
// Make newform 920.2.e.b in Magma, downloaded from the LMFDB on 17 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_920_e();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_920_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_920_2_e_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_920_2_e_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 := [8, -32, 64, -20, 24, -116, 297, -26, 2, -10, 39, -2, 0, 0, 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, 0], [1689305356572, 32069661208, 22675714854, -402160406420, 11737106209660, -814144658387, 28178713856, 134953581114, 1559311271490, -68154663521, 2084956408, 13894530424, 40698207136, 224809861], [-2008645915756, -142401498584, -143702096254, 1957301556084, -12743422120058, 3717676805391, -253853562284, 36371917978, -1634888076832, 450158077433, -19980270148, -9824626316, -39917569898, 9599816455], [2294924313380, -1305274242300, 4594222559616, 2505098929650, 11022845347922, -4962097019689, 1031632771786, 503279347384, 1401214875276, -612714646635, 49350875674, 22691432254, 33377065138, -14185734445], [-327723885618, 1216849046036, 378394875852, -24147116423, -863452980582, 5814921705639, -433265659128, 2129167227, -8586142534, 769444956141, -37882617844, 390318619, 4912313158, 19958784949], [-1596402425868, -1092034671500, 4857036973804, -615006277590, -9375688488996, -10634500106239, 1553791371930, -274689612780, -1462473090366, -1462932942489, 91091509614, -16903078854, -48607321956, -40039329007], [982823726512, -472073315000, -557261024458, 6809297955588, 20232610940731, 12498302486205, -1082405618534, 1361220760372, 3072231013943, 1799215942067, -86278427068, 57732941236, 95455044155, 51158604007], [-1737381486820, -65148969984, 3333018633676, 4086990094642, -10243935308455, -16317016456265, 1922577441112, 385087447514, -1671410135475, -2271165378531, 134629395734, -866985826, -58787127771, -61621117035], [1230815216570, -4712322126264, 2137159584915, -1242235247009, 3382407351091, -19510883885937, 1819608044077, -162746449503, 89277697645, -2575176731663, 135188822801, -4521994539, -15046471791, -66772094683], [-2143125728988, 8601416001444, -3062961226104, -2574515483268, -14884213113334, 39984967995781, -3645608595402, -460738908234, -1301055484020, 5252053841299, -277803595150, -18595581648, 166717918, 134719081377], [2488082785940, -8398081464788, 3312052402844, -394657434132, -2875620487424, -45390689597815, 4139003457850, -257699180354, -1259279988702, -6056933221517, 