LMFDB - Embedded newform 23.12.a.a.1.5

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			1.5
Root			$-9.43512$ 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)

$f(q)$	$=$	$q+14.8702 q^{2} +536.788 q^{3} -1826.88 q^{4} +2118.93 q^{5} +7982.17 q^{6} -68580.9 q^{7} -57620.3 q^{8} +110995. q^{9} +O(q^{10})$	\(q+14.8702 q^{2} +536.788 q^{3} -1826.88 q^{4} +2118.93 q^{5} +7982.17 q^{6} -68580.9 q^{7} -57620.3 q^{8} +110995. q^{9} +31509.0 q^{10} -573905. q^{11} -980645. q^{12} -209778. q^{13} -1.01981e6 q^{14} +1.13742e6 q^{15} +2.88461e6 q^{16} +7.02586e6 q^{17} +1.65052e6 q^{18} -1.37759e7 q^{19} -3.87103e6 q^{20} -3.68134e7 q^{21} -8.53411e6 q^{22} +6.43634e6 q^{23} -3.09299e7 q^{24} -4.43382e7 q^{25} -3.11945e6 q^{26} -3.55098e7 q^{27} +1.25289e8 q^{28} -1.90149e8 q^{29} +1.69137e7 q^{30} +1.11283e8 q^{31} +1.60901e8 q^{32} -3.08066e8 q^{33} +1.04476e8 q^{34} -1.45318e8 q^{35} -2.02773e8 q^{36} +3.70363e8 q^{37} -2.04850e8 q^{38} -1.12606e8 q^{39} -1.22094e8 q^{40} -4.78021e8 q^{41} -5.47424e8 q^{42} +1.32298e9 q^{43} +1.04845e9 q^{44} +2.35190e8 q^{45} +9.57100e7 q^{46} +1.04187e9 q^{47} +1.54843e9 q^{48} +2.72601e9 q^{49} -6.59320e8 q^{50} +3.77140e9 q^{51} +3.83238e8 q^{52} -3.65404e9 q^{53} -5.28040e8 q^{54} -1.21607e9 q^{55} +3.95165e9 q^{56} -7.39472e9 q^{57} -2.82757e9 q^{58} +9.91859e8 q^{59} -2.07792e9 q^{60} -5.85821e6 q^{61} +1.65481e9 q^{62} -7.61210e9 q^{63} -3.51505e9 q^{64} -4.44505e8 q^{65} -4.58101e9 q^{66} +1.92877e10 q^{67} -1.28354e10 q^{68} +3.45495e9 q^{69} -2.16092e9 q^{70} -5.27575e9 q^{71} -6.39555e9 q^{72} -2.46322e10 q^{73} +5.50739e9 q^{74} -2.38002e10 q^{75} +2.51668e10 q^{76} +3.93589e10 q^{77} -1.67448e9 q^{78} +9.77106e9 q^{79} +6.11230e9 q^{80} -3.87236e10 q^{81} -7.10829e9 q^{82} -1.67805e10 q^{83} +6.72535e10 q^{84} +1.48873e10 q^{85} +1.96731e10 q^{86} -1.02070e11 q^{87} +3.30686e10 q^{88} -3.63782e10 q^{89} +3.49733e9 q^{90} +1.43867e10 q^{91} -1.17584e10 q^{92} +5.97355e10 q^{93} +1.54929e10 q^{94} -2.91901e10 q^{95} +8.63700e10 q^{96} +5.73626e10 q^{97} +4.05364e10 q^{98} -6.37004e10 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	14.8702	0.328589	0.164295	−	0.986411i	$-0.447465\pi$
$2$	14.8702	0.328589	0.164295	+	0.986411i	$0.447465\pi$
$3$	536.788	1.27537	0.637685	−	0.770297i	$-0.279892\pi$
$3$	536.788	1.27537	0.637685	+	0.770297i	$0.279892\pi$
$4$	−1826.88	−0.892029
$5$	2118.93	0.303237	0.151618	−	0.988439i	$-0.451551\pi$
$5$	2118.93	0.303237	0.151618	+	0.988439i	$0.451551\pi$
$6$	7982.17	0.419072
$7$	−68580.9	−1.54228	−0.771141	−	0.636665i	$-0.780314\pi$
$7$	−68580.9	−1.54228	−0.771141	+	0.636665i	$0.780314\pi$
$8$	−57620.3	−0.621700
$9$	110995.	0.626568
$10$	31509.0	0.0996403
$11$	−573905.	−1.07444	−0.537218	−	0.843444i	$-0.680525\pi$
$11$	−573905.	−1.07444	−0.537218	+	0.843444i	$0.680525\pi$
$12$	−980645.	−1.13767
$13$	−209778.	−0.156701	−0.0783504	−	0.996926i	$-0.524965\pi$
$13$	−209778.	−0.156701	−0.0783504	+	0.996926i	$0.524965\pi$
$14$	−1.01981e6	−0.506777
$15$	1.13742e6	0.386739
$16$	2.88461e6	0.687745
$17$	7.02586e6	1.20014	0.600068	−	0.799949i	$-0.295140\pi$
$17$	7.02586e6	1.20014	0.600068	+	0.799949i	$0.295140\pi$
$18$	1.65052e6	0.205883
$19$	−1.37759e7	−1.27636	−0.638181	−	0.769886i	$-0.720313\pi$
$19$	−1.37759e7	−1.27636	−0.638181	+	0.769886i	$0.720313\pi$
$20$	−3.87103e6	−0.270496
$21$	−3.68134e7	−1.96698
$22$	−8.53411e6	−0.353048
$23$	6.43634e6	0.208514
$23$	6.43634e6	0.208514
$24$	−3.09299e7	−0.792897
$25$	−4.43382e7	−0.908047
$26$	−3.11945e6	−0.0514902
$27$	−3.55098e7	−0.476264
$28$	1.25289e8	1.37576
$29$	−1.90149e8	−1.72150	−0.860748	−	0.509032i	$-0.830004\pi$
$29$	−1.90149e8	−1.72150	−0.860748	+	0.509032i	$0.830004\pi$
$30$	1.69137e7	0.127078
$31$	1.11283e8	0.698136	0.349068	−	0.937097i	$-0.386498\pi$
$31$	1.11283e8	0.698136	0.349068	+	0.937097i	$0.386498\pi$
$32$	1.60901e8	0.847686
$33$	−3.08066e8	−1.37030
$34$	1.04476e8	0.394351
$35$	−1.45318e8	−0.467677
$36$	−2.02773e8	−0.558917
$37$	3.70363e8	0.878047	0.439024	−	0.898475i	$-0.355324\pi$
$37$	3.70363e8	0.878047	0.439024	+	0.898475i	$0.355324\pi$
$38$	−2.04850e8	−0.419399
$39$	−1.12606e8	−0.199851
$40$	−1.22094e8	−0.188522
$41$	−4.78021e8	−0.644371	−0.322185	−	0.946677i	$-0.604417\pi$
$41$	−4.78021e8	−0.644371	−0.322185	+	0.946677i	$0.604417\pi$
$42$	−5.47424e8	−0.646328
$43$	1.32298e9	1.37239	0.686196	−	0.727417i	$-0.259279\pi$
$43$	1.32298e9	1.37239	0.686196	+	0.727417i	$0.259279\pi$
$44$	1.04845e9	0.958428
$45$	2.35190e8	0.189998
$46$	9.57100e7	0.0685155
$47$	1.04187e9	0.662639	0.331319	−	0.943519i	$-0.392506\pi$
$47$	1.04187e9	0.662639	0.331319	+	0.943519i	$0.392506\pi$
$48$	1.54843e9	0.877130
$49$	2.72601e9	1.37863
$50$	−6.59320e8	−0.298374
$51$	3.77140e9	1.53062
$52$	3.83238e8	0.139782
$53$	−3.65404e9	−1.20021	−0.600103	−	0.799923i	$-0.704874\pi$
$53$	−3.65404e9	−1.20021	−0.600103	+	0.799923i	$0.704874\pi$
$54$	−5.28040e8	−0.156495
$55$	−1.21607e9	−0.325809
$56$	3.95165e9	0.958837
$57$	−7.39472e9	−1.62783
$58$	−2.82757e9	−0.565665
$59$	9.91859e8	0.180619	0.0903096	−	0.995914i	$-0.471214\pi$
$59$	9.91859e8	0.180619	0.0903096	+	0.995914i	$0.471214\pi$
$60$	−2.07792e9	−0.344983
$61$	−5.85821e6	−0.000888077 0	−0.000444039	−	1.00000i	$-0.500141\pi$
$61$	−5.85821e6	−0.000888077 0	−0.000444039		1.00000i	$0.500141\pi$
$62$	1.65481e9	0.229400
$63$	−7.61210e9	−0.966344
$64$	−3.51505e9	−0.409205
$65$	−4.44505e8	−0.0475175
$66$	−4.58101e9	−0.450266
$67$	1.92877e10	1.74530	0.872650	−	0.488346i	$-0.162400\pi$
$67$	1.92877e10	1.74530	0.872650	+	0.488346i	$0.162400\pi$
$68$	−1.28354e10	−1.07056
$69$	3.45495e9	0.265933
$70$	−2.16092e9	−0.153673
$71$	−5.27575e9	−0.347027	−0.173514	−	0.984831i	$-0.555512\pi$
$71$	−5.27575e9	−0.347027	−0.173514	+	0.984831i	$0.555512\pi$
$72$	−6.39555e9	−0.389537
$73$	−2.46322e10	−1.39068	−0.695340	−	0.718681i	$-0.744746\pi$
$73$	−2.46322e10	−1.39068	−0.695340	+	0.718681i	$0.744746\pi$
$74$	5.50739e9	0.288517
$75$	−2.38002e10	−1.15810
$76$	2.51668e10	1.13855
$77$	3.93589e10	1.65708
$78$	−1.67448e9	−0.0656690
