LMFDB - Embedded Newform 4030.2.a.n.1.7

Newspace parameters

Level:	$ N $	=	$ 4030 = 2 \cdot 5 \cdot 13 \cdot 31 $
Weight:	$ k $	=	$ 2 $
Character orbit:	$[\chi]$	=	4030.a (trivial)

Newform invariants

Self dual:	Yes
Analytic conductor:	$32.1797120146$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-2.63528$
Character	$\chi$	=	4030.1

$q$-expansion

$f(q)$	$=$	$q+1.00000 q^{2} +2.63528 q^{3} +1.00000 q^{4} -1.00000 q^{5} +2.63528 q^{6} +0.203142 q^{7} +1.00000 q^{8} +3.94469 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +2.63528 q^{3} +1.00000 q^{4} -1.00000 q^{5} +2.63528 q^{6} +0.203142 q^{7} +1.00000 q^{8} +3.94469 q^{9} -1.00000 q^{10} +4.83652 q^{11} +2.63528 q^{12} -1.00000 q^{13} +0.203142 q^{14} -2.63528 q^{15} +1.00000 q^{16} -0.0683421 q^{17} +3.94469 q^{18} +3.00496 q^{19} -1.00000 q^{20} +0.535335 q^{21} +4.83652 q^{22} +4.98419 q^{23} +2.63528 q^{24} +1.00000 q^{25} -1.00000 q^{26} +2.48951 q^{27} +0.203142 q^{28} -4.67817 q^{29} -2.63528 q^{30} -1.00000 q^{31} +1.00000 q^{32} +12.7456 q^{33} -0.0683421 q^{34} -0.203142 q^{35} +3.94469 q^{36} +6.73026 q^{37} +3.00496 q^{38} -2.63528 q^{39} -1.00000 q^{40} -11.8907 q^{41} +0.535335 q^{42} -0.517514 q^{43} +4.83652 q^{44} -3.94469 q^{45} +4.98419 q^{46} +2.49431 q^{47} +2.63528 q^{48} -6.95873 q^{49} +1.00000 q^{50} -0.180100 q^{51} -1.00000 q^{52} +7.40140 q^{53} +2.48951 q^{54} -4.83652 q^{55} +0.203142 q^{56} +7.91891 q^{57} -4.67817 q^{58} -9.25770 q^{59} -2.63528 q^{60} +8.56694 q^{61} -1.00000 q^{62} +0.801330 q^{63} +1.00000 q^{64} +1.00000 q^{65} +12.7456 q^{66} -5.12785 q^{67} -0.0683421 q^{68} +13.1347 q^{69} -0.203142 q^{70} +1.20157 q^{71} +3.94469 q^{72} -2.64613 q^{73} +6.73026 q^{74} +2.63528 q^{75} +3.00496 q^{76} +0.982499 q^{77} -2.63528 q^{78} +7.88727 q^{79} -1.00000 q^{80} -5.27351 q^{81} -11.8907 q^{82} +14.3936 q^{83} +0.535335 q^{84} +0.0683421 q^{85} -0.517514 q^{86} -12.3283 q^{87} +4.83652 q^{88} +13.1568 q^{89} -3.94469 q^{90} -0.203142 q^{91} +4.98419 q^{92} -2.63528 q^{93} +2.49431 q^{94} -3.00496 q^{95} +2.63528 q^{96} -1.83994 q^{97} -6.95873 q^{98} +19.0786 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8q + 8q^{2} - q^{3} + 8q^{4} - 8q^{5} - q^{6} + q^{7} + 8q^{8} + 9q^{9} + O(q^{10}) $	$ 8q + 8q^{2} - q^{3} + 8q^{4} - 8q^{5} - q^{6} + q^{7} + 8q^{8} + 9q^{9} - 8q^{10} + 4q^{11} - q^{12} - 8q^{13} + q^{14} + q^{15} + 8q^{16} - 5q^{17} + 9q^{18} + 2q^{19} - 8q^{20} + 17q^{21} + 4q^{22} + 4q^{23} - q^{24} + 8q^{25} - 8q^{26} + 11q^{27} + q^{28} + 11q^{29} + q^{30} - 8q^{31} + 8q^{32} + 10q^{33} - 5q^{34} - q^{35} + 9q^{36} + 19q^{37} + 2q^{38} + q^{39} - 8q^{40} + 10q^{41} + 17q^{42} + 19q^{43} + 4q^{44} - 9q^{45} + 4q^{46} + 11q^{47} - q^{48} + 11q^{49} + 8q^{50} + 7q^{51} - 8q^{52} + 8q^{53} + 11q^{54} - 4q^{55} + q^{56} - 11q^{57} + 11q^{58} + 28q^{59} + q^{60} - 12q^{61} - 8q^{62} + 20q^{63} + 8q^{64} + 8q^{65} + 10q^{66} + 24q^{67} - 5q^{68} + 30q^{69} - q^{70} + 18q^{71} + 9q^{72} - 3q^{73} + 19q^{74} - q^{75} + 2q^{76} - 7q^{77} + q^{78} + 22q^{79} - 8q^{80} + 24q^{81} + 10q^{82} + 17q^{83} + 17q^{84} + 5q^{85} + 19q^{86} + 11q^{87} + 4q^{88} + 17q^{89} - 9q^{90} - q^{91} + 4q^{92} + q^{93} + 11q^{94} - 2q^{95} - q^{96} - 24q^{97} + 11q^{98} + 23q^{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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	2.63528	1.52148	0.760739	−	0.649058i	$-0.224837\pi$
$3$	2.63528	1.52148	0.760739	+	0.649058i	$0.224837\pi$
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	2.63528	1.07585
$7$	0.203142	0.0767803	0.0383902	−	0.999263i	$-0.487777\pi$
$7$	0.203142	0.0767803	0.0383902	+	0.999263i	$0.487777\pi$
$8$	1.00000	0.353553
$9$	3.94469	1.31490
$10$	−1.00000	−0.316228
$11$	4.83652	1.45827	0.729133	−	0.684372i	$-0.239923\pi$
$11$	4.83652	1.45827	0.729133	+	0.684372i	$0.239923\pi$
$12$	2.63528	0.760739
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0.203142	0.0542919
$15$	−2.63528	−0.680426
$16$	1.00000	0.250000
$17$	−0.0683421	−0.0165754	−0.00828770	−	0.999966i	$-0.502638\pi$
$17$	−0.0683421	−0.0165754	−0.00828770	+	0.999966i	$0.502638\pi$
$18$	3.94469	0.929771
$19$	3.00496	0.689386	0.344693	−	0.938715i	$-0.387983\pi$
$19$	3.00496	0.689386	0.344693	+	0.938715i	$0.387983\pi$
$20$	−1.00000	−0.223607
$21$	0.535335	0.116820
$22$	4.83652	1.03115
$23$	4.98419	1.03927	0.519637	−	0.854387i	$-0.326067\pi$
$23$	4.98419	1.03927	0.519637	+	0.854387i	$0.326067\pi$
$24$	2.63528	0.537924
$25$	1.00000	0.200000
$26$	−1.00000	−0.196116
$27$	2.48951	0.479106
$28$	0.203142	0.0383902
$29$	−4.67817	−0.868714	−0.434357	−	0.900741i	$-0.643024\pi$
$29$	−4.67817	−0.868714	−0.434357	+	0.900741i	$0.643024\pi$
$30$	−2.63528	−0.481134
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	1.00000	0.176777
$33$	12.7456	2.21872
$34$	−0.0683421	−0.0117206
$35$	−0.203142	−0.0343372
$36$	3.94469	0.657448
$37$	6.73026	1.10645	0.553224	−	0.833033i	$-0.313398\pi$
$37$	6.73026	1.10645	0.553224	+	0.833033i	$0.313398\pi$
$38$	3.00496	0.487470
$39$	−2.63528	−0.421982
$40$	−1.00000	−0.158114
$41$	−11.8907	−1.85701	−0.928505	−	0.371320i	$-0.878905\pi$
$41$	−11.8907	−1.85701	−0.928505	+	0.371320i	$0.878905\pi$
$42$	0.535335	0.0826039
$43$	−0.517514	−0.0789202	−0.0394601	−	0.999221i	$-0.512564\pi$
$43$	−0.517514	−0.0789202	−0.0394601	+	0.999221i	$0.512564\pi$
$44$	4.83652	0.729133
$45$	−3.94469	−0.588039
$46$	4.98419	0.734878
$47$	2.49431	0.363832	0.181916	−	0.983314i	$-0.441770\pi$
$47$	2.49431	0.363832	0.181916	+	0.983314i	$0.441770\pi$
$48$	2.63528	0.380369
$49$	−6.95873	−0.994105
$50$	1.00000	0.141421
$51$	−0.180100	−0.0252191
$52$	−1.00000	−0.138675
$53$	7.40140	1.01666	0.508330	−	0.861162i	$-0.330263\pi$
$53$	7.40140	1.01666	0.508330	+	0.861162i	$0.330263\pi$
$54$	2.48951	0.338779
$55$	−4.83652	−0.652157
$56$	0.203142	0.0271459
$57$	7.91891	1.04889
$58$	−4.67817	−0.614273
$59$	−9.25770	−1.20525	−0.602625	−	0.798025i	$-0.705878\pi$
$59$	−9.25770	−1.20525	−0.602625	+	0.798025i	$0.705878\pi$
$60$	−2.63528	−0.340213
$61$	8.56694	1.09688	0.548442	−	0.836188i	$-0.315221\pi$
$61$	8.56694	1.09688	0.548442	+	0.836188i	$0.315221\pi$
$62$	−1.00000	−0.127000
$63$	0.801330	0.100958
$64$	1.00000	0.125000
$65$	1.00000	0.124035
$66$	12.7456	1.56887
