LMFDB - Embedded newform 91.8.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [91,8,Mod(1,91)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(91, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 8, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("91.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 91 = 7 \cdot 13 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	91.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$28.4270373191$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	91.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+22.0000 q^{2} +21.0000 q^{3} +356.000 q^{4} +140.000 q^{5} +462.000 q^{6} -343.000 q^{7} +5016.00 q^{8} -1746.00 q^{9} +O(q^{10})$

\(q+22.0000 q^{2} +21.0000 q^{3} +356.000 q^{4} +140.000 q^{5} +462.000 q^{6} -343.000 q^{7} +5016.00 q^{8} -1746.00 q^{9} +3080.00 q^{10} +5051.00 q^{11} +7476.00 q^{12} -2197.00 q^{13} -7546.00 q^{14} +2940.00 q^{15} +64784.0 q^{16} +27384.0 q^{17} -38412.0 q^{18} -32690.0 q^{19} +49840.0 q^{20} -7203.00 q^{21} +111122. q^{22} -23085.0 q^{23} +105336. q^{24} -58525.0 q^{25} -48334.0 q^{26} -82593.0 q^{27} -122108. q^{28} -14068.0 q^{29} +64680.0 q^{30} -203007. q^{31} +783200. q^{32} +106071. q^{33} +602448. q^{34} -48020.0 q^{35} -621576. q^{36} +544041. q^{37} -719180. q^{38} -46137.0 q^{39} +702240. q^{40} -352079. q^{41} -158466. q^{42} +340412. q^{43} +1.79816e6 q^{44} -244440. q^{45} -507870. q^{46} +406329. q^{47} +1.36046e6 q^{48} +117649. q^{49} -1.28755e6 q^{50} +575064. q^{51} -782132. q^{52} -1.90968e6 q^{53} -1.81705e6 q^{54} +707140. q^{55} -1.72049e6 q^{56} -686490. q^{57} -309496. q^{58} -2.86721e6 q^{59} +1.04664e6 q^{60} -216419. q^{61} -4.46615e6 q^{62} +598878. q^{63} +8.93805e6 q^{64} -307580. q^{65} +2.33356e6 q^{66} +2.53804e6 q^{67} +9.74870e6 q^{68} -484785. q^{69} -1.05644e6 q^{70} -2.07187e6 q^{71} -8.75794e6 q^{72} +185913. q^{73} +1.19689e7 q^{74} -1.22902e6 q^{75} -1.16376e7 q^{76} -1.73249e6 q^{77} -1.01501e6 q^{78} -954631. q^{79} +9.06976e6 q^{80} +2.08405e6 q^{81} -7.74574e6 q^{82} +5.64967e6 q^{83} -2.56427e6 q^{84} +3.83376e6 q^{85} +7.48906e6 q^{86} -295428. q^{87} +2.53358e7 q^{88} -4.67383e6 q^{89} -5.37768e6 q^{90} +753571. q^{91} -8.21826e6 q^{92} -4.26315e6 q^{93} +8.93924e6 q^{94} -4.57660e6 q^{95} +1.64472e7 q^{96} -1.36866e7 q^{97} +2.58828e6 q^{98} -8.81905e6 q^{99} +O(q^{100})\)

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 $

$2$	22.0000	1.94454	0.972272	−	0.233854i	$-0.0751336\pi$
$2$	22.0000	1.94454	0.972272	+	0.233854i	$0.0751336\pi$
$3$	21.0000	0.449050	0.224525	−	0.974468i	$-0.427917\pi$
$3$	21.0000	0.449050	0.224525	+	0.974468i	$0.427917\pi$
$4$	356.000	2.78125
$5$	140.000	0.500879	0.250440	−	0.968132i	$-0.419425\pi$
$5$	140.000	0.500879	0.250440	+	0.968132i	$0.419425\pi$
$6$	462.000	0.873198
$7$	−343.000	−0.377964
$7$	−343.000	−0.377964
$8$	5016.00	3.46372
$9$	−1746.00	−0.798354
$10$	3080.00	0.973982
$11$	5051.00	1.14420	0.572101	−	0.820183i	$-0.306128\pi$
$11$	5051.00	1.14420	0.572101	+	0.820183i	$0.306128\pi$
$12$	7476.00	1.24892
$13$	−2197.00	−0.277350
$13$	−2197.00	−0.277350
$14$	−7546.00	−0.734968
$15$	2940.00	0.224920
$16$	64784.0	3.95410
$17$	27384.0	1.35184	0.675921	−	0.736974i	$-0.263746\pi$
$17$	27384.0	1.35184	0.675921	+	0.736974i	$0.263746\pi$
$18$	−38412.0	−1.55243
$19$	−32690.0	−1.09340	−0.546698	−	0.837330i	$-0.684115\pi$
$19$	−32690.0	−1.09340	−0.546698	+	0.837330i	$0.684115\pi$
$20$	49840.0	1.39307
$21$	−7203.00	−0.169725
$22$	111122.	2.22495
$23$	−23085.0	−0.395624	−0.197812	−	0.980240i	$-0.563384\pi$
$23$	−23085.0	−0.395624	−0.197812	+	0.980240i	$0.563384\pi$
$24$	105336.	1.55538
$25$	−58525.0	−0.749120
$26$	−48334.0	−0.539319
$27$	−82593.0	−0.807551
$28$	−122108.	−1.05121
$29$	−14068.0	−0.107112	−0.0535562	−	0.998565i	$-0.517056\pi$
$29$	−14068.0	−0.107112	−0.0535562	+	0.998565i	$0.517056\pi$
$30$	64680.0	0.437367
$31$	−203007.	−1.22390	−0.611949	−	0.790897i	$-0.709614\pi$
$31$	−203007.	−1.22390	−0.611949	+	0.790897i	$0.709614\pi$
$32$	783200.	4.22520
$33$	106071.	0.513804
$34$	602448.	2.62871
$35$	−48020.0	−0.189315
$36$	−621576.	−2.22042
$37$	544041.	1.76574	0.882868	−	0.469622i	$-0.155610\pi$
$37$	544041.	1.76574	0.882868	+	0.469622i	$0.155610\pi$
$38$	−719180.	−2.12616
$39$	−46137.0	−0.124544
$40$	702240.	1.73490
$41$	−352079.	−0.797805	−0.398902	−	0.916993i	$-0.630609\pi$
$41$	−352079.	−0.797805	−0.398902	+	0.916993i	$0.630609\pi$
$42$	−158466.	−0.330038
$43$	340412.	0.652928	0.326464	−	0.945210i	$-0.394143\pi$
$43$	340412.	0.652928	0.326464	+	0.945210i	$0.394143\pi$
$44$	1.79816e6	3.18231
$45$	−244440.	−0.399879
$46$	−507870.	−0.769308
$47$	406329.	0.570867	0.285434	−	0.958398i	$-0.407862\pi$
$47$	406329.	0.570867	0.285434	+	0.958398i	$0.407862\pi$
$48$	1.36046e6	1.77559
$49$	117649.	0.142857
$50$	−1.28755e6	−1.45670
$51$	575064.	0.607045
$52$	−782132.	−0.771380
$53$	−1.90968e6	−1.76196	−0.880978	−	0.473157i	$-0.843114\pi$
$53$	−1.90968e6	−1.76196	−0.880978	+	0.473157i	$0.843114\pi$
$54$	−1.81705e6	−1.57032
$55$	707140.	0.573107
$56$	−1.72049e6	−1.30916
$57$	−686490.	−0.490990
$58$	−309496.	−0.208285
$59$	−2.86721e6	−1.81752	−0.908758	−	0.417324i	$-0.862968\pi$
$59$	−2.86721e6	−1.81752	−0.908758	+	0.417324i	$0.862968\pi$
$60$	1.04664e6	0.625559
$61$	−216419.	−0.122079	−0.0610395	−	0.998135i	$-0.519442\pi$
$61$	−216419.	−0.122079	−0.0610395	+	0.998135i	$0.519442\pi$
$62$	−4.46615e6	−2.37992
$63$	598878.	0.301749
$64$	8.93805e6	4.26199
$65$	−307580.	−0.138919
$66$	2.33356e6	0.999115
$67$	2.53804e6	1.03095	0.515474	−	0.856905i	$-0.327616\pi$
$67$	2.53804e6	1.03095	0.515474	+	0.856905i	$0.327616\pi$
