LMFDB - Embedded newform 2401.4.a.d.1.32

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2401,4,Mod(1,2401)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2401.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2401, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 2401 = 7^{4} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2401.a (trivial)

Newform invariants

comment:select newform

sage:traces = [39,1,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$141.663585924$
Analytic rank:	$0$
Dimension:	$39$
Twist minimal:	no (minimal twist has level 49)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.32
Character	$\chi$	$=$	2401.1

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+4.01016 q^{2} +8.06567 q^{3} +8.08136 q^{4} +14.0552 q^{5} +32.3446 q^{6} +0.326265 q^{8} +38.0550 q^{9} +56.3637 q^{10} +53.6801 q^{11} +65.1815 q^{12} +64.5836 q^{13} +113.365 q^{15} -63.3425 q^{16} -52.7881 q^{17} +152.606 q^{18} -11.4048 q^{19} +113.585 q^{20} +215.265 q^{22} -98.9357 q^{23} +2.63155 q^{24} +72.5494 q^{25} +258.990 q^{26} +89.1656 q^{27} +42.5988 q^{29} +454.610 q^{30} -67.2545 q^{31} -256.624 q^{32} +432.965 q^{33} -211.689 q^{34} +307.536 q^{36} +428.543 q^{37} -45.7349 q^{38} +520.910 q^{39} +4.58573 q^{40} +45.7510 q^{41} +196.585 q^{43} +433.808 q^{44} +534.871 q^{45} -396.748 q^{46} -568.702 q^{47} -510.899 q^{48} +290.934 q^{50} -425.771 q^{51} +521.923 q^{52} +151.593 q^{53} +357.568 q^{54} +754.485 q^{55} -91.9871 q^{57} +170.828 q^{58} +171.731 q^{59} +916.141 q^{60} -5.02754 q^{61} -269.701 q^{62} -522.361 q^{64} +907.737 q^{65} +1736.26 q^{66} -196.310 q^{67} -426.600 q^{68} -797.982 q^{69} +79.6959 q^{71} +12.4160 q^{72} +719.113 q^{73} +1718.52 q^{74} +585.159 q^{75} -92.1661 q^{76} +2088.93 q^{78} +241.207 q^{79} -890.293 q^{80} -308.304 q^{81} +183.468 q^{82} -421.785 q^{83} -741.949 q^{85} +788.338 q^{86} +343.587 q^{87} +17.5139 q^{88} +628.269 q^{89} +2144.92 q^{90} -799.535 q^{92} -542.452 q^{93} -2280.59 q^{94} -160.297 q^{95} -2069.84 q^{96} -1684.67 q^{97} +2042.79 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 39 q + q^{2} + q^{3} + 145 q^{4} + 27 q^{5} + 41 q^{6} - 12 q^{8} + 312 q^{9} + 78 q^{10} + q^{11} - 91 q^{12} + 77 q^{13} - 161 q^{15} + 461 q^{16} + 211 q^{17} + 8 q^{18} + 314 q^{19} + 476 q^{20}+ \cdots + 5384 q^{99}+O(q^{100}) $ 39 * q + q^2 + q^3 + 145 * q^4 + 27 * q^5 + 41 * q^6 - 12 * q^8 + 312 * q^9 + 78 * q^10 + q^11 - 91 * q^12 + 77 * q^13 - 161 * q^15 + 461 * q^16 + 211 * q^17 + 8 * q^18 + 314 * q^19 + 476 * q^20 - 61 * q^22 - 69 * q^23 + 330 * q^24 + 606 * q^25 + 504 * q^26 - 50 * q^27 + 57 * q^29 + 42 * q^30 + 638 * q^31 - 1600 * q^32 + 1574 * q^33 + 1343 * q^34 + 782 * q^36 + 71 * q^37 + 1359 * q^38 - 84 * q^39 - 155 * q^40 + 1393 * q^41 - 125 * q^43 + 52 * q^44 + 1129 * q^45 - 1454 * q^46 + 1483 * q^47 - 974 * q^48 + 3074 * q^50 - 2044 * q^51 + 3899 * q^52 + 2213 * q^53 + 1142 * q^54 + 1604 * q^55 + 98 * q^57 + 2403 * q^58 + 2073 * q^59 - 1519 * q^60 + 2575 * q^61 + 1742 * q^62 + 1358 * q^64 + 1876 * q^65 - 48 * q^66 + 176 * q^67 + 3038 * q^68 - 638 * q^69 - 1259 * q^71 - 3799 * q^72 - 307 * q^73 + 3845 * q^74 + 131 * q^75 + 1974 * q^76 - 6041 * q^78 + 22 * q^79 - 804 * q^80 + 795 * q^81 + 8043 * q^82 + 6349 * q^83 + 3094 * q^85 + 1745 * q^86 + 9508 * q^87 + 1299 * q^88 + 2253 * q^89 + 11156 * q^90 + 1284 * q^92 - 3430 * q^93 + 2738 * q^94 - 3290 * q^95 + 3031 * q^96 - 770 * q^97 + 5384 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	4.01016	1.41780	0.708902	−	0.705307i	$-0.249191\pi$
$2$	4.01016	1.41780	0.708902	+	0.705307i	$0.249191\pi$
$3$	8.06567	1.55224	0.776119	−	0.630586i	$-0.217186\pi$
$3$	8.06567	1.55224	0.776119	+	0.630586i	$0.217186\pi$
$4$	8.08136	1.01017
$5$	14.0552	1.25714	0.628569	−	0.777754i	$-0.283641\pi$
$5$	14.0552	1.25714	0.628569	+	0.777754i	$0.283641\pi$
$6$	32.3446	2.20077
$7$	0	0
$7$	0	0
$8$	0.326265	0.0144190
$9$	38.0550	1.40944
$10$	56.3637	1.78238
$11$	53.6801	1.47138	0.735689	−	0.677320i	$-0.236859\pi$
$11$	53.6801	1.47138	0.735689	+	0.677320i	$0.236859\pi$
$12$	65.1815	1.56802
$13$	64.5836	1.37787	0.688934	−	0.724825i	$-0.258079\pi$
$13$	64.5836	1.37787	0.688934	+	0.724825i	$0.258079\pi$
$14$	0	0
$15$	113.365	1.95138
$16$	−63.3425	−0.989727
$17$	−52.7881	−0.753118	−0.376559	−	0.926393i	$-0.622893\pi$
$17$	−52.7881	−0.753118	−0.376559	+	0.926393i	$0.622893\pi$
$18$	152.606	1.99831
$19$	−11.4048	−0.137707	−0.0688535	−	0.997627i	$-0.521934\pi$
$19$	−11.4048	−0.137707	−0.0688535	+	0.997627i	$0.521934\pi$
$20$	113.585	1.26992
$21$	0	0
$22$	215.265	2.08613
$23$	−98.9357	−0.896935	−0.448468	−	0.893799i	$-0.648030\pi$
$23$	−98.9357	−0.896935	−0.448468	+	0.893799i	$0.648030\pi$
$24$	2.63155	0.0223818
$25$	72.5494	0.580395
$26$	258.990	1.95355
$27$	89.1656	0.635553
$28$	0	0
$29$	42.5988	0.272772	0.136386	−	0.990656i	$-0.456451\pi$
$29$	42.5988	0.272772	0.136386	+	0.990656i	$0.456451\pi$
$30$	454.610	2.76667
$31$	−67.2545	−0.389654	−0.194827	−	0.980838i	$-0.562414\pi$
$31$	−67.2545	−0.389654	−0.194827	+	0.980838i	$0.562414\pi$
$32$	−256.624	−1.41766
$33$	432.965	2.28393
$34$	−211.689	−1.06777
$35$	0	0
$36$	307.536	1.42378
$37$	428.543	1.90411	0.952054	−	0.305929i	$-0.0989669\pi$
$37$	428.543	1.90411	0.952054	+	0.305929i	$0.0989669\pi$
$38$	−45.7349	−0.195242
$39$	520.910	2.13878
$40$	4.58573	0.0181267
$41$	45.7510	0.174271	0.0871353	−	0.996196i	$-0.472229\pi$
$41$	45.7510	0.174271	0.0871353	+	0.996196i	$0.472229\pi$
$42$	0	0
$43$	196.585	0.697186	0.348593	−	0.937274i	$-0.386660\pi$
$43$	196.585	0.697186	0.348593	+	0.937274i	$0.386660\pi$
