LMFDB - Embedded newform 177.8.a.a.1.2

Newspace parameters

Level:	$ N $	$=$	$ 177 = 3 \cdot 59 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	177.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$55.2921495107$
Analytic rank:	$1$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
Defining polynomial:	$x^{16} - 6 x^{15} - 1493 x^{14} + 8791 x^{13} + 890490 x^{12} - 5107725 x^{11} - 269092298 x^{10} + 1488374176 x^{9} + 42885295136 x^{8} - 226132003872 x^{7} - 3353576629440 x^{6} + 16796366777600 x^{5} + 99470801612800 x^{4} - 494039551757568 x^{3} - 493048066650624 x^{2} + 3193975642099712 x - 2385018853548032$
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{9}\cdot 3^{5} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$19.0314$ of defining polynomial
Character	$\chi$	$=$	177.1

$q$-expansion

$f(q)$	$=$	$q-19.0314 q^{2} -27.0000 q^{3} +234.195 q^{4} +37.6075 q^{5} +513.848 q^{6} +1158.54 q^{7} -2021.05 q^{8} +729.000 q^{9} +O(q^{10})$	\(q-19.0314 q^{2} -27.0000 q^{3} +234.195 q^{4} +37.6075 q^{5} +513.848 q^{6} +1158.54 q^{7} -2021.05 q^{8} +729.000 q^{9} -715.724 q^{10} +1618.00 q^{11} -6323.27 q^{12} -7200.81 q^{13} -22048.6 q^{14} -1015.40 q^{15} +8486.39 q^{16} +25827.4 q^{17} -13873.9 q^{18} -45988.2 q^{19} +8807.49 q^{20} -31280.5 q^{21} -30792.9 q^{22} -53080.0 q^{23} +54568.2 q^{24} -76710.7 q^{25} +137042. q^{26} -19683.0 q^{27} +271324. q^{28} +164417. q^{29} +19324.6 q^{30} +69217.7 q^{31} +97185.7 q^{32} -43686.1 q^{33} -491532. q^{34} +43569.7 q^{35} +170728. q^{36} -382449. q^{37} +875221. q^{38} +194422. q^{39} -76006.5 q^{40} -531606. q^{41} +595313. q^{42} +562745. q^{43} +378929. q^{44} +27415.9 q^{45} +1.01019e6 q^{46} +1.23094e6 q^{47} -229133. q^{48} +518668. q^{49} +1.45991e6 q^{50} -697340. q^{51} -1.68639e6 q^{52} +273502. q^{53} +374596. q^{54} +60849.1 q^{55} -2.34146e6 q^{56} +1.24168e6 q^{57} -3.12909e6 q^{58} +205379. q^{59} -237802. q^{60} +2.63599e6 q^{61} -1.31731e6 q^{62} +844575. q^{63} -2.93584e6 q^{64} -270804. q^{65} +831409. q^{66} -4.66343e6 q^{67} +6.04865e6 q^{68} +1.43316e6 q^{69} -829194. q^{70} -5.88720e6 q^{71} -1.47334e6 q^{72} -3.00177e6 q^{73} +7.27855e6 q^{74} +2.07119e6 q^{75} -1.07702e7 q^{76} +1.87452e6 q^{77} -3.70012e6 q^{78} +5.62846e6 q^{79} +319152. q^{80} +531441. q^{81} +1.01172e7 q^{82} -5.80414e6 q^{83} -7.32575e6 q^{84} +971304. q^{85} -1.07098e7 q^{86} -4.43925e6 q^{87} -3.27006e6 q^{88} -5.58369e6 q^{89} -521763. q^{90} -8.34241e6 q^{91} -1.24311e7 q^{92} -1.86888e6 q^{93} -2.34265e7 q^{94} -1.72950e6 q^{95} -2.62401e6 q^{96} +3.69012e6 q^{97} -9.87100e6 q^{98} +1.17952e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16q - 6q^{2} - 432q^{3} + 974q^{4} - 68q^{5} + 162q^{6} - 2343q^{7} + 819q^{8} + 11664q^{9} + O(q^{10}) $	\( 16q - 6q^{2} - 432q^{3} + 974q^{4} - 68q^{5} + 162q^{6} - 2343q^{7} + 819q^{8} + 11664q^{9} - 3479q^{10} + 898q^{11} - 26298q^{12} - 8172q^{13} - 13315q^{14} + 1836q^{15} + 3138q^{16} - 44985q^{17} - 4374q^{18} - 40137q^{19} + 130657q^{20} + 63261q^{21} + 109394q^{22} - 2833q^{23} - 22113q^{24} + 285746q^{25} - 129420q^{26} - 314928q^{27} + 112890q^{28} + 144375q^{29} + 93933q^{30} - 141759q^{31} - 36224q^{32} - 24246q^{33} - 341332q^{34} - 78859q^{35} + 710046q^{36} - 297971q^{37} + 329075q^{38} + 220644q^{39} - 203048q^{40} + 659077q^{41} + 359505q^{42} - 1431608q^{43} + 254916q^{44} - 49572q^{45} + 873113q^{46} - 1574073q^{47} - 84726q^{48} + 1893545q^{49} + 302533q^{50} + 1214595q^{51} - 4972548q^{52} + 587736q^{53} + 118098q^{54} - 4624036q^{55} - 5798506q^{56} + 1083699q^{57} - 6991380q^{58} + 3286064q^{59} - 3527739q^{60} - 6117131q^{61} - 11570258q^{62} - 1708047q^{63} - 19063011q^{64} - 5335514q^{65} - 2953638q^{66} - 16518710q^{67} - 17284669q^{68} + 76491q^{69} - 39189486q^{70} - 10882582q^{71} + 597051q^{72} - 21097441q^{73} - 16717030q^{74} - 7715142q^{75} - 40864952q^{76} - 3404601q^{77} + 3494340q^{78} - 3784458q^{79} - 27466195q^{80} + 8503056q^{81} - 24990117q^{82} - 1951425q^{83} - 3048030q^{84} - 23238675q^{85} - 35910572q^{86} - 3898125q^{87} - 27843055q^{88} + 10499443q^{89} - 2536191q^{90} + 699217q^{91} - 20062766q^{92} + 3827493q^{93} - 59358988q^{94} - 29236333q^{95} + 978048q^{96} - 25158976q^{97} + 2120460q^{98} + 654642q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−19.0314	−1.68216	−0.841078	−	0.540914i	$-0.818079\pi$
$2$	−19.0314	−1.68216	−0.841078	+	0.540914i	$0.818079\pi$
$3$	−27.0000	−0.577350
$3$	−27.0000	−0.577350
$4$	234.195	1.82965
$5$	37.6075	0.134549	0.0672743	−	0.997735i	$-0.478570\pi$
$5$	37.6075	0.134549	0.0672743	+	0.997735i	$0.478570\pi$
$6$	513.848	0.971193
$7$	1158.54	1.27664	0.638318	−	0.769773i	$-0.279630\pi$
$7$	1158.54	1.27664	0.638318	+	0.769773i	$0.279630\pi$
$8$	−2021.05	−1.39560
$9$	729.000	0.333333
$10$	−715.724	−0.226332
$11$	1618.00	0.366526	0.183263	−	0.983064i	$-0.441334\pi$
$11$	1618.00	0.366526	0.183263	+	0.983064i	$0.441334\pi$
$12$	−6323.27	−1.05635
$13$	−7200.81	−0.909033	−0.454516	−	0.890738i	$-0.650188\pi$
$13$	−7200.81	−0.909033	−0.454516	+	0.890738i	$0.650188\pi$
$14$	−22048.6	−2.14750
$15$	−1015.40	−0.0776817
$16$	8486.39	0.517968
$17$	25827.4	1.27500	0.637499	−	0.770451i	$-0.279969\pi$
$17$	25827.4	1.27500	0.637499	+	0.770451i	$0.279969\pi$
$18$	−13873.9	−0.560719
$19$	−45988.2	−1.53819	−0.769093	−	0.639137i	$-0.779292\pi$
$19$	−45988.2	−1.53819	−0.769093	+	0.639137i	$0.779292\pi$
$20$	8807.49	0.246177
$21$	−31280.5	−0.737067
$22$	−30792.9	−0.616554
$23$	−53080.0	−0.909670	−0.454835	−	0.890576i	$-0.650302\pi$
$23$	−53080.0	−0.909670	−0.454835	+	0.890576i	$0.650302\pi$
$24$	54568.2	0.805750
$25$	−76710.7	−0.981897
$26$	137042.	1.52913
$27$	−19683.0	−0.192450
$28$	271324.	2.33580
$29$	164417.	1.25185	0.625926	−	0.779882i	$-0.284721\pi$
$29$	164417.	1.25185	0.625926	+	0.779882i	$0.284721\pi$
$30$	19324.6	0.130673
$31$	69217.7	0.417303	0.208651	−	0.977990i	$-0.433093\pi$
$31$	69217.7	0.417303	0.208651	+	0.977990i	$0.433093\pi$
$32$	97185.7	0.524297
$33$	−43686.1	−0.211614
$34$	−491532.	−2.14475
$35$	43569.7	0.171770
$36$	170728.	0.609883
$37$	−382449.	−1.24127	−0.620637	−	0.784098i	$-0.713126\pi$
$37$	−382449.	−1.24127	−0.620637	+	0.784098i	$0.713126\pi$
$38$	875221.	2.58747
$39$	194422.	0.524830
$40$	−76006.5	−0.187776
$41$	−531606.	−1.20461	−0.602305	−	0.798266i	$-0.705751\pi$
