LMFDB - Embedded newform 363.2.f.g.215.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [363,2,Mod(161,363)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(363, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([5, 7]))

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

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

chi := DirichletCharacter("363.161");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 363 = 3 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	363.f (of order $10$, degree $4$, not 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:	$2.89856959337$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$4$ over $\Q(\zeta_{10})$
Coefficient field:	16.0.6879707136000000000000.7
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4x^{14} + 15x^{12} - 56x^{10} + 209x^{8} - 56x^{6} + 15x^{4} - 4x^{2} + 1 $ x^16 - 4x^14 + 15x^12 - 56x^10 + 209x^8 - 56x^6 + 15x^4 - 4*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 11^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{10}]$

Embedding invariants

Embedding label			215.4
Root			$-0.492303 + 0.159959i$ of defining polynomial
Character	$\chi$	$=$	363.215
Dual form			363.2.f.g.233.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.40126 + 1.01807i) q^{3} +(1.61803 - 1.17557i) q^{4} +(1.35825 + 1.86947i) q^{7} +(0.927051 + 2.85317i) q^{9} +O(q^{10})$	$q+(1.40126 + 1.01807i) q^{3} +(1.61803 - 1.17557i) q^{4} +(1.35825 + 1.86947i) q^{7} +(0.927051 + 2.85317i) q^{9} +3.46410 q^{12} +(-6.00420 + 1.95088i) q^{13} +(1.23607 - 3.80423i) q^{16} +(5.06905 - 6.97695i) q^{19} +4.00240i q^{21} +(-4.04508 - 2.93893i) q^{25} +(-1.60570 + 4.94183i) q^{27} +(4.39538 + 1.42815i) q^{28} +(0.535233 + 1.64728i) q^{31} +(4.85410 + 3.52671i) q^{36} +(-4.20378 + 3.05422i) q^{37} +(-10.3996 - 3.37903i) q^{39} -1.69161i q^{43} +(5.60503 - 4.07230i) q^{48} +(0.513047 - 1.57900i) q^{49} +(-7.42160 + 10.2150i) q^{52} +(14.2061 - 4.61584i) q^{57} +(9.81072 + 3.18769i) q^{61} +(-4.07474 + 5.60840i) q^{63} +(-2.47214 - 7.60845i) q^{64} -12.1244 q^{67} +(-7.78554 - 10.7159i) q^{73} +(-2.67617 - 8.23639i) q^{75} -17.2480i q^{76} +(-0.588870 + 0.191335i) q^{79} +(-7.28115 + 5.29007i) q^{81} +(4.70511 + 6.47603i) q^{84} +(-11.8023 - 8.57488i) q^{91} +(-0.927051 + 2.85317i) q^{93} +(-1.54508 - 4.75528i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 8 q^{4} - 12 q^{9}+O(q^{10}) $ 16 * q + 8 * q^4 - 12 * q^9	$ 16 q + 8 q^{4} - 12 q^{9} - 16 q^{16} - 20 q^{25} + 24 q^{36} + 28 q^{49} + 32 q^{64} - 36 q^{81} + 4 q^{91} + 12 q^{93} + 20 q^{97}+O(q^{100}) $ 16 * q + 8 * q^4 - 12 * q^9 - 16 * q^16 - 20 * q^25 + 24 * q^36 + 28 * q^49 + 32 * q^64 - 36 * q^81 + 4 * q^91 + 12 * q^93 + 20 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/363\mathbb{Z}\right)^\times$.

$n$	$122$	$244$
$\chi(n)$	$-1$	$e\left(\frac{9}{10}\right)$

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		0.951057	−	0.309017i	$-0.100000\pi$
$2$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$3$	1.40126	+	1.01807i	0.809017	+	0.587785i
$3$	1.40126	+	1.01807i	0.809017	+	0.587785i
$4$	1.61803	−	1.17557i	0.809017	−	0.587785i
$5$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$5$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$6$		0			0
$7$	1.35825	+	1.86947i	0.513369	+	0.706592i	0.984483	−	0.175480i	$-0.0561478\pi$
$7$	1.35825	+	1.86947i	0.513369	+	0.706592i	−0.471114	+	0.882073i	$0.656148\pi$
$8$		0			0
$9$	0.927051	+	2.85317i	0.309017	+	0.951057i
$10$		0			0
$11$		0			0
$11$		0			0
$12$	3.46410			1.00000
$13$	−6.00420	+	1.95088i	−1.66527	+	0.541078i	−0.981966	−	0.189059i	$-0.939456\pi$
$13$	−6.00420	+	1.95088i	−1.66527	+	0.541078i	−0.683301	+	0.730137i	$0.739456\pi$
$14$		0			0
$15$		0			0
$16$	1.23607	−	3.80423i	0.309017	−	0.951057i
$17$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$17$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$18$		0			0
$19$	5.06905	−	6.97695i	1.16292	−	1.60062i	0.463091	−	0.886310i	$-0.346740\pi$
$19$	5.06905	−	6.97695i	1.16292	−	1.60062i	0.699828	−	0.714311i	$-0.253260\pi$
$20$		0			0
$21$			4.00240i			0.873396i
$22$		0			0
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$		0			0
$25$	−4.04508	−	2.93893i	−0.809017	−	0.587785i
$26$		0			0
$27$	−1.60570	+	4.94183i	−0.309017	+	0.951057i
$28$	4.39538	+	1.42815i	0.830649	+	0.269894i
$29$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$29$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$30$		0			0
$31$	0.535233	+	1.64728i	0.0961307	+	0.295860i	0.987547	−	0.157326i	$-0.0502874\pi$
$31$	0.535233	+	1.64728i	0.0961307	+	0.295860i	−0.891416	+	0.453186i	$0.850287\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	4.85410	+	3.52671i	0.809017	+	0.587785i
$37$	−4.20378	+	3.05422i	−0.691096	+	0.502111i	−0.877020	−	0.480453i	$-0.840472\pi$
$37$	−4.20378	+	3.05422i	−0.691096	+	0.502111i	0.185924	+	0.982564i	$0.440472\pi$
$38$		0			0
$39$	−10.3996	−	3.37903i	−1.66527	−	0.541078i
$40$		0			0
$41$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$41$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$42$		0			0
$43$		−	1.69161i		−	0.257969i	−0.991647	−	0.128984i	$-0.958828\pi$
