LMFDB - Embedded newform 91.8.a.b.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [91,8,Mod(1,91)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(91, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 8, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("91.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 91 = 7 \cdot 13 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	91.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:	$28.4270373191$
Analytic rank:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 4 x^{8} - 764 x^{7} + 1562 x^{6} + 176422 x^{5} + 56746 x^{4} - 13204236 x^{3} + \cdots + 176334338 $ x^9 - 4x^8 - 764x^7 + 1562x^6 + 176422x^5 + 56746x^4 - 13204236x^3 - 22500802x^2 + 176026849x + 176334338
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$17.5192$ of defining polynomial
Character	$\chi$	$=$	91.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+16.5192 q^{2} -52.3583 q^{3} +144.884 q^{4} +210.689 q^{5} -864.917 q^{6} -343.000 q^{7} +278.914 q^{8} +554.387 q^{9} +O(q^{10})$	\(q+16.5192 q^{2} -52.3583 q^{3} +144.884 q^{4} +210.689 q^{5} -864.917 q^{6} -343.000 q^{7} +278.914 q^{8} +554.387 q^{9} +3480.42 q^{10} +687.612 q^{11} -7585.89 q^{12} +2197.00 q^{13} -5666.09 q^{14} -11031.3 q^{15} -13937.7 q^{16} -11372.4 q^{17} +9158.03 q^{18} -48772.0 q^{19} +30525.5 q^{20} +17958.9 q^{21} +11358.8 q^{22} -12421.3 q^{23} -14603.4 q^{24} -33735.1 q^{25} +36292.7 q^{26} +85480.8 q^{27} -49695.3 q^{28} -80739.6 q^{29} -182229. q^{30} -53715.3 q^{31} -265941. q^{32} -36002.1 q^{33} -187862. q^{34} -72266.4 q^{35} +80321.9 q^{36} -371041. q^{37} -805675. q^{38} -115031. q^{39} +58764.1 q^{40} -441887. q^{41} +296666. q^{42} +242290. q^{43} +99624.1 q^{44} +116803. q^{45} -205189. q^{46} +915426. q^{47} +729756. q^{48} +117649. q^{49} -557277. q^{50} +595436. q^{51} +318311. q^{52} -1.48597e6 q^{53} +1.41207e6 q^{54} +144872. q^{55} -95667.5 q^{56} +2.55362e6 q^{57} -1.33375e6 q^{58} +2.76600e6 q^{59} -1.59826e6 q^{60} +1.96459e6 q^{61} -887334. q^{62} -190155. q^{63} -2.60911e6 q^{64} +462884. q^{65} -594727. q^{66} -2.77138e6 q^{67} -1.64767e6 q^{68} +650356. q^{69} -1.19378e6 q^{70} +1.45502e6 q^{71} +154626. q^{72} +596207. q^{73} -6.12930e6 q^{74} +1.76631e6 q^{75} -7.06629e6 q^{76} -235851. q^{77} -1.90022e6 q^{78} +2.71310e6 q^{79} -2.93653e6 q^{80} -5.68807e6 q^{81} -7.29962e6 q^{82} +4.86836e6 q^{83} +2.60196e6 q^{84} -2.39603e6 q^{85} +4.00244e6 q^{86} +4.22739e6 q^{87} +191784. q^{88} +3.85576e6 q^{89} +1.92950e6 q^{90} -753571. q^{91} -1.79965e6 q^{92} +2.81244e6 q^{93} +1.51221e7 q^{94} -1.02757e7 q^{95} +1.39242e7 q^{96} +3.90082e6 q^{97} +1.94347e6 q^{98} +381203. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 5 q^{2} - 26 q^{3} + 393 q^{4} - 181 q^{5} + 703 q^{6} - 3087 q^{7} + 1197 q^{8} + 3211 q^{9}+O(q^{10}) $ 9 * q - 5 * q^2 - 26 * q^3 + 393 * q^4 - 181 * q^5 + 703 * q^6 - 3087 * q^7 + 1197 * q^8 + 3211 * q^9	\( 9 q - 5 q^{2} - 26 q^{3} + 393 q^{4} - 181 q^{5} + 703 q^{6} - 3087 q^{7} + 1197 q^{8} + 3211 q^{9} - 5124 q^{10} - 9826 q^{11} - 20919 q^{12} + 19773 q^{13} + 1715 q^{14} - 20346 q^{15} + 31113 q^{16} - 22766 q^{17} - 12978 q^{18} - 17769 q^{19} - 44204 q^{20} + 8918 q^{21} - 203553 q^{22} - 49103 q^{23} + 52737 q^{24} + 227466 q^{25} - 10985 q^{26} + 103624 q^{27} - 134799 q^{28} - 487455 q^{29} - 287992 q^{30} - 63843 q^{31} - 587099 q^{32} - 314392 q^{33} - 576240 q^{34} + 62083 q^{35} - 1514926 q^{36} - 796926 q^{37} - 766702 q^{38} - 57122 q^{39} - 2887296 q^{40} - 1567546 q^{41} - 241129 q^{42} - 277899 q^{43} - 1281195 q^{44} - 1650593 q^{45} - 1907445 q^{46} + 1077367 q^{47} - 1110835 q^{48} + 1058841 q^{49} - 267459 q^{50} - 3054368 q^{51} + 863421 q^{52} - 7322659 q^{53} - 3355387 q^{54} - 2613324 q^{55} - 410571 q^{56} - 3751946 q^{57} - 2992332 q^{58} - 169804 q^{59} - 2754416 q^{60} - 6352284 q^{61} + 6001087 q^{62} - 1101373 q^{63} + 1657017 q^{64} - 397657 q^{65} - 5962713 q^{66} + 921120 q^{67} + 5615224 q^{68} - 5202780 q^{69} + 1757532 q^{70} + 3786654 q^{71} + 2229758 q^{72} + 5792889 q^{73} - 1991961 q^{74} + 145628 q^{75} - 2806026 q^{76} + 3370318 q^{77} + 1544491 q^{78} + 3464037 q^{79} + 15422512 q^{80} - 5010363 q^{81} - 12539943 q^{82} + 6834945 q^{83} + 7175217 q^{84} + 3880662 q^{85} - 7977524 q^{86} + 3727078 q^{87} + 7013709 q^{88} - 20408371 q^{89} + 34910060 q^{90} - 6782139 q^{91} - 3544371 q^{92} + 3121742 q^{93} + 61343967 q^{94} + 3360807 q^{95} + 23547905 q^{96} + 41644125 q^{97} - 588245 q^{98} + 50754068 q^{99}+O(q^{100}) \) 9 * q - 5 * q^2 - 26 * q^3 + 393 * q^4 - 181 * q^5 + 703 * q^6 - 3087 * q^7 + 1197 * q^8 + 3211 * q^9 - 5124 * q^10 - 9826 * q^11 - 20919 * q^12 + 19773 * q^13 + 1715 * q^14 - 20346 * q^15 + 31113 * q^16 - 22766 * q^17 - 12978 * q^18 - 17769 * q^19 - 44204 * q^20 + 8918 * q^21 - 203553 * q^22 - 49103 * q^23 + 52737 * q^24 + 227466 * q^25 - 10985 * q^26 + 103624 * q^27 - 134799 * q^28 - 487455 * q^29 - 287992 * q^30 - 63843 * q^31 - 587099 * q^32 - 314392 * q^33 - 576240 * q^34 + 62083 * q^35 - 1514926 * q^36 - 796926 * q^37 - 766702 * q^38 - 57122 * q^39 - 2887296 * q^40 - 1567546 * q^41 - 241129 * q^42 - 277899 * q^43 - 1281195 * q^44 - 1650593 * q^45 - 1907445 * q^46 + 1077367 * q^47 - 1110835 * q^48 + 1058841 * q^49 - 267459 * q^50 - 3054368 * q^51 + 863421 * q^52 - 7322659 * q^53 - 3355387 * q^54 - 2613324 * q^55 - 410571 * q^56 - 3751946 * q^57 - 2992332 * q^58 - 169804 * q^59 - 2754416 * q^60 - 6352284 * q^61 + 6001087 * q^62 - 1101373 * q^63 + 1657017 * q^64 - 397657 * q^65 - 5962713 * q^66 + 921120 * q^67 + 5615224 * q^68 - 5202780 * q^69 + 1757532 * q^70 + 3786654 * q^71 + 2229758 * q^72 + 5792889 * q^73 - 1991961 * q^74 + 145628 * q^75 - 2806026 * q^76 + 3370318 * q^77 + 1544491 * q^78 + 3464037 * q^79 + 15422512 * q^80 - 5010363 * q^81 - 12539943 * q^82 + 6834945 * q^83 + 7175217 * q^84 + 3880662 * q^85 - 7977524 * q^86 + 3727078 * q^87 + 7013709 * q^88 - 20408371 * q^89 + 34910060 * q^90 - 6782139 * q^91 - 3544371 * q^92 + 3121742 * q^93 + 61343967 * q^94 + 3360807 * q^95 + 23547905 * q^96 + 41644125 * q^97 - 588245 * q^98 + 50754068 * q^99

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 $

$2$	16.5192	1.46011	0.730053	−	0.683391i	$-0.239495\pi$
$2$	16.5192	1.46011	0.730053	+	0.683391i	$0.239495\pi$
$3$	−52.3583	−1.11959	−0.559797	−	0.828630i	$-0.689121\pi$
$3$	−52.3583	−1.11959	−0.559797	+	0.828630i	$0.689121\pi$
$4$	144.884	1.13191
$5$	210.689	0.753784	0.376892	−	0.926257i	$-0.376993\pi$
$5$	210.689	0.753784	0.376892	+	0.926257i	$0.376993\pi$
$6$	−864.917	−1.63473
$7$	−343.000	−0.377964
$7$	−343.000	−0.377964
$8$	278.914	0.192600
$9$	554.387	0.253492
$10$	3480.42	1.10060
$11$	687.612	0.155765	0.0778823	−	0.996963i	$-0.475184\pi$
