
// Make newform 3150.3.c.f in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3150_c();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_3150_c_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_3150_3_c_f();".
// 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_3150_3_c_f();".
// 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, 0, 0, 112, 0, 0, 0, 753, 0, 0, 0, 7, 0, 0, 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], [42056, 0, 0, 0, 15, 0, 0, 0, 105, 0, 0, 0, -1, 0, 0, 0], [-815, 0, 0, 0, -10752, 0, 0, 0, -96, 0, 0, 0, -14, 0, 0, 0], [-56, 0, 0, 0, -737, 0, 0, 0, -7, 0, 0, 0, -1, 0, 0, 0], [0, 0, 18304, 0, 0, 0, 25025, 0, 0, 0, 247, 0, 0, 0, 33, 0], [0, 39712, 0, -185272, 0, -278358, 0, 182157, 0, -2490, 0, 1419, 0, -374, 0, 245], [0, -39712, 0, -185272, 0, 278358, 0, 182157, 0, 2490, 0, 1419, 0, 374, 0, 245], [0, 0, 12704, 0, 0, 0, 17367, 0, 0, 0, 177, 0, 0, 0, 23, 0], [0, 0, -10112, 0, 0, 0, 23295, 0, 0, 0, 201, 0, 0, 0, 31, 0], [0, -14800, 0, -69044, 0, 103737, 0, 67887, 0, 927, 0, 537, 0, 137, 0, 91], [0, 14800, 0, -69044, 0, -103737, 0, 67887, 0, -927, 0, 537, 0, -137, 0, 91], [0, -149024, 0, 115960, 0, -92826, 0, 303675, 0, -1110, 0, 2925, 0, -122, 0, 403], [0, 149024, 0, 115960, 0, 92826, 0, 303675, 0, 1110, 0, 2925, 0, 122, 0, 403], [0, 0, -8096, 0, 0, 0, 18649, 0, 0, 0, 159, 0, 0, 0, 25, 0], [0, 499664, 0, 388916, 0, 311271, 0, 1018545, 0, 3201, 0, 9735, 0, 407, 0, 1349], [0, 499664, 0, -388916, 0, 311271, 0, -1018545, 0, 3201, 0, -9735, 0, 407, 0, -1349]];
    Rf_basisdens := [1, 9396, 2349, 36, 8352, 150336, 150336, 2592, 2592, 25056, 12528, 50112, 50112, 928, 75168, 37584];
    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_3150_c();"
function MakeCharacter_3150_c()
    N := 3150;
    order := 2;
    char_gens := [2801, 127, 451];
    v := [1, 1, 2];
    // 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_3150_c_Hecke();"
function MakeCharacter_3150_c_Hecke(Kf)
    N := 3150;
    order := 2;
    char_gens := [2801, 127, 451];
    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], [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 := 3;
    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, 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, 0, 0, 0, 0], [0, 0, 0, 0, 0, -1, 0, 2, 0, 0, 0, 0, 1, 0, 0, 0], [0, 0, 4, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0], [0, -1, 0, 0, 0, 0, 6, 0, 0, 0, 0, 2, 0, 1, 0, 0], [10, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, -1], [0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 3, 0, 0, 0, 0], [0, 0, 0, 2, 0, 13, 0, 2, 0, 0, 0, 0, -2, 0, 0, 0], [14, 0, 0, 0, 0, 0, 0, 0, -2, 3, 0, 0, 0, 0, 0, 