LMFDB - Embedded newform 2151.2.a.k.1.20

Newspace parameters

Level:	$ N $	$=$	$ 2151 = 3^{2} \cdot 239 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2151.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$17.1758214748$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
Defining polynomial:	$x^{20} - 4 x^{19} - 21 x^{18} + 96 x^{17} + 164 x^{16} - 936 x^{15} - 540 x^{14} + 4804 x^{13} + 229 x^{12} - 14020 x^{11} + 3356 x^{10} + 23404 x^{9} - 9429 x^{8} - 21252 x^{7} + 10479 x^{6} + 9108 x^{5} - 4844 x^{4} - 1184 x^{3} + 640 x^{2} - 56 x + 1$
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
Root			$2.67404$ of defining polynomial
Character	$\chi$	$=$	2151.1

$q$-expansion

$f(q)$	$=$	$q+2.67404 q^{2} +5.15050 q^{4} +0.519163 q^{5} +1.02094 q^{7} +8.42455 q^{8} +O(q^{10})$	\(q+2.67404 q^{2} +5.15050 q^{4} +0.519163 q^{5} +1.02094 q^{7} +8.42455 q^{8} +1.38826 q^{10} +5.14954 q^{11} -2.29568 q^{13} +2.73005 q^{14} +12.2266 q^{16} -1.92182 q^{17} +3.15077 q^{19} +2.67395 q^{20} +13.7701 q^{22} -6.68812 q^{23} -4.73047 q^{25} -6.13875 q^{26} +5.25837 q^{28} -7.68981 q^{29} -1.20467 q^{31} +15.8454 q^{32} -5.13902 q^{34} +0.530037 q^{35} -0.791859 q^{37} +8.42528 q^{38} +4.37372 q^{40} +2.49631 q^{41} +1.15197 q^{43} +26.5227 q^{44} -17.8843 q^{46} +8.92862 q^{47} -5.95767 q^{49} -12.6495 q^{50} -11.8239 q^{52} -7.62842 q^{53} +2.67345 q^{55} +8.60100 q^{56} -20.5629 q^{58} +9.24454 q^{59} +2.23081 q^{61} -3.22134 q^{62} +17.9179 q^{64} -1.19183 q^{65} +14.8406 q^{67} -9.89831 q^{68} +1.41734 q^{70} -4.46542 q^{71} -2.29621 q^{73} -2.11746 q^{74} +16.2280 q^{76} +5.25739 q^{77} -9.20506 q^{79} +6.34761 q^{80} +6.67524 q^{82} +13.9566 q^{83} -0.997737 q^{85} +3.08042 q^{86} +43.3826 q^{88} +7.81208 q^{89} -2.34376 q^{91} -34.4471 q^{92} +23.8755 q^{94} +1.63576 q^{95} -4.25925 q^{97} -15.9311 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20q + 4q^{2} + 18q^{4} + 16q^{5} - 4q^{7} + 12q^{8} + O(q^{10}) $	$ 20q + 4q^{2} + 18q^{4} + 16q^{5} - 4q^{7} + 12q^{8} + 4q^{10} + 12q^{11} - 4q^{13} + 20q^{14} + 22q^{16} + 24q^{17} - 4q^{19} + 40q^{20} - 6q^{22} + 12q^{23} + 22q^{25} + 30q^{26} - 12q^{28} + 24q^{29} - 4q^{31} + 28q^{32} + 8q^{34} + 20q^{35} - 10q^{37} + 26q^{38} + 6q^{40} + 66q^{41} + 8q^{43} + 36q^{44} - 12q^{46} + 28q^{47} + 18q^{49} + 28q^{50} - 18q^{52} + 28q^{53} - 4q^{55} + 60q^{56} + 54q^{59} - 4q^{61} + 20q^{62} + 22q^{64} + 42q^{65} + 12q^{67} + 12q^{68} + 20q^{70} + 36q^{71} + 14q^{73} - 50q^{76} + 8q^{77} - 12q^{79} + 88q^{80} - 8q^{82} + 20q^{83} + 4q^{85} + 18q^{86} - 10q^{88} + 130q^{89} - 6q^{91} - 46q^{92} - 26q^{94} - 2q^{97} + 12q^{98} + 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 $

$2$	2.67404	1.89083	0.945416	−	0.325865i	$-0.105656\pi$
$2$	2.67404	1.89083	0.945416	+	0.325865i	$0.105656\pi$
$3$	0	0
$3$	0	0
$4$	5.15050	2.57525
$5$	0.519163	0.232177	0.116088	−	0.993239i	$-0.462964\pi$
$5$	0.519163	0.232177	0.116088	+	0.993239i	$0.462964\pi$
$6$	0	0
$7$	1.02094	0.385881	0.192940	−	0.981211i	$-0.438198\pi$
$7$	1.02094	0.385881	0.192940	+	0.981211i	$0.438198\pi$
$8$	8.42455	2.97853
$9$	0	0
$10$	1.38826	0.439008
$11$	5.14954	1.55265	0.776323	−	0.630336i	$-0.217083\pi$
$11$	5.14954	1.55265	0.776323	+	0.630336i	$0.217083\pi$
$12$	0	0
$13$	−2.29568	−0.636708	−0.318354	−	0.947972i	$-0.603130\pi$
$13$	−2.29568	−0.636708	−0.318354	+	0.947972i	$0.603130\pi$
$14$	2.73005	0.729635
$15$	0	0
$16$	12.2266	3.05665
$17$	−1.92182	−0.466109	−0.233054	−	0.972464i	$-0.574872\pi$
$17$	−1.92182	−0.466109	−0.233054	+	0.972464i	$0.574872\pi$
$18$	0	0
$19$	3.15077	0.722835	0.361418	−	0.932404i	$-0.382293\pi$
$19$	3.15077	0.722835	0.361418	+	0.932404i	$0.382293\pi$
$20$	2.67395	0.597913
$21$	0	0
$22$	13.7701	2.93579
$23$	−6.68812	−1.39457	−0.697284	−	0.716795i	$-0.745609\pi$
$23$	−6.68812	−1.39457	−0.697284	+	0.716795i	$0.745609\pi$
$24$	0	0
$25$	−4.73047	−0.946094
$26$	−6.13875	−1.20391
$27$	0	0
$28$	5.25837	0.993738
$29$	−7.68981	−1.42796	−0.713981	−	0.700165i	$-0.753110\pi$
$29$	−7.68981	−1.42796	−0.713981	+	0.700165i	$0.753110\pi$
$30$	0	0
$31$	−1.20467	−0.216365	−0.108182	−	0.994131i	$-0.534503\pi$
$31$	−1.20467	−0.216365	−0.108182	+	0.994131i	$0.534503\pi$
$32$	15.8454	2.80109
$33$	0	0
$34$	−5.13902	−0.881334
$35$	0.530037	0.0895926
$36$	0	0
$37$	−0.791859	−0.130181	−0.0650904	−	0.997879i	$-0.520734\pi$
$37$	−0.791859	−0.130181	−0.0650904	+	0.997879i	$0.520734\pi$
$38$	8.42528	1.36676
$39$	0	0
$40$	4.37372	0.691546
$41$	2.49631	0.389858	0.194929	−	0.980817i	$-0.437552\pi$
$41$	2.49631	0.389858	0.194929	+	0.980817i	$0.437552\pi$
$42$	0	0
$43$	1.15197	0.175674	0.0878370	−	0.996135i	$-0.472005\pi$
$43$	1.15197	0.175674	0.0878370	+	0.996135i	$0.472005\pi$
$44$	26.5227	3.99845
$45$	0	0
$46$	−17.8843	−2.63690
$47$	8.92862	1.30237	0.651187	−	0.758918i	$-0.274272\pi$
$47$	8.92862	1.30237	0.651187	+	0.758918i	$0.274272\pi$
$48$	0	0
$49$	−5.95767	−0.851096
$50$	−12.6495	−1.78891
$51$	0	0
$52$	−11.8239	−1.63968
$53$	−7.62842	−1.04784	−0.523922	−	0.851766i	$-0.675532\pi$
