LMFDB - Embedded newform 117.2.i.a.8.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [117,2,Mod(8,117)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(117, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 1]))

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

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

chi := DirichletCharacter("117.8");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 117 = 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	117.i (of order $4$, 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:	$0.934249703649$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 16x^{10} + 103x^{8} - 260x^{6} + 259x^{4} + 356x^{2} + 169 $ x^12 - 16x^10 + 103x^8 - 260x^6 + 259x^4 + 356*x^2 + 169
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{5}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			8.4
Root			$-2.59708 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	117.8
Dual form			117.2.i.a.44.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.404460 - 0.404460i) q^{2} +1.67282i q^{4} +(1.88997 - 1.88997i) q^{5} +(-0.428007 + 0.428007i) q^{7} +(1.48551 + 1.48551i) q^{8} +O(q^{10})$	\(q+(0.404460 - 0.404460i) q^{2} +1.67282i q^{4} +(1.88997 - 1.88997i) q^{5} +(-0.428007 + 0.428007i) q^{7} +(1.48551 + 1.48551i) q^{8} -1.52884i q^{10} +(-1.88997 - 1.88997i) q^{11} +(3.24482 + 1.57199i) q^{13} +0.346223i q^{14} -2.14399 q^{16} -4.24264 q^{17} +(-4.24482 - 4.24482i) q^{19} +(3.16159 + 3.16159i) q^{20} -1.52884 q^{22} -5.39778 q^{23} -2.14399i q^{25} +(1.94821 - 0.676591i) q^{26} +(-0.715980 - 0.715980i) q^{28} +6.40474i q^{29} +(3.10083 + 3.10083i) q^{31} +(-3.83818 + 3.83818i) q^{32} +(-1.71598 + 1.71598i) q^{34} +1.61784i q^{35} +(1.00000 - 1.00000i) q^{37} -3.43372 q^{38} +5.61515 q^{40} +(6.13261 - 6.13261i) q^{41} -10.2017i q^{43} +(3.16159 - 3.16159i) q^{44} +(-2.18319 + 2.18319i) q^{46} +(-1.88997 - 1.88997i) q^{47} +6.63362i q^{49} +(-0.867157 - 0.867157i) q^{50} +(-2.62967 + 5.42801i) q^{52} -13.4204i q^{53} -7.14399 q^{55} -1.27162 q^{56} +(2.59046 + 2.59046i) q^{58} +(10.7215 + 10.7215i) q^{59} +6.48963 q^{61} +2.50833 q^{62} -1.18319i q^{64} +(9.10364 - 3.16159i) q^{65} +(5.57199 + 5.57199i) q^{67} -7.09719i q^{68} +(0.654353 + 0.654353i) q^{70} +(-5.32369 + 5.32369i) q^{71} +(-2.52884 + 2.52884i) q^{73} -0.808921i q^{74} +(7.10083 - 7.10083i) q^{76} +1.61784 q^{77} +3.05767 q^{79} +(-4.05207 + 4.05207i) q^{80} -4.96080i q^{82} +(-2.23620 + 2.23620i) q^{83} +(-8.01847 + 8.01847i) q^{85} +(-4.12617 - 4.12617i) q^{86} -5.61515i q^{88} +(3.50781 + 3.50781i) q^{89} +(-2.06163 + 0.715980i) q^{91} -9.02954i q^{92} -1.52884 q^{94} -16.0452 q^{95} +(-2.81681 - 2.81681i) q^{97} +(2.68304 + 2.68304i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 8 q^{7}+O(q^{10}) $ 12 * q - 8 * q^7	$ 12 q - 8 q^{7} - 4 q^{13} - 20 q^{16} - 8 q^{19} + 16 q^{22} - 12 q^{34} + 12 q^{37} + 96 q^{40} - 72 q^{46} + 40 q^{52} - 80 q^{55} - 92 q^{58} - 8 q^{61} + 64 q^{67} + 88 q^{70} + 4 q^{73} + 48 q^{76} - 32 q^{79} + 24 q^{85} + 64 q^{91} + 16 q^{94} + 12 q^{97}+O(q^{100}) $ 12 * q - 8 * q^7 - 4 * q^13 - 20 * q^16 - 8 * q^19 + 16 * q^22 - 12 * q^34 + 12 * q^37 + 96 * q^40 - 72 * q^46 + 40 * q^52 - 80 * q^55 - 92 * q^58 - 8 * q^61 + 64 * q^67 + 88 * q^70 + 4 * q^73 + 48 * q^76 - 32 * q^79 + 24 * q^85 + 64 * q^91 + 16 * q^94 + 12 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/117\mathbb{Z}\right)^\times$.

$n$	$28$	$92$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$-1$

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 $

$2$	0.404460	−	0.404460i	0.285997	−	0.285997i	−0.549498	−	0.835495i	$-0.685181\pi$
$2$	0.404460	−	0.404460i	0.285997	−	0.285997i	0.835495	+	0.549498i	$0.185181\pi$
$3$		0			0
$3$		0			0
$4$			1.67282i			0.836412i
$5$	1.88997	−	1.88997i	0.845221	−	0.845221i	−0.144311	−	0.989532i	$-0.546097\pi$
$5$	1.88997	−	1.88997i	0.845221	−	0.845221i	0.989532	+	0.144311i	$0.0460967\pi$
$6$		0			0
$7$	−0.428007	+	0.428007i	−0.161771	+	0.161771i	−0.783351	−	0.621580i	$-0.786491\pi$
$7$	−0.428007	+	0.428007i	−0.161771	+	0.161771i	0.621580	+	0.783351i	$0.286491\pi$
$8$	1.48551	+	1.48551i	0.525208	+	0.525208i
$9$		0			0
$10$		−	1.52884i		−	0.483461i
$11$	−1.88997	−	1.88997i	−0.569848	−	0.569848i	0.362238	−	0.932086i	$-0.382013\pi$
$11$	−1.88997	−	1.88997i	−0.569848	−	0.569848i	−0.932086	+	0.362238i	$0.882013\pi$
$12$		0			0
$13$	3.24482	+	1.57199i	0.899950	+	0.435992i
$13$	3.24482	+	1.57199i	0.899950	+	0.435992i
$14$			0.346223i			0.0925321i
$15$		0			0
$16$	−2.14399			−0.535997
$17$	−4.24264			−1.02899			−0.514496	−	0.857493i	$-0.672021\pi$
$17$	−4.24264			−1.02899			−0.514496	+	0.857493i	$0.672021\pi$
$18$		0			0
$19$	−4.24482	−	4.24482i	−0.973828	−	0.973828i	0.0258383	−	0.999666i	$-0.491774\pi$
$19$	−4.24482	−	4.24482i	−0.973828	−	0.973828i	−0.999666	+	0.0258383i	$0.991774\pi$
$20$	3.16159	+	3.16159i	0.706953	+	0.706953i
$21$		0			0
$22$	−1.52884			−0.325949
$23$	−5.39778			−1.12552			−0.562758	−	0.826622i	$-0.690260\pi$
$23$	−5.39778			−1.12552			−0.562758	+	0.826622i	$0.690260\pi$
$24$		0			0
$25$		−	2.14399i		−	0.428797i
$26$	1.94821	−	0.676591i	0.382075	−	0.132690i
$27$		0			0
$28$	−0.715980	−	0.715980i	−0.135307	−	0.135307i
$29$			6.40474i			1.18933i	0.803973	+	0.594665i	$0.202715\pi$
$29$			6.40474i			1.18933i	−0.803973	+	0.594665i	$0.797285\pi$
$30$		0			0
$31$	3.10083	+	3.10083i	0.556926	+	0.556926i	0.928431	−	0.371505i	$-0.121158\pi$
$31$	3.10083	+	3.10083i	0.556926	+	0.556926i	−0.371505	+	0.928431i	$0.621158\pi$
$32$	−3.83818	+	3.83818i	−0.678501	+	0.678501i
$33$		0			0
$34$	−1.71598	+	1.71598i	−0.294288	+	0.294288i
$35$			1.61784i			0.273465i
$36$		0			0
$37$	1.00000	−	1.00000i	0.164399	−	0.164399i	−0.620113	−	0.784512i	$-0.712913\pi$
$37$	1.00000	−	1.00000i	0.164399	−	0.164399i	0.784512	+	0.620113i	$0.212913\pi$
$38$	−3.43372			−0.557023
$39$		0			0
$40$	5.61515			0.887833
$41$	6.13261	−	6.13261i	0.957753	−	0.957753i	−0.0413899	−	0.999143i	$-0.513179\pi$
$41$	6.13261	−	6.13261i	0.957753	−	0.957753i	0.999143	+	0.0413899i	$0.0131786\pi$
$42$		0			0
$43$		−	10.2017i		−	1.55574i	−0.628426	−	0.777869i	$-0.716301\pi$
$43$		−	10.2017i		−	1.55574i	0.628426	−	0.777869i	$-0.283699\pi$
$44$	3.16159	−	3.16159i	0.476628	−	0.476628i
$45$		0			0
$46$	−2.18319	+	2.18319i	−0.321894	+	0.321894i
$47$	−1.88997	−	1.88997i	−0.275681	−	0.275681i	0.555701	−	0.831382i	$-0.312450\pi$
$47$	−1.88997	−	1.88997i	−0.275681	−	0.275681i	−0.831382	+	0.555701i	$0.812450\pi$
$48$		0			0
$49$			6.63362i			0.947660i
$50$	−0.867157	−	0.867157i	−0.122635	−	0.122635i
$51$		0			0
$52$	−2.62967	+	5.42801i	−0.364669	+	0.752729i
$53$		−	13.4204i		−	1.84343i	−0.387868	−	0.921715i	$-0.626789\pi$
$53$		−	13.4204i		−	1.84343i	0.387868	−	0.921715i	$-0.373211\pi$
$54$		0			0
$55$	−7.14399			−0.963295
$56$	−1.27162			−0.169927
$57$		0			0
$58$	2.59046	+	2.59046i	0.340145	+	0.340145i
$59$	10.7215	+	10.7215i	1.39582	+	1.39582i	0.811590	+	0.584227i	$0.198602\pi$
