LMFDB - Embedded newform 605.2.m.a.118.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [605,2,Mod(112,605)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(605, base_ring=CyclotomicField(20))

chi = DirichletCharacter(H, H._module([5, 2]))

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

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

chi := DirichletCharacter("605.112");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 605 = 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	605.m (of order $20$, degree $8$, 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:	$4.83094932229$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$2$ over $\Q(\zeta_{20})$
Coefficient field:	16.0.3429742096000000000000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 5x^{14} + 16x^{12} + 35x^{10} + 31x^{8} + 315x^{6} + 1296x^{4} + 3645x^{2} + 6561 $ x^16 + 5x^14 + 16x^12 + 35x^10 + 31x^8 + 315x^6 + 1296x^4 + 3645*x^2 + 6561
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 55)
Sato-Tate group:	$\mathrm{U}(1)[D_{20}]$

Embedding invariants

Embedding label			118.1
Root			$0.0369185 + 1.73166i$ of defining polynomial
Character	$\chi$	$=$	605.118
Dual form			605.2.m.a.282.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.743682 + 1.45956i) q^{3} +(-1.90211 - 0.618034i) q^{4} +(2.22328 - 0.238794i) q^{5} +(0.186107 + 0.256155i) q^{9} +(2.31662 - 2.31662i) q^{12} +(-1.30488 + 3.42260i) q^{15} +(3.23607 + 2.35114i) q^{16} +(-4.37651 - 0.919850i) q^{20} +(6.15831 + 6.15831i) q^{23} +(4.88595 - 1.06181i) q^{25} +(-5.36608 + 0.849903i) q^{27} +(-8.04962 + 5.84839i) q^{31} +(-0.195685 - 0.602256i) q^{36} +(5.44124 + 10.6790i) q^{37} +(0.474937 + 0.525063i) q^{45} +(-3.38125 - 1.72283i) q^{47} +(-5.83824 + 2.97473i) q^{48} +(-4.11450 + 5.66312i) q^{49} +(13.4557 + 2.13118i) q^{53} +(-3.15430 - 1.02489i) q^{59} +(4.59731 - 5.70370i) q^{60} +(-4.70228 - 6.47214i) q^{64} +(1.52506 - 1.52506i) q^{67} +(-13.5682 + 4.40859i) q^{69} +(2.42705 + 1.76336i) q^{71} +(-2.08382 + 7.92099i) q^{75} +(7.75613 + 4.45449i) q^{80} +(2.45665 - 7.56078i) q^{81} -9.00000i q^{89} +(-7.90776 - 15.5199i) q^{92} +(-2.54971 - 16.0982i) q^{93} +(2.98108 - 18.8218i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 2 q^{3} - 16 q^{12} + 8 q^{15} + 16 q^{16} - 12 q^{20} + 72 q^{23} - 2 q^{25} - 22 q^{27} - 24 q^{36} + 14 q^{37} - 72 q^{45} + 24 q^{47} - 8 q^{48} + 12 q^{53} + 28 q^{60} + 104 q^{67} + 12 q^{71}+ \cdots + 34 q^{97}+O(q^{100}) $ 16 * q + 2 * q^3 - 16 * q^12 + 8 * q^15 + 16 * q^16 - 12 * q^20 + 72 * q^23 - 2 * q^25 - 22 * q^27 - 24 * q^36 + 14 * q^37 - 72 * q^45 + 24 * q^47 - 8 * q^48 + 12 * q^53 + 28 * q^60 + 104 * q^67 + 12 * q^71 - 32 * q^75 + 8 * q^81 - 36 * q^92 - 66 * q^93 + 34 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/605\mathbb{Z}\right)^\times$.

$n$	$122$	$486$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$e\left(\frac{3}{10}\right)$

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		0.156434	−	0.987688i	$-0.450000\pi$
$2$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$3$	−0.743682	+	1.45956i	−0.429365	+	0.842677i	0.570408	+	0.821362i	$0.306785\pi$
$3$	−0.743682	+	1.45956i	−0.429365	+	0.842677i	−0.999773	+	0.0213149i	$0.993215\pi$
$4$	−1.90211	−	0.618034i	−0.951057	−	0.309017i
$5$	2.22328	−	0.238794i	0.994281	−	0.106792i
$5$	2.22328	−	0.238794i	0.994281	−	0.106792i
$6$		0			0
$7$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$7$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$8$		0			0
$9$	0.186107	+	0.256155i	0.0620358	+	0.0853849i
$10$		0			0
$11$		0			0
$11$		0			0
$12$	2.31662	−	2.31662i	0.668752	−	0.668752i
$13$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$13$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$14$		0			0
$15$	−1.30488	+	3.42260i	−0.336919	+	0.883710i
$16$	3.23607	+	2.35114i	0.809017	+	0.587785i
$17$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$17$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$18$		0			0
$19$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$19$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$20$	−4.37651	−	0.919850i	−0.978618	−	0.205685i
$21$		0			0
$22$		0			0
$23$	6.15831	+	6.15831i	1.28410	+	1.28410i	0.938315	+	0.345782i	$0.112386\pi$
$23$	6.15831	+	6.15831i	1.28410	+	1.28410i	0.345782	+	0.938315i	$0.387614\pi$
$24$		0			0
$25$	4.88595	−	1.06181i	0.977191	−	0.212362i
$26$		0			0
$27$	−5.36608	+	0.849903i	−1.03270	+	0.163564i
$28$		0			0
$29$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$29$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$30$		0			0
$31$	−8.04962	+	5.84839i	−1.44575	+	1.05040i	−0.458954	+	0.888460i	$0.651776\pi$
$31$	−8.04962	+	5.84839i	−1.44575	+	1.05040i	−0.986800	+	0.161942i	$0.948224\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−0.195685	−	0.602256i	−0.0326141	−	0.100376i
$37$	5.44124	+	10.6790i	0.894535	+	1.75562i	0.600038	+	0.799972i	$0.295152\pi$
$37$	5.44124	+	10.6790i	0.894535	+	1.75562i	0.294497	+	0.955652i	$0.404848\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$41$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$42$		0			0
$43$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$43$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$44$		0			0
$45$	0.474937	+	0.525063i	0.0707995	+	0.0782717i
$46$		0			0
