LMFDB - Embedded newform 22.8.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [22,8,Mod(1,22)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(22, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("22.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 22 = 2 \cdot 11 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	22.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:	$6.87247056065$
Analytic rank:	$1$
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	$\chi$	$=$	22.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-8.00000 q^{2} -19.0000 q^{3} +64.0000 q^{4} +317.000 q^{5} +152.000 q^{6} -1030.00 q^{7} -512.000 q^{8} -1826.00 q^{9} +O(q^{10})$

\(q-8.00000 q^{2} -19.0000 q^{3} +64.0000 q^{4} +317.000 q^{5} +152.000 q^{6} -1030.00 q^{7} -512.000 q^{8} -1826.00 q^{9} -2536.00 q^{10} +1331.00 q^{11} -1216.00 q^{12} -14676.0 q^{13} +8240.00 q^{14} -6023.00 q^{15} +4096.00 q^{16} -30058.0 q^{17} +14608.0 q^{18} +38056.0 q^{19} +20288.0 q^{20} +19570.0 q^{21} -10648.0 q^{22} -12911.0 q^{23} +9728.00 q^{24} +22364.0 q^{25} +117408. q^{26} +76247.0 q^{27} -65920.0 q^{28} -90480.0 q^{29} +48184.0 q^{30} -139023. q^{31} -32768.0 q^{32} -25289.0 q^{33} +240464. q^{34} -326510. q^{35} -116864. q^{36} +251511. q^{37} -304448. q^{38} +278844. q^{39} -162304. q^{40} -318192. q^{41} -156560. q^{42} +672430. q^{43} +85184.0 q^{44} -578842. q^{45} +103288. q^{46} -519096. q^{47} -77824.0 q^{48} +237357. q^{49} -178912. q^{50} +571102. q^{51} -939264. q^{52} +773570. q^{53} -609976. q^{54} +421927. q^{55} +527360. q^{56} -723064. q^{57} +723840. q^{58} +2.19417e6 q^{59} -385472. q^{60} +3.16318e6 q^{61} +1.11218e6 q^{62} +1.88078e6 q^{63} +262144. q^{64} -4.65229e6 q^{65} +202312. q^{66} -1.29356e6 q^{67} -1.92371e6 q^{68} +245309. q^{69} +2.61208e6 q^{70} -1.20724e6 q^{71} +934912. q^{72} -4.72477e6 q^{73} -2.01209e6 q^{74} -424916. q^{75} +2.43558e6 q^{76} -1.37093e6 q^{77} -2.23075e6 q^{78} -2.63810e6 q^{79} +1.29843e6 q^{80} +2.54477e6 q^{81} +2.54554e6 q^{82} -4.83096e6 q^{83} +1.25248e6 q^{84} -9.52839e6 q^{85} -5.37944e6 q^{86} +1.71912e6 q^{87} -681472. q^{88} -2.44823e6 q^{89} +4.63074e6 q^{90} +1.51163e7 q^{91} -826304. q^{92} +2.64144e6 q^{93} +4.15277e6 q^{94} +1.20638e7 q^{95} +622592. q^{96} +3.94860e6 q^{97} -1.89886e6 q^{98} -2.43041e6 q^{99} +O(q^{100})\)

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 $

$2$	−8.00000	−0.707107
$2$	−8.00000	−0.707107
$3$	−19.0000	−0.406284	−0.203142	−	0.979149i	$-0.565115\pi$
$3$	−19.0000	−0.406284	−0.203142	+	0.979149i	$0.565115\pi$
$4$	64.0000	0.500000
$5$	317.000	1.13413	0.567067	−	0.823672i	$-0.308078\pi$
$5$	317.000	1.13413	0.567067	+	0.823672i	$0.308078\pi$
$6$	152.000	0.287286
$7$	−1030.00	−1.13500	−0.567498	−	0.823375i	$-0.692088\pi$
$7$	−1030.00	−1.13500	−0.567498	+	0.823375i	$0.692088\pi$
$8$	−512.000	−0.353553
$9$	−1826.00	−0.834934
$10$	−2536.00	−0.801954
$11$	1331.00	0.301511
$11$	1331.00	0.301511
$12$	−1216.00	−0.203142
$13$	−14676.0	−1.85270	−0.926352	−	0.376659i	$-0.877073\pi$
$13$	−14676.0	−1.85270	−0.926352	+	0.376659i	$0.877073\pi$
$14$	8240.00	0.802563
$15$	−6023.00	−0.460780
$16$	4096.00	0.250000
$17$	−30058.0	−1.48385	−0.741923	−	0.670485i	$-0.766086\pi$
$17$	−30058.0	−1.48385	−0.741923	+	0.670485i	$0.766086\pi$
$18$	14608.0	0.590387
$19$	38056.0	1.27287	0.636437	−	0.771329i	$-0.280407\pi$
$19$	38056.0	1.27287	0.636437	+	0.771329i	$0.280407\pi$
$20$	20288.0	0.567067
$21$	19570.0	0.461130
$22$	−10648.0	−0.213201
$23$	−12911.0	−0.221265	−0.110632	−	0.993861i	$-0.535288\pi$
$23$	−12911.0	−0.221265	−0.110632	+	0.993861i	$0.535288\pi$
$24$	9728.00	0.143643
$25$	22364.0	0.286259
$26$	117408.	1.31006
$27$	76247.0	0.745503
$28$	−65920.0	−0.567498
$29$	−90480.0	−0.688905	−0.344453	−	0.938804i	$-0.611936\pi$
$29$	−90480.0	−0.688905	−0.344453	+	0.938804i	$0.611936\pi$
$30$	48184.0	0.325821
$31$	−139023.	−0.838148	−0.419074	−	0.907952i	$-0.637645\pi$
$31$	−139023.	−0.838148	−0.419074	+	0.907952i	$0.637645\pi$
$32$	−32768.0	−0.176777
$33$	−25289.0	−0.122499
$34$	240464.	1.04924
$35$	−326510.	−1.28724
$36$	−116864.	−0.417467
$37$	251511.	0.816302	0.408151	−	0.912914i	$-0.366174\pi$
$37$	251511.	0.816302	0.408151	+	0.912914i	$0.366174\pi$
$38$	−304448.	−0.900058
$39$	278844.	0.752723
$40$	−162304.	−0.400977
$41$	−318192.	−0.721017	−0.360509	−	0.932756i	$-0.617397\pi$
$41$	−318192.	−0.721017	−0.360509	+	0.932756i	$0.617397\pi$
$42$	−156560.	−0.326068
$43$	672430.	1.28976	0.644878	−	0.764286i	$-0.276908\pi$
$43$	672430.	1.28976	0.644878	+	0.764286i	$0.276908\pi$
$44$	85184.0	0.150756
$45$	−578842.	−0.946926
$46$	103288.	0.156458
$47$	−519096.	−0.729298	−0.364649	−	0.931145i	$-0.618811\pi$
$47$	−519096.	−0.729298	−0.364649	+	0.931145i	$0.618811\pi$
$48$	−77824.0	−0.101571
$49$	237357.	0.288214
$50$	−178912.	−0.202416
$51$	571102.	0.602862
$52$	−939264.	−0.926352
$53$	773570.	0.713730	0.356865	−	0.934156i	$-0.383846\pi$
$53$	773570.	0.713730	0.356865	+	0.934156i	$0.383846\pi$
$54$	−609976.	−0.527150
$55$	421927.	0.341954
$56$	527360.	0.401281
$57$	−723064.	−0.517148
$58$	723840.	0.487130
$59$	2.19417e6	1.39087	0.695437	−	0.718587i	$-0.255211\pi$
$59$	2.19417e6	1.39087	0.695437	+	0.718587i	$0.255211\pi$
$60$	−385472.	−0.230390
$61$	3.16318e6	1.78431	0.892153	−	0.451733i	$-0.149194\pi$
$61$	3.16318e6	1.78431	0.892153	+	0.451733i	$0.149194\pi$
$62$	1.11218e6	0.592660
$63$	1.88078e6	0.947646
$64$	262144.	0.125000
$65$	−4.65229e6	−2.10121
$66$	202312.	0.0866199
$67$	−1.29356e6	−0.525441	−0.262720	−	0.964872i	$-0.584620\pi$
$67$	−1.29356e6	−0.525441	−0.262720	+	0.964872i	$0.584620\pi$
$68$	−1.92371e6	−0.741923
$69$	245309.	0.0898963
$70$	2.61208e6	0.910214
