LMFDB - Embedded newform 462.8.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [462,8,Mod(1,462)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(462, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("462.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 462 = 2 \cdot 3 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	462.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:	$144.321881774$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	462.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} +182.000 q^{5} +216.000 q^{6} +343.000 q^{7} -512.000 q^{8} +729.000 q^{9} +O(q^{10})$

\(q-8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} +182.000 q^{5} +216.000 q^{6} +343.000 q^{7} -512.000 q^{8} +729.000 q^{9} -1456.00 q^{10} +1331.00 q^{11} -1728.00 q^{12} +13758.0 q^{13} -2744.00 q^{14} -4914.00 q^{15} +4096.00 q^{16} +14738.0 q^{17} -5832.00 q^{18} +34516.0 q^{19} +11648.0 q^{20} -9261.00 q^{21} -10648.0 q^{22} +49816.0 q^{23} +13824.0 q^{24} -45001.0 q^{25} -110064. q^{26} -19683.0 q^{27} +21952.0 q^{28} +141174. q^{29} +39312.0 q^{30} -86112.0 q^{31} -32768.0 q^{32} -35937.0 q^{33} -117904. q^{34} +62426.0 q^{35} +46656.0 q^{36} +535038. q^{37} -276128. q^{38} -371466. q^{39} -93184.0 q^{40} +708282. q^{41} +74088.0 q^{42} +162292. q^{43} +85184.0 q^{44} +132678. q^{45} -398528. q^{46} +397296. q^{47} -110592. q^{48} +117649. q^{49} +360008. q^{50} -397926. q^{51} +880512. q^{52} +224798. q^{53} +157464. q^{54} +242242. q^{55} -175616. q^{56} -931932. q^{57} -1.12939e6 q^{58} +694668. q^{59} -314496. q^{60} +50110.0 q^{61} +688896. q^{62} +250047. q^{63} +262144. q^{64} +2.50396e6 q^{65} +287496. q^{66} -3.22840e6 q^{67} +943232. q^{68} -1.34503e6 q^{69} -499408. q^{70} -1.87150e6 q^{71} -373248. q^{72} -937542. q^{73} -4.28030e6 q^{74} +1.21503e6 q^{75} +2.20902e6 q^{76} +456533. q^{77} +2.97173e6 q^{78} +1.26603e6 q^{79} +745472. q^{80} +531441. q^{81} -5.66626e6 q^{82} -2.54500e6 q^{83} -592704. q^{84} +2.68232e6 q^{85} -1.29834e6 q^{86} -3.81170e6 q^{87} -681472. q^{88} -9.90983e6 q^{89} -1.06142e6 q^{90} +4.71899e6 q^{91} +3.18822e6 q^{92} +2.32502e6 q^{93} -3.17837e6 q^{94} +6.28191e6 q^{95} +884736. q^{96} -1.40108e7 q^{97} -941192. q^{98} +970299. q^{99} +O(q^{100})\)

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
$5$	182.000	0.651143	0.325571	−	0.945517i	$-0.394443\pi$
$5$	182.000	0.651143	0.325571	+	0.945517i	$0.394443\pi$
$6$	216.000	0.408248
$7$	343.000	0.377964
$7$	343.000	0.377964
$8$	−512.000	−0.353553
$9$	729.000	0.333333
$10$	−1456.00	−0.460428
$11$	1331.00	0.301511
$11$	1331.00	0.301511
$12$	−1728.00	−0.288675
$13$	13758.0	1.73682	0.868408	−	0.495851i	$-0.165144\pi$
$13$	13758.0	1.73682	0.868408	+	0.495851i	$0.165144\pi$
$14$	−2744.00	−0.267261
$15$	−4914.00	−0.375938
$16$	4096.00	0.250000
$17$	14738.0	0.727558	0.363779	−	0.931485i	$-0.381486\pi$
$17$	14738.0	0.727558	0.363779	+	0.931485i	$0.381486\pi$
$18$	−5832.00	−0.235702
$19$	34516.0	1.15447	0.577235	−	0.816578i	$-0.304132\pi$
$19$	34516.0	1.15447	0.577235	+	0.816578i	$0.304132\pi$
$20$	11648.0	0.325571
$21$	−9261.00	−0.218218
$22$	−10648.0	−0.213201
$23$	49816.0	0.853732	0.426866	−	0.904315i	$-0.359618\pi$
$23$	49816.0	0.853732	0.426866	+	0.904315i	$0.359618\pi$
$24$	13824.0	0.204124
$25$	−45001.0	−0.576013
$26$	−110064.	−1.22811
$27$	−19683.0	−0.192450
$28$	21952.0	0.188982
$29$	141174.	1.07488	0.537442	−	0.843301i	$-0.319391\pi$
$29$	141174.	1.07488	0.537442	+	0.843301i	$0.319391\pi$
$30$	39312.0	0.265828
$31$	−86112.0	−0.519156	−0.259578	−	0.965722i	$-0.583583\pi$
$31$	−86112.0	−0.519156	−0.259578	+	0.965722i	$0.583583\pi$
$32$	−32768.0	−0.176777
$33$	−35937.0	−0.174078
$34$	−117904.	−0.514461
$35$	62426.0	0.246109
$36$	46656.0	0.166667
$37$	535038.	1.73652	0.868258	−	0.496114i	$-0.165240\pi$
$37$	535038.	1.73652	0.868258	+	0.496114i	$0.165240\pi$
$38$	−276128.	−0.816334
$39$	−371466.	−1.00275
$40$	−93184.0	−0.230214
$41$	708282.	1.60495	0.802477	−	0.596683i	$-0.203515\pi$
$41$	708282.	1.60495	0.802477	+	0.596683i	$0.203515\pi$
$42$	74088.0	0.154303
$43$	162292.	0.311285	0.155642	−	0.987813i	$-0.450255\pi$
$43$	162292.	0.311285	0.155642	+	0.987813i	$0.450255\pi$
$44$	85184.0	0.150756
$45$	132678.	0.217048
$46$	−398528.	−0.603680
$47$	397296.	0.558177	0.279088	−	0.960265i	$-0.409968\pi$
$47$	397296.	0.558177	0.279088	+	0.960265i	$0.409968\pi$
$48$	−110592.	−0.144338
$49$	117649.	0.142857
$50$	360008.	0.407303
$51$	−397926.	−0.420056
$52$	880512.	0.868408
$53$	224798.	0.207409	0.103704	−	0.994608i	$-0.466930\pi$
$53$	224798.	0.207409	0.103704	+	0.994608i	$0.466930\pi$
$54$	157464.	0.136083
$55$	242242.	0.196327
$56$	−175616.	−0.133631
$57$	−931932.	−0.666534
$58$	−1.12939e6	−0.760058
$59$	694668.	0.440347	0.220174	−	0.975461i	$-0.429338\pi$
$59$	694668.	0.440347	0.220174	+	0.975461i	$0.429338\pi$
$60$	−314496.	−0.187969
$61$	50110.0	0.0282664	0.0141332	−	0.999900i	$-0.495501\pi$
$61$	50110.0	0.0282664	0.0141332	+	0.999900i	$0.495501\pi$
$62$	688896.	0.367099
$63$	250047.	0.125988
$64$	262144.	0.125000
$65$	2.50396e6	1.13091
$66$	287496.	0.123091
$67$	−3.22840e6	−1.31137	−0.655686	−	0.755033i	$-0.727621\pi$
$67$	−3.22840e6	−1.31137	−0.655686	+	0.755033i	$0.727621\pi$
$68$	943232.	0.363779
$69$	−1.34503e6	−0.492902
$70$	−499408.	−0.174025
$71$	−1.87150e6	−0.620561	−0.310281	−	0.950645i	$-0.600423\pi$
$71$	−1.87150e6	−0.620561	−0.310281	+	0.950645i	$0.600423\pi$
$72$	−373248.	−0.117851
$73$	−937542.	−0.282072	−0.141036	−	0.990004i	$-0.545043\pi$
$73$	−937542.	−0.282072	−0.141036	+	0.990004i	$0.545043\pi$
$74$	−4.28030e6	−1.22790
$75$	1.21503e6	0.332561