$68$	9.74870e6	3.75981
$69$	−484785.	−0.177655
$70$	−1.05644e6	−0.368130
$71$	−2.07187e6	−0.687003	−0.343502	−	0.939152i	$-0.611613\pi$
$71$	−2.07187e6	−0.687003	−0.343502	+	0.939152i	$0.611613\pi$
$72$	−8.75794e6	−2.76527
$73$	185913.	0.0559345	0.0279672	−	0.999609i	$-0.491097\pi$
$73$	185913.	0.0559345	0.0279672	+	0.999609i	$0.491097\pi$
$74$	1.19689e7	3.43355
$75$	−1.22902e6	−0.336392
$76$	−1.16376e7	−3.04101
$77$	−1.73249e6	−0.432468
$78$	−1.01501e6	−0.242181
$79$	−954631.	−0.217842	−0.108921	−	0.994050i	$-0.534739\pi$
$79$	−954631.	−0.217842	−0.108921	+	0.994050i	$0.534739\pi$
$80$	9.06976e6	1.98053
$81$	2.08405e6	0.435723
$82$	−7.74574e6	−1.55137
$83$	5.64967e6	1.08455	0.542275	−	0.840201i	$-0.317563\pi$
$83$	5.64967e6	1.08455	0.542275	+	0.840201i	$0.317563\pi$
$84$	−2.56427e6	−0.472048
$85$	3.83376e6	0.677109
$86$	7.48906e6	1.26965
$87$	−295428.	−0.0480988
$88$	2.53358e7	3.96320
$89$	−4.67383e6	−0.702761	−0.351381	−	0.936233i	$-0.614288\pi$
$89$	−4.67383e6	−0.702761	−0.351381	+	0.936233i	$0.614288\pi$
$90$	−5.37768e6	−0.777582
$91$	753571.	0.104828
$92$	−8.21826e6	−1.10033
$93$	−4.26315e6	−0.549591
$94$	8.93924e6	1.11008
$95$	−4.57660e6	−0.547659
$96$	1.64472e7	1.89733
$97$	−1.36866e7	−1.52264	−0.761318	−	0.648379i	$-0.775447\pi$
$97$	−1.36866e7	−1.52264	−0.761318	+	0.648379i	$0.775447\pi$
$98$	2.58828e6	0.277792
$99$	−8.81905e6	−0.913479
$100$	−2.08349e7	−2.08349
$101$	−8.71886e6	−0.842044	−0.421022	−	0.907050i	$-0.638328\pi$
$101$	−8.71886e6	−0.842044	−0.421022	+	0.907050i	$0.638328\pi$
$102$	1.26514e7	1.18042
$103$	4.20925e6	0.379555	0.189778	−	0.981827i	$-0.439223\pi$
$103$	4.20925e6	0.379555	0.189778	+	0.981827i	$0.439223\pi$
$104$	−1.10202e7	−0.960663
$105$	−1.00842e6	−0.0850117
$106$	−4.20130e7	−3.42620
$107$	7.89777e6	0.623249	0.311624	−	0.950205i	$-0.399127\pi$
$107$	7.89777e6	0.623249	0.311624	+	0.950205i	$0.399127\pi$
$108$	−2.94031e7	−2.24600
$109$	−5.57069e6	−0.412018	−0.206009	−	0.978550i	$-0.566048\pi$
$109$	−5.57069e6	−0.412018	−0.206009	+	0.978550i	$0.566048\pi$
$110$	1.55571e7	1.11443
$111$	1.14249e7	0.792904
$112$	−2.22209e7	−1.49451
$113$	1.59538e7	1.04014	0.520069	−	0.854124i	$-0.325906\pi$
$113$	1.59538e7	1.04014	0.520069	+	0.854124i	$0.325906\pi$
$114$	−1.51028e7	−0.954751
$115$	−3.23190e6	−0.198160
$116$	−5.00821e6	−0.297906
$117$	3.83596e6	0.221424
$118$	−6.30787e7	−3.53424
$119$	−9.39271e6	−0.510948
$120$	1.47470e7	0.779059
$121$	6.02543e6	0.309200
$122$	−4.76122e6	−0.237388
$123$	−7.39366e6	−0.358254
$124$	−7.22705e7	−3.40397
$125$	−1.91310e7	−0.876098
$126$	1.31753e7	0.586765
$127$	2.62910e7	1.13892	0.569460	−	0.822019i	$-0.307152\pi$
$127$	2.62910e7	1.13892	0.569460	+	0.822019i	$0.307152\pi$
$128$	9.63875e7	4.06243
$129$	7.14865e6	0.293197
$130$	−6.76676e6	−0.270134
$131$	−4.04186e7	−1.57084	−0.785419	−	0.618965i	$-0.787552\pi$
$131$	−4.04186e7	−1.57084	−0.785419	+	0.618965i	$0.787552\pi$
$132$	3.77613e7	1.42902
$133$	1.12127e7	0.413265
$134$	5.58369e7	2.00473
$135$	−1.15630e7	−0.404486
$136$	1.37358e8	4.68240
$137$	4.50126e7	1.49559	0.747794	−	0.663931i	$-0.231113\pi$
$137$	4.50126e7	1.49559	0.747794	+	0.663931i	$0.231113\pi$
$138$	−1.06653e7	−0.345458
$139$	−4.22176e6	−0.133334	−0.0666671	−	0.997775i	$-0.521237\pi$
$139$	−4.22176e6	−0.133334	−0.0666671	+	0.997775i	$0.521237\pi$
$140$	−1.70951e7	−0.526531
$141$	8.53291e6	0.256348
$142$	−4.55812e7	−1.33591
$143$	−1.10970e7	−0.317345
$144$	−1.13113e8	−3.15677
$145$	−1.96952e6	−0.0536503
$146$	4.09009e6	0.108767
$147$	2.47063e6	0.0641500
$148$	1.93679e8	4.91095
$149$	6.22867e7	1.54257	0.771283	−	0.636493i	$-0.219615\pi$
$149$	6.22867e7	1.54257	0.771283	+	0.636493i	$0.219615\pi$
$150$	−2.70386e7	−0.654130
$151$	−6.86933e7	−1.62366	−0.811830	−	0.583894i	$-0.801529\pi$
$151$	−6.86933e7	−1.62366	−0.811830	+	0.583894i	$0.801529\pi$
$152$	−1.63973e8	−3.78721
$153$	−4.78125e7	−1.07925
$154$	−3.81148e7	−0.840953
$155$	−2.84210e7	−0.613025
$156$	−1.64248e7	−0.346388
$157$	9.19557e7	1.89640	0.948200	−	0.317675i	$-0.102902\pi$
$157$	9.19557e7	1.89640	0.948200	+	0.317675i	$0.102902\pi$
$158$	−2.10019e7	−0.423602
$159$	−4.01033e7	−0.791207
$160$	1.09648e8	2.11632
$161$	7.91815e6	0.149532
$162$	4.58491e7	0.847282
$163$	4.80673e7	0.869347	0.434673	−	0.900588i	$-0.356864\pi$
$163$	4.80673e7	0.869347	0.434673	+	0.900588i	$0.356864\pi$
$164$	−1.25340e8	−2.21889
$165$	1.48499e7	0.257354
$166$	1.24293e8	2.10896
$167$	3.12486e7	0.519185	0.259593	−	0.965718i	$-0.416412\pi$
$167$	3.12486e7	0.519185	0.259593	+	0.965718i	$0.416412\pi$
$168$	−3.61302e7	−0.587880
$169$	4.82681e6	0.0769231
$170$	8.43427e7	1.31667
$171$	5.70767e7	0.872917
$172$	1.21187e8	1.81596
$173$	−2.11198e7	−0.310119	−0.155060	−	0.987905i	$-0.549557\pi$
$173$	−2.11198e7	−0.310119	−0.155060	+	0.987905i	$0.549557\pi$
$174$	−6.49942e6	−0.0935302
$175$	2.00741e7	0.283141
$176$	3.27224e8	4.52429
$177$	−6.02115e7	−0.816156
$178$	−1.02824e8	−1.36655
$179$	2.93120e7	0.381997	0.190999	−	0.981590i	$-0.438827\pi$
$179$	2.93120e7	0.381997	0.190999	+	0.981590i	$0.438827\pi$
$180$	−8.70206e7	−1.11216
$181$	1.41828e8	1.77782	0.888910	−	0.458083i	$-0.151464\pi$
$181$	1.41828e8	1.77782	0.888910	+	0.458083i	$0.151464\pi$
$182$	1.65786e7	0.203844
$183$	−4.54480e6	−0.0548196
$184$	−1.15794e8	−1.37033
$185$	7.61657e7	0.884420
$186$	−9.37892e7	−1.06870
$187$	1.38317e8	1.54678
$188$	1.44653e8	1.58773
$189$	2.83294e7	0.305226
$190$	−1.00685e8	−1.06495
$191$	−2.65213e7	−0.275409	−0.137705	−	0.990473i	$-0.543972\pi$
$191$	−2.65213e7	−0.275409	−0.137705	+	0.990473i	$0.543972\pi$
$192$	1.87699e8	1.91385
$193$	−6.90595e7	−0.691469	−0.345734	−	0.938332i	$-0.612370\pi$
