LMFDB - Embedded newform 177.2.d.b.176.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [177,2,Mod(176,177)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(177, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("177.176");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 177 = 3 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	177.d (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$1.41335211578$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.149721291.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 7x^{3} + 27 $ x^6 - 7*x^3 + 27
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			176.6
Root			$1.66591 + 0.474089i$ of defining polynomial
Character	$\chi$	$=$	177.176
Dual form			177.2.d.b.176.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.66591 + 0.474089i) q^{3} -2.00000 q^{4} +4.10732i q^{5} -1.56266 q^{7} +(2.55048 + 1.57957i) q^{9} +O(q^{10})$	$q+(1.66591 + 0.474089i) q^{3} -2.00000 q^{4} +4.10732i q^{5} -1.56266 q^{7} +(2.55048 + 1.57957i) q^{9} +(-3.33181 - 0.948177i) q^{12} +(-1.94724 + 6.84241i) q^{15} +4.00000 q^{16} -7.68115i q^{17} +8.43277 q^{19} -8.21465i q^{20} +(-2.60324 - 0.740840i) q^{21} -11.8701 q^{25} +(3.50000 + 3.84057i) q^{27} +3.12532 q^{28} +6.95186i q^{29} -6.41835i q^{35} +(-5.10096 - 3.15915i) q^{36} -11.0592i q^{41} +(-6.48782 + 10.4756i) q^{45} +(6.66362 + 1.89635i) q^{48} -4.55809 q^{49} +(3.64154 - 12.7961i) q^{51} -5.37012i q^{53} +(14.0482 + 3.99788i) q^{57} +7.68115i q^{59} +(3.89447 - 13.6848i) q^{60} +(-3.98553 - 2.46834i) q^{63} -8.00000 q^{64} +15.3623i q^{68} -7.68115i q^{71} +(-19.7745 - 5.62748i) q^{75} -16.8655 q^{76} +3.74479 q^{79} +16.4293i q^{80} +(4.00990 + 8.05734i) q^{81} +(5.20649 + 1.48168i) q^{84} +31.5490 q^{85} +(-3.29580 + 11.5811i) q^{87} +34.6361i q^{95} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 12 q^{4}+O(q^{10}) $ 6 * q - 12 * q^4	$ 6 q - 12 q^{4} + 3 q^{15} + 24 q^{16} - 15 q^{21} - 30 q^{25} + 21 q^{27} - 33 q^{45} + 42 q^{49} + 39 q^{57} - 6 q^{60} + 12 q^{63} - 48 q^{64} - 24 q^{75} + 30 q^{84} - 51 q^{87}+O(q^{100}) $ 6 * q - 12 * q^4 + 3 * q^15 + 24 * q^16 - 15 * q^21 - 30 * q^25 + 21 * q^27 - 33 * q^45 + 42 * q^49 + 39 * q^57 - 6 * q^60 + 12 * q^63 - 48 * q^64 - 24 * q^75 + 30 * q^84 - 51 * q^87

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/177\mathbb{Z}\right)^\times$.

$n$	$61$	$119$
$\chi(n)$	$-1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0			−	1.00000i	$-0.5\pi$
$2$		0			0				1.00000i	$0.5\pi$
$3$	1.66591	+	0.474089i	0.961811	+	0.273715i
$3$	1.66591	+	0.474089i	0.961811	+	0.273715i
$4$	−2.00000			−1.00000
$5$			4.10732i			1.83685i	0.395594	+	0.918426i	$0.370539\pi$
$5$			4.10732i			1.83685i	−0.395594	+	0.918426i	$0.629461\pi$
$6$		0			0
$7$	−1.56266			−0.590630			−0.295315	−	0.955400i	$-0.595425\pi$
$7$	−1.56266			−0.590630			−0.295315	+	0.955400i	$0.595425\pi$
$8$		0			0
$9$	2.55048	+	1.57957i	0.850160	+	0.526524i
$10$		0			0
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$	−3.33181	−	0.948177i	−0.961811	−	0.273715i
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$		0			0
$15$	−1.94724	+	6.84241i	−0.502774	+	1.76670i
$16$	4.00000			1.00000
$17$		−	7.68115i		−	1.86295i	−0.363803	−	0.931476i	$-0.618522\pi$
$17$		−	7.68115i		−	1.86295i	0.363803	−	0.931476i	$-0.381478\pi$
$18$		0			0
$19$	8.43277			1.93461			0.967305	−	0.253616i	$-0.0816198\pi$
$19$	8.43277			1.93461			0.967305	+	0.253616i	$0.0816198\pi$
$20$		−	8.21465i		−	1.83685i
$21$	−2.60324	−	0.740840i	−0.568075	−	0.161664i
$22$		0			0
$23$		0			0			−	1.00000i	$-0.5\pi$
$23$		0			0				1.00000i	$0.5\pi$
$24$		0			0
$25$	−11.8701			−2.37402
$26$		0			0
$27$	3.50000	+	3.84057i	0.673575	+	0.739119i
$28$	3.12532			0.590630
$29$			6.95186i			1.29093i	0.763791	+	0.645464i	$0.223336\pi$
$29$			6.95186i			1.29093i	−0.763791	+	0.645464i	$0.776664\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		−	6.41835i		−	1.08490i
$36$	−5.10096	−	3.15915i	−0.850160	−	0.526524i
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	11.0592i		−	1.72715i	−0.504217	−	0.863577i	$-0.668219\pi$
$41$		−	11.0592i		−	1.72715i	0.504217	−	0.863577i	$-0.331781\pi$
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$		0			0
$45$	−6.48782	+	10.4756i	−0.967147	+	1.56162i
$46$		0			0
