LMFDB - Embedded newform 531.8.a.b.1.15

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:	$1$
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:	no (minimal twist has level 177)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Root			$19.0314$ of defining polynomial
Character	$\chi$	$=$	531.1

$q$-expansion

$f(q)$	$=$	$q+19.0314 q^{2} +234.195 q^{4} -37.6075 q^{5} +1158.54 q^{7} +2021.05 q^{8} +O(q^{10})$	\(q+19.0314 q^{2} +234.195 q^{4} -37.6075 q^{5} +1158.54 q^{7} +2021.05 q^{8} -715.724 q^{10} -1618.00 q^{11} -7200.81 q^{13} +22048.6 q^{14} +8486.39 q^{16} -25827.4 q^{17} -45988.2 q^{19} -8807.49 q^{20} -30792.9 q^{22} +53080.0 q^{23} -76710.7 q^{25} -137042. q^{26} +271324. q^{28} -164417. q^{29} +69217.7 q^{31} -97185.7 q^{32} -491532. q^{34} -43569.7 q^{35} -382449. q^{37} -875221. q^{38} -76006.5 q^{40} +531606. q^{41} +562745. q^{43} -378929. q^{44} +1.01019e6 q^{46} -1.23094e6 q^{47} +518668. q^{49} -1.45991e6 q^{50} -1.68639e6 q^{52} -273502. q^{53} +60849.1 q^{55} +2.34146e6 q^{56} -3.12909e6 q^{58} -205379. q^{59} +2.63599e6 q^{61} +1.31731e6 q^{62} -2.93584e6 q^{64} +270804. q^{65} -4.66343e6 q^{67} -6.04865e6 q^{68} -829194. q^{70} +5.88720e6 q^{71} -3.00177e6 q^{73} -7.27855e6 q^{74} -1.07702e7 q^{76} -1.87452e6 q^{77} +5.62846e6 q^{79} -319152. q^{80} +1.01172e7 q^{82} +5.80414e6 q^{83} +971304. q^{85} +1.07098e7 q^{86} -3.27006e6 q^{88} +5.58369e6 q^{89} -8.34241e6 q^{91} +1.24311e7 q^{92} -2.34265e7 q^{94} +1.72950e6 q^{95} +3.69012e6 q^{97} +9.87100e6 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16q + 6q^{2} + 974q^{4} + 68q^{5} - 2343q^{7} - 819q^{8} + O(q^{10}) $	$ 16q + 6q^{2} + 974q^{4} + 68q^{5} - 2343q^{7} - 819q^{8} - 3479q^{10} - 898q^{11} - 8172q^{13} + 13315q^{14} + 3138q^{16} + 44985q^{17} - 40137q^{19} - 130657q^{20} + 109394q^{22} + 2833q^{23} + 285746q^{25} + 129420q^{26} + 112890q^{28} - 144375q^{29} - 141759q^{31} + 36224q^{32} - 341332q^{34} + 78859q^{35} - 297971q^{37} - 329075q^{38} - 203048q^{40} - 659077q^{41} - 1431608q^{43} - 254916q^{44} + 873113q^{46} + 1574073q^{47} + 1893545q^{49} - 302533q^{50} - 4972548q^{52} - 587736q^{53} - 4624036q^{55} + 5798506q^{56} - 6991380q^{58} - 3286064q^{59} - 6117131q^{61} + 11570258q^{62} - 19063011q^{64} + 5335514q^{65} - 16518710q^{67} + 17284669q^{68} - 39189486q^{70} + 10882582q^{71} - 21097441q^{73} + 16717030q^{74} - 40864952q^{76} + 3404601q^{77} - 3784458q^{79} + 27466195q^{80} - 24990117q^{82} + 1951425q^{83} - 23238675q^{85} + 35910572q^{86} - 27843055q^{88} - 10499443q^{89} + 699217q^{91} + 20062766q^{92} - 59358988q^{94} + 29236333q^{95} - 25158976q^{97} - 2120460q^{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 $

$2$	19.0314	1.68216	0.841078	−	0.540914i	$-0.181921\pi$
$2$	19.0314	1.68216	0.841078	+	0.540914i	$0.181921\pi$
$3$	0	0
$3$	0	0
$4$	234.195	1.82965
$5$	−37.6075	−0.134549	−0.0672743	−	0.997735i	$-0.521430\pi$
$5$	−37.6075	−0.134549	−0.0672743	+	0.997735i	$0.521430\pi$
$6$	0	0
$7$	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
$9$	0	0
$10$	−715.724	−0.226332
$11$	−1618.00	−0.366526	−0.183263	−	0.983064i	$-0.558666\pi$
$11$	−1618.00	−0.366526	−0.183263	+	0.983064i	$0.558666\pi$
$12$	0	0
$13$	−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
$15$	0	0
$16$	8486.39	0.517968
$17$	−25827.4	−1.27500	−0.637499	−	0.770451i	$-0.720031\pi$
$17$	−25827.4	−1.27500	−0.637499	+	0.770451i	$0.720031\pi$
$18$	0	0
$19$	−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
$21$	0	0
$22$	−30792.9	−0.616554
$23$	53080.0	0.909670	0.454835	−	0.890576i	$-0.349698\pi$
$23$	53080.0	0.909670	0.454835	+	0.890576i	$0.349698\pi$
$24$	0	0
$25$	−76710.7	−0.981897
$26$	−137042.	−1.52913
$27$	0	0
$28$	271324.	2.33580
$29$	−164417.	−1.25185	−0.625926	−	0.779882i	$-0.715279\pi$
$29$	−164417.	−1.25185	−0.625926	+	0.779882i	$0.715279\pi$
$30$	0	0
$31$	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
$33$	0	0
$34$	−491532.	−2.14475
$35$	−43569.7	−0.171770
$36$	0	0
$37$	−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
$39$	0	0
$40$	−76006.5	−0.187776
$41$	531606.	1.20461	0.602305	−	0.798266i	$-0.294249\pi$
$41$	531606.	1.20461	0.602305	+	0.798266i	$0.294249\pi$
$42$	0	0
$43$	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
$45$	0	0
$46$	1.01019e6	1.53021
$47$	−1.23094e6	−1.72939	−0.864695	−	0.502298i	$-0.832488\pi$
$47$	−1.23094e6	−1.72939	−0.864695	+	0.502298i	$0.832488\pi$