$193$	−6.90595e7	−0.691469	−0.345734	+	0.938332i	$0.612370\pi$
$194$	−3.01106e8	−2.96083
$195$	−6.45918e6	−0.0623816
$196$	4.18830e7	0.397321
$197$	1.71612e7	0.159925	0.0799623	−	0.996798i	$-0.474520\pi$
$197$	1.71612e7	0.159925	0.0799623	+	0.996798i	$0.474520\pi$
$198$	−1.94019e8	−1.77630
$199$	1.18181e8	1.06307	0.531536	−	0.847036i	$-0.321615\pi$
$199$	1.18181e8	1.06307	0.531536	+	0.847036i	$0.321615\pi$
$200$	−2.93561e8	−2.59474
$201$	5.32989e7	0.462948
$202$	−1.91815e8	−1.63739
$203$	4.82532e6	0.0404846
$204$	2.04723e8	1.68834
$205$	−4.92911e7	−0.399604
$206$	9.26036e7	0.738061
$207$	4.03064e7	0.315848
$208$	−1.42330e8	−1.09667
$209$	−1.65117e8	−1.25107
$210$	−2.21852e7	−0.165309
$211$	3.96598e7	0.290645	0.145322	−	0.989384i	$-0.453578\pi$
$211$	3.96598e7	0.290645	0.145322	+	0.989384i	$0.453578\pi$
$212$	−6.79846e8	−4.90044
$213$	−4.35093e7	−0.308499
$214$	1.73751e8	1.21193
$215$	4.76577e7	0.327038
$216$	−4.14286e8	−2.79713
$217$	6.96314e7	0.462590
$218$	−1.22555e8	−0.801188
$219$	3.90417e6	0.0251174
$220$	2.51742e8	1.59395
$221$	−6.01626e7	−0.374933
$222$	2.51347e8	1.54184
$223$	2.06177e8	1.24501	0.622506	−	0.782615i	$-0.286115\pi$
$223$	2.06177e8	1.24501	0.622506	+	0.782615i	$0.286115\pi$
$224$	−2.68638e8	−1.59698
$225$	1.02185e8	0.598063
$226$	3.50985e8	2.02259
$227$	−2.88095e7	−0.163473	−0.0817363	−	0.996654i	$-0.526047\pi$
$227$	−2.88095e7	−0.163473	−0.0817363	+	0.996654i	$0.526047\pi$
$228$	−2.44390e8	−1.36556
$229$	7.04582e7	0.387711	0.193855	−	0.981030i	$-0.437901\pi$
$229$	7.04582e7	0.387711	0.193855	+	0.981030i	$0.437901\pi$
$230$	−7.11018e7	−0.385330
$231$	−3.63824e7	−0.194200
$232$	−7.05651e7	−0.371007
$233$	1.22216e8	0.632969	0.316485	−	0.948598i	$-0.397497\pi$
$233$	1.22216e8	0.632969	0.316485	+	0.948598i	$0.397497\pi$
$234$	8.43912e7	0.430568
$235$	5.68861e7	0.285936
$236$	−1.02073e9	−5.05496
$237$	−2.00473e7	−0.0978218
$238$	−2.06640e8	−0.993561
$239$	2.13572e8	1.01193	0.505966	−	0.862553i	$-0.331136\pi$
$239$	2.13572e8	1.01193	0.505966	+	0.862553i	$0.331136\pi$
$240$	1.90465e8	0.889356
$241$	−1.36380e8	−0.627610	−0.313805	−	0.949487i	$-0.601604\pi$
$241$	−1.36380e8	−0.627610	−0.313805	+	0.949487i	$0.601604\pi$
$242$	1.32559e8	0.601253
$243$	2.24396e8	1.00321
$244$	−7.70452e7	−0.339532
$245$	1.64709e7	0.0715542
$246$	−1.62660e8	−0.696641
$247$	7.18199e7	0.303253
$248$	−1.01828e9	−4.23924
$249$	1.18643e8	0.487018
$250$	−4.20882e8	−1.70361
$251$	4.02924e7	0.160829	0.0804146	−	0.996762i	$-0.474376\pi$
$251$	4.02924e7	0.160829	0.0804146	+	0.996762i	$0.474376\pi$
$252$	2.13201e8	0.839241
$253$	−1.16602e8	−0.452674
$254$	5.78401e8	2.21468
$255$	8.05090e7	0.304056
$256$	9.76454e8	3.63757
$257$	3.39786e7	0.124865	0.0624324	−	0.998049i	$-0.480114\pi$
$257$	3.39786e7	0.124865	0.0624324	+	0.998049i	$0.480114\pi$
$258$	1.57270e8	0.570135
$259$	−1.86606e8	−0.667385
$260$	−1.09498e8	−0.386368
$261$	2.45627e7	0.0855135
$262$	−8.89208e8	−3.05456
$263$	−8.15307e7	−0.276361	−0.138180	−	0.990407i	$-0.544125\pi$
$263$	−8.15307e7	−0.276361	−0.138180	+	0.990407i	$0.544125\pi$
$264$	5.32052e8	1.77967
$265$	−2.67355e8	−0.882527
$266$	2.46679e8	0.803611
$267$	−9.81504e7	−0.315575
$268$	9.03543e8	2.86733
$269$	2.75400e8	0.862644	0.431322	−	0.902198i	$-0.358047\pi$
$269$	2.75400e8	0.862644	0.431322	+	0.902198i	$0.358047\pi$
$270$	−2.54386e8	−0.786540
$271$	5.31521e8	1.62229	0.811144	−	0.584847i	$-0.198846\pi$
$271$	5.31521e8	1.62229	0.811144	+	0.584847i	$0.198846\pi$
$272$	1.77405e9	5.34532
$273$	1.58250e7	0.0470733
$274$	9.90277e8	2.90824
$275$	−2.95610e8	−0.857145
$276$	−1.72583e8	−0.494103
$277$	7.69395e7	0.217505	0.108753	−	0.994069i	$-0.465314\pi$
$277$	7.69395e7	0.217505	0.108753	+	0.994069i	$0.465314\pi$
$278$	−9.28786e7	−0.259274
$279$	3.54450e8	0.977103
$280$	−2.40868e8	−0.655732
$281$	5.91777e8	1.59106	0.795528	−	0.605916i	$-0.207193\pi$
$281$	5.91777e8	1.59106	0.795528	+	0.605916i	$0.207193\pi$
$282$	1.87724e8	0.498480
$283$	−3.02493e7	−0.0793346	−0.0396673	−	0.999213i	$-0.512630\pi$
$283$	−3.02493e7	−0.0793346	−0.0396673	+	0.999213i	$0.512630\pi$
$284$	−7.37586e8	−1.91073
$285$	−9.61086e7	−0.245926
$286$	−2.44135e8	−0.617091
$287$	1.20763e8	0.301542
$288$	−1.36747e9	−3.37321
$289$	3.39545e8	0.827474
$290$	−4.33294e7	−0.104325
$291$	−2.87420e8	−0.683740
$292$	6.61850e7	0.155568
$293$	−8.36753e8	−1.94339	−0.971696	−	0.236233i	$-0.924087\pi$
$293$	−8.36753e8	−1.94339	−0.971696	+	0.236233i	$0.924087\pi$
$294$	5.43538e7	0.124743
$295$	−4.01410e8	−0.910356
$296$	2.72891e9	6.11601
$297$	−4.17177e8	−0.924002
$298$	1.37031e9	2.99959
$299$	5.07177e7	0.109726
$300$	−4.37533e8	−0.935592
$301$	−1.16761e8	−0.246784
$302$	−1.51125e9	−3.15728
$303$	−1.83096e8	−0.378120
$304$	−2.11779e9	−4.32340
$305$	−3.02987e7	−0.0611468
$306$	−1.05187e9	−2.09864
$307$	−5.63859e8	−1.11221	−0.556104	−	0.831113i	$-0.687704\pi$
$307$	−5.63859e8	−1.11221	−0.556104	+	0.831113i	$0.687704\pi$
$308$	−6.16768e8	−1.20280
$309$	8.83943e7	0.170439
$310$	−6.25262e8	−1.19205
$311$	1.38923e6	0.00261887	0.00130944	−	0.999999i	$-0.499583\pi$
$311$	1.38923e6	0.00261887	0.00130944	+	0.999999i	$0.499583\pi$
$312$	−2.31423e8	−0.431386
$313$	−4.49942e8	−0.829376	−0.414688	−	0.909964i	$-0.636109\pi$
$313$	−4.49942e8	−0.829376	−0.414688	+	0.909964i	$0.636109\pi$
$314$	2.02303e9	3.68763
$315$	8.38429e7	0.151140
$316$	−3.39849e8	−0.605872
$317$	−5.00789e8	−0.882974	−0.441487	−	0.897268i	$-0.645549\pi$
$317$	−5.00789e8	−0.882974	−0.441487	+	0.897268i	$0.645549\pi$
$318$	−8.82272e8	−1.53854
$319$	−7.10575e7	−0.122558
$320$	1.25133e9	2.13474
$321$	1.65853e8	0.279870
$322$	1.74199e8	0.290771
$323$	−8.95183e8	−1.47810
$324$	7.41921e8	1.21185
$325$	1.28579e8	0.207769
