LMFDB - Embedded newform 539.4.a.l.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [539,4,Mod(1,539)] 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("539.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(539, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 539 = 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	539.a (trivial)

Newform invariants

comment:select newform

sage:traces = [10,-4,0,8,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$31.8020294931$
Analytic rank:	$1$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 4x^{9} - 46x^{8} + 148x^{7} + 614x^{6} - 1476x^{5} - 3064x^{4} + 4428x^{3} + 4321x^{2} - 3480x - 658 $ x^10 - 4x^9 - 46x^8 + 148x^7 + 614x^6 - 1476x^5 - 3064x^4 + 4428x^3 + 4321x^2 - 3480*x - 658
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.44050$ of defining polynomial
Character	$\chi$	$=$	539.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-4.85471 q^{2} +5.89940 q^{3} +15.5682 q^{4} -4.11981 q^{5} -28.6399 q^{6} -36.7416 q^{8} +7.80287 q^{9} +20.0005 q^{10} +11.0000 q^{11} +91.8432 q^{12} -44.4887 q^{13} -24.3044 q^{15} +53.8242 q^{16} +40.7793 q^{17} -37.8807 q^{18} +115.075 q^{19} -64.1382 q^{20} -53.4018 q^{22} -5.44165 q^{23} -216.753 q^{24} -108.027 q^{25} +215.980 q^{26} -113.251 q^{27} +5.46469 q^{29} +117.991 q^{30} -198.523 q^{31} +32.6321 q^{32} +64.8934 q^{33} -197.972 q^{34} +121.477 q^{36} -271.779 q^{37} -558.658 q^{38} -262.457 q^{39} +151.368 q^{40} -286.044 q^{41} +299.844 q^{43} +171.251 q^{44} -32.1463 q^{45} +26.4176 q^{46} -259.640 q^{47} +317.530 q^{48} +524.441 q^{50} +240.573 q^{51} -692.611 q^{52} +648.664 q^{53} +549.803 q^{54} -45.3179 q^{55} +678.875 q^{57} -26.5295 q^{58} +206.083 q^{59} -378.376 q^{60} -177.991 q^{61} +963.774 q^{62} -589.013 q^{64} +183.285 q^{65} -315.039 q^{66} -123.722 q^{67} +634.861 q^{68} -32.1024 q^{69} -516.226 q^{71} -286.690 q^{72} +431.312 q^{73} +1319.41 q^{74} -637.295 q^{75} +1791.52 q^{76} +1274.15 q^{78} -1120.78 q^{79} -221.745 q^{80} -878.793 q^{81} +1388.66 q^{82} +79.4687 q^{83} -168.003 q^{85} -1455.66 q^{86} +32.2384 q^{87} -404.158 q^{88} +1093.57 q^{89} +156.061 q^{90} -84.7169 q^{92} -1171.17 q^{93} +1260.48 q^{94} -474.089 q^{95} +192.510 q^{96} -1113.81 q^{97} +85.8316 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 4 q^{2} + 8 q^{4} - 84 q^{8} - 18 q^{9} + 110 q^{11} - 76 q^{15} + 40 q^{16} - 44 q^{22} - 384 q^{23} - 338 q^{25} - 340 q^{29} - 228 q^{30} - 276 q^{32} - 60 q^{36} - 1384 q^{37} - 1232 q^{39} - 100 q^{43}+ \cdots - 198 q^{99}+O(q^{100}) $ 10 * q - 4 * q^2 + 8 * q^4 - 84 * q^8 - 18 * q^9 + 110 * q^11 - 76 * q^15 + 40 * q^16 - 44 * q^22 - 384 * q^23 - 338 * q^25 - 340 * q^29 - 228 * q^30 - 276 * q^32 - 60 * q^36 - 1384 * q^37 - 1232 * q^39 - 100 * q^43 + 88 * q^44 - 936 * q^46 + 184 * q^50 - 1712 * q^51 - 416 * q^53 + 1008 * q^57 - 1588 * q^58 - 1308 * q^60 - 904 * q^64 - 1860 * q^65 - 1588 * q^67 - 2148 * q^71 - 2736 * q^72 + 272 * q^74 + 3040 * q^78 - 4024 * q^79 - 2858 * q^81 - 1236 * q^85 - 6324 * q^86 - 924 * q^88 - 464 * q^92 - 3452 * q^93 - 2448 * q^95 - 198 * 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 $

$2$	−4.85471	−1.71640	−0.858200	−	0.513315i	$-0.828417\pi$
$2$	−4.85471	−1.71640	−0.858200	+	0.513315i	$0.828417\pi$
$3$	5.89940	1.13534	0.567670	−	0.823256i	$-0.307845\pi$
$3$	5.89940	1.13534	0.567670	+	0.823256i	$0.307845\pi$
$4$	15.5682	1.94603
$5$	−4.11981	−0.368487	−0.184243	−	0.982881i	$-0.558983\pi$
$5$	−4.11981	−0.368487	−0.184243	+	0.982881i	$0.558983\pi$
$6$	−28.6399	−1.94870
$7$	0	0
$7$	0	0
$8$	−36.7416	−1.62377
$9$	7.80287	0.288995
$10$	20.0005	0.632471
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	91.8432	2.20940
$13$	−44.4887	−0.949150	−0.474575	−	0.880215i	$-0.657398\pi$
$13$	−44.4887	−0.949150	−0.474575	+	0.880215i	$0.657398\pi$
$14$	0	0
$15$	−24.3044	−0.418358
$16$	53.8242	0.841003
$17$	40.7793	0.581790	0.290895	−	0.956755i	$-0.406047\pi$
$17$	40.7793	0.581790	0.290895	+	0.956755i	$0.406047\pi$
$18$	−37.8807	−0.496032
$19$	115.075	1.38948	0.694740	−	0.719261i	$-0.255520\pi$
$19$	115.075	1.38948	0.694740	+	0.719261i	$0.255520\pi$
$20$	−64.1382	−0.717086
$21$	0	0
$22$	−53.4018	−0.517514
$23$	−5.44165	−0.0493331	−0.0246666	−	0.999696i	$-0.507852\pi$
$23$	−5.44165	−0.0493331	−0.0246666	+	0.999696i	$0.507852\pi$
$24$	−216.753	−1.84353
$25$	−108.027	−0.864217
$26$	215.980	1.62912
$27$	−113.251	−0.807232
$28$	0	0
$29$	5.46469	0.0349920	0.0174960	−	0.999847i	$-0.494431\pi$
$29$	5.46469	0.0349920	0.0174960	+	0.999847i	$0.494431\pi$
$30$	117.991	0.718069
$31$	−198.523	−1.15019	−0.575094	−	0.818087i	$-0.695035\pi$
$31$	−198.523	−1.15019	−0.575094	+	0.818087i	$0.695035\pi$
$32$	32.6321	0.180269
$33$	64.8934	0.342318
$34$	−197.972	−0.998584
$35$	0	0
$36$	121.477	0.562394
$37$	−271.779	−1.20757	−0.603786	−	0.797147i	$-0.706342\pi$
$37$	−271.779	−1.20757	−0.603786	+	0.797147i	$0.706342\pi$
$38$	−558.658	−2.38490
$39$	−262.457	−1.07761
$40$	151.368	0.598336
$41$	−286.044	−1.08957	−0.544787	−	0.838575i	$-0.683389\pi$
$41$	−286.044	−1.08957	−0.544787	+	0.838575i	$0.683389\pi$
$42$	0	0
$43$	299.844	1.06339	0.531695	−	0.846936i	$-0.321555\pi$
$43$	299.844	1.06339	0.531695	+	0.846936i	$0.321555\pi$
$44$	171.251	0.586750
$45$	−32.1463	−0.106491
$46$	26.4176	0.0846754
$47$	−259.640	−0.805797	−0.402898	−	0.915245i	$-0.631997\pi$
$47$	−259.640	−0.805797	−0.402898	+	0.915245i	$0.631997\pi$
$48$	317.530	0.954823
$49$	0	0
$50$	524.441	1.48334