$71$	−1.20724e6	−0.400305	−0.200153	−	0.979765i	$-0.564144\pi$
$71$	−1.20724e6	−0.400305	−0.200153	+	0.979765i	$0.564144\pi$
$72$	934912.	0.295194
$73$	−4.72477e6	−1.42151	−0.710757	−	0.703438i	$-0.751647\pi$
$73$	−4.72477e6	−1.42151	−0.710757	+	0.703438i	$0.751647\pi$
$74$	−2.01209e6	−0.577213
$75$	−424916.	−0.116302
$76$	2.43558e6	0.636437
$77$	−1.37093e6	−0.342214
$78$	−2.23075e6	−0.532256
$79$	−2.63810e6	−0.602000	−0.301000	−	0.953624i	$-0.597320\pi$
$79$	−2.63810e6	−0.602000	−0.301000	+	0.953624i	$0.597320\pi$
$80$	1.29843e6	0.283533
$81$	2.54477e6	0.532048
$82$	2.54554e6	0.509836
$83$	−4.83096e6	−0.927385	−0.463693	−	0.885996i	$-0.653476\pi$
$83$	−4.83096e6	−0.927385	−0.463693	+	0.885996i	$0.653476\pi$
$84$	1.25248e6	0.230565
$85$	−9.52839e6	−1.68288
$86$	−5.37944e6	−0.911995
$87$	1.71912e6	0.279891
$88$	−681472.	−0.106600
$89$	−2.44823e6	−0.368119	−0.184059	−	0.982915i	$-0.558924\pi$
$89$	−2.44823e6	−0.368119	−0.184059	+	0.982915i	$0.558924\pi$
$90$	4.63074e6	0.669578
$91$	1.51163e7	2.10281
$92$	−826304.	−0.110632
$93$	2.64144e6	0.340526
$94$	4.15277e6	0.515692
$95$	1.20638e7	1.44361
$96$	622592.	0.0718215
$97$	3.94860e6	0.439281	0.219640	−	0.975581i	$-0.429512\pi$
$97$	3.94860e6	0.439281	0.219640	+	0.975581i	$0.429512\pi$
$98$	−1.89886e6	−0.203798
$99$	−2.43041e6	−0.251742
$100$	1.43130e6	0.143130
$101$	−1.31330e7	−1.26835	−0.634177	−	0.773188i	$-0.718661\pi$
$101$	−1.31330e7	−1.26835	−0.634177	+	0.773188i	$0.718661\pi$
$102$	−4.56882e6	−0.426288
$103$	1.78361e7	1.60831	0.804156	−	0.594418i	$-0.202618\pi$
$103$	1.78361e7	1.60831	0.804156	+	0.594418i	$0.202618\pi$
$104$	7.51411e6	0.655030
$105$	6.20369e6	0.522983
$106$	−6.18856e6	−0.504683
$107$	−1.40759e7	−1.11079	−0.555397	−	0.831586i	$-0.687433\pi$
$107$	−1.40759e7	−1.11079	−0.555397	+	0.831586i	$0.687433\pi$
$108$	4.87981e6	0.372752
$109$	−6.60622e6	−0.488608	−0.244304	−	0.969699i	$-0.578559\pi$
$109$	−6.60622e6	−0.488608	−0.244304	+	0.969699i	$0.578559\pi$
$110$	−3.37542e6	−0.241798
$111$	−4.77871e6	−0.331650
$112$	−4.21888e6	−0.283749
$113$	−2.61980e6	−0.170802	−0.0854011	−	0.996347i	$-0.527217\pi$
$113$	−2.61980e6	−0.170802	−0.0854011	+	0.996347i	$0.527217\pi$
$114$	5.78451e6	0.365679
$115$	−4.09279e6	−0.250944
$116$	−5.79072e6	−0.344453
$117$	2.67984e7	1.54688
$118$	−1.75533e7	−0.983496
$119$	3.09597e7	1.68416
$120$	3.08378e6	0.162910
$121$	1.77156e6	0.0909091
$122$	−2.53054e7	−1.26170
$123$	6.04565e6	0.292937
$124$	−8.89747e6	−0.419074
$125$	−1.76762e7	−0.809477
$126$	−1.50462e7	−0.670087
$127$	−2.15976e7	−0.935607	−0.467803	−	0.883833i	$-0.654954\pi$
$127$	−2.15976e7	−0.935607	−0.467803	+	0.883833i	$0.654954\pi$
$128$	−2.09715e6	−0.0883883
$129$	−1.27762e7	−0.524007
$130$	3.72183e7	1.48578
$131$	−4.44904e6	−0.172909	−0.0864543	−	0.996256i	$-0.527554\pi$
$131$	−4.44904e6	−0.172909	−0.0864543	+	0.996256i	$0.527554\pi$
$132$	−1.61850e6	−0.0612495
$133$	−3.91977e7	−1.44471
$134$	1.03485e7	0.371543
$135$	2.41703e7	0.845500
$136$	1.53897e7	0.524619
$137$	−2.79123e7	−0.927415	−0.463707	−	0.885988i	$-0.653481\pi$
$137$	−2.79123e7	−0.927415	−0.463707	+	0.885988i	$0.653481\pi$
$138$	−1.96247e6	−0.0635663
$139$	3.76631e7	1.18950	0.594750	−	0.803910i	$-0.297251\pi$
$139$	3.76631e7	1.18950	0.594750	+	0.803910i	$0.297251\pi$
$140$	−2.08966e7	−0.643618
$141$	9.86282e6	0.296302
$142$	9.65796e6	0.283058
$143$	−1.95338e7	−0.558611
$144$	−7.47930e6	−0.208733
$145$	−2.86822e7	−0.781311
$146$	3.77982e7	1.00516
$147$	−4.50978e6	−0.117097
$148$	1.60967e7	0.408151
$149$	5.80005e7	1.43641	0.718207	−	0.695830i	$-0.244963\pi$
$149$	5.80005e7	1.43641	0.718207	+	0.695830i	$0.244963\pi$
$150$	3.39933e6	0.0822382
$151$	−2.11339e7	−0.499529	−0.249765	−	0.968307i	$-0.580353\pi$
$151$	−2.11339e7	−0.499529	−0.249765	+	0.968307i	$0.580353\pi$
$152$	−1.94847e7	−0.450029
$153$	5.48859e7	1.23891
$154$	1.09674e7	0.241982
$155$	−4.40703e7	−0.950572
$156$	1.78460e7	0.376361
$157$	9.80105e6	0.202127	0.101063	−	0.994880i	$-0.467776\pi$
$157$	9.80105e6	0.202127	0.101063	+	0.994880i	$0.467776\pi$
$158$	2.11048e7	0.425679
$159$	−1.46978e7	−0.289977
$160$	−1.03875e7	−0.200488
$161$	1.32983e7	0.251135
$162$	−2.03582e7	−0.376215
$163$	−6.41768e7	−1.16070	−0.580352	−	0.814366i	$-0.697085\pi$
$163$	−6.41768e7	−1.16070	−0.580352	+	0.814366i	$0.697085\pi$
$164$	−2.03643e7	−0.360509
$165$	−8.01661e6	−0.138930
$166$	3.86477e7	0.655761
$167$	2.38538e7	0.396323	0.198162	−	0.980169i	$-0.436503\pi$
$167$	2.38538e7	0.396323	0.198162	+	0.980169i	$0.436503\pi$
$168$	−1.00198e7	−0.163034
$169$	1.52636e8	2.43251
$170$	7.62271e7	1.18998
$171$	−6.94903e7	−1.06277
$172$	4.30355e7	0.644878
$173$	−1.21532e7	−0.178455	−0.0892277	−	0.996011i	$-0.528440\pi$
$173$	−1.21532e7	−0.178455	−0.0892277	+	0.996011i	$0.528440\pi$
$174$	−1.37530e7	−0.197913
$175$	−2.30349e7	−0.324903
$176$	5.45178e6	0.0753778
$177$	−4.16892e7	−0.565089
$178$	1.95859e7	0.260299
$179$	9.53183e7	1.24220	0.621099	−	0.783732i	$-0.286687\pi$
$179$	9.53183e7	1.24220	0.621099	+	0.783732i	$0.286687\pi$
$180$	−3.70459e7	−0.473463
$181$	−8.93211e7	−1.11964	−0.559821	−	0.828614i	$-0.689130\pi$
$181$	−8.93211e7	−1.11964	−0.559821	+	0.828614i	$0.689130\pi$
$182$	−1.20930e8	−1.48691
$183$	−6.01004e7	−0.724934
$184$	6.61043e6	0.0782289
$185$	7.97290e7	0.925796
$186$	−2.11315e7	−0.240788
$187$	−4.00072e7	−0.447396
$188$	−3.32221e7	−0.364649
$189$	−7.85344e7	−0.846143
$190$	−9.65100e7	−1.02079
$191$	−1.68125e8	−1.74588	−0.872941	−	0.487825i	$-0.837790\pi$
$191$	−1.68125e8	−1.74588	−0.872941	+	0.487825i	$0.837790\pi$
$192$	−4.98074e6	−0.0507854
$193$	1.30663e8	1.30828	0.654141	−	0.756373i	$-0.273030\pi$
$193$	1.30663e8	1.30828	0.654141	+	0.756373i	$0.273030\pi$
