LMFDB - Embedded newform 177.2.d.b.176.4

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			176.4
Root			$-0.422380 + 1.67976i$ of defining polynomial
Character	$\chi$	$=$	177.176
Dual form			177.2.d.b.176.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.422380 + 1.67976i) q^{3} -2.00000 q^{4} +0.521533i q^{5} -3.59686 q^{7} +(-2.64319 - 1.41899i) q^{9} +O(q^{10})$	$q+(-0.422380 + 1.67976i) q^{3} -2.00000 q^{4} +0.521533i q^{5} -3.59686 q^{7} +(-2.64319 - 1.41899i) q^{9} +(0.844760 - 3.35952i) q^{12} +(-0.876051 - 0.220285i) q^{15} +4.00000 q^{16} +7.68115i q^{17} -6.13114 q^{19} -1.04307i q^{20} +(1.51924 - 6.04187i) q^{21} +4.72800 q^{25} +(3.50000 - 3.84057i) q^{27} +7.19372 q^{28} +10.6001i q^{29} -1.87588i q^{35} +(5.28638 + 2.83799i) q^{36} -11.1216i q^{41} +(0.740053 - 1.37851i) q^{45} +(-1.68952 + 6.71904i) q^{48} +5.93742 q^{49} +(-12.9025 - 3.24436i) q^{51} +9.03550i q^{53} +(2.58967 - 10.2988i) q^{57} -7.68115i q^{59} +(1.75210 + 0.440570i) q^{60} +(9.50719 + 5.10392i) q^{63} -8.00000 q^{64} -15.3623i q^{68} +7.68115i q^{71} +(-1.99701 + 7.94191i) q^{75} +12.2623 q^{76} -16.9217 q^{79} +2.08613i q^{80} +(4.97291 + 7.50134i) q^{81} +(-3.03848 + 12.0837i) q^{84} -4.00597 q^{85} +(-17.8056 - 4.47727i) q^{87} -3.19759i q^{95} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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	$0.5\pi$
$3$	−0.422380	+	1.67976i	−0.243861	+	0.969810i
$3$	−0.422380	+	1.67976i	−0.243861	+	0.969810i
$4$	−2.00000			−1.00000
$5$			0.521533i			0.233237i	0.993177	+	0.116618i	$0.0372055\pi$
$5$			0.521533i			0.233237i	−0.993177	+	0.116618i	$0.962795\pi$
$6$		0			0
$7$	−3.59686			−1.35949			−0.679743	−	0.733450i	$-0.737909\pi$
$7$	−3.59686			−1.35949			−0.679743	+	0.733450i	$0.737909\pi$
$8$		0			0
$9$	−2.64319	−	1.41899i	−0.881064	−	0.472998i
$10$		0			0
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$	0.844760	−	3.35952i	0.243861	−	0.969810i
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$		0			0
$15$	−0.876051	−	0.220285i	−0.226195	−	0.0568774i
$16$	4.00000			1.00000
$17$			7.68115i			1.86295i	0.363803	+	0.931476i	$0.381478\pi$
$17$			7.68115i			1.86295i	−0.363803	+	0.931476i	$0.618522\pi$
$18$		0			0
$19$	−6.13114			−1.40658			−0.703290	−	0.710903i	$-0.748286\pi$
$19$	−6.13114			−1.40658			−0.703290	+	0.710903i	$0.748286\pi$
$20$		−	1.04307i		−	0.233237i
$21$	1.51924	−	6.04187i	0.331526	−	1.31844i
$22$		0			0
$23$		0			0			−	1.00000i	$-0.5\pi$
$23$		0			0				1.00000i	$0.5\pi$
$24$		0			0
$25$	4.72800			0.945601
$26$		0			0
$27$	3.50000	−	3.84057i	0.673575	−	0.739119i
$28$	7.19372			1.35949
$29$			10.6001i			1.96839i	0.177092	+	0.984194i	$0.443331\pi$
$29$			10.6001i			1.96839i	−0.177092	+	0.984194i	$0.556669\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		−	1.87588i		−	0.317082i
$36$	5.28638	+	2.83799i	0.881064	+	0.472998i
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	11.1216i		−	1.73691i	−0.495771	−	0.868453i	$-0.665114\pi$
$41$		−	11.1216i		−	1.73691i	0.495771	−	0.868453i	$-0.334886\pi$
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$		0			0
$45$	0.740053	−	1.37851i	0.110321	−	0.205496i
$46$		0			0
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$	−1.68952	+	6.71904i	−0.243861	+	0.969810i
$49$	5.93742			0.848202
$50$		0			0
$51$	−12.9025	−	3.24436i	−1.80671	−	0.454301i
$52$		0			0
$53$			9.03550i			1.24112i	0.784159	+	0.620560i	$0.213095\pi$
$53$			9.03550i			1.24112i	−0.784159	+	0.620560i	$0.786905\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	2.58967	−	10.2988i	0.343010	−	1.36412i
$58$		0			0
$59$		−	7.68115i		−	1.00000i
$59$		−	7.68115i		−	1.00000i
$60$	1.75210	+	0.440570i	0.226195	+	0.0568774i
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$		0			0
