LMFDB - Embedded newform 1638.2.j.a.235.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1638,2,Mod(235,1638)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1638, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 4, 0]))

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

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

chi := DirichletCharacter("1638.235");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1638.j (of order $3$, degree $2$, 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:	$13.0794958511$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 546)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			235.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	1638.235
Dual form			1638.2.j.a.1171.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-0.500000 - 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(-1.00000 - 1.73205i) q^{5} +(-2.50000 - 0.866025i) q^{7} +1.00000 q^{8} +(-1.00000 + 1.73205i) q^{10} +(1.50000 - 2.59808i) q^{11} -1.00000 q^{13} +(0.500000 + 2.59808i) q^{14} +(-0.500000 - 0.866025i) q^{16} +(2.50000 - 4.33013i) q^{17} +(-0.500000 - 0.866025i) q^{19} +2.00000 q^{20} -3.00000 q^{22} +(0.500000 - 0.866025i) q^{25} +(0.500000 + 0.866025i) q^{26} +(2.00000 - 1.73205i) q^{28} +1.00000 q^{29} +(-2.00000 + 3.46410i) q^{31} +(-0.500000 + 0.866025i) q^{32} -5.00000 q^{34} +(1.00000 + 5.19615i) q^{35} +(1.00000 + 1.73205i) q^{37} +(-0.500000 + 0.866025i) q^{38} +(-1.00000 - 1.73205i) q^{40} -10.0000 q^{41} -10.0000 q^{43} +(1.50000 + 2.59808i) q^{44} +(-0.500000 - 0.866025i) q^{47} +(5.50000 + 4.33013i) q^{49} -1.00000 q^{50} +(0.500000 - 0.866025i) q^{52} +(-1.50000 + 2.59808i) q^{53} -6.00000 q^{55} +(-2.50000 - 0.866025i) q^{56} +(-0.500000 - 0.866025i) q^{58} +(1.50000 - 2.59808i) q^{59} +(2.50000 + 4.33013i) q^{61} +4.00000 q^{62} +1.00000 q^{64} +(1.00000 + 1.73205i) q^{65} +(-2.50000 + 4.33013i) q^{67} +(2.50000 + 4.33013i) q^{68} +(4.00000 - 3.46410i) q^{70} +1.00000 q^{71} +(-6.00000 + 10.3923i) q^{73} +(1.00000 - 1.73205i) q^{74} +1.00000 q^{76} +(-6.00000 + 5.19615i) q^{77} +(3.00000 + 5.19615i) q^{79} +(-1.00000 + 1.73205i) q^{80} +(5.00000 + 8.66025i) q^{82} -16.0000 q^{83} -10.0000 q^{85} +(5.00000 + 8.66025i) q^{86} +(1.50000 - 2.59808i) q^{88} +(-7.00000 - 12.1244i) q^{89} +(2.50000 + 0.866025i) q^{91} +(-0.500000 + 0.866025i) q^{94} +(-1.00000 + 1.73205i) q^{95} +4.00000 q^{97} +(1.00000 - 6.92820i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} - q^{4} - 2 q^{5} - 5 q^{7} + 2 q^{8} - 2 q^{10} + 3 q^{11} - 2 q^{13} + q^{14} - q^{16} + 5 q^{17} - q^{19} + 4 q^{20} - 6 q^{22} + q^{25} + q^{26} + 4 q^{28} + 2 q^{29} - 4 q^{31} - q^{32}+ \cdots + 2 q^{98}+O(q^{100}) $ 2 * q - q^2 - q^4 - 2 * q^5 - 5 * q^7 + 2 * q^8 - 2 * q^10 + 3 * q^11 - 2 * q^13 + q^14 - q^16 + 5 * q^17 - q^19 + 4 * q^20 - 6 * q^22 + q^25 + q^26 + 4 * q^28 + 2 * q^29 - 4 * q^31 - q^32 - 10 * q^34 + 2 * q^35 + 2 * q^37 - q^38 - 2 * q^40 - 20 * q^41 - 20 * q^43 + 3 * q^44 - q^47 + 11 * q^49 - 2 * q^50 + q^52 - 3 * q^53 - 12 * q^55 - 5 * q^56 - q^58 + 3 * q^59 + 5 * q^61 + 8 * q^62 + 2 * q^64 + 2 * q^65 - 5 * q^67 + 5 * q^68 + 8 * q^70 + 2 * q^71 - 12 * q^73 + 2 * q^74 + 2 * q^76 - 12 * q^77 + 6 * q^79 - 2 * q^80 + 10 * q^82 - 32 * q^83 - 20 * q^85 + 10 * q^86 + 3 * q^88 - 14 * q^89 + 5 * q^91 - q^94 - 2 * q^95 + 8 * q^97 + 2 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1638\mathbb{Z}\right)^\times$.

$n$	$379$	$703$	$911$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$	$1$

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 $

$2$	−0.500000	−	0.866025i	−0.353553	−	0.612372i
$2$	−0.500000	−	0.866025i	−0.353553	−	0.612372i
$3$		0			0
$3$		0			0
$4$	−0.500000	+	0.866025i	−0.250000	+	0.433013i
$5$	−1.00000	−	1.73205i	−0.447214	−	0.774597i	0.550990	−	0.834512i	$-0.314250\pi$
$5$	−1.00000	−	1.73205i	−0.447214	−	0.774597i	−0.998203	+	0.0599153i	$0.980917\pi$
$6$		0			0
$7$	−2.50000	−	0.866025i	−0.944911	−	0.327327i
$7$	−2.50000	−	0.866025i	−0.944911	−	0.327327i
$8$	1.00000			0.353553
$9$		0			0
$10$	−1.00000	+	1.73205i	−0.316228	+	0.547723i
$11$	1.50000	−	2.59808i	0.452267	−	0.783349i	−0.546259	−	0.837616i	$-0.683949\pi$
$11$	1.50000	−	2.59808i	0.452267	−	0.783349i	0.998526	+	0.0542666i	$0.0172821\pi$
$12$		0			0
$13$	−1.00000			−0.277350
$13$	−1.00000			−0.277350
$14$	0.500000	+	2.59808i	0.133631	+	0.694365i
$15$		0			0
$16$	−0.500000	−	0.866025i	−0.125000	−	0.216506i
$17$	2.50000	−	4.33013i	0.606339	−	1.05021i	−0.385499	−	0.922708i	$-0.625971\pi$
$17$	2.50000	−	4.33013i	0.606339	−	1.05021i	0.991838	−	0.127502i	$-0.0406959\pi$
$18$		0			0
$19$	−0.500000	−	0.866025i	−0.114708	−	0.198680i	0.802955	−	0.596040i	$-0.203260\pi$
$19$	−0.500000	−	0.866025i	−0.114708	−	0.198680i	−0.917663	+	0.397360i	$0.869927\pi$
$20$	2.00000			0.447214
$21$		0			0
$22$	−3.00000			−0.639602
$23$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$23$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$24$		0			0
$25$	0.500000	−	0.866025i	0.100000	−	0.173205i
$26$	0.500000	+	0.866025i	0.0980581	+	0.169842i
$27$		0			0
$28$	2.00000	−	1.73205i	0.377964	−	0.327327i
$29$	1.00000			0.185695			0.0928477	−	0.995680i	$-0.470403\pi$
$29$	1.00000			0.185695			0.0928477	+	0.995680i	$0.470403\pi$
$30$		0			0
$31$	−2.00000	+	3.46410i	−0.359211	+	0.622171i	−0.987829	−	0.155543i	$-0.950287\pi$
$31$	−2.00000	+	3.46410i	−0.359211	+	0.622171i	0.628619	+	0.777714i	$0.283621\pi$
$32$	−0.500000	+	0.866025i	−0.0883883	+	0.153093i
$33$		0			0
$34$	−5.00000			−0.857493
$35$	1.00000	+	5.19615i	0.169031	+	0.878310i
$36$		0			0
$37$	1.00000	+	1.73205i	0.164399	+	0.284747i	0.936442	−	0.350823i	$-0.114098\pi$
$37$	1.00000	+	1.73205i	0.164399	+	0.284747i	−0.772043	+	0.635571i	$0.780765\pi$
$38$	−0.500000	+	0.866025i	−0.0811107	+	0.140488i
$39$		0			0
$40$	−1.00000	−	1.73205i	−0.158114	−	0.273861i
$41$	−10.0000			−1.56174			−0.780869	−	0.624695i	$-0.785223\pi$
$41$	−10.0000			−1.56174			−0.780869	+	0.624695i	$0.785223\pi$
$42$		0			0
$43$	−10.0000			−1.52499			−0.762493	−	0.646997i	$-0.776025\pi$
$43$	−10.0000			−1.52499			−0.762493	+	0.646997i	$0.776025\pi$
$44$	1.50000	+	2.59808i	0.226134	+	0.391675i
$45$		0			0
$46$		0			0
$47$	−0.500000	−	0.866025i	−0.0729325	−	0.126323i	0.827253	−	0.561830i	$-0.189902\pi$
$47$	−0.500000	−	0.866025i	−0.0729325	−	0.126323i	−0.900185	+	0.435507i	$0.856569\pi$
$48$		0			0
$49$	5.50000	+	4.33013i	0.785714	+	0.618590i
$50$	−1.00000			−0.141421
$51$		0			0
$52$	0.500000	−	0.866025i	0.0693375	−	0.120096i
$53$	−1.50000	+	2.59808i	−0.206041	+	0.356873i	−0.950464	−	0.310835i	$-0.899391\pi$
$53$	−1.50000	+	2.59808i	−0.206041	+	0.356873i	0.744423	+	0.667708i	$0.232725\pi$
$54$		0			0
$55$	−6.00000			−0.809040
$56$	−2.50000	−	0.866025i	−0.334077	−	0.115728i
$57$		0			0
$58$	−0.500000	−	0.866025i	−0.0656532	−	0.113715i
$59$	1.50000	−	2.59808i	0.195283	−	0.338241i	−0.751710	−	0.659494i	$-0.770771\pi$
$59$	1.50000	−	2.59808i	0.195283	−	0.338241i	0.946993	+	0.321253i	$0.104104\pi$
$60$		0			0
$61$	2.50000	+	4.33013i	0.320092	+	0.554416i	0.980507	−	0.196485i	$-0.0629528\pi$
