LMFDB - Embedded newform 23.12.a.a.1.2

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			1.2
Root			$32.6179$ 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)

$f(q)$	$=$	$q-69.2357 q^{2} -806.194 q^{3} +2745.59 q^{4} -10567.4 q^{5} +55817.4 q^{6} -39263.4 q^{7} -48297.9 q^{8} +472801. q^{9} +O(q^{10})$	\(q-69.2357 q^{2} -806.194 q^{3} +2745.59 q^{4} -10567.4 q^{5} +55817.4 q^{6} -39263.4 q^{7} -48297.9 q^{8} +472801. q^{9} +731642. q^{10} +562993. q^{11} -2.21347e6 q^{12} -714131. q^{13} +2.71843e6 q^{14} +8.51938e6 q^{15} -2.27902e6 q^{16} -4.57861e6 q^{17} -3.27348e7 q^{18} +4.10765e6 q^{19} -2.90137e7 q^{20} +3.16539e7 q^{21} -3.89793e7 q^{22} +6.43634e6 q^{23} +3.89375e7 q^{24} +6.28420e7 q^{25} +4.94434e7 q^{26} -2.38355e8 q^{27} -1.07801e8 q^{28} +1.74709e8 q^{29} -5.89845e8 q^{30} -2.09192e8 q^{31} +2.56704e8 q^{32} -4.53882e8 q^{33} +3.17004e8 q^{34} +4.14912e8 q^{35} +1.29812e9 q^{36} +3.14990e8 q^{37} -2.84396e8 q^{38} +5.75728e8 q^{39} +5.10384e8 q^{40} +5.92065e8 q^{41} -2.19158e9 q^{42} +4.99500e8 q^{43} +1.54575e9 q^{44} -4.99629e9 q^{45} -4.45625e8 q^{46} +2.05903e9 q^{47} +1.83733e9 q^{48} -4.35712e8 q^{49} -4.35091e9 q^{50} +3.69125e9 q^{51} -1.96071e9 q^{52} -1.05290e9 q^{53} +1.65027e10 q^{54} -5.94938e9 q^{55} +1.89634e9 q^{56} -3.31156e9 q^{57} -1.20961e10 q^{58} +9.13400e7 q^{59} +2.33907e10 q^{60} -9.65477e9 q^{61} +1.44836e10 q^{62} -1.85638e10 q^{63} -1.31056e10 q^{64} +7.54652e9 q^{65} +3.14248e10 q^{66} +1.15932e10 q^{67} -1.25710e10 q^{68} -5.18894e9 q^{69} -2.87268e10 q^{70} -5.03903e9 q^{71} -2.28353e10 q^{72} -1.30801e10 q^{73} -2.18086e10 q^{74} -5.06628e10 q^{75} +1.12779e10 q^{76} -2.21050e10 q^{77} -3.98610e10 q^{78} +3.77068e10 q^{79} +2.40833e10 q^{80} +1.08405e11 q^{81} -4.09921e10 q^{82} +8.51264e9 q^{83} +8.69086e10 q^{84} +4.83841e10 q^{85} -3.45833e10 q^{86} -1.40850e11 q^{87} -2.71914e10 q^{88} -5.24237e10 q^{89} +3.45922e11 q^{90} +2.80392e10 q^{91} +1.76715e10 q^{92} +1.68649e11 q^{93} -1.42558e11 q^{94} -4.34072e10 q^{95} -2.06953e11 q^{96} -7.95521e10 q^{97} +3.01668e10 q^{98} +2.66184e11 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−69.2357	−1.52991	−0.764954	−	0.644085i	$-0.777238\pi$
$2$	−69.2357	−1.52991	−0.764954	+	0.644085i	$0.777238\pi$
$3$	−806.194	−1.91546	−0.957729	−	0.287673i	$-0.907119\pi$
$3$	−806.194	−1.91546	−0.957729	+	0.287673i	$0.907119\pi$
$4$	2745.59	1.34062
$5$	−10567.4	−1.51228	−0.756142	−	0.654407i	$-0.772918\pi$
$5$	−10567.4	−1.51228	−0.756142	+	0.654407i	$0.772918\pi$
$6$	55817.4	2.93047
$7$	−39263.4	−0.882976	−0.441488	−	0.897267i	$-0.645549\pi$
$7$	−39263.4	−0.882976	−0.441488	+	0.897267i	$0.645549\pi$
$8$	−48297.9	−0.521115
$9$	472801.	2.66898
$10$	731642.	2.31366
$11$	562993.	1.05401	0.527004	−	0.849863i	$-0.323315\pi$
$11$	562993.	1.05401	0.527004	+	0.849863i	$0.323315\pi$
$12$	−2.21347e6	−2.56790
$13$	−714131.	−0.533445	−0.266722	−	0.963773i	$-0.585941\pi$
$13$	−714131.	−0.533445	−0.266722	+	0.963773i	$0.585941\pi$
$14$	2.71843e6	1.35087
$15$	8.51938e6	2.89672
$16$	−2.27902e6	−0.543361
$17$	−4.57861e6	−0.782105	−0.391052	−	0.920368i	$-0.627889\pi$
$17$	−4.57861e6	−0.782105	−0.391052	+	0.920368i	$0.627889\pi$
$18$	−3.27348e7	−4.08329
$19$	4.10765e6	0.380582	0.190291	−	0.981728i	$-0.439057\pi$
$19$	4.10765e6	0.380582	0.190291	+	0.981728i	$0.439057\pi$
$20$	−2.90137e7	−2.02740
$21$	3.16539e7	1.69130
$22$	−3.89793e7	−1.61253
$23$	6.43634e6	0.208514
$23$	6.43634e6	0.208514
$24$	3.89375e7	0.998173
$25$	6.28420e7	1.28700
$26$	4.94434e7	0.816122
$27$	−2.38355e8	−3.19686
$28$	−1.07801e8	−1.18373
$29$	1.74709e8	1.58171	0.790855	−	0.612003i	$-0.209636\pi$
$29$	1.74709e8	1.58171	0.790855	+	0.612003i	$0.209636\pi$
$30$	−5.89845e8	−4.43171
$31$	−2.09192e8	−1.31237	−0.656184	−	0.754601i	$-0.727831\pi$
$31$	−2.09192e8	−1.31237	−0.656184	+	0.754601i	$0.727831\pi$
$32$	2.56704e8	1.35241
$33$	−4.53882e8	−2.01891
$34$	3.17004e8	1.19655
$35$	4.14912e8	1.33531
$36$	1.29812e9	3.57808
$37$	3.14990e8	0.746772	0.373386	−	0.927676i	$-0.378197\pi$
$37$	3.14990e8	0.746772	0.373386	+	0.927676i	$0.378197\pi$
$38$	−2.84396e8	−0.582256
$39$	5.75728e8	1.02179
$40$	5.10384e8	0.788074
$41$	5.92065e8	0.798102	0.399051	−	0.916929i	$-0.369340\pi$
$41$	5.92065e8	0.798102	0.399051	+	0.916929i	$0.369340\pi$
$42$	−2.19158e9	−2.58754
$43$	4.99500e8	0.518154	0.259077	−	0.965857i	$-0.416582\pi$
$43$	4.99500e8	0.518154	0.259077	+	0.965857i	$0.416582\pi$
$44$	1.54575e9	1.41302
$45$	−4.99629e9	−4.03625
$46$	−4.45625e8	−0.319008
$47$	2.05903e9	1.30956	0.654778	−	0.755821i	$-0.272762\pi$
$47$	2.05903e9	1.30956	0.654778	+	0.755821i	$0.272762\pi$
$48$	1.83733e9	1.04078
$49$	−4.35712e8	−0.220354
$50$	−4.35091e9	−1.96900
$51$	3.69125e9	1.49809
$52$	−1.96071e9	−0.715146
$53$	−1.05290e9	−0.345836	−0.172918	−	0.984936i	$-0.555320\pi$
$53$	−1.05290e9	−0.345836	−0.172918	+	0.984936i	$0.555320\pi$
$54$	1.65027e10	4.89090
$55$	−5.94938e9	−1.59396
$56$	1.89634e9	0.460132
$57$	−3.31156e9	−0.728989
$58$	−1.20961e10	−2.41987
$59$	9.13400e7	0.0166332	0.00831659	−	0.999965i	$-0.497353\pi$
$59$	9.13400e7	0.0166332	0.00831659	+	0.999965i	$0.497353\pi$
$60$	2.33907e10	3.88339
$61$	−9.65477e9	−1.46362	−0.731809	−	0.681510i	$-0.761324\pi$
$61$	−9.65477e9	−1.46362	−0.731809	+	0.681510i	$0.761324\pi$
$62$	1.44836e10	2.00780
$63$	−1.85638e10	−2.35664
$64$	−1.31056e10	−1.52570
$65$	7.54652e9	0.806720
$66$	3.14248e10	3.08874
$67$	1.15932e10	1.04904	0.524521	−	0.851398i	$-0.324245\pi$
$67$	1.15932e10	1.04904	0.524521	+	0.851398i	$0.324245\pi$
$68$	−1.25710e10	−1.04850
$69$	−5.18894e9	−0.399401
$70$	−2.87268e10	−2.04290
$71$	−5.03903e9	−0.331456	−0.165728	−	0.986172i	$-0.552997\pi$
$71$	−5.03903e9	−0.331456	−0.165728	+	0.986172i	$0.552997\pi$
$72$	−2.28353e10	−1.39084
$73$	−1.30801e10	−0.738476	−0.369238	−	0.929335i	$-0.620381\pi$
$73$	−1.30801e10	−0.738476	−0.369238	+	0.929335i	$0.620381\pi$
$74$	−2.18086e10	−1.14249
$75$	−5.06628e10	−2.46520
$76$	1.12779e10	0.510216
$77$	−2.21050e10	−0.930663