$53$	−7.62842	−1.04784	−0.523922	+	0.851766i	$0.675532\pi$
$54$	0	0
$55$	2.67345	0.360488
$56$	8.60100	1.14936
$57$	0	0
$58$	−20.5629	−2.70004
$59$	9.24454	1.20354	0.601768	−	0.798671i	$-0.294463\pi$
$59$	9.24454	1.20354	0.601768	+	0.798671i	$0.294463\pi$
$60$	0	0
$61$	2.23081	0.285626	0.142813	−	0.989750i	$-0.454385\pi$
$61$	2.23081	0.285626	0.142813	+	0.989750i	$0.454385\pi$
$62$	−3.22134	−0.409110
$63$	0	0
$64$	17.9179	2.23974
$65$	−1.19183	−0.147829
$66$	0	0
$67$	14.8406	1.81307	0.906535	−	0.422130i	$-0.138717\pi$
$67$	14.8406	1.81307	0.906535	+	0.422130i	$0.138717\pi$
$68$	−9.89831	−1.20035
$69$	0	0
$70$	1.41734	0.169405
$71$	−4.46542	−0.529948	−0.264974	−	0.964256i	$-0.585363\pi$
$71$	−4.46542	−0.529948	−0.264974	+	0.964256i	$0.585363\pi$
$72$	0	0
$73$	−2.29621	−0.268751	−0.134376	−	0.990930i	$-0.542903\pi$
$73$	−2.29621	−0.268751	−0.134376	+	0.990930i	$0.542903\pi$
$74$	−2.11746	−0.246150
$75$	0	0
$76$	16.2280	1.86148
$77$	5.25739	0.599136
$78$	0	0
$79$	−9.20506	−1.03565	−0.517825	−	0.855487i	$-0.673258\pi$
$79$	−9.20506	−1.03565	−0.517825	+	0.855487i	$0.673258\pi$
$80$	6.34761	0.709684
$81$	0	0
$82$	6.67524	0.737157
$83$	13.9566	1.53193	0.765967	−	0.642880i	$-0.222261\pi$
$83$	13.9566	1.53193	0.765967	+	0.642880i	$0.222261\pi$
$84$	0	0
$85$	−0.997737	−0.108220
$86$	3.08042	0.332170
$87$	0	0
$88$	43.3826	4.62460
$89$	7.81208	0.828079	0.414039	−	0.910259i	$-0.364118\pi$
$89$	7.81208	0.828079	0.414039	+	0.910259i	$0.364118\pi$
$90$	0	0
$91$	−2.34376	−0.245693
$92$	−34.4471	−3.59136
$93$	0	0
$94$	23.8755	2.46257
$95$	1.63576	0.167826
$96$	0	0
$97$	−4.25925	−0.432461	−0.216231	−	0.976342i	$-0.569376\pi$
$97$	−4.25925	−0.432461	−0.216231	+	0.976342i	$0.569376\pi$
$98$	−15.9311	−1.60928
$99$	0	0
$100$	−24.3643	−2.43643
$101$	16.5592	1.64770	0.823851	−	0.566807i	$-0.191821\pi$
$101$	16.5592	1.64770	0.823851	+	0.566807i	$0.191821\pi$
$102$	0	0
$103$	−13.0011	−1.28103	−0.640517	−	0.767944i	$-0.721280\pi$
$103$	−13.0011	−1.28103	−0.640517	+	0.767944i	$0.721280\pi$
$104$	−19.3401	−1.89645
$105$	0	0
$106$	−20.3987	−1.98130
$107$	−5.26519	−0.509005	−0.254503	−	0.967072i	$-0.581912\pi$
$107$	−5.26519	−0.509005	−0.254503	+	0.967072i	$0.581912\pi$
$108$	0	0
$109$	−10.9722	−1.05095	−0.525475	−	0.850809i	$-0.676112\pi$
$109$	−10.9722	−1.05095	−0.525475	+	0.850809i	$0.676112\pi$
$110$	7.14892	0.681623
$111$	0	0
$112$	12.4827	1.17950
$113$	−8.33557	−0.784144	−0.392072	−	0.919934i	$-0.628242\pi$
$113$	−8.33557	−0.784144	−0.392072	+	0.919934i	$0.628242\pi$
$114$	0	0
$115$	−3.47223	−0.323787
$116$	−39.6064	−3.67736
$117$	0	0
$118$	24.7203	2.27569
$119$	−1.96207	−0.179862
$120$	0	0
$121$	15.5178	1.41071
$122$	5.96527	0.540070
$123$	0	0
$124$	−6.20464	−0.557193
$125$	−5.05170	−0.451838
$126$	0	0
$127$	10.2324	0.907979	0.453990	−	0.891007i	$-0.350000\pi$
$127$	10.2324	0.907979	0.453990	+	0.891007i	$0.350000\pi$
$128$	16.2225	1.43388
$129$	0	0
$130$	−3.18702	−0.279520
$131$	−11.8436	−1.03478	−0.517389	−	0.855750i	$-0.673096\pi$
$131$	−11.8436	−1.03478	−0.517389	+	0.855750i	$0.673096\pi$
$132$	0	0
$133$	3.21676	0.278928
$134$	39.6844	3.42821
$135$	0	0
$136$	−16.1904	−1.38832
$137$	−18.1801	−1.55323	−0.776614	−	0.629977i	$-0.783064\pi$
$137$	−18.1801	−1.55323	−0.776614	+	0.629977i	$0.783064\pi$
$138$	0	0
$139$	3.89676	0.330519	0.165259	−	0.986250i	$-0.447154\pi$
$139$	3.89676	0.330519	0.165259	+	0.986250i	$0.447154\pi$
$140$	2.72995	0.230723
$141$	0	0
$142$	−11.9407	−1.00204
$143$	−11.8217	−0.988582
$144$	0	0
$145$	−3.99227	−0.331540
$146$	−6.14016	−0.508163
$147$	0	0
$148$	−4.07846	−0.335248
$149$	2.58671	0.211911	0.105956	−	0.994371i	$-0.466210\pi$
$149$	2.58671	0.211911	0.105956	+	0.994371i	$0.466210\pi$
$150$	0	0
$151$	−17.0193	−1.38501	−0.692506	−	0.721412i	$-0.743494\pi$
$151$	−17.0193	−1.38501	−0.692506	+	0.721412i	$0.743494\pi$
$152$	26.5438	2.15299
$153$	0	0
$154$	14.0585	1.13286
$155$	−0.625420	−0.0502350
$156$	0	0
$157$	−9.59695	−0.765920	−0.382960	−	0.923765i	$-0.625095\pi$
$157$	−9.59695	−0.765920	−0.382960	+	0.923765i	$0.625095\pi$
$158$	−24.6147	−1.95824
$159$	0	0
$160$	8.22633	0.650349
$161$	−6.82819	−0.538137
$162$	0	0
$163$	−1.07845	−0.0844708	−0.0422354	−	0.999108i	$-0.513448\pi$
$163$	−1.07845	−0.0844708	−0.0422354	+	0.999108i	$0.513448\pi$
$164$	12.8572	1.00398
$165$	0	0
$166$	37.3205	2.89663
$167$	11.9566	0.925230	0.462615	−	0.886559i	$-0.346911\pi$
$167$	11.9566	0.925230	0.462615	+	0.886559i	$0.346911\pi$
$168$	0	0
$169$	−7.72984	−0.594603
$170$	−2.66799	−0.204625
$171$	0	0
$172$	5.93322	0.452404
$173$	16.3039	1.23956	0.619780	−	0.784776i	$-0.287222\pi$
$173$	16.3039	1.23956	0.619780	+	0.784776i	$0.287222\pi$
$174$	0	0
$175$	−4.82954	−0.365079
$176$	62.9615	4.74590
$177$	0	0
$178$	20.8898	1.56576
$179$	−0.105075	−0.00785365	−0.00392682	−	0.999992i	$-0.501250\pi$
$179$	−0.105075	−0.00785365	−0.00392682	+	0.999992i	$0.501250\pi$
$180$	0	0
$181$	−18.0099	−1.33866	−0.669332	−	0.742963i	$-0.733420\pi$
$181$	−18.0099	−1.33866	−0.669332	+	0.742963i	$0.733420\pi$
$182$	−6.26732	−0.464565
$183$	0	0
$184$	−56.3444	−4.15376
