LMFDB - Embedded newform 363.2.f.g.215.2

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			215.2
Root			$1.83730 - 0.596975i$ of defining polynomial
Character	$\chi$	$=$	363.215
Dual form			363.2.f.g.233.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.40126 - 1.01807i) q^{3} +(1.61803 - 1.17557i) q^{4} +(2.79802 + 3.85115i) q^{7} +(0.927051 + 2.85317i) q^{9} +O(q^{10})$	$q+(-1.40126 - 1.01807i) q^{3} +(1.61803 - 1.17557i) q^{4} +(2.79802 + 3.85115i) q^{7} +(0.927051 + 2.85317i) q^{9} -3.46410 q^{12} +(3.31421 - 1.07685i) q^{13} +(1.23607 - 3.80423i) q^{16} +(0.749728 - 1.03191i) q^{19} -8.24504i q^{21} +(-4.04508 - 2.93893i) q^{25} +(1.60570 - 4.94183i) q^{27} +(9.05459 + 2.94201i) q^{28} +(-0.535233 - 1.64728i) q^{31} +(4.85410 + 3.52671i) q^{36} +(4.20378 - 3.05422i) q^{37} +(-5.74038 - 1.86516i) q^{39} +13.0053i q^{43} +(-5.60503 + 4.07230i) q^{48} +(-4.83928 + 14.8938i) q^{49} +(4.09659 - 5.63847i) q^{52} +(-2.10112 + 0.682697i) q^{57} +(-11.1557 - 3.62471i) q^{61} +(-8.39406 + 11.5534i) q^{63} +(-2.47214 - 7.60845i) q^{64} +12.1244 q^{67} +(-6.34577 - 8.73420i) q^{73} +(2.67617 + 8.23639i) q^{75} -2.55103i q^{76} +(-16.8961 + 5.48987i) q^{79} +(-7.28115 + 5.29007i) q^{81} +(-9.69263 - 13.3408i) q^{84} +(13.4203 + 9.75045i) q^{91} +(-0.927051 + 2.85317i) q^{93} +(-1.54508 - 4.75528i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	2.79802	+	3.85115i	1.05755	+	1.45560i	0.882073	+	0.471114i	$0.156148\pi$
$7$	2.79802	+	3.85115i	1.05755	+	1.45560i	0.175480	+	0.984483i	$0.443852\pi$
$8$		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
$13$	3.31421	−	1.07685i	0.919196	−	0.298665i	0.189059	−	0.981966i	$-0.439456\pi$
$13$	3.31421	−	1.07685i	0.919196	−	0.298665i	0.730137	+	0.683301i	$0.239456\pi$
$14$		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
$19$	0.749728	−	1.03191i	0.171999	−	0.236737i	−0.714311	−	0.699828i	$-0.753260\pi$
$19$	0.749728	−	1.03191i	0.171999	−	0.236737i	0.886310	+	0.463091i	$0.153260\pi$
$20$		0			0
$21$		−	8.24504i		−	1.79922i
$22$		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
$28$	9.05459	+	2.94201i	1.71116	+	0.555988i
$29$		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	0.891416	−	0.453186i	$-0.149713\pi$
$31$	−0.535233	−	1.64728i	−0.0961307	−	0.295860i	−0.987547	+	0.157326i	$0.949713\pi$
$32$		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	−0.185924	−	0.982564i	$-0.559528\pi$
$37$	4.20378	−	3.05422i	0.691096	−	0.502111i	0.877020	+	0.480453i	$0.159528\pi$
$38$		0			0
$39$	−5.74038	−	1.86516i	−0.919196	−	0.298665i
$40$		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
$43$			13.0053i			1.98329i	0.128984	+	0.991647i	$0.458828\pi$
$43$			13.0053i			1.98329i	−0.128984	+	0.991647i	$0.541172\pi$
$44$		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
$49$	−4.83928	+	14.8938i	−0.691326	+	2.12768i
$50$		0			0
$51$		0			0
$52$	4.09659	−	5.63847i	0.568095	−	0.781915i
$53$		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
$57$	−2.10112	+	0.682697i	−0.278301	+	0.0904254i
$58$		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
$61$	−11.1557	−	3.62471i	−1.42834	−	0.464097i	−0.510100	−	0.860115i	$-0.670392\pi$
$61$	−11.1557	−	3.62471i	−1.42834	−	0.464097i	−0.918243	+	0.396018i	$0.870392\pi$
$62$		0			0
$63$	−8.39406	+	11.5534i	−1.05755	+	1.45560i
$64$	−2.47214	−	7.60845i	−0.309017	−	0.951057i
$65$		0			0
$66$		0			0
$67$	12.1244			1.48123			0.740613	−	0.671932i	$-0.234535\pi$
