LMFDB - Embedded newform 8001.2.a.z.1.22

Newspace parameters

Level:	$ N $	$=$	$ 8001 = 3^{2} \cdot 7 \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8001.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$63.8883066572$
Analytic rank:	$1$
Dimension:	$32$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.22
Character	$\chi$	$=$	8001.1

$q$-expansion

$f(q)$	$=$	$q+0.959867 q^{2} -1.07866 q^{4} +2.40466 q^{5} -1.00000 q^{7} -2.95510 q^{8} +O(q^{10})$	\(q+0.959867 q^{2} -1.07866 q^{4} +2.40466 q^{5} -1.00000 q^{7} -2.95510 q^{8} +2.30815 q^{10} +4.25741 q^{11} +3.54385 q^{13} -0.959867 q^{14} -0.679190 q^{16} +0.302404 q^{17} -6.75953 q^{19} -2.59380 q^{20} +4.08655 q^{22} -6.51256 q^{23} +0.782373 q^{25} +3.40162 q^{26} +1.07866 q^{28} -3.90770 q^{29} -5.48700 q^{31} +5.25827 q^{32} +0.290267 q^{34} -2.40466 q^{35} -6.55479 q^{37} -6.48825 q^{38} -7.10600 q^{40} -2.17456 q^{41} +9.47361 q^{43} -4.59228 q^{44} -6.25119 q^{46} -1.43429 q^{47} +1.00000 q^{49} +0.750974 q^{50} -3.82259 q^{52} -10.5740 q^{53} +10.2376 q^{55} +2.95510 q^{56} -3.75087 q^{58} +7.97145 q^{59} +1.05914 q^{61} -5.26679 q^{62} +6.40561 q^{64} +8.52173 q^{65} -4.72133 q^{67} -0.326189 q^{68} -2.30815 q^{70} -6.25244 q^{71} -11.4772 q^{73} -6.29172 q^{74} +7.29120 q^{76} -4.25741 q^{77} +4.55627 q^{79} -1.63322 q^{80} -2.08729 q^{82} -10.1365 q^{83} +0.727177 q^{85} +9.09340 q^{86} -12.5811 q^{88} -8.24382 q^{89} -3.54385 q^{91} +7.02481 q^{92} -1.37672 q^{94} -16.2543 q^{95} +1.74407 q^{97} +0.959867 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32q + 30q^{4} - 32q^{7} + O(q^{10}) $	$ 32q + 30q^{4} - 32q^{7} - 16q^{10} - 14q^{13} + 18q^{16} - 30q^{19} - 10q^{22} + 36q^{25} - 30q^{28} - 58q^{31} - 34q^{34} + 8q^{37} - 34q^{40} + 6q^{43} - 36q^{46} + 32q^{49} - 56q^{52} - 88q^{55} - 22q^{58} - 46q^{61} + 20q^{64} - 8q^{67} + 16q^{70} - 60q^{73} - 128q^{76} - 74q^{79} - 52q^{82} - 16q^{85} - 64q^{88} + 14q^{91} - 58q^{94} - 44q^{97} + 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 $

$2$	0.959867	0.678728	0.339364	−	0.940655i	$-0.389788\pi$
$2$	0.959867	0.678728	0.339364	+	0.940655i	$0.389788\pi$
$3$	0	0
$3$	0	0
$4$	−1.07866	−0.539328
$5$	2.40466	1.07540	0.537698	−	0.843138i	$-0.319294\pi$
$5$	2.40466	1.07540	0.537698	+	0.843138i	$0.319294\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−2.95510	−1.04479
$9$	0	0
$10$	2.30815	0.729901
$11$	4.25741	1.28366	0.641829	−	0.766848i	$-0.278176\pi$
$11$	4.25741	1.28366	0.641829	+	0.766848i	$0.278176\pi$
$12$	0	0
$13$	3.54385	0.982886	0.491443	−	0.870910i	$-0.336470\pi$
$13$	3.54385	0.982886	0.491443	+	0.870910i	$0.336470\pi$
$14$	−0.959867	−0.256535
$15$	0	0
$16$	−0.679190	−0.169797
$17$	0.302404	0.0733436	0.0366718	−	0.999327i	$-0.488324\pi$
$17$	0.302404	0.0733436	0.0366718	+	0.999327i	$0.488324\pi$
$18$	0	0
$19$	−6.75953	−1.55074	−0.775371	−	0.631506i	$-0.782437\pi$
$19$	−6.75953	−1.55074	−0.775371	+	0.631506i	$0.782437\pi$
$20$	−2.59380	−0.579991
$21$	0	0
$22$	4.08655	0.871255
$23$	−6.51256	−1.35796	−0.678981	−	0.734155i	$-0.737578\pi$
$23$	−6.51256	−1.35796	−0.678981	+	0.734155i	$0.737578\pi$
$24$	0	0
$25$	0.782373	0.156475
$26$	3.40162	0.667113
$27$	0	0
$28$	1.07866	0.203847
$29$	−3.90770	−0.725641	−0.362820	−	0.931859i	$-0.618186\pi$
$29$	−3.90770	−0.725641	−0.362820	+	0.931859i	$0.618186\pi$
$30$	0	0
$31$	−5.48700	−0.985495	−0.492748	−	0.870172i	$-0.664007\pi$
$31$	−5.48700	−0.985495	−0.492748	+	0.870172i	$0.664007\pi$
$32$	5.25827	0.929539
$33$	0	0
$34$	0.290267	0.0497804
$35$	−2.40466	−0.406461
$36$	0	0
$37$	−6.55479	−1.07760	−0.538800	−	0.842434i	$-0.681122\pi$
$37$	−6.55479	−1.07760	−0.538800	+	0.842434i	$0.681122\pi$
$38$	−6.48825	−1.05253
$39$	0	0
$40$	−7.10600	−1.12356
$41$	−2.17456	−0.339609	−0.169805	−	0.985478i	$-0.554314\pi$
$41$	−2.17456	−0.339609	−0.169805	+	0.985478i	$0.554314\pi$
$42$	0	0
$43$	9.47361	1.44471	0.722356	−	0.691521i	$-0.243059\pi$
$43$	9.47361	1.44471	0.722356	+	0.691521i	$0.243059\pi$
$44$	−4.59228	−0.692313
$45$	0	0
$46$	−6.25119	−0.921688
$47$	−1.43429	−0.209212	−0.104606	−	0.994514i	$-0.533358\pi$
$47$	−1.43429	−0.209212	−0.104606	+	0.994514i	$0.533358\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0.750974	0.106204
$51$	0	0
$52$	−3.82259	−0.530098
$53$	−10.5740	−1.45246	−0.726228	−	0.687454i	$-0.758728\pi$
$53$	−10.5740	−1.45246	−0.726228	+	0.687454i	$0.758728\pi$
$54$	0	0
$55$	10.2376	1.38044
$56$	2.95510	0.394892
$57$	0	0
$58$	−3.75087	−0.492513
$59$	7.97145	1.03779	0.518897	−	0.854837i	$-0.326343\pi$
$59$	7.97145	1.03779	0.518897	+	0.854837i	$0.326343\pi$
$60$	0	0
$61$	1.05914	0.135608	0.0678042	−	0.997699i	$-0.478401\pi$
$61$	1.05914	0.135608	0.0678042	+	0.997699i	$0.478401\pi$
$62$	−5.26679	−0.668883
$63$	0	0
$64$	6.40561	0.800702
$65$	8.52173	1.05699
$66$	0	0
$67$	−4.72133	−0.576802	−0.288401	−	0.957510i	$-0.593124\pi$
$67$	−4.72133	−0.576802	−0.288401	+	0.957510i	$0.593124\pi$
$68$	−0.326189	−0.0395563