$326$	1.05748e9	1.69048
$327$	−1.16985e8	−0.185017
$328$	−1.76603e9	−2.76337
$329$	−1.39371e8	−0.215768
$330$	3.26699e8	0.500436
$331$	−6.10963e8	−0.926012	−0.463006	−	0.886355i	$-0.653229\pi$
$331$	−6.10963e8	−0.926012	−0.463006	+	0.886355i	$0.653229\pi$
$332$	2.01128e9	3.01641
$333$	−9.49896e8	−1.40968
$334$	6.87468e8	1.00958
$335$	3.55326e8	0.516381
$336$	−4.66639e8	−0.671110
$337$	3.50593e8	0.498997	0.249499	−	0.968375i	$-0.419734\pi$
$337$	3.50593e8	0.498997	0.249499	+	0.968375i	$0.419734\pi$
$338$	1.06190e8	0.149580
$339$	3.35031e8	0.467074
$340$	1.36482e9	1.88321
$341$	−1.02539e9	−1.40039
$342$	1.25569e9	1.69742
$343$	−4.03536e7	−0.0539949
$344$	1.70751e9	2.26156
$345$	−6.78699e7	−0.0889837
$346$	−4.64636e8	−0.603040
$347$	−1.03916e8	−0.133515	−0.0667573	−	0.997769i	$-0.521265\pi$
$347$	−1.03916e8	−0.133515	−0.0667573	+	0.997769i	$0.521265\pi$
$348$	−1.05172e8	−0.133775
$349$	−2.74408e8	−0.345548	−0.172774	−	0.984962i	$-0.555273\pi$
$349$	−2.74408e8	−0.345548	−0.172774	+	0.984962i	$0.555273\pi$
$350$	4.41630e8	0.550580
$351$	1.81457e8	0.223974
$352$	3.95594e9	4.83449
$353$	1.41846e8	0.171635	0.0858176	−	0.996311i	$-0.472650\pi$
$353$	1.41846e8	0.171635	0.0858176	+	0.996311i	$0.472650\pi$
$354$	−1.32465e9	−1.58705
$355$	−2.90062e8	−0.344106
$356$	−1.66388e9	−1.95456
$357$	−1.97247e8	−0.229441
$358$	6.44864e8	0.742810
$359$	−1.61883e9	−1.84659	−0.923297	−	0.384087i	$-0.874516\pi$
$359$	−1.61883e9	−1.84659	−0.923297	+	0.384087i	$0.874516\pi$
$360$	−1.22611e9	−1.38507
$361$	1.74764e8	0.195514
$362$	3.12022e9	3.45705
$363$	1.26534e8	0.138846
$364$	2.68271e8	0.291554
$365$	2.60278e7	0.0280164
$366$	−9.99856e7	−0.106599
$367$	−6.75474e7	−0.0713309	−0.0356654	−	0.999364i	$-0.511355\pi$
$367$	−6.75474e7	−0.0713309	−0.0356654	+	0.999364i	$0.511355\pi$
$368$	−1.49554e9	−1.56434
$369$	6.14730e8	0.636931
$370$	1.67565e9	1.71979
$371$	6.55020e8	0.665957
$372$	−1.51768e9	−1.52855
$373$	−5.37622e7	−0.0536409	−0.0268204	−	0.999640i	$-0.508538\pi$
$373$	−5.37622e7	−0.0536409	−0.0268204	+	0.999640i	$0.508538\pi$
$374$	3.04296e9	3.00778
$375$	−4.01751e8	−0.393412
$376$	2.03815e9	1.97732
$377$	3.09074e7	0.0297076
$378$	6.23247e8	0.593525
$379$	−1.86197e9	−1.75685	−0.878426	−	0.477878i	$-0.841406\pi$
$379$	−1.86197e9	−1.75685	−0.878426	+	0.477878i	$0.841406\pi$
$380$	−1.62927e9	−1.52318
$381$	5.52110e8	0.511432
$382$	−5.83469e8	−0.535545
$383$	1.05639e9	0.960794	0.480397	−	0.877051i	$-0.340493\pi$
$383$	1.05639e9	0.960794	0.480397	+	0.877051i	$0.340493\pi$
$384$	2.02414e9	1.82423
$385$	−2.42549e8	−0.216614
$386$	−1.51931e9	−1.34459
$387$	−5.94359e8	−0.521268
$388$	−4.87245e9	−4.23483
$389$	−8.52713e8	−0.734479	−0.367239	−	0.930126i	$-0.619697\pi$
$389$	−8.52713e8	−0.734479	−0.367239	+	0.930126i	$0.619697\pi$
$390$	−1.42102e8	−0.121304
$391$	−6.32160e8	−0.534821
$392$	5.90127e8	0.494817
$393$	−8.48790e8	−0.705385
$394$	3.77546e8	0.310980
$395$	−1.33648e8	−0.109112
$396$	−3.13958e9	−2.54061
$397$	2.38800e9	1.91544	0.957718	−	0.287709i	$-0.0928937\pi$
$397$	2.38800e9	1.91544	0.957718	+	0.287709i	$0.0928937\pi$
$398$	2.59999e9	2.06719
$399$	2.35466e8	0.185577
$400$	−3.79148e9	−2.96210
$401$	2.68331e8	0.207810	0.103905	−	0.994587i	$-0.466866\pi$
$401$	2.68331e8	0.207810	0.103905	+	0.994587i	$0.466866\pi$
$402$	1.17258e9	0.900222
$403$	4.46006e8	0.339448
$404$	−3.10391e9	−2.34193
$405$	2.91767e8	0.218245
$406$	1.06157e8	0.0787242
$407$	2.74795e9	2.02036
$408$	2.88452e9	2.10263
$409$	1.41001e9	1.01904	0.509519	−	0.860459i	$-0.329823\pi$
$409$	1.41001e9	1.01904	0.509519	+	0.860459i	$0.329823\pi$
$410$	−1.08440e9	−0.777047
$411$	9.45264e8	0.671594
$412$	1.49849e9	1.05564
$413$	9.83454e8	0.686956
$414$	8.86741e8	0.614180
$415$	7.90954e8	0.543229
$416$	−1.72069e9	−1.17186
$417$	−8.86569e7	−0.0598737
$418$	−3.63258e9	−2.43275
$419$	6.76422e8	0.449230	0.224615	−	0.974448i	$-0.427888\pi$
$419$	6.76422e8	0.449230	0.224615	+	0.974448i	$0.427888\pi$
$420$	−3.58998e8	−0.236439
$421$	−2.38329e9	−1.55665	−0.778324	−	0.627863i	$-0.783930\pi$
$421$	−2.38329e9	−1.55665	−0.778324	+	0.627863i	$0.783930\pi$
$422$	8.72516e8	0.565171
$423$	−7.09450e8	−0.455754
$424$	−9.57895e9	−6.10292
$425$	−1.60265e9	−1.01269
$426$	−9.57205e8	−0.599890
$427$	7.42317e7	0.0461415
$428$	2.81161e9	1.73341
$429$	−2.33038e8	−0.142504
$430$	1.04847e9	0.635940
$431$	−9.67412e8	−0.582024	−0.291012	−	0.956719i	$-0.593992\pi$
$431$	−9.67412e8	−0.582024	−0.291012	+	0.956719i	$0.593992\pi$
$432$	−5.35070e9	−3.19314
$433$	1.68128e9	0.995252	0.497626	−	0.867392i	$-0.334205\pi$
$433$	1.68128e9	0.995252	0.497626	+	0.867392i	$0.334205\pi$
$434$	1.53189e9	0.899526
$435$	−4.13599e7	−0.0240917
$436$	−1.98317e9	−1.14593
$437$	7.54649e8	0.432573
$438$	8.58918e7	0.0488419
$439$	−3.46847e9	−1.95665	−0.978323	−	0.207085i	$-0.933602\pi$
$439$	−3.46847e9	−1.95665	−0.978323	+	0.207085i	$0.933602\pi$
$440$	3.54701e9	1.98508
$441$	−2.05415e8	−0.114051
$442$	−1.32358e9	−0.729074
$443$	−8.31727e8	−0.454536	−0.227268	−	0.973832i	$-0.572979\pi$
$443$	−8.31727e8	−0.454536	−0.227268	+	0.973832i	$0.572979\pi$
$444$	4.06725e9	2.20526
$445$	−6.54336e8	−0.351999
$446$	4.53590e9	2.42098
$447$	1.30802e9	0.692689
$448$	−3.06575e9	−1.61088
$449$	3.44721e9	1.79724	0.898620	−	0.438728i	$-0.144571\pi$
$449$	3.44721e9	1.79724	0.898620	+	0.438728i	$0.144571\pi$
$450$	2.24806e9	1.16296
$451$	−1.77835e9	−0.912850
$452$	5.67957e9	2.89288
$453$	−1.44256e9	−0.729105
$454$	−6.33809e8	−0.317880
$455$	1.05500e8	0.0525064
$456$	−3.44343e9	−1.70065
$457$	2.78531e9	1.36511	0.682553	−	0.730836i	$-0.260869\pi$
$457$	2.78531e9	1.36511	0.682553	+	0.730836i	$0.260869\pi$
$458$	1.55008e9	0.753920
