
// Make newform 1875.2.b.h in Magma, downloaded from the LMFDB on 16 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1875_b();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1875_b_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_1875_2_b_h();".
// 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_1875_2_b_h();".
// 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], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-8, 0, -245, 0, -1020, 0, -1518, 0, -1037, 0, -345, 0, -53, 0, -3, 0], [0, 45, 0, 179, 0, -483, 0, -1413, 0, -1183, 0, -430, 0, -69, 0, -4], [0, 47, 0, 595, 0, 2400, 0, 3583, 0, 2444, 0, 810, 0, 124, 0, 7], [-2, 0, -435, 0, -2527, 0, -4185, 0, -2991, 0, -1017, 0, -158, 0, -9, 0], [0, -19, 0, -675, 0, -3225, 0, -5013, 0, -3465, 0, -1154, 0, -177, 0, -10], [2, 0, 565, 0, 3223, 0, 5277, 0, 3731, 0, 1257, 0, 194, 0, 11, 0], [-13, 0, -865, 0, -4732, 0, -7680, 0, -5419, 0, -1826, 0, -282, 0, -16, 0], [0, 15, 0, -945, 0, -5557, 0, -9110, 0, -6440, 0, -2170, 0, -335, 0, -19], [0, 7, 0, -1041, 0, -6237, 0, -10347, 0, -7373, 0, -2498, 0, -387, 0, -22], [0, 40, 0, 1542, 0, 7957, 0, 12693, 0, 8884, 0, 2980, 0, 459, 0, 26], [0, 40, 0, 1785, 0, 8977, 0, 14211, 0, 9921, 0, 3325, 0, 512, 0, 29], [6, 0, 858, 0, 4731, 0, 7680, 0, 5419, 0, 1826, 0, 282, 0, 16, 0], [-23, 0, -2487, 0, -13514, 0, -21803, 0, -15324, 0, -5150, 0, -794, 0, -45, 0], [-19, 0, -2485, 0, -13514, 0, -21803, 0, -15324, 0, -5150, 0, -794, 0, -45, 0]];
    Rf_basisdens := [1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2];
    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_1875_b();"
function MakeCharacter_1875_b()
    N := 1875;
    order := 2;
    char_gens := [626, 1252];
    v := [2, 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_1875_b_Hecke();"
function MakeCharacter_1875_b_Hecke(Kf)
    N := 1875;
    order := 2;
    char_gens := [626, 1252];
    char_values := [[1, 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]];
    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, 0, 0, 0, 0, 0, 0, -1, 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, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 1, 1, 0, 1, 0, 0, -1, 1, 1, -1, 0, 0, 0], [2, 0, 1, 0, 0, 0, 0, -2, 1, 0, 0, 0, 0, -1, 0, 1], [0, 1, 0, 0, 0, 0, 1, 0, 0, 2, 2, 1, 0, 0, 0, 0], [0, 1, 0, -1, 0, 0, 0, 0, 0, -1, 1, -1, -1, 0, 0, 0], [1, 0, -2, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0], [0, -1, 