
// Make newform 49.10.a.g in Magma, downloaded from the LMFDB on 17 December 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_49_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_49_10_a_g();".
// 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_49_10_a_g();".
// 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 := [2560703544, 231136416, -141427156, -5545800, 1834350, 6784, -2635, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [2520646585114104, -9391130414192452, -178031669549712, 82819618113684, 197200820962, -125962207375, -44711660, 49274683], [-713494864834055640, 5425636900055300, -161043465144144, -54658270595412, 2987518494754, 80023892879, -1541690600, -31390019], [7847173210074337188, 336198997778796376, -191557113264727428, -1368097428062604, 280751654882687, 1788590423572, -105637852963, -716486374], [60793517008736736, 9127093962747908, -1340036189950360, -160390870160612, 3255370619424, 240317116963, -1746475430, -94108819], [123511682670591096, -174595680326490388, -8723551807935888, 4058161287570516, 9662840227138, -6172148161375, -2190871340, 2414459467], [6091718295349520208, -5574216390959130604, -107294263848140544, 48469835837937948, 119190301919344, -65830850904205, -30841390430, 24430655821], [9355249609767379332, 30368901031673179864, 732942332245656444, -679132223245005948, -697160602645609, 999588533933260, 94704282965, -378764652766]];
    Rf_basisdens := [1, 5827953264672240, 1165590652934448, 16318269141082272, 388530217644816, 5827953264672240, 11655906529344480, 81591345705411360];
    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_49_a();"
function MakeCharacter_49_a()
    N := 49;
    order := 1;
    char_gens := [3];
    v := [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_49_a_Hecke( : Kf := Kf)
    return MakeCharacter_49_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 := 10;
    raw_aps := [[-8, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, -1, 0, -1, 0, 0], [0, 0, 0, 2, 2, -2, 0, -1], [0, 0, 0, 0, 0, 0, 0, 0], [-10355, -368, -16, 0, 0, 0, -19, 0], [0, 0, 0, 734, 98, 572, 0, 43], [0, 0, 0, 508, -1432, 1791, 0, -236], [0, 0, 0, -765, 624, 1979, 0, 612], [-307024, -32200, -48, 0, 0, 0, -876, 0], [-2291393, 48520, -2598, 0, 0, 0, 1329, 0], [0, 0, 0, 11314, 11080, 48726, 0, -136], [-1068205, 215496, 12738, 0, 0, 0, 4701, 0], [0, 0, 0, 93916, -50344, -128007, 0, 9208], [-4569677, -217344, 31288, 0, 0, 0, 8139, 0], [0, 0, 0, -32294, 78672, 55234, 0, -8984], [-17540120, -123296, -36148, 0, 0, 0, -5062, 0], [0, 0, 0, 229647, -272344, -115825, 0, -67836], [0, 0, 0, -48650, 388750, -458266, 0, 29513], [57602738, 577272, -8560, 0, 0, 0, -79734, 0], [28696514, 6822416, -38712, 0, 0, 0, -19518, 0], [0, 0, 0, -146752, -20908, -1975161, 0, 212110], [-95900214, 12628944, 28840, 0, 0, 0, 64290, 0], [0, 0, 0, -1898049, -1131256, 3805271, 0, 187368], [0, 0, 0, -2718584, 228392, -1613733, 0, 312496], [0, 0, 0, -2277436, -1165108, -3569089, 0, -166586], [0, 0, 0, -2585094, -1037330, -7569110, 0, -1488963], [0, 0, 0, -3808082, 3188152, 3155762, 0, 237992], [-160538986, -27311416, -473584, 0, 0, 0, 982742, 0], [-913275727, -57626712, 766094, 0, 0, 0, -224073, 0], [-1441529077, 16491200, -605854, 0, 0, 0, -556945, 0], [944447586, -106861176, 