LMFDB - Embedded newform 363.2.f.g.161.2

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			161.2
Root			$0.304260 - 0.418778i$ of defining polynomial
Character	$\chi$	$=$	363.161
Dual form			363.2.f.g.239.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.535233 + 1.64728i) q^{3} +(-0.618034 - 1.90211i) q^{4} +(2.19769 - 0.714073i) q^{7} +(-2.42705 - 1.76336i) q^{9} +O(q^{10})$	$q+(-0.535233 + 1.64728i) q^{3} +(-0.618034 - 1.90211i) q^{4} +(2.19769 - 0.714073i) q^{7} +(-2.42705 - 1.76336i) q^{9} +3.46410 q^{12} +(3.71080 - 5.10748i) q^{13} +(-3.23607 + 2.35114i) q^{16} +(8.20189 + 2.66496i) q^{19} +4.00240i q^{21} +(1.54508 - 4.75528i) q^{25} +(4.20378 - 3.05422i) q^{27} +(-2.71650 - 3.73894i) q^{28} +(-1.40126 - 1.01807i) q^{31} +(-1.85410 + 5.70634i) q^{36} +(1.60570 + 4.94183i) q^{37} +(6.42730 + 8.84642i) q^{39} -1.69161i q^{43} +(-2.14093 - 6.58911i) q^{48} +(-1.34317 + 0.975873i) q^{49} +(-12.0084 - 3.90177i) q^{52} +(-8.77985 + 12.0844i) q^{57} +(-6.06336 - 8.34549i) q^{61} +(-6.59307 - 2.14222i) q^{63} +(6.47214 + 4.70228i) q^{64} -12.1244 q^{67} +(-12.5973 + 4.09310i) q^{73} +(7.00629 + 5.09037i) q^{75} -17.2480i q^{76} +(0.363941 - 0.500922i) q^{79} +(2.78115 + 8.55951i) q^{81} +(7.61302 - 2.47362i) q^{84} +(4.50808 - 13.8744i) q^{91} +(2.42705 - 1.76336i) q^{93} +(4.04508 + 2.93893i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/363\mathbb{Z}\right)^\times$.

$n$	$122$	$244$
$\chi(n)$	$-1$	$e\left(\frac{7}{10}\right)$

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		0.587785	−	0.809017i	$-0.300000\pi$
$2$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$3$	−0.535233	+	1.64728i	−0.309017	+	0.951057i
$3$	−0.535233	+	1.64728i	−0.309017	+	0.951057i
$4$	−0.618034	−	1.90211i	−0.309017	−	0.951057i
$5$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$5$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$6$		0			0
$7$	2.19769	−	0.714073i	0.830649	−	0.269894i	0.137330	−	0.990525i	$-0.456148\pi$
$7$	2.19769	−	0.714073i	0.830649	−	0.269894i	0.693319	+	0.720631i	$0.256148\pi$
$8$		0			0
$9$	−2.42705	−	1.76336i	−0.809017	−	0.587785i
$10$		0			0
$11$		0			0
$11$		0			0
$12$	3.46410			1.00000
$13$	3.71080	−	5.10748i	1.02919	−	1.41656i	0.123638	−	0.992327i	$-0.460544\pi$
$13$	3.71080	−	5.10748i	1.02919	−	1.41656i	0.905553	−	0.424233i	$-0.139456\pi$
$14$		0			0
$15$		0			0
$16$	−3.23607	+	2.35114i	−0.809017	+	0.587785i
$17$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$17$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$18$		0			0
$19$	8.20189	+	2.66496i	1.88164	+	0.611383i	0.986035	+	0.166541i	$0.0532599\pi$
$19$	8.20189	+	2.66496i	1.88164	+	0.611383i	0.895609	+	0.444842i	$0.146740\pi$
$20$		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
$25$	1.54508	−	4.75528i	0.309017	−	0.951057i
$26$		0			0
$27$	4.20378	−	3.05422i	0.809017	−	0.587785i
$28$	−2.71650	−	3.73894i	−0.513369	−	0.706592i
$29$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$29$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$30$		0			0
$31$	−1.40126	−	1.01807i	−0.251673	−	0.182851i	0.454795	−	0.890596i	$-0.349713\pi$
$31$	−1.40126	−	1.01807i	−0.251673	−	0.182851i	−0.706468	+	0.707745i	$0.749713\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−1.85410	+	5.70634i	−0.309017	+	0.951057i
$37$	1.60570	+	4.94183i	0.263975	+	0.812433i	0.991928	+	0.126805i	$0.0404722\pi$
$37$	1.60570	+	4.94183i	0.263975	+	0.812433i	−0.727952	+	0.685628i	$0.759528\pi$
$38$		0			0
$39$	6.42730	+	8.84642i	1.02919	+	1.41656i
$40$		0			0
$41$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$41$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$42$		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		0.309017	−	0.951057i	$-0.400000\pi$
$47$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$48$	−2.14093	−	6.58911i	−0.309017	−	0.951057i
$49$	−1.34317	+	0.975873i	−0.191882	+	0.139410i
$50$		0			0
$51$		0			0
$52$	−12.0084	−	3.90177i	−1.66527	−	0.541078i
$53$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$53$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−8.77985	+	12.0844i	−1.16292	+	1.60062i
$58$		0			0
$59$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$59$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$60$		0			0
$61$	−6.06336	−	8.34549i	−0.776333	−	1.06853i	−0.995677	−	0.0928833i	$-0.970392\pi$
$61$	−6.06336	−	8.34549i	−0.776333	−	1.06853i	0.219344	−	0.975648i	$-0.429608\pi$