$48$	0	0
$49$	518668.	0.629801
$50$	−1.45991e6	−1.65170
$51$	0	0
$52$	−1.68639e6	−1.66321
$53$	−273502.	−0.252345	−0.126173	−	0.992008i	$-0.540269\pi$
$53$	−273502.	−0.252345	−0.126173	+	0.992008i	$0.540269\pi$
$54$	0	0
$55$	60849.1	0.0493156
$56$	2.34146e6	1.78167
$57$	0	0
$58$	−3.12909e6	−2.10581
$59$	−205379.	−0.130189
$59$	−205379.	−0.130189
$60$	0	0
$61$	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
$63$	0	0
$64$	−2.93584e6	−1.39992
$65$	270804.	0.122309
$66$	0	0
$67$	−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
$69$	0	0
$70$	−829194.	−0.288944
$71$	5.88720e6	1.95211	0.976056	−	0.217521i	$-0.0697969\pi$
$71$	5.88720e6	1.95211	0.976056	+	0.217521i	$0.0697969\pi$
$72$	0	0
$73$	−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
$75$	0	0
$76$	−1.07702e7	−2.81434
$77$	−1.87452e6	−0.467921
$78$	0	0
$79$	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
$81$	0	0
$82$	1.01172e7	2.02634
$83$	5.80414e6	1.11420	0.557102	−	0.830444i	$-0.311913\pi$
$83$	5.80414e6	1.11420	0.557102	+	0.830444i	$0.311913\pi$
$84$	0	0
$85$	971304.	0.171549
$86$	1.07098e7	1.81568
$87$	0	0
$88$	−3.27006e6	−0.511524
$89$	5.58369e6	0.839569	0.419784	−	0.907624i	$-0.362106\pi$
$89$	5.58369e6	0.839569	0.419784	+	0.907624i	$0.362106\pi$
$90$	0	0
$91$	−8.34241e6	−1.16050
$92$	1.24311e7	1.66438
$93$	0	0
$94$	−2.34265e7	−2.90910
$95$	1.72950e6	0.206961
$96$	0	0
$97$	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
$99$	0	0
$100$	−1.79653e7	−1.79653
$101$	−1.00192e7	−0.967632	−0.483816	−	0.875170i	$-0.660750\pi$
$101$	−1.00192e7	−0.967632	−0.483816	+	0.875170i	$0.660750\pi$
$102$	0	0
$103$	−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
$105$	0	0
$106$	−5.20513e6	−0.424484
$107$	−3.46301e6	−0.273282	−0.136641	−	0.990621i	$-0.543631\pi$
$107$	−3.46301e6	−0.273282	−0.136641	+	0.990621i	$0.543631\pi$
$108$	0	0
$109$	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
$111$	0	0
$112$	9.83181e6	0.661257
$113$	−1.27204e6	−0.0829328	−0.0414664	−	0.999140i	$-0.513203\pi$
$113$	−1.27204e6	−0.0829328	−0.0414664	+	0.999140i	$0.513203\pi$
$114$	0	0
$115$	−1.99621e6	−0.122395
$116$	−3.85056e7	−2.29045
$117$	0	0
$118$	−3.90866e6	−0.218998
$119$	−2.99220e7	−1.62771
$120$	0	0
$121$	−1.68692e7	−0.865658
$122$	5.01667e7	2.50124
$123$	0	0
$124$	1.62104e7	0.763518
$125$	5.82298e6	0.266662
$126$	0	0
$127$	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
$129$	0	0
$130$	5.15379e6	0.205743
$131$	−3.44414e7	−1.33854	−0.669270	−	0.743019i	$-0.733393\pi$
$131$	−3.44414e7	−1.33854	−0.669270	+	0.743019i	$0.733393\pi$
$132$	0	0
$133$	−5.32791e7	−1.96370
$134$	−8.87517e7	−3.18647
$135$	0	0
$136$	−5.21983e7	−1.77939
$137$	4.41452e7	1.46677	0.733385	−	0.679814i	$-0.237940\pi$
$137$	4.41452e7	1.46677	0.733385	+	0.679814i	$0.237940\pi$
$138$	0	0
$139$	−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
$141$	0	0
$142$	1.12042e8	3.28376
$143$	1.16509e7	0.333184
$144$	0	0
$145$	6.18330e6	0.168435
$146$	−5.71279e7	−1.51920
$147$	0	0
$148$	−8.95677e7	−2.27109
$149$	−4.50817e7	−1.11647	−0.558237	−	0.829682i	$-0.688522\pi$
$149$	−4.50817e7	−1.11647	−0.558237	+	0.829682i	$0.688522\pi$
$150$	0	0
$151$	−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
$153$	0	0
$154$	−3.56748e7	−0.787116
$155$	−2.60310e6	−0.0561475
$156$	0	0
$157$	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
$159$	0	0
$160$	3.65491e6	0.0705434
$161$	6.14952e7	1.16132
$162$	0	0
$163$	−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
$165$	0	0
$166$	1.10461e8	1.87426
$167$	−3.03652e7	−0.504508	−0.252254	−	0.967661i	$-0.581172\pi$
$167$	−3.03652e7	−0.504508	−0.252254	+	0.967661i	$0.581172\pi$
$168$	0	0
$169$	−1.08969e7	−0.173660
$170$	1.84853e7	0.288573
$171$	0	0
$172$	1.31792e8	1.97488
$173$	9.23593e7	1.35619	0.678093	−	0.734976i	$-0.262807\pi$
$173$	9.23593e7	1.35619	0.678093	+	0.734976i	$0.262807\pi$
$174$	0	0
$175$	−8.88723e7	−1.25353
$176$	−1.37310e7	−0.189849
$177$	0	0
$178$	1.06266e8	1.41229
$179$	−6.85422e7	−0.893249	−0.446624	−	0.894722i	$-0.647374\pi$
$179$	−6.85422e7	−0.893249	−0.446624	+	0.894722i	$0.647374\pi$
$180$	0	0
$181$	−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
$183$	0	0
$184$	1.07277e8	1.26954
$185$	1.43829e7	0.167012
$186$	0	0
$187$	4.17888e7	0.467320
$188$	−2.88279e8	−3.16418
$189$	0	0
$190$	3.29149e7	0.348141
$191$	−2.83438e7	−0.294335	−0.147167	−	0.989112i	$-0.547016\pi$
