LMFDB - Embedded newform 363.8.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [363,8,Mod(1,363)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(363, 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("363.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 363 = 3 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	363.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:	$113.395764251$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 33)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	363.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-10.0000 q^{2} +27.0000 q^{3} -28.0000 q^{4} -410.000 q^{5} -270.000 q^{6} +1028.00 q^{7} +1560.00 q^{8} +729.000 q^{9} +O(q^{10})$

\(q-10.0000 q^{2} +27.0000 q^{3} -28.0000 q^{4} -410.000 q^{5} -270.000 q^{6} +1028.00 q^{7} +1560.00 q^{8} +729.000 q^{9} +4100.00 q^{10} -756.000 q^{12} -12958.0 q^{13} -10280.0 q^{14} -11070.0 q^{15} -12016.0 q^{16} -17062.0 q^{17} -7290.00 q^{18} +54168.0 q^{19} +11480.0 q^{20} +27756.0 q^{21} -11488.0 q^{23} +42120.0 q^{24} +89975.0 q^{25} +129580. q^{26} +19683.0 q^{27} -28784.0 q^{28} +186654. q^{29} +110700. q^{30} -188672. q^{31} -79520.0 q^{32} +170620. q^{34} -421480. q^{35} -20412.0 q^{36} +395886. q^{37} -541680. q^{38} -349866. q^{39} -639600. q^{40} +47546.0 q^{41} -277560. q^{42} -602088. q^{43} -298890. q^{45} +114880. q^{46} -647200. q^{47} -324432. q^{48} +233241. q^{49} -899750. q^{50} -460674. q^{51} +362824. q^{52} -1.31272e6 q^{53} -196830. q^{54} +1.60368e6 q^{56} +1.46254e6 q^{57} -1.86654e6 q^{58} -2.68114e6 q^{59} +309960. q^{60} -551190. q^{61} +1.88672e6 q^{62} +749412. q^{63} +2.33325e6 q^{64} +5.31278e6 q^{65} +459260. q^{67} +477736. q^{68} -310176. q^{69} +4.21480e6 q^{70} -18072.0 q^{71} +1.13724e6 q^{72} +426062. q^{73} -3.95886e6 q^{74} +2.42932e6 q^{75} -1.51670e6 q^{76} +3.49866e6 q^{78} -297764. q^{79} +4.92656e6 q^{80} +531441. q^{81} -475460. q^{82} -5.68403e6 q^{83} -777168. q^{84} +6.99542e6 q^{85} +6.02088e6 q^{86} +5.03966e6 q^{87} -6.34297e6 q^{89} +2.98890e6 q^{90} -1.33208e7 q^{91} +321664. q^{92} -5.09414e6 q^{93} +6.47200e6 q^{94} -2.22089e7 q^{95} -2.14704e6 q^{96} +1.66516e7 q^{97} -2.33241e6 q^{98} +O(q^{100})\)

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 $

$2$	−10.0000	−0.883883	−0.441942	−	0.897044i	$-0.645710\pi$
$2$	−10.0000	−0.883883	−0.441942	+	0.897044i	$0.645710\pi$
$3$	27.0000	0.577350
$3$	27.0000	0.577350
$4$	−28.0000	−0.218750
$5$	−410.000	−1.46686	−0.733430	−	0.679765i	$-0.762082\pi$
$5$	−410.000	−1.46686	−0.733430	+	0.679765i	$0.762082\pi$
$6$	−270.000	−0.510310
$7$	1028.00	1.13279	0.566396	−	0.824133i	$-0.308337\pi$
$7$	1028.00	1.13279	0.566396	+	0.824133i	$0.308337\pi$
$8$	1560.00	1.07723
$9$	729.000	0.333333
$10$	4100.00	1.29653
$11$	0	0
$11$	0	0
$12$	−756.000	−0.126295
$13$	−12958.0	−1.63582	−0.817911	−	0.575344i	$-0.804868\pi$
$13$	−12958.0	−1.63582	−0.817911	+	0.575344i	$0.804868\pi$
$14$	−10280.0	−1.00126
$15$	−11070.0	−0.846892
$16$	−12016.0	−0.733398
$17$	−17062.0	−0.842284	−0.421142	−	0.906995i	$-0.638371\pi$
$17$	−17062.0	−0.842284	−0.421142	+	0.906995i	$0.638371\pi$
$18$	−7290.00	−0.294628
$19$	54168.0	1.81178	0.905889	−	0.423514i	$-0.139204\pi$
$19$	54168.0	1.81178	0.905889	+	0.423514i	$0.139204\pi$
$20$	11480.0	0.320876
$21$	27756.0	0.654017
$22$	0	0
$23$	−11488.0	−0.196878	−0.0984390	−	0.995143i	$-0.531385\pi$
$23$	−11488.0	−0.196878	−0.0984390	+	0.995143i	$0.531385\pi$
$24$	42120.0	0.621941
$25$	89975.0	1.15168
$26$	129580.	1.44588
$27$	19683.0	0.192450
$28$	−28784.0	−0.247798
$29$	186654.	1.42116	0.710582	−	0.703614i	$-0.248432\pi$
$29$	186654.	1.42116	0.710582	+	0.703614i	$0.248432\pi$
$30$	110700.	0.748554
$31$	−188672.	−1.13747	−0.568737	−	0.822519i	$-0.692568\pi$
$31$	−188672.	−1.13747	−0.568737	+	0.822519i	$0.692568\pi$
$32$	−79520.0	−0.428994
$33$	0	0
$34$	170620.	0.744481
$35$	−421480.	−1.66165
$36$	−20412.0	−0.0729167
$37$	395886.	1.28488	0.642442	−	0.766334i	$-0.277921\pi$
$37$	395886.	1.28488	0.642442	+	0.766334i	$0.277921\pi$
$38$	−541680.	−1.60140
$39$	−349866.	−0.944443
$40$	−639600.	−1.58015
$41$	47546.0	0.107738	0.0538692	−	0.998548i	$-0.482845\pi$
$41$	47546.0	0.107738	0.0538692	+	0.998548i	$0.482845\pi$
$42$	−277560.	−0.578075
$43$	−602088.	−1.15484	−0.577418	−	0.816449i	$-0.695940\pi$
$43$	−602088.	−1.15484	−0.577418	+	0.816449i	$0.695940\pi$
$44$	0	0
$45$	−298890.	−0.488954
$46$	114880.	0.174017
$47$	−647200.	−0.909277	−0.454638	−	0.890676i	$-0.650231\pi$
$47$	−647200.	−0.909277	−0.454638	+	0.890676i	$0.650231\pi$
$48$	−324432.	−0.423428
$49$	233241.	0.283217
$50$	−899750.	−1.01795
$51$	−460674.	−0.486293
$52$	362824.	0.357836
$53$	−1.31272e6	−1.21118	−0.605588	−	0.795778i	$-0.707062\pi$
$53$	−1.31272e6	−1.21118	−0.605588	+	0.795778i	$0.707062\pi$
$54$	−196830.	−0.170103
$55$	0	0
$56$	1.60368e6	1.22028
$57$	1.46254e6	1.04603
$58$	−1.86654e6	−1.25614
$59$	−2.68114e6	−1.69956	−0.849782	−	0.527135i	$-0.823266\pi$
$59$	−2.68114e6	−1.69956	−0.849782	+	0.527135i	$0.823266\pi$
$60$	309960.	0.185258
$61$	−551190.	−0.310919	−0.155459	−	0.987842i	$-0.549686\pi$
$61$	−551190.	−0.310919	−0.155459	+	0.987842i	$0.549686\pi$
$62$	1.88672e6	1.00539
$63$	749412.	0.377597
$64$	2.33325e6	1.11258
$65$	5.31278e6	2.39952
$66$	0	0
$67$	459260.	0.186551	0.0932753	−	0.995640i	$-0.470266\pi$
$67$	459260.	0.186551	0.0932753	+	0.995640i	$0.470266\pi$
$68$	477736.	0.184250
$69$	−310176.	−0.113668
$70$	4.21480e6	1.46870
