
// Make newform 630.2.i.i in Magma, downloaded from the LMFDB on 06 January 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_630_i();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_630_i_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_i_i();".
// 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_i_i();".
// 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 := [6561, -2187, 1458, -972, 405, 54, -315, 243, -66, 81, -35, 2, 5, -4, 2, -1, 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], [-188811, 96957, 7209, 38637, -36378, 3951, 741, -408, -961, -1109, 1486, -287, 13, -161, 47, 6], [-448335, 412614, -767880, 664524, -621135, 260487, -149940, 46911, 10029, -22498, 17755, -14819, 5890, -2882, 520, -193], [3881925, 2188458, -1598940, -164592, -514215, 522819, -121320, -5433, 40233, -1846, 10315, -22943, 1870, 586, 1900, 119], [-39366, -175689, 88209, 13041, 36207, -36702, 5841, -717, -12, -1447, -899, 1474, -317, 37, -173, 53], [428652, -226719, -50058, -67959, 32346, 23688, -19377, 16206, -8238, 4303, -2242, -736, 209, -208, 341, -92], [3833811, -1296162, 260496, -459432, 445203, -39411, -38736, 48093, 12471, 14734, -12511, 767, -718, 1106, -532, 349], [2187, -1458, 972, -405, -54, 315, -243, 66, -81, 35, -2, -5, 4, -2, 1, -1], [-2777490, 1508301, -633015, 827091, -558765, 160938, -27765, 2739, -24, -19357, 20005, -7466, 3145, -3143, 1105, -247], [-174960, 748683, 239355, 132273, -80325, -66816, 6705, -17913, 14658, -3071, 4205, 32, -1345, -349, -385, 229], [234009, -1892484, -92826, 580446, -23193, 257463, -249354, 121413, -57327, 3608, -899, -3971, 9448, -4148, 982, -727], [-2219805, -1853118, -680400, 1028052, 397035, -1899, -103500, -26547, 11127, -33094, -935, 4903, 6550, 514, -1520, -619], [3171150, -2186271, 786105, -1062801, 556335, -19458, -44865, 63771, -55416, 41647, -24115, 4346, -55, 1193, 905, -503], [1570266, -629127, 840051, -487377, 180333, -182646, 59139, 15513, 636, 15379, -7801, 5522, -3523, 1091, -367, 499], [-640791, 4547502, -1334556, 577692, -967923, 163791, 119916, -84333, 106629, -37874, 33671, -11947, -2902, -286, -88, 1291], [-968841, 271917, 17739, 190107, -13338, -140499, 53361, -32178, 22839, -11159, 3176, 3413, -1717, 269, -883, 376]];