$47$	−3.38125	−	1.72283i	−0.493206	−	0.251301i	0.189653	−	0.981851i	$-0.439264\pi$
$47$	−3.38125	−	1.72283i	−0.493206	−	0.251301i	−0.682859	+	0.730550i	$0.739264\pi$
$48$	−5.83824	+	2.97473i	−0.842677	+	0.429365i
$49$	−4.11450	+	5.66312i	−0.587785	+	0.809017i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	13.4557	+	2.13118i	1.84829	+	0.292740i	0.979349	−	0.202178i	$-0.0648018\pi$
$53$	13.4557	+	2.13118i	1.84829	+	0.292740i	0.868940	+	0.494918i	$0.164802\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−3.15430	−	1.02489i	−0.410655	−	0.133430i	0.0964021	−	0.995342i	$-0.469267\pi$
$59$	−3.15430	−	1.02489i	−0.410655	−	0.133430i	−0.507057	+	0.861913i	$0.669267\pi$
$60$	4.59731	−	5.70370i	0.593510	−	0.736345i
$61$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$61$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$62$		0			0
$63$		0			0
$64$	−4.70228	−	6.47214i	−0.587785	−	0.809017i
$65$		0			0
$66$		0			0
$67$	1.52506	−	1.52506i	0.186316	−	0.186316i	−0.607785	−	0.794101i	$-0.707942\pi$
$67$	1.52506	−	1.52506i	0.186316	−	0.186316i	0.794101	+	0.607785i	$0.207942\pi$
$68$		0			0
$69$	−13.5682	+	4.40859i	−1.63342	+	0.530732i
$70$		0			0
$71$	2.42705	+	1.76336i	0.288038	+	0.209272i	0.722416	−	0.691459i	$-0.243032\pi$
$71$	2.42705	+	1.76336i	0.288038	+	0.209272i	−0.434378	+	0.900731i	$0.643032\pi$
$72$		0			0
$73$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$73$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$74$		0			0
$75$	−2.08382	+	7.92099i	−0.240619	+	0.914637i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$79$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$80$	7.75613	+	4.45449i	0.867161	+	0.498027i
$81$	2.45665	−	7.56078i	0.272961	−	0.840087i
$82$		0			0
$83$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$83$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	9.00000i		−	0.953998i	−0.878904	−	0.476999i	$-0.841725\pi$
$89$		−	9.00000i		−	0.953998i	0.878904	−	0.476999i	$-0.158275\pi$
$90$		0			0
$91$		0			0
$92$	−7.90776	−	15.5199i	−0.824441	−	1.61806i
$93$	−2.54971	−	16.0982i	−0.264393	−	1.66931i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	2.98108	−	18.8218i	0.302683	−	1.91107i	−0.0986273	−	0.995124i	$-0.531445\pi$
$97$	2.98108	−	18.8218i	0.302683	−	1.91107i	0.401310	−	0.915942i	$-0.368555\pi$
$98$		0			0
$99$		0			0
$100$	−9.94987	−	1.00000i	−0.994987	−	0.100000i
$101$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$101$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$102$		0			0
$103$	15.0577	−	7.67229i	1.48368	−	0.755973i	0.490378	−	0.871510i	$-0.336859\pi$
$103$	15.0577	−	7.67229i	1.48368	−	0.755973i	0.993303	+	0.115536i	$0.0368587\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$107$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$108$	10.7322	+	1.69981i	1.03270	+	0.163564i
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$		0			0
$111$	−19.6332			−1.86351
$112$		0			0
$113$	5.67671	−	11.1412i	0.534020	−	1.04807i	−0.453599	−	0.891206i	$-0.649860\pi$
$113$	5.67671	−	11.1412i	0.534020	−	1.04807i	0.987620	−	0.156868i	$-0.0501397\pi$
$114$		0			0
$115$	15.1622	+	12.2211i	1.41388	+	1.13962i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$		0			0
$122$		0			0
$123$		0			0
$124$	18.9258	−	6.14936i	1.69959	−	0.552229i
$125$	10.6093	−	3.52744i	0.948924	−	0.315504i
$126$		0			0
$127$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$127$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	−11.7273	+	3.17096i	−1.00933	+	0.272913i
$136$		0			0
$137$	−18.3095	+	2.89995i	−1.56429	+	0.247759i	−0.877673	−	0.479260i	$-0.840905\pi$
$137$	−18.3095	+	2.89995i	−1.56429	+	0.247759i	−0.686617	+	0.727019i	$0.740905\pi$
$138$		0			0
$139$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$139$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$140$		0			0
$141$	5.02915	−	3.65389i	0.423531	−	0.307713i
$142$		0			0
$143$		0			0
$144$			1.26650i			0.105542i
$145$		0			0
$146$		0			0
$147$	−5.20578	−	10.2169i	−0.429365	−	0.842677i
$148$	−3.74985	−	23.6756i	−0.308236	−	1.94612i
$149$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$149$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$150$		0			0
$151$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$151$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−16.5000	+	14.9248i	−1.32531	+	1.19879i
$156$		0			0
$157$	8.22206	+	4.18935i	0.656192	+	0.334346i	0.750186	−	0.661226i	$-0.229964\pi$
$157$	8.22206	+	4.18935i	0.656192	+	0.334346i	−0.0939948	+	0.995573i	$0.529964\pi$
$158$		0			0
$159$	−13.1174	+	18.0545i	−1.04028	+	1.43182i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−2.72359	−	0.431374i	−0.213328	−	0.0337878i	0.0488556	−	0.998806i	$-0.484443\pi$
$163$	−2.72359	−	0.431374i	−0.213328	−	0.0337878i	−0.262184	+	0.965018i	$0.584443\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$167$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$168$		0			0
$169$	12.3637	+	4.01722i	0.951057	+	0.309017i
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$173$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	3.84169	−	3.84169i	0.288759	−	0.288759i
$178$		0			0
$179$	−19.9722	+	6.48936i	−1.49279	+	0.485037i	−0.937906	−	0.346890i	$-0.887238\pi$
$179$	−19.9722	+	6.48936i	−1.49279	+	0.485037i	−0.554885	+	0.831927i	$0.687238\pi$
$180$	−0.578878	−	1.29226i	−0.0431470	−	0.0963191i
