LMFDB - Embedded newform 531.8.a.c.1.3

Newspace parameters

Level:	$ N $	$=$	$ 531 = 3^{2} \cdot 59 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	531.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$165.876448532$
Analytic rank:	$0$
Dimension:	$17$
Coefficient field:	$\mathbb{Q}[x]/(x^{17} - \cdots)$
Defining polynomial:	$x^{17} - 2 x^{16} - 1669 x^{15} + 2385 x^{14} + 1108684 x^{13} - 848131 x^{12} - 377920980 x^{11} + 12724944 x^{10} + 71331230512 x^{9} + 50741131904 x^{8} - 7480805165760 x^{7} - 10751966150272 x^{6} + 413177144536320 x^{5} + 886760582981376 x^{4} - 10454479722123264 x^{3} - 29180140031461376 x^{2} + 79787300207378432 x + 248723246810300416$
Coefficient ring:	$\Z[a_1, \ldots, a_{31}]$
Coefficient ring index:	multiple of $ 2^{10}\cdot 3^{5} $
Twist minimal:	no (minimal twist has level 177)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$16.5905$ of defining polynomial
Character	$\chi$	$=$	531.1

$q$-expansion

$f(q)$	$=$	$q-16.5905 q^{2} +147.246 q^{4} -174.998 q^{5} +1111.07 q^{7} -319.307 q^{8} +O(q^{10})$	\(q-16.5905 q^{2} +147.246 q^{4} -174.998 q^{5} +1111.07 q^{7} -319.307 q^{8} +2903.31 q^{10} -3459.76 q^{11} -368.734 q^{13} -18433.2 q^{14} -13550.0 q^{16} -12018.6 q^{17} -20398.0 q^{19} -25767.8 q^{20} +57399.3 q^{22} -27686.4 q^{23} -47500.8 q^{25} +6117.50 q^{26} +163601. q^{28} -141983. q^{29} +224521. q^{31} +265674. q^{32} +199395. q^{34} -194434. q^{35} +433168. q^{37} +338413. q^{38} +55878.1 q^{40} -286012. q^{41} -152357. q^{43} -509436. q^{44} +459333. q^{46} -449388. q^{47} +410931. q^{49} +788065. q^{50} -54294.7 q^{52} -141405. q^{53} +605449. q^{55} -354773. q^{56} +2.35558e6 q^{58} +205379. q^{59} -522147. q^{61} -3.72492e6 q^{62} -2.67327e6 q^{64} +64527.6 q^{65} +1.89319e6 q^{67} -1.76969e6 q^{68} +3.22578e6 q^{70} -3.16299e6 q^{71} +3.37201e6 q^{73} -7.18650e6 q^{74} -3.00352e6 q^{76} -3.84403e6 q^{77} -1.64311e6 q^{79} +2.37123e6 q^{80} +4.74509e6 q^{82} +6.36325e6 q^{83} +2.10322e6 q^{85} +2.52769e6 q^{86} +1.10473e6 q^{88} -5.43034e6 q^{89} -409689. q^{91} -4.07672e6 q^{92} +7.45560e6 q^{94} +3.56959e6 q^{95} -2.91053e6 q^{97} -6.81757e6 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 17q - 2q^{2} + 1166q^{4} + 318q^{5} + 3145q^{7} - 2355q^{8} + O(q^{10}) $	$ 17q - 2q^{2} + 1166q^{4} + 318q^{5} + 3145q^{7} - 2355q^{8} + 6521q^{10} + 1764q^{11} + 18192q^{13} + 7827q^{14} + 139226q^{16} + 15507q^{17} + 52083q^{19} - 721q^{20} - 234434q^{22} - 63823q^{23} + 202153q^{25} + 367956q^{26} + 182306q^{28} + 502955q^{29} + 347531q^{31} + 243908q^{32} - 330872q^{34} - 92641q^{35} + 447615q^{37} - 775669q^{38} + 2203270q^{40} - 940335q^{41} + 478562q^{43} + 596924q^{44} - 3078663q^{46} - 703121q^{47} + 1895082q^{49} + 876967q^{50} + 6278296q^{52} + 1005974q^{53} + 5212846q^{55} - 3425294q^{56} + 6710166q^{58} + 3491443q^{59} + 11510749q^{61} - 5996234q^{62} + 29496941q^{64} - 11094180q^{65} + 14007144q^{67} - 19688159q^{68} + 30909708q^{70} - 5229074q^{71} + 5452211q^{73} - 12819662q^{74} + 41929340q^{76} - 9930777q^{77} + 15275654q^{79} - 36576105q^{80} + 32025935q^{82} - 7826609q^{83} + 11836945q^{85} - 51649136q^{86} + 30223741q^{88} + 6436185q^{89} + 11633535q^{91} - 43357972q^{92} - 4494252q^{94} - 23741055q^{95} + 26377540q^{97} - 26517816q^{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 $

$2$	−16.5905	−1.46641	−0.733206	−	0.680007i	$-0.761977\pi$
$2$	−16.5905	−1.46641	−0.733206	+	0.680007i	$0.761977\pi$
$3$	0	0
$3$	0	0
$4$	147.246	1.15036
$5$	−174.998	−0.626091	−0.313045	−	0.949738i	$-0.601349\pi$
$5$	−174.998	−0.626091	−0.313045	+	0.949738i	$0.601349\pi$
$6$	0	0
$7$	1111.07	1.22433	0.612164	−	0.790731i	$-0.290299\pi$
$7$	1111.07	1.22433	0.612164	+	0.790731i	$0.290299\pi$
$8$	−319.307	−0.220493
$9$	0	0
$10$	2903.31	0.918107
$11$	−3459.76	−0.783738	−0.391869	−	0.920021i	$-0.628171\pi$
$11$	−3459.76	−0.783738	−0.391869	+	0.920021i	$0.628171\pi$
$12$	0	0
$13$	−368.734	−0.0465491	−0.0232746	−	0.999729i	$-0.507409\pi$
$13$	−368.734	−0.0465491	−0.0232746	+	0.999729i	$0.507409\pi$
$14$	−18433.2	−1.79537
$15$	0	0
$16$	−13550.0	−0.827029
$17$	−12018.6	−0.593310	−0.296655	−	0.954985i	$-0.595871\pi$
$17$	−12018.6	−0.593310	−0.296655	+	0.954985i	$0.595871\pi$
$18$	0	0
$19$	−20398.0	−0.682259	−0.341129	−	0.940016i	$-0.610809\pi$
$19$	−20398.0	−0.682259	−0.341129	+	0.940016i	$0.610809\pi$
$20$	−25767.8	−0.720231
$21$	0	0
$22$	57399.3	1.14928
$23$	−27686.4	−0.474481	−0.237241	−	0.971451i	$-0.576243\pi$
$23$	−27686.4	−0.474481	−0.237241	+	0.971451i	$0.576243\pi$
$24$	0	0
$25$	−47500.8	−0.608010
$26$	6117.50	0.0682602
$27$	0	0
$28$	163601.	1.40842
$29$	−141983.	−1.08105	−0.540523	−	0.841329i	$-0.681773\pi$
$29$	−141983.	−1.08105	−0.540523	+	0.841329i	$0.681773\pi$
$30$	0	0
$31$	224521.	1.35360	0.676801	−	0.736166i	$-0.263366\pi$
$31$	224521.	1.35360	0.676801	+	0.736166i	$0.263366\pi$
$32$	265674.	1.43326
$33$	0	0
$34$	199395.	0.870037
$35$	−194434.	−0.766541
$36$	0	0
$37$	433168.	1.40589	0.702944	−	0.711246i	$-0.251869\pi$
$37$	433168.	1.40589	0.702944	+	0.711246i	$0.251869\pi$
$38$	338413.	1.00047
$39$	0	0
$40$	55878.1	0.138048
$41$	−286012.	−0.648097	−0.324049	−	0.946040i	$-0.605044\pi$
$41$	−286012.	−0.648097	−0.324049	+	0.946040i	$0.605044\pi$
$42$	0	0
$43$	−152357.	−0.292229	−0.146115	−	0.989268i	$-0.546677\pi$
$43$	−152357.	−0.292229	−0.146115	+	0.989268i	$0.546677\pi$
$44$	−509436.	−0.901583
