
// Make newform 630.2.be.b in Magma, downloaded from the LMFDB on 27 June 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_630_be();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_630_be_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_630_2_be_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_630_2_be_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 := [1, 0, 0, 0, -1, 0, 0, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    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_630_be();"
function MakeCharacter_630_be()
    N := 630;
    order := 6;
    char_gens := [281, 127, 451];
    v := [3, 6, 5];
    // 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_630_be_Hecke();"
function MakeCharacter_630_be_Hecke( : Kf := false)
    N := 630;
    order := 6;
    char_gens := [281, 127, 451];
    char_values := [[-1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, -1, 0, 0, 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 := [[0, 0, -1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, -1, 0, 0, 0], [0, 0, 0, 3, 0, 0, 0, -2], [4, -1, 0, -1, -2, 1, 0, 1], [0, -1, 0, 1, 0, 0, 2, -1], [0, -2, -2, 2, 0, 2, -2, -2], [0, 2, 1, 2, 0, 0, 0, 0], [1, 0, -2, 0, 1, 0, 0, 0], [0, 2, 0, -2, 0, 2, -4, 4], [0, -2, 2, 0, 0, 0, -2, 2], [6, -1, 0, 1, -6, 0, 0, 0], [4, 3, 0, 3, 0, -2, 0, -1], [-2, 2, 0, 2, 0, 0, 0, -2], [-2, 4, 1, 0, 2, -4, -2, -4], [4, -2, -1, -6, -2, 6, 1, 2], [0, 5, -2, -2, -6, -2, -2, 5], [0, -2, 2, 4, 0, 6, 0, -6], [0, 6, -4, -6, -6, -6, -4, 6], [0, 2, 0, -2, 0, 4, 4, 6], [0, -4, 6, 0, 0, 0, -6, 4], [-6, 2, 4, 4, 6, -6, -8, -6], [-2, 6, 0, 6, 0, -8, 0, 2], [-4, -4, 0, -3, 4, 7, 0, 7], [6, 2, 0, -2, -12, -6, 6, -4], [0, -4, 6, 0, -2, 0, 6, -4], [0, -6, 4, -5, 0, 1, 0, -1], [-2, -6, 2, 0, -2, 6, 0, -6], [0, 6, 4, -8, 6, -8, 4, 6], [-4, 6, 0, -6, 8, 2, 6, 8], [-4, -2, -16, -2, 0, 1, 8, 1], [-8, 5, -2, -4, 8, -1, 4, -1], [0, 4, -4, -2, 0, 2, 4, -4], [-6, 8, 0, -8, 12, 0, 4, 8], [-6, -6, 8, -6, -6, 0, 0, 0], [0, -6, -2, -4, 0, -4, -2, -6], [12, -7, -8, -3, -6, 3, 8, 7], [-6, -4, 6, -6, 6, 10, -12, 10], [1, 8, 4, 8, 0, -8, -2, 0], [-2, -10, -1, 2, 2, 8, 2, 8], [0, -8, 6, -3, 0, 3, -6, 8], [-4, 6, 0, -6, 8, -4, 14, 2], [2, -8, -8, -6, 2, 2, 0, -2], [0, 0, -2, -6, 6, -6, -2, 0], [-1, -10, 0, 10, 2, 12, 2, 2], [-12, -2, 8, -6, 6, 6, -8, 2], [-5, -2, -8, -2, 0, -8, 4, 10], [-4, -11, 0, 11, 8, 1, 4, -10], [0, 0, 2, -10, 0, -10, 2, 0], [0, 14, 4, 12, 0, -2, 0, 2], [-2, -10, -10, -4, -2, 6, 0, -6], [6, 12, 0, -12, -12, -4, -8, 8], [0, 8, 3, 12, 0, -12, -3, -8], [2, 8, -8, 8, 0, -9, 4, 1], [-6, 10, 0, -4, 6, -6, 0, -6], [12, 0, -4, 4, -6, -4, 4, 0], [0, -8, -10, 6, 0, 6, -10, -8], [2, 0, -14, 0, 2, 0, 0, 0], [0, -7, -2, 8, 6, 8, -2, -7], [4, 7, 0, -7, -8, -2, -4, 5], [-12, 0, -2, 10, 6, -10, 2, 0], [13, 2, 4, 2, 0, -4, -2, 2], [4, -6, 0, 6, -8, 8, -2, 2], [0, 2, -4, -6, 10, -6, -4, 2], [-4, -14, -2, 2, -4, 16, 0, -16], [4, -8, -4, -8, 4, 0, 0, 0], [-12, -2, 5, 6, 12, -4, -10, -4], [-10, 0, 0, 0, 0, -6, 0, 6], [20, 0, -2, -2, -10, 2, 2, 0], [-4, 4, 0, -4, 8, -4, 12, 0], [0, 0, 2, 12, 6, 12, 2, 0], [-4, 0, 16, -2, -4, -2, 0, 2], [0, 10, 0, -3, 0, 3, 0, -10], [6, -20, -6, 5, -6, 15, 12, 15], [9, -6, -8, -6, 0, -4, 4, 10], [-2, 0, 7, -4, 2, 4, -14, 4], [4, -8, -2, -6, -2, 6, 2, 8], [10, -8, 6, -7, 10, 1, 0, -1], [-8, 1, -4, 7, -8, 6, 0, -6], [0, -4, 0, -20, 0, 20, 0, 4], [-4, 15, 4, 15, 0, 0, -2, -15], [16, 4, -8, 4, 0, 