$69$	0	0
$70$	−2.30815	−0.275877
$71$	−6.25244	−0.742028	−0.371014	−	0.928627i	$-0.620990\pi$
$71$	−6.25244	−0.742028	−0.371014	+	0.928627i	$0.620990\pi$
$72$	0	0
$73$	−11.4772	−1.34330	−0.671652	−	0.740867i	$-0.734415\pi$
$73$	−11.4772	−1.34330	−0.671652	+	0.740867i	$0.734415\pi$
$74$	−6.29172	−0.731398
$75$	0	0
$76$	7.29120	0.836358
$77$	−4.25741	−0.485177
$78$	0	0
$79$	4.55627	0.512620	0.256310	−	0.966595i	$-0.417493\pi$
$79$	4.55627	0.512620	0.256310	+	0.966595i	$0.417493\pi$
$80$	−1.63322	−0.182599
$81$	0	0
$82$	−2.08729	−0.230502
$83$	−10.1365	−1.11263	−0.556315	−	0.830972i	$-0.687785\pi$
$83$	−10.1365	−1.11263	−0.556315	+	0.830972i	$0.687785\pi$
$84$	0	0
$85$	0.727177	0.0788734
$86$	9.09340	0.980567
$87$	0	0
$88$	−12.5811	−1.34115
$89$	−8.24382	−0.873844	−0.436922	−	0.899499i	$-0.643931\pi$
$89$	−8.24382	−0.873844	−0.436922	+	0.899499i	$0.643931\pi$
$90$	0	0
$91$	−3.54385	−0.371496
$92$	7.02481	0.732387
$93$	0	0
$94$	−1.37672	−0.141998
$95$	−16.2543	−1.66766
$96$	0	0
$97$	1.74407	0.177084	0.0885419	−	0.996072i	$-0.471779\pi$
$97$	1.74407	0.177084	0.0885419	+	0.996072i	$0.471779\pi$
$98$	0.959867	0.0969612
$99$	0	0
$100$	−0.843912	−0.0843912
$101$	3.61578	0.359783	0.179892	−	0.983686i	$-0.442425\pi$
$101$	3.61578	0.359783	0.179892	+	0.983686i	$0.442425\pi$
$102$	0	0
$103$	5.92884	0.584186	0.292093	−	0.956390i	$-0.405648\pi$
$103$	5.92884	0.584186	0.292093	+	0.956390i	$0.405648\pi$
$104$	−10.4724	−1.02690
$105$	0	0
$106$	−10.1497	−0.985823
$107$	−6.45523	−0.624050	−0.312025	−	0.950074i	$-0.601007\pi$
$107$	−6.45523	−0.624050	−0.312025	+	0.950074i	$0.601007\pi$
$108$	0	0
$109$	20.5285	1.96628	0.983138	−	0.182865i	$-0.0585371\pi$
$109$	20.5285	1.96628	0.983138	+	0.182865i	$0.0585371\pi$
$110$	9.82675	0.936943
$111$	0	0
$112$	0.679190	0.0641774
$113$	15.2547	1.43504	0.717520	−	0.696538i	$-0.245277\pi$
$113$	15.2547	1.43504	0.717520	+	0.696538i	$0.245277\pi$
$114$	0	0
$115$	−15.6605	−1.46035
$116$	4.21506	0.391358
$117$	0	0
$118$	7.65153	0.704380
$119$	−0.302404	−0.0277213
$120$	0	0
$121$	7.12556	0.647778
$122$	1.01663	0.0920412
$123$	0	0
$124$	5.91859	0.531505
$125$	−10.1419	−0.907123
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	−4.36800	−0.386080
$129$	0	0
$130$	8.17973	0.717410
$131$	5.94653	0.519551	0.259775	−	0.965669i	$-0.416352\pi$
$131$	5.94653	0.519551	0.259775	+	0.965669i	$0.416352\pi$
$132$	0	0
$133$	6.75953	0.586125
$134$	−4.53185	−0.391492
$135$	0	0
$136$	−0.893633	−0.0766284
$137$	−1.37243	−0.117255	−0.0586273	−	0.998280i	$-0.518672\pi$
$137$	−1.37243	−0.117255	−0.0586273	+	0.998280i	$0.518672\pi$
$138$	0	0
$139$	3.13422	0.265841	0.132921	−	0.991127i	$-0.457564\pi$
$139$	3.13422	0.265841	0.132921	+	0.991127i	$0.457564\pi$
$140$	2.59380	0.219216
$141$	0	0
$142$	−6.00151	−0.503635
$143$	15.0876	1.26169
$144$	0	0
$145$	−9.39666	−0.780351
$146$	−11.0166	−0.911738
$147$	0	0
$148$	7.07036	0.581180
$149$	−13.3703	−1.09534	−0.547670	−	0.836694i	$-0.684485\pi$
$149$	−13.3703	−1.09534	−0.547670	+	0.836694i	$0.684485\pi$
$150$	0	0
$151$	−10.2915	−0.837513	−0.418757	−	0.908098i	$-0.637534\pi$
$151$	−10.2915	−0.837513	−0.418757	+	0.908098i	$0.637534\pi$
$152$	19.9751	1.62019
$153$	0	0
$154$	−4.08655	−0.329303
$155$	−13.1944	−1.05980
$156$	0	0
$157$	7.70342	0.614800	0.307400	−	0.951580i	$-0.400541\pi$
$157$	7.70342	0.614800	0.307400	+	0.951580i	$0.400541\pi$
$158$	4.37341	0.347930
$159$	0	0
$160$	12.6443	0.999622
$161$	6.51256	0.513262
$162$	0	0
$163$	6.54115	0.512342	0.256171	−	0.966631i	$-0.417539\pi$
$163$	6.54115	0.512342	0.256171	+	0.966631i	$0.417539\pi$
$164$	2.34560	0.183161
$165$	0	0
$166$	−9.72973	−0.755173
$167$	−2.06466	−0.159769	−0.0798843	−	0.996804i	$-0.525455\pi$
$167$	−2.06466	−0.159769	−0.0798843	+	0.996804i	$0.525455\pi$
$168$	0	0
$169$	−0.441156	−0.0339351
$170$	0.697993	0.0535336
$171$	0	0
$172$	−10.2188	−0.779174
$173$	11.0221	0.837991	0.418996	−	0.907988i	$-0.362382\pi$
$173$	11.0221	0.837991	0.418996	+	0.907988i	$0.362382\pi$
$174$	0	0
$175$	−0.782373	−0.0591419
$176$	−2.89159	−0.217962
$177$	0	0
$178$	−7.91297	−0.593102
$179$	−23.6676	−1.76900	−0.884501	−	0.466537i	$-0.845501\pi$
$179$	−23.6676	−1.76900	−0.884501	+	0.466537i	$0.845501\pi$
$180$	0	0
$181$	−15.1437	−1.12563	−0.562813	−	0.826584i	$-0.690281\pi$
$181$	−15.1437	−1.12563	−0.562813	+	0.826584i	$0.690281\pi$
$182$	−3.40162	−0.252145
$183$	0	0
$184$	19.2453	1.41878
$185$	−15.7620	−1.15885
$186$	0	0
$187$	1.28746	0.0941482
$188$	1.54710	0.112834
$189$	0	0
$190$	−15.6020	−1.13189
$191$	−11.4808	−0.830722	−0.415361	−	0.909657i	$-0.636345\pi$
$191$	−11.4808	−0.830722	−0.415361	+	0.909657i	$0.636345\pi$
$192$	0	0
$193$	−8.41906	−0.606017	−0.303009	−	0.952988i	$-0.597991\pi$
$193$	−8.41906	−0.606017	−0.303009	+	0.952988i	$0.597991\pi$
$194$	1.67408	0.120192
$195$	0	0
$196$	−1.07866	−0.0770468
$197$	−6.10162	−0.434723	−0.217361	−	0.976091i	$-0.569745\pi$
$197$	−6.10162	−0.434723	−0.217361	+	0.976091i	$0.569745\pi$
