
// Make newform 495.2.n.b 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_b();".
// 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_b();".
// 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 := [121, 143, 80, -7, 29, -13, 10, -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], [-118459, 33270, -59433, 32996, -16865, 3310, 341, -256], [-16104, -102225, 81515, -41955, 18870, -9045, 2532, -620], [-75020, -72556, 52625, -77175, 23975, -10810, 2845, -672], [-143748, -25003, -32510, -12015, 14238, -17191, 4445, -1187], [144881, 62530, -20331, 48690, -32635, 16325, -4751, 1188], [-249744, -187297, -33635, -27965, -2207, -7808, 1420, -1432]];
    Rf_basisdens := [1, 1, 171589, 171589, 171589, 171589, 171589, 171589];
    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, 1, 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+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, 0, 0, -1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0, 0, 0], [1, 0, -1, 1, -1, 0, 0, 1], [1, 0, 2, 2, 1, 0, -1, 1], [4, 0, 4, 3, -1, 0, -3, 0], [0, 2, 0, 1, -2, -2, 0, 2], [0, 0, 1, 1, 1, 1, -1, 0], [1, -2, -1, 0, 0, 1, 1, -2], [-4, -1, -7, -4, 1, 0, 0, -1], [-4, 1, -4, -6, -3, -1, 6, 0], [-5, -1, -3, -5, 0, 0, 0, 0], [0, 0, -1, -1, -1, -1, -2, 1], [-5, 1, -2, 0, 0, -1, 2, 1], [0, 0, -5, -5, 1, 1, 6, -3], [2, -3, 2, 3, 2, 3, -3, 0], [-2, 1, 3, -2, -2, 0, 0, 2], [-2, 3, 0, -3, -3, 0, 2, 0], [4, 0, 5, 0, 0, -1, -5, 0], [-4, 5, 0, -4, -5, -2, 4, 2], [-2, -3, -14, -2, 2, 0, 0, -2], [1, -4, 1, 6, 6, 4, -6, 0], [2, 3, 0, -4, -3, -2, -2, 2], [-1, 2, 1, 0, 0, -2, -1, 2], [6, 1, 6, 5, -2, -1, -5, 0], [1, -2, 1, -3, -1, 2, 3, 0], [5, -1, -4, 5, 1, 0, 0, -1], [0, 0, 7, 7, 3, 3, -3, -2], [-2, 5, 0, 0, 0, -3, 0, 5], [0, 0, 0, 0, 1, 1, 0, 4], [-8, 0, 0, -14, 0, 3, 8, -3], [-1, 1, 11, 0, 0, 0, -11, 1], [3, -1, 0, 3, 1, 5, -3, -5], [0, 0, 1, 0, -3, 0, 0, 3], [0, -3, 0, 4, 3, -1, 0, 1], [0, 0, 11, 11, 4, 4, -5, 4], [0, 0, -12, -12, -1, -1, 13, -2], [-7, 2, -7, -2, 2, -2, 2, 0], [0, 4, 0, -8, -10, -4, 8, 0], [0, 0, 2, 2, 4, 4, -8, 1], [0, 0, -7, -7, -1, -1, 13, 2], [11, -5, 0, 4, 5, 0, -11, 0], [0, 5, 9, 0, -3, 0, 0, 3], [-6, -2, 0, -9, 2, -4, 6, 4], [-3, 5, -4, 0, 0, -1, 4, 5], [-1, 1, -3, 0, 0, 6, 3, 1], [-5, 3, -5, -12, -2, -3, 12, 0], [0, 0, 7, 7, 2, 2, -5, -2], [10, -3, -1, 10, -4, 0, 0, 4], [-8, 3, -8, -8, 6, -3, 8, 0], [-7, 0, -7, 4, 3, 0, -4, 0], [0, 0, -9, -9, -1, -1, 0, -1], [-1, 0, 2, 0, 0, -4, -2, 0], [9, 2, 9, -5, -4, -2, 5, 0], [-7, -5, -10, -7, 7, 0, 0, -7], [-7, 5, -8, 0, 0, -5, 8, 5], [2, -2, 0, 14, 2, 1, -2, -1], [14, -3, 9, 14, -2, 0, 0, 2], [2, -1, 2, 12, -1, 1, -12, 0], [2, -6, 0, 7, 6, 0, -2, 0], [0, 0, -4, -4, -10, -10, 11, 3], [5, -9, 7, 5, 7, 0, 0, -7], [-14, 5, -7, 0, 0, 3, 7, 5], [0, 0, -7, -7, -1, -1, -3, 1], [1, -8, 0, 4, 8, 10, -1, -10], [4, 2, 4, -10, 4, -2, 10, 0], [1, -5, 5, 0, 0, -1, -5, -5], [3, 5, -2, 3, 1, 0, 0, -1], [7, -3, 0, -4, 3, 3, -7, -3], [0, 0, 6, 6, 5, 5, 0, 2], [7, 0, 11, 0, 0, 0, -11, 0], [17, -4, 9, 17, 3, 0, 0, -3], [7, 8, 11, 7, -3, 0, 0, 3], [-6, 3, 4, 0, 0, 10, -4, 3], [8, -2, 0, -5, 2, -3, -8, 3], [6, 1, 6, 3, -8, -1, -3, 0], [-7, 0, -12, -7, 3, 0, 0, -3], [-11, -4, -20, 0, 0, -11, 20, -4], [9, -3, 