$67$	12.1244			1.48123			0.740613	+	0.671932i	$0.234535\pi$
$68$		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
$73$	−6.34577	−	8.73420i	−0.742716	−	1.02226i	−0.998458	−	0.0555161i	$-0.982320\pi$
$73$	−6.34577	−	8.73420i	−0.742716	−	1.02226i	0.255741	−	0.966745i	$-0.417680\pi$
$74$		0			0
$75$	2.67617	+	8.23639i	0.309017	+	0.951057i
$76$		−	2.55103i		−	0.292623i
$77$		0			0
$78$		0			0
$79$	−16.8961	+	5.48987i	−1.90096	+	0.617659i	−0.939716	+	0.341956i	$0.888911\pi$
$79$	−16.8961	+	5.48987i	−1.90096	+	0.617659i	−0.961243	+	0.275703i	$0.911089\pi$
$80$		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	$0.100000\pi$
$84$	−9.69263	−	13.3408i	−1.05755	−	1.45560i
$85$		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
$91$	13.4203	+	9.75045i	1.40683	+	1.02212i
$92$		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
$109$			7.31130i			0.700296i	0.936694	+	0.350148i	$0.113869\pi$
$109$			7.31130i			0.700296i	−0.936694	+	0.350148i	$0.886131\pi$
$110$		0			0
$111$	−9.00000			−0.854242
$112$	18.1092	−	5.88403i	1.71116	−	0.555988i
$113$		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
$117$	6.14488	+	8.45770i	0.568095	+	0.781915i
$118$		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
$127$	14.4699	+	4.70156i	1.28400	+	0.417196i	0.869987	−	0.493075i	$-0.164127\pi$
$127$	14.4699	+	4.70156i	1.28400	+	0.417196i	0.414011	+	0.910272i	$0.364127\pi$
$128$		0			0
$129$	13.2404	−	18.2238i	1.16575	−	1.60452i
$130$		0			0
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$		0			0
$133$	6.07180			0.526492
$134$		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
$139$	−0.548839	−	0.755412i	−0.0465519	−	0.0640732i	0.785106	−	0.619361i	$-0.212608\pi$
$139$	−0.548839	−	0.755412i	−0.0465519	−	0.0640732i	−0.831658	+	0.555288i	$0.812608\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	12.0000			1.00000
$145$		0			0
$146$		0			0
$147$	21.9441	−	15.9433i	1.80992	−	1.31498i
$148$	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
$151$	−11.7409	+	16.1600i	−0.955463	+	1.31508i	−0.00640530	+	0.999979i	$0.502039\pi$
$151$	−11.7409	+	16.1600i	−0.955463	+	1.31508i	−0.949058	+	0.315102i	$0.897961\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−11.4808	+	3.73032i	−0.919196	+	0.298665i
$157$	−18.2164	−	13.2350i	−1.45382	−	1.05626i	−0.984919	−	0.173016i	$-0.944649\pi$
$157$	−18.2164	−	13.2350i	−1.45382	−	1.05626i	−0.468905	−	0.883249i	$-0.655351\pi$
$158$		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
$169$	−0.692847	+	0.503383i	−0.0532960	+	0.0387218i
$170$		0			0
$171$	3.63925	+	1.18247i	0.278301	+	0.0904254i
$172$	15.2887	+	21.0431i	1.16575	+	1.60452i
$173$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$173$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$174$		0			0
$175$		−	23.8014i		−	1.79922i
$176$		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
$183$	11.9418	+	16.4365i	0.882764	+	1.21502i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	23.5245	−	7.64358i	1.71116	−	0.555988i
$190$		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
$193$	−25.9507	−	8.43189i	−1.86797	−	0.606941i	−0.992272	−	0.124083i	$-0.960401\pi$
$193$	−25.9507	−	8.43189i	−1.86797	−	0.606941i	−0.875699	−	0.482857i	$-0.839599\pi$
$194$		0			0
$195$		0			0
$196$	9.67857	+	29.7876i	0.691326	+	2.12768i
$197$		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	$-0.872515\pi$
$199$	−25.9808			−1.84173			−0.920864	+	0.389885i	$0.872515\pi$