$43$		−	1.69161i		−	0.257969i	0.991647	−	0.128984i	$-0.0411717\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$47$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$48$	5.60503	−	4.07230i	0.809017	−	0.587785i
$49$	0.513047	−	1.57900i	0.0732924	−	0.225571i
$50$		0			0
$51$		0			0
$52$	−7.42160	+	10.2150i	−1.02919	+	1.41656i
$53$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$53$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	14.2061	−	4.61584i	1.88164	−	0.611383i
$58$		0			0
$59$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$59$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$60$		0			0
$61$	9.81072	+	3.18769i	1.25613	+	0.408142i	0.860115	−	0.510100i	$-0.170392\pi$
$61$	9.81072	+	3.18769i	1.25613	+	0.408142i	0.396018	+	0.918243i	$0.370392\pi$
$62$		0			0
$63$	−4.07474	+	5.60840i	−0.513369	+	0.706592i
$64$	−2.47214	−	7.60845i	−0.309017	−	0.951057i
$65$		0			0
$66$		0			0
$67$	−12.1244			−1.48123			−0.740613	−	0.671932i	$-0.765465\pi$
$67$	−12.1244			−1.48123			−0.740613	+	0.671932i	$0.765465\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$71$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$72$		0			0
$73$	−7.78554	−	10.7159i	−0.911229	−	1.25420i	−0.966745	−	0.255741i	$-0.917680\pi$
$73$	−7.78554	−	10.7159i	−0.911229	−	1.25420i	0.0555161	−	0.998458i	$-0.482320\pi$
$74$		0			0
$75$	−2.67617	−	8.23639i	−0.309017	−	0.951057i
$76$		−	17.2480i		−	1.97848i
$77$		0			0
$78$		0			0
$79$	−0.588870	+	0.191335i	−0.0662530	+	0.0215269i	−0.341956	−	0.939716i	$-0.611089\pi$
$79$	−0.588870	+	0.191335i	−0.0662530	+	0.0215269i	0.275703	+	0.961243i	$0.411089\pi$
$80$		0			0
$81$	−7.28115	+	5.29007i	−0.809017	+	0.587785i
$82$		0			0
$83$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$83$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$84$	4.70511	+	6.47603i	0.513369	+	0.706592i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$	−11.8023	−	8.57488i	−1.23722	−	0.898892i
$92$		0			0
$93$	−0.927051	+	2.85317i	−0.0961307	+	0.295860i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−1.54508	−	4.75528i	−0.156880	−	0.482826i	0.841467	−	0.540309i	$-0.181693\pi$
$97$	−1.54508	−	4.75528i	−0.156880	−	0.482826i	−0.998346	+	0.0574829i	$0.981693\pi$
$98$		0			0
$99$		0			0
$100$	−10.0000			−1.00000
$101$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$101$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$102$		0			0
$103$	−5.66312	+	4.11450i	−0.558004	+	0.405413i	−0.830728	−	0.556679i	$-0.812075\pi$
$103$	−5.66312	+	4.11450i	−0.558004	+	0.405413i	0.272724	+	0.962092i	$0.412075\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$107$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$108$	3.21140	+	9.88367i	0.309017	+	0.951057i
$109$			19.5588i			1.87339i	0.350148	+	0.936694i	$0.386131\pi$
$109$			19.5588i			1.87339i	−0.350148	+	0.936694i	$0.613869\pi$
$110$		0			0
$111$	−9.00000			−0.854242
$112$	8.79076	−	2.85629i	0.830649	−	0.269894i
$113$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$113$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−11.1324	−	15.3224i	−1.02919	−	1.41656i
$118$		0			0
$119$		0			0
$120$		0			0
$121$		0			0
$122$		0			0
$123$		0			0
$124$	2.80252	+	2.03615i	0.251673	+	0.182851i
$125$		0			0
$126$		0			0
$127$	−15.8149	−	5.13858i	−1.40335	−	0.455975i	−0.493075	−	0.869987i	$-0.664127\pi$
$127$	−15.8149	−	5.13858i	−1.40335	−	0.455975i	−0.910272	+	0.414011i	$0.864127\pi$
$128$		0			0
$129$	1.72219	−	2.37039i	0.151630	−	0.208701i
$130$		0			0
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$		0			0
$133$	19.9282			1.72799
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$137$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$138$		0			0
$139$	13.8489	+	19.0614i	1.17465	+	1.61676i	0.619361	+	0.785106i	$0.287392\pi$
$139$	13.8489	+	19.0614i	1.17465	+	1.61676i	0.555288	+	0.831658i	$0.312608\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	12.0000			1.00000
$145$		0			0
$146$		0			0
$147$	2.32645	−	1.69026i	0.191882	−	0.139410i
$148$	−3.21140	+	9.88367i	−0.263975	+	0.812433i
$149$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$149$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$150$		0			0
$151$	8.41591	−	11.5835i	0.684877	−	0.942652i	−0.315102	−	0.949058i	$-0.602039\pi$
$151$	8.41591	−	11.5835i	0.684877	−	0.942652i	0.999979	+	0.00640530i	$0.00203888\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−20.7992	+	6.75806i	−1.66527	+	0.541078i
$157$	18.2164	+	13.2350i	1.45382	+	1.05626i	0.984919	+	0.173016i	$0.0553513\pi$
$157$	18.2164	+	13.2350i	1.45382	+	1.05626i	0.468905	+	0.883249i	$0.344649\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	5.25329	+	16.1680i	0.411469	+	1.26637i	0.915371	+	0.402611i	$0.131897\pi$
$163$	5.25329	+	16.1680i	0.411469	+	1.26637i	−0.503902	+	0.863761i	$0.668103\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$167$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$168$		0			0
$169$	21.7273	−	15.7858i	1.67133	−	1.21429i
$170$		0			0
$171$	24.6057	+	7.99487i	1.88164	+	0.611383i
$172$	−1.98861	−	2.73709i	−0.151630	−	0.208701i
