
// Make newform 375.2.i.c in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_375_i();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_375_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_375_2_i_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_375_2_i_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 := [1, 0, 130, 0, 761, 0, 1440, 0, 1286, 0, 610, 0, 156, 0, 20, 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], [-2, -34, -565, 270, -3223, 2332, -5277, 4097, -3731, 2975, -1257, 1016, -194, 158, -11, 9], [2, -64, -339, 444, -1847, 3837, -2948, 6764, -2058, 4929, -689, 1688, -106, 263, -6, 15], [-2, 34, -565, -270, -3223, -2332, -5277, -4097, -3731, -2975, -1257, -1016, -194, -158, -11, -9], [2, 64, -339, -444, -1847, -3837, -2948, -6764, -2058, -4929, -689, -1688, -106, -263, -6, -15], [-2, -2, -1247, 0, -7066, 0, -11513, 0, -8128, 0, -2739, 0, -423, 0, -24, 0], [-2, 2, -1247, 0, -7066, 0, -11513, 0, -8128, 0, -2739, 0, -423, 0, -24, 0], [-12, 47, -1716, 595, -9462, 2400, -15360, 3583, -10838, 2444, -3652, 810, -564, 124, -32, 7], [-12, -47, -1716, -595, -9462, -2400, -15360, -3583, -10838, -2444, -3652, -810, -564, -124, -32, -7], [-17, -156, -2485, -330, -13514, 1920, -21803, 4594, -15324, 3668, -5150, 1308, -794, 208, -45, 12], [23, -19, 2487, -675, 13514, -3225, 21803, -5013, 15324, -3465, 5150, -1154, 794, -177, 45, -10], [-17, 88, -2485, 870, -13514, 2744, -21803, 3600, -15324, 2282, -5150, 724, -794, 108, -45, 6], [-8, 7, -1620, -1041, -8782, -6237, -14123, -10347, -9905, -7373, -3324, -2498, -512, -387, -29, -22], [23, 19, 2487, 675, 13514, 3225, 21803, 5013, 15324, 3465, 5150, 1154, 794, 177, 45, 10], [10, 40, 1716, 1785, 9462, 8977, 15360, 14211, 10838, 9921, 3652, 3325, 564, 512, 32, 29], [-8, -25, -904, -2487, -4923, -13514, -8032, -21803, -5701, -15324, -1930, -5150, -299, -794, -17, -45]];
    Rf_basisdens := [1, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4];
    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_375_i();"
function MakeCharacter_375_i()
    N := 375;
    order := 10;
    char_gens := [251, 127];
    v := [10, 7];
    // 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_375_i_Hecke();"