$200$		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
$208$		−	13.9391i		−	0.966500i
$209$		0			0
$210$		0			0
$211$	8.72954	−	2.83640i	0.600967	−	0.195266i	0.00729508	−	0.999973i	$-0.497678\pi$
$211$	8.72954	−	2.83640i	0.600967	−	0.195266i	0.593671	+	0.804708i	$0.297678\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	4.84632	−	6.67038i	0.328989	−	0.452815i
$218$		0			0
$219$			18.6993i			1.26358i
$220$		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	$0.300000\pi$
$228$	−2.59713	+	3.57465i	−0.171999	+	0.236737i
$229$	−6.42280	−	19.7673i	−0.424430	−	1.30626i	−0.903539	−	0.428507i	$-0.859040\pi$
$229$	−6.42280	−	19.7673i	−0.424430	−	1.30626i	0.479108	−	0.877756i	$-0.340960\pi$
$230$		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
$237$	29.2649	+	9.50874i	1.90096	+	0.617659i
$238$		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
$241$		−	29.4954i		−	1.89997i	−0.312301	−	0.949983i	$-0.601100\pi$
$241$		−	29.4954i		−	1.89997i	0.312301	−	0.949983i	$-0.398900\pi$
$242$		0			0
$243$	15.5885			1.00000
$244$	−22.3114	+	7.24942i	−1.42834	+	0.464097i
$245$		0			0
$246$		0			0
$247$	1.37354	−	4.22732i	0.0873962	−	0.268978i
$248$		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	$0.400000\pi$
$252$			28.5617i			1.79922i
$253$		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
$259$	23.5245	+	7.64358i	1.46174	+	0.474949i
$260$		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
$271$	−18.8364	−	25.9261i	−1.14423	−	1.57490i	−0.757673	−	0.652634i	$-0.773664\pi$
$271$	−18.8364	−	25.9261i	−1.14423	−	1.57490i	−0.386558	−	0.922265i	$-0.626336\pi$
$272$		0			0
$273$	−8.87869	−	27.3258i	−0.537363	−	1.65383i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	12.6938	−	4.12448i	0.762699	−	0.247816i	0.0982624	−	0.995161i	$-0.468672\pi$
$277$	12.6938	−	4.12448i	0.762699	−	0.247816i	0.664437	+	0.747345i	$0.268672\pi$
$278$		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
$283$	16.5872	−	22.8304i	0.986009	−	1.35713i	0.0524806	−	0.998622i	$-0.483287\pi$
$283$	16.5872	−	22.8304i	0.986009	−	1.35713i	0.933528	−	0.358503i	$-0.116713\pi$
$284$		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
$292$	−20.5353	−	6.67234i	−1.20174	−	0.390469i
$293$		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
$301$	−50.0854	+	36.3892i	−2.88687	+	2.09744i
$302$		0			0
$303$		0			0
$304$	−2.99891	−	4.12765i	−0.171999	−	0.236737i
$305$		0			0
$306$		0			0
$307$			34.2557i			1.95508i	0.210760	+	0.977538i	$0.432406\pi$
$307$			34.2557i			1.95508i	−0.210760	+	0.977538i	$0.567594\pi$
$308$		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	$0.386419\pi$
$313$	−8.56373	+	26.3565i	−0.484051	+	1.48975i	−0.833351	+	0.552744i	$0.813581\pi$
$314$		0			0
$315$		0			0
$316$	−20.8847	+	28.7453i	−1.17486	+	1.61705i
$317$		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
$325$	−16.5710	−	5.38426i	−0.919196	−	0.298665i
$326$		0			0
$327$	7.44345	−	10.2450i	0.411624	−	0.566551i
$328$		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
$336$	−31.3660	−	10.1914i	−1.71116	−	0.555988i
$337$	−13.0395	−	17.9473i	−0.710306	−	0.977653i	−0.999790	−	0.0204700i	$-0.993484\pi$
$337$	−13.0395	−	17.9473i	−0.710306	−	0.977653i	0.289484	−	0.957183i	$-0.406516\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−39.2075	+	12.7393i	−2.11701	+	0.687857i
$344$		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
$349$	−17.5379	+	24.1388i	−0.938780	+	1.29212i	0.0175547	+	0.999846i	$0.494412\pi$