$173$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$173$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$174$		0			0
$175$		−	11.5539i		−	0.873396i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$179$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$180$		0			0
$181$	5.87132	−	18.0701i	0.436412	−	1.34314i	−0.455221	−	0.890379i	$-0.650440\pi$
$181$	5.87132	−	18.0701i	0.436412	−	1.34314i	0.891633	−	0.452759i	$-0.149560\pi$
$182$		0			0
$183$	10.5020	+	14.4548i	0.776333	+	1.06853i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−11.4195	+	3.71043i	−0.830649	+	0.269894i
$190$		0			0
$191$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$191$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$192$	4.28187	−	13.1782i	0.309017	−	0.951057i
$193$	−4.98425	−	1.61948i	−0.358774	−	0.116573i	0.124083	−	0.992272i	$-0.460401\pi$
$193$	−4.98425	−	1.61948i	−0.358774	−	0.116573i	−0.482857	+	0.875699i	$0.660401\pi$
$194$		0			0
$195$		0			0
$196$	−1.02609	−	3.15799i	−0.0732924	−	0.225571i
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$		0			0
$199$	25.9808			1.84173			0.920864	−	0.389885i	$-0.127485\pi$
$199$	25.9808			1.84173			0.920864	+	0.389885i	$0.127485\pi$
$200$		0			0
$201$	−16.9894	−	12.3435i	−1.19834	−	0.870643i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$			25.2528i			1.75096i
$209$		0			0
$210$		0			0
$211$	−26.2145	+	8.51761i	−1.80468	+	0.586376i	−0.999973	−	0.00729508i	$-0.997678\pi$
$211$	−26.2145	+	8.51761i	−1.80468	+	0.586376i	−0.804708	+	0.593671i	$0.797678\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−2.35255	+	3.23801i	−0.159702	+	0.219811i
$218$		0			0
$219$		−	22.9420i		−	1.55028i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	18.6074	+	13.5191i	1.24604	+	0.905303i	0.997986	−	0.0634420i	$-0.0202078\pi$
$223$	18.6074	+	13.5191i	1.24604	+	0.905303i	0.248058	+	0.968745i	$0.420208\pi$
$224$		0			0
$225$	4.63525	−	14.2658i	0.309017	−	0.951057i
$226$		0			0
$227$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$227$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$228$	17.5597	−	24.1689i	1.16292	−	1.60062i
$229$	6.42280	+	19.7673i	0.424430	+	1.30626i	0.903539	+	0.428507i	$0.140960\pi$
$229$	6.42280	+	19.7673i	0.424430	+	1.30626i	−0.479108	+	0.877756i	$0.659040\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$233$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−1.01995	−	0.331402i	−0.0662530	−	0.0215269i
$238$		0			0
$239$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$239$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$240$		0			0
$241$			9.69642i			0.624602i	0.949983	+	0.312301i	$0.101100\pi$
$241$			9.69642i			0.624602i	−0.949983	+	0.312301i	$0.898900\pi$
$242$		0			0
$243$	−15.5885			−1.00000
$244$	19.6214	−	6.37539i	1.25613	−	0.408142i
$245$		0			0
$246$		0			0
$247$	−16.8244	+	51.7801i	−1.07051	+	3.29469i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$251$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$252$			13.8647i			0.873396i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−12.9443	−	9.40456i	−0.809017	−	0.587785i
$257$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$257$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$258$		0			0
$259$	−11.4195	−	3.71043i	−0.709576	−	0.230555i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0			−	1.00000i	$-0.5\pi$
$263$		0			0				1.00000i	$0.5\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−19.6176	+	14.2530i	−1.19834	+	0.870643i
$269$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$269$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$270$		0			0
$271$	−4.43868	−	6.10933i	−0.269631	−	0.371115i	0.652634	−	0.757673i	$-0.273664\pi$
$271$	−4.43868	−	6.10933i	−0.269631	−	0.371115i	−0.922265	+	0.386558i	$0.873664\pi$
$272$		0			0
$273$	−7.80823	−	24.0312i	−0.472575	−	1.45444i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	29.0011	−	9.42302i	1.74251	−	0.566174i	0.747345	−	0.664437i	$-0.231328\pi$
$277$	29.0011	−	9.42302i	1.74251	−	0.566174i	0.995161	+	0.0982624i	$0.0313285\pi$
$278$		0			0
$279$	−4.20378	+	3.05422i	−0.251673	+	0.182851i
$280$		0			0
$281$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$281$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$282$		0			0
$283$	−10.7685	+	14.8215i	−0.640119	+	0.881048i	−0.998622	−	0.0524806i	$-0.983287\pi$
$283$	−10.7685	+	14.8215i	−0.640119	+	0.881048i	0.358503	+	0.933528i	$0.383287\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	13.7533	+	9.99235i	0.809017	+	0.587785i
$290$		0			0
$291$	2.67617	−	8.23639i	0.156880	−	0.482826i
$292$	−25.1946	−	8.18621i	−1.47440	−	0.479061i
$293$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$293$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	−14.0126	−	10.1807i	−0.809017	−	0.587785i
$301$	3.16242	−	2.29763i	0.182279	−	0.132433i
$302$		0			0
$303$		0			0
$304$	−20.2762	−	27.9078i	−1.16292	−	1.60062i
$305$		0			0
$306$		0			0
$307$		−	7.38563i		−	0.421520i	−0.977538	−	0.210760i	$-0.932406\pi$
$307$		−	7.38563i		−	0.421520i	0.977538	−	0.210760i	$-0.0675939\pi$
$308$		0			0
$309$	−12.1244			−0.689730
$310$		0			0
$311$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$311$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$312$		0			0