$41$	−531606.	−1.20461	−0.602305	+	0.798266i	$0.705751\pi$
$42$	595313.	1.23986
$43$	562745.	1.07937	0.539687	−	0.841866i	$-0.318543\pi$
$43$	562745.	1.07937	0.539687	+	0.841866i	$0.318543\pi$
$44$	378929.	0.670615
$45$	27415.9	0.0448496
$46$	1.01019e6	1.53021
$47$	1.23094e6	1.72939	0.864695	−	0.502298i	$-0.167512\pi$
$47$	1.23094e6	1.72939	0.864695	+	0.502298i	$0.167512\pi$
$48$	−229133.	−0.299049
$49$	518668.	0.629801
$50$	1.45991e6	1.65170
$51$	−697340.	−0.736120
$52$	−1.68639e6	−1.66321
$53$	273502.	0.252345	0.126173	−	0.992008i	$-0.459731\pi$
$53$	273502.	0.252345	0.126173	+	0.992008i	$0.459731\pi$
$54$	374596.	0.323731
$55$	60849.1	0.0493156
$56$	−2.34146e6	−1.78167
$57$	1.24168e6	0.888072
$58$	−3.12909e6	−2.10581
$59$	205379.	0.130189
$59$	205379.	0.130189
$60$	−237802.	−0.142130
$61$	2.63599e6	1.48693	0.743464	−	0.668776i	$-0.233182\pi$
$61$	2.63599e6	1.48693	0.743464	+	0.668776i	$0.233182\pi$
$62$	−1.31731e6	−0.701968
$63$	844575.	0.425546
$64$	−2.93584e6	−1.39992
$65$	−270804.	−0.122309
$66$	831409.	0.355968
$67$	−4.66343e6	−1.89428	−0.947138	−	0.320825i	$-0.896040\pi$
$67$	−4.66343e6	−1.89428	−0.947138	+	0.320825i	$0.896040\pi$
$68$	6.04865e6	2.33280
$69$	1.43316e6	0.525198
$70$	−829194.	−0.288944
$71$	−5.88720e6	−1.95211	−0.976056	−	0.217521i	$-0.930203\pi$
$71$	−5.88720e6	−1.95211	−0.976056	+	0.217521i	$0.930203\pi$
$72$	−1.47334e6	−0.465200
$73$	−3.00177e6	−0.903124	−0.451562	−	0.892240i	$-0.649133\pi$
$73$	−3.00177e6	−0.903124	−0.451562	+	0.892240i	$0.649133\pi$
$74$	7.27855e6	2.08802
$75$	2.07119e6	0.566898
$76$	−1.07702e7	−2.81434
$77$	1.87452e6	0.467921
$78$	−3.70012e6	−0.882846
$79$	5.62846e6	1.28438	0.642192	−	0.766544i	$-0.278025\pi$
$79$	5.62846e6	1.28438	0.642192	+	0.766544i	$0.278025\pi$
$80$	319152.	0.0696919
$81$	531441.	0.111111
$82$	1.01172e7	2.02634
$83$	−5.80414e6	−1.11420	−0.557102	−	0.830444i	$-0.688087\pi$
$83$	−5.80414e6	−1.11420	−0.557102	+	0.830444i	$0.688087\pi$
$84$	−7.32575e6	−1.34857
$85$	971304.	0.171549
$86$	−1.07098e7	−1.81568
$87$	−4.43925e6	−0.722757
$88$	−3.27006e6	−0.511524
$89$	−5.58369e6	−0.839569	−0.419784	−	0.907624i	$-0.637894\pi$
$89$	−5.58369e6	−0.839569	−0.419784	+	0.907624i	$0.637894\pi$
$90$	−521763.	−0.0754440
$91$	−8.34241e6	−1.16050
$92$	−1.24311e7	−1.66438
$93$	−1.86888e6	−0.240930
$94$	−2.34265e7	−2.90910
$95$	−1.72950e6	−0.206961
$96$	−2.62401e6	−0.302703
$97$	3.69012e6	0.410525	0.205262	−	0.978707i	$-0.434195\pi$
$97$	3.69012e6	0.410525	0.205262	+	0.978707i	$0.434195\pi$
$98$	−9.87100e6	−1.05942
$99$	1.17952e6	0.122175
$100$	−1.79653e7	−1.79653
$101$	1.00192e7	0.967632	0.483816	−	0.875170i	$-0.339250\pi$
$101$	1.00192e7	0.967632	0.483816	+	0.875170i	$0.339250\pi$
$102$	1.32714e7	1.23827
$103$	−8.73833e6	−0.787949	−0.393974	−	0.919121i	$-0.628900\pi$
$103$	−8.73833e6	−0.787949	−0.393974	+	0.919121i	$0.628900\pi$
$104$	1.45532e7	1.26865
$105$	−1.17638e6	−0.0991713
$106$	−5.20513e6	−0.424484
$107$	3.46301e6	0.273282	0.136641	−	0.990621i	$-0.456369\pi$
$107$	3.46301e6	0.273282	0.136641	+	0.990621i	$0.456369\pi$
$108$	−4.60966e6	−0.352116
$109$	108153.	0.00799916	0.00399958	−	0.999992i	$-0.498727\pi$
$109$	108153.	0.00799916	0.00399958	+	0.999992i	$0.498727\pi$
$110$	−1.15804e6	−0.0829566
$111$	1.03261e7	0.716649
$112$	9.83181e6	0.661257
$113$	1.27204e6	0.0829328	0.0414664	−	0.999140i	$-0.486797\pi$
$113$	1.27204e6	0.0829328	0.0414664	+	0.999140i	$0.486797\pi$
$114$	−2.36310e7	−1.49388
$115$	−1.99621e6	−0.122395
$116$	3.85056e7	2.29045
$117$	−5.24939e6	−0.303011
$118$	−3.90866e6	−0.218998
$119$	2.99220e7	1.62771
$120$	2.05217e6	0.108413
$121$	−1.68692e7	−0.865658
$122$	−5.01667e7	−2.50124
$123$	1.43534e7	0.695482
$124$	1.62104e7	0.763518
$125$	−5.82298e6	−0.266662
$126$	−1.60735e7	−0.715834
$127$	1.46482e7	0.634560	0.317280	−	0.948332i	$-0.397230\pi$
$127$	1.46482e7	0.634560	0.317280	+	0.948332i	$0.397230\pi$
$128$	4.34335e7	1.83058
$129$	−1.51941e7	−0.623177
$130$	5.15379e6	0.205743
$131$	3.44414e7	1.33854	0.669270	−	0.743019i	$-0.266607\pi$
$131$	3.44414e7	1.33854	0.669270	+	0.743019i	$0.266607\pi$
$132$	−1.02311e7	−0.387180
$133$	−5.32791e7	−1.96370
$134$	8.87517e7	3.18647
$135$	−740228.	−0.0258939
$136$	−5.21983e7	−1.77939
$137$	−4.41452e7	−1.46677	−0.733385	−	0.679814i	$-0.762060\pi$
$137$	−4.41452e7	−1.46677	−0.733385	+	0.679814i	$0.762060\pi$
$138$	−2.72751e7	−0.883465
$139$	−1.22144e7	−0.385762	−0.192881	−	0.981222i	$-0.561783\pi$
$139$	−1.22144e7	−0.385762	−0.192881	+	0.981222i	$0.561783\pi$
$140$	1.02038e7	0.314279
$141$	−3.32353e7	−0.998463
$142$	1.12042e8	3.28376
$143$	−1.16509e7	−0.333184
$144$	6.18658e6	0.172656
$145$	6.18330e6	0.168435
$146$	5.71279e7	1.51920
$147$	−1.40040e7	−0.363616
$148$	−8.95677e7	−2.27109
$149$	4.50817e7	1.11647	0.558237	−	0.829682i	$-0.311478\pi$
$149$	4.50817e7	1.11647	0.558237	+	0.829682i	$0.311478\pi$
$150$	−3.94177e7	−0.953612
$151$	−2.91236e7	−0.688376	−0.344188	−	0.938901i	$-0.611846\pi$
$151$	−2.91236e7	−0.688376	−0.344188	+	0.938901i	$0.611846\pi$
$152$	9.29443e7	2.14669
$153$	1.88282e7	0.424999
$154$	−3.56748e7	−0.787116
$155$	2.60310e6	0.0561475
$156$	4.55326e7	0.960255
$157$	7.43102e7	1.53250	0.766249	−	0.642544i	$-0.222121\pi$
$157$	7.43102e7	1.53250	0.766249	+	0.642544i	$0.222121\pi$
$158$	−1.07118e8	−2.16053
$159$	−7.38455e6	−0.145692
$160$	3.65491e6	0.0705434
$161$	−6.14952e7	−1.16132
$162$	−1.01141e7	−0.186906
$163$	−5.15149e7	−0.931700	−0.465850	−	0.884864i	$-0.654251\pi$
$163$	−5.15149e7	−0.931700	−0.465850	+	0.884864i	$0.654251\pi$
$164$	−1.24500e8	−2.20401
$165$	−1.64292e6	−0.0284724
$166$	1.10461e8	1.87426
$167$	3.03652e7	0.504508	0.252254	−	0.967661i	$-0.418828\pi$
$167$	3.03652e7	0.504508	0.252254	+	0.967661i	$0.418828\pi$
$168$	6.32194e7	1.02865
$169$	−1.08969e7	−0.173660
$170$	−1.84853e7	−0.288573
$171$	−3.35254e7	−0.512729
$172$	1.31792e8	1.97488
$173$	−9.23593e7	−1.35619	−0.678093	−	0.734976i	$-0.737193\pi$
$173$	−9.23593e7	−1.35619	−0.678093	+	0.734976i	$0.737193\pi$
$174$	8.44853e7	1.21579
$175$	−8.88723e7	−1.25353
$176$	1.37310e7	0.189849
$177$	−5.54523e6	−0.0751646
$178$	1.06266e8	1.41229
$179$	6.85422e7	0.893249	0.446624	−	0.894722i	$-0.352626\pi$
$179$	6.85422e7	0.893249	0.446624	+	0.894722i	$0.352626\pi$
$180$	6.42066e6	0.0820590
$181$	−3.27553e7	−0.410588	−0.205294	−	0.978700i	$-0.565815\pi$
