LMFDB - Embedded newform 5077.2.a.c.1.14

Newspace parameters

Level:	$ N $	$=$	$ 5077 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5077.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$40.5400491062$
Analytic rank:	$0$
Dimension:	$216$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Character	$\chi$	$=$	5077.1

$q$-expansion

$f(q)$	$=$	$q-2.46114 q^{2} +2.54758 q^{3} +4.05723 q^{4} +0.498025 q^{5} -6.26997 q^{6} -4.84312 q^{7} -5.06313 q^{8} +3.49018 q^{9} +O(q^{10})$	\(q-2.46114 q^{2} +2.54758 q^{3} +4.05723 q^{4} +0.498025 q^{5} -6.26997 q^{6} -4.84312 q^{7} -5.06313 q^{8} +3.49018 q^{9} -1.22571 q^{10} -3.11702 q^{11} +10.3361 q^{12} -3.68171 q^{13} +11.9196 q^{14} +1.26876 q^{15} +4.34664 q^{16} +1.96034 q^{17} -8.58983 q^{18} -2.10999 q^{19} +2.02060 q^{20} -12.3383 q^{21} +7.67144 q^{22} -5.10126 q^{23} -12.8988 q^{24} -4.75197 q^{25} +9.06122 q^{26} +1.24877 q^{27} -19.6497 q^{28} -9.54412 q^{29} -3.12260 q^{30} +5.27605 q^{31} -0.571449 q^{32} -7.94087 q^{33} -4.82469 q^{34} -2.41200 q^{35} +14.1605 q^{36} +10.8609 q^{37} +5.19299 q^{38} -9.37947 q^{39} -2.52157 q^{40} +6.08868 q^{41} +30.3662 q^{42} +6.41401 q^{43} -12.6465 q^{44} +1.73820 q^{45} +12.5549 q^{46} -5.11266 q^{47} +11.0734 q^{48} +16.4559 q^{49} +11.6953 q^{50} +4.99414 q^{51} -14.9375 q^{52} +5.59624 q^{53} -3.07341 q^{54} -1.55236 q^{55} +24.5214 q^{56} -5.37538 q^{57} +23.4894 q^{58} +5.97964 q^{59} +5.14765 q^{60} -10.2278 q^{61} -12.9851 q^{62} -16.9034 q^{63} -7.28687 q^{64} -1.83359 q^{65} +19.5436 q^{66} +4.03308 q^{67} +7.95356 q^{68} -12.9959 q^{69} +5.93627 q^{70} +14.4935 q^{71} -17.6712 q^{72} +0.0705208 q^{73} -26.7302 q^{74} -12.1060 q^{75} -8.56072 q^{76} +15.0961 q^{77} +23.0842 q^{78} +3.20700 q^{79} +2.16474 q^{80} -7.28919 q^{81} -14.9851 q^{82} -3.85529 q^{83} -50.0591 q^{84} +0.976300 q^{85} -15.7858 q^{86} -24.3144 q^{87} +15.7819 q^{88} +1.63395 q^{89} -4.27795 q^{90} +17.8310 q^{91} -20.6970 q^{92} +13.4412 q^{93} +12.5830 q^{94} -1.05083 q^{95} -1.45581 q^{96} +9.03075 q^{97} -40.5002 q^{98} -10.8790 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 216q + 25q^{2} + 62q^{3} + 223q^{4} + 46q^{5} + 26q^{6} + 30q^{7} + 75q^{8} + 234q^{9} + O(q^{10}) $	\( 216q + 25q^{2} + 62q^{3} + 223q^{4} + 46q^{5} + 26q^{6} + 30q^{7} + 75q^{8} + 234q^{9} + 24q^{10} + 89q^{11} + 114q^{12} + 34q^{13} + 53q^{14} + 61q^{15} + 229q^{16} + 76q^{17} + 57q^{18} + 54q^{19} + 118q^{20} + 25q^{21} + 26q^{22} + 109q^{23} + 65q^{24} + 232q^{25} + 58q^{26} + 236q^{27} + 57q^{28} + 54q^{29} + 6q^{30} + 77q^{31} + 155q^{32} + 80q^{33} + 28q^{34} + 137q^{35} + 257q^{36} + 42q^{37} + 104q^{38} + 46q^{39} + 47q^{40} + 109q^{41} + 27q^{42} + 68q^{43} + 145q^{44} + 109q^{45} - 7q^{46} + 264q^{47} + 198q^{48} + 222q^{49} + 86q^{50} + 57q^{51} + 68q^{52} + 95q^{53} + 79q^{54} + 50q^{55} + 108q^{56} + 55q^{57} + 38q^{58} + 292q^{59} + 91q^{60} + 16q^{61} + 91q^{62} + 113q^{63} + 231q^{64} + 68q^{65} - 15q^{66} + 152q^{67} + 199q^{68} + 83q^{69} + 24q^{70} + 131q^{71} + 162q^{72} + 71q^{73} + 10q^{74} + 232q^{75} + 60q^{76} + 131q^{77} + 102q^{78} + 10q^{79} + 236q^{80} + 268q^{81} + 54q^{82} + 299q^{83} - 9q^{85} + 35q^{86} + 103q^{87} + 45q^{88} + 134q^{89} + 8q^{90} + 79q^{91} + 206q^{92} + 95q^{93} + 18q^{94} + 119q^{95} + 77q^{96} + 129q^{97} + 150q^{98} + 221q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