$349$	−17.5379	+	24.1388i	−0.938780	+	1.29212i	−0.956335	+	0.292274i	$0.905588\pi$
$350$		0			0
$351$		−	18.1074i		−	0.966500i
$352$		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
$361$	5.36857	+	16.5228i	0.282556	+	0.869619i
$362$		0			0
$363$		0			0
$364$	33.1769			1.73894
$365$		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
$373$			3.14299i			0.162738i	0.996684	+	0.0813690i	$0.0259292\pi$
$373$			3.14299i			0.162738i	−0.996684	+	0.0813690i	$0.974071\pi$
$374$		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
$381$	−15.4896	−	21.3196i	−0.793554	−	1.09223i
$382$		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
$387$	−37.1064	+	12.0566i	−1.88622	+	0.612871i
$388$	−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
$399$	−8.50816	−	6.18154i	−0.425941	−	0.309464i
$400$	−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
$403$	−3.54775	−	4.88306i	−0.176726	−	0.243242i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	38.3195	−	12.4508i	1.89478	−	0.615650i	0.920273	−	0.391277i	$-0.127967\pi$
$409$	38.3195	−	12.4508i	1.89478	−	0.615650i	0.974503	−	0.224373i	$-0.0720334\pi$
$410$		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
$417$			1.61729i			0.0791988i
$418$		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	$0.0198943\pi$
$421$	28.0252	+	20.3615i	1.36586	+	0.992358i	0.367816	+	0.929899i	$0.380106\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−17.2546	−	53.1043i	−0.835010	−	2.56990i
$428$		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
$436$	8.59495	+	11.8299i	0.411624	+	0.566551i
$437$		0			0
$438$		0			0
$439$			22.8677i			1.09141i	0.837976	+	0.545707i	$0.183739\pi$
$439$			22.8677i			1.09141i	−0.837976	+	0.545707i	$0.816261\pi$
$440$		0			0
$441$	−46.9808			−2.23718
$442$		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
$448$	22.3842	−	30.8092i	1.05755	−	1.45560i
$449$		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
$453$	32.9041	−	10.6912i	1.54597	−	0.502317i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	34.6802	+	11.2683i	1.62227	+	0.527108i	0.972477	−	0.233001i	$-0.0748544\pi$
$457$	34.6802	+	11.2683i	1.62227	+	0.527108i	0.649796	+	0.760109i	$0.274854\pi$
$458$		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	$0.600000\pi$
$468$	19.8853	+	6.46111i	0.919196	+	0.298665i
$469$	33.9242	+	46.6927i	1.56647	+	2.15607i
$470$		0			0
$471$	12.0517	+	37.0912i	0.555311	+	1.70907i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−6.06542	+	1.97078i	−0.278301	+	0.0904254i
$476$		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
$481$	10.6432	−	14.6492i	0.485290	−	0.667945i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−2.80252	−	2.03615i	−0.126994	−	0.0922667i	0.522475	−	0.852655i	$-0.325009\pi$
$487$	−2.80252	−	2.03615i	−0.126994	−	0.0922667i	−0.649469	+	0.760388i	$0.725009\pi$
$488$		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	$-0.279187\pi$
$499$	35.0315	−	25.4518i	1.56822	−	1.13938i	0.928833	−	0.370499i	$-0.120813\pi$
$500$		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
$507$	1.48334			0.0658774
$508$	28.9398	−	9.40313i	1.28400	−	0.417196i
$509$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$509$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$510$		0			0
$511$	15.8811	−	48.8770i	0.702538	−	2.16219i
$512$		0			0
$513$	−3.89570	−	5.36197i	−0.171999	−	0.236737i
$514$		0			0
$515$		0			0
$516$		−	45.0518i		−	1.98329i
$517$		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
