LMFDB - Embedded newform 2151.2.b.a.2150.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2151,2,Mod(2150,2151)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2151, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("2151.2150");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2151 = 3^{2} \cdot 239 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2151.b (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$17.1758214748$
Analytic rank:	$0$
Dimension:	$80$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2150.19
Character	$\chi$	$=$	2151.2150
Dual form			2151.2.b.a.2150.62

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73293i q^{2} -1.00304 q^{4} -1.48937i q^{5} -1.62864i q^{7} -1.72766i q^{8} +O(q^{10})$	\(q-1.73293i q^{2} -1.00304 q^{4} -1.48937i q^{5} -1.62864i q^{7} -1.72766i q^{8} -2.58096 q^{10} +4.73925i q^{11} -6.66359i q^{13} -2.82232 q^{14} -4.99999 q^{16} -5.60764i q^{17} -0.237326i q^{19} +1.49389i q^{20} +8.21277 q^{22} -0.0913956 q^{23} +2.78179 q^{25} -11.5475 q^{26} +1.63359i q^{28} -0.768311i q^{29} +7.99330 q^{31} +5.20930i q^{32} -9.71763 q^{34} -2.42565 q^{35} +3.00118i q^{37} -0.411269 q^{38} -2.57312 q^{40} -11.7971 q^{41} +6.56850i q^{43} -4.75365i q^{44} +0.158382i q^{46} +5.97086 q^{47} +4.34752 q^{49} -4.82064i q^{50} +6.68384i q^{52} -13.3910 q^{53} +7.05848 q^{55} -2.81374 q^{56} -1.33143 q^{58} -9.35109 q^{59} +2.37159 q^{61} -13.8518i q^{62} -0.972638 q^{64} -9.92452 q^{65} -10.6262 q^{67} +5.62468i q^{68} +4.20347i q^{70} +6.42158i q^{71} -15.9016i q^{73} +5.20084 q^{74} +0.238048i q^{76} +7.71855 q^{77} -6.09288i q^{79} +7.44682i q^{80} +20.4435i q^{82} +1.27230i q^{83} -8.35183 q^{85} +11.3827 q^{86} +8.18781 q^{88} +8.90207 q^{89} -10.8526 q^{91} +0.0916733 q^{92} -10.3471i q^{94} -0.353466 q^{95} -3.35958i q^{97} -7.53394i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 80 q - 80 q^{4}+O(q^{10}) $ 80 * q - 80 * q^4	$ 80 q - 80 q^{4} + 16 q^{10} + 56 q^{16} + 40 q^{22} - 64 q^{25} - 8 q^{31} + 32 q^{34} - 24 q^{40} - 104 q^{49} - 24 q^{55} + 56 q^{58} + 40 q^{61} - 80 q^{64} - 8 q^{67} - 8 q^{85} - 120 q^{88} + 32 q^{91}+O(q^{100}) $ 80 * q - 80 * q^4 + 16 * q^10 + 56 * q^16 + 40 * q^22 - 64 * q^25 - 8 * q^31 + 32 * q^34 - 24 * q^40 - 104 * q^49 - 24 * q^55 + 56 * q^58 + 40 * q^61 - 80 * q^64 - 8 * q^67 - 8 * q^85 - 120 * q^88 + 32 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2151\mathbb{Z}\right)^\times$.

$n$	$479$	$1441$
$\chi(n)$	$-1$	$-1$

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 $

$2$		−	1.73293i		−	1.22537i	−0.790329	−	0.612683i	$-0.790090\pi$
$2$		−	1.73293i		−	1.22537i	0.790329	−	0.612683i	$-0.209910\pi$
$3$		0			0
$3$		0			0
$4$	−1.00304			−0.501520
$5$		−	1.48937i		−	0.666065i	−0.942915	−	0.333032i	$-0.891928\pi$
$5$		−	1.48937i		−	0.666065i	0.942915	−	0.333032i	$-0.108072\pi$
$6$		0			0
$7$		−	1.62864i		−	0.615569i	−0.951456	−	0.307785i	$-0.900412\pi$
$7$		−	1.62864i		−	0.615569i	0.951456	−	0.307785i	$-0.0995877\pi$
$8$		−	1.72766i		−	0.610820i
$9$		0			0
$10$	−2.58096			−0.816173
$11$			4.73925i			1.42894i	0.699668	+	0.714468i	$0.253331\pi$
$11$			4.73925i			1.42894i	−0.699668	+	0.714468i	$0.746669\pi$
$12$		0			0
$13$		−	6.66359i		−	1.84815i	−0.382216	−	0.924073i	$-0.624839\pi$
$13$		−	6.66359i		−	1.84815i	0.382216	−	0.924073i	$-0.375161\pi$
$14$	−2.82232			−0.754297
$15$		0			0
$16$	−4.99999			−1.25000
$17$		−	5.60764i		−	1.36005i	−0.733188	−	0.680026i	$-0.761968\pi$
$17$		−	5.60764i		−	1.36005i	0.733188	−	0.680026i	$-0.238032\pi$
$18$		0			0
$19$		−	0.237326i		−	0.0544464i	−0.999629	−	0.0272232i	$-0.991334\pi$
$19$		−	0.237326i		−	0.0544464i	0.999629	−	0.0272232i	$-0.00866648\pi$
$20$			1.49389i			0.334045i
$21$		0			0
$22$	8.21277			1.75097
$23$	−0.0913956			−0.0190573			−0.00952865	−	0.999955i	$-0.503033\pi$
$23$	−0.0913956			−0.0190573			−0.00952865	+	0.999955i	$0.503033\pi$
$24$		0			0
$25$	2.78179			0.556358
$26$	−11.5475			−2.26465
$27$		0			0
$28$			1.63359i			0.308720i
$29$		−	0.768311i		−	0.142672i	−0.997452	−	0.0713359i	$-0.977274\pi$
$29$		−	0.768311i		−	0.142672i	0.997452	−	0.0713359i	$-0.0227262\pi$
$30$		0			0
$31$	7.99330			1.43564			0.717820	−	0.696229i	$-0.245140\pi$
$31$	7.99330			1.43564			0.717820	+	0.696229i	$0.245140\pi$
$32$			5.20930i			0.920883i
$33$		0			0
$34$	−9.71763			−1.66656
$35$	−2.42565			−0.410009
$36$		0			0
$37$			3.00118i			0.493392i	0.969093	+	0.246696i	$0.0793449\pi$
$37$			3.00118i			0.493392i	−0.969093	+	0.246696i	$0.920655\pi$
$38$	−0.411269			−0.0667167
$39$		0			0
$40$	−2.57312			−0.406846
$41$	−11.7971			−1.84240			−0.921198	−	0.389095i	$-0.872788\pi$
$41$	−11.7971			−1.84240			−0.921198	+	0.389095i	$0.872788\pi$
$42$		0			0
$43$			6.56850i			1.00169i	0.865538	+	0.500843i	$0.166977\pi$
$43$			6.56850i			1.00169i	−0.865538	+	0.500843i	$0.833023\pi$
$44$		−	4.75365i		−	0.716640i
$45$		0			0
$46$			0.158382i			0.0233521i
$47$	5.97086			0.870940			0.435470	−	0.900203i	$-0.356582\pi$
$47$	5.97086			0.870940			0.435470	+	0.900203i	$0.356582\pi$
$48$		0			0
$49$	4.34752			0.621074
$50$		−	4.82064i		−	0.681741i
$51$		0			0
$52$			6.68384i			0.926882i
$53$	−13.3910			−1.83939			−0.919694	−	0.392635i	$-0.871564\pi$
$53$	−13.3910			−1.83939			−0.919694	+	0.392635i	$0.871564\pi$
$54$		0			0
$55$	7.05848			0.951765
$56$	−2.81374			−0.376002
$57$		0			0
$58$	−1.33143			−0.174825
$59$	−9.35109			−1.21741			−0.608704	−	0.793397i	$-0.708310\pi$
$59$	−9.35109			−1.21741			−0.608704	+	0.793397i	$0.708310\pi$
$60$		0			0
$61$	2.37159			0.303651			0.151825	−	0.988407i	$-0.451485\pi$
$61$	2.37159			0.303651			0.151825	+	0.988407i	$0.451485\pi$
