
// Make newform 370.2.m.c in Magma, downloaded from the LMFDB on 20 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_370_m();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_370_m_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_370_2_m_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_370_2_m_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 := [256, 0, 25392, 0, 58656, 0, 48248, 0, 19684, 0, 4431, 0, 559, 0, 37, 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], [-128, 2272, 0, 8872, 0, 12700, 0, 7734, 0, 2361, 0, 381, 0, 31, 0, 1], [-384, -736, -1232, -8168, 3304, -12636, 5252, -7734, 2538, -2361, 546, -381, 54, -31, 2, -1], [384, -736, 1232, -8168, -3304, -12636, -5252, -7734, -2538, -2361, -546, -381, -54, -31, -2, -1], [0, 5632, 10640, 9576, 24280, 2188, 20652, -2018, 8286, -1237, 1670, -273, 162, -27, 6, -1], [0, 5632, -10640, 9576, -24280, 2188, -20652, -2018, -8286, -1237, -1670, -273, -162, -27, -6, -1], [768, 6528, 16240, 15864, 36184, 12708, 27876, 5682, 10102, 1637, 1870, 289, 170, 27, 6, 1], [768, -6528, 16240, -15864, 36184, -12708, 27876, -5682, 10102, -1637, 1870, -289, 170, -27, 6, -1], [-384, -1952, 15536, -9800, 36120, -13084, 27876, -7766, 10102, -2361, 1870, -381, 170, -31, 6, -1], [-384, 1952, 15536, 9800, 36120, 13084, 27876, 7766, 10102, 2361, 1870, 381, 170, 31, 6, 1], [128, 64, 11560, 0, 24892, 0, 17774, 0, 5975, 0, 1035, 0, 89, 0, 3, 0], [-128, 64, -11560, 0, -24892, 0, -17774, 0, -5975, 0, -1035, 0, -89, 0, -3, 0], [256, 1664, 16112, 7016, 36184, 12588, 27876, 10358, 10102, 4143, 1870, 835, 170, 81, 6, 3], [-128, 0, -8056, 0, -18092, 0, -13938, 0, -5051, 0, -935, 0, -85, 0, -3, 0], [320, 496, 10336, 440, 21684, 112, 15872, 8, 5513, 0, 985, 0, 87, 0, 3, 0], [-320, 496, -10336, 440, -21684, 112, -15872, 8, -5513, 0, -985, 0, -87, 0, -3, 0]];
    Rf_basisdens := [1, 256, 256, 256, 256, 256, 256, 256, 256, 256, 128, 128, 256, 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_370_m();"
function MakeCharacter_370_m()
    N := 370;
    order := 6;
    char_gens := [297, 261];
    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_370_m_Hecke();"
function MakeCharacter_370_m_Hecke(Kf)
    N := 370;
    order := 6;
    char_gens := [297, 261];
    char_values := [[-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]];