$459$	−2.26173e9	−1.09168
$460$	−1.15056e9	−0.551132
$461$	2.57092e9	1.22218	0.611091	−	0.791560i	$-0.290731\pi$
$461$	2.57092e9	1.22218	0.611091	+	0.791560i	$0.290731\pi$
$462$	−8.00412e8	−0.377630
$463$	1.47649e9	0.691349	0.345674	−	0.938355i	$-0.387650\pi$
$463$	1.47649e9	0.691349	0.345674	+	0.938355i	$0.387650\pi$
$464$	−9.11381e8	−0.423533
$465$	−5.96841e8	−0.275279
$466$	2.68875e9	1.23084
$467$	−3.53173e9	−1.60464	−0.802322	−	0.596892i	$-0.796402\pi$
$467$	−3.53173e9	−1.60464	−0.802322	+	0.596892i	$0.796402\pi$
$468$	1.36560e9	0.615834
$469$	−8.70549e8	−0.389662
$470$	1.25149e9	0.556014
$471$	1.93107e9	0.851579
$472$	−1.43819e10	−6.29536
$473$	1.71942e9	0.747082
$474$	−4.41040e8	−0.190219
$475$	1.91318e9	0.819085
$476$	−3.34381e9	−1.42107
$477$	3.33430e9	1.40666
$478$	4.69858e9	1.96775
$479$	2.18478e9	0.908309	0.454155	−	0.890923i	$-0.349941\pi$
$479$	2.18478e9	0.908309	0.454155	+	0.890923i	$0.349941\pi$
$480$	2.30261e9	0.950333
$481$	−1.19526e9	−0.489727
$482$	−3.00035e9	−1.22042
$483$	1.66281e8	0.0671473
$484$	2.14505e9	0.859962
$485$	−1.91613e9	−0.762656
$486$	4.93671e9	1.95079
$487$	−3.55449e9	−1.39453	−0.697263	−	0.716816i	$-0.745599\pi$
$487$	−3.55449e9	−1.39453	−0.697263	+	0.716816i	$0.745599\pi$
$488$	−1.08556e9	−0.422847
$489$	1.00941e9	0.390380
$490$	3.62359e8	0.139140
$491$	−1.40832e9	−0.536930	−0.268465	−	0.963289i	$-0.586516\pi$
$491$	−1.40832e9	−0.536930	−0.268465	+	0.963289i	$0.586516\pi$
$492$	−2.63214e9	−0.996395
$493$	−3.85238e8	−0.144799
$494$	1.58004e9	0.589689
$495$	−1.23467e9	−0.457543
$496$	−1.31516e10	−4.83942
$497$	7.10652e8	0.259663
$498$	2.61015e9	0.947027
$499$	−7.45093e8	−0.268447	−0.134223	−	0.990951i	$-0.542854\pi$
$499$	−7.45093e8	−0.268447	−0.134223	+	0.990951i	$0.542854\pi$
$500$	−6.81064e9	−2.43665
$501$	6.56220e8	0.233140
$502$	8.86432e8	0.312739
$503$	−2.40385e9	−0.842207	−0.421103	−	0.907013i	$-0.638357\pi$
$503$	−2.40385e9	−0.842207	−0.421103	+	0.907013i	$0.638357\pi$
$504$	3.00397e9	1.04517
$505$	−1.22064e9	−0.421762
$506$	−2.56525e9	−0.880244
$507$	1.01363e8	0.0345423
$508$	9.35958e9	3.16762
$509$	−2.33067e9	−0.783374	−0.391687	−	0.920099i	$-0.628108\pi$
$509$	−2.33067e9	−0.783374	−0.391687	+	0.920099i	$0.628108\pi$
$510$	1.77120e9	0.591250
$511$	−6.37682e7	−0.0211413
$512$	9.14439e9	3.01099
$513$	2.69997e9	0.882973
$514$	7.47530e8	0.242805
$515$	5.89296e8	0.190111
$516$	2.54492e9	0.815455
$517$	2.05237e9	0.653188
$518$	−4.10533e9	−1.29776
$519$	−4.43516e8	−0.139259
$520$	−1.54282e9	−0.481176
$521$	5.59606e9	1.73361	0.866803	−	0.498651i	$-0.166171\pi$
$521$	5.59606e9	1.73361	0.866803	+	0.498651i	$0.166171\pi$
$522$	5.40380e8	0.166285
$523$	5.27571e9	1.61259	0.806296	−	0.591512i	$-0.201469\pi$
$523$	5.27571e9	1.61259	0.806296	+	0.591512i	$0.201469\pi$
$524$	−1.43890e10	−4.36889
$525$	4.21556e8	0.127144
$526$	−1.79368e9	−0.537395
$527$	−5.55914e9	−1.65452
$528$	6.87170e9	2.03164
$529$	−2.87191e9	−0.843482
$530$	−5.88181e9	−1.71611
$531$	5.00616e9	1.45102
$532$	3.99171e9	1.14939
$533$	7.73518e8	0.221271
$534$	−2.15931e9	−0.613650
$535$	1.10569e9	0.312172
$536$	1.27308e10	3.57092
$537$	6.15552e8	0.171536
$538$	6.05881e9	1.67745
$539$	5.94245e8	0.163458
$540$	−4.11644e9	−1.12498
$541$	−5.09138e9	−1.38244	−0.691218	−	0.722646i	$-0.742926\pi$
$541$	−5.09138e9	−1.38244	−0.691218	+	0.722646i	$0.742926\pi$
$542$	1.16935e10	3.15461
$543$	2.97839e9	0.798330
$544$	2.14471e10	5.71181
$545$	−7.79897e8	−0.206371
$546$	3.48150e8	0.0915360
$547$	9.82088e8	0.256563	0.128282	−	0.991738i	$-0.459054\pi$
$547$	9.82088e8	0.256563	0.128282	+	0.991738i	$0.459054\pi$
$548$	1.60245e10	4.15960
$549$	3.77868e8	0.0974622
$550$	−6.50342e9	−1.66676
$551$	4.59883e8	0.117116
$552$	−2.43168e9	−0.615347
$553$	3.27438e8	0.0823364
$554$	1.69267e9	0.422949
$555$	1.59948e9	0.397149
$556$	−1.50295e9	−0.370836
$557$	−4.93899e9	−1.21100	−0.605501	−	0.795844i	$-0.707027\pi$
$557$	−4.93899e9	−1.21100	−0.605501	+	0.795844i	$0.707027\pi$
$558$	7.79790e9	1.90002
$559$	−7.47885e8	−0.181090
$560$	−3.11093e9	−0.748569
$561$	2.90465e9	0.694582
$562$	1.30191e10	3.09388
$563$	−5.92949e9	−1.40036	−0.700178	−	0.713969i	$-0.746896\pi$
$563$	−5.92949e9	−1.40036	−0.700178	+	0.713969i	$0.746896\pi$
$564$	3.03772e9	0.712968
$565$	2.23354e9	0.520983
$566$	−6.65484e8	−0.154270
$567$	−7.14829e8	−0.164688
$568$	−1.03925e10	−2.37959
$569$	−3.42693e9	−0.779851	−0.389926	−	0.920846i	$-0.627499\pi$
$569$	−3.42693e9	−0.779851	−0.389926	+	0.920846i	$0.627499\pi$
$570$	−2.11439e9	−0.478215
$571$	−2.35816e9	−0.530086	−0.265043	−	0.964237i	$-0.585386\pi$
$571$	−2.35816e9	−0.530086	−0.265043	+	0.964237i	$0.585386\pi$
$572$	−3.95055e9	−0.882615
$573$	−5.56948e8	−0.123672
$574$	2.65679e9	0.586361
$575$	1.35105e9	0.296370
$576$	−1.56058e10	−3.40258
$577$	2.41978e9	0.524397	0.262199	−	0.965014i	$-0.415552\pi$
$577$	2.41978e9	0.524397	0.262199	+	0.965014i	$0.415552\pi$
$578$	7.46999e9	1.60906
$579$	−1.45025e9	−0.310504
$580$	−7.01149e8	−0.149215
$581$	−1.93784e9	−0.409922
$582$	−6.32323e9	−1.32956
$583$	−9.64579e9	−2.01603
$584$	9.32540e8	0.193741
$585$	5.37035e8	0.110906
$586$	−1.84086e10	−3.77901
$587$	−4.70780e9	−0.960692	−0.480346	−	0.877079i	$-0.659489\pi$
$587$	−4.70780e9	−0.960692	−0.480346	+	0.877079i	$0.659489\pi$
$588$	8.79544e8	0.178417
$589$	6.63630e9	1.33820
$590$	−8.83102e9	−1.77023
$591$	3.60385e8	0.0718142
$592$	3.52452e10	6.98190
$593$	3.08990e9	0.608489	0.304244	−	0.952594i	$-0.401596\pi$
$593$	3.08990e9	0.608489	0.304244	+	0.952594i	$0.401596\pi$
$594$	−9.17790e9	−1.79676
$595$	−1.31498e9	−0.255923
$596$	2.21741e10	4.29026
$597$	2.48181e9	0.477373
$598$	1.11579e9	0.213368
