
// Make newform 585.2.j.h in Magma, downloaded from the LMFDB on 20 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_585_j();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_585_j_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_585_2_j_h();".
// 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_585_2_j_h();".
// 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 := [36, -36, 96, -12, 124, -42, 50, -8, 10, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [-55404, 124656, -156238, 143476, -87134, 39184, -18528, 7398, -2199, 355], [-235800, 136368, -289782, 721480, -318192, 231068, -64307, 50191, -12311, 4282], [99432, -339018, -642880, -494008, 44032, -263193, 2209, -53189, 8818, -6550], [1426266, -218016, 463284, -959878, 508704, -369416, 81275, -80242, 19682, -8029], [-408504, 907392, -732992, -621080, 123782, -354900, 36452, -89335, 17684, -9476], [-953850, -2260776, -1844240, -1168190, -700936, -519015, -139765, -72631, -7555, -10145], [-297390, -311802, -551312, -414542, -19472, -217077, -3447, -43172, 6361, -5301], [675982, -256176, 544374, -956678, 597744, -434076, 111339, -94287, 23127, -9385]];
    Rf_basisdens := [1, 1, 40746, 448206, 448206, 448206, 448206, 448206, 149402, 149402];
    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_585_j();"
function MakeCharacter_585_j()
    N := 585;
    order := 3;
    char_gens := [326, 352, 496];
    v := [3, 3, 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_585_j_Hecke();"
function MakeCharacter_585_j_Hecke(Kf)
    N := 585;
    order := 3;
    char_gens := [326, 352, 496];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 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, 1, 0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 1, 0, 0], [0, 1, 0, 1, -1, 0, 0, -1, 0, -1], [0, 1, -1, 0, 0, 0, 1, 0, -1, 0], [0, 1, 0, 0, 0, 0, 0, 0, -2, 0], [1, 0, 0, 0, -1, 0, 0, 0, -1, 0], [0, 0, 1, -1, 1, -1, 0, 0, -1, 0], [0, 0, 1, 0, -3, 0, 0, -1, 0, -1], [2, 0, -1, 1, 0, 0, 1, 0, 0, 1], [0, -1, 1, -2, 0, -1, 0, 1, -1, 1], [0, 0, 1, 2, -1, 2, 0, 1, 2, 1], [-2, -3, 0, 0, 2, 0, 0, 0, 1, 0], [3, 0, -1, 2, 0, 1, 1, 0, 0, -2], [-4, 0, 0, 2, 0, 2, 0, 0, 0, 2], [-3, 0, 0, 0, 3, 0, 1, -1, 2, 0], [-2, -1, 0, 0, 2, 0, 1, -1, 0, 0], [0, -2, -1, -2, -2, 0, 0, 1, 0, 1], [-2, -1, 0, 0, 2, 0, -1, 0, 2, 0], [-2, 0, -1, -6, 0, -1, 1, 0, 0, 0], [-2, 0, 2, 3, 0, -1, -2, 0, 0, 0], [3, 0, -3, -1, 0, -1, 3, 0, 0, 0], [0, -3, 1, -3, -2, 0, 0, 0, 0, 0], [2, -1, 0, 0, -2, 0, -2, -2, 1, 0], [0, -5, 0, -3, 5, 2, 0, -1, 2, -1], [6, 0, -2, -4, 0, -3, 2, 0, 0, 0], [0, -1, -1, 0, -5, 1, 0, 2, 1, 2], [3, 0, 1, 6, 0, 1, -1, 0, 0, -3], [0, 0, 0, 0, 0, 0, 4, 2, 0, 0], [0, 7, -1, 6, 4, -1, 0, -2, -1, -2], [-7, 0, -3, -4, 0, 2, 3, 0, 