0, -2, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 1, 0, -2, 1, 0, 0, 0, 0, -1, -2, 0], [-2, 0, 0, 0, 0, 1, 0, -1, 0, 0, 0, 0, 0, 1, 0, 2], [0, -1, 0, -1, -2, 0, 1, 0, 0, -3, 1, -3, 1, 0, 0, 0], [0, 0, -1, 0, 0, 1, 0, 0, -3, 0, 0, 0, 0, 0, 2, 1], [0, 2, 0, 2, -1, 0, 2, 0, 0, 4, 0, -2, 0, 0, 0, 0], [0, 0, 0, 3, -2, 0, 2, 0, 0, -2, -2, 0, -3, 0, 0, 0], [0, -1, 0, -1, -2, 0, -2, 0, 0, 7, -1, -3, -1, 0, 0, 0], [-4, 0, -2, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 1, 2, -2], [-5, 0, -2, 0, 0, 4, 0, -2, -4, 0, 0, 0, 0, 2, 2, 0], [0, 0, 0, 0, -1, 0, 4, 0, 0, -4, 4, 2, 0, 0, 0, 0], [2, 0, -1, 0, 0, -1, 0, 2, 1, 0, 0, 0, 0, 0, -4, 0], [0, 4, 0, 0, 2, 0, 4, 0, 0, 4, 4, 0, -1, 0, 0, 0], [5, 0, 2, 0, 0, 3, 0, -3, -2, 0, 0, 0, 0, 1, -1, 3], [0, 0, 0, 1, 2, 0, 0, 0, 0, 0, -1, 3, 3, 0, 0, 0], [-1, 0, 1, 0, 0, -1, 0, -2, 2, 0, 0, 0, 0, 0, -4, 0], [0, 2, 0, 0, -3, 0, 0, 0, 0, 0, 0, -2, 3, 0, 0, 0], [-2, 0, 1, 0, 0, -1, 0, -2, 1, 0, 0, 0, 0, 6, -2, -2], [0, -3, 0, -2, 2, 0, -1, 0, 0, -2, 0, 1, 4, 0, 0, 0], [0, 3, 0, 0, 2, 0, 0, 0, 0, -3, -1, 1, 2, 0, 0, 0], [-1, 0, 0, 0, 0, 1, 0, -1, 0, 0, 0, 0, 0, 7, 3, 1], [0, 0, 0, 2, 2, 0, 2, 0, 0, 4, -3, 0, -4, 0, 0, 0], [0, -1, 0, -3, -2, 0, -3, 0, 0, 1, 1, -5, 1, 0, 0, 0], [3, 0, 2, 0, 0, -2, 0, -2, 1, 0, 0, 0, 0, -6, 0, -6], [0, 3, 0, 2, 4, 0, 0, 0, 0, 3, -2, 1, 4, 0, 0, 0], [2, 0, -4, 0, 0, 2, 0, 2, -4, 0, 0, 0, 0, 2, 4, 2], [-2, 0, 0, 0, 0, 1, 0, 0, -2, 0, 0, 0, 0, 3, 2, 5], [-7, 0, -2, 0, 0, -2, 0, 6, -2, 0, 0, 0, 0, -5, 2, -4], [0, 2, 0, 1, -3, 0, 2, 0, 0, 1, 5, 0, 0, 0, 0, 0], [0, -3, 0, 3, 1, 0, -3, 0, 0, -3, -1, -1, -1, 0, 0, 0], [0, 1, 0, 0, 4, 0, 0, 0, 0, -3, 2, 6, 2, 0, 0, 0], [0, 2, 0, 4, -2, 0, 0, 0, 0, 2, -3, -1, -2, 0, 0, 0], [4, 0, 2, 0, 0, -1, 0, -2, 6, 0, 0, 0, 0, -5, 0, 4], [-6, 0, 0, 0, 0, 1, 0, -1, 4, 0, 0, 0, 0, 7, 0, -4], [5, 0, 1, 0, 0, -3, 0, 0, 8, 0, 0, 0, 0, -6, -2, -6], [0, 1, 0, -6, 2, 0, -3, 0, 0, 0, 4, 1, 0, 0, 0, 0], [0, 3, 0, 0, -2, 0, -2, 0, 0, -9, 1, 3, -4, 0, 0, 0], [15, 0, 4, 0, 0, -3, 0, 1, 10, 0, 0, 0, 0, -2, -4, 2], [-3, 0, 2, 0, 0, -7, 0, -3, 2, 0, 0, 0, 0, -4, -4, -4], [0, 3, 0, 0, -4, 0, -13, 0, 0, 6, 0, 3, -2, 0, 0, 0], [0, -4, 0, -1, -8, 0, 2, 0, 0, -4, -2, -4, 3, 0, 0, 0], [3, 0, 2, 0, 0, -3, 0, -1, -2, 0, 0, 0, 0, 0, 2, 2], [0, -5, 0, -2, -4, 0, -2, 0, 0, -1, -1, 0, 8, 0, 0, 0], [5, 0, 5, 0, 0, -4, 0, 2, 2, 0, 0, 0, 0, -11, 4, 3], [0, 0, 4, 0, 0, 5, 0, 5, -2, 0, 0, 0, 0, 0, 3, 7], [9, 0, -5, 0, 0, 4, 0, 6, 0, 0, 0, 0, 