$76$	2.20902e6	0.577235
$77$	456533.	0.113961
$78$	2.97173e6	0.709052
$79$	1.26603e6	0.288902	0.144451	−	0.989512i	$-0.453858\pi$
$79$	1.26603e6	0.288902	0.144451	+	0.989512i	$0.453858\pi$
$80$	745472.	0.162786
$81$	531441.	0.111111
$82$	−5.66626e6	−1.13487
$83$	−2.54500e6	−0.488557	−0.244278	−	0.969705i	$-0.578551\pi$
$83$	−2.54500e6	−0.488557	−0.244278	+	0.969705i	$0.578551\pi$
$84$	−592704.	−0.109109
$85$	2.68232e6	0.473744
$86$	−1.29834e6	−0.220111
$87$	−3.81170e6	−0.620585
$88$	−681472.	−0.106600
$89$	−9.90983e6	−1.49005	−0.745026	−	0.667036i	$-0.767563\pi$
$89$	−9.90983e6	−1.49005	−0.745026	+	0.667036i	$0.767563\pi$
$90$	−1.06142e6	−0.153476
$91$	4.71899e6	0.656454
$92$	3.18822e6	0.426866
$93$	2.32502e6	0.299735
$94$	−3.17837e6	−0.394691
$95$	6.28191e6	0.751725
$96$	884736.	0.102062
$97$	−1.40108e7	−1.55870	−0.779350	−	0.626588i	$-0.784451\pi$
$97$	−1.40108e7	−1.55870	−0.779350	+	0.626588i	$0.784451\pi$
$98$	−941192.	−0.101015
$99$	970299.	0.100504
$100$	−2.88006e6	−0.288006
$101$	−7.34281e6	−0.709149	−0.354574	−	0.935028i	$-0.615374\pi$
$101$	−7.34281e6	−0.709149	−0.354574	+	0.935028i	$0.615374\pi$
$102$	3.18341e6	0.297024
$103$	−8.86030e6	−0.798947	−0.399473	−	0.916745i	$-0.630807\pi$
$103$	−8.86030e6	−0.798947	−0.399473	+	0.916745i	$0.630807\pi$
$104$	−7.04410e6	−0.614057
$105$	−1.68550e6	−0.142091
$106$	−1.79838e6	−0.146660
$107$	−1.50185e7	−1.18518	−0.592588	−	0.805506i	$-0.701894\pi$
$107$	−1.50185e7	−1.18518	−0.592588	+	0.805506i	$0.701894\pi$
$108$	−1.25971e6	−0.0962250
$109$	1.06251e7	0.785851	0.392925	−	0.919570i	$-0.371463\pi$
$109$	1.06251e7	0.785851	0.392925	+	0.919570i	$0.371463\pi$
$110$	−1.93794e6	−0.138824
$111$	−1.44460e7	−1.00258
$112$	1.40493e6	0.0944911
$113$	−7.09310e6	−0.462447	−0.231223	−	0.972901i	$-0.574273\pi$
$113$	−7.09310e6	−0.462447	−0.231223	+	0.972901i	$0.574273\pi$
$114$	7.45546e6	0.471311
$115$	9.06651e6	0.555901
$116$	9.03514e6	0.537442
$117$	1.00296e7	0.578938
$118$	−5.55734e6	−0.311373
$119$	5.05513e6	0.274991
$120$	2.51597e6	0.132914
$121$	1.77156e6	0.0909091
$122$	−400880.	−0.0199873
$123$	−1.91236e7	−0.926621
$124$	−5.51117e6	−0.259578
$125$	−2.24089e7	−1.02621
$126$	−2.00038e6	−0.0890871
$127$	3.60394e7	1.56122	0.780611	−	0.625017i	$-0.214908\pi$
$127$	3.60394e7	1.56122	0.780611	+	0.625017i	$0.214908\pi$
$128$	−2.09715e6	−0.0883883
$129$	−4.38188e6	−0.179720
$130$	−2.00316e7	−0.799678
$131$	2.25637e7	0.876923	0.438462	−	0.898750i	$-0.355523\pi$
$131$	2.25637e7	0.876923	0.438462	+	0.898750i	$0.355523\pi$
$132$	−2.29997e6	−0.0870388
$133$	1.18390e7	0.436349
$134$	2.58272e7	0.927280
$135$	−3.58231e6	−0.125313
$136$	−7.54586e6	−0.257230
$137$	1.24461e7	0.413535	0.206768	−	0.978390i	$-0.433706\pi$
$137$	1.24461e7	0.413535	0.206768	+	0.978390i	$0.433706\pi$
$138$	1.07603e7	0.348535
$139$	1.63300e6	0.0515746	0.0257873	−	0.999667i	$-0.491791\pi$
$139$	1.63300e6	0.0515746	0.0257873	+	0.999667i	$0.491791\pi$
$140$	3.99526e6	0.123054
$141$	−1.07270e7	−0.322263
$142$	1.49720e7	0.438803
$143$	1.83119e7	0.523669
$144$	2.98598e6	0.0833333
$145$	2.56937e7	0.699903
$146$	7.50034e6	0.199455
$147$	−3.17652e6	−0.0824786
$148$	3.42424e7	0.868258
$149$	6.75131e7	1.67200	0.835999	−	0.548730i	$-0.184889\pi$
$149$	6.75131e7	1.67200	0.835999	+	0.548730i	$0.184889\pi$
$150$	−9.72022e6	−0.235156
$151$	−6.23218e7	−1.47306	−0.736530	−	0.676405i	$-0.763537\pi$
$151$	−6.23218e7	−1.47306	−0.736530	+	0.676405i	$0.763537\pi$
$152$	−1.76722e7	−0.408167
$153$	1.07440e7	0.242519
$154$	−3.65226e6	−0.0805823
$155$	−1.56724e7	−0.338045
$156$	−2.37738e7	−0.501375
$157$	−8.02772e7	−1.65555	−0.827777	−	0.561058i	$-0.810395\pi$
$157$	−8.02772e7	−1.65555	−0.827777	+	0.561058i	$0.810395\pi$
$158$	−1.01283e7	−0.204284
$159$	−6.06955e6	−0.119747
$160$	−5.96378e6	−0.115107
$161$	1.70869e7	0.322680
$162$	−4.25153e6	−0.0785674
$163$	−1.17849e7	−0.213143	−0.106571	−	0.994305i	$-0.533987\pi$
$163$	−1.17849e7	−0.213143	−0.106571	+	0.994305i	$0.533987\pi$
$164$	4.53300e7	0.802477
$165$	−6.54053e6	−0.113349
$166$	2.03600e7	0.345462
$167$	−5.51803e7	−0.916805	−0.458402	−	0.888745i	$-0.651578\pi$
$167$	−5.51803e7	−0.916805	−0.458402	+	0.888745i	$0.651578\pi$
$168$	4.74163e6	0.0771517
$169$	1.26534e8	2.01653
$170$	−2.14585e7	−0.334988
$171$	2.51622e7	0.384824
$172$	1.03867e7	0.155642
$173$	5.63952e7	0.828096	0.414048	−	0.910255i	$-0.364114\pi$
$173$	5.63952e7	0.828096	0.414048	+	0.910255i	$0.364114\pi$
$174$	3.04936e7	0.438820
$175$	−1.54353e7	−0.217712
$176$	5.45178e6	0.0753778
$177$	−1.87560e7	−0.254235
$178$	7.92786e7	1.05363
$179$	1.03312e7	0.134637	0.0673185	−	0.997732i	$-0.478556\pi$
$179$	1.03312e7	0.134637	0.0673185	+	0.997732i	$0.478556\pi$
$180$	8.49139e6	0.108524
$181$	6.67241e7	0.836388	0.418194	−	0.908358i	$-0.362663\pi$
$181$	6.67241e7	0.836388	0.418194	+	0.908358i	$0.362663\pi$
$182$	−3.77520e7	−0.464183
$183$	−1.35297e6	−0.0163196
$184$	−2.55058e7	−0.301840
$185$	9.73769e7	1.13072
$186$	−1.86002e7	−0.211944
$187$	1.96163e7	0.219367
$188$	2.54269e7	0.279088
$189$	−6.75127e6	−0.0727393
$190$	−5.02553e7	−0.531550
$191$	−4.53862e7	−0.471310	−0.235655	−	0.971837i	$-0.575724\pi$
$191$	−4.53862e7	−0.471310	−0.235655	+	0.971837i	$0.575724\pi$
$192$	−7.07789e6	−0.0721688
$193$	−2.92030e7	−0.292399	−0.146200	−	0.989255i	$-0.546704\pi$
$193$	−2.92030e7	−0.292399	−0.146200	+	0.989255i	$0.546704\pi$
$194$	1.12087e8	1.10217
$195$	−6.76068e7	−0.652934
$196$	7.52954e6	0.0714286
$197$	1.17638e8	1.09626	0.548131	−	0.836393i	$-0.315340\pi$
$197$	1.17638e8	1.09626	0.548131	+	0.836393i	$0.315340\pi$
$198$	−7.76239e6	−0.0710669
$199$	−6.21347e7	−0.558918	−0.279459	−	0.960158i	$-0.590155\pi$