$45$	0	0
$46$	459333.	0.695785
$47$	−449388.	−0.631363	−0.315682	−	0.948865i	$-0.602233\pi$
$47$	−449388.	−0.631363	−0.315682	+	0.948865i	$0.602233\pi$
$48$	0	0
$49$	410931.	0.498980
$50$	788065.	0.891593
$51$	0	0
$52$	−54294.7	−0.0535483
$53$	−141405.	−0.130467	−0.0652335	−	0.997870i	$-0.520779\pi$
$53$	−141405.	−0.130467	−0.0652335	+	0.997870i	$0.520779\pi$
$54$	0	0
$55$	605449.	0.490691
$56$	−354773.	−0.269955
$57$	0	0
$58$	2.35558e6	1.58526
$59$	205379.	0.130189
$59$	205379.	0.130189
$60$	0	0
$61$	−522147.	−0.294536	−0.147268	−	0.989097i	$-0.547048\pi$
$61$	−522147.	−0.294536	−0.147268	+	0.989097i	$0.547048\pi$
$62$	−3.72492e6	−1.98494
$63$	0	0
$64$	−2.67327e6	−1.27472
$65$	64527.6	0.0291440
$66$	0	0
$67$	1.89319e6	0.769010	0.384505	−	0.923123i	$-0.374372\pi$
$67$	1.89319e6	0.769010	0.384505	+	0.923123i	$0.374372\pi$
$68$	−1.76969e6	−0.682522
$69$	0	0
$70$	3.22578e6	1.12406
$71$	−3.16299e6	−1.04880	−0.524400	−	0.851472i	$-0.675711\pi$
$71$	−3.16299e6	−1.04880	−0.524400	+	0.851472i	$0.675711\pi$
$72$	0	0
$73$	3.37201e6	1.01452	0.507258	−	0.861794i	$-0.330659\pi$
$73$	3.37201e6	1.01452	0.507258	+	0.861794i	$0.330659\pi$
$74$	−7.18650e6	−2.06161
$75$	0	0
$76$	−3.00352e6	−0.784844
$77$	−3.84403e6	−0.959553
$78$	0	0
$79$	−1.64311e6	−0.374949	−0.187475	−	0.982269i	$-0.560030\pi$
$79$	−1.64311e6	−0.374949	−0.187475	+	0.982269i	$0.560030\pi$
$80$	2.37123e6	0.517795
$81$	0	0
$82$	4.74509e6	0.950377
$83$	6.36325e6	1.22153	0.610767	−	0.791810i	$-0.290861\pi$
$83$	6.36325e6	1.22153	0.610767	+	0.791810i	$0.290861\pi$
$84$	0	0
$85$	2.10322e6	0.371466
$86$	2.52769e6	0.428528
$87$	0	0
$88$	1.10473e6	0.172809
$89$	−5.43034e6	−0.816511	−0.408256	−	0.912868i	$-0.633863\pi$
$89$	−5.43034e6	−0.816511	−0.408256	+	0.912868i	$0.633863\pi$
$90$	0	0
$91$	−409689.	−0.0569914
$92$	−4.07672e6	−0.545825
$93$	0	0
$94$	7.45560e6	0.925838
$95$	3.56959e6	0.427156
$96$	0	0
$97$	−2.91053e6	−0.323796	−0.161898	−	0.986807i	$-0.551762\pi$
$97$	−2.91053e6	−0.323796	−0.161898	+	0.986807i	$0.551762\pi$
$98$	−6.81757e6	−0.731709
$99$	0	0
$100$	−6.99432e6	−0.699432
$101$	1.20381e7	1.16261	0.581305	−	0.813686i	$-0.302542\pi$
$101$	1.20381e7	1.16261	0.581305	+	0.813686i	$0.302542\pi$
$102$	0	0
$103$	−4.50334e6	−0.406074	−0.203037	−	0.979171i	$-0.565081\pi$
$103$	−4.50334e6	−0.406074	−0.203037	+	0.979171i	$0.565081\pi$
$104$	117739.	0.0102637
$105$	0	0
$106$	2.34599e6	0.191318
$107$	1.38674e6	0.109434	0.0547171	−	0.998502i	$-0.482574\pi$
$107$	1.38674e6	0.109434	0.0547171	+	0.998502i	$0.482574\pi$
$108$	0	0
$109$	−1.51322e7	−1.11920	−0.559602	−	0.828761i	$-0.689046\pi$
$109$	−1.51322e7	−1.11920	−0.559602	+	0.828761i	$0.689046\pi$
$110$	−1.00447e7	−0.719555
$111$	0	0
$112$	−1.50550e7	−1.01256
$113$	1.74988e7	1.14086	0.570432	−	0.821345i	$-0.306776\pi$
$113$	1.74988e7	1.14086	0.570432	+	0.821345i	$0.306776\pi$
$114$	0	0
$115$	4.84506e6	0.297068
$116$	−2.09065e7	−1.24359
$117$	0	0
$118$	−3.40735e6	−0.190910
$119$	−1.33535e7	−0.726407
$120$	0	0
$121$	−7.51726e6	−0.385754
$122$	8.66270e6	0.431911
$123$	0	0
$124$	3.30599e7	1.55713
$125$	2.19842e7	1.00676
$126$	0	0
$127$	−2.00254e7	−0.867495	−0.433748	−	0.901034i	$-0.642809\pi$
$127$	−2.00254e7	−0.867495	−0.433748	+	0.901034i	$0.642809\pi$
$128$	1.03448e7	0.436000
$129$	0	0
$130$	−1.07055e6	−0.0427370
$131$	−1.77758e7	−0.690845	−0.345422	−	0.938447i	$-0.612264\pi$
$131$	−1.77758e7	−0.690845	−0.345422	+	0.938447i	$0.612264\pi$
$132$	0	0
$133$	−2.26635e7	−0.835308
$134$	−3.14090e7	−1.12769
$135$	0	0
$136$	3.83762e6	0.130821
$137$	−4.37881e7	−1.45490	−0.727451	−	0.686159i	$-0.759295\pi$
$137$	−4.37881e7	−1.45490	−0.727451	+	0.686159i	$0.759295\pi$
$138$	0	0
$139$	−2.03284e7	−0.642024	−0.321012	−	0.947075i	$-0.604023\pi$
$139$	−2.03284e7	−0.642024	−0.321012	+	0.947075i	$0.604023\pi$
$140$	−2.86298e7	−0.881799
$141$	0	0
$142$	5.24757e7	1.53797
$143$	1.27573e6	0.0364823
$144$	0	0
$145$	2.48467e7	0.676832
$146$	−5.59435e7	−1.48770
$147$	0	0
$148$	6.37824e7	1.61728
$149$	−6.14768e7	−1.52251	−0.761254	−	0.648454i	$-0.775416\pi$
$149$	−6.14768e7	−1.52251	−0.761254	+	0.648454i	$0.775416\pi$
$150$	0	0
$151$	6.11417e6	0.144517	0.0722583	−	0.997386i	$-0.476979\pi$
$151$	6.11417e6	0.144517	0.0722583	+	0.997386i	$0.476979\pi$
$152$	6.51322e6	0.150433
$153$	0	0
$154$	6.37745e7	1.40710
$155$	−3.92906e7	−0.847477
$156$	0	0
$157$	3.59609e7	0.741621	0.370810	−	0.928709i	$-0.379080\pi$
$157$	3.59609e7	0.741621	0.370810	+	0.928709i	$0.379080\pi$
$158$	2.72601e7	0.549830
$159$	0	0
$160$	−4.64923e7	−0.897349
$161$	−3.07615e7	−0.580921
$162$	0	0
$163$	−3.88451e7	−0.702553	−0.351277	−	0.936272i	$-0.614252\pi$
$163$	−3.88451e7	−0.702553	−0.351277	+	0.936272i	$0.614252\pi$
$164$	−4.21142e7	−0.745547
$165$	0	0
$166$	−1.05570e8	−1.79127
$167$	8.48288e7	1.40940	0.704702	−	0.709503i	$-0.251081\pi$
$167$	8.48288e7	1.40940	0.704702	+	0.709503i	$0.251081\pi$
$168$	0	0
$169$	−6.26126e7	−0.997833
$170$	−3.48936e7	−0.544722
$171$	0	0
$172$	−2.24341e7	−0.336169
$173$	2.04205e7	0.299850	0.149925	−	0.988697i	$-0.452097\pi$
$173$	2.04205e7	0.299850	0.149925	+	0.988697i	$0.452097\pi$
$174$	0	0
$175$	−5.27767e7	−0.744404
$176$	4.68799e7	0.648175
$177$	0	0
$178$	9.00924e7	1.19734
$179$	2.15476e7	0.280811	0.140405	−	0.990094i	$-0.455159\pi$
$179$	2.15476e7	0.280811	0.140405	+	0.990094i	$0.455159\pi$
$180$	0	0
$181$	−6.19200e7	−0.776168	−0.388084	−	0.921624i	$-0.626863\pi$