$67$	−5.12785	−0.626467	−0.313234	−	0.949676i	$-0.601412\pi$
$67$	−5.12785	−0.626467	−0.313234	+	0.949676i	$0.601412\pi$
$68$	−0.0683421	−0.00828770
$69$	13.1347	1.58123
$70$	−0.203142	−0.0242801
$71$	1.20157	0.142600	0.0712998	−	0.997455i	$-0.477285\pi$
$71$	1.20157	0.142600	0.0712998	+	0.997455i	$0.477285\pi$
$72$	3.94469	0.464886
$73$	−2.64613	−0.309707	−0.154853	−	0.987937i	$-0.549490\pi$
$73$	−2.64613	−0.309707	−0.154853	+	0.987937i	$0.549490\pi$
$74$	6.73026	0.782376
$75$	2.63528	0.304296
$76$	3.00496	0.344693
$77$	0.982499	0.111966
$78$	−2.63528	−0.298386
$79$	7.88727	0.887387	0.443694	−	0.896178i	$-0.353668\pi$
$79$	7.88727	0.887387	0.443694	+	0.896178i	$0.353668\pi$
$80$	−1.00000	−0.111803
$81$	−5.27351	−0.585946
$82$	−11.8907	−1.31310
$83$	14.3936	1.57990	0.789952	−	0.613169i	$-0.210105\pi$
$83$	14.3936	1.57990	0.789952	+	0.613169i	$0.210105\pi$
$84$	0.535335	0.0584098
$85$	0.0683421	0.00741275
$86$	−0.517514	−0.0558050
$87$	−12.3283	−1.32173
$88$	4.83652	0.515575
$89$	13.1568	1.39461	0.697307	−	0.716773i	$-0.254382\pi$
$89$	13.1568	1.39461	0.697307	+	0.716773i	$0.254382\pi$
$90$	−3.94469	−0.415806
$91$	−0.203142	−0.0212950
$92$	4.98419	0.519637
$93$	−2.63528	−0.273266
$94$	2.49431	0.257268
$95$	−3.00496	−0.308303
$96$	2.63528	0.268962
$97$	−1.83994	−0.186817	−0.0934086	−	0.995628i	$-0.529776\pi$
$97$	−1.83994	−0.186817	−0.0934086	+	0.995628i	$0.529776\pi$
$98$	−6.95873	−0.702938
$99$	19.0786	1.91747
$100$	1.00000	0.100000
$101$	2.38178	0.236996	0.118498	−	0.992954i	$-0.462192\pi$
$101$	2.38178	0.236996	0.118498	+	0.992954i	$0.462192\pi$
$102$	−0.180100	−0.0178326
$103$	−6.44296	−0.634844	−0.317422	−	0.948284i	$-0.602817\pi$
$103$	−6.44296	−0.634844	−0.317422	+	0.948284i	$0.602817\pi$
$104$	−1.00000	−0.0980581
$105$	−0.535335	−0.0522433
$106$	7.40140	0.718887
$107$	−4.19546	−0.405591	−0.202795	−	0.979221i	$-0.565003\pi$
$107$	−4.19546	−0.405591	−0.202795	+	0.979221i	$0.565003\pi$
$108$	2.48951	0.239553
$109$	−7.15776	−0.685589	−0.342794	−	0.939410i	$-0.611373\pi$
$109$	−7.15776	−0.685589	−0.342794	+	0.939410i	$0.611373\pi$
$110$	−4.83652	−0.461144
$111$	17.7361	1.68344
$112$	0.203142	0.0191951
$113$	0.127104	0.0119569	0.00597845	−	0.999982i	$-0.498097\pi$
$113$	0.127104	0.0119569	0.00597845	+	0.999982i	$0.498097\pi$
$114$	7.91891	0.741674
$115$	−4.98419	−0.464778
$116$	−4.67817	−0.434357
$117$	−3.94469	−0.364686
$118$	−9.25770	−0.852240
$119$	−0.0138831	−0.00127267
$120$	−2.63528	−0.240567
$121$	12.3920	1.12654
$122$	8.56694	0.775615
$123$	−31.3352	−2.82540
$124$	−1.00000	−0.0898027
$125$	−1.00000	−0.0894427
$126$	0.801330	0.0713882
$127$	6.91730	0.613811	0.306905	−	0.951740i	$-0.400706\pi$
$127$	6.91730	0.613811	0.306905	+	0.951740i	$0.400706\pi$
$128$	1.00000	0.0883883
$129$	−1.36379	−0.120075
$130$	1.00000	0.0877058
$131$	−12.1974	−1.06569	−0.532844	−	0.846213i	$-0.678877\pi$
$131$	−12.1974	−1.06569	−0.532844	+	0.846213i	$0.678877\pi$
$132$	12.7456	1.10936
$133$	0.610433	0.0529313
$134$	−5.12785	−0.442979
$135$	−2.48951	−0.214263
$136$	−0.0683421	−0.00586029
$137$	4.52125	0.386277	0.193138	−	0.981172i	$-0.438133\pi$
$137$	4.52125	0.386277	0.193138	+	0.981172i	$0.438133\pi$
$138$	13.1347	1.11810
$139$	−10.2832	−0.872206	−0.436103	−	0.899897i	$-0.643642\pi$
$139$	−10.2832	−0.872206	−0.436103	+	0.899897i	$0.643642\pi$
$140$	−0.203142	−0.0171686
$141$	6.57320	0.553563
$142$	1.20157	0.100833
$143$	−4.83652	−0.404450
$144$	3.94469	0.328724
$145$	4.67817	0.388501
$146$	−2.64613	−0.218996
$147$	−18.3382	−1.51251
$148$	6.73026	0.553224
$149$	19.4531	1.59366	0.796829	−	0.604204i	$-0.206509\pi$
$149$	19.4531	1.59366	0.796829	+	0.604204i	$0.206509\pi$
$150$	2.63528	0.215169
$151$	12.0592	0.981364	0.490682	−	0.871339i	$-0.336748\pi$
$151$	12.0592	0.981364	0.490682	+	0.871339i	$0.336748\pi$
$152$	3.00496	0.243735
$153$	−0.269588	−0.0217949
$154$	0.982499	0.0791721
$155$	1.00000	0.0803219
$156$	−2.63528	−0.210991
$157$	−19.6491	−1.56817	−0.784084	−	0.620655i	$-0.786867\pi$
$157$	−19.6491	−1.56817	−0.784084	+	0.620655i	$0.786867\pi$
$158$	7.88727	0.627478
$159$	19.5047	1.54683
$160$	−1.00000	−0.0790569
$161$	1.01250	0.0797959
$162$	−5.27351	−0.414326
$163$	−8.98152	−0.703487	−0.351743	−	0.936096i	$-0.614411\pi$
$163$	−8.98152	−0.703487	−0.351743	+	0.936096i	$0.614411\pi$
$164$	−11.8907	−0.928505
$165$	−12.7456	−0.992242
$166$	14.3936	1.11716
$167$	−13.4277	−1.03906	−0.519532	−	0.854451i	$-0.673894\pi$
$167$	−13.4277	−1.03906	−0.519532	+	0.854451i	$0.673894\pi$
$168$	0.535335	0.0413020
$169$	1.00000	0.0769231
$170$	0.0683421	0.00524160
$171$	11.8536	0.906470
$172$	−0.517514	−0.0394601
$173$	15.0496	1.14420	0.572100	−	0.820184i	$-0.306129\pi$
$173$	15.0496	1.14420	0.572100	+	0.820184i	$0.306129\pi$
$174$	−12.3283	−0.934603
$175$	0.203142	0.0153561
$176$	4.83652	0.364567
$177$	−24.3966	−1.83376
$178$	13.1568	0.986140
$179$	−10.0195	−0.748891	−0.374445	−	0.927249i	$-0.622167\pi$
$179$	−10.0195	−0.748891	−0.374445	+	0.927249i	$0.622167\pi$
$180$	−3.94469	−0.294020
$181$	8.45118	0.628171	0.314086	−	0.949395i	$-0.398302\pi$
$181$	8.45118	0.628171	0.314086	+	0.949395i	$0.398302\pi$
$182$	−0.203142	−0.0150579
$183$	22.5763	1.66889
$184$	4.98419	0.367439
$185$	−6.73026	−0.494818
$186$	−2.63528	−0.193228
$187$	−0.330538	−0.0241714
$188$	2.49431	0.181916
$189$	0.505723	0.0367859
$190$	−3.00496	−0.218003
$191$	0.282464	0.0204384	0.0102192	−	0.999948i	$-0.496747\pi$
$191$	0.282464	0.0204384	0.0102192	+	0.999948i	$0.496747\pi$
$192$	2.63528	0.190185
$193$	3.11170	0.223985	0.111992	−	0.993709i	$-0.464277\pi$
$193$	3.11170	0.223985	0.111992	+	0.993709i	$0.464277\pi$
$194$	−1.83994	−0.132100
$195$	2.63528	0.188716
$196$	−6.95873	−0.497052
$197$	−0.893060	−0.0636279	−0.0318140	−	0.999494i	$-0.510128\pi$
$197$	−0.893060	−0.0636279	−0.0318140	+	0.999494i	$0.510128\pi$
$198$	19.0786	1.35585
$199$	5.49310	0.389396	0.194698	−	0.980863i	$-0.437627\pi$
$199$	5.49310	0.389396	0.194698	+	0.980863i	$0.437627\pi$
$200$	1.00000	0.0707107
$201$	−13.5133	−0.953156
$202$	2.38178	0.167581
$203$	−0.950330	−0.0667001
$204$	−0.180100	−0.0126096
$205$	11.8907	0.830480
$206$	−6.44296	−0.448902
$207$	19.6611	1.36654
$208$	−1.00000	−0.0693375
$209$	14.5336	1.00531
$210$	−0.535335	−0.0369416
$211$	−12.1262	−0.834805	−0.417403	−	0.908722i	$-0.637059\pi$
