LMFDB - Embedded newform 354.8.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [354,8,Mod(1,354)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(354, 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("354.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 354 = 2 \cdot 3 \cdot 59 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	354.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:	$110.584299021$
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	$\chi$	$=$	354.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} -320.000 q^{5} +216.000 q^{6} -505.000 q^{7} +512.000 q^{8} +729.000 q^{9} +O(q^{10})$

\(q+8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} -320.000 q^{5} +216.000 q^{6} -505.000 q^{7} +512.000 q^{8} +729.000 q^{9} -2560.00 q^{10} +2559.00 q^{11} +1728.00 q^{12} +7037.00 q^{13} -4040.00 q^{14} -8640.00 q^{15} +4096.00 q^{16} -551.000 q^{17} +5832.00 q^{18} -22144.0 q^{19} -20480.0 q^{20} -13635.0 q^{21} +20472.0 q^{22} -35966.0 q^{23} +13824.0 q^{24} +24275.0 q^{25} +56296.0 q^{26} +19683.0 q^{27} -32320.0 q^{28} +102278. q^{29} -69120.0 q^{30} -91872.0 q^{31} +32768.0 q^{32} +69093.0 q^{33} -4408.00 q^{34} +161600. q^{35} +46656.0 q^{36} +250189. q^{37} -177152. q^{38} +189999. q^{39} -163840. q^{40} -418053. q^{41} -109080. q^{42} -40225.0 q^{43} +163776. q^{44} -233280. q^{45} -287728. q^{46} +25656.0 q^{47} +110592. q^{48} -568518. q^{49} +194200. q^{50} -14877.0 q^{51} +450368. q^{52} -1.68159e6 q^{53} +157464. q^{54} -818880. q^{55} -258560. q^{56} -597888. q^{57} +818224. q^{58} +205379. q^{59} -552960. q^{60} -1.85765e6 q^{61} -734976. q^{62} -368145. q^{63} +262144. q^{64} -2.25184e6 q^{65} +552744. q^{66} +3.84328e6 q^{67} -35264.0 q^{68} -971082. q^{69} +1.29280e6 q^{70} -3.10461e6 q^{71} +373248. q^{72} -4.32839e6 q^{73} +2.00151e6 q^{74} +655425. q^{75} -1.41722e6 q^{76} -1.29229e6 q^{77} +1.51999e6 q^{78} -4.67906e6 q^{79} -1.31072e6 q^{80} +531441. q^{81} -3.34442e6 q^{82} -8.91025e6 q^{83} -872640. q^{84} +176320. q^{85} -321800. q^{86} +2.76151e6 q^{87} +1.31021e6 q^{88} +785984. q^{89} -1.86624e6 q^{90} -3.55369e6 q^{91} -2.30182e6 q^{92} -2.48054e6 q^{93} +205248. q^{94} +7.08608e6 q^{95} +884736. q^{96} -9.84140e6 q^{97} -4.54814e6 q^{98} +1.86551e6 q^{99} +O(q^{100})\)

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
$5$	−320.000	−1.14487	−0.572433	−	0.819951i	$-0.694000\pi$
$5$	−320.000	−1.14487	−0.572433	+	0.819951i	$0.694000\pi$
$6$	216.000	0.408248
$7$	−505.000	−0.556478	−0.278239	−	0.960512i	$-0.589751\pi$
$7$	−505.000	−0.556478	−0.278239	+	0.960512i	$0.589751\pi$
$8$	512.000	0.353553
$9$	729.000	0.333333
$10$	−2560.00	−0.809543
$11$	2559.00	0.579690	0.289845	−	0.957074i	$-0.406396\pi$
$11$	2559.00	0.579690	0.289845	+	0.957074i	$0.406396\pi$
$12$	1728.00	0.288675
$13$	7037.00	0.888354	0.444177	−	0.895939i	$-0.353496\pi$
$13$	7037.00	0.888354	0.444177	+	0.895939i	$0.353496\pi$
$14$	−4040.00	−0.393490
$15$	−8640.00	−0.660989
$16$	4096.00	0.250000
$17$	−551.000	−0.0272007	−0.0136004	−	0.999908i	$-0.504329\pi$
$17$	−551.000	−0.0272007	−0.0136004	+	0.999908i	$0.504329\pi$
$18$	5832.00	0.235702
$19$	−22144.0	−0.740659	−0.370330	−	0.928900i	$-0.620755\pi$
$19$	−22144.0	−0.740659	−0.370330	+	0.928900i	$0.620755\pi$
$20$	−20480.0	−0.572433
$21$	−13635.0	−0.321283
$22$	20472.0	0.409903
$23$	−35966.0	−0.616375	−0.308187	−	0.951326i	$-0.599722\pi$
$23$	−35966.0	−0.616375	−0.308187	+	0.951326i	$0.599722\pi$
$24$	13824.0	0.204124
$25$	24275.0	0.310720
$26$	56296.0	0.628161
$27$	19683.0	0.192450
$28$	−32320.0	−0.278239
$29$	102278.	0.778734	0.389367	−	0.921083i	$-0.372694\pi$
$29$	102278.	0.778734	0.389367	+	0.921083i	$0.372694\pi$
$30$	−69120.0	−0.467390
$31$	−91872.0	−0.553882	−0.276941	−	0.960887i	$-0.589321\pi$
$31$	−91872.0	−0.553882	−0.276941	+	0.960887i	$0.589321\pi$
$32$	32768.0	0.176777
$33$	69093.0	0.334684
$34$	−4408.00	−0.0192338
$35$	161600.	0.637094
$36$	46656.0	0.166667
$37$	250189.	0.812011	0.406006	−	0.913871i	$-0.366921\pi$
$37$	250189.	0.812011	0.406006	+	0.913871i	$0.366921\pi$
$38$	−177152.	−0.523725
$39$	189999.	0.512891
$40$	−163840.	−0.404772
$41$	−418053.	−0.947301	−0.473650	−	0.880713i	$-0.657064\pi$
$41$	−418053.	−0.947301	−0.473650	+	0.880713i	$0.657064\pi$
$42$	−109080.	−0.227181
$43$	−40225.0	−0.0771536	−0.0385768	−	0.999256i	$-0.512282\pi$
$43$	−40225.0	−0.0771536	−0.0385768	+	0.999256i	$0.512282\pi$
$44$	163776.	0.289845
$45$	−233280.	−0.381622
$46$	−287728.	−0.435843
$47$	25656.0	0.0360451	0.0180226	−	0.999838i	$-0.494263\pi$
$47$	25656.0	0.0360451	0.0180226	+	0.999838i	$0.494263\pi$
$48$	110592.	0.144338
$49$	−568518.	−0.690332
$50$	194200.	0.219712
$51$	−14877.0	−0.0157043
$52$	450368.	0.444177
$53$	−1.68159e6	−1.55151	−0.775754	−	0.631035i	$-0.782630\pi$
$53$	−1.68159e6	−1.55151	−0.775754	+	0.631035i	$0.782630\pi$
$54$	157464.	0.136083
$55$	−818880.	−0.663668
$56$	−258560.	−0.196745
$57$	−597888.	−0.427620
$58$	818224.	0.550648
$59$	205379.	0.130189
$59$	205379.	0.130189
$60$	−552960.	−0.330495
$61$	−1.85765e6	−1.04787	−0.523937	−	0.851757i	$-0.675537\pi$
$61$	−1.85765e6	−1.04787	−0.523937	+	0.851757i	$0.675537\pi$
$62$	−734976.	−0.391654
$63$	−368145.	−0.185493
$64$	262144.	0.125000
$65$	−2.25184e6	−1.01705
$66$	552744.	0.236657
$67$	3.84328e6	1.56114	0.780568	−	0.625071i	$-0.214930\pi$
$67$	3.84328e6	1.56114	0.780568	+	0.625071i	$0.214930\pi$
$68$	−35264.0	−0.0136004
$69$	−971082.	−0.355864
$70$	1.29280e6	0.450493
$71$	−3.10461e6	−1.02944	−0.514722	−	0.857357i	$-0.672105\pi$
$71$	−3.10461e6	−1.02944	−0.514722	+	0.857357i	$0.672105\pi$
$72$	373248.	0.117851
$73$	−4.32839e6	−1.30226	−0.651128	−	0.758968i	$-0.725704\pi$