0, -5, 2, -3], [0, -3, 0, -1, 2, 0, -4, 0, 0, 3, -1, -6, -5, 0, 0, 0], [0, 1, 0, -1, 6, 0, -6, 0, 0, 5, -2, 0, 3, 0, 0, 0], [2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 6, 4, 6], [-5, 0, 0, 0, 0, 5, 0, -3, -2, 0, 0, 0, 0, 1, -1, 5], [0, 1, 0, 2, 9, 0, 3, 0, 0, -6, 4, 5, -3, 0, 0, 0], [2, 0, -6, 0, 0, 3, 0, 6, -6, 0, 0, 0, 0, 3, 2, 6], [0, 1, 0, 2, -2, 0, -9, 0, 0, 2, 4, 3, -6, 0, 0, 0], [0, -2, 0, -4, -4, 0, -16, 0, 0, 12, -8, -4, 2, 0, 0, 0], [0, -4, 0, 1, -1, 0, -10, 0, 0, 5, -7, -2, 2, 0, 0, 0], [-5, 0, -8, 0, 0, -5, 0, 0, 1, 0, 0, 0, 0, -1, -6, -11], [0, 0, 0, -2, -4, 0, -2, 0, 0, 2, 2, 0, 1, 0, 0, 0], [0, -6, 0, 1, 0, 0, -4, 0, 0, 0, -4, -6, -5, 0, 0, 0], [2, 0, -2, 0, 0, -1, 0, 5, 0, 0, 0, 0, 0, -3, 5, 1], [0, 4, 0, 3, 4, 0, 12, 0, 0, -15, 1, 6, 5, 0, 0, 0], [0, -14, 0, -6, 0, 0, -6, 0, 0, -8, -7, -5, 2, 0, 0, 0], [8, 0, 0, 0, 0, 8, 0, 0, 4, 0, 0, 0, 0, -2, -9, -1], [0, -4, 0, 1, 4, 0, -2, 0, 0, -4, -9, -4, -3, 0, 0, 0], [-3, 0, -4, 0, 0, 2, 0, -4, -1, 0, 0, 0, 0, -2, -4, -5], [0, 5, 0, -2, 4, 0, -11, 0, 0, 10, 4, -1, -3, 0, 0, 0], [0, 0, 0, 3, 4, 0, 10, 0, 0, 3, -1, 0, 7, 0, 0, 0], [-17, 0, -4, 0, 0, -1, 0, 1, -4, 0, 0, 0, 0, 1, 9, -9], [0, -5, 0, -4, 2, 0, -12, 0, 0, -3, -5, -7, 4, 0, 0, 0], [-11, 0, -8, 0, 0, 3, 0, -2, -1, 0, 0, 0, 0, 11, 0, -4], [0, -1, 0, -2, 7, 0, -7, 0, 0, 4, -10, -3, 5, 0, 0, 0], [6, 0, 7, 0, 0, 4, 0, -6, -1, 0, 0, 0, 0, -5, 4, 5], [16, 0, 0, 0, 0, 0, 0, -6, 6, 0, 0, 0, 0, -4, 3, 3], [-10, 0, 1, 0, 0, -3, 0, 4, 1, 0, 0, 0, 0, -6, -4, -2], [8, 0, 4, 0, 0, -1, 0, -7, 4, 0, 0, 0, 0, -8, 3, -1], [13, 0, 9, 0, 0, 7, 0, -2, 2, 0, 0, 0, 0, 0, -4, 4], [0, -6, 0, 1, -8, 0, 10, 0, 0, -5, -1, -4, -2, 0, 0, 0], [8, 0, 2, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, -5, 4, -2], [0, 3, 0, 5, 2, 0, 0, 0, 0, -3, -2, 2, 5, 0, 0, 0], [-4, 0, 1, 0, 0, -3, 0, -2, -3, 0, 0, 0, 0, -8, 0, -10], [0, 7, 0, 2, -6, 0, -3, 0, 0, 2, 2, 1, -2, 0, 0, 0], [7, 0, 5, 0, 0, -8, 0, 0, 6, 0, 0, 0, 0, -1, -12, 2], [0, -4, 0, 2, -1, 0, -8, 0, 0, -2, 0, 4, 3, 0, 0, 0], [0, 3, 0, -1, 0, 0, -6, 0, 0, 15, -3, -3, 9, 0, 0, 0], [-9, 0, 3, 0, 0, -6, 0, 0, 2, 0, 0, 0, 0, 9, 4, -12], [0, 1, 0, -2, 0, 0, -5, 0, 0, 14, -4, -9, -9, 0, 0, 0], [4, 0, -1, 0, 0, 0, 0, -4, -3, 0, 0, 0, 0, 3, 2, 12], [8, 0, 4, 0, 0, 6, 0, -4, -2, 0, 0, 0, 0, -6, 2, 16], [0, 1, 0, 2, 0, 0, -8, 0, 0, -1, 3, -5, -2, 0, 0, 0], [-8, 0, 0, 0, 0, 7, 0, -2, 4, 0, 0, 0, 0, 7, 0, 2], [-1, 0, -5, 0, 0, 1, 0, -8, 0, 0, 0, 0, 0, 4, -2, 