
// Make newform 1210.3.d.c in Magma, downloaded from the LMFDB on 21 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_1210_d();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1210_d_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_1210_3_d_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_1210_3_d_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 := [339856, -1437952, 3026408, -3906048, 3947928, -2898272, 1876224, -924384, 423553, -147672, 49836, -12280, 3078, -504, 92, -8, 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], [504479976, -562402864, -103456032, 1761270816, -2044156418, 1655832332, -1010873414, 447109587, -181766829, 51088887, -14744955, 2593985, -536177, 47929, -6847, 0], [120629080, -740544624, 1846412444, -2628630822, 2425780539, -1533383522, 861443717, -336239873, 130733321, -33809419, 9511204, -1570227, 320536, -27153, 3879, 0], [2438433712, -7845001032, 15524320008, -18174156156, 16612923328, -10437600250, 5953906664, -2387837003, 943586351, -252027133, 71798033, -12241685, 2517773, -220381, 31483, 0], [14465132, -68821248, 136210210, -159558816, 146063689, -92056712, 52719277, -21288328, 8446186, -2273630, 649767, -111664, 23009, -2030, 290, 0], [1018240960, -3545899104, 7154754608, -8631102060, 8109606064, -5306651383, 3098791452, -1290572435, 518791610, -143104318, 41233831, -7228931, 1495764, -134281, 19183, 0], [-3587134496, 12381687712, -22896801856, 24628950564, -21917871024, 13275894434, -7507448955, 2974171305, -1175790672, 314682485, -90037978, 15526187, -3207332, 285985, -40855, 0], [-172033840, -3822269408, 9729524408, -14422322660, 14239013312, -9905604016, 5904435179, -2535831402, 1031609721, -290825792, 84477033, -15099442, 3139499, -287504, 41072, 0], [2203083416072, -7434463439008, 10499754196416, -12026863826132, 8661204282334, -5825590764308, 2646660360138, -1179812859509, 345513449337, -108327547723, 19857702799, -4457724181, 426079069, -67445001, 811875, -108250], [-677093564048, 1894315499960, -1762353077880, 1663630512420, -787613616480, 424954426628, -111617732880, 33498802385, 318966375, -1901136125, 1291731639, -416432975, 97310265, -19989215, 2150745, -286766], [19099979970720, -66367262111184, 96618001246844, -107190271002216, 72785135201834, -47257083446168, 20312303824849, -9036688782526, 2663868516519, -865804982228, 176016706921, -42831068498, 5620459767, -1023489448, 68058570, -9074476], [6234316637296, -22483109359304, 34961365535290, -40800611453100, 30216979825876, -20509478102342, 9543713656314, -4404842878959, 1395187239861, -467068277547, 101582546285, -25289136315, 3552968393, -656746807, 47201265, -6293502], [12389117708, 873976224, -117328742654, 230197357680, -292213712659, 257683672408, -170680049684, 94514939280, -39817446453, 15266511168, -4278312126, 1176406260, -210917668, 41411952, -3841320, 512176], [22363264616, -13702928288, -145000690310, 290995902992, -369642174953, 318352830296, -204918929942, 111110330224, -45507746145, 17198203436, -4709727880, 1283617556, -225878822, 44156068, -4030920, 