$71$	−18072.0	−0.00599242	−0.00299621	−	0.999996i	$-0.500954\pi$
$71$	−18072.0	−0.00599242	−0.00299621	+	0.999996i	$0.500954\pi$
$72$	1.13724e6	0.359078
$73$	426062.	0.128187	0.0640933	−	0.997944i	$-0.479584\pi$
$73$	426062.	0.128187	0.0640933	+	0.997944i	$0.479584\pi$
$74$	−3.95886e6	−1.13569
$75$	2.42932e6	0.664923
$76$	−1.51670e6	−0.396327
$77$	0	0
$78$	3.49866e6	0.834777
$79$	−297764.	−0.0679481	−0.0339741	−	0.999423i	$-0.510816\pi$
$79$	−297764.	−0.0679481	−0.0339741	+	0.999423i	$0.510816\pi$
$80$	4.92656e6	1.07579
$81$	531441.	0.111111
$82$	−475460.	−0.0952282
$83$	−5.68403e6	−1.09115	−0.545573	−	0.838063i	$-0.683688\pi$
$83$	−5.68403e6	−1.09115	−0.545573	+	0.838063i	$0.683688\pi$
$84$	−777168.	−0.143066
$85$	6.99542e6	1.23551
$86$	6.02088e6	1.02074
$87$	5.03966e6	0.820510
$88$	0	0
$89$	−6.34297e6	−0.953734	−0.476867	−	0.878975i	$-0.658228\pi$
$89$	−6.34297e6	−0.953734	−0.476867	+	0.878975i	$0.658228\pi$
$90$	2.98890e6	0.432178
$91$	−1.33208e7	−1.85305
$92$	321664.	0.0430670
$93$	−5.09414e6	−0.656721
$94$	6.47200e6	0.803695
$95$	−2.22089e7	−2.65763
$96$	−2.14704e6	−0.247680
$97$	1.66516e7	1.85248	0.926242	−	0.376929i	$-0.123020\pi$
$97$	1.66516e7	1.85248	0.926242	+	0.376929i	$0.123020\pi$
$98$	−2.33241e6	−0.250330
$99$	0	0
$100$	−2.51930e6	−0.251930
$101$	2.08327e6	0.201197	0.100598	−	0.994927i	$-0.467924\pi$
$101$	2.08327e6	0.201197	0.100598	+	0.994927i	$0.467924\pi$
$102$	4.60674e6	0.429826
$103$	−2.39046e6	−0.215552	−0.107776	−	0.994175i	$-0.534373\pi$
$103$	−2.39046e6	−0.215552	−0.107776	+	0.994175i	$0.534373\pi$
$104$	−2.02145e7	−1.76216
$105$	−1.13800e7	−0.959352
$106$	1.31272e7	1.07054
$107$	1.40615e7	1.10965	0.554827	−	0.831966i	$-0.312785\pi$
$107$	1.40615e7	1.10965	0.554827	+	0.831966i	$0.312785\pi$
$108$	−551124.	−0.0420985
$109$	1.11321e7	0.823347	0.411674	−	0.911331i	$-0.364944\pi$
$109$	1.11321e7	0.823347	0.411674	+	0.911331i	$0.364944\pi$
$110$	0	0
$111$	1.06889e7	0.741828
$112$	−1.23524e7	−0.830788
$113$	5.66903e6	0.369602	0.184801	−	0.982776i	$-0.440836\pi$
$113$	5.66903e6	0.369602	0.184801	+	0.982776i	$0.440836\pi$
$114$	−1.46254e7	−0.924570
$115$	4.71008e6	0.288792
$116$	−5.22631e6	−0.310880
$117$	−9.44638e6	−0.545274
$118$	2.68114e7	1.50222
$119$	−1.75397e7	−0.954132
$120$	−1.72692e7	−0.912300
$121$	0	0
$122$	5.51190e6	0.274816
$123$	1.28374e6	0.0622028
$124$	5.28282e6	0.248822
$125$	−4.85850e6	−0.222493
$126$	−7.49412e6	−0.333752
$127$	2.09170e7	0.906123	0.453061	−	0.891479i	$-0.350332\pi$
$127$	2.09170e7	0.906123	0.453061	+	0.891479i	$0.350332\pi$
$128$	−1.31539e7	−0.554396
$129$	−1.62564e7	−0.666745
$130$	−5.31278e7	−2.12090
$131$	1.12649e7	0.437802	0.218901	−	0.975747i	$-0.429753\pi$
$131$	1.12649e7	0.437802	0.218901	+	0.975747i	$0.429753\pi$
$132$	0	0
$133$	5.56847e7	2.05237
$134$	−4.59260e6	−0.164889
$135$	−8.07003e6	−0.282297
$136$	−2.66167e7	−0.907336
$137$	444290.	0.0147620	0.00738099	−	0.999973i	$-0.497651\pi$
$137$	444290.	0.0147620	0.00738099	+	0.999973i	$0.497651\pi$
$138$	3.10176e6	0.100469
$139$	−3.42613e7	−1.08206	−0.541030	−	0.841003i	$-0.681966\pi$
$139$	−3.42613e7	−1.08206	−0.541030	+	0.841003i	$0.681966\pi$
$140$	1.18014e7	0.363485
$141$	−1.74744e7	−0.524971
$142$	180720.	0.00529660
$143$	0	0
$144$	−8.75966e6	−0.244466
$145$	−7.65281e7	−2.08465
$146$	−4.26062e6	−0.113302
$147$	6.29751e6	0.163515
$148$	−1.10848e7	−0.281068
$149$	4.82211e7	1.19422	0.597112	−	0.802158i	$-0.296315\pi$
$149$	4.82211e7	1.19422	0.597112	+	0.802158i	$0.296315\pi$
$150$	−2.42932e7	−0.587714
$151$	4.48693e7	1.06055	0.530273	−	0.847827i	$-0.322089\pi$
$151$	4.48693e7	1.06055	0.530273	+	0.847827i	$0.322089\pi$
$152$	8.45021e7	1.95171
$153$	−1.24382e7	−0.280761
$154$	0	0
$155$	7.73555e7	1.66852
$156$	9.79625e6	0.206597
$157$	−5.38907e6	−0.111139	−0.0555693	−	0.998455i	$-0.517697\pi$
$157$	−5.38907e6	−0.111139	−0.0555693	+	0.998455i	$0.517697\pi$
$158$	2.97764e6	0.0600582
$159$	−3.54435e7	−0.699273
$160$	3.26032e7	0.629275
$161$	−1.18097e7	−0.223022
$162$	−5.31441e6	−0.0982093
$163$	9.81674e7	1.77546	0.887730	−	0.460365i	$-0.152281\pi$
$163$	9.81674e7	1.77546	0.887730	+	0.460365i	$0.152281\pi$
$164$	−1.33129e6	−0.0235678
$165$	0	0
$166$	5.68403e7	0.964446
$167$	4.40611e7	0.732062	0.366031	−	0.930603i	$-0.380716\pi$
$167$	4.40611e7	0.732062	0.366031	+	0.930603i	$0.380716\pi$
$168$	4.32994e7	0.704529
$169$	1.05161e8	1.67592
$170$	−6.99542e7	−1.09205
$171$	3.94885e7	0.603926
$172$	1.68585e7	0.252620
$173$	−6.71087e7	−0.985411	−0.492706	−	0.870196i	$-0.663992\pi$
$173$	−6.71087e7	−0.985411	−0.492706	+	0.870196i	$0.663992\pi$
$174$	−5.03966e7	−0.725235
$175$	9.24943e7	1.30461
$176$	0	0
$177$	−7.23908e7	−0.981244
$178$	6.34297e7	0.842990
$179$	4.34929e6	0.0566804	0.0283402	−	0.999598i	$-0.490978\pi$
$179$	4.34929e6	0.0566804	0.0283402	+	0.999598i	$0.490978\pi$
$180$	8.36892e6	0.106959
$181$	−1.20238e7	−0.150719	−0.0753593	−	0.997156i	$-0.524010\pi$
$181$	−1.20238e7	−0.150719	−0.0753593	+	0.997156i	$0.524010\pi$
$182$	1.33208e8	1.63788
$183$	−1.48821e7	−0.179509
$184$	−1.79213e7	−0.212083
$185$	−1.62313e8	−1.88475
$186$	5.09414e7	0.580465
$187$	0	0
$188$	1.81216e7	0.198904
$189$	2.02341e7	0.218006
$190$	2.22089e8	2.34903
$191$	5.96399e7	0.619327	0.309664	−	0.950846i	$-0.399784\pi$
$191$	5.96399e7	0.619327	0.309664	+	0.950846i	$0.399784\pi$
$192$	6.29977e7	0.642348
$193$	9.81036e7	0.982278	0.491139	−	0.871081i	$-0.336581\pi$
$193$	9.81036e7	0.982278	0.491139	+	0.871081i	$0.336581\pi$
$194$	−1.66516e8	−1.63738
$195$	1.43445e8	1.38537
