LMFDB - Embedded newform 2151.2.a.k.1.9

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			1.9
Root			$0.0242456$ of defining polynomial
Character	$\chi$	$=$	2151.1

$q$-expansion

$f(q)$	$=$	$q+0.0242456 q^{2} -1.99941 q^{4} -2.24441 q^{5} -4.21532 q^{7} -0.0969680 q^{8} +O(q^{10})$	\(q+0.0242456 q^{2} -1.99941 q^{4} -2.24441 q^{5} -4.21532 q^{7} -0.0969680 q^{8} -0.0544170 q^{10} -2.12483 q^{11} -3.04809 q^{13} -0.102203 q^{14} +3.99647 q^{16} -4.57587 q^{17} -0.932781 q^{19} +4.48751 q^{20} -0.0515178 q^{22} -4.32903 q^{23} +0.0373862 q^{25} -0.0739027 q^{26} +8.42816 q^{28} -5.55445 q^{29} -1.27333 q^{31} +0.290833 q^{32} -0.110944 q^{34} +9.46091 q^{35} -8.37064 q^{37} -0.0226158 q^{38} +0.217636 q^{40} +10.8223 q^{41} -9.30747 q^{43} +4.24842 q^{44} -0.104960 q^{46} -7.05894 q^{47} +10.7689 q^{49} +0.000906450 q^{50} +6.09439 q^{52} +10.9362 q^{53} +4.76900 q^{55} +0.408751 q^{56} -0.134671 q^{58} +14.2846 q^{59} -4.93532 q^{61} -0.0308727 q^{62} -7.98590 q^{64} +6.84118 q^{65} +11.3100 q^{67} +9.14905 q^{68} +0.229385 q^{70} +5.92285 q^{71} -13.7138 q^{73} -0.202951 q^{74} +1.86501 q^{76} +8.95685 q^{77} +4.19498 q^{79} -8.96973 q^{80} +0.262392 q^{82} -17.1030 q^{83} +10.2701 q^{85} -0.225665 q^{86} +0.206041 q^{88} +10.9105 q^{89} +12.8487 q^{91} +8.65552 q^{92} -0.171148 q^{94} +2.09355 q^{95} -3.59611 q^{97} +0.261098 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.0242456	0.0171442	0.00857210	−	0.999963i	$-0.497271\pi$
$2$	0.0242456	0.0171442	0.00857210	+	0.999963i	$0.497271\pi$
$3$	0	0
$3$	0	0
$4$	−1.99941	−0.999706
$5$	−2.24441	−1.00373	−0.501866	−	0.864945i	$-0.667353\pi$
$5$	−2.24441	−1.00373	−0.501866	+	0.864945i	$0.667353\pi$
$6$	0	0
$7$	−4.21532	−1.59324	−0.796620	−	0.604480i	$-0.793381\pi$
$7$	−4.21532	−1.59324	−0.796620	+	0.604480i	$0.793381\pi$
$8$	−0.0969680	−0.0342834
$9$	0	0
$10$	−0.0544170	−0.0172082
$11$	−2.12483	−0.640661	−0.320331	−	0.947306i	$-0.603794\pi$
$11$	−2.12483	−0.640661	−0.320331	+	0.947306i	$0.603794\pi$
$12$	0	0
$13$	−3.04809	−0.845389	−0.422694	−	0.906272i	$-0.638916\pi$
$13$	−3.04809	−0.845389	−0.422694	+	0.906272i	$0.638916\pi$
$14$	−0.102203	−0.0273148
$15$	0	0
$16$	3.99647	0.999118
$17$	−4.57587	−1.10981	−0.554906	−	0.831913i	$-0.687246\pi$
$17$	−4.57587	−1.10981	−0.554906	+	0.831913i	$0.687246\pi$
$18$	0	0
$19$	−0.932781	−0.213995	−0.106997	−	0.994259i	$-0.534124\pi$
$19$	−0.932781	−0.213995	−0.106997	+	0.994259i	$0.534124\pi$
$20$	4.48751	1.00344
$21$	0	0
$22$	−0.0515178	−0.0109836
$23$	−4.32903	−0.902666	−0.451333	−	0.892356i	$-0.649051\pi$
$23$	−4.32903	−0.902666	−0.451333	+	0.892356i	$0.649051\pi$
$24$	0	0
$25$	0.0373862	0.00747724
$26$	−0.0739027	−0.0144935
$27$	0	0
$28$	8.42816	1.59277
$29$	−5.55445	−1.03144	−0.515718	−	0.856758i	$-0.672475\pi$
$29$	−5.55445	−1.03144	−0.515718	+	0.856758i	$0.672475\pi$
$30$	0	0
$31$	−1.27333	−0.228697	−0.114349	−	0.993441i	$-0.536478\pi$
$31$	−1.27333	−0.228697	−0.114349	+	0.993441i	$0.536478\pi$
$32$	0.290833	0.0514125
$33$	0	0
$34$	−0.110944	−0.0190268
$35$	9.46091	1.59919
$36$	0	0
$37$	−8.37064	−1.37612	−0.688062	−	0.725652i	$-0.741538\pi$
$37$	−8.37064	−1.37612	−0.688062	+	0.725652i	$0.741538\pi$
$38$	−0.0226158	−0.00366877
$39$	0	0
$40$	0.217636	0.0344113
$41$	10.8223	1.69016	0.845078	−	0.534643i	$-0.179554\pi$
$41$	10.8223	1.69016	0.845078	+	0.534643i	$0.179554\pi$
$42$	0	0
$43$	−9.30747	−1.41938	−0.709688	−	0.704516i	$-0.751164\pi$
$43$	−9.30747	−1.41938	−0.709688	+	0.704516i	$0.751164\pi$
$44$	4.24842	0.640473
$45$	0	0
$46$	−0.104960	−0.0154755
$47$	−7.05894	−1.02965	−0.514826	−	0.857295i	$-0.672143\pi$
$47$	−7.05894	−1.02965	−0.514826	+	0.857295i	$0.672143\pi$
$48$	0	0
$49$	10.7689	1.53842
$50$	0.000906450 0	0.000128191 0
$51$	0	0
$52$	6.09439	0.845140
$53$	10.9362	1.50220	0.751101	−	0.660188i	$-0.229523\pi$
$53$	10.9362	1.50220	0.751101	+	0.660188i	$0.229523\pi$
$54$	0	0
$55$	4.76900	0.643052
$56$	0.408751	0.0546217
$57$	0	0
$58$	−0.134671	−0.0176832
$59$	14.2846	1.85969	0.929846	−	0.367948i	$-0.119940\pi$
$59$	14.2846	1.85969	0.929846	+	0.367948i	$0.119940\pi$
$60$	0	0
$61$	−4.93532	−0.631903	−0.315951	−	0.948775i	$-0.602324\pi$
$61$	−4.93532	−0.631903	−0.315951	+	0.948775i	$0.602324\pi$
$62$	−0.0308727	−0.00392083
$63$	0	0
$64$	−7.98590	−0.998237
$65$	6.84118	0.848544
$66$	0	0
$67$	11.3100	1.38174	0.690871	−	0.722978i	$-0.257227\pi$
$67$	11.3100	1.38174	0.690871	+	0.722978i	$0.257227\pi$
$68$	9.14905	1.10948
$69$	0	0
$70$	0.229385	0.0274168
$71$	5.92285	0.702913	0.351456	−	0.936204i	$-0.385687\pi$
$71$	5.92285	0.702913	0.351456	+	0.936204i	$0.385687\pi$
$72$	0	0
$73$	−13.7138	−1.60508	−0.802540	−	0.596598i	$-0.796519\pi$
$73$	−13.7138	−1.60508	−0.802540	+	0.596598i	$0.796519\pi$
$74$	−0.202951	−0.0235926
$75$	0	0
$76$	1.86501	0.213932
$77$	8.95685	1.02073
$78$	0	0
$79$	4.19498	0.471972	0.235986	−	0.971756i	$-0.424168\pi$
$79$	4.19498	0.471972	0.235986	+	0.971756i	$0.424168\pi$
$80$	−8.96973	−1.00285
$81$	0	0