$79$	9.77106e9	0.357267	0.178633	−	0.983916i	$-0.442832\pi$
$79$	9.77106e9	0.357267	0.178633	+	0.983916i	$0.442832\pi$
$80$	6.11230e9	0.208550
$81$	−3.87236e10	−1.23398
$82$	−7.10829e9	−0.211733
$83$	−1.67805e10	−0.467602	−0.233801	−	0.972285i	$-0.575116\pi$
$83$	−1.67805e10	−0.467602	−0.233801	+	0.972285i	$0.575116\pi$
$84$	6.72535e10	1.75460
$85$	1.48873e10	0.363926
$86$	1.96731e10	0.450953
$87$	−1.02070e11	−2.19554
$88$	3.30686e10	0.667977
$89$	−3.63782e10	−0.690551	−0.345275	−	0.938501i	$-0.612214\pi$
$89$	−3.63782e10	−0.690551	−0.345275	+	0.938501i	$0.612214\pi$
$90$	3.49733e9	0.0624314
$91$	1.43867e10	0.241677
$92$	−1.17584e10	−0.186001
$93$	5.97355e10	0.890381
$94$	1.54929e10	0.217736
$95$	−2.91901e10	−0.387040
$96$	8.63700e10	1.08111
$97$	5.73626e10	0.678241	0.339120	−	0.940743i	$-0.389871\pi$
$97$	5.73626e10	0.678241	0.339120	+	0.940743i	$0.389871\pi$
$98$	4.05364e10	0.453004
$99$	−6.37004e10	−0.673207
$100$	8.10005e10	0.810005
$101$	1.17247e11	1.11003	0.555014	−	0.831841i	$-0.312713\pi$
$101$	1.17247e11	1.11003	0.555014	+	0.831841i	$0.312713\pi$
$102$	5.60816e10	0.502944
$103$	−8.05838e10	−0.684925	−0.342463	−	0.939531i	$-0.611261\pi$
$103$	−8.05838e10	−0.684925	−0.342463	+	0.939531i	$0.611261\pi$
$104$	1.20875e10	0.0974209
$105$	−7.80051e10	−0.596461
$106$	−5.43364e10	−0.394374
$107$	5.11195e10	0.352351	0.176176	−	0.984359i	$-0.443627\pi$
$107$	5.11195e10	0.352351	0.176176	+	0.984359i	$0.443627\pi$
$108$	6.48721e10	0.424842
$109$	−2.34009e11	−1.45675	−0.728377	−	0.685177i	$-0.759725\pi$
$109$	−2.34009e11	−1.45675	−0.728377	+	0.685177i	$0.759725\pi$
$110$	−1.80832e10	−0.107057
$111$	1.98806e11	1.11984
$112$	−1.97829e11	−1.06070
$113$	−9.72965e10	−0.496782	−0.248391	−	0.968660i	$-0.579902\pi$
$113$	−9.72965e10	−0.496782	−0.248391	+	0.968660i	$0.579902\pi$
$114$	−1.09961e11	−0.534888
$115$	1.36382e10	0.0632293
$116$	3.47379e11	1.53562
$117$	−2.32842e10	−0.0981837
$118$	1.47492e10	0.0593495
$119$	−4.81840e11	−1.85095
$120$	−6.55384e10	−0.240436
$121$	4.40555e10	0.154412
$122$	−8.71130e7	−0.000291812 0
$123$	−2.56596e11	−0.821811
$124$	−2.03300e11	−0.622758
$125$	−1.97413e11	−0.578591
$126$	−1.13194e11	−0.317530
$127$	−3.16122e11	−0.849051	−0.424525	−	0.905416i	$-0.639559\pi$
$127$	−3.16122e11	−0.849051	−0.424525	+	0.905416i	$0.639559\pi$
$128$	−3.81796e11	−0.982146
$129$	7.10163e11	1.75031
$130$	−6.60990e9	−0.0156137
$131$	−2.48312e11	−0.562348	−0.281174	−	0.959657i	$-0.590724\pi$
$131$	−2.48312e11	−0.562348	−0.281174	+	0.959657i	$0.590724\pi$
$132$	5.62797e11	1.22235
$133$	9.44760e11	1.96851
$134$	2.86813e11	0.573486
$135$	−7.52430e10	−0.144421
$136$	−4.04832e11	−0.746125
$137$	2.69695e11	0.477429	0.238715	−	0.971090i	$-0.423274\pi$
$137$	2.69695e11	0.477429	0.238715	+	0.971090i	$0.423274\pi$
$138$	5.13760e10	0.0873826
$139$	−9.35942e11	−1.52992	−0.764958	−	0.644081i	$-0.777240\pi$
$139$	−9.35942e11	−1.52992	−0.764958	+	0.644081i	$0.777240\pi$
$140$	2.65478e11	0.417181
$141$	5.59265e11	0.845109
$142$	−7.84517e10	−0.114029
$143$	1.20393e11	0.168365
$144$	3.20176e11	0.430919
$145$	−4.02914e11	−0.522021
$146$	−3.66287e11	−0.456962
$147$	1.46329e12	1.75827
$148$	−6.76607e11	−0.783244
$149$	−1.52904e12	−1.70566	−0.852831	−	0.522186i	$-0.825117\pi$
$149$	−1.52904e12	−1.70566	−0.852831	+	0.522186i	$0.825117\pi$
$150$	−3.53915e11	−0.380538
$151$	4.04164e11	0.418972	0.209486	−	0.977812i	$-0.432821\pi$
$151$	4.04164e11	0.418972	0.209486	+	0.977812i	$0.432821\pi$
$152$	7.93770e11	0.793514
$153$	7.79832e11	0.751966
$154$	5.85276e11	0.544499
$155$	2.35801e11	0.211701
$156$	2.05718e11	0.178273
$157$	1.85179e12	1.54933	0.774665	−	0.632372i	$-0.217919\pi$
$157$	1.85179e12	1.54933	0.774665	+	0.632372i	$0.217919\pi$
$158$	1.45298e11	0.117394
$159$	−1.96144e12	−1.53071
$160$	3.40939e11	0.257050
$161$	−4.41410e11	−0.321588
$162$	−5.75830e11	−0.405472
$163$	1.39551e12	0.949950	0.474975	−	0.879999i	$-0.342457\pi$
$163$	1.39551e12	0.949950	0.474975	+	0.879999i	$0.342457\pi$
$164$	8.73285e11	0.574798
$165$	−6.52770e11	−0.415526
$166$	−2.49530e11	−0.153649
$167$	2.13124e12	1.26967	0.634836	−	0.772647i	$-0.281068\pi$
$167$	2.13124e12	1.26967	0.634836	+	0.772647i	$0.281068\pi$
$168$	2.12120e12	1.22287
$169$	−1.74815e12	−0.975445
$170$	2.21378e11	0.119582
$171$	−1.52905e12	−0.799727
$172$	−2.41693e12	−1.22421
$173$	1.26158e12	0.618960	0.309480	−	0.950906i	$-0.399845\pi$
$173$	1.26158e12	0.618960	0.309480	+	0.950906i	$0.399845\pi$
$174$	−1.51780e12	−0.721431
$175$	3.04076e12	1.40046
$176$	−1.65549e12	−0.738938
$177$	5.32418e11	0.230356
$178$	−5.40952e11	−0.226907
$179$	−4.34895e12	−1.76886	−0.884429	−	0.466674i	$-0.845452\pi$
$179$	−4.34895e12	−1.76886	−0.884429	+	0.466674i	$0.845452\pi$
$180$	−4.29663e11	−0.169484
$181$	−4.47831e12	−1.71349	−0.856747	−	0.515737i	$-0.827518\pi$
$181$	−4.47831e12	−1.71349	−0.856747	+	0.515737i	$0.827518\pi$
$182$	2.13934e11	0.0794123
$183$	−3.14462e9	−0.00113263
$184$	−3.70864e11	−0.129633
$185$	7.84774e11	0.266256
$186$	8.88281e11	0.292569
$187$	−4.03218e12	−1.28947
$188$	−1.90337e12	−0.591093
$189$	2.43530e12	0.734534
$190$	−4.34064e11	−0.127177
$191$	1.33967e12	0.381341	0.190671	−	0.981654i	$-0.438934\pi$
$191$	1.33967e12	0.381341	0.190671	+	0.981654i	$0.438934\pi$
$192$	−1.88684e12	−0.521888
$193$	−3.43747e12	−0.924002	−0.462001	−	0.886879i	$-0.652868\pi$
$193$	−3.43747e12	−0.924002	−0.462001	+	0.886879i	$0.652868\pi$
$194$	8.52995e11	0.222862
$195$	−2.38605e11	−0.0606023
$196$	−4.98008e12	−1.22978
$197$	4.01962e12	0.965207	0.482603	−	0.875839i	$-0.339691\pi$
$197$	4.01962e12	0.965207	0.482603	+	0.875839i	$0.339691\pi$
$198$	−9.47240e11	−0.221208
$199$	1.99472e12	0.453097	0.226548	−	0.974000i	$-0.427256\pi$
$199$	1.99472e12	0.453097	0.226548	+	0.974000i	$0.427256\pi$
$200$	2.55478e12	0.564533
$201$	1.03534e13	2.22590
$202$	1.74349e12	0.364743
$203$	1.30406e13	2.65503
$204$	−6.88988e12	−1.36536
$205$	−1.01289e12	−0.195397
$206$	−1.19830e12	−0.225059
$207$	7.14399e11	0.130648
$208$	−6.05128e11	−0.107770
$209$	7.90604e12	1.37137
$210$	−1.15995e12	−0.195990
