LMFDB - Embedded newform 114.8.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [114,8,Mod(1,114)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(114, 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("114.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 114 = 2 \cdot 3 \cdot 19 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	114.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:	$35.6118929052$
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	$\chi$	$=$	114.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} -75.0000 q^{5} +216.000 q^{6} -497.000 q^{7} +512.000 q^{8} +729.000 q^{9} +O(q^{10})$

\(q+8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} -75.0000 q^{5} +216.000 q^{6} -497.000 q^{7} +512.000 q^{8} +729.000 q^{9} -600.000 q^{10} -8411.00 q^{11} +1728.00 q^{12} -3750.00 q^{13} -3976.00 q^{14} -2025.00 q^{15} +4096.00 q^{16} -4409.00 q^{17} +5832.00 q^{18} +6859.00 q^{19} -4800.00 q^{20} -13419.0 q^{21} -67288.0 q^{22} -53036.0 q^{23} +13824.0 q^{24} -72500.0 q^{25} -30000.0 q^{26} +19683.0 q^{27} -31808.0 q^{28} +10806.0 q^{29} -16200.0 q^{30} +46386.0 q^{31} +32768.0 q^{32} -227097. q^{33} -35272.0 q^{34} +37275.0 q^{35} +46656.0 q^{36} -46736.0 q^{37} +54872.0 q^{38} -101250. q^{39} -38400.0 q^{40} -123680. q^{41} -107352. q^{42} +502779. q^{43} -538304. q^{44} -54675.0 q^{45} -424288. q^{46} -154445. q^{47} +110592. q^{48} -576534. q^{49} -580000. q^{50} -119043. q^{51} -240000. q^{52} -580534. q^{53} +157464. q^{54} +630825. q^{55} -254464. q^{56} +185193. q^{57} +86448.0 q^{58} -57584.0 q^{59} -129600. q^{60} -460705. q^{61} +371088. q^{62} -362313. q^{63} +262144. q^{64} +281250. q^{65} -1.81678e6 q^{66} +934320. q^{67} -282176. q^{68} -1.43197e6 q^{69} +298200. q^{70} -1.85396e6 q^{71} +373248. q^{72} -5.08645e6 q^{73} -373888. q^{74} -1.95750e6 q^{75} +438976. q^{76} +4.18027e6 q^{77} -810000. q^{78} -3.68108e6 q^{79} -307200. q^{80} +531441. q^{81} -989440. q^{82} +4.45257e6 q^{83} -858816. q^{84} +330675. q^{85} +4.02223e6 q^{86} +291762. q^{87} -4.30643e6 q^{88} +5.89220e6 q^{89} -437400. q^{90} +1.86375e6 q^{91} -3.39430e6 q^{92} +1.25242e6 q^{93} -1.23556e6 q^{94} -514425. q^{95} +884736. q^{96} +9.29363e6 q^{97} -4.61227e6 q^{98} -6.13162e6 q^{99} +O(q^{100})\)

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
$3$	27.0000	0.577350
$3$	27.0000	0.577350
$4$	64.0000	0.500000
$5$	−75.0000	−0.268328	−0.134164	−	0.990959i	$-0.542835\pi$
$5$	−75.0000	−0.268328	−0.134164	+	0.990959i	$0.542835\pi$
$6$	216.000	0.408248
$7$	−497.000	−0.547663	−0.273831	−	0.961778i	$-0.588291\pi$
$7$	−497.000	−0.547663	−0.273831	+	0.961778i	$0.588291\pi$
$8$	512.000	0.353553
$9$	729.000	0.333333
$10$	−600.000	−0.189737
$11$	−8411.00	−1.90534	−0.952672	−	0.304001i	$-0.901677\pi$
$11$	−8411.00	−1.90534	−0.952672	+	0.304001i	$0.901677\pi$
$12$	1728.00	0.288675
$13$	−3750.00	−0.473401	−0.236701	−	0.971583i	$-0.576066\pi$
$13$	−3750.00	−0.473401	−0.236701	+	0.971583i	$0.576066\pi$
$14$	−3976.00	−0.387256
$15$	−2025.00	−0.154919
$16$	4096.00	0.250000
$17$	−4409.00	−0.217655	−0.108828	−	0.994061i	$-0.534710\pi$
$17$	−4409.00	−0.217655	−0.108828	+	0.994061i	$0.534710\pi$
$18$	5832.00	0.235702
$19$	6859.00	0.229416
$19$	6859.00	0.229416
$20$	−4800.00	−0.134164
$21$	−13419.0	−0.316193
$22$	−67288.0	−1.34728
$23$	−53036.0	−0.908915	−0.454458	−	0.890768i	$-0.650167\pi$
$23$	−53036.0	−0.908915	−0.454458	+	0.890768i	$0.650167\pi$
$24$	13824.0	0.204124
$25$	−72500.0	−0.928000
$26$	−30000.0	−0.334745
$27$	19683.0	0.192450
$28$	−31808.0	−0.273831
$29$	10806.0	0.0822758	0.0411379	−	0.999153i	$-0.486902\pi$
$29$	10806.0	0.0822758	0.0411379	+	0.999153i	$0.486902\pi$
$30$	−16200.0	−0.109545
$31$	46386.0	0.279654	0.139827	−	0.990176i	$-0.455345\pi$
$31$	46386.0	0.279654	0.139827	+	0.990176i	$0.455345\pi$
$32$	32768.0	0.176777
$33$	−227097.	−1.10005
$34$	−35272.0	−0.153905
$35$	37275.0	0.146953
$36$	46656.0	0.166667
$37$	−46736.0	−0.151686	−0.0758430	−	0.997120i	$-0.524165\pi$
$37$	−46736.0	−0.151686	−0.0758430	+	0.997120i	$0.524165\pi$
$38$	54872.0	0.162221
$39$	−101250.	−0.273318
$40$	−38400.0	−0.0948683
$41$	−123680.	−0.280257	−0.140128	−	0.990133i	$-0.544752\pi$
$41$	−123680.	−0.280257	−0.140128	+	0.990133i	$0.544752\pi$
$42$	−107352.	−0.223582
$43$	502779.	0.964356	0.482178	−	0.876073i	$-0.339846\pi$
$43$	502779.	0.964356	0.482178	+	0.876073i	$0.339846\pi$
$44$	−538304.	−0.952672
$45$	−54675.0	−0.0894427
$46$	−424288.	−0.642700
$47$	−154445.	−0.216986	−0.108493	−	0.994097i	$-0.534602\pi$
$47$	−154445.	−0.216986	−0.108493	+	0.994097i	$0.534602\pi$
$48$	110592.	0.144338
$49$	−576534.	−0.700065
$50$	−580000.	−0.656195
$51$	−119043.	−0.125663
$52$	−240000.	−0.236701
$53$	−580534.	−0.535627	−0.267813	−	0.963471i	$-0.586301\pi$
$53$	−580534.	−0.535627	−0.267813	+	0.963471i	$0.586301\pi$
$54$	157464.	0.136083
$55$	630825.	0.511257
$56$	−254464.	−0.193628
$57$	185193.	0.132453
$58$	86448.0	0.0581778
$59$	−57584.0	−0.0365023	−0.0182511	−	0.999833i	$-0.505810\pi$
$59$	−57584.0	−0.0365023	−0.0182511	+	0.999833i	$0.505810\pi$
$60$	−129600.	−0.0774597
$61$	−460705.	−0.259877	−0.129939	−	0.991522i	$-0.541478\pi$
$61$	−460705.	−0.259877	−0.129939	+	0.991522i	$0.541478\pi$
$62$	371088.	0.197745
$63$	−362313.	−0.182554
$64$	262144.	0.125000
$65$	281250.	0.127027
$66$	−1.81678e6	−0.777853
$67$	934320.	0.379519	0.189760	−	0.981831i	$-0.439229\pi$
$67$	934320.	0.379519	0.189760	+	0.981831i	$0.439229\pi$
$68$	−282176.	−0.108828
$69$	−1.43197e6	−0.524762
$70$	298200.	0.103912
$71$	−1.85396e6	−0.614745	−0.307373	−	0.951589i	$-0.599450\pi$
$71$	−1.85396e6	−0.614745	−0.307373	+	0.951589i	$0.599450\pi$
