
// Make newform 350.2.h.c in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_350_h();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_350_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_350_2_h_c();".
// 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_350_2_h_c();".
// 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 := [16, 0, 480, 0, 2156, 0, 3600, 0, 2841, 0, 1145, 0, 241, 0, 25, 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], [448, 976, 14272, 10332, 55152, 30074, 65168, 33847, 31568, 17143, 7248, 4215, 784, 487, 32, 21], [192, 0, 4132, 0, 16022, 0, 19079, 0, 9191, 0, 2087, 0, 223, 0, 9, 0], [-448, 976, -14272, 10332, -55152, 30074, -65168, 33847, -31568, 17143, -7248, 4215, -784, 487, -32, 21], [-224, 136, -9820, -3388, -39346, -12254, -48649, -14488, -25289, -7608, -6249, -1928, -721, -228, -31, -10], [224, 136, 9820, -3388, 39346, -12254, 48649, -14488, 25289, -7608, 6249, -1928, 721, -228, 31, -10], [384, 1112, 14272, 21360, 55152, 78308, 65168, 92911, 31568, 46191, 7248, 10943, 784, 1219, 32, 51], [1248, -616, 31360, 7952, 112736, 33396, 129148, 39391, 61372, 18975, 13884, 4335, 1484, 467, 60, 19], [-1248, -616, -31360, 7952, -112736, 33396, -129148, 39391, -61372, 18975, -13884, 4335, -1484, 467, -60, 19], [944, 1248, 24748, 6052, 90562, 7966, 107399, 3683, 53799, 163, 12871, -253, 1447, -53, 61, -3], [1456, -376, 40400, 18000, 147680, 85148, 174674, 108901, 87186, 56101, 20786, 13589, 2330, 1537, 98, 65], [-1456, -376, -40400, 18000, -147680, 85148, -174674, 108901, -87186, 56101, -20786, 13589, -2330, 1537, -98, 65], [-1232, 2280, -37008, 30664, -145680, 94576, -180982, 103931, -94774, 49467, -23574, 11371, -2734, 1239, -118, 51], [104, 764, -384, 25714, -700, 97957, -983, 116869, -1015, 58117, -391, 13757, -59, 1531, -3, 64], [-912, 204, -24748, -9794, -90562, -46225, -107399, -59696, -53799, -31504, -12871, -7832, -1447, -906, -61, -39], [-920, -596, -24332, -26098, -89862, -98657, -106416, -117852, -52784, -59132, -12480, -14148, -1388, -1590, -58, -67]];
    Rf_basisdens := [1, 128, 32, 128, 64, 64, 128, 128, 128, 64, 128, 128, 128, 64, 64, 64];
