
// Make newform 495.2.n.a in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_495_n();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_495_n_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_495_2_n_a();".
// 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_495_2_n_a();".
// 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 := [1, 3, 5, 3, 4, -3, 5, -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], [-7, -12, -7, -4, 0, -3, 2, -1], [9, 6, 19, -16, 20, -7, 0, 1], [-1, -10, 13, -16, 12, -9, 4, -1], [-5, -8, -13, -4, 8, -17, 10, -3], [-1, 6, 1, 16, -20, 23, -12, 3], [-7, -12, -11, -28, 32, -35, 18, -5]];
    Rf_basisdens := [1, 1, 8, 8, 8, 8, 8, 8];
    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_495_n();"
function MakeCharacter_495_n()
    N := 495;
    order := 5;
    char_gens := [56, 397, 46];
    v := [5, 5, 4];
    // 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_495_n_Hecke();"
function MakeCharacter_495_n_Hecke(Kf)
    N := 495;
    order := 5;
    char_gens := [56, 397, 46];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -1, 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+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, -1, -1, 1, -1, 1, 1, -1], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -1, 0, 0, 0], [0, 1, 0, 2, 0, -1, -1, 0], [0, 1, 0, 1, 1, -3, -1, -1], [1, -1, -1, -1, 2, 1, 1, 2], [0, 0, -2, 0, 1, 2, 2, 0], [0, 1, -1, 0, 0, 0, -1, -2], [-2, 4, 4, 1, 0, -3, 0, -1], [-3, 6, 0, -3, -3, -1, -1, 0], [4, -4, -5, -4, -1, 4, 5, -1], [1, 3, 0, 4, 1, -6, -6, 0], [0, 1, 0, 4, -4, 0, -1, 4], [-4, 5, 5, 3, 0, -5, 0, -3], [0, -1, 4, 2, -2, 0, 1, 0], [1, -3, 0, -1, 7, 3, 0, 7], [-2, 5, 0, -4, -2, -2, -2, 0], [5, 0, 1, 0, 2, -1, -7, 5], [3, 0, 0, -7, 0, 3, 0, 7], [5, 0, 5, 0, 9, -5, -3, 5], [-4, 5, 0, 1, -4, 0, 0, 0], [-1, -2, 2, 1, 4, 2, -2, 4], [5, 0, 3, 0, 5, -3, -1, 5], [5, -4, -4, -3, 0, 8, 0, 3], [7, 5, 4, -7, 5, -5, -4, 5], [0, 3, 1, 0, -12, -3, -1, -12], [-2, 4, 0, -4, -2, 5, 5, 0], [0, 3, -5, 4, -4, 0, -3, 2], [-1, -1, -1, 7, 0, 1, 0, -7], [0, 5, 5, -13, 13, 0, -5, 5], [-1, 0, -1, 0, 6, 1, 2, -1], [9, -1, -1, -4, 0, 6, 0, 4], [1, 0, 0, 0, 2, 0, -11, 1], [-3, -11, 0, -8, -3, 13, 13, 0], [6, 0, -4, 0, 7, 4, -3, 6], [0, -4, -8, 13, -13, 0, 4, -7], [0, -3, 1, -9, 9, 0, 3, 4], [-5, -6, 0, 5, 4, 6, 0, 4], [-6, 10, 6, 6, 6, -10, -6, 6], [0, -4, -3, 14, -14, 0, 4, -7], [0, 5, -1, 4, -4, 0, -5, 2], [-6, 0, 5, 0, -15, -5, 1, -6], [-7, 4, 0, -1, -7, 3, 3, 0], [-12, 0, -8, 0, -9, 8, 12, -12], [0, -9, -9, 3, 0, 9, 0, -3], [7, 5, 5, -12, 0, 8, 0, 12], [-6, 9, 0, 6, -4, -9, 0, -4], [0, 0, 4, -11, 11, 0, 0, 7], [2, 7, 0, -6, 2, -6, -6, 0], [7, -3, 0, -7, 12, 3, 0, 12], [-2, -9, -11, 2, -13, 9, 11, -13], [0, -11, 2, -4, 4, 0, 11, 4], [-13, 0, 0, 6, 0, 0, 0, -6], [-9, -4, -2, 9, -9, 4, 2, -9], [-6, -4, 0, -4, -6, 1, 1, 0], [-16, 7, 7, 7, 0, -11, 0, -7], [1, 0, 5, 0, 6, -5, 6, 1], [-18, 9, 0, 0, -18, -4, -4, 0], [-16, 8, 1, 16, -15, -8, -1, -15], [-4, 0, 2, 0, 9, -2, 12, -4], [0, 6, 7, 2, -2, 0, -6, 2], [6, 10, 0, 5, 6, -13, -13, 0], [17, -7, -7, 2, 0, 13, 0, -2], [0, 1, 6, 10, -10, 0, -1, -3], [1, 0, -2, 0, -8, 2, -6, 1], [10, -2, 4, -10, 10, 2, -4, 10], [0, 5, 5, 8, 0, -5, 0, -8], [-8, -6, 0, 4, -8, -3, -3, 0], [17, 0, -6, 0, -5, 6, 7, 17], [0, -3, -3, 3, -3, 0, 3, -9], [5, -10, -10, -9, 0, 20, 0, 9], [2, -19, 0, -12, 2, 19, 19, 0], [4, -5, 0, 12, 4, -7, -7, 0], [12, -9, -9, 3, 0, 4, 0, -3], [13, 0, -5, 0, 19, 5, 0, 13], [15, -9, 6, -15, 13, 9, -6, 13], [2, -9, 