$72$	373248.	0.117851
$73$	−5.08645e6	−1.53033	−0.765165	−	0.643835i	$-0.777342\pi$
$73$	−5.08645e6	−1.53033	−0.765165	+	0.643835i	$0.777342\pi$
$74$	−373888.	−0.107258
$75$	−1.95750e6	−0.535781
$76$	438976.	0.114708
$77$	4.18027e6	1.04349
$78$	−810000.	−0.193265
$79$	−3.68108e6	−0.840002	−0.420001	−	0.907524i	$-0.637970\pi$
$79$	−3.68108e6	−0.840002	−0.420001	+	0.907524i	$0.637970\pi$
$80$	−307200.	−0.0670820
$81$	531441.	0.111111
$82$	−989440.	−0.198171
$83$	4.45257e6	0.854747	0.427374	−	0.904075i	$-0.359439\pi$
$83$	4.45257e6	0.854747	0.427374	+	0.904075i	$0.359439\pi$
$84$	−858816.	−0.158097
$85$	330675.	0.0584030
$86$	4.02223e6	0.681903
$87$	291762.	0.0475019
$88$	−4.30643e6	−0.673641
$89$	5.89220e6	0.885957	0.442978	−	0.896532i	$-0.353922\pi$
$89$	5.89220e6	0.885957	0.442978	+	0.896532i	$0.353922\pi$
$90$	−437400.	−0.0632456
$91$	1.86375e6	0.259264
$92$	−3.39430e6	−0.454458
$93$	1.25242e6	0.161458
$94$	−1.23556e6	−0.153432
$95$	−514425.	−0.0615587
$96$	884736.	0.102062
$97$	9.29363e6	1.03391	0.516957	−	0.856011i	$-0.327065\pi$
$97$	9.29363e6	1.03391	0.516957	+	0.856011i	$0.327065\pi$
$98$	−4.61227e6	−0.495021
$99$	−6.13162e6	−0.635114
$100$	−4.64000e6	−0.464000
$101$	1.09178e7	1.05441	0.527207	−	0.849737i	$-0.323239\pi$
$101$	1.09178e7	1.05441	0.527207	+	0.849737i	$0.323239\pi$
$102$	−952344.	−0.0888573
$103$	5.57914e6	0.503080	0.251540	−	0.967847i	$-0.419063\pi$
$103$	5.57914e6	0.503080	0.251540	+	0.967847i	$0.419063\pi$
$104$	−1.92000e6	−0.167373
$105$	1.00642e6	0.0848436
$106$	−4.64427e6	−0.378745
$107$	−1.80544e7	−1.42476	−0.712378	−	0.701796i	$-0.752382\pi$
$107$	−1.80544e7	−1.42476	−0.712378	+	0.701796i	$0.752382\pi$
$108$	1.25971e6	0.0962250
$109$	−3.40124e6	−0.251561	−0.125781	−	0.992058i	$-0.540144\pi$
$109$	−3.40124e6	−0.251561	−0.125781	+	0.992058i	$0.540144\pi$
$110$	5.04660e6	0.361513
$111$	−1.26187e6	−0.0875760
$112$	−2.03571e6	−0.136916
$113$	−1.21749e7	−0.793763	−0.396882	−	0.917870i	$-0.629908\pi$
$113$	−1.21749e7	−0.793763	−0.396882	+	0.917870i	$0.629908\pi$
$114$	1.48154e6	0.0936586
$115$	3.97770e6	0.243888
$116$	691584.	0.0411379
$117$	−2.73375e6	−0.157800
$118$	−460672.	−0.0258110
$119$	2.19127e6	0.119202
$120$	−1.03680e6	−0.0547723
$121$	5.12578e7	2.63033
$122$	−3.68564e6	−0.183761
$123$	−3.33936e6	−0.161806
$124$	2.96870e6	0.139827
$125$	1.12969e7	0.517337
$126$	−2.89850e6	−0.129085
$127$	3.56219e7	1.54313	0.771567	−	0.636148i	$-0.219473\pi$
$127$	3.56219e7	1.54313	0.771567	+	0.636148i	$0.219473\pi$
$128$	2.09715e6	0.0883883
$129$	1.35750e7	0.556771
$130$	2.25000e6	0.0898216
$131$	1.09100e7	0.424009	0.212004	−	0.977269i	$-0.432001\pi$
$131$	1.09100e7	0.424009	0.212004	+	0.977269i	$0.432001\pi$
$132$	−1.45342e7	−0.550025
$133$	−3.40892e6	−0.125642
$134$	7.47456e6	0.268361
$135$	−1.47622e6	−0.0516398
$136$	−2.25741e6	−0.0769527
$137$	2.31048e7	0.767679	0.383839	−	0.923400i	$-0.374602\pi$
$137$	2.31048e7	0.767679	0.383839	+	0.923400i	$0.374602\pi$
$138$	−1.14558e7	−0.371063
$139$	4.07189e7	1.28601	0.643005	−	0.765862i	$-0.277688\pi$
$139$	4.07189e7	1.28601	0.643005	+	0.765862i	$0.277688\pi$
$140$	2.38560e6	0.0734767
$141$	−4.17002e6	−0.125277
$142$	−1.48316e7	−0.434691
$143$	3.15412e7	0.901992
$144$	2.98598e6	0.0833333
$145$	−810450.	−0.0220769
$146$	−4.06916e7	−1.08211
$147$	−1.55664e7	−0.404183
$148$	−2.99110e6	−0.0758430
$149$	1.42480e6	0.0352858	0.0176429	−	0.999844i	$-0.494384\pi$
$149$	1.42480e6	0.0352858	0.0176429	+	0.999844i	$0.494384\pi$
$150$	−1.56600e7	−0.378854
$151$	3.57984e7	0.846145	0.423072	−	0.906096i	$-0.360952\pi$
$151$	3.57984e7	0.846145	0.423072	+	0.906096i	$0.360952\pi$
$152$	3.51181e6	0.0811107
$153$	−3.21416e6	−0.0725517
$154$	3.34421e7	0.737856
$155$	−3.47895e6	−0.0750390
$156$	−6.48000e6	−0.136659
$157$	6.86282e7	1.41532	0.707659	−	0.706554i	$-0.249751\pi$
$157$	6.86282e7	1.41532	0.707659	+	0.706554i	$0.249751\pi$
$158$	−2.94486e7	−0.593971
$159$	−1.56744e7	−0.309244
$160$	−2.45760e6	−0.0474342
$161$	2.63589e7	0.497779
$162$	4.25153e6	0.0785674
$163$	−1.54547e7	−0.279515	−0.139757	−	0.990186i	$-0.544632\pi$
$163$	−1.54547e7	−0.279515	−0.139757	+	0.990186i	$0.544632\pi$
$164$	−7.91552e6	−0.140128
$165$	1.70323e7	0.295175
$166$	3.56206e7	0.604397
$167$	−8.57737e6	−0.142510	−0.0712552	−	0.997458i	$-0.522700\pi$
$167$	−8.57737e6	−0.142510	−0.0712552	+	0.997458i	$0.522700\pi$
$168$	−6.87053e6	−0.111791
$169$	−4.86860e7	−0.775891
$170$	2.64540e6	0.0412972
$171$	5.00021e6	0.0764719
$172$	3.21779e7	0.482178
$173$	−1.32929e8	−1.95191	−0.975954	−	0.217976i	$-0.930055\pi$
$173$	−1.32929e8	−1.95191	−0.975954	+	0.217976i	$0.930055\pi$
$174$	2.33410e6	0.0335889
$175$	3.60325e7	0.508231
$176$	−3.44515e7	−0.476336
$177$	−1.55477e6	−0.0210746
$178$	4.71376e7	0.626466
$179$	−7.49290e7	−0.976482	−0.488241	−	0.872709i	$-0.662361\pi$
$179$	−7.49290e7	−0.976482	−0.488241	+	0.872709i	$0.662361\pi$
$180$	−3.49920e6	−0.0447214
$181$	−2.39442e7	−0.300140	−0.150070	−	0.988675i	$-0.547950\pi$
$181$	−2.39442e7	−0.300140	−0.150070	+	0.988675i	$0.547950\pi$
$182$	1.49100e7	0.183328
$183$	−1.24390e7	−0.150040
$184$	−2.71544e7	−0.321350
$185$	3.50520e6	0.0407016
$186$	1.00194e7	0.114168
$187$	3.70841e7	0.414708
$188$	−9.88448e6	−0.108493
$189$	−9.78245e6	−0.105398
$190$	−4.11540e6	−0.0435286
$191$	−1.15491e8	−1.19931	−0.599655	−	0.800259i	$-0.704695\pi$
$191$	−1.15491e8	−1.19931	−0.599655	+	0.800259i	$0.704695\pi$
$192$	7.07789e6	0.0721688
$193$	1.19376e8	1.19527	0.597635	−	0.801768i	$-0.296107\pi$
$193$	1.19376e8	1.19527	0.597635	+	0.801768i	$0.296107\pi$
$194$	7.43490e7	0.731087
$195$	7.59375e6	0.0733390
$196$	−3.68982e7	−0.350033
$197$	−1.68512e8	−1.57036	−0.785178	−	0.619270i	$-0.787429\pi$