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 $

$2$	−2.46114	−1.74029	−0.870146	−	0.492795i	$-0.835975\pi$
$2$	−2.46114	−1.74029	−0.870146	+	0.492795i	$0.835975\pi$
$3$	2.54758	1.47085	0.735424	−	0.677607i	$-0.236983\pi$
$3$	2.54758	1.47085	0.735424	+	0.677607i	$0.236983\pi$
$4$	4.05723	2.02861
$5$	0.498025	0.222724	0.111362	−	0.993780i	$-0.464479\pi$
$5$	0.498025	0.222724	0.111362	+	0.993780i	$0.464479\pi$
$6$	−6.26997	−2.55970
$7$	−4.84312	−1.83053	−0.915264	−	0.402854i	$-0.868018\pi$
$7$	−4.84312	−1.83053	−0.915264	+	0.402854i	$0.868018\pi$
$8$	−5.06313	−1.79009
$9$	3.49018	1.16339
$10$	−1.22571	−0.387604
$11$	−3.11702	−0.939818	−0.469909	−	0.882715i	$-0.655713\pi$
$11$	−3.11702	−0.939818	−0.469909	+	0.882715i	$0.655713\pi$
$12$	10.3361	2.98378
$13$	−3.68171	−1.02112	−0.510562	−	0.859841i	$-0.670562\pi$
$13$	−3.68171	−1.02112	−0.510562	+	0.859841i	$0.670562\pi$
$14$	11.9196	3.18565
$15$	1.26876	0.327593
$16$	4.34664	1.08666
$17$	1.96034	0.475453	0.237726	−	0.971332i	$-0.423598\pi$
$17$	1.96034	0.475453	0.237726	+	0.971332i	$0.423598\pi$
$18$	−8.58983	−2.02464
$19$	−2.10999	−0.484065	−0.242033	−	0.970268i	$-0.577814\pi$
$19$	−2.10999	−0.484065	−0.242033	+	0.970268i	$0.577814\pi$
$20$	2.02060	0.451820
$21$	−12.3383	−2.69243
$22$	7.67144	1.63556
$23$	−5.10126	−1.06369	−0.531844	−	0.846843i	$-0.678501\pi$
$23$	−5.10126	−1.06369	−0.531844	+	0.846843i	$0.678501\pi$
$24$	−12.8988	−2.63295
$25$	−4.75197	−0.950394
$26$	9.06122	1.77705
$27$	1.24877	0.240326
$28$	−19.6497	−3.71344
$29$	−9.54412	−1.77230	−0.886149	−	0.463400i	$-0.846629\pi$
$29$	−9.54412	−1.77230	−0.886149	+	0.463400i	$0.846629\pi$
$30$	−3.12260	−0.570106
$31$	5.27605	0.947607	0.473803	−	0.880631i	$-0.342881\pi$
$31$	5.27605	0.947607	0.473803	+	0.880631i	$0.342881\pi$
$32$	−0.571449	−0.101019
$33$	−7.94087	−1.38233
$34$	−4.82469	−0.827427
$35$	−2.41200	−0.407702
$36$	14.1605	2.36008
$37$	10.8609	1.78552	0.892759	−	0.450534i	$-0.148767\pi$
$37$	10.8609	1.78552	0.892759	+	0.450534i	$0.148767\pi$
$38$	5.19299	0.842415
$39$	−9.37947	−1.50192
$40$	−2.52157	−0.398695
$41$	6.08868	0.950891	0.475446	−	0.879745i	$-0.342287\pi$
$41$	6.08868	0.950891	0.475446	+	0.879745i	$0.342287\pi$
$42$	30.3662	4.68561
$43$	6.41401	0.978128	0.489064	−	0.872248i	$-0.337338\pi$
$43$	6.41401	0.978128	0.489064	+	0.872248i	$0.337338\pi$
$44$	−12.6465	−1.90653
$45$	1.73820	0.259115
$46$	12.5549	1.85113
$47$	−5.11266	−0.745758	−0.372879	−	0.927880i	$-0.621629\pi$
$47$	−5.11266	−0.745758	−0.372879	+	0.927880i	$0.621629\pi$
$48$	11.0734	1.59831
$49$	16.4559	2.35084
$50$	11.6953	1.65396
$51$	4.99414	0.699319
$52$	−14.9375	−2.07147
$53$	5.59624	0.768702	0.384351	−	0.923187i	$-0.374425\pi$
$53$	5.59624	0.768702	0.384351	+	0.923187i	$0.374425\pi$
$54$	−3.07341	−0.418238
$55$	−1.55236	−0.209320
$56$	24.5214	3.27681
$57$	−5.37538	−0.711986
$58$	23.4894	3.08432
$59$	5.97964	0.778483	0.389241	−	0.921136i	$-0.372737\pi$
$59$	5.97964	0.778483	0.389241	+	0.921136i	$0.372737\pi$
$60$	5.14765	0.664559
$61$	−10.2278	−1.30953	−0.654767	−	0.755831i	$-0.727233\pi$
$61$	−10.2278	−1.30953	−0.654767	+	0.755831i	$0.727233\pi$
$62$	−12.9851	−1.64911
$63$	−16.9034	−2.12962
$64$	−7.28687	−0.910859
$65$	−1.83359	−0.227428
$66$	19.5436	2.40565
$67$	4.03308	0.492720	0.246360	−	0.969178i	$-0.420765\pi$
$67$	4.03308	0.492720	0.246360	+	0.969178i	$0.420765\pi$
$68$	7.95356	0.964511
$69$	−12.9959	−1.56452
$70$	5.93627	0.709520
$71$	14.4935	1.72006	0.860029	−	0.510245i	$-0.170445\pi$
$71$	14.4935	1.72006	0.860029	+	0.510245i	$0.170445\pi$
$72$	−17.6712	−2.08258
$73$	0.0705208	0.00825384	0.00412692	−	0.999991i	$-0.498686\pi$
$73$	0.0705208	0.00825384	0.00412692	+	0.999991i	$0.498686\pi$
$74$	−26.7302	−3.10732
$75$	−12.1060	−1.39789
$76$	−8.56072	−0.981982
$77$	15.0961	1.72036
$78$	23.0842	2.61377
$79$	3.20700	0.360815	0.180408	−	0.983592i	$-0.442258\pi$
$79$	3.20700	0.360815	0.180408	+	0.983592i	$0.442258\pi$
$80$	2.16474	0.242025
$81$	−7.28919	−0.809910
$82$	−14.9851	−1.65483
$83$	−3.85529	−0.423173	−0.211587	−	0.977359i	$-0.567863\pi$
$83$	−3.85529	−0.423173	−0.211587	+	0.977359i	$0.567863\pi$
$84$	−50.0591	−5.46190
$85$	0.976300	0.105895
$86$	−15.7858	−1.70223
$87$	−24.3144	−2.60678
$88$	15.7819	1.68236
$89$	1.63395	0.173199	0.0865993	−	0.996243i	$-0.472400\pi$
$89$	1.63395	0.173199	0.0865993	+	0.996243i	$0.472400\pi$
$90$	−4.27795	−0.450936
$91$	17.8310	1.86920
$92$	−20.6970	−2.15781
$93$	13.4412	1.39379
$94$	12.5830	1.29784
$95$	−1.05083	−0.107813
$96$	−1.45581	−0.148583
$97$	9.03075	0.916934	0.458467	−	0.888712i	$-0.348399\pi$
$97$	9.03075	0.916934	0.458467	+	0.888712i	$0.348399\pi$
$98$	−40.5002	−4.09114
$99$	−10.8790	−1.09338
$100$	−19.2798	−1.92798
$101$	8.75427	0.871083	0.435541	−	0.900169i	$-0.356557\pi$
$101$	8.75427	0.871083	0.435541	+	0.900169i	$0.356557\pi$
$102$	−12.2913	−1.21702
$103$	4.69400	0.462514	0.231257	−	0.972893i	$-0.425716\pi$
$103$	4.69400	0.462514	0.231257	+	0.972893i	$0.425716\pi$
$104$	18.6410	1.82790
$105$	−6.14476	−0.599668
$106$	−13.7731	−1.33777
$107$	14.2147	1.37419	0.687095	−	0.726568i	$-0.258886\pi$
$107$	14.2147	1.37419	0.687095	+	0.726568i	$0.258886\pi$
$108$	5.06655	0.487529
$109$	−10.9113	−1.04511	−0.522555	−	0.852606i	$-0.675021\pi$
$109$	−10.9113	−1.04511	−0.522555	+	0.852606i	$0.675021\pi$
$110$	3.82057	0.364277
$111$	27.6690	2.62623
$112$	−21.0513	−1.98916
$113$	2.90892	0.273648	0.136824	−	0.990595i	$-0.456310\pi$
$113$	2.90892	0.273648	0.136824	+	0.990595i	$0.456310\pi$
$114$	13.2296	1.23906
$115$	−2.54056	−0.236908
$116$	−38.7227	−3.59531
$117$	−12.8498	−1.18797
$118$	−14.7168	−1.35479
$119$	−9.49418	−0.870330
$120$	−6.42390	−0.586420
$121$	−1.28417	−0.116743
$122$	25.1720	2.27897
$123$	15.5114	1.39862
$124$	21.4061	1.92233
$125$	−4.85673	−0.434399
$126$	41.6016	3.70617
$127$	12.9759	1.15143	0.575713	−	0.817652i	$-0.304725\pi$
$127$	12.9759	1.15143	0.575713	+	0.817652i	$0.304725\pi$
$128$	19.0769	1.68618
$129$	16.3402	1.43868
$130$	4.51272	0.395791
$131$	−4.12382	−0.360300	−0.180150	−	0.983639i	$-0.557658\pi$
$131$	−4.12382	−0.360300	−0.180150	+	0.983639i	$0.557658\pi$
$132$	−32.2179	−2.80421
$133$	10.2190	0.886095
$134$	−9.92600	−0.857476
$135$	0.621920	0.0535263
$136$	−9.92548	−0.851103
$137$	3.38638	0.289318	0.144659	−	0.989482i	$-0.453792\pi$
$137$	3.38638	0.289318	0.144659	+	0.989482i	$0.453792\pi$
$138$	31.9848	2.72272
$139$	5.81049	0.492839	0.246420	−	0.969163i	$-0.420746\pi$
$139$	5.81049	0.492839	0.246420	+	0.969163i	$0.420746\pi$
$140$	−9.78602	−0.827070
$141$	−13.0249	−1.09690
$142$	−35.6705	−2.99340
$143$	11.4760	0.959670
$144$	15.1706	1.26421
$145$	−4.75321	−0.394733
$146$	−0.173562	−0.0143641