$63$	9.50719	+	5.10392i	1.19779	+	0.643034i
$64$	−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	$0.150644\pi$
$71$			7.68115i			0.911584i	−0.890086	+	0.455792i	$0.849356\pi$
$72$		0			0
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$		0			0
$75$	−1.99701	+	7.94191i	−0.230595	+	0.917053i
$76$	12.2623			1.40658
$77$		0			0
$78$		0			0
$79$	−16.9217			−1.90384			−0.951922	−	0.306342i	$-0.900895\pi$
$79$	−16.9217			−1.90384			−0.951922	+	0.306342i	$0.900895\pi$
$80$			2.08613i			0.233237i
$81$	4.97291	+	7.50134i	0.552546	+	0.833482i
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$	−3.03848	+	12.0837i	−0.331526	+	1.31844i
$85$	−4.00597			−0.434509
$86$		0			0
$87$	−17.8056	−	4.47727i	−1.90896	−	0.480013i
$88$		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
$95$		−	3.19759i		−	0.328066i
$96$		0			0
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$		0			0
$99$		0			0
$100$	−9.45601			−0.945601
$101$		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
$105$	3.15104	+	0.792335i	0.307510	+	0.0773240i
$106$		0			0
$107$			20.6787i			1.99908i	0.0303041	+	0.999541i	$0.490352\pi$
$107$			20.6787i			1.99908i	−0.0303041	+	0.999541i	$0.509648\pi$
$108$	−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
$112$	−14.3874			−1.35949
$113$		0			0			−	1.00000i	$-0.5\pi$
$113$		0			0				1.00000i	$0.5\pi$
$114$		0			0
$115$		0			0
$116$		−	21.2002i		−	1.96839i
$117$		0			0
$118$		0			0
$119$		−	27.6280i		−	2.53266i
$120$		0			0
$121$	−11.0000			−1.00000
$122$		0			0
$123$	18.6817	+	4.69755i	1.68447	+	0.423564i
$124$		0			0
$125$			5.07348i			0.453786i
$126$		0			0
$127$	20.5186			1.82073			0.910365	−	0.413806i	$-0.135801\pi$
$127$	20.5186			1.82073			0.910365	+	0.413806i	$0.135801\pi$
$128$		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
$133$	22.0529			1.91223
$134$		0			0
$135$	2.00299	+	1.82537i	0.172390	+	0.157103i
$136$		0			0
$137$		−	12.6862i		−	1.08386i	−0.840424	−	0.541929i	$-0.817694\pi$
$137$		−	12.6862i		−	1.08386i	0.840424	−	0.541929i	$-0.182306\pi$
$138$		0			0
$139$	5.00000			0.424094			0.212047	−	0.977259i	$-0.431987\pi$
$139$	5.00000			0.424094			0.212047	+	0.977259i	$0.431987\pi$
$140$			3.75177i			0.317082i
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−10.5728	−	5.67598i	−0.881064	−	0.472998i
$145$	−5.52830			−0.459101
$146$		0			0
$147$	−2.50784	+	9.97344i	−0.206844	+	0.822595i
$148$		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
$153$	10.8995	−	20.3027i	0.881172	−	1.64138i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		1.00000			$0$
$157$		0			0		−1.00000			$\pi$
$158$		0			0
$159$	−15.1775	−	3.81641i	−1.20365	−	0.302661i
$160$		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	$0.358234\pi$
$164$			22.2433i			1.73691i
$165$		0			0
$166$		0			0
$167$			7.47090i			0.578115i	0.957312	+	0.289058i	$0.0933419\pi$
$167$			7.47090i			0.578115i	−0.957312	+	0.289058i	$0.906658\pi$
$168$		0			0
$169$	13.0000			1.00000
$170$		0			0
$171$	16.2058	+	8.70005i	1.23929	+	0.665310i
$172$		0			0
$173$		0			0			−	1.00000i	$-0.5\pi$
$173$		0			0				1.00000i	$0.5\pi$
$174$		0			0
$175$	−17.0060			−1.28553
$176$		0			0
$177$	12.9025	+	3.24436i	0.969810	+	0.243861i
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$	−1.48011	+	2.75702i	−0.110321	+	0.205496i
$181$	14.7966			1.09982			0.549910	−	0.835224i	$-0.314662\pi$
$181$	14.7966			1.09982			0.549910	+	0.835224i	$0.314662\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−12.5890	+	13.8140i	−0.915716	+	1.00482i
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$	3.37904	−	13.4381i	0.243861	−	0.969810i
$193$	25.5871			1.84180			0.920902	−	0.389795i	$-0.127454\pi$