$62$		−	13.8518i		−	1.75918i
$63$		0			0
$64$	−0.972638			−0.121580
$65$	−9.92452			−1.23099
$66$		0			0
$67$	−10.6262			−1.29820			−0.649098	−	0.760705i	$-0.724854\pi$
$67$	−10.6262			−1.29820			−0.649098	+	0.760705i	$0.724854\pi$
$68$			5.62468i			0.682093i
$69$		0			0
$70$			4.20347i			0.502411i
$71$			6.42158i			0.762102i	0.924554	+	0.381051i	$0.124438\pi$
$71$			6.42158i			0.762102i	−0.924554	+	0.381051i	$0.875562\pi$
$72$		0			0
$73$		−	15.9016i		−	1.86114i	−0.366114	−	0.930570i	$-0.619312\pi$
$73$		−	15.9016i		−	1.86114i	0.366114	−	0.930570i	$-0.380688\pi$
$74$	5.20084			0.604585
$75$		0			0
$76$			0.238048i			0.0273059i
$77$	7.71855			0.879610
$78$		0			0
$79$		−	6.09288i		−	0.685503i	−0.939426	−	0.342752i	$-0.888641\pi$
$79$		−	6.09288i		−	0.685503i	0.939426	−	0.342752i	$-0.111359\pi$
$80$			7.44682i			0.832580i
$81$		0			0
$82$			20.4435i			2.25761i
$83$			1.27230i			0.139653i	0.997559	+	0.0698265i	$0.0222446\pi$
$83$			1.27230i			0.139653i	−0.997559	+	0.0698265i	$0.977755\pi$
$84$		0			0
$85$	−8.35183			−0.905883
$86$	11.3827			1.22743
$87$		0			0
$88$	8.18781			0.872824
$89$	8.90207			0.943617			0.471809	−	0.881701i	$-0.343601\pi$
$89$	8.90207			0.943617			0.471809	+	0.881701i	$0.343601\pi$
$90$		0			0
$91$	−10.8526			−1.13766
$92$	0.0916733			0.00955761
$93$		0			0
$94$		−	10.3471i		−	1.06722i
$95$	−0.353466			−0.0362648
$96$		0			0
$97$		−	3.35958i		−	0.341114i	−0.985348	−	0.170557i	$-0.945443\pi$
$97$		−	3.35958i		−	0.341114i	0.985348	−	0.170557i	$-0.0545567\pi$
$98$		−	7.53394i		−	0.761043i
$99$		0			0
$100$	−2.79024			−0.279024
$101$		−	11.4243i		−	1.13676i	−0.822766	−	0.568381i	$-0.807570\pi$
$101$		−	11.4243i		−	1.13676i	0.822766	−	0.568381i	$-0.192430\pi$
$102$		0			0
$103$			3.77283i			0.371748i	0.982574	+	0.185874i	$0.0595117\pi$
$103$			3.77283i			0.371748i	−0.982574	+	0.185874i	$0.940488\pi$
$104$	−11.5124			−1.12889
$105$		0			0
$106$			23.2056i			2.25392i
$107$	10.5357			1.01852			0.509260	−	0.860612i	$-0.329919\pi$
$107$	10.5357			1.01852			0.509260	+	0.860612i	$0.329919\pi$
$108$		0			0
$109$	9.37374			0.897841			0.448921	−	0.893572i	$-0.351809\pi$
$109$	9.37374			0.897841			0.448921	+	0.893572i	$0.351809\pi$
$110$		−	12.2318i		−	1.16626i
$111$		0			0
$112$			8.14320i			0.769460i
$113$			11.6054i			1.09174i	0.837868	+	0.545872i	$0.183802\pi$
$113$			11.6054i			1.09174i	−0.837868	+	0.545872i	$0.816198\pi$
$114$		0			0
$115$			0.136121i			0.0126934i
$116$			0.770646i			0.0715527i
$117$		0			0
$118$			16.2048i			1.49177i
$119$	−9.13285			−0.837207
$120$		0			0
$121$	−11.4605			−1.04186
$122$		−	4.10979i		−	0.372083i
$123$		0			0
$124$	−8.01760			−0.720001
$125$		−	11.5899i		−	1.03664i
$126$		0			0
$127$	2.87869			0.255442			0.127721	−	0.991810i	$-0.459234\pi$
$127$	2.87869			0.255442			0.127721	+	0.991810i	$0.459234\pi$
$128$			12.1041i			1.06986i
$129$		0			0
$130$			17.1985i			1.50841i
$131$	−6.94181			−0.606509			−0.303254	−	0.952910i	$-0.598073\pi$
$131$	−6.94181			−0.606509			−0.303254	+	0.952910i	$0.598073\pi$
$132$		0			0
$133$	−0.386520			−0.0335155
$134$			18.4144i			1.59076i
$135$		0			0
$136$	−9.68810			−0.830748
$137$	−12.3270			−1.05317			−0.526583	−	0.850124i	$-0.676527\pi$
$137$	−12.3270			−1.05317			−0.526583	+	0.850124i	$0.676527\pi$
$138$		0			0
$139$			14.1468i			1.19992i	0.800030	+	0.599960i	$0.204817\pi$
$139$			14.1468i			1.19992i	−0.800030	+	0.599960i	$0.795183\pi$
$140$	2.43302			0.205628
$141$		0			0
$142$	11.1281			0.933853
$143$	31.5804			2.64088
$144$		0			0
$145$	−1.14430			−0.0950286
$146$	−27.5563			−2.28058
$147$		0			0
$148$		−	3.01031i		−	0.247446i
$149$	−18.9985			−1.55642			−0.778209	−	0.628005i	$-0.783872\pi$
$149$	−18.9985			−1.55642			−0.778209	+	0.628005i	$0.783872\pi$
$150$		0			0
$151$			12.1119i			0.985651i	0.870128	+	0.492826i	$0.164036\pi$
$151$			12.1119i			0.985651i	−0.870128	+	0.492826i	$0.835964\pi$
$152$	−0.410020			−0.0332570
$153$		0			0
$154$		−	13.3757i		−	1.07784i
$155$		−	11.9050i		−	0.956229i
$156$		0			0
$157$	1.80461			0.144024			0.0720119	−	0.997404i	$-0.477058\pi$
$157$	1.80461			0.144024			0.0720119	+	0.997404i	$0.477058\pi$
$158$	−10.5585			−0.839991
$159$		0			0
$160$	7.75856			0.613368
$161$			0.148851i			0.0117311i
$162$		0			0
$163$	8.90497			0.697491			0.348745	−	0.937218i	$-0.386608\pi$
$163$	8.90497			0.697491			0.348745	+	0.937218i	$0.386608\pi$
$164$	11.8329			0.923997
$165$		0			0
$166$	2.20480			0.171126
$167$	14.8987			1.15290			0.576448	−	0.817134i	$-0.304438\pi$
$167$	14.8987			1.15290			0.576448	+	0.817134i	$0.304438\pi$
$168$		0			0
$169$	−31.4034			−2.41564
$170$			14.4731i			1.11004i
$171$		0			0
$172$		−	6.58846i		−	0.502365i
$173$	−3.78195			−0.287536			−0.143768	−	0.989611i	$-0.545922\pi$
$173$	−3.78195			−0.287536			−0.143768	+	0.989611i	$0.545922\pi$
$174$		0			0
$175$		−	4.53054i		−	0.342477i
$176$		−	23.6962i		−	1.78617i
$177$		0			0
$178$		−	15.4266i		−	1.15628i
$179$	−5.10846			−0.381824			−0.190912	−	0.981607i	$-0.561145\pi$
$179$	−5.10846			−0.381824			−0.190912	+	0.981607i	$0.561145\pi$
$180$		0			0
$181$		−	7.69353i		−	0.571855i	−0.958251	−	0.285928i	$-0.907698\pi$
$181$		−	7.69353i		−	0.571855i	0.958251	−	0.285928i	$-0.0923017\pi$
$182$			18.8068i			1.39405i
$183$		0			0
$184$			0.157901i			0.0116406i
$185$	4.46986			0.328631
$186$		0			0
$187$	26.5760			1.94343
$188$	−5.98901			−0.436793
$189$		0			0
$190$			0.612531i			0.0444377i
$191$	1.01696			0.0735844			0.0367922	−	0.999323i	$-0.488286\pi$
$191$	1.01696			0.0735844			0.0367922	+	0.999323i	$0.488286\pi$
$192$		0			0
$193$	14.6982			1.05800			0.528998	−	0.848623i	$-0.322568\pi$
$193$	14.6982			1.05800			0.528998	+	0.848623i	$0.322568\pi$