0, -1], [-9, 3, 0, 0, 9, 0, 1, 2, 3, 0], [5, 3, 0, 0, -5, 0, -3, -1, -1, 0], [-7, 3, 0, 0, 7, 0, 4, -3, -4, 0], [-3, 0, -3, -5, 0, 4, 3, 0, 0, 1], [4, 0, -1, 6, 0, 2, 1, 0, 0, 3], [-10, 0, 0, 0, 10, 0, 0, -1, -4, 0], [0, -6, -2, -4, -4, 2, 0, -2, 2, -2], [8, -1, 0, 0, -8, 0, 2, 2, -4, 0], [0, 3, -5, 5, 4, 2, 0, 0, 2, 0], [-7, 0, -1, 1, 0, 6, 1, 0, 0, 1], [-4, 5, 0, 0, 4, 0, -1, -4, 2, 0], [0, -2, -4, 0, 2, 2, 0, -1, 2, -1], [0, 0, 1, -1, -3, -1, 0, -4, -1, -4], [7, -4, 0, 0, -7, 0, -2, -4, 0, 0], [0, -2, 2, -4, 1, -2, 0, -2, -2, -2], [0, -7, 1, -2, 0, 5, 0, 1, 5, 1], [-9, -1, 0, 0, 9, 0, -1, -2, 3, 0], [-5, 0, 1, -7, 0, 0, -1, 0, 0, 1], [11, 0, 1, 3, 0, 1, -1, 0, 0, -2], [7, 0, 2, 1, 0, 0, -2, 0, 0, -1], [3, 1, 0, 0, -3, 0, 5, -1, 4, 0], [11, -9, 0, 0, -11, 0, -2, 3, 2, 0], [0, 5, 5, 8, -1, 3, 0, 0, 3, 0], [0, 3, -2, -3, -4, -6, 0, 4, -6, 4], [-2, 1, 0, 0, 2, 0, -3, 2, -4, 0], [0, -6, 0, 0, 1, 6, 0, 0, 6, 0], [-2, -3, 0, 0, 2, 0, -3, -3, 1, 0], [1, 0, 1, 2, 0, 6, -1, 0, 0, -3], [0, 3, 3, 6, -14, 3, 0, 0, 3, 0], [-9, 3, 0, 0, 9, 0, 1, 2, -3, 0], [0, 0, 1, -4, 0, 0, -1, 0, 0, -3], [14, 0, 3, 5, 0, 2, -3, 0, 0, 2], [-2, 0, 1, 2, 0, -6, -1, 0, 0, 1], [-5, 0, 3, -3, 0, 3, -3, 0, 0, 4], [3, 5, 0, 0, -3, 0, 1, -3, -1, 0], [-6, 0, 4, 2, 0, 3, -4, 0, 0, 0], [-8, -6, 0, 0, 8, 0, -2, -2, 4, 0], [0, -2, 2, -7, 2, -5, 0, 2, -5, 2], [0, 6, 1, -1, -1, -7, 0, 4, -7, 4], [7, 0, 3, 8, 0, 2, -3, 0, 0, 1], [0, -3, 0, -10, -10, -7, 0, 4, -7, 4], [2, -1, 0, 0, -2, 0, 1, 4, 1, 0], [0, 4, -4, 9, 8, 5, 0, 2, 5, 2], [-7, -1, 0, 0, 7, 0, -1, 2, -1, 0], [13, 0, 0, -1, 0, 2, 0, 0, 0, 1], [16, -4, 0, 0, -16, 0, -5, -1, 2, 0], [0, 12, 4, 10, 6, -2, 0, 2, -2, 2], [-6, 5, 0, 0, 6, 0, 1, 3, 2, 0], [0, -5, -3, -11, 6, -6, 0, 0, -6, 0], [12, 0, -4, -7, 0, 1, 4, 0, 0, 2], [0, 11, -4, 5, -1, -6, 0, -3, -6, -3], [-2, 0, 0, 0, 2, 0, 0, 3, -8, 0], [0, -9, -3, -11, -4, -2, 0, 1, -2, 1], [6, 0, -4, 3, 0, -4, 4, 0, 0, -2], [-10, 4, 0, 0, 10, 0, 0, -2, 2, 0], [0, 5, -2, -4, -14, -9, 0, 2, -9, 2], [-10, -3, 0, 0, 10, 0, -5, 0, 6, 0], [6, 0, 3, 2, 0, 2, -3, 0, 0, -5], [9, 0, -3, -4, 0, -1, 3, 0, 0, 0], [0, -3, -5, -7, -12, -4, 0, -4, -4, -4], [-2, 10, 0, 0, 2, 0, -1, 4, -6, 0], [0, -14, -3, -6, -11, 8, 0, 1, 8, 1], [7, 0, -3, -5, 0, -2, 3, 0, 0, 5], [-20, -8, 0, 0, 20, 0, -6, 2, 2, 0], [0, -7, 2, -5, -5, 2, 0, -1, 2, -1], [-14, 0, -3, -9, 0, 6, 3, 0, 0, 4], [0, -4, 4, -6, -4, -2, 0, 6, -2, 6], [-9, 0, 0, 0, 0, 0, 0, 0, 0, -2], [16, 0, 4, 10, 0, -1, -4, 0, 0, -2], [0, -14, -3, -13, -3, 1, 0, 6, 1, 6], [16, -6, 0, 0, -16, 0, -2, -2, 4, 0], [0, 7, 3, 5, 12, -2, 0, 0, -2, 0], [-5, 0, 4, 7, 0, 5, -4, 0, 0, -2], [-2, 0, -2, 2, 0, -6, 2, 0, 0, -2], [0, -11, -9, -4, -5, 7, 0, 4, 7, 4], [-6, 0, 2, -6, 0, 2, -2, 0, 0, -2], [-16, 0, 3, -7, 