
// Make newform 637.2.e.n in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_637_e();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_637_e_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_637_2_e_n();".
// 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_637_2_e_n();".
// 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, 6, 43, -26, 92, -2, 85, -18, 34, -6, 9, -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, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [9713364, 5740174, 40367568, 109670152, 58644524, 44372910, 17140164, 17395862, 4369060, 3065721, 5336, 512378], [115458972, 898922, 6182709, -55980756, 3886852, 15296370, -16319384, 8967106, -4394050, 4377607, -1551017, 517714], [103693971, 10958926, 76649413, 33981855, 95992492, 109881238, -10138403, 52032662, -6968746, 18068728, -4668361, 2297118], [-66913, -17454, -122643, -233985, -174444, -141030, -38216, -57462, -9570, -12173, 1119, -1878], [159636402, 50284591, 352920684, 510469395, 486894758, 431005791, 87381018, 184020371, 12543658, 48698198, -7815976, 6705029], [10820061, -552502129, 383580396, -854628237, 167410082, -813418056, 220945734, -318884784, 69981075, -86127174, 20220925, -9197775], [52540010, 377307084, -362217616, 834984964, -63799638, 808495776, -192236414, 325885316, -61345905, 87414940, -19939106, 9713364], [17525090, -609066000, 621612052, -1381879675, 273803355, -1313722872, 359867408, -526956324, 113931690, -138901006, 31867988, -14791857], [-20852300, 943653761, -747084060, 1591586825, -492235424, 1478901160, -473129740, 577528481, -148661200, 151870530, -36300822, 15232235], [95366656, 686031637, -701960443, 1560299776, -186243800, 1572618642, -401612992, 634374770, -127060870, 171764159, -39883548, 18914350]];
    Rf_basisdens := [1, 1, 62842357, 62842357, 62842357, 45971, 62842357, 62842357, 62842357, 62842357, 62842357, 62842357];
    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_637_e();"
function MakeCharacter_637_e()
    N := 637;
    order := 3;
    char_gens := [248, 197];
    v := [1, 3];
    // 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_637_e_Hecke();"
function MakeCharacter_637_e_Hecke(Kf)
    N := 637;
    order := 3;
    char_gens := [248, 197];