$147$	41.9227	3.45772
$148$	44.0651	3.62213
$149$	5.66001	0.463686	0.231843	−	0.972753i	$-0.425524\pi$
$149$	5.66001	0.463686	0.231843	+	0.972753i	$0.425524\pi$
$150$	29.7947	2.43273
$151$	4.89829	0.398618	0.199309	−	0.979937i	$-0.436130\pi$
$151$	4.89829	0.398618	0.199309	+	0.979937i	$0.436130\pi$
$152$	10.6832	0.866520
$153$	6.84195	0.553139
$154$	−37.1537	−2.99393
$155$	2.62761	0.211054
$156$	−38.0546	−3.04681
$157$	−17.0242	−1.35868	−0.679339	−	0.733825i	$-0.737733\pi$
$157$	−17.0242	−1.35868	−0.679339	+	0.733825i	$0.737733\pi$
$158$	−7.89288	−0.627924
$159$	14.2569	1.13064
$160$	−0.284596	−0.0224993
$161$	24.7061	1.94711
$162$	17.9397	1.40948
$163$	8.55783	0.670301	0.335151	−	0.942165i	$-0.391213\pi$
$163$	8.55783	0.670301	0.335151	+	0.942165i	$0.391213\pi$
$164$	24.7031	1.92899
$165$	−3.95475	−0.307877
$166$	9.48843	0.736445
$167$	−8.77925	−0.679359	−0.339679	−	0.940541i	$-0.610319\pi$
$167$	−8.77925	−0.679359	−0.339679	+	0.940541i	$0.610319\pi$
$168$	62.4703	4.81969
$169$	0.555006	0.0426927
$170$	−2.40281	−0.184287
$171$	−7.36425	−0.563158
$172$	26.0231	1.98424
$173$	9.75543	0.741692	0.370846	−	0.928694i	$-0.379068\pi$
$173$	9.75543	0.741692	0.370846	+	0.928694i	$0.379068\pi$
$174$	59.8413	4.53656
$175$	23.0144	1.73972
$176$	−13.5486	−1.02126
$177$	15.2336	1.14503
$178$	−4.02139	−0.301416
$179$	17.1359	1.28080	0.640400	−	0.768042i	$-0.278769\pi$
$179$	17.1359	1.28080	0.640400	+	0.768042i	$0.278769\pi$
$180$	7.05226	0.525645
$181$	−9.50242	−0.706310	−0.353155	−	0.935565i	$-0.614891\pi$
$181$	−9.50242	−0.706310	−0.353155	+	0.935565i	$0.614891\pi$
$182$	−43.8846	−3.25295
$183$	−26.0561	−1.92612
$184$	25.8284	1.90409
$185$	5.40899	0.397677
$186$	−33.0807	−2.42559
$187$	−6.11043	−0.446839
$188$	−20.7432	−1.51285
$189$	−6.04796	−0.439924
$190$	2.58624	0.187626
$191$	−4.68020	−0.338647	−0.169324	−	0.985560i	$-0.554158\pi$
$191$	−4.68020	−0.338647	−0.169324	+	0.985560i	$0.554158\pi$
$192$	−18.5639	−1.33973
$193$	26.0836	1.87754	0.938771	−	0.344543i	$-0.111966\pi$
$193$	26.0836	1.87754	0.938771	+	0.344543i	$0.111966\pi$
$194$	−22.2260	−1.59573
$195$	−4.67121	−0.334512
$196$	66.7652	4.76894
$197$	3.91475	0.278915	0.139457	−	0.990228i	$-0.455464\pi$
$197$	3.91475	0.278915	0.139457	+	0.990228i	$0.455464\pi$
$198$	26.7747	1.90280
$199$	−15.1426	−1.07343	−0.536716	−	0.843763i	$-0.680335\pi$
$199$	−15.1426	−1.07343	−0.536716	+	0.843763i	$0.680335\pi$
$200$	24.0599	1.70129
$201$	10.2746	0.724716
$202$	−21.5455	−1.51594
$203$	46.2234	3.24424
$204$	20.2623	1.41865
$205$	3.03231	0.211786
$206$	−11.5526	−0.804909
$207$	−17.8043	−1.23749
$208$	−16.0031	−1.10962
$209$	6.57689	0.454933
$210$	15.1231	1.04360
$211$	−27.8910	−1.92010	−0.960048	−	0.279834i	$-0.909721\pi$
$211$	−27.8910	−1.92010	−0.960048	+	0.279834i	$0.909721\pi$
$212$	22.7052	1.55940
$213$	36.9233	2.52994
$214$	−34.9845	−2.39149
$215$	3.19434	0.217852
$216$	−6.32270	−0.430205
$217$	−25.5526	−1.73462
$218$	26.8542	1.81879
$219$	0.179658	0.0121401
$220$	−6.29826	−0.424629
$221$	−7.21742	−0.485496
$222$	−68.0974	−4.57040
$223$	−21.3522	−1.42985	−0.714926	−	0.699201i	$-0.753539\pi$
$223$	−21.3522	−1.42985	−0.714926	+	0.699201i	$0.753539\pi$
$224$	2.76760	0.184918
$225$	−16.5852	−1.10568
$226$	−7.15927	−0.476228
$227$	−1.24150	−0.0824010	−0.0412005	−	0.999151i	$-0.513118\pi$
$227$	−1.24150	−0.0824010	−0.0412005	+	0.999151i	$0.513118\pi$
$228$	−21.8091	−1.44435
$229$	24.5978	1.62547	0.812734	−	0.582635i	$-0.197978\pi$
$229$	24.5978	1.62547	0.812734	+	0.582635i	$0.197978\pi$
$230$	6.25268	0.412289
$231$	38.4586	2.53039
$232$	48.3232	3.17257
$233$	17.6700	1.15760	0.578801	−	0.815469i	$-0.303521\pi$
$233$	17.6700	1.15760	0.578801	+	0.815469i	$0.303521\pi$
$234$	31.6253	2.06741
$235$	−2.54623	−0.166098
$236$	24.2608	1.57924
$237$	8.17009	0.530704
$238$	23.3666	1.51463
$239$	21.3992	1.38420	0.692101	−	0.721801i	$-0.256685\pi$
$239$	21.3992	1.38420	0.692101	+	0.721801i	$0.256685\pi$
$240$	5.51485	0.355982
$241$	17.4458	1.12378	0.561892	−	0.827211i	$-0.310074\pi$
$241$	17.4458	1.12378	0.561892	+	0.827211i	$0.310074\pi$
$242$	3.16053	0.203167
$243$	−22.3161	−1.43158
$244$	−41.4965	−2.65654
$245$	8.19543	0.523587
$246$	−38.1758	−2.43400
$247$	7.76838	0.494290
$248$	−26.7134	−1.69630
$249$	−9.82168	−0.622423
$250$	11.9531	0.755981
$251$	25.5777	1.61445	0.807224	−	0.590246i	$-0.200969\pi$
$251$	25.5777	1.61445	0.807224	+	0.590246i	$0.200969\pi$
$252$	−68.5808	−4.32019
$253$	15.9008	0.999672
$254$	−31.9356	−2.00382
$255$	2.48721	0.155755
$256$	−32.3773	−2.02358
$257$	−18.8741	−1.17733	−0.588666	−	0.808376i	$-0.700347\pi$
$257$	−18.8741	−1.17733	−0.588666	+	0.808376i	$0.700347\pi$
$258$	−40.2157	−2.50372
$259$	−52.6006	−3.26844
$260$	−7.43927	−0.461364
$261$	−33.3107	−2.06188
$262$	10.1493	0.627026
$263$	−21.8899	−1.34979	−0.674895	−	0.737914i	$-0.735811\pi$
$263$	−21.8899	−1.34979	−0.674895	+	0.737914i	$0.735811\pi$
$264$	40.2057	2.47449
$265$	2.78707	0.171208
$266$	−25.1503	−1.54206
$267$	4.16263	0.254749
$268$	16.3631	0.999538
$269$	−25.5912	−1.56033	−0.780163	−	0.625576i	$-0.784864\pi$
$269$	−25.5912	−1.56033	−0.780163	+	0.625576i	$0.784864\pi$
$270$	−1.53063	−0.0931514
$271$	17.2043	1.04509	0.522543	−	0.852613i	$-0.324984\pi$
$271$	17.2043	1.04509	0.522543	+	0.852613i	$0.324984\pi$
$272$	8.52091	0.516656
$273$	45.4259	2.74930
$274$	−8.33436	−0.503497
$275$	14.8120	0.893197
$276$	−52.7273	−3.17381
$277$	−25.6496	−1.54113	−0.770567	−	0.637359i	$-0.780027\pi$
$277$	−25.6496	−1.54113	−0.770567	+	0.637359i	$0.780027\pi$
$278$	−14.3005	−0.857684
$279$	18.4144	1.10244
$280$	12.2123	0.729823
$281$	−19.4675	−1.16134	−0.580668	−	0.814140i	$-0.697209\pi$
$281$	−19.4675	−1.16134	−0.580668	+	0.814140i	$0.697209\pi$
$282$	32.0562	1.90892
$283$	−0.968842	−0.0575917	−0.0287958	−	0.999585i	$-0.509167\pi$
$283$	−0.968842	−0.0575917	−0.0287958	+	0.999585i	$0.509167\pi$
$284$	58.8033	3.48934
$285$	−2.67707	−0.158576
$286$	−28.2440	−1.67010
$287$	−29.4882	−1.74063
$288$	−1.99446	−0.117525
$289$	−13.1571	−0.773944
$290$	11.6983	0.686950
$291$	23.0066	1.34867
$292$	0.286119	0.0167438
$293$	12.1498	0.709800	0.354900	−	0.934904i	$-0.384515\pi$
$293$	12.1498	0.709800	0.354900	+	0.934904i	$0.384515\pi$
$294$	−103.178	−6.01744
$295$	2.97801	0.173387
$296$	−54.9901	−3.19624
$297$	−3.89245	−0.225863
$298$	−13.9301	−0.806949
$299$	18.7814	1.08616
$300$	−49.1170	−2.83577
$301$	−31.0639	−1.79049
$302$	−12.0554	−0.693711
$303$	22.3022	1.28123
$304$	−9.17138	−0.526015
$305$	−5.09369	−0.291664
$306$	−16.8390	−0.962622
$307$	0.314156	0.0179298	0.00896492	−	0.999960i	$-0.497146\pi$
$307$	0.314156	0.0179298	0.00896492	+	0.999960i	$0.497146\pi$
$308$	61.2484	3.48995
$309$	11.9584	0.680287