$73$	−4.32839e6	−1.30226	−0.651128	+	0.758968i	$0.725704\pi$
$74$	2.00151e6	0.574179
$75$	655425.	0.179394
$76$	−1.41722e6	−0.370330
$77$	−1.29229e6	−0.322585
$78$	1.51999e6	0.362669
$79$	−4.67906e6	−1.06774	−0.533868	−	0.845568i	$-0.679262\pi$
$79$	−4.67906e6	−1.06774	−0.533868	+	0.845568i	$0.679262\pi$
$80$	−1.31072e6	−0.286217
$81$	531441.	0.111111
$82$	−3.34442e6	−0.669843
$83$	−8.91025e6	−1.71047	−0.855237	−	0.518236i	$-0.826589\pi$
$83$	−8.91025e6	−1.71047	−0.855237	+	0.518236i	$0.826589\pi$
$84$	−872640.	−0.160641
$85$	176320.	0.0311412
$86$	−321800.	−0.0545559
$87$	2.76151e6	0.449602
$88$	1.31021e6	0.204951
$89$	785984.	0.118181	0.0590906	−	0.998253i	$-0.481180\pi$
$89$	785984.	0.118181	0.0590906	+	0.998253i	$0.481180\pi$
$90$	−1.86624e6	−0.269848
$91$	−3.55369e6	−0.494349
$92$	−2.30182e6	−0.308187
$93$	−2.48054e6	−0.319784
$94$	205248.	0.0254877
$95$	7.08608e6	0.847956
$96$	884736.	0.102062
$97$	−9.84140e6	−1.09485	−0.547426	−	0.836854i	$-0.684392\pi$
$97$	−9.84140e6	−1.09485	−0.547426	+	0.836854i	$0.684392\pi$
$98$	−4.54814e6	−0.488138
$99$	1.86551e6	0.193230
$100$	1.55360e6	0.155360
$101$	1.67727e7	1.61986	0.809930	−	0.586527i	$-0.199505\pi$
$101$	1.67727e7	1.61986	0.809930	+	0.586527i	$0.199505\pi$
$102$	−119016.	−0.0111046
$103$	−1.10331e7	−0.994874	−0.497437	−	0.867500i	$-0.665725\pi$
$103$	−1.10331e7	−0.994874	−0.497437	+	0.867500i	$0.665725\pi$
$104$	3.60294e6	0.314080
$105$	4.36320e6	0.367826
$106$	−1.34527e7	−1.09708
$107$	1.44995e6	0.114422	0.0572109	−	0.998362i	$-0.481779\pi$
$107$	1.44995e6	0.114422	0.0572109	+	0.998362i	$0.481779\pi$
$108$	1.25971e6	0.0962250
$109$	−7.50085e6	−0.554776	−0.277388	−	0.960758i	$-0.589469\pi$
$109$	−7.50085e6	−0.554776	−0.277388	+	0.960758i	$0.589469\pi$
$110$	−6.55104e6	−0.469284
$111$	6.75510e6	0.468815
$112$	−2.06848e6	−0.139120
$113$	−2.10913e7	−1.37508	−0.687540	−	0.726146i	$-0.741309\pi$
$113$	−2.10913e7	−1.37508	−0.687540	+	0.726146i	$0.741309\pi$
$114$	−4.78310e6	−0.302373
$115$	1.15091e7	0.705667
$116$	6.54579e6	0.389367
$117$	5.12997e6	0.296118
$118$	1.64303e6	0.0920575
$119$	278255.	0.0151366
$120$	−4.42368e6	−0.233695
$121$	−1.29387e7	−0.663959
$122$	−1.48612e7	−0.740960
$123$	−1.12874e7	−0.546924
$124$	−5.87981e6	−0.276941
$125$	1.72320e7	0.789134
$126$	−2.94516e6	−0.131163
$127$	3.14214e7	1.36117	0.680585	−	0.732669i	$-0.261726\pi$
$127$	3.14214e7	1.36117	0.680585	+	0.732669i	$0.261726\pi$
$128$	2.09715e6	0.0883883
$129$	−1.08608e6	−0.0445447
$130$	−1.80147e7	−0.719160
$131$	−2.08045e6	−0.0808552	−0.0404276	−	0.999182i	$-0.512872\pi$
$131$	−2.08045e6	−0.0808552	−0.0404276	+	0.999182i	$0.512872\pi$
$132$	4.42195e6	0.167342
$133$	1.11827e7	0.412161
$134$	3.07463e7	1.10389
$135$	−6.29856e6	−0.220330
$136$	−282112.	−0.00961691
$137$	4.26138e7	1.41589	0.707943	−	0.706270i	$-0.249624\pi$
$137$	4.26138e7	1.41589	0.707943	+	0.706270i	$0.249624\pi$
$138$	−7.76866e6	−0.251634
$139$	−7.40028e6	−0.233720	−0.116860	−	0.993148i	$-0.537283\pi$
$139$	−7.40028e6	−0.233720	−0.116860	+	0.993148i	$0.537283\pi$
$140$	1.03424e7	0.318547
$141$	692712.	0.0208107
$142$	−2.48369e7	−0.727926
$143$	1.80077e7	0.514970
$144$	2.98598e6	0.0833333
$145$	−3.27290e7	−0.891547
$146$	−3.46271e7	−0.920834
$147$	−1.53500e7	−0.398563
$148$	1.60121e7	0.406006
$149$	1.02946e7	0.254951	0.127475	−	0.991842i	$-0.459313\pi$
$149$	1.02946e7	0.254951	0.127475	+	0.991842i	$0.459313\pi$
$150$	5.24340e6	0.126851
$151$	−4.05839e7	−0.959255	−0.479628	−	0.877472i	$-0.659228\pi$
$151$	−4.05839e7	−0.959255	−0.479628	+	0.877472i	$0.659228\pi$
$152$	−1.13377e7	−0.261863
$153$	−401679.	−0.00906691
$154$	−1.03384e7	−0.228102
$155$	2.93990e7	0.634121
$156$	1.21599e7	0.256446
$157$	−3.27228e7	−0.674842	−0.337421	−	0.941354i	$-0.609555\pi$
$157$	−3.27228e7	−0.674842	−0.337421	+	0.941354i	$0.609555\pi$
$158$	−3.74325e7	−0.755003
$159$	−4.54029e7	−0.895764
$160$	−1.04858e7	−0.202386
$161$	1.81628e7	0.342999
$162$	4.25153e6	0.0785674
$163$	3.45304e7	0.624517	0.312259	−	0.949997i	$-0.398914\pi$
$163$	3.45304e7	0.624517	0.312259	+	0.949997i	$0.398914\pi$
$164$	−2.67554e7	−0.473650
$165$	−2.21098e7	−0.383169
$166$	−7.12820e7	−1.20949
$167$	−4.86030e7	−0.807524	−0.403762	−	0.914864i	$-0.632298\pi$
$167$	−4.86030e7	−0.807524	−0.403762	+	0.914864i	$0.632298\pi$
$168$	−6.98112e6	−0.113591
$169$	−1.32291e7	−0.210828
$170$	1.41056e6	0.0220202
$171$	−1.61430e7	−0.246886
$172$	−2.57440e6	−0.0385768
$173$	1.18906e7	0.174599	0.0872997	−	0.996182i	$-0.472176\pi$
$173$	1.18906e7	0.174599	0.0872997	+	0.996182i	$0.472176\pi$
$174$	2.20920e7	0.317917
$175$	−1.22589e7	−0.172909
$176$	1.04817e7	0.144923
$177$	5.54523e6	0.0751646
$178$	6.28787e6	0.0835668
$179$	3.58712e7	0.467477	0.233738	−	0.972300i	$-0.424904\pi$
$179$	3.58712e7	0.467477	0.233738	+	0.972300i	$0.424904\pi$
$180$	−1.49299e7	−0.190811
$181$	−5.32395e7	−0.667358	−0.333679	−	0.942687i	$-0.608290\pi$
$181$	−5.32395e7	−0.667358	−0.333679	+	0.942687i	$0.608290\pi$
$182$	−2.84295e7	−0.349558
$183$	−5.01566e7	−0.604991
$184$	−1.84146e7	−0.217921
$185$	−8.00605e7	−0.929645
$186$	−1.98444e7	−0.226121
$187$	−1.41001e6	−0.0157680
$188$	1.64198e6	0.0180226
$189$	−9.93992e6	−0.107094
$190$	5.66886e7	0.599596
$191$	2.12088e7	0.220242	0.110121	−	0.993918i	$-0.464876\pi$
$191$	2.12088e7	0.220242	0.110121	+	0.993918i	$0.464876\pi$
$192$	7.07789e6	0.0721688
$193$	3.97511e7	0.398015	0.199007	−	0.979998i	$-0.436228\pi$
$193$	3.97511e7	0.398015	0.199007	+	0.979998i	$0.436228\pi$
$194$	−7.87312e7	−0.774178
$195$	−6.07997e7	−0.587192
$196$	−3.63852e7	−0.345166
$197$	−1.13852e8	−1.06098	−0.530492	−	0.847690i	$-0.677993\pi$