$51$	240.573	0.660528
$52$	−692.611	−1.84707
$53$	648.664	1.68115	0.840574	−	0.541697i	$-0.182218\pi$
$53$	648.664	1.68115	0.840574	+	0.541697i	$0.182218\pi$
$54$	549.803	1.38553
$55$	−45.3179	−0.111103
$56$	0	0
$57$	678.875	1.57753
$58$	−26.5295	−0.0600603
$59$	206.083	0.454741	0.227371	−	0.973808i	$-0.426987\pi$
$59$	206.083	0.454741	0.227371	+	0.973808i	$0.426987\pi$
$60$	−378.376	−0.814136
$61$	−177.991	−0.373596	−0.186798	−	0.982398i	$-0.559811\pi$
$61$	−177.991	−0.373596	−0.186798	+	0.982398i	$0.559811\pi$
$62$	963.774	1.97418
$63$	0	0
$64$	−589.013	−1.15042
$65$	183.285	0.349749
$66$	−315.039	−0.587554
$67$	−123.722	−0.225597	−0.112799	−	0.993618i	$-0.535982\pi$
$67$	−123.722	−0.225597	−0.112799	+	0.993618i	$0.535982\pi$
$68$	634.861	1.13218
$69$	−32.1024	−0.0560099
$70$	0	0
$71$	−516.226	−0.862884	−0.431442	−	0.902141i	$-0.641995\pi$
$71$	−516.226	−0.862884	−0.431442	+	0.902141i	$0.641995\pi$
$72$	−286.690	−0.469261
$73$	431.312	0.691524	0.345762	−	0.938322i	$-0.387620\pi$
$73$	431.312	0.691524	0.345762	+	0.938322i	$0.387620\pi$
$74$	1319.41	2.07268
$75$	−637.295	−0.981180
$76$	1791.52	2.70397
$77$	0	0
$78$	1274.15	1.84961
$79$	−1120.78	−1.59617	−0.798086	−	0.602543i	$-0.794154\pi$
$79$	−1120.78	−1.59617	−0.798086	+	0.602543i	$0.794154\pi$
$80$	−221.745	−0.309898
$81$	−878.793	−1.20548
$82$	1388.66	1.87014
$83$	79.4687	0.105094	0.0525471	−	0.998618i	$-0.483266\pi$
$83$	79.4687	0.105094	0.0525471	+	0.998618i	$0.483266\pi$
$84$	0	0
$85$	−168.003	−0.214382
$86$	−1455.66	−1.82520
$87$	32.2384	0.0397278
$88$	−404.158	−0.489584
$89$	1093.57	1.30245	0.651224	−	0.758886i	$-0.274256\pi$
$89$	1093.57	1.30245	0.651224	+	0.758886i	$0.274256\pi$
$90$	156.061	0.182781
$91$	0	0
$92$	−84.7169	−0.0960038
$93$	−1171.17	−1.30585
$94$	1260.48	1.38307
$95$	−474.089	−0.512005
$96$	192.510	0.204666
$97$	−1113.81	−1.16588	−0.582941	−	0.812515i	$-0.698098\pi$
$97$	−1113.81	−1.16588	−0.582941	+	0.812515i	$0.698098\pi$
$98$	0	0
$99$	85.8316	0.0871354
$100$	−1681.79	−1.68179
$101$	−844.439	−0.831929	−0.415964	−	0.909381i	$-0.636556\pi$
$101$	−844.439	−0.831929	−0.415964	+	0.909381i	$0.636556\pi$
$102$	−1167.91	−1.13373
$103$	−1851.34	−1.77105	−0.885523	−	0.464596i	$-0.846199\pi$
$103$	−1851.34	−1.77105	−0.885523	+	0.464596i	$0.846199\pi$
$104$	1634.59	1.54120
$105$	0	0
$106$	−3149.08	−2.88552
$107$	2053.86	1.85564	0.927822	−	0.373022i	$-0.121678\pi$
$107$	2053.86	1.85564	0.927822	+	0.373022i	$0.121678\pi$
$108$	−1763.13	−1.57090
$109$	−1922.51	−1.68938	−0.844691	−	0.535254i	$-0.820216\pi$
$109$	−1922.51	−1.68938	−0.844691	+	0.535254i	$0.820216\pi$
$110$	220.005	0.190697
$111$	−1603.33	−1.37100
$112$	0	0
$113$	−1834.00	−1.52680	−0.763400	−	0.645926i	$-0.776471\pi$
$113$	−1834.00	−1.52680	−0.763400	+	0.645926i	$0.776471\pi$
$114$	−3295.75	−2.70767
$115$	22.4185	0.0181786
$116$	85.0756	0.0680955
$117$	−347.140	−0.274300
$118$	−1000.47	−0.780518
$119$	0	0
$120$	892.983	0.679315
$121$	121.000	0.0909091
$122$	864.094	0.641241
$123$	−1687.48	−1.23704
$124$	−3090.66	−2.23830
$125$	960.027	0.686940
$126$	0	0
$127$	−1633.03	−1.14101	−0.570505	−	0.821294i	$-0.693252\pi$
$127$	−1633.03	−1.14101	−0.570505	+	0.821294i	$0.693252\pi$
$128$	2598.43	1.79431
$129$	1768.90	1.20731
$130$	−889.796	−0.600310
$131$	−1274.62	−0.850106	−0.425053	−	0.905169i	$-0.639744\pi$
$131$	−1274.62	−0.850106	−0.425053	+	0.905169i	$0.639744\pi$
$132$	1010.28	0.666160
$133$	0	0
$134$	600.634	0.387215
$135$	466.574	0.297454
$136$	−1498.30	−0.944690
$137$	−2229.26	−1.39021	−0.695104	−	0.718909i	$-0.744642\pi$
$137$	−2229.26	−1.39021	−0.695104	+	0.718909i	$0.744642\pi$
$138$	155.848	0.0961353
$139$	649.837	0.396536	0.198268	−	0.980148i	$-0.436468\pi$
$139$	649.837	0.396536	0.198268	+	0.980148i	$0.436468\pi$
$140$	0	0
$141$	−1531.72	−0.914853
$142$	2506.13	1.48105
$143$	−489.376	−0.286180
$144$	419.983	0.243046
$145$	−22.5135	−0.0128941
$146$	−2093.90	−1.18693
$147$	0	0
$148$	−4231.12	−2.34997
$149$	1510.31	0.830397	0.415198	−	0.909731i	$-0.363712\pi$
$149$	1510.31	0.830397	0.415198	+	0.909731i	$0.363712\pi$
$150$	3093.89	1.68410
$151$	654.054	0.352491	0.176246	−	0.984346i	$-0.443605\pi$
$151$	654.054	0.352491	0.176246	+	0.984346i	$0.443605\pi$
$152$	−4228.06	−2.25619
$153$	318.195	0.168134
$154$	0	0
$155$	817.878	0.423829
$156$	−4085.99	−2.09706
$157$	2856.39	1.45201	0.726003	−	0.687692i	$-0.241376\pi$
$157$	2856.39	1.45201	0.726003	+	0.687692i	$0.241376\pi$
$158$	5441.07	2.73967
$159$	3826.73	1.90867
$160$	−134.438	−0.0664267
$161$	0	0
$162$	4266.29	2.06908
$163$	−2834.74	−1.36217	−0.681085	−	0.732204i	$-0.738492\pi$
$163$	−2834.74	−1.36217	−0.681085	+	0.732204i	$0.738492\pi$
$164$	−4453.20	−2.12034
$165$	−267.348	−0.126140
$166$	−385.798	−0.180384
$167$	−2532.48	−1.17347	−0.586733	−	0.809780i	$-0.699586\pi$
$167$	−2532.48	−1.17347	−0.586733	+	0.809780i	$0.699586\pi$
$168$	0	0
$169$	−217.753	−0.0991139
$170$	815.605	0.367965
$171$	897.919	0.401553
$172$	4668.04	2.06939
$173$	−573.147	−0.251882	−0.125941	−	0.992038i	$-0.540195\pi$
$173$	−573.147	−0.251882	−0.125941	+	0.992038i	$0.540195\pi$
$174$	−156.508	−0.0681888
$175$	0	0
$176$	592.066	0.253572
$177$	1215.77	0.516285
$178$	−5308.95	−2.23552
$179$	−274.651	−0.114684	−0.0573419	−	0.998355i	$-0.518263\pi$
$179$	−274.651	−0.114684	−0.0573419	+	0.998355i	$0.518263\pi$
$180$	−500.462	−0.207235
$181$	3958.51	1.62560	0.812801	−	0.582541i	$-0.197942\pi$
$181$	3958.51	1.62560	0.812801	+	0.582541i	$0.197942\pi$
$182$	0	0
$183$	−1050.04	−0.424159