$181$	−8.04962	−	5.84839i	−0.598323	−	0.434707i	0.246960	−	0.969026i	$-0.420568\pi$
$181$	−8.04962	−	5.84839i	−0.598323	−	0.434707i	−0.845283	+	0.534318i	$0.820568\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	14.6475	+	22.4432i	1.07691	+	1.65006i
$186$		0			0
$187$		0			0
$188$	5.36675	+	5.36675i	0.391411	+	0.391411i
$189$		0			0
$190$		0			0
$191$	−7.17425	+	22.0801i	−0.519111	+	1.59766i	0.256565	+	0.966527i	$0.417409\pi$
$191$	−7.17425	+	22.0801i	−0.519111	+	1.59766i	−0.775676	+	0.631132i	$0.782591\pi$
$192$	12.9435	−	2.05004i	0.934114	−	0.147949i
$193$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$193$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$194$		0			0
$195$		0			0
$196$	11.3262	−	8.22899i	0.809017	−	0.587785i
$197$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$197$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$198$		0			0
$199$		−	19.8997i		−	1.41066i	−0.708881	−	0.705328i	$-0.750800\pi$
$199$		−	19.8997i		−	1.41066i	0.708881	−	0.705328i	$-0.249200\pi$
$200$		0			0
$201$	1.09176	+	3.36008i	0.0770066	+	0.237002i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−0.431374	+	2.72359i	−0.0299826	+	0.189303i
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$211$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$212$	−24.2772	−	12.3699i	−1.66737	−	0.849565i
$213$	−4.37868	+	2.23105i	−0.300022	+	0.152869i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	9.90334	−	19.4364i	0.663177	−	1.30156i	−0.277000	−	0.960870i	$-0.589340\pi$
$223$	9.90334	−	19.4364i	0.663177	−	1.30156i	0.940177	−	0.340687i	$-0.110660\pi$
$224$		0			0
$225$	1.18130	+	1.05395i	0.0787534	+	0.0702633i
$226$		0			0
$227$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$227$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$228$		0			0
$229$	−17.5452	−	24.1489i	−1.15942	−	1.59580i	−0.713362	−	0.700796i	$-0.752828\pi$
$229$	−17.5452	−	24.1489i	−1.15942	−	1.59580i	−0.446055	−	0.895005i	$-0.647172\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$233$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$234$		0			0
$235$	−7.92887	−	3.02292i	−0.517222	−	0.197194i
$236$	5.36641	+	3.89893i	0.349324	+	0.253798i
$237$		0			0
$238$		0			0
$239$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$239$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$240$	−12.2697	+	8.00779i	−0.792005	+	0.516901i
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$		0			0
$243$	−2.31662	−	2.31662i	−0.148612	−	0.148612i
$244$		0			0
$245$	−7.79536	+	13.5732i	−0.498027	+	0.867161i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−21.8435	+	15.8702i	−1.37875	+	1.00172i	−0.381751	+	0.924265i	$0.624679\pi$
$251$	−21.8435	+	15.8702i	−1.37875	+	1.00172i	−0.996996	+	0.0774530i	$0.975321\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	4.94427	+	15.2169i	0.309017	+	0.951057i
$257$	2.73926	+	5.37610i	0.170870	+	0.335352i	0.960522	−	0.278203i	$-0.0897388\pi$
$257$	2.73926	+	5.37610i	0.170870	+	0.335352i	−0.789652	+	0.613555i	$0.789739\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$263$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$264$		0			0
$265$	30.4248	+	1.52506i	1.86898	+	0.0936839i
$266$		0			0
$267$	13.1360	+	6.69314i	0.803912	+	0.409614i
$268$	−3.84338	+	1.95830i	−0.234772	+	0.119622i
$269$	7.79785	−	10.7328i	0.475443	−	0.654392i	−0.502178	−	0.864764i	$-0.667468\pi$
$269$	7.79785	−	10.7328i	0.475443	−	0.654392i	0.977621	+	0.210373i	$0.0674677\pi$
$270$		0			0
$271$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$271$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$	28.5330			1.71748
$277$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$277$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$278$		0			0
$279$	−2.99619	−	0.973520i	−0.179377	−	0.0582831i
$280$		0			0
$281$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$281$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$282$		0			0
$283$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$283$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$284$	−3.52671	−	4.85410i	−0.209272	−	0.288038i
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	16.1680	−	5.25329i	0.951057	−	0.309017i
$290$		0			0
$291$	25.2546	+	18.3485i	1.48045	+	1.07561i
$292$		0			0
$293$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$293$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$294$		0			0
$295$	−7.25763	−	1.52540i	−0.422555	−	0.0888121i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	8.85910	−	13.7787i	0.511481	−	0.795516i
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$307$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$308$		0			0
$309$			27.6834i			1.57485i
$310$		0			0
$311$	3.70820	+	11.4127i	0.210273	+	0.647154i	0.999456	+	0.0329949i	$0.0105045\pi$
$311$	3.70820	+	11.4127i	0.210273	+	0.647154i	−0.789183	+	0.614159i	$0.789495\pi$
$312$		0			0
$313$	1.20014	+	7.57739i	0.0678360	+	0.428299i	0.998111	+	0.0614365i	$0.0195682\pi$
$313$	1.20014	+	7.57739i	0.0678360	+	0.428299i	−0.930275	+	0.366863i	$0.880432\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	0.418529	−	2.64249i	0.0235069	−	0.148417i	−0.973143	−	0.230201i	$-0.926062\pi$
$317$	0.418529	−	2.64249i	0.0235069	−	0.148417i	0.996650	+	0.0817838i	$0.0260617\pi$
$318$		0			0
$319$		0			0
