LMFDB - Embedded newform 546.8.a.c.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:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	546.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} -10.0000 q^{5} +216.000 q^{6} -343.000 q^{7} +512.000 q^{8} +729.000 q^{9} +O(q^{10})$

\(q+8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} -10.0000 q^{5} +216.000 q^{6} -343.000 q^{7} +512.000 q^{8} +729.000 q^{9} -80.0000 q^{10} +1508.00 q^{11} +1728.00 q^{12} +2197.00 q^{13} -2744.00 q^{14} -270.000 q^{15} +4096.00 q^{16} -1042.00 q^{17} +5832.00 q^{18} -9068.00 q^{19} -640.000 q^{20} -9261.00 q^{21} +12064.0 q^{22} -98988.0 q^{23} +13824.0 q^{24} -78025.0 q^{25} +17576.0 q^{26} +19683.0 q^{27} -21952.0 q^{28} -213642. q^{29} -2160.00 q^{30} -22048.0 q^{31} +32768.0 q^{32} +40716.0 q^{33} -8336.00 q^{34} +3430.00 q^{35} +46656.0 q^{36} +418246. q^{37} -72544.0 q^{38} +59319.0 q^{39} -5120.00 q^{40} -76414.0 q^{41} -74088.0 q^{42} -177524. q^{43} +96512.0 q^{44} -7290.00 q^{45} -791904. q^{46} +631916. q^{47} +110592. q^{48} +117649. q^{49} -624200. q^{50} -28134.0 q^{51} +140608. q^{52} -982354. q^{53} +157464. q^{54} -15080.0 q^{55} -175616. q^{56} -244836. q^{57} -1.70914e6 q^{58} -596384. q^{59} -17280.0 q^{60} +1.86341e6 q^{61} -176384. q^{62} -250047. q^{63} +262144. q^{64} -21970.0 q^{65} +325728. q^{66} -1.84565e6 q^{67} -66688.0 q^{68} -2.67268e6 q^{69} +27440.0 q^{70} +1.63228e6 q^{71} +373248. q^{72} +216650. q^{73} +3.34597e6 q^{74} -2.10668e6 q^{75} -580352. q^{76} -517244. q^{77} +474552. q^{78} -6.46017e6 q^{79} -40960.0 q^{80} +531441. q^{81} -611312. q^{82} +3.59215e6 q^{83} -592704. q^{84} +10420.0 q^{85} -1.42019e6 q^{86} -5.76833e6 q^{87} +772096. q^{88} -9.48266e6 q^{89} -58320.0 q^{90} -753571. q^{91} -6.33523e6 q^{92} -595296. q^{93} +5.05533e6 q^{94} +90680.0 q^{95} +884736. q^{96} +840226. q^{97} +941192. q^{98} +1.09933e6 q^{99} +O(q^{100})\)

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
$5$	−10.0000	−0.0357771	−0.0178885	−	0.999840i	$-0.505694\pi$
$5$	−10.0000	−0.0357771	−0.0178885	+	0.999840i	$0.505694\pi$
$6$	216.000	0.408248
$7$	−343.000	−0.377964
$7$	−343.000	−0.377964
$8$	512.000	0.353553
$9$	729.000	0.333333
$10$	−80.0000	−0.0252982
$11$	1508.00	0.341607	0.170804	−	0.985305i	$-0.445364\pi$
$11$	1508.00	0.341607	0.170804	+	0.985305i	$0.445364\pi$
$12$	1728.00	0.288675
$13$	2197.00	0.277350
$13$	2197.00	0.277350
$14$	−2744.00	−0.267261
$15$	−270.000	−0.0206559
$16$	4096.00	0.250000
$17$	−1042.00	−0.0514395	−0.0257197	−	0.999669i	$-0.508188\pi$
$17$	−1042.00	−0.0514395	−0.0257197	+	0.999669i	$0.508188\pi$
$18$	5832.00	0.235702
$19$	−9068.00	−0.303301	−0.151651	−	0.988434i	$-0.548459\pi$
$19$	−9068.00	−0.303301	−0.151651	+	0.988434i	$0.548459\pi$
$20$	−640.000	−0.0178885
$21$	−9261.00	−0.218218
$22$	12064.0	0.241553
$23$	−98988.0	−1.69643	−0.848213	−	0.529655i	$-0.822322\pi$
$23$	−98988.0	−1.69643	−0.848213	+	0.529655i	$0.822322\pi$
$24$	13824.0	0.204124
$25$	−78025.0	−0.998720
$26$	17576.0	0.196116
$27$	19683.0	0.192450
$28$	−21952.0	−0.188982
$29$	−213642.	−1.62665	−0.813324	−	0.581811i	$-0.802344\pi$
$29$	−213642.	−1.62665	−0.813324	+	0.581811i	$0.802344\pi$
$30$	−2160.00	−0.0146059
$31$	−22048.0	−0.132924	−0.0664620	−	0.997789i	$-0.521171\pi$
$31$	−22048.0	−0.132924	−0.0664620	+	0.997789i	$0.521171\pi$
$32$	32768.0	0.176777
$33$	40716.0	0.197227
$34$	−8336.00	−0.0363732
$35$	3430.00	0.0135225
$36$	46656.0	0.166667
$37$	418246.	1.35746	0.678728	−	0.734390i	$-0.262532\pi$
$37$	418246.	1.35746	0.678728	+	0.734390i	$0.262532\pi$
$38$	−72544.0	−0.214466
$39$	59319.0	0.160128
$40$	−5120.00	−0.0126491
$41$	−76414.0	−0.173153	−0.0865764	−	0.996245i	$-0.527593\pi$
$41$	−76414.0	−0.173153	−0.0865764	+	0.996245i	$0.527593\pi$
$42$	−74088.0	−0.154303
$43$	−177524.	−0.340500	−0.170250	−	0.985401i	$-0.554458\pi$
$43$	−177524.	−0.340500	−0.170250	+	0.985401i	$0.554458\pi$
$44$	96512.0	0.170804
$45$	−7290.00	−0.0119257
$46$	−791904.	−1.19955
$47$	631916.	0.887803	0.443902	−	0.896075i	$-0.353594\pi$
$47$	631916.	0.887803	0.443902	+	0.896075i	$0.353594\pi$
$48$	110592.	0.144338
$49$	117649.	0.142857
$50$	−624200.	−0.706202
$51$	−28134.0	−0.0296986
$52$	140608.	0.138675
$53$	−982354.	−0.906364	−0.453182	−	0.891418i	$-0.649711\pi$
$53$	−982354.	−0.906364	−0.453182	+	0.891418i	$0.649711\pi$
$54$	157464.	0.136083
$55$	−15080.0	−0.0122217
$56$	−175616.	−0.133631
$57$	−244836.	−0.175111
$58$	−1.70914e6	−1.15021
$59$	−596384.	−0.378045	−0.189023	−	0.981973i	$-0.560532\pi$
$59$	−596384.	−0.378045	−0.189023	+	0.981973i	$0.560532\pi$
$60$	−17280.0	−0.0103280
$61$	1.86341e6	1.05112	0.525561	−	0.850756i	$-0.323855\pi$
$61$	1.86341e6	1.05112	0.525561	+	0.850756i	$0.323855\pi$
$62$	−176384.	−0.0939914
$63$	−250047.	−0.125988
$64$	262144.	0.125000
$65$	−21970.0	−0.00992278
$66$	325728.	0.139461
$67$	−1.84565e6	−0.749701	−0.374850	−	0.927085i	$-0.622306\pi$
$67$	−1.84565e6	−0.749701	−0.374850	+	0.927085i	$0.622306\pi$
$68$	−66688.0	−0.0257197
$69$	−2.67268e6	−0.979432
$70$	27440.0	0.00956183
$71$	1.63228e6	0.541241	0.270620	−	0.962686i	$-0.412771\pi$
$71$	1.63228e6	0.541241	0.270620	+	0.962686i	$0.412771\pi$
$72$	373248.	0.117851
$73$	216650.	0.0651822	0.0325911	−	0.999469i	$-0.489624\pi$
$73$	216650.	0.0651822	0.0325911	+	0.999469i	$0.489624\pi$
$74$	3.34597e6	0.959866
$75$	−2.10668e6	−0.576611
$76$	−580352.	−0.151651
$77$	−517244.	−0.129115
$78$	474552.	0.113228
$79$	−6.46017e6	−1.47417	−0.737087	−	0.675797i	$-0.763799\pi$
$79$	−6.46017e6	−1.47417	−0.737087	+	0.675797i	$0.763799\pi$
$80$	−40960.0	−0.00894427
$81$	531441.	0.111111