$211$	−12.1262	−0.834805	−0.417403	+	0.908722i	$0.637059\pi$
$212$	7.40140	0.508330
$213$	3.16646	0.216962
$214$	−4.19546	−0.286796
$215$	0.517514	0.0352942
$216$	2.48951	0.169390
$217$	−0.203142	−0.0137902
$218$	−7.15776	−0.484784
$219$	−6.97330	−0.471212
$220$	−4.83652	−0.326078
$221$	0.0683421	0.00459719
$222$	17.7361	1.19037
$223$	4.43362	0.296897	0.148449	−	0.988920i	$-0.452572\pi$
$223$	4.43362	0.296897	0.148449	+	0.988920i	$0.452572\pi$
$224$	0.203142	0.0135730
$225$	3.94469	0.262979
$226$	0.127104	0.00845481
$227$	−25.4808	−1.69122	−0.845610	−	0.533802i	$-0.820763\pi$
$227$	−25.4808	−1.69122	−0.845610	+	0.533802i	$0.820763\pi$
$228$	7.91891	0.524443
$229$	−7.75619	−0.512543	−0.256272	−	0.966605i	$-0.582494\pi$
$229$	−7.75619	−0.512543	−0.256272	+	0.966605i	$0.582494\pi$
$230$	−4.98419	−0.328648
$231$	2.58916	0.170354
$232$	−4.67817	−0.307137
$233$	−16.0043	−1.04848	−0.524239	−	0.851571i	$-0.675650\pi$
$233$	−16.0043	−1.04848	−0.524239	+	0.851571i	$0.675650\pi$
$234$	−3.94469	−0.257872
$235$	−2.49431	−0.162711
$236$	−9.25770	−0.602625
$237$	20.7851	1.35014
$238$	−0.0138831	−0.000899910 0
$239$	−10.1862	−0.658891	−0.329446	−	0.944175i	$-0.606862\pi$
$239$	−10.1862	−0.658891	−0.329446	+	0.944175i	$0.606862\pi$
$240$	−2.63528	−0.170106
$241$	−11.5154	−0.741770	−0.370885	−	0.928679i	$-0.620946\pi$
$241$	−11.5154	−0.741770	−0.370885	+	0.928679i	$0.620946\pi$
$242$	12.3920	0.796585
$243$	−21.3657	−1.37061
$244$	8.56694	0.548442
$245$	6.95873	0.444577
$246$	−31.3352	−1.99786
$247$	−3.00496	−0.191201
$248$	−1.00000	−0.0635001
$249$	37.9311	2.40379
$250$	−1.00000	−0.0632456
$251$	−6.50524	−0.410607	−0.205303	−	0.978698i	$-0.565818\pi$
$251$	−6.50524	−0.410607	−0.205303	+	0.978698i	$0.565818\pi$
$252$	0.801330	0.0504791
$253$	24.1061	1.51554
$254$	6.91730	0.434030
$255$	0.180100	0.0112783
$256$	1.00000	0.0625000
$257$	−10.5507	−0.658134	−0.329067	−	0.944307i	$-0.606734\pi$
$257$	−10.5507	−0.658134	−0.329067	+	0.944307i	$0.606734\pi$
$258$	−1.36379	−0.0849061
$259$	1.36720	0.0849534
$260$	1.00000	0.0620174
$261$	−18.4539	−1.14227
$262$	−12.1974	−0.753556
$263$	1.12009	0.0690677	0.0345339	−	0.999404i	$-0.489005\pi$
$263$	1.12009	0.0690677	0.0345339	+	0.999404i	$0.489005\pi$
$264$	12.7456	0.784436
$265$	−7.40140	−0.454664
$266$	0.610433	0.0374281
$267$	34.6717	2.12187
$268$	−5.12785	−0.313234
$269$	−20.5157	−1.25087	−0.625433	−	0.780278i	$-0.715078\pi$
$269$	−20.5157	−1.25087	−0.625433	+	0.780278i	$0.715078\pi$
$270$	−2.48951	−0.151507
$271$	15.3800	0.934269	0.467134	−	0.884186i	$-0.345286\pi$
$271$	15.3800	0.934269	0.467134	+	0.884186i	$0.345286\pi$
$272$	−0.0683421	−0.00414385
$273$	−0.535335	−0.0323999
$274$	4.52125	0.273139
$275$	4.83652	0.291653
$276$	13.1347	0.790617
$277$	−4.93138	−0.296298	−0.148149	−	0.988965i	$-0.547331\pi$
$277$	−4.93138	−0.296298	−0.148149	+	0.988965i	$0.547331\pi$
$278$	−10.2832	−0.616742
$279$	−3.94469	−0.236162
$280$	−0.203142	−0.0121400
$281$	16.0892	0.959800	0.479900	−	0.877323i	$-0.340673\pi$
$281$	16.0892	0.959800	0.479900	+	0.877323i	$0.340673\pi$
$282$	6.57320	0.391428
$283$	27.9171	1.65950	0.829749	−	0.558137i	$-0.188484\pi$
$283$	27.9171	1.65950	0.829749	+	0.558137i	$0.188484\pi$
$284$	1.20157	0.0712998
$285$	−7.91891	−0.469076
$286$	−4.83652	−0.285990
$287$	−2.41549	−0.142582
$288$	3.94469	0.232443
$289$	−16.9953	−0.999725
$290$	4.67817	0.274711
$291$	−4.84874	−0.284238
$292$	−2.64613	−0.154853
$293$	1.98261	0.115825	0.0579127	−	0.998322i	$-0.481556\pi$
$293$	1.98261	0.115825	0.0579127	+	0.998322i	$0.481556\pi$
$294$	−18.3382	−1.06951
$295$	9.25770	0.539004
$296$	6.73026	0.391188
$297$	12.0406	0.698665
$298$	19.4531	1.12689
$299$	−4.98419	−0.288243
$300$	2.63528	0.152148
$301$	−0.105129	−0.00605952
$302$	12.0592	0.693929
$303$	6.27664	0.360584
$304$	3.00496	0.172346
$305$	−8.56694	−0.490542
$306$	−0.269588	−0.0154113
$307$	−2.21971	−0.126685	−0.0633426	−	0.997992i	$-0.520176\pi$
$307$	−2.21971	−0.126685	−0.0633426	+	0.997992i	$0.520176\pi$
$308$	0.982499	0.0559831
$309$	−16.9790	−0.965901
$310$	1.00000	0.0567962
$311$	0.631295	0.0357975	0.0178987	−	0.999840i	$-0.494302\pi$
$311$	0.631295	0.0357975	0.0178987	+	0.999840i	$0.494302\pi$
$312$	−2.63528	−0.149193
$313$	−2.04157	−0.115396	−0.0576981	−	0.998334i	$-0.518376\pi$
$313$	−2.04157	−0.115396	−0.0576981	+	0.998334i	$0.518376\pi$
$314$	−19.6491	−1.10886
$315$	−0.801330	−0.0451498
$316$	7.88727	0.443694
$317$	3.86021	0.216811	0.108406	−	0.994107i	$-0.465426\pi$
$317$	3.86021	0.216811	0.108406	+	0.994107i	$0.465426\pi$
$318$	19.5047	1.09377
$319$	−22.6261	−1.26682
$320$	−1.00000	−0.0559017
$321$	−11.0562	−0.617097
$322$	1.01250	0.0564242
$323$	−0.205366	−0.0114269
$324$	−5.27351	−0.292973
$325$	−1.00000	−0.0554700
$326$	−8.98152	−0.497440
$327$	−18.8627	−1.04311
$328$	−11.8907	−0.656552
$329$	0.506698	0.0279352
$330$	−12.7456	−0.701621
$331$	5.36697	0.294995	0.147498	−	0.989062i	$-0.452878\pi$
$331$	5.36697	0.294995	0.147498	+	0.989062i	$0.452878\pi$
$332$	14.3936	0.789952
$333$	26.5487	1.45486
$334$	−13.4277	−0.734729
$335$	5.12785	0.280165
$336$	0.535335	0.0292049
$337$	−14.5518	−0.792685	−0.396342	−	0.918103i	$-0.629721\pi$
$337$	−14.5518	−0.792685	−0.396342	+	0.918103i	$0.629721\pi$
$338$	1.00000	0.0543928
$339$	0.334953	0.0181922
$340$	0.0683421	0.00370637
$341$	−4.83652	−0.261912
$342$	11.8536	0.640971
$343$	−2.83560	−0.153108
$344$	−0.517514	−0.0279025
$345$	−13.1347	−0.707149
$346$	15.0496	0.809071
$347$	−34.5256	−1.85343	−0.926716	−	0.375763i	$-0.877381\pi$
$347$	−34.5256	−1.85343	−0.926716	+	0.375763i	$0.877381\pi$
$348$	−12.3283	−0.660864
$349$	12.4174	0.664687	0.332343	−	0.943158i	$-0.392161\pi$
$349$	12.4174	0.664687	0.332343	+	0.943158i	$0.392161\pi$
$350$	0.203142	0.0108584
$351$	−2.48951	−0.132880
$352$	4.83652	0.257788
$353$	−4.79438	−0.255179	−0.127590	−	0.991827i	$-0.540724\pi$
$353$	−4.79438	−0.255179	−0.127590	+	0.991827i	$0.540724\pi$
$354$	−24.3966	−1.29666
$355$	−1.20157	−0.0637725
$356$	13.1568	0.697307
$357$	−0.0365859	−0.00193633
$358$	−10.0195	−0.529546
$359$	−11.3590	−0.599504	−0.299752	−	0.954017i	$-0.596904\pi$
$359$	−11.3590	−0.599504	−0.299752	+	0.954017i	$0.596904\pi$
$360$	−3.94469	−0.207903
$361$	−9.97019	−0.524747