$313$	8.56373	−	26.3565i	0.484051	−	1.48975i	−0.349300	−	0.937011i	$-0.613581\pi$
$313$	8.56373	−	26.3565i	0.484051	−	1.48975i	0.833351	−	0.552744i	$-0.186419\pi$
$314$		0			0
$315$		0			0
$316$	−0.727883	+	1.00184i	−0.0409466	+	0.0563582i
$317$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$317$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−5.56231	+	17.1190i	−0.309017	+	0.951057i
$325$	30.0210	+	9.75442i	1.66527	+	0.541078i
$326$		0			0
$327$	−19.9123	+	27.4069i	−1.10115	+	1.51560i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−31.0000			−1.70391			−0.851957	−	0.523612i	$-0.824584\pi$
$331$	−31.0000			−1.70391			−0.851957	+	0.523612i	$0.824584\pi$
$332$		0			0
$333$	−12.6113	−	9.16267i	−0.691096	−	0.502111i
$334$		0			0
$335$		0			0
$336$	15.2260	+	4.94724i	0.830649	+	0.269894i
$337$	17.1958	+	23.6679i	0.936713	+	1.28927i	0.957183	+	0.289484i	$0.0934837\pi$
$337$	17.1958	+	23.6679i	0.936713	+	1.28927i	−0.0204700	+	0.999790i	$0.506516\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	19.0326	−	6.18405i	1.02766	−	0.333908i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$347$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$348$		0			0
$349$	−13.2185	+	18.1938i	−0.707572	+	0.973889i	0.292274	+	0.956335i	$0.405588\pi$
$349$	−13.2185	+	18.1938i	−0.707572	+	0.973889i	−0.999846	+	0.0175547i	$0.994412\pi$
$350$		0			0
$351$		−	32.8043i		−	1.75096i
$352$		0			0
$353$		0			0		1.00000			$0$
$353$		0			0		−1.00000			$\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$359$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$360$		0			0
$361$	−17.1112	−	52.6629i	−0.900590	−	2.77173i
$362$		0			0
$363$		0			0
$364$	−29.1769			−1.52929
$365$		0			0
$366$		0			0
$367$	3.23607	−	2.35114i	0.168921	−	0.122729i	−0.500113	−	0.865960i	$-0.666708\pi$
$367$	3.23607	−	2.35114i	0.168921	−	0.122729i	0.669034	+	0.743232i	$0.266708\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	1.85410	+	5.70634i	0.0961307	+	0.295860i
$373$		−	38.4983i		−	1.99337i	−0.0813690	−	0.996684i	$-0.525929\pi$
$373$		−	38.4983i		−	1.99337i	0.0813690	−	0.996684i	$-0.474071\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	2.47214	−	7.60845i	0.126985	−	0.390820i	−0.867272	−	0.497834i	$-0.834129\pi$
$379$	2.47214	−	7.60845i	0.126985	−	0.390820i	0.994258	+	0.107014i	$0.0341289\pi$
$380$		0			0
$381$	−16.9293	−	23.3012i	−0.867316	−	1.19376i
$382$		0			0
$383$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$383$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	4.82646	−	1.56821i	0.245343	−	0.0797168i
$388$	−8.09017	−	5.87785i	−0.410716	−	0.298403i
$389$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$389$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	35.0000			1.75660			0.878300	−	0.478110i	$-0.158678\pi$
$397$	35.0000			1.75660			0.878300	+	0.478110i	$0.158678\pi$
$398$		0			0
$399$	27.9246	+	20.2884i	1.39798	+	1.01569i
$400$	−16.1803	+	11.7557i	−0.809017	+	0.587785i
$401$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$401$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$402$		0			0
$403$	−6.42730	−	8.84642i	−0.320166	−	0.440671i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	3.37543	−	1.09674i	0.166904	−	0.0542305i	−0.224373	−	0.974503i	$-0.572033\pi$
$409$	3.37543	−	1.09674i	0.166904	−	0.0542305i	0.391277	+	0.920273i	$0.372033\pi$
$410$		0			0
$411$		0			0
$412$	−4.32624	+	13.3148i	−0.213138	+	0.655973i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$			40.8091i			1.99843i
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$	−28.0252	−	20.3615i	−1.36586	−	0.992358i	−0.998048	−	0.0624590i	$-0.980106\pi$
$421$	−28.0252	−	20.3615i	−1.36586	−	0.992358i	−0.367816	−	0.929899i	$-0.619894\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	7.36609	+	22.6705i	0.356470	+	1.09710i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$431$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$432$	16.8151	+	12.2169i	0.809017	+	0.587785i
$433$	29.9336	−	21.7481i	1.43852	−	1.04514i	0.450169	−	0.892943i	$-0.351364\pi$
$433$	29.9336	−	21.7481i	1.43852	−	1.04514i	0.988350	−	0.152201i	$-0.0486362\pi$
$434$		0			0
$435$		0			0
$436$	22.9927	+	31.6467i	1.10115	+	1.51560i
$437$		0			0
$438$		0			0
$439$			35.1151i			1.67595i	0.545707	+	0.837976i	$0.316261\pi$
$439$			35.1151i			1.67595i	−0.545707	+	0.837976i	$0.683739\pi$
$440$		0			0
$441$	4.98076			0.237179
$442$		0			0
$443$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$443$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$444$	−14.5623	+	10.5801i	−0.691096	+	0.502111i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	10.8660	−	14.9557i	0.513369	−	0.706592i
$449$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$449$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	23.5857	−	7.66347i	1.10815	−	0.360061i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−21.2303	−	6.89813i	−0.993109	−	0.322681i	−0.233001	−	0.972477i	$-0.574854\pi$
$457$	−21.2303	−	6.89813i	−0.993109	−	0.322681i	−0.760109	+	0.649796i	$0.774854\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$	−20.0000			−0.929479			−0.464739	−	0.885448i	$-0.653852\pi$