2], [0, 0, 10, 0, 8, 0, 0, 0, 0, 0, 3, 0, 0, 0, -4, 0], [0, 0, 0, -1, 0, 20, 0, -6, 0, 0, 0, 0, -6, 0, 0, 0], [0, 0, -2, 0, 4, 0, 0, 0, 0, 0, 2, 0, 0, 0, 6, 0], [0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, -3, 0, 0], [0, -7, 0, 0, 0, 0, 15, 0, 0, 0, 0, -3, 0, 2, 0, 0], [0, 0, 0, 2, 0, 2, 0, -9, 0, 0, 0, 0, -12, 0, 0, 0], [-18, 0, 0, 0, 0, 0, 0, 0, 6, -6, 0, 0, 0, 0, 0, 2], [0, 0, 14, 0, -4, 0, 0, 0, 0, 0, -1, 0, 0, 0, -4, 0], [0, 0, 0, 0, 0, 7, 0, 4, 0, 0, 0, 0, -13, 0, 0, 0], [0, 0, 24, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 3, 0], [8, 0, 0, 0, 0, 0, 0, 0, -12, -14, 0, 0, 0, 0, 0, -2], [0, 23, 0, 0, 0, 0, -16, 0, 0, 0, 0, 10, 0, -1, 0, 0], [0, 0, 0, -4, 0, -26, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 32, 0, -2, 0, 0, 0, 0, 0, -2, 0, 0, 0, -5, 0], [0, 0, 0, -3, 0, -48, 0, 0, 0, 0, 0, 0, -18, 0, 0, 0], [0, 0, -8, 0, -6, 0, 0, 0, 0, 0, -12, 0, 0, 0, -6, 0], [0, -17, 0, 0, 0, 0, -10, 0, 0, 0, 0, -1, 0, 2, 0, 0], [-74, 0, 0, 0, 0, 0, 0, 0, -4, 16, 0, 0, 0, 0, 0, 0], [0, -11, 0, 0, 0, 0, 21, 0, 0, 0, 0, 29, 0, 0, 0, 0], [0, 0, 44, 0, 20, 0, 0, 0, 0, 0, -17, 0, 0, 0, 6, 0], [0, 0, 0, 6, 0, -4, 0, 15, 0, 0, 0, 0, 2, 0, 0, 0], [0, 23, 0, 0, 0, 0, 61, 0, 0, 0, 0, -3, 0, 0, 0, 0], [62, 0, 0, 0, 0, 0, 0, 0, -16, 25, 0, 0, 0, 0, 0, 2], [0, 0, 0, -2, 0, 29, 0, 34, 0, 0, 0, 0, -4, 0, 0, 0], [34, 0, 0, 0, 0, 0, 0, 0, 25, -2, 0, 0, 0, 0, 0, 8], [0, 0, 60, 0, -10, 0, 0, 0, 0, 0, 7, 0, 0, 0, 11, 0], [0, 0, -16, 0, 18, 0, 0, 0, 0, 0, 7, 0, 0, 0, 10, 0], [0, -24, 0, 0, 0, 0, -48, 0, 0, 0, 0, 20, 0, 8, 0, 0], [0, -9, 0, 0, 0, 0, -74, 0, 0, 0, 0, 6, 0, -2, 0, 0], [0, 0, 0, -12, 0, 65, 0, -38, 0, 0, 0, 0, -27, 0, 0, 0], [14, 0, 0, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, -4], [0, 0, 0, 12, 0, -7, 0, -32, 0, 0, 0, 0, -9, 0, 0, 0], [0, 0, -26, 0, -46, 0, 0, 0, 0, 0, 17, 0, 0, 0, -22, 0], [0, 1, 0, 0, 0, 0, 49, 0, 0, 0, 0, 17, 0, 16, 0, 0], [18, 0, 0, 0, 0, 0, 0, 0, -14, 31, 0, 0, 0, 0, 0, -7], [-94, 0, 0, 0, 0, 0, 0, 0, 21, 40, 0, 0, 0, 0, 0, 2], [0, 0, 28, 0, 10, 0, 0, 0, 0, 0, 2, 0, 0, 0, 18, 0], [0, 23, 0, 0, 0, 0, -36, 0, 0, 0, 0, 0, 0, 5, 0, 0], [42, 0, 0, 0, 0, 0, 0, 0, -54, 24, 0, 0, 0, 0, 0, 4], [0, 39, 0, 0, 0, 0, -21, 0, 0, 0, 0, 17, 0, -4, 0, 0], [0, 0, 0, -12, 0, -183, 0, 2, 0, 0, 0, 0, -13, 0, 0, 0], [58, 0, 0, 0, 0, 0, 0, 0, 6, -40, 0, 0, 0, 0, 0, -10], [0, 0, 0, 8, 0, 62, 0, -53, 0, 0, 0, 0, 16, 0, 0, 0], [0, 19, 0, 0, 0, 0, -128, 0, 0, 0, 0, 28, 0, -3, 0, 0], [0, -27, 0, 0, 0, 0, 184, 0, 0, 0, 0, 23, 0, -6, 0, 0], [0, 0, 0, -3, 0, 90, 0, -30, 0, 0, 0, 0, 6, 0, 0, 0], [-30, 0, 0, 0, 0, 0, 0, 0, 4, 7, 0, 0, 0, 0, 0, -6], [0, 0, 72, 0, 137, 0, 0, 0, 0, 0, -10, 0, 0, 0, -2, 0], [0, 0, 