$82$	0.262392	0.0289764
$83$	−17.1030	−1.87730	−0.938652	−	0.344867i	$-0.887924\pi$
$83$	−17.1030	−1.87730	−0.938652	+	0.344867i	$0.887924\pi$
$84$	0	0
$85$	10.2701	1.11395
$86$	−0.225665	−0.0243341
$87$	0	0
$88$	0.206041	0.0219640
$89$	10.9105	1.15651	0.578253	−	0.815858i	$-0.303735\pi$
$89$	10.9105	1.15651	0.578253	+	0.815858i	$0.303735\pi$
$90$	0	0
$91$	12.8487	1.34691
$92$	8.65552	0.902400
$93$	0	0
$94$	−0.171148	−0.0176526
$95$	2.09355	0.214793
$96$	0	0
$97$	−3.59611	−0.365130	−0.182565	−	0.983194i	$-0.558440\pi$
$97$	−3.59611	−0.365130	−0.182565	+	0.983194i	$0.558440\pi$
$98$	0.261098	0.0263749
$99$	0	0
$100$	−0.0747504	−0.00747504
$101$	−1.12250	−0.111693	−0.0558465	−	0.998439i	$-0.517786\pi$
$101$	−1.12250	−0.111693	−0.0558465	+	0.998439i	$0.517786\pi$
$102$	0	0
$103$	10.3522	1.02003	0.510015	−	0.860165i	$-0.329640\pi$
$103$	10.3522	1.02003	0.510015	+	0.860165i	$0.329640\pi$
$104$	0.295567	0.0289828
$105$	0	0
$106$	0.265154	0.0257540
$107$	−2.57093	−0.248541	−0.124271	−	0.992248i	$-0.539659\pi$
$107$	−2.57093	−0.248541	−0.124271	+	0.992248i	$0.539659\pi$
$108$	0	0
$109$	−4.60875	−0.441438	−0.220719	−	0.975337i	$-0.570840\pi$
$109$	−4.60875	−0.441438	−0.220719	+	0.975337i	$0.570840\pi$
$110$	0.115627	0.0110246
$111$	0	0
$112$	−16.8464	−1.59184
$113$	19.4118	1.82611	0.913057	−	0.407833i	$-0.133715\pi$
$113$	19.4118	1.82611	0.913057	+	0.407833i	$0.133715\pi$
$114$	0	0
$115$	9.71613	0.906034
$116$	11.1056	1.03113
$117$	0	0
$118$	0.346337	0.0318829
$119$	19.2887	1.76820
$120$	0	0
$121$	−6.48509	−0.589553
$122$	−0.119660	−0.0108335
$123$	0	0
$124$	2.54592	0.228630
$125$	11.1382	0.996227
$126$	0	0
$127$	−19.0061	−1.68652	−0.843260	−	0.537506i	$-0.819367\pi$
$127$	−19.0061	−1.68652	−0.843260	+	0.537506i	$0.819367\pi$
$128$	−0.775288	−0.0685264
$129$	0	0
$130$	0.165868	0.0145476
$131$	−0.993956	−0.0868424	−0.0434212	−	0.999057i	$-0.513826\pi$
$131$	−0.993956	−0.0868424	−0.0434212	+	0.999057i	$0.513826\pi$
$132$	0	0
$133$	3.93197	0.340945
$134$	0.274218	0.0236889
$135$	0	0
$136$	0.443713	0.0380481
$137$	−10.2766	−0.877989	−0.438994	−	0.898490i	$-0.644665\pi$
$137$	−10.2766	−0.877989	−0.438994	+	0.898490i	$0.644665\pi$
$138$	0	0
$139$	−9.12331	−0.773829	−0.386915	−	0.922116i	$-0.626459\pi$
$139$	−9.12331	−0.773829	−0.386915	+	0.922116i	$0.626459\pi$
$140$	−18.9163	−1.59872
$141$	0	0
$142$	0.143603	0.0120509
$143$	6.47669	0.541608
$144$	0	0
$145$	12.4665	1.03529
$146$	−0.332499	−0.0275178
$147$	0	0
$148$	16.7364	1.37572
$149$	−8.76418	−0.717989	−0.358995	−	0.933340i	$-0.616880\pi$
$149$	−8.76418	−0.717989	−0.358995	+	0.933340i	$0.616880\pi$
$150$	0	0
$151$	2.25182	0.183251	0.0916253	−	0.995794i	$-0.470794\pi$
$151$	2.25182	0.183251	0.0916253	+	0.995794i	$0.470794\pi$
$152$	0.0904499	0.00733646
$153$	0	0
$154$	0.217164	0.0174996
$155$	2.85788	0.229551
$156$	0	0
$157$	−9.16321	−0.731304	−0.365652	−	0.930752i	$-0.619154\pi$
$157$	−9.16321	−0.731304	−0.365652	+	0.930752i	$0.619154\pi$
$158$	0.101710	0.00809158
$159$	0	0
$160$	−0.652749	−0.0516043
$161$	18.2483	1.43816
$162$	0	0
$163$	4.26808	0.334302	0.167151	−	0.985931i	$-0.446543\pi$
$163$	4.26808	0.334302	0.167151	+	0.985931i	$0.446543\pi$
$164$	−21.6382	−1.68966
$165$	0	0
$166$	−0.414673	−0.0321849
$167$	−21.3942	−1.65553	−0.827766	−	0.561074i	$-0.810388\pi$
$167$	−21.3942	−1.65553	−0.827766	+	0.561074i	$0.810388\pi$
$168$	0	0
$169$	−3.70913	−0.285318
$170$	0.249005	0.0190978
$171$	0	0
$172$	18.6095	1.41896
$173$	4.73652	0.360111	0.180056	−	0.983656i	$-0.442372\pi$
$173$	4.73652	0.360111	0.180056	+	0.983656i	$0.442372\pi$
$174$	0	0
$175$	−0.157595	−0.0119130
$176$	−8.49184	−0.640096
$177$	0	0
$178$	0.264530	0.0198274
$179$	−20.8872	−1.56118	−0.780591	−	0.625043i	$-0.785082\pi$
$179$	−20.8872	−1.56118	−0.780591	+	0.625043i	$0.785082\pi$
$180$	0	0
$181$	2.27241	0.168907	0.0844535	−	0.996427i	$-0.473086\pi$
$181$	2.27241	0.168907	0.0844535	+	0.996427i	$0.473086\pi$
$182$	0.311524	0.0230917
$183$	0	0
$184$	0.419778	0.0309464
$185$	18.7872	1.38126
$186$	0	0
$187$	9.72295	0.711013
$188$	14.1137	1.02935
$189$	0	0
$190$	0.0507592	0.00368246
$191$	14.4450	1.04521	0.522603	−	0.852576i	$-0.324961\pi$
$191$	14.4450	1.04521	0.522603	+	0.852576i	$0.324961\pi$
$192$	0	0
$193$	13.2203	0.951618	0.475809	−	0.879549i	$-0.342155\pi$
$193$	13.2203	0.951618	0.475809	+	0.879549i	$0.342155\pi$
$194$	−0.0871897	−0.00625986
$195$	0	0
$196$	−21.5315	−1.53796
$197$	−14.8080	−1.05503	−0.527513	−	0.849547i	$-0.676875\pi$
$197$	−14.8080	−1.05503	−0.527513	+	0.849547i	$0.676875\pi$
$198$	0	0
$199$	−9.69564	−0.687306	−0.343653	−	0.939097i	$-0.611664\pi$
$199$	−9.69564	−0.687306	−0.343653	+	0.939097i	$0.611664\pi$
$200$	−0.00362527	−0.000256345 0
$201$	0	0
$202$	−0.0272156	−0.00191489
$203$	23.4138	1.64333
$204$	0	0
$205$	−24.2896	−1.69646
$206$	0.250994	0.0174876
$207$	0	0
$208$	−12.1816	−0.844644
$209$	1.98200	0.137098
$210$	0	0
$211$	−3.95231	−0.272088	−0.136044	−	0.990703i	$-0.543439\pi$