$463$	−20.0000			−0.929479			−0.464739	+	0.885448i	$0.653852\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$467$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$468$	−36.0252	−	11.7053i	−1.66527	−	0.541078i
$469$	−16.4679	−	22.6661i	−0.760416	−	1.04662i
$470$		0			0
$471$	12.0517	+	37.0912i	0.555311	+	1.70907i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−41.0095	+	13.3248i	−1.88164	+	0.611383i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$479$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$480$		0			0
$481$	19.2819	−	26.5392i	0.879179	−	1.21009i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	2.80252	+	2.03615i	0.126994	+	0.0922667i	0.649469	−	0.760388i	$-0.274991\pi$
$487$	2.80252	+	2.03615i	0.126994	+	0.0922667i	−0.522475	+	0.852655i	$0.674991\pi$
$488$		0			0
$489$	−9.09896	+	28.0037i	−0.411469	+	1.26637i
$490$		0			0
$491$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$491$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	6.92820			0.311086
$497$		0			0
$498$		0			0
$499$	−35.0315	+	25.4518i	−1.56822	+	1.13938i	−0.639391	+	0.768882i	$0.720813\pi$
$499$	−35.0315	+	25.4518i	−1.56822	+	1.13938i	−0.928833	+	0.370499i	$0.879187\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$503$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	46.5167			2.06588
$508$	−31.6298	+	10.2772i	−1.40335	+	0.455975i
$509$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$509$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$510$		0			0
$511$	9.45830	−	29.1096i	0.418410	−	1.28774i
$512$		0			0
$513$	26.3396	+	36.2533i	1.16292	+	1.60062i
$514$		0			0
$515$		0			0
$516$		−	5.85993i		−	0.257969i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$521$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$522$		0			0
$523$	39.4007	+	12.8020i	1.72287	+	0.559794i	0.992389	−	0.123139i	$-0.0392961\pi$
$523$	39.4007	+	12.8020i	1.72287	+	0.559794i	0.730481	+	0.682933i	$0.239296\pi$
$524$		0			0
$525$	11.7628	−	16.1901i	0.513369	−	0.706592i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	23.0000			1.00000
$530$		0			0
$531$		0			0
$532$	32.2445	−	23.4270i	1.39798	−	1.01569i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−35.5941	+	11.5652i	−1.53031	+	0.497228i	−0.948684	−	0.316227i	$-0.897584\pi$
$541$	−35.5941	+	11.5652i	−1.53031	+	0.497228i	−0.581628	+	0.813455i	$0.697584\pi$
$542$		0			0
$543$	26.6239	−	19.3434i	1.14254	−	0.830105i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	26.7035	−	36.7542i	1.14176	−	1.57150i	0.378245	−	0.925706i	$-0.376528\pi$
$547$	26.7035	−	36.7542i	1.14176	−	1.57150i	0.763514	−	0.645791i	$-0.223472\pi$
$548$		0			0
$549$			30.9468i			1.32078i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−1.15753	−	0.840992i	−0.0492230	−	0.0357626i
$554$		0			0
$555$		0			0
$556$	44.8160	+	14.5616i	1.90062	+	0.617549i
$557$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$557$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$558$		0			0
$559$	3.30014	+	10.1568i	0.139581	+	0.429587i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$563$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−19.7792	−	6.42666i	−0.830649	−	0.269894i
$568$		0			0
$569$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$569$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$570$		0			0
$571$		−	27.1103i		−	1.13453i	−0.823535	−	0.567265i	$-0.808001\pi$
$571$		−	27.1103i		−	1.13453i	0.823535	−	0.567265i	$-0.191999\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	19.4164	−	14.1068i	0.809017	−	0.587785i
$577$	14.4513	−	44.4765i	0.601615	−	1.85158i	0.0830461	−	0.996546i	$-0.473535\pi$
$577$	14.4513	−	44.4765i	0.601615	−	1.85158i	0.518569	−	0.855036i	$-0.326465\pi$
$578$		0			0
$579$	−5.33547	−	7.34365i	−0.221735	−	0.305192i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$587$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$588$	1.77725	−	5.46980i	0.0732924	−	0.225571i
$589$	14.2061	+	4.61584i	0.585352	+	0.190192i
$590$		0			0
$591$		0			0
$592$	6.42280	+	19.7673i	0.263975	+	0.812433i
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	36.4058	+	26.4503i	1.48999	+	1.08254i
$598$		0			0
$599$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$599$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$600$		0			0
$601$	19.6458	+	27.0402i	0.801370	+	1.10299i	0.992598	+	0.121446i	$0.0387531\pi$
$601$	19.6458	+	27.0402i	0.801370	+	1.10299i	−0.191228	+	0.981546i	$0.561247\pi$
$602$		0			0
$603$	−11.2399	−	34.5928i	−0.457724	−	1.40873i
$604$		−	28.6360i		−	1.16518i
$605$		0			0
$606$		0			0
$607$	−16.8349	+	5.46998i	−0.683306	+	0.222020i	−0.630042	−	0.776561i	$-0.716962\pi$
$607$	−16.8349	+	5.46998i	−0.683306	+	0.222020i	−0.0532640	+	0.998580i	$0.516962\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−29.0560	+	39.9922i	−1.17356	+	1.61527i	−0.539519	+	0.841973i	$0.681394\pi$
$613$	−29.0560	+	39.9922i	−1.17356	+	1.61527i	−0.634044	+	0.773297i	$0.718606\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$		0			0
$619$	−25.8885	−	18.8091i	−1.04055	−	0.756003i	−0.0701559	−	0.997536i	$-0.522350\pi$