$197$	−1.13852e8	−1.06098	−0.530492	+	0.847690i	$0.677993\pi$
$198$	1.49241e7	0.136634
$199$	−2.20884e6	−0.0198691	−0.00993454	−	0.999951i	$-0.503162\pi$
$199$	−2.20884e6	−0.0198691	−0.00993454	+	0.999951i	$0.503162\pi$
$200$	1.24288e7	0.109856
$201$	1.03769e8	0.901322
$202$	1.34181e8	1.14541
$203$	−5.16504e7	−0.433349
$204$	−952128.	−0.00785217
$205$	1.33777e8	1.08453
$206$	−8.82649e7	−0.703482
$207$	−2.62192e7	−0.205458
$208$	2.88236e7	0.222088
$209$	−5.66665e7	−0.429353
$210$	3.49056e7	0.260092
$211$	2.32902e8	1.70681	0.853404	−	0.521250i	$-0.174534\pi$
$211$	2.32902e8	1.70681	0.853404	+	0.521250i	$0.174534\pi$
$212$	−1.07622e8	−0.775754
$213$	−8.38244e7	−0.594349
$214$	1.15996e7	0.0809084
$215$	1.28720e7	0.0883306
$216$	1.00777e7	0.0680414
$217$	4.63954e7	0.308223
$218$	−6.00068e7	−0.392286
$219$	−1.16867e8	−0.751858
$220$	−5.24083e7	−0.331834
$221$	−3.87739e6	−0.0241639
$222$	5.40408e7	0.331502
$223$	−1.49202e8	−0.900963	−0.450481	−	0.892786i	$-0.648748\pi$
$223$	−1.49202e8	−0.900963	−0.450481	+	0.892786i	$0.648748\pi$
$224$	−1.65478e7	−0.0983724
$225$	1.76965e7	0.103573
$226$	−1.68730e8	−0.972329
$227$	−1.86027e8	−1.05557	−0.527783	−	0.849379i	$-0.676977\pi$
$227$	−1.86027e8	−1.05557	−0.527783	+	0.849379i	$0.676977\pi$
$228$	−3.82648e7	−0.213810
$229$	2.88120e8	1.58544	0.792718	−	0.609588i	$-0.208665\pi$
$229$	2.88120e8	1.58544	0.792718	+	0.609588i	$0.208665\pi$
$230$	9.20730e7	0.498982
$231$	−3.48920e7	−0.186245
$232$	5.23663e7	0.275324
$233$	1.20813e8	0.625703	0.312851	−	0.949802i	$-0.398716\pi$
$233$	1.20813e8	0.625703	0.312851	+	0.949802i	$0.398716\pi$
$234$	4.10398e7	0.209387
$235$	−8.20992e6	−0.0412669
$236$	1.31443e7	0.0650945
$237$	−1.26335e8	−0.616457
$238$	2.22604e6	0.0107032
$239$	6.27874e7	0.297495	0.148748	−	0.988875i	$-0.452476\pi$
$239$	6.27874e7	0.297495	0.148748	+	0.988875i	$0.452476\pi$
$240$	−3.53894e7	−0.165247
$241$	−1.53976e8	−0.708588	−0.354294	−	0.935134i	$-0.615279\pi$
$241$	−1.53976e8	−0.708588	−0.354294	+	0.935134i	$0.615279\pi$
$242$	−1.03510e8	−0.469490
$243$	1.43489e7	0.0641500
$244$	−1.18890e8	−0.523937
$245$	1.81926e8	0.790338
$246$	−9.02994e7	−0.386734
$247$	−1.55827e8	−0.657967
$248$	−4.70385e7	−0.195827
$249$	−2.40577e8	−0.987543
$250$	1.37856e8	0.558002
$251$	−1.74311e8	−0.695772	−0.347886	−	0.937537i	$-0.613100\pi$
$251$	−1.74311e8	−0.695772	−0.347886	+	0.937537i	$0.613100\pi$
$252$	−2.35613e7	−0.0927464
$253$	−9.20370e7	−0.357306
$254$	2.51371e8	0.962492
$255$	4.76064e6	0.0179794
$256$	1.67772e7	0.0625000
$257$	−2.43978e8	−0.896573	−0.448286	−	0.893890i	$-0.647966\pi$
$257$	−2.43978e8	−0.896573	−0.448286	+	0.893890i	$0.647966\pi$
$258$	−8.68860e6	−0.0314978
$259$	−1.26345e8	−0.451867
$260$	−1.44118e8	−0.508523
$261$	7.45607e7	0.259578
$262$	−1.66436e7	−0.0571733
$263$	−1.88526e8	−0.639036	−0.319518	−	0.947580i	$-0.603521\pi$
$263$	−1.88526e8	−0.639036	−0.319518	+	0.947580i	$0.603521\pi$
$264$	3.53756e7	0.118329
$265$	5.38108e8	1.77627
$266$	8.94618e7	0.291442
$267$	2.12216e7	0.0682320
$268$	2.45970e8	0.780568
$269$	4.70558e8	1.47394	0.736970	−	0.675925i	$-0.236256\pi$
$269$	4.70558e8	1.47394	0.736970	+	0.675925i	$0.236256\pi$
$270$	−5.03885e7	−0.155797
$271$	−6.37027e6	−0.0194431	−0.00972155	−	0.999953i	$-0.503095\pi$
$271$	−6.37027e6	−0.0194431	−0.00972155	+	0.999953i	$0.503095\pi$
$272$	−2.25690e6	−0.00680018
$273$	−9.59495e7	−0.285413
$274$	3.40910e8	1.00118
$275$	6.21197e7	0.180121
$276$	−6.21492e7	−0.177932
$277$	−1.43815e8	−0.406561	−0.203280	−	0.979121i	$-0.565160\pi$
$277$	−1.43815e8	−0.406561	−0.203280	+	0.979121i	$0.565160\pi$
$278$	−5.92022e7	−0.165265
$279$	−6.69747e7	−0.184627
$280$	8.27392e7	0.225247
$281$	1.60084e8	0.430404	0.215202	−	0.976570i	$-0.430959\pi$
$281$	1.60084e8	0.430404	0.215202	+	0.976570i	$0.430959\pi$
$282$	5.54170e6	0.0147154
$283$	5.01911e8	1.31636	0.658180	−	0.752861i	$-0.271327\pi$
$283$	5.01911e8	1.31636	0.658180	+	0.752861i	$0.271327\pi$
$284$	−1.98695e8	−0.514722
$285$	1.91324e8	0.489568
$286$	1.44061e8	0.364139
$287$	2.11117e8	0.527152
$288$	2.38879e7	0.0589256
$289$	−4.10035e8	−0.999260
$290$	−2.61832e8	−0.630419
$291$	−2.65718e8	−0.632113
$292$	−2.77017e8	−0.651128
$293$	3.71183e8	0.862087	0.431044	−	0.902331i	$-0.358146\pi$
$293$	3.71183e8	0.862087	0.431044	+	0.902331i	$0.358146\pi$
$294$	−1.22800e8	−0.281827
$295$	−6.57213e7	−0.149049
$296$	1.28097e8	0.287089
$297$	5.03688e7	0.111561
$298$	8.23566e7	0.180278
$299$	−2.53093e8	−0.547559
$300$	4.19472e7	0.0896971
$301$	2.03136e7	0.0429343
$302$	−3.24671e8	−0.678296
$303$	4.52862e8	0.935226
$304$	−9.07018e7	−0.185165
$305$	5.94448e8	1.19968
$306$	−3.21343e6	−0.00641127
$307$	9.10907e7	0.179676	0.0898379	−	0.995956i	$-0.471365\pi$
$307$	9.10907e7	0.179676	0.0898379	+	0.995956i	$0.471365\pi$
$308$	−8.27069e7	−0.161292
$309$	−2.97894e8	−0.574391
$310$	2.35192e8	0.448391
$311$	4.85846e8	0.915876	0.457938	−	0.888984i	$-0.348588\pi$
$311$	4.85846e8	0.915876	0.457938	+	0.888984i	$0.348588\pi$
$312$	9.72795e7	0.181334
$313$	−5.70870e8	−1.05228	−0.526141	−	0.850397i	$-0.676362\pi$
$313$	−5.70870e8	−1.05228	−0.526141	+	0.850397i	$0.676362\pi$
$314$	−2.61783e8	−0.477185
$315$	1.17806e8	0.212365
$316$	−2.99460e8	−0.533868
$317$	−2.54614e8	−0.448926	−0.224463	−	0.974483i	$-0.572063\pi$
$317$	−2.54614e8	−0.448926	−0.224463	+	0.974483i	$0.572063\pi$
$318$	−3.63223e8	−0.633401
$319$	2.61729e8	0.451424
$320$	−8.38861e7	−0.143108
$321$	3.91485e7	0.0660614
$322$	1.45303e8	0.242537
$323$	1.22013e7	0.0201465
$324$	3.40122e7	0.0555556
$325$	1.70823e8	0.276029
$326$	2.76243e8	0.441600
$327$	−2.02523e8	−0.320300
$328$	−2.14043e8	−0.334921
$329$	−1.29563e7	−0.0200583
$330$	−1.76878e8	−0.270941
