LMFDB - Embedded newform 546.8.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [546,8,Mod(1,546)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(546, 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("546.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 546 = 2 \cdot 3 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	546.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:	$170.562223914$
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	$\chi$	$=$	546.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} -390.000 q^{5} +216.000 q^{6} -343.000 q^{7} +512.000 q^{8} +729.000 q^{9} +O(q^{10})$

\(q+8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} -390.000 q^{5} +216.000 q^{6} -343.000 q^{7} +512.000 q^{8} +729.000 q^{9} -3120.00 q^{10} -388.000 q^{11} +1728.00 q^{12} -2197.00 q^{13} -2744.00 q^{14} -10530.0 q^{15} +4096.00 q^{16} -37294.0 q^{17} +5832.00 q^{18} +35164.0 q^{19} -24960.0 q^{20} -9261.00 q^{21} -3104.00 q^{22} -102980. q^{23} +13824.0 q^{24} +73975.0 q^{25} -17576.0 q^{26} +19683.0 q^{27} -21952.0 q^{28} +224826. q^{29} -84240.0 q^{30} -150552. q^{31} +32768.0 q^{32} -10476.0 q^{33} -298352. q^{34} +133770. q^{35} +46656.0 q^{36} +306058. q^{37} +281312. q^{38} -59319.0 q^{39} -199680. q^{40} +784994. q^{41} -74088.0 q^{42} -771532. q^{43} -24832.0 q^{44} -284310. q^{45} -823840. q^{46} +653976. q^{47} +110592. q^{48} +117649. q^{49} +591800. q^{50} -1.00694e6 q^{51} -140608. q^{52} -6646.00 q^{53} +157464. q^{54} +151320. q^{55} -175616. q^{56} +949428. q^{57} +1.79861e6 q^{58} +1.37660e6 q^{59} -673920. q^{60} -1.21549e6 q^{61} -1.20442e6 q^{62} -250047. q^{63} +262144. q^{64} +856830. q^{65} -83808.0 q^{66} -3.04181e6 q^{67} -2.38682e6 q^{68} -2.78046e6 q^{69} +1.07016e6 q^{70} +611256. q^{71} +373248. q^{72} +3.53169e6 q^{73} +2.44846e6 q^{74} +1.99733e6 q^{75} +2.25050e6 q^{76} +133084. q^{77} -474552. q^{78} -1.35179e6 q^{79} -1.59744e6 q^{80} +531441. q^{81} +6.27995e6 q^{82} -4.88222e6 q^{83} -592704. q^{84} +1.45447e7 q^{85} -6.17226e6 q^{86} +6.07030e6 q^{87} -198656. q^{88} +6.89375e6 q^{89} -2.27448e6 q^{90} +753571. q^{91} -6.59072e6 q^{92} -4.06490e6 q^{93} +5.23181e6 q^{94} -1.37140e7 q^{95} +884736. q^{96} +3.76813e6 q^{97} +941192. q^{98} -282852. q^{99} +O(q^{100})\)

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
$5$	−390.000	−1.39531	−0.697653	−	0.716436i	$-0.745772\pi$
$5$	−390.000	−1.39531	−0.697653	+	0.716436i	$0.745772\pi$
$6$	216.000	0.408248
$7$	−343.000	−0.377964
$7$	−343.000	−0.377964
$8$	512.000	0.353553
$9$	729.000	0.333333
$10$	−3120.00	−0.986631
$11$	−388.000	−0.0878936	−0.0439468	−	0.999034i	$-0.513993\pi$
$11$	−388.000	−0.0878936	−0.0439468	+	0.999034i	$0.513993\pi$
$12$	1728.00	0.288675
$13$	−2197.00	−0.277350
$13$	−2197.00	−0.277350
$14$	−2744.00	−0.267261
$15$	−10530.0	−0.805581
$16$	4096.00	0.250000
$17$	−37294.0	−1.84106	−0.920530	−	0.390673i	$-0.872242\pi$
$17$	−37294.0	−1.84106	−0.920530	+	0.390673i	$0.872242\pi$
$18$	5832.00	0.235702
$19$	35164.0	1.17614	0.588072	−	0.808808i	$-0.299887\pi$
$19$	35164.0	1.17614	0.588072	+	0.808808i	$0.299887\pi$
$20$	−24960.0	−0.697653
$21$	−9261.00	−0.218218
$22$	−3104.00	−0.0621502
$23$	−102980.	−1.76484	−0.882420	−	0.470462i	$-0.844087\pi$
$23$	−102980.	−1.76484	−0.882420	+	0.470462i	$0.844087\pi$
$24$	13824.0	0.204124
$25$	73975.0	0.946880
$26$	−17576.0	−0.196116
$27$	19683.0	0.192450
$28$	−21952.0	−0.188982
$29$	224826.	1.71180	0.855901	−	0.517140i	$-0.173003\pi$
$29$	224826.	1.71180	0.855901	+	0.517140i	$0.173003\pi$
$30$	−84240.0	−0.569631
$31$	−150552.	−0.907655	−0.453827	−	0.891090i	$-0.649942\pi$
$31$	−150552.	−0.907655	−0.453827	+	0.891090i	$0.649942\pi$
$32$	32768.0	0.176777
$33$	−10476.0	−0.0507454
$34$	−298352.	−1.30183
$35$	133770.	0.527376
$36$	46656.0	0.166667
$37$	306058.	0.993339	0.496670	−	0.867940i	$-0.334556\pi$
$37$	306058.	0.993339	0.496670	+	0.867940i	$0.334556\pi$
$38$	281312.	0.831660
$39$	−59319.0	−0.160128
$40$	−199680.	−0.493315
$41$	784994.	1.77878	0.889391	−	0.457147i	$-0.151129\pi$
$41$	784994.	1.77878	0.889391	+	0.457147i	$0.151129\pi$
$42$	−74088.0	−0.154303
$43$	−771532.	−1.47984	−0.739919	−	0.672696i	$-0.765136\pi$
$43$	−771532.	−1.47984	−0.739919	+	0.672696i	$0.765136\pi$
$44$	−24832.0	−0.0439468
$45$	−284310.	−0.465102
$46$	−823840.	−1.24793
$47$	653976.	0.918796	0.459398	−	0.888230i	$-0.348065\pi$
$47$	653976.	0.918796	0.459398	+	0.888230i	$0.348065\pi$
$48$	110592.	0.144338
$49$	117649.	0.142857
$50$	591800.	0.669545
$51$	−1.00694e6	−1.06294
$52$	−140608.	−0.138675
$53$	−6646.00	−0.00613190	−0.00306595	−	0.999995i	$-0.500976\pi$
$53$	−6646.00	−0.00613190	−0.00306595	+	0.999995i	$0.500976\pi$
$54$	157464.	0.136083
$55$	151320.	0.122639
$56$	−175616.	−0.133631
$57$	949428.	0.679047
$58$	1.79861e6	1.21043
$59$	1.37660e6	0.872621	0.436311	−	0.899796i	$-0.356285\pi$
$59$	1.37660e6	0.872621	0.436311	+	0.899796i	$0.356285\pi$
$60$	−673920.	−0.402790
$61$	−1.21549e6	−0.685644	−0.342822	−	0.939400i	$-0.611383\pi$
$61$	−1.21549e6	−0.685644	−0.342822	+	0.939400i	$0.611383\pi$
$62$	−1.20442e6	−0.641809
$63$	−250047.	−0.125988
$64$	262144.	0.125000
$65$	856830.	0.386988
$66$	−83808.0	−0.0358824
$67$	−3.04181e6	−1.23558	−0.617789	−	0.786344i	$-0.711971\pi$
$67$	−3.04181e6	−1.23558	−0.617789	+	0.786344i	$0.711971\pi$
$68$	−2.38682e6	−0.920530
$69$	−2.78046e6	−1.01893
$70$	1.07016e6	0.372911
$71$	611256.	0.202684	0.101342	−	0.994852i	$-0.467686\pi$
$71$	611256.	0.202684	0.101342	+	0.994852i	$0.467686\pi$
$72$	373248.	0.117851
$73$	3.53169e6	1.06256	0.531278	−	0.847197i	$-0.321712\pi$