$11$	687.612	0.155765	0.0778823	+	0.996963i	$0.475184\pi$
$12$	−7585.89	−1.26728
$13$	2197.00	0.277350
$13$	2197.00	0.277350
$14$	−5666.09	−0.551868
$15$	−11031.3	−0.843933
$16$	−13937.7	−0.850692
$17$	−11372.4	−0.561409	−0.280704	−	0.959794i	$-0.590568\pi$
$17$	−11372.4	−0.561409	−0.280704	+	0.959794i	$0.590568\pi$
$18$	9158.03	0.370125
$19$	−48772.0	−1.63130	−0.815648	−	0.578548i	$-0.803619\pi$
$19$	−48772.0	−1.63130	−0.815648	+	0.578548i	$0.803619\pi$
$20$	30525.5	0.853215
$21$	17958.9	0.423167
$22$	11358.8	0.227433
$23$	−12421.3	−0.212872	−0.106436	−	0.994320i	$-0.533944\pi$
$23$	−12421.3	−0.212872	−0.106436	+	0.994320i	$0.533944\pi$
$24$	−14603.4	−0.215633
$25$	−33735.1	−0.431809
$26$	36292.7	0.404960
$27$	85480.8	0.835786
$28$	−49695.3	−0.427821
$29$	−80739.6	−0.614743	−0.307372	−	0.951590i	$-0.599449\pi$
$29$	−80739.6	−0.614743	−0.307372	+	0.951590i	$0.599449\pi$
$30$	−182229.	−1.23223
$31$	−53715.3	−0.323841	−0.161921	−	0.986804i	$-0.551769\pi$
$31$	−53715.3	−0.323841	−0.161921	+	0.986804i	$0.551769\pi$
$32$	−265941.	−1.43470
$33$	−36002.1	−0.174393
$34$	−187862.	−0.819716
$35$	−72266.4	−0.284904
$36$	80321.9	0.286930
$37$	−371041.	−1.20425	−0.602123	−	0.798403i	$-0.705678\pi$
$37$	−371041.	−1.20425	−0.602123	+	0.798403i	$0.705678\pi$
$38$	−805675.	−2.38186
$39$	−115031.	−0.310520
$40$	58764.1	0.145179
$41$	−441887.	−1.00131	−0.500654	−	0.865647i	$-0.666907\pi$
$41$	−441887.	−1.00131	−0.500654	+	0.865647i	$0.666907\pi$
$42$	296666.	0.617868
$43$	242290.	0.464725	0.232363	−	0.972629i	$-0.425354\pi$
$43$	242290.	0.464725	0.232363	+	0.972629i	$0.425354\pi$
$44$	99624.1	0.176311
$45$	116803.	0.191078
$46$	−205189.	−0.310815
$47$	915426.	1.28612	0.643059	−	0.765817i	$-0.277665\pi$
$47$	915426.	1.28612	0.643059	+	0.765817i	$0.277665\pi$
$48$	729756.	0.952430
$49$	117649.	0.142857
$50$	−557277.	−0.630487
$51$	595436.	0.628550
$52$	318311.	0.313935
$53$	−1.48597e6	−1.37102	−0.685512	−	0.728062i	$-0.740422\pi$
$53$	−1.48597e6	−1.37102	−0.685512	+	0.728062i	$0.740422\pi$
$54$	1.41207e6	1.22034
$55$	144872.	0.117413
$56$	−95667.5	−0.0727958
$57$	2.55362e6	1.82639
$58$	−1.33375e6	−0.897590
$59$	2.76600e6	1.75335	0.876677	−	0.481080i	$-0.159755\pi$
$59$	2.76600e6	1.75335	0.876677	+	0.481080i	$0.159755\pi$
$60$	−1.59826e6	−0.955254
$61$	1.96459e6	1.10820	0.554100	−	0.832450i	$-0.313062\pi$
$61$	1.96459e6	1.10820	0.554100	+	0.832450i	$0.313062\pi$
$62$	−887334.	−0.472842
$63$	−190155.	−0.0958110
$64$	−2.60911e6	−1.24412
$65$	462884.	0.209062
$66$	−594727.	−0.254632
$67$	−2.77138e6	−1.12573	−0.562866	−	0.826548i	$-0.690301\pi$
$67$	−2.77138e6	−1.12573	−0.562866	+	0.826548i	$0.690301\pi$
$68$	−1.64767e6	−0.635463
$69$	650356.	0.238330
$70$	−1.19378e6	−0.415989
$71$	1.45502e6	0.482464	0.241232	−	0.970467i	$-0.422448\pi$
$71$	1.45502e6	0.482464	0.241232	+	0.970467i	$0.422448\pi$
$72$	154626.	0.0488224
$73$	596207.	0.179377	0.0896885	−	0.995970i	$-0.471413\pi$
$73$	596207.	0.179377	0.0896885	+	0.995970i	$0.471413\pi$
$74$	−6.12930e6	−1.75833
$75$	1.76631e6	0.483451
$76$	−7.06629e6	−1.84648
$77$	−235851.	−0.0588735
$78$	−1.90022e6	−0.453391
$79$	2.71310e6	0.619115	0.309558	−	0.950881i	$-0.399819\pi$
$79$	2.71310e6	0.619115	0.309558	+	0.950881i	$0.399819\pi$
$80$	−2.93653e6	−0.641239
$81$	−5.68807e6	−1.18923
$82$	−7.29962e6	−1.46202
$83$	4.86836e6	0.934566	0.467283	−	0.884108i	$-0.345233\pi$
$83$	4.86836e6	0.934566	0.467283	+	0.884108i	$0.345233\pi$
$84$	2.60196e6	0.478986
$85$	−2.39603e6	−0.423181
$86$	4.00244e6	0.678548
$87$	4.22739e6	0.688263
$88$	191784.	0.0300002
$89$	3.85576e6	0.579755	0.289878	−	0.957064i	$-0.406385\pi$
$89$	3.85576e6	0.579755	0.289878	+	0.957064i	$0.406385\pi$
$90$	1.92950e6	0.278994
$91$	−753571.	−0.104828
$92$	−1.79965e6	−0.240951
$93$	2.81244e6	0.362571
$94$	1.51221e7	1.87787
$95$	−1.02757e7	−1.22965
$96$	1.39242e7	1.60628
$97$	3.90082e6	0.433965	0.216982	−	0.976176i	$-0.430379\pi$
$97$	3.90082e6	0.433965	0.216982	+	0.976176i	$0.430379\pi$
$98$	1.94347e6	0.208586
$99$	381203.	0.0394851
$100$	−4.88768e6	−0.488768
$101$	13839.7	0.00133660	0.000668301	−	1.00000i	$-0.499787\pi$
$101$	13839.7	0.00133660	0.000668301		1.00000i	$0.499787\pi$
$102$	9.83614e6	0.917749
$103$	−8.35289e6	−0.753194	−0.376597	−	0.926377i	$-0.622906\pi$
$103$	−8.35289e6	−0.753194	−0.376597	+	0.926377i	$0.622906\pi$
$104$	612774.	0.0534175
$105$	3.78374e6	0.318977
$106$	−2.45471e7	−2.00184
$107$	744784.	0.0587743	0.0293871	−	0.999568i	$-0.490644\pi$
$107$	744784.	0.0587743	0.0293871	+	0.999568i	$0.490644\pi$
$108$	1.23848e7	0.946033
$109$	1.71903e7	1.27142	0.635712	−	0.771927i	$-0.280707\pi$
$109$	1.71903e7	1.27142	0.635712	+	0.771927i	$0.280707\pi$
$110$	2.39318e6	0.171435
$111$	1.94270e7	1.34827
$112$	4.78065e6	0.321531
$113$	−8.97124e6	−0.584895	−0.292448	−	0.956282i	$-0.594470\pi$
$113$	−8.97124e6	−0.584895	−0.292448	+	0.956282i	$0.594470\pi$
$114$	4.21837e7	2.66672
$115$	−2.61703e6	−0.160460
$116$	−1.16979e7	−0.695833
$117$	1.21799e6	0.0703060
$118$	4.56921e7	2.56008
$119$	3.90072e6	0.212192
$120$	−3.07679e6	−0.162541
$121$	−1.90144e7	−0.975737
$122$	3.24535e7	1.61809
$123$	2.31364e7	1.12106
$124$	−7.78250e6	−0.366558
$125$	−2.35677e7	−1.07928
$126$	−3.14121e6	−0.139894
$127$	3.25129e6	0.140846	0.0704228	−	0.997517i	$-0.477565\pi$
$127$	3.25129e6	0.140846	0.0704228	+	0.997517i	$0.477565\pi$
$128$	−9.05994e6	−0.381848
$129$	−1.26859e7	−0.520304
$130$	7.64648e6	0.305253
$131$	2.34813e7	0.912583	0.456292	−	0.889830i	$-0.349177\pi$
$131$	2.34813e7	0.912583	0.456292	+	0.889830i	$0.349177\pi$
$132$	−5.21614e6	−0.197397
$133$	1.67288e7	0.616572
$134$	−4.57811e7	−1.64369
$135$	1.80099e7	0.630003
$136$	−3.17191e6	−0.108127
$137$	−4.50922e7	−1.49823	−0.749116	−	0.662439i	$-0.769521\pi$
$137$	−4.50922e7	−1.49823	−0.749116	+	0.662439i	$0.769521\pi$
$138$	1.07434e7	0.347987
$139$	−4.32634e7	−1.36637	−0.683187	−	0.730244i	$-0.739406\pi$
$139$	−4.32634e7	−1.36637	−0.683187	+	0.730244i	$0.739406\pi$
$140$	−1.04703e7	−0.322485
$141$	−4.79301e7	−1.43993
$142$	2.40358e7	0.704449
$143$	1.51068e6	0.0432013
$144$	−7.72690e6	−0.215644
$145$	−1.70110e7	−0.463384
$146$	9.84886e6	0.261909
$147$	−6.15990e6	−0.159942
$148$	−5.37579e7	−1.36310
$149$	1.10108e7	0.272689	0.136345	−	0.990661i	$-0.456465\pi$
$149$	1.10108e7	0.272689	0.136345	+	0.990661i	$0.456465\pi$
$150$	2.91780e7	0.705890
$151$	−6.27230e7	−1.48254	−0.741271	−	0.671205i	$-0.765777\pi$
$151$	−6.27230e7	−1.48254	−0.741271	+	0.671205i	$0.765777\pi$
$152$	−1.36032e7	−0.314187
$153$	−6.30468e6	−0.142313
$154$	−3.89607e6	−0.0859615
$155$	−1.13172e7	−0.244106
$156$	−1.66662e7	−0.351480
$157$	−8.65488e7	−1.78489	−0.892446	−	0.451154i	$-0.851013\pi$
$157$	−8.65488e7	−1.78489	−0.892446	+	0.451154i	$0.851013\pi$
$158$	4.48183e7	0.903974
$159$	7.78029e7	1.53499
$160$	−5.60310e7	−1.08145