$196$	−6.53075e6	−0.0619536
$197$	1.09317e8	1.01872	0.509361	−	0.860553i	$-0.329882\pi$
$197$	1.09317e8	1.01872	0.509361	+	0.860553i	$0.329882\pi$
$198$	0	0
$199$	−3.64317e7	−0.327713	−0.163857	−	0.986484i	$-0.552393\pi$
$199$	−3.64317e7	−0.327713	−0.163857	+	0.986484i	$0.552393\pi$
$200$	1.40361e8	1.24063
$201$	1.24000e7	0.107705
$202$	−2.08327e7	−0.177834
$203$	1.91880e8	1.60988
$204$	1.28989e7	0.106377
$205$	−1.94939e7	−0.158037
$206$	2.39046e7	0.190523
$207$	−8.37475e6	−0.0656260
$208$	1.55703e8	1.19971
$209$	0	0
$210$	1.13800e8	0.847956
$211$	−1.38637e7	−0.101599	−0.0507997	−	0.998709i	$-0.516177\pi$
$211$	−1.38637e7	−0.101599	−0.0507997	+	0.998709i	$0.516177\pi$
$212$	3.67562e7	0.264945
$213$	−487944.	−0.00345972
$214$	−1.40615e8	−0.980805
$215$	2.46856e8	1.69398
$216$	3.07055e7	0.207314
$217$	−1.93955e8	−1.28852
$218$	−1.11321e8	−0.727743
$219$	1.15037e7	0.0740086
$220$	0	0
$221$	2.21089e8	1.37783
$222$	−1.06889e8	−0.655690
$223$	1.35935e8	0.820850	0.410425	−	0.911894i	$-0.365380\pi$
$223$	1.35935e8	0.820850	0.410425	+	0.911894i	$0.365380\pi$
$224$	−8.17466e7	−0.485961
$225$	6.55918e7	0.383893
$226$	−5.66903e7	−0.326685
$227$	−2.82203e7	−0.160129	−0.0800646	−	0.996790i	$-0.525513\pi$
$227$	−2.82203e7	−0.160129	−0.0800646	+	0.996790i	$0.525513\pi$
$228$	−4.09510e7	−0.228819
$229$	−5.31215e7	−0.292312	−0.146156	−	0.989262i	$-0.546690\pi$
$229$	−5.31215e7	−0.292312	−0.146156	+	0.989262i	$0.546690\pi$
$230$	−4.71008e7	−0.255259
$231$	0	0
$232$	2.91180e8	1.53093
$233$	−1.54589e8	−0.800631	−0.400316	−	0.916377i	$-0.631099\pi$
$233$	−1.54589e8	−0.800631	−0.400316	+	0.916377i	$0.631099\pi$
$234$	9.44638e7	0.481959
$235$	2.65352e8	1.33378
$236$	7.50719e7	0.371780
$237$	−8.03963e6	−0.0392299
$238$	1.75397e8	0.843342
$239$	1.86143e8	0.881972	0.440986	−	0.897514i	$-0.354629\pi$
$239$	1.86143e8	0.881972	0.440986	+	0.897514i	$0.354629\pi$
$240$	1.33017e8	0.621110
$241$	−2.62107e8	−1.20620	−0.603100	−	0.797666i	$-0.706068\pi$
$241$	−2.62107e8	−1.20620	−0.603100	+	0.797666i	$0.706068\pi$
$242$	0	0
$243$	1.43489e7	0.0641500
$244$	1.54333e7	0.0680135
$245$	−9.56288e7	−0.415439
$246$	−1.28374e7	−0.0549800
$247$	−7.01909e8	−2.96375
$248$	−2.94328e8	−1.22532
$249$	−1.53469e8	−0.629973
$250$	4.85850e7	0.196658
$251$	−2.75827e8	−1.10098	−0.550489	−	0.834842i	$-0.685559\pi$
$251$	−2.75827e8	−1.10098	−0.550489	+	0.834842i	$0.685559\pi$
$252$	−2.09835e7	−0.0825994
$253$	0	0
$254$	−2.09170e8	−0.800907
$255$	1.88876e8	0.713324
$256$	−1.67117e8	−0.622558
$257$	1.06856e6	0.00392675	0.00196338	−	0.999998i	$-0.499375\pi$
$257$	1.06856e6	0.00392675	0.00196338	+	0.999998i	$0.499375\pi$
$258$	1.62564e8	0.589325
$259$	4.06971e8	1.45551
$260$	−1.48758e8	−0.524896
$261$	1.36071e8	0.473721
$262$	−1.12649e8	−0.386966
$263$	7.92924e7	0.268774	0.134387	−	0.990929i	$-0.457094\pi$
$263$	7.92924e7	0.268774	0.134387	+	0.990929i	$0.457094\pi$
$264$	0	0
$265$	5.38216e8	1.77663
$266$	−5.56847e8	−1.81405
$267$	−1.71260e8	−0.550639
$268$	−1.28593e7	−0.0408080
$269$	2.10170e8	0.658321	0.329160	−	0.944274i	$-0.393234\pi$
$269$	2.10170e8	0.658321	0.329160	+	0.944274i	$0.393234\pi$
$270$	8.07003e7	0.249518
$271$	2.65510e8	0.810378	0.405189	−	0.914233i	$-0.367206\pi$
$271$	2.65510e8	0.810378	0.405189	+	0.914233i	$0.367206\pi$
$272$	2.05017e8	0.617730
$273$	−3.59662e8	−1.06986
$274$	−4.44290e6	−0.0130479
$275$	0	0
$276$	8.68493e6	0.0248648
$277$	6.23529e8	1.76270	0.881349	−	0.472466i	$-0.156636\pi$
$277$	6.23529e8	1.76270	0.881349	+	0.472466i	$0.156636\pi$
$278$	3.42613e8	0.956415
$279$	−1.37542e8	−0.379158
$280$	−6.57509e8	−1.78998
$281$	−1.30611e8	−0.351162	−0.175581	−	0.984465i	$-0.556180\pi$
$281$	−1.30611e8	−0.351162	−0.175581	+	0.984465i	$0.556180\pi$
$282$	1.74744e8	0.464013
$283$	2.20874e7	0.0579283	0.0289642	−	0.999580i	$-0.490779\pi$
$283$	2.20874e7	0.0579283	0.0289642	+	0.999580i	$0.490779\pi$
$284$	506016.	0.00131084
$285$	−5.99640e8	−1.53438
$286$	0	0
$287$	4.88773e7	0.122045
$288$	−5.79701e7	−0.142998
$289$	−1.19227e8	−0.290557
$290$	7.65281e8	1.84259
$291$	4.49593e8	1.06953
$292$	−1.19297e7	−0.0280408
$293$	−2.00188e8	−0.464944	−0.232472	−	0.972603i	$-0.574681\pi$
$293$	−2.00188e8	−0.464944	−0.232472	+	0.972603i	$0.574681\pi$
$294$	−6.29751e7	−0.144528
$295$	1.09927e9	2.49302
$296$	6.17582e8	1.38412
$297$	0	0
$298$	−4.82211e8	−1.05555
$299$	1.48862e8	0.322057
$300$	−6.80211e7	−0.145452
$301$	−6.18946e8	−1.30819
$302$	−4.48693e8	−0.937399
$303$	5.62483e7	0.116161
$304$	−6.50883e8	−1.32876
$305$	2.25988e8	0.456074
$306$	1.24382e8	0.248160
$307$	4.79736e8	0.946276	0.473138	−	0.880988i	$-0.343121\pi$
$307$	4.79736e8	0.946276	0.473138	+	0.880988i	$0.343121\pi$
$308$	0	0
$309$	−6.45425e7	−0.124449
$310$	−7.73555e8	−1.47477
$311$	−5.19734e8	−0.979761	−0.489880	−	0.871790i	$-0.662960\pi$
$311$	−5.19734e8	−0.979761	−0.489880	+	0.871790i	$0.662960\pi$
$312$	−5.45791e8	−1.01738
$313$	9.69759e8	1.78755	0.893776	−	0.448514i	$-0.148047\pi$
$313$	9.69759e8	1.78755	0.893776	+	0.448514i	$0.148047\pi$
$314$	5.38907e7	0.0982335
$315$	−3.07259e8	−0.553882
$316$	8.33739e6	0.0148636
$317$	7.56875e8	1.33450	0.667248	−	0.744836i	$-0.267472\pi$
$317$	7.56875e8	1.33450	0.667248	+	0.744836i	$0.267472\pi$
$318$	3.54435e8	0.618076
$319$	0	0
$320$	−9.56632e8	−1.63200
$321$	3.79660e8	0.640659
$322$	1.18097e8	0.197125
$323$	−9.24214e8	−1.52603
$324$	−1.48803e7	−0.0243056
$325$	−1.16590e9	−1.88394
$326$	−9.81674e8	−1.56930
$327$	3.00566e8	0.475360
$328$	7.41718e7	0.116059
$329$	−6.65322e8	−1.03002