    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_350_h();"
function MakeCharacter_350_h()
    N := 350;
    order := 5;
    char_gens := [127, 101];
    v := [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_350_h_Hecke();"
function MakeCharacter_350_h_Hecke(Kf)
    N := 350;
    order := 5;
    char_gens := [127, 101];
    char_values := [[0, -1, 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]];
    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, 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, -1, 0, 0, 0, -1, 0, 0, -1, 0, 0, 0, -1, 0, 1, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [2, -1, 0, 1, 1, -1, 1, 1, -1, -1, 0, 0, -1, -1, 1, 0], [-1, -1, -1, -1, -1, -2, -1, -1, -1, 2, 1, 1, -1, 1, 1, 1], [-1, 0, 0, -3, -1, 0, -1, 0, 0, 2, 0, 2, 2, 0, -1, 0], [-3, 0, -1, 0, -1, -2, -3, -2, -2, 1, 0, 2, -1, 0, 1, -1], [0, 1, 0, 1, 0, 1, 0, -1, -1, -1, 0, 0, 0, -1, 1, -3], [0, 1, 0, -1, -1, -1, -1, -1, -1, 0, 0, -1, 0, 0, 1, 1], [4, 0, 1, 4, 2, -1, 4, 2, 2, -2, 0, -2, 0, -3, 0, 1], [0, -2, -3, 0, 1, -1, 1, -1, -2, 1, -1, 3, 0, 1, 0, -1], [-1, 5, -1, -1, 2, 1, -4, 2, -1, -1, -3, 1, -1, -2, 1, -3], [-1, 0, 1, 0, -1, 1, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0], [0, -3, 0, 4, -1, -1, 4, -1, -2, 0, -1, -1, 0, -1, 1, 1], [1, -5, 0, 2, 1, 0, 2, 1, 2, -1, -1, -1, 0, 1, -1, 1], [-1, 2, -2, -1, 2, 2, 0, -1, 0, -2, -2, 2, 1, -1, -3, -2], [4, -3, 4, 1, 1, -2, 3, 2, -1, -1, 4, 0, -1, -2, 2, 3], [0, 0, -1, -1, 3, 1, 0, 3, 3, -3, 0, -1, 0, -2, 0, -1], [2, 0, 0, -1, 1, -1, -1, 0, -1, -2, -1, 1, 1, -2, -1, -1], [0, -3, -1, -3, 1, -2, -1, 2, 3, 3, -1, 0, -1, 2, -2, 0], [-1, 0, 0, -4, 1, 2, -4, 1, 1, 1, -4, 1, 0, 0, -1, -1], [6, 0, 2, 4, 2, 1, 6, 4, 4, -2, 0, -4, 2, -3, -2, 2], [0, -2, 1, -2, -3, 3, 3, -3, 2, 2, 1, 0, 3, 0, 0, 0], [-2, 1, 0, 0, -1, 1, 0, -1, 2, 2, -1, 1, 0, 3, 1, -1], [6, -3, -3, 3, 0, -1, 0, 0, -1, 0, -1, 1, -1, 1, 1, 0], [-3, -3, 0, -2, -3, 0, -2, -1, -2, 3, 1, 3, -2, 1, 3, -3], [9, 2, 2, -1, 1, 2, 1, 1, 2, -1, 1, -1, 3, -3, -2, 0], [-10, 8, -6, -10, 0, -2, -10, -2, -2, 2, -2, 6, -2, 0, 0, -2], [-2, 2, -1, -2, 1, 1, -1, 2, 0, -1, 2, 1, 1, 0, 1, 2], [6, -4, 3, 2, -1, -3, 3, 3, -2, -2, 