$59$	10.7215	+	10.7215i	1.39582	+	1.39582i	0.584227	+	0.811590i	$0.301398\pi$
$60$		0			0
$61$	6.48963			0.830912			0.415456	−	0.909613i	$-0.363622\pi$
$61$	6.48963			0.830912			0.415456	+	0.909613i	$0.363622\pi$
$62$	2.50833			0.318558
$63$		0			0
$64$		−	1.18319i		−	0.147899i
$65$	9.10364	−	3.16159i	1.12917	−	0.392147i
$66$		0			0
$67$	5.57199	+	5.57199i	0.680727	+	0.680727i	0.960164	−	0.279437i	$-0.0901477\pi$
$67$	5.57199	+	5.57199i	0.680727	+	0.680727i	−0.279437	+	0.960164i	$0.590148\pi$
$68$		−	7.09719i		−	0.860661i
$69$		0			0
$70$	0.654353	+	0.654353i	0.0782101	+	0.0782101i
$71$	−5.32369	+	5.32369i	−0.631806	+	0.631806i	−0.948521	−	0.316715i	$-0.897420\pi$
$71$	−5.32369	+	5.32369i	−0.631806	+	0.631806i	0.316715	+	0.948521i	$0.397420\pi$
$72$		0			0
$73$	−2.52884	+	2.52884i	−0.295978	+	0.295978i	−0.839436	−	0.543458i	$-0.817115\pi$
$73$	−2.52884	+	2.52884i	−0.295978	+	0.295978i	0.543458	+	0.839436i	$0.317115\pi$
$74$		−	0.808921i		−	0.0940351i
$75$		0			0
$76$	7.10083	−	7.10083i	0.814521	−	0.814521i
$77$	1.61784			0.184370
$78$		0			0
$79$	3.05767			0.344015			0.172008	−	0.985096i	$-0.444975\pi$
$79$	3.05767			0.344015			0.172008	+	0.985096i	$0.444975\pi$
$80$	−4.05207	+	4.05207i	−0.453036	+	0.453036i
$81$		0			0
$82$		−	4.96080i		−	0.547828i
$83$	−2.23620	+	2.23620i	−0.245454	+	0.245454i	−0.819102	−	0.573648i	$-0.805528\pi$
$83$	−2.23620	+	2.23620i	−0.245454	+	0.245454i	0.573648	+	0.819102i	$0.305528\pi$
$84$		0			0
$85$	−8.01847	+	8.01847i	−0.869725	+	0.869725i
$86$	−4.12617	−	4.12617i	−0.444936	−	0.444936i
$87$		0			0
$88$		−	5.61515i		−	0.598577i
$89$	3.50781	+	3.50781i	0.371827	+	0.371827i	0.868142	−	0.496315i	$-0.165314\pi$
$89$	3.50781	+	3.50781i	0.371827	+	0.371827i	−0.496315	+	0.868142i	$0.665314\pi$
$90$		0			0
$91$	−2.06163	+	0.715980i	−0.216117	+	0.0750551i
$92$		−	9.02954i		−	0.941395i
$93$		0			0
$94$	−1.52884			−0.157688
$95$	−16.0452			−1.64620
$96$		0			0
$97$	−2.81681	−	2.81681i	−0.286004	−	0.286004i	0.549494	−	0.835498i	$-0.314821\pi$
$97$	−2.81681	−	2.81681i	−0.286004	−	0.286004i	−0.835498	+	0.549494i	$0.814821\pi$
$98$	2.68304	+	2.68304i	0.271028	+	0.271028i
$99$		0			0
$100$	3.58651			0.358651
$101$	1.15514			0.114941			0.0574706	−	0.998347i	$-0.481696\pi$
$101$	1.15514			0.114941			0.0574706	+	0.998347i	$0.481696\pi$
$102$		0			0
$103$		−	5.83528i		−	0.574967i	−0.957786	−	0.287484i	$-0.907181\pi$
$103$		−	5.83528i		−	0.574967i	0.957786	−	0.287484i	$-0.0928187\pi$
$104$	2.48500	+	7.15543i	0.243674	+	0.701647i
$105$		0			0
$106$	−5.42801	−	5.42801i	−0.527215	−	0.527215i
$107$			15.5009i			1.49853i	0.662271	+	0.749265i	$0.269593\pi$
$107$			15.5009i			1.49853i	−0.662271	+	0.749265i	$0.730407\pi$
$108$		0			0
$109$	−2.52884	−	2.52884i	−0.242219	−	0.242219i	0.575549	−	0.817767i	$-0.304788\pi$
$109$	−2.52884	−	2.52884i	−0.242219	−	0.242219i	−0.817767	+	0.575549i	$0.804788\pi$
$110$	−2.88946	+	2.88946i	−0.275499	+	0.275499i
$111$		0			0
$112$	0.917641	−	0.917641i	0.0867089	−	0.0867089i
$113$			6.40474i			0.602508i	0.953544	+	0.301254i	$0.0974051\pi$
$113$			6.40474i			0.602508i	−0.953544	+	0.301254i	$0.902595\pi$
$114$		0			0
$115$	−10.2017	+	10.2017i	−0.951310	+	0.951310i
$116$	−10.7140			−0.994770
$117$		0			0
$118$	8.67282			0.798398
$119$	1.81588	−	1.81588i	0.166461	−	0.166461i
$120$		0			0
$121$		−	3.85601i		−	0.350547i
$122$	2.62480	−	2.62480i	0.237638	−	0.237638i
$123$		0			0
$124$	−5.18714	+	5.18714i	−0.465819	+	0.465819i
$125$	5.39778	+	5.39778i	0.482793	+	0.482793i
$126$		0			0
$127$		−	4.48963i		−	0.398391i	−0.979960	−	0.199195i	$-0.936167\pi$
$127$		−	4.48963i		−	0.398391i	0.979960	−	0.199195i	$-0.0638328\pi$
$128$	−8.15491	−	8.15491i	−0.720799	−	0.720799i
$129$		0			0
$130$	2.40332	−	4.96080i	0.210785	−	0.435091i
$131$			2.16210i			0.188904i	0.995529	+	0.0944519i	$0.0301098\pi$
$131$			2.16210i			0.188904i	−0.995529	+	0.0944519i	$0.969890\pi$
$132$		0			0
$133$	3.63362			0.315075
$134$	4.50730			0.389371
$135$		0			0
$136$	−6.30249	−	6.30249i	−0.540434	−	0.540434i
$137$	−6.13261	−	6.13261i	−0.523944	−	0.523944i	0.394816	−	0.918760i	$-0.370808\pi$
$137$	−6.13261	−	6.13261i	−0.523944	−	0.523944i	−0.918760	+	0.394816i	$0.870808\pi$
$138$		0			0
$139$	−20.0369			−1.69951			−0.849756	−	0.527177i	$-0.823251\pi$
$139$	−20.0369			−1.69951			−0.849756	+	0.527177i	$0.823251\pi$
$140$	−2.70636			−0.228729
$141$		0			0
$142$			4.30644i			0.361389i
$143$	−3.16159	−	9.10364i	−0.264385	−	0.761284i
$144$		0			0
$145$	12.1048	+	12.1048i	1.00525	+	1.00525i
$146$			2.04563i			0.169297i
$147$		0			0
$148$	1.67282	+	1.67282i	0.137505	+	0.137505i
$149$	6.13261	−	6.13261i	0.502403	−	0.502403i	−0.409781	−	0.912184i	$-0.634395\pi$
$149$	6.13261	−	6.13261i	0.502403	−	0.502403i	0.912184	+	0.409781i	$0.134395\pi$
$150$		0			0
$151$	12.5328	−	12.5328i	1.01990	−	1.01990i	0.0201061	−	0.999798i	$-0.493600\pi$
$151$	12.5328	−	12.5328i	1.01990	−	1.01990i	0.999798	−	0.0201061i	$-0.00640041\pi$
$152$		−	12.6114i		−	1.02292i
$153$		0			0
$154$	0.654353	−	0.654353i	0.0527292	−	0.0527292i
$155$	11.7210			0.941450
$156$		0			0
$157$	−5.92159			−0.472595			−0.236297	−	0.971681i	$-0.575934\pi$
$157$	−5.92159			−0.472595			−0.236297	+	0.971681i	$0.575934\pi$
$158$	1.23671	−	1.23671i	0.0983872	−	0.0983872i
$159$		0			0
$160$			14.5081i			1.14697i
$161$	2.31029	−	2.31029i	0.182076	−	0.182076i
$162$		0			0
$163$	7.30249	−	7.30249i	0.571975	−	0.571975i	−0.360705	−	0.932680i	$-0.617464\pi$
$163$	7.30249	−	7.30249i	0.571975	−	0.571975i	0.932680	+	0.360705i	$0.117464\pi$
$164$	10.2588	+	10.2588i	0.801076	+	0.801076i
$165$		0			0
$166$			1.80890i			0.140398i
$167$	1.19752	+	1.19752i	0.0926673	+	0.0926673i	0.751921	−	0.659254i	$-0.229128\pi$
$167$	1.19752	+	1.19752i	0.0926673	+	0.0926673i	−0.659254	+	0.751921i	$0.729128\pi$
$168$		0			0
$169$	8.05767	+	10.2017i	0.619821	+	0.784743i
$170$			6.48631i			0.497477i
$171$		0			0
$172$	17.0656			1.30124
$173$	−3.69838			−0.281183			−0.140591	−	0.990068i	$-0.544900\pi$
$173$	−3.69838			−0.281183			−0.140591	+	0.990068i	$0.544900\pi$
$174$		0			0
$175$	0.917641	+	0.917641i	0.0693671	+	0.0693671i
$176$	4.05207	+	4.05207i	0.305437	+	0.305437i
$177$		0			0
$178$	2.83754			0.212683
$179$	8.48528			0.634220			0.317110	−	0.948389i	$-0.397288\pi$
$179$	8.48528			0.634220			0.317110	+	0.948389i	$0.397288\pi$
$180$		0			0
$181$			16.0369i			1.19202i	0.802978	+	0.596008i	$0.203247\pi$
$181$			16.0369i			1.19202i	−0.802978	+	0.596008i	$0.796753\pi$
$182$	−0.544261	+	1.12343i	−0.0403433	+	0.0832743i
$183$		0			0
$184$	−8.01847	−	8.01847i	−0.591130	−	0.591130i
$185$		−	3.77994i		−	0.277907i
$186$		0			0
$187$	8.01847	+	8.01847i	0.586369	+	0.586369i
$188$	3.16159	−	3.16159i	0.230583	−	0.230583i
$189$		0			0
$190$	−6.48963	+	6.48963i	−0.470808	+	0.470808i
$191$		−	9.17773i		−	0.664077i	−0.943266	−	0.332039i	$-0.892264\pi$
$191$		−	9.17773i		−	0.664077i	0.943266	−	0.332039i	$-0.107736\pi$
$192$		0			0
$193$	9.69129	−	9.69129i	0.697595	−	0.697595i	−0.266296	−	0.963891i	$-0.585800\pi$