$194$	−5.82191			−0.417989
$195$		0			0
$196$	−4.36073			−0.311481
$197$			15.0503i			1.07229i	0.844125	+	0.536146i	$0.180120\pi$
$197$			15.0503i			1.07229i	−0.844125	+	0.536146i	$0.819880\pi$
$198$		0			0
$199$			2.85630i			0.202478i	0.994862	+	0.101239i	$0.0322806\pi$
$199$			2.85630i			0.202478i	−0.994862	+	0.101239i	$0.967719\pi$
$200$		−	4.80599i		−	0.339835i
$201$		0			0
$202$	−19.7975			−1.39295
$203$	−1.25130			−0.0878243
$204$		0			0
$205$			17.5702i			1.22715i
$206$	6.53805			0.455527
$207$		0			0
$208$			33.3179i			2.31018i
$209$	1.12475			0.0778005
$210$		0			0
$211$	−25.7886			−1.77536			−0.887679	−	0.460463i	$-0.847683\pi$
$211$	−25.7886			−1.77536			−0.887679	+	0.460463i	$0.847683\pi$
$212$	13.4317			0.922490
$213$		0			0
$214$		−	18.2575i		−	1.24806i
$215$	9.78290			0.667188
$216$		0			0
$217$		−	13.0182i		−	0.883736i
$218$		−	16.2440i		−	1.10018i
$219$		0			0
$220$	−7.07993			−0.477329
$221$	−37.3670			−2.51358
$222$		0			0
$223$		−	25.8818i		−	1.73317i	−0.499027	−	0.866586i	$-0.666309\pi$
$223$		−	25.8818i		−	1.73317i	0.499027	−	0.866586i	$-0.333691\pi$
$224$	8.48410			0.566868
$225$		0			0
$226$	20.1113			1.33779
$227$	19.7295			1.30950			0.654748	−	0.755848i	$-0.272775\pi$
$227$	19.7295			1.30950			0.654748	+	0.755848i	$0.272775\pi$
$228$		0			0
$229$			4.52132i			0.298777i	0.988779	+	0.149389i	$0.0477306\pi$
$229$			4.52132i			0.298777i	−0.988779	+	0.149389i	$0.952269\pi$
$230$	0.235889			0.0155540
$231$		0			0
$232$	−1.32738			−0.0871468
$233$	15.4765			1.01390			0.506950	−	0.861975i	$-0.330773\pi$
$233$	15.4765			1.01390			0.506950	+	0.861975i	$0.330773\pi$
$234$		0			0
$235$		−	8.89280i		−	0.580103i
$236$	9.37951			0.610554
$237$		0			0
$238$			15.8266i			1.02588i
$239$	14.7497	−	4.63092i	0.954081	−	0.299549i
$239$	14.7497	−	4.63092i	0.954081	−	0.299549i
$240$		0			0
$241$	2.62019			0.168781			0.0843906	−	0.996433i	$-0.473106\pi$
$241$	2.62019			0.168781			0.0843906	+	0.996433i	$0.473106\pi$
$242$			19.8602i			1.27666i
$243$		0			0
$244$	−2.37880			−0.152287
$245$		−	6.47505i		−	0.413676i
$246$		0			0
$247$	−1.58144			−0.100625
$248$		−	13.8097i		−	0.876918i
$249$		0			0
$250$	−20.0845			−1.27026
$251$		−	13.8326i		−	0.873108i	−0.899678	−	0.436554i	$-0.856199\pi$
$251$		−	13.8326i		−	0.873108i	0.899678	−	0.436554i	$-0.143801\pi$
$252$		0			0
$253$		−	0.433146i		−	0.0272317i
$254$		−	4.98856i		−	0.313010i
$255$		0			0
$256$	19.0303			1.18939
$257$		−	5.76476i		−	0.359596i	−0.983704	−	0.179798i	$-0.942456\pi$
$257$		−	5.76476i		−	0.359596i	0.983704	−	0.179798i	$-0.0575444\pi$
$258$		0			0
$259$	4.88786			0.303717
$260$	9.95468			0.617363
$261$		0			0
$262$			12.0297i			0.743195i
$263$		−	10.2744i		−	0.633548i	−0.948501	−	0.316774i	$-0.897400\pi$
$263$		−	10.2744i		−	0.633548i	0.948501	−	0.316774i	$-0.102600\pi$
$264$		0			0
$265$			19.9440i			1.22515i
$266$			0.669811i			0.0410688i
$267$		0			0
$268$	10.6585			0.651071
$269$		−	7.36697i		−	0.449172i	−0.974454	−	0.224586i	$-0.927897\pi$
$269$		−	7.36697i		−	0.449172i	0.974454	−	0.224586i	$-0.0721030\pi$
$270$		0			0
$271$	7.65277			0.464873			0.232436	−	0.972612i	$-0.425330\pi$
$271$	7.65277			0.464873			0.232436	+	0.972612i	$0.425330\pi$
$272$			28.0381i			1.70006i
$273$		0			0
$274$			21.3618i			1.29051i
$275$			13.1836i			0.795000i
$276$		0			0
$277$			4.78059i			0.287238i	0.989633	+	0.143619i	$0.0458739\pi$
$277$			4.78059i			0.287238i	−0.989633	+	0.143619i	$0.954126\pi$
$278$	24.5155			1.47034
$279$		0			0
$280$			4.19070i			0.250442i
$281$	24.9727			1.48975			0.744873	−	0.667206i	$-0.232510\pi$
$281$	24.9727			1.48975			0.744873	+	0.667206i	$0.232510\pi$
$282$		0			0
$283$	6.20762			0.369005			0.184502	−	0.982832i	$-0.440933\pi$
$283$	6.20762			0.369005			0.184502	+	0.982832i	$0.440933\pi$
$284$		−	6.44110i		−	0.382209i
$285$		0			0
$286$		−	54.7265i		−	3.23605i
$287$			19.2133i			1.13412i
$288$		0			0
$289$	−14.4456			−0.849742
$290$			1.98298i			0.116445i
$291$		0			0
$292$			15.9499i			0.933398i
$293$			7.48226i			0.437119i	0.975824	+	0.218559i	$0.0701357\pi$
$293$			7.48226i			0.437119i	−0.975824	+	0.218559i	$0.929864\pi$
$294$		0			0
$295$			13.9272i			0.810873i
$296$	5.18503			0.301374
$297$		0			0
$298$			32.9231i			1.90718i
$299$			0.609022i			0.0352207i
$300$		0			0
$301$	10.6977			0.616607
$302$	20.9890			1.20778
$303$		0			0
$304$			1.18663i			0.0680579i
$305$		−	3.53216i		−	0.202251i
$306$		0			0
$307$	−29.8082			−1.70124			−0.850621	−	0.525779i	$-0.823774\pi$
$307$	−29.8082			−1.70124			−0.850621	+	0.525779i	$0.823774\pi$
$308$	−7.74200			−0.441142
$309$		0			0
$310$	−20.6304			−1.17173
$311$			23.8662i			1.35333i	0.736291	+	0.676665i	$0.236575\pi$
$311$			23.8662i			1.35333i	−0.736291	+	0.676665i	$0.763425\pi$
$312$		0			0
$313$		−	25.1517i		−	1.42166i	−0.703364	−	0.710830i	$-0.748319\pi$
$313$		−	25.1517i		−	1.42166i	0.703364	−	0.710830i	$-0.251681\pi$
$314$		−	3.12726i		−	0.176482i
$315$		0			0
$316$			6.11140i			0.343793i
$317$	26.1271			1.46744			0.733721	−	0.679451i	$-0.237782\pi$
$317$	26.1271			1.46744			0.733721	+	0.679451i	$0.237782\pi$
$318$		0			0
$319$	3.64121			0.203869
$320$			1.44861i			0.0809800i
$321$		0			0
$322$	0.257948			0.0143749
$323$	−1.33084			−0.0740499
$324$		0			0
$325$		−	18.5367i		−	1.02823i
$326$		−	15.4317i		−	0.854681i
$327$		0			0
$328$			20.3814i			1.12537i
$329$		−	9.72441i		−	0.536124i
$330$		0			0
$331$		−	28.1858i		−	1.54923i	−0.632433	−	0.774615i	$-0.717944\pi$
$331$		−	28.1858i		−	1.54923i	0.632433	−	0.774615i	$-0.282056\pi$
$332$		−	1.27617i		−	0.0700388i
$333$		0			0
$334$		−	25.8184i		−	1.41272i
$335$			15.8263i			0.864683i
