
// Make newform 4032.2.a.y in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4032_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_4032_2_a_y();".
// 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_4032_2_a_y();".
// 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_4032_a();"
function MakeCharacter_4032_a()
    N := 4032;
    order := 1;
    char_gens := [127, 3781, 1793, 577];
    v := [1, 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_4032_a_Hecke(Kf)
    return MakeCharacter_4032_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 := 2;
    raw_aps := [[0], [0], [0], [1], [4], [-2], [-4], [0], [-4], [-4], [-8], [2], [4], [-8], [-8], [4], [8], [2], [-8], [-12], [6], [-8], [16], [12], [-2], [0], [-8], [4], [-10], [0], [-8], [0], [-16], [-16], [12], [-8], [18], [-8], [8], [-24], [12], [6], [-12], [2], [-12], [8], [8], [-24], [-8], [-10], [-24], [12], [14], [-8], [28], [-4], [24], [-8], [-2], [24], [-32], [-16], [16], [24], [10], [20], [-24], [-18], [-36], [-14], [-12], [20], [-8], [18], [-8], [32], [36], [34], [-16], [22], [-24], [-18], [20], [-14], [24], [12], [24], [22], [-8], [-24], [-24], [-24], [24], [-20], [40], [-16], [-40], [-20], [16], [6], [-8], [12], [-40], [0], [-40], [34], [-24], [36], [-12], [-22], [-24], [-14], [8], [32], [40], [-32], [-16], [24], [4], [36], [42], [-30], [-24], [12], [32], [-36], [46], [-48], [8], [-2], [40], [4], [-24], [-34], [36], [-30], [48], [32], [8], [8], [16], [52], [40], [-12], [30], [-40], [-6], [-12], [-32], [-36], [-22], [20], [56], [40], [8], [-44], [40], [36], [26], [-16], [4], [-8], [8], [0], [-32], [-48], [-8], [58], [-50], [-40], [36], [18], [-12], [-42], [8], [-44], [8], [-12], [8], [50], [8], [48], [-34], [20], [12], [-24], [-10], [-48], [-26], [0], [-34], [36], [40], [-8], [4], [36], [-30], [22], [12], [-56], [60], [-8], [-10], [62], [48], [16], [56], [-12], [0], [-32], [34], [32], [-56], [0], [-8], [58], [-40], [12], [12], [52], [46], [8], [24], [24], [-32], [-46], [-20], [-20], [-24], [-20], [-42], [48], [-56], [16], [-16], [64], [-18], [12], [-4], [16], [12], [0], [-24], [10], [28], [8], [8], [24], [24], [-36], [-10], [-76], [68], [-58], [16], [20], [14], [-16], [-32], [42], [24], [-68], [38], [2], [44], [-64], [-52], [-12], [-72], [-12], [66], [-8], [22], [-56], [-50], [-56], [-60], [-70], [10], [-8], [-16], [-40], [16], [-70], [-16], [84], [62], [44], [-56], [-36], [-20], [0], [24], [-8], [26], [-56], [-56], [24], [-8], [-80], [58], [36], [56], [20], [-88], [-50], [-52], [14], [-44], [18], [-48], [-36], [32], [-88], [44], [38], [-16], [-84], [-34], [-88], [0], [-54], [-16], [-8], [24], [-50], [-8], [16], [68], [20], [38], [-12], [72], [8], [8], [-88], [-38], [-36], [70], [8], [82], [-48], [56], [-72], [-4], [36], [-10], [72], [-48], [80], [-80], [-58], [36], [24], [-14], [84], [48], [0], [24], [84], [2], [-36], [-12], [84], [-80], [10], [56], [-40], [10], [-52], [64], [4], [-28], [24], [-58], [64], [-20], [34], [-52], [-42], [8], [-56], [-88], [80], [16], [-96], [-8], [22], [-56], [-56], [14], [64], [-12], [16], [-12], [-26], [88], [-84], [-8], [-28], [66], [-24], [72], [-92], [40], [-8], [-74], [-24], [-16], [40], [-78], [-76], [76], [-40], [-10], [72], [-84], [-24], [12], [-72], [20], [-38], [-12], [96], [-6], [-64], [-52], [56], [0], [-24]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4032_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4032_2_a_y();".
// 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_4032_2_a_y(:prec:=1)
    chi := MakeCharacter_4032_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    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_4032_2_a_y();".
// 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_4032_2_a_y( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4032_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![0, 1]>,<11,R![-4, 1]>,<13,R![2, 1]>,<17,R![4, 1]>,<19,R![0, 1]>],Snew);
    return Vf;
end function;