$181$	−6.19200e7	−0.776168	−0.388084	+	0.921624i	$0.626863\pi$
$182$	6.79696e6	0.0835728
$183$	0	0
$184$	8.84047e6	0.104620
$185$	−7.58034e7	−0.880213
$186$	0	0
$187$	4.15814e7	0.465000
$188$	−6.61708e7	−0.726296
$189$	0	0
$190$	−5.92215e7	−0.626386
$191$	1.83254e8	1.90300	0.951498	−	0.307656i	$-0.0995445\pi$
$191$	1.83254e8	1.90300	0.951498	+	0.307656i	$0.0995445\pi$
$192$	0	0
$193$	−2.51674e7	−0.251993	−0.125996	−	0.992031i	$-0.540213\pi$
$193$	−2.51674e7	−0.251993	−0.125996	+	0.992031i	$0.540213\pi$
$194$	4.82873e7	0.474818
$195$	0	0
$196$	6.05081e7	0.574007
$197$	−4.11501e7	−0.383477	−0.191738	−	0.981446i	$-0.561412\pi$
$197$	−4.11501e7	−0.383477	−0.191738	+	0.981446i	$0.561412\pi$
$198$	0	0
$199$	1.20872e8	1.08728	0.543638	−	0.839320i	$-0.317047\pi$
$199$	1.20872e8	1.08728	0.543638	+	0.839320i	$0.317047\pi$
$200$	1.51674e7	0.134062
$201$	0	0
$202$	−1.99719e8	−1.70486
$203$	−1.57753e8	−1.32355
$204$	0	0
$205$	5.00514e7	0.405768
$206$	7.47129e7	0.595471
$207$	0	0
$208$	4.99636e6	0.0384975
$209$	7.05720e7	0.534712
$210$	0	0
$211$	−1.24225e8	−0.910379	−0.455189	−	0.890395i	$-0.650428\pi$
$211$	−1.24225e8	−0.910379	−0.455189	+	0.890395i	$0.650428\pi$
$212$	−2.08214e7	−0.150084
$213$	0	0
$214$	−2.30068e7	−0.160475
$215$	2.66622e7	0.182962
$216$	0	0
$217$	2.49458e8	1.65725
$218$	2.51052e8	1.64121
$219$	0	0
$220$	8.91502e7	0.564473
$221$	4.43166e6	0.0276181
$222$	0	0
$223$	−4.32331e6	−0.0261065	−0.0130533	−	0.999915i	$-0.504155\pi$
$223$	−4.32331e6	−0.0261065	−0.0130533	+	0.999915i	$0.504155\pi$
$224$	2.95182e8	1.75478
$225$	0	0
$226$	−2.90315e8	−1.67298
$227$	1.29928e8	0.737244	0.368622	−	0.929579i	$-0.379830\pi$
$227$	1.29928e8	0.737244	0.368622	+	0.929579i	$0.379830\pi$
$228$	0	0
$229$	−6.03820e7	−0.332264	−0.166132	−	0.986103i	$-0.553128\pi$
$229$	−6.03820e7	−0.332264	−0.166132	+	0.986103i	$0.553128\pi$
$230$	−8.03821e7	−0.435624
$231$	0	0
$232$	4.53363e7	0.238362
$233$	2.38228e8	1.23381	0.616904	−	0.787038i	$-0.288387\pi$
$233$	2.38228e8	1.23381	0.616904	+	0.787038i	$0.288387\pi$
$234$	0	0
$235$	7.86419e7	0.395291
$236$	3.02413e7	0.149764
$237$	0	0
$238$	2.21541e8	1.06521
$239$	3.08535e7	0.146188	0.0730940	−	0.997325i	$-0.476713\pi$
$239$	3.08535e7	0.146188	0.0730940	+	0.997325i	$0.476713\pi$
$240$	0	0
$241$	9.76207e6	0.0449244	0.0224622	−	0.999748i	$-0.492849\pi$
$241$	9.76207e6	0.0449244	0.0224622	+	0.999748i	$0.492849\pi$
$242$	1.24715e8	0.565674
$243$	0	0
$244$	−7.68842e7	−0.338823
$245$	−7.19120e7	−0.312407
$246$	0	0
$247$	7.52142e6	0.0317585
$248$	−7.16912e7	−0.298459
$249$	0	0
$250$	−3.64730e8	−1.47632
$251$	−2.04882e8	−0.817798	−0.408899	−	0.912580i	$-0.634087\pi$
$251$	−2.04882e8	−0.817798	−0.408899	+	0.912580i	$0.634087\pi$
$252$	0	0
$253$	9.57882e7	0.371869
$254$	3.32232e8	1.27211
$255$	0	0
$256$	1.70553e8	0.635360
$257$	7.58629e7	0.278781	0.139391	−	0.990237i	$-0.455486\pi$
$257$	7.58629e7	0.278781	0.139391	+	0.990237i	$0.455486\pi$
$258$	0	0
$259$	4.81280e8	1.72127
$260$	9.50145e6	0.0335261
$261$	0	0
$262$	2.94911e8	1.01306
$263$	3.76186e8	1.27514	0.637569	−	0.770393i	$-0.279940\pi$
$263$	3.76186e8	1.27514	0.637569	+	0.770393i	$0.279940\pi$
$264$	0	0
$265$	2.47456e7	0.0816842
$266$	3.76000e8	1.22491
$267$	0	0
$268$	2.78765e8	0.884640
$269$	5.43241e8	1.70161	0.850804	−	0.525483i	$-0.176115\pi$
$269$	5.43241e8	1.70161	0.850804	+	0.525483i	$0.176115\pi$
$270$	0	0
$271$	5.59119e8	1.70652	0.853261	−	0.521484i	$-0.174621\pi$
$271$	5.59119e8	1.70652	0.853261	+	0.521484i	$0.174621\pi$
$272$	1.62852e8	0.490685
$273$	0	0
$274$	7.26468e8	2.13349
$275$	1.64341e8	0.476521
$276$	0	0
$277$	4.08022e8	1.15346	0.576732	−	0.816933i	$-0.304328\pi$
$277$	4.08022e8	1.15346	0.576732	+	0.816933i	$0.304328\pi$
$278$	3.37259e8	0.941472
$279$	0	0
$280$	6.20844e7	0.169017
$281$	−3.46395e8	−0.931321	−0.465660	−	0.884964i	$-0.654183\pi$
$281$	−3.46395e8	−0.931321	−0.465660	+	0.884964i	$0.654183\pi$
$282$	0	0
$283$	1.32435e8	0.347337	0.173669	−	0.984804i	$-0.444438\pi$
$283$	1.32435e8	0.347337	0.173669	+	0.984804i	$0.444438\pi$
$284$	−4.65738e8	−1.20650
$285$	0	0
$286$	−2.11651e7	−0.0534981
$287$	−3.17779e8	−0.793484
$288$	0	0
$289$	−2.65892e8	−0.647983
$290$	−4.12221e8	−0.992515
$291$	0	0
$292$	4.96516e8	1.16706
$293$	6.11894e8	1.42115	0.710574	−	0.703622i	$-0.248435\pi$
$293$	6.11894e8	1.42115	0.710574	+	0.703622i	$0.248435\pi$
$294$	0	0
$295$	−3.59408e7	−0.0815101
$296$	−1.38314e8	−0.309988
$297$	0	0
$298$	1.01993e9	2.23262
$299$	1.02089e7	0.0220867
$300$	0	0
$301$	−1.69279e8	−0.357785
$302$	−1.01437e8	−0.211921
$303$	0	0
$304$	2.76393e8	0.564248
$305$	9.13745e7	0.184406
$306$	0	0
$307$	1.61294e8	0.318151	0.159075	−	0.987266i	$-0.449149\pi$
$307$	1.61294e8	0.318151	0.159075	+	0.987266i	$0.449149\pi$
$308$	−5.66019e8	−1.10383
$309$	0	0
$310$	6.51853e8	1.24275
$311$	−5.52921e8	−1.04232	−0.521160	−	0.853459i	$-0.674501\pi$
$311$	−5.52921e8	−1.04232	−0.521160	+	0.853459i	$0.674501\pi$
$312$	0	0
$313$	3.45307e8	0.636503	0.318252	−	0.948006i	$-0.396904\pi$
$313$	3.45307e8	0.636503	0.318252	+	0.948006i	$0.396904\pi$
$314$	−5.96611e8	−1.08752
$315$	0	0
$316$	−2.41942e8	−0.431327
$317$	5.61715e8	0.990395	0.495198	−	0.868780i	$-0.335096\pi$
$317$	5.61715e8	0.990395	0.495198	+	0.868780i	$0.335096\pi$
$318$	0	0
$319$	4.91227e8	0.847257
$320$	4.67817e8	0.798088
$321$	0	0
$322$	5.10350e8	0.851869
$323$	2.45154e8	0.404791
$324$	0	0
$325$	1.75152e7	0.0283023
$326$	6.44461e8	1.03023