    Rf_basisdens := [1, 43740, 787320, 787320, 131220, 196830, 787320, 2187, 393660, 393660, 787320, 787320, 393660, 393660, 787320, 196830];
    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_i();"
function MakeCharacter_630_i()
    N := 630;
    order := 3;
    char_gens := [281, 127, 451];
    v := [1, 3, 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_630_i_Hecke();"
function MakeCharacter_630_i_Hecke( : Kf := false)
    N := 630;
    order := 3;
    char_gens := [281, 127, 451];
    char_values := [[0, 0, 0, 0, -1, 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, 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 := 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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [1, 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, 1, 0, 0, 0], [0, 0, 0, -1, 0, 0, 0, 1, 0, 0, 0, 0, 0, -1, 0, 1], [-1, 1, -1, 0, 1, -2, 1, -2, 0, 0, 0, -1, 0, 0, -1, 0], [0, 1, -1, 1, 0, -2, 2, 0, 2, -1, 0, -1, 0, 0, -1, 0], [0, 2, 2, -2, 1, 3, 0, -1, -1, 1, 1, 1, 1, 0, 3, -1], [0, 0, 2, -1, 0, 1, 0, 1, 0, 1, 0, 2, 1, 0, 3, -2], [2, -1, -1, 0, -2, 0, 1, 1, -1, -1, -1, -1, -2, 1, -1, -1], [3, 0, 0, 0, -1, 1, -1, 0, 0, 1, 0, 1, 0, 1, 0, 0], [-2, 0, -1, 2, 2, -5, 2, -2, 1, -1, 0, -3, 0, 1, -3, 0], [0, -4, -1, 1, -1, -1, 0, 5, 1, -1, -2, -1, 0, -2, -3, 3], [-1, 0, 0, 1, 1, -2, -1, -1, 1, -1, 0, -1, 0, 0, -3, 1], [0, 1, -1, 1, 1, -2, 1, -2, 0, -2, 1, -1, 0, 0, 1, 1], [2, 0, -3, 1, -2, -3, 2, 1, 2, -2, -1, -3, -2, 1, -5, 1], [-2, 0, 0, 0, 3, -1, 3, 1, 2, -1, -2, -2, -1, 0, -1, 0], [2, 2, -1, -1, 1, 0, 1, 1, 1, 0, -2, -1, 0, 0, 0, -1], [0, -1, 3, 0, 1, 0, 1, 0, -1, -2, 1, 1, 1, -1, 0, 0], [-3, 2, 0, -2, 2, 2, 2, 2, 0, 1, -2, -2, -2, 0, 0, -2], [-1, 0, 1, -2, 1, 3, -1, 2, -1, 2, 0, 1, 2, 0, 3, 0], [3, -1, -1, 1, 0, -2, 0, -1, 0, 0, 1, -1, 0, -1, 0, 1], [4, 1, 0, -1, -4, 3, -3, 0, -1, 2, 1, 2, 1, -1, 1, 4], [2, 0, 3, -2, 0, 5, -2, 3, 0, 0, 1, 3, -1, 2, 5, -2], [4, 1, 3, 1, -4, 0, -3, -2, 0, 0, 1, 1, 3, -1, 1, 1], [1, 1, -1, 0, 2, -3, -1, 0, 1, -1, 2, 1, -2, 1, 0, 0], [1, -2, -1, -3, -1, 5, -2, 1, -4, 3, 0, 1, -1, 0, 2, 1], [0, 1, -2, 2, 1, -4, 0, -4, 1, -1, 0, -1, -3, 4, -1, -3], [5, 0, -3, 1, -5, 0, -2, -3, -1, 0, 1, -2, -1, -1, -4, 