
// Make newform 1900.3.e.e in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1900_e();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1900_e_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_1900_3_e_e();".
// 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_1900_3_e_e();".
// 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 := [687241, 0, 71664, 0, 36676, 0, 270, 0, 44, 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], [11929187385, 0, -167525143, 0, -2760266, 0, -7852919, 0, 464639, 0, -1092, 0], [558303007, 0, 1825258945, 0, -54167333, 0, 4391039, 0, -524952, 0, 28618, 0], [21947164823, 0, -517724371, 0, 100053868, 0, -32340835, 0, 2443821, 0, -60512, 0], [-184041084331, 0, -157720051429, 0, -1329918273, 0, -153849803, 0, 85163504, 0, -4088716, 0], [473221869537, 0, 260828374443, 0, 10398083744, 0, 139206261, 0, -172547115, 0, 8648182, 0], [0, -2065201213063, 0, -354532580162, 0, 3883239581, 0, -813261775, 0, 195485288, 0, -9729001], [0, 36331199656762, 0, -12940529662267, 0, 162835339255, 0, -28805936576, 0, 5048505213, 0, -194997239], [0, 25942480991957, 0, -17994962875673, 0, -565819505556, 0, -65363306369, 0, 15674641213, 0, -651724712], [0, -10974976155720, 0, -1532495976047, 0, -52572021371, 0, 6471056538, 0, 503811445, 0, -39207097], [0, 134361173655723, 0, -57879968630252, 0, -107857958145, 0, -155977137845, 0, 35580028314, 0, -1586126027], [0, 34021640801649, 0, 4527798347604, 0, 240525327873, 0, -24848763895, 0, -992670198, 0, 108734495]];
    Rf_basisdens := [1, 1817684474, 908842237, 1817684474, 17268002503, 34536005006, 1506860428946, 28630348149974, 28630348149974, 1506860428946, 28630348149974, 1506860428946];
    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_1900_e();"
function MakeCharacter_1900_e()
    N := 1900;
    order := 2;
    char_gens := [951, 77, 401];
    v := [2, 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_1900_e_Hecke();"
function MakeCharacter_1900_e_Hecke(Kf)
    N := 1900;
    order := 2;
    char_gens := [951, 77, 401];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 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, 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, 0, 0, 1, 0, 0, 0], [-2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -1, 0, 0, 1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0], [-6, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -5, 0, 0, 0], [0, 0, -1, 0, 0, -1, 0, 0, 0, 0, 0, 0], [0, 0, -1, 0, -1, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, -1], [0, 0, -3, 0, -1, -1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -6, 1, 0, 1, 0], [0, 0, 0, 0, 0, 0, 0, 3, -1, 0, 3, 0], [0, 0, 0, 0, 0, 0, 7, 0, 0, -1, 0, 0], [0, 0, 2, 0, 3, 0, 0, 0, 0, 0, 0, 0], [-50, 7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -5, 0, 0, 0, 0, -2], [0, 0, -3, 0, 1, -1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 7, -1, 0, -2, 0], [0, 0, -6, 0, 0, 2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0], [0, 0, 1, 0, -3, -1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 15, 0, 0, 4, 0, 1], [-36, -15, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -5, 0, 0, 7, 0, -1], [0, 0, 0, 0, 0, 0, 11, 0, 0, 4, 0, 0], [0, 0, -1, 0, 0, -5, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -17, 0, 0, -11, 