$211$	−4.55008e12	−0.748972	−0.374486	−	0.927233i	$-0.622181\pi$
$211$	−4.55008e12	−0.748972	−0.374486	+	0.927233i	$0.622181\pi$
$212$	6.67547e12	1.07062
$213$	−2.83196e12	−0.442588
$214$	7.60160e11	0.115779
$215$	2.80332e12	0.416160
$216$	2.04609e12	0.296093
$217$	−7.63189e12	−1.07672
$218$	−3.47976e12	−0.478673
$219$	−1.32223e13	−1.77363
$220$	2.22160e12	0.290631
$221$	−1.47387e12	−0.188062
$222$	2.95630e12	0.367965
$223$	−6.35776e12	−0.772018	−0.386009	−	0.922495i	$-0.626147\pi$
$223$	−6.35776e12	−0.772018	−0.386009	+	0.922495i	$0.626147\pi$
$224$	−1.10348e13	−1.30737
$225$	−4.92131e12	−0.568953
$226$	−1.44682e12	−0.163237
$227$	1.62023e13	1.78416	0.892080	−	0.451878i	$-0.149246\pi$
$227$	1.62023e13	1.78416	0.892080	+	0.451878i	$0.149246\pi$
$228$	1.35092e13	1.45207
$229$	2.41365e12	0.253267	0.126633	−	0.991950i	$-0.459583\pi$
$229$	2.41365e12	0.253267	0.126633	+	0.991950i	$0.459583\pi$
$230$	2.02803e11	0.0207764
$231$	2.11274e13	2.11339
$232$	1.09565e13	1.07025
$233$	−1.51989e13	−1.44995	−0.724976	−	0.688774i	$-0.758149\pi$
$233$	−1.51989e13	−1.44995	−0.724976	+	0.688774i	$0.758149\pi$
$234$	−3.46242e11	−0.0322621
$235$	2.20766e12	0.200937
$236$	−1.81200e12	−0.161118
$237$	5.24499e12	0.455647
$238$	−7.16507e12	−0.608201
$239$	1.02197e13	0.847717	0.423858	−	0.905728i	$-0.360675\pi$
$239$	1.02197e13	0.847717	0.423858	+	0.905728i	$0.360675\pi$
$240$	3.28101e12	0.265978
$241$	−6.38405e12	−0.505827	−0.252914	−	0.967489i	$-0.581389\pi$
$241$	−6.38405e12	−0.505827	−0.252914	+	0.967489i	$0.581389\pi$
$242$	6.55115e11	0.0507380
$243$	−1.44959e13	−1.09752
$244$	1.07022e10	0.000792191 0
$245$	5.77623e12	0.418052
$246$	−3.81565e12	−0.270038
$247$	2.88987e12	0.200007
$248$	−6.41217e12	−0.434031
$249$	−9.00759e12	−0.596365
$250$	−2.93558e12	−0.190118
$251$	4.94359e12	0.313211	0.156605	−	0.987661i	$-0.449945\pi$
$251$	4.94359e12	0.313211	0.156605	+	0.987661i	$0.449945\pi$
$252$	1.39064e13	0.862007
$253$	−3.69385e12	−0.224035
$254$	−4.70080e12	−0.278989
$255$	7.99134e12	0.464140
$256$	1.52142e12	0.0864829
$257$	−2.89848e13	−1.61264	−0.806320	−	0.591480i	$-0.798544\pi$
$257$	−2.89848e13	−1.61264	−0.806320	+	0.591480i	$0.798544\pi$
$258$	1.05603e13	0.575132
$259$	−2.53998e13	−1.35420
$260$	8.12056e11	0.0423870
$261$	−2.11056e13	−1.07863
$262$	−3.69246e12	−0.184782
$263$	−3.37198e12	−0.165245	−0.0826224	−	0.996581i	$-0.526330\pi$
$263$	−3.37198e12	−0.165245	−0.0826224	+	0.996581i	$0.526330\pi$
$264$	1.77508e13	0.851917
$265$	−7.74266e12	−0.363947
$266$	1.40488e13	0.646831
$267$	−1.95274e13	−0.880708
$268$	−3.52363e13	−1.55686
$269$	−1.90487e13	−0.824572	−0.412286	−	0.911055i	$-0.635269\pi$
$269$	−1.90487e13	−0.824572	−0.412286	+	0.911055i	$0.635269\pi$
$270$	−1.11888e12	−0.0474551
$271$	2.01073e13	0.835647	0.417823	−	0.908528i	$-0.362793\pi$
$271$	2.01073e13	0.835647	0.417823	+	0.908528i	$0.362793\pi$
$272$	2.02669e13	0.825388
$273$	7.72264e12	0.308227
$274$	4.01042e12	0.156878
$275$	2.54459e13	0.975638
$276$	−6.31177e12	−0.237220
$277$	−3.85765e13	−1.42130	−0.710648	−	0.703548i	$-0.751598\pi$
$277$	−3.85765e13	−1.42130	−0.710648	+	0.703548i	$0.751598\pi$
$278$	−1.39177e13	−0.502713
$279$	1.23518e13	0.437429
$280$	8.37329e12	0.290755
$281$	−1.54736e13	−0.526873	−0.263436	−	0.964677i	$-0.584856\pi$
$281$	−1.54736e13	−0.526873	−0.263436	+	0.964677i	$0.584856\pi$
$282$	8.31641e12	0.277694
$283$	−1.52107e13	−0.498110	−0.249055	−	0.968489i	$-0.580120\pi$
$283$	−1.52107e13	−0.498110	−0.249055	+	0.968489i	$0.580120\pi$
$284$	9.63815e12	0.309558
$285$	−1.56689e13	−0.493619
$286$	1.79027e12	0.0553229
$287$	3.27831e13	0.993801
$288$	1.78592e13	0.531132
$289$	1.50908e13	0.440326
$290$	−5.99142e12	−0.171530
$291$	3.07915e13	0.865008
$292$	4.50000e13	1.24053
$293$	6.67427e13	1.80564	0.902822	−	0.430014i	$-0.141491\pi$
$293$	6.67427e13	1.80564	0.902822	+	0.430014i	$0.141491\pi$
$294$	2.17595e13	0.577747
$295$	2.10168e12	0.0547704
$296$	−2.13404e13	−0.545882
$297$	2.03793e13	0.511715
$298$	−2.27371e13	−0.560462
$299$	−1.35020e12	−0.0326744
$300$	4.34801e13	1.03306
$301$	−9.07314e13	−2.11662
$302$	6.01002e12	0.137670
$303$	6.29367e13	1.41570
$304$	−3.97380e13	−0.877812
$305$	−1.24131e10	−0.000269298 0
$306$	1.15963e13	0.247088
$307$	−9.49792e12	−0.198778	−0.0993888	−	0.995049i	$-0.531689\pi$
$307$	−9.49792e12	−0.198778	−0.0993888	+	0.995049i	$0.531689\pi$
$308$	−7.19038e13	−1.47817
$309$	−4.32564e13	−0.873533
$310$	3.50642e12	0.0695625
$311$	1.92082e13	0.374373	0.187186	−	0.982324i	$-0.440063\pi$
$311$	1.92082e13	0.374373	0.187186	+	0.982324i	$0.440063\pi$
$312$	6.48841e12	0.124248
$313$	9.70177e13	1.82540	0.912698	−	0.408635i	$-0.133995\pi$
$313$	9.70177e13	1.82540	0.912698	+	0.408635i	$0.133995\pi$
$314$	2.75366e13	0.509093
$315$	−1.61295e13	−0.293031
$316$	−1.78505e13	−0.318692
$317$	−2.10018e13	−0.368494	−0.184247	−	0.982880i	$-0.558985\pi$
$317$	−2.10018e13	−0.368494	−0.184247	+	0.982880i	$0.558985\pi$
$318$	−2.91672e13	−0.502973
$319$	1.09128e14	1.84964
$320$	−7.44815e12	−0.124086
$321$	2.74404e13	0.449378
$322$	−6.56387e12	−0.105670
$323$	−9.67872e13	−1.53181
$324$	7.07432e13	1.10075
$325$	9.30118e12	0.142292
$326$	2.07516e13	0.312143
$327$	−1.25613e14	−1.85790
$328$	2.75437e13	0.400605
$329$	−7.14526e13	−1.02198
$330$	−9.70685e12	−0.136537
$331$	−3.65032e12	−0.0504983	−0.0252491	−	0.999681i	$-0.508038\pi$
$331$	−3.65032e12	−0.0504983	−0.0252491	+	0.999681i	$0.508038\pi$
$332$	3.06559e13	0.417114
$333$	4.11083e13	0.550156
$334$	3.16920e13	0.417200
$335$	4.08694e13	0.529239
$336$	−1.06192e14	−1.35278
$337$	−1.07930e14	−1.35263	−0.676314	−	0.736613i	$-0.736424\pi$
$337$	−1.07930e14	−1.35263	−0.676314	+	0.736613i	$0.736424\pi$
$338$	−2.59955e13	−0.320520
$339$	−5.22276e13	−0.633581
$340$	−2.71973e13	−0.324632
$341$	−6.38659e13	−0.750102
$342$	−2.27373e13	−0.262782
$343$	−5.13452e13	−0.583958
$344$	−7.62308e13	−0.853216
$345$	7.32081e12	0.0806407
$346$	1.87601e13	0.203384
$347$	−2.46710e13	−0.263253	−0.131627	−	0.991299i	$-0.542020\pi$
$347$	−2.46710e13	−0.263253	−0.131627	+	0.991299i	$0.542020\pi$