$198$	0	0
$199$	−10.7103	−0.759231	−0.379616	−	0.925144i	$-0.623944\pi$
$199$	−10.7103	−0.759231	−0.379616	+	0.925144i	$0.623944\pi$
$200$	−2.31199	−0.163482
$201$	0	0
$202$	3.47066	0.244195
$203$	3.90770	0.274266
$204$	0	0
$205$	−5.22907	−0.365214
$206$	5.69089	0.396503
$207$	0	0
$208$	−2.40694	−0.166891
$209$	−28.7781	−1.99062
$210$	0	0
$211$	−13.8349	−0.952436	−0.476218	−	0.879327i	$-0.657993\pi$
$211$	−13.8349	−0.952436	−0.476218	+	0.879327i	$0.657993\pi$
$212$	11.4057	0.783350
$213$	0	0
$214$	−6.19616	−0.423561
$215$	22.7808	1.55364
$216$	0	0
$217$	5.48700	0.372482
$218$	19.7046	1.33457
$219$	0	0
$220$	−11.0429	−0.744510
$221$	1.07167	0.0720884
$222$	0	0
$223$	−5.00800	−0.335360	−0.167680	−	0.985841i	$-0.553628\pi$
$223$	−5.00800	−0.335360	−0.167680	+	0.985841i	$0.553628\pi$
$224$	−5.25827	−0.351333
$225$	0	0
$226$	14.6425	0.974002
$227$	23.4386	1.55567	0.777837	−	0.628467i	$-0.216317\pi$
$227$	23.4386	1.55567	0.777837	+	0.628467i	$0.216317\pi$
$228$	0	0
$229$	−3.95298	−0.261220	−0.130610	−	0.991434i	$-0.541694\pi$
$229$	−3.95298	−0.261220	−0.130610	+	0.991434i	$0.541694\pi$
$230$	−15.0320	−0.991179
$231$	0	0
$232$	11.5476	0.758139
$233$	−1.27357	−0.0834346	−0.0417173	−	0.999129i	$-0.513283\pi$
$233$	−1.27357	−0.0834346	−0.0417173	+	0.999129i	$0.513283\pi$
$234$	0	0
$235$	−3.44897	−0.224986
$236$	−8.59845	−0.559712
$237$	0	0
$238$	−0.290267	−0.0188152
$239$	−8.13559	−0.526248	−0.263124	−	0.964762i	$-0.584753\pi$
$239$	−8.13559	−0.526248	−0.263124	+	0.964762i	$0.584753\pi$
$240$	0	0
$241$	19.4833	1.25503	0.627513	−	0.778606i	$-0.284073\pi$
$241$	19.4833	1.25503	0.627513	+	0.778606i	$0.284073\pi$
$242$	6.83959	0.439665
$243$	0	0
$244$	−1.14244	−0.0731374
$245$	2.40466	0.153628
$246$	0	0
$247$	−23.9547	−1.52420
$248$	16.2146	1.02963
$249$	0	0
$250$	−9.73491	−0.615690
$251$	−2.25176	−0.142130	−0.0710650	−	0.997472i	$-0.522640\pi$
$251$	−2.25176	−0.142130	−0.0710650	+	0.997472i	$0.522640\pi$
$252$	0	0
$253$	−27.7267	−1.74316
$254$	−0.959867	−0.0602274
$255$	0	0
$256$	−17.0039	−1.06275
$257$	−15.5072	−0.967310	−0.483655	−	0.875259i	$-0.660691\pi$
$257$	−15.5072	−0.967310	−0.483655	+	0.875259i	$0.660691\pi$
$258$	0	0
$259$	6.55479	0.407295
$260$	−9.19202	−0.570065
$261$	0	0
$262$	5.70787	0.352634
$263$	−3.83687	−0.236591	−0.118296	−	0.992978i	$-0.537743\pi$
$263$	−3.83687	−0.236591	−0.118296	+	0.992978i	$0.537743\pi$
$264$	0	0
$265$	−25.4269	−1.56196
$266$	6.48825	0.397820
$267$	0	0
$268$	5.09269	0.311086
$269$	−15.6381	−0.953470	−0.476735	−	0.879047i	$-0.658180\pi$
$269$	−15.6381	−0.953470	−0.476735	+	0.879047i	$0.658180\pi$
$270$	0	0
$271$	23.4561	1.42485	0.712427	−	0.701746i	$-0.247596\pi$
$271$	23.4561	1.42485	0.712427	+	0.701746i	$0.247596\pi$
$272$	−0.205389	−0.0124536
$273$	0	0
$274$	−1.31735	−0.0795840
$275$	3.33089	0.200860
$276$	0	0
$277$	26.8299	1.61205	0.806025	−	0.591881i	$-0.201615\pi$
$277$	26.8299	1.61205	0.806025	+	0.591881i	$0.201615\pi$
$278$	3.00844	0.180434
$279$	0	0
$280$	7.10600	0.424665
$281$	20.9721	1.25109	0.625545	−	0.780188i	$-0.284877\pi$
$281$	20.9721	1.25109	0.625545	+	0.780188i	$0.284877\pi$
$282$	0	0
$283$	−16.5527	−0.983959	−0.491979	−	0.870607i	$-0.663726\pi$
$283$	−16.5527	−0.983959	−0.491979	+	0.870607i	$0.663726\pi$
$284$	6.74423	0.400196
$285$	0	0
$286$	14.4821	0.856344
$287$	2.17456	0.128360
$288$	0	0
$289$	−16.9086	−0.994621
$290$	−9.01955	−0.529646
$291$	0	0
$292$	12.3799	0.724481
$293$	16.6113	0.970443	0.485221	−	0.874391i	$-0.338739\pi$
$293$	16.6113	0.970443	0.485221	+	0.874391i	$0.338739\pi$
$294$	0	0
$295$	19.1686	1.11604
$296$	19.3701	1.12586
$297$	0	0
$298$	−12.8337	−0.743439
$299$	−23.0795	−1.33472
$300$	0	0
$301$	−9.47361	−0.546050
$302$	−9.87850	−0.568444
$303$	0	0
$304$	4.59100	0.263312
$305$	2.54686	0.145833
$306$	0	0
$307$	−25.6493	−1.46388	−0.731942	−	0.681367i	$-0.761386\pi$
$307$	−25.6493	−1.46388	−0.731942	+	0.681367i	$0.761386\pi$
$308$	4.59228	0.261670
$309$	0	0
$310$	−12.6648	−0.719314
$311$	−6.28790	−0.356554	−0.178277	−	0.983980i	$-0.557052\pi$
$311$	−6.28790	−0.356554	−0.178277	+	0.983980i	$0.557052\pi$
$312$	0	0
$313$	−24.0220	−1.35781	−0.678903	−	0.734228i	$-0.737544\pi$
$313$	−24.0220	−1.35781	−0.678903	+	0.734228i	$0.737544\pi$
$314$	7.39425	0.417282
$315$	0	0
$316$	−4.91464	−0.276470
$317$	11.7159	0.658031	0.329016	−	0.944324i	$-0.393283\pi$
$317$	11.7159	0.658031	0.329016	+	0.944324i	$0.393283\pi$
$318$	0	0
$319$	−16.6367	−0.931475
$320$	15.4033	0.861071
$321$	0	0
$322$	6.25119	0.348365
$323$	−2.04411	−0.113737
$324$	0	0
$325$	2.77261	0.153797
$326$	6.27863	0.347741
$327$	0	0
$328$	6.42604	0.354819
$329$	1.43429	0.0790748
$330$	0	0
$331$	−14.4126	−0.792187	−0.396094	−	0.918210i	$-0.629634\pi$
$331$	−14.4126	−0.792187	−0.396094	+	0.918210i	$0.629634\pi$
$332$	10.9338	0.600072
$333$	0	0
$334$	−1.98180	−0.108439
$335$	−11.3532	−0.620290
$336$	0	0
$337$	−14.1181	−0.769061	−0.384531	−	0.923112i	$-0.625637\pi$