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$	6.66362	+	1.89635i	0.961811	+	0.273715i
$49$	−4.55809			−0.651156
$50$		0			0
$51$	3.64154	−	12.7961i	0.509918	−	1.79181i
$52$		0			0
$53$		−	5.37012i		−	0.737642i	−0.929500	−	0.368821i	$-0.879762\pi$
$53$		−	5.37012i		−	0.737642i	0.929500	−	0.368821i	$-0.120238\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	14.0482	+	3.99788i	1.86073	+	0.529532i
$58$		0			0
$59$			7.68115i			1.00000i
$59$			7.68115i			1.00000i
$60$	3.89447	−	13.6848i	0.502774	−	1.76670i
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$		0			0
$63$	−3.98553	−	2.46834i	−0.502130	−	0.310981i
$64$	−8.00000			−1.00000
$65$		0			0
$66$		0			0
$67$		0			0		1.00000			$0$
$67$		0			0		−1.00000			$\pi$
$68$			15.3623i			1.86295i
$69$		0			0
$70$		0			0
$71$		−	7.68115i		−	0.911584i	−0.890086	−	0.455792i	$-0.849356\pi$
$71$		−	7.68115i		−	0.911584i	0.890086	−	0.455792i	$-0.150644\pi$
$72$		0			0
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$		0			0
$75$	−19.7745	−	5.62748i	−2.28336	−	0.649806i
$76$	−16.8655			−1.93461
$77$		0			0
$78$		0			0
$79$	3.74479			0.421322			0.210661	−	0.977559i	$-0.432438\pi$
$79$	3.74479			0.421322			0.210661	+	0.977559i	$0.432438\pi$
$80$			16.4293i			1.83685i
$81$	4.00990	+	8.05734i	0.445544	+	0.895260i
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$	5.20649	+	1.48168i	0.568075	+	0.161664i
$85$	31.5490			3.42196
$86$		0			0
$87$	−3.29580	+	11.5811i	−0.353346	+	1.24163i
$88$		0			0
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$			34.6361i			3.55359i
$96$		0			0
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$		0			0
$99$		0			0
$100$	23.7402			2.37402
$101$		0			0			−	1.00000i	$-0.5\pi$
$101$		0			0				1.00000i	$0.5\pi$
$102$		0			0
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		0			0
$105$	3.04287	−	10.6924i	0.296954	−	1.04347i
$106$		0			0
$107$			9.79639i			0.947053i	0.880780	+	0.473526i	$0.157019\pi$
$107$			9.79639i			0.947053i	−0.880780	+	0.473526i	$0.842981\pi$
$108$	−7.00000	−	7.68115i	−0.673575	−	0.739119i
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$	−6.25064			−0.590630
$113$		0			0			−	1.00000i	$-0.5\pi$
$113$		0			0				1.00000i	$0.5\pi$
$114$		0			0
$115$		0			0
$116$		−	13.9037i		−	1.29093i
$117$		0			0
$118$		0			0
$119$			12.0030i			1.10032i
$120$		0			0
$121$	−11.0000			−1.00000
$122$		0			0
$123$	5.24303	−	18.4235i	0.472748	−	1.66120i
$124$		0			0
$125$		−	28.2178i		−	2.52387i
$126$		0			0
$127$	−2.18213			−0.193633			−0.0968163	−	0.995302i	$-0.530866\pi$
$127$	−2.18213			−0.193633			−0.0968163	+	0.995302i	$0.530866\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$		0			0
$133$	−13.1776			−1.14264
$134$		0			0
$135$	−15.7745	+	14.3756i	−1.35765	+	1.23726i
$136$		0			0
$137$		−	23.3812i		−	1.99759i	−0.0491120	−	0.998793i	$-0.515639\pi$
$137$		−	23.3812i		−	1.99759i	0.0491120	−	0.998793i	$-0.484361\pi$
$138$		0			0
$139$	5.00000			0.424094			0.212047	−	0.977259i	$-0.431987\pi$
$139$	5.00000			0.424094			0.212047	+	0.977259i	$0.431987\pi$
$140$			12.8367i			1.08490i
$141$		0			0
$142$		0			0
$143$		0			0
$144$	10.2019	+	6.31829i	0.850160	+	0.526524i
$145$	−28.5535			−2.37124
$146$		0			0
$147$	−7.59335	−	2.16094i	−0.626289	−	0.178231i
$148$		0			0
$149$		0			0			−	1.00000i	$-0.5\pi$
$149$		0			0				1.00000i	$0.5\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$		0			0
$153$	12.1329	−	19.5906i	0.980889	−	1.58381i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		1.00000			$0$
$157$		0			0		−1.00000			$\pi$
$158$		0			0
$159$	2.54591	−	8.94610i	0.201904	−	0.709472i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	11.0000			0.861586			0.430793	−	0.902451i	$-0.358234\pi$
$163$	11.0000			0.861586			0.430793	+	0.902451i	$0.358234\pi$
$164$			22.1184i			1.72715i
$165$		0			0
$166$		0			0
$167$		−	17.6921i		−	1.36905i	−0.728987	−	0.684527i	$-0.760009\pi$
$167$		−	17.6921i		−	1.36905i	0.728987	−	0.684527i	$-0.239991\pi$
$168$		0			0
$169$	13.0000			1.00000
$170$		0			0
$171$	21.5076	+	13.3202i	1.64473	+	1.01862i
$172$		0			0
$173$		0			0			−	1.00000i	$-0.5\pi$
$173$		0			0				1.00000i	$0.5\pi$
$174$		0			0
$175$	18.5490			1.40217
$176$		0			0
$177$	−3.64154	+	12.7961i	−0.273715	+	0.961811i
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$	12.9756	−	20.9513i	0.967147	−	1.56162i
$181$	−26.8610			−1.99656			−0.998280	−	0.0586239i	$-0.981329\pi$