$191$	−2.83438e7	−0.294335	−0.147167	+	0.989112i	$0.547016\pi$
$192$	0	0
$193$	−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
$195$	0	0
$196$	1.21470e8	1.15232
$197$	1.98897e7	0.185352	0.0926760	−	0.995696i	$-0.470458\pi$
$197$	1.98897e7	0.185352	0.0926760	+	0.995696i	$0.470458\pi$
$198$	0	0
$199$	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
$201$	0	0
$202$	−1.90680e8	−1.62771
$203$	−1.90483e8	−1.59816
$204$	0	0
$205$	−1.99924e7	−0.162079
$206$	−1.66303e8	−1.32545
$207$	0	0
$208$	−6.11089e7	−0.470850
$209$	7.44091e7	0.563786
$210$	0	0
$211$	−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
$213$	0	0
$214$	−6.59061e7	−0.459703
$215$	−2.11634e7	−0.145228
$216$	0	0
$217$	8.01913e7	0.532744
$218$	2.05830e6	0.0134558
$219$	0	0
$220$	1.42506e7	0.0902303
$221$	1.85978e8	1.15901
$222$	0	0
$223$	−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
$225$	0	0
$226$	−2.42087e7	−0.139506
$227$	3.00342e7	0.170422	0.0852109	−	0.996363i	$-0.472844\pi$
$227$	3.00342e7	0.170422	0.0852109	+	0.996363i	$0.472844\pi$
$228$	0	0
$229$	−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
$231$	0	0
$232$	−3.32294e8	−1.74709
$233$	1.17471e8	0.608396	0.304198	−	0.952609i	$-0.401612\pi$
$233$	1.17471e8	0.608396	0.304198	+	0.952609i	$0.401612\pi$
$234$	0	0
$235$	4.62924e7	0.232687
$236$	−4.80988e7	−0.238200
$237$	0	0
$238$	−5.69459e8	−2.73806
$239$	−1.02120e8	−0.483859	−0.241930	−	0.970294i	$-0.577780\pi$
$239$	−1.02120e8	−0.483859	−0.241930	+	0.970294i	$0.577780\pi$
$240$	0	0
$241$	−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
$243$	0	0
$244$	6.17337e8	2.72056
$245$	−1.95058e7	−0.0847389
$246$	0	0
$247$	3.31152e8	1.39826
$248$	1.39892e8	0.582388
$249$	0	0
$250$	1.10820e8	0.448566
$251$	4.09066e8	1.63281	0.816405	−	0.577480i	$-0.195964\pi$
$251$	4.09066e8	1.63281	0.816405	+	0.577480i	$0.195964\pi$
$252$	0	0
$253$	−8.58837e7	−0.333418
$254$	2.78777e8	1.06743
$255$	0	0
$256$	−4.50813e8	−1.67941
$257$	5.16514e8	1.89809	0.949043	−	0.315147i	$-0.102054\pi$
$257$	5.16514e8	1.89809	0.949043	+	0.315147i	$0.102054\pi$
$258$	0	0
$259$	−4.43082e8	−1.58465
$260$	6.34211e7	0.223783
$261$	0	0
$262$	−6.55469e8	−2.25163
$263$	5.42714e8	1.83961	0.919805	−	0.392377i	$-0.128347\pi$
$263$	5.42714e8	1.83961	0.919805	+	0.392377i	$0.128347\pi$
$264$	0	0
$265$	1.02857e7	0.0339527
$266$	−1.01398e9	−3.30326
$267$	0	0
$268$	−1.09215e9	−3.46586
$269$	−1.04494e8	−0.327309	−0.163655	−	0.986518i	$-0.552328\pi$
$269$	−1.04494e8	−0.327309	−0.163655	+	0.986518i	$0.552328\pi$
$270$	0	0
$271$	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
$273$	0	0
$274$	8.40147e8	2.46733
$275$	1.24118e8	0.359891
$276$	0	0
$277$	−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
$279$	0	0
$280$	−8.80564e7	−0.239722
$281$	6.50736e7	0.174958	0.0874788	−	0.996166i	$-0.472119\pi$
$281$	6.50736e7	0.174958	0.0874788	+	0.996166i	$0.472119\pi$
$282$	0	0
$283$	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
$285$	0	0
$286$	2.21734e8	0.560468
$287$	6.15886e8	1.53785
$288$	0	0
$289$	2.56716e8	0.625620
$290$	1.17677e8	0.283334
$291$	0	0
$292$	−7.03000e8	−1.65240
$293$	−2.50544e7	−0.0581900	−0.0290950	−	0.999577i	$-0.509263\pi$
$293$	−2.50544e7	−0.0581900	−0.0290950	+	0.999577i	$0.509263\pi$
$294$	0	0
$295$	7.72379e6	0.0175167
$296$	−7.72946e8	−1.73232
$297$	0	0
$298$	−8.57969e8	−1.87808
$299$	−3.82219e8	−0.826919
$300$	0	0
$301$	6.51961e8	1.37797
$302$	−5.54264e8	−1.15796
$303$	0	0
$304$	−3.90274e8	−0.796732
$305$	−9.91331e7	−0.200064
$306$	0	0
$307$	−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
$309$	0	0
$310$	−4.95408e7	−0.0944489
$311$	1.38483e8	0.261058	0.130529	−	0.991445i	$-0.458332\pi$
$311$	1.38483e8	0.261058	0.130529	+	0.991445i	$0.458332\pi$
$312$	0	0
$313$	−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
$315$	0	0
$316$	1.31816e9	2.34997
$317$	31157.4	5.49355e−5 0	2.74678e−5	−	1.00000i	$-0.499991\pi$
$317$	31157.4	5.49355e−5 0	2.74678e−5		1.00000i	$0.499991\pi$
$318$	0	0
$319$	2.66027e8	0.458837
$320$	1.10410e8	0.188357
$321$	0	0
$322$	1.17034e9	1.95352
$323$	1.18776e9	1.96118
$324$	0	0
$325$	5.52379e8	0.892576
$326$	−9.80402e8	−1.56727
$327$	0	0
$328$	1.07440e9	1.68115
$329$	−1.42609e9	−2.20780
$330$	0	0
$331$	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
$333$	0	0
$334$	−5.77893e8	−0.848662
$335$	1.75380e8	0.254872
$336$	0	0
$337$	−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
