LMFDB - Embedded newform 117.2.i.a.44.6

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			44.6
Root			$1.64111 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	117.44
Dual form			117.2.i.a.8.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.77776 + 1.77776i) q^{2} +4.32088i q^{4} +(-2.34822 - 2.34822i) q^{5} +(1.51414 + 1.51414i) q^{7} +(-4.12598 + 4.12598i) q^{8} +O(q^{10})$	\(q+(1.77776 + 1.77776i) q^{2} +4.32088i q^{4} +(-2.34822 - 2.34822i) q^{5} +(1.51414 + 1.51414i) q^{7} +(-4.12598 + 4.12598i) q^{8} -8.34916i q^{10} +(2.34822 - 2.34822i) q^{11} +(-0.806748 - 3.51414i) q^{13} +5.38355i q^{14} -6.02827 q^{16} -4.24264 q^{17} +(-0.193252 + 0.193252i) q^{19} +(10.1464 - 10.1464i) q^{20} +8.34916 q^{22} -2.41461 q^{23} +6.02827i q^{25} +(4.81310 - 7.68151i) q^{26} +(-6.54241 + 6.54241i) q^{28} +7.56485i q^{29} +(-4.83502 + 4.83502i) q^{31} +(-2.46488 - 2.46488i) q^{32} +(-7.54241 - 7.54241i) q^{34} -7.11105i q^{35} +(1.00000 + 1.00000i) q^{37} -0.687114 q^{38} +19.3774 q^{40} +(1.89442 + 1.89442i) q^{41} -5.67004i q^{43} +(10.1464 + 10.1464i) q^{44} +(-4.29261 - 4.29261i) q^{46} +(2.34822 - 2.34822i) q^{47} -2.41478i q^{49} +(-10.7168 + 10.7168i) q^{50} +(15.1842 - 3.48586i) q^{52} +1.96081i q^{53} -11.0283 q^{55} -12.4946 q^{56} +(-13.4485 + 13.4485i) q^{58} +(0.753507 - 0.753507i) q^{59} -1.61350 q^{61} -17.1910 q^{62} +3.29261i q^{64} +(-6.35755 + 10.1464i) q^{65} +(7.51414 - 7.51414i) q^{67} -18.3320i q^{68} +(12.6418 - 12.6418i) q^{70} +(1.66111 + 1.66111i) q^{71} +(7.34916 + 7.34916i) q^{73} +3.55553i q^{74} +(-0.835021 - 0.835021i) q^{76} +7.11105 q^{77} -16.6983 q^{79} +(14.1557 + 14.1557i) q^{80} +6.73566i q^{82} +(7.73177 + 7.73177i) q^{83} +(9.96265 + 9.96265i) q^{85} +(10.0800 - 10.0800i) q^{86} +19.3774i q^{88} +(4.76283 - 4.76283i) q^{89} +(4.09936 - 6.54241i) q^{91} -10.4333i q^{92} +8.34916 q^{94} +0.907598 q^{95} +(-0.707389 + 0.707389i) q^{97} +(4.29290 - 4.29290i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/117\mathbb{Z}\right)^\times$.

$n$	$28$	$92$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$-1$

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 $

$2$	1.77776	+	1.77776i	1.25707	+	1.25707i	0.952486	+	0.304583i	$0.0985169\pi$
$2$	1.77776	+	1.77776i	1.25707	+	1.25707i	0.304583	+	0.952486i	$0.401483\pi$
$3$		0			0
$3$		0			0
$4$			4.32088i			2.16044i
$5$	−2.34822	−	2.34822i	−1.05016	−	1.05016i	−0.998674	−	0.0514820i	$-0.983606\pi$
$5$	−2.34822	−	2.34822i	−1.05016	−	1.05016i	−0.0514820	−	0.998674i	$-0.516394\pi$
$6$		0			0
$7$	1.51414	+	1.51414i	0.572290	+	0.572290i	0.932768	−	0.360478i	$-0.117386\pi$
$7$	1.51414	+	1.51414i	0.572290	+	0.572290i	−0.360478	+	0.932768i	$0.617386\pi$
$8$	−4.12598	+	4.12598i	−1.45876	+	1.45876i
$9$		0			0
$10$		−	8.34916i		−	2.64024i
$11$	2.34822	−	2.34822i	0.708015	−	0.708015i	−0.258103	−	0.966118i	$-0.583097\pi$
$11$	2.34822	−	2.34822i	0.708015	−	0.708015i	0.966118	+	0.258103i	$0.0830972\pi$
$12$		0			0
$13$	−0.806748	−	3.51414i	−0.223752	−	0.974646i
$13$	−0.806748	−	3.51414i	−0.223752	−	0.974646i
$14$			5.38355i			1.43882i
$15$		0			0
$16$	−6.02827			−1.50707
$17$	−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
$19$	−0.193252	+	0.193252i	−0.0443351	+	0.0443351i	−0.728927	−	0.684592i	$-0.759981\pi$
$19$	−0.193252	+	0.193252i	−0.0443351	+	0.0443351i	0.684592	+	0.728927i	$0.259981\pi$
$20$	10.1464	−	10.1464i	2.26880	−	2.26880i
$21$		0			0
$22$	8.34916			1.78005
$23$	−2.41461			−0.503482			−0.251741	−	0.967795i	$-0.581003\pi$
$23$	−2.41461			−0.503482			−0.251741	+	0.967795i	$0.581003\pi$
$24$		0			0
$25$			6.02827i			1.20565i
$26$	4.81310	−	7.68151i	0.943926	−	1.50647i
$27$		0			0
$28$	−6.54241	+	6.54241i	−1.23640	+	1.23640i
$29$			7.56485i			1.40476i	0.711803	+	0.702379i	$0.247879\pi$
$29$			7.56485i			1.40476i	−0.711803	+	0.702379i	$0.752121\pi$
$30$		0			0
$31$	−4.83502	+	4.83502i	−0.868395	+	0.868395i	−0.992295	−	0.123899i	$-0.960460\pi$
$31$	−4.83502	+	4.83502i	−0.868395	+	0.868395i	0.123899	+	0.992295i	$0.460460\pi$
$32$	−2.46488	−	2.46488i	−0.435733	−	0.435733i
$33$		0			0
$34$	−7.54241	−	7.54241i	−1.29351	−	1.29351i
$35$		−	7.11105i		−	1.20199i
$36$		0			0
$37$	1.00000	+	1.00000i	0.164399	+	0.164399i	0.784512	−	0.620113i	$-0.212913\pi$
$37$	1.00000	+	1.00000i	0.164399	+	0.164399i	−0.620113	+	0.784512i	$0.712913\pi$
$38$	−0.687114			−0.111465
$39$		0			0
$40$	19.3774			3.06384
$41$	1.89442	+	1.89442i	0.295859	+	0.295859i	0.839389	−	0.543531i	$-0.182913\pi$
$41$	1.89442	+	1.89442i	0.295859	+	0.295859i	−0.543531	+	0.839389i	$0.682913\pi$
$42$		0			0
$43$		−	5.67004i		−	0.864673i	−0.901712	−	0.432337i	$-0.857689\pi$
$43$		−	5.67004i		−	0.864673i	0.901712	−	0.432337i	$-0.142311\pi$
$44$	10.1464	+	10.1464i	1.52963	+	1.52963i
$45$		0			0
$46$	−4.29261	−	4.29261i	−0.632911	−	0.632911i
$47$	2.34822	−	2.34822i	0.342523	−	0.342523i	−0.514792	−	0.857315i	$-0.672131\pi$
$47$	2.34822	−	2.34822i	0.342523	−	0.342523i	0.857315	+	0.514792i	$0.172131\pi$
$48$		0			0
$49$		−	2.41478i		−	0.344968i
$50$	−10.7168	+	10.7168i	−1.51559	+	1.51559i
$51$		0			0
$52$	15.1842	−	3.48586i	2.10567	−	0.483402i
$53$			1.96081i			0.269339i	0.990891	+	0.134669i	$0.0429972\pi$
$53$			1.96081i			0.269339i	−0.990891	+	0.134669i	$0.957003\pi$
$54$		0			0
$55$	−11.0283			−1.48705
$56$	−12.4946			−1.66966
$57$		0			0
$58$	−13.4485	+	13.4485i	−1.76588	+	1.76588i
$59$	0.753507	−	0.753507i	0.0980983	−	0.0980983i	−0.656354	−	0.754453i	$-0.727902\pi$
$59$	0.753507	−	0.753507i	0.0980983	−	0.0980983i	0.754453	+	0.656354i	$0.227902\pi$
$60$		0			0
$61$	−1.61350			−0.206587			−0.103293	−	0.994651i	$-0.532938\pi$
$61$	−1.61350			−0.206587			−0.103293	+	0.994651i	$0.532938\pi$
$62$	−17.1910			−2.18327
$63$		0			0
$64$			3.29261i			0.411576i
$65$	−6.35755	+	10.1464i	−0.788556	+	1.25850i
$66$		0			0
$67$	7.51414	−	7.51414i	0.917998	−	0.917998i	−0.0788857	−	0.996884i	$-0.525136\pi$
$67$	7.51414	−	7.51414i	0.917998	−	0.917998i	0.996884	+	0.0788857i	$0.0251362\pi$
$68$		−	18.3320i		−	2.22308i
$69$		0			0
$70$	12.6418	−	12.6418i	1.51098	−	1.51098i
$71$	1.66111	+	1.66111i	0.197137	+	0.197137i	0.798772	−	0.601635i	$-0.205484\pi$
$71$	1.66111	+	1.66111i	0.197137	+	0.197137i	−0.601635	+	0.798772i	$0.705484\pi$
$72$		0			0
$73$	7.34916	+	7.34916i	0.860154	+	0.860154i	0.991356	−	0.131202i	$-0.0418836\pi$
$73$	7.34916	+	7.34916i	0.860154	+	0.860154i	−0.131202	+	0.991356i	$0.541884\pi$
$74$			3.55553i			0.413322i
$75$		0			0
$76$	−0.835021	−	0.835021i	−0.0957835	−	0.0957835i
$77$	7.11105			0.810380
$78$		0			0
$79$	−16.6983			−1.87871			−0.939354	−	0.342950i	$-0.888574\pi$
$79$	−16.6983			−1.87871			−0.939354	+	0.342950i	$0.888574\pi$