$331$	−4.66712e8	−0.707378	−0.353689	−	0.935363i	$-0.615073\pi$
$331$	−4.66712e8	−0.707378	−0.353689	+	0.935363i	$0.615073\pi$
$332$	−5.70256e8	−0.855237
$333$	1.82388e8	0.270670
$334$	−3.88824e8	−0.571005
$335$	−1.22985e9	−1.78729
$336$	−5.58490e7	−0.0803207
$337$	−6.01441e8	−0.856029	−0.428014	−	0.903772i	$-0.640787\pi$
$337$	−6.01441e8	−0.856029	−0.428014	+	0.903772i	$0.640787\pi$
$338$	−1.05833e8	−0.149078
$339$	−5.69464e8	−0.793903
$340$	1.12845e7	0.0155706
$341$	−2.35100e8	−0.321080
$342$	−1.29144e8	−0.174575
$343$	7.02991e8	0.940633
$344$	−2.05952e7	−0.0272779
$345$	3.10746e8	0.407417
$346$	9.51249e7	0.123460
$347$	−2.48673e8	−0.319503	−0.159752	−	0.987157i	$-0.551069\pi$
$347$	−2.48673e8	−0.319503	−0.159752	+	0.987157i	$0.551069\pi$
$348$	1.76736e8	0.224801
$349$	8.84811e8	1.11420	0.557098	−	0.830447i	$-0.311915\pi$
$349$	8.84811e8	1.11420	0.557098	+	0.830447i	$0.311915\pi$
$350$	−9.80710e7	−0.122265
$351$	1.38509e8	0.170964
$352$	8.38533e7	0.102476
$353$	7.59457e8	0.918949	0.459475	−	0.888191i	$-0.348038\pi$
$353$	7.59457e8	0.918949	0.459475	+	0.888191i	$0.348038\pi$
$354$	4.43619e7	0.0531494
$355$	9.93474e8	1.17858
$356$	5.03030e7	0.0590906
$357$	7.51288e6	0.00873913
$358$	2.86969e8	0.330556
$359$	−1.66585e8	−0.190023	−0.0950113	−	0.995476i	$-0.530289\pi$
$359$	−1.66585e8	−0.190023	−0.0950113	+	0.995476i	$0.530289\pi$
$360$	−1.19439e8	−0.134924
$361$	−4.03515e8	−0.451424
$362$	−4.25916e8	−0.471894
$363$	−3.49345e8	−0.383337
$364$	−2.27436e8	−0.247175
$365$	1.38509e9	1.49091
$366$	−4.01252e8	−0.427793
$367$	−2.85455e8	−0.301444	−0.150722	−	0.988576i	$-0.548160\pi$
$367$	−2.85455e8	−0.301444	−0.150722	+	0.988576i	$0.548160\pi$
$368$	−1.47317e8	−0.154094
$369$	−3.04761e8	−0.315767
$370$	−6.40484e8	−0.657358
$371$	8.49202e8	0.863381
$372$	−1.58755e8	−0.159892
$373$	9.12950e8	0.910890	0.455445	−	0.890264i	$-0.349480\pi$
$373$	9.12950e8	0.910890	0.455445	+	0.890264i	$0.349480\pi$
$374$	−1.12801e7	−0.0111497
$375$	4.65264e8	0.455607
$376$	1.31359e7	0.0127439
$377$	7.19730e8	0.691791
$378$	−7.95193e7	−0.0757271
$379$	−3.98102e8	−0.375627	−0.187813	−	0.982205i	$-0.560140\pi$
$379$	−3.98102e8	−0.375627	−0.187813	+	0.982205i	$0.560140\pi$
$380$	4.53509e8	0.423978
$381$	8.48377e8	0.785872
$382$	1.69671e8	0.155734
$383$	−2.68242e8	−0.243967	−0.121984	−	0.992532i	$-0.538926\pi$
$383$	−2.68242e8	−0.243967	−0.121984	+	0.992532i	$0.538926\pi$
$384$	5.66231e7	0.0510310
$385$	4.13534e8	0.369317
$386$	3.18009e8	0.281439
$387$	−2.93240e7	−0.0257179
$388$	−6.29849e8	−0.547426
$389$	1.58995e9	1.36949	0.684746	−	0.728782i	$-0.259913\pi$
$389$	1.58995e9	1.36949	0.684746	+	0.728782i	$0.259913\pi$
$390$	−4.86397e8	−0.415207
$391$	1.98173e7	0.0167658
$392$	−2.91081e8	−0.244069
$393$	−5.61722e7	−0.0466818
$394$	−9.10817e8	−0.750230
$395$	1.49730e9	1.22241
$396$	1.19393e8	0.0966150
$397$	−1.04070e9	−0.834755	−0.417377	−	0.908733i	$-0.637051\pi$
$397$	−1.04070e9	−0.834755	−0.417377	+	0.908733i	$0.637051\pi$
$398$	−1.76707e7	−0.0140496
$399$	3.01933e8	0.237961
$400$	9.94304e7	0.0776800
$401$	1.62005e9	1.25465	0.627326	−	0.778757i	$-0.284149\pi$
$401$	1.62005e9	1.25465	0.627326	+	0.778757i	$0.284149\pi$
$402$	8.30149e8	0.637331
$403$	−6.46503e8	−0.492043
$404$	1.07345e9	0.809930
$405$	−1.70061e8	−0.127207
$406$	−4.13203e8	−0.306424
$407$	6.40234e8	0.470715
$408$	−7.61702e6	−0.00555232
$409$	8.60192e8	0.621675	0.310838	−	0.950463i	$-0.399390\pi$
$409$	8.60192e8	0.621675	0.310838	+	0.950463i	$0.399390\pi$
$410$	1.07022e9	0.766881
$411$	1.15057e9	0.817462
$412$	−7.06119e8	−0.497437
$413$	−1.03716e8	−0.0724473
$414$	−2.09754e8	−0.145281
$415$	2.85128e9	1.95827
$416$	2.30588e8	0.157040
$417$	−1.99807e8	−0.134938
$418$	−4.53332e8	−0.303598
$419$	−1.18864e9	−0.789405	−0.394703	−	0.918809i	$-0.629152\pi$
$419$	−1.18864e9	−0.789405	−0.394703	+	0.918809i	$0.629152\pi$
$420$	2.79245e8	0.183913
$421$	8.23852e8	0.538099	0.269049	−	0.963126i	$-0.413290\pi$
$421$	8.23852e8	0.538099	0.269049	+	0.963126i	$0.413290\pi$
$422$	1.86322e9	1.20690
$423$	1.87032e7	0.0120150
$424$	−8.60973e8	−0.548541
$425$	−1.33755e7	−0.00845181
$426$	−6.70595e8	−0.420268
$427$	9.38113e8	0.583120
$428$	9.27965e7	0.0572109
$429$	4.86207e8	0.297318
$430$	1.02976e8	0.0624592
$431$	7.98124e8	0.480175	0.240088	−	0.970751i	$-0.422824\pi$
$431$	7.98124e8	0.480175	0.240088	+	0.970751i	$0.422824\pi$
$432$	8.06216e7	0.0481125
$433$	1.98393e9	1.17441	0.587204	−	0.809439i	$-0.300229\pi$
$433$	1.98393e9	1.17441	0.587204	+	0.809439i	$0.300229\pi$
$434$	3.71163e8	0.217947
$435$	−8.83682e8	−0.514735
$436$	−4.80054e8	−0.277388
$437$	7.96431e8	0.456524
$438$	−9.34933e8	−0.531644
$439$	3.41572e8	0.192689	0.0963444	−	0.995348i	$-0.469285\pi$
$439$	3.41572e8	0.192689	0.0963444	+	0.995348i	$0.469285\pi$
$440$	−4.19267e8	−0.234642
$441$	−4.14450e8	−0.230111
$442$	−3.10191e7	−0.0170864
$443$	2.86979e9	1.56833	0.784166	−	0.620552i	$-0.213091\pi$
$443$	2.86979e9	1.56833	0.784166	+	0.620552i	$0.213091\pi$
$444$	4.32327e8	0.234408
$445$	−2.51515e8	−0.135302
$446$	−1.19361e9	−0.637077
$447$	2.77954e8	0.147196
$448$	−1.32383e8	−0.0695598
$449$	3.75880e9	1.95969	0.979845	−	0.199759i	$-0.0640159\pi$
$449$	3.75880e9	1.95969	0.979845	+	0.199759i	$0.0640159\pi$
$450$	1.41572e8	0.0732374
$451$	−1.06980e9	−0.549141
$452$	−1.34984e9	−0.687540
$453$	−1.09576e9	−0.553826
$454$	−1.48822e9	−0.746398
$455$	1.13718e9	0.565964
$456$	−3.06119e8	−0.151186
$457$	4.53063e8	0.222050	0.111025	−	0.993818i	$-0.464587\pi$
$457$	4.53063e8	0.222050	0.111025	+	0.993818i	$0.464587\pi$
$458$	2.30496e9	1.12107
$459$	−1.08453e7	−0.00523478
$460$	7.36584e8	0.352833
$461$	3.49355e9	1.66078	0.830392	−	0.557179i	$-0.188116\pi$