$199$	−6.21347e7	−0.558918	−0.279459	+	0.960158i	$0.590155\pi$
$200$	2.30405e7	0.203651
$201$	8.71669e7	0.757121
$202$	5.87425e7	0.501444
$203$	4.84227e7	0.406268
$204$	−2.54673e7	−0.210028
$205$	1.28907e8	1.04505
$206$	7.08824e7	0.564941
$207$	3.63159e7	0.284577
$208$	5.63528e7	0.434204
$209$	4.59408e7	0.348086
$210$	1.34840e7	0.100474
$211$	−1.84022e8	−1.34860	−0.674299	−	0.738459i	$-0.735554\pi$
$211$	−1.84022e8	−1.34860	−0.674299	+	0.738459i	$0.735554\pi$
$212$	1.43871e7	0.103704
$213$	5.05304e7	0.358281
$214$	1.20148e8	0.838046
$215$	2.95371e7	0.202691
$216$	1.00777e7	0.0680414
$217$	−2.95364e7	−0.196222
$218$	−8.50008e7	−0.555680
$219$	2.53136e7	0.162855
$220$	1.55035e7	0.0981635
$221$	2.02765e8	1.26363
$222$	1.15568e8	0.708929
$223$	8.81072e7	0.532040	0.266020	−	0.963967i	$-0.414291\pi$
$223$	8.81072e7	0.532040	0.266020	+	0.963967i	$0.414291\pi$
$224$	−1.12394e7	−0.0668153
$225$	−3.28057e7	−0.192004
$226$	5.67448e7	0.326999
$227$	2.75878e8	1.56541	0.782703	−	0.622395i	$-0.213840\pi$
$227$	2.75878e8	1.56541	0.782703	+	0.622395i	$0.213840\pi$
$228$	−5.96436e7	−0.333267
$229$	−8.28916e7	−0.456128	−0.228064	−	0.973646i	$-0.573239\pi$
$229$	−8.28916e7	−0.456128	−0.228064	+	0.973646i	$0.573239\pi$
$230$	−7.25321e7	−0.393082
$231$	−1.23264e7	−0.0657952
$232$	−7.22811e7	−0.380029
$233$	−2.36402e8	−1.22435	−0.612175	−	0.790722i	$-0.709705\pi$
$233$	−2.36402e8	−1.22435	−0.612175	+	0.790722i	$0.709705\pi$
$234$	−8.02367e7	−0.409371
$235$	7.23079e7	0.363453
$236$	4.44588e7	0.220174
$237$	−3.41829e7	−0.166797
$238$	−4.04411e7	−0.194448
$239$	2.78052e8	1.31745	0.658725	−	0.752384i	$-0.271096\pi$
$239$	2.78052e8	1.31745	0.658725	+	0.752384i	$0.271096\pi$
$240$	−2.01277e7	−0.0939844
$241$	−1.48269e8	−0.682324	−0.341162	−	0.940004i	$-0.610821\pi$
$241$	−1.48269e8	−0.682324	−0.341162	+	0.940004i	$0.610821\pi$
$242$	−1.41725e7	−0.0642824
$243$	−1.43489e7	−0.0641500
$244$	3.20704e6	0.0141332
$245$	2.14121e7	0.0930204
$246$	1.52989e8	0.655220
$247$	4.74871e8	2.00510
$248$	4.40893e7	0.183549
$249$	6.87151e7	0.282068
$250$	1.79271e8	0.725640
$251$	2.89064e8	1.15381	0.576906	−	0.816810i	$-0.304260\pi$
$251$	2.89064e8	1.15381	0.576906	+	0.816810i	$0.304260\pi$
$252$	1.60030e7	0.0629941
$253$	6.63051e7	0.257410
$254$	−2.88315e8	−1.10395
$255$	−7.24225e7	−0.273516
$256$	1.67772e7	0.0625000
$257$	−1.99415e8	−0.732813	−0.366406	−	0.930455i	$-0.619412\pi$
$257$	−1.99415e8	−0.732813	−0.366406	+	0.930455i	$0.619412\pi$
$258$	3.50551e7	0.127081
$259$	1.83518e8	0.656341
$260$	1.60253e8	0.565457
$261$	1.02916e8	0.358295
$262$	−1.80510e8	−0.620079
$263$	−4.04763e8	−1.37200	−0.686002	−	0.727600i	$-0.740636\pi$
$263$	−4.04763e8	−1.37200	−0.686002	+	0.727600i	$0.740636\pi$
$264$	1.83997e7	0.0615457
$265$	4.09132e7	0.135053
$266$	−9.47119e7	−0.308545
$267$	2.67565e8	0.860281
$268$	−2.06618e8	−0.655686
$269$	4.35356e7	0.136368	0.0681838	−	0.997673i	$-0.478280\pi$
$269$	4.35356e7	0.136368	0.0681838	+	0.997673i	$0.478280\pi$
$270$	2.86584e7	0.0886093
$271$	−2.64981e8	−0.808765	−0.404382	−	0.914590i	$-0.632514\pi$
$271$	−2.64981e8	−0.808765	−0.404382	+	0.914590i	$0.632514\pi$
$272$	6.03668e7	0.181889
$273$	−1.27413e8	−0.379004
$274$	−9.95691e7	−0.292414
$275$	−5.98963e7	−0.173674
$276$	−8.60820e7	−0.246451
$277$	6.49499e8	1.83611	0.918057	−	0.396449i	$-0.129758\pi$
$277$	6.49499e8	1.83611	0.918057	+	0.396449i	$0.129758\pi$
$278$	−1.30640e7	−0.0364687
$279$	−6.27756e7	−0.173052
$280$	−3.19621e7	−0.0870126
$281$	1.75806e8	0.472674	0.236337	−	0.971671i	$-0.424053\pi$
$281$	1.75806e8	0.472674	0.236337	+	0.971671i	$0.424053\pi$
$282$	8.58159e7	0.227875
$283$	−1.21587e8	−0.318885	−0.159443	−	0.987207i	$-0.550970\pi$
$283$	−1.21587e8	−0.318885	−0.159443	+	0.987207i	$0.550970\pi$
$284$	−1.19776e8	−0.310281
$285$	−1.69612e8	−0.434009
$286$	−1.46495e8	−0.370290
$287$	2.42941e8	0.606616
$288$	−2.38879e7	−0.0589256
$289$	−1.93130e8	−0.470660
$290$	−2.05549e8	−0.494906
$291$	3.78292e8	0.899916
$292$	−6.00027e7	−0.141036
$293$	5.27431e8	1.22498	0.612490	−	0.790478i	$-0.290168\pi$
$293$	5.27431e8	1.22498	0.612490	+	0.790478i	$0.290168\pi$
$294$	2.54122e7	0.0583212
$295$	1.26430e8	0.286729
$296$	−2.73939e8	−0.613951
$297$	−2.61981e7	−0.0580259
$298$	−5.40105e8	−1.18228
$299$	6.85369e8	1.48277
$300$	7.77617e7	0.166281
$301$	5.56662e7	0.117654
$302$	4.98574e8	1.04161
$303$	1.98256e8	0.409427
$304$	1.41378e8	0.288618
$305$	9.12002e6	0.0184054
$306$	−8.59520e7	−0.171487
$307$	1.23804e8	0.244203	0.122101	−	0.992518i	$-0.461037\pi$
$307$	1.23804e8	0.244203	0.122101	+	0.992518i	$0.461037\pi$
$308$	2.92181e7	0.0569803
$309$	2.39228e8	0.461272
$310$	1.25379e8	0.239034
$311$	5.17448e8	0.975451	0.487726	−	0.872997i	$-0.337827\pi$
$311$	5.17448e8	0.975451	0.487726	+	0.872997i	$0.337827\pi$
$312$	1.90191e8	0.354526
$313$	−9.84678e7	−0.181505	−0.0907526	−	0.995873i	$-0.528927\pi$
$313$	−9.84678e7	−0.181505	−0.0907526	+	0.995873i	$0.528927\pi$
$314$	6.42217e8	1.17065
$315$	4.55086e7	0.0820363
$316$	8.10260e7	0.144451
$317$	3.52106e8	0.620820	0.310410	−	0.950603i	$-0.399534\pi$
$317$	3.52106e8	0.620820	0.310410	+	0.950603i	$0.399534\pi$
$318$	4.85564e7	0.0846742
$319$	1.87903e8	0.324090
$320$	4.77102e7	0.0813929
$321$	4.05499e8	0.684261
$322$	−1.36695e8	−0.228169
$323$	5.08697e8	0.839944
$324$	3.40122e7	0.0555556
$325$	−6.19124e8	−1.00043
$326$	9.42795e7	0.150715
$327$	−2.86878e8	−0.453711
$328$	−3.62640e8	−0.567437
$329$	1.36273e8	0.210971
$330$	5.23243e7	0.0801502
$331$	−5.36200e8	−0.812697	−0.406349	−	0.913718i	$-0.633198\pi$
$331$	−5.36200e8	−0.812697	−0.406349	+	0.913718i	$0.633198\pi$
$332$	−1.62880e8	−0.244278