$320$	−12.0000	−	13.2665i	−0.670820	−	0.741620i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−9.34564	+	12.8632i	−0.519202	+	0.714620i
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	9.94987			0.546895			0.273447	−	0.961887i	$-0.411836\pi$
$331$	9.94987			0.546895			0.273447	+	0.961887i	$0.411836\pi$
$332$		0			0
$333$	−1.72283	+	3.38125i	−0.0944107	+	0.185291i
$334$		0			0
$335$	3.02647	−	3.75482i	0.165354	−	0.205148i
$336$		0			0
$337$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$337$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$338$		0			0
$339$	12.0395	+	16.5710i	0.653898	+	0.900013i
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$	−29.1133	+	13.0415i	−1.56741	+	0.702133i
$346$		0			0
$347$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$347$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$348$		0			0
$349$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$349$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	22.7414	+	22.7414i	1.21040	+	1.21040i	0.970894	+	0.239511i	$0.0769871\pi$
$353$	22.7414	+	22.7414i	1.21040	+	1.21040i	0.239511	+	0.970894i	$0.423013\pi$
$354$		0			0
$355$	5.81709	+	3.34087i	0.308739	+	0.177315i
$356$	−5.56231	+	17.1190i	−0.294802	+	0.907306i
$357$		0			0
$358$		0			0
$359$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$359$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$360$		0			0
$361$	15.3713	−	11.1679i	0.809017	−	0.587785i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	15.0718	+	29.5801i	0.786744	+	1.54407i	0.838179	+	0.545395i	$0.183620\pi$
$367$	15.0718	+	29.5801i	0.786744	+	1.54407i	−0.0514358	+	0.998676i	$0.516380\pi$
$368$	5.44966	+	34.4078i	0.284083	+	1.79363i
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−5.09942	+	32.1965i	−0.264393	+	1.66931i
$373$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$373$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$374$		0			0
$375$	−2.74144	+	18.1082i	−0.141567	+	0.935103i
$376$		0			0
$377$		0			0
$378$		0			0
$379$	17.5452	−	24.1489i	0.901235	−	1.24044i	−0.0688378	−	0.997628i	$-0.521929\pi$
$379$	17.5452	−	24.1489i	0.901235	−	1.24044i	0.970073	−	0.242815i	$-0.0780709\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−29.5540	−	4.68089i	−1.51014	−	0.239182i	−0.654224	−	0.756301i	$-0.727005\pi$
$383$	−29.5540	−	4.68089i	−1.51014	−	0.239182i	−0.855914	+	0.517119i	$0.827005\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	−17.3029	+	33.9588i	−0.878421	+	1.72400i
$389$	−34.6973	−	11.2738i	−1.75922	−	0.571606i	−0.762102	−	0.647456i	$-0.775833\pi$
$389$	−34.6973	−	11.2738i	−1.75922	−	0.571606i	−0.997119	+	0.0758507i	$0.975833\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	18.8997	−	18.8997i	0.948551	−	0.948551i	−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	18.8997	−	18.8997i	0.948551	−	0.948551i	0.998740	+	0.0501886i	$0.0159822\pi$
$398$		0			0
$399$		0			0
$400$	18.3078	+	8.05147i	0.915388	+	0.402574i
$401$	−21.4656	−	15.5957i	−1.07194	−	0.778812i	−0.0956827	−	0.995412i	$-0.530503\pi$
$401$	−21.4656	−	15.5957i	−1.07194	−	0.778812i	−0.976261	+	0.216600i	$0.930503\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	3.65635	−	17.3964i	0.181685	−	0.864433i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$409$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$410$		0			0
$411$	9.38384	−	28.8805i	0.462871	−	1.42457i
$412$	−33.3832	+	5.28738i	−1.64467	+	0.260491i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	24.0000i		−	1.17248i	−0.810139	−	0.586238i	$-0.800608\pi$
$419$		−	24.0000i		−	1.17248i	0.810139	−	0.586238i	$-0.199392\pi$
$420$		0			0
$421$	−12.2987	−	37.8516i	−0.599403	−	1.84477i	−0.531460	−	0.847084i	$-0.678356\pi$
$421$	−12.2987	−	37.8516i	−0.599403	−	1.84477i	−0.0679432	−	0.997689i	$-0.521644\pi$
$422$		0			0
$423$	−0.187964	−	1.18676i	−0.00913910	−	0.0577020i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$431$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$432$	−19.3632	−	9.86606i	−0.931614	−	0.474681i
$433$	−0.535294	+	0.272746i	−0.0257246	+	0.0131073i	−0.466805	−	0.884360i	$-0.654595\pi$
$433$	−0.535294	+	0.272746i	−0.0257246	+	0.0131073i	0.441081	+	0.897467i	$0.354595\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$	−2.21637			−0.105542
$442$		0			0
$443$	−4.97031	+	9.75478i	−0.236146	+	0.463464i	−0.978418	−	0.206636i	$-0.933748\pi$
$443$	−4.97031	+	9.75478i	−0.236146	+	0.463464i	0.742271	+	0.670099i	$0.233748\pi$
$444$	37.3447	+	12.1340i	1.77230	+	0.575855i
$445$	−2.14915	−	20.0095i	−0.101879	−	0.948543i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	22.9236	+	31.5517i	1.08183	+	1.48902i	0.857487	+	0.514505i	$0.172024\pi$
$449$	22.9236	+	31.5517i	1.08183	+	1.48902i	0.224346	+	0.974510i	$0.427976\pi$
$450$		0			0
$451$		0			0
$452$	−17.6834	+	17.6834i	−0.831756	+	0.831756i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$457$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$458$		0			0
$459$		0			0
$460$	−21.2872	−	32.6167i	−0.992522	−	1.52076i
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$	−30.4248	−	30.4248i	−1.41396	−	1.41396i	−0.720346	−	0.693615i	$-0.756017\pi$
$463$	−30.4248	−	30.4248i	−1.41396	−	1.41396i	−0.693615	−	0.720346i	$-0.743983\pi$
$464$		0			0
$465$	−9.51288	−	35.1820i	−0.441149	−	1.63153i
$466$		0			0
$467$	28.0171	−	4.43748i	1.29648	−	0.205342i	0.530212	−	0.847865i	$-0.322112\pi$
