
// Make newform 275.2.h.d in Magma, downloaded from the LMFDB on 29 December 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_275_h();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_275_h_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_275_2_h_d();".
// 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_275_2_h_d();".
// 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 := [121, 0, 33, 0, 235, 0, 303, 0, 194, 0, 57, 0, 25, 0, 7, 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, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 7498263355, 0, 7929326490, 0, 3810530967, 0, 1500352252, 0, 557238290, 0, 152757677, 0, 24476532, 0, 1829973], [31775447, 0, -726449607, 0, 74558203, 0, 9430964, 0, -2560494, 0, 9344761, 0, 4779625, 0, 608673, 0], [-771926881, 0, -3415114020, 0, -1632321387, 0, -814713604, 0, -276654262, 0, -99837091, 0, -22433610, 0, -2744937, 0], [0, 771926881, 0, 3415114020, 0, 1632321387, 0, 814713604, 0, 276654262, 0, 99837091, 0, 22433610, 0, 2744937], [236512683, 0, -719547006, 0, -3520413317, 0, -1053314964, 0, -488836470, 0, -193036323, 0, -43736988, 0, -2924455, 0], [-629543387, 0, -664868085, 0, -304413167, 0, 42401020, 0, -13223046, 0, 7678521, 0, 5558819, 0, 1351905, 0], [1738523017, 0, 1844303058, 0, 797037357, 0, -665668084, 0, -38446538, 0, -61040185, 0, -30587176, 0, -6960689, 0], [-383811549, 0, 742925482, 0, 1781437793, 0, 833575532, 0, 271533274, 0, 118526613, 0, 31992860, 0, 3962283, 0], [0, -2315780643, 0, -7806763412, 0, -4896964161, 0, -2444140812, 0, -829962786, 0, -299511273, 0, -67300830, 0, -8234811], [243303327, 0, -21444456, 0, 152673185, 0, 126275508, 0, 28071690, 0, 13758165, 0, 4704418, 0, 770947, 0], [0, -4303783, 0, -4900964984, 0, -5195196973, 0, -2481864668, 0, -819720810, 0, -336890317, 0, -86419330, 0, -10669503], [0, 352179707, 0, 1528330963, 0, 2199079808, 0, 1573766376, 0, 429083674, 0, 198965251, 0, 58503253, 0, 8815096], [0, -4755604557, 0, -8290421402, 0, -10096782513, 0, -4886486364, 0, -1627587846, 0, -643887411, 0, -162786416, 0, -19639683], [0, 172220447, 0, 33071203, 0, -268358314, 0, -228914540, 0, -60570826, 0, -31032777, 0, -9707543, 0, -1496498]];
    Rf_basisdens := [1, 1, 2438578648, 609644662, 2438578648, 2438578648, 2438578648, 609644662, 2438578648, 1219289324, 2438578648, 221688968, 2438578648, 1219289324, 2438578648, 110844484];
    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_275_h();"
function MakeCharacter_275_h()
    N := 275;
    order := 5;
    char_gens := [177, 101];