$62$		0			0
$63$	−6.59307	−	2.14222i	−0.830649	−	0.269894i
$64$	6.47214	+	4.70228i	0.809017	+	0.587785i
$65$		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		0.809017	−	0.587785i	$-0.200000\pi$
$71$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$72$		0			0
$73$	−12.5973	+	4.09310i	−1.47440	+	0.479061i	−0.932434	−	0.361339i	$-0.882320\pi$
$73$	−12.5973	+	4.09310i	−1.47440	+	0.479061i	−0.541965	+	0.840401i	$0.682320\pi$
$74$		0			0
$75$	7.00629	+	5.09037i	0.809017	+	0.587785i
$76$		−	17.2480i		−	1.97848i
$77$		0			0
$78$		0			0
$79$	0.363941	−	0.500922i	0.0409466	−	0.0563582i	−0.788053	−	0.615608i	$-0.788911\pi$
$79$	0.363941	−	0.500922i	0.0409466	−	0.0563582i	0.828999	+	0.559250i	$0.188911\pi$
$80$		0			0
$81$	2.78115	+	8.55951i	0.309017	+	0.951057i
$82$		0			0
$83$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$83$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$84$	7.61302	−	2.47362i	0.830649	−	0.269894i
$85$		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
$91$	4.50808	−	13.8744i	0.472575	−	1.45444i
$92$		0			0
$93$	2.42705	−	1.76336i	0.251673	−	0.182851i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	4.04508	+	2.93893i	0.410716	+	0.298403i	0.773892	−	0.633318i	$-0.218307\pi$
$97$	4.04508	+	2.93893i	0.410716	+	0.298403i	−0.363176	+	0.931721i	$0.618307\pi$
$98$		0			0
$99$		0			0
$100$	−10.0000			−1.00000
$101$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$101$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$102$		0			0
$103$	2.16312	+	6.65740i	0.213138	+	0.655973i	0.999281	+	0.0379269i	$0.0120754\pi$
$103$	2.16312	+	6.65740i	0.213138	+	0.655973i	−0.786142	+	0.618046i	$0.787925\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$107$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$108$	−8.40755	−	6.10844i	−0.809017	−	0.587785i
$109$			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
$112$	−5.43299	+	7.47787i	−0.513369	+	0.706592i
$113$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$113$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−18.0126	+	5.85265i	−1.66527	+	0.541078i
$118$		0			0
$119$		0			0
$120$		0			0
$121$		0			0
$122$		0			0
$123$		0			0
$124$	−1.07047	+	3.29456i	−0.0961307	+	0.295860i
$125$		0			0
$126$		0			0
$127$	9.77416	+	13.4530i	0.867316	+	1.19376i	0.979775	+	0.200102i	$0.0641274\pi$
$127$	9.77416	+	13.4530i	0.867316	+	1.19376i	−0.112459	+	0.993656i	$0.535873\pi$
$128$		0			0
$129$	2.78656	+	0.905408i	0.245343	+	0.0797168i
$130$		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		0.809017	−	0.587785i	$-0.200000\pi$
$137$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$138$		0			0
$139$	22.4080	−	7.28080i	1.90062	−	0.617549i	0.938074	−	0.346436i	$-0.112608\pi$
$139$	22.4080	−	7.28080i	1.90062	−	0.617549i	0.962547	−	0.271113i	$-0.0873918\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	12.0000			1.00000
$145$		0			0
$146$		0			0
$147$	−0.888623	−	2.73490i	−0.0732924	−	0.225571i
$148$	8.40755	−	6.10844i	0.691096	−	0.502111i
$149$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$149$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$150$		0			0
$151$	13.6172	+	4.42451i	1.10815	+	0.360061i	0.805235	−	0.592955i	$-0.202039\pi$
$151$	13.6172	+	4.42451i	1.10815	+	0.360061i	0.302919	+	0.953016i	$0.402039\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	12.8546	−	17.6928i	1.02919	−	1.41656i
$157$	−6.95803	+	21.4146i	−0.555311	+	1.70907i	0.139808	+	0.990179i	$0.455351\pi$
$157$	−6.95803	+	21.4146i	−0.555311	+	1.70907i	−0.695120	+	0.718894i	$0.744649\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−13.7533	−	9.99235i	−1.07724	−	0.782661i	−0.100041	−	0.994983i	$-0.531897\pi$
$163$	−13.7533	−	9.99235i	−1.07724	−	0.782661i	−0.977200	+	0.212322i	$0.931897\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$167$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$168$		0			0
$169$	−8.29909	−	25.5420i	−0.638391	−	1.96477i
$170$		0			0
$171$	−15.2071	−	20.9308i	−1.16292	−	1.60062i
$172$	−3.21764	+	1.04548i	−0.245343	+	0.0797168i
$173$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$173$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$174$		0			0
$175$		−	11.5539i		−	0.873396i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$179$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$180$		0			0
$181$	−15.3713	+	11.1679i	−1.14254	+	0.830105i	−0.987471	−	0.157799i	$-0.949560\pi$