$330$	0	0
$331$	−1.79867e8	−0.272618	−0.136309	−	0.990666i	$-0.543524\pi$
$331$	−1.79867e8	−0.272618	−0.136309	+	0.990666i	$0.543524\pi$
$332$	1.59153e8	0.238688
$333$	2.88601e8	0.428295
$334$	−4.40611e8	−0.647058
$335$	−1.88297e8	−0.273644
$336$	−3.33516e8	−0.479655
$337$	1.38092e9	1.96546	0.982728	−	0.185054i	$-0.0592461\pi$
$337$	1.38092e9	1.96546	0.982728	+	0.185054i	$0.0592461\pi$
$338$	−1.05161e9	−1.48131
$339$	1.53064e8	0.213390
$340$	−1.95872e8	−0.270269
$341$	0	0
$342$	−3.94885e8	−0.533800
$343$	−6.06830e8	−0.811966
$344$	−9.39257e8	−1.24403
$345$	1.27172e8	0.166734
$346$	6.71087e8	0.870989
$347$	−7.66253e8	−0.984507	−0.492254	−	0.870452i	$-0.663827\pi$
$347$	−7.66253e8	−0.984507	−0.492254	+	0.870452i	$0.663827\pi$
$348$	−1.41110e8	−0.179486
$349$	2.68852e8	0.338552	0.169276	−	0.985569i	$-0.445857\pi$
$349$	2.68852e8	0.338552	0.169276	+	0.985569i	$0.445857\pi$
$350$	−9.24943e8	−1.15313
$351$	−2.55052e8	−0.314814
$352$	0	0
$353$	−3.95002e8	−0.477956	−0.238978	−	0.971025i	$-0.576812\pi$
$353$	−3.95002e8	−0.477956	−0.238978	+	0.971025i	$0.576812\pi$
$354$	7.23908e8	0.867305
$355$	7.40952e6	0.00879004
$356$	1.77603e8	0.208629
$357$	−4.73573e8	−0.550869
$358$	−4.34929e7	−0.0500989
$359$	4.25768e7	0.0485671	0.0242836	−	0.999705i	$-0.492270\pi$
$359$	4.25768e7	0.0485671	0.0242836	+	0.999705i	$0.492270\pi$
$360$	−4.66268e8	−0.526717
$361$	2.04030e9	2.28254
$362$	1.20238e8	0.133218
$363$	0	0
$364$	3.72983e8	0.405354
$365$	−1.74685e8	−0.188032
$366$	1.48821e8	0.158665
$367$	1.85295e9	1.95673	0.978366	−	0.206882i	$-0.0663315\pi$
$367$	1.85295e9	1.95673	0.978366	+	0.206882i	$0.0663315\pi$
$368$	1.38040e8	0.144390
$369$	3.46610e7	0.0359128
$370$	1.62313e9	1.66590
$371$	−1.34948e9	−1.37201
$372$	1.42636e8	0.143658
$373$	4.83602e7	0.0482511	0.0241256	−	0.999709i	$-0.492320\pi$
$373$	4.83602e7	0.0482511	0.0241256	+	0.999709i	$0.492320\pi$
$374$	0	0
$375$	−1.31180e8	−0.128457
$376$	−1.00963e9	−0.979503
$377$	−2.41866e9	−2.32477
$378$	−2.02341e8	−0.192692
$379$	−2.26078e8	−0.213315	−0.106658	−	0.994296i	$-0.534015\pi$
$379$	−2.26078e8	−0.213315	−0.106658	+	0.994296i	$0.534015\pi$
$380$	6.21849e8	0.581356
$381$	5.64760e8	0.523150
$382$	−5.96399e8	−0.547413
$383$	−1.35198e9	−1.22963	−0.614815	−	0.788671i	$-0.710769\pi$
$383$	−1.35198e9	−1.22963	−0.614815	+	0.788671i	$0.710769\pi$
$384$	−3.55156e8	−0.320081
$385$	0	0
$386$	−9.81036e8	−0.868219
$387$	−4.38922e8	−0.384945
$388$	−4.66244e8	−0.405231
$389$	−1.09107e9	−0.939789	−0.469894	−	0.882723i	$-0.655708\pi$
$389$	−1.09107e9	−0.939789	−0.469894	+	0.882723i	$0.655708\pi$
$390$	−1.43445e9	−1.22450
$391$	1.96008e8	0.165827
$392$	3.63856e8	0.305090
$393$	3.04152e8	0.252765
$394$	−1.09317e9	−0.900431
$395$	1.22083e8	0.0996704
$396$	0	0
$397$	−6.97868e8	−0.559766	−0.279883	−	0.960034i	$-0.590296\pi$
$397$	−6.97868e8	−0.559766	−0.279883	+	0.960034i	$0.590296\pi$
$398$	3.64317e8	0.289660
$399$	1.50349e9	1.18494
$400$	−1.08114e9	−0.844640
$401$	1.74689e9	1.35288	0.676441	−	0.736497i	$-0.263521\pi$
$401$	1.74689e9	1.35288	0.676441	+	0.736497i	$0.263521\pi$
$402$	−1.24000e8	−0.0951987
$403$	2.44481e9	1.86071
$404$	−5.83316e7	−0.0440118
$405$	−2.17891e8	−0.162985
$406$	−1.91880e9	−1.42295
$407$	0	0
$408$	−7.18651e8	−0.523851
$409$	1.30304e9	0.941729	0.470865	−	0.882205i	$-0.343942\pi$
$409$	1.30304e9	0.941729	0.470865	+	0.882205i	$0.343942\pi$
$410$	1.94939e8	0.139686
$411$	1.19958e7	0.00852283
$412$	6.69330e7	0.0471520
$413$	−2.75621e9	−1.92525
$414$	8.37475e7	0.0580057
$415$	2.33045e9	1.60056
$416$	1.03042e9	0.701759
$417$	−9.25054e8	−0.624728
$418$	0	0
$419$	2.87139e9	1.90697	0.953484	−	0.301443i	$-0.0974684\pi$
$419$	2.87139e9	1.90697	0.953484	+	0.301443i	$0.0974684\pi$
$420$	3.18639e8	0.209858
$421$	1.15946e9	0.757299	0.378650	−	0.925540i	$-0.376389\pi$
$421$	1.15946e9	0.757299	0.378650	+	0.925540i	$0.376389\pi$
$422$	1.38637e8	0.0898020
$423$	−4.71809e8	−0.303092
$424$	−2.04785e9	−1.30472
$425$	−1.53515e9	−0.970042
$426$	4.87944e6	0.00305799
$427$	−5.66623e8	−0.352206
$428$	−3.93721e8	−0.242737
$429$	0	0
$430$	−2.46856e9	−1.49728
$431$	1.66703e9	1.00294	0.501468	−	0.865176i	$-0.332793\pi$
$431$	1.66703e9	1.00294	0.501468	+	0.865176i	$0.332793\pi$
$432$	−2.36511e8	−0.141143
$433$	6.34094e8	0.375358	0.187679	−	0.982230i	$-0.439903\pi$
$433$	6.34094e8	0.375358	0.187679	+	0.982230i	$0.439903\pi$
$434$	1.93955e9	1.13890
$435$	−2.06626e9	−1.20357
$436$	−3.11698e8	−0.180107
$437$	−6.22282e8	−0.356699
$438$	−1.15037e8	−0.0654150
$439$	1.22368e9	0.690307	0.345154	−	0.938546i	$-0.387827\pi$
$439$	1.22368e9	0.690307	0.345154	+	0.938546i	$0.387827\pi$
$440$	0	0
$441$	1.70033e8	0.0944055
$442$	−2.21089e9	−1.21784
$443$	−1.23213e9	−0.673355	−0.336677	−	0.941620i	$-0.609303\pi$
$443$	−1.23213e9	−0.673355	−0.336677	+	0.941620i	$0.609303\pi$
$444$	−2.99290e8	−0.162275
$445$	2.60062e9	1.39900
$446$	−1.35935e9	−0.725536
$447$	1.30197e9	0.689485
$448$	2.39858e9	1.26032
$449$	−3.07511e9	−1.60324	−0.801621	−	0.597833i	$-0.796029\pi$
$449$	−3.07511e9	−1.60324	−0.801621	+	0.597833i	$0.796029\pi$
$450$	−6.55918e8	−0.339317
$451$	0	0
$452$	−1.58733e8	−0.0808505
$453$	1.21147e9	0.612307
$454$	2.82203e8	0.141536
$455$	5.46154e9	2.71816
$456$	2.28156e9	1.12682
$457$	2.44730e9	1.19945	0.599723	−	0.800207i	$-0.295277\pi$
$457$	2.44730e9	1.19945	0.599723	+	0.800207i	$0.295277\pi$
$458$	5.31215e8	0.258370
$459$	−3.35831e8	−0.162098
$460$	−1.31882e8	−0.0631733
$461$	−9.52419e8	−0.452767	−0.226383	−	0.974038i	$-0.572690\pi$