$181$	−3.27553e7	−0.410588	−0.205294	+	0.978700i	$0.565815\pi$
$182$	1.58768e8	1.95215
$183$	−7.11718e7	−0.858478
$184$	1.07277e8	1.26954
$185$	−1.43829e7	−0.167012
$186$	3.55674e7	0.405282
$187$	4.17888e7	0.467320
$188$	2.88279e8	3.16418
$189$	−2.28035e7	−0.245689
$190$	3.29149e7	0.348141
$191$	2.83438e7	0.294335	0.147167	−	0.989112i	$-0.452984\pi$
$191$	2.83438e7	0.294335	0.147167	+	0.989112i	$0.452984\pi$
$192$	7.92677e7	0.808243
$193$	−1.12860e8	−1.13003	−0.565013	−	0.825082i	$-0.691129\pi$
$193$	−1.12860e8	−1.13003	−0.565013	+	0.825082i	$0.691129\pi$
$194$	−7.02282e7	−0.690567
$195$	7.31172e6	0.0706152
$196$	1.21470e8	1.15232
$197$	−1.98897e7	−0.185352	−0.0926760	−	0.995696i	$-0.529542\pi$
$197$	−1.98897e7	−0.185352	−0.0926760	+	0.995696i	$0.529542\pi$
$198$	−2.24480e7	−0.205518
$199$	1.48348e8	1.33443	0.667217	−	0.744863i	$-0.267485\pi$
$199$	1.48348e8	1.33443	0.667217	+	0.744863i	$0.267485\pi$
$200$	1.55036e8	1.37034
$201$	1.25913e8	1.09366
$202$	−1.90680e8	−1.62771
$203$	1.90483e8	1.59816
$204$	−1.63314e8	−1.34684
$205$	−1.99924e7	−0.162079
$206$	1.66303e8	1.32545
$207$	−3.86953e7	−0.303223
$208$	−6.11089e7	−0.470850
$209$	−7.44091e7	−0.563786
$210$	2.23882e7	0.166822
$211$	−2.67425e8	−1.95981	−0.979905	−	0.199465i	$-0.936079\pi$
$211$	−2.67425e8	−1.95981	−0.979905	+	0.199465i	$0.936079\pi$
$212$	6.40528e7	0.461703
$213$	1.58954e8	1.12705
$214$	−6.59061e7	−0.459703
$215$	2.11634e7	0.145228
$216$	3.97802e7	0.268583
$217$	8.01913e7	0.532744
$218$	−2.05830e6	−0.0134558
$219$	8.10478e7	0.521419
$220$	1.42506e7	0.0902303
$221$	−1.85978e8	−1.15901
$222$	−1.96521e8	−1.20552
$223$	−2.62238e8	−1.58354	−0.791768	−	0.610822i	$-0.790839\pi$
$223$	−2.62238e8	−1.58354	−0.791768	+	0.610822i	$0.790839\pi$
$224$	1.12593e8	0.669337
$225$	−5.59221e7	−0.327299
$226$	−2.42087e7	−0.139506
$227$	−3.00342e7	−0.170422	−0.0852109	−	0.996363i	$-0.527156\pi$
$227$	−3.00342e7	−0.170422	−0.0852109	+	0.996363i	$0.527156\pi$
$228$	2.90796e8	1.62486
$229$	−1.22290e8	−0.672925	−0.336463	−	0.941697i	$-0.609231\pi$
$229$	−1.22290e8	−0.672925	−0.336463	+	0.941697i	$0.609231\pi$
$230$	3.79907e7	0.205887
$231$	−5.06120e7	−0.270154
$232$	−3.32294e8	−1.74709
$233$	−1.17471e8	−0.608396	−0.304198	−	0.952609i	$-0.598388\pi$
$233$	−1.17471e8	−0.608396	−0.304198	+	0.952609i	$0.598388\pi$
$234$	9.99033e7	0.509712
$235$	4.62924e7	0.232687
$236$	4.80988e7	0.238200
$237$	−1.51968e8	−0.741539
$238$	−5.69459e8	−2.73806
$239$	1.02120e8	0.483859	0.241930	−	0.970294i	$-0.422220\pi$
$239$	1.02120e8	0.483859	0.241930	+	0.970294i	$0.422220\pi$
$240$	−8.61710e6	−0.0402367
$241$	−1.01574e8	−0.467439	−0.233719	−	0.972304i	$-0.575090\pi$
$241$	−1.01574e8	−0.467439	−0.233719	+	0.972304i	$0.575090\pi$
$242$	3.21046e8	1.45617
$243$	−1.43489e7	−0.0641500
$244$	6.17337e8	2.72056
$245$	1.95058e7	0.0847389
$246$	−2.73165e8	−1.16991
$247$	3.31152e8	1.39826
$248$	−1.39892e8	−0.582388
$249$	1.56712e8	0.643286
$250$	1.10820e8	0.448566
$251$	−4.09066e8	−1.63281	−0.816405	−	0.577480i	$-0.804036\pi$
$251$	−4.09066e8	−1.63281	−0.816405	+	0.577480i	$0.804036\pi$
$252$	1.97795e8	0.778599
$253$	−8.58837e7	−0.333418
$254$	−2.78777e8	−1.06743
$255$	−2.62252e7	−0.0990440
$256$	−4.50813e8	−1.67941
$257$	−5.16514e8	−1.89809	−0.949043	−	0.315147i	$-0.897946\pi$
$257$	−5.16514e8	−1.89809	−0.949043	+	0.315147i	$0.897946\pi$
$258$	2.89166e8	1.04828
$259$	−4.43082e8	−1.58465
$260$	−6.34211e7	−0.223783
$261$	1.19860e8	0.417284
$262$	−6.55469e8	−2.25163
$263$	−5.42714e8	−1.83961	−0.919805	−	0.392377i	$-0.871653\pi$
$263$	−5.42714e8	−1.83961	−0.919805	+	0.392377i	$0.871653\pi$
$264$	8.82916e7	0.295329
$265$	1.02857e7	0.0339527
$266$	1.01398e9	3.30326
$267$	1.50760e8	0.484725
$268$	−1.09215e9	−3.46586
$269$	1.04494e8	0.327309	0.163655	−	0.986518i	$-0.447672\pi$
$269$	1.04494e8	0.327309	0.163655	+	0.986518i	$0.447672\pi$
$270$	1.40876e7	0.0435576
$271$	6.80419e7	0.207675	0.103837	−	0.994594i	$-0.466888\pi$
$271$	6.80419e7	0.207675	0.103837	+	0.994594i	$0.466888\pi$
$272$	2.19181e8	0.660408
$273$	2.25245e8	0.670017
$274$	8.40147e8	2.46733
$275$	−1.24118e8	−0.359891
$276$	3.35639e8	0.960928
$277$	−2.90966e8	−0.822553	−0.411276	−	0.911511i	$-0.634917\pi$
$277$	−2.90966e8	−0.822553	−0.411276	+	0.911511i	$0.634917\pi$
$278$	2.32457e8	0.648913
$279$	5.04597e7	0.139101
$280$	−8.80564e7	−0.239722
$281$	−6.50736e7	−0.174958	−0.0874788	−	0.996166i	$-0.527881\pi$
$281$	−6.50736e7	−0.174958	−0.0874788	+	0.996166i	$0.527881\pi$
$282$	6.32514e8	1.67957
$283$	5.18818e8	1.36070	0.680350	−	0.732887i	$-0.261828\pi$
$283$	5.18818e8	1.36070	0.680350	+	0.732887i	$0.261828\pi$
$284$	−1.37875e9	−3.57168
$285$	4.66965e7	0.119489
$286$	2.21734e8	0.560468
$287$	−6.15886e8	−1.53785
$288$	7.08483e7	0.174766
$289$	2.56716e8	0.625620
$290$	−1.17677e8	−0.283334
$291$	−9.96332e7	−0.237017
$292$	−7.03000e8	−1.65240
$293$	2.50544e7	0.0581900	0.0290950	−	0.999577i	$-0.490737\pi$
$293$	2.50544e7	0.0581900	0.0290950	+	0.999577i	$0.490737\pi$
$294$	2.66517e8	0.611659
$295$	7.72379e6	0.0175167
$296$	7.72946e8	1.73232
$297$	−3.18472e7	−0.0705380
$298$	−8.57969e8	−1.87808
$299$	3.82219e8	0.826919
$300$	4.85062e8	1.03723
$301$	6.51961e8	1.37797
$302$	5.54264e8	1.15796
$303$	−2.70520e8	−0.558662
$304$	−3.90274e8	−0.796732
$305$	9.91331e7	0.200064
$306$	−3.58327e8	−0.714915
$307$	−3.47020e8	−0.684496	−0.342248	−	0.939610i	$-0.611188\pi$
$307$	−3.47020e8	−0.684496	−0.342248	+	0.939610i	$0.611188\pi$
$308$	4.39003e8	0.856131
$309$	2.35935e8	0.454922
$310$	−4.95408e7	−0.0944489
$311$	−1.38483e8	−0.261058	−0.130529	−	0.991445i	$-0.541668\pi$
$311$	−1.38483e8	−0.261058	−0.130529	+	0.991445i	$0.541668\pi$
$312$	−3.92935e8	−0.732453
$313$	−3.52364e8	−0.649511	−0.324756	−	0.945798i	$-0.605282\pi$
$313$	−3.52364e8	−0.649511	−0.324756	+	0.945798i	$0.605282\pi$
$314$	−1.41423e9	−2.57790
$315$	3.17623e7	0.0572566
$316$	1.31816e9	2.34997
$317$	−31157.4	−5.49355e−5 0	−2.74678e−5	−	1.00000i	$-0.500009\pi$
$317$	−31157.4	−5.49355e−5 0	−2.74678e−5		1.00000i	$0.500009\pi$
$318$	1.40539e8	0.245076
$319$	2.66027e8	0.458837
$320$	−1.10410e8	−0.188357
$321$	−9.35014e7	−0.157779
$322$	1.17034e9	1.95352
$323$	−1.18776e9	−1.96118
$324$	1.24461e8	0.203294
$325$	5.52379e8	0.892576
$326$	9.80402e8	1.56727
$327$	−2.92012e6	−0.00461832
$328$	1.07440e9	1.68115
$329$	1.42609e9	2.20780