$61$	2.50000	+	4.33013i	0.320092	+	0.554416i	−0.660415	+	0.750901i	$0.729619\pi$
$62$	4.00000			0.508001
$63$		0			0
$64$	1.00000			0.125000
$65$	1.00000	+	1.73205i	0.124035	+	0.214834i
$66$		0			0
$67$	−2.50000	+	4.33013i	−0.305424	+	0.529009i	−0.977356	−	0.211604i	$-0.932131\pi$
$67$	−2.50000	+	4.33013i	−0.305424	+	0.529009i	0.671932	+	0.740613i	$0.265465\pi$
$68$	2.50000	+	4.33013i	0.303170	+	0.525105i
$69$		0			0
$70$	4.00000	−	3.46410i	0.478091	−	0.414039i
$71$	1.00000			0.118678			0.0593391	−	0.998238i	$-0.481101\pi$
$71$	1.00000			0.118678			0.0593391	+	0.998238i	$0.481101\pi$
$72$		0			0
$73$	−6.00000	+	10.3923i	−0.702247	+	1.21633i	0.265429	+	0.964130i	$0.414486\pi$
$73$	−6.00000	+	10.3923i	−0.702247	+	1.21633i	−0.967676	+	0.252197i	$0.918847\pi$
$74$	1.00000	−	1.73205i	0.116248	−	0.201347i
$75$		0			0
$76$	1.00000			0.114708
$77$	−6.00000	+	5.19615i	−0.683763	+	0.592157i
$78$		0			0
$79$	3.00000	+	5.19615i	0.337526	+	0.584613i	0.983967	−	0.178352i	$-0.0570765\pi$
$79$	3.00000	+	5.19615i	0.337526	+	0.584613i	−0.646440	+	0.762964i	$0.723743\pi$
$80$	−1.00000	+	1.73205i	−0.111803	+	0.193649i
$81$		0			0
$82$	5.00000	+	8.66025i	0.552158	+	0.956365i
$83$	−16.0000			−1.75623			−0.878114	−	0.478451i	$-0.841198\pi$
$83$	−16.0000			−1.75623			−0.878114	+	0.478451i	$0.841198\pi$
$84$		0			0
$85$	−10.0000			−1.08465
$86$	5.00000	+	8.66025i	0.539164	+	0.933859i
$87$		0			0
$88$	1.50000	−	2.59808i	0.159901	−	0.276956i
$89$	−7.00000	−	12.1244i	−0.741999	−	1.28518i	−0.951584	−	0.307389i	$-0.900545\pi$
$89$	−7.00000	−	12.1244i	−0.741999	−	1.28518i	0.209585	−	0.977790i	$-0.432789\pi$
$90$		0			0
$91$	2.50000	+	0.866025i	0.262071	+	0.0907841i
$92$		0			0
$93$		0			0
$94$	−0.500000	+	0.866025i	−0.0515711	+	0.0893237i
$95$	−1.00000	+	1.73205i	−0.102598	+	0.177705i
$96$		0			0
$97$	4.00000			0.406138			0.203069	−	0.979164i	$-0.434908\pi$
$97$	4.00000			0.406138			0.203069	+	0.979164i	$0.434908\pi$
$98$	1.00000	−	6.92820i	0.101015	−	0.699854i
$99$		0			0
$100$	0.500000	+	0.866025i	0.0500000	+	0.0866025i
$101$	1.00000	−	1.73205i	0.0995037	−	0.172345i	−0.811976	−	0.583691i	$-0.801608\pi$
$101$	1.00000	−	1.73205i	0.0995037	−	0.172345i	0.911479	+	0.411346i	$0.134941\pi$
$102$		0			0
$103$	1.00000	+	1.73205i	0.0985329	+	0.170664i	0.911078	−	0.412235i	$-0.135252\pi$
$103$	1.00000	+	1.73205i	0.0985329	+	0.170664i	−0.812545	+	0.582899i	$0.801918\pi$
$104$	−1.00000			−0.0980581
$105$		0			0
$106$	3.00000			0.291386
$107$	6.00000	+	10.3923i	0.580042	+	1.00466i	0.995474	+	0.0950377i	$0.0302972\pi$
$107$	6.00000	+	10.3923i	0.580042	+	1.00466i	−0.415432	+	0.909624i	$0.636370\pi$
$108$		0			0
$109$	8.00000	−	13.8564i	0.766261	−	1.32720i	−0.173316	−	0.984866i	$-0.555448\pi$
$109$	8.00000	−	13.8564i	0.766261	−	1.32720i	0.939577	−	0.342337i	$-0.111218\pi$
$110$	3.00000	+	5.19615i	0.286039	+	0.495434i
$111$		0			0
$112$	0.500000	+	2.59808i	0.0472456	+	0.245495i
$113$	−15.0000			−1.41108			−0.705541	−	0.708669i	$-0.749296\pi$
$113$	−15.0000			−1.41108			−0.705541	+	0.708669i	$0.749296\pi$
$114$		0			0
$115$		0			0
$116$	−0.500000	+	0.866025i	−0.0464238	+	0.0804084i
$117$		0			0
$118$	−3.00000			−0.276172
$119$	−10.0000	+	8.66025i	−0.916698	+	0.793884i
$120$		0			0
$121$	1.00000	+	1.73205i	0.0909091	+	0.157459i
$122$	2.50000	−	4.33013i	0.226339	−	0.392031i
$123$		0			0
$124$	−2.00000	−	3.46410i	−0.179605	−	0.311086i
$125$	−12.0000			−1.07331
$126$		0			0
$127$		0			0			−	1.00000i	$-0.5\pi$
$127$		0			0				1.00000i	$0.5\pi$
$128$	−0.500000	−	0.866025i	−0.0441942	−	0.0765466i
$129$		0			0
$130$	1.00000	−	1.73205i	0.0877058	−	0.151911i
$131$	−2.00000	−	3.46410i	−0.174741	−	0.302660i	0.765331	−	0.643637i	$-0.222575\pi$
$131$	−2.00000	−	3.46410i	−0.174741	−	0.302660i	−0.940072	+	0.340977i	$0.889242\pi$
$132$		0			0
$133$	0.500000	+	2.59808i	0.0433555	+	0.225282i
$134$	5.00000			0.431934
$135$		0			0
$136$	2.50000	−	4.33013i	0.214373	−	0.371305i
$137$	9.00000	−	15.5885i	0.768922	−	1.33181i	−0.169226	−	0.985577i	$-0.554127\pi$
$137$	9.00000	−	15.5885i	0.768922	−	1.33181i	0.938148	−	0.346235i	$-0.112540\pi$
$138$		0			0
$139$	2.00000			0.169638			0.0848189	−	0.996396i	$-0.472969\pi$
$139$	2.00000			0.169638			0.0848189	+	0.996396i	$0.472969\pi$
$140$	−5.00000	−	1.73205i	−0.422577	−	0.146385i
$141$		0			0
$142$	−0.500000	−	0.866025i	−0.0419591	−	0.0726752i
$143$	−1.50000	+	2.59808i	−0.125436	+	0.217262i
$144$		0			0
$145$	−1.00000	−	1.73205i	−0.0830455	−	0.143839i
$146$	12.0000			0.993127
$147$		0			0
$148$	−2.00000			−0.164399
$149$	4.00000	+	6.92820i	0.327693	+	0.567581i	0.982054	−	0.188602i	$-0.0603956\pi$
$149$	4.00000	+	6.92820i	0.327693	+	0.567581i	−0.654361	+	0.756182i	$0.727062\pi$
$150$		0			0
$151$	−9.50000	+	16.4545i	−0.773099	+	1.33905i	0.162758	+	0.986666i	$0.447961\pi$
$151$	−9.50000	+	16.4545i	−0.773099	+	1.33905i	−0.935857	+	0.352381i	$0.885372\pi$
$152$	−0.500000	−	0.866025i	−0.0405554	−	0.0702439i
$153$		0			0
$154$	7.50000	+	2.59808i	0.604367	+	0.209359i
$155$	8.00000			0.642575
$156$		0			0
$157$	−10.5000	+	18.1865i	−0.837991	+	1.45144i	0.0535803	+	0.998564i	$0.482937\pi$
$157$	−10.5000	+	18.1865i	−0.837991	+	1.45144i	−0.891572	+	0.452880i	$0.850397\pi$
$158$	3.00000	−	5.19615i	0.238667	−	0.413384i
$159$		0			0
$160$	2.00000			0.158114
$161$		0			0
$162$		0			0
$163$	−9.50000	−	16.4545i	−0.744097	−	1.28881i	−0.950615	−	0.310372i	$-0.899546\pi$
$163$	−9.50000	−	16.4545i	−0.744097	−	1.28881i	0.206518	−	0.978443i	$-0.433787\pi$
$164$	5.00000	−	8.66025i	0.390434	−	0.676252i
$165$		0			0
$166$	8.00000	+	13.8564i	0.620920	+	1.07547i
$167$	−21.0000			−1.62503			−0.812514	−	0.582941i	$-0.801902\pi$
$167$	−21.0000			−1.62503			−0.812514	+	0.582941i	$0.801902\pi$
$168$		0			0
$169$	1.00000			0.0769231
$170$	5.00000	+	8.66025i	0.383482	+	0.664211i
$171$		0			0
$172$	5.00000	−	8.66025i	0.381246	−	0.660338i
$173$	1.50000	+	2.59808i	0.114043	+	0.197528i	0.917397	−	0.397974i	$-0.130287\pi$
$173$	1.50000	+	2.59808i	0.114043	+	0.197528i	−0.803354	+	0.595502i	$0.796953\pi$
$174$		0			0
$175$	−2.00000	+	1.73205i	−0.151186	+	0.130931i
$176$	−3.00000			−0.226134
$177$		0			0
$178$	−7.00000	+	12.1244i	−0.524672	+	0.908759i
$179$	1.00000	−	1.73205i	0.0747435	−	0.129460i	−0.826231	−	0.563331i	$-0.809520\pi$
$179$	1.00000	−	1.73205i	0.0747435	−	0.129460i	0.900975	+	0.433872i	$0.142853\pi$
$180$		0			0
$181$	3.00000			0.222988			0.111494	−	0.993765i	$-0.464436\pi$
$181$	3.00000			0.222988			0.111494	+	0.993765i	$0.464436\pi$
$182$	−0.500000	−	2.59808i	−0.0370625	−	0.192582i
$183$		0			0
$184$		0			0
$185$	2.00000	−	3.46410i	0.147043	−	0.254686i
$186$		0			0
$187$	−7.50000	−	12.9904i	−0.548454	−	0.949951i
$188$	1.00000			0.0729325
$189$		0			0
$190$	2.00000			0.145095
$191$	10.0000	+	17.3205i	0.723575	+	1.25327i	0.959558	+	0.281511i	$0.0908356\pi$
$191$	10.0000	+	17.3205i	0.723575	+	1.25327i	−0.235983	+	0.971757i	$0.575831\pi$
$192$		0			0
$193$	7.00000	−	12.1244i	0.503871	−	0.872730i	−0.496119	−	0.868255i	$-0.665242\pi$
$193$	7.00000	−	12.1244i	0.503871	−	0.872730i	0.999990	−	0.00447566i	$-0.00142465\pi$
$194$	−2.00000	−	3.46410i	−0.143592	−	0.248708i