$619$	−25.8885	−	18.8091i	−1.04055	−	0.756003i	−0.970393	+	0.241533i	$0.922350\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	−25.7092	+	35.3857i	−1.02919	+	1.41656i
$625$	7.72542	+	23.7764i	0.309017	+	0.951057i
$626$		0			0
$627$		0			0
$628$	45.0333			1.79703
$629$		0			0
$630$		0			0
$631$	−19.6176	+	14.2530i	−0.780965	+	0.567404i	−0.905269	−	0.424840i	$-0.860330\pi$
$631$	−19.6176	+	14.2530i	−0.780965	+	0.567404i	0.124303	+	0.992244i	$0.460330\pi$
$632$		0			0
$633$	−45.4049	−	14.7529i	−1.80468	−	0.586376i
$634$		0			0
$635$		0			0
$636$		0			0
$637$			10.4815i			0.415292i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$641$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$642$		0			0
$643$	−14.5238	+	44.6997i	−0.572763	+	1.76278i	0.0709114	+	0.997483i	$0.477409\pi$
$643$	−14.5238	+	44.6997i	−0.572763	+	1.76278i	−0.643674	+	0.765300i	$0.722591\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$647$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−6.59307	+	2.14222i	−0.258403	+	0.0839602i
$652$	27.5066	+	19.9847i	1.07724	+	0.782661i
$653$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$653$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	23.3566	−	32.1476i	0.911229	−	1.25420i
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$	−50.2295			−1.95370			−0.976850	−	0.213925i	$-0.931375\pi$
$661$	−50.2295			−1.95370			−0.976850	+	0.213925i	$0.931375\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	12.3104	+	37.8874i	0.475946	+	1.46481i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	25.0368	−	8.13494i	0.965096	−	0.313579i	0.216262	−	0.976335i	$-0.430614\pi$
$673$	25.0368	−	8.13494i	0.965096	−	0.313579i	0.748835	+	0.662757i	$0.230614\pi$
$674$		0			0
$675$	21.0189	−	15.2711i	0.809017	−	0.587785i
$676$	16.5982	−	51.0839i	0.638391	−	1.96477i
$677$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$677$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$678$		0			0
$679$	6.79124	−	9.34734i	0.260624	−	0.358718i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		1.00000			$0$
$683$		0			0		−1.00000			$\pi$
$684$	49.2114	−	15.9897i	1.88164	−	0.611383i
$685$		0			0
$686$		0			0
$687$	−11.1246	+	34.2380i	−0.424430	+	1.30626i
$688$	−6.43529	−	2.09095i	−0.245343	−	0.0797168i
$689$		0			0
$690$		0			0
$691$	−15.1418	−	46.6018i	−0.576022	−	1.77281i	−0.632671	−	0.774421i	$-0.718041\pi$
$691$	−15.1418	−	46.6018i	−0.576022	−	1.77281i	0.0566486	−	0.998394i	$-0.481959\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−13.5825	−	18.6947i	−0.513369	−	0.706592i
$701$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$701$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$702$		0			0
$703$			44.8115i			1.69010i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−14.9865	+	46.1238i	−0.562831	+	1.73222i	0.111479	+	0.993767i	$0.464441\pi$
$709$	−14.9865	+	46.1238i	−0.562831	+	1.73222i	−0.674310	+	0.738448i	$0.735559\pi$
$710$		0			0
$711$	−1.09182	−	1.50277i	−0.0409466	−	0.0563582i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$719$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$720$		0			0
$721$	−15.3838	−	4.99851i	−0.572924	−	0.186154i
$722$		0			0
$723$	−9.87168	+	13.5872i	−0.367132	+	0.505313i
$724$	−11.7426	−	36.1401i	−0.436412	−	1.34314i
$725$		0			0
$726$		0			0
$727$	−31.1769			−1.15629			−0.578144	−	0.815935i	$-0.696223\pi$
$727$	−31.1769			−1.15629			−0.578144	+	0.815935i	$0.696223\pi$
$728$		0			0
$729$	−21.8435	−	15.8702i	−0.809017	−	0.587785i
$730$		0			0
$731$		0			0
$732$	33.9853	+	11.0425i	1.25613	+	0.408142i
$733$	−29.4200	−	40.4931i	−1.08665	−	1.49565i	−0.851980	−	0.523574i	$-0.824598\pi$
$733$	−29.4200	−	40.4931i	−1.08665	−	1.49565i	−0.234672	−	0.972075i	$-0.575402\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	43.7960	−	14.2302i	1.61106	−	0.523466i	0.641253	−	0.767329i	$-0.278415\pi$
$739$	43.7960	−	14.2302i	1.61106	−	0.523466i	0.969810	+	0.243863i	$0.0784147\pi$
$740$		0			0
$741$	−76.2913	+	55.4289i	−2.80263	+	2.03623i
$742$		0			0
$743$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$743$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−43.4390	−	31.5603i	−1.58511	−	1.15165i	−0.910523	−	0.413458i	$-0.864321\pi$
$751$	−43.4390	−	31.5603i	−1.58511	−	1.15165i	−0.674589	−	0.738194i	$-0.735679\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	−14.1153	+	19.4281i	−0.513369	+	0.706592i
$757$	0.535233	+	1.64728i	0.0194534	+	0.0598713i	0.960312	−	0.278928i	$-0.0899792\pi$
$757$	0.535233	+	1.64728i	0.0194534	+	0.0598713i	−0.940859	+	0.338800i	$0.889979\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$761$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$762$		0			0
$763$	−36.5645	+	26.5656i	−1.32372	+	0.961740i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−8.56373	−	26.3565i	−0.309017	−	0.951057i
$769$		−	54.0547i		−	1.94926i	−0.223819	−	0.974631i	$-0.571852\pi$
$769$		−	54.0547i		−	1.94926i	0.223819	−	0.974631i	$-0.428148\pi$
$770$		0			0
$771$		0			0
$772$	−9.96850	+	3.23896i	−0.358774	+	0.116573i
$773$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$773$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$774$		0			0