$310$	−6.46692	−0.367296
$311$	−1.50810	−0.0855166	−0.0427583	−	0.999085i	$-0.513615\pi$
$311$	−1.50810	−0.0855166	−0.0427583	+	0.999085i	$0.513615\pi$
$312$	47.4895	2.68856
$313$	10.4059	0.588176	0.294088	−	0.955778i	$-0.404984\pi$
$313$	10.4059	0.588176	0.294088	+	0.955778i	$0.404984\pi$
$314$	41.8990	2.36450
$315$	−8.41830	−0.474318
$316$	13.0115	0.731955
$317$	21.7763	1.22308	0.611538	−	0.791215i	$-0.290551\pi$
$317$	21.7763	1.22308	0.611538	+	0.791215i	$0.290551\pi$
$318$	−35.0882	−1.96765
$319$	29.7492	1.66564
$320$	−3.62905	−0.202870
$321$	36.2132	2.02122
$322$	−60.8051	−3.38854
$323$	−4.13631	−0.230150
$324$	−29.5739	−1.64299
$325$	17.4954	0.970470
$326$	−21.0621	−1.16652
$327$	−27.7973	−1.53720
$328$	−30.8278	−1.70218
$329$	24.7612	1.36513
$330$	9.73322	0.535796
$331$	2.69039	0.147877	0.0739386	−	0.997263i	$-0.476443\pi$
$331$	2.69039	0.147877	0.0739386	+	0.997263i	$0.476443\pi$
$332$	−15.6418	−0.858455
$333$	37.9064	2.07726
$334$	21.6070	1.18228
$335$	2.00858	0.109740
$336$	−53.6300	−2.92576
$337$	−14.7306	−0.802424	−0.401212	−	0.915985i	$-0.631411\pi$
$337$	−14.7306	−0.802424	−0.401212	+	0.915985i	$0.631411\pi$
$338$	−1.36595	−0.0742978
$339$	7.41072	0.402495
$340$	3.96107	0.214819
$341$	−16.4456	−0.890577
$342$	18.1245	0.980059
$343$	−45.7959	−2.47274
$344$	−32.4750	−1.75094
$345$	−6.47228	−0.348456
$346$	−24.0095	−1.29076
$347$	−9.84756	−0.528645	−0.264322	−	0.964434i	$-0.585148\pi$
$347$	−9.84756	−0.528645	−0.264322	+	0.964434i	$0.585148\pi$
$348$	−98.6492	−5.28815
$349$	−27.9788	−1.49767	−0.748836	−	0.662755i	$-0.769387\pi$
$349$	−27.9788	−1.49767	−0.748836	+	0.662755i	$0.769387\pi$
$350$	−56.6417	−3.02763
$351$	−4.59762	−0.245403
$352$	1.78122	0.0949392
$353$	23.5344	1.25261	0.626306	−	0.779577i	$-0.284566\pi$
$353$	23.5344	1.25261	0.626306	+	0.779577i	$0.284566\pi$
$354$	−37.4922	−1.99269
$355$	7.21811	0.383098
$356$	6.62932	0.351353
$357$	−24.1872	−1.28012
$358$	−42.1740	−2.22896
$359$	−0.0937245	−0.00494659	−0.00247329	−	0.999997i	$-0.500787\pi$
$359$	−0.0937245	−0.00494659	−0.00247329	+	0.999997i	$0.500787\pi$
$360$	−8.80072	−0.463839
$361$	−14.5479	−0.765681
$362$	23.3868	1.22918
$363$	−3.27153	−0.171711
$364$	72.3444	3.79188
$365$	0.0351211	0.00183832
$366$	64.1279	3.35202
$367$	30.6013	1.59737	0.798687	−	0.601747i	$-0.205528\pi$
$367$	30.6013	1.59737	0.798687	+	0.601747i	$0.205528\pi$
$368$	−22.1734	−1.15587
$369$	21.2506	1.10626
$370$	−13.3123	−0.692074
$371$	−27.1033	−1.40713
$372$	54.5339	2.82745
$373$	−17.9035	−0.927009	−0.463504	−	0.886095i	$-0.653408\pi$
$373$	−17.9035	−0.927009	−0.463504	+	0.886095i	$0.653408\pi$
$374$	15.0387	0.777630
$375$	−12.3729	−0.638935
$376$	25.8861	1.33497
$377$	35.1387	1.80974
$378$	14.8849	0.765596
$379$	−15.5647	−0.799503	−0.399752	−	0.916624i	$-0.630904\pi$
$379$	−15.5647	−0.799503	−0.399752	+	0.916624i	$0.630904\pi$
$380$	−4.26345	−0.218710
$381$	33.0572	1.69357
$382$	11.5186	0.589345
$383$	6.51278	0.332788	0.166394	−	0.986059i	$-0.446788\pi$
$383$	6.51278	0.332788	0.166394	+	0.986059i	$0.446788\pi$
$384$	48.6001	2.48011
$385$	7.51825	0.383166
$386$	−64.1956	−3.26747
$387$	22.3861	1.13795
$388$	36.6398	1.86010
$389$	21.7179	1.10114	0.550571	−	0.834789i	$-0.314410\pi$
$389$	21.7179	1.10114	0.550571	+	0.834789i	$0.314410\pi$
$390$	11.4965	0.582149
$391$	−10.0002	−0.505733
$392$	−83.3182	−4.20820
$393$	−10.5058	−0.529946
$394$	−9.63477	−0.485393
$395$	1.59716	0.0803621
$396$	−44.1384	−2.21804
$397$	−9.37765	−0.470651	−0.235326	−	0.971917i	$-0.575616\pi$
$397$	−9.37765	−0.470651	−0.235326	+	0.971917i	$0.575616\pi$
$398$	37.2681	1.86808
$399$	26.0336	1.30331
$400$	−20.6551	−1.03276
$401$	−4.99848	−0.249612	−0.124806	−	0.992181i	$-0.539831\pi$
$401$	−4.99848	−0.249612	−0.124806	+	0.992181i	$0.539831\pi$
$402$	−25.2873	−1.26122
$403$	−19.4249	−0.967623
$404$	35.5181	1.76709
$405$	−3.63020	−0.180386
$406$	−113.762	−5.64593
$407$	−33.8536	−1.67806
$408$	−25.2860	−1.25184
$409$	−37.0960	−1.83428	−0.917138	−	0.398569i	$-0.869507\pi$
$409$	−37.0960	−1.83428	−0.917138	+	0.398569i	$0.869507\pi$
$410$	−7.46296	−0.368569
$411$	8.62708	0.425542
$412$	19.0446	0.938262
$413$	−28.9601	−1.42504
$414$	43.8190	2.15359
$415$	−1.92003	−0.0942507
$416$	2.10391	0.103153
$417$	14.8027	0.724892
$418$	−16.1867	−0.791716
$419$	34.8618	1.70311	0.851555	−	0.524266i	$-0.175660\pi$
$419$	34.8618	1.70311	0.851555	+	0.524266i	$0.175660\pi$
$420$	−24.9307	−1.21649
$421$	−32.8749	−1.60223	−0.801113	−	0.598513i	$-0.795758\pi$
$421$	−32.8749	−1.60223	−0.801113	+	0.598513i	$0.795758\pi$
$422$	68.6438	3.34153
$423$	−17.8441	−0.867609
$424$	−28.3345	−1.37604
$425$	−9.31549	−0.451868
$426$	−90.8736	−4.40284
$427$	49.5344	2.39714
$428$	57.6724	2.78770
$429$	29.2360	1.41153
$430$	−7.86173	−0.379126
$431$	−38.9186	−1.87464	−0.937320	−	0.348469i	$-0.886702\pi$
$431$	−38.9186	−1.87464	−0.937320	+	0.348469i	$0.886702\pi$
$432$	5.42797	0.261153
$433$	−5.96585	−0.286700	−0.143350	−	0.989672i	$-0.545788\pi$
$433$	−5.96585	−0.286700	−0.143350	+	0.989672i	$0.545788\pi$
$434$	62.8885	3.01875
$435$	−12.1092	−0.580592
$436$	−44.2695	−2.12012
$437$	10.7636	0.514894
$438$	−0.442163	−0.0211274
$439$	8.26299	0.394371	0.197185	−	0.980366i	$-0.436820\pi$
$439$	8.26299	0.394371	0.197185	+	0.980366i	$0.436820\pi$
$440$	7.85978	0.374701
$441$	57.4339	2.73495
$442$	17.7631	0.844905
$443$	2.38384	0.113260	0.0566299	−	0.998395i	$-0.481964\pi$
$443$	2.38384	0.113260	0.0566299	+	0.998395i	$0.481964\pi$
$444$	112.259	5.32760
$445$	0.813749	0.0385754
$446$	52.5509	2.48836
$447$	14.4193	0.682011
$448$	35.2912	1.66735
$449$	−20.6263	−0.973417	−0.486709	−	0.873564i	$-0.661803\pi$
$449$	−20.6263	−0.973417	−0.486709	+	0.873564i	$0.661803\pi$
$450$	40.8186	1.92421
$451$	−18.9785	−0.893665
$452$	11.8022	0.555127
$453$	12.4788	0.586306
$454$	3.05550	0.143402
$455$	8.88028	0.416314
$456$	27.2163	1.27452
$457$	−2.56153	−0.119823	−0.0599116	−	0.998204i	$-0.519082\pi$
$457$	−2.56153	−0.119823	−0.0599116	+	0.998204i	$0.519082\pi$
$458$	−60.5387	−2.82879
$459$	2.44802	0.114264
$460$	−10.3076	−0.480595
$461$	15.9974	0.745073	0.372536	−	0.928018i	$-0.378488\pi$
$461$	15.9974	0.745073	0.372536	+	0.928018i	$0.378488\pi$
$462$	−94.6522	−4.40362
$463$	14.4785	0.672873	0.336437	−	0.941706i	$-0.390778\pi$
$463$	14.4785	0.672873	0.336437	+	0.941706i	$0.390778\pi$
$464$	−41.4849	−1.92589
$465$	6.69404	0.310429
$466$	−43.4885	−2.01456
$467$	18.7824	0.869144	0.434572	−	0.900637i	$-0.356900\pi$
$467$	18.7824	0.869144	0.434572	+	0.900637i	$0.356900\pi$
$468$	−52.1347	−2.40993
$469$	−19.5327	−0.901938
$470$	6.26664	0.289059
$471$	−43.3705	−1.99841
$472$	−30.2757	−1.39355
$473$	−19.9926	−0.919262
