
// Make newform 6336.2.a.o in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6336_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_6336_2_a_o();".
// 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_6336_2_a_o();".
// 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_6336_a();"
function MakeCharacter_6336_a()
    N := 6336;
    order := 1;
    char_gens := [4159, 4357, 3521, 1729];
    v := [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_6336_a_Hecke(Kf)
    return MakeCharacter_6336_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], [0], [-2], [-4], [1], [-6], [-6], [-8], [0], [-6], [0], [-6], [10], [-8], [0], [6], [-4], [2], [-12], [-8], [2], [4], [12], [6], [2], [2], [0], [-12], [-6], [-10], [-12], [-12], [-18], [-16], [-6], [-4], [2], [-4], [0], [-22], [-4], [10], [16], [2], [18], [8], [-8], [-8], [-4], [26], [26], [-24], [-22], [4], [-18], [0], [30], [-20], [-30], [10], [24], [-6], [8], [-16], [-22], [22], [4], [-6], [4], [18], [14], [8], [-8], [26], [4], [-24], [14], [-14], [-10], [18], [-36], [-22], [-8], [2], [20], [-20], [-2], [-14], [-14], [-24], [-12], [-40], [-24], [-28], [4], [16], [-2], [14], [40], [-22], [-32], [-30], [-12], [-30], [8], [34], [28], [-14], [16], [-38], [-44], [-6], [-18], [-44], [-8], [14], [-4], [0], [30], [-12], [-22], [34], [34], [-12], [12], [18], [-6], [0], [-8], [-38], [32], [-24], [32], [-6], [50], [-14], [6], [-32], [-42], [-30], [32], [-46], [16], [36], [-14], [-16], [-54], [34], [4], [24], [18], [22], [-44], [-24], [4], [48], [20], [54], [-38], [-6], [12], [2], [-4], [60], [-42], [0], [-16], [-22], [-22], [-26], [-36], [2], [-8], [-14], [-24], [30], [-8], [22], [-36], [50], [-48], [-44], [-54], [50], [-16], [-42], [26], [12], [18], [-8], [-14], [36], [20], [-10], [-44], [-10], [-30], [-62], [42], [40], [-30], [-28], [10], [-6], [20], [38], [-8], [20], [2], [28], [-30], [6], [-56], [44], [40], [-38], [-52], [2], [24], [-2], [-46], [60], [6], [-40], [-28], [26], [-66], [-40], [20], [36], [34], [-56], [68], [-30], [44], [32], [34], [-54], [-12], [-48], [52], [-8], [-56], [-14], [2], [-72], [48], [12], [-40], [-64], [-38], [66], [8], [-6], [18], [44], [58], [16], [-34], [42], [-36], [4], [-14], [18], [-34], [20], [-50], [14], [0], [66], [50], [60], [-22], [-20], [-22], [24], [-12], [2], [-70], [52], [-8], [40], [8], [10], [-24], [72], [-46], [18], [72], [-46], [-10], [4], [-38], [-52], [66], [-30], [-64], [22], [-28], [56], [10], [-62], [-44], [52], [-68], [18], [84], [34], [16], [58], [-56], [14], [-78], [-12], [-80], [34], [-60], [32], [82], [-22], [-24], [-38], [66], [-16], [-22], [82], [36], [-68], [-72], [34], [-6], [-42], [20], [-4], [24], [-68], [-30], [82], [42], [-28], [10], [-10], [74], [-32], [-34], [-20], [-38], [-60], [0], [-42], [8], [-86], [-34], [-60], [42], [34], [-96], [84], [42], [24], [10], [10], [72], [-60], [-28], [10], [50], [-52], [26], [-60], [-12], [96], [50], [-60], [-14], [-20], [-24], [-14], [90], [2], [6], [-74], [76], [10], [-16], [-24], [96], [-70], [-24], [32], [34], [86], [68], [-12], [48], [-46], [52], [-58], [-12], [34], [50], [54], [-28], [-10], [10], [-52], [2], [-30], [20], [12], [98], [42], [84], [0], [-14], [-10], [72], [-16], [62], [80], [30], [90], [48], [-60], [10], [-82], [44], [-54], [68], [-32]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6336_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6336_2_a_o();".
// 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_6336_2_a_o(:prec:=1)
    chi := MakeCharacter_6336_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_6336_2_a_o();".
// 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_6336_2_a_o( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6336_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![2, 1]>,<7,R![4, 1]>,<13,R![6, 1]>,<17,R![6, 1]>,<19,R![8, 1]>,<23,R![0, 1]>,<47,R![0, 1]>],Snew);
    return Vf;
end function;