$330$	3.12672e7	0.0478950
$331$	1.16652e9	1.76805	0.884026	−	0.467438i	$-0.154823\pi$
$331$	1.16652e9	1.76805	0.884026	+	0.467438i	$0.154823\pi$
$332$	−1.35930e9	−2.03860
$333$	−2.78805e8	−0.413758
$334$	−5.77893e8	−0.848662
$335$	−1.75380e8	−0.254872
$336$	−2.65459e8	−0.381777
$337$	−4.15334e8	−0.591143	−0.295572	−	0.955321i	$-0.595510\pi$
$337$	−4.15334e8	−0.591143	−0.295572	+	0.955321i	$0.595510\pi$
$338$	2.07383e8	0.292123
$339$	−3.43451e7	−0.0478813
$340$	2.27475e8	0.313875
$341$	1.11994e8	0.152952
$342$	6.38036e8	0.862490
$343$	−3.53209e8	−0.472609
$344$	−1.13733e9	−1.50637
$345$	5.38976e7	0.0706647
$346$	1.75773e9	2.28132
$347$	7.92787e8	1.01860	0.509300	−	0.860589i	$-0.329904\pi$
$347$	7.92787e8	1.01860	0.509300	+	0.860589i	$0.329904\pi$
$348$	−1.03965e9	−1.32239
$349$	3.35615e7	0.0422622	0.0211311	−	0.999777i	$-0.493273\pi$
$349$	3.35615e7	0.0422622	0.0211311	+	0.999777i	$0.493273\pi$
$350$	1.69137e9	2.10863
$351$	1.41733e8	0.174943
$352$	1.57247e8	0.192169
$353$	−8.45067e8	−1.02254	−0.511269	−	0.859421i	$-0.670825\pi$
$353$	−8.45067e8	−1.02254	−0.511269	+	0.859421i	$0.670825\pi$
$354$	1.05534e8	0.126439
$355$	−2.21403e8	−0.262654
$356$	−1.30767e9	−1.53612
$357$	−8.07895e8	−0.939758
$358$	−1.30446e9	−1.50258
$359$	−9.14770e8	−1.04347	−0.521736	−	0.853107i	$-0.674716\pi$
$359$	−9.14770e8	−1.04347	−0.521736	+	0.853107i	$0.674716\pi$
$360$	−5.54087e7	−0.0625921
$361$	1.22104e9	1.36602
$362$	6.23380e8	0.690673
$363$	4.55469e8	0.499788
$364$	−1.95375e9	−2.12332
$365$	−1.12889e8	−0.121514
$366$	1.35450e9	1.44409
$367$	−1.55832e9	−1.64561	−0.822805	−	0.568324i	$-0.807592\pi$
$367$	−1.55832e9	−1.64561	−0.822805	+	0.568324i	$0.807592\pi$
$368$	−4.50458e8	−0.471180
$369$	−3.87541e8	−0.401537
$370$	2.73728e8	0.280940
$371$	3.16863e8	0.322153
$372$	−4.37682e8	−0.440817
$373$	1.33970e9	1.33667	0.668337	−	0.743858i	$-0.267006\pi$
$373$	1.33970e9	1.33667	0.668337	+	0.743858i	$0.267006\pi$
$374$	−7.95301e8	−0.786106
$375$	1.57221e8	0.153957
$376$	−2.48778e9	−2.41354
$377$	−1.18393e9	−1.13797
$378$	4.33983e8	0.413287
$379$	−3.03138e8	−0.286024	−0.143012	−	0.989721i	$-0.545679\pi$
$379$	−3.03138e8	−0.286024	−0.143012	+	0.989721i	$0.545679\pi$
$380$	−4.05041e8	−0.378666
$381$	−3.95503e8	−0.366363
$382$	−5.39423e8	−0.495117
$383$	−7.20550e8	−0.655342	−0.327671	−	0.944792i	$-0.606264\pi$
$383$	−7.20550e8	−0.655342	−0.327671	+	0.944792i	$0.606264\pi$
$384$	−1.17270e9	−1.05689
$385$	7.04960e7	0.0629581
$386$	2.14788e9	1.90088
$387$	4.10241e8	0.359791
$388$	8.64208e8	0.751117
$389$	−5.84956e8	−0.503848	−0.251924	−	0.967747i	$-0.581063\pi$
$389$	−5.84956e8	−0.503848	−0.251924	+	0.967747i	$0.581063\pi$
$390$	−1.39152e8	−0.118786
$391$	−1.37092e9	−1.15983
$392$	−1.04825e9	−0.878951
$393$	−9.29918e8	−0.772807
$394$	3.78530e8	0.311791
$395$	2.11672e8	0.172812
$396$	2.76239e8	0.223538
$397$	7.98494e8	0.640479	0.320240	−	0.947337i	$-0.396237\pi$
$397$	7.98494e8	0.640479	0.320240	+	0.947337i	$0.396237\pi$
$398$	−2.82328e9	−2.24473
$399$	1.43854e9	1.13375
$400$	−6.50997e8	−0.508591
$401$	−1.07381e9	−0.831618	−0.415809	−	0.909452i	$-0.636501\pi$
$401$	−1.07381e9	−0.831618	−0.415809	+	0.909452i	$0.636501\pi$
$402$	−2.39630e9	−1.83971
$403$	−4.98423e8	−0.379342
$404$	2.34646e9	1.77043
$405$	1.99862e7	0.0149499
$406$	−3.62517e9	−2.68836
$407$	−6.18804e8	−0.454959
$408$	1.40936e9	1.02733
$409$	−6.01255e8	−0.434537	−0.217269	−	0.976112i	$-0.569715\pi$
$409$	−6.01255e8	−0.434537	−0.217269	+	0.976112i	$0.569715\pi$
$410$	3.80483e8	0.272642
$411$	1.19192e9	0.846840
$412$	−2.04647e9	−1.44167
$413$	2.37939e8	0.166204
$414$	7.36427e8	0.510069
$415$	−2.18279e8	−0.149915
$416$	−6.99815e8	−0.476603
$417$	3.29788e8	0.222720
$418$	1.41611e9	0.948376
$419$	1.42631e9	0.947251	0.473626	−	0.880726i	$-0.342945\pi$
$419$	1.42631e9	0.947251	0.473626	+	0.880726i	$0.342945\pi$
$420$	−2.75503e8	−0.181449
$421$	−6.51184e8	−0.425320	−0.212660	−	0.977126i	$-0.568213\pi$
$421$	−6.51184e8	−0.425320	−0.212660	+	0.977126i	$0.568213\pi$
$422$	5.08948e9	3.29671
$423$	8.97352e8	0.576463
$424$	−5.52760e8	−0.352173
$425$	−1.98124e9	−1.25192
$426$	−3.02513e9	−1.89588
$427$	3.05390e9	1.89827
$428$	8.11021e8	0.500010
$429$	3.14575e8	0.192364
$430$	−4.02770e8	−0.244297
$431$	1.75470e9	1.05568	0.527841	−	0.849343i	$-0.323002\pi$
$431$	1.75470e9	1.05568	0.527841	+	0.849343i	$0.323002\pi$
$432$	−1.67038e8	−0.0996830
$433$	9.35278e8	0.553647	0.276824	−	0.960921i	$-0.410718\pi$
$433$	9.35278e8	0.553647	0.276824	+	0.960921i	$0.410718\pi$
$434$	−1.52616e9	−0.896158
$435$	−1.66949e8	−0.0972461
$436$	2.53288e7	0.0146357
$437$	2.44106e9	1.39924
$438$	−1.54245e9	−0.877108
$439$	8.68080e8	0.489704	0.244852	−	0.969560i	$-0.421261\pi$
$439$	8.68080e8	0.489704	0.244852	+	0.969560i	$0.421261\pi$
$440$	−1.22979e8	−0.0688249
$441$	3.78109e8	0.209934
$442$	3.53943e9	1.94964
$443$	8.22011e8	0.449226	0.224613	−	0.974448i	$-0.427888\pi$
$443$	8.22011e8	0.449226	0.224613	+	0.974448i	$0.427888\pi$
$444$	2.41833e9	1.31122
$445$	−2.09989e8	−0.112963
$446$	4.99076e9	2.66376
$447$	−1.21721e9	−0.644596
$448$	−3.40128e9	−1.78719
$449$	1.57362e9	0.820422	0.410211	−	0.911991i	$-0.365455\pi$
$449$	1.57362e9	0.820422	0.410211	+	0.911991i	$0.365455\pi$
$450$	1.06428e9	0.550568
$451$	−8.60140e8	−0.441521
$452$	2.97906e8	0.151738
$453$	7.86338e8	0.397434
$454$	5.71593e8	0.286676
$455$	−3.13737e8	−0.156144
$456$	−2.50950e9	−1.23939
$457$	2.23657e9	1.09616	0.548081	−	0.836425i	$-0.315358\pi$
$457$	2.23657e9	1.09616	0.548081	+	0.836425i	$0.315358\pi$
$458$	2.32735e9	1.13197
$459$	−5.08361e8	−0.245373
$460$	−4.67502e8	−0.223940
$461$	3.55618e9	1.69056	0.845280	−	0.534324i	$-0.179434\pi$
$461$	3.55618e9	1.69056	0.845280	+	0.534324i	$0.179434\pi$
$462$	9.63219e8	0.454442
$463$	−3.48898e9	−1.63367	−0.816836	−	0.576870i	$-0.804274\pi$
$463$	−3.48898e9	−1.63367	−0.816836	+	0.576870i	$0.804274\pi$
$464$	1.39531e9	0.648420
$465$	−7.02838e7	−0.0324168
$466$	2.23565e9	1.02342
$467$	−9.44654e8	−0.429204	−0.214602	−	0.976702i	$-0.568845\pi$
$467$	−9.44654e8	−0.429204	−0.214602	+	0.976702i	$0.568845\pi$
$468$	−1.22938e9	−0.554404
$469$	−5.40276e9	−2.41830
$470$	−8.81010e8	−0.391416
$471$	−2.00638e9	−0.884788
$472$	−4.15080e8	−0.181692
$473$	9.10523e8	0.395619
$474$	2.89217e9	1.24738
$475$	3.52779e9	1.51034
$476$	7.00760e9	2.97814
$477$	1.99383e8	0.0841150
$478$	−1.94349e9	−0.813927