$73$	3.53169e6	1.06256	0.531278	+	0.847197i	$0.321712\pi$
$74$	2.44846e6	0.702397
$75$	1.99733e6	0.546681
$76$	2.25050e6	0.588072
$77$	133084.	0.0332207
$78$	−474552.	−0.113228
$79$	−1.35179e6	−0.308472	−0.154236	−	0.988034i	$-0.549292\pi$
$79$	−1.35179e6	−0.308472	−0.154236	+	0.988034i	$0.549292\pi$
$80$	−1.59744e6	−0.348827
$81$	531441.	0.111111
$82$	6.27995e6	1.25779
$83$	−4.88222e6	−0.937225	−0.468612	−	0.883404i	$-0.655246\pi$
$83$	−4.88222e6	−0.937225	−0.468612	+	0.883404i	$0.655246\pi$
$84$	−592704.	−0.109109
$85$	1.45447e7	2.56884
$86$	−6.17226e6	−1.04640
$87$	6.07030e6	0.988309
$88$	−198656.	−0.0310751
$89$	6.89375e6	1.03655	0.518276	−	0.855214i	$-0.326574\pi$
$89$	6.89375e6	1.03655	0.518276	+	0.855214i	$0.326574\pi$
$90$	−2.27448e6	−0.328877
$91$	753571.	0.104828
$92$	−6.59072e6	−0.882420
$93$	−4.06490e6	−0.524035
$94$	5.23181e6	0.649687
$95$	−1.37140e7	−1.64108
$96$	884736.	0.102062
$97$	3.76813e6	0.419203	0.209601	−	0.977787i	$-0.432783\pi$
$97$	3.76813e6	0.419203	0.209601	+	0.977787i	$0.432783\pi$
$98$	941192.	0.101015
$99$	−282852.	−0.0292979
$100$	4.73440e6	0.473440
$101$	1.81150e7	1.74950	0.874750	−	0.484575i	$-0.161026\pi$
$101$	1.81150e7	1.74950	0.874750	+	0.484575i	$0.161026\pi$
$102$	−8.05550e6	−0.751609
$103$	−3.80619e6	−0.343210	−0.171605	−	0.985166i	$-0.554895\pi$
$103$	−3.80619e6	−0.343210	−0.171605	+	0.985166i	$0.554895\pi$
$104$	−1.12486e6	−0.0980581
$105$	3.61179e6	0.304481
$106$	−53168.0	−0.00433591
$107$	1.52015e7	1.19962	0.599808	−	0.800144i	$-0.295244\pi$
$107$	1.52015e7	1.19962	0.599808	+	0.800144i	$0.295244\pi$
$108$	1.25971e6	0.0962250
$109$	−1.09744e7	−0.811685	−0.405842	−	0.913943i	$-0.633022\pi$
$109$	−1.09744e7	−0.811685	−0.405842	+	0.913943i	$0.633022\pi$
$110$	1.21056e6	0.0867185
$111$	8.26357e6	0.573505
$112$	−1.40493e6	−0.0944911
$113$	1.08721e7	0.708826	0.354413	−	0.935089i	$-0.384681\pi$
$113$	1.08721e7	0.708826	0.354413	+	0.935089i	$0.384681\pi$
$114$	7.59542e6	0.480159
$115$	4.01622e7	2.46249
$116$	1.43889e7	0.855901
$117$	−1.60161e6	−0.0924500
$118$	1.10128e7	0.617036
$119$	1.27918e7	0.695855
$120$	−5.39136e6	−0.284816
$121$	−1.93366e7	−0.992275
$122$	−9.72395e6	−0.484823
$123$	2.11948e7	1.02698
$124$	−9.63533e6	−0.453827
$125$	1.61850e6	0.0741187
$126$	−2.00038e6	−0.0890871
$127$	3.84450e7	1.66543	0.832715	−	0.553702i	$-0.186785\pi$
$127$	3.84450e7	1.66543	0.832715	+	0.553702i	$0.186785\pi$
$128$	2.09715e6	0.0883883
$129$	−2.08314e7	−0.854385
$130$	6.85464e6	0.273642
$131$	1.18236e7	0.459514	0.229757	−	0.973248i	$-0.426207\pi$
$131$	1.18236e7	0.459514	0.229757	+	0.973248i	$0.426207\pi$
$132$	−670464.	−0.0253727
$133$	−1.20613e7	−0.444541
$134$	−2.43345e7	−0.873685
$135$	−7.67637e6	−0.268527
$136$	−1.90945e7	−0.650913
$137$	3.42043e7	1.13647	0.568236	−	0.822866i	$-0.307626\pi$
$137$	3.42043e7	1.13647	0.568236	+	0.822866i	$0.307626\pi$
$138$	−2.22437e7	−0.720493
$139$	3.66211e7	1.15659	0.578295	−	0.815828i	$-0.303718\pi$
$139$	3.66211e7	1.15659	0.578295	+	0.815828i	$0.303718\pi$
$140$	8.56128e6	0.263688
$141$	1.76574e7	0.530467
$142$	4.89005e6	0.143319
$143$	852436.	0.0243773
$144$	2.98598e6	0.0833333
$145$	−8.76821e7	−2.38849
$146$	2.82535e7	0.751341
$147$	3.17652e6	0.0824786
$148$	1.95877e7	0.496670
$149$	3.72332e7	0.922101	0.461051	−	0.887374i	$-0.347473\pi$
$149$	3.72332e7	0.922101	0.461051	+	0.887374i	$0.347473\pi$
$150$	1.59786e7	0.386562
$151$	−3.54939e7	−0.838947	−0.419473	−	0.907768i	$-0.637785\pi$
$151$	−3.54939e7	−0.838947	−0.419473	+	0.907768i	$0.637785\pi$
$152$	1.80040e7	0.415830
$153$	−2.71873e7	−0.613686
$154$	1.06467e6	0.0234906
$155$	5.87153e7	1.26646
$156$	−3.79642e6	−0.0800641
$157$	6.56180e7	1.35324	0.676619	−	0.736334i	$-0.263445\pi$
$157$	6.56180e7	1.35324	0.676619	+	0.736334i	$0.263445\pi$
$158$	−1.08143e7	−0.218122
$159$	−179442.	−0.00354025
$160$	−1.27795e7	−0.246658
$161$	3.53221e7	0.667047
$162$	4.25153e6	0.0785674
$163$	−5.55343e7	−1.00439	−0.502197	−	0.864753i	$-0.667475\pi$
$163$	−5.55343e7	−1.00439	−0.502197	+	0.864753i	$0.667475\pi$
$164$	5.02396e7	0.889391
$165$	4.08564e6	0.0708054
$166$	−3.90577e7	−0.662718
$167$	3.83624e7	0.637379	0.318690	−	0.947859i	$-0.396757\pi$
$167$	3.83624e7	0.637379	0.318690	+	0.947859i	$0.396757\pi$
$168$	−4.74163e6	−0.0771517
$169$	4.82681e6	0.0769231
$170$	1.16357e8	1.81645
$171$	2.56346e7	0.392048
$172$	−4.93780e7	−0.739919
$173$	−3.42583e7	−0.503043	−0.251521	−	0.967852i	$-0.580931\pi$
$173$	−3.42583e7	−0.503043	−0.251521	+	0.967852i	$0.580931\pi$
$174$	4.85624e7	0.698840
$175$	−2.53734e7	−0.357887
$176$	−1.58925e6	−0.0219734
$177$	3.71682e7	0.503808
$178$	5.51500e7	0.732952
$179$	3.11130e7	0.405468	0.202734	−	0.979234i	$-0.435017\pi$
$179$	3.11130e7	0.405468	0.202734	+	0.979234i	$0.435017\pi$
$180$	−1.81958e7	−0.232551
$181$	6.48305e7	0.812652	0.406326	−	0.913728i	$-0.366810\pi$
$181$	6.48305e7	0.812652	0.406326	+	0.913728i	$0.366810\pi$
$182$	6.02857e6	0.0741249
$183$	−3.28183e7	−0.395856
$184$	−5.27258e7	−0.623965
$185$	−1.19363e8	−1.38601
$186$	−3.25192e7	−0.370548
$187$	1.44701e7	0.161817
$188$	4.18545e7	0.459398
$189$	−6.75127e6	−0.0727393
$190$	−1.09712e8	−1.16042
$191$	−5.79611e7	−0.601894	−0.300947	−	0.953641i	$-0.597303\pi$
$191$	−5.79611e7	−0.601894	−0.300947	+	0.953641i	$0.597303\pi$
$192$	7.07789e6	0.0721688
$193$	7.42947e7	0.743887	0.371943	−	0.928255i	$-0.378692\pi$
$193$	7.42947e7	0.743887	0.371943	+	0.928255i	$0.378692\pi$
$194$	3.01450e7	0.296421
$195$	2.31344e7	0.223428
$196$	7.52954e6	0.0714286
$197$	−1.77434e8	−1.65351	−0.826753	−	0.562565i	$-0.809815\pi$
$197$	−1.77434e8	−1.65351	−0.826753	+	0.562565i	$0.809815\pi$
$198$	−2.26282e6	−0.0207167
