LMFDB - Embedded newform 23.12.a.a.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [23,12,Mod(1,23)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(23, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 12, names="a")

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

chi := DirichletCharacter("23.1");

S:= CuspForms(chi, 12);

N := Newforms(S);

Level:	$ N $	$=$	$ 23 $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	23.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:	$17.6718931529$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2672x^{6} - 1234x^{5} + 2202967x^{4} + 2386582x^{3} - 543567396x^{2} - 1204011928x + 23305583840 $ x^8 - 2672x^6 - 1234x^5 + 2202967x^4 + 2386582x^3 - 543567396x^2 - 1204011928x + 23305583840
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{11}\cdot 3^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-16.3063$ of defining polynomial
Character	$\chi$	$=$	23.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+28.6126 q^{2} +141.482 q^{3} -1229.32 q^{4} -2645.71 q^{5} +4048.18 q^{6} +76456.7 q^{7} -93772.7 q^{8} -157130. q^{9} +O(q^{10})$	\(q+28.6126 q^{2} +141.482 q^{3} -1229.32 q^{4} -2645.71 q^{5} +4048.18 q^{6} +76456.7 q^{7} -93772.7 q^{8} -157130. q^{9} -75700.7 q^{10} -603251. q^{11} -173926. q^{12} -941600. q^{13} +2.18763e6 q^{14} -374320. q^{15} -165445. q^{16} -5.54744e6 q^{17} -4.49590e6 q^{18} -8.15289e6 q^{19} +3.25241e6 q^{20} +1.08173e7 q^{21} -1.72606e7 q^{22} +6.43634e6 q^{23} -1.32672e7 q^{24} -4.18283e7 q^{25} -2.69417e7 q^{26} -4.72942e7 q^{27} -9.39895e7 q^{28} +1.14880e8 q^{29} -1.07103e7 q^{30} -1.09756e8 q^{31} +1.87313e8 q^{32} -8.53492e7 q^{33} -1.58727e8 q^{34} -2.02282e8 q^{35} +1.93162e8 q^{36} -3.97244e8 q^{37} -2.33276e8 q^{38} -1.33220e8 q^{39} +2.48095e8 q^{40} +7.31306e8 q^{41} +3.09510e8 q^{42} +7.98600e8 q^{43} +7.41587e8 q^{44} +4.15720e8 q^{45} +1.84161e8 q^{46} -4.30123e8 q^{47} -2.34075e7 q^{48} +3.86830e9 q^{49} -1.19682e9 q^{50} -7.84863e8 q^{51} +1.15752e9 q^{52} +4.54586e9 q^{53} -1.35321e9 q^{54} +1.59603e9 q^{55} -7.16955e9 q^{56} -1.15349e9 q^{57} +3.28701e9 q^{58} -7.83815e8 q^{59} +4.60158e8 q^{60} -7.23765e9 q^{61} -3.14042e9 q^{62} -1.20136e10 q^{63} +5.69834e9 q^{64} +2.49120e9 q^{65} -2.44207e9 q^{66} -1.06037e10 q^{67} +6.81956e9 q^{68} +9.10627e8 q^{69} -5.78783e9 q^{70} -1.08505e10 q^{71} +1.47345e10 q^{72} +3.25510e10 q^{73} -1.13662e10 q^{74} -5.91796e9 q^{75} +1.00225e10 q^{76} -4.61226e10 q^{77} -3.81176e9 q^{78} -5.15171e10 q^{79} +4.37720e8 q^{80} +2.11438e10 q^{81} +2.09246e10 q^{82} +6.27091e10 q^{83} -1.32978e10 q^{84} +1.46769e10 q^{85} +2.28501e10 q^{86} +1.62534e10 q^{87} +5.65685e10 q^{88} -2.11172e10 q^{89} +1.18948e10 q^{90} -7.19916e10 q^{91} -7.91230e9 q^{92} -1.55285e10 q^{93} -1.23070e10 q^{94} +2.15702e10 q^{95} +2.65014e10 q^{96} -1.36592e11 q^{97} +1.10682e11 q^{98} +9.47888e10 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 32 q^{2} - 992 q^{3} + 5120 q^{4} - 11466 q^{5} + 34618 q^{6} - 54118 q^{7} - 155568 q^{8} + 648814 q^{9}+O(q^{10}) $ 8 * q - 32 * q^2 - 992 * q^3 + 5120 * q^4 - 11466 * q^5 + 34618 * q^6 - 54118 * q^7 - 155568 * q^8 + 648814 * q^9	\( 8 q - 32 q^{2} - 992 q^{3} + 5120 q^{4} - 11466 q^{5} + 34618 q^{6} - 54118 q^{7} - 155568 q^{8} + 648814 q^{9} + 517892 q^{10} - 291462 q^{11} - 884188 q^{12} - 2211306 q^{13} - 939584 q^{14} - 2205330 q^{15} - 8561344 q^{16} - 5775330 q^{17} - 51349034 q^{18} - 21015588 q^{19} - 65503576 q^{20} - 36171230 q^{21} - 83047784 q^{22} + 51490744 q^{23} - 129286728 q^{24} - 36491644 q^{25} - 119299562 q^{26} - 394617320 q^{27} - 392796032 q^{28} - 322285430 q^{29} - 646885140 q^{30} - 415184840 q^{31} + 31831744 q^{32} - 549306602 q^{33} - 28224252 q^{34} + 603721008 q^{35} + 690703676 q^{36} + 176642018 q^{37} + 554685496 q^{38} + 2251149264 q^{39} + 1337904816 q^{40} + 357962218 q^{41} + 340644280 q^{42} + 2500461376 q^{43} + 5064743472 q^{44} + 385017072 q^{45} - 205962976 q^{46} + 261795200 q^{47} + 4421752784 q^{48} + 2656605924 q^{49} + 1642758328 q^{50} + 6771514570 q^{51} + 3841657212 q^{52} + 3542935060 q^{53} + 18173306686 q^{54} - 10100187604 q^{55} + 7995463104 q^{56} - 14761628752 q^{57} - 9113565454 q^{58} + 930905396 q^{59} + 19344914040 q^{60} - 25338655048 q^{61} + 4385691666 q^{62} - 25499316044 q^{63} - 34067008768 q^{64} - 25954746658 q^{65} + 13172584012 q^{66} - 3123467482 q^{67} - 37358480280 q^{68} - 6384852256 q^{69} - 35719175696 q^{70} - 52612263236 q^{71} - 9100886376 q^{72} - 67014176274 q^{73} + 10171443276 q^{74} - 87540153860 q^{75} + 17955918576 q^{76} - 44516617816 q^{77} - 25596104778 q^{78} - 27683357604 q^{79} + 74357773216 q^{80} + 55141240264 q^{81} + 73615849126 q^{82} - 12253964262 q^{83} + 168565479344 q^{84} + 58779027600 q^{85} + 90522557252 q^{86} - 129275944888 q^{87} + 33736356800 q^{88} + 10662817760 q^{89} + 450294422856 q^{90} - 28336741418 q^{91} + 32954076160 q^{92} + 164368292014 q^{93} + 285145948346 q^{94} - 64104297380 q^{95} + 208023008864 q^{96} - 124519454530 q^{97} + 215615498272 q^{98} + 186256571332 q^{99}+O(q^{100}) \) 8 * q - 32 * q^2 - 992 * q^3 + 5120 * q^4 - 11466 * q^5 + 34618 * q^6 - 54118 * q^7 - 155568 * q^8 + 648814 * q^9 + 517892 * q^10 - 291462 * q^11 - 884188 * q^12 - 2211306 * q^13 - 939584 * q^14 - 2205330 * q^15 - 8561344 * q^16 - 5775330 * q^17 - 51349034 * q^18 - 21015588 * q^19 - 65503576 * q^20 - 36171230 * q^21 - 83047784 * q^22 + 51490744 * q^23 - 129286728 * q^24 - 36491644 * q^25 - 119299562 * q^26 - 394617320 * q^27 - 392796032 * q^28 - 322285430 * q^29 - 646885140 * q^30 - 415184840 * q^31 + 31831744 * q^32 - 549306602 * q^33 - 28224252 * q^34 + 603721008 * q^35 + 690703676 * q^36 + 176642018 * q^37 + 554685496 * q^38 + 2251149264 * q^39 + 1337904816 * q^40 + 357962218 * q^41 + 340644280 * q^42 + 2500461376 * q^43 + 5064743472 * q^44 + 385017072 * q^45 - 205962976 * q^46 + 261795200 * q^47 + 4421752784 * q^48 + 2656605924 * q^49 + 1642758328 * q^50 + 6771514570 * q^51 + 3841657212 * q^52 + 3542935060 * q^53 + 18173306686 * q^54 - 10100187604 * q^55 + 7995463104 * q^56 - 14761628752 * q^57 - 9113565454 * q^58 + 930905396 * q^59 + 19344914040 * q^60 - 25338655048 * q^61 + 4385691666 * q^62 - 25499316044 * q^63 - 34067008768 * q^64 - 25954746658 * q^65 + 13172584012 * q^66 - 3123467482 * q^67 - 37358480280 * q^68 - 6384852256 * q^69 - 35719175696 * q^70 - 52612263236 * q^71 - 9100886376 * q^72 - 67014176274 * q^73 + 10171443276 * q^74 - 87540153860 * q^75 + 17955918576 * q^76 - 44516617816 * q^77 - 25596104778 * q^78 - 27683357604 * q^79 + 74357773216 * q^80 + 55141240264 * q^81 + 73615849126 * q^82 - 12253964262 * q^83 + 168565479344 * q^84 + 58779027600 * q^85 + 90522557252 * q^86 - 129275944888 * q^87 + 33736356800 * q^88 + 10662817760 * q^89 + 450294422856 * q^90 - 28336741418 * q^91 + 32954076160 * q^92 + 164368292014 * q^93 + 285145948346 * q^94 - 64104297380 * q^95 + 208023008864 * q^96 - 124519454530 * q^97 + 215615498272 * q^98 + 186256571332 * 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 $

$2$	28.6126	0.632256	0.316128	−	0.948717i	$-0.397617\pi$