2, 4, -6], [-16, -10, -6, 6, 8, -6, 6, 10], [6, 2, 0, -2, -12, -2, -2, 0], [-4, 16, -4, 12, -4, -4, 0, 4], [2, 2, -4, -4, 2, -6, 0, 6], [0, 10, 0, -10, 0, -13, 0, -3], [-6, -14, 6, 8, 6, 6, -12, 6], [-2, -4, 4, -4, 0, 16, -2, -12], [4, -11, -8, -11, 0, -6, 4, 17], [6, -2, 0, -2, -6, 4, 0, 4], [0, -12, -6, 8, -4, 8, -6, -12], [0, 11, 12, -6, 4, -6, 12, 11], [-6, 5, 0, -5, 12, -7, 6, -2], [12, 8, -14, 4, -12, -12, 28, -12], [0, -12, 4, -12, 0, 20, -2, -8], [-16, -10, -8, 10, 16, 0, 16, 0], [0, 14, 8, -5, -4, -5, 8, 14], [-6, -16, -12, -2, -6, 14, 0, -14], [2, -10, -2, -12, -2, 22, 4, 22], [-4, -10, 28, -10, 0, 8, -14, 2], [-28, -10, 3, 2, 14, -2, -3, 10], [0, 2, 10, 16, -4, 16, 10, 2], [-4, 5, 16, 23, -4, 18, 0, -18], [0, -8, -6, -8, -4, -8, -6, -8], [-4, -6, 24, 4, 2, -4, -24, 6], [-22, -20, 24, -20, 0, 4, -12, 16], [-12, 10, 6, 6, 12, -16, -12, -16], [-8, 14, -6, -4, 4, 4, 6, -14], [8, 12, 0, -12, -16, -4, -22, 8], [4, -1, 28, 2, 4, 3, 0, -3], [0, -9, 4, 5, 6, 5, 4, -9], [-14, 6, 0, -6, 28, 0, 4, 6], [22, -12, -8, -18, -11, 18, 8, 12], [0, -12, 28, -12, 0, -4, -14, 16], [16, -2, 16, -7, -8, 7, -16, 2], [8, 0, 0, 0, -16, -6, -8, -6], [0, 20, 3, -16, 14, -16, 3, 20], [-5, 18, -18, 8, -5, -10, 0, 10], [6, -5, 0, 5, -12, -9, -6, -14], [4, -2, -10, -4, -2, 4, 10, 2], [-6, 6, 12, 6, 0, -10, -6, 4], [-15, 14, -2, -16, 15, 2, 4, 2], [-4, 24, 6, 0, 2, 0, -6, -24], [-4, -12, 12, -16, -4, -4, 0, 4], [6, -18, 0, 18, -12, 8, 6, -10], [2, -12, -2, 26, -2, -14, 4, -14], [6, -2, -10, 12, -6, -10, 20, -10], [4, -11, 0, 11, -8, 22, 0, 11], [2, -21, 12, -13, 2, 8, 0, -8], [0, -2, 11, -6, -12, -6, 11, -2], [10, -12, 0, 12, -20, 8, -3, -4], [8, -6, -12, 0, -8, 6, 24, 6], [-14, 15, 12, 15, 0, -11, -6, -4], [12, 19, 8, -9, -12, -10, -16, -10], [8, 4, 0, -4, -16, 4, -27, 8], [0, -24, 10, 6, 9, 6, 10, -24], [8, 16, 8, 14, -4, -14, -8, -16], [16, -16, -32, -16, 0, 8, 16, 8], [8, 11, -16, 21, -4, -21, 16, -11], [-5, -22, 0, 22, 10, 8, 20, -14], [0, -2, 16, 2, 0, 4, 0, -4], [0, -11, -8, 6, 8, 6, -8, -11], [8, -14, 0, 14, -16, 10, -6, -4], [12, -6, -16, -2, -6, 2, 16, 6], [2, -18, -8, -18, 0, 28, 4, -10], [-2, 12, 0, -12, 4, -15, -4, -3], [0, 12, -16, -2, -12, -2, -16, 12], [2, 28, 16, 16, 2, -12, 0, 12], [3, -16, -14, -12, 3, 4, 0, -4], [-8, -1, -18, 12, 8, -11, 36, -11], [16, -2, 32, -2, 0, -5, -16, 7], [10, 16, 16, 16, 0, -12, -8, -4], [8, 0, 12, -8, -8, 8, -24, 8], [0, 8, 10, -12, 16, -12, 10, 8], [0, 0, 0, 0, 0, -26, 8, -26], [18, -4, 2, 30, -18, -26, -4, -26], [8, 5, -16, -6, -8, 1, 32, 1], [-8, 14, 0, -14, 16, -20, -22, -6], [0, 10, 0, -20, -26, -20, 0, 10], [16, -2, 4, 6, 16, 8, 0, -8], [-8, 6, 0, -6, 16, -30, -12, -24], [12, 9, 8, 9, 0, -11, -4, 2], [-4, -19, 10, 8, 4, 11, -20, 11], [-4, 6, 0, 12, 2, -12, 0, -6], [0, -8, 3, 16, -2, 16, 3, -8], [0, 2, 8, -2, -4, -2, 8, 2], [-4, -29, -2, 0, 2, 0, 2, 29]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_630_be_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_630_2_be_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_630_2_be_b(:prec:=8)
    chi := MakeCharacter_630_be();
    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_630_2_be_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_630_2_be_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_630_be();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<11,R![9409, -23280, 25796, -16320, 6447, -1632, 260, -24, 1]>],Snew);
    return Vf;
end function;