
// Make newform 3150.3.c.c in Magma, downloaded from the LMFDB on 23 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_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_3150_3_c_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 := [16, 0, 0, 0, 1, 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], [1, 0, 0, 0, 2, 0, 0, 0], [0, 0, 5, 0, 0, 0, 1, 0], [0, -4, 0, 7, 0, 4, 0, -1], [0, 0, 3, 0, 0, 0, -1, 0], [0, 10, 0, -5, 0, 2, 0, -1], [0, 4, 0, 7, 0, -4, 0, -1], [0, 10, 0, 5, 0, 2, 0, 1]];
    Rf_basisdens := [1, 3, 12, 24, 4, 12, 24, 12];
    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], [-1, 0, 0, 0, 0, 0, 0, 0], [1, 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, -1, 0, 0, 0, 0, 0, 0], [0, 0, 0, -9, 0, 0, 0, 7], [0, -1, 3, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -4, -19, 0], [15, 0, 0, 0, 5, 0, 0, 0], [0, 0, 0, 0, 0, -13, -7, 0], [0, 0, 0, -15, 0, 0, 0, 7], [4, 0, 0, 0, 8, 0, 0, 0], [0, 5, 28, 0, 0, 0, 0, 0], [0, 0, 0, -6, 0, 0, 0, 18], [0, 6, 55, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 30, 27, 0], [0, 0, 0, 0, 0, 28, 37, 0], [0, 0, 0, -33, 0, 0, 0, 36], [-22, 0, 0, 0, 24, 0, 0, 0], [0, -7, -38, 0, 0, 0, 0, 0], [0, 0, 0, 6, 0, 0, 0, -43], [0, 35, -21, 0, 0, 0, 0, 0], [-36, 0, 0, 0, -19, 0, 0, 0], [0, 0, 0, 0, 0, 6, -96, 0], [0, 0, 0, 39, 0, 0, 0, 34], [0, 36, 38, 0, 0, 0, 0, 0], [0, 0, 0, -51, 0, 0, 0, 30], [0, 11, -93, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -36, -33, 0], [52, 0, 0, 0, 15, 0, 0, 0], [0, 0, 0, 0, 0, -21, 78, 0], [0, -21, 32, 0, 0, 0, 0, 0], [0, 0, 0, 87, 0, 0, 0, -44], [0, 0, 0, 0, 0, 22, 1, 0], [-83, 0, 0, 0, 21, 0, 0, 0], [0, 0, 0, -42, 0, 0, 0, 77], [213, 0, 0, 0, 22, 0, 0, 0], [0, 2, -188, 0, 0, 0, 0, 0], [0, 72, 130, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 106, 19, 0], [0, 0, 0, 0, 0, -102, -51, 0], [0, 0, 0, -159, 0, 0, 0, 76], [61, 0, 0, 0, -55, 0, 0, 0], [0, 0, 0, 15, 0, 0, 0, -92], [0, 75, 76, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 57, 87, 0], [-52, 0, 0, 0, -2, 0, 0, 0], [-38, 0, 0, 0, 104, 0, 0, 0], [0, 117, -77, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 40, 13, 0], [-104, 0, 0, 0, -60, 0, 0, 0], [0, 0, 0, 0, 0, -179, -53, 0], [0, 0, 0, -129, 0, 0, 0, 168], [3, 0, 0, 0, 91, 0, 0, 0], [0, 0, 0, -153, 0, 0, 0, 120], [0, 0, 0, 0, 0, -86, -74, 0], [0, 0, 0, 0, 0, -23, 46, 0], [0, 0, 0, 114, 0, 0, 0, -30], [45, 0, 0, 0, 13, 0, 0, 0], [0, -20, -48, 0, 0, 0, 0, 0], [0, 0, 0, 165, 0, 0, 0, -45], [0, -10, 222, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -96, -147, 0], [0, -147, 107, 0, 0, 0, 0, 0], [0, 0, 0, 117, 0, 0, 0, 26], [0, 181, -19, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1, -41, 0], [-431, 0, 0, 0, -10, 0, 0, 0], [0, -172, -50, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -19, 59, 0], [-233, 0, 0, 0, 33, 0, 0, 0], [0, 0, 0, 0, 0, -88, -226, 0], [0, 0, 0, 180, 0, 0, 0, 39], [0, 40, 480, 0, 0, 0, 0, 0], [0, -108, 133, 0, 