    v := [5, 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_275_h_Hecke();"
function MakeCharacter_275_h_Hecke( : Kf := false)
    N := 275;
    order := 5;
    char_gens := [177, 101];
    char_values := [[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]];
    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, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0], [0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 1], [-2, 0, 0, 1, 1, 0, 1, 0, 1, -2, 0, 2, 0, 0, 0, 0], [0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0], [0, -1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 1, 0, -3, 0], [-1, 0, 0, -1, 1, 0, 0, -2, 0, 0, 0, 1, 0, 0, 0, 0], [0, 3, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, -3, -2, 0], [1, 0, 0, -1, 0, 0, -1, 1, 0, 3, 0, 0, 0, 0, 0, 0], [2, 0, 0, -2, -3, 0, -1, 0, 0, 2, 0, 1, 0, 0, 0, 0], [0, 2, -1, 0, 0, -1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 2], [-5, 0, 0, 2, 5, 0, 0, -3, -2, 0, 0, -2, 0, 0, 0, 0], [0, -4, 0, 0, 0, -1, 0, 0, 0, 0, -2, 0, -1, 2, 0, -2], [0, -5, 3, 0, 0, -4, 0, 0, 0, 0, 3, 0, 0, 4, 0, 1], [0, -8, 3, 0, 0, -3, 0, 0, 0, 0, 0, 0, 5, 5, 1, -1], [-3, 0, 0, 3, 0, 0, -2, -3, 0, -2, 0, 0, 0, 0, 0, 0], [-4, 0, 0, 0, 3, 0, 0, -4, 4, 1, 0, -4, 0, 0, 0, 0], [0, -2, 1, 0, 0, -2, 0, 0, 0, 0, -3, 0, -1, -1, 1, -3], [9, 0, 0, -6, -9, 0, -6, 3, -9, 6, 0, 3, 0, 0, 0, 0], [0, 5, -4, 0, 0, -1, 0, 0, 0, 0, 1, 0, -5, -1, 1, 5], [-1, 0, 0, -5, -1, 0, -1, 0, 0, -1, 0, 1, 0, 0, 0, 0], [0, 2, 0, 0, 0, -1, 0, 0, 0, 0, -1, 0, -2, 0, -1, 0], [-1, 0, 0, 0, -6, 0, -6, 6, -2, -4, 0, 2, 0, 0, 0, 0], [0, -4, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 4, 1, -1], [-8, 0, 0, 7, -2, 0, 4, 0, 0, -8, 0, -4, 0, 0, 0, 0], [0, -3, 8, 0, 0, -5, 0, 0, 0, 0, 5, 0, 3, 0, 5, -3], [0, 2, -3, 0, 0, 6, 0, 0, 0, 0, -3, 0, 0, -6, 0, -2], [-9, 0, 0, 0, 5, 0, 5, -5, 13, -8, 0, 5, 0, 0, 0, 0], [0, -3, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, -5], [0, 3, 0, 0, 0, -7, 0, 0, 0, 0, 4, 0, -3, 0, 3, 0], [6, 0, 0, 0, 3, 0, 3, -3, 5, -2, 0, 3, 0, 0, 0, 0], [0, -5, 0, 0, 0, -4, 0, 0, 0, 0, -3, 0, 5, 0, 0, 0], [2, 0, 0, -2, -4, 0, 3, 2, -4, 8, 0, 0, 0, 0, 0, 0], [-2, 0, 0, 5, 12, 0, 5, -8, 2, -4, 0, -8, 0, 0, 0, 0], [-2, 0, 0, -5, 2, 0, 0, 6, -3, 0, 0, 5, 0, 0, 0, 0], [0, 3, -6, 0, 0, 8, 0, 0, 0, 0, -6, 0, 0, -8, 0, -4], [0, 6, -6, 0, 0, 7, 0, 0, 0, 0, 0, 0, 1, 1, -6, 6], [0, -4, 9, 0, 0, -2, 0, 0, 0, 0, 0, 0, 2, 2, 4, -4], [0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3], [2, 0, 0, -6, -2, 0, 0, 8, -1, 0, 0, 6, 0, 0, 0, 0], [2, 0, 0, 6, -2, 0, 6, -3, -2, 5, 0, -3, 0, 0, 0, 0], [1, 