
// Make newform 396.2.i.e in Magma, downloaded from the LMFDB on 30 January 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_396_i();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_396_i_Hecke();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_396_2_i_e();".
// 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_396_2_i_e();".
// 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 := [81, -81, 9, 27, -23, 9, 1, -3, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0], [54, -99, 30, 14, -15, 8, -3, -1], [-81, 9, 27, -23, 9, 1, -3, 1], [135, -99, 3, 56, -15, 5, 6, -4], [108, 45, -93, 70, -21, -14, 12, -5], [-27, -18, 30, -19, 3, 2, -3, 2], [-540, 315, 6, -71, 78, -29, -6, 7]];
    Rf_basisdens := [1, 1, 81, 27, 81, 81, 27, 81];
    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_396_i();"
function MakeCharacter_396_i()
    N := 396;
    order := 3;
    char_gens := [199, 353, 145];
    v := [3, 2, 3];
    // 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_396_i_Hecke();"
function MakeCharacter_396_i_Hecke(Kf)
    N := 396;
    order := 3;
    char_gens := [199, 353, 145];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := true, Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
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 := [[0, 0, 0, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, 0, 0], [0, 0, 1, -1, 0, 0, 1, 0], [1, 2, 1, 2, 0, 1, 1, 0], [1, 0, 1, 0, 0, 0, 0, 0], [0, -1, 0, 0, -1, 1, 0, -1], [-2, 1, -1, -1, 2, 0, 0, -1], [3, 0, 0, 1, 0, 1, 1, 0], [0, 1, 1, -1, -1, -1, 1, -1], [-3, -1, -2, -1, 1, 1, -2, -2], [0, -3, 1, 1, 0, 3, -1, 0], [-5, 1, -1, 0, 2, 1, 1, -1], [0, -2, -5, -2, 1, 2, 2, 1], [0, -2, -1, -2, -1, 1, -3, 2], [3, 2, 2, 2, -1, 1, 1, 2], [1, 1, 0, 2, 0, 3, 3, 0], [0, 3, 4, -3, 1, -3, 3, 1], [-3, 0, -3, 0, 0, -1, 1, 0], [0, 0, 8, 2, -1, 0, -2, -1], [-3, -1, -1, -4, 2, -5, -5, -1], [10, -3, 2, 2, -4, -1, -1, 2], [-11, 1, -11, 1, 0, 2, -1, 0], [-5, -5, -6, -5, -1, -3, -2, 2], [-7, 2, -1, 1, 2, 3, 3, -1], [0, -1, -1, -1, -1, 0, -1, 2], [11, -1, 9, -1, -2, 0, -1, 4], [0, 1, 4, 2, 0, -1, -2, 0], [-10, 1, -1, -3, 2, -2, -2, -1], [4, -1, -1, 1, 2, 0, 0, -1], [0, -4, -4, 4, -2, 4, -4, -2], [1, 3, -2, -1, 4, 2, 2, -2], [0, -2, -2, 2, 2, 2, -2, 2], [11, 1, 9, 1, -2, -2, 3, 4], [0, 5, 9, 2, -2, -5, -2, -2], [0, -1, -2, 1, 3, 1, -1, 3], [-6, 