0, -10, -3, 0, 0, 2], [0, -12, 0, -7, -1, 5, 0, 4, 5, 4], [-18, 4, 0, 0, 18, 0, -1, -4, 10, 0], [0, 19, 2, 20, 10, 1, 0, -2, 1, -2], [-15, 6, 0, 0, 15, 0, 1, 2, -7, 0], [-9, 0, -1, -4, 0, -11, 1, 0, 0, 5], [0, -10, 0, 0, 0, 0, 4, 0, 1, 0], [-1, -15, 0, 0, 1, 0, -2, -3, 4, 0], [4, 13, 0, 0, -4, 0, 5, -2, -3, 0], [0, 10, 3, 1, 11, -9, 0, 8, -9, 8], [0, 3, 4, -1, 6, -4, 0, -4, -4, -4], [-10, -10, 0, 0, 10, 0, 2, -4, 2, 0], [0, -3, 3, -6, -13, -3, 0, 1, -3, 1], [0, -14, -4, -10, -16, 4, 0, -1, 4, -1], [7, 0, 3, 15, 0, 3, -3, 0, 0, 4], [-6, 1, 0, 0, 6, 0, 0, 0, 6, 0], [0, 0, 6, -1, 4, -1, 0, 6, -1, 6], [24, 0, 4, 6, 0, -2, -4, 0, 0, 2], [-5, 3, 0, 0, 5, 0, 3, 3, 1, 0], [5, 12, 0, 0, -5, 0, 1, -3, -6, 0], [14, 0, 3, 14, 0, -4, -3, 0, 0, 3], [-2, 0, 4, 0, 0, 1, -4, 0, 0, -2], [0, -13, -5, -12, -10, 1, 0, -3, 1, -3], [0, 9, 4, 11, -24, 2, 0, -4, 2, -4], [0, -7, 1, -13, 1, -6, 0, -1, -6, -1], [0, -6, -3, -6, 6, 0, 0, -4, 0, -4], [13, -11, 0, 0, -13, 0, 0, -1, -4, 0], [0, 2, 4, 5, 31, 3, 0, 2, 3, 2], [-11, 4, 0, 0, 11, 0, 3, -4, 3, 0], [16, -9, 0, 0, -16, 0, 3, 0, -9, 0], [-3, 9, 0, 0, 3, 0, 1, 2, -13, 0], [0, -6, -2, 4, -2, 10, 0, 0, 10, 0], [5, 0, 0, 2, 0, 2, 0, 0, 0, -6], [0, 6, 10, 6, -16, 0, 0, -2, 0, -2], [-18, 5, 0, 0, 18, 0, 1, 3, -3, 0], [9, 0, -3, 4, 0, 1, 3, 0, 0, 0], [0, -4, -2, -5, 0, -1, 0, 10, -1, 10], [-11, -3, 0, 0, 11, 0, -2, 1, 4, 0], [4, 0, -1, 6, 0, -7, 1, 0, 0, -6], [11, 0, 3, -6, 0, -3, -3, 0, 0, -10], [-22, 0, -8, 3, 0, -5, 8, 0, 0, -4], [0, 0, 6, -12, 0, 0, -6, 0, 0, -6], [4, 18, 0, 0, -4, 0, 0, -6, -4, 0], [0, -12, -8, -18, 10, -6, 0, 6, -6, 6], [2, 0, 3, 0, 0, 6, -3, 0, 0, -9], [0, 7, 6, 5, 16, -2, 0, 4, -2, 4], [0, -16, -5, -10, 4, 6, 0, 2, 6, 2], [0, 0, 0, -6, 0, 6, 0, 0, 0, -6], [19, -1, 0, 0, -19, 0, -5, 5, 0, 0], [-8, 8, 0, 0, 8, 0, -4, 0, -6, 0], [10, 0, 5, 6, 0, 5, -5, 0, 0, -3], [13, 0, 3, -6, 0, -12, -3, 0, 0, -5], [0, -13, -4, -5, -26, 8, 0, 2, 8, 2], [-13, 2, 0, 0, 13, 0, 5, -10, -1, 0], [-18, 0, 0, -8, 0, 4, 0, 0, 0, 10], [-9, 7, 0, 0, 9, 0, 6, -9, -2, 0], [0, 0, -2, 2, -12, 2, 0, 2, 2, 2], [5, 0, 3, -7, 0, -1, -3, 0, 0, 8], [0, -3, -3, 7, 11, 10, 0, 1, 10, 1], [-30, 0, 0, 0, 30, 0, -6, 1, 6, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_585_j_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_585_2_j_h();".
// 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_585_2_j_h(:prec:=10)
    chi := MakeCharacter_585_j();
    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_585_2_j_h();".
// 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_585_2_j_h( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_585_j();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![36, 36, 96, 12, 124, 42, 50, 8, 10, 2, 1]>],Snew);
    return Vf;
end function;