$479$	3.10871e8	0.129243	0.0646213	−	0.997910i	$-0.479416\pi$
$479$	3.10871e8	0.129243	0.0646213	+	0.997910i	$0.479416\pi$
$480$	−9.86825e7	−0.0407283
$481$	2.75394e9	1.12836
$482$	1.93311e9	0.786305
$483$	1.66037e9	0.670487
$484$	−3.95069e9	−1.58385
$485$	1.38776e8	0.0552356
$486$	2.73080e8	0.107910
$487$	−2.73852e9	−1.07440	−0.537198	−	0.843456i	$-0.680517\pi$
$487$	−2.73852e9	−1.07440	−0.537198	+	0.843456i	$0.680517\pi$
$488$	−5.32746e9	−2.07516
$489$	1.39090e9	0.537917
$490$	−3.71224e8	−0.142544
$491$	−3.51022e8	−0.133829	−0.0669143	−	0.997759i	$-0.521315\pi$
$491$	−3.51022e8	−0.133829	−0.0669143	+	0.997759i	$0.521315\pi$
$492$	3.36149e9	1.27249
$493$	4.24646e9	1.59611
$494$	−6.30230e9	−2.35209
$495$	4.43590e7	0.0164385
$496$	5.87408e8	0.216150
$497$	−6.82055e9	−2.49214
$498$	−2.98245e9	−1.08211
$499$	−4.09829e9	−1.47656	−0.738280	−	0.674495i	$-0.764362\pi$
$499$	−4.09829e9	−1.47656	−0.738280	+	0.674495i	$0.764362\pi$
$500$	−1.36371e9	−0.487897
$501$	−8.19860e8	−0.291278
$502$	7.78511e9	2.74664
$503$	−3.66315e9	−1.28341	−0.641707	−	0.766950i	$-0.721774\pi$
$503$	−3.66315e9	−1.28341	−0.641707	+	0.766950i	$0.721774\pi$
$504$	−1.70692e9	−0.593892
$505$	3.76799e8	0.130194
$506$	1.63449e9	0.560861
$507$	2.94216e8	0.100263
$508$	3.43055e9	1.16102
$509$	−2.18735e9	−0.735202	−0.367601	−	0.929984i	$-0.619821\pi$
$509$	−2.18735e9	−0.735202	−0.367601	+	0.929984i	$0.619821\pi$
$510$	4.99103e8	0.166608
$511$	−3.47767e9	−1.15296
$512$	3.02013e9	0.994446
$513$	9.05186e8	0.296024
$514$	9.82999e9	3.19288
$515$	−3.28627e8	−0.106017
$516$	−3.55839e9	−1.14019
$517$	1.99166e9	0.633867
$518$	8.43248e9	2.66564
$519$	2.49370e9	0.782994
$520$	5.47308e8	0.170695
$521$	2.54763e9	0.789231	0.394615	−	0.918846i	$-0.370878\pi$
$521$	2.54763e9	0.789231	0.394615	+	0.918846i	$0.370878\pi$
$522$	−2.28110e9	−0.701937
$523$	1.00934e9	0.308519	0.154259	−	0.988030i	$-0.450701\pi$
$523$	1.00934e9	0.308519	0.154259	+	0.988030i	$0.450701\pi$
$524$	8.06601e9	2.44906
$525$	2.39955e9	0.723723
$526$	1.03286e10	3.09451
$527$	1.78771e9	0.532060
$528$	−3.70737e8	−0.109609
$529$	−5.87336e8	−0.172501
$530$	−1.95752e8	−0.0571138
$531$	1.49721e8	0.0433963
$532$	−1.24777e10	−3.59289
$533$	3.82799e9	1.09503
$534$	−2.86917e9	−0.815384
$535$	1.30235e8	0.0367697
$536$	9.42500e9	2.64365
$537$	−1.85064e9	−0.515717
$538$	−1.98867e9	−0.550585
$539$	8.39207e8	0.230839
$540$	−1.73358e8	−0.0473768
$541$	−9.25297e8	−0.251241	−0.125621	−	0.992078i	$-0.540092\pi$
$541$	−9.25297e8	−0.251241	−0.125621	+	0.992078i	$0.540092\pi$
$542$	−1.29493e9	−0.349341
$543$	8.84393e8	0.237053
$544$	2.51005e9	0.668477
$545$	4.06735e6	0.00107628
$546$	−4.28674e9	−1.12707
$547$	−5.27884e9	−1.37906	−0.689530	−	0.724257i	$-0.742183\pi$
$547$	−5.27884e9	−1.37906	−0.689530	+	0.724257i	$0.742183\pi$
$548$	−1.03386e10	−2.68367
$549$	1.92164e9	0.495642
$550$	2.36215e9	0.605393
$551$	−7.56123e9	−1.92558
$552$	−2.89648e9	−0.732967
$553$	6.52079e9	1.63969
$554$	5.53751e9	1.38366
$555$	3.88339e8	0.0964242
$556$	−2.86055e9	−0.705810
$557$	3.38179e9	0.829190	0.414595	−	0.910006i	$-0.363923\pi$
$557$	3.38179e9	0.829190	0.414595	+	0.910006i	$0.363923\pi$
$558$	−9.60320e8	−0.233989
$559$	−4.05222e9	−0.981186
$560$	3.69750e8	0.0889713
$561$	−1.12830e9	−0.269807
$562$	1.23844e9	0.294306
$563$	−8.68855e7	−0.0205196	−0.0102598	−	0.999947i	$-0.503266\pi$
$563$	−8.68855e7	−0.0205196	−0.0102598	+	0.999947i	$0.503266\pi$
$564$	−7.78354e9	−1.82684
$565$	4.78383e7	0.0111585
$566$	−9.87385e9	−2.28891
$567$	6.15695e8	0.141849
$568$	1.18983e10	2.72437
$569$	3.35876e9	0.764338	0.382169	−	0.924092i	$-0.375177\pi$
$569$	3.35876e9	0.764338	0.382169	+	0.924092i	$0.375177\pi$
$570$	−8.88702e8	−0.200999
$571$	−4.94562e9	−1.11172	−0.555859	−	0.831277i	$-0.687610\pi$
$571$	−4.94562e9	−1.11172	−0.555859	+	0.831277i	$0.687610\pi$
$572$	−2.72859e9	−0.609611
$573$	−7.65283e8	−0.169934
$574$	1.17212e10	2.58690
$575$	4.07180e9	0.893202
$576$	−2.14023e9	−0.466639
$577$	−6.72074e9	−1.45647	−0.728235	−	0.685327i	$-0.759659\pi$
$577$	−6.72074e9	−1.45647	−0.728235	+	0.685327i	$0.759659\pi$
$578$	−4.88567e9	−1.05239
$579$	3.04721e9	0.652421
$580$	1.44810e9	0.308177
$581$	−6.72432e9	−1.42243
$582$	1.89616e9	0.398699
$583$	4.42527e8	0.0924911
$584$	6.06671e9	1.26040
$585$	−1.97416e8	−0.0407697
$586$	−4.76822e8	−0.0978846
$587$	−9.21166e9	−1.87977	−0.939885	−	0.341491i	$-0.889068\pi$
$587$	−9.21166e9	−1.87977	−0.939885	+	0.341491i	$0.889068\pi$
$588$	−3.27968e9	−0.665290
$589$	−3.18320e9	−0.641889
$590$	−1.46995e8	−0.0294659
$591$	5.37023e8	0.107013
$592$	−3.24561e9	−0.642940
$593$	−5.16913e9	−1.01795	−0.508974	−	0.860782i	$-0.669975\pi$
$593$	−5.16913e9	−1.01795	−0.508974	+	0.860782i	$0.669975\pi$
$594$	6.06097e8	0.118656
$595$	1.12529e9	0.219006
$596$	1.05579e10	2.04275
$597$	−4.00541e9	−0.770436
$598$	−7.27417e9	−1.39101
$599$	−2.29633e9	−0.436557	−0.218278	−	0.975887i	$-0.570044\pi$
$599$	−2.29633e9	−0.436557	−0.218278	+	0.975887i	$0.570044\pi$
$600$	−4.18597e9	−0.791163
$601$	4.46476e9	0.838952	0.419476	−	0.907766i	$-0.362214\pi$
$601$	4.46476e9	0.838952	0.419476	+	0.907766i	$0.362214\pi$
$602$	−1.24078e10	−2.31796
$603$	−3.39964e9	−0.631426
$604$	−6.82061e9	−1.25949
$605$	−6.34410e8	−0.116473
$606$	5.14837e9	0.939757
$607$	−2.99837e9	−0.544158	−0.272079	−	0.962275i	$-0.587711\pi$
$607$	−2.99837e9	−0.544158	−0.272079	+	0.962275i	$0.587711\pi$
$608$	−4.46939e9	−0.806466
$609$	−5.14305e9	−0.922699
$610$	−1.88664e9	−0.336539
$611$	−8.86373e9	−1.57207
$612$	4.40947e9	0.777600
$613$	8.71253e9	1.52768	0.763840	−	0.645405i	$-0.223312\pi$
$613$	8.71253e9	1.52768	0.763840	+	0.645405i	$0.223312\pi$
$614$	6.60429e9	1.15143
$615$	5.39794e8	0.0935761
$616$	−3.78849e9	−0.653031
$617$	3.68960e9	0.632385	0.316192	−	0.948695i	$-0.397595\pi$
$617$	3.68960e9	0.632385	0.316192	+	0.948695i	$0.397595\pi$
$618$	−4.49018e9	−0.765251
$619$	2.78943e9	0.472713	0.236357	−	0.971666i	$-0.424047\pi$
$619$	2.78943e9	0.472713	0.236357	+	0.971666i	$0.424047\pi$
$620$	6.09634e8	0.102730
$621$	1.04477e9	0.175066
$622$	2.63554e9	0.439140
$623$	−6.46892e9	−1.07182
$624$	1.64994e9	0.271845
$625$	5.77403e9	0.946018
$626$	6.70599e9	1.09258
$627$	2.00905e9	0.325502
$628$	1.74031e10	2.80393
$629$	−9.87766e9	−1.58262
$630$	−6.04483e8	−0.0963145
$631$	1.40846e9	0.223173	0.111587	−	0.993755i	$-0.464407\pi$