0, -18, 0, -105, 0, 14, 0, 0, 0, 0, 14, 0, 0, 0], [0, 0, -4, 0, -10, 0, 0, 0, 0, 0, -19, 0, 0, 0, -20, 0], [0, 55, 0, 0, 0, 0, 88, 0, 0, 0, 0, -4, 0, 1, 0, 0], [0, 0, 64, 0, 64, 0, 0, 0, 0, 0, 11, 0, 0, 0, 10, 0], [0, 0, 0, 2, 0, 132, 0, 63, 0, 0, 0, 0, -2, 0, 0, 0], [0, 0, -28, 0, -26, 0, 0, 0, 0, 0, -39, 0, 0, 0, 1, 0], [0, -55, 0, 0, 0, 0, 131, 0, 0, 0, 0, 57, 0, -20, 0, 0], [-26, 0, 0, 0, 0, 0, 0, 0, -19, 8, 0, 0, 0, 0, 0, 18], [0, 0, 102, 0, -136, 0, 0, 0, 0, 0, 13, 0, 0, 0, 14, 0], [0, 7, 0, 0, 0, 0, 24, 0, 0, 0, 0, 63, 0, -8, 0, 0], [62, 0, 0, 0, 0, 0, 0, 0, -40, 28, 0, 0, 0, 0, 0, -1], [0, -43, 0, 0, 0, 0, -176, 0, 0, 0, 0, 56, 0, 32, 0, 0], [0, 0, 0, 12, 0, 123, 0, 94, 0, 0, 0, 0, -11, 0, 0, 0], [0, 0, 40, 0, 86, 0, 0, 0, 0, 0, 16, 0, 0, 0, -26, 0], [0, 0, 0, 0, 109, 0, 0, 0, 0, 0, 54, 0, 0, 0, -12, 0], [50, 0, 0, 0, 0, 0, 0, 0, -53, 30, 0, 0, 0, 0, 0, 16], [0, -28, 0, 0, 0, 0, 28, 0, 0, 0, 0, 50, 0, 8, 0, 0], [0, 0, 0, 28, 0, -75, 0, -14, 0, 0, 0, 0, 72, 0, 0, 0], [0, 0, -40, 0, -134, 0, 0, 0, 0, 0, -31, 0, 0, 0, 21, 0], [0, 0, 0, -24, 0, -149, 0, -102, 0, 0, 0, 0, 64, 0, 0, 0], [302, 0, 0, 0, 0, 0, 0, 0, -62, 108, 0, 0, 0, 0, 0, 13], [0, 0, 0, 22, 0, -28, 0, 34, 0, 0, 0, 0, 44, 0, 0, 0], [-198, 0, 0, 0, 0, 0, 0, 0, 0, 44, 0, 0, 0, 0, 0, 6], [0, 0, 0, -24, 0, 33, 0, 60, 0, 0, 0, 0, -3, 0, 0, 0], [0, 0, -32, 0, -104, 0, 0, 0, 0, 0, -9, 0, 0, 0, -43, 0], [90, 0, 0, 0, 0, 0, 0, 0, 16, 9, 0, 0, 0, 0, 0, -21], [0, 47, 0, 0, 0, 0, -156, 0, 0, 0, 0, 67, 0, 12, 0, 0], [0, 0, 0, 26, 0, -111, 0, 30, 0, 0, 0, 0, 4, 0, 0, 0], [0, 0, 46, 0, -34, 0, 0, 0, 0, 0, -47, 0, 0, 0, 4, 0], [0, 0, 0, -4, 0, 262, 0, 3, 0, 0, 0, 0, 80, 0, 0, 0], [0, 0, -80, 0, -126, 0, 0, 0, 0, 0, -29, 0, 0, 0, 2, 0], [0, 89, 0, 0, 0, 0, -36, 0, 0, 0, 0, 76, 0, 9, 0, 0], [0, 0, 0, -14, 0, 272, 0, 7, 0, 0, 0, 0, 86, 0, 0, 0], [0, 0, 98, 0, 56, 0, 0, 0, 0, 0, 18, 0, 0, 0, -2, 0], [0, 0, 0, 24, 0, -123, 0, -78, 0, 0, 0, 0, -15, 0, 0, 0], [-66, 0, 0, 0, 0, 0, 0, 0, -51, -34, 0, 0, 0, 0, 0, -10], [0, -8, 0, 0, 0, 0, 140, 0, 0, 0, 0, 78, 0, 6, 0, 0], [0, 0, 0, 3, 0, -102, 0, -88, 0, 0, 0, 0, 74, 0, 0, 0], [0, 0, 0, -23, 0, -74, 0, -26, 0, 0, 0, 0, 130, 0, 0, 0], [0, 0, 88, 0, -6, 0, 0, 0, 0, 0, 15, 0, 0, 0, 24, 0], [-210, 0, 0, 0, 0, 0, 0, 0, -52, 184, 0, 0, 0, 0, 0, -10], [0, 0, -30, 0, -114, 0, 0, 0, 0, 0, -5, 0, 0, 0, -44, 0], [0, 75, 0, 0, 0, 0, 85, 0, 0, 0, 0, 3, 0, -2, 0, 0], [0, -18, 0, 0, 0, 0, 76, 0, 0, 0, 0, 26, 0, -30, 0, 0], [0, 0, 0, 16, 0, 269, 0, -98, 0, 0, 0, 0, 116, 0, 0, 0], [-116, 0, 0, 0, 0, 0, 0, 0, -118, 48, 0, 0, 0, 0, 0, -4], [0, 0, -64, 0, 48, 0, 0, 0, 0, 0, 105, 0, 0, 0, 33, 0], [0, -44, 0, 0, 0, 0, -104, 0, 0, 0, 0, -22, 0, 16, 0, 0], [0, -21, 