$181$	−15.3713	+	11.1679i	−1.14254	+	0.830105i	−0.155070	+	0.987903i	$0.549560\pi$
$182$		0			0
$183$	16.9927	−	5.52125i	1.25613	−	0.408142i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	7.05766	−	9.71404i	0.513369	−	0.706592i
$190$		0			0
$191$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$191$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$192$	−11.2101	+	8.14459i	−0.809017	+	0.587785i
$193$	3.08044	+	4.23986i	0.221735	+	0.305192i	0.905363	−	0.424639i	$-0.139599\pi$
$193$	3.08044	+	4.23986i	0.221735	+	0.305192i	−0.683628	+	0.729831i	$0.739599\pi$
$194$		0			0
$195$		0			0
$196$	2.68635	+	1.95175i	0.191882	+	0.139410i
$197$		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
$201$	6.48936	−	19.9722i	0.457724	−	1.40873i
$202$		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
$211$	16.2015	−	22.2994i	1.11535	−	1.53515i	0.302071	−	0.953286i	$-0.402322\pi$
$211$	16.2015	−	22.2994i	1.11535	−	1.53515i	0.813283	−	0.581868i	$-0.197678\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−3.80651	−	1.23681i	−0.258403	−	0.0839602i
$218$		0			0
$219$		−	22.9420i		−	1.55028i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−7.10739	+	21.8743i	−0.475946	+	1.46481i	0.368731	+	0.929536i	$0.379792\pi$
$223$	−7.10739	+	21.8743i	−0.475946	+	1.46481i	−0.844678	+	0.535275i	$0.820208\pi$
$224$		0			0
$225$	−12.1353	+	8.81678i	−0.809017	+	0.587785i
$226$		0			0
$227$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$227$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$228$	28.4122	+	9.23168i	1.88164	+	0.611383i
$229$	−16.8151	−	12.2169i	−1.11117	−	0.807315i	−0.128325	−	0.991732i	$-0.540960\pi$
$229$	−16.8151	−	12.2169i	−1.11117	−	0.807315i	−0.982848	+	0.184418i	$0.940960\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$233$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	0.630365	+	0.867623i	0.0409466	+	0.0563582i
$238$		0			0
$239$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$239$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$240$		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
$244$	−12.1267	+	16.6910i	−0.776333	+	1.06853i
$245$		0			0
$246$		0			0
$247$	44.0468	−	32.0019i	2.80263	−	2.03623i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$251$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$252$			13.8647i			0.873396i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	4.94427	−	15.2169i	0.309017	−	0.951057i
$257$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$257$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$258$		0			0
$259$	7.05766	+	9.71404i	0.438542	+	0.603601i
$260$		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
$268$	7.49326	+	23.0619i	0.457724	+	1.40873i
$269$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$269$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$270$		0			0
$271$	−7.18194	+	2.33355i	−0.436272	+	0.141753i	−0.518915	−	0.854826i	$-0.673664\pi$
$271$	−7.18194	+	2.33355i	−0.436272	+	0.141753i	0.0826430	+	0.996579i	$0.473664\pi$
$272$		0			0
$273$	20.4422	+	14.8521i	1.23722	+	0.898892i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−17.9236	+	24.6698i	−1.07693	+	1.48226i	−0.214068	+	0.976819i	$0.568672\pi$
$277$	−17.9236	+	24.6698i	−1.07693	+	1.48226i	−0.862859	+	0.505445i	$0.831328\pi$
$278$		0			0
$279$	1.60570	+	4.94183i	0.0961307	+	0.295860i
$280$		0			0
$281$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$281$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$282$		0			0
$283$	−17.4237	−	5.66132i	−1.03573	−	0.336530i	−0.258679	−	0.965963i	$-0.583287\pi$
$283$	−17.4237	−	5.66132i	−1.03573	−	0.336530i	−0.777055	+	0.629433i	$0.783287\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−5.25329	+	16.1680i	−0.309017	+	0.951057i
$290$		0			0
$291$	−7.00629	+	5.09037i	−0.410716	+	0.298403i
$292$	15.5711	+	21.4318i	0.911229	+	1.25420i
$293$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$293$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	5.35233	−	16.4728i	0.309017	−	0.951057i
$301$	−1.20794	−	3.71765i	−0.0696243	−	0.214282i
$302$		0			0
$303$		0			0
$304$	−32.8076	+	10.6598i	−1.88164	+	0.611383i
$305$		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		0.309017	−	0.951057i	$-0.400000\pi$
$311$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$312$		0			0
$313$	−22.4201	+	16.2892i	−1.26726	+	0.920719i	−0.999090	−	0.0426523i	$-0.986419\pi$