$348$	1.86469e14	1.95849
$349$	1.80182e14	1.86282	0.931409	−	0.363973i	$-0.118580\pi$
$349$	1.80182e14	1.86282	0.931409	+	0.363973i	$0.118580\pi$
$350$	4.52168e13	0.460177
$351$	7.44918e12	0.0746310
$352$	−9.23421e13	−0.910784
$353$	1.75917e14	1.70823	0.854117	−	0.520081i	$-0.174098\pi$
$353$	1.75917e14	1.70823	0.854117	+	0.520081i	$0.174098\pi$
$354$	7.91719e12	0.0756925
$355$	−1.11790e13	−0.105231
$356$	6.64584e13	0.615992
$357$	−2.58646e14	−2.36064
$358$	−6.46700e13	−0.581228
$359$	−1.07222e14	−0.948993	−0.474496	−	0.880257i	$-0.657370\pi$
$359$	−1.07222e14	−0.948993	−0.474496	+	0.880257i	$0.657370\pi$
$360$	−1.35517e13	−0.118122
$361$	7.32840e13	0.629100
$362$	−6.65936e13	−0.563035
$363$	2.36484e13	0.196932
$364$	−2.62828e13	−0.215583
$365$	−5.21940e13	−0.421706
$366$	−4.67612e10	−0.000372169 0
$367$	3.35932e13	0.263384	0.131692	−	0.991291i	$-0.457959\pi$
$367$	3.35932e13	0.263384	0.131692	+	0.991291i	$0.457959\pi$
$368$	1.85664e13	0.143405
$369$	−5.30577e13	−0.403742
$370$	1.16698e13	0.0874889
$371$	2.50597e14	1.85106
$372$	−1.09129e14	−0.794246
$373$	7.35501e13	0.527454	0.263727	−	0.964597i	$-0.415048\pi$
$373$	7.35501e13	0.527454	0.263727	+	0.964597i	$0.415048\pi$
$374$	−5.99594e13	−0.423705
$375$	−1.05969e14	−0.737917
$376$	−6.00331e13	−0.411963
$377$	3.98891e13	0.269760
$378$	3.62134e13	0.241360
$379$	2.26744e14	1.48943	0.744714	−	0.667384i	$-0.232586\pi$
$379$	2.26744e14	1.48943	0.744714	+	0.667384i	$0.232586\pi$
$380$	5.33267e13	0.345251
$381$	−1.69690e14	−1.08285
$382$	1.99212e13	0.125305
$383$	1.59732e14	0.990375	0.495187	−	0.868786i	$-0.335099\pi$
$383$	1.59732e14	0.990375	0.495187	+	0.868786i	$0.335099\pi$
$384$	−2.04943e14	−1.25260
$385$	8.33989e13	0.502489
$386$	−5.11159e13	−0.303617
$387$	1.46844e14	0.859897
$388$	−1.04794e14	−0.605011
$389$	3.08043e14	1.75343	0.876714	−	0.481012i	$-0.159731\pi$
$389$	3.08043e14	1.75343	0.876714	+	0.481012i	$0.159731\pi$
$390$	−3.54812e12	−0.0199133
$391$	4.52208e13	0.250246
$392$	−1.57073e14	−0.857096
$393$	−1.33291e14	−0.717202
$394$	5.97726e13	0.317156
$395$	2.07042e13	0.108337
$396$	1.16373e14	0.600520
$397$	3.32327e14	1.69129	0.845645	−	0.533746i	$-0.179216\pi$
$397$	3.32327e14	1.69129	0.845645	+	0.533746i	$0.179216\pi$
$398$	2.96620e13	0.148883
$399$	5.07136e14	2.51058
$400$	−1.27899e14	−0.624505
$401$	8.82588e13	0.425074	0.212537	−	0.977153i	$-0.431827\pi$
$401$	8.82588e13	0.425074	0.212537	+	0.977153i	$0.431827\pi$
$402$	1.53958e14	0.731407
$403$	−2.33447e13	−0.109398
$404$	−2.14195e14	−0.990177
$405$	−8.20527e13	−0.374189
$406$	1.93917e14	0.872414
$407$	−2.12553e14	−0.943405
$408$	−2.17309e14	−0.951585
$409$	−1.46893e14	−0.634633	−0.317317	−	0.948320i	$-0.602782\pi$
$409$	−1.46893e14	−0.634633	−0.317317	+	0.948320i	$0.602782\pi$
$410$	−1.50620e13	−0.0642053
$411$	1.44769e14	0.608899
$412$	1.47217e14	0.610973
$413$	−6.80225e13	−0.278566
$414$	1.06233e13	0.0429296
$415$	−3.55568e13	−0.141794
$416$	−3.37535e13	−0.132833
$417$	−5.02402e14	−1.95121
$418$	1.17565e14	0.450617
$419$	−1.82043e14	−0.688647	−0.344323	−	0.938851i	$-0.611892\pi$
$419$	−1.82043e14	−0.688647	−0.344323	+	0.938851i	$0.611892\pi$
$420$	1.42506e14	0.532061
$421$	−3.53391e14	−1.30228	−0.651139	−	0.758958i	$-0.725709\pi$
$421$	−3.53391e14	−1.30228	−0.651139	+	0.758958i	$0.725709\pi$
$422$	−6.76608e13	−0.246104
$423$	1.15642e14	0.415188
$424$	2.10547e14	0.746168
$425$	−3.11514e14	−1.08978
$426$	−4.21120e13	−0.145430
$427$	4.01761e11	0.00136966
$428$	−9.33890e13	−0.314308
$429$	6.46253e13	0.214727
$430$	4.16860e13	0.136746
$431$	−8.37205e13	−0.271148	−0.135574	−	0.990767i	$-0.543288\pi$
$431$	−8.37205e13	−0.271148	−0.135574	+	0.990767i	$0.543288\pi$
$432$	−1.02432e14	−0.327549
$433$	2.73556e13	0.0863700	0.0431850	−	0.999067i	$-0.486250\pi$
$433$	2.73556e13	0.0863700	0.0431850	+	0.999067i	$0.486250\pi$
$434$	−1.13488e14	−0.353799
$435$	−2.16279e14	−0.665770
$436$	4.27505e14	1.29947
$437$	−8.86661e13	−0.266140
$438$	−1.96618e14	−0.582796
$439$	1.60445e14	0.469646	0.234823	−	0.972038i	$-0.424549\pi$
$439$	1.60445e14	0.469646	0.234823	+	0.972038i	$0.424549\pi$
$440$	7.00702e13	0.202555
$441$	3.02572e14	0.863807
$442$	−2.19168e13	−0.0617952
$443$	3.29401e14	0.917284	0.458642	−	0.888621i	$-0.348336\pi$
$443$	3.29401e14	0.917284	0.458642	+	0.888621i	$0.348336\pi$
$444$	−3.63195e14	−0.998926
$445$	−7.70829e13	−0.209401
$446$	−9.45414e13	−0.253677
$447$	−8.20768e14	−2.17535
$448$	2.41065e14	0.631110
$449$	−4.42737e14	−1.14496	−0.572481	−	0.819918i	$-0.694019\pi$
$449$	−4.42737e14	−1.14496	−0.572481	+	0.819918i	$0.694019\pi$
$450$	−7.31810e13	−0.186952
$451$	2.74339e14	0.692335
$452$	1.77749e14	0.443144
$453$	2.16951e14	0.534344
$454$	2.40932e14	0.586255
$455$	3.04845e13	0.0732853
$456$	4.26086e14	1.01202
$457$	−1.58449e14	−0.371834	−0.185917	−	0.982565i	$-0.559526\pi$
$457$	−1.58449e14	−0.371834	−0.185917	+	0.982565i	$0.559526\pi$
$458$	3.58915e13	0.0832207
$459$	−2.49487e14	−0.571582
$460$	−2.49153e13	−0.0564024
$461$	−2.05796e14	−0.460343	−0.230172	−	0.973150i	$-0.573929\pi$
$461$	−2.05796e14	−0.460343	−0.230172	+	0.973150i	$0.573929\pi$
$462$	3.14169e14	0.694437
$463$	4.02346e14	0.878828	0.439414	−	0.898285i	$-0.355186\pi$
$463$	4.02346e14	0.878828	0.439414	+	0.898285i	$0.355186\pi$
$464$	−5.48507e14	−1.18395
$465$	1.26575e14	0.269997
$466$	−2.26011e14	−0.476438
$467$	5.67176e14	1.18161	0.590806	−	0.806814i	$-0.298810\pi$
$467$	5.67176e14	1.18161	0.590806	+	0.806814i	$0.298810\pi$
$468$	4.25374e13	0.0875827
$469$	−1.32277e15	−2.69174
$470$	3.28284e13	0.0660255
$471$	9.94020e14	1.97597
$472$	−5.71513e13	−0.112291
$473$	−7.59268e14	−1.47455
$474$	7.79943e13	0.149721
$475$	6.10797e14	1.15900
$476$	8.80261e14	1.65110
$477$	−4.05578e14	−0.752010
$478$	1.51970e14	0.278550
$479$	−3.20416e14	−0.580589	−0.290294	−	0.956937i	$-0.593753\pi$
$479$	−3.20416e14	−0.580589	−0.290294	+	0.956937i	$0.593753\pi$
$480$	1.83012e14	0.327833
$481$	−7.76939e13	−0.137591
$482$	−9.49323e13	−0.166209
$483$	−2.36944e14	−0.410144
$484$	−8.04839e13	−0.137740