1278560, 0, 0, 0, 1651758, 0], [0, 0, 0, -22485847, -13737976, 1304473, 0, 571760], [-657903310, 28272848, 4651004, 0, 0, 0, 416462, 0], [0, 0, 0, -3524937, -7444080, 20436775, 0, -4015596], [-2139553622, -110850320, 4313944, 0, 0, 0, -538232, 0], [-363991038, -96493032, -7790736, 0, 0, 0, -4160082, 0], [0, 0, 0, 46710074, 9570734, -76507556, 0, -1646819], [-642989213, -202852272, -10848152, 0, 0, 0, 3446115, 0], [0, 0, 0, 20551330, 31150560, 142790482, 0, 2496520], [0, 0, 0, 40370298, -17152810, 32378696, 0, 2362617], [-7128065110, 275825000, -4794048, 0, 0, 0, -3809934, 0], [0, 0, 0, -47061314, -36247610, -7363340, 0, -4827523], [-8027996106, -20498544, 24287856, 0, 0, 0, 1775502, 0], [-709950043, -286677744, -7611274, 0, 0, 0, 14950749, 0], [-12731108536, 32756720, 26515180, 0, 0, 0, -2637986, 0], [0, 0, 0, -48466006, 58306472, -77332266, 0, 7294000], [-20377811110, 408501432, -16182888, 0, 0, 0, -20584782, 0], [0, 0, 0, -2092316, -42897176, -276361360, 0, -8626840], [0, 0, 0, 75212377, 2113600, 201890897, 0, -63130196], [0, 0, 0, 204115010, 139677566, 420015928, 0, 20283613], [-17663354628, -199370736, -21576184, 0, 0, 0, -22310716, 0], [-23467100712, 691238712, 11204864, 0, 0, 0, 19879124, 0], [0, 0, 0, -107874732, -140832060, -28885479, 0, 21185730], [0, 0, 0, -323008865, 51125144, 262584279, 0, -22437488], [0, 0, 0, 173117224, 17177984, -778224231, 0, 10540384], [17463423370, 1424745376, -84035912, 0, 0, 0, -20830670, 0], [0, 0, 0, 349890278, 87091474, 709939284, 0, 6621611], [0, 0, 0, -75301300, -134372248, -427046464, 0, -52740536], [46830643314, 1938931920, 66752320, 0, 0, 0, -37164180, 0], [-13046523128, -2567935568, 116950928, 0, 0, 0, -45548920, 0], [0, 0, 0, 277237647, 180230544, 566278143, 0, 40303296], [0, 0, 0, 115521434, 231572838, -928282, 0, -106506463], [0, 0, 0, 36271577, 335676152, -1089946263, 0, 136360120], [0, 0, 0, -1109418264, 35935504, 1205892376, 0, -25464192], [0, 0, 0, -621747604, -480453784, 851277609, 0, -93774608], [-10091148032, -4647497216, -8207740, 0, 0, 0, -62953258, 0], [7708729775, 1135540608, -355726776, 0, 0, 0, 28489335, 0], [135342494831, -755032896, -44995066, 0, 0, 0, 110566053, 0], [-73502492617, -5405599840, 139584048, 0, 0, 0, -25230225, 0], [0, 0, 0, 467200086, 94974558, -2521228028, 0, 29631417], [0, 0, 0, -1125065104, 845862484, 660720135, 0, -16478230], [-102644065150, -1097631016, 679648160, 0, 0, 0, 21594254, 0], [0, 0, 0, 71011384, 542146648, -771038300, 0, 45756392], [227524123306, 3215310864, 163793960, 0, 0, 0, 76056312, 0], [-27643871715, 17296315008, 166744144, 0, 0, 0, 51789717, 0], [0, 0, 0, -1003158458, -766864368, -8197652762, 0, -33977984], [-12097410765, 5338115160, -346945606, 0, 0, 0, -145187755, 0], [0, 0, 0, -2418542782, 298807154, 1632197612, 0, -82851161], [-123293896887, -10712477520, 222812450, 0, 0, 0, -98900917, 0], [0, 0, 0, -1431608188, -1412118796, -7941287293, 0, -149684162], [0, 0, 0, 162869257, -907054848, 9574997689, 0, -553880], [441965362090, -2507805168, -127140176, 0, 0, 0, -45788892, 0], [-302953996816, 3291855656, -438017552, 0, 0, 0, 193611988, 0], [0, 0, 0, 1053195868, 101753008, -1918264165, 0, 2313944], [0, 0, 0, 4613894688, 3727052664, 1377273668, 0, -89097312], [-667754373222, -2711795448, -872862576, 