$78$	−3.98610e10	−1.56325
$79$	3.77068e10	1.37870	0.689351	−	0.724427i	$-0.257896\pi$
$79$	3.77068e10	1.37870	0.689351	+	0.724427i	$0.257896\pi$
$80$	2.40833e10	0.821716
$81$	1.08405e11	3.45447
$82$	−4.09921e10	−1.22102
$83$	8.51264e9	0.237211	0.118606	−	0.992941i	$-0.462158\pi$
$83$	8.51264e9	0.237211	0.118606	+	0.992941i	$0.462158\pi$
$84$	8.69086e10	2.26739
$85$	4.83841e10	1.18276
$86$	−3.45833e10	−0.792729
$87$	−1.40850e11	−3.02970
$88$	−2.71914e10	−0.549259
$89$	−5.24237e10	−0.995137	−0.497569	−	0.867425i	$-0.665774\pi$
$89$	−5.24237e10	−0.995137	−0.497569	+	0.867425i	$0.665774\pi$
$90$	3.45922e11	6.17510
$91$	2.80392e10	0.471019
$92$	1.76715e10	0.279538
$93$	1.68649e11	2.51379
$94$	−1.42558e11	−2.00350
$95$	−4.34072e10	−0.575549
$96$	−2.06953e11	−2.59048
$97$	−7.95521e10	−0.940605	−0.470303	−	0.882505i	$-0.655855\pi$
$97$	−7.95521e10	−0.940605	−0.470303	+	0.882505i	$0.655855\pi$
$98$	3.01668e10	0.337121
$99$	2.66184e11	2.81312
$100$	1.72538e11	1.72538
$101$	1.10019e11	1.04160	0.520799	−	0.853679i	$-0.325634\pi$
$101$	1.10019e11	1.04160	0.520799	+	0.853679i	$0.325634\pi$
$102$	−2.55566e11	−2.29194
$103$	1.94821e11	1.65589	0.827943	−	0.560813i	$-0.189511\pi$
$103$	1.94821e11	1.65589	0.827943	+	0.560813i	$0.189511\pi$
$104$	3.44911e10	0.277986
$105$	−3.34500e11	−2.55773
$106$	7.28984e10	0.529098
$107$	−2.19752e10	−0.151469	−0.0757343	−	0.997128i	$-0.524130\pi$
$107$	−2.19752e10	−0.151469	−0.0757343	+	0.997128i	$0.524130\pi$
$108$	−6.54424e11	−4.28577
$109$	−2.27512e11	−1.41631	−0.708156	−	0.706056i	$-0.750473\pi$
$109$	−2.27512e11	−1.41631	−0.708156	+	0.706056i	$0.750473\pi$
$110$	4.11910e11	2.43861
$111$	−2.53943e11	−1.43041
$112$	8.94821e10	0.479774
$113$	−3.02801e10	−0.154606	−0.0773029	−	0.997008i	$-0.524631\pi$
$113$	−3.02801e10	−0.154606	−0.0773029	+	0.997008i	$0.524631\pi$
$114$	2.29278e11	1.11529
$115$	−6.80155e10	−0.315333
$116$	4.79680e11	2.12047
$117$	−3.37642e11	−1.42375
$118$	−6.32400e9	−0.0254472
$119$	1.79772e11	0.690580
$120$	−4.11468e11	−1.50952
$121$	3.16500e10	0.110931
$122$	6.68455e11	2.23920
$123$	−4.77319e11	−1.52873
$124$	−5.74355e11	−1.75939
$125$	−1.48090e11	−0.434032
$126$	1.28528e12	3.60545
$127$	−2.79673e11	−0.751157	−0.375578	−	0.926791i	$-0.622556\pi$
$127$	−2.79673e11	−0.751157	−0.375578	+	0.926791i	$0.622556\pi$
$128$	3.81649e11	0.981769
$129$	−4.02694e11	−0.992503
$130$	−5.22489e11	−1.23421
$131$	−3.28862e10	−0.0744769	−0.0372385	−	0.999306i	$-0.511856\pi$
$131$	−3.28862e10	−0.0744769	−0.0372385	+	0.999306i	$0.511856\pi$
$132$	−1.24617e12	−2.70658
$133$	−1.61280e11	−0.336045
$134$	−8.02665e11	−1.60494
$135$	2.51879e12	4.83456
$136$	2.21138e11	0.407566
$137$	−3.36553e11	−0.595787	−0.297893	−	0.954599i	$-0.596284\pi$
$137$	−3.36553e11	−0.595787	−0.297893	+	0.954599i	$0.596284\pi$
$138$	3.59260e11	0.611046
$139$	5.60014e11	0.915414	0.457707	−	0.889103i	$-0.348671\pi$
$139$	5.60014e11	0.915414	0.457707	+	0.889103i	$0.348671\pi$
$140$	1.13918e12	1.79014
$141$	−1.65998e12	−2.50840
$142$	3.48881e11	0.507097
$143$	−4.02051e11	−0.562255
$144$	−1.07752e12	−1.45022
$145$	−1.84622e12	−2.39200
$146$	9.05612e11	1.12980
$147$	3.51268e11	0.422078
$148$	8.64833e11	1.00114
$149$	1.14842e12	1.28108	0.640541	−	0.767924i	$-0.278710\pi$
$149$	1.14842e12	1.28108	0.640541	+	0.767924i	$0.278710\pi$
$150$	3.50768e12	3.77153
$151$	−1.11140e12	−1.15212	−0.576058	−	0.817409i	$-0.695410\pi$
$151$	−1.11140e12	−1.15212	−0.576058	+	0.817409i	$0.695410\pi$
$152$	−1.98391e11	−0.198327
$153$	−2.16478e12	−2.08742
$154$	1.53046e12	1.42383
$155$	2.21062e12	1.98467
$156$	1.58071e12	1.36983
$157$	2.67536e11	0.223838	0.111919	−	0.993717i	$-0.464300\pi$
$157$	2.67536e11	0.223838	0.111919	+	0.993717i	$0.464300\pi$
$158$	−2.61066e12	−2.10929
$159$	8.48843e11	0.662435
$160$	−2.71269e12	−2.04522
$161$	−2.52713e11	−0.184113
$162$	−7.50549e12	−5.28501
$163$	2.58698e11	0.176101	0.0880503	−	0.996116i	$-0.471936\pi$
$163$	2.58698e11	0.176101	0.0880503	+	0.996116i	$0.471936\pi$
$164$	1.62557e12	1.06995
$165$	4.79635e12	3.05316
$166$	−5.89379e11	−0.362911
$167$	1.82419e12	1.08675	0.543376	−	0.839489i	$-0.317146\pi$
$167$	1.82419e12	1.08675	0.543376	+	0.839489i	$0.317146\pi$
$168$	−1.52882e12	−0.881363
$169$	−1.28218e12	−0.715436
$170$	−3.34991e12	−1.80952
$171$	1.94210e12	1.01577
$172$	1.37142e12	0.694647
$173$	1.47286e12	0.722616	0.361308	−	0.932447i	$-0.382330\pi$
$173$	1.47286e12	0.722616	0.361308	+	0.932447i	$0.382330\pi$
$174$	9.75182e12	4.63516
$175$	−2.46739e12	−1.13639
$176$	−1.28307e12	−0.572706
$177$	−7.36378e10	−0.0318601
$178$	3.62960e12	1.52247
$179$	−1.42396e12	−0.579170	−0.289585	−	0.957152i	$-0.593517\pi$
$179$	−1.42396e12	−0.579170	−0.289585	+	0.957152i	$0.593517\pi$
$180$	−1.37177e13	−5.41108
$181$	2.22964e12	0.853107	0.426553	−	0.904462i	$-0.359728\pi$
$181$	2.22964e12	0.853107	0.426553	+	0.904462i	$0.359728\pi$
$182$	−1.94132e12	−0.720616
$183$	7.78362e12	2.80350
$184$	−3.10862e11	−0.108660
$185$	−3.32863e12	−1.12933
$186$	−1.16766e13	−3.84586
$187$	−2.57773e12	−0.824344
$188$	5.65324e12	1.75562
$189$	9.35862e12	2.82275
$190$	3.00533e12	0.880537
$191$	−1.57809e12	−0.449210	−0.224605	−	0.974450i	$-0.572109\pi$
$191$	−1.57809e12	−0.449210	−0.224605	+	0.974450i	$0.572109\pi$
$192$	1.05657e13	2.92241
$193$	−4.81902e12	−1.29537	−0.647685	−	0.761908i	$-0.724263\pi$
$193$	−4.81902e12	−1.29537	−0.647685	+	0.761908i	$0.724263\pi$
$194$	5.50785e12	1.43904
$195$	−6.08396e12	−1.54524
$196$	−1.19628e12	−0.295410
$197$	2.22478e12	0.534223	0.267111	−	0.963666i	$-0.413931\pi$
$197$	2.22478e12	0.534223	0.267111	+	0.963666i	$0.413931\pi$
$198$	−1.84295e13	−4.30382
$199$	3.46090e12	0.786134	0.393067	−	0.919510i	$-0.371414\pi$
$199$	3.46090e12	0.786134	0.393067	+	0.919510i	$0.371414\pi$
$200$	−3.03514e12	−0.670677
$201$	−9.34638e12	−2.00939
$202$	−7.61724e12	−1.59355
$203$	−6.85968e12	−1.39661
$204$	1.01346e13	2.00837
$205$	−6.25660e12	−1.20696
$206$	−1.34886e13	−2.53335
$207$	3.04311e12	0.556520
$208$	1.62752e12	0.289853