$485$	1.21547e14	0.205668
$486$	−2.15558e14	−0.360632
$487$	−4.43098e14	−0.732977	−0.366488	−	0.930423i	$-0.619440\pi$
$487$	−4.43098e14	−0.732977	−0.366488	+	0.930423i	$0.619440\pi$
$488$	3.37552e11	0.000552118 0
$489$	7.49093e14	1.21154
$490$	8.58939e13	0.137367
$491$	−1.03321e15	−1.63396	−0.816979	−	0.576668i	$-0.804353\pi$
$491$	−1.03321e15	−1.63396	−0.816979	+	0.576668i	$0.804353\pi$
$492$	4.68769e14	0.733079
$493$	−1.33596e15	−2.06603
$494$	4.29731e13	0.0657201
$495$	−1.34977e14	−0.204141
$496$	3.21009e14	0.480140
$497$	3.61816e14	0.535214
$498$	−1.33945e14	−0.195959
$499$	2.42908e14	0.351471	0.175735	−	0.984437i	$-0.443770\pi$
$499$	2.42908e14	0.351471	0.175735	+	0.984437i	$0.443770\pi$
$500$	3.60650e14	0.516120
$501$	1.14402e15	1.61930
$502$	7.35123e13	0.102918
$503$	1.21106e15	1.67703	0.838514	−	0.544880i	$-0.183425\pi$
$503$	1.21106e15	1.67703	0.838514	+	0.544880i	$0.183425\pi$
$504$	4.38612e14	0.600776
$505$	2.48438e14	0.336601
$506$	−5.49284e13	−0.0736155
$507$	−9.38388e14	−1.24405
$508$	5.77515e14	0.757378
$509$	3.69683e14	0.479603	0.239801	−	0.970822i	$-0.422918\pi$
$509$	3.69683e14	0.479603	0.239801	+	0.970822i	$0.422918\pi$
$510$	1.18833e14	0.152511
$511$	1.68930e15	2.14482
$512$	8.04541e14	1.01056
$513$	4.89178e14	0.607886
$514$	−4.31010e14	−0.529896
$515$	−1.70752e14	−0.207695
$516$	−1.29738e15	−1.56133
$517$	−5.97937e14	−0.711963
$518$	−3.77701e14	−0.444974
$519$	6.77204e14	0.789403
$520$	2.56125e13	0.0295416
$521$	−1.66121e14	−0.189591	−0.0947955	−	0.995497i	$-0.530220\pi$
$521$	−1.66121e14	−0.189591	−0.0947955	+	0.995497i	$0.530220\pi$
$522$	−3.13845e14	−0.354427
$523$	1.37994e14	0.154206	0.0771030	−	0.997023i	$-0.475433\pi$
$523$	1.37994e14	0.154206	0.0771030	+	0.997023i	$0.475433\pi$
$524$	4.53635e14	0.501631
$525$	1.63224e15	1.78611
$526$	−5.01421e13	−0.0542977
$527$	7.81860e14	0.837858
$528$	−8.88650e14	−0.942419
$529$	4.14265e13	0.0434783
$530$	−1.15135e14	−0.119589
$531$	1.10091e14	0.113170
$532$	−1.72596e15	−1.75597
$533$	1.00278e14	0.100973
$534$	−2.90377e14	−0.289391
$535$	1.08319e14	0.106846
$536$	−1.11137e15	−1.08505
$537$	−2.33447e15	−2.25595
$538$	−2.83259e14	−0.270945
$539$	−1.56447e15	−1.48125
$540$	1.37460e14	0.128828
$541$	−1.11102e15	−1.03071	−0.515357	−	0.856976i	$-0.672341\pi$
$541$	−1.11102e15	−1.03071	−0.515357	+	0.856976i	$0.672341\pi$
$542$	2.99001e14	0.274584
$543$	−2.40391e15	−2.18534
$544$	1.13047e15	1.01734
$545$	−4.95848e14	−0.441742
$546$	1.14837e14	0.101280
$547$	−1.17036e15	−1.02185	−0.510925	−	0.859625i	$-0.670697\pi$
$547$	−1.17036e15	−1.02185	−0.510925	+	0.859625i	$0.670697\pi$
$548$	−4.92699e14	−0.425881
$549$	−6.50229e11	−0.000556440 0
$550$	3.78387e14	0.320584
$551$	2.61947e15	2.19725
$552$	−1.99076e14	−0.165331
$553$	−6.70108e14	−0.551006
$554$	−5.73642e14	−0.467022
$555$	4.21257e14	0.339575
$556$	1.70985e15	1.36473
$557$	1.83380e15	1.44927	0.724634	−	0.689133i	$-0.242009\pi$
$557$	1.83380e15	1.44927	0.724634	+	0.689133i	$0.242009\pi$
$558$	1.83675e14	0.143734
$559$	−2.77533e14	−0.215055
$560$	−4.19187e14	−0.321643
$561$	−2.16443e15	−1.64455
$562$	−2.30096e14	−0.173125
$563$	−8.49508e14	−0.632953	−0.316477	−	0.948600i	$-0.602500\pi$
$563$	−8.49508e14	−0.632953	−0.316477	+	0.948600i	$0.602500\pi$
$564$	−1.02171e15	−0.753862
$565$	−2.06165e14	−0.150643
$566$	−2.26187e14	−0.163673
$567$	2.65570e15	1.90315
$568$	3.03991e14	0.215747
$569$	−1.97273e15	−1.38660	−0.693298	−	0.720651i	$-0.743843\pi$
$569$	−1.97273e15	−1.38660	−0.693298	+	0.720651i	$0.743843\pi$
$570$	−2.33000e14	−0.162198
$571$	1.62788e15	1.12234	0.561168	−	0.827702i	$-0.310352\pi$
$571$	1.62788e15	1.12234	0.561168	+	0.827702i	$0.310352\pi$
$572$	−2.19942e14	−0.150186
$573$	7.19118e14	0.486351
$574$	4.87493e14	0.326552
$575$	−2.85376e14	−0.189341
$576$	−3.90151e14	−0.256395
$577$	1.53888e15	1.00170	0.500851	−	0.865534i	$-0.333021\pi$
$577$	1.53888e15	1.00170	0.500851	+	0.865534i	$0.333021\pi$
$578$	2.24404e14	0.144686
$579$	−1.84519e15	−1.17844
$580$	7.36073e14	0.465658
$581$	1.15082e15	0.721173
$582$	4.57878e14	0.284232
$583$	2.09707e15	1.28954
$584$	1.41932e15	0.864586
$585$	−4.93377e13	−0.0297729
$586$	9.92481e14	0.593315
$587$	−1.62013e15	−0.959487	−0.479743	−	0.877409i	$-0.659270\pi$
$587$	−1.62013e15	−0.959487	−0.479743	+	0.877409i	$0.659270\pi$
$588$	−2.67325e15	−1.56842
$589$	−1.53302e15	−0.891074
$590$	3.12525e13	0.0179970
$591$	2.15768e15	1.23100
$592$	1.06835e15	0.603873
$593$	1.70158e15	0.952910	0.476455	−	0.879199i	$-0.341922\pi$
$593$	1.70158e15	0.952910	0.476455	+	0.879199i	$0.341922\pi$
$594$	3.03045e14	0.168144
$595$	−1.02099e15	−0.561276
$596$	2.79336e15	1.52150
$597$	1.07074e15	0.577866
$598$	−2.00778e13	−0.0107364
$599$	−2.43837e15	−1.29197	−0.645983	−	0.763351i	$-0.723553\pi$
$599$	−2.43837e15	−1.29197	−0.645983	+	0.763351i	$0.723553\pi$
$600$	1.37138e15	0.719988
$601$	−3.03781e15	−1.58034	−0.790171	−	0.612887i	$-0.790008\pi$
$601$	−3.03781e15	−1.58034	−0.790171	+	0.612887i	$0.790008\pi$
$602$	−1.34920e15	−0.695497
$603$	2.14084e15	1.09355
$604$	−7.38358e14	−0.373735
$605$	9.33505e13	0.0468233
$606$	9.35884e14	0.465182
$607$	−2.46255e15	−1.21296	−0.606482	−	0.795097i	$-0.707420\pi$
$607$	−2.46255e15	−1.21296	−0.606482	+	0.795097i	$0.707420\pi$
$608$	−2.21655e15	−1.08195
$609$	7.00004e15	3.38615
$610$	−1.84586e11	−8.84883e−5 0
$611$	−2.18562e14	−0.103836
$612$	−1.42466e15	−0.670776
$613$	8.07779e13	0.0376929	0.0188465	−	0.999822i	$-0.494001\pi$
$613$	8.07779e13	0.0376929	0.0188465	+	0.999822i	$0.494001\pi$
$614$	−1.41236e14	−0.0653162
$615$	−5.43710e14	−0.249203
$616$	−2.26787e15	−1.03021
$617$	2.12085e14	0.0954864	0.0477432	−	0.998860i	$-0.484797\pi$
$617$	2.12085e14	0.0954864	0.0477432	+	0.998860i	$0.484797\pi$
$618$	−6.43234e14	−0.287033
$619$	−2.96321e15	−1.31058	−0.655291	−	0.755377i	$-0.727454\pi$
$619$	−2.96321e15	−1.31058	−0.655291	+	0.755377i	$0.727454\pi$
$620$	−4.30780e14	−0.188843
$621$	−2.28553e14	−0.0993080
$622$	2.85630e14	0.123015
$623$	2.49485e15	1.06502
$624$	−3.24826e14	−0.137447