$184$	199.935	0.0801055
$185$	1119.68	0.444974
$186$	5685.69	2.24137
$187$	448.572	0.175416
$188$	−4042.14	−1.56810
$189$	0	0
$190$	2301.56	0.878805
$191$	−1726.80	−0.654171	−0.327086	−	0.944995i	$-0.606067\pi$
$191$	−1726.80	−0.654171	−0.327086	+	0.944995i	$0.606067\pi$
$192$	−3474.82	−1.30611
$193$	−155.412	−0.0579627	−0.0289813	−	0.999580i	$-0.509226\pi$
$193$	−155.412	−0.0579627	−0.0289813	+	0.999580i	$0.509226\pi$
$194$	5407.24	2.00112
$195$	1081.27	0.397084
$196$	0	0
$197$	−889.441	−0.321675	−0.160838	−	0.986981i	$-0.551420\pi$
$197$	−889.441	−0.321675	−0.160838	+	0.986981i	$0.551420\pi$
$198$	−416.688	−0.149559
$199$	−613.641	−0.218592	−0.109296	−	0.994009i	$-0.534860\pi$
$199$	−613.641	−0.218592	−0.109296	+	0.994009i	$0.534860\pi$
$200$	3969.10	1.40329
$201$	−729.884	−0.256130
$202$	4099.51	1.42792
$203$	0	0
$204$	3745.30	1.28541
$205$	1178.44	0.401493
$206$	8987.71	3.03982
$207$	−42.4605	−0.0142570
$208$	−2394.57	−0.798238
$209$	1265.83	0.418944
$210$	0	0
$211$	−6093.90	−1.98825	−0.994127	−	0.108218i	$-0.965486\pi$
$211$	−6093.90	−1.98825	−0.994127	+	0.108218i	$0.965486\pi$
$212$	10098.6	3.27156
$213$	−3045.42	−0.979666
$214$	−9970.89	−3.18503
$215$	−1235.30	−0.391845
$216$	4161.04	1.31076
$217$	0	0
$218$	9333.21	2.89966
$219$	2544.48	0.785114
$220$	−705.520	−0.216210
$221$	−1814.22	−0.552206
$222$	7783.71	2.35319
$223$	−1881.69	−0.565054	−0.282527	−	0.959259i	$-0.591173\pi$
$223$	−1881.69	−0.565054	−0.282527	+	0.959259i	$0.591173\pi$
$224$	0	0
$225$	−842.922	−0.249755
$226$	8903.55	2.62060
$227$	3129.25	0.914960	0.457480	−	0.889220i	$-0.348752\pi$
$227$	3129.25	0.914960	0.457480	+	0.889220i	$0.348752\pi$
$228$	10568.9	3.06992
$229$	242.244	0.0699036	0.0349518	−	0.999389i	$-0.488872\pi$
$229$	242.244	0.0699036	0.0349518	+	0.999389i	$0.488872\pi$
$230$	−108.836	−0.0312018
$231$	0	0
$232$	−200.782	−0.0568188
$233$	−1436.14	−0.403796	−0.201898	−	0.979407i	$-0.564711\pi$
$233$	−1436.14	−0.403796	−0.201898	+	0.979407i	$0.564711\pi$
$234$	1685.26	0.470809
$235$	1069.67	0.296925
$236$	3208.35	0.884940
$237$	−6611.93	−1.81220
$238$	0	0
$239$	2855.04	0.772709	0.386354	−	0.922350i	$-0.373734\pi$
$239$	2855.04	0.772709	0.386354	+	0.922350i	$0.373734\pi$
$240$	−1308.16	−0.351840
$241$	−2302.62	−0.615455	−0.307727	−	0.951475i	$-0.599568\pi$
$241$	−2302.62	−0.615455	−0.307727	+	0.951475i	$0.599568\pi$
$242$	−587.420	−0.156036
$243$	−2126.56	−0.561394
$244$	−2771.00	−0.727030
$245$	0	0
$246$	8192.25	2.12325
$247$	−5119.56	−1.31882
$248$	7294.07	1.86764
$249$	468.817	0.119318
$250$	−4660.66	−1.17906
$251$	−2175.15	−0.546988	−0.273494	−	0.961874i	$-0.588179\pi$
$251$	−2175.15	−0.546988	−0.273494	+	0.961874i	$0.588179\pi$
$252$	0	0
$253$	−59.8581	−0.0148745
$254$	7927.91	1.95843
$255$	−991.114	−0.243396
$256$	−7902.54	−1.92933
$257$	4748.60	1.15257	0.576284	−	0.817250i	$-0.304502\pi$
$257$	4748.60	1.15257	0.576284	+	0.817250i	$0.304502\pi$
$258$	−8587.50	−2.07223
$259$	0	0
$260$	2853.42	0.680623
$261$	42.6403	0.0101125
$262$	6187.90	1.45912
$263$	2900.00	0.679930	0.339965	−	0.940438i	$-0.389585\pi$
$263$	2900.00	0.679930	0.339965	+	0.940438i	$0.389585\pi$
$264$	−2384.29	−0.555844
$265$	−2672.37	−0.619481
$266$	0	0
$267$	6451.39	1.47872
$268$	−1926.13	−0.439019
$269$	2748.37	0.622940	0.311470	−	0.950256i	$-0.399179\pi$
$269$	2748.37	0.622940	0.311470	+	0.950256i	$0.399179\pi$
$270$	−2265.08	−0.510550
$271$	−2535.68	−0.568383	−0.284191	−	0.958768i	$-0.591725\pi$
$271$	−2535.68	−0.568383	−0.284191	+	0.958768i	$0.591725\pi$
$272$	2194.91	0.489287
$273$	0	0
$274$	10822.4	2.38615
$275$	−1188.30	−0.260571
$276$	−499.779	−0.108997
$277$	1557.73	0.337889	0.168944	−	0.985626i	$-0.445964\pi$
$277$	1557.73	0.337889	0.168944	+	0.985626i	$0.445964\pi$
$278$	−3154.77	−0.680614
$279$	−1549.05	−0.332399
$280$	0	0
$281$	−6642.94	−1.41027	−0.705133	−	0.709075i	$-0.749113\pi$
$281$	−6642.94	−1.41027	−0.705133	+	0.709075i	$0.749113\pi$
$282$	7436.07	1.57025
$283$	3932.86	0.826093	0.413046	−	0.910710i	$-0.364465\pi$
$283$	3932.86	0.826093	0.413046	+	0.910710i	$0.364465\pi$
$284$	−8036.73	−1.67920
$285$	−2796.84	−0.581299
$286$	2375.78	0.491199
$287$	0	0
$288$	254.624	0.0520968
$289$	−3250.05	−0.661521
$290$	109.296	0.0221314
$291$	−6570.82	−1.32367
$292$	6714.77	1.34573
$293$	−6137.97	−1.22384	−0.611918	−	0.790921i	$-0.709602\pi$
$293$	−6137.97	−1.22384	−0.611918	+	0.790921i	$0.709602\pi$
$294$	0	0
$295$	−849.022	−0.167566
$296$	9985.60	1.96081
$297$	−1245.77	−0.243389
$298$	−7332.10	−1.42529
$299$	242.092	0.0468246
$300$	−9921.56	−1.90941
$301$	0	0
$302$	−3175.25	−0.605016
$303$	−4981.68	−0.944521
$304$	6193.84	1.16856
$305$	733.288	0.137665
$306$	−1544.75	−0.288586
$307$	5838.84	1.08547	0.542736	−	0.839903i	$-0.317388\pi$
$307$	5838.84	1.08547	0.542736	+	0.839903i	$0.317388\pi$
$308$	0	0
$309$	−10921.8	−2.01074
$310$	−3970.56	−0.727461
$311$	7761.86	1.41522	0.707612	−	0.706601i	$-0.249773\pi$
$311$	7761.86	1.41522	0.707612	+	0.706601i	$0.249773\pi$
$312$	9643.09	1.74978
$313$	6181.00	1.11620	0.558100	−	0.829774i	$-0.311531\pi$
$313$	6181.00	1.11620	0.558100	+	0.829774i	$0.311531\pi$
$314$	−13867.0	−2.49222
$315$	0	0
$316$	−17448.6	−3.10620
$317$	5737.86	1.01663	0.508313	−	0.861173i	$-0.330269\pi$
$317$	5737.86	1.01663	0.508313	+	0.861173i	$0.330269\pi$
$318$	−18577.7	−3.27605
$319$	60.1116	0.0105505
$320$	2426.62	0.423913
$321$	12116.5	2.10679
$322$	0	0
$323$	4692.69	0.808385
$324$	−13681.3	−2.34589
$325$	4805.99	0.820272
$326$	13761.8	2.33803