$631$	1.40846e9	0.223173	0.111587	+	0.993755i	$0.464407\pi$
$632$	−1.13754e10	−1.79249
$633$	7.22048e9	1.13150
$634$	592969.	9.24101e−5 0
$635$	5.50884e8	0.0853792
$636$	−1.72943e9	−0.266564
$637$	−3.73483e9	−0.572510
$638$	−5.06287e9	−0.771835
$639$	−4.29177e9	−0.650704
$640$	1.63342e9	0.246303
$641$	−8.08641e9	−1.21270	−0.606349	−	0.795199i	$-0.707367\pi$
$641$	−8.08641e9	−1.21270	−0.606349	+	0.795199i	$0.707367\pi$
$642$	1.77946e9	0.265410
$643$	−5.86883e9	−0.870590	−0.435295	−	0.900288i	$-0.643356\pi$
$643$	−5.86883e9	−0.870590	−0.435295	+	0.900288i	$0.643356\pi$
$644$	−1.44019e10	−2.12480
$645$	−5.71412e8	−0.0838476
$646$	2.26047e10	3.29902
$647$	3.25260e9	0.472135	0.236067	−	0.971737i	$-0.424141\pi$
$647$	3.25260e9	0.472135	0.236067	+	0.971737i	$0.424141\pi$
$648$	−1.07407e9	−0.155067
$649$	3.32304e8	0.0477177
$650$	−1.05126e10	−1.50145
$651$	−2.16517e9	−0.307580
$652$	−1.20645e10	−1.70469
$653$	−1.64182e9	−0.230744	−0.115372	−	0.993322i	$-0.536806\pi$
$653$	−1.64182e9	−0.230744	−0.115372	+	0.993322i	$0.536806\pi$
$654$	5.55741e7	0.00776874
$655$	1.29526e9	0.180099
$656$	−4.51142e9	−0.623950
$657$	−2.18829e9	−0.301041
$658$	−2.71404e10	−3.71387
$659$	4.63845e9	0.631355	0.315678	−	0.948866i	$-0.397768\pi$
$659$	4.63845e9	0.631355	0.315678	+	0.948866i	$0.397768\pi$
$660$	−3.84765e8	−0.0520945
$661$	9.44802e9	1.27244	0.636218	−	0.771510i	$-0.280498\pi$
$661$	9.44802e9	1.27244	0.636218	+	0.771510i	$0.280498\pi$
$662$	−2.22006e10	−2.97414
$663$	5.02141e9	0.669157
$664$	1.17304e10	1.55498
$665$	−2.00369e9	−0.264214
$666$	5.30606e9	0.696005
$667$	−8.72725e9	−1.13877
$668$	7.11138e9	0.923073
$669$	7.08042e9	0.914255
$670$	3.33773e9	0.428735
$671$	4.26505e9	0.544998
$672$	−3.04002e9	−0.386442
$673$	8.85543e9	1.11984	0.559921	−	0.828546i	$-0.310831\pi$
$673$	8.85543e9	1.11984	0.559921	+	0.828546i	$0.310831\pi$
$674$	7.90440e9	0.994395
$675$	1.50990e9	0.188966
$676$	−2.55200e9	−0.317737
$677$	1.90990e8	0.0236565	0.0118283	−	0.999930i	$-0.496235\pi$
$677$	1.90990e8	0.0236565	0.0118283	+	0.999930i	$0.496235\pi$
$678$	6.53636e8	0.0805438
$679$	4.27515e9	0.524091
$680$	−1.96305e9	−0.239414
$681$	8.10922e8	0.0983930
$682$	−2.13141e9	−0.257290
$683$	−3.13279e9	−0.376235	−0.188117	−	0.982147i	$-0.560239\pi$
$683$	−3.13279e9	−0.376235	−0.188117	+	0.982147i	$0.560239\pi$
$684$	−7.85149e9	−0.938114
$685$	−1.66019e9	−0.197352
$686$	6.72207e9	0.795003
$687$	3.30183e9	0.388513
$688$	4.77567e9	0.559081
$689$	−1.96944e9	−0.229390
$690$	−1.02575e9	−0.118869
$691$	1.83456e9	0.211524	0.105762	−	0.994391i	$-0.466272\pi$
$691$	1.83456e9	0.211524	0.105762	+	0.994391i	$0.466272\pi$
$692$	−2.16301e10	−2.48135
$693$	1.36652e9	0.155974
$694$	−1.50879e10	−1.71344
$695$	−4.59352e8	−0.0519038
$696$	8.97193e9	1.00868
$697$	−1.37300e10	−1.53587
$698$	−6.38723e8	−0.0710917
$699$	3.17173e9	0.351258
$700$	−2.08135e10	−2.29351
$701$	5.98660e9	0.656398	0.328199	−	0.944609i	$-0.393558\pi$
$701$	5.98660e9	0.656398	0.328199	+	0.944609i	$0.393558\pi$
$702$	−2.69739e9	−0.294282
$703$	1.75881e10	1.90931
$704$	−4.75020e9	−0.513107
$705$	−1.24989e9	−0.134342
$706$	1.60828e10	1.72007
$707$	1.16077e10	1.23531
$708$	−1.29867e9	−0.137525
$709$	6.41985e9	0.676493	0.338246	−	0.941058i	$-0.390166\pi$
$709$	6.41985e9	0.676493	0.338246	+	0.941058i	$0.390166\pi$
$710$	4.21361e9	0.441825
$711$	4.10315e9	0.428128
$712$	1.12849e10	1.17170
$713$	−3.67408e9	−0.379608
$714$	1.53754e10	1.58082
$715$	−4.38162e8	−0.0448295
$716$	1.60523e10	1.63433
$717$	−2.75725e9	−0.279356
$718$	1.74094e10	1.75528
$719$	−1.53454e10	−1.53967	−0.769836	−	0.638242i	$-0.779662\pi$
$719$	−1.53454e10	−1.53967	−0.769836	+	0.638242i	$0.779662\pi$
$720$	2.32662e8	0.0232306
$721$	−1.01237e10	−1.00592
$722$	−2.32382e10	−2.29785
$723$	2.74251e9	0.269876
$724$	−7.67113e9	−0.751232
$725$	−1.26125e10	−1.22919
$726$	−8.66823e9	−0.840722
$727$	5.06244e8	0.0488640	0.0244320	−	0.999701i	$-0.492222\pi$
$727$	5.06244e8	0.0488640	0.0244320	+	0.999701i	$0.492222\pi$
$728$	1.68604e10	1.61960
$729$	3.87420e8	0.0370370
$730$	2.14844e9	0.204406
$731$	1.45342e10	1.37620
$732$	−1.66681e10	−1.57071
$733$	−1.29579e10	−1.21527	−0.607633	−	0.794218i	$-0.707881\pi$
$733$	−1.29579e10	−1.21527	−0.607633	+	0.794218i	$0.707881\pi$
$734$	2.96571e10	2.76817
$735$	−5.26657e8	−0.0489240
$736$	−5.15862e9	−0.476937
$737$	−7.54544e9	−0.694302
$738$	7.37545e9	0.675447
$739$	−3.46591e9	−0.315909	−0.157955	−	0.987446i	$-0.550490\pi$
$739$	−3.46591e9	−0.315909	−0.157955	+	0.987446i	$0.550490\pi$
$740$	−3.36842e9	−0.305573
$741$	−8.94111e9	−0.807287
$742$	−6.03035e9	−0.541912
$743$	4.65023e9	0.415924	0.207962	−	0.978137i	$-0.433317\pi$
$743$	4.65023e9	0.415924	0.207962	+	0.978137i	$0.433317\pi$
$744$	3.77709e9	0.336242
$745$	1.69541e9	0.150220
$746$	−2.54963e10	−2.24850
$747$	−4.23122e9	−0.371401
$748$	9.78674e9	0.855032
$749$	4.01204e9	0.348882
$750$	−2.99213e9	−0.258980
$751$	2.20118e10	1.89634	0.948169	−	0.317766i	$-0.102933\pi$
$751$	2.20118e10	1.89634	0.948169	+	0.317766i	$0.102933\pi$
$752$	1.04462e10	0.895769
$753$	1.10448e10	0.942703
$754$	2.25319e10	1.91425
$755$	−1.09527e9	−0.0926201
$756$	−5.34047e9	−0.449525
$757$	5.77225e9	0.483626	0.241813	−	0.970323i	$-0.422258\pi$
$757$	5.77225e9	0.483626	0.241813	+	0.970323i	$0.422258\pi$
$758$	5.76914e9	0.481137
$759$	2.31886e9	0.192499
$760$	3.49540e9	0.288835
$761$	5.11102e9	0.420398	0.210199	−	0.977659i	$-0.432589\pi$
$761$	5.11102e9	0.420398	0.210199	+	0.977659i	$0.432589\pi$
$762$	7.52698e9	0.616280
$763$	1.25299e8	0.0102120
$764$	6.63798e9	0.538529
$765$	7.08081e8	0.0571831
$766$	1.37131e10	1.10239
$767$	−1.47889e9	−0.118346
$768$	1.21720e10	0.969607
$769$	2.04941e9	0.162512	0.0812560	−	0.996693i	$-0.474107\pi$
$769$	2.04941e9	0.162512	0.0812560	+	0.996693i	$0.474107\pi$
$770$	−1.34164e9	−0.105905
$771$	1.39459e10	1.09586
$772$	−2.64312e10	−2.06755
$773$	2.05915e10	1.60346	0.801732	−	0.597683i	$-0.203912\pi$
$773$	2.05915e10	1.60346	0.801732	+	0.597683i	$0.203912\pi$
$774$	−7.80747e9	−0.605225
$775$	−5.30974e9	−0.409748
$776$	−7.45790e9	−0.572929
$777$	1.19632e10	0.914901
$778$	1.11326e10	0.847552
$779$	2.44476e10	1.85291
$780$	1.71237e9	0.129201
$781$	−9.52551e9	−0.715500
$782$	2.60905e10	1.95101
$783$	−3.23622e9	−0.240919
$784$	4.40162e9	0.326217
$785$	2.79462e9	0.206196
$786$	1.76977e10	1.29998