$197$	−1.68512e8	−1.57036	−0.785178	+	0.619270i	$0.787429\pi$
$198$	−4.90530e7	−0.449094
$199$	−4.05839e7	−0.365063	−0.182531	−	0.983200i	$-0.558429\pi$
$199$	−4.05839e7	−0.365063	−0.182531	+	0.983200i	$0.558429\pi$
$200$	−3.71200e7	−0.328098
$201$	2.52266e7	0.219116
$202$	8.73425e7	0.745583
$203$	−5.37058e6	−0.0450594
$204$	−7.61875e6	−0.0628316
$205$	9.27600e6	0.0752008
$206$	4.46331e7	0.355731
$207$	−3.86632e7	−0.302972
$208$	−1.53600e7	−0.118350
$209$	−5.76910e7	−0.437116
$210$	8.05140e6	0.0599935
$211$	−3.76660e7	−0.276033	−0.138016	−	0.990430i	$-0.544073\pi$
$211$	−3.76660e7	−0.276033	−0.138016	+	0.990430i	$0.544073\pi$
$212$	−3.71542e7	−0.267813
$213$	−5.00568e7	−0.354923
$214$	−1.44435e8	−1.00745
$215$	−3.77084e7	−0.258764
$216$	1.00777e7	0.0680414
$217$	−2.30538e7	−0.153156
$218$	−2.72099e7	−0.177881
$219$	−1.37334e8	−0.883536
$220$	4.03728e7	0.255629
$221$	1.65338e7	0.103038
$222$	−1.00950e7	−0.0619256
$223$	7.66674e7	0.462960	0.231480	−	0.972840i	$-0.425643\pi$
$223$	7.66674e7	0.462960	0.231480	+	0.972840i	$0.425643\pi$
$224$	−1.62857e7	−0.0968140
$225$	−5.28525e7	−0.309333
$226$	−9.73992e7	−0.561275
$227$	−2.93240e8	−1.66392	−0.831960	−	0.554835i	$-0.812781\pi$
$227$	−2.93240e8	−1.66392	−0.831960	+	0.554835i	$0.812781\pi$
$228$	1.18524e7	0.0662266
$229$	−3.04228e8	−1.67408	−0.837038	−	0.547145i	$-0.815714\pi$
$229$	−3.04228e8	−1.67408	−0.837038	+	0.547145i	$0.815714\pi$
$230$	3.18216e7	0.172455
$231$	1.12867e8	0.602457
$232$	5.53267e6	0.0290889
$233$	−2.78288e8	−1.44128	−0.720640	−	0.693309i	$-0.756152\pi$
$233$	−2.78288e8	−1.44128	−0.720640	+	0.693309i	$0.756152\pi$
$234$	−2.18700e7	−0.111582
$235$	1.15834e7	0.0582234
$236$	−3.68538e6	−0.0182511
$237$	−9.93892e7	−0.484976
$238$	1.75302e7	0.0842883
$239$	−7.74940e7	−0.367177	−0.183588	−	0.983003i	$-0.558771\pi$
$239$	−7.74940e7	−0.367177	−0.183588	+	0.983003i	$0.558771\pi$
$240$	−8.29440e6	−0.0387298
$241$	2.74030e8	1.26107	0.630534	−	0.776161i	$-0.282836\pi$
$241$	2.74030e8	1.26107	0.630534	+	0.776161i	$0.282836\pi$
$242$	4.10062e8	1.85993
$243$	1.43489e7	0.0641500
$244$	−2.94851e7	−0.129939
$245$	4.32400e7	0.187847
$246$	−2.67149e7	−0.114414
$247$	−2.57212e7	−0.108606
$248$	2.37496e7	0.0988726
$249$	1.20219e8	0.493488
$250$	9.03750e7	0.365812
$251$	1.38006e8	0.550858	0.275429	−	0.961321i	$-0.411180\pi$
$251$	1.38006e8	0.550858	0.275429	+	0.961321i	$0.411180\pi$
$252$	−2.31880e7	−0.0912771
$253$	4.46086e8	1.73180
$254$	2.84975e8	1.09116
$255$	8.92822e6	0.0337190
$256$	1.67772e7	0.0625000
$257$	−2.67662e8	−0.983605	−0.491802	−	0.870707i	$-0.663662\pi$
$257$	−2.67662e8	−0.983605	−0.491802	+	0.870707i	$0.663662\pi$
$258$	1.08600e8	0.393697
$259$	2.32278e7	0.0830728
$260$	1.80000e7	0.0635135
$261$	7.87757e6	0.0274253
$262$	8.72800e7	0.299820
$263$	3.18989e8	1.08126	0.540630	−	0.841260i	$-0.318186\pi$
$263$	3.18989e8	1.08126	0.540630	+	0.841260i	$0.318186\pi$
$264$	−1.16274e8	−0.388927
$265$	4.35401e7	0.143724
$266$	−2.72714e7	−0.0888426
$267$	1.59089e8	0.511507
$268$	5.97965e7	0.189760
$269$	−4.01878e8	−1.25881	−0.629406	−	0.777077i	$-0.716702\pi$
$269$	−4.01878e8	−1.25881	−0.629406	+	0.777077i	$0.716702\pi$
$270$	−1.18098e7	−0.0365148
$271$	−1.19505e8	−0.364748	−0.182374	−	0.983229i	$-0.558378\pi$
$271$	−1.19505e8	−0.364748	−0.182374	+	0.983229i	$0.558378\pi$
$272$	−1.80593e7	−0.0544138
$273$	5.03213e7	0.149686
$274$	1.84838e8	0.542831
$275$	6.09798e8	1.76816
$276$	−9.16462e7	−0.262381
$277$	4.33996e8	1.22689	0.613446	−	0.789737i	$-0.289783\pi$
$277$	4.33996e8	1.22689	0.613446	+	0.789737i	$0.289783\pi$
$278$	3.25751e8	0.909346
$279$	3.38154e7	0.0932180
$280$	1.90848e7	0.0519559
$281$	−4.22654e8	−1.13635	−0.568177	−	0.822907i	$-0.692351\pi$
$281$	−4.22654e8	−1.13635	−0.568177	+	0.822907i	$0.692351\pi$
$282$	−3.33601e7	−0.0885841
$283$	6.05768e6	0.0158874	0.00794372	−	0.999968i	$-0.497471\pi$
$283$	6.05768e6	0.0158874	0.00794372	+	0.999968i	$0.497471\pi$
$284$	−1.18653e8	−0.307373
$285$	−1.38895e7	−0.0355409
$286$	2.52330e8	0.637805
$287$	6.14690e7	0.153486
$288$	2.38879e7	0.0589256
$289$	−3.90899e8	−0.952626
$290$	−6.48360e6	−0.0156107
$291$	2.50928e8	0.596930
$292$	−3.25533e8	−0.765165
$293$	5.32768e8	1.23737	0.618687	−	0.785637i	$-0.287665\pi$
$293$	5.32768e8	1.23737	0.618687	+	0.785637i	$0.287665\pi$
$294$	−1.24531e8	−0.285801
$295$	4.31880e6	0.00979458
$296$	−2.39288e7	−0.0536291
$297$	−1.65554e8	−0.366683
$298$	1.13984e7	0.0249509
$299$	1.98885e8	0.430282
$300$	−1.25280e8	−0.267891
$301$	−2.49881e8	−0.528142
$302$	2.86388e8	0.598315
$303$	2.94781e8	0.608766
$304$	2.80945e7	0.0573539
$305$	3.45529e7	0.0697324
$306$	−2.57133e7	−0.0513018
$307$	−1.79710e8	−0.354476	−0.177238	−	0.984168i	$-0.556716\pi$
$307$	−1.79710e8	−0.354476	−0.177238	+	0.984168i	$0.556716\pi$
$308$	2.67537e8	0.521743
$309$	1.50637e8	0.290453
$310$	−2.78316e7	−0.0530606
$311$	5.09031e8	0.959584	0.479792	−	0.877382i	$-0.340712\pi$
$311$	5.09031e8	0.959584	0.479792	+	0.877382i	$0.340712\pi$
$312$	−5.18400e7	−0.0966327
$313$	−2.42989e8	−0.447900	−0.223950	−	0.974601i	$-0.571895\pi$
$313$	−2.42989e8	−0.447900	−0.223950	+	0.974601i	$0.571895\pi$
$314$	5.49026e8	1.00078
$315$	2.71735e7	0.0489845
$316$	−2.35589e8	−0.420001
$317$	−7.73286e7	−0.136343	−0.0681715	−	0.997674i	$-0.521716\pi$
$317$	−7.73286e7	−0.136343	−0.0681715	+	0.997674i	$0.521716\pi$
$318$	−1.25395e8	−0.218669
$319$	−9.08893e7	−0.156764
$320$	−1.96608e7	−0.0335410
$321$	−4.87469e8	−0.822583
$322$	2.10871e8	0.351983
$323$	−3.02413e7	−0.0499335
$324$	3.40122e7	0.0555556
$325$	2.71875e8	0.439316
$326$	−1.23638e8	−0.197647
$327$	−9.18334e7	−0.145239
$328$	−6.33242e7	−0.0990857
$329$	7.67592e7	0.118835
$330$	1.36258e8	0.208720