$193$	25.5871			1.84180			0.920902	+	0.389795i	$0.127454\pi$
$194$		0			0
$195$		0			0
$196$	−11.8748			−0.848202
$197$			7.68115i			0.547259i	0.961835	+	0.273629i	$0.0882242\pi$
$197$			7.68115i			0.547259i	−0.961835	+	0.273629i	$0.911776\pi$
$198$		0			0
$199$	−11.8532			−0.840249			−0.420124	−	0.907466i	$-0.638014\pi$
$199$	−11.8532			−0.840249			−0.420124	+	0.907466i	$0.638014\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	38.1271i		−	2.67600i
$204$	25.8050	+	6.48872i	1.80671	+	0.454301i
$205$	5.80030			0.405111
$206$		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			$\pi$
$212$		−	18.0710i		−	1.24112i
$213$	−12.9025	−	3.24436i	−0.884064	−	0.222300i
$214$		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
$225$	−12.4970	−	6.70901i	−0.833134	−	0.447267i
$226$		0			0
$227$		0			0			−	1.00000i	$-0.5\pi$
$227$		0			0				1.00000i	$0.5\pi$
$228$	−5.17934	+	20.5977i	−0.343010	+	1.36412i
$229$		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
$237$	7.14740	−	28.4244i	0.464273	−	1.84637i
$238$		0			0
$239$			30.7572i			1.98952i	0.102241	+	0.994760i	$0.467399\pi$
$239$			30.7572i			1.98952i	−0.102241	+	0.994760i	$0.532601\pi$
$240$	−3.50420	−	0.881141i	−0.226195	−	0.0568774i
$241$	−30.2466			−1.94835			−0.974177	−	0.225785i	$-0.927505\pi$
$241$	−30.2466			−1.94835			−0.974177	+	0.225785i	$0.927505\pi$
$242$		0			0
$243$	−14.7009	+	5.18489i	−0.943064	+	0.332611i
$244$		0			0
$245$			3.09656i			0.197832i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$			13.7293i			0.866586i	0.901253	+	0.433293i	$0.142648\pi$
$251$			13.7293i			0.866586i	−0.901253	+	0.433293i	$0.857352\pi$
$252$	−19.0144	−	10.2078i	−1.19779	−	0.643034i
$253$		0			0
$254$		0			0
$255$	1.69204	−	6.72908i	0.105960	−	0.421391i
$256$	16.0000			1.00000
$257$		−	29.7142i		−	1.85352i	−0.375656	−	0.926759i	$-0.622583\pi$
$257$		−	29.7142i		−	1.85352i	0.375656	−	0.926759i	$-0.377417\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	15.0415	−	28.0181i	0.931044	−	1.73428i
$262$		0			0
$263$			17.5495i			1.08215i	0.840976	+	0.541073i	$0.181982\pi$
$263$			17.5495i			1.08215i	−0.840976	+	0.541073i	$0.818018\pi$
$264$		0			0
$265$	−4.71231			−0.289475
$266$		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
$271$	−32.7809			−1.99130			−0.995648	−	0.0931932i	$-0.970293\pi$
$271$	−32.7809			−1.99130			−0.995648	+	0.0931932i	$0.970293\pi$
$272$			30.7246i			1.86295i
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	15.4500			0.928302			0.464151	−	0.885756i	$-0.346359\pi$
$277$	15.4500			0.928302			0.464151	+	0.885756i	$0.346359\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			29.1926i			1.74149i	0.491739	+	0.870743i	$0.336362\pi$
$281$			29.1926i			1.74149i	−0.491739	+	0.870743i	$0.663638\pi$
$282$		0			0
$283$		0			0		1.00000			$0$
$283$		0			0		−1.00000			$\pi$
$284$		−	15.3623i		−	0.911584i
$285$	5.37119	+	1.35060i	0.318162	+	0.0800026i
$286$		0			0
$287$			40.0030i			2.36130i
$288$		0			0
$289$	−42.0000			−2.47059
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	6.42783i		−	0.375518i	−0.982215	−	0.187759i	$-0.939878\pi$
$293$		−	6.42783i		−	0.375518i	0.982215	−	0.187759i	$-0.0601224\pi$
$294$		0			0
$295$	4.00597			0.233237
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	3.99403	−	15.8838i	0.230595	−	0.917053i
$301$		0			0
$302$		0			0
$303$		0			0
$304$	−24.5246			−1.40658
$305$		0			0
$306$		0			0
$307$	28.1214			1.60497			0.802487	−	0.596669i	$-0.203510\pi$
$307$	28.1214			1.60497			0.802487	+	0.596669i	$0.203510\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			23.8079i			1.35002i	0.737809	+	0.675010i	$0.235861\pi$
$311$			23.8079i			1.35002i	−0.737809	+	0.675010i	$0.764139\pi$
$312$		0			0
$313$		0			0		1.00000			$0$
$313$		0			0		−1.00000			$\pi$
