
// Make newform 585.2.bu.b in Magma, downloaded from the LMFDB on 21 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_585_bu();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_585_bu_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_bu_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_585_2_bu_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 := [13, -38, 70, -74, 63, -34, 16, -4, 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], [-159, 235, -426, 221, -172, 32, -15, -1], [-70, 298, -390, 256, -146, 22, -8, -3], [-181, 335, -427, 256, -146, 22, -8, -3], [-152, 294, -498, 373, -261, 89, -29, 3], [97, -266, 366, -378, 215, -114, 28, -8], [39, 38, -84, 276, -184, 159, -41, 17]];
    Rf_basisdens := [1, 1, 37, 37, 37, 37, 37, 37];
    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_bu();"
function MakeCharacter_585_bu()
    N := 585;
    order := 6;
    char_gens := [326, 352, 496];
    v := [6, 6, 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_bu_Hecke();"
function MakeCharacter_585_bu_Hecke( : Kf := false)
    N := 585;
    order := 6;
    char_gens := [326, 352, 496];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, -1, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := ISA(Type(Kf), Fld), 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 := [[-1, 0, -1, 0, 0, 1, 1, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, -1, 0, 0], [1, -1, -1, 0, -2, 1, 1, 0], [-1, 0, 1, -2, 1, -4, 1, -1], [1, -1, -1, 0, 0, 2, 2, -2], [0, -1, 1, 2, -2, 3, -2, 2], [2, -1, -1, -1, 0, 1, 1, 1], [1, -2, -1, 0, -1, 2, -1, 1], [2, -4, -2, 0, -1, 0, 5, -5], [4, -3, 0, -1, -1, 0, -3, -2], [-2, 0, 2, 0, 0, -2, 0, 0], [8, 0, 4, -6, 3, -4, -3, -3], [4, -1, 1, -4, -1, 1, 0, -4], [-1, 0, 0, 1, -10, 11, 0, 2], [4, 2, 4, -2, 4, -2, -2, 0], [2, 0, 0, 1, -4, 0, 3, -1], [-5, 3, -3, 7, -2, 7, -2, 7], [5, 0, 5, 0, 0, -6, -5, 0], [-3, 3, 3, 0, 1, -3, -3, 0], [4, 1, 0, 0, 6, -6, -9, 0], [8, -3, -6, 1, -1, 6, 3, 0], [4, 3, 0, -3, 8, -11, -5, -6], [1, 0, 3, 4, -2, -6, -4, 2], [2, -1, -1, -4, -1, 1, -2, 4], [3, -6, -3, 0, -7, 8, 1, 3], [-1, -1, -2, 0, -3, -1, 1, 0], [-2, 4, 2, 0, 7, -7, -8, -3], [-1, 2, 0, 0, 4, -4, 0, 0], [-4, 4, -4, 4, -2, 8, 0, 4], [3, -6, -3, 0, -14, 7, 4, -6], [2, -6, -12, 5, -5, 12, 6, 0], [1, -2, -2, -5, 5, 2, -4, 5], [9, -2, 2, -2, 4, -10, -7, -2], [5, -7, -7, 1, -13, 7, 6, -1], [-3, 5, 0, 1, 