$82$	−611312.	−0.122437
$83$	3.59215e6	0.689575	0.344787	−	0.938681i	$-0.387951\pi$
$83$	3.59215e6	0.689575	0.344787	+	0.938681i	$0.387951\pi$
$84$	−592704.	−0.109109
$85$	10420.0	0.00184035
$86$	−1.42019e6	−0.240770
$87$	−5.76833e6	−0.939146
$88$	772096.	0.120776
$89$	−9.48266e6	−1.42582	−0.712911	−	0.701255i	$-0.752624\pi$
$89$	−9.48266e6	−1.42582	−0.712911	+	0.701255i	$0.752624\pi$
$90$	−58320.0	−0.00843274
$91$	−753571.	−0.104828
$92$	−6.33523e6	−0.848213
$93$	−595296.	−0.0767437
$94$	5.05533e6	0.627772
$95$	90680.0	0.0108512
$96$	884736.	0.102062
$97$	840226.	0.0934749	0.0467375	−	0.998907i	$-0.485118\pi$
$97$	840226.	0.0934749	0.0467375	+	0.998907i	$0.485118\pi$
$98$	941192.	0.101015
$99$	1.09933e6	0.113869
$100$	−4.99360e6	−0.499360
$101$	−8.66694e6	−0.837030	−0.418515	−	0.908210i	$-0.637449\pi$
$101$	−8.66694e6	−0.837030	−0.418515	+	0.908210i	$0.637449\pi$
$102$	−225072.	−0.0210001
$103$	−9.63078e6	−0.868422	−0.434211	−	0.900811i	$-0.642973\pi$
$103$	−9.63078e6	−0.868422	−0.434211	+	0.900811i	$0.642973\pi$
$104$	1.12486e6	0.0980581
$105$	92610.0	0.00780720
$106$	−7.85883e6	−0.640896
$107$	3.51576e6	0.277444	0.138722	−	0.990331i	$-0.455700\pi$
$107$	3.51576e6	0.277444	0.138722	+	0.990331i	$0.455700\pi$
$108$	1.25971e6	0.0962250
$109$	1.15869e6	0.0856991	0.0428495	−	0.999082i	$-0.486356\pi$
$109$	1.15869e6	0.0856991	0.0428495	+	0.999082i	$0.486356\pi$
$110$	−120640.	−0.00864205
$111$	1.12926e7	0.783728
$112$	−1.40493e6	−0.0944911
$113$	−7.50697e6	−0.489429	−0.244715	−	0.969595i	$-0.578694\pi$
$113$	−7.50697e6	−0.489429	−0.244715	+	0.969595i	$0.578694\pi$
$114$	−1.95869e6	−0.123822
$115$	989880.	0.0606932
$116$	−1.36731e7	−0.813324
$117$	1.60161e6	0.0924500
$118$	−4.77107e6	−0.267318
$119$	357406.	0.0194423
$120$	−138240.	−0.00730297
$121$	−1.72131e7	−0.883305
$122$	1.49072e7	0.743255
$123$	−2.06318e6	−0.0999698
$124$	−1.41107e6	−0.0664620
$125$	1.56150e6	0.0715084
$126$	−2.00038e6	−0.0890871
$127$	−2.50594e7	−1.08557	−0.542784	−	0.839872i	$-0.682630\pi$
$127$	−2.50594e7	−1.08557	−0.542784	+	0.839872i	$0.682630\pi$
$128$	2.09715e6	0.0883883
$129$	−4.79315e6	−0.196588
$130$	−175760.	−0.00701646
$131$	1.65072e7	0.641540	0.320770	−	0.947157i	$-0.396058\pi$
$131$	1.65072e7	0.641540	0.320770	+	0.947157i	$0.396058\pi$
$132$	2.60582e6	0.0986135
$133$	3.11032e6	0.114637
$134$	−1.47652e7	−0.530119
$135$	−196830.	−0.00688530
$136$	−533504.	−0.0181866
$137$	−2.64726e6	−0.0879578	−0.0439789	−	0.999032i	$-0.514003\pi$
$137$	−2.64726e6	−0.0879578	−0.0439789	+	0.999032i	$0.514003\pi$
$138$	−2.13814e7	−0.692563
$139$	−4.79145e7	−1.51327	−0.756634	−	0.653839i	$-0.773157\pi$
$139$	−4.79145e7	−1.51327	−0.756634	+	0.653839i	$0.773157\pi$
$140$	219520.	0.00676123
$141$	1.70617e7	0.512574
$142$	1.30582e7	0.382715
$143$	3.31308e6	0.0947448
$144$	2.98598e6	0.0833333
$145$	2.13642e6	0.0581967
$146$	1.73320e6	0.0460907
$147$	3.17652e6	0.0824786
$148$	2.67677e7	0.678728
$149$	−1.62956e7	−0.403569	−0.201784	−	0.979430i	$-0.564674\pi$
$149$	−1.62956e7	−0.403569	−0.201784	+	0.979430i	$0.564674\pi$
$150$	−1.68534e7	−0.407726
$151$	6.98150e7	1.65017	0.825086	−	0.565007i	$-0.191126\pi$
$151$	6.98150e7	1.65017	0.825086	+	0.565007i	$0.191126\pi$
$152$	−4.64282e6	−0.107233
$153$	−759618.	−0.0171465
$154$	−4.13795e6	−0.0912983
$155$	220480.	0.00475563
$156$	3.79642e6	0.0800641
$157$	−3.05107e7	−0.629221	−0.314611	−	0.949221i	$-0.601874\pi$
$157$	−3.05107e7	−0.629221	−0.314611	+	0.949221i	$0.601874\pi$
$158$	−5.16813e7	−1.04240
$159$	−2.65236e7	−0.523289
$160$	−327680.	−0.00632456
$161$	3.39529e7	0.641189
$162$	4.25153e6	0.0785674
$163$	1.03245e6	0.0186730	0.00933648	−	0.999956i	$-0.497028\pi$
$163$	1.03245e6	0.0186730	0.00933648	+	0.999956i	$0.497028\pi$
$164$	−4.89050e6	−0.0865764
$165$	−407160.	−0.00705621
$166$	2.87372e7	0.487603
$167$	−5.59050e7	−0.928845	−0.464423	−	0.885614i	$-0.653738\pi$
$167$	−5.59050e7	−0.928845	−0.464423	+	0.885614i	$0.653738\pi$
$168$	−4.74163e6	−0.0771517
$169$	4.82681e6	0.0769231
$170$	83360.0	0.00130133
$171$	−6.61057e6	−0.101100
$172$	−1.13615e7	−0.170250
$173$	−4.10563e7	−0.602863	−0.301431	−	0.953488i	$-0.597464\pi$
$173$	−4.10563e7	−0.602863	−0.301431	+	0.953488i	$0.597464\pi$
$174$	−4.61467e7	−0.664076
$175$	2.67626e7	0.377481
$176$	6.17677e6	0.0854018
$177$	−1.61024e7	−0.218265
$178$	−7.58613e7	−1.00821
$179$	−7.65930e7	−0.998167	−0.499083	−	0.866554i	$-0.666330\pi$
$179$	−7.65930e7	−0.998167	−0.499083	+	0.866554i	$0.666330\pi$
$180$	−466560.	−0.00596285
$181$	−7.49751e7	−0.939815	−0.469907	−	0.882716i	$-0.655713\pi$
$181$	−7.49751e7	−0.939815	−0.469907	+	0.882716i	$0.655713\pi$
$182$	−6.02857e6	−0.0741249
$183$	5.03120e7	0.606865
$184$	−5.06819e7	−0.599777
$185$	−4.18246e6	−0.0485658
$186$	−4.76237e6	−0.0542660
$187$	−1.57134e6	−0.0175721
$188$	4.04426e7	0.443902
$189$	−6.75127e6	−0.0727393
$190$	725440.	0.00767298
$191$	1.50516e8	1.56303	0.781514	−	0.623888i	$-0.214448\pi$
$191$	1.50516e8	1.56303	0.781514	+	0.623888i	$0.214448\pi$
$192$	7.07789e6	0.0721688
$193$	1.26286e8	1.26445	0.632227	−	0.774783i	$-0.282141\pi$
$193$	1.26286e8	1.26445	0.632227	+	0.774783i	$0.282141\pi$
$194$	6.72181e6	0.0660967
$195$	−593190.	−0.00572892
$196$	7.52954e6	0.0714286
$197$	−8.50143e7	−0.792246	−0.396123	−	0.918197i	$-0.629645\pi$
$197$	−8.50143e7	−0.792246	−0.396123	+	0.918197i	$0.629645\pi$
$198$	8.79466e6	0.0805176
$199$	−7.77938e7	−0.699776	−0.349888	−	0.936792i	$-0.613780\pi$
$199$	−7.77938e7	−0.699776	−0.349888	+	0.936792i	$0.613780\pi$
$200$	−3.99488e7	−0.353101
$201$	−4.98326e7	−0.432840
$202$	−6.93355e7	−0.591870
$203$	7.32792e7	0.614815