$337$	−14.1181	−0.769061	−0.384531	+	0.923112i	$0.625637\pi$
$338$	−0.423451	−0.0230327
$339$	0	0
$340$	−0.784373	−0.0425386
$341$	−23.3604	−1.26504
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−27.9955	−1.50941
$345$	0	0
$346$	10.5797	0.568768
$347$	28.8833	1.55053	0.775267	−	0.631633i	$-0.217615\pi$
$347$	28.8833	1.55053	0.775267	+	0.631633i	$0.217615\pi$
$348$	0	0
$349$	24.3596	1.30394	0.651971	−	0.758244i	$-0.273943\pi$
$349$	24.3596	1.30394	0.651971	+	0.758244i	$0.273943\pi$
$350$	−0.750974	−0.0401413
$351$	0	0
$352$	22.3866	1.19321
$353$	−22.1309	−1.17791	−0.588955	−	0.808166i	$-0.700461\pi$
$353$	−22.1309	−1.17791	−0.588955	+	0.808166i	$0.700461\pi$
$354$	0	0
$355$	−15.0350	−0.797973
$356$	8.89225	0.471288
$357$	0	0
$358$	−22.7178	−1.20067
$359$	−34.4554	−1.81848	−0.909242	−	0.416268i	$-0.863338\pi$
$359$	−34.4554	−1.81848	−0.909242	+	0.416268i	$0.863338\pi$
$360$	0	0
$361$	26.6912	1.40480
$362$	−14.5360	−0.763994
$363$	0	0
$364$	3.82259	0.200358
$365$	−27.5987	−1.44458
$366$	0	0
$367$	−29.6470	−1.54756	−0.773780	−	0.633455i	$-0.781636\pi$
$367$	−29.6470	−1.54756	−0.773780	+	0.633455i	$0.781636\pi$
$368$	4.42326	0.230579
$369$	0	0
$370$	−15.1294	−0.786542
$371$	10.5740	0.548977
$372$	0	0
$373$	−24.6924	−1.27852	−0.639262	−	0.768989i	$-0.720760\pi$
$373$	−24.6924	−1.27852	−0.639262	+	0.768989i	$0.720760\pi$
$374$	1.23579	0.0639010
$375$	0	0
$376$	4.23846	0.218582
$377$	−13.8483	−0.713222
$378$	0	0
$379$	−37.1608	−1.90882	−0.954412	−	0.298493i	$-0.903516\pi$
$379$	−37.1608	−1.90882	−0.954412	+	0.298493i	$0.903516\pi$
$380$	17.5328	0.899416
$381$	0	0
$382$	−11.0200	−0.563834
$383$	−16.4370	−0.839894	−0.419947	−	0.907549i	$-0.637951\pi$
$383$	−16.4370	−0.839894	−0.419947	+	0.907549i	$0.637951\pi$
$384$	0	0
$385$	−10.2376	−0.521757
$386$	−8.08118	−0.411321
$387$	0	0
$388$	−1.88125	−0.0955062
$389$	−3.32747	−0.168709	−0.0843547	−	0.996436i	$-0.526883\pi$
$389$	−3.32747	−0.168709	−0.0843547	+	0.996436i	$0.526883\pi$
$390$	0	0
$391$	−1.96942	−0.0995979
$392$	−2.95510	−0.149255
$393$	0	0
$394$	−5.85675	−0.295059
$395$	10.9563	0.551269
$396$	0	0
$397$	4.88603	0.245223	0.122611	−	0.992455i	$-0.460873\pi$
$397$	4.88603	0.245223	0.122611	+	0.992455i	$0.460873\pi$
$398$	−10.2804	−0.515312
$399$	0	0
$400$	−0.531380	−0.0265690
$401$	18.0905	0.903396	0.451698	−	0.892171i	$-0.350818\pi$
$401$	18.0905	0.903396	0.451698	+	0.892171i	$0.350818\pi$
$402$	0	0
$403$	−19.4451	−0.968629
$404$	−3.90018	−0.194041
$405$	0	0
$406$	3.75087	0.186152
$407$	−27.9064	−1.38327
$408$	0	0
$409$	13.1826	0.651837	0.325918	−	0.945398i	$-0.394327\pi$
$409$	13.1826	0.651837	0.325918	+	0.945398i	$0.394327\pi$
$410$	−5.01921	−0.247881
$411$	0	0
$412$	−6.39518	−0.315068
$413$	−7.97145	−0.392249
$414$	0	0
$415$	−24.3749	−1.19652
$416$	18.6345	0.913631
$417$	0	0
$418$	−27.6231	−1.35109
$419$	33.2567	1.62470	0.812348	−	0.583173i	$-0.198189\pi$
$419$	33.2567	1.62470	0.812348	+	0.583173i	$0.198189\pi$
$420$	0	0
$421$	−4.43830	−0.216309	−0.108155	−	0.994134i	$-0.534494\pi$
$421$	−4.43830	−0.216309	−0.108155	+	0.994134i	$0.534494\pi$
$422$	−13.2797	−0.646445
$423$	0	0
$424$	31.2473	1.51750
$425$	0.236592	0.0114764
$426$	0	0
$427$	−1.05914	−0.0512552
$428$	6.96297	0.336568
$429$	0	0
$430$	21.8665	1.05450
$431$	19.7379	0.950743	0.475371	−	0.879785i	$-0.342314\pi$
$431$	19.7379	0.950743	0.475371	+	0.879785i	$0.342314\pi$
$432$	0	0
$433$	29.6295	1.42390	0.711952	−	0.702228i	$-0.247811\pi$
$433$	29.6295	1.42390	0.711952	+	0.702228i	$0.247811\pi$
$434$	5.26679	0.252814
$435$	0	0
$436$	−22.1432	−1.06047
$437$	44.0218	2.10585
$438$	0	0
$439$	−41.2017	−1.96645	−0.983224	−	0.182403i	$-0.941612\pi$
$439$	−41.2017	−1.96645	−0.983224	+	0.182403i	$0.941612\pi$
$440$	−30.2532	−1.44226
$441$	0	0
$442$	1.02866	0.0489285
$443$	24.9988	1.18773	0.593863	−	0.804566i	$-0.297602\pi$
$443$	24.9988	1.18773	0.593863	+	0.804566i	$0.297602\pi$
$444$	0	0
$445$	−19.8236	−0.939727
$446$	−4.80701	−0.227619
$447$	0	0
$448$	−6.40561	−0.302637
$449$	−4.39785	−0.207548	−0.103774	−	0.994601i	$-0.533092\pi$
$449$	−4.39785	−0.207548	−0.103774	+	0.994601i	$0.533092\pi$
$450$	0	0
$451$	−9.25800	−0.435942
$452$	−16.4545	−0.773957
$453$	0	0
$454$	22.4979	1.05588
$455$	−8.52173	−0.399505
$456$	0	0
$457$	13.3749	0.625653	0.312826	−	0.949810i	$-0.398724\pi$
$457$	13.3749	0.625653	0.312826	+	0.949810i	$0.398724\pi$
$458$	−3.79433	−0.177298
$459$	0	0
$460$	16.8923	0.787606
$461$	10.2563	0.477685	0.238842	−	0.971058i	$-0.423232\pi$
$461$	10.2563	0.477685	0.238842	+	0.971058i	$0.423232\pi$
$462$	0	0
$463$	−16.2905	−0.757084	−0.378542	−	0.925584i	$-0.623575\pi$
$463$	−16.2905	−0.757084	−0.378542	+	0.925584i	$0.623575\pi$
$464$	2.65407	0.123212
$465$	0	0
$466$	−1.22246	−0.0566294
$467$	8.99794	0.416375	0.208188	−	0.978089i	$-0.433244\pi$
$467$	8.99794	0.416375	0.208188	+	0.978089i	$0.433244\pi$
$468$	0	0
$469$	4.72133	0.218011
$470$	−3.31055	−0.152704
$471$	0	0
$472$	−23.5564	−1.08427
$473$	40.3331	1.85452
