
// Make newform 6384.2.a.a 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_a();".
// 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_a();".
// 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], [4], [-1], [6], [-2], [-4], [2], [-2], [8], [-4], [-10], [4], [6], [10], [16], [-6], [10], [12], [6], [16], [-12], [-4], [-8], [10], [-18], [-2], [0], [-2], [-20], [-18], [2], [-22], [24], [0], [-22], [-8], [16], [6], [2], [-18], [16], [2], [-16], [-4], [22], [-18], [6], [-4], [8], [10], [-30], [-6], [-8], [22], [6], [4], [10], [32], [-32], [-22], [18], [-10], [-18], [10], [10], [12], [-22], [-8], [22], [14], [0], [-6], [-30], [-6], [-24], [24], [22], [0], [16], [-28], [-18], [-10], [6], [-12], [-8], [-12], [0], [-10], [-30], [0], [24], [-30], [6], [16], [-46], [18], [-18], [-36], [26], [-20], [-30], [12], [24], [40], [24], [-32], [14], [14], [4], [20], [-34], [4], [32], [42], [-40], [-16], [30], [-26], [-36], [-4], [34], [10], [-48], [-16], [-46], [-44], [36], [-26], [30], [24], [-22], [-38], [48], [42], [34], [0], [6], [16], [0], [36], [16], [-26], [-18], [36], [-4], [18], [-44], [-40], [32], [30], [56], [48], [20], [22], [-22], [42], [-2], [28], [-36], [42], [0], [-22], [34], [-38], [36], [4], [-16], [-54], [34], [-32], [-16], [-28], [34], [16], [-10], [-20], [-52], [42], [-6], [10], [12], [-58], [4], [-42], [-52], [28], [18], [-30], [-50], [18], [14], [-22], [-26], [12], [-16], [-50], [48], [48], [-60], [52], [-60], [-64], [-12], [-6], [-24], [-34], [-68], [56], [-4], [-24], [20], [32], [46], [-24], [-54], [-62], [16], [18], [-56], [-12], [-10], [34], [28], [20], [-22], [18], [-64], [38], [-38], [12], [-4], [-22], [46], [-34], [0], [60], [28], [32], [-10], [18], [48], [-16], [20], [22], [-62], [-62], [-48], [-10], [60], [-32], [54], [-38], [4], [30], [22], [54], [-36], [-22], [20], [-32], [-20], [46], [48], [-46], [-6], [48], [-34], [34], [-36], [32], [-40], [-18], [70], [-30], [-36], [16], [-16], [-72], [28], [44], [-30], [14], [-42], [8], [-62], [-2], [4], [-42], [4], [54], [0], [-60], [-52], [-12], [44], [-34], [-50], [-76], [-20], [56], [-26], [-36], [-4], [-18], [-30], [-76], [10], [30], [82], [-18], [-64], [28], [-80], [-26], [-82], [-52], [54], [-2], [26], [42], [52], [14], [72], [-64], [-86], [-62], [18], [-8], [32], [8], [12], [-70], [6], [-26], [48], [-90], [-82], [86], [14], [-62], [-24], [82], [-30], [-32], [68], [-60], [-46], [70], [80], [6], [-42], [-36], [32], [-22], [8], [-70], [32], [-24], [76], [12], [14], [24], [86], [-78], [-18], [-76], [-54], [-62], [-64], [-38], [-84], [22], [-70], [0], [12], [46], [-78], [20], [-54], [44], [-56], [12], [-82], [-64], [-40], [46], [14], [42], [-92], [44], [46], [-60], [-54], [-74], [18], [20], [58], [-94], [82], [-54], [-8], [-82], [50], [-48], [52], [-40], [-14], [96], [64], [6], [-48], [12], [-88], [4], [24], [18], [80], [18], [-44], [64], [34], [-32], [102], [-52], [100]];
    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_a();".
// 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_a(: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_a();".
// 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_a( : 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![-4, 1]>,<23,R![-6, 1]>],Snew);
    return Vf;
end function;