$313$	−22.4201	+	16.2892i	−1.26726	+	0.920719i	−0.268171	+	0.963371i	$0.586419\pi$
$314$		0			0
$315$		0			0
$316$	−1.17774	−	0.382671i	−0.0662530	−	0.0215269i
$317$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$317$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	14.5623	−	10.5801i	0.809017	−	0.587785i
$325$	−18.5540	−	25.5374i	−1.02919	−	1.41656i
$326$		0			0
$327$	−32.2187	−	10.4685i	−1.78170	−	0.578909i
$328$		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
$333$	4.81710	−	14.8255i	0.263975	−	0.812433i
$334$		0			0
$335$		0			0
$336$	−9.41022	−	12.9521i	−0.513369	−	0.706592i
$337$	27.8233	−	9.04035i	1.51563	−	0.492459i	0.571101	−	0.820879i	$-0.306516\pi$
$337$	27.8233	−	9.04035i	1.51563	−	0.492459i	0.944532	+	0.328420i	$0.106516\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−11.7628	+	16.1901i	−0.635130	+	0.874181i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$347$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$348$		0			0
$349$	−21.3880	−	6.94940i	−1.14488	−	0.371993i	−0.325665	−	0.945485i	$-0.605588\pi$
$349$	−21.3880	−	6.94940i	−1.14488	−	0.371993i	−0.819211	+	0.573493i	$0.805588\pi$
$350$		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		0.951057	−	0.309017i	$-0.100000\pi$
$359$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$360$		0			0
$361$	44.7978	+	32.5475i	2.35778	+	1.71302i
$362$		0			0
$363$		0			0
$364$	−29.1769			−1.52929
$365$		0			0
$366$		0			0
$367$	−1.23607	−	3.80423i	−0.0645222	−	0.198579i	0.913598	−	0.406618i	$-0.133292\pi$
$367$	−1.23607	−	3.80423i	−0.0645222	−	0.198579i	−0.978121	+	0.208039i	$0.933292\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−4.85410	−	3.52671i	−0.251673	−	0.182851i
$373$		−	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
$379$	−6.47214	+	4.70228i	−0.332451	+	0.241540i	−0.741470	−	0.670986i	$-0.765871\pi$
$379$	−6.47214	+	4.70228i	−0.332451	+	0.241540i	0.409019	+	0.912526i	$0.365871\pi$
$380$		0			0
$381$	−27.3922	+	8.90028i	−1.40335	+	0.455975i
$382$		0			0
$383$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$383$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−2.98292	+	4.10564i	−0.151630	+	0.208701i
$388$	3.09017	−	9.51057i	0.156880	−	0.482826i
$389$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$389$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$390$		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
$399$	−10.6662	+	32.8273i	−0.533980	+	1.64342i
$400$	6.18034	+	19.0211i	0.309017	+	0.951057i
$401$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$401$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$402$		0			0
$403$	−10.3996	+	3.37903i	−0.518040	+	0.168321i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−2.08613	+	2.87131i	−0.103153	+	0.141977i	−0.857473	−	0.514529i	$-0.827967\pi$
$409$	−2.08613	+	2.87131i	−0.103153	+	0.141977i	0.754320	+	0.656507i	$0.227967\pi$
$410$		0			0
$411$		0			0
$412$	11.3262	−	8.22899i	0.558004	−	0.405413i
$413$		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
$421$	10.7047	−	32.9456i	0.521713	−	1.60567i	−0.249012	−	0.968501i	$-0.580106\pi$
$421$	10.7047	−	32.9456i	0.521713	−	1.60567i	0.770725	−	0.637168i	$-0.219894\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−19.2847	−	14.0111i	−0.933251	−	0.678046i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$431$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$432$	−6.42280	+	19.7673i	−0.309017	+	0.951057i
$433$	−11.4336	−	35.1891i	−0.549465	−	1.69108i	−0.710130	−	0.704071i	$-0.751364\pi$
$433$	−11.4336	−	35.1891i	−0.549465	−	1.69108i	0.160665	−	0.987009i	$-0.448636\pi$
$434$		0			0
$435$		0			0
$436$	37.2030	−	12.0880i	1.78170	−	0.578909i
$437$		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		0.309017	−	0.951057i	$-0.400000\pi$
$443$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$444$	5.56231	+	17.1190i	0.263975	+	0.812433i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	17.5815	+	5.71258i	0.830649	+	0.269894i
$449$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$449$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−14.5768	+	20.0632i	−0.684877	+	0.942652i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	13.1210	+	18.0595i	0.613775	+	0.844789i	0.996881	−	0.0789151i	$-0.0251456\pi$
$457$	13.1210	+	18.0595i	0.613775	+	0.844789i	−0.383106	+	0.923704i	$0.625146\pi$