$199$	−1.27747e8	−1.14912	−0.574561	−	0.818462i	$-0.694827\pi$
$199$	−1.27747e8	−1.14912	−0.574561	+	0.818462i	$0.694827\pi$
$200$	3.78752e7	0.334773
$201$	−8.21288e7	−0.713361
$202$	1.44920e8	1.23708
$203$	−7.71153e7	−0.647000
$204$	−6.44440e7	−0.531468
$205$	−3.06148e8	−2.48195
$206$	−3.04495e7	−0.242686
$207$	−7.50724e7	−0.588280
$208$	−8.99891e6	−0.0693375
$209$	−1.36436e7	−0.103376
$210$	2.88943e7	0.215300
$211$	−7.24196e7	−0.530722	−0.265361	−	0.964149i	$-0.585491\pi$
$211$	−7.24196e7	−0.530722	−0.265361	+	0.964149i	$0.585491\pi$
$212$	−425344.	−0.00306595
$213$	1.65039e7	0.117020
$214$	1.21612e8	0.848257
$215$	3.00897e8	2.06483
$216$	1.00777e7	0.0680414
$217$	5.16393e7	0.343061
$218$	−8.77951e7	−0.573948
$219$	9.53555e7	0.613467
$220$	9.68448e6	0.0613193
$221$	8.19349e7	0.510618
$222$	6.61085e7	0.405529
$223$	1.05028e8	0.634217	0.317108	−	0.948389i	$-0.397288\pi$
$223$	1.05028e8	0.634217	0.317108	+	0.948389i	$0.397288\pi$
$224$	−1.12394e7	−0.0668153
$225$	5.39278e7	0.315627
$226$	8.69770e7	0.501216
$227$	−6.28932e7	−0.356873	−0.178436	−	0.983951i	$-0.557104\pi$
$227$	−6.28932e7	−0.356873	−0.178436	+	0.983951i	$0.557104\pi$
$228$	6.07634e7	0.339524
$229$	2.51053e8	1.38147	0.690736	−	0.723107i	$-0.257287\pi$
$229$	2.51053e8	1.38147	0.690736	+	0.723107i	$0.257287\pi$
$230$	3.21298e8	1.74125
$231$	3.59327e6	0.0191800
$232$	1.15111e8	0.605213
$233$	1.47539e8	0.764121	0.382061	−	0.924137i	$-0.375215\pi$
$233$	1.47539e8	0.764121	0.382061	+	0.924137i	$0.375215\pi$
$234$	−1.28129e7	−0.0653720
$235$	−2.55051e8	−1.28200
$236$	8.81024e7	0.436311
$237$	−3.64984e7	−0.178096
$238$	1.02335e8	0.492044
$239$	−3.36886e7	−0.159621	−0.0798105	−	0.996810i	$-0.525432\pi$
$239$	−3.36886e7	−0.159621	−0.0798105	+	0.996810i	$0.525432\pi$
$240$	−4.31309e7	−0.201395
$241$	−1.19476e8	−0.549821	−0.274911	−	0.961470i	$-0.588648\pi$
$241$	−1.19476e8	−0.549821	−0.274911	+	0.961470i	$0.588648\pi$
$242$	−1.54693e8	−0.701644
$243$	1.43489e7	0.0641500
$244$	−7.77916e7	−0.342822
$245$	−4.58831e7	−0.199329
$246$	1.69559e8	0.726185
$247$	−7.72553e7	−0.326204
$248$	−7.70826e7	−0.320904
$249$	−1.31820e8	−0.541107
$250$	1.29480e7	0.0524098
$251$	8.44956e7	0.337269	0.168634	−	0.985679i	$-0.446064\pi$
$251$	8.44956e7	0.337269	0.168634	+	0.985679i	$0.446064\pi$
$252$	−1.60030e7	−0.0629941
$253$	3.99562e7	0.155118
$254$	3.07560e8	1.17764
$255$	3.92706e8	1.48312
$256$	1.67772e7	0.0625000
$257$	4.84275e8	1.77961	0.889807	−	0.456337i	$-0.150839\pi$
$257$	4.84275e8	1.77961	0.889807	+	0.456337i	$0.150839\pi$
$258$	−1.66651e8	−0.604142
$259$	−1.04978e8	−0.375447
$260$	5.48371e7	0.193494
$261$	1.63898e8	0.570601
$262$	9.45885e7	0.324926
$263$	−3.07390e8	−1.04194	−0.520972	−	0.853574i	$-0.674430\pi$
$263$	−3.07390e8	−1.04194	−0.520972	+	0.853574i	$0.674430\pi$
$264$	−5.36371e6	−0.0179412
$265$	2.59194e6	0.00855587
$266$	−9.64900e7	−0.314338
$267$	1.86131e8	0.598453
$268$	−1.94676e8	−0.617789
$269$	4.72062e8	1.47865	0.739326	−	0.673348i	$-0.235144\pi$
$269$	4.72062e8	1.47865	0.739326	+	0.673348i	$0.235144\pi$
$270$	−6.14110e7	−0.189877
$271$	5.41087e8	1.65148	0.825742	−	0.564048i	$-0.190757\pi$
$271$	5.41087e8	1.65148	0.825742	+	0.564048i	$0.190757\pi$
$272$	−1.52756e8	−0.460265
$273$	2.03464e7	0.0605228
$274$	2.73634e8	0.803607
$275$	−2.87023e7	−0.0832247
$276$	−1.77949e8	−0.509466
$277$	−1.87522e8	−0.530117	−0.265059	−	0.964232i	$-0.585391\pi$
$277$	−1.87522e8	−0.530117	−0.265059	+	0.964232i	$0.585391\pi$
$278$	2.92969e8	0.817832
$279$	−1.09752e8	−0.302552
$280$	6.84902e7	0.186456
$281$	3.11535e8	0.837596	0.418798	−	0.908080i	$-0.362452\pi$
$281$	3.11535e8	0.837596	0.418798	+	0.908080i	$0.362452\pi$
$282$	1.41259e8	0.375097
$283$	−5.67573e8	−1.48857	−0.744285	−	0.667862i	$-0.767210\pi$
$283$	−5.67573e8	−1.48857	−0.744285	+	0.667862i	$0.767210\pi$
$284$	3.91204e7	0.101342
$285$	−3.70277e8	−0.947479
$286$	6.81949e6	0.0172374
$287$	−2.69253e8	−0.672317
$288$	2.38879e7	0.0589256
$289$	9.80504e8	2.38950
$290$	−7.01457e8	−1.68892
$291$	1.01739e8	0.242027
$292$	2.26028e8	0.531278
$293$	−6.85385e8	−1.59183	−0.795917	−	0.605406i	$-0.793011\pi$
$293$	−6.85385e8	−1.59183	−0.795917	+	0.605406i	$0.793011\pi$
$294$	2.54122e7	0.0583212
$295$	−5.36874e8	−1.21757
$296$	1.56702e8	0.351199
$297$	−7.63700e6	−0.0169151
$298$	2.97866e8	0.652024
$299$	2.26247e8	0.489479
$300$	1.27829e8	0.273341
$301$	2.64635e8	0.559326
$302$	−2.83951e8	−0.593225
$303$	4.89105e8	1.01007
$304$	1.44032e8	0.294036
$305$	4.74043e8	0.956683
$306$	−2.17499e8	−0.433942
$307$	3.80243e8	0.750027	0.375013	−	0.927019i	$-0.377638\pi$
$307$	3.80243e8	0.750027	0.375013	+	0.927019i	$0.377638\pi$
$308$	8.51738e6	0.0166103
$309$	−1.02767e8	−0.198152
$310$	4.69722e8	0.895520
$311$	2.29090e8	0.431863	0.215931	−	0.976409i	$-0.430721\pi$
$311$	2.29090e8	0.431863	0.215931	+	0.976409i	$0.430721\pi$
$312$	−3.03713e7	−0.0566139
$313$	−6.04313e8	−1.11393	−0.556964	−	0.830537i	$-0.688034\pi$
$313$	−6.04313e8	−1.11393	−0.556964	+	0.830537i	$0.688034\pi$
$314$	5.24944e8	0.956883
$315$	9.75183e7	0.175792
$316$	−8.65147e7	−0.154236
$317$	−5.39157e8	−0.950622	−0.475311	−	0.879818i	$-0.657664\pi$
$317$	−5.39157e8	−0.950622	−0.475311	+	0.879818i	$0.657664\pi$
$318$	−1.43554e6	−0.00250334
$319$	−8.72325e7	−0.150456
$320$	−1.02236e8	−0.174413
$321$	4.10440e8	0.692599
$322$	2.82577e8	0.471673
$323$	−1.31141e9	−2.16535
$324$	3.40122e7	0.0555556
$325$	−1.62523e8	−0.262617
$326$	−4.44274e8	−0.710214
$327$	−2.96308e8	−0.468626
$328$	4.01917e8	0.628895
$329$	−2.24314e8	−0.347272
$330$	3.26851e7	0.0500670
$331$	4.16412e8	0.631139	0.315569	−	0.948903i	$-0.397804\pi$
