
// Make newform 1845.4.a.s in Magma, downloaded from the LMFDB on 19 June 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_1845_a();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_1845_4_a_s();".
// 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_1845_4_a_s();".
// 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 := [-2084736, -6471808, 4115176, 6666900, -2212114, -1883652, 491813, 233727, -53983, -14427, 3100, 433, -89, -5, 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, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-15, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-4, -21, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-519434938283760, -2101097502208768, 3011759338730948, 1997652450632428, -1813377285141140, -422204442714922, 354109651849449, 28373495967648, -29914321892827, -58295352122, 1130531683500, -51087891293, -15450685482, 1158660373], [12785114211284832, 40733801869988704, -28648764981732236, -37633681075295194, 16969153888941362, 8287252230531847, -3477723882831288, -666615990718791, 305143427074492, 17120846438492, -11831201101287, 202342223366, 164383165437, -9818086552], [-1303685395977568, -5150300735909648, 1871297092875032, 5508318827822172, -655293637616910, -1545104356112642, 93006169378375, 186638836861704, -5987583699167, -11073820482584, 172652643864, 317002499801, -1822236778, -3489037715], [-92279627000577888, -252818081698663696, 135165363116484704, 267285418166824096, -48989755221668978, -73387650530712568, 7503528349488327, 8619824407854054, -541003392502303, -494564475657728, 18059329631838, 13619338066861, -224181829428, -143848027787], [75931380225662976, 240047622493010392, -98181288212558468, -240644285275875802, 29627037971117636, 67450897558947541, -3747442122818379, -8274641489429763, 220511265528691, 500393949646376, -5829760706691, -14580750021967, 55900337691, 162889120919], [-11884310800593552, -41917444449173704, 15272946750287836, 44578326496938994, -3903114745636552, -12744961613215757, 373522019129993, 1582300010926431, -13261093483657, -96679916498312, 39787564727, 2846999838569, 3702723093, -32140534573], [-29213073337537472, -86019873810870384, 39467228622713856, 91339691310488524, -10191608899283002, -25976889565286842, 962984473252683, 3223344713525376, -32709382930567, -197551226036652, 19481419752, 5848090022749, 10871461998, -66422412883], [127494491910101952, 355541854040728784, -152595679878912256, -384158318977060304, 34609618755080662, 111931339345325312, -2264787253941213, -14219702224666146, -33387975454003, 891209553231712, 7119061792398, -26948150596079, -152206758708, 311987939473], [11692960525787808, 41003108377512016, -13283816794569056, -45050630002575166, 2755029469564346, 13119154147064065, -123674894428725, -1664946069607863, -10935097445177, 104332835777426, 993517505445, -3155780850049, -19171216785, 36552017459], [-16757124524242896, -55480744014221680, 21331747467850580, 56268232250347270, -5980723774474112, -15562653655314253, 671916846497013, 1882980244214895, -33028435936093, -112822661611928, 636570591375, 3274821255973, -2971363179, -36587580245]];