0, 0, 0, 0, -336, 0, 0, 0, 0, 24, 0, 18, 0, 0], [0, 0, 0, -40, 0, 77, 0, -44, 0, 0, 0, 0, -17, 0, 0, 0], [302, 0, 0, 0, 0, 0, 0, 0, -54, 26, 0, 0, 0, 0, 0, 16], [0, 0, 36, 0, -158, 0, 0, 0, 0, 0, -31, 0, 0, 0, -16, 0], [0, 0, 104, 0, -151, 0, 0, 0, 0, 0, -48, 0, 0, 0, -6, 0], [0, -101, 0, 0, 0, 0, -137, 0, 0, 0, 0, 165, 0, 14, 0, 0], [470, 0, 0, 0, 0, 0, 0, 0, -38, -25, 0, 0, 0, 0, 0, 2], [-12, 0, 0, 0, 0, 0, 0, 0, -32, -214, 0, 0, 0, 0, 0, 8], [0, 0, 0, 20, 0, 9, 0, 32, 0, 0, 0, 0, 18, 0, 0, 0], [0, 0, -252, 0, 60, 0, 0, 0, 0, 0, -48, 0, 0, 0, 4, 0], [0, -30, 0, 0, 0, 0, 448, 0, 0, 0, 0, -48, 0, 4, 0, 0], [0, 1, 0, 0, 0, 0, -63, 0, 0, 0, 0, 55, 0, -40, 0, 0], [0, 0, 0, -16, 0, -199, 0, 54, 0, 0, 0, 0, 153, 0, 0, 0], [166, 0, 0, 0, 0, 0, 0, 0, -2, -178, 0, 0, 0, 0, 0, -8], [0, 0, -6, 0, -278, 0, 0, 0, 0, 0, -7, 0, 0, 0, -44, 0], [0, 7, 0, 0, 0, 0, -724, 0, 0, 0, 0, 28, 0, -5, 0, 0], [0, -153, 0, 0, 0, 0, 256, 0, 0, 0, 0, -63, 0, -2, 0, 0], [-314, 0, 0, 0, 0, 0, 0, 0, 110, 155, 0, 0, 0, 0, 0, 6], [0, 0, 0, -16, 0, -139, 0, -28, 0, 0, 0, 0, 122, 0, 0, 0], [220, 0, 0, 0, 0, 0, 0, 0, 62, -88, 0, 0, 0, 0, 0, -22], [0, 0, 0, -8, 0, -334, 0, -131, 0, 0, 0, 0, -40, 0, 0, 0], [0, 0, 12, 0, 40, 0, 0, 0, 0, 0, 69, 0, 0, 0, -62, 0], [0, 0, -180, 0, 62, 0, 0, 0, 0, 0, 76, 0, 0, 0, -17, 0], [-320, 0, 0, 0, 0, 0, 0, 0, 58, -4, 0, 0, 0, 0, 0, 0], [0, 81, 0, 0, 0, 0, 12, 0, 0, 0, 0, 177, 0, -24, 0, 0], [186, 0, 0, 0, 0, 0, 0, 0, 21, 56, 0, 0, 0, 0, 0, -48], [0, 0, -214, 0, -72, 0, 0, 0, 0, 0, 69, 0, 0, 0, -6, 0], [0, 0, 0, 50, 0, 18, 0, 51, 0, 0, 0, 0, -32, 0, 0, 0], [202, 0, 0, 0, 0, 0, 0, 0, -22, 38, 0, 0, 0, 0, 0, 44], [0, 67, 0, 0, 0, 0, -58, 0, 0, 0, 0, -150, 0, -6, 0, 0], [0, 0, 64, 0, -50, 0, 0, 0, 0, 0, 14, 0, 0, 0, -2, 0], [0, 99, 0, 0, 0, 0, -42, 0, 0, 0, 0, -138, 0, 24, 0, 0], [0, 0, 0, 32, 0, 117, 0, -30, 0, 0, 0, 0, 238, 0, 0, 0], [-46, 0, 0, 0, 0, 0, 0, 0, 110, 57, 0, 0, 0, 0, 0, -32], [0, 0, 0, 12, 0, 319, 0, 206, 0, 0, 0, 0, -54, 0, 0, 0], [0, 0, -212, 0, -124, 0, 0, 0, 0, 0, 85, 0, 0, 0, 18, 0], [0, -49, 0, 0, 0, 0, -186, 0, 0, 0, 0, -219, 0, 20, 0, 0], [618, 0, 0, 0, 0, 0, 0, 0, 178, 52, 0, 0, 0, 0, 0, -7], [0, 0, 0, -60, 0, 82, 0, 60, 0, 0, 0, 0, 118, 0, 0, 0], [0, 0, 36, 0, -48, 0, 0, 0, 0, 0, -5, 0, 0, 0, 15, 0], [0, -203, 0, 0, 0, 0, 362, 0, 0, 0, 0, 6, 0, 6, 0, 0], [814, 0, 0, 0, 0, 0, 0, 0, 42, -131, 0, 0, 0, 0, 0, 5], [0, 77, 0, 0, 0, 0, -114, 0, 0, 0, 0, -195, 0, 50, 0, 0], [0, 0, -192, 0, 179, 0, 0, 0, 0, 0, -24, 0, 0, 0, -30, 0], [0, 0, 0, -30, 0, 260, 0, -39, 0, 0, 0, 0, 8, 0, 0, 0], [0, 0, -28, 0, -422, 0, 0, 0, 0, 0, -56, 0, 0, 0, 48, 0], [0, 85, 0, 0, 0, 0, -332, 0, 0, 0, 0, -330, 0, 29, 0, 0], [0, 0, -88, 