$599$	−3.00562e9	−0.571401	−0.285700	−	0.958319i	$-0.592226\pi$
$599$	−3.00562e9	−0.571401	−0.285700	+	0.958319i	$0.592226\pi$
$600$	−6.16479e9	−1.16517
$601$	−6.45447e9	−1.21283	−0.606415	−	0.795149i	$-0.707393\pi$
$601$	−6.45447e9	−1.21283	−0.606415	+	0.795149i	$0.707393\pi$
$602$	−2.56875e9	−0.479881
$603$	−4.43142e9	−0.823062
$604$	−2.44548e10	−4.51581
$605$	8.43560e8	0.154872
$606$	−4.02811e9	−0.735271
$607$	−3.25412e9	−0.590572	−0.295286	−	0.955409i	$-0.595415\pi$
$607$	−3.25412e9	−0.590572	−0.295286	+	0.955409i	$0.595415\pi$
$608$	−2.56028e10	−4.61982
$609$	1.01332e8	0.0181796
$610$	−6.66571e8	−0.118903
$611$	−8.92705e8	−0.158330
$612$	−1.70212e10	−3.00166
$613$	4.93923e9	0.866059	0.433029	−	0.901380i	$-0.357445\pi$
$613$	4.93923e9	0.866059	0.433029	+	0.901380i	$0.357445\pi$
$614$	−1.24049e10	−2.16274
$615$	−1.03511e9	−0.179442
$616$	−8.69018e9	−1.49795
$617$	8.53048e9	1.46209	0.731047	−	0.682327i	$-0.239032\pi$
$617$	8.53048e9	1.46209	0.731047	+	0.682327i	$0.239032\pi$
$618$	1.94468e9	0.331427
$619$	1.16054e10	1.96672	0.983360	−	0.181667i	$-0.0581493\pi$
$619$	1.16054e10	1.96672	0.983360	+	0.181667i	$0.0581493\pi$
$620$	−1.01179e10	−1.70498
$621$	1.90666e9	0.319487
$622$	3.05631e7	0.00509251
$623$	1.60312e9	0.265619
$624$	−2.98894e9	−0.492460
$625$	1.89393e9	0.310301
$626$	−9.89873e9	−1.61276
$627$	−3.46746e9	−0.561792
$628$	3.27362e10	5.27436
$629$	1.48980e10	2.38699
$630$	1.84454e9	0.293898
$631$	−2.09336e8	−0.0331696	−0.0165848	−	0.999862i	$-0.505279\pi$
$631$	−2.09336e8	−0.0331696	−0.0165848	+	0.999862i	$0.505279\pi$
$632$	−4.78843e9	−0.754542
$633$	8.32856e8	0.130514
$634$	−1.10174e10	−1.71698
$635$	3.68073e9	0.570462
$636$	−1.42768e10	−2.20054
$637$	−2.58475e8	−0.0396214
$638$	−1.56326e9	−0.238320
$639$	3.61749e9	0.548472
$640$	1.34942e10	2.03479
$641$	−6.38080e9	−0.956912	−0.478456	−	0.878111i	$-0.658803\pi$
$641$	−6.38080e9	−0.956912	−0.478456	+	0.878111i	$0.658803\pi$
$642$	3.64877e9	0.544219
$643$	3.22657e9	0.478632	0.239316	−	0.970942i	$-0.423077\pi$
$643$	3.22657e9	0.478632	0.239316	+	0.970942i	$0.423077\pi$
$644$	2.81886e9	0.415885
$645$	1.00081e9	0.146857
$646$	−1.96940e10	−2.87422
$647$	2.70657e9	0.392875	0.196437	−	0.980516i	$-0.437063\pi$
$647$	2.70657e9	0.392875	0.196437	+	0.980516i	$0.437063\pi$
$648$	1.04536e10	1.50922
$649$	−1.44823e10	−2.07961
$650$	2.82875e9	0.404015
$651$	1.46226e9	0.207726
$652$	1.71120e10	2.41787
$653$	3.68952e9	0.518529	0.259265	−	0.965806i	$-0.416520\pi$
$653$	3.68952e9	0.518529	0.259265	+	0.965806i	$0.416520\pi$
$654$	−2.57366e9	−0.359773
$655$	−5.65860e9	−0.786800
$656$	−2.28091e10	−3.15460
$657$	−3.24604e8	−0.0446555
$658$	−3.06616e9	−0.419570
$659$	−7.16541e9	−0.975308	−0.487654	−	0.873037i	$-0.662147\pi$
$659$	−7.16541e9	−0.975308	−0.487654	+	0.873037i	$0.662147\pi$
$660$	5.28658e9	0.715766
$661$	−7.04455e9	−0.948742	−0.474371	−	0.880325i	$-0.657325\pi$
$661$	−7.04455e9	−0.948742	−0.474371	+	0.880325i	$0.657325\pi$
$662$	−1.34412e10	−1.80067
$663$	−1.26342e9	−0.168364
$664$	2.83388e10	3.75658
$665$	1.56977e9	0.206996
$666$	−2.08977e10	−2.74119
$667$	3.24760e8	0.0423762
$668$	1.11245e10	1.44398
$669$	4.32972e9	0.559073
$670$	7.81717e9	1.00413
$671$	−1.09313e9	−0.139683
$672$	−5.64139e9	−0.717123
$673$	−2.72024e9	−0.343997	−0.171999	−	0.985097i	$-0.555022\pi$
$673$	−2.72024e9	−0.343997	−0.171999	+	0.985097i	$0.555022\pi$
$674$	7.71304e9	0.970322
$675$	4.83376e9	0.604953
$676$	1.71834e9	0.213942
$677$	5.92835e7	0.00734300	0.00367150	−	0.999993i	$-0.498831\pi$
$677$	5.92835e7	0.00734300	0.00367150	+	0.999993i	$0.498831\pi$
$678$	7.37068e9	0.908246
$679$	4.69452e9	0.575502
$680$	1.92301e10	2.34532
$681$	−6.04999e8	−0.0734074
$682$	−2.25585e10	−2.72311
$683$	−9.30031e9	−1.11693	−0.558464	−	0.829529i	$-0.688609\pi$
$683$	−9.30031e9	−1.11693	−0.558464	+	0.829529i	$0.688609\pi$
$684$	2.03193e10	2.42780
$685$	6.30176e9	0.749109
$686$	−8.87779e8	−0.104995
$687$	1.47962e9	0.174102
$688$	2.20533e10	2.58174
$689$	4.19557e9	0.488679
$690$	−1.49314e9	−0.173033
$691$	−9.39647e9	−1.08341	−0.541704	−	0.840570i	$-0.682220\pi$
$691$	−9.39647e9	−1.08341	−0.541704	+	0.840570i	$0.682220\pi$
$692$	−7.51865e9	−0.862519
$693$	3.02493e9	0.345262
$694$	−2.28615e9	−0.259625
$695$	−5.91046e8	−0.0667843
$696$	−1.48187e9	−0.166601
$697$	−9.64133e9	−1.07851
$698$	−6.03698e9	−0.671932
$699$	2.56654e9	0.284235
$700$	7.14637e9	0.787485
$701$	1.49982e10	1.64447	0.822234	−	0.569150i	$-0.192728\pi$
$701$	1.49982e10	1.64447	0.822234	+	0.569150i	$0.192728\pi$
$702$	3.99205e9	0.435528
$703$	−1.77847e10	−1.93065
$704$	4.51461e10	4.87658
$705$	1.19461e9	0.128399
$706$	3.12062e9	0.333752
$707$	2.99057e9	0.318263
$708$	−2.14353e10	−2.26993
$709$	6.03501e9	0.635940	0.317970	−	0.948101i	$-0.396999\pi$
$709$	6.03501e9	0.635940	0.317970	+	0.948101i	$0.396999\pi$
$710$	−6.38137e9	−0.669128
$711$	1.66679e9	0.173915
$712$	−2.34439e10	−2.43417
$713$	4.68642e9	0.484203
$714$	−4.33943e9	−0.446159
$715$	−1.55359e9	−0.158951
$716$	1.04351e10	1.06243
$717$	4.48501e9	0.454409
$718$	−3.56143e10	−3.59078
$719$	7.21183e9	0.723594	0.361797	−	0.932257i	$-0.382163\pi$
$719$	7.21183e9	0.723594	0.361797	+	0.932257i	$0.382163\pi$
$720$	−1.58358e10	−1.58116
$721$	−1.44377e9	−0.143458
$722$	3.84482e9	0.380185
$723$	−2.86397e9	−0.281829
$724$	5.04908e10	4.94456
$725$	8.23330e8	0.0802400
$726$	2.78375e9	0.269993
$727$	−5.39439e9	−0.520682	−0.260341	−	0.965517i	$-0.583835\pi$
$727$	−5.39439e9	−0.520682	−0.260341	+	0.965517i	$0.583835\pi$
$728$	3.77991e9	0.363096
$729$	1.54499e8	0.0147700
$730$	5.72612e8	0.0544792
$731$	9.32184e9	0.882655
$732$	−1.61795e9	−0.152467
$733$	−1.00977e10	−0.947017	−0.473508	−	0.880789i	$-0.657013\pi$