$327$	0	0
$328$	9.13256e7	0.142901
$329$	−4.99302e8	−0.772996
$330$	0	0
$331$	−7.48834e8	−1.13498	−0.567489	−	0.823381i	$-0.692085\pi$
$331$	−7.48834e8	−1.13498	−0.567489	+	0.823381i	$0.692085\pi$
$332$	9.36965e8	1.40521
$333$	0	0
$334$	−1.40736e9	−2.06677
$335$	−3.31304e8	−0.481470
$336$	0	0
$337$	3.77520e8	0.537322	0.268661	−	0.963235i	$-0.413419\pi$
$337$	3.77520e8	0.537322	0.268661	+	0.963235i	$0.413419\pi$
$338$	1.03878e9	1.46323
$339$	0	0
$340$	3.09692e8	0.427321
$341$	−7.76787e8	−1.06087
$342$	0	0
$343$	−4.58440e8	−0.613413
$344$	4.86488e7	0.0644344
$345$	0	0
$346$	−3.38787e8	−0.439703
$347$	−2.25075e8	−0.289185	−0.144592	−	0.989491i	$-0.546187\pi$
$347$	−2.25075e8	−0.289185	−0.144592	+	0.989491i	$0.546187\pi$
$348$	0	0
$349$	2.00272e8	0.252192	0.126096	−	0.992018i	$-0.459755\pi$
$349$	2.00272e8	0.252192	0.126096	+	0.992018i	$0.459755\pi$
$350$	8.75594e8	1.09160
$351$	0	0
$352$	−9.19168e8	−1.12330
$353$	9.69527e6	0.0117314	0.00586568	−	0.999983i	$-0.498133\pi$
$353$	9.69527e6	0.0117314	0.00586568	+	0.999983i	$0.498133\pi$
$354$	0	0
$355$	5.53515e8	0.656645
$356$	−7.99598e8	−0.939284
$357$	0	0
$358$	−3.57487e8	−0.411784
$359$	−1.59435e9	−1.81867	−0.909334	−	0.416067i	$-0.863408\pi$
$359$	−1.59435e9	−1.81867	−0.909334	+	0.416067i	$0.863408\pi$
$360$	0	0
$361$	−4.77795e8	−0.534523
$362$	1.02729e9	1.13818
$363$	0	0
$364$	−6.03252e7	−0.0655607
$365$	−5.90094e8	−0.635179
$366$	0	0
$367$	4.36226e8	0.460660	0.230330	−	0.973113i	$-0.426019\pi$
$367$	4.36226e8	0.460660	0.230330	+	0.973113i	$0.426019\pi$
$368$	3.75152e8	0.392410
$369$	0	0
$370$	1.25762e9	1.29075
$371$	−1.57111e8	−0.159734
$372$	0	0
$373$	8.71065e8	0.869100	0.434550	−	0.900648i	$-0.356907\pi$
$373$	8.71065e8	0.869100	0.434550	+	0.900648i	$0.356907\pi$
$374$	−6.89858e8	−0.681882
$375$	0	0
$376$	1.43493e8	0.139211
$377$	5.23540e7	0.0503217
$378$	0	0
$379$	1.10371e9	1.04140	0.520702	−	0.853739i	$-0.325670\pi$
$379$	1.10371e9	1.04140	0.520702	+	0.853739i	$0.325670\pi$
$380$	5.25610e8	0.491384
$381$	0	0
$382$	−3.04029e9	−2.79057
$383$	−3.23348e8	−0.294086	−0.147043	−	0.989130i	$-0.546976\pi$
$383$	−3.23348e8	−0.294086	−0.147043	+	0.989130i	$0.546976\pi$
$384$	0	0
$385$	6.72696e8	0.600767
$386$	4.17541e8	0.369525
$387$	0	0
$388$	−4.28565e8	−0.372483
$389$	−1.09065e9	−0.939423	−0.469711	−	0.882820i	$-0.655642\pi$
$389$	−1.09065e9	−0.939423	−0.469711	+	0.882820i	$0.655642\pi$
$390$	0	0
$391$	3.32751e8	0.281515
$392$	−1.31213e8	−0.110021
$393$	0	0
$394$	6.82702e8	0.562334
$395$	2.87541e8	0.234752
$396$	0	0
$397$	−2.19755e9	−1.76268	−0.881338	−	0.472487i	$-0.843356\pi$
$397$	−2.19755e9	−1.76268	−0.881338	+	0.472487i	$0.843356\pi$
$398$	−2.00533e9	−1.59439
$399$	0	0
$400$	6.43638e8	0.502842
$401$	2.55403e9	1.97797	0.988987	−	0.148004i	$-0.0472848\pi$
$401$	2.55403e9	1.97797	0.988987	+	0.148004i	$0.0472848\pi$
$402$	0	0
$403$	−8.27885e7	−0.0630089
$404$	1.77257e9	1.33742
$405$	0	0
$406$	2.61721e9	1.94087
$407$	−1.49866e9	−1.10185
$408$	0	0
$409$	−2.88636e8	−0.208602	−0.104301	−	0.994546i	$-0.533261\pi$
$409$	−2.88636e8	−0.208602	−0.104301	+	0.994546i	$0.533261\pi$
$410$	−8.30380e8	−0.595022
$411$	0	0
$412$	−6.63101e8	−0.467132
$413$	2.28190e8	0.159394
$414$	0	0
$415$	−1.11355e9	−0.764791
$416$	−9.79631e7	−0.0667169
$417$	0	0
$418$	−1.17083e9	−0.784108
$419$	5.30993e8	0.352647	0.176323	−	0.984332i	$-0.443580\pi$
$419$	5.30993e8	0.352647	0.176323	+	0.984332i	$0.443580\pi$
$420$	0	0
$421$	−2.15095e9	−1.40489	−0.702445	−	0.711738i	$-0.747908\pi$
$421$	−2.15095e9	−1.40489	−0.702445	+	0.711738i	$0.747908\pi$
$422$	2.06097e9	1.33499
$423$	0	0
$424$	4.51518e7	0.0287670
$425$	5.70892e8	0.360739
$426$	0	0
$427$	−5.80141e8	−0.360609
$428$	2.04193e8	0.125889
$429$	0	0
$430$	−4.42340e8	−0.268298
$431$	3.30976e8	0.199125	0.0995624	−	0.995031i	$-0.468256\pi$
$431$	3.30976e8	0.199125	0.0995624	+	0.995031i	$0.468256\pi$
$432$	0	0
$433$	2.05100e9	1.21411	0.607056	−	0.794659i	$-0.292350\pi$
$433$	2.05100e9	1.21411	0.607056	+	0.794659i	$0.292350\pi$
$434$	−4.13865e9	−2.43021
$435$	0	0
$436$	−2.22816e9	−1.28749
$437$	5.64746e8	0.323719
$438$	0	0
$439$	1.57643e9	0.889299	0.444649	−	0.895705i	$-0.353328\pi$
$439$	1.57643e9	0.889299	0.444649	+	0.895705i	$0.353328\pi$
$440$	−1.93324e8	−0.108194
$441$	0	0
$442$	−7.35237e7	−0.0404995
$443$	−3.27614e8	−0.179040	−0.0895199	−	0.995985i	$-0.528533\pi$
$443$	−3.27614e8	−0.179040	−0.0895199	+	0.995985i	$0.528533\pi$
$444$	0	0
$445$	9.50297e8	0.511210
$446$	7.17261e7	0.0382829
$447$	0	0
$448$	−2.97019e9	−1.56067
$449$	3.76746e7	0.0196420	0.00982101	−	0.999952i	$-0.496874\pi$
$449$	3.76746e7	0.0196420	0.00982101	+	0.999952i	$0.496874\pi$
$450$	0	0
$451$	9.89531e8	0.507939
$452$	2.57664e9	1.31241
$453$	0	0
$454$	−2.15557e9	−1.08110
$455$	7.16946e7	0.0356818
$456$	0	0
$457$	1.95746e8	0.0959372	0.0479686	−	0.998849i	$-0.484725\pi$
$457$	1.95746e8	0.0959372	0.0479686	+	0.998849i	$0.484725\pi$
$458$	1.00177e9	0.487236
$459$	0	0
$460$	7.13417e8	0.341736
$461$	1.06215e9	0.504931	0.252465	−	0.967606i	$-0.418759\pi$
$461$	1.06215e9	0.504931	0.252465	+	0.967606i	$0.418759\pi$
$462$	0	0
$463$	1.12424e9	0.526412	0.263206	−	0.964740i	$-0.415220\pi$
$463$	1.12424e9	0.526412	0.263206	+	0.964740i	$0.415220\pi$
$464$	1.92388e9	0.894056
$465$	0	0
$466$	−3.95234e9	−1.80927
$467$	1.93669e9	0.879936	0.439968	−	0.898013i	$-0.354990\pi$
$467$	1.93669e9	0.879936	0.439968	+	0.898013i	$0.354990\pi$
$468$	0	0