$195$		0			0
$196$	−6.50000	+	2.59808i	−0.464286	+	0.185577i
$197$	24.0000			1.70993			0.854965	−	0.518686i	$-0.173579\pi$
$197$	24.0000			1.70993			0.854965	+	0.518686i	$0.173579\pi$
$198$		0			0
$199$	10.0000	−	17.3205i	0.708881	−	1.22782i	−0.256391	−	0.966573i	$-0.582534\pi$
$199$	10.0000	−	17.3205i	0.708881	−	1.22782i	0.965272	−	0.261245i	$-0.0841331\pi$
$200$	0.500000	−	0.866025i	0.0353553	−	0.0612372i
$201$		0			0
$202$	−2.00000			−0.140720
$203$	−2.50000	−	0.866025i	−0.175466	−	0.0607831i
$204$		0			0
$205$	10.0000	+	17.3205i	0.698430	+	1.20972i
$206$	1.00000	−	1.73205i	0.0696733	−	0.120678i
$207$		0			0
$208$	0.500000	+	0.866025i	0.0346688	+	0.0600481i
$209$	−3.00000			−0.207514
$210$		0			0
$211$	−16.0000			−1.10149			−0.550743	−	0.834675i	$-0.685655\pi$
$211$	−16.0000			−1.10149			−0.550743	+	0.834675i	$0.685655\pi$
$212$	−1.50000	−	2.59808i	−0.103020	−	0.178437i
$213$		0			0
$214$	6.00000	−	10.3923i	0.410152	−	0.710403i
$215$	10.0000	+	17.3205i	0.681994	+	1.18125i
$216$		0			0
$217$	8.00000	−	6.92820i	0.543075	−	0.470317i
$218$	−16.0000			−1.08366
$219$		0			0
$220$	3.00000	−	5.19615i	0.202260	−	0.350325i
$221$	−2.50000	+	4.33013i	−0.168168	+	0.291276i
$222$		0			0
$223$	11.0000			0.736614			0.368307	−	0.929704i	$-0.379937\pi$
$223$	11.0000			0.736614			0.368307	+	0.929704i	$0.379937\pi$
$224$	2.00000	−	1.73205i	0.133631	−	0.115728i
$225$		0			0
$226$	7.50000	+	12.9904i	0.498893	+	0.864107i
$227$	4.00000	−	6.92820i	0.265489	−	0.459841i	−0.702202	−	0.711977i	$-0.747800\pi$
$227$	4.00000	−	6.92820i	0.265489	−	0.459841i	0.967692	+	0.252136i	$0.0811332\pi$
$228$		0			0
$229$	−3.00000	−	5.19615i	−0.198246	−	0.343371i	0.749714	−	0.661762i	$-0.230191\pi$
$229$	−3.00000	−	5.19615i	−0.198246	−	0.343371i	−0.947960	+	0.318390i	$0.896858\pi$
$230$		0			0
$231$		0			0
$232$	1.00000			0.0656532
$233$	5.50000	+	9.52628i	0.360317	+	0.624087i	0.988013	−	0.154371i	$-0.0493352\pi$
$233$	5.50000	+	9.52628i	0.360317	+	0.624087i	−0.627696	+	0.778459i	$0.716002\pi$
$234$		0			0
$235$	−1.00000	+	1.73205i	−0.0652328	+	0.112987i
$236$	1.50000	+	2.59808i	0.0976417	+	0.169120i
$237$		0			0
$238$	12.5000	+	4.33013i	0.810255	+	0.280680i
$239$	−21.0000			−1.35838			−0.679189	−	0.733964i	$-0.737668\pi$
$239$	−21.0000			−1.35838			−0.679189	+	0.733964i	$0.737668\pi$
$240$		0			0
$241$	−2.00000	+	3.46410i	−0.128831	+	0.223142i	−0.923224	−	0.384262i	$-0.874456\pi$
$241$	−2.00000	+	3.46410i	−0.128831	+	0.223142i	0.794393	+	0.607404i	$0.207789\pi$
$242$	1.00000	−	1.73205i	0.0642824	−	0.111340i
$243$		0			0
$244$	−5.00000			−0.320092
$245$	2.00000	−	13.8564i	0.127775	−	0.885253i
$246$		0			0
$247$	0.500000	+	0.866025i	0.0318142	+	0.0551039i
$248$	−2.00000	+	3.46410i	−0.127000	+	0.219971i
$249$		0			0
$250$	6.00000	+	10.3923i	0.379473	+	0.657267i
$251$	18.0000			1.13615			0.568075	−	0.822977i	$-0.307688\pi$
$251$	18.0000			1.13615			0.568075	+	0.822977i	$0.307688\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−0.500000	+	0.866025i	−0.0312500	+	0.0541266i
$257$	3.00000	+	5.19615i	0.187135	+	0.324127i	0.944294	−	0.329104i	$-0.106747\pi$
$257$	3.00000	+	5.19615i	0.187135	+	0.324127i	−0.757159	+	0.653231i	$0.773413\pi$
$258$		0			0
$259$	−1.00000	−	5.19615i	−0.0621370	−	0.322873i
$260$	−2.00000			−0.124035
$261$		0			0
$262$	−2.00000	+	3.46410i	−0.123560	+	0.214013i
$263$	−3.00000	+	5.19615i	−0.184988	+	0.320408i	−0.943572	−	0.331166i	$-0.892558\pi$
$263$	−3.00000	+	5.19615i	−0.184988	+	0.320408i	0.758585	+	0.651575i	$0.225891\pi$
$264$		0			0
$265$	6.00000			0.368577
$266$	2.00000	−	1.73205i	0.122628	−	0.106199i
$267$		0			0
$268$	−2.50000	−	4.33013i	−0.152712	−	0.264505i
$269$	−4.50000	+	7.79423i	−0.274370	+	0.475223i	−0.969976	−	0.243201i	$-0.921803\pi$
$269$	−4.50000	+	7.79423i	−0.274370	+	0.475223i	0.695606	+	0.718423i	$0.255136\pi$
$270$		0			0
$271$	−5.50000	−	9.52628i	−0.334101	−	0.578680i	0.649211	−	0.760609i	$-0.275099\pi$
$271$	−5.50000	−	9.52628i	−0.334101	−	0.578680i	−0.983312	+	0.181928i	$0.941766\pi$
$272$	−5.00000			−0.303170
$273$		0			0
$274$	−18.0000			−1.08742
$275$	−1.50000	−	2.59808i	−0.0904534	−	0.156670i
$276$		0			0
$277$	13.5000	−	23.3827i	0.811136	−	1.40493i	−0.100933	−	0.994893i	$-0.532183\pi$
$277$	13.5000	−	23.3827i	0.811136	−	1.40493i	0.912069	−	0.410036i	$-0.134484\pi$
$278$	−1.00000	−	1.73205i	−0.0599760	−	0.103882i
$279$		0			0
$280$	1.00000	+	5.19615i	0.0597614	+	0.310530i
$281$	−16.0000			−0.954480			−0.477240	−	0.878773i	$-0.658363\pi$
$281$	−16.0000			−0.954480			−0.477240	+	0.878773i	$0.658363\pi$
$282$		0			0
$283$	−2.00000	+	3.46410i	−0.118888	+	0.205919i	−0.919327	−	0.393494i	$-0.871266\pi$
$283$	−2.00000	+	3.46410i	−0.118888	+	0.205919i	0.800439	+	0.599414i	$0.204600\pi$
$284$	−0.500000	+	0.866025i	−0.0296695	+	0.0513892i
$285$		0			0
$286$	3.00000			0.177394
$287$	25.0000	+	8.66025i	1.47570	+	0.511199i
$288$		0			0
$289$	−4.00000	−	6.92820i	−0.235294	−	0.407541i
$290$	−1.00000	+	1.73205i	−0.0587220	+	0.101710i
$291$		0			0
$292$	−6.00000	−	10.3923i	−0.351123	−	0.608164i
$293$	12.0000			0.701047			0.350524	−	0.936554i	$-0.386004\pi$
$293$	12.0000			0.701047			0.350524	+	0.936554i	$0.386004\pi$
$294$		0			0
$295$	−6.00000			−0.349334
$296$	1.00000	+	1.73205i	0.0581238	+	0.100673i
$297$		0			0
$298$	4.00000	−	6.92820i	0.231714	−	0.401340i
$299$		0			0
$300$		0			0
$301$	25.0000	+	8.66025i	1.44098	+	0.499169i
$302$	19.0000			1.09333
$303$		0			0
$304$	−0.500000	+	0.866025i	−0.0286770	+	0.0496700i
$305$	5.00000	−	8.66025i	0.286299	−	0.495885i
$306$		0			0
$307$	−27.0000			−1.54097			−0.770486	−	0.637457i	$-0.779986\pi$
$307$	−27.0000			−1.54097			−0.770486	+	0.637457i	$0.779986\pi$
$308$	−1.50000	−	7.79423i	−0.0854704	−	0.444117i
$309$		0			0
$310$	−4.00000	−	6.92820i	−0.227185	−	0.393496i
$311$	7.00000	−	12.1244i	0.396934	−	0.687509i	−0.596412	−	0.802678i	$-0.703408\pi$
$311$	7.00000	−	12.1244i	0.396934	−	0.687509i	0.993346	+	0.115169i	$0.0367410\pi$
$312$		0			0
$313$	−17.0000	−	29.4449i	−0.960897	−	1.66432i	−0.720257	−	0.693708i	$-0.755976\pi$
$313$	−17.0000	−	29.4449i	−0.960897	−	1.66432i	−0.240640	−	0.970614i	$-0.577357\pi$
$314$	21.0000			1.18510
$315$		0			0
$316$	−6.00000			−0.337526
$317$	−3.00000	−	5.19615i	−0.168497	−	0.291845i	0.769395	−	0.638774i	$-0.220558\pi$
$317$	−3.00000	−	5.19615i	−0.168497	−	0.291845i	−0.937892	+	0.346929i	$0.887225\pi$
$318$		0			0
$319$	1.50000	−	2.59808i	0.0839839	−	0.145464i
$320$	−1.00000	−	1.73205i	−0.0559017	−	0.0968246i
$321$		0			0
$322$		0			0
$323$	−5.00000			−0.278207
$324$		0			0
$325$	−0.500000	+	0.866025i	−0.0277350	+	0.0480384i
$326$	−9.50000	+	16.4545i	−0.526156	+	0.911330i
$327$		0			0
$328$	−10.0000			−0.552158
$329$	0.500000	+	2.59808i	0.0275659	+	0.143237i
$330$		0			0
$331$	10.0000	+	17.3205i	0.549650	+	0.952021i	0.998298	+	0.0583130i	$0.0185721\pi$
$331$	10.0000	+	17.3205i	0.549650	+	0.952021i	−0.448649	+	0.893708i	$0.648095\pi$
$332$	8.00000	−	13.8564i	0.439057	−	0.760469i
$333$		0			0
$334$	10.5000	+	18.1865i	0.574534	+	0.995123i
$335$	10.0000			0.546358
$336$		0			0