$339$	0	0
$340$	2.27475e8	0.313875
$341$	−1.11994e8	−0.152952
$342$	0	0
$343$	−3.53209e8	−0.472609
$344$	1.13733e9	1.50637
$345$	0	0
$346$	1.75773e9	2.28132
$347$	−7.92787e8	−1.01860	−0.509300	−	0.860589i	$-0.670096\pi$
$347$	−7.92787e8	−1.01860	−0.509300	+	0.860589i	$0.670096\pi$
$348$	0	0
$349$	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
$351$	0	0
$352$	1.57247e8	0.192169
$353$	8.45067e8	1.02254	0.511269	−	0.859421i	$-0.329175\pi$
$353$	8.45067e8	1.02254	0.511269	+	0.859421i	$0.329175\pi$
$354$	0	0
$355$	−2.21403e8	−0.262654
$356$	1.30767e9	1.53612
$357$	0	0
$358$	−1.30446e9	−1.50258
$359$	9.14770e8	1.04347	0.521736	−	0.853107i	$-0.325284\pi$
$359$	9.14770e8	1.04347	0.521736	+	0.853107i	$0.325284\pi$
$360$	0	0
$361$	1.22104e9	1.36602
$362$	−6.23380e8	−0.690673
$363$	0	0
$364$	−1.95375e9	−2.12332
$365$	1.12889e8	0.121514
$366$	0	0
$367$	−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
$369$	0	0
$370$	2.73728e8	0.280940
$371$	−3.16863e8	−0.322153
$372$	0	0
$373$	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
$375$	0	0
$376$	−2.48778e9	−2.41354
$377$	1.18393e9	1.13797
$378$	0	0
$379$	−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
$381$	0	0
$382$	−5.39423e8	−0.495117
$383$	7.20550e8	0.655342	0.327671	−	0.944792i	$-0.393736\pi$
$383$	7.20550e8	0.655342	0.327671	+	0.944792i	$0.393736\pi$
$384$	0	0
$385$	7.04960e7	0.0629581
$386$	−2.14788e9	−1.90088
$387$	0	0
$388$	8.64208e8	0.751117
$389$	5.84956e8	0.503848	0.251924	−	0.967747i	$-0.418937\pi$
$389$	5.84956e8	0.503848	0.251924	+	0.967747i	$0.418937\pi$
$390$	0	0
$391$	−1.37092e9	−1.15983
$392$	1.04825e9	0.878951
$393$	0	0
$394$	3.78530e8	0.311791
$395$	−2.11672e8	−0.172812
$396$	0	0
$397$	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
$399$	0	0
$400$	−6.50997e8	−0.508591
$401$	1.07381e9	0.831618	0.415809	−	0.909452i	$-0.363499\pi$
$401$	1.07381e9	0.831618	0.415809	+	0.909452i	$0.363499\pi$
$402$	0	0
$403$	−4.98423e8	−0.379342
$404$	−2.34646e9	−1.77043
$405$	0	0
$406$	−3.62517e9	−2.68836
$407$	6.18804e8	0.454959
$408$	0	0
$409$	−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
$411$	0	0
$412$	−2.04647e9	−1.44167
$413$	−2.37939e8	−0.166204
$414$	0	0
$415$	−2.18279e8	−0.149915
$416$	6.99815e8	0.476603
$417$	0	0
$418$	1.41611e9	0.948376
$419$	−1.42631e9	−0.947251	−0.473626	−	0.880726i	$-0.657055\pi$
$419$	−1.42631e9	−0.947251	−0.473626	+	0.880726i	$0.657055\pi$
$420$	0	0
$421$	−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
$423$	0	0
$424$	−5.52760e8	−0.352173
$425$	1.98124e9	1.25192
$426$	0	0
$427$	3.05390e9	1.89827
$428$	−8.11021e8	−0.500010
$429$	0	0
$430$	−4.02770e8	−0.244297
$431$	−1.75470e9	−1.05568	−0.527841	−	0.849343i	$-0.676998\pi$
$431$	−1.75470e9	−1.05568	−0.527841	+	0.849343i	$0.676998\pi$
$432$	0	0
$433$	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
$435$	0	0
$436$	2.53288e7	0.0146357
$437$	−2.44106e9	−1.39924
$438$	0	0
$439$	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
$441$	0	0
$442$	3.53943e9	1.94964
$443$	−8.22011e8	−0.449226	−0.224613	−	0.974448i	$-0.572112\pi$
$443$	−8.22011e8	−0.449226	−0.224613	+	0.974448i	$0.572112\pi$
$444$	0	0
$445$	−2.09989e8	−0.112963
$446$	−4.99076e9	−2.66376
$447$	0	0
$448$	−3.40128e9	−1.78719
$449$	−1.57362e9	−0.820422	−0.410211	−	0.911991i	$-0.634545\pi$
$449$	−1.57362e9	−0.820422	−0.410211	+	0.911991i	$0.634545\pi$
$450$	0	0
$451$	−8.60140e8	−0.441521
$452$	−2.97906e8	−0.151738
$453$	0	0
$454$	5.71593e8	0.286676
$455$	3.13737e8	0.156144
$456$	0	0
$457$	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
$459$	0	0
$460$	−4.67502e8	−0.223940
$461$	−3.55618e9	−1.69056	−0.845280	−	0.534324i	$-0.820566\pi$
$461$	−3.55618e9	−1.69056	−0.845280	+	0.534324i	$0.820566\pi$
$462$	0	0
$463$	−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
$465$	0	0
$466$	2.23565e9	1.02342
$467$	9.44654e8	0.429204	0.214602	−	0.976702i	$-0.431155\pi$
$467$	9.44654e8	0.429204	0.214602	+	0.976702i	$0.431155\pi$
$468$	0	0
$469$	−5.40276e9	−2.41830
$470$	8.81010e8	0.391416
$471$	0	0
$472$	−4.15080e8	−0.181692
$473$	−9.10523e8	−0.395619
$474$	0	0
$475$	3.52779e9	1.51034
$476$	−7.00760e9	−2.97814
$477$	0	0
$478$	−1.94349e9	−0.813927
$479$	−3.10871e8	−0.129243	−0.0646213	−	0.997910i	$-0.520584\pi$
$479$	−3.10871e8	−0.129243	−0.0646213	+	0.997910i	$0.520584\pi$
$480$	0	0
$481$	2.75394e9	1.12836
$482$	−1.93311e9	−0.786305