$474$	−20.1078	−0.923580
$475$	10.0266	0.460053
$476$	−38.5201	−1.76556
$477$	19.5319	0.894303
$478$	−52.6666	−2.40891
$479$	14.0936	0.643955	0.321977	−	0.946747i	$-0.395653\pi$
$479$	14.0936	0.643955	0.321977	+	0.946747i	$0.395653\pi$
$480$	−0.725031	−0.0330930
$481$	−39.9866	−1.82323
$482$	−42.9366	−1.95571
$483$	62.9407	2.86390
$484$	−5.21018	−0.236826
$485$	4.49754	0.204223
$486$	54.9232	2.49137
$487$	−2.21478	−0.100361	−0.0501806	−	0.998740i	$-0.515980\pi$
$487$	−2.21478	−0.100361	−0.0501806	+	0.998740i	$0.515980\pi$
$488$	51.7846	2.34418
$489$	21.8018	0.985911
$490$	−20.1701	−0.911193
$491$	−38.2578	−1.72655	−0.863275	−	0.504733i	$-0.831591\pi$
$491$	−38.2578	−1.72655	−0.863275	+	0.504733i	$0.831591\pi$
$492$	62.9333	2.83725
$493$	−18.7097	−0.842645
$494$	−19.1191	−0.860209
$495$	−5.41800	−0.243521
$496$	22.9331	1.02973
$497$	−70.1937	−3.14862
$498$	24.1726	1.08320
$499$	29.4056	1.31638	0.658188	−	0.752853i	$-0.271323\pi$
$499$	29.4056	1.31638	0.658188	+	0.752853i	$0.271323\pi$
$500$	−19.7048	−0.881228
$501$	−22.3659	−0.999233
$502$	−62.9503	−2.80961
$503$	37.8702	1.68855	0.844275	−	0.535910i	$-0.180031\pi$
$503$	37.8702	1.68855	0.844275	+	0.535910i	$0.180031\pi$
$504$	85.5840	3.81222
$505$	4.35985	0.194011
$506$	−39.1340	−1.73972
$507$	1.41392	0.0627945
$508$	52.6462	2.33580
$509$	−24.2543	−1.07505	−0.537527	−	0.843247i	$-0.680641\pi$
$509$	−24.2543	−1.07505	−0.537527	+	0.843247i	$0.680641\pi$
$510$	−6.12137	−0.271059
$511$	−0.341541	−0.0151089
$512$	41.5314	1.83545
$513$	−2.63490	−0.116334
$514$	46.4518	2.04890
$515$	2.33773	0.103013
$516$	66.2960	2.91852
$517$	15.9363	0.700876
$518$	129.458	5.68804
$519$	24.8528	1.09092
$520$	9.28369	0.407117
$521$	−1.09893	−0.0481452	−0.0240726	−	0.999710i	$-0.507663\pi$
$521$	−1.09893	−0.0481452	−0.0240726	+	0.999710i	$0.507663\pi$
$522$	81.9824	3.58827
$523$	−40.2783	−1.76125	−0.880623	−	0.473818i	$-0.842875\pi$
$523$	−40.2783	−1.76125	−0.880623	+	0.473818i	$0.842875\pi$
$524$	−16.7313	−0.730909
$525$	58.6311	2.55887
$526$	53.8742	2.34903
$527$	10.3429	0.450542
$528$	−34.5162	−1.50212
$529$	3.02289	0.131430
$530$	−6.85937	−0.297952
$531$	20.8700	0.905682
$532$	41.4606	1.79755
$533$	−22.4168	−0.970977
$534$	−10.2448	−0.443337
$535$	7.07929	0.306064
$536$	−20.4201	−0.882012
$537$	43.6552	1.88386
$538$	62.9837	2.71542
$539$	−51.2933	−2.20936
$540$	2.52327	0.108584
$541$	−15.6569	−0.673143	−0.336572	−	0.941658i	$-0.609267\pi$
$541$	−15.6569	−0.673143	−0.336572	+	0.941658i	$0.609267\pi$
$542$	−42.3422	−1.81875
$543$	−24.2082	−1.03887
$544$	−1.12024	−0.0480297
$545$	−5.43408	−0.232770
$546$	−111.800	−4.78459
$547$	26.3164	1.12521	0.562604	−	0.826727i	$-0.309800\pi$
$547$	26.3164	1.12521	0.562604	+	0.826727i	$0.309800\pi$
$548$	13.7393	0.586914
$549$	−35.6968	−1.52350
$550$	−36.4545	−1.55442
$551$	20.1380	0.857908
$552$	65.8000	2.80063
$553$	−15.5319	−0.660483
$554$	63.1273	2.68202
$555$	13.7799	0.584922
$556$	23.5745	0.999781
$557$	−10.2631	−0.434860	−0.217430	−	0.976076i	$-0.569767\pi$
$557$	−10.2631	−0.434860	−0.217430	+	0.976076i	$0.569767\pi$
$558$	−45.3204	−1.91857
$559$	−23.6146	−0.998789
$560$	−10.4841	−0.443034
$561$	−15.5668	−0.657232
$562$	47.9124	2.02106
$563$	13.9941	0.589782	0.294891	−	0.955531i	$-0.404717\pi$
$563$	13.9941	0.589782	0.294891	+	0.955531i	$0.404717\pi$
$564$	−52.8451	−2.22518
$565$	1.44872	0.0609479
$566$	2.38446	0.100226
$567$	35.3024	1.48256
$568$	−73.3824	−3.07906
$569$	−4.63507	−0.194312	−0.0971561	−	0.995269i	$-0.530975\pi$
$569$	−4.63507	−0.194312	−0.0971561	+	0.995269i	$0.530975\pi$
$570$	6.58866	0.275969
$571$	−18.6791	−0.781696	−0.390848	−	0.920455i	$-0.627818\pi$
$571$	−18.6791	−0.781696	−0.390848	+	0.920455i	$0.627818\pi$
$572$	46.5607	1.94680
$573$	−11.9232	−0.498099
$574$	72.5747	3.02921
$575$	24.2411	1.01092
$576$	−25.4325	−1.05969
$577$	4.43894	0.184796	0.0923978	−	0.995722i	$-0.470547\pi$
$577$	4.43894	0.184796	0.0923978	+	0.995722i	$0.470547\pi$
$578$	32.3814	1.34689
$579$	66.4502	2.76158
$580$	−19.2849	−0.800760
$581$	18.6717	0.774631
$582$	−56.6225	−2.34708
$583$	−17.4436	−0.722440
$584$	−0.357056	−0.0147751
$585$	−6.39954	−0.264588
$586$	−29.9025	−1.23526
$587$	24.8269	1.02471	0.512357	−	0.858772i	$-0.328772\pi$
$587$	24.8269	1.02471	0.512357	+	0.858772i	$0.328772\pi$
$588$	170.090	7.01438
$589$	−11.1324	−0.458703
$590$	−7.32931	−0.301743
$591$	9.97316	0.410241
$592$	47.2084	1.94025
$593$	29.5417	1.21313	0.606566	−	0.795033i	$-0.292546\pi$
$593$	29.5417	1.21313	0.606566	+	0.795033i	$0.292546\pi$
$594$	9.57988	0.393067
$595$	−4.72834	−0.193843
$596$	22.9639	0.940640
$597$	−38.5771	−1.57885
$598$	−46.2237	−1.89023
$599$	20.8607	0.852345	0.426173	−	0.904642i	$-0.359862\pi$
$599$	20.8607	0.852345	0.426173	+	0.904642i	$0.359862\pi$
$600$	61.2945	2.50234
$601$	26.1219	1.06553	0.532767	−	0.846262i	$-0.321152\pi$
$601$	26.1219	1.06553	0.532767	+	0.846262i	$0.321152\pi$
$602$	76.4526	3.11598
$603$	14.0762	0.573227
$604$	19.8735	0.808641
$605$	−0.639550	−0.0260014
$606$	−54.8890	−2.22971
$607$	14.7912	0.600355	0.300178	−	0.953883i	$-0.402954\pi$
$607$	14.7912	0.600355	0.300178	+	0.953883i	$0.402954\pi$
$608$	1.20575	0.0488997
$609$	117.758	4.77179
$610$	12.5363	0.507580
$611$	18.8233	0.761511
$612$	27.7593	1.12210
$613$	0.585736	0.0236577	0.0118288	−	0.999930i	$-0.496235\pi$
$613$	0.585736	0.0236577	0.0118288	+	0.999930i	$0.496235\pi$
$614$	−0.773184	−0.0312032
$615$	7.72507	0.311505
$616$	−76.4337	−3.07960
$617$	−11.1345	−0.448258	−0.224129	−	0.974560i	$-0.571954\pi$
$617$	−11.1345	−0.448258	−0.224129	+	0.974560i	$0.571954\pi$
$618$	−29.4312	−1.18390
$619$	26.6873	1.07265	0.536326	−	0.844011i	$-0.319812\pi$
$619$	26.6873	1.07265	0.536326	+	0.844011i	$0.319812\pi$
$620$	10.6608	0.428148
$621$	−6.37031	−0.255632
$622$	3.71165	0.148824
$623$	−7.91343	−0.317045
$624$	−40.7692	−1.63207
$625$	21.3411	0.853643
$626$	−25.6104	−1.02360
$627$	16.7552	0.669137
$628$	−69.0710	−2.75623
$629$	21.2911	0.848930
$630$	20.7187	0.825451
$631$	3.13409	0.124766	0.0623831	−	0.998052i	$-0.480130\pi$
$631$	3.13409	0.124766	0.0623831	+	0.998052i	$0.480130\pi$
$632$	−16.2375	−0.645891
$633$	−71.0547	−2.82417
$634$	−53.5945	−2.12851
$635$	6.46233	0.256450
$636$	57.8434	2.29364
$637$	−60.5857	−2.40049
$638$	−73.2171	−2.89869
$639$	50.5848	2.00110
$640$	9.50079	0.375552
$641$	−24.7216	−0.976444	−0.488222	−	0.872719i	$-0.662354\pi$
$641$	−24.7216	−0.976444	−0.488222	+	0.872719i	$0.662354\pi$
$642$	−89.1259	−3.51752
$643$	29.5599	1.16573	0.582864	−	0.812570i	$-0.301932\pi$
$643$	29.5599	1.16573	0.582864	+	0.812570i	$0.301932\pi$
$644$	100.238	3.94993
$645$	8.13785	0.320427
$646$	10.1800	0.400529