$461$	−9.52419e8	−0.452767	−0.226383	+	0.974038i	$0.572690\pi$
$462$	0	0
$463$	−6.05200e8	−0.283378	−0.141689	−	0.989911i	$-0.545253\pi$
$463$	−6.05200e8	−0.283378	−0.141689	+	0.989911i	$0.545253\pi$
$464$	−2.24283e9	−1.04228
$465$	2.08860e9	0.963318
$466$	1.54589e9	0.707665
$467$	−1.37708e9	−0.625676	−0.312838	−	0.949806i	$-0.601280\pi$
$467$	−1.37708e9	−0.625676	−0.312838	+	0.949806i	$0.601280\pi$
$468$	2.64499e8	0.119279
$469$	4.72119e8	0.211323
$470$	−2.65352e9	−1.17891
$471$	−1.45505e8	−0.0641659
$472$	−4.18258e9	−1.83083
$473$	0	0
$474$	8.03963e7	0.0346746
$475$	4.87377e9	2.08659
$476$	4.91113e8	0.208716
$477$	−9.56974e8	−0.403725
$478$	−1.86143e9	−0.779561
$479$	4.00222e9	1.66390	0.831949	−	0.554851i	$-0.187225\pi$
$479$	4.00222e9	1.66390	0.831949	+	0.554851i	$0.187225\pi$
$480$	8.80286e8	0.363312
$481$	−5.12989e9	−2.10184
$482$	2.62107e9	1.06614
$483$	−3.18861e8	−0.128762
$484$	0	0
$485$	−6.82715e9	−2.71734
$486$	−1.43489e8	−0.0567012
$487$	−2.88677e9	−1.13256	−0.566279	−	0.824214i	$-0.691617\pi$
$487$	−2.88677e9	−1.13256	−0.566279	+	0.824214i	$0.691617\pi$
$488$	−8.59856e8	−0.334932
$489$	2.65052e9	1.02506
$490$	9.56288e8	0.367200
$491$	−1.19743e8	−0.0456525	−0.0228262	−	0.999739i	$-0.507266\pi$
$491$	−1.19743e8	−0.0456525	−0.0228262	+	0.999739i	$0.507266\pi$
$492$	−3.59448e7	−0.0136069
$493$	−3.18469e9	−1.19702
$494$	7.01909e9	2.61961
$495$	0	0
$496$	2.26708e9	0.834222
$497$	−1.85780e7	−0.00678816
$498$	1.53469e9	0.556823
$499$	−4.78950e9	−1.72559	−0.862796	−	0.505552i	$-0.831289\pi$
$499$	−4.78950e9	−1.72559	−0.862796	+	0.505552i	$0.831289\pi$
$500$	1.36038e8	0.0486704
$501$	1.18965e9	0.422656
$502$	2.75827e9	0.973137
$503$	−3.83047e9	−1.34203	−0.671017	−	0.741442i	$-0.734142\pi$
$503$	−3.83047e9	−1.34203	−0.671017	+	0.741442i	$0.734142\pi$
$504$	1.16908e9	0.406760
$505$	−8.54141e8	−0.295127
$506$	0	0
$507$	2.83935e9	0.967591
$508$	−5.85677e8	−0.198214
$509$	−2.34385e9	−0.787803	−0.393902	−	0.919153i	$-0.628875\pi$
$509$	−2.34385e9	−0.787803	−0.393902	+	0.919153i	$0.628875\pi$
$510$	−1.88876e9	−0.630495
$511$	4.37992e8	0.145209
$512$	3.35487e9	1.10466
$513$	1.06619e9	0.348677
$514$	−1.06856e7	−0.00347079
$515$	9.80090e8	0.316185
$516$	4.55179e8	0.145850
$517$	0	0
$518$	−4.06971e9	−1.28650
$519$	−1.81194e9	−0.568928
$520$	8.28794e9	2.58485
$521$	5.77085e9	1.78775	0.893877	−	0.448313i	$-0.147975\pi$
$521$	5.77085e9	1.78775	0.893877	+	0.448313i	$0.147975\pi$
$522$	−1.36071e9	−0.418715
$523$	3.49411e8	0.106802	0.0534012	−	0.998573i	$-0.482994\pi$
$523$	3.49411e8	0.106802	0.0534012	+	0.998573i	$0.482994\pi$
$524$	−3.15417e8	−0.0957691
$525$	2.49735e9	0.753219
$526$	−7.92924e8	−0.237565
$527$	3.21912e9	0.958077
$528$	0	0
$529$	−3.27285e9	−0.961239
$530$	−5.38216e9	−1.57033
$531$	−1.95455e9	−0.566521
$532$	−1.55917e9	−0.448955
$533$	−6.16101e8	−0.176241
$534$	1.71260e9	0.486700
$535$	−5.76520e9	−1.62771
$536$	7.16446e8	0.200959
$537$	1.17431e8	0.0327244
$538$	−2.10170e9	−0.581879
$539$	0	0
$540$	2.25961e8	0.0617526
$541$	5.10025e9	1.38484	0.692422	−	0.721493i	$-0.256544\pi$
$541$	5.10025e9	1.38484	0.692422	+	0.721493i	$0.256544\pi$
$542$	−2.65510e9	−0.716280
$543$	−3.24643e8	−0.0870175
$544$	1.35677e9	0.361335
$545$	−4.56415e9	−1.20774
$546$	3.59662e9	0.945629
$547$	−4.96217e9	−1.29633	−0.648166	−	0.761499i	$-0.724464\pi$
$547$	−4.96217e9	−1.29633	−0.648166	+	0.761499i	$0.724464\pi$
$548$	−1.24401e7	−0.00322918
$549$	−4.01818e8	−0.103640
$550$	0	0
$551$	1.01107e10	2.57484
$552$	−4.83875e8	−0.122446
$553$	−3.06101e8	−0.0769710
$554$	−6.23529e9	−1.55802
$555$	−4.38246e9	−1.08816
$556$	9.59315e8	0.236701
$557$	−1.42590e9	−0.349620	−0.174810	−	0.984602i	$-0.555931\pi$
$557$	−1.42590e9	−0.349620	−0.174810	+	0.984602i	$0.555931\pi$
$558$	1.37542e9	0.335132
$559$	7.80186e9	1.88911
$560$	5.06450e9	1.21865
$561$	0	0
$562$	1.30611e9	0.310386
$563$	−5.96929e9	−1.40975	−0.704876	−	0.709330i	$-0.748998\pi$
$563$	−5.96929e9	−1.40975	−0.704876	+	0.709330i	$0.748998\pi$
$564$	4.89283e8	0.114837
$565$	−2.32430e9	−0.542155
$566$	−2.20874e8	−0.0512019
$567$	5.46321e8	0.125866
$568$	−2.81923e7	−0.00645523
$569$	3.51616e9	0.800158	0.400079	−	0.916481i	$-0.368983\pi$
$569$	3.51616e9	0.800158	0.400079	+	0.916481i	$0.368983\pi$
$570$	5.99640e9	1.35621
$571$	−6.44706e8	−0.144922	−0.0724611	−	0.997371i	$-0.523085\pi$
$571$	−6.44706e8	−0.144922	−0.0724611	+	0.997371i	$0.523085\pi$
$572$	0	0
$573$	1.61028e9	0.357569
$574$	−4.88773e8	−0.107874
$575$	−1.03363e9	−0.226740
$576$	1.70094e9	0.370860
$577$	−2.63322e9	−0.570652	−0.285326	−	0.958430i	$-0.592102\pi$
$577$	−2.63322e9	−0.570652	−0.285326	+	0.958430i	$0.592102\pi$
$578$	1.19227e9	0.256819
$579$	2.64880e9	0.567118
$580$	2.14279e9	0.456017
$581$	−5.84318e9	−1.23604
$582$	−4.49593e9	−0.945342
$583$	0	0
$584$	6.64657e8	0.138087
$585$	3.87302e9	0.799841
$586$	2.00188e9	0.410956
$587$	6.76347e9	1.38018	0.690090	−	0.723723i	$-0.257571\pi$
$587$	6.76347e9	1.38018	0.690090	+	0.723723i	$0.257571\pi$
$588$	−1.76330e8	−0.0357689
$589$	−1.02200e10	−2.06085
$590$	−1.09927e10	−2.20354
$591$	2.95156e9	0.588159
$592$	−4.75697e9	−0.942332
$593$	4.22718e9	0.832452	0.416226	−	0.909261i	$-0.363352\pi$
$593$	4.22718e9	0.832452	0.416226	+	0.909261i	$0.363352\pi$
$594$	0	0
$595$	7.19129e9	1.39958
$596$	−1.35019e9	−0.261236
$597$	−9.83657e8	−0.189205
$598$	−1.48862e9	−0.284661
$599$	4.00299e9	0.761010	0.380505	−	0.924779i	$-0.375750\pi$
$599$	4.00299e9	0.761010	0.380505	+	0.924779i	$0.375750\pi$
$600$	3.78975e9	0.716277