317230204266, -17491051676, -74011296796, -159011401463], [-2076944162144, 7131200437916, -9490182700870, -5587980665236, -4899429442643, 46296345417282, -5300077616756, -731148740058, 182163846443, 6182243364690, -359941008954, -17266584944, 45302004725, 161362783524], [-88794449364, 8104580101084, -3654530344182, 4612113196920, 11212902114921, 53149974193276, -4799923558176, 1049309031078, 2529556762223, 7164617072456, -369867084278, 47907054048, 115971282613, 189780871574], [1060356158372, -10569330546208, 6729521670402, 3205954983134, -2758530555507, -64065095492413, 6202649999094, 226661713146, -1551259260459, -8578405340227, 462876963284, -6621708034, -97924060431, -225258249983]];
    Rf_basisdens := [1, 732246302876, 732246302876, 732246302876, 366123151438, 732246302876, 732246302876, 732246302876, 366123151438, 732246302876, 732246302876, 732246302876, 732246302876, 732246302876];
    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_920_e();"
function MakeCharacter_920_e()
    N := 920;
    order := 2;
    char_gens := [231, 461, 737, 281];
    v := [2, 2, 1, 2];
    // 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_920_e_Hecke();"
function MakeCharacter_920_e_Hecke(Kf)
    N := 920;
    order := 2;
    char_gens := [231, 461, 737, 281];
    char_values := [[1, 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], [-1, 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]];
    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, 0, 0, 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, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, -1], [0, 0, -1, -1, 0, 1, -1, 0, 0, 0, -1, 0, -1, 0], [0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, -1], [0, 0, 0, 0, 1, 1, 0, -2, -1, 0, 0, 0, 0, 1], [2, 2, 1, -1, 0, 1, -1, 0, 0, 1, -1, 0, -2, 0], [0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, -2, -2, 0, 0, 0, 0, 0, 0, -2, 1, 0, 3, 0], [1, 1, -2, -1, 0, 1, 3, 0, 0, 1, 0, 0, -1, 0], [0, 0, 0, 0, -1, 0, 0, 0, -2, -2, 2, 1, 0, 0], [-1, 1, 0, -2, 0, 2, 1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 1, -1, -1, 0, 1, 1, -1, 1, 0, 0, 1], [0, 0, 0, 2, 1, 1, 0, 1, -1, 0, 0, 3, 0, 0], [0, 0, 0, 0, 2, 2, 0, 1, -1, -1, 1, 3, 0, -3], [5, 1, -5, -2, 0, 2, 3, 0, 0, -1, 1, 0, 2, 0], [-2, -2, 1, 2, 0, -2, -1, 0, 0, -4, 1, 0, 5, 0], [0, 0, 0, -2, 3, 0, 0, -4, -2, 2, -2, -3, 0, 2], [-5, -1, 2, 1, 0, -1, -3, 0, 0, 1, 1, 0, 0, 0], [0, 0, 0, -1, 3, 0, 0, -1, -1, -1, 1, 1, 0, 0], [6, 1, 1, -1, 0, 1, 2, 0, 0, 3, -1, 0, -4, 0], [0, 0, 0, 1, -4, 1, 0, 1, 5, 1, -1, -1, 0, -1], [2, -1, 3, 1, 0, -1, 0, 0, 0, 1, -1, 