$787$	−1.82710e10	−1.33613	−0.668067	−	0.744101i	$-0.732878\pi$
$787$	−1.82710e10	−1.33613	−0.668067	+	0.744101i	$0.732878\pi$
$788$	−4.65808e9	−0.339129
$789$	1.46533e10	1.06210
$790$	−4.02842e9	−0.290697
$791$	1.47371e9	0.105875
$792$	−2.38387e9	−0.170508
$793$	−1.89813e10	−1.35167
$794$	−1.51965e10	−1.07739
$795$	−2.77715e8	−0.0196026
$796$	3.47425e10	2.44155
$797$	2.36801e10	1.65684	0.828418	−	0.560110i	$-0.189241\pi$
$797$	2.36801e10	1.65684	0.828418	+	0.560110i	$0.189241\pi$
$798$	−2.73774e10	−1.90714
$799$	3.17919e10	2.20497
$800$	−7.45518e9	−0.514805
$801$	−4.07051e9	−0.279856
$802$	2.04362e10	1.39891
$803$	−4.85687e9	−0.331019
$804$	2.94881e10	2.00102
$805$	−2.31268e9	−0.156254
$806$	9.48570e9	0.638112
$807$	−2.82134e9	−0.188972
$808$	−2.02493e10	−1.35043
$809$	1.22571e10	0.813897	0.406949	−	0.913451i	$-0.366593\pi$
$809$	1.22571e10	0.813897	0.406949	+	0.913451i	$0.366593\pi$
$810$	−3.80365e8	−0.0251480
$811$	−7.84590e9	−0.516499	−0.258250	−	0.966078i	$-0.583146\pi$
$811$	−7.84590e9	−0.516499	−0.258250	+	0.966078i	$0.583146\pi$
$812$	4.46102e10	2.92407
$813$	−1.83713e9	−0.119901
$814$	1.17767e10	0.765312
$815$	−1.93735e9	−0.125359
$816$	−5.91790e9	−0.381287
$817$	−2.58796e10	−1.66028
$818$	1.14427e10	0.730960
$819$	−6.08162e9	−0.386835
$820$	−4.68212e9	−0.296547
$821$	−8.10180e9	−0.510952	−0.255476	−	0.966815i	$-0.582232\pi$
$821$	−8.10180e9	−0.510952	−0.255476	+	0.966815i	$0.582232\pi$
$822$	−2.26840e10	−1.42452
$823$	1.64265e9	0.102718	0.0513588	−	0.998680i	$-0.483645\pi$
$823$	1.64265e9	0.102718	0.0513588	+	0.998680i	$0.483645\pi$
$824$	1.76606e10	1.09966
$825$	3.35119e9	0.207783
$826$	−4.52833e9	−0.279581
$827$	−2.72238e10	−1.67371	−0.836854	−	0.547427i	$-0.815607\pi$
$827$	−2.72238e10	−1.67371	−0.836854	+	0.547427i	$0.815607\pi$
$828$	−9.06226e9	−0.554792
$829$	1.28481e10	0.783245	0.391623	−	0.920126i	$-0.371914\pi$
$829$	1.28481e10	0.783245	0.391623	+	0.920126i	$0.371914\pi$
$830$	4.15416e9	0.252180
$831$	7.85609e9	0.474901
$832$	2.11404e10	1.27257
$833$	1.33959e10	0.802995
$834$	−6.27634e9	−0.374650
$835$	1.14196e9	0.0678809
$836$	−1.74263e10	−1.03153
$837$	−1.36241e9	−0.0803099
$838$	−2.71447e10	−1.59342
$839$	−2.14886e10	−1.25615	−0.628073	−	0.778154i	$-0.716156\pi$
$839$	−2.14886e10	−1.25615	−0.628073	+	0.778154i	$0.716156\pi$
$840$	2.37752e9	0.138404
$841$	9.78301e9	0.567135
$842$	1.23930e10	0.715455
$843$	1.75699e9	0.101012
$844$	−6.26297e10	−3.58577
$845$	−4.09805e8	−0.0233657
$846$	−1.70779e10	−0.969701
$847$	−1.95437e10	−1.10513
$848$	2.32105e9	0.130707
$849$	−1.40081e10	−0.785601
$850$	3.77058e10	2.10592
$851$	2.03004e10	1.12915
$852$	3.72264e10	2.06211
$853$	7.06458e9	0.389731	0.194865	−	0.980830i	$-0.437573\pi$
$853$	7.06458e9	0.389731	0.194865	+	0.980830i	$0.437573\pi$
$854$	−5.81200e10	−3.19318
$855$	−1.26081e9	−0.0689870
$856$	−6.99891e9	−0.381393
$857$	−1.97164e10	−1.07003	−0.535015	−	0.844843i	$-0.679694\pi$
$857$	−1.97164e10	−1.07003	−0.535015	+	0.844843i	$0.679694\pi$
$858$	−5.98681e9	−0.323586
$859$	−1.48500e10	−0.799376	−0.399688	−	0.916651i	$-0.630882\pi$
$859$	−1.48500e10	−0.799376	−0.399688	+	0.916651i	$0.630882\pi$
$860$	4.95637e9	0.265717
$861$	1.66289e10	0.887877
$862$	−3.33945e10	−1.77582
$863$	1.20569e10	0.638552	0.319276	−	0.947662i	$-0.396560\pi$
$863$	1.20569e10	0.638552	0.319276	+	0.947662i	$0.396560\pi$
$864$	−1.91291e9	−0.100901
$865$	−3.47340e9	−0.182473
$866$	−1.77997e10	−0.931321
$867$	−6.93133e9	−0.361202
$868$	1.87804e10	0.974735
$869$	9.10687e9	0.470760
$870$	3.17728e9	0.163583
$871$	3.35804e10	1.72196
$872$	−2.18581e8	−0.0111636
$873$	2.69010e9	0.136842
$874$	−4.64568e10	−2.35374
$875$	−6.74615e9	−0.340430
$876$	1.89810e10	0.954014
$877$	3.18390e10	1.59390	0.796949	−	0.604046i	$-0.206446\pi$
$877$	3.18390e10	1.59390	0.796949	+	0.604046i	$0.206446\pi$
$878$	−1.65208e10	−0.823759
$879$	−6.76470e8	−0.0335960
$880$	5.16389e8	0.0255439
$881$	−2.68011e10	−1.32050	−0.660248	−	0.751048i	$-0.729549\pi$
$881$	−2.68011e10	−1.32050	−0.660248	+	0.751048i	$0.729549\pi$
$882$	−7.19596e9	−0.353141
$883$	−3.21062e10	−1.56937	−0.784686	−	0.619893i	$-0.787176\pi$
$883$	−3.21062e10	−1.56937	−0.784686	+	0.619893i	$0.787176\pi$
$884$	−4.35552e10	−2.12059
$885$	−2.08542e8	−0.0101133
$886$	−1.56440e10	−0.755667
$887$	1.26881e10	0.610468	0.305234	−	0.952277i	$-0.401265\pi$
$887$	1.26881e10	0.610468	0.305234	+	0.952277i	$0.401265\pi$
$888$	−2.08696e10	−1.00016
$889$	1.69706e10	0.810103
$890$	3.99638e9	0.190021
$891$	8.59874e8	0.0407251
$892$	−6.14148e10	−2.89732
$893$	−5.66085e10	−2.66012
$894$	2.31652e10	1.08431
$895$	2.57770e9	0.120185
$896$	5.03193e10	2.33699
$897$	−1.03199e10	−0.477422
$898$	−2.99482e10	−1.38008
$899$	1.13805e10	0.522401
$900$	−1.30967e10	−0.598842
$901$	7.06385e9	0.321740
$902$	1.63697e10	0.742708
$903$	−1.76030e10	−0.795570
$904$	−2.57085e9	−0.115741
$905$	−1.23184e9	−0.0552441
$906$	−1.49651e10	−0.668546
$907$	−1.35224e10	−0.601767	−0.300884	−	0.953661i	$-0.597282\pi$
$907$	−1.35224e10	−0.601767	−0.300884	+	0.953661i	$0.597282\pi$
$908$	−7.03385e9	−0.311812
$909$	7.30403e9	0.322544
$910$	5.97087e9	0.262659
$911$	2.02421e10	0.887035	0.443517	−	0.896266i	$-0.353730\pi$
$911$	2.02421e10	0.887035	0.443517	+	0.896266i	$0.353730\pi$
$912$	1.05374e10	0.459993
$913$	−9.39112e9	−0.408385
$914$	−4.25650e10	−1.84392
$915$	−2.67659e9	−0.115507
$916$	−2.86397e10	−1.23122
$917$	3.99017e10	1.70883
$918$	9.67483e9	0.412756
$919$	3.59827e10	1.52929	0.764644	−	0.644453i	$-0.222915\pi$
$919$	3.59827e10	1.52929	0.764644	+	0.644453i	$0.222915\pi$
$920$	4.03442e9	0.170814
$921$	9.36955e9	0.395194
$922$	−6.76791e10	−2.84378
$923$	4.23926e10	1.77453
$924$	−1.18531e10	−0.494288
$925$	2.93379e10	1.21880
$926$	6.64002e10	2.74809
$927$	−6.37024e9	−0.262650
$928$	1.59790e10	0.656342
$929$	3.99605e10	1.63522	0.817609	−	0.575774i	$-0.195299\pi$
$929$	3.99605e10	1.63522	0.817609	+	0.575774i	$0.195299\pi$
$930$	1.33760e9	0.0545301
$931$	−2.38526e10	−0.968751
$932$	−2.75112e10	−1.11315
$933$	3.73905e9	0.150722
$934$	1.79781e10	0.721988
$935$	1.57157e9	0.0628773
$936$	1.06093e10	0.422882
$937$	2.19504e10	0.871674	0.435837	−	0.900026i	$-0.356452\pi$
$937$	2.19504e10	0.871674	0.435837	+	0.900026i	$0.356452\pi$
$938$	1.02822e11	4.06796
$939$	9.51384e9	0.374996
$940$	1.08415e10	0.425736
$941$	−2.26436e10	−0.885896	−0.442948	−	0.896547i	$-0.646067\pi$