$194$	−3.15888e7	−0.310618
$195$	8.83935e7	0.853688
$196$	1.51908e7	0.144107
$197$	−7.99839e7	−0.745368	−0.372684	−	0.927958i	$-0.621562\pi$
$197$	−7.99839e7	−0.745368	−0.372684	+	0.927958i	$0.621562\pi$
$198$	1.94432e7	0.178008
$199$	6.30116e7	0.566807	0.283403	−	0.959001i	$-0.408536\pi$
$199$	6.30116e7	0.566807	0.283403	+	0.959001i	$0.408536\pi$
$200$	−1.14504e7	−0.101208
$201$	2.45776e7	0.213478
$202$	1.05064e8	0.896862
$203$	9.31944e7	0.781904
$204$	3.65505e7	0.301431
$205$	−1.00867e8	−0.817730
$206$	−1.42689e8	−1.13725
$207$	2.35755e7	0.184741
$208$	−6.01129e7	−0.463176
$209$	5.06525e7	0.383786
$210$	−4.96295e7	−0.369805
$211$	−1.34758e8	−0.987566	−0.493783	−	0.869585i	$-0.664386\pi$
$211$	−1.34758e8	−0.987566	−0.493783	+	0.869585i	$0.664386\pi$
$212$	4.95085e7	0.356865
$213$	2.29377e7	0.162637
$214$	1.12607e8	0.785449
$215$	2.13160e8	1.46276
$216$	−3.90385e7	−0.263575
$217$	1.43194e8	0.951294
$218$	5.28498e7	0.345498
$219$	8.97707e7	0.577537
$220$	2.70033e7	0.170977
$221$	4.41131e8	2.74913
$222$	3.82297e7	0.234512
$223$	−2.47808e8	−1.49640	−0.748201	−	0.663472i	$-0.769082\pi$
$223$	−2.47808e8	−1.49640	−0.748201	+	0.663472i	$0.769082\pi$
$224$	3.37510e7	0.200641
$225$	−4.08367e7	−0.239007
$226$	2.09584e7	0.120775
$227$	2.29740e8	1.30361	0.651803	−	0.758388i	$-0.274013\pi$
$227$	2.29740e8	1.30361	0.651803	+	0.758388i	$0.274013\pi$
$228$	−4.62761e7	−0.258574
$229$	−2.31280e8	−1.27266	−0.636331	−	0.771416i	$-0.719549\pi$
$229$	−2.31280e8	−1.27266	−0.636331	+	0.771416i	$0.719549\pi$
$230$	3.27423e7	0.177444
$231$	2.60477e7	0.139036
$232$	4.63258e7	0.243565
$233$	−1.23167e8	−0.637892	−0.318946	−	0.947773i	$-0.603329\pi$
$233$	−1.23167e8	−0.637892	−0.318946	+	0.947773i	$0.603329\pi$
$234$	−2.14387e8	−1.09381
$235$	−1.64553e8	−0.827122
$236$	1.40427e8	0.695437
$237$	5.01239e7	0.244583
$238$	−2.47678e8	−1.19088
$239$	1.92817e8	0.913594	0.456797	−	0.889571i	$-0.348996\pi$
$239$	1.92817e8	0.913594	0.456797	+	0.889571i	$0.348996\pi$
$240$	−2.46702e7	−0.115195
$241$	−8.01808e7	−0.368987	−0.184493	−	0.982834i	$-0.559064\pi$
$241$	−8.01808e7	−0.368987	−0.184493	+	0.982834i	$0.559064\pi$
$242$	−1.41725e7	−0.0642824
$243$	−2.15103e8	−0.961666
$244$	2.02444e8	0.892153
$245$	7.52422e7	0.326874
$246$	−4.83652e7	−0.207138
$247$	−5.58510e8	−2.35826
$248$	7.11798e7	0.296330
$249$	9.17883e7	0.376781
$250$	1.41410e8	0.572387
$251$	−2.87910e8	−1.14921	−0.574604	−	0.818432i	$-0.694844\pi$
$251$	−2.87910e8	−1.14921	−0.574604	+	0.818432i	$0.694844\pi$
$252$	1.20370e8	0.473823
$253$	−1.71845e7	−0.0667139
$254$	1.72781e8	0.661574
$255$	1.81039e8	0.683726
$256$	1.67772e7	0.0625000
$257$	−3.55063e8	−1.30479	−0.652393	−	0.757881i	$-0.726235\pi$
$257$	−3.55063e8	−1.30479	−0.652393	+	0.757881i	$0.726235\pi$
$258$	1.02209e8	0.370529
$259$	−2.59056e8	−0.926499
$260$	−2.97747e8	−1.05061
$261$	1.65216e8	0.575190
$262$	3.55923e7	0.122265
$263$	−5.34564e7	−0.181199	−0.0905993	−	0.995887i	$-0.528878\pi$
$263$	−5.34564e7	−0.181199	−0.0905993	+	0.995887i	$0.528878\pi$
$264$	1.29480e7	0.0433100
$265$	2.45222e8	0.809465
$266$	3.13581e8	1.02156
$267$	4.65164e7	0.149561
$268$	−8.27876e7	−0.262720
$269$	1.81725e8	0.569223	0.284612	−	0.958643i	$-0.408135\pi$
$269$	1.81725e8	0.569223	0.284612	+	0.958643i	$0.408135\pi$
$270$	−1.93362e8	−0.597859
$271$	−1.47558e8	−0.450370	−0.225185	−	0.974316i	$-0.572299\pi$
$271$	−1.47558e8	−0.450370	−0.225185	+	0.974316i	$0.572299\pi$
$272$	−1.23118e8	−0.370962
$273$	−2.87209e8	−0.854337
$274$	2.23299e8	0.655781
$275$	2.97665e7	0.0863104
$276$	1.56998e7	0.0449481
$277$	−2.32680e8	−0.657779	−0.328890	−	0.944368i	$-0.606674\pi$
$277$	−2.32680e8	−0.657779	−0.328890	+	0.944368i	$0.606674\pi$
$278$	−3.01305e8	−0.841104
$279$	2.53856e8	0.699798
$280$	1.67173e8	0.455107
$281$	−2.62549e8	−0.705892	−0.352946	−	0.935644i	$-0.614820\pi$
$281$	−2.62549e8	−0.705892	−0.352946	+	0.935644i	$0.614820\pi$
$282$	−7.89026e7	−0.209517
$283$	6.56671e8	1.72225	0.861123	−	0.508396i	$-0.169761\pi$
$283$	6.56671e8	1.72225	0.861123	+	0.508396i	$0.169761\pi$
$284$	−7.72637e7	−0.200153
$285$	−2.29211e8	−0.586515
$286$	1.56270e8	0.394998
$287$	3.27738e8	0.818351
$288$	5.98344e7	0.147597
$289$	4.93145e8	1.20180
$290$	2.29457e8	0.552470
$291$	−7.50234e7	−0.178473
$292$	−3.02385e8	−0.710757
$293$	−5.24346e8	−1.21782	−0.608908	−	0.793241i	$-0.708392\pi$
$293$	−5.24346e8	−1.21782	−0.608908	+	0.793241i	$0.708392\pi$
$294$	3.60783e7	0.0827999
$295$	6.95551e8	1.57744
$296$	−1.28774e8	−0.288606
$297$	1.01485e8	0.224778
$298$	−4.64004e8	−1.01570
$299$	1.89482e8	0.409938
$300$	−2.71946e7	−0.0581512
$301$	−6.92603e8	−1.46387
$302$	1.69071e8	0.353220
$303$	2.49528e8	0.515311
$304$	1.55877e8	0.318219
$305$	1.00273e9	2.02364
$306$	−4.39087e8	−0.876044
$307$	7.18657e8	1.41755	0.708774	−	0.705436i	$-0.249249\pi$
$307$	7.18657e8	1.41755	0.708774	+	0.705436i	$0.249249\pi$
$308$	−8.77395e7	−0.171107
$309$	−3.38886e8	−0.653431
$310$	3.52562e8	0.672156
$311$	−4.07844e8	−0.768834	−0.384417	−	0.923160i	$-0.625597\pi$
$311$	−4.07844e8	−0.768834	−0.384417	+	0.923160i	$0.625597\pi$
$312$	−1.42768e8	−0.266128
$313$	−2.89586e8	−0.533793	−0.266896	−	0.963725i	$-0.585998\pi$
$313$	−2.89586e8	−0.533793	−0.266896	+	0.963725i	$0.585998\pi$
$314$	−7.84084e7	−0.142925
$315$	5.96207e8	1.07476
$316$	−1.68839e8	−0.301000
$317$	−7.71959e7	−0.136109	−0.0680545	−	0.997682i	$-0.521679\pi$
$317$	−7.71959e7	−0.136109	−0.0680545	+	0.997682i	$0.521679\pi$
$318$	1.17583e8	0.205045
$319$	−1.20429e8	−0.207713
$320$	8.30996e7	0.141767
$321$	2.67442e8	0.451297
$322$	−1.06387e8	−0.177579
$323$	−1.14389e9	−1.88875
$324$	1.62865e8	0.266024
$325$	−3.28214e8	−0.530353
$326$	5.13414e8	0.820741
$327$	1.25518e8	0.198513