$327$	−11341.6	−1.91802
$328$	10509.7	1.76921
$329$	0	0
$330$	1297.90	0.216506
$331$	−6488.71	−1.07750	−0.538749	−	0.842466i	$-0.681103\pi$
$331$	−6488.71	−1.07750	−0.538749	+	0.842466i	$0.681103\pi$
$332$	1237.19	0.204517
$333$	−2120.66	−0.348983
$334$	12294.4	2.01414
$335$	509.710	0.0831297
$336$	0	0
$337$	5300.69	0.856816	0.428408	−	0.903585i	$-0.359075\pi$
$337$	5300.69	0.856816	0.428408	+	0.903585i	$0.359075\pi$
$338$	1057.13	0.170119
$339$	−10819.5	−1.73344
$340$	−2615.51	−0.417193
$341$	−2183.76	−0.346795
$342$	−4359.14	−0.689226
$343$	0	0
$344$	−11016.8	−1.72670
$345$	132.256	0.0206389
$346$	2782.46	0.432330
$347$	6265.02	0.969233	0.484617	−	0.874727i	$-0.338959\pi$
$347$	6265.02	0.969233	0.484617	+	0.874727i	$0.338959\pi$
$348$	501.895	0.0773115
$349$	8120.80	1.24555	0.622774	−	0.782401i	$-0.286005\pi$
$349$	8120.80	1.24555	0.622774	+	0.782401i	$0.286005\pi$
$350$	0	0
$351$	5038.41	0.766184
$352$	358.954	0.0543531
$353$	10721.1	1.61650	0.808249	−	0.588840i	$-0.200415\pi$
$353$	10721.1	1.61650	0.808249	+	0.588840i	$0.200415\pi$
$354$	−5902.19	−0.886152
$355$	2126.75	0.317961
$356$	17024.9	2.53460
$357$	0	0
$358$	1333.35	0.196843
$359$	−6877.75	−1.01112	−0.505562	−	0.862790i	$-0.668715\pi$
$359$	−6877.75	−1.01112	−0.505562	+	0.862790i	$0.668715\pi$
$360$	1181.11	0.172916
$361$	6383.35	0.930654
$362$	−19217.4	−2.79018
$363$	713.827	0.103213
$364$	0	0
$365$	−1776.92	−0.254817
$366$	5097.63	0.728026
$367$	−3386.40	−0.481659	−0.240829	−	0.970567i	$-0.577419\pi$
$367$	−3386.40	−0.481659	−0.240829	+	0.970567i	$0.577419\pi$
$368$	−292.892	−0.0414893
$369$	−2231.96	−0.314882
$370$	−5435.71	−0.763754
$371$	0	0
$372$	−18233.0	−2.54123
$373$	−4673.85	−0.648801	−0.324401	−	0.945920i	$-0.605163\pi$
$373$	−4673.85	−0.648801	−0.324401	+	0.945920i	$0.605163\pi$
$374$	−2177.69	−0.301084
$375$	5663.58	0.779909
$376$	9539.61	1.30843
$377$	−243.117	−0.0332127
$378$	0	0
$379$	−3746.77	−0.507806	−0.253903	−	0.967230i	$-0.581714\pi$
$379$	−3746.77	−0.507806	−0.253903	+	0.967230i	$0.581714\pi$
$380$	−7380.73	−0.996377
$381$	−9633.91	−1.29543
$382$	8383.11	1.12282
$383$	−7041.24	−0.939402	−0.469701	−	0.882826i	$-0.655638\pi$
$383$	−7041.24	−0.939402	−0.469701	+	0.882826i	$0.655638\pi$
$384$	15329.2	2.03715
$385$	0	0
$386$	754.480	0.0994872
$387$	2339.65	0.307315
$388$	−17340.1	−2.26884
$389$	3770.10	0.491393	0.245696	−	0.969347i	$-0.420983\pi$
$389$	3770.10	0.491393	0.245696	+	0.969347i	$0.420983\pi$
$390$	−5249.26	−0.681555
$391$	−221.906	−0.0287015
$392$	0	0
$393$	−7519.47	−0.965158
$394$	4317.98	0.552124
$395$	4617.40	0.588169
$396$	1336.25	0.169568
$397$	−4230.65	−0.534836	−0.267418	−	0.963581i	$-0.586171\pi$
$397$	−4230.65	−0.534836	−0.267418	+	0.963581i	$0.586171\pi$
$398$	2979.05	0.375192
$399$	0	0
$400$	−5814.47	−0.726809
$401$	−4455.02	−0.554796	−0.277398	−	0.960755i	$-0.589472\pi$
$401$	−4455.02	−0.554796	−0.277398	+	0.960755i	$0.589472\pi$
$402$	3543.38	0.439621
$403$	8832.05	1.09170
$404$	−13146.4	−1.61896
$405$	3620.46	0.444202
$406$	0	0
$407$	−2989.57	−0.364097
$408$	−8839.04	−1.07254
$409$	5021.55	0.607090	0.303545	−	0.952817i	$-0.401830\pi$
$409$	5021.55	0.607090	0.303545	+	0.952817i	$0.401830\pi$
$410$	−5721.01	−0.689123
$411$	−13151.3	−1.57836
$412$	−28822.1	−3.44651
$413$	0	0
$414$	206.134	0.0244708
$415$	−327.396	−0.0387259
$416$	−1451.76	−0.171102
$417$	3833.64	0.450202
$418$	−6145.24	−0.719075
$419$	5218.89	0.608495	0.304247	−	0.952593i	$-0.401595\pi$
$419$	5218.89	0.608495	0.304247	+	0.952593i	$0.401595\pi$
$420$	0	0
$421$	3993.60	0.462319	0.231159	−	0.972916i	$-0.425748\pi$
$421$	3993.60	0.462319	0.231159	+	0.972916i	$0.425748\pi$
$422$	29584.2	3.41264
$423$	−2025.94	−0.232871
$424$	−23833.0	−2.72979
$425$	−4405.27	−0.502793
$426$	14784.6	1.68150
$427$	0	0
$428$	31975.0	3.61114
$429$	−2887.02	−0.324911
$430$	5997.03	0.672564
$431$	5398.58	0.603342	0.301671	−	0.953412i	$-0.402456\pi$
$431$	5398.58	0.603342	0.301671	+	0.953412i	$0.402456\pi$
$432$	−6095.67	−0.678884
$433$	14819.2	1.64472	0.822361	−	0.568966i	$-0.192657\pi$
$433$	14819.2	1.64472	0.822361	+	0.568966i	$0.192657\pi$
$434$	0	0
$435$	−132.816	−0.0146392
$436$	−29930.0	−3.28759
$437$	−626.200	−0.0685474
$438$	−12352.7	−1.34757
$439$	9137.14	0.993376	0.496688	−	0.867929i	$-0.334549\pi$
$439$	9137.14	0.993376	0.496688	+	0.867929i	$0.334549\pi$
$440$	1665.05	0.180405
$441$	0	0
$442$	8807.50	0.947806
$443$	−5432.36	−0.582616	−0.291308	−	0.956629i	$-0.594091\pi$
$443$	−5432.36	−0.582616	−0.291308	+	0.956629i	$0.594091\pi$
$444$	−24961.0	−2.66801
$445$	−4505.29	−0.479935
$446$	9135.05	0.969859
$447$	8909.90	0.942782
$448$	0	0
$449$	−15009.5	−1.57760	−0.788802	−	0.614648i	$-0.789298\pi$
$449$	−15009.5	−1.57760	−0.788802	+	0.614648i	$0.789298\pi$
$450$	4092.15	0.428679
$451$	−3146.48	−0.328519
$452$	−28552.2	−2.97120
$453$	3858.53	0.400197
$454$	−15191.6	−1.57044
$455$	0	0
$456$	−24943.0	−2.56154
$457$	14612.6	1.49573	0.747865	−	0.663850i	$-0.231079\pi$
$457$	14612.6	1.49573	0.747865	+	0.663850i	$0.231079\pi$
$458$	−1176.02	−0.119982
$459$	−4618.31	−0.469639
$460$	349.017	0.0353761
$461$	17012.4	1.71876	0.859380	−	0.511337i	$-0.170850\pi$
$461$	17012.4	1.71876	0.859380	+	0.511337i	$0.170850\pi$
$462$	0	0
$463$	6094.36	0.611726	0.305863	−	0.952076i	$-0.401055\pi$
$463$	6094.36	0.611726	0.305863	+	0.952076i	$0.401055\pi$
$464$	294.133	0.0294284
$465$	4824.99	0.481190
$466$	6972.04	0.693076
$467$	−11622.7	−1.15168	−0.575838	−	0.817564i	$-0.695324\pi$