    Rf_basisdens := [1, 1, 1, 1, 18051785519568, 90258927597840, 12034523679712, 180517855195680, 180517855195680, 30086309199280, 60172618398560, 180517855195680, 18051785519568, 18051785519568];
    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_1845_a();"
function MakeCharacter_1845_a()
    N := 1845;
    order := 1;
    char_gens := [821, 1477, 1441];
    v := [1, 1, 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;

function MakeCharacter_1845_a_Hecke( : Kf := Kf)
    return MakeCharacter_1845_a();
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 := 4;
    raw_aps := [[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], [-5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [3, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0], [-3, 0, -1, 1, 0, 0, 1, 0, 0, -1, 1, 0, 0, 0], [6, 1, 0, 0, -1, 0, 0, 0, 1, 0, 0, 0, 0, 0], [-9, -5, 3, -1, 1, 0, -1, 0, 1, 1, -1, 0, 0, 0], [23, 0, 0, 0, 0, -1, 0, 1, 0, -1, 0, 0, 0, -1], [-38, 3, 1, 1, -2, -1, -1, -1, 2, 2, 0, -1, 1, -1], [-27, 5, 1, 1, -2, -2, 1, 0, 2, 1, 0, 1, 0, 0], [62, -5, 3, -1, 0, 1, -3, -1, 2, 0, 0, 1, 0, -1], [48, -9, 1, -1, -4, 0, -2, 0, 2, 3, 4, -1, 0, 0], [41, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [28, -9, -5, -7, -4, -6, -2, 0, 2, 1, -4, -1, 2, 0], [-45, -11, 7, -3, 0, 3, 2, 1, -2, 0, 0, -3, 0, -1], [68, -38, 23, 1, -3, -3, 1, -5, -1, -2, 0, -1, -1, -3], [18, 6, -3, 5, -4, -6, 2, 4, 0, -1, 7, 0, 1, 0], [-48, 20, -16, 6, 10, 11, 1, 3, -2, -5, -4, 0, -1, 1], [72, -29, -6, 12, 6, -1, 15, -5, -2, 7, 5, -1, -2, 1], [-4, -1, 14, 14, -2, -5, 11, -7, 2, -1, -1, -3, -3, 3], [-106, 29, -28, 8, -6, -2, 6, -6, 4, 4, 3, -3, -1, -4], [140, -40, 14, 2, 12, 12, -12, 10, 2, -4, -8, 0, -6, 0], [52, -2, 24, 2, -22, -1, -13, -3, 6, 1, -6, 0, 2, -3], [-84, 30, -13, -9, 16, 13, -15, 19, -2, -2, -13, 0, 3, -1], [197, -66, -29, 21, -23, -23, -2, -15, 7, 4, 6, -7, 5, -1], [23, 17, 59, 1, 10, 3, 10, 3, -6, 6, -4, 3, -2, -1], [270, -81, 4, 2, 0, -10, 2, -10, 4, 2, 5, 7, -11, 0], [-128, 41, 10, -4, 2, 3, -9, -7, 16, 13, 9, 3, -4, 1], [577, -146, 46, 8, 2, -1, -4, 11, -8, -7, 10, 2, -6, 1], [-94, 12, -3, 15, 4, 5, 13, -3, -4, 16, -5, -4, 3, 1], [200, -216, -18, -4, 2, 6, 24, 2, -12, -6, 10, 0, 0, 2], [-40, 136, -9, 11, 14, 12, 26, -6, 4, -17, -3, 2, 9, 12], [-301, 15, -1, -3, -5, -8, 55, -8, -13, -25, 7, -4, -6, -2], [168, -138, 64, -4, -12, 23, -7, 1, 10, 23, -14, -8, 14, 5], [-337, 101, -91, -13, -18, -14, 21, -14, -4, -11, 10, 1, -3, -4], [770, -208, 22, 16, 4, 30, -8, 0, -2, -8, -14, 14, 2, 12], [42, -138, -110, 12, 17, -10, -24, -4, 5, 14, -1, -1, -16, -6], [94, 69, -54, 38, -14, -2, 0, -32, 12, 20, -5, -11, 5, -2], [11, 168, -53, 3, 6, 41, -32, 27, 6, -10, 3, 14, 17, -3], [-62, -125, -20, -10, 30, 37, 13, 13, -20, 11, -29, 17, 0, -3], [649, -93, -101, -27, 28, -1, 32, -1, 0, -14, 20, 3, 6, 19], [625, -260, -66, 14, 24, 27, 28, -1, 28, -9, 8, -6, 22, 1], [-430, 346, -23, -27, -26, -25, -87, 33, 34, 30, -5, -12, -3, -9], [361, 157, 26, 8, -13, 19, -14, 45, 5, -25, 24, 18, 8, -15], [10, -40, -84, -44, 