$331$	4.16412e8	0.631139	0.315569	+	0.948903i	$0.397804\pi$
$332$	−3.12462e8	−0.468612
$333$	2.23116e8	0.331113
$334$	3.06899e8	0.450695
$335$	1.18631e9	1.72401
$336$	−3.79331e7	−0.0545545
$337$	7.74406e8	1.10221	0.551104	−	0.834436i	$-0.314207\pi$
$337$	7.74406e8	1.10221	0.551104	+	0.834436i	$0.314207\pi$
$338$	3.86145e7	0.0543928
$339$	2.93547e8	0.409241
$340$	9.30858e8	1.28442
$341$	5.84142e7	0.0797770
$342$	2.05076e8	0.277220
$343$	−4.03536e7	−0.0539949
$344$	−3.95024e8	−0.523202
$345$	1.08438e9	1.42172
$346$	−2.74067e8	−0.355705
$347$	5.81832e8	0.747558	0.373779	−	0.927518i	$-0.378062\pi$
$347$	5.81832e8	0.747558	0.373779	+	0.927518i	$0.378062\pi$
$348$	3.88499e8	0.494155
$349$	−1.49462e9	−1.88209	−0.941047	−	0.338275i	$-0.890157\pi$
$349$	−1.49462e9	−1.88209	−0.941047	+	0.338275i	$0.890157\pi$
$350$	−2.02987e8	−0.253064
$351$	−4.32436e7	−0.0533761
$352$	−1.27140e7	−0.0155375
$353$	1.48958e7	0.0180241	0.00901203	−	0.999959i	$-0.497131\pi$
$353$	1.48958e7	0.0180241	0.00901203	+	0.999959i	$0.497131\pi$
$354$	2.97346e8	0.356246
$355$	−2.38390e8	−0.282806
$356$	4.41200e8	0.518276
$357$	3.45380e8	0.401752
$358$	2.48904e8	0.286709
$359$	−9.94146e7	−0.113402	−0.0567008	−	0.998391i	$-0.518058\pi$
$359$	−9.94146e7	−0.113402	−0.0567008	+	0.998391i	$0.518058\pi$
$360$	−1.45567e8	−0.164438
$361$	3.42635e8	0.383316
$362$	5.18644e8	0.574632
$363$	−5.22089e8	−0.572890
$364$	4.82285e7	0.0524142
$365$	−1.37736e9	−1.48259
$366$	−2.62547e8	−0.279913
$367$	−8.91006e7	−0.0940913	−0.0470456	−	0.998893i	$-0.514981\pi$
$367$	−8.91006e7	−0.0940913	−0.0470456	+	0.998893i	$0.514981\pi$
$368$	−4.21806e8	−0.441210
$369$	5.72261e8	0.592927
$370$	−9.54901e8	−0.980059
$371$	2.27958e6	0.00231764
$372$	−2.60154e8	−0.262017
$373$	−1.51116e8	−0.150775	−0.0753877	−	0.997154i	$-0.524019\pi$
$373$	−1.51116e8	−0.150775	−0.0753877	+	0.997154i	$0.524019\pi$
$374$	1.15761e8	0.114422
$375$	4.36995e7	0.0427924
$376$	3.34836e8	0.324844
$377$	−4.93943e8	−0.474768
$378$	−5.40102e7	−0.0514344
$379$	−1.75411e9	−1.65508	−0.827540	−	0.561407i	$-0.810260\pi$
$379$	−1.75411e9	−1.65508	−0.827540	+	0.561407i	$0.810260\pi$
$380$	−8.77693e8	−0.820541
$381$	1.03801e9	0.961537
$382$	−4.63689e8	−0.425603
$383$	3.22844e8	0.293628	0.146814	−	0.989164i	$-0.453098\pi$
$383$	3.22844e8	0.293628	0.146814	+	0.989164i	$0.453098\pi$
$384$	5.66231e7	0.0510310
$385$	−5.19028e7	−0.0463530
$386$	5.94357e8	0.526008
$387$	−5.62447e8	−0.493280
$388$	2.41160e8	0.209601
$389$	−7.86628e8	−0.677557	−0.338778	−	0.940866i	$-0.610014\pi$
$389$	−7.86628e8	−0.677557	−0.338778	+	0.940866i	$0.610014\pi$
$390$	1.85075e8	0.157987
$391$	3.84054e9	3.24918
$392$	6.02363e7	0.0505076
$393$	3.19236e8	0.265301
$394$	−1.41947e9	−1.16921
$395$	5.27199e8	0.430412
$396$	−1.81025e7	−0.0146489
$397$	−2.50277e8	−0.200749	−0.100375	−	0.994950i	$-0.532004\pi$
$397$	−2.50277e8	−0.200749	−0.100375	+	0.994950i	$0.532004\pi$
$398$	−1.02198e9	−0.812552
$399$	−3.25654e8	−0.256656
$400$	3.03002e8	0.236720
$401$	9.70974e8	0.751973	0.375987	−	0.926625i	$-0.377304\pi$
$401$	9.70974e8	0.751973	0.375987	+	0.926625i	$0.377304\pi$
$402$	−6.57031e8	−0.504422
$403$	3.30763e8	0.251738
$404$	1.15936e9	0.874750
$405$	−2.07262e8	−0.155034
$406$	−6.16923e8	−0.457498
$407$	−1.18751e8	−0.0873082
$408$	−5.15552e8	−0.375805
$409$	6.22290e7	0.0449739	0.0224870	−	0.999747i	$-0.492842\pi$
$409$	6.22290e7	0.0449739	0.0224870	+	0.999747i	$0.492842\pi$
$410$	−2.44918e9	−1.75500
$411$	9.23516e8	0.656142
$412$	−2.43596e8	−0.171605
$413$	−4.72174e8	−0.329820
$414$	−6.00579e8	−0.415977
$415$	1.90406e9	1.30772
$416$	−7.19913e7	−0.0490290
$417$	9.88769e8	0.667757
$418$	−1.09149e8	−0.0730976
$419$	1.66947e9	1.10874	0.554371	−	0.832270i	$-0.312959\pi$
$419$	1.66947e9	1.10874	0.554371	+	0.832270i	$0.312959\pi$
$420$	2.31155e8	0.152240
$421$	−1.12539e9	−0.735049	−0.367525	−	0.930014i	$-0.619795\pi$
$421$	−1.12539e9	−0.735049	−0.367525	+	0.930014i	$0.619795\pi$
$422$	−5.79357e8	−0.375277
$423$	4.76749e8	0.306265
$424$	−3.40275e6	−0.00216795
$425$	−2.75882e9	−1.74326
$426$	1.32031e8	0.0827453
$427$	4.16914e8	0.259149
$428$	9.72894e8	0.599808
$429$	2.30158e7	0.0140742
$430$	2.40718e9	1.46005
$431$	8.33512e8	0.501466	0.250733	−	0.968056i	$-0.419328\pi$
$431$	8.33512e8	0.501466	0.250733	+	0.968056i	$0.419328\pi$
$432$	8.06216e7	0.0481125
$433$	−3.20560e8	−0.189759	−0.0948794	−	0.995489i	$-0.530247\pi$
$433$	−3.20560e8	−0.189759	−0.0948794	+	0.995489i	$0.530247\pi$
$434$	4.13115e8	0.242581
$435$	−2.36742e9	−1.37899
$436$	−7.02360e8	−0.405842
$437$	−3.62119e9	−2.07571
$438$	7.62844e8	0.433787
$439$	7.91342e8	0.446415	0.223207	−	0.974771i	$-0.428347\pi$
$439$	7.91342e8	0.446415	0.223207	+	0.974771i	$0.428347\pi$
$440$	7.74758e7	0.0433593
$441$	8.57661e7	0.0476190
$442$	6.55479e8	0.361061
$443$	−6.89560e8	−0.376842	−0.188421	−	0.982088i	$-0.560337\pi$
$443$	−6.89560e8	−0.376842	−0.188421	+	0.982088i	$0.560337\pi$
$444$	5.28868e8	0.286752
$445$	−2.68856e9	−1.44631
$446$	8.40223e8	0.448459
$447$	1.00530e9	0.532375
$448$	−8.99154e7	−0.0472456
$449$	−2.09202e9	−1.09069	−0.545347	−	0.838210i	$-0.683602\pi$
$449$	−2.09202e9	−1.09069	−0.545347	+	0.838210i	$0.683602\pi$
$450$	4.31422e8	0.223182
$451$	−3.04578e8	−0.156344
$452$	6.95816e8	0.354413
$453$	−9.58335e8	−0.484366
$454$	−5.03146e8	−0.252347
$455$	−2.93893e8	−0.146268
$456$	4.86107e8	0.240079
$457$	9.10740e8	0.446362	0.223181	−	0.974777i	$-0.428356\pi$
$457$	9.10740e8	0.446362	0.223181	+	0.974777i	$0.428356\pi$
$458$	2.00843e9	0.976848
$459$	−7.34058e8	−0.354312
$460$	2.57038e9	1.23125
$461$	−7.39494e8	−0.351545	−0.175773	−	0.984431i	$-0.556242\pi$