$469$	2.10346e9	0.941521
$470$	−1.30471e9	−0.579659
$471$	0	0
$472$	−6.55790e7	−0.0287057
$473$	5.27119e8	0.229031
$474$	0	0
$475$	9.68919e8	0.414820
$476$	−1.96625e9	−0.835631
$477$	0	0
$478$	−5.11877e8	−0.214372
$479$	−2.67237e9	−1.11102	−0.555510	−	0.831510i	$-0.687477\pi$
$479$	−2.67237e9	−1.11102	−0.555510	+	0.831510i	$0.687477\pi$
$480$	0	0
$481$	−1.59724e8	−0.0654428
$482$	−1.61958e8	−0.0658777
$483$	0	0
$484$	−1.10689e9	−0.443757
$485$	5.09337e8	0.202726
$486$	0	0
$487$	9.86225e8	0.386923	0.193462	−	0.981108i	$-0.438029\pi$
$487$	9.86225e8	0.386923	0.193462	+	0.981108i	$0.438029\pi$
$488$	1.66725e8	0.0649430
$489$	0	0
$490$	1.19306e9	0.458116
$491$	3.33799e9	1.27262	0.636312	−	0.771431i	$-0.280459\pi$
$491$	3.33799e9	1.27262	0.636312	+	0.771431i	$0.280459\pi$
$492$	0	0
$493$	1.70644e9	0.641395
$494$	−1.24784e8	−0.0465711
$495$	0	0
$496$	−3.04227e9	−1.11947
$497$	−3.51430e9	−1.28408
$498$	0	0
$499$	4.31622e9	1.55508	0.777539	−	0.628835i	$-0.216468\pi$
$499$	4.31622e9	1.55508	0.777539	+	0.628835i	$0.216468\pi$
$500$	3.23710e9	1.15814
$501$	0	0
$502$	3.39911e9	1.19923
$503$	3.79261e9	1.32877	0.664386	−	0.747389i	$-0.268693\pi$
$503$	3.79261e9	1.32877	0.664386	+	0.747389i	$0.268693\pi$
$504$	0	0
$505$	−2.10664e9	−0.727899
$506$	−1.58918e9	−0.545313
$507$	0	0
$508$	−2.94866e9	−0.997934
$509$	−1.85516e9	−0.623548	−0.311774	−	0.950156i	$-0.600923\pi$
$509$	−1.85516e9	−0.623548	−0.311774	+	0.950156i	$0.600923\pi$
$510$	0	0
$511$	3.74653e9	1.24210
$512$	−4.15370e9	−1.36770
$513$	0	0
$514$	−1.25861e9	−0.408808
$515$	7.88075e8	0.254239
$516$	0	0
$517$	1.55477e9	0.494824
$518$	−7.98470e9	−2.52409
$519$	0	0
$520$	−2.06041e7	−0.00642603
$521$	2.97325e8	0.0921083	0.0460542	−	0.998939i	$-0.485335\pi$
$521$	2.97325e8	0.0921083	0.0460542	+	0.998939i	$0.485335\pi$
$522$	0	0
$523$	2.48461e9	0.759456	0.379728	−	0.925098i	$-0.376018\pi$
$523$	2.48461e9	0.759456	0.379728	+	0.925098i	$0.376018\pi$
$524$	−2.61743e9	−0.794722
$525$	0	0
$526$	−6.24113e9	−1.86988
$527$	−2.69842e9	−0.803106
$528$	0	0
$529$	−2.63829e9	−0.774867
$530$	−4.10544e8	−0.119783
$531$	0	0
$532$	−3.33712e9	−0.960907
$533$	1.05462e8	0.0301684
$534$	0	0
$535$	−2.42677e8	−0.0685157
$536$	−6.04509e8	−0.169561
$537$	0	0
$538$	−9.01267e9	−2.49526
$539$	−1.42172e9	−0.391070
$540$	0	0
$541$	−4.16187e9	−1.13005	−0.565026	−	0.825073i	$-0.691134\pi$
$541$	−4.16187e9	−1.13005	−0.565026	+	0.825073i	$0.691134\pi$
$542$	−9.27610e9	−2.50246
$543$	0	0
$544$	−3.19303e9	−0.850367
$545$	2.64810e9	0.700724
$546$	0	0
$547$	6.75557e9	1.76484	0.882422	−	0.470459i	$-0.155912\pi$
$547$	6.75557e9	1.76484	0.882422	+	0.470459i	$0.155912\pi$
$548$	−6.44764e9	−1.67367
$549$	0	0
$550$	−2.72651e9	−0.698776
$551$	2.89617e9	0.737552
$552$	0	0
$553$	−1.82561e9	−0.459061
$554$	−6.76931e9	−1.69145
$555$	0	0
$556$	−2.99328e9	−0.738560
$557$	−3.44498e9	−0.844681	−0.422341	−	0.906437i	$-0.638791\pi$
$557$	−3.44498e9	−0.844681	−0.422341	+	0.906437i	$0.638791\pi$
$558$	0	0
$559$	5.61793e7	0.0136030
$560$	2.63460e9	0.633951
$561$	0	0
$562$	5.74688e9	1.36570
$563$	2.05492e9	0.485306	0.242653	−	0.970113i	$-0.421982\pi$
$563$	2.05492e9	0.485306	0.242653	+	0.970113i	$0.421982\pi$
$564$	0	0
$565$	−3.06225e9	−0.714284
$566$	−2.19717e9	−0.509339
$567$	0	0
$568$	1.00996e9	0.231253
$569$	−7.09854e8	−0.161538	−0.0807692	−	0.996733i	$-0.525738\pi$
$569$	−7.09854e8	−0.161538	−0.0807692	+	0.996733i	$0.525738\pi$
$570$	0	0
$571$	−3.56358e7	−0.00801051	−0.00400525	−	0.999992i	$-0.501275\pi$
$571$	−3.56358e7	−0.00801051	−0.00400525	+	0.999992i	$0.501275\pi$
$572$	1.87847e8	0.0419679
$573$	0	0
$574$	5.27212e9	1.16357
$575$	1.31513e9	0.288490
$576$	0	0
$577$	−1.37806e9	−0.298644	−0.149322	−	0.988789i	$-0.547709\pi$
$577$	−1.37806e9	−0.298644	−0.149322	+	0.988789i	$0.547709\pi$
$578$	4.41130e9	0.950209
$579$	0	0
$580$	3.65859e9	0.778602
$581$	7.07001e9	1.49556
$582$	0	0
$583$	4.89228e8	0.102252
$584$	−1.07671e9	−0.223693
$585$	0	0
$586$	−1.01517e10	−2.08399
$587$	−2.82850e9	−0.577195	−0.288597	−	0.957451i	$-0.593189\pi$
$587$	−2.82850e9	−0.577195	−0.288597	+	0.957451i	$0.593189\pi$
$588$	0	0
$589$	−4.57977e9	−0.923506
$590$	5.96278e8	0.119527
$591$	0	0
$592$	−5.86945e9	−1.16271
$593$	4.23981e9	0.834940	0.417470	−	0.908691i	$-0.362917\pi$
$593$	4.23981e9	0.834940	0.417470	+	0.908691i	$0.362917\pi$
$594$	0	0
$595$	2.33683e9	0.454797
$596$	−9.05224e9	−1.75143
$597$	0	0
$598$	−1.69372e8	−0.0323882
$599$	7.47951e9	1.42193	0.710967	−	0.703225i	$-0.248258\pi$
$599$	7.47951e9	1.42193	0.710967	+	0.703225i	$0.248258\pi$
$600$	0	0
$601$	2.42694e9	0.456034	0.228017	−	0.973657i	$-0.426776\pi$
$601$	2.42694e9	0.456034	0.228017	+	0.973657i	$0.426776\pi$
$602$	2.80844e9	0.524659
$603$	0	0
$604$	9.00288e8	0.166246
$605$	1.31550e9	0.241517
$606$	0	0
$607$	−6.09411e9	−1.10599	−0.552993	−	0.833186i	$-0.686514\pi$
$607$	−6.09411e9	−1.10599	−0.552993	+	0.833186i	$0.686514\pi$
$608$	−5.41921e9	−0.977852
$609$	0	0
$610$	−1.51595e9	−0.270415
$611$	1.65705e8	0.0293894
$612$	0	0
$613$	−6.89396e9	−1.20881	−0.604403	−	0.796678i	$-0.706588\pi$
$613$	−6.89396e9	−1.20881	−0.604403	+	0.796678i	$0.706588\pi$
$614$	−2.67595e9	−0.466540
$615$	0	0
$616$	1.22743e9	0.211574
$617$	1.05989e10	1.81661	0.908304	−	0.418311i	$-0.137378\pi$
$617$	1.05989e10	1.81661	0.908304	+	0.418311i	$0.137378\pi$
$618$	0	0
$619$	7.11095e9	1.20507	0.602533	−	0.798094i	$-0.294158\pi$