$161$	4.26049e6	0.0804580
$162$	−9.39624e7	−1.73641
$163$	1.14772e7	0.207577	0.103789	−	0.994599i	$-0.466903\pi$
$163$	1.14772e7	0.207577	0.103789	+	0.994599i	$0.466903\pi$
$164$	−6.40224e7	−1.13339
$165$	−7.58526e6	−0.131455
$166$	8.04215e7	1.36456
$167$	−1.64249e7	−0.272894	−0.136447	−	0.990647i	$-0.543568\pi$
$167$	−1.64249e7	−0.272894	−0.136447	+	0.990647i	$0.543568\pi$
$168$	5.00898e6	0.0815018
$169$	4.82681e6	0.0769231
$170$	−3.95805e7	−0.617889
$171$	−2.70386e7	−0.413521
$172$	3.51040e7	0.526026
$173$	6.34723e7	0.932016	0.466008	−	0.884781i	$-0.345692\pi$
$173$	6.34723e7	0.932016	0.466008	+	0.884781i	$0.345692\pi$
$174$	6.98331e7	1.00494
$175$	1.15711e7	0.163209
$176$	−9.58375e6	−0.132508
$177$	−1.44823e8	−1.96305
$178$	6.36940e7	0.846504
$179$	4.98047e7	0.649059	0.324530	−	0.945876i	$-0.394794\pi$
$179$	4.98047e7	0.649059	0.324530	+	0.945876i	$0.394794\pi$
$180$	1.69230e7	0.216283
$181$	4.37655e7	0.548601	0.274301	−	0.961644i	$-0.411554\pi$
$181$	4.37655e7	0.548601	0.274301	+	0.961644i	$0.411554\pi$
$182$	−1.24484e7	−0.153061
$183$	−1.02863e8	−1.24074
$184$	−3.46446e6	−0.0409990
$185$	−7.81742e7	−0.907742
$186$	4.64592e7	0.529391
$187$	−7.81976e6	−0.0874476
$188$	1.32631e8	1.45577
$189$	−2.93199e7	−0.315898
$190$	−1.69747e8	−1.79541
$191$	1.19641e8	1.24240	0.621202	−	0.783651i	$-0.286645\pi$
$191$	1.19641e8	1.24240	0.621202	+	0.783651i	$0.286645\pi$
$192$	1.36609e8	1.39291
$193$	−4.96587e7	−0.497215	−0.248608	−	0.968604i	$-0.579973\pi$
$193$	−4.96587e7	−0.497215	−0.248608	+	0.968604i	$0.579973\pi$
$194$	6.44384e7	0.633634
$195$	−2.42358e7	−0.234065
$196$	1.70455e7	0.161701
$197$	1.22250e6	0.0113925	0.00569623	−	0.999984i	$-0.498187\pi$
$197$	1.22250e6	0.0113925	0.00569623	+	0.999984i	$0.498187\pi$
$198$	6.29717e6	0.0576524
$199$	−1.29052e8	−1.16086	−0.580428	−	0.814312i	$-0.697115\pi$
$199$	−1.29052e8	−1.16086	−0.580428	+	0.814312i	$0.697115\pi$
$200$	−9.40919e6	−0.0831663
$201$	1.45105e8	1.26036
$202$	228621.	0.00195158
$203$	2.76937e7	0.232351
$204$	8.62693e7	0.711461
$205$	−9.31008e7	−0.754770
$206$	−1.37983e8	−1.09974
$207$	−6.88619e6	−0.0539613
$208$	−3.06212e7	−0.235940
$209$	−3.35362e7	−0.254098
$210$	6.25044e7	0.465740
$211$	−4.78231e7	−0.350469	−0.175234	−	0.984527i	$-0.556068\pi$
$211$	−4.78231e7	−0.350469	−0.175234	+	0.984527i	$0.556068\pi$
$212$	−2.15294e8	−1.55187
$213$	−7.61824e7	−0.540164
$214$	1.23032e7	0.0858166
$215$	5.10479e7	0.350303
$216$	2.38418e7	0.160972
$217$	1.84243e7	0.122400
$218$	2.83970e8	1.85641
$219$	−3.12163e7	−0.200830
$220$	2.09897e7	0.132901
$221$	−2.49851e7	−0.155707
$222$	3.20919e8	1.96861
$223$	1.30010e7	0.0785073	0.0392536	−	0.999229i	$-0.487502\pi$
$223$	1.30010e7	0.0785073	0.0392536	+	0.999229i	$0.487502\pi$
$224$	9.12179e7	0.542266
$225$	−1.87023e7	−0.109460
$226$	−1.48198e8	−0.854008
$227$	−1.83498e7	−0.104122	−0.0520609	−	0.998644i	$-0.516579\pi$
$227$	−1.83498e7	−0.104122	−0.0520609	+	0.998644i	$0.516579\pi$
$228$	3.69979e8	2.06731
$229$	−1.97509e7	−0.108683	−0.0543415	−	0.998522i	$-0.517306\pi$
$229$	−1.97509e7	−0.108683	−0.0543415	+	0.998522i	$0.517306\pi$
$230$	−4.32312e7	−0.234288
$231$	1.23487e7	0.0659144
$232$	−2.25194e7	−0.118399
$233$	2.94047e8	1.52290	0.761449	−	0.648225i	$-0.224489\pi$
$233$	2.94047e8	1.52290	0.761449	+	0.648225i	$0.224489\pi$
$234$	2.01202e7	0.102654
$235$	1.92870e8	0.969455
$236$	4.00749e8	1.98463
$237$	−1.42053e8	−0.693158
$238$	6.44367e7	0.309823
$239$	1.48371e8	0.703000	0.351500	−	0.936188i	$-0.385672\pi$
$239$	1.48371e8	0.703000	0.351500	+	0.936188i	$0.385672\pi$
$240$	1.53752e8	0.717927
$241$	−3.40713e8	−1.56794	−0.783969	−	0.620800i	$-0.786808\pi$
$241$	−3.40713e8	−1.56794	−0.783969	+	0.620800i	$0.786808\pi$
$242$	−3.14102e8	−1.42468
$243$	1.10871e8	0.495673
$244$	2.84639e8	1.25438
$245$	2.47874e7	0.107683
$246$	3.82195e8	1.63686
$247$	−1.07152e8	−0.452440
$248$	−1.49819e7	−0.0623716
$249$	−2.54899e8	−1.04633
$250$	−3.89320e8	−1.57586
$251$	−2.37078e8	−0.946309	−0.473154	−	0.880980i	$-0.656885\pi$
$251$	−2.37078e8	−0.946309	−0.473154	+	0.880980i	$0.656885\pi$
$252$	−2.75504e7	−0.108449
$253$	−8.54101e6	−0.0331579
$254$	5.37088e7	0.205649
$255$	1.25452e8	0.473791
$256$	1.84303e8	0.686583
$257$	−2.77628e8	−1.02023	−0.510114	−	0.860107i	$-0.670397\pi$
$257$	−2.77628e8	−1.02023	−0.510114	+	0.860107i	$0.670397\pi$
$258$	−2.09561e8	−0.759698
$259$	1.27267e8	0.455162
$260$	6.70646e7	0.236639
$261$	−4.47610e7	−0.155832
$262$	3.87892e8	1.33247
$263$	2.17434e8	0.737025	0.368513	−	0.929623i	$-0.379867\pi$
$263$	2.17434e8	0.737025	0.368513	+	0.929623i	$0.379867\pi$
$264$	−1.00415e7	−0.0335881
$265$	−3.13078e8	−1.03346
$266$	2.76346e8	0.900260
$267$	−2.01881e8	−0.649091
$268$	−4.01530e8	−1.27422
$269$	1.35225e8	0.423569	0.211785	−	0.977316i	$-0.432072\pi$
$269$	1.35225e8	0.423569	0.211785	+	0.977316i	$0.432072\pi$
$270$	2.97509e8	0.919870
$271$	−5.66490e8	−1.72902	−0.864509	−	0.502617i	$-0.832371\pi$
$271$	−5.66490e8	−1.72902	−0.864509	+	0.502617i	$0.832371\pi$
$272$	1.58505e8	0.477586
$273$	3.94557e7	0.117365
$274$	−7.44887e8	−2.18758
$275$	−2.31966e7	−0.0672606
$276$	9.42263e7	0.269768
$277$	−3.18921e8	−0.901580	−0.450790	−	0.892630i	$-0.648858\pi$
$277$	−3.18921e8	−0.901580	−0.450790	+	0.892630i	$0.648858\pi$
$278$	−7.14678e8	−1.99505
$279$	−2.97790e7	−0.0820911
$280$	−2.01561e7	−0.0548723
$281$	−400230.	−0.00107606	−0.000538031	−	1.00000i	$-0.500171\pi$
$281$	−400230.	−0.00107606	−0.000538031		1.00000i	$0.500171\pi$
$282$	−7.91767e8	−2.10245
$283$	4.58601e8	1.20277	0.601385	−	0.798959i	$-0.294616\pi$
$283$	4.58601e8	1.20277	0.601385	+	0.798959i	$0.294616\pi$
$284$	2.10810e8	0.546105
$285$	5.38019e8	1.37670
$286$	2.49553e7	0.0630785
$287$	1.51567e8	0.378459
$288$	−1.47434e8	−0.363685
$289$	−2.81008e8	−0.684820
$290$	−2.81008e8	−0.676589
$291$	−2.04240e8	−0.485865
$292$	8.63809e7	0.203038
$293$	1.47017e8	0.341453	0.170726	−	0.985318i	$-0.445389\pi$
$293$	1.47017e8	0.341453	0.170726	+	0.985318i	$0.445389\pi$
$294$	−1.01757e8	−0.233532
$295$	5.82765e8	1.32165
$296$	−1.03488e8	−0.231937
$297$	5.87776e7	0.130186
$298$	1.81890e8	0.398155
$299$	−2.72895e7	−0.0590401
$300$	2.55911e8	0.547222
$301$	−8.31055e7	−0.175650
$302$	−1.03613e9	−2.16467
$303$	−724623.	−0.00149645
$304$	6.79771e8	1.38773
$305$	4.13919e8	0.835344
$306$	−1.04148e8	−0.207791
$307$	7.40579e8	1.46079	0.730394	−	0.683026i	$-0.239337\pi$
$307$	7.40579e8	1.46079	0.730394	+	0.683026i	$0.239337\pi$
$308$	−3.41711e7	−0.0666394
$309$	4.37343e8	0.843272
$310$	−1.86952e8	−0.356421
$311$	4.38098e8	0.825867	0.412933	−	0.910761i	$-0.364504\pi$
$311$	4.38098e8	0.825867	0.412933	+	0.910761i	$0.364504\pi$
$312$	−3.20838e7	−0.0598060
$313$	9.14427e8	1.68556	0.842779	−	0.538259i	$-0.180918\pi$
$313$	9.14427e8	1.68556	0.842779	+	0.538259i	$0.180918\pi$
$314$	−1.42972e9	−2.60613