$625$	1.74665e15	0.732597
$626$	1.44268e15	0.599805
$627$	4.24387e15	1.74900
$628$	−3.38299e15	−1.38205
$629$	2.60212e15	1.05378
$630$	−2.39850e14	−0.0962868
$631$	2.88905e15	1.14972	0.574862	−	0.818250i	$-0.305056\pi$
$631$	2.88905e15	1.14972	0.574862	+	0.818250i	$0.305056\pi$
$632$	−5.63012e14	−0.222113
$633$	−2.44243e15	−0.955216
$634$	−3.12302e14	−0.121083
$635$	−6.69840e14	−0.257464
$636$	3.58332e15	1.36543
$637$	−5.71856e14	−0.216033
$638$	1.62276e15	0.607770
$639$	−5.85580e14	−0.217436
$640$	−8.08999e14	−0.297823
$641$	3.29686e15	1.20332	0.601660	−	0.798753i	$-0.294506\pi$
$641$	3.29686e15	1.20332	0.601660	+	0.798753i	$0.294506\pi$
$642$	4.08045e14	0.147661
$643$	−4.92409e15	−1.76671	−0.883355	−	0.468704i	$-0.844721\pi$
$643$	−4.92409e15	−1.76671	−0.883355	+	0.468704i	$0.844721\pi$
$644$	8.06401e14	0.286866
$645$	1.50479e15	0.530758
$646$	−1.43925e15	−0.503335
$647$	4.52679e15	1.56970	0.784851	−	0.619685i	$-0.212740\pi$
$647$	4.52679e15	1.56970	0.784851	+	0.619685i	$0.212740\pi$
$648$	2.23127e15	0.767166
$649$	−5.69233e14	−0.194064
$650$	1.38311e14	0.0467555
$651$	−4.09671e15	−1.37322
$652$	−2.54942e15	−0.847383
$653$	−3.02494e15	−0.996998	−0.498499	−	0.866890i	$-0.666115\pi$
$653$	−3.02494e15	−0.996998	−0.498499	+	0.866890i	$0.666115\pi$
$654$	−1.86790e15	−0.610485
$655$	−5.26156e14	−0.170525
$656$	−1.37891e15	−0.443163
$657$	−2.73404e15	−0.871355
$658$	−1.06252e15	−0.335810
$659$	1.43759e15	0.450573	0.225287	−	0.974293i	$-0.427668\pi$
$659$	1.43759e15	0.450573	0.225287	+	0.974293i	$0.427668\pi$
$660$	1.19253e15	0.370662
$661$	−3.08710e15	−0.951575	−0.475788	−	0.879560i	$-0.657837\pi$
$661$	−3.08710e15	−0.951575	−0.475788	+	0.879560i	$0.657837\pi$
$662$	−5.42811e13	−0.0165932
$663$	−7.91156e14	−0.239849
$664$	9.66899e14	0.290708
$665$	2.00188e15	0.596925
$666$	6.11290e14	0.180775
$667$	−1.22387e15	−0.358957
$668$	−3.89351e15	−1.13258
$669$	−3.41277e15	−0.984608
$670$	6.07738e14	0.173902
$671$	3.36206e12	0.000954182 0
$672$	−5.92333e15	−1.66738
$673$	4.91496e15	1.37226	0.686131	−	0.727478i	$-0.259308\pi$
$673$	4.91496e15	1.37226	0.686131	+	0.727478i	$0.259308\pi$
$674$	−1.60495e15	−0.444459
$675$	1.57444e15	0.432470
$676$	3.19366e15	0.870125
$677$	−1.19609e14	−0.0323240	−0.0161620	−	0.999869i	$-0.505145\pi$
$677$	−1.19609e14	−0.0323240	−0.0161620	+	0.999869i	$0.505145\pi$
$678$	−7.76637e14	−0.208188
$679$	−3.93397e15	−1.04604
$680$	−8.57813e14	−0.226253
$681$	8.69719e15	2.27546
$682$	−9.49702e14	−0.246475
$683$	−6.17337e15	−1.58931	−0.794655	−	0.607061i	$-0.792348\pi$
$683$	−6.17337e15	−1.58931	−0.794655	+	0.607061i	$0.792348\pi$
$684$	2.79338e15	0.713380
$685$	5.71465e14	0.144774
$686$	−7.63515e14	−0.191882
$687$	1.29562e15	0.323009
$688$	3.81630e15	0.943857
$689$	7.66536e14	0.188073
$690$	1.08862e14	0.0264976
$691$	−4.07938e15	−0.985066	−0.492533	−	0.870294i	$-0.663929\pi$
$691$	−4.07938e15	−0.985066	−0.492533	+	0.870294i	$0.663929\pi$
$692$	−2.30476e15	−0.552131
$693$	4.36863e15	1.03827
$694$	−3.66863e14	−0.0865021
$695$	−1.98320e15	−0.463927
$696$	5.88131e15	1.36497
$697$	−3.35851e15	−0.773332
$698$	2.67934e15	0.612102
$699$	−8.15857e15	−1.84922
$700$	−5.55508e15	−1.24926
$701$	7.86951e15	1.75590	0.877948	−	0.478756i	$-0.158912\pi$
$701$	7.86951e15	1.75590	0.877948	+	0.478756i	$0.158912\pi$
$702$	1.10771e14	0.0245229
$703$	−5.10207e15	−1.12071
$704$	2.01730e15	0.439665
$705$	1.18505e15	0.256268
$706$	2.61593e15	0.561307
$707$	−8.04089e15	−1.71197
$708$	−9.72662e14	−0.205485
$709$	1.46428e15	0.306951	0.153475	−	0.988152i	$-0.450953\pi$
$709$	1.46428e15	0.306951	0.153475	+	0.988152i	$0.450953\pi$
$710$	−1.66234e14	−0.0345779
$711$	1.08453e15	0.223852
$712$	2.09612e15	0.429315
$713$	7.16256e14	0.145571
$714$	−3.84612e15	−0.775681
$715$	2.55104e14	0.0510545
$716$	7.94500e15	1.57787
$717$	5.48583e15	1.08115
$718$	−1.59441e15	−0.311829
$719$	−5.54772e15	−1.07673	−0.538364	−	0.842713i	$-0.680957\pi$
$719$	−5.54772e15	−1.07673	−0.538364	+	0.842713i	$0.680957\pi$
$720$	6.78432e14	0.130671
$721$	5.52651e15	1.05635
$722$	1.08975e15	0.206715
$723$	−3.42688e15	−0.645117
$724$	8.18133e15	1.52849
$725$	8.43089e15	1.56320
$726$	3.51658e14	0.0647097
$727$	−3.53006e15	−0.644677	−0.322339	−	0.946624i	$-0.604469\pi$
$727$	−3.53006e15	−0.644677	−0.322339	+	0.946624i	$0.604469\pi$
$728$	−8.28969e14	−0.150250
$729$	−9.21467e14	−0.165759
$730$	−7.76137e14	−0.138568
$731$	9.29510e15	1.64706
$732$	5.74482e12	0.00101034
$733$	−3.42088e15	−0.597126	−0.298563	−	0.954390i	$-0.596507\pi$
$733$	−3.42088e15	−0.597126	−0.298563	+	0.954390i	$0.596507\pi$
$734$	4.99540e14	0.0865449
$735$	3.10061e15	0.533171
$736$	1.03562e15	0.176755
$737$	−1.10693e16	−1.87521
$738$	−7.88982e14	−0.132665
$739$	9.78682e15	1.63342	0.816709	−	0.577050i	$-0.195796\pi$
$739$	9.78682e15	1.63342	0.816709	+	0.577050i	$0.195796\pi$
$740$	−1.43368e15	−0.237509
$741$	1.55125e15	0.255083
$742$	3.72644e15	0.608237
$743$	2.89890e15	0.469672	0.234836	−	0.972035i	$-0.424545\pi$
$743$	2.89890e15	0.469672	0.234836	+	0.972035i	$0.424545\pi$
$744$	−3.44198e15	−0.553550
$745$	−3.23992e15	−0.517220
$746$	1.09371e15	0.173316
$747$	−1.86255e15	−0.292984
$748$	7.36629e15	1.15024
$749$	−3.50582e15	−0.543425
$750$	−1.57579e15	−0.242471
$751$	−4.79339e15	−0.732188	−0.366094	−	0.930578i	$-0.619305\pi$
$751$	−4.79339e15	−0.732188	−0.366094	+	0.930578i	$0.619305\pi$
$752$	3.00540e15	0.455727
$753$	2.65366e15	0.399459
$754$	5.93161e14	0.0886401
$755$	8.56397e14	0.127048
$756$	−4.44898e15	−0.655225
$757$	−6.22282e15	−0.909830	−0.454915	−	0.890535i	$-0.650330\pi$
$757$	−6.22282e15	−0.909830	−0.454915	+	0.890535i	$0.650330\pi$
$758$	3.37173e15	0.489410
$759$	−1.98282e15	−0.285728
$760$	1.68194e15	0.240623
$761$	−6.95542e15	−0.987887	−0.493944	−	0.869494i	$-0.664445\pi$
$761$	−6.95542e15	−0.987887	−0.493944	+	0.869494i	$0.664445\pi$
$762$	−2.52334e15	−0.355814
$763$	1.60485e16	2.24672
$764$	−2.44741e15	−0.340168
$765$	1.65241e15	0.228024
$766$	2.37526e15	0.325426
$767$	−2.08070e14	−0.0283032
$768$	8.16682e14	0.110298
