LMFDB - Embedded newform 605.2.m.a.403.2

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			403.2
Root			$0.987975 - 1.42264i$ of defining polynomial
Character	$\chi$	$=$	605.403
Dual form			605.2.m.a.602.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.45956 - 0.743682i) q^{3} +(1.90211 - 0.618034i) q^{4} +(0.459925 - 2.18826i) q^{5} +(-0.186107 + 0.256155i) q^{9} +(2.31662 - 2.31662i) q^{12} +(-0.956081 - 3.53593i) q^{15} +(3.23607 - 2.35114i) q^{16} +(-0.477588 - 4.44656i) q^{20} +(6.15831 + 6.15831i) q^{23} +(-4.57694 - 2.01287i) q^{25} +(-0.849903 + 5.36608i) q^{27} +(-8.04962 - 5.84839i) q^{31} +(-0.195685 + 0.602256i) q^{36} +(-10.6790 - 5.44124i) q^{37} +(0.474937 + 0.525063i) q^{45} +(1.72283 + 3.38125i) q^{47} +(2.97473 - 5.83824i) q^{48} +(4.11450 + 5.66312i) q^{49} +(2.13118 + 13.4557i) q^{53} +(3.15430 - 1.02489i) q^{59} +(-4.00390 - 6.13484i) q^{60} +(4.70228 - 6.47214i) q^{64} +(1.52506 - 1.52506i) q^{67} +(13.5682 + 4.40859i) q^{69} +(2.42705 - 1.76336i) q^{71} +(-8.17724 + 0.465888i) q^{75} +(-3.65655 - 8.16270i) q^{80} +(2.45665 + 7.56078i) q^{81} -9.00000i q^{89} +(15.5199 + 7.90776i) q^{92} +(-16.0982 - 2.54971i) q^{93} +(18.8218 - 2.98108i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/605\mathbb{Z}\right)^\times$.

$n$	$122$	$486$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$e\left(\frac{7}{10}\right)$

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		0.987688	−	0.156434i	$-0.0500000\pi$
$2$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$3$	1.45956	−	0.743682i	0.842677	−	0.429365i	0.0213149	−	0.999773i	$-0.493215\pi$
$3$	1.45956	−	0.743682i	0.842677	−	0.429365i	0.821362	+	0.570408i	$0.193215\pi$
$4$	1.90211	−	0.618034i	0.951057	−	0.309017i
$5$	0.459925	−	2.18826i	0.205685	−	0.978618i
$5$	0.459925	−	2.18826i	0.205685	−	0.978618i
$6$		0			0
$7$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$7$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$8$		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		−0.156434	−	0.987688i	$-0.550000\pi$
$13$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$14$		0			0
$15$	−0.956081	−	3.53593i	−0.246859	−	0.912973i
$16$	3.23607	−	2.35114i	0.809017	−	0.587785i
$17$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$17$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$18$		0			0
$19$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$19$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$20$	−0.477588	−	4.44656i	−0.106792	−	0.994281i
$21$		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
$25$	−4.57694	−	2.01287i	−0.915388	−	0.402574i
$26$		0			0
$27$	−0.849903	+	5.36608i	−0.163564	+	1.03270i
$28$		0			0
$29$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$29$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$30$		0			0
$31$	−8.04962	−	5.84839i	−1.44575	−	1.05040i	−0.986800	−	0.161942i	$-0.948224\pi$
$31$	−8.04962	−	5.84839i	−1.44575	−	1.05040i	−0.458954	−	0.888460i	$-0.651776\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−0.195685	+	0.602256i	−0.0326141	+	0.100376i
$37$	−10.6790	−	5.44124i	−1.75562	−	0.894535i	−0.955652	−	0.294497i	$-0.904848\pi$
$37$	−10.6790	−	5.44124i	−1.75562	−	0.894535i	−0.799972	−	0.600038i	$-0.795152\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$41$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$42$		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
$47$	1.72283	+	3.38125i	0.251301	+	0.493206i	0.981851	−	0.189653i	$-0.0607363\pi$
$47$	1.72283	+	3.38125i	0.251301	+	0.493206i	−0.730550	+	0.682859i	$0.760736\pi$
$48$	2.97473	−	5.83824i	0.429365	−	0.842677i
$49$	4.11450	+	5.66312i	0.587785	+	0.809017i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	2.13118	+	13.4557i	0.292740	+	1.84829i	0.494918	+	0.868940i	$0.335198\pi$
$53$	2.13118	+	13.4557i	0.292740	+	1.84829i	−0.202178	+	0.979349i	$0.564802\pi$