0, 260, 0, 0, 0, 0, 0, -48, 0, 0, 0, 28, 0], [0, 0, 0, 20, 0, -113, 0, -106, 0, 0, 0, 0, -95, 0, 0, 0], [114, 0, 0, 0, 0, 0, 0, 0, 171, 268, 0, 0, 0, 0, 0, -6], [0, 0, 0, 40, 0, -150, 0, 53, 0, 0, 0, 0, 148, 0, 0, 0], [0, 0, -196, 0, 176, 0, 0, 0, 0, 0, -99, 0, 0, 0, -17, 0], [0, 0, 0, 58, 0, 352, 0, -53, 0, 0, 0, 0, 4, 0, 0, 0], [0, 155, 0, 0, 0, 0, 60, 0, 0, 0, 0, -213, 0, 20, 0, 0], [0, -35, 0, 0, 0, 0, -55, 0, 0, 0, 0, -203, 0, 52, 0, 0], [0, 0, -266, 0, 140, 0, 0, 0, 0, 0, 67, 0, 0, 0, 64, 0], [0, 0, 0, -32, 0, 254, 0, -8, 0, 0, 0, 0, -34, 0, 0, 0], [0, 15, 0, 0, 0, 0, 399, 0, 0, 0, 0, 33, 0, 54, 0, 0], [0, 129, 0, 0, 0, 0, -176, 0, 0, 0, 0, 144, 0, 1, 0, 0], [-166, 0, 0, 0, 0, 0, 0, 0, 205, 136, 0, 0, 0, 0, 0, 34], [0, 0, -344, 0, 130, 0, 0, 0, 0, 0, -71, 0, 0, 0, -73, 0], [-612, 0, 0, 0, 0, 0, 0, 0, 94, 24, 0, 0, 0, 0, 0, -30], [0, -89, 0, 0, 0, 0, -344, 0, 0, 0, 0, -76, 0, -11, 0, 0], [0, 0, 0, 56, 0, -665, 0, -18, 0, 0, 0, 0, -79, 0, 0, 0], [-854, 0, 0, 0, 0, 0, 0, 0, -86, -254, 0, 0, 0, 0, 0, 41], [0, 0, 0, -4, 0, -93, 0, 0, 0, 0, 0, 0, -233, 0, 0, 0], [0, 0, -480, 0, -119, 0, 0, 0, 0, 0, -20, 0, 0, 0, 20, 0], [898, 0, 0, 0, 0, 0, 0, 0, -96, 33, 0, 0, 0, 0, 0, -9], [0, 0, 0, 17, 0, -346, 0, -84, 0, 0, 0, 0, 18, 0, 0, 0], [-72, 0, 0, 0, 0, 0, 0, 0, -28, -30, 0, 0, 0, 0, 0, 18], [0, 0, 0, 12, 0, -207, 0, -30, 0, 0, 0, 0, 324, 0, 0, 0], [0, 0, 124, 0, -432, 0, 0, 0, 0, 0, 34, 0, 0, 0, 44, 0], [766, 0, 0, 0, 0, 0, 0, 0, 52, -4, 0, 0, 0, 0, 0, -54], [0, 0, -62, 0, -22, 0, 0, 0, 0, 0, 93, 0, 0, 0, -16, 0], [0, 0, 0, -40, 0, -442, 0, 76, 0, 0, 0, 0, -362, 0, 0, 0], [0, 0, -168, 0, -279, 0, 0, 0, 0, 0, -14, 0, 0, 0, 132, 0], [0, -137, 0, 0, 0, 0, -254, 0, 0, 0, 0, -146, 0, -69, 0, 0], [0, 71, 0, 0, 0, 0, 208, 0, 0, 0, 0, -191, 0, 34, 0, 0], [0, 0, 0, -7, 0, 776, 0, 208, 0, 0, 0, 0, 58, 0, 0, 0], [0, 0, -42, 0, -192, 0, 0, 0, 0, 0, 39, 0, 0, 0, -184, 0], [0, 0, -128, 0, 678, 0, 0, 0, 0, 0, 100, 0, 0, 0, 50, 0], [-124, 0, 0, 0, 0, 0, 0, 0, -70, -200, 0, 0, 0, 0, 0, -10], [0, 0, 0, -62, 0, 10, 0, 23, 0, 0, 0, 0, -112, 0, 0, 0], [0, 0, 40, 0, -246, 0, 0, 0, 0, 0, 62, 0, 0, 0, 1, 0], [0, -171, 0, 0, 0, 0, -526, 0, 0, 0, 0, 71, 0, 16, 0, 0], [370, 0, 0, 0, 0, 0, 0, 0, 209, -118, 0, 0, 0, 0, 0, 0], [0, 0, 0, -42, 0, -66, 0, -307, 0, 0, 0, 0, -40, 0, 0, 0], [0, -71, 0, 0, 0, 0, 44, 0, 0, 0, 0, -281, 0, -62, 0, 0], [0, 123, 0, 0, 0, 0, -792, 0, 0, 0, 0, 48, 0, -48, 0, 0], [-638, 0, 0, 0, 0, 0, 0, 0, -44, 392, 0, 0, 0, 0, 0, -14], [0, 0, 192, 0, -75, 0, 0, 0, 0, 0, -6, 0, 0, 0, -56, 0], [0, 205, 0, 0, 0, 0, 320, 0, 0, 0, 0, 28, 0, -65, 0, 0], [0, 38, 0, 0, 0, 0, -688, 0, 0, 0, 0, -296, 0, 26, 0, 0], [0, 0, 0, -38, 0, 679, 0, -32, 