$204$	−1.80058e6	−0.0148493
$205$	764140.	0.00619490
$206$	−7.70462e7	−0.614067
$207$	−7.21623e7	−0.565476
$208$	8.99891e6	0.0693375
$209$	−1.36745e7	−0.103610
$210$	740880.	0.00552052
$211$	1.51198e8	1.10805	0.554023	−	0.832501i	$-0.313092\pi$
$211$	1.51198e8	1.10805	0.554023	+	0.832501i	$0.313092\pi$
$212$	−6.28707e7	−0.453182
$213$	4.40716e7	0.312485
$214$	2.81261e7	0.196183
$215$	1.77524e6	0.0121821
$216$	1.00777e7	0.0680414
$217$	7.56246e6	0.0502405
$218$	9.26955e6	0.0605984
$219$	5.84955e6	0.0376329
$220$	−965120.	−0.00611085
$221$	−2.28927e6	−0.0142667
$222$	9.03411e7	0.554179
$223$	−9.82783e7	−0.593459	−0.296729	−	0.954962i	$-0.595896\pi$
$223$	−9.82783e7	−0.593459	−0.296729	+	0.954962i	$0.595896\pi$
$224$	−1.12394e7	−0.0668153
$225$	−5.68802e7	−0.332907
$226$	−6.00557e7	−0.346079
$227$	−1.80043e8	−1.02161	−0.510806	−	0.859696i	$-0.670653\pi$
$227$	−1.80043e8	−1.02161	−0.510806	+	0.859696i	$0.670653\pi$
$228$	−1.56695e7	−0.0875555
$229$	3.10223e7	0.170707	0.0853533	−	0.996351i	$-0.472798\pi$
$229$	3.10223e7	0.170707	0.0853533	+	0.996351i	$0.472798\pi$
$230$	7.91904e6	0.0429166
$231$	−1.39656e7	−0.0745448
$232$	−1.09385e8	−0.575107
$233$	−8.98629e7	−0.465409	−0.232704	−	0.972548i	$-0.574757\pi$
$233$	−8.98629e7	−0.465409	−0.232704	+	0.972548i	$0.574757\pi$
$234$	1.28129e7	0.0653720
$235$	−6.31916e6	−0.0317630
$236$	−3.81686e7	−0.189023
$237$	−1.74425e8	−0.851115
$238$	2.85925e6	0.0137478
$239$	5.22535e6	0.0247584	0.0123792	−	0.999923i	$-0.496059\pi$
$239$	5.22535e6	0.0247584	0.0123792	+	0.999923i	$0.496059\pi$
$240$	−1.10592e6	−0.00516398
$241$	4.35619e7	0.200469	0.100234	−	0.994964i	$-0.468041\pi$
$241$	4.35619e7	0.200469	0.100234	+	0.994964i	$0.468041\pi$
$242$	−1.37705e8	−0.624591
$243$	1.43489e7	0.0641500
$244$	1.19258e8	0.525561
$245$	−1.17649e6	−0.00511101
$246$	−1.65054e7	−0.0706893
$247$	−1.99224e7	−0.0841206
$248$	−1.12886e7	−0.0469957
$249$	9.69881e7	0.398126
$250$	1.24920e7	0.0505641
$251$	3.21053e8	1.28150	0.640750	−	0.767750i	$-0.278624\pi$
$251$	3.21053e8	1.28150	0.640750	+	0.767750i	$0.278624\pi$
$252$	−1.60030e7	−0.0629941
$253$	−1.49274e8	−0.579512
$254$	−2.00475e8	−0.767613
$255$	281340.	0.00106253
$256$	1.67772e7	0.0625000
$257$	5.27063e7	0.193685	0.0968427	−	0.995300i	$-0.469126\pi$
$257$	5.27063e7	0.193685	0.0968427	+	0.995300i	$0.469126\pi$
$258$	−3.83452e7	−0.139009
$259$	−1.43458e8	−0.513070
$260$	−1.40608e6	−0.00496139
$261$	−1.55745e8	−0.542216
$262$	1.32058e8	0.453638
$263$	−4.90103e8	−1.66128	−0.830638	−	0.556812i	$-0.812024\pi$
$263$	−4.90103e8	−1.66128	−0.830638	+	0.556812i	$0.812024\pi$
$264$	2.08466e7	0.0697303
$265$	9.82354e6	0.0324271
$266$	2.48826e7	0.0810606
$267$	−2.56032e8	−0.823199
$268$	−1.18122e8	−0.374850
$269$	3.43105e8	1.07472	0.537358	−	0.843354i	$-0.319422\pi$
$269$	3.43105e8	1.07472	0.537358	+	0.843354i	$0.319422\pi$
$270$	−1.57464e6	−0.00486864
$271$	1.83046e8	0.558686	0.279343	−	0.960191i	$-0.409883\pi$
$271$	1.83046e8	0.558686	0.279343	+	0.960191i	$0.409883\pi$
$272$	−4.26803e6	−0.0128599
$273$	−2.03464e7	−0.0605228
$274$	−2.11781e7	−0.0621955
$275$	−1.17662e8	−0.341170
$276$	−1.71051e8	−0.489716
$277$	3.26261e8	0.922328	0.461164	−	0.887315i	$-0.347432\pi$
$277$	3.26261e8	0.922328	0.461164	+	0.887315i	$0.347432\pi$
$278$	−3.83316e8	−1.07004
$279$	−1.60730e7	−0.0443080
$280$	1.75616e6	0.00478091
$281$	−3.37042e8	−0.906173	−0.453087	−	0.891466i	$-0.649677\pi$
$281$	−3.37042e8	−0.906173	−0.453087	+	0.891466i	$0.649677\pi$
$282$	1.36494e8	0.362444
$283$	3.08528e8	0.809175	0.404587	−	0.914499i	$-0.367415\pi$
$283$	3.08528e8	0.809175	0.404587	+	0.914499i	$0.367415\pi$
$284$	1.04466e8	0.270620
$285$	2.44836e6	0.00626496
$286$	2.65046e7	0.0669947
$287$	2.62100e7	0.0654456
$288$	2.38879e7	0.0589256
$289$	−4.09253e8	−0.997354
$290$	1.70914e7	0.0411513
$291$	2.26861e7	0.0539678
$292$	1.38656e7	0.0325911
$293$	4.83962e8	1.12402	0.562011	−	0.827130i	$-0.310028\pi$
$293$	4.83962e8	1.12402	0.562011	+	0.827130i	$0.310028\pi$
$294$	2.54122e7	0.0583212
$295$	5.96384e6	0.0135254
$296$	2.14142e8	0.479933
$297$	2.96820e7	0.0657423
$298$	−1.30365e8	−0.285366
$299$	−2.17477e8	−0.470504
$300$	−1.34827e8	−0.288306
$301$	6.08907e7	0.128697
$302$	5.58520e8	1.16685
$303$	−2.34007e8	−0.483260
$304$	−3.71425e7	−0.0758253
$305$	−1.86341e7	−0.0376061
$306$	−6.07694e6	−0.0121244
$307$	−6.16455e8	−1.21595	−0.607977	−	0.793955i	$-0.708019\pi$
$307$	−6.16455e8	−1.21595	−0.607977	+	0.793955i	$0.708019\pi$
$308$	−3.31036e7	−0.0645577
$309$	−2.60031e8	−0.501384
$310$	1.76384e6	0.00336274
$311$	4.63646e7	0.0874028	0.0437014	−	0.999045i	$-0.486085\pi$
$311$	4.63646e7	0.0874028	0.0437014	+	0.999045i	$0.486085\pi$
$312$	3.03713e7	0.0566139
$313$	−8.55946e8	−1.57776	−0.788881	−	0.614546i	$-0.789339\pi$
$313$	−8.55946e8	−1.57776	−0.788881	+	0.614546i	$0.789339\pi$
$314$	−2.44086e8	−0.444927
$315$	2.50047e6	0.00450749
$316$	−4.13451e8	−0.737087
$317$	3.66851e8	0.646819	0.323409	−	0.946259i	$-0.395171\pi$
$317$	3.66851e8	0.646819	0.323409	+	0.946259i	$0.395171\pi$
$318$	−2.12188e8	−0.370021
$319$	−3.22172e8	−0.555675
$320$	−2.62144e6	−0.00447214
$321$	9.49255e7	0.160183
$322$	2.71623e8	0.453389
$323$	9.44886e6	0.0156016
$324$	3.40122e7	0.0555556
$325$	−1.71421e8	−0.276995
$326$	8.25962e6	0.0132038
$327$	3.12847e7	0.0494784
$328$	−3.91240e7	−0.0612187
$329$	−2.16747e8	−0.335558
$330$	−3.25728e6	−0.00498949
$331$	3.26194e8	0.494399	0.247200	−	0.968965i	$-0.420490\pi$
$331$	3.26194e8	0.494399	0.247200	+	0.968965i	$0.420490\pi$
$332$	2.29898e8	0.344787
$333$	3.04901e8	0.452485
$334$	−4.47240e8	−0.656793
$335$	1.84565e7	0.0268221