3, 0, 1, -2, 2, 4], [4, 0, 2, 7, -3, 1, 4, 1, 1, 1, 0, -3, 2, 0, -4, 2], [-2, 6, 0, -2, 2, 4, -2, 1, 2, -4, -1, 0, 0, -4, -3, -1], [0, 1, 0, 1, -1, 1, 5, -1, -1, -1, 0, 0, 1, 1, -1, 2], [-10, 1, -1, -5, -3, -3, -4, -4, -2, 4, 2, -2, -6, 6, 2, 0], [10, -3, 3, 6, 0, 3, 3, 4, -1, -3, 1, -1, 3, -3, 0, 0], [0, -5, -3, 2, -2, -2, -3, -2, -2, 3, -3, 3, -4, 4, 1, 0], [-5, -2, 2, -5, -4, 0, 2, 2, 4, 0, 2, -2, -1, -1, 2, 2], [-6, 0, -7, -5, 0, -2, -6, -4, -4, -1, 0, 2, -5, 2, 4, -7], [-2, 4, 4, 2, -2, 1, 2, -1, -2, -2, 4, 0, 2, -3, 3, -1], [-3, 8, 0, -4, 1, 4, -4, -2, 1, 3, -1, 1, 3, 0, -1, -1], [6, 0, -2, 3, -1, -3, 6, -5, -5, 1, 0, 1, -4, 2, 4, -2], [-9, 1, -1, -9, -1, -1, -2, -2, 0, 1, 0, 1, 1, 2, -1, 0], [-4, -4, 4, 4, -4, 4, 0, -5, 5, 0, 8, -8, 0, 0, 0, 0], [1, -1, 0, 9, 0, -1, 9, 1, -2, -1, -3, -2, -1, -2, 0, 2], [13, 0, 9, 3, 2, 0, 3, 1, 1, -3, 2, -2, 2, -2, 1, 0], [0, -1, -3, -1, 0, 2, -2, -2, 1, 1, -3, 0, 0, 1, -1, 2], [-2, 3, -2, 1, 1, -3, -1, 3, -1, -1, -2, 0, -1, -3, 3, 2], [1, -1, -1, 0, -1, 3, 0, -3, 0, 0, -1, 0, 1, -1, 1, -6], [1, -6, 0, 3, 0, -1, 3, 0, -1, -1, 0, 3, 0, -1, 0, -3], [0, 0, -7, -13, 0, -5, 0, -3, -3, 1, 0, 11, -2, -2, 3, -7], [0, -1, -2, -1, 5, -1, -11, 1, 1, 1, -2, 0, -5, 2, -2, 1], [-16, 11, -8, -16, 3, -2, -13, 2, -5, 2, -7, 8, -4, -2, 4, -7], [-2, 1, 2, -1, -2, 1, 0, -4, 3, 0, 4, -4, -1, 1, 1, 0], [10, -2, 0, 5, 1, 1, 3, -2, 4, -3, -2, 2, 2, -2, 1, 0], [-2, 11, -4, -2, 0, 2, -9, -1, 2, -2, -3, 4, 2, 0, -3, -3], [-5, 0, 2, 10, 1, -2, -5, 0, 0, -1, 0, -2, -1, -2, 1, 2], [1, -5, 0, -1, 0, -1, -1, 1, 4, -1, 5, -2, -1, 4, 0, 2], [5, -2, 3, 3, 0, 0, 7, 0, -3, -3, 3, 0, 0, 0, 0, 1], [-1, 0, -1, -8, 0, 2, -1, 2, 2, 3, 0, 4, 5, 0, -2, -1], [3, 0, 0, 1, 3, -2, 3, 0, 0, 4, 0, 3, 4, -2, 3, 0], [6, 5, -1, -6, 2, -2, -1, 0, 0, 1, 0, 0, 0, 0, -1, 0], [3, -7, 4, 7, -2, 0, 0, -1, -1, 0, 4, -4, -2, 2, 2, 0], [-9, 4, 3, -5, -4, 4, -1, -4, 5, 5, 3, 0, 4, 2, -2, -1], [-13, 10, -9, -13, 1, 1, -9, -2, 0, -1, -12, 9, -1, -2, -3, -12], [-5, 0, 1, -6, -1, 2, -5, 2, 2, -1, 0, -6, 1, 0, -3, 1], [5, 0, 2, 11, 3, -2, 5, -2, -2, -2, 0, 0, -4, 0, 5, 2], [18, -1, -2, 18, 6, 4, 5, 2, -2, -4, -6, 2, 4, 0, -2, -6], [2, -1, 