$483$	0	0
$484$	−3.95069e9	−1.58385
$485$	−1.38776e8	−0.0552356
$486$	0	0
$487$	−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
$489$	0	0
$490$	−3.71224e8	−0.142544
$491$	3.51022e8	0.133829	0.0669143	−	0.997759i	$-0.478685\pi$
$491$	3.51022e8	0.133829	0.0669143	+	0.997759i	$0.478685\pi$
$492$	0	0
$493$	4.24646e9	1.59611
$494$	6.30230e9	2.35209
$495$	0	0
$496$	5.87408e8	0.216150
$497$	6.82055e9	2.49214
$498$	0	0
$499$	−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
$501$	0	0
$502$	7.78511e9	2.74664
$503$	3.66315e9	1.28341	0.641707	−	0.766950i	$-0.278226\pi$
$503$	3.66315e9	1.28341	0.641707	+	0.766950i	$0.278226\pi$
$504$	0	0
$505$	3.76799e8	0.130194
$506$	−1.63449e9	−0.560861
$507$	0	0
$508$	3.43055e9	1.16102
$509$	2.18735e9	0.735202	0.367601	−	0.929984i	$-0.380179\pi$
$509$	2.18735e9	0.735202	0.367601	+	0.929984i	$0.380179\pi$
$510$	0	0
$511$	−3.47767e9	−1.15296
$512$	−3.02013e9	−0.994446
$513$	0	0
$514$	9.82999e9	3.19288
$515$	3.28627e8	0.106017
$516$	0	0
$517$	1.99166e9	0.633867
$518$	−8.43248e9	−2.66564
$519$	0	0
$520$	5.47308e8	0.170695
$521$	−2.54763e9	−0.789231	−0.394615	−	0.918846i	$-0.629122\pi$
$521$	−2.54763e9	−0.789231	−0.394615	+	0.918846i	$0.629122\pi$
$522$	0	0
$523$	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
$525$	0	0
$526$	1.03286e10	3.09451
$527$	−1.78771e9	−0.532060
$528$	0	0
$529$	−5.87336e8	−0.172501
$530$	1.95752e8	0.0571138
$531$	0	0
$532$	−1.24777e10	−3.59289
$533$	−3.82799e9	−1.09503
$534$	0	0
$535$	1.30235e8	0.0367697
$536$	−9.42500e9	−2.64365
$537$	0	0
$538$	−1.98867e9	−0.550585
$539$	−8.39207e8	−0.230839
$540$	0	0
$541$	−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
$543$	0	0
$544$	2.51005e9	0.668477
$545$	−4.06735e6	−0.00107628
$546$	0	0
$547$	−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
$549$	0	0
$550$	2.36215e9	0.605393
$551$	7.56123e9	1.92558
$552$	0	0
$553$	6.52079e9	1.63969
$554$	−5.53751e9	−1.38366
$555$	0	0
$556$	−2.86055e9	−0.705810
$557$	−3.38179e9	−0.829190	−0.414595	−	0.910006i	$-0.636077\pi$
$557$	−3.38179e9	−0.829190	−0.414595	+	0.910006i	$0.636077\pi$
$558$	0	0
$559$	−4.05222e9	−0.981186
$560$	−3.69750e8	−0.0889713
$561$	0	0
$562$	1.23844e9	0.294306
$563$	8.68855e7	0.0205196	0.0102598	−	0.999947i	$-0.496734\pi$
$563$	8.68855e7	0.0205196	0.0102598	+	0.999947i	$0.496734\pi$
$564$	0	0
$565$	4.78383e7	0.0111585
$566$	9.87385e9	2.28891
$567$	0	0
$568$	1.18983e10	2.72437
$569$	−3.35876e9	−0.764338	−0.382169	−	0.924092i	$-0.624823\pi$
$569$	−3.35876e9	−0.764338	−0.382169	+	0.924092i	$0.624823\pi$
$570$	0	0
$571$	−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
$573$	0	0
$574$	1.17212e10	2.58690
$575$	−4.07180e9	−0.893202
$576$	0	0
$577$	−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
$579$	0	0
$580$	1.44810e9	0.308177
$581$	6.72432e9	1.42243
$582$	0	0
$583$	4.42527e8	0.0924911
$584$	−6.06671e9	−1.26040
$585$	0	0
$586$	−4.76822e8	−0.0978846
$587$	9.21166e9	1.87977	0.939885	−	0.341491i	$-0.110932\pi$
$587$	9.21166e9	1.87977	0.939885	+	0.341491i	$0.110932\pi$
$588$	0	0
$589$	−3.18320e9	−0.641889
$590$	1.46995e8	0.0294659
$591$	0	0
$592$	−3.24561e9	−0.642940
$593$	5.16913e9	1.01795	0.508974	−	0.860782i	$-0.330025\pi$
$593$	5.16913e9	1.01795	0.508974	+	0.860782i	$0.330025\pi$
$594$	0	0
$595$	1.12529e9	0.219006
$596$	−1.05579e10	−2.04275
$597$	0	0
$598$	−7.27417e9	−1.39101
$599$	2.29633e9	0.436557	0.218278	−	0.975887i	$-0.429956\pi$
$599$	2.29633e9	0.436557	0.218278	+	0.975887i	$0.429956\pi$
$600$	0	0
$601$	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
$603$	0	0
$604$	−6.82061e9	−1.25949
$605$	6.34410e8	0.116473
$606$	0	0
$607$	−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
$609$	0	0
$610$	−1.88664e9	−0.336539
$611$	8.86373e9	1.57207
$612$	0	0
$613$	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
$615$	0	0
$616$	−3.78849e9	−0.653031
$617$	−3.68960e9	−0.632385	−0.316192	−	0.948695i	$-0.602405\pi$
$617$	−3.68960e9	−0.632385	−0.316192	+	0.948695i	$0.602405\pi$
$618$	0	0
$619$	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
$621$	0	0
$622$	2.63554e9	0.439140
$623$	6.46892e9	1.07182
$624$	0	0
$625$	5.77403e9	0.946018
$626$	−6.70599e9	−1.09258
$627$	0	0
$628$	1.74031e10	2.80393
$629$	9.87766e9	1.58262
$630$	0	0
$631$	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
$633$	0	0
$634$	592969.	9.24101e−5 0
$635$	−5.50884e8	−0.0853792
$636$	0	0
$637$	−3.73483e9	−0.572510