$336$	−3.79331e7	−0.0545545
$337$	7.22325e8	1.02808	0.514042	−	0.857765i	$-0.328148\pi$
$337$	7.22325e8	1.02808	0.514042	+	0.857765i	$0.328148\pi$
$338$	3.86145e7	0.0543928
$339$	−2.02688e8	−0.282572
$340$	666880.	0.000920177 0
$341$	−3.32484e7	−0.0454078
$342$	−5.28846e7	−0.0714887
$343$	−4.03536e7	−0.0539949
$344$	−9.08923e7	−0.120385
$345$	2.67268e7	0.0350412
$346$	−3.28450e8	−0.426288
$347$	4.03337e8	0.518220	0.259110	−	0.965848i	$-0.416571\pi$
$347$	4.03337e8	0.518220	0.259110	+	0.965848i	$0.416571\pi$
$348$	−3.69173e8	−0.469573
$349$	6.62276e8	0.833969	0.416985	−	0.908914i	$-0.363087\pi$
$349$	6.62276e8	0.833969	0.416985	+	0.908914i	$0.363087\pi$
$350$	2.14101e8	0.266919
$351$	4.32436e7	0.0533761
$352$	4.94141e7	0.0603882
$353$	−1.33432e8	−0.161453	−0.0807267	−	0.996736i	$-0.525724\pi$
$353$	−1.33432e8	−0.161453	−0.0807267	+	0.996736i	$0.525724\pi$
$354$	−1.28819e8	−0.154336
$355$	−1.63228e7	−0.0193640
$356$	−6.06890e8	−0.712911
$357$	9.64996e6	0.0112250
$358$	−6.12744e8	−0.705811
$359$	−3.89725e8	−0.444557	−0.222279	−	0.974983i	$-0.571349\pi$
$359$	−3.89725e8	−0.444557	−0.222279	+	0.974983i	$0.571349\pi$
$360$	−3.73248e6	−0.00421637
$361$	−8.11643e8	−0.908008
$362$	−5.99801e8	−0.664549
$363$	−4.64754e8	−0.509976
$364$	−4.82285e7	−0.0524142
$365$	−2.16650e6	−0.00233203
$366$	4.02496e8	0.429119
$367$	7.13690e8	0.753666	0.376833	−	0.926281i	$-0.377013\pi$
$367$	7.13690e8	0.753666	0.376833	+	0.926281i	$0.377013\pi$
$368$	−4.05455e8	−0.424107
$369$	−5.57058e7	−0.0577176
$370$	−3.34597e7	−0.0343412
$371$	3.36947e8	0.342573
$372$	−3.80989e7	−0.0383718
$373$	−8.19941e8	−0.818091	−0.409046	−	0.912514i	$-0.634138\pi$
$373$	−8.19941e8	−0.818091	−0.409046	+	0.912514i	$0.634138\pi$
$374$	−1.25707e7	−0.0124253
$375$	4.21605e7	0.0412854
$376$	3.23541e8	0.313886
$377$	−4.69371e8	−0.451151
$378$	−5.40102e7	−0.0514344
$379$	3.57866e8	0.337663	0.168831	−	0.985645i	$-0.446001\pi$
$379$	3.57866e8	0.337663	0.168831	+	0.985645i	$0.446001\pi$
$380$	5.80352e6	0.00542561
$381$	−6.76603e8	−0.626753
$382$	1.20413e9	1.10523
$383$	8.40713e8	0.764632	0.382316	−	0.924032i	$-0.375127\pi$
$383$	8.40713e8	0.764632	0.382316	+	0.924032i	$0.375127\pi$
$384$	5.66231e7	0.0510310
$385$	5.17244e6	0.00461937
$386$	1.01029e9	0.894105
$387$	−1.29415e8	−0.113500
$388$	5.37745e7	0.0467375
$389$	1.30488e9	1.12395	0.561975	−	0.827154i	$-0.310042\pi$
$389$	1.30488e9	1.12395	0.561975	+	0.827154i	$0.310042\pi$
$390$	−4.74552e6	−0.00405096
$391$	1.03145e8	0.0872633
$392$	6.02363e7	0.0505076
$393$	4.45695e8	0.370393
$394$	−6.80115e8	−0.560203
$395$	6.46017e7	0.0527417
$396$	7.03572e7	0.0569345
$397$	−1.07773e9	−0.864458	−0.432229	−	0.901764i	$-0.642273\pi$
$397$	−1.07773e9	−0.864458	−0.432229	+	0.901764i	$0.642273\pi$
$398$	−6.22350e8	−0.494816
$399$	8.39787e7	0.0661857
$400$	−3.19590e8	−0.249680
$401$	−1.80045e9	−1.39436	−0.697181	−	0.716895i	$-0.745563\pi$
$401$	−1.80045e9	−1.39436	−0.697181	+	0.716895i	$0.745563\pi$
$402$	−3.98661e8	−0.306064
$403$	−4.84395e7	−0.0368665
$404$	−5.54684e8	−0.418515
$405$	−5.31441e6	−0.00397523
$406$	5.86234e8	0.434740
$407$	6.30715e8	0.463717
$408$	−1.44046e7	−0.0105000
$409$	−2.50445e9	−1.81001	−0.905004	−	0.425402i	$-0.860133\pi$
$409$	−2.50445e9	−1.81001	−0.905004	+	0.425402i	$0.860133\pi$
$410$	6.11312e6	0.00438046
$411$	−7.14760e7	−0.0507825
$412$	−6.16370e8	−0.434211
$413$	2.04560e8	0.142888
$414$	−5.77298e8	−0.399852
$415$	−3.59215e7	−0.0246710
$416$	7.19913e7	0.0490290
$417$	−1.29369e9	−0.873685
$418$	−1.09396e8	−0.0732632
$419$	−4.49332e8	−0.298414	−0.149207	−	0.988806i	$-0.547672\pi$
$419$	−4.49332e8	−0.298414	−0.149207	+	0.988806i	$0.547672\pi$
$420$	5.92704e6	0.00390360
$421$	7.67923e8	0.501569	0.250784	−	0.968043i	$-0.419311\pi$
$421$	7.67923e8	0.501569	0.250784	+	0.968043i	$0.419311\pi$
$422$	1.20959e9	0.783507
$423$	4.60667e8	0.295934
$424$	−5.02965e8	−0.320448
$425$	8.13020e7	0.0513736
$426$	3.52572e8	0.220961
$427$	−6.39148e8	−0.397287
$428$	2.25009e8	0.138722
$429$	8.94531e7	0.0547009
$430$	1.42019e7	0.00861405
$431$	−7.03828e8	−0.423444	−0.211722	−	0.977330i	$-0.567907\pi$
$431$	−7.03828e8	−0.423444	−0.211722	+	0.977330i	$0.567907\pi$
$432$	8.06216e7	0.0481125
$433$	1.67590e9	0.992063	0.496032	−	0.868304i	$-0.334790\pi$
$433$	1.67590e9	0.992063	0.496032	+	0.868304i	$0.334790\pi$
$434$	6.04997e7	0.0355254
$435$	5.76833e7	0.0335999
$436$	7.41564e7	0.0428495
$437$	8.97623e8	0.514528
$438$	4.67964e7	0.0266105
$439$	2.12336e9	1.19784	0.598918	−	0.800811i	$-0.295598\pi$
$439$	2.12336e9	1.19784	0.598918	+	0.800811i	$0.295598\pi$
$440$	−7.72096e6	−0.00432103
$441$	8.57661e7	0.0476190
$442$	−1.83142e7	−0.0100881
$443$	1.19425e9	0.652654	0.326327	−	0.945257i	$-0.394189\pi$
$443$	1.19425e9	0.652654	0.326327	+	0.945257i	$0.394189\pi$
$444$	7.22729e8	0.391864
$445$	9.48266e7	0.0510118
$446$	−7.86226e8	−0.419639
$447$	−4.39980e8	−0.233001
$448$	−8.99154e7	−0.0472456
$449$	1.62441e9	0.846903	0.423451	−	0.905919i	$-0.360818\pi$
$449$	1.62441e9	0.846903	0.423451	+	0.905919i	$0.360818\pi$
$450$	−4.55042e8	−0.235401
$451$	−1.15232e8	−0.0591502
$452$	−4.80446e8	−0.244715
$453$	1.88500e9	0.952728
$454$	−1.44034e9	−0.722388
$455$	7.53571e6	0.00375046
$456$	−1.25356e8	−0.0619111
$457$	1.96761e8	0.0964343	0.0482172	−	0.998837i	$-0.484646\pi$
$457$	1.96761e8	0.0964343	0.0482172	+	0.998837i	$0.484646\pi$
$458$	2.48179e8	0.120708
$459$	−2.05097e7	−0.00989953
$460$	6.33523e7	0.0303466
$461$	1.32676e9	0.630722	0.315361	−	0.948972i	$-0.397874\pi$
$461$	1.32676e9	0.630722	0.315361	+	0.948972i	$0.397874\pi$
$462$	−1.11725e8	−0.0527111
$463$	2.87807e9	1.34762	0.673810	−	0.738904i	$-0.264656\pi$