    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, -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, -1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0], [1, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, -1, 0, 0, -1, -1, 0, 0, 0, 0, -1, 1], [0, -2, 0, 0, 0, -1, 1, 0, 0, -1, 0, 0, -1, -1, -1, 0], [-1, -1, 0, 0, -1, 0, 0, -1, 1, 0, 0, 0, 1, 0, 0, -1], [-1, -1, 1, 0, 0, 1, -1, 0, 0, -1, -1, 0, 1, -1, -1, 0], [2, 0, 0, 0, 1, -1, 1, 1, 1, 1, 1, -1, 0, 1, -1, 1], [-2, -4, 0, 0, -1, -1, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0], [1, 2, 1, 1, 0, 0, 0, 0, 0, 0, -1, -1, 0, 0, 0, 0], [0, 0, -1, 0, 1, -1, 0, 2, 0, 0, 1, 1, 0, 2, 0, 0], [-1, -4, 1, 2, 2, 1, 3, 0, 2, -1, -1, -2, -1, -1, -3, 0], [2, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, -2, 0, 0], [-2, -4, 1, 1, 1, 1, 1, -1, 2, -2, -2, -2, 2, 1, 0, 0], [-1, -1, 0, 2, 1, -1, 0, -1, 0, -1, 0, -1, -1, -2, 0, 1], [2, 0, 0, -2, 1, -2, -1, -1, -1, -2, 0, 0, 0, 0, 1, 0], [4, 2, 0, 0, 1, -2, 2, 2, -1, 1, 0, 0, 0, 0, 0, 2], [1, 2, -3, 0, 1, -1, 3, 2, -1, 0, 0, 0, -1, 1, 1, 2], [-1, -2, 1, 2, 1, 1, 1, 1, 1, 0, -1, -2, -2, -2, -2, -1], [1, 2, 1, 1, 0, 0, -3, 3, -2, 2, 1, 1, 2, 1, 0, 0], [1, 0, 1, 0, 1, -1, 1, 2, -1, 0, 2, 0, -1, 1, -1, 2], [1, 0, 0, -1, -1, -1, 1, 1, -2, -1, 0, 3, 0, 0, -1, 0], [0, 1, 0, -1, -3, 3, -1, 1, 0, 3, 0, 2, 0, 0, 1, -2], [-4, 0, 1, -1, -1, 1, 2, 2, 1, 1, 1, -1, 0, 2, -1, 1], [1, 0, 0, 0, 1, -1, 2, 2, -3, -3, 1, -1, 0, 0, -1, 1], [-6, 0, 0, 0, -1, 1, -1, -1, -1, -1, -1, 1, 0, -1, 3, -3], [8, 4, 0, 0, 2, 0, 0, 0, -2, -2, -1, 0, -2, 2, 0, 0], [0, 2, 0, -2, -2, 1, -4, -4, -1, 1, 0, 4, -2, -4, 4, 0], [4, 4, 0, 0, -2, 0, 2, -2, 2, 0, 0, 0, 0, 0, 2, 0], [-1, 1, 0, 0, 0, 1, -1, -2, 1, 1, 0, 2, -3, -6, 1, 1], [-1, 0, 0, 1, 1, -1, 1, 1, 0, -1, 0, -1, 2, 4, -1, 0], [1, 2, -4, -4, -5, -5, -1, 1, -1, 1, 4, 4, 2, 1, -1, -1], [-3, -4, 2, 1, -2, -1, 0, 2, 1, 1, 2, 1, 0, 0, 0, 2], [-6, 0, -2, 2, -1, 1, 2, 2, 2, 2, 2, -2, 0, 0, -2, 2], [-1, 1, 1, 2, 2, 4, -2, 1, 2, 2, 0, 0, 3, 3, 1, -1], [1, -2, 0, 1, 0, 2, -2, -2, 2, 2, 0, -4, 0, 0, 2, 0], [0, 0, 0, 0, 2, 2, -2, 2, 0, -2, 2, 1, 2, 0, -2, 0], [2, 4, -2, -4, 2, 2, 2, 0, 2, 0, 0, 0, 2, 2, -2, 0], [-1, -1, 0, 2, 2, 0, 4, 2, 2, 0, 0, -1, 2, 4, -4, 2], [-3, -6, 5, 5, -1, -1, -1, 1, -1, 1, -3, -3, -2, -1, -3, -3], [0, 1, 0, 0, 2, -1, 3, 3, 2, -3, -1, -2, 4, 4, -6, -3], [-3, -6, -1, -1, -4, -4, -1, 1, -2, 2, -2, -2, 2, 1, -2, -2], [1, 0, 0, 0, -3, 3, -3, -3, 1, 1, 2, -2, 0, 1, 5, -5], [1, 1, 0, -2, -1, 0, -2, -5, -1, 0, 0, 1, -3, -6, 2, 3], [5, 10, -3, -3, -2, -2, 1, -1, -2, 2, -4, -4, -2, -1, 0, 0], [-3, 0, 2, -2, 1, -1, -1, -1, 0, 0, -2, 2, 0, 1, 0, 0], [-2, -4, 4, 4, 1, 1, 5, -5, 5, -5, 1, 1, -6, -3, 