
// Make newform 6384.2.a.s 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_s();".
// 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_s();".
// 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], [6], [1], [0], [-10], [4], [2], [10], [-4], [-12], [-2], [4], [-2], [-8], [0], [-6], [8], [4], [10], [-10], [-18], [20], [12], [-14], [-14], [8], [-20], [-6], [4], [-14], [0], [-2], [-4], [-24], [14], [12], [-26], [8], [-22], [18], [-16], [-16], [-4], [12], [-26], [2], [0], [-18], [-12], [-22], [32], [-18], [-16], [-10], [10], [4], [6], [-28], [-12], [10], [-2], [8], [-22], [24], [-2], [-34], [24], [-16], [-6], [-8], [0], [18], [-18], [34], [30], [20], [10], [-32], [22], [-4], [-16], [34], [-22], [14], [16], [-28], [-28], [24], [-32], [20], [36], [-2], [-6], [-20], [38], [8], [-6], [28], [-38], [-12], [2], [4], [6], [24], [-26], [-28], [-18], [34], [36], [8], [18], [20], [28], [2], [-4], [14], [34], [6], [12], [36], [26], [-10], [-36], [0], [46], [4], [16], [0], [-2], [22], [2], [-26], [-20], [-2], [34], [44], [-22], [-16], [-12], [-10], [-40], [-42], [10], [-44], [-40], [-38], [14], [-28], [48], [-40], [32], [16], [6], [2], [-18], [8], [-22], [32], [4], [-30], [0], [16], [-42], [10], [54], [12], [62], [-16], [-54], [-60], [30], [-36], [-22], [36], [14], [8], [20], [-38], [-14], [40], [-42], [42], [-60], [-38], [-20], [-10], [-32], [-8], [-34], [-32], [42], [18], [22], [30], [-36], [22], [4], [14], [-10], [-28], [-42], [40], [52], [10], [-44], [34], [-66], [8], [60], [40], [-10], [-8], [-62], [-40], [42], [-6], [36], [-30], [24], [36], [22], [66], [8], [-28], [24], [-10], [12], [48], [-54], [-12], [44], [34], [74], [-24], [72], [44], [44], [8], [10], [-6], [-60], [-40], [52], [16], [32], [22], [6], [-24], [46], [-42], [-56], [-10], [4], [-10], [66], [-8], [-4], [-10], [38], [-18], [-76], [-10], [78], [8], [18], [-34], [-8], [-38], [-8], [-66], [56], [40], [58], [10], [12], [24], [-40], [36], [-34], [-60], [-48], [18], [-58], [80], [42], [58], [-12], [42], [28], [26], [-2], [-20], [-58], [-76], [-12], [82], [6], [-64], [-52], [-52], [2], [44], [22], [88], [-10], [44], [-6], [-14], [80], [8], [54], [-28], [-40], [-70], [-22], [-20], [-86], [14], [32], [-30], [62], [48], [52], [-72], [-6], [-42], [-90], [40], [-28], [52], [28], [-22], [-62], [34], [-8], [74], [90], [-58], [-16], [-26], [12], [70], [16], [-48], [46], [-52], [-46], [-6], [-84], [74], [74], [-60], [36], [74], [64], [14], [-18], [-16], [60], [-28], [50], [-50], [0], [-30], [-48], [-4], [80], [-82], [88], [-50], [-84], [88], [2], [-2], [-42], [-2], [34], [-16], [18], [-76], [-48], [-32], [-74], [-36], [-48], [26], [-42], [-96], [-44], [-8], [-86], [-76], [-46], [-88], [-30], [-42], [-6], [-32], [-78], [-90], [-64], [62], [-30], [92], [-76], [-34], [66], [60], [-20], [-6], [78], [-72], [12], [54], [80], [-10], [-34], [72], [-52], [-18], [-50], [4], [66], [28], [-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_s();".
// 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_s(: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_s();".
// 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_s( : 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![-6, 1]>,<23,R![0, 1]>],Snew);
    return Vf;
end function;