$461$	3.49355e9	1.66078	0.830392	+	0.557179i	$0.188116\pi$
$462$	−2.79136e8	−0.131695
$463$	−2.52696e9	−1.18322	−0.591610	−	0.806224i	$-0.701507\pi$
$463$	−2.52696e9	−1.18322	−0.591610	+	0.806224i	$0.701507\pi$
$464$	4.18931e8	0.194684
$465$	7.93774e8	0.366110
$466$	9.66504e8	0.442439
$467$	−1.23913e8	−0.0562997	−0.0281499	−	0.999604i	$-0.508962\pi$
$467$	−1.23913e8	−0.0562997	−0.0281499	+	0.999604i	$0.508962\pi$
$468$	3.28318e8	0.148059
$469$	−1.94086e9	−0.868738
$470$	−6.56794e7	−0.0291801
$471$	−8.83516e8	−0.389620
$472$	1.05154e8	0.0460287
$473$	−1.02936e8	−0.0447252
$474$	−1.01068e9	−0.435901
$475$	−5.37546e8	−0.230138
$476$	1.78083e7	0.00756830
$477$	−1.22588e9	−0.517169
$478$	5.02299e8	0.210361
$479$	−2.23111e9	−0.927572	−0.463786	−	0.885947i	$-0.653509\pi$
$479$	−2.23111e9	−0.927572	−0.463786	+	0.885947i	$0.653509\pi$
$480$	−2.83116e8	−0.116847
$481$	1.76058e9	0.721353
$482$	−1.23181e9	−0.501048
$483$	4.90396e8	0.198031
$484$	−8.28076e8	−0.331980
$485$	3.14925e9	1.25346
$486$	1.14791e8	0.0453609
$487$	−4.81183e8	−0.188781	−0.0943907	−	0.995535i	$-0.530090\pi$
$487$	−4.81183e8	−0.188781	−0.0943907	+	0.995535i	$0.530090\pi$
$488$	−9.51117e8	−0.370480
$489$	9.32319e8	0.360565
$490$	1.45541e9	0.558853
$491$	−1.54422e9	−0.588742	−0.294371	−	0.955691i	$-0.595110\pi$
$491$	−1.54422e9	−0.588742	−0.294371	+	0.955691i	$0.595110\pi$
$492$	−7.22396e8	−0.273462
$493$	−5.63552e7	−0.0211821
$494$	−1.24662e9	−0.465253
$495$	−5.96964e8	−0.221223
$496$	−3.76308e8	−0.138470
$497$	1.56783e9	0.572863
$498$	−1.92461e9	−0.698298
$499$	−2.79695e9	−1.00770	−0.503851	−	0.863790i	$-0.668084\pi$
$499$	−2.79695e9	−1.00770	−0.503851	+	0.863790i	$0.668084\pi$
$500$	1.10285e9	0.394567
$501$	−1.31228e9	−0.466224
$502$	−1.39449e9	−0.491985
$503$	−1.53779e9	−0.538777	−0.269388	−	0.963032i	$-0.586822\pi$
$503$	−1.53779e9	−0.538777	−0.269388	+	0.963032i	$0.586822\pi$
$504$	−1.88490e8	−0.0655816
$505$	−5.36725e9	−1.85452
$506$	−7.36296e8	−0.252654
$507$	−3.57187e8	−0.121722
$508$	2.01097e9	0.680585
$509$	−5.12686e8	−0.172321	−0.0861607	−	0.996281i	$-0.527460\pi$
$509$	−5.12686e8	−0.172321	−0.0861607	+	0.996281i	$0.527460\pi$
$510$	3.80851e7	0.0127133
$511$	2.18584e9	0.724678
$512$	1.34218e8	0.0441942
$513$	−4.35860e8	−0.142540
$514$	−1.95183e9	−0.633973
$515$	3.53060e9	1.13900
$516$	−6.95088e7	−0.0222723
$517$	6.56537e7	0.0208950
$518$	−1.01076e9	−0.319518
$519$	3.21046e8	0.100805
$520$	−1.15294e9	−0.359580
$521$	−2.14172e9	−0.663483	−0.331742	−	0.943370i	$-0.607636\pi$
$521$	−2.14172e9	−0.663483	−0.331742	+	0.943370i	$0.607636\pi$
$522$	5.96485e8	0.183549
$523$	2.06866e9	0.632314	0.316157	−	0.948707i	$-0.397607\pi$
$523$	2.06866e9	0.632314	0.316157	+	0.948707i	$0.397607\pi$
$524$	−1.33149e8	−0.0404276
$525$	−3.30990e8	−0.0998290
$526$	−1.50820e9	−0.451866
$527$	5.06215e7	0.0150660
$528$	2.83005e8	0.0836711
$529$	−2.11127e9	−0.620082
$530$	4.30487e9	1.25601
$531$	1.49721e8	0.0433963
$532$	7.15694e8	0.206080
$533$	−2.94184e9	−0.841538
$534$	1.69773e8	0.0482473
$535$	−4.63983e8	−0.130998
$536$	1.96776e9	0.551945
$537$	9.68521e8	0.269898
$538$	3.76446e9	1.04223
$539$	−1.45484e9	−0.400179
$540$	−4.03108e8	−0.110165
$541$	−4.98061e9	−1.35236	−0.676180	−	0.736736i	$-0.736366\pi$
$541$	−4.98061e9	−1.35236	−0.676180	+	0.736736i	$0.736366\pi$
$542$	−5.09622e7	−0.0137483
$543$	−1.43747e9	−0.385299
$544$	−1.80552e7	−0.00480845
$545$	2.40027e9	0.635145
$546$	−7.67596e8	−0.201817
$547$	1.40132e9	0.366084	0.183042	−	0.983105i	$-0.441406\pi$
$547$	1.40132e9	0.366084	0.183042	+	0.983105i	$0.441406\pi$
$548$	2.72728e9	0.707943
$549$	−1.35423e9	−0.349292
$550$	4.96958e8	0.127365
$551$	−2.26484e9	−0.576777
$552$	−4.97194e8	−0.125817
$553$	2.36292e9	0.594172
$554$	−1.15052e9	−0.287482
$555$	−2.16163e9	−0.536731
$556$	−4.73618e8	−0.116860
$557$	3.06917e9	0.752538	0.376269	−	0.926510i	$-0.377207\pi$
$557$	3.06917e9	0.752538	0.376269	+	0.926510i	$0.377207\pi$
$558$	−5.35798e8	−0.130551
$559$	−2.83063e8	−0.0685397
$560$	6.61914e8	0.159273
$561$	−3.80702e7	−0.00910365
$562$	1.28067e9	0.304341
$563$	3.76717e9	0.889684	0.444842	−	0.895609i	$-0.353260\pi$
$563$	3.76717e9	0.889684	0.444842	+	0.895609i	$0.353260\pi$
$564$	4.43336e7	0.0104053
$565$	6.74921e9	1.57428
$566$	4.01529e9	0.930806
$567$	−2.68378e8	−0.0618309
$568$	−1.58956e9	−0.363963
$569$	−6.11843e9	−1.39235	−0.696173	−	0.717874i	$-0.745115\pi$
$569$	−6.11843e9	−1.39235	−0.696173	+	0.717874i	$0.745115\pi$
$570$	1.53059e9	0.346177
$571$	3.99117e8	0.0897168	0.0448584	−	0.998993i	$-0.485716\pi$
$571$	3.99117e8	0.0897168	0.0448584	+	0.998993i	$0.485716\pi$
$572$	1.15249e9	0.257485
$573$	5.72638e8	0.127157
$574$	1.68893e9	0.372753
$575$	−8.73075e8	−0.191520
$576$	1.91103e8	0.0416667
$577$	−7.74287e9	−1.67798	−0.838990	−	0.544148i	$-0.816853\pi$
$577$	−7.74287e9	−1.67798	−0.838990	+	0.544148i	$0.816853\pi$
$578$	−3.28028e9	−0.706584
$579$	1.07328e9	0.229794
$580$	−2.09465e9	−0.445773
$581$	4.49968e9	0.951842
$582$	−2.12574e9	−0.446972
$583$	−4.30318e9	−0.899394
$584$	−2.21614e9	−0.460417
$585$	−1.64159e9	−0.339015
$586$	2.96946e9	0.609588
$587$	4.03848e9	0.824110	0.412055	−	0.911159i	$-0.364811\pi$
$587$	4.03848e9	0.824110	0.412055	+	0.911159i	$0.364811\pi$
$588$	−9.82399e8	−0.199282
$589$	2.03441e9	0.410238
$590$	−5.25770e8	−0.105394
$591$	−3.07401e9	−0.612560
$592$	1.02477e9	0.203003
$593$	−3.44082e9	−0.677595	−0.338797	−	0.940859i	$-0.610020\pi$
$593$	−3.44082e9	−0.677595	−0.338797	+	0.940859i	$0.610020\pi$
$594$	4.02950e8	0.0788858
$595$	−8.90416e7	−0.0173294
$596$	6.58853e8	0.127475
$597$	−5.96386e7	−0.0114714
$598$	−2.02474e9	−0.387182
$599$	1.94448e9	0.369666	0.184833	−	0.982770i	$-0.440826\pi$