18, 1, -25, -13, -24, 33, -26, -8, -14, -1], [-8, 26, 38, -18, -10, -48, -8, -30, -14, 20, 24, 22, -16, 10], [-191, 16, -175, -11, 0, -15, -24, 7, -4, 16, -53, 8, -11, -11], [-36, 178, 102, -38, -12, -38, 28, -6, 42, 32, 0, 10, -6, 20], [-80, 71, -145, 5, -38, -20, -36, -32, 4, 65, -20, -3, -20, 12], [45, -26, 95, 25, -2, -9, -4, -65, -6, 12, 13, -8, -7, -9], [73, -342, -149, -13, -19, -20, 53, 4, 35, 35, 2, -19, 33, -14], [1236, 15, 125, -93, 12, 13, -161, 13, -10, 62, -84, 7, -19, -3], [192, 141, -169, 43, -14, -65, 1, -9, -46, 12, -2, 13, -30, -3], [152, 226, 16, 22, -20, -28, 6, -38, -8, 36, -38, 18, -14, -20], [941, -117, -139, 43, 45, 10, 71, 34, 7, -1, 35, 28, 18, -6], [-289, 69, -167, 105, -78, -15, 20, 23, 22, -6, 18, 51, 16, 1], [1274, 182, -29, 5, -50, -90, 18, -46, 24, 1, 19, -12, -5, -6], [-6, 322, -79, 19, -104, -11, -143, 19, 46, 50, -11, -20, -24, -21], [62, -153, 201, -41, -40, -34, -56, 16, 34, 39, -78, -23, 20, -20], [318, 89, -109, 7, -44, -92, 20, -56, 4, -13, 14, 1, -54, 4], [868, 109, 290, -114, 20, -40, 22, 24, -12, 28, -53, -9, 19, 12], [1090, -455, -5, 35, 41, 37, 55, -9, -33, -18, 33, 32, -8, 31], [-912, 125, 65, 41, 96, 48, 2, -8, -62, -35, -16, 37, -2, -18], [-526, 321, -132, -120, -44, -69, -129, 79, 12, 7, -67, -23, 30, -9], [1867, 696, 102, 48, 1, -28, 5, 10, -41, -48, 33, -17, 36, 18], [-774, 429, -369, -11, -75, -23, 47, -49, -5, 28, -5, -42, -4, -3], [1989, 230, 146, -106, 16, 57, -228, 15, 30, 83, -68, -28, 4, 35], [-312, -123, 176, -52, 16, -24, 150, -52, 16, 46, 25, -11, 37, 8], [1924, 231, -167, -29, 0, -62, -198, -6, 64, 29, -64, -11, -70, -36], [964, 88, 162, -204, 90, 45, -25, 93, -58, -49, -22, -16, -13, 19], [4, -224, 85, 107, 32, 45, 135, -19, -20, -40, 87, 0, -41, -31], [998, 716, 14, -104, 88, 2, -134, -16, -28, 24, -46, -12, -34, 10], [-1236, 403, 182, 104, -30, -8, 154, -76, -36, -46, 85, -1, -83, 4], [56, 647, 88, 110, 36, -22, 60, 2, -4, 30, -95, -7, -51, -4], [366, 589, -398, -96, 16, -85, -135, -69, 70, 55, -3, 9, -6, -21], [-215, -239, 133, 75, -78, -19, -94, -77, 22, 4, 0, -79, 82, 3], [490, 1292, 141, 123, 44, -24, 220, 48, -24, -67, 1, -26, 13, -22], [-1434, 76, 168, -24, -83, 55, 47, 35, 91, 51, -43, 13, 60, 3], [1772, -574, 448, 130, 40, -80, 108, -30, -32, -8, 2, 22, -70, -8], [1370, 850, -373, 167, 84, 101, 75, 67, -86, -50, 65, -22, -56, -19], [84, 198, 119, -11, -40, -148, 150, -114, -8, 89, 27, -22, -63, 48], [4497, 1010, 197, 31, 52, -57, 4, 19, 94, 118, 69, 4, 75, 9], [-632, 94, 32, 56, 0, 32, -314, 38, 100, 154, -108, -40, -18, -42], [912, 151, -244, 114, -6, -16, 182, 12, -28, -72, 75, 25, -29, 60], [3146, 692, 118, 120, -42, 8, 10, -146, -40, -26, 4, -56, 24, 18], [-1548, -252, 274, -20, 120, 160, -14, 124, 24, -208, -24, 40, -26, -18], [-784, 458, -270, -164, -110, 2, -262, 40, 16, -52, -22, -74, 18, -40], [1147, 79, 69, 37, -91, -133, 152, -55, 5, 66, -69, 46, 57, -11], [1890, -740, 415, -225, -6, 12, -280, -60, -124, -107, 51, -40, 27, 26], [1323, -407, 139, -175, 10, 16, -429, 146, 52, 47, -94, -1, -31, -84], [2222, 1051, 80, -34, -82, -73, 3, 3, -60, 