$733$	−1.00977e10	−0.947017	−0.473508	+	0.880789i	$0.657013\pi$
$734$	−1.48604e9	−0.138706
$735$	3.45888e8	0.0321314
$736$	−1.80802e10	−1.67159
$737$	1.28197e10	1.17961
$738$	1.35241e10	1.23854
$739$	−1.91801e10	−1.74822	−0.874111	−	0.485727i	$-0.838555\pi$
$739$	−1.91801e10	−1.74822	−0.874111	+	0.485727i	$0.838555\pi$
$740$	2.71150e10	2.45979
$741$	1.50822e9	0.136176
$742$	1.44104e10	1.29498
$743$	6.37179e9	0.569902	0.284951	−	0.958542i	$-0.408023\pi$
$743$	6.37179e9	0.569902	0.284951	+	0.958542i	$0.408023\pi$
$744$	−2.13839e10	−1.90363
$745$	8.72014e9	0.772639
$746$	−1.18277e9	−0.104307
$747$	−9.86433e9	−0.865855
$748$	4.92407e10	4.30198
$749$	−2.70894e9	−0.235566
$750$	−8.83852e9	−0.765007
$751$	−9.59820e9	−0.826894	−0.413447	−	0.910528i	$-0.635675\pi$
$751$	−9.59820e9	−0.826894	−0.413447	+	0.910528i	$0.635675\pi$
$752$	2.63236e10	2.25727
$753$	8.46139e8	0.0722203
$754$	6.79963e8	0.0577677
$755$	−9.61706e9	−0.813258
$756$	1.00853e10	0.848909
$757$	4.00641e9	0.335676	0.167838	−	0.985815i	$-0.446321\pi$
$757$	4.00641e9	0.335676	0.167838	+	0.985815i	$0.446321\pi$
$758$	−4.09633e10	−3.41628
$759$	−2.44865e9	−0.203273
$760$	−2.29562e10	−1.89694
$761$	−7.15961e9	−0.588902	−0.294451	−	0.955667i	$-0.595137\pi$
$761$	−7.15961e9	−0.588902	−0.294451	+	0.955667i	$0.595137\pi$
$762$	1.21464e10	0.994503
$763$	1.91075e9	0.155728
$764$	−9.44159e9	−0.765981
$765$	−6.69374e9	−0.540573
$766$	2.32407e10	1.86831
$767$	6.29927e9	0.504088
$768$	2.05055e10	1.63345
$769$	1.80694e10	1.43285	0.716427	−	0.697662i	$-0.245776\pi$
$769$	1.80694e10	1.43285	0.716427	+	0.697662i	$0.245776\pi$
$770$	−5.33608e9	−0.421216
$771$	7.13551e8	0.0560705
$772$	−2.45852e10	−1.92315
$773$	−7.43025e9	−0.578596	−0.289298	−	0.957239i	$-0.593422\pi$
$773$	−7.43025e9	−0.578596	−0.289298	+	0.957239i	$0.593422\pi$
$774$	−1.30759e10	−1.01363
$775$	1.18810e10	0.916846
$776$	−6.86522e10	−5.27398
$777$	−3.91873e9	−0.299689
$778$	−1.87597e10	−1.42823
$779$	1.15095e10	0.872316
$780$	−2.29947e9	−0.173499
$781$	−1.04650e10	−0.786071
$782$	−1.39075e10	−1.03998
$783$	1.16192e9	0.0864987
$784$	7.62177e9	0.564872
$785$	1.28738e10	0.949867
$786$	−1.86734e10	−1.37165
$787$	−7.02736e9	−0.513903	−0.256951	−	0.966424i	$-0.582718\pi$
$787$	−7.02736e9	−0.513903	−0.256951	+	0.966424i	$0.582718\pi$
$788$	6.10938e9	0.444790
$789$	−1.71214e9	−0.124100
$790$	−2.94026e9	−0.212174
$791$	−5.47217e9	−0.393135
$792$	−4.42363e10	−3.16403
$793$	4.75473e8	0.0338586
$794$	5.25360e10	3.72465
$795$	−5.61446e9	−0.396299
$796$	4.20725e10	2.95667
$797$	9.94013e9	0.695485	0.347743	−	0.937590i	$-0.386948\pi$
$797$	9.94013e9	0.695485	0.347743	+	0.937590i	$0.386948\pi$
$798$	5.18025e9	0.360862
$799$	1.11269e10	0.771722
$800$	−4.58368e10	−3.16519
$801$	8.16051e9	0.561052
$802$	5.90329e9	0.404095
$803$	9.39047e8	0.0640004
$804$	1.89744e10	1.28757
$805$	1.10854e9	0.0748974
$806$	9.81214e9	0.660072
$807$	5.78341e9	0.387370
$808$	−4.37338e10	−2.91660
$809$	8.69558e9	0.577403	0.288701	−	0.957419i	$-0.406777\pi$
$809$	8.69558e9	0.577403	0.288701	+	0.957419i	$0.406777\pi$
$810$	6.41887e9	0.424386
$811$	−4.02087e9	−0.264696	−0.132348	−	0.991203i	$-0.542252\pi$
$811$	−4.02087e9	−0.264696	−0.132348	+	0.991203i	$0.542252\pi$
$812$	1.71782e9	0.112598
$813$	1.11619e10	0.728488
$814$	6.04549e10	3.92868
$815$	6.72942e9	0.435438
$816$	3.72549e10	2.40032
$817$	−1.11281e10	−0.713909
$818$	3.10202e10	1.98156
$819$	−1.31573e9	−0.0836902
$820$	−1.75476e10	−1.11140
$821$	−2.39905e10	−1.51300	−0.756499	−	0.653995i	$-0.773092\pi$
$821$	−2.39905e10	−1.51300	−0.756499	+	0.653995i	$0.773092\pi$
$822$	2.07958e10	1.30594
$823$	1.19596e10	0.747852	0.373926	−	0.927458i	$-0.378011\pi$
$823$	1.19596e10	0.747852	0.373926	+	0.927458i	$0.378011\pi$
$824$	2.11136e10	1.31467
$825$	−6.20781e9	−0.384901
$826$	2.16360e10	1.33582
$827$	2.77336e10	1.70505	0.852525	−	0.522687i	$-0.175070\pi$
$827$	2.77336e10	1.70505	0.852525	+	0.522687i	$0.175070\pi$
$828$	1.43491e10	0.878452
$829$	−5.49238e9	−0.334827	−0.167413	−	0.985887i	$-0.553541\pi$
$829$	−5.49238e9	−0.334827	−0.167413	+	0.985887i	$0.553541\pi$
$830$	1.74010e10	1.05633
$831$	1.61573e9	0.0976709
$832$	−1.96369e10	−1.18206
$833$	3.22170e9	0.193120
$834$	−1.95045e9	−0.116427
$835$	4.37480e9	0.260049
$836$	−5.87817e10	−3.47953
$837$	1.67670e10	0.988360
$838$	1.48813e10	0.873547
$839$	−2.81719e10	−1.64683	−0.823416	−	0.567438i	$-0.807935\pi$
$839$	−2.81719e10	−1.64683	−0.823416	+	0.567438i	$0.807935\pi$
$840$	−5.05823e9	−0.294457
$841$	−1.70520e10	−0.988527
$842$	−5.24325e10	−3.02697
$843$	1.24273e10	0.714464
$844$	1.41189e10	0.808355
$845$	6.75753e8	0.0385292
$846$	−1.56079e10	−0.886234
$847$	−2.06672e9	−0.116867
$848$	−1.23717e11	−6.96695
$849$	−6.35235e8	−0.0356252
$850$	−3.52583e10	−1.96922
$851$	−1.25592e10	−0.698567
$852$	−1.54893e10	−0.858013
$853$	7.20058e9	0.397233	0.198617	−	0.980077i	$-0.436355\pi$
$853$	7.20058e9	0.397233	0.198617	+	0.980077i	$0.436355\pi$
$854$	1.63310e9	0.0897242
$855$	7.99074e9	0.437226
$856$	3.96152e10	2.15876
$857$	1.24962e10	0.678178	0.339089	−	0.940754i	$-0.389881\pi$
$857$	1.24962e10	0.678178	0.339089	+	0.940754i	$0.389881\pi$
$858$	−5.12684e9	−0.277105
$859$	−8.08532e9	−0.435232	−0.217616	−	0.976034i	$-0.569828\pi$
$859$	−8.08532e9	−0.435232	−0.217616	+	0.976034i	$0.569828\pi$
$860$	1.69661e10	0.909575
$861$	2.53603e9	0.135407
$862$	−2.12831e10	−1.13177
$863$	1.19317e10	0.631922	0.315961	−	0.948772i	$-0.397673\pi$
$863$	1.19317e10	0.631922	0.315961	+	0.948772i	$0.397673\pi$
$864$	−6.46868e10	−3.41207
$865$	−2.95677e9	−0.155332
$866$	3.69882e10	1.93531
$867$	7.13044e9	0.371578
$868$	2.47888e10	1.28658
$869$	−4.82184e9	−0.249255