$315$	−4.00635e7	−0.0722208
$316$	3.93086e8	0.700782
$317$	−5.05516e8	−0.891308	−0.445654	−	0.895205i	$-0.647029\pi$
$317$	−5.05516e8	−0.891308	−0.445654	+	0.895205i	$0.647029\pi$
$318$	1.28524e9	2.24125
$319$	−5.55175e7	−0.0957552
$320$	−5.49711e8	−0.937799
$321$	−3.89956e7	−0.0658034
$322$	7.03800e7	0.117477
$323$	5.54652e8	0.915824
$324$	−8.24111e8	−1.34610
$325$	−7.41160e7	−0.119762
$326$	1.89594e8	0.303084
$327$	−9.00053e8	−1.42348
$328$	−1.23248e8	−0.192851
$329$	−3.13991e8	−0.486107
$330$	−1.25302e8	−0.191938
$331$	−9.26963e8	−1.40496	−0.702481	−	0.711703i	$-0.747924\pi$
$331$	−9.26963e8	−1.40496	−0.702481	+	0.711703i	$0.747924\pi$
$332$	7.05349e8	1.05784
$333$	−2.05700e8	−0.305267
$334$	−2.71326e8	−0.398455
$335$	−5.83901e8	−0.848559
$336$	−2.50306e8	−0.359985
$337$	8.86499e8	1.26175	0.630875	−	0.775884i	$-0.282696\pi$
$337$	8.86499e8	1.26175	0.630875	+	0.775884i	$0.282696\pi$
$338$	7.97351e7	0.112316
$339$	4.69719e8	0.654845
$340$	−3.47147e8	−0.479002
$341$	−3.69352e7	−0.0504430
$342$	−4.46655e8	−0.603784
$343$	−4.03536e7	−0.0539949
$344$	6.75781e7	0.0895059
$345$	1.37023e8	0.179650
$346$	1.04851e9	1.36084
$347$	5.14839e8	0.661482	0.330741	−	0.943722i	$-0.392701\pi$
$347$	5.14839e8	0.661482	0.330741	+	0.943722i	$0.392701\pi$
$348$	6.12482e8	0.779051
$349$	−1.33575e9	−1.68204	−0.841022	−	0.541001i	$-0.818046\pi$
$349$	−1.33575e9	−1.68204	−0.841022	+	0.541001i	$0.818046\pi$
$350$	1.91146e8	0.238302
$351$	1.87801e8	0.231805
$352$	−1.82864e8	−0.223476
$353$	−1.25324e9	−1.51643	−0.758213	−	0.652006i	$-0.773928\pi$
$353$	−1.25324e9	−1.51643	−0.758213	+	0.652006i	$0.773928\pi$
$354$	−2.39236e9	−2.86625
$355$	3.06557e8	0.363674
$356$	5.58638e8	0.656229
$357$	−2.04235e8	−0.237570
$358$	8.22733e8	0.947695
$359$	−4.53958e8	−0.517827	−0.258914	−	0.965901i	$-0.583364\pi$
$359$	−4.53958e8	−0.517827	−0.258914	+	0.965901i	$0.583364\pi$
$360$	3.25781e7	0.0368016
$361$	1.48483e9	1.66113
$362$	7.22971e8	0.801016
$363$	9.95559e8	1.09243
$364$	−1.09181e8	−0.118656
$365$	1.25614e8	0.135212
$366$	−1.69921e9	−1.81160
$367$	−9.50902e8	−1.00416	−0.502082	−	0.864820i	$-0.667432\pi$
$367$	−9.50902e8	−1.00416	−0.502082	+	0.864820i	$0.667432\pi$
$368$	1.73124e8	0.181089
$369$	−2.44976e8	−0.253824
$370$	−1.29138e9	−1.32540
$371$	5.09688e8	0.518198
$372$	4.07478e8	0.410397
$373$	1.35526e9	1.35221	0.676103	−	0.736807i	$-0.263668\pi$
$373$	1.35526e9	1.35221	0.676103	+	0.736807i	$0.263668\pi$
$374$	−1.29176e8	−0.127683
$375$	1.23396e9	1.20835
$376$	2.55325e8	0.247706
$377$	−1.77385e8	−0.170499
$378$	−4.84342e8	−0.461244
$379$	2.46368e8	0.232460	0.116230	−	0.993222i	$-0.462919\pi$
$379$	2.46368e8	0.232460	0.116230	+	0.993222i	$0.462919\pi$
$380$	−1.48879e9	−1.39185
$381$	−1.70232e8	−0.157690
$382$	1.97637e9	1.81404
$383$	8.56138e8	0.778660	0.389330	−	0.921098i	$-0.372707\pi$
$383$	8.56138e8	0.778660	0.389330	+	0.921098i	$0.372707\pi$
$384$	4.74363e8	0.427515
$385$	−4.96912e7	−0.0443779
$386$	−8.20322e8	−0.725987
$387$	1.34322e8	0.117804
$388$	5.65167e8	0.491208
$389$	−1.48753e9	−1.28128	−0.640638	−	0.767843i	$-0.721330\pi$
$389$	−1.48753e9	−1.28128	−0.640638	+	0.767843i	$0.721330\pi$
$390$	−4.00356e8	−0.341759
$391$	1.41259e8	0.119508
$392$	3.28140e7	0.0275142
$393$	−1.22944e9	−1.02172
$394$	2.01947e7	0.0166342
$395$	5.71621e8	0.466679
$396$	5.52303e7	0.0446935
$397$	1.41259e9	1.13305	0.566526	−	0.824044i	$-0.308287\pi$
$397$	1.41259e9	1.13305	0.566526	+	0.824044i	$0.308287\pi$
$398$	−2.13183e9	−1.69497
$399$	−8.75890e8	−0.690311
$400$	4.70191e8	0.367337
$401$	−1.35108e9	−1.04635	−0.523175	−	0.852225i	$-0.675253\pi$
$401$	−1.35108e9	−1.04635	−0.523175	+	0.852225i	$0.675253\pi$
$402$	2.39702e9	1.84026
$403$	−1.18012e8	−0.0898173
$404$	2.00516e6	0.00151291
$405$	−1.19841e9	−0.896426
$406$	4.57478e8	0.339257
$407$	−2.55132e8	−0.187579
$408$	1.66076e8	0.121058
$409$	9.32581e8	0.673992	0.336996	−	0.941506i	$-0.390589\pi$
$409$	9.32581e8	0.673992	0.336996	+	0.941506i	$0.390589\pi$
$410$	−1.53795e9	−1.10204
$411$	2.36095e9	1.67741
$412$	−1.21020e9	−0.852546
$413$	−9.48737e8	−0.662705
$414$	−1.13754e8	−0.0787892
$415$	1.02571e9	0.704461
$416$	−5.84273e8	−0.397914
$417$	2.26520e9	1.52978
$418$	−5.53991e8	−0.371010
$419$	−8.78711e8	−0.583575	−0.291788	−	0.956483i	$-0.594250\pi$
$419$	−8.78711e8	−0.583575	−0.291788	+	0.956483i	$0.594250\pi$
$420$	5.48204e8	0.361052
$421$	8.26105e8	0.539570	0.269785	−	0.962921i	$-0.413047\pi$
$421$	8.26105e8	0.539570	0.269785	+	0.962921i	$0.413047\pi$
$422$	−7.90000e8	−0.511721
$423$	5.07500e8	0.326020
$424$	−4.14458e8	−0.264059
$425$	3.83647e8	0.242421
$426$	−1.25847e9	−0.788697
$427$	−6.73856e8	−0.418860
$428$	1.07907e8	0.0665271
$429$	−7.90967e7	−0.0483680
$430$	8.43271e8	0.511479
$431$	1.95027e9	1.17334	0.586671	−	0.809825i	$-0.300438\pi$
$431$	1.95027e9	1.17334	0.586671	+	0.809825i	$0.300438\pi$
$432$	−1.19141e9	−0.710997
$433$	1.98444e9	1.17471	0.587355	−	0.809330i	$-0.300169\pi$
$433$	1.98444e9	1.17471	0.587355	+	0.809330i	$0.300169\pi$
$434$	3.04355e8	0.178718
$435$	8.90664e8	0.518802
$436$	2.49060e9	1.43913
$437$	6.05810e8	0.347257
$438$	−5.15669e8	−0.293232
$439$	2.69998e8	0.152312	0.0761562	−	0.997096i	$-0.475735\pi$
$439$	2.69998e8	0.152312	0.0761562	+	0.997096i	$0.475735\pi$
$440$	4.04069e7	0.0226137
$441$	6.52231e7	0.0362131
$442$	−4.12733e8	−0.227348
$443$	−3.05521e9	−1.66966	−0.834830	−	0.550508i	$-0.814434\pi$
$443$	−3.05521e9	−1.66966	−0.834830	+	0.550508i	$0.814434\pi$
$444$	2.81467e9	1.52612
$445$	8.12366e8	0.437010
$446$	2.14766e8	0.114629
$447$	−5.76507e8	−0.305301
$448$	8.94925e8	0.470234
$449$	−3.41546e9	−1.78068	−0.890341	−	0.455294i	$-0.849534\pi$
$449$	−3.41546e9	−1.78068	−0.890341	+	0.455294i	$0.849534\pi$
$450$	−3.08947e8	−0.159823
$451$	−3.03847e8	−0.155968
$452$	−1.29979e9	−0.662047
$453$	3.28406e9	1.65985
$454$	−3.03125e8	−0.152029
$455$	−1.58769e8	−0.0790181
$456$	7.12239e8	0.351762
$457$	−1.56276e9	−0.765923	−0.382961	−	0.923764i	$-0.625096\pi$
$457$	−1.56276e9	−0.765923	−0.382961	+	0.923764i	$0.625096\pi$
$458$	−3.26269e8	−0.158689
$459$	−9.72117e8	−0.469218
$460$	−3.79166e8	−0.181625
$461$	7.99223e8	0.379940	0.189970	−	0.981790i	$-0.439161\pi$
$461$	7.99223e8	0.379940	0.189970	+	0.981790i	$0.439161\pi$
$462$	2.03991e8	0.0962420
$463$	2.13320e9	0.998844	0.499422	−	0.866359i	$-0.333546\pi$
$463$	2.13320e9	0.998844	0.499422	+	0.866359i	$0.333546\pi$
$464$	1.12533e9	0.522957
$465$	5.92550e8	0.273300
$466$	4.85742e9	2.22359
$467$	1.02179e9	0.464249	0.232124	−	0.972686i	$-0.425432\pi$
$467$	1.02179e9	0.464249	0.232124	+	0.972686i	$0.425432\pi$
$468$	1.76467e8	0.0795800
$469$	9.50585e8	0.425487
$470$	3.18606e9	1.41551
$471$	4.53154e9	1.99836
$472$	7.71475e8	0.337695
$473$	1.66602e8	0.0723877
$474$	−2.34661e9	−1.01208
$475$	1.64533e9	0.704409
$476$	5.65152e8	0.240182