25, -69, -33, 40, 3], [278, 314, 239, -57, -66, 37, 169, -139, -86, -84, 113, 4, 20, 13], [290, -201, -204, 98, 146, 268, -120, 44, -18, -24, -73, 37, -11, -70], [-192, -1350, 236, -116, 80, 88, 142, -2, -16, -114, 108, 24, -112, 46], [1667, 1268, 75, 163, 66, -19, 122, -61, 10, -62, -85, -74, 15, -23], [-805, -911, 95, 77, -112, -191, -60, -181, 58, 150, 116, 65, -120, 9], [-997, 46, -295, -19, -90, -74, -159, 46, 186, 49, -7, 90, 20, -48], [-2940, -720, -40, 24, -22, 73, -5, -27, 92, 75, -30, 36, -73, 39], [1788, -338, 303, -165, -106, -32, -260, -34, 108, 119, -29, -22, -13, 16], [576, 1102, -462, -34, -242, -33, 107, 17, 38, 5, 114, -4, -70, -77], [4458, 404, 166, -48, 10, -1, -177, -3, -10, -41, 130, 60, 16, 5], [3156, -467, 330, 0, 67, 15, 231, -239, 63, 107, 30, -38, 7, -19], [502, -1393, 543, 229, -70, -54, 200, 130, 40, -73, 28, 165, 40, -58], [-5558, 972, -228, 0, -78, -66, -110, 42, 72, 42, 18, 116, -84, -12], [4835, -104, -126, -66, -210, -113, -312, -223, -24, 169, -72, -96, 66, 13], [-45, 501, -658, -128, -129, -163, 30, -25, 101, 45, 180, -82, 48, 63], [2248, -397, -163, -167, -36, -186, 66, 14, -52, 51, 34, -145, 46, -50], [1093, -1583, 103, -157, -203, -52, 101, 32, 5, -7, -127, -14, 82, -32], [-3200, -426, -654, 142, -210, -72, -24, 32, 128, 14, 82, 122, -32, -70], [3044, 538, 46, 294, 120, -48, 346, -76, 162, -102, 172, 8, -100, -10], [-838, 346, 428, -66, 134, 104, 448, 36, -34, 2, 154, -68, 124, 54], [-2134, 654, 72, -30, -176, -104, -216, -64, -52, 78, -116, -148, 22, 18], [-4730, 536, -816, 208, 162, 106, 128, 24, -82, -52, -140, 32, -162, -104], [5334, 1197, 1176, -70, 190, 114, -104, -10, 24, 70, -43, -53, -119, -82], [-1766, 997, -481, 219, 64, -94, 114, 20, -60, 99, 128, 91, -20, -16], [-502, -1497, -71, -21, -86, -87, -153, -13, 66, -88, 246, 31, -30, 23], [-1150, -1565, 354, 148, 138, -13, 181, -229, -54, -101, 77, 19, -48, 105], [4668, 11, 148, -14, -218, -265, -147, -37, 96, 197, -73, -25, -10, 39], [-4954, 381, -278, 18, -119, -157, 413, -203, -41, 173, 102, -4, 107, 5], [-795, 114, 301, 279, -16, -202, 509, 60, -40, -39, 141, 74, -51, 26], [4606, -223, -30, 126, 76, 99, 15, -5, 126, -99, 83, 119, 120, -7], [-1259, 26, 554, -318, -19, 107, -36, -57, -183, -43, -139, -123, 214, 5], [4352, -1237, 622, 148, -76, 29, 169, -97, 82, 175, 21, -39, 204, 55], [-2152, 458, -255, 7, 232, -86, -58, -118, -124, -133, -33, -20, -19, -32], [-3863, 2372, 71, -131, 26, 71, -558, 165, -80, -86, 41, 86, 87, 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-41, 160, -363, 566, 536, -138, -533], [1274, -11855, -1820, -678, -826, -1411, -173, -1447, 6, 1623, -551, -923, -102, 375], [-55407, -4289, -1335, -573, 1356, 515, 484, -339, 870, 450, -728, 515, -770, 55], [-8056, -2320, 66, -674, 1130, 280, 912, 114, 46, 180, 214, 1232, -908, 450], [157, 4144, -2388, 566, -2394, -385, -188, 471, 1562, 387, 966, 82, 798, -845], [20230, 11169, -964, -1206, -66, -917, -1101, -2201, 544, 1617, -563, -181, 588, 399], [2118, -4007, -1617, 1215, -1346, -1653, 1589, -1149, -218, 1732, 932, 459, -551, -285], [12310, -2594, 