$523$	18.4342	+	5.98964i	0.806072	+	0.261909i	0.682933	−	0.730481i	$-0.260704\pi$
$523$	18.4342	+	5.98964i	0.806072	+	0.261909i	0.123139	+	0.992389i	$0.460704\pi$
$524$		0			0
$525$	−24.2316	+	33.3519i	−1.05755	+	1.45560i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	23.0000			1.00000
$530$		0			0
$531$		0			0
$532$	9.82437	−	7.13783i	0.425941	−	0.309464i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−26.2757	+	8.53750i	−1.12968	+	0.367056i	−0.813455	−	0.581628i	$-0.802416\pi$
$541$	−26.2757	+	8.53750i	−1.12968	+	0.367056i	−0.316227	+	0.948684i	$0.602416\pi$
$542$		0			0
$543$	−26.6239	+	19.3434i	−1.14254	+	0.830105i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	6.54666	−	9.01070i	0.279915	−	0.385270i	−0.645791	−	0.763514i	$-0.723472\pi$
$547$	6.54666	−	9.01070i	0.279915	−	0.385270i	0.925706	+	0.378245i	$0.123472\pi$
$548$		0			0
$549$		−	35.1894i		−	1.50185i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−68.4179	−	49.7085i	−2.90943	−	2.11382i
$554$		0			0
$555$		0			0
$556$	−1.77608	−	0.577083i	−0.0753225	−	0.0244738i
$557$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$557$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$558$		0			0
$559$	14.0048	+	43.1024i	0.592340	+	1.82304i
$560$		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
$567$	−40.7456	−	13.2391i	−1.71116	−	0.555988i
$568$		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
$571$		−	39.3577i		−	1.64707i	−0.567265	−	0.823535i	$-0.691999\pi$
$571$		−	39.3577i		−	1.64707i	0.567265	−	0.823535i	$-0.308001\pi$
$572$		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	$0.526465\pi$
$577$	−14.4513	+	44.4765i	−0.601615	+	1.85158i	−0.518569	+	0.855036i	$0.673535\pi$
$578$		0			0
$579$	27.7793	+	38.2350i	1.15447	+	1.58899i
$580$		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	$0.800000\pi$
$588$	16.7638	−	51.5936i	0.691326	−	2.12768i
$589$	−2.10112	−	0.682697i	−0.0865753	−	0.0281300i
$590$		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
$601$	21.0856	+	29.0218i	0.860100	+	1.18383i	0.981546	+	0.191228i	$0.0612469\pi$
$601$	21.0856	+	29.0218i	0.860100	+	1.18383i	−0.121446	+	0.992598i	$0.538753\pi$
$602$		0			0
$603$	11.2399	+	34.5928i	0.457724	+	1.40873i
$604$			39.9497i			1.62553i
$605$		0			0
$606$		0			0
$607$	43.7348	−	14.2103i	1.77514	−	0.576778i	0.776561	−	0.630042i	$-0.216962\pi$
$607$	43.7348	−	14.2103i	1.77514	−	0.576778i	0.998580	+	0.0532640i	$0.0169625\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−1.70034	+	2.34032i	−0.0686763	+	0.0945248i	−0.841973	−	0.539519i	$-0.818606\pi$
$613$	−1.70034	+	2.34032i	−0.0686763	+	0.0945248i	0.773297	+	0.634044i	$0.218606\pi$
$614$		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
$624$	−14.1910	+	19.5322i	−0.568095	+	0.781915i
$625$	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	−0.124303	−	0.992244i	$-0.539670\pi$
$631$	19.6176	−	14.2530i	0.780965	−	0.567404i	0.905269	+	0.424840i	$0.139670\pi$
$632$		0			0
$633$	−15.1200	−	4.91279i	−0.600967	−	0.195266i
$634$		0			0
$635$		0			0
$636$		0			0
$637$			54.5723i			2.16223i
$638$		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
$651$	−13.5819	+	4.41302i	−0.532316	+	0.172960i
$652$	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
$657$	19.0373	−	26.2026i	0.742716	−	1.02226i
$658$		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	$-0.0686249\pi$
$661$	50.2295			1.95370			0.976850	+	0.213925i	$0.0686249\pi$