$44$	433.808	1.48634
$45$	534.871	1.77186
$46$	−396.748	−1.27168
$47$	−568.702	−1.76497	−0.882487	−	0.470337i	$-0.844132\pi$
$47$	−568.702	−1.76497	−0.882487	+	0.470337i	$0.844132\pi$
$48$	−510.899	−1.53629
$49$	0	0
$50$	290.934	0.822887
$51$	−425.771	−1.16902
$52$	521.923	1.39188
$53$	151.593	0.392884	0.196442	−	0.980515i	$-0.437061\pi$
$53$	151.593	0.392884	0.196442	+	0.980515i	$0.437061\pi$
$54$	357.568	0.901090
$55$	754.485	1.84972
$56$	0	0
$57$	−91.9871	−0.213754
$58$	170.828	0.386738
$59$	171.731	0.378940	0.189470	−	0.981887i	$-0.439323\pi$
$59$	171.731	0.378940	0.189470	+	0.981887i	$0.439323\pi$
$60$	916.141	1.97122
$61$	−5.02754	−0.0105526	−0.00527631	−	0.999986i	$-0.501680\pi$
$61$	−5.02754	−0.0105526	−0.00527631	+	0.999986i	$0.501680\pi$
$62$	−269.701	−0.552453
$63$	0	0
$64$	−522.361	−1.02024
$65$	907.737	1.73217
$66$	1736.26	3.23816
$67$	−196.310	−0.357956	−0.178978	−	0.983853i	$-0.557279\pi$
$67$	−196.310	−0.357956	−0.178978	+	0.983853i	$0.557279\pi$
$68$	−426.600	−0.760777
$69$	−797.982	−1.39226
$70$	0	0
$71$	79.6959	0.133214	0.0666068	−	0.997779i	$-0.478783\pi$
$71$	79.6959	0.133214	0.0666068	+	0.997779i	$0.478783\pi$
$72$	12.4160	0.0203228
$73$	719.113	1.15296	0.576478	−	0.817112i	$-0.304426\pi$
$73$	719.113	1.15296	0.576478	+	0.817112i	$0.304426\pi$
$74$	1718.52	2.69965
$75$	585.159	0.900911
$76$	−92.1661	−0.139108
$77$	0	0
$78$	2088.93	3.03237
$79$	241.207	0.343518	0.171759	−	0.985139i	$-0.445055\pi$
$79$	241.207	0.343518	0.171759	+	0.985139i	$0.445055\pi$
$80$	−890.293	−1.24422
$81$	−308.304	−0.422913
$82$	183.468	0.247082
$83$	−421.785	−0.557794	−0.278897	−	0.960321i	$-0.589969\pi$
$83$	−421.785	−0.557794	−0.278897	+	0.960321i	$0.589969\pi$
$84$	0	0
$85$	−741.949	−0.946772
$86$	788.338	0.988473
$87$	343.587	0.423407
$88$	17.5139	0.0212158
$89$	628.269	0.748274	0.374137	−	0.927373i	$-0.377939\pi$
$89$	628.269	0.748274	0.374137	+	0.927373i	$0.377939\pi$
$90$	2144.92	2.51216
$91$	0	0
$92$	−799.535	−0.906057
$93$	−542.452	−0.604835
$94$	−2280.59	−2.50239
$95$	−160.297	−0.173117
$96$	−2069.84	−2.20054
$97$	−1684.67	−1.76343	−0.881715	−	0.471783i	$-0.843611\pi$
$97$	−1684.67	−1.76343	−0.881715	+	0.471783i	$0.843611\pi$
$98$	0	0
$99$	2042.79	2.07382
$100$	586.298	0.586298
$101$	127.424	0.125536	0.0627680	−	0.998028i	$-0.480007\pi$
$101$	127.424	0.125536	0.0627680	+	0.998028i	$0.480007\pi$
$102$	−1707.41	−1.65744
$103$	−537.160	−0.513863	−0.256932	−	0.966430i	$-0.582712\pi$
$103$	−537.160	−0.513863	−0.256932	+	0.966430i	$0.582712\pi$
$104$	21.0714	0.0198675
$105$	0	0
$106$	607.910	0.557033
$107$	−716.356	−0.647222	−0.323611	−	0.946190i	$-0.604897\pi$
$107$	−716.356	−0.647222	−0.323611	+	0.946190i	$0.604897\pi$
$108$	720.579	0.642017
$109$	−2213.74	−1.94530	−0.972652	−	0.232266i	$-0.925386\pi$
$109$	−2213.74	−1.94530	−0.972652	+	0.232266i	$0.925386\pi$
$110$	3025.61	2.62255
$111$	3456.48	2.95563
$112$	0	0
$113$	−1314.10	−1.09398	−0.546990	−	0.837139i	$-0.684226\pi$
$113$	−1314.10	−1.09398	−0.546990	+	0.837139i	$0.684226\pi$
$114$	−368.883	−0.303062
$115$	−1390.56	−1.12757
$116$	344.256	0.275546
$117$	2457.73	1.94203
$118$	688.667	0.537262
$119$	0	0
$120$	36.9870	0.0281370
$121$	1550.55	1.16495
$122$	−20.1612	−0.0149616
$123$	369.012	0.270510
$124$	−543.508	−0.393616
$125$	−737.205	−0.527501
$126$	0	0
$127$	−928.573	−0.648800	−0.324400	−	0.945920i	$-0.605162\pi$
$127$	−928.573	−0.648800	−0.324400	+	0.945920i	$0.605162\pi$
$128$	−41.7598	−0.0288366
$129$	1585.59	1.08220
$130$	3640.17	2.45588
$131$	2258.19	1.50610	0.753050	−	0.657964i	$-0.228582\pi$
$131$	2258.19	1.50610	0.753050	+	0.657964i	$0.228582\pi$
$132$	3498.95	2.30716
$133$	0	0
$134$	−787.233	−0.507512
$135$	1253.24	0.798978
$136$	−17.2229	−0.0108592
$137$	−1577.45	−0.983730	−0.491865	−	0.870671i	$-0.663685\pi$
$137$	−1577.45	−0.983730	−0.491865	+	0.870671i	$0.663685\pi$
$138$	−3200.03	−1.97395
$139$	−1525.01	−0.930576	−0.465288	−	0.885159i	$-0.654049\pi$
$139$	−1525.01	−0.930576	−0.465288	+	0.885159i	$0.654049\pi$
$140$	0	0
$141$	−4586.96	−2.73966
$142$	319.593	0.188871
$143$	3466.85	2.02736
$144$	−2410.50	−1.39496
$145$	598.735	0.342912
$146$	2883.76	1.63467
$147$	0	0
$148$	3463.21	1.92347
$149$	−2095.67	−1.15224	−0.576120	−	0.817365i	$-0.695434\pi$
$149$	−2095.67	−1.15224	−0.576120	+	0.817365i	$0.695434\pi$
$150$	2346.58	1.27732
$151$	−1256.02	−0.676912	−0.338456	−	0.940982i	$-0.609905\pi$
$151$	−1256.02	−0.676912	−0.338456	+	0.940982i	$0.609905\pi$
$152$	−3.72098	−0.00198560
$153$	−2008.85	−1.06148
$154$	0	0
$155$	−945.277	−0.489848
$156$	4209.66	2.16053
$157$	664.243	0.337658	0.168829	−	0.985645i	$-0.446001\pi$
$157$	664.243	0.337658	0.168829	+	0.985645i	$0.446001\pi$
$158$	967.279	0.487042
$159$	1222.70	0.609849
$160$	−3606.90	−1.78219
$161$	0	0
$162$	−1236.35	−0.599609
$163$	433.990	0.208544	0.104272	−	0.994549i	$-0.466749\pi$
$163$	433.990	0.208544	0.104272	+	0.994549i	$0.466749\pi$
$164$	369.730	0.176043
$165$	6085.43	2.87121
$166$	−1691.42	−0.790843
$167$	1788.14	0.828566	0.414283	−	0.910148i	$-0.364032\pi$
$167$	1788.14	0.828566	0.414283	+	0.910148i	$0.364032\pi$
$168$	0	0
$169$	1974.04	0.898518
$170$	−2975.33	−1.34234
$171$	−434.008	−0.194090
$172$	1588.68	0.704276
$173$	−891.072	−0.391601	−0.195800	−	0.980644i	$-0.562730\pi$
$173$	−891.072	−0.391601	−0.195800	+	0.980644i	$0.562730\pi$
$174$	1377.84	0.600309
$175$	0	0
$176$	−3400.23	−1.45626
$177$	1385.12	0.588205
$178$	2519.46	1.06091
$179$	2744.98	1.14620	0.573100	−	0.819486i	$-0.305741\pi$
$179$	2744.98	1.14620	0.573100	+	0.819486i	$0.305741\pi$
$180$	4322.49	1.78988
$181$	4144.28	1.70189	0.850944	−	0.525257i	$-0.176031\pi$