$328$	1.62914e8	0.254918
$329$	5.34669e8	0.827750
$330$	6.41329e7	0.0982386
$331$	4.42024e8	0.669959	0.334979	−	0.942225i	$-0.391271\pi$
$331$	4.42024e8	0.669959	0.334979	+	0.942225i	$0.391271\pi$
$332$	−3.09182e8	−0.463693
$333$	−4.59259e8	−0.681558
$334$	−1.90830e8	−0.280243
$335$	−4.10058e8	−0.595920
$336$	8.01587e7	0.115282
$337$	1.12580e8	0.160234	0.0801171	−	0.996785i	$-0.474471\pi$
$337$	1.12580e8	0.160234	0.0801171	+	0.996785i	$0.474471\pi$
$338$	−1.22109e9	−1.72005
$339$	4.97762e7	0.0693941
$340$	−6.09817e8	−0.841440
$341$	−1.85040e8	−0.252711
$342$	5.55922e8	0.751489
$343$	6.03772e8	0.807873
$344$	−3.44284e8	−0.455998
$345$	7.77630e7	0.101954
$346$	9.72257e7	0.126187
$347$	8.61296e8	1.10662	0.553311	−	0.832975i	$-0.313364\pi$
$347$	8.61296e8	1.10662	0.553311	+	0.832975i	$0.313364\pi$
$348$	1.10024e8	0.139945
$349$	−9.16905e8	−1.15461	−0.577305	−	0.816528i	$-0.695896\pi$
$349$	−9.16905e8	−1.15461	−0.577305	+	0.816528i	$0.695896\pi$
$350$	1.84279e8	0.229741
$351$	−1.11900e9	−1.38120
$352$	−4.36142e7	−0.0533002
$353$	−5.97949e8	−0.723523	−0.361762	−	0.932271i	$-0.617825\pi$
$353$	−5.97949e8	−0.723523	−0.361762	+	0.932271i	$0.617825\pi$
$354$	3.33513e8	0.399578
$355$	−3.82697e8	−0.454000
$356$	−1.56687e8	−0.184059
$357$	−5.88235e8	−0.684246
$358$	−7.62546e8	−0.878366
$359$	−6.43323e8	−0.733836	−0.366918	−	0.930253i	$-0.619587\pi$
$359$	−6.43323e8	−0.733836	−0.366918	+	0.930253i	$0.619587\pi$
$360$	2.96367e8	0.334789
$361$	5.54387e8	0.620209
$362$	7.14569e8	0.791707
$363$	−3.36597e7	−0.0369349
$364$	9.67442e8	1.05141
$365$	−1.49775e9	−1.61219
$366$	4.80803e8	0.512606
$367$	1.10619e9	1.16815	0.584077	−	0.811698i	$-0.301457\pi$
$367$	1.10619e9	1.16815	0.584077	+	0.811698i	$0.301457\pi$
$368$	−5.28835e7	−0.0553162
$369$	5.81019e8	0.602002
$370$	−6.37832e8	−0.654636
$371$	−7.96777e8	−0.810080
$372$	1.69052e8	0.170263
$373$	6.69823e8	0.668312	0.334156	−	0.942518i	$-0.391549\pi$
$373$	6.69823e8	0.668312	0.334156	+	0.942518i	$0.391549\pi$
$374$	3.20058e8	0.316357
$375$	3.35849e8	0.328877
$376$	2.65777e8	0.257846
$377$	1.32788e9	1.27634
$378$	6.28275e8	0.598313
$379$	1.26107e9	1.18988	0.594939	−	0.803771i	$-0.297176\pi$
$379$	1.26107e9	1.18988	0.594939	+	0.803771i	$0.297176\pi$
$380$	7.72080e8	0.721805
$381$	4.10355e8	0.380122
$382$	1.34500e9	1.23453
$383$	−7.25195e8	−0.659567	−0.329784	−	0.944057i	$-0.606976\pi$
$383$	−7.25195e8	−0.659567	−0.329784	+	0.944057i	$0.606976\pi$
$384$	3.98459e7	0.0359107
$385$	−4.34585e8	−0.388116
$386$	−1.04530e9	−0.925095
$387$	−1.22786e9	−1.07686
$388$	2.52710e8	0.219640
$389$	5.21503e8	0.449193	0.224596	−	0.974452i	$-0.427894\pi$
$389$	5.21503e8	0.449193	0.224596	+	0.974452i	$0.427894\pi$
$390$	−7.07148e8	−0.603649
$391$	3.88079e8	0.328323
$392$	−1.21527e8	−0.101899
$393$	8.45317e7	0.0702499
$394$	6.39871e8	0.527055
$395$	−8.36278e8	−0.682749
$396$	−1.55546e8	−0.125871
$397$	−1.31997e9	−1.05876	−0.529381	−	0.848384i	$-0.677576\pi$
$397$	−1.31997e9	−1.05876	−0.529381	+	0.848384i	$0.677576\pi$
$398$	−5.04093e8	−0.400793
$399$	7.44756e8	0.586960
$400$	9.16029e7	0.0715648
$401$	8.58639e8	0.664975	0.332487	−	0.943108i	$-0.392112\pi$
$401$	8.58639e8	0.664975	0.332487	+	0.943108i	$0.392112\pi$
$402$	−1.96621e8	−0.150952
$403$	2.04030e9	1.55284
$404$	−8.40514e8	−0.634177
$405$	8.06692e8	0.603414
$406$	−7.45555e8	−0.552890
$407$	3.34761e8	0.246124
$408$	−2.92404e8	−0.213144
$409$	1.72609e9	1.24747	0.623736	−	0.781635i	$-0.285614\pi$
$409$	1.72609e9	1.24747	0.623736	+	0.781635i	$0.285614\pi$
$410$	8.06935e8	0.578223
$411$	5.30334e8	0.376793
$412$	1.14151e9	0.804156
$413$	−2.25999e9	−1.57863
$414$	−1.88604e8	−0.130632
$415$	−1.53141e9	−1.05178
$416$	4.80903e8	0.327515
$417$	−7.15600e8	−0.483275
$418$	−4.05220e8	−0.271378
$419$	−8.74420e8	−0.580726	−0.290363	−	0.956917i	$-0.593776\pi$
$419$	−8.74420e8	−0.580726	−0.290363	+	0.956917i	$0.593776\pi$
$420$	3.97036e8	0.261491
$421$	−4.30242e8	−0.281013	−0.140506	−	0.990080i	$-0.544873\pi$
$421$	−4.30242e8	−0.281013	−0.140506	+	0.990080i	$0.544873\pi$
$422$	1.07806e9	0.698314
$423$	9.47869e8	0.608916
$424$	−3.96068e8	−0.252342
$425$	−6.72217e8	−0.424765
$426$	−1.83501e8	−0.115002
$427$	−3.25808e9	−2.02518
$428$	−9.00858e8	−0.555397
$429$	3.71141e8	0.226955
$430$	−1.70528e9	−1.03432
$431$	−1.91181e9	−1.15020	−0.575101	−	0.818083i	$-0.695037\pi$
$431$	−1.91181e9	−1.15020	−0.575101	+	0.818083i	$0.695037\pi$
$432$	3.12308e8	0.186376
$433$	2.07982e9	1.23117	0.615587	−	0.788069i	$-0.288919\pi$
$433$	2.07982e9	1.23117	0.615587	+	0.788069i	$0.288919\pi$
$434$	−1.14555e9	−0.672667
$435$	5.44961e8	0.317434
$436$	−4.22798e8	−0.244304
$437$	−4.91341e8	−0.281642
$438$	−7.18165e8	−0.408381
$439$	−7.53463e8	−0.425046	−0.212523	−	0.977156i	$-0.568168\pi$
$439$	−7.53463e8	−0.425046	−0.212523	+	0.977156i	$0.568168\pi$
$440$	−2.16027e8	−0.120899
$441$	−4.33414e8	−0.240640
$442$	−3.52905e9	−1.94393
$443$	1.44182e9	0.787946	0.393973	−	0.919122i	$-0.371100\pi$
$443$	1.44182e9	0.787946	0.393973	+	0.919122i	$0.371100\pi$
$444$	−3.05837e8	−0.165825
$445$	−7.76090e8	−0.417496
$446$	1.98246e9	1.05812
$447$	−1.10201e9	−0.583591
$448$	−2.70008e8	−0.141874
$449$	2.72928e9	1.42294	0.711469	−	0.702717i	$-0.248030\pi$
$449$	2.72928e9	1.42294	0.711469	+	0.702717i	$0.248030\pi$
$450$	3.26693e8	0.169004
$451$	−4.23514e8	−0.217395
$452$	−1.67667e8	−0.0854011
$453$	4.01545e8	0.202950
$454$	−1.83792e9	−0.921788
$455$	4.79186e9	2.38487
$456$	3.70209e8	0.182839
$457$	2.53387e9	1.24187	0.620936	−	0.783861i	$-0.286752\pi$
$457$	2.53387e9	1.24187	0.620936	+	0.783861i	$0.286752\pi$
$458$	1.85024e9	0.899908
$459$	−2.29183e9	−1.10621
$460$	−2.61938e8	−0.125472
$461$	3.20595e9	1.52406	0.762031	−	0.647540i	$-0.224202\pi$