0, -2, -1, -3, -2, 3, -4, -2, 5, 3, -4, -3, 1, -3], [11, -4, -1, 7, 1, 2, 3, 1, 2, -3, 3, -3, 3, -3, 0, 0], [0, 14, 0, -2, 4, 4, -2, -2, 2, 0, -2, 2, 6, -2, -4, -2], [-10, 9, -1, -10, 3, -7, -16, -3, -10, 7, -7, 1, -8, -1, 4, -7], [-8, 0, 1, 1, -1, 3, -8, 3, 3, 1, 0, 3, 4, 0, -4, 1], [-4, 2, 3, -2, -4, 3, -6, -3, 2, 2, 3, 0, 4, -3, 3, 6], [4, -3, 0, 2, 5, 1, 2, 1, 2, -4, 1, -1, 4, -3, -5, 1], [8, 0, -1, 5, -1, 5, 8, 1, 1, -3, 0, -5, -2, 4, -2, -1], [11, -12, 4, -1, 1, 1, 10, -1, 1, 1, 4, 0, -1, -4, 4, 1], [2, 9, 0, 6, 2, 0, 6, -1, 3, -2, -1, 0, 3, 1, -2, 0], [0, -10, -1, 5, -4, -3, -5, -2, -5, 5, 1, -1, -7, 7, 2, 0], [-7, 17, -6, -7, -1, 0, -17, -1, 1, 0, -4, 6, -3, -3, -1, -4], [4, 0, 1, 4, 1, 4, 4, 1, 1, 2, 0, 4, 3, 3, 0, 1], [-4, 7, 0, -6, 0, 4, -6, -5, -1, 4, 4, 0, 5, -1, 0, 0], [3, 0, 3, -3, -3, 4, 3, 3, 3, 6, 0, -3, 9, 1, -6, 3], [8, 0, -2, -8, -2, 0, 8, 0, 0, 2, 0, 0, 2, 0, -2, -2], [-6, 1, 0, -5, 2, 1, -4, -1, 5, 5, 0, 0, -2, 0, 0, 4], [9, -11, 7, 13, 3, -2, 2, 1, 0, -2, 3, -3, 1, -1, 1, 0], [-13, 7, 5, -3, -1, 7, 4, 3, 3, -4, 0, 0, 6, -6, -2, 0], [-1, -3, 4, 8, 3, -3, 5, 2, -2, -5, 8, -8, 0, 0, 5, 0], [-14, 11, -2, -3, -1, 1, -9, -1, 3, 3, -2, 0, 1, -1, 1, 0], [-13, 14, 5, -13, 3, 1, -13, 0, -2, -1, 4, -5, 0, -1, -1, 4], [1, 0, -2, -5, 1, -2, 1, 4, 4, -2, 0, -2, 2, -6, -3, -2], [2, 0, 0, -1, -2, -4, -1, 2, -5, -2, -1, 2, -4, -3, 2, -2], [15, -5, 6, 15, 5, 6, 11, 6, 1, -6, 1, -6, 0, -6, 0, 1], [1, -6, -6, 1, 0, -8, -2, -5, -8, 8, 1, 6, -5, 3, 3, 1], [-9, 13, -9, -15, -1, 0, -2, 1, -2, 2, -3, 3, -1, 1, -1, 0], [2, -8, 0, 6, 10, 8, 6, 6, 8, -2, -4, 4, 4, -2, -10, -4], [8, -4, -5, 8, 5, 5, 9, 4, 0, -5, -2, 5, 2, -3, -1, -2], [-6, -10, 0, -2, -2, 4, -2, -2, 0, 6, 10, -2, 0, 2, 2, 2], [0, 0, 3, 0, 0, 3, -14, -3, 0, 0, 3, 0, 0, 7, -7, 3], [-2, 7, -2, -2, -1, -6, -13, -3, -5, 6, 4, 2, -2, 4, 3, 4], [1, 0, 0, 5, -2, -3, 5, 4, 3, -1, 1, -10, -6, 5, 2, 10], [14, -14, 8, 14, 0, 2, 0, -2, 4, 0, -2, 2, 2, -2, -2, 0], [3, 2, -2, 3, 1, -3, -5, -2, -4, 3, 5, 2, 1, 4, 1, 5], [5, 0, 3, 18, -4, 1, 5, -4, -4, 0, 0, -2, -4, 5, 0, 3], [2, -21, 0, 13, -6, -8, 13, -3, -8, -2, 9, -2, -3, -2, 6, 2], [5, -11, 0, -6, -1, 0, -4, 0, 6, 