1, 1], [0, -4, 0, 0, -2, 0, -2, 3, -2, 2, 0, 0, 1, 1, -1, -3], [2, 1, -2, -4, 0, -1, 3, -3, 0, -1, -1, -2, -2, -2, 0, 3], [-3, -6, 0, 0, -1, -1, 1, -1, 3, -3, -4, -4, 2, 1, -3, -3], [-4, 2, 0, 2, 4, -4, 1, 0, 0, -4, 0, 1, -1, -2, -1, 1], [3, 1, 1, 0, -6, 5, -1, -2, 6, 1, 3, 0, 1, -1, 1, -2], [3, 6, 0, 0, 1, 1, 1, -1, -4, 4, -2, -2, -6, -3, 4, 4], [1, 3, -4, -2, -1, 2, 2, 1, 3, -2, 0, 0, 3, 0, 2, 3], [-3, -4, 5, 0, 5, -2, -2, -1, -5, -3, 0, 0, -1, 1, -1, -1], [-1, 0, 0, 0, 2, -2, -2, -2, -1, -1, -1, 1, 0, 8, 3, -3], [2, 3, -2, -1, -2, 2, 0, 2, 4, -2, 2, 1, 6, 0, 0, 2], [-1, 7, 1, 2, -1, 0, 0, 4, -1, 1, -1, -2, -2, -2, -4, -4], [8, -5, 0, -3, -1, -3, -2, -1, -4, -3, 0, 8, 1, 2, 2, -1], [2, -2, -2, -4, -2, 0, -4, 2, -2, 2, 3, 6, 0, 0, 2, -2], [10, 4, 2, 0, -3, 1, 1, 0, 3, 2, 2, 0, -1, 1, 1, 0], [2, 4, 0, 0, -6, -6, 0, 0, -2, 2, 1, 1, 4, 2, 2, 2], [-2, 5, 0, -3, -4, 4, -5, -2, 0, 4, 0, -4, 3, 6, 5, -3], [3, 2, 2, 1, -3, -6, -3, 3, -3, 6, 6, 3, 0, 0, -3, 0], [1, 0, 0, -1, -4, 2, -4, 0, -2, 2, 0, 4, 2, 4, 4, -4], [2, -2, 0, 0, 1, -2, -1, 3, -1, -2, 0, 0, 0, 0, 1, -4], [-1, 0, -1, 0, 2, 1, -1, 1, -2, -3, 1, 0, 0, 0, -2, 1], [0, 0, -2, 2, 6, -6, 5, 5, -4, -4, -1, 1, 0, 7, -2, 2], [8, 12, -8, -4, 0, -3, 2, 2, -3, 3, 4, 2, -4, 0, 2, 4], [13, 13, 0, 0, 5, 2, 0, -3, -3, -2, -8, -4, -1, 0, 0, -3], [13, 0, -2, 2, 4, -4, 1, 1, -1, -1, 1, -1, 0, 5, 1, -1], [-3, -3, 3, 0, 2, 1, -5, -4, -2, -3, -9, 0, -1, 1, -1, -4], [-1, 0, -1, 0, -4, 2, 0, 2, 4, 2, 8, 0, 2, -2, -2, 2], [13, 13, 0, 0, 1, -1, 4, 1, -2, 1, -4, -2, -5, 0, 4, 5], [-2, -4, 2, 4, -2, -1, -3, 0, -2, 1, -3, -6, 1, 1, 3, 0], [-2, -2, 2, 0, -5, 0, 2, 2, 5, 5, 2, 0, -2, 2, 0, 2], [0, 0, -2, -2, -2, -2, -3, 3, -1, 1, 2, 2, 2, 1, -3, -3], [7, 14, -2, -2, -3, -3, -2, 2, -4, 4, -5, -5, 4, 2, -2, -2], [1, 0, 0, -1, 1, 4, 2, 1, 5, 4, 0, -3, 1, 2, -2, 1], [1, 2, -2, -1, 0, -3, 0, 0, -3, 3, 2, 1, 0, 0, 0, 0], [-7, -14, 8, 8, 1, 1, 0, 0, 3, -3, 5, 5, 0, 0, -3, -3], [-14, -9, 4, 0, -3, 7, -1, -6, 3, -4, -7, 0, 1, -1, 5, -6], [0, 0, -1, -1, -3, -3, 4, -4, -1, 1, 3, 3, -4, -2, 5, 5], [1, 3, -5, 0, 0, 2, -4, -4, 0, -2, 3, 0, 0, 0, 0, -4], [-2, -4, -2, -2, -2, -2, 3, -3, 2, -2, 0, 0, -2, -1, -2, -2], [-9, -4, -1, 0, 1, 1, 1, -3, -1, -2, 2, 0, 0, 0, 4, -3], [-1, -4, 1, 2, -2, -7, 3, -1, -2, -5, 5, 10, -4, -4, -2, 1], [2, -4, 0, 2, 6, 0, 2, 0, 6, 0, 0, 2, 0, 0, -2, 2], [-3, -8, 3, 6, 7, 7, 1, -1, 7, 0, -1, -2, -2, -2, 0, 1], [-12, 0, -1, 1, -8, 8, -4, -4, 5, 5, 0, 0, 0, 6, 7, -7], [-3, 7, 0, -4, -3, 1, 0, -3, -2, 1, 0, 1, -1, -2, 