$2$	28.6126	0.632256	0.316128	+	0.948717i	$0.397617\pi$
$3$	141.482	0.336151	0.168076	−	0.985774i	$-0.446245\pi$
$3$	141.482	0.336151	0.168076	+	0.985774i	$0.446245\pi$
$4$	−1229.32	−0.600252
$5$	−2645.71	−0.378623	−0.189312	−	0.981917i	$-0.560626\pi$
$5$	−2645.71	−0.378623	−0.189312	+	0.981917i	$0.560626\pi$
$6$	4048.18	0.212534
$7$	76456.7	1.71940	0.859699	−	0.510801i	$-0.170651\pi$
$7$	76456.7	1.71940	0.859699	+	0.510801i	$0.170651\pi$
$8$	−93772.7	−1.01177
$9$	−157130.	−0.887002
$10$	−75700.7	−0.239387
$11$	−603251.	−1.12938	−0.564688	−	0.825305i	$-0.691003\pi$
$11$	−603251.	−1.12938	−0.564688	+	0.825305i	$0.691003\pi$
$12$	−173926.	−0.201775
$13$	−941600.	−0.703361	−0.351680	−	0.936120i	$-0.614390\pi$
$13$	−941600.	−0.703361	−0.351680	+	0.936120i	$0.614390\pi$
$14$	2.18763e6	1.08710
$15$	−374320.	−0.127275
$16$	−165445.	−0.0394452
$17$	−5.54744e6	−0.947596	−0.473798	−	0.880633i	$-0.657117\pi$
$17$	−5.54744e6	−0.947596	−0.473798	+	0.880633i	$0.657117\pi$
$18$	−4.49590e6	−0.560813
$19$	−8.15289e6	−0.755382	−0.377691	−	0.925932i	$-0.623282\pi$
$19$	−8.15289e6	−0.755382	−0.377691	+	0.925932i	$0.623282\pi$
$20$	3.25241e6	0.227269
$21$	1.08173e7	0.577977
$22$	−1.72606e7	−0.714055
$23$	6.43634e6	0.208514
$23$	6.43634e6	0.208514
$24$	−1.32672e7	−0.340107
$25$	−4.18283e7	−0.856645
$26$	−2.69417e7	−0.444704
$27$	−4.72942e7	−0.634318
$28$	−9.39895e7	−1.03207
$29$	1.14880e8	1.04005	0.520025	−	0.854151i	$-0.325923\pi$
$29$	1.14880e8	1.04005	0.520025	+	0.854151i	$0.325923\pi$
$30$	−1.07103e7	−0.0804701
$31$	−1.09756e8	−0.688557	−0.344278	−	0.938868i	$-0.611876\pi$
$31$	−1.09756e8	−0.688557	−0.344278	+	0.938868i	$0.611876\pi$
$32$	1.87313e8	0.986830
$33$	−8.53492e7	−0.379641
$34$	−1.58727e8	−0.599124
$35$	−2.02282e8	−0.651004
$36$	1.93162e8	0.532425
$37$	−3.97244e8	−0.941778	−0.470889	−	0.882193i	$-0.656067\pi$
$37$	−3.97244e8	−0.941778	−0.470889	+	0.882193i	$0.656067\pi$
$38$	−2.33276e8	−0.477595
$39$	−1.33220e8	−0.236435
$40$	2.48095e8	0.383079
$41$	7.31306e8	0.985798	0.492899	−	0.870087i	$-0.335937\pi$
$41$	7.31306e8	0.985798	0.492899	+	0.870087i	$0.335937\pi$
$42$	3.09510e8	0.365430
$43$	7.98600e8	0.828424	0.414212	−	0.910180i	$-0.364057\pi$
$43$	7.98600e8	0.828424	0.414212	+	0.910180i	$0.364057\pi$
$44$	7.41587e8	0.677910
$45$	4.15720e8	0.335840
$46$	1.84161e8	0.131835
$47$	−4.30123e8	−0.273561	−0.136781	−	0.990601i	$-0.543676\pi$
$47$	−4.30123e8	−0.273561	−0.136781	+	0.990601i	$0.543676\pi$
$48$	−2.34075e7	−0.0132596
$49$	3.86830e9	1.95633
$50$	−1.19682e9	−0.541619
$51$	−7.84863e8	−0.318536
$52$	1.15752e9	0.422194
$53$	4.54586e9	1.49314	0.746568	−	0.665310i	$-0.231700\pi$
$53$	4.54586e9	1.49314	0.746568	+	0.665310i	$0.231700\pi$
$54$	−1.35321e9	−0.401051
$55$	1.59603e9	0.427608
$56$	−7.16955e9	−1.73963
$57$	−1.15349e9	−0.253922
$58$	3.28701e9	0.657578
$59$	−7.83815e8	−0.142734	−0.0713670	−	0.997450i	$-0.522736\pi$
$59$	−7.83815e8	−0.142734	−0.0713670	+	0.997450i	$0.522736\pi$
$60$	4.60158e8	0.0763968
$61$	−7.23765e9	−1.09719	−0.548597	−	0.836087i	$-0.684838\pi$
$61$	−7.23765e9	−1.09719	−0.548597	+	0.836087i	$0.684838\pi$
$62$	−3.14042e9	−0.435344
$63$	−1.20136e10	−1.52511
$64$	5.69834e9	0.663374
$65$	2.49120e9	0.266309
$66$	−2.44207e9	−0.240030
$67$	−1.06037e10	−0.959501	−0.479751	−	0.877405i	$-0.659273\pi$
$67$	−1.06037e10	−0.959501	−0.479751	+	0.877405i	$0.659273\pi$
$68$	6.81956e9	0.568797
$69$	9.10627e8	0.0700923
$70$	−5.78783e9	−0.411601
$71$	−1.08505e10	−0.713722	−0.356861	−	0.934158i	$-0.616153\pi$
$71$	−1.08505e10	−0.713722	−0.356861	+	0.934158i	$0.616153\pi$
$72$	1.47345e10	0.897442
$73$	3.25510e10	1.83776	0.918879	−	0.394539i	$-0.129096\pi$
$73$	3.25510e10	1.83776	0.918879	+	0.394539i	$0.129096\pi$
$74$	−1.13662e10	−0.595445
$75$	−5.91796e9	−0.287962
$76$	1.00225e10	0.453420
$77$	−4.61226e10	−1.94185
$78$	−3.81176e9	−0.149488
$79$	−5.15171e10	−1.88366	−0.941830	−	0.336091i	$-0.890895\pi$
$79$	−5.15171e10	−1.88366	−0.941830	+	0.336091i	$0.890895\pi$
$80$	4.37720e8	0.0149349
$81$	2.11438e10	0.673776
$82$	2.09246e10	0.623277
$83$	6.27091e10	1.74744	0.873718	−	0.486433i	$-0.161702\pi$
$83$	6.27091e10	1.74744	0.873718	+	0.486433i	$0.161702\pi$
$84$	−1.32978e10	−0.346932
$85$	1.46769e10	0.358782
$86$	2.28501e10	0.523776
$87$	1.62534e10	0.349614
$88$	5.65685e10	1.14267
$89$	−2.11172e10	−0.400858	−0.200429	−	0.979708i	$-0.564234\pi$
$89$	−2.11172e10	−0.400858	−0.200429	+	0.979708i	$0.564234\pi$
$90$	1.18948e10	0.212337
$91$	−7.19916e10	−1.20936
$92$	−7.91230e9	−0.125161
$93$	−1.55285e10	−0.231459
$94$	−1.23070e10	−0.172961
$95$	2.15702e10	0.286005
$96$	2.65014e10	0.331724
$97$	−1.36592e11	−1.61503	−0.807514	−	0.589848i	$-0.799188\pi$
$97$	−1.36592e11	−1.61503	−0.807514	+	0.589848i	$0.799188\pi$
$98$	1.10682e11	1.23690
$99$	9.47888e10	1.00176
$100$	5.14203e10	0.514203
$101$	−6.21955e10	−0.588832	−0.294416	−	0.955677i	$-0.595125\pi$
$101$	−6.21955e10	−0.588832	−0.294416	+	0.955677i	$0.595125\pi$
$102$	−2.24570e10	−0.201396
$103$	6.30585e10	0.535968	0.267984	−	0.963423i	$-0.413643\pi$
$103$	6.30585e10	0.535968	0.267984	+	0.963423i	$0.413643\pi$
$104$	8.82964e10	0.711639
$105$	−2.86193e10	−0.218836
$106$	1.30069e11	0.944044
$107$	−4.40935e10	−0.303923	−0.151962	−	0.988386i	$-0.548559\pi$
$107$	−4.40935e10	−0.303923	−0.151962	+	0.988386i	$0.548559\pi$
$108$	5.81395e10	0.380751
$109$	−1.57897e10	−0.0982941	−0.0491471	−	0.998792i	$-0.515650\pi$
$109$	−1.57897e10	−0.0982941	−0.0491471	+	0.998792i	$0.515650\pi$
$110$	4.56666e10	0.270358
$111$	−5.62030e10	−0.316580
$112$	−1.26494e10	−0.0678220
$113$	−2.66856e11	−1.36253	−0.681263	−	0.732039i	$-0.738569\pi$
$113$	−2.66856e11	−1.36253	−0.681263	+	0.732039i	$0.738569\pi$
$114$	−3.30043e10	−0.160544
$115$	−1.70287e10	−0.0789484
$116$	−1.41224e11	−0.624293
$117$	1.47953e11	0.623883
$118$	−2.24270e10	−0.0902445
$119$	−4.24139e11	−1.62930
$120$	3.51010e10	0.128772
$121$	7.86004e10	0.275489
$122$	−2.07088e11	−0.693707
$123$	1.03467e11	0.331377
$124$	1.34925e11	0.413308
$125$	2.39851e11	0.702968
$126$	−3.43742e11	−0.964260
$127$	−3.75812e11	−1.00937	−0.504685	−	0.863303i	$-0.668391\pi$
$127$	−3.75812e11	−1.00937	−0.504685	+	0.863303i	$0.668391\pi$
$128$	−2.20572e11	−0.567407
$129$	1.12988e11	0.278476
$130$	7.12798e10	0.168375
$131$	−1.55539e11	−0.352246	−0.176123	−	0.984368i	$-0.556356\pi$
$131$	−1.55539e11	−0.352246	−0.176123	+	0.984368i	$0.556356\pi$
$132$	1.04921e11	0.227880
$133$	−6.23343e11	−1.29880
$134$	−3.03400e11	−0.606651
$135$	1.25127e11	0.240167
$136$	5.20198e11	0.958749
$137$	6.46330e11	1.14417	0.572086	−	0.820194i	$-0.306134\pi$
$137$	6.46330e11	1.14417	0.572086	+	0.820194i	$0.306134\pi$
$138$	2.60555e10	0.0443163
$139$	1.91182e11	0.312512	0.156256	−	0.987717i	$-0.450058\pi$
$139$	1.91182e11	0.312512	0.156256	+	0.987717i	$0.450058\pi$
$140$	2.48669e11	0.390766
$141$	−6.08547e10	−0.0919580
$142$	−3.10462e11	−0.451255
$143$	5.68022e11	0.794359
$144$	2.59964e10	0.0349880
$145$	−3.03938e11	−0.393787
$146$	9.31370e11	1.16193
$147$	5.47295e11	0.657622
$148$	4.88339e11	0.565304
$149$	1.30853e11	0.145969	0.0729843	−	0.997333i	$-0.476748\pi$
$149$	1.30853e11	0.145969	0.0729843	+	0.997333i	$0.476748\pi$
$150$	−1.69329e11	−0.182066