$599$	1.94448e9	0.369666	0.184833	+	0.982770i	$0.440826\pi$
$600$	3.35578e8	0.0634255
$601$	−5.82716e9	−1.09495	−0.547477	−	0.836820i	$-0.684412\pi$
$601$	−5.82716e9	−1.09495	−0.547477	+	0.836820i	$0.684412\pi$
$602$	1.62509e8	0.0303592
$603$	2.80175e9	0.520379
$604$	−2.59737e9	−0.479628
$605$	4.14038e9	0.760145
$606$	3.62290e9	0.661305
$607$	2.96885e9	0.538800	0.269400	−	0.963028i	$-0.413175\pi$
$607$	2.96885e9	0.538800	0.269400	+	0.963028i	$0.413175\pi$
$608$	−7.25615e8	−0.130931
$609$	−1.39456e9	−0.250194
$610$	4.75558e9	0.848300
$611$	1.80541e8	0.0320208
$612$	−2.57075e7	−0.00453345
$613$	6.09640e9	1.06896	0.534480	−	0.845181i	$-0.320507\pi$
$613$	6.09640e9	1.06896	0.534480	+	0.845181i	$0.320507\pi$
$614$	7.28725e8	0.127050
$615$	3.61198e9	0.626155
$616$	−6.61655e8	−0.114051
$617$	−3.76189e9	−0.644775	−0.322388	−	0.946608i	$-0.604485\pi$
$617$	−3.76189e9	−0.644775	−0.322388	+	0.946608i	$0.604485\pi$
$618$	−2.38315e9	−0.406155
$619$	−5.86183e9	−0.993381	−0.496691	−	0.867928i	$-0.665452\pi$
$619$	−5.86183e9	−0.993381	−0.496691	+	0.867928i	$0.665452\pi$
$620$	1.88154e9	0.317061
$621$	−7.07919e8	−0.118621
$622$	3.88676e9	0.647622
$623$	−3.96922e8	−0.0657653
$624$	7.78236e8	0.128223
$625$	−7.41072e9	−1.21417
$626$	−4.56696e9	−0.744076
$627$	−1.53000e9	−0.247887
$628$	−2.09426e9	−0.337421
$629$	−1.37854e8	−0.0220873
$630$	9.42451e8	0.150164
$631$	−6.25936e9	−0.991807	−0.495903	−	0.868378i	$-0.665163\pi$
$631$	−6.25936e9	−0.991807	−0.495903	+	0.868378i	$0.665163\pi$
$632$	−2.39568e9	−0.377502
$633$	6.28835e9	0.985426
$634$	−2.03691e9	−0.317438
$635$	−1.00548e10	−1.55836
$636$	−2.90578e9	−0.447882
$637$	−4.00066e9	−0.613259
$638$	2.09384e9	0.319205
$639$	−2.26326e9	−0.343148
$640$	−6.71089e8	−0.101193
$641$	−3.83899e9	−0.575723	−0.287861	−	0.957672i	$-0.592944\pi$
$641$	−3.83899e9	−0.575723	−0.287861	+	0.957672i	$0.592944\pi$
$642$	3.13188e8	0.0467125
$643$	−4.65813e8	−0.0690993	−0.0345496	−	0.999403i	$-0.511000\pi$
$643$	−4.65813e8	−0.0690993	−0.0345496	+	0.999403i	$0.511000\pi$
$644$	1.16242e9	0.171500
$645$	3.47544e8	0.0509977
$646$	9.76108e7	0.0142457
$647$	5.64076e9	0.818791	0.409395	−	0.912357i	$-0.365740\pi$
$647$	5.64076e9	0.818791	0.409395	+	0.912357i	$0.365740\pi$
$648$	2.72098e8	0.0392837
$649$	5.25565e8	0.0754692
$650$	1.36659e9	0.195182
$651$	1.25267e9	0.177953
$652$	2.20994e9	0.312259
$653$	8.78094e9	1.23408	0.617042	−	0.786930i	$-0.288331\pi$
$653$	8.78094e9	1.23408	0.617042	+	0.786930i	$0.288331\pi$
$654$	−1.62018e9	−0.226486
$655$	6.65745e8	0.0925685
$656$	−1.71235e9	−0.236825
$657$	−3.15540e9	−0.434086
$658$	−1.03650e8	−0.0141834
$659$	4.16573e9	0.567012	0.283506	−	0.958970i	$-0.408502\pi$
$659$	4.16573e9	0.567012	0.283506	+	0.958970i	$0.408502\pi$
$660$	−1.41502e9	−0.191584
$661$	−1.31151e10	−1.76631	−0.883156	−	0.469080i	$-0.844586\pi$
$661$	−1.31151e10	−1.76631	−0.883156	+	0.469080i	$0.844586\pi$
$662$	−3.73370e9	−0.500192
$663$	−1.04689e8	−0.0139510
$664$	−4.56205e9	−0.604744
$665$	−3.57847e9	−0.471869
$666$	1.45910e9	0.191393
$667$	−3.67853e9	−0.479992
$668$	−3.11059e9	−0.403762
$669$	−4.02845e9	−0.520171
$670$	−9.83881e9	−1.26381
$671$	−4.75373e9	−0.607443
$672$	−4.46792e8	−0.0567953
$673$	−1.33093e10	−1.68307	−0.841536	−	0.540201i	$-0.818348\pi$
$673$	−1.33093e10	−1.68307	−0.841536	+	0.540201i	$0.818348\pi$
$674$	−4.81153e9	−0.605304
$675$	4.77805e8	0.0597981
$676$	−8.46665e8	−0.105414
$677$	−2.58996e9	−0.320799	−0.160399	−	0.987052i	$-0.551278\pi$
$677$	−2.58996e9	−0.320799	−0.160399	+	0.987052i	$0.551278\pi$
$678$	−4.55571e9	−0.561374
$679$	4.96990e9	0.609262
$680$	9.02758e7	0.0110101
$681$	−5.02273e9	−0.609431
$682$	−1.88080e9	−0.227038
$683$	5.80536e9	0.697198	0.348599	−	0.937272i	$-0.386657\pi$
$683$	5.80536e9	0.697198	0.348599	+	0.937272i	$0.386657\pi$
$684$	−1.03315e9	−0.123443
$685$	−1.36364e10	−1.62100
$686$	5.62393e9	0.665128
$687$	7.77924e9	0.915352
$688$	−1.64762e8	−0.0192884
$689$	−1.18333e10	−1.37829
$690$	2.48597e9	0.288087
$691$	−5.62008e9	−0.647992	−0.323996	−	0.946058i	$-0.605026\pi$
$691$	−5.62008e9	−0.647992	−0.323996	+	0.946058i	$0.605026\pi$
$692$	7.60999e8	0.0872997
$693$	−9.42083e8	−0.107528
$694$	−1.98938e9	−0.225923
$695$	2.36809e9	0.267578
$696$	1.41389e9	0.158958
$697$	2.30347e8	0.0257673
$698$	7.07849e9	0.787855
$699$	3.26195e9	0.361250
$700$	−7.84568e8	−0.0864545
$701$	−4.17407e9	−0.457664	−0.228832	−	0.973466i	$-0.573491\pi$
$701$	−4.17407e9	−0.457664	−0.228832	+	0.973466i	$0.573491\pi$
$702$	1.10807e9	0.120890
$703$	−5.54019e9	−0.601424
$704$	6.70826e8	0.0724613
$705$	−2.21668e8	−0.0238254
$706$	6.07565e9	0.649795
$707$	−8.47020e9	−0.901417
$708$	3.54895e8	0.0375823
$709$	−2.79727e9	−0.294762	−0.147381	−	0.989080i	$-0.547084\pi$
$709$	−2.79727e9	−0.294762	−0.147381	+	0.989080i	$0.547084\pi$
$710$	7.94779e9	0.833379
$711$	−3.41103e9	−0.355912
$712$	4.02424e8	0.0417834
$713$	3.30427e9	0.341399
$714$	6.01031e7	0.00617950
$715$	−5.76246e9	−0.589572
$716$	2.29575e9	0.233738
$717$	1.69526e9	0.171759
$718$	−1.33268e9	−0.134366
$719$	1.58465e10	1.58994	0.794971	−	0.606647i	$-0.207486\pi$
$719$	1.58465e10	1.58994	0.794971	+	0.606647i	$0.207486\pi$
$720$	−9.55515e8	−0.0954056
$721$	5.57172e9	0.553626
$722$	−3.22812e9	−0.319205
$723$	−4.15736e9	−0.409104
$724$	−3.40733e9	−0.333679
$725$	2.48280e9	0.241968
$726$	−2.79476e9	−0.271060
$727$	−5.34483e9	−0.515897	−0.257949	−	0.966159i	$-0.583047\pi$
$727$	−5.34483e9	−0.515897	−0.257949	+	0.966159i	$0.583047\pi$
$728$	−1.81949e9	−0.174779
$729$	3.87420e8	0.0370370
$730$	1.10807e10	1.05423
$731$	2.21640e7	0.00209863
$732$	−3.21002e9	−0.302495
$733$	−1.25294e10	−1.17508	−0.587540	−	0.809195i	$-0.699903\pi$