$662$		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
$673$	−42.5217	+	13.8161i	−1.63909	+	0.532573i	−0.976335	−	0.216262i	$-0.930614\pi$
$673$	−42.5217	+	13.8161i	−1.63909	+	0.532573i	−0.662757	+	0.748835i	$0.730614\pi$
$674$		0			0
$675$	−21.0189	+	15.2711i	−0.809017	+	0.587785i
$676$	−0.529288	+	1.62898i	−0.0203572	+	0.0626531i
$677$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$677$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$678$		0			0
$679$	13.9901	−	19.2557i	0.536891	−	0.738967i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		1.00000			$0$
$683$		0			0		−1.00000			$\pi$
$684$	7.27851	−	2.36493i	0.278301	−	0.0904254i
$685$		0			0
$686$		0			0
$687$	−11.1246	+	34.2380i	−0.424430	+	1.30626i
$688$	49.4752	+	16.0755i	1.88622	+	0.612871i
$689$		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
$700$	−27.9802	−	38.5115i	−1.05755	−	1.45560i
$701$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$701$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$702$		0			0
$703$		−	6.62776i		−	0.249971i
$704$		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	$-0.535559\pi$
$709$	14.9865	−	46.1238i	0.562831	−	1.73222i	0.674310	−	0.738448i	$-0.264441\pi$
$710$		0			0
$711$	−31.3271	−	43.1180i	−1.17486	−	1.61705i
$712$		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
$721$	−31.6911	−	10.2970i	−1.18024	−	0.383482i
$722$		0			0
$723$	−30.0285	+	41.3307i	−1.11677	+	1.53711i
$724$	−11.7426	−	36.1401i	−0.436412	−	1.34314i
$725$		0			0
$726$		0			0
$727$	31.1769			1.15629			0.578144	−	0.815935i	$-0.303777\pi$
$727$	31.1769			1.15629			0.578144	+	0.815935i	$0.303777\pi$
$728$		0			0
$729$	−21.8435	−	15.8702i	−0.809017	−	0.587785i
$730$		0			0
$731$		0			0
$732$	38.6445	+	12.5564i	1.42834	+	0.464097i
$733$	−12.1427	−	16.7130i	−0.448501	−	0.617309i	0.523574	−	0.851980i	$-0.324598\pi$
$733$	−12.1427	−	16.7130i	−0.448501	−	0.617309i	−0.972075	+	0.234672i	$0.924598\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	27.4888	−	8.93166i	1.01119	−	0.328556i	0.243863	−	0.969810i	$-0.421585\pi$
$739$	27.4888	−	8.93166i	1.01119	−	0.328556i	0.767329	+	0.641253i	$0.221585\pi$
$740$		0			0
$741$	−6.22840	+	4.52520i	−0.228806	+	0.166237i
$742$		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	$0.135679\pi$
$751$	43.4390	+	31.5603i	1.58511	+	1.15165i	0.674589	+	0.738194i	$0.264321\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	29.0779	−	40.0223i	1.05755	−	1.45560i
$757$	−0.535233	−	1.64728i	−0.0194534	−	0.0598713i	0.940859	−	0.338800i	$-0.110021\pi$
$757$	−0.535233	−	1.64728i	−0.0194534	−	0.0598713i	−0.960312	+	0.278928i	$0.910021\pi$
$758$		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
$763$	−28.1569	+	20.4572i	−1.01935	+	0.740600i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	8.56373	+	26.3565i	0.309017	+	0.951057i
$769$		−	12.4134i		−	0.447637i	−0.974631	−	0.223819i	$-0.928148\pi$
$769$		−	12.4134i		−	0.447637i	0.974631	−	0.223819i	$-0.0718523\pi$
$770$		0			0
$771$		0			0
$772$	−51.9014	+	16.8638i	−1.86797	+	0.606941i
$773$		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
$777$	−25.1822	−	34.6603i	−0.903406	−	1.24343i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	50.6776	+	36.8195i	1.80992	+	1.31498i
$785$		0			0
$786$		0			0
$787$	−35.3303	−	11.4795i	−1.25939	−	0.409201i	−0.398113	−	0.917336i	$-0.630335\pi$
$787$	−35.3303	−	11.4795i	−1.25939	−	0.409201i	−0.861277	+	0.508136i	$0.830335\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−40.8756			−1.45154