7], [0, -14, 0, -5, 6, 0, -16, 0, 0, -3, -5, -2, 1, 0, 0, 0], [15, 0, 0, 0, 0, -2, 0, 8, 0, 0, 0, 0, 0, -1, -2, 8], [0, 4, 0, -2, 4, 0, -6, 0, 0, -2, 12, 12, 5, 0, 0, 0], [0, -8, 0, 1, 12, 0, 0, 0, 0, -10, 0, 1, -1, 0, 0, 0], [0, 13, 0, 8, 10, 0, 14, 0, 0, 1, 5, 9, -4, 0, 0, 0], [-5, 0, -2, 0, 0, -5, 0, 10, 3, 0, 0, 0, 0, 3, 2, -5], [-4, 0, -2, 0, 0, -3, 0, 1, -10, 0, 0, 0, 0, 3, -2, 2], [0, -1, 0, 4, -9, 0, -1, 0, 0, 0, -6, -5, -3, 0, 0, 0], [0, -8, 0, -9, -10, 0, -8, 0, 0, 0, -9, -9, 7, 0, 0, 0], [0, 1, 0, 5, 0, 0, 14, 0, 0, -3, 7, 4, -5, 0, 0, 0], [-12, 0, 2, 0, 0, 8, 0, -4, 0, 0, 0, 0, 0, 16, 0, -1], [4, 0, -6, 0, 0, -2, 0, 0, 4, 0, 0, 0, 0, -3, 6, 2], [0, 10, 0, 0, -1, 0, -6, 0, 0, 6, 0, 0, 3, 0, 0, 0], [0, 9, 0, -2, -10, 0, 7, 0, 0, 8, 4, 5, -3, 0, 0, 0], [0, -9, 0, 1, -4, 0, -2, 0, 0, -13, 0, 1, -3, 0, 0, 0], [-3, 0, -2, 0, 0, 8, 0, -4, -8, 0, 0, 0, 0, 7, -4, -8], [-12, 0, -12, 0, 0, -9, 0, 5, -4, 0, 0, 0, 0, 4, 4, -12], [5, 0, 0, 0, 0, -7, 0, -2, 7, 0, 0, 0, 0, 13, 0, -10], [0, -7, 0, -3, -1, 0, -7, 0, 0, -3, 3, 1, -6, 0, 0, 0], [0, 2, 0, 1, 6, 0, 0, 0, 0, 20, 2, -3, 3, 0, 0, 0], [0, -4, 0, -7, -2, 0, -2, 0, 0, 18, 1, -4, -1, 0, 0, 0], [-3, 0, 4, 0, 0, -4, 0, -4, 1, 0, 0, 0, 0, -2, -10, 1], [18, 0, 6, 0, 0, 4, 0, -4, 2, 0, 0, 0, 0, 2, -10, 12], [0, -15, 0, -5, -1, 0, 1, 0, 0, -21, 1, -5, -1, 0, 0, 0], [0, -9, 0, -11, 2, 0, -6, 0, 0, -9, -1, -9, 15, 0, 0, 0], [0, 19, 0, 5, 6, 0, 12, 0, 0, 9, 12, 7, -11, 0, 0, 0], [0, 0, -4, 0, 0, 1, 0, -5, -8, 0, 0, 0, 0, -9, 13, -5], [-2, 0, 2, 0, 0, -12, 0, -10, 8, 0, 0, 0, 0, -2, -2, -4], [-1, 0, 0, 0, 0, -10, 0, 12, 8, 0, 0, 0, 0, -13, -2, -14], [-6, 0, -3, 0, 0, -5, 0, -6, 7, 0, 0, 0, 0, 0, -12, -3], [0, 17, 0, 1, 1, 0, 9, 0, 0, -5, 7, 1, -12, 0, 0, 0], [0, 6, 0, -3, -1, 0, 10, 0, 0, -11, -3, 0, -2, 0, 0, 0], [-9, 0, 0, 0, 0, -12, 0, 2, 2, 0, 0, 0, 0, 1, 7, -7], [0, 7, 0, 4, 12, 0, -2, 0, 0, -9, 3, 13, -12, 0, 0, 0], [6, 0, 6, 0, 0, 3, 0, -1, 12, 0, 0, 0, 0, 1, -19, 1], [0, 2, 0, -6, 7, 0, 14, 0, 0, -20, 8, 8, -2, 0, 0, 0], [-15, 0, -13, 0, 0, -8, 0, 10, 2, 0, 0, 0, 0, 5, 6, -15], [-9, 0, 0, 0, 0, -3, 0, 7, -6, 0, 0, 0, 0, 0, -1, 9], [0, 6, 0, 10, 2, 0, 16, 0, 0, 4, 12, 5, -6, 0, 0, 0], [0, 8, 0, 8, 5, 0, -2, 0, 0, 8, 4, 10, -10, 0, 0, 0], [0, -16, 0, -10, -8, 0, -14, 0, 0, -10, -8, -11, 4, 0, 0, 0], [-20, 0, -2, 0, 0, -8, 0, 8, -8, 0, 0, 0, 0, 14, 10, -8], [12, 0, 8, 0, 0, 3, 0, -5, 8, 0, 0, 0, 0, -23, 8, -4], [4, 0, 2, 0, 0, -10, 0, 10, 4, 0, 0, 0, 