$181$	4144.28	1.70189	0.850944	+	0.525257i	$0.176031\pi$
$182$	0	0
$183$	−40.5504	−0.0163802
$184$	−32.2793	−0.0129329
$185$	6023.27	2.39373
$186$	−2175.32	−0.857538
$187$	−2833.67	−1.10812
$188$	−4595.89	−1.78292
$189$	0	0
$190$	−642.815	−0.245446
$191$	4540.25	1.72001	0.860003	−	0.510288i	$-0.170461\pi$
$191$	4540.25	1.72001	0.860003	+	0.510288i	$0.170461\pi$
$192$	−4213.19	−1.58365
$193$	−2821.79	−1.05242	−0.526210	−	0.850354i	$-0.676388\pi$
$193$	−2821.79	−1.05242	−0.526210	+	0.850354i	$0.676388\pi$
$194$	−6755.80	−2.50020
$195$	7321.51	2.68874
$196$	0	0
$197$	−2402.69	−0.868958	−0.434479	−	0.900682i	$-0.643067\pi$
$197$	−2402.69	−0.868958	−0.434479	+	0.900682i	$0.643067\pi$
$198$	8191.92	2.94027
$199$	5226.81	1.86190	0.930952	−	0.365142i	$-0.118980\pi$
$199$	5226.81	1.86190	0.930952	+	0.365142i	$0.118980\pi$
$200$	23.6704	0.00836873
$201$	−1583.37	−0.555633
$202$	510.989	0.177985
$203$	0	0
$204$	−3440.81	−1.18091
$205$	643.040	0.219082
$206$	−2154.10	−0.728558
$207$	−3764.99	−1.26418
$208$	−4090.89	−1.36371
$209$	−612.209	−0.202619
$210$	0	0
$211$	2243.91	0.732118	0.366059	−	0.930592i	$-0.380707\pi$
$211$	2243.91	0.732118	0.366059	+	0.930592i	$0.380707\pi$
$212$	1225.07	0.396880
$213$	642.801	0.206779
$214$	−2872.70	−0.917634
$215$	2763.05	0.876458
$216$	29.0917	0.00916406
$217$	0	0
$218$	−8877.46	−2.75806
$219$	5800.13	1.78966
$220$	6097.27	1.86854
$221$	−3409.25	−1.03770
$222$	13861.0	4.19051
$223$	2317.42	0.695900	0.347950	−	0.937513i	$-0.386878\pi$
$223$	2317.42	0.695900	0.347950	+	0.937513i	$0.386878\pi$
$224$	0	0
$225$	2760.86	0.818034
$226$	−5269.73	−1.55105
$227$	392.392	0.114731	0.0573656	−	0.998353i	$-0.481730\pi$
$227$	392.392	0.114731	0.0573656	+	0.998353i	$0.481730\pi$
$228$	−743.381	−0.215928
$229$	425.559	0.122802	0.0614011	−	0.998113i	$-0.480443\pi$
$229$	425.559	0.122802	0.0614011	+	0.998113i	$0.480443\pi$
$230$	−5576.38	−1.59868
$231$	0	0
$232$	13.8985	0.00393311
$233$	2386.26	0.670940	0.335470	−	0.942051i	$-0.391105\pi$
$233$	2386.26	0.670940	0.335470	+	0.942051i	$0.391105\pi$
$234$	9855.87	2.75341
$235$	−7993.24	−2.21881
$236$	1387.82	0.382794
$237$	1945.50	0.533222
$238$	0	0
$239$	3849.04	1.04173	0.520866	−	0.853639i	$-0.325609\pi$
$239$	3849.04	1.04173	0.520866	+	0.853639i	$0.325609\pi$
$240$	−7180.81	−1.93133
$241$	761.264	0.203474	0.101737	−	0.994811i	$-0.467560\pi$
$241$	761.264	0.203474	0.101737	+	0.994811i	$0.467560\pi$
$242$	6217.95	1.65167
$243$	−4894.15	−1.29202
$244$	−40.6293	−0.0106599
$245$	0	0
$246$	1479.80	0.383530
$247$	−736.561	−0.189742
$248$	−21.9428	−0.00561843
$249$	−3401.98	−0.865829
$250$	−2956.31	−0.747894
$251$	2479.68	0.623569	0.311785	−	0.950153i	$-0.399073\pi$
$251$	2479.68	0.623569	0.311785	+	0.950153i	$0.399073\pi$
$252$	0	0
$253$	−5310.87	−1.31973
$254$	−3723.73	−0.919871
$255$	−5984.31	−1.46962
$256$	4011.42	0.979351
$257$	1276.68	0.309873	0.154936	−	0.987924i	$-0.450483\pi$
$257$	1276.68	0.309873	0.154936	+	0.987924i	$0.450483\pi$
$258$	6358.47	1.53435
$259$	0	0
$260$	7335.75	1.74978
$261$	1621.09	0.384457
$262$	9055.70	2.13535
$263$	1700.49	0.398694	0.199347	−	0.979929i	$-0.436118\pi$
$263$	1700.49	0.398694	0.199347	+	0.979929i	$0.436118\pi$
$264$	141.262	0.0329320
$265$	2130.67	0.493909
$266$	0	0
$267$	5067.41	1.16150
$268$	−1586.45	−0.361597
$269$	−1037.68	−0.235198	−0.117599	−	0.993061i	$-0.537520\pi$
$269$	−1037.68	−0.235198	−0.117599	+	0.993061i	$0.537520\pi$
$270$	5025.70	1.13279
$271$	−4672.86	−1.04744	−0.523720	−	0.851891i	$-0.675456\pi$
$271$	−4672.86	−1.04744	−0.523720	+	0.851891i	$0.675456\pi$
$272$	3343.73	0.745380
$273$	0	0
$274$	−6325.84	−1.39474
$275$	3894.46	0.853980
$276$	−6448.78	−1.40642
$277$	−6819.03	−1.47912	−0.739559	−	0.673092i	$-0.764966\pi$
$277$	−6819.03	−1.47912	−0.739559	+	0.673092i	$0.764966\pi$
$278$	−6115.55	−1.31937
$279$	−2559.37	−0.549195
$280$	0	0
$281$	5370.77	1.14019	0.570095	−	0.821579i	$-0.306906\pi$
$281$	5370.77	1.14019	0.570095	+	0.821579i	$0.306906\pi$
$282$	−18394.4	−3.88430
$283$	−403.722	−0.0848014	−0.0424007	−	0.999101i	$-0.513501\pi$
$283$	−403.722	−0.0848014	−0.0424007	+	0.999101i	$0.513501\pi$
$284$	644.051	0.134568
$285$	−1292.90	−0.268718
$286$	13902.6	2.87440
$287$	0	0
$288$	−9765.80	−1.99811
$289$	−2126.41	−0.432814
$290$	2401.02	0.486182
$291$	−13588.0	−2.73726
$292$	5811.41	1.16468
$293$	−3581.72	−0.714152	−0.357076	−	0.934075i	$-0.616226\pi$
$293$	−3581.72	−0.714152	−0.357076	+	0.934075i	$0.616226\pi$
$294$	0	0
$295$	2413.72	0.476379
$296$	139.819	0.0274554
$297$	4786.42	0.935138
$298$	−8403.95	−1.63365
$299$	−6389.62	−1.23586
$300$	4728.88	0.910074
$301$	0	0
$302$	−5036.85	−0.959729
$303$	1027.76	0.194862
$304$	722.407	0.136292
$305$	−70.6632	−0.0132661
$306$	−8055.80	−1.50497
$307$	−1486.48	−0.276345	−0.138173	−	0.990408i	$-0.544123\pi$
$307$	−1486.48	−0.276345	−0.138173	+	0.990408i	$0.544123\pi$
$308$	0	0
$309$	−4332.55	−0.797638
$310$	−3790.71	−0.694509
$311$	−1574.74	−0.287123	−0.143561	−	0.989641i	$-0.545855\pi$
$311$	−1574.74	−0.287123	−0.143561	+	0.989641i	$0.545855\pi$
$312$	169.955	0.0308391
$313$	1514.08	0.273421	0.136711	−	0.990611i	$-0.456347\pi$
$313$	1514.08	0.273421	0.136711	+	0.990611i	$0.456347\pi$
$314$	2663.72	0.478734
$315$	0	0
$316$	1949.28	0.347012
$317$	−3669.78	−0.650206	−0.325103	−	0.945679i	$-0.605399\pi$
$317$	−3669.78	−0.650206	−0.325103	+	0.945679i	$0.605399\pi$
$318$	4903.20	0.864647
$319$	2286.70	0.401351
$320$	−7341.90	−1.28258
$321$	−5777.88	−1.00464
$322$	0	0
$323$	602.036	0.103710
$324$	−2491.51	−0.427214
$325$	4685.50	0.799707
$326$	1740.37	0.295675
$327$	−17855.3	−3.01958