$638$	5.06287e9	0.771835
$639$	0	0
$640$	1.63342e9	0.246303
$641$	8.08641e9	1.21270	0.606349	−	0.795199i	$-0.292633\pi$
$641$	8.08641e9	1.21270	0.606349	+	0.795199i	$0.292633\pi$
$642$	0	0
$643$	−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
$645$	0	0
$646$	2.26047e10	3.29902
$647$	−3.25260e9	−0.472135	−0.236067	−	0.971737i	$-0.575859\pi$
$647$	−3.25260e9	−0.472135	−0.236067	+	0.971737i	$0.575859\pi$
$648$	0	0
$649$	3.32304e8	0.0477177
$650$	1.05126e10	1.50145
$651$	0	0
$652$	−1.20645e10	−1.70469
$653$	1.64182e9	0.230744	0.115372	−	0.993322i	$-0.463194\pi$
$653$	1.64182e9	0.230744	0.115372	+	0.993322i	$0.463194\pi$
$654$	0	0
$655$	1.29526e9	0.180099
$656$	4.51142e9	0.623950
$657$	0	0
$658$	−2.71404e10	−3.71387
$659$	−4.63845e9	−0.631355	−0.315678	−	0.948866i	$-0.602232\pi$
$659$	−4.63845e9	−0.631355	−0.315678	+	0.948866i	$0.602232\pi$
$660$	0	0
$661$	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
$663$	0	0
$664$	1.17304e10	1.55498
$665$	2.00369e9	0.264214
$666$	0	0
$667$	−8.72725e9	−1.13877
$668$	−7.11138e9	−0.923073
$669$	0	0
$670$	3.33773e9	0.428735
$671$	−4.26505e9	−0.544998
$672$	0	0
$673$	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
$675$	0	0
$676$	−2.55200e9	−0.317737
$677$	−1.90990e8	−0.0236565	−0.0118283	−	0.999930i	$-0.503765\pi$
$677$	−1.90990e8	−0.0236565	−0.0118283	+	0.999930i	$0.503765\pi$
$678$	0	0
$679$	4.27515e9	0.524091
$680$	1.96305e9	0.239414
$681$	0	0
$682$	−2.13141e9	−0.257290
$683$	3.13279e9	0.376235	0.188117	−	0.982147i	$-0.439761\pi$
$683$	3.13279e9	0.376235	0.188117	+	0.982147i	$0.439761\pi$
$684$	0	0
$685$	−1.66019e9	−0.197352
$686$	−6.72207e9	−0.795003
$687$	0	0
$688$	4.77567e9	0.559081
$689$	1.96944e9	0.229390
$690$	0	0
$691$	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
$693$	0	0
$694$	−1.50879e10	−1.71344
$695$	4.59352e8	0.0519038
$696$	0	0
$697$	−1.37300e10	−1.53587
$698$	6.38723e8	0.0710917
$699$	0	0
$700$	−2.08135e10	−2.29351
$701$	−5.98660e9	−0.656398	−0.328199	−	0.944609i	$-0.606442\pi$
$701$	−5.98660e9	−0.656398	−0.328199	+	0.944609i	$0.606442\pi$
$702$	0	0
$703$	1.75881e10	1.90931
$704$	4.75020e9	0.513107
$705$	0	0
$706$	1.60828e10	1.72007
$707$	−1.16077e10	−1.23531
$708$	0	0
$709$	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
$711$	0	0
$712$	1.12849e10	1.17170
$713$	3.67408e9	0.379608
$714$	0	0
$715$	−4.38162e8	−0.0448295
$716$	−1.60523e10	−1.63433
$717$	0	0
$718$	1.74094e10	1.75528
$719$	1.53454e10	1.53967	0.769836	−	0.638242i	$-0.220338\pi$
$719$	1.53454e10	1.53967	0.769836	+	0.638242i	$0.220338\pi$
$720$	0	0
$721$	−1.01237e10	−1.00592
$722$	2.32382e10	2.29785
$723$	0	0
$724$	−7.67113e9	−0.751232
$725$	1.26125e10	1.22919
$726$	0	0
$727$	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
$729$	0	0
$730$	2.14844e9	0.204406
$731$	−1.45342e10	−1.37620
$732$	0	0
$733$	−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
$735$	0	0
$736$	−5.15862e9	−0.476937
$737$	7.54544e9	0.694302
$738$	0	0
$739$	−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
$741$	0	0
$742$	−6.03035e9	−0.541912
$743$	−4.65023e9	−0.415924	−0.207962	−	0.978137i	$-0.566683\pi$
$743$	−4.65023e9	−0.415924	−0.207962	+	0.978137i	$0.566683\pi$
$744$	0	0
$745$	1.69541e9	0.150220
$746$	2.54963e10	2.24850
$747$	0	0
$748$	9.78674e9	0.855032
$749$	−4.01204e9	−0.348882
$750$	0	0
$751$	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
$753$	0	0
$754$	2.25319e10	1.91425
$755$	1.09527e9	0.0926201
$756$	0	0
$757$	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
$759$	0	0
$760$	3.49540e9	0.288835
$761$	−5.11102e9	−0.420398	−0.210199	−	0.977659i	$-0.567411\pi$
$761$	−5.11102e9	−0.420398	−0.210199	+	0.977659i	$0.567411\pi$
$762$	0	0
$763$	1.25299e8	0.0102120
$764$	−6.63798e9	−0.538529
$765$	0	0
$766$	1.37131e10	1.10239
$767$	1.47889e9	0.118346
$768$	0	0
$769$	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
$771$	0	0
$772$	−2.64312e10	−2.06755
$773$	−2.05915e10	−1.60346	−0.801732	−	0.597683i	$-0.796088\pi$
$773$	−2.05915e10	−1.60346	−0.801732	+	0.597683i	$0.796088\pi$
$774$	0	0
$775$	−5.30974e9	−0.409748
$776$	7.45790e9	0.572929
$777$	0	0
$778$	1.11326e10	0.847552
$779$	−2.44476e10	−1.85291
$780$	0	0
$781$	−9.52551e9	−0.715500
$782$	−2.60905e10	−1.95101
$783$	0	0
$784$	4.40162e9	0.326217
$785$	−2.79462e9	−0.206196
$786$	0	0
$787$	−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
$789$	0	0
$790$	−4.02842e9	−0.290697