6, 0, 0, 1, 6, -6, -5], [-13, 3, -2, -13, -1, -2, -5, -3, -1, 2, -7, 2, -5, -3, -1, -7], [9, 1, 5, 8, 2, 7, 9, 7, 2, -9, -1, 1, 9, -9, 0, 0], [-8, 9, -12, -19, 0, -4, -10, -4, 0, 10, -6, 6, -4, 4, -6, 0], [5, -4, 2, 11, 3, 0, 7, -1, 4, -7, 5, -5, 3, -3, 4, 0], [1, -4, 6, 3, -1, -2, -1, 1, -4, 1, 3, -3, -3, 3, 2, 0], [-8, 7, -7, -8, -4, -5, -12, -9, -1, 5, -4, 7, 0, 5, -4, -4], [0, 0, 2, -14, 6, 2, 0, 5, 5, -3, 0, -7, 2, -3, 1, 2], [1, 0, 5, 13, 1, 4, 1, 2, 2, -4, 0, -6, -2, 2, -1, 5], [-8, 0, -7, 8, -5, -5, -8, -8, -8, 2, 0, 2, -6, 3, 3, -7], [13, -10, 9, 13, -6, -1, 9, 6, 5, 1, 9, -9, 5, 6, 7, 9], [-3, -2, 13, 4, -3, 1, 2, 1, -3, -2, 7, -7, -2, 2, 4, 0], [3, 12, 0, 6, 10, 7, 6, 8, 2, -3, -8, -4, 2, -8, -10, 4], [-4, 2, 0, 3, -5, -1, 3, -2, 0, 4, 16, -9, -3, 5, 5, 9], [13, -2, 1, 13, -1, 11, 13, 3, 12, -11, -3, -1, 9, -2, -8, -3], [-5, 4, 4, -1, -2, 2, 13, -2, 1, 1, 4, 0, 2, -2, 2, 2], [2, 0, 0, 2, -5, 4, -6, -4, -2, -2, 0, 0, 5, 0, 0, -3], [0, 2, -3, 2, -2, 2, -4, -2, -2, -2, -3, 0, 2, -8, 8, -9], [-2, 0, -1, 3, 2, 6, -2, 4, 4, -5, 0, -12, -1, 2, -2, -1], [1, 20, 4, 1, 5, 11, -9, 7, 6, -11, -2, -4, 3, -8, -4, -2], [-3, 10, -4, -9, 5, 4, 1, 7, 2, -1, -15, 15, 9, -9, -8, 0], [5, 1, 17, 5, -2, 7, 6, 3, 9, -7, 12, -17, 7, 0, -4, 12], [2, 0, 2, 16, 1, 5, 2, 1, 1, 2, 0, -9, 3, 4, 0, 2], [2, -3, -4, -1, -1, 1, 23, -1, 1, 1, -4, 0, 1, 7, -7, 10], [16, 0, 1, 6, -1, 1, 16, -1, -1, 0, 0, -3, -1, 2, 0, 1], [-17, 16, 5, -1, -3, 2, -16, -2, 1, 1, 5, 0, 3, -1, 1, 7], [5, 5, 11, -11, 3, -4, -6, -2, 1, 6, -1, 1, -1, 1, -5, 0], [20, -3, 2, 9, -2, 7, 6, 4, 1, -6, 3, -3, 5, -5, 1, 0], [-9, 9, 0, -5, 4, -2, 4, 1, 1, -4, 2, -2, 2, -2, 2, 0], [-22, 17, 2, -5, -4, 1, -10, -1, 5, 5, 2, 0, 4, 6, -6, 2], [-4, 0, -1, -10, 3, -8, -4, -5, -5, -8, 0, -3, -13, -3, 8, -1], [18, -10, -3, 8, 9, -9, 19, 9, -8, -8, -3, 0, -9, -6, 6, 3], [-2, -9, 6, -2, 3, 6, 15, 10, 3, -6, 9, -6, 0, -6, 4, 9], [-3, -23, 0, -3, 1, 4, -3, -3, -1, 3, -5, 0, 4, -2, -1, 0], [-26, -2, -8, -26, -8, -12, -10, -4, -4, 12, -4, 8, -12, 0, 8, -4], [-3, 4, 0, 1, -2, 5, -15, -5, -1, -1, 0, 0, 2, 5, -5, -4], [1, 5, 0, 0, 1, 0, 0, 1, -2, -1, -9, 7, 0, -3, -1, -7], [-2, 4, -6, 2, -6, 