$794$		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
$811$	−33.3754	+	45.9373i	−1.17197	+	1.61308i	−0.522695	+	0.852520i	$0.675073\pi$
$811$	−33.3754	+	45.9373i	−1.17197	+	1.61308i	−0.649273	+	0.760555i	$0.724927\pi$
$812$		0			0
$813$			55.5061i			1.94668i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	13.4203	+	9.75045i	0.469518	+	0.341125i
$818$		0			0
$819$	−15.3783	+	47.3297i	−0.537363	+	1.65383i
$820$		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
$831$	−21.9864	−	7.14381i	−0.762699	−	0.247816i
$832$	−16.3864	−	22.5539i	−0.568095	−	0.781915i
$833$		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
$844$	10.7903	−	14.8516i	0.371418	−	0.511213i
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−46.4860	+	15.1042i	−1.59540	+	0.518376i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−51.5763	−	16.7582i	−1.76594	−	0.573788i	−0.768152	−	0.640268i	$-0.778823\pi$
$853$	−51.5763	−	16.7582i	−1.76594	−	0.573788i	−0.997788	+	0.0664795i	$0.978823\pi$
$854$		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
$868$		−	16.4901i		−	0.559710i
$869$		0			0
$870$		0			0
$871$	40.1827	−	13.0561i	1.36154	−	0.442390i
$872$		0			0
$873$	12.1353	−	8.81678i	0.410716	−	0.298403i
$874$		0			0
$875$		0			0
$876$	21.9824	+	30.2562i	0.742716	+	1.02226i
$877$	−26.6817	+	36.7241i	−0.900975	+	1.24009i	0.0691806	+	0.997604i	$0.477962\pi$
$877$	−26.6817	+	36.7241i	−0.900975	+	1.24009i	−0.970156	+	0.242482i	$0.922038\pi$
$878$		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	0.237450	−	0.971400i	$-0.423688\pi$
$883$	−18.2164	−	13.2350i	−0.613029	−	0.445392i	−0.850480	+	0.526008i	$0.823688\pi$
$884$		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
$889$	22.3807	+	68.8809i	0.750626	+	2.31019i
$890$		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
$903$	107.229			3.56837
$904$		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	$0.400000\pi$
$912$			8.83701i			0.292623i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−33.6302	−	24.4338i	−1.11117	−	0.807315i
$917$		0			0
$918$		0			0
$919$	40.0956	+	13.0278i	1.32263	+	0.429749i	0.883397	−	0.468625i	$-0.155250\pi$
$919$	40.0956	+	13.0278i	1.32263	+	0.429749i	0.439233	+	0.898373i	$0.355250\pi$
$920$		0			0
$921$	34.8748	−	48.0011i	1.14916	−	1.58169i
$922$		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
$931$	11.7409	+	16.1600i	0.384793	+	0.529623i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−57.3167	+	18.6233i	−1.87245	+	0.608397i	−0.881871	+	0.471491i	$0.843716\pi$
$937$	−57.3167	+	18.6233i	−1.87245	+	0.608397i	−0.990584	+	0.136906i	$0.956284\pi$
$938$		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			$\pi$
$948$	58.5298	−	19.0175i	1.90096	−	0.617659i
$949$	−30.4367	−	22.1135i	−0.988016	−	0.717835i
$950$		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
$964$	−34.6739	−	47.7246i	−1.11677	−	1.53711i
$965$		0			0
$966$		0			0
$967$		−	4.07673i		−	0.131099i	−0.997849	−	0.0655495i	$-0.979120\pi$
$967$		−	4.07673i		−	0.131099i	0.997849	−	0.0655495i	$-0.0208800\pi$
$968$		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
$973$	1.37354	−	4.22732i	0.0440336	−	0.135522i
$974$		0			0
$975$	17.7387	+	24.4153i	0.568095	+	0.781915i
$976$	−27.5784	+	37.9585i	−0.882764	+	1.21502i
$977$		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
$981$	−20.8604	+	6.77795i	−0.666021	+	0.216403i
$982$		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