$331$	−4.39033e8	−0.665425	−0.332713	−	0.943028i	$-0.607964\pi$
$331$	−4.39033e8	−0.665425	−0.332713	+	0.943028i	$0.607964\pi$
$332$	2.84965e8	0.427374
$333$	−3.40705e7	−0.0505620
$334$	−6.86189e7	−0.100770
$335$	−7.00740e7	−0.101836
$336$	−5.49642e7	−0.0790483
$337$	−1.14625e9	−1.63146	−0.815729	−	0.578434i	$-0.803664\pi$
$337$	−1.14625e9	−1.63146	−0.815729	+	0.578434i	$0.803664\pi$
$338$	−3.89488e8	−0.548638
$339$	−3.28722e8	−0.458279
$340$	2.11632e7	0.0292015
$341$	−3.90153e8	−0.532837
$342$	4.00017e7	0.0540738
$343$	6.95838e8	0.931063
$344$	2.57423e8	0.340951
$345$	1.07398e8	0.140809
$346$	−1.06343e9	−1.38021
$347$	−4.28422e8	−0.550450	−0.275225	−	0.961380i	$-0.588752\pi$
$347$	−4.28422e8	−0.550450	−0.275225	+	0.961380i	$0.588752\pi$
$348$	1.86728e7	0.0237510
$349$	5.42106e8	0.682646	0.341323	−	0.939946i	$-0.389125\pi$
$349$	5.42106e8	0.682646	0.341323	+	0.939946i	$0.389125\pi$
$350$	2.88260e8	0.359374
$351$	−7.38112e7	−0.0911061
$352$	−2.75612e8	−0.336820
$353$	−1.20260e9	−1.45515	−0.727576	−	0.686027i	$-0.759353\pi$
$353$	−1.20260e9	−1.45515	−0.727576	+	0.686027i	$0.759353\pi$
$354$	−1.24381e7	−0.0149020
$355$	1.39047e8	0.164953
$356$	3.77101e8	0.442978
$357$	5.91644e7	0.0688211
$358$	−5.99432e8	−0.690477
$359$	7.38775e8	0.842717	0.421359	−	0.906894i	$-0.361553\pi$
$359$	7.38775e8	0.842717	0.421359	+	0.906894i	$0.361553\pi$
$360$	−2.79936e7	−0.0316228
$361$	4.70459e7	0.0526316
$362$	−1.91553e8	−0.212231
$363$	1.38396e9	1.51862
$364$	1.19280e8	0.129632
$365$	3.81484e8	0.410630
$366$	−9.95123e7	−0.106095
$367$	1.02906e9	1.08670	0.543349	−	0.839507i	$-0.317156\pi$
$367$	1.02906e9	1.08670	0.543349	+	0.839507i	$0.317156\pi$
$368$	−2.17235e8	−0.227229
$369$	−9.01627e7	−0.0934189
$370$	2.80416e7	0.0287804
$371$	2.88525e8	0.293343
$372$	8.01550e7	0.0807291
$373$	−4.45581e8	−0.444576	−0.222288	−	0.974981i	$-0.571353\pi$
$373$	−4.45581e8	−0.444576	−0.222288	+	0.974981i	$0.571353\pi$
$374$	2.96673e8	0.293243
$375$	3.05016e8	0.298684
$376$	−7.90758e7	−0.0767161
$377$	−4.05225e7	−0.0389495
$378$	−7.82596e7	−0.0745275
$379$	−9.60280e8	−0.906068	−0.453034	−	0.891493i	$-0.649658\pi$
$379$	−9.60280e8	−0.906068	−0.453034	+	0.891493i	$0.649658\pi$
$380$	−3.29232e7	−0.0307794
$381$	9.61790e8	0.890929
$382$	−9.23928e8	−0.848040
$383$	−2.11976e9	−1.92793	−0.963963	−	0.266036i	$-0.914286\pi$
$383$	−2.11976e9	−1.92793	−0.963963	+	0.266036i	$0.914286\pi$
$384$	5.66231e7	0.0510310
$385$	−3.13520e8	−0.279997
$386$	9.55007e8	0.845183
$387$	3.66526e8	0.321452
$388$	5.94792e8	0.516957
$389$	4.07822e8	0.351275	0.175638	−	0.984455i	$-0.443801\pi$
$389$	4.07822e8	0.351275	0.175638	+	0.984455i	$0.443801\pi$
$390$	6.07500e7	0.0518585
$391$	2.33836e8	0.197830
$392$	−2.95185e8	−0.247511
$393$	2.94570e8	0.244802
$394$	−1.34809e9	−1.11041
$395$	2.76081e8	0.225396
$396$	−3.92424e8	−0.317557
$397$	−1.25577e9	−1.00726	−0.503632	−	0.863918i	$-0.668003\pi$
$397$	−1.25577e9	−1.00726	−0.503632	+	0.863918i	$0.668003\pi$
$398$	−3.24671e8	−0.258139
$399$	−9.20409e7	−0.0725397
$400$	−2.96960e8	−0.232000
$401$	7.70856e8	0.596991	0.298496	−	0.954411i	$-0.403515\pi$
$401$	7.70856e8	0.596991	0.298496	+	0.954411i	$0.403515\pi$
$402$	2.01813e8	0.154938
$403$	−1.73948e8	−0.132389
$404$	6.98740e8	0.527207
$405$	−3.98581e7	−0.0298142
$406$	−4.29647e7	−0.0318618
$407$	3.93096e8	0.289014
$408$	−6.09500e7	−0.0444287
$409$	−1.14560e9	−0.827942	−0.413971	−	0.910290i	$-0.635858\pi$
$409$	−1.14560e9	−0.827942	−0.413971	+	0.910290i	$0.635858\pi$
$410$	7.42080e7	0.0531750
$411$	6.23829e8	0.443220
$412$	3.57065e8	0.251540
$413$	2.86192e7	0.0199909
$414$	−3.09306e8	−0.214233
$415$	−3.33943e8	−0.229353
$416$	−1.22880e8	−0.0836863
$417$	1.09941e9	0.742478
$418$	−4.61528e8	−0.309087
$419$	1.78101e9	1.18282	0.591408	−	0.806372i	$-0.298572\pi$
$419$	1.78101e9	1.18282	0.591408	+	0.806372i	$0.298572\pi$
$420$	6.44112e7	0.0424218
$421$	−3.72186e8	−0.243093	−0.121547	−	0.992586i	$-0.538785\pi$
$421$	−3.72186e8	−0.243093	−0.121547	+	0.992586i	$0.538785\pi$
$422$	−3.01328e8	−0.195185
$423$	−1.12590e8	−0.0723286
$424$	−2.97233e8	−0.189373
$425$	3.19652e8	0.201984
$426$	−4.00454e8	−0.250969
$427$	2.28970e8	0.142325
$428$	−1.15548e9	−0.712378
$429$	8.51614e8	0.520765
$430$	−3.01667e8	−0.182974
$431$	−1.79492e9	−1.07988	−0.539939	−	0.841704i	$-0.681553\pi$
$431$	−1.79492e9	−1.07988	−0.539939	+	0.841704i	$0.681553\pi$
$432$	8.06216e7	0.0481125
$433$	−1.76107e9	−1.04248	−0.521242	−	0.853409i	$-0.674531\pi$
$433$	−1.76107e9	−1.04248	−0.521242	+	0.853409i	$0.674531\pi$
$434$	−1.84431e8	−0.108298
$435$	−2.18822e7	−0.0127461
$436$	−2.17679e8	−0.125781
$437$	−3.63774e8	−0.208519
$438$	−1.09867e9	−0.624754
$439$	2.17049e9	1.22442	0.612211	−	0.790694i	$-0.290280\pi$
$439$	2.17049e9	1.22442	0.612211	+	0.790694i	$0.290280\pi$
$440$	3.22982e8	0.180757
$441$	−4.20293e8	−0.233355
$442$	1.32270e8	0.0728590
$443$	−7.98483e8	−0.436368	−0.218184	−	0.975908i	$-0.570013\pi$
$443$	−7.98483e8	−0.436368	−0.218184	+	0.975908i	$0.570013\pi$
$444$	−8.07598e7	−0.0437880
$445$	−4.41915e8	−0.237727
$446$	6.13339e8	0.327362
$447$	3.84695e7	0.0203723
$448$	−1.30286e8	−0.0684579
$449$	−7.27113e8	−0.379088	−0.189544	−	0.981872i	$-0.560701\pi$
$449$	−7.27113e8	−0.379088	−0.189544	+	0.981872i	$0.560701\pi$
$450$	−4.22820e8	−0.218732
$451$	1.04027e9	0.533985
$452$	−7.79194e8	−0.396882
$453$	9.66558e8	0.488522
$454$	−2.34592e9	−1.17657
$455$	−1.39781e8	−0.0695679
$456$	9.48188e7	0.0468293
$457$	−1.22842e9	−0.602061	−0.301031	−	0.953614i	$-0.597331\pi$
$457$	−1.22842e9	−0.602061	−0.301031	+	0.953614i	$0.597331\pi$
$458$	−2.43382e9	−1.18375
$459$	−8.67823e7	−0.0418877
$460$	2.54573e8	0.121944
$461$	−1.75044e9	−0.832133	−0.416067	−	0.909334i	$-0.636592\pi$
