
// Make newform 6384.2.a.n 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_n();".
// 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_n();".
// 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], [-8], [2], [-4], [2], [6], [12], [8], [10], [4], [-10], [-4], [-8], [10], [4], [0], [6], [-6], [10], [4], [-12], [-6], [6], [20], [8], [-6], [4], [-6], [-4], [14], [-12], [16], [-14], [20], [-14], [8], [26], [-14], [16], [20], [4], [-4], [-2], [-6], [16], [10], [8], [6], [8], [18], [-8], [-10], [-26], [-20], [-6], [-28], [16], [10], [18], [28], [-6], [0], [-18], [10], [16], [8], [2], [-20], [-24], [10], [-2], [6], [18], [0], [10], [24], [18], [-20], [0], [30], [-22], [42], [24], [32], [-40], [4], [24], [-36], [-40], [-30], [38], [20], [-26], [20], [2], [4], [-10], [28], [-14], [0], [-14], [40], [-14], [-12], [14], [34], [-36], [48], [38], [-12], [24], [-14], [12], [-30], [10], [42], [28], [-20], [-6], [-26], [-24], [-40], [30], [-28], [-16], [20], [-2], [-38], [18], [-6], [-44], [2], [-14], [28], [18], [-48], [-20], [10], [-24], [-50], [6], [-4], [-24], [-6], [42], [-36], [-48], [-28], [-16], [48], [26], [-22], [-46], [-48], [38], [8], [36], [54], [-40], [-60], [-26], [34], [-30], [12], [-46], [0], [26], [36], [58], [-20], [10], [4], [-10], [8], [-12], [-14], [-2], [-8], [-14], [-22], [-28], [10], [-16], [66], [24], [20], [18], [-8], [-42], [-62], [-26], [-62], [0], [18], [20], [-38], [-62], [8], [-14], [-48], [-60], [66], [-60], [2], [-14], [0], [68], [24], [-38], [-64], [30], [-48], [-30], [18], [-20], [6], [24], [36], [-74], [-18], [56], [44], [-48], [6], [-36], [44], [-18], [-28], [-48], [34], [-22], [-24], [0], [60], [-12], [56], [-30], [-10], [-8], [8], [-28], [20], [-48], [38], [-14], [48], [10], [50], [-8], [-26], [28], [-6], [42], [76], [20], [-74], [2], [-6], [4], [-14], [58], [20], [-54], [-78], [4], [-54], [-80], [18], [-48], [48], [34], [58], [8], [24], [40], [-56], [10], [28], [0], [2], [-54], [-8], [-26], [-46], [-32], [22], [76], [-78], [-54], [-20], [58], [44], [12], [-54], [18], [80], [-4], [52], [34], [20], [26], [-64], [38], [-40], [18], [38], [-20], [48], [10], [72], [-88], [18], [-54], [-68], [10], [-30], [-36], [-54], [-38], [4], [-84], [16], [-38], [6], [2], [-8], [-80], [-76], [-24], [90], [14], [90], [-48], [50], [26], [-6], [-52], [-62], [-84], [86], [68], [-40], [-54], [-36], [-38], [-6], [-84], [34], [-10], [-96], [-72], [66], [72], [78], [82], [24], [92], [20], [-70], [-62], [60], [50], [88], [20], [48], [10], [-32], [46], [-84], [80], [-78], [90], [-62], [18], [50], [16], [-78], [-28], [-24], [0], [-34], [-76], [56], [-6], [10], [-40], [28], [-88], [26], [-84], [-66], [100], [90], [66], [-86], [-52], [-98], [-38], [72], [-50], [30], [-20], [-8], [26], [-22], [-36], [-20], [-86], [66], [-96], [28], [34], [-88], [-62], [-78], [-104], [36], [-6], [-46], [28], [-6], [100], [-24]];
    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_n();".
// 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_n(: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_n();".
// 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_n( : 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![8, 1]>],Snew);
    return Vf;
end function;