$181$	−26.8610			−1.99656			−0.998280	+	0.0586239i	$0.981329\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−5.46931	−	6.00151i	−0.397834	−	0.436546i
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$	−13.3272	−	3.79271i	−0.961811	−	0.273715i
$193$	−22.1730			−1.59605			−0.798023	−	0.602627i	$-0.794121\pi$
$193$	−22.1730			−1.59605			−0.798023	+	0.602627i	$0.794121\pi$
$194$		0			0
$195$		0			0
$196$	9.11618			0.651156
$197$		−	7.68115i		−	0.547259i	−0.961835	−	0.273629i	$-0.911776\pi$
$197$		−	7.68115i		−	0.547259i	0.961835	−	0.273629i	$-0.0882242\pi$
$198$		0			0
$199$	−16.2461			−1.15165			−0.575827	−	0.817572i	$-0.695320\pi$
$199$	−16.2461			−1.15165			−0.575827	+	0.817572i	$0.695320\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	10.8634i		−	0.762461i
$204$	−7.28309	+	25.5921i	−0.509918	+	1.79181i
$205$	45.4236			3.17252
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$			10.7402i			0.737642i
$213$	3.64154	−	12.7961i	0.249514	−	0.876772i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−19.0000			−1.27233			−0.636167	−	0.771551i	$-0.719481\pi$
$223$	−19.0000			−1.27233			−0.636167	+	0.771551i	$0.719481\pi$
$224$		0			0
$225$	−30.2745	−	18.7497i	−2.01830	−	1.24998i
$226$		0			0
$227$		0			0			−	1.00000i	$-0.5\pi$
$227$		0			0				1.00000i	$0.5\pi$
$228$	−28.0964	−	7.99576i	−1.86073	−	0.529532i
$229$		0			0		1.00000			$0$
$229$		0			0		−1.00000			$\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0			−	1.00000i	$-0.5\pi$
$233$		0			0				1.00000i	$0.5\pi$
$234$		0			0
$235$		0			0
$236$		−	15.3623i		−	1.00000i
$237$	6.23846	+	1.77536i	0.405232	+	0.115322i
$238$		0			0
$239$			12.6409i			0.817673i	0.912608	+	0.408837i	$0.134065\pi$
$239$			12.6409i			0.817673i	−0.912608	+	0.408837i	$0.865935\pi$
$240$	−7.78894	+	27.3696i	−0.502774	+	1.76670i
$241$	9.05224			0.583106			0.291553	−	0.956555i	$-0.405828\pi$
$241$	9.05224			0.583106			0.291553	+	0.956555i	$0.405828\pi$
$242$		0			0
$243$	2.86021	+	15.3238i	0.183483	+	0.983023i
$244$		0			0
$245$		−	18.7216i		−	1.19608i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$			31.5958i			1.99431i	0.0753841	+	0.997155i	$0.475982\pi$
$251$			31.5958i			1.99431i	−0.0753841	+	0.997155i	$0.524018\pi$
$252$	7.97107	+	4.93667i	0.502130	+	0.310981i
$253$		0			0
$254$		0			0
$255$	52.5576	+	14.9570i	3.29128	+	0.936644i
$256$	16.0000			1.00000
$257$		−	4.42627i		−	0.276103i	−0.990425	−	0.138052i	$-0.955916\pi$
$257$		−	4.42627i		−	0.276103i	0.990425	−	0.138052i	$-0.0440840\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−10.9810	+	17.7306i	−0.679705	+	1.09749i
$262$		0			0
$263$		−	14.8476i		−	0.915540i	−0.889071	−	0.457770i	$-0.848648\pi$
$263$		−	14.8476i		−	0.915540i	0.889071	−	0.457770i	$-0.151352\pi$
$264$		0			0
$265$	22.0568			1.35494
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$	19.0477			1.15706			0.578532	−	0.815660i	$-0.303626\pi$
$271$	19.0477			1.15706			0.578532	+	0.815660i	$0.303626\pi$
$272$		−	30.7246i		−	1.86295i
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	17.8087			1.07002			0.535012	−	0.844845i	$-0.320307\pi$
$277$	17.8087			1.07002			0.535012	+	0.844845i	$0.320307\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			0.318947i			0.0190268i	0.999955	+	0.00951340i	$0.00302825\pi$
$281$			0.318947i			0.0190268i	−0.999955	+	0.00951340i	$0.996972\pi$
$282$		0			0
$283$		0			0		1.00000			$0$
$283$		0			0		−1.00000			$\pi$
$284$			15.3623i			0.911584i
$285$	−16.4206	+	57.7005i	−0.972672	+	3.41788i
$286$		0			0
$287$			17.2817i			1.02011i
$288$		0			0
$289$	−42.0000			−2.47059
$290$		0			0
$291$		0			0
$292$		0			0
$293$			25.9067i			1.51349i	0.653712	+	0.756744i	$0.273211\pi$
$293$			25.9067i			1.51349i	−0.653712	+	0.756744i	$0.726789\pi$
$294$		0			0
$295$	−31.5490			−1.83685
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	39.5490	+	11.2550i	2.28336	+	0.649806i
$301$		0			0
$302$		0			0
$303$		0			0
$304$	33.7311			1.93461
$305$		0			0
$306$		0			0
$307$	−32.1684			−1.83595			−0.917974	−	0.396640i	$-0.870176\pi$
$307$	−32.1684			−1.83595			−0.917974	+	0.396640i	$0.870176\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			34.4403i			1.95293i	0.215671	+	0.976466i	$0.430806\pi$
$311$			34.4403i			1.95293i	−0.215671	+	0.976466i	$0.569194\pi$
$312$		0			0
$313$		0			0		1.00000			$0$