$601$	−6.67554e9	−1.25437	−0.627185	−	0.778870i	$-0.715793\pi$
$601$	−6.67554e9	−1.25437	−0.627185	+	0.778870i	$0.715793\pi$
$602$	6.18946e9	1.15629
$603$	3.34801e8	0.0621836
$604$	−1.25634e9	−0.231994
$605$	0	0
$606$	−5.62483e8	−0.102673
$607$	5.30634e9	0.963018	0.481509	−	0.876441i	$-0.340089\pi$
$607$	5.30634e9	0.963018	0.481509	+	0.876441i	$0.340089\pi$
$608$	−4.30744e9	−0.777243
$609$	5.18077e9	0.929466
$610$	−2.25988e9	−0.403117
$611$	8.38642e9	1.48742
$612$	3.48270e8	0.0614166
$613$	8.65802e9	1.51812	0.759061	−	0.651019i	$-0.225658\pi$
$613$	8.65802e9	1.51812	0.759061	+	0.651019i	$0.225658\pi$
$614$	−4.79736e9	−0.836398
$615$	−5.26334e8	−0.0912428
$616$	0	0
$617$	−7.38891e9	−1.26643	−0.633217	−	0.773974i	$-0.718266\pi$
$617$	−7.38891e9	−1.26643	−0.633217	+	0.773974i	$0.718266\pi$
$618$	6.45425e8	0.109998
$619$	9.99141e9	1.69321	0.846603	−	0.532225i	$-0.178644\pi$
$619$	9.99141e9	1.69321	0.846603	+	0.532225i	$0.178644\pi$
$620$	−2.16595e9	−0.364988
$621$	−2.26118e8	−0.0378892
$622$	5.19734e9	0.865994
$623$	−6.52057e9	−1.08038
$624$	4.20399e9	0.692653
$625$	−5.03731e9	−0.825313
$626$	−9.69759e9	−1.57999
$627$	0	0
$628$	1.50894e8	0.0243116
$629$	−6.75461e9	−1.08224
$630$	3.07259e9	0.489567
$631$	−3.29834e9	−0.522628	−0.261314	−	0.965254i	$-0.584156\pi$
$631$	−3.29834e9	−0.522628	−0.261314	+	0.965254i	$0.584156\pi$
$632$	−4.64512e8	−0.0731959
$633$	−3.74320e8	−0.0586584
$634$	−7.56875e9	−1.17954
$635$	−8.57598e9	−1.32916
$636$	9.92418e8	0.152966
$637$	−3.02234e9	−0.463292
$638$	0	0
$639$	−1.31745e7	−0.00199747
$640$	5.39311e9	0.813222
$641$	−9.76971e9	−1.46514	−0.732569	−	0.680692i	$-0.761679\pi$
$641$	−9.76971e9	−1.46514	−0.732569	+	0.680692i	$0.761679\pi$
$642$	−3.79660e9	−0.566268
$643$	−4.18444e9	−0.620724	−0.310362	−	0.950618i	$-0.600450\pi$
$643$	−4.18444e9	−0.620724	−0.310362	+	0.950618i	$0.600450\pi$
$644$	3.30671e8	0.0487860
$645$	6.66511e9	0.978022
$646$	9.24214e9	1.34884
$647$	−6.96085e8	−0.101041	−0.0505204	−	0.998723i	$-0.516088\pi$
$647$	−6.96085e8	−0.101041	−0.0505204	+	0.998723i	$0.516088\pi$
$648$	8.29048e8	0.119693
$649$	0	0
$650$	1.16590e10	1.66519
$651$	−5.23678e9	−0.743928
$652$	−2.74869e9	−0.388382
$653$	6.20046e9	0.871420	0.435710	−	0.900087i	$-0.356497\pi$
$653$	6.20046e9	0.871420	0.435710	+	0.900087i	$0.356497\pi$
$654$	−3.00566e9	−0.420163
$655$	−4.61861e9	−0.642194
$656$	−5.71313e8	−0.0790152
$657$	3.10599e8	0.0427289
$658$	6.65322e9	0.910418
$659$	1.11404e10	1.51636	0.758178	−	0.652047i	$-0.226090\pi$
$659$	1.11404e10	1.51636	0.758178	+	0.652047i	$0.226090\pi$
$660$	0	0
$661$	4.56096e9	0.614258	0.307129	−	0.951668i	$-0.400632\pi$
$661$	4.56096e9	0.614258	0.307129	+	0.951668i	$0.400632\pi$
$662$	1.79867e9	0.240963
$663$	5.96941e9	0.795489
$664$	−8.86708e9	−1.17542
$665$	−2.28307e10	−3.01054
$666$	−2.88601e9	−0.378563
$667$	−2.14428e9	−0.279796
$668$	−1.23371e9	−0.160139
$669$	3.67024e9	0.473918
$670$	1.88297e9	0.241869
$671$	0	0
$672$	−2.20716e9	−0.280570
$673$	−5.82879e9	−0.737099	−0.368550	−	0.929608i	$-0.620146\pi$
$673$	−5.82879e9	−0.737099	−0.368550	+	0.929608i	$0.620146\pi$
$674$	−1.38092e10	−1.73723
$675$	1.77098e9	0.221641
$676$	−2.94451e9	−0.366607
$677$	−4.99624e9	−0.618846	−0.309423	−	0.950924i	$-0.600136\pi$
$677$	−4.99624e9	−0.618846	−0.309423	+	0.950924i	$0.600136\pi$
$678$	−1.53064e9	−0.188612
$679$	1.71178e10	2.09848
$680$	1.09129e10	1.33094
$681$	−7.61947e8	−0.0924507
$682$	0	0
$683$	1.21371e10	1.45762	0.728808	−	0.684718i	$-0.240075\pi$
$683$	1.21371e10	1.45762	0.728808	+	0.684718i	$0.240075\pi$
$684$	−1.10568e9	−0.132109
$685$	−1.82159e8	−0.0216538
$686$	6.06830e9	0.717684
$687$	−1.43428e9	−0.168766
$688$	7.23469e9	0.846955
$689$	1.70103e10	1.98127
$690$	−1.27172e9	−0.147374
$691$	−9.23403e9	−1.06468	−0.532339	−	0.846531i	$-0.678687\pi$
$691$	−9.23403e9	−1.06468	−0.532339	+	0.846531i	$0.678687\pi$
$692$	1.87904e9	0.215559
$693$	0	0
$694$	7.66253e9	0.870190
$695$	1.40471e10	1.58723
$696$	7.86187e9	0.883880
$697$	−8.11230e8	−0.0907464
$698$	−2.68852e9	−0.299240
$699$	−4.17390e9	−0.462245
$700$	−2.58984e9	−0.285384
$701$	−4.74530e9	−0.520296	−0.260148	−	0.965569i	$-0.583771\pi$
$701$	−4.74530e9	−0.520296	−0.260148	+	0.965569i	$0.583771\pi$
$702$	2.55052e9	0.278259
$703$	2.14444e10	2.32793
$704$	0	0
$705$	7.16450e9	0.770059
$706$	3.95002e9	0.422458
$707$	2.14160e9	0.227914
$708$	2.02694e9	0.214647
$709$	1.34547e10	1.41779	0.708894	−	0.705315i	$-0.249195\pi$
$709$	1.34547e10	1.41779	0.708894	+	0.705315i	$0.249195\pi$
$710$	−7.40952e7	−0.00776937
$711$	−2.17070e8	−0.0226494
$712$	−9.89503e9	−1.02739
$713$	2.16746e9	0.223944
$714$	4.73573e9	0.486904
$715$	0	0
$716$	−1.21780e8	−0.0123988
$717$	5.02587e9	0.509207
$718$	−4.25768e8	−0.0429277
$719$	2.63976e9	0.264858	0.132429	−	0.991192i	$-0.457722\pi$
$719$	2.63976e9	0.264858	0.132429	+	0.991192i	$0.457722\pi$
$720$	3.59146e9	0.358598
$721$	−2.45740e9	−0.244175
$722$	−2.04030e10	−2.01750
$723$	−7.07689e9	−0.696400
$724$	3.36667e8	0.0329697
$725$	1.67942e10	1.63673
$726$	0	0
$727$	−3.52707e9	−0.340442	−0.170221	−	0.985406i	$-0.554448\pi$
$727$	−3.52707e9	−0.340442	−0.170221	+	0.985406i	$0.554448\pi$
$728$	−2.07805e10	−1.99616
$729$	3.87420e8	0.0370370
$730$	1.74685e9	0.166198
$731$	1.02728e10	0.972700
$732$	4.16700e8	0.0392676
$733$	1.03828e10	0.973760	0.486880	−	0.873469i	$-0.338135\pi$
$733$	1.03828e10	0.973760	0.486880	+	0.873469i	$0.338135\pi$
$734$	−1.85295e10	−1.72952
$735$	−2.58198e9	−0.239854
$736$	9.13526e8	0.0844595