$209$	2.31258e12	0.401137
$210$	2.31593e13	3.91309
$211$	1.57826e12	0.259792	0.129896	−	0.991528i	$-0.458536\pi$
$211$	1.57826e12	0.259792	0.129896	+	0.991528i	$0.458536\pi$
$212$	−2.89083e12	−0.463635
$213$	4.06243e12	0.634890
$214$	1.52147e12	0.231733
$215$	−5.27842e12	−0.783597
$216$	1.15120e13	1.66593
$217$	8.21359e12	1.15879
$218$	1.57520e13	2.16683
$219$	1.05451e13	1.41452
$220$	−1.63345e13	−2.13689
$221$	3.26973e12	0.417210
$222$	1.75819e13	2.18839
$223$	1.20487e13	1.46307	0.731535	−	0.681804i	$-0.238804\pi$
$223$	1.20487e13	1.46307	0.731535	+	0.681804i	$0.238804\pi$
$224$	−1.00791e13	−1.19414
$225$	2.97118e13	3.43499
$226$	2.09646e12	0.236533
$227$	−9.10510e12	−1.00263	−0.501317	−	0.865264i	$-0.667151\pi$
$227$	−9.10510e12	−1.00263	−0.501317	+	0.865264i	$0.667151\pi$
$228$	−9.09218e12	−0.977297
$229$	2.96148e12	0.310752	0.155376	−	0.987855i	$-0.450341\pi$
$229$	2.96148e12	0.310752	0.155376	+	0.987855i	$0.450341\pi$
$230$	4.70910e12	0.482431
$231$	1.78209e13	1.78265
$232$	−8.43810e12	−0.824253
$233$	−1.06908e13	−1.01989	−0.509943	−	0.860208i	$-0.670333\pi$
$233$	−1.06908e13	−1.01989	−0.509943	+	0.860208i	$0.670333\pi$
$234$	2.33769e13	2.17821
$235$	−2.17586e13	−1.98042
$236$	2.50782e11	0.0222987
$237$	−3.03990e13	−2.64085
$238$	−1.24466e13	−1.05652
$239$	−4.11412e12	−0.341262	−0.170631	−	0.985335i	$-0.554581\pi$
$239$	−4.11412e12	−0.341262	−0.170631	+	0.985335i	$0.554581\pi$
$240$	−1.94158e13	−1.57396
$241$	−1.51688e12	−0.120187	−0.0600937	−	0.998193i	$-0.519140\pi$
$241$	−1.51688e12	−0.120187	−0.0600937	+	0.998193i	$0.519140\pi$
$242$	−2.19131e12	−0.169714
$243$	−4.51714e13	−3.42003
$244$	−2.65080e13	−1.96215
$245$	4.60434e12	0.333238
$246$	3.30476e13	2.33882
$247$	−2.93340e12	−0.203020
$248$	1.01035e13	0.683895
$249$	−6.86284e12	−0.454368
$250$	1.02531e13	0.664029
$251$	1.85498e13	1.17526	0.587629	−	0.809131i	$-0.300062\pi$
$251$	1.85498e13	1.17526	0.587629	+	0.809131i	$0.300062\pi$
$252$	−5.09685e13	−3.15936
$253$	3.62362e12	0.219776
$254$	1.93634e13	1.14920
$255$	−3.90070e13	−2.26554
$256$	4.16585e11	0.0236801
$257$	−2.57532e13	−1.43284	−0.716422	−	0.697667i	$-0.754221\pi$
$257$	−2.57532e13	−1.43284	−0.716422	+	0.697667i	$0.754221\pi$
$258$	2.78808e13	1.51844
$259$	−1.23676e13	−0.659381
$260$	2.07196e13	1.08150
$261$	8.26028e13	4.22155
$262$	2.27690e12	0.113943
$263$	−3.05942e12	−0.149928	−0.0749638	−	0.997186i	$-0.523884\pi$
$263$	−3.05942e12	−0.149928	−0.0749638	+	0.997186i	$0.523884\pi$
$264$	2.19215e13	1.05208
$265$	1.11264e13	0.523003
$266$	1.11664e13	0.514118
$267$	4.22637e13	1.90614
$268$	3.18302e13	1.40636
$269$	−2.01029e13	−0.870205	−0.435102	−	0.900381i	$-0.643288\pi$
$269$	−2.01029e13	−0.870205	−0.435102	+	0.900381i	$0.643288\pi$
$270$	−1.74390e14	−7.39643
$271$	1.23621e13	0.513761	0.256880	−	0.966443i	$-0.417305\pi$
$271$	1.23621e13	0.513761	0.256880	+	0.966443i	$0.417305\pi$
$272$	1.04348e13	0.424965
$273$	−2.26051e13	−0.902217
$274$	2.33015e13	0.911499
$275$	3.53796e13	1.35651
$276$	−1.42467e13	−0.535444
$277$	−2.36412e13	−0.871027	−0.435514	−	0.900182i	$-0.643433\pi$
$277$	−2.36412e13	−0.871027	−0.435514	+	0.900182i	$0.643433\pi$
$278$	−3.87730e13	−1.40050
$279$	−9.89063e13	−3.50268
$280$	−2.00394e13	−0.695850
$281$	2.33490e13	0.795029	0.397515	−	0.917596i	$-0.369873\pi$
$281$	2.33490e13	0.795029	0.397515	+	0.917596i	$0.369873\pi$
$282$	1.14930e14	3.83762
$283$	1.17433e12	0.0384562	0.0192281	−	0.999815i	$-0.493879\pi$
$283$	1.17433e12	0.0384562	0.0192281	+	0.999815i	$0.493879\pi$
$284$	−1.38351e13	−0.444356
$285$	3.49946e13	1.10244
$286$	2.78363e13	0.860198
$287$	−2.32465e13	−0.704705
$288$	1.21370e14	3.60954
$289$	−1.33082e13	−0.388312
$290$	1.27825e14	3.65953
$291$	6.41344e13	1.80169
$292$	−3.59126e13	−0.990015
$293$	5.10962e13	1.38235	0.691173	−	0.722690i	$-0.257094\pi$
$293$	5.10962e13	1.38235	0.691173	+	0.722690i	$0.257094\pi$
$294$	−2.43203e13	−0.645741
$295$	−9.65228e11	−0.0251541
$296$	−1.52134e13	−0.389154
$297$	−1.34192e14	−3.36951
$298$	−7.95118e13	−1.95994
$299$	−4.59639e12	−0.111231
$300$	−1.39099e14	−3.30490
$301$	−1.96121e13	−0.457518
$302$	7.69484e13	1.76263
$303$	−8.86966e13	−1.99514
$304$	−9.36142e12	−0.206793
$305$	1.02026e14	2.21341
$306$	1.49880e14	3.19356
$307$	7.28874e13	1.52543	0.762713	−	0.646737i	$-0.223867\pi$
$307$	7.28874e13	1.52543	0.762713	+	0.646737i	$0.223867\pi$
$308$	−6.06913e13	−1.24766
$309$	−1.57063e14	−3.17178
$310$	−1.53054e14	−3.03637
$311$	3.35375e13	0.653656	0.326828	−	0.945084i	$-0.394020\pi$
$311$	3.35375e13	0.653656	0.326828	+	0.945084i	$0.394020\pi$
$312$	−2.78065e13	−0.532471
$313$	−6.27990e13	−1.18157	−0.590784	−	0.806830i	$-0.701181\pi$
$313$	−6.27990e13	−1.18157	−0.590784	+	0.806830i	$0.701181\pi$
$314$	−1.85231e13	−0.342452
$315$	1.96171e14	3.56391
$316$	1.03527e14	1.84831
$317$	−6.58350e13	−1.15513	−0.577565	−	0.816345i	$-0.695997\pi$
$317$	−6.58350e13	−1.15513	−0.577565	+	0.816345i	$0.695997\pi$
$318$	−5.87703e13	−1.01346
$319$	9.83602e13	1.66713
$320$	1.38493e14	2.30729
$321$	1.77163e13	0.290132
$322$	1.74968e13	0.281676
$323$	−1.88073e13	−0.297655
$324$	2.97635e14	4.63112
$325$	−4.48774e13	−0.686546
$326$	−1.79111e13	−0.269418
$327$	1.83419e14	2.71289
$328$	−2.85955e13	−0.415903
$329$	−8.08445e13	−1.15631
$330$	−3.32079e14	−4.67105
$331$	−5.14877e13	−0.712278	−0.356139	−	0.934433i	$-0.615907\pi$
$331$	−5.14877e13	−0.712278	−0.356139	+	0.934433i	$0.615907\pi$
$332$	2.33722e13	0.318009
$333$	1.48928e14	1.99312
$334$	−1.26299e14	−1.66263
$335$	−1.22510e14	−1.58645
$336$	−7.21399e13	−0.918987
$337$	−7.77567e13	−0.974481	−0.487241	−	0.873268i	$-0.661997\pi$
$337$	−7.77567e13	−0.974481	−0.487241	+	0.873268i	$0.661997\pi$
$338$	8.87725e13	1.09455
$339$	2.44116e13	0.296141
$340$	1.32843e14	1.58564
$341$	−1.17774e14	−1.38325
$342$	−1.34463e14	−1.55403
$343$	9.47441e13	1.07754
$344$	−2.41248e13	−0.270018
$345$	5.48336e13	0.604007
$346$	−1.01974e14	−1.10554
$347$	5.41968e13	0.578311	0.289155	−	0.957282i	$-0.406626\pi$
$347$	5.41968e13	0.578311	0.289155	+	0.957282i	$0.406626\pi$