$211$	−3.95231	−0.272088	−0.136044	+	0.990703i	$0.543439\pi$
$212$	−21.8660	−1.50176
$213$	0	0
$214$	−0.0623336	−0.00426104
$215$	20.8898	1.42467
$216$	0	0
$217$	5.36750	0.364370
$218$	−0.111742	−0.00756810
$219$	0	0
$220$	−9.53520	−0.642863
$221$	13.9477	0.938222
$222$	0	0
$223$	−27.4364	−1.83728	−0.918639	−	0.395098i	$-0.870710\pi$
$223$	−27.4364	−1.83728	−0.918639	+	0.395098i	$0.870710\pi$
$224$	−1.22595	−0.0819124
$225$	0	0
$226$	0.470651	0.0313073
$227$	−2.57694	−0.171037	−0.0855187	−	0.996337i	$-0.527255\pi$
$227$	−2.57694	−0.171037	−0.0855187	+	0.996337i	$0.527255\pi$
$228$	0	0
$229$	−14.8499	−0.981306	−0.490653	−	0.871355i	$-0.663242\pi$
$229$	−14.8499	−0.981306	−0.490653	+	0.871355i	$0.663242\pi$
$230$	0.235573	0.0155332
$231$	0	0
$232$	0.538604	0.0353611
$233$	6.79867	0.445396	0.222698	−	0.974888i	$-0.428514\pi$
$233$	6.79867	0.445396	0.222698	+	0.974888i	$0.428514\pi$
$234$	0	0
$235$	15.8432	1.03349
$236$	−28.5607	−1.85915
$237$	0	0
$238$	0.467666	0.0303143
$239$	1.00000	0.0646846
$239$	1.00000	0.0646846
$240$	0	0
$241$	−14.4562	−0.931208	−0.465604	−	0.884993i	$-0.654163\pi$
$241$	−14.4562	−0.931208	−0.465604	+	0.884993i	$0.654163\pi$
$242$	−0.157235	−0.0101074
$243$	0	0
$244$	9.86774	0.631717
$245$	−24.1699	−1.54416
$246$	0	0
$247$	2.84320	0.180909
$248$	0.123473	0.00784051
$249$	0	0
$250$	0.270051	0.0170795
$251$	24.6887	1.55834	0.779170	−	0.626813i	$-0.215641\pi$
$251$	24.6887	1.55834	0.779170	+	0.626813i	$0.215641\pi$
$252$	0	0
$253$	9.19847	0.578303
$254$	−0.460814	−0.0289140
$255$	0	0
$256$	15.9530	0.997062
$257$	−16.5700	−1.03361	−0.516805	−	0.856103i	$-0.672879\pi$
$257$	−16.5700	−1.03361	−0.516805	+	0.856103i	$0.672879\pi$
$258$	0	0
$259$	35.2849	2.19250
$260$	−13.6783	−0.848294
$261$	0	0
$262$	−0.0240990	−0.00148884
$263$	−9.61196	−0.592699	−0.296349	−	0.955080i	$-0.595769\pi$
$263$	−9.61196	−0.592699	−0.296349	+	0.955080i	$0.595769\pi$
$264$	0	0
$265$	−24.5453	−1.50781
$266$	0.0953328	0.00584523
$267$	0	0
$268$	−22.6134	−1.38134
$269$	22.2724	1.35797	0.678987	−	0.734151i	$-0.262419\pi$
$269$	22.2724	1.35797	0.678987	+	0.734151i	$0.262419\pi$
$270$	0	0
$271$	−8.41793	−0.511353	−0.255676	−	0.966762i	$-0.582298\pi$
$271$	−8.41793	−0.511353	−0.255676	+	0.966762i	$0.582298\pi$
$272$	−18.2873	−1.10883
$273$	0	0
$274$	−0.249162	−0.0150524
$275$	−0.0794394	−0.00479038
$276$	0	0
$277$	12.8383	0.771382	0.385691	−	0.922628i	$-0.373963\pi$
$277$	12.8383	0.771382	0.385691	+	0.922628i	$0.373963\pi$
$278$	−0.221200	−0.0132667
$279$	0	0
$280$	−0.917406	−0.0548255
$281$	4.62713	0.276031	0.138016	−	0.990430i	$-0.455928\pi$
$281$	4.62713	0.276031	0.138016	+	0.990430i	$0.455928\pi$
$282$	0	0
$283$	−31.0298	−1.84453	−0.922264	−	0.386560i	$-0.873663\pi$
$283$	−31.0298	−1.84453	−0.922264	+	0.386560i	$0.873663\pi$
$284$	−11.8422	−0.702706
$285$	0	0
$286$	0.157031	0.00928543
$287$	−45.6193	−2.69282
$288$	0	0
$289$	3.93857	0.231681
$290$	0.302257	0.0177491
$291$	0	0
$292$	27.4196	1.60461
$293$	−2.36334	−0.138068	−0.0690340	−	0.997614i	$-0.521992\pi$
$293$	−2.36334	−0.138068	−0.0690340	+	0.997614i	$0.521992\pi$
$294$	0	0
$295$	−32.0605	−1.86663
$296$	0.811684	0.0471782
$297$	0	0
$298$	−0.212492	−0.0123094
$299$	13.1953	0.763104
$300$	0	0
$301$	39.2340	2.26141
$302$	0.0545967	0.00314169
$303$	0	0
$304$	−3.72784	−0.213806
$305$	11.0769	0.634261
$306$	0	0
$307$	9.68688	0.552859	0.276430	−	0.961034i	$-0.410849\pi$
$307$	9.68688	0.552859	0.276430	+	0.961034i	$0.410849\pi$
$308$	−17.9084	−1.02043
$309$	0	0
$310$	0.0692910	0.00393546
$311$	−20.7787	−1.17825	−0.589125	−	0.808042i	$-0.700527\pi$
$311$	−20.7787	−1.17825	−0.589125	+	0.808042i	$0.700527\pi$
$312$	0	0
$313$	33.5440	1.89602	0.948009	−	0.318245i	$-0.103093\pi$
$313$	33.5440	1.89602	0.948009	+	0.318245i	$0.103093\pi$
$314$	−0.222167	−0.0125376
$315$	0	0
$316$	−8.38749	−0.471833
$317$	−1.53143	−0.0860137	−0.0430069	−	0.999075i	$-0.513694\pi$
$317$	−1.53143	−0.0860137	−0.0430069	+	0.999075i	$0.513694\pi$
$318$	0	0
$319$	11.8023	0.660801
$320$	17.9236	1.00196
$321$	0	0
$322$	0.442439	0.0246562
$323$	4.26828	0.237494
$324$	0	0
$325$	−0.113957	−0.00632118
$326$	0.103482	0.00573134
$327$	0	0
$328$	−1.04941	−0.0579442
$329$	29.7557	1.64048
$330$	0	0
$331$	17.6141	0.968160	0.484080	−	0.875024i	$-0.339154\pi$
$331$	17.6141	0.968160	0.484080	+	0.875024i	$0.339154\pi$
$332$	34.1960	1.87675
$333$	0	0
$334$	−0.518714	−0.0283828
$335$	−25.3844	−1.38690
$336$	0	0
$337$	1.96831	0.107221	0.0536103	−	0.998562i	$-0.482927\pi$
$337$	1.96831	0.107221	0.0536103	+	0.998562i	$0.482927\pi$
$338$	−0.0899299	−0.00489154
$339$	0	0
$340$	−20.5342	−1.11363
$341$	2.70562	0.146517
$342$	0	0
$343$	−15.8872	−0.857827
$344$	0.902527	0.0486610
$345$	0	0
$346$	0.114840	0.00617382
$347$	−35.6033	−1.91128	−0.955642	−	0.294532i	$-0.904836\pi$
$347$	−35.6033	−1.91128	−0.955642	+	0.294532i	$0.904836\pi$
$348$	0	0
$349$	−21.4549	−1.14845	−0.574226	−	0.818697i	$-0.694697\pi$
$349$	−21.4549	−1.14845	−0.574226	+	0.818697i	$0.694697\pi$