$193$	9.69129	−	9.69129i	0.697595	−	0.697595i	0.963891	+	0.266296i	$0.0858000\pi$
$194$	−2.27858			−0.163592
$195$		0			0
$196$	−11.0969			−0.792634
$197$	4.97747	−	4.97747i	0.354630	−	0.354630i	−0.507199	−	0.861829i	$-0.669319\pi$
$197$	4.97747	−	4.97747i	0.354630	−	0.354630i	0.861829	+	0.507199i	$0.169319\pi$
$198$		0			0
$199$			11.5473i			0.818567i	0.912407	+	0.409283i	$0.134221\pi$
$199$			11.5473i			0.818567i	−0.912407	+	0.409283i	$0.865779\pi$
$200$	3.18492	−	3.18492i	0.225208	−	0.225208i
$201$		0			0
$202$	0.467210	−	0.467210i	0.0328728	−	0.0328728i
$203$	−2.74127	−	2.74127i	−0.192400	−	0.192400i
$204$		0			0
$205$		−	23.1809i		−	1.61903i
$206$	−2.36014	−	2.36014i	−0.164439	−	0.164439i
$207$		0			0
$208$	−6.95684	−	3.37033i	−0.482370	−	0.233691i
$209$			16.0452i			1.10987i
$210$		0			0
$211$	−12.9793			−0.893530			−0.446765	−	0.894651i	$-0.647424\pi$
$211$	−12.9793			−0.893530			−0.446765	+	0.894651i	$0.647424\pi$
$212$	22.4499			1.54187
$213$		0			0
$214$	6.26950	+	6.26950i	0.428574	+	0.428574i
$215$	−19.2809	−	19.2809i	−1.31494	−	1.31494i
$216$		0			0
$217$	−2.65435			−0.180189
$218$	−2.04563			−0.138547
$219$		0			0
$220$		−	11.9506i		−	0.805711i
$221$	−13.7666	−	6.66940i	−0.926041	−	0.448633i
$222$		0			0
$223$	−2.61120	−	2.61120i	−0.174859	−	0.174859i	0.614252	−	0.789110i	$-0.289458\pi$
$223$	−2.61120	−	2.61120i	−0.174859	−	0.174859i	−0.789110	+	0.614252i	$0.789458\pi$
$224$		−	3.28553i		−	0.219524i
$225$		0			0
$226$	2.59046	+	2.59046i	0.172315	+	0.172315i
$227$	−4.05207	+	4.05207i	−0.268946	+	0.268946i	−0.828675	−	0.559730i	$-0.810905\pi$
$227$	−4.05207	+	4.05207i	−0.268946	+	0.268946i	0.559730	+	0.828675i	$0.310905\pi$
$228$		0			0
$229$	1.67282	−	1.67282i	0.110543	−	0.110543i	−0.649672	−	0.760215i	$-0.725094\pi$
$229$	1.67282	−	1.67282i	0.110543	−	0.110543i	0.760215	+	0.649672i	$0.225094\pi$
$230$			8.25233i			0.544143i
$231$		0			0
$232$	−9.51432	+	9.51432i	−0.624646	+	0.624646i
$233$	−11.9507			−0.782917			−0.391459	−	0.920196i	$-0.628029\pi$
$233$	−11.9507			−0.782917			−0.391459	+	0.920196i	$0.628029\pi$
$234$		0			0
$235$	−7.14399			−0.466022
$236$	−17.9351	+	17.9351i	−1.16748	+	1.16748i
$237$		0			0
$238$		−	1.46890i		−	0.0952148i
$239$	−4.77943	+	4.77943i	−0.309156	+	0.309156i	−0.844582	−	0.535426i	$-0.820151\pi$
$239$	−4.77943	+	4.77943i	−0.309156	+	0.309156i	0.535426	+	0.844582i	$0.320151\pi$
$240$		0			0
$241$	10.8168	−	10.8168i	0.696772	−	0.696772i	−0.266941	−	0.963713i	$-0.586013\pi$
$241$	10.8168	−	10.8168i	0.696772	−	0.696772i	0.963713	+	0.266941i	$0.0860129\pi$
$242$	−1.55960	−	1.55960i	−0.100255	−	0.100255i
$243$		0			0
$244$			10.8560i			0.694985i
$245$	12.5374	+	12.5374i	0.800982	+	0.800982i
$246$		0			0
$247$	−7.10083	−	20.4465i	−0.451815	−	1.30098i
$248$			9.21264i			0.585003i
$249$		0			0
$250$	4.36638			0.276154
$251$	11.3398			0.715764			0.357882	−	0.933767i	$-0.383499\pi$
$251$	11.3398			0.715764			0.357882	+	0.933767i	$0.383499\pi$
$252$		0			0
$253$	10.2017	+	10.2017i	0.641373	+	0.641373i
$254$	−1.81588	−	1.81588i	−0.113938	−	0.113938i
$255$		0			0
$256$	−4.23030			−0.264394
$257$	−9.64043			−0.601353			−0.300677	−	0.953726i	$-0.597212\pi$
$257$	−9.64043			−0.601353			−0.300677	+	0.953726i	$0.597212\pi$
$258$		0			0
$259$			0.856013i			0.0531901i
$260$	5.28878	+	15.2288i	0.327996	+	0.944449i
$261$		0			0
$262$	0.874485	+	0.874485i	0.0540258	+	0.0540258i
$263$		−	12.0323i		−	0.741942i	−0.928644	−	0.370971i	$-0.879025\pi$
$263$		−	12.0323i		−	0.741942i	0.928644	−	0.370971i	$-0.120975\pi$
$264$		0			0
$265$	−25.3641	−	25.3641i	−1.55811	−	1.55811i
$266$	1.46966	−	1.46966i	0.0901103	−	0.0901103i
$267$		0			0
$268$	−9.32096	+	9.32096i	−0.569368	+	0.569368i
$269$			19.7435i			1.20379i	0.798577	+	0.601893i	$0.205587\pi$
$269$			19.7435i			1.20379i	−0.798577	+	0.601893i	$0.794413\pi$
$270$		0			0
$271$	2.71598	−	2.71598i	0.164984	−	0.164984i	−0.619786	−	0.784770i	$-0.712781\pi$
$271$	2.71598	−	2.71598i	0.164984	−	0.164984i	0.784770	+	0.619786i	$0.212781\pi$
$272$	9.09616			0.551536
$273$		0			0
$274$	−4.96080			−0.299693
$275$	−4.05207	+	4.05207i	−0.244349	+	0.244349i
$276$		0			0
$277$		−	13.6336i		−	0.819165i	−0.912273	−	0.409582i	$-0.865674\pi$
$277$		−	13.6336i		−	0.819165i	0.912273	−	0.409582i	$-0.134326\pi$
$278$	−8.10415	+	8.10415i	−0.486054	+	0.486054i
$279$		0			0
$280$	−2.40332	+	2.40332i	−0.143626	+	0.143626i
$281$	−8.98716	−	8.98716i	−0.536129	−	0.536129i	0.386261	−	0.922390i	$-0.373767\pi$
$281$	−8.98716	−	8.98716i	−0.536129	−	0.536129i	−0.922390	+	0.386261i	$0.873767\pi$
$282$		0			0
$283$			24.1233i			1.43398i	0.697084	+	0.716989i	$0.254480\pi$
$283$			24.1233i			1.43398i	−0.697084	+	0.716989i	$0.745520\pi$
$284$	−8.90560	−	8.90560i	−0.528450	−	0.528450i
$285$		0			0
$286$	−4.96080	−	2.40332i	−0.293338	−	0.142111i
$287$			5.24960i			0.309874i
$288$		0			0
$289$	1.00000			0.0588235
$290$	9.79181			0.574995
$291$		0			0
$292$	−4.23030	−	4.23030i	−0.247560	−	0.247560i
$293$	−15.1622	−	15.1622i	−0.885783	−	0.885783i	0.108332	−	0.994115i	$-0.465449\pi$
$293$	−15.1622	−	15.1622i	−0.885783	−	0.885783i	−0.994115	+	0.108332i	$0.965449\pi$
$294$		0			0
$295$	40.5266			2.35955
$296$	2.97102			0.172687
$297$		0			0
$298$		−	4.96080i		−	0.287371i
$299$	−17.5148	−	8.48528i	−1.01291	−	0.490716i
$300$		0			0
$301$	4.36638	+	4.36638i	0.251674	+	0.251674i
$302$		−	10.1380i		−	0.583378i
$303$		0			0
$304$	9.10083	+	9.10083i	0.521968	+	0.521968i
$305$	12.2652	−	12.2652i	0.702305	−	0.702305i
$306$		0			0
$307$	−0.715980	+	0.715980i	−0.0408631	+	0.0408631i	−0.727243	−	0.686380i	$-0.759199\pi$
$307$	−0.715980	+	0.715980i	−0.0408631	+	0.0408631i	0.686380	+	0.727243i	$0.259199\pi$
$308$			2.70636i			0.154209i
$309$		0			0
$310$	4.74066	−	4.74066i	0.269252	−	0.269252i
$311$	34.2524			1.94228			0.971139	−	0.238515i	$-0.0766606\pi$
$311$	34.2524			1.94228			0.971139	+	0.238515i	$0.0766606\pi$
$312$		0			0
$313$	−27.5473			−1.55707			−0.778533	−	0.627604i	$-0.784036\pi$
$313$	−27.5473			−1.55707			−0.778533	+	0.627604i	$0.784036\pi$
$314$	−2.39505	+	2.39505i	−0.135160	+	0.135160i
$315$		0			0
$316$			5.11495i			0.287738i
$317$	−17.3909	+	17.3909i	−0.976769	+	0.976769i	−0.999736	−	0.0229671i	$-0.992689\pi$
$317$	−17.3909	+	17.3909i	−0.976769	+	0.976769i	0.0229671	+	0.999736i	$0.492689\pi$
$318$		0			0
$319$	12.1048	−	12.1048i	0.677738	−	0.677738i
$320$	−2.23620	−	2.23620i	−0.125007	−	0.125007i
$321$		0			0
$322$		−	1.86884i		−	0.104146i
$323$	18.0092	+	18.0092i	1.00206	+	1.00206i
$324$		0			0
$325$	3.37033	−	6.95684i	0.186952	−	0.385896i
$326$		−	5.90714i		−	0.327166i
$327$		0			0
$328$	18.2201			1.00604
$329$	1.61784			0.0891945
$330$		0			0
$331$	23.4073	+	23.4073i	1.28658	+	1.28658i	0.936851	+	0.349729i	$0.113726\pi$
$331$	23.4073	+	23.4073i	1.28658	+	1.28658i	0.349729	+	0.936851i	$0.386274\pi$
$332$	−3.74076	−	3.74076i	−0.205301	−	0.205301i
$333$		0			0
$334$	0.968703			0.0530050
$335$	21.0618			1.15073
$336$		0			0