4, -5, -4, -2, -2, -6, 0, 6, 3, -3, 1], [-1, -2, 1, -3, 0, 0, 5, 0, 3, 3, 1, 0, 0, -2, 2, -5], [6, -27, 0, 13, -2, -8, 13, 3, 4, -6, 3, -4, -5, 6, 2, 4], [3, 3, -9, 6, 9, -7, -18, 7, -6, -6, -9, 0, -9, -1, 1, -5], [1, -6, 0, 16, -2, -3, 16, 8, -6, -1, -3, -7, -10, -4, 2, 7], [-4, 2, 12, 2, 4, 8, 4, 6, 6, -4, 2, -2, 12, -12, -8, 0], [-13, -16, -13, -13, -11, -15, 1, -11, -4, 15, 2, 13, 1, 16, 4, 2], [20, -9, 3, 20, 4, 1, 10, 7, -3, -1, 2, -3, -1, -2, 6, 2], [-13, 7, 7, -6, 4, 3, -21, -3, 6, 6, 7, 0, -4, 4, -4, 9], [10, 0, -3, -14, 6, 1, 10, 6, 6, -2, 0, 8, 4, -5, 0, -3], [18, 0, -3, 12, 1, 4, 18, -1, -1, 0, 0, -5, -1, 5, 2, -3], [-7, 24, 1, -7, 3, -1, -25, -5, -4, 1, -1, -1, -7, -6, -4, -1], [1, 14, 11, -13, 4, 4, 1, 9, -1, -1, -8, 8, 8, -8, -7, 0], [0, -5, 15, -5, -7, 2, 2, -2, 5, 5, 15, 0, 7, -5, 5, 3], [14, 0, 1, 10, 5, -4, 14, 9, 9, -4, 0, 1, 5, -13, -4, 1], [-1, -3, 0, 9, 0, 1, 9, -7, -8, 1, -7, 6, 7, -8, 0, -6], [7, -10, -1, 7, -3, -7, 3, -10, -4, 7, -2, 1, -1, 6, -3, -2], [-16, 24, 1, -16, 3, -3, -27, -4, -6, 3, -5, -1, -4, -1, -1, -5], [-2, -15, 0, 8, -10, -8, 8, -9, -9, 2, 6, -2, -1, 1, 10, 2], [6, 14, 0, 8, 5, -1, 8, -3, 1, -6, -6, -3, 8, -4, -5, 3], [-13, 0, -14, -9, -5, -9, -13, -11, -11, 6, 0, 11, -5, 2, 6, -14], [-13, 5, 0, -16, -9, 4, -16, -11, -6, 13, 8, 7, 2, 3, 9, -7], [-7, 8, -5, 1, 3, -6, -15, 6, -1, -1, -5, 0, -3, -4, 4, 0], [-3, 0, -5, 9, 3, 1, -3, 0, 0, -4, 0, -6, -4, 1, 3, -5], [12, 16, 3, 12, 7, 15, -1, 5, 8, -15, -3, -3, 1, -14, -10, -3], [-6, 10, 0, 3, -5, 1, 3, 0, 4, 6, 5, -9, -5, 9, 5, 9]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_350_h_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_350_2_h_c();".
// 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_350_2_h_c(:prec:=16)
    chi := MakeCharacter_350_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_350_2_h_c();".
// 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_350_2_h_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_350_h();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![25, 125, 2400, -3025, 3385, -795, 1325, 660, 1496, 686, 387, -22, 25, 12, 7, -1, 1]>],Snew);
    return Vf;
end function;