$941$	−2.26436e10	−0.885896	−0.442948	+	0.896547i	$0.646067\pi$
$942$	3.81842e10	1.48835
$943$	2.82177e10	1.09580
$944$	1.74293e9	0.0674337
$945$	−8.57583e8	−0.0330571
$946$	−1.73286e10	−0.665493
$947$	7.27658e9	0.278421	0.139211	−	0.990263i	$-0.455543\pi$
$947$	7.27658e9	0.278421	0.139211	+	0.990263i	$0.455543\pi$
$948$	−3.55903e10	−1.35676
$949$	2.16152e10	0.820969
$950$	−6.71388e10	−2.54063
$951$	841249.	3.17170e−5 0
$952$	−6.04738e10	−2.27163
$953$	4.70442e10	1.76068	0.880340	−	0.474343i	$-0.157314\pi$
$953$	4.70442e10	1.76068	0.880340	+	0.474343i	$0.157314\pi$
$954$	−3.79454e9	−0.141495
$955$	1.06594e9	0.0396023
$956$	2.39161e10	0.885293
$957$	−7.18273e9	−0.264910
$958$	−5.91631e9	−0.217406
$959$	−5.11439e10	−1.87253
$960$	2.98106e9	0.108748
$961$	−2.27215e10	−0.825859
$962$	−5.24114e10	−1.89807
$963$	2.52454e9	0.0910940
$964$	−2.37882e10	−0.855249
$965$	−4.24437e9	−0.152043
$966$	−3.15992e10	−1.12786
$967$	−2.01441e8	−0.00716400	−0.00358200	−	0.999994i	$-0.501140\pi$
$967$	−2.01441e8	−0.00716400	−0.00358200	+	0.999994i	$0.501140\pi$
$968$	3.40935e10	1.20811
$969$	3.20694e10	1.13229
$970$	−2.64111e9	−0.0929149
$971$	4.43321e10	1.55400	0.777001	−	0.629500i	$-0.216740\pi$
$971$	4.43321e10	1.55400	0.777001	+	0.629500i	$0.216740\pi$
$972$	−3.36044e9	−0.117372
$973$	−1.41508e10	−0.492478
$974$	5.21179e10	1.80730
$975$	−1.49142e10	−0.515329
$976$	2.23701e10	0.770181
$977$	4.78091e10	1.64013	0.820067	−	0.572268i	$-0.193936\pi$
$977$	4.78091e10	1.64013	0.820067	+	0.572268i	$0.193936\pi$
$978$	−2.64709e10	−0.904861
$979$	−9.03443e9	−0.307724
$980$	4.56817e9	0.155043
$981$	7.88433e7	0.00266639
$982$	6.68045e9	0.225121
$983$	−3.22996e10	−1.08458	−0.542288	−	0.840193i	$-0.682442\pi$
$983$	−3.22996e10	−1.08458	−0.542288	+	0.840193i	$0.682442\pi$
$984$	−2.90088e10	−0.970615
$985$	−7.48003e8	−0.0249389
$986$	−8.08162e10	−2.68491
$987$	−3.85043e10	−1.27467
$988$	7.75543e10	2.55833
$989$	−2.98705e10	−0.981873
$990$	−8.44215e8	−0.0276522
$991$	−5.62073e10	−1.83457	−0.917286	−	0.398230i	$-0.869625\pi$
$991$	−5.62073e10	−1.83457	−0.917286	+	0.398230i	$0.869625\pi$
$992$	6.72697e9	0.218790
$993$	−3.14961e10	−1.02079
$994$	1.29805e11	4.19216
$995$	5.57901e9	0.179546
$996$	3.67011e10	1.17699
$997$	−6.21852e9	−0.198726	−0.0993629	−	0.995051i	$-0.531680\pi$
$997$	−6.21852e9	−0.198726	−0.0993629	+	0.995051i	$0.531680\pi$
$998$	7.79963e10	2.48380
$999$	7.52774e9	0.238883

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.8.a.a.1.2	✓	16
3.2	odd	2		531.8.a.b.1.15		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.8.a.a.1.2	✓	16	1.1	even	1	trivial
531.8.a.b.1.15		16	3.2	odd	2

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:	\( N \)	\(=\)	\( 177 = 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	177.a (trivial)

\(f(q)\)	\(=\)	\(q-19.0314 q^{2} -27.0000 q^{3} +234.195 q^{4} +37.6075 q^{5} +513.848 q^{6} +1158.54 q^{7} -2021.05 q^{8} +729.000 q^{9} +O(q^{10})\)	\(q-19.0314 q^{2} -27.0000 q^{3} +234.195 q^{4} +37.6075 q^{5} +513.848 q^{6} +1158.54 q^{7} -2021.05 q^{8} +729.000 q^{9} -715.724 q^{10} +1618.00 q^{11} -6323.27 q^{12} -7200.81 q^{13} -22048.6 q^{14} -1015.40 q^{15} +8486.39 q^{16} +25827.4 q^{17} -13873.9 q^{18} -45988.2 q^{19} +8807.49 q^{20} -31280.5 q^{21} -30792.9 q^{22} -53080.0 q^{23} +54568.2 q^{24} -76710.7 q^{25} +137042. q^{26} -19683.0 q^{27} +271324. q^{28} +164417. q^{29} +19324.6 q^{30} +69217.7 q^{31} +97185.7 q^{32} -43686.1 q^{33} -491532. q^{34} +43569.7 q^{35} +170728. q^{36} -382449. q^{37} +875221. q^{38} +194422. q^{39} -76006.5 q^{40} -531606. q^{41} +595313. q^{42} +562745. q^{43} +378929. q^{44} +27415.9 q^{45} +1.01019e6 q^{46} +1.23094e6 q^{47} -229133. q^{48} +518668. q^{49} +1.45991e6 q^{50} -697340. q^{51} -1.68639e6 q^{52} +273502. q^{53} +374596. q^{54} +60849.1 q^{55} -2.34146e6 q^{56} +1.24168e6 q^{57} -3.12909e6 q^{58} +205379. q^{59} -237802. q^{60} +2.63599e6 q^{61} -1.31731e6 q^{62} +844575. q^{63} -2.93584e6 q^{64} -270804. q^{65} +831409. q^{66} -4.66343e6 q^{67} +6.04865e6 q^{68} +1.43316e6 q^{69} -829194. q^{70} -5.88720e6 q^{71} -1.47334e6 q^{72} -3.00177e6 q^{73} +7.27855e6 q^{74} +2.07119e6 q^{75} -1.07702e7 q^{76} +1.87452e6 q^{77} -3.70012e6 q^{78} +5.62846e6 q^{79} +319152. q^{80} +531441. q^{81} +1.01172e7 q^{82} -5.80414e6 q^{83} -7.32575e6 q^{84} +971304. q^{85} -1.07098e7 q^{86} -4.43925e6 q^{87} -3.27006e6 q^{88} -5.58369e6 q^{89} -521763. q^{90} -8.34241e6 q^{91} -1.24311e7 q^{92} -1.86888e6 q^{93} -2.34265e7 q^{94} -1.72950e6 q^{95} -2.62401e6 q^{96} +3.69012e6 q^{97} -9.87100e6 q^{98} +1.17952e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16q - 6q^{2} - 432q^{3} + 974q^{4} - 68q^{5} + 162q^{6} - 2343q^{7} + 819q^{8} + 11664q^{9} + O(q^{10}) \)	\( 16q - 6q^{2} - 432q^{3} + 974q^{4} - 68q^{5} + 162q^{6} - 2343q^{7} + 819q^{8} + 11664q^{9} - 3479q^{10} + 898q^{11} - 26298q^{12} - 8172q^{13} - 13315q^{14} + 1836q^{15} + 3138q^{16} - 44985q^{17} - 4374q^{18} - 40137q^{19} + 130657q^{20} + 63261q^{21} + 109394q^{22} - 2833q^{23} - 22113q^{24} + 285746q^{25} - 129420q^{26} - 314928q^{27} + 112890q^{28} + 144375q^{29} + 93933q^{30} - 141759q^{31} - 36224q^{32} - 24246q^{33} - 341332q^{34} - 78859q^{35} + 710046q^{36} - 297971q^{37} + 329075q^{38} + 220644q^{39} - 203048q^{40} + 659077q^{41} + 359505q^{42} - 1431608q^{43} + 254916q^{44} - 49572q^{45} + 873113q^{46} - 1574073q^{47} - 84726q^{48} + 1893545q^{49} + 302533q^{50} + 1214595q^{51} - 4972548q^{52} + 587736q^{53} + 118098q^{54} - 4624036q^{55} - 5798506q^{56} + 1083699q^{57} - 6991380q^{58} + 3286064q^{59} - 3527739q^{60} - 6117131q^{61} - 11570258q^{62} - 1708047q^{63} - 19063011q^{64} - 5335514q^{65} - 2953638q^{66} - 16518710q^{67} - 17284669q^{68} + 76491q^{69} - 39189486q^{70} - 10882582q^{71} + 597051q^{72} - 21097441q^{73} - 16717030q^{74} - 7715142q^{75} - 40864952q^{76} - 3404601q^{77} + 3494340q^{78} - 3784458q^{79} - 27466195q^{80} + 8503056q^{81} - 24990117q^{82} - 1951425q^{83} - 3048030q^{84} - 23238675q^{85} - 35910572q^{86} - 3898125q^{87} - 27843055q^{88} + 10499443q^{89} - 2536191q^{90} + 699217q^{91} - 20062766q^{92} + 3827493q^{93} - 59358988q^{94} - 29236333q^{95} + 978048q^{96} - 25158976q^{97} + 2120460q^{98} + 654642q^{99} + O(q^{100}) \)

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