    char_values := [[-1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 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, -1, 0, 0, 0, 0, 0, 0, 0, 1], [0, 1, 1, 0, 0, 0, 0, 0, -1, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0, -2, 0, 0, -1, -2, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -2, -2, 0, 1, -1, -1, -1, -3, -1, -1, -1], [0, 0, 0, -1, 0, 0, 0, -1, 0, 1, 1, 1], [0, 1, 0, 0, 0, 0, 0, 1, 0, -1, -2, 0], [-1, 0, 0, 0, -2, -2, 0, 0, 0, 0, 0, 0], [0, 1, 1, 0, 3, 0, 1, 0, -1, -3, 1, -2], [0, -2, 0, 2, 0, 0, 0, 2, 0, -1, -2, -2], [0, 0, 3, -2, 0, -2, -1, 0, 0, 0, 0, 0], [1, 0, 4, 0, 2, 0, 2, 0, 0, 0, 0, 0], [-6, 0, 0, 3, 0, 0, 0, 3, 6, -1, -3, -3], [0, -1, -1, 0, 1, 1, 0, 1, 2, -1, 0, -4], [0, -1, -1, 0, -2, -3, 2, -3, -5, 2, 2, 1], [-2, 3, 0, 1, 0, 0, 0, 2, 2, 1, -3, -1], [0, -1, -1, 0, -1, 1, 3, 1, -4, 1, 3, 1], [2, 0, -1, -4, 5, 5, -3, 0, 0, 0, 0, 0], [0, -2, -2, 0, 1, 1, -5, 1, 2, -1, -5, -1], [3, 2, 0, -3, 0, 0, 0, -3, -3, -3, 0, 3], [7, 0, -2, 0, 1, 0, 2, 0, 0, 0, 0, 0], [-4, 1, 0, 1, 0, 0, 0, 1, 4, -3, 2, -1], [-4, 0, -3, -3, -1, 5, 2, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 3, -1, 3, 1, -1, -1, -1], [-4, 3, 0, 3, 0, 0, 0, -1, 4, -3, -1, -3], [1, -1, 0, 6, 0, 0, 0, -3, -1, 3, -2, -6], [0, 1, 1, 0, 0, 1, 3, 1, -2, 0, 3, 3], [-6, 0, -3, -1, -4, 2, 0, 0, 0, 0, 0, 0], [-7, 0, -1, 0, -3, 3, 6, 0, 0, 0, 0, 0], [-4, -6, 0, -2, 0, 0, 0, -4, 4, 2, -2, 2], [0, -3, -3, 0, 6, 1, -5, 1, 0, -6, -5, -3], [-3, 0, 3, 4, 2, 2, 2, 0, 0, 0, 0, 0], [-2, 2, 0, 7, 0, 0, 0, 0, 2, -4, -4, -7], [0, -1, -1, 0, 4, 5, -3, 5, -4, -4, -3, -5], [0, -5, -5, 0, -6, -1, -4, -1, -1, 6, -4, -2], [0, 9, 0, -3, 0, 0, 0, 1, 0, -3, -1, 3], [8, 0, 2, -1, 1, -5, -1, 0, 0, 0, 0, 0], [-1, 1, 0, -10, 0, 0, 0, 2, 1, 4, -2, 10], [0, 2, 2, 0, -6, -4, 0, -4, 7, 6, 0, 4], [-9, 0, -2, -1, 5, 6, -3, 0, 0, 0, 0, 0], [5, 7, 0, -2, 0, 0, 0, -5, -5, 1, 4, 2], [0, 0, 0, 0, 2, 8, 4, 8, 2, -2, 4, -2], [0, 0, -2, -1, 1, -4, 2, 0, 0, 0, 0, 0], [0, -1, -1, 0, -8, -1, -2, -1, -3, 8, -2, -2], [4, 0, 7, 1, -4, 2, 0, 0, 0, 0, 0, 0], [1, 0, 5, 6, -3, -4, -3, 0, 0, 0, 0, 0], [0, 0, 0, 0, 4, 4, 2, 4, -10, -4, 2, -4], [7, -3, 0, -5, 0, 0, 0, 5, -7, 0, -2, 5], [-1, -2, 0, -6, 0, 0, 0, 0, 1, 0, 4, 6], [4, 0, -2, 1, -1, 2, 0, 0, 0, 0, 0, 0], [0, 13, 13, 0, 5, 6, 1, 6, 5, -5, 1, 6], [13, 0, -1, 0, -2, -4, 2, 0, 0, 0, 0, 0], [1, -2, 0, 7, 0, 0, 0, -8, -1, 3, -3, -7], [0, -2, -2, 0, 3, -1, 6, -1, -5, -3, 6, 3], [0, -4, -4, 0, 0, 1, -2, 1, 0, 0, -2, 6], [6, 6, 0, -2, 0, 0, 0, 0, -6, 0, 2, 2], [0, 4, 4, 0, 1, 3, 2, 3, 3, -1, 2, -3], [10, 0, 4, 7, 0, 0, -2, 0, 0, 0, 0, 0], [0, 1, 1, 0, -1, 5, 3, 5, -10, 1, 3, 7], [-2, 0, -8, 1, -3, -3, -1, 0, 0, 0, 0, 0], [9, 0, 5, -6, 3, -2, -5, 0, 0, 0, 0, 0], [0, -2, -2, 0, -1, 8, 1, 8, -9, 1, 1, 3], [-5, 0, 0, 5, 0, 0, 0, -2, 5, -1, 1, -5], [-6, 5, 0, -4, 0, 0, 0, 3, 6, 4, 3, 4], [-4, -5, 0, 6, 0, 0, 0, 1, 4, 