// Make newform 75.9.c.e in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_75_c();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_75_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_75_9_c_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_75_9_c_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 := [105274575720, 0, 10575941812, 0, 148116566, 0, 776307, 0, 1634, 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], [-420291052310412, 0, -46859095322612, 0, -444620825535, 0, -1265751468, 0, -872557, 0], [0, 3248388872882004, 0, 435449159046332, 0, 4207972253529, 0, 12128076852, 0, 8400715], [8500823024651488512, 333149360354096228, 838390872436304960, 7688439646206644, 8070520521247776, 45386771554953, 23429801734080, 95509278940, 16283051488, 57764587], [-3166797504195456336, 302148258954712068, -294739647408790352, 25589598548147796, -2784656217757620, 239395694736177, -7973245106928, 683931862716, -5511595228, 472531299], [78686308946433819024, 1406792850841336212, 7306496448374739344, 114013619651256356, 68650021977427524, 1062323361567957, 195884281212336, 3031560854316, 135239021452, 2093833615], [-2376421310019837708, -79972255476863300, -232437279369669340, -11263359272015892, -2222840829944163, -111468906693249, -6426141790740, -326570626108, -4459205273, -227599251], [6149466470074792824, 2502732354612363948, 470492136055365208, 126525776892571548, 4181709941725758, 1168104544393851, 11462785669512, 3397055634708, 7788738074, 2367048033], [-2305416434330119500, 1081393765837086024, -244869910854985916, 101849890279564424, -2330474252110563, 968434359217698, -6744083301684, 2788945929432, -4685079769, 1932377350], [-744725103888164832, 2509334301004612548, -53505191980737856, 197852659386244756, -485940762965256, 1824598860588897, -1377320091264, 5148112932156, -949770776, 3538913459]]; Rf_basisdens := [1, 157481360352, 153386844982848, 818063173241856, 2454189519725568, 2454189519725568, 204515793310464, 1227094759862784, 613547379931392, 613547379931392]; 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_75_c();" function MakeCharacter_75_c() N := 75; order := 2; char_gens := [26, 52]; v := [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_75_c_Hecke();" function MakeCharacter_75_c_Hecke(Kf) N := 75; order := 2; char_gens := [26, 52]; char_values := [[-1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 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 := 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 := 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 := 9; raw_aps := [[0, 0, 1, 0, 0, 0, 0, 0, 0, 0], [-2, 0, -1, 0, 1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [204, -5, -5, 0, 12, 0, -1, 0, -1, 0], [27, 3, -227, 1, 62, 3, 0, -1, 2, 5], [-1782, 55, 55, 9, -133, 14, -9, 0, 0, 0], [-65, 0, -1146, -31, -119, -12, -45, 11, 3, 11], [-14219, 7, -103, 51, 274, -83, -69, 0, -18, 0], [30, -15, 3399, -74, 2, -6, -90, -13, 29, -43], [-2242, 33, 13887, -42, -4446, -102, 198, -61, -179, 5], [-300298, -1371, -579, -102, 1322, 236, -47, 0, -149, 0], [305580, 2860, -4160, -654, 10012, -794, 532, 0, -122, 0], [-1817, 252, 73222, 569, -2897, 42, 945, -19, -357, 485], [1165014, 5535, 6475, 1170, -15408, 510, -781, 0, 389, 0], [-12930, 210, -74702, 730, -25378, -552, 2250, -148, -1372, 272], [16110, -945, 205525, 750, 29856, 564, -900, 471, 