$80$	14.1557	+	14.1557i	1.58266	+	1.58266i
$81$		0			0
$82$			6.73566i			0.743830i
$83$	7.73177	+	7.73177i	0.848672	+	0.848672i	0.989968	−	0.141295i	$-0.0451266\pi$
$83$	7.73177	+	7.73177i	0.848672	+	0.848672i	−0.141295	+	0.989968i	$0.545127\pi$
$84$		0			0
$85$	9.96265	+	9.96265i	1.08060	+	1.08060i
$86$	10.0800	−	10.0800i	1.08695	−	1.08695i
$87$		0			0
$88$			19.3774i			2.06564i
$89$	4.76283	−	4.76283i	0.504859	−	0.504859i	−0.408085	−	0.912944i	$-0.633803\pi$
$89$	4.76283	−	4.76283i	0.504859	−	0.504859i	0.912944	+	0.408085i	$0.133803\pi$
$90$		0			0
$91$	4.09936	−	6.54241i	0.429730	−	0.685831i
$92$		−	10.4333i		−	1.08774i
$93$		0			0
$94$	8.34916			0.861150
$95$	0.907598			0.0931176
$96$		0			0
$97$	−0.707389	+	0.707389i	−0.0718245	+	0.0718245i	−0.742106	−	0.670282i	$-0.766173\pi$
$97$	−0.707389	+	0.707389i	−0.0718245	+	0.0718245i	0.670282	+	0.742106i	$0.266173\pi$
$98$	4.29290	−	4.29290i	0.433649	−	0.433649i
$99$		0			0
$100$	−26.0475			−2.60475
$101$	−1.82803			−0.181896			−0.0909478	−	0.995856i	$-0.528990\pi$
$101$	−1.82803			−0.181896			−0.0909478	+	0.995856i	$0.528990\pi$
$102$		0			0
$103$		−	14.2553i		−	1.40461i	−0.711875	−	0.702306i	$-0.752154\pi$
$103$		−	14.2553i		−	1.40461i	0.711875	−	0.702306i	$-0.247846\pi$
$104$	17.8279	+	11.1706i	1.74817	+	1.09537i
$105$		0			0
$106$	−3.48586	+	3.48586i	−0.338577	+	0.338577i
$107$		−	18.0109i		−	1.74118i	−0.492006	−	0.870592i	$-0.663736\pi$
$107$		−	18.0109i		−	1.74118i	0.492006	−	0.870592i	$-0.336264\pi$
$108$		0			0
$109$	7.34916	−	7.34916i	0.703922	−	0.703922i	−0.261328	−	0.965250i	$-0.584161\pi$
$109$	7.34916	−	7.34916i	0.703922	−	0.703922i	0.965250	+	0.261328i	$0.0841605\pi$
$110$	−19.6057	−	19.6057i	−1.86933	−	1.86933i
$111$		0			0
$112$	−9.12763	−	9.12763i	−0.862480	−	0.862480i
$113$			7.56485i			0.711641i	0.934554	+	0.355821i	$0.115799\pi$
$113$			7.56485i			0.711641i	−0.934554	+	0.355821i	$0.884201\pi$
$114$		0			0
$115$	5.67004	+	5.67004i	0.528734	+	0.528734i
$116$	−32.6869			−3.03490
$117$		0			0
$118$	2.67912			0.246633
$119$	−6.42394	−	6.42394i	−0.588882	−	0.588882i
$120$		0			0
$121$		−	0.0282739i		−	0.00257035i
$122$	−2.86841	−	2.86841i	−0.259694	−	0.259694i
$123$		0			0
$124$	−20.8916	−	20.8916i	−1.87612	−	1.87612i
$125$	2.41461	−	2.41461i	0.215970	−	0.215970i
$126$		0			0
$127$		−	3.61350i		−	0.320646i	−0.987065	−	0.160323i	$-0.948746\pi$
$127$		−	3.61350i		−	0.320646i	0.987065	−	0.160323i	$-0.0512536\pi$
$128$	−10.7832	+	10.7832i	−0.953113	+	0.953113i
$129$		0			0
$130$	−29.3401	+	6.73566i	−2.57330	+	0.590757i
$131$			11.8075i			1.03163i	0.856701	+	0.515813i	$0.172510\pi$
$131$			11.8075i			1.03163i	−0.856701	+	0.515813i	$0.827490\pi$
$132$		0			0
$133$	−0.585221			−0.0507451
$134$	26.7167			2.30797
$135$		0			0
$136$	17.5051	−	17.5051i	1.50105	−	1.50105i
$137$	−1.89442	+	1.89442i	−0.161851	+	0.161851i	−0.783386	−	0.621535i	$-0.786509\pi$
$137$	−1.89442	+	1.89442i	−0.161851	+	0.161851i	0.621535	+	0.783386i	$0.286509\pi$
$138$		0			0
$139$	15.9253			1.35077			0.675383	−	0.737467i	$-0.263978\pi$
$139$	15.9253			1.35077			0.675383	+	0.737467i	$0.263978\pi$
$140$	30.7260			2.59682
$141$		0			0
$142$			5.90611i			0.495629i
$143$	−10.1464	−	6.35755i	−0.848484	−	0.531645i
$144$		0			0
$145$	17.7639	−	17.7639i	1.47521	−	1.47521i
$146$			26.1301i			2.16254i
$147$		0			0
$148$	−4.32088	+	4.32088i	−0.355175	+	0.355175i
$149$	1.89442	+	1.89442i	0.155197	+	0.155197i	0.780435	−	0.625238i	$-0.214998\pi$
$149$	1.89442	+	1.89442i	0.155197	+	0.155197i	−0.625238	+	0.780435i	$0.714998\pi$
$150$		0			0
$151$	16.2498	+	16.2498i	1.32239	+	1.32239i	0.911837	+	0.410553i	$0.134664\pi$
$151$	16.2498	+	16.2498i	1.32239	+	1.32239i	0.410553	+	0.911837i	$0.365336\pi$
$152$		−	1.59471i		−	0.129348i
$153$		0			0
$154$	12.6418	+	12.6418i	1.01870	+	1.01870i
$155$	22.7074			1.82390
$156$		0			0
$157$	−9.47133			−0.755894			−0.377947	−	0.925827i	$-0.623370\pi$
$157$	−9.47133			−0.755894			−0.377947	+	0.925827i	$0.623370\pi$
$158$	−29.6857	−	29.6857i	−2.36166	−	2.36166i
$159$		0			0
$160$			11.5761i			0.915175i
$161$	−3.65605	−	3.65605i	−0.288137	−	0.288137i
$162$		0			0
$163$	−16.5051	−	16.5051i	−1.29278	−	1.29278i	−0.933063	−	0.359714i	$-0.882874\pi$
$163$	−16.5051	−	16.5051i	−1.29278	−	1.29278i	−0.359714	−	0.933063i	$-0.617126\pi$
$164$	−8.18557	+	8.18557i	−0.639186	+	0.639186i
$165$		0			0
$166$			27.4905i			2.13368i
$167$	8.41889	−	8.41889i	0.651473	−	0.651473i	−0.301875	−	0.953348i	$-0.597612\pi$
$167$	8.41889	−	8.41889i	0.651473	−	0.651473i	0.953348	+	0.301875i	$0.0976124\pi$
$168$		0			0
$169$	−11.6983	+	5.67004i	−0.899871	+	0.436157i
$170$			35.4225i			2.71678i
$171$		0			0
$172$	24.4996			1.86808
$173$	−23.1612			−1.76091			−0.880456	−	0.474127i	$-0.842764\pi$
$173$	−23.1612			−1.76091			−0.880456	+	0.474127i	$0.842764\pi$
$174$		0			0
$175$	−9.12763	+	9.12763i	−0.689984	+	0.689984i
$176$	−14.1557	+	14.1557i	−1.06703	+	1.06703i
$177$		0			0
$178$	16.9344			1.26929
$179$	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
$181$			19.9253i			1.48104i	0.672036	+	0.740518i	$0.265420\pi$
$181$			19.9253i			1.48104i	−0.672036	+	0.740518i	$0.734580\pi$
$182$	18.9185	−	4.34317i	1.40234	−	0.321937i
$183$		0			0
$184$	9.96265	−	9.96265i	0.734457	−	0.734457i
$185$		−	4.69644i		−	0.345289i
$186$		0			0
$187$	−9.96265	+	9.96265i	−0.728541	+	0.728541i
$188$	10.1464	+	10.1464i	0.740001	+	0.740001i
$189$		0			0
$190$	1.61350	+	1.61350i	0.117055	+	0.117055i
$191$		−	2.28183i		−	0.165107i	−0.996587	−	0.0825536i	$-0.973692\pi$
$191$		−	2.28183i		−	0.165107i	0.996587	−	0.0825536i	$-0.0263076\pi$
$192$		0			0
$193$	−14.2835	−	14.2835i	−1.02815	−	1.02815i	−0.999592	−	0.0285595i	$-0.990908\pi$
$193$	−14.2835	−	14.2835i	−1.02815	−	1.02815i	−0.0285595	−	0.999592i	$-0.509092\pi$
$194$	−2.51514			−0.180577
$195$		0			0
$196$	10.4340			0.745284
$197$	3.72245	+	3.72245i	0.265213	+	0.265213i	0.827168	−	0.561955i	$-0.189950\pi$
$197$	3.72245	+	3.72245i	0.265213	+	0.265213i	−0.561955	+	0.827168i	$0.689950\pi$
$198$		0			0
$199$			16.3118i			1.15631i	0.815926	+	0.578157i	$0.196228\pi$
$199$			16.3118i			1.15631i	−0.815926	+	0.578157i	$0.803772\pi$
$200$	−24.8726	−	24.8726i	−1.75876	−	1.75876i
$201$		0			0
$202$	−3.24980	−	3.24980i	−0.228655	−	0.228655i
$203$	−11.4542	+	11.4542i	−0.803929	+	0.803929i
$204$		0			0
$205$		−	8.89703i		−	0.621396i
$206$	25.3425	−	25.3425i	1.76569	−	1.76569i
$207$		0			0
$208$	4.86330	+	21.1842i	0.337209	+	1.46886i
$209$			0.907598i			0.0627799i
$210$		0			0
$211$	3.22699			0.222155			0.111078	−	0.993812i	$-0.464570\pi$
$211$	3.22699			0.222155			0.111078	+	0.993812i	$0.464570\pi$
$212$	−8.47245			−0.581890
$213$		0			0