$151$	−1.49673e12	−1.55156	−0.775781	−	0.631003i	$-0.782644\pi$
$151$	−1.49673e12	−1.55156	−0.775781	+	0.631003i	$0.782644\pi$
$152$	7.64518e11	0.764272
$153$	8.71668e11	0.840520
$154$	−1.31969e12	−1.22774
$155$	2.90383e11	0.260703
$156$	1.63769e11	0.141921
$157$	2.09173e12	1.75008	0.875038	−	0.484054i	$-0.160836\pi$
$157$	2.09173e12	1.75008	0.875038	+	0.484054i	$0.160836\pi$
$158$	−1.47404e12	−1.19096
$159$	6.43158e11	0.501919
$160$	−4.95575e11	−0.373637
$161$	4.92102e11	0.358519
$162$	6.04980e11	0.425999
$163$	8.32399e11	0.566630	0.283315	−	0.959027i	$-0.408566\pi$
$163$	8.32399e11	0.566630	0.283315	+	0.959027i	$0.408566\pi$
$164$	−8.99006e11	−0.591727
$165$	2.25809e11	0.143741
$166$	1.79427e12	1.10483
$167$	−4.92669e11	−0.293504	−0.146752	−	0.989173i	$-0.546882\pi$
$167$	−4.92669e11	−0.293504	−0.146752	+	0.989173i	$0.546882\pi$
$168$	−1.01436e12	−0.584780
$169$	−9.05549e11	−0.505284
$170$	4.19945e11	0.226842
$171$	1.28106e12	0.670026
$172$	−9.81732e11	−0.497263
$173$	−1.53081e12	−0.751047	−0.375524	−	0.926813i	$-0.622537\pi$
$173$	−1.53081e12	−0.751047	−0.375524	+	0.926813i	$0.622537\pi$
$174$	4.65053e11	0.221046
$175$	−3.19806e12	−1.47291
$176$	9.98050e10	0.0445485
$177$	−1.10896e11	−0.0479802
$178$	−6.04218e11	−0.253445
$179$	−1.49680e12	−0.608797	−0.304398	−	0.952545i	$-0.598455\pi$
$179$	−1.49680e12	−0.608797	−0.304398	+	0.952545i	$0.598455\pi$
$180$	−5.11051e11	−0.201588
$181$	−2.28005e12	−0.872394	−0.436197	−	0.899851i	$-0.643675\pi$
$181$	−2.28005e12	−0.872394	−0.436197	+	0.899851i	$0.643675\pi$
$182$	−2.05987e12	−0.764623
$183$	−1.02400e12	−0.368823
$184$	−6.03553e11	−0.210968
$185$	1.05099e12	0.356579
$186$	−4.44312e11	−0.146341
$187$	3.34650e12	1.07019
$188$	5.28758e11	0.164206
$189$	−3.61596e12	−1.09064
$190$	6.17179e11	0.180828
$191$	4.28631e12	1.22011	0.610057	−	0.792358i	$-0.291147\pi$
$191$	4.28631e12	1.22011	0.610057	+	0.792358i	$0.291147\pi$
$192$	8.06213e11	0.222994
$193$	−8.89609e11	−0.239130	−0.119565	−	0.992826i	$-0.538150\pi$
$193$	−8.89609e11	−0.239130	−0.119565	+	0.992826i	$0.538150\pi$
$194$	−3.90825e12	−1.02111
$195$	3.52460e11	0.0895199
$196$	−4.75536e12	−1.17429
$197$	5.79895e12	1.39247	0.696233	−	0.717815i	$-0.254858\pi$
$197$	5.79895e12	1.39247	0.696233	+	0.717815i	$0.254858\pi$
$198$	2.71216e12	0.633368
$199$	−2.25183e12	−0.511497	−0.255749	−	0.966743i	$-0.582322\pi$
$199$	−2.25183e12	−0.511497	−0.255749	+	0.966743i	$0.582322\pi$
$200$	3.92236e12	0.866727
$201$	−1.50023e12	−0.322537
$202$	−1.77958e12	−0.372293
$203$	8.78333e12	1.78826
$204$	9.64845e11	0.191202
$205$	−1.93482e12	−0.373246
$206$	1.80427e12	0.338869
$207$	−1.01134e12	−0.184953
$208$	1.55783e11	0.0277442
$209$	4.91824e12	0.853110
$210$	−8.18874e11	−0.138360
$211$	8.85142e11	0.145700	0.0728500	−	0.997343i	$-0.476791\pi$
$211$	8.85142e11	0.145700	0.0728500	+	0.997343i	$0.476791\pi$
$212$	−5.58830e12	−0.896258
$213$	−1.53515e12	−0.239918
$214$	−1.26163e12	−0.192157
$215$	−2.11286e12	−0.313660
$216$	4.43490e12	0.641783
$217$	−8.39160e12	−1.18390
$218$	−4.51784e11	−0.0621471
$219$	4.60538e12	0.617764
$220$	−1.96202e12	−0.256672
$221$	5.22347e12	0.666502
$222$	−1.60812e12	−0.200159
$223$	6.00962e11	0.0729744	0.0364872	−	0.999334i	$-0.488383\pi$
$223$	6.00962e11	0.0729744	0.0364872	+	0.999334i	$0.488383\pi$
$224$	1.43213e13	1.69675
$225$	6.57248e12	0.759846
$226$	−7.63544e12	−0.861466
$227$	5.41331e12	0.596102	0.298051	−	0.954550i	$-0.403663\pi$
$227$	5.41331e12	0.596102	0.298051	+	0.954550i	$0.403663\pi$
$228$	1.41800e12	0.152417
$229$	3.47550e11	0.0364689	0.0182344	−	0.999834i	$-0.494195\pi$
$229$	3.47550e11	0.0364689	0.0182344	+	0.999834i	$0.494195\pi$
$230$	−4.87236e11	−0.0499156
$231$	−6.52552e12	−0.652754
$232$	−1.07726e13	−1.05229
$233$	1.80276e13	1.71981	0.859906	−	0.510453i	$-0.170522\pi$
$233$	1.80276e13	1.71981	0.859906	+	0.510453i	$0.170522\pi$
$234$	4.23334e12	0.394454
$235$	1.13798e12	0.103577
$236$	9.63557e11	0.0856764
$237$	−7.28874e12	−0.633194
$238$	−1.21357e13	−1.03013
$239$	−1.45740e13	−1.20890	−0.604452	−	0.796642i	$-0.706608\pi$
$239$	−1.45740e13	−1.20890	−0.604452	+	0.796642i	$0.706608\pi$
$240$	6.19295e10	0.00502037
$241$	−8.66132e12	−0.686263	−0.343131	−	0.939287i	$-0.611488\pi$
$241$	−8.66132e12	−0.686263	−0.343131	+	0.939287i	$0.611488\pi$
$242$	2.24896e12	0.174180
$243$	1.13695e13	0.860808
$244$	8.89736e12	0.658593
$245$	−1.02344e13	−0.740711
$246$	2.96046e12	0.209515
$247$	7.67676e12	0.531306
$248$	1.02921e13	0.696661
$249$	8.87222e12	0.587402
$250$	6.86276e12	0.444456
$251$	7.50674e12	0.475604	0.237802	−	0.971314i	$-0.423573\pi$
$251$	7.50674e12	0.475604	0.237802	+	0.971314i	$0.423573\pi$
$252$	1.47685e13	0.915450
$253$	−3.88273e12	−0.235491
$254$	−1.07530e13	−0.638181
$255$	2.07652e12	0.120605
$256$	−1.79813e13	−1.02212
$257$	−2.60152e13	−1.44742	−0.723710	−	0.690104i	$-0.757565\pi$
$257$	−2.60152e13	−1.44742	−0.723710	+	0.690104i	$0.757565\pi$
$258$	3.23287e12	0.176068
$259$	−3.03720e13	−1.61929
$260$	−3.06247e12	−0.159852
$261$	−1.80510e13	−0.922527
$262$	−4.45037e12	−0.222710
$263$	1.24017e13	0.607748	0.303874	−	0.952712i	$-0.401720\pi$
$263$	1.24017e13	0.607748	0.303874	+	0.952712i	$0.401720\pi$
$264$	8.00343e12	0.384109
$265$	−1.20270e13	−0.565335
$266$	−1.78355e13	−0.821175
$267$	−2.98770e12	−0.134749
$268$	1.30353e13	0.575943
$269$	−3.06825e13	−1.32817	−0.664085	−	0.747657i	$-0.731179\pi$
$269$	−3.06825e13	−1.32817	−0.664085	+	0.747657i	$0.731179\pi$
$270$	3.58020e12	0.151847
$271$	−2.41558e13	−1.00390	−0.501951	−	0.864896i	$-0.667384\pi$
$271$	−2.41558e13	−1.00390	−0.501951	+	0.864896i	$0.667384\pi$
$272$	9.17797e11	0.0373782
$273$	−1.01855e13	−0.406527
$274$	1.84932e13	0.723410
$275$	2.52330e13	0.967474
$276$	−1.11945e12	−0.0420731
$277$	2.63550e13	0.971011	0.485505	−	0.874234i	$-0.338636\pi$
$277$	2.63550e13	0.971011	0.485505	+	0.874234i	$0.338636\pi$
$278$	5.47023e12	0.197587
$279$	1.72460e13	0.610752
$280$	1.89685e13	0.658665
$281$	−2.22980e13	−0.759243	−0.379622	−	0.925142i	$-0.623946\pi$
$281$	−2.22980e13	−0.759243	−0.379622	+	0.925142i	$0.623946\pi$
$282$	−1.74122e12	−0.0581410
$283$	−1.47725e13	−0.483758	−0.241879	−	0.970306i	$-0.577764\pi$
$283$	−1.47725e13	−0.483758	−0.241879	+	0.970306i	$0.577764\pi$
$284$	1.33387e13	0.428413
$285$	3.05179e12	0.0961409
$286$	1.62526e13	0.502238
$287$	5.59132e13	1.69498
$288$	−2.94324e13	−0.875320
$289$	−3.49782e12	−0.102061
$290$	−8.69648e12	−0.248974
$291$	−1.93253e13	−0.542894
$292$	−4.00155e13	−1.10312
$293$	−1.05930e13	−0.286580	−0.143290	−	0.989681i	$-0.545768\pi$
$293$	−1.05930e13	−0.286580	−0.143290	+	0.989681i	$0.545768\pi$
$294$	1.56596e13	0.415785
$295$	2.07375e12	0.0540424
$296$	3.72507e13	0.952862
$297$	2.85303e13	0.716383
$298$	3.74405e12	0.0922896
$299$	−6.06046e12	−0.146661
$300$	7.27505e12	0.172850
$301$	6.10583e13	1.42439
$302$	−4.28253e13	−0.980984
$303$	−8.79955e12	−0.197937
$304$	1.34886e12	0.0297962
$305$	1.91487e13	0.415423
$306$	2.49407e13	0.531424
$307$	8.91638e13	1.86607	0.933035	−	0.359787i	$-0.117151\pi$
$307$	8.91638e13	1.86607	0.933035	+	0.359787i	$0.117151\pi$
$308$	5.66993e13	1.16560
$309$	8.92165e12	0.180166
$310$	8.30863e12	0.164831