$333$	3.90043e8	0.578838
$334$	4.41443e8	0.648279
$335$	−5.87570e8	−0.853891
$336$	−3.79331e7	−0.0545545
$337$	−7.28923e8	−1.03747	−0.518737	−	0.854934i	$-0.673598\pi$
$337$	−7.28923e8	−1.03747	−0.518737	+	0.854934i	$0.673598\pi$
$338$	−1.01227e9	−1.42590
$339$	1.91514e8	0.266994
$340$	1.71668e8	0.236872
$341$	−1.14615e8	−0.156531
$342$	−2.01297e8	−0.272111
$343$	4.03536e7	0.0539949
$344$	−8.30935e7	−0.110056
$345$	−2.44796e8	−0.320950
$346$	−4.51162e8	−0.585553
$347$	8.08560e8	1.03887	0.519433	−	0.854511i	$-0.326143\pi$
$347$	8.08560e8	1.03887	0.519433	+	0.854511i	$0.326143\pi$
$348$	−2.43949e8	−0.310292
$349$	−6.65358e8	−0.837850	−0.418925	−	0.908021i	$-0.637593\pi$
$349$	−6.65358e8	−0.837850	−0.418925	+	0.908021i	$0.637593\pi$
$350$	1.23483e8	0.153946
$351$	−2.70799e8	−0.334250
$352$	−4.36142e7	−0.0533002
$353$	−6.66617e7	−0.0806613	−0.0403306	−	0.999186i	$-0.512841\pi$
$353$	−6.66617e7	−0.0806613	−0.0403306	+	0.999186i	$0.512841\pi$
$354$	1.50048e8	0.179771
$355$	−3.40612e8	−0.404074
$356$	−6.34229e8	−0.745026
$357$	−1.36489e8	−0.158766
$358$	−8.26494e7	−0.0952027
$359$	−4.00001e8	−0.456279	−0.228140	−	0.973628i	$-0.573264\pi$
$359$	−4.00001e8	−0.456279	−0.228140	+	0.973628i	$0.573264\pi$
$360$	−6.79311e7	−0.0767379
$361$	2.97483e8	0.332802
$362$	−5.33793e8	−0.591416
$363$	−4.78321e7	−0.0524864
$364$	3.02016e8	0.328227
$365$	−1.70633e8	−0.183670
$366$	1.08238e7	0.0115397
$367$	−3.49648e8	−0.369232	−0.184616	−	0.982811i	$-0.559104\pi$
$367$	−3.49648e8	−0.369232	−0.184616	+	0.982811i	$0.559104\pi$
$368$	2.04046e8	0.213433
$369$	5.16338e8	0.534985
$370$	−7.79015e8	−0.799540
$371$	7.71057e7	0.0783931
$372$	1.48802e8	0.149867
$373$	3.23621e8	0.322891	0.161445	−	0.986882i	$-0.448384\pi$
$373$	3.23621e8	0.322891	0.161445	+	0.986882i	$0.448384\pi$
$374$	−1.56930e8	−0.155116
$375$	6.05041e8	0.592482
$376$	−2.03416e8	−0.197345
$377$	1.94227e9	1.86688
$378$	5.40102e7	0.0514344
$379$	−6.49660e8	−0.612984	−0.306492	−	0.951873i	$-0.599155\pi$
$379$	−6.49660e8	−0.612984	−0.306492	+	0.951873i	$0.599155\pi$
$380$	4.02042e8	0.375863
$381$	−9.73064e8	−0.901372
$382$	3.63090e8	0.333267
$383$	1.11697e9	1.01589	0.507946	−	0.861389i	$-0.330405\pi$
$383$	1.11697e9	1.01589	0.507946	+	0.861389i	$0.330405\pi$
$384$	5.66231e7	0.0510310
$385$	8.30890e7	0.0742046
$386$	2.33624e8	0.206758
$387$	1.18311e8	0.103762
$388$	−8.96693e8	−0.779350
$389$	1.18772e9	1.02303	0.511517	−	0.859273i	$-0.329084\pi$
$389$	1.18772e9	1.02303	0.511517	+	0.859273i	$0.329084\pi$
$390$	5.40854e8	0.461694
$391$	7.34188e8	0.621139
$392$	−6.02363e7	−0.0505076
$393$	−6.09221e8	−0.506292
$394$	−9.41100e8	−0.775174
$395$	2.30418e8	0.188116
$396$	6.20991e7	0.0502519
$397$	2.20243e9	1.76659	0.883293	−	0.468822i	$-0.155321\pi$
$397$	2.20243e9	1.76659	0.883293	+	0.468822i	$0.155321\pi$
$398$	4.97078e8	0.395215
$399$	−3.19653e8	−0.251926
$400$	−1.84324e8	−0.144003
$401$	8.22757e8	0.637186	0.318593	−	0.947892i	$-0.396790\pi$
$401$	8.22757e8	0.637186	0.318593	+	0.947892i	$0.396790\pi$
$402$	−6.97335e8	−0.535366
$403$	−1.18473e9	−0.901678
$404$	−4.69940e8	−0.354574
$405$	9.67223e7	0.0723492
$406$	−3.87381e8	−0.287275
$407$	7.12136e8	0.523579
$408$	2.03738e8	0.148512
$409$	9.01095e8	0.651237	0.325618	−	0.945501i	$-0.394428\pi$
$409$	9.01095e8	0.651237	0.325618	+	0.945501i	$0.394428\pi$
$410$	−1.03126e9	−0.738965
$411$	−3.36046e8	−0.238755
$412$	−5.67059e8	−0.399473
$413$	2.38271e8	0.166436
$414$	−2.90527e8	−0.201227
$415$	−4.63191e8	−0.318120
$416$	−4.50822e8	−0.307028
$417$	−4.40911e7	−0.0297766
$418$	−3.67526e8	−0.246134
$419$	−1.76829e9	−1.17437	−0.587184	−	0.809453i	$-0.699764\pi$
$419$	−1.76829e9	−1.17437	−0.587184	+	0.809453i	$0.699764\pi$
$420$	−1.07872e8	−0.0710455
$421$	4.56926e8	0.298441	0.149220	−	0.988804i	$-0.452324\pi$
$421$	4.56926e8	0.298441	0.149220	+	0.988804i	$0.452324\pi$
$422$	1.47218e9	0.953602
$423$	2.89629e8	0.186059
$424$	−1.15097e8	−0.0733300
$425$	−6.63225e8	−0.419082
$426$	−4.04243e8	−0.253343
$427$	1.71877e7	0.0106837
$428$	−9.61182e8	−0.592588
$429$	−4.94421e8	−0.302341
$430$	−2.36297e8	−0.143324
$431$	−1.56635e9	−0.942366	−0.471183	−	0.882036i	$-0.656173\pi$
$431$	−1.56635e9	−0.942366	−0.471183	+	0.882036i	$0.656173\pi$
$432$	−8.06216e7	−0.0481125
$433$	−1.70307e9	−1.00815	−0.504075	−	0.863660i	$-0.668166\pi$
$433$	−1.70307e9	−1.00815	−0.504075	+	0.863660i	$0.668166\pi$
$434$	2.36291e8	0.138750
$435$	−6.93729e8	−0.404089
$436$	6.80006e8	0.392925
$437$	1.71945e9	0.985608
$438$	−2.02509e8	−0.115156
$439$	1.64535e9	0.928179	0.464090	−	0.885788i	$-0.346382\pi$
$439$	1.64535e9	0.928179	0.464090	+	0.885788i	$0.346382\pi$
$440$	−1.24028e8	−0.0694121
$441$	8.57661e7	0.0476190
$442$	−1.62212e9	−0.893523
$443$	1.37306e9	0.750372	0.375186	−	0.926950i	$-0.377579\pi$
$443$	1.37306e9	0.750372	0.375186	+	0.926950i	$0.377579\pi$
$444$	−9.24546e8	−0.501289
$445$	−1.80359e9	−0.970236
$446$	−7.04858e8	−0.376209
$447$	−1.82285e9	−0.965329
$448$	8.99154e7	0.0472456
$449$	2.61157e9	1.36157	0.680785	−	0.732483i	$-0.261639\pi$
$449$	2.61157e9	1.36157	0.680785	+	0.732483i	$0.261639\pi$
$450$	2.62446e8	0.135768
$451$	9.42723e8	0.483912
$452$	−4.53959e8	−0.231223
$453$	1.68269e9	0.850472
$454$	−2.20703e9	−1.10691
$455$	8.58857e8	0.427446
$456$	4.77149e8	0.235655
$457$	−3.13057e9	−1.53432	−0.767162	−	0.641453i	$-0.778332\pi$
$457$	−3.13057e9	−1.53432	−0.767162	+	0.641453i	$0.778332\pi$
$458$	6.63133e8	0.322531
$459$	−2.90088e8	−0.140019
$460$	5.80257e8	0.277951
$461$	−1.69540e9	−0.805968	−0.402984	−	0.915207i	$-0.632027\pi$
$461$	−1.69540e9	−0.805968	−0.402984	+	0.915207i	$0.632027\pi$
$462$	9.86111e7	0.0465242