$474$	0	0
$475$	−5.28847	−0.242652
$476$	0.326189	0.0149509
$477$	0	0
$478$	−7.80908	−0.357179
$479$	14.2639	0.651733	0.325866	−	0.945416i	$-0.394344\pi$
$479$	14.2639	0.651733	0.325866	+	0.945416i	$0.394344\pi$
$480$	0	0
$481$	−23.2292	−1.05916
$482$	18.7013	0.851822
$483$	0	0
$484$	−7.68603	−0.349365
$485$	4.19390	0.190435
$486$	0	0
$487$	−10.7755	−0.488284	−0.244142	−	0.969740i	$-0.578506\pi$
$487$	−10.7755	−0.488284	−0.244142	+	0.969740i	$0.578506\pi$
$488$	−3.12985	−0.141682
$489$	0	0
$490$	2.30815	0.104272
$491$	6.80357	0.307041	0.153520	−	0.988145i	$-0.450939\pi$
$491$	6.80357	0.307041	0.153520	+	0.988145i	$0.450939\pi$
$492$	0	0
$493$	−1.18170	−0.0532211
$494$	−22.9933	−1.03452
$495$	0	0
$496$	3.72672	0.167335
$497$	6.25244	0.280460
$498$	0	0
$499$	−30.2923	−1.35607	−0.678035	−	0.735030i	$-0.737168\pi$
$499$	−30.2923	−1.35607	−0.678035	+	0.735030i	$0.737168\pi$
$500$	10.9397	0.489237
$501$	0	0
$502$	−2.16139	−0.0964677
$503$	−14.4114	−0.642571	−0.321286	−	0.946982i	$-0.604115\pi$
$503$	−14.4114	−0.642571	−0.321286	+	0.946982i	$0.604115\pi$
$504$	0	0
$505$	8.69470	0.386909
$506$	−26.6139	−1.18313
$507$	0	0
$508$	1.07866	0.0478576
$509$	0.467272	0.0207115	0.0103557	−	0.999946i	$-0.496704\pi$
$509$	0.467272	0.0207115	0.0103557	+	0.999946i	$0.496704\pi$
$510$	0	0
$511$	11.4772	0.507721
$512$	−7.58551	−0.335235
$513$	0	0
$514$	−14.8848	−0.656541
$515$	14.2568	0.628230
$516$	0	0
$517$	−6.10635	−0.268557
$518$	6.29172	0.276442
$519$	0	0
$520$	−25.1826	−1.10433
$521$	34.7244	1.52130	0.760651	−	0.649160i	$-0.224880\pi$
$521$	34.7244	1.52130	0.760651	+	0.649160i	$0.224880\pi$
$522$	0	0
$523$	−9.13868	−0.399607	−0.199803	−	0.979836i	$-0.564030\pi$
$523$	−9.13868	−0.399607	−0.199803	+	0.979836i	$0.564030\pi$
$524$	−6.41426	−0.280208
$525$	0	0
$526$	−3.68288	−0.160581
$527$	−1.65929	−0.0722798
$528$	0	0
$529$	19.4134	0.844063
$530$	−24.4065	−1.06015
$531$	0	0
$532$	−7.29120	−0.316114
$533$	−7.70631	−0.333797
$534$	0	0
$535$	−15.5226	−0.671101
$536$	13.9520	0.602635
$537$	0	0
$538$	−15.0105	−0.647147
$539$	4.25741	0.183380
$540$	0	0
$541$	8.80281	0.378462	0.189231	−	0.981933i	$-0.439400\pi$
$541$	8.80281	0.378462	0.189231	+	0.981933i	$0.439400\pi$
$542$	22.5147	0.967089
$543$	0	0
$544$	1.59012	0.0681758
$545$	49.3641	2.11452
$546$	0	0
$547$	2.47099	0.105652	0.0528258	−	0.998604i	$-0.483177\pi$
$547$	2.47099	0.105652	0.0528258	+	0.998604i	$0.483177\pi$
$548$	1.48038	0.0632386
$549$	0	0
$550$	3.19721	0.136329
$551$	26.4142	1.12528
$552$	0	0
$553$	−4.55627	−0.193752
$554$	25.7531	1.09414
$555$	0	0
$556$	−3.38075	−0.143376
$557$	−32.2057	−1.36460	−0.682299	−	0.731073i	$-0.739020\pi$
$557$	−32.2057	−1.36460	−0.682299	+	0.731073i	$0.739020\pi$
$558$	0	0
$559$	33.5730	1.41999
$560$	1.63322	0.0690161
$561$	0	0
$562$	20.1304	0.849150
$563$	−13.9684	−0.588696	−0.294348	−	0.955698i	$-0.595102\pi$
$563$	−13.9684	−0.588696	−0.294348	+	0.955698i	$0.595102\pi$
$564$	0	0
$565$	36.6823	1.54323
$566$	−15.8884	−0.667841
$567$	0	0
$568$	18.4766	0.775260
$569$	18.2780	0.766252	0.383126	−	0.923696i	$-0.374847\pi$
$569$	18.2780	0.766252	0.383126	+	0.923696i	$0.374847\pi$
$570$	0	0
$571$	40.3925	1.69037	0.845187	−	0.534470i	$-0.179489\pi$
$571$	40.3925	1.69037	0.845187	+	0.534470i	$0.179489\pi$
$572$	−16.2743	−0.680464
$573$	0	0
$574$	2.08729	0.0871217
$575$	−5.09525	−0.212487
$576$	0	0
$577$	4.42495	0.184213	0.0921065	−	0.995749i	$-0.470640\pi$
$577$	4.42495	0.184213	0.0921065	+	0.995749i	$0.470640\pi$
$578$	−16.2300	−0.675077
$579$	0	0
$580$	10.1358	0.420865
$581$	10.1365	0.420534
$582$	0	0
$583$	−45.0180	−1.86446
$584$	33.9162	1.40346
$585$	0	0
$586$	15.9446	0.658667
$587$	22.3504	0.922500	0.461250	−	0.887270i	$-0.347401\pi$
$587$	22.3504	0.922500	0.461250	+	0.887270i	$0.347401\pi$
$588$	0	0
$589$	37.0896	1.52825
$590$	18.3993	0.757487
$591$	0	0
$592$	4.45194	0.182974
$593$	39.1966	1.60961	0.804805	−	0.593539i	$-0.202270\pi$
$593$	39.1966	1.60961	0.804805	+	0.593539i	$0.202270\pi$
$594$	0	0
$595$	−0.727177	−0.0298113
$596$	14.4220	0.590748
$597$	0	0
$598$	−22.1533	−0.905914
$599$	20.9632	0.856534	0.428267	−	0.903652i	$-0.359124\pi$
$599$	20.9632	0.856534	0.428267	+	0.903652i	$0.359124\pi$
$600$	0	0
$601$	39.3899	1.60675	0.803375	−	0.595474i	$-0.203036\pi$
$601$	39.3899	1.60675	0.803375	+	0.595474i	$0.203036\pi$
$602$	−9.09340	−0.370620
$603$	0	0
$604$	11.1010	0.451694
$605$	17.1345	0.696618
$606$	0	0
$607$	−6.62814	−0.269028	−0.134514	−	0.990912i	$-0.542947\pi$
$607$	−6.62814	−0.269028	−0.134514	+	0.990912i	$0.542947\pi$
$608$	−35.5434	−1.44148
$609$	0	0
$610$	2.44464	0.0989807
$611$	−5.08289	−0.205632
$612$	0	0
$613$	−22.9071	−0.925210	−0.462605	−	0.886565i	$-0.653085\pi$
$613$	−22.9071	−0.925210	−0.462605	+	0.886565i	$0.653085\pi$
$614$	−24.6199	−0.993579
$615$	0	0
$616$	12.5811	0.506906
$617$	40.6908	1.63815	0.819076	−	0.573685i	$-0.194487\pi$
$617$	40.6908	1.63815	0.819076	+	0.573685i	$0.194487\pi$
$618$	0	0