$185$	−0.411104	−0.0302250
$186$	0	0
$187$	−9.89647	−0.723702
$188$	45.9868	3.35393
$189$	0	0
$190$	4.37410	0.317330
$191$	−1.54242	−0.111606	−0.0558028	−	0.998442i	$-0.517772\pi$
$191$	−1.54242	−0.111606	−0.0558028	+	0.998442i	$0.517772\pi$
$192$	0	0
$193$	9.49555	0.683505	0.341752	−	0.939790i	$-0.388980\pi$
$193$	9.49555	0.683505	0.341752	+	0.939790i	$0.388980\pi$
$194$	−11.3894	−0.817712
$195$	0	0
$196$	−30.6850	−2.19178
$197$	4.42454	0.315235	0.157618	−	0.987500i	$-0.449619\pi$
$197$	4.42454	0.315235	0.157618	+	0.987500i	$0.449619\pi$
$198$	0	0
$199$	−14.1893	−1.00585	−0.502927	−	0.864329i	$-0.667744\pi$
$199$	−14.1893	−1.00585	−0.502927	+	0.864329i	$0.667744\pi$
$200$	−39.8521	−2.81797
$201$	0	0
$202$	44.2800	3.11553
$203$	−7.85087	−0.551023
$204$	0	0
$205$	1.29599	0.0905161
$206$	−34.7654	−2.42222
$207$	0	0
$208$	−28.0684	−1.94620
$209$	16.2250	1.12231
$210$	0	0
$211$	0.0479471	0.00330081	0.00165041	−	0.999999i	$-0.499475\pi$
$211$	0.0479471	0.00330081	0.00165041	+	0.999999i	$0.499475\pi$
$212$	−39.2901	−2.69846
$213$	0	0
$214$	−14.0793	−0.962444
$215$	0.598061	0.0407874
$216$	0	0
$217$	−1.22990	−0.0834910
$218$	−29.3402	−1.98717
$219$	0	0
$220$	13.7696	0.928347
$221$	4.41188	0.296775
$222$	0	0
$223$	−22.1434	−1.48283	−0.741415	−	0.671047i	$-0.765845\pi$
$223$	−22.1434	−1.48283	−0.741415	+	0.671047i	$0.765845\pi$
$224$	16.1772	1.08089
$225$	0	0
$226$	−22.2896	−1.48269
$227$	25.9911	1.72509	0.862546	−	0.505978i	$-0.168868\pi$
$227$	25.9911	1.72509	0.862546	+	0.505978i	$0.168868\pi$
$228$	0	0
$229$	−16.4825	−1.08919	−0.544596	−	0.838699i	$-0.683317\pi$
$229$	−16.4825	−1.08919	−0.544596	+	0.838699i	$0.683317\pi$
$230$	−9.28487	−0.612226
$231$	0	0
$232$	−64.7833	−4.25323
$233$	8.28328	0.542656	0.271328	−	0.962487i	$-0.412537\pi$
$233$	8.28328	0.542656	0.271328	+	0.962487i	$0.412537\pi$
$234$	0	0
$235$	4.63541	0.302381
$236$	47.6140	3.09940
$237$	0	0
$238$	−5.24665	−0.340090
$239$	1.00000	0.0646846
$239$	1.00000	0.0646846
$240$	0	0
$241$	18.7517	1.20790	0.603951	−	0.797021i	$-0.293592\pi$
$241$	18.7517	1.20790	0.603951	+	0.797021i	$0.293592\pi$
$242$	41.4952	2.66741
$243$	0	0
$244$	11.4898	0.735557
$245$	−3.09301	−0.197605
$246$	0	0
$247$	−7.23316	−0.460235
$248$	−10.1488	−0.644450
$249$	0	0
$250$	−13.5085	−0.854350
$251$	8.36672	0.528103	0.264051	−	0.964509i	$-0.414941\pi$
$251$	8.36672	0.528103	0.264051	+	0.964509i	$0.414941\pi$
$252$	0	0
$253$	−34.4407	−2.16527
$254$	27.3619	1.71684
$255$	0	0
$256$	7.54385	0.471491
$257$	−8.28841	−0.517017	−0.258509	−	0.966009i	$-0.583231\pi$
$257$	−8.28841	−0.517017	−0.258509	+	0.966009i	$0.583231\pi$
$258$	0	0
$259$	−0.808443	−0.0502342
$260$	−6.13854	−0.380696
$261$	0	0
$262$	−31.6702	−1.95659
$263$	−12.7232	−0.784543	−0.392272	−	0.919849i	$-0.628311\pi$
$263$	−12.7232	−0.784543	−0.392272	+	0.919849i	$0.628311\pi$
$264$	0	0
$265$	−3.96040	−0.243285
$266$	8.60174	0.527406
$267$	0	0
$268$	76.4366	4.66911
$269$	7.72368	0.470921	0.235461	−	0.971884i	$-0.424340\pi$
$269$	7.72368	0.470921	0.235461	+	0.971884i	$0.424340\pi$
$270$	0	0
$271$	23.2037	1.40953	0.704764	−	0.709442i	$-0.251053\pi$
$271$	23.2037	1.40953	0.704764	+	0.709442i	$0.251053\pi$
$272$	−23.4973	−1.42473
$273$	0	0
$274$	−48.6142	−2.93689
$275$	−24.3597	−1.46895
$276$	0	0
$277$	−5.69405	−0.342123	−0.171061	−	0.985260i	$-0.554720\pi$
$277$	−5.69405	−0.342123	−0.171061	+	0.985260i	$0.554720\pi$
$278$	10.4201	0.624956
$279$	0	0
$280$	4.46532	0.266854
$281$	26.2890	1.56827	0.784134	−	0.620592i	$-0.213108\pi$
$281$	26.2890	1.56827	0.784134	+	0.620592i	$0.213108\pi$
$282$	0	0
$283$	21.4629	1.27584	0.637918	−	0.770104i	$-0.279796\pi$
$283$	21.4629	1.27584	0.637918	+	0.770104i	$0.279796\pi$
$284$	−22.9991	−1.36475
$285$	0	0
$286$	−31.6118	−1.86924
$287$	2.54859	0.150439
$288$	0	0
$289$	−13.3066	−0.782742
$290$	−10.6755	−0.626887
$291$	0	0
$292$	−11.8266	−0.692101
$293$	−15.8224	−0.924356	−0.462178	−	0.886787i	$-0.652932\pi$
$293$	−15.8224	−0.924356	−0.462178	+	0.886787i	$0.652932\pi$
$294$	0	0
$295$	4.79943	0.279433
$296$	−6.67106	−0.387747
$297$	0	0
$298$	6.91696	0.400689
$299$	15.3538	0.887933
$300$	0	0
$301$	1.17610	0.0677891
$302$	−45.5104	−2.61883
$303$	0	0
$304$	38.5232	2.20946
$305$	1.15815	0.0663157
$306$	0	0
$307$	−11.3738	−0.649137	−0.324569	−	0.945862i	$-0.605219\pi$
$307$	−11.3738	−0.649137	−0.324569	+	0.945862i	$0.605219\pi$
$308$	27.0782	1.54292
$309$	0	0
$310$	−1.67240	−0.0949859
$311$	19.7338	1.11900	0.559501	−	0.828830i	$-0.310993\pi$
$311$	19.7338	1.11900	0.559501	+	0.828830i	$0.310993\pi$
$312$	0	0
$313$	7.24433	0.409474	0.204737	−	0.978817i	$-0.434366\pi$
$313$	7.24433	0.409474	0.204737	+	0.978817i	$0.434366\pi$
$314$	−25.6626	−1.44823
$315$	0	0
$316$	−47.4106	−2.66705
$317$	11.0779	0.622199	0.311099	−	0.950377i	$-0.399303\pi$
$317$	11.0779	0.622199	0.311099	+	0.950377i	$0.399303\pi$
$318$	0	0
$319$	−39.5990	−2.21712
$320$	9.30233	0.520016
$321$	0	0
$322$	−18.2589	−1.01753
$323$	−6.05519	−0.336920
$324$	0	0
$325$	10.8597	0.602386
$326$	−2.88382	−0.159720
$327$	0	0
$328$	21.0303	1.16120
$329$	9.11562	0.502560