537456], [4694721525472, -19515868005560, 38741745924992, -56639200125648, 57001127298464, -46236853476713, 27962891605420, -14893226840091, 5963862146844, -2240020456248, 607602356273, -165240999869, 28946116862, -5654937033, 515009265, -68667902], [100756609096, -596290010584, 1555637438300, -2408763795044, 2549578977330, -2089296933402, 1276130920725, -679659582858, 271881095325, -101818595050, 27470325817, -7444602322, 1293955065, -252113280, 22732830, -3031044]];
    Rf_basisdens := [1, 187906840, 93953420, 187906840, 1540220, 93953420, 187906840, 187906840, 1147922885560, 229584577112, 1147922885560, 573961442780, 9409203980, 9409203980, 573961442780, 18818407960];
    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_1210_d();"
function MakeCharacter_1210_d()
    N := 1210;
    order := 2;
    char_gens := [727, 1091];
    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_1210_d_Hecke();"
function MakeCharacter_1210_d_Hecke( : Kf := false)
    N := 1210;
    order := 2;
    char_gens := [727, 1091];
    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 := ISA(Type(Kf), Fld), 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]];
    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, -1, 0, 0, 0, 0, 0, 0], [-1, 0, 1, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 1, -3, 1, 1, 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, -2, 2, -1, -2, 1, -5, 6, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -3, 1, 9, -1, -2, -6], [0, 0, 0, 0, 0, 0, 0, 0, 3, -9, -3, 0, -3, 3, -2, 3], [6, 2, -1, 3, -3, 2, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -12, 2, 0, -1, 5, -8, 4, 0], [-22, 4, 4, 0, -11, 1, -1, -2, 0, 0, 0, 0, 0, 0, 0, 0], [-7, -12, 7, 6, -2, -2, 0, -3, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -2, -2, 5, 3, 29, -7, -6, -6], [0, 0, 0, 0, 0, 0, 0, 0, 3, 12, -7, -7, 8, 17, 2, 0], [-33, 4, -3, -2, -13, 10, -5, -4, 0, 0, 0, 0, 0, 0, 0, 0], [-34, -8, 13, -6, 6, -10, -10, -3, 0, 0, 0, 0, 0, 0, 0, 0], [0, 14, -2, 27, 14, 14, -5, 10, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -14, -2, 0, 4, 15, -12, -20, -2], [7, 12, -8, 15, -35, 3, 9, -2, 0, 0, 0, 0, 0, 0, 0, 0], [27, -6, -8, 11, 7, -9, -1, -14, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 4, 30, 1, -6, -4, 26, 4, 6], [0, 0, 0, 0, 0, 0, 0, 0, -11, -25, -15, -2, 23, -37, 0, -1], [0, 0, 0, 0, 0, 0, 0, 0, 14, 1, 2, -9, -7, 8, -54, -7], [1, -10, -10, 26, 24, -24, -2, -2, 0, 0, 0, 0, 0, 0, 0, 0], [13, 4, -6, 12, -18, -18, 6, 12, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 8, 6, 19, -21, -14, 18, -20, -2], [-9, -4, 4, 37, 12, -10, -22, 14, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 27, 25, -3, 0, 13, -14, -25, -27], [0, 0, 0, 0, 0, 0, 0, 0, 6, 28, -5, 30, 4, -30, -4, 4], [5, -14, -25, -8, -39, 18, 6, -23, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -5, 8, -5, -12, -14, -3, 12, 3], [0, 0, 0, 0, 0, 0, 0, 0, 3, 82, -15, -5, 10, 43, 24, -2], [13, 7, 17, -34, -2, 21, -5, -38, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 20, 27, 8, 3, 5, -3, -11, 3], [0, 0, 0, 0, 0, 0, 