$458$		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		0.809017	−	0.587785i	$-0.200000\pi$
$467$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$468$	22.2648	+	30.6449i	1.02919	+	1.41656i
$469$	−26.6456	+	8.65768i	−1.23038	+	0.399774i
$470$		0			0
$471$	−31.5517	−	22.9236i	−1.45382	−	1.05626i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	25.3452	−	34.8847i	1.16292	−	1.60062i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$479$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$480$		0			0
$481$	31.1988	+	10.1371i	1.42254	+	0.462212i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−1.07047	+	3.29456i	−0.0485075	+	0.149291i	−0.972376	−	0.233418i	$-0.925009\pi$
$487$	−1.07047	+	3.29456i	−0.0485075	+	0.149291i	0.923869	+	0.382709i	$0.125009\pi$
$488$		0			0
$489$	23.8214	−	17.3073i	1.07724	−	0.782661i
$490$		0			0
$491$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$491$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	6.92820			0.311086
$497$		0			0
$498$		0			0
$499$	13.3808	+	41.1820i	0.599008	+	1.84356i	0.533667	+	0.845694i	$0.320813\pi$
$499$	13.3808	+	41.1820i	0.599008	+	1.84356i	0.0653408	+	0.997863i	$0.479187\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$503$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	46.5167			2.06588
$508$	19.5483	−	26.9059i	0.867316	−	1.19376i
$509$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$509$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$510$		0			0
$511$	−24.7621	+	17.9908i	−1.09541	+	0.795864i
$512$		0			0
$513$	42.6183	−	13.8475i	1.88164	−	0.611383i
$514$		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		−0.309017	−	0.951057i	$-0.600000\pi$
$521$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$522$		0			0
$523$	−24.3509	−	33.5162i	−1.06479	−	1.46556i	−0.875239	−	0.483690i	$-0.839296\pi$
$523$	−24.3509	−	33.5162i	−1.06479	−	1.46556i	−0.189553	−	0.981870i	$-0.560704\pi$
$524$		0			0
$525$	19.0326	+	6.18405i	0.830649	+	0.269894i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	23.0000			1.00000
$530$		0			0
$531$		0			0
$532$	−12.3163	−	37.9057i	−0.533980	−	1.64342i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	21.9984	−	30.2782i	0.945785	−	1.30176i	−0.00759004	−	0.999971i	$-0.502416\pi$
$541$	21.9984	−	30.2782i	0.945785	−	1.30176i	0.953375	−	0.301790i	$-0.0975840\pi$
$542$		0			0
$543$	−10.1694	−	31.2983i	−0.436412	−	1.34314i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	43.2072	+	14.0389i	1.84740	+	0.600258i	0.997282	+	0.0736732i	$0.0234722\pi$
$547$	43.2072	+	14.0389i	1.84740	+	0.600258i	0.850122	+	0.526585i	$0.176528\pi$
$548$		0			0
$549$			30.9468i			1.32078i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	0.442136	−	1.36075i	0.0188015	−	0.0578651i
$554$		0			0
$555$		0			0
$556$	−27.6978	−	38.1228i	−1.17465	−	1.61676i
$557$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$557$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$558$		0			0
$559$	−8.63989	−	6.27725i	−0.365428	−	0.265499i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$563$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	12.2242	+	16.8252i	0.513369	+	0.706592i
$568$		0			0
$569$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$569$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$570$		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
$576$	−7.41641	−	22.8254i	−0.309017	−	0.951057i
$577$	−37.8340	+	27.4880i	−1.57505	+	1.14434i	−0.652941	+	0.757409i	$0.726465\pi$
$577$	−37.8340	+	27.4880i	−1.57505	+	1.14434i	−0.922109	+	0.386931i	$0.873535\pi$
$578$		0			0
$579$	−8.63298	+	2.80502i	−0.358774	+	0.116573i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$587$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$588$	−4.65289	+	3.38052i	−0.191882	+	0.139410i
$589$	−8.77985	−	12.0844i	−0.361767	−	0.497930i
$590$		0			0
$591$		0			0
$592$	−16.8151	−	12.2169i	−0.691096	−	0.502111i
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−13.9058	+	42.7975i	−0.569125	+	1.75159i
$598$		0			0
$599$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$599$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$600$		0			0
$601$	31.7876	−	10.3284i	1.29664	−	0.421305i	0.422232	−	0.906488i	$-0.361247\pi$
$601$	31.7876	−	10.3284i	1.29664	−	0.421305i	0.874413	+	0.485183i	$0.161247\pi$
$602$		0			0
$603$	29.4264	+	21.3796i	1.19834	+	0.870643i
$604$		−	28.6360i		−	1.16518i
$605$		0			0
$606$		0			0