function MakeCharacter_375_i_Hecke(Kf)
    N := 375;
    order := 10;
    char_gens := [251, 127];
    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, -1, 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, 1, 0, -1, 1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 1], [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, 0, 0], [0, -1, 0, 1, 0, 0, 0, -1, 1, -1, 1, 1, 0, -1, 0, 0], [-2, 1, 0, -1, 0, 1, -1, -1, 0, 1, -4, 1, -1, 3, 0, 1], [-2, -1, -1, 3, -1, 1, -1, -1, -1, 2, 0, 1, 0, 1, 0, 0], [1, 0, 0, 0, -2, -2, 1, 1, 0, -2, 2, -2, 2, 0, -2, -1], [-4, 0, -1, 1, -2, 2, 0, -4, 0, -1, 1, 2, -1, 2, -2, 0], [0, 2, -1, -4, 2, 2, -1, -1, -1, -1, -2, -1, -2, 3, 0, 1], [0, -1, -2, 1, 1, 1, 1, 0, -2, 0, 1, -1, 0, 2, 2, 1], [2, 0, 1, -1, 0, 0, -2, 2, -1, -1, 1, 0, -1, 0, 0, -2], [-5, 0, -1, -1, -1, 3, 0, -4, -3, 1, 0, 4, -2, 3, -2, -1], [0, 1, 0, 1, 0, 1, 1, 1, -2, 1, 2, -2, 1, -1, 2, -1], [-4, -2, 2, 2, -4, -2, 4, -7, -1, -2, 0, 2, 2, 0, -4, 2], [2, 1, -2, -3, 1, -1, -3, 4, -2, 0, -1, -3, 2, -2, 2, -3], [-2, 1, 0, 1, 1, 1, 1, -1, -4, 2, 5, 0, 1, 1, 0, 0], [2, 2, 1, 0, -1, 1, -1, -1, 1, -2, 3, -1, 2, -3, -2, 0], [3, 6, -2, -4, 2, 2, 0, 3, 1, 0, -6, 0, 2, 4, 3, -2], [4, -6, 2, 10, -4, -4, 4, 1, 3, 0, 8, 2, 0, -2, 1, -4], [-2, -4, -1, 10, -2, -1, 2, 1, -1, 1, 6, 1, 0, -5, 0, -1], [9, 0, 4, -4, 0, 0, -4, 3, 2, -4, 0, -4, 0, 0, 2, -4], [3, -1, 1, -1, 2, 0, -1, 4, 1, -2, -1, -3, 3, -2, 0, 3], [-2, -1, 2, 1, 2, 2, -1, -5, 0, 1, -2, 3, -1, -1, 2, 1], [2, 1, 0, -5, 0, 0, -1, 3, 2, 1, -4, 0, -1, -1, 2, 1], [-2, 0, -2, 0, 2, 2, 0, 3, 4, 4, -4, 2, 0, 2, 6, 2], [-4, 0, 0, 4, 0, 1, 1, -8, -8, 2, 2, 2, 0, 2, 0, 0], [5, -1, 3, 2, -3, -1, 2, 0, 7, -4, 3, -1, 0, -2, 0, -1], [0, -6, 1, 6, 1, -4, 2, -1, 1, 0, 5, 0, -2, -5, -4, -2], [5, 1, 0, 2, 0, 0, 1, -1, 6, 1, 4, 1, 0, -2, 5, 1], [0, 5, 1, 1, 1, -1, 2, 3, 5, -3, -3, -3, 0, -1, 2, 2], [8, 3, 3, -2, 2, -2, -6, 4, 5, -6, -2, -1, -3, -3, -2, -4], [-3, -1, 0, -1, 1, -1, 6, -7, -2, 2, 1, -2, 2, -2, -2, 6], [12, 2, 3, -2, 3, -1, -1, 9, 1, -2, -3, -3, -2, -1, 4, 2], [-2, -2, 4, 6, 2, -2, 2, -4, -2, -2, 2, 2, -2, 0, -6, -4], [-5, 4, -2, -4, -1, 1, 0, -4, -4, -3, -1, -3, -1, 1, 