$311$	−6.29673e13	−1.22725	−0.613625	−	0.789598i	$-0.710289\pi$
$311$	−6.29673e13	−1.22725	−0.613625	+	0.789598i	$0.710289\pi$
$312$	1.24924e13	0.239218
$313$	−9.95803e13	−1.87361	−0.936805	−	0.349851i	$-0.886232\pi$
$313$	−9.95803e13	−1.87361	−0.936805	+	0.349851i	$0.886232\pi$
$314$	5.98499e13	1.10650
$315$	3.17846e13	0.577442
$316$	6.33308e13	1.13067
$317$	−9.82735e13	−1.72429	−0.862146	−	0.506661i	$-0.830880\pi$
$317$	−9.82735e13	−1.72429	−0.862146	+	0.506661i	$0.830880\pi$
$318$	1.84025e13	0.317341
$319$	−6.93013e13	−1.17461
$320$	−1.50762e13	−0.251169
$321$	−6.23844e12	−0.102164
$322$	1.40803e13	0.226676
$323$	4.52276e13	0.715797
$324$	−2.59924e13	−0.404435
$325$	3.93856e13	0.602530
$326$	2.38171e13	0.358255
$327$	−2.23396e12	−0.0330417
$328$	−6.85765e13	−0.997400
$329$	−3.28858e13	−0.470361
$330$	6.46100e12	0.0908810
$331$	−1.21614e14	−1.68241	−0.841204	−	0.540718i	$-0.818153\pi$
$331$	−1.21614e14	−1.68241	−0.841204	+	0.540718i	$0.818153\pi$
$332$	−7.70894e13	−1.04890
$333$	6.24190e13	0.835359
$334$	−1.40966e13	−0.185570
$335$	2.80543e13	0.363289
$336$	−1.78966e12	−0.0227984
$337$	−1.09684e14	−1.37461	−0.687307	−	0.726367i	$-0.741207\pi$
$337$	−1.09684e14	−1.37461	−0.687307	+	0.726367i	$0.741207\pi$
$338$	−2.59102e13	−0.319469
$339$	−3.77553e13	−0.458015
$340$	−1.80426e13	−0.215360
$341$	6.62106e13	0.777639
$342$	3.66546e13	0.423628
$343$	1.44578e14	1.64431
$344$	−7.48868e13	−0.838174
$345$	−2.40925e12	−0.0265386
$346$	−4.38005e13	−0.474854
$347$	−1.40419e13	−0.149835	−0.0749175	−	0.997190i	$-0.523869\pi$
$347$	−1.40419e13	−0.149835	−0.0749175	+	0.997190i	$0.523869\pi$
$348$	−1.99806e13	−0.209857
$349$	4.33882e13	0.448571	0.224286	−	0.974523i	$-0.427995\pi$
$349$	4.33882e13	0.448571	0.224286	+	0.974523i	$0.427995\pi$
$350$	−9.15049e13	−0.931258
$351$	4.45322e13	0.446154
$352$	−1.12997e14	−1.11450
$353$	−6.45069e13	−0.626390	−0.313195	−	0.949689i	$-0.601399\pi$
$353$	−6.45069e13	−0.626390	−0.313195	+	0.949689i	$0.601399\pi$
$354$	−3.17302e12	−0.0303358
$355$	2.87073e13	0.270231
$356$	2.59597e13	0.240616
$357$	−6.00080e13	−0.547689
$358$	−4.28274e13	−0.384915
$359$	1.57557e14	1.39450	0.697250	−	0.716828i	$-0.254407\pi$
$359$	1.57557e14	1.39450	0.697250	+	0.716828i	$0.254407\pi$
$360$	−3.89832e13	−0.339792
$361$	−5.00207e13	−0.429398
$362$	−6.52384e13	−0.551577
$363$	1.11205e13	0.0926061
$364$	8.85005e13	0.725919
$365$	−8.61204e13	−0.695818
$366$	−2.92993e13	−0.233191
$367$	1.60580e14	1.25901	0.629505	−	0.776996i	$-0.283258\pi$
$367$	1.60580e14	1.25901	0.629505	+	0.776996i	$0.283258\pi$
$368$	−1.06486e12	−0.00822490
$369$	−1.14910e14	−0.874405
$370$	3.00717e13	0.225449
$371$	3.47562e14	2.56729
$372$	1.90895e13	0.138934
$373$	1.38579e14	0.993801	0.496900	−	0.867808i	$-0.334471\pi$
$373$	1.38579e14	0.993801	0.496900	+	0.867808i	$0.334471\pi$
$374$	9.57522e13	0.676636
$375$	3.39346e13	0.236304
$376$	4.03338e13	0.276781
$377$	−1.08171e14	−0.731531
$378$	−1.03462e14	−0.689567
$379$	−3.96029e13	−0.260143	−0.130071	−	0.991505i	$-0.541521\pi$
$379$	−3.96029e13	−0.260143	−0.130071	+	0.991505i	$0.541521\pi$
$380$	−2.65166e13	−0.171675
$381$	−5.31707e13	−0.339301
$382$	1.22643e14	0.771424
$383$	6.34150e13	0.393187	0.196593	−	0.980485i	$-0.437012\pi$
$383$	6.34150e13	0.393187	0.196593	+	0.980485i	$0.437012\pi$
$384$	−3.12069e13	−0.190735
$385$	1.22027e14	0.735228
$386$	−2.54541e13	−0.151191
$387$	−1.25484e14	−0.734814
$388$	1.67915e14	0.969424
$389$	−6.75375e13	−0.384435	−0.192217	−	0.981352i	$-0.561568\pi$
$389$	−6.75375e13	−0.384435	−0.192217	+	0.981352i	$0.561568\pi$
$390$	1.00848e13	0.0565995
$391$	−3.57052e13	−0.197588
$392$	−3.62741e14	−1.97935
$393$	−2.20059e13	−0.118408
$394$	1.65923e14	0.880396
$395$	1.36299e14	0.713197
$396$	−1.16525e14	−0.601308
$397$	1.35168e14	0.687901	0.343950	−	0.938988i	$-0.388235\pi$
$397$	1.35168e14	0.687901	0.343950	+	0.938988i	$0.388235\pi$
$398$	−6.44308e13	−0.323397
$399$	−8.81918e13	−0.436594
$400$	6.92030e12	0.0337905
$401$	2.68773e13	0.129447	0.0647235	−	0.997903i	$-0.479383\pi$
$401$	2.68773e13	0.129447	0.0647235	+	0.997903i	$0.479383\pi$
$402$	−4.29256e13	−0.203926
$403$	1.03346e14	0.484304
$404$	7.64580e13	0.353448
$405$	−5.59403e13	−0.255107
$406$	2.51314e14	1.13064
$407$	2.39638e14	1.06362
$408$	7.35987e13	0.322284
$409$	3.06320e14	1.32342	0.661710	−	0.749760i	$-0.269831\pi$
$409$	3.06320e14	1.32342	0.661710	+	0.749760i	$0.269831\pi$
$410$	−5.53604e13	−0.235987
$411$	9.14441e13	0.384615
$412$	−7.75188e13	−0.321716
$413$	−5.99279e13	−0.245417
$414$	−2.89372e13	−0.116938
$415$	−1.65910e14	−0.661619
$416$	−1.76374e14	−0.694097
$417$	2.70489e13	0.105051
$418$	1.40724e14	0.539384
$419$	−2.32390e14	−0.879104	−0.439552	−	0.898217i	$-0.644863\pi$
$419$	−2.32390e14	−0.879104	−0.439552	+	0.898217i	$0.644863\pi$
$420$	3.51822e13	0.131356
$421$	−2.34718e14	−0.864957	−0.432478	−	0.901644i	$-0.642361\pi$
$421$	−2.34718e14	−0.864957	−0.432478	+	0.901644i	$0.642361\pi$
$422$	2.53263e13	0.0921197
$423$	6.75852e13	0.242650
$424$	−4.26278e14	−1.51071
$425$	2.32040e14	0.811753
$426$	−4.39247e13	−0.151690
$427$	−5.53367e14	−1.88651
$428$	5.42049e13	0.182431
$429$	8.03649e13	0.267024
$430$	−6.04546e13	−0.198314
$431$	−1.34894e14	−0.436886	−0.218443	−	0.975850i	$-0.570098\pi$
$431$	−1.34894e14	−0.436886	−0.218443	+	0.975850i	$0.570098\pi$
$432$	7.82460e12	0.0250208
$433$	2.17558e14	0.686896	0.343448	−	0.939172i	$-0.388405\pi$
$433$	2.17558e14	0.686896	0.343448	+	0.939172i	$0.388405\pi$
$434$	−2.40106e14	−0.748530
$435$	−4.30018e13	−0.132372
$436$	1.94105e13	0.0590012
$437$	−5.24748e13	−0.157508
$438$	1.31772e14	0.390585
$439$	2.19895e14	0.643666	0.321833	−	0.946796i	$-0.395701\pi$
$439$	2.19895e14	0.643666	0.321833	+	0.946796i	$0.395701\pi$
$440$	−1.49664e14	−0.432640
$441$	−6.07825e14	−1.73527
$442$	1.49457e14	0.421400
$443$	4.11878e14	1.14696	0.573480	−	0.819219i	$-0.305593\pi$
$443$	4.11878e14	1.14696	0.573480	+	0.819219i	$0.305593\pi$
$444$	6.90912e13	0.190028
$445$	5.58699e13	0.151774
$446$	1.71951e13	0.0461385
$447$	1.85134e13	0.0490675
$448$	4.35676e14	1.14060
$449$	−7.08008e14	−1.83098	−0.915490	−	0.402341i	$-0.868197\pi$
$449$	−7.08008e14	−1.83098	−0.915490	+	0.402341i	$0.868197\pi$
$450$	1.88056e14	0.480417
$451$	−4.41161e14	−1.11334
$452$	3.28050e14	0.817859
$453$	−2.11760e14	−0.521559
$454$	1.54889e14	0.376889
$455$	1.90469e14	0.457890
$456$	1.08166e14	0.256911
$457$	1.17302e14	0.275275	0.137637	−	0.990483i	$-0.456049\pi$
$457$	1.17302e14	0.275275	0.137637	+	0.990483i	$0.456049\pi$
$458$	9.94433e12	0.0230577
$459$	2.62362e14	0.601077
$460$	2.09336e13	0.0473889
$461$	1.41935e14	0.317493	0.158747	−	0.987319i	$-0.449255\pi$
$461$	1.41935e14	0.317493	0.158747	+	0.987319i	$0.449255\pi$
$462$	−1.86712e14	−0.412707
$463$	−4.90646e14	−1.07170	−0.535850	−	0.844313i	$-0.680009\pi$
$463$	−4.90646e14	−1.07170	−0.535850	+	0.844313i	$0.680009\pi$
$464$	−1.90063e13	−0.0410250
$465$	4.10840e13	0.0876357
$466$	5.15818e14	1.08736
$467$	−6.71957e14	−1.39990	−0.699952	−	0.714190i	$-0.746795\pi$
$467$	−6.71957e14	−1.39990	−0.699952	+	0.714190i	$0.746795\pi$
$468$	−1.81882e14	−0.374487
$469$	−8.10723e14	−1.64976
$470$	3.25607e13	0.0654870
