
// Make newform 630.2.bo.b in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_630_bo();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_630_bo_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_bo_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_bo_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 := [390625, -468750, 328125, -168750, 70625, -21000, 4650, -420, -6, -84, 186, -168, 113, -54, 21, -6, 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, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [608281250, -834078125, 555896875, -173418750, 73399500, -12526500, -726180, 1002264, -1213884, -397554, 232392, -241632, 175846, -67399, 11789, -384], [3815703125, -1186125000, 326318750, -87703125, 98359500, -97970400, 37261770, 11261652, 7153068, -99702, 1092156, -550416, 294133, -24132, 5902, -8097], [2128046875, -1142515625, 473087500, -179800625, 21773250, 14289450, -1325940, 6197028, -1809558, -1618968, 1464834, -519294, 210377, -42523, 20678, 6137], [91640625, 48828125, 347887500, -274375, -20291250, -2914800, -1181070, 3451632, 311568, 1248438, 51246, 60504, 93033, -14237, 7932, 6613], [1565390625, 257328125, 246750000, 193373750, -62958000, 42895650, 4944330, 3490098, 2521482, 1336152, 412344, 118266, -47583, 66757, -4902, 17122], [468750, -328125, 168750, -70625, 21000, -4650, 420, 6, 84, -186, 168, -113, 54, -21, 6, -1], [3913984375, -2635218750, 1557531250, -737662500, 287925000, -65558250, 10333650, -43854, -1282116, -369666, 1301988, -982878, 546479, -238236, 79646, -22146], [6441171875, -3831734375, 3073906250, -1441943750, 538473750, -111582750, 27087000, 1076184, 1105386, -2772714, 2447502, -1435662, 973141, -411169, 147184, -31084], [-2969453125, 3278171875, -2090471875, 984950000, -375636000, 133123800, -23957910, 3919626, 2003274, 1395684, -1131672, 1218072, -699881, 303959, -117899, 29284], [14823593750, -9888984375, 6006781250, -2661213750, 965847000, -179819100, 30074280, 12645312, 4871838, -2122542, 4898436, -3542736, 2102878, -764517, 298312, -52692], [-6022343750, 5965500000, -3731140625, 1569078750, -606171000, 130483800, -12184890, -12714198, 2591088, 3038328, -1794444, 1770924, -1115872, 537918, -173323, 53838], [-27148437500, 22579546875, -12666678125, 