$461$	−1.75044e9	−0.832133	−0.416067	+	0.909334i	$0.636592\pi$
$462$	9.02938e8	0.426001
$463$	2.60512e9	1.21982	0.609908	−	0.792472i	$-0.291206\pi$
$463$	2.60512e9	1.21982	0.609908	+	0.792472i	$0.291206\pi$
$464$	4.42614e7	0.0205689
$465$	−9.39316e7	−0.0433238
$466$	−2.22630e9	−1.01914
$467$	1.58937e9	0.722133	0.361066	−	0.932540i	$-0.382413\pi$
$467$	1.58937e9	0.722133	0.361066	+	0.932540i	$0.382413\pi$
$468$	−1.74960e8	−0.0789002
$469$	−4.64357e8	−0.207849
$470$	9.26670e7	0.0411702
$471$	1.85296e9	0.817134
$472$	−2.94830e7	−0.0129055
$473$	−4.22887e9	−1.83743
$474$	−7.95113e8	−0.342929
$475$	−4.97278e8	−0.212898
$476$	1.40241e8	0.0596008
$477$	−4.23209e8	−0.178542
$478$	−6.19952e8	−0.259633
$479$	1.42213e9	0.591241	0.295620	−	0.955306i	$-0.404474\pi$
$479$	1.42213e9	0.591241	0.295620	+	0.955306i	$0.404474\pi$
$480$	−6.63552e7	−0.0273861
$481$	1.75260e8	0.0718084
$482$	2.19224e9	0.891710
$483$	7.11690e8	0.287393
$484$	3.28050e9	1.31517
$485$	−6.97022e8	−0.277428
$486$	1.14791e8	0.0453609
$487$	4.40488e9	1.72816	0.864078	−	0.503357i	$-0.167902\pi$
$487$	4.40488e9	1.72816	0.864078	+	0.503357i	$0.167902\pi$
$488$	−2.35881e8	−0.0918805
$489$	−4.17277e8	−0.161378
$490$	3.45920e8	0.132828
$491$	1.09697e9	0.418226	0.209113	−	0.977891i	$-0.432942\pi$
$491$	1.09697e9	0.418226	0.209113	+	0.977891i	$0.432942\pi$
$492$	−2.13719e8	−0.0809031
$493$	−4.76437e7	−0.0179077
$494$	−2.05770e8	−0.0767958
$495$	4.59871e8	0.170419
$496$	1.89997e8	0.0699135
$497$	9.21416e8	0.336673
$498$	9.61756e8	0.348949
$499$	3.59404e9	1.29488	0.647442	−	0.762115i	$-0.275839\pi$
$499$	3.59404e9	1.29488	0.647442	+	0.762115i	$0.275839\pi$
$500$	7.23000e8	0.258668
$501$	−2.31589e8	−0.0822784
$502$	1.10405e9	0.389515
$503$	−1.61555e9	−0.566020	−0.283010	−	0.959117i	$-0.591333\pi$
$503$	−1.61555e9	−0.566020	−0.283010	+	0.959117i	$0.591333\pi$
$504$	−1.85504e8	−0.0645427
$505$	−8.18836e8	−0.282929
$506$	3.56869e9	1.22456
$507$	−1.31452e9	−0.447961
$508$	2.27980e9	0.771567
$509$	−1.57842e9	−0.530530	−0.265265	−	0.964176i	$-0.585459\pi$
$509$	−1.57842e9	−0.530530	−0.265265	+	0.964176i	$0.585459\pi$
$510$	7.14258e7	0.0238429
$511$	2.52797e9	0.838104
$512$	1.34218e8	0.0441942
$513$	1.35006e8	0.0441511
$514$	−2.14130e9	−0.695514
$515$	−4.18436e8	−0.134991
$516$	8.68802e8	0.278386
$517$	1.29904e9	0.413432
$518$	1.85822e8	0.0587413
$519$	−3.58909e9	−1.12693
$520$	1.44000e8	0.0449108
$521$	1.05135e9	0.325698	0.162849	−	0.986651i	$-0.447932\pi$
$521$	1.05135e9	0.325698	0.162849	+	0.986651i	$0.447932\pi$
$522$	6.30206e7	0.0193926
$523$	−3.27513e9	−1.00109	−0.500545	−	0.865711i	$-0.666867\pi$
$523$	−3.27513e9	−1.00109	−0.500545	+	0.865711i	$0.666867\pi$
$524$	6.98240e8	0.212004
$525$	9.72878e8	0.293427
$526$	2.55191e9	0.764566
$527$	−2.04516e8	−0.0608681
$528$	−9.30189e8	−0.275013
$529$	−5.92008e8	−0.173873
$530$	3.48320e8	0.101628
$531$	−4.19787e7	−0.0121674
$532$	−2.18171e8	−0.0628212
$533$	4.63800e8	0.132674
$534$	1.27272e9	0.361690
$535$	1.35408e9	0.382302
$536$	4.78372e8	0.134180
$537$	−2.02308e9	−0.563772
$538$	−3.21502e9	−0.890115
$539$	4.84923e9	1.33387
$540$	−9.44784e7	−0.0258199
$541$	3.38518e9	0.919160	0.459580	−	0.888136i	$-0.348000\pi$
$541$	3.38518e9	0.919160	0.459580	+	0.888136i	$0.348000\pi$
$542$	−9.56038e8	−0.257916
$543$	−6.46492e8	−0.173286
$544$	−1.44474e8	−0.0384764
$545$	2.55093e8	0.0675010
$546$	4.02570e8	0.105844
$547$	−3.07554e9	−0.803463	−0.401732	−	0.915757i	$-0.631592\pi$
$547$	−3.07554e9	−0.803463	−0.401732	+	0.915757i	$0.631592\pi$
$548$	1.47871e9	0.383839
$549$	−3.35854e8	−0.0866258
$550$	4.87838e9	1.25028
$551$	7.41184e7	0.0188754
$552$	−7.33170e8	−0.185532
$553$	1.82950e9	0.460038
$554$	3.47197e9	0.867544
$555$	9.46404e7	0.0234991
$556$	2.60601e9	0.643005
$557$	−3.53567e9	−0.866919	−0.433460	−	0.901173i	$-0.642707\pi$
$557$	−3.53567e9	−0.866919	−0.433460	+	0.901173i	$0.642707\pi$
$558$	2.70523e8	0.0659151
$559$	−1.88542e9	−0.456528
$560$	1.52678e8	0.0367383
$561$	1.00127e9	0.239432
$562$	−3.38124e9	−0.803523
$563$	1.17546e9	0.277607	0.138803	−	0.990320i	$-0.455674\pi$
$563$	1.17546e9	0.277607	0.138803	+	0.990320i	$0.455674\pi$
$564$	−2.66881e8	−0.0626384
$565$	9.13118e8	0.212989
$566$	4.84614e7	0.0112341
$567$	−2.64126e8	−0.0608514
$568$	−9.49225e8	−0.217345
$569$	7.22118e9	1.64329	0.821647	−	0.569997i	$-0.193056\pi$
$569$	7.22118e9	1.64329	0.821647	+	0.569997i	$0.193056\pi$
$570$	−1.11116e8	−0.0251312
$571$	−3.83286e9	−0.861581	−0.430791	−	0.902452i	$-0.641765\pi$
$571$	−3.83286e9	−0.861581	−0.430791	+	0.902452i	$0.641765\pi$
$572$	2.01864e9	0.450996
$573$	−3.11826e9	−0.692422
$574$	4.91752e8	0.108531
$575$	3.84511e9	0.843473
$576$	1.91103e8	0.0416667
$577$	4.24249e9	0.919401	0.459701	−	0.888074i	$-0.347957\pi$
$577$	4.24249e9	0.919401	0.459701	+	0.888074i	$0.347957\pi$
$578$	−3.12720e9	−0.673608
$579$	3.22315e9	0.690089
$580$	−5.18688e7	−0.0110385
$581$	−2.21293e9	−0.468113
$582$	2.00742e9	0.422094
$583$	4.88287e9	1.02055
$584$	−2.60426e9	−0.541053
$585$	2.05031e8	0.0423423
$586$	4.26214e9	0.874956
$587$	−2.79676e9	−0.570718	−0.285359	−	0.958421i	$-0.592113\pi$
$587$	−2.79676e9	−0.570718	−0.285359	+	0.958421i	$0.592113\pi$
$588$	−9.96251e8	−0.202091
$589$	3.18162e8	0.0641570
$590$	3.45504e7	0.00692582
$591$	−4.54981e9	−0.906645
$592$	−1.91431e8	−0.0379215
$593$	−1.77136e8	−0.0348832	−0.0174416	−	0.999848i	$-0.505552\pi$
$593$	−1.77136e8	−0.0348832	−0.0174416	+	0.999848i	$0.505552\pi$
$594$	−1.32443e9	−0.259284
$595$	−1.64345e8	−0.0319851
$596$	9.11869e7	0.0176429
$597$	−1.09576e9	−0.210769
$598$	1.59108e9	0.304255
$599$	−6.64127e9	−1.26258	−0.631288	−	0.775549i	$-0.717473\pi$