$607$	10.4045	−	14.3206i	0.422307	−	0.581255i	−0.543859	−	0.839176i	$-0.683038\pi$
$607$	10.4045	−	14.3206i	0.422307	−	0.581255i	0.966166	+	0.257921i	$0.0830375\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−47.0137	−	15.2757i	−1.89886	−	0.616978i	−0.967485	−	0.252929i	$-0.918606\pi$
$613$	−47.0137	−	15.2757i	−1.89886	−	0.616978i	−0.931380	−	0.364049i	$-0.881394\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$		0			0
$619$	9.88854	−	30.4338i	0.397454	−	1.22324i	−0.529580	−	0.848260i	$-0.677650\pi$
$619$	9.88854	−	30.4338i	0.397454	−	1.22324i	0.927034	−	0.374978i	$-0.122350\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	−41.5983	−	13.5161i	−1.66527	−	0.541078i
$625$	−20.2254	−	14.6946i	−0.809017	−	0.587785i
$626$		0			0
$627$		0			0
$628$	45.0333			1.79703
$629$		0			0
$630$		0			0
$631$	7.49326	+	23.0619i	0.298302	+	0.918080i	0.982092	+	0.188401i	$0.0603304\pi$
$631$	7.49326	+	23.0619i	0.298302	+	0.918080i	−0.683790	+	0.729679i	$0.739670\pi$
$632$		0			0
$633$	28.0617	+	38.6237i	1.11535	+	1.53515i
$634$		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		0.309017	−	0.951057i	$-0.400000\pi$
$641$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$642$		0			0
$643$	38.0238	−	27.6259i	1.49951	−	1.08946i	0.528937	−	0.848661i	$-0.322591\pi$
$643$	38.0238	−	27.6259i	1.49951	−	1.08946i	0.970575	−	0.240798i	$-0.0774093\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$647$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	4.07474	−	5.60840i	0.159702	−	0.219811i
$652$	−10.5066	+	32.3359i	−0.411469	+	1.26637i
$653$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$653$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	37.7918	+	12.2793i	1.47440	+	0.479061i
$658$		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
$669$	−32.2289	−	23.4157i	−1.24604	−	0.905303i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−15.4736	+	21.2975i	−0.596462	+	0.820960i	−0.995379	−	0.0960273i	$-0.969386\pi$
$673$	−15.4736	+	21.2975i	−0.596462	+	0.820960i	0.398916	+	0.916987i	$0.369386\pi$
$674$		0			0
$675$	−8.02850	−	24.7092i	−0.309017	−	0.951057i
$676$	−43.4546	+	31.5716i	−1.67133	+	1.21429i
$677$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$677$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$678$		0			0
$679$	10.9885	+	3.57037i	0.421698	+	0.137018i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		1.00000			$0$
$683$		0			0		−1.00000			$\pi$
$684$	−30.4143	+	41.8617i	−1.16292	+	1.60062i
$685$		0			0
$686$		0			0
$687$	29.1246	−	21.1603i	1.11117	−	0.807315i
$688$	3.97723	+	5.47418i	0.151630	+	0.208701i
$689$		0			0
$690$		0			0
$691$	39.6418	+	28.8015i	1.50805	+	1.09566i	0.967035	+	0.254645i	$0.0819585\pi$
$691$	39.6418	+	28.8015i	1.50805	+	1.09566i	0.541012	+	0.841015i	$0.318041\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−21.9769	+	7.14073i	−0.830649	+	0.269894i
$701$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$701$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$702$		0			0
$703$			44.8115i			1.69010i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	39.2352	−	28.5061i	1.47351	−	1.07057i	0.493933	−	0.869500i	$-0.335559\pi$
$709$	39.2352	−	28.5061i	1.47351	−	1.07057i	0.979577	−	0.201068i	$-0.0644412\pi$
$710$		0			0
$711$	−1.76661	+	0.574006i	−0.0662530	+	0.0215269i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$719$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$720$		0			0
$721$	9.50773	+	13.0863i	0.354087	+	0.487358i
$722$		0			0
$723$	−15.9727	−	5.18985i	−0.594031	−	0.193012i
$724$	30.7426	+	22.3358i	1.14254	+	0.830105i
$725$		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
$729$	8.34346	−	25.6785i	0.309017	−	0.951057i
$730$		0			0
$731$		0			0
$732$	−21.0041	−	28.9096i	−0.776333	−	1.06853i
$733$	−47.6025	+	15.4670i	−1.75824	+	0.571287i	−0.997016	−	0.0772015i	$-0.975402\pi$
$733$	−47.6025	+	15.4670i	−1.75824	+	0.571287i	−0.761225	+	0.648488i	$0.775402\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−27.0674	+	37.2551i	−0.995692	+	1.37045i	−0.0677601	+	0.997702i	$0.521585\pi$
$739$	−27.0674	+	37.2551i	−0.995692	+	1.37045i	−0.927932	+	0.372750i	$0.878415\pi$
$740$		0			0
$741$	29.1407	+	89.6858i	1.07051	+	3.29469i
$742$		0			0
$743$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$743$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	16.5922	−	51.0656i	0.605459	−	1.86341i	0.111854	−	0.993725i	$-0.464321\pi$