$330$	0	0
$331$	11.7234	0.644374	0.322187	−	0.946676i	$-0.395582\pi$
$331$	11.7234	0.644374	0.322187	+	0.946676i	$0.395582\pi$
$332$	71.8833	3.94511
$333$	0	0
$334$	31.9724	1.74945
$335$	7.70471	0.420953
$336$	0	0
$337$	−28.6376	−1.55999	−0.779995	−	0.625786i	$-0.784778\pi$
$337$	−28.6376	−1.55999	−0.779995	+	0.625786i	$0.784778\pi$
$338$	−20.6699	−1.12429
$339$	0	0
$340$	−5.13884	−0.278693
$341$	−6.20349	−0.335938
$342$	0	0
$343$	−13.2291	−0.714302
$344$	9.70484	0.523250
$345$	0	0
$346$	43.5972	2.34380
$347$	−1.96621	−0.105552	−0.0527760	−	0.998606i	$-0.516807\pi$
$347$	−1.96621	−0.105552	−0.0527760	+	0.998606i	$0.516807\pi$
$348$	0	0
$349$	−1.25673	−0.0672713	−0.0336357	−	0.999434i	$-0.510709\pi$
$349$	−1.25673	−0.0672713	−0.0336357	+	0.999434i	$0.510709\pi$
$350$	−12.9144	−0.690304
$351$	0	0
$352$	81.5963	4.34910
$353$	21.0991	1.12299	0.561495	−	0.827480i	$-0.310226\pi$
$353$	21.0991	1.12299	0.561495	+	0.827480i	$0.310226\pi$
$354$	0	0
$355$	−2.31828	−0.123042
$356$	40.2361	2.13251
$357$	0	0
$358$	−0.280974	−0.0148499
$359$	17.5335	0.925382	0.462691	−	0.886520i	$-0.346884\pi$
$359$	17.5335	0.925382	0.462691	+	0.886520i	$0.346884\pi$
$360$	0	0
$361$	−9.07267	−0.477509
$362$	−48.1592	−2.53119
$363$	0	0
$364$	−12.0715	−0.632721
$365$	−1.19211	−0.0623978
$366$	0	0
$367$	28.4166	1.48334	0.741668	−	0.670767i	$-0.234035\pi$
$367$	28.4166	1.48334	0.741668	+	0.670767i	$0.234035\pi$
$368$	−81.7730	−4.26271
$369$	0	0
$370$	−1.09931	−0.0571503
$371$	−7.78819	−0.404343
$372$	0	0
$373$	22.4497	1.16240	0.581201	−	0.813760i	$-0.302583\pi$
$373$	22.4497	1.16240	0.581201	+	0.813760i	$0.302583\pi$
$374$	−26.4636	−1.36840
$375$	0	0
$376$	75.2197	3.87916
$377$	17.6534	0.909195
$378$	0	0
$379$	16.8689	0.866499	0.433250	−	0.901274i	$-0.357367\pi$
$379$	16.8689	0.866499	0.433250	+	0.901274i	$0.357367\pi$
$380$	8.42499	0.432193
$381$	0	0
$382$	−4.12449	−0.211027
$383$	−15.0924	−0.771184	−0.385592	−	0.922669i	$-0.626003\pi$
$383$	−15.0924	−0.771184	−0.385592	+	0.922669i	$0.626003\pi$
$384$	0	0
$385$	2.72945	0.139105
$386$	25.3915	1.29239
$387$	0	0
$388$	−21.9373	−1.11370
$389$	4.28146	0.217079	0.108539	−	0.994092i	$-0.465383\pi$
$389$	4.28146	0.217079	0.108539	+	0.994092i	$0.465383\pi$
$390$	0	0
$391$	12.8533	0.650021
$392$	−50.1907	−2.53502
$393$	0	0
$394$	11.8314	0.596057
$395$	−4.77893	−0.240454
$396$	0	0
$397$	−26.2285	−1.31637	−0.658186	−	0.752855i	$-0.728676\pi$
$397$	−26.2285	−1.31637	−0.658186	+	0.752855i	$0.728676\pi$
$398$	−37.9429	−1.90190
$399$	0	0
$400$	−57.8376	−2.89188
$401$	−1.34368	−0.0671001	−0.0335500	−	0.999437i	$-0.510681\pi$
$401$	−1.34368	−0.0671001	−0.0335500	+	0.999437i	$0.510681\pi$
$402$	0	0
$403$	2.76554	0.137761
$404$	85.2881	4.24324
$405$	0	0
$406$	−20.9935	−1.04189
$407$	−4.07771	−0.202124
$408$	0	0
$409$	14.8377	0.733677	0.366838	−	0.930285i	$-0.380440\pi$
$409$	14.8377	0.733677	0.366838	+	0.930285i	$0.380440\pi$
$410$	3.46554	0.171151
$411$	0	0
$412$	−66.9620	−3.29898
$413$	9.43816	0.464421
$414$	0	0
$415$	7.24575	0.355680
$416$	−36.3759	−1.78348
$417$	0	0
$418$	43.3863	2.12209
$419$	33.9802	1.66004	0.830021	−	0.557731i	$-0.188328\pi$
$419$	33.9802	1.66004	0.830021	+	0.557731i	$0.188328\pi$
$420$	0	0
$421$	40.1479	1.95669	0.978344	−	0.206985i	$-0.0663652\pi$
$421$	40.1479	1.95669	0.978344	+	0.206985i	$0.0663652\pi$
$422$	0.128213	0.00624129
$423$	0	0
$424$	−64.2660	−3.12103
$425$	9.09109	0.440983
$426$	0	0
$427$	2.27753	0.110217
$428$	−27.1183	−1.31081
$429$	0	0
$430$	1.59924	0.0771222
$431$	25.2309	1.21533	0.607665	−	0.794194i	$-0.292106\pi$
$431$	25.2309	1.21533	0.607665	+	0.794194i	$0.292106\pi$
$432$	0	0
$433$	−31.2782	−1.50313	−0.751567	−	0.659657i	$-0.770702\pi$
$433$	−31.2782	−1.50313	−0.751567	+	0.659657i	$0.770702\pi$
$434$	−3.28880	−0.157868
$435$	0	0
$436$	−56.5124	−2.70645
$437$	−21.0727	−1.00804
$438$	0	0
$439$	−4.15467	−0.198291	−0.0991457	−	0.995073i	$-0.531611\pi$
$439$	−4.15467	−0.198291	−0.0991457	+	0.995073i	$0.531611\pi$
$440$	22.5227	1.07373
$441$	0	0
$442$	11.7976	0.561152
$443$	−20.0339	−0.951840	−0.475920	−	0.879489i	$-0.657885\pi$
$443$	−20.0339	−0.951840	−0.475920	+	0.879489i	$0.657885\pi$
$444$	0	0
$445$	4.05575	0.192261
$446$	−59.2123	−2.80378
$447$	0	0
$448$	18.2932	0.864272
$449$	15.7631	0.743906	0.371953	−	0.928252i	$-0.378688\pi$
$449$	15.7631	0.743906	0.371953	+	0.928252i	$0.378688\pi$
$450$	0	0
$451$	12.8549	0.605312
$452$	−42.9323	−2.01937
$453$	0	0
$454$	69.5014	3.26186
$455$	−1.21680	−0.0570443
$456$	0	0
$457$	14.3920	0.673227	0.336614	−	0.941643i	$-0.390718\pi$
$457$	14.3920	0.673227	0.336614	+	0.941643i	$0.390718\pi$
$458$	−44.0748	−2.05948
$459$	0	0
$460$	−17.8837	−0.833831
$461$	9.25187	0.430903	0.215451	−	0.976515i	$-0.430878\pi$
$461$	9.25187	0.430903	0.215451	+	0.976515i	$0.430878\pi$
$462$	0	0
$463$	9.95525	0.462660	0.231330	−	0.972875i	$-0.425692\pi$
$463$	9.95525	0.462660	0.231330	+	0.972875i	$0.425692\pi$
$464$	−94.0204	−4.36479
$465$	0	0
$466$	22.1498	1.02607
$467$	−18.9730	−0.877964	−0.438982	−	0.898496i	$-0.644661\pi$