$769$	−2.09625e15	−0.281092	−0.140546	−	0.990074i	$-0.544886\pi$
$769$	−2.09625e15	−0.281092	−0.140546	+	0.990074i	$0.544886\pi$
$770$	1.24016e15	0.165112
$771$	−1.55587e16	−2.05671
$772$	6.27982e15	0.824237
$773$	−6.55473e15	−0.854216	−0.427108	−	0.904201i	$-0.640468\pi$
$773$	−6.55473e15	−0.854216	−0.427108	+	0.904201i	$0.640468\pi$
$774$	2.18361e15	0.282553
$775$	−4.93410e15	−0.633940
$776$	−3.30525e15	−0.421662
$777$	−1.36343e16	−1.72710
$778$	4.58067e15	0.576157
$779$	6.58515e15	0.822450
$780$	4.35902e14	0.0540591
$781$	3.02778e15	0.372858
$782$	6.72445e14	0.0822280
$783$	6.75217e15	0.819887
$784$	7.86348e15	0.948148
$785$	3.92382e15	0.469814
$786$	−1.98207e15	−0.235665
$787$	−1.24777e16	−1.47324	−0.736620	−	0.676307i	$-0.763579\pi$
$787$	−1.24777e16	−1.47324	−0.736620	+	0.676307i	$0.763579\pi$
$788$	−7.34334e15	−0.860992
$789$	−1.81004e15	−0.210748
$790$	3.07877e14	0.0355982
$791$	6.67268e15	0.766178
$792$	3.67044e15	0.418533
$793$	1.22892e12	0.000139162 0
$794$	4.94178e15	0.555739
$795$	−4.15617e15	−0.464167
$796$	−3.64411e15	−0.404176
$797$	3.38386e15	0.372728	0.186364	−	0.982481i	$-0.440330\pi$
$797$	3.38386e15	0.372728	0.186364	+	0.982481i	$0.440330\pi$
$798$	7.54124e15	0.824948
$799$	7.32006e15	0.795257
$800$	−7.13408e15	−0.769739
$801$	−4.03778e15	−0.432677
$802$	1.31243e15	0.139674
$803$	1.41365e16	1.49420
$804$	−1.89144e16	−1.98557
$805$	−9.35318e14	−0.0975174
$806$	−3.47142e14	−0.0359471
$807$	−1.02251e16	−1.05163
$808$	−6.75580e15	−0.690104
$809$	−1.06483e16	−1.08035	−0.540173	−	0.841554i	$-0.681641\pi$
$809$	−1.06483e16	−1.08035	−0.540173	+	0.841554i	$0.681641\pi$
$810$	−1.22014e15	−0.122954
$811$	8.28981e15	0.829716	0.414858	−	0.909886i	$-0.363831\pi$
$811$	8.28981e15	0.829716	0.414858	+	0.909886i	$0.363831\pi$
$812$	−2.38236e16	−2.36837
$813$	1.07934e16	1.06576
$814$	−3.16072e15	−0.309993
$815$	2.95699e15	0.288060
$816$	1.08790e16	1.05267
$817$	−1.82252e16	−1.75167
$818$	−2.18433e15	−0.208534
$819$	1.59685e15	0.151427
$820$	1.85043e15	0.174300
$821$	−2.00661e16	−1.87748	−0.938741	−	0.344624i	$-0.888006\pi$
$821$	−2.00661e16	−1.87748	−0.938741	+	0.344624i	$0.888006\pi$
$822$	2.15275e15	0.200077
$823$	1.00366e16	0.926592	0.463296	−	0.886203i	$-0.346667\pi$
$823$	1.00366e16	0.926592	0.463296	+	0.886203i	$0.346667\pi$
$824$	4.64327e15	0.425818
$825$	1.36591e16	1.24430
$826$	−1.01151e15	−0.0915336
$827$	9.11917e15	0.819738	0.409869	−	0.912144i	$-0.365574\pi$
$827$	9.11917e15	0.819738	0.409869	+	0.912144i	$0.365574\pi$
$828$	−1.30512e15	−0.116542
$829$	1.06230e16	0.942316	0.471158	−	0.882049i	$-0.343836\pi$
$829$	1.06230e16	0.942316	0.471158	+	0.882049i	$0.343836\pi$
$830$	−5.28738e14	−0.0465920
$831$	−2.07074e16	−1.81268
$832$	7.37379e14	0.0641228
$833$	1.91525e16	1.65455
$834$	−7.47084e15	−0.641145
$835$	4.51595e15	0.385011
$836$	−1.44433e16	−1.22330
$837$	−3.95165e15	−0.332497
$838$	−2.70702e15	−0.226282
$839$	−7.33970e15	−0.609520	−0.304760	−	0.952429i	$-0.598576\pi$
$839$	−7.33970e15	−0.609520	−0.304760	+	0.952429i	$0.598576\pi$
$840$	4.49468e15	0.370820
$841$	2.39563e16	1.96355
$842$	−5.25501e15	−0.427914
$843$	−8.30603e15	−0.671957
$844$	8.31243e15	0.668105
$845$	−3.70422e15	−0.295791
$846$	1.71963e15	0.136426
$847$	−3.02136e15	−0.238146
$848$	−1.05405e16	−0.825436
$849$	−8.16495e15	−0.635274
$850$	−4.63229e15	−0.358090
$851$	2.38378e15	0.183086
$852$	5.17364e15	0.394801
$853$	2.62255e16	1.98841	0.994203	−	0.107523i	$-0.0342918\pi$
$853$	2.62255e16	1.98841	0.994203	+	0.107523i	$0.0342918\pi$
$854$	5.97428e12	0.000450057 0
$855$	−3.23994e15	−0.242507
$856$	−2.94552e15	−0.219057
$857$	−5.55008e15	−0.410114	−0.205057	−	0.978750i	$-0.565738\pi$
$857$	−5.55008e15	−0.410114	−0.205057	+	0.978750i	$0.565738\pi$
$858$	9.60994e14	0.0705571
$859$	7.00228e14	0.0510831	0.0255415	−	0.999674i	$-0.491869\pi$
$859$	7.00228e14	0.0510831	0.0255415	+	0.999674i	$0.491869\pi$
$860$	−5.12131e15	−0.371227
$861$	1.75976e16	1.26746
$862$	−1.24494e15	−0.0890964
$863$	−1.74210e16	−1.23884	−0.619419	−	0.785061i	$-0.712632\pi$
$863$	−1.74210e16	−1.23884	−0.619419	+	0.785061i	$0.712632\pi$
$864$	−5.71358e15	−0.403722
$865$	2.67321e15	0.187692
$866$	4.06785e14	0.0283802
$867$	8.10057e15	0.561579
$868$	1.39425e16	0.960468
$869$	−5.60766e15	−0.383860
$870$	−3.21613e15	−0.218765
$871$	−4.04614e15	−0.273490
$872$	1.34837e16	0.905664
$873$	6.36693e15	0.424964
$874$	−1.31849e15	−0.0874506
$875$	1.35388e16	0.892350
$876$	2.41554e16	1.58213
$877$	−1.23403e16	−0.803205	−0.401602	−	0.915814i	$-0.631547\pi$
$877$	−1.23403e16	−0.803205	−0.401602	+	0.915814i	$0.631547\pi$
$878$	2.38585e15	0.154320
$879$	3.58267e16	2.30286
$880$	−3.50788e15	−0.224073
$881$	2.58898e16	1.64347	0.821735	−	0.569870i	$-0.193007\pi$
$881$	2.58898e16	1.64347	0.821735	+	0.569870i	$0.193007\pi$
$882$	4.49932e15	0.283837
$883$	−3.56947e14	−0.0223779	−0.0111890	−	0.999937i	$-0.503562\pi$
$883$	−3.56947e14	−0.0223779	−0.0111890	+	0.999937i	$0.503562\pi$
$884$	2.69258e15	0.167757
$885$	1.12816e15	0.0698525
$886$	4.89827e15	0.301409
$887$	4.35822e15	0.266519	0.133260	−	0.991081i	$-0.457456\pi$
$887$	4.35822e15	0.266519	0.133260	+	0.991081i	$0.457456\pi$
$888$	−1.14553e16	−0.696201
$889$	2.16799e16	1.30948
$890$	−1.14624e15	−0.0688067
$891$	2.22237e16	1.32583
$892$	1.16148e16	0.688663
$893$	−1.43527e16	−0.845767
$894$	−1.22050e16	−0.714796
$895$	−9.21514e15	−0.536383
$896$	2.61839e16	1.51475
$897$	−7.24773e14	−0.0416719
$898$	−6.58361e15	−0.376222
$899$	−2.11604e16	−1.20184
$900$	8.99061e15	0.507523
$901$	−2.56728e16	−1.44041
$902$	4.07948e15	0.227494
$903$	−4.87036e16	−2.69947
$904$	5.60626e15	0.308849
$905$	−9.48925e15	−0.519595
$906$	3.22611e15	0.175580
$907$	−2.75194e15	−0.148867	−0.0744335	−	0.997226i	$-0.523715\pi$
$907$	−2.75194e15	−0.148867	−0.0744335	+	0.997226i	$0.523715\pi$
$908$	−2.95995e16	−1.59152
$909$	1.30138e16	0.695507
$910$	4.53313e14	0.0240808
$911$	−1.35179e15	−0.0713770	−0.0356885	−	0.999363i	$-0.511362\pi$
$911$	−1.35179e15	−0.0713770	−0.0356885	+	0.999363i	$0.511362\pi$
$912$	−2.13309e16	−1.11953
$913$	9.63043e15	0.502408