$619$	7.11095e9	1.20507	0.602533	+	0.798094i	$0.294158\pi$
$620$	−5.78540e9	−0.974905
$621$	0	0
$622$	9.17326e9	1.52847
$623$	−6.03349e9	−0.999678
$624$	0	0
$625$	−1.36187e8	−0.0223130
$626$	−5.72884e9	−0.933375
$627$	0	0
$628$	5.29511e9	0.853133
$629$	−5.20607e9	−0.834128
$630$	0	0
$631$	−3.97705e9	−0.630170	−0.315085	−	0.949063i	$-0.602033\pi$
$631$	−3.97705e9	−0.630170	−0.315085	+	0.949063i	$0.602033\pi$
$632$	5.24658e8	0.0826735
$633$	0	0
$634$	−9.31916e9	−1.45233
$635$	3.50439e9	0.543131
$636$	0	0
$637$	−1.51524e8	−0.0232271
$638$	−8.14973e9	−1.24243
$639$	0	0
$640$	−1.81031e9	−0.272976
$641$	−5.39534e8	−0.0809125	−0.0404563	−	0.999181i	$-0.512881\pi$
$641$	−5.39534e8	−0.0809125	−0.0404563	+	0.999181i	$0.512881\pi$
$642$	0	0
$643$	6.57420e8	0.0975225	0.0487612	−	0.998810i	$-0.484473\pi$
$643$	6.57420e8	0.0975225	0.0487612	+	0.998810i	$0.484473\pi$
$644$	−4.52952e9	−0.668269
$645$	0	0
$646$	−4.06725e9	−0.593590
$647$	2.49688e9	0.362437	0.181219	−	0.983443i	$-0.441996\pi$
$647$	2.49688e9	0.362437	0.181219	+	0.983443i	$0.441996\pi$
$648$	0	0
$649$	−7.10561e8	−0.102034
$650$	−2.90586e8	−0.0415029
$651$	0	0
$652$	−5.71979e9	−0.808191
$653$	9.29237e9	1.30596	0.652981	−	0.757374i	$-0.273518\pi$
$653$	9.29237e9	1.30596	0.652981	+	0.757374i	$0.273518\pi$
$654$	0	0
$655$	3.11073e9	0.432532
$656$	3.87547e9	0.535995
$657$	0	0
$658$	8.28369e9	1.13353
$659$	6.58505e9	0.896315	0.448157	−	0.893955i	$-0.352080\pi$
$659$	6.58505e9	0.896315	0.448157	+	0.893955i	$0.352080\pi$
$660$	0	0
$661$	−3.64989e9	−0.491558	−0.245779	−	0.969326i	$-0.579044\pi$
$661$	−3.64989e9	−0.491558	−0.245779	+	0.969326i	$0.579044\pi$
$662$	1.24236e10	1.66434
$663$	0	0
$664$	−2.03183e9	−0.269339
$665$	3.96607e9	0.522979
$666$	0	0
$667$	3.93100e9	0.512936
$668$	1.24907e10	1.62132
$669$	0	0
$670$	5.49651e9	0.706033
$671$	1.80650e9	0.230839
$672$	0	0
$673$	7.02119e9	0.887887	0.443944	−	0.896055i	$-0.353579\pi$
$673$	7.02119e9	0.887887	0.443944	+	0.896055i	$0.353579\pi$
$674$	−6.26326e9	−0.787936
$675$	0	0
$676$	−9.21947e9	−1.14787
$677$	−8.84175e9	−1.09516	−0.547580	−	0.836753i	$-0.684451\pi$
$677$	−8.84175e9	−1.09516	−0.547580	+	0.836753i	$0.684451\pi$
$678$	0	0
$679$	−3.23380e9	−0.396433
$680$	−6.71575e8	−0.0819055
$681$	0	0
$682$	1.28873e10	1.55567
$683$	−3.28325e9	−0.394304	−0.197152	−	0.980373i	$-0.563169\pi$
$683$	−3.28325e9	−0.394304	−0.197152	+	0.980373i	$0.563169\pi$
$684$	0	0
$685$	7.66281e9	0.910901
$686$	7.60577e9	0.899516
$687$	0	0
$688$	2.06445e9	0.241682
$689$	5.21410e7	0.00607312
$690$	0	0
$691$	3.72206e9	0.429151	0.214576	−	0.976707i	$-0.431163\pi$
$691$	3.72206e9	0.429151	0.214576	+	0.976707i	$0.431163\pi$
$692$	3.00684e9	0.344936
$693$	0	0
$694$	3.73413e9	0.424063
$695$	3.55742e9	0.401965
$696$	0	0
$697$	3.43745e9	0.384523
$698$	−3.32263e9	−0.369818
$699$	0	0
$700$	−7.77117e9	−0.856334
$701$	−2.70122e9	−0.296174	−0.148087	−	0.988974i	$-0.547312\pi$
$701$	−2.70122e9	−0.296174	−0.148087	+	0.988974i	$0.547312\pi$
$702$	0	0
$703$	−8.83575e9	−0.959179
$704$	9.24887e9	0.999044
$705$	0	0
$706$	−1.60850e8	−0.0172030
$707$	1.33752e10	1.42342
$708$	0	0
$709$	4.56678e8	0.0481225	0.0240613	−	0.999710i	$-0.492340\pi$
$709$	4.56678e8	0.0481225	0.0240613	+	0.999710i	$0.492340\pi$
$710$	−9.18312e9	−0.962911
$711$	0	0
$712$	1.73395e9	0.180035
$713$	−6.21617e9	−0.642258
$714$	0	0
$715$	−2.23250e8	−0.0228412
$716$	3.17281e9	0.323034
$717$	0	0
$718$	2.64512e10	2.66692
$719$	−4.28395e9	−0.429827	−0.214913	−	0.976633i	$-0.568947\pi$
$719$	−4.28395e9	−0.429827	−0.214913	+	0.976633i	$0.568947\pi$
$720$	0	0
$721$	−5.00352e9	−0.497167
$722$	7.92689e9	0.783831
$723$	0	0
$724$	−9.11749e9	−0.892874
$725$	6.74432e9	0.657287
$726$	0	0
$727$	9.46884e9	0.913958	0.456979	−	0.889477i	$-0.348931\pi$
$727$	9.46884e9	0.913958	0.456979	+	0.889477i	$0.348931\pi$
$728$	1.30817e8	0.0125662
$729$	0	0
$730$	9.78998e9	0.931433
$731$	1.83112e9	0.173383
$732$	0	0
$733$	−7.31501e9	−0.686043	−0.343021	−	0.939328i	$-0.611450\pi$
$733$	−7.31501e9	−0.686043	−0.343021	+	0.939328i	$0.611450\pi$
$734$	−7.23723e9	−0.675517
$735$	0	0
$736$	−7.35556e9	−0.680054
$737$	−6.54997e9	−0.602703
$738$	0	0
$739$	−9.28262e9	−0.846087	−0.423043	−	0.906109i	$-0.639038\pi$
$739$	−9.28262e9	−0.846087	−0.423043	+	0.906109i	$0.639038\pi$
$740$	−1.11618e10	−1.01256
$741$	0	0
$742$	2.60656e9	0.234236
$743$	8.15117e9	0.729053	0.364526	−	0.931193i	$-0.381231\pi$
$743$	8.15117e9	0.729053	0.364526	+	0.931193i	$0.381231\pi$
$744$	0	0
$745$	1.07583e10	0.953228
$746$	−1.44515e10	−1.27446
$747$	0	0
$748$	6.12270e9	0.534919
$749$	1.54077e9	0.133983
$750$	0	0
$751$	−1.21007e10	−1.04248	−0.521242	−	0.853409i	$-0.674531\pi$
$751$	−1.21007e10	−1.04248	−0.521242	+	0.853409i	$0.674531\pi$
$752$	6.08923e9	0.522156
$753$	0	0
$754$	−8.68582e8	−0.0737923
$755$	−1.06996e9	−0.0904805
$756$	0	0
$757$	−2.15810e10	−1.80816	−0.904078	−	0.427367i	$-0.859441\pi$
$757$	−2.15810e10	−1.80816	−0.904078	+	0.427367i	$0.859441\pi$
$758$	−1.83112e10	−1.52713
$759$	0	0
$760$	−1.13980e9	−0.0941847
$761$	−2.06853e9	−0.170144	−0.0850719	−	0.996375i	$-0.527112\pi$
$761$	−2.06853e9	−0.170144	−0.0850719	+	0.996375i	$0.527112\pi$
$762$	0	0
$763$	−1.68129e10	−1.37027
$764$	2.69835e10	2.18913
$765$	0	0
$766$	5.36452e9	0.431251
$767$	−7.57302e7	−0.00606018
$768$	0	0
$769$	1.90437e10	1.51011	0.755055	−	0.655662i	$-0.227610\pi$
$769$	1.90437e10	1.51011	0.755055	+	0.655662i	$0.227610\pi$
$770$	−1.11604e10	−0.880972