0, 1], [-7, -6, 0, -6, 4, 2, -2, 5, 5, 4, -6, 4, -4, 2, 0, 4], [-6, 3, -4, -3, -1, 3, 2, -4, -8, -1, 0, 1, 5, 0, -2, 5], [-8, -2, -5, -1, -5, 3, 6, -6, -6, -1, 8, 0, 2, 3, 1, 3], [-1, -7, 6, 7, -2, -2, 2, -6, 2, -1, 2, 5, 1, -3, -2, -2], [2, -9, 2, 9, -5, -5, 2, 0, 4, 2, 5, 3, 4, -6, -2, -3], [1, 0, -3, -2, -3, -5, 8, -6, -3, -2, 3, 1, 4, 1, 2, -1], [-8, -4, 1, 3, -4, 4, 4, -2, -3, -1, 5, 0, -1, 0, 6, 4], [-4, -2, 3, 0, -3, -4, -3, -4, 1, -6, 1, 5, -1, -2, -6, 1], [3, 10, -10, -10, 7, 3, 0, 8, -2, 6, -3, -6, 3, 3, 12, 3], [-14, 2, -2, 0, -2, 2, 0, -9, -10, 0, -12, 3, -1, 5, -2, 3], [20, 0, 6, -2, -1, -3, 4, 1, 1, -8, 5, -8, 7, -9, 0, -7], [3, 0, 1, -4, -7, -5, 5, -2, 3, -5, 1, -4, 4, -6, 2, 2], [-4, -6, -3, 16, -3, 3, 3, -5, 1, 3, 3, 0, 0, -10, -4, 0], [0, 3, 1, -5, 2, -3, -2, 4, 9, -2, -2, 2, -1, 2, -8, 0], [4, -7, 7, 4, -3, -5, -2, 3, 7, -2, 1, 3, 2, -6, -5, -3], [5, -6, 2, 4, -1, -1, -1, -4, 6, 3, -1, 5, -3, -1, 10, 1], [-8, -9, 2, 13, -7, 1, 2, -10, 6, -2, 11, 3, 0, 0, 4, 5], [-6, 5, -2, -2, 2, -2, 5, 1, -3, 7, -4, 3, -4, 16, -3, 1], [28, -8, 5, -4, 1, -9, -5, 19, 19, -2, -3, -2, 4, -11, 0, -4], [1, 0, 0, 0, -5, 2, 3, -6, 8, 1, 3, -1, 5, -3, 12, 5], [16, -4, 4, 2, 4, -3, -2, 17, 8, -2, 2, -2, -1, -3, 6, -1], [8, -10, 2, 6, 0, 0, -6, 14, 8, 0, 4, -2, 0, -6, 8, -6], [1, -1, -3, -1, 0, 0, 3, -8, -3, 0, 1, -5, 5, 0, 4, 5], [3, 8, 1, -4, 3, 3, 1, -2, -8, 1, -3, -4, 2, 0, 9, 4], [0, 2, -3, 2, 0, 0, -6, 0, 5, -3, -4, 6, -3, 4, -6, -6], [6, -4, 1, -5, -2, -2, 4, 8, 3, 1, -7, 2, -1, 8, -2, -4], [6, 6, 4, -6, 0, -6, 2, 10, 2, 4, 2, -4, -4, -2, 4, -4], [2, -1, 8, 1, -3, -3, -2, 0, 4, -1, 4, 1, -5, -4, -6, -5], [14, 6, 2, -12, 4, -13, 4, 10, 4, -4, -4, -11, 2, -10, 2, -8], [-3, -4, 0, 4, 0, 0, -2, -6, -3, 2, 2, 2, 0, 0, -4, -2], [-13, 5, -7, -9, 5, 5, -4, 5, -7, 7, -9, 0, -7, 12, -4, 4], [-3, -10, -3, 10, -2, 2, -1, -5, -3, 1, 2, 2, 1, -3, -4, -1], [11, 11, 0, -18, 0, -2, -1, 5, -4, 1, -12, -5, 2, -2, 4, 3], [10, 0, 7, 6, -8, -1, 6, -4, -1, -9, 8, -8, 7, -8, -4, -8], [24, -9, 4, 9, -8, -12, 0, 14, 14, -3, 17, -3, 12, -7, 0, -12], [7, 3, -5, -1, -4, 4, -1, -1, -9, 0, 2, -5, 9, -2, -6, -5], [-2, -17, 7, 13, -7, -3, 2, -5, -1, -5, 7, 1, 4, -19, -6, -2], [-2, 