$328$	14.9270	0.00251281
$329$	0	0
$330$	24403.5	4.07082
$331$	−4571.30	−0.759098	−0.379549	−	0.925172i	$-0.623921\pi$
$331$	−4571.30	−0.759098	−0.379549	+	0.925172i	$0.623921\pi$
$332$	−3408.60	−0.563467
$333$	16308.2	2.68373
$334$	7170.73	1.17474
$335$	−2759.18	−0.450000
$336$	0	0
$337$	4183.06	0.676160	0.338080	−	0.941117i	$-0.390223\pi$
$337$	4183.06	0.676160	0.338080	+	0.941117i	$0.390223\pi$
$338$	7916.22	1.27392
$339$	−10599.1	−1.69812
$340$	−5995.96	−0.956401
$341$	−3610.23	−0.573327
$342$	−1740.44	−0.275182
$343$	0	0
$344$	64.1390	0.0100527
$345$	−11215.8	−1.75026
$346$	−3573.34	−0.555214
$347$	2545.96	0.393874	0.196937	−	0.980416i	$-0.436901\pi$
$347$	2545.96	0.393874	0.196937	+	0.980416i	$0.436901\pi$
$348$	2776.65	0.427713
$349$	2294.54	0.351931	0.175965	−	0.984396i	$-0.443695\pi$
$349$	2294.54	0.351931	0.175965	+	0.984396i	$0.443695\pi$
$350$	0	0
$351$	5758.64	0.875708
$352$	−13775.6	−2.08591
$353$	−1427.38	−0.215218	−0.107609	−	0.994193i	$-0.534319\pi$
$353$	−1427.38	−0.215218	−0.107609	+	0.994193i	$0.534319\pi$
$354$	5554.56	0.833959
$355$	1120.14	0.167468
$356$	5077.27	0.755884
$357$	0	0
$358$	11007.8	1.62509
$359$	−10036.8	−1.47555	−0.737776	−	0.675046i	$-0.764124\pi$
$359$	−10036.8	−1.47555	−0.737776	+	0.675046i	$0.764124\pi$
$360$	174.510	0.0255486
$361$	−6728.93	−0.981037
$362$	16619.2	2.41294
$363$	12506.2	1.80828
$364$	0	0
$365$	10107.3	1.44943
$366$	−162.614	−0.0232239
$367$	826.066	0.117494	0.0587470	−	0.998273i	$-0.481289\pi$
$367$	826.066	0.117494	0.0587470	+	0.998273i	$0.481289\pi$
$368$	6266.83	0.887721
$369$	1741.05	0.245625
$370$	24154.3	3.39384
$371$	0	0
$372$	−4383.75	−0.610986
$373$	−7560.64	−1.04953	−0.524766	−	0.851247i	$-0.675847\pi$
$373$	−7560.64	−1.04953	−0.524766	+	0.851247i	$0.675847\pi$
$374$	−11363.5	−1.57110
$375$	−5946.05	−0.818807
$376$	−185.548	−0.0254492
$377$	2751.18	0.375844
$378$	0	0
$379$	−2347.05	−0.318099	−0.159050	−	0.987271i	$-0.550843\pi$
$379$	−2347.05	−0.318099	−0.159050	+	0.987271i	$0.550843\pi$
$380$	−1295.42	−0.174877
$381$	−7489.56	−1.00709
$382$	18207.1	2.43863
$383$	−6196.76	−0.826736	−0.413368	−	0.910564i	$-0.635648\pi$
$383$	−6196.76	−0.826736	−0.413368	+	0.910564i	$0.635648\pi$
$384$	−336.821	−0.0447612
$385$	0	0
$386$	−11315.8	−1.49213
$387$	7481.05	0.982643
$388$	−13614.5	−1.78136
$389$	12844.6	1.67415	0.837076	−	0.547086i	$-0.184263\pi$
$389$	12844.6	1.67415	0.837076	+	0.547086i	$0.184263\pi$
$390$	29360.4	3.81211
$391$	5222.63	0.675498
$392$	0	0
$393$	18213.8	2.33782
$394$	−9635.17	−1.23201
$395$	3390.22	0.431850
$396$	16508.5	2.09491
$397$	−8684.23	−1.09786	−0.548928	−	0.835870i	$-0.684964\pi$
$397$	−8684.23	−1.09786	−0.548928	+	0.835870i	$0.684964\pi$
$398$	20960.3	2.63982
$399$	0	0
$400$	−4595.46	−0.574432
$401$	−9281.58	−1.15586	−0.577930	−	0.816086i	$-0.696139\pi$
$401$	−9281.58	−1.15586	−0.577930	+	0.816086i	$0.696139\pi$
$402$	−6349.56	−0.787779
$403$	−4343.54	−0.536891
$404$	1029.76	0.126813
$405$	−4333.28	−0.531660
$406$	0	0
$407$	23004.2	2.80166
$408$	−138.914	−0.0168561
$409$	−10083.7	−1.21908	−0.609541	−	0.792755i	$-0.708646\pi$
$409$	−10083.7	−1.21908	−0.609541	+	0.792755i	$0.708646\pi$
$410$	2578.69	0.310616
$411$	−12723.2	−1.52698
$412$	−4340.98	−0.519089
$413$	0	0
$414$	−15098.2	−1.79236
$415$	−5928.28	−0.701224
$416$	−16573.7	−1.95334
$417$	−12300.3	−1.44448
$418$	−2455.05	−0.287274
$419$	−13931.9	−1.62438	−0.812192	−	0.583391i	$-0.801726\pi$
$419$	−13931.9	−1.62438	−0.812192	+	0.583391i	$0.801726\pi$
$420$	0	0
$421$	13906.0	1.60983	0.804914	−	0.593391i	$-0.202211\pi$
$421$	13906.0	1.60983	0.804914	+	0.593391i	$0.202211\pi$
$422$	8998.42	1.03800
$423$	−21641.9	−2.48763
$424$	49.4594	0.00566500
$425$	−3829.75	−0.437106
$426$	2577.73	0.293173
$427$	0	0
$428$	−5789.13	−0.653804
$429$	27962.5	3.14695
$430$	11080.3	1.24265
$431$	33.5963	0.00375470	0.00187735	−	0.999998i	$-0.499402\pi$
$431$	33.5963	0.00375470	0.00187735	+	0.999998i	$0.499402\pi$
$432$	−5647.97	−0.629024
$433$	6033.58	0.669643	0.334821	−	0.942282i	$-0.391324\pi$
$433$	6033.58	0.669643	0.334821	+	0.942282i	$0.391324\pi$
$434$	0	0
$435$	4829.20	0.532281
$436$	−17890.1	−1.96509
$437$	1128.34	0.123514
$438$	23259.4	2.53739
$439$	4860.19	0.528392	0.264196	−	0.964469i	$-0.414893\pi$
$439$	4860.19	0.528392	0.264196	+	0.964469i	$0.414893\pi$
$440$	246.163	0.0266712
$441$	0	0
$442$	−13671.6	−1.47125
$443$	5937.54	0.636797	0.318399	−	0.947957i	$-0.396855\pi$
$443$	5937.54	0.636797	0.318399	+	0.947957i	$0.396855\pi$
$444$	27933.1	2.98569
$445$	8830.46	0.940683
$446$	9293.21	0.986651
$447$	−16902.9	−1.78855
$448$	0	0
$449$	1354.76	0.142394	0.0711970	−	0.997462i	$-0.477318\pi$
$449$	1354.76	0.142394	0.0711970	+	0.997462i	$0.477318\pi$
$450$	11071.5	1.15981
$451$	2455.91	0.256418
$452$	−10619.7	−1.10511
$453$	−10130.7	−1.05073
$454$	1573.55	0.162666
$455$	0	0
$456$	−30.0122	−0.00308213
$457$	−496.606	−0.0508320	−0.0254160	−	0.999677i	$-0.508091\pi$
$457$	−496.606	−0.0508320	−0.0254160	+	0.999677i	$0.508091\pi$
$458$	1706.56	0.174110
$459$	−4706.89	−0.478646
$460$	−11237.6	−1.13904
$461$	−253.212	−0.0255819	−0.0127909	−	0.999918i	$-0.504072\pi$
$461$	−253.212	−0.0255819	−0.0127909	+	0.999918i	$0.504072\pi$
$462$	0	0
$463$	−10623.6	−1.06635	−0.533174	−	0.846006i	$-0.679001\pi$
$463$	−10623.6	−1.06635	−0.533174	+	0.846006i	$0.679001\pi$
$464$	−2698.31	−0.269970
$465$	−7624.29	−0.760361
$466$	9569.27	0.951262
$467$	−7415.64	−0.734807	−0.367404	−	0.930062i	$-0.619753\pi$
$467$	−7415.64	−0.734807	−0.367404	+	0.930062i	$0.619753\pi$
$468$	19861.8	1.96178
$469$	0	0
$470$	−32054.1	−3.14585
$471$	5357.56	0.524126