1179, -1419], [-35864, -174, 112314, -2384, -69742, -3474, -2214, 2508, -2678, 2160], [149352, 21081, -3551, -753, 9273, -4506, 2297, 0, 1544, 0], [-1601149, 685, -28425, -12243, 64192, 7897, 10745, 0, -1498, 0], [42754, 1491, 851161, 4522, 92384, 2496, 180, 2403, 1939, 5385], [-5296635, 32675, 48955, -18639, -125393, 20866, 31267, 0, 12628, 0], [847336, -98948, 41072, 8376, -97546, 3242, -6484, 0, 1892, 0], [-56115, 3150, 946122, -17685, -102372, -4623, -18630, 408, 537, 6708], [201699, -4851, 626361, 3729, 389847, 10464, -14031, 1002, 16758, -8700], [-40438498, 158945, 139485, 24933, -330143, 1828, -12727, 0, 12206, 0], [239900, -6195, 2037055, 36840, 452700, 20130, 34740, -8515, 10615, -20905], [-616214, 272920, -40840, 102, 91302, 33672, -21800, 0, -21698, 0], [93485, -12285, -33467, 36559, 163370, 729, 27720, 13255, -4416, -11315], [106288656, 25709, 210685, -22245, -532455, 111810, 33005, 0, 10760, 0], [85005, 14835, -2510381, -8945, 205775, 11910, -10755, -8740, 10550, 20930], [-108151942, 467070, -547830, -23646, 1307812, 11254, -76234, 0, -99880, 0], [411496, -14061, -11738191, 29188, 807596, 12474, -22140, 26787, 24541, -1335], [-890865, 13305, 7025409, -97883, -1692943, -80130, -93285, 48872, -53470, 75482], [131935131, -70086, -278120, -16635, 692690, -142645, 28050, 0, 11415, 0], [-1060270, 12390, 3465810, 12630, -2098740, -50910, 124560, -15370, -96560, 9410], [-12919808, 1463091, 493019, -87768, -1244632, 219164, 151367, 0, 63599, 0], [-540042246, 518785, -739835, -185655, 1733737, 23680, 139609, 0, -46046, 0], [-363107947, -1085935, 1689735, 423399, -3983708, 65119, -365539, 0, 57860, 0], [-2658650, -44685, 27086105, -88570, -5419856, -192924, 86940, 31499, -207009, -57871], [2264300, 40515, -6065459, 200448, 4430778, 259020, 261450, -219367, 158365, -138337], [1188073, -23772, -3215208, 167383, 2433554, 29583, 1728, 128724, 36931, 81180], [-117496118, 1825321, 2291789, 379917, -5469867, 227784, -240623, 0, 139294, 0], [3391338, 2097, 39641567, 21834, 6540558, 304962, -51030, -248409, 321273, -244215], [-38963101, -1937830, -1105770, -105840, 2837242, -1022810, 357314, 0, 251474, 0], [724150, 175410, 10436042, 251474, 1683814, 243480, 535770, -290716, 6420, 60104], [197539636, -3842511, 1315325, 546000, -3070090, 112550, -655555, 0, -109555, 0], [-499818493, -4299211, 4516671, -371127, -11027468, 566161, 1270673, 0, 899546, 0], [735373060, 1993905, -1698355, 835596, 4227682, 77446, -1707603, 0, -872007, 0], [-3259060, -283905, 33802125, -909776, -7420168, -361518, -888840, -50333, 29397, -618143], [584631816, 4764459, 4519975, 1485375, -10370465, -902000, -1111805, 0, 373570, 0], [5493540, 422535, -24140775, 494488, 12289082, 491436, 164250, 34397, 295841, 879467], [9319714, -84771, 76769631, 1120274, 18760132, 500664, 324504, 375017, 420753, 205475], [864670047, 3908734, 6566026, -19212, -16167058, 2023766, 534438, 0, 515226, 0], [4657705, -214500, 69401600, 412855, 9046130, 105375, -207360, 374220, 245995, -54780], [-1138180, 458670, 6670114, 195740, -871772, 79212, 361260, 28578, -194098, 945918], [2373790, -266685, -78405995, -72354, 4074558, 378, -432990, 127033, 233603, -406337], [-11793988, 440187, 97049693, -2071248, -22686114, -635568, -1305378, -418699, -347171, 461675], [2205694002, -1863626, -7902254, 1977798, 19950392, -2947114, -3566082, 0, -1588284, 0], [3062007558, -13275825, -15856985, 