$313$		0			0		−1.00000			$\pi$
$314$		0			0
$315$	10.1383	−	16.3699i	0.571226	−	0.922338i
$316$	−7.48958			−0.421322
$317$		−	30.7246i		−	1.72566i	−0.505490	−	0.862832i	$-0.668688\pi$
$317$		−	30.7246i		−	1.72566i	0.505490	−	0.862832i	$-0.331312\pi$
$318$		0			0
$319$		0			0
$320$		−	32.8586i		−	1.83685i
$321$	−4.64436	+	16.3199i	−0.259223	+	0.910885i
$322$		0			0
$323$		−	64.7733i		−	3.60408i
$324$	−8.01979	−	16.1147i	−0.445544	−	0.895260i
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−17.4850			−0.961063			−0.480531	−	0.876977i	$-0.659556\pi$
$331$	−17.4850			−0.961063			−0.480531	+	0.876977i	$0.659556\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$	−10.4130	−	2.96336i	−0.568075	−	0.161664i
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$		0			0
$339$		0			0
$340$	−63.0979			−3.42196
$341$		0			0
$342$		0			0
$343$	18.0614			0.975223
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0			−	1.00000i	$-0.5\pi$
$347$		0			0				1.00000i	$0.5\pi$
$348$	6.59159	−	23.1623i	0.353346	−	1.24163i
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0			−	1.00000i	$-0.5\pi$
$353$		0			0				1.00000i	$0.5\pi$
$354$		0			0
$355$	31.5490			1.67444
$356$		0			0
$357$	−5.69050	+	19.9959i	−0.301173	+	1.05830i
$358$		0			0
$359$			7.89570i			0.416719i	0.978052	+	0.208360i	$0.0668124\pi$
$359$			7.89570i			0.416719i	−0.978052	+	0.208360i	$0.933188\pi$
$360$		0			0
$361$	52.1116			2.74272
$362$		0			0
$363$	−18.3250	−	5.21497i	−0.961811	−	0.273715i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		1.00000			$0$
$367$		0			0		−1.00000			$\pi$
$368$		0			0
$369$	17.4688	−	28.2062i	0.909389	−	1.46836i
$370$		0			0
$371$			8.39167i			0.435674i
$372$		0			0
$373$	−31.0000			−1.60512			−0.802560	−	0.596572i	$-0.796529\pi$
$373$	−31.0000			−1.60512			−0.802560	+	0.596572i	$0.796529\pi$
$374$		0			0
$375$	13.3777	−	47.0081i	0.690823	−	2.42749i
$376$		0			0
$377$		0			0
$378$		0			0
$379$	38.4191			1.97345			0.986727	−	0.162386i	$-0.0519189\pi$
$379$	38.4191			1.97345			0.986727	+	0.162386i	$0.0519189\pi$
$380$		−	69.2722i		−	3.55359i
$381$	−3.63522	−	1.03452i	−0.186238	−	0.0530002i
$382$		0			0
$383$			15.3623i			0.784976i	0.919757	+	0.392488i	$0.128386\pi$
$383$			15.3623i			0.784976i	−0.919757	+	0.392488i	$0.871614\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$			38.4057i			1.94725i	0.228159	+	0.973624i	$0.426729\pi$
$389$			38.4057i			1.94725i	−0.228159	+	0.973624i	$0.573271\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$			15.3811i			0.773905i
$396$		0			0
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$		0			0
$399$	−21.9526	−	6.24733i	−1.09900	−	0.312758i
$400$	−47.4804			−2.37402
$401$		0			0			−	1.00000i	$-0.5\pi$
$401$		0			0				1.00000i	$0.5\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−33.0941	+	16.4699i	−1.64446	+	0.818398i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$		0			0
$411$	11.0847	−	38.9508i	0.546770	−	1.92130i
$412$		0			0
$413$		−	12.0030i		−	0.590630i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	8.32953	+	2.37044i	0.407899	+	0.116081i
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$	−6.08574	+	21.3847i	−0.296954	+	1.04347i
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			91.1760i			4.42269i
$426$		0			0
$427$		0			0
$428$		−	19.5928i		−	0.947053i
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$	14.0000	+	15.3623i	0.673575	+	0.739119i
$433$	27.8042			1.33618			0.668091	−	0.744079i	$-0.267111\pi$
$433$	27.8042			1.33618			0.668091	+	0.744079i	$0.267111\pi$
$434$		0			0
$435$	−47.5675	−	13.5369i	−2.28069	−	0.649045i
$436$		0			0
$437$		0			0
$438$		0			0
$439$	35.0000			1.67046			0.835229	−	0.549902i	$-0.185335\pi$
$439$	35.0000			1.67046			0.835229	+	0.549902i	$0.185335\pi$
$440$		0			0
$441$	−11.6253	−	7.19984i	−0.553587	−	0.342850i
$442$		0			0
$443$		0			0			−	1.00000i	$-0.5\pi$
$443$		0			0				1.00000i	$0.5\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	12.5013			0.590630
$449$		−	7.27080i		−	0.343130i	−0.985173	−	0.171565i	$-0.945118\pi$
$449$		−	7.27080i		−	0.343130i	0.985173	−	0.171565i	$-0.0548824\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		1.00000			$0$
$457$		0			0		−1.00000			$\pi$
$458$		0			0
$459$	29.5000	−	26.8840i	1.37694	−	1.25484i
$460$		0			0
$461$		−	30.7246i		−	1.43099i	−0.698620	−	0.715493i	$-0.746202\pi$