$619$	−24.2607	−0.975118	−0.487559	−	0.873090i	$-0.662113\pi$
$619$	−24.2607	−0.975118	−0.487559	+	0.873090i	$0.662113\pi$
$620$	14.2322	0.571578
$621$	0	0
$622$	−6.03554	−0.242003
$623$	8.24382	0.330282
$624$	0	0
$625$	−28.2998	−1.13199
$626$	−23.0579	−0.921581
$627$	0	0
$628$	−8.30933	−0.331579
$629$	−1.98219	−0.0790351
$630$	0	0
$631$	−36.8846	−1.46835	−0.734177	−	0.678959i	$-0.762432\pi$
$631$	−36.8846	−1.46835	−0.734177	+	0.678959i	$0.762432\pi$
$632$	−13.4642	−0.535578
$633$	0	0
$634$	11.2457	0.446624
$635$	−2.40466	−0.0954259
$636$	0	0
$637$	3.54385	0.140412
$638$	−15.9690	−0.632218
$639$	0	0
$640$	−10.5035	−0.415189
$641$	37.0112	1.46186	0.730928	−	0.682455i	$-0.239088\pi$
$641$	37.0112	1.46186	0.730928	+	0.682455i	$0.239088\pi$
$642$	0	0
$643$	−19.7735	−0.779790	−0.389895	−	0.920859i	$-0.627489\pi$
$643$	−19.7735	−0.779790	−0.389895	+	0.920859i	$0.627489\pi$
$644$	−7.02481	−0.276816
$645$	0	0
$646$	−1.96207	−0.0771966
$647$	8.97670	0.352910	0.176455	−	0.984309i	$-0.443537\pi$
$647$	8.97670	0.352910	0.176455	+	0.984309i	$0.443537\pi$
$648$	0	0
$649$	33.9378	1.33217
$650$	2.66134	0.104386
$651$	0	0
$652$	−7.05565	−0.276321
$653$	−11.3033	−0.442331	−0.221166	−	0.975236i	$-0.570986\pi$
$653$	−11.3033	−0.442331	−0.221166	+	0.975236i	$0.570986\pi$
$654$	0	0
$655$	14.2994	0.558722
$656$	1.47694	0.0576648
$657$	0	0
$658$	1.37672	0.0536703
$659$	−44.1617	−1.72029	−0.860147	−	0.510046i	$-0.829629\pi$
$659$	−44.1617	−1.72029	−0.860147	+	0.510046i	$0.829629\pi$
$660$	0	0
$661$	35.8877	1.39587	0.697936	−	0.716160i	$-0.254102\pi$
$661$	35.8877	1.39587	0.697936	+	0.716160i	$0.254102\pi$
$662$	−13.8342	−0.537680
$663$	0	0
$664$	29.9545	1.16246
$665$	16.2543	0.630316
$666$	0	0
$667$	25.4491	0.985393
$668$	2.22706	0.0861676
$669$	0	0
$670$	−10.8975	−0.421009
$671$	4.50918	0.174075
$672$	0	0
$673$	30.0012	1.15646	0.578230	−	0.815873i	$-0.303743\pi$
$673$	30.0012	1.15646	0.578230	+	0.815873i	$0.303743\pi$
$674$	−13.5515	−0.521984
$675$	0	0
$676$	0.475856	0.0183022
$677$	12.7361	0.489488	0.244744	−	0.969588i	$-0.421296\pi$
$677$	12.7361	0.489488	0.244744	+	0.969588i	$0.421296\pi$
$678$	0	0
$679$	−1.74407	−0.0669314
$680$	−2.14888	−0.0824058
$681$	0	0
$682$	−22.4229	−0.858618
$683$	−21.9682	−0.840591	−0.420295	−	0.907387i	$-0.638074\pi$
$683$	−21.9682	−0.840591	−0.420295	+	0.907387i	$0.638074\pi$
$684$	0	0
$685$	−3.30022	−0.126095
$686$	−0.959867	−0.0366479
$687$	0	0
$688$	−6.43438	−0.245308
$689$	−37.4728	−1.42760
$690$	0	0
$691$	34.3489	1.30669	0.653347	−	0.757058i	$-0.273364\pi$
$691$	34.3489	1.30669	0.653347	+	0.757058i	$0.273364\pi$
$692$	−11.8890	−0.451952
$693$	0	0
$694$	27.7241	1.05239
$695$	7.53673	0.285884
$696$	0	0
$697$	−0.657595	−0.0249082
$698$	23.3820	0.885022
$699$	0	0
$700$	0.843912	0.0318969
$701$	−44.3643	−1.67562	−0.837809	−	0.545963i	$-0.816164\pi$
$701$	−44.3643	−1.67562	−0.837809	+	0.545963i	$0.816164\pi$
$702$	0	0
$703$	44.3073	1.67108
$704$	27.2713	1.02783
$705$	0	0
$706$	−21.2427	−0.799481
$707$	−3.61578	−0.135985
$708$	0	0
$709$	26.5449	0.996913	0.498457	−	0.866915i	$-0.333900\pi$
$709$	26.5449	0.996913	0.498457	+	0.866915i	$0.333900\pi$
$710$	−14.4316	−0.541607
$711$	0	0
$712$	24.3613	0.912979
$713$	35.7345	1.33827
$714$	0	0
$715$	36.2805	1.35681
$716$	25.5292	0.954073
$717$	0	0
$718$	−33.0725	−1.23426
$719$	−29.8919	−1.11478	−0.557389	−	0.830251i	$-0.688197\pi$
$719$	−29.8919	−1.11478	−0.557389	+	0.830251i	$0.688197\pi$
$720$	0	0
$721$	−5.92884	−0.220801
$722$	25.6200	0.953478
$723$	0	0
$724$	16.3349	0.607081
$725$	−3.05728	−0.113544
$726$	0	0
$727$	−42.0562	−1.55978	−0.779889	−	0.625918i	$-0.784725\pi$
$727$	−42.0562	−1.55978	−0.779889	+	0.625918i	$0.784725\pi$
$728$	10.4724	0.388134
$729$	0	0
$730$	−26.4911	−0.980478
$731$	2.86485	0.105960
$732$	0	0
$733$	11.2662	0.416128	0.208064	−	0.978115i	$-0.433284\pi$
$733$	11.2662	0.416128	0.208064	+	0.978115i	$0.433284\pi$
$734$	−28.4571	−1.05037
$735$	0	0
$736$	−34.2448	−1.26228
$737$	−20.1006	−0.740417
$738$	0	0
$739$	30.3213	1.11539	0.557694	−	0.830046i	$-0.311686\pi$
$739$	30.3213	1.11539	0.557694	+	0.830046i	$0.311686\pi$
$740$	17.0018	0.624998
$741$	0	0
$742$	10.1497	0.372606
$743$	39.3851	1.44490	0.722450	−	0.691423i	$-0.243016\pi$
$743$	39.3851	1.44490	0.722450	+	0.691423i	$0.243016\pi$
$744$	0	0
$745$	−32.1511	−1.17792
$746$	−23.7014	−0.867771
$747$	0	0
$748$	−1.38872	−0.0507767
$749$	6.45523	0.235869
$750$	0	0
$751$	1.95873	0.0714751	0.0357375	−	0.999361i	$-0.488622\pi$
$751$	1.95873	0.0714751	0.0357375	+	0.999361i	$0.488622\pi$
$752$	0.974153	0.0355237
$753$	0	0
$754$	−13.2925	−0.484084
$755$	−24.7476	−0.900658
$756$	0	0
$757$	29.2998	1.06492	0.532460	−	0.846455i	$-0.321268\pi$
$757$	29.2998	1.06492	0.532460	+	0.846455i	$0.321268\pi$
$758$	−35.6694	−1.29557
$759$	0	0
$760$	48.0332	1.74235
$761$	2.27331	0.0824075	0.0412038	−	0.999151i	$-0.486881\pi$
$761$	2.27331	0.0824075	0.0412038	+	0.999151i	$0.486881\pi$
$762$	0	0
$763$	−20.5285	−0.743183