$350$	−0.00382097	−0.000204240 0
$351$	0	0
$352$	−0.617971	−0.0329380
$353$	7.36184	0.391831	0.195916	−	0.980621i	$-0.437232\pi$
$353$	7.36184	0.391831	0.195916	+	0.980621i	$0.437232\pi$
$354$	0	0
$355$	−13.2933	−0.705536
$356$	−21.8145	−1.15617
$357$	0	0
$358$	−0.506421	−0.0267652
$359$	7.46205	0.393832	0.196916	−	0.980420i	$-0.436907\pi$
$359$	7.46205	0.393832	0.196916	+	0.980420i	$0.436907\pi$
$360$	0	0
$361$	−18.1299	−0.954206
$362$	0.0550959	0.00289577
$363$	0	0
$364$	−25.6898	−1.34651
$365$	30.7794	1.61107
$366$	0	0
$367$	30.8625	1.61101	0.805505	−	0.592589i	$-0.201894\pi$
$367$	30.8625	1.61101	0.805505	+	0.592589i	$0.201894\pi$
$368$	−17.3009	−0.901870
$369$	0	0
$370$	0.455505	0.0236806
$371$	−46.0995	−2.39337
$372$	0	0
$373$	17.3634	0.899043	0.449521	−	0.893270i	$-0.351595\pi$
$373$	17.3634	0.899043	0.449521	+	0.893270i	$0.351595\pi$
$374$	0.235738	0.0121897
$375$	0	0
$376$	0.684491	0.0352999
$377$	16.9305	0.871965
$378$	0	0
$379$	14.2565	0.732306	0.366153	−	0.930555i	$-0.380675\pi$
$379$	14.2565	0.732306	0.366153	+	0.930555i	$0.380675\pi$
$380$	−4.18586	−0.214730
$381$	0	0
$382$	0.350228	0.0179192
$383$	12.8620	0.657217	0.328608	−	0.944466i	$-0.393420\pi$
$383$	12.8620	0.657217	0.328608	+	0.944466i	$0.393420\pi$
$384$	0	0
$385$	−20.1029	−1.02454
$386$	0.320534	0.0163147
$387$	0	0
$388$	7.19011	0.365022
$389$	−17.6690	−0.895856	−0.447928	−	0.894070i	$-0.647838\pi$
$389$	−17.6690	−0.895856	−0.447928	+	0.894070i	$0.647838\pi$
$390$	0	0
$391$	19.8091	1.00179
$392$	−1.04424	−0.0527421
$393$	0	0
$394$	−0.359028	−0.0180876
$395$	−9.41526	−0.473733
$396$	0	0
$397$	7.56134	0.379493	0.189747	−	0.981833i	$-0.439233\pi$
$397$	7.56134	0.379493	0.189747	+	0.981833i	$0.439233\pi$
$398$	−0.235076	−0.0117833
$399$	0	0
$400$	0.149413	0.00747065
$401$	20.5825	1.02784	0.513921	−	0.857837i	$-0.328192\pi$
$401$	20.5825	1.02784	0.513921	+	0.857837i	$0.328192\pi$
$402$	0	0
$403$	3.88124	0.193338
$404$	2.24434	0.111660
$405$	0	0
$406$	0.567681	0.0281735
$407$	17.7862	0.881629
$408$	0	0
$409$	30.2928	1.49788	0.748940	−	0.662637i	$-0.230563\pi$
$409$	30.2928	1.49788	0.748940	+	0.662637i	$0.230563\pi$
$410$	−0.588916	−0.0290845
$411$	0	0
$412$	−20.6983	−1.01973
$413$	−60.2140	−2.96294
$414$	0	0
$415$	38.3863	1.88431
$416$	−0.886485	−0.0434635
$417$	0	0
$418$	0.0480548	0.00235044
$419$	4.89531	0.239152	0.119576	−	0.992825i	$-0.461847\pi$
$419$	4.89531	0.239152	0.119576	+	0.992825i	$0.461847\pi$
$420$	0	0
$421$	−21.9505	−1.06980	−0.534900	−	0.844915i	$-0.679651\pi$
$421$	−21.9505	−1.06980	−0.534900	+	0.844915i	$0.679651\pi$
$422$	−0.0958260	−0.00466474
$423$	0	0
$424$	−1.06046	−0.0515005
$425$	−0.171074	−0.00829832
$426$	0	0
$427$	20.8039	1.00677
$428$	5.14035	0.248468
$429$	0	0
$430$	0.506485	0.0244249
$431$	−29.7089	−1.43103	−0.715513	−	0.698600i	$-0.753807\pi$
$431$	−29.7089	−1.43103	−0.715513	+	0.698600i	$0.753807\pi$
$432$	0	0
$433$	−7.35974	−0.353686	−0.176843	−	0.984239i	$-0.556589\pi$
$433$	−7.35974	−0.353686	−0.176843	+	0.984239i	$0.556589\pi$
$434$	0.130138	0.00624683
$435$	0	0
$436$	9.21479	0.441308
$437$	4.03804	0.193166
$438$	0	0
$439$	19.8181	0.945865	0.472933	−	0.881099i	$-0.343195\pi$
$439$	19.8181	0.945865	0.472933	+	0.881099i	$0.343195\pi$
$440$	−0.462440	−0.0220460
$441$	0	0
$442$	0.338169	0.0160851
$443$	40.2158	1.91071	0.955356	−	0.295456i	$-0.0954716\pi$
$443$	40.2158	1.91071	0.955356	+	0.295456i	$0.0954716\pi$
$444$	0	0
$445$	−24.4876	−1.16082
$446$	−0.665211	−0.0314987
$447$	0	0
$448$	33.6631	1.59043
$449$	31.7168	1.49681	0.748403	−	0.663244i	$-0.230821\pi$
$449$	31.7168	1.49681	0.748403	+	0.663244i	$0.230821\pi$
$450$	0	0
$451$	−22.9955	−1.08282
$452$	−38.8123	−1.82558
$453$	0	0
$454$	−0.0624793	−0.00293230
$455$	−28.8377	−1.35193
$456$	0	0
$457$	21.4539	1.00357	0.501786	−	0.864992i	$-0.332676\pi$
$457$	21.4539	1.00357	0.501786	+	0.864992i	$0.332676\pi$
$458$	−0.360043	−0.0168237
$459$	0	0
$460$	−19.4266	−0.905768
$461$	−16.1130	−0.750459	−0.375230	−	0.926932i	$-0.622436\pi$
$461$	−16.1130	−0.750459	−0.375230	+	0.926932i	$0.622436\pi$
$462$	0	0
$463$	−12.7421	−0.592178	−0.296089	−	0.955160i	$-0.595682\pi$
$463$	−12.7421	−0.592178	−0.296089	+	0.955160i	$0.595682\pi$
$464$	−22.1982	−1.03053
$465$	0	0
$466$	0.164838	0.00763595
$467$	−2.54331	−0.117691	−0.0588453	−	0.998267i	$-0.518742\pi$
$467$	−2.54331	−0.117691	−0.0588453	+	0.998267i	$0.518742\pi$
$468$	0	0
$469$	−47.6755	−2.20145
$470$	0.384126	0.0177184
$471$	0	0
$472$	−1.38515	−0.0637565
$473$	19.7768	0.909339
$474$	0	0
$475$	−0.0348732	−0.00160009
$476$	−38.5661	−1.76768
$477$	0	0
$478$	0.0242456	0.00110897
$479$	−9.60241	−0.438745	−0.219373	−	0.975641i	$-0.570401\pi$
$479$	−9.60241	−0.438745	−0.219373	+	0.975641i	$0.570401\pi$
$480$	0	0
$481$	25.5145	1.16336
$482$	−0.350500	−0.0159648
$483$	0	0
$484$	12.9664	0.589380
$485$	8.07115	0.366492
$486$	0	0
$487$	−36.2997	−1.64490	−0.822448	−	0.568841i	$-0.807392\pi$
$487$	−36.2997	−1.64490	−0.822448	+	0.568841i	$0.807392\pi$