5893719375, -1937064000, 383712300, -62378340, -24049518, 2446458, 7522398, -9146904, 7691484, -4301152, 1698663, -499393, 115983], [16796953125, -13068156250, 8129756250, -3519160000, 1302099000, -220676100, 70889400, -4597266, 1398786, -4476264, 6163452, -4526112, 2840991, -1026644, 358884, -74084], [36593515625, -29359171875, 17728671875, -8041421250, 2960143500, -720224100, 136998150, 15497646, 6178134, -9796716, 13033788, -10614828, 6076429, -2551911, 901171, -182946]];
    Rf_basisdens := [1, 1, 568125000, 3408750000, 1704375000, 1704375000, 1704375000, 78125, 1704375000, 1704375000, 1136250000, 1704375000, 1704375000, 3408750000, 1704375000, 3408750000];
    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_630_bo();"
function MakeCharacter_630_bo()
    N := 630;
    order := 6;
    char_gens := [281, 127, 451];
    v := [3, 3, 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_bo_Hecke();"
function MakeCharacter_630_bo_Hecke(Kf)
    N := 630;
    order := 6;
    char_gens := [281, 127, 451];
    char_values := [[-1, 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], [1, 0, 0, 0, 0, 0, 0, 0, -1, 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 := [[1, 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, 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, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, 0, 0, 0, -1, 0, 0, -1, 0, -1], [0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, -1], [1, 1, -2, -1, 0, 2, -1, 1, -1, -1, 0, -2, 0, -1, 0, 1], [-2, 1, 0, 0, -1, -1, 1, 1, 2, 0, -1, 1, -1, 0, -1, -1], [0, -1, -2, -2, -1, 1, -1, -1, 0, 2, -1, -1, 1, -2, -1, 1], [2, 2, 0, 0, 1, -2, 2, -2, -2, -1, 0, 0, 0, 0, 0, 0], [-1, -1, 4, 2, 0, -4, 2, -1, 1, 1, 0, 4, 0, 2, 0, -2], [0, 0, 0, 1, 0, 3, 0, 0, 0, 0, 1, 0, 0, 1, 0, -1], [0, 0, 0, -1, -1, -2, -2, 0, 0, -1, -1, 0, 0, -2, 0, -1], [-2, -2, 0, -1, -1, 3, -3, 2, 2, 1, 1, 0, 0, 1, 0, 0], [-4, 1, 0, 0, -1, 0, 0, 1, -2, 0, 0, 0, 0, 0, 0, 0], [1, -1, 0, -1, 0, 0, -1, -1, -2, 1, 0, 0, 2, -1, 0, 1], [-1, -1, 0, 0, 0, 2, -1, 1, 1, 1, 0, 0, 0, 0, 0, 0], [5, -1, 0, 0, 1, 1, -2, -1, -1, 0, 1, 0, 2, 0, 2, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 2, 0, 2], [2, 2, 0, -1, 1, 1, -1, -2, -2, -1, 1, 0, 0, -3, 