2749, -761, -932, -1186, 424, -490, -276, -561, 1415, 256, -131, -292], [-3540, 14225, -1616, 834, 1054, 365, 1619, 1287, -1054, -551, -559, 125, -455, 47], [-1298, 1253, -659, 129, -1018, -1542, 628, 566, 1092, 67, 1386, 81, -306, -584], [19405, -7038, 5066, -2134, -302, -457, -640, 583, -552, 395, -132, -168, 740, 659], [-25806, 17364, -4363, -1133, -1130, -913, -37, -5, 1306, -506, 849, -494, 503, -65], [29786, -7624, -2744, -930, -1250, -514, -2990, -202, 1134, 1760, -1264, -364, 696, 356], [-16678, -7747, -4368, 2798, 772, 248, 1860, 372, -638, -1084, 1503, 607, -63, -348], [32794, -15212, 3734, 1132, -1023, 610, -992, 52, -895, -1614, 949, -147, 230, -202], [-38710, 2370, 748, -1376, 1504, 168, 326, 166, -820, 390, -560, 824, -1124, -166], [27991, 5170, 5637, -1429, 1874, 1364, -2899, 870, -1152, -717, -33, 62, -367, -66], [4254, 11582, -2876, -228, 258, 804, 1450, -606, -1344, -134, 102, 492, -760, 208], [-21990, 3101, -754, 640, 1252, 1511, 1367, 1523, -274, -1525, 1265, 1157, -244, 583], [248, 15138, 2346, 664, -654, 8, 602, -1990, 692, 114, -10, 154, -348, 150], [-14448, -2622, -768, 1412, 1514, 1139, 1503, 125, 682, 55, -820, -88, 112, 399], [14136, 7275, -4085, -785, 594, -402, -308, -1226, -476, 73, -986, -87, -268, 736], [10512, -10217, 107, -943, 544, -1099, 1151, 613, -1030, 728, -1374, 387, -705, 169], [-25466, 12598, -1072, 1560, -1192, -687, 4543, -557, -518, 109, 642, -320, 512, -441], [27363, 5610, 3, -1209, 1350, 1387, 1192, 485, -170, 1944, -575, 728, -9, 351], [-45972, -2961, -496, -768, -114, -884, -334, -812, -1366, -846, 1077, -183, 383, -16], [-55637, -3626, 5811, -657, 378, -170, 2629, -1162, -450, 1367, 411, -514, 186, 172], [-1864, -14525, 5662, 510, -1236, -1902, 464, -490, 530, -1118, 1505, 79, -67, 316], [-57875, 7202, -6690, -626, 1448, 569, -1878, 521, 846, 89, -762, 730, 108, 225], [-73633, 6624, -4099, 677, 887, 956, -169, 436, 21, -775, -1696, 729, -249, 158], [-23640, -598, -2810, 2922, -290, -58, 2964, -1368, 228, 534, 904, 172, 48, 270], [-17206, -9282, -331, 1205, -2, -212, -400, 1370, -448, -1159, -1267, 104, 679, -760], [-17660, -1831, 1653, -819, 2570, 1579, 3687, 1117, 26, -1180, -24, 237, 785, 557], [44306, -2888, 3991, -197, 1044, 393, -337, 855, 142, 18, -435, -628, 1049, 179], [-29100, 9823, 1827, -1631, -504, -990, -3268, 306, 792, -1437, -998, -515, 910, 276], [-44452, 8581, 6067, 491, -494, 239, -387, 147, 916, -1032, 56, 353, 183, 789], [-16671, -2011, 1031, -55, -568, -299, 866, 363, -162, 410, 292, -1159, 1170, -859], [-8792, -3562, -1564, -218, -1228, -1346, -3588, 2576, 1172, -386, 510, 586, 124, -1238], [43594, 9771, 2766, -136, -2154, -295, -4433, 261, -202, 365, -1515, -53, -194, -505], [17401, 5240, 2358, -738, -1709, -1197, -2332, 1527, 779, 789, 865, 305, -492, -751], [38566, 2650, -2451, 537, 1089, -393, 33, -851, -1149, 1404, 92, -455, -1273, -55], [-33714, -7609, -1252, -1248, 100, -1094, 90, -1770, 262, 1224, 95, 807, 241, -8], [31394, -6404, 415, 1171, 1048, -863, 1183, -1093, -1310, 286, 151, -176, -1322, 441], [8607, -4426, -313, 865, -398, -64, -1467, -1326, 