$733$	−1.25294e10	−1.17508	−0.587540	+	0.809195i	$0.699903\pi$
$734$	−2.28364e9	−0.213153
$735$	4.91200e9	0.456302
$736$	−1.17853e9	−0.108961
$737$	9.83496e9	0.904975
$738$	−2.43809e9	−0.223281
$739$	−1.87539e10	−1.70937	−0.854685	−	0.519147i	$-0.826250\pi$
$739$	−1.87539e10	−1.70937	−0.854685	+	0.519147i	$0.826250\pi$
$740$	−5.12387e9	−0.464823
$741$	−4.20734e9	−0.379878
$742$	6.79362e9	0.610502
$743$	−3.97664e8	−0.0355676	−0.0177838	−	0.999842i	$-0.505661\pi$
$743$	−3.97664e8	−0.0355676	−0.0177838	+	0.999842i	$0.505661\pi$
$744$	−1.27004e9	−0.113061
$745$	−3.29426e9	−0.291885
$746$	7.30360e9	0.644097
$747$	−6.49557e9	−0.570158
$748$	−9.02406e7	−0.00788399
$749$	−7.32223e8	−0.0636732
$750$	3.72211e9	0.322163
$751$	1.54462e10	1.33071	0.665353	−	0.746529i	$-0.268281\pi$
$751$	1.54462e10	1.33071	0.665353	+	0.746529i	$0.268281\pi$
$752$	1.05087e8	0.00901128
$753$	−4.70640e9	−0.401704
$754$	5.75784e9	0.489170
$755$	1.29868e10	1.09822
$756$	−6.36155e8	−0.0535471
$757$	−1.40161e8	−0.0117434	−0.00587168	−	0.999983i	$-0.501869\pi$
$757$	−1.40161e8	−0.0117434	−0.00587168	+	0.999983i	$0.501869\pi$
$758$	−3.18481e9	−0.265608
$759$	−2.48500e9	−0.206291
$760$	3.62807e9	0.299798
$761$	2.13197e9	0.175362	0.0876809	−	0.996149i	$-0.472054\pi$
$761$	2.13197e9	0.175362	0.0876809	+	0.996149i	$0.472054\pi$
$762$	6.78702e9	0.555695
$763$	3.78793e9	0.308721
$764$	1.35736e9	0.110121
$765$	1.28537e8	0.0103804
$766$	−2.14594e9	−0.172511
$767$	1.44525e9	0.115654
$768$	4.52985e8	0.0360844
$769$	−7.52218e9	−0.596488	−0.298244	−	0.954490i	$-0.596401\pi$
$769$	−7.52218e9	−0.596488	−0.298244	+	0.954490i	$0.596401\pi$
$770$	3.30828e9	0.261146
$771$	−6.58742e9	−0.517637
$772$	2.54407e9	0.199007
$773$	−7.45592e9	−0.580595	−0.290297	−	0.956936i	$-0.593754\pi$
$773$	−7.45592e9	−0.580595	−0.290297	+	0.956936i	$0.593754\pi$
$774$	−2.34592e8	−0.0181853
$775$	−2.23019e9	−0.172102
$776$	−5.03879e9	−0.387089
$777$	−3.41133e9	−0.260885
$778$	1.27196e10	0.968377
$779$	9.25737e9	0.701627
$780$	−3.89118e9	−0.293596
$781$	−7.94469e9	−0.596758
$782$	1.58538e8	0.0118552
$783$	2.01314e9	0.149867
$784$	−2.32865e9	−0.172583
$785$	1.04713e10	0.772604
$786$	−4.49378e8	−0.0330090
$787$	−3.99517e9	−0.292162	−0.146081	−	0.989273i	$-0.546666\pi$
$787$	−3.99517e9	−0.292162	−0.146081	+	0.989273i	$0.546666\pi$
$788$	−7.28653e9	−0.530492
$789$	−5.09019e9	−0.368947
$790$	1.19784e10	0.864378
$791$	1.06511e10	0.765202
$792$	9.55142e8	0.0683171
$793$	−1.30723e10	−0.930883
$794$	−8.32560e9	−0.590261
$795$	1.45289e10	1.02553
$796$	−1.41366e8	−0.00993454
$797$	7.26131e9	0.508055	0.254027	−	0.967197i	$-0.418245\pi$
$797$	7.26131e9	0.508055	0.254027	+	0.967197i	$0.418245\pi$
$798$	2.41547e9	0.168264
$799$	−1.41365e7	−0.000980453 0
$800$	7.95443e8	0.0549281
$801$	5.72982e8	0.0393938
$802$	1.29604e10	0.887172
$803$	−1.10764e10	−0.754905
$804$	6.64119e9	0.450661
$805$	−5.81211e9	−0.392688
$806$	−5.17203e9	−0.347927
$807$	1.27051e10	0.850980
$808$	8.58761e9	0.572707
$809$	−8.45718e9	−0.561573	−0.280786	−	0.959770i	$-0.590595\pi$
$809$	−8.45718e9	−0.561573	−0.280786	+	0.959770i	$0.590595\pi$
$810$	−1.36049e9	−0.0899492
$811$	9.47721e8	0.0623889	0.0311945	−	0.999513i	$-0.490069\pi$
$811$	9.47721e8	0.0623889	0.0311945	+	0.999513i	$0.490069\pi$
$812$	−3.30562e9	−0.216674
$813$	−1.71997e8	−0.0112255
$814$	5.12187e9	0.332846
$815$	−1.10497e10	−0.714989
$816$	−6.09362e7	−0.00392609
$817$	8.90742e8	0.0571446
$818$	6.88153e9	0.439591
$819$	−2.59064e9	−0.164783
$820$	8.56173e9	0.542267
$821$	2.84875e10	1.79661	0.898305	−	0.439372i	$-0.144799\pi$
$821$	2.84875e10	1.79661	0.898305	+	0.439372i	$0.144799\pi$
$822$	9.20457e9	0.578033
$823$	2.69700e10	1.68648	0.843240	−	0.537537i	$-0.180645\pi$
$823$	2.69700e10	1.68648	0.843240	+	0.537537i	$0.180645\pi$
$824$	−5.64896e9	−0.351741
$825$	1.67723e9	0.103993
$826$	−8.29731e8	−0.0512280
$827$	−4.54676e9	−0.279533	−0.139766	−	0.990185i	$-0.544635\pi$
$827$	−4.54676e9	−0.279533	−0.139766	+	0.990185i	$0.544635\pi$
$828$	−1.67803e9	−0.102729
$829$	1.05419e10	0.642653	0.321326	−	0.946969i	$-0.395871\pi$
$829$	1.05419e10	0.642653	0.321326	+	0.946969i	$0.395871\pi$
$830$	2.28102e10	1.38470
$831$	−3.88301e9	−0.234728
$832$	1.84471e9	0.111044
$833$	3.13253e8	0.0187775
$834$	−1.59846e9	−0.0954158
$835$	1.55530e10	0.924507
$836$	−3.62666e9	−0.214676
$837$	−1.80832e9	−0.106595
$838$	−9.50909e9	−0.558194
$839$	2.98507e10	1.74497	0.872483	−	0.488644i	$-0.162508\pi$
$839$	2.98507e10	1.74497	0.872483	+	0.488644i	$0.162508\pi$
$840$	2.23396e9	0.130046
$841$	−6.78909e9	−0.393573
$842$	6.59082e9	0.380493
$843$	4.32227e9	0.248494
$844$	1.49057e10	0.853404
$845$	4.23333e9	0.241370
$846$	1.49626e8	0.00849592
$847$	6.53404e9	0.369479
$848$	−6.88778e9	−0.387877
$849$	1.35516e10	0.760000
$850$	−1.07004e8	−0.00597633
$851$	−8.99830e9	−0.500503
$852$	−5.36476e9	−0.297175
$853$	2.95767e10	1.63165	0.815826	−	0.578298i	$-0.196283\pi$
$853$	2.95767e10	1.63165	0.815826	+	0.578298i	$0.196283\pi$
$854$	7.50491e9	0.412328
$855$	5.16575e9	0.282652
$856$	7.42372e8	0.0404542
$857$	5.01794e9	0.272328	0.136164	−	0.990686i	$-0.456523\pi$
$857$	5.01794e9	0.272328	0.136164	+	0.990686i	$0.456523\pi$
$858$	3.88966e9	0.210236
$859$	−8.17300e9	−0.439952	−0.219976	−	0.975505i	$-0.570598\pi$
$859$	−8.17300e9	−0.439952	−0.219976	+	0.975505i	$0.570598\pi$
$860$	8.23808e8	0.0441653
$861$	5.70015e9	0.304352
$862$	6.38499e9	0.339535
$863$	2.90590e8	0.0153901	0.00769507	−	0.999970i	$-0.497551\pi$
$863$	2.90590e8	0.0153901	0.00769507	+	0.999970i	$0.497551\pi$
$864$	6.44973e8	0.0340207
$865$	−3.80500e9	−0.199893
$866$	1.58714e10	0.830431
$867$	−1.10709e10	−0.576923
$868$	2.96930e9	0.154112
$869$	−1.19737e10	−0.618956