$348$	−3.86715e14	−4.06167
$349$	1.80204e13	0.186305	0.0931527	−	0.995652i	$-0.470306\pi$
$349$	1.80204e13	0.186305	0.0931527	+	0.995652i	$0.470306\pi$
$350$	1.70832e14	1.73858
$351$	1.70217e14	1.70535
$352$	1.44523e14	1.42545
$353$	−1.69096e14	−1.64200	−0.820998	−	0.570931i	$-0.806583\pi$
$353$	−1.69096e14	−1.64200	−0.820998	+	0.570931i	$0.806583\pi$
$354$	5.09837e12	0.0487431
$355$	5.32494e13	0.501255
$356$	−1.43934e14	−1.33410
$357$	−1.44931e14	−1.32278
$358$	9.85889e13	0.886077
$359$	−6.74493e13	−0.596977	−0.298489	−	0.954413i	$-0.596483\pi$
$359$	−6.74493e13	−0.596977	−0.298489	+	0.954413i	$0.596483\pi$
$360$	2.41310e14	2.10335
$361$	−9.96175e13	−0.855157
$362$	−1.54371e14	−1.30518
$363$	−2.55160e13	−0.212484
$364$	7.69841e13	0.631457
$365$	1.38223e14	1.11679
$366$	−5.38904e14	−4.28910
$367$	−4.22638e13	−0.331364	−0.165682	−	0.986179i	$-0.552983\pi$
$367$	−4.22638e13	−0.331364	−0.165682	+	0.986179i	$0.552983\pi$
$368$	−1.46686e13	−0.113299
$369$	2.79929e14	2.13012
$370$	2.30460e14	1.72777
$371$	4.13405e13	0.305365
$372$	4.63041e14	3.37003
$373$	1.39214e14	0.998350	0.499175	−	0.866501i	$-0.333636\pi$
$373$	1.39214e14	0.998350	0.499175	+	0.866501i	$0.333636\pi$
$374$	1.78471e14	1.26117
$375$	1.19390e14	0.831370
$376$	−9.94468e13	−0.682429
$377$	−1.24765e14	−0.843756
$378$	−6.47951e14	−4.31854
$379$	−2.32076e14	−1.52446	−0.762229	−	0.647308i	$-0.775895\pi$
$379$	−2.32076e14	−1.52446	−0.762229	+	0.647308i	$0.775895\pi$
$380$	−1.19178e14	−0.771591
$381$	2.25471e14	1.43881
$382$	1.09260e14	0.687249
$383$	−2.50863e14	−1.55541	−0.777703	−	0.628631i	$-0.783615\pi$
$383$	−2.50863e14	−1.55541	−0.777703	+	0.628631i	$0.783615\pi$
$384$	−3.07683e14	−1.88054
$385$	2.33593e14	1.40743
$386$	3.33649e14	1.98180
$387$	2.36164e14	1.38294
$388$	−2.18417e14	−1.26099
$389$	−2.35447e14	−1.34021	−0.670103	−	0.742269i	$-0.733750\pi$
$389$	−2.35447e14	−1.34021	−0.670103	+	0.742269i	$0.733750\pi$
$390$	4.21227e14	2.36407
$391$	−2.94695e13	−0.163080
$392$	2.10440e13	0.114830
$393$	2.65127e13	0.142657
$394$	−1.54034e14	−0.817312
$395$	−3.98463e14	−2.08499
$396$	7.30832e14	3.77132
$397$	1.67185e14	0.850843	0.425421	−	0.904995i	$-0.360126\pi$
$397$	1.67185e14	0.850843	0.425421	+	0.904995i	$0.360126\pi$
$398$	−2.39618e14	−1.20271
$399$	1.30023e14	0.643680
$400$	−1.43218e14	−0.699307
$401$	2.69325e14	1.29713	0.648565	−	0.761160i	$-0.275370\pi$
$401$	2.69325e14	1.29713	0.648565	+	0.761160i	$0.275370\pi$
$402$	6.47104e14	3.07419
$403$	1.49391e14	0.700076
$404$	3.02067e14	1.39639
$405$	−1.14556e15	−5.22413
$406$	4.74935e14	2.13669
$407$	1.77338e14	0.787103
$408$	−1.78280e14	−0.780676
$409$	−3.92321e14	−1.69498	−0.847489	−	0.530814i	$-0.821886\pi$
$409$	−3.92321e14	−1.69498	−0.847489	+	0.530814i	$0.821886\pi$
$410$	4.33180e14	1.84653
$411$	2.71327e14	1.14120
$412$	5.34897e14	2.21991
$413$	−3.58632e12	−0.0146867
$414$	−2.10692e14	−0.851425
$415$	−8.99566e13	−0.358731
$416$	−1.83320e14	−0.721434
$417$	−4.51480e14	−1.75344
$418$	−1.60113e14	−0.613702
$419$	−3.38199e14	−1.27937	−0.639683	−	0.768639i	$-0.720934\pi$
$419$	−3.38199e14	−1.27937	−0.639683	+	0.768639i	$0.720934\pi$
$420$	−9.18398e14	−3.42894
$421$	7.80395e12	0.0287583	0.0143791	−	0.999897i	$-0.495423\pi$
$421$	7.80395e12	0.0287583	0.0143791	+	0.999897i	$0.495423\pi$
$422$	−1.09272e14	−0.397458
$423$	9.73512e14	3.49518
$424$	5.08530e13	0.180220
$425$	−2.87729e14	−1.00657
$426$	−2.81265e14	−0.971323
$427$	3.79079e14	1.29234
$428$	−6.03349e13	−0.203062
$429$	3.24131e14	1.07698
$430$	3.65455e14	1.19883
$431$	−3.57028e14	−1.15632	−0.578160	−	0.815924i	$-0.696229\pi$
$431$	−3.57028e14	−1.15632	−0.578160	+	0.815924i	$0.696229\pi$
$432$	5.43215e14	1.73705
$433$	−1.63092e14	−0.514930	−0.257465	−	0.966288i	$-0.582887\pi$
$433$	−1.63092e14	−0.514930	−0.257465	+	0.966288i	$0.582887\pi$
$434$	−5.68674e14	−1.77284
$435$	1.48841e15	4.58177
$436$	−6.24655e14	−1.89873
$437$	2.64382e13	0.0793569
$438$	−7.30099e14	−2.16408
$439$	−2.28549e14	−0.668998	−0.334499	−	0.942396i	$-0.608567\pi$
$439$	−2.28549e14	−0.668998	−0.334499	+	0.942396i	$0.608567\pi$
$440$	2.87343e14	0.830636
$441$	−2.06005e14	−0.588120
$442$	−2.26382e14	−0.638293
$443$	−1.76794e14	−0.492320	−0.246160	−	0.969229i	$-0.579169\pi$
$443$	−1.76794e14	−0.492320	−0.246160	+	0.969229i	$0.579169\pi$
$444$	−6.97223e14	−1.91763
$445$	5.53983e14	1.50493
$446$	−8.34204e14	−2.23836
$447$	−9.25850e14	−2.45386
$448$	5.14572e14	1.34715
$449$	1.04168e14	0.269388	0.134694	−	0.990887i	$-0.456995\pi$
$449$	1.04168e14	0.269388	0.134694	+	0.990887i	$0.456995\pi$
$450$	−2.05712e15	−5.25521
$451$	3.33329e14	0.841205
$452$	−8.31366e13	−0.207267
$453$	8.96001e14	2.20683
$454$	6.30398e14	1.53394
$455$	−2.96302e14	−0.712315
$456$	1.59942e14	0.379887
$457$	7.06823e14	1.65871	0.829357	−	0.558718i	$-0.188707\pi$
$457$	7.06823e14	1.65871	0.829357	+	0.558718i	$0.188707\pi$
$458$	−2.05040e14	−0.475422
$459$	1.09133e15	2.50028
$460$	−1.86742e14	−0.422741
$461$	8.16200e13	0.182575	0.0912875	−	0.995825i	$-0.470902\pi$
$461$	8.16200e13	0.182575	0.0912875	+	0.995825i	$0.470902\pi$
$462$	−1.23385e15	−2.72728
$463$	2.90931e14	0.635469	0.317734	−	0.948180i	$-0.397078\pi$
$463$	2.90931e14	0.635469	0.317734	+	0.948180i	$0.397078\pi$
$464$	−3.98166e14	−0.859439
$465$	−1.78219e15	−3.80156
$466$	7.40183e14	1.56033
$467$	−1.21733e14	−0.253610	−0.126805	−	0.991928i	$-0.540472\pi$
$467$	−1.21733e14	−0.253610	−0.126805	+	0.991928i	$0.540472\pi$
$468$	−9.27026e14	−1.90871
$469$	−4.55189e14	−0.926278
$470$	1.50647e15	3.02986
$471$	−2.15686e14	−0.428753
$472$	−4.41153e12	−0.00866779
$473$	2.81215e14	0.546138
$474$	2.10470e15	4.04025
$475$	2.58133e14	0.489811
$476$	4.93580e14	0.925804
$477$	−4.97814e14	−0.923030
$478$	2.84844e14	0.522100
$479$	1.19562e14	0.216645	0.108322	−	0.994116i	$-0.465452\pi$
$479$	1.19562e14	0.216645	0.108322	+	0.994116i	$0.465452\pi$
$480$	2.18696e15	3.91754
$481$	−2.24944e14	−0.398362
$482$	1.05023e14	0.183876
$483$	2.03735e14	0.352661
$484$	8.68977e13	0.148716