0, 4, 6, 1], [0, 14, 0, 14, -3, 0, 20, 0, 0, 4, 6, 4, 4, 0, 0, 0], [0, 17, 0, 5, -6, 0, -10, 0, 0, 5, -2, -3, 7, 0, 0, 0], [-6, 0, 2, 0, 0, -13, 0, 3, 2, 0, 0, 0, 0, -5, -14, -6], [2, 0, -2, 0, 0, -7, 0, 14, 2, 0, 0, 0, 0, -9, -2, -5], [0, -10, 0, 3, -2, 0, -2, 0, 0, 17, -1, -4, 1, 0, 0, 0], [0, 7, 0, 0, 0, 0, 12, 0, 0, -15, -6, 7, -8, 0, 0, 0], [2, 0, 0, 0, 0, 1, 0, -1, -4, 0, 0, 0, 0, 4, 3, 9], [0, -4, 0, -1, -2, 0, 2, 0, 0, -12, -1, -8, 15, 0, 0, 0], [0, 3, 0, -4, -8, 0, -1, 0, 0, -8, 2, -3, -7, 0, 0, 0], [8, 0, 8, 0, 0, 4, 0, -6, -6, 0, 0, 0, 0, -4, 10, 6], [0, -7, 0, 7, -2, 0, 11, 0, 0, -17, 1, 15, -4, 0, 0, 0], [0, -15, 0, -2, 2, 0, 4, 0, 0, -3, 1, 4, 2, 0, 0, 0], [0, -5, 0, -6, 7, 0, 5, 0, 0, 4, 0, -1, 15, 0, 0, 0], [2, 0, -4, 0, 0, 8, 0, -6, -8, 0, 0, 0, 0, 8, 16, 19], [20, 0, 8, 0, 0, -10, 0, -2, 6, 0, 0, 0, 0, -15, -4, 0], [8, 0, 0, 0, 0, 11, 0, -2, -4, 0, 0, 0, 0, 1, -12, -3], [0, 2, 0, -7, 11, 0, -10, 0, 0, -21, -1, 12, -6, 0, 0, 0], [-17, 0, -4, 0, 0, 9, 0, 16, -1, 0, 0, 0, 0, 13, 0, 8], [0, 20, 0, -2, 8, 0, -12, 0, 0, 6, 7, 4, -2, 0, 0, 0], [0, 10, 0, 0, 2, 0, 28, 0, 0, -10, 10, 6, 0, 0, 0, 0], [0, -6, 0, 1, -4, 0, 2, 0, 0, 1, -13, -10, 0, 0, 0, 0], [7, 0, -2, 0, 0, 0, 0, 12, 15, 0, 0, 0, 0, -10, -6, -2], [0, -8, 0, -1, -4, 0, 10, 0, 0, -10, -14, -17, 11, 0, 0, 0], [0, -13, 0, -3, 8, 0, 10, 0, 0, -21, -4, -11, -7, 0, 0, 0], [-14, 0, -12, 0, 0, 6, 0, 2, -8, 0, 0, 0, 0, 25, 0, 4], [0, -6, 0, 0, -2, 0, 14, 0, 0, -18, 8, 4, 1, 0, 0, 0], [-7, 0, -2, 0, 0, 0, 0, 10, 4, 0, 0, 0, 0, -5, 12, -2], [0, 2, 0, -2, 2, 0, -6, 0, 0, 20, -5, -5, 6, 0, 0, 0], [4, 0, 2, 0, 0, -1, 0, -10, -6, 0, 0, 0, 0, 5, -2, 15], [9, 0, -6, 0, 0, 14, 0, 2, -16, 0, 0, 0, 0, 1, 16, 6], [-2, 0, -2, 0, 0, 5, 0, 14, 4, 0, 0, 0, 0, -3, 0, 9], [0, -6, 0, 0, -5, 0, -12, 0, 0, 16, -12, -20, -1, 0, 0, 0], [6, 0, -4, 0, 0, -1, 0, -9, 10, 0, 0, 0, 0, 11, 5, -9], [9, 0, 17, 0, 0, -3, 0, -10, 10, 0, 0, 0, 0, 0, -4, 10], [-7, 0, -6, 0, 0, 1, 0, 1, -16, 0, 0, 0, 0, 8, 1, 23], [4, 0, 20, 0, 0, 0, 0, -10, 8, 0, 0, 0, 0, 8, -12, 0], [0, 2, 0, -7, -10, 0, 14, 0, 0, 3, 9, -12, -5, 0, 0, 0], [15, 0, -8, 0, 0, 1, 0, -9, -8, 0, 0, 0, 0, -13, 16, -4], [0, 15, 0, 5, 6, 0, 9, 0, 0, -1, 11, 9, 12, 0, 0, 0], [10, 0, -4, 0, 0, 10, 0, -4, -6, 0, 0, 0, 0, 4, 2, -1], [0, 13, 0, 6, 13, 0, 23, 0, 0, 4, 8, 17, 2, 0, 0, 0], [0, 4, 0, 0, 16, 0, 30, 0, 0, -12, 6, 6, -2, 0, 0, 0], [0, 8, 0, 2, -6, 0, -2, 0, 0, 2, -4, 0, 8, 0, 0, 0], [-25, 0, -3, 0, 0, -2, 0, 