0, 0, 0, 0, 58, 0, 0, 0], [-946, 0, 0, 0, 0, 0, 0, 0, 86, -171, 0, 0, 0, 0, 0, -47], [0, 0, -64, 0, -42, 0, 0, 0, 0, 0, 106, 0, 0, 0, -19, 0], [1602, 0, 0, 0, 0, 0, 0, 0, 56, 74, 0, 0, 0, 0, 0, 9], [0, 0, 0, 104, 0, -586, 0, -224, 0, 0, 0, 0, -230, 0, 0, 0], [0, 79, 0, 0, 0, 0, 190, 0, 0, 0, 0, -90, 0, 50, 0, 0], [598, 0, 0, 0, 0, 0, 0, 0, 262, -207, 0, 0, 0, 0, 0, 35], [0, -51, 0, 0, 0, 0, -906, 0, 0, 0, 0, 99, 0, 48, 0, 0], [0, 0, 0, 30, 0, -649, 0, -80, 0, 0, 0, 0, 186, 0, 0, 0], [1362, 0, 0, 0, 0, 0, 0, 0, -128, 53, 0, 0, 0, 0, 0, -1], [0, 0, -164, 0, 5, 0, 0, 0, 0, 0, 92, 0, 0, 0, 150, 0], [0, 0, 0, 75, 0, 450, 0, -30, 0, 0, 0, 0, -222, 0, 0, 0], [0, 0, 188, 0, -266, 0, 0, 0, 0, 0, 50, 0, 0, 0, -136, 0], [0, -211, 0, 0, 0, 0, -624, 0, 0, 0, 0, -54, 0, 23, 0, 0], [0, 0, 0, -44, 0, 890, 0, 40, 0, 0, 0, 0, -34, 0, 0, 0], [546, 0, 0, 0, 0, 0, 0, 0, 36, -4, 0, 0, 0, 0, 0, -75], [0, 0, -94, 0, 192, 0, 0, 0, 0, 0, -276, 0, 0, 0, -42, 0], [0, 0, 0, -98, 0, 424, 0, -157, 0, 0, 0, 0, 8, 0, 0, 0], [0, 93, 0, 0, 0, 0, 304, 0, 0, 0, 0, -155, 0, -64, 0, 0], [0, -181, 0, 0, 0, 0, 475, 0, 0, 0, 0, 85, 0, -104, 0, 0], [356, 0, 0, 0, 0, 0, 0, 0, -210, 48, 0, 0, 0, 0, 0, 66], [-922, 0, 0, 0, 0, 0, 0, 0, -150, 383, 0, 0, 0, 0, 0, -12], [0, 0, 0, 30, 0, 1281, 0, -72, 0, 0, 0, 0, 234, 0, 0, 0], [0, 0, -56, 0, 104, 0, 0, 0, 0, 0, -306, 0, 0, 0, -32, 0], [0, -262, 0, 0, 0, 0, 336, 0, 0, 0, 0, -4, 0, 34, 0, 0], [-362, 0, 0, 0, 0, 0, 0, 0, 2, -336, 0, 0, 0, 0, 0, 64], [0, 155, 0, 0, 0, 0, -276, 0, 0, 0, 0, 352, 0, 21, 0, 0], [0, 0, 0, -16, 0, -109, 0, -64, 0, 0, 0, 0, -385, 0, 0, 0], [0, 0, 160, 0, 350, 0, 0, 0, 0, 0, -2, 0, 0, 0, 166, 0], [0, 0, 0, -24, 0, -481, 0, 270, 0, 0, 0, 0, -229, 0, 0, 0], [0, 0, 50, 0, -56, 0, 0, 0, 0, 0, 245, 0, 0, 0, 2, 0], [1522, 0, 0, 0, 0, 0, 0, 0, 150, -341, 0, 0, 0, 0, 0, 35], [382, 0, 0, 0, 0, 0, 0, 0, -39, -130, 0, 0, 0, 0, 0, -34], [0, 0, 0, -50, 0, 47, 0, 88, 0, 0, 0, 0, 290, 0, 0, 0], [0, 0, 192, 0, 52, 0, 0, 0, 0, 0, -34, 0, 0, 0, -164, 0], [0, -155, 0, 0, 0, 0, 680, 0, 0, 0, 0, -156, 0, 9, 0, 0], [242, 0, 0, 0, 0, 0, 0, 0, -182, -20, 0, 0, 0, 0, 0, -15], [0, -183, 0, 0, 0, 0, -887, 0, 0, 0, 0, -303, 0, -86, 0, 0], [0, 0, 0, 44, 0, -749, 0, 24, 0, 0, 0, 0, -313, 0, 0, 0], [0, 0, 0, -4, 0, 382, 0, -69, 0, 0, 0, 0, 356, 0, 0, 0], [0, 205, 0, 0, 0, 0, 1066, 0, 0, 0, 0, -49, 0, 16, 0, 0], [834, 0, 0, 0, 0, 0, 0, 0, -108, -613, 0, 0, 0, 0, 0, -30], [0, 0, -92, 0, 484, 0, 0, 0, 0, 0, 103, 0, 0, 0, -38, 0], [1082, 0, 0, 0, 0, 0, 0, 0, -256, -152, 0, 0, 0, 0, 0, -40], [0, 145, 0, 0, 0, 0, 854, 0, 0, 0, 0, -142, 0, -58, 0, 0], [0, 0, 0, -20, 0, -26, 0, 92, 0, 0, 0, 0, -202, 0, 0, 0], [0, 0, 116, 0, -276, 0, 0, 0, 