$488$	0.478568	0.0216638
$489$	0	0
$490$	−0.586012	−0.0264733
$491$	−7.26687	−0.327949	−0.163975	−	0.986465i	$-0.552431\pi$
$491$	−7.26687	−0.327949	−0.163975	+	0.986465i	$0.552431\pi$
$492$	0	0
$493$	25.4165	1.14470
$494$	0.0689351	0.00310154
$495$	0	0
$496$	−5.08884	−0.228496
$497$	−24.9667	−1.11991
$498$	0	0
$499$	−26.2796	−1.17644	−0.588218	−	0.808703i	$-0.700170\pi$
$499$	−26.2796	−1.17644	−0.588218	+	0.808703i	$0.700170\pi$
$500$	−22.2698	−0.995934
$501$	0	0
$502$	0.598592	0.0267165
$503$	−34.2338	−1.52641	−0.763206	−	0.646156i	$-0.776376\pi$
$503$	−34.2338	−1.52641	−0.763206	+	0.646156i	$0.776376\pi$
$504$	0	0
$505$	2.51935	0.112110
$506$	0.223022	0.00991454
$507$	0	0
$508$	38.0011	1.68602
$509$	12.0294	0.533194	0.266597	−	0.963808i	$-0.414101\pi$
$509$	12.0294	0.533194	0.266597	+	0.963808i	$0.414101\pi$
$510$	0	0
$511$	57.8081	2.55728
$512$	1.93737	0.0856203
$513$	0	0
$514$	−0.401750	−0.0177204
$515$	−23.2346	−1.02384
$516$	0	0
$517$	14.9991	0.659658
$518$	0.855503	0.0375886
$519$	0	0
$520$	−0.663375	−0.0290909
$521$	−2.94578	−0.129057	−0.0645285	−	0.997916i	$-0.520554\pi$
$521$	−2.94578	−0.129057	−0.0645285	+	0.997916i	$0.520554\pi$
$522$	0	0
$523$	−9.40392	−0.411205	−0.205602	−	0.978636i	$-0.565915\pi$
$523$	−9.40392	−0.411205	−0.205602	+	0.978636i	$0.565915\pi$
$524$	1.98733	0.0868168
$525$	0	0
$526$	−0.233047	−0.0101613
$527$	5.82660	0.253811
$528$	0	0
$529$	−4.25948	−0.185195
$530$	−0.595115	−0.0258502
$531$	0	0
$532$	−7.86163	−0.340845
$533$	−32.9873	−1.42884
$534$	0	0
$535$	5.77023	0.249469
$536$	−1.09671	−0.0473708
$537$	0	0
$538$	0.540007	0.0232814
$539$	−22.8821	−0.985603
$540$	0	0
$541$	13.6171	0.585447	0.292723	−	0.956197i	$-0.405438\pi$
$541$	13.6171	0.585447	0.292723	+	0.956197i	$0.405438\pi$
$542$	−0.204097	−0.00876674
$543$	0	0
$544$	−1.33081	−0.0570581
$545$	10.3439	0.443085
$546$	0	0
$547$	−20.3413	−0.869730	−0.434865	−	0.900496i	$-0.643204\pi$
$547$	−20.3413	−0.869730	−0.434865	+	0.900496i	$0.643204\pi$
$548$	20.5471	0.877731
$549$	0	0
$550$	−0.00192605	−8.21272e−5 0
$551$	5.18109	0.220722
$552$	0	0
$553$	−17.6832	−0.751965
$554$	0.311273	0.0132247
$555$	0	0
$556$	18.2413	0.773602
$557$	6.44485	0.273077	0.136539	−	0.990635i	$-0.456402\pi$
$557$	6.44485	0.273077	0.136539	+	0.990635i	$0.456402\pi$
$558$	0	0
$559$	28.3700	1.19993
$560$	37.8103	1.59778
$561$	0	0
$562$	0.112187	0.00473234
$563$	40.0936	1.68974	0.844871	−	0.534970i	$-0.179677\pi$
$563$	40.0936	1.68974	0.844871	+	0.534970i	$0.179677\pi$
$564$	0	0
$565$	−43.5682	−1.83293
$566$	−0.752334	−0.0316230
$567$	0	0
$568$	−0.574327	−0.0240982
$569$	−12.5303	−0.525296	−0.262648	−	0.964892i	$-0.584596\pi$
$569$	−12.5303	−0.525296	−0.262648	+	0.964892i	$0.584596\pi$
$570$	0	0
$571$	−29.8993	−1.25125	−0.625623	−	0.780126i	$-0.715155\pi$
$571$	−29.8993	−1.25125	−0.625623	+	0.780126i	$0.715155\pi$
$572$	−12.9496	−0.541449
$573$	0	0
$574$	−1.10607	−0.0461663
$575$	−0.161846	−0.00674945
$576$	0	0
$577$	20.1769	0.839974	0.419987	−	0.907530i	$-0.362035\pi$
$577$	20.1769	0.839974	0.419987	+	0.907530i	$0.362035\pi$
$578$	0.0954928	0.00397198
$579$	0	0
$580$	−24.9256	−1.03498
$581$	72.0948	2.99100
$582$	0	0
$583$	−23.2376	−0.962402
$584$	1.32980	0.0550276
$585$	0	0
$586$	−0.0573006	−0.00236706
$587$	−14.8609	−0.613373	−0.306687	−	0.951811i	$-0.599220\pi$
$587$	−14.8609	−0.613373	−0.306687	+	0.951811i	$0.599220\pi$
$588$	0	0
$589$	1.18774	0.0489400
$590$	−0.777324	−0.0320019
$591$	0	0
$592$	−33.4530	−1.37491
$593$	1.07945	0.0443276	0.0221638	−	0.999754i	$-0.492944\pi$
$593$	1.07945	0.0443276	0.0221638	+	0.999754i	$0.492944\pi$
$594$	0	0
$595$	−43.2919	−1.77479
$596$	17.5232	0.717778
$597$	0	0
$598$	0.319927	0.0130828
$599$	5.03917	0.205895	0.102948	−	0.994687i	$-0.467173\pi$
$599$	5.03917	0.205895	0.102948	+	0.994687i	$0.467173\pi$
$600$	0	0
$601$	16.2608	0.663291	0.331645	−	0.943404i	$-0.392396\pi$
$601$	16.2608	0.663291	0.331645	+	0.943404i	$0.392396\pi$
$602$	0.951250	0.0387700
$603$	0	0
$604$	−4.50232	−0.183197
$605$	14.5552	0.591753
$606$	0	0
$607$	20.0565	0.814069	0.407034	−	0.913413i	$-0.366563\pi$
$607$	20.0565	0.814069	0.407034	+	0.913413i	$0.366563\pi$
$608$	−0.271283	−0.0110020
$609$	0	0
$610$	0.268565	0.0108739
$611$	21.5163	0.870456
$612$	0	0
$613$	12.3418	0.498479	0.249240	−	0.968442i	$-0.419819\pi$
$613$	12.3418	0.498479	0.249240	+	0.968442i	$0.419819\pi$
$614$	0.234864	0.00947833
$615$	0	0
$616$	−0.868527	−0.0349940
$617$	−33.6404	−1.35431	−0.677156	−	0.735840i	$-0.736788\pi$
$617$	−33.6404	−1.35431	−0.677156	+	0.735840i	$0.736788\pi$
$618$	0	0
$619$	2.36911	0.0952227	0.0476113	−	0.998866i	$-0.484839\pi$
$619$	2.36911	0.0952227	0.0476113	+	0.998866i	$0.484839\pi$
$620$	−5.71409	−0.229483
$621$	0	0
$622$	−0.503790	−0.0202002
$623$	−45.9910	−1.84259
$624$	0	0
$625$	−25.1855	−1.00742
$626$	0.813292	0.0325057
$627$	0	0
$628$	18.3210	0.731089
$629$	38.3029	1.52724
$630$	0	0
$631$	−5.90443	−0.235052	−0.117526	−	0.993070i	$-0.537496\pi$