$214$	32.0192	−	32.0192i	2.18879	−	2.18879i
$215$	−13.3145	+	13.3145i	−0.908042	+	0.908042i
$216$		0			0
$217$	−14.6418			−0.993948
$218$	26.1301			1.76976
$219$		0			0
$220$		−	47.6519i		−	3.21269i
$221$	3.42274	+	14.9092i	0.230238	+	1.00290i
$222$		0			0
$223$	−2.77847	+	2.77847i	−0.186060	+	0.186060i	−0.793991	−	0.607930i	$-0.792000\pi$
$223$	−2.77847	+	2.77847i	−0.186060	+	0.186060i	0.607930	+	0.793991i	$0.292000\pi$
$224$		−	7.46432i		−	0.498731i
$225$		0			0
$226$	−13.4485	+	13.4485i	−0.894582	+	0.894582i
$227$	14.1557	+	14.1557i	0.939548	+	0.939548i	0.998274	−	0.0587264i	$-0.0187040\pi$
$227$	14.1557	+	14.1557i	0.939548	+	0.939548i	−0.0587264	+	0.998274i	$0.518704\pi$
$228$		0			0
$229$	−4.32088	−	4.32088i	−0.285532	−	0.285532i	0.549778	−	0.835311i	$-0.314712\pi$
$229$	−4.32088	−	4.32088i	−0.285532	−	0.285532i	−0.835311	+	0.549778i	$0.814712\pi$
$230$			20.1600i			1.32931i
$231$		0			0
$232$	−31.2125	−	31.2125i	−2.04920	−	2.04920i
$233$	−3.00120			−0.196615			−0.0983075	−	0.995156i	$-0.531343\pi$
$233$	−3.00120			−0.196615			−0.0983075	+	0.995156i	$0.531343\pi$
$234$		0			0
$235$	−11.0283			−0.719405
$236$	3.25582	+	3.25582i	0.211936	+	0.211936i
$237$		0			0
$238$		−	22.8405i		−	1.48053i
$239$	−17.2574	−	17.2574i	−1.11629	−	1.11629i	−0.992281	−	0.124010i	$-0.960424\pi$
$239$	−17.2574	−	17.2574i	−1.11629	−	1.11629i	−0.124010	−	0.992281i	$-0.539576\pi$
$240$		0			0
$241$	8.70739	+	8.70739i	0.560892	+	0.560892i	0.929561	−	0.368669i	$-0.120186\pi$
$241$	8.70739	+	8.70739i	0.560892	+	0.560892i	−0.368669	+	0.929561i	$0.620186\pi$
$242$	0.0502642	−	0.0502642i	0.00323111	−	0.00323111i
$243$		0			0
$244$		−	6.97173i		−	0.446319i
$245$	−5.67043	+	5.67043i	−0.362271	+	0.362271i
$246$		0			0
$247$	0.835021	+	0.523210i	0.0531311	+	0.0332910i
$248$		−	39.8984i		−	2.53355i
$249$		0			0
$250$	8.58522			0.542977
$251$	−14.0893			−0.889310			−0.444655	−	0.895702i	$-0.646674\pi$
$251$	−14.0893			−0.889310			−0.444655	+	0.895702i	$0.646674\pi$
$252$		0			0
$253$	−5.67004	+	5.67004i	−0.356473	+	0.356473i
$254$	6.42394	−	6.42394i	0.403074	−	0.403074i
$255$		0			0
$256$	−31.7549			−1.98468
$257$	−6.65725			−0.415268			−0.207634	−	0.978207i	$-0.566576\pi$
$257$	−6.65725			−0.415268			−0.207634	+	0.978207i	$0.566576\pi$
$258$		0			0
$259$			3.02827i			0.188168i
$260$	−43.8414	−	27.4702i	−2.71893	−	1.70363i
$261$		0			0
$262$	−20.9909	+	20.9909i	−1.29682	+	1.29682i
$263$		−	24.8564i		−	1.53271i	−0.642416	−	0.766356i	$-0.722068\pi$
$263$		−	24.8564i		−	1.53271i	0.642416	−	0.766356i	$-0.277932\pi$
$264$		0			0
$265$	4.60442	−	4.60442i	0.282847	−	0.282847i
$266$	−1.04038	−	1.04038i	−0.0637901	−	0.0637901i
$267$		0			0
$268$	32.4677	+	32.4677i	1.98328	+	1.98328i
$269$		−	22.2536i		−	1.35683i	−0.734681	−	0.678413i	$-0.762668\pi$
$269$		−	22.2536i		−	1.35683i	0.734681	−	0.678413i	$-0.237332\pi$
$270$		0			0
$271$	8.54241	+	8.54241i	0.518915	+	0.518915i	0.917243	−	0.398328i	$-0.130410\pi$
$271$	8.54241	+	8.54241i	0.518915	+	0.518915i	−0.398328	+	0.917243i	$0.630410\pi$
$272$	25.5758			1.55076
$273$		0			0
$274$	−6.73566			−0.406916
$275$	14.1557	+	14.1557i	0.853622	+	0.853622i
$276$		0			0
$277$			9.41478i			0.565679i	0.959167	+	0.282840i	$0.0912764\pi$
$277$			9.41478i			0.565679i	−0.959167	+	0.282840i	$0.908724\pi$
$278$	28.3114	+	28.3114i	1.69801	+	1.69801i
$279$		0			0
$280$	29.3401	+	29.3401i	1.75341	+	1.75341i
$281$	20.6802	−	20.6802i	1.23368	−	1.23368i	0.271135	−	0.962541i	$-0.412601\pi$
$281$	20.6802	−	20.6802i	1.23368	−	1.23368i	0.962541	−	0.271135i	$-0.0873989\pi$
$282$		0			0
$283$		−	11.8013i		−	0.701513i	−0.936467	−	0.350757i	$-0.885924\pi$
$283$		−	11.8013i		−	0.701513i	0.936467	−	0.350757i	$-0.114076\pi$
$284$	−7.17745	+	7.17745i	−0.425903	+	0.425903i
$285$		0			0
$286$	−6.73566	−	29.3401i	−0.398288	−	1.73492i
$287$			5.73682i			0.338634i
$288$		0			0
$289$	1.00000			0.0588235
$290$	63.1601			3.70889
$291$		0			0
$292$	−31.7549	+	31.7549i	−1.85831	+	1.85831i
$293$	8.53884	−	8.53884i	0.498845	−	0.498845i	−0.412234	−	0.911078i	$-0.635251\pi$
$293$	8.53884	−	8.53884i	0.498845	−	0.498845i	0.911078	+	0.412234i	$0.135251\pi$
$294$		0			0
$295$	−3.53880			−0.206037
$296$	−8.25197			−0.479636
$297$		0			0
$298$			6.73566i			0.390186i
$299$	1.94798	+	8.48528i	0.112655	+	0.490716i
$300$		0			0
$301$	8.58522	−	8.58522i	0.494844	−	0.494844i
$302$			57.7766i			3.32467i
$303$		0			0
$304$	1.16498	−	1.16498i	0.0668161	−	0.0668161i
$305$	3.78884	+	3.78884i	0.216948	+	0.216948i
$306$		0			0
$307$	−6.54241	−	6.54241i	−0.373395	−	0.373395i	0.495317	−	0.868712i	$-0.335052\pi$
$307$	−6.54241	−	6.54241i	−0.373395	−	0.373395i	−0.868712	+	0.495317i	$0.835052\pi$
$308$			30.7260i			1.75078i
$309$		0			0
$310$	40.3684	+	40.3684i	2.29277	+	2.29277i
$311$	−13.6227			−0.772472			−0.386236	−	0.922400i	$-0.626225\pi$
$311$	−13.6227			−0.772472			−0.386236	+	0.922400i	$0.626225\pi$
$312$		0			0
$313$	0.311812			0.0176246			0.00881232	−	0.999961i	$-0.497195\pi$
$313$	0.311812			0.0176246			0.00881232	+	0.999961i	$0.497195\pi$
$314$	−16.8378	−	16.8378i	−0.950211	−	0.950211i
$315$		0			0
$316$		−	72.1515i		−	4.05884i
$317$	−15.6627	−	15.6627i	−0.879706	−	0.879706i	0.113798	−	0.993504i	$-0.463698\pi$
$317$	−15.6627	−	15.6627i	−0.879706	−	0.879706i	−0.993504	+	0.113798i	$0.963698\pi$
$318$		0			0
$319$	17.7639	+	17.7639i	0.994590	+	0.994590i
$320$	7.73177	−	7.73177i	0.432219	−	0.432219i
$321$		0			0
$322$		−	12.9992i		−	0.724417i
$323$	0.819901	−	0.819901i	0.0456205	−	0.0456205i
$324$		0			0
$325$	21.1842	−	4.86330i	1.17509	−	0.269767i
$326$		−	58.6842i		−	3.25022i
$327$		0			0
$328$	−15.6327			−0.863171
$329$	7.11105			0.392045
$330$		0			0
$331$	5.25887	−	5.25887i	0.289054	−	0.289054i	−0.547652	−	0.836706i	$-0.684478\pi$
$331$	5.25887	−	5.25887i	0.289054	−	0.289054i	0.836706	+	0.547652i	$0.184478\pi$
$332$	−33.4081	+	33.4081i	−1.83351	+	1.83351i
$333$		0			0
$334$	29.9336			1.63789
$335$	−35.2897			−1.92808
$336$		0			0
$337$			18.3684i			1.00059i	0.865856	+	0.500294i	$0.166775\pi$
$337$			18.3684i			1.00059i	−0.865856	+	0.500294i	$0.833225\pi$
$338$	−30.8768	−	10.7168i	−1.67948	−	0.582919i
$339$		0			0
$340$	−43.0475	+	43.0475i	−2.33458	+	2.33458i
$341$			22.7074i			1.22967i
$342$		0			0
$343$	14.2553	−	14.2553i	0.769712	−	0.769712i
$344$	23.3945	+	23.3945i	1.26135	+	1.26135i
$345$		0			0
$346$	−41.1751	−	41.1751i	−2.21359	−	2.21359i
$347$			6.20345i			0.333019i	0.986040	+	0.166509i	$0.0532496\pi$
$347$			6.20345i			0.333019i	−0.986040	+	0.166509i	$0.946750\pi$
$348$		0			0
$349$	−24.0283	−	24.0283i	−1.28620	−	1.28620i	−0.937074	−	0.349130i	$-0.886477\pi$
$349$	−24.0283	−	24.0283i	−1.28620	−	1.28620i	−0.349130	−	0.937074i	$-0.613523\pi$