$362$	8.45118	0.444184
$363$	32.6562	1.71401
$364$	−0.203142	−0.0106475
$365$	2.64613	0.138505
$366$	22.5763	1.18008
$367$	−14.6997	−0.767316	−0.383658	−	0.923475i	$-0.625336\pi$
$367$	−14.6997	−0.767316	−0.383658	+	0.923475i	$0.625336\pi$
$368$	4.98419	0.259819
$369$	−46.9049	−2.44177
$370$	−6.73026	−0.349889
$371$	1.50353	0.0780595
$372$	−2.63528	−0.136633
$373$	−20.0256	−1.03689	−0.518443	−	0.855112i	$-0.673488\pi$
$373$	−20.0256	−1.03689	−0.518443	+	0.855112i	$0.673488\pi$
$374$	−0.330538	−0.0170917
$375$	−2.63528	−0.136085
$376$	2.49431	0.128634
$377$	4.67817	0.240938
$378$	0.505723	0.0260116
$379$	10.5500	0.541915	0.270957	−	0.962591i	$-0.412660\pi$
$379$	10.5500	0.541915	0.270957	+	0.962591i	$0.412660\pi$
$380$	−3.00496	−0.154151
$381$	18.2290	0.933900
$382$	0.282464	0.0144521
$383$	−22.0178	−1.12506	−0.562529	−	0.826778i	$-0.690171\pi$
$383$	−22.0178	−1.12506	−0.562529	+	0.826778i	$0.690171\pi$
$384$	2.63528	0.134481
$385$	−0.982499	−0.0500728
$386$	3.11170	0.158381
$387$	−2.04143	−0.103772
$388$	−1.83994	−0.0934086
$389$	9.92601	0.503269	0.251634	−	0.967822i	$-0.419032\pi$
$389$	9.92601	0.503269	0.251634	+	0.967822i	$0.419032\pi$
$390$	2.63528	0.133442
$391$	−0.340630	−0.0172264
$392$	−6.95873	−0.351469
$393$	−32.1434	−1.62142
$394$	−0.893060	−0.0449917
$395$	−7.88727	−0.396852
$396$	19.0786	0.958734
$397$	−30.3617	−1.52381	−0.761905	−	0.647689i	$-0.775736\pi$
$397$	−30.3617	−1.52381	−0.761905	+	0.647689i	$0.775736\pi$
$398$	5.49310	0.275344
$399$	1.60866	0.0805338
$400$	1.00000	0.0500000
$401$	−21.9321	−1.09524	−0.547619	−	0.836728i	$-0.684465\pi$
$401$	−21.9321	−1.09524	−0.547619	+	0.836728i	$0.684465\pi$
$402$	−13.5133	−0.673983
$403$	1.00000	0.0498135
$404$	2.38178	0.118498
$405$	5.27351	0.262043
$406$	−0.950330	−0.0471641
$407$	32.5510	1.61350
$408$	−0.180100	−0.00891630
$409$	−2.08788	−0.103239	−0.0516194	−	0.998667i	$-0.516438\pi$
$409$	−2.08788	−0.103239	−0.0516194	+	0.998667i	$0.516438\pi$
$410$	11.8907	0.587238
$411$	11.9148	0.587712
$412$	−6.44296	−0.317422
$413$	−1.88062	−0.0925395
$414$	19.6611	0.966288
$415$	−14.3936	−0.706554
$416$	−1.00000	−0.0490290
$417$	−27.0990	−1.32704
$418$	14.5336	0.710860
$419$	−34.6798	−1.69422	−0.847110	−	0.531418i	$-0.821659\pi$
$419$	−34.6798	−1.69422	−0.847110	+	0.531418i	$0.821659\pi$
$420$	−0.535335	−0.0261217
$421$	−0.223194	−0.0108778	−0.00543889	−	0.999985i	$-0.501731\pi$
$421$	−0.223194	−0.0108778	−0.00543889	+	0.999985i	$0.501731\pi$
$422$	−12.1262	−0.590296
$423$	9.83927	0.478402
$424$	7.40140	0.359444
$425$	−0.0683421	−0.00331508
$426$	3.16646	0.153415
$427$	1.74030	0.0842192
$428$	−4.19546	−0.202795
$429$	−12.7456	−0.615362
$430$	0.517514	0.0249568
$431$	−20.9742	−1.01029	−0.505145	−	0.863035i	$-0.668561\pi$
$431$	−20.9742	−1.01029	−0.505145	+	0.863035i	$0.668561\pi$
$432$	2.48951	0.119777
$433$	−14.6915	−0.706028	−0.353014	−	0.935618i	$-0.614843\pi$
$433$	−14.6915	−0.706028	−0.353014	+	0.935618i	$0.614843\pi$
$434$	−0.203142	−0.00975111
$435$	12.3283	0.591095
$436$	−7.15776	−0.342794
$437$	14.9773	0.716461
$438$	−6.97330	−0.333197
$439$	−38.7604	−1.84993	−0.924966	−	0.380049i	$-0.875907\pi$
$439$	−38.7604	−1.84993	−0.924966	+	0.380049i	$0.875907\pi$
$440$	−4.83652	−0.230572
$441$	−27.4500	−1.30714
$442$	0.0683421	0.00325070
$443$	−28.2306	−1.34128	−0.670639	−	0.741784i	$-0.733980\pi$
$443$	−28.2306	−1.34128	−0.670639	+	0.741784i	$0.733980\pi$
$444$	17.7361	0.841718
$445$	−13.1568	−0.623690
$446$	4.43362	0.209938
$447$	51.2643	2.42472
$448$	0.203142	0.00959754
$449$	36.3549	1.71570	0.857848	−	0.513904i	$-0.171801\pi$
$449$	36.3549	1.71570	0.857848	+	0.513904i	$0.171801\pi$
$450$	3.94469	0.185954
$451$	−57.5095	−2.70802
$452$	0.127104	0.00597845
$453$	31.7793	1.49312
$454$	−25.4808	−1.19587
$455$	0.203142	0.00952343
$456$	7.91891	0.370837
$457$	31.8140	1.48820	0.744099	−	0.668069i	$-0.232879\pi$
$457$	31.8140	1.48820	0.744099	+	0.668069i	$0.232879\pi$
$458$	−7.75619	−0.362423
$459$	−0.170138	−0.00794138
$460$	−4.98419	−0.232389
$461$	−15.5327	−0.723431	−0.361716	−	0.932288i	$-0.617809\pi$
$461$	−15.5327	−0.723431	−0.361716	+	0.932288i	$0.617809\pi$
$462$	2.58916	0.120459
$463$	27.7525	1.28977	0.644885	−	0.764279i	$-0.276905\pi$
$463$	27.7525	1.28977	0.644885	+	0.764279i	$0.276905\pi$
$464$	−4.67817	−0.217178
$465$	2.63528	0.122208
$466$	−16.0043	−0.741386
$467$	−20.6674	−0.956375	−0.478188	−	0.878258i	$-0.658706\pi$
$467$	−20.6674	−0.956375	−0.478188	+	0.878258i	$0.658706\pi$
$468$	−3.94469	−0.182343
$469$	−1.04168	−0.0481004
$470$	−2.49431	−0.115054
$471$	−51.7808	−2.38593
$472$	−9.25770	−0.426120
$473$	−2.50297	−0.115087
$474$	20.7851	0.954693
$475$	3.00496	0.137877
$476$	−0.0138831	−0.000636333 0
$477$	29.1962	1.33680
$478$	−10.1862	−0.465906
$479$	15.9286	0.727798	0.363899	−	0.931438i	$-0.381445\pi$
$479$	15.9286	0.727798	0.363899	+	0.931438i	$0.381445\pi$
$480$	−2.63528	−0.120283
$481$	−6.73026	−0.306873
$482$	−11.5154	−0.524511
$483$	2.66821	0.121408
$484$	12.3920	0.563271
$485$	1.83994	0.0835472
$486$	−21.3657	−0.969167
$487$	−20.3387	−0.921633	−0.460817	−	0.887495i	$-0.652443\pi$
$487$	−20.3387	−0.921633	−0.460817	+	0.887495i	$0.652443\pi$
$488$	8.56694	0.387807
$489$	−23.6688	−1.07034
$490$	6.95873	0.314364
$491$	29.8096	1.34529	0.672645	−	0.739965i	$-0.265158\pi$
$491$	29.8096	1.34529	0.672645	+	0.739965i	$0.265158\pi$
$492$	−31.3352	−1.41270
$493$	0.319716	0.0143993
$494$	−3.00496	−0.135200
$495$	−19.0786	−0.857518
$496$	−1.00000	−0.0449013
$497$	0.244088	0.0109488
$498$	37.9311	1.69974
$499$	2.18894	0.0979904	0.0489952	−	0.998799i	$-0.484398\pi$
$499$	2.18894	0.0979904	0.0489952	+	0.998799i	$0.484398\pi$
$500$	−1.00000	−0.0447214
$501$	−35.3856	−1.58091
$502$	−6.50524	−0.290343
$503$	−15.7771	−0.703465	−0.351732	−	0.936101i	$-0.614407\pi$
$503$	−15.7771	−0.703465	−0.351732	+	0.936101i	$0.614407\pi$
$504$	0.801330	0.0356941
$505$	−2.38178	−0.105988
$506$	24.1061	1.07165
$507$	2.63528	0.117037
$508$	6.91730	0.306905
$509$	27.9587	1.23925	0.619623	−	0.784899i	$-0.287285\pi$
$509$	27.9587	1.23925	0.619623	+	0.784899i	$0.287285\pi$
$510$	0.180100	0.00797498
$511$	−0.537540	−0.0237794
$512$	1.00000	0.0441942
$513$	7.48089	0.330289
$514$	−10.5507	−0.465371
$515$	6.44296	0.283911
$516$	−1.36379	−0.0600377