$337$	−17.0000			−0.926049			−0.463025	−	0.886345i	$-0.653236\pi$
$337$	−17.0000			−0.926049			−0.463025	+	0.886345i	$0.653236\pi$
$338$	−0.500000	−	0.866025i	−0.0271964	−	0.0471056i
$339$		0			0
$340$	5.00000	−	8.66025i	0.271163	−	0.469668i
$341$	6.00000	+	10.3923i	0.324918	+	0.562775i
$342$		0			0
$343$	−10.0000	−	15.5885i	−0.539949	−	0.841698i
$344$	−10.0000			−0.539164
$345$		0			0
$346$	1.50000	−	2.59808i	0.0806405	−	0.139673i
$347$	1.00000	−	1.73205i	0.0536828	−	0.0929814i	−0.837935	−	0.545770i	$-0.816237\pi$
$347$	1.00000	−	1.73205i	0.0536828	−	0.0929814i	0.891618	+	0.452788i	$0.149571\pi$
$348$		0			0
$349$	36.0000			1.92704			0.963518	−	0.267644i	$-0.0862451\pi$
$349$	36.0000			1.92704			0.963518	+	0.267644i	$0.0862451\pi$
$350$	2.50000	+	0.866025i	0.133631	+	0.0462910i
$351$		0			0
$352$	1.50000	+	2.59808i	0.0799503	+	0.138478i
$353$	7.00000	−	12.1244i	0.372572	−	0.645314i	−0.617388	−	0.786659i	$-0.711809\pi$
$353$	7.00000	−	12.1244i	0.372572	−	0.645314i	0.989960	+	0.141344i	$0.0451425\pi$
$354$		0			0
$355$	−1.00000	−	1.73205i	−0.0530745	−	0.0919277i
$356$	14.0000			0.741999
$357$		0			0
$358$	−2.00000			−0.105703
$359$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$359$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$360$		0			0
$361$	9.00000	−	15.5885i	0.473684	−	0.820445i
$362$	−1.50000	−	2.59808i	−0.0788382	−	0.136552i
$363$		0			0
$364$	−2.00000	+	1.73205i	−0.104828	+	0.0907841i
$365$	24.0000			1.25622
$366$		0			0
$367$	−13.0000	+	22.5167i	−0.678594	+	1.17536i	0.296810	+	0.954937i	$0.404077\pi$
$367$	−13.0000	+	22.5167i	−0.678594	+	1.17536i	−0.975404	+	0.220423i	$0.929256\pi$
$368$		0			0
$369$		0			0
$370$	−4.00000			−0.207950
$371$	6.00000	−	5.19615i	0.311504	−	0.269771i
$372$		0			0
$373$	−6.50000	−	11.2583i	−0.336557	−	0.582934i	0.647225	−	0.762299i	$-0.275929\pi$
$373$	−6.50000	−	11.2583i	−0.336557	−	0.582934i	−0.983783	+	0.179364i	$0.942596\pi$
$374$	−7.50000	+	12.9904i	−0.387816	+	0.671717i
$375$		0			0
$376$	−0.500000	−	0.866025i	−0.0257855	−	0.0446619i
$377$	−1.00000			−0.0515026
$378$		0			0
$379$	28.0000			1.43826			0.719132	−	0.694874i	$-0.244540\pi$
$379$	28.0000			1.43826			0.719132	+	0.694874i	$0.244540\pi$
$380$	−1.00000	−	1.73205i	−0.0512989	−	0.0888523i
$381$		0			0
$382$	10.0000	−	17.3205i	0.511645	−	0.886194i
$383$	−16.0000	−	27.7128i	−0.817562	−	1.41606i	−0.907474	−	0.420109i	$-0.861992\pi$
$383$	−16.0000	−	27.7128i	−0.817562	−	1.41606i	0.0899119	−	0.995950i	$-0.471341\pi$
$384$		0			0
$385$	15.0000	+	5.19615i	0.764471	+	0.264820i
$386$	−14.0000			−0.712581
$387$		0			0
$388$	−2.00000	+	3.46410i	−0.101535	+	0.175863i
$389$	4.50000	−	7.79423i	0.228159	−	0.395183i	−0.729103	−	0.684403i	$-0.760063\pi$
$389$	4.50000	−	7.79423i	0.228159	−	0.395183i	0.957263	+	0.289220i	$0.0933960\pi$
$390$		0			0
$391$		0			0
$392$	5.50000	+	4.33013i	0.277792	+	0.218704i
$393$		0			0
$394$	−12.0000	−	20.7846i	−0.604551	−	1.04711i
$395$	6.00000	−	10.3923i	0.301893	−	0.522894i
$396$		0			0
$397$	−18.0000	−	31.1769i	−0.903394	−	1.56472i	−0.823058	−	0.567957i	$-0.807734\pi$
$397$	−18.0000	−	31.1769i	−0.903394	−	1.56472i	−0.0803356	−	0.996768i	$-0.525599\pi$
$398$	−20.0000			−1.00251
$399$		0			0
$400$	−1.00000			−0.0500000
$401$	−18.0000	−	31.1769i	−0.898877	−	1.55690i	−0.828932	−	0.559350i	$-0.811051\pi$
$401$	−18.0000	−	31.1769i	−0.898877	−	1.55690i	−0.0699455	−	0.997551i	$-0.522283\pi$
$402$		0			0
$403$	2.00000	−	3.46410i	0.0996271	−	0.172559i
$404$	1.00000	+	1.73205i	0.0497519	+	0.0861727i
$405$		0			0
$406$	0.500000	+	2.59808i	0.0248146	+	0.128940i
$407$	6.00000			0.297409
$408$		0			0
$409$	15.0000	−	25.9808i	0.741702	−	1.28467i	−0.210017	−	0.977698i	$-0.567352\pi$
$409$	15.0000	−	25.9808i	0.741702	−	1.28467i	0.951720	−	0.306968i	$-0.0993146\pi$
$410$	10.0000	−	17.3205i	0.493865	−	0.855399i
$411$		0			0
$412$	−2.00000			−0.0985329
$413$	−6.00000	+	5.19615i	−0.295241	+	0.255686i
$414$		0			0
$415$	16.0000	+	27.7128i	0.785409	+	1.36037i
$416$	0.500000	−	0.866025i	0.0245145	−	0.0424604i
$417$		0			0
$418$	1.50000	+	2.59808i	0.0733674	+	0.127076i
$419$	−10.0000			−0.488532			−0.244266	−	0.969708i	$-0.578547\pi$
$419$	−10.0000			−0.488532			−0.244266	+	0.969708i	$0.578547\pi$
$420$		0			0
$421$	8.00000			0.389896			0.194948	−	0.980814i	$-0.437546\pi$
$421$	8.00000			0.389896			0.194948	+	0.980814i	$0.437546\pi$
$422$	8.00000	+	13.8564i	0.389434	+	0.674519i
$423$		0			0
$424$	−1.50000	+	2.59808i	−0.0728464	+	0.126174i
$425$	−2.50000	−	4.33013i	−0.121268	−	0.210042i
$426$		0			0
$427$	−2.50000	−	12.9904i	−0.120983	−	0.628649i
$428$	−12.0000			−0.580042
$429$		0			0
$430$	10.0000	−	17.3205i	0.482243	−	0.835269i
$431$	−4.00000	+	6.92820i	−0.192673	+	0.333720i	−0.946135	−	0.323772i	$-0.895049\pi$
$431$	−4.00000	+	6.92820i	−0.192673	+	0.333720i	0.753462	+	0.657491i	$0.228382\pi$
$432$		0			0
$433$	−19.0000			−0.913082			−0.456541	−	0.889702i	$-0.650912\pi$
$433$	−19.0000			−0.913082			−0.456541	+	0.889702i	$0.650912\pi$
$434$	−10.0000	−	3.46410i	−0.480015	−	0.166282i
$435$		0			0
$436$	8.00000	+	13.8564i	0.383131	+	0.663602i
$437$		0			0
$438$		0			0
$439$	−20.0000	−	34.6410i	−0.954548	−	1.65333i	−0.735399	−	0.677634i	$-0.763005\pi$
$439$	−20.0000	−	34.6410i	−0.954548	−	1.65333i	−0.219149	−	0.975691i	$-0.570328\pi$
$440$	−6.00000			−0.286039
$441$		0			0
$442$	5.00000			0.237826
$443$	−9.00000	−	15.5885i	−0.427603	−	0.740630i	0.569057	−	0.822298i	$-0.307309\pi$
$443$	−9.00000	−	15.5885i	−0.427603	−	0.740630i	−0.996660	+	0.0816684i	$0.973975\pi$
$444$		0			0
$445$	−14.0000	+	24.2487i	−0.663664	+	1.14950i
$446$	−5.50000	−	9.52628i	−0.260433	−	0.451082i
$447$		0			0
$448$	−2.50000	−	0.866025i	−0.118114	−	0.0409159i
$449$	8.00000			0.377543			0.188772	−	0.982021i	$-0.439549\pi$
$449$	8.00000			0.377543			0.188772	+	0.982021i	$0.439549\pi$
$450$		0			0
$451$	−15.0000	+	25.9808i	−0.706322	+	1.22339i
$452$	7.50000	−	12.9904i	0.352770	−	0.611016i
$453$		0			0
$454$	−8.00000			−0.375459
$455$	−1.00000	−	5.19615i	−0.0468807	−	0.243599i
$456$		0			0
$457$	8.00000	+	13.8564i	0.374224	+	0.648175i	0.990211	−	0.139581i	$-0.0445757\pi$
$457$	8.00000	+	13.8564i	0.374224	+	0.648175i	−0.615986	+	0.787757i	$0.711242\pi$
$458$	−3.00000	+	5.19615i	−0.140181	+	0.242800i
$459$		0			0
$460$		0			0
$461$	−20.0000			−0.931493			−0.465746	−	0.884918i	$-0.654214\pi$
$461$	−20.0000			−0.931493			−0.465746	+	0.884918i	$0.654214\pi$
$462$		0			0
$463$	8.00000			0.371792			0.185896	−	0.982569i	$-0.440481\pi$
$463$	8.00000			0.371792			0.185896	+	0.982569i	$0.440481\pi$
$464$	−0.500000	−	0.866025i	−0.0232119	−	0.0402042i
$465$		0			0
$466$	5.50000	−	9.52628i	0.254783	−	0.441296i
$467$	−12.0000	−	20.7846i	−0.555294	−	0.961797i	−0.997881	−	0.0650714i	$-0.979272\pi$
$467$	−12.0000	−	20.7846i	−0.555294	−	0.961797i	0.442587	−	0.896726i	$-0.354061\pi$
$468$		0			0
$469$	10.0000	−	8.66025i	0.461757	−	0.399893i
$470$	2.00000			0.0922531
$471$		0			0
$472$	1.50000	−	2.59808i	0.0690431	−	0.119586i
$473$	−15.0000	+	25.9808i	−0.689701	+	1.19460i
$474$		0			0
$475$	−1.00000			−0.0458831
$476$	−2.50000	−	12.9904i	−0.114587	−	0.595413i
$477$		0			0