$764$	12.3838	0.448031
$765$	0	0
$766$	−15.7774	−0.570060
$767$	28.2496	1.02003
$768$	0	0
$769$	−37.8564	−1.36514	−0.682568	−	0.730822i	$-0.739137\pi$
$769$	−37.8564	−1.36514	−0.682568	+	0.730822i	$0.739137\pi$
$770$	−9.82675	−0.354131
$771$	0	0
$772$	9.08127	0.326842
$773$	−31.5316	−1.13411	−0.567056	−	0.823679i	$-0.691918\pi$
$773$	−31.5316	−1.13411	−0.567056	+	0.823679i	$0.691918\pi$
$774$	0	0
$775$	−4.29289	−0.154205
$776$	−5.15391	−0.185015
$777$	0	0
$778$	−3.19393	−0.114508
$779$	14.6990	0.526646
$780$	0	0
$781$	−26.6192	−0.952510
$782$	−1.89038	−0.0675999
$783$	0	0
$784$	−0.679190	−0.0242568
$785$	18.5241	0.661152
$786$	0	0
$787$	−5.10263	−0.181889	−0.0909446	−	0.995856i	$-0.528989\pi$
$787$	−5.10263	−0.181889	−0.0909446	+	0.995856i	$0.528989\pi$
$788$	6.58155	0.234458
$789$	0	0
$790$	10.5165	0.374162
$791$	−15.2547	−0.542394
$792$	0	0
$793$	3.75341	0.133288
$794$	4.68994	0.166440
$795$	0	0
$796$	11.5527	0.409475
$797$	20.3107	0.719441	0.359720	−	0.933060i	$-0.382872\pi$
$797$	20.3107	0.719441	0.359720	+	0.933060i	$0.382872\pi$
$798$	0	0
$799$	−0.433733	−0.0153444
$800$	4.11393	0.145449
$801$	0	0
$802$	17.3645	0.613161
$803$	−48.8631	−1.72434
$804$	0	0
$805$	15.6605	0.551959
$806$	−18.6647	−0.657436
$807$	0	0
$808$	−10.6850	−0.375896
$809$	41.8654	1.47191	0.735955	−	0.677030i	$-0.236733\pi$
$809$	41.8654	1.47191	0.735955	+	0.677030i	$0.236733\pi$
$810$	0	0
$811$	1.27580	0.0447992	0.0223996	−	0.999749i	$-0.492869\pi$
$811$	1.27580	0.0447992	0.0223996	+	0.999749i	$0.492869\pi$
$812$	−4.21506	−0.147920
$813$	0	0
$814$	−26.7865	−0.938865
$815$	15.7292	0.550970
$816$	0	0
$817$	−64.0371	−2.24038
$818$	12.6535	0.442420
$819$	0	0
$820$	5.64037	0.196970
$821$	−41.2734	−1.44045	−0.720226	−	0.693740i	$-0.755962\pi$
$821$	−41.2734	−1.44045	−0.720226	+	0.693740i	$0.755962\pi$
$822$	0	0
$823$	37.3455	1.30178	0.650891	−	0.759171i	$-0.274396\pi$
$823$	37.3455	1.30178	0.650891	+	0.759171i	$0.274396\pi$
$824$	−17.5203	−0.610349
$825$	0	0
$826$	−7.65153	−0.266231
$827$	5.74511	0.199777	0.0998886	−	0.994999i	$-0.468151\pi$
$827$	5.74511	0.199777	0.0998886	+	0.994999i	$0.468151\pi$
$828$	0	0
$829$	46.6365	1.61975	0.809876	−	0.586601i	$-0.199534\pi$
$829$	46.6365	1.61975	0.809876	+	0.586601i	$0.199534\pi$
$830$	−23.3966	−0.812110
$831$	0	0
$832$	22.7005	0.786999
$833$	0.302404	0.0104777
$834$	0	0
$835$	−4.96481	−0.171814
$836$	31.0417	1.07360
$837$	0	0
$838$	31.9220	1.10273
$839$	−31.6074	−1.09121	−0.545605	−	0.838043i	$-0.683700\pi$
$839$	−31.6074	−1.09121	−0.545605	+	0.838043i	$0.683700\pi$
$840$	0	0
$841$	−13.7299	−0.473446
$842$	−4.26017	−0.146815
$843$	0	0
$844$	14.9231	0.513675
$845$	−1.06083	−0.0364937
$846$	0	0
$847$	−7.12556	−0.244837
$848$	7.18178	0.246623
$849$	0	0
$850$	0.227097	0.00778937
$851$	42.6885	1.46334
$852$	0	0
$853$	−35.7198	−1.22302	−0.611512	−	0.791235i	$-0.709438\pi$
$853$	−35.7198	−1.22302	−0.611512	+	0.791235i	$0.709438\pi$
$854$	−1.01663	−0.0347883
$855$	0	0
$856$	19.0758	0.651999
$857$	11.5668	0.395113	0.197556	−	0.980292i	$-0.436699\pi$
$857$	11.5668	0.395113	0.197556	+	0.980292i	$0.436699\pi$
$858$	0	0
$859$	−34.8509	−1.18910	−0.594549	−	0.804059i	$-0.702669\pi$
$859$	−34.8509	−1.18910	−0.594549	+	0.804059i	$0.702669\pi$
$860$	−24.5726	−0.837920
$861$	0	0
$862$	18.9458	0.645296
$863$	−11.4883	−0.391068	−0.195534	−	0.980697i	$-0.562644\pi$
$863$	−11.4883	−0.391068	−0.195534	+	0.980697i	$0.562644\pi$
$864$	0	0
$865$	26.5043	0.901172
$866$	28.4404	0.966444
$867$	0	0
$868$	−5.91859	−0.200890
$869$	19.3979	0.658029
$870$	0	0
$871$	−16.7317	−0.566931
$872$	−60.6638	−2.05434
$873$	0	0
$874$	42.2551	1.42930
$875$	10.1419	0.342860
$876$	0	0
$877$	−7.92650	−0.267659	−0.133829	−	0.991004i	$-0.542727\pi$
$877$	−7.92650	−0.267659	−0.133829	+	0.991004i	$0.542727\pi$
$878$	−39.5481	−1.33468
$879$	0	0
$880$	−6.95328	−0.234395
$881$	31.4350	1.05907	0.529536	−	0.848287i	$-0.322366\pi$
$881$	31.4350	1.05907	0.529536	+	0.848287i	$0.322366\pi$
$882$	0	0
$883$	1.77775	0.0598260	0.0299130	−	0.999553i	$-0.490477\pi$
$883$	1.77775	0.0598260	0.0299130	+	0.999553i	$0.490477\pi$
$884$	−1.15596	−0.0388793
$885$	0	0
$886$	23.9955	0.806144
$887$	−10.3134	−0.346292	−0.173146	−	0.984896i	$-0.555393\pi$
$887$	−10.3134	−0.346292	−0.173146	+	0.984896i	$0.555393\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	−19.0280	−0.637819
$891$	0	0
$892$	5.40191	0.180869
$893$	9.69510	0.324434
$894$	0	0
$895$	−56.9126	−1.90238
$896$	4.36800	0.145925
$897$	0	0
$898$	−4.22135	−0.140868
$899$	21.4415	0.715115
$900$	0	0
$901$	−3.19763	−0.106528
$902$	−8.88644	−0.295886
$903$	0	0
$904$	−45.0791	−1.49931
$905$	−36.4155	−1.21049
$906$	0	0
$907$	2.28463	0.0758598	0.0379299	−	0.999280i	$-0.487924\pi$
$907$	2.28463	0.0758598	0.0379299	+	0.999280i	$0.487924\pi$
$908$	−25.2822	−0.839018
$909$	0	0
$910$	−8.17973	−0.271155
$911$	5.90967	0.195796	0.0978980	−	0.995196i	$-0.468788\pi$
$911$	5.90967	0.195796	0.0978980	+	0.995196i	$0.468788\pi$