$751$	16.5922	−	51.0656i	0.605459	−	1.86341i	0.493604	−	0.869687i	$-0.335679\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	−22.8391	−	7.42087i	−0.830649	−	0.269894i
$757$	−1.40126	−	1.01807i	−0.0509296	−	0.0370025i	0.562029	−	0.827117i	$-0.310021\pi$
$757$	−1.40126	−	1.01807i	−0.0509296	−	0.0370025i	−0.612959	+	0.790115i	$0.710021\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$761$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$762$		0			0
$763$	13.9664	+	42.9841i	0.505617	+	1.55613i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	22.4201	+	16.2892i	0.809017	+	0.587785i
$769$		−	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
$772$	6.16087	−	8.47972i	0.221735	−	0.305192i
$773$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$773$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$774$		0			0
$775$	−7.00629	+	5.09037i	−0.251673	+	0.182851i
$776$		0			0
$777$	−19.7792	+	6.42666i	−0.709576	+	0.230555i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	2.05219	−	6.31598i	0.0732924	−	0.225571i
$785$		0			0
$786$		0			0
$787$	24.7149	+	34.0171i	0.880990	+	1.21258i	0.976146	+	0.217117i	$0.0696653\pi$
$787$	24.7149	+	34.0171i	0.880990	+	1.21258i	−0.0951552	+	0.995462i	$0.530335\pi$
$788$		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
$796$	−16.0570	−	49.4183i	−0.569125	−	1.75159i
$797$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$797$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$798$		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		−0.587785	−	0.809017i	$-0.700000\pi$
$809$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$810$		0			0
$811$	4.23759	+	1.37688i	0.148802	+	0.0483487i	0.382471	−	0.923967i	$-0.375073\pi$
$811$	4.23759	+	1.37688i	0.148802	+	0.0483487i	−0.233669	+	0.972316i	$0.575073\pi$
$812$		0			0
$813$		−	13.0797i		−	0.458723i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	4.50808	−	13.8744i	0.157718	−	0.485405i
$818$		0			0
$819$	−35.4069	+	25.7246i	−1.23722	+	0.898892i
$820$		0			0
$821$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$821$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$822$		0			0
$823$	4.04508	+	2.93893i	0.141003	+	0.102445i	0.656051	−	0.754717i	$-0.272226\pi$
$823$	4.04508	+	2.93893i	0.141003	+	0.102445i	−0.515048	+	0.857161i	$0.672226\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$827$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$828$		0			0
$829$	2.16312	+	6.65740i	0.0751282	+	0.231221i	0.981568	−	0.191115i	$-0.0612103\pi$
$829$	2.16312	+	6.65740i	0.0751282	+	0.231221i	−0.906439	+	0.422336i	$0.861210\pi$
$830$		0			0
$831$	−31.0447	−	42.7293i	−1.07693	−	1.48226i
$832$	48.0336	−	15.6071i	1.66527	−	0.541078i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−9.00000			−0.311086
$838$		0			0
$839$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$839$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$840$		0			0
$841$	23.4615	−	17.0458i	0.809017	−	0.587785i
$842$		0			0
$843$		0			0
$844$	−52.4290	−	17.0352i	−1.80468	−	0.586376i
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	18.6515	−	25.6716i	0.640119	−	0.881048i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−12.7571	−	17.5586i	−0.436794	−	0.601195i	0.532702	−	0.846303i	$-0.321177\pi$
$853$	−12.7571	−	17.5586i	−0.436794	−	0.601195i	−0.969496	+	0.245108i	$0.921177\pi$
$854$		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		0.809017	−	0.587785i	$-0.200000\pi$
$863$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−23.8214	−	17.3073i	−0.809017	−	0.587785i
$868$			8.00481i			0.271701i
$869$		0			0
$870$		0			0
$871$	−44.9911	+	61.9249i	−1.52446	+	2.09825i
$872$		0			0
$873$	−4.63525	−	14.2658i	−0.156880	−	0.482826i
$874$		0			0
$875$		0			0
$876$	−43.6382	+	14.1789i	−1.47440	+	0.479061i
$877$	−36.1830	−	11.7566i	−1.22181	−	0.396991i	−0.374071	−	0.927400i	$-0.622038\pi$
$877$	−36.1830	−	11.7566i	−1.22181	−	0.396991i	−0.847742	+	0.530409i	$0.822038\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$		0			0
$883$	−6.95803	+	21.4146i	−0.234156	+	0.720659i	0.763076	+	0.646309i	$0.223688\pi$
$883$	−6.95803	+	21.4146i	−0.234156	+	0.720659i	−0.997232	+	0.0743502i	$0.976312\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$887$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$888$		0			0
$889$	31.0870	+	22.5860i	1.04262	+	0.757511i
$890$		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