$472$	56.0298	0.00546394
$473$	10552.7	1.02582
$474$	7801.75	0.756005
$475$	−827.409	−0.0799245
$476$	0	0
$477$	5768.85	0.553747
$478$	15435.3	1.47697
$479$	17499.2	1.66923	0.834614	−	0.550836i	$-0.185691\pi$
$479$	17499.2	1.66923	0.834614	+	0.550836i	$0.185691\pi$
$480$	−29092.1	−2.76639
$481$	27676.9	2.62361
$482$	3052.79	0.288487
$483$	0	0
$484$	12530.5	1.17680
$485$	−23678.5	−2.21687
$486$	−19626.3	−1.83183
$487$	−2800.72	−0.260602	−0.130301	−	0.991475i	$-0.541594\pi$
$487$	−2800.72	−0.260602	−0.130301	+	0.991475i	$0.541594\pi$
$488$	−1.64031	−0.000152159 0
$489$	3500.42	0.323711
$490$	0	0
$491$	−1083.45	−0.0995834	−0.0497917	−	0.998760i	$-0.515856\pi$
$491$	−1083.45	−0.0995834	−0.0497917	+	0.998760i	$0.515856\pi$
$492$	2982.12	0.273261
$493$	−2248.71	−0.205430
$494$	−2953.73	−0.269017
$495$	28711.9	2.60708
$496$	4260.07	0.385651
$497$	0	0
$498$	−13642.5	−1.22758
$499$	9101.64	0.816524	0.408262	−	0.912865i	$-0.366135\pi$
$499$	9101.64	0.816524	0.408262	+	0.912865i	$0.366135\pi$
$500$	−5957.62	−0.532866
$501$	14422.5	1.28613
$502$	9943.90	0.884099
$503$	5111.20	0.453076	0.226538	−	0.974002i	$-0.427259\pi$
$503$	5111.20	0.453076	0.226538	+	0.974002i	$0.427259\pi$
$504$	0	0
$505$	1790.97	0.157816
$506$	−21297.4	−1.87112
$507$	15922.0	1.39471
$508$	−7504.14	−0.655398
$509$	5897.73	0.513580	0.256790	−	0.966467i	$-0.417335\pi$
$509$	5897.73	0.513580	0.256790	+	0.966467i	$0.417335\pi$
$510$	−23998.0	−2.08363
$511$	0	0
$512$	16420.5	1.41736
$513$	−1016.91	−0.0875202
$514$	5119.70	0.439339
$515$	−7549.90	−0.645997
$516$	12813.7	1.09320
$517$	−30528.0	−2.59694
$518$	0	0
$519$	−7187.09	−0.607858
$520$	296.163	0.0249762
$521$	15211.3	1.27911	0.639556	−	0.768744i	$-0.279118\pi$
$521$	15211.3	1.27911	0.639556	+	0.768744i	$0.279118\pi$
$522$	6500.84	0.545085
$523$	9441.78	0.789408	0.394704	−	0.918808i	$-0.370847\pi$
$523$	9441.78	0.789408	0.394704	+	0.918808i	$0.370847\pi$
$524$	18249.2	1.52142
$525$	0	0
$526$	6819.22	0.565271
$527$	3550.24	0.293455
$528$	−27425.1	−2.26046
$529$	−2378.73	−0.195507
$530$	8544.32	0.700267
$531$	6535.21	0.534094
$532$	0	0
$533$	2954.76	0.240122
$534$	20321.1	1.64678
$535$	−10068.5	−0.813647
$536$	−64.0491	−0.00516138
$537$	22140.1	1.77917
$538$	−4161.24	−0.333465
$539$	0	0
$540$	10127.9	0.807103
$541$	−5373.52	−0.427035	−0.213517	−	0.976939i	$-0.568492\pi$
$541$	−5373.52	−0.427035	−0.213517	+	0.976939i	$0.568492\pi$
$542$	−18738.9	−1.48506
$543$	33426.4	2.64173
$544$	13546.7	1.06766
$545$	−31114.7	−2.44552
$546$	0	0
$547$	−10492.2	−0.820133	−0.410066	−	0.912056i	$-0.634495\pi$
$547$	−10492.2	−0.820133	−0.410066	+	0.912056i	$0.634495\pi$
$548$	−12748.0	−0.993735
$549$	−191.323	−0.0148733
$550$	15617.4	1.21078
$551$	−485.829	−0.0375627
$552$	−260.354	−0.0200750
$553$	0	0
$554$	−27345.4	−2.09710
$555$	48581.7	3.71563
$556$	−12324.2	−0.940040
$557$	−14248.6	−1.08390	−0.541949	−	0.840411i	$-0.682313\pi$
$557$	−14248.6	−1.08390	−0.541949	+	0.840411i	$0.682313\pi$
$558$	−10263.5	−0.778651
$559$	12696.2	0.960629
$560$	0	0
$561$	−22855.4	−1.72007
$562$	21537.6	1.61657
$563$	14263.4	1.06773	0.533865	−	0.845570i	$-0.320739\pi$
$563$	14263.4	1.06773	0.533865	+	0.845570i	$0.320739\pi$
$564$	−37068.9	−2.76752
$565$	−18469.9	−1.37528
$566$	−1618.99	−0.120232
$567$	0	0
$568$	26.0020	0.00192081
$569$	−2648.01	−0.195098	−0.0975488	−	0.995231i	$-0.531100\pi$
$569$	−2648.01	−0.195098	−0.0975488	+	0.995231i	$0.531100\pi$
$570$	−5184.73	−0.380990
$571$	−20313.0	−1.48874	−0.744371	−	0.667767i	$-0.767250\pi$
$571$	−20313.0	−1.48874	−0.744371	+	0.667767i	$0.767250\pi$
$572$	28016.9	2.04798
$573$	36620.2	2.66986
$574$	0	0
$575$	−7177.72	−0.520577
$576$	−19878.4	−1.43796
$577$	−7003.51	−0.505303	−0.252652	−	0.967557i	$-0.581303\pi$
$577$	−7003.51	−0.505303	−0.252652	+	0.967557i	$0.581303\pi$
$578$	−8527.26	−0.613646
$579$	−22759.7	−1.63361
$580$	4838.60	0.346400
$581$	0	0
$582$	−54490.1	−3.88090
$583$	8137.50	0.578080
$584$	234.622	0.0166245
$585$	34543.9	2.44139
$586$	−14363.3	−1.01253
$587$	13324.8	0.936919	0.468459	−	0.883485i	$-0.344809\pi$
$587$	13324.8	0.936919	0.468459	+	0.883485i	$0.344809\pi$
$588$	0	0
$589$	767.022	0.0536581
$590$	9679.38	0.675413
$591$	−19379.3	−1.34883
$592$	−27145.0	−1.88455
$593$	−14345.7	−0.993436	−0.496718	−	0.867912i	$-0.665462\pi$
$593$	−14345.7	−0.993436	−0.496718	+	0.867912i	$0.665462\pi$
$594$	19194.3	1.32584
$595$	0	0
$596$	−16935.8	−1.16396
$597$	42157.7	2.89012
$598$	−25623.4	−1.75220
$599$	−19690.2	−1.34310	−0.671552	−	0.740958i	$-0.734372\pi$
$599$	−19690.2	−1.34310	−0.671552	+	0.740958i	$0.734372\pi$
$600$	190.917	0.0129903
$601$	16484.9	1.11886	0.559429	−	0.828878i	$-0.311020\pi$
$601$	16484.9	1.11886	0.559429	+	0.828878i	$0.311020\pi$
$602$	0	0
$603$	−7470.56	−0.504519
$604$	−10150.4	−0.683796
$605$	21793.3	1.46450
$606$	4121.47	0.276276
$607$	−10083.7	−0.674277	−0.337138	−	0.941455i	$-0.609459\pi$
$607$	−10083.7	−0.674277	−0.337138	+	0.941455i	$0.609459\pi$
$608$	2926.73	0.195222
$609$	0	0
$610$	−283.370	−0.0188087
$611$	−36728.9	−2.43190
$612$	−16234.2	−1.07227
$613$	−21009.7	−1.38430	−0.692149	−	0.721755i	$-0.743336\pi$
$613$	−21009.7	−1.38430	−0.692149	+	0.721755i	$0.743336\pi$
$614$	−5961.03	−0.391804
$615$	5186.55	0.340068
$616$	0	0
$617$	−28167.9	−1.83792	−0.918962	−	0.394347i	$-0.870971\pi$
$617$	−28167.9	−1.83792	−0.918962	+	0.394347i	$0.870971\pi$
$618$	−17374.2	−1.13090
$619$	−21371.3	−1.38770	−0.693849	−	0.720121i	$-0.744086\pi$
$619$	−21371.3	−1.38770	−0.693849	+	0.720121i	$0.744086\pi$
$620$	−7639.12	−0.494830
$621$	−8821.66	−0.570050
$622$	−6314.95	−0.407084
$623$	0	0
$624$	−32995.7	−2.11681