18, -2, 0, 0, 0, 0, 11, -2, -22], [0, 20, 0, 8, 9, 0, -4, 0, 0, -10, 12, 10, 8, 0, 0, 0], [0, -1, 0, -5, -3, 0, 25, 0, 0, -27, -7, 1, -2, 0, 0, 0], [2, 0, 6, 0, 0, 4, 0, -20, -4, 0, 0, 0, 0, -4, 2, 2], [-30, 0, -11, 0, 0, -15, 0, 14, 9, 0, 0, 0, 0, 14, -4, -13], [0, 25, 0, 10, 10, 0, 25, 0, 0, -2, 10, 9, -9, 0, 0, 0], [0, 0, 0, -1, -4, 0, -20, 0, 0, 10, 3, 9, 5, 0, 0, 0], [-34, 0, -10, 0, 0, 1, 0, 3, -10, 0, 0, 0, 0, 32, 8, -14], [-35, 0, -10, 0, 0, 12, 0, 10, -1, 0, 0, 0, 0, 20, 2, -18], [0, 11, 0, 3, -6, 0, 10, 0, 0, -11, 5, 3, 9, 0, 0, 0], [0, 8, 0, -2, -8, 0, 20, 0, 0, -2, 1, -8, 2, 0, 0, 0], [21, 0, -2, 0, 0, 4, 0, 2, 4, 0, 0, 0, 0, -12, 10, -8], [0, 18, 0, -1, 15, 0, -10, 0, 0, 13, -3, 8, 2, 0, 0, 0], [0, -2, 0, 0, -2, 0, 8, 0, 0, -2, 2, 5, -10, 0, 0, 0], [0, -17, 0, 5, 0, 0, -4, 0, 0, -19, -6, 4, -1, 0, 0, 0], [7, 0, 3, 0, 0, -22, 0, 4, 14, 0, 0, 0, 0, 7, -14, -19], [-7, 0, 2, 0, 0, -2, 0, -8, 16, 0, 0, 0, 0, -1, -8, 4], [0, -10, 0, -6, 4, 0, 8, 0, 0, -2, -8, -4, 1, 0, 0, 0], [15, 0, -4, 0, 0, 2, 0, -8, 4, 0, 0, 0, 0, -5, -6, -4], [-2, 0, -7, 0, 0, 24, 0, 0, -15, 0, 0, 0, 0, 13, 10, 5], [0, 9, 0, 6, 4, 0, 6, 0, 0, 27, -1, -8, 0, 0, 0, 0], [8, 0, -2, 0, 0, 25, 0, 11, -8, 0, 0, 0, 0, 25, 4, 14], [0, 7, 0, 3, 12, 0, -10, 0, 0, 3, -5, 7, -5, 0, 0, 0], [9, 0, 7, 0, 0, 6, 0, 4, 14, 0, 0, 0, 0, 15, -10, 2], [9, 0, 6, 0, 0, -17, 0, -7, 14, 0, 0, 0, 0, -9, -9, -3], [0, 7, 0, 15, 4, 0, 21, 0, 0, -11, 5, -3, -2, 0, 0, 0], [4, 0, -6, 0, 0, 6, 0, -10, 2, 0, 0, 0, 0, 30, -6, 2], [0, 5, 0, 4, 7, 0, -19, 0, 0, 14, -12, -7, 6, 0, 0, 0], [0, 4, 0, 0, 0, 0, -26, 0, 0, 30, 3, 5, -14, 0, 0, 0], [-6, 0, 6, 0, 0, 10, 0, -10, -10, 0, 0, 0, 0, 14, 12, 10], [10, 0, 10, 0, 0, -6, 0, -14, 10, 0, 0, 0, 0, -13, -5, 15], [0, -21, 0, -1, 5, 0, -15, 0, 0, -11, -1, 9, 7, 0, 0, 0], [1, 0, 0, 0, 0, 2, 0, 22, 11, 0, 0, 0, 0, 10, 0, 15], [0, 0, 0, -6, 8, 0, -14, 0, 0, 14, 1, -2, 14, 0, 0, 0], [0, 3, 0, -1, -16, 0, 22, 0, 0, 5, 12, -1, -9, 0, 0, 0], [-2, 0, 0, 0, 0, 12, 0, 2, -10, 0, 0, 0, 0, 13, 11, 7], [-5, 0, 0, 0, 0, -2, 0, -2, 6, 0, 0, 0, 0, -12, -10, -14], [12, 0, -5, 0, 0, 18, 0, 2, -7, 0, 0, 0, 0, 25, 6, 22], [0, 15, 0, 8, 11, 0, -15, 0, 0, 10, 6, 5, 6, 0, 0, 0], [0, -9, 0, 6, 10, 0, -2, 0, 0, 11, 12, 10, 4, 0, 0, 0], [4, 0, 8, 0, 0, 3, 0, 3, -12, 0, 0, 0, 0, -6, 9, 1], [0, 3, 0, -3, -10, 0, 32, 0, 0, -7, -1, -10, 5, 0, 0, 0], [-32, 0, 6, 0, 0, -4, 0, -10, 2, 0, 0, 0, 0, 4, -2, 1], [0, -5, 0, -4, -21, 0, -19, 0, 0, 6, -6, -13, 