4, 5, -6], [0, 0, -3, 9, 2, -4, 0, 0, 0, 0, 0, 0], [0, 4, 4, 0, -5, -7, 0, -7, 4, 5, 0, 7], [-6, 0, -2, 1, -3, -9, -3, 0, 0, 0, 0, 0], [0, 7, 7, 0, -2, 1, -2, 1, -3, 2, -2, -9], [-16, 8, 0, -1, 0, 0, 0, -2, 16, -4, 2, 1], [0, 2, 2, 0, 7, 1, -1, 1, 15, -7, -1, 3], [11, -1, 0, 4, 0, 0, 0, 3, -11, -3, 2, -4], [6, 0, 1, -2, -3, -7, -3, 0, 0, 0, 0, 0], [-14, 2, 0, -6, 0, 0, 0, -2, 14, 2, -6, 6], [0, 5, 5, 0, -2, -14, 2, -14, 1, 2, 2, 3], [4, 9, 0, 3, 0, 0, 0, 11, -4, -3, -4, -3], [0, -10, 0, 3, 0, 0, 0, 4, 0, -11, -8, -3], [0, -12, -12, 0, -3, 7, -3, 7, 0, 3, -3, -1], [13, 0, 0, -5, -1, -10, 1, 0, 0, 0, 0, 0], [0, 0, 5, 1, 0, -1, -9, 0, 0, 0, 0, 0], [0, -1, -1, 0, 8, -3, -3, -3, -12, -8, -3, -10], [-6, 0, -9, -1, 5, -2, 7, 0, 0, 0, 0, 0], [10, -8, 0, -2, 0, 0, 0, -3, -10, 4, -2, 2], [10, -9, 0, -2, 0, 0, 0, -5, -10, 7, 0, 2], [-4, 0, -7, -1, -1, -13, -5, 0, 0, 0, 0, 0], [6, -1, 0, 6, 0, 0, 0, -3, -6, 5, 1, -6], [-8, 0, 0, 12, 0, 4, 8, 0, 0, 0, 0, 0], [10, 0, -5, -7, 8, 11, 1, 0, 0, 0, 0, 0], [-9, 5, 0, -6, 0, 0, 0, 8, 9, 0, 12, 6], [0, 8, 8, 0, -5, -4, 0, -4, 11, 5, 0, -2], [0, 10, 10, 0, -2, -2, -8, -2, 4, 2, -8, 10], [-11, 0, -1, 7, 0, -2, -2, 0, 0, 0, 0, 0], [2, 4, 0, -17, 0, 0, 0, -8, -2, 10, 4, 17], [14, 0, 11, 1, 1, 10, -3, 0, 0, 0, 0, 0], [-2, -6, 0, -3, 0, 0, 0, 15, 2, 3, -1, 3], [0, 3, 3, 0, -5, -11, 7, -11, -4, 5, 7, 5], [6, -6, 0, 0, 0, 0, 0, 1, -6, -6, -8, 0], [-6, 13, 0, -4, 0, 0, 0, 5, 6, 1, -1, 4], [-4, 0, -11, -13, 4, 2, -6, 0, 0, 0, 0, 0], [0, -9, -9, 0, -3, 1, -3, 1, 8, 3, -3, -7], [0, -1, -1, 0, -5, -4, 5, -4, -16, 5, 5, -1], [-9, 10, 0, 1, 0, 0, 0, 5, 9, -5, 2, -1], [0, -9, -9, 0, -1, -1, 4, -1, -2, 1, 4, -2], [0, 1, 1, 0, 6, 0, -3, 0, 14, -6, -3, 4], [15, 0, 4, 6, -7, -6, 2, 0, 0, 0, 0, 0], [9, -5, 0, 12, 0, 0, 0, -2, -9, -7, -1, -12], [0, 12, 12, 0, 8, 6, -10, 6, 17, -8, -10, 0], [-4, 0, -3, -11, -5, -3, -1, 0, 0, 0, 0, 0], [-1, -5, 0, 2, 0, 0, 0, -11, 1, 6, 0, -2], [0, 6, 6, 0, 8, 4, 2, 4, 0, -8, 2, 8], [-2, 0, -4, -8, 6, -10, 8, 0, 0, 0, 0, 0], [0, -4, -4, 0, -2, 10, 2, 10, 12, 2, 2, 10], [-2, 0, -9, 6, 8, -1, 7, 0, 0, 0, 0, 0], [0, 10, 10, 0, 2, 2, 0, 2, -1, -2, 0, 12], [4, 0, -13, -12, 4, 8, -9, 0, 0, 0, 0, 0], [0, 12, 12, 0, 5, 1, -11, 1, 7, -5, -11, 3], [4, -12, 0, -1, 0, 0, 0, -1, -4, 1, -6, 1], [-4, 0, 1, 0, 1, -11, 4, 0, 0, 0, 0, 0], [0, -2, -2, 0, 1, -10, -2, -10, 5, -1, -2, 10], [0, 0, 11, 4, -3, 9, 6, 0, 0, 0, 0, 0], [11, 3, 0, 0, 0, 0, 0, -16, -11, -2, 2, 0], [0, 9, 9, 0, 10, -1, 3, -1, -2, -10, 3, 4], [4, 2, 0, 5, 0, 0, 0, -5, -4, 1, 7, -5], [-13, 0, 4, 5, 1, -1, -2, 0, 0, 