
// Make newform 784.4.a.w in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_784_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_784_4_a_w();".
// 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_784_4_a_w();".
// 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 := [-2, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    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_784_a();"
function MakeCharacter_784_a()
    N := 784;
    order := 1;
    char_gens := [687, 197, 689];
    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_784_a_Hecke(Kf)
    return MakeCharacter_784_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 := 4;
    raw_aps := [[0, 0], [0, 1], [0, 2], [0, 0], [26, 0], [0, 24], [0, -73], [0, -67], [148, 0], [-118, 0], [0, 210], [-254, 0], [0, 65], [-122, 0], [0, -218], [-170, 0], [0, -215], [0, -430], [-420, 0], [-420, 0], [0, 575], [-1052, 0], [0, 1025], [0, -725], [0, 223], [0, 1014], [0, -26], [-884, 0], [598, 0], [-1548, 0], [-2080, 0], [0, -1521], [-2100, 0], [0, -247], [-2990, 0], [-656, 0], [0, -1520], [1742, 0], [0, 998], [0, 876], [4620, 0], [0, 1560], [676, 0], [800, 0], [-1270, 0], [0, 1890], [1236, 0], [0, -3456], [0, -3657], [0, -2340], [840, 0], [2028, 0], [0, 1401], [0, 5369], [0, 3705], [-2172, 0], [0, 3952], [0, 4160], [4246, 0], [-1664, 0], [0, -4199], [0, -3806], [0, 2863], [0, -3016], [0, -351], [3874, 0], [1790, 0], [9794, 0], [10086, 0], [0, 1992], [0, -1729], [7040, 0], [0, 7852], [-2990, 0], [11570, 0], [0, 4082], [-9838, 0], [0, 10312], [-9906, 0], [0, -2013], [0, 1495], [1686, 0], [-8844, 0], [0, 195], [0, 8172], [5212, 0], [-3042, 0], [-360, 0], [0, 1300], [4160, 0], [0, -7007], [0, -2642], [-11964, 0], [9636, 0], [12172, 0], [0, -2716], [0, -12248], [0, -2041], [0, -4785], [11746, 0], [-16990, 0], [12090, 0], [0, -14495], [-8390, 0], [6170, 0], [0, 9633], [0, 12309], [0, -15701], [-25232, 0], [0, -5019], [0, -10452], [-11622, 0], [16246, 0], [0, -4693], [-21404, 0], [-3622, 0], [0, 8489], [0, 13650], [-19578, 0], [1962, 0], [0, 1830], [-26460, 0], [0, 14508], [-5460, 0], [0, -10545], [-15306, 0], [-29822, 0], [0, 7462], [0, 3830], [0, -17106], [35958, 0], [-23920, 0], [30724, 0], [22010, 0], [0, 6211], [0, 11373], [0, -25454], [0, 9097], [0, 5208], [-15952, 0], [0, 8541], [-7674, 0], [5720, 0], [-41964, 0], [0, 19370], [0, -16022], [0, 1752], [0, -25355], [0, 18355], [-39260, 0], [46874, 0], [0, 3765], [10540, 0], [0, -4030], [9524, 0], [-15560, 0], [-22048, 0], [0, -14545], [0, -31433], [0, 5930], [-9990, 0], [7990, 0], [26364, 0], [0, 34945], [13884, 0], [0, -43066], [-16640, 0], [0, -598], [34528, 0], [0, 4360], [-39156, 0], [0, -8164], [876, 0], [-18248, 0], [0, -25298], [0, 10985], [2964, 0], [-43206, 0], [0, 23712], [0, -3770], [48620, 0], [0, -20605], [-35802, 0], [0, 41611], [46280, 0], [0, -46592], [54270, 0], [0, 15741], [-68452, 0], [0, -3130], [0, 24777], [55332, 0], [-3524, 0], [0, 30020], [-39060, 0], [0, -46561], [-24414, 0], [44530, 0], [0, -38763], [0, -24584], [-63330, 0], [0, -29886], [0, -5930], [0, 9373], [0, -8993], [0, 20878], [0, -13418], [46710, 0], [-46276, 0], [0, 2085], [57130, 0], [0, 11946], [19760, 0], [0, 30817], [0, -34552], [0, -11645], [81184, 0], [0, 