$467$	28.0171	−	4.43748i	1.29648	−	0.205342i	0.766267	+	0.642523i	$0.222112\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−12.2292	+	8.88503i	−0.563492	+	0.409401i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	1.95830	+	3.84338i	0.0896645	+	0.175976i
$478$		0			0
$479$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$479$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$	2.13325	−	42.5581i	0.0968659	−	1.93246i
$486$		0			0
$487$	20.8228	+	10.6097i	0.943571	+	0.480773i	0.856911	−	0.515465i	$-0.172381\pi$
$487$	20.8228	+	10.6097i	0.943571	+	0.480773i	0.0866600	+	0.996238i	$0.472381\pi$
$488$		0			0
$489$	2.65510	−	3.65443i	0.120068	−	0.165259i
$490$		0			0
$491$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$491$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−39.7995			−1.78705
$497$		0			0
$498$		0			0
$499$	−18.9258	−	6.14936i	−0.847235	−	0.275283i	−0.146947	−	0.989144i	$-0.546945\pi$
$499$	−18.9258	−	6.14936i	−0.847235	−	0.275283i	−0.700287	+	0.713861i	$0.746945\pi$
$500$	−22.3602	+	0.152689i	−0.999977	+	0.00682845i
$501$		0			0
$502$		0			0
$503$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$503$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−15.0581	+	15.0581i	−0.668752	+	0.668752i
$508$		0			0
$509$	3.15430	−	1.02489i	0.139812	−	0.0454276i	−0.238275	−	0.971198i	$-0.576582\pi$
$509$	3.15430	−	1.02489i	0.139812	−	0.0454276i	0.378087	+	0.925770i	$0.376582\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	31.6454	−	20.6534i	1.39446	−	0.910095i
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	13.3236	−	41.0059i	0.583718	−	1.79650i	−0.0206400	−	0.999787i	$-0.506570\pi$
$521$	13.3236	−	41.0059i	0.583718	−	1.79650i	0.604358	−	0.796713i	$-0.293430\pi$
$522$		0			0
$523$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$523$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$			52.8496i			2.29781i
$530$		0			0
$531$	−0.324507	−	0.998729i	−0.0140824	−	0.0433411i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	5.38136	−	33.9766i	0.232223	−	1.46620i
$538$		0			0
$539$		0			0
$540$	24.2665	+	1.21637i	1.04426	+	0.0523444i
$541$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$541$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$542$		0			0
$543$	14.5224	−	7.39955i	0.623217	−	0.317545i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$547$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$548$	36.6191	+	5.79989i	1.56429	+	0.247759i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−43.6502	+	4.68830i	−1.85285	+	0.199007i
$556$		0			0
$557$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$557$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$563$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$564$	−11.8242	+	3.84193i	−0.497891	+	0.161774i
$565$	9.96048	−	26.1255i	0.419041	−	1.09911i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$569$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$570$		0			0
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$		0			0
$573$	−26.8918	−	26.8918i	−1.12342	−	1.12342i
$574$		0			0
$575$	36.6282	+	23.5503i	1.52750	+	0.982114i
$576$	0.782740	−	2.40902i	0.0326141	−	0.100376i
$577$	39.7739	−	6.29956i	1.65581	−	0.262254i	0.742596	−	0.669740i	$-0.233594\pi$
$577$	39.7739	−	6.29956i	1.65581	−	0.262254i	0.913212	+	0.407486i	$0.133594\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−17.5384	−	34.4210i	−0.723885	−	1.42070i	−0.899801	−	0.436300i	$-0.856289\pi$
$587$	−17.5384	−	34.4210i	−0.723885	−	1.42070i	0.175916	−	0.984405i	$-0.443711\pi$
$588$	3.58758	+	22.6511i	0.147949	+	0.934114i
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−7.49970	+	47.3512i	−0.308236	+	1.94612i
$593$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$593$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	29.0449	+	14.7991i	1.18873	+	0.605686i
$598$		0			0
$599$	21.1603	−	29.1246i	0.864585	−	1.19000i	−0.115872	−	0.993264i	$-0.536966\pi$
$599$	21.1603	−	29.1246i	0.864585	−	1.19000i	0.980457	−	0.196735i	$-0.0630339\pi$
$600$		0			0
$601$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$601$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$602$		0			0
$603$	0.674478	+	0.106827i	0.0274669	+	0.00435032i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$607$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$613$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−34.2665	+	34.2665i	−1.37952	+	1.37952i	−0.534089	+	0.845428i	$0.679345\pi$
$617$	−34.2665	+	34.2665i	−1.37952	+	1.37952i	−0.845428	+	0.534089i	$0.820655\pi$
$618$		0			0
$619$	−0.951057	+	0.309017i	−0.0382262	+	0.0124204i	−0.328068	−	0.944654i	$-0.606397\pi$
$619$	−0.951057	+	0.309017i	−0.0382262	+	0.0124204i	0.289841	+	0.957075i	$0.406397\pi$
$620$	40.6089	−	18.1911i	1.63089	−	0.730573i
$621$	−38.2800	−	27.8120i	−1.53612	−	1.11606i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	22.7451	−	10.3759i	0.909804	−	0.415037i
$626$		0			0
$627$		0			0
$628$	−13.0501	−	13.0501i	−0.520757	−	0.520757i
$629$		0			0
$630$		0			0
$631$	2.16312	−	6.65740i	0.0861124	−	0.265027i	−0.898723	−	0.438516i	$-0.855504\pi$
$631$	2.16312	−	6.65740i	0.0861124	−	0.265027i	0.984836	+	0.173489i	$0.0555042\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$	36.1091	−	26.2348i	1.43182	−	1.04028i
$637$		0			0