2, -5, 2, 1, -3, 5, -2], [0, -1, 4, -1, -4, 1, 1, -4], [6, 1, -1, -4, 2, -3, -3, -1], [0, 4, 5, 1, -1, -4, -1, -1], [-7, -2, -5, -2, 2, 1, -3, -4], [-2, -6, -1, 4, 2, -2, -2, -1], [-9, -2, -1, -1, 2, -3, -3, -1], [0, -1, 3, -1, 3, 1, -2, -6], [0, 0, -10, 2, 2, 0, -2, 2], [2, -3, 0, 8, 0, 5, 5, 0], [5, 4, 1, 1, -2, 5, 5, 1], [0, 3, 13, -6, 2, -3, 6, 2], [2, 0, 1, 0, -1, 4, -4, 2], [3, -2, 2, -2, -1, -4, 2, 2], [0, 5, 9, -7, 1, -5, 7, 1], [-7, 3, -4, -1, 8, 2, 2, -4], [0, -1, -15, -5, 2, 1, 5, 2], [-15, 1, -13, 1, 2, 4, -3, -4], [-17, -1, -1, 3, 2, 2, 2, -1], [0, -4, -3, 1, -1, 4, -1, -1], [-3, 9, -4, 9, -1, 3, 6, 2], [-2, -4, 1, 0, -2, -4, -4, 1], [5, -9, 0, 3, 0, -6, -6, 0], [10, 6, 9, 6, -1, -3, 9, 2], [-16, -10, -17, -10, -1, -3, -7, 2], [0, 3, -2, -4, 1, -3, 4, 1], [0, 3, 3, 1, -2, -3, -1, -2], [0, 6, 1, -11, -2, -5, -5, 1], [0, -3, 10, 1, 2, 3, -1, 2], [7, -4, 5, -4, -2, -8, 4, 4], [15, 6, 16, 6, 1, 3, 3, -2], [-8, 3, -4, 3, 4, 0, 3, -8], [0, 6, 9, -5, -1, -6, 5, -1], [0, 3, 4, 5, -3, -3, -5, -3], [8, -4, 4, -4, -4, -4, 0, 8], [3, -1, -1, -1, -4, 1, -2, 8], [-12, 3, -2, -12, 4, -9, -9, -2], [4, 4, 6, 4, 2, -1, 5, -4], [0, -2, -7, -3, -1, 2, 3, -1], [8, 3, -3, 0, 6, 3, 3, -3], [0, -2, -5, 3, 6, 2, -3, 6], [3, 0, -4, 0, -7, 0, 0, 14], [6, 3, 5, -2, -10, 1, 1, 5], [0, 0, -15, -2, 4, 0, 2, 4], [0, 6, -3, 2, 3, -6, -2, 3], [0, 9, 5, 0, -1, -9, 0, -1], [10, -7, 8, -7, -2, 4, -11, 4], [-3, 9, 0, -3, 0, 6, 6, 0], [-16, -3, 1, 0, -2, -3, -3, 1], [-6, -6, -6, -6, 0, 2, -8, 0], [-5, 5, -3, 5, 2, -4, 9, -4], [13, -2, 1, 8, -2, 6, 6, 1], [0, -3, 0, -3, 0, -1, -2, 0], [-16, 3, -16, 3, 0, 5, -2, 0], [0, -6, -1, 1, -2, 6, -1, -2], [-15, 4, -3, 1, 6, 5, 5, -3], [6, 0, 10, 0, 4, 0, 0, -8], [5, 6, 3, 0, -6, 6, 6, 3], [0, 6, 6, -13, 2, -6, 13, 2], [0, 0, -16, 3, -3, 0, -3, -3], [-5, 8, 1, -6, -2, 2, 2, 1], [0, -7, -21, 4, -1, 7, -4, -1], [-6, 5, -2, -10, 4, -5, -5, -2], [-13, 13, -1, -8, 2, 5, 5, -1], [18, -5, 3, 11, -6, 6, 6, 3], [-14, -5, -12, -5, 2, -1, -4, -4], [12, 9, 0, -6, 0, 3, 3, 0], [0, 3, -10, -2, 3, -3, 2, 3], [28, -2, 26, -2, -2, -6, 4, 4], [0, 2, 2, 1, -4, -2, -1, -4], [14, 2, 4, 5, -8, 7, 7, 4], [6, -4, 10, -4, 4, -5, 1, -8], [-3, 4, -3, -2, 6, 2, 2, -3], [0, -9, 3, 0, 0, 9, 0, 0], [9, -1, 8, -1, -1, -3, 2, 2], [0, 0, 8, 5, -4, 0, -5, -4], [-29, -3, -1, 2, 2, -1, -1, -1], [0, 2, 3, -2, 8, -2, 