$870$	−7.06946e9	−0.363973
$871$	2.70452e10	1.38684
$872$	−3.84044e9	−0.196143
$873$	−7.17438e9	−0.364951
$874$	6.37145e9	0.322811
$875$	−8.70216e9	−0.439136
$876$	−7.47946e9	−0.375929
$877$	9.15984e9	0.458553	0.229276	−	0.973361i	$-0.426364\pi$
$877$	9.15984e9	0.458553	0.229276	+	0.973361i	$0.426364\pi$
$878$	2.73258e9	0.136252
$879$	1.00219e10	0.497726
$880$	−3.35413e9	−0.165917
$881$	3.55103e10	1.74960	0.874800	−	0.484484i	$-0.160993\pi$
$881$	3.55103e10	1.74960	0.874800	+	0.484484i	$0.160993\pi$
$882$	−3.31560e9	−0.162713
$883$	2.15421e10	1.05300	0.526498	−	0.850176i	$-0.323505\pi$
$883$	2.15421e10	1.05300	0.526498	+	0.850176i	$0.323505\pi$
$884$	−2.48153e8	−0.0120819
$885$	−1.77447e9	−0.0860535
$886$	2.29584e10	1.10898
$887$	−2.69917e10	−1.29867	−0.649334	−	0.760503i	$-0.724952\pi$
$887$	−2.69917e10	−1.29867	−0.649334	+	0.760503i	$0.724952\pi$
$888$	3.45861e9	0.165751
$889$	−1.58678e10	−0.757461
$890$	−2.01212e9	−0.0956728
$891$	1.35996e9	0.0644100
$892$	−9.54891e9	−0.450481
$893$	−5.68126e8	−0.0266972
$894$	2.22363e9	0.104083
$895$	−1.14788e10	−0.535198
$896$	−1.05906e9	−0.0491862
$897$	−6.83350e9	−0.316133
$898$	3.00704e10	1.38571
$899$	−9.39648e9	−0.431327
$900$	1.13257e9	0.0517867
$901$	9.26555e8	0.0422021
$902$	−8.55838e9	−0.388301
$903$	5.48468e8	0.0247881
$904$	−1.07987e10	−0.486164
$905$	1.70366e10	0.764036
$906$	−8.76612e9	−0.391614
$907$	−2.16552e10	−0.963690	−0.481845	−	0.876256i	$-0.660033\pi$
$907$	−2.16552e10	−0.963690	−0.481845	+	0.876256i	$0.660033\pi$
$908$	−1.19057e10	−0.527783
$909$	1.22273e10	0.539953
$910$	9.09743e9	0.400197
$911$	−1.58468e10	−0.694430	−0.347215	−	0.937786i	$-0.612873\pi$
$911$	−1.58468e10	−0.694430	−0.347215	+	0.937786i	$0.612873\pi$
$912$	−2.44895e9	−0.106905
$913$	−2.28013e10	−0.991545
$914$	3.62450e9	0.157013
$915$	1.60501e10	0.692634
$916$	1.84397e10	0.792718
$917$	1.05063e9	0.0449942
$918$	−8.67627e7	−0.00370155
$919$	8.19603e8	0.0348337	0.0174168	−	0.999848i	$-0.494456\pi$
$919$	8.19603e8	0.0348337	0.0174168	+	0.999848i	$0.494456\pi$
$920$	5.89267e9	0.249491
$921$	2.45945e9	0.103736
$922$	2.79484e10	1.17435
$923$	−2.18471e10	−0.914510
$924$	−2.23309e9	−0.0931223
$925$	6.07334e9	0.252308
$926$	−2.02157e10	−0.836663
$927$	−8.04314e9	−0.331625
$928$	3.35145e9	0.137662
$929$	7.74167e9	0.316796	0.158398	−	0.987375i	$-0.449367\pi$
$929$	7.74167e9	0.316796	0.158398	+	0.987375i	$0.449367\pi$
$930$	6.35019e9	0.258879
$931$	1.25893e10	0.511301
$932$	7.73203e9	0.312851
$933$	1.31178e10	0.528782
$934$	−9.91301e8	−0.0398099
$935$	4.51203e8	0.0180522
$936$	2.62655e9	0.104693
$937$	−7.68509e9	−0.305183	−0.152592	−	0.988289i	$-0.548762\pi$
$937$	−7.68509e9	−0.305183	−0.152592	+	0.988289i	$0.548762\pi$
$938$	−1.55269e10	−0.614291
$939$	−1.54135e10	−0.607535
$940$	−5.25435e8	−0.0206334
$941$	−2.36009e10	−0.923346	−0.461673	−	0.887050i	$-0.652751\pi$
$941$	−2.36009e10	−0.923346	−0.461673	+	0.887050i	$0.652751\pi$
$942$	−7.06813e9	−0.275503
$943$	1.50357e10	0.583892
$944$	8.41232e8	0.0325472
$945$	3.18077e9	0.122609
$946$	−8.23486e8	−0.0316255
$947$	3.69258e10	1.41288	0.706439	−	0.707774i	$-0.250301\pi$
$947$	3.69258e10	1.41288	0.706439	+	0.707774i	$0.250301\pi$
$948$	−8.08541e9	−0.308229
$949$	−3.04589e10	−1.15686
$950$	−4.30036e9	−0.162732
$951$	−6.87457e9	−0.259187
$952$	1.42467e8	0.00535160
$953$	−1.37393e10	−0.514209	−0.257105	−	0.966384i	$-0.582769\pi$
$953$	−1.37393e10	−0.514209	−0.257105	+	0.966384i	$0.582769\pi$
$954$	−9.80702e9	−0.365694
$955$	−6.78682e9	−0.252148
$956$	4.01840e9	0.148748
$957$	7.06669e9	0.260630
$958$	−1.78489e10	−0.655892
$959$	−2.15200e10	−0.787909
$960$	−2.26492e9	−0.0826236
$961$	−1.90721e10	−0.693215
$962$	1.40846e10	0.510074
$963$	1.05701e9	0.0381406
$964$	−9.85448e9	−0.354294
$965$	−1.27204e10	−0.455674
$966$	3.92317e9	0.140029
$967$	−1.57250e10	−0.559241	−0.279620	−	0.960111i	$-0.590209\pi$
$967$	−1.57250e10	−0.559241	−0.279620	+	0.960111i	$0.590209\pi$
$968$	−6.62461e9	−0.234745
$969$	3.29436e8	0.0116316
$970$	2.51940e10	0.886330
$971$	−2.98875e10	−1.04767	−0.523833	−	0.851821i	$-0.675498\pi$
$971$	−2.98875e10	−1.04767	−0.523833	+	0.851821i	$0.675498\pi$
$972$	9.18330e8	0.0320750
$973$	3.73714e9	0.130060
$974$	−3.84947e9	−0.133489
$975$	4.61223e9	0.159366
$976$	−7.60893e9	−0.261969
$977$	−7.70363e9	−0.264280	−0.132140	−	0.991231i	$-0.542185\pi$
$977$	−7.70363e9	−0.264280	−0.132140	+	0.991231i	$0.542185\pi$
$978$	7.45856e9	0.254958
$979$	2.01133e9	0.0685085
$980$	1.16432e10	0.395169
$981$	−5.46812e9	−0.184925
$982$	−1.23538e10	−0.416304
$983$	2.40391e10	0.807200	0.403600	−	0.914936i	$-0.367759\pi$
$983$	2.40391e10	0.807200	0.403600	+	0.914936i	$0.367759\pi$
$984$	−5.77916e9	−0.193367
$985$	3.64327e10	1.21469
$986$	−4.50841e8	−0.0149780
$987$	−3.49820e8	−0.0115807
$988$	−9.97295e9	−0.328984
$989$	1.44673e9	0.0475555
$990$	−4.77571e9	−0.156428
$991$	1.40302e10	0.457938	0.228969	−	0.973434i	$-0.426465\pi$
$991$	1.40302e10	0.457938	0.228969	+	0.973434i	$0.426465\pi$
$992$	−3.01046e9	−0.0979134
$993$	−1.26012e10	−0.408405
$994$	1.25426e10	0.405075
$995$	7.06828e8	0.0227474
$996$	−1.53969e10	−0.493772
$997$	−1.91677e10	−0.612543	−0.306271	−	0.951944i	$-0.599081\pi$
$997$	−1.91677e10	−0.612543	−0.306271	+	0.951944i	$0.599081\pi$
$998$	−2.23756e10	−0.712554
$999$	4.92447e9	0.156272

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	354.8.a.a.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
354.8.a.a.1.1	✓	1	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 354 = 2 \cdot 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	354.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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