0, -1], [0, 0, 0, 0, 0, 0, -29, 0, 0, -2, 0, -1], [-56, -8, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -7, 1, 0, 4, 0], [54, -2, 0, -4, 0, 0, 0, 0, 0, 0, 0, 0], [34, 22, 0, -8, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -4, 0, -6, -8, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 5, 15, 0, -5, 0], [0, 0, 0, 0, 0, 0, 0, 2, -25, 0, 5, 0], [0, 0, 0, 0, 0, 0, 21, 0, 0, -9, 0, 3], [0, 0, 0, 0, 0, 0, -59, 0, 0, 0, 0, 1], [0, 0, 2, 0, -6, -4, 0, 0, 0, 0, 0, 0], [0, 0, -8, 0, 0, 2, 0, 0, 0, 0, 0, 0], [25, 30, 0, -17, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 21, 0, 0, -6, 0, 3], [0, 0, 0, 0, 0, 0, 0, 21, -30, 0, -8, 0], [-9, -20, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -2, 0, -11, -8, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 41, 0, 0, 20, 0, 1], [0, 0, 0, 0, 0, 0, 7, 0, 0, 2, 0, 0], [-20, -34, 0, -14, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -2, -16, 0, 4, 0], [75, -6, 0, -11, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -14, 0, 4, 2, 0, 0, 0, 0, 0, 0], [86, 3, 0, -12, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 25, 0, 0, -7, 0, 3], [0, 0, 0, 0, 0, 0, 0, 11, 21, 0, 7, 0], [0, 0, -6, 0, 8, 12, 0, 0, 0, 0, 0, 0], [53, 56, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 7, 29, 0, 5, 0], [0, 0, -11, 0, 5, 7, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 20, -26, 0, 2, 0], [0, 0, 0, 0, 0, 0, -9, 0, 0, 19, 0, 4], [0, 0, 0, 0, 0, 0, -1, 0, 0, -6, 0, -7], [-67, -12, 0, 23, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 11, 5, 0, 18, 0], [0, 0, 0, 0, 0, 0, 5, 0, 0, -11, 0, 0], [0, 0, 12, 0, -1, 6, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 29, 0, 0, 5, 0, -5], [0, 0, 0, 0, 0, 0, 0, 42, 36, 0, 8, 0], [-170, -52, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 13, 35, 0, -10, 0], [121, 14, 0, 31, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -41, -35, 0, 9, 0], [0, 0, 0, 0, 0, 0, 19, 0, 0, 13, 0, 2], [0, 0, 6, 0, -3, -8, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 35, 0, 0, -9, 0, -13], [-56, 11, 0, -6, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -17, 21, 0, 13, 0], [0, 0, 13, 0, 5, -9, 0, 0, 0, 0, 0, 0], [0, 0, -9, 0, -15, -11, 0, 0, 0, 0, 0, 0], [200, -37, 0, 26, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 11, 0, -12, 7, 0, 0, 0, 0, 0, 0], [0, 0, 3, 0, 13, 9, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 3, 0, 0, 7, 0, -11], [0, 0, -3, 0, -5, 23, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 86, 21, 0, 7, 0], [0, 0, 20, 0, 2, -16, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 47, -81, 0, -24, 0], [206, 59, 0, 12, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -27, -109, 0, -1, 0], [0, 0, 0, 0, 0, 0, 0, -24, 1, 0, 11, 0], [-146, 60, 0, 28, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -25, 0, 0, 2, 0, -5], [384, -55, 0, -34, 0, 0, 0, 0, 0, 0, 0, 0], [80, -83, 0, 30, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 44, 55, 0, -18, 0], [0, 0, -8, 0, -4, 10, 0, 0, 0, 0, 0, 0], [0, 0, 2, 0, 4, 10, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 125, 0, 0, 6, 0, -4], [106, -27, 0, 12, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 111, 0, 0, 0, 0, -9], [0, 0, 0, 0, 0, 0, 0, -49, -41, 0, -11, 0], [0, 0, 0, 0, 0, 0, 5, 0, 0, -14, 0, -9], [0, 0, 28, 0, -6, -4, 