
// Make newform 6384.2.a.q in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6384_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_6384_2_a_q();".
// 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_6384_2_a_q();".
// 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_6384_a();"
function MakeCharacter_6384_a()
    N := 6384;
    order := 1;
    char_gens := [799, 4789, 2129, 913, 1009];
    v := [1, 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_6384_a_Hecke(Kf)
    return MakeCharacter_6384_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], [-1], [4], [1], [2], [4], [0], [1], [6], [10], [0], [6], [-10], [-8], [-12], [-6], [12], [-2], [2], [12], [-6], [-2], [0], [-2], [-12], [-4], [16], [-4], [6], [-14], [18], [-4], [-18], [-12], [-6], [-2], [10], [-16], [-16], [2], [4], [-8], [-10], [2], [-14], [-24], [-22], [-4], [12], [14], [-18], [6], [8], [-4], [-22], [26], [-6], [8], [22], [26], [-4], [2], [8], [24], [-22], [6], [-2], [-2], [-2], [34], [16], [-14], [0], [26], [6], [24], [30], [-30], [-18], [-4], [-36], [-22], [12], [4], [0], [-6], [18], [22], [20], [32], [-24], [-8], [26], [-2], [-16], [24], [26], [30], [-16], [2], [2], [34], [-4], [22], [-44], [10], [8], [36], [-12], [-12], [-20], [-26], [6], [-4], [-44], [-6], [-44], [16], [-42], [12], [-32], [22], [-18], [24], [4], [6], [-30], [24], [-8], [-30], [12], [24], [50], [38], [-36], [26], [26], [-40], [-38], [18], [-8], [50], [24], [36], [-28], [-40], [-10], [22], [4], [-24], [-50], [-24], [-56], [-32], [-2], [-12], [16], [32], [54], [50], [46], [-22], [44], [36], [30], [0], [-26], [-14], [-14], [-28], [24], [-24], [18], [-46], [-20], [44], [44], [-2], [-60], [22], [28], [12], [-34], [-22], [-54], [44], [10], [-52], [-26], [44], [64], [6], [10], [-18], [-34], [54], [26], [30], [0], [-24], [-46], [-20], [-48], [-56], [40], [-52], [0], [-24], [42], [8], [-66], [-12], [8], [20], [16], [16], [16], [70], [-4], [-42], [14], [-52], [-34], [40], [4], [-10], [-38], [-8], [-32], [30], [-14], [-16], [58], [22], [-36], [-36], [10], [-46], [-30], [40], [-24], [-12], [0], [10], [-54], [-16], [32], [28], [14], [-54], [2], [12], [-10], [-32], [-56], [-38], [50], [-36], [30], [-34], [-14], [40], [26], [20], [-28], [-44], [66], [-20], [-22], [46], [40], [-74], [10], [52], [76], [-8], [2], [-54], [-38], [40], [-56], [24], [40], [-68], [-68], [18], [-2], [-54], [-32], [18], [-62], [8], [-6], [84], [34], [-32], [8], [-4], [4], [-68], [30], [-54], [-76], [32], [-48], [30], [24], [60], [-82], [58], [-36], [-10], [-54], [66], [70], [-52], [-72], [-12], [-74], [78], [28], [-74], [78], [-2], [-6], [16], [14], [64], [-20], [54], [82], [-66], [-56], [-60], [80], [-64], [38], [-42], [-58], [56], [10], [38], [54], [26], [-34], [44], [-14], [-46], [24], [92], [68], [-54], [34], [36], [-38], [-34], [-36], [-60], [-22], [36], [-6], [28], [-12], [16], [-20], [46], [56], [26], [18], [-6], [-60], [-54], [6], [-32], [-14], [-76], [38], [2], [36], [-24], [86], [-30], [92], [-22], [84], [-8], [52], [-98], [88], [72], [22], [54], [-66], [28], [64], [70], [8], [-54], [-66], [-18], [20], [2], [-34], [-54], [-102], [-40], [30], [86], [-32], [48], [-76], [38], [-20], [32], [-42], [-48], [-24], [52], [0], [0], [30], [16], [58], [-44], [-28], [90], [-52], [-18], [-84], [-12]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6384_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6384_2_a_q();".
// 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_6384_2_a_q(:prec:=1)
    chi := MakeCharacter_6384_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_6384_2_a_q();".
// 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_6384_2_a_q( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6384_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-4, 1]>,<11,R![-2, 1]>,<13,R![-4, 1]>,<17,R![0, 1]>,<23,R![-6, 1]>],Snew);
    return Vf;
end function;