0, 1, -11, -6, -6, 4, -1, -3, -7, -5, -6, 7, -2, -9, -4], [-6, -7, 2, -6, -1, -3, 1, 9, -2, 1, 1, 1, 3, 5, 4, 0], [-1, 9, 1, -17, 4, 8, -9, -4, 1, 4, -13, 1, -9, 16, 2, -1], [-12, 3, -3, -15, 1, 1, -5, -6, -5, 1, -13, -2, -1, 14, -2, 5], [18, 9, 1, -9, 10, -5, -2, 20, 3, 2, -14, -4, -3, 4, 2, -8], [1, 5, -9, 2, -9, 11, 2, -7, -3, 2, 7, 1, 10, 6, -4, 1], [-13, 0, -8, 12, -11, 4, 7, -6, -6, 3, 7, 3, 3, 1, 0, -3], [-13, 0, -3, 3, 3, 3, 0, -6, -7, 3, 6, 6, 0, -9, 6, 9], [-1, 7, -2, 1, 0, 0, 2, -4, -16, 3, -8, -1, 3, 3, -2, 2], [8, 3, 1, -13, 4, 0, -1, 5, 1, -4, -7, 1, -1, 10, 3, -1], [20, 6, 16, -4, 7, -7, -4, 13, 20, -7, -9, -9, -7, -5, 2, -4], [5, 0, -5, 10, 2, 2, -1, 10, 3, 7, 8, 2, -7, 1, 7, 1], [-8, 0, -4, 4, 0, 2, -2, -6, -5, 2, 4, -2, 2, 4, -1, 2], [-6, -11, 7, 11, -3, -5, 1, -11, -1, -5, 9, 5, -4, -9, -20, -4], [-16, -6, -4, 10, 2, 7, 1, 0, 0, 14, -2, 14, -6, 10, 0, 6], [-1, -4, 0, 4, -1, -5, 6, 4, -6, -2, 9, 2, 1, -9, 0, 1], [-6, -10, 5, -2, -7, -3, 2, -8, -3, -2, 7, 2, -3, -14, -8, 5], [-2, -8, -6, 2, -6, -2, 6, -10, -12, 0, 14, -4, 8, -2, -1, 2], [1, 8, -2, -14, 2, 2, -3, -9, -2, -4, -8, -6, 4, -8, 8, 3], [-8, 9, 7, -13, 9, 6, -13, 1, 7, -3, -13, 6, -12, 9, -18, -6], [-5, -1, 7, 9, 7, 3, -3, 9, 18, -4, -2, 5, -10, 3, 0, -2], [-7, 10, -12, -8, 12, 13, 0, 2, -10, 12, -6, -3, 1, 3, 10, 10], [-2, 10, -6, -2, -10, 0, 4, 2, 2, -10, 8, -10, 4, 0, 0, -4], [10, 10, 2, -2, 10, -4, -2, 11, 2, 6, -20, -1, -3, 9, 0, -1], [-6, -14, 5, 14, -5, -5, 7, -11, 5, 2, 7, 11, 0, -9, -12, 6], [-9, -20, -6, 14, -11, -6, 12, -8, -6, 5, 13, -1, 7, 5, 2, 1], [-12, 3, -8, 2, 9, 3, 3, -7, -14, 3, -9, 5, -5, -3, 6, 12], [-7, 18, -10, -20, 10, 10, -2, 2, -9, 8, -20, -2, 0, 26, 16, 8], [-5, 4, 8, 2, -2, -6, 2, -5, 14, -8, -10, 6, -12, 10, -1, 6], [1, -2, 1, 2, 0, 8, -9, -11, 13, -1, 2, 1, -1, -2, 0, -1], [2, 9, 0, -7, 0, 4, -5, 12, 6, 5, -14, -4, -4, 4, 10, 8], [-12, -7, -7, 15, -5, 5, 5, -4, -7, 5, -3, 2, 5, 0, 6, 5], [5, -21, 11, 24, -7, -5, -6, 4, 11, 2, 9, 3, 2, -16, -4, -3], [5, -10, 3, 10, -5, -9, 1, 0, 12, 1, 5, 6, 4, -8, -9, -4], [16, 1, 10, -1, 10, -9, -1, -1, 2, -9, -2, 0, -1, -9, -10, -9], [-3, 2, -2, -2, -1, 6, -3, -4, -2, -5, -9, 5, 3, 9, 