0, 4], [2, 2, 0, -2, 0, -4, 2, -2, -2, -2, 0, 0, 0, -2, 0, 2], [1, -1, -4, 0, -1, 4, -2, -1, -1, 2, 0, -2, -2, 0, 2, 0], [-2, 0, 2, 1, 2, -1, 1, 0, 8, -2, 1, 0, 0, 1, 0, 0], [-1, -1, 0, -2, -1, 0, 0, 1, 1, 0, 1, 0, 0, 2, 0, 1], [0, 0, 0, -1, 2, -1, -1, 0, 0, 2, -1, 0, 0, -1, 0, 0], [-2, -2, 0, 0, 0, 0, 0, 2, 2, 2, 0, 0, 0, 0, 0, 0], [-1, -1, 0, 0, -1, -2, 4, 1, 1, 0, 1, 0, 0, 0, 0, 1], [-6, -2, -4, -2, -2, 3, -2, -2, 6, 4, -1, -2, 0, -2, 0, 1], [0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0], [-6, 0, 4, 2, 0, -6, 2, 0, 0, 0, -2, 8, 0, 2, 0, -4], [-4, -4, 0, -1, -2, 0, 0, 4, 4, 2, 1, 0, 0, 0, 0, 1], [-1, -1, 0, 1, -1, -1, 2, 1, 1, 0, 1, 0, 0, -1, 0, 1], [-2, 2, -4, 0, 0, 0, 0, 2, -2, -2, -4, 4, -4, 0, 0, -4], [2, 0, 4, 2, 2, -2, 2, 0, 0, -2, 2, 0, 0, 2, 0, 0], [0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0], [-1, -1, 0, -2, -2, 0, -2, -1, 9, 3, 0, 0, 4, -2, 0, 2], [1, 1, 0, 2, 0, -8, 4, -1, -1, -1, 1, 0, 0, 3, 0, -3], [-1, -1, 0, 2, 1, 3, 0, 1, 1, 2, 1, 0, 0, 2, 0, -1], [7, -1, 3, 0, 5, -2, 1, -1, -3, -4, 1, 0, 2, 0, -1, 1], [1, 2, 0, 0, -2, -1, 0, 2, 5, 0, -1, 2, 0, 0, 0, -1], [1, 1, 0, 0, 2, 0, -2, -1, -1, 1, 3, 0, 0, 3, 0, 3], [6, -2, 2, -2, 0, -2, 0, -2, -10, 2, 0, 0, 4, -2, -2, 2], [0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 3, 0, 0, -3, 0, 0, 0, 0, 0, 0, -3, 0, -3], [-15, 3, 3, 0, -3, -6, 3, 3, 3, 0, -3, 6, 0, 0, -3, -3], [-4, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0], [-3, 0, 2, 2, 1, -2, 0, 0, -2, -1, 0, 4, 0, 2, 2, -2], [0, 0, 0, -3, 0, 0, 0, 0, 0, 0, -3, 0, 0, 1, 0, 4], [4, -2, 0, -3, -2, -2, 1, -2, 0, 4, -2, 0, 2, -3, -4, 1], [-6, 2, 0, 0, -2, -2, 0, 2, -2, 0, -2, 4, 0, 0, 0, -2], [3, -1, 0, 2, -1, 1, 0, -1, -3, 2, 1, 0, -2, 2, 2, -1], [2, 2, 0, 1, 1, -11, 11, -2, -2, -1, -1, 0, 0, -3, 0, 2], [2, 1, -5, -1, 1, 6, -3, 1, -2, -2, 1, -5, -1, -1, 2, 2], [0, 0, 0, -1, 0, -9, -9, 0, 0, 0, -1, 0, 0, -2, 0, -1], [-10, 0, 0, 0, 0, -2, 2, 0, -6, 0, -2, 2, -2, 0, -2, -2], [-1, 1, 0, 1, 0, 0, 1, 1, -13, -1, 0, 0, -2, 1, 0, -1], [1, 1, 0, 2, 0, -4, 2, -1, -1, -1, 2, 0, 0, 4, 0, -4], [-3, 1, 0, 0, -1, -4, 0, 1, -1, 0, -4, 8, 0, 0, 0, -4], [-3, -3, 0, -1, -6, 0, -6, 3, 3, -3, -2, 0, 0, -1, 0, -1], [6, 6, 0, 1, 3, 4, -4, -6, -6, -3, -1, 0, 0, 0, 