0, 0, 0, -219444510, 0], [86029028246, 11820289760, 1300154336, 0, 0, 0, 358700648, 0], [-533039915595, 27247697376, 136854054, 0, 0, 0, -478955979, 0], [0, 0, 0, -2446073642, -484641738, -5593553168, 0, 758652277], [-513136333460, -11904561216, -833088736, 0, 0, 0, 161582004, 0], [0, 0, 0, -2133032857, -1078762240, -4932595145, 0, 15149972], [0, 0, 0, 3512170054, 1270067320, 3141274586, 0, -1361995448], [-842793935092, 6327196296, -1896953504, 0, 0, 0, 402230328, 0], [-1071459864398, -2272854872, -366095704, 0, 0, 0, 172803482, 0], [-770455388838, 3630570408, 1600781768, 0, 0, 0, -72192678, 0], [0, 0, 0, 3274130880, 3700807992, 2735644908, 0, 773693208], [0, 0, 0, 5273158938, 2314737986, 23247925580, 0, 661324695], [0, 0, 0, 396146004, -4434892372, -14050741705, 0, 415331634], [0, 0, 0, 7004934833, -5118986152, -11360228359, 0, 303595180], [468736865702, -36514850304, -802574632, 0, 0, 0, 53330796, 0], [-1086326761235, -21885047088, -2089855800, 0, 0, 0, -564287091, 0], [1783601721380, -15306411184, 713365828, 0, 0, 0, 1102367338, 0], [0, 0, 0, -16581908193, 413811896, 2294872079, 0, 1147649604], [1010283947467, 37922429296, 2052511574, 0, 0, 0, -1624147711, 0], [200745127641, 3668864832, 2360020208, 0, 0, 0, -993792543, 0], [0, 0, 0, -2841323568, 10341811260, -21901273463, 0, 406087974], [0, 0, 0, -19448854677, 4050252304, -21358340789, 0, -1568328468], [0, 0, 0, -7864159952, -3694394408, 22931393531, 0, -551677316], [-861699033452, 49827890320, 2081824208, 0, 0, 0, -1484777428, 0], [0, 0, 0, 16459451336, 316243928, 1422544821, 0, -2538705056], [0, 0, 0, -1499996892, 351143472, -49667504620, 0, -2425536096], [676436329895, 19772032728, -1921169966, 0, 0, 0, -319891119, 0], [2868622458317, -27599360608, -1426339070, 0, 0, 0, -493330721, 0], [0, 0, 0, -15372392691, -404925648, 29663859021, 0, 710132448], [-2582024131882, -55094214816, 123965120, 0, 0, 0, 2628082974, 0], [-756450172557, -142233674640, 1787959174, 0, 0, 0, -395342087, 0], [0, 0, 0, -1586507633, -5867828792, -2306884297, 0, 5181484676], [0, 0, 0, -17520872518, -15103317672, 22295988662, 0, 556783088], [6453589260481, -40241986520, -974367634, 0, 0, 0, 437364983, 0], [-1029754979623, 104657048240, -3037769400, 0, 0, 0, 1560402633, 0], [0, 0, 0, 7969406798, 3080159594, -13022657358, 0, -3472390793], [631033537526, 187499320224, 324583460, 0, 0, 0, 1376501082, 0], [0, 0, 0, -12025897682, 12259630974, 71457940024, 0, -1259424143], [2412212333978, -70357521256, -5023578072, 0, 0, 0, -147546918, 0], [0, 0, 0, -15734378007, -14416661880, -54983455079, 0, 3196897752], [2395117866545, 209196318392, -8599093290, 0, 0, 0, -1815154065, 0], [2992776551123, 242530679016, 5390300818, 0, 0, 0, 5325156429, 0], [0, 0, 0, 31968343450, 5114834144, 22533699234, 0, -995013488], [0, 0, 0, 21225732098, 12782997920, 102908589202, 0, -4074075968], [0, 0, 0, 60165600394, -14907800174, 24250146170, 0, 98962007], [613288338983, 164467234896, -2401357832, 0, 0, 0, 688525047, 0], [-549947093638, -32199855016, -9095908640, 0, 0, 0, -3453182810, 0], [-8153229719496, 128612495304, 6883933600, 0, 0, 0, 127975980, 0], [2967488197255, 22131068088, 8782532786, 0, 0, 0, -3676140399, 0], [0, 0, 0, -40477729660, -547463920, 99504304555, 0, 1321431452], [0, 0, 