
// Make newform 6384.2.a.m in Magma, downloaded from the LMFDB on 24 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_m();".
// 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_m();".
// 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], [2], [1], [0], [2], [-2], [-1], [-4], [-2], [0], [-2], [-6], [-4], [0], [-10], [-12], [-10], [8], [0], [-6], [4], [-4], [10], [-2], [2], [-8], [4], [-10], [2], [-4], [-12], [2], [20], [6], [12], [14], [20], [0], [6], [12], [2], [-12], [-14], [-10], [-16], [8], [8], [-12], [14], [18], [-12], [6], [12], [10], [-12], [6], [-16], [-10], [-6], [-12], [-2], [4], [0], [26], [30], [8], [-22], [-24], [-2], [-18], [-28], [-32], [6], [-8], [0], [-18], [30], [18], [22], [-28], [-10], [-24], [-2], [8], [-32], [-14], [26], [10], [-16], [-4], [-8], [-12], [40], [4], [32], [6], [-22], [-36], [30], [8], [-2], [44], [10], [-12], [34], [-44], [-42], [-16], [-10], [8], [6], [-30], [12], [-16], [-46], [20], [-16], [6], [-36], [2], [10], [-10], [-4], [12], [6], [22], [24], [8], [-50], [-4], [24], [12], [22], [-50], [-14], [-18], [52], [6], [-6], [44], [6], [8], [-12], [-14], [-16], [30], [-6], [28], [48], [-58], [-18], [4], [24], [8], [-24], [40], [-2], [34], [-10], [0], [-54], [-32], [-36], [50], [-16], [36], [54], [-46], [34], [12], [-22], [-12], [26], [40], [30], [-12], [-50], [16], [38], [-40], [-44], [46], [18], [-4], [50], [30], [60], [50], [32], [38], [-56], [56], [62], [-32], [-22], [-30], [-2], [-2], [16], [46], [40], [18], [-10], [-44], [42], [-16], [36], [-6], [-4], [-62], [-46], [-40], [-4], [16], [14], [-64], [-46], [0], [54], [30], [0], [2], [0], [36], [70], [34], [16], [40], [-8], [46], [36], [60], [-54], [-44], [16], [-46], [6], [48], [40], [-28], [-28], [32], [-34], [-38], [32], [-64], [4], [40], [-52], [-66], [-42], [12], [-34], [-46], [72], [-58], [52], [54], [34], [44], [-44], [38], [34], [6], [-20], [14], [30], [-32], [-18], [-38], [32], [26], [8], [30], [-16], [-64], [-34], [-38], [-36], [24], [-40], [16], [-22], [36], [60], [34], [22], [0], [-22], [-66], [-36], [-6], [4], [-34], [18], [80], [2], [-76], [-44], [50], [46], [-8], [84], [-52], [-86], [-52], [82], [-36], [-26], [48], [-2], [-78], [-56], [-20], [6], [12], [-56], [26], [-70], [44], [26], [-34], [-76], [58], [22], [0], [-4], [40], [-34], [46], [30], [-64], [36], [28], [-44], [-26], [66], [-62], [64], [38], [-46], [-10], [-76], [-2], [60], [22], [-64], [-40], [-78], [-68], [-70], [-34], [24], [-74], [-22], [0], [-20], [18], [80], [-26], [-2], [-24], [44], [92], [82], [-62], [-60], [-6], [-24], [-28], [76], [54], [-88], [-98], [-76], [20], [34], [-42], [30], [78], [26], [80], [50], [-20], [-72], [40], [-2], [-36], [88], [74], [-90], [-24], [-28], [8], [-14], [64], [18], [-104], [-2], [66], [66], [-76], [50], [-18], [-56], [-18], [-46], [4], [60], [-2], [62], [-44], [52], [58], [-14], [16], [-48], [62], [-40], [-34], [-46], [60], [52], [6], [102], [76], [50], [-4], [-24]];
    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_m();".
// 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_m(: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_m();".
// 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_m( : 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![0, 1]>,<13,R![-2, 1]>,<17,R![2, 1]>,<23,R![4, 1]>],Snew);
    return Vf;
end function;