$461$	3.20595e9	1.52406	0.762031	+	0.647540i	$0.224202\pi$
$462$	−2.08381e8	−0.0983132
$463$	1.38946e9	0.650598	0.325299	−	0.945611i	$-0.394535\pi$
$463$	1.38946e9	0.650598	0.325299	+	0.945611i	$0.394535\pi$
$464$	−3.70606e8	−0.172226
$465$	8.37336e8	0.386202
$466$	9.85332e8	0.451058
$467$	−4.27451e9	−1.94213	−0.971063	−	0.238824i	$-0.923238\pi$
$467$	−4.27451e9	−1.94213	−0.971063	+	0.238824i	$0.923238\pi$
$468$	1.71510e9	0.773442
$469$	1.33236e9	0.596373
$470$	1.31643e9	0.584863
$471$	−1.86220e8	−0.0821208
$472$	−1.12341e9	−0.491748
$473$	8.95004e8	0.388876
$474$	−4.00992e8	−0.172946
$475$	8.51084e8	0.364372
$476$	1.98142e9	0.842079
$477$	−1.41254e9	−0.595917
$478$	−1.54254e9	−0.646009
$479$	−1.24454e9	−0.517411	−0.258706	−	0.965956i	$-0.583296\pi$
$479$	−1.24454e9	−0.517411	−0.258706	+	0.965956i	$0.583296\pi$
$480$	1.97362e8	0.0814551
$481$	−3.69118e9	−1.51237
$482$	6.41446e8	0.260913
$483$	−2.52668e8	−0.102032
$484$	1.13380e8	0.0454545
$485$	1.25171e9	0.498203
$486$	1.72082e9	0.680000
$487$	−7.49172e8	−0.293921	−0.146960	−	0.989142i	$-0.546949\pi$
$487$	−7.49172e8	−0.293921	−0.146960	+	0.989142i	$0.546949\pi$
$488$	−1.61955e9	−0.630848
$489$	1.21936e9	0.471575
$490$	−6.01937e8	−0.231135
$491$	−1.01082e9	−0.385380	−0.192690	−	0.981260i	$-0.561721\pi$
$491$	−1.01082e9	−0.385380	−0.192690	+	0.981260i	$0.561721\pi$
$492$	3.86921e8	0.146469
$493$	2.71965e9	1.02223
$494$	4.46808e9	1.66754
$495$	−7.70439e8	−0.285509
$496$	−5.69438e8	−0.209537
$497$	1.24346e9	0.454345
$498$	−7.34306e8	−0.266425
$499$	1.49520e9	0.538702	0.269351	−	0.963042i	$-0.413191\pi$
$499$	1.49520e9	0.538702	0.269351	+	0.963042i	$0.413191\pi$
$500$	−1.13128e9	−0.404739
$501$	−4.53222e8	−0.161020
$502$	2.30328e9	0.812612
$503$	−3.09414e9	−1.08406	−0.542028	−	0.840361i	$-0.682343\pi$
$503$	−3.09414e9	−1.08406	−0.542028	+	0.840361i	$0.682343\pi$
$504$	−9.62959e8	−0.335043
$505$	−4.16317e9	−1.43848
$506$	1.37476e8	0.0471738
$507$	−2.90009e9	−0.988289
$508$	−1.38225e9	−0.467803
$509$	−3.27586e9	−1.10106	−0.550532	−	0.834814i	$-0.685575\pi$
$509$	−3.27586e9	−1.10106	−0.550532	+	0.834814i	$0.685575\pi$
$510$	−1.44831e9	−0.483468
$511$	4.86652e9	1.61341
$512$	−1.34218e8	−0.0441942
$513$	2.90166e9	0.948932
$514$	2.84050e9	0.922623
$515$	5.65405e9	1.82404
$516$	−8.17675e8	−0.262003
$517$	−6.90917e8	−0.219892
$518$	2.07245e9	0.655134
$519$	2.30911e8	0.0725035
$520$	2.38197e9	0.742891
$521$	−4.24693e9	−1.31566	−0.657829	−	0.753168i	$-0.728525\pi$
$521$	−4.24693e9	−1.31566	−0.657829	+	0.753168i	$0.728525\pi$
$522$	−1.32173e9	−0.406721
$523$	−2.54849e9	−0.778981	−0.389490	−	0.921031i	$-0.627349\pi$
$523$	−2.54849e9	−0.778981	−0.389490	+	0.921031i	$0.627349\pi$
$524$	−2.84738e8	−0.0864543
$525$	4.37663e8	0.132003
$526$	4.27651e8	0.128127
$527$	4.17875e9	1.24368
$528$	−1.03584e8	−0.0306248
$529$	−3.23813e9	−0.951042
$530$	−1.96177e9	−0.572379
$531$	−4.00655e9	−1.16129
$532$	−2.50865e9	−0.722353
$533$	4.66979e9	1.33583
$534$	−3.72131e8	−0.105755
$535$	−4.46206e9	−1.25979
$536$	6.62301e8	0.185771
$537$	−1.81105e9	−0.504684
$538$	−1.45380e9	−0.402502
$539$	3.15922e8	0.0868999
$540$	1.54690e9	0.422750
$541$	−2.54208e9	−0.690239	−0.345119	−	0.938559i	$-0.612162\pi$
$541$	−2.54208e9	−0.690239	−0.345119	+	0.938559i	$0.612162\pi$
$542$	1.18046e9	0.318460
$543$	1.69710e9	0.454892
$544$	9.84941e8	0.262309
$545$	−2.09417e9	−0.554147
$546$	2.29767e9	0.604108
$547$	6.64245e9	1.73529	0.867646	−	0.497182i	$-0.165632\pi$
$547$	6.64245e9	1.73529	0.867646	+	0.497182i	$0.165632\pi$
$548$	−1.78639e9	−0.463707
$549$	−5.77597e9	−1.48978
$550$	−2.38132e8	−0.0610307
$551$	−3.44331e9	−0.876890
$552$	−1.25598e8	−0.0317831
$553$	2.71725e9	0.683268
$554$	1.86144e9	0.465120
$555$	−1.51485e9	−0.376136
$556$	2.41044e9	0.594750
$557$	−5.80408e8	−0.142312	−0.0711558	−	0.997465i	$-0.522669\pi$
$557$	−5.80408e8	−0.142312	−0.0711558	+	0.997465i	$0.522669\pi$
$558$	−2.03085e9	−0.494832
$559$	−9.86858e9	−2.38954
$560$	−1.33738e9	−0.321809
$561$	7.60137e8	0.181770
$562$	2.10039e9	0.499141
$563$	6.41406e9	1.51479	0.757397	−	0.652954i	$-0.226471\pi$
$563$	6.41406e9	1.51479	0.757397	+	0.652954i	$0.226471\pi$
$564$	6.31221e8	0.148151
$565$	−8.30476e8	−0.193713
$566$	−5.25337e9	−1.21781
$567$	−2.62111e9	−0.603872
$568$	6.18109e8	0.141529
$569$	1.24745e9	0.283877	0.141939	−	0.989875i	$-0.454666\pi$
$569$	1.24745e9	0.283877	0.141939	+	0.989875i	$0.454666\pi$
$570$	1.83369e9	0.414729
$571$	7.00233e9	1.57404	0.787021	−	0.616926i	$-0.211622\pi$
$571$	7.00233e9	1.57404	0.787021	+	0.616926i	$0.211622\pi$
$572$	−1.25016e9	−0.279306
$573$	3.19437e9	0.709323
$574$	−2.62190e9	−0.578662
$575$	−2.88742e8	−0.0633391
$576$	−4.78675e8	−0.104367
$577$	3.51543e9	0.761838	0.380919	−	0.924608i	$-0.375608\pi$
$577$	3.51543e9	0.761838	0.380919	+	0.924608i	$0.375608\pi$
$578$	−3.94516e9	−0.849800
$579$	−2.48259e9	−0.531534
$580$	−1.83566e9	−0.390655
$581$	4.97589e9	1.05258
$582$	6.00187e8	0.126199
$583$	1.02962e9	0.215198
$584$	2.41908e9	0.502581
$585$	8.49509e9	1.75437
$586$	4.19477e9	0.861126
$587$	−8.87055e9	−1.81016	−0.905080	−	0.425241i	$-0.860189\pi$
$587$	−8.87055e9	−1.81016	−0.905080	+	0.425241i	$0.860189\pi$
$588$	−2.88626e8	−0.0585484
$589$	−5.29066e9	−1.06686
$590$	−5.56441e9	−1.11542
$591$	1.51969e9	0.302831
$592$	1.03019e9	0.204076
$593$	1.94809e9	0.383635	0.191817	−	0.981431i	$-0.438562\pi$
$593$	1.94809e9	0.383635	0.191817	+	0.981431i	$0.438562\pi$
$594$	−8.11878e8	−0.158942
$595$	9.81424e9	1.91006
$596$	3.71203e9	0.718207
$597$	−1.19722e9	−0.230284
$598$	−1.51585e9	−0.289870
$599$	7.04896e9	1.34008	0.670041	−	0.742324i	$-0.266277\pi$
$599$	7.04896e9	1.34008	0.670041	+	0.742324i	$0.266277\pi$
$600$	2.17557e8	0.0411191