$461$		−	30.7246i		−	1.43099i	0.698620	−	0.715493i	$-0.253798\pi$
$462$		0			0
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$			27.8074i			1.29093i
$465$		0			0
$466$		0			0
$467$		0			0			−	1.00000i	$-0.5\pi$
$467$		0			0				1.00000i	$0.5\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−100.098			−4.59281
$476$		−	24.0060i		−	1.10032i
$477$	8.48249	−	13.6964i	0.388387	−	0.627114i
$478$		0			0
$479$			38.4057i			1.75480i	0.479757	+	0.877401i	$0.340725\pi$
$479$			38.4057i			1.75480i	−0.479757	+	0.877401i	$0.659275\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	22.0000			1.00000
$485$		0			0
$486$		0			0
$487$	43.7265			1.98144			0.990719	−	0.135928i	$-0.0434016\pi$
$487$	43.7265			1.98144			0.990719	+	0.135928i	$0.0434016\pi$
$488$		0			0
$489$	18.3250	+	5.21497i	0.828683	+	0.235829i
$490$		0			0
$491$		−	41.3922i		−	1.86800i	−0.357269	−	0.934002i	$-0.616292\pi$
$491$		−	41.3922i		−	1.86800i	0.357269	−	0.934002i	$-0.383708\pi$
$492$	−10.4861	+	36.8471i	−0.472748	+	1.66120i
$493$	53.3982			2.40493
$494$		0			0
$495$		0			0
$496$		0			0
$497$			12.0030i			0.538409i
$498$		0			0
$499$	−41.5444			−1.85978			−0.929891	−	0.367835i	$-0.880099\pi$
$499$	−41.5444			−1.85978			−0.929891	+	0.367835i	$0.880099\pi$
$500$			56.4355i			2.52387i
$501$	8.38762	−	29.4733i	0.374731	−	1.31677i
$502$		0			0
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	21.6568	+	6.16315i	0.961811	+	0.273715i
$508$	4.36426			0.193633
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	29.5147	+	32.3867i	1.30311	+	1.42991i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	7.68115i		−	0.336517i	−0.985743	−	0.168259i	$-0.946186\pi$
$521$		−	7.68115i		−	0.336517i	0.985743	−	0.168259i	$-0.0538144\pi$
$522$		0			0
$523$	−37.4759			−1.63871			−0.819353	−	0.573290i	$-0.805667\pi$
$523$	−37.4759			−1.63871			−0.819353	+	0.573290i	$0.805667\pi$
$524$		0			0
$525$	30.9008	+	8.79385i	1.34862	+	0.383795i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−23.0000			−1.00000
$530$		0			0
$531$	−12.1329	+	19.5906i	−0.526524	+	0.850160i
$532$	26.3551			1.14264
$533$		0			0
$534$		0			0
$535$	−40.2369			−1.73959
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$	31.5490	−	28.7513i	1.35765	−	1.23726i
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$	−44.7478	−	12.7345i	−1.92031	−	0.546489i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	8.00000			0.342055			0.171028	−	0.985266i	$-0.445291\pi$
$547$	8.00000			0.342055			0.171028	+	0.985266i	$0.445291\pi$
$548$			46.7623i			1.99759i
$549$		0			0
$550$		0			0
$551$			58.6234i			2.49744i
$552$		0			0
$553$	−5.85183			−0.248845
$554$		0			0
$555$		0			0
$556$	−10.0000			−0.424094
$557$			38.2287i			1.61980i	0.586566	+	0.809901i	$0.300479\pi$
$557$			38.2287i			1.61980i	−0.586566	+	0.809901i	$0.699521\pi$
$558$		0			0
$559$		0			0
$560$		−	25.6734i		−	1.08490i
$561$		0			0
$562$		0			0
$563$		0			0			−	1.00000i	$-0.5\pi$
$563$		0			0				1.00000i	$0.5\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−6.26611	−	12.5909i	−0.263152	−	0.528768i
$568$		0			0
$569$		0			0			−	1.00000i	$-0.5\pi$
$569$		0			0				1.00000i	$0.5\pi$
$570$		0			0
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−20.4038	−	12.6366i	−0.850160	−	0.526524i
$577$	−5.63118			−0.234429			−0.117214	−	0.993107i	$-0.537396\pi$
$577$	−5.63118			−0.234429			−0.117214	+	0.993107i	$0.537396\pi$
$578$		0			0
$579$	−36.9381	−	10.5120i	−1.53509	−	0.436862i
$580$	57.1070			2.37124
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$	15.1867	+	4.32188i	0.626289	+	0.178231i
$589$		0			0
$590$		0			0
$591$	3.64154	−	12.7961i	0.149793	−	0.526360i
$592$		0			0
$593$			40.1294i			1.64792i	0.566650	+	0.823958i	$0.308239\pi$
$593$			40.1294i			1.64792i	−0.566650	+	0.823958i	$0.691761\pi$
$594$		0			0
$595$	−49.3003			−2.02112
$596$		0			0
$597$	−27.0644	−	7.70208i	−1.10767	−	0.315225i
$598$		0			0
$599$		−	48.0251i		−	1.96225i	−0.193369	−	0.981126i	$-0.561941\pi$
$599$		−	48.0251i		−	1.96225i	0.193369	−	0.981126i	$-0.438059\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		−	45.1806i		−	1.83685i
$606$		0			0
$607$	48.4145			1.96508			0.982542	−	0.186042i	$-0.0595660\pi$
$607$	48.4145			1.96508			0.982542	+	0.186042i	$0.0595660\pi$
$608$		0			0
$609$	5.15021	−	18.0974i	0.208697	−	0.733343i