0, -1], [-1, -1, 0, 5, 0, 2, -1, 1, 1, 1, -2, 0, 0, 3, 0, -3], [-3, -1, 11, 4, 0, -6, 5, -1, 7, 1, 5, 0, 2, 4, -1, 1], [0, 0, 0, 3, -1, -7, -7, 0, 0, -1, 3, 0, 0, 1, 0, -2], [4, 4, 0, -2, 0, -8, 4, -4, -4, -4, -2, 0, 0, -4, 0, 4], [4, 4, 0, 2, 4, -7, 14, -4, -4, 0, -3, 0, 0, -2, 0, -3], [-9, -1, -4, -8, -1, 0, -2, -1, 9, 2, -4, -2, 6, -8, -6, 4], [7, 0, -6, 0, 0, 6, -3, 0, -7, 0, 0, -3, -3, 0, 3, 0], [0, 0, 0, -1, 0, -3, 3, 0, 0, 0, 1, 0, 0, 5, 0, -4], [6, -4, 2, -2, 2, 0, -2, -4, -12, 2, 2, -2, 6, -2, 0, 4], [0, 0, -8, -4, 2, 4, -4, 0, 4, -2, -4, 0, 0, -4, 0, 0], [8, 2, -2, 2, 2, 4, -2, 2, -6, -4, 2, -2, -2, 2, 4, 0], [0, 0, 0, 4, 0, 8, 0, 0, 0, 0, 6, 0, 0, 4, 0, -6], [-1, -1, 0, -5, 0, -4, 2, 1, 1, 1, 3, 0, 0, -2, 0, 2], [3, 3, 0, -2, -3, -6, 0, -3, -3, -6, -1, 0, 0, -2, 0, 1], [-5, 6, 4, 1, -4, -7, 3, 6, 2, -2, -3, 8, 0, 1, -2, -4], [-1, 4, 0, 0, -4, 3, 0, 4, 7, 0, 3, -6, 0, 0, 0, 3], [2, 2, 0, 0, 4, 0, -8, -2, -2, 2, -2, 0, 0, -2, 0, -2], [3, 3, 0, 2, 3, 6, -12, -3, -3, 0, 3, 0, 0, -2, 0, 3], [0, 0, 0, 3, 0, -4, 0, 0, 0, 0, 3, 0, 0, 3, 0, -3], [14, -4, 0, -3, -2, -2, 1, -4, -4, 6, -2, 0, 2, -3, -4, 1], [0, 0, 0, -3, 3, 2, 2, 0, 0, 3, -3, 0, 0, 0, 0, 3], [-10, -2, -2, -1, 1, 3, -1, -2, 0, 1, 1, -4, 0, -1, 0, 2], [0, 0, 0, -6, 0, 0, 4, 0, 0, 0, -6, 0, 0, 0, 0, 0], [0, 0, 0, -2, 2, 8, 8, 0, 0, 2, -2, 0, 0, -6, 0, -4], [1, -3, 0, 0, 3, 0, -2, -3, -9, 0, 0, 2, 2, 0, 2, 0], [12, 2, -4, 2, 2, 5, -2, 2, -12, -4, 1, -2, -4, 2, 4, -1], [-2, -2, 0, -7, -1, 1, -1, 2, 2, 1, 7, 0, 0, 6, 0, 1], [-2, -2, 0, 0, 2, 21, 0, 2, 2, 4, 3, 0, 0, 0, 0, -3], [0, 0, 0, -2, -4, -6, -6, 0, 0, -4, -2, 0, 0, 2, 0, 4], [4, 4, 0, -4, 2, 0, 0, -4, -4, -2, 4, 0, 0, -1, 0, 5], [-14, 4, 0, 0, -4, -6, 2, 4, -2, 0, -6, 10, -2, 0, -2, -6], [1, 1, 0, -1, 0, -6, 3, -1, -1, -1, -4, 0, 0, -5, 0, 5], [0, 0, 0, 1, 0, 0, 0, 0, 0, 0, -4, 0, 0, -5, 0, -5], [-10, -10, 0, -4, -5, 19, -19, 10, 10, 5, 4, 0, 0, 6, 0, -2], [-14, -4, -4, -4, -4, 2, -2, -4, 14, 8, -2, -2, 2, -4, -2, 2], [6, 4, -6, 3, 3, 5, -1, 4, -10, -7, -1, 0, -8, 3, 4, -4], [-3, -3, 0, 2, -3, 4, -8, 3, 3, 0, 2, 0, 0, -2, 0, 2], [0, 0, 0, 3, 0, 10, -5, 0, 0, 0, 3, 0, 0, 6, 0, -6], [5, 5, 0, -4, 5, -5, 10, -5, -5, 0, 9, 0, 0, 4, 0, 9], [19, 7, -8, 2, 7, 9, -4, 7, -19, -14, 1, -4, -6, 