0, 3], [-10, -7, -2, 15, -10, -6, 2, -12, -12, 3, 10, 3, 8, -6, 0, -8], [28, 10, 10, -10, 8, -8, -6, 17, 17, -12, -14, -12, 2, -18, 0, -2], [-3, -4, 0, 4, 0, -10, 10, -2, -4, 0, 10, 0, 0, -10, -6, 0], [-3, 3, 7, 3, 7, -3, -11, -11, 2, 4, -14, -1, -4, -3, -10, -2], [-1, -9, 0, 7, 1, 1, 0, 7, 4, 10, -3, 10, -10, 16, 2, 0], [-4, 16, 6, -16, 6, -6, 8, -5, 10, -8, -6, 2, -8, 6, -3, 8], [-32, -4, -21, 5, -8, 8, 12, -19, -16, 9, 7, 12, 9, 8, 1, 12], [23, 2, 3, -2, -3, -3, -7, 0, 23, -10, 5, 0, 0, -7, 18, -3], [7, 11, -6, -11, 4, 10, -8, 12, 2, 3, -1, -3, 2, 1, 18, 2], [13, 1, 0, -1, 6, -1, -1, 9, 8, 5, 12, 1, -2, 2, -2, 1], [-11, -5, 5, 7, -2, 2, 7, -13, -11, 0, 22, 3, -1, -2, -6, 3], [4, -5, -5, -5, 5, 1, -4, 7, 1, 1, -11, -3, -4, 1, -2, -2], [-22, -4, 2, -6, 0, 10, -6, -16, -12, 6, -8, 12, -8, 0, -14, 6], [8, 27, 1, -22, 16, 0, -6, 0, 2, -6, -16, -1, -5, 5, -1, 10], [-10, 19, -3, -13, 8, 2, -8, 12, -5, -8, -8, 1, -11, 8, 2, 0], [9, 5, 2, 7, -7, -7, 12, 5, -4, -14, 19, -12, 14, -18, -12, -12], [-1, 1, 6, 12, -6, -2, 1, 3, -8, -5, 24, -7, 4, -20, -2, -9], [-4, 16, 6, 4, 10, 6, 2, -6, -6, -2, -8, -2, -4, 0, 0, 4], [3, 16, 4, -18, -4, -4, -10, 2, 1, -14, -10, -2, 0, 20, -12, -6], [-16, 17, -8, -11, -10, 10, 3, -11, -16, 5, 4, 3, 5, 15, 2, 3], [25, 11, 7, 2, -10, 0, -6, -5, 6, -6, 10, -7, 1, 1, 1, -16], [11, 6, 3, 4, 0, 0, -3, 3, 4, -9, 10, -6, 9, -5, -4, 3], [11, -7, 5, 14, -10, -12, 10, 1, 3, -10, 14, -7, 5, -5, -2, 0], [18, -4, 1, 4, 9, -4, -6, 15, 21, -4, 9, 4, -10, -9, 16, -10], [15, -7, 6, 9, -9, -9, 6, 2, 2, -7, 28, -7, 15, -2, 0, -15], [8, 0, -14, 0, 14, 6, -6, 15, 1, 6, -6, -6, 0, 6, 16, 0], [30, -3, 6, 9, 7, -21, 6, 24, 10, -6, 13, -15, 0, -10, 4, -13], [15, 8, -3, 0, 6, -6, -9, 18, -3, 3, -14, 0, 3, 1, 3, -9], [30, -2, 17, -12, 6, -5, -7, 11, 15, -7, -6, -12, 10, -11, 2, -1], [-11, -6, -4, 4, -4, 2, 0, -2, 3, 4, 6, -6, 2, 2, 5, 8], [1, 4, 8, 10, -7, 2, 10, -7, -4, -5, -1, -6, 8, -2, 2, -6], [36, 26, 0, -16, 0, 2, -8, 14, 20, -8, 2, -16, 2, 14, 10, -14], [-12, 12, -15, -4, 3, 11, -11, -1, 7, 11, -13, -4, 4, 24, 22, 8], [-6, 1, 8, 3, -9, -11, 3, -1, 8, -2, 3, 10, 1, -13, -16, -10], [14, -12, 16, 2, -8, -10, -2, 20, 19, -2, 8, -6, 