$485$	8.40660e14	1.42246
$486$	3.12748e15	5.23233
$487$	−1.54437e14	−0.255471	−0.127735	−	0.991808i	$-0.540771\pi$
$487$	−1.54437e14	−0.255471	−0.127735	+	0.991808i	$0.540771\pi$
$488$	4.66305e14	0.762713
$489$	−2.08561e14	−0.337313
$490$	−3.18785e14	−0.509823
$491$	8.04840e14	1.27280	0.636402	−	0.771358i	$-0.280422\pi$
$491$	8.04840e14	1.27280	0.636402	+	0.771358i	$0.280422\pi$
$492$	−1.31052e15	−2.04944
$493$	−7.99927e14	−1.23706
$494$	2.03096e14	0.310602
$495$	−2.81288e15	−4.25424
$496$	4.76753e14	0.713089
$497$	1.97849e14	0.292667
$498$	4.75154e14	0.695141
$499$	−1.09080e15	−1.57830	−0.789152	−	0.614197i	$-0.789480\pi$
$499$	−1.09080e15	−1.57830	−0.789152	+	0.614197i	$0.789480\pi$
$500$	−4.06595e14	−0.581871
$501$	−1.47065e15	−2.08163
$502$	−1.28431e15	−1.79804
$503$	−9.15682e14	−1.26800	−0.634002	−	0.773331i	$-0.718589\pi$
$503$	−9.15682e14	−1.26800	−0.634002	+	0.773331i	$0.718589\pi$
$504$	8.96593e14	1.22808
$505$	−1.16262e15	−1.57519
$506$	−2.50884e14	−0.336237
$507$	1.03368e15	1.37039
$508$	−7.67867e14	−1.00701
$509$	1.22289e15	1.58650	0.793250	−	0.608897i	$-0.208388\pi$
$509$	1.22289e15	1.58650	0.793250	+	0.608897i	$0.208388\pi$
$510$	2.70067e15	3.46606
$511$	5.13571e14	0.652056
$512$	−8.10460e14	−1.01800
$513$	−9.79078e14	−1.21667
$514$	1.78304e15	2.19212
$515$	−2.05875e15	−2.50417
$516$	−1.10563e15	−1.33057
$517$	1.15922e15	1.38028
$518$	8.56279e14	1.00879
$519$	−1.18741e15	−1.38414
$520$	−3.64481e14	−0.420394
$521$	4.66839e14	0.532795	0.266397	−	0.963863i	$-0.414167\pi$
$521$	4.66839e14	0.532795	0.266397	+	0.963863i	$0.414167\pi$
$522$	−5.71907e15	−6.45859
$523$	1.03452e15	1.15606	0.578032	−	0.816014i	$-0.303821\pi$
$523$	1.03452e15	1.15606	0.578032	+	0.816014i	$0.303821\pi$
$524$	−9.02920e13	−0.0998452
$525$	1.98920e15	2.17671
$526$	2.11821e14	0.229376
$527$	9.57809e14	1.02641
$528$	1.03441e15	1.09699
$529$	4.14265e13	0.0434783
$530$	−7.70347e14	−0.800146
$531$	4.31857e13	0.0443936
$532$	−4.42809e14	−0.450508
$533$	−4.22812e14	−0.425743
$534$	−2.92616e15	−2.91622
$535$	2.32221e14	0.229063
$536$	−5.59928e14	−0.546671
$537$	1.14799e15	1.10938
$538$	1.39184e15	1.33133
$539$	−2.45303e14	−0.232255
$540$	6.91556e15	6.48130
$541$	7.00694e14	0.650045	0.325023	−	0.945706i	$-0.394628\pi$
$541$	7.00694e14	0.650045	0.325023	+	0.945706i	$0.394628\pi$
$542$	−8.55899e14	−0.786007
$543$	−1.79753e15	−1.63409
$544$	−1.17535e15	−1.05772
$545$	2.40422e15	2.14187
$546$	1.56508e15	1.38031
$547$	2.24513e14	0.196025	0.0980123	−	0.995185i	$-0.468752\pi$
$547$	2.24513e14	0.196025	0.0980123	+	0.995185i	$0.468752\pi$
$548$	−9.24036e14	−0.798723
$549$	−4.56479e15	−3.90637
$550$	−2.44953e15	−2.07534
$551$	7.17645e14	0.601971
$552$	2.50615e14	0.208134
$553$	−1.48050e15	−1.21736
$554$	1.63682e15	1.33259
$555$	2.68352e15	2.16319
$556$	1.53757e15	1.22722
$557$	−2.99544e14	−0.236732	−0.118366	−	0.992970i	$-0.537766\pi$
$557$	−2.99544e14	−0.236732	−0.118366	+	0.992970i	$0.537766\pi$
$558$	6.84785e15	5.35878
$559$	−3.56709e14	−0.276407
$560$	−9.45594e14	−0.725555
$561$	2.07815e15	1.57900
$562$	−1.61658e15	−1.21632
$563$	−5.52077e13	−0.0411343	−0.0205671	−	0.999788i	$-0.506547\pi$
$563$	−5.52077e13	−0.0411343	−0.0205671	+	0.999788i	$0.506547\pi$
$564$	−4.55761e15	−3.36281
$565$	3.19982e14	0.233808
$566$	−8.13059e13	−0.0588345
$567$	−4.25634e15	−3.05021
$568$	2.43374e14	0.172727
$569$	−3.85034e14	−0.270633	−0.135317	−	0.990802i	$-0.543205\pi$
$569$	−3.85034e14	−0.270633	−0.135317	+	0.990802i	$0.543205\pi$
$570$	−2.42288e15	−1.68663
$571$	−6.87913e14	−0.474280	−0.237140	−	0.971475i	$-0.576210\pi$
$571$	−6.87913e14	−0.474280	−0.237140	+	0.971475i	$0.576210\pi$
$572$	−1.10387e15	−0.753769
$573$	1.27225e15	0.860442
$574$	1.60949e15	1.07813
$575$	4.04473e14	0.268359
$576$	−6.19636e15	−4.07205
$577$	1.40215e15	0.912699	0.456350	−	0.889801i	$-0.349157\pi$
$577$	1.40215e15	0.912699	0.456350	+	0.889801i	$0.349157\pi$
$578$	9.21402e14	0.594082
$579$	3.88507e15	2.48123
$580$	−5.06897e15	−3.20675
$581$	−3.34235e14	−0.209452
$582$	−4.44040e15	−2.75642
$583$	−5.92777e14	−0.364514
$584$	6.31743e14	0.384831
$585$	3.56800e15	2.15312
$586$	−3.53768e15	−2.11486
$587$	2.73266e15	1.61836	0.809182	−	0.587558i	$-0.199911\pi$
$587$	2.73266e15	1.61836	0.809182	+	0.587558i	$0.199911\pi$
$588$	9.64437e14	0.565846
$589$	−8.59287e14	−0.499464
$590$	6.68282e13	0.0384834
$591$	−1.79360e15	−1.02328
$592$	−7.17869e14	−0.405766
$593$	−3.87793e14	−0.217170	−0.108585	−	0.994087i	$-0.534632\pi$
$593$	−3.87793e14	−0.217170	−0.108585	+	0.994087i	$0.534632\pi$
$594$	9.29089e15	5.15504
$595$	−1.89972e15	−1.04435
$596$	3.15309e15	1.71744
$597$	−2.79015e15	−1.50581
$598$	3.18235e14	0.170173
$599$	−1.97431e15	−1.04609	−0.523043	−	0.852306i	$-0.675203\pi$
$599$	−1.97431e15	−1.04609	−0.523043	+	0.852306i	$0.675203\pi$
$600$	2.44691e15	1.28465
$601$	−3.62753e15	−1.88713	−0.943565	−	0.331188i	$-0.892550\pi$
$601$	−3.62753e15	−1.88713	−0.943565	+	0.331188i	$0.892550\pi$
$602$	1.35786e15	0.699960
$603$	5.48129e15	2.79987
$604$	−3.05144e15	−1.54455
$605$	−3.34458e14	−0.167759
$606$	6.14097e15	3.05237
$607$	2.73498e15	1.34715	0.673575	−	0.739119i	$-0.264758\pi$
$607$	2.73498e15	1.34715	0.673575	+	0.739119i	$0.264758\pi$
$608$	1.05445e15	0.514702
$609$	5.53023e15	2.67515
$610$	−7.06384e15	−3.38631
$611$	−1.47042e15	−0.698576
$612$	−5.94358e15	−2.79843
$613$	1.77110e15	0.826436	0.413218	−	0.910632i	$-0.364405\pi$
$613$	1.77110e15	0.826436	0.413218	+	0.910632i	$0.364405\pi$
$614$	−5.04641e15	−2.33376
$615$	5.04403e15	2.31188
$616$	1.06763e15	0.484982
$617$	−4.18057e13	−0.0188221	−0.00941103	−	0.999956i	$-0.502996\pi$
$617$	−4.18057e13	−0.0188221	−0.00941103	+	0.999956i	$0.502996\pi$
$618$	1.08744e16	4.85253
$619$	7.93007e14	0.350734	0.175367	−	0.984503i	$-0.443889\pi$
$619$	7.93007e14	0.350734	0.175367	+	0.984503i	$0.443889\pi$
$620$	6.06944e15	2.66069
$621$	−1.53413e15	−0.666591
$622$	−2.32200e15	−1.00003
$623$	2.05833e15	0.878682
$624$	−1.31210e15	−0.555201
$625$	−1.50353e15	−0.630624