222789, 38768227, -3334526, -2194049, 0, -1971260, 0], [-4307244, -376671, -27371241, 1448048, -8597774, -665976, 1262610, 1146727, -961289, 393385], [-1285922693, -17474225, 12471965, 1630953, -30055318, 2243963, -1012717, 0, 618236, 0], [7468820, -249330, -180935018, -813804, 13627830, 501606, -1255050, -561820, 1003066, -1060480], [6156539559, 16749895, 5373565, 1798485, -12949678, 2064095, -2628501, 0, -830016, 0], [-7336228, 1302558, 43456198, -251604, -13552578, 1069878, 2850750, -2787796, -314558, -182680], [5273445258, 13306205, 12134185, -842931, -30140943, 4251564, 2074901, 0, 1231970, 0], [-8164120, -86250, 466173042, -2329008, -16697400, -734148, -2269800, -110410, 51202, -282910], [1107579307, -22687534, -28211056, -3473343, 66896518, 576739, -315558, 0, -3788901, 0], [152383065, -27323410, 16768390, -3969114, -42577658, 7698496, 6624862, 0, 2655748, 0], [58612455, 1132980, 92929828, 2321225, 121365592, 3748683, -2687310, 741742, 4266793, 3007702], [4821999216, -6030916, -17829800, -3103350, 43578880, -7040150, 4215460, 0, 1112110, 0], [-34434420, 1474905, -242476569, -4951304, -67518106, -908508, -715770, -3073981, -1161553, -124171], [34347036, -1204779, -353247081, 2418296, 65059108, 1959066, 117396, -20627, 2321527, -2430185], [7893328236, 20752605, 39173425, 3032712, -93852038, 2211802, 2097625, 0, 5130337, 0], [2948646630, 17193805, -7602815, -4675599, 17996777, -3425704, 7330677, 0, 2655078, 0], [-3509491959, -12763617, 73775161, 10973643, -175151778, 1430601, -3375037, 0, 7598606, 0], [-30886460, 1863270, -468900682, 3510596, -54452774, -1568310, 5954490, 1730336, -4174230, 5456876], [746374, 1066449, 815626811, -5197506, 4073952, 624, -6336756, -618143, 1757393, 1514755], [702208872, 21774255, 17064595, 7587903, -37067253, -11850972, -3987337, 0, 3600566, 0], [-53079305, 9990, 885899000, -2365855, -106743335, -3144600, 2095155, -614695, -3846315, -594715], [9873479701, 12459542, -19290058, 2477286, 47466434, -3374188, -6471994, 0, -3994708, 0], [-7187833, -2203473, 726775053, -3912843, -18162204, -2694003, -7165818, 2823881, 429094, -1583065], [-8818737428, 66535626, -54248086, -1451358, 132472838, -12838516, -3392038, 0, -4843396, 0], [-70917664, -340641, 512339869, -13247492, -142560184, -5952666, -11558160, 297067, -1986399, -384215], [71059597, -50087320, 20420300, 10956738, -46246308, -2855322, -12304592, 0, -1347854, 0], [19269123966, -91386597, -63167009, -10645542, 150142692, -3354894, 5741423, 0, -4904119, 0], [-27873475, 2446020, 783229696, -6958685, -57454858, 1985943, 1375020, -9045968, 712263, -4153928], [-115463145, 4740975, -1632904025, 4356165, -218530125, -4132860, 17844165, -1826760, -11661240, 7655190], [-563047271, 44399105, -104983795, -17430675, 247291377, 5917980, 4525129, 0, -12905546, 0], [-93205868, 879183, 1577506313, -2862424, -184482938, -5269032, 5112630, -708751, -7266303, 1049615], [13551156248, -14631510, -67882230, -13584816, 162315272, -10574416, 12389026, 0, -1195790, 0], [-14280600, -943935, -1838759245, -9510500, -33532216, -1355694, -9495360, -2139211, 2124071, -4027081], [53559938, 2333598, 1113913522, 26291478, 114100644, 6938298, 30226608, -20026, -3915104, 4647170], [-9684982250, 101922645, 35907245, 929346, -87543038, 7757596, 2591457, 0, 3520803, 0], [62685388, 4247382, -970373738, -6222756, 143003178, 2190882, -17832150, 4890256, 