$599$	−6.64127e9	−1.26258	−0.631288	+	0.775549i	$0.717473\pi$
$600$	−1.00224e9	−0.189427
$601$	−3.77732e9	−0.709779	−0.354889	−	0.934908i	$-0.615481\pi$
$601$	−3.77732e9	−0.709779	−0.354889	+	0.934908i	$0.615481\pi$
$602$	−1.99905e9	−0.373453
$603$	6.81119e8	0.126506
$604$	2.29110e9	0.423072
$605$	−3.84433e9	−0.705792
$606$	2.35825e9	0.430463
$607$	−2.17939e9	−0.395526	−0.197763	−	0.980250i	$-0.563368\pi$
$607$	−2.17939e9	−0.395526	−0.197763	+	0.980250i	$0.563368\pi$
$608$	2.24756e8	0.0405554
$609$	−1.45006e8	−0.0260150
$610$	2.76423e8	0.0493083
$611$	5.79169e8	0.102721
$612$	−2.05706e8	−0.0362759
$613$	−4.13485e9	−0.725017	−0.362509	−	0.931980i	$-0.618080\pi$
$613$	−4.13485e9	−0.725017	−0.362509	+	0.931980i	$0.618080\pi$
$614$	−1.43768e9	−0.250653
$615$	2.50452e8	0.0434172
$616$	2.14030e9	0.368928
$617$	−5.84763e9	−1.00226	−0.501132	−	0.865371i	$-0.667083\pi$
$617$	−5.84763e9	−1.00226	−0.501132	+	0.865371i	$0.667083\pi$
$618$	1.20509e9	0.205382
$619$	−6.59455e9	−1.11755	−0.558776	−	0.829318i	$-0.688729\pi$
$619$	−6.59455e9	−1.11755	−0.558776	+	0.829318i	$0.688729\pi$
$620$	−2.22653e8	−0.0375195
$621$	−1.04391e9	−0.174921
$622$	4.07225e9	0.678528
$623$	−2.92842e9	−0.485206
$624$	−4.14720e8	−0.0683296
$625$	4.81680e9	0.789184
$626$	−1.94391e9	−0.316713
$627$	−1.55766e9	−0.252369
$628$	4.39221e9	0.707659
$629$	2.06059e8	0.0330152
$630$	2.17388e8	0.0346372
$631$	3.57471e9	0.566419	0.283210	−	0.959058i	$-0.408601\pi$
$631$	3.57471e9	0.566419	0.283210	+	0.959058i	$0.408601\pi$
$632$	−1.88471e9	−0.296986
$633$	−1.01698e9	−0.159368
$634$	−6.18629e8	−0.0964090
$635$	−2.67164e9	−0.414066
$636$	−1.00316e9	−0.154622
$637$	2.16200e9	0.331412
$638$	−7.27114e8	−0.110849
$639$	−1.35153e9	−0.204915
$640$	−1.57286e8	−0.0237171
$641$	−5.76145e9	−0.864029	−0.432015	−	0.901867i	$-0.642197\pi$
$641$	−5.76145e9	−0.864029	−0.432015	+	0.901867i	$0.642197\pi$
$642$	−3.89975e9	−0.581654
$643$	−8.33303e9	−1.23613	−0.618065	−	0.786127i	$-0.712083\pi$
$643$	−8.33303e9	−1.23613	−0.618065	+	0.786127i	$0.712083\pi$
$644$	1.68697e9	0.248890
$645$	−1.01813e9	−0.149397
$646$	−2.41931e8	−0.0353083
$647$	1.01794e10	1.47761	0.738804	−	0.673921i	$-0.235391\pi$
$647$	1.01794e10	1.47761	0.738804	+	0.673921i	$0.235391\pi$
$648$	2.72098e8	0.0392837
$649$	4.84339e8	0.0695493
$650$	2.17500e9	0.310644
$651$	−6.22454e8	−0.0884247
$652$	−9.89102e8	−0.139757
$653$	1.13418e10	1.59399	0.796993	−	0.603989i	$-0.206423\pi$
$653$	1.13418e10	1.59399	0.796993	+	0.603989i	$0.206423\pi$
$654$	−7.34667e8	−0.102700
$655$	−8.18250e8	−0.113774
$656$	−5.06593e8	−0.0700642
$657$	−3.70802e9	−0.510110
$658$	6.14073e8	0.0840291
$659$	9.34774e9	1.27235	0.636176	−	0.771544i	$-0.280515\pi$
$659$	9.34774e9	1.27235	0.636176	+	0.771544i	$0.280515\pi$
$660$	1.09007e9	0.147587
$661$	−1.16280e10	−1.56603	−0.783015	−	0.622003i	$-0.786319\pi$
$661$	−1.16280e10	−1.56603	−0.783015	+	0.622003i	$0.786319\pi$
$662$	−3.51226e9	−0.470527
$663$	4.46411e8	0.0594892
$664$	2.27972e9	0.302199
$665$	2.55669e8	0.0337134
$666$	−2.72564e8	−0.0357527
$667$	−5.73107e8	−0.0747817
$668$	−5.48951e8	−0.0712552
$669$	2.07002e9	0.267290
$670$	−5.60592e8	−0.0720087
$671$	3.87499e9	0.495156
$672$	−4.39714e8	−0.0558956
$673$	3.34838e9	0.423430	0.211715	−	0.977331i	$-0.432095\pi$
$673$	3.34838e9	0.423430	0.211715	+	0.977331i	$0.432095\pi$
$674$	−9.17003e9	−1.15362
$675$	−1.42702e9	−0.178594
$676$	−3.11591e9	−0.387946
$677$	−1.39797e10	−1.73156	−0.865782	−	0.500421i	$-0.833179\pi$
$677$	−1.39797e10	−1.73156	−0.865782	+	0.500421i	$0.833179\pi$
$678$	−2.62978e9	−0.324052
$679$	−4.61893e9	−0.566236
$680$	1.69306e8	0.0206486
$681$	−7.91748e9	−0.960665
$682$	−3.12122e9	−0.376773
$683$	5.82014e9	0.698974	0.349487	−	0.936941i	$-0.386356\pi$
$683$	5.82014e9	0.698974	0.349487	+	0.936941i	$0.386356\pi$
$684$	3.20014e8	0.0382360
$685$	−1.73286e9	−0.205990
$686$	5.56671e9	0.658361
$687$	−8.21416e9	−0.966528
$688$	2.05938e9	0.241089
$689$	2.17700e9	0.253566
$690$	8.59183e8	0.0995667
$691$	5.51192e9	0.635521	0.317761	−	0.948171i	$-0.397069\pi$
$691$	5.51192e9	0.635521	0.317761	+	0.948171i	$0.397069\pi$
$692$	−8.50748e9	−0.975954
$693$	3.04741e9	0.347829
$694$	−3.42737e9	−0.389227
$695$	−3.05392e9	−0.345072
$696$	1.49382e8	0.0167945
$697$	5.45305e8	0.0609993
$698$	4.33685e9	0.482704
$699$	−7.51377e9	−0.832123
$700$	2.30608e9	0.254116
$701$	−1.25751e10	−1.37879	−0.689393	−	0.724387i	$-0.742123\pi$
$701$	−1.25751e10	−1.37879	−0.689393	+	0.724387i	$0.742123\pi$
$702$	−5.90490e8	−0.0644218
$703$	−3.20562e8	−0.0347992
$704$	−2.20489e9	−0.238168
$705$	3.12751e8	0.0336153
$706$	−9.62077e9	−1.02895
$707$	−5.42616e9	−0.577463
$708$	−9.95052e7	−0.0105373
$709$	1.57667e10	1.66141	0.830707	−	0.556710i	$-0.187937\pi$
$709$	1.57667e10	1.66141	0.830707	+	0.556710i	$0.187937\pi$
$710$	1.11237e9	0.116640
$711$	−2.68351e9	−0.280001
$712$	3.01681e9	0.313233
$713$	−2.46013e9	−0.254182
$714$	4.73315e8	0.0486639
$715$	−2.36559e9	−0.242030
$716$	−4.79546e9	−0.488241
$717$	−2.09234e9	−0.211990
$718$	5.91020e9	0.595891
$719$	−8.81671e9	−0.884617	−0.442309	−	0.896863i	$-0.645840\pi$
$719$	−8.81671e9	−0.884617	−0.442309	+	0.896863i	$0.645840\pi$
$720$	−2.23949e8	−0.0223607
$721$	−2.77283e9	−0.275518
$722$	3.76367e8	0.0372161
$723$	7.39882e9	0.728078
$724$	−1.53243e9	−0.150070
$725$	−7.83435e8	−0.0763519
$726$	1.10717e10	1.07383
$727$	1.54272e10	1.48908	0.744538	−	0.667580i	$-0.232670\pi$
$727$	1.54272e10	1.48908	0.744538	+	0.667580i	$0.232670\pi$
$728$	9.54240e8	0.0916638
$729$	3.87420e8	0.0370370
$730$	3.05187e9	0.290360
$731$	−2.21675e9	−0.209897
$732$	−7.96098e8	−0.0750201
$733$	−1.38466e10	−1.29861	−0.649304	−	0.760529i	$-0.724940\pi$
$733$	−1.38466e10	−1.29861	−0.649304	+	0.760529i	$0.724940\pi$