0, 0, 0], [6, 0, 0, -8, 0, 0, 0, -2, -6, -3, 4, 8], [5, -11, 0, -4, 0, 0, 0, 9, -5, 4, 4, 4], [-4, 0, -14, -2, -4, 4, 0, 0, 0, 0, 0, 0], [-3, -8, 0, 10, 0, 0, 0, 2, 3, 13, 2, -10], [0, 6, 6, 0, 3, -20, 4, -20, 0, -3, 4, -4], [4, 0, 17, 0, 10, 9, 1, 0, 0, 0, 0, 0], [5, -4, 0, -2, 0, 0, 0, 4, -5, 10, 2, 2], [-5, 0, 4, 5, -11, 5, -2, 0, 0, 0, 0, 0], [11, 6, 0, 6, 0, 0, 0, -6, -11, -13, 8, -6], [-1, 0, -13, -4, 9, 4, -9, 0, 0, 0, 0, 0], [0, -5, -5, 0, 0, 12, -7, 12, 0, 0, -7, -6], [0, -8, -8, 0, 11, 0, 0, 0, -15, -11, 0, -18], [10, 0, -3, -11, -3, -1, 1, 0, 0, 0, 0, 0], [0, 4, 4, 0, 5, 5, -10, 5, 21, -5, -10, -3], [14, 0, 9, -6, 2, 8, -5, 0, 0, 0, 0, 0], [6, 10, 0, -9, 0, 0, 0, 18, -6, -9, 0, 9], [0, 1, 1, 0, -12, 4, 4, 4, -9, 12, 4, 1], [8, 0, 3, -3, -3, -9, -7, 0, 0, 0, 0, 0], [0, 6, 6, 0, -11, -4, 11, -4, 5, 11, 11, 13], [4, 0, -9, 2, -2, 4, -3, 0, 0, 0, 0, 0], [11, 0, -9, 0, -1, -4, 5, 0, 0, 0, 0, 0], [0, -7, -7, 0, 5, 8, 9, 8, -4, -5, 9, -13], [-4, 11, 0, -5, 0, 0, 0, 3, 4, 3, 13, 5], [-16, 2, 0, -10, 0, 0, 0, 20, 16, 0, 6, 10], [2, 10, 0, 0, 0, 0, 0, 0, -2, -11, -4, 0], [-4, 0, 8, 6, 4, 7, -16, 0, 0, 0, 0, 0], [5, 0, -7, 8, -9, -15, 10, 0, 0, 0, 0, 0], [6, -9, 0, 15, 0, 0, 0, 3, -6, -5, -7, -15], [0, 3, 3, 0, 1, 7, -8, 7, 14, -1, -8, 6], [-1, 0, -14, -1, -11, 1, -4, 0, 0, 0, 0, 0], [4, 6, 0, -5, 0, 0, 0, -11, -4, 1, -12, 5], [19, 5, 0, -4, 0, 0, 0, 12, -19, -7, -11, 4], [15, 0, -15, 2, -6, -4, -4, 0, 0, 0, 0, 0], [0, 8, 8, 0, -1, 2, -16, 2, -1, 1, -16, 0], [8, 4, 0, -1, 0, 0, 0, 14, -8, -11, 0, 1], [11, 0, 12, -4, 10, 2, -8, 0, 0, 0, 0, 0], [-18, 0, -10, 8, -1, 4, -2, 0, 0, 0, 0, 0], [-10, 9, 0, -7, 0, 0, 0, -4, 10, 3, -13, 7], [0, -9, -9, 0, -7, -7, -7, -7, 12, 7, -7, 4], [0, 6, 6, 0, 15, 13, -1, 13, -8, -15, -1, -17], [0, -5, -5, 0, 0, -2, -4, -2, 11, 0, -4, -15], [0, 5, 5, 0, 7, -3, 3, -3, 6, -7, 3, 3]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_637_e_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_637_2_e_n();".
// 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_637_2_e_n(:prec:=12)
    chi := MakeCharacter_637_e();
    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_637_2_e_n();".
// 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_637_2_e_n( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_637_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![4, 8, 44, -56, 180, -64, 108, -4, 50, 0, 8, 0, 1]>,<3,R![4, -16, 88, 48, 200, 192, 452, 424, 316, 136, 44, 8, 1]>,<5,R![961, 806, 1637, 682, 1430, 670, 637, 278, 200, 78, 31, 6, 1]>],Snew);
    return Vf;
end function;