$54$		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	$-0.530733\pi$
$59$	3.15430	−	1.02489i	0.410655	−	0.133430i	0.507057	+	0.861913i	$0.330733\pi$
$60$	−4.00390	−	6.13484i	−0.516901	−	0.792005i
$61$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$61$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$62$		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	−0.434378	−	0.900731i	$-0.643032\pi$
$71$	2.42705	−	1.76336i	0.288038	−	0.209272i	0.722416	+	0.691459i	$0.243032\pi$
$72$		0			0
$73$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$73$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$74$		0			0
$75$	−8.17724	+	0.465888i	−0.944227	+	0.0537961i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$79$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$80$	−3.65655	−	8.16270i	−0.408815	−	0.912617i
$81$	2.45665	+	7.56078i	0.272961	+	0.840087i
$82$		0			0
$83$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$83$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$84$		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
$92$	15.5199	+	7.90776i	1.61806	+	0.824441i
$93$	−16.0982	−	2.54971i	−1.66931	−	0.264393i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	18.8218	−	2.98108i	1.91107	−	0.302683i	0.915942	−	0.401310i	$-0.131445\pi$
$97$	18.8218	−	2.98108i	1.91107	−	0.302683i	0.995124	+	0.0986273i	$0.0314452\pi$
$98$		0			0
$99$		0			0
$100$	−9.94987	−	1.00000i	−0.994987	−	0.100000i
$101$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$101$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$102$		0			0
$103$	−7.67229	+	15.0577i	−0.755973	+	1.48368i	0.115536	+	0.993303i	$0.463141\pi$
$103$	−7.67229	+	15.0577i	−0.755973	+	1.48368i	−0.871510	+	0.490378i	$0.836859\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$107$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$108$	1.69981	+	10.7322i	0.163564	+	1.03270i
$109$		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
$113$	−11.1412	+	5.67671i	−1.04807	+	0.534020i	−0.891206	−	0.453599i	$-0.850140\pi$
$113$	−11.1412	+	5.67671i	−1.04807	+	0.534020i	−0.156868	+	0.987620i	$0.550140\pi$
$114$		0			0
$115$	16.3083	−	10.6436i	1.52076	−	0.992522i
$116$		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
$125$	−6.50972	+	9.08975i	−0.582247	+	0.813012i
$126$		0			0
$127$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$127$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$128$		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
$135$	11.3515	+	4.32780i	0.976979	+	0.372478i
$136$		0			0
$137$	−2.89995	+	18.3095i	−0.247759	+	1.56429i	0.479260	+	0.877673i	$0.340905\pi$
$137$	−2.89995	+	18.3095i	−0.247759	+	1.56429i	−0.727019	+	0.686617i	$0.759095\pi$
$138$		0			0
$139$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$139$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$140$		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
$147$	10.2169	+	5.20578i	0.842677	+	0.429365i
$148$	−23.6756	−	3.74985i	−1.94612	−	0.308236i
$149$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$149$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$150$		0			0
$151$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$151$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−16.5000	+	14.9248i	−1.32531	+	1.19879i
$156$		0			0
$157$	−4.18935	−	8.22206i	−0.334346	−	0.656192i	0.661226	−	0.750186i	$-0.270036\pi$
$157$	−4.18935	−	8.22206i	−0.334346	−	0.656192i	−0.995573	+	0.0939948i	$0.970036\pi$
$158$		0			0
$159$	13.1174	+	18.0545i	1.04028	+	1.43182i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−0.431374	−	2.72359i	−0.0337878	−	0.213328i	0.965018	−	0.262184i	$-0.0844426\pi$
$163$	−0.431374	−	2.72359i	−0.0337878	−	0.213328i	−0.998806	+	0.0488556i	$0.984443\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$167$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$168$		0			0