0, 0, 0, 0, 0, 0], [358, 31, 0, 24, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -15, 29, 0, 20, 0], [0, 0, 0, 0, 0, 0, 0, 14, -69, 0, 7, 0], [0, 0, 0, 0, 0, 0, 0, -24, -40, 0, 2, 0], [0, 0, 27, 0, 5, 1, 0, 0, 0, 0, 0, 0], [0, 0, 35, 0, 21, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -193, 0, 0, 10, 0, 9], [0, 0, 0, 0, 0, 0, 0, -45, -76, 0, -12, 0], [0, 0, 0, 0, 0, 0, 0, 8, 86, 0, 10, 0], [-128, -11, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0], [-144, 56, 0, 34, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -5, 0, -33, -3, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 72, 44, 0, -10, 0], [0, 0, 0, 0, 0, 0, 0, -14, 49, 0, -6, 0], [0, 0, 0, 0, 0, 0, 0, -21, -61, 0, 19, 0], [0, 0, 26, 0, -5, 20, 0, 0, 0, 0, 0, 0], [0, 0, -9, 0, -10, 11, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -131, 0, 0, -28, 0, 3], [0, 0, 0, 0, 0, 0, 1, 0, 0, -47, 0, -4], [0, 0, 0, 0, 0, 0, -75, 0, 0, 54, 0, 3], [284, 94, 0, -10, 0, 0, 0, 0, 0, 0, 0, 0], [96, -64, 0, 58, 0, 0, 0, 0, 0, 0, 0, 0], [216, 28, 0, 34, 0, 0, 0, 0, 0, 0, 0, 0], [589, 2, 0, 67, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 84, 129, 0, -6, 0], [0, 0, 0, 0, 0, 0, 0, 59, 35, 0, 29, 0], [-456, 37, 0, 58, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -129, 0, 0, 15, 0, 21], [0, 0, 17, 0, 1, 11, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -55, 21, 0, 11, 0], [-119, 103, 0, 27, 0, 0, 0, 0, 0, 0, 0, 0], [-317, 103, 0, -47, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -185, 0, 0, -39, 0, -2], [0, 0, 0, 0, 0, 0, 71, 0, 0, 40, 0, -6], [0, 0, 0, 0, 0, 0, -135, 0, 0, -47, 0, -8], [-325, 57, 0, -19, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -32, 0, -3, 38, 0, 0, 0, 0, 0, 0], [310, -40, 0, 28, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -28, 21, 0, -44, 0], [0, 0, 0, 0, 0, 0, -69, 0, 0, 62, 0, -16], [0, 0, 11, 0, -34, -1, 0, 0, 0, 0, 0, 0], [0, 0, 60, 0, 4, -12, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 91, 85, 0, 15, 0], [0, 0, 0, 0, 0, 0, -69, 0, 0, 18, 0, 15], [-152, 51, 0, -54, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -253, 0, 0, -63, 0, -1], [0, 0, 0, 0, 0, 0, 229, 0, 0, 51, 0, 16], [284, -24, 0, -82, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 48, -64, 0, 20, 0], [0, 0, 0, 0, 0, 0, 13, 0, 0, -23, 0, -7], [0, 0, 0, 0, 0, 0, 139, 0, 0, -40, 0, -6], [0, 0, -12, 0, 28, 36, 0, 0, 0, 0, 0, 0], [135, -64, 0, -47, 0, 0, 0, 0, 0, 0, 0, 0], [935, 29, 0, -55, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -47, -219, 0, 12, 0], [0, 0, 35, 0, 34, 7, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -82, 20, 0, 40, 0], [0, 0, 0, 0, 0, 0, 127, 0, 0, 19, 0, -19], [0, 0, 0, 0, 0, 0, 0, -1, 65, 0, -19, 0], [0, 0, 40, 0, 18, -42, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -219, 0, 0, -23, 0, -11], [0, 0, 0, 0, 0, 0, 317, 0, 0, -12, 0, -7], [0, 0, -39, 0, -13, -21, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 13, -204, 0, -18, 0], [0, 0, -26, 0, -8, -38, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -7, 76, 0, 42, 0], [0, 0, 0, 0, -1, 46, 0, 0, 0, 0, 0, 0], [0, 0, 39, 0, 20, -21, 0, 0, 0, 0, 0, 0], [-224, 80, 0, 114, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 7, 165, 0, 2, 0], [0, 0, -25, 0, -29, -51, 0, 0, 0, 0, 0, 0], [-944, -156, 0, 46, 0, 0, 0, 0, 