0, 3], [0, 0, 1, -1, -4, 4, -6, -6, 1, 1, -3, 3, 0, 4, 7, -7], [-4, 0, 0, 0, 5, -5, 8, 8, -1, -1, 2, -2, 0, 0, -9, 9], [-8, -6, 4, 0, 4, -4, -4, 0, -4, 0, -2, 0, -2, 2, -4, 0], [-11, -11, 0, 0, -3, -9, 1, -1, -6, 9, -2, -1, -2, 0, 1, 0], [-7, -13, 12, 6, 3, 8, 3, -5, 5, -8, -10, -5, -2, 0, 3, -2], [0, 7, 0, 0, 3, -2, 0, -5, 3, -5, 1, 2, -7, -7, 5, 5], [-3, 4, 3, 6, 5, 7, 5, -2, 5, 2, -3, -6, -5, -5, -3, 2], [-2, -4, 6, 6, -1, -1, -6, 6, -4, 4, 4, 4, 0, 0, -2, -2], [18, 0, -4, 4, 8, -8, 9, 9, -4, -4, 4, -4, 0, 1, -6, 6], [5, 4, 2, 1, 6, 2, 2, 2, -4, -2, 4, 2, -2, 0, 2, 4], [-2, 0, -4, 4, -4, 4, 5, 5, 6, 6, 7, -7, 0, -5, -4, 4], [5, 10, 4, 4, -6, -6, -2, 2, -6, 6, 0, 0, 0, 0, 0, 0], [5, 3, 4, 2, 0, -1, -5, 0, -1, 1, -12, -6, 5, 0, -5, -5], [16, 15, 2, 1, 4, 2, -3, 4, -2, -2, 2, 1, 5, 0, -3, 1], [-3, -12, 3, 6, 5, -1, 5, 0, 5, -6, -5, -10, 1, 1, -5, 0], [9, 18, -4, -4, 5, 5, -3, 3, -1, 1, -4, -4, 14, 7, 3, 3], [3, -5, -3, -6, 4, 2, -2, 0, 4, -2, -3, -6, 2, 2, 2, 0], [2, 3, -2, -1, -1, -5, -4, 1, -4, 5, 4, 2, -3, 0, -4, -3], [5, 5, 0, 0, -3, -5, 3, -5, -2, 5, -2, -1, -4, 0, 3, -2], [-4, 5, 0, -1, -7, -5, 0, -3, -12, -5, 0, 5, -3, -6, 0, 3], [0, 4, 0, -4, -2, 4, -6, 0, 2, 4, 0, -2, 0, 0, 6, -6], [-10, 0, -1, 1, -6, 6, -2, -2, 5, 5, -3, 3, 0, -4, 1, -1], [-5, -1, 0, 6, 3, -1, -1, 1, 2, -1, 0, -4, 0, 0, 1, -2], [-8, -7, -2, -1, 3, 3, 2, -3, 0, -3, 0, 0, -1, 0, 2, -1], [-16, 0, 2, -2, -2, 2, -1, -1, 2, 2, 4, -4, 0, 3, 0, 0], [0, 17, 0, 0, -1, -2, -2, -1, -1, -1, -2, -4, 7, 7, 3, 1], [-1, 8, 1, 2, 2, 4, 0, -2, 2, 2, 0, 0, 0, 0, 2, 2], [1, -3, -1, -2, 2, 3, 1, 2, 2, 1, -1, -2, 3, 3, -3, -2], [6, 1, 4, 0, 2, 1, -5, -5, -2, -3, -9, 0, 4, -4, 0, -5], [31, 19, -7, 0, 0, -2, 8, 9, 0, 2, -1, 0, -1, 1, -1, 9], [1, 2, 1, 1, 2, 2, -7, 7, -1, 1, -3, -3, -2, -1, -3, -3], [-2, -2, 0, 0, -4, 0, 2, 4, 4, 0, 2, 1, -4, 0, 2, 6], [-2, -1, -2, -1, 6, 8, 3, 2, 2, -8, 2, 1, 3, 0, 3, 5], [-10, 4, 0, 6, -6, 6, 0, -2, 0, 6, 0, -6, -2, -4, 0, 2], [-14, -28, 2, 2, 0, 0, 2, -2, 8, -8, 0, 0, -4, -2, -8, -8], [-2, -3, 2, 1, -4, 2, 3, -2, 6, -2, 4, 2, 3, 0, 3, 1], [5, 23, -5, -10, -4, -3, -5, -6, -4, 1, 1, 2, -3, -3, 11, 6], [13, 11, -9, 0, 2, -5, 7, 4, -2, 3, 1, 0, -3, 3, 3, 4], [4, 0, 5, -5, 1, -1, -4, -4, -10, -10, -7, 7, 0, 2, 6, -6], [-11, -8, 5, 0, -7, 4, -2, -5, 7, 3, -2, 0, -1, 1, 3, -5], [-19, 0, 1, -1, -2, 2, -3, -3, 2, 2, 2, -2, 0, 1, 6, -6], [8, 6, 4, 2, -1, -7, -4, 9, -6, 7, 8, 4, -1, 0, -4, 5], [10, 11, -2, -1, 7, -1, 3, 5, -8, 1, 8, 4, -6, 0, 3, 8], [-8, -16, 1, 1, -2, -2, -5, 