$337$			17.2593i			0.940176i	0.882620	+	0.470088i	$0.155778\pi$
$337$			17.2593i			0.940176i	−0.882620	+	0.470088i	$0.844222\pi$
$338$	7.38518	+	0.867157i	0.401701	+	0.0471671i
$339$		0			0
$340$	−13.4135	−	13.4135i	−0.727449	−	0.727449i
$341$		−	11.7210i		−	0.634726i
$342$		0			0
$343$	−5.83528	−	5.83528i	−0.315076	−	0.315076i
$344$	15.1547	−	15.1547i	0.817086	−	0.817086i
$345$		0			0
$346$	−1.49585	+	1.49585i	−0.0804173	+	0.0804173i
$347$		−	17.6630i		−	0.948200i	−0.880471	−	0.474100i	$-0.842774\pi$
$347$		−	17.6630i		−	0.948200i	0.880471	−	0.474100i	$-0.157226\pi$
$348$		0			0
$349$	−20.1440	+	20.1440i	−1.07828	+	1.07828i	−0.0816193	+	0.996664i	$0.526009\pi$
$349$	−20.1440	+	20.1440i	−1.07828	+	1.07828i	−0.996664	+	0.0816193i	$0.973991\pi$
$350$	0.742298			0.0396775
$351$		0			0
$352$	14.5081			0.773285
$353$	11.5304	−	11.5304i	0.613701	−	0.613701i	−0.330207	−	0.943908i	$-0.607119\pi$
$353$	11.5304	−	11.5304i	0.613701	−	0.613701i	0.943908	+	0.330207i	$0.107119\pi$
$354$		0			0
$355$			20.1233i			1.06803i
$356$	−5.86795	+	5.86795i	−0.311001	+	0.311001i
$357$		0			0
$358$	3.43196	−	3.43196i	0.181385	−	0.181385i
$359$	−21.1708	−	21.1708i	−1.11735	−	1.11735i	−0.992128	−	0.125225i	$-0.960035\pi$
$359$	−21.1708	−	21.1708i	−1.11735	−	1.11735i	−0.125225	−	0.992128i	$-0.539965\pi$
$360$		0			0
$361$			17.0369i			0.896681i
$362$	6.48631	+	6.48631i	0.340913	+	0.340913i
$363$		0			0
$364$	−1.19771	−	3.44874i	−0.0627770	−	0.180763i
$365$			9.55886i			0.500334i
$366$		0			0
$367$	−11.0577			−0.577206			−0.288603	−	0.957449i	$-0.593191\pi$
$367$	−11.0577			−0.577206			−0.288603	+	0.957449i	$0.593191\pi$
$368$	11.5728			0.603273
$369$		0			0
$370$	−1.52884	−	1.52884i	−0.0794805	−	0.0794805i
$371$	5.74401	+	5.74401i	0.298214	+	0.298214i
$372$		0			0
$373$	−27.5473			−1.42635			−0.713173	−	0.700988i	$-0.752743\pi$
$373$	−27.5473			−1.42635			−0.713173	+	0.700988i	$0.752743\pi$
$374$	6.48631			0.335399
$375$		0			0
$376$		−	5.61515i		−	0.289579i
$377$	−10.0682	+	20.7822i	−0.518539	+	1.07034i
$378$		0			0
$379$	−8.73445	−	8.73445i	−0.448659	−	0.448659i	0.446250	−	0.894908i	$-0.352759\pi$
$379$	−8.73445	−	8.73445i	−0.448659	−	0.448659i	−0.894908	+	0.446250i	$0.852759\pi$
$380$		−	26.8407i		−	1.37690i
$381$		0			0
$382$	−3.71203	−	3.71203i	−0.189924	−	0.189924i
$383$	13.2298	−	13.2298i	0.676011	−	0.676011i	−0.283084	−	0.959095i	$-0.591357\pi$
$383$	13.2298	−	13.2298i	0.676011	−	0.676011i	0.959095	+	0.283084i	$0.0913575\pi$
$384$		0			0
$385$	3.05767	−	3.05767i	0.155834	−	0.155834i
$386$		−	7.83949i		−	0.399019i
$387$		0			0
$388$	4.71203	−	4.71203i	0.239217	−	0.239217i
$389$	−13.8164			−0.700522			−0.350261	−	0.936652i	$-0.613907\pi$
$389$	−13.8164			−0.700522			−0.350261	+	0.936652i	$0.613907\pi$
$390$		0			0
$391$	22.9009			1.15815
$392$	−9.85432	+	9.85432i	−0.497718	+	0.497718i
$393$		0			0
$394$		−	4.02638i		−	0.202846i
$395$	5.77892	−	5.77892i	0.290769	−	0.290769i
$396$		0			0
$397$	20.0577	−	20.0577i	1.00667	−	1.00667i	0.00668818	−	0.999978i	$-0.497871\pi$
$397$	20.0577	−	20.0577i	1.00667	−	1.00667i	0.999978	−	0.00668818i	$-0.00212893\pi$
$398$	4.67043	+	4.67043i	0.234107	+	0.234107i
$399$		0			0
$400$			4.59668i			0.229834i
$401$	8.90560	+	8.90560i	0.444724	+	0.444724i	0.893596	−	0.448872i	$-0.148174\pi$
$401$	8.90560	+	8.90560i	0.444724	+	0.444724i	−0.448872	+	0.893596i	$0.648174\pi$
$402$		0			0
$403$	5.18714	+	14.9361i	0.258390	+	0.744021i
$404$			1.93235i			0.0961381i
$405$		0			0
$406$	−2.21747			−0.110051
$407$	−3.77994			−0.187365
$408$		0			0
$409$	−4.16246	−	4.16246i	−0.205820	−	0.205820i	0.596668	−	0.802488i	$-0.296491\pi$
$409$	−4.16246	−	4.16246i	−0.205820	−	0.205820i	−0.802488	+	0.596668i	$0.796491\pi$
$410$	−9.37577	−	9.37577i	−0.463036	−	0.463036i
$411$		0			0
$412$	9.76140			0.480910
$413$	−9.17773			−0.451606
$414$		0			0
$415$			8.45269i			0.414926i
$416$	−18.4878	+	6.42060i	−0.906438	+	0.314796i
$417$		0			0
$418$	6.48963	+	6.48963i	0.317418	+	0.317418i
$419$			19.1327i			0.934692i	0.884075	+	0.467346i	$0.154790\pi$
$419$			19.1327i			0.934692i	−0.884075	+	0.467346i	$0.845210\pi$
$420$		0			0
$421$	−7.75914	−	7.75914i	−0.378157	−	0.378157i	0.492280	−	0.870437i	$-0.336164\pi$
$421$	−7.75914	−	7.75914i	−0.378157	−	0.378157i	−0.870437	+	0.492280i	$0.836164\pi$
$422$	−5.24960	+	5.24960i	−0.255546	+	0.255546i
$423$		0			0
$424$	19.9361	−	19.9361i	0.968183	−	0.968183i
$425$			9.09616i			0.441229i
$426$		0			0
$427$	−2.77761	+	2.77761i	−0.134418	+	0.134418i
$428$	−25.9303			−1.25339
$429$		0			0
$430$	−15.5967			−0.752139
$431$	1.34571	−	1.34571i	0.0648206	−	0.0648206i	−0.673953	−	0.738774i	$-0.735405\pi$
$431$	1.34571	−	1.34571i	0.0648206	−	0.0648206i	0.738774	+	0.673953i	$0.235405\pi$
$432$		0			0
$433$			4.48963i			0.215758i	0.994164	+	0.107879i	$0.0344059\pi$
$433$			4.48963i			0.215758i	−0.994164	+	0.107879i	$0.965594\pi$
$434$	−1.07358	+	1.07358i	−0.0515335	+	0.0515335i
$435$		0			0
$436$	4.23030	−	4.23030i	0.202595	−	0.202595i
$437$	22.9126	+	22.9126i	1.09606	+	1.09606i
$438$		0			0
$439$		−	10.2017i		−	0.486899i	−0.969914	−	0.243449i	$-0.921721\pi$
$439$		−	10.2017i		−	0.486899i	0.969914	−	0.243449i	$-0.0782790\pi$
$440$	−10.6125	−	10.6125i	−0.505930	−	0.505930i
$441$		0			0
$442$	−8.26555	+	2.87053i	−0.393152	+	0.136537i
$443$			38.1806i			1.81401i	0.421116	+	0.907007i	$0.361639\pi$
$443$			38.1806i			1.81401i	−0.421116	+	0.907007i	$0.638361\pi$
$444$		0			0
$445$	13.2593			0.628553
$446$	−2.11225			−0.100018
$447$		0			0
$448$	0.506413	+	0.506413i	0.0239258	+	0.0239258i
$449$	−1.34571	−	1.34571i	−0.0635080	−	0.0635080i	0.674639	−	0.738147i	$-0.264299\pi$
$449$	−1.34571	−	1.34571i	−0.0635080	−	0.0635080i	−0.738147	+	0.674639i	$0.764299\pi$
$450$		0			0
$451$	−23.1809			−1.09155
$452$	−10.7140			−0.503944
$453$		0			0
$454$			3.27781i			0.153835i
$455$	−2.54324	+	5.24960i	−0.119229	+	0.246105i
$456$		0			0
$457$	−22.2857	−	22.2857i	−1.04248	−	1.04248i	−0.999057	−	0.0434249i	$-0.986173\pi$
$457$	−22.2857	−	22.2857i	−1.04248	−	1.04248i	−0.0434249	−	0.999057i	$-0.513827\pi$
$458$		−	1.35318i		−	0.0632300i
$459$		0			0
$460$	−17.0656	−	17.0656i	−0.795687	−	0.795687i
$461$	−0.420316	+	0.420316i	−0.0195761	+	0.0195761i	−0.716827	−	0.697251i	$-0.754406\pi$
$461$	−0.420316	+	0.420316i	−0.0195761	+	0.0195761i	0.697251	+	0.716827i	$0.254406\pi$
$462$		0			0
$463$	−0.551261	+	0.551261i	−0.0256193	+	0.0256193i	−0.719800	−	0.694181i	$-0.755767\pi$
$463$	−0.551261	+	0.551261i	−0.0256193	+	0.0256193i	0.694181	+	0.719800i	$0.255767\pi$
$464$		−	13.7317i		−	0.637477i
$465$		0			0
$466$	−4.83359	+	4.83359i	−0.223912	+	0.223912i
$467$	28.3104			1.31005			0.655024	−	0.755608i	$-0.272658\pi$
$467$	28.3104			1.31005			0.655024	+	0.755608i	$0.272658\pi$
$468$		0			0
$469$	−4.76970			−0.220244
$470$	−2.88946	+	2.88946i	−0.133281	+	0.133281i
$471$		0			0
$472$			31.8538i			1.46619i
$473$	−19.2809	+	19.2809i	−0.886534	+	0.886534i
$474$		0			0
$475$	−9.10083	+	9.10083i	−0.417575	+	0.417575i