$471$	2.95942e14	0.588290
$472$	7.35005e13	0.144414
$473$	−4.81756e14	−0.935602
$474$	−2.08550e14	−0.400341
$475$	3.41022e14	0.647094
$476$	5.21401e14	0.977988
$477$	−7.14291e14	−1.32441
$478$	−4.17002e14	−0.764337
$479$	4.42888e14	0.802506	0.401253	−	0.915967i	$-0.368575\pi$
$479$	4.42888e14	0.802506	0.401253	+	0.915967i	$0.368575\pi$
$480$	−7.01149e13	−0.125598
$481$	3.74046e14	0.662409
$482$	−2.47823e14	−0.433894
$483$	6.96235e13	0.120517
$484$	−9.66247e13	−0.165363
$485$	3.61382e14	0.611487
$486$	3.25311e14	0.544251
$487$	1.02696e15	1.69881	0.849404	−	0.527743i	$-0.176962\pi$
$487$	1.02696e15	1.69881	0.849404	+	0.527743i	$0.176962\pi$
$488$	6.78694e14	1.11011
$489$	1.17769e14	0.190473
$490$	−2.92833e14	−0.468319
$491$	9.31399e14	1.47295	0.736474	−	0.676465i	$-0.236489\pi$
$491$	9.31399e14	1.47295	0.736474	+	0.676465i	$0.236489\pi$
$492$	−1.27193e14	−0.198910
$493$	−6.37288e14	−0.985548
$494$	2.19652e14	0.335921
$495$	−2.50783e14	−0.379289
$496$	1.81586e13	0.0271603
$497$	−8.29593e14	−1.22717
$498$	2.53858e14	0.371389
$499$	4.36788e14	0.632001	0.316000	−	0.948759i	$-0.397660\pi$
$499$	4.36788e14	0.632001	0.316000	+	0.948759i	$0.397660\pi$
$500$	−2.94852e14	−0.421958
$501$	−6.97038e13	−0.0986618
$502$	2.14788e14	0.300704
$503$	2.38355e14	0.330066	0.165033	−	0.986288i	$-0.447227\pi$
$503$	2.38355e14	0.330066	0.165033	+	0.986288i	$0.447227\pi$
$504$	1.12655e15	1.54306
$505$	1.64551e14	0.222946
$506$	−1.11095e14	−0.148891
$507$	−1.28119e14	−0.169852
$508$	4.61992e14	0.605877
$509$	−1.39566e15	−1.81064	−0.905321	−	0.424728i	$-0.860370\pi$
$509$	−1.39566e15	−1.81064	−0.905321	+	0.424728i	$0.860370\pi$
$510$	5.94147e13	0.0762532
$511$	2.48874e15	3.15984
$512$	−6.27632e13	−0.0788352
$513$	3.85584e14	0.479152
$514$	−7.44363e14	−0.915140
$515$	−1.66834e14	−0.202930
$516$	−1.38897e14	−0.167156
$517$	2.59472e14	0.308954
$518$	−8.69023e14	−1.02381
$519$	−2.16582e14	−0.252465
$520$	−2.33607e14	−0.269443
$521$	1.74707e15	1.99390	0.996950	−	0.0780489i	$-0.0248690\pi$
$521$	1.74707e15	1.99390	0.996950	+	0.0780489i	$0.0248690\pi$
$522$	−5.16488e14	−0.583274
$523$	−8.11334e14	−0.906651	−0.453326	−	0.891345i	$-0.649762\pi$
$523$	−8.11334e14	−0.906651	−0.453326	+	0.891345i	$0.649762\pi$
$524$	1.91206e14	0.211436
$525$	−4.52468e14	−0.495121
$526$	3.54845e14	0.384252
$527$	6.08866e14	0.652474
$528$	1.41206e13	0.0149750
$529$	4.14265e13	0.0434783
$530$	−3.44125e14	−0.357437
$531$	1.23161e14	0.126605
$532$	7.66285e14	0.779609
$533$	−6.88598e14	−0.693372
$534$	−8.54860e13	−0.0851958
$535$	1.16659e14	0.115072
$536$	9.94336e14	0.970794
$537$	−2.11770e14	−0.204648
$538$	−8.77908e14	−0.839744
$539$	−2.33356e15	−2.20943
$540$	−1.53820e14	−0.144161
$541$	−3.04767e14	−0.282737	−0.141369	−	0.989957i	$-0.545150\pi$
$541$	−3.04767e14	−0.282737	−0.141369	+	0.989957i	$0.545150\pi$
$542$	−6.91163e14	−0.634723
$543$	−3.22587e14	−0.293256
$544$	−1.03911e15	−0.935116
$545$	4.17749e13	0.0372164
$546$	−2.91435e14	−0.257029
$547$	−5.77587e14	−0.504298	−0.252149	−	0.967688i	$-0.581137\pi$
$547$	−5.77587e14	−0.504298	−0.252149	+	0.967688i	$0.581137\pi$
$548$	−7.94544e14	−0.686792
$549$	1.13725e15	0.973214
$550$	7.21983e14	0.611691
$551$	−9.36601e14	−0.785635
$552$	−8.53920e13	−0.0709173
$553$	−3.93883e15	−3.23876
$554$	7.54086e14	0.613928
$555$	1.48697e14	0.119864
$556$	−2.35024e14	−0.187586
$557$	−5.76692e14	−0.455765	−0.227882	−	0.973689i	$-0.573180\pi$
$557$	−5.76692e14	−0.455765	−0.227882	+	0.973689i	$0.573180\pi$
$558$	4.93453e14	0.386151
$559$	−7.51962e14	−0.582681
$560$	3.34666e13	0.0256790
$561$	4.73470e14	0.359746
$562$	−6.38005e14	−0.480036
$563$	2.00152e15	1.49129	0.745647	−	0.666341i	$-0.232141\pi$
$563$	2.00152e15	1.49129	0.745647	+	0.666341i	$0.232141\pi$
$564$	7.48097e13	0.0551980
$565$	7.06022e14	0.515884
$566$	−4.22680e14	−0.305859
$567$	1.61659e15	1.15849
$568$	1.01748e15	0.722122
$569$	9.99467e14	0.702508	0.351254	−	0.936280i	$-0.385755\pi$
$569$	9.99467e14	0.702508	0.351254	+	0.936280i	$0.385755\pi$
$570$	8.73198e13	0.0607857
$571$	−1.09316e14	−0.0753680	−0.0376840	−	0.999290i	$-0.511998\pi$
$571$	−1.09316e14	−0.0753680	−0.0376840	+	0.999290i	$0.511998\pi$
$572$	−6.98278e14	−0.476815
$573$	6.06436e14	0.410142
$574$	1.59983e15	1.07166
$575$	−2.69222e14	−0.178623
$576$	−8.95380e14	−0.588415
$577$	−2.58096e14	−0.168002	−0.0840009	−	0.996466i	$-0.526770\pi$
$577$	−2.58096e14	−0.168002	−0.0840009	+	0.996466i	$0.526770\pi$
$578$	−1.00082e14	−0.0645286
$579$	−1.25864e14	−0.0803838
$580$	3.73636e14	0.236372
$581$	4.79453e15	3.00454
$582$	−5.52948e14	−0.343248
$583$	−2.74230e15	−1.68631
$584$	−3.05239e15	−1.85939
$585$	−3.91442e14	−0.236216
$586$	−3.03092e14	−0.181192
$587$	−8.11186e14	−0.480409	−0.240204	−	0.970722i	$-0.577214\pi$
$587$	−8.11186e14	−0.480409	−0.240204	+	0.970722i	$0.577214\pi$
$588$	−6.72799e14	−0.394739
$589$	8.94830e14	0.520123
$590$	5.93354e13	0.0341686
$591$	8.20447e14	0.468079
$592$	6.57222e13	0.0371486
$593$	4.86245e14	0.272304	0.136152	−	0.990688i	$-0.456526\pi$
$593$	4.86245e14	0.272304	0.136152	+	0.990688i	$0.456526\pi$
$594$	8.16326e14	0.452938
$595$	1.12215e15	0.616889
$596$	−1.60860e14	−0.0876180
$597$	−3.18593e14	−0.171940
$598$	−1.73406e14	−0.0927272
$599$	7.21096e14	0.382072	0.191036	−	0.981583i	$-0.438815\pi$
$599$	7.21096e14	0.382072	0.191036	+	0.981583i	$0.438815\pi$
$600$	5.54943e14	0.291351
$601$	4.88518e14	0.254139	0.127069	−	0.991894i	$-0.459443\pi$
$601$	4.88518e14	0.254139	0.127069	+	0.991894i	$0.459443\pi$
$602$	1.74704e15	0.900579
$603$	1.66616e15	0.851080
$604$	1.83995e15	0.931328
$605$	−2.07954e14	−0.104307
$606$	−2.51779e14	−0.125147
$607$	−2.24836e15	−1.10746	−0.553729	−	0.832697i	$-0.686796\pi$
$607$	−2.24836e15	−1.10746	−0.553729	+	0.832697i	$0.686796\pi$
$608$	−1.52714e15	−0.745433
$609$	1.24268e15	0.601126
$610$	5.47895e14	0.262654
$611$	4.05004e14	0.192412
$612$	−1.07156e15	−0.504524
$613$	1.41355e15	0.659595	0.329797	−	0.944052i	$-0.393020\pi$
$613$	1.41355e15	0.659595	0.329797	+	0.944052i	$0.393020\pi$
$614$	2.55121e15	1.17983
$615$	−2.73743e14	−0.125467
$616$	4.32504e15	1.96470
$617$	1.04942e15	0.472476	0.236238	−	0.971695i	$-0.424085\pi$
$617$	1.04942e15	0.472476	0.236238	+	0.971695i	$0.424085\pi$
$618$	2.55272e14	0.113911
$619$	−2.76905e15	−1.22471	−0.612354	−	0.790583i	$-0.709777\pi$
$619$	−2.76905e15	−1.22471	−0.612354	+	0.790583i	$0.709777\pi$
$620$	−3.56973e14	−0.156488
$621$	−3.04402e14	−0.132264
$622$	−1.80166e15	−0.775936
$623$	−1.61455e15	−0.689235
$624$	2.20405e13	0.00932625
$625$	1.40782e15	0.590485
$626$	−2.84925e15	−1.18460
$627$	6.95842e14	0.286774
$628$	−2.57139e15	−1.05049
$629$	2.20369e15	0.892425
$630$	9.09441e14	0.365091
$631$	−3.93884e15	−1.56750	−0.783749	−	0.621078i	$-0.786695\pi$
$631$	−3.93884e15	−1.56750	−0.783749	+	0.621078i	$0.786695\pi$
$632$	4.83090e15	1.90583
$633$	1.25232e14	0.0489772
$634$	−2.81187e15	−1.09019
$635$	9.94290e14	0.382171
$636$	−7.90645e14	−0.301278
$637$	−3.64239e15	−1.37600
$638$	−1.98290e15	−0.742653
$639$	1.70494e15	0.633073
$640$	5.83568e14	0.214833
$641$	3.32893e15	1.21502	0.607512	−	0.794310i	$-0.292168\pi$
$641$	3.32893e15	1.21502	0.607512	+	0.794310i	$0.292168\pi$
$642$	−1.78498e14	−0.0645939