$477$	−8.23803e8	−0.347543
$478$	2.45097e9	1.02645
$479$	−1.90064e9	−0.790181	−0.395091	−	0.918642i	$-0.629287\pi$
$479$	−1.90064e9	−0.790181	−0.395091	+	0.918642i	$0.629287\pi$
$480$	2.93368e9	1.21079
$481$	−8.15176e8	−0.333998
$482$	−5.62831e9	−2.28936
$483$	−2.23072e8	−0.0900804
$484$	−2.75488e9	−1.10444
$485$	8.21860e8	0.327116
$486$	1.83150e9	0.723735
$487$	−9.73325e8	−0.381862	−0.190931	−	0.981603i	$-0.561151\pi$
$487$	−9.73325e8	−0.381862	−0.190931	+	0.981603i	$0.561151\pi$
$488$	5.47953e8	0.213439
$489$	−6.00926e8	−0.232402
$490$	4.09468e8	0.157229
$491$	−3.24434e9	−1.23692	−0.618459	−	0.785817i	$-0.712243\pi$
$491$	−3.24434e9	−1.23692	−0.618459	+	0.785817i	$0.712243\pi$
$492$	3.35210e9	1.26894
$493$	9.18199e8	0.345122
$494$	−1.77007e9	−0.660610
$495$	8.03153e7	0.0297632
$496$	7.48670e8	0.275489
$497$	−4.99072e8	−0.182354
$498$	−4.21073e9	−1.52776
$499$	−3.98126e9	−1.43439	−0.717197	−	0.696871i	$-0.754575\pi$
$499$	−3.98126e9	−1.43439	−0.717197	+	0.696871i	$0.754575\pi$
$500$	−3.41459e9	−1.22164
$501$	8.59978e8	0.305531
$502$	−3.91634e9	−1.38171
$503$	−7.51330e8	−0.263235	−0.131617	−	0.991301i	$-0.542017\pi$
$503$	−7.51330e8	−0.263235	−0.131617	+	0.991301i	$0.542017\pi$
$504$	−5.30368e7	−0.0184532
$505$	2.91588e6	0.00100751
$506$	−1.41091e8	−0.0484141
$507$	−2.52723e8	−0.0861227
$508$	4.71061e8	0.159424
$509$	−2.00038e9	−0.672357	−0.336178	−	0.941798i	$-0.609134\pi$
$509$	−2.00038e9	−0.672357	−0.336178	+	0.941798i	$0.609134\pi$
$510$	2.07237e9	0.691785
$511$	−2.04499e8	−0.0677981
$512$	4.20421e9	1.38433
$513$	−4.16907e9	−1.36342
$514$	−4.58619e9	−1.48964
$515$	−1.75986e9	−0.567746
$516$	−1.83799e9	−0.588936
$517$	6.29457e8	0.200332
$518$	2.10235e9	0.664585
$519$	−3.32330e9	−1.04348
$520$	1.29105e8	0.0402653
$521$	−3.49972e9	−1.08418	−0.542090	−	0.840320i	$-0.682367\pi$
$521$	−3.49972e9	−1.08418	−0.542090	+	0.840320i	$0.682367\pi$
$522$	−7.39416e8	−0.227532
$523$	1.06541e9	0.325656	0.162828	−	0.986654i	$-0.447938\pi$
$523$	1.06541e9	0.325656	0.162828	+	0.986654i	$0.447938\pi$
$524$	3.40207e9	1.03296
$525$	−6.05844e8	−0.182727
$526$	3.59184e9	1.07613
$527$	6.10869e8	0.181807
$528$	5.01789e8	0.148355
$529$	−3.25054e9	−0.954686
$530$	−5.17180e9	−1.50896
$531$	1.53343e9	0.444461
$532$	2.42374e9	0.697903
$533$	−9.70826e8	−0.277713
$534$	−3.33491e9	−0.947741
$535$	1.56918e8	0.0443031
$536$	−7.72978e8	−0.216815
$537$	−2.60768e9	−0.726683
$538$	2.23381e9	0.618456
$539$	8.08968e7	0.0222521
$540$	2.60935e9	0.713105
$541$	−1.03768e9	−0.281755	−0.140878	−	0.990027i	$-0.544992\pi$
$541$	−1.03768e9	−0.281755	−0.140878	+	0.990027i	$0.544992\pi$
$542$	−9.35797e9	−2.52455
$543$	−2.29148e9	−0.614211
$544$	3.02438e9	0.805453
$545$	3.62181e9	0.958379
$546$	6.51776e8	0.171366
$547$	−5.61635e9	−1.46723	−0.733616	−	0.679565i	$-0.762169\pi$
$547$	−5.61635e9	−1.46723	−0.733616	+	0.679565i	$0.762169\pi$
$548$	−6.53314e9	−1.69586
$549$	1.08915e9	0.280920
$550$	−3.83190e8	−0.0982075
$551$	3.93783e9	1.00283
$552$	1.81393e8	0.0459023
$553$	−9.30595e8	−0.234004
$554$	−5.26833e9	−1.31640
$555$	4.09307e9	1.01630
$556$	−6.26819e9	−1.54661
$557$	−2.69594e9	−0.661025	−0.330512	−	0.943802i	$-0.607222\pi$
$557$	−2.69594e9	−0.661025	−0.330512	+	0.943802i	$0.607222\pi$
$558$	−4.91926e8	−0.119862
$559$	5.32311e8	0.128892
$560$	1.00723e9	0.242365
$561$	4.09429e8	0.0979058
$562$	−6.61148e6	−0.00157116
$563$	−8.10327e9	−1.91373	−0.956865	−	0.290532i	$-0.906168\pi$
$563$	−8.10327e9	−1.91373	−0.956865	+	0.290532i	$0.906168\pi$
$564$	−6.94431e9	−1.62987
$565$	−1.89014e9	−0.440885
$566$	7.57573e9	1.75617
$567$	1.95101e9	0.449488
$568$	4.05826e8	0.0929224
$569$	2.85648e9	0.650037	0.325019	−	0.945708i	$-0.394629\pi$
$569$	2.85648e9	0.650037	0.325019	+	0.945708i	$0.394629\pi$
$570$	8.88765e9	2.01013
$571$	4.37481e7	0.00983407	0.00491703	−	0.999988i	$-0.498435\pi$
$571$	4.37481e7	0.00983407	0.00491703	+	0.999988i	$0.498435\pi$
$572$	2.18874e8	0.0488999
$573$	−6.26418e9	−1.39099
$574$	2.50377e9	0.552590
$575$	4.19032e8	0.0919200
$576$	−1.44646e9	−0.315375
$577$	−3.46429e9	−0.750756	−0.375378	−	0.926872i	$-0.622487\pi$
$577$	−3.46429e9	−0.750756	−0.375378	+	0.926872i	$0.622487\pi$
$578$	−4.64203e9	−0.999910
$579$	2.60004e9	0.556680
$580$	−2.46462e9	−0.524508
$581$	−1.66985e9	−0.353233
$582$	−3.37388e9	−0.709414
$583$	−1.02177e9	−0.213557
$584$	1.66290e8	0.0345479
$585$	2.56617e8	0.0529956
$586$	2.42860e9	0.498557
$587$	5.09133e9	1.03896	0.519479	−	0.854483i	$-0.326126\pi$
$587$	5.09133e9	1.03896	0.519479	+	0.854483i	$0.326126\pi$
$588$	−8.92472e8	−0.181040
$589$	2.61980e9	0.528281
$590$	9.62682e9	1.92975
$591$	−6.40080e7	−0.0127549
$592$	5.17147e9	1.02444
$593$	8.39572e9	1.65336	0.826678	−	0.562675i	$-0.190228\pi$
$593$	8.39572e9	1.65336	0.826678	+	0.562675i	$0.190228\pi$
$594$	9.70959e8	0.190085
$595$	8.21839e8	0.159947
$596$	1.59529e9	0.308659
$597$	6.75693e9	1.29969
$598$	−4.50801e8	−0.0862047
$599$	7.00541e9	1.33180	0.665901	−	0.746040i	$-0.268047\pi$
$599$	7.00541e9	1.33180	0.665901	+	0.746040i	$0.268047\pi$
$600$	4.92649e8	0.0931125
$601$	2.31926e9	0.435801	0.217900	−	0.975971i	$-0.430079\pi$
$601$	2.31926e9	0.435801	0.217900	+	0.975971i	$0.430079\pi$
$602$	−1.37284e9	−0.256467
$603$	−1.53642e9	−0.285364
$604$	−9.08757e9	−1.67810
$605$	−4.00612e9	−0.735496
$606$	−1.19702e7	−0.00218498
$607$	−3.18733e8	−0.0578452	−0.0289226	−	0.999582i	$-0.509208\pi$
$607$	−3.18733e8	−0.0578452	−0.0289226	+	0.999582i	$0.509208\pi$
$608$	1.29705e10	2.34042
$609$	−1.44999e9	−0.260139
$610$	6.83761e9	1.21969
$611$	2.01119e9	0.356705
$612$	−9.13449e8	−0.161085
$613$	6.63909e9	1.16412	0.582058	−	0.813147i	$-0.302247\pi$
$613$	6.63909e9	1.16412	0.582058	+	0.813147i	$0.302247\pi$
$614$	1.22338e10	2.13290
$615$	4.87459e9	0.845037
$616$	−6.57821e7	−0.0113390
$617$	−7.83771e9	−1.34336	−0.671678	−	0.740844i	$-0.734426\pi$
$617$	−7.83771e9	−1.34336	−0.671678	+	0.740844i	$0.734426\pi$
$618$	7.22456e9	1.23127
$619$	7.53785e9	1.27741	0.638705	−	0.769452i	$-0.279471\pi$
$619$	7.53785e9	1.27741	0.638705	+	0.769452i	$0.279471\pi$
$620$	−1.63969e9	−0.276306
$621$	−1.06178e9	−0.177915
$622$	7.23703e9	1.20585
$623$	−1.32252e9	−0.219127
$624$	1.60327e9	0.264157
$625$	−2.32991e9	−0.381732
$626$	1.51056e10	2.46109
$627$	1.75590e9	0.284487
$628$	−1.25396e10	−2.02033
$629$	4.21960e9	0.676074
$630$	−6.61818e8	−0.105450
$631$	1.18993e10	1.88547	0.942735	−	0.333542i	$-0.108244\pi$
$631$	1.18993e10	1.88547	0.942735	+	0.333542i	$0.108244\pi$
$632$	7.56723e8	0.119241
$633$	2.50393e9	0.392383
$634$	−8.35073e9	−1.30140
$635$	6.85012e8	0.106167
$636$	1.12724e10	1.73747
$637$	2.58475e8	0.0396214
$638$	−9.17105e8	−0.139813
$639$	8.06645e8	0.122301
$640$	−1.90883e9	−0.287831
$641$	−6.39177e9	−0.958557	−0.479279	−	0.877663i	$-0.659102\pi$
$641$	−6.39177e9	−0.958557	−0.479279	+	0.877663i	$0.659102\pi$