$601$	−3.40333e9	−0.639504	−0.319752	−	0.947501i	$-0.603600\pi$
$601$	−3.40333e9	−0.639504	−0.319752	+	0.947501i	$0.603600\pi$
$602$	5.54082e9	1.03511
$603$	2.36204e9	0.438708
$604$	−1.35257e9	−0.249765
$605$	5.61585e8	0.103103
$606$	−1.99622e9	−0.364380
$607$	6.62144e9	1.20169	0.600845	−	0.799366i	$-0.294831\pi$
$607$	6.62144e9	1.20169	0.600845	+	0.799366i	$0.294831\pi$
$608$	−1.24702e9	−0.225015
$609$	−1.77069e9	−0.317675
$610$	−8.02182e9	−1.43093
$611$	7.61825e9	1.35117
$612$	3.51270e9	0.619457
$613$	3.03636e9	0.532404	0.266202	−	0.963917i	$-0.414231\pi$
$613$	3.03636e9	0.532404	0.266202	+	0.963917i	$0.414231\pi$
$614$	−5.74926e9	−1.00236
$615$	1.91647e9	0.332230
$616$	7.01916e8	0.120991
$617$	−2.83998e9	−0.486763	−0.243381	−	0.969931i	$-0.578257\pi$
$617$	−2.83998e9	−0.486763	−0.243381	+	0.969931i	$0.578257\pi$
$618$	2.71109e9	0.462045
$619$	2.24244e9	0.380018	0.190009	−	0.981782i	$-0.439148\pi$
$619$	2.24244e9	0.380018	0.190009	+	0.981782i	$0.439148\pi$
$620$	−2.82050e9	−0.475286
$621$	−9.84425e8	−0.164954
$622$	3.26275e9	0.543648
$623$	2.52168e9	0.417813
$624$	1.14215e9	0.188181
$625$	−7.35055e9	−1.20431
$626$	2.31669e9	0.377449
$627$	−9.62398e8	−0.155926
$628$	6.27267e8	0.101063
$629$	−7.55992e9	−1.21127
$630$	−4.76966e9	−0.759968
$631$	−5.57095e9	−0.882727	−0.441363	−	0.897328i	$-0.645505\pi$
$631$	−5.57095e9	−0.882727	−0.441363	+	0.897328i	$0.645505\pi$
$632$	1.35071e9	0.212839
$633$	2.56040e9	0.401232
$634$	6.17567e8	0.0962436
$635$	−6.84645e9	−1.06110
$636$	−9.40661e8	−0.144988
$637$	−3.48345e9	−0.533976
$638$	9.63431e8	0.146875
$639$	2.20443e9	0.334228
$640$	−6.64797e8	−0.100244
$641$	−6.61251e9	−0.991661	−0.495831	−	0.868419i	$-0.665136\pi$
$641$	−6.61251e9	−0.991661	−0.495831	+	0.868419i	$0.665136\pi$
$642$	−2.13954e9	−0.319115
$643$	−3.07161e9	−0.455647	−0.227823	−	0.973702i	$-0.573161\pi$
$643$	−3.07161e9	−0.455647	−0.227823	+	0.973702i	$0.573161\pi$
$644$	8.51093e8	0.125567
$645$	−4.05005e9	−0.594293
$646$	9.15110e9	1.33555
$647$	−2.81527e9	−0.408654	−0.204327	−	0.978903i	$-0.565501\pi$
$647$	−2.81527e9	−0.408654	−0.204327	+	0.978903i	$0.565501\pi$
$648$	−1.30292e9	−0.188107
$649$	2.92044e9	0.419364
$650$	2.62571e9	0.375017
$651$	−2.72068e9	−0.386495
$652$	−4.10731e9	−0.580352
$653$	5.66090e9	0.795590	0.397795	−	0.917474i	$-0.369775\pi$
$653$	5.66090e9	0.795590	0.397795	+	0.917474i	$0.369775\pi$
$654$	−1.00415e9	−0.140370
$655$	−1.41035e9	−0.196101
$656$	−1.30331e9	−0.180254
$657$	8.62743e9	1.18687
$658$	−4.27735e9	−0.585308
$659$	−6.05155e9	−0.823697	−0.411848	−	0.911252i	$-0.635117\pi$
$659$	−6.05155e9	−0.823697	−0.411848	+	0.911252i	$0.635117\pi$
$660$	−5.13063e8	−0.0694652
$661$	1.96771e9	0.265006	0.132503	−	0.991183i	$-0.457699\pi$
$661$	1.96771e9	0.265006	0.132503	+	0.991183i	$0.457699\pi$
$662$	−3.53619e9	−0.473732
$663$	−8.38149e9	−1.11693
$664$	2.47345e9	0.327880
$665$	−1.24257e10	−1.63849
$666$	3.67407e9	0.481934
$667$	1.16819e9	0.152431
$668$	1.52664e9	0.198162
$669$	4.70835e9	0.607964
$670$	3.28046e9	0.421379
$671$	4.21019e9	0.537989
$672$	−6.41270e8	−0.0815170
$673$	−6.59401e9	−0.833867	−0.416934	−	0.908937i	$-0.636895\pi$
$673$	−6.59401e9	−0.833867	−0.416934	+	0.908937i	$0.636895\pi$
$674$	−9.00637e8	−0.113303
$675$	1.70519e9	0.213407
$676$	9.76873e9	1.21626
$677$	−2.72382e9	−0.337379	−0.168689	−	0.985669i	$-0.553953\pi$
$677$	−2.72382e9	−0.337379	−0.168689	+	0.985669i	$0.553953\pi$
$678$	−3.98209e8	−0.0490690
$679$	−4.06706e9	−0.498582
$680$	4.87853e9	0.594988
$681$	−4.36506e9	−0.529634
$682$	1.48032e9	0.178694
$683$	4.52062e9	0.542907	0.271453	−	0.962452i	$-0.412496\pi$
$683$	4.52062e9	0.542907	0.271453	+	0.962452i	$0.412496\pi$
$684$	−4.44738e9	−0.531383
$685$	−8.84821e9	−1.05181
$686$	−4.83017e9	−0.571253
$687$	4.39431e9	0.517062
$688$	2.75427e9	0.322439
$689$	−1.13529e10	−1.32233
$690$	−6.22104e8	−0.0720926
$691$	−1.98258e9	−0.228590	−0.114295	−	0.993447i	$-0.536461\pi$
$691$	−1.98258e9	−0.228590	−0.114295	+	0.993447i	$0.536461\pi$
$692$	−7.77806e8	−0.0892277
$693$	2.50332e9	0.285726
$694$	−6.89037e9	−0.782500
$695$	1.19392e10	1.34905
$696$	−8.80189e8	−0.0989564
$697$	9.56422e9	1.06988
$698$	7.33524e9	0.816433
$699$	2.34016e9	0.259165
$700$	−1.47423e9	−0.162451
$701$	8.01665e9	0.878981	0.439491	−	0.898247i	$-0.355159\pi$
$701$	8.01665e9	0.878981	0.439491	+	0.898247i	$0.355159\pi$
$702$	8.95201e9	0.976654
$703$	9.57150e9	1.03905
$704$	3.48914e8	0.0376889
$705$	3.12652e9	0.336046
$706$	4.78359e9	0.511608
$707$	1.35270e10	1.43958
$708$	−2.66811e9	−0.282544
$709$	1.02912e10	1.08444	0.542221	−	0.840236i	$-0.317584\pi$
$709$	1.02912e10	1.08444	0.542221	+	0.840236i	$0.317584\pi$
$710$	3.06157e9	0.321026
$711$	4.81717e9	0.502630
$712$	1.25350e9	0.130150
$713$	1.79493e9	0.185453
$714$	4.70588e9	0.483835
$715$	−6.19220e9	−0.633540
$716$	6.10037e9	0.621099
$717$	−3.66353e9	−0.371178
$718$	5.14659e9	0.518900
$719$	−1.29702e10	−1.30136	−0.650680	−	0.759352i	$-0.725516\pi$
$719$	−1.29702e10	−1.30136	−0.650680	+	0.759352i	$0.725516\pi$
$720$	−2.37094e9	−0.236732
$721$	−1.83712e10	−1.82543
$722$	−4.43510e9	−0.438554
$723$	1.52344e9	0.149913
$724$	−5.71655e9	−0.559821
$725$	−2.02349e9	−0.197206
$726$	2.69277e8	0.0261169
$727$	7.55298e9	0.729035	0.364517	−	0.931197i	$-0.381234\pi$
$727$	7.55298e9	0.729035	0.364517	+	0.931197i	$0.381234\pi$
$728$	−7.73954e9	−0.743456
$729$	−1.47846e9	−0.141339
$730$	1.19820e10	1.13999
$731$	−2.02119e10	−1.91380
$732$	−3.84643e9	−0.362467
$733$	1.24600e9	0.116857	0.0584284	−	0.998292i	$-0.481391\pi$
$733$	1.24600e9	0.116857	0.0584284	+	0.998292i	$0.481391\pi$
$734$	−8.84956e9	−0.826010
$735$	−1.42960e9	−0.132803
$736$	4.23068e8	0.0391145
$737$	−1.72172e9	−0.158426