$775$	2.67617	−	8.23639i	0.0961307	−	0.295860i
$776$		0			0
$777$	−12.2242	−	16.8252i	−0.438542	−	0.603601i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−5.37269	−	3.90349i	−0.191882	−	0.139410i
$785$		0			0
$786$		0			0
$787$	−39.9895	−	12.9934i	−1.42547	−	0.463164i	−0.508136	−	0.861277i	$-0.669665\pi$
$787$	−39.9895	−	12.9934i	−1.42547	−	0.463164i	−0.917336	+	0.398113i	$0.869665\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−65.1244			−2.31263
$794$		0			0
$795$		0			0
$796$	42.0378	−	30.5422i	1.48999	−	1.08254i
$797$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$797$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$	−42.0000			−1.48123
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$809$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$810$		0			0
$811$	2.61898	−	3.60471i	0.0919647	−	0.126579i	−0.760555	−	0.649273i	$-0.775073\pi$
$811$	2.61898	−	3.60471i	0.0919647	−	0.126579i	0.852520	+	0.522695i	$0.175073\pi$
$812$		0			0
$813$		−	13.0797i		−	0.458723i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−11.8023	−	8.57488i	−0.412911	−	0.299997i
$818$		0			0
$819$	13.5242	−	41.6233i	0.472575	−	1.45444i
$820$		0			0
$821$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$821$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$822$		0			0
$823$	−1.54508	−	4.75528i	−0.0538583	−	0.165759i	0.920509	−	0.390721i	$-0.127774\pi$
$823$	−1.54508	−	4.75528i	−0.0538583	−	0.165759i	−0.974368	+	0.224962i	$0.927774\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$827$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$828$		0			0
$829$	−5.66312	+	4.11450i	−0.196688	+	0.142902i	−0.681770	−	0.731566i	$-0.738790\pi$
$829$	−5.66312	+	4.11450i	−0.196688	+	0.142902i	0.485082	+	0.874469i	$0.338790\pi$
$830$		0			0
$831$	50.2313	+	16.3211i	1.74251	+	0.566174i
$832$	29.6864	+	40.8598i	1.02919	+	1.41656i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−9.00000			−0.311086
$838$		0			0
$839$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$839$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$840$		0			0
$841$	−8.96149	+	27.5806i	−0.309017	+	0.951057i
$842$		0			0
$843$		0			0
$844$	−32.4029	+	44.5988i	−1.11535	+	1.53515i
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−30.1788	+	9.80569i	−1.03573	+	0.336530i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	20.6414	+	6.70679i	0.706748	+	0.229636i	0.640268	−	0.768152i	$-0.278823\pi$
$853$	20.6414	+	6.70679i	0.706748	+	0.229636i	0.0664795	+	0.997788i	$0.478823\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0			−	1.00000i	$-0.5\pi$
$857$		0			0				1.00000i	$0.5\pi$
$858$		0			0
$859$	13.0000			0.443554			0.221777	−	0.975097i	$-0.428814\pi$
$859$	13.0000			0.443554			0.221777	+	0.975097i	$0.428814\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$863$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	9.09896	+	28.0037i	0.309017	+	0.951057i
$868$			8.00481i			0.271701i
$869$		0			0
$870$		0			0
$871$	72.7971	−	23.6532i	2.46664	−	0.801458i
$872$		0			0
$873$	12.1353	−	8.81678i	0.410716	−	0.298403i
$874$		0			0
$875$		0			0
$876$	−26.9699	−	37.1209i	−0.911229	−	1.25420i
$877$	−22.3623	+	30.7791i	−0.755122	+	1.03934i	0.242482	+	0.970156i	$0.422038\pi$
$877$	−22.3623	+	30.7791i	−0.755122	+	1.03934i	−0.997604	+	0.0691806i	$0.977962\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$		0			0
$883$	18.2164	+	13.2350i	0.613029	+	0.445392i	0.850480	−	0.526008i	$-0.176312\pi$
$883$	18.2164	+	13.2350i	0.613029	+	0.445392i	−0.237450	+	0.971400i	$0.576312\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$887$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$888$		0			0
$889$	−11.8742	−	36.5449i	−0.398247	−	1.22568i
$890$		0			0
$891$		0			0
$892$	46.0000			1.54019
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−9.27051	−	28.5317i	−0.309017	−	0.951057i
$901$		0			0
$902$		0			0
$903$	6.77053			0.225309
$904$		0			0
$905$		0			0
$906$		0			0
$907$	5.87132	−	18.0701i	0.194954	−	0.600007i	−0.805023	−	0.593244i	$-0.797847\pi$
$907$	5.87132	−	18.0701i	0.194954	−	0.600007i	0.999977	−	0.00676342i	$-0.00215288\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$911$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$912$		−	59.7487i		−	1.97848i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	33.6302	+	24.4338i	1.11117	+	0.807315i
$917$		0			0
$918$		0			0
$919$	−41.4406	−	13.4649i	−1.36700	−	0.444164i	−0.468625	−	0.883397i	$-0.655250\pi$
$919$	−41.4406	−	13.4649i	−1.36700	−	0.444164i	−0.898373	+	0.439233i	$0.855250\pi$
$920$		0			0
$921$	7.51912	−	10.3492i	0.247763	−	0.341017i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	25.9808			0.854242
$926$		0			0
$927$	−16.9894	−	12.3435i	−0.558004	−	0.405413i
$928$		0			0
$929$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$929$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$930$		0			0
$931$	−8.41591	−	11.5835i	−0.275820	−	0.379634i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	10.2418	−	3.32776i	0.334585	−	0.108713i	−0.136906	−	0.990584i	$-0.543716\pi$
$937$	10.2418	−	3.32776i	0.334585	−	0.108713i	0.471491	+	0.881871i	$0.343716\pi$
$938$		0			0
