
// Make newform 3344.2.a.f in Magma, downloaded from the LMFDB on 23 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_f();".
// 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_f();".
// 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], [4], [1], [2], [2], [-1], [0], [2], [0], [10], [-2], [4], [-8], [2], [-8], [10], [8], [0], [2], [-8], [4], [-6], [2], [10], [8], [12], [-14], [10], [8], [4], [10], [-4], [10], [-8], [14], [12], [-16], [2], [24], [10], [-16], [22], [-6], [24], [-12], [24], [-12], [14], [10], [-20], [-2], [4], [18], [-4], [-30], [-12], [10], [6], [4], [-6], [-4], [0], [-6], [18], [24], [22], [28], [-14], [34], [-36], [16], [10], [-8], [-32], [-34], [-2], [34], [-26], [-12], [-30], [8], [-14], [-8], [-28], [-6], [2], [2], [-16], [-28], [4], [0], [4], [-12], [-36], [-30], [-6], [20], [-38], [-4], [18], [-4], [30], [-12], [2], [-12], [-14], [24], [22], [8], [34], [-6], [-4], [-8], [-6], [4], [8], [38], [-20], [26], [14], [-6], [0], [-4], [-6], [-34], [-24], [-16], [42], [36], [-24], [-24], [-18], [42], [-6], [-6], [4], [-6], [18], [-28], [-22], [8], [-36], [18], [-40], [-14], [-10], [52], [-24], [50], [18], [20], [-24], [-56], [32], [-44], [-30], [-30], [-30], [-36], [30], [-4], [0], [2], [-16], [-8], [42], [-18], [-2], [-28], [10], [-60], [-38], [32], [-54], [-52], [-50], [-64], [-30], [16], [-4], [26], [-18], [32], [30], [2], [64], [14], [-4], [-54], [4], [-32], [-14], [60], [-6], [42], [-34], [2], [-36], [50], [40], [50], [54], [60], [-34], [-56], [12], [66], [40], [10], [-26], [-48], [40], [40], [-46], [68], [-42], [-24], [62], [26], [-56], [50], [16], [20], [-38], [34], [8], [48], [60], [38], [60], [32], [70], [20], [12], [-30], [-70], [20], [24], [-24], [-36], [-56], [42], [46], [44], [72], [24], [52], [20], [66], [50], [24], [-38], [-54], [-60], [-10], [-20], [-30], [-38], [16], [68], [-46], [-30], [18], [48], [2], [26], [-36], [-30], [-38], [56], [-70], [-12], [-66], [56], [-28], [-14], [70], [-28], [0], [-40], [-12], [-38], [-68], [48], [-30], [-6], [0], [-66], [78], [-12], [6], [12], [-46], [-14], [80], [70], [44], [-68], [-70], [18], [-28], [-48], [-52], [58], [-48], [82], [56], [-6], [-60], [-18], [22], [64], [12], [22], [28], [-24], [66], [-6], [-60], [74], [18], [32], [-6], [58], [8], [-56], [-24], [-46], [66], [2], [-28], [-36], [20], [-92], [-62], [-42], [42], [76], [-6], [-22], [42], [-40], [-6], [-36], [-10], [-32], [72], [-18], [4], [2], [-50], [-24], [10], [38], [48], [-20], [-30], [8], [82], [-14], [16], [60], [-76], [66], [-54], [80], [-50], [68], [-48], [-60], [-54], [-100], [78], [-48], [60], [-46], [-70], [-42], [-14], [10], [-44], [-14], [-4], [-24], [-88], [-42], [-52], [24], [-70], [-54], [20], [-20], [-16], [30], [8], [-78], [-40], [-14], [-14], [50], [-48], [-30], [-6], [-68], [94], [30], [-24], [-36], [38], [-22], [-72], [-36], [50], [46], [48], [-32], [-30], [16], [98], [-102], [24], [76], [-30], [-30], [40], [-86], [-44], [84]];
    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_f();".
// 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_f(: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_f();".
// 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_f( : 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![-4, 1]>],Snew);
    return Vf;
end function;