$467$	−11622.7	−1.15168	−0.575838	+	0.817564i	$0.695324\pi$
$468$	−5404.36	−0.533796
$469$	0	0
$470$	−5192.93	−0.509643
$471$	16851.0	1.64852
$472$	−7571.83	−0.738393
$473$	3298.28	0.320624
$474$	32099.0	3.11046
$475$	−12431.3	−1.20081
$476$	0	0
$477$	5061.44	0.485844
$478$	−13860.4	−1.32628
$479$	−20039.2	−1.91151	−0.955755	−	0.294164i	$-0.904959\pi$
$479$	−20039.2	−1.91151	−0.955755	+	0.294164i	$0.904959\pi$
$480$	−793.104	−0.0754168
$481$	12091.1	1.14617
$482$	11178.5	1.05637
$483$	0	0
$484$	1883.76	0.176912
$485$	4588.69	0.429612
$486$	10323.8	0.963577
$487$	−16120.0	−1.49994	−0.749968	−	0.661474i	$-0.769931\pi$
$487$	−16120.0	−1.49994	−0.749968	+	0.661474i	$0.769931\pi$
$488$	6539.67	0.606633
$489$	−16723.2	−1.54653
$490$	0	0
$491$	−2949.34	−0.271083	−0.135542	−	0.990772i	$-0.543277\pi$
$491$	−2949.34	−0.271083	−0.135542	+	0.990772i	$0.543277\pi$
$492$	−26271.2	−2.40731
$493$	222.846	0.0203580
$494$	24854.0	2.26363
$495$	−353.610	−0.0321082
$496$	−10685.4	−0.967312
$497$	0	0
$498$	−2275.97	−0.204797
$499$	−6590.50	−0.591245	−0.295623	−	0.955305i	$-0.595527\pi$
$499$	−6590.50	−0.591245	−0.295623	+	0.955305i	$0.595527\pi$
$500$	14945.9	1.33680
$501$	−14940.1	−1.33228
$502$	10559.7	0.938850
$503$	16436.2	1.45697	0.728485	−	0.685062i	$-0.240225\pi$
$503$	16436.2	1.45697	0.728485	+	0.685062i	$0.240225\pi$
$504$	0	0
$505$	3478.93	0.306555
$506$	290.594	0.0255306
$507$	−1284.61	−0.112528
$508$	−25423.5	−2.22044
$509$	−5359.31	−0.466694	−0.233347	−	0.972394i	$-0.574968\pi$
$509$	−5359.31	−0.466694	−0.233347	+	0.972394i	$0.574968\pi$
$510$	4811.58	0.417765
$511$	0	0
$512$	17577.1	1.51720
$513$	−13032.5	−1.12163
$514$	−23053.1	−1.97827
$515$	7627.15	0.652607
$516$	27538.6	2.34946
$517$	−2856.04	−0.242957
$518$	0	0
$519$	−3381.22	−0.285971
$520$	−6734.19	−0.567911
$521$	14787.0	1.24344	0.621718	−	0.783241i	$-0.286435\pi$
$521$	14787.0	1.24344	0.621718	+	0.783241i	$0.286435\pi$
$522$	−207.006	−0.0173571
$523$	11638.8	0.973094	0.486547	−	0.873654i	$-0.338256\pi$
$523$	11638.8	0.973094	0.486547	+	0.873654i	$0.338256\pi$
$524$	−19843.5	−1.65433
$525$	0	0
$526$	−14078.7	−1.16703
$527$	−8095.64	−0.669168
$528$	3492.83	0.287890
$529$	−12137.4	−0.997566
$530$	12973.6	1.06328
$531$	1608.04	0.131418
$532$	0	0
$533$	12725.7	1.03417
$534$	−31319.6	−2.53808
$535$	−8461.50	−0.683781
$536$	4545.74	0.366317
$537$	−1620.28	−0.130205
$538$	−13342.5	−1.06921
$539$	0	0
$540$	7263.74	0.578855
$541$	16465.4	1.30851	0.654255	−	0.756274i	$-0.272982\pi$
$541$	16465.4	1.30851	0.654255	+	0.756274i	$0.272982\pi$
$542$	12310.0	0.975573
$543$	23352.8	1.84561
$544$	1330.71	0.104878
$545$	7920.35	0.622515
$546$	0	0
$547$	20673.9	1.61600	0.807998	−	0.589185i	$-0.200551\pi$
$547$	20673.9	1.61600	0.807998	+	0.589185i	$0.200551\pi$
$548$	−34705.6	−2.70539
$549$	−1388.84	−0.107968
$550$	5768.85	0.447245
$551$	628.852	0.0486207
$552$	1179.50	0.0909469
$553$	0	0
$554$	−7562.36	−0.579953
$555$	6605.41	0.505197
$556$	10116.8	0.771670
$557$	−3023.82	−0.230024	−0.115012	−	0.993364i	$-0.536691\pi$
$557$	−3023.82	−0.230024	−0.115012	+	0.993364i	$0.536691\pi$
$558$	7520.21	0.570530
$559$	−13339.7	−1.00932
$560$	0	0
$561$	2646.30	0.199157
$562$	32249.6	2.42058
$563$	3319.87	0.248518	0.124259	−	0.992250i	$-0.460345\pi$
$563$	3319.87	0.248518	0.124259	+	0.992250i	$0.460345\pi$
$564$	−23846.2	−1.78033
$565$	7555.74	0.562606
$566$	−19092.9	−1.41791
$567$	0	0
$568$	18967.0	1.40112
$569$	13382.1	0.985955	0.492978	−	0.870042i	$-0.335908\pi$
$569$	13382.1	0.985955	0.492978	+	0.870042i	$0.335908\pi$
$570$	13577.8	0.997742
$571$	13975.4	1.02426	0.512131	−	0.858908i	$-0.328856\pi$
$571$	13975.4	1.02426	0.512131	+	0.858908i	$0.328856\pi$
$572$	−7618.72	−0.556914
$573$	−10187.1	−0.742706
$574$	0	0
$575$	587.846	0.0426346
$576$	−4595.99	−0.332465
$577$	17492.0	1.26205	0.631023	−	0.775764i	$-0.282635\pi$
$577$	17492.0	1.26205	0.631023	+	0.775764i	$0.282635\pi$
$578$	15778.1	1.13543
$579$	−916.837	−0.0658073
$580$	−350.495	−0.0250923
$581$	0	0
$582$	31899.5	2.27195
$583$	7135.30	0.506885
$584$	−15847.1	−1.12287
$585$	1430.15	0.101076
$586$	29798.1	2.10059
$587$	19568.3	1.37593	0.687964	−	0.725745i	$-0.258505\pi$
$587$	19568.3	1.37593	0.687964	+	0.725745i	$0.258505\pi$
$588$	0	0
$589$	−22845.2	−1.59816
$590$	4121.76	0.287610
$591$	−5247.17	−0.365211
$592$	−14628.3	−1.01557
$593$	−8768.80	−0.607237	−0.303618	−	0.952794i	$-0.598195\pi$
$593$	−8768.80	−0.607237	−0.303618	+	0.952794i	$0.598195\pi$
$594$	6047.84	0.417754
$595$	0	0
$596$	23512.8	1.61598
$597$	−3620.11	−0.248176
$598$	−1175.29	−0.0803697
$599$	2894.13	0.197414	0.0987069	−	0.995117i	$-0.468529\pi$
$599$	2894.13	0.197414	0.0987069	+	0.995117i	$0.468529\pi$
$600$	23415.3	1.59321
$601$	−1745.49	−0.118469	−0.0592346	−	0.998244i	$-0.518866\pi$
$601$	−1745.49	−0.118469	−0.0592346	+	0.998244i	$0.518866\pi$
$602$	0	0
$603$	−965.386	−0.0651966
$604$	10182.5	0.685959
$605$	−498.497	−0.0334988
$606$	24184.6	1.62118
$607$	−16026.7	−1.07167	−0.535836	−	0.844322i	$-0.680004\pi$
$607$	−16026.7	−1.07167	−0.535836	+	0.844322i	$0.680004\pi$
$608$	3755.16	0.250480
$609$	0	0
$610$	−3559.90	−0.236289
$611$	11551.1	0.764822
$612$	4953.74	0.327195
$613$	−10412.1	−0.686035	−0.343017	−	0.939329i	$-0.611449\pi$
$613$	−10412.1	−0.686035	−0.343017	+	0.939329i	$0.611449\pi$
$614$	−28345.9	−1.86311
$615$	6952.11	0.455831
$616$	0	0
$617$	−91.3588	−0.00596105	−0.00298052	−	0.999996i	$-0.500949\pi$
$617$	−91.3588	−0.00596105	−0.00298052	+	0.999996i	$0.500949\pi$
$618$	53022.1	3.45123