$350$	−32.4535			−1.73471
$351$		0			0
$352$	−11.5761			−0.617011
$353$	4.30903	+	4.30903i	0.229347	+	0.229347i	0.812420	−	0.583073i	$-0.198150\pi$
$353$	4.30903	+	4.30903i	0.229347	+	0.229347i	−0.583073	+	0.812420i	$0.698150\pi$
$354$		0			0
$355$		−	7.80128i		−	0.414049i
$356$	20.5797	+	20.5797i	1.09072	+	1.09072i
$357$		0			0
$358$	15.0848	+	15.0848i	0.797258	+	0.797258i
$359$	−10.9663	+	10.9663i	−0.578779	+	0.578779i	−0.934567	−	0.355788i	$-0.884212\pi$
$359$	−10.9663	+	10.9663i	−0.578779	+	0.578779i	0.355788	+	0.934567i	$0.384212\pi$
$360$		0			0
$361$			18.9253i			0.996069i
$362$	−35.4225	+	35.4225i	−1.86176	+	1.86176i
$363$		0			0
$364$	28.2690	+	17.7129i	1.48170	+	0.928406i
$365$		−	34.5149i		−	1.80659i
$366$		0			0
$367$	8.69832			0.454048			0.227024	−	0.973889i	$-0.427100\pi$
$367$	8.69832			0.454048			0.227024	+	0.973889i	$0.427100\pi$
$368$	14.5559			0.758781
$369$		0			0
$370$	8.34916	−	8.34916i	0.434052	−	0.434052i
$371$	−2.96894	+	2.96894i	−0.154140	+	0.154140i
$372$		0			0
$373$	0.311812			0.0161450			0.00807250	−	0.999967i	$-0.497430\pi$
$373$	0.311812			0.0161450			0.00807250	+	0.999967i	$0.497430\pi$
$374$	−35.4225			−1.83165
$375$		0			0
$376$			19.3774i			0.999315i
$377$	26.5839	−	6.10293i	1.36914	−	0.314317i
$378$		0			0
$379$	3.42024	−	3.42024i	0.175686	−	0.175686i	−0.613786	−	0.789472i	$-0.710354\pi$
$379$	3.42024	−	3.42024i	0.175686	−	0.175686i	0.789472	+	0.613786i	$0.210354\pi$
$380$			3.92163i			0.201175i
$381$		0			0
$382$	4.05655	−	4.05655i	0.207551	−	0.207551i
$383$	−16.4375	−	16.4375i	−0.839919	−	0.839919i	0.148929	−	0.988848i	$-0.452417\pi$
$383$	−16.4375	−	16.4375i	−0.839919	−	0.839919i	−0.988848	+	0.148929i	$0.952417\pi$
$384$		0			0
$385$	−16.6983	−	16.6983i	−0.851025	−	0.851025i
$386$		−	50.7855i		−	2.58491i
$387$		0			0
$388$	−3.05655	−	3.05655i	−0.155173	−	0.155173i
$389$	25.1092			1.27309			0.636543	−	0.771241i	$-0.280364\pi$
$389$	25.1092			1.27309			0.636543	+	0.771241i	$0.280364\pi$
$390$		0			0
$391$	10.2443			0.518078
$392$	9.96334	+	9.96334i	0.503224	+	0.503224i
$393$		0			0
$394$			13.2353i			0.666783i
$395$	39.2113	+	39.2113i	1.97294	+	1.97294i
$396$		0			0
$397$	0.301683	+	0.301683i	0.0151410	+	0.0151410i	0.714637	−	0.699496i	$-0.246592\pi$
$397$	0.301683	+	0.301683i	0.0151410	+	0.0151410i	−0.699496	+	0.714637i	$0.746592\pi$
$398$	−28.9985	+	28.9985i	−1.45357	+	1.45357i
$399$		0			0
$400$		−	36.3401i		−	1.81700i
$401$	7.17745	−	7.17745i	0.358425	−	0.358425i	−0.504807	−	0.863232i	$-0.668436\pi$
$401$	7.17745	−	7.17745i	0.358425	−	0.358425i	0.863232	+	0.504807i	$0.168436\pi$
$402$		0			0
$403$	20.8916	+	13.0903i	1.04068	+	0.652074i
$404$		−	7.89870i		−	0.392975i
$405$		0			0
$406$	−40.7258			−2.02119
$407$	4.69644			0.232794
$408$		0			0
$409$	9.93438	−	9.93438i	0.491223	−	0.491223i	−0.417468	−	0.908692i	$-0.637082\pi$
$409$	9.93438	−	9.93438i	0.491223	−	0.491223i	0.908692	+	0.417468i	$0.137082\pi$
$410$	15.8168	−	15.8168i	0.781137	−	0.781137i
$411$		0			0
$412$	61.5953			3.03459
$413$	2.28183			0.112281
$414$		0			0
$415$		−	36.3118i		−	1.78248i
$416$	−6.67338	+	10.6505i	−0.327190	+	0.522181i
$417$		0			0
$418$	−1.61350	+	1.61350i	−0.0789186	+	0.0789186i
$419$		−	5.16307i		−	0.252232i	−0.992015	−	0.126116i	$-0.959749\pi$
$419$		−	5.16307i		−	0.252232i	0.992015	−	0.126116i	$-0.0402512\pi$
$420$		0			0
$421$	−25.4057	+	25.4057i	−1.23820	+	1.23820i	−0.277462	+	0.960737i	$0.589493\pi$
$421$	−25.4057	+	25.4057i	−1.23820	+	1.23820i	−0.960737	+	0.277462i	$0.910507\pi$
$422$	5.73682	+	5.73682i	0.279264	+	0.279264i
$423$		0			0
$424$	−8.09029	−	8.09029i	−0.392899	−	0.392899i
$425$		−	25.5758i		−	1.24061i
$426$		0			0
$427$	−2.44305	−	2.44305i	−0.118228	−	0.118228i
$428$	77.8232			3.76173
$429$		0			0
$430$	−47.3401			−2.28294
$431$	16.5703	+	16.5703i	0.798165	+	0.798165i	0.982806	−	0.184641i	$-0.0591124\pi$
$431$	16.5703	+	16.5703i	0.798165	+	0.798165i	−0.184641	+	0.982806i	$0.559112\pi$
$432$		0			0
$433$			3.61350i			0.173653i	0.996223	+	0.0868267i	$0.0276727\pi$
$433$			3.61350i			0.173653i	−0.996223	+	0.0868267i	$0.972327\pi$
$434$	−26.0296	−	26.0296i	−1.24946	−	1.24946i
$435$		0			0
$436$	31.7549	+	31.7549i	1.52078	+	1.52078i
$437$	0.466630	−	0.466630i	0.0223219	−	0.0223219i
$438$		0			0
$439$		−	5.67004i		−	0.270616i	−0.990804	−	0.135308i	$-0.956798\pi$
$439$		−	5.67004i		−	0.270616i	0.990804	−	0.135308i	$-0.0432025\pi$
$440$	45.5025	−	45.5025i	2.16925	−	2.16925i
$441$		0			0
$442$	−20.4202	+	32.5899i	−0.971292	+	1.55014i
$443$			10.1677i			0.483082i	0.970391	+	0.241541i	$0.0776528\pi$
$443$			10.1677i			0.483082i	−0.970391	+	0.241541i	$0.922347\pi$
$444$		0			0
$445$	−22.3684			−1.06036
$446$	−9.87894			−0.467781
$447$		0			0
$448$	−4.98546	+	4.98546i	−0.235541	+	0.235541i
$449$	−16.5703	+	16.5703i	−0.782002	+	0.782002i	−0.980168	−	0.198166i	$-0.936501\pi$
$449$	−16.5703	+	16.5703i	−0.782002	+	0.782002i	0.198166	+	0.980168i	$0.436501\pi$
$450$		0			0
$451$	8.89703			0.418945
$452$	−32.6869			−1.53746
$453$		0			0
$454$			50.3310i			2.36215i
$455$	−24.9892	+	5.73682i	−1.17151	+	0.268946i
$456$		0			0
$457$	4.13310	−	4.13310i	0.193338	−	0.193338i	−0.603799	−	0.797137i	$-0.706347\pi$
$457$	4.13310	−	4.13310i	0.193338	−	0.193338i	0.797137	+	0.603799i	$0.206347\pi$
$458$		−	15.3630i		−	0.717867i
$459$		0			0
$460$	−24.4996	+	24.4996i	−1.14230	+	1.14230i
$461$	1.30784	+	1.30784i	0.0609120	+	0.0609120i	0.736907	−	0.675995i	$-0.236286\pi$
$461$	1.30784	+	1.30784i	0.0609120	+	0.0609120i	−0.675995	+	0.736907i	$0.736286\pi$
$462$		0			0
$463$	13.7129	+	13.7129i	0.637290	+	0.637290i	0.949886	−	0.312596i	$-0.101199\pi$
$463$	13.7129	+	13.7129i	0.637290	+	0.637290i	−0.312596	+	0.949886i	$0.601199\pi$
$464$		−	45.6030i		−	2.11707i
$465$		0			0
$466$	−5.33542	−	5.33542i	−0.247159	−	0.247159i
$467$	2.88124			0.133328			0.0666640	−	0.997775i	$-0.478764\pi$
$467$	2.88124			0.133328			0.0666640	+	0.997775i	$0.478764\pi$
$468$		0			0
$469$	22.7549			1.05072
$470$	−19.6057	−	19.6057i	−0.904342	−	0.904342i
$471$		0			0
$472$			6.21792i			0.286203i
$473$	−13.3145	−	13.3145i	−0.612202	−	0.612202i
$474$		0			0
$475$	−1.16498	−	1.16498i	−0.0534529	−	0.0534529i
$476$	27.7571	−	27.7571i	1.27224	−	1.27224i
$477$		0			0
$478$		−	61.3593i		−	2.80651i
$479$	1.52832	−	1.52832i	0.0698307	−	0.0698307i	−0.671329	−	0.741160i	$-0.734276\pi$
$479$	1.52832	−	1.52832i	0.0698307	−	0.0698307i	0.741160	+	0.671329i	$0.234276\pi$
$480$		0			0
$481$	2.70739	−	4.32088i	0.123446	−	0.197015i
$482$			30.9594i			1.41016i
$483$		0			0
$484$	0.122168			0.00555309
$485$	3.32221			0.150854
$486$		0			0
$487$	−14.5798	+	14.5798i	−0.660672	+	0.660672i	−0.955538	−	0.294867i	$-0.904725\pi$