$642$	−6.44176e8	−0.0960798
$643$	−9.10080e9	−1.35002	−0.675011	−	0.737808i	$-0.735861\pi$
$643$	−9.10080e9	−1.35002	−0.675011	+	0.737808i	$0.735861\pi$
$644$	6.17278e8	0.0910711
$645$	−2.67278e9	−0.392197
$646$	9.16241e9	1.33720
$647$	−7.91727e9	−1.14924	−0.574620	−	0.818420i	$-0.694850\pi$
$647$	−7.91727e9	−1.14924	−0.574620	+	0.818420i	$0.694850\pi$
$648$	−1.58648e9	−0.229046
$649$	1.90193e9	0.273110
$650$	−1.22434e9	−0.174866
$651$	−9.64666e8	−0.137039
$652$	1.66287e9	0.234958
$653$	−1.90120e9	−0.267197	−0.133599	−	0.991036i	$-0.542653\pi$
$653$	−1.90120e9	−0.267197	−0.133599	+	0.991036i	$0.542653\pi$
$654$	−1.48682e10	−2.07843
$655$	4.94725e9	0.687891
$656$	6.15891e9	0.851805
$657$	3.30529e8	0.0454706
$658$	−5.18688e9	−0.709767
$659$	−3.96408e9	−0.539564	−0.269782	−	0.962921i	$-0.586952\pi$
$659$	−3.96408e9	−0.539564	−0.269782	+	0.962921i	$0.586952\pi$
$660$	−1.09898e9	−0.148795
$661$	−9.74900e9	−1.31297	−0.656485	−	0.754339i	$-0.727958\pi$
$661$	−9.74900e9	−1.31297	−0.656485	+	0.754339i	$0.727958\pi$
$662$	−1.53127e10	−2.05139
$663$	1.30817e9	0.174328
$664$	1.35786e9	0.179997
$665$	3.52457e9	0.464762
$666$	−3.39800e9	−0.445722
$667$	1.00289e9	0.130862
$668$	−2.37971e9	−0.308891
$669$	−6.80710e8	−0.0878963
$670$	−9.64558e9	−1.23899
$671$	1.35088e9	0.172618
$672$	−4.77601e9	−0.607118
$673$	−1.09522e10	−1.38500	−0.692498	−	0.721420i	$-0.743490\pi$
$673$	−1.09522e10	−1.38500	−0.692498	+	0.721420i	$0.743490\pi$
$674$	1.46443e10	1.84229
$675$	−2.88370e9	−0.360900
$676$	6.99328e8	0.0870698
$677$	1.17826e10	1.45943	0.729713	−	0.683754i	$-0.239654\pi$
$677$	1.17826e10	1.45943	0.729713	+	0.683754i	$0.239654\pi$
$678$	7.75938e9	0.956143
$679$	−1.33798e9	−0.164023
$680$	−6.68286e8	−0.0815045
$681$	9.60765e8	0.116574
$682$	−6.10141e8	−0.0736521
$683$	2.71563e9	0.326136	0.163068	−	0.986615i	$-0.447861\pi$
$683$	2.71563e9	0.326136	0.163068	+	0.986615i	$0.447861\pi$
$684$	−3.91746e9	−0.468067
$685$	−9.50043e9	−1.12934
$686$	−6.66610e8	−0.0788383
$687$	1.03412e9	0.121681
$688$	−3.37698e9	−0.395338
$689$	−3.26468e9	−0.380254
$690$	2.26351e9	0.262307
$691$	8.95931e9	1.03300	0.516501	−	0.856286i	$-0.327234\pi$
$691$	8.95931e9	1.03300	0.516501	+	0.856286i	$0.327234\pi$
$692$	9.19614e9	1.05496
$693$	−1.30753e8	−0.0149240
$694$	8.50473e9	0.965834
$695$	−9.11514e9	−1.02995
$696$	1.17908e9	0.132559
$697$	5.02529e9	0.562143
$698$	−2.20656e10	−2.45596
$699$	−1.53958e10	−1.70503
$700$	1.67647e9	0.184737
$701$	−1.06385e10	−1.16645	−0.583225	−	0.812310i	$-0.698210\pi$
$701$	−1.06385e10	−1.16645	−0.583225	+	0.812310i	$0.698210\pi$
$702$	3.10233e9	0.338460
$703$	1.80964e10	1.96448
$704$	−1.79406e9	−0.193790
$705$	−1.00984e10	−1.08540
$706$	−2.07025e10	−2.21414
$707$	−4.74702e6	−0.000505188 0
$708$	−2.09825e10	−2.22199
$709$	1.72488e10	1.81760	0.908798	−	0.417237i	$-0.137001\pi$
$709$	1.72488e10	1.81760	0.908798	+	0.417237i	$0.137001\pi$
$710$	5.06408e9	0.531002
$711$	1.50411e9	0.156941
$712$	1.07542e9	0.111661
$713$	6.67212e8	0.0689367
$714$	−3.37380e9	−0.346877
$715$	3.18284e8	0.0325645
$716$	7.21591e9	0.734675
$717$	−7.76843e9	−0.787075
$718$	−7.49902e9	−0.756082
$719$	−4.55235e9	−0.456756	−0.228378	−	0.973573i	$-0.573342\pi$
$719$	−4.55235e9	−0.456756	−0.228378	+	0.973573i	$0.573342\pi$
$720$	−1.62797e9	−0.162549
$721$	2.86504e9	0.284680
$722$	2.45283e10	2.42542
$723$	1.78391e10	1.75546
$724$	6.34093e9	0.620966
$725$	2.72376e9	0.265452
$726$	1.64458e10	1.59506
$727$	−1.17226e10	−1.13150	−0.565748	−	0.824579i	$-0.691412\pi$
$727$	−1.17226e10	−1.13150	−0.565748	+	0.824579i	$0.691412\pi$
$728$	−2.10181e8	−0.0201899
$729$	6.63480e9	0.634281
$730$	2.07505e9	0.197423
$731$	−2.75541e9	−0.260901
$732$	−1.49032e10	−1.40440
$733$	3.62086e9	0.339585	0.169792	−	0.985480i	$-0.445690\pi$
$733$	3.62086e9	0.339585	0.169792	+	0.985480i	$0.445690\pi$
$734$	−1.57082e10	−1.46619
$735$	−1.29782e9	−0.120562
$736$	3.30333e9	0.305407
$737$	−1.90564e9	−0.175349
$738$	−4.04682e9	−0.370609
$739$	−3.13654e9	−0.285888	−0.142944	−	0.989731i	$-0.545657\pi$
$739$	−3.13654e9	−0.285888	−0.142944	+	0.989731i	$0.545657\pi$
$740$	−1.13262e10	−1.02748
$741$	5.61029e9	0.506550
$742$	8.41965e9	0.756624
$743$	−4.08464e9	−0.365336	−0.182668	−	0.983175i	$-0.558473\pi$
$743$	−4.08464e9	−0.365336	−0.182668	+	0.983175i	$0.558473\pi$
$744$	7.84428e8	0.0698310
$745$	2.31986e9	0.205549
$746$	2.23879e10	1.97436
$747$	2.69896e9	0.236905
$748$	−1.13296e9	−0.0989826
$749$	−2.55461e8	−0.0222146
$750$	2.03841e10	1.76432
$751$	1.72129e10	1.48291	0.741453	−	0.671005i	$-0.234137\pi$
$751$	1.72129e10	1.48291	0.741453	+	0.671005i	$0.234137\pi$
$752$	−1.27590e10	−1.09409
$753$	1.24130e10	1.05948
$754$	−2.93026e9	−0.248947
$755$	−1.32150e10	−1.11752
$756$	−4.24799e9	−0.357567
$757$	−2.46798e9	−0.206778	−0.103389	−	0.994641i	$-0.532969\pi$
$757$	−2.46798e9	−0.206778	−0.103389	+	0.994641i	$0.532969\pi$
$758$	4.06981e9	0.339416
$759$	4.47192e8	0.0371234
$760$	−2.86604e9	−0.236829
$761$	−2.10854e10	−1.73434	−0.867171	−	0.498011i	$-0.834064\pi$
$761$	−2.10854e10	−1.73434	−0.867171	+	0.498011i	$0.834064\pi$
$762$	−2.81210e9	−0.230244
$763$	−5.89627e9	−0.480553
$764$	1.73341e10	1.40629
$765$	−1.32833e9	−0.107273
$766$	1.41427e10	1.13693
$767$	6.07689e9	0.486293
$768$	−9.64979e9	−0.768694
$769$	−1.71440e10	−1.35947	−0.679736	−	0.733457i	$-0.737906\pi$
$769$	−1.71440e10	−1.35947	−0.679736	+	0.733457i	$0.737906\pi$
$770$	−8.20859e8	−0.0647964
$771$	1.45361e10	1.14224
$772$	−7.19476e9	−0.562802
$773$	−1.01968e10	−0.794031	−0.397016	−	0.917812i	$-0.629954\pi$
$773$	−1.01968e10	−0.794031	−0.397016	+	0.917812i	$0.629954\pi$
$774$	2.21890e9	0.172006
$775$	1.81209e9	0.139838
$776$	1.08799e9	0.0835815
$777$	−6.66347e9	−0.509597
$778$	−2.45729e10	−1.87080
$779$	2.15517e10	1.63343
$780$	−3.51139e9	−0.264940
$781$	1.00049e9	0.0751508
$782$	2.33349e9	0.174494
$783$	−6.90169e9	−0.513794
$784$	−1.63976e9	−0.121527
$785$	−1.82349e10	−1.34542
$786$	−2.03094e10	−1.49182
$787$	−1.66624e10	−1.21850	−0.609249	−	0.792979i	$-0.708529\pi$
$787$	−1.66624e10	−1.21850	−0.609249	+	0.792979i	$0.708529\pi$
$788$	1.77121e8	0.0128952
$789$	−1.13845e10	−0.825169
$790$	9.44273e9	0.681401
$791$	3.07714e9	0.221070
$792$	1.06323e8	0.00760481
$793$	4.31621e9	0.307360
$794$	2.33349e10	1.65437
$795$	1.63922e10	1.15705
$796$	−1.86976e10	−1.31398
$797$	1.48558e10	1.03942	0.519710	−	0.854343i	$-0.326040\pi$
$797$	1.48558e10	1.03942	0.519710	+	0.854343i	$0.326040\pi$
$798$	−1.44690e10	−1.00793
$799$	−1.04105e10	−0.722037
$800$	8.97156e9	0.619517
$801$	2.13758e9	0.146963
$802$	−2.23188e10	−1.52778
$803$	4.09959e8	0.0279406
$804$	2.10234e10	1.42662
$805$	8.97640e8	0.0606480
$806$	−1.94947e9	−0.131143
$807$	−7.08015e9	−0.474226
$808$	3.86009e6	0.000257429 0
$809$	−1.90191e10	−1.26290	−0.631452	−	0.775415i	$-0.717541\pi$