$336$		0			0
$337$	4.18541			0.227994			0.113997	−	0.993481i	$-0.463635\pi$
$337$	4.18541			0.227994			0.113997	+	0.993481i	$0.463635\pi$
$338$			54.4198i			2.96005i
$339$		0			0
$340$	8.37721			0.454318
$341$			37.8822i			2.05144i
$342$		0			0
$343$		−	18.4811i		−	0.997884i
$344$	11.3481			0.611850
$345$		0			0
$346$			6.55385i			0.352337i
$347$			13.7519i			0.738243i	0.929381	+	0.369121i	$0.120341\pi$
$347$			13.7519i			0.738243i	−0.929381	+	0.369121i	$0.879659\pi$
$348$		0			0
$349$	23.4792			1.25681			0.628406	−	0.777886i	$-0.283708\pi$
$349$	23.4792			1.25681			0.628406	+	0.777886i	$0.283708\pi$
$350$	−7.85110			−0.419659
$351$		0			0
$352$	−24.6882			−1.31588
$353$	13.2919			0.707456			0.353728	−	0.935348i	$-0.384914\pi$
$353$	13.2919			0.707456			0.353728	+	0.935348i	$0.384914\pi$
$354$		0			0
$355$	9.56409			0.507609
$356$	−8.92912			−0.473243
$357$		0			0
$358$			8.85259i			0.467874i
$359$			12.3842i			0.653615i	0.945091	+	0.326808i	$0.105973\pi$
$359$			12.3842i			0.653615i	−0.945091	+	0.326808i	$0.894027\pi$
$360$		0			0
$361$	18.9437			0.997036
$362$	−13.3323			−0.700732
$363$		0			0
$364$	10.8856			0.570560
$365$	−23.6833			−1.23964
$366$		0			0
$367$	24.2727			1.26702			0.633511	−	0.773733i	$-0.281613\pi$
$367$	24.2727			1.26702			0.633511	+	0.773733i	$0.281613\pi$
$368$	0.456977			0.0238216
$369$		0			0
$370$		−	7.74595i		−	0.402693i
$371$			21.8091i			1.13227i
$372$		0			0
$373$	16.6007			0.859551			0.429775	−	0.902936i	$-0.358593\pi$
$373$	16.6007			0.859551			0.429775	+	0.902936i	$0.358593\pi$
$374$		−	46.0543i		−	2.38141i
$375$		0			0
$376$		−	10.3156i		−	0.531988i
$377$	−5.11970			−0.263678
$378$		0			0
$379$		−	28.1547i		−	1.44621i	−0.690738	−	0.723105i	$-0.742714\pi$
$379$		−	28.1547i		−	1.44621i	0.690738	−	0.723105i	$-0.257286\pi$
$380$	0.354540			0.0181875
$381$		0			0
$382$		−	1.76231i		−	0.0901678i
$383$		−	27.6821i		−	1.41449i	−0.706969	−	0.707244i	$-0.749938\pi$
$383$		−	27.6821i		−	1.41449i	0.706969	−	0.707244i	$-0.250062\pi$
$384$		0			0
$385$		−	11.4957i		−	0.585877i
$386$		−	25.4709i		−	1.29643i
$387$		0			0
$388$			3.36979i			0.171075i
$389$		−	35.2232i		−	1.78589i	−0.450171	−	0.892943i	$-0.648637\pi$
$389$		−	35.2232i		−	1.78589i	0.450171	−	0.892943i	$-0.351363\pi$
$390$		0			0
$391$			0.512513i			0.0259189i
$392$		−	7.51104i		−	0.379365i
$393$		0			0
$394$	26.0811			1.31395
$395$	−9.07454			−0.456590
$396$		0			0
$397$		−	19.8743i		−	0.997464i	−0.866756	−	0.498732i	$-0.833799\pi$
$397$		−	19.8743i		−	0.997464i	0.866756	−	0.498732i	$-0.166201\pi$
$398$	4.94976			0.248109
$399$		0			0
$400$	−13.9089			−0.695446
$401$		−	10.9070i		−	0.544672i	−0.962202	−	0.272336i	$-0.912204\pi$
$401$		−	10.9070i		−	0.544672i	0.962202	−	0.272336i	$-0.0877962\pi$
$402$		0			0
$403$		−	53.2641i		−	2.65327i
$404$			11.4590i			0.570108i
$405$		0			0
$406$			2.16842i			0.107617i
$407$	−14.2234			−0.705026
$408$		0			0
$409$	−5.13882			−0.254098			−0.127049	−	0.991896i	$-0.540551\pi$
$409$	−5.13882			−0.254098			−0.127049	+	0.991896i	$0.540551\pi$
$410$	30.4479			1.50371
$411$		0			0
$412$		−	3.78430i		−	0.186439i
$413$			15.2296i			0.749399i
$414$		0			0
$415$	1.89492			0.0930180
$416$	34.7126			1.70193
$417$		0			0
$418$		−	1.94911i		−	0.0953340i
$419$		−	25.9355i		−	1.26703i	−0.773730	−	0.633516i	$-0.781611\pi$
$419$		−	25.9355i		−	1.26703i	0.773730	−	0.633516i	$-0.218389\pi$
$420$		0			0
$421$	29.2439			1.42526			0.712631	−	0.701539i	$-0.247503\pi$
$421$	29.2439			1.42526			0.712631	+	0.701539i	$0.247503\pi$
$422$			44.6897i			2.17546i
$423$		0			0
$424$			23.1350i			1.12354i
$425$		−	15.5993i		−	0.756675i
$426$		0			0
$427$		−	3.86247i		−	0.186918i
$428$	−10.5677			−0.510808
$429$		0			0
$430$		−	16.9531i		−	0.817549i
$431$		−	18.3159i		−	0.882246i	−0.897447	−	0.441123i	$-0.854580\pi$
$431$		−	18.3159i		−	0.882246i	0.897447	−	0.441123i	$-0.145420\pi$
$432$		0			0
$433$			9.34122i			0.448910i	0.974484	+	0.224455i	$0.0720602\pi$
$433$			9.34122i			0.448910i	−0.974484	+	0.224455i	$0.927940\pi$
$434$	−22.5597			−1.08290
$435$		0			0
$436$	−9.40223			−0.450285
$437$			0.0216906i			0.00103760i
$438$		0			0
$439$	−27.8044			−1.32703			−0.663515	−	0.748163i	$-0.730936\pi$
$439$	−27.8044			−1.32703			−0.663515	+	0.748163i	$0.730936\pi$
$440$		−	12.1947i		−	0.581357i
$441$		0			0
$442$			64.7543i			3.08005i
$443$		−	10.9599i		−	0.520719i	−0.965512	−	0.260360i	$-0.916159\pi$
$443$		−	10.9599i		−	0.520719i	0.965512	−	0.260360i	$-0.0838412\pi$
$444$		0			0
$445$		−	13.2584i		−	0.628510i
$446$	−44.8513			−2.12377
$447$		0			0
$448$			1.58408i			0.0748408i
$449$	33.8707			1.59846			0.799228	−	0.601027i	$-0.205242\pi$
$449$	33.8707			1.59846			0.799228	+	0.601027i	$0.205242\pi$
$450$		0			0
$451$		−	55.9093i		−	2.63267i
$452$		−	11.6407i		−	0.547531i
$453$		0			0
$454$		−	34.1899i		−	1.60461i
$455$			16.1635i			0.757757i
$456$		0			0
$457$	11.8489			0.554270			0.277135	−	0.960831i	$-0.410615\pi$
$457$	11.8489			0.554270			0.277135	+	0.960831i	$0.410615\pi$
$458$	7.83513			0.366111
$459$		0			0
$460$		−	0.136535i		−	0.00636599i
$461$	3.24929			0.151334			0.0756672	−	0.997133i	$-0.475891\pi$
$461$	3.24929			0.151334			0.0756672	+	0.997133i	$0.475891\pi$
$462$		0			0
$463$			17.0389i			0.791864i	0.918280	+	0.395932i	$0.129578\pi$
$463$			17.0389i			0.791864i	−0.918280	+	0.395932i	$0.870422\pi$
$464$			3.84155i			0.178339i
$465$		0			0
$466$		−	26.8197i		−	1.24240i
$467$	2.07426			0.0959854			0.0479927	−	0.998848i	$-0.484718\pi$
$467$	2.07426			0.0959854			0.0479927	+	0.998848i	$0.484718\pi$
$468$		0			0
$469$			17.3063i			0.799130i
$470$	−15.4106			−0.710837
$471$		0			0
$472$			16.1555i			0.743618i