$914$	−2.35617e15	−0.122181
$915$	−6.66323e12	−0.000343454 0
$916$	−4.40943e15	−0.225921
$917$	1.70294e16	0.867300
$918$	−3.70993e15	−0.187816
$919$	−1.00536e16	−0.505927	−0.252963	−	0.967476i	$-0.581405\pi$
$919$	−1.00536e16	−0.505927	−0.252963	+	0.967476i	$0.581405\pi$
$920$	−7.85836e14	−0.0393096
$921$	−5.09837e15	−0.253515
$922$	−3.06024e15	−0.151264
$923$	1.10674e15	0.0543794
$924$	−3.85971e16	−1.88521
$925$	−1.64212e16	−0.797309
$926$	5.98298e15	0.288773
$927$	−8.94437e15	−0.429152
$928$	−3.05953e16	−1.45929
$929$	1.88670e16	0.894575	0.447288	−	0.894390i	$-0.352390\pi$
$929$	1.88670e16	0.894575	0.447288	+	0.894390i	$0.352390\pi$
$930$	1.88221e15	0.0887179
$931$	−3.75531e16	−1.75963
$932$	2.77664e16	1.29340
$933$	1.03107e16	0.477464
$934$	8.43404e15	0.388265
$935$	−8.54391e15	−0.391015
$936$	1.34164e15	0.0610408
$937$	−2.56131e16	−1.15849	−0.579247	−	0.815152i	$-0.696653\pi$
$937$	−2.56131e16	−1.15849	−0.579247	+	0.815152i	$0.696653\pi$
$938$	−1.96699e16	−0.884477
$939$	5.20780e16	2.32805
$940$	−4.03312e15	−0.179241
$941$	−6.57238e15	−0.290389	−0.145194	−	0.989403i	$-0.546381\pi$
$941$	−6.57238e15	−0.290389	−0.145194	+	0.989403i	$0.546381\pi$
$942$	1.47813e16	0.649281
$943$	−3.07671e15	−0.134361
$944$	2.86113e15	0.124220
$945$	5.16023e15	0.222738
$946$	−1.12905e16	−0.484520
$947$	3.14635e16	1.34240	0.671201	−	0.741276i	$-0.265779\pi$
$947$	3.14635e16	1.34240	0.671201	+	0.741276i	$0.265779\pi$
$948$	−9.58195e15	−0.406451
$949$	5.16729e15	0.217921
$950$	9.08270e15	0.380834
$951$	−1.12735e16	−0.469967
$952$	2.77638e16	1.15073
$953$	−2.72372e16	−1.12241	−0.561205	−	0.827677i	$-0.689662\pi$
$953$	−2.72372e16	−1.12241	−0.561205	+	0.827677i	$0.689662\pi$
$954$	−6.03105e15	−0.247102
$955$	2.83867e15	0.115637
$956$	−1.86702e16	−0.756188
$957$	5.85785e16	2.35897
$958$	−4.76466e15	−0.190775
$959$	−1.84959e16	−0.736331
$960$	−3.99808e15	−0.158256
$961$	−1.30245e16	−0.512606
$962$	−1.15533e15	−0.0452108
$963$	5.67399e15	0.220772
$964$	1.16629e16	0.451213
$965$	−7.28376e15	−0.280192
$966$	−3.52341e15	−0.134769
$967$	2.83130e16	1.07681	0.538407	−	0.842685i	$-0.319026\pi$
$967$	2.83130e16	1.07681	0.538407	+	0.842685i	$0.319026\pi$
$968$	−2.53849e15	−0.0959977
$969$	−5.19543e16	−1.95362
$970$	1.80744e15	0.0675801
$971$	−2.55713e16	−0.950708	−0.475354	−	0.879795i	$-0.657680\pi$
$971$	−2.55713e16	−0.950708	−0.475354	+	0.879795i	$0.657680\pi$
$972$	2.64822e16	0.979017
$973$	6.41877e16	2.35956
$974$	−6.58897e15	−0.240848
$975$	4.99277e15	0.181475
$976$	−1.68987e13	−0.000610771 0
$977$	−1.73935e16	−0.625125	−0.312563	−	0.949897i	$-0.601187\pi$
$977$	−1.73935e16	−0.625125	−0.312563	+	0.949897i	$0.601187\pi$
$978$	1.11392e16	0.398098
$979$	2.08776e16	0.741952
$980$	−1.05524e16	−0.372915
$981$	−2.59737e16	−0.912755
$982$	−1.53641e16	−0.536900
$983$	2.03114e16	0.705822	0.352911	−	0.935657i	$-0.385192\pi$
$983$	2.03114e16	0.705822	0.352911	+	0.935657i	$0.385192\pi$
$984$	1.47852e16	0.510920
$985$	8.51729e15	0.292686
$986$	−1.98661e16	−0.678874
$987$	−3.83549e16	−1.30340
$988$	−5.27943e15	−0.178412
$989$	8.51518e15	0.286164
$990$	−2.00714e15	−0.0670785
$991$	−1.72761e16	−0.574172	−0.287086	−	0.957905i	$-0.592686\pi$
$991$	−1.72761e16	−0.574172	−0.287086	+	0.957905i	$0.592686\pi$
$992$	1.79056e16	0.591800
$993$	−1.95945e15	−0.0644040
$994$	5.38029e15	0.175865
$995$	4.22669e15	0.137396
$996$	1.64557e16	0.531975
$997$	−2.58661e16	−0.831588	−0.415794	−	0.909459i	$-0.636496\pi$
$997$	−2.58661e16	−0.831588	−0.415794	+	0.909459i	$0.636496\pi$
$998$	3.61211e15	0.115489
$999$	−1.31515e16	−0.418183

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.12.a.a.1.5	✓	8
3.2	odd	2		207.12.a.a.1.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.12.a.a.1.5	✓	8	1.1	even	1	trivial
207.12.a.a.1.4		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)

\(f(q)\)	\(=\)	\(q+14.8702 q^{2} +536.788 q^{3} -1826.88 q^{4} +2118.93 q^{5} +7982.17 q^{6} -68580.9 q^{7} -57620.3 q^{8} +110995. q^{9} +O(q^{10})\)	\(q+14.8702 q^{2} +536.788 q^{3} -1826.88 q^{4} +2118.93 q^{5} +7982.17 q^{6} -68580.9 q^{7} -57620.3 q^{8} +110995. q^{9} +31509.0 q^{10} -573905. q^{11} -980645. q^{12} -209778. q^{13} -1.01981e6 q^{14} +1.13742e6 q^{15} +2.88461e6 q^{16} +7.02586e6 q^{17} +1.65052e6 q^{18} -1.37759e7 q^{19} -3.87103e6 q^{20} -3.68134e7 q^{21} -8.53411e6 q^{22} +6.43634e6 q^{23} -3.09299e7 q^{24} -4.43382e7 q^{25} -3.11945e6 q^{26} -3.55098e7 q^{27} +1.25289e8 q^{28} -1.90149e8 q^{29} +1.69137e7 q^{30} +1.11283e8 q^{31} +1.60901e8 q^{32} -3.08066e8 q^{33} +1.04476e8 q^{34} -1.45318e8 q^{35} -2.02773e8 q^{36} +3.70363e8 q^{37} -2.04850e8 q^{38} -1.12606e8 q^{39} -1.22094e8 q^{40} -4.78021e8 q^{41} -5.47424e8 q^{42} +1.32298e9 q^{43} +1.04845e9 q^{44} +2.35190e8 q^{45} +9.57100e7 q^{46} +1.04187e9 q^{47} +1.54843e9 q^{48} +2.72601e9 q^{49} -6.59320e8 q^{50} +3.77140e9 q^{51} +3.83238e8 q^{52} -3.65404e9 q^{53} -5.28040e8 q^{54} -1.21607e9 q^{55} +3.95165e9 q^{56} -7.39472e9 q^{57} -2.82757e9 q^{58} +9.91859e8 q^{59} -2.07792e9 q^{60} -5.85821e6 q^{61} +1.65481e9 q^{62} -7.61210e9 q^{63} -3.51505e9 q^{64} -4.44505e8 q^{65} -4.58101e9 q^{66} +1.92877e10 q^{67} -1.28354e10 q^{68} +3.45495e9 q^{69} -2.16092e9 q^{70} -5.27575e9 q^{71} -6.39555e9 q^{72} -2.46322e10 q^{73} +5.50739e9 q^{74} -2.38002e10 q^{75} +2.51668e10 q^{76} +3.93589e10 q^{77} -1.67448e9 q^{78} +9.77106e9 q^{79} +6.11230e9 q^{80} -3.87236e10 q^{81} -7.10829e9 q^{82} -1.67805e10 q^{83} +6.72535e10 q^{84} +1.48873e10 q^{85} +1.96731e10 q^{86} -1.02070e11 q^{87} +3.30686e10 q^{88} -3.63782e10 q^{89} +3.49733e9 q^{90} +1.43867e10 q^{91} -1.17584e10 q^{92} +5.97355e10 q^{93} +1.54929e10 q^{94} -2.91901e10 q^{95} +8.63700e10 q^{96} +5.73626e10 q^{97} +4.05364e10 q^{98} -6.37004e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 32 q^{2} - 992 q^{3} + 5120 q^{4} - 11466 q^{5} + 34618 q^{6} - 54118 q^{7} - 155568 q^{8} + 648814 q^{9}+O(q^{10}) \) 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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