$631$	−5.90443	−0.235052	−0.117526	+	0.993070i	$0.537496\pi$
$632$	−0.406779	−0.0161808
$633$	0	0
$634$	−0.0371304	−0.00147464
$635$	42.6576	1.69281
$636$	0	0
$637$	−32.8247	−1.30056
$638$	0.286153	0.0113289
$639$	0	0
$640$	1.74007	0.0687821
$641$	−40.4846	−1.59904	−0.799522	−	0.600637i	$-0.794914\pi$
$641$	−40.4846	−1.59904	−0.799522	+	0.600637i	$0.794914\pi$
$642$	0	0
$643$	−16.3242	−0.643762	−0.321881	−	0.946780i	$-0.604315\pi$
$643$	−16.3242	−0.643762	−0.321881	+	0.946780i	$0.604315\pi$
$644$	−36.4858	−1.43774
$645$	0	0
$646$	0.103487	0.00407164
$647$	9.53248	0.374760	0.187380	−	0.982287i	$-0.440000\pi$
$647$	9.53248	0.374760	0.187380	+	0.982287i	$0.440000\pi$
$648$	0	0
$649$	−30.3523	−1.19143
$650$	−0.00276294	−0.000108372 0
$651$	0	0
$652$	−8.53365	−0.334204
$653$	−31.1156	−1.21765	−0.608825	−	0.793305i	$-0.708359\pi$
$653$	−31.1156	−1.21765	−0.608825	+	0.793305i	$0.708359\pi$
$654$	0	0
$655$	2.23085	0.0871664
$656$	43.2509	1.68867
$657$	0	0
$658$	0.721443	0.0281248
$659$	−1.01441	−0.0395160	−0.0197580	−	0.999805i	$-0.506290\pi$
$659$	−1.01441	−0.0395160	−0.0197580	+	0.999805i	$0.506290\pi$
$660$	0	0
$661$	8.64358	0.336196	0.168098	−	0.985770i	$-0.446237\pi$
$661$	8.64358	0.336196	0.168098	+	0.985770i	$0.446237\pi$
$662$	0.427065	0.0165983
$663$	0	0
$664$	1.65845	0.0643603
$665$	−8.82496	−0.342217
$666$	0	0
$667$	24.0454	0.931042
$668$	42.7758	1.65505
$669$	0	0
$670$	−0.615459	−0.0237773
$671$	10.4867	0.404835
$672$	0	0
$673$	2.37311	0.0914768	0.0457384	−	0.998953i	$-0.485436\pi$
$673$	2.37311	0.0914768	0.0457384	+	0.998953i	$0.485436\pi$
$674$	0.0477227	0.00183821
$675$	0	0
$676$	7.41608	0.285234
$677$	33.5812	1.29063	0.645315	−	0.763916i	$-0.276726\pi$
$677$	33.5812	1.29063	0.645315	+	0.763916i	$0.276726\pi$
$678$	0	0
$679$	15.1588	0.581740
$680$	−0.995874	−0.0381900
$681$	0	0
$682$	0.0655992	0.00251192
$683$	−4.70697	−0.180107	−0.0900536	−	0.995937i	$-0.528704\pi$
$683$	−4.70697	−0.180107	−0.0900536	+	0.995937i	$0.528704\pi$
$684$	0	0
$685$	23.0649	0.881265
$686$	−0.385193	−0.0147068
$687$	0	0
$688$	−37.1971	−1.41812
$689$	−33.3345	−1.26994
$690$	0	0
$691$	14.1647	0.538851	0.269426	−	0.963021i	$-0.413166\pi$
$691$	14.1647	0.538851	0.269426	+	0.963021i	$0.413166\pi$
$692$	−9.47026	−0.360005
$693$	0	0
$694$	−0.863221	−0.0327674
$695$	20.4765	0.776717
$696$	0	0
$697$	−49.5213	−1.87575
$698$	−0.520185	−0.0196893
$699$	0	0
$700$	0.315097	0.0119095
$701$	−27.8527	−1.05198	−0.525991	−	0.850490i	$-0.676305\pi$
$701$	−27.8527	−1.05198	−0.525991	+	0.850490i	$0.676305\pi$
$702$	0	0
$703$	7.80798	0.294483
$704$	16.9687	0.639531
$705$	0	0
$706$	0.178492	0.00671764
$707$	4.73170	0.177954
$708$	0	0
$709$	2.50868	0.0942155	0.0471077	−	0.998890i	$-0.485000\pi$
$709$	2.50868	0.0942155	0.0471077	+	0.998890i	$0.485000\pi$
$710$	−0.322304	−0.0120959
$711$	0	0
$712$	−1.05796	−0.0396489
$713$	5.51230	0.206437
$714$	0	0
$715$	−14.5364	−0.543629
$716$	41.7621	1.56072
$717$	0	0
$718$	0.180922	0.00675193
$719$	−28.4290	−1.06022	−0.530112	−	0.847928i	$-0.677850\pi$
$719$	−28.4290	−1.06022	−0.530112	+	0.847928i	$0.677850\pi$
$720$	0	0
$721$	−43.6377	−1.62515
$722$	−0.439570	−0.0163591
$723$	0	0
$724$	−4.54348	−0.168857
$725$	−0.207660	−0.00771230
$726$	0	0
$727$	44.1617	1.63787	0.818933	−	0.573888i	$-0.194566\pi$
$727$	44.1617	1.63787	0.818933	+	0.573888i	$0.194566\pi$
$728$	−1.24591	−0.0461765
$729$	0	0
$730$	0.746265	0.0276205
$731$	42.5898	1.57524
$732$	0	0
$733$	48.2081	1.78061	0.890303	−	0.455368i	$-0.150492\pi$
$733$	48.2081	1.78061	0.890303	+	0.455368i	$0.150492\pi$
$734$	0.748279	0.0276195
$735$	0	0
$736$	−1.25902	−0.0464083
$737$	−24.0320	−0.885228
$738$	0	0
$739$	13.7530	0.505911	0.252956	−	0.967478i	$-0.418597\pi$
$739$	13.7530	0.505911	0.252956	+	0.967478i	$0.418597\pi$
$740$	−37.5633	−1.38085
$741$	0	0
$742$	−1.11771	−0.0410324
$743$	22.2915	0.817796	0.408898	−	0.912580i	$-0.365913\pi$
$743$	22.2915	0.817796	0.408898	+	0.912580i	$0.365913\pi$
$744$	0	0
$745$	19.6704	0.720669
$746$	0.420985	0.0154134
$747$	0	0
$748$	−19.4402	−0.710804
$749$	10.8373	0.395986
$750$	0	0
$751$	−0.504426	−0.0184068	−0.00920338	−	0.999958i	$-0.502930\pi$
$751$	−0.504426	−0.0184068	−0.00920338	+	0.999958i	$0.502930\pi$
$752$	−28.2109	−1.02874
$753$	0	0
$754$	0.410489	0.0149491
$755$	−5.05401	−0.183934
$756$	0	0
$757$	27.5960	1.00299	0.501497	−	0.865159i	$-0.332783\pi$
$757$	27.5960	1.00299	0.501497	+	0.865159i	$0.332783\pi$
$758$	0.345656	0.0125548
$759$	0	0
$760$	−0.203007	−0.00736384
$761$	45.1173	1.63550	0.817751	−	0.575572i	$-0.195220\pi$
$761$	45.1173	1.63550	0.817751	+	0.575572i	$0.195220\pi$
$762$	0	0
$763$	19.4273	0.703317
$764$	−28.8816	−1.04490
$765$	0	0
$766$	0.311846	0.0112675
$767$	−43.5407	−1.57216
$768$	0	0
$769$	−35.6644	−1.28609	−0.643045	−	0.765829i	$-0.722329\pi$
$769$	−35.6644	−1.28609	−0.643045	+	0.765829i	$0.722329\pi$
$770$	−0.487405	−0.0175649
$771$	0	0
$772$	−26.4328	−0.951339
$773$	31.4644	1.13170	0.565848	−	0.824510i	$-0.308549\pi$