$771$	0	0
$772$	−3.70581e9	−0.289883
$773$	−7.43503e9	−0.578968	−0.289484	−	0.957183i	$-0.593484\pi$
$773$	−7.43503e9	−0.578968	−0.289484	+	0.957183i	$0.593484\pi$
$774$	0	0
$775$	−1.06649e10	−0.823003
$776$	9.29355e8	0.0713946
$777$	0	0
$778$	1.80945e10	1.37758
$779$	5.83405e9	0.442170
$780$	0	0
$781$	1.09432e10	0.821986
$782$	−5.52053e9	−0.412816
$783$	0	0
$784$	−5.56814e9	−0.412671
$785$	−6.29308e9	−0.464322
$786$	0	0
$787$	9.17424e9	0.670901	0.335451	−	0.942058i	$-0.391111\pi$
$787$	9.17424e9	0.670901	0.335451	+	0.942058i	$0.391111\pi$
$788$	−6.05920e9	−0.441137
$789$	0	0
$790$	−4.77046e9	−0.344243
$791$	1.94424e10	1.39679
$792$	0	0
$793$	1.92533e8	0.0137104
$794$	3.64586e10	2.58481
$795$	0	0
$796$	1.77979e10	1.25076
$797$	2.11317e10	1.47853	0.739265	−	0.673415i	$-0.235173\pi$
$797$	2.11317e10	1.47853	0.739265	+	0.673415i	$0.235173\pi$
$798$	0	0
$799$	5.40101e9	0.374594
$800$	−1.26197e10	−0.871436
$801$	0	0
$802$	−4.23727e10	−2.90052
$803$	−1.16663e10	−0.795115
$804$	0	0
$805$	5.38319e9	0.363709
$806$	1.37351e9	0.0923970
$807$	0	0
$808$	−3.84386e9	−0.256347
$809$	2.83173e9	0.188032	0.0940161	−	0.995571i	$-0.470029\pi$
$809$	2.83173e9	0.188032	0.0940161	+	0.995571i	$0.470029\pi$
$810$	0	0
$811$	2.43178e10	1.60085	0.800427	−	0.599430i	$-0.204606\pi$
$811$	2.43178e10	1.60085	0.800427	+	0.599430i	$0.204606\pi$
$812$	−2.32286e10	−1.52257
$813$	0	0
$814$	2.48635e10	1.61576
$815$	6.79780e9	0.439862
$816$	0	0
$817$	3.10778e9	0.199376
$818$	4.78863e9	0.305897
$819$	0	0
$820$	7.36988e9	0.466780
$821$	2.00983e10	1.26753	0.633765	−	0.773525i	$-0.281509\pi$
$821$	2.00983e10	1.26753	0.633765	+	0.773525i	$0.281509\pi$
$822$	0	0
$823$	1.63406e10	1.02180	0.510902	−	0.859639i	$-0.329312\pi$
$823$	1.63406e10	1.02180	0.510902	+	0.859639i	$0.329312\pi$
$824$	1.43795e9	0.0895362
$825$	0	0
$826$	−3.78580e9	−0.233737
$827$	9.17251e9	0.563922	0.281961	−	0.959426i	$-0.409015\pi$
$827$	9.17251e9	0.563922	0.281961	+	0.959426i	$0.409015\pi$
$828$	0	0
$829$	−2.43239e10	−1.48283	−0.741417	−	0.671045i	$-0.765846\pi$
$829$	−2.43239e10	−1.48283	−0.741417	+	0.671045i	$0.765846\pi$
$830$	1.84745e10	1.12150
$831$	0	0
$832$	9.85727e8	0.0593369
$833$	−4.93881e9	−0.296050
$834$	0	0
$835$	−1.48448e10	−0.882415
$836$	1.03915e10	0.615113
$837$	0	0
$838$	−8.80946e9	−0.517125
$839$	−1.35860e9	−0.0794193	−0.0397096	−	0.999211i	$-0.512643\pi$
$839$	−1.35860e9	−0.0794193	−0.0397096	+	0.999211i	$0.512643\pi$
$840$	0	0
$841$	2.90934e9	0.168659
$842$	3.56854e10	2.06015
$843$	0	0
$844$	−1.82917e10	−1.04726
$845$	1.09571e10	0.624734
$846$	0	0
$847$	−8.35219e9	−0.472290
$848$	1.91605e9	0.107900
$849$	0	0
$850$	−9.47142e9	−0.528992
$851$	−1.19929e10	−0.667067
$852$	0	0
$853$	−1.40851e10	−0.777031	−0.388516	−	0.921442i	$-0.627012\pi$
$853$	−1.40851e10	−0.777031	−0.388516	+	0.921442i	$0.627012\pi$
$854$	9.62486e9	0.528800
$855$	0	0
$856$	−4.42797e8	−0.0241294
$857$	−2.25854e10	−1.22573	−0.612865	−	0.790188i	$-0.709983\pi$
$857$	−2.25854e10	−1.22573	−0.612865	+	0.790188i	$0.709983\pi$
$858$	0	0
$859$	3.28621e10	1.76897	0.884483	−	0.466572i	$-0.154511\pi$
$859$	3.28621e10	1.76897	0.884483	+	0.466572i	$0.154511\pi$
$860$	3.92591e9	0.210473
$861$	0	0
$862$	−5.49107e9	−0.291999
$863$	7.68268e9	0.406888	0.203444	−	0.979087i	$-0.434787\pi$
$863$	7.68268e9	0.406888	0.203444	+	0.979087i	$0.434787\pi$
$864$	0	0
$865$	−3.57353e9	−0.187733
$866$	−3.40273e10	−1.78039
$867$	0	0
$868$	3.67318e10	1.90644
$869$	5.68477e9	0.293862
$870$	0	0
$871$	−6.98083e8	−0.0357967
$872$	4.83183e9	0.246776
$873$	0	0
$874$	−9.36945e9	−0.474705
$875$	2.44260e10	1.23261
$876$	0	0
$877$	3.95364e10	1.97924	0.989621	−	0.143702i	$-0.0459008\pi$
$877$	3.95364e10	1.97924	0.989621	+	0.143702i	$0.0459008\pi$
$878$	−2.61538e10	−1.30408
$879$	0	0
$880$	−8.20387e9	−0.405816
$881$	−1.90800e10	−0.940074	−0.470037	−	0.882647i	$-0.655759\pi$
$881$	−1.90800e10	−0.940074	−0.470037	+	0.882647i	$0.655759\pi$
$882$	0	0
$883$	−4.60772e9	−0.225228	−0.112614	−	0.993639i	$-0.535922\pi$
$883$	−4.60772e9	−0.225228	−0.112614	+	0.993639i	$0.535922\pi$
$884$	6.52546e8	0.0317708
$885$	0	0
$886$	5.43530e9	0.262546
$887$	1.44107e10	0.693349	0.346674	−	0.937986i	$-0.387311\pi$
$887$	1.44107e10	0.693349	0.346674	+	0.937986i	$0.387311\pi$
$888$	0	0
$889$	−2.22495e10	−1.06210
$890$	−1.57660e10	−0.749644
$891$	0	0
$892$	−6.36591e8	−0.0300320
$893$	9.16660e9	0.430753
$894$	0	0
$895$	−3.77078e9	−0.175813
$896$	1.14938e10	0.533807
$897$	0	0
$898$	−6.25042e8	−0.0288033
$899$	−3.18782e10	−1.46330
$900$	0	0
$901$	1.69949e9	0.0774074
$902$	−1.64169e10	−0.744847
$903$	0	0
$904$	−5.58750e9	−0.251552
$905$	1.08358e10	0.485951
$906$	0	0
$907$	3.03408e10	1.35021	0.675105	−	0.737721i	$-0.264098\pi$
$907$	3.03408e10	1.35021	0.675105	+	0.737721i	$0.264098\pi$
$908$	1.91314e10	0.848098
$909$	0	0
$910$	−1.18945e9	−0.0523242
$911$	3.82951e9	0.167814	0.0839072	−	0.996474i	$-0.473260\pi$
$911$	3.82951e9	0.167814	0.0839072	+	0.996474i	$0.473260\pi$
$912$	0	0
$913$	−2.20153e10	−0.957363
$914$	−3.24754e9	−0.140683
$915$	0	0
$916$	−8.89104e9	−0.382224
$917$	−1.97502e10	−0.845821
$918$	0	0
$919$	2.78817e10	1.18499	0.592495	−	0.805574i	$-0.298143\pi$
$919$	2.78817e10	1.18499	0.592495	+	0.805574i	$0.298143\pi$
$920$	−1.54706e9	−0.0655014
$921$	0	0
$922$	−1.76216e10	−0.740436
$923$	1.16630e9	0.0488208
$924$	0	0
$925$	−2.05758e10	−0.854794
$926$	−1.86518e10	−0.771936
$927$	0	0
$928$	−3.77212e10	−1.54942