$169$	−12.3637	+	4.01722i	−0.951057	+	0.309017i
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$173$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$174$		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	$-0.112762\pi$
$179$	19.9722	+	6.48936i	1.49279	+	0.485037i	0.554885	+	0.831927i	$0.312762\pi$
$180$	1.22789	+	0.705201i	0.0915216	+	0.0525626i
$181$	−8.04962	+	5.84839i	−0.598323	+	0.434707i	−0.845283	−	0.534318i	$-0.820568\pi$
$181$	−8.04962	+	5.84839i	−0.598323	+	0.434707i	0.246960	+	0.969026i	$0.420568\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−16.8184	+	20.8659i	−1.23651	+	1.53409i
$186$		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	−0.775676	−	0.631132i	$-0.782591\pi$
$191$	−7.17425	−	22.0801i	−0.519111	−	1.59766i	0.256565	−	0.966527i	$-0.417409\pi$
$192$	2.05004	−	12.9435i	0.147949	−	0.934114i
$193$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$193$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$194$		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
$207$	−2.72359	+	0.431374i	−0.189303	+	0.0299826i
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$211$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$212$	12.3699	+	24.2772i	0.849565	+	1.66737i
$213$	2.23105	−	4.37868i	0.152869	−	0.300022i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−19.4364	+	9.90334i	−1.30156	+	0.663177i	−0.960870	−	0.277000i	$-0.910660\pi$
$223$	−19.4364	+	9.90334i	−1.30156	+	0.663177i	−0.340687	+	0.940177i	$0.610660\pi$
$224$		0			0
$225$	1.36741	−	0.797795i	0.0911605	−	0.0531863i
$226$		0			0
$227$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$227$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$228$		0			0
$229$	17.5452	−	24.1489i	1.15942	−	1.59580i	0.446055	−	0.895005i	$-0.352828\pi$
$229$	17.5452	−	24.1489i	1.15942	−	1.59580i	0.713362	−	0.700796i	$-0.247172\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$233$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$234$		0			0
$235$	8.19142	−	2.21488i	0.534349	−	0.144483i
$236$	5.36641	−	3.89893i	0.349324	−	0.253798i
$237$		0			0
$238$		0			0
$239$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$239$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$240$	−11.4074	−	9.19462i	−0.736345	−	0.593510i
$241$		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
$245$	14.2847	−	6.39897i	0.912617	−	0.408815i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−21.8435	−	15.8702i	−1.37875	−	1.00172i	−0.996996	−	0.0774530i	$-0.975321\pi$
$251$	−21.8435	−	15.8702i	−1.37875	−	1.00172i	−0.381751	−	0.924265i	$-0.624679\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	4.94427	−	15.2169i	0.309017	−	0.951057i
$257$	−5.37610	−	2.73926i	−0.335352	−	0.170870i	0.278203	−	0.960522i	$-0.410261\pi$
$257$	−5.37610	−	2.73926i	−0.335352	−	0.170870i	−0.613555	+	0.789652i	$0.710261\pi$
$258$		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
$267$	−6.69314	−	13.1360i	−0.409614	−	0.803912i
$268$	1.95830	−	3.84338i	0.119622	−	0.234772i
$269$	−7.79785	−	10.7328i	−0.475443	−	0.654392i	0.502178	−	0.864764i	$-0.332532\pi$
$269$	−7.79785	−	10.7328i	−0.475443	−	0.654392i	−0.977621	+	0.210373i	$0.932532\pi$
$270$		0			0
$271$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$271$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$	28.5330			1.71748
$277$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$277$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$278$		0			0
$279$	2.99619	−	0.973520i	0.179377	−	0.0582831i
$280$		0			0
$281$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$281$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$282$		0			0
$283$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$283$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$284$	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		0.453990	−	0.891007i	$-0.350000\pi$