$626$	4.34793e15	1.80769
$627$	−1.86439e15	−0.768360
$628$	7.34544e14	0.300082
$629$	−1.44222e15	−0.584054
$630$	−1.35821e16	−5.45246
$631$	2.63678e15	1.04933	0.524665	−	0.851309i	$-0.324191\pi$
$631$	2.63678e15	1.04933	0.524665	+	0.851309i	$0.324191\pi$
$632$	−1.82116e15	−0.718462
$633$	−1.27239e15	−0.497621
$634$	4.55813e15	1.76724
$635$	2.95542e15	1.13596
$636$	2.33057e15	0.888072
$637$	3.11155e14	0.117547
$638$	−6.81004e15	−2.55056
$639$	−2.38246e15	−0.884648
$640$	−4.03304e15	−1.48471
$641$	1.43484e15	0.523703	0.261852	−	0.965108i	$-0.415667\pi$
$641$	1.43484e15	0.523703	0.261852	+	0.965108i	$0.415667\pi$
$642$	−1.22660e15	−0.443875
$643$	4.75693e15	1.70674	0.853368	−	0.521309i	$-0.174556\pi$
$643$	4.75693e15	1.70674	0.853368	+	0.521309i	$0.174556\pi$
$644$	−6.93845e14	−0.246826
$645$	4.25543e15	1.50095
$646$	1.30214e15	0.455385
$647$	4.15344e15	1.44024	0.720120	−	0.693850i	$-0.244087\pi$
$647$	4.15344e15	1.44024	0.720120	+	0.693850i	$0.244087\pi$
$648$	−5.23573e15	−1.80017
$649$	5.14238e13	0.0175315
$650$	3.10712e15	1.05035
$651$	−6.62175e15	−2.21961
$652$	7.10277e14	0.236084
$653$	−2.49976e15	−0.823904	−0.411952	−	0.911206i	$-0.635153\pi$
$653$	−2.49976e15	−0.823904	−0.411952	+	0.911206i	$0.635153\pi$
$654$	−1.26991e16	−4.15047
$655$	3.47522e14	0.112630
$656$	−1.34933e15	−0.433657
$657$	−6.18431e15	−1.97098
$658$	5.59733e15	1.76904
$659$	3.45445e14	0.108270	0.0541351	−	0.998534i	$-0.482760\pi$
$659$	3.45445e14	0.108270	0.0541351	+	0.998534i	$0.482760\pi$
$660$	1.31688e16	4.09312
$661$	5.21703e15	1.60811	0.804054	−	0.594556i	$-0.202672\pi$
$661$	5.21703e15	1.60811	0.804054	+	0.594556i	$0.202672\pi$
$662$	3.56479e15	1.08972
$663$	−2.63604e15	−0.799148
$664$	−4.11143e14	−0.123614
$665$	1.70432e15	0.508196
$666$	−1.03111e16	−3.04929
$667$	1.12449e15	0.329809
$668$	5.00848e15	1.45692
$669$	−9.71362e15	−2.80245
$670$	8.48209e15	2.42712
$671$	−5.43557e15	−1.54266
$672$	8.12568e15	2.28733
$673$	−3.29011e15	−0.918602	−0.459301	−	0.888281i	$-0.651900\pi$
$673$	−3.29011e15	−0.918602	−0.459301	+	0.888281i	$0.651900\pi$
$674$	5.38354e15	1.49087
$675$	−1.49787e16	−4.11437
$676$	−3.52033e15	−0.959127
$677$	−4.28992e15	−1.15934	−0.579670	−	0.814851i	$-0.696819\pi$
$677$	−4.28992e15	−1.15934	−0.579670	+	0.814851i	$0.696819\pi$
$678$	−1.69016e15	−0.453068
$679$	3.12349e15	0.830532
$680$	−2.33685e15	−0.616356
$681$	7.34048e15	1.92050
$682$	8.15415e15	2.11624
$683$	−6.04706e15	−1.55679	−0.778395	−	0.627774i	$-0.783966\pi$
$683$	−6.04706e15	−1.55679	−0.778395	+	0.627774i	$0.783966\pi$
$684$	5.33221e15	1.36175
$685$	3.55650e15	0.900999
$686$	−6.55968e15	−1.64854
$687$	−2.38753e15	−0.595232
$688$	−1.13837e15	−0.281545
$689$	7.51910e14	0.184485
$690$	−3.79645e15	−0.924075
$691$	−4.52667e15	−1.09307	−0.546537	−	0.837435i	$-0.684054\pi$
$691$	−4.52667e15	−1.09307	−0.546537	+	0.837435i	$0.684054\pi$
$692$	4.04386e15	0.968752
$693$	−1.04513e16	−2.48392
$694$	−3.75235e15	−0.884762
$695$	−5.91790e15	−1.38437
$696$	6.80274e15	1.57882
$697$	−2.71084e15	−0.624199
$698$	−1.24766e15	−0.285030
$699$	8.61883e15	1.95355
$700$	−6.77444e15	−1.52347
$701$	−3.85742e15	−0.860691	−0.430346	−	0.902664i	$-0.641608\pi$
$701$	−3.85742e15	−0.860691	−0.430346	+	0.902664i	$0.641608\pi$
$702$	−1.17851e16	−2.60902
$703$	1.29387e15	0.284208
$704$	−7.37839e15	−1.60810
$705$	1.75417e16	3.79341
$706$	1.17075e16	2.51210
$707$	−4.31972e15	−0.919705
$708$	−2.02179e14	−0.0427123
$709$	−1.46357e15	−0.306802	−0.153401	−	0.988164i	$-0.549023\pi$
$709$	−1.46357e15	−0.306802	−0.153401	+	0.988164i	$0.549023\pi$
$710$	−3.68676e15	−0.766875
$711$	1.78278e16	3.67973
$712$	2.53196e15	0.518581
$713$	−1.34643e15	−0.273648
$714$	1.00344e16	2.02373
$715$	4.24864e15	0.850289
$716$	−3.90961e15	−0.776446
$717$	3.31678e15	0.653673
$718$	4.66990e15	0.913320
$719$	−6.67579e15	−1.29567	−0.647834	−	0.761782i	$-0.724325\pi$
$719$	−6.67579e15	−1.29567	−0.647834	+	0.761782i	$0.724325\pi$
$720$	1.13866e16	2.19314
$721$	−7.64932e15	−1.46211
$722$	6.89709e15	1.30831
$723$	1.22290e15	0.230214
$724$	6.12168e15	1.14369
$725$	1.09791e16	2.03567
$726$	1.76662e15	0.325081
$727$	−1.01637e16	−1.85614	−0.928070	−	0.372405i	$-0.878533\pi$
$727$	−1.01637e16	−1.85614	−0.928070	+	0.372405i	$0.878533\pi$
$728$	−1.35424e15	−0.245455
$729$	1.72134e16	3.09645
$730$	−9.56998e15	−1.70858
$731$	−2.28702e15	−0.405251
$732$	2.13706e16	3.75842
$733$	−4.29078e15	−0.748970	−0.374485	−	0.927233i	$-0.622180\pi$
$733$	−4.29078e15	−0.748970	−0.374485	+	0.927233i	$0.622180\pi$
$734$	2.92616e15	0.506956
$735$	−3.71199e15	−0.638303
$736$	1.65223e15	0.281996
$737$	6.52691e15	1.10570
$738$	−1.93811e16	−3.25888
$739$	−8.54063e15	−1.42543	−0.712715	−	0.701454i	$-0.752535\pi$
$739$	−8.54063e15	−1.42543	−0.712715	+	0.701454i	$0.752535\pi$
$740$	−9.13905e15	−1.51400
$741$	2.36489e15	0.388876
$742$	−2.86224e15	−0.467180
$743$	2.00951e15	0.325575	0.162787	−	0.986661i	$-0.447952\pi$
$743$	2.00951e15	0.325575	0.162787	+	0.986661i	$0.447952\pi$
$744$	−8.14541e15	−1.30997
$745$	−1.21358e16	−1.93736
$746$	−9.63855e15	−1.52738
$747$	4.02479e15	0.633111
$748$	−7.07738e15	−1.10513
$749$	8.62822e14	0.133743
$750$	−8.26602e15	−1.27192
$751$	8.33073e15	1.27252	0.636258	−	0.771476i	$-0.280481\pi$
$751$	8.33073e15	1.27252	0.636258	+	0.771476i	$0.280481\pi$
$752$	−4.69257e15	−0.711562
$753$	−1.49547e16	−2.25116
$754$	8.63822e15	1.29087
$755$	1.17446e16	1.74233
$756$	2.56949e16	3.78423
$757$	−1.23043e16	−1.79899	−0.899494	−	0.436932i	$-0.856065\pi$
$757$	−1.23043e16	−1.79899	−0.899494	+	0.436932i	$0.856065\pi$
$758$	1.60680e16	2.33228
$759$	−2.92134e15	−0.420971
$760$	2.09648e15	0.299927
$761$	1.27350e16	1.80877	0.904387	−	0.426713i	$-0.140329\pi$
$761$	1.27350e16	1.80877	0.904387	+	0.426713i	$0.140329\pi$
$762$	−1.56106e16	−2.20124
$763$	8.93291e15	1.25057
$764$	−4.33279e15	−0.602219
$765$	2.28761e16	3.15677
$766$	1.73687e16	2.37963
$767$	−6.52288e13	−0.00887288
$768$	−3.35848e14	−0.0453582
$769$	−1.45826e15	−0.195542	−0.0977709	−	0.995209i	$-0.531171\pi$