$929$	−1.44369e10	−0.590770	−0.295385	−	0.955378i	$-0.595448\pi$
$929$	−1.44369e10	−0.590770	−0.295385	+	0.955378i	$0.595448\pi$
$930$	0	0
$931$	−8.38216e9	−0.340433
$932$	3.50782e10	1.41933
$933$	0	0
$934$	−3.21308e10	−1.29035
$935$	−7.27664e9	−0.291132
$936$	0	0
$937$	−3.25299e10	−1.29180	−0.645898	−	0.763424i	$-0.723517\pi$
$937$	−3.25299e10	−1.29180	−0.645898	+	0.763424i	$0.723517\pi$
$938$	−3.48976e10	−1.38066
$939$	0	0
$940$	1.15797e10	0.454727
$941$	1.74946e9	0.0684448	0.0342224	−	0.999414i	$-0.489105\pi$
$941$	1.74946e9	0.0684448	0.0342224	+	0.999414i	$0.489105\pi$
$942$	0	0
$943$	7.91863e9	0.307510
$944$	−2.78290e9	−0.107670
$945$	0	0
$946$	−8.74520e9	−0.335854
$947$	2.65135e10	1.01448	0.507239	−	0.861805i	$-0.330666\pi$
$947$	2.65135e10	1.01448	0.507239	+	0.861805i	$0.330666\pi$
$948$	0	0
$949$	−1.24337e9	−0.0472248
$950$	−1.60749e10	−0.608297
$951$	0	0
$952$	4.26386e9	0.160167
$953$	−3.31597e10	−1.24104	−0.620519	−	0.784191i	$-0.713078\pi$
$953$	−3.31597e10	−1.24104	−0.620519	+	0.784191i	$0.713078\pi$
$954$	0	0
$955$	−3.20691e10	−1.19145
$956$	4.54307e9	0.168169
$957$	0	0
$958$	4.43361e10	1.62921
$959$	−4.86516e10	−1.78128
$960$	0	0
$961$	2.28970e10	0.832236
$962$	2.64991e9	0.0959661
$963$	0	0
$964$	1.43743e9	0.0516793
$965$	4.40424e9	0.157770
$966$	0	0
$967$	1.58334e9	0.0563095	0.0281547	−	0.999604i	$-0.491037\pi$
$967$	1.58334e9	0.0563095	0.0281547	+	0.999604i	$0.491037\pi$
$968$	2.40032e9	0.0850559
$969$	0	0
$970$	−8.45017e9	−0.297279
$971$	7.55884e9	0.264965	0.132482	−	0.991185i	$-0.457705\pi$
$971$	7.55884e9	0.264965	0.132482	+	0.991185i	$0.457705\pi$
$972$	0	0
$973$	−2.25863e10	−0.786048
$974$	−1.63620e10	−0.567388
$975$	0	0
$976$	7.07511e9	0.243590
$977$	−1.24479e9	−0.0427036	−0.0213518	−	0.999772i	$-0.506797\pi$
$977$	−1.24479e9	−0.0427036	−0.0213518	+	0.999772i	$0.506797\pi$
$978$	0	0
$979$	1.87877e10	0.639931
$980$	−1.05888e10	−0.359381
$981$	0	0
$982$	−5.53792e10	−1.86619
$983$	−1.03091e10	−0.346166	−0.173083	−	0.984907i	$-0.555373\pi$
$983$	−1.03091e10	−0.346166	−0.173083	+	0.984907i	$0.555373\pi$
$984$	0	0
$985$	7.20117e9	0.240091
$986$	−2.83107e10	−0.940550
$987$	0	0
$988$	1.10750e9	0.0365338
$989$	4.21823e9	0.138657
$990$	0	0
$991$	−1.42062e10	−0.463683	−0.231842	−	0.972754i	$-0.574475\pi$
$991$	−1.42062e10	−0.463683	−0.231842	+	0.972754i	$0.574475\pi$
$992$	5.96494e10	1.94006
$993$	0	0
$994$	5.83041e10	1.88298
$995$	−2.11523e10	−0.680733
$996$	0	0
$997$	8.79297e9	0.280998	0.140499	−	0.990081i	$-0.455129\pi$
$997$	8.79297e9	0.280998	0.140499	+	0.990081i	$0.455129\pi$
$998$	−7.16085e10	−2.28038
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.8.a.c.1.3		17
3.2	odd	2		177.8.a.c.1.15	✓	17

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.8.a.c.1.15	✓	17	3.2	odd	2
531.8.a.c.1.3		17	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:	\( N \)	\(=\)	\( 531 = 3^{2} \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	531.a (trivial)

\(f(q)\)	\(=\)	\(q-16.5905 q^{2} +147.246 q^{4} -174.998 q^{5} +1111.07 q^{7} -319.307 q^{8} +O(q^{10})\)	\(q-16.5905 q^{2} +147.246 q^{4} -174.998 q^{5} +1111.07 q^{7} -319.307 q^{8} +2903.31 q^{10} -3459.76 q^{11} -368.734 q^{13} -18433.2 q^{14} -13550.0 q^{16} -12018.6 q^{17} -20398.0 q^{19} -25767.8 q^{20} +57399.3 q^{22} -27686.4 q^{23} -47500.8 q^{25} +6117.50 q^{26} +163601. q^{28} -141983. q^{29} +224521. q^{31} +265674. q^{32} +199395. q^{34} -194434. q^{35} +433168. q^{37} +338413. q^{38} +55878.1 q^{40} -286012. q^{41} -152357. q^{43} -509436. q^{44} +459333. q^{46} -449388. q^{47} +410931. q^{49} +788065. q^{50} -54294.7 q^{52} -141405. q^{53} +605449. q^{55} -354773. q^{56} +2.35558e6 q^{58} +205379. q^{59} -522147. q^{61} -3.72492e6 q^{62} -2.67327e6 q^{64} +64527.6 q^{65} +1.89319e6 q^{67} -1.76969e6 q^{68} +3.22578e6 q^{70} -3.16299e6 q^{71} +3.37201e6 q^{73} -7.18650e6 q^{74} -3.00352e6 q^{76} -3.84403e6 q^{77} -1.64311e6 q^{79} +2.37123e6 q^{80} +4.74509e6 q^{82} +6.36325e6 q^{83} +2.10322e6 q^{85} +2.52769e6 q^{86} +1.10473e6 q^{88} -5.43034e6 q^{89} -409689. q^{91} -4.07672e6 q^{92} +7.45560e6 q^{94} +3.56959e6 q^{95} -2.91053e6 q^{97} -6.81757e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 17q - 2q^{2} + 1166q^{4} + 318q^{5} + 3145q^{7} - 2355q^{8} + O(q^{10}) \)	\( 17q - 2q^{2} + 1166q^{4} + 318q^{5} + 3145q^{7} - 2355q^{8} + 6521q^{10} + 1764q^{11} + 18192q^{13} + 7827q^{14} + 139226q^{16} + 15507q^{17} + 52083q^{19} - 721q^{20} - 234434q^{22} - 63823q^{23} + 202153q^{25} + 367956q^{26} + 182306q^{28} + 502955q^{29} + 347531q^{31} + 243908q^{32} - 330872q^{34} - 92641q^{35} + 447615q^{37} - 775669q^{38} + 2203270q^{40} - 940335q^{41} + 478562q^{43} + 596924q^{44} - 3078663q^{46} - 703121q^{47} + 1895082q^{49} + 876967q^{50} + 6278296q^{52} + 1005974q^{53} + 5212846q^{55} - 3425294q^{56} + 6710166q^{58} + 3491443q^{59} + 11510749q^{61} - 5996234q^{62} + 29496941q^{64} - 11094180q^{65} + 14007144q^{67} - 19688159q^{68} + 30909708q^{70} - 5229074q^{71} + 5452211q^{73} - 12819662q^{74} + 41929340q^{76} - 9930777q^{77} + 15275654q^{79} - 36576105q^{80} + 32025935q^{82} - 7826609q^{83} + 11836945q^{85} - 51649136q^{86} + 30223741q^{88} + 6436185q^{89} + 11633535q^{91} - 43357972q^{92} - 4494252q^{94} - 23741055q^{95} + 26377540q^{97} - 26517816q^{98} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

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