0, 0, -1, 9, 0, 6, 1, 9, 2, 0, 0, 0, 0, 0, 0], [0, -2, 0, 0, 0, -5, 0, 0, 0, 0, -7, 0, 2, 0, 1, 0], [0, 2, 7, 0, 0, -16, 0, 0, 0, 0, 4, 0, -9, 2, 7, 4], [10, 0, 0, 0, 1, 0, 1, -1, -2, 3, 0, -8, 0, 0, 0, 0], [2, 0, 0, -8, -2, 0, 1, 0, 0, 2, 0, -1, 0, 0, 0, 0], [0, 2, -2, 0, 0, 8, 0, 0, 0, 0, -2, 0, 0, -8, 0, 2], [0, 5, -7, 0, 0, 2, 0, 0, 0, 0, -2, 0, 2, 6, -2, 5], [7, 0, 0, -3, 8, 0, 6, 0, 0, 7, 0, -6, 0, 0, 0, 0], [0, -4, 1, 0, 0, -6, 0, 0, 0, 0, 0, 0, -2, -2, 1, -1], [4, 0, 0, -4, -4, 0, 0, -1, -17, 0, 0, 4, 0, 0, 0, 0], [-4, 0, 0, 0, -10, 0, -10, 10, 1, -11, 0, -2, 0, 0, 0, 0], [-6, 0, 0, -11, -4, 0, -11, 0, 0, -6, 0, 11, 0, 0, 0, 0], [0, -1, 10, 0, 0, -9, 0, 0, 0, 0, 9, 0, -4, 12, 9, -1], [0, 8, 5, 0, 0, -10, 0, 0, 0, 0, 5, 0, -5, -3, 5, 5], [-10, 0, 0, 4, 6, 0, 4, -9, 10, 3, 0, -9, 0, 0, 0, 0], [9, 0, 0, -9, -16, 0, -8, 9, -16, 9, 0, 0, 0, 0, 0, 0], [0, 1, -11, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, -1, -3, 3], [-13, 0, 0, 11, 12, 0, 11, -6, 13, -6, 0, -6, 0, 0, 0, 0], [0, 5, -2, 0, 0, 4, 0, 0, 0, 0, -2, 0, 0, -4, 0, -11], [0, -1, 9, 0, 0, -8, 0, 0, 0, 0, 8, 0, -1, 10, 8, -1], [0, 6, -5, 0, 0, 2, 0, 0, 0, 0, 8, 0, -3, 2, -5, 8], [-2, 0, 0, 0, 2, 0, 0, 1, 8, 0, 0, 0, 0, 0, 0, 0], [0, -3, 0, 0, 0, -5, 0, 0, 0, 0, 4, 0, 3, 0, 4, 0], [0, 13, -4, 0, 0, 3, 0, 0, 0, 0, 0, 0, -10, -10, -2, 2], [16, 0, 0, 0, 4, 0, 4, -4, -6, 10, 0, -4, 0, 0, 0, 0], [0, -2, 1, 0, 0, 1, 0, 0, 0, 0, -1, 0, -14, -3, -1, -2], [0, -4, 0, 0, 0, 1, 0, 0, 0, 0, -6, 0, 4, 0, 3, 0], [9, 0, 0, -7, -9, 0, 0, 7, 2, 0, 0, 7, 0, 0, 0, 0], [0, -15, -1, 0, 0, 11, 0, 0, 0, 0, -6, 0, 10, 9, -1, -6], [-6, 0, 0, 6, 16, 0, 12, -6, 16, 9, 0, 0, 0, 0, 0, 0], [0, -1, -4, 0, 0, 5, 0, 0, 0, 0, -5, 0, 6, -5, -5, -1], [0, -4, 4, 0, 0, 0, 0, 0, 0, 0, -5, 0, 4, -1, 4, -5], [4, 0, 0, -18, -13, 0, -18, 14, -4, -1, 0, 14, 0, 0, 0, 0], [0, 4, 5, 0, 0, -3, 0, 0, 0, 0, 0, 0, -7, -7, -3, 3], [0, 0, 0, 0, -12, 0, 10, 0, -12, 13, 0, 0, 0, 0, 0, 0], [0, -9, 2, 0, 0, -1, 0, 0, 0, 0, 3, 0, 1, 12, 2, 3], [-16, 0, 0, 1, 6, 0, 1, 3, 16, -9, 0, 3, 0, 0, 0, 0], [-21, 0, 0, 24, 6, 0, 0, 0, 0, -21, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 8, 0, 8, -8, 0, 8, 0, -1, 0, 0, 0, 0], [-2, 0, 0, -7, 2, 0, 0, 5, -3, 0, 0, 7, 0, 0, 0, 0], [2, 0, 0, 11, -24, 0, -8, 0, 0, 2, 0, 8, 0, 0, 0, 0], [0, 7, -1, 0, 0, -6, 0, 0, 0, 0, 6, 0, -12, 4, 6, 7], [22, 0, 0, 0, -5, 0, -5, 5, -12, 7, 0, -14, 0, 0, 0, 0], [0, -7, -2, 0, 0, -8, 0, 0, 0, 0, -2, 0, 0, 8, 0, 3], [-2, 0, 0, -11, 1, 0, -17, 0, 0, -2, 0, 17, 0, 0, 0, 0], [0, -6, 0, 0, 0, -12, 0, 0, 0, 0, -8, 0, 6, 0, -7, 