$791$	−1.47371e9	−0.105875
$792$	0	0
$793$	−1.89813e10	−1.35167
$794$	1.51965e10	1.07739
$795$	0	0
$796$	3.47425e10	2.44155
$797$	−2.36801e10	−1.65684	−0.828418	−	0.560110i	$-0.810759\pi$
$797$	−2.36801e10	−1.65684	−0.828418	+	0.560110i	$0.810759\pi$
$798$	0	0
$799$	3.17919e10	2.20497
$800$	7.45518e9	0.514805
$801$	0	0
$802$	2.04362e10	1.39891
$803$	4.85687e9	0.331019
$804$	0	0
$805$	−2.31268e9	−0.156254
$806$	−9.48570e9	−0.638112
$807$	0	0
$808$	−2.02493e10	−1.35043
$809$	−1.22571e10	−0.813897	−0.406949	−	0.913451i	$-0.633407\pi$
$809$	−1.22571e10	−0.813897	−0.406949	+	0.913451i	$0.633407\pi$
$810$	0	0
$811$	−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
$813$	0	0
$814$	1.17767e10	0.765312
$815$	1.93735e9	0.125359
$816$	0	0
$817$	−2.58796e10	−1.66028
$818$	−1.14427e10	−0.730960
$819$	0	0
$820$	−4.68212e9	−0.296547
$821$	8.10180e9	0.510952	0.255476	−	0.966815i	$-0.417768\pi$
$821$	8.10180e9	0.510952	0.255476	+	0.966815i	$0.417768\pi$
$822$	0	0
$823$	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
$825$	0	0
$826$	−4.52833e9	−0.279581
$827$	2.72238e10	1.67371	0.836854	−	0.547427i	$-0.184393\pi$
$827$	2.72238e10	1.67371	0.836854	+	0.547427i	$0.184393\pi$
$828$	0	0
$829$	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
$831$	0	0
$832$	2.11404e10	1.27257
$833$	−1.33959e10	−0.802995
$834$	0	0
$835$	1.14196e9	0.0678809
$836$	1.74263e10	1.03153
$837$	0	0
$838$	−2.71447e10	−1.59342
$839$	2.14886e10	1.25615	0.628073	−	0.778154i	$-0.283844\pi$
$839$	2.14886e10	1.25615	0.628073	+	0.778154i	$0.283844\pi$
$840$	0	0
$841$	9.78301e9	0.567135
$842$	−1.23930e10	−0.715455
$843$	0	0
$844$	−6.26297e10	−3.58577
$845$	4.09805e8	0.0233657
$846$	0	0
$847$	−1.95437e10	−1.10513
$848$	−2.32105e9	−0.130707
$849$	0	0
$850$	3.77058e10	2.10592
$851$	−2.03004e10	−1.12915
$852$	0	0
$853$	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
$855$	0	0
$856$	−6.99891e9	−0.381393
$857$	1.97164e10	1.07003	0.535015	−	0.844843i	$-0.320306\pi$
$857$	1.97164e10	1.07003	0.535015	+	0.844843i	$0.320306\pi$
$858$	0	0
$859$	−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
$861$	0	0
$862$	−3.33945e10	−1.77582
$863$	−1.20569e10	−0.638552	−0.319276	−	0.947662i	$-0.603440\pi$
$863$	−1.20569e10	−0.638552	−0.319276	+	0.947662i	$0.603440\pi$
$864$	0	0
$865$	−3.47340e9	−0.182473
$866$	1.77997e10	0.931321
$867$	0	0
$868$	1.87804e10	0.974735
$869$	−9.10687e9	−0.470760
$870$	0	0
$871$	3.35804e10	1.72196
$872$	2.18581e8	0.0111636
$873$	0	0
$874$	−4.64568e10	−2.35374
$875$	6.74615e9	0.340430
$876$	0	0
$877$	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
$879$	0	0
$880$	5.16389e8	0.0255439
$881$	2.68011e10	1.32050	0.660248	−	0.751048i	$-0.270451\pi$
$881$	2.68011e10	1.32050	0.660248	+	0.751048i	$0.270451\pi$
$882$	0	0
$883$	−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
$885$	0	0
$886$	−1.56440e10	−0.755667
$887$	−1.26881e10	−0.610468	−0.305234	−	0.952277i	$-0.598735\pi$
$887$	−1.26881e10	−0.610468	−0.305234	+	0.952277i	$0.598735\pi$
$888$	0	0
$889$	1.69706e10	0.810103
$890$	−3.99638e9	−0.190021
$891$	0	0
$892$	−6.14148e10	−2.89732
$893$	5.66085e10	2.66012
$894$	0	0
$895$	2.57770e9	0.120185
$896$	−5.03193e10	−2.33699
$897$	0	0
$898$	−2.99482e10	−1.38008
$899$	−1.13805e10	−0.522401
$900$	0	0
$901$	7.06385e9	0.321740
$902$	−1.63697e10	−0.742708
$903$	0	0
$904$	−2.57085e9	−0.115741
$905$	1.23184e9	0.0552441
$906$	0	0
$907$	−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
$909$	0	0
$910$	5.97087e9	0.262659
$911$	−2.02421e10	−0.887035	−0.443517	−	0.896266i	$-0.646270\pi$
$911$	−2.02421e10	−0.887035	−0.443517	+	0.896266i	$0.646270\pi$
$912$	0	0
$913$	−9.39112e9	−0.408385
$914$	4.25650e10	1.84392
$915$	0	0
$916$	−2.86397e10	−1.23122
$917$	−3.99017e10	−1.70883
$918$	0	0
$919$	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
$921$	0	0
$922$	−6.76791e10	−2.84378
$923$	−4.23926e10	−1.77453
$924$	0	0
$925$	2.93379e10	1.21880
$926$	−6.64002e10	−2.74809
$927$	0	0
$928$	1.59790e10	0.656342
$929$	−3.99605e10	−1.63522	−0.817609	−	0.575774i	$-0.804701\pi$
$929$	−3.99605e10	−1.63522	−0.817609	+	0.575774i	$0.804701\pi$
$930$	0	0
$931$	−2.38526e10	−0.968751
$932$	2.75112e10	1.11315
$933$	0	0
$934$	1.79781e10	0.721988
$935$	−1.57157e9	−0.0628773
$936$	0	0
$937$	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
$939$	0	0
$940$	1.08415e10	0.425736
$941$	2.26436e10	0.885896	0.442948	−	0.896547i	$-0.353933\pi$