$912$	0	0
$913$	−43.1554	−1.42824
$914$	12.8381	0.424648
$915$	0	0
$916$	4.26390	0.140883
$917$	−5.94653	−0.196372
$918$	0	0
$919$	−45.1535	−1.48947	−0.744737	−	0.667358i	$-0.767425\pi$
$919$	−45.1535	−1.48947	−0.744737	+	0.667358i	$0.767425\pi$
$920$	46.2783	1.52575
$921$	0	0
$922$	9.84471	0.324218
$923$	−22.1577	−0.729329
$924$	0	0
$925$	−5.12829	−0.168617
$926$	−15.6367	−0.513855
$927$	0	0
$928$	−20.5477	−0.674511
$929$	−43.7874	−1.43662	−0.718309	−	0.695724i	$-0.755084\pi$
$929$	−43.7874	−1.43662	−0.718309	+	0.695724i	$0.755084\pi$
$930$	0	0
$931$	−6.75953	−0.221535
$932$	1.37375	0.0449986
$933$	0	0
$934$	8.63683	0.282606
$935$	3.09589	0.101246
$936$	0	0
$937$	39.0370	1.27528	0.637642	−	0.770333i	$-0.279910\pi$
$937$	39.0370	1.27528	0.637642	+	0.770333i	$0.279910\pi$
$938$	4.53185	0.147970
$939$	0	0
$940$	3.72025	0.121341
$941$	−15.5982	−0.508488	−0.254244	−	0.967140i	$-0.581827\pi$
$941$	−15.5982	−0.508488	−0.254244	+	0.967140i	$0.581827\pi$
$942$	0	0
$943$	14.1620	0.461177
$944$	−5.41413	−0.176215
$945$	0	0
$946$	38.7144	1.25871
$947$	51.9065	1.68673	0.843367	−	0.537338i	$-0.180570\pi$
$947$	51.9065	1.68673	0.843367	+	0.537338i	$0.180570\pi$
$948$	0	0
$949$	−40.6734	−1.32031
$950$	−5.07623	−0.164695
$951$	0	0
$952$	0.893633	0.0289628
$953$	9.02726	0.292422	0.146211	−	0.989253i	$-0.453292\pi$
$953$	9.02726	0.292422	0.146211	+	0.989253i	$0.453292\pi$
$954$	0	0
$955$	−27.6074	−0.893354
$956$	8.77550	0.283820
$957$	0	0
$958$	13.6914	0.442349
$959$	1.37243	0.0443180
$960$	0	0
$961$	−0.892778	−0.0287993
$962$	−22.2969	−0.718881
$963$	0	0
$964$	−21.0157	−0.676871
$965$	−20.2449	−0.651708
$966$	0	0
$967$	6.81098	0.219027	0.109513	−	0.993985i	$-0.465071\pi$
$967$	6.81098	0.219027	0.109513	+	0.993985i	$0.465071\pi$
$968$	−21.0567	−0.676789
$969$	0	0
$970$	4.02558	0.129254
$971$	−24.2046	−0.776764	−0.388382	−	0.921499i	$-0.626966\pi$
$971$	−24.2046	−0.776764	−0.388382	+	0.921499i	$0.626966\pi$
$972$	0	0
$973$	−3.13422	−0.100479
$974$	−10.3430	−0.331412
$975$	0	0
$976$	−0.719354	−0.0230260
$977$	25.4883	0.815443	0.407722	−	0.913106i	$-0.366323\pi$
$977$	25.4883	0.815443	0.407722	+	0.913106i	$0.366323\pi$
$978$	0	0
$979$	−35.0974	−1.12172
$980$	−2.59380	−0.0828558
$981$	0	0
$982$	6.53052	0.208397
$983$	21.3210	0.680035	0.340017	−	0.940419i	$-0.389567\pi$
$983$	21.3210	0.680035	0.340017	+	0.940419i	$0.389567\pi$
$984$	0	0
$985$	−14.6723	−0.467499
$986$	−1.13428	−0.0361227
$987$	0	0
$988$	25.8389	0.822045
$989$	−61.6975	−1.96187
$990$	0	0
$991$	54.5238	1.73200	0.866002	−	0.500040i	$-0.166681\pi$
$991$	54.5238	1.73200	0.866002	+	0.500040i	$0.166681\pi$
$992$	−28.8521	−0.916056
$993$	0	0
$994$	6.00151	0.190356
$995$	−25.7545	−0.816473
$996$	0	0
$997$	−54.8655	−1.73761	−0.868804	−	0.495155i	$-0.835111\pi$
$997$	−54.8655	−1.73761	−0.868804	+	0.495155i	$0.835111\pi$
$998$	−29.0766	−0.920403
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.z.1.22	yes	32
3.2	odd	2	inner	8001.2.a.z.1.11	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8001.2.a.z.1.11	✓	32	3.2	odd	2	inner
8001.2.a.z.1.22	yes	32	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:	\( N \)	\(=\)	\( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8001.a (trivial)

\(f(q)\)	\(=\)	\(q+0.959867 q^{2} -1.07866 q^{4} +2.40466 q^{5} -1.00000 q^{7} -2.95510 q^{8} +O(q^{10})\)	\(q+0.959867 q^{2} -1.07866 q^{4} +2.40466 q^{5} -1.00000 q^{7} -2.95510 q^{8} +2.30815 q^{10} +4.25741 q^{11} +3.54385 q^{13} -0.959867 q^{14} -0.679190 q^{16} +0.302404 q^{17} -6.75953 q^{19} -2.59380 q^{20} +4.08655 q^{22} -6.51256 q^{23} +0.782373 q^{25} +3.40162 q^{26} +1.07866 q^{28} -3.90770 q^{29} -5.48700 q^{31} +5.25827 q^{32} +0.290267 q^{34} -2.40466 q^{35} -6.55479 q^{37} -6.48825 q^{38} -7.10600 q^{40} -2.17456 q^{41} +9.47361 q^{43} -4.59228 q^{44} -6.25119 q^{46} -1.43429 q^{47} +1.00000 q^{49} +0.750974 q^{50} -3.82259 q^{52} -10.5740 q^{53} +10.2376 q^{55} +2.95510 q^{56} -3.75087 q^{58} +7.97145 q^{59} +1.05914 q^{61} -5.26679 q^{62} +6.40561 q^{64} +8.52173 q^{65} -4.72133 q^{67} -0.326189 q^{68} -2.30815 q^{70} -6.25244 q^{71} -11.4772 q^{73} -6.29172 q^{74} +7.29120 q^{76} -4.25741 q^{77} +4.55627 q^{79} -1.63322 q^{80} -2.08729 q^{82} -10.1365 q^{83} +0.727177 q^{85} +9.09340 q^{86} -12.5811 q^{88} -8.24382 q^{89} -3.54385 q^{91} +7.02481 q^{92} -1.37672 q^{94} -16.2543 q^{95} +1.74407 q^{97} +0.959867 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32q + 30q^{4} - 32q^{7} + O(q^{10}) \)	\( 32q + 30q^{4} - 32q^{7} - 16q^{10} - 14q^{13} + 18q^{16} - 30q^{19} - 10q^{22} + 36q^{25} - 30q^{28} - 58q^{31} - 34q^{34} + 8q^{37} - 34q^{40} + 6q^{43} - 36q^{46} + 32q^{49} - 56q^{52} - 88q^{55} - 22q^{58} - 46q^{61} + 20q^{64} - 8q^{67} + 16q^{70} - 60q^{73} - 128q^{76} - 74q^{79} - 52q^{82} - 16q^{85} - 64q^{88} + 14q^{91} - 58q^{94} - 44q^{97} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

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