$487$	−14.5798	+	14.5798i	−0.660672	+	0.660672i	0.294867	+	0.955538i	$0.404725\pi$
$488$	6.65725	−	6.65725i	0.301360	−	0.301360i
$489$		0			0
$490$	−20.1614			−0.910798
$491$	35.8891			1.61965			0.809826	−	0.586670i	$-0.199561\pi$
$491$	35.8891			1.61965			0.809826	+	0.586670i	$0.199561\pi$
$492$		0			0
$493$		−	32.0950i		−	1.44548i
$494$	0.554328	+	2.41461i	0.0249404	+	0.108639i
$495$		0			0
$496$	29.1468	−	29.1468i	1.30873	−	1.30873i
$497$			5.03028i			0.225639i
$498$		0			0
$499$	0.287147	−	0.287147i	0.0128544	−	0.0128544i	−0.700650	−	0.713505i	$-0.747107\pi$
$499$	0.287147	−	0.287147i	0.0128544	−	0.0128544i	0.713505	+	0.700650i	$0.247107\pi$
$500$	10.4333	+	10.4333i	0.466590	+	0.466590i
$501$		0			0
$502$	−25.0475	−	25.0475i	−1.11792	−	1.11792i
$503$			35.6235i			1.58837i	0.607673	+	0.794187i	$0.292103\pi$
$503$			35.6235i			1.58837i	−0.607673	+	0.794187i	$0.707897\pi$
$504$		0			0
$505$	4.29261	+	4.29261i	0.191019	+	0.191019i
$506$	−20.1600			−0.896221
$507$		0			0
$508$	15.6135			0.692737
$509$	20.2264	+	20.2264i	0.896519	+	0.896519i	0.995126	−	0.0986078i	$-0.0314389\pi$
$509$	20.2264	+	20.2264i	0.896519	+	0.896519i	−0.0986078	+	0.995126i	$0.531439\pi$
$510$		0			0
$511$			22.2553i			0.984515i
$512$	−34.8862	−	34.8862i	−1.54176	−	1.54176i
$513$		0			0
$514$	−11.8350	−	11.8350i	−0.522020	−	0.522020i
$515$	−33.4745	+	33.4745i	−1.47506	+	1.47506i
$516$		0			0
$517$		−	11.0283i		−	0.485023i
$518$	−5.38355	+	5.38355i	−0.236540	+	0.236540i
$519$		0			0
$520$	−15.6327	−	68.0950i	−0.685539	−	2.98616i
$521$			7.56485i			0.331422i	0.986174	+	0.165711i	$0.0529919\pi$
$521$			7.56485i			0.331422i	−0.986174	+	0.165711i	$0.947008\pi$
$522$		0			0
$523$	−23.2270			−1.01565			−0.507823	−	0.861462i	$-0.669549\pi$
$523$	−23.2270			−1.01565			−0.507823	+	0.861462i	$0.669549\pi$
$524$	−51.0188			−2.22877
$525$		0			0
$526$	44.1888	−	44.1888i	1.92673	−	1.92673i
$527$	20.5133	−	20.5133i	0.893572	−	0.893572i
$528$		0			0
$529$	−17.1696			−0.746506
$530$	16.3711			0.711117
$531$		0			0
$532$		−	2.52867i		−	0.109632i
$533$	5.12893	−	8.18557i	0.222159	−	0.354557i
$534$		0			0
$535$	−42.2937	+	42.2937i	−1.82851	+	1.82851i
$536$			62.0064i			2.67827i
$537$		0			0
$538$	39.5616	−	39.5616i	1.70562	−	1.70562i
$539$	−5.67043	−	5.67043i	−0.244243	−	0.244243i
$540$		0			0
$541$	−10.6508	−	10.6508i	−0.457915	−	0.457915i	0.440055	−	0.897971i	$-0.354959\pi$
$541$	−10.6508	−	10.6508i	−0.457915	−	0.457915i	−0.897971	+	0.440055i	$0.854959\pi$
$542$			30.3728i			1.30462i
$543$		0			0
$544$	10.4576	+	10.4576i	0.448365	+	0.448365i
$545$	−34.5149			−1.47846
$546$		0			0
$547$	27.9253			1.19400			0.597000	−	0.802241i	$-0.296359\pi$
$547$	27.9253			1.19400			0.597000	+	0.802241i	$0.296359\pi$
$548$	−8.18557	−	8.18557i	−0.349670	−	0.349670i
$549$		0			0
$550$			50.3310i			2.14612i
$551$	−1.46193	−	1.46193i	−0.0622801	−	0.0622801i
$552$		0			0
$553$	−25.2835	−	25.2835i	−1.07517	−	1.07517i
$554$	−16.7372	+	16.7372i	−0.711098	+	0.711098i
$555$		0			0
$556$			68.8114i			2.91825i
$557$	10.2469	−	10.2469i	0.434176	−	0.434176i	−0.455870	−	0.890046i	$-0.650672\pi$
$557$	10.2469	−	10.2469i	0.434176	−	0.434176i	0.890046	+	0.455870i	$0.150672\pi$
$558$		0			0
$559$	−19.9253	+	4.57429i	−0.842751	+	0.193472i
$560$			42.8674i			1.81148i
$561$		0			0
$562$	73.5289			3.10163
$563$	26.1624			1.10261			0.551307	−	0.834303i	$-0.314129\pi$
$563$	26.1624			1.10261			0.551307	+	0.834303i	$0.314129\pi$
$564$		0			0
$565$	17.7639	−	17.7639i	0.747334	−	0.747334i
$566$	20.9799	−	20.9799i	0.881850	−	0.881850i
$567$		0			0
$568$	−13.7074			−0.575149
$569$	34.8359			1.46040			0.730198	−	0.683235i	$-0.239428\pi$
$569$	34.8359			1.46040			0.730198	+	0.683235i	$0.239428\pi$
$570$		0			0
$571$		−	17.2726i		−	0.722836i	−0.932404	−	0.361418i	$-0.882293\pi$
$571$		−	17.2726i		−	0.722836i	0.932404	−	0.361418i	$-0.117707\pi$
$572$	27.4702	−	43.8414i	1.14859	−	1.83310i
$573$		0			0
$574$	−10.1987	+	10.1987i	−0.425686	+	0.425686i
$575$		−	14.5559i		−	0.607025i
$576$		0			0
$577$	−3.84049	+	3.84049i	−0.159882	+	0.159882i	−0.782514	−	0.622633i	$-0.786063\pi$
$577$	−3.84049	+	3.84049i	−0.159882	+	0.159882i	0.622633	+	0.782514i	$0.286063\pi$
$578$	1.77776	+	1.77776i	0.0739452	+	0.0739452i
$579$		0			0
$580$	76.7559	+	76.7559i	3.18712	+	3.18712i
$581$			23.4139i			0.971373i
$582$		0			0
$583$	4.60442	+	4.60442i	0.190696	+	0.190696i
$584$	−60.6450			−2.50951
$585$		0			0
$586$	30.3601			1.25416
$587$	−28.9321	−	28.9321i	−1.19416	−	1.19416i	−0.975889	−	0.218269i	$-0.929959\pi$
$587$	−28.9321	−	28.9321i	−1.19416	−	1.19416i	−0.218269	−	0.975889i	$-0.570041\pi$
$588$		0			0
$589$		−	1.86876i		−	0.0770009i
$590$	−6.29115	−	6.29115i	−0.259003	−	0.259003i
$591$		0			0
$592$	−6.02827	−	6.02827i	−0.247761	−	0.247761i
$593$	−22.1744	+	22.1744i	−0.910592	+	0.910592i	−0.996319	−	0.0857267i	$-0.972679\pi$
$593$	−22.1744	+	22.1744i	−0.910592	+	0.910592i	0.0857267	+	0.996319i	$0.472679\pi$
$594$		0			0
$595$			30.1696i			1.23683i
$596$	−8.18557	+	8.18557i	−0.335294	+	0.335294i
$597$		0			0
$598$	−11.6218	+	18.5479i	−0.475249	+	0.758479i
$599$		−	20.7337i		−	0.847158i	−0.905859	−	0.423579i	$-0.860774\pi$
$599$		−	20.7337i		−	0.847158i	0.905859	−	0.423579i	$-0.139226\pi$
$600$		0			0
$601$	6.14217			0.250544			0.125272	−	0.992122i	$-0.460020\pi$
$601$	6.14217			0.250544			0.125272	+	0.992122i	$0.460020\pi$
$602$	30.5250			1.24411
$603$		0			0
$604$	−70.2135	+	70.2135i	−2.85695	+	2.85695i
$605$	−0.0663932	+	0.0663932i	−0.00269927	+	0.00269927i
$606$		0			0
$607$	−19.1523			−0.777368			−0.388684	−	0.921371i	$-0.627070\pi$
$607$	−19.1523			−0.777368			−0.388684	+	0.921371i	$0.627070\pi$
$608$	0.952687			0.0386366
$609$		0			0
$610$			13.4713i			0.545438i
$611$	−10.1464	−	6.35755i	−0.410479	−	0.257199i
$612$		0			0
$613$	−18.9253	+	18.9253i	−0.764386	+	0.764386i	−0.977112	−	0.212726i	$-0.931766\pi$
$613$	−18.9253	+	18.9253i	−0.764386	+	0.764386i	0.212726	+	0.977112i	$0.431766\pi$
$614$		−	23.2617i		−	0.938766i
$615$		0			0
$616$	−29.3401	+	29.3401i	−1.18215	+	1.18215i
$617$	−0.986822	−	0.986822i	−0.0397280	−	0.0397280i	0.686964	−	0.726692i	$-0.258943\pi$
$617$	−0.986822	−	0.986822i	−0.0397280	−	0.0397280i	−0.726692	+	0.686964i	$0.758943\pi$
$618$		0			0
$619$	19.5141	+	19.5141i	0.784339	+	0.784339i	0.980560	−	0.196220i	$-0.0628668\pi$
$619$	19.5141	+	19.5141i	0.784339	+	0.784339i	−0.196220	+	0.980560i	$0.562867\pi$
$620$			98.1160i			3.94043i
$621$		0			0
$622$	−24.2179	−	24.2179i	−0.971050	−	0.971050i
$623$	14.4232			0.577852
$624$		0			0
$625$	18.8013			0.752051
$626$	0.554328	+	0.554328i	0.0221554	+	0.0221554i
$627$		0			0