$478$	10.5000	+	18.1865i	0.480259	+	0.831833i
$479$	−16.5000	+	28.5788i	−0.753904	+	1.30580i	0.192013	+	0.981392i	$0.438498\pi$
$479$	−16.5000	+	28.5788i	−0.753904	+	1.30580i	−0.945917	+	0.324408i	$0.894835\pi$
$480$		0			0
$481$	−1.00000	−	1.73205i	−0.0455961	−	0.0789747i
$482$	4.00000			0.182195
$483$		0			0
$484$	−2.00000			−0.0909091
$485$	−4.00000	−	6.92820i	−0.181631	−	0.314594i
$486$		0			0
$487$	−5.50000	+	9.52628i	−0.249229	+	0.431677i	−0.963312	−	0.268384i	$-0.913510\pi$
$487$	−5.50000	+	9.52628i	−0.249229	+	0.431677i	0.714083	+	0.700061i	$0.246844\pi$
$488$	2.50000	+	4.33013i	0.113170	+	0.196016i
$489$		0			0
$490$	−13.0000	+	5.19615i	−0.587280	+	0.234738i
$491$	−22.0000			−0.992846			−0.496423	−	0.868081i	$-0.665354\pi$
$491$	−22.0000			−0.992846			−0.496423	+	0.868081i	$0.665354\pi$
$492$		0			0
$493$	2.50000	−	4.33013i	0.112594	−	0.195019i
$494$	0.500000	−	0.866025i	0.0224961	−	0.0389643i
$495$		0			0
$496$	4.00000			0.179605
$497$	−2.50000	−	0.866025i	−0.112140	−	0.0388465i
$498$		0			0
$499$	14.0000	+	24.2487i	0.626726	+	1.08552i	0.988204	+	0.153141i	$0.0489388\pi$
$499$	14.0000	+	24.2487i	0.626726	+	1.08552i	−0.361478	+	0.932381i	$0.617728\pi$
$500$	6.00000	−	10.3923i	0.268328	−	0.464758i
$501$		0			0
$502$	−9.00000	−	15.5885i	−0.401690	−	0.695747i
$503$	12.0000			0.535054			0.267527	−	0.963550i	$-0.413794\pi$
$503$	12.0000			0.535054			0.267527	+	0.963550i	$0.413794\pi$
$504$		0			0
$505$	−4.00000			−0.177998
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−14.0000	−	24.2487i	−0.620539	−	1.07481i	−0.989385	−	0.145315i	$-0.953580\pi$
$509$	−14.0000	−	24.2487i	−0.620539	−	1.07481i	0.368846	−	0.929490i	$-0.379753\pi$
$510$		0			0
$511$	24.0000	−	20.7846i	1.06170	−	0.919457i
$512$	1.00000			0.0441942
$513$		0			0
$514$	3.00000	−	5.19615i	0.132324	−	0.229192i
$515$	2.00000	−	3.46410i	0.0881305	−	0.152647i
$516$		0			0
$517$	−3.00000			−0.131940
$518$	−4.00000	+	3.46410i	−0.175750	+	0.152204i
$519$		0			0
$520$	1.00000	+	1.73205i	0.0438529	+	0.0759555i
$521$	21.0000	−	36.3731i	0.920027	−	1.59353i	0.120656	−	0.992694i	$-0.461500\pi$
$521$	21.0000	−	36.3731i	0.920027	−	1.59353i	0.799370	−	0.600839i	$-0.205167\pi$
$522$		0			0
$523$	−3.00000	−	5.19615i	−0.131181	−	0.227212i	0.792951	−	0.609285i	$-0.208544\pi$
$523$	−3.00000	−	5.19615i	−0.131181	−	0.227212i	−0.924132	+	0.382073i	$0.875210\pi$
$524$	4.00000			0.174741
$525$		0			0
$526$	6.00000			0.261612
$527$	10.0000	+	17.3205i	0.435607	+	0.754493i
$528$		0			0
$529$	11.5000	−	19.9186i	0.500000	−	0.866025i
$530$	−3.00000	−	5.19615i	−0.130312	−	0.225706i
$531$		0			0
$532$	−2.50000	−	0.866025i	−0.108389	−	0.0375470i
$533$	10.0000			0.433148
$534$		0			0
$535$	12.0000	−	20.7846i	0.518805	−	0.898597i
$536$	−2.50000	+	4.33013i	−0.107984	+	0.187033i
$537$		0			0
$538$	9.00000			0.388018
$539$	19.5000	−	7.79423i	0.839924	−	0.335721i
$540$		0			0
$541$	17.0000	+	29.4449i	0.730887	+	1.26593i	0.956504	+	0.291718i	$0.0942267\pi$
$541$	17.0000	+	29.4449i	0.730887	+	1.26593i	−0.225617	+	0.974216i	$0.572440\pi$
$542$	−5.50000	+	9.52628i	−0.236245	+	0.409189i
$543$		0			0
$544$	2.50000	+	4.33013i	0.107187	+	0.185653i
$545$	−32.0000			−1.37073
$546$		0			0
$547$	28.0000			1.19719			0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000			1.19719			0.598597	+	0.801050i	$0.295725\pi$
$548$	9.00000	+	15.5885i	0.384461	+	0.665906i
$549$		0			0
$550$	−1.50000	+	2.59808i	−0.0639602	+	0.110782i
$551$	−0.500000	−	0.866025i	−0.0213007	−	0.0368939i
$552$		0			0
$553$	−3.00000	−	15.5885i	−0.127573	−	0.662889i
$554$	−27.0000			−1.14712
$555$		0			0
$556$	−1.00000	+	1.73205i	−0.0424094	+	0.0734553i
$557$	9.00000	−	15.5885i	0.381342	−	0.660504i	−0.609912	−	0.792469i	$-0.708795\pi$
$557$	9.00000	−	15.5885i	0.381342	−	0.660504i	0.991254	+	0.131965i	$0.0421286\pi$
$558$		0			0
$559$	10.0000			0.422955
$560$	4.00000	−	3.46410i	0.169031	−	0.146385i
$561$		0			0
$562$	8.00000	+	13.8564i	0.337460	+	0.584497i
$563$	12.0000	−	20.7846i	0.505740	−	0.875967i	−0.494238	−	0.869326i	$-0.664553\pi$
$563$	12.0000	−	20.7846i	0.505740	−	0.875967i	0.999978	−	0.00664037i	$-0.00211371\pi$
$564$		0			0
$565$	15.0000	+	25.9808i	0.631055	+	1.09302i
$566$	4.00000			0.168133
$567$		0			0
$568$	1.00000			0.0419591
$569$	9.50000	+	16.4545i	0.398261	+	0.689808i	0.993511	−	0.113732i	$-0.0362806\pi$
$569$	9.50000	+	16.4545i	0.398261	+	0.689808i	−0.595251	+	0.803540i	$0.702947\pi$
$570$		0			0
$571$	−18.0000	+	31.1769i	−0.753277	+	1.30471i	0.192950	+	0.981209i	$0.438194\pi$
$571$	−18.0000	+	31.1769i	−0.753277	+	1.30471i	−0.946227	+	0.323505i	$0.895139\pi$
$572$	−1.50000	−	2.59808i	−0.0627182	−	0.108631i
$573$		0			0
$574$	−5.00000	−	25.9808i	−0.208696	−	1.08442i
$575$		0			0
$576$		0			0
$577$	−19.0000	+	32.9090i	−0.790980	+	1.37002i	0.134380	+	0.990930i	$0.457096\pi$
$577$	−19.0000	+	32.9090i	−0.790980	+	1.37002i	−0.925361	+	0.379088i	$0.876238\pi$
$578$	−4.00000	+	6.92820i	−0.166378	+	0.288175i
$579$		0			0
$580$	2.00000			0.0830455
$581$	40.0000	+	13.8564i	1.65948	+	0.574861i
$582$		0			0
$583$	4.50000	+	7.79423i	0.186371	+	0.322804i
$584$	−6.00000	+	10.3923i	−0.248282	+	0.430037i
$585$		0			0
$586$	−6.00000	−	10.3923i	−0.247858	−	0.429302i
$587$	−31.0000			−1.27951			−0.639753	−	0.768580i	$-0.720964\pi$
$587$	−31.0000			−1.27951			−0.639753	+	0.768580i	$0.720964\pi$
$588$		0			0
$589$	4.00000			0.164817
$590$	3.00000	+	5.19615i	0.123508	+	0.213922i
$591$		0			0
$592$	1.00000	−	1.73205i	0.0410997	−	0.0711868i
$593$	7.00000	+	12.1244i	0.287456	+	0.497888i	0.973202	−	0.229953i	$-0.0738573\pi$
$593$	7.00000	+	12.1244i	0.287456	+	0.497888i	−0.685746	+	0.727841i	$0.740524\pi$
$594$		0			0
$595$	25.0000	+	8.66025i	1.02490	+	0.355036i
$596$	−8.00000			−0.327693
$597$		0			0
$598$		0			0
$599$	−19.0000	+	32.9090i	−0.776319	+	1.34462i	0.157731	+	0.987482i	$0.449582\pi$
$599$	−19.0000	+	32.9090i	−0.776319	+	1.34462i	−0.934050	+	0.357142i	$0.883751\pi$
$600$		0			0
$601$	−19.0000			−0.775026			−0.387513	−	0.921864i	$-0.626666\pi$
$601$	−19.0000			−0.775026			−0.387513	+	0.921864i	$0.626666\pi$
$602$	−5.00000	−	25.9808i	−0.203785	−	1.05890i
$603$		0			0
$604$	−9.50000	−	16.4545i	−0.386550	−	0.669523i
$605$	2.00000	−	3.46410i	0.0813116	−	0.140836i
$606$		0			0
$607$	6.00000	+	10.3923i	0.243532	+	0.421811i	0.961718	−	0.274041i	$-0.0883604\pi$
$607$	6.00000	+	10.3923i	0.243532	+	0.421811i	−0.718186	+	0.695852i	$0.755027\pi$
$608$	1.00000			0.0405554
$609$		0			0
$610$	−10.0000			−0.404888
$611$	0.500000	+	0.866025i	0.0202278	+	0.0350356i
$612$		0			0
$613$	8.00000	−	13.8564i	0.323117	−	0.559655i	−0.658012	−	0.753007i	$-0.728603\pi$
$613$	8.00000	−	13.8564i	0.323117	−	0.559655i	0.981129	+	0.193352i	$0.0619359\pi$
$614$	13.5000	+	23.3827i	0.544816	+	0.943648i
$615$		0			0
$616$	−6.00000	+	5.19615i	−0.241747	+	0.209359i
$617$	−34.0000			−1.36879			−0.684394	−	0.729112i	$-0.739933\pi$
$617$	−34.0000			−1.36879			−0.684394	+	0.729112i	$0.739933\pi$
$618$		0			0
$619$	2.00000	−	3.46410i	0.0803868	−	0.139234i	−0.823029	−	0.567999i	$-0.807718\pi$
$619$	2.00000	−	3.46410i	0.0803868	−	0.139234i	0.903416	+	0.428765i	$0.141051\pi$
