
// Make newform 2394.2.a.l in Magma, downloaded from the LMFDB on 01 January 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_2394_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_2394_2_a_l();".
// 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_2394_2_a_l();".
// 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_2394_a();"
function MakeCharacter_2394_a()
    N := 2394;
    order := 1;
    char_gens := [533, 1711, 1009];
    v := [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_2394_a_Hecke( : Kf := Kf)
    return MakeCharacter_2394_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]];
    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 := [[1], [0], [0], [1], [6], [2], [0], [1], [-6], [6], [8], [-10], [-6], [-4], [6], [6], [12], [2], [-4], [0], [2], [-4], [6], [-6], [14], [12], [-4], [0], [2], [6], [-16], [18], [-12], [-4], [0], [-4], [-22], [-4], [0], [-18], [0], [2], [6], [-22], [12], [-16], [-4], [-28], [0], [26], [-24], [6], [-10], [-18], [-18], [6], [30], [8], [26], [18], [-4], [6], [-16], [-18], [-10], [-6], [8], [14], [6], [-34], [0], [-18], [8], [14], [-16], [36], [-12], [14], [-18], [-10], [-30], [26], [24], [26], [-28], [-18], [-6], [2], [-12], [-40], [6], [-30], [-16], [-42], [-28], [-30], [6], [42], [44], [38], [-16], [0], [24], [-6], [-4], [14], [-18], [-36], [36], [2], [32], [26], [-36], [-4], [-40], [-18], [44], [-42], [-48], [-36], [14], [26], [-6], [0], [-28], [-24], [-10], [6], [32], [14], [-28], [24], [-4], [-22], [48], [14], [6], [-28], [6], [24], [-16], [-24], [56], [0], [2], [-12], [-10], [18], [-28], [0], [-10], [48], [20], [0], [-52], [12], [32], [12], [26], [-42], [-42], [-6], [32], [48], [54], [-24], [8], [50], [-58], [36], [0], [-58], [42], [-10], [32], [-12], [20], [0], [-40], [-34], [-64], [0], [26], [54], [18], [-12], [26], [-28], [62], [30], [38], [6], [-40], [-6], [-42], [-6], [50], [-10], [12], [-6], [54], [56], [26], [-34], [-54], [36], [-64], [-48], [60], [8], [-10], [-24], [8], [12], [-36], [-46], [32], [-6], [24], [-36], [50], [20], [54], [-16], [72], [-46], [-66], [-24], [8], [-18], [50], [56], [32], [54], [-52], [54], [-22], [36], [-18], [36], [48], [20], [32], [38], [-42], [-6], [-40], [48], [32], [-6], [2], [-12], [42], [-34], [0], [-42], [38], [-40], [66], [-34], [44], [36], [-58], [38], [-12], [68], [6], [12], [-40], [36], [14], [44], [-10], [8], [-58], [56], [-78], [-58], [50], [-18], [-12], [-40], [18], [62], [-52], [66], [-34], [-54], [-40], [-42], [24], [6], [66], [-84], [38], [72], [-52], [-24], [72], [44], [-34], [-18], [-16], [-24], [-76], [62], [-84], [50], [-30], [38], [-6], [12], [-42], [32], [66], [-34], [42], [12], [50], [36], [-28], [-22], [-54], [44], [-84], [-34], [-64], [-52], [24], [-72], [50], [-42], [-64], [18], [-52], [78], [38], [-6], [38], [32], [-34], [-12], [-78], [80], [-18], [36], [74], [44], [48], [12], [-28], [-34], [60], [8], [-10], [-54], [42], [-18], [60], [-96], [-58], [-24], [-48], [48], [68], [-22], [60], [-16], [-22], [-30], [-16], [-6], [78], [32], [-82], [0], [18], [-46], [60], [-82], [42], [-72], [-64], [-72], [-4], [-60], [-16], [74], [-52], [48], [-58], [-6], [30], [68], [96], [38], [32], [78], [20], [-12], [-58], [0], [-76], [78], [-90], [80], [-34], [42], [-64], [42], [-46], [60], [12], [92], [26], [-60], [60], [8], [-24], [12], [102], [-82], [42], [0], [2], [-66], [72], [-96], [-52], [-42]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2394_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2394_2_a_l();".
// 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_2394_2_a_l(:prec:=1)
    chi := MakeCharacter_2394_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_2394_2_a_l();".
// 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_2394_2_a_l( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2394_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![0, 1]>,<11,R![-6, 1]>,<13,R![-2, 1]>,<17,R![0, 1]>],Snew);
    return Vf;
end function;