6719138, 13385020], [39893030226, 64359167, 35298067, -3781614, -84532076, -4698038, 14989411, 0, 11207797, 0], [-78624180, -296775, 2381051303, -4051512, -154404210, -6726612, -1737450, 3734475, -6048537, 3140925], [-197139234, -7414824, 445199164, -11452894, -409455242, -18034884, -4435974, 7067698, -14084618, -7761950], [-7408391, -4732824, -2580784154, 16033207, -31541311, 1098606, 23824575, -2526057, -5265311, -11991705], [25676468825, 172424905, 47223975, -2457969, -111926928, -9511029, 17181957, 0, 14723988, 0], [-7010431068, -74485681, -35363669, -11546097, 82383167, 902096, 11040183, 0, -505914, 0], [11619354437, -124374580, 149565330, 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[10181145202, 58546214, 138951026, 53715588, -306197178, -82710834, -25070402, 0, 28645186, 0], [191701412590, -258491220, 489322460, 45728736, -1168841448, 18066456, 17471932, 0, 63200668, 0], [-783560360, 9008205, 5086459811, -24533540, -1571616370, -33349680, 65008710, -31520265, -58021985, -13503855], [211850515, -22500030, 11639597558, 49388341, 360701756, 9801255, 41379390, 4500816, 3508135, -40499244], [-55445776283, 411396484, 470372906, 234903, -1158189368, 146706871, 35015988, 0, 35250891, 0], [27965590, 20690145, -1245295025, 18750590, 117163450, 10281450, 24306030, -838165, -4717275, 40542125], [143578542316, 412001167, -753683173, -105819009, 1797155239, -47397128, 37532631, 0, -68286378, 0], [400805912, -12926463, -12093628157, 14705612, 759699256, 22795122, -13897728, -3133179, 31736519, -28986105], [20995555518, -294611445, 456602335, 88207734, -1091803408, 67996754, -77010609, 0, 11197125, 0], [43485934342, 130278930, 476655710, 115454148, -1105790378, -75462482, -59210502, 0, 56243646, 0], [69698480786, 321668764, 545288100, 60937350, -1306629750, 50628900, -8693700, 0, 52243650, 0], [-798745970, -20926470, 2899190214, -52851782, -1653515404, -62139618, -6665400, 6755946, -52942328, -35096994], [122194684722, 360605016, 97911064, -146096958, -261287622, -14591616, 259173992, 0, 113077034, 0], [-51675503836, 174541935, -207646245, -122438085, 464631307, 46160320, 135907559, 0, 13469474, 0], [-757002883, -27794097, 3618748423, -71839409, -1564428673, -74727522, -59003955, 32959384, -45794838, -22628810], [-81758612949, -1098430598, -866363018, -150796044, 2073752374, -123201698, 117281746, 0, -33514298, 0], [465490870, -15484995, 8917446563, 101473550, 903369274, 27006786, 75693870, 18842519, 6937161, -12127471], [-73039177730, 33682835, 209039255, 189410946, -487445868, 118169226, -308517353, 0, -119106407, 0], [525028230, -58379280, 4831019132, 19841914, 946656434, -13491120, -86796810, 71737814, 34900910, -45020746], [-720955924, 33788361, -1456436121, -75883624, -1459117442, 7424316, 69788286, -118570677, -27101233, -50993955], [-215749112258, -238343329, 590983019, 266290182, -1358647652, -32267786, -294960573, 0, -28670391, 0], [-705533886, -4509624, 935919876, 52136702, -1435967846, -32012364, 141025590, -11371202, -77517686, -20390450], [-318602606220, 417875905, -331686755, -26910684, 831906682, -206203034, 59200957, 0, 32290273, 0], [1481596075, -15705825, -5957786139, 147986865, 2936697240, 91451925, 46015650, 20622565, 81348650, -10789085], [-137046252034, -1041663618, -420638238, -163154304, 1016570654, -198348718, 256258726, 0, 93104422, 0], [1204466664, -23757711, 