$517$	12.0638	0.530565
$518$	1.36720	0.0600711
$519$	39.6598	1.74087
$520$	1.00000	0.0438529
$521$	36.2792	1.58942	0.794709	−	0.606990i	$-0.207623\pi$
$521$	36.2792	1.58942	0.794709	+	0.606990i	$0.207623\pi$
$522$	−18.4539	−0.807705
$523$	22.8142	0.997595	0.498798	−	0.866718i	$-0.333775\pi$
$523$	22.8142	0.997595	0.498798	+	0.866718i	$0.333775\pi$
$524$	−12.1974	−0.532844
$525$	0.535335	0.0233639
$526$	1.12009	0.0488382
$527$	0.0683421	0.00297703
$528$	12.7456	0.554680
$529$	1.84212	0.0800921
$530$	−7.40140	−0.321496
$531$	−36.5187	−1.58478
$532$	0.610433	0.0264656
$533$	11.8907	0.515042
$534$	34.6717	1.50039
$535$	4.19546	0.181386
$536$	−5.12785	−0.221490
$537$	−26.4041	−1.13942
$538$	−20.5157	−0.884496
$539$	−33.6561	−1.44967
$540$	−2.48951	−0.107131
$541$	38.4605	1.65355	0.826773	−	0.562536i	$-0.190174\pi$
$541$	38.4605	1.65355	0.826773	+	0.562536i	$0.190174\pi$
$542$	15.3800	0.660628
$543$	22.2712	0.955749
$544$	−0.0683421	−0.00293015
$545$	7.15776	0.306605
$546$	−0.535335	−0.0229102
$547$	−31.3197	−1.33913	−0.669567	−	0.742752i	$-0.733520\pi$
$547$	−31.3197	−1.33913	−0.669567	+	0.742752i	$0.733520\pi$
$548$	4.52125	0.193138
$549$	33.7939	1.44229
$550$	4.83652	0.206230
$551$	−14.0577	−0.598879
$552$	13.1347	0.559051
$553$	1.60223	0.0681339
$554$	−4.93138	−0.209514
$555$	−17.7361	−0.752855
$556$	−10.2832	−0.436103
$557$	10.9919	0.465742	0.232871	−	0.972508i	$-0.425188\pi$
$557$	10.9919	0.465742	0.232871	+	0.972508i	$0.425188\pi$
$558$	−3.94469	−0.166992
$559$	0.517514	0.0218885
$560$	−0.203142	−0.00858430
$561$	−0.871060	−0.0367762
$562$	16.0892	0.678681
$563$	11.7230	0.494064	0.247032	−	0.969007i	$-0.420545\pi$
$563$	11.7230	0.494064	0.247032	+	0.969007i	$0.420545\pi$
$564$	6.57320	0.276782
$565$	−0.127104	−0.00534729
$566$	27.9171	1.17344
$567$	−1.07127	−0.0449891
$568$	1.20157	0.0504166
$569$	43.2745	1.81416	0.907081	−	0.420957i	$-0.138306\pi$
$569$	43.2745	1.81416	0.907081	+	0.420957i	$0.138306\pi$
$570$	−7.91891	−0.331687
$571$	10.7786	0.451070	0.225535	−	0.974235i	$-0.427587\pi$
$571$	10.7786	0.451070	0.225535	+	0.974235i	$0.427587\pi$
$572$	−4.83652	−0.202225
$573$	0.744372	0.0310966
$574$	−2.41549	−0.100821
$575$	4.98419	0.207855
$576$	3.94469	0.164362
$577$	−41.9527	−1.74651	−0.873256	−	0.487262i	$-0.837996\pi$
$577$	−41.9527	−1.74651	−0.873256	+	0.487262i	$0.837996\pi$
$578$	−16.9953	−0.706913
$579$	8.20018	0.340788
$580$	4.67817	0.194250
$581$	2.92394	0.121306
$582$	−4.84874	−0.200987
$583$	35.7970	1.48256
$584$	−2.64613	−0.109498
$585$	3.94469	0.163093
$586$	1.98261	0.0819009
$587$	−0.719381	−0.0296920	−0.0148460	−	0.999890i	$-0.504726\pi$
$587$	−0.719381	−0.0296920	−0.0148460	+	0.999890i	$0.504726\pi$
$588$	−18.3382	−0.756254
$589$	−3.00496	−0.123817
$590$	9.25770	0.381133
$591$	−2.35346	−0.0968085
$592$	6.73026	0.276612
$593$	32.1995	1.32227	0.661137	−	0.750266i	$-0.270074\pi$
$593$	32.1995	1.32227	0.661137	+	0.750266i	$0.270074\pi$
$594$	12.0406	0.494030
$595$	0.0138831	0.000569153 0
$596$	19.4531	0.796829
$597$	14.4758	0.592457
$598$	−4.98419	−0.203819
$599$	−6.63600	−0.271140	−0.135570	−	0.990768i	$-0.543286\pi$
$599$	−6.63600	−0.271140	−0.135570	+	0.990768i	$0.543286\pi$
$600$	2.63528	0.107585
$601$	41.9368	1.71064	0.855318	−	0.518103i	$-0.173362\pi$
$601$	41.9368	1.71064	0.855318	+	0.518103i	$0.173362\pi$
$602$	−0.105129	−0.00428473
$603$	−20.2278	−0.823739
$604$	12.0592	0.490682
$605$	−12.3920	−0.503805
$606$	6.27664	0.254971
$607$	17.1288	0.695235	0.347617	−	0.937636i	$-0.386991\pi$
$607$	17.1288	0.695235	0.347617	+	0.937636i	$0.386991\pi$
$608$	3.00496	0.121867
$609$	−2.50438	−0.101483
$610$	−8.56694	−0.346865
$611$	−2.49431	−0.100909
$612$	−0.269588	−0.0108975
$613$	31.0071	1.25236	0.626182	−	0.779677i	$-0.284617\pi$
$613$	31.0071	1.25236	0.626182	+	0.779677i	$0.284617\pi$
$614$	−2.21971	−0.0895800
$615$	31.3352	1.26356
$616$	0.982499	0.0395860
$617$	19.1720	0.771837	0.385918	−	0.922533i	$-0.373885\pi$
$617$	19.1720	0.771837	0.385918	+	0.922533i	$0.373885\pi$
$618$	−16.9790	−0.682995
$619$	−10.9693	−0.440894	−0.220447	−	0.975399i	$-0.570752\pi$
$619$	−10.9693	−0.440894	−0.220447	+	0.975399i	$0.570752\pi$
$620$	1.00000	0.0401610
$621$	12.4082	0.497923
$622$	0.631295	0.0253126
$623$	2.67269	0.107079
$624$	−2.63528	−0.105496
$625$	1.00000	0.0400000
$626$	−2.04157	−0.0815975
$627$	38.3000	1.52955
$628$	−19.6491	−0.784084
$629$	−0.459960	−0.0183398
$630$	−0.801330	−0.0319258
$631$	−11.8054	−0.469965	−0.234982	−	0.972000i	$-0.575503\pi$
$631$	−11.8054	−0.469965	−0.234982	+	0.972000i	$0.575503\pi$
$632$	7.88727	0.313739
$633$	−31.9560	−1.27014
$634$	3.86021	0.153309
$635$	−6.91730	−0.274505
$636$	19.5047	0.773413
$637$	6.95873	0.275715
$638$	−22.6261	−0.895774
$639$	4.73980	0.187504
$640$	−1.00000	−0.0395285
$641$	−14.2924	−0.564514	−0.282257	−	0.959339i	$-0.591083\pi$
$641$	−14.2924	−0.564514	−0.282257	+	0.959339i	$0.591083\pi$
$642$	−11.0562	−0.436354
$643$	9.11564	0.359486	0.179743	−	0.983714i	$-0.442473\pi$
$643$	9.11564	0.359486	0.179743	+	0.983714i	$0.442473\pi$
$644$	1.01250	0.0398979
$645$	1.36379	0.0536993
$646$	−0.205366	−0.00808000
$647$	3.54098	0.139210	0.0696051	−	0.997575i	$-0.477826\pi$
$647$	3.54098	0.139210	0.0696051	+	0.997575i	$0.477826\pi$
$648$	−5.27351	−0.207163
$649$	−44.7751	−1.75758
$650$	−1.00000	−0.0392232
$651$	−0.535335	−0.0209814
$652$	−8.98152	−0.351743
$653$	45.8053	1.79250	0.896250	−	0.443550i	$-0.146281\pi$
$653$	45.8053	1.79250	0.896250	+	0.443550i	$0.146281\pi$
$654$	−18.8627	−0.737589
$655$	12.1974	0.476591
$656$	−11.8907	−0.464253
$657$	−10.4382	−0.407232
$658$	0.506698	0.0197532
$659$	10.2160	0.397958	0.198979	−	0.980004i	$-0.436237\pi$
$659$	10.2160	0.397958	0.198979	+	0.980004i	$0.436237\pi$
$660$	−12.7456	−0.496121
$661$	−20.3176	−0.790264	−0.395132	−	0.918624i	$-0.629301\pi$
$661$	−20.3176	−0.790264	−0.395132	+	0.918624i	$0.629301\pi$
$662$	5.36697	0.208593
$663$	0.180100	0.00699452
$664$	14.3936	0.558580
$665$	−0.610433	−0.0236716
$666$	26.5487	1.02874
$667$	−23.3169	−0.902832
$668$	−13.4277	−0.519532
$669$	11.6838	0.451722
$670$	5.12785	0.198106
$671$	41.4342	1.59955
$672$	0.535335	0.0206510
$673$	−37.3963	−1.44152	−0.720761	−	0.693184i	$-0.756207\pi$
$673$	−37.3963	−1.44152	−0.720761	+	0.693184i	$0.756207\pi$