38155], [-48984, 0], [-31818, 0], [-41170, 0], [0, 20592], [78702, 0], [-48572, 0], [0, 10837], [35950, 0], [0, -18629], [-532, 0], [0, 21008], [-72150, 0], [68510, 0], [0, 31173], [91520, 0], [-82254, 0], [0, -30731], [0, -45888], [0, -69225], [-33670, 0], [-44468, 0], [0, 71356], [32370, 0], [0, -66945], [0, 6084], [117750, 0], [0, -28381], [0, -64202], [0, 7660], [0, 43781], [-45842, 0], [-16232, 0], [-93574, 0], [0, 1599], [-18740, 0], [0, 50435], [0, 21490], [5252, 0], [-121850, 0], [0, 47645], [0, 39182], [0, 33893], [47468, 0], [116414, 0], [0, -22410], [0, 56914], [0, 13065], [0, -51897], [29922, 0], [0, -1701], [24700, 0], [16198, 0], [0, -58500], [-32470, 0], [0, -87321], [-304, 0], [0, 63935], [0, -101036], [33510, 0], [-12526, 0], [51116, 0], [0, 91], [0, -40964], [-25096, 0], [0, -51610], [0, -33126], [0, 102557], [15384, 0], [135838, 0], [-109174, 0], [0, 3790], [0, -67735], [68134, 0], [0, -37037], [36950, 0], [0, -54265], [68342, 0], [0, 101920], [0, 40496], [0, -30004], [0, 26755], [0, 26387], [0, -19461], [105606, 0], [-34728, 0], [29550, 0], [-2666, 0], [31044, 0], [-133146, 0], [0, -22190], [-40020, 0], [98350, 0], [0, -32800], [98202, 0], [42092, 0], [171302, 0], [72924, 0], [0, -104195], [0, -14207], [-169624, 0], [0, 2051], [138312, 0], [0, -52819], [-36140, 0], [0, -71006], [-49140, 0], [31512, 0], [0, 87125], [-61022, 0], [0, 11295], [120640, 0], [143390, 0], [-69030, 0], [-56330, 0], [0, -71942], [0, 3559], [54340, 0], [0, -127073], [126198, 0], [0, 7345], [0, -29211], [0, 78798], [-78238, 0], [12350, 0], [0, 62608], [-6276, 0], [125342, 0], [70330, 0], [0, -141804], [92930, 0], [0, -153416], [0, -15522], [0, 32385], [-129610, 0], [80206, 0], [0, -9014], [-134758, 0], [0, 98345], [0, -44158], [0, 27419], [13026, 0], [-160108, 0], [-178910, 0], [0, 26135], [-66600, 0], [142788, 0], [0, 43485], [-15302, 0], [0, -7628], [-71812, 0], [63224, 0], [104364, 0], [0, -141375], [-195280, 0], [76818, 0], [0, -78636], [53754, 0], [0, -5235], [-127764, 0], [0, -3861], [0, 10257], [0, 45843], [0, 124576], [28048, 0], [-51240, 0], [85466, 0], [0, 177635], [0, 69924], [-173676, 0], [0, 163878], [-185020, 0], [0, -179022], [-64798, 0], [0, -177474], [7852, 0], [0, 18525], [-160576, 0], [-86100, 0], [0, -55250], [0, 30415], [-47436, 0], [-155870, 0], [0, -184290], [232110, 0], [149344, 0], [0, -88397], [0, 37320], [0, -47306], [81094, 0], [-123296, 0], [0, -115905], [0, -8655], [0, -66651], [31734, 0], [79962, 0], [-127660, 0], [-283616, 0], [0, 94860], [-72228, 0], [0, 125866], [0, -75095], [0, -134204], [-120882, 0], [0, 40226], [-228024, 0], [0, 16615], [0, 102999], [0, 6658], [45690, 0], [-84760, 0], [0, 71815], [0, 132690]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_784_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_784_4_a_w();".
// 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_784_4_a_w(:prec:=2)
    chi := MakeCharacter_784_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_784_4_a_w();".
// 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_784_4_a_w( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_784_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<3,R![-2, 0, 1]>,<5,R![-8, 0, 1]>],Snew);
    return Vf;
end function;