$625$	−19430.3	−1.24354
$626$	6071.70	0.387658
$627$	−4937.87	−0.314513
$628$	5367.99	0.341092
$629$	−22622.0	−1.43402
$630$	0	0
$631$	701.447	0.0442538	0.0221269	−	0.999755i	$-0.492956\pi$
$631$	701.447	0.0442538	0.0221269	+	0.999755i	$0.492956\pi$
$632$	78.6976	0.00495320
$633$	18098.6	1.13642
$634$	−14716.4	−0.921865
$635$	−13051.3	−0.815631
$636$	9881.04	0.616051
$637$	0	0
$638$	9170.05	0.569037
$639$	3032.83	0.187757
$640$	−586.944	−0.0362515
$641$	20210.7	1.24536	0.622679	−	0.782477i	$-0.286044\pi$
$641$	20210.7	1.24536	0.622679	+	0.782477i	$0.286044\pi$
$642$	−23170.2	−1.42439
$643$	13798.0	0.846252	0.423126	−	0.906071i	$-0.360933\pi$
$643$	13798.0	0.846252	0.423126	+	0.906071i	$0.360933\pi$
$644$	0	0
$645$	22285.8	1.36047
$646$	2414.26	0.147040
$647$	15004.1	0.911702	0.455851	−	0.890056i	$-0.349335\pi$
$647$	15004.1	0.911702	0.455851	+	0.890056i	$0.349335\pi$
$648$	−100.589	−0.00609800
$649$	9218.52	0.557563
$650$	18789.6	1.13383
$651$	0	0
$652$	3507.23	0.210665
$653$	20498.1	1.22841	0.614207	−	0.789145i	$-0.289476\pi$
$653$	20498.1	1.22841	0.614207	+	0.789145i	$0.289476\pi$
$654$	−71602.6	−4.28117
$655$	31739.4	1.89337
$656$	−2897.98	−0.172480
$657$	27365.8	1.62503
$658$	0	0
$659$	18620.9	1.10071	0.550355	−	0.834931i	$-0.314493\pi$
$659$	18620.9	1.10071	0.550355	+	0.834931i	$0.314493\pi$
$660$	49178.5	2.90041
$661$	12018.2	0.707194	0.353597	−	0.935398i	$-0.384958\pi$
$661$	12018.2	0.707194	0.353597	+	0.935398i	$0.384958\pi$
$662$	−18331.6	−1.07625
$663$	−27497.8	−1.61075
$664$	−137.614	−0.00804285
$665$	0	0
$666$	65398.4	3.80501
$667$	−4214.54	−0.244659
$668$	14450.6	0.836992
$669$	18691.5	1.08020
$670$	−11064.7	−0.638012
$671$	−269.879	−0.0155269
$672$	0	0
$673$	16441.9	0.941737	0.470869	−	0.882203i	$-0.343941\pi$
$673$	16441.9	0.941737	0.470869	+	0.882203i	$0.343941\pi$
$674$	16774.7	0.958663
$675$	6468.91	0.368872
$676$	15953.0	0.907655
$677$	−6907.91	−0.392160	−0.196080	−	0.980588i	$-0.562821\pi$
$677$	−6907.91	−0.392160	−0.196080	+	0.980588i	$0.562821\pi$
$678$	−42503.9	−2.40760
$679$	0	0
$680$	−242.072	−0.0136515
$681$	3164.91	0.178090
$682$	−14477.6	−0.812866
$683$	−3413.85	−0.191255	−0.0956276	−	0.995417i	$-0.530486\pi$
$683$	−3413.85	−0.191255	−0.0956276	+	0.995417i	$0.530486\pi$
$684$	−3507.38	−0.196064
$685$	−22171.5	−1.23668
$686$	0	0
$687$	3432.41	0.190618
$688$	−12452.2	−0.690023
$689$	9790.40	0.541342
$690$	−44977.2	−2.48153
$691$	9907.08	0.545417	0.272709	−	0.962097i	$-0.412081\pi$
$691$	9907.08	0.545417	0.272709	+	0.962097i	$0.412081\pi$
$692$	−7201.08	−0.395584
$693$	0	0
$694$	10209.7	0.558436
$695$	−21434.4	−1.16986
$696$	112.101	0.00610512
$697$	−2415.11	−0.131246
$698$	9201.47	0.498969
$699$	19246.8	1.04146
$700$	0	0
$701$	−29483.5	−1.58855	−0.794276	−	0.607557i	$-0.792150\pi$
$701$	−29483.5	−1.58855	−0.794276	+	0.607557i	$0.792150\pi$
$702$	23093.0	1.24158
$703$	−4887.44	−0.262209
$704$	−28040.3	−1.50115
$705$	−64470.8	−3.44413
$706$	−5724.03	−0.305137
$707$	0	0
$708$	11193.7	0.594187
$709$	−20145.8	−1.06712	−0.533562	−	0.845761i	$-0.679147\pi$
$709$	−20145.8	−1.06712	−0.533562	+	0.845761i	$0.679147\pi$
$710$	4491.95	0.237437
$711$	9179.13	0.484169
$712$	204.982	0.0107894
$713$	6653.87	0.349494
$714$	0	0
$715$	48727.4	2.54867
$716$	22183.2	1.15786
$717$	31045.1	1.61702
$718$	−40249.2	−2.09204
$719$	−29999.6	−1.55604	−0.778022	−	0.628237i	$-0.783777\pi$
$719$	−29999.6	−1.55604	−0.778022	+	0.628237i	$0.783777\pi$
$720$	−33880.1	−1.75366
$721$	0	0
$722$	−26984.1	−1.39092
$723$	6140.10	0.315840
$724$	33491.4	1.71920
$725$	3090.51	0.158316
$726$	50151.9	2.56379
$727$	29092.6	1.48416	0.742079	−	0.670312i	$-0.233840\pi$
$727$	29092.6	1.48416	0.742079	+	0.670312i	$0.233840\pi$
$728$	0	0
$729$	−31150.4	−1.58260
$730$	40531.9	2.05500
$731$	−10377.4	−0.525063
$732$	−327.703	−0.0165468
$733$	−2752.47	−0.138697	−0.0693483	−	0.997593i	$-0.522092\pi$
$733$	−2752.47	−0.138697	−0.0693483	+	0.997593i	$0.522092\pi$
$734$	3312.66	0.166584
$735$	0	0
$736$	25389.2	1.27155
$737$	−10537.9	−0.526689
$738$	6981.89	0.348248
$739$	33147.8	1.65002	0.825009	−	0.565120i	$-0.191170\pi$
$739$	33147.8	1.65002	0.825009	+	0.565120i	$0.191170\pi$
$740$	48676.2	2.41807
$741$	−5940.86	−0.294525
$742$	0	0
$743$	25438.4	1.25605	0.628023	−	0.778194i	$-0.283864\pi$
$743$	25438.4	1.25605	0.628023	+	0.778194i	$0.283864\pi$
$744$	−176.983	−0.00872114
$745$	−29455.1	−1.44852
$746$	−30319.4	−1.48803
$747$	−16051.0	−0.786179
$748$	−22899.9	−1.11939
$749$	0	0
$750$	−23844.6	−1.16091
$751$	−34097.2	−1.65676	−0.828379	−	0.560169i	$-0.810736\pi$
$751$	−34097.2	−1.65676	−0.828379	+	0.560169i	$0.810736\pi$
$752$	36023.0	1.74684
$753$	20000.3	0.967928
$754$	11032.7	0.532873
$755$	−17653.7	−0.850972
$756$	0	0
$757$	25616.0	1.22990	0.614948	−	0.788568i	$-0.289177\pi$
$757$	25616.0	1.22990	0.614948	+	0.788568i	$0.289177\pi$
$758$	−9412.03	−0.451003
$759$	−42835.7	−2.04854
$760$	−52.2993	−0.00249618
$761$	7942.56	0.378341	0.189170	−	0.981944i	$-0.439420\pi$
$761$	7942.56	0.378341	0.189170	+	0.981944i	$0.439420\pi$
$762$	−30034.3	−1.42786
$763$	0	0
$764$	36691.4	1.73750
$765$	−28234.8	−1.33442
$766$	−24850.0	−1.17215
$767$	11091.0	0.522129
$768$	32354.8	1.52019
$769$	−21172.1	−0.992830	−0.496415	−	0.868085i	$-0.665351\pi$
$769$	−21172.1	−0.992830	−0.496415	+	0.868085i	$0.665351\pi$
$770$	0	0
$771$	10297.3	0.480996
$772$	−22803.9	−1.06312
$773$	−5444.18	−0.253316	−0.126658	−	0.991946i	$-0.540425\pi$
$773$	−5444.18	−0.253316	−0.126658	+	0.991946i	$0.540425\pi$
$774$	30000.2	1.39320
$775$	−4879.27	−0.226153
$776$	−549.651	−0.0254269
$777$	0	0
$778$	51508.7	2.37362