14, -10, -3, -10], [0, -12, 6, 34, -19, -8, -1, 13, 12, -1, 19, 2, 5, -21, -6, -20], [5, -27, 13, 44, -5, -7, -6, -2, 13, -2, 19, 13, -6, -24, -10, -13], [2, -16, 4, 14, -3, -9, 5, -3, 6, -5, 31, -5, 1, -21, 2, -2], [31, 5, 9, -3, 6, -12, -3, 23, 17, -6, 19, -13, 1, -8, 6, -13], [21, -21, 6, 21, 7, -13, 0, 28, 14, 0, 3, 0, -13, -3, 16, -13], [-2, 9, -8, -6, 8, 8, 1, 3, -5, 9, -4, -3, 0, 6, 9, 5], [5, 3, -2, -9, -2, 6, 3, -5, -18, -1, 0, 7, -4, 6, -2, -1], [13, 14, 1, -2, -1, 1, -5, -6, 9, -5, 1, -2, -4, 2, -8, -6], [28, -14, 12, -2, 14, -14, -6, 20, 18, -4, 2, -8, -4, -18, 0, -6], [5, -16, 8, 27, 6, 6, 1, 6, 10, 14, 5, 22, -14, 5, 5, -1], [-5, 3, -14, 13, -14, -5, 13, -8, -12, 1, 10, -3, 19, -5, 2, -2], [-10, 14, -6, -14, 5, 5, -4, -12, -8, 6, -19, -6, 1, 19, -18, 1], [-6, -5, 15, 21, 4, 4, 0, -6, 5, 0, -16, 19, -15, 22, -10, -4], [10, 2, -2, 0, 2, -2, 0, 5, 7, 10, -4, -8, 10, -4, 3, 0], [3, 7, -7, 3, 14, 14, -7, 15, -7, 0, -2, -11, -3, 0, 10, 11], [35, -9, 10, -24, 10, -20, -9, 39, 34, -1, 0, -13, 10, -20, 11, -7], [-7, 6, 1, -6, -1, 3, 9, -12, 1, 8, -15, 8, 4, 27, -12, 11], [-37, 10, -13, 2, 1, 15, 2, -25, -21, 16, -27, 22, -7, 32, -12, 22], [-14, -28, 0, 0, 4, 0, 0, -6, 0, 4, 4, 10, -10, -18, -2, 10], [-9, 18, -11, -16, 14, 14, -13, 4, -7, 7, -12, -4, -7, 9, 5, 13], [-7, 21, -17, -3, 0, 2, 15, -6, -17, 2, 6, 0, -2, 1, 12, 0], [-6, -3, 2, -5, -6, -16, -3, 13, 6, -3, 6, 14, -1, 9, -4, -9], [-37, -1, -8, 11, 1, 1, -4, -1, -16, 10, 9, 2, -10, 16, -18, 4], [11, -6, 2, 6, -2, 10, 6, -10, 9, 4, 0, -2, 12, -10, 13, 8], [15, -14, 14, 0, 4, -14, 0, 11, 16, -10, -2, -6, -8, -20, 3, -6]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_375_i_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_375_2_i_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_375_2_i_c(:prec:=16)
    chi := MakeCharacter_375_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_375_2_i_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_375_2_i_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_375_i();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, 10, 85, -360, 226, 410, -365, -30, 381, -20, -95, 50, 26, -10, -5, 0, 1]>],Snew);
    return Vf;
end function;