5], [-1, 2, 0, -2, 4, 3, 1, -5, -2, 2, -3, -2, 0, 0, 1, -2], [-3, 2, 1, 0, 4, -2, 4, -1, 0, 2, 0, -2, 0, -1, 1, 0], [2, 2, 2, -2, -2, 7, -4, 2, -3, 3, 1, 5, 2, -1, 6, 4], [-2, -2, -3, 2, -3, -3, 2, 1, 1, -1, -1, 0, 1, 0, -3, 1], [4, 0, 0, 4, -3, 0, -4, -2, -1, 1, 2, 1, 0, -2, -2, 1], [3, 1, -2, -1, -3, 3, -1, 0, -1, 2, 1, 0, 1, -1, 1, 4], [4, -1, 3, -2, -9, 7, -4, 3, -2, 2, -1, 1, 0, -4, 2, 2], [2, 5, 0, 0, 4, 0, 4, -2, 1, -2, -1, -1, -1, 3, 2, 0], [5, 0, 1, 0, 0, 5, -5, 0, -1, 1, 0, 3, -3, 1, 3, -2], [0, 1, 4, 0, -2, 4, 0, 0, -1, 1, 2, 1, 3, 0, 3, -1], [4, 5, 2, -2, -1, 4, -1, -1, -3, 0, 1, 3, 1, 2, 3, -2], [5, 0, 0, 0, -5, 3, -3, 0, -3, 0, 0, -2, 2, 0, 0, 3], [-3, 0, 0, 1, 4, -3, 4, -2, -1, -2, 2, -4, -2, -2, 1, 1], [4, 8, 1, -3, -3, 9, -3, -2, -5, 4, 2, 4, -2, 6, 5, -3], [4, -1, 0, -1, -3, 4, -3, 1, -2, 1, 1, 1, -2, 1, 0, -1], [-5, -4, 1, 0, -5, 0, -5, 1, -2, 4, 2, 3, 3, -3, -1, 0], [-8, 2, 0, 1, 8, -2, 4, 5, 3, -3, -2, 1, -2, 2, 0, -2], [2, -3, -2, 1, -8, -1, -2, 4, 1, -1, -1, 1, -2, -1, -2, 2], [8, -1, -3, -3, -8, 3, -2, -4, -1, 6, 3, 3, -1, -3, 1, 4], [2, -1, -1, 1, 0, 4, -2, 2, 0, 0, 1, 6, 0, 2, 3, -2], [-1, 1, 4, -5, 1, 6, -1, 4, 0, 6, 1, 6, 5, -1, 10, -2], [1, 1, 4, -2, 10, 5, -1, 1, -1, 1, 1, 1, 1, 0, 4, -1], [-4, 7, 0, 0, 10, -3, 4, -12, -2, 2, 2, -3, 3, 1, 0, -3], [2, -7, 1, 2, -8, 0, -2, 11, 3, -3, -1, 1, -2, 1, -1, 2], [9, -2, -1, 1, 1, 1, 1, 1, 3, -2, -2, -1, -3, 3, -2, 1], [-6, -1, -1, -2, 6, 0, 3, -1, 1, 3, 1, 0, 1, -1, 3, -3], [-1, -2, 4, -1, 9, 7, 1, 1, -1, 1, -4, -2, 1, 2, 3, -3], [6, -1, 1, 1, -6, -1, -2, -2, -1, -1, 0, -4, 3, 0, -2, 0], [0, 2, 0, -7, -10, 7, 0, 2, -2, 2, -1, 4, 1, -3, 5, 1], [-1, -1, -4, 5, 0, -12, 1, -2, 2, -2, 1, -5, -1, -2, -9, 4], [10, -4, -1, -2, -10, -1, -3, -2, -1, 2, 0, 1, -4, 0, -5, 0], [-5, 3, -2, 1, 5, -5, 5, -1, 4, -4, -3, -3, 1, 0, 0, 1], [0, 2, 0, -1, 6, -1, 0, 1, 1, -1, 4, 4, 0, 2, 3, -1], [5, -5, 2, -1, -3, 7, -3, 3, -2, 2, 1, 1, -5, 2, -2, -1], [3, 3, 2, 0, -1, 1, -1, -3, -3, 4, 3, 3, 2, 1, 3, 0], [1, 0, 2, -3, -3, 6, -3, 3, -3, 0, 0, 2, -1, 0, 0, -3], [-11, -12, 0, 3, 1, -10, 1, 4, 5, -1, -2, -3, 4, -9, -7, 3], [1, -10, -3, 2, -4, -1, -4, 2, 0, -1, 2, -3, -4, -3, -4, 2], [0, 4, -3, 2, 2, -5, 0, -9, -2, 2, 1, -3, 1, -4, -5, 