$737$	0	0
$738$	−3.46610e8	−0.0317427
$739$	−2.05418e9	−0.187233	−0.0936164	−	0.995608i	$-0.529843\pi$
$739$	−2.05418e9	−0.187233	−0.0936164	+	0.995608i	$0.529843\pi$
$740$	4.54477e9	0.412288
$741$	−1.89515e10	−1.71112
$742$	1.34948e10	1.21270
$743$	−4.87476e9	−0.436006	−0.218003	−	0.975948i	$-0.569954\pi$
$743$	−4.87476e9	−0.436006	−0.218003	+	0.975948i	$0.569954\pi$
$744$	−7.94686e9	−0.707442
$745$	−1.97707e10	−1.75176
$746$	−4.83602e8	−0.0426484
$747$	−4.14366e9	−0.363715
$748$	0	0
$749$	1.44552e10	1.25701
$750$	1.31180e9	0.113541
$751$	1.15809e10	0.997705	0.498853	−	0.866687i	$-0.333755\pi$
$751$	1.15809e10	0.997705	0.498853	+	0.866687i	$0.333755\pi$
$752$	7.77676e9	0.666862
$753$	−7.44733e9	−0.635650
$754$	2.41866e10	2.05483
$755$	−1.83964e10	−1.55567
$756$	−5.66555e8	−0.0476888
$757$	3.46735e9	0.290511	0.145255	−	0.989394i	$-0.453600\pi$
$757$	3.46735e9	0.290511	0.145255	+	0.989394i	$0.453600\pi$
$758$	2.26078e9	0.188546
$759$	0	0
$760$	−3.46459e10	−2.86288
$761$	1.14023e10	0.937877	0.468938	−	0.883231i	$-0.344637\pi$
$761$	1.14023e10	0.937877	0.468938	+	0.883231i	$0.344637\pi$
$762$	−5.64760e9	−0.462404
$763$	1.14438e10	0.932681
$764$	−1.66992e9	−0.135478
$765$	5.09966e9	0.411838
$766$	1.35198e10	1.08685
$767$	3.47422e10	2.78018
$768$	−4.51215e9	−0.359434
$769$	−2.30715e10	−1.82951	−0.914754	−	0.404012i	$-0.867616\pi$
$769$	−2.30715e10	−1.82951	−0.914754	+	0.404012i	$0.867616\pi$
$770$	0	0
$771$	2.88512e7	0.00226711
$772$	−2.74690e9	−0.214873
$773$	2.15091e10	1.67492	0.837461	−	0.546497i	$-0.184039\pi$
$773$	2.15091e10	1.67492	0.837461	+	0.546497i	$0.184039\pi$
$774$	4.38922e9	0.340247
$775$	−1.69758e10	−1.31001
$776$	2.59765e10	1.99556
$777$	1.09882e10	0.840337
$778$	1.09107e10	0.830664
$779$	2.57547e9	0.195198
$780$	−4.01646e9	−0.303049
$781$	0	0
$782$	−1.96008e9	−0.146572
$783$	3.67391e9	0.273503
$784$	−2.80262e9	−0.207711
$785$	2.20952e9	0.163025
$786$	−3.04152e9	−0.223415
$787$	4.46678e9	0.326651	0.163325	−	0.986572i	$-0.447778\pi$
$787$	4.46678e9	0.326651	0.163325	+	0.986572i	$0.447778\pi$
$788$	−3.06087e9	−0.222845
$789$	2.14090e9	0.155176
$790$	−1.22083e9	−0.0880970
$791$	5.82777e9	0.418682
$792$	0	0
$793$	7.14232e9	0.508608
$794$	6.97868e9	0.494768
$795$	1.45318e10	1.02574
$796$	1.02009e9	0.0716873
$797$	2.43899e10	1.70650	0.853248	−	0.521505i	$-0.174629\pi$
$797$	2.43899e10	1.70650	0.853248	+	0.521505i	$0.174629\pi$
$798$	−1.50349e10	−1.04734
$799$	1.10425e10	0.765869
$800$	−7.15481e9	−0.494064
$801$	−4.62402e9	−0.317911
$802$	−1.74689e10	−1.19579
$803$	0	0
$804$	−3.47201e8	−0.0235605
$805$	4.84196e9	0.327142
$806$	−2.44481e10	−1.64465
$807$	5.67459e9	0.380082
$808$	3.24990e9	0.216736
$809$	−9.88857e9	−0.656620	−0.328310	−	0.944570i	$-0.606479\pi$
$809$	−9.88857e9	−0.656620	−0.328310	+	0.944570i	$0.606479\pi$
$810$	2.17891e9	0.144059
$811$	1.15204e10	0.758395	0.379198	−	0.925316i	$-0.376200\pi$
$811$	1.15204e10	0.758395	0.379198	+	0.925316i	$0.376200\pi$
$812$	−5.37265e9	−0.352162
$813$	7.16876e9	0.467872
$814$	0	0
$815$	−4.02487e10	−2.60435
$816$	5.53546e9	0.356647
$817$	−3.26139e10	−2.09231
$818$	−1.30304e10	−0.832379
$819$	−9.71088e9	−0.617682
$820$	5.45828e8	0.0345706
$821$	−2.63516e9	−0.166191	−0.0830953	−	0.996542i	$-0.526481\pi$
$821$	−2.63516e9	−0.166191	−0.0830953	+	0.996542i	$0.526481\pi$
$822$	−1.19958e8	−0.00753319
$823$	−1.27039e10	−0.794400	−0.397200	−	0.917732i	$-0.630018\pi$
$823$	−1.27039e10	−0.794400	−0.397200	+	0.917732i	$0.630018\pi$
$824$	−3.72912e9	−0.232200
$825$	0	0
$826$	2.75621e10	1.70170
$827$	1.11339e10	0.684504	0.342252	−	0.939608i	$-0.388810\pi$
$827$	1.11339e10	0.684504	0.342252	+	0.939608i	$0.388810\pi$
$828$	2.34493e8	0.0143557
$829$	2.79852e10	1.70604	0.853018	−	0.521881i	$-0.174770\pi$
$829$	2.79852e10	1.70604	0.853018	+	0.521881i	$0.174770\pi$
$830$	−2.33045e10	−1.41471
$831$	1.68353e10	1.01769
$832$	−3.02342e10	−1.81998
$833$	−3.97956e9	−0.238549
$834$	9.25054e9	0.552187
$835$	−1.80651e10	−1.07383
$836$	0	0
$837$	−3.71363e9	−0.218907
$838$	−2.87139e10	−1.68554
$839$	2.71170e9	0.158517	0.0792583	−	0.996854i	$-0.474745\pi$
$839$	2.71170e9	0.158517	0.0792583	+	0.996854i	$0.474745\pi$
$840$	−1.77527e10	−1.03345
$841$	1.75898e10	1.01971
$842$	−1.15946e10	−0.669364
$843$	−3.52650e9	−0.202744
$844$	3.88184e8	0.0222249
$845$	−4.31161e10	−2.45834
$846$	4.71809e9	0.267898
$847$	0	0
$848$	1.57737e10	0.888275
$849$	5.96359e8	0.0334449
$850$	1.53515e10	0.857404
$851$	−4.54794e9	−0.252965
$852$	1.36624e7	0.000756814 0
$853$	−1.97175e10	−1.08775	−0.543877	−	0.839165i	$-0.683044\pi$
$853$	−1.97175e10	−1.08775	−0.543877	+	0.839165i	$0.683044\pi$
$854$	5.66623e9	0.311309
$855$	−1.61903e10	−0.885876
$856$	2.19359e10	1.19536
$857$	−1.89411e10	−1.02795	−0.513976	−	0.857804i	$-0.671828\pi$
$857$	−1.89411e10	−1.02795	−0.513976	+	0.857804i	$0.671828\pi$
$858$	0	0
$859$	−6.77637e9	−0.364772	−0.182386	−	0.983227i	$-0.558382\pi$
$859$	−6.77637e9	−0.364772	−0.182386	+	0.983227i	$0.558382\pi$
$860$	−6.91197e9	−0.370559
$861$	1.31969e9	0.0704628
$862$	−1.66703e10	−0.886479
$863$	2.80635e10	1.48629	0.743146	−	0.669129i	$-0.233333\pi$
$863$	2.80635e10	1.48629	0.743146	+	0.669129i	$0.233333\pi$
$864$	−1.56519e9	−0.0825600
$865$	2.75146e10	1.44546
$866$	−6.34094e9	−0.331773
$867$	−3.21912e9	−0.167753
$868$	5.43073e9	0.281864
$869$	0	0
$870$	2.06626e10	1.06382
$871$	−5.95109e9	−0.305164
$872$	1.73660e10	0.886937
$873$	1.21390e10	0.617495
$874$	6.22282e9	0.315281
$875$	−4.99454e9	−0.252039
$876$	−3.22103e8	−0.0161894