5, -5, 5, 0, 0, 6, 3, -5, -5], [-4, -6, 8, 0, -1, 5, -7, -12, 1, -4, -2, 0, -1, 1, 5, -12], [4, 8, -3, -3, -6, -6, 0, 0, -7, 7, 6, 6, -8, -4, 5, 5], [3, 6, -6, -3, 2, 1, -4, 4, -1, -1, 4, 2, 12, 0, -4, 0], [5, 3, -1, 0, 0, -1, 3, 5, 0, 1, 3, 0, 4, -4, -2, 5], [-1, -9, 1, 2, 2, -5, 1, -2, 2, -7, -5, -10, -1, -1, 1, 2], [0, 10, 0, 0, 0, -4, 0, 2, 0, -4, 8, 16, 2, 2, -2, -2], [2, 3, 0, -5, 2, -6, -6, -4, -4, -6, 0, 6, 2, 4, 6, -2], [-5, -4, 5, 10, 3, -3, 7, -2, 3, -6, 4, 8, -11, -11, -5, 2], [-7, 5, 0, 2, -1, 3, -1, 1, 2, 3, 0, -5, 0, 0, 1, -2], [-5, -3, -4, -2, -1, 1, -7, 5, 2, -1, 4, 2, 10, 0, -7, -2], [29, 26, 6, 3, 4, 2, 2, 0, -2, -2, -8, -4, -8, 0, 2, 2], [3, 0, 0, 0, 13, -13, 3, 3, -7, -7, 2, -2, 0, 7, -3, 3], [15, 30, -4, -4, 1, 1, -1, 1, 0, 0, -2, -2, 6, 3, 8, 8], [4, -1, 0, -3, 9, -10, 10, 1, -1, -10, 0, 1, -3, -6, -10, 9], [-6, -12, -3, -3, 0, 0, -2, 2, 0, 0, 10, 10, -4, -2, 2, 2], [3, 7, -3, -6, -3, 1, -3, 3, -3, 4, -6, -12, 2, 2, 0, -3], [5, 6, -5, -10, -1, 3, -3, -2, -1, 4, 3, 6, 3, 3, 5, 2], [-2, -4, 1, 1, 4, 4, -10, 10, -5, 5, -3, -3, 12, 6, -9, -9], [-3, -6, 3, 6, 1, 5, -1, 0, 1, 4, 0, 0, -3, -3, 1, 0], [5, 10, -12, -12, 0, 0, 6, -6, 1, -1, -2, -2, -8, -4, 9, 9], [-4, -8, 3, 3, 2, 2, -4, 4, -5, 5, 7, 7, 4, 2, 1, 1], [4, 11, -4, -8, -8, 0, -6, -6, -8, 8, 0, 0, -10, -10, 12, 6], [-1, 0, 0, 1, -3, 0, -5, -1, -3, 0, 0, -1, 0, 0, 5, -4], [-5, -9, 8, 4, 5, 2, 5, -3, -3, -2, -18, -9, -6, 0, 5, 2], [17, 16, 2, 1, 3, -1, -1, 5, -4, 1, 0, 0, 6, 0, -1, 4], [10, -15, 0, 5, 4, -4, 9, 4, 0, -4, 0, -11, 3, 6, -9, 5], [-5, -7, 5, 10, 4, 5, 11, -4, 4, 1, -3, -6, -13, -13, -7, 4], [-31, -34, 6, 3, -6, 3, -4, -6, 9, -3, -6, -3, 8, 0, -4, -10], [16, 14, 4, 2, 8, 0, 2, 4, -8, 0, -4, -2, -2, 0, 2, 6], [-18, 13, 0, 5, 1, 4, 0, -1, 5, 4, 0, -7, 5, 10, 0, 1], [13, 26, -4, -4, -6, -6, 2, -2, -5, 5, -4, -4, -8, -4, -3, -3], [1, -2, 6, 3, 0, 1, 0, 0, 1, -1, 4, 2, -2, 0, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_370_m_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_370_2_m_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_370_2_m_c(:prec:=16)
    chi := MakeCharacter_370_m();
    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_370_2_m_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_370_2_m_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_370_m();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![256, -4800, 32304, -43200, 1776, 30096, -4132, -13308, 3208, 3318, -771, -561, 154, 51, -14, -3, 1]>],Snew);
    return Vf;
end function;