$638$		0			0
$639$			0.949874i			0.0375764i
$640$		0			0
$641$	−7.17425	−	22.0801i	−0.283366	−	0.872111i	−0.986884	−	0.161433i	$-0.948388\pi$
$641$	−7.17425	−	22.0801i	−0.283366	−	0.872111i	0.703518	−	0.710678i	$-0.251612\pi$
$642$		0			0
$643$	7.83709	+	49.4815i	0.309065	+	1.95136i	0.308086	+	0.951359i	$0.400312\pi$
$643$	7.83709	+	49.4815i	0.309065	+	1.95136i	0.000979141		1.00000i	$0.499688\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	7.75596	−	48.9692i	0.304918	−	1.92518i	−0.0687910	−	0.997631i	$-0.521914\pi$
$647$	7.75596	−	48.9692i	0.304918	−	1.92518i	0.373709	−	0.927546i	$-0.378086\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$	4.91397	+	2.50379i	0.192446	+	0.0980561i
$653$	−30.0423	+	15.3073i	−1.17565	+	0.599021i	−0.928998	−	0.370084i	$-0.879329\pi$
$653$	−30.0423	+	15.3073i	−1.17565	+	0.599021i	−0.246647	+	0.969105i	$0.579329\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$	−13.0000			−0.505641			−0.252821	−	0.967513i	$-0.581358\pi$
$661$	−13.0000			−0.505641			−0.252821	+	0.967513i	$0.581358\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	21.0036	+	28.9090i	0.812047	+	1.11769i
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$673$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$674$		0			0
$675$	−25.3160	+	9.85035i	−0.974412	+	0.379140i
$676$	−21.0344	−	15.2824i	−0.809017	−	0.587785i
$677$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$677$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−11.2164	−	11.2164i	−0.429183	−	0.429183i	0.459167	−	0.888350i	$-0.348148\pi$
$683$	−11.2164	−	11.2164i	−0.429183	−	0.429183i	−0.888350	+	0.459167i	$0.848148\pi$
$684$		0			0
$685$	−40.0148	+	10.8196i	−1.52889	+	0.413396i
$686$		0			0
$687$	48.2947	−	7.64913i	1.84256	−	0.291832i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−13.7533	+	9.99235i	−0.523200	+	0.380127i	−0.817808	−	0.575491i	$-0.804811\pi$
$691$	−13.7533	+	9.99235i	−0.523200	+	0.380127i	0.294608	+	0.955618i	$0.404811\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$701$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$	10.3087	−	9.32456i	0.388248	−	0.351183i
$706$		0			0
$707$		0			0
$708$	−9.68162	+	4.93303i	−0.363857	+	0.185395i
$709$	−11.1679	+	15.3713i	−0.419420	+	0.577282i	−0.965484	−	0.260461i	$-0.916125\pi$
$709$	−11.1679	+	15.3713i	−0.419420	+	0.577282i	0.546064	+	0.837743i	$0.316125\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−85.5883	−	13.5558i	−3.20531	−	0.507671i
$714$		0			0
$715$		0			0
$716$	42.0000			1.56961
$717$		0			0
$718$		0			0
$719$	48.5039	+	15.7599i	1.80889	+	0.587744i	1.00000		0.000216702i	$-6.89785e-5\pi$
$719$	48.5039	+	15.7599i	1.80889	+	0.587744i	0.808890	+	0.587961i	$0.200069\pi$
$720$	0.302432	+	2.81578i	0.0112710	+	0.104938i
$721$		0			0
$722$		0			0
$723$		0			0
$724$	11.6968	+	16.0992i	0.434707	+	0.598323i
$725$		0			0
$726$		0			0
$727$	21.5251	−	21.5251i	0.798320	−	0.798320i	−0.184510	−	0.982831i	$-0.559070\pi$
$727$	21.5251	−	21.5251i	0.798320	−	0.798320i	0.982831	+	0.184510i	$0.0590699\pi$
$728$		0			0
$729$	27.7864	−	9.02836i	1.02913	−	0.334384i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$733$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$734$		0			0
$735$	−14.0136	−	21.4720i	−0.516901	−	0.792005i
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$739$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$740$	−13.9906	−	51.7421i	−0.514303	−	1.90208i
$741$		0			0
$742$		0			0
$743$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$743$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−7.10739	−	21.8743i	−0.259352	−	0.798205i	−0.992941	−	0.118611i	$-0.962156\pi$
$751$	−7.10739	−	21.8743i	−0.259352	−	0.798205i	0.733588	−	0.679594i	$-0.237844\pi$
$752$	−6.89133	−	13.5250i	−0.251301	−	0.493206i
$753$	−6.91890	−	43.6842i	−0.252139	−	1.59194i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−8.60586	+	54.3353i	−0.312785	+	1.97485i	−0.133554	+	0.991042i	$0.542639\pi$
$757$	−8.60586	+	54.3353i	−0.312785	+	1.97485i	−0.179232	+	0.983807i	$0.557361\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$761$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$762$		0			0
$763$		0			0
$764$	27.2925	−	37.5649i	0.987407	−	1.35905i
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−25.8869	−	4.10009i	−0.934114	−	0.147949i
$769$		0			0			−	1.00000i	$-0.5\pi$
$769$		0			0				1.00000i	$0.5\pi$
$770$		0			0
$771$	−9.88388			−0.355959
$772$		0			0
$773$	−13.0763	+	25.6636i	−0.470320	+	0.923056i	0.526998	+	0.849867i	$0.323318\pi$
$773$	−13.0763	+	25.6636i	−0.470320	+	0.923056i	−0.997318	+	0.0731890i	$0.976682\pi$
$774$		0			0
$775$	−33.1202	+	37.1221i	−1.18971	+	1.33347i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−26.6296	+	8.65248i	−0.951057	+	0.309017i
$785$	19.2803	+	7.35072i	0.688145	+	0.262358i
$786$		0			0
$787$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$787$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$	−24.8523	+	43.2726i	−0.881421	+	1.53472i
$796$	−12.2987	+	37.8516i	−0.435917	+	1.34161i
$797$	−41.4729	+	6.56866i	−1.46905	+	0.232674i	−0.839105	−	0.543970i	$-0.816921\pi$
$797$	−41.4729	+	6.56866i	−1.46905	+	0.232674i	−0.629940	+	0.776644i	$0.716921\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	2.30539	−	1.67497i	0.0814571	−	0.0591820i
$802$		0			0
$803$		0			0
$804$		−	7.06600i		−	0.249198i