$773$	31.4644	1.13170	0.565848	+	0.824510i	$0.308549\pi$
$774$	0	0
$775$	−0.0476051	−0.00171002
$776$	0.348708	0.0125179
$777$	0	0
$778$	−0.428396	−0.0153587
$779$	−10.0948	−0.361684
$780$	0	0
$781$	−12.5851	−0.450329
$782$	0.480282	0.0171749
$783$	0	0
$784$	43.0377	1.53706
$785$	20.5660	0.734033
$786$	0	0
$787$	−0.598640	−0.0213392	−0.0106696	−	0.999943i	$-0.503396\pi$
$787$	−0.598640	−0.0213392	−0.0106696	+	0.999943i	$0.503396\pi$
$788$	29.6073	1.05472
$789$	0	0
$790$	−0.228278	−0.00812178
$791$	−81.8271	−2.90944
$792$	0	0
$793$	15.0433	0.534204
$794$	0.183329	0.00650610
$795$	0	0
$796$	19.3856	0.687104
$797$	−9.41471	−0.333486	−0.166743	−	0.986000i	$-0.553325\pi$
$797$	−9.41471	−0.333486	−0.166743	+	0.986000i	$0.553325\pi$
$798$	0	0
$799$	32.3008	1.14272
$800$	0.0108731	0.000384423 0
$801$	0	0
$802$	0.499035	0.0176215
$803$	29.1395	1.02831
$804$	0	0
$805$	−40.9566	−1.44353
$806$	0.0941028	0.00331463
$807$	0	0
$808$	0.108847	0.00382921
$809$	16.7127	0.587588	0.293794	−	0.955869i	$-0.405082\pi$
$809$	16.7127	0.587588	0.293794	+	0.955869i	$0.405082\pi$
$810$	0	0
$811$	−33.3891	−1.17245	−0.586225	−	0.810148i	$-0.699387\pi$
$811$	−33.3891	−1.17245	−0.586225	+	0.810148i	$0.699387\pi$
$812$	−46.8138	−1.64284
$813$	0	0
$814$	0.431237	0.0151148
$815$	−9.57933	−0.335549
$816$	0	0
$817$	8.68184	0.303739
$818$	0.734465	0.0256800
$819$	0	0
$820$	48.5650	1.69596
$821$	−36.8927	−1.28756	−0.643782	−	0.765209i	$-0.722636\pi$
$821$	−36.8927	−1.28756	−0.643782	+	0.765209i	$0.722636\pi$
$822$	0	0
$823$	−17.1641	−0.598304	−0.299152	−	0.954205i	$-0.596704\pi$
$823$	−17.1641	−0.598304	−0.299152	+	0.954205i	$0.596704\pi$
$824$	−1.00383	−0.0349701
$825$	0	0
$826$	−1.45992	−0.0507972
$827$	−12.8940	−0.448369	−0.224185	−	0.974547i	$-0.571972\pi$
$827$	−12.8940	−0.448369	−0.224185	+	0.974547i	$0.571972\pi$
$828$	0	0
$829$	−8.56457	−0.297460	−0.148730	−	0.988878i	$-0.547518\pi$
$829$	−8.56457	−0.297460	−0.148730	+	0.988878i	$0.547518\pi$
$830$	0.930697	0.0323050
$831$	0	0
$832$	24.3418	0.843898
$833$	−49.2771	−1.70735
$834$	0	0
$835$	48.0174	1.66171
$836$	−3.96284	−0.137058
$837$	0	0
$838$	0.118690	0.00410007
$839$	−27.2656	−0.941312	−0.470656	−	0.882317i	$-0.655983\pi$
$839$	−27.2656	−0.941312	−0.470656	+	0.882317i	$0.655983\pi$
$840$	0	0
$841$	1.85197	0.0638610
$842$	−0.532202	−0.0183409
$843$	0	0
$844$	7.90230	0.272008
$845$	8.32481	0.286382
$846$	0	0
$847$	27.3367	0.939301
$848$	43.7062	1.50088
$849$	0	0
$850$	−0.00414779	−0.000142268 0
$851$	36.2368	1.24218
$852$	0	0
$853$	−17.4725	−0.598248	−0.299124	−	0.954214i	$-0.596694\pi$
$853$	−17.4725	−0.598248	−0.299124	+	0.954214i	$0.596694\pi$
$854$	0.504403	0.0172603
$855$	0	0
$856$	0.249298	0.00852083
$857$	20.8610	0.712597	0.356298	−	0.934372i	$-0.384039\pi$
$857$	20.8610	0.712597	0.356298	+	0.934372i	$0.384039\pi$
$858$	0	0
$859$	−39.4728	−1.34679	−0.673397	−	0.739281i	$-0.735166\pi$
$859$	−39.4728	−1.34679	−0.673397	+	0.739281i	$0.735166\pi$
$860$	−41.7673	−1.42425
$861$	0	0
$862$	−0.720308	−0.0245338
$863$	37.2894	1.26935	0.634673	−	0.772781i	$-0.281135\pi$
$863$	37.2894	1.26935	0.634673	+	0.772781i	$0.281135\pi$
$864$	0	0
$865$	−10.6307	−0.361455
$866$	−0.178441	−0.00606367
$867$	0	0
$868$	−10.7319	−0.364263
$869$	−8.91362	−0.302374
$870$	0	0
$871$	−34.4741	−1.16811
$872$	0.446901	0.0151340
$873$	0	0
$874$	0.0979046	0.00331167
$875$	−46.9509	−1.58723
$876$	0	0
$877$	−51.4108	−1.73602	−0.868010	−	0.496548i	$-0.834601\pi$
$877$	−51.4108	−1.73602	−0.868010	+	0.496548i	$0.834601\pi$
$878$	0.480501	0.0162161
$879$	0	0
$880$	19.0592	0.642485
$881$	−4.11918	−0.138779	−0.0693893	−	0.997590i	$-0.522105\pi$
$881$	−4.11918	−0.138779	−0.0693893	+	0.997590i	$0.522105\pi$
$882$	0	0
$883$	55.1078	1.85453	0.927263	−	0.374410i	$-0.122155\pi$
$883$	55.1078	1.85453	0.927263	+	0.374410i	$0.122155\pi$
$884$	−27.8871	−0.937946
$885$	0	0
$886$	0.975056	0.0327576
$887$	7.97495	0.267773	0.133886	−	0.990997i	$-0.457254\pi$
$887$	7.97495	0.267773	0.133886	+	0.990997i	$0.457254\pi$
$888$	0	0
$889$	80.1168	2.68703
$890$	−0.593714	−0.0199014
$891$	0	0
$892$	54.8567	1.83674
$893$	6.58445	0.220340
$894$	0	0
$895$	46.8794	1.56701
$896$	3.26809	0.109179
$897$	0	0
$898$	0.768991	0.0256616
$899$	7.07267	0.235887
$900$	0	0
$901$	−50.0426	−1.66716
$902$	−0.557539	−0.0185640
$903$	0	0
$904$	−1.88233	−0.0626053
$905$	−5.10023	−0.169537
$906$	0	0
$907$	−50.2402	−1.66820	−0.834099	−	0.551615i	$-0.814012\pi$
$907$	−50.2402	−1.66820	−0.834099	+	0.551615i	$0.814012\pi$
$908$	5.15236	0.170987
$909$	0	0
$910$	−0.699187	−0.0231778
$911$	−43.3795	−1.43723	−0.718613	−	0.695410i	$-0.755223\pi$
$911$	−43.3795	−1.43723	−0.718613	+	0.695410i	$0.755223\pi$
$912$	0	0
$913$	36.3411	1.20271
$914$	0.520163	0.0172054
$915$	0	0
$916$	29.6910	0.981018
$917$	4.18984	0.138361
$918$	0	0
$919$	9.92184	0.327291	0.163646	−	0.986519i	$-0.447675\pi$
$919$	9.92184	0.327291	0.163646	+	0.986519i	$0.447675\pi$
$920$	−0.942154	−0.0310619
$921$	0	0
$922$	−0.390670	−0.0128660