2, 6, -1], [6, 6, 0, 1, 3, -15, 15, -6, -6, -3, -1, 0, 0, -5, 0, 4], [11, -3, 0, 1, 8, 0, 1, -3, -10, -5, 0, 0, -2, 1, 0, -1], [-20, -2, 0, -3, -2, -2, 1, -2, 12, 4, -2, 0, 2, -3, -4, 1], [-1, -1, 0, -5, 1, 5, 0, 1, 1, 2, -6, 0, 0, -5, 0, 6], [11, -3, 4, 3, 8, 0, 3, -3, -9, -5, 4, -4, -2, 3, 0, 1], [-2, -2, 0, 1, 0, -14, 7, 2, 2, 2, 0, 0, 0, 1, 0, -1], [6, 0, -2, -1, 0, 1, -1, 0, -12, 0, -1, 0, 0, -1, 0, 0], [9, -1, 0, 0, 1, -1, -4, -1, -1, 0, -1, 6, 4, 0, 4, -1], [-2, -2, 0, 0, -4, 0, -14, 2, 2, -2, -4, 0, 0, -4, 0, -4], [6, 0, -2, -1, -4, 1, -1, 0, -20, 4, -1, 0, 0, -1, 0, 0], [-3, -3, 0, -5, -3, 12, -24, 3, 3, 0, 5, 0, 0, 5, 0, 5], [-2, -2, 0, 1, -4, 0, -9, 2, 2, -2, 3, 0, 0, 2, 0, 2], [20, 0, -4, -3, -1, 5, -1, 0, 2, 1, 1, -8, 0, -3, -2, 4], [29, 1, -6, 0, 2, 8, -4, 1, -16, -3, 2, -6, -2, 0, 4, 2], [-24, 6, -4, -4, -5, 4, 0, 6, 4, -1, 0, -8, 0, -4, -4, 4], [0, 0, 0, -9, 0, 0, -5, 0, 0, 0, -1, 0, 0, 8, 0, 8], [0, 0, 0, 0, -2, -2, -2, 0, 0, -2, 0, 0, 0, -2, 0, -2], [27, -3, 4, 8, 4, 0, 0, -3, -14, -1, 4, 4, -4, 8, 8, -4], [-6, -5, 18, 4, -5, -16, 9, -5, 6, 10, 2, 9, 5, 4, -5, -2], [-2, -2, 0, 7, -1, -2, 2, 2, 2, 1, -7, 0, 0, -3, 0, -4], [-1, 5, -4, 1, 2, 6, -3, 5, -3, -7, 2, -4, -2, 1, 4, 1], [0, 0, 0, -3, 0, 15, -15, 0, 0, 0, 3, 0, 0, 1, 0, 2], [-4, -5, 0, 0, 5, 6, -3, -5, -20, 0, 6, -9, 3, 0, 3, 6], [-4, 4, -2, 3, 0, 0, 3, 4, 8, -4, -2, 2, -8, 3, 0, -5], [-10, 2, 0, 0, -2, -2, 4, 2, 2, 0, -2, 0, -4, 0, -4, -2], [-2, -2, 0, 2, -1, 16, -16, 2, 2, 1, -2, 0, 0, 0, 0, -2], [-20, 0, 12, 8, 0, -8, 6, 0, 20, 0, 4, 6, -2, 8, 2, -4], [-4, -4, 0, -6, -4, 0, 0, 4, 4, 0, 0, 0, 0, 6, 0, 0], [0, 0, 0, -4, -7, -4, -4, 0, 0, -7, -4, 0, 0, -7, 0, -3], [-8, -8, 0, 3, -8, -1, 2, 8, 8, 0, 1, 0, 0, -3, 0, 1], [8, -5, -1, -4, 3, 0, -4, -5, -6, 2, -1, 1, 7, -4, 0, 3], [3, -4, -6, -3, 2, 9, -3, -4, 0, 2, 3, -12, 0, -3, 0, 6], [1, -3, -12, -8, -3, 8, -6, -3, -1, 6, -4, -6, 2, -8, -2, 4], [-8, -8, 0, -2, -4, 9, -9, 8, 8, 4, 2, 0, 0, 6, 0, -4], [-1, -1, 0, -3, -1, -7, 14, 1, 1, 0, 0, 0, 0, 3, 0, 0], [-13, 3, 1, 5, -7, 1, 2, 3, 9, 4, 2, 0, -6, 5, 3, -3], [-30, -3, 5, -2, -3, -8, 4, -3, 18, 6, -3, 5, 3, -2, -6, -1], [-1, -1, 0, 5, 0, -22, 11, 1, 1, 1, -2, 0, 0, 3, 0, -3], [-8, -2, -2, -4, -7, 0, -2, -2, 4, 9, -2, 0, 