0, -2, 0], [0, 0, 0, -1, 0, 0, 0, 4, -1, 0, 0, 3, 0, -5], [-1, 5, 3, -1, 0, 1, -1, 0, 0, 0, -4, 0, -4, 0], [0, 0, 0, -2, -1, 1, 0, -5, 0, 3, -3, -1, 0, 2], [0, 0, 0, -1, -3, 3, 0, 0, 2, 0, 0, 1, 0, -4], [4, 0, 1, 1, 0, -1, -3, 0, 0, -3, -3, 0, 0, 0], [0, 0, 0, 2, 1, -4, 0, 4, 0, -2, 2, -1, 0, 2], [0, 0, 0, -1, 1, -4, 0, 1, 3, 1, -1, -5, 0, 2], [0, 1, -5, -1, 0, 1, 3, 0, 0, -2, 3, 0, 5, 0], [0, 0, 0, -2, 5, -2, 0, 3, -2, 0, 0, -3, 0, -3], [5, 0, -4, 0, 0, 0, -2, 0, 0, -2, 1, 0, 3, 0], [10, 1, -2, -2, 0, 2, 3, 0, 0, 0, 0, 0, 0, 0], [2, 6, 3, 1, 0, -1, -2, 0, 0, 2, -2, 0, -4, 0], [0, 0, 0, -1, -7, -1, 0, 2, 4, -2, 2, 1, 0, -2], [0, 0, 0, -3, -4, 4, 0, -4, -1, 5, -5, 2, 0, -3], [0, 0, 0, -1, 4, 3, 0, -2, 0, 1, -1, 0, 0, -2], [0, 0, 0, 0, 7, -1, 0, -3, 0, 2, -2, -1, 0, 4], [6, -1, -5, 2, 0, -2, -1, 0, 0, 1, -2, 0, -3, 0], [0, 0, -1, 2, 0, -2, 3, 0, 0, -1, -3, 0, -2, 0], [-4, 2, -2, 1, 0, -1, 0, 0, 0, -2, -2, 0, 0, 0], [0, 0, 0, -7, 7, 0, 0, -5, -7, -1, 1, -1, 0, -2], [0, 0, 0, 0, -4, 0, 0, -2, 5, 2, -2, 0, 0, 2], [2, -5, -3, 4, 0, -4, -2, 0, 0, -3, 1, 0, 4, 0], [-7, -11, -3, 3, 0, -3, -1, 0, 0, -2, 2, 0, 4, 0], [0, 0, 0, 0, -8, -4, 0, -2, -2, -2, 2, 0, 0, 6], [0, 0, 0, 2, 7, -2, 0, 3, -1, -5, 5, 4, 0, 1], [4, 1, -1, 1, 0, -1, 0, 0, 0, 3, -5, 0, -8, 0], [0, 0, 0, 1, 5, -2, 0, -1, 1, 7, -7, -3, 0, 4], [-3, -2, 8, 1, 0, -1, 0, 0, 0, -3, -2, 0, 1, 0], [6, 6, -2, -1, 0, 1, 6, 0, 0, 0, 4, 0, 4, 0], [-4, -1, 0, -1, 0, 1, 1, 0, 0, 0, -6, 0, -6, 0], [0, 0, 0, 0, -3, 3, 0, -7, 5, 2, -2, -1, 0, 4], [0, 0, 0, 0, -4, -1, 0, 5, -4, 1, -1, -4, 0, -4], [9, 0, 2, -1, 0, 1, -2, 0, 0, 1, 0, 0, -1, 0], [-3, -2, 3, 3, 0, -3, 0, 0, 0, 2, 0, 0, -2, 0], [0, 0, 0, -4, -5, 3, 0, -5, 3, 4, -4, 1, 0, -2], [-8, -1, -1, 0, 0, 0, 0, 0, 0, 5, 1, 0, -4, 0], [0, 0, 0, -3, -4, -1, 0, -1, 1, 1, -1, 1, 0, -1], [0, 0, 0, 2, -2, 6, 0, 3, 3, 1, -1, 1, 0, -7], [0, 0, 0, 0, 13, -1, 0, -5, 2, 0, 0, 1, 0, 6], [-6, -1, 3, -2, 0, 2, 1, 0, 0, 5, 2, 0, -3, 0], [0, 0, 0, 2, 3, 0, 0, 4, -3, -1, 1, 6, 0, -2], [0, 0, 0, 3, -5, 5, 0, -3, 0, 3, -3, 3, 0, 1], [-3, -2, 2, 4, 0, -4, -2, 0, 0, -6, -1, 0, 5, 0], [0, 0, 0, 1, -10, 1, 0, 2, 11, 1, -1, 3, 0, -2], [0, 0, 0, 3, -5, 0, 0, -3, 6, 3, -3, -7, 0, 6], [7, 4, 4, 3, 0, -3, 4, 0, 0, -5, 4, 0, 9, 0], [0, 0, 0, -1, 3, 6, 0, -5, -5, 3, -3, 1, 0, -2], [2, -1, 1, -3, 0, 3, -8, 0, 0, -3, -9, 0, -6, 0], [0, 0, 0, 7, 12, -5, 0, 1, 3, -3, 3, 1, 0, 11], [0, 0, 0, 3, 16, -5, 0, 0, -8, -4, 4, 0, 0, 8], [-6, -9, 4, 5, 0, -5, -3, 0, 0, -1, 2, 0, 3, 0], [0, 0, 0, 1, -5, -1, 0, -2, -2, -6, 6, -3, 0, 4], [-2, -5, 0, 3, 