9, -8, -1, 2], [-7, 3, 6, -4, -3, -13, -3, 0], [-1, -1, -1, -2, 14, 1, -3, 2], [0, 0, -8, 4, -2, 4, 2, 2], [12, -3, 3, -8, -4, 5, -4, -8], [-1, 2, 1, 0, 8, -6, -4, -2], [1, 1, 2, -5, 7, 0, -1, 0], [5, -1, 1, 4, -1, 1, -9, 4], [1, 0, -5, 12, -6, 17, -4, 6], [3, 0, -7, 2, -1, 6, 1, 1], [-7, 0, 0, 12, -8, 16, -5, 12], [-2, 4, 2, 0, -12, 4, 7, 0], [4, 0, -4, 0, 0, 12, 0, 0], [5, -6, -6, -1, -15, 6, 4, 1], [-9, -1, 0, 3, 1, 2, 13, 6], [-14, 1, 2, -3, 2, -3, -1, 0], [-2, 5, 0, -1, 4, -5, 1, -2], [4, 5, 5, -5, -2, -5, -1, 5], [-9, 7, -7, 9, -3, 13, 0, 9], [-2, 4, 2, 0, -1, -3, 6, -3], [-1, 2, 1, 0, 6, -9, 3, -10], [-3, -5, 5, -8, 1, -7, 11, -8], [-10, 0, -4, 12, -6, 4, 1, 6], [-8, 4, -4, -8, 0, 4, 16, -8], [-16, 12, 0, 5, 13, -8, 10, 10], [3, -6, -3, 0, -18, 11, -8, -2], [3, 8, 8, -7, 9, -8, -4, 7], [5, -13, 0, 0, 3, -3, 3, 0], [-2, 1, 2, -4, 5, -1, -1, 0], [-1, 3, 6, -2, 3, -5, -3, 0], [0, -3, 0, -1, 8, -9, 5, -2], [9, -8, -8, 4, -4, 8, 13, -4], [11, 1, 2, 0, 7, 5, -1, 0], [2, -4, 4, -4, -10, 16, 2, -4], [-21, 0, 5, 6, -3, 3, 5, 3], [9, 0, -1, 10, -5, 10, -9, 5], [0, 10, 0, 7, -9, 16, -24, 14], [-1, 2, 1, 0, -8, 4, -11, 2], [-7, -1, 1, -2, 10, -21, 9, -2], [13, 0, -13, 6, -3, 17, -3, 3], [5, 6, 6, 3, 5, -6, 8, -3], [10, -5, -10, 3, -14, -1, 5, 0], [0, 7, 7, -2, -5, -7, -2, 2], [-4, 0, 8, 8, -4, -6, -6, 4], [-1, -3, -3, 5, 8, 3, 4, -5], [1, -2, -1, 0, 4, 4, -2, 10], [-2, -3, 0, 9, 5, 4, -11, 18], [-3, 0, -9, 14, -7, 24, -1, 7], [-11, -7, 7, -6, 0, -7, 17, -6], [-9, 18, 9, 0, 1, -13, 5, -7], [11, -5, -10, 14, -6, 18, 5, 0], [0, 4, 4, -4, 10, -4, -4, 4], [-5, 0, 11, 4, -2, 8, -5, 2], [-5, -13, -13, 8, -19, 13, 3, -8], [-5, 17, 0, 2, -3, 5, -11, 4], [-1, 4, 8, -1, -10, -19, -4, 0], [-11, 0, 3, -8, 4, 4, 8, -4], [-11, -6, -6, 4, -9, 6, -7, -4], [-4, 8, 4, 0, 17, -12, 15, 1], [-5, 3, 0, 5, -3, 8, -3, 10], [-2, 6, -6, -2, 8, -10, 4, -2], [-3, 0, 19, -22, 11, -36, 3, -11], [-20, -3, -6, 12, -9, 9, 3, 0], [8, -16, -8, 0, -14, 10, -4, -10], [-11, 16, 0, 2, -8, 10, 2, 4], [-3, 1, 2, -2, -3, -7, -1, 0], [31, 0, 5, -10, 5, -18, -13, -5], [6, -10, 10, -10, 12, -34, 4, -10], [-7, 14, 7, 0, 4, -10, -22, -2], [3, -5, -10, -9, 11, 12, 5, 0], [10, 0, 0, 4, -6, 10, -28, 8], [8, 0, 14, -10, 5, -25, -6, -5], [-4, -8, 0, 10, -12, 22, -4, 20], [-10, 5, 10, -2, 1, -11, -5, 0], [-9, 18, 9, 0, 1, -13, 16, -7], [-15, 8, -8, 8, 5, -2, 7, 8], [-23, 0, 1, 0, 0, 8, 11, 