$628$		−	40.9245i		−	1.63307i
$629$	−4.24264	−	4.24264i	−0.169165	−	0.169165i
$630$		0			0
$631$	4.64723	+	4.64723i	0.185003	+	0.185003i	0.793532	−	0.608529i	$-0.208240\pi$
$631$	4.64723	+	4.64723i	0.185003	+	0.185003i	−0.608529	+	0.793532i	$0.708240\pi$
$632$	68.8970	−	68.8970i	2.74057	−	2.74057i
$633$		0			0
$634$		−	55.6892i		−	2.21170i
$635$	−8.48528	+	8.48528i	−0.336728	+	0.336728i
$636$		0			0
$637$	−8.48586	+	1.94812i	−0.336222	+	0.0771872i
$638$			63.1601i			2.50053i
$639$		0			0
$640$	50.6428			2.00183
$641$	−32.8879			−1.29899			−0.649497	−	0.760364i	$-0.725021\pi$
$641$	−32.8879			−1.29899			−0.649497	+	0.760364i	$0.725021\pi$
$642$		0			0
$643$	27.4394	−	27.4394i	1.08211	−	1.08211i	0.0857931	−	0.996313i	$-0.472658\pi$
$643$	27.4394	−	27.4394i	1.08211	−	1.08211i	0.996313	−	0.0857931i	$-0.0273424\pi$
$644$	15.7974	−	15.7974i	0.622504	−	0.622504i
$645$		0			0
$646$	2.91518			0.114696
$647$	24.0560			0.945737			0.472869	−	0.881133i	$-0.343219\pi$
$647$	24.0560			0.945737			0.472869	+	0.881133i	$0.343219\pi$
$648$		0			0
$649$		−	3.53880i		−	0.138910i
$650$	46.3062	+	29.0147i	1.81628	+	1.13805i
$651$		0			0
$652$	71.3165	−	71.3165i	2.79297	−	2.79297i
$653$		−	26.6162i		−	1.04157i	−0.853687	−	0.520786i	$-0.825639\pi$
$653$		−	26.6162i		−	1.04157i	0.853687	−	0.520786i	$-0.174361\pi$
$654$		0			0
$655$	27.7266	−	27.7266i	1.08337	−	1.08337i
$656$	−11.4201	−	11.4201i	−0.445879	−	0.445879i
$657$		0			0
$658$	12.6418	+	12.6418i	0.492827	+	0.492827i
$659$		−	10.7671i		−	0.419427i	−0.977763	−	0.209713i	$-0.932747\pi$
$659$		−	10.7671i		−	0.419427i	0.977763	−	0.209713i	$-0.0672531\pi$
$660$		0			0
$661$	31.4996	+	31.4996i	1.22519	+	1.22519i	0.965762	+	0.259431i	$0.0835349\pi$
$661$	31.4996	+	31.4996i	1.22519	+	1.22519i	0.259431	+	0.965762i	$0.416465\pi$
$662$	18.6981			0.726721
$663$		0			0
$664$	−63.8023			−2.47601
$665$	1.37423	+	1.37423i	0.0532903	+	0.0532903i
$666$		0			0
$667$		−	18.2662i		−	0.707270i
$668$	36.3770	+	36.3770i	1.40747	+	1.40747i
$669$		0			0
$670$	−62.7367	−	62.7367i	−2.42373	−	2.42373i
$671$	−3.78884	+	3.78884i	−0.146267	+	0.146267i
$672$		0			0
$673$			39.5844i			1.52587i	0.646477	+	0.762934i	$0.276242\pi$
$673$			39.5844i			1.52587i	−0.646477	+	0.762934i	$0.723758\pi$
$674$	−32.6546	+	32.6546i	−1.25781	+	1.25781i
$675$		0			0
$676$	−24.4996	−	50.5471i	−0.942292	−	1.94412i
$677$		−	2.40178i		−	0.0923080i	−0.998934	−	0.0461540i	$-0.985304\pi$
$677$		−	2.40178i		−	0.0923080i	0.998934	−	0.0461540i	$-0.0146965\pi$
$678$		0			0
$679$	−2.14217			−0.0822089
$680$	−82.2115			−3.15267
$681$		0			0
$682$	−40.3684	+	40.3684i	−1.54578	+	1.54578i
$683$	−5.20380	+	5.20380i	−0.199118	+	0.199118i	−0.799622	−	0.600504i	$-0.794967\pi$
$683$	−5.20380	+	5.20380i	−0.199118	+	0.199118i	0.600504	+	0.799622i	$0.294967\pi$
$684$		0			0
$685$	8.89703			0.339938
$686$	50.6850			1.93516
$687$		0			0
$688$			34.1806i			1.30312i
$689$	6.89057	−	1.58188i	0.262510	−	0.0602649i
$690$		0			0
$691$	13.8825	−	13.8825i	0.528115	−	0.528115i	−0.391895	−	0.920010i	$-0.628180\pi$
$691$	13.8825	−	13.8825i	0.528115	−	0.528115i	0.920010	+	0.391895i	$0.128180\pi$
$692$		−	100.077i		−	3.80435i
$693$		0			0
$694$	−11.0283	+	11.0283i	−0.418628	+	0.418628i
$695$	−37.3961	−	37.3961i	−1.41852	−	1.41852i
$696$		0			0
$697$	−8.03735	−	8.03735i	−0.304436	−	0.304436i
$698$		−	85.4332i		−	3.23369i
$699$		0			0
$700$	−39.4394	−	39.4394i	−1.49067	−	1.49067i
$701$	−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
$703$	−0.386505			−0.0145773
$704$	7.73177	+	7.73177i	0.291402	+	0.291402i
$705$		0			0
$706$			15.3209i			0.576609i
$707$	−2.76788	−	2.76788i	−0.104097	−	0.104097i
$708$		0			0
$709$	−4.45213	−	4.45213i	−0.167203	−	0.167203i	0.618546	−	0.785749i	$-0.287722\pi$
$709$	−4.45213	−	4.45213i	−0.167203	−	0.167203i	−0.785749	+	0.618546i	$0.787722\pi$
$710$	13.8688	−	13.8688i	0.520488	−	0.520488i
$711$		0			0
$712$			39.3027i			1.47293i
$713$	11.6747	−	11.6747i	0.437221	−	0.437221i
$714$		0			0
$715$	8.89703	+	38.7549i	0.332730	+	1.44935i
$716$			36.6639i			1.37020i
$717$		0			0
$718$	−38.9909			−1.45513
$719$	−20.0917			−0.749295			−0.374647	−	0.927167i	$-0.622236\pi$
$719$	−20.0917			−0.749295			−0.374647	+	0.927167i	$0.622236\pi$
$720$		0			0
$721$	21.5844	−	21.5844i	0.803846	−	0.803846i
$722$	−33.6447	+	33.6447i	−1.25213	+	1.25213i
$723$		0			0
$724$	−86.0950			−3.19969
$725$	−45.6030			−1.69365
$726$		0			0
$727$		−	16.9717i		−	0.629446i	−0.949183	−	0.314723i	$-0.898088\pi$
$727$		−	16.9717i		−	0.629446i	0.949183	−	0.314723i	$-0.101912\pi$
$728$	10.0800	+	43.9078i	0.373589	+	1.62733i
$729$		0			0
$730$	61.3593	−	61.3593i	2.27101	−	2.27101i
$731$			24.0560i			0.889742i
$732$		0			0
$733$	−27.4623	+	27.4623i	−1.01434	+	1.01434i	−0.0144458	+	0.999896i	$0.504598\pi$
$733$	−27.4623	+	27.4623i	−1.01434	+	1.01434i	−0.999896	+	0.0144458i	$0.995402\pi$
$734$	15.4635	+	15.4635i	0.570770	+	0.570770i
$735$		0			0
$736$	5.95173	+	5.95173i	0.219384	+	0.219384i
$737$		−	35.2897i		−	1.29991i
$738$		0			0
$739$	−27.1660	−	27.1660i	−0.999319	−	0.999319i	0.000681035	−	1.00000i	$-0.499783\pi$
$739$	−27.1660	−	27.1660i	−0.999319	−	0.999319i	−1.00000		0.000681035i	$0.999783\pi$
$740$	20.2928			0.745977
$741$		0			0
$742$	−10.5561			−0.387528
$743$	9.67976	+	9.67976i	0.355116	+	0.355116i	0.862009	−	0.506893i	$-0.169206\pi$
$743$	9.67976	+	9.67976i	0.355116	+	0.355116i	−0.506893	+	0.862009i	$0.669206\pi$
$744$		0			0
$745$		−	8.89703i		−	0.325962i
$746$	0.554328	+	0.554328i	0.0202954	+	0.0202954i
$747$		0			0
$748$	−43.0475	−	43.0475i	−1.57397	−	1.57397i
$749$	27.2710	−	27.2710i	0.996462	−	0.996462i
$750$		0			0
$751$			15.9144i			0.580724i	0.956917	+	0.290362i	$0.0937757\pi$
$751$			15.9144i			0.580724i	−0.956917	+	0.290362i	$0.906224\pi$
$752$	−14.1557	+	14.1557i	−0.516206	+	0.516206i
$753$		0			0
$754$	58.1095	+	36.4104i	2.11622	+	1.32599i
$755$		−	76.3162i		−	2.77743i
$756$		0			0
$757$	46.5561			1.69211			0.846056	−	0.533094i	$-0.178971\pi$
$757$	46.5561			1.69211			0.846056	+	0.533094i	$0.178971\pi$
$758$	12.1608			0.441699
$759$		0			0
$760$	−3.74474	+	3.74474i	−0.135836	+	0.135836i
$761$	−24.0024	+	24.0024i	−0.870086	+	0.870086i	−0.992481	−	0.122395i	$-0.960942\pi$
$761$	−24.0024	+	24.0024i	−0.870086	+	0.870086i	0.122395	+	0.992481i	$0.460942\pi$
$762$		0			0
$763$	22.2553			0.805695
$764$	9.85951			0.356705
$765$		0			0
$766$		−	58.4441i		−	2.11167i
$767$	−3.25582	−	2.04004i	−0.117561	−	0.0736615i
$768$		0			0
$769$	−7.25526	+	7.25526i	−0.261632	+	0.261632i	−0.825717	−	0.564085i	$-0.809229\pi$
$769$	−7.25526	+	7.25526i	−0.261632	+	0.261632i	0.564085	+	0.825717i	$0.309229\pi$
$770$		−	59.3713i		−	2.13959i
$771$		0			0
