
// Make newform 3344.2.a.g in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3344_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_3344_2_a_g();".
// 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_3344_2_a_g();".
// 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_3344_a();"
function MakeCharacter_3344_a()
    N := 3344;
    order := 1;
    char_gens := [2927, 837, 2433, 705];
    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_3344_a_Hecke(Kf)
    return MakeCharacter_3344_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], [-2], [-1], [-2], [6], [-1], [8], [-6], [-6], [8], [6], [8], [8], [12], [0], [-8], [8], [6], [-14], [12], [12], [2], [-2], [0], [-14], [12], [10], [18], [8], [-4], [22], [-16], [0], [-12], [-2], [-12], [-8], [14], [-20], [8], [-12], [-18], [12], [24], [12], [10], [4], [2], [18], [-6], [-22], [-12], [26], [6], [-16], [6], [8], [26], [-28], [10], [-4], [28], [6], [12], [12], [-14], [-24], [-8], [-2], [14], [4], [-34], [4], [6], [-6], [38], [-10], [2], [12], [8], [20], [22], [24], [-36], [6], [-26], [-16], [-8], [-20], [-14], [18], [8], [-36], [26], [24], [-34], [-28], [-8], [-4], [-8], [-28], [-10], [12], [-2], [-4], [-30], [-30], [-22], [-28], [4], [-18], [4], [8], [30], [-12], [32], [42], [-20], [-28], [14], [-30], [20], [-20], [-20], [26], [44], [4], [-48], [20], [-48], [6], [-6], [22], [-18], [-8], [-20], [40], [-18], [-20], [8], [-16], [12], [-20], [30], [-40], [-14], [-28], [54], [22], [-42], [36], [0], [32], [2], [-38], [-34], [18], [-34], [-12], [38], [50], [36], [62], [-10], [-22], [48], [-38], [-6], [12], [-40], [10], [34], [14], [-10], [-40], [-14], [28], [12], [52], [44], [52], [14], [-52], [-6], [46], [-4], [6], [-38], [42], [-44], [-28], [-4], [-12], [-22], [-30], [-50], [-54], [-14], [-34], [40], [28], [-10], [12], [-18], [-20], [-52], [-6], [-4], [14], [-18], [12], [-4], [64], [22], [22], [66], [2], [-50], [46], [8], [-50], [-24], [-12], [4], [66], [30], [-12], [-36], [58], [-20], [32], [-26], [-28], [2], [30], [-20], [60], [-6], [-12], [72], [-40], [-48], [46], [34], [-12], [-76], [-4], [54], [-4], [58], [40], [10], [0], [-48], [-2], [76], [24], [-58], [28], [12], [56], [-38], [-14], [-60], [0], [74], [36], [-48], [-56], [68], [-38], [18], [-78], [44], [-20], [14], [54], [36], [8], [-80], [-62], [-38], [16], [0], [78], [46], [-72], [18], [-26], [12], [-22], [-84], [-14], [-36], [74], [-26], [52], [20], [14], [-18], [-70], [12], [-52], [22], [80], [20], [40], [52], [6], [-22], [34], [56], [62], [70], [28], [24], [-62], [-62], [-28], [22], [78], [-2], [-26], [-26], [-40], [-64], [-12], [-36], [-12], [20], [-38], [-60], [-12], [52], [-68], [34], [78], [38], [60], [66], [-74], [-46], [68], [-84], [-50], [-4], [0], [90], [-20], [-66], [-82], [96], [-86], [-54], [68], [24], [54], [-22], [20], [10], [-30], [28], [4], [-6], [-84], [-56], [-22], [60], [-56], [34], [-78], [2], [54], [-48], [-90], [2], [30], [-42], [84], [58], [-70], [-6], [60], [-50], [88], [-42], [24], [-2], [34], [-24], [-44], [76], [42], [6], [-68], [-34], [-52], [24], [22], [62], [12], [78], [66], [-94], [-58], [90], [36], [60], [-26], [12], [52], [64], [-94], [78], [-24], [2], [58], [24], [-48], [18], [-28], [36], [-90], [16], [12], [-22], [28], [-38]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3344_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3344_2_a_g();".
// 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_3344_2_a_g(:prec:=1)
    chi := MakeCharacter_3344_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_3344_2_a_g();".
// 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_3344_2_a_g( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3344_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![0, 1]>,<5,R![-2, 1]>,<7,R![2, 1]>],Snew);
    return Vf;
end function;