0, 3, 3, 1, -9, -1], [2, -4, 2, 18, -4, 4, -18, 0], [-19, 5, 1, 0, 0, 3, -1, 5], [0, 0, 1, 1, -1, -1, -7, 13], [3, 2, 3, 16, 8, -2, -16, 0], [-14, 1, -8, -14, 5, 0, 0, -5], [-8, 10, 1, 0, 0, 1, -1, 10], [0, 0, 23, 23, 0, 0, -12, 0], [-7, 9, -7, -4, -5, -9, 4, 0], [18, -1, 0, 0, 1, -2, -18, 2], [0, 0, 7, 0, 0, 0, -7, 0], [5, -4, 3, 0, 0, 0, -3, -4], [-12, -1, 0, -2, 1, 1, 12, -1], [-7, -5, 0, -9, 5, 6, 7, -6], [0, 0, 1, 1, 2, 2, -24, -8], [-4, -2, -1, -4, -3, 0, 0, 3], [6, -7, -19, 6, 7, 0, 0, -7], [0, 0, -10, -10, -10, -10, 13, 2], [0, 0, 5, 5, 4, 4, -7, 1], [-6, 3, 6, -6, 6, 0, 0, -6], [0, 15, 0, -19, -15, -8, 0, 8], [-11, 4, -11, 0, -5, -4, 0, 0], [0, 0, 1, 1, 6, 6, 1, -8], [15, 6, 15, 15, -3, 0, 0, 3], [-4, -7, -4, -17, 0, 7, 17, 0], [0, 0, 15, 15, 7, 7, -27, -8], [0, -11, -11, 0, 0, 4, 11, -11], [-7, 3, 0, -27, -3, -2, 7, 2], [7, 0, 5, 7, 3, 0, 0, -3], [24, -2, -3, 0, 0, 4, 3, -2], [-1, -6, 0, 19, 6, -2, 1, 2], [-8, 1, -20, -8, 2, 0, 0, -2], [19, -6, 19, 9, -1, 6, -9, 0], [0, 0, 5, 5, 0, 0, 18, 11], [10, -1, 20, 0, 0, 4, -20, -1], [0, 0, 6, 6, 2, 2, 11, -2], [10, 1, 1, 10, 6, 0, 0, -6], [0, 0, -10, -10, -11, -11, 18, 1], [-5, 8, 0, -5, -8, -4, 5, 4], [13, 7, 13, 10, 1, -7, -10, 0], [3, -2, -34, 3, 0, 0, 0, 0], [-21, 0, 16, 0, 0, 2, -16, 0], [-11, 8, 18, 0, 0, -12, -18, 8], [25, -1, 25, 24, 7, 1, -24, 0], [10, -14, 0, 19, 14, 5, -10, -5], [0, -2, -17, 0, 0, -14, 17, -2], [13, -6, 13, 12, -7, 6, -12, 0], [0, 0, 0, 0, -2, -2, 7, 1], [-9, 6, 0, 4, -6, -10, 9, 10], [2, -16, -4, 2, 3, 0, 0, -3], [22, 8, 12, 0, 0, 2, -12, 8], [18, -3, 3, 18, 9, 0, 0, -9], [-2, 9, -2, 1, 0, -9, -1, 0], [-13, 3, 0, -10, -3, -14, 13, 14], [0, 0, 6, 6, 14, 14, -15, -7], [-4, 2, -4, -20, -10, -2, 20, 0], [-22, 1, -22, -36, -5, -1, 36, 0], [-17, -2, -14, 0, 0, 4, 14, -2], [0, 0, 9, 9, 7, 7, -8, 10], [1, -5, 0, 12, 5, -10, -1, 10], [-1, 5, 0, -4, -5, -4, 1, 4], [17, -5, 0, 15, 5, 0, -17, 0], [0, 0, 8, 8, 0, 0, 10, 11], [-5, 1, -12, -5, -8, 0, 0, 8], [12, -6, 12, 6, 13, 6, -6, 0], [13, -1, 13, 26, 1, 1, -26, 0], [16, -17, 20, 16, 9, 0, 0, -9], [0, 0, 2, 2, 0, 0, -5, 6], [0, 7, 0, -18, -7, 3, 0, -3], [9, 7, -15, 0, 0, -7, 15, 7], [-14, 0, 10, 0, 0, 7, -10, 0], [-5, 17, 0, -20, -17, -9, 5, 9], [0, 0, -2, -2, -11, -11, -3, 4], [-12, -5, -20, 0, 0, -12, 20, -5], [0, 0, 19, 19, 4, 4, 2, 4], [-18, -1, -23, -18, 0, 0, 0, 0], [-28, -2, 0, -27, 2, 2, 28, -2], [-12, 2, -12, -23, 12, -2, 23, 0], [-17, 9, 0, -5, -9, -8, 17, 8], [-1, -9, 0, 0, 9, 12, 1, -12], [3, 4, 3, 7, -15, -4, -7, 0], [-4, -4, 0, 5, 4, 8, 4, -8], [-17, 1, -38, 0, 0, 3, 38, 1], [-5, 15, 18, -5, -2, 0, 0, 2], [10, -6, 9, 0, 0, -5, -9, -6], [0, 0, -16, -16, 0, 0, -8, 6], [6, 1, 6, 8, 4, -1, -8, 0], [7, -8, 3, 7, 13, 0, 0, -13], [1, -3, 21, 0, 0, 10, -21, -3], [2, -4, 16, 2, 2, 0, 0, -2]];
    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_b();".
// 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_b(: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_b();".
// 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_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_495_n();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![121, -77, 135, 3, 4, -3, 5, 2, 1]>],Snew);
    return Vf;
end function;