$647$	35.7825	1.40675	0.703377	−	0.710817i	$-0.251675\pi$
$647$	35.7825	1.40675	0.703377	+	0.710817i	$0.251675\pi$
$648$	36.9061	1.44981
$649$	−18.6387	−0.731632
$650$	−43.0587	−1.68890
$651$	−65.0973	−2.55136
$652$	34.7211	1.35978
$653$	36.1882	1.41615	0.708076	−	0.706136i	$-0.249563\pi$
$653$	36.1882	1.41615	0.708076	+	0.706136i	$0.249563\pi$
$654$	68.4133	2.67517
$655$	−2.05377	−0.0802472
$656$	26.4653	1.03330
$657$	0.246130	0.00960246
$658$	−60.9410	−2.37573
$659$	30.5084	1.18844	0.594219	−	0.804304i	$-0.297461\pi$
$659$	30.5084	1.18844	0.594219	+	0.804304i	$0.297461\pi$
$660$	−16.0453	−0.624564
$661$	−8.13789	−0.316527	−0.158264	−	0.987397i	$-0.550590\pi$
$661$	−8.13789	−0.316527	−0.158264	+	0.987397i	$0.550590\pi$
$662$	−6.62144	−0.257350
$663$	−18.3870	−0.714091
$664$	19.5199	0.757518
$665$	5.08929	0.197354
$666$	−93.2932	−3.61504
$667$	48.6871	1.88517
$668$	−35.6194	−1.37816
$669$	−54.3966	−2.10309
$670$	−4.94340	−0.190980
$671$	31.8802	1.23072
$672$	7.05068	0.271986
$673$	−3.18796	−0.122887	−0.0614435	−	0.998111i	$-0.519570\pi$
$673$	−3.18796	−0.122887	−0.0614435	+	0.998111i	$0.519570\pi$
$674$	36.2540	1.39645
$675$	−5.93413	−0.228405
$676$	2.25178	0.0866071
$677$	48.7601	1.87400	0.937002	−	0.349324i	$-0.113589\pi$
$677$	48.7601	1.87400	0.937002	+	0.349324i	$0.113589\pi$
$678$	−18.2388	−0.700459
$679$	−43.7370	−1.67847
$680$	−4.94314	−0.189561
$681$	−3.16281	−0.121199
$682$	40.4749	1.54986
$683$	37.0621	1.41814	0.709071	−	0.705137i	$-0.249114\pi$
$683$	37.0621	1.41814	0.709071	+	0.705137i	$0.249114\pi$
$684$	−29.8784	−1.14243
$685$	1.68650	0.0644379
$686$	112.710	4.30330
$687$	62.6649	2.39082
$688$	27.8794	1.06289
$689$	−20.6037	−0.784940
$690$	15.9292	0.606415
$691$	−6.91861	−0.263196	−0.131598	−	0.991303i	$-0.542011\pi$
$691$	−6.91861	−0.263196	−0.131598	+	0.991303i	$0.542011\pi$
$692$	39.5800	1.50461
$693$	52.6882	2.00146
$694$	24.2363	0.919996
$695$	2.89377	0.109767
$696$	123.107	4.66637
$697$	11.9359	0.452104
$698$	68.8599	2.60639
$699$	45.0158	1.70266
$700$	93.3746	3.52923
$701$	16.9589	0.640527	0.320264	−	0.947328i	$-0.396229\pi$
$701$	16.9589	0.640527	0.320264	+	0.947328i	$0.396229\pi$
$702$	11.3154	0.427072
$703$	−22.9164	−0.864307
$704$	22.7133	0.856041
$705$	−6.48674	−0.244305
$706$	−57.9217	−2.17991
$707$	−42.3980	−1.59454
$708$	61.8063	2.32282
$709$	−11.5856	−0.435106	−0.217553	−	0.976048i	$-0.569808\pi$
$709$	−11.5856	−0.435106	−0.217553	+	0.976048i	$0.569808\pi$
$710$	−17.7648	−0.666702
$711$	11.1930	0.419770
$712$	−8.27292	−0.310041
$713$	−26.9145	−1.00796
$714$	59.5282	2.22779
$715$	5.71533	0.213741
$716$	69.5244	2.59825
$717$	54.5163	2.03595
$718$	0.230669	0.00860851
$719$	10.6469	0.397062	0.198531	−	0.980095i	$-0.436383\pi$
$719$	10.6469	0.397062	0.198531	+	0.980095i	$0.436383\pi$
$720$	7.55532	0.281570
$721$	−22.7336	−0.846645
$722$	35.8046	1.33251
$723$	44.4446	1.65291
$724$	−38.5535	−1.43283
$725$	45.3534	1.68438
$726$	8.05171	0.298827
$727$	−36.5695	−1.35629	−0.678144	−	0.734929i	$-0.737215\pi$
$727$	−36.5695	−1.35629	−0.678144	+	0.734929i	$0.737215\pi$
$728$	−90.2807	−3.34603
$729$	−34.9846	−1.29573
$730$	−0.0864382	−0.00319922
$731$	12.5737	0.465054
$732$	−105.716	−3.90736
$733$	−17.8068	−0.657709	−0.328855	−	0.944381i	$-0.606663\pi$
$733$	−17.8068	−0.657709	−0.328855	+	0.944381i	$0.606663\pi$
$734$	−75.3141	−2.77990
$735$	20.8785	0.770116
$736$	2.91511	0.107452
$737$	−12.5712	−0.463067
$738$	−52.3007	−1.92522
$739$	41.6790	1.53319	0.766594	−	0.642132i	$-0.221950\pi$
$739$	41.6790	1.53319	0.766594	+	0.642132i	$0.221950\pi$
$740$	21.9455	0.806733
$741$	19.7906	0.727026
$742$	66.7050	2.44882
$743$	−20.7703	−0.761987	−0.380994	−	0.924578i	$-0.624418\pi$
$743$	−20.7703	−0.761987	−0.380994	+	0.924578i	$0.624418\pi$
$744$	−68.0545	−2.49500
$745$	2.81883	0.103274
$746$	44.0631	1.61327
$747$	−13.4557	−0.492317
$748$	−24.7914	−0.906464
$749$	−68.8437	−2.51549
$750$	30.4515	1.11193
$751$	9.65977	0.352490	0.176245	−	0.984346i	$-0.443605\pi$
$751$	9.65977	0.352490	0.176245	+	0.984346i	$0.443605\pi$
$752$	−22.2229	−0.810386
$753$	65.1612	2.37461
$754$	−86.4814	−3.14947
$755$	2.43947	0.0887815
$756$	−24.5379	−0.892436
$757$	6.28118	0.228293	0.114147	−	0.993464i	$-0.463587\pi$
$757$	6.28118	0.228293	0.114147	+	0.993464i	$0.463587\pi$
$758$	38.3069	1.39137
$759$	40.5085	1.47036
$760$	5.32049	0.192994
$761$	6.56520	0.237988	0.118994	−	0.992895i	$-0.462033\pi$
$761$	6.56520	0.237988	0.118994	+	0.992895i	$0.462033\pi$
$762$	−81.3586	−2.94731
$763$	52.8446	1.91310
$764$	−18.9886	−0.686985
$765$	3.40746	0.123197
$766$	−16.0289	−0.579147
$767$	−22.0153	−0.794927
$768$	−82.4840	−2.97638
$769$	5.63828	0.203321	0.101661	−	0.994819i	$-0.467584\pi$
$769$	5.63828	0.203321	0.101661	+	0.994819i	$0.467584\pi$
$770$	−18.5035	−0.666820
$771$	−48.0833	−1.73168
$772$	105.827	3.80881
$773$	4.65494	0.167427	0.0837133	−	0.996490i	$-0.473322\pi$
$773$	4.65494	0.167427	0.0837133	+	0.996490i	$0.473322\pi$
$774$	−55.0953	−1.98036
$775$	−25.0716	−0.900600
$776$	−45.7239	−1.64139
$777$	−134.004	−4.80738
$778$	−53.4509	−1.91631
$779$	−12.8471	−0.460293
$780$	−18.9522	−0.678596
$781$	−45.1765	−1.61654
$782$	24.6120	0.880123
$783$	−11.9184	−0.425930
$784$	71.5277	2.55456
$785$	−8.47847	−0.302610
$786$	25.8562	0.922260
$787$	−40.5724	−1.44625	−0.723124	−	0.690718i	$-0.757295\pi$
$787$	−40.5724	−1.44625	−0.723124	+	0.690718i	$0.757295\pi$
$788$	15.8830	0.565810
$789$	−55.7664	−1.98533
$790$	−3.93085	−0.139853
$791$	−14.0883	−0.500921
$792$	55.0817	1.95724
$793$	37.6558	1.33720
$794$	23.0798	0.819070
$795$	7.10028	0.251821
$796$	−61.4370	−2.17758
$797$	44.0804	1.56141	0.780704	−	0.624901i	$-0.214861\pi$
$797$	44.0804	1.56141	0.780704	+	0.624901i	$0.214861\pi$
$798$	−64.0725	−2.26814
$799$	−10.0226	−0.354573
$800$	2.71551	0.0960077
$801$	5.70278	0.201498
$802$	12.3020	0.434398
$803$	−0.219815	−0.00775710
$804$	41.6865	1.47017
$805$	12.3042	0.433667
$806$	47.8075	1.68395
$807$	−65.1958	−2.29500
$808$	−44.3241	−1.55931
$809$	−30.2995	−1.06527	−0.532637	−	0.846344i	$-0.678799\pi$
$809$	−30.2995	−1.06527	−0.532637	+	0.846344i	$0.678799\pi$
$810$	8.93444	0.313924
$811$	14.6943	0.515986	0.257993	−	0.966147i	$-0.416939\pi$
$811$	14.6943	0.515986	0.257993	+	0.966147i	$0.416939\pi$
$812$	187.539	6.58132
$813$	43.8293	1.53716
$814$	83.3186	2.92032
$815$	4.26202	0.149292
$816$	21.7077	0.759923
$817$	−13.5335	−0.473478
$818$	91.2985	3.19218
$819$	62.2333	2.17461
$820$	12.3028	0.429632
$821$	6.23239	0.217512	0.108756	−	0.994068i	$-0.465313\pi$
$821$	6.23239	0.217512	0.108756	+	0.994068i	$0.465313\pi$
$822$	−21.2325	−0.740568
$823$	−40.2644	−1.40353	−0.701765	−	0.712409i	$-0.747604\pi$
$823$	−40.2644	−1.40353	−0.701765	+	0.712409i	$0.747604\pi$