$769$	−1.45826e15	−0.195542	−0.0977709	+	0.995209i	$0.531171\pi$
$770$	−1.61730e16	−2.15323
$771$	2.07621e16	2.74455
$772$	−1.32310e16	−1.73660
$773$	−6.87012e15	−0.895317	−0.447658	−	0.894205i	$-0.647742\pi$
$773$	−6.87012e15	−0.895317	−0.447658	+	0.894205i	$0.647742\pi$
$774$	−1.63510e16	−2.11578
$775$	−1.31460e16	−1.68902
$776$	3.84220e15	0.490163
$777$	9.97068e15	1.26302
$778$	1.63014e16	2.05039
$779$	2.43200e15	0.303744
$780$	−1.67040e16	−2.07158
$781$	−2.83694e15	−0.349357
$782$	2.04034e15	0.249498
$783$	−4.16428e16	−5.05650
$784$	9.92995e14	0.119732
$785$	−2.82716e15	−0.338507
$786$	−1.83562e15	−0.218253
$787$	−1.41118e16	−1.66618	−0.833092	−	0.553135i	$-0.813432\pi$
$787$	−1.41118e16	−1.66618	−0.833092	+	0.553135i	$0.813432\pi$
$788$	6.10832e15	0.716189
$789$	2.46648e15	0.287180
$790$	2.75879e16	3.18984
$791$	1.18890e15	0.136513
$792$	−1.28561e16	−1.46596
$793$	6.89477e15	0.780760
$794$	−1.15752e16	−1.30171
$795$	−8.97007e15	−1.00179
$796$	9.50219e15	1.05391
$797$	1.33247e16	1.46769	0.733846	−	0.679315i	$-0.237723\pi$
$797$	1.33247e16	1.46769	0.733846	+	0.679315i	$0.237723\pi$
$798$	−9.00225e15	−0.984771
$799$	−9.42750e15	−1.02421
$800$	1.61318e16	1.74055
$801$	−2.47860e16	−2.65600
$802$	−1.86469e16	−1.98449
$803$	−7.36403e15	−0.778359
$804$	−2.56613e16	−2.69383
$805$	2.67052e15	0.278431
$806$	−1.03432e16	−1.07105
$807$	1.62068e16	1.66684
$808$	−5.31369e15	−0.542792
$809$	7.90622e15	0.802143	0.401072	−	0.916047i	$-0.368638\pi$
$809$	7.90622e15	0.802143	0.401072	+	0.916047i	$0.368638\pi$
$810$	7.93135e16	7.99245
$811$	−1.14946e16	−1.15048	−0.575238	−	0.817986i	$-0.695091\pi$
$811$	−1.14946e16	−1.15048	−0.575238	+	0.817986i	$0.695091\pi$
$812$	−1.88339e16	−1.87232
$813$	−9.96624e15	−0.984087
$814$	−1.22781e16	−1.20419
$815$	−2.73377e15	−0.266314
$816$	−8.41243e15	−0.814002
$817$	2.05177e15	0.197200
$818$	2.71627e16	2.59316
$819$	1.32570e16	1.25714
$820$	−1.71780e16	−1.61807
$821$	1.25751e16	1.17659	0.588295	−	0.808646i	$-0.299799\pi$
$821$	1.25751e16	1.17659	0.588295	+	0.808646i	$0.299799\pi$
$822$	−1.87855e16	−1.74594
$823$	3.42171e15	0.315896	0.157948	−	0.987447i	$-0.449512\pi$
$823$	3.42171e15	0.315896	0.157948	+	0.987447i	$0.449512\pi$
$824$	−9.40943e15	−0.862906
$825$	−2.85228e16	−2.59834
$826$	2.48302e14	0.0224693
$827$	2.58917e14	0.0232745	0.0116373	−	0.999932i	$-0.496296\pi$
$827$	2.58917e14	0.0232745	0.0116373	+	0.999932i	$0.496296\pi$
$828$	8.35513e15	0.746082
$829$	7.22947e15	0.641293	0.320647	−	0.947199i	$-0.396100\pi$
$829$	7.22947e15	0.641293	0.320647	+	0.947199i	$0.396100\pi$
$830$	6.22821e15	0.548825
$831$	1.90594e16	1.66842
$832$	9.35915e15	0.813875
$833$	1.99496e15	0.172340
$834$	3.12585e16	2.68260
$835$	−1.92770e16	−1.64348
$836$	6.34939e15	0.537771
$837$	4.98619e16	4.19545
$838$	2.34154e16	1.95731
$839$	−3.81690e15	−0.316972	−0.158486	−	0.987361i	$-0.550661\pi$
$839$	−3.81690e15	−0.316972	−0.158486	+	0.987361i	$0.550661\pi$
$840$	1.61556e16	1.33287
$841$	1.83228e16	1.50181
$842$	−5.40312e14	−0.0439975
$843$	−1.88238e16	−1.52284
$844$	4.33326e15	0.348282
$845$	1.35493e16	1.08194
$846$	−6.74018e16	−5.34730
$847$	−1.24269e15	−0.0979495
$848$	2.39958e15	0.187914
$849$	−9.46741e14	−0.0736612
$850$	1.99211e16	1.53996
$851$	2.02739e15	0.155713
$852$	1.11538e16	0.851145
$853$	8.54280e15	0.647710	0.323855	−	0.946107i	$-0.395021\pi$
$853$	8.54280e15	0.647710	0.323855	+	0.946107i	$0.395021\pi$
$854$	−2.62458e16	−1.97716
$855$	−2.05230e16	−1.53613
$856$	1.06136e15	0.0789325
$857$	1.67856e16	1.24034	0.620171	−	0.784467i	$-0.287063\pi$
$857$	1.67856e16	1.24034	0.620171	+	0.784467i	$0.287063\pi$
$858$	−2.24415e16	−1.64767
$859$	−7.95759e14	−0.0580523	−0.0290261	−	0.999579i	$-0.509241\pi$
$859$	−7.95759e14	−0.0580523	−0.0290261	+	0.999579i	$0.509241\pi$
$860$	−1.44924e16	−1.05050
$861$	1.87412e16	1.34983
$862$	2.47191e16	1.76906
$863$	−1.80554e16	−1.28395	−0.641974	−	0.766727i	$-0.721884\pi$
$863$	−1.80554e16	−1.28395	−0.641974	+	0.766727i	$0.721884\pi$
$864$	−6.11866e16	−4.32345
$865$	−1.55643e16	−1.09280
$866$	1.12918e16	0.787796
$867$	1.07290e16	0.743795
$868$	2.25511e16	1.55349
$869$	2.12287e16	1.45316
$870$	−1.03051e17	−7.00968
$871$	−8.27908e15	−0.559606
$872$	1.09884e16	0.738061
$873$	−3.76124e16	−2.51045
$874$	−1.83047e15	−0.121409
$875$	5.81453e15	0.383240
$876$	2.89525e16	1.89633
$877$	−1.15931e16	−0.754574	−0.377287	−	0.926096i	$-0.623143\pi$
$877$	−1.15931e16	−0.754574	−0.377287	+	0.926096i	$0.623143\pi$
$878$	1.58238e16	1.02350
$879$	−4.11934e16	−2.64782
$880$	1.35588e16	0.866094
$881$	8.88390e15	0.563944	0.281972	−	0.959423i	$-0.409011\pi$
$881$	8.88390e15	0.563944	0.281972	+	0.959423i	$0.409011\pi$
$882$	1.42629e16	0.899769
$883$	−4.84045e15	−0.303460	−0.151730	−	0.988422i	$-0.548484\pi$
$883$	−4.84045e15	−0.303460	−0.151730	+	0.988422i	$0.548484\pi$
$884$	8.97733e15	0.559319
$885$	7.78161e14	0.0481816
$886$	1.22405e16	0.753204
$887$	1.23707e16	0.756507	0.378254	−	0.925702i	$-0.376525\pi$
$887$	1.23707e16	0.756507	0.378254	+	0.925702i	$0.376525\pi$
$888$	1.22649e16	0.745408
$889$	1.09809e16	0.663253
$890$	−3.83554e16	−2.30240
$891$	6.10312e16	3.64103
$892$	3.30809e16	1.96142
$893$	8.45777e15	0.498394
$894$	6.41019e16	3.75418
$895$	1.50476e16	0.875870
$896$	−1.49848e16	−0.866879
$897$	3.70558e15	0.213058
$898$	−7.21213e15	−0.412139
$899$	−3.65478e16	−2.07579
$900$	8.15763e16	4.60501
$901$	4.82083e15	0.270480
$902$	−2.30783e16	−1.28697
$903$	1.58111e16	0.876356
$904$	1.46246e15	0.0805673
$905$	−2.35616e16	−1.29014
$906$	−6.20353e16	−3.37624
$907$	−2.44403e16	−1.32211	−0.661054	−	0.750338i	$-0.729891\pi$
$907$	−2.44403e16	−1.32211	−0.661054	+	0.750338i	$0.729891\pi$
$908$	−2.49988e16	−1.34415
$909$	5.20171e16	2.78000
$910$	2.05147e16	1.08978
$911$	3.13603e16	1.65588	0.827940	−	0.560817i	$-0.189513\pi$
$911$	3.13603e16	1.65588	0.827940	+	0.560817i	$0.189513\pi$
$912$	7.54712e15	0.396104
$913$	4.79256e15	0.250022
$914$	−4.89374e16	−2.53768
$915$	−8.22527e16	−4.23969
$916$	8.13100e15	0.416599