$734$	8.23246e9	0.768411
$735$	1.16748e9	0.108454
$736$	−1.73788e9	−0.160675
$737$	−7.85857e9	−0.723115
$738$	−7.21302e8	−0.0660571
$739$	1.82404e10	1.66257	0.831284	−	0.555849i	$-0.187607\pi$
$739$	1.82404e10	1.66257	0.831284	+	0.555849i	$0.187607\pi$
$740$	2.24333e8	0.0203508
$741$	−6.94474e8	−0.0627035
$742$	2.30820e9	0.207425
$743$	1.84123e10	1.64682	0.823410	−	0.567447i	$-0.192069\pi$
$743$	1.84123e10	1.64682	0.823410	+	0.567447i	$0.192069\pi$
$744$	6.41240e8	0.0570841
$745$	−1.06860e8	−0.00946818
$746$	−3.56465e9	−0.314363
$747$	3.24592e9	0.284916
$748$	2.37338e9	0.207354
$749$	8.97305e9	0.780285
$750$	2.44012e9	0.211202
$751$	1.29306e9	0.111399	0.0556993	−	0.998448i	$-0.482261\pi$
$751$	1.29306e9	0.111399	0.0556993	+	0.998448i	$0.482261\pi$
$752$	−6.32607e8	−0.0542465
$753$	3.72616e9	0.318038
$754$	−3.24180e8	−0.0275414
$755$	−2.68488e9	−0.227045
$756$	−6.26077e8	−0.0526989
$757$	−2.29690e9	−0.192445	−0.0962225	−	0.995360i	$-0.530676\pi$
$757$	−2.29690e9	−0.192445	−0.0962225	+	0.995360i	$0.530676\pi$
$758$	−7.68224e9	−0.640687
$759$	1.20443e10	0.999853
$760$	−2.63386e8	−0.0217643
$761$	−3.04544e9	−0.250497	−0.125249	−	0.992125i	$-0.539973\pi$
$761$	−3.04544e9	−0.250497	−0.125249	+	0.992125i	$0.539973\pi$
$762$	7.69432e9	0.629982
$763$	1.69041e9	0.137771
$764$	−7.39142e9	−0.599655
$765$	2.41062e8	0.0194677
$766$	−1.69581e10	−1.36325
$767$	2.15940e8	0.0172802
$768$	4.52985e8	0.0360844
$769$	6.80258e9	0.539425	0.269713	−	0.962941i	$-0.413071\pi$
$769$	6.80258e9	0.539425	0.269713	+	0.962941i	$0.413071\pi$
$770$	−2.50816e9	−0.197987
$771$	−7.22687e9	−0.567884
$772$	7.64005e9	0.597635
$773$	1.36210e10	1.06067	0.530337	−	0.847787i	$-0.322065\pi$
$773$	1.36210e10	1.06067	0.530337	+	0.847787i	$0.322065\pi$
$774$	2.93221e9	0.227301
$775$	−3.36299e9	−0.259519
$776$	4.75834e9	0.365544
$777$	6.27150e8	0.0479621
$778$	3.26258e9	0.248389
$779$	−8.48321e8	−0.0642953
$780$	4.86000e8	0.0366695
$781$	1.55936e10	1.17130
$782$	1.87069e9	0.139887
$783$	2.12694e8	0.0158340
$784$	−2.36148e9	−0.175016
$785$	−5.14712e9	−0.379770
$786$	2.35656e9	0.173101
$787$	−8.58172e9	−0.627571	−0.313785	−	0.949494i	$-0.601597\pi$
$787$	−8.58172e9	−0.627571	−0.313785	+	0.949494i	$0.601597\pi$
$788$	−1.07847e10	−0.785178
$789$	8.61270e9	0.624266
$790$	2.20865e9	0.159379
$791$	6.05093e9	0.434715
$792$	−3.13939e9	−0.224547
$793$	1.72764e9	0.123026
$794$	−1.00462e10	−0.712244
$795$	1.17558e9	0.0829789
$796$	−2.59737e9	−0.182531
$797$	1.91948e10	1.34301	0.671506	−	0.741000i	$-0.265648\pi$
$797$	1.91948e10	1.34301	0.671506	+	0.741000i	$0.265648\pi$
$798$	−7.36327e8	−0.0512933
$799$	6.80948e8	0.0472281
$800$	−2.37568e9	−0.164049
$801$	4.29542e9	0.295319
$802$	6.16685e9	0.422136
$803$	4.27821e10	2.91580
$804$	1.61450e9	0.109558
$805$	−1.97692e9	−0.133568
$806$	−1.39158e9	−0.0936129
$807$	−1.08507e10	−0.726776
$808$	5.58992e9	0.372792
$809$	−1.54837e10	−1.02814	−0.514072	−	0.857747i	$-0.671864\pi$
$809$	−1.54837e10	−1.02814	−0.514072	+	0.857747i	$0.671864\pi$
$810$	−3.18865e8	−0.0210819
$811$	1.81577e9	0.119533	0.0597666	−	0.998212i	$-0.480964\pi$
$811$	1.81577e9	0.119533	0.0597666	+	0.998212i	$0.480964\pi$
$812$	−3.43717e8	−0.0225297
$813$	−3.22663e9	−0.210587
$814$	3.14477e9	0.204364
$815$	1.15910e9	0.0750016
$816$	−4.87600e8	−0.0314158
$817$	3.44856e9	0.221239
$818$	−9.16477e9	−0.585443
$819$	1.35867e9	0.0864214
$820$	5.93664e8	0.0376004
$821$	2.03054e10	1.28059	0.640295	−	0.768129i	$-0.278812\pi$
$821$	2.03054e10	1.28059	0.640295	+	0.768129i	$0.278812\pi$
$822$	4.99063e9	0.313404
$823$	−5.08303e9	−0.317851	−0.158925	−	0.987291i	$-0.550803\pi$
$823$	−5.08303e9	−0.317851	−0.158925	+	0.987291i	$0.550803\pi$
$824$	2.85652e9	0.177866
$825$	1.64645e10	1.02085
$826$	2.28954e8	0.0141357
$827$	−1.93189e9	−0.118772	−0.0593859	−	0.998235i	$-0.518914\pi$
$827$	−1.93189e9	−0.118772	−0.0593859	+	0.998235i	$0.518914\pi$
$828$	−2.47445e9	−0.151486
$829$	1.44409e10	0.880343	0.440172	−	0.897914i	$-0.354918\pi$
$829$	1.44409e10	0.880343	0.440172	+	0.897914i	$0.354918\pi$
$830$	−2.67154e9	−0.162177
$831$	1.17179e10	0.708346
$832$	−9.83040e8	−0.0591752
$833$	2.54194e9	0.152373
$834$	8.79528e9	0.525011
$835$	6.43302e8	0.0382395
$836$	−3.69223e9	−0.218558
$837$	9.13016e8	0.0538194
$838$	1.42481e10	0.836378
$839$	2.92644e10	1.71069	0.855347	−	0.518055i	$-0.173344\pi$
$839$	2.92644e10	1.71069	0.855347	+	0.518055i	$0.173344\pi$
$840$	5.15290e8	0.0299967
$841$	−1.71331e10	−0.993231
$842$	−2.97749e9	−0.171893
$843$	−1.14117e10	−0.656074
$844$	−2.41062e9	−0.138016
$845$	3.65145e9	0.208193
$846$	−9.00723e8	−0.0511440
$847$	−2.54751e10	−1.44054
$848$	−2.37787e9	−0.133907
$849$	1.63557e8	0.00917261
$850$	2.55722e9	0.142824
$851$	2.47869e9	0.137870
$852$	−3.20364e9	−0.177462
$853$	−1.46705e10	−0.809328	−0.404664	−	0.914465i	$-0.632612\pi$
$853$	−1.46705e10	−0.809328	−0.404664	+	0.914465i	$0.632612\pi$
$854$	1.83176e9	0.100639
$855$	−3.75016e8	−0.0205196
$856$	−9.24386e9	−0.503727
$857$	6.48965e9	0.352199	0.176100	−	0.984372i	$-0.443652\pi$
$857$	6.48965e9	0.352199	0.176100	+	0.984372i	$0.443652\pi$
$858$	6.81291e9	0.368237
$859$	−1.72702e10	−0.929654	−0.464827	−	0.885402i	$-0.653883\pi$
$859$	−1.72702e10	−0.929654	−0.464827	+	0.885402i	$0.653883\pi$
$860$	−2.41334e9	−0.129382
$861$	1.65966e9	0.0886153
$862$	−1.43594e10	−0.763589
$863$	−1.97542e10	−1.04622	−0.523108	−	0.852267i	$-0.675227\pi$
$863$	−1.97542e10	−1.04622	−0.523108	+	0.852267i	$0.675227\pi$
$864$	6.44973e8	0.0340207
$865$	9.96970e9	0.523752
$866$	−1.40886e10	−0.737147
$867$	−1.05543e10	−0.549999
$868$	−1.47545e9	−0.0765780
$869$	3.09616e10	1.60049
$870$	−1.75057e8	−0.00901286
$871$	−3.50370e9	−0.179665