$923$	−18.0534	−0.594235
$924$	0	0
$925$	−0.312946	−0.0102896
$926$	−0.308941	−0.0101524
$927$	0	0
$928$	−1.61542	−0.0530287
$929$	−22.2900	−0.731312	−0.365656	−	0.930750i	$-0.619155\pi$
$929$	−22.2900	−0.731312	−0.365656	+	0.930750i	$0.619155\pi$
$930$	0	0
$931$	−10.0450	−0.329213
$932$	−13.5933	−0.445265
$933$	0	0
$934$	−0.0616641	−0.00201771
$935$	−21.8223	−0.713666
$936$	0	0
$937$	−46.5712	−1.52141	−0.760707	−	0.649096i	$-0.775147\pi$
$937$	−46.5712	−1.52141	−0.760707	+	0.649096i	$0.775147\pi$
$938$	−1.15592	−0.0377421
$939$	0	0
$940$	−31.6770	−1.03319
$941$	20.6597	0.673486	0.336743	−	0.941597i	$-0.390675\pi$
$941$	20.6597	0.673486	0.336743	+	0.941597i	$0.390675\pi$
$942$	0	0
$943$	−46.8500	−1.52565
$944$	57.0879	1.85805
$945$	0	0
$946$	0.479500	0.0155899
$947$	39.3901	1.28001	0.640003	−	0.768373i	$-0.278933\pi$
$947$	39.3901	1.28001	0.640003	+	0.768373i	$0.278933\pi$
$948$	0	0
$949$	41.8010	1.35692
$950$	−0.000845519 0	−2.74323e−5 0
$951$	0	0
$952$	−1.87039	−0.0606197
$953$	32.7355	1.06041	0.530203	−	0.847871i	$-0.322116\pi$
$953$	32.7355	1.06041	0.530203	+	0.847871i	$0.322116\pi$
$954$	0	0
$955$	−32.4206	−1.04911
$956$	−1.99941	−0.0646656
$957$	0	0
$958$	−0.232816	−0.00752194
$959$	43.3191	1.39885
$960$	0	0
$961$	−29.3786	−0.947698
$962$	0.618613	0.0199449
$963$	0	0
$964$	28.9040	0.930934
$965$	−29.6718	−0.955169
$966$	0	0
$967$	−14.7746	−0.475118	−0.237559	−	0.971373i	$-0.576347\pi$
$967$	−14.7746	−0.475118	−0.237559	+	0.971373i	$0.576347\pi$
$968$	0.628846	0.0202119
$969$	0	0
$970$	0.195690	0.00628322
$971$	38.8307	1.24614	0.623069	−	0.782167i	$-0.285886\pi$
$971$	38.8307	1.24614	0.623069	+	0.782167i	$0.285886\pi$
$972$	0	0
$973$	38.4577	1.23290
$974$	−0.880106	−0.0282004
$975$	0	0
$976$	−19.7239	−0.631346
$977$	−7.75740	−0.248181	−0.124091	−	0.992271i	$-0.539601\pi$
$977$	−7.75740	−0.248181	−0.124091	+	0.992271i	$0.539601\pi$
$978$	0	0
$979$	−23.1829	−0.740928
$980$	48.3256	1.54370
$981$	0	0
$982$	−0.176189	−0.00562242
$983$	0.802036	0.0255810	0.0127905	−	0.999918i	$-0.495929\pi$
$983$	0.802036	0.0255810	0.0127905	+	0.999918i	$0.495929\pi$
$984$	0	0
$985$	33.2352	1.05896
$986$	0.616236	0.0196250
$987$	0	0
$988$	−5.68474	−0.180856
$989$	40.2924	1.28122
$990$	0	0
$991$	−13.3518	−0.424135	−0.212068	−	0.977255i	$-0.568020\pi$
$991$	−13.3518	−0.424135	−0.212068	+	0.977255i	$0.568020\pi$
$992$	−0.370327	−0.0117579
$993$	0	0
$994$	−0.605332	−0.0192000
$995$	21.7610	0.689870
$996$	0	0
$997$	−4.98041	−0.157731	−0.0788655	−	0.996885i	$-0.525130\pi$
$997$	−4.98041	−0.157731	−0.0788655	+	0.996885i	$0.525130\pi$
$998$	−0.637163	−0.0201690
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.2.a.k.1.9	yes	20
3.2	odd	2		2151.2.a.j.1.12	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2151.2.a.j.1.12	✓	20	3.2	odd	2
2151.2.a.k.1.9	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)

\(f(q)\)	\(=\)	\(q+0.0242456 q^{2} -1.99941 q^{4} -2.24441 q^{5} -4.21532 q^{7} -0.0969680 q^{8} +O(q^{10})\)	\(q+0.0242456 q^{2} -1.99941 q^{4} -2.24441 q^{5} -4.21532 q^{7} -0.0969680 q^{8} -0.0544170 q^{10} -2.12483 q^{11} -3.04809 q^{13} -0.102203 q^{14} +3.99647 q^{16} -4.57587 q^{17} -0.932781 q^{19} +4.48751 q^{20} -0.0515178 q^{22} -4.32903 q^{23} +0.0373862 q^{25} -0.0739027 q^{26} +8.42816 q^{28} -5.55445 q^{29} -1.27333 q^{31} +0.290833 q^{32} -0.110944 q^{34} +9.46091 q^{35} -8.37064 q^{37} -0.0226158 q^{38} +0.217636 q^{40} +10.8223 q^{41} -9.30747 q^{43} +4.24842 q^{44} -0.104960 q^{46} -7.05894 q^{47} +10.7689 q^{49} +0.000906450 q^{50} +6.09439 q^{52} +10.9362 q^{53} +4.76900 q^{55} +0.408751 q^{56} -0.134671 q^{58} +14.2846 q^{59} -4.93532 q^{61} -0.0308727 q^{62} -7.98590 q^{64} +6.84118 q^{65} +11.3100 q^{67} +9.14905 q^{68} +0.229385 q^{70} +5.92285 q^{71} -13.7138 q^{73} -0.202951 q^{74} +1.86501 q^{76} +8.95685 q^{77} +4.19498 q^{79} -8.96973 q^{80} +0.262392 q^{82} -17.1030 q^{83} +10.2701 q^{85} -0.225665 q^{86} +0.206041 q^{88} +10.9105 q^{89} +12.8487 q^{91} +8.65552 q^{92} -0.171148 q^{94} +2.09355 q^{95} -3.59611 q^{97} +0.261098 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20q + 4q^{2} + 18q^{4} + 16q^{5} - 4q^{7} + 12q^{8} + O(q^{10}) \)	\( 20q + 4q^{2} + 18q^{4} + 16q^{5} - 4q^{7} + 12q^{8} + 4q^{10} + 12q^{11} - 4q^{13} + 20q^{14} + 22q^{16} + 24q^{17} - 4q^{19} + 40q^{20} - 6q^{22} + 12q^{23} + 22q^{25} + 30q^{26} - 12q^{28} + 24q^{29} - 4q^{31} + 28q^{32} + 8q^{34} + 20q^{35} - 10q^{37} + 26q^{38} + 6q^{40} + 66q^{41} + 8q^{43} + 36q^{44} - 12q^{46} + 28q^{47} + 18q^{49} + 28q^{50} - 18q^{52} + 28q^{53} - 4q^{55} + 60q^{56} + 54q^{59} - 4q^{61} + 20q^{62} + 22q^{64} + 42q^{65} + 12q^{67} + 12q^{68} + 20q^{70} + 36q^{71} + 14q^{73} - 50q^{76} + 8q^{77} - 12q^{79} + 88q^{80} - 8q^{82} + 20q^{83} + 4q^{85} + 18q^{86} - 10q^{88} + 130q^{89} - 6q^{91} - 46q^{92} - 26q^{94} - 2q^{97} + 12q^{98} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

Related objects

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