671639264660222890891519919933624861827955236059], [11756891255082457831687297342177689114141159763583053999948655711885312, -19028251349698148068670898744710787735104152717055269320595084384152576, -14897591244908884280125410317379838234406283462538579562976977801502720, 23279882053219460087737317265511124015933828930710146183235748315128832, -1440545700304135540446678783546099180457250279253778255202817424566016, -4206542674229413200573577620159997957484356736018220571401176557910016, 489699078735765953516785870765007460707899050387939829381420344280448, 350378533306332628820073685379956325898594447043687272212372780852832, -38548677360511517712303132926560283891532542017651422262192323852256, -16310753599870799903457839878036030208361968976027877306124403000328, 1436062412623149395746767709497123209160036380561608803863738925914, 450332428225432241411185805254729028793886710106338373284073679503, -28832948946686238952210426834975940759286325050383513657712830949, -7453817337969773340745730840189348204962811355784881378595269655, 318401300779506288002765289079031116692925500042908282554586163, 72111625497554507325377701635450788461386776832856784295369632, -1814887247071040661505919313904454669104634227595269702685247, -374232363343025297285942034071959664567114655399111798707055, 4162819260408430149127606830192319201811593825258290280055, 802180398821095487965729575283401782475830543657466309503], [20429011176878606024800002370240966418050086937584900012817661006600192, -81486199582004680957292466177274160914742301899097247560606026257716224, -14933262243241670290680224661341457648056992769869540585476072085977088, 68930614294534043829912266776368314186674829817121575402901100678778880, -7229510208066382450383669120067759569902304279479405659039226604371200, -13158047848462636662224228438432846035702467102406711545999229203665920, 1732385336143623396958851762625313111895987231034774434242360209830272, 1163042614206892452539828020081649044977074257829902022913309614565728, -133811955200997528822457053458253491497415488626883232085632437313152, -56244086185271983475090812899473578997224187463477192279725452803256, 4993037104374415773159323400180691853298549984448103738975118087930, 1585063831157104070079592448769255607518724825662325183064519241041, -100482511283924992656744327981717644277398426611098956753937938275, -26499728517814154354996078146955286722487004045146091748723810457, 1111032075067097912402504026552522190342181585607394790170747001, 257515703043865815509250334369916852607818013199679545099835204, -6335933388745526385441114908116819566270689383195355581940273, -1338647000246724445329515634011881436505012969341042988238089, 14533272580422207684301390153278304509016663644283950930849, 2870380005950056813701754307986222713728633142163932993509]];
    Rf_basisdens := [1, 1, 1, 3095786373319275371773656083462092919902869546860544, 197296752855642593502161928304282315806769441428841967053440913637376, 394593505711285187004323856608564631613538882857683934106881827274752, 1183780517133855561012971569825693894840616648573051802320645481824256, 98648376427821296751080964152141157903384720714420983526720456818688, 1183780517133855561012971569825693894840616648573051802320645481824256, 107616410648532323728451960893244899530965149870277436574604134711296, 1183780517133855561012971569825693894840616648573051802320645481824256, 147972564641731945126621446228211736855077081071631475290080685228032, 295945129283463890253242892456423473710154162143262950580161370456064, 1183780517133855561012971569825693894840616648573051802320645481824256, 295945129283463890253242892456423473710154162143262950580161370456064, 394593505711285187004323856608564631613538882857683934106881827274752, 1183780517133855561012971569825693894840616648573051802320645481824256, 2242008555177756744342749185275935406895107288964113261970919473152, 17936068441422053954741993482207483255160858311712906095767355785216, 49324188213910648375540482076070578951692360357210491763360228409344];
    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_605_a();"
function MakeCharacter_605_a()
    N := 605;
    order := 1;
    char_gens := [122, 486];
    v := [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_605_a_Hecke(Kf)