$467$	−18.9730	−0.877964	−0.438982	+	0.898496i	$0.644661\pi$
$468$	0	0
$469$	15.1514	0.699629
$470$	12.3953	0.571752
$471$	0	0
$472$	77.8811	3.58477
$473$	5.93212	0.272759
$474$	0	0
$475$	−14.9046	−0.683870
$476$	−10.1056	−0.463190
$477$	0	0
$478$	2.67404	0.122308
$479$	−33.1598	−1.51511	−0.757555	−	0.652771i	$-0.773606\pi$
$479$	−33.1598	−1.51511	−0.757555	+	0.652771i	$0.773606\pi$
$480$	0	0
$481$	1.81786	0.0828871
$482$	50.1428	2.28394
$483$	0	0
$484$	79.9242	3.63292
$485$	−2.21125	−0.100408
$486$	0	0
$487$	−37.0030	−1.67677	−0.838384	−	0.545080i	$-0.816499\pi$
$487$	−37.0030	−1.67677	−0.838384	+	0.545080i	$0.816499\pi$
$488$	18.7936	0.850745
$489$	0	0
$490$	−8.27083	−0.373638
$491$	−15.5770	−0.702980	−0.351490	−	0.936192i	$-0.614325\pi$
$491$	−15.5770	−0.702980	−0.351490	+	0.936192i	$0.614325\pi$
$492$	0	0
$493$	14.7784	0.665586
$494$	−19.3418	−0.870227
$495$	0	0
$496$	−14.7290	−0.661353
$497$	−4.55895	−0.204497
$498$	0	0
$499$	−3.63382	−0.162672	−0.0813360	−	0.996687i	$-0.525919\pi$
$499$	−3.63382	−0.162672	−0.0813360	+	0.996687i	$0.525919\pi$
$500$	−26.0188	−1.16360
$501$	0	0
$502$	22.3730	0.998554
$503$	−42.5761	−1.89838	−0.949188	−	0.314710i	$-0.898093\pi$
$503$	−42.5761	−1.89838	−0.949188	+	0.314710i	$0.898093\pi$
$504$	0	0
$505$	8.59693	0.382558
$506$	−92.0959	−4.09416
$507$	0	0
$508$	52.7020	2.33827
$509$	35.1080	1.55614	0.778068	−	0.628180i	$-0.216200\pi$
$509$	35.1080	1.55614	0.778068	+	0.628180i	$0.216200\pi$
$510$	0	0
$511$	−2.34430	−0.103706
$512$	−12.2725	−0.542371
$513$	0	0
$514$	−22.1636	−0.977593
$515$	−6.74968	−0.297426
$516$	0	0
$517$	45.9783	2.02212
$518$	−2.16181	−0.0949845
$519$	0	0
$520$	−10.0407	−0.440313
$521$	36.0412	1.57899	0.789497	−	0.613754i	$-0.210341\pi$
$521$	36.0412	1.57899	0.789497	+	0.613754i	$0.210341\pi$
$522$	0	0
$523$	0.568975	0.0248795	0.0124398	−	0.999923i	$-0.496040\pi$
$523$	0.568975	0.0248795	0.0124398	+	0.999923i	$0.496040\pi$
$524$	−61.0003	−2.66481
$525$	0	0
$526$	−34.0222	−1.48344
$527$	2.31515	0.100850
$528$	0	0
$529$	21.7309	0.944822
$530$	−10.5903	−0.460012
$531$	0	0
$532$	16.5679	0.718309
$533$	−5.73074	−0.248226
$534$	0	0
$535$	−2.73349	−0.118179
$536$	125.026	5.40029
$537$	0	0
$538$	20.6534	0.890433
$539$	−30.6793	−1.32145
$540$	0	0
$541$	12.1080	0.520565	0.260283	−	0.965533i	$-0.416184\pi$
$541$	12.1080	0.520565	0.260283	+	0.965533i	$0.416184\pi$
$542$	62.0478	2.66518
$543$	0	0
$544$	−30.4519	−1.30561
$545$	−5.69638	−0.244006
$546$	0	0
$547$	40.4919	1.73131	0.865654	−	0.500642i	$-0.166903\pi$
$547$	40.4919	1.73131	0.865654	+	0.500642i	$0.166903\pi$
$548$	−93.6363	−3.99995
$549$	0	0
$550$	−65.1390	−2.77753
$551$	−24.2288	−1.03218
$552$	0	0
$553$	−9.39785	−0.399637
$554$	−15.2261	−0.646896
$555$	0	0
$556$	20.0702	0.851168
$557$	−26.3904	−1.11820	−0.559098	−	0.829101i	$-0.688853\pi$
$557$	−26.3904	−1.11820	−0.559098	+	0.829101i	$0.688853\pi$
$558$	0	0
$559$	−2.64456	−0.111853
$560$	6.48055	0.273853
$561$	0	0
$562$	70.2977	2.96533
$563$	−30.2542	−1.27506	−0.637532	−	0.770424i	$-0.720045\pi$
$563$	−30.2542	−1.27506	−0.637532	+	0.770424i	$0.720045\pi$
$564$	0	0
$565$	−4.32752	−0.182060
$566$	57.3927	2.41239
$567$	0	0
$568$	−37.6192	−1.57847
$569$	4.56920	0.191551	0.0957754	−	0.995403i	$-0.469467\pi$
$569$	4.56920	0.191551	0.0957754	+	0.995403i	$0.469467\pi$
$570$	0	0
$571$	−10.4991	−0.439375	−0.219687	−	0.975570i	$-0.570504\pi$
$571$	−10.4991	−0.439375	−0.219687	+	0.975570i	$0.570504\pi$
$572$	−60.8877	−2.54584
$573$	0	0
$574$	6.81505	0.284455
$575$	31.6379	1.31939
$576$	0	0
$577$	−3.62882	−0.151070	−0.0755349	−	0.997143i	$-0.524066\pi$
$577$	−3.62882	−0.151070	−0.0755349	+	0.997143i	$0.524066\pi$
$578$	−35.5825	−1.48003
$579$	0	0
$580$	−20.5622	−0.853798
$581$	14.2489	0.591143
$582$	0	0
$583$	−39.2829	−1.62693
$584$	−19.3446	−0.800483
$585$	0	0
$586$	−42.3098	−1.74780
$587$	22.5623	0.931244	0.465622	−	0.884984i	$-0.345831\pi$
$587$	22.5623	0.931244	0.465622	+	0.884984i	$0.345831\pi$
$588$	0	0
$589$	−3.79563	−0.156396
$590$	12.8339	0.528362
$591$	0	0
$592$	−9.68175	−0.397917
$593$	−26.6646	−1.09498	−0.547492	−	0.836811i	$-0.684417\pi$
$593$	−26.6646	−1.09498	−0.547492	+	0.836811i	$0.684417\pi$
$594$	0	0
$595$	−1.01863	−0.0417599
$596$	13.3228	0.545724
$597$	0	0
$598$	41.0567	1.67893
$599$	29.4480	1.20321	0.601605	−	0.798793i	$-0.294528\pi$
$599$	29.4480	1.20321	0.601605	+	0.798793i	$0.294528\pi$
$600$	0	0
$601$	−33.9235	−1.38377	−0.691884	−	0.722009i	$-0.743219\pi$
$601$	−33.9235	−1.38377	−0.691884	+	0.722009i	$0.743219\pi$
$602$	3.14493	0.128178
$603$	0	0
$604$	−87.6579	−3.56675
$605$	8.05626	0.327534
$606$	0	0
$607$	0.484162	0.0196515	0.00982577	−	0.999952i	$-0.496872\pi$
$607$	0.484162	0.0196515	0.00982577	+	0.999952i	$0.496872\pi$
$608$	49.9250	2.02473
$609$	0	0
$610$	3.09695	0.125392
$611$	−20.4973	−0.829231
$612$	0	0
$613$	−20.8655	−0.842750	−0.421375	−	0.906886i	$-0.638452\pi$
$613$	−20.8655	−0.842750	−0.421375	+	0.906886i	$0.638452\pi$
$614$	−30.4140	−1.22741
$615$	0	0
$616$	44.2912	1.78454
$617$	−13.7272	−0.552637	−0.276319	−	0.961066i	$-0.589115\pi$