$674$	−14.5518	−0.560513
$675$	2.48951	0.0958213
$676$	1.00000	0.0384615
$677$	−1.79070	−0.0688224	−0.0344112	−	0.999408i	$-0.510956\pi$
$677$	−1.79070	−0.0688224	−0.0344112	+	0.999408i	$0.510956\pi$
$678$	0.334953	0.0128638
$679$	−0.373768	−0.0143439
$680$	0.0683421	0.00262080
$681$	−67.1489	−2.57315
$682$	−4.83652	−0.185200
$683$	18.4293	0.705177	0.352588	−	0.935779i	$-0.385302\pi$
$683$	18.4293	0.705177	0.352588	+	0.935779i	$0.385302\pi$
$684$	11.8536	0.453235
$685$	−4.52125	−0.172748
$686$	−2.83560	−0.108264
$687$	−20.4397	−0.779823
$688$	−0.517514	−0.0197300
$689$	−7.40140	−0.281971
$690$	−13.1347	−0.500030
$691$	−27.3619	−1.04090	−0.520448	−	0.853893i	$-0.674235\pi$
$691$	−27.3619	−1.04090	−0.520448	+	0.853893i	$0.674235\pi$
$692$	15.0496	0.572100
$693$	3.87565	0.147224
$694$	−34.5256	−1.31057
$695$	10.2832	0.390062
$696$	−12.3283	−0.467302
$697$	0.812634	0.0307807
$698$	12.4174	0.470005
$699$	−42.1758	−1.59524
$700$	0.203142	0.00767803
$701$	−12.3238	−0.465462	−0.232731	−	0.972541i	$-0.574766\pi$
$701$	−12.3238	−0.465462	−0.232731	+	0.972541i	$0.574766\pi$
$702$	−2.48951	−0.0939605
$703$	20.2242	0.762769
$704$	4.83652	0.182283
$705$	−6.57320	−0.247561
$706$	−4.79438	−0.180439
$707$	0.483838	0.0181966
$708$	−24.3966	−0.916880
$709$	−7.38825	−0.277471	−0.138736	−	0.990329i	$-0.544304\pi$
$709$	−7.38825	−0.277471	−0.138736	+	0.990329i	$0.544304\pi$
$710$	−1.20157	−0.0450940
$711$	31.1128	1.16682
$712$	13.1568	0.493070
$713$	−4.98419	−0.186659
$714$	−0.0365859	−0.00136919
$715$	4.83652	0.180876
$716$	−10.0195	−0.374445
$717$	−26.8435	−1.00249
$718$	−11.3590	−0.423914
$719$	35.9631	1.34120	0.670600	−	0.741820i	$-0.266037\pi$
$719$	35.9631	1.34120	0.670600	+	0.741820i	$0.266037\pi$
$720$	−3.94469	−0.147010
$721$	−1.30883	−0.0487435
$722$	−9.97019	−0.371052
$723$	−30.3462	−1.12859
$724$	8.45118	0.314086
$725$	−4.67817	−0.173743
$726$	32.6562	1.21199
$727$	13.2386	0.490994	0.245497	−	0.969397i	$-0.421049\pi$
$727$	13.2386	0.490994	0.245497	+	0.969397i	$0.421049\pi$
$728$	−0.203142	−0.00752893
$729$	−40.4840	−1.49941
$730$	2.64613	0.0979378
$731$	0.0353680	0.00130813
$732$	22.5763	0.834443
$733$	15.1478	0.559497	0.279749	−	0.960073i	$-0.409749\pi$
$733$	15.1478	0.559497	0.279749	+	0.960073i	$0.409749\pi$
$734$	−14.6997	−0.542574
$735$	18.3382	0.676414
$736$	4.98419	0.183720
$737$	−24.8010	−0.913556
$738$	−46.9049	−1.72659
$739$	12.4022	0.456223	0.228111	−	0.973635i	$-0.426745\pi$
$739$	12.4022	0.456223	0.228111	+	0.973635i	$0.426745\pi$
$740$	−6.73026	−0.247409
$741$	−7.91891	−0.290909
$742$	1.50353	0.0551964
$743$	9.69652	0.355731	0.177865	−	0.984055i	$-0.443081\pi$
$743$	9.69652	0.355731	0.177865	+	0.984055i	$0.443081\pi$
$744$	−2.63528	−0.0966139
$745$	−19.4531	−0.712706
$746$	−20.0256	−0.733190
$747$	56.7783	2.07741
$748$	−0.330538	−0.0120857
$749$	−0.852274	−0.0311414
$750$	−2.63528	−0.0962267
$751$	9.26706	0.338160	0.169080	−	0.985602i	$-0.445920\pi$
$751$	9.26706	0.338160	0.169080	+	0.985602i	$0.445920\pi$
$752$	2.49431	0.0909581
$753$	−17.1431	−0.624729
$754$	4.67817	0.170369
$755$	−12.0592	−0.438879
$756$	0.505723	0.0183930
$757$	−40.4386	−1.46977	−0.734883	−	0.678194i	$-0.762763\pi$
$757$	−40.4386	−1.46977	−0.734883	+	0.678194i	$0.762763\pi$
$758$	10.5500	0.383191
$759$	63.5263	2.30586
$760$	−3.00496	−0.109001
$761$	−52.0192	−1.88569	−0.942847	−	0.333227i	$-0.891862\pi$
$761$	−52.0192	−1.88569	−0.942847	+	0.333227i	$0.891862\pi$
$762$	18.2290	0.660367
$763$	−1.45404	−0.0526397
$764$	0.282464	0.0102192
$765$	0.269588	0.00974698
$766$	−22.0178	−0.795536
$767$	9.25770	0.334276
$768$	2.63528	0.0950924
$769$	33.4408	1.20591	0.602953	−	0.797777i	$-0.293991\pi$
$769$	33.4408	1.20591	0.602953	+	0.797777i	$0.293991\pi$
$770$	−0.982499	−0.0354068
$771$	−27.8040	−1.00134
$772$	3.11170	0.111992
$773$	40.3103	1.44986	0.724930	−	0.688823i	$-0.241872\pi$
$773$	40.3103	1.44986	0.724930	+	0.688823i	$0.241872\pi$
$774$	−2.04143	−0.0733777
$775$	−1.00000	−0.0359211
$776$	−1.83994	−0.0660498
$777$	3.60294	0.129255
$778$	9.92601	0.355865
$779$	−35.7310	−1.28020
$780$	2.63528	0.0943581
$781$	5.81140	0.207948
$782$	−0.340630	−0.0121809
$783$	−11.6463	−0.416206
$784$	−6.95873	−0.248526
$785$	19.6491	0.701306
$786$	−32.1434	−1.14652
$787$	−4.16425	−0.148440	−0.0742198	−	0.997242i	$-0.523647\pi$
$787$	−4.16425	−0.148440	−0.0742198	+	0.997242i	$0.523647\pi$
$788$	−0.893060	−0.0318140
$789$	2.95175	0.105085
$790$	−7.88727	−0.280617
$791$	0.0258201	0.000918055 0
$792$	19.0786	0.677927
$793$	−8.56694	−0.304221
$794$	−30.3617	−1.07750
$795$	−19.5047	−0.691762
$796$	5.49310	0.194698
$797$	4.73708	0.167796	0.0838981	−	0.996474i	$-0.473263\pi$
$797$	4.73708	0.167796	0.0838981	+	0.996474i	$0.473263\pi$
$798$	1.60866	0.0569460
$799$	−0.170466	−0.00603067
$800$	1.00000	0.0353553
$801$	51.8993	1.83377
$802$	−21.9321	−0.774450
$803$	−12.7981	−0.451635
$804$	−13.5133	−0.476578
$805$	−1.01250	−0.0356858
$806$	1.00000	0.0352235
$807$	−54.0647	−1.90317
$808$	2.38178	0.0837906
$809$	29.6006	1.04070	0.520351	−	0.853952i	$-0.325801\pi$
$809$	29.6006	1.04070	0.520351	+	0.853952i	$0.325801\pi$
$810$	5.27351	0.185292
$811$	−1.15683	−0.0406219	−0.0203110	−	0.999794i	$-0.506466\pi$
$811$	−1.15683	−0.0406219	−0.0203110	+	0.999794i	$0.506466\pi$
$812$	−0.950330	−0.0333501
$813$	40.5306	1.42147
$814$	32.5510	1.14091
$815$	8.98152	0.314609
$816$	−0.180100	−0.00630478
$817$	−1.55511	−0.0544065
$818$	−2.08788	−0.0730009
$819$	−0.801330	−0.0280007
$820$	11.8907	0.415240
$821$	4.74940	0.165755	0.0828776	−	0.996560i	$-0.473589\pi$
$821$	4.74940	0.165755	0.0828776	+	0.996560i	$0.473589\pi$
$822$	11.9148	0.415575
$823$	−21.9283	−0.764373	−0.382186	−	0.924085i	$-0.624829\pi$
$823$	−21.9283	−0.764373	−0.382186	+	0.924085i	$0.624829\pi$
$824$	−6.44296	−0.224451
$825$	12.7456	0.443744
$826$	−1.88062	−0.0654353
$827$	51.0416	1.77489	0.887445	−	0.460914i	$-0.152478\pi$
$827$	51.0416	1.77489	0.887445	+	0.460914i	$0.152478\pi$
$828$	19.6611	0.683269
$829$	34.7117	1.20559	0.602793	−	0.797897i	$-0.294054\pi$
$829$	34.7117	1.20559	0.602793	+	0.797897i	$0.294054\pi$
$830$	−14.3936	−0.499609
$831$	−12.9955	−0.450811
$832$	−1.00000	−0.0346688
$833$	0.475575	0.0164777
$834$	−27.0990	−0.938360
$835$	13.4277	0.464683
$836$	14.5336	0.502654