$463$	2.87807e9	1.34762	0.673810	+	0.738904i	$0.264656\pi$
$464$	−8.75078e8	−0.406662
$465$	5.95296e6	0.00274567
$466$	−7.18903e8	−0.329094
$467$	2.01274e9	0.914487	0.457244	−	0.889342i	$-0.348837\pi$
$467$	2.01274e9	0.914487	0.457244	+	0.889342i	$0.348837\pi$
$468$	1.02503e8	0.0462250
$469$	6.33059e8	0.283360
$470$	−5.05533e7	−0.0224598
$471$	−8.23789e8	−0.363281
$472$	−3.05349e8	−0.133659
$473$	−2.67706e8	−0.116317
$474$	−1.39540e9	−0.601829
$475$	7.07531e8	0.302913
$476$	2.28740e7	0.00972115
$477$	−7.16136e8	−0.302121
$478$	4.18028e7	0.0175068
$479$	−1.72294e9	−0.716302	−0.358151	−	0.933664i	$-0.616593\pi$
$479$	−1.72294e9	−0.716302	−0.358151	+	0.933664i	$0.616593\pi$
$480$	−8.84736e6	−0.00365148
$481$	9.18886e8	0.376491
$482$	3.48495e8	0.141753
$483$	9.16728e8	0.370191
$484$	−1.10164e9	−0.441652
$485$	−8.40226e6	−0.00334426
$486$	1.14791e8	0.0453609
$487$	−3.30049e9	−1.29487	−0.647437	−	0.762119i	$-0.724159\pi$
$487$	−3.30049e9	−1.29487	−0.647437	+	0.762119i	$0.724159\pi$
$488$	9.54064e8	0.371628
$489$	2.78762e7	0.0107808
$490$	−9.41192e6	−0.00361403
$491$	−2.46868e8	−0.0941194	−0.0470597	−	0.998892i	$-0.514985\pi$
$491$	−2.46868e8	−0.0941194	−0.0470597	+	0.998892i	$0.514985\pi$
$492$	−1.32043e8	−0.0499849
$493$	2.22615e8	0.0836739
$494$	−1.59379e8	−0.0594822
$495$	−1.09933e7	−0.00407390
$496$	−9.03086e7	−0.0332310
$497$	−5.59872e8	−0.204570
$498$	7.75905e8	0.281518
$499$	3.62691e8	0.130673	0.0653364	−	0.997863i	$-0.479188\pi$
$499$	3.62691e8	0.130673	0.0653364	+	0.997863i	$0.479188\pi$
$500$	9.99360e7	0.0357542
$501$	−1.50944e9	−0.536269
$502$	2.56842e9	0.906157
$503$	−1.48371e9	−0.519829	−0.259914	−	0.965632i	$-0.583694\pi$
$503$	−1.48371e9	−0.519829	−0.259914	+	0.965632i	$0.583694\pi$
$504$	−1.28024e8	−0.0445435
$505$	8.66694e7	0.0299465
$506$	−1.19419e9	−0.409777
$507$	1.30324e8	0.0444116
$508$	−1.60380e9	−0.542784
$509$	−4.96412e8	−0.166851	−0.0834257	−	0.996514i	$-0.526586\pi$
$509$	−4.96412e8	−0.166851	−0.0834257	+	0.996514i	$0.526586\pi$
$510$	2.25072e6	0.000751322 0
$511$	−7.43110e7	−0.0246365
$512$	1.34218e8	0.0441942
$513$	−1.78485e8	−0.0583703
$514$	4.21651e8	0.136956
$515$	9.63078e7	0.0310696
$516$	−3.06761e8	−0.0982940
$517$	9.52929e8	0.303280
$518$	−1.14767e9	−0.362795
$519$	−1.10852e9	−0.348063
$520$	−1.12486e7	−0.00350823
$521$	9.52711e8	0.295141	0.147570	−	0.989052i	$-0.452855\pi$
$521$	9.52711e8	0.295141	0.147570	+	0.989052i	$0.452855\pi$
$522$	−1.24596e9	−0.383405
$523$	−4.02099e9	−1.22907	−0.614535	−	0.788890i	$-0.710656\pi$
$523$	−4.02099e9	−1.22907	−0.614535	+	0.788890i	$0.710656\pi$
$524$	1.05646e9	0.320770
$525$	7.22590e8	0.217939
$526$	−3.92082e9	−1.17470
$527$	2.29740e7	0.00683754
$528$	1.66773e8	0.0493067
$529$	6.39380e9	1.87786
$530$	7.85883e7	0.0229294
$531$	−4.34764e8	−0.126015
$532$	1.99061e8	0.0573185
$533$	−1.67882e8	−0.0480239
$534$	−2.04825e9	−0.582089
$535$	−3.51576e7	−0.00992615
$536$	−9.44974e8	−0.265059
$537$	−2.06801e9	−0.576292
$538$	2.74484e9	0.759939
$539$	1.77415e8	0.0488010
$540$	−1.25971e7	−0.00344265
$541$	1.54687e9	0.420013	0.210006	−	0.977700i	$-0.432652\pi$
$541$	1.54687e9	0.420013	0.210006	+	0.977700i	$0.432652\pi$
$542$	1.46437e9	0.395051
$543$	−2.02433e9	−0.542602
$544$	−3.41443e7	−0.00909330
$545$	−1.15869e7	−0.00306606
$546$	−1.62771e8	−0.0427960
$547$	−2.54655e9	−0.665267	−0.332634	−	0.943056i	$-0.607937\pi$
$547$	−2.54655e9	−0.665267	−0.332634	+	0.943056i	$0.607937\pi$
$548$	−1.69425e8	−0.0439789
$549$	1.35842e9	0.350374
$550$	−9.41294e8	−0.241244
$551$	1.93731e9	0.493364
$552$	−1.36841e9	−0.346282
$553$	2.21584e9	0.557186
$554$	2.61008e9	0.652184
$555$	−1.12926e8	−0.0280395
$556$	−3.06653e9	−0.756634
$557$	6.56728e9	1.61025	0.805123	−	0.593107i	$-0.202099\pi$
$557$	6.56728e9	1.61025	0.805123	+	0.593107i	$0.202099\pi$
$558$	−1.28584e8	−0.0313305
$559$	−3.90020e8	−0.0944378
$560$	1.40493e7	0.00338062
$561$	−4.24261e7	−0.0101453
$562$	−2.69633e9	−0.640761
$563$	−1.05889e9	−0.250075	−0.125038	−	0.992152i	$-0.539905\pi$
$563$	−1.05889e9	−0.250075	−0.125038	+	0.992152i	$0.539905\pi$
$564$	1.09195e9	0.256287
$565$	7.50697e7	0.0175104
$566$	2.46823e9	0.572173
$567$	−1.82284e8	−0.0419961
$568$	8.35727e8	0.191357
$569$	3.21980e9	0.732717	0.366358	−	0.930474i	$-0.380604\pi$
$569$	3.21980e9	0.732717	0.366358	+	0.930474i	$0.380604\pi$
$570$	1.95869e7	0.00443000
$571$	−1.98851e9	−0.446995	−0.223497	−	0.974705i	$-0.571747\pi$
$571$	−1.98851e9	−0.446995	−0.223497	+	0.974705i	$0.571747\pi$
$572$	2.12037e8	0.0473724
$573$	4.06394e9	0.902415
$574$	2.09680e8	0.0462770
$575$	7.72354e9	1.69426
$576$	1.91103e8	0.0416667
$577$	4.08039e9	0.884272	0.442136	−	0.896948i	$-0.354221\pi$
$577$	4.08039e9	0.884272	0.442136	+	0.896948i	$0.354221\pi$
$578$	−3.27402e9	−0.705236
$579$	3.40971e9	0.730033
$580$	1.36731e8	0.0290984
$581$	−1.23211e9	−0.260635
$582$	1.81489e8	0.0381610
$583$	−1.48139e9	−0.309620
$584$	1.10925e8	0.0230454
$585$	−1.60161e7	−0.00330759
$586$	3.87170e9	0.794803
$587$	−2.96773e9	−0.605606	−0.302803	−	0.953053i	$-0.597922\pi$
$587$	−2.96773e9	−0.605606	−0.302803	+	0.953053i	$0.597922\pi$
$588$	2.03297e8	0.0412393
$589$	1.99931e8	0.0403160
$590$	4.77107e7	0.00956388
$591$	−2.29539e9	−0.457404
$592$	1.71314e9	0.339364
$593$	−8.04326e9	−1.58395	−0.791973	−	0.610556i	$-0.790946\pi$
$593$	−8.04326e9	−1.58395	−0.791973	+	0.610556i	$0.790946\pi$
$594$	2.37456e8	0.0464868
$595$	−3.57406e6	−0.000695589 0
$596$	−1.04292e9	−0.201784
$597$	−2.10043e9	−0.404016
$598$	−1.73981e9	−0.332697
$599$	4.02244e9	0.764708	0.382354	−	0.924016i	$-0.375113\pi$
$599$	4.02244e9	0.764708	0.382354	+	0.924016i	$0.375113\pi$