0, 0, 6, 16, 16, -9, -56, 70, 0, -8], [0, 0, 0, 0, 0, 0, 0, 0, -7, -69, 17, -16, -45, 44, -11, 5], [-43, -3, -27, 0, -29, -49, 19, 26, 0, 0, 0, 0, 0, 0, 0, 0], [-124, 0, 25, 43, 14, 3, -8, 32, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 17, 18, 11, 17, -58, -2, 45, 6], [0, 0, 0, 0, 0, 0, 0, 0, 28, 24, -5, -4, -45, -30, -12, 34], [65, 44, 20, 15, 72, 16, -16, 22, 0, 0, 0, 0, 0, 0, 0, 0], [-61, -1, 27, -22, 57, 15, 25, 20, 0, 0, 0, 0, 0, 0, 0, 0], [-13, -20, -36, 17, -57, -41, 51, 26, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 38, -4, 17, -13, -34, 53, -30, 18], [0, 0, 0, 0, 0, 0, 0, 0, 6, -4, -13, -5, 72, 52, -16, -58], [8, 22, -7, 69, 74, -13, 28, 54, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 33, 21, -15, 7, 111, 14, 3, -26], [128, 2, 56, -50, 28, 12, 22, -40, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -21, -39, 25, -32, -15, 38, 73, 65], [-111, -8, 27, -52, -43, -52, 10, -29, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 20, 70, -45, 38, 82, -3, -30, 10], [0, 0, 0, 0, 0, 0, 0, 0, 50, 20, -12, -9, -120, -29, 33, -27], [0, 0, 0, 0, 0, 0, 0, 0, 14, -38, 60, -40, -57, 8, 48, 10], [-47, 34, 45, -59, 39, 24, 28, 10, 0, 0, 0, 0, 0, 0, 0, 0], [141, 24, -1, 6, 27, -44, -20, 47, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 19, -83, -11, 13, -17, -26, -115, 22], [-5, 5, 57, -6, -5, -13, -21, 50, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -17, 46, -47, 13, 22, -78, 45, -2], [0, 0, 0, 0, 0, 0, 0, 0, -54, -116, -8, -48, 25, 72, -94, -28], [0, 0, 0, 0, 0, 0, 0, 0, 22, -86, -24, 9, 58, 87, -20, -34], [0, 0, 0, 0, 0, 0, 0, 0, 49, 188, 21, -59, -132, 84, -55, -36], [0, 0, 0, 0, 0, 0, 0, 0, -14, 58, -25, -10, 86, 33, -12, -38], [0, 0, 0, 0, 0, 0, 0, 0, -10, -141, -22, 3, 97, 47, -41, 19], [-93, -40, 23, 98, -5, 72, -51, -28, 0, 0, 0, 0, 0, 0, 0, 0], [53, -53, -40, 12, 162, -33, 17, 1, 0, 0, 0, 0, 0, 0, 0, 0], [34, 14, -51, 48, 107, 46, -18, 23, 0, 0, 0, 0, 0, 0, 0, 0], [28, 46, -64, 64, 80, 146, -40, 6, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -8, -44, 22, 48, 90, -197, -46, -50], [0, 0, 0, 0, 0, 0, 0, 0, 21, -86, 63, -15, -58, 6, -11, 26], [0, 0, 0, 0, 0, 0, 0, 0, 44, 58, -5, -32, 71, 67, 128, -24], [17, 19, -37, 26, -18, 105, -37, 84, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 88, -93, -2, -10, 7, 139, 109, 60], [-63, 44, -33, -130, -106, 101, -20, -40, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 14, -26, -116, 17, 176, -162, 84, -28], [-294, -26, 14, 33, -129, 45, 10, -46, 0, 0, 0, 0, 0, 0, 0, 0], [-40, -96, -43, 114, -12, -57, 29, 24, 0, 0, 0, 0, 0, 0, 0, 0], [57, -46, 14, 72, 158, 0, -46, 62, 0, 0, 0, 0, 0, 0, 0, 0], [-49, 156, 44, -56, -63, 150, -24, -20, 0, 0, 0, 0, 0, 0, 0, 0], [177, -26, -95, -86, 96, 8, 46, -23, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 114, 222, -28, 36, 68, -23, 102, -30], [157, -38, -15, 81, -26, -79, -11, -94, 0, 0, 0, 0, 0, 0, 0, 0], [46, 107, -61, 72, 120, -27, 25, 50, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -70, 193, 0, -27, -67, -113, 45, 49], [-95, 61, -12, -78, 