$872$	−1.74143e9	−0.0889404
$873$	6.77506e9	0.344638
$874$	−2.91019e9	−0.147446
$875$	−5.61455e9	−0.283326
$876$	−8.78939e9	−0.441768
$877$	1.01565e10	0.508448	0.254224	−	0.967145i	$-0.418180\pi$
$877$	1.01565e10	0.508448	0.254224	+	0.967145i	$0.418180\pi$
$878$	1.73639e10	0.865798
$879$	1.43847e10	0.714399
$880$	2.58386e9	0.127814
$881$	1.26367e9	0.0622615	0.0311307	−	0.999515i	$-0.490089\pi$
$881$	1.26367e9	0.0622615	0.0311307	+	0.999515i	$0.490089\pi$
$882$	−3.36235e9	−0.165007
$883$	−3.01971e10	−1.47605	−0.738027	−	0.674771i	$-0.764242\pi$
$883$	−3.01971e10	−1.47605	−0.738027	+	0.674771i	$0.764242\pi$
$884$	1.05816e9	0.0515191
$885$	1.16608e8	0.00565491
$886$	−6.38786e9	−0.308558
$887$	−1.56437e10	−0.752675	−0.376338	−	0.926483i	$-0.622817\pi$
$887$	−1.56437e10	−0.752675	−0.376338	+	0.926483i	$0.622817\pi$
$888$	−6.46078e8	−0.0309628
$889$	−1.77041e10	−0.845117
$890$	−3.53532e9	−0.168099
$891$	−4.46995e9	−0.211705
$892$	4.90671e9	0.231480
$893$	−1.05934e9	−0.0497800
$894$	3.07756e8	0.0144054
$895$	5.61968e9	0.262018
$896$	−1.04228e9	−0.0484070
$897$	5.36990e9	0.248423
$898$	−5.81690e9	−0.268055
$899$	5.01247e8	0.0230087
$900$	−3.38256e9	−0.154667
$901$	2.55957e9	0.116582
$902$	8.32218e9	0.377585
$903$	−6.74679e9	−0.304923
$904$	−6.23355e9	−0.280638
$905$	1.79581e9	0.0805361
$906$	7.73246e9	0.345437
$907$	2.21854e10	0.987282	0.493641	−	0.869666i	$-0.335666\pi$
$907$	2.21854e10	0.987282	0.493641	+	0.869666i	$0.335666\pi$
$908$	−1.87674e10	−0.831960
$909$	7.95909e9	0.351471
$910$	−1.11825e9	−0.0491919
$911$	−1.42496e10	−0.624437	−0.312218	−	0.950010i	$-0.601072\pi$
$911$	−1.42496e10	−0.624437	−0.312218	+	0.950010i	$0.601072\pi$
$912$	7.58551e8	0.0331133
$913$	−3.74506e10	−1.62859
$914$	−9.82737e9	−0.425722
$915$	9.32928e8	0.0402600
$916$	−1.94706e10	−0.837038
$917$	−5.42227e9	−0.232214
$918$	−6.94259e8	−0.0296191
$919$	7.22447e9	0.307045	0.153522	−	0.988145i	$-0.450938\pi$
$919$	7.22447e9	0.307045	0.153522	+	0.988145i	$0.450938\pi$
$920$	2.03658e9	0.0862273
$921$	−4.85216e9	−0.204657
$922$	−1.40035e10	−0.588407
$923$	6.95234e9	0.291021
$924$	7.22350e9	0.301228
$925$	3.38836e9	0.140765
$926$	2.08410e10	0.862540
$927$	4.06719e9	0.167693
$928$	3.54091e8	0.0145444
$929$	−1.55001e10	−0.634280	−0.317140	−	0.948379i	$-0.602722\pi$
$929$	−1.55001e10	−0.634280	−0.317140	+	0.948379i	$0.602722\pi$
$930$	−7.51453e8	−0.0306346
$931$	−3.95445e9	−0.160606
$932$	−1.78104e10	−0.720640
$933$	1.37438e10	0.554016
$934$	1.27150e10	0.510625
$935$	−2.78131e9	−0.111278
$936$	−1.39968e9	−0.0557909
$937$	−8.20635e9	−0.325883	−0.162941	−	0.986636i	$-0.552098\pi$
$937$	−8.20635e9	−0.325883	−0.162941	+	0.986636i	$0.552098\pi$
$938$	−3.71486e9	−0.146971
$939$	−6.56069e9	−0.258595
$940$	7.41336e8	0.0291117
$941$	3.00301e10	1.17488	0.587440	−	0.809268i	$-0.300136\pi$
$941$	3.00301e10	1.17488	0.587440	+	0.809268i	$0.300136\pi$
$942$	1.48237e10	0.577801
$943$	6.55949e9	0.254730
$944$	−2.35864e8	−0.00912557
$945$	7.33684e8	0.0282812
$946$	−3.38310e10	−1.29926
$947$	1.29562e10	0.495739	0.247870	−	0.968793i	$-0.420270\pi$
$947$	1.29562e10	0.495739	0.247870	+	0.968793i	$0.420270\pi$
$948$	−6.36091e9	−0.242488
$949$	1.90742e10	0.724460
$950$	−3.97822e9	−0.150541
$951$	−2.08787e9	−0.0787176
$952$	1.12193e9	0.0421441
$953$	−3.34776e10	−1.25294	−0.626469	−	0.779447i	$-0.715500\pi$
$953$	−3.34776e10	−1.25294	−0.626469	+	0.779447i	$0.715500\pi$
$954$	−3.38567e9	−0.126248
$955$	8.66182e9	0.321809
$956$	−4.95962e9	−0.183588
$957$	−2.45401e9	−0.0905075
$958$	1.13770e10	0.418070
$959$	−1.14831e10	−0.420429
$960$	−5.30842e8	−0.0193649
$961$	−2.53610e10	−0.921794
$962$	1.40208e9	0.0507762
$963$	−1.31617e10	−0.474918
$964$	1.75379e10	0.630534
$965$	−8.95319e9	−0.320725
$966$	5.69352e9	0.203217
$967$	−3.99922e10	−1.42227	−0.711137	−	0.703054i	$-0.751819\pi$
$967$	−3.99922e10	−1.42227	−0.711137	+	0.703054i	$0.751819\pi$
$968$	2.62440e10	0.929963
$969$	−8.16516e8	−0.0288291
$970$	−5.57618e9	−0.196171
$971$	−1.62336e10	−0.569045	−0.284523	−	0.958669i	$-0.591835\pi$
$971$	−1.62336e10	−0.569045	−0.284523	+	0.958669i	$0.591835\pi$
$972$	9.18330e8	0.0320750
$973$	−2.02373e10	−0.704299
$974$	3.52391e10	1.22199
$975$	7.34062e9	0.253639
$976$	−1.88705e9	−0.0649693
$977$	2.48812e10	0.853572	0.426786	−	0.904353i	$-0.359646\pi$
$977$	2.48812e10	0.853572	0.426786	+	0.904353i	$0.359646\pi$
$978$	−3.33822e9	−0.114111
$979$	−4.95593e10	−1.68805
$980$	2.76736e9	0.0939236
$981$	−2.47950e9	−0.0838538
$982$	8.77580e9	0.295731
$983$	5.40801e10	1.81593	0.907966	−	0.419043i	$-0.137634\pi$
$983$	5.40801e10	1.81593	0.907966	+	0.419043i	$0.137634\pi$
$984$	−1.70975e9	−0.0572072
$985$	1.26384e10	0.421371
$986$	−3.81149e8	−0.0126627
$987$	2.07250e9	0.0686095
$988$	−1.64616e9	−0.0543029
$989$	−2.66654e10	−0.876518
$990$	3.67897e9	0.120504
$991$	−5.30250e10	−1.73070	−0.865352	−	0.501165i	$-0.832905\pi$
$991$	−5.30250e10	−1.73070	−0.865352	+	0.501165i	$0.832905\pi$
$992$	1.51998e9	0.0494363
$993$	−1.18539e10	−0.384183
$994$	7.37133e9	0.238064
$995$	3.04379e9	0.0979567
$996$	7.69404e9	0.246744
$997$	−5.51918e10	−1.76377	−0.881884	−	0.471467i	$-0.843725\pi$
$997$	−5.51918e10	−1.76377	−0.881884	+	0.471467i	$0.843725\pi$
$998$	2.87523e10	0.915621
$999$	−9.19905e8	−0.0291920

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	114.8.a.e.1.1	✓	1
3.2	odd	2		342.8.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
114.8.a.e.1.1	✓	1	1.1	even	1	trivial
342.8.a.b.1.1		1	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 114 = 2 \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	114.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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