$314$		0			0
$315$	−2.66187	+	4.95832i	−0.149979	+	0.279370i
$316$	33.8435			1.90384
$317$			30.7246i			1.72566i	0.505490	+	0.862832i	$0.331312\pi$
$317$			30.7246i			1.72566i	−0.505490	+	0.862832i	$0.668688\pi$
$318$		0			0
$319$		0			0
$320$		−	4.17227i		−	0.233237i
$321$	−34.7352	−	8.73425i	−1.93873	−	0.487498i
$322$		0			0
$323$		−	47.0942i		−	2.62039i
$324$	−9.94583	−	15.0027i	−0.552546	−	0.833482i
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	36.3777			1.99950			0.999751	−	0.0223363i	$-0.00711045\pi$
$331$	36.3777			1.99950			0.999751	+	0.0223363i	$0.00711045\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$	6.07697	−	24.1675i	0.331526	−	1.31844i
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$		0			0
$339$		0			0
$340$	8.01195			0.434509
$341$		0			0
$342$		0			0
$343$	3.82197			0.206367
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0			−	1.00000i	$-0.5\pi$
$347$		0			0				1.00000i	$0.5\pi$
$348$	35.6112	+	8.95453i	1.90896	+	0.480013i
$349$		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
$355$	−4.00597			−0.212615
$356$		0			0
$357$	46.4085	+	11.6695i	2.45620	+	0.617616i
$358$		0			0
$359$		−	28.1496i		−	1.48568i	−0.669471	−	0.742838i	$-0.733479\pi$
$359$		−	28.1496i		−	1.48568i	0.669471	−	0.742838i	$-0.266521\pi$
$360$		0			0
$361$	18.5909			0.978468
$362$		0			0
$363$	4.64618	−	18.4774i	0.243861	−	0.969810i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		1.00000			$0$
$367$		0			0		−1.00000			$\pi$
$368$		0			0
$369$	−15.7815	+	29.3966i	−0.821553	+	1.53033i
$370$		0			0
$371$		−	32.4994i		−	1.68729i
$372$		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
$375$	−8.52223	−	2.14293i	−0.440086	−	0.110661i
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−13.7340			−0.705467			−0.352734	−	0.935724i	$-0.614748\pi$
$379$	−13.7340			−0.705467			−0.352734	+	0.935724i	$0.614748\pi$
$380$			6.39519i			0.328066i
$381$	−8.66664	+	34.4663i	−0.444005	+	1.76576i
$382$		0			0
$383$		−	15.3623i		−	0.784976i	−0.919757	−	0.392488i	$-0.871614\pi$
$383$		−	15.3623i		−	0.784976i	0.919757	−	0.392488i	$-0.128386\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	38.4057i		−	1.94725i	−0.228159	−	0.973624i	$-0.573271\pi$
$389$		−	38.4057i		−	1.94725i	0.228159	−	0.973624i	$-0.426729\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	8.82525i		−	0.444046i
$396$		0			0
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$		0			0
$399$	−9.31469	+	37.0435i	−0.466318	+	1.85450i
$400$	18.9120			0.945601
$401$		0			0			−	1.00000i	$-0.5\pi$
$401$		0			0				1.00000i	$0.5\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−3.91220	+	2.59354i	−0.194399	+	0.128874i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$		0			0
$411$	21.3098	+	5.35841i	1.05114	+	0.264311i
$412$		0			0
$413$			27.6280i			1.35949i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−2.11190	+	8.39880i	−0.103420	+	0.411291i
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$	−6.30207	−	1.58467i	−0.307510	−	0.0773240i
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			36.3165i			1.76161i
$426$		0			0
$427$		0			0
$428$		−	41.3573i		−	1.99908i
$429$		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
$433$	12.9158			0.620691			0.310346	−	0.950624i	$-0.399555\pi$
$433$	12.9158			0.620691			0.310346	+	0.950624i	$0.399555\pi$
$434$		0			0
$435$	2.33504	−	9.28623i	0.111957	−	0.445241i
$436$		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
$441$	−15.6937	−	8.42516i	−0.747320	−	0.401198i
$442$		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
$448$	28.7749			1.35949
$449$		−	39.7927i		−	1.87793i	−0.344007	−	0.938967i	$-0.611784\pi$
$449$		−	39.7927i		−	1.87793i	0.344007	−	0.938967i	$-0.388216\pi$