43, -97, 119, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 33, -48, -83, -42, 58, 76, -103, 25], [269, 12, -74, 90, 280, 12, 27, 146, 0, 0, 0, 0, 0, 0, 0, 0], [253, 109, 20, 12, 67, -3, -105, 29, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -164, 50, -10, 164, 71, 8, 22], [0, 0, 0, 0, 0, 0, 0, 0, 30, -180, -31, -97, -113, 114, -38, 116], [-105, -82, 11, -187, -141, -30, 86, -6, 0, 0, 0, 0, 0, 0, 0, 0], [-108, 12, -8, 23, 187, 87, 20, 54, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -38, 138, -6, 59, -128, -17, 131, 121], [102, 38, -7, -38, 59, 8, 4, 124, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 13, 87, 79, 99, -139, -98, 97, -74], [14, 54, -89, 80, 81, 212, -54, 44, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -28, 106, -10, -23, -50, -105, -145, -71], [-290, -31, -21, 170, -122, -37, 49, -36, 0, 0, 0, 0, 0, 0, 0, 0], [4, -35, -85, -32, 126, -163, 21, -64, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 139, 140, 67, 108, 164, -42, 3, 3], [0, 0, 0, 0, 0, 0, 0, 0, -4, -96, -16, 16, -39, 118, -84, -4], [0, 0, 0, 0, 0, 0, 0, 0, 33, -114, -11, 82, -216, 144, -45, -35], [0, 0, 0, 0, 0, 0, 0, 0, 168, 56, 18, 1, -119, 112, 80, 14], [0, 0, 0, 0, 0, 0, 0, 0, -1, 105, 33, -128, 49, 23, -174, -121], [0, 0, 0, 0, 0, 0, 0, 0, -84, -14, -11, -109, 80, -45, 32, 10], [0, 0, 0, 0, 0, 0, 0, 0, 131, 332, 91, 29, -34, 129, 106, -18], [-240, -90, 45, -36, -15, -22, 20, 91, 0, 0, 0, 0, 0, 0, 0, 0], [269, -116, 33, 0, 204, -175, 16, -56, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -12, -84, 72, 63, -80, -179, -114, -58], [-61, -104, -63, -18, -199, -254, 79, -40, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -96, -94, 13, -20, -41, -173, 78, 114], [0, 0, 0, 0, 0, 0, 0, 0, 28, -76, -26, 19, 134, 145, -165, -113], [0, 0, 0, 0, 0, 0, 0, 0, 144, 174, 5, 113, 100, 177, 156, -46], [-30, -32, 97, 102, 132, -120, -22, 77, 0, 0, 0, 0, 0, 0, 0, 0], [87, 86, 153, -169, -103, 96, 96, -12, 0, 0, 0, 0, 0, 0, 0, 0], [-221, 22, 78, 61, -150, 44, -156, 62, 0, 0, 0, 0, 0, 0, 0, 0], [-291, -6, 122, 166, -316, -16, -32, 58, 0, 0, 0, 0, 0, 0, 0, 0], [145, 88, 153, -14, -2, 163, -172, -56, 0, 0, 0, 0, 0, 0, 0, 0], [-31, 66, -20, 128, -297, -11, 16, 22, 0, 0, 0, 0, 0, 0, 0, 0], [73, 124, 31, 54, -185, -12, -108, 55, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 7, 313, 25, -5, 57, 303, 164, 14], [143, -127, -118, 112, 214, -137, -99, -67, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -26, -236, 52, 37, 131, -89, -296, -152], [0, 0, 0, 0, 0, 0, 0, 0, -2, 88, -26, -42, 65, 17, -136, -108], [235, 66, -19, 36, -261, -72, 23, 24, 0, 0, 0, 0, 0, 0, 0, 0], [422, 86, 85, 142, 107, 166, -216, 76, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -6, 112, -115, 35, -47, 245, -186, 42], [-62, 41, 106, 80, 360, 19, -125, 23, 0, 0, 0, 0, 0, 0, 0, 0], [-237, -94, 213, -148, -165, -222, 175, -34, 0, 0, 0, 0, 0, 0, 0, 0], [23, -52, 110, -47, 290, 98, 40, 44, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 64, 426, 148, 26, 117, 42, 142, 50], [0, 0, 0, 0, 0, 0, 0, 0, -191, -82, -103, -6, 