$620$	−4.00000	+	6.92820i	−0.160644	+	0.278243i
$621$		0			0
$622$	−14.0000			−0.561349
$623$	7.00000	+	36.3731i	0.280449	+	1.45726i
$624$		0			0
$625$	9.50000	+	16.4545i	0.380000	+	0.658179i
$626$	−17.0000	+	29.4449i	−0.679457	+	1.17685i
$627$		0			0
$628$	−10.5000	−	18.1865i	−0.418996	−	0.725722i
$629$	10.0000			0.398726
$630$		0			0
$631$	48.0000			1.91085			0.955425	−	0.295234i	$-0.0953977\pi$
$631$	48.0000			1.91085			0.955425	+	0.295234i	$0.0953977\pi$
$632$	3.00000	+	5.19615i	0.119334	+	0.206692i
$633$		0			0
$634$	−3.00000	+	5.19615i	−0.119145	+	0.206366i
$635$		0			0
$636$		0			0
$637$	−5.50000	−	4.33013i	−0.217918	−	0.171566i
$638$	−3.00000			−0.118771
$639$		0			0
$640$	−1.00000	+	1.73205i	−0.0395285	+	0.0684653i
$641$	15.0000	−	25.9808i	0.592464	−	1.02618i	−0.401435	−	0.915888i	$-0.631488\pi$
$641$	15.0000	−	25.9808i	0.592464	−	1.02618i	0.993899	−	0.110291i	$-0.0351782\pi$
$642$		0			0
$643$	−19.0000			−0.749287			−0.374643	−	0.927169i	$-0.622235\pi$
$643$	−19.0000			−0.749287			−0.374643	+	0.927169i	$0.622235\pi$
$644$		0			0
$645$		0			0
$646$	2.50000	+	4.33013i	0.0983612	+	0.170367i
$647$	−19.0000	+	32.9090i	−0.746967	+	1.29378i	0.202303	+	0.979323i	$0.435157\pi$
$647$	−19.0000	+	32.9090i	−0.746967	+	1.29378i	−0.949270	+	0.314462i	$0.898176\pi$
$648$		0			0
$649$	−4.50000	−	7.79423i	−0.176640	−	0.305950i
$650$	1.00000			0.0392232
$651$		0			0
$652$	19.0000			0.744097
$653$	9.00000	+	15.5885i	0.352197	+	0.610023i	0.986634	−	0.162951i	$-0.0521013\pi$
$653$	9.00000	+	15.5885i	0.352197	+	0.610023i	−0.634437	+	0.772975i	$0.718768\pi$
$654$		0			0
$655$	−4.00000	+	6.92820i	−0.156293	+	0.270707i
$656$	5.00000	+	8.66025i	0.195217	+	0.338126i
$657$		0			0
$658$	2.00000	−	1.73205i	0.0779681	−	0.0675224i
$659$	−6.00000			−0.233727			−0.116863	−	0.993148i	$-0.537284\pi$
$659$	−6.00000			−0.233727			−0.116863	+	0.993148i	$0.537284\pi$
$660$		0			0
$661$	13.0000	−	22.5167i	0.505641	−	0.875797i	−0.494337	−	0.869270i	$-0.664589\pi$
$661$	13.0000	−	22.5167i	0.505641	−	0.875797i	0.999979	−	0.00652642i	$-0.00207744\pi$
$662$	10.0000	−	17.3205i	0.388661	−	0.673181i
$663$		0			0
$664$	−16.0000			−0.620920
$665$	4.00000	−	3.46410i	0.155113	−	0.134332i
$666$		0			0
$667$		0			0
$668$	10.5000	−	18.1865i	0.406257	−	0.703658i
$669$		0			0
$670$	−5.00000	−	8.66025i	−0.193167	−	0.334575i
$671$	15.0000			0.579069
$672$		0			0
$673$	14.0000			0.539660			0.269830	−	0.962908i	$-0.413032\pi$
$673$	14.0000			0.539660			0.269830	+	0.962908i	$0.413032\pi$
$674$	8.50000	+	14.7224i	0.327408	+	0.567087i
$675$		0			0
$676$	−0.500000	+	0.866025i	−0.0192308	+	0.0333087i
$677$	−1.50000	−	2.59808i	−0.0576497	−	0.0998522i	0.835760	−	0.549095i	$-0.185027\pi$
$677$	−1.50000	−	2.59808i	−0.0576497	−	0.0998522i	−0.893410	+	0.449242i	$0.851694\pi$
$678$		0			0
$679$	−10.0000	−	3.46410i	−0.383765	−	0.132940i
$680$	−10.0000			−0.383482
$681$		0			0
$682$	6.00000	−	10.3923i	0.229752	−	0.397942i
$683$	−6.00000	+	10.3923i	−0.229584	+	0.397650i	−0.957685	−	0.287819i	$-0.907070\pi$
$683$	−6.00000	+	10.3923i	−0.229584	+	0.397650i	0.728101	+	0.685470i	$0.240403\pi$
$684$		0			0
$685$	−36.0000			−1.37549
$686$	−8.50000	+	16.4545i	−0.324532	+	0.628235i
$687$		0			0
$688$	5.00000	+	8.66025i	0.190623	+	0.330169i
$689$	1.50000	−	2.59808i	0.0571454	−	0.0989788i
$690$		0			0
$691$	8.50000	+	14.7224i	0.323355	+	0.560068i	0.981178	−	0.193105i	$-0.0618558\pi$
$691$	8.50000	+	14.7224i	0.323355	+	0.560068i	−0.657823	+	0.753173i	$0.728522\pi$
$692$	−3.00000			−0.114043
$693$		0			0
$694$	−2.00000			−0.0759190
$695$	−2.00000	−	3.46410i	−0.0758643	−	0.131401i
$696$		0			0
$697$	−25.0000	+	43.3013i	−0.946943	+	1.64015i
$698$	−18.0000	−	31.1769i	−0.681310	−	1.18006i
$699$		0			0
$700$	−0.500000	−	2.59808i	−0.0188982	−	0.0981981i
$701$	50.0000			1.88847			0.944237	−	0.329267i	$-0.106802\pi$
$701$	50.0000			1.88847			0.944237	+	0.329267i	$0.106802\pi$
$702$		0			0
$703$	1.00000	−	1.73205i	0.0377157	−	0.0653255i
$704$	1.50000	−	2.59808i	0.0565334	−	0.0979187i
$705$		0			0
$706$	−14.0000			−0.526897
$707$	−4.00000	+	3.46410i	−0.150435	+	0.130281i
$708$		0			0
$709$	−10.0000	−	17.3205i	−0.375558	−	0.650485i	0.614852	−	0.788642i	$-0.289216\pi$
$709$	−10.0000	−	17.3205i	−0.375558	−	0.650485i	−0.990410	+	0.138157i	$0.955882\pi$
$710$	−1.00000	+	1.73205i	−0.0375293	+	0.0650027i
$711$		0			0
$712$	−7.00000	−	12.1244i	−0.262336	−	0.454379i
$713$		0			0
$714$		0			0
$715$	6.00000			0.224387
$716$	1.00000	+	1.73205i	0.0373718	+	0.0647298i
$717$		0			0
$718$		0			0
$719$	−9.00000	−	15.5885i	−0.335643	−	0.581351i	0.647965	−	0.761670i	$-0.275620\pi$
$719$	−9.00000	−	15.5885i	−0.335643	−	0.581351i	−0.983608	+	0.180319i	$0.942287\pi$
$720$		0			0
$721$	−1.00000	−	5.19615i	−0.0372419	−	0.193515i
$722$	−18.0000			−0.669891
$723$		0			0
$724$	−1.50000	+	2.59808i	−0.0557471	+	0.0965567i
$725$	0.500000	−	0.866025i	0.0185695	−	0.0321634i
$726$		0			0
$727$	12.0000			0.445055			0.222528	−	0.974926i	$-0.428569\pi$
$727$	12.0000			0.445055			0.222528	+	0.974926i	$0.428569\pi$
$728$	2.50000	+	0.866025i	0.0926562	+	0.0320970i
$729$		0			0
$730$	−12.0000	−	20.7846i	−0.444140	−	0.769273i
$731$	−25.0000	+	43.3013i	−0.924658	+	1.60156i
$732$		0			0
$733$	20.0000	+	34.6410i	0.738717	+	1.27950i	0.953073	+	0.302740i	$0.0979013\pi$
$733$	20.0000	+	34.6410i	0.738717	+	1.27950i	−0.214356	+	0.976756i	$0.568765\pi$
$734$	26.0000			0.959678
$735$		0			0
$736$		0			0
$737$	7.50000	+	12.9904i	0.276266	+	0.478507i
$738$		0			0
$739$	−20.0000	+	34.6410i	−0.735712	+	1.27429i	0.218698	+	0.975793i	$0.429819\pi$
$739$	−20.0000	+	34.6410i	−0.735712	+	1.27429i	−0.954410	+	0.298498i	$0.903514\pi$
$740$	2.00000	+	3.46410i	0.0735215	+	0.127343i
$741$		0			0
$742$	−7.50000	−	2.59808i	−0.275334	−	0.0953784i
$743$	3.00000			0.110059			0.0550297	−	0.998485i	$-0.482475\pi$
$743$	3.00000			0.110059			0.0550297	+	0.998485i	$0.482475\pi$
$744$		0			0
$745$	8.00000	−	13.8564i	0.293097	−	0.507659i
$746$	−6.50000	+	11.2583i	−0.237982	+	0.412197i
$747$		0			0
$748$	15.0000			0.548454
$749$	−6.00000	−	31.1769i	−0.219235	−	1.13918i
$750$		0			0
$751$	6.00000	+	10.3923i	0.218943	+	0.379221i	0.954485	−	0.298259i	$-0.0964058\pi$
$751$	6.00000	+	10.3923i	0.218943	+	0.379221i	−0.735542	+	0.677479i	$0.763072\pi$
$752$	−0.500000	+	0.866025i	−0.0182331	+	0.0315807i
$753$		0			0
$754$	0.500000	+	0.866025i	0.0182089	+	0.0315388i
$755$	38.0000			1.38296
$756$		0			0
$757$	−37.0000			−1.34479			−0.672394	−	0.740193i	$-0.734734\pi$
$757$	−37.0000			−1.34479			−0.672394	+	0.740193i	$0.734734\pi$
$758$	−14.0000	−	24.2487i	−0.508503	−	0.880753i
$759$		0			0
$760$	−1.00000	+	1.73205i	−0.0362738	+	0.0628281i
$761$	6.00000	+	10.3923i	0.217500	+	0.376721i	0.954043	−	0.299670i	$-0.0968765\pi$
$761$	6.00000	+	10.3923i	0.217500	+	0.376721i	−0.736543	+	0.676391i	$0.763543\pi$
$762$		0			0
$763$	−32.0000	+	27.7128i	−1.15848	+	1.00327i
$764$	−20.0000			−0.723575
$765$		0			0
$766$	−16.0000	+	27.7128i	−0.578103	+	1.00130i
$767$	−1.50000	+	2.59808i	−0.0541619	+	0.0938111i
$768$		0			0
$769$	−44.0000			−1.58668			−0.793340	−	0.608778i	$-0.791660\pi$