$476$	3.03765	+	3.03765i	0.139230	+	0.139230i
$477$		0			0
$478$			3.86618i			0.176835i
$479$	−19.8992	−	19.8992i	−0.909218	−	0.909218i	0.0869912	−	0.996209i	$-0.472275\pi$
$479$	−19.8992	−	19.8992i	−0.909218	−	0.909218i	−0.996209	+	0.0869912i	$0.972275\pi$
$480$		0			0
$481$	4.81681	−	1.67282i	0.219628	−	0.0762742i
$482$		−	8.74994i		−	0.398549i
$483$		0			0
$484$	6.45043			0.293201
$485$	−10.6474			−0.483473
$486$		0			0
$487$	−26.7345	−	26.7345i	−1.21145	−	1.21145i	−0.970549	−	0.240905i	$-0.922556\pi$
$487$	−26.7345	−	26.7345i	−1.21145	−	1.21145i	−0.240905	−	0.970549i	$-0.577444\pi$
$488$	9.64043	+	9.64043i	0.436402	+	0.436402i
$489$		0			0
$490$	10.1417			0.458156
$491$	16.4263			0.741309			0.370654	−	0.928771i	$-0.379133\pi$
$491$	16.4263			0.741309			0.370654	+	0.928771i	$0.379133\pi$
$492$		0			0
$493$		−	27.1730i		−	1.22381i
$494$	−11.1418	−	5.39778i	−0.501293	−	0.242858i
$495$		0			0
$496$	−6.64814	−	6.64814i	−0.298510	−	0.298510i
$497$		−	4.55715i		−	0.204416i
$498$		0			0
$499$	14.5513	+	14.5513i	0.651404	+	0.651404i	0.953331	−	0.301927i	$-0.0976299\pi$
$499$	14.5513	+	14.5513i	0.651404	+	0.651404i	−0.301927	+	0.953331i	$0.597630\pi$
$500$	−9.02954	+	9.02954i	−0.403813	+	0.403813i
$501$		0			0
$502$	4.58651	−	4.58651i	0.204706	−	0.204706i
$503$			12.7247i			0.567367i	0.958918	+	0.283684i	$0.0915566\pi$
$503$			12.7247i			0.567367i	−0.958918	+	0.283684i	$0.908443\pi$
$504$		0			0
$505$	2.18319	−	2.18319i	0.0971507	−	0.0971507i
$506$	8.25233			0.366861
$507$		0			0
$508$	7.51037			0.333219
$509$	−0.964577	+	0.964577i	−0.0427541	+	0.0427541i	−0.728161	−	0.685407i	$-0.759625\pi$
$509$	−0.964577	+	0.964577i	−0.0427541	+	0.0427541i	0.685407	+	0.728161i	$0.259625\pi$
$510$		0			0
$511$		−	2.16472i		−	0.0957615i
$512$	14.5988	−	14.5988i	0.645184	−	0.645184i
$513$		0			0
$514$	−3.89917	+	3.89917i	−0.171985	+	0.171985i
$515$	−11.0285	−	11.0285i	−0.485975	−	0.485975i
$516$		0			0
$517$			7.14399i			0.314192i
$518$	0.346223	+	0.346223i	0.0152122	+	0.0152122i
$519$		0			0
$520$	18.2201	+	8.82698i	0.799006	+	0.387089i
$521$			6.40474i			0.280597i	0.990109	+	0.140298i	$0.0448062\pi$
$521$			6.40474i			0.280597i	−0.990109	+	0.140298i	$0.955194\pi$
$522$		0			0
$523$	−7.02073			−0.306995			−0.153498	−	0.988149i	$-0.549054\pi$
$523$	−7.02073			−0.306995			−0.153498	+	0.988149i	$0.549054\pi$
$524$	−3.61682			−0.158001
$525$		0			0
$526$	−4.86658	−	4.86658i	−0.212193	−	0.212193i
$527$	−13.1557	−	13.1557i	−0.573072	−	0.573072i
$528$		0			0
$529$	6.13608			0.266786
$530$	−20.5176			−0.891226
$531$		0			0
$532$			6.07841i			0.263532i
$533$	29.5396	−	10.2588i	1.27950	−	0.444357i
$534$		0			0
$535$	29.2963	+	29.2963i	1.26659	+	1.26659i
$536$			16.5545i			0.715046i
$537$		0			0
$538$	7.98548	+	7.98548i	0.344279	+	0.344279i
$539$	12.5374	−	12.5374i	0.540022	−	0.540022i
$540$		0			0
$541$	−20.5288	+	20.5288i	−0.882604	+	0.882604i	−0.993799	−	0.111195i	$-0.964532\pi$
$541$	−20.5288	+	20.5288i	−0.882604	+	0.882604i	0.111195	+	0.993799i	$0.464532\pi$
$542$		−	2.19701i		−	0.0943698i
$543$		0			0
$544$	16.2840	−	16.2840i	0.698172	−	0.698172i
$545$	−9.55886			−0.409457
$546$		0			0
$547$	−8.03694			−0.343635			−0.171817	−	0.985129i	$-0.554964\pi$
$547$	−8.03694			−0.343635			−0.171817	+	0.985129i	$0.554964\pi$
$548$	10.2588	−	10.2588i	0.438233	−	0.438233i
$549$		0			0
$550$			3.27781i			0.139766i
$551$	27.1870	−	27.1870i	1.15820	−	1.15820i
$552$		0			0
$553$	−1.30871	+	1.30871i	−0.0556518	+	0.0556518i
$554$	−5.51426	−	5.51426i	−0.234278	−	0.234278i
$555$		0			0
$556$		−	33.5183i		−	1.42149i
$557$	0.0423807	+	0.0423807i	0.00179573	+	0.00179573i	0.708004	−	0.706208i	$-0.249596\pi$
$557$	0.0423807	+	0.0423807i	0.00179573	+	0.00179573i	−0.706208	+	0.708004i	$0.749596\pi$
$558$		0			0
$559$	16.0369	−	33.1025i	0.678290	−	1.40009i
$560$		−	3.46863i		−	0.146576i
$561$		0			0
$562$	−7.26990			−0.306662
$563$	15.6491			0.659531			0.329765	−	0.944063i	$-0.393030\pi$
$563$	15.6491			0.659531			0.329765	+	0.944063i	$0.393030\pi$
$564$		0			0
$565$	12.1048	+	12.1048i	0.509252	+	0.509252i
$566$	9.75690	+	9.75690i	0.410113	+	0.410113i
$567$		0			0
$568$	−15.8168			−0.663659
$569$	−13.0392			−0.546633			−0.273317	−	0.961924i	$-0.588121\pi$
$569$	−13.0392			−0.546633			−0.273317	+	0.961924i	$0.588121\pi$
$570$		0			0
$571$			26.0448i			1.08994i	0.838455	+	0.544971i	$0.183460\pi$
$571$			26.0448i			1.08994i	−0.838455	+	0.544971i	$0.816540\pi$
$572$	15.2288	−	5.28878i	0.636747	−	0.221135i
$573$		0			0
$574$	2.12325	+	2.12325i	0.0886229	+	0.0886229i
$575$			11.5728i			0.482618i
$576$		0			0
$577$	20.4689	+	20.4689i	0.852132	+	0.852132i	0.990395	−	0.138264i	$-0.0441522\pi$
$577$	20.4689	+	20.4689i	0.852132	+	0.852132i	−0.138264	+	0.990395i	$0.544152\pi$
$578$	0.404460	−	0.404460i	0.0168233	−	0.0168233i
$579$		0			0
$580$	−20.2492	+	20.2492i	−0.840801	+	0.840801i
$581$		−	1.91421i		−	0.0794149i
$582$		0			0
$583$	−25.3641	+	25.3641i	−1.05047	+	1.05047i
$584$	−7.51323			−0.310900
$585$		0			0
$586$	−12.2650			−0.506662
$587$	11.9582	−	11.9582i	0.493567	−	0.493567i	−0.415861	−	0.909428i	$-0.636520\pi$
$587$	11.9582	−	11.9582i	0.493567	−	0.493567i	0.909428	+	0.415861i	$0.136520\pi$
$588$		0			0
$589$		−	26.3249i		−	1.08470i
$590$	16.3914	−	16.3914i	0.674823	−	0.674823i
$591$		0			0
$592$	−2.14399	+	2.14399i	−0.0881173	+	0.0881173i
$593$	18.4794	+	18.4794i	0.758858	+	0.758858i	0.976115	−	0.217256i	$-0.0697108\pi$
$593$	18.4794	+	18.4794i	0.758858	+	0.758858i	−0.217256	+	0.976115i	$0.569711\pi$
$594$		0			0
$595$		−	6.86392i		−	0.281393i
$596$	10.2588	+	10.2588i	0.420216	+	0.420216i
$597$		0			0
$598$	−10.5160	+	3.65209i	−0.430032	+	0.149345i
$599$		−	32.6346i		−	1.33341i	−0.745320	−	0.666707i	$-0.767703\pi$
$599$		−	32.6346i		−	1.33341i	0.745320	−	0.666707i	$-0.232297\pi$
$600$		0			0
$601$	1.58877			0.0648074			0.0324037	−	0.999475i	$-0.489684\pi$
$601$	1.58877			0.0648074			0.0324037	+	0.999475i	$0.489684\pi$
$602$	3.53205			0.143956
$603$		0			0
$604$	20.9651	+	20.9651i	0.853060	+	0.853060i
$605$	−7.28776	−	7.28776i	−0.296289	−	0.296289i
$606$		0			0
$607$	33.0162			1.34009			0.670043	−	0.742322i	$-0.266276\pi$
$607$	33.0162			1.34009			0.670043	+	0.742322i	$0.266276\pi$
$608$	32.5847			1.32149
$609$		0			0
$610$		−	9.92159i		−	0.401714i
$611$	−3.16159	−	9.10364i	−0.127904	−	0.368294i
$612$		0			0
$613$	17.0369	+	17.0369i	0.688116	+	0.688116i	0.961815	−	0.273700i	$-0.0882474\pi$
$613$	17.0369	+	17.0369i	0.688116	+	0.688116i	−0.273700	+	0.961815i	$0.588247\pi$
$614$			0.579171i			0.0233734i
$615$		0			0
$616$	2.40332	+	2.40332i	0.0968326	+	0.0968326i
$617$	−22.1778	+	22.1778i	−0.892844	+	0.892844i	−0.994790	−	0.101946i	$-0.967493\pi$
$617$	−22.1778	+	22.1778i	−0.892844	+	0.892844i	0.101946	+	0.994790i	$0.467493\pi$
$618$		0			0
$619$	17.5720	−	17.5720i	0.706278	−	0.706278i	−0.259472	−	0.965751i	$-0.583549\pi$
$619$	17.5720	−	17.5720i	0.706278	−	0.706278i	0.965751	+	0.259472i	$0.0835487\pi$
$620$			19.6071i			0.787440i
$621$		0			0