0, -3, 5, 0, 0, 1, 6, 0, 5, 0], [0, 0, 0, -1, -4, -3, 0, -4, -1, 1, -1, -6, 0, 6], [-2, 4, 0, -3, 0, 3, 2, 0, 0, 6, 4, 0, -2, 0], [-5, 1, -6, -3, 0, 3, 3, 0, 0, 3, 0, 0, -3, 0], [6, 3, 9, -1, 0, 1, -2, 0, 0, 5, -5, 0, -10, 0], [12, 8, -1, -4, 0, 4, 1, 0, 0, 1, -3, 0, -4, 0], [-4, -5, -1, 3, 0, -3, -2, 0, 0, -7, -3, 0, 4, 0], [0, 0, 0, 3, -5, 1, 0, 4, 7, 3, -3, -4, 0, -2], [-6, -2, -5, 1, 0, -1, 4, 0, 0, 2, 9, 0, 7, 0], [0, 0, 0, 4, -2, 3, 0, 0, -3, -6, 6, 0, 0, 1], [17, 8, -1, 0, 0, 0, 6, 0, 0, 7, 3, 0, -4, 0], [0, 0, 0, -1, 4, -7, 0, 5, -1, -7, 7, -3, 0, 1], [-2, -3, -11, -5, 0, 5, -3, 0, 0, 2, -3, 0, -5, 0], [0, 0, 0, -2, -12, 4, 0, 4, 6, 6, -6, 0, 0, -10], [0, 0, 0, 0, -6, -4, 0, -1, -1, -1, 1, -5, 0, 5], [4, -2, -2, -1, 0, 1, -2, 0, 0, -6, 0, 0, 6, 0], [0, 0, 0, -2, -7, 3, 0, 7, -3, -2, 2, 5, 0, -12], [-17, -8, 6, 6, 0, -6, -4, 0, 0, -8, 1, 0, 9, 0], [-1, 6, -4, -3, 0, 3, 4, 0, 0, 1, 4, 0, 3, 0], [0, 0, 0, 4, -1, 0, 0, -4, -5, -3, 3, -2, 0, 8], [2, 5, 7, -7, 0, 7, -3, 0, 0, 6, 1, 0, -5, 0], [0, 2, -4, 2, 0, -2, 10, 0, 0, -2, 4, 0, 6, 0], [0, 0, 0, -2, -7, 0, 0, 1, -1, -3, 3, 2, 0, -3], [-4, -9, -9, -2, 0, 2, 1, 0, 0, 1, 0, 0, -1, 0], [0, 0, 0, -4, -2, -3, 0, 4, 7, 4, -4, -4, 0, -5], [0, 0, 0, 3, -13, -5, 0, 10, 2, 0, 0, -3, 0, -2], [0, 0, 0, 2, -10, 4, 0, -1, 7, 9, -9, 1, 0, -1], [-4, -4, 8, -3, 0, 3, 0, 0, 0, -2, 0, 0, 2, 0], [10, -3, 2, 2, 0, -2, -5, 0, 0, 1, -4, 0, -5, 0], [0, 0, 0, -6, 3, -1, 0, -5, -3, 2, -2, -5, 0, 0], [0, 0, 0, 6, -6, 4, 0, 8, 9, 2, -2, 8, 0, -6], [0, 0, 0, 8, -2, 0, 0, 4, 8, -2, 2, 4, 0, 4], [6, 3, -4, -6, 0, 6, 1, 0, 0, 4, -1, 0, -5, 0], [-7, -5, -6, 4, 0, -4, -1, 0, 0, 2, 9, 0, 7, 0], [0, 0, 0, 5, 4, 3, 0, -4, 4, -3, 3, 6, 0, 6], [0, 0, 0, -5, 16, -1, 0, -8, -2, -6, 6, -4, 0, 4], [0, 0, 0, -2, 18, -1, 0, -7, -8, 3, -3, 2, 0, 6], [-8, 1, -4, 4, 0, -4, -3, 0, 0, -4, 2, 0, 6, 0], [8, 3, 5, 1, 0, -1, 0, 0, 0, 7, 5, 0, -2, 0], [-16, -9, 3, 5, 0, -5, -10, 0, 0, -5, -3, 0, 2, 0], [0, 0, 0, 5, -14, -5, 0, 3, 7, 5, -5, -7, 0, 7], [0, 0, 0, -3, -3, 0, 0, -9, 1, 11, -11, -5, 0, 6], [0, 0, 0, 8, 2, -5, 0, 10, -1, -10, 10, 6, 0, 3], [-14, -4, -8, 5, 0, -5, 0, 0, 0, -2, 2, 0, 4, 0], [8, -3, -6, -3, 0, 3, 5, 0, 0, -3, -8, 0, -5, 0], [0, 0, 0, -7, -7, -4, 0, -11, 7, 7, -7, -5, 0, 8], [0, 0, 0, 6, -28, -4, 0, 11, 5, 3, -3, 3, 0, -1], [0, 0, 0, -1, 4, -2, 0, -2, 5, -5, 5, 4, 0, 3], [-4, 4, 2, -8, 0, 8, 2, 0, 0, 8, 2, 0, -6, 0], [-14, 0, 1, 6, 0, -6, -11, 0, 0, 1, 5, 0, 4, 0], [-12, 2, 1, -5, 0, 5, 5, 0, 0, 2, 1, 0, -1, 0], [5, 2, 3, 1, 0, -1, 4, 0, 0, -4, 6, 0, 10, 0], [0, 0, 0, 