$643$	2.65204e15	0.951522	0.475761	−	0.879575i	$-0.342173\pi$
$643$	2.65204e15	0.951522	0.475761	+	0.879575i	$0.342173\pi$
$644$	−6.04948e14	−0.215202
$645$	−2.98932e14	−0.105437
$646$	1.29408e15	0.452567
$647$	−2.30227e15	−0.798330	−0.399165	−	0.916879i	$-0.630700\pi$
$647$	−2.30227e15	−0.798330	−0.399165	+	0.916879i	$0.630700\pi$
$648$	−1.98271e15	−0.681706
$649$	4.72837e14	0.161200
$650$	1.12693e15	0.380953
$651$	−1.18726e15	−0.397970
$652$	−1.02328e15	−0.340121
$653$	−2.41936e15	−0.797402	−0.398701	−	0.917081i	$-0.630539\pi$
$653$	−2.41936e15	−0.797402	−0.398701	+	0.917081i	$0.630539\pi$
$654$	−6.39194e13	−0.0208908
$655$	4.11510e14	0.133368
$656$	−1.20991e14	−0.0388850
$657$	−5.11473e15	−1.63010
$658$	−9.40950e14	−0.297389
$659$	4.51424e15	1.41486	0.707432	−	0.706781i	$-0.249853\pi$
$659$	4.51424e15	1.41486	0.707432	+	0.706781i	$0.249853\pi$
$660$	−2.77591e14	−0.0862807
$661$	2.28878e15	0.705499	0.352750	−	0.935718i	$-0.385247\pi$
$661$	2.28878e15	0.705499	0.352750	+	0.935718i	$0.385247\pi$
$662$	−3.47971e15	−1.06371
$663$	7.39027e14	0.224045
$664$	−5.88040e15	−1.76800
$665$	1.64918e15	0.491756
$666$	1.78597e15	0.528161
$667$	7.39405e14	0.216866
$668$	6.05646e14	0.176177
$669$	8.50253e13	0.0245304
$670$	8.02707e14	0.229692
$671$	4.36612e15	1.23914
$672$	2.02621e15	0.570365
$673$	1.42254e14	0.0397174	0.0198587	−	0.999803i	$-0.493678\pi$
$673$	1.42254e14	0.0397174	0.0198587	+	0.999803i	$0.493678\pi$
$674$	−3.13836e15	−0.869108
$675$	1.97824e15	0.543385
$676$	1.11321e15	0.303298
$677$	−2.33537e15	−0.631129	−0.315564	−	0.948904i	$-0.602194\pi$
$677$	−2.33537e15	−0.631129	−0.315564	+	0.948904i	$0.602194\pi$
$678$	−1.08028e15	−0.289583
$679$	−1.04434e16	−2.77688
$680$	−1.37629e15	−0.363004
$681$	7.65886e14	0.200380
$682$	1.89446e15	0.491667
$683$	1.58630e15	0.408386	0.204193	−	0.978931i	$-0.434543\pi$
$683$	1.58630e15	0.408386	0.204193	+	0.978931i	$0.434543\pi$
$684$	−1.57483e15	−0.402184
$685$	−1.71000e15	−0.433210
$686$	4.13675e15	1.03962
$687$	4.91721e13	0.0122591
$688$	−1.32125e14	−0.0326774
$689$	−4.28039e15	−1.05021
$690$	−6.89351e13	−0.0167792
$691$	−6.98178e15	−1.68592	−0.842960	−	0.537976i	$-0.819189\pi$
$691$	−6.98178e15	−1.68592	−0.842960	+	0.537976i	$0.819189\pi$
$692$	1.88185e15	0.450818
$693$	7.24724e15	1.72242
$694$	−4.01775e14	−0.0947341
$695$	−5.05813e14	−0.118324
$696$	−1.52413e15	−0.353729
$697$	−4.05687e15	−0.934139
$698$	1.24145e15	0.283612
$699$	2.55058e15	0.578116
$700$	3.93142e15	0.884119
$701$	5.56706e15	1.24216	0.621079	−	0.783748i	$-0.286694\pi$
$701$	5.56706e15	1.24216	0.621079	+	0.783748i	$0.286694\pi$
$702$	1.27418e15	0.282084
$703$	3.23869e15	0.711402
$704$	−3.43753e15	−0.749199
$705$	1.61004e14	0.0348174
$706$	−1.84571e15	−0.396039
$707$	−4.75527e15	−1.01244
$708$	1.36326e14	0.0288002
$709$	−1.49580e15	−0.313559	−0.156780	−	0.987634i	$-0.550111\pi$
$709$	−1.49580e15	−0.313559	−0.156780	+	0.987634i	$0.550111\pi$
$710$	8.21391e14	0.170855
$711$	8.09487e15	1.67081
$712$	1.98021e15	0.405576
$713$	−7.06429e14	−0.143574
$714$	−1.71699e15	−0.346280
$715$	−1.50282e15	−0.300762
$716$	1.84004e15	0.365431
$717$	−2.06197e15	−0.406374
$718$	4.50813e15	0.881681
$719$	−1.35574e15	−0.263128	−0.131564	−	0.991308i	$-0.542000\pi$
$719$	−1.35574e15	−0.263128	−0.131564	+	0.991308i	$0.542000\pi$
$720$	−6.87789e13	−0.0132473
$721$	4.82124e15	0.921542
$722$	−1.43123e15	−0.271490
$723$	−1.22542e15	−0.230688
$724$	2.80291e15	0.523657
$725$	−4.80523e15	−0.890954
$726$	3.18188e14	0.0585508
$727$	−8.72778e15	−1.59391	−0.796956	−	0.604037i	$-0.793558\pi$
$727$	−8.72778e15	−1.59391	−0.796956	+	0.604037i	$0.793558\pi$
$728$	6.75085e15	1.22359
$729$	−2.13698e15	−0.384414
$730$	−2.46413e15	−0.439935
$731$	−4.43018e15	−0.785012
$732$	1.25882e15	0.221387
$733$	2.05348e14	0.0358442	0.0179221	−	0.999839i	$-0.494295\pi$
$733$	2.05348e14	0.0358442	0.0179221	+	0.999839i	$0.494295\pi$
$734$	4.59463e15	0.796017
$735$	−1.44798e15	−0.248991
$736$	1.20561e15	0.205768
$737$	6.39669e15	1.08364
$738$	−3.28788e15	−0.552848
$739$	8.69240e15	1.45076	0.725379	−	0.688350i	$-0.241664\pi$
$739$	8.69240e15	1.45076	0.725379	+	0.688350i	$0.241664\pi$
$740$	−1.29200e15	−0.214037
$741$	1.08612e15	0.178599
$742$	9.94466e15	1.62319
$743$	−9.70377e14	−0.157218	−0.0786090	−	0.996906i	$-0.525048\pi$
$743$	−9.70377e14	−0.157218	−0.0786090	+	0.996906i	$0.525048\pi$
$744$	1.45615e15	0.234183
$745$	−3.46199e14	−0.0552671
$746$	3.96512e15	0.628337
$747$	−9.85347e15	−1.54998
$748$	−4.11391e15	−0.642385
$749$	−3.37125e15	−0.522565
$750$	9.70958e14	0.149404
$751$	−4.00402e14	−0.0611612	−0.0305806	−	0.999532i	$-0.509736\pi$
$751$	−4.00402e14	−0.0611612	−0.0305806	+	0.999532i	$0.509736\pi$
$752$	7.11619e13	0.0107907
$753$	1.06207e15	0.159875
$754$	−3.09505e15	−0.462515
$755$	3.95990e15	0.587457
$756$	4.44515e15	0.654662
$757$	1.11279e16	1.62699	0.813497	−	0.581570i	$-0.197561\pi$
$757$	1.11279e16	1.62699	0.813497	+	0.581570i	$0.197561\pi$
$758$	−1.13314e15	−0.164477
$759$	−5.49337e14	−0.0791606
$760$	−2.02269e15	−0.289371
$761$	6.97593e15	0.990802	0.495401	−	0.868665i	$-0.335021\pi$
$761$	6.97593e15	0.990802	0.495401	+	0.868665i	$0.335021\pi$
$762$	−1.52135e15	−0.214525
$763$	−1.20723e15	−0.169007
$764$	−5.26923e15	−0.732376
$765$	−2.30618e15	−0.318240
$766$	1.81447e15	0.248595
$767$	7.38041e14	0.100394
$768$	−2.54404e15	−0.343587
$769$	−5.49087e14	−0.0736285	−0.0368143	−	0.999322i	$-0.511721\pi$
$769$	−5.49087e14	−0.0736285	−0.0368143	+	0.999322i	$0.511721\pi$
$770$	3.49151e15	0.464852
$771$	−3.68068e15	−0.486552
$772$	1.09361e15	0.143538
$773$	6.26563e15	0.816540	0.408270	−	0.912861i	$-0.366132\pi$
$773$	6.26563e15	0.816540	0.408270	+	0.912861i	$0.366132\pi$
$774$	−3.59042e15	−0.464591
$775$	4.59092e15	0.589848
$776$	1.28086e16	1.63404
$777$	−4.29709e15	−0.544326
$778$	−1.93243e15	−0.243061
$779$	−5.96225e15	−0.744654
$780$	−4.33285e14	−0.0537345
$781$	6.54558e15	0.806060
$782$	−1.02162e15	−0.124926
$783$	−5.43314e15	−0.659723
$784$	−6.39992e14	−0.0771678
$785$	−5.53410e15	−0.662619
$786$	−6.29647e14	−0.0748641
$787$	−2.55700e14	−0.0301904	−0.0150952	−	0.999886i	$-0.504805\pi$
$787$	−2.55700e14	−0.0301904	−0.0150952	+	0.999886i	$0.504805\pi$
$788$	−7.12874e15	−0.835831
$789$	1.75461e15	0.204295
$790$	3.89988e15	0.450923
$791$	−2.04029e16	−2.34272
$792$	−8.88860e15	−1.01355
$793$	6.81497e15	0.771723
$794$	3.86751e15	0.434929
$795$	−1.70161e15	−0.190038
$796$	2.76821e15	0.307027
$797$	−1.18231e16	−1.30230	−0.651149	−	0.758950i	$-0.725713\pi$
$797$	−1.18231e16	−1.30230	−0.651149	+	0.758950i	$0.725713\pi$
$798$	−2.52340e15	−0.276039
$799$	2.38608e15	0.259226
$800$	−7.83498e15	−0.845362
$801$	3.31814e15	0.355562
$802$	7.69031e14	0.0818436
$803$	−1.96364e16	−2.07552
$804$	1.84426e15	0.193604
$805$	−1.30196e15	−0.135744
$806$	2.95702e15	0.306204
$807$	−4.34103e15	−0.446466
$808$	5.83224e15	0.595763
$809$	1.78304e16	1.80903	0.904514	−	0.426444i	$-0.140234\pi$
$809$	1.78304e16	1.80903	0.904514	+	0.426444i	$0.140234\pi$
$810$	−1.60060e15	−0.161293
$811$	−3.99987e15	−0.400342	−0.200171	−	0.979761i	$-0.564150\pi$
$811$	−3.99987e15	−0.400342	−0.200171	+	0.979761i	$0.564150\pi$
$812$	−1.07975e16	−1.07341
