
// Make newform 4004.2.a.c in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4004_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_4004_2_a_c();".
// 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_4004_2_a_c();".
// 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_4004_a();"
function MakeCharacter_4004_a()
    N := 4004;
    order := 1;
    char_gens := [2003, 3433, 365, 925];
    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_4004_a_Hecke(Kf)
    return MakeCharacter_4004_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], [2], [4], [1], [1], [-1], [2], [4], [8], [-10], [8], [2], [-10], [-10], [-12], [-10], [12], [2], [-4], [0], [2], [10], [-4], [12], [-8], [10], [-14], [2], [16], [-6], [-14], [0], [2], [-12], [-4], [-16], [22], [4], [0], [-18], [-4], [2], [12], [12], [12], [-26], [-10], [-4], [4], [16], [6], [8], [22], [2], [-18], [18], [-10], [-16], [22], [12], [-28], [-18], [-20], [10], [22], [10], [28], [18], [-18], [-10], [-12], [-4], [2], [6], [-12], [-8], [22], [32], [-30], [-6], [2], [-38], [-4], [-26], [24], [36], [10], [-8], [-18], [-16], [-6], [8], [-16], [-42], [36], [-40], [0], [6], [16], [-8], [10], [-20], [-32], [-10], [46], [-16], [-16], [18], [12], [-26], [-12], [-20], [-22], [8], [32], [-18], [-8], [-26], [34], [-6], [-40], [10], [-6], [-4], [20], [10], [42], [34], [26], [-10], [-20], [20], [-8], [-22], [26], [10], [-40], [44], [42], [-34], [-12], [-24], [-24], [36], [54], [-48], [-6], [10], [14], [-24], [-24], [30], [16], [20], [40], [-36], [-46], [4], [42], [22], [36], [54], [-40], [-54], [42], [-36], [-28], [58], [-16], [-26], [-52], [4], [-38], [4], [50], [-6], [-60], [10], [-12], [-50], [-40], [56], [-10], [-2], [8], [-22], [2], [28], [24], [24], [-42], [28], [4], [20], [22], [54], [52], [22], [-14], [36], [8], [-12], [-28], [66], [-12], [18], [-4], [14], [64], [-2], [-2], [-22], [-56], [0], [-56], [-8], [-42], [-42], [-48], [18], [10], [-24], [-66], [8], [-4], [-34], [-62], [32], [32], [12], [-62], [72], [-72], [42], [-14], [72], [64], [36], [36], [14], [-60], [0], [14], [10], [-50], [-60], [52], [-56], [44], [-66], [32], [-46], [48], [74], [-70], [-76], [2], [-76], [18], [-2], [-10], [-6], [-10], [-30], [-24], [50], [14], [0], [76], [46], [-10], [-44], [40], [-10], [62], [-40], [-60], [64], [20], [0], [36], [-16], [-28], [26], [52], [-4], [30], [-8], [-52], [30], [-46], [34], [32], [-12], [6], [-50], [66], [2], [48], [52], [74], [44], [-10], [-36], [4], [-50], [-12], [50], [0], [2], [44], [-74], [38], [64], [-16], [70], [-12], [-24], [-44], [-46], [48], [90], [78], [-40], [-28], [82], [60], [-28], [34], [-42], [48], [26], [80], [84], [-56], [72], [-42], [46], [-16], [48], [-38], [22], [-14], [52], [-70], [-62], [-10], [68], [88], [-54], [20], [-54], [22], [64], [78], [-46], [-36], [-60], [30], [32], [-36], [58], [-52], [84], [26], [6], [-30], [-52], [18], [-4], [-50], [20], [90], [24], [2], [28], [66], [-18], [-34], [-10], [-44], [-14], [-4], [72], [-44], [12], [40], [70], [-32], [-66], [66], [12], [44], [46], [24], [-86], [-48], [-54], [36], [24], [-10], [-66], [-20], [-44], [-30], [20], [26], [52], [-28], [44], [-62], [90], [-64], [50], [-50], [-62], [-36], [14], [-48], [-40], [14], [-26], [-72], [-4], [60], [48], [20], [48], [-16], [-12]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4004_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4004_2_a_c();".
// 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_4004_2_a_c(:prec:=1)
    chi := MakeCharacter_4004_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_4004_2_a_c();".
// 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_4004_2_a_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4004_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-2, 1]>],Snew);
    return Vf;
end function;