82, -302, 57, -13], [0, 0, 0, 0, 0, 0, 0, 0, -84, -382, 80, -22, -186, -58, -118, 128], [159, 208, -18, -154, -301, 283, -50, -46, 0, 0, 0, 0, 0, 0, 0, 0], [132, 54, -32, -136, -170, 18, 36, -176, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -30, 394, 17, 19, -328, 51, -116, -32], [0, 0, 0, 0, 0, 0, 0, 0, 2, 144, 92, -14, 49, -85, -162, -150], [228, 187, 45, -92, -336, 145, 37, -8, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 8, 63, 18, -66, -271, 164, 82, 140], [492, 61, 26, -292, 317, -181, -37, 57, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 42, 262, -6, -18, 155, 19, -112, -22], [0, 0, 0, 0, 0, 0, 0, 0, 28, -13, 80, 37, 1, 79, 19, 43], [0, 0, 0, 0, 0, 0, 0, 0, 112, 100, -33, -68, 52, 321, -30, -94], [-346, -74, 180, -177, -305, -89, 130, 80, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -10, 227, 78, -11, 1, 305, 399, 115], [111, -130, -63, 172, 218, 130, 12, 45, 0, 0, 0, 0, 0, 0, 0, 0], [-334, -154, 39, 73, 124, -163, -82, -12, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 14, -254, 8, 65, 35, -348, -500, -174], [0, 0, 0, 0, 0, 0, 0, 0, 70, 440, 42, 107, -256, -362, 254, -52], [-381, -202, 79, -2, -59, -52, -35, -20, 0, 0, 0, 0, 0, 0, 0, 0], [-223, -216, 168, -228, -561, -119, 68, -26, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -106, 10, 50, -90, 317, -113, -48, 0], [137, 89, -158, 210, 65, 47, -223, -7, 0, 0, 0, 0, 0, 0, 0, 0], [406, -186, 192, -361, 210, -146, -17, -6, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 7, 358, -55, -78, -118, 241, 74, -109], [-694, -48, -88, -108, -127, 73, 29, -214, 0, 0, 0, 0, 0, 0, 0, 0], [-68, -2, 92, 238, -64, 60, -154, 96, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 78, -110, -84, 30, -282, -92, -6, 90], [-123, 75, 252, -44, -461, 33, -109, -133, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -252, -280, 51, -19, 361, -30, 326, 100], [0, 0, 0, 0, 0, 0, 0, 0, 254, 296, -26, 181, 82, 7, -284, -126], [-421, -58, 82, -266, 182, -94, -101, -50, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 136, 132, -144, -29, 16, 430, 44, 38], [0, 0, 0, 0, 0, 0, 0, 0, -23, 297, 107, -145, -35, 395, 406, -74], [113, 104, -22, -22, 588, 102, -33, -94, 0, 0, 0, 0, 0, 0, 0, 0], [-74, 58, -143, 210, 270, -216, -30, -73, 0, 0, 0, 0, 0, 0, 0, 0], [439, -30, -120, -150, -161, -239, 50, -66, 0, 0, 0, 0, 0, 0, 0, 0], [679, -80, -11, -238, 225, -200, 59, 60, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -112, -374, 175, -130, 32, 247, -204, -88], [0, 0, 0, 0, 0, 0, 0, 0, 36, -478, 22, 149, 64, -77, -292, -80], [-827, -29, -148, -116, -260, -263, 23, -87, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -40, 116, -40, -258, -66, 33, 327, -88], [-214, -69, -32, -398, -148, -19, 197, -295, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 117, -52, -1, -148, 124, 252, -43, -175], [0, 0, 0, 0, 0, 0, 0, 0, -76, -224, -156, 71, 193, -83, 168, 192], [-8, 242, -26, -108, 139, 5, 15, -62, 0, 0, 0, 0, 0, 0, 0, 0], [134, -35, -49, 262, 204, 43, -137, 132, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -47, -350, -5, -156, -266, 