$837$	−2.48951	−0.0860500
$838$	−34.6798	−1.19799
$839$	15.5885	0.538175	0.269088	−	0.963116i	$-0.413278\pi$
$839$	15.5885	0.538175	0.269088	+	0.963116i	$0.413278\pi$
$840$	−0.535335	−0.0184708
$841$	−7.11476	−0.245337
$842$	−0.223194	−0.00769176
$843$	42.3994	1.46031
$844$	−12.1262	−0.417403
$845$	−1.00000	−0.0344010
$846$	9.83927	0.338281
$847$	2.51732	0.0864962
$848$	7.40140	0.254165
$849$	73.5692	2.52489
$850$	−0.0683421	−0.00234412
$851$	33.5449	1.14990
$852$	3.16646	0.108481
$853$	−39.7065	−1.35952	−0.679762	−	0.733432i	$-0.737917\pi$
$853$	−39.7065	−1.35952	−0.679762	+	0.733432i	$0.737917\pi$
$854$	1.74030	0.0595519
$855$	−11.8536	−0.405386
$856$	−4.19546	−0.143398
$857$	10.1410	0.346408	0.173204	−	0.984886i	$-0.444588\pi$
$857$	10.1410	0.346408	0.173204	+	0.984886i	$0.444588\pi$
$858$	−12.7456	−0.435127
$859$	−24.3423	−0.830550	−0.415275	−	0.909696i	$-0.636315\pi$
$859$	−24.3423	−0.830550	−0.415275	+	0.909696i	$0.636315\pi$
$860$	0.517514	0.0176471
$861$	−6.36549	−0.216935
$862$	−20.9742	−0.714383
$863$	46.1396	1.57061	0.785306	−	0.619108i	$-0.212506\pi$
$863$	46.1396	1.57061	0.785306	+	0.619108i	$0.212506\pi$
$864$	2.48951	0.0846948
$865$	−15.0496	−0.511701
$866$	−14.6915	−0.499237
$867$	−44.7874	−1.52106
$868$	−0.203142	−0.00689508
$869$	38.1470	1.29405
$870$	12.3283	0.417967
$871$	5.12785	0.173751
$872$	−7.15776	−0.242392
$873$	−7.25797	−0.245645
$874$	14.9773	0.506615
$875$	−0.203142	−0.00686744
$876$	−6.97330	−0.235606
$877$	−25.3312	−0.855373	−0.427687	−	0.903927i	$-0.640671\pi$
$877$	−25.3312	−0.855373	−0.427687	+	0.903927i	$0.640671\pi$
$878$	−38.7604	−1.30810
$879$	5.22473	0.176226
$880$	−4.83652	−0.163039
$881$	53.1604	1.79102	0.895510	−	0.445041i	$-0.146811\pi$
$881$	53.1604	1.79102	0.895510	+	0.445041i	$0.146811\pi$
$882$	−27.4500	−0.924290
$883$	−10.0176	−0.337119	−0.168560	−	0.985691i	$-0.553912\pi$
$883$	−10.0176	−0.337119	−0.168560	+	0.985691i	$0.553912\pi$
$884$	0.0683421	0.00229859
$885$	24.3966	0.820083
$886$	−28.2306	−0.948427
$887$	−4.54847	−0.152723	−0.0763613	−	0.997080i	$-0.524330\pi$
$887$	−4.54847	−0.152723	−0.0763613	+	0.997080i	$0.524330\pi$
$888$	17.7361	0.595184
$889$	1.40519	0.0471286
$890$	−13.1568	−0.441015
$891$	−25.5055	−0.854465
$892$	4.43362	0.148449
$893$	7.49531	0.250821
$894$	51.2643	1.71453
$895$	10.0195	0.334914
$896$	0.203142	0.00678649
$897$	−13.1347	−0.438555
$898$	36.3549	1.21318
$899$	4.67817	0.156026
$900$	3.94469	0.131490
$901$	−0.505827	−0.0168516
$902$	−57.5095	−1.91486
$903$	−0.277043	−0.00921942
$904$	0.127104	0.00422741
$905$	−8.45118	−0.280927
$906$	31.7793	1.05580
$907$	27.3180	0.907078	0.453539	−	0.891236i	$-0.350161\pi$
$907$	27.3180	0.907078	0.453539	+	0.891236i	$0.350161\pi$
$908$	−25.4808	−0.845610
$909$	9.39536	0.311625
$910$	0.203142	0.00673408
$911$	−50.0911	−1.65959	−0.829797	−	0.558066i	$-0.811544\pi$
$911$	−50.0911	−1.65959	−0.829797	+	0.558066i	$0.811544\pi$
$912$	7.91891	0.262221
$913$	69.6150	2.30392
$914$	31.8140	1.05232
$915$	−22.5763	−0.746348
$916$	−7.75619	−0.256272
$917$	−2.47779	−0.0818240
$918$	−0.170138	−0.00561540
$919$	21.5037	0.709342	0.354671	−	0.934991i	$-0.384593\pi$
$919$	21.5037	0.709342	0.354671	+	0.934991i	$0.384593\pi$
$920$	−4.98419	−0.164324
$921$	−5.84954	−0.192749
$922$	−15.5327	−0.511543
$923$	−1.20157	−0.0395500
$924$	2.58916	0.0851771
$925$	6.73026	0.221289
$926$	27.7525	0.912005
$927$	−25.4154	−0.834753
$928$	−4.67817	−0.153568
$929$	43.3836	1.42337	0.711686	−	0.702498i	$-0.247932\pi$
$929$	43.3836	1.42337	0.711686	+	0.702498i	$0.247932\pi$
$930$	2.63528	0.0864141
$931$	−20.9107	−0.685322
$932$	−16.0043	−0.524239
$933$	1.66364	0.0544650
$934$	−20.6674	−0.676259
$935$	0.330538	0.0108098
$936$	−3.94469	−0.128936
$937$	−43.1554	−1.40983	−0.704913	−	0.709294i	$-0.749014\pi$
$937$	−43.1554	−1.40983	−0.704913	+	0.709294i	$0.749014\pi$
$938$	−1.04168	−0.0340121
$939$	−5.38010	−0.175573
$940$	−2.49431	−0.0813554
$941$	11.4513	0.373303	0.186652	−	0.982426i	$-0.440236\pi$
$941$	11.4513	0.373303	0.186652	+	0.982426i	$0.440236\pi$
$942$	−51.7808	−1.68711
$943$	−59.2653	−1.92994
$944$	−9.25770	−0.301312
$945$	−0.505723	−0.0164512
$946$	−2.50297	−0.0813786
$947$	22.5810	0.733785	0.366893	−	0.930263i	$-0.380422\pi$
$947$	22.5810	0.733785	0.366893	+	0.930263i	$0.380422\pi$
$948$	20.7851	0.675070
$949$	2.64613	0.0858972
$950$	3.00496	0.0974939
$951$	10.1727	0.329873
$952$	−0.0138831	−0.000449955 0
$953$	−32.2718	−1.04539	−0.522693	−	0.852521i	$-0.675073\pi$
$953$	−32.2718	−1.04539	−0.522693	+	0.852521i	$0.675073\pi$
$954$	29.1962	0.945262
$955$	−0.282464	−0.00914033
$956$	−10.1862	−0.329446
$957$	−59.6259	−1.92743
$958$	15.9286	0.514631
$959$	0.918455	0.0296585
$960$	−2.63528	−0.0850532
$961$	1.00000	0.0322581
$962$	−6.73026	−0.216992
$963$	−16.5498	−0.533309
$964$	−11.5154	−0.370885
$965$	−3.11170	−0.100169
$966$	2.66821	0.0858482
$967$	47.0898	1.51431	0.757154	−	0.653237i	$-0.226589\pi$
$967$	47.0898	1.51431	0.757154	+	0.653237i	$0.226589\pi$
$968$	12.3920	0.398292
$969$	−0.541195	−0.0173857
$970$	1.83994	0.0590768
$971$	1.02586	0.0329215	0.0164607	−	0.999865i	$-0.494760\pi$
$971$	1.02586	0.0329215	0.0164607	+	0.999865i	$0.494760\pi$
$972$	−21.3657	−0.685305
$973$	−2.08894	−0.0669682
$974$	−20.3387	−0.651693
$975$	−2.63528	−0.0843964
$976$	8.56694	0.274221
$977$	18.3514	0.587113	0.293556	−	0.955942i	$-0.405161\pi$
$977$	18.3514	0.587113	0.293556	+	0.955942i	$0.405161\pi$
$978$	−23.6688	−0.756844
$979$	63.6329	2.03372
$980$	6.95873	0.222289
$981$	−28.2351	−0.901477
$982$	29.8096	0.951264
$983$	26.5509	0.846844	0.423422	−	0.905933i	$-0.360829\pi$
$983$	26.5509	0.846844	0.423422	+	0.905933i	$0.360829\pi$
$984$	−31.3352	−0.998930
$985$	0.893060	0.0284553
$986$	0.319716	0.0101818
$987$	1.33529	0.0425028
$988$	−3.00496	−0.0956006
$989$	−2.57939	−0.0820198
$990$	−19.0786	−0.606357
$991$	−39.3894	−1.25125	−0.625623	−	0.780125i	$-0.715155\pi$
$991$	−39.3894	−1.25125	−0.625623	+	0.780125i	$0.715155\pi$
$992$	−1.00000	−0.0317500
$993$	14.1434	0.448829
$994$	0.244088	0.00774200
$995$	−5.49310	−0.174143
$996$	37.9311	1.20189
$997$	−50.6245	−1.60330	−0.801648	−	0.597797i	$-0.796043\pi$
$997$	−50.6245	−1.60330	−0.801648	+	0.597797i	$0.796043\pi$
$998$	2.18894	0.0692897
$999$	16.7550	0.530106