0], [8, 0, 0, 0, 14, 0, 14, -14, 3, 11, 0, -13, 0, 0, 0, 0], [0, -4, 4, 0, 0, -2, 0, 0, 0, 0, -5, 0, 2, -1, 4, -5], [0, 7, 0, 0, 0, -5, 0, 0, 0, 0, 13, 0, -7, 0, 2, 0], [-25, 0, 0, 7, 32, 0, 7, -1, 25, -31, 0, -1, 0, 0, 0, 0], [0, 10, 1, 0, 0, 10, 0, 0, 0, 0, 1, 0, 0, -10, 0, 12], [-9, 0, 0, 9, 1, 0, 8, -9, 1, -4, 0, 0, 0, 0, 0, 0], [15, 0, 0, -15, -15, 0, -25, 15, -15, 5, 0, 0, 0, 0, 0, 0], [0, 7, -4, 0, 0, 13, 0, 0, 0, 0, -4, 0, 0, -13, 0, -10], [0, 0, 0, -3, 0, 0, 0, -15, 1, 0, 0, 3, 0, 0, 0, 0], [1, 0, 0, -1, -26, 0, -10, 1, -26, 13, 0, 0, 0, 0, 0, 0], [0, 11, 0, 0, 0, -1, 0, 0, 0, 0, 6, 0, -11, 0, -10, 0], [-5, 0, 0, -8, 19, 0, -8, 0, 0, -5, 0, 8, 0, 0, 0, 0], [0, 16, -14, 0, 0, 11, 0, 0, 0, 0, -14, 0, 0, -11, 0, -10], [0, -15, 14, 0, 0, 1, 0, 0, 0, 0, -1, 0, 12, -3, -1, -15], [0, -4, 10, 0, 0, -1, 0, 0, 0, 0, 0, 0, 3, 3, -1, 1], [4, 0, 0, 8, -4, 0, 0, -1, -24, 0, 0, -8, 0, 0, 0, 0], [-8, 0, 0, 0, -1, 0, -1, 1, -11, 10, 0, -1, 0, 0, 0, 0], [0, -1, 0, 0, 0, 14, 0, 0, 0, 0, 2, 0, 1, 0, 4, 0], [0, -5, -8, 0, 0, 13, 0, 0, 0, 0, -13, 0, 4, 1, -13, -5], [0, -1, -6, 0, 0, 1, 0, 0, 0, 0, 6, 0, -5, 7, -6, 6], [-12, 0, 0, 5, 1, 0, 5, -9, 12, 8, 0, -9, 0, 0, 0, 0], [15, 0, 0, -15, 5, 0, -2, 15, 5, -7, 0, 0, 0, 0, 0, 0], [0, -25, 24, 0, 0, -19, 0, 0, 0, 0, 0, 0, 6, 6, 12, -12], [0, -9, 1, 0, 0, -10, 0, 0, 0, 0, 1, 0, 0, 10, 0, 13], [0, 20, -9, 0, 0, 10, 0, 0, 0, 0, 12, 0, 1, -8, -9, 12], [-26, 0, 0, 15, 26, 0, 0, -18, 2, 0, 0, -15, 0, 0, 0, 0], [12, 0, 0, -12, -14, 0, -4, 12, -14, 15, 0, 0, 0, 0, 0, 0], [10, 0, 0, 4, -10, 0, 0, 4, -13, 0, 0, -4, 0, 0, 0, 0], [0, 6, 0, 0, 0, -4, 0, 0, 0, 0, -9, 0, -6, 0, 15, 0], [0, -7, -3, 0, 0, -2, 0, 0, 0, 0, 0, 0, 5, 5, -6, 6], [0, 13, -12, 0, 0, -1, 0, 0, 0, 0, 1, 0, 8, 0, 1, 13], [-2, 0, 0, 0, 3, 0, 3, -3, 4, -1, 0, 13, 0, 0, 0, 0], [-2, 0, 0, 0, -5, 0, -5, 5, -14, 9, 0, 4, 0, 0, 0, 0], [0, -25, 15, 0, 0, 1, 0, 0, 0, 0, 0, 0, 26, 26, -1, 1], [0, -15, 0, 0, 0, 5, 0, 0, 0, 0, -5, 0, 15, 0, -1, 0], [0, 17, -3, 0, 0, -11, 0, 0, 0, 0, 9, 0, -14, -8, -3, 9], [-14, 0, 0, -1, 29, 0, 3, 0, 0, -14, 0, -3, 0, 0, 0, 0], [-5, 0, 0, -22, 5, 0, 0, -3, 12, 0, 0, 22, 0, 0, 0, 0], [-16, 0, 0, 7, -7, 0, 7, 6, 16, 1, 0, 6, 0, 0, 0, 0], [4, 0, 0, -4, -15, 0, -1, 4, -15, 20, 0, 0, 0, 0, 0, 0], [0, -10, -5, 0, 0, -11, 0, 0, 0, 0, -5, 0, -16, 5, -5, -5], [0, -13, 1, 0, 0, 12, 0, 0, 0, 0, -12, 0, -1, -17, -12, -13], [5, 0, 0, 1, 18, 0, 11, 0, 0, 5, 0, -11, 0, 0, 0, 0], [0, 4, 0, 0, 0, 2, 0, 0, 0, 0, 10, 0, -4, 0, 8, 0], [-3, 0, 0, 24, 3, 0, 0, -18, -3, 0, 0, -24, 0, 0, 0, 0], [0, -10, 6, 0, 0, 6, 0, 0, 0, 