$617$	−13.7272	−0.552637	−0.276319	+	0.961066i	$0.589115\pi$
$618$	0	0
$619$	−45.3078	−1.82107	−0.910536	−	0.413429i	$-0.864331\pi$
$619$	−45.3078	−1.82107	−0.910536	+	0.413429i	$0.864331\pi$
$620$	−3.22122	−0.129367
$621$	0	0
$622$	52.7690	2.11585
$623$	7.97569	0.319539
$624$	0	0
$625$	21.0297	0.841187
$626$	19.3716	0.774247
$627$	0	0
$628$	−49.4290	−1.97243
$629$	1.52181	0.0606784
$630$	0	0
$631$	27.0302	1.07605	0.538027	−	0.842927i	$-0.319170\pi$
$631$	27.0302	1.07605	0.538027	+	0.842927i	$0.319170\pi$
$632$	−77.5485	−3.08471
$633$	0	0
$634$	29.6228	1.17647
$635$	5.31229	0.210812
$636$	0	0
$637$	13.6769	0.541900
$638$	−105.889	−4.19220
$639$	0	0
$640$	8.42214	0.332914
$641$	44.9994	1.77737	0.888684	−	0.458520i	$-0.151620\pi$
$641$	44.9994	1.77737	0.888684	+	0.458520i	$0.151620\pi$
$642$	0	0
$643$	−43.2140	−1.70419	−0.852097	−	0.523384i	$-0.824669\pi$
$643$	−43.2140	−1.70419	−0.852097	+	0.523384i	$0.824669\pi$
$644$	−35.1686	−1.38584
$645$	0	0
$646$	−16.1918	−0.637059
$647$	17.6690	0.694642	0.347321	−	0.937746i	$-0.387091\pi$
$647$	17.6690	0.694642	0.347321	+	0.937746i	$0.387091\pi$
$648$	0	0
$649$	47.6051	1.86867
$650$	29.0392	1.13901
$651$	0	0
$652$	−5.55456	−0.217533
$653$	38.1713	1.49376	0.746879	−	0.664960i	$-0.231551\pi$
$653$	38.1713	1.49376	0.746879	+	0.664960i	$0.231551\pi$
$654$	0	0
$655$	−6.14876	−0.240252
$656$	30.5214	1.19166
$657$	0	0
$658$	24.3755	0.950258
$659$	13.7251	0.534655	0.267328	−	0.963606i	$-0.413859\pi$
$659$	13.7251	0.534655	0.267328	+	0.963606i	$0.413859\pi$
$660$	0	0
$661$	31.8679	1.23952	0.619759	−	0.784793i	$-0.287231\pi$
$661$	31.8679	1.23952	0.619759	+	0.784793i	$0.287231\pi$
$662$	31.3488	1.21840
$663$	0	0
$664$	117.578	4.56291
$665$	1.67002	0.0647607
$666$	0	0
$667$	51.4304	1.99139
$668$	61.5824	2.38270
$669$	0	0
$670$	20.6027	0.795952
$671$	11.4876	0.443475
$672$	0	0
$673$	−33.3721	−1.28640	−0.643200	−	0.765698i	$-0.722394\pi$
$673$	−33.3721	−1.28640	−0.643200	+	0.765698i	$0.722394\pi$
$674$	−76.5781	−2.94968
$675$	0	0
$676$	−39.8125	−1.53125
$677$	−36.3936	−1.39872	−0.699359	−	0.714770i	$-0.746531\pi$
$677$	−36.3936	−1.39872	−0.699359	+	0.714770i	$0.746531\pi$
$678$	0	0
$679$	−4.34846	−0.166878
$680$	−8.40549	−0.322336
$681$	0	0
$682$	−16.5884	−0.635203
$683$	22.4689	0.859748	0.429874	−	0.902889i	$-0.358558\pi$
$683$	22.4689	0.859748	0.429874	+	0.902889i	$0.358558\pi$
$684$	0	0
$685$	−9.43842	−0.360624
$686$	−35.3750	−1.35063
$687$	0	0
$688$	14.0847	0.536974
$689$	17.5124	0.667171
$690$	0	0
$691$	−19.7002	−0.749430	−0.374715	−	0.927140i	$-0.622260\pi$
$691$	−19.7002	−0.749430	−0.374715	+	0.927140i	$0.622260\pi$
$692$	83.9730	3.19217
$693$	0	0
$694$	−5.25774	−0.199581
$695$	2.02305	0.0767388
$696$	0	0
$697$	−4.79745	−0.181716
$698$	−3.36055	−0.127199
$699$	0	0
$700$	−24.8745	−0.940169
$701$	−26.1013	−0.985833	−0.492917	−	0.870077i	$-0.664069\pi$
$701$	−26.1013	−0.985833	−0.492917	+	0.870077i	$0.664069\pi$
$702$	0	0
$703$	−2.49496	−0.0940992
$704$	92.2690	3.47752
$705$	0	0
$706$	56.4197	2.12339
$707$	16.9060	0.635816
$708$	0	0
$709$	38.7834	1.45654	0.728270	−	0.685291i	$-0.240325\pi$
$709$	38.7834	1.45654	0.728270	+	0.685291i	$0.240325\pi$
$710$	−6.19919	−0.232651
$711$	0	0
$712$	65.8133	2.46646
$713$	8.05697	0.301736
$714$	0	0
$715$	−6.13740	−0.229526
$716$	−0.541186	−0.0202251
$717$	0	0
$718$	46.8852	1.74974
$719$	−28.2950	−1.05522	−0.527612	−	0.849485i	$-0.676913\pi$
$719$	−28.2950	−1.05522	−0.527612	+	0.849485i	$0.676913\pi$
$720$	0	0
$721$	−13.2734	−0.494326
$722$	−24.2607	−0.902890
$723$	0	0
$724$	−92.7598	−3.44739
$725$	36.3764	1.35099
$726$	0	0
$727$	−5.95426	−0.220831	−0.110416	−	0.993886i	$-0.535218\pi$
$727$	−5.95426	−0.220831	−0.110416	+	0.993886i	$0.535218\pi$
$728$	−19.7452	−0.731805
$729$	0	0
$730$	−3.18775	−0.117984
$731$	−2.21388	−0.0818832
$732$	0	0
$733$	38.6216	1.42652	0.713260	−	0.700900i	$-0.247218\pi$
$733$	38.6216	1.42652	0.713260	+	0.700900i	$0.247218\pi$
$734$	75.9873	2.80474
$735$	0	0
$736$	−105.976	−3.90631
$737$	76.4224	2.81506
$738$	0	0
$739$	52.1755	1.91931	0.959654	−	0.281183i	$-0.0907266\pi$
$739$	52.1755	1.91931	0.959654	+	0.281183i	$0.0907266\pi$
$740$	−2.11739	−0.0778368
$741$	0	0
$742$	−20.8259	−0.764544
$743$	10.6094	0.389223	0.194611	−	0.980880i	$-0.437655\pi$
$743$	10.6094	0.389223	0.194611	+	0.980880i	$0.437655\pi$
$744$	0	0
$745$	1.34292	0.0492009
$746$	60.0315	2.19791
$747$	0	0
$748$	−50.9717	−1.86371
$749$	−5.37546	−0.196415
$750$	0	0
$751$	1.96839	0.0718277	0.0359139	−	0.999355i	$-0.488566\pi$
$751$	1.96839	0.0718277	0.0359139	+	0.999355i	$0.488566\pi$
$752$	109.167	3.98090
$753$	0	0
$754$	47.2059	1.71914
$755$	−8.83581	−0.321568
$756$	0	0
$757$	42.9685	1.56172	0.780859	−	0.624707i	$-0.214782\pi$
$757$	42.9685	1.56172	0.780859	+	0.624707i	$0.214782\pi$
$758$	45.1082	1.63840
$759$	0	0
$760$	13.7806	0.499874
$761$	4.94581	0.179285	0.0896427	−	0.995974i	$-0.471427\pi$
$761$	4.94581	0.179285	0.0896427	+	0.995974i	$0.471427\pi$
$762$	0	0
$763$	−11.2020	−0.405541
$764$	−7.94423	−0.287412
$765$	0	0
$766$	−40.3576	−1.45818
$767$	−21.2225	−0.766301