$824$	−23.7664	−0.827940
$825$	37.7348	1.31376
$826$	71.2751	2.47998
$827$	23.4708	0.816159	0.408080	−	0.912946i	$-0.366199\pi$
$827$	23.4708	0.816159	0.408080	+	0.912946i	$0.366199\pi$
$828$	−72.2362	−2.51038
$829$	−43.1027	−1.49702	−0.748509	−	0.663124i	$-0.769230\pi$
$829$	−43.1027	−1.49702	−0.748509	+	0.663124i	$0.769230\pi$
$830$	4.72548	0.164024
$831$	−65.3445	−2.26677
$832$	26.8282	0.930099
$833$	32.2591	1.11771
$834$	−36.4316	−1.26152
$835$	−4.37229	−0.151309
$836$	26.6839	0.922884
$837$	6.58858	0.227735
$838$	−85.7998	−2.96391
$839$	−18.2383	−0.629657	−0.314828	−	0.949149i	$-0.601947\pi$
$839$	−18.2383	−0.629657	−0.314828	+	0.949149i	$0.601947\pi$
$840$	31.1118	1.07346
$841$	62.0902	2.14104
$842$	80.9099	2.78834
$843$	−49.5952	−1.70815
$844$	−113.160	−3.89514
$845$	0.276407	0.00950868
$846$	43.9169	1.50989
$847$	6.21940	0.213701
$848$	24.3249	0.835319
$849$	−2.46821	−0.0847086
$850$	22.9268	0.786382
$851$	−55.4042	−1.89923
$852$	149.806	5.13228
$853$	−6.43860	−0.220453	−0.110227	−	0.993906i	$-0.535158\pi$
$853$	−6.43860	−0.220453	−0.110227	+	0.993906i	$0.535158\pi$
$854$	−121.911	−4.17172
$855$	−3.66758	−0.125429
$856$	−71.9711	−2.45992
$857$	0.451812	0.0154336	0.00771680	−	0.999970i	$-0.497544\pi$
$857$	0.451812	0.0154336	0.00771680	+	0.999970i	$0.497544\pi$
$858$	−71.9540	−2.45647
$859$	−0.702645	−0.0239739	−0.0119870	−	0.999928i	$-0.503816\pi$
$859$	−0.702645	−0.0239739	−0.0119870	+	0.999928i	$0.503816\pi$
$860$	12.9602	0.441938
$861$	−75.1237	−2.56021
$862$	95.7842	3.26242
$863$	3.86126	0.131439	0.0657193	−	0.997838i	$-0.479066\pi$
$863$	3.86126	0.131439	0.0657193	+	0.997838i	$0.479066\pi$
$864$	−0.713609	−0.0242775
$865$	4.85845	0.165192
$866$	14.6828	0.498942
$867$	−33.5187	−1.13835
$868$	−103.673	−3.51888
$869$	−9.99628	−0.339100
$870$	29.8025	1.01040
$871$	−14.8487	−0.503128
$872$	55.2452	1.87084
$873$	31.5189	1.06675
$874$	−26.4908	−0.896065
$875$	23.5217	0.795180
$876$	0.728912	0.0246277
$877$	19.6597	0.663861	0.331930	−	0.943304i	$-0.392300\pi$
$877$	19.6597	0.663861	0.331930	+	0.943304i	$0.392300\pi$
$878$	−20.3364	−0.686320
$879$	30.9527	1.04401
$880$	−6.74754	−0.227459
$881$	22.1688	0.746885	0.373442	−	0.927653i	$-0.378177\pi$
$881$	22.1688	0.746885	0.373442	+	0.927653i	$0.378177\pi$
$882$	−141.353	−4.75960
$883$	30.1890	1.01594	0.507970	−	0.861375i	$-0.330396\pi$
$883$	30.1890	1.01594	0.507970	+	0.861375i	$0.330396\pi$
$884$	−29.2827	−0.984884
$885$	7.58673	0.255025
$886$	−5.86698	−0.197105
$887$	1.25332	0.0420822	0.0210411	−	0.999779i	$-0.493302\pi$
$887$	1.25332	0.0420822	0.0210411	+	0.999779i	$0.493302\pi$
$888$	−140.092	−4.70118
$889$	−62.8440	−2.10772
$890$	−2.00275	−0.0671324
$891$	22.7206	0.761167
$892$	−86.6309	−2.90062
$893$	10.7877	0.360995
$894$	−35.4881	−1.18690
$895$	8.53412	0.285264
$896$	−92.3920	−3.08660
$897$	47.8471	1.59757
$898$	50.7644	1.69403
$899$	−50.3553	−1.67944
$900$	−67.2901	−2.24300
$901$	10.9705	0.365482
$902$	46.7089	1.55524
$903$	−79.1378	−2.63354
$904$	−14.7283	−0.489855
$905$	−4.73245	−0.157312
$906$	−30.7121	−1.02034
$907$	11.7446	0.389973	0.194986	−	0.980806i	$-0.437534\pi$
$907$	11.7446	0.389973	0.194986	+	0.980806i	$0.437534\pi$
$908$	−5.03703	−0.167160
$909$	30.5540	1.01341
$910$	−21.8556	−0.724508
$911$	−57.3044	−1.89858	−0.949290	−	0.314400i	$-0.898197\pi$
$911$	−57.3044	−1.89858	−0.949290	+	0.314400i	$0.898197\pi$
$912$	−23.3649	−0.773688
$913$	12.0170	0.397706
$914$	6.30429	0.208527
$915$	−12.9766	−0.428993
$916$	99.7988	3.29745
$917$	19.9722	0.659539
$918$	−6.02493	−0.198852
$919$	37.2022	1.22719	0.613594	−	0.789621i	$-0.289723\pi$
$919$	37.2022	1.22719	0.613594	+	0.789621i	$0.289723\pi$
$920$	12.8632	0.424087
$921$	0.800339	0.0263721
$922$	−39.3719	−1.29664
$923$	−53.3608	−1.75639
$924$	156.035	5.13319
$925$	−51.6106	−1.69695
$926$	−35.6337	−1.17100
$927$	16.3829	0.538085
$928$	5.45397	0.179035
$929$	5.10800	0.167588	0.0837941	−	0.996483i	$-0.473296\pi$
$929$	5.10800	0.167588	0.0837941	+	0.996483i	$0.473296\pi$
$930$	−16.4750	−0.540237
$931$	−34.7217	−1.13796
$932$	71.6913	2.34833
$933$	−3.84201	−0.125782
$934$	−46.2261	−1.51256
$935$	−3.04315	−0.0995216
$936$	65.0604	2.12657
$937$	7.37990	0.241091	0.120545	−	0.992708i	$-0.461536\pi$
$937$	7.37990	0.241091	0.120545	+	0.992708i	$0.461536\pi$
$938$	48.0729	1.56963
$939$	26.5099	0.865117
$940$	−10.3306	−0.336948
$941$	−20.6212	−0.672231	−0.336115	−	0.941821i	$-0.609113\pi$
$941$	−20.6212	−0.672231	−0.336115	+	0.941821i	$0.609113\pi$
$942$	106.741	3.47781
$943$	−31.0599	−1.01145
$944$	25.9914	0.845947
$945$	−3.01203	−0.0979815
$946$	49.2047	1.59978
$947$	−20.2395	−0.657696	−0.328848	−	0.944383i	$-0.606660\pi$
$947$	−20.2395	−0.657696	−0.328848	+	0.944383i	$0.606660\pi$
$948$	33.1479	1.07659
$949$	−0.259637	−0.00842818
$950$	−24.6769	−0.800626
$951$	55.4768	1.79896
$952$	48.0703	1.55797
$953$	28.5283	0.924122	0.462061	−	0.886848i	$-0.347110\pi$
$953$	28.5283	0.924122	0.462061	+	0.886848i	$0.347110\pi$
$954$	−48.0707	−1.55635
$955$	−2.33086	−0.0754248
$956$	86.8216	2.80801
$957$	75.7886	2.44990
$958$	−34.6865	−1.12067
$959$	−16.4006	−0.529605
$960$	−9.24529	−0.298391
$961$	−3.16329	−0.102042
$962$	98.4129	3.17296
$963$	49.6119	1.59872
$964$	70.7816	2.27972
$965$	12.9903	0.418173
$966$	−154.906	−4.98402
$967$	34.5664	1.11158	0.555790	−	0.831323i	$-0.312416\pi$
$967$	34.5664	1.11158	0.555790	+	0.831323i	$0.312416\pi$
$968$	6.50193	0.208980
$969$	−10.5376	−0.338516
$970$	−11.0691	−0.355407
$971$	43.3283	1.39047	0.695236	−	0.718781i	$-0.255300\pi$
$971$	43.3283	1.39047	0.695236	+	0.718781i	$0.255300\pi$
$972$	−90.5416	−2.90412
$973$	−28.1409	−0.902157
$974$	5.45088	0.174658
$975$	44.5710	1.42741
$976$	−44.4565	−1.42302
$977$	−27.3148	−0.873879	−0.436940	−	0.899491i	$-0.643938\pi$
$977$	−27.3148	−0.873879	−0.436940	+	0.899491i	$0.643938\pi$
$978$	−53.6573	−1.71577
$979$	−5.09306	−0.162775
$980$	33.2507	1.06216
$981$	−38.0823	−1.21587
$982$	94.1580	3.00470
$983$	21.7810	0.694706	0.347353	−	0.937734i	$-0.387080\pi$
$983$	21.7810	0.694706	0.347353	+	0.937734i	$0.387080\pi$
$984$	−78.5363	−2.50365
$985$	1.94964	0.0621209
$986$	46.0474	1.46645
$987$	63.0813	2.00790
$988$	31.5181	1.00272
$989$	−32.7196	−1.04042
$990$	13.3345	0.423797
$991$	−14.0850	−0.447423	−0.223712	−	0.974655i	$-0.571817\pi$
$991$	−14.0850	−0.447423	−0.223712	+	0.974655i	$0.571817\pi$
$992$	−3.01499	−0.0957261
$993$	6.85399	0.217505
$994$	172.757	5.47951
$995$	−7.54140	−0.239078
$996$	−39.8488	−1.26266
$997$	−32.1257	−1.01743	−0.508716	−	0.860934i	$-0.669880\pi$
$997$	−32.1257	−1.01743	−0.508716	+	0.860934i	$0.669880\pi$
$998$	−72.3715	−2.29088
$999$	13.5628	0.429107