Introduction	Features
Universe	News

Level:	\( N \)	=	\( 4030 = 2 \cdot 5 \cdot 13 \cdot 31 \)
Weight:	\( k \)	=	\( 2 \)
Character orbit:	\([\chi]\)	=	4030.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +2.63528 q^{3} +1.00000 q^{4} -1.00000 q^{5} +2.63528 q^{6} +0.203142 q^{7} +1.00000 q^{8} +3.94469 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +2.63528 q^{3} +1.00000 q^{4} -1.00000 q^{5} +2.63528 q^{6} +0.203142 q^{7} +1.00000 q^{8} +3.94469 q^{9} -1.00000 q^{10} +4.83652 q^{11} +2.63528 q^{12} -1.00000 q^{13} +0.203142 q^{14} -2.63528 q^{15} +1.00000 q^{16} -0.0683421 q^{17} +3.94469 q^{18} +3.00496 q^{19} -1.00000 q^{20} +0.535335 q^{21} +4.83652 q^{22} +4.98419 q^{23} +2.63528 q^{24} +1.00000 q^{25} -1.00000 q^{26} +2.48951 q^{27} +0.203142 q^{28} -4.67817 q^{29} -2.63528 q^{30} -1.00000 q^{31} +1.00000 q^{32} +12.7456 q^{33} -0.0683421 q^{34} -0.203142 q^{35} +3.94469 q^{36} +6.73026 q^{37} +3.00496 q^{38} -2.63528 q^{39} -1.00000 q^{40} -11.8907 q^{41} +0.535335 q^{42} -0.517514 q^{43} +4.83652 q^{44} -3.94469 q^{45} +4.98419 q^{46} +2.49431 q^{47} +2.63528 q^{48} -6.95873 q^{49} +1.00000 q^{50} -0.180100 q^{51} -1.00000 q^{52} +7.40140 q^{53} +2.48951 q^{54} -4.83652 q^{55} +0.203142 q^{56} +7.91891 q^{57} -4.67817 q^{58} -9.25770 q^{59} -2.63528 q^{60} +8.56694 q^{61} -1.00000 q^{62} +0.801330 q^{63} +1.00000 q^{64} +1.00000 q^{65} +12.7456 q^{66} -5.12785 q^{67} -0.0683421 q^{68} +13.1347 q^{69} -0.203142 q^{70} +1.20157 q^{71} +3.94469 q^{72} -2.64613 q^{73} +6.73026 q^{74} +2.63528 q^{75} +3.00496 q^{76} +0.982499 q^{77} -2.63528 q^{78} +7.88727 q^{79} -1.00000 q^{80} -5.27351 q^{81} -11.8907 q^{82} +14.3936 q^{83} +0.535335 q^{84} +0.0683421 q^{85} -0.517514 q^{86} -12.3283 q^{87} +4.83652 q^{88} +13.1568 q^{89} -3.94469 q^{90} -0.203142 q^{91} +4.98419 q^{92} -2.63528 q^{93} +2.49431 q^{94} -3.00496 q^{95} +2.63528 q^{96} -1.83994 q^{97} -6.95873 q^{98} +19.0786 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8q + 8q^{2} - q^{3} + 8q^{4} - 8q^{5} - q^{6} + q^{7} + 8q^{8} + 9q^{9} + O(q^{10}) \)	\( 8q + 8q^{2} - q^{3} + 8q^{4} - 8q^{5} - q^{6} + q^{7} + 8q^{8} + 9q^{9} - 8q^{10} + 4q^{11} - q^{12} - 8q^{13} + q^{14} + q^{15} + 8q^{16} - 5q^{17} + 9q^{18} + 2q^{19} - 8q^{20} + 17q^{21} + 4q^{22} + 4q^{23} - q^{24} + 8q^{25} - 8q^{26} + 11q^{27} + q^{28} + 11q^{29} + q^{30} - 8q^{31} + 8q^{32} + 10q^{33} - 5q^{34} - q^{35} + 9q^{36} + 19q^{37} + 2q^{38} + q^{39} - 8q^{40} + 10q^{41} + 17q^{42} + 19q^{43} + 4q^{44} - 9q^{45} + 4q^{46} + 11q^{47} - q^{48} + 11q^{49} + 8q^{50} + 7q^{51} - 8q^{52} + 8q^{53} + 11q^{54} - 4q^{55} + q^{56} - 11q^{57} + 11q^{58} + 28q^{59} + q^{60} - 12q^{61} - 8q^{62} + 20q^{63} + 8q^{64} + 8q^{65} + 10q^{66} + 24q^{67} - 5q^{68} + 30q^{69} - q^{70} + 18q^{71} + 9q^{72} - 3q^{73} + 19q^{74} - q^{75} + 2q^{76} - 7q^{77} + q^{78} + 22q^{79} - 8q^{80} + 24q^{81} + 10q^{82} + 17q^{83} + 17q^{84} + 5q^{85} + 19q^{86} + 11q^{87} + 4q^{88} + 17q^{89} - 9q^{90} - q^{91} + 4q^{92} + q^{93} + 11q^{94} - 2q^{95} - q^{96} - 24q^{97} + 11q^{98} + 23q^{99} + O(q^{100}) \)

Introduction and more

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Modular Forms

Varieties

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Representations

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Knowledge

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$-expansion

Coefficient data

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	4030.2.a.n.1.7

Level	4030
Weight	2
Character	4030.1
Self dual	Yes
Analytic conductor	32.180
Analytic rank	0
Dimension	8
CM	No
Inner twists	1

Self dual:	Yes
Analytic conductor:	\(32.1797120146\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 1 \)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$