$768$	0	0
$769$	−20.4312	−0.736769	−0.368385	−	0.929674i	$-0.620089\pi$
$769$	−20.4312	−0.736769	−0.368385	+	0.929674i	$0.620089\pi$
$770$	7.29865	0.263025
$771$	0	0
$772$	48.9068	1.76019
$773$	−5.93494	−0.213465	−0.106733	−	0.994288i	$-0.534039\pi$
$773$	−5.93494	−0.213465	−0.106733	+	0.994288i	$0.534039\pi$
$774$	0	0
$775$	5.69865	0.204702
$776$	−35.8823	−1.28810
$777$	0	0
$778$	11.4488	0.410460
$779$	7.86529	0.281803
$780$	0	0
$781$	−22.9949	−0.822821
$782$	34.3703	1.22908
$783$	0	0
$784$	−72.8422	−2.60151
$785$	−4.98238	−0.177829
$786$	0	0
$787$	26.9085	0.959184	0.479592	−	0.877492i	$-0.340785\pi$
$787$	26.9085	0.959184	0.479592	+	0.877492i	$0.340785\pi$
$788$	22.7886	0.811809
$789$	0	0
$790$	−12.7790	−0.454658
$791$	−8.51015	−0.302586
$792$	0	0
$793$	−5.12123	−0.181860
$794$	−70.1362	−2.48904
$795$	0	0
$796$	−73.0821	−2.59033
$797$	−0.527733	−0.0186933	−0.00934663	−	0.999956i	$-0.502975\pi$
$797$	−0.527733	−0.0186933	−0.00934663	+	0.999956i	$0.502975\pi$
$798$	0	0
$799$	−17.1592	−0.607048
$800$	−74.9560	−2.65009
$801$	0	0
$802$	−3.59305	−0.126875
$803$	−11.8244	−0.417275
$804$	0	0
$805$	−3.54495	−0.124943
$806$	7.39517	0.260484
$807$	0	0
$808$	139.504	4.90773
$809$	29.8806	1.05055	0.525273	−	0.850934i	$-0.323963\pi$
$809$	29.8806	1.05055	0.525273	+	0.850934i	$0.323963\pi$
$810$	0	0
$811$	−0.241148	−0.00846783	−0.00423392	−	0.999991i	$-0.501348\pi$
$811$	−0.241148	−0.00846783	−0.00423392	+	0.999991i	$0.501348\pi$
$812$	−40.4359	−1.41902
$813$	0	0
$814$	−10.9040	−0.382184
$815$	−0.559893	−0.0196122
$816$	0	0
$817$	3.62959	0.126983
$818$	39.6766	1.38726
$819$	0	0
$820$	6.67501	0.233101
$821$	−12.4693	−0.435183	−0.217592	−	0.976040i	$-0.569820\pi$
$821$	−12.4693	−0.435183	−0.217592	+	0.976040i	$0.569820\pi$
$822$	0	0
$823$	−17.5737	−0.612582	−0.306291	−	0.951938i	$-0.599088\pi$
$823$	−17.5737	−0.612582	−0.306291	+	0.951938i	$0.599088\pi$
$824$	−109.528	−3.81560
$825$	0	0
$826$	25.2380	0.878143
$827$	27.0166	0.939458	0.469729	−	0.882811i	$-0.344352\pi$
$827$	27.0166	0.939458	0.469729	+	0.882811i	$0.344352\pi$
$828$	0	0
$829$	−34.3187	−1.19194	−0.595969	−	0.803008i	$-0.703232\pi$
$829$	−34.3187	−1.19194	−0.595969	+	0.803008i	$0.703232\pi$
$830$	19.3754	0.672531
$831$	0	0
$832$	−41.1339	−1.42606
$833$	11.4496	0.396704
$834$	0	0
$835$	6.20743	0.214817
$836$	83.5668	2.89022
$837$	0	0
$838$	90.8645	3.13886
$839$	−57.5893	−1.98820	−0.994101	−	0.108454i	$-0.965410\pi$
$839$	−57.5893	−1.98820	−0.994101	+	0.108454i	$0.965410\pi$
$840$	0	0
$841$	30.1332	1.03908
$842$	107.357	3.69977
$843$	0	0
$844$	0.246951	0.00850042
$845$	−4.01305	−0.138053
$846$	0	0
$847$	15.8428	0.544364
$848$	−93.2697	−3.20290
$849$	0	0
$850$	24.3100	0.833825
$851$	5.29604	0.181546
$852$	0	0
$853$	4.45708	0.152607	0.0763037	−	0.997085i	$-0.475688\pi$
$853$	4.45708	0.152607	0.0763037	+	0.997085i	$0.475688\pi$
$854$	6.09021	0.208403
$855$	0	0
$856$	−44.3569	−1.51609
$857$	14.3007	0.488503	0.244252	−	0.969712i	$-0.421458\pi$
$857$	14.3007	0.488503	0.244252	+	0.969712i	$0.421458\pi$
$858$	0	0
$859$	56.6289	1.93215	0.966077	−	0.258256i	$-0.0831477\pi$
$859$	56.6289	1.93215	0.966077	+	0.258256i	$0.0831477\pi$
$860$	3.08031	0.105038
$861$	0	0
$862$	67.4685	2.29799
$863$	21.5070	0.732106	0.366053	−	0.930594i	$-0.380709\pi$
$863$	21.5070	0.732106	0.366053	+	0.930594i	$0.380709\pi$
$864$	0	0
$865$	8.46437	0.287797
$866$	−83.6392	−2.84217
$867$	0	0
$868$	−6.33459	−0.215010
$869$	−47.4018	−1.60800
$870$	0	0
$871$	−34.0694	−1.15440
$872$	−92.4362	−3.13028
$873$	0	0
$874$	−56.3492	−1.90604
$875$	−5.15751	−0.174356
$876$	0	0
$877$	−9.70284	−0.327642	−0.163821	−	0.986490i	$-0.552382\pi$
$877$	−9.70284	−0.327642	−0.163821	+	0.986490i	$0.552382\pi$
$878$	−11.1098	−0.374936
$879$	0	0
$880$	32.6873	1.10189
$881$	10.7139	0.360960	0.180480	−	0.983579i	$-0.442235\pi$
$881$	10.7139	0.360960	0.180480	+	0.983579i	$0.442235\pi$
$882$	0	0
$883$	16.9553	0.570592	0.285296	−	0.958440i	$-0.407908\pi$
$883$	16.9553	0.570592	0.285296	+	0.958440i	$0.407908\pi$
$884$	22.7234	0.764270
$885$	0	0
$886$	−53.5715	−1.79977
$887$	13.5417	0.454687	0.227343	−	0.973815i	$-0.426996\pi$
$887$	13.5417	0.454687	0.227343	+	0.973815i	$0.426996\pi$
$888$	0	0
$889$	10.4467	0.350371
$890$	10.8452	0.363533
$891$	0	0
$892$	−114.049	−3.81865
$893$	28.1320	0.941401
$894$	0	0
$895$	−0.0545509	−0.00182344
$896$	16.5623	0.553307
$897$	0	0
$898$	42.1511	1.40660
$899$	9.26368	0.308961
$900$	0	0
$901$	14.6604	0.488409
$902$	34.3744	1.14454
$903$	0	0
$904$	−70.2234	−2.33560
$905$	−9.35007	−0.310807
$906$	0	0
$907$	6.47529	0.215008	0.107504	−	0.994205i	$-0.465714\pi$
$907$	6.47529	0.215008	0.107504	+	0.994205i	$0.465714\pi$
$908$	133.867	4.44254
$909$	0	0
$910$	−3.25376	−0.107861
$911$	32.6698	1.08240	0.541199	−	0.840895i	$-0.317971\pi$
$911$	32.6698	1.08240	0.541199	+	0.840895i	$0.317971\pi$
$912$	0	0
$913$	71.8700	2.37855
$914$	38.4847	1.27296
$915$	0	0
$916$	−84.8928	−2.80494
$917$	−12.0916	−0.399301
$918$	0	0
$919$	−1.45226	−0.0479055	−0.0239527	−	0.999713i	$-0.507625\pi$