0, 0, 0, 0], [18, 157, 0, -64, 0, 0, 0, 0, 0, 0, 0, 0], [-262, -42, 0, 68, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -209, 0, 0, 71, 0, -3], [76, 147, 0, 58, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -42, 81, 0, 40, 0], [0, 0, -74, 0, -19, -12, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 265, 0, 0, 14, 0, -15], [0, 0, 0, 0, 0, 0, 71, 0, 0, 58, 0, -21], [0, 0, 0, 0, 0, 0, 0, 28, 19, 0, -12, 0], [-612, 72, 0, 70, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -79, 0, 0, 51, 0, 14], [0, 0, 0, 0, 0, 0, -271, 0, 0, -64, 0, 7], [0, 0, -37, 0, -45, 25, 0, 0, 0, 0, 0, 0], [-157, -28, 0, -39, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -129, 0, 0, -1, 0, -13], [0, 0, 0, 0, 0, 0, 0, 56, -126, 0, 20, 0], [0, 0, 2, 0, -8, -16, 0, 0, 0, 0, 0, 0], [0, 0, 37, 0, -26, -31, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 10, -30, 0, 44, 0], [0, 0, 0, 0, 0, 0, 83, 0, 0, 27, 0, 11], [-983, 7, 0, 7, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 107, 35, 0, -25, 0], [0, 0, 0, 0, 0, 0, 0, 89, 285, 0, -10, 0], [0, 0, 0, 0, 0, 0, 0, 24, 9, 0, 54, 0], [0, 0, 70, 0, 30, -36, 0, 0, 0, 0, 0, 0], [0, 0, -18, 0, -14, -46, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 229, 0, 0, 25, 0, -10], [0, 0, -81, 0, 31, 13, 0, 0, 0, 0, 0, 0], [836, -168, 0, 42, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 69, -49, 0, -65, 0], [855, 36, 0, -19, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 229, 0, 0, 44, 0, -12], [217, -163, 0, 27, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -8, 0, 17, -42, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -145, -135, 0, 4, 0], [888, 112, 0, -62, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -55, -91, 0, -23, 0], [0, 0, 0, 0, 0, 0, 119, 0, 0, -60, 0, -22], [0, 0, 41, 0, -9, 23, 0, 0, 0, 0, 0, 0], [0, 0, -131, 0, 7, 7, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -46, 95, 0, -58, 0], [0, 0, -48, 0, -22, -88, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -109, 0, 0, 10, 0, 45], [0, 0, 0, 0, 0, 0, 0, -1, -159, 0, -49, 0], [0, 0, -42, 0, 2, -8, 0, 0, 0, 0, 0, 0], [0, 0, 6, 0, 58, 6, 0, 0, 0, 0, 0, 0], [0, 0, 47, 0, 27, -11, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -42, -149, 0, -36, 0], [0, 0, 0, 0, 0, 0, -257, 0, 0, 74, 0, -12], [-1412, -6, 0, -42, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -55, 0, 0, 78, 0, -7], [0, 0, -67, 0, -59, 31, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 73, 0, 0, -1, 0, 3], [116, 90, 0, -54, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -227, 16, 0, -12, 0], [0, 0, -92, 0, -9, 30, 0, 0, 0, 0, 0, 0], [0, 0, -8, 0, 50, 88, 0, 0, 0, 0, 0, 0], [0, 0, 40, 0, -46, -32, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 96, -34, 0, 64, 0], [0, 0, 0, 0, 0, 0, 0, -20, -109, 0, 64, 0], [-716, 38, 0, -14, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 33, 226, 0, -34, 0], [-1452, -14, 0, 74, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -83, 0, 3, -13, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -75, 0, 0, -12, 0, 15], [-1132, -129, 0, 66, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 22, 345, 0, -42, 0], [0, 0, -29, 0, -22, 19, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 135, 0, 0, 59, 0, 41], [778, 280, 0, 12, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 