
// Make newform 2394.2.a.ba in Magma, downloaded from the LMFDB on 25 April 2024.
// 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_ba();".
// 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_ba();".
// 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
        poly := [4, -5, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0], [0, 1, 0], [-4, 0, 1]];
    Rf_basisdens := [1, 1, 1];
    Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf);
    Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]];
    inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf);
    return Eltseq(inp_vec);
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)
    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+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 := [[-1, 0, 0], [0, 0, 0], [-2, 1, 0], [-1, 0, 0], [0, -2, 1], [-2, 2, 0], [-2, 2, 2], [-1, 0, 0], [0, 2, 0], [-2, -1, -2], [0, 2, -2], [2, -1, -2], [2, 1, 0], [0, 1, 2], [4, 0, 1], [-2, 0, -3], [0, 3, 0], [2, 2, 1], [4, 2, 2], [-4, 2, -1], [2, -4, 2], [4, 1, -2], [4, 4, 0], [2, -2, 1], [-2, 2, 1], [2, -8, 0], [0, 2, -4], [12, -2, 4], [10, 0, 3], [-2, -6, -2], [-4, 2, 1], [12, 0, 2], [-6, 1, -6], [4, 0, 4], [10, -2, -2], [8, -4, 4], [2, 2, 5], [8, 2, 3], [8, -2, 6], [2, 4, -2], [4, 6, -4], [6, 0, 6], [8, 6, -4], [-6, 4, 2], [-6, 4, 4], [12, -5, -4], [-12, 4, -4], [8, 0, -6], [4, -4, -4], [18, -1, 0], [-6, 4, 1], [-8, 4, -2], [-18, 1, -4], [4, -8, 2], [10, -3, 0], [0, 8, 6], [-6, 4, 4], [-4, -3, 4], [-2, -4, -10], [22, -2, 2], [-12, 4, -4], [10, -2, -4], [0, -13, 0], [-4, -4, 1], [10, 6, 6], [14, -4, 7], [12, -2, -6], [-6, -2, -8], [4, -4, 0], [6, 8, 0], [-2, -2, 6], [-16, 2, -8], [-4, -4, -1], [10, -7, -6], [4, 4, 2], [-8, -6, -4], [10, 4, 4], [-14, 10, 5], [-2, 10, 6], [6, 2, 5], [4, -6, -10], [-26, 4, 0], [-20, 2, -15], [-18, 1, -4], [-8, -6, 2], [-8, 9, -2], [6, -10, 0], [-10, 12, 3], [-2, -2, -9], [8, 4, 6], [-4, 6, 8], [-12, 3, 8], [12, -15, -10], [-12, 8, 8], [16, 3, -6], [28, -5, 4], [2, 8, -2], [-10, 4, 8], [-28, -4, -4], [22, 0, -2], [-20, 6, -12], [-6, -14, 4], [8, -16, -3], [-2, 8, -2], [-8, -2, -3], [-14, -16, 4], [20, 6, 0], [-10, 0, 2], [-4, 6, -1], [-6, 4, 4], [-24, 8, 2], [-26, -2, -6], [2, -5, -10], [12, -10, 8], [8, 12, 4], [-10, -4, -14], [12, 4, -4], [4, -7, -4], [-6, -18, 4], [-4, -4, -12], [14, -16, -4], [-22, -14, 4], [-14, -4, 8], [-12, 6, 6], [20, 2, 12], [-6, 0, 10], [14, -6, 8], [16, 0, 0], [-12, -12, -1], [-6, -13, -4], [8, -7, 6], [-36, 1, -2], [12, 3, -2], [6, 10, 2], [6, -8, -4], [42, 4, 2], [-6, -4, -10], [16, 3, -4], [2, -12, 10], [10, -3, 10], [16, 12, 3], [26, -4, -8], [-8, -10, 2], [12, 10, -4], [-10, 6, -14], [8, 6, -2], [-22, 5, -8], [-10, 0, 0], [-4, 2, -8], [4, -7, -6], [10, 4, -1], [-10, 10, -12], [24, 2, -9], [-24, 10, -10], [20, -4, -2], [-12, 5, -14], [0, -26, 4], [22, -14, 4], [10, 2, 18], [-14, -2, -8], [32, 2, -1], [-42, 8, 4], [8, 12, -12], [8, -20, 3], [-10, 8, 2], [8, -4, 0], [-4, -14, -3], [26, 9, 8], [26, -10, -8], [18, 0, 4], [44, -6, 4], [14, 4, 2], [-16, 10, -4], [6, -5, 10], [16, 2, 6], [6, -4, 12], [-8, 3, -6], [2, -16, -2], [-40, 2, -10], [-6, -9, -20], [24, -18, -4], [16, -4, -9], [18, 12, -5], [2, 2, 9], [40, -2, 12], [6, 2, -1], [18, 9, 2], [-8, -11, 4], [18, 8, 10], [-4, -4, -17], [-14, 0, -4], [16, -5, -14], [4, 30, -6], [-30, -2, 4], [24, 10, -3], [-46, -3, 4], [-10, 12, 1], [22, 2, 4], [14, 6, 2], [-36, -9, -4], [18, 16, -4], [8, -6, -4], [22, -10, 12], [-18, 2, -15], [-20, -8, 4], [6, -10, 9], [-20, 12, -5], [12, -2, 6], [6, 16, 4], [0, -4, 11], [22, -3, 6], [-26, 10, -5], [0, -24, 2], [16, 19, 4], [-16, 