$919$	−1.45226	−0.0479055	−0.0239527	+	0.999713i	$0.507625\pi$
$920$	−29.2520	−0.964408
$921$	0	0
$922$	24.7399	0.814765
$923$	10.2512	0.337422
$924$	0	0
$925$	3.74586	0.123163
$926$	26.6207	0.874812
$927$	0	0
$928$	−121.848	−3.99985
$929$	−37.4415	−1.22842	−0.614208	−	0.789144i	$-0.710524\pi$
$929$	−37.4415	−1.22842	−0.614208	+	0.789144i	$0.710524\pi$
$930$	0	0
$931$	−18.7712	−0.615202
$932$	42.6630	1.39747
$933$	0	0
$934$	−50.7345	−1.66008
$935$	−5.13789	−0.168027
$936$	0	0
$937$	−15.6947	−0.512724	−0.256362	−	0.966581i	$-0.582524\pi$
$937$	−15.6947	−0.512724	−0.256362	+	0.966581i	$0.582524\pi$
$938$	40.5156	1.32288
$939$	0	0
$940$	23.8747	0.778706
$941$	25.7719	0.840140	0.420070	−	0.907492i	$-0.362006\pi$
$941$	25.7719	0.840140	0.420070	+	0.907492i	$0.362006\pi$
$942$	0	0
$943$	−16.6956	−0.543684
$944$	113.029	3.67879
$945$	0	0
$946$	15.8627	0.515742
$947$	−53.4780	−1.73780	−0.868900	−	0.494988i	$-0.835173\pi$
$947$	−53.4780	−1.73780	−0.868900	+	0.494988i	$0.835173\pi$
$948$	0	0
$949$	5.27137	0.171116
$950$	−39.8555	−1.29308
$951$	0	0
$952$	−16.5295	−0.535725
$953$	−24.6066	−0.797085	−0.398542	−	0.917150i	$-0.630484\pi$
$953$	−24.6066	−0.797085	−0.398542	+	0.917150i	$0.630484\pi$
$954$	0	0
$955$	−0.800768	−0.0259122
$956$	5.15050	0.166579
$957$	0	0
$958$	−88.6707	−2.86482
$959$	−18.5608	−0.599360
$960$	0	0
$961$	−29.5488	−0.953186
$962$	4.86102	0.156726
$963$	0	0
$964$	96.5805	3.11065
$965$	4.92974	0.158694
$966$	0	0
$967$	−22.5510	−0.725192	−0.362596	−	0.931946i	$-0.618110\pi$
$967$	−22.5510	−0.725192	−0.362596	+	0.931946i	$0.618110\pi$
$968$	130.730	4.20183
$969$	0	0
$970$	−5.91297	−0.189854
$971$	2.61563	0.0839396	0.0419698	−	0.999119i	$-0.486637\pi$
$971$	2.61563	0.0839396	0.0419698	+	0.999119i	$0.486637\pi$
$972$	0	0
$973$	3.97837	0.127541
$974$	−98.9477	−3.17049
$975$	0	0
$976$	27.2752	0.873059
$977$	44.8697	1.43551	0.717755	−	0.696296i	$-0.245170\pi$
$977$	44.8697	1.43551	0.717755	+	0.696296i	$0.245170\pi$
$978$	0	0
$979$	40.2286	1.28571
$980$	−15.9305	−0.508882
$981$	0	0
$982$	−41.6536	−1.32922
$983$	27.0373	0.862355	0.431177	−	0.902267i	$-0.358098\pi$
$983$	27.0373	0.862355	0.431177	+	0.902267i	$0.358098\pi$
$984$	0	0
$985$	2.29706	0.0731903
$986$	39.5181	1.25851
$987$	0	0
$988$	−37.2544	−1.18522
$989$	−7.70452	−0.244989
$990$	0	0
$991$	−29.3172	−0.931293	−0.465646	−	0.884971i	$-0.654178\pi$
$991$	−29.3172	−0.931293	−0.465646	+	0.884971i	$0.654178\pi$
$992$	−19.0884	−0.606058
$993$	0	0
$994$	−12.1908	−0.386669
$995$	−7.36658	−0.233536
$996$	0	0
$997$	57.2679	1.81369	0.906847	−	0.421460i	$-0.138482\pi$
$997$	57.2679	1.81369	0.906847	+	0.421460i	$0.138482\pi$
$998$	−9.71698	−0.307585
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.2.a.k.1.20	yes	20
3.2	odd	2		2151.2.a.j.1.1	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2151.2.a.j.1.1	✓	20	3.2	odd	2
2151.2.a.k.1.20	yes	20	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2151 = 3^{2} \cdot 239 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2151.a (trivial)

\(f(q)\)	\(=\)	\(q+2.67404 q^{2} +5.15050 q^{4} +0.519163 q^{5} +1.02094 q^{7} +8.42455 q^{8} +O(q^{10})\)	\(q+2.67404 q^{2} +5.15050 q^{4} +0.519163 q^{5} +1.02094 q^{7} +8.42455 q^{8} +1.38826 q^{10} +5.14954 q^{11} -2.29568 q^{13} +2.73005 q^{14} +12.2266 q^{16} -1.92182 q^{17} +3.15077 q^{19} +2.67395 q^{20} +13.7701 q^{22} -6.68812 q^{23} -4.73047 q^{25} -6.13875 q^{26} +5.25837 q^{28} -7.68981 q^{29} -1.20467 q^{31} +15.8454 q^{32} -5.13902 q^{34} +0.530037 q^{35} -0.791859 q^{37} +8.42528 q^{38} +4.37372 q^{40} +2.49631 q^{41} +1.15197 q^{43} +26.5227 q^{44} -17.8843 q^{46} +8.92862 q^{47} -5.95767 q^{49} -12.6495 q^{50} -11.8239 q^{52} -7.62842 q^{53} +2.67345 q^{55} +8.60100 q^{56} -20.5629 q^{58} +9.24454 q^{59} +2.23081 q^{61} -3.22134 q^{62} +17.9179 q^{64} -1.19183 q^{65} +14.8406 q^{67} -9.89831 q^{68} +1.41734 q^{70} -4.46542 q^{71} -2.29621 q^{73} -2.11746 q^{74} +16.2280 q^{76} +5.25739 q^{77} -9.20506 q^{79} +6.34761 q^{80} +6.67524 q^{82} +13.9566 q^{83} -0.997737 q^{85} +3.08042 q^{86} +43.3826 q^{88} +7.81208 q^{89} -2.34376 q^{91} -34.4471 q^{92} +23.8755 q^{94} +1.63576 q^{95} -4.25925 q^{97} -15.9311 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20q + 4q^{2} + 18q^{4} + 16q^{5} - 4q^{7} + 12q^{8} + O(q^{10}) \)	\( 20q + 4q^{2} + 18q^{4} + 16q^{5} - 4q^{7} + 12q^{8} + 4q^{10} + 12q^{11} - 4q^{13} + 20q^{14} + 22q^{16} + 24q^{17} - 4q^{19} + 40q^{20} - 6q^{22} + 12q^{23} + 22q^{25} + 30q^{26} - 12q^{28} + 24q^{29} - 4q^{31} + 28q^{32} + 8q^{34} + 20q^{35} - 10q^{37} + 26q^{38} + 6q^{40} + 66q^{41} + 8q^{43} + 36q^{44} - 12q^{46} + 28q^{47} + 18q^{49} + 28q^{50} - 18q^{52} + 28q^{53} - 4q^{55} + 60q^{56} + 54q^{59} - 4q^{61} + 20q^{62} + 22q^{64} + 42q^{65} + 12q^{67} + 12q^{68} + 20q^{70} + 36q^{71} + 14q^{73} - 50q^{76} + 8q^{77} - 12q^{79} + 88q^{80} - 8q^{82} + 20q^{83} + 4q^{85} + 18q^{86} - 10q^{88} + 130q^{89} - 6q^{91} - 46q^{92} - 26q^{94} - 2q^{97} + 12q^{98} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

Related objects

Downloads

Learn more about