$877$	1.01559e10	0.508418	0.254209	−	0.967149i	$-0.418185\pi$
$877$	1.01559e10	0.508418	0.254209	+	0.967149i	$0.418185\pi$
$878$	−1.22368e10	−0.610151
$879$	−5.40507e9	−0.268436
$880$	0	0
$881$	−2.78023e10	−1.36982	−0.684912	−	0.728626i	$-0.740159\pi$
$881$	−2.78023e10	−1.36982	−0.684912	+	0.728626i	$0.740159\pi$
$882$	−1.70033e9	−0.0834435
$883$	2.15199e10	1.05191	0.525954	−	0.850513i	$-0.323709\pi$
$883$	2.15199e10	1.05191	0.525954	+	0.850513i	$0.323709\pi$
$884$	−6.19050e9	−0.301400
$885$	2.96802e10	1.43935
$886$	1.23213e10	0.595167
$887$	3.13376e10	1.50776	0.753881	−	0.657011i	$-0.228180\pi$
$887$	3.13376e10	1.50776	0.753881	+	0.657011i	$0.228180\pi$
$888$	1.66747e10	0.799122
$889$	2.15027e10	1.02645
$890$	−2.60062e10	−1.23655
$891$	0	0
$892$	−3.80618e9	−0.179561
$893$	−3.50575e10	−1.64741
$894$	−1.30197e10	−0.609424
$895$	−1.78321e9	−0.0831423
$896$	−1.35222e10	−0.628015
$897$	4.01926e9	0.185940
$898$	3.07511e10	1.41708
$899$	−3.52164e10	−1.61654
$900$	−1.83657e9	−0.0839767
$901$	2.23977e10	1.02015
$902$	0	0
$903$	−1.67116e10	−0.755283
$904$	8.84369e9	0.398148
$905$	4.92976e9	0.221083
$906$	−1.21147e10	−0.541208
$907$	−1.62459e10	−0.722966	−0.361483	−	0.932379i	$-0.617730\pi$
$907$	−1.62459e10	−0.722966	−0.361483	+	0.932379i	$0.617730\pi$
$908$	7.90168e8	0.0350283
$909$	1.51870e9	0.0670656
$910$	−5.46154e10	−2.40254
$911$	−3.15726e9	−0.138356	−0.0691778	−	0.997604i	$-0.522038\pi$
$911$	−3.15726e9	−0.138356	−0.0691778	+	0.997604i	$0.522038\pi$
$912$	−1.75738e10	−0.767158
$913$	0	0
$914$	−2.44730e10	−1.06017
$915$	6.10167e9	0.263315
$916$	1.48740e9	0.0639432
$917$	1.15803e10	0.495938
$918$	3.35831e9	0.143275
$919$	−2.53655e9	−0.107805	−0.0539026	−	0.998546i	$-0.517166\pi$
$919$	−2.53655e9	−0.107805	−0.0539026	+	0.998546i	$0.517166\pi$
$920$	7.34772e9	0.311097
$921$	1.29529e10	0.546333
$922$	9.52419e9	0.400193
$923$	2.34177e8	0.00980253
$924$	0	0
$925$	3.56198e10	1.47978
$926$	6.05200e9	0.250473
$927$	−1.74265e9	−0.0718506
$928$	−1.48427e10	−0.609671
$929$	−2.10282e10	−0.860494	−0.430247	−	0.902711i	$-0.641574\pi$
$929$	−2.10282e10	−0.860494	−0.430247	+	0.902711i	$0.641574\pi$
$930$	−2.08860e10	−0.851461
$931$	1.26342e10	0.513126
$932$	4.32849e9	0.175138
$933$	−1.40328e10	−0.565665
$934$	1.37708e10	0.553025
$935$	0	0
$936$	−1.47364e10	−0.587387
$937$	−3.12893e10	−1.24253	−0.621265	−	0.783601i	$-0.713381\pi$
$937$	−3.12893e10	−1.24253	−0.621265	+	0.783601i	$0.713381\pi$
$938$	−4.72119e9	−0.186785
$939$	2.61835e10	1.03204
$940$	−7.42986e9	−0.291765
$941$	−7.91706e9	−0.309742	−0.154871	−	0.987935i	$-0.549496\pi$
$941$	−7.91706e9	−0.309742	−0.154871	+	0.987935i	$0.549496\pi$
$942$	1.45505e9	0.0567152
$943$	−5.46208e8	−0.0212113
$944$	3.22166e10	1.24646
$945$	−8.29599e9	−0.319784
$946$	0	0
$947$	8.55849e9	0.327471	0.163735	−	0.986504i	$-0.447646\pi$
$947$	8.55849e9	0.327471	0.163735	+	0.986504i	$0.447646\pi$
$948$	2.25110e8	0.00858153
$949$	−5.52091e9	−0.209691
$950$	−4.87377e10	−1.84430
$951$	2.04356e10	0.770471
$952$	−2.73620e10	−1.02782
$953$	8.49661e9	0.317995	0.158998	−	0.987279i	$-0.449174\pi$
$953$	8.49661e9	0.317995	0.158998	+	0.987279i	$0.449174\pi$
$954$	9.56974e9	0.356846
$955$	−2.44524e10	−0.908466
$956$	−5.21202e9	−0.192931
$957$	0	0
$958$	−4.00222e10	−1.47069
$959$	4.56730e8	0.0167222
$960$	−2.58291e10	−0.942235
$961$	8.08451e9	0.293847
$962$	5.12989e10	1.85778
$963$	1.02508e10	0.369885
$964$	7.33900e9	0.263856
$965$	−4.02225e10	−1.44086
$966$	3.18861e9	0.113810
$967$	1.47988e10	0.526300	0.263150	−	0.964755i	$-0.415239\pi$
$967$	1.47988e10	0.526300	0.263150	+	0.964755i	$0.415239\pi$
$968$	0	0
$969$	−2.49538e10	−0.881056
$970$	6.82715e10	2.40181
$971$	−2.86157e10	−1.00308	−0.501542	−	0.865133i	$-0.667234\pi$
$971$	−2.86157e10	−1.00308	−0.501542	+	0.865133i	$0.667234\pi$
$972$	−4.01769e8	−0.0140328
$973$	−3.52206e10	−1.22575
$974$	2.88677e10	1.00105
$975$	−3.14792e10	−1.08770
$976$	6.62310e9	0.228027
$977$	3.37991e10	1.15951	0.579755	−	0.814791i	$-0.303148\pi$
$977$	3.37991e10	1.15951	0.579755	+	0.814791i	$0.303148\pi$
$978$	−2.65052e10	−0.906036
$979$	0	0
$980$	2.67761e9	0.0908773
$981$	8.11528e9	0.274449
$982$	1.19743e9	0.0403515
$983$	1.03134e9	0.0346308	0.0173154	−	0.999850i	$-0.494488\pi$
$983$	1.03134e9	0.0346308	0.0173154	+	0.999850i	$0.494488\pi$
$984$	2.00264e9	0.0670069
$985$	−4.48199e10	−1.49432
$986$	3.18469e10	1.05803
$987$	−1.79637e10	−0.594683
$988$	1.96535e10	0.648320
$989$	6.91679e9	0.227362
$990$	0	0
$991$	−5.63139e10	−1.83805	−0.919027	−	0.394195i	$-0.871023\pi$
$991$	−5.63139e10	−1.83805	−0.919027	+	0.394195i	$0.871023\pi$
$992$	1.50032e10	0.487970
$993$	−4.85642e9	−0.157396
$994$	1.85780e8	0.00599994
$995$	1.49370e10	0.480710
$996$	4.29713e9	0.137807
$997$	−2.55531e10	−0.816603	−0.408301	−	0.912847i	$-0.633879\pi$
$997$	−2.55531e10	−0.816603	−0.408301	+	0.912847i	$0.633879\pi$
$998$	4.78950e10	1.52522
$999$	7.79222e9	0.247276

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	363.8.a.a.1.1		1
11.10	odd	2		33.8.a.a.1.1	✓	1
33.32	even	2		99.8.a.a.1.1		1
44.43	even	2		528.8.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.8.a.a.1.1	✓	1	11.10	odd	2
99.8.a.a.1.1		1	33.32	even	2
363.8.a.a.1.1		1	1.1	even	1	trivial
528.8.a.a.1.1		1	44.43	even	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:	\( N \)	\(=\)	\( 363 = 3 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	363.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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