2], [-7, 3, -1, -4, 1, 8, 1, 6, 3, -2, -7, -1, -5, 5, -2, -4], [1, -5, -4, 2, -1, -6, -1, -11, -1, 0, 2, -6, -3, -2, -9, 0], [1, 6, 1, -3, 6, 2, -1, -8, -3, 3, 1, -1, 0, -2, 2, -1], [0, 2, -1, 4, 5, -5, 0, -5, -1, 1, 3, -2, 3, -4, -5, 3], [1, -1, 5, -6, 5, 7, -1, 7, 1, -1, -1, 2, -3, 4, 8, -3], [-6, -2, -1, 1, 6, -1, 3, -7, -2, 1, 2, -5, 3, -2, 3, -4], [-5, -8, 0, 2, 0, -3, 0, 4, 6, -2, -4, 0, 1, -1, -6, 2], [-2, 1, -5, 2, 2, -5, 1, 3, 4, -4, -2, -6, -2, 2, -11, 5], [2, 5, -1, -6, 1, 8, -2, -3, -2, 2, 1, 8, -2, 3, 9, -5], [-3, 0, 6, -6, 9, 12, 3, 1, -2, 2, -5, -1, 2, 3, 8, -5], [15, -5, -3, -1, -15, 1, 0, -2, -5, 0, -1, -2, -6, 1, -3, -2], [3, -5, 1, 3, -5, -3, -5, -1, 2, -1, 1, 6, 6, 0, -2, 3], [-5, -1, 2, 2, 3, -2, 5, 0, 2, -2, 0, -2, 2, 6, 1, -4], [-7, 2, -3, 7, 7, -11, 7, -7, 6, -6, 1, -9, 2, -1, -6, -2], [-5, 3, 8, -4, 5, 6, -4, 11, 1, 2, -2, 6, 7, 2, 8, -1], [2, -5, -5, 3, -8, -5, -8, -2, 0, 3, 3, 2, 4, -3, -1, 3], [7, 8, -2, -1, -3, 5, -3, -6, -7, 2, 6, 1, -3, 4, 7, -1], [-18, -7, 0, 4, 4, -11, 4, 1, 7, -3, -3, -2, 5, -5, -5, 4], [-10, 7, -5, -2, 1, 2, 1, -1, -1, -4, -1, -4, -4, 2, 3, -2], [7, 6, 2, -5, -2, 10, -2, 3, -3, 5, -2, 3, -2, 4, 2, -5], [-2, 6, 2, 1, 2, -1, 1, 1, 6, 1, 2, 3, 6, -2, 4, 3], [4, -13, -2, 8, -13, -6, -4, 15, 4, -4, -3, -3, -2, -5, -11, 9], [1, 3, -2, 0, -4, 0, -1, 0, 0, 0, 6, 8, 2, 0, 3, 0], [0, 5, 2, -2, 4, 7, 0, -7, -3, 3, 1, 3, 2, 2, 6, -5], [1, 3, -2, 6, -1, -7, 1, -5, 4, -5, 1, -1, -3, -1, -7, 4], [1, 4, -1, 5, 9, -3, -1, -12, -3, 3, 1, -4, 2, -3, -5, 0], [1, -1, -1, 5, -1, -1, -3, -6, -5, -5, 0, -5, -2, 0, -5, 3], [8, -2, -3, 5, -6, -4, -6, -6, 1, 3, 4, 5, 4, 3, 1, 5], [-17, 5, -2, 2, 17, -3, 6, 9, 6, -5, -3, -2, 0, 3, -2, 1], [8, -2, -1, 1, -8, 1, 1, -7, -4, 1, 2, 0, -2, -2, 3, -2], [4, 9, 2, -4, -14, 11, -4, -12, -5, 5, 2, 6, -1, 3, 10, -8], [0, -7, -9, 8, -2, -12, 0, 2, 3, -3, -4, -4, -3, -2, -13, 5], [6, -3, 2, -1, -6, 3, -8, -8, -5, 4, 3, 2, 1, -3, 0, 3], [-12, -4, 0, -10, 12, 8, 3, 16, -2, 5, -5, 4, -3, 5, 6, -7], [-2, -2, 0, -1, -5, 1, -5, 0, -5, 5, 4, 1, 1, -4, 1, -1], [-7, -11, 3, 1, -7, 3, -7, 2, -4, -3, 5, 3, -1, -3, -3, 1], [-3, -1, 1, -1, -3, -4, -3, 1, -2, 5, 1, 2, 7, -7, 0, -1], [2, 4, 2, -5, -2, 9, -5, 7, 0, 6, 1, 6, 6, -1, 7, 7], [-9, 