$450$		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	$0.253798\pi$
$461$			30.7246i			1.43099i	−0.698620	+	0.715493i	$0.746202\pi$
$462$		0			0
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$			42.4004i			1.96839i
$465$		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
$475$	−28.9881			−1.33006
$476$			55.2560i			2.53266i
$477$	12.8213	−	23.8825i	0.587048	−	1.09351i
$478$		0			0
$479$		−	38.4057i		−	1.75480i	−0.479757	−	0.877401i	$-0.659275\pi$
$479$		−	38.4057i		−	1.75480i	0.479757	−	0.877401i	$-0.340725\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	22.0000			1.00000
$485$		0			0
$486$		0			0
$487$	−27.0588			−1.22615			−0.613077	−	0.790023i	$-0.710068\pi$
$487$	−27.0588			−1.22615			−0.613077	+	0.790023i	$0.710068\pi$
$488$		0			0
$489$	−4.64618	+	18.4774i	−0.210107	+	0.835575i
$490$		0			0
$491$		−	34.4080i		−	1.55281i	−0.630235	−	0.776405i	$-0.717041\pi$
$491$		−	34.4080i		−	1.55281i	0.630235	−	0.776405i	$-0.282959\pi$
$492$	−37.3633	−	9.39510i	−1.68447	−	0.423564i
$493$	−81.4209			−3.66701
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	27.6280i		−	1.23929i
$498$		0			0
$499$	6.54025			0.292782			0.146391	−	0.989227i	$-0.453234\pi$
$499$	6.54025			0.292782			0.146391	+	0.989227i	$0.453234\pi$
$500$		−	10.1470i		−	0.453786i
$501$	−12.5493	−	3.15556i	−0.560662	−	0.140980i
$502$		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
$507$	−5.49094	+	21.8369i	−0.243861	+	0.969810i
$508$	−41.0372			−1.82073
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	−21.4590	+	23.5471i	−0.947438	+	1.03963i
$514$		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	$0.0538144\pi$
$521$			7.68115i			0.336517i	−0.985743	+	0.168259i	$0.946186\pi$
$522$		0			0
$523$	41.4463			1.81232			0.906160	−	0.422935i	$-0.139000\pi$
$523$	41.4463			1.81232			0.906160	+	0.422935i	$0.139000\pi$
$524$		0			0
$525$	7.18298	−	28.5660i	0.313491	−	1.24672i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−23.0000			−1.00000
$530$		0			0
$531$	−10.8995	+	20.3027i	−0.472998	+	0.881064i
$532$	−44.1057			−1.91223
$533$		0			0
$534$		0			0
$535$	−10.7846			−0.466259
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$	−4.00597	−	3.65073i	−0.172390	−	0.157103i
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$	−6.24977	+	24.8547i	−0.268203	+	1.06662i
$544$		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	$0.445291\pi$
$548$			25.3725i			1.08386i
$549$		0			0
$550$		0			0
$551$		−	64.9907i		−	2.76870i
$552$		0			0
$553$	60.8651			2.58825
$554$		0			0
$555$		0			0
$556$	−10.0000			−0.424094
$557$		−	4.86323i		−	0.206062i	−0.994678	−	0.103031i	$-0.967146\pi$
$557$		−	4.86323i		−	0.206062i	0.994678	−	0.103031i	$-0.0328540\pi$
$558$		0			0
$559$		0			0
$560$		−	7.50354i		−	0.317082i
$561$		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
$567$	−17.8869	−	26.9813i	−0.751178	−	1.13311i
$568$		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
$576$	21.1455	+	11.3520i	0.881064	+	0.472998i
$577$	−38.5029			−1.60290			−0.801448	−	0.598064i	$-0.795937\pi$
$577$	−38.5029			−1.60290			−0.801448	+	0.598064i	$0.795937\pi$
$578$		0			0
$579$	−10.8075	+	42.9803i	−0.449144	+	1.78620i
$580$	11.0566			0.459101
$581$		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	$0.5\pi$
$588$	5.01569	−	19.9469i	0.206844	−	0.822595i
$589$		0			0
$590$		0			0
$591$	−12.9025	−	3.24436i	−0.530737	−	0.133455i
$592$		0			0
$593$			43.9650i			1.80543i	0.430244	+	0.902713i	$0.358428\pi$
$593$			43.9650i			1.80543i	−0.430244	+	0.902713i	$0.641572\pi$
$594$		0			0
$595$	14.4089			0.590709
$596$		0			0
$597$	5.00654	−	19.9105i	0.204904	−	0.814882i
$598$		0			0
$599$		−	15.8154i		−	0.646201i	−0.946365	−	0.323100i	$-0.895275\pi$