$779$	−521.779	−0.0239983
$780$	59167.7	2.71608
$781$	4278.08	0.196007
$782$	20943.6	0.957724
$783$	3798.35	0.173361
$784$	0	0
$785$	9336.09	0.424483
$786$	73040.2	3.31458
$787$	−38282.2	−1.73394	−0.866971	−	0.498359i	$-0.833936\pi$
$787$	−38282.2	−1.73394	−0.866971	+	0.498359i	$0.833936\pi$
$788$	−19417.0	−0.877795
$789$	13715.6	0.618868
$790$	13595.3	0.612279
$791$	0	0
$792$	666.493	0.0299025
$793$	−324.697	−0.0145401
$794$	−34825.1	−1.55655
$795$	17185.3	0.766665
$796$	42239.7	1.88084
$797$	−39274.1	−1.74550	−0.872748	−	0.488171i	$-0.837664\pi$
$797$	−39274.1	−1.74550	−0.872748	+	0.488171i	$0.837664\pi$
$798$	0	0
$799$	30020.7	1.32923
$800$	−18617.9	−0.822802
$801$	23908.8	1.05465
$802$	−37220.6	−1.63878
$803$	38602.0	1.69643
$804$	−12795.8	−0.561284
$805$	0	0
$806$	−17418.3	−0.761206
$807$	−8369.55	−0.365083
$808$	41.5739	0.00181011
$809$	7762.88	0.337365	0.168683	−	0.985670i	$-0.446049\pi$
$809$	7762.88	0.337365	0.168683	+	0.985670i	$0.446049\pi$
$810$	−17377.1	−0.753790
$811$	12842.8	0.556071	0.278035	−	0.960571i	$-0.410317\pi$
$811$	12842.8	0.556071	0.278035	+	0.960571i	$0.410317\pi$
$812$	0	0
$813$	−37689.7	−1.62588
$814$	92250.5	3.97221
$815$	6099.83	0.262169
$816$	26969.4	1.15701
$817$	−2242.01	−0.0960074
$818$	−40437.0	−1.72842
$819$	0	0
$820$	5196.64	0.221310
$821$	−23797.3	−1.01161	−0.505804	−	0.862648i	$-0.668804\pi$
$821$	−23797.3	−1.01161	−0.505804	+	0.862648i	$0.668804\pi$
$822$	−51022.1	−2.16496
$823$	34893.3	1.47789	0.738946	−	0.673764i	$-0.235324\pi$
$823$	34893.3	1.47789	0.738946	+	0.673764i	$0.235324\pi$
$824$	−175.257	−0.00740941
$825$	31411.4	1.32558
$826$	0	0
$827$	20251.8	0.851540	0.425770	−	0.904831i	$-0.360003\pi$
$827$	20251.8	0.851540	0.425770	+	0.904831i	$0.360003\pi$
$828$	−30426.3	−1.27704
$829$	15197.0	0.636687	0.318343	−	0.947975i	$-0.396874\pi$
$829$	15197.0	0.636687	0.318343	+	0.947975i	$0.396874\pi$
$830$	−23773.3	−0.994199
$831$	−55000.0	−2.29594
$832$	−33735.9	−1.40575
$833$	0	0
$834$	−49326.0	−2.04798
$835$	25132.7	1.04162
$836$	−4947.48	−0.204680
$837$	−5996.79	−0.247646
$838$	−55869.0	−2.30306
$839$	−18324.1	−0.754014	−0.377007	−	0.926210i	$-0.623047\pi$
$839$	−18324.1	−0.754014	−0.377007	+	0.926210i	$0.623047\pi$
$840$	0	0
$841$	−22574.3	−0.925595
$842$	55765.3	2.28242
$843$	43318.8	1.76985
$844$	18133.8	0.739564
$845$	27745.6	1.12956
$846$	−86787.6	−3.52697
$847$	0	0
$848$	−9602.26	−0.388848
$849$	−3256.29	−0.131632
$850$	−15357.9	−0.619731
$851$	−42398.2	−1.70786
$852$	5194.70	0.208882
$853$	1968.65	0.0790215	0.0395107	−	0.999219i	$-0.487420\pi$
$853$	1968.65	0.0790215	0.0395107	+	0.999219i	$0.487420\pi$
$854$	0	0
$855$	−6100.08	−0.243998
$856$	−233.722	−0.00933231
$857$	39077.0	1.55758	0.778789	−	0.627285i	$-0.215834\pi$
$857$	39077.0	1.55758	0.778789	+	0.627285i	$0.215834\pi$
$858$	112134.	4.46176
$859$	31855.3	1.26530	0.632648	−	0.774439i	$-0.281968\pi$
$859$	31855.3	1.26530	0.632648	+	0.774439i	$0.281968\pi$
$860$	22329.2	0.885372
$861$	0	0
$862$	134.726	0.00532343
$863$	−39089.1	−1.54184	−0.770920	−	0.636932i	$-0.780203\pi$
$863$	−39089.1	−1.54184	−0.770920	+	0.636932i	$0.780203\pi$
$864$	−22882.0	−0.900997
$865$	−12524.2	−0.492296
$866$	24195.6	0.949423
$867$	−17151.0	−0.671830
$868$	0	0
$869$	12948.0	0.505445
$870$	19365.8	0.754671
$871$	−12678.4	−0.493216
$872$	−722.268	−0.0280494
$873$	−64110.2	−2.48545
$874$	4524.82	0.175119
$875$	0	0
$876$	46872.9	1.80786
$877$	3408.38	0.131235	0.0656174	−	0.997845i	$-0.479098\pi$
$877$	3408.38	0.131235	0.0656174	+	0.997845i	$0.479098\pi$
$878$	19490.1	0.749157
$879$	−28889.0	−1.10853
$880$	−47791.0	−1.83072
$881$	15980.2	0.611110	0.305555	−	0.952174i	$-0.401158\pi$
$881$	15980.2	0.611110	0.305555	+	0.952174i	$0.401158\pi$
$882$	0	0
$883$	11495.7	0.438122	0.219061	−	0.975711i	$-0.429701\pi$
$883$	11495.7	0.438122	0.219061	+	0.975711i	$0.429701\pi$
$884$	−27551.4	−1.04825
$885$	19468.2	0.739454
$886$	23810.5	0.902854
$887$	40892.6	1.54796	0.773979	−	0.633212i	$-0.218264\pi$
$887$	40892.6	1.54796	0.773979	+	0.633212i	$0.218264\pi$
$888$	1127.73	0.0426173
$889$	0	0
$890$	35411.5	1.33371
$891$	−16549.8	−0.622265
$892$	18727.9	0.702978
$893$	6485.92	0.243049
$894$	−67783.4	−2.53581
$895$	38581.4	1.44093
$896$	0	0
$897$	−51536.6	−1.91835
$898$	5432.79	0.201887
$899$	−2864.96	−0.106287
$900$	22311.5	0.826353
$901$	−8002.29	−0.295888
$902$	9848.60	0.363550
$903$	0	0
$904$	−428.744	−0.0157741
$905$	58248.8	2.13951
$906$	−40625.6	−1.48973
$907$	460.820	0.0168702	0.00843510	−	0.999964i	$-0.497315\pi$
$907$	460.820	0.0168702	0.00843510	+	0.999964i	$0.497315\pi$
$908$	3171.06	0.115898
$909$	4849.10	0.176936
$910$	0	0
$911$	−7770.81	−0.282611	−0.141305	−	0.989966i	$-0.545130\pi$
$911$	−7770.81	−0.282611	−0.141305	+	0.989966i	$0.545130\pi$
$912$	5826.69	0.211558
$913$	−22641.4	−0.820725
$914$	−1991.47	−0.0720699
$915$	−569.946	−0.0205922
$916$	3439.09	0.124051
$917$	0	0
$918$	−18875.3	−0.678627
$919$	52718.2	1.89229	0.946143	−	0.323748i	$-0.104943\pi$
$919$	52718.2	1.89229	0.946143	+	0.323748i	$0.104943\pi$
$920$	−453.693	−0.0162585
$921$	−11989.5	−0.428954
$922$	−1015.42	−0.0362701
$923$	5147.05	0.183551
$924$	0	0
$925$	31090.5	1.10514
$926$	−42602.2	−1.51187
$927$	−20441.6	−0.724261
$928$	−10931.8	−0.386698
$929$	5250.87	0.185442	0.0927209	−	0.995692i	$-0.470444\pi$
$929$	5250.87	0.185442	0.0927209	+	0.995692i	$0.470444\pi$
$930$	−30574.6	−1.07804
$931$	0	0
$932$	19284.2	0.677763
$933$	−12701.3	−0.445683
$934$	−29737.9	−1.04181
$935$	−39827.9	−1.39306
$936$	801.871	0.0280021
$937$	40404.4	1.40870	0.704352	−	0.709851i	$-0.251238\pi$
$937$	40404.4	1.40870	0.704352	+	0.709851i	$0.251238\pi$