$473$	−31.1297			−1.43135
$474$		0			0
$475$		−	0.660192i		−	0.0302917i
$476$	9.16060			0.419875
$477$		0			0
$478$	−8.02504	−	25.5602i	−0.367057	−	1.16910i
$479$			14.2272i			0.650057i	0.945704	+	0.325028i	$0.105374\pi$
$479$			14.2272i			0.650057i	−0.945704	+	0.325028i	$0.894626\pi$
$480$		0			0
$481$	19.9987			0.911860
$482$		−	4.54060i		−	0.206819i
$483$		0			0
$484$	11.4953			0.522514
$485$	−5.00365			−0.227204
$486$		0			0
$487$	−36.7719			−1.66630			−0.833148	−	0.553051i	$-0.813464\pi$
$487$	−36.7719			−1.66630			−0.833148	+	0.553051i	$0.813464\pi$
$488$		−	4.09730i		−	0.185476i
$489$		0			0
$490$	−11.2208			−0.506904
$491$	−18.9553			−0.855441			−0.427720	−	0.903911i	$-0.640683\pi$
$491$	−18.9553			−0.855441			−0.427720	+	0.903911i	$0.640683\pi$
$492$		0			0
$493$	−4.30841			−0.194041
$494$			2.74053i			0.123302i
$495$		0			0
$496$	−39.9664			−1.79455
$497$	10.4585			0.469127
$498$		0			0
$499$			2.10862i			0.0943948i	0.998886	+	0.0471974i	$0.0150290\pi$
$499$			2.10862i			0.0943948i	−0.998886	+	0.0471974i	$0.984971\pi$
$500$			11.6252i			0.519893i
$501$		0			0
$502$	−23.9710			−1.06988
$503$		−	29.1037i		−	1.29767i	−0.760929	−	0.648836i	$-0.775256\pi$
$503$		−	29.1037i		−	1.29767i	0.760929	−	0.648836i	$-0.224744\pi$
$504$		0			0
$505$	−17.0150			−0.757157
$506$	−0.750611			−0.0333687
$507$		0			0
$508$	−2.88744			−0.128109
$509$		−	10.9913i		−	0.487180i	−0.969878	−	0.243590i	$-0.921675\pi$
$509$		−	10.9913i		−	0.487180i	0.969878	−	0.243590i	$-0.0783251\pi$
$510$		0			0
$511$	−25.8980			−1.14566
$512$		−	8.76988i		−	0.387578i
$513$		0			0
$514$	−9.98991			−0.440636
$515$	5.61913			0.247609
$516$		0			0
$517$			28.2974i			1.24452i
$518$		−	8.47031i		−	0.372164i
$519$		0			0
$520$			17.1462i			0.751911i
$521$	−37.5268			−1.64408			−0.822039	−	0.569431i	$-0.807164\pi$
$521$	−37.5268			−1.64408			−0.822039	+	0.569431i	$0.807164\pi$
$522$		0			0
$523$	39.3617			1.72117			0.860584	−	0.509309i	$-0.170099\pi$
$523$	39.3617			1.72117			0.860584	+	0.509309i	$0.170099\pi$
$524$	6.96291			0.304176
$525$		0			0
$526$	−17.8048			−0.776328
$527$		−	44.8235i		−	1.95254i
$528$		0			0
$529$	−22.9916			−0.999637
$530$	34.5616			1.50126
$531$		0			0
$532$	0.387695			0.0168087
$533$			78.6109i			3.40502i
$534$		0			0
$535$		−	15.6915i		−	0.678401i
$536$			18.3585i			0.792965i
$537$		0			0
$538$	−12.7664			−0.550400
$539$			20.6040i			0.887476i
$540$		0			0
$541$			19.9144i			0.856187i	0.903734	+	0.428094i	$0.140815\pi$
$541$			19.9144i			0.856187i	−0.903734	+	0.428094i	$0.859185\pi$
$542$		−	13.2617i		−	0.569639i
$543$		0			0
$544$	29.2119			1.25245
$545$		−	13.9609i		−	0.598021i
$546$		0			0
$547$			37.0039i			1.58217i	0.611704	+	0.791086i	$0.290484\pi$
$547$			37.0039i			1.58217i	−0.611704	+	0.791086i	$0.709516\pi$
$548$	12.3644			0.528183
$549$		0			0
$550$	22.8462			0.974165
$551$	−0.182340			−0.00776796
$552$		0			0
$553$	−9.92314			−0.421975
$554$	8.28442			0.351971
$555$		0			0
$556$		−	14.1898i		−	0.601783i
$557$	−11.1203			−0.471184			−0.235592	−	0.971852i	$-0.575703\pi$
$557$	−11.1203			−0.471184			−0.235592	+	0.971852i	$0.575703\pi$
$558$		0			0
$559$	43.7697			1.85126
$560$	12.1282			0.512511
$561$		0			0
$562$		−	43.2759i		−	1.82548i
$563$		−	39.1687i		−	1.65076i	−0.564575	−	0.825381i	$-0.690960\pi$
$563$		−	39.1687i		−	1.65076i	0.564575	−	0.825381i	$-0.309040\pi$
$564$		0			0
$565$	17.2847			0.727173
$566$		−	10.7574i		−	0.452166i
$567$		0			0
$568$	11.0943			0.465507
$569$			0.120136i			0.00503638i	0.999997	+	0.00251819i	$0.000801565\pi$
$569$			0.120136i			0.00503638i	−0.999997	+	0.00251819i	$0.999198\pi$
$570$		0			0
$571$	30.2918			1.26767			0.633837	−	0.773467i	$-0.281479\pi$
$571$	30.2918			1.26767			0.633837	+	0.773467i	$0.281479\pi$
$572$	−31.6764			−1.32446
$573$		0			0
$574$	33.2952			1.38971
$575$	−0.254243			−0.0106027
$576$		0			0
$577$	−34.4019			−1.43217			−0.716085	−	0.698013i	$-0.754068\pi$
$577$	−34.4019			−1.43217			−0.716085	+	0.698013i	$0.754068\pi$
$578$			25.0332i			1.04124i
$579$		0			0
$580$	1.14777			0.0476587
$581$	2.07212			0.0859662
$582$		0			0
$583$		−	63.4630i		−	2.62837i
$584$	−27.4726			−1.13682
$585$		0			0
$586$	12.9662			0.535630
$587$			7.20698i			0.297464i	0.988878	+	0.148732i	$0.0475192\pi$
$587$			7.20698i			0.297464i	−0.988878	+	0.148732i	$0.952481\pi$
$588$		0			0
$589$		−	1.89702i		−	0.0781654i
$590$	24.1348			0.993615
$591$		0			0
$592$		−	15.0059i		−	0.616739i
$593$	19.1581			0.786730			0.393365	−	0.919382i	$-0.371311\pi$
$593$	19.1581			0.786730			0.393365	+	0.919382i	$0.371311\pi$
$594$		0			0
$595$			13.6022i			0.557634i
$596$	19.0563			0.780574
$597$		0			0
$598$	1.05539			0.0431582
$599$			26.6696i			1.08969i	0.838536	+	0.544846i	$0.183412\pi$
$599$			26.6696i			1.08969i	−0.838536	+	0.544846i	$0.816588\pi$
$600$		0			0
$601$		−	21.4806i		−	0.876213i	−0.898923	−	0.438107i	$-0.855649\pi$
$601$		−	21.4806i		−	0.876213i	0.898923	−	0.438107i	$-0.144351\pi$
$602$		−	18.5384i		−	0.755569i
$603$		0			0
$604$		−	12.1487i		−	0.494323i
$605$			17.0688i			0.693947i
$606$		0			0
$607$			30.2385i			1.22734i	0.789561	+	0.613672i	$0.210308\pi$
$607$			30.2385i			1.22734i	−0.789561	+	0.613672i	$0.789692\pi$
$608$	1.23630			0.0501388
$609$		0			0
$610$	−6.12099			−0.247831
$611$		−	39.7874i		−	1.60962i
$612$		0			0
$613$	41.3975			1.67203			0.836014	−	0.548708i	$-0.184880\pi$
$613$	41.3975			1.67203			0.836014	+	0.548708i	$0.184880\pi$
$614$			51.6554i			2.08464i
$615$		0			0
$616$		−	13.3350i		−	0.537284i
$617$	−32.0498			−1.29028			−0.645139	−	0.764065i	$-0.723200\pi$
$617$	−32.0498			−1.29028			−0.645139	+	0.764065i	$0.723200\pi$