$813$	−3.41762e15	−0.337463
$814$	6.85668e15	0.672481
$815$	−2.20228e15	−0.214539
$816$	1.29852e14	0.0125647
$817$	−6.51089e15	−0.625776
$818$	8.76463e15	0.836740
$819$	1.13120e16	1.07270
$820$	2.37851e15	0.224042
$821$	−4.81174e15	−0.450210	−0.225105	−	0.974335i	$-0.572272\pi$
$821$	−4.81174e15	−0.450210	−0.225105	+	0.974335i	$0.572272\pi$
$822$	2.61646e15	0.243175
$823$	9.82975e15	0.907493	0.453747	−	0.891131i	$-0.350087\pi$
$823$	9.82975e15	0.907493	0.453747	+	0.891131i	$0.350087\pi$
$824$	−5.91317e15	−0.542276
$825$	3.57002e15	0.325217
$826$	−1.71470e15	−0.155166
$827$	−2.21484e15	−0.199096	−0.0995481	−	0.995033i	$-0.531740\pi$
$827$	−2.21484e15	−0.199096	−0.0995481	+	0.995033i	$0.531740\pi$
$828$	1.24326e15	0.111018
$829$	−4.47815e15	−0.397236	−0.198618	−	0.980077i	$-0.563645\pi$
$829$	−4.47815e15	−0.397236	−0.198618	+	0.980077i	$0.563645\pi$
$830$	−4.74713e15	−0.418313
$831$	3.72876e15	0.326406
$832$	−5.36556e15	−0.466592
$833$	−2.14592e16	−1.85381
$834$	7.73940e14	0.0664192
$835$	1.30346e15	0.111127
$836$	−6.04607e15	−0.512081
$837$	5.19083e15	0.436764
$838$	−6.64929e15	−0.555819
$839$	1.73763e16	1.44300	0.721502	−	0.692413i	$-0.243452\pi$
$839$	1.73763e16	1.44300	0.721502	+	0.692413i	$0.243452\pi$
$840$	2.68371e15	0.221411
$841$	9.96847e14	0.0817053
$842$	−6.71590e15	−0.546874
$843$	−3.15477e15	−0.255220
$844$	−1.08812e15	−0.0874567
$845$	2.39582e15	0.191312
$846$	1.93379e15	0.153417
$847$	6.00952e15	0.473676
$848$	−7.52092e14	−0.0588970
$849$	−2.09004e15	−0.162616
$850$	6.63928e15	0.513236
$851$	−2.55680e15	−0.196374
$852$	1.88719e15	0.144011
$853$	2.53524e16	1.92220	0.961102	−	0.276192i	$-0.0890727\pi$
$853$	2.53524e16	1.92220	0.961102	+	0.276192i	$0.0890727\pi$
$854$	−1.58333e16	−1.19276
$855$	−3.38932e15	−0.253687
$856$	4.13477e15	0.307500
$857$	−3.01213e15	−0.222576	−0.111288	−	0.993788i	$-0.535498\pi$
$857$	−3.01213e15	−0.222576	−0.111288	+	0.993788i	$0.535498\pi$
$858$	2.29945e15	0.168828
$859$	−1.98534e16	−1.44835	−0.724175	−	0.689616i	$-0.757779\pi$
$859$	−1.98534e16	−1.44835	−0.724175	+	0.689616i	$0.757779\pi$
$860$	2.59738e15	0.188275
$861$	7.91072e15	0.569769
$862$	−3.85968e15	−0.276224
$863$	1.40444e16	0.998720	0.499360	−	0.866395i	$-0.333569\pi$
$863$	1.40444e16	0.998720	0.499360	+	0.866395i	$0.333569\pi$
$864$	−8.85880e15	−0.625964
$865$	4.05007e15	0.284364
$866$	6.22490e15	0.434294
$867$	−4.94879e14	−0.0343079
$868$	1.03159e16	0.710640
$869$	3.10777e16	2.12736
$870$	−1.23040e15	−0.0836930
$871$	9.98444e15	0.674876
$872$	1.48064e15	0.0994510
$873$	2.14627e16	1.43253
$874$	−1.50144e15	−0.0995854
$875$	1.83382e16	1.20868
$876$	−5.66147e15	−0.370814
$877$	2.32617e15	0.151407	0.0757033	−	0.997130i	$-0.475880\pi$
$877$	2.32617e15	0.151407	0.0757033	+	0.997130i	$0.475880\pi$
$878$	6.29178e15	0.406962
$879$	−1.49871e15	−0.0963340
$880$	−2.64055e14	−0.0168671
$881$	5.60487e15	0.355793	0.177897	−	0.984049i	$-0.443071\pi$
$881$	5.60487e15	0.355793	0.177897	+	0.984049i	$0.443071\pi$
$882$	−1.73915e16	−1.09713
$883$	1.92902e16	1.20935	0.604676	−	0.796472i	$-0.293303\pi$
$883$	1.92902e16	1.20935	0.604676	+	0.796472i	$0.293303\pi$
$884$	−6.42130e15	−0.400069
$885$	2.93398e14	0.0181664
$886$	1.17849e16	0.725173
$887$	1.62581e16	0.994239	0.497120	−	0.867682i	$-0.334391\pi$
$887$	1.62581e16	0.994239	0.497120	+	0.867682i	$0.334391\pi$
$888$	5.27030e15	0.320305
$889$	−2.87334e16	−1.73551
$890$	1.59859e15	0.0959601
$891$	−1.27550e16	−0.760946
$892$	−7.38772e14	−0.0438030
$893$	3.50675e15	0.206643
$894$	5.29716e14	0.0310232
$895$	3.96010e15	0.230504
$896$	−1.68642e16	−0.975599
$897$	−8.57447e14	−0.0493002
$898$	−2.02580e16	−1.15765
$899$	−1.26088e16	−0.716134
$900$	−8.07966e15	−0.456099
$901$	−2.52179e16	−1.41489
$902$	−1.26228e16	−0.703914
$903$	8.63865e15	0.478810
$904$	2.50238e16	1.37856
$905$	6.03236e15	0.330309
$906$	−6.05901e15	−0.329759
$907$	2.90821e16	1.57321	0.786604	−	0.617458i	$-0.211837\pi$
$907$	2.90821e16	1.57321	0.786604	+	0.617458i	$0.211837\pi$
$908$	−6.65467e15	−0.357811
$909$	9.77277e15	0.522296
$910$	5.44982e15	0.289504
$911$	1.28854e15	0.0680373	0.0340186	−	0.999421i	$-0.489169\pi$
$911$	1.28854e15	0.0680373	0.0340186	+	0.999421i	$0.489169\pi$
$912$	1.90839e14	0.0100160
$913$	−3.78294e16	−1.97351
$914$	3.35632e15	0.174044
$915$	2.70920e15	0.139645
$916$	−4.27249e14	−0.0218905
$917$	−1.18920e16	−0.605651
$918$	7.50686e15	0.380035
$919$	−1.41190e16	−0.710508	−0.355254	−	0.934770i	$-0.615606\pi$
$919$	−1.41190e16	−0.710508	−0.355254	+	0.934770i	$0.615606\pi$
$920$	1.59683e15	0.0798775
$921$	1.26151e16	0.627281
$922$	4.06114e15	0.200737
$923$	1.02168e16	0.502004
$924$	8.02193e15	0.391817
$925$	1.66161e16	0.806769
$926$	−1.40387e16	−0.677589
$927$	−9.90837e15	−0.475405
$928$	2.15184e16	1.02635
$929$	−1.04551e16	−0.495726	−0.247863	−	0.968795i	$-0.579728\pi$
$929$	−1.04551e16	−0.495726	−0.247863	+	0.968795i	$0.579728\pi$
$930$	1.17552e15	0.0554082
$931$	−3.15378e16	−1.47777
$932$	−2.21616e16	−1.03232
$933$	−8.90875e15	−0.412541
$934$	−1.92265e16	−0.885098
$935$	−8.85386e15	−0.405200
$936$	−1.38740e16	−0.631225
$937$	1.43609e16	0.649552	0.324776	−	0.945791i	$-0.394711\pi$
$937$	1.43609e16	0.649552	0.324776	+	0.945791i	$0.394711\pi$
$938$	−2.31969e16	−1.04307
$939$	−1.40888e16	−0.629816
$940$	−1.39894e15	−0.0621721
$941$	−1.45554e16	−0.643105	−0.321552	−	0.946892i	$-0.604205\pi$
$941$	−1.45554e16	−0.643105	−0.321552	+	0.946892i	$0.604205\pi$
$942$	8.46768e15	0.371950
$943$	4.70694e15	0.205553
$944$	1.29678e14	0.00563018
$945$	9.56677e15	0.412943
$946$	−1.37843e16	−0.591540
$947$	−2.59516e16	−1.10723	−0.553616	−	0.832772i	$-0.686753\pi$
$947$	−2.59516e16	−1.10723	−0.553616	+	0.832772i	$0.686753\pi$
$948$	8.96017e15	0.380076
$949$	−3.06500e16	−1.29261
$950$	9.75754e15	0.409129
$951$	−1.39039e16	−0.579622
$952$	3.97726e16	1.64847
$953$	−5.79427e15	−0.238774	−0.119387	−	0.992848i	$-0.538093\pi$
$953$	−5.79427e15	−0.238774	−0.119387	+	0.992848i	$0.538093\pi$
$954$	−2.04378e16	−0.837369
$955$	−1.13403e16	−0.461963
$956$	1.79161e16	0.725647
$957$	−9.80490e15	−0.394846
$958$	1.26722e16	0.507390
$959$	4.94163e16	1.96729
$960$	−2.13301e15	−0.0844307
$961$	−1.33621e16	−0.525890
$962$	1.07024e16	0.418812
$963$	6.92841e15	0.269581
$964$	1.06475e16	0.411931
$965$	2.35365e15	0.0905401
$966$	1.99211e15	0.0761974
$967$	3.66543e16	1.39405	0.697026	−	0.717046i	$-0.254506\pi$
$967$	3.66543e16	1.39405	0.697026	+	0.717046i	$0.254506\pi$
$968$	−7.37057e15	−0.278732
$969$	6.39890e15	0.240616
$970$	1.03401e16	0.386616
$971$	2.83402e16	1.05365	0.526826	−	0.849973i	$-0.323382\pi$
$971$	2.83402e16	1.05365	0.526826	+	0.849973i	$0.323382\pi$
$972$	−1.39767e16	−0.516702
$973$	1.46172e16	0.537332
$974$	2.93841e16	1.07408
$975$	5.57235e15	0.202541
$976$	1.19743e15	0.0432790
$977$	−1.77407e16	−0.637603	−0.318802	−	0.947821i	$-0.603280\pi$
$977$	−1.77407e16	−0.637603	−0.318802	+	0.947821i	$0.603280\pi$
$978$	3.36970e15	0.120428
$979$	1.27390e16	0.452720
$980$	1.25813e16	0.444613
$981$	2.48103e15	0.0871871
$982$	2.66498e16	0.931281
$983$	−4.29162e16	−1.49134	−0.745670	−	0.666316i	$-0.767870\pi$
$983$	−4.29162e16	−1.49134	−0.745670	+	0.666316i	$0.767870\pi$