$988$	−2.74708	−	8.45463i	−0.0873962	−	0.268978i
$989$		0			0
$990$		0			0
$991$	−45.0333			−1.43053			−0.715265	−	0.698853i	$-0.753694\pi$
$991$	−45.0333			−1.43053			−0.715265	+	0.698853i	$0.753694\pi$
$992$		0			0
$993$	43.4390	+	31.5603i	1.37850	+	1.00154i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	36.9231	+	50.8203i	1.16937	+	1.60950i	0.667687	+	0.744442i	$0.267285\pi$
$997$	36.9231	+	50.8203i	1.16937	+	1.60950i	0.501680	+	0.865053i	$0.332715\pi$
$998$		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.215.2		16
3.2	odd	2	CM	363.2.f.g.215.2		16
11.2	odd	10	inner	363.2.f.g.233.2		16
11.3	even	5		363.2.d.d.362.3	✓	4
11.4	even	5	inner	363.2.f.g.239.3		16
11.5	even	5	inner	363.2.f.g.161.4		16
11.6	odd	10	inner	363.2.f.g.161.3		16
11.7	odd	10	inner	363.2.f.g.239.4		16
11.8	odd	10		363.2.d.d.362.4	yes	4
11.9	even	5	inner	363.2.f.g.233.1		16
11.10	odd	2	inner	363.2.f.g.215.1		16
33.2	even	10	inner	363.2.f.g.233.2		16
33.5	odd	10	inner	363.2.f.g.161.4		16
33.8	even	10		363.2.d.d.362.4	yes	4
33.14	odd	10		363.2.d.d.362.3	✓	4
33.17	even	10	inner	363.2.f.g.161.3		16
33.20	odd	10	inner	363.2.f.g.233.1		16
33.26	odd	10	inner	363.2.f.g.239.3		16
33.29	even	10	inner	363.2.f.g.239.4		16
33.32	even	2	inner	363.2.f.g.215.1		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
363.2.d.d.362.3	✓	4	11.3	even	5
363.2.d.d.362.3	✓	4	33.14	odd	10
363.2.d.d.362.4	yes	4	11.8	odd	10
363.2.d.d.362.4	yes	4	33.8	even	10
363.2.f.g.161.3		16	11.6	odd	10	inner
363.2.f.g.161.3		16	33.17	even	10	inner
363.2.f.g.161.4		16	11.5	even	5	inner
363.2.f.g.161.4		16	33.5	odd	10	inner
363.2.f.g.215.1		16	11.10	odd	2	inner
363.2.f.g.215.1		16	33.32	even	2	inner
363.2.f.g.215.2		16	1.1	even	1	trivial
363.2.f.g.215.2		16	3.2	odd	2	CM
363.2.f.g.233.1		16	11.9	even	5	inner
363.2.f.g.233.1		16	33.20	odd	10	inner
363.2.f.g.233.2		16	11.2	odd	10	inner
363.2.f.g.233.2		16	33.2	even	10	inner
363.2.f.g.239.3		16	11.4	even	5	inner
363.2.f.g.239.3		16	33.26	odd	10	inner
363.2.f.g.239.4		16	11.7	odd	10	inner
363.2.f.g.239.4		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)

\(f(q)\)	\(=\)	\(q+(-1.40126 - 1.01807i) q^{3} +(1.61803 - 1.17557i) q^{4} +(2.79802 + 3.85115i) q^{7} +(0.927051 + 2.85317i) q^{9} +O(q^{10})\)	\(q+(-1.40126 - 1.01807i) q^{3} +(1.61803 - 1.17557i) q^{4} +(2.79802 + 3.85115i) q^{7} +(0.927051 + 2.85317i) q^{9} -3.46410 q^{12} +(3.31421 - 1.07685i) q^{13} +(1.23607 - 3.80423i) q^{16} +(0.749728 - 1.03191i) q^{19} -8.24504i q^{21} +(-4.04508 - 2.93893i) q^{25} +(1.60570 - 4.94183i) q^{27} +(9.05459 + 2.94201i) q^{28} +(-0.535233 - 1.64728i) q^{31} +(4.85410 + 3.52671i) q^{36} +(4.20378 - 3.05422i) q^{37} +(-5.74038 - 1.86516i) q^{39} +13.0053i q^{43} +(-5.60503 + 4.07230i) q^{48} +(-4.83928 + 14.8938i) q^{49} +(4.09659 - 5.63847i) q^{52} +(-2.10112 + 0.682697i) q^{57} +(-11.1557 - 3.62471i) q^{61} +(-8.39406 + 11.5534i) q^{63} +(-2.47214 - 7.60845i) q^{64} +12.1244 q^{67} +(-6.34577 - 8.73420i) q^{73} +(2.67617 + 8.23639i) q^{75} -2.55103i q^{76} +(-16.8961 + 5.48987i) q^{79} +(-7.28115 + 5.29007i) q^{81} +(-9.69263 - 13.3408i) q^{84} +(13.4203 + 9.75045i) q^{91} +(-0.927051 + 2.85317i) q^{93} +(-1.54508 - 4.75528i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 8 q^{4} - 12 q^{9}+O(q^{10}) \) 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

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Database

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