4, -4, -2, 2], [2, 2, 0, 3, 4, 0, -14, -2, -2, 2, 5, 0, 0, 2, 0, 2], [-1, 6, -4, 7, -2, 5, 1, 6, -8, -4, 1, 0, -12, 7, 6, -6], [-1, -1, 0, -6, 1, -10, 0, 1, 1, 2, 0, 0, 0, -6, 0, 0], [5, 5, 0, 3, 10, 0, 18, -5, -5, 5, 4, 0, 0, 1, 0, 1], [30, 2, -4, 1, 2, 9, -5, 2, -6, -4, 5, -8, 0, 1, 6, 4], [-1, 3, 2, 7, 4, 2, -1, 3, -3, -7, 4, 2, -4, 7, 8, -3], [0, 0, 0, 1, -1, 9, 9, 0, 0, -1, 1, 0, 0, -1, 0, -2], [0, 0, 0, 5, -5, -2, -2, 0, 0, -5, 5, 0, 0, 8, 0, 3], [35, 11, -10, 1, 4, 14, -7, 11, -25, -15, 4, -10, -4, 1, 8, 3], [-17, 13, 0, 0, -13, -1, 0, 13, 9, 0, -1, 2, 0, 0, 0, -1], [3, 4, -10, -8, 4, 6, -5, 4, -3, -8, -4, -5, 3, -8, -3, 4], [2, 2, 0, 1, -2, 24, 0, -2, -2, -4, -1, 0, 0, 1, 0, 1], [0, 0, 0, -2, 2, 2, 2, 0, 0, 2, -2, 0, 0, -1, 0, 1], [8, 8, 0, 0, 4, -12, 12, -8, -8, -4, 0, 0, 0, 0, 0, 0], [-25, 7, 0, 0, -7, 3, 0, 7, -11, 0, 3, -6, 0, 0, 0, 3], [3, 3, 0, 11, 6, 0, 3, -3, -3, 3, 10, 0, 0, -1, 0, -1], [6, 6, 0, 5, 3, 1, -1, -6, -6, -3, -5, 0, 0, 3, 0, -8], [46, 6, -12, 0, 6, 12, -6, 6, -46, -12, 0, -6, -6, 0, 6, 0], [2, 2, 0, 3, 2, 1, -2, -2, -2, 0, -15, 0, 0, -3, 0, -15], [0, 0, 0, 3, 2, 5, 5, 0, 0, 2, 3, 0, 0, 1, 0, -2], [3, 3, 0, 4, 0, -12, 6, -3, -3, -3, -8, 0, 0, -4, 0, 4], [-24, 0, 24, 12, 0, -18, 12, 0, 24, 0, 6, 12, 0, 12, 0, -6], [18, -10, 2, 1, 5, -3, 1, -10, 0, 5, -1, 4, 0, 1, 0, -2], [12, 12, 0, 5, 6, -12, 12, -12, -12, -6, -5, 0, 0, -1, 0, -4], [-1, -1, 0, 1, -1, 9, -18, 1, 1, 0, -5, 0, 0, -1, 0, -5], [-15, 7, 4, 5, -8, 0, 5, 7, -3, 1, 4, -4, -6, 5, 0, -1], [-6, 12, 6, -3, -4, -10, 5, 12, -1, -8, -4, 6, 4, -3, -8, -1], [-25, 9, -2, 0, -16, 0, 0, 9, -3, 7, -2, 2, -2, 0, 0, -2], [-1, -1, 0, -1, 0, -28, 14, 1, 1, 1, -2, 0, 0, -3, 0, 3]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_630_bo_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_630_2_bo_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_bo_b(:prec:=16)
    chi := MakeCharacter_630_bo();
    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_bo_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_bo_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_630_bo();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<17,R![61504, -17856, -8688, 3024, 1516, 0, -42, 0, 1]>],Snew);
    return Vf;
end function;