$939$	38.8328	−	28.2137i	1.26726	−	0.920719i
$940$		0			0
$941$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$941$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		1.00000			$0$
$947$		0			0		−1.00000			$\pi$
$948$	−2.03990	+	0.662805i	−0.0662530	+	0.0215269i
$949$	67.6514	+	49.1516i	2.19606	+	1.59553i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$953$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	22.6525	−	16.4580i	0.730725	−	0.530903i
$962$		0			0
$963$		0			0
$964$	11.3988	+	15.6891i	0.367132	+	0.505313i
$965$		0			0
$966$		0			0
$967$			62.0595i			1.99570i	0.0655495	+	0.997849i	$0.479120\pi$
$967$			62.0595i			1.99570i	−0.0655495	+	0.997849i	$0.520880\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$971$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$972$	−25.2227	+	18.3253i	−0.809017	+	0.587785i
$973$	−16.8244	+	51.7801i	−0.539365	+	1.66000i
$974$		0			0
$975$	32.1365	+	44.2321i	1.02919	+	1.41656i
$976$	24.2534	−	33.3820i	0.776333	−	1.06853i
$977$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$977$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−55.8044	+	18.1320i	−1.78170	+	0.578909i
$982$		0			0
$983$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$983$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	33.6488	+	103.560i	1.07051	+	3.29469i
$989$		0			0
$990$		0			0
$991$	45.0333			1.43053			0.715265	−	0.698853i	$-0.246306\pi$
$991$	45.0333			1.43053			0.715265	+	0.698853i	$0.246306\pi$
$992$		0			0
$993$	−43.4390	−	31.5603i	−1.37850	−	1.00154i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	3.80832	+	5.24170i	0.120611	+	0.166006i	0.865053	−	0.501680i	$-0.167285\pi$
$997$	3.80832	+	5.24170i	0.120611	+	0.166006i	−0.744442	+	0.667687i	$0.767285\pi$
$998$		0			0
$999$	−8.34346	−	25.6785i	−0.263975	−	0.812433i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	363.2.f.g.215.4		16
3.2	odd	2	CM	363.2.f.g.215.4		16
11.2	odd	10	inner	363.2.f.g.233.4		16
11.3	even	5		363.2.d.d.362.1	✓	4
11.4	even	5	inner	363.2.f.g.239.1		16
11.5	even	5	inner	363.2.f.g.161.2		16
11.6	odd	10	inner	363.2.f.g.161.1		16
11.7	odd	10	inner	363.2.f.g.239.2		16
11.8	odd	10		363.2.d.d.362.2	yes	4
11.9	even	5	inner	363.2.f.g.233.3		16
11.10	odd	2	inner	363.2.f.g.215.3		16
33.2	even	10	inner	363.2.f.g.233.4		16
33.5	odd	10	inner	363.2.f.g.161.2		16
33.8	even	10		363.2.d.d.362.2	yes	4
33.14	odd	10		363.2.d.d.362.1	✓	4
33.17	even	10	inner	363.2.f.g.161.1		16
33.20	odd	10	inner	363.2.f.g.233.3		16
33.26	odd	10	inner	363.2.f.g.239.1		16
33.29	even	10	inner	363.2.f.g.239.2		16
33.32	even	2	inner	363.2.f.g.215.3		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
363.2.d.d.362.1	✓	4	11.3	even	5
363.2.d.d.362.1	✓	4	33.14	odd	10
363.2.d.d.362.2	yes	4	11.8	odd	10
363.2.d.d.362.2	yes	4	33.8	even	10
363.2.f.g.161.1		16	11.6	odd	10	inner
363.2.f.g.161.1		16	33.17	even	10	inner
363.2.f.g.161.2		16	11.5	even	5	inner
363.2.f.g.161.2		16	33.5	odd	10	inner
363.2.f.g.215.3		16	11.10	odd	2	inner
363.2.f.g.215.3		16	33.32	even	2	inner
363.2.f.g.215.4		16	1.1	even	1	trivial
363.2.f.g.215.4		16	3.2	odd	2	CM
363.2.f.g.233.3		16	11.9	even	5	inner
363.2.f.g.233.3		16	33.20	odd	10	inner
363.2.f.g.233.4		16	11.2	odd	10	inner
363.2.f.g.233.4		16	33.2	even	10	inner
363.2.f.g.239.1		16	11.4	even	5	inner
363.2.f.g.239.1		16	33.26	odd	10	inner
363.2.f.g.239.2		16	11.7	odd	10	inner
363.2.f.g.239.2		16	33.29	even	10	inner

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:	\( N \)	\(=\)	\( 363 = 3 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	363.f (of order \(10\), degree \(4\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.40126 + 1.01807i) q^{3} +(1.61803 - 1.17557i) q^{4} +(1.35825 + 1.86947i) q^{7} +(0.927051 + 2.85317i) q^{9} +O(q^{10})\)	\(q+(1.40126 + 1.01807i) q^{3} +(1.61803 - 1.17557i) q^{4} +(1.35825 + 1.86947i) q^{7} +(0.927051 + 2.85317i) q^{9} +3.46410 q^{12} +(-6.00420 + 1.95088i) q^{13} +(1.23607 - 3.80423i) q^{16} +(5.06905 - 6.97695i) q^{19} +4.00240i q^{21} +(-4.04508 - 2.93893i) q^{25} +(-1.60570 + 4.94183i) q^{27} +(4.39538 + 1.42815i) q^{28} +(0.535233 + 1.64728i) q^{31} +(4.85410 + 3.52671i) q^{36} +(-4.20378 + 3.05422i) q^{37} +(-10.3996 - 3.37903i) q^{39} -1.69161i q^{43} +(5.60503 - 4.07230i) q^{48} +(0.513047 - 1.57900i) q^{49} +(-7.42160 + 10.2150i) q^{52} +(14.2061 - 4.61584i) q^{57} +(9.81072 + 3.18769i) q^{61} +(-4.07474 + 5.60840i) q^{63} +(-2.47214 - 7.60845i) q^{64} -12.1244 q^{67} +(-7.78554 - 10.7159i) q^{73} +(-2.67617 - 8.23639i) q^{75} -17.2480i q^{76} +(-0.588870 + 0.191335i) q^{79} +(-7.28115 + 5.29007i) q^{81} +(4.70511 + 6.47603i) q^{84} +(-11.8023 - 8.57488i) q^{91} +(-0.927051 + 2.85317i) q^{93} +(-1.54508 - 4.75528i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 8 q^{4} - 12 q^{9}+O(q^{10}) \) 16 * q + 8 * q^4 - 12 * q^9	\( 16 q + 8 q^{4} - 12 q^{9} - 16 q^{16} - 20 q^{25} + 24 q^{36} + 28 q^{49} + 32 q^{64} - 36 q^{81} + 4 q^{91} + 12 q^{93} + 20 q^{97}+O(q^{100}) \) 16 * q + 8 * q^4 - 12 * q^9 - 16 * q^16 - 20 * q^25 + 24 * q^36 + 28 * q^49 + 32 * q^64 - 36 * q^81 + 4 * q^91 + 12 * q^93 + 20 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more