2, 8], [-5, -6, -2, -6, 3, 6, -12, -6], [9, -7, -1, 9, 2, 2, 2, -1], [30, -4, 28, -4, -2, 1, -5, 4], [0, 2, 6, -7, 1, -2, 7, 1], [-26, -2, 4, -1, -8, -3, -3, 4], [0, 9, 19, -15, 4, -9, 15, 4], [0, 12, -4, 12, -4, 0, 12, 8], [0, -1, -15, -3, 8, 1, 3, 8], [-2, 11, 6, 11, 8, 7, 4, -16], [1, 6, -1, 6, -2, -5, 11, 4], [3, -4, 0, 2, 0, -2, -2, 0], [2, 4, 0, 4, -2, 3, 1, 4], [-15, 4, -2, -17, 4, -13, -13, -2], [23, 0, 23, 0, 0, -2, 2, 0], [0, 13, -2, -5, 4, 8, 8, -2], [14, 5, 9, 5, -5, 2, 3, 10], [0, -1, -24, -4, 1, 1, 4, 1], [5, -9, 4, 3, -8, -6, -6, 4], [0, -2, 24, 2, 4, 2, -2, 4], [0, -7, -10, 5, -3, 7, -5, -3], [-10, 6, 0, -15, 0, -9, -9, 0], [0, -12, 7, -6, -5, 12, 6, -5], [0, -20, -16, 0, -4, 20, 0, -4], [12, -7, -2, 8, 4, 1, 1, -2], [0, 3, 11, -7, -4, -3, 7, -4], [0, 15, -2, 0, 1, -15, 0, 1], [2, -9, 4, -4, -8, -13, -13, 4], [14, -2, 9, -5, -18, -7, -7, 9], [-26, -1, -26, -1, 0, 5, -6, 0], [0, 7, -3, 7, 1, -7, -7, 1], [28, -1, 0, 6, 0, 5, 5, 0], [-24, 4, -1, 2, 2, 6, 6, -1], [-18, -8, -20, -8, -2, -1, -7, 4], [-1, 9, -1, 9, 0, 10, -1, 0], [5, 3, 4, 3, -1, -7, 10, 2], [0, 0, -6, -1, 9, 0, 1, 9], [-17, 2, 3, 6, -6, 8, 8, 3], [0, 2, -14, -9, 0, -2, 9, 0], [-4, 5, -5, -2, 10, 3, 3, -5], [-36, 2, -6, -9, 12, -7, -7, -6], [0, -5, -36, -8, 3, 5, 8, 3], [-6, -14, -2, -14, 4, -8, -6, -8], [6, 10, 12, 10, 6, 11, -1, -12], [11, 6, 3, 6, -6, 12, 12, 3], [3, -3, 4, -3, 1, -6, 3, -2], [-15, 12, -6, -5, 12, 7, 7, -6], [0, 7, 6, -8, -3, -7, 8, -3], [-18, 2, -16, 2, 2, -11, 13, -4], [12, -6, -1, 9, 2, 3, 3, -1], [0, 8, 30, -5, 6, -8, 5, 6], [18, 0, 6, 15, -12, 15, 15, 6], [0, -1, -10, 4, -2, 1, -4, -2], [-9, 7, -12, 7, -3, 5, 2, 6], [8, -19, 1, 5, -2, -14, -14, 1], [-12, -19, -8, -19, 4, -9, -10, -8]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_396_i_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_396_2_i_e();".
// 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_396_2_i_e(:prec:=8)
    chi := MakeCharacter_396_i();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(997) - 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_396_2_i_e();".
// 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_396_2_i_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_396_i();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![9, -63, 456, 81, 106, 22, 21, 4, 1]>],Snew);
    return Vf;
end function;