$463$	−8.34761e8	−0.390867	−0.195433	−	0.980717i	$-0.562611\pi$
$463$	−8.34761e8	−0.390867	−0.195433	+	0.980717i	$0.562611\pi$
$464$	5.78249e8	0.268721
$465$	4.23154e8	0.195170
$466$	1.89122e9	0.865746
$467$	−1.12459e9	−0.510960	−0.255480	−	0.966814i	$-0.582233\pi$
$467$	−1.12459e9	−0.510960	−0.255480	+	0.966814i	$0.582233\pi$
$468$	6.41893e8	0.289469
$469$	−1.10734e9	−0.495652
$470$	−5.78463e8	−0.257000
$471$	2.16748e9	0.955834
$472$	−3.55670e8	−0.155686
$473$	2.16011e8	0.0938558
$474$	2.73463e8	0.117944
$475$	−1.55325e9	−0.664990
$476$	3.23529e8	0.137495
$477$	1.63878e8	0.0691362
$478$	−2.22442e9	−0.931578
$479$	−3.77226e9	−1.56830	−0.784148	−	0.620574i	$-0.786899\pi$
$479$	−3.77226e9	−1.56830	−0.784148	+	0.620574i	$0.786899\pi$
$480$	1.61022e8	0.0664570
$481$	7.36105e9	3.01601
$482$	1.18615e9	0.482476
$483$	−4.61346e8	−0.186300
$484$	1.13380e8	0.0454545
$485$	−2.54997e9	−1.01494
$486$	1.14791e8	0.0453609
$487$	−4.40349e9	−1.72761	−0.863805	−	0.503827i	$-0.831925\pi$
$487$	−4.40349e9	−1.72761	−0.863805	+	0.503827i	$0.831925\pi$
$488$	−2.56563e7	−0.00999367
$489$	3.18193e8	0.123058
$490$	−1.71297e8	−0.0657754
$491$	−2.61532e9	−0.997102	−0.498551	−	0.866860i	$-0.666134\pi$
$491$	−2.61532e9	−0.997102	−0.498551	+	0.866860i	$0.666134\pi$
$492$	−1.22391e9	−0.463310
$493$	2.08062e9	0.782040
$494$	−3.79897e9	−1.41782
$495$	1.76594e8	0.0654423
$496$	−3.52715e8	−0.129789
$497$	−6.41923e8	−0.234550
$498$	−5.49721e8	−0.199453
$499$	2.15541e9	0.776566	0.388283	−	0.921540i	$-0.373068\pi$
$499$	2.15541e9	0.776566	0.388283	+	0.921540i	$0.373068\pi$
$500$	−1.43417e9	−0.513105
$501$	1.48987e9	0.529317
$502$	−2.31251e9	−0.815869
$503$	−4.75527e9	−1.66605	−0.833023	−	0.553238i	$-0.813392\pi$
$503$	−4.75527e9	−1.66605	−0.833023	+	0.553238i	$0.813392\pi$
$504$	−1.28024e8	−0.0445435
$505$	−1.33639e9	−0.461757
$506$	−5.30441e8	−0.182016
$507$	−3.41642e9	−1.16424
$508$	2.30652e9	0.780611
$509$	−3.04257e9	−1.02265	−0.511326	−	0.859387i	$-0.670845\pi$
$509$	−3.04257e9	−1.02265	−0.511326	+	0.859387i	$0.670845\pi$
$510$	5.79380e8	0.193405
$511$	−3.21577e8	−0.106613
$512$	−1.34218e8	−0.0441942
$513$	−6.79378e8	−0.222178
$514$	1.59532e9	0.518177
$515$	−1.61257e9	−0.520229
$516$	−2.80441e8	−0.0898601
$517$	5.28801e8	0.168297
$518$	−1.46814e9	−0.464103
$519$	−1.52267e9	−0.478102
$520$	−1.28203e9	−0.399839
$521$	−1.81380e9	−0.561897	−0.280949	−	0.959723i	$-0.590649\pi$
$521$	−1.81380e9	−0.561897	−0.280949	+	0.959723i	$0.590649\pi$
$522$	−8.23327e8	−0.253353
$523$	4.85164e8	0.148297	0.0741485	−	0.997247i	$-0.476376\pi$
$523$	4.85164e8	0.148297	0.0741485	+	0.997247i	$0.476376\pi$
$524$	1.44408e9	0.438462
$525$	4.16754e8	0.125696
$526$	3.23810e9	0.970153
$527$	−1.26912e9	−0.377716
$528$	−1.47198e8	−0.0435194
$529$	−9.23192e8	−0.271142
$530$	−3.27306e8	−0.0954967
$531$	5.06413e8	0.146782
$532$	7.57695e8	0.218174
$533$	9.74454e9	2.78751
$534$	−2.14052e9	−0.608311
$535$	−2.73336e9	−0.771719
$536$	1.65294e9	0.463640
$537$	−2.78942e8	−0.0777327
$538$	−3.48285e8	−0.0964265
$539$	1.56591e8	0.0430730
$540$	−2.29268e8	−0.0626563
$541$	−4.45922e9	−1.21079	−0.605394	−	0.795926i	$-0.706985\pi$
$541$	−4.45922e9	−1.21079	−0.605394	+	0.795926i	$0.706985\pi$
$542$	2.11985e9	0.571883
$543$	−1.80155e9	−0.482889
$544$	−4.82935e8	−0.128615
$545$	1.93377e9	0.511701
$546$	1.01930e9	0.267996
$547$	−3.95780e9	−1.03395	−0.516973	−	0.856002i	$-0.672941\pi$
$547$	−3.95780e9	−1.03395	−0.516973	+	0.856002i	$0.672941\pi$
$548$	7.96553e8	0.206768
$549$	3.65302e7	0.00942212
$550$	4.79171e8	0.122806
$551$	4.87276e9	1.24092
$552$	6.88656e8	0.174267
$553$	4.34249e8	0.109195
$554$	−5.19599e9	−1.29833
$555$	−2.62918e9	−0.652821
$556$	1.04512e8	0.0257873
$557$	2.30018e9	0.563987	0.281994	−	0.959416i	$-0.409004\pi$
$557$	2.30018e9	0.563987	0.281994	+	0.959416i	$0.409004\pi$
$558$	5.02205e8	0.122366
$559$	2.23281e9	0.540644
$560$	2.55697e8	0.0615272
$561$	−5.29640e8	−0.126652
$562$	−1.40645e9	−0.334231
$563$	3.77133e9	0.890667	0.445333	−	0.895365i	$-0.353085\pi$
$563$	3.77133e9	0.890667	0.445333	+	0.895365i	$0.353085\pi$
$564$	−6.86527e8	−0.161132
$565$	−1.29094e9	−0.301119
$566$	9.72695e8	0.225486
$567$	1.82284e8	0.0419961
$568$	9.58206e8	0.219402
$569$	−3.86882e9	−0.880412	−0.440206	−	0.897897i	$-0.645095\pi$
$569$	−3.86882e9	−0.880412	−0.440206	+	0.897897i	$0.645095\pi$
$570$	1.35689e9	0.306891
$571$	2.65511e9	0.596837	0.298418	−	0.954435i	$-0.403541\pi$
$571$	2.65511e9	0.596837	0.298418	+	0.954435i	$0.403541\pi$
$572$	1.17196e9	0.261835
$573$	1.22543e9	0.272111
$574$	−1.94353e9	−0.428942
$575$	−2.24177e9	−0.491760
$576$	1.91103e8	0.0416667
$577$	−6.50529e9	−1.40978	−0.704890	−	0.709317i	$-0.749004\pi$
$577$	−6.50529e9	−1.40978	−0.704890	+	0.709317i	$0.749004\pi$
$578$	1.54504e9	0.332807
$579$	7.88480e8	0.168817
$580$	1.64439e9	0.349952
$581$	−8.72936e8	−0.184657
$582$	−3.02634e9	−0.636337
$583$	2.99206e8	0.0625361
$584$	4.80022e8	0.0997277
$585$	1.82538e9	0.376972
$586$	−4.21945e9	−0.866192
$587$	5.82831e9	1.18935	0.594675	−	0.803966i	$-0.297281\pi$
$587$	5.82831e9	1.18935	0.594675	+	0.803966i	$0.297281\pi$
$588$	−2.03297e8	−0.0412393
$589$	−2.97224e9	−0.599350
$590$	−1.01144e9	−0.202748
$591$	−3.17621e9	−0.632927
$592$	2.19152e9	0.434129
$593$	9.21565e9	1.81482	0.907412	−	0.420243i	$-0.138055\pi$
$593$	9.21565e9	1.81482	0.907412	+	0.420243i	$0.138055\pi$
$594$	2.09585e8	0.0410305
$595$	9.20034e8	0.179058
$596$	4.32084e9	0.835999
$597$	1.67764e9	0.322692
$598$	−5.48295e9	−1.04848
$599$	8.32457e9	1.58259	0.791294	−	0.611436i	$-0.209408\pi$
$599$	8.32457e9	1.58259	0.791294	+	0.611436i	$0.209408\pi$
$600$	−6.22094e8	−0.117578