$610$		0			0
$611$		0			0
$612$	−24.2659	+	39.1812i	−0.980889	+	1.58381i
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$		0			0
$615$	75.6715	+	21.5348i	3.05137	+	0.868368i
$616$		0			0
$617$		−	9.15849i		−	0.368707i	−0.982860	−	0.184354i	$-0.940981\pi$
$617$		−	9.15849i		−	0.368707i	0.982860	−	0.184354i	$-0.0590191\pi$
$618$		0			0
$619$	−22.7925			−0.916106			−0.458053	−	0.888925i	$-0.651453\pi$
$619$	−22.7925			−0.916106			−0.458053	+	0.888925i	$0.651453\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	56.5490			2.26196
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	20.0000			0.796187			0.398094	−	0.917345i	$-0.369672\pi$
$631$	20.0000			0.796187			0.398094	+	0.917345i	$0.369672\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		−	8.96270i		−	0.355674i
$636$	−5.09182	+	17.8922i	−0.201904	+	0.709472i
$637$		0			0
$638$		0			0
$639$	12.1329	−	19.5906i	0.479971	−	0.774992i
$640$		0			0
$641$			38.4057i			1.51694i	0.651711	+	0.758468i	$0.274052\pi$
$641$			38.4057i			1.51694i	−0.651711	+	0.758468i	$0.725948\pi$
$642$		0			0
$643$	37.7996			1.49067			0.745335	−	0.666690i	$-0.232289\pi$
$643$	37.7996			1.49067			0.745335	+	0.666690i	$0.232289\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	50.8696i		−	1.99989i	−0.0104225	−	0.999946i	$-0.503318\pi$
$647$		−	50.8696i		−	1.99989i	0.0104225	−	0.999946i	$-0.496682\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$	−22.0000			−0.861586
$653$			49.6068i			1.94127i	0.240566	+	0.970633i	$0.422667\pi$
$653$			49.6068i			1.94127i	−0.240566	+	0.970633i	$0.577333\pi$
$654$		0			0
$655$		0			0
$656$		−	44.2367i		−	1.72715i
$657$		0			0
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$	−30.9295			−1.20302			−0.601509	−	0.798866i	$-0.705434\pi$
$661$	−30.9295			−1.20302			−0.601509	+	0.798866i	$0.705434\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	54.1245i		−	2.09886i
$666$		0			0
$667$		0			0
$668$			35.3842i			1.36905i
$669$	−31.6522	−	9.00768i	−1.22374	−	0.348257i
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$		0			0
$675$	−41.5454	−	45.5880i	−1.59908	−	1.75468i
$676$	−26.0000			−1.00000
$677$		−	30.7246i		−	1.18084i	−0.807096	−	0.590421i	$-0.798962\pi$
$677$		−	30.7246i		−	1.18084i	0.807096	−	0.590421i	$-0.201038\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0			−	1.00000i	$-0.5\pi$
$683$		0			0				1.00000i	$0.5\pi$
$684$	−43.0152	−	26.6404i	−1.64473	−	1.01862i
$685$	96.0340			3.66927
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0		1.00000			$0$
$691$		0			0		−1.00000			$\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			20.5366i			0.778998i
$696$		0			0
$697$	−84.9472			−3.21760
$698$		0			0
$699$		0			0
$700$	−37.0979			−1.40217
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$	7.28309	−	25.5921i	0.273715	−	0.961811i
$709$	7.19384			0.270170			0.135085	−	0.990834i	$-0.456869\pi$
$709$	7.19384			0.270170			0.135085	+	0.990834i	$0.456869\pi$
$710$		0			0
$711$	9.55101	+	5.91517i	0.358191	+	0.221836i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−5.99292	+	21.0586i	−0.223810	+	0.786447i
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$	−25.9513	+	41.9026i	−0.967147	+	1.56162i
$721$		0			0
$722$		0			0
$723$	15.0802	+	4.29156i	0.560838	+	0.159605i
$724$	53.7219			1.99656
$725$		−	82.5193i		−	3.06469i
$726$		0			0
$727$	−28.0000			−1.03846			−0.519231	−	0.854634i	$-0.673782\pi$
$727$	−28.0000			−1.03846			−0.519231	+	0.854634i	$0.673782\pi$
$728$		0			0
$729$	−2.50000	+	26.8840i	−0.0925926	+	0.995704i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−49.0000			−1.80986			−0.904928	−	0.425564i	$-0.860076\pi$
$733$	−49.0000			−1.80986			−0.904928	+	0.425564i	$0.860076\pi$
$734$		0			0
$735$	8.87568	−	31.1883i	0.327384	−	1.15040i
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		1.00000			$0$
$739$		0			0		−1.00000			$\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	53.7680i		−	1.97256i	−0.165089	−	0.986279i	$-0.552791\pi$
$743$		−	53.7680i		−	1.97256i	0.165089	−	0.986279i	$-0.447209\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	15.3084i		−	0.559358i
$750$		0			0
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$		0			0
$753$	−14.9792	+	52.6356i	−0.545873	+	1.91815i
$754$		0			0
$755$		0			0
$756$	10.9386	+	12.0030i	0.397834	+	0.436546i