16, 0, 0, 0], [18, 0, 10, 0, 0, -2, 0, -12, 2, 0, 0, 0, 0, -22, -6, 2], [0, -7, 0, 8, -8, 0, 15, 0, 0, 0, -8, -1, -12, 0, 0, 0], [9, 0, 16, 0, 0, 4, 0, -6, 12, 0, 0, 0, 0, 10, -20, 0], [11, 0, 14, 0, 0, -4, 0, 4, 12, 0, 0, 0, 0, -18, -2, 4], [-9, 0, -5, 0, 0, 3, 0, -10, -16, 0, 0, 0, 0, 22, 0, 6], [0, 16, 0, -4, 10, 0, -8, 0, 0, 0, 14, 2, 9, 0, 0, 0], [0, 12, 0, 10, 12, 0, 22, 0, 0, 4, 3, 7, 0, 0, 0, 0], [-3, 0, -2, 0, 0, -8, 0, -2, 6, 0, 0, 0, 0, -14, -9, -3], [0, 6, 0, 9, 0, 0, -16, 0, 0, 10, 8, 0, -3, 0, 0, 0], [7, 0, 2, 0, 0, 5, 0, 0, 17, 0, 0, 0, 0, 1, -6, 1], [-9, 0, -2, 0, 0, 13, 0, 8, -11, 0, 0, 0, 0, 15, 8, -10], [0, 16, 0, -7, 6, 0, 0, 0, 0, -4, 6, 4, -1, 0, 0, 0], [17, 0, 14, 0, 0, 5, 0, -7, 0, 0, 0, 0, 0, -3, -8, 18], [0, 11, 0, 7, 4, 0, -19, 0, 0, 9, 7, -9, -7, 0, 0, 0], [-3, 0, -6, 0, 0, 10, 0, -6, -10, 0, 0, 0, 0, 22, 12, 16], [0, -3, 0, 0, 12, 0, -6, 0, 0, -9, -7, -3, 6, 0, 0, 0], [15, 0, 5, 0, 0, 12, 0, 14, 0, 0, 0, 0, 0, -7, -4, 13], [0, -7, 0, -5, 4, 0, -13, 0, 0, 9, 1, 13, 11, 0, 0, 0], [15, 0, 3, 0, 0, 6, 0, -4, -22, 0, 0, 0, 0, 1, 14, 4], [-1, 0, -10, 0, 0, 1, 0, 23, -4, 0, 0, 0, 0, 11, 8, 6], [0, -2, 0, 19, 16, 0, -10, 0, 0, -6, -10, 9, -13, 0, 0, 0], [0, 1, 0, -4, -15, 0, 13, 0, 0, -2, -2, -3, -3, 0, 0, 0], [-20, 0, -2, 0, 0, 9, 0, -4, 6, 0, 0, 0, 0, 13, 14, 12], [0, -26, 0, -10, -8, 0, 6, 0, 0, -18, -4, 0, 4, 0, 0, 0], [33, 0, 10, 0, 0, 0, 0, -10, 8, 0, 0, 0, 0, -9, 1, 5], [0, 5, 0, 14, 14, 0, 10, 0, 0, -19, 4, 9, -22, 0, 0, 0], [-6, 0, -5, 0, 0, 1, 0, -2, -3, 0, 0, 0, 0, -10, 0, -9], [23, 0, 14, 0, 0, 11, 0, -7, 4, 0, 0, 0, 0, -9, 11, 9], [0, -10, 0, -1, 10, 0, -2, 0, 0, 9, -15, -8, -19, 0, 0, 0], [0, -1, 0, -2, -4, 0, -8, 0, 0, 25, 1, 6, 6, 0, 0, 0], [0, 1, 0, 13, 2, 0, 9, 0, 0, 11, -1, 3, -14, 0, 0, 0], [0, 5, 0, 10, 6, 0, -1, 0, 0, -4, 10, 19, 6, 0, 0, 0], [0, 16, 0, 7, 12, 0, -4, 0, 0, 4, 2, 17, -17, 0, 0, 0], [-28, 0, -10, 0, 0, 2, 0, 0, -20, 0, 0, 0, 0, -9, 25, -3], [0, 13, 0, -7, 1, 0, -11, 0, 0, 41, 5, -11, 12, 0, 0, 0], [0, 18, 0, 20, 2, 0, 20, 0, 0, 6, 15, 12, -4, 0, 0, 0], [1, 0, 4, 0, 0, 4, 0, -8, -18, 0, 0, 0, 0, -3, -3, 5], [12, 0, -16, 0, 0, -6, 0, 12, -14, 0, 0, 0, 0, 0, 4, -2], [-25, 0, -3, 0, 0, 4, 0, -10, 4, 0, 0, 0, 0, 17, 16, 0], [0, 6, 0, -8, 5, 0, -28, 0, 0, 12, 2, -12, 15, 0, 0, 0], [0, 9, 0, 5, -2, 0, 6, 0, 0, -9, 4, 21, -1, 0, 0, 0], [-7, 0, -16, 0, 0, 17, 0, 11, -6, 0, 0, 0, 0, 10, 5, 3], [0, -2, 0, -6, 18, 0, 10, 0, 0, 22, 8, 0, 8, 0, 0, 