0, -35060197332, 29158994508, -10036175571, 0, 235303998], [0, 0, 0, 13282062130, -2503074990, 25543331134, 0, -12799615313], [0, 0, 0, 47069075031, 14639480016, -55902739777, 0, -9916280724], [0, 0, 0, 63272956270, 12712930258, 530496860, 0, 3125089171], [-8837123498297, -207967458272, -12796494286, 0, 0, 0, -2601021985, 0], [0, 0, 0, 5760765099, -8451297456, 90320846251, 0, 2896092768], [-3149497042926, -429504623616, 13179606440, 0, 0, 0, 9905801492, 0], [538131772960, 6743333952, 24658172280, 0, 0, 0, 3106593024, 0], [-7845262869106, -189857466904, -219938488, 0, 0, 0, -440352178, 0], [0, 0, 0, 30735680430, 32634005070, -38905268466, 0, 6847167753], [0, 0, 0, 55596252554, 15785473032, -39443755426, 0, 4103753840], [0, 0, 0, 84291933714, -53271747018, 8822449748, 0, 2813265033], [0, 0, 0, -1921274380, -4447908436, -36806581979, 0, 7145096066], [0, 0, 0, 19826004901, -56253083288, 114370256285, 0, -747544912], [-2161198211918, 557251092544, 10453767520, 0, 0, 0, -5177859902, 0], [-6672855649769, 72059767320, -17800407470, 0, 0, 0, 3557277777, 0], [0, 0, 0, -101753425256, -13468801948, -69188979747, 0, -9172514750], [363243455902, 466306040856, 9704928320, 0, 0, 0, 2534223870, 0], [0, 0, 0, 73918511662, -18430376928, 142395712006, 0, 198178672], [2314802711498, -263783309832, -36410646456, 0, 0, 0, -441880686, 0], [-8314684590606, 15598254264, -4971632880, 0, 0, 0, -4765976058, 0], [-799009885836, -441401605824, -10422080072, 0, 0, 0, 1569051156, 0], [0, 0, 0, -104123611644, 7261392312, -309426896649, 0, -16805671620], [0, 0, 0, -9422954216, 6598832884, 148630569999, 0, -27551629570], [0, 0, 0, -109868461334, -56742147058, -205375465650, 0, 7363958869], [-9724801829287, 150534002864, 9456153224, 0, 0, 0, 5420946329, 0], [-10871953408780, 930018529760, 39513948820, 0, 0, 0, -6712272914, 0], [8302434841268, 964129790424, -11692230320, 0, 0, 0, 5078324376, 0], [0, 0, 0, -30499075545, 92647496840, -79204821721, 0, 7359284340], [-5110242606974, -33768887552, -2031511316, 0, 0, 0, -6548434754, 0], [0, 0, 0, 8517170598, 10066910728, 460896822322, 0, -14172169704], 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-124066066404, 0], [1310565857410920, 620456903328, 1892374483508, 0, 0, 0, -76994471986, 0], [0, 0, 0, -13721574649595, 225608558056, 12193885375629, 0, 449666783756], [0, 0, 0, -5021260796800, 6552394768552, -4848065733684, 0, 322894788848], [-1015618763248874, 38856007156128, -3757404483336, 0, 0, 0, 324647016918, 0], [0, 0, 0, 6371001916610, -3063395966758, -9111580236526, 0, 743458568227], [-1167978738745222, -213172825368, -1996960356344, 0, 0, 0, -1702825457070, 0], [0, 0, 0, 3069998675762, -4086668457816, -2795965623130, 0, -2707021044592], [-337040522451121, -16710277743744, -3005904633978, 0, 0, 0, -248191349883, 0], [0, 0, 0, -3720894504778, -4626438693794, -28254943352478, 0, -135374236231], [2021861190386938, 18808614274792, 2428209729912, 0, 0, 0, -437467269750, 0], [0, 0, 0, -3047407765717, 1933589098544, -4425508680685, 0, 3733997937664], [-1059701007190306, 2424992575688, 1721427113232, 0, 0, 0, -51278774622, 0], [812034151382178, 12627185931216, 174767279660, 0, 0, 0, 788905733262, 0], [0, 0, 0, 15266894459134, -4883139784424, 204991881754, 0, -241925426320], [0, 0, 0, 3439858117564, 