
// Make newform 6384.2.a.v 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_v();".
// 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_v();".
// 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], [4], [2], [-6], [-1], [0], [-10], [-8], [-2], [6], [-4], [8], [6], [4], [-10], [-8], [-12], [-14], [4], [4], [6], [14], [6], [-16], [0], [-10], [2], [-20], [-4], [10], [-4], [-10], [20], [14], [-4], [-24], [18], [0], [18], [-16], [-14], [-26], [0], [0], [-8], [4], [6], [-6], [-8], [22], [-20], [22], [8], [-6], [-8], [-10], [2], [4], [-22], [-12], [24], [10], [-18], [-32], [-22], [-12], [-34], [26], [16], [0], [-26], [-16], [-32], [-10], [-34], [-6], [22], [36], [6], [12], [22], [-40], [12], [18], [-6], [-34], [8], [-36], [0], [28], [20], [-28], [0], [-6], [14], [36], [-2], [-8], [-18], [28], [18], [28], [-14], [-44], [18], [20], [-2], [8], [38], [42], [36], [8], [10], [20], [8], [46], [-8], [10], [26], [18], [32], [-28], [-2], [6], [24], [16], [22], [-12], [-36], [20], [6], [18], [-14], [-54], [-4], [18], [-38], [4], [-42], [16], [-32], [-30], [24], [14], [6], [-20], [44], [-10], [-22], [-28], [32], [-16], [12], [40], [-30], [-46], [-38], [-12], [-54], [8], [-28], [26], [32], [-52], [38], [50], [54], [-24], [2], [-48], [58], [-40], [-6], [-4], [38], [8], [-26], [0], [12], [-2], [22], [-32], [-10], [62], [-44], [-46], [0], [38], [44], [-32], [42], [-44], [46], [50], [-18], [50], [-40], [-18], [40], [-62], [6], [-36], [62], [-32], [48], [26], [44], [2], [-2], [-56], [-12], [-8], [38], [-32], [-10], [28], [14], [-66], [-8], [34], [48], [-36], [-26], [14], [12], [0], [20], [62], [36], [-44], [58], [-12], [0], [-54], [-10], [44], [56], [24], [28], [-48], [46], [54], [56], [56], [-12], [56], [-64], [-2], [18], [8], [-2], [54], [-36], [22], [-44], [2], [66], [12], [-72], [-2], [26], [2], [-12], [46], [42], [-64], [54], [-62], [-24], [-46], [56], [30], [-8], [-60], [-18], [-38], [-68], [-40], [64], [-48], [-62], [44], [-8], [-30], [38], [-24], [-42], [14], [-60], [66], [-60], [-2], [6], [-64], [-2], [84], [44], [-62], [22], [-32], [-16], [68], [42], [-80], [-14], [24], [6], [-64], [-26], [-70], [0], [48], [-10], [20], [-44], [34], [2], [-84], [-54], [58], [-68], [42], [-10], [16], [76], [-20], [-26], [46], [-42], [24], [36], [36], [4], [86], [-34], [26], [-8], [-58], [-54], [42], [-36], [22], [-24], [30], [8], [8], [-10], [28], [90], [-82], [24], [22], [-82], [0], [-12], [18], [36], [-26], [66], [12], [-88], [20], [-30], [-18], [-60], [58], [-60], [20], [-24], [6], [16], [94], [-4], [72], [-14], [-30], [-50], [-70], [-6], [-48], [-94], [84], [24], [-56], [-42], [52], [-64], [10], [26], [-68], [68], [60], [50], [32], [-42], [-24], [-90], [42], [66], [-28], [78], [34], [-8], [-82], [-14], [44], [-4], [-58], [-90], [72], [-60], [-6], [-34], [28], [104], [74], [0], [-10], [-22], [24], [12], [30], [34], [16], [-70], [100], [16]];
    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_v();".
// 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_v(: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_v();".
// 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_v( : 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![6, 1]>,<23,R![0, 1]>],Snew);
    return Vf;
end function;