$870$	−9.09918e8	−0.0468473
$871$	−5.57608e9	−0.285934
$872$	−2.79426e10	−1.42712
$873$	2.38969e10	1.21560
$874$	1.66023e10	0.841158
$875$	6.56193e9	0.331134
$876$	1.38989e9	0.0698578
$877$	2.61118e10	1.30719	0.653593	−	0.756846i	$-0.273261\pi$
$877$	2.61118e10	1.30719	0.653593	+	0.756846i	$0.273261\pi$
$878$	−7.63064e10	−3.80478
$879$	−1.75718e10	−0.872681
$880$	4.58114e10	2.26612
$881$	1.20446e10	0.593441	0.296721	−	0.954964i	$-0.404107\pi$
$881$	1.20446e10	0.593441	0.296721	+	0.954964i	$0.404107\pi$
$882$	−4.51913e9	−0.221776
$883$	−9.62926e9	−0.470685	−0.235343	−	0.971912i	$-0.575621\pi$
$883$	−9.62926e9	−0.470685	−0.235343	+	0.971912i	$0.575621\pi$
$884$	−2.14179e10	−1.04278
$885$	−8.42961e9	−0.408795
$886$	−1.82980e10	−0.883864
$887$	−1.78707e10	−0.859825	−0.429912	−	0.902871i	$-0.641456\pi$
$887$	−1.78707e10	−0.859825	−0.429912	+	0.902871i	$0.641456\pi$
$888$	5.73071e10	2.74640
$889$	−9.01780e9	−0.430471
$890$	−1.43954e10	−0.684477
$891$	1.05265e10	0.498555
$892$	7.33990e10	3.46269
$893$	−1.32829e10	−0.624184
$894$	2.87765e10	1.34696
$895$	4.10368e9	0.191334
$896$	−3.30609e10	−1.53545
$897$	1.06507e9	0.0492726
$898$	7.58387e10	3.49481
$899$	2.85590e9	0.131094
$900$	3.63777e10	1.66336
$901$	−5.22947e10	−2.38188
$902$	−3.91237e10	−1.77508
$903$	−2.45199e9	−0.110818
$904$	8.00245e10	3.60274
$905$	1.98559e10	0.890473
$906$	−3.17363e10	−1.41778
$907$	−2.77751e10	−1.23603	−0.618016	−	0.786166i	$-0.712063\pi$
$907$	−2.77751e10	−1.23603	−0.618016	+	0.786166i	$0.712063\pi$
$908$	−1.02562e10	−0.454658
$909$	1.52231e10	0.672249
$910$	2.32100e9	0.102101
$911$	1.05850e10	0.463850	0.231925	−	0.972734i	$-0.425498\pi$
$911$	1.05850e10	0.463850	0.231925	+	0.972734i	$0.425498\pi$
$912$	−4.44736e10	−1.94142
$913$	2.85365e10	1.24095
$914$	6.12768e10	2.65451
$915$	−6.36272e8	−0.0274580
$916$	2.50831e10	1.07832
$917$	1.38636e10	0.593721
$918$	−4.97580e10	−2.12282
$919$	5.46662e8	0.0232335	0.0116167	−	0.999933i	$-0.496302\pi$
$919$	5.46662e8	0.0232335	0.0116167	+	0.999933i	$0.496302\pi$
$920$	−1.62112e10	−0.686370
$921$	−1.18410e10	−0.499437
$922$	5.65603e10	2.37659
$923$	4.55190e9	0.190540
$924$	−1.29521e10	−0.540118
$925$	−3.18400e10	−1.32275
$926$	3.24828e10	1.34436
$927$	−7.34936e9	−0.303019
$928$	−1.10181e10	−0.452571
$929$	1.32799e10	0.543424	0.271712	−	0.962379i	$-0.412410\pi$
$929$	1.32799e10	0.543424	0.271712	+	0.962379i	$0.412410\pi$
$930$	−1.31305e10	−0.535292
$931$	−3.84595e9	−0.156199
$932$	4.35089e10	1.76045
$933$	2.91739e7	0.00117600
$934$	−7.76981e10	−3.12030
$935$	1.93643e10	0.774750
$936$	1.92412e10	0.766949
$937$	−4.29710e10	−1.70642	−0.853212	−	0.521564i	$-0.825349\pi$
$937$	−4.29710e10	−1.70642	−0.853212	+	0.521564i	$0.825349\pi$
$938$	−1.91521e10	−0.757715
$939$	−9.44878e9	−0.372431
$940$	2.02514e10	0.795259
$941$	2.91760e10	1.14147	0.570733	−	0.821136i	$-0.306659\pi$
$941$	2.91760e10	1.14147	0.570733	+	0.821136i	$0.306659\pi$
$942$	4.24835e10	1.65593
$943$	8.12774e9	0.315631
$944$	−1.85750e11	−7.18664
$945$	3.96612e9	0.152881
$946$	3.78273e10	1.45273
$947$	−3.90819e10	−1.49538	−0.747689	−	0.664049i	$-0.768837\pi$
$947$	−3.90819e10	−1.49538	−0.747689	+	0.664049i	$0.768837\pi$
$948$	−7.13682e9	−0.272067
$949$	−4.08451e8	−0.0155134
$950$	4.20900e10	1.59275
$951$	−1.05166e10	−0.396499
$952$	−4.71138e10	−1.76978
$953$	−4.09337e9	−0.153199	−0.0765994	−	0.997062i	$-0.524406\pi$
$953$	−4.09337e9	−0.153199	−0.0765994	+	0.997062i	$0.524406\pi$
$954$	7.33546e10	2.73532
$955$	−3.71298e9	−0.137947
$956$	7.60316e10	2.81444
$957$	−1.49221e9	−0.0550348
$958$	4.80652e10	1.76625
$959$	−1.54393e10	−0.565279
$960$	2.62779e10	0.958607
$961$	1.36992e10	0.497925
$962$	−2.62957e10	−0.952295
$963$	−1.37895e10	−0.497573
$964$	−4.85512e10	−1.74554
$965$	−9.66833e9	−0.346342
$966$	3.65819e9	0.130571
$967$	3.00399e10	1.06833	0.534165	−	0.845380i	$-0.320626\pi$
$967$	3.00399e10	1.06833	0.534165	+	0.845380i	$0.320626\pi$
$968$	3.02236e10	1.07098
$969$	−1.87988e10	−0.663740
$970$	−4.21549e10	−1.48302
$971$	−3.26535e10	−1.14462	−0.572312	−	0.820036i	$-0.693953\pi$
$971$	−3.26535e10	−1.14462	−0.572312	+	0.820036i	$0.693953\pi$
$972$	7.98849e10	2.79019
$973$	1.44806e9	0.0503956
$974$	−7.81989e10	−2.71172
$975$	2.70017e9	0.0932985
$976$	−1.40205e10	−0.482713
$977$	4.17986e10	1.43394	0.716969	−	0.697105i	$-0.245529\pi$
$977$	4.17986e10	1.43394	0.716969	+	0.697105i	$0.245529\pi$
$978$	2.22071e10	0.759112
$979$	−2.36075e10	−0.804101
$980$	5.86363e9	0.199010
$981$	9.72643e9	0.328936
$982$	−3.09831e10	−1.04408
$983$	1.33316e10	0.447656	0.223828	−	0.974629i	$-0.428145\pi$
$983$	1.33316e10	0.447656	0.223828	+	0.974629i	$0.428145\pi$
$984$	−3.70866e10	−1.24089
$985$	2.40257e9	0.0801029
$986$	−8.47524e9	−0.281568
$987$	−2.92679e9	−0.0968905
$988$	2.55679e10	0.843423
$989$	−7.85841e9	−0.258314
$990$	−2.71627e10	−0.889711
$991$	−2.42613e9	−0.0791873	−0.0395936	−	0.999216i	$-0.512606\pi$
$991$	−2.42613e9	−0.0791873	−0.0395936	+	0.999216i	$0.512606\pi$
$992$	−1.58995e11	−5.17122
$993$	−1.28302e10	−0.415826
$994$	1.56343e10	0.504926
$995$	1.65454e10	0.532471
$996$	4.22369e10	1.35452
$997$	3.27152e10	1.04548	0.522741	−	0.852492i	$-0.324909\pi$
$997$	3.27152e10	1.04548	0.522741	+	0.852492i	$0.324909\pi$
$998$	−1.63920e10	−0.522007
$999$	−4.49340e10	−1.42592

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	91.8.a.a.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.8.a.a.1.1	✓	1	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:	\( N \)	\(=\)	\( 91 = 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	91.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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