$738$	−4.64815e9	−0.425680
$739$	−6.57696e8	−0.0599473	−0.0299736	−	0.999551i	$-0.509542\pi$
$739$	−6.57696e8	−0.0599473	−0.0299736	+	0.999551i	$0.509542\pi$
$740$	5.10266e9	0.462898
$741$	1.06117e10	0.958122
$742$	6.37422e9	0.572813
$743$	9.78668e9	0.875335	0.437668	−	0.899137i	$-0.355805\pi$
$743$	9.78668e9	0.875335	0.437668	+	0.899137i	$0.355805\pi$
$744$	−1.35242e9	−0.120394
$745$	1.83861e10	1.62908
$746$	−5.35858e9	−0.472568
$747$	8.82134e9	0.774305
$748$	−2.56046e9	−0.223698
$749$	1.44982e10	1.26074
$750$	−2.68679e9	−0.232551
$751$	−6.07010e9	−0.522945	−0.261472	−	0.965211i	$-0.584208\pi$
$751$	−6.07010e9	−0.522945	−0.261472	+	0.965211i	$0.584208\pi$
$752$	−2.12622e9	−0.182325
$753$	5.47028e9	0.466904
$754$	−1.06231e10	−0.902507
$755$	−6.69945e9	−0.566533
$756$	−5.02620e9	−0.423071
$757$	−1.51175e10	−1.26661	−0.633307	−	0.773901i	$-0.718303\pi$
$757$	−1.51175e10	−1.26661	−0.633307	+	0.773901i	$0.718303\pi$
$758$	−1.00886e10	−0.841371
$759$	3.26506e8	0.0271047
$760$	−6.17664e9	−0.510393
$761$	−9.81734e9	−0.807509	−0.403755	−	0.914867i	$-0.632295\pi$
$761$	−9.81734e9	−0.807509	−0.403755	+	0.914867i	$0.632295\pi$
$762$	−3.28284e9	−0.268787
$763$	6.80441e9	0.554568
$764$	−1.07600e10	−0.872941
$765$	1.73988e10	1.40509
$766$	5.80156e9	0.466385
$767$	−3.22016e10	−2.57688
$768$	−3.18767e8	−0.0253927
$769$	−1.27830e10	−1.01366	−0.506828	−	0.862047i	$-0.669182\pi$
$769$	−1.27830e10	−1.01366	−0.506828	+	0.862047i	$0.669182\pi$
$770$	3.47668e9	0.274440
$771$	6.74619e9	0.530113
$772$	8.36242e9	0.654141
$773$	1.94821e10	1.51708	0.758538	−	0.651629i	$-0.225914\pi$
$773$	1.94821e10	1.51708	0.758538	+	0.651629i	$0.225914\pi$
$774$	9.82286e9	0.761455
$775$	−3.10911e9	−0.239928
$776$	−2.02168e9	−0.155309
$777$	4.92207e9	0.376421
$778$	−4.17202e9	−0.317627
$779$	−1.21091e10	−0.917765
$780$	5.65719e9	0.426844
$781$	−1.60684e9	−0.120697
$782$	−3.10463e9	−0.232159
$783$	−6.89883e9	−0.513581
$784$	9.72214e8	0.0720536
$785$	3.10693e9	0.229239
$786$	−6.76254e8	−0.0496742
$787$	−5.09817e9	−0.372823	−0.186412	−	0.982472i	$-0.559686\pi$
$787$	−5.09817e9	−0.372823	−0.186412	+	0.982472i	$0.559686\pi$
$788$	−5.11897e9	−0.372684
$789$	1.01567e9	0.0736180
$790$	6.69023e9	0.482776
$791$	2.69839e9	0.193860
$792$	1.24437e9	0.0890042
$793$	−4.64228e10	−3.30579
$794$	1.05598e10	0.748658
$795$	−4.65921e9	−0.328872
$796$	4.03275e9	0.283403
$797$	2.76875e10	1.93722	0.968610	−	0.248585i	$-0.0799655\pi$
$797$	2.76875e10	1.93722	0.968610	+	0.248585i	$0.0799655\pi$
$798$	−5.95805e9	−0.415044
$799$	1.56030e10	1.08217
$800$	−7.32824e8	−0.0506040
$801$	4.47047e9	0.307355
$802$	−6.86911e9	−0.470208
$803$	−6.28867e9	−0.428602
$804$	1.57297e9	0.106739
$805$	4.21557e9	0.284820
$806$	−1.63224e10	−1.09802
$807$	−3.45278e9	−0.231266
$808$	6.72412e9	0.448431
$809$	2.37439e10	1.57664	0.788320	−	0.615265i	$-0.210951\pi$
$809$	2.37439e10	1.57664	0.788320	+	0.615265i	$0.210951\pi$
$810$	−6.45353e9	−0.426678
$811$	7.75950e9	0.510812	0.255406	−	0.966834i	$-0.417791\pi$
$811$	7.75950e9	0.510812	0.255406	+	0.966834i	$0.417791\pi$
$812$	5.96444e9	0.390952
$813$	2.80360e9	0.182978
$814$	−2.67809e9	−0.174036
$815$	−2.03440e10	−1.31639
$816$	2.33923e9	0.150716
$817$	2.55900e10	1.64170
$818$	−1.38087e10	−0.882096
$819$	−2.76023e10	−1.75571
$820$	−6.45548e9	−0.408865
$821$	2.83557e10	1.78830	0.894148	−	0.447771i	$-0.147782\pi$
$821$	2.83557e10	1.78830	0.894148	+	0.447771i	$0.147782\pi$
$822$	−4.24267e9	−0.266433
$823$	1.73646e10	1.08584	0.542920	−	0.839784i	$-0.317319\pi$
$823$	1.73646e10	1.08584	0.542920	+	0.839784i	$0.317319\pi$
$824$	−9.13210e9	−0.568624
$825$	−5.65563e8	−0.0350665
$826$	1.80799e10	1.11626
$827$	−9.26570e9	−0.569651	−0.284825	−	0.958579i	$-0.591936\pi$
$827$	−9.26570e9	−0.569651	−0.284825	+	0.958579i	$0.591936\pi$
$828$	1.50883e9	0.0923707
$829$	−2.10376e10	−1.28250	−0.641248	−	0.767334i	$-0.721583\pi$
$829$	−2.10376e10	−1.28250	−0.641248	+	0.767334i	$0.721583\pi$
$830$	1.22513e10	0.743720
$831$	4.42092e9	0.267245
$832$	−3.84723e9	−0.231588
$833$	−7.13448e9	−0.427666
$834$	5.72480e9	0.341727
$835$	7.56165e9	0.449484
$836$	3.24176e9	0.191893
$837$	−1.06001e10	−0.624842
$838$	6.99536e9	0.410635
$839$	1.27028e10	0.742564	0.371282	−	0.928520i	$-0.378918\pi$
$839$	1.27028e10	0.742564	0.371282	+	0.928520i	$0.378918\pi$
$840$	−3.17629e9	−0.184902
$841$	−9.06325e9	−0.525409
$842$	3.44194e9	0.198706
$843$	4.98843e9	0.286792
$844$	−8.62451e9	−0.493783
$845$	4.83858e10	2.75879
$846$	−7.58295e9	−0.430568
$847$	−1.82471e9	−0.103181
$848$	3.16854e9	0.178433
$849$	−1.24767e10	−0.699720
$850$	5.37774e9	0.300354
$851$	−3.24726e9	−0.180619
$852$	1.46801e9	0.0813187
$853$	−1.62891e10	−0.898617	−0.449309	−	0.893377i	$-0.648330\pi$
$853$	−1.62891e10	−0.898617	−0.449309	+	0.893377i	$0.648330\pi$
$854$	2.60646e10	1.43202
$855$	−2.20284e10	−1.20532
$856$	7.20686e9	0.392725
$857$	−9.87796e9	−0.536086	−0.268043	−	0.963407i	$-0.586377\pi$
$857$	−9.87796e9	−0.536086	−0.268043	+	0.963407i	$0.586377\pi$
$858$	−2.96913e9	−0.160481
$859$	−7.28022e9	−0.391894	−0.195947	−	0.980614i	$-0.562778\pi$
$859$	−7.28022e9	−0.391894	−0.195947	+	0.980614i	$0.562778\pi$
$860$	1.36423e10	0.731378
$861$	−6.22702e9	−0.332483
$862$	1.52945e10	0.813315
$863$	−1.14668e10	−0.607303	−0.303651	−	0.952783i	$-0.598206\pi$
$863$	−1.14668e10	−0.607303	−0.303651	+	0.952783i	$0.598206\pi$
$864$	−2.49846e9	−0.131788
$865$	−3.85257e9	−0.202392
$866$	−1.66386e10	−0.870571
$867$	−9.36975e9	−0.488271
$868$	9.16440e9	0.475647
$869$	−3.51131e9	−0.181510
$870$	−4.35969e9	−0.224460
$871$	1.89842e10	0.973486
$872$	3.38239e9	0.172749
$873$	−7.21015e9	−0.366770
$874$	3.93073e9	0.199151
$875$	1.82065e10	0.918753
$876$	5.74532e9	0.288769