$805$		0			0
$806$		0			0
$807$	9.86606	+	19.3632i	0.347302	+	0.681618i
$808$		0			0
$809$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$809$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$810$		0			0
$811$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$811$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−6.15831	−	0.308689i	−0.215716	−	0.0108129i
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$821$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$822$		0			0
$823$	−13.3746	−	2.11834i	−0.466211	−	0.0738405i	−0.0810902	−	0.996707i	$-0.525840\pi$
$823$	−13.3746	−	2.11834i	−0.466211	−	0.0738405i	−0.385121	+	0.922866i	$0.625840\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$827$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$828$	2.50379	−	4.91397i	0.0870128	−	0.170772i
$829$	−27.5806	−	8.96149i	−0.957915	−	0.311246i	−0.211987	−	0.977272i	$-0.567994\pi$
$829$	−27.5806	−	8.96149i	−0.957915	−	0.311246i	−0.745928	+	0.666027i	$0.767994\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	38.2243	−	38.2243i	1.32123	−	1.32123i
$838$		0			0
$839$	34.6973	−	11.2738i	1.19788	−	0.389216i	0.358901	−	0.933376i	$-0.383151\pi$
$839$	34.6973	−	11.2738i	1.19788	−	0.389216i	0.838982	+	0.544160i	$0.183151\pi$
$840$		0			0
$841$	23.4615	+	17.0458i	0.809017	+	0.587785i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	28.4473	+	5.97903i	0.978618	+	0.205685i
$846$		0			0
$847$		0			0
$848$	38.5330	+	38.5330i	1.32323	+	1.32323i
$849$		0			0
$850$		0			0
$851$	−32.2560	+	99.2738i	−1.10572	+	3.40306i
$852$	9.70760	−	1.53753i	0.332577	−	0.0526750i
$853$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$853$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$857$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$858$		0			0
$859$			31.0000i			1.05771i	0.848713	+	0.528853i	$0.177378\pi$
$859$			31.0000i			1.05771i	−0.848713	+	0.528853i	$0.822622\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	9.11837	+	57.5711i	0.310393	+	1.95974i	0.279210	+	0.960230i	$0.409928\pi$
$863$	9.11837	+	57.5711i	0.310393	+	1.95974i	0.0311832	+	0.999514i	$0.490072\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−4.35634	+	27.5049i	−0.147949	+	0.934114i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	5.37610	−	2.73926i	0.181954	−	0.0927099i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$877$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	57.0000			1.92038			0.960189	−	0.279350i	$-0.0901189\pi$
$881$	57.0000			1.92038			0.960189	+	0.279350i	$0.0901189\pi$
$882$		0			0
$883$	11.5889	−	22.7445i	0.389997	−	0.765413i	−0.609631	−	0.792686i	$-0.708682\pi$
$883$	11.5889	−	22.7445i	0.389997	−	0.765413i	0.999628	+	0.0272727i	$0.00868225\pi$
$884$		0			0
$885$	7.62378	−	9.45852i	0.256271	−	0.317945i
$886$		0			0
$887$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$887$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$	−30.8496	+	30.8496i	−1.03292	+	1.03292i
$893$		0			0
$894$		0			0
$895$	−42.8542	+	19.1969i	−1.43246	+	0.641682i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−1.59559	−	2.73482i	−0.0531863	−	0.0911605i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−19.2931	−	11.0804i	−0.641325	−	0.368325i
$906$		0			0
$907$	36.1068	−	5.71876i	1.19891	−	0.189888i	0.475126	−	0.879918i	$-0.342403\pi$
$907$	36.1068	−	5.71876i	1.19891	−	0.189888i	0.723781	+	0.690030i	$0.242403\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	5.36641	−	3.89893i	0.177797	−	0.129177i	−0.495327	−	0.868706i	$-0.664952\pi$
$911$	5.36641	−	3.89893i	0.177797	−	0.129177i	0.673124	+	0.739529i	$0.264952\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	18.4481	+	56.7774i	0.609542	+	1.87598i
$917$		0			0
$918$		0			0
$919$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$919$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	37.9248	+	46.3997i	1.24696	+	1.52561i
$926$		0			0
$927$	4.76765	+	2.42924i	0.156590	+	0.0797866i
$928$		0			0
$929$	−31.1914	+	42.9313i	−1.02336	+	1.40853i	−0.113534	+	0.993534i	$0.536217\pi$
$929$	−31.1914	+	42.9313i	−1.02336	+	1.40853i	−0.909823	+	0.414996i	$0.863783\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	−19.4152	−	3.07507i	−0.635625	−	0.100673i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$937$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$938$		0			0
$939$	−11.9522	−	3.88349i	−0.390044	−	0.126733i
$940$	13.2133	+	10.6502i	0.430972	+	0.347373i
$941$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$941$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$942$		0			0
$943$		0			0
$944$	−7.79785	−	10.7328i	−0.253798	−	0.349324i
$945$		0			0
$946$		0			0
$947$	−16.8918	+	16.8918i	−0.548910	+	0.548910i	−0.926126	−	0.377215i	$-0.876882\pi$
$947$	−16.8918	+	16.8918i	−0.548910	+	0.548910i	0.377215	+	0.926126i	$0.376882\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	3.54562	+	2.57604i	0.114974	+	0.0835338i
$952$		0			0
$953$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$953$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$954$		0			0
$955$	−10.6778	+	50.8034i	−0.345525	+	1.64396i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$	28.2874	−	7.64864i	0.912973	−	0.246859i
$961$	21.0132	−	64.6718i	0.677844	−	2.08619i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$967$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	13.3236	+	41.0059i	0.427575	+	1.31594i	0.900507	+	0.434842i	$0.143196\pi$