$941$	2.26436e10	0.885896	0.442948	+	0.896547i	$0.353933\pi$
$942$	0	0
$943$	2.82177e10	1.09580
$944$	−1.74293e9	−0.0674337
$945$	0	0
$946$	−1.73286e10	−0.665493
$947$	−7.27658e9	−0.278421	−0.139211	−	0.990263i	$-0.544457\pi$
$947$	−7.27658e9	−0.278421	−0.139211	+	0.990263i	$0.544457\pi$
$948$	0	0
$949$	2.16152e10	0.820969
$950$	6.71388e10	2.54063
$951$	0	0
$952$	−6.04738e10	−2.27163
$953$	−4.70442e10	−1.76068	−0.880340	−	0.474343i	$-0.842686\pi$
$953$	−4.70442e10	−1.76068	−0.880340	+	0.474343i	$0.842686\pi$
$954$	0	0
$955$	1.06594e9	0.0396023
$956$	−2.39161e10	−0.885293
$957$	0	0
$958$	−5.91631e9	−0.217406
$959$	5.11439e10	1.87253
$960$	0	0
$961$	−2.27215e10	−0.825859
$962$	5.24114e10	1.89807
$963$	0	0
$964$	−2.37882e10	−0.855249
$965$	4.24437e9	0.152043
$966$	0	0
$967$	−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
$969$	0	0
$970$	−2.64111e9	−0.0929149
$971$	−4.43321e10	−1.55400	−0.777001	−	0.629500i	$-0.783260\pi$
$971$	−4.43321e10	−1.55400	−0.777001	+	0.629500i	$0.783260\pi$
$972$	0	0
$973$	−1.41508e10	−0.492478
$974$	−5.21179e10	−1.80730
$975$	0	0
$976$	2.23701e10	0.770181
$977$	−4.78091e10	−1.64013	−0.820067	−	0.572268i	$-0.806064\pi$
$977$	−4.78091e10	−1.64013	−0.820067	+	0.572268i	$0.806064\pi$
$978$	0	0
$979$	−9.03443e9	−0.307724
$980$	−4.56817e9	−0.155043
$981$	0	0
$982$	6.68045e9	0.225121
$983$	3.22996e10	1.08458	0.542288	−	0.840193i	$-0.317558\pi$
$983$	3.22996e10	1.08458	0.542288	+	0.840193i	$0.317558\pi$
$984$	0	0
$985$	−7.48003e8	−0.0249389
$986$	8.08162e10	2.68491
$987$	0	0
$988$	7.75543e10	2.55833
$989$	2.98705e10	0.981873
$990$	0	0
$991$	−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
$993$	0	0
$994$	1.29805e11	4.19216
$995$	−5.57901e9	−0.179546
$996$	0	0
$997$	−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
$999$	0	0

Twists

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

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

\(f(q)\)	\(=\)	\(q+19.0314 q^{2} +234.195 q^{4} -37.6075 q^{5} +1158.54 q^{7} +2021.05 q^{8} +O(q^{10})\)	\(q+19.0314 q^{2} +234.195 q^{4} -37.6075 q^{5} +1158.54 q^{7} +2021.05 q^{8} -715.724 q^{10} -1618.00 q^{11} -7200.81 q^{13} +22048.6 q^{14} +8486.39 q^{16} -25827.4 q^{17} -45988.2 q^{19} -8807.49 q^{20} -30792.9 q^{22} +53080.0 q^{23} -76710.7 q^{25} -137042. q^{26} +271324. q^{28} -164417. q^{29} +69217.7 q^{31} -97185.7 q^{32} -491532. q^{34} -43569.7 q^{35} -382449. q^{37} -875221. q^{38} -76006.5 q^{40} +531606. q^{41} +562745. q^{43} -378929. q^{44} +1.01019e6 q^{46} -1.23094e6 q^{47} +518668. q^{49} -1.45991e6 q^{50} -1.68639e6 q^{52} -273502. q^{53} +60849.1 q^{55} +2.34146e6 q^{56} -3.12909e6 q^{58} -205379. q^{59} +2.63599e6 q^{61} +1.31731e6 q^{62} -2.93584e6 q^{64} +270804. q^{65} -4.66343e6 q^{67} -6.04865e6 q^{68} -829194. q^{70} +5.88720e6 q^{71} -3.00177e6 q^{73} -7.27855e6 q^{74} -1.07702e7 q^{76} -1.87452e6 q^{77} +5.62846e6 q^{79} -319152. q^{80} +1.01172e7 q^{82} +5.80414e6 q^{83} +971304. q^{85} +1.07098e7 q^{86} -3.27006e6 q^{88} +5.58369e6 q^{89} -8.34241e6 q^{91} +1.24311e7 q^{92} -2.34265e7 q^{94} +1.72950e6 q^{95} +3.69012e6 q^{97} +9.87100e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16q + 6q^{2} + 974q^{4} + 68q^{5} - 2343q^{7} - 819q^{8} + O(q^{10}) \)	\( 16q + 6q^{2} + 974q^{4} + 68q^{5} - 2343q^{7} - 819q^{8} - 3479q^{10} - 898q^{11} - 8172q^{13} + 13315q^{14} + 3138q^{16} + 44985q^{17} - 40137q^{19} - 130657q^{20} + 109394q^{22} + 2833q^{23} + 285746q^{25} + 129420q^{26} + 112890q^{28} - 144375q^{29} - 141759q^{31} + 36224q^{32} - 341332q^{34} + 78859q^{35} - 297971q^{37} - 329075q^{38} - 203048q^{40} - 659077q^{41} - 1431608q^{43} - 254916q^{44} + 873113q^{46} + 1574073q^{47} + 1893545q^{49} - 302533q^{50} - 4972548q^{52} - 587736q^{53} - 4624036q^{55} + 5798506q^{56} - 6991380q^{58} - 3286064q^{59} - 6117131q^{61} + 11570258q^{62} - 19063011q^{64} + 5335514q^{65} - 16518710q^{67} + 17284669q^{68} - 39189486q^{70} + 10882582q^{71} - 21097441q^{73} + 16717030q^{74} - 40864952q^{76} + 3404601q^{77} - 3784458q^{79} + 27466195q^{80} - 24990117q^{82} + 1951425q^{83} - 23238675q^{85} + 35910572q^{86} - 27843055q^{88} - 10499443q^{89} + 699217q^{91} + 20062766q^{92} - 59358988q^{94} + 29236333q^{95} - 25158976q^{97} - 2120460q^{98} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

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