28, 125, 81], [303, 115, 250, 74, 185, 349, -125, 29, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 56, 213, -98, -6, 549, 165, -33, -204], [0, 0, 0, 0, 0, 0, 0, 0, 344, 82, 15, -21, -365, 359, -124, -28], [-10, 154, -207, -27, 429, 80, 153, -110, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 251, 196, 267, 76, -442, 121, 48, -183], [-421, -263, 240, -192, -27, -97, 89, -241, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -60, 42, -236, 120, 79, -55, -48, 108], [135, 2, 116, 76, -165, 33, -32, -200, 0, 0, 0, 0, 0, 0, 0, 0], [14, -208, -180, 202, 360, -34, 92, -126, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -182, 246, 45, 236, 263, -520, -160, -252], [-308, 50, -38, -36, 214, -202, 40, 192, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 60, 234, 49, 403, 56, -252, 102, 164], [0, 0, 0, 0, 0, 0, 0, 0, -140, -640, 354, -63, -408, -66, -152, 15], [388, -237, -147, 60, 610, -163, 115, -72, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -39, 371, -209, -108, -313, -55, 294, 5], [-51, 12, 223, 246, 226, 33, -34, -112, 0, 0, 0, 0, 0, 0, 0, 0], [754, 162, 44, -80, 583, -14, -102, -154, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -95, -570, -57, 85, 272, 401, -200, -214], [-623, -121, -108, -420, -645, -29, -69, -181, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -50, -276, 11, -177, -84, 205, -150, -174], [-400, 37, 289, 162, -585, 37, -25, -190, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 24, -614, -192, 172, 307, -52, 114, 156], [0, 0, 0, 0, 0, 0, 0, 0, 54, 380, 84, 47, -360, 509, 249, 259], [0, 0, 0, 0, 0, 0, 0, 0, -42, 322, 64, -43, -388, 25, 52, 48], [0, 0, 0, 0, 0, 0, 0, 0, -264, -214, 126, -162, -66, 224, 494, 136], [-1004, -81, 93, -4, -232, -115, -171, 138, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -62, 28, 244, 33, -35, -608, 326, -38], [-376, 0, -134, -93, 140, 150, 53, -208, 0, 0, 0, 0, 0, 0, 0, 0], [-267, -141, -129, -150, -563, -135, 235, -86, 0, 0, 0, 0, 0, 0, 0, 0], [-1111, -90, 413, -377, -480, -361, 113, -44, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -307, -81, -271, 53, 239, -358, 225, -38], [0, 0, 0, 0, 0, 0, 0, 0, -250, -216, 134, -194, 12, -55, 50, 94], [132, 104, 124, -108, -255, 229, -99, -68, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 98, -668, -205, 55, -138, 122, -284, 266], [-658, -164, -153, 168, 448, -240, -76, -77, 0, 0, 0, 0, 0, 0, 0, 0], [402, 170, -69, -65, 338, 15, 64, 116, 0, 0, 0, 0, 0, 0, 0, 0], [-140, -74, -58, -106, 48, -196, 266, -232, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 374, 312, -152, 258, -48, 409, 101, 96], [369, 79, -114, 214, 631, 215, -121, -183, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 212, 164, 32, 153, 196, 271, -117, 17], [0, 0, 0, 0, 0, 0, 0, 0, 394, 706, 171, 182, -163, 172, 224, 52], [-320, -162, -60, 178, -803, 233, 75, -252, 0, 0, 0, 0, 0, 0, 0, 0], [-702, 210, 253, -262, -398, -194, -32, -71, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -28, -132, -45, 21, 182, -254, 526, -44], [0, 0, 0, 0, 0, 0, 0, 