0], [-13, 17, 17, 1, 1, -17, -12, -1], [13, -18, 0, -10, -4, -6, 12, -20], [1, -1, -1, 7, -4, 1, 8, -7], [-23, 1, -1, 11, -11, 23, 12, 11], [5, 13, 13, -2, 8, -13, 3, 2], [5, -10, -5, 0, -9, 4, -3, -11], [-3, 6, 3, 0, -16, 11, -11, 12], [14, 2, -2, -8, -6, 14, -6, -8], [12, 0, 10, -8, 4, -16, -7, -4], [-1, 2, 1, 0, 18, -13, -8, -6], [-22, -5, -10, 15, -20, 5, 5, 0], [-4, 7, 7, 2, -2, -7, -2, -2], [-17, 0, 9, 14, -7, 27, -3, 7], [34, -4, -8, 0, -2, 6, 4, 0], [-1, 18, 18, -8, 4, -18, -9, 8], [20, -2, 2, 7, -8, 14, -27, 7], [-7, 13, 26, 0, 5, -21, -13, 0], [-9, 7, 0, 4, 19, -15, 3, 8], [-22, 0, 6, -12, 6, -12, 14, -6], [1, 0, 5, 4, -2, -9, -5, 2], [9, -18, -9, 0, -17, 9, 0, -17], [0, 0, 0, 0, -4, 3, 7, 2], [3, 1, 1, 3, 13, -1, 6, -3], [29, 0, 21, -18, 9, -11, -16, -9], [-3, 5, 5, -11, -1, -5, -14, 11], [7, -1, -1, -10, 14, 1, -3, 10], [-3, 8, -8, 13, -17, 42, -10, 13], [6, -12, -6, 0, -14, 8, 30, -10], [7, -14, 0, -2, 22, -24, 4, -4], [-28, 0, -24, 16, -8, 12, 18, 8], [-20, 4, -4, -8, 10, -16, 28, -8], [-9, -6, 0, 1, -6, 7, 22, 2], [5, -10, -5, 0, 35, -17, 25, -9], [13, -9, -9, -5, 1, 9, 8, 5], [-6, 5, 0, 8, -8, 16, -9, 16], [-13, 2, 4, -7, -10, -21, -2, 0], [4, -5, -10, 19, -17, 12, 5, 0], [8, -12, 0, -2, 0, -2, 0, -4], [-16, -4, -4, 0, 20, 4, -16, 0], [2, -4, -2, 0, 22, -12, 14, -6], [15, 1, 2, -14, -1, -17, -1, 0], [-7, 14, 7, 0, -4, 1, -35, 12], [4, -8, -4, 0, -26, 11, 19, -12], [20, 8, 16, -12, 8, -20, -8, 0], [8, 5, -5, 3, -2, 9, -11, 3], [6, -6, -6, -2, -2, 6, 4, 2], [-10, -4, -8, 8, -6, 10, 4, 0], [-4, 0, 0, 9, 17, -8, -10, 18], [5, 0, -11, 16, -8, 35, -5, 8], [-3, 7, -7, 15, -17, 41, -12, 15], [10, -12, 0, -4, 16, -20, 0, -8], [29, -7, 7, -19, -1, -5, -10, -19], [-16, 0, 12, 0, 0, 18, 2, 0], [8, 9, 0, -3, 12, -15, -19, -6], [-5, 10, 5, 0, 11, -11, -8, -1], [9, -7, 7, -12, -18, 29, 3, -12]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_585_bu_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_585_2_bu_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_585_2_bu_b(:prec:=8)
    chi := MakeCharacter_585_bu();
    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_bu_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_585_2_bu_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_585_bu();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![4, 24, 48, 0, -22, 0, 12, 6, 1]>],Snew);
    return Vf;
end function;