405049376508, 28442586470815, 0, -3102199604078], [1742253715258882, -31212538069560, 3630939374816, 0, 0, 0, -473600465214, 0], [1735197170200315, -108176939151224, -682607998462, 0, 0, 0, 727938297845, 0], [0, 0, 0, 19989580385798, 7858350000146, -3465394559750, 0, 865024607467], [65792417448854, -28690037702176, -3885709561568, 0, 0, 0, -1742672480456, 0], [390279912768944, -106206183457392, 1879490483360, 0, 0, 0, -722526741456, 0], [0, 0, 0, -6497892828692, -1406458524380, -6567221499325, 0, 1963655511286], [0, 0, 0, -9598746574326, 3460298180166, 7086300775908, 0, -922485109239], [0, 0, 0, -4938301831578, 1367305984680, 52555991702674, 0, -1510645236096], [1080251066543038, -91821202847424, -4069443802304, 0, 0, 0, -323595805344, 0], [-1724562668076563, 28438107017200, -3315014695162, 0, 0, 0, -54811595323, 0], [0, 0, 0, -8108236218567, 4650430246656, -4886978956367, 0, 263074278888], [0, 0, 0, 28803460528671, 3760482351560, -3118510160113, 0, -832972318068], [0, 0, 0, 15623041088080, 564705550132, -24191692115043, 0, -732287498374], [1335802953281737, -143028657120968, 247461819190, 0, 0, 0, 889387859431, 0], [1266540595268097, -74623236635808, -309624185472, 0, 0, 0, 181523098281, 0], [3155562916815902, -32003001575640, 917497687832, 0, 0, 0, 1703271291150, 0], [2277761025517328, -104417524852176, -566410767280, 0, 0, 0, 1756007390304, 0], [0, 0, 0, -18855351921758, -14654663657766, -40928387411264, 0, -1436032478381], [-889252758302438, 15034787761792, -3831607351672, 0, 0, 0, 99250767914, 0], [0, 0, 0, -2424680946450, 11234098752816, 41488588664254, 0, -1778177522304], [0, 0, 0, 685279637948, 4295875216276, 11958153922041, 0, 4224498072218], [0, 0, 0, 5018959446312, -10207141221384, -30938031293892, 0, 2496056592744], [-1197193137413171, 79174191484888, -931324377458, 0, 0, 0, -783463621133, 0], [0, 0, 0, -2999037819726, 7068087556674, 36713007199998, 0, -915729422337], [-655826151678844, 13645223544032, 1771896044008, 0, 0, 0, 1691342371228, 0], [0, 0, 0, 8861008913953, -11907073247720, 64001781356873, 0, -2522382967496], [0, 0, 0, -11260606526884, -6832796202568, -22919527791969, 0, -2227733122040], [0, 0, 0, -1833935445730, 1295309611438, -10423640347770, 0, -2531454959143], [-17445066459035, 76510567537120, 7076531760760, 0, 0, 0, 201136259581, 0], [-2683281125311164, -89560790636976, 2714738295624, 0, 0, 0, -834726802212, 0], [0, 0, 0, 3177000813961, -1798990683512, -26260303145511, 0, 5952555443552], [0, 0, 0, -10075752012594, 6837096257472, -790722238170, 0, -3526688761224]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_49_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_49_10_a_g();".
// 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_49_10_a_g(:prec:=8)
    chi := MakeCharacter_49_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 10));
    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_49_10_a_g();".
// 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_49_10_a_g( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_49_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,10,sign)));
    Vf := Kernel([<2,R![-42752, -20856, -906, 33, 1]>,<3,R![7015444108029504, 0, -11516433342120, 0, 1908561916, 0, -92702, 0, 1]>],Snew);
    return Vf;
end function;