$619$	−21103.7	−1.37032	−0.685160	−	0.728392i	$-0.740268\pi$
$619$	−21103.7	−1.37032	−0.685160	+	0.728392i	$0.740268\pi$
$620$	12732.9	0.824785
$621$	616.275	0.0398233
$622$	−37681.6	−2.42909
$623$	0	0
$624$	−14126.5	−0.906271
$625$	9548.27	0.611089
$626$	−30007.0	−1.91585
$627$	7467.63	0.475643
$628$	44469.0	2.82565
$629$	−11082.9	−0.702552
$630$	0	0
$631$	18083.2	1.14086	0.570430	−	0.821347i	$-0.306777\pi$
$631$	18083.2	1.14086	0.570430	+	0.821347i	$0.306777\pi$
$632$	41179.3	2.59181
$633$	−35950.4	−2.25734
$634$	−27855.7	−1.74494
$635$	6727.78	0.420447
$636$	59575.4	3.71434
$637$	0	0
$638$	−291.825	−0.0181089
$639$	−4028.05	−0.249369
$640$	−10705.0	−0.661178
$641$	−2999.52	−0.184827	−0.0924133	−	0.995721i	$-0.529458\pi$
$641$	−2999.52	−0.184827	−0.0924133	+	0.995721i	$0.529458\pi$
$642$	−58822.3	−3.61609
$643$	20905.2	1.28215	0.641075	−	0.767478i	$-0.278489\pi$
$643$	20905.2	1.28215	0.641075	+	0.767478i	$0.278489\pi$
$644$	0	0
$645$	−7287.52	−0.444877
$646$	−22781.7	−1.38751
$647$	23032.1	1.39951	0.699755	−	0.714383i	$-0.253292\pi$
$647$	23032.1	1.39951	0.699755	+	0.714383i	$0.253292\pi$
$648$	32288.3	1.95741
$649$	2266.91	0.137110
$650$	−23331.7	−1.40792
$651$	0	0
$652$	−44131.9	−2.65082
$653$	5176.54	0.310220	0.155110	−	0.987897i	$-0.450427\pi$
$653$	5176.54	0.310220	0.155110	+	0.987897i	$0.450427\pi$
$654$	55060.3	3.29209
$655$	5251.18	0.313253
$656$	−15396.1	−0.916334
$657$	3365.47	0.199847
$658$	0	0
$659$	−3226.07	−0.190698	−0.0953490	−	0.995444i	$-0.530397\pi$
$659$	−3226.07	−0.190698	−0.0953490	+	0.995444i	$0.530397\pi$
$660$	−4162.14	−0.245471
$661$	−22662.6	−1.33355	−0.666773	−	0.745261i	$-0.732325\pi$
$661$	−22662.6	−1.33355	−0.666773	+	0.745261i	$0.732325\pi$
$662$	31500.8	1.84942
$663$	−10702.8	−0.626941
$664$	−2919.81	−0.170649
$665$	0	0
$666$	10295.2	0.598994
$667$	−29.7369	−0.00172626
$668$	−39426.2	−2.28360
$669$	−11100.8	−0.641528
$670$	−2474.50	−0.142684
$671$	−1957.90	−0.112644
$672$	0	0
$673$	25252.6	1.44639	0.723193	−	0.690646i	$-0.242674\pi$
$673$	25252.6	1.44639	0.723193	+	0.690646i	$0.242674\pi$
$674$	−25733.3	−1.47064
$675$	12234.2	0.697624
$676$	−3390.03	−0.192879
$677$	−964.426	−0.0547502	−0.0273751	−	0.999625i	$-0.508715\pi$
$677$	−964.426	−0.0547502	−0.0273751	+	0.999625i	$0.508715\pi$
$678$	52525.6	2.97527
$679$	0	0
$680$	6172.69	0.348106
$681$	18460.7	1.03879
$682$	10601.5	0.595239
$683$	−855.516	−0.0479288	−0.0239644	−	0.999713i	$-0.507629\pi$
$683$	−855.516	−0.0479288	−0.0239644	+	0.999713i	$0.507629\pi$
$684$	13979.0	0.781434
$685$	9184.12	0.512273
$686$	0	0
$687$	1429.09	0.0793642
$688$	16138.9	0.894314
$689$	−28858.2	−1.59566
$690$	−642.064	−0.0354246
$691$	−12028.3	−0.662197	−0.331098	−	0.943596i	$-0.607419\pi$
$691$	−12028.3	−0.662197	−0.331098	+	0.943596i	$0.607419\pi$
$692$	−8922.89	−0.490170
$693$	0	0
$694$	−30414.9	−1.66359
$695$	−2677.20	−0.146118
$696$	−1184.49	−0.0645086
$697$	−11664.6	−0.633902
$698$	−39424.2	−2.13786
$699$	−8472.35	−0.458446
$700$	0	0
$701$	17608.5	0.948738	0.474369	−	0.880326i	$-0.342676\pi$
$701$	17608.5	0.948738	0.474369	+	0.880326i	$0.342676\pi$
$702$	−24460.0	−1.31508
$703$	−31275.1	−1.67790
$704$	−6479.14	−0.346864
$705$	6310.40	0.337111
$706$	−52047.6	−2.77456
$707$	0	0
$708$	18927.3	1.00471
$709$	−15103.8	−0.800050	−0.400025	−	0.916504i	$-0.630999\pi$
$709$	−15103.8	−0.800050	−0.400025	+	0.916504i	$0.630999\pi$
$710$	−10324.8	−0.545749
$711$	−8745.31	−0.461286
$712$	−40179.4	−2.11487
$713$	1080.29	0.0567424
$714$	0	0
$715$	2016.14	0.105453
$716$	−4275.84	−0.223178
$717$	16843.0	0.877287
$718$	33389.5	1.73550
$719$	−20022.6	−1.03855	−0.519275	−	0.854607i	$-0.673798\pi$
$719$	−20022.6	−1.03855	−0.519275	+	0.854607i	$0.673798\pi$
$720$	−1730.25	−0.0895592
$721$	0	0
$722$	−30989.3	−1.59737
$723$	−13584.0	−0.698750
$724$	61627.1	3.16347
$725$	−590.335	−0.0302407
$726$	−3465.42	−0.177154
$727$	−23331.6	−1.19026	−0.595131	−	0.803629i	$-0.702900\pi$
$727$	−23331.6	−1.19026	−0.595131	+	0.803629i	$0.702900\pi$
$728$	0	0
$729$	11182.0	0.568105
$730$	8626.45	0.437369
$731$	12227.4	0.618669
$732$	−16347.2	−0.825425
$733$	−12218.0	−0.615663	−0.307832	−	0.951441i	$-0.599603\pi$
$733$	−12218.0	−0.615663	−0.307832	+	0.951441i	$0.599603\pi$
$734$	16440.0	0.826719
$735$	0	0
$736$	−177.573	−0.00889323
$737$	−1360.94	−0.0680202
$738$	10835.5	0.540463
$739$	34196.9	1.70224	0.851118	−	0.524975i	$-0.175925\pi$
$739$	34196.9	1.70224	0.851118	+	0.524975i	$0.175925\pi$
$740$	17431.4	0.865933
$741$	−30202.3	−1.49731
$742$	0	0
$743$	−33416.9	−1.65000	−0.824998	−	0.565135i	$-0.808824\pi$
$743$	−33416.9	−1.65000	−0.824998	+	0.565135i	$0.808824\pi$
$744$	43030.6	2.12040
$745$	−6222.17	−0.305990
$746$	22690.2	1.11360
$747$	620.084	0.0303718
$748$	6983.47	0.341365
$749$	0	0
$750$	−27495.1	−1.33864
$751$	−31791.8	−1.54474	−0.772369	−	0.635174i	$-0.780929\pi$
$751$	−31791.8	−1.54474	−0.772369	+	0.635174i	$0.780929\pi$
$752$	−13974.9	−0.677677
$753$	−12832.0	−0.621017
$754$	1180.26	0.0570062
$755$	−2694.58	−0.129888
$756$	0	0
$757$	−6779.94	−0.325523	−0.162762	−	0.986665i	$-0.552040\pi$
$757$	−6779.94	−0.325523	−0.162762	+	0.986665i	$0.552040\pi$
$758$	18189.5	0.871599
$759$	−353.127	−0.0168876
$760$	17418.8	0.831376
$761$	−7330.25	−0.349174	−0.174587	−	0.984642i	$-0.555859\pi$
$761$	−7330.25	−0.349174	−0.174587	+	0.984642i	$0.555859\pi$
$762$	46769.9	2.22348
$763$	0	0
$764$	−26883.2	−1.27304
$765$	−1310.90	−0.0619553
$766$	34183.2	1.61239
$767$	−9168.37	−0.431618
$768$	−46620.2	−2.19045