$769$	−44.0000			−1.58668			−0.793340	+	0.608778i	$0.791660\pi$
$770$	−3.00000	−	15.5885i	−0.108112	−	0.561769i
$771$		0			0
$772$	7.00000	+	12.1244i	0.251936	+	0.436365i
$773$	13.0000	−	22.5167i	0.467578	−	0.809868i	−0.531736	−	0.846910i	$-0.678460\pi$
$773$	13.0000	−	22.5167i	0.467578	−	0.809868i	0.999314	+	0.0370420i	$0.0117935\pi$
$774$		0			0
$775$	2.00000	+	3.46410i	0.0718421	+	0.124434i
$776$	4.00000			0.143592
$777$		0			0
$778$	−9.00000			−0.322666
$779$	5.00000	+	8.66025i	0.179144	+	0.310286i
$780$		0			0
$781$	1.50000	−	2.59808i	0.0536742	−	0.0929665i
$782$		0			0
$783$		0			0
$784$	1.00000	−	6.92820i	0.0357143	−	0.247436i
$785$	42.0000			1.49904
$786$		0			0
$787$	2.50000	−	4.33013i	0.0891154	−	0.154352i	−0.818022	−	0.575187i	$-0.804929\pi$
$787$	2.50000	−	4.33013i	0.0891154	−	0.154352i	0.907137	+	0.420834i	$0.138263\pi$
$788$	−12.0000	+	20.7846i	−0.427482	+	0.740421i
$789$		0			0
$790$	−12.0000			−0.426941
$791$	37.5000	+	12.9904i	1.33335	+	0.461885i
$792$		0			0
$793$	−2.50000	−	4.33013i	−0.0887776	−	0.153767i
$794$	−18.0000	+	31.1769i	−0.638796	+	1.10643i
$795$		0			0
$796$	10.0000	+	17.3205i	0.354441	+	0.613909i
$797$	−2.00000			−0.0708436			−0.0354218	−	0.999372i	$-0.511277\pi$
$797$	−2.00000			−0.0708436			−0.0354218	+	0.999372i	$0.511277\pi$
$798$		0			0
$799$	−5.00000			−0.176887
$800$	0.500000	+	0.866025i	0.0176777	+	0.0306186i
$801$		0			0
$802$	−18.0000	+	31.1769i	−0.635602	+	1.10090i
$803$	18.0000	+	31.1769i	0.635206	+	1.10021i
$804$		0			0
$805$		0			0
$806$	−4.00000			−0.140894
$807$		0			0
$808$	1.00000	−	1.73205i	0.0351799	−	0.0609333i
$809$	−2.50000	+	4.33013i	−0.0878953	+	0.152239i	−0.906621	−	0.421945i	$-0.861347\pi$
$809$	−2.50000	+	4.33013i	−0.0878953	+	0.152239i	0.818726	+	0.574184i	$0.194681\pi$
$810$		0			0
$811$	−28.0000			−0.983213			−0.491606	−	0.870817i	$-0.663590\pi$
$811$	−28.0000			−0.983213			−0.491606	+	0.870817i	$0.663590\pi$
$812$	2.00000	−	1.73205i	0.0701862	−	0.0607831i
$813$		0			0
$814$	−3.00000	−	5.19615i	−0.105150	−	0.182125i
$815$	−19.0000	+	32.9090i	−0.665541	+	1.15275i
$816$		0			0
$817$	5.00000	+	8.66025i	0.174928	+	0.302984i
$818$	−30.0000			−1.04893
$819$		0			0
$820$	−20.0000			−0.698430
$821$	21.0000	+	36.3731i	0.732905	+	1.26943i	0.955636	+	0.294549i	$0.0951694\pi$
$821$	21.0000	+	36.3731i	0.732905	+	1.26943i	−0.222731	+	0.974880i	$0.571497\pi$
$822$		0			0
$823$	15.0000	−	25.9808i	0.522867	−	0.905632i	−0.476779	−	0.879023i	$-0.658196\pi$
$823$	15.0000	−	25.9808i	0.522867	−	0.905632i	0.999646	−	0.0266091i	$-0.00847095\pi$
$824$	1.00000	+	1.73205i	0.0348367	+	0.0603388i
$825$		0			0
$826$	7.50000	+	2.59808i	0.260958	+	0.0903986i
$827$	35.0000			1.21707			0.608535	−	0.793527i	$-0.291758\pi$
$827$	35.0000			1.21707			0.608535	+	0.793527i	$0.291758\pi$
$828$		0			0
$829$	9.50000	−	16.4545i	0.329949	−	0.571488i	−0.652553	−	0.757743i	$-0.726302\pi$
$829$	9.50000	−	16.4545i	0.329949	−	0.571488i	0.982501	+	0.186256i	$0.0596352\pi$
$830$	16.0000	−	27.7128i	0.555368	−	0.961926i
$831$		0			0
$832$	−1.00000			−0.0346688
$833$	32.5000	−	12.9904i	1.12606	−	0.450090i
$834$		0			0
$835$	21.0000	+	36.3731i	0.726735	+	1.25874i
$836$	1.50000	−	2.59808i	0.0518786	−	0.0898563i
$837$		0			0
$838$	5.00000	+	8.66025i	0.172722	+	0.299164i
$839$	9.00000			0.310715			0.155357	−	0.987858i	$-0.450347\pi$
$839$	9.00000			0.310715			0.155357	+	0.987858i	$0.450347\pi$
$840$		0			0
$841$	−28.0000			−0.965517
$842$	−4.00000	−	6.92820i	−0.137849	−	0.238762i
$843$		0			0
$844$	8.00000	−	13.8564i	0.275371	−	0.476957i
$845$	−1.00000	−	1.73205i	−0.0344010	−	0.0595844i
$846$		0			0
$847$	−1.00000	−	5.19615i	−0.0343604	−	0.178542i
$848$	3.00000			0.103020
$849$		0			0
$850$	−2.50000	+	4.33013i	−0.0857493	+	0.148522i
$851$		0			0
$852$		0			0
$853$	−40.0000			−1.36957			−0.684787	−	0.728743i	$-0.740105\pi$
$853$	−40.0000			−1.36957			−0.684787	+	0.728743i	$0.740105\pi$
$854$	−10.0000	+	8.66025i	−0.342193	+	0.296348i
$855$		0			0
$856$	6.00000	+	10.3923i	0.205076	+	0.355202i
$857$	1.50000	−	2.59808i	0.0512390	−	0.0887486i	−0.839268	−	0.543718i	$-0.817016\pi$
$857$	1.50000	−	2.59808i	0.0512390	−	0.0887486i	0.890507	+	0.454969i	$0.150350\pi$
$858$		0			0
$859$	20.0000	+	34.6410i	0.682391	+	1.18194i	0.974249	+	0.225475i	$0.0723932\pi$
$859$	20.0000	+	34.6410i	0.682391	+	1.18194i	−0.291858	+	0.956462i	$0.594273\pi$
$860$	−20.0000			−0.681994
$861$		0			0
$862$	8.00000			0.272481
$863$	−22.0000	−	38.1051i	−0.748889	−	1.29711i	−0.948356	−	0.317209i	$-0.897254\pi$
$863$	−22.0000	−	38.1051i	−0.748889	−	1.29711i	0.199467	−	0.979905i	$-0.436079\pi$
$864$		0			0
$865$	3.00000	−	5.19615i	0.102003	−	0.176674i
$866$	9.50000	+	16.4545i	0.322823	+	0.559146i
$867$		0			0
$868$	2.00000	+	10.3923i	0.0678844	+	0.352738i
$869$	18.0000			0.610608
$870$		0			0
$871$	2.50000	−	4.33013i	0.0847093	−	0.146721i
$872$	8.00000	−	13.8564i	0.270914	−	0.469237i
$873$		0			0
$874$		0			0
$875$	30.0000	+	10.3923i	1.01419	+	0.351324i
$876$		0			0
$877$	−11.0000	−	19.0526i	−0.371444	−	0.643359i	0.618344	−	0.785907i	$-0.287804\pi$
$877$	−11.0000	−	19.0526i	−0.371444	−	0.643359i	−0.989788	+	0.142548i	$0.954470\pi$
$878$	−20.0000	+	34.6410i	−0.674967	+	1.16908i
$879$		0			0
$880$	3.00000	+	5.19615i	0.101130	+	0.175162i
$881$	26.0000			0.875962			0.437981	−	0.898984i	$-0.355694\pi$
$881$	26.0000			0.875962			0.437981	+	0.898984i	$0.355694\pi$
$882$		0			0
$883$	−38.0000			−1.27880			−0.639401	−	0.768874i	$-0.720818\pi$
$883$	−38.0000			−1.27880			−0.639401	+	0.768874i	$0.720818\pi$
$884$	−2.50000	−	4.33013i	−0.0840841	−	0.145638i
$885$		0			0
$886$	−9.00000	+	15.5885i	−0.302361	+	0.523704i
$887$	−7.00000	−	12.1244i	−0.235037	−	0.407096i	0.724246	−	0.689541i	$-0.242188\pi$
$887$	−7.00000	−	12.1244i	−0.235037	−	0.407096i	−0.959283	+	0.282445i	$0.908854\pi$
$888$		0			0
$889$		0			0
$890$	28.0000			0.938562
$891$		0			0
$892$	−5.50000	+	9.52628i	−0.184154	+	0.318963i
$893$	−0.500000	+	0.866025i	−0.0167319	+	0.0289804i
$894$		0			0
$895$	−4.00000			−0.133705
$896$	0.500000	+	2.59808i	0.0167038	+	0.0867956i
$897$		0			0
$898$	−4.00000	−	6.92820i	−0.133482	−	0.231197i
$899$	−2.00000	+	3.46410i	−0.0667037	+	0.115534i
$900$		0			0
$901$	7.50000	+	12.9904i	0.249861	+	0.432772i
$902$	30.0000			0.998891
$903$		0			0
$904$	−15.0000			−0.498893
$905$	−3.00000	−	5.19615i	−0.0997234	−	0.172726i
$906$		0			0
$907$	−14.0000	+	24.2487i	−0.464862	+	0.805165i	−0.999195	−	0.0401089i	$-0.987230\pi$
$907$	−14.0000	+	24.2487i	−0.464862	+	0.805165i	0.534333	+	0.845274i	$0.320563\pi$
$908$	4.00000	+	6.92820i	0.132745	+	0.229920i
$909$		0			0
$910$	−4.00000	+	3.46410i	−0.132599	+	0.114834i
$911$	−26.0000			−0.861418			−0.430709	−	0.902491i	$-0.641737\pi$
$911$	−26.0000			−0.861418			−0.430709	+	0.902491i	$0.641737\pi$
$912$		0			0
$913$	−24.0000	+	41.5692i	−0.794284	+	1.37574i
$914$	8.00000	−	13.8564i	0.264616	−	0.458329i
$915$		0			0
$916$	6.00000			0.198246
$917$	2.00000	+	10.3923i	0.0660458	+	0.343184i
$918$		0			0
$919$	1.00000	+	1.73205i	0.0329870	+	0.0571351i	0.882048	−	0.471160i	$-0.156165\pi$