$461$	−7.39494e8	−0.351545	−0.175773	+	0.984431i	$0.556242\pi$
$462$	2.87461e7	0.0135623
$463$	1.91078e9	0.894702	0.447351	−	0.894359i	$-0.352368\pi$
$463$	1.91078e9	0.894702	0.447351	+	0.894359i	$0.352368\pi$
$464$	9.20887e8	0.427951
$465$	1.58531e9	0.731189
$466$	1.18032e9	0.540315
$467$	−1.00162e9	−0.455085	−0.227542	−	0.973768i	$-0.573069\pi$
$467$	−1.00162e9	−0.455085	−0.227542	+	0.973768i	$0.573069\pi$
$468$	−1.02503e8	−0.0462250
$469$	1.04334e9	0.467004
$470$	−2.04041e9	−0.906513
$471$	1.77169e9	0.781292
$472$	7.04819e8	0.308518
$473$	2.99354e8	0.130068
$474$	−2.91987e8	−0.125933
$475$	2.60126e9	1.11367
$476$	8.18678e8	0.347927
$477$	−4.84493e6	−0.00204397
$478$	−2.69509e8	−0.112869
$479$	−3.25226e9	−1.35211	−0.676053	−	0.736853i	$-0.736311\pi$
$479$	−3.25226e9	−1.35211	−0.676053	+	0.736853i	$0.736311\pi$
$480$	−3.45047e8	−0.142408
$481$	−6.72409e8	−0.275503
$482$	−9.55809e8	−0.388782
$483$	9.53698e8	0.385120
$484$	−1.23754e9	−0.496137
$485$	−1.46957e9	−0.584917
$486$	1.14791e8	0.0453609
$487$	−2.91217e9	−1.14252	−0.571261	−	0.820768i	$-0.693546\pi$
$487$	−2.91217e9	−1.14252	−0.571261	+	0.820768i	$0.693546\pi$
$488$	−6.22333e8	−0.242412
$489$	−1.49942e9	−0.579887
$490$	−3.67065e8	−0.140947
$491$	4.82601e9	1.83994	0.919968	−	0.391994i	$-0.128214\pi$
$491$	4.82601e9	1.83994	0.919968	+	0.391994i	$0.128214\pi$
$492$	1.35647e9	0.513490
$493$	−8.38466e9	−3.15153
$494$	−6.18042e8	−0.230661
$495$	1.10312e8	0.0408795
$496$	−6.16661e8	−0.226914
$497$	−2.09661e8	−0.0766073
$498$	−1.05456e9	−0.382620
$499$	2.46895e9	0.889529	0.444764	−	0.895648i	$-0.353287\pi$
$499$	2.46895e9	0.889529	0.444764	+	0.895648i	$0.353287\pi$
$500$	1.03584e8	0.0370593
$501$	1.03578e9	0.367991
$502$	6.75965e8	0.238485
$503$	−2.98872e9	−1.04712	−0.523561	−	0.851988i	$-0.675397\pi$
$503$	−2.98872e9	−1.04712	−0.523561	+	0.851988i	$0.675397\pi$
$504$	−1.28024e8	−0.0445435
$505$	−7.06486e9	−2.44109
$506$	3.19650e8	0.109685
$507$	1.30324e8	0.0444116
$508$	2.46048e9	0.832715
$509$	3.41170e9	1.14672	0.573361	−	0.819303i	$-0.305639\pi$
$509$	3.41170e9	1.14672	0.573361	+	0.819303i	$0.305639\pi$
$510$	3.14165e9	1.04873
$511$	−1.21137e9	−0.401609
$512$	1.34218e8	0.0441942
$513$	6.92133e8	0.226349
$514$	3.87420e9	1.25838
$515$	1.48441e9	0.478883
$516$	−1.33321e9	−0.427193
$517$	−2.53743e8	−0.0807563
$518$	−8.39823e8	−0.265481
$519$	−9.24975e8	−0.290432
$520$	4.38697e8	0.136821
$521$	1.57030e9	0.486464	0.243232	−	0.969968i	$-0.421792\pi$
$521$	1.57030e9	0.486464	0.243232	+	0.969968i	$0.421792\pi$
$522$	1.31119e9	0.403476
$523$	2.07833e9	0.635272	0.317636	−	0.948213i	$-0.397111\pi$
$523$	2.07833e9	0.635272	0.317636	+	0.948213i	$0.397111\pi$
$524$	7.56708e8	0.229757
$525$	−6.85082e8	−0.206626
$526$	−2.45912e9	−0.736765
$527$	5.61469e9	1.67105
$528$	−4.29097e7	−0.0126864
$529$	7.20005e9	2.11466
$530$	2.07355e7	0.00604992
$531$	1.00354e9	0.290874
$532$	−7.71920e8	−0.222270
$533$	−1.72463e9	−0.493345
$534$	1.48905e9	0.423170
$535$	−5.92857e9	−1.67383
$536$	−1.55741e9	−0.436843
$537$	8.40052e8	0.234097
$538$	3.77649e9	1.04556
$539$	−4.56478e7	−0.0125562
$540$	−4.91288e8	−0.134263
$541$	4.44722e8	0.120753	0.0603765	−	0.998176i	$-0.480770\pi$
$541$	4.44722e8	0.120753	0.0603765	+	0.998176i	$0.480770\pi$
$542$	4.32869e9	1.16778
$543$	1.75042e9	0.469185
$544$	−1.22205e9	−0.325456
$545$	4.28001e9	1.13255
$546$	1.62771e8	0.0427960
$547$	−6.37551e8	−0.166556	−0.0832778	−	0.996526i	$-0.526539\pi$
$547$	−6.37551e8	−0.166556	−0.0832778	+	0.996526i	$0.526539\pi$
$548$	2.18907e9	0.568236
$549$	−8.86095e8	−0.228548
$550$	−2.29618e8	−0.0588488
$551$	7.90578e9	2.01333
$552$	−1.42360e9	−0.360247
$553$	4.63665e8	0.116591
$554$	−1.50017e9	−0.374849
$555$	−3.22279e9	−0.800215
$556$	2.34375e9	0.578295
$557$	−3.98359e9	−0.976745	−0.488373	−	0.872635i	$-0.662409\pi$
$557$	−3.98359e9	−0.976745	−0.488373	+	0.872635i	$0.662409\pi$
$558$	−8.78019e8	−0.213936
$559$	1.69506e9	0.410433
$560$	5.47922e8	0.131844
$561$	3.90692e8	0.0934253
$562$	2.49228e9	0.592269
$563$	1.06186e9	0.250778	0.125389	−	0.992108i	$-0.459982\pi$
$563$	1.06186e9	0.250778	0.125389	+	0.992108i	$0.459982\pi$
$564$	1.13007e9	0.265234
$565$	−4.24013e9	−0.989030
$566$	−4.54058e9	−1.05258
$567$	−1.82284e8	−0.0419961
$568$	3.12963e8	0.0716595
$569$	6.71593e9	1.52832	0.764158	−	0.645029i	$-0.223155\pi$
$569$	6.71593e9	1.52832	0.764158	+	0.645029i	$0.223155\pi$
$570$	−2.96222e9	−0.669969
$571$	−2.16218e9	−0.486032	−0.243016	−	0.970022i	$-0.578137\pi$
$571$	−2.16218e9	−0.486032	−0.243016	+	0.970022i	$0.578137\pi$
$572$	5.45559e7	0.0121887
$573$	−1.56495e9	−0.347503
$574$	−2.15402e9	−0.475400
$575$	−7.61795e9	−1.67109
$576$	1.91103e8	0.0416667
$577$	−5.04871e9	−1.09412	−0.547060	−	0.837094i	$-0.684253\pi$
$577$	−5.04871e9	−1.09412	−0.547060	+	0.837094i	$0.684253\pi$
$578$	7.84403e9	1.68963
$579$	2.00596e9	0.429483
$580$	−5.61166e9	−1.19424
$581$	1.67460e9	0.354238
$582$	8.13915e8	0.171139
$583$	2.57865e6	0.000538955 0
$584$	1.80822e9	0.375670
$585$	6.24629e8	0.128996
$586$	−5.48308e9	−1.12560
$587$	−5.15780e8	−0.105252	−0.0526261	−	0.998614i	$-0.516759\pi$
$587$	−5.15780e8	−0.105252	−0.0526261	+	0.998614i	$0.516759\pi$
$588$	2.03297e8	0.0412393
$589$	−5.29401e9	−1.06753
$590$	−4.29499e9	−0.860955
$591$	−4.79073e9	−0.954652
$592$	1.25361e9	0.248335
$593$	−1.65236e8	−0.0325396	−0.0162698	−	0.999868i	$-0.505179\pi$
$593$	−1.65236e8	−0.0325396	−0.0162698	+	0.999868i	$0.505179\pi$
$594$	−6.10960e7	−0.0119608
$595$	−4.98882e9	−0.970931
$596$	2.38293e9	0.461051
$597$	−3.44918e9	−0.663446
$598$	1.80998e9	0.346114
$599$	−6.29377e9	−1.19651	−0.598256	−	0.801305i	$-0.704139\pi$
$599$	−6.29377e9	−1.19651	−0.598256	+	0.801305i	$0.704139\pi$