$622$	13.8538	−	13.8538i	0.555485	−	0.555485i
$623$	−3.00274			−0.120302
$624$		0			0
$625$	31.1233			1.24493
$626$	−11.1418	+	11.1418i	−0.445316	+	0.445316i
$627$		0			0
$628$		−	9.90578i		−	0.395284i
$629$	−4.24264	+	4.24264i	−0.169165	+	0.169165i
$630$		0			0
$631$	−23.7137	+	23.7137i	−0.944028	+	0.944028i	−0.998515	−	0.0544863i	$-0.982648\pi$
$631$	−23.7137	+	23.7137i	−0.944028	+	0.944028i	0.0544863	+	0.998515i	$0.482648\pi$
$632$	4.54221	+	4.54221i	0.180679	+	0.180679i
$633$		0			0
$634$			14.0678i			0.558705i
$635$	−8.48528	−	8.48528i	−0.336728	−	0.336728i
$636$		0			0
$637$	−10.4280	+	21.5249i	−0.413173	+	0.852847i
$638$		−	9.79181i		−	0.387661i
$639$		0			0
$640$	−30.8251			−1.21847
$641$	−4.47559			−0.176775			−0.0883875	−	0.996086i	$-0.528171\pi$
$641$	−4.47559			−0.176775			−0.0883875	+	0.996086i	$0.528171\pi$
$642$		0			0
$643$	−10.4649	−	10.4649i	−0.412697	−	0.412697i	0.469980	−	0.882677i	$-0.344261\pi$
$643$	−10.4649	−	10.4649i	−0.412697	−	0.412697i	−0.882677	+	0.469980i	$0.844261\pi$
$644$	3.86470	+	3.86470i	0.152291	+	0.152291i
$645$		0			0
$646$	14.5680			0.573172
$647$	−43.2820			−1.70159			−0.850795	−	0.525498i	$-0.823879\pi$
$647$	−43.2820			−1.70159			−0.850795	+	0.525498i	$0.823879\pi$
$648$		0			0
$649$		−	40.5266i		−	1.59081i
$650$	−1.45060	−	4.17693i	−0.0568973	−	0.163833i
$651$		0			0
$652$	12.2158	+	12.2158i	0.478407	+	0.478407i
$653$			7.62651i			0.298448i	0.988803	+	0.149224i	$0.0476776\pi$
$653$			7.62651i			0.298448i	−0.988803	+	0.149224i	$0.952322\pi$
$654$		0			0
$655$	4.08631	+	4.08631i	0.159665	+	0.159665i
$656$	−13.1482	+	13.1482i	−0.513352	+	0.513352i
$657$		0			0
$658$	0.654353	−	0.654353i	0.0255093	−	0.0255093i
$659$		−	0.692447i		−	0.0269739i	−0.999909	−	0.0134870i	$-0.995707\pi$
$659$		−	0.692447i		−	0.0269739i	0.999909	−	0.0134870i	$-0.00429316\pi$
$660$		0			0
$661$	24.0656	−	24.0656i	0.936043	−	0.936043i	−0.0620316	−	0.998074i	$-0.519758\pi$
$661$	24.0656	−	24.0656i	0.936043	−	0.936043i	0.998074	+	0.0620316i	$0.0197580\pi$
$662$	18.9346			0.735915
$663$		0			0
$664$	−6.64379			−0.257829
$665$	6.86744	−	6.86744i	0.266308	−	0.266308i
$666$		0			0
$667$		−	34.5714i		−	1.33861i
$668$	−2.00325	+	2.00325i	−0.0775080	+	0.0775080i
$669$		0			0
$670$	8.51867	−	8.51867i	0.329105	−	0.329105i
$671$	−12.2652	−	12.2652i	−0.473494	−	0.473494i
$672$		0			0
$673$		−	20.4975i		−	0.790122i	−0.918655	−	0.395061i	$-0.870723\pi$
$673$		−	20.4975i		−	0.790122i	0.918655	−	0.395061i	$-0.129277\pi$
$674$	6.98072	+	6.98072i	0.268887	+	0.268887i
$675$		0			0
$676$	−17.0656	+	13.4791i	−0.656368	+	0.518426i
$677$		−	25.5374i		−	0.981482i	−0.871305	−	0.490741i	$-0.836726\pi$
$677$		−	25.5374i		−	0.981482i	0.871305	−	0.490741i	$-0.163274\pi$
$678$		0			0
$679$	2.41123			0.0925344
$680$	−23.8231			−0.913573
$681$		0			0
$682$	−4.74066	−	4.74066i	−0.181529	−	0.181529i
$683$	35.4500	+	35.4500i	1.35646	+	1.35646i	0.878246	+	0.478209i	$0.158714\pi$
$683$	35.4500	+	35.4500i	1.35646	+	1.35646i	0.478209	+	0.878246i	$0.341286\pi$
$684$		0			0
$685$	−23.1809			−0.885698
$686$	−4.72028			−0.180221
$687$		0			0
$688$			21.8722i			0.833871i
$689$	21.0967	−	43.5466i	0.803721	−	1.65899i
$690$		0			0
$691$	−23.6873	−	23.6873i	−0.901109	−	0.901109i	0.0944232	−	0.995532i	$-0.469899\pi$
$691$	−23.6873	−	23.6873i	−0.901109	−	0.901109i	−0.995532	+	0.0944232i	$0.969899\pi$
$692$		−	6.18674i		−	0.235184i
$693$		0			0
$694$	−7.14399	−	7.14399i	−0.271182	−	0.271182i
$695$	−37.8693	+	37.8693i	−1.43646	+	1.43646i
$696$		0			0
$697$	−26.0185	+	26.0185i	−0.985520	+	0.985520i
$698$			16.2949i			0.616771i
$699$		0			0
$700$	−1.53505	+	1.53505i	−0.0580195	+	0.0580195i
$701$	−29.6985			−1.12170			−0.560848	−	0.827919i	$-0.689525\pi$
$701$	−29.6985			−1.12170			−0.560848	+	0.827919i	$0.689525\pi$
$702$		0			0
$703$	−8.48963			−0.320193
$704$	−2.23620	+	2.23620i	−0.0842798	+	0.0842798i
$705$		0			0
$706$		−	9.32718i		−	0.351033i
$707$	−0.494409	+	0.494409i	−0.0185942	+	0.0185942i
$708$		0			0
$709$	−26.6521	+	26.6521i	−1.00094	+	1.00094i	−0.000940752		1.00000i	$0.500299\pi$
$709$	−26.6521	+	26.6521i	−1.00094	+	1.00094i	−1.00000		0.000940752i	$0.999701\pi$
$710$	8.13906	+	8.13906i	0.305453	+	0.305453i
$711$		0			0
$712$			10.4218i			0.390573i
$713$	−16.7376	−	16.7376i	−0.626829	−	0.626829i
$714$		0			0
$715$	−23.1809	−	11.2303i	−0.866918	−	0.419989i
$716$			14.1944i			0.530469i
$717$		0			0
$718$	−17.1255			−0.639119
$719$	−12.5616			−0.468469			−0.234234	−	0.972180i	$-0.575258\pi$
$719$	−12.5616			−0.468469			−0.234234	+	0.972180i	$0.575258\pi$
$720$		0			0
$721$	2.49754	+	2.49754i	0.0930132	+	0.0930132i
$722$	6.89077	+	6.89077i	0.256448	+	0.256448i
$723$		0			0
$724$	−26.8270			−0.997017
$725$	13.7317			0.509982
$726$		0			0
$727$			20.8560i			0.773507i	0.922183	+	0.386753i	$0.126404\pi$
$727$			20.8560i			0.773507i	−0.922183	+	0.386753i	$0.873596\pi$
$728$	−4.12617	−	1.99897i	−0.152926	−	0.0740869i
$729$		0			0
$730$	3.86618	+	3.86618i	0.143094	+	0.143094i
$731$			43.2820i			1.60084i
$732$		0			0
$733$	−2.04711	−	2.04711i	−0.0756117	−	0.0756117i	0.668290	−	0.743901i	$-0.267027\pi$
$733$	−2.04711	−	2.04711i	−0.0756117	−	0.0756117i	−0.743901	+	0.668290i	$0.767027\pi$
$734$	−4.47239	+	4.47239i	−0.165079	+	0.165079i
$735$		0			0
$736$	20.7177	−	20.7177i	0.763664	−	0.763664i
$737$		−	21.0618i		−	0.775822i
$738$		0			0
$739$	34.3786	−	34.3786i	1.26464	−	1.26464i	0.315819	−	0.948819i	$-0.397721\pi$
$739$	34.3786	−	34.3786i	1.26464	−	1.26464i	0.948819	−	0.315819i	$-0.102279\pi$
$740$	6.32318			0.232445
$741$		0			0
$742$	4.64645			0.170576
$743$	−19.7510	+	19.7510i	−0.724595	+	0.724595i	−0.969538	−	0.244943i	$-0.921231\pi$
$743$	−19.7510	+	19.7510i	−0.724595	+	0.724595i	0.244943	+	0.969538i	$0.421231\pi$
$744$		0			0
$745$		−	23.1809i		−	0.849284i
$746$	−11.1418	+	11.1418i	−0.407930	+	0.407930i
$747$		0			0
$748$	−13.4135	+	13.4135i	−0.490446	+	0.490446i
$749$	−6.63449	−	6.63449i	−0.242419	−	0.242419i
$750$		0			0
$751$		−	12.6992i		−	0.463400i	−0.972787	−	0.231700i	$-0.925571\pi$
$751$		−	12.6992i		−	0.463400i	0.972787	−	0.231700i	$-0.0744288\pi$
$752$	4.05207	+	4.05207i	0.147764	+	0.147764i
$753$		0			0
$754$	4.33339	+	12.4778i	0.157813	+	0.454414i
$755$		−	47.3732i		−	1.72409i
$756$		0			0
$757$	31.3536			1.13956			0.569782	−	0.821796i	$-0.307028\pi$
$757$	31.3536			1.13956			0.569782	+	0.821796i	$0.307028\pi$
$758$	−7.06548			−0.256630
$759$		0			0
$760$	−23.8353	−	23.8353i	−0.864597	−	0.864597i
$761$	19.6345	+	19.6345i	0.711752	+	0.711752i	0.966902	−	0.255150i	$-0.0821248\pi$
$761$	19.6345	+	19.6345i	0.711752	+	0.711752i	−0.255150	+	0.966902i	$0.582125\pi$
$762$		0			0
$763$	2.16472			0.0783681
$764$	15.3527			0.555442
$765$		0			0
$766$		−	10.7019i		−	0.386674i
$767$	17.9351	+	51.6433i	0.647600	+	1.86473i
$768$		0			0
$769$	12.8353	+	12.8353i	0.462852	+	0.462852i	0.899589	−	0.436737i	$-0.143866\pi$
$769$	12.8353	+	12.8353i	0.462852	+	0.462852i	−0.436737	+	0.899589i	$0.643866\pi$
$770$		−	2.47342i		−	0.0891357i