$601$	−1.80533e9	−0.339230	−0.169615	−	0.985510i	$-0.554252\pi$
$601$	−1.80533e9	−0.339230	−0.169615	+	0.985510i	$0.554252\pi$
$602$	−4.45329e8	−0.0831943
$603$	−2.35351e9	−0.437124
$604$	−3.98859e9	−0.736530
$605$	3.22424e8	0.0591948
$606$	−1.58605e9	−0.289509
$607$	9.91327e9	1.79911	0.899553	−	0.436812i	$-0.143892\pi$
$607$	9.91327e9	1.79911	0.899553	+	0.436812i	$0.143892\pi$
$608$	−1.13102e9	−0.204083
$609$	−1.30741e9	−0.234559
$610$	−7.29602e7	−0.0130146
$611$	5.46600e9	0.969450
$612$	6.87616e8	0.121260
$613$	−1.21157e9	−0.212440	−0.106220	−	0.994343i	$-0.533875\pi$
$613$	−1.21157e9	−0.212440	−0.106220	+	0.994343i	$0.533875\pi$
$614$	−9.90432e8	−0.172677
$615$	−3.48050e9	−0.603363
$616$	−2.33745e8	−0.0402911
$617$	−9.57641e8	−0.164136	−0.0820681	−	0.996627i	$-0.526153\pi$
$617$	−9.57641e8	−0.164136	−0.0820681	+	0.996627i	$0.526153\pi$
$618$	−1.91382e9	−0.326169
$619$	−4.81235e9	−0.815531	−0.407765	−	0.913087i	$-0.633692\pi$
$619$	−4.81235e9	−0.815531	−0.407765	+	0.913087i	$0.633692\pi$
$620$	−1.00303e9	−0.169022
$621$	−9.80528e8	−0.164301
$622$	−4.13959e9	−0.689748
$623$	−3.39907e9	−0.563186
$624$	−1.52152e9	−0.250688
$625$	−5.62722e8	−0.0921965
$626$	7.87743e8	0.128344
$627$	−1.24040e9	−0.200968
$628$	−5.13774e9	−0.827777
$629$	7.88539e9	1.26341
$630$	−3.64068e8	−0.0580084
$631$	−5.18280e9	−0.821224	−0.410612	−	0.911810i	$-0.634685\pi$
$631$	−5.18280e9	−0.821224	−0.410612	+	0.911810i	$0.634685\pi$
$632$	−6.48208e8	−0.102142
$633$	4.96861e9	0.778613
$634$	−2.81685e9	−0.438986
$635$	6.55917e9	1.01658
$636$	−3.88451e8	−0.0598737
$637$	1.61861e9	0.248116
$638$	−1.50322e9	−0.229166
$639$	−1.36432e9	−0.206854
$640$	−3.81682e8	−0.0575535
$641$	−2.63106e9	−0.394574	−0.197287	−	0.980346i	$-0.563213\pi$
$641$	−2.63106e9	−0.394574	−0.197287	+	0.980346i	$0.563213\pi$
$642$	−3.24399e9	−0.483846
$643$	−4.32469e9	−0.641530	−0.320765	−	0.947159i	$-0.603940\pi$
$643$	−4.32469e9	−0.641530	−0.320765	+	0.947159i	$0.603940\pi$
$644$	1.09356e9	0.161340
$645$	−7.97503e8	−0.117024
$646$	−4.06957e9	−0.593930
$647$	−4.34323e9	−0.630446	−0.315223	−	0.949018i	$-0.602079\pi$
$647$	−4.34323e9	−0.630446	−0.315223	+	0.949018i	$0.602079\pi$
$648$	−2.72098e8	−0.0392837
$649$	9.24603e8	0.132770
$650$	4.95299e9	0.707409
$651$	7.97483e8	0.113289
$652$	−7.54236e8	−0.106571
$653$	4.59143e9	0.645285	0.322643	−	0.946521i	$-0.395429\pi$
$653$	4.59143e9	0.645285	0.322643	+	0.946521i	$0.395429\pi$
$654$	2.29502e9	0.320822
$655$	4.10660e9	0.571003
$656$	2.90112e9	0.401239
$657$	−6.83468e8	−0.0940242
$658$	−1.09018e9	−0.149179
$659$	3.99244e9	0.543424	0.271712	−	0.962379i	$-0.412410\pi$
$659$	3.99244e9	0.543424	0.271712	+	0.962379i	$0.412410\pi$
$660$	−4.18594e8	−0.0566747
$661$	−4.95805e9	−0.667737	−0.333869	−	0.942620i	$-0.608354\pi$
$661$	−4.95805e9	−0.667737	−0.333869	+	0.942620i	$0.608354\pi$
$662$	4.28960e9	0.574664
$663$	−5.47467e9	−0.729559
$664$	1.30304e9	0.172731
$665$	2.15470e9	0.284125
$666$	−3.12034e9	−0.409301
$667$	7.03272e9	0.917663
$668$	−3.53154e9	−0.458402
$669$	−2.37889e9	−0.307174
$670$	4.70056e9	0.603792
$671$	6.66964e7	0.00852263
$672$	3.03464e8	0.0385758
$673$	2.86855e8	0.0362751	0.0181376	−	0.999836i	$-0.494226\pi$
$673$	2.86855e8	0.0362751	0.0181376	+	0.999836i	$0.494226\pi$
$674$	5.83138e9	0.733605
$675$	8.85755e8	0.110854
$676$	8.09818e9	1.00826
$677$	7.36156e8	0.0911821	0.0455911	−	0.998960i	$-0.485483\pi$
$677$	7.36156e8	0.0911821	0.0455911	+	0.998960i	$0.485483\pi$
$678$	−1.53211e9	−0.188793
$679$	−4.80571e9	−0.589134
$680$	−1.37335e9	−0.167494
$681$	−7.44872e9	−0.903788
$682$	9.16921e8	0.110684
$683$	2.38050e8	0.0285888	0.0142944	−	0.999898i	$-0.495450\pi$
$683$	2.38050e8	0.0285888	0.0142944	+	0.999898i	$0.495450\pi$
$684$	1.61038e9	0.192412
$685$	2.26520e9	0.269271
$686$	−3.22829e8	−0.0381802
$687$	2.23807e9	0.263345
$688$	6.64748e8	0.0778211
$689$	3.09277e9	0.360230
$690$	1.95837e9	0.226946
$691$	2.18434e9	0.251852	0.125926	−	0.992040i	$-0.459810\pi$
$691$	2.18434e9	0.251852	0.125926	+	0.992040i	$0.459810\pi$
$692$	3.60929e9	0.414048
$693$	3.32813e8	0.0379869
$694$	−6.46848e9	−0.734589
$695$	2.97207e8	0.0335824
$696$	1.95159e9	0.219410
$697$	1.04387e10	1.16770
$698$	5.32286e9	0.592449
$699$	6.38286e9	0.706879
$700$	−9.87862e8	−0.108856
$701$	1.33891e10	1.46804	0.734019	−	0.679129i	$-0.237643\pi$
$701$	1.33891e10	1.46804	0.734019	+	0.679129i	$0.237643\pi$
$702$	2.16639e9	0.236351
$703$	1.84674e10	2.00476
$704$	3.48914e8	0.0376889
$705$	−1.95231e9	−0.209840
$706$	5.33294e8	0.0570361
$707$	−2.51858e9	−0.268033
$708$	−1.20039e9	−0.127117
$709$	1.20468e10	1.26944	0.634718	−	0.772744i	$-0.281116\pi$
$709$	1.20468e10	1.26944	0.634718	+	0.772744i	$0.281116\pi$
$710$	2.72490e9	0.285724
$711$	9.22937e8	0.0963005
$712$	5.07383e9	0.526813
$713$	−4.28976e9	−0.443220
$714$	1.09191e9	0.112265
$715$	3.33277e9	0.340984
$716$	6.61196e8	0.0673185
$717$	−7.50742e9	−0.760630
$718$	3.20001e9	0.322638
$719$	−1.40604e9	−0.141074	−0.0705368	−	0.997509i	$-0.522471\pi$
$719$	−1.40604e9	−0.141074	−0.0705368	+	0.997509i	$0.522471\pi$
$720$	5.43449e8	0.0542619
$721$	−3.03908e9	−0.301974
$722$	−2.37986e9	−0.235327
$723$	4.00327e9	0.393940
$724$	4.27034e9	0.418194
$725$	−6.35297e9	−0.619147
$726$	3.82657e8	0.0371135
$727$	2.21138e9	0.213448	0.106724	−	0.994289i	$-0.465964\pi$
$727$	2.21138e9	0.213448	0.106724	+	0.994289i	$0.465964\pi$
$728$	−2.41612e9	−0.232092
$729$	3.87420e8	0.0370370
$730$	1.36506e9	0.129874
$731$	2.39186e9	0.226477
$732$	−8.65901e7	−0.00815980
$733$	−6.28279e9	−0.589235	−0.294618	−	0.955615i	$-0.595192\pi$
$733$	−6.28279e9	−0.589235	−0.294618	+	0.955615i	$0.595192\pi$
$734$	2.79718e9	0.261087