20, -6], [14, 3, -16], [40, -2, -8], [18, 11, 16], [20, -9, 2], [2, -10, 2], [2, -4, -9], [0, 0, -6], [30, -10, -2], [16, 12, 12], [-8, -8, -3], [6, -4, -4], [-14, 5, -8], [16, 0, 8], [8, -20, 4], [24, -19, 2], [-26, 10, -4], [16, 0, 25], [4, -6, -17], [-18, 2, 0], [-20, 10, 4], [20, 3, 0], [34, 2, -6], [-6, -12, 16], [32, -9, 6], [0, 14, 8], [4, -20, -2], [-12, 24, 0], [-40, 8, 8], [42, -12, -5], [-34, 16, 4], [36, 4, -1], [36, -17, -12], [-12, 0, 4], [4, -24, -6], [16, 4, 8], [14, 2, 6], [-2, -12, 12], [16, 8, -6], [-54, -4, 4], [-6, 4, -4], [16, 5, -10], [22, -26, -6], [32, -8, 3], [-14, -14, -14], [26, -12, 0], [12, 7, -10], [36, -18, 4], [-6, -2, -7], [14, 0, 2], [-26, 10, 8], [8, 8, 11], [18, -24, -20], [14, 16, -6], [-36, -8, 8], [18, 14, 8], [22, -12, 16], [20, 4, -8], [-14, -4, -14], [-24, 0, 10], [-2, 9, -20], [-4, 7, -4], [-44, 4, 4], [-38, -5, -2], [10, 18, -6], [-36, 2, -12], [0, 14, 6], [-8, -12, -4], [-44, -4, -5], [-10, 10, 12], [20, 8, 12], [8, -24, 14], [-2, 8, 15], [14, 25, -10], [52, 4, 7], [14, 24, -12], [18, -14, 26], [36, 2, 14], [62, 0, 6], [-32, -1, 0], [30, 0, 16], [-34, 11, -20], [24, -10, 28], [-10, -21, 16], [-16, -20, 5], [-4, -4, -8], [18, -22, 0], [2, 12, -16], [0, -2, -18], [-12, 10, 8], [32, -3, 2], [34, -14, 10], [12, -16, 2], [38, 2, -10], [-8, -12, 8], [6, -26, -2], [-52, 1, -4], [26, -12, 12], [-2, 6, 6], [36, 4, 8], [32, -10, -12], [-10, 17, 4], [20, -6, 14], [20, 9, 14], [-14, 4, 16], [-22, 8, -13], [16, -20, 13], [54, -17, -2], [10, -16, -4], [-36, -2, -7], [-26, -24, 12], [-2, -11, -12], [-20, 16, -2], [8, -21, -24], [-12, -30, 7], [-30, 26, 16], [14, -16, -24], [-10, 13, -14], [-44, 0, 9], [20, -24, -24], [-8, -1, 10], [-4, 0, 20], [42, -4, -5], [-70, 6, -15], [-22, 28, -12], [-44, 8, -3], [-38, -12, -11], [26, 12, -3], [2, 20, -12], [8, 4, -20], [-2, 23, 14], [-12, 20, 22], [18, 6, -5], [-4, 18, 10], [0, 6, -6], [-10, -9, -16], [-24, 1, -32], [10, 16, 24], [-46, -2, -26], [8, -4, -2], [-22, 33, 10], [10, 6, -21], [68, 8, -3], [-4, -26, 6], [42, -20, 15], [-20, 2, 3], [-34, 12, -6], [-42, -8, -16], [16, 20, 16], [-36, 16, -10], [28, -14, -6], [-22, -20, 12], [38, 5, 20], [68, -1, 2], [-6, 14, 6], [-24, 19, 22], [-24, -9, 20], [8, -14, -28], [-22, 12, -32], [8, 0, -8], [30, 10, -6], [0, 36, -21], [16, -22, -18], [26, 6, 4], [-2, 36, -4], [-42, -11, -8], [26, 8, 12], [-22, -8, -7], [-32, -18, 12], [-6, 7, -18], [24, -8, -1], [0, 6, 0], [32, 10, 10], [58, -7, 4], [28, 8, -8], [8, -30, 16], [34, -24, 16], [2, -22, -2], [-8, 13, -10], [12, 40, -4], [-60, 7, -6], [26, -2, 6], [-56, 2, -16], [26, 22, -9], [12, 12, 20], [10, -14, -2], [38, -8, -16], [-14, -24, 9], [-12, 6, -5], [-14, 10, -19], [-14, 26, -10], [36, 5, -4], [46, -14, -14], [14, -20, 20], [28, 16, 8], [-12, 16, 16], [-50, 2, 13], [-46, -2, 4], [-44, 14, -18], [-8, -30, 19], [30, -23, 22], [-18, -26, -13], [-16, -28, 16], [0, 42, -20], [30, -16, -8], [16, 10, 20], [-10, -20, -11], [-18, 26, 2], [16, 10, -2], [-20, -8, -4], [14, 10, -5], [2, 2, 8], [36, -26, -16], [-6, -11, -22], [-20, -16, -6], [8, -20, 28]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2394_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2394_2_a_ba();".
// 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_ba(:prec:=3)
    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_ba();".
// 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_ba( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2394_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-2, 3, 5, 1]>,<11,R![-76, -25, 3, 1]>,<13,R![-8, -16, 4, 1]>,<17,R![-224, -40, 6, 1]>],Snew);
    return Vf;
end function;