$771$		0			0
$772$	16.2118	+	16.2118i	0.583476	+	0.583476i
$773$	8.44290	−	8.44290i	0.303670	−	0.303670i	−0.538778	−	0.842448i	$-0.681114\pi$
$773$	8.44290	−	8.44290i	0.303670	−	0.303670i	0.842448	+	0.538778i	$0.181114\pi$
$774$		0			0
$775$	6.64814	−	6.64814i	0.238808	−	0.238808i
$776$		−	8.36881i		−	0.300423i
$777$		0			0
$778$	−5.58820	+	5.58820i	−0.200347	+	0.200347i
$779$	−52.0636			−1.86537
$780$		0			0
$781$	20.1233			0.720067
$782$	9.26249	−	9.26249i	0.331226	−	0.331226i
$783$		0			0
$784$		−	14.2224i		−	0.507943i
$785$	−11.1916	+	11.1916i	−0.399447	+	0.399447i
$786$		0			0
$787$	−2.51432	+	2.51432i	−0.0896258	+	0.0896258i	−0.750498	−	0.660872i	$-0.770186\pi$
$787$	−2.51432	+	2.51432i	−0.0896258	+	0.0896258i	0.660872	+	0.750498i	$0.270186\pi$
$788$	8.32643	+	8.32643i	0.296617	+	0.296617i
$789$		0			0
$790$		−	4.67469i		−	0.166318i
$791$	−2.74127	−	2.74127i	−0.0974685	−	0.0974685i
$792$		0			0
$793$	21.0577	+	10.2017i	0.747780	+	0.362272i
$794$		−	16.2251i		−	0.575806i
$795$		0			0
$796$	−19.3166			−0.684659
$797$	−16.1267			−0.571238			−0.285619	−	0.958343i	$-0.592199\pi$
$797$	−16.1267			−0.571238			−0.285619	+	0.958343i	$0.592199\pi$
$798$		0			0
$799$	8.01847	+	8.01847i	0.283673	+	0.283673i
$800$	8.22901	+	8.22901i	0.290939	+	0.290939i
$801$		0			0
$802$	7.20392			0.254379
$803$	9.55886			0.337325
$804$		0			0
$805$		−	8.73276i		−	0.307789i
$806$	8.13906	+	3.94307i	0.286686	+	0.138889i
$807$		0			0
$808$	1.71598	+	1.71598i	0.0603680	+	0.0603680i
$809$		−	14.1976i		−	0.499160i	−0.968354	−	0.249580i	$-0.919707\pi$
$809$		−	14.1976i		−	0.499160i	0.968354	−	0.249580i	$-0.0802926\pi$
$810$		0			0
$811$	10.3232	+	10.3232i	0.362497	+	0.362497i	0.864732	−	0.502234i	$-0.167488\pi$
$811$	10.3232	+	10.3232i	0.362497	+	0.362497i	−0.502234	+	0.864732i	$0.667488\pi$
$812$	4.58567	−	4.58567i	0.160925	−	0.160925i
$813$		0			0
$814$	−1.52884	+	1.52884i	−0.0535857	+	0.0535857i
$815$		−	27.6030i		−	0.966891i
$816$		0			0
$817$	−43.3042	+	43.3042i	−1.51502	+	1.51502i
$818$	−3.36710			−0.117728
$819$		0			0
$820$	38.7776			1.35417
$821$	20.0157	−	20.0157i	0.698552	−	0.698552i	−0.265546	−	0.964098i	$-0.585552\pi$
$821$	20.0157	−	20.0157i	0.698552	−	0.698552i	0.964098	+	0.265546i	$0.0855523\pi$
$822$		0			0
$823$			3.33774i			0.116346i	0.998307	+	0.0581732i	$0.0185276\pi$
$823$			3.33774i			0.116346i	−0.998307	+	0.0581732i	$0.981472\pi$
$824$	8.66838	−	8.66838i	0.301977	−	0.301977i
$825$		0			0
$826$	−3.71203	+	3.71203i	−0.129158	+	0.129158i
$827$	−16.3173	−	16.3173i	−0.567408	−	0.567408i	0.363993	−	0.931402i	$-0.381413\pi$
$827$	−16.3173	−	16.3173i	−0.567408	−	0.567408i	−0.931402	+	0.363993i	$0.881413\pi$
$828$		0			0
$829$			2.52658i			0.0877516i	0.999037	+	0.0438758i	$0.0139706\pi$
$829$			2.52658i			0.0877516i	−0.999037	+	0.0438758i	$0.986029\pi$
$830$	3.41878	+	3.41878i	0.118668	+	0.118668i
$831$		0			0
$832$	1.85997	−	3.83923i	0.0644827	−	0.133101i
$833$		−	28.1441i		−	0.975134i
$834$		0			0
$835$	4.52658			0.156649
$836$	−26.8407			−0.928306
$837$		0			0
$838$	7.73840	+	7.73840i	0.267319	+	0.267319i
$839$	16.8965	+	16.8965i	0.583331	+	0.583331i	0.935817	−	0.352486i	$-0.114664\pi$
$839$	16.8965	+	16.8965i	0.583331	+	0.583331i	−0.352486	+	0.935817i	$0.614664\pi$
$840$		0			0
$841$	−12.0207			−0.414508
$842$	−6.27653			−0.216303
$843$		0			0
$844$		−	21.7120i		−	0.747359i
$845$	34.5096	+	4.05207i	1.18717	+	0.139396i
$846$		0			0
$847$	1.65040	+	1.65040i	0.0567084	+	0.0567084i
$848$			28.7731i			0.988072i
$849$		0			0
$850$	3.67904	+	3.67904i	0.126190	+	0.126190i
$851$	−5.39778	+	5.39778i	−0.185034	+	0.185034i
$852$		0			0
$853$	11.9714	−	11.9714i	0.409892	−	0.409892i	−0.471809	−	0.881701i	$-0.656399\pi$
$853$	11.9714	−	11.9714i	0.409892	−	0.409892i	0.881701	+	0.471809i	$0.156399\pi$
$854$			2.24686i			0.0768861i
$855$		0			0
$856$	−23.0268	+	23.0268i	−0.787039	+	0.787039i
$857$	18.9813			0.648388			0.324194	−	0.945991i	$-0.394907\pi$
$857$	18.9813			0.648388			0.324194	+	0.945991i	$0.394907\pi$
$858$		0			0
$859$	9.92159			0.338520			0.169260	−	0.985571i	$-0.445862\pi$
$859$	9.92159			0.338520			0.169260	+	0.985571i	$0.445862\pi$
$860$	32.2535	−	32.2535i	1.09983	−	1.09983i
$861$		0			0
$862$		−	1.08857i		−	0.0370770i
$863$	33.2879	−	33.2879i	1.13313	−	1.13313i	0.143479	−	0.989653i	$-0.454171\pi$
$863$	33.2879	−	33.2879i	1.13313	−	1.13313i	0.989653	−	0.143479i	$-0.0458288\pi$
$864$		0			0
$865$	−6.98983	+	6.98983i	−0.237661	+	0.237661i
$866$	1.81588	+	1.81588i	0.0617061	+	0.0617061i
$867$		0			0
$868$		−	4.44026i		−	0.150712i
$869$	−5.77892	−	5.77892i	−0.196036	−	0.196036i
$870$		0			0
$871$	9.32096	+	26.8392i	0.315829	+	0.909413i
$872$		−	7.51323i		−	0.254430i
$873$		0			0
$874$	18.5345			0.626938
$875$	−4.62058			−0.156204
$876$		0			0
$877$	−40.6706	−	40.6706i	−1.37335	−	1.37335i	−0.855432	−	0.517916i	$-0.826708\pi$
$877$	−40.6706	−	40.6706i	−1.37335	−	1.37335i	−0.517916	−	0.855432i	$-0.673292\pi$
$878$	−4.12617	−	4.12617i	−0.139251	−	0.139251i
$879$		0			0
$880$	15.3166			0.516323
$881$	34.6304			1.16673			0.583363	−	0.812211i	$-0.301736\pi$
$881$	34.6304			1.16673			0.583363	+	0.812211i	$0.301736\pi$
$882$		0			0
$883$			11.5473i			0.388598i	0.980942	+	0.194299i	$0.0622432\pi$
$883$			11.5473i			0.388598i	−0.980942	+	0.194299i	$0.937757\pi$
$884$	11.1567	−	23.0291i	0.375242	−	0.774552i
$885$		0			0
$886$	15.4425	+	15.4425i	0.518802	+	0.518802i
$887$			26.8407i			0.901224i	0.892720	+	0.450612i	$0.148794\pi$
$887$			26.8407i			0.901224i	−0.892720	+	0.450612i	$0.851206\pi$
$888$		0			0
$889$	1.92159	+	1.92159i	0.0644482	+	0.0644482i
$890$	5.36287	−	5.36287i	0.179764	−	0.179764i
$891$		0			0
$892$	4.36807	−	4.36807i	0.146254	−	0.146254i
$893$			16.0452i			0.536931i
$894$		0			0
$895$	16.0369	−	16.0369i	0.536056	−	0.536056i
$896$	6.98072			0.233209
$897$		0			0
$898$	−1.08857			−0.0363262
$899$	−19.8600	+	19.8600i	−0.662369	+	0.662369i
$900$		0			0
$901$			56.9378i			1.89687i
$902$	−9.37577	+	9.37577i	−0.312179	+	0.312179i
$903$		0			0
$904$	−9.51432	+	9.51432i	−0.316442	+	0.316442i
$905$	30.3094	+	30.3094i	1.00752	+	1.00752i
$906$		0			0
$907$			17.6415i			0.585777i	0.956147	+	0.292889i	$0.0946165\pi$
$907$			17.6415i			0.585777i	−0.956147	+	0.292889i	$0.905383\pi$
$908$	−6.77841	−	6.77841i	−0.224949	−	0.224949i
$909$		0			0
$910$	1.09462	+	3.15189i	0.0362862	+	0.104484i
$911$		−	7.10039i		−	0.235246i	−0.993058	−	0.117623i	$-0.962473\pi$
$911$		−	7.10039i		−	0.235246i	0.993058	−	0.117623i	$-0.0375275\pi$
$912$		0			0
$913$	8.45269			0.279743
$914$	−18.0274			−0.596292
$915$		0			0
$916$	2.79834	+	2.79834i	0.0924597	+	0.0924597i
$917$	−0.925394	−	0.925394i	−0.0305592	−	0.0305592i
$918$		0			0
$919$	−3.80624			−0.125556			−0.0627782	−	0.998028i	$-0.519996\pi$
$919$	−3.80624			−0.125556			−0.0627782	+	0.998028i	$0.519996\pi$
$920$	−30.3094			−0.999270
$921$		0			0
$922$			0.340003i			0.0111974i
$923$	−25.6432	+	8.90560i	−0.844057	+	0.293131i
$924$		0			0
$925$	−2.14399	−	2.14399i	−0.0704938	−	0.0704938i