738, 993, -1683, -852, 475, 440], [25163, -15592, -4980, 1766, 1952, 1879, 1640, -123, -22, -199, 1722, 420, -610, 803], [-30130, -5072, 2801, -2575, -786, -986, 1030, 1158, 996, -1007, 753, 38, -69, 0], [-47471, 17043, 3365, 891, -3194, -3351, 1638, -191, 666, 142, 628, -475, 426, -135], [-18006, 4640, 901, -3153, 918, 1058, -2028, 2350, -364, -2289, -1053, 210, 359, 46], [-36410, 16414, 2132, -784, 641, 1087, -553, 407, 2091, 1655, -601, -625, 126, -21], [12944, -11338, -731, -323, 1590, 1828, -758, 594, -24, -1695, -203, 252, 307, -176], [13754, -9452, 5372, 2034, 1720, -298, 2828, -1514, -140, 1636, 570, 854, -32, 312], [-39196, -7420, -4814, -1218, -1072, 234, 2568, 850, 600, 728, -166, 836, 314, 194], [-43068, -4143, -4405, 297, 68, 681, -2075, -353, -872, -2462, 1436, 139, -182, 259], [-38687, 5151, 1243, 639, 223, -2175, 3480, -321, -321, -1270, 535, 1488, -1435, 611], [-27973, 7240, 3733, 1093, 1208, 1710, 1213, 570, -1160, -843, 113, 622, 836, 448], [17822, 2054, 8795, -343, 1264, 381, -515, -663, -324, -102, -209, -318, 1535, 193], [7866, 3839, 7030, -4002, -772, -1822, -3878, -570, 1144, 2130, -2195, -795, 861, 494], [-30156, 5289, -1027, 1549, -2590, -2303, 2885, -693, 178, 434, 1416, -111, -589, 311], [38185, -3386, 6862, -756, 832, 1079, -2314, 1477, 102, -209, -538, 194, -1182, -293], [-16698, -5822, -7374, 132, -1072, 569, 987, 71, -752, 81, 102, 926, 264, -285], [-8710, -18334, 1788, 1966, -458, 626, 2228, -1396, 210, 2018, 1032, 544, -1086, -302], [-112364, 6104, 1588, -3122, -799, 38, -5418, 1020, 2101, 968, -529, 393, 420, -22], [49802, 329, 3740, -2302, 1126, 710, -2222, 86, -1410, 266, -1719, 739, 735, 364], [22503, -4748, 3853, -2369, -394, -725, -2470, 873, 166, 660, -1545, -468, 501, 481], [-5400, 12088, 6014, -796, 210, -1450, -3128, -14, 244, 942, 14, 470, 446, 162], [66912, -9083, 6315, -443, 61, 1199, -2125, 965, -237, -1674, 639, 306, -684, 307], [1257, -10788, 7479, -133, 2204, 2093, -534, 669, -378, -1472, -209, 440, 19, 1297], [35316, -2612, 11031, 1329, 424, 1900, 1026, -688, -896, 1551, -245, -176, 197, 298], [10972, -7710, -3182, 546, 906, -60, -4538, -630, 864, 1940, 72, 44, -970, 668], [36339, -14132, 5684, -1196, -741, -861, 206, -1257, 487, -1557, -745, 301, 734, 93], [-7528, -5682, -2200, -4334, 1122, -682, 350, -364, -334, 2118, -552, -172, 228, -246], [-15846, 19612, -6888, -3192, -24, -252, 230, 286, 468, -314, -106, -686, 678, 378], [37976, 1163, -252, -1222, 678, 832, 1528, -68, -264, -688, 47, 1077, 371, 656], [-13007, -12193, -6925, 1013, 68, 1184, 4677, 1364, 1426, -579, 536, 1629, 39, 46], [-84928, -6077, 478, 1928, -1914, -1994, -20, -1302, -872, 440, 1109, -311, -135, -518], [-59281, -16396, 67, -395, 1021, 432, -2465, 240, -1527, -1235, 830, 171, -219, 306], [29952, 6455, 9516, -1116, -2144, -1488, -3426, -1140, 40, -1330, 43, 25, -579, -336], [38189, 18632, 1507, 693, -261, 582, 211, -690, -865, 99, 840, -939, 61, -236], [-20970, -9188, -5952, 592, -4, 1292, 4564, -140, 1264, 844, 864, 204, 1558, -212], [13096, -21614, 994, -2636, -1024, 140, -1912, 494, 874, 1778, 