0, -191, 217, 65, -267, 139, 130, -129, 12], [-358, -77, -49, -308, 602, 273, -91, -28, 0, 0, 0, 0, 0, 0, 0, 0], [-1063, -246, 103, -48, -988, -251, -38, 176, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -97, -414, -253, -129, -372, 341, -92, 170], [0, 0, 0, 0, 0, 0, 0, 0, 208, -550, 6, -85, 22, 408, 94, -224], [211, -177, 117, 488, 229, -291, 135, -24, 0, 0, 0, 0, 0, 0, 0, 0], [-29, -56, 93, -43, 159, 214, -72, -70, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 107, 722, -333, 50, -166, -586, 607, -83], [0, 0, 0, 0, 0, 0, 0, 0, -339, -739, 257, -105, -21, -86, -465, -124], [-134, -500, 28, -10, 316, -242, -150, -60, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -18, 248, -216, -51, 596, -387, -43, -145], [0, 0, 0, 0, 0, 0, 0, 0, 10, -488, -204, 152, -200, -727, 55, -38], [0, 0, 0, 0, 0, 0, 0, 0, -50, 450, 122, -98, -14, -577, 210, 348], [104, -150, -6, 80, -56, 488, -356, 8, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -40, 103, 108, -538, -325, 113, -367, -10], [-628, 410, -177, 58, -718, 498, -390, 73, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -10, -92, -188, 229, -103, -258, 520, 236], [71, -232, 316, -184, 1248, 300, -175, 202, 0, 0, 0, 0, 0, 0, 0, 0], [-920, 136, 70, -51, -50, 514, -227, -130, 0, 0, 0, 0, 0, 0, 0, 0], [207, 162, -8, 562, 263, 171, -100, 264, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -112, -168, 180, -298, 760, 579, -237, -40], [469, 220, -219, 481, -517, -114, 208, 308, 0, 0, 0, 0, 0, 0, 0, 0], [302, 47, -73, 96, 164, 407, -99, 58, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -52, -126, -329, 44, 698, -14, 224, 136], [0, 0, 0, 0, 0, 0, 0, 0, -431, -346, 195, 103, -152, -678, -65, 234], [138, -104, 197, -462, 123, -10, 244, -210, 0, 0, 0, 0, 0, 0, 0, 0], [233, 74, -131, 20, -24, -119, 210, 258, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -170, 53, -154, 272, 207, -121, -863, -62], [0, 0, 0, 0, 0, 0, 0, 0, 6, 374, 0, -322, -126, 56, 124, 154], [0, 0, 0, 0, 0, 0, 0, 0, 38, -194, -114, 346, -377, -87, 66, 44], [0, 0, 0, 0, 0, 0, 0, 0, -486, 60, -10, 78, 80, 331, -72, 104], [672, 384, 94, 195, 641, 199, -60, 238, 0, 0, 0, 0, 0, 0, 0, 0], [-214, 303, -18, -134, -269, 621, -255, 161, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 288, -750, 38, 335, -441, -44, -40, 52], [0, 0, 0, 0, 0, 0, 0, 0, 52, -494, -54, 192, 248, -7, -921, -172], [95, 241, 329, -170, 258, -163, 93, 448, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -107, -677, -275, 110, 761, -103, -658, -195], [1075, 2, 267, -136, -210, -404, 220, -71, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 12, 230, 68, -276, 804, 200, -138, 12], [0, 0, 0, 0, 0, 0, 0, 0, 114, 337, -150, -16, -479, -174, 318, 40], [0, 0, 0, 0, 0, 0, 0, 0, 18, -1216, 66, 101, 12, 372, -374, -277], [0, 0, 0, 0, 0, 0, 0, 0, 92, 182, 435, -292, -146, 435, -2, 300], [0, 0, 0, 0, 0, 0, 0, 0, -130, -332, 217, 434, 522, -13, -342, -226], [183, 366, -38, 392, -76, 618, -164, 122, 0, 0, 0, 0, 0, 0, 0, 0], [-203, 384, -13, -424, -841, 482, 61, -202, 0, 0, 0, 0, 0, 0, 0, 0], [220, -201, 