$600$	−1.07862e9	−0.203863
$601$	8.88903e9	1.67030	0.835149	−	0.550025i	$-0.185382\pi$
$601$	8.88903e9	1.67030	0.835149	+	0.550025i	$0.185382\pi$
$602$	4.87126e8	0.0910025
$603$	−1.34548e9	−0.249900
$604$	4.46816e9	0.825086
$605$	1.72131e8	0.0316021
$606$	−1.87206e9	−0.341716
$607$	−1.99678e9	−0.362384	−0.181192	−	0.983448i	$-0.557996\pi$
$607$	−1.99678e9	−0.362384	−0.181192	+	0.983448i	$0.557996\pi$
$608$	−2.97140e8	−0.0536166
$609$	1.97854e9	0.354964
$610$	−1.49072e8	−0.0265915
$611$	1.38832e9	0.246232
$612$	−4.86156e7	−0.00857325
$613$	2.57276e9	0.451115	0.225557	−	0.974230i	$-0.427580\pi$
$613$	2.57276e9	0.451115	0.225557	+	0.974230i	$0.427580\pi$
$614$	−4.93164e9	−0.859809
$615$	2.06318e7	0.00357663
$616$	−2.64829e8	−0.0456492
$617$	−2.39839e9	−0.411076	−0.205538	−	0.978649i	$-0.565894\pi$
$617$	−2.39839e9	−0.411076	−0.205538	+	0.978649i	$0.565894\pi$
$618$	−2.08025e9	−0.354532
$619$	7.51379e9	1.27333	0.636666	−	0.771140i	$-0.280313\pi$
$619$	7.51379e9	1.27333	0.636666	+	0.771140i	$0.280313\pi$
$620$	1.41107e7	0.00237782
$621$	−1.94838e9	−0.326477
$622$	3.70917e8	0.0618031
$623$	3.25255e9	0.538910
$624$	2.42971e8	0.0400320
$625$	6.08009e9	0.996162
$626$	−6.84757e9	−1.11565
$627$	−3.69213e8	−0.0598191
$628$	−1.95269e9	−0.314611
$629$	−4.35812e8	−0.0698268
$630$	2.00038e7	0.00318728
$631$	1.12189e10	1.77766	0.888828	−	0.458241i	$-0.151520\pi$
$631$	1.12189e10	1.77766	0.888828	+	0.458241i	$0.151520\pi$
$632$	−3.30761e9	−0.521200
$633$	4.08235e9	0.639731
$634$	2.93481e9	0.457370
$635$	2.50594e8	0.0388385
$636$	−1.69751e9	−0.261645
$637$	2.58475e8	0.0396214
$638$	−2.57738e9	−0.392921
$639$	1.18993e9	0.180414
$640$	−2.09715e7	−0.00316228
$641$	−4.73617e9	−0.710271	−0.355135	−	0.934815i	$-0.615565\pi$
$641$	−4.73617e9	−0.710271	−0.355135	+	0.934815i	$0.615565\pi$
$642$	7.59404e8	0.113266
$643$	2.70605e9	0.401418	0.200709	−	0.979651i	$-0.435675\pi$
$643$	2.70605e9	0.401418	0.200709	+	0.979651i	$0.435675\pi$
$644$	2.17298e9	0.320595
$645$	4.79315e7	0.00703334
$646$	7.55908e7	0.0110320
$647$	1.87404e9	0.272028	0.136014	−	0.990707i	$-0.456571\pi$
$647$	1.87404e9	0.272028	0.136014	+	0.990707i	$0.456571\pi$
$648$	2.72098e8	0.0392837
$649$	−8.99347e8	−0.129143
$650$	−1.37137e9	−0.195865
$651$	2.04187e8	0.0290064
$652$	6.60769e7	0.00933648
$653$	4.27498e9	0.600811	0.300406	−	0.953812i	$-0.402878\pi$
$653$	4.27498e9	0.600811	0.300406	+	0.953812i	$0.402878\pi$
$654$	2.50278e8	0.0349865
$655$	−1.65072e8	−0.0229524
$656$	−3.12992e8	−0.0432882
$657$	1.57938e8	0.0217274
$658$	−1.73398e9	−0.237275
$659$	5.27465e9	0.717951	0.358975	−	0.933347i	$-0.383126\pi$
$659$	5.27465e9	0.717951	0.358975	+	0.933347i	$0.383126\pi$
$660$	−2.60582e7	−0.00352810
$661$	8.72606e9	1.17520	0.587602	−	0.809150i	$-0.300072\pi$
$661$	8.72606e9	1.17520	0.587602	+	0.809150i	$0.300072\pi$
$662$	2.60955e9	0.349593
$663$	−6.18104e7	−0.00823691
$664$	1.83918e9	0.243802
$665$	−3.11032e7	−0.00410138
$666$	2.43921e9	0.319955
$667$	2.11480e10	2.75949
$668$	−3.57792e9	−0.464423
$669$	−2.65351e9	−0.342634
$670$	1.47652e8	0.0189661
$671$	2.81002e9	0.359071
$672$	−3.03464e8	−0.0385758
$673$	1.36345e10	1.72420	0.862098	−	0.506742i	$-0.169150\pi$
$673$	1.36345e10	1.72420	0.862098	+	0.506742i	$0.169150\pi$
$674$	5.77860e9	0.726965
$675$	−1.53577e9	−0.192204
$676$	3.08916e8	0.0384615
$677$	4.89007e9	0.605696	0.302848	−	0.953039i	$-0.402063\pi$
$677$	4.89007e9	0.605696	0.302848	+	0.953039i	$0.402063\pi$
$678$	−1.62150e9	−0.199809
$679$	−2.88198e8	−0.0353302
$680$	5.33504e6	0.000650664 0
$681$	−4.86116e9	−0.589828
$682$	−2.65987e8	−0.0321081
$683$	−7.61689e9	−0.914756	−0.457378	−	0.889272i	$-0.651211\pi$
$683$	−7.61689e9	−0.914756	−0.457378	+	0.889272i	$0.651211\pi$
$684$	−4.23077e8	−0.0505502
$685$	2.64726e7	0.00314687
$686$	−3.22829e8	−0.0381802
$687$	8.37603e8	0.0985575
$688$	−7.27138e8	−0.0851251
$689$	−2.15823e9	−0.251380
$690$	2.13814e8	0.0247779
$691$	1.12162e10	1.29322	0.646611	−	0.762820i	$-0.276186\pi$
$691$	1.12162e10	1.29322	0.646611	+	0.762820i	$0.276186\pi$
$692$	−2.62760e9	−0.301431
$693$	−3.77071e8	−0.0430385
$694$	3.22669e9	0.366437
$695$	4.79145e8	0.0541403
$696$	−2.95339e9	−0.332038
$697$	7.96234e7	0.00890689
$698$	5.29821e9	0.589705
$699$	−2.42630e9	−0.268704
$700$	1.71280e9	0.188740
$701$	−8.80671e8	−0.0965607	−0.0482804	−	0.998834i	$-0.515374\pi$
$701$	−8.80671e8	−0.0965607	−0.0482804	+	0.998834i	$0.515374\pi$
$702$	3.45948e8	0.0377426
$703$	−3.79265e9	−0.411718
$704$	3.95313e8	0.0427009
$705$	−1.70617e8	−0.0183384
$706$	−1.06745e9	−0.114165
$707$	2.97276e9	0.316368
$708$	−1.03055e9	−0.109132
$709$	−6.35616e9	−0.669781	−0.334890	−	0.942257i	$-0.608699\pi$
$709$	−6.35616e9	−0.669781	−0.334890	+	0.942257i	$0.608699\pi$
$710$	−1.30582e8	−0.0136924
$711$	−4.70946e9	−0.491392
$712$	−4.85512e9	−0.504104
$713$	2.18249e9	0.225496
$714$	7.71997e7	0.00793728
$715$	−3.31308e7	−0.00338969
$716$	−4.90195e9	−0.499083
$717$	1.41085e8	0.0142943
$718$	−3.11780e9	−0.314349
$719$	3.57315e9	0.358509	0.179255	−	0.983803i	$-0.442631\pi$
$719$	3.57315e9	0.358509	0.179255	+	0.983803i	$0.442631\pi$
$720$	−2.98598e7	−0.00298142
$721$	3.30336e9	0.328233
$722$	−6.49314e9	−0.642059
$723$	1.17617e9	0.115741
$724$	−4.79841e9	−0.469907
$725$	1.66694e10	1.62457
$726$	−3.71803e9	−0.360608
$727$	−6.46393e9	−0.623916	−0.311958	−	0.950096i	$-0.600985\pi$
$727$	−6.46393e9	−0.623916	−0.311958	+	0.950096i	$0.600985\pi$
$728$	−3.85828e8	−0.0370625
$729$	3.87420e8	0.0370370
$730$	−1.73320e7	−0.00164899
$731$	1.84980e8	0.0175152
$732$	3.21997e9	0.303433
$733$	5.70905e9	0.535426	0.267713	−	0.963499i	$-0.413732\pi$