$984$	−9.70235e15	−0.335277
$985$	−1.53423e16	−0.527220
$986$	−1.82345e16	−0.623119
$987$	−4.65275e15	−0.158112
$988$	−9.43717e15	−0.318918
$989$	5.14006e15	0.172738
$990$	−7.17558e15	−0.239808
$991$	−3.95958e16	−1.31596	−0.657982	−	0.753034i	$-0.728590\pi$
$991$	−3.95958e16	−1.31596	−0.657982	+	0.753034i	$0.728590\pi$
$992$	−2.05587e16	−0.679488
$993$	−1.72063e16	−0.565543
$994$	−2.37369e16	−0.775886
$995$	5.95768e15	0.193665
$996$	−1.09068e16	−0.352590
$997$	2.11136e16	0.678796	0.339398	−	0.940643i	$-0.389777\pi$
$997$	2.11136e16	0.678796	0.339398	+	0.940643i	$0.389777\pi$
$998$	1.24977e16	0.399586
$999$	1.87873e16	0.597386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.12.a.a.1.6	✓	8
3.2	odd	2		207.12.a.a.1.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.12.a.a.1.6	✓	8	1.1	even	1	trivial
207.12.a.a.1.3		8	3.2	odd	2

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:	\( N \)	\(=\)	\( 23 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	23.a (trivial)

\(f(q)\)	\(=\)	\(q+28.6126 q^{2} +141.482 q^{3} -1229.32 q^{4} -2645.71 q^{5} +4048.18 q^{6} +76456.7 q^{7} -93772.7 q^{8} -157130. q^{9} +O(q^{10})\)	\(q+28.6126 q^{2} +141.482 q^{3} -1229.32 q^{4} -2645.71 q^{5} +4048.18 q^{6} +76456.7 q^{7} -93772.7 q^{8} -157130. q^{9} -75700.7 q^{10} -603251. q^{11} -173926. q^{12} -941600. q^{13} +2.18763e6 q^{14} -374320. q^{15} -165445. q^{16} -5.54744e6 q^{17} -4.49590e6 q^{18} -8.15289e6 q^{19} +3.25241e6 q^{20} +1.08173e7 q^{21} -1.72606e7 q^{22} +6.43634e6 q^{23} -1.32672e7 q^{24} -4.18283e7 q^{25} -2.69417e7 q^{26} -4.72942e7 q^{27} -9.39895e7 q^{28} +1.14880e8 q^{29} -1.07103e7 q^{30} -1.09756e8 q^{31} +1.87313e8 q^{32} -8.53492e7 q^{33} -1.58727e8 q^{34} -2.02282e8 q^{35} +1.93162e8 q^{36} -3.97244e8 q^{37} -2.33276e8 q^{38} -1.33220e8 q^{39} +2.48095e8 q^{40} +7.31306e8 q^{41} +3.09510e8 q^{42} +7.98600e8 q^{43} +7.41587e8 q^{44} +4.15720e8 q^{45} +1.84161e8 q^{46} -4.30123e8 q^{47} -2.34075e7 q^{48} +3.86830e9 q^{49} -1.19682e9 q^{50} -7.84863e8 q^{51} +1.15752e9 q^{52} +4.54586e9 q^{53} -1.35321e9 q^{54} +1.59603e9 q^{55} -7.16955e9 q^{56} -1.15349e9 q^{57} +3.28701e9 q^{58} -7.83815e8 q^{59} +4.60158e8 q^{60} -7.23765e9 q^{61} -3.14042e9 q^{62} -1.20136e10 q^{63} +5.69834e9 q^{64} +2.49120e9 q^{65} -2.44207e9 q^{66} -1.06037e10 q^{67} +6.81956e9 q^{68} +9.10627e8 q^{69} -5.78783e9 q^{70} -1.08505e10 q^{71} +1.47345e10 q^{72} +3.25510e10 q^{73} -1.13662e10 q^{74} -5.91796e9 q^{75} +1.00225e10 q^{76} -4.61226e10 q^{77} -3.81176e9 q^{78} -5.15171e10 q^{79} +4.37720e8 q^{80} +2.11438e10 q^{81} +2.09246e10 q^{82} +6.27091e10 q^{83} -1.32978e10 q^{84} +1.46769e10 q^{85} +2.28501e10 q^{86} +1.62534e10 q^{87} +5.65685e10 q^{88} -2.11172e10 q^{89} +1.18948e10 q^{90} -7.19916e10 q^{91} -7.91230e9 q^{92} -1.55285e10 q^{93} -1.23070e10 q^{94} +2.15702e10 q^{95} +2.65014e10 q^{96} -1.36592e11 q^{97} +1.10682e11 q^{98} +9.47888e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 32 q^{2} - 992 q^{3} + 5120 q^{4} - 11466 q^{5} + 34618 q^{6} - 54118 q^{7} - 155568 q^{8} + 648814 q^{9}+O(q^{10}) \) 8 * q - 32 * q^2 - 992 * q^3 + 5120 * q^4 - 11466 * q^5 + 34618 * q^6 - 54118 * q^7 - 155568 * q^8 + 648814 * q^9	\( 8 q - 32 q^{2} - 992 q^{3} + 5120 q^{4} - 11466 q^{5} + 34618 q^{6} - 54118 q^{7} - 155568 q^{8} + 648814 q^{9} + 517892 q^{10} - 291462 q^{11} - 884188 q^{12} - 2211306 q^{13} - 939584 q^{14} - 2205330 q^{15} - 8561344 q^{16} - 5775330 q^{17} - 51349034 q^{18} - 21015588 q^{19} - 65503576 q^{20} - 36171230 q^{21} - 83047784 q^{22} + 51490744 q^{23} - 129286728 q^{24} - 36491644 q^{25} - 119299562 q^{26} - 394617320 q^{27} - 392796032 q^{28} - 322285430 q^{29} - 646885140 q^{30} - 415184840 q^{31} + 31831744 q^{32} - 549306602 q^{33} - 28224252 q^{34} + 603721008 q^{35} + 690703676 q^{36} + 176642018 q^{37} + 554685496 q^{38} + 2251149264 q^{39} + 1337904816 q^{40} + 357962218 q^{41} + 340644280 q^{42} + 2500461376 q^{43} + 5064743472 q^{44} + 385017072 q^{45} - 205962976 q^{46} + 261795200 q^{47} + 4421752784 q^{48} + 2656605924 q^{49} + 1642758328 q^{50} + 6771514570 q^{51} + 3841657212 q^{52} + 3542935060 q^{53} + 18173306686 q^{54} - 10100187604 q^{55} + 7995463104 q^{56} - 14761628752 q^{57} - 9113565454 q^{58} + 930905396 q^{59} + 19344914040 q^{60} - 25338655048 q^{61} + 4385691666 q^{62} - 25499316044 q^{63} - 34067008768 q^{64} - 25954746658 q^{65} + 13172584012 q^{66} - 3123467482 q^{67} - 37358480280 q^{68} - 6384852256 q^{69} - 35719175696 q^{70} - 52612263236 q^{71} - 9100886376 q^{72} - 67014176274 q^{73} + 10171443276 q^{74} - 87540153860 q^{75} + 17955918576 q^{76} - 44516617816 q^{77} - 25596104778 q^{78} - 27683357604 q^{79} + 74357773216 q^{80} + 55141240264 q^{81} + 73615849126 q^{82} - 12253964262 q^{83} + 168565479344 q^{84} + 58779027600 q^{85} + 90522557252 q^{86} - 129275944888 q^{87} + 33736356800 q^{88} + 10662817760 q^{89} + 450294422856 q^{90} - 28336741418 q^{91} + 32954076160 q^{92} + 164368292014 q^{93} + 285145948346 q^{94} - 64104297380 q^{95} + 208023008864 q^{96} - 124519454530 q^{97} + 215615498272 q^{98} + 186256571332 q^{99}+O(q^{100}) \) 8 * q - 32 * q^2 - 992 * q^3 + 5120 * q^4 - 11466 * q^5 + 34618 * q^6 - 54118 * q^7 - 155568 * q^8 + 648814 * q^9 + 517892 * q^10 - 291462 * q^11 - 884188 * q^12 - 2211306 * q^13 - 939584 * q^14 - 2205330 * q^15 - 8561344 * q^16 - 5775330 * q^17 - 51349034 * q^18 - 21015588 * q^19 - 65503576 * q^20 - 36171230 * q^21 - 83047784 * q^22 + 51490744 * q^23 - 129286728 * q^24 - 36491644 * q^25 - 119299562 * q^26 - 394617320 * q^27 - 392796032 * q^28 - 322285430 * q^29 - 646885140 * q^30 - 415184840 * q^31 + 31831744 * q^32 - 549306602 * q^33 - 28224252 * q^34 + 603721008 * q^35 + 690703676 * q^36 + 176642018 * q^37 + 554685496 * q^38 + 2251149264 * q^39 + 1337904816 * q^40 + 357962218 * q^41 + 340644280 * q^42 + 2500461376 * q^43 + 5064743472 * q^44 + 385017072 * q^45 - 205962976 * q^46 + 261795200 * q^47 + 4421752784 * q^48 + 2656605924 * q^49 + 1642758328 * q^50 + 6771514570 * q^51 + 3841657212 * q^52 + 3542935060 * q^53 + 18173306686 * q^54 - 10100187604 * q^55 + 7995463104 * q^56 - 14761628752 * q^57 - 9113565454 * q^58 + 930905396 * q^59 + 19344914040 * q^60 - 25338655048 * q^61 + 4385691666 * q^62 - 25499316044 * q^63 - 34067008768 * q^64 - 25954746658 * q^65 + 13172584012 * q^66 - 3123467482 * q^67 - 37358480280 * q^68 - 6384852256 * q^69 - 35719175696 * q^70 - 52612263236 * q^71 - 9100886376 * q^72 - 67014176274 * q^73 + 10171443276 * q^74 - 87540153860 * q^75 + 17955918576 * q^76 - 44516617816 * q^77 - 25596104778 * q^78 - 27683357604 * q^79 + 74357773216 * q^80 + 55141240264 * q^81 + 73615849126 * q^82 - 12253964262 * q^83 + 168565479344 * q^84 + 58779027600 * q^85 + 90522557252 * q^86 - 129275944888 * q^87 + 33736356800 * q^88 + 10662817760 * q^89 + 450294422856 * q^90 - 28336741418 * q^91 + 32954076160 * q^92 + 164368292014 * q^93 + 285145948346 * q^94 - 64104297380 * q^95 + 208023008864 * q^96 - 124519454530 * q^97 + 215615498272 * q^98 + 186256571332 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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