0, -9, -7, 0, 6, 11, 0, 0, -12, 0, 1], [0, 0, 0, -5, 8, 1, 0, -3, 3, -7, 7, 5, 0, -3], [-9, 4, 4, 8, 0, -8, 0, 0, 0, 6, 9, 0, 3, 0], [0, 0, 0, -1, 5, -7, 0, -1, 8, -2, 2, 1, 0, 7], [-2, -7, 9, 1, 0, -1, -4, 0, 0, -5, -3, 0, 2, 0], [0, 0, 0, -4, -25, -2, 0, 2, 16, 2, -2, -1, 0, -4], [-3, -1, -6, -3, 0, 3, 3, 0, 0, 5, -2, 0, -7, 0], [-10, 0, -6, -1, 0, 1, 8, 0, 0, 0, 10, 0, 10, 0], [0, 0, 0, -5, 11, -3, 0, -5, 1, -1, 1, -4, 0, 3], [0, 0, 0, -10, 2, 0, 0, -13, 1, 9, -9, -1, 0, 3], [0, 0, 0, -3, -13, 7, 0, 0, -12, -2, 2, 9, 0, -10], [-15, -4, -1, 8, 0, -8, 0, 0, 0, -7, -4, 0, 3, 0], [15, 4, -8, -13, 0, 13, 2, 0, 0, 1, -6, 0, -7, 0], [2, 4, 0, 2, 0, -2, 14, 0, 0, 4, 10, 0, 6, 0], [0, 0, 0, -3, 7, -6, 0, -7, 1, 9, -9, -5, 0, 10], [0, 0, 0, -8, 16, -4, 0, -5, -11, -3, 3, 5, 0, 1], [-1, 9, 13, 5, 0, -5, -3, 0, 0, 6, -6, 0, -12, 0], [-6, 11, 15, 6, 0, -6, 6, 0, 0, 5, 5, 0, 0, 0], [0, 0, 0, 1, 7, -3, 0, 1, 8, 5, -5, -7, 0, 3], [0, 0, 0, -5, 19, 2, 0, -5, 11, 3, -3, 5, 0, -2], [-7, 8, -2, 1, 0, -1, -2, 0, 0, 3, -6, 0, -9, 0], [0, 0, 0, -1, -1, 2, 0, 1, 5, 5, -5, -1, 0, -4], [0, 0, 0, -4, -19, 3, 0, 9, 8, 2, -2, -3, 0, -16], [-6, -10, -4, -3, 0, 3, -2, 0, 0, 0, -4, 0, -4, 0], [0, 0, 0, -1, -15, -3, 0, 7, 16, 1, -1, -11, 0, -5], [0, 0, 0, 8, -5, -3, 0, -1, 13, 8, -8, -9, 0, 12], [0, 0, 0, -5, -3, 9, 0, -8, 2, 10, -10, 1, 0, -6], [-12, 3, 13, 5, 0, -5, -10, 0, 0, 3, -11, 0, -14, 0], [-4, -1, -7, -4, 0, 4, -2, 0, 0, 9, 1, 0, -8, 0], [19, 6, 0, -1, 0, 1, 0, 0, 0, -11, -10, 0, 1, 0], [0, 0, 0, 6, 8, 2, 0, -6, 3, -5, 5, -7, 0, 10], [-18, 1, 14, 4, 0, -4, -5, 0, 0, 5, -2, 0, -7, 0], [0, 0, 0, 5, -24, 0, 0, 6, 1, -7, 7, -4, 0, -1], [0, 0, 0, -11, -3, 0, 0, -11, 4, 7, -7, -7, 0, 0], [0, 0, 0, -3, 19, -6, 0, 1, -13, -3, 3, -1, 0, 2], [-2, 5, -8, 5, 0, -5, 9, 0, 0, -1, 2, 0, 3, 0], [0, 0, 0, -2, 10, 8, 0, -7, -14, 0, 0, 2, 0, -3], [0, 0, 0, -13, 7, 8, 0, -16, -9, 8, -8, -2, 0, -5], [9, -4, -15, -6, 0, 6, 6, 0, 0, 3, -2, 0, -5, 0], [0, 0, 0, 5, 4, 1, 0, -2, -2, -3, 3, 6, 0, 6]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_920_e_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_920_2_e_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_920_2_e_b(:prec:=14)
    chi := MakeCharacter_920_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_920_2_e_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_920_2_e_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_920_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![4, 0, 149, 0, 1095, 0, 1431, 0, 766, 0, 195, 0, 23, 0, 1]>],Snew);
    return Vf;
end function;