$600$	1.02263e9	0.193281
$601$	−9.47857e9	−1.78108	−0.890538	−	0.454910i	$-0.849672\pi$
$601$	−9.47857e9	−1.78108	−0.890538	+	0.454910i	$0.849672\pi$
$602$	2.11708e9	0.395504
$603$	−2.21748e9	−0.411859
$604$	−2.27161e9	−0.419473
$605$	7.54128e9	1.38453
$606$	3.91284e9	0.714230
$607$	3.69476e9	0.670541	0.335271	−	0.942122i	$-0.391172\pi$
$607$	3.69476e9	0.670541	0.335271	+	0.942122i	$0.391172\pi$
$608$	1.15225e9	0.207915
$609$	−2.08211e9	−0.373546
$610$	3.79234e9	0.676477
$611$	−1.43679e9	−0.254828
$612$	−1.73999e9	−0.306843
$613$	2.84324e9	0.498542	0.249271	−	0.968434i	$-0.419809\pi$
$613$	2.84324e9	0.498542	0.249271	+	0.968434i	$0.419809\pi$
$614$	3.04194e9	0.530349
$615$	−8.26599e9	−1.43295
$616$	6.81390e7	0.0117453
$617$	−9.26077e9	−1.58726	−0.793631	−	0.608399i	$-0.791812\pi$
$617$	−9.26077e9	−1.58726	−0.793631	+	0.608399i	$0.791812\pi$
$618$	−8.22137e8	−0.140115
$619$	2.62159e8	0.0444271	0.0222135	−	0.999753i	$-0.492929\pi$
$619$	2.62159e8	0.0444271	0.0222135	+	0.999753i	$0.492929\pi$
$620$	3.75778e9	0.633228
$621$	−2.02696e9	−0.339644
$622$	1.83272e9	0.305373
$623$	−2.36456e9	−0.391780
$624$	−2.42971e8	−0.0400320
$625$	−6.41051e9	−1.05030
$626$	−4.83451e9	−0.787666
$627$	−3.68378e8	−0.0596839
$628$	4.19955e9	0.676619
$629$	−1.14141e10	−1.82880
$630$	7.80147e8	0.124304
$631$	−4.89819e9	−0.776127	−0.388064	−	0.921633i	$-0.626856\pi$
$631$	−4.89819e9	−0.776127	−0.388064	+	0.921633i	$0.626856\pi$
$632$	−6.92118e8	−0.109061
$633$	−1.95533e9	−0.306413
$634$	−4.31326e9	−0.672191
$635$	−1.49935e10	−2.32379
$636$	−1.14843e7	−0.00177013
$637$	−2.58475e8	−0.0396214
$638$	−6.97860e8	−0.106389
$639$	4.45606e8	0.0675612
$640$	−8.17889e8	−0.123329
$641$	−7.14167e9	−1.07102	−0.535509	−	0.844530i	$-0.679880\pi$
$641$	−7.14167e9	−1.07102	−0.535509	+	0.844530i	$0.679880\pi$
$642$	3.28352e9	0.489741
$643$	8.74662e8	0.129748	0.0648742	−	0.997893i	$-0.479335\pi$
$643$	8.74662e8	0.129748	0.0648742	+	0.997893i	$0.479335\pi$
$644$	2.26062e9	0.333524
$645$	8.12423e9	1.19213
$646$	−1.04912e10	−1.53113
$647$	−2.44888e9	−0.355470	−0.177735	−	0.984078i	$-0.556877\pi$
$647$	−2.44888e9	−0.355470	−0.177735	+	0.984078i	$0.556877\pi$
$648$	2.72098e8	0.0392837
$649$	−5.34121e8	−0.0766978
$650$	−1.30018e9	−0.185698
$651$	1.39426e9	0.198066
$652$	−3.55419e9	−0.502197
$653$	3.15396e9	0.443262	0.221631	−	0.975131i	$-0.428862\pi$
$653$	3.15396e9	0.443262	0.221631	+	0.975131i	$0.428862\pi$
$654$	−2.37047e9	−0.331369
$655$	−4.61119e9	−0.641163
$656$	3.21534e9	0.444696
$657$	2.57460e9	0.354186
$658$	−1.79451e9	−0.245559
$659$	−1.05992e10	−1.44269	−0.721344	−	0.692577i	$-0.756475\pi$
$659$	−1.05992e10	−1.44269	−0.721344	+	0.692577i	$0.756475\pi$
$660$	2.61481e8	0.0354027
$661$	−9.10735e8	−0.122655	−0.0613277	−	0.998118i	$-0.519533\pi$
$661$	−9.10735e8	−0.122655	−0.0613277	+	0.998118i	$0.519533\pi$
$662$	3.33129e9	0.446282
$663$	2.21224e9	0.294805
$664$	−2.49969e9	−0.331359
$665$	4.70389e9	0.620271
$666$	1.78493e9	0.234132
$667$	−2.31526e10	−3.02106
$668$	2.45519e9	0.318690
$669$	2.83575e9	0.366165
$670$	9.49044e9	1.21906
$671$	4.71612e8	0.0602637
$672$	−3.03464e8	−0.0385758
$673$	7.11216e9	0.899391	0.449696	−	0.893182i	$-0.351532\pi$
$673$	7.11216e9	0.899391	0.449696	+	0.893182i	$0.351532\pi$
$674$	6.19524e9	0.779379
$675$	1.45605e9	0.182227
$676$	3.08916e8	0.0384615
$677$	−9.12863e9	−1.13069	−0.565347	−	0.824853i	$-0.691258\pi$
$677$	−9.12863e9	−1.13069	−0.565347	+	0.824853i	$0.691258\pi$
$678$	2.34838e9	0.289377
$679$	−1.29247e9	−0.158444
$680$	7.44687e9	0.908223
$681$	−1.69812e9	−0.206041
$682$	4.67313e8	0.0564109
$683$	3.29313e9	0.395491	0.197745	−	0.980253i	$-0.436638\pi$
$683$	3.29313e9	0.395491	0.197745	+	0.980253i	$0.436638\pi$
$684$	1.64061e9	0.196024
$685$	−1.33397e10	−1.58573
$686$	−3.22829e8	−0.0381802
$687$	6.77844e9	0.797593
$688$	−3.16020e9	−0.369960
$689$	1.46013e7	0.00170068
$690$	8.67504e9	1.00531
$691$	4.44040e9	0.511975	0.255987	−	0.966680i	$-0.417599\pi$
$691$	4.44040e9	0.511975	0.255987	+	0.966680i	$0.417599\pi$
$692$	−2.19253e9	−0.251521
$693$	9.70182e7	0.0110736
$694$	4.65466e9	0.528603
$695$	−1.42822e10	−1.61380
$696$	3.10799e9	0.349420
$697$	−2.92756e10	−3.27484
$698$	−1.19569e10	−1.33084
$699$	3.98356e9	0.441166
$700$	−1.62390e9	−0.178944
$701$	9.52436e9	1.04429	0.522147	−	0.852855i	$-0.325131\pi$
$701$	9.52436e9	1.04429	0.522147	+	0.852855i	$0.325131\pi$
$702$	−3.45948e8	−0.0377426
$703$	1.07622e10	1.16831
$704$	−1.01712e8	−0.0109867
$705$	−6.88637e9	−0.740165
$706$	1.19166e8	0.0127449
$707$	−6.21345e9	−0.661249
$708$	2.37876e9	0.251904
$709$	−1.53006e10	−1.61230	−0.806150	−	0.591711i	$-0.798452\pi$
$709$	−1.53006e10	−1.61230	−0.806150	+	0.591711i	$0.798452\pi$
$710$	−1.90712e9	−0.199974
$711$	−9.85456e8	−0.102824
$712$	3.52960e9	0.366476
$713$	1.55038e10	1.60187
$714$	2.76304e9	0.284082
$715$	−3.32450e8	−0.0340138
$716$	1.99123e9	0.202734
$717$	−9.09591e8	−0.0921572
$718$	−7.95316e8	−0.0801871
$719$	1.39474e9	0.139940	0.0699700	−	0.997549i	$-0.477710\pi$
$719$	1.39474e9	0.139940	0.0699700	+	0.997549i	$0.477710\pi$
$720$	−1.16453e9	−0.116276
$721$	1.30552e9	0.129721
$722$	2.74108e9	0.271045
$723$	−3.22585e9	−0.317439
$724$	4.14915e9	0.406326
$725$	1.66315e10	1.62087
$726$	−4.17671e9	−0.405094
$727$	1.29373e10	1.24874	0.624371	−	0.781128i	$-0.285355\pi$
$727$	1.29373e10	1.24874	0.624371	+	0.781128i	$0.285355\pi$
$728$	3.85828e8	0.0370625
$729$	3.87420e8	0.0370370
$730$	−1.10189e10	−1.04835
$731$	2.87735e10	2.72447
$732$	−2.10037e9	−0.197928
$733$	1.08671e10	1.01917	0.509587	−	0.860419i	$-0.329798\pi$
$733$	1.08671e10	1.01917	0.509587	+	0.860419i	$0.329798\pi$