$772$	61.7175	−	61.7175i	2.22126	−	2.22126i
$773$	−1.76163	−	1.76163i	−0.0633616	−	0.0633616i	0.674716	−	0.738078i	$-0.264266\pi$
$773$	−1.76163	−	1.76163i	−0.0633616	−	0.0633616i	−0.738078	+	0.674716i	$0.764266\pi$
$774$		0			0
$775$	−29.1468	−	29.1468i	−1.04699	−	1.04699i
$776$		−	5.83735i		−	0.209549i
$777$		0			0
$778$	44.6382	+	44.6382i	1.60036	+	1.60036i
$779$	−0.732203			−0.0262339
$780$		0			0
$781$	7.80128			0.279152
$782$	18.2120	+	18.2120i	0.651260	+	0.651260i
$783$		0			0
$784$			14.5569i			0.519891i
$785$	22.2408	+	22.2408i	0.793807	+	0.793807i
$786$		0			0
$787$	−24.2125	−	24.2125i	−0.863081	−	0.863081i	0.128614	−	0.991695i	$-0.458947\pi$
$787$	−24.2125	−	24.2125i	−0.863081	−	0.863081i	−0.991695	+	0.128614i	$0.958947\pi$
$788$	−16.0843	+	16.0843i	−0.572978	+	0.572978i
$789$		0			0
$790$			139.417i			4.96023i
$791$	−11.4542	+	11.4542i	−0.407265	+	0.407265i
$792$		0			0
$793$	1.30168	+	5.67004i	0.0462241	+	0.201349i
$794$			1.07264i			0.0380667i
$795$		0			0
$796$	−70.4815			−2.49815
$797$	28.7652			1.01892			0.509458	−	0.860495i	$-0.329846\pi$
$797$	28.7652			1.01892			0.509458	+	0.860495i	$0.329846\pi$
$798$		0			0
$799$	−9.96265	+	9.96265i	−0.352453	+	0.352453i
$800$	14.8590	−	14.8590i	0.525343	−	0.525343i
$801$		0			0
$802$	25.5196			0.901128
$803$	34.5149			1.21800
$804$		0			0
$805$			17.1704i			0.605179i
$806$	13.8688	+	60.4117i	0.488509	+	2.12791i
$807$		0			0
$808$	7.54241	−	7.54241i	0.265341	−	0.265341i
$809$			11.6875i			0.410912i	0.978666	+	0.205456i	$0.0658677\pi$
$809$			11.6875i			0.410912i	−0.978666	+	0.205456i	$0.934132\pi$
$810$		0			0
$811$	2.72193	−	2.72193i	0.0955797	−	0.0955797i	−0.657700	−	0.753280i	$-0.728471\pi$
$811$	2.72193	−	2.72193i	0.0955797	−	0.0955797i	0.753280	+	0.657700i	$0.228471\pi$
$812$	−49.4924	−	49.4924i	−1.73684	−	1.73684i
$813$		0			0
$814$	8.34916	+	8.34916i	0.292638	+	0.292638i
$815$			77.5150i			2.71523i
$816$		0			0
$817$	1.09575	+	1.09575i	0.0383354	+	0.0383354i
$818$	35.3220			1.23500
$819$		0			0
$820$	38.4431			1.34249
$821$	12.7943	+	12.7943i	0.446525	+	0.446525i	0.894197	−	0.447673i	$-0.147747\pi$
$821$	12.7943	+	12.7943i	0.446525	+	0.446525i	−0.447673	+	0.894197i	$0.647747\pi$
$822$		0			0
$823$			35.8397i			1.24929i	0.780908	+	0.624646i	$0.214757\pi$
$823$			35.8397i			1.24929i	−0.780908	+	0.624646i	$0.785243\pi$
$824$	58.8170	+	58.8170i	2.04899	+	2.04899i
$825$		0			0
$826$	4.05655	+	4.05655i	0.141145	+	0.141145i
$827$	10.3669	−	10.3669i	0.360491	−	0.360491i	−0.503502	−	0.863994i	$-0.667955\pi$
$827$	10.3669	−	10.3669i	0.360491	−	0.360491i	0.863994	+	0.503502i	$0.167955\pi$
$828$		0			0
$829$			41.5388i			1.44270i	0.692570	+	0.721351i	$0.256479\pi$
$829$			41.5388i			1.44270i	−0.692570	+	0.721351i	$0.743521\pi$
$830$	64.5538	−	64.5538i	2.24070	−	2.24070i
$831$		0			0
$832$	11.5707	−	2.65631i	0.401141	−	0.0920908i
$833$			10.2450i			0.354970i
$834$		0			0
$835$	−39.5388			−1.36830
$836$	−3.92163			−0.135632
$837$		0			0
$838$	9.17872	−	9.17872i	0.317073	−	0.317073i
$839$	12.8948	−	12.8948i	0.445179	−	0.445179i	−0.448569	−	0.893748i	$-0.648066\pi$
$839$	12.8948	−	12.8948i	0.445179	−	0.445179i	0.893748	+	0.448569i	$0.148066\pi$
$840$		0			0
$841$	−28.2270			−0.973344
$842$	−90.3307			−3.11300
$843$		0			0
$844$			13.9435i			0.479953i
$845$	40.7847	+	14.1557i	1.40304	+	0.486971i
$846$		0			0
$847$	0.0428105	−	0.0428105i	0.00147099	−	0.00147099i
$848$		−	11.8203i		−	0.405912i
$849$		0			0
$850$	45.4677	−	45.4677i	1.55953	−	1.55953i
$851$	−2.41461	−	2.41461i	−0.0827719	−	0.0827719i
$852$		0			0
$853$	−31.4249	−	31.4249i	−1.07597	−	1.07597i	−0.996867	−	0.0791018i	$-0.974795\pi$
$853$	−31.4249	−	31.4249i	−1.07597	−	1.07597i	−0.0791018	−	0.996867i	$-0.525205\pi$
$854$		−	8.68634i		−	0.297240i
$855$		0			0
$856$	74.3129	+	74.3129i	2.53996	+	2.53996i
$857$	−51.3398			−1.75374			−0.876868	−	0.480732i	$-0.840371\pi$
$857$	−51.3398			−1.75374			−0.876868	+	0.480732i	$0.840371\pi$
$858$		0			0
$859$	13.4713			0.459636			0.229818	−	0.973234i	$-0.426187\pi$
$859$	13.4713			0.459636			0.229818	+	0.973234i	$0.426187\pi$
$860$	−57.5305	−	57.5305i	−1.96177	−	1.96177i
$861$		0			0
$862$			58.9162i			2.00669i
$863$	6.60369	+	6.60369i	0.224792	+	0.224792i	0.810513	−	0.585721i	$-0.199188\pi$
$863$	6.60369	+	6.60369i	0.224792	+	0.224792i	−0.585721	+	0.810513i	$0.699188\pi$
$864$		0			0
$865$	54.3876	+	54.3876i	1.84923	+	1.84923i
$866$	−6.42394	+	6.42394i	−0.218294	+	0.218294i
$867$		0			0
$868$		−	63.2654i		−	2.14737i
$869$	−39.2113	+	39.2113i	−1.33015	+	1.33015i
$870$		0			0
$871$	−32.4677	−	20.3437i	−1.10013	−	0.689320i
$872$			60.6450i			2.05370i
$873$		0			0
$874$	1.65911			0.0561204
$875$	7.31211			0.247194
$876$		0			0
$877$	−0.489472	+	0.489472i	−0.0165283	+	0.0165283i	−0.715323	−	0.698794i	$-0.753720\pi$
$877$	−0.489472	+	0.489472i	−0.0165283	+	0.0165283i	0.698794	+	0.715323i	$0.253720\pi$
$878$	10.0800	−	10.0800i	0.340183	−	0.340183i
$879$		0			0
$880$	66.4815			2.24109
$881$	−25.1774			−0.848250			−0.424125	−	0.905604i	$-0.639418\pi$
$881$	−25.1774			−0.848250			−0.424125	+	0.905604i	$0.639418\pi$
$882$		0			0
$883$			16.3118i			0.548936i	0.961596	+	0.274468i	$0.0885018\pi$
$883$			16.3118i			0.548936i	−0.961596	+	0.274468i	$0.911498\pi$
$884$	−64.4210	+	14.7893i	−2.16671	+	0.497417i
$885$		0			0
$886$	−18.0757	+	18.0757i	−0.607267	+	0.607267i
$887$		−	3.92163i		−	0.131675i	−0.997830	−	0.0658377i	$-0.979028\pi$
$887$		−	3.92163i		−	0.131675i	0.997830	−	0.0658377i	$-0.0209720\pi$
$888$		0			0
$889$	5.47133	−	5.47133i	0.183502	−	0.183502i
$890$	−39.7656	−	39.7656i	−1.33295	−	1.33295i
$891$		0			0
$892$	−12.0055	−	12.0055i	−0.401973	−	0.401973i
$893$			0.907598i			0.0303716i
$894$		0			0
$895$	−19.9253	−	19.9253i	−0.666030	−	0.666030i
$896$	−32.6546			−1.09091
$897$		0			0
$898$	−58.9162			−1.96606
$899$	−36.5762	−	36.5762i	−1.21989	−	1.21989i
$900$		0			0
$901$		−	8.31903i		−	0.277147i
$902$	15.8168	+	15.8168i	0.526642	+	0.526642i
$903$		0			0
$904$	−31.2125	−	31.2125i	−1.03811	−	1.03811i
$905$	46.7890	−	46.7890i	1.55532	−	1.55532i
$906$		0			0
$907$		−	40.6127i		−	1.34852i	−0.738493	−	0.674261i	$-0.764462\pi$
$907$		−	40.6127i		−	1.34852i	0.738493	−	0.674261i	$-0.235538\pi$
$908$	−61.1652	+	61.1652i	−2.02984	+	2.02984i
$909$		0			0
$910$	−54.6236	−	34.2262i	−1.81076	−	1.13459i
$911$			30.0195i			0.994590i	0.867581	+	0.497295i	$0.165673\pi$
$911$			30.0195i			0.994590i	−0.867581	+	0.497295i	$0.834327\pi$
$912$		0			0
$913$	36.3118			1.20175
$914$	14.6953			0.486078
$915$		0			0
$916$	18.6700	−	18.6700i	0.616876	−	0.616876i
$917$	−17.8782	+	17.8782i	−0.590389	+	0.590389i
$918$		0			0