0, 0, 98, 0, 0, 0, 160, 0], [0, 0, 0, 10, 0, 248, 0, -26, 0, 0, 0, 0, -184, 0, 0, 0], [-1432, 0, 0, 0, 0, 0, 0, 0, 178, -418, 0, 0, 0, 0, 0, -34], [0, -33, 0, 0, 0, 0, 874, 0, 0, 0, 0, -367, 0, 0, 0, 0], [0, 0, -2, 0, 302, 0, 0, 0, 0, 0, 55, 0, 0, 0, -236, 0], [0, 0, 0, 140, 0, 388, 0, -217, 0, 0, 0, 0, 172, 0, 0, 0], [0, 131, 0, 0, 0, 0, 586, 0, 0, 0, 0, 71, 0, -22, 0, 0], [-2062, 0, 0, 0, 0, 0, 0, 0, 128, 90, 0, 0, 0, 0, 0, 42], [0, 395, 0, 0, 0, 0, -398, 0, 0, 0, 0, -202, 0, 62, 0, 0], [0, 0, 0, 0, 0, 815, 0, 216, 0, 0, 0, 0, -139, 0, 0, 0], [-860, 0, 0, 0, 0, 0, 0, 0, -182, -152, 0, 0, 0, 0, 0, -10], [0, 0, 300, 0, -38, 0, 0, 0, 0, 0, -88, 0, 0, 0, 170, 0], [0, 149, 0, 0, 0, 0, 358, 0, 0, 0, 0, 242, 0, 128, 0, 0], [0, 0, -100, 0, 318, 0, 0, 0, 0, 0, 37, 0, 0, 0, -91, 0], [0, 0, 424, 0, 458, 0, 0, 0, 0, 0, -99, 0, 0, 0, -2, 0], [0, 279, 0, 0, 0, 0, -440, 0, 0, 0, 0, -45, 0, 100, 0, 0], [1266, 0, 0, 0, 0, 0, 0, 0, -138, 24, 0, 0, 0, 0, 0, -38], [0, 0, -128, 0, 320, 0, 0, 0, 0, 0, 113, 0, 0, 0, 227, 0], [0, -131, 0, 0, 0, 0, -556, 0, 0, 0, 0, 476, 0, 44, 0, 0], [466, 0, 0, 0, 0, 0, 0, 0, -38, 249, 0, 0, 0, 0, 0, 98], [0, 0, 0, -148, 0, -599, 0, -238, 0, 0, 0, 0, 70, 0, 0, 0], [0, 0, 0, -65, 0, -722, 0, 118, 0, 0, 0, 0, 190, 0, 0, 0], [0, 0, -82, 0, -1228, 0, 0, 0, 0, 0, -77, 0, 0, 0, 8, 0], [0, -153, 0, 0, 0, 0, 75, 0, 0, 0, 0, 93, 0, -204, 0, 0], [262, 0, 0, 0, 0, 0, 0, 0, -22, 354, 0, 0, 0, 0, 0, 71], [0, 0, 182, 0, 250, 0, 0, 0, 0, 0, 228, 0, 0, 0, 62, 0], [0, 0, 212, 0, -12, 0, 0, 0, 0, 0, 70, 0, 0, 0, -257, 0], [1406, 0, 0, 0, 0, 0, 0, 0, -87, 480, 0, 0, 0, 0, 0, 50], [0, 0, -652, 0, 92, 0, 0, 0, 0, 0, -12, 0, 0, 0, -29, 0], [0, 0, 164, 0, 728, 0, 0, 0, 0, 0, 68, 0, 0, 0, 16, 0], [0, -171, 0, 0, 0, 0, -658, 0, 0, 0, 0, -313, 0, -44, 0, 0], [-1640, 0, 0, 0, 0, 0, 0, 0, -186, 204, 0, 0, 0, 0, 0, -48], [928, 0, 0, 0, 0, 0, 0, 0, -202, 168, 0, 0, 0, 0, 0, -34], [0, 0, 0, -16, 0, -30, 0, 13, 0, 0, 0, 0, -124, 0, 0, 0], [0, -117, 0, 0, 0, 0, -1300, 0, 0, 0, 0, 100, 0, 63, 0, 0], [-412, 0, 0, 0, 0, 0, 0, 0, -114, 270, 0, 0, 0, 0, 0, 42], [0, 53, 0, 0, 0, 0, 796, 0, 0, 0, 0, 316, 0, -35, 0, 0], [-750, 0, 0, 0, 0, 0, 0, 0, 330, 334, 0, 0, 0, 0, 0, 43], [0, 0, 288, 0, -730, 0, 0, 0, 0, 0, -141, 0, 0, 0, -48, 0], [0, 0, 0, 4, 0, 197, 0, -308, 0, 0, 0, 0, 119, 0, 0, 0], [0, 0, 350, 0, -82, 0, 0, 0, 0, 0, -239, 0, 0, 0, 80, 0], [0, -177, 0, 0, 0, 0, 249, 0, 0, 0, 0, 467, 0, -64, 0, 0], [1394, 0, 0, 0, 0, 0, 0, 0, 88, -505, 0, 0, 0, 0, 0, -4], [0, 0, 0, -8, 0, 218, 0, 157, 0, 0, 0, 0, -172, 0, 0, 0], [0, 0, 0, -36, 0, -599, 0, -154, 0, 0, 0, 0, -18, 0, 0, 0], [0, -27, 0, 0, 0, 0, 572, 0, 0, 0, 0, 14, 0, -53, 0, 0], [0, 77, 0, 0, 0, 