$769$	−19265.9	−0.903439	−0.451720	−	0.892160i	$-0.649189\pi$
$769$	−19265.9	−0.903439	−0.451720	+	0.892160i	$0.649189\pi$
$770$	0	0
$771$	28013.9	1.30855
$772$	−2419.49	−0.112797
$773$	37972.7	1.76686	0.883429	−	0.468565i	$-0.155229\pi$
$773$	37972.7	1.76686	0.883429	+	0.468565i	$0.155229\pi$
$774$	−11358.3	−0.527475
$775$	21445.9	0.994013
$776$	40923.3	1.89312
$777$	0	0
$778$	−18302.8	−0.843427
$779$	−32916.6	−1.51394
$780$	16833.5	0.772738
$781$	−5678.49	−0.260169
$782$	1077.29	0.0492633
$783$	−618.884	−0.0282466
$784$	0	0
$785$	−11767.8	−0.535045
$786$	36504.9	1.65660
$787$	27617.9	1.25092	0.625458	−	0.780258i	$-0.284912\pi$
$787$	27617.9	1.25092	0.625458	+	0.780258i	$0.284912\pi$
$788$	−13847.0	−0.625990
$789$	17108.2	0.771951
$790$	−22416.2	−1.00953
$791$	0	0
$792$	−3153.59	−0.141487
$793$	7918.58	0.354599
$794$	20538.6	0.917993
$795$	−15765.4	−0.703321
$796$	−9553.30	−0.425387
$797$	2043.47	0.0908200	0.0454100	−	0.998968i	$-0.485541\pi$
$797$	2043.47	0.0908200	0.0454100	+	0.998968i	$0.485541\pi$
$798$	0	0
$799$	−10587.9	−0.468804
$800$	−3525.16	−0.155791
$801$	8532.97	0.376401
$802$	21627.8	0.952251
$803$	4744.43	0.208502
$804$	−11363.0	−0.498436
$805$	0	0
$806$	−42877.1	−1.87380
$807$	16213.7	0.707249
$808$	31026.1	1.35086
$809$	−26909.6	−1.16946	−0.584729	−	0.811228i	$-0.698799\pi$
$809$	−26909.6	−1.16946	−0.584729	+	0.811228i	$0.698799\pi$
$810$	−17576.3	−0.762429
$811$	−2152.16	−0.0931844	−0.0465922	−	0.998914i	$-0.514836\pi$
$811$	−2152.16	−0.0931844	−0.0465922	+	0.998914i	$0.514836\pi$
$812$	0	0
$813$	−14959.0	−0.645307
$814$	14513.5	0.624935
$815$	11678.6	0.501942
$816$	12948.6	0.555506
$817$	34504.7	1.47756
$818$	−24378.2	−1.04201
$819$	0	0
$820$	18346.3	0.781318
$821$	−37914.9	−1.61174	−0.805871	−	0.592091i	$-0.798302\pi$
$821$	−37914.9	−1.61174	−0.805871	+	0.592091i	$0.798302\pi$
$822$	63845.7	2.70909
$823$	39023.8	1.65284	0.826419	−	0.563055i	$-0.190374\pi$
$823$	39023.8	1.65284	0.826419	+	0.563055i	$0.190374\pi$
$824$	68021.1	2.87576
$825$	−7010.25	−0.295837
$826$	0	0
$827$	14151.2	0.595025	0.297513	−	0.954718i	$-0.403843\pi$
$827$	14151.2	0.595025	0.297513	+	0.954718i	$0.403843\pi$
$828$	−661.035	−0.0277446
$829$	32259.7	1.35154	0.675770	−	0.737113i	$-0.263811\pi$
$829$	32259.7	1.35154	0.675770	+	0.737113i	$0.263811\pi$
$830$	1589.41	0.0664691
$831$	9189.69	0.383619
$832$	26204.4	1.09192
$833$	0	0
$834$	−18611.2	−0.772728
$835$	10433.3	0.432407
$836$	19706.7	0.815277
$837$	22483.1	0.928469
$838$	−25336.2	−1.04442
$839$	−27554.4	−1.13383	−0.566915	−	0.823776i	$-0.691863\pi$
$839$	−27554.4	−1.13383	−0.566915	+	0.823776i	$0.691863\pi$
$840$	0	0
$841$	−24359.1	−0.998776
$842$	−19387.8	−0.793524
$843$	−39189.4	−1.60113
$844$	−94871.4	−3.86920
$845$	897.101	0.0365222
$846$	9835.36	0.399701
$847$	0	0
$848$	34913.8	1.41385
$849$	23201.5	0.937896
$850$	21386.3	0.862993
$851$	1478.92	0.0595733
$852$	−47411.8	−1.90646
$853$	−17185.1	−0.689808	−0.344904	−	0.938638i	$-0.612089\pi$
$853$	−17185.1	−0.689808	−0.344904	+	0.938638i	$0.612089\pi$
$854$	0	0
$855$	−3699.25	−0.147967
$856$	−75462.1	−3.01313
$857$	10479.5	0.417705	0.208853	−	0.977947i	$-0.433027\pi$
$857$	10479.5	0.417705	0.208853	+	0.977947i	$0.433027\pi$
$858$	14015.7	0.557677
$859$	−39134.8	−1.55444	−0.777219	−	0.629231i	$-0.783370\pi$
$859$	−39134.8	−1.55444	−0.777219	+	0.629231i	$0.783370\pi$
$860$	−19231.4	−0.762543
$861$	0	0
$862$	−26208.6	−1.03558
$863$	41738.1	1.64633	0.823163	−	0.567805i	$-0.192207\pi$
$863$	41738.1	1.64633	0.823163	+	0.567805i	$0.192207\pi$
$864$	−3695.64	−0.145519
$865$	2361.26	0.0928151
$866$	−71942.9	−2.82300
$867$	−19173.3	−0.751051
$868$	0	0
$869$	−12328.6	−0.481264
$870$	644.783	0.0251267
$871$	5504.23	0.214126
$872$	70636.0	2.74316
$873$	−8690.94	−0.336934
$874$	3040.02	0.117655
$875$	0	0
$876$	39613.1	1.52786
$877$	−44718.3	−1.72181	−0.860907	−	0.508763i	$-0.830103\pi$
$877$	−44718.3	−1.72181	−0.860907	+	0.508763i	$0.830103\pi$
$878$	−44358.2	−1.70503
$879$	−36210.3	−1.38947
$880$	−2439.20	−0.0934379
$881$	−35092.6	−1.34200	−0.670999	−	0.741459i	$-0.734134\pi$
$881$	−35092.6	−1.34200	−0.670999	+	0.741459i	$0.734134\pi$
$882$	0	0
$883$	29569.8	1.12696	0.563479	−	0.826130i	$-0.309462\pi$
$883$	29569.8	1.12696	0.563479	+	0.826130i	$0.309462\pi$
$884$	−28244.2	−1.07461
$885$	−5008.72	−0.190244
$886$	26372.5	1.00000
$887$	39879.6	1.50961	0.754807	−	0.655947i	$-0.227731\pi$
$887$	39879.6	1.50961	0.754807	+	0.655947i	$0.227731\pi$
$888$	58909.0	2.22619
$889$	0	0
$890$	21871.9	0.823760
$891$	−9666.72	−0.363465
$892$	−29294.5	−1.09961
$893$	−29878.2	−1.11964
$894$	−43255.0	−1.61819
$895$	1131.51	0.0422595
$896$	0	0
$897$	1428.20	0.0531618
$898$	72867.0	2.70780
$899$	−1084.87	−0.0402474
$900$	−13122.8	−0.486030
$901$	26452.0	0.978074
$902$	15275.3	0.563870
$903$	0	0
$904$	67384.2	2.47917
$905$	−16308.3	−0.599013
$906$	−18732.0	−0.686899
$907$	−28212.9	−1.03285	−0.516425	−	0.856333i	$-0.672737\pi$
$907$	−28212.9	−1.03285	−0.516425	+	0.856333i	$0.672737\pi$
$908$	48717.0	1.78054
$909$	−6589.05	−0.240423
$910$	0	0
$911$	216.357	0.00786851	0.00393426	−	0.999992i	$-0.498748\pi$
$911$	216.357	0.00786851	0.00393426	+	0.999992i	$0.498748\pi$
$912$	36539.9	1.32671
$913$	874.156	0.0316871
$914$	−70940.0	−2.56727
$915$	4325.95	0.156297
$916$	3771.31	0.136034
$917$	0	0
$918$	22420.6	0.806088
$919$	23788.4	0.853869	0.426935	−	0.904282i	$-0.359593\pi$
$919$	23788.4	0.853869	0.426935	+	0.904282i	$0.359593\pi$
$920$	−823.694	−0.0295178
$921$	34445.6	1.23238