$877$	2.98990e9	0.149678	0.0748390	−	0.997196i	$-0.476156\pi$
$877$	2.98990e9	0.149678	0.0748390	+	0.997196i	$0.476156\pi$
$878$	6.02771e9	0.300553
$879$	9.96258e9	0.494778
$880$	1.72821e9	0.0854885
$881$	−7.27001e9	−0.358195	−0.179097	−	0.983831i	$-0.557318\pi$
$881$	−7.27001e9	−0.358195	−0.179097	+	0.983831i	$0.557318\pi$
$882$	3.46731e9	0.170158
$883$	2.14877e10	1.05034	0.525168	−	0.850999i	$-0.324003\pi$
$883$	2.14877e10	1.05034	0.525168	+	0.850999i	$0.324003\pi$
$884$	2.82324e10	1.37456
$885$	−1.32155e10	−0.640886
$886$	−1.15345e10	−0.557162
$887$	−1.89790e10	−0.913145	−0.456573	−	0.889686i	$-0.650923\pi$
$887$	−1.89790e10	−0.913145	−0.456573	+	0.889686i	$0.650923\pi$
$888$	2.44670e9	0.117256
$889$	2.22456e10	1.06191
$890$	6.20872e9	0.295214
$891$	3.38709e9	0.160419
$892$	−1.58597e10	−0.748201
$893$	−1.97547e10	−0.928305
$894$	8.81607e9	0.412661
$895$	3.02159e10	1.40882
$896$	2.16007e9	0.100320
$897$	−3.60015e9	−0.166551
$898$	−2.18343e10	−1.00617
$899$	1.25788e10	0.577405
$900$	−2.61355e9	−0.119504
$901$	−2.32520e10	−1.05907
$902$	3.38811e9	0.153721
$903$	1.31595e10	0.594745
$904$	1.34134e9	0.0603877
$905$	−2.83148e10	−1.26982
$906$	−3.21236e9	−0.143508
$907$	5.81525e8	0.0258787	0.0129394	−	0.999916i	$-0.495881\pi$
$907$	5.81525e8	0.0258787	0.0129394	+	0.999916i	$0.495881\pi$
$908$	1.47034e10	0.651803
$909$	2.39809e10	1.05899
$910$	−3.83349e10	−1.68636
$911$	−3.06924e10	−1.34498	−0.672490	−	0.740106i	$-0.734775\pi$
$911$	−3.06924e10	−1.34498	−0.672490	+	0.740106i	$0.734775\pi$
$912$	−2.96167e9	−0.129287
$913$	−6.43001e9	−0.279617
$914$	−2.02709e10	−0.878137
$915$	−1.90518e10	−0.822172
$916$	−1.48019e10	−0.636331
$917$	4.58251e9	0.196250
$918$	1.83347e10	0.782210
$919$	−2.70441e10	−1.14939	−0.574697	−	0.818366i	$-0.694880\pi$
$919$	−2.70441e10	−1.14939	−0.574697	+	0.818366i	$0.694880\pi$
$920$	2.09551e9	0.0887221
$921$	−1.36545e10	−0.575926
$922$	−2.56476e10	−1.07768
$923$	1.77175e10	0.741647
$924$	1.66705e9	0.0695179
$925$	5.62479e9	0.233674
$926$	−1.11157e10	−0.460042
$927$	−3.25688e10	−1.34283
$928$	2.96485e9	0.121782
$929$	−6.17228e8	−0.0252575	−0.0126288	−	0.999920i	$-0.504020\pi$
$929$	−6.17228e8	−0.0252575	−0.0126288	+	0.999920i	$0.504020\pi$
$930$	−6.69868e9	−0.273086
$931$	9.03286e9	0.366861
$932$	−7.88266e9	−0.318946
$933$	7.74903e9	0.312364
$934$	3.41961e10	1.37329
$935$	−1.26823e10	−0.507407
$936$	−1.37208e10	−0.546906
$937$	−1.63038e10	−0.647440	−0.323720	−	0.946153i	$-0.604934\pi$
$937$	−1.63038e10	−0.647440	−0.323720	+	0.946153i	$0.604934\pi$
$938$	−1.06589e10	−0.421699
$939$	5.50214e9	0.216871
$940$	−1.05314e10	−0.413561
$941$	−1.16423e10	−0.455486	−0.227743	−	0.973721i	$-0.573135\pi$
$941$	−1.16423e10	−0.455486	−0.227743	+	0.973721i	$0.573135\pi$
$942$	1.48976e9	0.0580682
$943$	4.10818e9	0.159536
$944$	8.98731e9	0.347718
$945$	−2.48954e10	−0.959639
$946$	−7.16003e9	−0.274977
$947$	8.03011e9	0.307253	0.153627	−	0.988129i	$-0.450905\pi$
$947$	8.03011e9	0.307253	0.153627	+	0.988129i	$0.450905\pi$
$948$	3.20793e9	0.122291
$949$	6.93408e10	2.63364
$950$	−6.80868e9	−0.257650
$951$	1.46672e9	0.0552988
$952$	−1.58514e10	−0.595440
$953$	2.93973e10	1.10023	0.550114	−	0.835089i	$-0.314584\pi$
$953$	2.93973e10	1.10023	0.550114	+	0.835089i	$0.314584\pi$
$954$	1.13003e10	0.421377
$955$	−5.32956e10	−1.98006
$956$	1.23403e10	0.456797
$957$	2.28815e9	0.0843903
$958$	9.95635e9	0.365865
$959$	2.87497e10	1.05261
$960$	−1.57889e9	−0.0575975
$961$	−8.18522e9	−0.297508
$962$	2.95294e10	1.06940
$963$	2.57026e10	0.927439
$964$	−5.13157e9	−0.184493
$965$	4.14201e10	1.48377
$966$	2.02135e9	0.0721474
$967$	3.31262e10	1.17809	0.589045	−	0.808100i	$-0.299504\pi$
$967$	3.31262e10	1.17809	0.589045	+	0.808100i	$0.299504\pi$
$968$	−9.07039e8	−0.0321412
$969$	2.17339e10	0.767368
$970$	−1.00137e10	−0.352283
$971$	1.89899e10	0.665664	0.332832	−	0.942986i	$-0.391996\pi$
$971$	1.89899e10	0.665664	0.332832	+	0.942986i	$0.391996\pi$
$972$	−1.37666e10	−0.480833
$973$	−3.87930e10	−1.35008
$974$	5.99338e9	0.207833
$975$	6.23607e9	0.215474
$976$	1.29564e10	0.446077
$977$	3.70878e10	1.27233	0.636165	−	0.771553i	$-0.280520\pi$
$977$	3.70878e10	1.27233	0.636165	+	0.771553i	$0.280520\pi$
$978$	−9.75487e9	−0.333454
$979$	−3.25860e9	−0.110992
$980$	4.81550e9	0.163437
$981$	1.20630e10	0.407955
$982$	8.08657e9	0.272505
$983$	2.82981e10	0.950211	0.475106	−	0.879929i	$-0.342410\pi$
$983$	2.82981e10	0.950211	0.475106	+	0.879929i	$0.342410\pi$
$984$	−3.09537e9	−0.103569
$985$	−2.53549e10	−0.845347
$986$	−2.17572e10	−0.722825
$987$	−1.01587e10	−0.336301
$988$	−3.57446e10	−1.17913
$989$	−8.68174e9	−0.285378
$990$	6.16351e9	0.201885
$991$	1.13820e10	0.371503	0.185751	−	0.982597i	$-0.440528\pi$
$991$	1.13820e10	0.371503	0.185751	+	0.982597i	$0.440528\pi$
$992$	4.55551e9	0.148165
$993$	−8.39846e9	−0.272193
$994$	−9.94770e9	−0.321270
$995$	1.99747e10	0.642835
$996$	5.87445e9	0.188391
$997$	−5.47116e10	−1.74842	−0.874211	−	0.485546i	$-0.838621\pi$
$997$	−5.47116e10	−1.74842	−0.874211	+	0.485546i	$0.838621\pi$
$998$	−1.19616e10	−0.380920
$999$	1.91770e10	0.608556

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	22.8.a.a.1.1	✓	1
3.2	odd	2		198.8.a.c.1.1		1
4.3	odd	2		176.8.a.b.1.1		1
5.4	even	2		550.8.a.c.1.1		1
11.10	odd	2		242.8.a.d.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
22.8.a.a.1.1	✓	1	1.1	even	1	trivial
176.8.a.b.1.1		1	4.3	odd	2
198.8.a.c.1.1		1	3.2	odd	2
242.8.a.d.1.1		1	11.10	odd	2
550.8.a.c.1.1		1	5.4	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:	\( N \)	\(=\)	\( 22 = 2 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	22.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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