0, -1203, 0, 0, 0, 0, -185, 0, 18, 0, 0], [0, 0, 0, 2, 0, 1140, 0, -118, 0, 0, 0, 0, -276, 0, 0, 0], [0, 0, 364, 0, -611, 0, 0, 0, 0, 0, 148, 0, 0, 0, 184, 0], [0, 0, 0, 4, 0, 922, 0, 35, 0, 0, 0, 0, -400, 0, 0, 0], [-274, 0, 0, 0, 0, 0, 0, 0, 96, 319, 0, 0, 0, 0, 0, -37], [0, -285, 0, 0, 0, 0, 956, 0, 0, 0, 0, -188, 0, 72, 0, 0], [0, 0, 0, 36, 0, 1442, 0, 85, 0, 0, 0, 0, -216, 0, 0, 0], [0, 0, 608, 0, 418, 0, 0, 0, 0, 0, 28, 0, 0, 0, -38, 0], [0, 0, -72, 0, 430, 0, 0, 0, 0, 0, -5, 0, 0, 0, 71, 0], [0, -57, 0, 0, 0, 0, -649, 0, 0, 0, 0, 473, 0, 70, 0, 0], [64, 0, 0, 0, 0, 0, 0, 0, -150, 272, 0, 0, 0, 0, 0, 124], [0, 281, 0, 0, 0, 0, 124, 0, 0, 0, 0, 21, 0, 120, 0, 0], [1248, 0, 0, 0, 0, 0, 0, 0, 440, -268, 0, 0, 0, 0, 0, 28], [0, 0, -246, 0, 350, 0, 0, 0, 0, 0, -169, 0, 0, 0, 46, 0], [0, 61, 0, 0, 0, 0, 142, 0, 0, 0, 0, 353, 0, -14, 0, 0], [-458, 0, 0, 0, 0, 0, 0, 0, -116, 142, 0, 0, 0, 0, 0, -101], [0, 0, 0, 156, 0, 753, 0, 120, 0, 0, 0, 0, -243, 0, 0, 0], [0, 0, 94, 0, -66, 0, 0, 0, 0, 0, 19, 0, 0, 0, 194, 0], [0, -3, 0, 0, 0, 0, -1092, 0, 0, 0, 0, -156, 0, -117, 0, 0], [0, 0, 0, 76, 0, -867, 0, 96, 0, 0, 0, 0, -82, 0, 0, 0], [0, 0, 0, -100, 0, 509, 0, 314, 0, 0, 0, 0, 68, 0, 0, 0], [0, 0, 452, 0, -714, 0, 0, 0, 0, 0, 196, 0, 0, 0, 112, 0], [0, -133, 0, 0, 0, 0, 922, 0, 0, 0, 0, -247, 0, 164, 0, 0], [2298, 0, 0, 0, 0, 0, 0, 0, -166, 76, 0, 0, 0, 0, 0, -32], [0, 0, 0, 18, 0, -1284, 0, 81, 0, 0, 0, 0, -366, 0, 0, 0], [0, 0, 0, -160, 0, -745, 0, 136, 0, 0, 0, 0, -89, 0, 0, 0], [0, 0, 20, 0, 316, 0, 0, 0, 0, 0, -2, 0, 0, 0, -9, 0], [0, 0, 0, -10, 0, -1027, 0, -88, 0, 0, 0, 0, -316, 0, 0, 0], [470, 0, 0, 0, 0, 0, 0, 0, 442, 85, 0, 0, 0, 0, 0, -41], [0, 0, 138, 0, 136, 0, 0, 0, 0, 0, -163, 0, 0, 0, -172, 0], [0, 0, 0, -18, 0, -370, 0, 165, 0, 0, 0, 0, 704, 0, 0, 0], [0, 0, -244, 0, -734, 0, 0, 0, 0, 0, -38, 0, 0, 0, -60, 0], [0, 245, 0, 0, 0, 0, 511, 0, 0, 0, 0, 465, 0, 114, 0, 0], [-298, 0, 0, 0, 0, 0, 0, 0, -32, -654, 0, 0, 0, 0, 0, -31]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3150_c_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3150_3_c_f();".
// 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_3150_3_c_f(:prec:=16)
    chi := MakeCharacter_3150_c();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 3));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(2161) - 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_3150_3_c_f();".
// 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_3150_3_c_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3150_c();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<11,R![15872256, 0, 1673216, 0, 44832, 0, 384, 0, 1]>],Snew);
    return Vf;
end function;