53, -62, -101, -211, 159, -166, 0, 0, 0, 0, 0, 0, 0, 0], [136, 452, 15, 48, 211, 84, 170, 113, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 54, 216, 24, -178, 578, -266, 32, 212], [0, 0, 0, 0, 0, 0, 0, 0, 22, 678, 198, 47, -246, 189, 488, -206], [-725, 282, -184, 48, -406, 106, 314, -104, 0, 0, 0, 0, 0, 0, 0, 0], [-1062, 366, 174, -247, 277, 339, 92, 278, 0, 0, 0, 0, 0, 0, 0, 0], [199, 214, -361, 70, 546, 694, -94, 39, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 47, 233, -93, 6, -999, -49, -344, 149], [0, 0, 0, 0, 0, 0, 0, 0, 178, 226, -96, -202, 440, 117, -778, 44], [734, 288, -104, 596, 444, 44, -310, 258, 0, 0, 0, 0, 0, 0, 0, 0], [280, 130, -361, 351, 697, 258, -169, 32, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -240, 790, 347, -374, 25, 224, -364, 94], [0, 0, 0, 0, 0, 0, 0, 0, 262, -610, -76, 218, 280, 311, 130, 18], [0, 0, 0, 0, 0, 0, 0, 0, 199, -122, 11, -251, 344, 99, -150, 256], [0, 0, 0, 0, 0, 0, 0, 0, 150, 31, -242, 38, 51, -1025, -17, 14], [-181, -262, -285, 727, -598, -229, -341, -22, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 284, 212, 16, 743, 372, 240, -120, -23], [0, 0, 0, 0, 0, 0, 0, 0, -132, 178, -32, 61, 119, -229, 282, 316], [0, 0, 0, 0, 0, 0, 0, 0, 80, -129, 26, -115, 91, -401, -859, -197], [-550, -506, 23, 37, 381, 266, -331, 78, 0, 0, 0, 0, 0, 0, 0, 0], [-1191, -120, -288, -400, -244, -474, 178, 8, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -76, -6, -29, -233, 563, -270, -70, -342], [-1156, 118, 214, 181, -713, 543, 114, 208, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 174, 688, 121, -307, -196, 1206, 612, -300], [-684, 160, 63, -414, 66, -68, 10, 51, 0, 0, 0, 0, 0, 0, 0, 0], [890, 150, -237, 427, 890, 145, -54, 692, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 184, -30, -36, 9, 99, -725, -376, -68], [0, 0, 0, 0, 0, 0, 0, 0, -140, 1, -384, 2, 749, -21, 321, -52], [0, 0, 0, 0, 0, 0, 0, 0, 86, 144, -270, -143, 81, 369, 98, 32], [0, 0, 0, 0, 0, 0, 0, 0, 74, 500, -366, -433, 153, 544, -218, 30], [-124, -200, -417, -39, -440, -415, 102, -154, 0, 0, 0, 0, 0, 0, 0, 0], [-402, 47, 346, -22, 994, 625, -37, 13, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 25, 389, 269, 142, 225, -1134, 731, 191], [0, 0, 0, 0, 0, 0, 0, 0, 287, -975, 267, 63, -631, 130, -71, 270], [0, 0, 0, 0, 0, 0, 0, 0, 184, 134, 56, -133, 351, 318, 724, -258], [0, 0, 0, 0, 0, 0, 0, 0, 230, -324, -35, -283, 255, -106, -64, 198], [0, 0, 0, 0, 0, 0, 0, 0, 333, 172, 113, -236, -164, -681, -734, -165]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1210_d_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1210_3_d_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_1210_3_d_c(:prec:=16)
    chi := MakeCharacter_1210_d();
    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_1210_3_d_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_1210_3_d_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1210_d();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<3,R![601, -1374, -271, 1032, 4, -186, -14, 8, 1]>],Snew);
    return Vf;
end function;