5809362071, -15305876, 2390948402, 40120524, -174950766, 53288207, 106356683, 5772785], [301132431426, 29717485, -378922775, -203210283, 860777937, 39973332, 233362565, 0, 30152282, 0], [-747684515, 54467475, 16696643859, 107521855, -1301192497, -27983658, 184330845, 30774108, -107583098, 139709058], [165491020581, 569492298, 194778556, 44632503, -471055398, 62566821, -57483022, 0, -12850519, 0], [-2409895370, 53496060, 955080592, -350344502, -4723192786, -140933676, -191532870, -63890226, -94921406, 43101894], [-159775702292, 134131855, -1144756405, -221256261, 2725658147, -112470136, 170046791, 0, -51209470, 0], [-1936365564, -63370566, -2390457386, -12238332, -4006557444, -157158396, 101266740, 56285382, -169790454, -70455750], [-113470741423, 523213035, 11638365, 104754183, -54681258, 238500093, -260679057, 0, -155924874, 0], [207763400, 68235060, 17751935812, -39317080, 462132196, 92764884, 73144260, -158099784, 45638444, -21629664], [-140488025626, 1265397000, 329358960, 73173810, -808639738, 164064980, -116832296, 0, -43658486, 0], [-60391068, 102721938, 8078161078, -11482204, 51978832, 81064008, 115648920, -135404846, 8273722, 70039030], [-183115181574, -226841511, 528519985, 58233330, -1318070790, 306347460, -110008315, 0, -51774985, 0], [-2942292452, 36275913, 19925817607, -111542872, -5759905586, -182056452, 100130958, 15851579, -227525409, 88403405], [30073033415, 1926843730, -1578896730, -256274304, 3690783022, 236571346, -368998, 0, -256643302, 0], [-1167872062, -373728, 5935364592, -92230866, -2303816892, -93267138, -48205080, 33048416, -77074202, 32300960], [2943676440, 14598465, -22386427533, 33389804, 6022268824, 143859090, -295085520, 91120549, 237354775, 120317479], [110892055, 83408205, -12371484939, -100241139, 539815629, -8658678, -197379945, 67806904, 29331902, 234623314], [-467364328144, -771130590, 1499914050, -379762488, -3758895408, 415745952, 741304410, 0, 361541922, 0], [-303332541, -18049944, 9252812136, -49196323, -500191316, -95078829, -181136610, 160623598, -28683311, 124523710], [-1382506115, -78396615, 7826349105, -164672281, -2998551323, -124434648, -102030885, 1650752, -64292148, -155142478], [1715336740, 23585910, 9181741382, 69004900, 3279533258, 207327162, 137156490, -221897952, 153746362, -174726132], [530538895272, -88136909, -1833173141, 15623352, 4481662288, -394610396, -233787833, 0, -218164481, 0], [512875846652, 1284195170, 1861263330, 220987638, -4447728558, 124470468, -31115742, 0, 189871896, 0]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_75_c_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_75_9_c_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_75_9_c_e(:prec:=10) chi := MakeCharacter_75_c(); f_vec := qexpCoeffs(); Kf := Universe(f_vec); // SetVerbose("ModularForms", true); // SetVerbose("ModularSymbols", true); S := CuspidalSubspace(ModularForms(chi, 9)); S := BaseChange(S, Kf); maxprec := NextPrime(997) - 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_75_9_c_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_75_9_c_e( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_75_c(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,9,sign))); Vf := Kernel([<2,R![409079808000, 0, 48388216960, 0, 442732064, 0, 1478418, 0, 2057, 0, 1]>,<7,R![525098700000000, 8426583900000, -7054500600, -16170189, -980, 1]>],Snew); return Vf; end function;