$809$	−1.90191e10	−1.26290	−0.631452	+	0.775415i	$0.717541\pi$
$810$	−1.97969e10	−1.30888
$811$	−1.04660e10	−0.688980	−0.344490	−	0.938790i	$-0.611948\pi$
$811$	−1.04660e10	−0.688980	−0.344490	+	0.938790i	$0.611948\pi$
$812$	4.01238e9	0.263000
$813$	2.96604e10	1.93580
$814$	−4.21458e9	−0.273885
$815$	2.41812e9	0.156468
$816$	−8.29904e9	−0.534703
$817$	−1.18170e10	−0.758104
$818$	1.54055e10	0.984100
$819$	−4.17770e8	−0.0265732
$820$	−1.34888e10	−0.854331
$821$	2.76554e9	0.174413	0.0872067	−	0.996190i	$-0.472206\pi$
$821$	2.76554e9	0.174413	0.0872067	+	0.996190i	$0.472206\pi$
$822$	3.90010e10	2.44920
$823$	1.48219e10	0.926839	0.463419	−	0.886139i	$-0.346622\pi$
$823$	1.48219e10	0.926839	0.463419	+	0.886139i	$0.346622\pi$
$824$	−2.32974e9	−0.145065
$825$	1.21454e9	0.0753046
$826$	−1.56724e10	−0.967620
$827$	−2.04045e10	−1.25446	−0.627228	−	0.778835i	$-0.715811\pi$
$827$	−2.04045e10	−1.25446	−0.627228	+	0.778835i	$0.715811\pi$
$828$	−9.97700e8	−0.0610793
$829$	−4.50375e8	−0.0274558	−0.0137279	−	0.999906i	$-0.504370\pi$
$829$	−4.50375e8	−0.0274558	−0.0137279	+	0.999906i	$0.504370\pi$
$830$	1.69439e10	1.02859
$831$	1.66982e10	1.00940
$832$	−5.73222e9	−0.345057
$833$	−1.33795e9	−0.0802012
$834$	3.74193e10	2.23365
$835$	−3.46054e9	−0.205704
$836$	−4.85886e9	−0.287616
$837$	−4.59162e9	−0.270662
$838$	−1.45156e10	−0.852081
$839$	−5.84044e9	−0.341412	−0.170706	−	0.985322i	$-0.554605\pi$
$839$	−5.84044e9	−0.341412	−0.170706	+	0.985322i	$0.554605\pi$
$840$	1.05534e9	0.0614348
$841$	−1.07310e10	−0.622091
$842$	1.36466e10	0.787830
$843$	2.09553e7	0.00120475
$844$	−6.92881e9	−0.396698
$845$	1.01696e9	0.0579834
$846$	8.38350e9	0.476024
$847$	6.52193e9	0.368794
$848$	2.07111e10	1.16632
$849$	−2.40116e10	−1.34662
$850$	6.33755e9	0.353961
$851$	4.60879e9	0.256350
$852$	−1.10376e10	−0.611416
$853$	−2.26912e10	−1.25180	−0.625901	−	0.779903i	$-0.715269\pi$
$853$	−2.26912e10	−1.25180	−0.625901	+	0.779903i	$0.715269\pi$
$854$	−1.11316e10	−0.611580
$855$	−5.69673e9	−0.311705
$856$	2.07731e8	0.0113199
$857$	7.94601e9	0.431237	0.215619	−	0.976478i	$-0.430823\pi$
$857$	7.94601e9	0.431237	0.215619	+	0.976478i	$0.430823\pi$
$858$	−1.30662e9	−0.0706223
$859$	1.83953e10	0.990218	0.495109	−	0.868831i	$-0.335128\pi$
$859$	1.83953e10	0.990218	0.495109	+	0.868831i	$0.335128\pi$
$860$	7.39603e9	0.396510
$861$	−7.93580e9	−0.423720
$862$	3.22169e10	1.71320
$863$	1.82955e10	0.968959	0.484480	−	0.874803i	$-0.339009\pi$
$863$	1.82955e10	0.968959	0.484480	+	0.874803i	$0.339009\pi$
$864$	−2.27329e10	−1.19910
$865$	1.33729e10	0.702539
$866$	3.27814e10	1.71520
$867$	1.47131e10	0.766721
$868$	2.66940e9	0.138546
$869$	1.86556e9	0.0964363
$870$	1.47131e10	0.757506
$871$	−6.08873e9	−0.312222
$872$	4.79461e9	0.244876
$873$	2.16256e9	0.110007
$874$	1.00075e10	0.507032
$875$	8.08372e9	0.407928
$876$	−4.52275e9	−0.227321
$877$	−3.30162e10	−1.65283	−0.826415	−	0.563061i	$-0.809624\pi$
$877$	−3.30162e10	−1.65283	−0.826415	+	0.563061i	$0.809624\pi$
$878$	4.46016e9	0.222392
$879$	−7.69755e9	−0.382289
$880$	−2.01919e9	−0.0998823
$881$	6.60658e9	0.325508	0.162754	−	0.986667i	$-0.447962\pi$
$881$	6.60658e9	0.325508	0.162754	+	0.986667i	$0.447962\pi$
$882$	1.07743e9	0.0528750
$883$	3.55658e10	1.73848	0.869241	−	0.494389i	$-0.164608\pi$
$883$	3.55658e10	1.73848	0.869241	+	0.494389i	$0.164608\pi$
$884$	−3.61994e9	−0.176246
$885$	−3.05126e10	−1.47971
$886$	−5.04696e10	−2.43788
$887$	−9.78272e9	−0.470681	−0.235341	−	0.971913i	$-0.575621\pi$
$887$	−9.78272e9	−0.470681	−0.235341	+	0.971913i	$0.575621\pi$
$888$	5.41847e9	0.259676
$889$	−1.11519e9	−0.0532346
$890$	1.34196e10	0.638081
$891$	−3.91118e9	−0.185241
$892$	1.88364e9	0.0888630
$893$	−4.46471e10	−2.09804
$894$	−9.52344e9	−0.445772
$895$	1.04933e10	0.489251
$896$	3.10756e9	0.144325
$897$	1.42883e9	0.0661009
$898$	−5.64206e10	−2.59998
$899$	4.33695e9	0.199079
$900$	−2.70967e9	−0.123899
$901$	1.68990e10	0.769704
$902$	−5.01931e9	−0.227730
$903$	4.35126e9	0.196656
$904$	−2.50220e9	−0.112651
$905$	9.22091e9	0.413527
$906$	5.42501e10	2.42355
$907$	2.36003e10	1.05025	0.525124	−	0.851026i	$-0.324019\pi$
$907$	2.36003e10	1.05025	0.525124	+	0.851026i	$0.324019\pi$
$908$	−2.65860e9	−0.117856
$909$	7.67256e6	0.000338818 0
$910$	−2.62274e9	−0.115375
$911$	1.33238e10	0.583868	0.291934	−	0.956438i	$-0.405701\pi$
$911$	1.33238e10	0.583868	0.291934	+	0.956438i	$0.405701\pi$
$912$	−3.55916e10	−1.55370
$913$	3.34754e9	0.145572
$914$	−2.58155e10	−1.11833
$915$	−2.16721e10	−0.935247
$916$	−2.86159e9	−0.123019
$917$	−8.05408e9	−0.344924
$918$	−1.60586e10	−0.685107
$919$	3.06892e10	1.30431	0.652155	−	0.758086i	$-0.273865\pi$
$919$	3.06892e10	1.30431	0.652155	+	0.758086i	$0.273865\pi$
$920$	−7.29925e8	−0.0309044
$921$	−3.87754e10	−1.63549
$922$	1.32025e10	0.554752
$923$	3.19668e9	0.133811
$924$	1.78914e9	0.0746091
$925$	1.25171e10	0.520005
$926$	3.52387e10	1.45842
$927$	−4.63074e9	−0.190929
$928$	2.14720e10	0.881972
$929$	2.42640e10	0.992905	0.496452	−	0.868064i	$-0.334636\pi$
$929$	2.42640e10	0.992905	0.496452	+	0.868064i	$0.334636\pi$
$930$	9.78846e9	0.399047
$931$	−5.73798e9	−0.233042
$932$	4.26027e10	1.72378
$933$	−2.29381e10	−0.924636
$934$	1.68791e10	0.677852
$935$	−1.64754e9	−0.0659166
$936$	3.39714e8	0.0135409
$937$	−9.57434e8	−0.0380207	−0.0190104	−	0.999819i	$-0.506052\pi$
$937$	−9.57434e8	−0.0380207	−0.0190104	+	0.999819i	$0.506052\pi$
$938$	1.57029e10	0.621255
$939$	−4.78778e10	−1.88714
$940$	2.79439e10	1.09733
$941$	−1.66015e10	−0.649508	−0.324754	−	0.945799i	$-0.605282\pi$
$941$	−1.66015e10	−0.649508	−0.324754	+	0.945799i	$0.605282\pi$
$942$	7.48575e10	2.91781
$943$	5.48879e9	0.213150
$944$	−3.85517e10	−1.49156
$945$	−6.17739e9	−0.238119
$946$	2.75212e9	0.105694
$947$	−2.54825e9	−0.0975028	−0.0487514	−	0.998811i	$-0.515524\pi$
$947$	−2.54825e9	−0.0975028	−0.0487514	+	0.998811i	$0.515524\pi$
$948$	−2.05813e10	−0.784591
$949$	1.30987e9	0.0497502
$950$	2.71795e10	1.02851
$951$	2.64679e10	0.997903
$952$	1.08796e9	0.0408682
$953$	1.63601e10	0.612294	0.306147	−	0.951984i	$-0.400960\pi$
$953$	1.63601e10	0.612294	0.306147	+	0.951984i	$0.400960\pi$
$954$	−1.36086e10	−0.507450
$955$	2.52070e10	0.936504
$956$	2.14966e10	0.795732
$957$	2.90680e9	0.107207
$958$	−3.13971e10	−1.15375
$959$	1.54666e10	0.566278
$960$	2.87819e10	1.04995
$961$	−2.46273e10	−0.895127
$962$	−1.34661e10	−0.487672
$963$	4.12899e8	0.0148988
$964$	−4.93639e10	−1.77476
$965$	−1.04625e10	−0.374793
$966$	−3.68497e9	−0.131527
$967$	4.09400e10	1.45598	0.727990	−	0.685588i	$-0.240455\pi$
$967$	4.09400e10	1.45598	0.727990	+	0.685588i	$0.240455\pi$
$968$	−5.30337e9	−0.187927
$969$	−2.90406e10	−1.02535
$970$	1.35765e10	0.477624
$971$	−2.64816e10	−0.928276	−0.464138	−	0.885763i	$-0.653636\pi$
$971$	−2.64816e10	−0.928276	−0.464138	+	0.885763i	$0.653636\pi$