$734$	−7.12805e8	−0.0665326
$735$	−1.23884e9	−0.115083
$736$	−3.37445e9	−0.311983
$737$	1.18022e9	0.108599
$738$	4.57809e9	0.419263
$739$	1.71536e9	0.156351	0.0781756	−	0.996940i	$-0.475091\pi$
$739$	1.71536e9	0.156351	0.0781756	+	0.996940i	$0.475091\pi$
$740$	−7.63921e9	−0.693006
$741$	−2.08589e9	−0.188334
$742$	1.82366e7	0.00163882
$743$	9.06488e9	0.810777	0.405388	−	0.914145i	$-0.367136\pi$
$743$	9.06488e9	0.810777	0.405388	+	0.914145i	$0.367136\pi$
$744$	−2.08123e9	−0.185274
$745$	−1.45210e10	−1.28661
$746$	−1.20893e9	−0.106614
$747$	−3.55914e9	−0.312408
$748$	9.26085e8	0.0809087
$749$	−5.21410e9	−0.453412
$750$	3.49596e8	0.0302588
$751$	−6.63350e9	−0.571482	−0.285741	−	0.958307i	$-0.592240\pi$
$751$	−6.63350e9	−0.571482	−0.285741	+	0.958307i	$0.592240\pi$
$752$	2.67869e9	0.229699
$753$	2.28138e9	0.194722
$754$	−3.95154e9	−0.335712
$755$	1.38426e10	1.17059
$756$	−4.32081e8	−0.0363696
$757$	2.08461e10	1.74658	0.873291	−	0.487199i	$-0.161981\pi$
$757$	2.08461e10	1.74658	0.873291	+	0.487199i	$0.161981\pi$
$758$	−1.40329e10	−1.17032
$759$	1.07882e9	0.0895575
$760$	−7.02155e9	−0.580210
$761$	−1.58054e10	−1.30004	−0.650022	−	0.759916i	$-0.725240\pi$
$761$	−1.58054e10	−1.30004	−0.650022	+	0.759916i	$0.725240\pi$
$762$	8.30411e9	0.679909
$763$	3.76421e9	0.306788
$764$	−3.70951e9	−0.300947
$765$	1.06031e10	0.856281
$766$	2.58275e9	0.207626
$767$	−3.02439e9	−0.242022
$768$	4.52985e8	0.0360844
$769$	1.45886e10	1.15684	0.578419	−	0.815740i	$-0.303670\pi$
$769$	1.45886e10	1.15684	0.578419	+	0.815740i	$0.303670\pi$
$770$	−4.15222e8	−0.0327765
$771$	1.30754e10	1.02746
$772$	4.75486e9	0.371943
$773$	−1.74429e10	−1.35828	−0.679142	−	0.734007i	$-0.737648\pi$
$773$	−1.74429e10	−1.35828	−0.679142	+	0.734007i	$0.737648\pi$
$774$	−4.49957e9	−0.348801
$775$	−1.11371e10	−0.859440
$776$	1.92928e9	0.148211
$777$	−2.83440e9	−0.216764
$778$	−6.29302e9	−0.479105
$779$	2.76035e10	2.09211
$780$	1.48060e9	0.111714
$781$	−2.37167e8	−0.0178146
$782$	3.07243e10	2.29751
$783$	4.42525e9	0.329436
$784$	4.81890e8	0.0357143
$785$	−2.55910e10	−1.88818
$786$	2.55389e9	0.187596
$787$	−2.05678e10	−1.50410	−0.752051	−	0.659105i	$-0.770935\pi$
$787$	−2.05678e10	−1.50410	−0.752051	+	0.659105i	$0.770935\pi$
$788$	−1.13558e10	−0.826753
$789$	−8.29952e9	−0.601566
$790$	4.21759e9	0.304347
$791$	−3.72914e9	−0.267911
$792$	−1.44820e8	−0.0103584
$793$	2.67044e9	0.190163
$794$	−2.00221e9	−0.141951
$795$	6.99824e7	0.00493974
$796$	−8.17583e9	−0.574561
$797$	1.42711e10	0.998513	0.499257	−	0.866454i	$-0.333607\pi$
$797$	1.42711e10	0.998513	0.499257	+	0.866454i	$0.333607\pi$
$798$	−2.60523e9	−0.181483
$799$	−2.43894e10	−1.69156
$800$	2.42401e9	0.167386
$801$	5.02555e9	0.345517
$802$	7.76779e9	0.531725
$803$	−1.37029e9	−0.0933919
$804$	−5.25624e9	−0.356680
$805$	−1.37756e10	−0.930735
$806$	2.64610e9	0.178006
$807$	1.27457e10	0.853700
$808$	9.27489e9	0.618542
$809$	2.27230e10	1.50885	0.754426	−	0.656385i	$-0.227915\pi$
$809$	2.27230e10	1.50885	0.754426	+	0.656385i	$0.227915\pi$
$810$	−1.65810e9	−0.109626
$811$	−8.54871e8	−0.0562766	−0.0281383	−	0.999604i	$-0.508958\pi$
$811$	−8.54871e8	−0.0562766	−0.0281383	+	0.999604i	$0.508958\pi$
$812$	−4.93538e9	−0.323500
$813$	1.46093e10	0.953484
$814$	−9.50004e8	−0.0617362
$815$	2.16584e10	1.40144
$816$	−4.12442e9	−0.265734
$817$	−2.71302e10	−1.74050
$818$	4.97832e8	0.0318014
$819$	5.49353e8	0.0349428
$820$	−1.95935e10	−1.24097
$821$	2.42850e10	1.53157	0.765784	−	0.643098i	$-0.222351\pi$
$821$	2.42850e10	1.53157	0.765784	+	0.643098i	$0.222351\pi$
$822$	7.38812e9	0.463962
$823$	1.79215e10	1.12067	0.560333	−	0.828268i	$-0.310673\pi$
$823$	1.79215e10	1.12067	0.560333	+	0.828268i	$0.310673\pi$
$824$	−1.94877e9	−0.121343
$825$	−7.74962e8	−0.0480498
$826$	−3.77739e9	−0.233218
$827$	−7.25505e9	−0.446037	−0.223019	−	0.974814i	$-0.571591\pi$
$827$	−7.25505e9	−0.446037	−0.223019	+	0.974814i	$0.571591\pi$
$828$	−4.80463e9	−0.294140
$829$	−3.51522e9	−0.214295	−0.107147	−	0.994243i	$-0.534172\pi$
$829$	−3.51522e9	−0.214295	−0.107147	+	0.994243i	$0.534172\pi$
$830$	1.52325e10	0.924694
$831$	−5.06308e9	−0.306063
$832$	−5.75930e8	−0.0346688
$833$	−4.38760e9	−0.263008
$834$	7.91015e9	0.472176
$835$	−1.49613e10	−0.889339
$836$	−8.73192e8	−0.0516878
$837$	−2.96332e9	−0.174678
$838$	1.33558e10	0.783999
$839$	1.30593e10	0.763401	0.381700	−	0.924286i	$-0.375339\pi$
$839$	1.30593e10	0.763401	0.381700	+	0.924286i	$0.375339\pi$
$840$	1.84924e9	0.107650
$841$	3.32969e10	1.93027
$842$	−9.00314e9	−0.519758
$843$	8.41144e9	0.483586
$844$	−4.63485e9	−0.265361
$845$	−1.88246e9	−0.107331
$846$	3.81399e9	0.216562
$847$	6.63246e9	0.375045
$848$	−2.72220e7	−0.00153297
$849$	−1.53245e10	−0.859426
$850$	−2.20706e10	−1.23267
$851$	−3.15179e10	−1.75309
$852$	1.05625e9	0.0585098
$853$	−2.98281e10	−1.64552	−0.822761	−	0.568388i	$-0.807567\pi$
$853$	−2.98281e10	−1.64552	−0.822761	+	0.568388i	$0.807567\pi$
$854$	3.33532e9	0.183246
$855$	−9.99748e9	−0.547027
$856$	7.78315e9	0.424128
$857$	1.13861e10	0.617933	0.308966	−	0.951073i	$-0.400017\pi$
$857$	1.13861e10	0.617933	0.308966	+	0.951073i	$0.400017\pi$
$858$	1.84126e8	0.00995199
$859$	2.50748e10	1.34978	0.674889	−	0.737920i	$-0.264192\pi$
$859$	2.50748e10	1.34978	0.674889	+	0.737920i	$0.264192\pi$
$860$	1.92574e10	1.03241
$861$	−7.26983e9	−0.388162
$862$	6.66810e9	0.354590
$863$	−2.73456e10	−1.44827	−0.724134	−	0.689659i	$-0.757760\pi$
$863$	−2.73456e10	−1.44827	−0.724134	+	0.689659i	$0.757760\pi$
$864$	6.44973e8	0.0340207
$865$	1.33607e10	0.701899
$866$	−2.56448e9	−0.134180
$867$	2.64736e10	1.37958
$868$	3.30492e9	0.171531
$869$	5.24495e8	0.0271127