0, 0, 0, 0], [-285, 0, 0, 0, -22, 0, 0, 0], [0, 0, 0, 0, 0, -34, -79, 0], [0, 0, 0, -162, 0, 0, 0, -23], [0, -142, 82, 0, 0, 0, 0, 0], [0, 0, 0, -417, 0, 0, 0, 59], [-123, 0, 0, 0, 11, 0, 0, 0], [0, 0, 0, -210, 0, 0, 0, 80], [-111, 0, 0, 0, -50, 0, 0, 0], [0, 0, 0, -171, 0, 0, 0, -78], [0, -82, 300, 0, 0, 0, 0, 0], [513, 0, 0, 0, -13, 0, 0, 0], [0, 0, 0, 0, 0, 156, 369, 0], [0, 0, 0, 522, 0, 0, 0, -61], [0, -113, -462, 0, 0, 0, 0, 0], [0, 0, 0, 225, 0, 0, 0, -284], [0, -26, -436, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 222, 168, 0], [0, 0, 0, -351, 0, 0, 0, 188], [0, 160, -105, 0, 0, 0, 0, 0], [0, 0, 0, 276, 0, 0, 0, 159], [578, 0, 0, 0, 10, 0, 0, 0], [0, 0, 0, 0, 0, -256, 218, 0], [0, 0, 0, 327, 0, 0, 0, -78], [0, 0, 0, -318, 0, 0, 0, -90], [0, -190, -348, 0, 0, 0, 0, 0], [143, 0, 0, 0, 114, 0, 0, 0], [0, 143, -188, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -381, -132, 0], [0, 0, 0, 0, 0, -254, -182, 0], [0, 0, 0, -249, 0, 0, 0, -79], [222, 0, 0, 0, 157, 0, 0, 0], [0, -131, 27, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -498, -129, 0], [0, 0, 0, 0, 0, -318, -510, 0], [0, 0, 0, -267, 0, 0, 0, 189], [432, 0, 0, 0, -50, 0, 0, 0], [0, -11, -123, 0, 0, 0, 0, 0], [0, -296, 245, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 47, 344, 0], [559, 0, 0, 0, 177, 0, 0, 0], [818, 0, 0, 0, -123, 0, 0, 0], [0, 0, 0, 624, 0, 0, 0, 23], [0, -30, -506, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 144, 12, 0], [0, 0, 0, 0, 0, 4, -89, 0], [0, 0, 0, 225, 0, 0, 0, -366], [-997, 0, 0, 0, -57, 0, 0, 0], [0, -14, -664, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -68, -86, 0], [0, 0, 0, 0, 0, -141, -21, 0], [-866, 0, 0, 0, 110, 0, 0, 0], [0, 0, 0, 123, 0, 0, 0, -378], [-1038, 0, 0, 0, -16, 0, 0, 0], [0, 0, 0, 756, 0, 0, 0, -100], [0, -51, -457, 0, 0, 0, 0, 0], [0, -154, -558, 0, 0, 0, 0, 0], [-170, 0, 0, 0, -387, 0, 0, 0], [0, 0, 0, 0, 0, 396, 453, 0], [558, 0, 0, 0, -158, 0, 0, 0], [0, 85, -726, 0, 0, 0, 0, 0], [0, 0, 0, 174, 0, 0, 0, -130], [-216, 0, 0, 0, -340, 0, 0, 0], [0, 0, 0, 0, 0, 338, 428, 0], [0, -231, 887, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -78, 183, 0], [0, 0, 0, -423, 0, 0, 0, -205], [-408, 0, 0, 0, 208, 0, 0, 0], [0, 0, 0, -213, 0, 0, 0, 236], [0, -23, 326, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 289, -194, 0], [1331, 0, 0, 0, 97, 0, 0, 0], [0, 0, 0, -255, 0, 0, 0, 542], [0, -406, 162, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -484, -598, 0], [384, 0, 0, 0, -352, 0, 0, 0], [0, 0, 0, 0, 0, -59, 154, 0], [0, 72, 1376, 0, 0, 0, 0, 0], [0, 0, 0, 666, 0, 0, 0, -224], [0, -228, 829, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -148, 332, 0], [0, 106, 186, 0, 0, 0, 0, 0], [0, 0, 0, 663, 0, 0, 0, -95], [465, 0, 0, 0, -484, 0, 0, 0], [0, 0, 0, 129, 0, 0, 0, 466], [0, 220, 72, 0, 0, 0, 0, 0], [0, 0, 0, -846, 0, 0, 0, -162], [0, 0, 0, 0, 0, 190, -443, 0], [0, 0, 0, 0, 0, -47, 736, 0], [0, -164, -852, 