$618$		0			0
$619$		−	22.2258i		−	0.893332i	−0.894701	−	0.446666i	$-0.852611\pi$
$619$		−	22.2258i		−	0.893332i	0.894701	−	0.446666i	$-0.147389\pi$
$620$			11.9411i			0.479568i
$621$		0			0
$622$	41.3584			1.65832
$623$		−	14.4983i		−	0.580862i
$624$		0			0
$625$	−3.35272			−0.134109
$626$	−43.5861			−1.74205
$627$		0			0
$628$	−1.81010			−0.0722307
$629$	16.8296			0.671038
$630$		0			0
$631$	−15.4733			−0.615983			−0.307992	−	0.951389i	$-0.599657\pi$
$631$	−15.4733			−0.615983			−0.307992	+	0.951389i	$0.599657\pi$
$632$	−10.5264			−0.418719
$633$		0			0
$634$		−	45.2763i		−	1.79815i
$635$		−	4.28742i		−	0.170141i
$636$		0			0
$637$		−	28.9701i		−	1.14784i
$638$		−	6.30996i		−	0.249814i
$639$		0			0
$640$	18.0275			0.712598
$641$		−	10.9532i		−	0.432625i	−0.976324	−	0.216312i	$-0.930597\pi$
$641$		−	10.9532i		−	0.432625i	0.976324	−	0.216312i	$-0.0694029\pi$
$642$		0			0
$643$	−36.4625			−1.43794			−0.718970	−	0.695042i	$-0.755386\pi$
$643$	−36.4625			−1.43794			−0.718970	+	0.695042i	$0.755386\pi$
$644$		−	0.149303i		−	0.00588337i
$645$		0			0
$646$			2.30625i			0.0907382i
$647$			22.7507i			0.894421i	0.894429	+	0.447210i	$0.147582\pi$
$647$			22.7507i			0.894421i	−0.894429	+	0.447210i	$0.852418\pi$
$648$		0			0
$649$		−	44.3171i		−	1.73960i
$650$	−32.1227			−1.25996
$651$		0			0
$652$	−8.93203			−0.349805
$653$	41.4549			1.62226			0.811128	−	0.584868i	$-0.198854\pi$
$653$	41.4549			1.62226			0.811128	+	0.584868i	$0.198854\pi$
$654$		0			0
$655$			10.3389i			0.403974i
$656$	58.9853			2.30299
$657$		0			0
$658$	−16.8517			−0.656948
$659$	7.68129			0.299221			0.149610	−	0.988745i	$-0.452198\pi$
$659$	7.68129			0.299221			0.149610	+	0.988745i	$0.452198\pi$
$660$		0			0
$661$	−23.5641			−0.916537			−0.458269	−	0.888814i	$-0.651530\pi$
$661$	−23.5641			−0.916537			−0.458269	+	0.888814i	$0.651530\pi$
$662$	−48.8439			−1.89837
$663$		0			0
$664$	2.19810			0.0853030
$665$			0.575670i			0.0223235i
$666$		0			0
$667$			0.0702202i			0.00271894i
$668$	−14.9440			−0.578200
$669$		0			0
$670$	27.4258			1.05955
$671$			11.2395i			0.433898i
$672$		0			0
$673$		−	2.34943i		−	0.0905640i	−0.998974	−	0.0452820i	$-0.985581\pi$
$673$		−	2.34943i		−	0.0905640i	0.998974	−	0.0452820i	$-0.0144186\pi$
$674$		−	7.25302i		−	0.279376i
$675$		0			0
$676$	31.4988			1.21149
$677$	15.3498			0.589941			0.294970	−	0.955506i	$-0.404690\pi$
$677$	15.3498			0.589941			0.294970	+	0.955506i	$0.404690\pi$
$678$		0			0
$679$	−5.47156			−0.209979
$680$			14.4291i			0.553332i
$681$		0			0
$682$	65.6472			2.51376
$683$	−29.2811			−1.12041			−0.560205	−	0.828354i	$-0.689278\pi$
$683$	−29.2811			−1.12041			−0.560205	+	0.828354i	$0.689278\pi$
$684$		0			0
$685$			18.3594i			0.701476i
$686$	−32.0264			−1.22277
$687$		0			0
$688$		−	32.8424i		−	1.25211i
$689$			89.2318i			3.39946i
$690$		0			0
$691$	−23.3913			−0.889848			−0.444924	−	0.895568i	$-0.646769\pi$
$691$	−23.3913			−0.889848			−0.444924	+	0.895568i	$0.646769\pi$
$692$	3.79344			0.144205
$693$		0			0
$694$	23.8311			0.904617
$695$	21.0698			0.799225
$696$		0			0
$697$			66.1538i			2.50575i
$698$		−	40.6877i		−	1.54005i
$699$		0			0
$700$			4.54431i			0.171759i
$701$	−4.20354			−0.158766			−0.0793829	−	0.996844i	$-0.525295\pi$
$701$	−4.20354			−0.158766			−0.0793829	+	0.996844i	$0.525295\pi$
$702$		0			0
$703$	0.712260			0.0268634
$704$		−	4.60957i		−	0.173730i
$705$		0			0
$706$		−	23.0339i		−	0.866892i
$707$	−18.6061			−0.699756
$708$		0			0
$709$			40.0652i			1.50468i	0.658776	+	0.752339i	$0.271075\pi$
$709$			40.0652i			1.50468i	−0.658776	+	0.752339i	$0.728925\pi$
$710$		−	16.5739i		−	0.622007i
$711$		0			0
$712$		−	15.3798i		−	0.576381i
$713$	−0.730552			−0.0273594
$714$		0			0
$715$		−	47.0348i		−	1.75900i
$716$	5.12398			0.191492
$717$		0			0
$718$	21.4610			0.800917
$719$			27.1620i			1.01297i	0.862248	+	0.506486i	$0.169056\pi$
$719$			27.1620i			1.01297i	−0.862248	+	0.506486i	$0.830944\pi$
$720$		0			0
$721$	6.14460			0.228837
$722$		−	32.8280i		−	1.22173i
$723$		0			0
$724$			7.71691i			0.286797i
$725$		−	2.13728i		−	0.0793765i
$726$		0			0
$727$	−18.8586			−0.699427			−0.349713	−	0.936857i	$-0.613721\pi$
$727$	−18.8586			−0.699427			−0.349713	+	0.936857i	$0.613721\pi$
$728$			18.7496i			0.694908i
$729$		0			0
$730$			41.0414i			1.51901i
$731$	36.8337			1.36235
$732$		0			0
$733$	−40.0571			−1.47954			−0.739771	−	0.672859i	$-0.765066\pi$
$733$	−40.0571			−1.47954			−0.739771	+	0.672859i	$0.765066\pi$
$734$		−	42.0628i		−	1.55257i
$735$		0			0
$736$		−	0.476107i		−	0.0175495i
$737$		−	50.3602i		−	1.85504i
$738$		0			0
$739$	33.9048			1.24721			0.623605	−	0.781740i	$-0.285667\pi$
$739$	33.9048			1.24721			0.623605	+	0.781740i	$0.285667\pi$
$740$	−4.48345			−0.164815
$741$		0			0
$742$	37.7936			1.38745
$743$	16.9031			0.620116			0.310058	−	0.950718i	$-0.399652\pi$
$743$	16.9031			0.620116			0.310058	+	0.950718i	$0.399652\pi$
$744$		0			0
$745$			28.2957i			1.03668i
$746$		−	28.7678i		−	1.05326i
$747$		0			0
$748$	−26.6568			−0.974668
$749$		−	17.1588i		−	0.626970i
$750$		0			0
$751$	5.25406			0.191723			0.0958617	−	0.995395i	$-0.469439\pi$
$751$	5.25406			0.191723			0.0958617	+	0.995395i	$0.469439\pi$
$752$	−29.8543			−1.08867
$753$		0			0
$754$			8.87208i			0.323102i
$755$	18.0390			0.656508
$756$		0			0
$757$	9.97514			0.362553			0.181276	−	0.983432i	$-0.441977\pi$
$757$	9.97514			0.362553			0.181276	+	0.983432i	$0.441977\pi$
$758$	−48.7901			−1.77214
$759$		0			0
$760$			0.610669i			0.0221513i
$761$			13.1739i			0.477553i	0.971075	+	0.238777i	$0.0767464\pi$
$761$			13.1739i			0.477553i	−0.971075	+	0.238777i	$0.923254\pi$
$762$		0			0
$763$		−	15.2665i		−	0.552684i