$870$	−1.89393e10	−0.975096
$871$	6.68285e9	0.342688
$872$	−5.61888e9	−0.286974
$873$	2.74696e9	0.139734
$874$	−2.89695e10	−1.46775
$875$	−5.55146e8	−0.0280142
$876$	6.10275e9	0.306734
$877$	7.19979e9	0.360430	0.180215	−	0.983627i	$-0.442321\pi$
$877$	7.19979e9	0.360430	0.180215	+	0.983627i	$0.442321\pi$
$878$	6.33074e9	0.315663
$879$	−1.85054e10	−0.919045
$880$	6.19807e8	0.0306596
$881$	−2.09219e10	−1.03083	−0.515413	−	0.856942i	$-0.672361\pi$
$881$	−2.09219e10	−1.03083	−0.515413	+	0.856942i	$0.672361\pi$
$882$	6.86129e8	0.0336718
$883$	8.29723e9	0.405575	0.202787	−	0.979223i	$-0.435000\pi$
$883$	8.29723e9	0.405575	0.202787	+	0.979223i	$0.435000\pi$
$884$	5.24383e9	0.255309
$885$	−1.44956e10	−0.702967
$886$	−5.51648e9	−0.266467
$887$	2.98584e10	1.43659	0.718297	−	0.695737i	$-0.244922\pi$
$887$	2.98584e10	1.43659	0.718297	+	0.695737i	$0.244922\pi$
$888$	4.23095e9	0.202765
$889$	−1.31866e10	−0.629474
$890$	−2.15085e10	−1.02269
$891$	−2.06199e8	−0.00976596
$892$	6.72178e9	0.317108
$893$	2.29964e10	1.08064
$894$	8.04238e9	0.376446
$895$	−1.21341e10	−0.565752
$896$	−7.19323e8	−0.0334077
$897$	6.10867e9	0.282601
$898$	−1.67361e10	−0.771238
$899$	−3.38480e10	−1.55372
$900$	3.45138e9	0.157813
$901$	2.47856e8	0.0112892
$902$	−2.43662e9	−0.110552
$903$	7.14516e9	0.322927
$904$	5.56653e9	0.250608
$905$	−2.52839e10	−1.13390
$906$	−7.66668e9	−0.342499
$907$	3.82024e10	1.70006	0.850032	−	0.526731i	$-0.176583\pi$
$907$	3.82024e10	1.70006	0.850032	+	0.526731i	$0.176583\pi$
$908$	−4.02517e9	−0.178436
$909$	1.32058e10	0.583167
$910$	−2.35114e9	−0.103427
$911$	3.80287e9	0.166647	0.0833235	−	0.996523i	$-0.473447\pi$
$911$	3.80287e9	0.166647	0.0833235	+	0.996523i	$0.473447\pi$
$912$	3.88886e9	0.169762
$913$	1.89430e9	0.0823761
$914$	7.28592e9	0.315626
$915$	1.27992e10	0.552341
$916$	1.60674e10	0.690736
$917$	−4.05548e9	−0.173680
$918$	−5.87246e9	−0.250536
$919$	−4.60553e9	−0.195738	−0.0978690	−	0.995199i	$-0.531203\pi$
$919$	−4.60553e9	−0.195738	−0.0978690	+	0.995199i	$0.531203\pi$
$920$	2.05630e10	0.870623
$921$	1.02666e10	0.433028
$922$	−5.91595e9	−0.248580
$923$	−1.34293e9	−0.0562144
$924$	2.29969e8	0.00958998
$925$	2.26406e10	0.940573
$926$	1.52863e10	0.632650
$927$	−2.77471e9	−0.114403
$928$	7.36710e9	0.302607
$929$	4.16289e10	1.70349	0.851746	−	0.523955i	$-0.175544\pi$
$929$	4.16289e10	1.70349	0.851746	+	0.523955i	$0.175544\pi$
$930$	1.26825e10	0.517029
$931$	4.13701e9	0.168021
$932$	9.44252e9	0.382061
$933$	6.18544e9	0.249336
$934$	−8.01293e9	−0.321793
$935$	−5.64333e9	−0.225785
$936$	−8.20026e8	−0.0326860
$937$	3.05557e10	1.21340	0.606700	−	0.794931i	$-0.292493\pi$
$937$	3.05557e10	1.21340	0.606700	+	0.794931i	$0.292493\pi$
$938$	8.34672e9	0.330222
$939$	−1.63165e10	−0.643126
$940$	−1.63232e10	−0.641001
$941$	3.14782e10	1.23153	0.615767	−	0.787928i	$-0.288846\pi$
$941$	3.14782e10	1.23153	0.615767	+	0.787928i	$0.288846\pi$
$942$	1.41735e10	0.552457
$943$	−8.08387e10	−3.13927
$944$	5.63855e9	0.218155
$945$	2.63299e9	0.101494
$946$	2.39484e9	0.0919722
$947$	1.26722e10	0.484873	0.242437	−	0.970167i	$-0.422053\pi$
$947$	1.26722e10	0.484873	0.242437	+	0.970167i	$0.422053\pi$
$948$	−2.33590e9	−0.0890481
$949$	−7.75911e9	−0.294700
$950$	2.08101e10	0.787482
$951$	−1.45572e10	−0.548842
$952$	6.54942e9	0.246022
$953$	−4.00804e10	−1.50005	−0.750026	−	0.661408i	$-0.769959\pi$
$953$	−4.00804e10	−1.50005	−0.750026	+	0.661408i	$0.769959\pi$
$954$	−3.87595e7	−0.00144530
$955$	2.26048e10	0.839826
$956$	−2.15607e9	−0.0798105
$957$	−2.35528e9	−0.0868661
$958$	−2.60181e10	−0.956083
$959$	−1.17321e10	−0.429546
$960$	−2.76038e9	−0.100698
$961$	−4.84671e9	−0.176163
$962$	−5.37928e9	−0.194810
$963$	1.10819e10	0.399872
$964$	−7.64647e9	−0.274911
$965$	−2.89749e10	−1.03795
$966$	7.62958e9	0.272321
$967$	1.66530e9	0.0592244	0.0296122	−	0.999561i	$-0.490573\pi$
$967$	1.66530e9	0.0592244	0.0296122	+	0.999561i	$0.490573\pi$
$968$	−9.90035e9	−0.350822
$969$	−3.54080e10	−1.25017
$970$	−1.17566e10	−0.413598
$971$	−3.54297e10	−1.24194	−0.620969	−	0.783835i	$-0.713261\pi$
$971$	−3.54297e10	−1.24194	−0.620969	+	0.783835i	$0.713261\pi$
$972$	9.18330e8	0.0320750
$973$	−1.25610e10	−0.437150
$974$	−2.32973e10	−0.807886
$975$	−4.38812e9	−0.151622
$976$	−4.97866e9	−0.171411
$977$	3.48134e10	1.19431	0.597153	−	0.802127i	$-0.296299\pi$
$977$	3.48134e10	1.19431	0.597153	+	0.802127i	$0.296299\pi$
$978$	−1.19954e10	−0.410042
$979$	−2.67478e9	−0.0911062
$980$	−2.93652e9	−0.0996647
$981$	−8.00032e9	−0.270562
$982$	3.86081e10	1.30103
$983$	2.78673e9	0.0935746	0.0467873	−	0.998905i	$-0.485102\pi$
$983$	2.78673e9	0.0935746	0.0467873	+	0.998905i	$0.485102\pi$
$984$	1.08518e10	0.363092
$985$	6.91994e10	2.30715
$986$	−6.70773e10	−2.22847
$987$	−6.05647e9	−0.200498
$988$	−4.94434e9	−0.163102
$989$	7.94524e10	2.61168
$990$	8.82498e8	0.0289062
$991$	2.45762e10	0.802152	0.401076	−	0.916045i	$-0.368636\pi$
$991$	2.45762e10	0.802152	0.401076	+	0.916045i	$0.368636\pi$
$992$	−4.93329e9	−0.160452
$993$	1.12431e10	0.364388
$994$	−1.67729e9	−0.0541695
$995$	4.98215e10	1.60338
$996$	−8.43647e9	−0.270553
$997$	−4.45770e10	−1.42455	−0.712276	−	0.701900i	$-0.752335\pi$
$997$	−4.45770e10	−1.42455	−0.712276	+	0.701900i	$0.752335\pi$
$998$	1.97516e10	0.628992
$999$	6.02414e9	0.191168

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	546.8.a.b.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
546.8.a.b.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:	\( N \)	\(=\)	\( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	546.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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