    return MakeCharacter_605_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 := [[0, 1, 0, 0, 0, 0, 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], [-25, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-8, -1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-42, -8, -1, -9, 1, 0, 1, 0, -1, -1, 0, -2, 1, 1, 1, 0, 0, 1, -1, 0], [148, 1, 4, -1, 11, 3, -1, -1, 1, 2, -1, 2, 2, 1, 3, -2, 0, 1, -2, 0], [-279, -12, -13, 12, 15, -3, 2, -1, 3, 2, 2, 3, -3, -1, 0, 0, 2, -1, -1, 0], [-10, -79, -11, 35, 12, 1, -4, 1, -9, 1, 3, 8, -5, 1, 1, -1, 0, -2, -3, 2], [-569, 39, -13, 0, -10, -1, 1, -2, -6, 4, 3, -10, 0, 4, 3, -3, -6, 1, -2, 6], [-605, -28, 13, -55, -13, 2, 2, 1, -7, 15, -4, 10, -3, -3, 6, 2, 4, -1, 9, -3], [-190, -137, -8, -100, -87, 9, 9, 13, -25, -6, 5, 3, -5, 0, -9, -1, -10, 6, 2, 3], [101, -139, -111, -33, -22, -13, 7, 27, 21, -45, -24, -12, 14, -2, -16, 0, 0, -3, 18, -9], [-2226, -38, -122, -175, 48, -18, -21, -6, 31, 13, -3, 23, 9, -13, -10, 10, 10, -9, 18, 1], [1689, -499, 35, 109, 111, -24, -4, 4, 1, -27, 7, 35, -1, -16, -29, 35, -10, -3, 26, -12], [1471, -910, 26, 82, 223, 32, -22, 2, 38, -42, -11, -60, -4, -10, -22, -10, 10, -23, -13, 16], [-2611, -636, -153, 221, -244, 17, 63, -7, -4, -21, 19, 28, -28, -10, 14, 28, 20, -8, 11, -1], [-4219, 1360, -12, 261, 204, 66, -19, -35, -2, 45, -32, -15, -4, 59, 51, -48, 0, -14, -91, 59], [-1493, -275, -332, -102, -229, -34, 55, -57, -98, -74, -1, -1, -119, 6, -2, -30, 30, -52, -11, 10], [-5444, -1566, -31, -344, 654, -13, -35, -5, 107, -108, 26, -139, 24, -46, -13, -3, -6, 3, 19, -31], [-6261, 632, -17, -587, 673, -38, 10, 17, 75, -54, 10, -173, -25, -18, -2, -4, 0, -25, 76, -27], [-10307, -3468, -77, 483, 337, -201, 48, 49, -57, -40, -56, 4, -46, -34, 0, 31, 16, -10, 27, -55], [-3145, 2092, 381, 944, 903, -161, -31, 1, -100, -27, 25, -9, -177, -29, -76, 57, 30, -3, 71, -22], [13251, -3282, -64, -779, -258, 10, 60, -36, -166, -104, -4, -295, 33, 51, 30, 25, -18, 107, -94, 81], [3638, -4737, -443, 2077, 1303, 63, 3, -11, -265, -355, 59, -76, 74, -48, 42, -26, -70, 68, 59, -35], [-2914, -2223, 331, -238, 84, -163, 181, -49, -19, -213, -36, 9, -225, 97, -106, -83, 18, -52, -124, 38], [-2286, -5345, -65, 2188, -362, 16, -91, -10, -335, 145, 55, 30, -175, -128, -116, 65, 40, -14, 128, -34], [-23419, 928, 886, 605, -900, -133, 155, 80, 230, 183, -154, -179, 287, 135, 9, -31, 10, -84, -38, -102], [-23486, 5414, -160, 97, -1934, -168, 139, -6, -13, -432, -25, -22, -33, 10, -62, 158, -100, 89, 17, -73], [-32183, -5231, -700, -3016, 801, -83, -4, -32, -423, 303, -111, -101, -226, 58, 96, -32, -80, 37, -147, 111], [-28684, 2303, -520, 1213, -253, -137, 5, -91, 103, -392, -68, 222, 80, -125, -129, 86, 160, 68, 107, -64], [-14798, 943, -530, -1799, 1490, -33, -196, 123, 246, -283, -41, 426, 28, -149, -75, -229, -144, 92, 59, -210], [-50525, -823, -303, -1531, -1025, 503, 35, -248, -558, -90, 272, 501, -161, 87, 457, -28, 30, 107, -139, 95], [-31336, 2627, 1557, -5788, -715, -74, 9, 271, -382, 125, 4, -663, -301, -123, -197, -10, -66, -170, -115, -1], [32822, 1489, 2643, 5877, -4698, 867, -122, -139, -196, 933, 129, 1189, 636, 71, 480, -249, -62, 199, -41, 217], [-68976, -6950, 1313, 7185, 2217, 483, -236, -351, -742, 815, 239, -454, 167, 253, 561, -21, 84, -30, -335, 276], [-33000, 17982, -1307, -9830, -2051, -538, 38, -193, -943, -1136, -54, -650, -700, -315, -286, -55, -230, -30, 270, -152], [-19267, 7781, 342, 3668, 1219, -429, 601, 169, -220, -1091, 393, 3, -4, -1, 0, 257, -50, 273, 19, 234], [-26169, -1695, -1654, 6742, -3156, 159, -294, -331, 113, 502, -144, -2347, -144, 313, -176, -92, -70, 23, -161, -4], [-47728, 13231, 3095, 3768, -287, 511, -387, -232, 646, 508, -404, 614, 30, -174, -87, -75, 260, -372, 347, 114], [23161, 8204, 1540, -11393, -3895, -293, -254, -161, 502, 368, 38, -1266, -711, 222, -252, 187, 188, -402, -191, -263], [-79593, 24351, 2214, 94, -14112, 239, -1005, -450, -740, 131, 199, -533, -155, 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-199075, 759431, -222141, -477322, -1307795, 78605, -266523, -610553, -8161, 267098, 88755, 594970, -453545, 138653, 33580], [-614486, 10742981, 606625, -1730498, -7376532, 664119, -747982, -189894, -406397, 33588, 93662, 2405072, 350845, 82594, -18321, -234769, 38074, 134770, -26741, 420295], [-74312903, -16556295, -1347902, -1476409, 9168230, 100184, 104967, -428799, 695810, -1296122, -387049, -862718, -104962, 250963, 107562, -335819, 26552, -373199, -676771, 60905], [-208427790, -5695204, 2602490, -627116, -3563306, 530405, 129666, -53876, -1818698, -1107563, 380448, -1853708, -1047933, 31387, 623074, -304309, -434032, 290877, -369722, 448258]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_605_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_605_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_605_6_a_p(:prec:=20)
    chi := MakeCharacter_605_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_605_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_605_6_a_p( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_605_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,6,sign)));
    Vf := Kernel([<2,R![-32708279373824, 157475390795776, -51220983567360, -163614980036608, 88321308161536, 22119016261376, -15019635572992, -1362443421632, 1089062915552, 48021607312, -42703408468, -1043264932, 987264735, 14112735, -13821739, -115347, 115018, 521, -523, -1, 1]>],Snew);
    return Vf;
end function;