$917$	1.29123e15	0.0657613
$918$	−7.55594e16	−3.82519
$919$	−1.94154e16	−0.977035	−0.488517	−	0.872554i	$-0.662462\pi$
$919$	−1.94154e16	−0.977035	−0.488517	+	0.872554i	$0.662462\pi$
$920$	3.28501e15	0.164325
$921$	−5.87613e16	−2.92189
$922$	−5.65102e15	−0.279323
$923$	3.59853e15	0.176813
$924$	4.89290e16	2.38985
$925$	1.97946e16	0.961098
$926$	−2.01428e16	−0.972209
$927$	9.21115e16	4.41952
$928$	4.48485e16	2.13912
$929$	2.27110e16	1.07684	0.538419	−	0.842677i	$-0.319022\pi$
$929$	2.27110e16	1.07684	0.538419	+	0.842677i	$0.319022\pi$
$930$	1.23391e17	5.81604
$931$	−1.78975e15	−0.0838628
$932$	−2.93524e16	−1.36728
$933$	−2.70378e16	−1.25205
$934$	8.42829e15	0.388000
$935$	2.72399e16	1.24664
$936$	1.63074e16	0.741939
$937$	−1.44175e16	−0.652113	−0.326057	−	0.945350i	$-0.605720\pi$
$937$	−1.44175e16	−0.652113	−0.326057	+	0.945350i	$0.605720\pi$
$938$	3.15154e16	1.41712
$939$	5.06281e16	2.26324
$940$	−5.97401e16	−2.65499
$941$	−9.08880e15	−0.401572	−0.200786	−	0.979635i	$-0.564350\pi$
$941$	−9.08880e15	−0.401572	−0.200786	+	0.979635i	$0.564350\pi$
$942$	1.49332e16	0.655952
$943$	3.81074e15	0.166416
$944$	−2.08166e14	−0.00903781
$945$	−9.88964e16	−4.26880
$946$	−1.94702e16	−0.835542
$947$	−8.11454e15	−0.346210	−0.173105	−	0.984903i	$-0.555380\pi$
$947$	−8.11454e15	−0.346210	−0.173105	+	0.984903i	$0.555380\pi$
$948$	−8.34630e16	−3.54037
$949$	9.34093e15	0.393936
$950$	−1.78720e16	−0.749366
$951$	5.30757e16	2.21260
$952$	−8.68261e15	−0.359871
$953$	−3.54669e16	−1.46154	−0.730772	−	0.682621i	$-0.760840\pi$
$953$	−3.54669e16	−1.46154	−0.730772	+	0.682621i	$0.760840\pi$
$954$	3.44665e16	1.41215
$955$	1.66764e16	0.679333
$956$	−1.12957e16	−0.457503
$957$	−7.92974e16	−3.19333
$958$	−8.27796e15	−0.331446
$959$	1.32142e16	0.526065
$960$	−1.11652e17	−4.41951
$961$	1.83528e16	0.722310
$962$	1.55742e16	0.609456
$963$	−1.03899e16	−0.404266
$964$	−4.16474e15	−0.161125
$965$	5.09246e16	1.95897
$966$	−1.41058e16	−0.539539
$967$	4.17412e16	1.58752	0.793760	−	0.608231i	$-0.208121\pi$
$967$	4.17412e16	1.58752	0.793760	+	0.608231i	$0.208121\pi$
$968$	−1.52863e15	−0.0578079
$969$	1.51624e16	0.570146
$970$	−5.82037e16	−2.17624
$971$	−3.11241e16	−1.15715	−0.578576	−	0.815628i	$-0.696391\pi$
$971$	−3.11241e16	−1.15715	−0.578576	+	0.815628i	$0.696391\pi$
$972$	−1.24022e17	−4.58495
$973$	−2.19881e16	−0.808288
$974$	1.06925e16	0.390847
$975$	3.61799e16	1.31505
$976$	2.20034e16	0.795273
$977$	−3.17947e16	−1.14271	−0.571354	−	0.820704i	$-0.693582\pi$
$977$	−3.17947e16	−1.14271	−0.571354	+	0.820704i	$0.693582\pi$
$978$	1.44398e16	0.516058
$979$	−2.95142e16	−1.04888
$980$	1.26416e16	0.446745
$981$	−1.07568e17	−3.78011
$982$	−5.57237e16	−1.94727
$983$	5.13332e16	1.78383	0.891916	−	0.452202i	$-0.149361\pi$
$983$	5.13332e16	1.78383	0.891916	+	0.452202i	$0.149361\pi$
$984$	2.30535e16	0.796644
$985$	−2.35101e16	−0.807897
$986$	5.53835e16	1.89259
$987$	6.51763e16	2.21486
$988$	−8.05391e15	−0.272172
$989$	3.21495e15	0.108043
$990$	1.94752e17	6.50860
$991$	−3.98741e16	−1.32521	−0.662606	−	0.748968i	$-0.730550\pi$
$991$	−3.98741e16	−1.32521	−0.662606	+	0.748968i	$0.730550\pi$
$992$	−5.37004e16	−1.77486
$993$	4.15091e16	1.36434
$994$	−1.36982e16	−0.447754
$995$	−3.65727e16	−1.18886
$996$	−1.88425e16	−0.609134
$997$	1.12012e16	0.360114	0.180057	−	0.983656i	$-0.442372\pi$
$997$	1.12012e16	0.360114	0.180057	+	0.983656i	$0.442372\pi$
$998$	7.55221e16	2.41466
$999$	−7.50795e16	−2.38732

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.12.a.a.1.2	✓	8
3.2	odd	2		207.12.a.a.1.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.12.a.a.1.2	✓	8	1.1	even	1	trivial
207.12.a.a.1.7		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)

\(f(q)\)	\(=\)	\(q-69.2357 q^{2} -806.194 q^{3} +2745.59 q^{4} -10567.4 q^{5} +55817.4 q^{6} -39263.4 q^{7} -48297.9 q^{8} +472801. q^{9} +O(q^{10})\)	\(q-69.2357 q^{2} -806.194 q^{3} +2745.59 q^{4} -10567.4 q^{5} +55817.4 q^{6} -39263.4 q^{7} -48297.9 q^{8} +472801. q^{9} +731642. q^{10} +562993. q^{11} -2.21347e6 q^{12} -714131. q^{13} +2.71843e6 q^{14} +8.51938e6 q^{15} -2.27902e6 q^{16} -4.57861e6 q^{17} -3.27348e7 q^{18} +4.10765e6 q^{19} -2.90137e7 q^{20} +3.16539e7 q^{21} -3.89793e7 q^{22} +6.43634e6 q^{23} +3.89375e7 q^{24} +6.28420e7 q^{25} +4.94434e7 q^{26} -2.38355e8 q^{27} -1.07801e8 q^{28} +1.74709e8 q^{29} -5.89845e8 q^{30} -2.09192e8 q^{31} +2.56704e8 q^{32} -4.53882e8 q^{33} +3.17004e8 q^{34} +4.14912e8 q^{35} +1.29812e9 q^{36} +3.14990e8 q^{37} -2.84396e8 q^{38} +5.75728e8 q^{39} +5.10384e8 q^{40} +5.92065e8 q^{41} -2.19158e9 q^{42} +4.99500e8 q^{43} +1.54575e9 q^{44} -4.99629e9 q^{45} -4.45625e8 q^{46} +2.05903e9 q^{47} +1.83733e9 q^{48} -4.35712e8 q^{49} -4.35091e9 q^{50} +3.69125e9 q^{51} -1.96071e9 q^{52} -1.05290e9 q^{53} +1.65027e10 q^{54} -5.94938e9 q^{55} +1.89634e9 q^{56} -3.31156e9 q^{57} -1.20961e10 q^{58} +9.13400e7 q^{59} +2.33907e10 q^{60} -9.65477e9 q^{61} +1.44836e10 q^{62} -1.85638e10 q^{63} -1.31056e10 q^{64} +7.54652e9 q^{65} +3.14248e10 q^{66} +1.15932e10 q^{67} -1.25710e10 q^{68} -5.18894e9 q^{69} -2.87268e10 q^{70} -5.03903e9 q^{71} -2.28353e10 q^{72} -1.30801e10 q^{73} -2.18086e10 q^{74} -5.06628e10 q^{75} +1.12779e10 q^{76} -2.21050e10 q^{77} -3.98610e10 q^{78} +3.77068e10 q^{79} +2.40833e10 q^{80} +1.08405e11 q^{81} -4.09921e10 q^{82} +8.51264e9 q^{83} +8.69086e10 q^{84} +4.83841e10 q^{85} -3.45833e10 q^{86} -1.40850e11 q^{87} -2.71914e10 q^{88} -5.24237e10 q^{89} +3.45922e11 q^{90} +2.80392e10 q^{91} +1.76715e10 q^{92} +1.68649e11 q^{93} -1.42558e11 q^{94} -4.34072e10 q^{95} -2.06953e11 q^{96} -7.95521e10 q^{97} +3.01668e10 q^{98} +2.66184e11 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 32 q^{2} - 992 q^{3} + 5120 q^{4} - 11466 q^{5} + 34618 q^{6} - 54118 q^{7} - 155568 q^{8} + 648814 q^{9}+O(q^{10}) \) 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

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Groups

Database

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