
// Make newform 6384.2.a.t in Magma, downloaded from the LMFDB on 19 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_t();".
// 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_t();".
// 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], [-2], [-6], [-4], [-1], [4], [-2], [6], [-4], [6], [4], [6], [6], [-4], [-6], [14], [-8], [10], [0], [-8], [6], [16], [-2], [10], [12], [16], [-2], [0], [-4], [10], [20], [0], [8], [-10], [16], [-16], [-14], [-24], [-6], [12], [14], [-16], [-20], [10], [-14], [12], [-26], [14], [0], [-4], [-12], [-6], [-12], [-26], [24], [14], [-10], [-4], [6], [20], [2], [10], [-6], [34], [6], [18], [22], [-24], [32], [16], [-8], [26], [-32], [-8], [18], [18], [-16], [24], [0], [24], [32], [6], [-26], [34], [-26], [34], [8], [8], [-2], [-20], [30], [20], [42], [6], [-42], [-20], [18], [6], [-32], [4], [30], [-24], [-38], [28], [-36], [24], [44], [-42], [-6], [-34], [-4], [-8], [2], [-20], [6], [36], [16], [-2], [50], [42], [-12], [-20], [16], [-34], [-30], [44], [30], [-12], [-24], [-12], [-6], [8], [10], [-26], [-48], [-30], [22], [-4], [20], [-16], [-48], [46], [-32], [26], [54], [-20], [8], [-20], [24], [4], [36], [42], [-48], [32], [-36], [-58], [-50], [38], [-10], [40], [36], [-30], [-36], [-20], [38], [-10], [-54], [20], [-14], [60], [6], [30], [28], [60], [-28], [-14], [-22], [-24], [-20], [16], [58], [-52], [18], [-36], [28], [-30], [-6], [-8], [54], [-10], [-46], [18], [-34], [62], [34], [-48], [30], [30], [-18], [-34], [24], [-12], [-18], [28], [-60], [18], [-56], [-10], [22], [72], [-28], [24], [20], [32], [-10], [-32], [-64], [-32], [50], [30], [8], [12], [-10], [2], [32], [42], [-54], [-30], [-16], [-28], [54], [-44], [-30], [-14], [-36], [-54], [56], [-8], [28], [-8], [12], [-70], [66], [-12], [-20], [46], [48], [58], [-12], [-48], [40], [14], [-74], [-58], [48], [34], [62], [-16], [32], [-26], [-74], [0], [44], [6], [52], [50], [24], [18], [22], [18], [-32], [28], [-48], [-14], [60], [-14], [-12], [-12], [24], [-6], [-34], [-12], [-8], [30], [-22], [-20], [-54], [48], [-24], [54], [20], [20], [-78], [10], [-30], [36], [36], [46], [-18], [-64], [-48], [52], [2], [-4], [62], [40], [62], [-30], [76], [82], [10], [8], [-76], [-24], [8], [-34], [6], [88], [-74], [30], [-4], [22], [20], [-14], [-16], [64], [44], [26], [38], [-60], [-32], [-16], [32], [-36], [22], [-58], [20], [-64], [90], [42], [-48], [-66], [-96], [26], [-74], [24], [54], [16], [62], [24], [-78], [64], [-58], [-10], [-60], [-58], [-56], [-38], [-28], [80], [-20], [20], [-34], [66], [24], [46], [34], [68], [48], [94], [32], [-2], [-36], [-20], [34], [80], [68], [-30], [22], [8], [18], [68], [-4], [16], [-62], [20], [64], [62], [-30], [-50], [4], [56], [90], [-10], [-46], [66], [56], [-82], [78], [-28], [82], [30], [4], [-74], [30], [-32], [16], [-92], [8], [52], [24], [-58], [98], [16], [38], [-60], [-100], [-34], [-10], [4], [-84], [-40], [82], [-52], [78], [60], [-14]];
    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_t();".
// 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_t(: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_t();".
// 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_t( : 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![2, 1]>,<13,R![6, 1]>,<17,R![4, 1]>,<23,R![-4, 1]>],Snew);
    return Vf;
end function;