$599$		−	15.8154i		−	0.646201i	0.946365	−	0.323100i	$-0.104725\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		−	5.73687i		−	0.233237i
$606$		0			0
$607$	−16.2683			−0.660308			−0.330154	−	0.943927i	$-0.607101\pi$
$607$	−16.2683			−0.660308			−0.330154	+	0.943927i	$0.607101\pi$
$608$		0			0
$609$	64.0444	+	16.1041i	2.59521	+	0.652571i
$610$		0			0
$611$		0			0
$612$	−21.7990	+	40.6055i	−0.881172	+	1.64138i
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$		0			0
$615$	−2.44993	+	9.74312i	−0.0987907	+	0.392880i
$616$		0			0
$617$			37.7066i			1.51801i	0.651085	+	0.759005i	$0.274314\pi$
$617$			37.7066i			1.51801i	−0.651085	+	0.759005i	$0.725686\pi$
$618$		0			0
$619$	49.7026			1.99772			0.998858	−	0.0477768i	$-0.0152136\pi$
$619$	49.7026			1.99772			0.998858	+	0.0477768i	$0.0152136\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	20.9940			0.839761
$626$		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
$635$			10.7011i			0.424661i
$636$	30.3549	+	7.63282i	1.20365	+	0.302661i
$637$		0			0
$638$		0			0
$639$	10.8995	−	20.3027i	0.431177	−	0.803164i
$640$		0			0
$641$		−	38.4057i		−	1.51694i	−0.651711	−	0.758468i	$-0.725948\pi$
$641$		−	38.4057i		−	1.51694i	0.651711	−	0.758468i	$-0.274052\pi$
$642$		0			0
$643$	10.3815			0.409405			0.204703	−	0.978824i	$-0.434377\pi$
$643$	10.3815			0.409405			0.204703	+	0.978824i	$0.434377\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	25.8940i		−	1.01800i	−0.860767	−	0.508999i	$-0.830016\pi$
$647$		−	25.8940i		−	1.01800i	0.860767	−	0.508999i	$-0.169984\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$	−22.0000			−0.861586
$653$			35.4510i			1.38731i	0.720309	+	0.693653i	$0.244000\pi$
$653$			35.4510i			1.38731i	−0.720309	+	0.693653i	$0.756000\pi$
$654$		0			0
$655$		0			0
$656$		−	44.4865i		−	1.73691i
$657$		0			0
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$	−20.1095			−0.782168			−0.391084	−	0.920355i	$-0.627900\pi$
$661$	−20.1095			−0.782168			−0.391084	+	0.920355i	$0.627900\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			11.5013i			0.446002i
$666$		0			0
$667$		0			0
$668$		−	14.9418i		−	0.578115i
$669$	8.02522	−	31.9154i	0.310273	−	1.23392i
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$		0			0
$675$	16.5480	−	18.1582i	0.636933	−	0.698911i
$676$	−26.0000			−1.00000
$677$			30.7246i			1.18084i	0.807096	+	0.590421i	$0.201038\pi$
$677$			30.7246i			1.18084i	−0.807096	+	0.590421i	$0.798962\pi$
$678$		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	$0.5\pi$
$684$	−32.4115	−	17.4001i	−1.23929	−	0.665310i
$685$	6.61629			0.252796
$686$		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
$695$			2.60767i			0.0989145i
$696$		0			0
$697$	85.4269			3.23577
$698$		0			0
$699$		0			0
$700$	34.0119			1.28553
$701$		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
$708$	−25.8050	−	6.48872i	−0.969810	−	0.243861i
$709$	42.0998			1.58109			0.790545	−	0.612404i	$-0.209797\pi$
$709$	42.0998			1.58109			0.790545	+	0.612404i	$0.209797\pi$
$710$		0			0
$711$	44.7274	+	24.0118i	1.67741	+	0.900514i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−51.6648	−	12.9912i	−1.92946	−	0.485166i
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$	2.96021	−	5.51405i	0.110321	−	0.205496i
$721$		0			0
$722$		0			0
$723$	12.7755	−	50.8070i	0.475128	−	1.88953i
$724$	−29.5931			−1.09982
$725$			50.1173i			1.86131i
$726$		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
$735$	−5.20148	−	1.30792i	−0.191860	−	0.0482435i
$736$		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	$0.447209\pi$
$743$			53.7680i			1.97256i	−0.165089	+	0.986279i	$0.552791\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	74.3783i		−	2.71772i
$750$		0			0