1268, -968, 404, -306], [-85690, -9295, 2494, -752, -780, -612, 26, 1542, 212, -1128, 671, 405, -1145, 824], [-47922, -15836, 6987, -2563, -748, -1395, -715, 273, -512, 1066, -1685, -886, 1230, -813], [-5006, 11694, -5444, 1298, -22, 1089, -367, 387, -1398, 455, 290, -386, 128, 283], [-31566, 24886, -2856, -1834, -3354, -1202, -2198, 1656, 1444, 1502, -772, -854, 1330, -1100], [-40118, -1457, 5708, -2574, -76, 1148, -1624, 864, 1940, 732, -955, -535, 1049, -874], [29728, -1206, 4448, -260, 216, -821, 109, -981, 146, 1273, -856, -666, 370, 67], [18888, -9226, -3075, 2249, 1614, 938, -1594, 1862, 1208, 681, -1033, -154, -347, -308], [16683, 5434, -4755, -1047, -1982, -763, 1166, -1637, 1528, 52, 395, 498, -765, -589], [-34225, 5022, -1224, -1832, -747, -1713, -1086, -285, -335, -893, -1617, -1655, 1096, -587], [-5348, -7678, 1055, 2679, -2426, -3280, -970, -176, 1004, 165, 381, -1410, -375, -1582], [8626, 18282, 3008, -1370, -564, -708, -3218, 484, 504, -1106, 1500, -376, 692, -98], [-62307, -2323, 10180, 1000, 1708, 144, -1451, -1142, -1676, 408, 1065, -883, -1058, -96], [-18855, 16799, 4069, 413, 747, 82, -1697, -514, 43, -1167, 931, -146, 362, 70], [17032, 9930, 6301, 905, -280, -1925, -953, -1415, 1126, 254, 1001, 326, -507, 285], [73248, -3304, 8035, 673, 624, 772, -838, -1772, -1312, -2365, 955, 978, -339, 854], [-59030, 3406, 5212, -3482, 187, 410, -4300, 1320, -1609, 1032, -2953, 653, -246, -110], [-28744, -4250, 4800, -1358, 2872, 1739, 1031, -1163, -640, 1865, -1572, 294, 506, 1337], [17843, -5631, -931, -511, 2270, -16, -275, -266, -1216, -1737, -220, 663, -1868, 340], [33396, 14109, -5318, -992, -974, -1400, -6, 1764, 1614, 1114, -777, 553, 149, 62], [19908, -1010, -2680, 56, 3430, 2439, 3593, 1887, -1600, -515, -402, 284, 70, 55], [20064, -18995, 1972, 1358, -1926, 312, -2196, -592, 1692, 1836, 153, -265, 451, -1304], [26012, -3425, 2561, -367, -3410, -3916, -246, -2896, 1546, 2889, 1336, 329, 502, -176], [22503, -14062, 5264, 562, -428, 1053, 1482, -1145, -448, -413, -556, -244, 818, -865], [-39572, 2273, -1427, 1425, 2812, 1418, 452, 1754, -2372, 163, 744, 685, 114, -526]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1845_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1845_4_a_s();".
// 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_1845_4_a_s(:prec:=14)
    chi := MakeCharacter_1845_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 4));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(2999) - 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_1845_4_a_s();".
// 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_1845_4_a_s( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1845_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<2,R![-2084736, 6471808, 4115176, -6666900, -2212114, 1883652, 491813, -233727, -53983, 14427, 3100, -433, -89, 5, 1]>,<7,R![22394911116019712, 8183919734131712, -4978797090417920, 415659148133696, 54361462989960, -7228705705200, -144731460784, 40052922917, -248121664, -97509755, 1643997, 105673, -2347, -40, 1]>],Snew);
    return Vf;
end function;