$900$	24.2705	+	17.6336i	0.809017	+	0.587785i
$901$		0			0
$902$		0			0
$903$	6.77053			0.225309
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−15.3713	+	11.1679i	−0.510396	+	0.370825i	−0.812974	−	0.582300i	$-0.802153\pi$
$907$	−15.3713	+	11.1679i	−0.510396	+	0.370825i	0.302578	+	0.953125i	$0.402153\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$911$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$912$		−	59.7487i		−	1.97848i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−12.8456	+	39.5347i	−0.424430	+	1.30626i
$917$		0			0
$918$		0			0
$919$	25.6117	+	35.2514i	0.844851	+	1.16284i	0.984974	+	0.172704i	$0.0552503\pi$
$919$	25.6117	+	35.2514i	0.844851	+	1.16284i	−0.140123	+	0.990134i	$0.544750\pi$
$920$		0			0
$921$	12.1662	+	3.95304i	0.400890	+	0.130257i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	25.9808			0.854242
$926$		0			0
$927$	6.48936	−	19.9722i	0.213138	−	0.655973i
$928$		0			0
$929$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$929$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$930$		0			0
$931$	−13.6172	+	4.42451i	−0.446287	+	0.145007i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−6.32978	+	8.71219i	−0.206785	+	0.284615i	−0.899795	−	0.436313i	$-0.856284\pi$
$937$	−6.32978	+	8.71219i	−0.206785	+	0.284615i	0.693010	+	0.720928i	$0.256284\pi$
$938$		0			0
$939$	−14.8328	−	45.6507i	−0.484051	−	1.48975i
$940$		0			0
$941$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$941$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$942$		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			$\pi$
$948$	1.26073	−	1.73525i	0.0409466	−	0.0563582i
$949$	−25.8406	+	79.5290i	−0.838820	+	2.58162i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$953$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−8.65248	−	26.6296i	−0.279112	−	0.859019i
$962$		0			0
$963$		0			0
$964$	18.4437	−	5.99272i	0.594031	−	0.193012i
$965$		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		0.309017	−	0.951057i	$-0.400000\pi$
$971$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$972$	9.63420	+	29.6510i	0.309017	+	0.951057i
$973$	44.0468	−	32.0019i	1.41208	−	1.02593i
$974$		0			0
$975$	51.9979	−	16.8952i	1.66527	−	0.541078i
$976$	39.2429	+	12.7508i	1.25613	+	0.408142i
$977$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$977$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	34.4890	−	47.4701i	1.10115	−	1.51560i
$982$		0			0
$983$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$983$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	−88.0936	−	64.0038i	−2.80263	−	2.03623i
$989$		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
$993$	16.5922	−	51.0656i	0.526538	−	1.62052i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	6.16199	−	2.00215i	0.195152	−	0.0634088i	−0.209810	−	0.977742i	$-0.567285\pi$
$997$	6.16199	−	2.00215i	0.195152	−	0.0634088i	0.404962	+	0.914333i	$0.367285\pi$
$998$		0			0
$999$	21.8435	+	15.8702i	0.691096	+	0.502111i

Twists

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

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

\(f(q)\)	\(=\)	\(q+(-0.535233 + 1.64728i) q^{3} +(-0.618034 - 1.90211i) q^{4} +(2.19769 - 0.714073i) q^{7} +(-2.42705 - 1.76336i) q^{9} +O(q^{10})\)	\(q+(-0.535233 + 1.64728i) q^{3} +(-0.618034 - 1.90211i) q^{4} +(2.19769 - 0.714073i) q^{7} +(-2.42705 - 1.76336i) q^{9} +3.46410 q^{12} +(3.71080 - 5.10748i) q^{13} +(-3.23607 + 2.35114i) q^{16} +(8.20189 + 2.66496i) q^{19} +4.00240i q^{21} +(1.54508 - 4.75528i) q^{25} +(4.20378 - 3.05422i) q^{27} +(-2.71650 - 3.73894i) q^{28} +(-1.40126 - 1.01807i) q^{31} +(-1.85410 + 5.70634i) q^{36} +(1.60570 + 4.94183i) q^{37} +(6.42730 + 8.84642i) q^{39} -1.69161i q^{43} +(-2.14093 - 6.58911i) q^{48} +(-1.34317 + 0.975873i) q^{49} +(-12.0084 - 3.90177i) q^{52} +(-8.77985 + 12.0844i) q^{57} +(-6.06336 - 8.34549i) q^{61} +(-6.59307 - 2.14222i) q^{63} +(6.47214 + 4.70228i) q^{64} -12.1244 q^{67} +(-12.5973 + 4.09310i) q^{73} +(7.00629 + 5.09037i) q^{75} -17.2480i q^{76} +(0.363941 - 0.500922i) q^{79} +(2.78115 + 8.55951i) q^{81} +(7.61302 - 2.47362i) q^{84} +(4.50808 - 13.8744i) q^{91} +(2.42705 - 1.76336i) q^{93} +(4.04508 + 2.93893i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 8 q^{4} - 12 q^{9}+O(q^{10}) \) 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

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