
// Make newform 432.4.a.g in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_432_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_432_4_a_g();".
// 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_432_4_a_g();".
// 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
        Kf := Rationals();
    end if;
    return [Kf!elt[1] : elt in input];
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_432_a();"
function MakeCharacter_432_a()
    N := 432;
    order := 1;
    char_gens := [271, 325, 353];
    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_432_a_Hecke(Kf)
    return MakeCharacter_432_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], [-17], [0], [89], [0], [-107], [0], [0], [-308], [-433], [0], [520], [0], [0], [0], [-901], [-1007], [0], [-271], [-503], [0], [0], [1853], [0], [19], [0], [-646], [0], [-380], [0], [0], [3043], [0], [3709], [-3850], [379], [0], [0], [0], [-4699], [0], [5111], [0], [-4373], [2071], [3220], [0], [4466], [0], [0], [4769], [0], [0], [0], [0], [8101], [-4030], [0], [-5600], [0], [-10640], [0], [-9109], [0], [10891], [12293], [0], [1367], [0], [0], [-9413], [-12601], [-14687], [0], [0], [1190], [0], [8297], [0], [-6679], [0], [-2590], [-14924], [0], [0], [12710], [0], [-11969], [0], [0], [6103], [0], [15136], [0], [0], [0], [11881], [1889], [-21293], [0], [0], [0], [23941], [27323], [0], [0], [0], [-29302], [-6137], [12851], [0], [37], [-26459], [0], [-13160], [0], [0], [0], [33731], [9899], [0], [0], [16072], [0], [-8623], [0], [10780], [15050], [-31376], [0], [17569], [24137], [0], [-34417], [0], [14653], [0], [0], [-39368], [0], [-45611], [0], [30077], [0], [8279], [0], [49807], [0], [-36793], [0], [-52109], [0], [-4607], [0], [-2756], [0], [15227], [0], [0], [0], [-53927], [0], [0], [0], [-59669], [28910], [8693], [0], [0], [-40138], [0], [-48259], [65683], [0], [60352], [0], [19441], [39494], [67537], [0], [61490], [0], [0], [0], [-70363], [-42569], [52037], [0], [-76519], [0], [-80081], [0], [0], [0], [75419], [-25579], [0], [0], [0], [82549], [19637], [88037], [0], [0], [70957], [0], [0], [-92591], [-1513], [0], [-45431], [0], [0], [11951], [-15860], [0], [0], [0], [-73891], [-29036], [0], [77380], [0], [-108037], [0], [0], [-67517], [0], [-91279], [69496], [103588], [0], [50311], [0], [-36307], [0], [0], [0], [0], [-84131], [109261], [-71497], [0], [0], [-81863], [0], [118816], [0], [-114550], [0], [0], [41813], [0], [0], [-7939], [-121520], [0], [4970], [-127889], [0], [96857], [37259], [0], [-64064], [0], [0], [-142760], [0], [-107749], [-51353], [51590], [13033], [-71890], [-5021], [0], [-111274], [5402], [0], [0], [59092], [0], [-111169], [-141353], [0], [-71119], [0], [150967], [0], [0], [0], [0], [0], [-121210], [0], [-146249], [0], [0], [60517], [101771], [0], [-171236], [0], [10711], [166430], [0], [164033], [0], [122039], [0], [0], [0], [-184661], [0], [-183274], [0], [0], [-31570], [0], [-2879], [194867], [0], [-187220], [0], [-19171], [106216], [52360], [0], [0], [62441], [0], [-207107], [0], [-103952], [0], [-190513], [0], [47558], [129220], [-31249], [0], [0], [220681], [0], [0], [-114478], [-188480], [0], [0], [-58571], [-226423], [0], [-178220], [182447], [0], [0], [0], [0], [0], [-211933], [0], [0], [0], [160543], [-243361], [0], [-147041], [-65449], [0], [-245537], [0], [0], [117199], [-93367], [0], [0], [-257689], [0], [248957], [0], [0], [-257363], [0], [-171323], [0], [267388], [37133], [-277829], [0], [-261826], [0], [0], [115813], [0], [20609], [-65123], [0], [-272792], [0], [-112714], [0], [-191900], [0], [0], [-8372], [-232723], [0], [281251], [0], [108290], [0], [0], [-49589], [241793], [0], [0], [284293], [0], [0], [0], [-263410], [0], [0], [277130], [0], [0], [0], [323512], [0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_432_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_432_4_a_g();".
// 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_432_4_a_g(:prec:=1)
    chi := MakeCharacter_432_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_432_4_a_g();".
// 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_432_4_a_g( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_432_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<5,R![0, 1]>,<7,R![17, 1]>],Snew);
    return Vf;
end function;