$764$	−1.02005			−0.0369040
$765$		0			0
$766$	−47.9711			−1.73327
$767$			62.3118i			2.24995i
$768$		0			0
$769$			1.47313i			0.0531224i	0.999647	+	0.0265612i	$0.00845569\pi$
$769$			1.47313i			0.0531224i	−0.999647	+	0.0265612i	$0.991544\pi$
$770$	−19.9213			−0.717914
$771$		0			0
$772$	−14.7428			−0.530606
$773$	−50.1191			−1.80266			−0.901329	−	0.433134i	$-0.857408\pi$
$773$	−50.1191			−1.80266			−0.901329	+	0.433134i	$0.857408\pi$
$774$		0			0
$775$	22.2357			0.798729
$776$	−5.80422			−0.208359
$777$		0			0
$778$	−61.0392			−2.18836
$779$			2.79976i			0.100312i
$780$		0			0
$781$	−30.4335			−1.08900
$782$	0.888149			0.0317601
$783$		0			0
$784$	−21.7376			−0.776341
$785$		−	2.68773i		−	0.0959291i
$786$		0			0
$787$			20.5001i			0.730749i	0.930861	+	0.365375i	$0.119059\pi$
$787$			20.5001i			0.730749i	−0.930861	+	0.365375i	$0.880941\pi$
$788$		−	15.0961i		−	0.537775i
$789$		0			0
$790$			15.7255i			0.559489i
$791$	18.9011			0.672045
$792$		0			0
$793$		−	15.8033i		−	0.561191i
$794$	−34.4408			−1.22226
$795$		0			0
$796$		−	2.86498i		−	0.101547i
$797$			0.522896i			0.0185219i	0.999957	+	0.00926097i	$0.00294790\pi$
$797$			0.522896i			0.0185219i	−0.999957	+	0.00926097i	$0.997052\pi$
$798$		0			0
$799$		−	33.4824i		−	1.18452i
$800$			14.4912i			0.512340i
$801$		0			0
$802$	−18.9011			−0.667422
$803$	75.3615			2.65945
$804$		0			0
$805$	0.221693			0.00781367
$806$	−92.3028			−3.25123
$807$		0			0
$808$	−19.7373			−0.694357
$809$	36.6238			1.28762			0.643812	−	0.765183i	$-0.277352\pi$
$809$	36.6238			1.28762			0.643812	+	0.765183i	$0.277352\pi$
$810$		0			0
$811$			19.5301i			0.685796i	0.939373	+	0.342898i	$0.111409\pi$
$811$			19.5301i			0.685796i	−0.939373	+	0.342898i	$0.888591\pi$
$812$	1.25511			0.0440456
$813$		0			0
$814$			24.6481i			0.863914i
$815$		−	13.2628i		−	0.464574i
$816$		0			0
$817$	1.55888			0.0545382
$818$			8.90520i			0.311363i
$819$		0			0
$820$		−	17.6236i		−	0.615442i
$821$	20.1340			0.702682			0.351341	−	0.936248i	$-0.385726\pi$
$821$	20.1340			0.702682			0.351341	+	0.936248i	$0.385726\pi$
$822$		0			0
$823$		−	33.7611i		−	1.17684i	−0.808557	−	0.588419i	$-0.799751\pi$
$823$		−	33.7611i		−	1.17684i	0.808557	−	0.588419i	$-0.200249\pi$
$824$	6.51818			0.227072
$825$		0			0
$826$	26.3918			0.918288
$827$		−	52.8165i		−	1.83661i	−0.395875	−	0.918304i	$-0.629559\pi$
$827$		−	52.8165i		−	1.83661i	0.395875	−	0.918304i	$-0.370441\pi$
$828$		0			0
$829$			41.1364i			1.42872i	0.699776	+	0.714362i	$0.253283\pi$
$829$			41.1364i			1.42872i	−0.699776	+	0.714362i	$0.746717\pi$
$830$		−	3.28376i		−	0.113981i
$831$		0			0
$832$			6.48126i			0.224697i
$833$		−	24.3793i		−	0.844693i
$834$		0			0
$835$		−	22.1896i		−	0.767904i
$836$	−1.12817			−0.0390185
$837$		0			0
$838$	−44.9443			−1.55258
$839$			17.3973i			0.600622i	0.953841	+	0.300311i	$0.0970904\pi$
$839$			17.3973i			0.600622i	−0.953841	+	0.300311i	$0.902910\pi$
$840$		0			0
$841$	28.4097			0.979645
$842$		−	50.6777i		−	1.74647i
$843$		0			0
$844$	25.8669			0.890377
$845$			46.7711i			1.60898i
$846$		0			0
$847$			18.6650i			0.641338i
$848$	66.9546			2.29923
$849$		0			0
$850$	−27.0324			−0.927203
$851$		−	0.274295i		−	0.00940271i
$852$		0			0
$853$	14.6009			0.499924			0.249962	−	0.968256i	$-0.419582\pi$
$853$	14.6009			0.499924			0.249962	+	0.968256i	$0.419582\pi$
$854$	−6.69339			−0.229043
$855$		0			0
$856$		−	18.2021i		−	0.622133i
$857$	25.2072			0.861061			0.430530	−	0.902576i	$-0.358327\pi$
$857$	25.2072			0.861061			0.430530	+	0.902576i	$0.358327\pi$
$858$		0			0
$859$	42.0489			1.43469			0.717344	−	0.696719i	$-0.245357\pi$
$859$	42.0489			1.43469			0.717344	+	0.696719i	$0.245357\pi$
$860$	−9.81263			−0.334608
$861$		0			0
$862$	−31.7401			−1.08107
$863$	46.5997			1.58627			0.793136	−	0.609045i	$-0.208447\pi$
$863$	46.5997			1.58627			0.793136	+	0.609045i	$0.208447\pi$
$864$		0			0
$865$			5.63271i			0.191518i
$866$	16.1877			0.550079
$867$		0			0
$868$			13.0578i			0.443211i
$869$	28.8757			0.979541
$870$		0			0
$871$			70.8086i			2.39926i
$872$		−	16.1946i		−	0.548420i
$873$		0			0
$874$	0.0375882			0.00127144
$875$	−18.8759			−0.638121
$876$		0			0
$877$	0.412885			0.0139421			0.00697106	−	0.999976i	$-0.497781\pi$
$877$	0.412885			0.0139421			0.00697106	+	0.999976i	$0.497781\pi$
$878$			48.1830i			1.62610i
$879$		0			0
$880$	−35.2923			−1.18970
$881$	23.4525			0.790135			0.395068	−	0.918652i	$-0.370721\pi$
$881$	23.4525			0.790135			0.395068	+	0.918652i	$0.370721\pi$
$882$		0			0
$883$	13.0624			0.439586			0.219793	−	0.975547i	$-0.429462\pi$
$883$	13.0624			0.439586			0.219793	+	0.975547i	$0.429462\pi$
$884$	37.4805			1.26061
$885$		0			0
$886$	−18.9927			−0.638071
$887$		−	15.2216i		−	0.511092i	−0.966797	−	0.255546i	$-0.917745\pi$
$887$		−	15.2216i		−	0.511092i	0.966797	−	0.255546i	$-0.0822552\pi$
$888$		0			0
$889$		−	4.68836i		−	0.157242i
$890$	−22.9759			−0.770155
$891$		0			0
$892$			25.9604i			0.869220i
$893$		−	1.41704i		−	0.0474195i
$894$		0			0
$895$			7.60837i			0.254320i
$896$	19.7133			0.658575
$897$		0			0
$898$		−	58.6955i		−	1.95869i
$899$		−	6.14134i		−	0.204825i
$900$		0			0
$901$			75.0916i			2.50166i
$902$	−96.8868			−3.22598
$903$		0			0
$904$	20.0502			0.666860
$905$	−11.4585			−0.380893
$906$		0			0
$907$		−	14.7039i		−	0.488234i	−0.969746	−	0.244117i	$-0.921502\pi$
$907$		−	14.7039i		−	0.488234i	0.969746	−	0.244117i	$-0.0784981\pi$
$908$	−19.7895			−0.656737
$909$		0			0
$910$	28.0102			0.928529
$911$	10.9783			0.363729			0.181864	−	0.983324i	$-0.441787\pi$
$911$	10.9783			0.363729			0.181864	+	0.983324i	$0.441787\pi$
$912$		0			0
$913$	−6.02974			−0.199555
$914$		−	20.5334i		−	0.679184i