$919$	1.00000	+	1.73205i	0.0329870	+	0.0571351i	−0.849061	+	0.528295i	$0.822831\pi$
$920$		0			0
$921$		0			0
$922$	10.0000	+	17.3205i	0.329332	+	0.570421i
$923$	−1.00000			−0.0329154
$924$		0			0
$925$	2.00000			0.0657596
$926$	−4.00000	−	6.92820i	−0.131448	−	0.227675i
$927$		0			0
$928$	−0.500000	+	0.866025i	−0.0164133	+	0.0284287i
$929$	−23.0000	−	39.8372i	−0.754606	−	1.30702i	−0.945570	−	0.325418i	$-0.894495\pi$
$929$	−23.0000	−	39.8372i	−0.754606	−	1.30702i	0.190965	−	0.981597i	$-0.438838\pi$
$930$		0			0
$931$	1.00000	−	6.92820i	0.0327737	−	0.227063i
$932$	−11.0000			−0.360317
$933$		0			0
$934$	−12.0000	+	20.7846i	−0.392652	+	0.680093i
$935$	−15.0000	+	25.9808i	−0.490552	+	0.849662i
$936$		0			0
$937$	−5.00000			−0.163343			−0.0816714	−	0.996659i	$-0.526026\pi$
$937$	−5.00000			−0.163343			−0.0816714	+	0.996659i	$0.526026\pi$
$938$	−12.5000	−	4.33013i	−0.408139	−	0.141384i
$939$		0			0
$940$	−1.00000	−	1.73205i	−0.0326164	−	0.0564933i
$941$	−12.0000	+	20.7846i	−0.391189	+	0.677559i	−0.992607	−	0.121376i	$-0.961269\pi$
$941$	−12.0000	+	20.7846i	−0.391189	+	0.677559i	0.601418	+	0.798935i	$0.294603\pi$
$942$		0			0
$943$		0			0
$944$	−3.00000			−0.0976417
$945$		0			0
$946$	30.0000			0.975384
$947$	−12.5000	−	21.6506i	−0.406195	−	0.703551i	0.588264	−	0.808669i	$-0.299811\pi$
$947$	−12.5000	−	21.6506i	−0.406195	−	0.703551i	−0.994460	+	0.105118i	$0.966478\pi$
$948$		0			0
$949$	6.00000	−	10.3923i	0.194768	−	0.337348i
$950$	0.500000	+	0.866025i	0.0162221	+	0.0280976i
$951$		0			0
$952$	−10.0000	+	8.66025i	−0.324102	+	0.280680i
$953$	−19.0000			−0.615470			−0.307735	−	0.951472i	$-0.599571\pi$
$953$	−19.0000			−0.615470			−0.307735	+	0.951472i	$0.599571\pi$
$954$		0			0
$955$	20.0000	−	34.6410i	0.647185	−	1.12096i
$956$	10.5000	−	18.1865i	0.339594	−	0.588195i
$957$		0			0
$958$	33.0000			1.06618
$959$	−36.0000	+	31.1769i	−1.16250	+	1.00676i
$960$		0			0
$961$	7.50000	+	12.9904i	0.241935	+	0.419045i
$962$	−1.00000	+	1.73205i	−0.0322413	+	0.0558436i
$963$		0			0
$964$	−2.00000	−	3.46410i	−0.0644157	−	0.111571i
$965$	−28.0000			−0.901352
$966$		0			0
$967$	−31.0000			−0.996893			−0.498446	−	0.866921i	$-0.666096\pi$
$967$	−31.0000			−0.996893			−0.498446	+	0.866921i	$0.666096\pi$
$968$	1.00000	+	1.73205i	0.0321412	+	0.0556702i
$969$		0			0
$970$	−4.00000	+	6.92820i	−0.128432	+	0.222451i
$971$	−7.00000	−	12.1244i	−0.224641	−	0.389089i	0.731571	−	0.681765i	$-0.238788\pi$
$971$	−7.00000	−	12.1244i	−0.224641	−	0.389089i	−0.956212	+	0.292676i	$0.905454\pi$
$972$		0			0
$973$	−5.00000	−	1.73205i	−0.160293	−	0.0555270i
$974$	11.0000			0.352463
$975$		0			0
$976$	2.50000	−	4.33013i	0.0800230	−	0.138604i
$977$	31.0000	−	53.6936i	0.991778	−	1.71781i	0.385063	−	0.922890i	$-0.374180\pi$
$977$	31.0000	−	53.6936i	0.991778	−	1.71781i	0.606715	−	0.794919i	$-0.292487\pi$
$978$		0			0
$979$	−42.0000			−1.34233
$980$	11.0000	+	8.66025i	0.351382	+	0.276642i
$981$		0			0
$982$	11.0000	+	19.0526i	0.351024	+	0.607992i
$983$	−29.5000	+	51.0955i	−0.940904	+	1.62969i	−0.177152	+	0.984184i	$0.556688\pi$
$983$	−29.5000	+	51.0955i	−0.940904	+	1.62969i	−0.763752	+	0.645510i	$0.776645\pi$
$984$		0			0
$985$	−24.0000	−	41.5692i	−0.764704	−	1.32451i
$986$	−5.00000			−0.159232
$987$		0			0
$988$	−1.00000			−0.0318142
$989$		0			0
$990$		0			0
$991$	24.0000	−	41.5692i	0.762385	−	1.32049i	−0.179233	−	0.983807i	$-0.557362\pi$
$991$	24.0000	−	41.5692i	0.762385	−	1.32049i	0.941618	−	0.336683i	$-0.109305\pi$
$992$	−2.00000	−	3.46410i	−0.0635001	−	0.109985i
$993$		0			0
$994$	0.500000	+	2.59808i	0.0158590	+	0.0824060i
$995$	−40.0000			−1.26809
$996$		0			0
$997$	−9.50000	+	16.4545i	−0.300868	+	0.521119i	−0.976333	−	0.216274i	$-0.930610\pi$
$997$	−9.50000	+	16.4545i	−0.300868	+	0.521119i	0.675465	+	0.737392i	$0.263943\pi$
$998$	14.0000	−	24.2487i	0.443162	−	0.767580i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1638.2.j.a.235.1		2
3.2	odd	2		546.2.i.e.235.1	yes	2
7.2	even	3	inner	1638.2.j.a.1171.1		2
21.2	odd	6		546.2.i.e.79.1	✓	2
21.11	odd	6		3822.2.a.k.1.1		1
21.17	even	6		3822.2.a.i.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
546.2.i.e.79.1	✓	2	21.2	odd	6
546.2.i.e.235.1	yes	2	3.2	odd	2
1638.2.j.a.235.1		2	1.1	even	1	trivial
1638.2.j.a.1171.1		2	7.2	even	3	inner
3822.2.a.i.1.1		1	21.17	even	6
3822.2.a.k.1.1		1	21.11	odd	6

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:	\( N \)	\(=\)	\( 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1638.j (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 - 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(-1.00000 - 1.73205i) q^{5} +(-2.50000 - 0.866025i) q^{7} +1.00000 q^{8} +(-1.00000 + 1.73205i) q^{10} +(1.50000 - 2.59808i) q^{11} -1.00000 q^{13} +(0.500000 + 2.59808i) q^{14} +(-0.500000 - 0.866025i) q^{16} +(2.50000 - 4.33013i) q^{17} +(-0.500000 - 0.866025i) q^{19} +2.00000 q^{20} -3.00000 q^{22} +(0.500000 - 0.866025i) q^{25} +(0.500000 + 0.866025i) q^{26} +(2.00000 - 1.73205i) q^{28} +1.00000 q^{29} +(-2.00000 + 3.46410i) q^{31} +(-0.500000 + 0.866025i) q^{32} -5.00000 q^{34} +(1.00000 + 5.19615i) q^{35} +(1.00000 + 1.73205i) q^{37} +(-0.500000 + 0.866025i) q^{38} +(-1.00000 - 1.73205i) q^{40} -10.0000 q^{41} -10.0000 q^{43} +(1.50000 + 2.59808i) q^{44} +(-0.500000 - 0.866025i) q^{47} +(5.50000 + 4.33013i) q^{49} -1.00000 q^{50} +(0.500000 - 0.866025i) q^{52} +(-1.50000 + 2.59808i) q^{53} -6.00000 q^{55} +(-2.50000 - 0.866025i) q^{56} +(-0.500000 - 0.866025i) q^{58} +(1.50000 - 2.59808i) q^{59} +(2.50000 + 4.33013i) q^{61} +4.00000 q^{62} +1.00000 q^{64} +(1.00000 + 1.73205i) q^{65} +(-2.50000 + 4.33013i) q^{67} +(2.50000 + 4.33013i) q^{68} +(4.00000 - 3.46410i) q^{70} +1.00000 q^{71} +(-6.00000 + 10.3923i) q^{73} +(1.00000 - 1.73205i) q^{74} +1.00000 q^{76} +(-6.00000 + 5.19615i) q^{77} +(3.00000 + 5.19615i) q^{79} +(-1.00000 + 1.73205i) q^{80} +(5.00000 + 8.66025i) q^{82} -16.0000 q^{83} -10.0000 q^{85} +(5.00000 + 8.66025i) q^{86} +(1.50000 - 2.59808i) q^{88} +(-7.00000 - 12.1244i) q^{89} +(2.50000 + 0.866025i) q^{91} +(-0.500000 + 0.866025i) q^{94} +(-1.00000 + 1.73205i) q^{95} +4.00000 q^{97} +(1.00000 - 6.92820i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} - q^{4} - 2 q^{5} - 5 q^{7} + 2 q^{8} - 2 q^{10} + 3 q^{11} - 2 q^{13} + q^{14} - q^{16} + 5 q^{17} - q^{19} + 4 q^{20} - 6 q^{22} + q^{25} + q^{26} + 4 q^{28} + 2 q^{29} - 4 q^{31} - q^{32}+ \cdots + 2 q^{98}+O(q^{100}) \) 2 * q - q^2 - q^4 - 2 * q^5 - 5 * q^7 + 2 * q^8 - 2 * q^10 + 3 * q^11 - 2 * q^13 + q^14 - q^16 + 5 * q^17 - q^19 + 4 * q^20 - 6 * q^22 + q^25 + q^26 + 4 * q^28 + 2 * q^29 - 4 * q^31 - q^32 - 10 * q^34 + 2 * q^35 + 2 * q^37 - q^38 - 2 * q^40 - 20 * q^41 - 20 * q^43 + 3 * q^44 - q^47 + 11 * q^49 - 2 * q^50 + q^52 - 3 * q^53 - 12 * q^55 - 5 * q^56 - q^58 + 3 * q^59 + 5 * q^61 + 8 * q^62 + 2 * q^64 + 2 * q^65 - 5 * q^67 + 5 * q^68 + 8 * q^70 + 2 * q^71 - 12 * q^73 + 2 * q^74 + 2 * q^76 - 12 * q^77 + 6 * q^79 - 2 * q^80 + 10 * q^82 - 32 * q^83 - 20 * q^85 + 10 * q^86 + 3 * q^88 - 14 * q^89 + 5 * q^91 - q^94 - 2 * q^95 + 8 * q^97 + 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more