0, 0, 0, 0, 0], [0, 0, 0, 291, 0, 0, 0, -310], [0, 0, 0, 0, 0, 195, 69, 0], [0, 0, 0, 0, 0, 132, -18, 0], [93, 0, 0, 0, -452, 0, 0, 0], [0, 31, 1047, 0, 0, 0, 0, 0], [-399, 0, 0, 0, 416, 0, 0, 0], [0, 0, 0, 0, 0, -78, -1191, 0], [0, 0, 0, -390, 0, 0, 0, 91], [1701, 0, 0, 0, -17, 0, 0, 0], [0, 0, 0, -21, 0, 0, 0, -553], [0, -376, -402, 0, 0, 0, 0, 0], [73, 0, 0, 0, -717, 0, 0, 0], [0, 0, 0, 147, 0, 0, 0, -762], [162, 0, 0, 0, 37, 0, 0, 0], [0, 0, 0, 69, 0, 0, 0, -378], [0, 121, -1189, 0, 0, 0, 0, 0], [46, 0, 0, 0, -498, 0, 0, 0], [0, -205, -1562, 0, 0, 0, 0, 0], [0, 0, 0, -870, 0, 0, 0, 108], [0, 146, -1017, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -452, -512, 0], [0, 0, 0, 0, 0, 9, 705, 0], [0, 0, 0, 480, 0, 0, 0, -114], [0, 219, -922, 0, 0, 0, 0, 0], [0, 58, 1316, 0, 0, 0, 0, 0], [1063, 0, 0, 0, 90, 0, 0, 0], [0, 0, 0, -603, 0, 0, 0, -384], [0, -132, 1060, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 757, -11, 0], [-761, 0, 0, 0, -184, 0, 0, 0], [0, 0, 0, 378, 0, 0, 0, 528], [0, 0, 0, 0, 0, 979, 232, 0], [0, 0, 0, 0, 0, 420, -738, 0], [1322, 0, 0, 0, -117, 0, 0, 0], [0, 66, 1123, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -608, 487, 0], [0, 0, 0, 0, 0, 276, -294, 0], [0, 0, 0, 969, 0, 0, 0, 108], [36, 0, 0, 0, -368, 0, 0, 0], [0, -162, -298, 0, 0, 0, 0, 0], [756, 0, 0, 0, -244, 0, 0, 0], [0, 0, 0, 252, 0, 0, 0, 642], [0, 0, 0, 0, 0, 632, 728, 0], [-1338, 0, 0, 0, 248, 0, 0, 0], [0, 0, 0, 0, 0, 984, 681, 0], [0, 0, 0, 522, 0, 0, 0, 475], [-1894, 0, 0, 0, -78, 0, 0, 0], [0, 114, 1157, 0, 0, 0, 0, 0], [0, 0, 0, -459, 0, 0, 0, 858], [0, -322, -1041, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -194, 256, 0], [0, 0, 0, -645, 0, 0, 0, -166], [-1095, 0, 0, 0, 289, 0, 0, 0], [0, -80, 177, 0, 0, 0, 0, 0], [0, 0, 0, -765, 0, 0, 0, 1092], [0, 0, 0, 0, 0, 437, -847, 0], [0, 0, 0, 0, 0, -915, -423, 0], [-892, 0, 0, 0, 225, 0, 0, 0], [329, 0, 0, 0, 445, 0, 0, 0], [0, 0, 0, -102, 0, 0, 0, -45], [0, -88, 1905, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 542, 269, 0], [-328, 0, 0, 0, 490, 0, 0, 0], [0, 0, 0, 0, 0, 174, 705, 0], [0, 0, 0, -882, 0, 0, 0, -429], [0, 1009, 81, 0, 0, 0, 0, 0], [0, 0, 0, 42, 0, 0, 0, 163], [0, -457, 342, 0, 0, 0, 0, 0], [618, 0, 0, 0, -142, 0, 0, 0], [744, 0, 0, 0, 40, 0, 0, 0], [0, 0, 0, -1236, 0, 0, 0, 143], [0, 631, -943, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -454, -1282, 0], [-604, 0, 0, 0, 616, 0, 0, 0], [0, 0, 0, 0, 0, -939, 252, 0], [0, 0, 0, 24, 0, 0, 0, 541], [0, 0, 0, -1062, 0, 0, 0, 916], [0, 0, 0, 0, 0, 315, 654, 0], [1491, 0, 0, 0, -233, 0, 0, 0], [0, 88, 932, 0, 0, 0, 0, 0], [1108, 0, 0, 0, -375, 0, 0, 0], [0, 0, 0, 0, 0, -30, 39, 0], [0, 0, 0, -813, 0, 0, 0, 426], [0, -142, -368, 0, 0, 0, 0, 0], [0, 0, 0, -1410, 0, 0, 0, 296], [-1790, 0, 0, 0, 180, 0, 0, 0], [0, 0, 0, 0, 0, -122, -575, 0], [0, -32, -2010, 0, 0, 0, 0, 0], [0, 0, 0, -72, 0, 0, 0, -812], [0, 0, 0, 0, 0, -136, 281, 0], [2039, 0, 0, 