$915$		0			0
$916$		−	4.53506i		−	0.149843i
$917$			11.3057i			0.373348i
$918$		0			0
$919$	18.1585			0.598993			0.299497	−	0.954097i	$-0.403181\pi$
$919$	18.1585			0.598993			0.299497	+	0.954097i	$0.403181\pi$
$920$	0.235172			0.00775338
$921$		0			0
$922$		−	5.63078i		−	0.185440i
$923$	42.7908			1.40848
$924$		0			0
$925$			8.34866i			0.274502i
$926$	29.5271			0.970322
$927$		0			0
$928$	4.00236			0.131384
$929$	18.2202			0.597784			0.298892	−	0.954287i	$-0.403383\pi$
$929$	18.2202			0.597784			0.298892	+	0.954287i	$0.403383\pi$
$930$		0			0
$931$		−	1.03178i		−	0.0338153i
$932$	−15.5236			−0.508491
$933$		0			0
$934$		−	3.59455i		−	0.117617i
$935$		−	39.5814i		−	1.29445i
$936$		0			0
$937$	−40.6389			−1.32761			−0.663807	−	0.747904i	$-0.731061\pi$
$937$	−40.6389			−1.32761			−0.663807	+	0.747904i	$0.731061\pi$
$938$	29.9905			0.979226
$939$		0			0
$940$			8.91983i			0.290933i
$941$	−20.7108			−0.675154			−0.337577	−	0.941298i	$-0.609607\pi$
$941$	−20.7108			−0.675154			−0.337577	+	0.941298i	$0.609607\pi$
$942$		0			0
$943$	1.07820			0.0351111
$944$	46.7554			1.52176
$945$		0			0
$946$			53.9456i			1.75392i
$947$	−17.0166			−0.552966			−0.276483	−	0.961019i	$-0.589169\pi$
$947$	−17.0166			−0.552966			−0.276483	+	0.961019i	$0.589169\pi$
$948$		0			0
$949$	−105.962			−3.43966
$950$	−1.14406			−0.0371184
$951$		0			0
$952$			15.7785i			0.511383i
$953$	−25.9474			−0.840519			−0.420260	−	0.907404i	$-0.638061\pi$
$953$	−25.9474			−0.840519			−0.420260	+	0.907404i	$0.638061\pi$
$954$		0			0
$955$		−	1.51462i		−	0.0490120i
$956$	−14.7946	+	4.64499i	−0.478490	+	0.150230i
$957$		0			0
$958$	24.6547			0.796557
$959$			20.0763i			0.648296i
$960$		0			0
$961$	32.8929			1.06106
$962$		−	34.6562i		−	1.11736i
$963$		0			0
$964$	−2.62815			−0.0846471
$965$		−	21.8909i		−	0.704695i
$966$		0			0
$967$	−38.3740			−1.23403			−0.617013	−	0.786953i	$-0.711657\pi$
$967$	−38.3740			−1.23403			−0.617013	+	0.786953i	$0.711657\pi$
$968$			19.7998i			0.636390i
$969$		0			0
$970$			8.67096i			0.278408i
$971$			57.3961i			1.84193i	0.389647	+	0.920964i	$0.372597\pi$
$971$			57.3961i			1.84193i	−0.389647	+	0.920964i	$0.627403\pi$
$972$		0			0
$973$	23.0402			0.738634
$974$			63.7231i			2.04182i
$975$		0			0
$976$	−11.8579			−0.379563
$977$	−4.35694			−0.139391			−0.0696954	−	0.997568i	$-0.522203\pi$
$977$	−4.35694			−0.139391			−0.0696954	+	0.997568i	$0.522203\pi$
$978$		0			0
$979$			42.1891i			1.34837i
$980$			6.49473i			0.207466i
$981$		0			0
$982$			32.8482i			1.04823i
$983$			31.5235i			1.00544i	0.864448	+	0.502722i	$0.167668\pi$
$983$			31.5235i			1.00544i	−0.864448	+	0.502722i	$0.832332\pi$
$984$		0			0
$985$	22.4155			0.714216
$986$			7.46616i			0.237771i
$987$		0			0
$988$	1.58625			0.0504654
$989$		−	0.600331i		−	0.0190894i
$990$		0			0
$991$			24.0048i			0.762536i	0.924464	+	0.381268i	$0.124513\pi$
$991$			24.0048i			0.762536i	−0.924464	+	0.381268i	$0.875487\pi$
$992$			41.6395i			1.32206i
$993$		0			0
$994$		−	18.1238i		−	0.574851i
$995$	4.25408			0.134863
$996$		0			0
$997$			41.5736i			1.31665i	0.752734	+	0.658325i	$0.228735\pi$
$997$			41.5736i			1.31665i	−0.752734	+	0.658325i	$0.771265\pi$
$998$	3.65409			0.115668
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.2.b.a.2150.19	✓	80
3.2	odd	2	inner	2151.2.b.a.2150.61	yes	80
239.238	odd	2	inner	2151.2.b.a.2150.20	yes	80
717.716	even	2	inner	2151.2.b.a.2150.62	yes	80

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2151.2.b.a.2150.19	✓	80	1.1	even	1	trivial
2151.2.b.a.2150.20	yes	80	239.238	odd	2	inner
2151.2.b.a.2150.61	yes	80	3.2	odd	2	inner
2151.2.b.a.2150.62	yes	80	717.716	even	2	inner

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:	\( N \)	\(=\)	\( 2151 = 3^{2} \cdot 239 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2151.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.73293i q^{2} -1.00304 q^{4} -1.48937i q^{5} -1.62864i q^{7} -1.72766i q^{8} +O(q^{10})\)	\(q-1.73293i q^{2} -1.00304 q^{4} -1.48937i q^{5} -1.62864i q^{7} -1.72766i q^{8} -2.58096 q^{10} +4.73925i q^{11} -6.66359i q^{13} -2.82232 q^{14} -4.99999 q^{16} -5.60764i q^{17} -0.237326i q^{19} +1.49389i q^{20} +8.21277 q^{22} -0.0913956 q^{23} +2.78179 q^{25} -11.5475 q^{26} +1.63359i q^{28} -0.768311i q^{29} +7.99330 q^{31} +5.20930i q^{32} -9.71763 q^{34} -2.42565 q^{35} +3.00118i q^{37} -0.411269 q^{38} -2.57312 q^{40} -11.7971 q^{41} +6.56850i q^{43} -4.75365i q^{44} +0.158382i q^{46} +5.97086 q^{47} +4.34752 q^{49} -4.82064i q^{50} +6.68384i q^{52} -13.3910 q^{53} +7.05848 q^{55} -2.81374 q^{56} -1.33143 q^{58} -9.35109 q^{59} +2.37159 q^{61} -13.8518i q^{62} -0.972638 q^{64} -9.92452 q^{65} -10.6262 q^{67} +5.62468i q^{68} +4.20347i q^{70} +6.42158i q^{71} -15.9016i q^{73} +5.20084 q^{74} +0.238048i q^{76} +7.71855 q^{77} -6.09288i q^{79} +7.44682i q^{80} +20.4435i q^{82} +1.27230i q^{83} -8.35183 q^{85} +11.3827 q^{86} +8.18781 q^{88} +8.90207 q^{89} -10.8526 q^{91} +0.0916733 q^{92} -10.3471i q^{94} -0.353466 q^{95} -3.35958i q^{97} -7.53394i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - 80 q^{4}+O(q^{10}) \) 80 * q - 80 * q^4	\( 80 q - 80 q^{4} + 16 q^{10} + 56 q^{16} + 40 q^{22} - 64 q^{25} - 8 q^{31} + 32 q^{34} - 24 q^{40} - 104 q^{49} - 24 q^{55} + 56 q^{58} + 40 q^{61} - 80 q^{64} - 8 q^{67} - 8 q^{85} - 120 q^{88} + 32 q^{91}+O(q^{100}) \) 80 * q - 80 * q^4 + 16 * q^10 + 56 * q^16 + 40 * q^22 - 64 * q^25 - 8 * q^31 + 32 * q^34 - 24 * q^40 - 104 * q^49 - 24 * q^55 + 56 * q^58 + 40 * q^61 - 80 * q^64 - 8 * q^67 - 8 * q^85 - 120 * q^88 + 32 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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