
// Make newform 6384.2.a.u 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_u();".
// 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_u();".
// 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], [-10], [-10], [-4], [-12], [-6], [4], [14], [4], [16], [10], [-16], [-8], [6], [18], [-10], [0], [-12], [-10], [14], [-8], [0], [2], [-12], [-18], [-24], [-2], [-20], [-24], [-6], [12], [-10], [20], [18], [6], [-16], [-28], [-8], [20], [6], [-6], [-12], [-14], [0], [22], [-4], [-6], [-24], [-26], [14], [-20], [18], [4], [12], [10], [-22], [-12], [10], [-16], [-2], [2], [-20], [24], [-26], [36], [24], [6], [-18], [30], [26], [-16], [-2], [24], [-22], [16], [0], [-26], [26], [-2], [-24], [8], [-4], [-16], [0], [-4], [-12], [42], [-26], [-44], [30], [-20], [38], [36], [-18], [12], [-14], [0], [26], [40], [-30], [16], [6], [2], [4], [-40], [-34], [-12], [28], [-26], [-12], [46], [34], [-38], [20], [4], [-42], [-10], [-28], [-32], [-2], [44], [-48], [48], [22], [42], [18], [-14], [28], [-54], [10], [12], [46], [40], [52], [46], [48], [-26], [22], [20], [-24], [-2], [-14], [12], [-8], [-52], [-32], [24], [-6], [26], [-6], [24], [46], [-24], [-36], [-18], [40], [8], [-26], [10], [6], [4], [38], [20], [42], [0], [50], [-12], [-34], [40], [30], [-56], [-60], [30], [62], [12], [6], [-42], [12], [2], [-12], [18], [-16], [28], [2], [-24], [6], [50], [-34], [-54], [36], [-30], [-16], [22], [34], [-40], [-2], [-24], [-20], [-30], [-4], [34], [62], [16], [-12], [48], [-30], [-24], [-34], [64], [46], [-18], [64], [-18], [-16], [-44], [22], [62], [-24], [-56], [-48], [14], [4], [8], [54], [52], [-28], [-46], [54], [0], [-48], [-4], [-4], [0], [-2], [6], [-60], [32], [-44], [-28], [-4], [-2], [-70], [-12], [-38], [54], [24], [6], [-4], [42], [26], [16], [-12], [-58], [6], [42], [-20], [-30], [2], [-44], [38], [-10], [20], [74], [16], [-14], [-40], [-56], [-50], [42], [-24], [40], [40], [-12], [30], [60], [60], [-78], [-46], [8], [-10], [-10], [-24], [62], [20], [-66], [70], [-8], [46], [-36], [60], [-54], [50], [-40], [-36], [-12], [-22], [28], [-66], [36], [70], [-84], [-42], [30], [-68], [36], [-70], [-48], [48], [34], [-6], [12], [-70], [-14], [-64], [50], [34], [-4], [36], [24], [-50], [46], [-14], [32], [-48], [-68], [-16], [46], [46], [58], [88], [-58], [-78], [-38], [-80], [2], [36], [38], [60], [-16], [-90], [92], [-62], [14], [-64], [-58], [6], [-84], [-16], [-6], [16], [-26], [90], [72], [-28], [28], [90], [-2], [-48], [26], [32], [60], [-36], [-70], [-32], [-82], [-12], [-4], [34], [42], [-38], [66], [-22], [-48], [50], [44], [80], [8], [86], [52], [0], [26], [90], [-16], [-52], [-48], [-30], [-80], [-66], [-4], [-42], [-10], [-46], [0], [-50], [18], [32], [-2], [-90], [52], [-16], [26], [-90], [12], [-28], [-22], [-50], [48], [8], [-78], [96], [-30], [-2], [-36], [-68], [34], [-38], [-36], [-102], [52], [28]];
    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_u();".
// 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_u(: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_u();".
// 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_u( : 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;