$733$	5.70905e9	0.535426	0.267713	+	0.963499i	$0.413732\pi$
$734$	5.70952e9	0.532922
$735$	−3.17652e7	−0.00295084
$736$	−3.24364e9	−0.299889
$737$	−2.78324e9	−0.256103
$738$	−4.45646e8	−0.0408125
$739$	−1.05430e10	−0.960966	−0.480483	−	0.877004i	$-0.659539\pi$
$739$	−1.05430e10	−0.960966	−0.480483	+	0.877004i	$0.659539\pi$
$740$	−2.67677e8	−0.0242829
$741$	−5.37905e8	−0.0485670
$742$	2.69558e9	0.242236
$743$	−1.55085e9	−0.138710	−0.0693551	−	0.997592i	$-0.522094\pi$
$743$	−1.55085e9	−0.138710	−0.0693551	+	0.997592i	$0.522094\pi$
$744$	−3.04792e8	−0.0271330
$745$	1.62956e8	0.0144385
$746$	−6.55953e9	−0.578478
$747$	2.61868e9	0.229858
$748$	−1.00566e8	−0.00878605
$749$	−1.20591e9	−0.104864
$750$	3.37284e8	0.0291932
$751$	−9.50917e8	−0.0819224	−0.0409612	−	0.999161i	$-0.513042\pi$
$751$	−9.50917e8	−0.0819224	−0.0409612	+	0.999161i	$0.513042\pi$
$752$	2.58833e9	0.221951
$753$	8.66842e9	0.739874
$754$	−3.75497e9	−0.319012
$755$	−6.98150e8	−0.0590384
$756$	−4.32081e8	−0.0363696
$757$	1.00840e10	0.844887	0.422443	−	0.906389i	$-0.361173\pi$
$757$	1.00840e10	0.844887	0.422443	+	0.906389i	$0.361173\pi$
$758$	2.86293e9	0.238763
$759$	−4.03040e9	−0.334581
$760$	4.64282e7	0.00383649
$761$	−7.21075e9	−0.593109	−0.296554	−	0.955016i	$-0.595838\pi$
$761$	−7.21075e9	−0.593109	−0.296554	+	0.955016i	$0.595838\pi$
$762$	−5.41283e9	−0.443182
$763$	−3.97432e8	−0.0323912
$764$	9.63304e9	0.781514
$765$	7.59618e6	0.000613451 0
$766$	6.72571e9	0.540676
$767$	−1.31026e9	−0.104851
$768$	4.52985e8	0.0360844
$769$	1.73356e10	1.37467	0.687334	−	0.726341i	$-0.258781\pi$
$769$	1.73356e10	1.37467	0.687334	+	0.726341i	$0.258781\pi$
$770$	4.13795e7	0.00326639
$771$	1.42307e9	0.111824
$772$	8.08228e9	0.632227
$773$	−1.81631e10	−1.41436	−0.707182	−	0.707031i	$-0.750034\pi$
$773$	−1.81631e10	−1.41436	−0.707182	+	0.707031i	$0.750034\pi$
$774$	−1.03532e9	−0.0802567
$775$	1.72030e9	0.132754
$776$	4.30196e8	0.0330484
$777$	−3.87338e9	−0.296221
$778$	1.04390e10	0.794753
$779$	6.92922e8	0.0525174
$780$	−3.79642e7	−0.00286446
$781$	2.46148e9	0.184892
$782$	8.25164e8	0.0617045
$783$	−4.20512e9	−0.313049
$784$	4.81890e8	0.0357143
$785$	3.05107e8	0.0225117
$786$	3.56556e9	0.261908
$787$	1.72323e10	1.26018	0.630090	−	0.776522i	$-0.283018\pi$
$787$	1.72323e10	1.26018	0.630090	+	0.776522i	$0.283018\pi$
$788$	−5.44092e9	−0.396123
$789$	−1.32328e10	−0.959139
$790$	5.16813e8	0.0372940
$791$	2.57489e9	0.184987
$792$	5.62858e8	0.0402588
$793$	4.09390e9	0.291529
$794$	−8.62185e9	−0.611264
$795$	2.65236e8	0.0187218
$796$	−4.97880e9	−0.349888
$797$	−1.51855e9	−0.106249	−0.0531246	−	0.998588i	$-0.516918\pi$
$797$	−1.51855e9	−0.106249	−0.0531246	+	0.998588i	$0.516918\pi$
$798$	6.71830e8	0.0468004
$799$	−6.58456e8	−0.0456681
$800$	−2.55672e9	−0.176550
$801$	−6.91286e9	−0.475274
$802$	−1.44036e10	−0.985963
$803$	3.26708e8	0.0222667
$804$	−3.18929e9	−0.216420
$805$	−3.39529e8	−0.0229399
$806$	−3.87516e8	−0.0260685
$807$	9.26383e9	0.620488
$808$	−4.43747e9	−0.295935
$809$	−2.11304e10	−1.40310	−0.701548	−	0.712623i	$-0.747507\pi$
$809$	−2.11304e10	−1.40310	−0.701548	+	0.712623i	$0.747507\pi$
$810$	−4.25153e7	−0.00281091
$811$	−1.32957e10	−0.875260	−0.437630	−	0.899155i	$-0.644182\pi$
$811$	−1.32957e10	−0.875260	−0.437630	+	0.899155i	$0.644182\pi$
$812$	4.68987e9	0.307408
$813$	4.94225e9	0.322558
$814$	5.04572e9	0.327897
$815$	−1.03245e7	−0.000668064 0
$816$	−1.15237e8	−0.00742465
$817$	1.60979e9	0.103274
$818$	−2.00356e10	−1.27987
$819$	−5.49353e8	−0.0349428
$820$	4.89050e7	0.00309745
$821$	1.02411e10	0.645869	0.322934	−	0.946421i	$-0.395331\pi$
$821$	1.02411e10	0.645869	0.322934	+	0.946421i	$0.395331\pi$
$822$	−5.71808e8	−0.0359086
$823$	1.53728e9	0.0961285	0.0480643	−	0.998844i	$-0.484695\pi$
$823$	1.53728e9	0.0961285	0.0480643	+	0.998844i	$0.484695\pi$
$824$	−4.93096e9	−0.307034
$825$	−3.17687e9	−0.196975
$826$	1.63648e9	0.101037
$827$	1.29037e10	0.793316	0.396658	−	0.917966i	$-0.370170\pi$
$827$	1.29037e10	0.793316	0.396658	+	0.917966i	$0.370170\pi$
$828$	−4.61838e9	−0.282738
$829$	1.47174e10	0.897200	0.448600	−	0.893733i	$-0.351923\pi$
$829$	1.47174e10	0.897200	0.448600	+	0.893733i	$0.351923\pi$
$830$	−2.87372e8	−0.0174450
$831$	8.80903e9	0.532506
$832$	5.75930e8	0.0346688
$833$	−1.22590e8	−0.00734850
$834$	−1.03495e10	−0.617789
$835$	5.59050e8	0.0332314
$836$	−8.75171e8	−0.0518049
$837$	−4.33971e8	−0.0255812
$838$	−3.59466e9	−0.211010
$839$	−1.05302e10	−0.615557	−0.307778	−	0.951458i	$-0.599586\pi$
$839$	−1.05302e10	−0.615557	−0.307778	+	0.951458i	$0.599586\pi$
$840$	4.74163e7	0.00276026
$841$	2.83930e10	1.64598
$842$	6.14339e9	0.354663
$843$	−9.10012e9	−0.523179
$844$	9.67668e9	0.554023
$845$	−4.82681e7	−0.00275208
$846$	3.68533e9	0.209257
$847$	5.90410e9	0.333858
$848$	−4.02372e9	−0.226591
$849$	8.33026e9	0.467177
$850$	6.50416e8	0.0363266
$851$	−4.14013e10	−2.30282
$852$	2.82058e9	0.156243
$853$	−2.70736e10	−1.49357	−0.746784	−	0.665067i	$-0.768403\pi$
$853$	−2.70736e10	−1.49357	−0.746784	+	0.665067i	$0.768403\pi$
$854$	−5.11319e9	−0.280924
$855$	6.61057e7	0.00361708
$856$	1.80007e9	0.0980914
$857$	2.17919e10	1.18267	0.591333	−	0.806427i	$-0.298602\pi$
$857$	2.17919e10	1.18267	0.591333	+	0.806427i	$0.298602\pi$
$858$	7.15624e8	0.0386794
$859$	−7.41255e8	−0.0399017	−0.0199509	−	0.999801i	$-0.506351\pi$
$859$	−7.41255e8	−0.0399017	−0.0199509	+	0.999801i	$0.506351\pi$
$860$	1.13615e8	0.00609105
$861$	7.07670e8	0.0377850
$862$	−5.63063e9	−0.299420
$863$	−1.06003e10	−0.561410	−0.280705	−	0.959794i	$-0.590568\pi$
$863$	−1.06003e10	−0.561410	−0.280705	+	0.959794i	$0.590568\pi$
$864$	6.44973e8	0.0340207