0, -353, 0, 0, 0], [0, 0, 0, 0, 0, -1390, -337, 0], [0, 0, 0, -249, 0, 0, 0, 664], [1185, 0, 0, 0, -602, 0, 0, 0], [0, -47, 1785, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 896, 1427, 0], [0, -519, -49, 0, 0, 0, 0, 0], [0, 524, -36, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -117, 759, 0], [712, 0, 0, 0, 678, 0, 0, 0], [0, -318, -1040, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -174, -477, 0], [198, 0, 0, 0, -382, 0, 0, 0], [0, 0, 0, 342, 0, 0, 0, -1073], [0, 0, 0, -1065, 0, 0, 0, -492], [0, 105, 934, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 735, 1086, 0], [-948, 0, 0, 0, 778, 0, 0, 0], [0, -302, -2106, 0, 0, 0, 0, 0], [0, 198, -344, 0, 0, 0, 0, 0], [-89, 0, 0, 0, 378, 0, 0, 0], [0, 121, 393, 0, 0, 0, 0, 0], [0, 35, 1719, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -671, -1082, 0], [346, 0, 0, 0, 288, 0, 0, 0], [-2154, 0, 0, 0, -125, 0, 0, 0], [0, 0, 0, -393, 0, 0, 0, 920], [0, 0, 0, 0, 0, 368, -271, 0], [68, 0, 0, 0, 360, 0, 0, 0], [0, 0, 0, 0, 0, 66, 1731, 0], [19, 0, 0, 0, 125, 0, 0, 0], [0, 387, -613, 0, 0, 0, 0, 0], [0, 0, 0, -1050, 0, 0, 0, -193], [0, 129, -1460, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 391, -959, 0], [592, 0, 0, 0, 474, 0, 0, 0], [0, 0, 0, 360, 0, 0, 0, -1330], [0, 0, 0, -165, 0, 0, 0, 725], [0, 0, 0, 0, 0, 1240, -164, 0], [0, 0, 0, 0, 0, -693, -738, 0], [0, 0, 0, 813, 0, 0, 0, -94], [0, 1228, 575, 0, 0, 0, 0, 0], [0, 0, 0, 1455, 0, 0, 0, -720], [-674, 0, 0, 0, -406, 0, 0, 0], [0, 0, 0, 0, 0, -724, -415, 0], [0, 0, 0, 1338, 0, 0, 0, -26], [0, 96, 1370, 0, 0, 0, 0, 0], [0, -19, -183, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 820, -917, 0], [2720, 0, 0, 0, -113, 0, 0, 0], [0, 0, 0, 0, 0, 307, 1480, 0], [-422, 0, 0, 0, 521, 0, 0, 0], [0, -1172, -540, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 105, 1716, 0], [1073, 0, 0, 0, 895, 0, 0, 0], [0, 0, 0, 1773, 0, 0, 0, -1329], [0, 928, -394, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -846, -1365, 0], [0, 0, 0, -831, 0, 0, 0, -296], [0, 0, 0, -1338, 0, 0, 0, 1683], [0, -66, -3374, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1043, -85, 0], [1507, 0, 0, 0, 195, 0, 0, 0], [0, 0, 0, -48, 0, 0, 0, 336], [0, 0, 0, -1179, 0, 0, 0, 90], [0, 624, 1456, 0, 0, 0, 0, 0], [0, 0, 0, -1203, 0, 0, 0, -585], [-3927, 0, 0, 0, 55, 0, 0, 0], [0, -614, -1188, 0, 0, 0, 0, 0], [0, 0, 0, 177, 0, 0, 0, 328], [0, 1212, 589, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -1349, -212, 0], [-86, 0, 0, 0, -10, 0, 0, 0]];
    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_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_3150_3_c_c(:prec:=8)
    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_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_3150_3_c_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3150_c();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<11,R![12321, 0, 464, 0, 1]>],Snew);
    return Vf;
end function;