$735$	−5.78127e8	−0.0537054
$736$	−1.63237e9	−0.150920
$737$	−4.29701e9	−0.395394
$738$	−4.13070e9	−0.378291
$739$	−2.04236e10	−1.86156	−0.930779	−	0.365582i	$-0.880870\pi$
$739$	−2.04236e10	−1.86156	−0.930779	+	0.365582i	$0.880870\pi$
$740$	6.23212e9	0.565360
$741$	−1.28215e10	−1.15765
$742$	−6.16846e8	−0.0554323
$743$	−1.04573e10	−0.935314	−0.467657	−	0.883910i	$-0.654902\pi$
$743$	−1.04573e10	−0.935314	−0.467657	+	0.883910i	$0.654902\pi$
$744$	−1.19041e9	−0.105972
$745$	1.22874e10	1.08871
$746$	−2.58897e9	−0.228318
$747$	−1.85531e9	−0.162852
$748$	1.25544e9	0.109683
$749$	−5.15134e9	−0.447954
$750$	−4.84033e9	−0.418948
$751$	−3.96979e9	−0.342001	−0.171001	−	0.985271i	$-0.554700\pi$
$751$	−3.96979e9	−0.342001	−0.171001	+	0.985271i	$0.554700\pi$
$752$	1.62732e9	0.139544
$753$	−7.80472e9	−0.666154
$754$	−1.55382e10	−1.32008
$755$	−1.13426e10	−0.959173
$756$	−4.32081e8	−0.0363696
$757$	−2.09521e9	−0.175546	−0.0877731	−	0.996140i	$-0.527975\pi$
$757$	−2.09521e9	−0.175546	−0.0877731	+	0.996140i	$0.527975\pi$
$758$	5.19728e9	0.433445
$759$	−1.79024e9	−0.148616
$760$	−3.21634e9	−0.265775
$761$	2.98963e9	0.245907	0.122954	−	0.992412i	$-0.460763\pi$
$761$	2.98963e9	0.245907	0.122954	+	0.992412i	$0.460763\pi$
$762$	7.78451e9	0.637366
$763$	3.64441e9	0.297024
$764$	−2.90472e9	−0.235655
$765$	1.95541e9	0.157915
$766$	−8.93580e9	−0.718344
$767$	9.55724e9	0.764802
$768$	−4.52985e8	−0.0360844
$769$	2.28863e10	1.81482	0.907411	−	0.420245i	$-0.138056\pi$
$769$	2.28863e10	1.81482	0.907411	+	0.420245i	$0.138056\pi$
$770$	−6.64712e8	−0.0524706
$771$	5.38422e9	0.423090
$772$	−1.86899e9	−0.146200
$773$	1.50121e10	1.16900	0.584499	−	0.811394i	$-0.301291\pi$
$773$	1.50121e10	1.16900	0.584499	+	0.811394i	$0.301291\pi$
$774$	−9.46487e8	−0.0733705
$775$	3.87513e9	0.299040
$776$	7.17354e9	0.551084
$777$	−4.95499e9	−0.378939
$778$	−9.50175e9	−0.723394
$779$	2.44471e10	1.85287
$780$	−4.32684e9	−0.326467
$781$	−2.49096e9	−0.187106
$782$	−5.87351e9	−0.439212
$783$	−2.77873e9	−0.206862
$784$	4.81890e8	0.0357143
$785$	−1.46104e10	−1.07800
$786$	4.87377e9	0.358003
$787$	1.23722e10	0.904762	0.452381	−	0.891825i	$-0.350575\pi$
$787$	1.23722e10	0.904762	0.452381	+	0.891825i	$0.350575\pi$
$788$	7.52880e9	0.548131
$789$	1.09286e10	0.792127
$790$	−1.84334e9	−0.133018
$791$	−2.43293e9	−0.174788
$792$	−4.96793e8	−0.0355335
$793$	6.89413e8	0.0490934
$794$	−1.76194e10	−1.24916
$795$	−1.10466e9	−0.0779727
$796$	−3.97662e9	−0.279459
$797$	1.59988e10	1.11939	0.559696	−	0.828698i	$-0.310918\pi$
$797$	1.59988e10	1.11939	0.559696	+	0.828698i	$0.310918\pi$
$798$	2.55722e9	0.178139
$799$	5.85535e9	0.406106
$800$	1.47459e9	0.101826
$801$	−7.22427e9	−0.496684
$802$	−6.58206e9	−0.450559
$803$	−1.24787e9	−0.0850481
$804$	5.57868e9	0.378561
$805$	3.10981e9	0.210111
$806$	9.47783e9	0.637582
$807$	−1.17546e9	−0.0787319
$808$	3.75952e9	0.250722
$809$	1.19166e10	0.791281	0.395641	−	0.918405i	$-0.370523\pi$
$809$	1.19166e10	0.791281	0.395641	+	0.918405i	$0.370523\pi$
$810$	−7.73778e8	−0.0511586
$811$	−2.03252e10	−1.33802	−0.669010	−	0.743254i	$-0.733282\pi$
$811$	−2.03252e10	−1.33802	−0.669010	+	0.743254i	$0.733282\pi$
$812$	3.09905e9	0.203134
$813$	7.15449e9	0.466941
$814$	−5.69708e9	−0.370226
$815$	−2.14486e9	−0.138786
$816$	−1.62990e9	−0.105014
$817$	5.60167e9	0.359369
$818$	−7.20876e9	−0.460494
$819$	3.44015e9	0.218818
$820$	8.25007e9	0.522527
$821$	−8.13232e9	−0.512877	−0.256439	−	0.966561i	$-0.582549\pi$
$821$	−8.13232e9	−0.512877	−0.256439	+	0.966561i	$0.582549\pi$
$822$	2.68837e9	0.168825
$823$	1.27514e10	0.797366	0.398683	−	0.917089i	$-0.369467\pi$
$823$	1.27514e10	0.797366	0.398683	+	0.917089i	$0.369467\pi$
$824$	4.53647e9	0.282470
$825$	1.61720e9	0.100271
$826$	−1.90617e9	−0.117688
$827$	−3.43433e9	−0.211141	−0.105570	−	0.994412i	$-0.533667\pi$
$827$	−3.43433e9	−0.211141	−0.105570	+	0.994412i	$0.533667\pi$
$828$	2.32422e9	0.142289
$829$	3.10696e10	1.89407	0.947033	−	0.321136i	$-0.104065\pi$
$829$	3.10696e10	1.89407	0.947033	+	0.321136i	$0.104065\pi$
$830$	3.70553e9	0.224945
$831$	−1.75365e10	−1.06008
$832$	3.60658e9	0.217102
$833$	1.73391e9	0.103937
$834$	3.52729e8	0.0210552
$835$	−1.00428e10	−0.596971
$836$	2.94021e9	0.174043
$837$	1.69494e9	0.0999116
$838$	1.41463e10	0.830404
$839$	−2.69261e10	−1.57401	−0.787003	−	0.616949i	$-0.788368\pi$
$839$	−2.69261e10	−1.57401	−0.787003	+	0.616949i	$0.788368\pi$
$840$	8.62977e8	0.0502368
$841$	2.68022e9	0.155376
$842$	−3.65541e9	−0.211030
$843$	−4.74676e9	−0.272898
$844$	−1.17774e10	−0.674299
$845$	2.30292e10	1.31305
$846$	−2.31703e9	−0.131564
$847$	6.07645e8	0.0343604
$848$	9.20773e8	0.0518522
$849$	3.28285e9	0.184108
$850$	5.30580e9	0.296336
$851$	2.66535e10	1.48252
$852$	3.23395e9	0.179141
$853$	−2.28862e10	−1.26256	−0.631281	−	0.775554i	$-0.717470\pi$
$853$	−2.28862e10	−1.26256	−0.631281	+	0.775554i	$0.717470\pi$
$854$	−1.37502e8	−0.00755450
$855$	4.57951e9	0.250575
$856$	7.68946e9	0.419023
$857$	−2.94221e10	−1.59676	−0.798382	−	0.602151i	$-0.794311\pi$
$857$	−2.94221e10	−1.59676	−0.798382	+	0.602151i	$0.794311\pi$
$858$	3.95537e9	0.213787
$859$	−1.03932e10	−0.559465	−0.279733	−	0.960078i	$-0.590246\pi$
$859$	−1.03932e10	−0.559465	−0.279733	+	0.960078i	$0.590246\pi$
$860$	1.89038e9	0.101345
$861$	−6.55940e9	−0.350230
$862$	1.25308e10	0.666353
$863$	2.21900e10	1.17522	0.587610	−	0.809144i	$-0.300069\pi$
$863$	2.21900e10	1.17522	0.587610	+	0.809144i	$0.300069\pi$
$864$	6.44973e8	0.0340207
$865$	1.02639e10	0.539209
$866$	1.36246e10	0.712870
$867$	5.21451e9	0.271736
$868$	−1.89033e9	−0.0981112
$869$	1.68509e9	0.0871071
$870$	5.54983e9	0.285734
$871$	−4.44164e10	−2.27761
