
// Make newform 6384.2.a.f in Magma, downloaded from the LMFDB on 23 December 2025.
// 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_f();".
// 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_f();".
// 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 := 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]];
    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], [-2], [-1], [4], [2], [2], [1], [0], [-2], [8], [-2], [-2], [-4], [0], [-2], [12], [14], [-8], [12], [-14], [4], [4], [-18], [-10], [-10], [-16], [-8], [6], [-6], [12], [12], [10], [4], [-10], [20], [6], [-4], [8], [18], [8], [2], [0], [-14], [-10], [16], [-16], [-8], [28], [14], [10], [8], [14], [12], [30], [8], [26], [8], [22], [-22], [-4], [26], [-20], [0], [26], [22], [16], [-22], [4], [-10], [18], [16], [0], [6], [32], [0], [-26], [22], [18], [-2], [20], [-10], [-12], [14], [-40], [12], [-6], [-6], [-2], [-40], [12], [-24], [12], [20], [-28], [-24], [26], [6], [28], [-2], [-8], [14], [36], [-6], [-20], [18], [-12], [-22], [-36], [-26], [24], [-10], [42], [-4], [8], [2], [-4], [0], [-18], [-16], [-38], [-22], [18], [8], [-20], [-2], [22], [0], [-48], [30], [20], [-28], [20], [22], [42], [2], [-6], [52], [18], [-6], [-20], [22], [-16], [24], [-14], [8], [22], [30], [4], [20], [22], [-30], [4], [0], [-32], [-28], [56], [-6], [-30], [42], [-12], [50], [-40], [12], [-14], [-16], [-20], [-18], [34], [-42], [0], [34], [48], [-38], [24], [50], [44], [-26], [-24], [30], [32], [36], [14], [-34], [32], [22], [-50], [-20], [50], [40], [14], [-36], [64], [-54], [-28], [-42], [18], [-2], [26], [-16], [6], [-56], [-30], [14], [12], [-18], [-16], [24], [-54], [52], [-30], [30], [-8], [-20], [24], [-50], [64], [-18], [36], [-2], [-50], [40], [26], [-16], [-60], [22], [22], [36], [32], [4], [-18], [-20], [52], [18], [28], [-72], [26], [-58], [-52], [72], [-16], [4], [48], [-18], [-34], [0], [-40], [12], [-40], [-48], [-66], [42], [40], [-26], [6], [76], [-58], [28], [18], [2], [28], [0], [38], [-38], [42], [-68], [6], [-30], [48], [6], [-30], [72], [-46], [-40], [-74], [24], [-12], [46], [-6], [44], [72], [32], [-24], [34], [-44], [72], [-30], [-18], [-40], [30], [-18], [68], [42], [-84], [-34], [6], [64], [14], [-36], [-44], [2], [-2], [-16], [24], [-28], [10], [-72], [-30], [40], [38], [-24], [-10], [-46], [-16], [-32], [14], [-44], [12], [50], [-30], [36], [10], [42], [76], [-6], [-82], [-48], [-76], [20], [6], [-50], [94], [-56], [-44], [68], [52], [54], [-58], [10], [-8], [-26], [-22], [42], [92], [62], [0], [86], [8], [40], [-58], [20], [-22], [30], [24], [54], [-74], [-40], [-12], [66], [-20], [-58], [-54], [-44], [0], [76], [-46], [-66], [36], [-38], [-44], [28], [8], [-34], [32], [62], [-60], [72], [50], [-86], [-26], [-70], [90], [-48], [-30], [28], [40], [8], [-18], [36], [0], [-22], [-86], [-84], [28], [-28], [-46], [32], [78], [40], [54], [74], [-62], [4], [70], [-94], [-88], [46], [90], [36], [44], [78], [54], [-16], [20], [26], [-50], [20], [24], [82], [-32], [30], [26], [-72], [-60], [86], [-14], [-24], [106], [-52], [8]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6384_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6384_2_a_f();".
// 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_f(: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_f();".
// 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_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6384_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![2, 1]>,<11,R![-4, 1]>,<13,R![-2, 1]>,<17,R![-2, 1]>,<23,R![0, 1]>],Snew);
    return Vf;
end function;