$757$	−51.5398			−1.87325			−0.936623	−	0.350338i	$-0.886067\pi$
$757$	−51.5398			−1.87325			−0.936623	+	0.350338i	$0.886067\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			17.3731i			0.629776i	0.949129	+	0.314888i	$0.101967\pi$
$761$			17.3731i			0.629776i	−0.949129	+	0.314888i	$0.898033\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	80.4650	+	49.8339i	2.90922	+	1.80175i
$766$		0			0
$767$		0			0
$768$	26.6545	+	7.58542i	0.961811	+	0.273715i
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$	2.09844	−	7.37375i	0.0755736	−	0.265559i
$772$	44.3460			1.59605
$773$		0			0			−	1.00000i	$-0.5\pi$
$773$		0			0				1.00000i	$0.5\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	93.2595i		−	3.34137i
$780$		0			0
$781$		0			0
$782$		0			0
$783$	−26.6991	+	24.3315i	−0.954148	+	0.869537i
$784$	−18.2324			−0.651156
$785$		0			0
$786$		0			0
$787$	32.0000			1.14068			0.570338	−	0.821410i	$-0.306812\pi$
$787$	32.0000			1.14068			0.570338	+	0.821410i	$0.306812\pi$
$788$			15.3623i			0.547259i
$789$	7.03906	−	24.7346i	0.250597	−	0.880576i
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$	36.7445	+	10.4569i	1.30319	+	0.370867i
$796$	32.4921			1.15165
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0			−	1.00000i	$-0.5\pi$
$809$		0			0				1.00000i	$0.5\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$			21.7268i			0.762461i
$813$	31.7316	+	9.03028i	1.11288	+	0.316706i
$814$		0			0
$815$			45.1806i			1.58261i
$816$	14.5662	−	51.1842i	0.509918	−	1.79181i
$817$		0			0
$818$		0			0
$819$		0			0
$820$	−90.8473			−3.17252
$821$		0			0			−	1.00000i	$-0.5\pi$
$821$		0			0				1.00000i	$0.5\pi$
$822$		0			0
$823$		0			0		1.00000			$0$
$823$		0			0		−1.00000			$\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	7.68115i		−	0.267100i	−0.991042	−	0.133550i	$-0.957362\pi$
$827$		−	7.68115i		−	0.267100i	0.991042	−	0.133550i	$-0.0426376\pi$
$828$		0			0
$829$	−42.7833			−1.48593			−0.742963	−	0.669332i	$-0.766580\pi$
$829$	−42.7833			−1.48593			−0.742963	+	0.669332i	$0.766580\pi$
$830$		0			0
$831$	29.6677	+	8.44292i	1.02916	+	0.292882i
$832$		0			0
$833$			35.0114i			1.21307i
$834$		0			0
$835$	72.6671			2.51475
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	−19.3283			−0.666493
$842$		0			0
$843$	−0.151209	+	0.531336i	−0.00520792	+	0.0183002i
$844$		0			0
$845$			53.3952i			1.83685i
$846$		0			0
$847$	17.1893			0.590630
$848$		−	21.4805i		−	0.737642i
$849$		0			0
$850$		0			0
$851$		0			0
$852$	−7.28309	+	25.5921i	−0.249514	+	0.876772i
$853$	53.1025			1.81819			0.909097	−	0.416584i	$-0.136773\pi$
$853$	53.1025			1.81819			0.909097	+	0.416584i	$0.136773\pi$
$854$		0			0
$855$	−54.7103	+	88.3387i	−1.87105	+	3.02112i
$856$		0			0
$857$		0			0			−	1.00000i	$-0.5\pi$
$857$		0			0				1.00000i	$0.5\pi$
$858$		0			0
$859$		0			0		1.00000			$0$
$859$		0			0		−1.00000			$\pi$
$860$		0			0
$861$	−8.19308	+	28.7897i	−0.279219	+	0.981152i
$862$		0			0
$863$		0			0			−	1.00000i	$-0.5\pi$
$863$		0			0				1.00000i	$0.5\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−69.9680	−	19.9117i	−2.37624	−	0.676238i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$			44.0948i			1.49068i
$876$		0			0
$877$	−25.6220			−0.865195			−0.432597	−	0.901587i	$-0.642403\pi$
$877$	−25.6220			−0.865195			−0.432597	+	0.901587i	$0.642403\pi$
$878$		0			0
$879$	−12.2821	+	43.1582i	−0.414265	+	1.45569i
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$		0			0
$883$	29.6626			0.998225			0.499112	−	0.866537i	$-0.333659\pi$
$883$	29.6626			0.998225			0.499112	+	0.866537i	$0.333659\pi$
$884$		0			0
$885$	−52.5576	−	14.9570i	−1.76670	−	0.502774i
$886$		0			0
$887$		0			0			−	1.00000i	$-0.5\pi$
$887$		0			0				1.00000i	$0.5\pi$
$888$		0			0
$889$	3.40992			0.114365
$890$		0			0
$891$		0			0
$892$	38.0000			1.27233
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	60.5490	+	37.4994i	2.01830	+	1.24998i
$901$	−41.2486			−1.37419
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	110.327i		−	3.66738i
$906$		0			0
$907$	47.7950			1.58701			0.793504	−	0.608565i	$-0.208255\pi$
$907$	47.7950			1.58701			0.793504	+	0.608565i	$0.208255\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			50.5507i			1.67482i	0.546576	+	0.837409i	$0.315931\pi$
$911$			50.5507i			1.67482i	−0.546576	+	0.837409i	$0.684069\pi$
$912$	56.1928	+	15.9915i	1.86073	+	0.529532i