0], [0, 17, 0, -3, 5, 0, 19, 0, 0, -29, 9, 11, 7, 0, 0, 0], [-9, 0, -10, 0, 0, 2, 0, 4, 0, 0, 0, 0, 0, 13, -25, -3], [0, -11, 0, -5, 1, 0, 19, 0, 0, 1, 7, -11, 4, 0, 0, 0], [0, -11, 0, -7, -22, 0, 5, 0, 0, -25, -7, -1, 13, 0, 0, 0], [0, 0, 0, -2, -4, 0, -22, 0, 0, 18, -5, 12, 12, 0, 0, 0], [8, 0, -4, 0, 0, -17, 0, -19, -12, 0, 0, 0, 0, -26, 4, -10], [-31, 0, -6, 0, 0, -1, 0, -1, -16, 0, 0, 0, 0, 18, 10, -10], [26, 0, 11, 0, 0, -3, 0, -6, 13, 0, 0, 0, 0, -22, 0, -4], [0, 3, 0, -7, 0, 0, -8, 0, 0, -21, -4, -14, -5, 0, 0, 0], [-2, 0, -8, 0, 0, -11, 0, 17, 2, 0, 0, 0, 0, -4, -4, -14], [0, -1, 0, -5, -14, 0, 16, 0, 0, -3, 1, -10, 15, 0, 0, 0], [0, 0, -4, 0, 0, -7, 0, 1, 4, 0, 0, 0, 0, -23, -11, 3], [0, 1, 0, -3, -6, 0, -15, 0, 0, 35, 13, 1, 0, 0, 0, 0], [3, 0, -5, 0, 0, 6, 0, 4, -10, 0, 0, 0, 0, 11, 14, -13], [0, 22, 0, -3, -5, 0, 16, 0, 0, 15, -3, -10, -4, 0, 0, 0], [0, -10, 0, -15, -10, 0, -10, 0, 0, -10, 6, -13, 19, 0, 0, 0], [21, 0, 8, 0, 0, 2, 0, 6, 4, 0, 0, 0, 0, -11, -6, 10], [21, 0, 9, 0, 0, 18, 0, 6, 4, 0, 0, 0, 0, 13, -20, 15], [-38, 0, -2, 0, 0, -5, 0, -2, -8, 0, 0, 0, 0, 3, 6, -7], [0, 6, 0, 4, 8, 0, 10, 0, 0, 20, 4, 5, -18, 0, 0, 0], [0, 9, 0, -6, 6, 0, 16, 0, 0, 7, -2, -8, 12, 0, 0, 0], [24, 0, -15, 0, 0, 2, 0, 12, 1, 0, 0, 0, 0, -17, -2, -7], [0, -4, 0, 7, 22, 0, 16, 0, 0, 9, 11, 8, 5, 0, 0, 0], [28, 0, 1, 0, 0, -5, 0, 6, -1, 0, 0, 0, 0, -22, -4, 7], [0, 0, 18, 0, 0, -1, 0, -3, 8, 0, 0, 0, 0, -18, 8, -16], [0, 6, 0, 9, -4, 0, -26, 0, 0, 28, -7, 1, 17, 0, 0, 0], [-11, 0, 12, 0, 0, -17, 0, 4, 13, 0, 0, 0, 0, 3, -18, -2], [0, -3, 0, 10, 5, 0, 9, 0, 0, -10, -2, 21, -12, 0, 0, 0], [0, -16, 0, -7, 4, 0, 0, 0, 0, -11, -23, -16, 3, 0, 0, 0], [0, -16, 0, -7, -8, 0, -20, 0, 0, -6, -30, -26, 9, 0, 0, 0], [21, 0, 0, 0, 0, -10, 0, 10, 2, 0, 0, 0, 0, 5, 4, 8]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1875_b_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1875_2_b_h();".
// 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_1875_2_b_h(:prec:=16)
    chi := MakeCharacter_1875_b();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(1999) - 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_1875_2_b_h();".
// 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_1875_2_b_h( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1875_b();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, 0, 130, 0, 761, 0, 1440, 0, 1286, 0, 610, 0, 156, 0, 20, 0, 1]>],Snew);
    return Vf;
end function;