$926$			0.445927i			0.0146541i
$927$		0			0
$928$	−24.5826	−	24.5826i	−0.806962	−	0.806962i
$929$	24.8692	−	24.8692i	0.815932	−	0.815932i	−0.169584	−	0.985516i	$-0.554242\pi$
$929$	24.8692	−	24.8692i	0.815932	−	0.815932i	0.985516	+	0.169584i	$0.0542423\pi$
$930$		0			0
$931$	28.1585	−	28.1585i	0.922858	−	0.922858i
$932$		−	19.9914i		−	0.654841i
$933$		0			0
$934$	11.4504	−	11.4504i	0.374670	−	0.374670i
$935$	30.3094			0.991222
$936$		0			0
$937$	38.5635			1.25982			0.629908	−	0.776670i	$-0.283093\pi$
$937$	38.5635			1.25982			0.629908	+	0.776670i	$0.283093\pi$
$938$	−1.92915	+	1.92915i	−0.0629891	+	0.0629891i
$939$		0			0
$940$		−	11.9506i		−	0.389787i
$941$	21.1042	−	21.1042i	0.687977	−	0.687977i	−0.273807	−	0.961785i	$-0.588283\pi$
$941$	21.1042	−	21.1042i	0.687977	−	0.687977i	0.961785	+	0.273807i	$0.0882831\pi$
$942$		0			0
$943$	−33.1025	+	33.1025i	−1.07797	+	1.07797i
$944$	−22.9867	−	22.9867i	−0.748153	−	0.748153i
$945$		0			0
$946$			15.5967i			0.507092i
$947$	−0.851301	−	0.851301i	−0.0276636	−	0.0276636i	0.693140	−	0.720803i	$-0.256227\pi$
$947$	−0.851301	−	0.851301i	−0.0276636	−	0.0276636i	−0.720803	+	0.693140i	$0.756227\pi$
$948$		0			0
$949$	−12.1809	+	4.23030i	−0.395410	+	0.137321i
$950$			7.36185i			0.238850i
$951$		0			0
$952$	5.39502			0.174854
$953$	−38.2621			−1.23943			−0.619716	−	0.784826i	$-0.712752\pi$
$953$	−38.2621			−1.23943			−0.619716	+	0.784826i	$0.712752\pi$
$954$		0			0
$955$	−17.3456	−	17.3456i	−0.561292	−	0.561292i
$956$	−7.99515	−	7.99515i	−0.258581	−	0.258581i
$957$		0			0
$958$	−16.0969			−0.520067
$959$	5.24960			0.169518
$960$		0			0
$961$		−	11.7697i		−	0.379668i
$962$	1.27162	−	2.62480i	0.0409986	−	0.0846269i
$963$		0			0
$964$	18.0946	+	18.0946i	0.582788	+	0.582788i
$965$		−	36.6325i		−	1.17924i
$966$		0			0
$967$	−36.4834	−	36.4834i	−1.17323	−	1.17323i	−0.981436	−	0.191792i	$-0.938570\pi$
$967$	−36.4834	−	36.4834i	−1.17323	−	1.17323i	−0.191792	−	0.981436i	$-0.561430\pi$
$968$	5.72815	−	5.72815i	0.184110	−	0.184110i
$969$		0			0
$970$	−4.30644	+	4.30644i	−0.138272	+	0.138272i
$971$			9.01460i			0.289292i	0.989483	+	0.144646i	$0.0462044\pi$
$971$			9.01460i			0.289292i	−0.989483	+	0.144646i	$0.953796\pi$
$972$		0			0
$973$	8.57595	−	8.57595i	0.274932	−	0.274932i
$974$	−21.6260			−0.692943
$975$		0			0
$976$	−13.9137			−0.445366
$977$	11.5970	−	11.5970i	0.371022	−	0.371022i	−0.496828	−	0.867849i	$-0.665502\pi$
$977$	11.5970	−	11.5970i	0.371022	−	0.371022i	0.867849	+	0.496828i	$0.165502\pi$
$978$		0			0
$979$		−	13.2593i		−	0.423770i
$980$	−20.9728	+	20.9728i	−0.669951	+	0.669951i
$981$		0			0
$982$	6.64379	−	6.64379i	0.212012	−	0.212012i
$983$	7.63398	+	7.63398i	0.243486	+	0.243486i	0.818291	−	0.574805i	$-0.194922\pi$
$983$	7.63398	+	7.63398i	0.243486	+	0.243486i	−0.574805	+	0.818291i	$0.694922\pi$
$984$		0			0
$985$		−	18.8145i		−	0.599481i
$986$	−10.9904	−	10.9904i	−0.350006	−	0.350006i
$987$		0			0
$988$	34.2034	−	11.8784i	1.08815	−	0.377903i
$989$			55.0664i			1.75101i
$990$		0			0
$991$	−54.9378			−1.74516			−0.872578	−	0.488474i	$-0.837554\pi$
$991$	−54.9378			−1.74516			−0.872578	+	0.488474i	$0.837554\pi$
$992$	−23.8031			−0.755749
$993$		0			0
$994$	−1.84319	−	1.84319i	−0.0584623	−	0.0584623i
$995$	21.8241	+	21.8241i	0.691870	+	0.691870i
$996$		0			0
$997$	22.9793			0.727761			0.363880	−	0.931446i	$-0.381452\pi$
$997$	22.9793			0.727761			0.363880	+	0.931446i	$0.381452\pi$
$998$	11.7708			0.372599
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	117.2.i.a.8.4	yes	12
3.2	odd	2	inner	117.2.i.a.8.3	✓	12
4.3	odd	2		1872.2.bi.f.593.5		12
12.11	even	2		1872.2.bi.f.593.2		12
13.5	odd	4	inner	117.2.i.a.44.3	yes	12
13.8	odd	4		1521.2.i.g.746.4		12
13.12	even	2		1521.2.i.g.944.3		12
39.5	even	4	inner	117.2.i.a.44.4	yes	12
39.8	even	4		1521.2.i.g.746.3		12
39.38	odd	2		1521.2.i.g.944.4		12
52.31	even	4		1872.2.bi.f.161.2		12
156.83	odd	4		1872.2.bi.f.161.5		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
117.2.i.a.8.3	✓	12	3.2	odd	2	inner
117.2.i.a.8.4	yes	12	1.1	even	1	trivial
117.2.i.a.44.3	yes	12	13.5	odd	4	inner
117.2.i.a.44.4	yes	12	39.5	even	4	inner
1521.2.i.g.746.3		12	39.8	even	4
1521.2.i.g.746.4		12	13.8	odd	4
1521.2.i.g.944.3		12	13.12	even	2
1521.2.i.g.944.4		12	39.38	odd	2
1872.2.bi.f.161.2		12	52.31	even	4
1872.2.bi.f.161.5		12	156.83	odd	4
1872.2.bi.f.593.2		12	12.11	even	2
1872.2.bi.f.593.5		12	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 117 = 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	117.i (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.404460 - 0.404460i) q^{2} +1.67282i q^{4} +(1.88997 - 1.88997i) q^{5} +(-0.428007 + 0.428007i) q^{7} +(1.48551 + 1.48551i) q^{8} +O(q^{10})\)	\(q+(0.404460 - 0.404460i) q^{2} +1.67282i q^{4} +(1.88997 - 1.88997i) q^{5} +(-0.428007 + 0.428007i) q^{7} +(1.48551 + 1.48551i) q^{8} -1.52884i q^{10} +(-1.88997 - 1.88997i) q^{11} +(3.24482 + 1.57199i) q^{13} +0.346223i q^{14} -2.14399 q^{16} -4.24264 q^{17} +(-4.24482 - 4.24482i) q^{19} +(3.16159 + 3.16159i) q^{20} -1.52884 q^{22} -5.39778 q^{23} -2.14399i q^{25} +(1.94821 - 0.676591i) q^{26} +(-0.715980 - 0.715980i) q^{28} +6.40474i q^{29} +(3.10083 + 3.10083i) q^{31} +(-3.83818 + 3.83818i) q^{32} +(-1.71598 + 1.71598i) q^{34} +1.61784i q^{35} +(1.00000 - 1.00000i) q^{37} -3.43372 q^{38} +5.61515 q^{40} +(6.13261 - 6.13261i) q^{41} -10.2017i q^{43} +(3.16159 - 3.16159i) q^{44} +(-2.18319 + 2.18319i) q^{46} +(-1.88997 - 1.88997i) q^{47} +6.63362i q^{49} +(-0.867157 - 0.867157i) q^{50} +(-2.62967 + 5.42801i) q^{52} -13.4204i q^{53} -7.14399 q^{55} -1.27162 q^{56} +(2.59046 + 2.59046i) q^{58} +(10.7215 + 10.7215i) q^{59} +6.48963 q^{61} +2.50833 q^{62} -1.18319i q^{64} +(9.10364 - 3.16159i) q^{65} +(5.57199 + 5.57199i) q^{67} -7.09719i q^{68} +(0.654353 + 0.654353i) q^{70} +(-5.32369 + 5.32369i) q^{71} +(-2.52884 + 2.52884i) q^{73} -0.808921i q^{74} +(7.10083 - 7.10083i) q^{76} +1.61784 q^{77} +3.05767 q^{79} +(-4.05207 + 4.05207i) q^{80} -4.96080i q^{82} +(-2.23620 + 2.23620i) q^{83} +(-8.01847 + 8.01847i) q^{85} +(-4.12617 - 4.12617i) q^{86} -5.61515i q^{88} +(3.50781 + 3.50781i) q^{89} +(-2.06163 + 0.715980i) q^{91} -9.02954i q^{92} -1.52884 q^{94} -16.0452 q^{95} +(-2.81681 - 2.81681i) q^{97} +(2.68304 + 2.68304i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 8 q^{7}+O(q^{10}) \) 12 * q - 8 * q^7	\( 12 q - 8 q^{7} - 4 q^{13} - 20 q^{16} - 8 q^{19} + 16 q^{22} - 12 q^{34} + 12 q^{37} + 96 q^{40} - 72 q^{46} + 40 q^{52} - 80 q^{55} - 92 q^{58} - 8 q^{61} + 64 q^{67} + 88 q^{70} + 4 q^{73} + 48 q^{76} - 32 q^{79} + 24 q^{85} + 64 q^{91} + 16 q^{94} + 12 q^{97}+O(q^{100}) \) 12 * q - 8 * q^7 - 4 * q^13 - 20 * q^16 - 8 * q^19 + 16 * q^22 - 12 * q^34 + 12 * q^37 + 96 * q^40 - 72 * q^46 + 40 * q^52 - 80 * q^55 - 92 * q^58 - 8 * q^61 + 64 * q^67 + 88 * q^70 + 4 * q^73 + 48 * q^76 - 32 * q^79 + 24 * q^85 + 64 * q^91 + 16 * q^94 + 12 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more