0, 0, 0, 16, 16, -2, 2], [13, 0, 0, -7, -4, 0, -8, 0, 0, 13, 0, 8, 0, 0, 0, 0], [-5, 0, 0, 0, -8, 0, -8, 8, 8, -16, 0, 10, 0, 0, 0, 0], [0, 5, 0, 0, 0, -8, 0, 0, 0, 0, 0, 0, 0, 8, 0, 9], [0, 27, 0, 0, 0, -13, 0, 0, 0, 0, 11, 0, -27, 0, 8, 0], [0, -1, 0, 0, 0, 23, 0, 0, 0, 0, -4, 0, 1, 0, -9, 0], [7, 0, 0, -10, 5, 0, -10, 16, -7, -21, 0, 16, 0, 0, 0, 0], [19, 0, 0, 12, -19, 0, 0, -5, -15, 0, 0, -12, 0, 0, 0, 0], [-3, 0, 0, 3, 22, 0, 14, -3, 22, -20, 0, 0, 0, 0, 0, 0], [0, -23, 13, 0, 0, -22, 0, 0, 0, 0, 0, 0, 1, 1, 17, -17], [0, 24, -15, 0, 0, 9, 0, 0, 0, 0, 0, 0, -15, -15, -13, 13], [-14, 0, 0, 14, 10, 0, 23, -14, 10, -21, 0, 0, 0, 0, 0, 0], [25, 0, 0, 7, -25, 0, 0, 2, -15, 0, 0, -7, 0, 0, 0, 0], [0, 10, 0, 0, 0, -12, 0, 0, 0, 0, 9, 0, -10, 0, 13, 0], [0, 6, 12, 0, 0, -24, 0, 0, 0, 0, 19, 0, -12, 13, 12, 19], [19, 0, 0, 0, -4, 0, -4, 4, 7, -11, 0, -5, 0, 0, 0, 0], [0, 10, 0, 0, 0, 12, 0, 0, 0, 0, 2, 0, -10, 0, -9, 0], [0, -10, 25, 0, 0, -15, 0, 0, 0, 0, 25, 0, 0, 15, 0, 5], [10, 0, 0, 0, 1, 0, 1, -1, -31, 32, 0, 2, 0, 0, 0, 0], [0, -5, 9, 0, 0, 5, 0, 0, 0, 0, 9, 0, 0, -5, 0, 6], [0, 14, -27, 0, 0, 13, 0, 0, 0, 0, -13, 0, 7, -8, -13, 14], [0, -10, 0, 0, 0, 9, 0, 0, 0, 0, -10, 0, 10, 0, -6, 0], [-14, 0, 0, 14, 10, 0, 16, 0, 0, -14, 0, -16, 0, 0, 0, 0], [2, 0, 0, -24, 4, 0, -24, -6, -2, 2, 0, -6, 0, 0, 0, 0], [-11, 0, 0, 7, 27, 0, 7, -12, 11, -15, 0, -12, 0, 0, 0, 0], [0, -37, 22, 0, 0, -19, 0, 0, 0, 0, 0, 0, 18, 18, 16, -16], [-3, 0, 0, -7, -10, 0, -7, 14, 3, -4, 0, 14, 0, 0, 0, 0], [0, -9, 18, 0, 0, -18, 0, 0, 0, 0, -7, 0, 0, 2, 18, -7], [0, 6, -17, 0, 0, 11, 0, 0, 0, 0, -11, 0, 21, -6, -11, 6], [0, -15, 1, 0, 0, 15, 0, 0, 0, 0, 2, 0, 16, 17, 1, 2], [-16, 0, 0, -7, 16, 0, 0, -23, 31, 0, 0, 7, 0, 0, 0, 0], [0, 2, -3, 0, 0, 1, 0, 0, 0, 0, 0, 0, -1, -1, 11, -11], [0, 10, -17, 0, 0, 7, 0, 0, 0, 0, -7, 0, -2, 11, -7, 10], [-9, 0, 0, 0, 2, 0, 2, -2, 27, -25, 0, 1, 0, 0, 0, 0], [0, -5, -5, 0, 0, 10, 0, 0, 0, 0, -10, 0, 11, -8, -10, -5]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_275_h_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_275_2_h_d();".
// 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_275_2_h_d(:prec:=16)
    chi := MakeCharacter_275_h();
    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_275_2_h_d();".
// 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_275_2_h_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_275_h();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![121, 0, 308, 0, 345, 0, 138, 0, 159, 0, 62, 0, 15, 0, 2, 0, 1]>],Snew);
    return Vf;
end function;