0, 5, 2, -5, -5, 0], [8, -4, -4, 12, 0, -1, 0, -12], [1, 0, 4, 0, 6, -4, 3, 1], [2, -4, 4, -2, 2, 4, -4, 2], [-22, 7, 7, 16, 0, 5, 0, -16], [0, 9, 10, -8, 8, 0, -9, 5], [9, 4, 8, -9, -4, -4, -8, -4], [9, 4, 0, -10, 9, 7, 7, 0], [19, 4, 4, -13, 0, -5, 0, 13], [0, -10, -10, -1, 1, 0, 10, -4], [15, -14, 7, -15, 9, 14, -7, 9], [7, 0, 15, 0, 19, -15, -3, 7], [12, -2, -2, 5, 0, -20, 0, -5], [23, -16, -16, 3, 0, -2, 0, -3], [-16, 0, 4, 0, -21, -4, 19, -16], [-12, 0, -3, 0, -6, 3, 5, -12], [0, 10, 0, -9, 9, 0, -10, -6], [7, -3, 0, -18, 7, 3, 3, 0], [-21, -2, 0, 17, -21, 1, 1, 0], [0, -8, 12, 0, 0, 0, 8, -11], [0, 2, -25, 9, -9, 0, -2, -16], [-10, 1, 0, 19, -10, 2, 2, 0], [3, 0, 7, 0, -2, -7, -5, 3], [-8, -3, 3, 8, 9, 3, -3, 9], [0, -4, 10, -9, 9, 0, 4, 23], [14, 3, 0, 8, 14, -11, -11, 0], [5, 3, -2, -5, 17, -3, 2, 17], [0, 1, 1, -6, 6, 0, -1, 16], [18, 1, 1, -4, 0, -14, 0, 4], [-12, 0, 1, 0, -18, -1, 17, -12], [-12, -13, 0, -18, -12, 15, 15, 0], [20, -10, -10, 11, 0, 6, 0, -11], [-5, 0, 6, 0, -27, -6, 0, -5], [8, -23, 0, -7, 8, 12, 12, 0], [0, 19, 9, 0, -8, -19, -9, -8], [0, -10, -19, 5, -5, 0, 10, -17], [-6, 11, 11, 17, 0, -12, 0, -17], [0, -6, -8, 4, -4, 0, 6, -31], [-18, 25, 0, -4, -18, 0, 0, 0], [0, -3, -14, 7, -7, 0, 3, -14], [17, 0, 10, 0, 21, -10, 8, 17], [-13, 16, -5, 13, -9, -16, 5, -9], [-7, 4, 0, 8, -7, 0, 0, 0], [-3, 12, 12, -4, 0, -8, 0, 4], [-15, 0, 0, 34, 0, -12, 0, -34], [-3, -26, -25, 3, -7, 26, 25, -7], [27, 0, 7, 0, 11, -7, -30, 27], [22, -26, -26, 5, 0, 16, 0, -5], [-24, 5, 9, 24, -9, -5, -9, -9], [0, -2, -11, 8, -8, 0, 2, 12], [-5, 0, 4, 0, -14, -4, 22, -5], [19, -9, 0, 9, 19, -7, -7, 0], [-28, 0, 0, -4, 0, 10, 0, 4], [-7, 20, 0, 1, -7, -19, -19, 0], [9, -19, -2, -9, -17, 19, 2, -17], [8, 0, -7, 0, -9, 7, 5, 8], [0, -8, -13, 30, -30, 0, 8, -8], [-12, 16, 10, 12, -2, -16, -10, -2], [-16, -12, -17, 16, -27, 12, 17, -27], [-9, -2, -2, -4, 0, 12, 0, 4], [0, -7, 3, -26, 26, 0, 7, 9], [-12, 0, -1, 0, -25, 1, 3, -12], [0, 0, 5, 0, 3, -5, 3, 0], [-18, 0, 3, 0, 18, -3, -7, -18], [0, 6, 3, -18, 18, 0, -6, 27], [9, -11, 0, -7, 9, 6, 6, 0], [-35, 5, 5, 35, -23, -5, -5, -23], [-41, -4, -5, 41, -33, 4, 5, -33], [3, -6, 0, -6, 3, 7, 7, 0], [0, -20, 0, 14, -14, 0, 20, -29], [-10, 0, 8, 0, -27, -8, -1, -10], [14, -3, -3, -14, 0, 13, 0, 14], [-9, -2, -2, -4, 0, -11, 0, 4], [17, 0, 2, 0, 23, -2, -9, 17], [0, -1, 21, -9, 9, 0, 1, 16], [6, 15, 15, -27, 0, 0, 0, 27], [0, 4, 22, -17, 17, 0, -4, 6], [-14, 5, 0, -2, -14, -2, -2, 0], [16, 0, -2, 0, -3, 2, 8, 16], [-10, 10, 14, 10, -7, -10, -14, -7], [12, 0, -9, 0, -2, 9, -1, 12], [20, 0, -5, 0, -9, 5, 19, 20], [0, -29, -29, 0, -4, 29, 29, -4], [4, 0, 4, 0, 17, -4, -10, 4], [10, 11, 11, 1, 0, -3, 0, -1], [-21, -13, 0, -3, -21, 14, 14, 0], [5, -10, -10, -17, 0, 3, 0, 17], [0, 24, 6, 4, -4, 0, -24, 6], [-5, -27, -8, 5, -12, 27, 8, -12], [-6, 17, 0, -6, -6, 1, 1, 0], [13, -19, -19, -12, 0, 8, 0, 12], [-12, -10, 0, 10, -12, -6, -6, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_495_n_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_495_2_n_a();".
// 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_495_2_n_a(:prec:=8)
    chi := MakeCharacter_495_n();
    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_495_2_n_a();".
// 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_495_2_n_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_495_n();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, 7, 21, 21, 24, 13, 9, 4, 1]>],Snew);
    return Vf;
end function;