$972$	1.60634e10	0.561057
$973$	1.48394e10	0.516441
$974$	−1.60785e10	−0.557559
$975$	3.88058e9	0.134085
$976$	−2.73820e10	−0.942738
$977$	4.06202e10	1.39351	0.696756	−	0.717308i	$-0.254626\pi$
$977$	4.06202e10	1.39351	0.696756	+	0.717308i	$0.254626\pi$
$978$	−9.92682e9	−0.339332
$979$	2.65126e9	0.0903053
$980$	3.59130e9	0.121888
$981$	9.53007e9	0.322296
$982$	−5.35939e10	−1.80603
$983$	4.75132e10	1.59543	0.797714	−	0.603036i	$-0.206042\pi$
$983$	4.75132e10	1.59543	0.797714	+	0.603036i	$0.206042\pi$
$984$	6.45307e9	0.215915
$985$	2.57568e8	0.00858746
$986$	1.51679e10	0.503915
$987$	1.64400e10	0.544242
$988$	−1.55246e10	−0.512121
$989$	−3.00955e9	−0.0989269
$990$	1.32675e9	0.0434575
$991$	9.60220e9	0.313410	0.156705	−	0.987645i	$-0.449913\pi$
$991$	9.60220e9	0.313410	0.156705	+	0.987645i	$0.449913\pi$
$992$	1.42851e10	0.464615
$993$	4.85342e10	1.57299
$994$	−8.24428e9	−0.266257
$995$	−2.71898e10	−0.875035
$996$	−3.69309e10	−1.18435
$997$	8.05835e9	0.257521	0.128761	−	0.991676i	$-0.458900\pi$
$997$	8.05835e9	0.257521	0.128761	+	0.991676i	$0.458900\pi$
$998$	−6.57672e10	−2.09437
$999$	−3.17168e10	−1.00649

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	91.8.a.b.1.8	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.8.a.b.1.8	✓	9	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:	\( N \)	\(=\)	\( 91 = 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	91.a (trivial)

\(f(q)\)	\(=\)	\(q+16.5192 q^{2} -52.3583 q^{3} +144.884 q^{4} +210.689 q^{5} -864.917 q^{6} -343.000 q^{7} +278.914 q^{8} +554.387 q^{9} +O(q^{10})\)	\(q+16.5192 q^{2} -52.3583 q^{3} +144.884 q^{4} +210.689 q^{5} -864.917 q^{6} -343.000 q^{7} +278.914 q^{8} +554.387 q^{9} +3480.42 q^{10} +687.612 q^{11} -7585.89 q^{12} +2197.00 q^{13} -5666.09 q^{14} -11031.3 q^{15} -13937.7 q^{16} -11372.4 q^{17} +9158.03 q^{18} -48772.0 q^{19} +30525.5 q^{20} +17958.9 q^{21} +11358.8 q^{22} -12421.3 q^{23} -14603.4 q^{24} -33735.1 q^{25} +36292.7 q^{26} +85480.8 q^{27} -49695.3 q^{28} -80739.6 q^{29} -182229. q^{30} -53715.3 q^{31} -265941. q^{32} -36002.1 q^{33} -187862. q^{34} -72266.4 q^{35} +80321.9 q^{36} -371041. q^{37} -805675. q^{38} -115031. q^{39} +58764.1 q^{40} -441887. q^{41} +296666. q^{42} +242290. q^{43} +99624.1 q^{44} +116803. q^{45} -205189. q^{46} +915426. q^{47} +729756. q^{48} +117649. q^{49} -557277. q^{50} +595436. q^{51} +318311. q^{52} -1.48597e6 q^{53} +1.41207e6 q^{54} +144872. q^{55} -95667.5 q^{56} +2.55362e6 q^{57} -1.33375e6 q^{58} +2.76600e6 q^{59} -1.59826e6 q^{60} +1.96459e6 q^{61} -887334. q^{62} -190155. q^{63} -2.60911e6 q^{64} +462884. q^{65} -594727. q^{66} -2.77138e6 q^{67} -1.64767e6 q^{68} +650356. q^{69} -1.19378e6 q^{70} +1.45502e6 q^{71} +154626. q^{72} +596207. q^{73} -6.12930e6 q^{74} +1.76631e6 q^{75} -7.06629e6 q^{76} -235851. q^{77} -1.90022e6 q^{78} +2.71310e6 q^{79} -2.93653e6 q^{80} -5.68807e6 q^{81} -7.29962e6 q^{82} +4.86836e6 q^{83} +2.60196e6 q^{84} -2.39603e6 q^{85} +4.00244e6 q^{86} +4.22739e6 q^{87} +191784. q^{88} +3.85576e6 q^{89} +1.92950e6 q^{90} -753571. q^{91} -1.79965e6 q^{92} +2.81244e6 q^{93} +1.51221e7 q^{94} -1.02757e7 q^{95} +1.39242e7 q^{96} +3.90082e6 q^{97} +1.94347e6 q^{98} +381203. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 5 q^{2} - 26 q^{3} + 393 q^{4} - 181 q^{5} + 703 q^{6} - 3087 q^{7} + 1197 q^{8} + 3211 q^{9}+O(q^{10}) \) 9 * q - 5 * q^2 - 26 * q^3 + 393 * q^4 - 181 * q^5 + 703 * q^6 - 3087 * q^7 + 1197 * q^8 + 3211 * q^9	\( 9 q - 5 q^{2} - 26 q^{3} + 393 q^{4} - 181 q^{5} + 703 q^{6} - 3087 q^{7} + 1197 q^{8} + 3211 q^{9} - 5124 q^{10} - 9826 q^{11} - 20919 q^{12} + 19773 q^{13} + 1715 q^{14} - 20346 q^{15} + 31113 q^{16} - 22766 q^{17} - 12978 q^{18} - 17769 q^{19} - 44204 q^{20} + 8918 q^{21} - 203553 q^{22} - 49103 q^{23} + 52737 q^{24} + 227466 q^{25} - 10985 q^{26} + 103624 q^{27} - 134799 q^{28} - 487455 q^{29} - 287992 q^{30} - 63843 q^{31} - 587099 q^{32} - 314392 q^{33} - 576240 q^{34} + 62083 q^{35} - 1514926 q^{36} - 796926 q^{37} - 766702 q^{38} - 57122 q^{39} - 2887296 q^{40} - 1567546 q^{41} - 241129 q^{42} - 277899 q^{43} - 1281195 q^{44} - 1650593 q^{45} - 1907445 q^{46} + 1077367 q^{47} - 1110835 q^{48} + 1058841 q^{49} - 267459 q^{50} - 3054368 q^{51} + 863421 q^{52} - 7322659 q^{53} - 3355387 q^{54} - 2613324 q^{55} - 410571 q^{56} - 3751946 q^{57} - 2992332 q^{58} - 169804 q^{59} - 2754416 q^{60} - 6352284 q^{61} + 6001087 q^{62} - 1101373 q^{63} + 1657017 q^{64} - 397657 q^{65} - 5962713 q^{66} + 921120 q^{67} + 5615224 q^{68} - 5202780 q^{69} + 1757532 q^{70} + 3786654 q^{71} + 2229758 q^{72} + 5792889 q^{73} - 1991961 q^{74} + 145628 q^{75} - 2806026 q^{76} + 3370318 q^{77} + 1544491 q^{78} + 3464037 q^{79} + 15422512 q^{80} - 5010363 q^{81} - 12539943 q^{82} + 6834945 q^{83} + 7175217 q^{84} + 3880662 q^{85} - 7977524 q^{86} + 3727078 q^{87} + 7013709 q^{88} - 20408371 q^{89} + 34910060 q^{90} - 6782139 q^{91} - 3544371 q^{92} + 3121742 q^{93} + 61343967 q^{94} + 3360807 q^{95} + 23547905 q^{96} + 41644125 q^{97} - 588245 q^{98} + 50754068 q^{99}+O(q^{100}) \) 9 * q - 5 * q^2 - 26 * q^3 + 393 * q^4 - 181 * q^5 + 703 * q^6 - 3087 * q^7 + 1197 * q^8 + 3211 * q^9 - 5124 * q^10 - 9826 * q^11 - 20919 * q^12 + 19773 * q^13 + 1715 * q^14 - 20346 * q^15 + 31113 * q^16 - 22766 * q^17 - 12978 * q^18 - 17769 * q^19 - 44204 * q^20 + 8918 * q^21 - 203553 * q^22 - 49103 * q^23 + 52737 * q^24 + 227466 * q^25 - 10985 * q^26 + 103624 * q^27 - 134799 * q^28 - 487455 * q^29 - 287992 * q^30 - 63843 * q^31 - 587099 * q^32 - 314392 * q^33 - 576240 * q^34 + 62083 * q^35 - 1514926 * q^36 - 796926 * q^37 - 766702 * q^38 - 57122 * q^39 - 2887296 * q^40 - 1567546 * q^41 - 241129 * q^42 - 277899 * q^43 - 1281195 * q^44 - 1650593 * q^45 - 1907445 * q^46 + 1077367 * q^47 - 1110835 * q^48 + 1058841 * q^49 - 267459 * q^50 - 3054368 * q^51 + 863421 * q^52 - 7322659 * q^53 - 3355387 * q^54 - 2613324 * q^55 - 410571 * q^56 - 3751946 * q^57 - 2992332 * q^58 - 169804 * q^59 - 2754416 * q^60 - 6352284 * q^61 + 6001087 * q^62 - 1101373 * q^63 + 1657017 * q^64 - 397657 * q^65 - 5962713 * q^66 + 921120 * q^67 + 5615224 * q^68 - 5202780 * q^69 + 1757532 * q^70 + 3786654 * q^71 + 2229758 * q^72 + 5792889 * q^73 - 1991961 * q^74 + 145628 * q^75 - 2806026 * q^76 + 3370318 * q^77 + 1544491 * q^78 + 3464037 * q^79 + 15422512 * q^80 - 5010363 * q^81 - 12539943 * q^82 + 6834945 * q^83 + 7175217 * q^84 + 3880662 * q^85 - 7977524 * q^86 + 3727078 * q^87 + 7013709 * q^88 - 20408371 * q^89 + 34910060 * q^90 - 6782139 * q^91 - 3544371 * q^92 + 3121742 * q^93 + 61343967 * q^94 + 3360807 * q^95 + 23547905 * q^96 + 41644125 * q^97 - 588245 * q^98 + 50754068 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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