$919$	−46.8680			−1.54603			−0.773016	−	0.634387i	$-0.781253\pi$
$919$	−46.8680			−1.54603			−0.773016	+	0.634387i	$0.781253\pi$
$920$	−46.7890			−1.54259
$921$		0			0
$922$			4.65004i			0.153141i
$923$	4.49726	−	7.17745i	0.148029	−	0.236249i
$924$		0			0
$925$	−6.02827	+	6.02827i	−0.198208	+	0.198208i
$926$			48.7564i			1.60224i
$927$		0			0
$928$	18.6464	−	18.6464i	0.612099	−	0.612099i
$929$	34.1275	+	34.1275i	1.11969	+	1.11969i	0.991787	+	0.127899i	$0.0408233\pi$
$929$	34.1275	+	34.1275i	1.11969	+	1.11969i	0.127899	+	0.991787i	$0.459177\pi$
$930$		0			0
$931$	0.466662	+	0.466662i	0.0152942	+	0.0152942i
$932$		−	12.9678i		−	0.424776i
$933$		0			0
$934$	5.12217	+	5.12217i	0.167602	+	0.167602i
$935$	46.7890			1.53016
$936$		0			0
$937$	−41.4641			−1.35457			−0.677287	−	0.735719i	$-0.736844\pi$
$937$	−41.4641			−1.35457			−0.677287	+	0.735719i	$0.736844\pi$
$938$	40.4528	+	40.4528i	1.32083	+	1.32083i
$939$		0			0
$940$		−	47.6519i		−	1.55423i
$941$	−25.0428	−	25.0428i	−0.816371	−	0.816371i	0.169209	−	0.985580i	$-0.445879\pi$
$941$	−25.0428	−	25.0428i	−0.816371	−	0.816371i	−0.985580	+	0.169209i	$0.945879\pi$
$942$		0			0
$943$	−4.57429	−	4.57429i	−0.148959	−	0.148959i
$944$	−4.54235	+	4.54235i	−0.147841	+	0.147841i
$945$		0			0
$946$		−	47.3401i		−	1.53916i
$947$	−13.8024	+	13.8024i	−0.448519	+	0.448519i	−0.894862	−	0.446343i	$-0.852726\pi$
$947$	−13.8024	+	13.8024i	−0.448519	+	0.448519i	0.446343	+	0.894862i	$0.352726\pi$
$948$		0			0
$949$	19.8970	−	31.7549i	0.645885	−	1.03081i
$950$		−	4.14211i		−	0.134388i
$951$		0			0
$952$	53.0101			1.71807
$953$	38.0253			1.23176			0.615880	−	0.787840i	$-0.288800\pi$
$953$	38.0253			1.23176			0.615880	+	0.787840i	$0.288800\pi$
$954$		0			0
$955$	−5.35823	+	5.35823i	−0.173388	+	0.173388i
$956$	74.5674	−	74.5674i	2.41168	−	2.41168i
$957$		0			0
$958$	5.43398			0.175564
$959$	−5.73682			−0.185252
$960$		0			0
$961$		−	15.7549i		−	0.508221i
$962$	12.4946	−	2.86841i	0.402842	−	0.0924813i
$963$		0			0
$964$	−37.6236	+	37.6236i	−1.21178	+	1.21178i
$965$			67.0818i			2.15944i
$966$		0			0
$967$	19.4021	−	19.4021i	0.623929	−	0.623929i	−0.322604	−	0.946534i	$-0.604558\pi$
$967$	19.4021	−	19.4021i	0.623929	−	0.623929i	0.946534	+	0.322604i	$0.104558\pi$
$968$	0.116657	+	0.116657i	0.00374951	+	0.00374951i
$969$		0			0
$970$	5.90611	+	5.90611i	0.189634	+	0.189634i
$971$		−	53.4334i		−	1.71476i	−0.514684	−	0.857380i	$-0.672091\pi$
$971$		−	53.4334i		−	1.71476i	0.514684	−	0.857380i	$-0.327909\pi$
$972$		0			0
$973$	24.1131	+	24.1131i	0.773030	+	0.773030i
$974$	−51.8387			−1.66102
$975$		0			0
$976$	9.72659			0.311341
$977$	40.3181	+	40.3181i	1.28989	+	1.28989i	0.934852	+	0.355039i	$0.115532\pi$
$977$	40.3181	+	40.3181i	1.28989	+	1.28989i	0.355039	+	0.934852i	$0.384468\pi$
$978$		0			0
$979$		−	22.3684i		−	0.714896i
$980$	−24.5013	−	24.5013i	−0.782665	−	0.782665i
$981$		0			0
$982$	63.8023	+	63.8023i	2.03601	+	2.03601i
$983$	−5.31716	+	5.31716i	−0.169591	+	0.169591i	−0.786800	−	0.617209i	$-0.788263\pi$
$983$	−5.31716	+	5.31716i	−0.169591	+	0.169591i	0.617209	+	0.786800i	$0.288263\pi$
$984$		0			0
$985$		−	17.4823i		−	0.557031i
$986$	57.0572	−	57.0572i	1.81707	−	1.81707i
$987$		0			0
$988$	−2.26073	+	3.60803i	−0.0719233	+	0.114787i
$989$			13.6910i			0.435347i
$990$		0			0
$991$	−6.31903			−0.200731			−0.100365	−	0.994951i	$-0.532001\pi$
$991$	−6.31903			−0.200731			−0.100365	+	0.994951i	$0.532001\pi$
$992$	23.8355			0.756777
$993$		0			0
$994$	−8.94265	+	8.94265i	−0.283644	+	0.283644i
$995$	38.3037	−	38.3037i	1.21431	−	1.21431i
$996$		0			0
$997$	6.77301			0.214503			0.107252	−	0.994232i	$-0.465795\pi$
$997$	6.77301			0.214503			0.107252	+	0.994232i	$0.465795\pi$
$998$	1.02096			0.0323178
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	117.2.i.a.44.6	yes	12
3.2	odd	2	inner	117.2.i.a.44.1	yes	12
4.3	odd	2		1872.2.bi.f.161.1		12
12.11	even	2		1872.2.bi.f.161.6		12
13.5	odd	4		1521.2.i.g.944.6		12
13.8	odd	4	inner	117.2.i.a.8.1	✓	12
13.12	even	2		1521.2.i.g.746.1		12
39.5	even	4		1521.2.i.g.944.1		12
39.8	even	4	inner	117.2.i.a.8.6	yes	12
39.38	odd	2		1521.2.i.g.746.6		12
52.47	even	4		1872.2.bi.f.593.6		12
156.47	odd	4		1872.2.bi.f.593.1		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
117.2.i.a.8.1	✓	12	13.8	odd	4	inner
117.2.i.a.8.6	yes	12	39.8	even	4	inner
117.2.i.a.44.1	yes	12	3.2	odd	2	inner
117.2.i.a.44.6	yes	12	1.1	even	1	trivial
1521.2.i.g.746.1		12	13.12	even	2
1521.2.i.g.746.6		12	39.38	odd	2
1521.2.i.g.944.1		12	39.5	even	4
1521.2.i.g.944.6		12	13.5	odd	4
1872.2.bi.f.161.1		12	4.3	odd	2
1872.2.bi.f.161.6		12	12.11	even	2
1872.2.bi.f.593.1		12	156.47	odd	4
1872.2.bi.f.593.6		12	52.47	even	4

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)

\(f(q)\)	\(=\)	\(q+(1.77776 + 1.77776i) q^{2} +4.32088i q^{4} +(-2.34822 - 2.34822i) q^{5} +(1.51414 + 1.51414i) q^{7} +(-4.12598 + 4.12598i) q^{8} +O(q^{10})\)	\(q+(1.77776 + 1.77776i) q^{2} +4.32088i q^{4} +(-2.34822 - 2.34822i) q^{5} +(1.51414 + 1.51414i) q^{7} +(-4.12598 + 4.12598i) q^{8} -8.34916i q^{10} +(2.34822 - 2.34822i) q^{11} +(-0.806748 - 3.51414i) q^{13} +5.38355i q^{14} -6.02827 q^{16} -4.24264 q^{17} +(-0.193252 + 0.193252i) q^{19} +(10.1464 - 10.1464i) q^{20} +8.34916 q^{22} -2.41461 q^{23} +6.02827i q^{25} +(4.81310 - 7.68151i) q^{26} +(-6.54241 + 6.54241i) q^{28} +7.56485i q^{29} +(-4.83502 + 4.83502i) q^{31} +(-2.46488 - 2.46488i) q^{32} +(-7.54241 - 7.54241i) q^{34} -7.11105i q^{35} +(1.00000 + 1.00000i) q^{37} -0.687114 q^{38} +19.3774 q^{40} +(1.89442 + 1.89442i) q^{41} -5.67004i q^{43} +(10.1464 + 10.1464i) q^{44} +(-4.29261 - 4.29261i) q^{46} +(2.34822 - 2.34822i) q^{47} -2.41478i q^{49} +(-10.7168 + 10.7168i) q^{50} +(15.1842 - 3.48586i) q^{52} +1.96081i q^{53} -11.0283 q^{55} -12.4946 q^{56} +(-13.4485 + 13.4485i) q^{58} +(0.753507 - 0.753507i) q^{59} -1.61350 q^{61} -17.1910 q^{62} +3.29261i q^{64} +(-6.35755 + 10.1464i) q^{65} +(7.51414 - 7.51414i) q^{67} -18.3320i q^{68} +(12.6418 - 12.6418i) q^{70} +(1.66111 + 1.66111i) q^{71} +(7.34916 + 7.34916i) q^{73} +3.55553i q^{74} +(-0.835021 - 0.835021i) q^{76} +7.11105 q^{77} -16.6983 q^{79} +(14.1557 + 14.1557i) q^{80} +6.73566i q^{82} +(7.73177 + 7.73177i) q^{83} +(9.96265 + 9.96265i) q^{85} +(10.0800 - 10.0800i) q^{86} +19.3774i q^{88} +(4.76283 - 4.76283i) q^{89} +(4.09936 - 6.54241i) q^{91} -10.4333i q^{92} +8.34916 q^{94} +0.907598 q^{95} +(-0.707389 + 0.707389i) q^{97} +(4.29290 - 4.29290i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 8 q^{7}+O(q^{10}) \) 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

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Database

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