$872$	−5.44005e9	−0.277840
$873$	−1.02139e10	−0.519567
$874$	−1.37556e10	−0.696930
$875$	−7.68626e9	−0.387871
$876$	1.62007e9	0.0814273
$877$	3.59473e10	1.79957	0.899783	−	0.436337i	$-0.143724\pi$
$877$	3.59473e10	1.79957	0.899783	+	0.436337i	$0.143724\pi$
$878$	−1.31628e10	−0.656322
$879$	−1.42406e10	−0.707243
$880$	9.92223e8	0.0490817
$881$	−2.76295e10	−1.36131	−0.680655	−	0.732605i	$-0.738305\pi$
$881$	−2.76295e10	−1.36131	−0.680655	+	0.732605i	$0.738305\pi$
$882$	−6.86129e8	−0.0336718
$883$	−2.32225e10	−1.13513	−0.567567	−	0.823328i	$-0.692115\pi$
$883$	−2.32225e10	−1.13513	−0.567567	+	0.823328i	$0.692115\pi$
$884$	1.29770e10	0.631816
$885$	−3.41360e9	−0.165543
$886$	−1.09845e10	−0.530593
$887$	2.93323e10	1.41128	0.705641	−	0.708569i	$-0.250659\pi$
$887$	2.93323e10	1.41128	0.705641	+	0.708569i	$0.250659\pi$
$888$	7.39637e9	0.354465
$889$	1.23615e10	0.590086
$890$	1.44287e10	0.686061
$891$	7.07348e8	0.0335013
$892$	5.63886e9	0.266020
$893$	1.37131e10	0.644399
$894$	1.45828e10	0.682591
$895$	1.88027e9	0.0876679
$896$	−7.19323e8	−0.0334077
$897$	−1.85050e10	−0.856080
$898$	−2.08926e10	−0.962775
$899$	−1.21568e10	−0.558032
$900$	−2.09957e9	−0.0960021
$901$	3.31307e9	0.150902
$902$	−7.54179e9	−0.342177
$903$	−1.50299e9	−0.0679279
$904$	3.63167e9	0.163500
$905$	1.21438e10	0.544608
$906$	−1.34615e10	−0.601374
$907$	−1.38883e10	−0.618050	−0.309025	−	0.951054i	$-0.600003\pi$
$907$	−1.38883e10	−0.618050	−0.309025	+	0.951054i	$0.600003\pi$
$908$	1.76562e10	0.782703
$909$	−5.35291e9	−0.236383
$910$	−6.87086e9	−0.302250
$911$	4.83193e9	0.211742	0.105871	−	0.994380i	$-0.466237\pi$
$911$	4.83193e9	0.211742	0.105871	+	0.994380i	$0.466237\pi$
$912$	−3.81719e9	−0.166633
$913$	−3.38740e9	−0.147305
$914$	2.50446e10	1.08493
$915$	−2.46241e8	−0.0106264
$916$	−5.30506e9	−0.228064
$917$	7.73937e9	0.331446
$918$	2.32070e9	0.0990080
$919$	1.95192e10	0.829578	0.414789	−	0.909918i	$-0.363855\pi$
$919$	1.95192e10	0.829578	0.414789	+	0.909918i	$0.363855\pi$
$920$	−4.64205e9	−0.196541
$921$	−3.34271e9	−0.140990
$922$	1.35632e10	0.569906
$923$	−2.57480e10	−1.07780
$924$	−7.88889e8	−0.0328976
$925$	−2.40772e10	−1.00025
$926$	6.67809e9	0.276385
$927$	−6.45916e9	−0.266316
$928$	−4.62599e9	−0.190014
$929$	−3.96997e10	−1.62455	−0.812274	−	0.583277i	$-0.801770\pi$
$929$	−3.96997e10	−1.62455	−0.812274	+	0.583277i	$0.801770\pi$
$930$	−3.38523e9	−0.138006
$931$	4.06077e9	0.164924
$932$	−1.51297e10	−0.612175
$933$	−1.39711e10	−0.563177
$934$	8.99675e9	0.361303
$935$	3.57016e9	0.142839
$936$	−5.13515e9	−0.204686
$937$	3.04155e9	0.120783	0.0603916	−	0.998175i	$-0.480765\pi$
$937$	3.04155e9	0.120783	0.0603916	+	0.998175i	$0.480765\pi$
$938$	8.85874e9	0.350479
$939$	2.65863e9	0.104792
$940$	4.62770e9	0.181726
$941$	−7.60593e9	−0.297570	−0.148785	−	0.988870i	$-0.547536\pi$
$941$	−7.60593e9	−0.297570	−0.148785	+	0.988870i	$0.547536\pi$
$942$	−1.73399e10	−0.675877
$943$	3.52838e10	1.37020
$944$	2.84536e9	0.110087
$945$	−1.22873e9	−0.0473637
$946$	−1.72809e9	−0.0663661
$947$	2.22782e10	0.852424	0.426212	−	0.904623i	$-0.359848\pi$
$947$	2.22782e10	0.852424	0.426212	+	0.904623i	$0.359848\pi$
$948$	−2.18770e9	−0.0833987
$949$	−1.28987e10	−0.489908
$950$	1.24260e10	0.470219
$951$	−9.50685e9	−0.358431
$952$	−2.58823e9	−0.0972240
$953$	−4.93836e10	−1.84824	−0.924119	−	0.382106i	$-0.875199\pi$
$953$	−4.93836e10	−1.84824	−0.924119	+	0.382106i	$0.875199\pi$
$954$	−1.31102e9	−0.0488867
$955$	−8.26029e9	−0.306891
$956$	1.77954e10	0.658725
$957$	−5.07337e9	−0.187113
$958$	3.01781e10	1.10895
$959$	4.26903e9	0.156302
$960$	−1.28818e9	−0.0469922
$961$	−2.00973e10	−0.730477
$962$	−5.88884e10	−2.13264
$963$	−1.09485e10	−0.395058
$964$	−9.48922e9	−0.341162
$965$	−5.31494e9	−0.190394
$966$	3.69077e9	0.131734
$967$	−4.59604e8	−0.0163452	−0.00817261	−	0.999967i	$-0.502601\pi$
$967$	−4.59604e8	−0.0163452	−0.00817261	+	0.999967i	$0.502601\pi$
$968$	−9.07039e8	−0.0321412
$969$	−1.37348e10	−0.484942
$970$	2.03998e10	0.717669
$971$	2.51853e9	0.0882836	0.0441418	−	0.999025i	$-0.485945\pi$
$971$	2.51853e9	0.0882836	0.0441418	+	0.999025i	$0.485945\pi$
$972$	−9.18330e8	−0.0320750
$973$	5.60120e8	0.0194933
$974$	3.52279e10	1.22160
$975$	1.67163e10	0.577597
$976$	2.05251e8	0.00706659
$977$	2.47811e10	0.850137	0.425069	−	0.905161i	$-0.360250\pi$
$977$	2.47811e10	0.850137	0.425069	+	0.905161i	$0.360250\pi$
$978$	−2.54555e9	−0.0870151
$979$	−1.31900e10	−0.449267
$980$	1.37038e9	0.0465102
$981$	7.74569e9	0.261950
$982$	2.09226e10	0.705058
$983$	4.99754e10	1.67810	0.839052	−	0.544052i	$-0.183110\pi$
$983$	4.99754e10	1.67810	0.839052	+	0.544052i	$0.183110\pi$
$984$	9.79129e9	0.327610
$985$	2.14100e10	0.713823
$986$	−1.66450e10	−0.552986
$987$	−3.67936e9	−0.121804
$988$	3.03918e10	1.00255
$989$	8.08474e9	0.265753
$990$	−1.41276e9	−0.0462747
$991$	2.07635e9	0.0677708	0.0338854	−	0.999426i	$-0.489212\pi$
$991$	2.07635e9	0.0677708	0.0338854	+	0.999426i	$0.489212\pi$
$992$	2.82172e9	0.0917747
$993$	1.44774e10	0.469211
$994$	5.13539e9	0.165852
$995$	−1.13085e10	−0.363936
$996$	4.39777e9	0.141034
$997$	−1.92580e10	−0.615428	−0.307714	−	0.951479i	$-0.599564\pi$
$997$	−1.92580e10	−0.615428	−0.307714	+	0.951479i	$0.599564\pi$
$998$	−1.72433e10	−0.549115
$999$	−1.05312e10	−0.334193

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	462.8.a.a.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
462.8.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:	\( N \)	\(=\)	\( 462 = 2 \cdot 3 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	462.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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