7, -231, 0, -57, 0], [0, 0, -54, 0, 60, 20, 0, 0, 0, 0, 0, 0], [0, 0, -16, 0, 66, -2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 90, 209, 0, 20, 0], [0, 0, 0, 0, 0, 0, 0, -111, -80, 0, -28, 0], [439, -337, 0, 37, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 186, 55, 0, -18, 0], [0, 0, -15, 0, -91, -69, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 49, 184, 0, -72, 0], [-344, 345, 0, 42, 0, 0, 0, 0, 0, 0, 0, 0], [1506, -10, 0, 56, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 11, 0, 0, 80, 0, 13], [0, 0, 0, 0, 0, 0, 359, 0, 0, 40, 0, 15], [0, 0, 0, 0, 0, 0, 0, -52, 250, 0, -34, 0], [0, 0, 0, 0, 0, 0, 331, 0, 0, -65, 0, -1], [0, 0, 0, 0, 0, 0, 167, 0, 0, 36, 0, 29], [756, -1, 0, 146, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -225, 0, 0, -37, 0, 44], [0, 0, 0, 0, 0, 0, 0, 42, 346, 0, -62, 0], [0, 0, -10, 0, -75, -96, 0, 0, 0, 0, 0, 0], [0, 0, 105, 0, -36, 37, 0, 0, 0, 0, 0, 0], [-1323, 155, 0, 103, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 177, 0, 0, 78, 0, 12], [0, 0, 0, 0, 0, 0, 0, -35, 34, 0, 54, 0], [0, 0, 37, 0, -32, 5, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -253, 0, 0, 14, 0, -8], [0, 0, 0, 0, 0, 0, 0, 253, 181, 0, -28, 0], [-1487, 90, 0, -89, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 81, 0, 0, -111, 0, 21], [0, 0, 0, 0, 0, 0, 0, 4, -125, 0, 48, 0], [0, 0, 0, 0, 0, 0, 0, 16, -249, 0, -55, 0], [0, 0, 45, 0, 92, 9, 0, 0, 0, 0, 0, 0], [0, 0, 2, 0, -50, 6, 0, 0, 0, 0, 0, 0], [762, 52, 0, -44, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -203, 0, 0, -80, 0, 11], [665, 166, 0, 35, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 80, 71, 0, 76, 0], [0, 0, 41, 0, 58, 77, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -60, -36, 0, 18, 0], [-610, -328, 0, 80, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 31, 309, 0, -64, 0], [0, 0, 0, 0, 0, 0, -45, 0, 0, -100, 0, 5], [787, -336, 0, -195, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 104, 0, -20, -44, 0, 0, 0, 0, 0, 0], [1784, -3, 0, 54, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -146, -164, 0, 64, 0], [0, 0, 0, 0, 0, 0, 315, 0, 0, 77, 0, 47], [0, 0, 62, 0, -48, 4, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 449, 0, 0, 62, 0, 25], [-1296, 265, 0, -82, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 78, 0, 4, 46, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -243, 89, 0, 25, 0], [0, 0, 60, 0, 13, 42, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -142, 259, 0, 29, 0], [0, 0, 0, 0, 0, 0, 0, -172, -244, 0, 68, 0], [0, 0, 0, 0, 0, 0, 533, 0, 0, 145, 0, 18], [-1603, 152, 0, -189, 0, 0, 0, 0, 0, 0, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1900_e_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1900_3_e_e();".
// 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_1900_3_e_e(:prec:=12)
    chi := MakeCharacter_1900_e();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 3));
    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_1900_3_e_e();".
// 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_1900_3_e_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1900_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<3,R![1280, 0, 497, 0, 46, 0, 1]>,<7,R![-9408, 0, 2393, 0, -94, 0, 1]>],Snew);
    return Vf;
end function;