1, 1, -3, 9, 7, 6, 10, -3, 0, -3, 0, 2, 3, 11, -6], [-3, -13, -3, 3, -4, 1, 3, 12, 3, -3, -11, -4, -1, 1, -6, 2], [9, -4, 2, 1, -7, 7, -7, -3, -7, -3, 8, 4, -4, 3, 2, 1], [-4, 3, 11, -3, 4, 3, -5, 0, 3, 6, 3, 8, 12, -3, 12, -3], [3, -3, 4, -4, -3, 4, 2, 5, -3, 2, -2, 3, 0, 2, 8, -9], [-15, -3, 2, -4, 15, 0, -2, 1, 1, 4, 0, 2, 1, 0, 0, -3], [7, 2, 2, -4, 3, 12, -7, 2, -3, 3, 2, 9, -1, -2, 8, -1], [-10, -4, -1, 2, -1, -10, 10, 7, 4, -4, 1, -4, 7, -1, -6, 5], [2, -7, 6, 0, -1, 3, -1, 1, -5, 1, 5, 1, -2, -3, -1, 0], [2, -6, -1, 12, -1, -7, -2, 4, 6, -6, 4, 4, -4, 13, -1, -7], [4, 7, 12, -10, 9, 16, -4, 0, -5, 5, 4, 3, 3, -4, 13, -1], [9, 0, -1, -5, -9, 9, -7, -3, -4, 9, 4, 5, 3, -4, 5, 8], [18, 0, 4, 1, -18, -1, -8, -3, 0, 0, 1, 4, 0, -1, -4, 4], [8, 0, 5, 0, -8, 1, 1, -2, -1, 1, 1, 2, 4, -1, 8, -6], [9, -10, 6, 2, -6, 6, -6, 1, -2, -9, 4, 7, -1, 3, -3, 2], [3, 1, -5, 4, -3, 1, -8, -17, -4, 3, 7, 0, -1, -7, -5, 14], [-4, -7, 1, -3, 2, 2, 2, 8, 0, 0, -3, -6, -5, -6, -5, -3], [3, 1, -3, 7, -3, -11, 11, -2, 5, -9, -2, -4, -7, 2, -5, -7], [17, 0, -7, -2, -4, 1, -4, 2, -2, 5, 0, -5, -4, -3, 0, -2], [-10, -2, 2, -2, 7, -3, 7, 4, 0, -3, -2, -7, -2, -7, -2, -2], [-4, 7, 4, -1, 2, -3, 2, -2, 0, -2, -1, 5, 9, -1, 3, -1], [-10, -13, -1, 4, -2, -9, 10, 15, 9, -9, -7, -6, -2, 12, -4, -3], [14, 0, -2, 6, -14, -3, 2, -6, -3, -6, 0, -7, -1, 0, -3, 0], [-11, -2, -8, 10, 11, -19, 7, -4, 7, -14, -4, -18, -6, 4, -24, -1], [2, -9, -5, -3, -2, -3, -1, -12, -3, 6, 3, -4, -4, -3, -6, -3], [-4, 5, -2, -8, 0, 2, 4, -4, -2, 2, -2, 0, 1, -2, 3, 0], [4, 0, 0, 4, -4, -11, 7, -2, 7, -5, -1, -3, -2, 1, -5, -7], [11, 3, 2, -3, -4, 6, -4, -1, -8, 3, 5, 2, -1, -1, 4, -3], [-10, 5, -4, 6, 10, -5, 5, 9, 4, -11, -5, -3, -7, 5, -9, 4], [-4, -2, -9, 11, -2, -17, 4, -12, 1, -1, -5, -12, -1, -1, -17, 2], [13, -1, -2, 3, -13, -5, 6, -12, 1, 1, 4, -2, 0, -4, 3, -5], [-1, 10, -3, -1, 2, 0, 2, 0, 7, -2, -8, 3, 4, 7, 1, -1], [-3, -10, -10, 3, -9, -2, 3, 4, 4, -4, -13, -1, -8, 10, -4, -6], [18, 3, 4, -2, -18, 8, -10, -11, -3, 10, 8, 10, 7, -8, 10, 9], [-1, -7, -3, 6, -6, -7, -6, -3, 5, 0, 1, 5, 6, 1, -3, 6], [-20, 2, -4, 0, 1, -5, 1, 0, 2, 5, -2, -3, 2, -3, 0, 0], [2, 5, -2, -2, 5, -2, 5, 2, 3, 6, -5, -4, 0, -1, 0, -2], [-20, 3, -4, 