$938$	0	0
$939$	12212.1	0.424415
$940$	−64596.2	−2.24138
$941$	27813.0	0.963525	0.481762	−	0.876302i	$-0.339997\pi$
$941$	27813.0	0.963525	0.481762	+	0.876302i	$0.339997\pi$
$942$	21484.7	0.743109
$943$	−4526.40	−0.156310
$944$	−10877.9	−0.375047
$945$	0	0
$946$	42318.0	1.45442
$947$	38207.2	1.31105	0.655527	−	0.755172i	$-0.272447\pi$
$947$	38207.2	1.31105	0.655527	+	0.755172i	$0.272447\pi$
$948$	15722.3	0.538645
$949$	46442.9	1.58862
$950$	−3318.04	−0.113317
$951$	−29599.2	−1.00927
$952$	0	0
$953$	−8477.38	−0.288152	−0.144076	−	0.989567i	$-0.546021\pi$
$953$	−8477.38	−0.288152	−0.144076	+	0.989567i	$0.546021\pi$
$954$	23134.0	0.785106
$955$	63814.3	2.16229
$956$	31105.5	1.05233
$957$	18443.8	0.622992
$958$	70174.6	2.36664
$959$	0	0
$960$	−59217.3	−1.99086
$961$	−25267.8	−0.848170
$962$	110989.	3.71976
$963$	−27260.9	−0.912222
$964$	6152.04	0.205544
$965$	−39661.0	−1.32304
$966$	0	0
$967$	−29009.6	−0.964721	−0.482360	−	0.875973i	$-0.660220\pi$
$967$	−29009.6	−0.964721	−0.482360	+	0.875973i	$0.660220\pi$
$968$	505.891	0.0167975
$969$	4855.82	0.160982
$970$	−94954.4	−3.14309
$971$	−24942.3	−0.824343	−0.412172	−	0.911106i	$-0.635230\pi$
$971$	−24942.3	−0.824343	−0.412172	+	0.911106i	$0.635230\pi$
$972$	−39551.4	−1.30516
$973$	0	0
$974$	−11231.3	−0.369482
$975$	37791.7	1.24134
$976$	318.457	0.0104442
$977$	−31422.9	−1.02897	−0.514487	−	0.857498i	$-0.672017\pi$
$977$	−31422.9	−1.02897	−0.514487	+	0.857498i	$0.672017\pi$
$978$	14037.2	0.458958
$979$	33725.5	1.10099
$980$	0	0
$981$	−84243.9	−2.74180
$982$	−4344.81	−0.141190
$983$	5168.21	0.167691	0.0838455	−	0.996479i	$-0.473280\pi$
$983$	5168.21	0.167691	0.0838455	+	0.996479i	$0.473280\pi$
$984$	120.396	0.00390049
$985$	−33770.4	−1.09240
$986$	−9017.68	−0.291259
$987$	0	0
$988$	−5952.42	−0.191672
$989$	−19449.3	−0.625330
$990$	115139.	3.69633
$991$	−967.858	−0.0310242	−0.0155121	−	0.999880i	$-0.504938\pi$
$991$	−967.858	−0.0310242	−0.0155121	+	0.999880i	$0.504938\pi$
$992$	17259.1	0.552396
$993$	−36870.6	−1.17830
$994$	0	0
$995$	73464.0	2.34067
$996$	−27492.6	−0.874635
$997$	−45786.0	−1.45442	−0.727210	−	0.686415i	$-0.759183\pi$
$997$	−45786.0	−1.45442	−0.727210	+	0.686415i	$0.759183\pi$
$998$	36499.0	1.15767
$999$	38211.3	1.21016

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2401.4.a.d.1.32		39
7.6	odd	2		2401.4.a.c.1.32		39
49.8	even	7		49.4.e.a.15.2	✓	78
49.43	even	7		49.4.e.a.36.2	yes	78

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.4.e.a.15.2	✓	78	49.8	even	7
49.4.e.a.36.2	yes	78	49.43	even	7
2401.4.a.c.1.32		39	7.6	odd	2
2401.4.a.d.1.32		39	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2401 = 7^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2401.a (trivial)

\(f(q)\)	\(=\)	\(q+4.01016 q^{2} +8.06567 q^{3} +8.08136 q^{4} +14.0552 q^{5} +32.3446 q^{6} +0.326265 q^{8} +38.0550 q^{9} +56.3637 q^{10} +53.6801 q^{11} +65.1815 q^{12} +64.5836 q^{13} +113.365 q^{15} -63.3425 q^{16} -52.7881 q^{17} +152.606 q^{18} -11.4048 q^{19} +113.585 q^{20} +215.265 q^{22} -98.9357 q^{23} +2.63155 q^{24} +72.5494 q^{25} +258.990 q^{26} +89.1656 q^{27} +42.5988 q^{29} +454.610 q^{30} -67.2545 q^{31} -256.624 q^{32} +432.965 q^{33} -211.689 q^{34} +307.536 q^{36} +428.543 q^{37} -45.7349 q^{38} +520.910 q^{39} +4.58573 q^{40} +45.7510 q^{41} +196.585 q^{43} +433.808 q^{44} +534.871 q^{45} -396.748 q^{46} -568.702 q^{47} -510.899 q^{48} +290.934 q^{50} -425.771 q^{51} +521.923 q^{52} +151.593 q^{53} +357.568 q^{54} +754.485 q^{55} -91.9871 q^{57} +170.828 q^{58} +171.731 q^{59} +916.141 q^{60} -5.02754 q^{61} -269.701 q^{62} -522.361 q^{64} +907.737 q^{65} +1736.26 q^{66} -196.310 q^{67} -426.600 q^{68} -797.982 q^{69} +79.6959 q^{71} +12.4160 q^{72} +719.113 q^{73} +1718.52 q^{74} +585.159 q^{75} -92.1661 q^{76} +2088.93 q^{78} +241.207 q^{79} -890.293 q^{80} -308.304 q^{81} +183.468 q^{82} -421.785 q^{83} -741.949 q^{85} +788.338 q^{86} +343.587 q^{87} +17.5139 q^{88} +628.269 q^{89} +2144.92 q^{90} -799.535 q^{92} -542.452 q^{93} -2280.59 q^{94} -160.297 q^{95} -2069.84 q^{96} -1684.67 q^{97} +2042.79 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 39 q + q^{2} + q^{3} + 145 q^{4} + 27 q^{5} + 41 q^{6} - 12 q^{8} + 312 q^{9} + 78 q^{10} + q^{11} - 91 q^{12} + 77 q^{13} - 161 q^{15} + 461 q^{16} + 211 q^{17} + 8 q^{18} + 314 q^{19} + 476 q^{20}+ \cdots + 5384 q^{99}+O(q^{100}) \) 39 * q + q^2 + q^3 + 145 * q^4 + 27 * q^5 + 41 * q^6 - 12 * q^8 + 312 * q^9 + 78 * q^10 + q^11 - 91 * q^12 + 77 * q^13 - 161 * q^15 + 461 * q^16 + 211 * q^17 + 8 * q^18 + 314 * q^19 + 476 * q^20 - 61 * q^22 - 69 * q^23 + 330 * q^24 + 606 * q^25 + 504 * q^26 - 50 * q^27 + 57 * q^29 + 42 * q^30 + 638 * q^31 - 1600 * q^32 + 1574 * q^33 + 1343 * q^34 + 782 * q^36 + 71 * q^37 + 1359 * q^38 - 84 * q^39 - 155 * q^40 + 1393 * q^41 - 125 * q^43 + 52 * q^44 + 1129 * q^45 - 1454 * q^46 + 1483 * q^47 - 974 * q^48 + 3074 * q^50 - 2044 * q^51 + 3899 * q^52 + 2213 * q^53 + 1142 * q^54 + 1604 * q^55 + 98 * q^57 + 2403 * q^58 + 2073 * q^59 - 1519 * q^60 + 2575 * q^61 + 1742 * q^62 + 1358 * q^64 + 1876 * q^65 - 48 * q^66 + 176 * q^67 + 3038 * q^68 - 638 * q^69 - 1259 * q^71 - 3799 * q^72 - 307 * q^73 + 3845 * q^74 + 131 * q^75 + 1974 * q^76 - 6041 * q^78 + 22 * q^79 - 804 * q^80 + 795 * q^81 + 8043 * q^82 + 6349 * q^83 + 3094 * q^85 + 1745 * q^86 + 9508 * q^87 + 1299 * q^88 + 2253 * q^89 + 11156 * q^90 + 1284 * q^92 - 3430 * q^93 + 2738 * q^94 - 3290 * q^95 + 3031 * q^96 - 770 * q^97 + 5384 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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