$865$	4.10563e8	0.0215687
$866$	1.34072e10	0.701495
$867$	−1.10498e10	−0.575823
$868$	4.83998e8	0.0251203
$869$	−9.74193e9	−0.503589
$870$	4.61467e8	0.0237587
$871$	−4.05490e9	−0.207930
$872$	5.93251e8	0.0302992
$873$	6.12525e8	0.0311583
$874$	7.18099e9	0.363826
$875$	−5.35594e8	−0.0270276
$876$	3.74371e8	0.0188165
$877$	1.79764e10	0.899921	0.449960	−	0.893049i	$-0.351438\pi$
$877$	1.79764e10	0.899921	0.449960	+	0.893049i	$0.351438\pi$
$878$	1.69868e10	0.846997
$879$	1.30670e10	0.648954
$880$	−6.17677e7	−0.00305543
$881$	−3.82110e10	−1.88266	−0.941332	−	0.337482i	$-0.890425\pi$
$881$	−3.82110e10	−1.88266	−0.941332	+	0.337482i	$0.890425\pi$
$882$	6.86129e8	0.0336718
$883$	1.27600e10	0.623718	0.311859	−	0.950128i	$-0.399048\pi$
$883$	1.27600e10	0.623718	0.311859	+	0.950128i	$0.399048\pi$
$884$	−1.46514e8	−0.00713337
$885$	1.61024e8	0.00780887
$886$	9.55402e9	0.461496
$887$	−2.94961e10	−1.41916	−0.709582	−	0.704623i	$-0.751116\pi$
$887$	−2.94961e10	−1.41916	−0.709582	+	0.704623i	$0.751116\pi$
$888$	5.78183e9	0.277090
$889$	8.59537e9	0.410306
$890$	7.58613e8	0.0360708
$891$	8.01413e8	0.0379563
$892$	−6.28981e9	−0.296729
$893$	−5.73021e9	−0.269272
$894$	−3.51984e9	−0.164756
$895$	7.65930e8	0.0357115
$896$	−7.19323e8	−0.0334077
$897$	−5.87187e9	−0.271646
$898$	1.29953e10	0.598851
$899$	4.71038e9	0.216221
$900$	−3.64033e9	−0.166453
$901$	1.02361e9	0.0466229
$902$	−9.21858e8	−0.0418255
$903$	1.64405e9	0.0743033
$904$	−3.84357e9	−0.173039
$905$	7.49751e8	0.0336238
$906$	1.50800e10	0.673680
$907$	8.81674e9	0.392358	0.196179	−	0.980568i	$-0.437147\pi$
$907$	8.81674e9	0.392358	0.196179	+	0.980568i	$0.437147\pi$
$908$	−1.15228e10	−0.510806
$909$	−6.31820e9	−0.279010
$910$	6.02857e7	0.00265197
$911$	−4.40436e10	−1.93005	−0.965026	−	0.262156i	$-0.915567\pi$
$911$	−4.40436e10	−1.93005	−0.965026	+	0.262156i	$0.915567\pi$
$912$	−1.00285e9	−0.0437777
$913$	5.41697e9	0.235564
$914$	1.57409e9	0.0681894
$915$	−5.03120e8	−0.0217119
$916$	1.98543e9	0.0853533
$917$	−5.66197e9	−0.242479
$918$	−1.64077e8	−0.00700003
$919$	1.62626e10	0.691171	0.345586	−	0.938387i	$-0.387680\pi$
$919$	1.62626e10	0.691171	0.345586	+	0.938387i	$0.387680\pi$
$920$	5.06819e8	0.0214583
$921$	−1.66443e10	−0.702031
$922$	1.06141e10	0.445988
$923$	3.58612e9	0.150113
$924$	−8.93798e8	−0.0372724
$925$	−3.26336e10	−1.35572
$926$	2.30246e10	0.952912
$927$	−7.02084e9	−0.289474
$928$	−7.00062e9	−0.287553
$929$	−2.37038e10	−0.969979	−0.484989	−	0.874520i	$-0.661176\pi$
$929$	−2.37038e10	−0.969979	−0.484989	+	0.874520i	$0.661176\pi$
$930$	4.76237e7	0.00194148
$931$	−1.06684e9	−0.0433287
$932$	−5.75122e9	−0.232704
$933$	1.25184e9	0.0504620
$934$	1.61019e10	0.646640
$935$	1.57134e7	0.000628678 0
$936$	8.20026e8	0.0326860
$937$	1.56896e10	0.623052	0.311526	−	0.950238i	$-0.399160\pi$
$937$	1.56896e10	0.623052	0.311526	+	0.950238i	$0.399160\pi$
$938$	5.06447e9	0.200366
$939$	−2.31106e10	−0.910921
$940$	−4.04426e8	−0.0158815
$941$	1.84030e10	0.719986	0.359993	−	0.932955i	$-0.382779\pi$
$941$	1.84030e10	0.719986	0.359993	+	0.932955i	$0.382779\pi$
$942$	−6.59031e9	−0.256879
$943$	7.56407e9	0.293741
$944$	−2.44279e9	−0.0945113
$945$	6.75127e7	0.00260240
$946$	−2.14165e9	−0.0822488
$947$	−1.22185e10	−0.467513	−0.233756	−	0.972295i	$-0.575102\pi$
$947$	−1.22185e10	−0.467513	−0.233756	+	0.972295i	$0.575102\pi$
$948$	−1.11632e10	−0.425558
$949$	4.75980e8	0.0180783
$950$	5.66025e9	0.214192
$951$	9.90498e9	0.373441
$952$	1.82992e8	0.00687389
$953$	3.37697e10	1.26387	0.631934	−	0.775022i	$-0.282261\pi$
$953$	3.37697e10	1.26387	0.631934	+	0.775022i	$0.282261\pi$
$954$	−5.72909e9	−0.213632
$955$	−1.50516e9	−0.0559206
$956$	3.34423e8	0.0123792
$957$	−8.69865e9	−0.320819
$958$	−1.37835e10	−0.506502
$959$	9.08009e8	0.0332449
$960$	−7.07789e7	−0.00258199
$961$	−2.70265e10	−0.982331
$962$	7.35109e9	0.266219
$963$	2.56299e9	0.0924815
$964$	2.78796e9	0.100234
$965$	−1.26286e9	−0.0452385
$966$	7.33382e9	0.261764
$967$	−3.09840e10	−1.10191	−0.550954	−	0.834536i	$-0.685736\pi$
$967$	−3.09840e10	−1.10191	−0.550954	+	0.834536i	$0.685736\pi$
$968$	−8.81311e9	−0.312295
$969$	2.55119e8	0.00900761
$970$	−6.72181e7	−0.00236475
$971$	−4.82560e10	−1.69155	−0.845773	−	0.533542i	$-0.820860\pi$
$971$	−4.82560e10	−1.69155	−0.845773	+	0.533542i	$0.820860\pi$
$972$	9.18330e8	0.0320750
$973$	1.64347e10	0.571961
$974$	−2.64039e10	−0.915614
$975$	−4.62836e9	−0.159923
$976$	7.63251e9	0.262780
$977$	−1.49627e10	−0.513308	−0.256654	−	0.966503i	$-0.582620\pi$
$977$	−1.49627e10	−0.513308	−0.256654	+	0.966503i	$0.582620\pi$
$978$	2.23010e8	0.00762321
$979$	−1.42999e10	−0.487071
$980$	−7.52954e7	−0.00255551
$981$	8.44688e8	0.0285664
$982$	−1.97494e9	−0.0665525
$983$	−2.24569e10	−0.754071	−0.377036	−	0.926199i	$-0.623057\pi$
$983$	−2.24569e10	−0.754071	−0.377036	+	0.926199i	$0.623057\pi$
$984$	−1.05635e9	−0.0353447
$985$	8.50143e8	0.0283443
$986$	1.78092e9	0.0591664
$987$	−5.85217e9	−0.193735
$988$	−1.27503e9	−0.0420603
$989$	1.75727e10	0.577634
$990$	−8.79466e7	−0.00288068
$991$	7.33554e9	0.239427	0.119714	−	0.992808i	$-0.461802\pi$
$991$	7.33554e9	0.239427	0.119714	+	0.992808i	$0.461802\pi$
$992$	−7.22469e8	−0.0234979
$993$	8.80723e9	0.285441
$994$	−4.47898e9	−0.144653
$995$	7.77938e8	0.0250359
$996$	6.20724e9	0.199063
$997$	−2.09002e10	−0.667909	−0.333955	−	0.942589i	$-0.608383\pi$
$997$	−2.09002e10	−0.667909	−0.333955	+	0.942589i	$0.608383\pi$
$998$	2.90153e9	0.0923996
$999$	8.23234e9	0.261243

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
546.8.a.c.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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