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$		0			0
$753$	−23.0619	−	5.79898i	−0.840423	−	0.211326i
$754$		0			0
$755$		0			0
$756$	25.1780	−	27.6280i	0.915716	−	1.00482i
$757$	9.07453			0.329819			0.164910	−	0.986309i	$-0.447267\pi$
$757$	9.07453			0.329819			0.164910	+	0.986309i	$0.447267\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	36.6635i		−	1.32905i	−0.747265	−	0.664526i	$-0.768634\pi$
$761$		−	36.6635i		−	1.32905i	0.747265	−	0.664526i	$-0.231366\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	10.5886	+	5.68445i	0.382830	+	0.205522i
$766$		0			0
$767$		0			0
$768$	−6.75808	+	26.8762i	−0.243861	+	0.969810i
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$	49.9127	+	12.5507i	1.79756	+	0.452001i
$772$	−51.1743			−1.84180
$773$		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
$779$			68.1883i			2.44310i
$780$		0			0
$781$		0			0
$782$		0			0
$783$	40.7104	+	37.1003i	1.45487	+	1.32586i
$784$	23.7497			0.848202
$785$		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
$789$	−29.4789	−	7.41254i	−1.04948	−	0.263893i
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$	1.99039	−	7.91556i	0.0705917	−	0.280736i
$796$	23.7063			0.840249
$797$		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			$\pi$
$812$			76.2542i			2.67600i
$813$	13.8460	−	55.0640i	0.485600	−	1.93118i
$814$		0			0
$815$			5.73687i			0.200954i
$816$	−51.6099	−	12.9774i	−1.80671	−	0.454301i
$817$		0			0
$818$		0			0
$819$		0			0
$820$	−11.6006			−0.405111
$821$		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	$0.0426376\pi$
$827$			7.68115i			0.267100i	−0.991042	+	0.133550i	$0.957362\pi$
$828$		0			0
$829$	54.7712			1.90228			0.951140	−	0.308759i	$-0.0999135\pi$
$829$	54.7712			1.90228			0.951140	+	0.308759i	$0.0999135\pi$
$830$		0			0
$831$	−6.52578	+	25.9524i	−0.226377	+	0.900277i
$832$		0			0
$833$			45.6062i			1.58016i
$834$		0			0
$835$	−3.89632			−0.134838
$836$		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
$841$	−83.3620			−2.87455
$842$		0			0
$843$	−49.0366	−	12.3304i	−1.68891	−	0.424681i
$844$		0			0
$845$			6.77993i			0.233237i
$846$		0			0
$847$	39.5655			1.35949
$848$			36.1420i			1.24112i
$849$		0			0
$850$		0			0
$851$		0			0
$852$	25.8050	+	6.48872i	0.884064	+	0.222300i
$853$	−5.47767			−0.187552			−0.0937759	−	0.995593i	$-0.529894\pi$
$853$	−5.47767			−0.187552			−0.0937759	+	0.995593i	$0.529894\pi$
$854$		0			0
$855$	−4.53737	+	8.45185i	−0.155175	+	0.289047i
$856$		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
$861$	−67.1954	−	16.8964i	−2.29001	−	0.575829i
$862$		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
$867$	17.7400	−	70.5499i	0.602480	−	2.39600i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		−	18.2486i		−	0.616915i
$876$		0			0
$877$	−33.4343			−1.12900			−0.564499	−	0.825434i	$-0.690931\pi$
$877$	−33.4343			−1.12900			−0.564499	+	0.825434i	$0.690931\pi$
$878$		0			0
$879$	10.7972	+	2.71499i	0.364181	+	0.0915742i
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$		0			0
$883$	−59.4306			−2.00000			−0.999999	−	0.00102460i	$-0.999674\pi$
$883$	−59.4306			−2.00000			−0.999999	+	0.00102460i	$0.999674\pi$
$884$		0			0
$885$	−1.69204	+	6.72908i	−0.0568774	+	0.226195i
$886$		0			0
$887$		0			0			−	1.00000i	$-0.5\pi$
$887$		0			0				1.00000i	$0.5\pi$
$888$		0			0
$889$	−73.8025			−2.47526
$890$		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
$900$	24.9940	+	13.4180i	0.833134	+	0.447267i
$901$	−69.4030			−2.31215
$902$		0			0
$903$		0			0
$904$		0			0
$905$			7.71690i			0.256518i
$906$		0			0
$907$	7.84720			0.260562			0.130281	−	0.991477i	$-0.458412\pi$
$907$	7.84720			0.260562			0.130281	+	0.991477i	$0.458412\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	3.29863i		−	0.109288i	−0.998506	−	0.0546442i	$-0.982598\pi$