$971$	13.3236	+	41.0059i	0.427575	+	1.31594i	−0.472932	+	0.881099i	$0.656804\pi$
$972$	2.97473	+	5.83824i	0.0954145	+	0.187261i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−3.25018	+	20.5209i	−0.103983	+	0.656520i	0.879552	+	0.475802i	$0.157842\pi$
$977$	−3.25018	+	20.5209i	−0.103983	+	0.656520i	−0.983535	+	0.180718i	$0.942158\pi$
$978$		0			0
$979$		0			0
$980$	23.2164	−	21.0000i	0.741620	−	0.670820i
$981$		0			0
$982$		0			0
$983$	−9.14632	+	4.66028i	−0.291722	+	0.148640i	−0.593727	−	0.804666i	$-0.702344\pi$
$983$	−9.14632	+	4.66028i	−0.291722	+	0.148640i	0.302005	+	0.953306i	$0.402344\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	59.6992			1.89641			0.948205	−	0.317660i	$-0.102897\pi$
$991$	59.6992			1.89641			0.948205	+	0.317660i	$0.102897\pi$
$992$		0			0
$993$	−7.39955	+	14.5224i	−0.234817	+	0.460855i
$994$		0			0
$995$	−4.75194	−	44.2427i	−0.150647	−	1.40259i
$996$		0			0
$997$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$997$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$998$		0			0
$999$	−38.2743	−	52.6800i	−1.21095	−	1.66672i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	605.2.m.a.118.1		16
5.2	odd	4	inner	605.2.m.a.602.2		16
11.2	odd	10		55.2.e.b.43.2	yes	4
11.3	even	5	inner	605.2.m.a.578.1		16
11.4	even	5	inner	605.2.m.a.403.2		16
11.5	even	5	inner	605.2.m.a.233.1		16
11.6	odd	10	inner	605.2.m.a.233.1		16
11.7	odd	10	inner	605.2.m.a.403.2		16
11.8	odd	10	inner	605.2.m.a.578.1		16
11.9	even	5		55.2.e.b.43.2	yes	4
11.10	odd	2	CM	605.2.m.a.118.1		16
33.2	even	10		495.2.k.a.208.2		4
33.20	odd	10		495.2.k.a.208.2		4
44.31	odd	10		880.2.bd.d.593.1		4
44.35	even	10		880.2.bd.d.593.1		4
55.2	even	20		55.2.e.b.32.2	✓	4
55.7	even	20	inner	605.2.m.a.282.1		16
55.9	even	10		275.2.e.a.43.1		4
55.13	even	20		275.2.e.a.32.1		4
55.17	even	20	inner	605.2.m.a.112.1		16
55.24	odd	10		275.2.e.a.43.1		4
55.27	odd	20	inner	605.2.m.a.112.1		16
55.32	even	4	inner	605.2.m.a.602.2		16
55.37	odd	20	inner	605.2.m.a.282.1		16
55.42	odd	20		55.2.e.b.32.2	✓	4
55.47	odd	20	inner	605.2.m.a.457.1		16
55.52	even	20	inner	605.2.m.a.457.1		16
55.53	odd	20		275.2.e.a.32.1		4
165.2	odd	20		495.2.k.a.307.2		4
165.152	even	20		495.2.k.a.307.2		4
220.167	odd	20		880.2.bd.d.417.1		4
220.207	even	20		880.2.bd.d.417.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
55.2.e.b.32.2	✓	4	55.2	even	20
55.2.e.b.32.2	✓	4	55.42	odd	20
55.2.e.b.43.2	yes	4	11.2	odd	10
55.2.e.b.43.2	yes	4	11.9	even	5
275.2.e.a.32.1		4	55.13	even	20
275.2.e.a.32.1		4	55.53	odd	20
275.2.e.a.43.1		4	55.9	even	10
275.2.e.a.43.1		4	55.24	odd	10
495.2.k.a.208.2		4	33.2	even	10
495.2.k.a.208.2		4	33.20	odd	10
495.2.k.a.307.2		4	165.2	odd	20
495.2.k.a.307.2		4	165.152	even	20
605.2.m.a.112.1		16	55.17	even	20	inner
605.2.m.a.112.1		16	55.27	odd	20	inner
605.2.m.a.118.1		16	1.1	even	1	trivial
605.2.m.a.118.1		16	11.10	odd	2	CM
605.2.m.a.233.1		16	11.5	even	5	inner
605.2.m.a.233.1		16	11.6	odd	10	inner
605.2.m.a.282.1		16	55.7	even	20	inner
605.2.m.a.282.1		16	55.37	odd	20	inner
605.2.m.a.403.2		16	11.4	even	5	inner
605.2.m.a.403.2		16	11.7	odd	10	inner
605.2.m.a.457.1		16	55.47	odd	20	inner
605.2.m.a.457.1		16	55.52	even	20	inner
605.2.m.a.578.1		16	11.3	even	5	inner
605.2.m.a.578.1		16	11.8	odd	10	inner
605.2.m.a.602.2		16	5.2	odd	4	inner
605.2.m.a.602.2		16	55.32	even	4	inner
880.2.bd.d.417.1		4	220.167	odd	20
880.2.bd.d.417.1		4	220.207	even	20
880.2.bd.d.593.1		4	44.31	odd	10
880.2.bd.d.593.1		4	44.35	even	10

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 605 = 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	605.m (of order \(20\), degree \(8\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.743682 + 1.45956i) q^{3} +(-1.90211 - 0.618034i) q^{4} +(2.22328 - 0.238794i) q^{5} +(0.186107 + 0.256155i) q^{9} +(2.31662 - 2.31662i) q^{12} +(-1.30488 + 3.42260i) q^{15} +(3.23607 + 2.35114i) q^{16} +(-4.37651 - 0.919850i) q^{20} +(6.15831 + 6.15831i) q^{23} +(4.88595 - 1.06181i) q^{25} +(-5.36608 + 0.849903i) q^{27} +(-8.04962 + 5.84839i) q^{31} +(-0.195685 - 0.602256i) q^{36} +(5.44124 + 10.6790i) q^{37} +(0.474937 + 0.525063i) q^{45} +(-3.38125 - 1.72283i) q^{47} +(-5.83824 + 2.97473i) q^{48} +(-4.11450 + 5.66312i) q^{49} +(13.4557 + 2.13118i) q^{53} +(-3.15430 - 1.02489i) q^{59} +(4.59731 - 5.70370i) q^{60} +(-4.70228 - 6.47214i) q^{64} +(1.52506 - 1.52506i) q^{67} +(-13.5682 + 4.40859i) q^{69} +(2.42705 + 1.76336i) q^{71} +(-2.08382 + 7.92099i) q^{75} +(7.75613 + 4.45449i) q^{80} +(2.45665 - 7.56078i) q^{81} -9.00000i q^{89} +(-7.90776 - 15.5199i) q^{92} +(-2.54971 - 16.0982i) q^{93} +(2.98108 - 18.8218i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 2 q^{3} - 16 q^{12} + 8 q^{15} + 16 q^{16} - 12 q^{20} + 72 q^{23} - 2 q^{25} - 22 q^{27} - 24 q^{36} + 14 q^{37} - 72 q^{45} + 24 q^{47} - 8 q^{48} + 12 q^{53} + 28 q^{60} + 104 q^{67} + 12 q^{71}+ \cdots + 34 q^{97}+O(q^{100}) \) 16 * q + 2 * q^3 - 16 * q^12 + 8 * q^15 + 16 * q^16 - 12 * q^20 + 72 * q^23 - 2 * q^25 - 22 * q^27 - 24 * q^36 + 14 * q^37 - 72 * q^45 + 24 * q^47 - 8 * q^48 + 12 * q^53 + 28 * q^60 + 104 * q^67 + 12 * q^71 - 32 * q^75 + 8 * q^81 - 36 * q^92 - 66 * q^93 + 34 * q^97

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