2, 3, -2, 3, -3, 3, -3, -1, -3, -3, 4, 1, 2], [0, 0, -6, 4, 0, -6, 1, 0, 2, -6, -2, -9, -3, 2, -13, 5], [-11, 2, -2, 7, 11, -13, 8, 5, 8, -12, -5, -11, -1, 5, -12, -4], [13, -3, 6, -2, -13, 8, -4, -3, -9, 3, 1, 4, 2, -1, 11, -5], [3, -2, 3, -4, 1, 7, -3, 6, -2, 2, -2, 0, 1, -7, 1, 5], [-7, -4, 6, 1, 7, -1, -7, -17, -4, 5, 6, 1, 6, -6, 4, -3], [2, -3, -2, -1, 8, -1, -2, 7, 2, -2, 0, 3, -3, 0, 0, 2], [-13, -1, 1, -7, 13, 3, 5, 20, 3, 0, -7, 1, -1, 7, 2, -7], [9, -4, 0, 2, -1, -3, -1, -1, 0, 6, 2, 0, 1, -2, -1, 2], [7, -2, 3, 2, 2, 1, 2, -1, 2, -7, 0, 2, -1, 4, -1, 2], [-8, 1, 5, 5, 8, -10, -1, 0, 6, -7, -2, -2, 1, 2, -8, -3], [2, -5, 8, 1, -6, -3, -2, 12, 2, -2, 3, -6, 2, -7, -4, 9], [16, -5, 2, -7, -16, 8, -6, 4, -6, 6, -1, 4, -1, 1, 3, -1], [3, -3, 2, -4, 7, 5, -3, 14, 3, -3, 4, 10, -2, 3, 8, 0], [7, -1, 6, -1, 4, 3, 4, 4, 4, 0, -5, 3, 1, 3, -3, -1], [-8, 1, 4, 0, 1, 3, 1, -2, -3, 2, 3, 2, -1, 2, 2, 0], [-2, -2, 4, -2, -13, 11, -13, 0, -8, 5, 6, 11, 2, 3, 2, -2], [-1, -4, -1, 3, -2, -1, -2, -6, -7, 0, 10, -3, -5, -2, 3, 3], [-5, 5, 6, -6, -2, 13, -2, 4, -5, 2, -1, 5, -2, 4, 2, -6], [6, 12, 1, 1, -3, -4, -6, -15, -3, 3, 10, 2, -2, -1, 2, -2], [-2, -3, 1, 6, -9, -12, 2, 11, 5, -5, 11, 3, 6, -3, -5, 8], [6, 3, -11, -1, 0, -3, 0, 2, 4, 5, -5, -8, -4, -1, -1, -1], [0, 4, -4, 12, 2, -16, 0, -14, 1, -1, 6, -5, 0, 2, -10, -1], [-14, -4, -11, -4, 14, 4, 8, 0, -4, 4, 0, -3, -8, 0, 1, -1], [0, -3, 4, 2, 5, 8, 0, 0, -2, 2, -4, -2, 3, 0, 1, -2]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_630_i_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_630_2_i_i();".
// 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_i_i(:prec:=16)
    chi := MakeCharacter_630_i();
    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_i_i();".
// 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_i_i( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_630_i();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<11,R![2916, 28188, 188406, 627426, 1493181, 1987308, 1937466, 1194714, 562494, 156501, 39862, 4688, 2018, 116, 53, -1, 1]>,<13,R![2298256, 6900832, 15234300, 16719280, 14639257, 6671493, 3205075, 684178, 433476, 52582, 34540, 477, 1834, -8, 54, -2, 1]>],Snew);
    return Vf;
end function;