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$	−53.5895	−	15.2507i	−1.76584	−	0.502527i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0			−	1.00000i	$-0.5\pi$
$929$		0			0				1.00000i	$0.5\pi$
$930$		0			0
$931$	−38.4373			−1.25973
$932$		0			0
$933$	−16.3278	+	57.3743i	−0.534547	+	1.87835i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0			−	1.00000i	$-0.5\pi$
$941$		0			0				1.00000i	$0.5\pi$
$942$		0			0
$943$		0			0
$944$			30.7246i			1.00000i
$945$	24.6502	−	22.4642i	0.801870	−	0.730762i
$946$		0			0
$947$		−	36.6470i		−	1.19087i	−0.803405	−	0.595433i	$-0.796980\pi$
$947$		−	36.6470i		−	1.19087i	0.803405	−	0.595433i	$-0.203020\pi$
$948$	−12.4769	−	3.55072i	−0.405232	−	0.115322i
$949$		0			0
$950$		0			0
$951$	14.5662	−	51.1842i	0.472341	−	1.65976i
$952$		0			0
$953$			61.4492i			1.99053i	0.0971795	+	0.995267i	$0.469018\pi$
$953$			61.4492i			1.99053i	−0.0971795	+	0.995267i	$0.530982\pi$
$954$		0			0
$955$		0			0
$956$		−	25.2818i		−	0.817673i
$957$		0			0
$958$		0			0
$959$			36.5368i			1.17983i
$960$	15.5779	−	54.7393i	0.502774	−	1.76670i
$961$	31.0000			1.00000
$962$		0			0
$963$	−15.4741	+	24.9855i	−0.498646	+	0.805146i
$964$	−18.1045			−0.583106
$965$		−	91.0716i		−	2.93170i
$966$		0			0
$967$		0			0		1.00000			$0$
$967$		0			0		−1.00000			$\pi$
$968$		0			0
$969$	30.7083	−	107.906i	0.986493	−	3.46645i
$970$		0			0
$971$			61.9288i			1.98739i	0.112117	+	0.993695i	$0.464237\pi$
$971$			61.9288i			1.98739i	−0.112117	+	0.993695i	$0.535763\pi$
$972$	−5.72043	−	30.6476i	−0.183483	−	0.983023i
$973$	−7.81330			−0.250483
$974$		0			0
$975$		0			0
$976$		0			0
$977$		0			0			−	1.00000i	$-0.5\pi$
$977$		0			0				1.00000i	$0.5\pi$
$978$		0			0
$979$		0			0
$980$			37.4431i			1.19608i
$981$		0			0
$982$		0			0
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$		0			0
$985$	31.5490			1.00523
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$		0			0
$993$	−29.1284	−	8.28944i	−0.924361	−	0.263058i
$994$		0			0
$995$		−	66.7279i		−	2.11542i
$996$		0			0
$997$	−28.0999			−0.889933			−0.444966	−	0.895547i	$-0.646784\pi$
$997$	−28.0999			−0.889933			−0.444966	+	0.895547i	$0.646784\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.d.b.176.6	yes	6
3.2	odd	2	inner	177.2.d.b.176.5	✓	6
59.58	odd	2	CM	177.2.d.b.176.6	yes	6
177.176	even	2	inner	177.2.d.b.176.5	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.d.b.176.5	✓	6	3.2	odd	2	inner
177.2.d.b.176.5	✓	6	177.176	even	2	inner
177.2.d.b.176.6	yes	6	1.1	even	1	trivial
177.2.d.b.176.6	yes	6	59.58	odd	2	CM

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:	\( N \)	\(=\)	\( 177 = 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	177.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(1.66591 + 0.474089i) q^{3} -2.00000 q^{4} +4.10732i q^{5} -1.56266 q^{7} +(2.55048 + 1.57957i) q^{9} +O(q^{10})\)	\(q+(1.66591 + 0.474089i) q^{3} -2.00000 q^{4} +4.10732i q^{5} -1.56266 q^{7} +(2.55048 + 1.57957i) q^{9} +(-3.33181 - 0.948177i) q^{12} +(-1.94724 + 6.84241i) q^{15} +4.00000 q^{16} -7.68115i q^{17} +8.43277 q^{19} -8.21465i q^{20} +(-2.60324 - 0.740840i) q^{21} -11.8701 q^{25} +(3.50000 + 3.84057i) q^{27} +3.12532 q^{28} +6.95186i q^{29} -6.41835i q^{35} +(-5.10096 - 3.15915i) q^{36} -11.0592i q^{41} +(-6.48782 + 10.4756i) q^{45} +(6.66362 + 1.89635i) q^{48} -4.55809 q^{49} +(3.64154 - 12.7961i) q^{51} -5.37012i q^{53} +(14.0482 + 3.99788i) q^{57} +7.68115i q^{59} +(3.89447 - 13.6848i) q^{60} +(-3.98553 - 2.46834i) q^{63} -8.00000 q^{64} +15.3623i q^{68} -7.68115i q^{71} +(-19.7745 - 5.62748i) q^{75} -16.8655 q^{76} +3.74479 q^{79} +16.4293i q^{80} +(4.00990 + 8.05734i) q^{81} +(5.20649 + 1.48168i) q^{84} +31.5490 q^{85} +(-3.29580 + 11.5811i) q^{87} +34.6361i q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 12 q^{4}+O(q^{10}) \) 6 * q - 12 * q^4	\( 6 q - 12 q^{4} + 3 q^{15} + 24 q^{16} - 15 q^{21} - 30 q^{25} + 21 q^{27} - 33 q^{45} + 42 q^{49} + 39 q^{57} - 6 q^{60} + 12 q^{63} - 48 q^{64} - 24 q^{75} + 30 q^{84} - 51 q^{87}+O(q^{100}) \) 6 * q - 12 * q^4 + 3 * q^15 + 24 * q^16 - 15 * q^21 - 30 * q^25 + 21 * q^27 - 33 * q^45 + 42 * q^49 + 39 * q^57 - 6 * q^60 + 12 * q^63 - 48 * q^64 - 24 * q^75 + 30 * q^84 - 51 * q^87

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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