
// Make newform 6336.2.a.bw 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_bw();".
// 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_bw();".
// 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], [-2], [-4], [-4], [6], [0], [-6], [6], [-4], [12], [2], [12], [14], [-4], [12], [-6], [-4], [4], [-10], [-14], [14], [0], [4], [6], [-2], [12], [4], [-2], [4], [-10], [4], [10], [20], [0], [-10], [20], [2], [12], [10], [-2], [-16], [4], [0], [12], [-14], [-10], [-8], [10], [-4], [-2], [24], [26], [20], [-26], [22], [-4], [22], [-4], [-4], [26], [18], [-20], [18], [-4], [6], [-18], [0], [-16], [14], [28], [4], [-30], [-22], [38], [-14], [-12], [10], [0], [2], [4], [28], [22], [34], [14], [-24], [12], [0], [-16], [-28], [4], [-32], [-22], [-18], [-20], [38], [28], [30], [-20], [-10], [20], [-46], [-36], [30], [-36], [10], [28], [6], [22], [4], [-8], [-42], [28], [28], [18], [-36], [18], [26], [46], [12], [12], [30], [18], [-12], [0], [22], [-44], [32], [32], [-38], [-42], [10], [18], [20], [-54], [-10], [28], [-2], [-40], [-52], [-46], [-12], [6], [-26], [-4], [-20], [-42], [14], [-4], [0], [-28], [-12], [-4], [-42], [-38], [-42], [12], [6], [-44], [12], [-26], [-36], [32], [14], [10], [-22], [-52], [10], [48], [-22], [48], [38], [52], [42], [12], [22], [64], [36], [-6], [54], [36], [26], [-2], [-36], [-30], [32], [-46], [60], [-60], [-54], [-4], [22], [-30], [-38], [-66], [-24], [46], [52], [2], [-30], [-36], [2], [56], [4], [6], [4], [42], [-6], [-56], [-60], [-48], [-70], [-68], [14], [-4], [34], [38], [4], [-10], [-32], [44], [30], [-10], [-36], [52], [-60], [-62], [68], [20], [-58], [-20], [-48], [50], [-42], [60], [-28], [12], [-20], [-8], [10], [-2], [16], [-72], [-20], [-44], [-8], [14], [30], [68], [-6], [14], [-12], [58], [-52], [-6], [-78], [-44], [-12], [-66], [6], [6], [-20], [34], [54], [-12], [-34], [10], [12], [10], [28], [-14], [-16], [52], [-2], [-38], [44], [48], [64], [0], [-2], [36], [-60], [-62], [-18], [48], [14], [26], [-12], [-42], [-12], [-10], [14], [-8], [50], [12], [52], [-46], [-2], [4], [60], [20], [18], [84], [-14], [-12], [-50], [-24], [42], [78], [-12], [-40], [-38], [44], [64], [66], [-18], [-84], [-38], [-2], [40], [86], [50], [-36], [-68], [24], [-2], [30], [-6], [68], [-4], [60], [36], [42], [-50], [42], [28], [-46], [78], [70], [-8], [-54], [52], [50], [28], [40], [-62], [-92], [26], [-6], [12], [6], [38], [-44], [-76], [-18], [-28], [70], [-58], [20], [-28], [28], [-22], [14], [-12], [50], [36], [-84], [-48], [30], [-60], [-14], [-100], [8], [98], [14], [-94], [-22], [-18], [-28], [-50], [-44], [84], [-80], [74], [-36], [-84], [82], [-6], [-20], [36], [4], [-94], [-12], [62], [60], [30], [14], [54], [-92], [-82], [70], [-52], [-62], [30], [-4], [76], [-46], [70], [-60], [12], [10], [50], [88], [-32], [-74], [96], [-30], [6], [100], [-12], [26], [-6], [-36], [70], [-4], [-72]];
    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_bw();".
// 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_bw(: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_bw();".
// 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_bw( : 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![2, 1]>,<19,R![4, 1]>,<23,R![4, 1]>,<47,R![-12, 1]>],Snew);
    return Vf;
end function;