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5077.2.a.c.1.14	✓	216

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5077.2.a.c.1.14	✓	216	1.1	even	1	trivial

Introduction	Features
Universe	News

Level:	\( N \)	\(=\)	\( 5077 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5077.a (trivial)

\(f(q)\)	\(=\)	\(q-2.46114 q^{2} +2.54758 q^{3} +4.05723 q^{4} +0.498025 q^{5} -6.26997 q^{6} -4.84312 q^{7} -5.06313 q^{8} +3.49018 q^{9} +O(q^{10})\)	\(q-2.46114 q^{2} +2.54758 q^{3} +4.05723 q^{4} +0.498025 q^{5} -6.26997 q^{6} -4.84312 q^{7} -5.06313 q^{8} +3.49018 q^{9} -1.22571 q^{10} -3.11702 q^{11} +10.3361 q^{12} -3.68171 q^{13} +11.9196 q^{14} +1.26876 q^{15} +4.34664 q^{16} +1.96034 q^{17} -8.58983 q^{18} -2.10999 q^{19} +2.02060 q^{20} -12.3383 q^{21} +7.67144 q^{22} -5.10126 q^{23} -12.8988 q^{24} -4.75197 q^{25} +9.06122 q^{26} +1.24877 q^{27} -19.6497 q^{28} -9.54412 q^{29} -3.12260 q^{30} +5.27605 q^{31} -0.571449 q^{32} -7.94087 q^{33} -4.82469 q^{34} -2.41200 q^{35} +14.1605 q^{36} +10.8609 q^{37} +5.19299 q^{38} -9.37947 q^{39} -2.52157 q^{40} +6.08868 q^{41} +30.3662 q^{42} +6.41401 q^{43} -12.6465 q^{44} +1.73820 q^{45} +12.5549 q^{46} -5.11266 q^{47} +11.0734 q^{48} +16.4559 q^{49} +11.6953 q^{50} +4.99414 q^{51} -14.9375 q^{52} +5.59624 q^{53} -3.07341 q^{54} -1.55236 q^{55} +24.5214 q^{56} -5.37538 q^{57} +23.4894 q^{58} +5.97964 q^{59} +5.14765 q^{60} -10.2278 q^{61} -12.9851 q^{62} -16.9034 q^{63} -7.28687 q^{64} -1.83359 q^{65} +19.5436 q^{66} +4.03308 q^{67} +7.95356 q^{68} -12.9959 q^{69} +5.93627 q^{70} +14.4935 q^{71} -17.6712 q^{72} +0.0705208 q^{73} -26.7302 q^{74} -12.1060 q^{75} -8.56072 q^{76} +15.0961 q^{77} +23.0842 q^{78} +3.20700 q^{79} +2.16474 q^{80} -7.28919 q^{81} -14.9851 q^{82} -3.85529 q^{83} -50.0591 q^{84} +0.976300 q^{85} -15.7858 q^{86} -24.3144 q^{87} +15.7819 q^{88} +1.63395 q^{89} -4.27795 q^{90} +17.8310 q^{91} -20.6970 q^{92} +13.4412 q^{93} +12.5830 q^{94} -1.05083 q^{95} -1.45581 q^{96} +9.03075 q^{97} -40.5002 q^{98} -10.8790 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 216q + 25q^{2} + 62q^{3} + 223q^{4} + 46q^{5} + 26q^{6} + 30q^{7} + 75q^{8} + 234q^{9} + O(q^{10}) \)	\( 216q + 25q^{2} + 62q^{3} + 223q^{4} + 46q^{5} + 26q^{6} + 30q^{7} + 75q^{8} + 234q^{9} + 24q^{10} + 89q^{11} + 114q^{12} + 34q^{13} + 53q^{14} + 61q^{15} + 229q^{16} + 76q^{17} + 57q^{18} + 54q^{19} + 118q^{20} + 25q^{21} + 26q^{22} + 109q^{23} + 65q^{24} + 232q^{25} + 58q^{26} + 236q^{27} + 57q^{28} + 54q^{29} + 6q^{30} + 77q^{31} + 155q^{32} + 80q^{33} + 28q^{34} + 137q^{35} + 257q^{36} + 42q^{37} + 104q^{38} + 46q^{39} + 47q^{40} + 109q^{41} + 27q^{42} + 68q^{43} + 145q^{44} + 109q^{45} - 7q^{46} + 264q^{47} + 198q^{48} + 222q^{49} + 86q^{50} + 57q^{51} + 68q^{52} + 95q^{53} + 79q^{54} + 50q^{55} + 108q^{56} + 55q^{57} + 38q^{58} + 292q^{59} + 91q^{60} + 16q^{61} + 91q^{62} + 113q^{63} + 231q^{64} + 68q^{65} - 15q^{66} + 152q^{67} + 199q^{68} + 83q^{69} + 24q^{70} + 131q^{71} + 162q^{72} + 71q^{73} + 10q^{74} + 232q^{75} + 60q^{76} + 131q^{77} + 102q^{78} + 10q^{79} + 236q^{80} + 268q^{81} + 54q^{82} + 299q^{83} - 9q^{85} + 35q^{86} + 103q^{87} + 45q^{88} + 134q^{89} + 8q^{90} + 79q^{91} + 206q^{92} + 95q^{93} + 18q^{94} + 119q^{95} + 77q^{96} + 129q^{97} + 150q^{98} + 221q^{99} + O(q^{100}) \)

Introduction and more

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Knowledge

Properties

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