$911$		−	3.29863i		−	0.109288i	0.998506	−	0.0546442i	$-0.0174025\pi$
$912$	10.3587	−	41.1954i	0.343010	−	1.36412i
$913$		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
$921$	−11.8779	+	47.2373i	−0.391391	+	1.55652i
$922$		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
$931$	−36.4031			−1.19306
$932$		0			0
$933$	−39.9915	−	10.0560i	−1.30926	−	0.329217i
$934$		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
$945$	−7.20447	−	6.56559i	−0.234361	−	0.213579i
$946$		0			0
$947$			24.4988i			0.796105i	0.917363	+	0.398052i	$0.130314\pi$
$947$			24.4988i			0.796105i	−0.917363	+	0.398052i	$0.869686\pi$
$948$	−14.2948	+	56.8489i	−0.464273	+	1.84637i
$949$		0			0
$950$		0			0
$951$	−51.6099	−	12.9774i	−1.67357	−	0.420822i
$952$		0			0
$953$		−	61.4492i		−	1.99053i	−0.0971795	−	0.995267i	$-0.530982\pi$
$953$		−	61.4492i		−	1.99053i	0.0971795	−	0.995267i	$-0.469018\pi$
$954$		0			0
$955$		0			0
$956$		−	61.5144i		−	1.98952i
$957$		0			0
$958$		0			0
$959$			45.6306i			1.47349i
$960$	7.00841	+	1.76228i	0.226195	+	0.0568774i
$961$	31.0000			1.00000
$962$		0			0
$963$	29.3429	−	54.6576i	0.945561	−	1.76132i
$964$	60.4932			1.94835
$965$			13.3446i			0.429576i
$966$		0			0
$967$		0			0		1.00000			$0$
$967$		0			0		−1.00000			$\pi$
$968$		0			0
$969$	79.1070	+	19.8916i	2.54128	+	0.639011i
$970$		0			0
$971$			37.0156i			1.18789i	0.804506	+	0.593944i	$0.202430\pi$
$971$			37.0156i			1.18789i	−0.804506	+	0.593944i	$0.797570\pi$
$972$	29.4018	−	10.3698i	0.943064	−	0.332611i
$973$	−17.9843			−0.576551
$974$		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
$980$		−	6.19312i		−	0.197832i
$981$		0			0
$982$		0			0
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$		0			0
$985$	−4.00597			−0.127641
$986$		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
$993$	−15.3652	+	61.1059i	−0.487600	+	1.93914i
$994$		0			0
$995$		−	6.18182i		−	0.195977i
$996$		0			0
$997$	63.0275			1.99610			0.998050	−	0.0624214i	$-0.0198823\pi$
$997$	63.0275			1.99610			0.998050	+	0.0624214i	$0.0198823\pi$
$998$		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.176.4	yes	6
3.2	odd	2	inner	177.2.d.b.176.3	✓	6
59.58	odd	2	CM	177.2.d.b.176.4	yes	6
177.176	even	2	inner	177.2.d.b.176.3	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.d.b.176.3	✓	6	3.2	odd	2	inner
177.2.d.b.176.3	✓	6	177.176	even	2	inner
177.2.d.b.176.4	yes	6	1.1	even	1	trivial
177.2.d.b.176.4	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)

\(f(q)\)	\(=\)	\(q+(-0.422380 + 1.67976i) q^{3} -2.00000 q^{4} +0.521533i q^{5} -3.59686 q^{7} +(-2.64319 - 1.41899i) q^{9} +O(q^{10})\)	\(q+(-0.422380 + 1.67976i) q^{3} -2.00000 q^{4} +0.521533i q^{5} -3.59686 q^{7} +(-2.64319 - 1.41899i) q^{9} +(0.844760 - 3.35952i) q^{12} +(-0.876051 - 0.220285i) q^{15} +4.00000 q^{16} +7.68115i q^{17} -6.13114 q^{19} -1.04307i q^{20} +(1.51924 - 6.04187i) q^{21} +4.72800 q^{25} +(3.50000 - 3.84057i) q^{27} +7.19372 q^{28} +10.6001i q^{29} -1.87588i q^{35} +(5.28638 + 2.83799i) q^{36} -11.1216i q^{41} +(0.740053 - 1.37851i) q^{45} +(-1.68952 + 6.71904i) q^{48} +5.93742 q^{49} +(-12.9025 - 3.24436i) q^{51} +9.03550i q^{53} +(2.58967 - 10.2988i) q^{57} -7.68115i q^{59} +(1.75210 + 0.440570i) q^{60} +(9.50719 + 5.10392i) q^{63} -8.00000 q^{64} -15.3623i q^{68} +7.68115i q^{71} +(-1.99701 + 7.94191i) q^{75} +12.2623 q^{76} -16.9217 q^{79} +2.08613i q^{80} +(4.97291 + 7.50134i) q^{81} +(-3.03848 + 12.0837i) q^{84} -4.00597 q^{85} +(-17.8056 - 4.47727i) q^{87} -3.19759i q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 12 q^{4}+O(q^{10}) \) 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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