$922$	−82590.5	−2.95008
$923$	22966.2	0.819006
$924$	0	0
$925$	29359.5	1.04360
$926$	−29586.4	−1.04997
$927$	−14445.7	−0.511824
$928$	178.325	0.00630796
$929$	41261.9	1.45722	0.728611	−	0.684928i	$-0.240166\pi$
$929$	41261.9	1.45722	0.728611	+	0.684928i	$0.240166\pi$
$930$	−23423.9	−0.825915
$931$	0	0
$932$	−22358.1	−0.785800
$933$	45790.3	1.60676
$934$	56424.7	1.97674
$935$	−1848.03	−0.0646385
$936$	12754.5	0.445399
$937$	−2424.87	−0.0845431	−0.0422716	−	0.999106i	$-0.513459\pi$
$937$	−2424.87	−0.0845431	−0.0422716	+	0.999106i	$0.513459\pi$
$938$	0	0
$939$	36464.1	1.26727
$940$	16652.9	0.577826
$941$	−25122.6	−0.870321	−0.435160	−	0.900353i	$-0.643308\pi$
$941$	−25122.6	−0.870321	−0.435160	+	0.900353i	$0.643308\pi$
$942$	−81806.7	−2.82952
$943$	1556.55	0.0537521
$944$	11092.2	0.382438
$945$	0	0
$946$	−16012.2	−0.550320
$947$	−38241.4	−1.31223	−0.656114	−	0.754662i	$-0.727801\pi$
$947$	−38241.4	−1.31223	−0.656114	+	0.754662i	$0.727801\pi$
$948$	−102936.	−3.52659
$949$	−19188.5	−0.656360
$950$	60350.3	2.06108
$951$	33849.9	1.15421
$952$	0	0
$953$	−9595.98	−0.326175	−0.163087	−	0.986612i	$-0.552145\pi$
$953$	−9595.98	−0.326175	−0.163087	+	0.986612i	$0.552145\pi$
$954$	−24571.9	−0.833903
$955$	7114.07	0.241053
$956$	44448.0	1.50371
$957$	354.622	0.0119784
$958$	97284.5	3.28092
$959$	0	0
$960$	14315.6	0.481285
$961$	9620.54	0.322934
$962$	−58698.8	−1.96728
$963$	16026.0	0.536273
$964$	−35847.7	−1.19769
$965$	640.267	0.0213585
$966$	0	0
$967$	−14335.7	−0.476739	−0.238369	−	0.971175i	$-0.576613\pi$
$967$	−14335.7	−0.476739	−0.238369	+	0.971175i	$0.576613\pi$
$968$	−4445.74	−0.147615
$969$	27684.0	0.917791
$970$	−22276.8	−0.737386
$971$	16995.1	0.561686	0.280843	−	0.959754i	$-0.409386\pi$
$971$	16995.1	0.561686	0.280843	+	0.959754i	$0.409386\pi$
$972$	−33106.8	−1.09249
$973$	0	0
$974$	78258.2	2.57449
$975$	28352.4	0.931287
$976$	−9580.20	−0.314196
$977$	−9992.06	−0.327200	−0.163600	−	0.986527i	$-0.552311\pi$
$977$	−9992.06	−0.327200	−0.163600	+	0.986527i	$0.552311\pi$
$978$	81186.5	2.65446
$979$	12029.2	0.392703
$980$	0	0
$981$	−15001.1	−0.488224
$982$	14318.2	0.465287
$983$	−3773.62	−0.122441	−0.0612206	−	0.998124i	$-0.519499\pi$
$983$	−3773.62	−0.122441	−0.0612206	+	0.998124i	$0.519499\pi$
$984$	62000.9	2.00866
$985$	3664.33	0.118533
$986$	−1081.85	−0.0349424
$987$	0	0
$988$	−79702.5	−2.56647
$989$	−1631.65	−0.0524604
$990$	1716.67	0.0551106
$991$	25846.6	0.828501	0.414251	−	0.910163i	$-0.364044\pi$
$991$	25846.6	0.828501	0.414251	+	0.910163i	$0.364044\pi$
$992$	−6478.24	−0.207343
$993$	−38279.5	−1.22333
$994$	0	0
$995$	2528.08	0.0805483
$996$	7298.66	0.232196
$997$	−30809.7	−0.978688	−0.489344	−	0.872091i	$-0.662764\pi$
$997$	−30809.7	−0.978688	−0.489344	+	0.872091i	$0.662764\pi$
$998$	31995.0	1.01481
$999$	30779.3	0.974790

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	539.4.a.l.1.2	yes	10
7.6	odd	2	inner	539.4.a.l.1.1	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
539.4.a.l.1.1	✓	10	7.6	odd	2	inner
539.4.a.l.1.2	yes	10	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:	\( N \)	\(=\)	\( 539 = 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	539.a (trivial)

\(f(q)\)	\(=\)	\(q-4.85471 q^{2} +5.89940 q^{3} +15.5682 q^{4} -4.11981 q^{5} -28.6399 q^{6} -36.7416 q^{8} +7.80287 q^{9} +20.0005 q^{10} +11.0000 q^{11} +91.8432 q^{12} -44.4887 q^{13} -24.3044 q^{15} +53.8242 q^{16} +40.7793 q^{17} -37.8807 q^{18} +115.075 q^{19} -64.1382 q^{20} -53.4018 q^{22} -5.44165 q^{23} -216.753 q^{24} -108.027 q^{25} +215.980 q^{26} -113.251 q^{27} +5.46469 q^{29} +117.991 q^{30} -198.523 q^{31} +32.6321 q^{32} +64.8934 q^{33} -197.972 q^{34} +121.477 q^{36} -271.779 q^{37} -558.658 q^{38} -262.457 q^{39} +151.368 q^{40} -286.044 q^{41} +299.844 q^{43} +171.251 q^{44} -32.1463 q^{45} +26.4176 q^{46} -259.640 q^{47} +317.530 q^{48} +524.441 q^{50} +240.573 q^{51} -692.611 q^{52} +648.664 q^{53} +549.803 q^{54} -45.3179 q^{55} +678.875 q^{57} -26.5295 q^{58} +206.083 q^{59} -378.376 q^{60} -177.991 q^{61} +963.774 q^{62} -589.013 q^{64} +183.285 q^{65} -315.039 q^{66} -123.722 q^{67} +634.861 q^{68} -32.1024 q^{69} -516.226 q^{71} -286.690 q^{72} +431.312 q^{73} +1319.41 q^{74} -637.295 q^{75} +1791.52 q^{76} +1274.15 q^{78} -1120.78 q^{79} -221.745 q^{80} -878.793 q^{81} +1388.66 q^{82} +79.4687 q^{83} -168.003 q^{85} -1455.66 q^{86} +32.2384 q^{87} -404.158 q^{88} +1093.57 q^{89} +156.061 q^{90} -84.7169 q^{92} -1171.17 q^{93} +1260.48 q^{94} -474.089 q^{95} +192.510 q^{96} -1113.81 q^{97} +85.8316 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 4 q^{2} + 8 q^{4} - 84 q^{8} - 18 q^{9} + 110 q^{11} - 76 q^{15} + 40 q^{16} - 44 q^{22} - 384 q^{23} - 338 q^{25} - 340 q^{29} - 228 q^{30} - 276 q^{32} - 60 q^{36} - 1384 q^{37} - 1232 q^{39} - 100 q^{43}+ \cdots - 198 q^{99}+O(q^{100}) \) 10 * q - 4 * q^2 + 8 * q^4 - 84 * q^8 - 18 * q^9 + 110 * q^11 - 76 * q^15 + 40 * q^16 - 44 * q^22 - 384 * q^23 - 338 * q^25 - 340 * q^29 - 228 * q^30 - 276 * q^32 - 60 * q^36 - 1384 * q^37 - 1232 * q^39 - 100 * q^43 + 88 * q^44 - 936 * q^46 + 184 * q^50 - 1712 * q^51 - 416 * q^53 + 1008 * q^57 - 1588 * q^58 - 1308 * q^60 - 904 * q^64 - 1860 * q^65 - 1588 * q^67 - 2148 * q^71 - 2736 * q^72 + 272 * q^74 + 3040 * q^78 - 4024 * q^79 - 2858 * q^81 - 1236 * q^85 - 6324 * q^86 - 924 * q^88 - 464 * q^92 - 3452 * q^93 - 2448 * q^95 - 198 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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