$293$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$294$		0			0
$295$	−0.791990	−	7.37379i	−0.0461114	−	0.429319i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	−15.2661	+	5.93999i	−0.881389	+	0.342945i
$301$		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	−0.789183	−	0.614159i	$-0.789495\pi$
$311$	3.70820	−	11.4127i	0.210273	−	0.647154i	0.999456	−	0.0329949i	$-0.0105045\pi$
$312$		0			0
$313$	7.57739	+	1.20014i	0.428299	+	0.0678360i	0.366863	−	0.930275i	$-0.380432\pi$
$313$	7.57739	+	1.20014i	0.428299	+	0.0678360i	0.0614365	+	0.998111i	$0.480432\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	2.64249	−	0.418529i	0.148417	−	0.0235069i	−0.0817838	−	0.996650i	$-0.526062\pi$
$317$	2.64249	−	0.418529i	0.148417	−	0.0235069i	0.230201	+	0.973143i	$0.426062\pi$
$318$		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
$333$	3.38125	−	1.72283i	0.185291	−	0.0944107i
$334$		0			0
$335$	−2.63581	−	4.03864i	−0.144010	−	0.220655i
$336$		0			0
$337$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$337$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$338$		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
$345$	15.8875	−	27.6632i	0.855355	−	1.48934i
$346$		0			0
$347$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$347$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$348$		0			0
$349$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$349$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$350$		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
$355$	−2.74241	−	6.12202i	−0.145552	−	0.324923i
$356$	−5.56231	−	17.1190i	−0.294802	−	0.907306i
$357$		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
$361$	15.3713	+	11.1679i	0.809017	+	0.587785i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−29.5801	−	15.0718i	−1.54407	−	0.786744i	−0.545395	−	0.838179i	$-0.683620\pi$
$367$	−29.5801	−	15.0718i	−1.54407	−	0.786744i	−0.998676	+	0.0514358i	$0.983620\pi$
$368$	34.4078	+	5.44966i	1.79363	+	0.284083i
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−32.1965	+	5.09942i	−1.66931	+	0.264393i
$373$		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	−0.970073	−	0.242815i	$-0.921929\pi$
$379$	−17.5452	−	24.1489i	−0.901235	−	1.24044i	0.0688378	−	0.997628i	$-0.478071\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−4.68089	−	29.5540i	−0.239182	−	1.51014i	−0.756301	−	0.654224i	$-0.772995\pi$
$383$	−4.68089	−	29.5540i	−0.239182	−	1.51014i	0.517119	−	0.855914i	$-0.327005\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	33.9588	−	17.3029i	1.72400	−	0.878421i
$389$	34.6973	−	11.2738i	1.75922	−	0.571606i	0.762102	−	0.647456i	$-0.224167\pi$
$389$	34.6973	−	11.2738i	1.75922	−	0.571606i	0.997119	+	0.0758507i	$0.0241672\pi$
$390$		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
$400$	−19.5438	+	4.24725i	−0.977191	+	0.212362i
$401$	−21.4656	+	15.5957i	−1.07194	+	0.778812i	−0.976261	−	0.216600i	$-0.930503\pi$
$401$	−21.4656	+	15.5957i	−1.07194	+	0.778812i	−0.0956827	+	0.995412i	$0.530503\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	17.6748	−	1.89838i	0.878268	−	0.0943313i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$409$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$410$		0			0
$411$	9.38384	+	28.8805i	0.462871	+	1.42457i
$412$	−5.28738	+	33.3832i	−0.260491	+	1.64467i
$413$		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	−0.0679432	+	0.997689i	$0.521644\pi$
$421$	−12.2987	+	37.8516i	−0.599403	+	1.84477i	−0.531460	+	0.847084i	$0.678356\pi$
$422$		0			0
$423$	−1.18676	−	0.187964i	−0.0577020	−	0.00913910i
$424$		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	$-0.800000\pi$
$431$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$432$	9.86606	+	19.3632i	0.474681	+	0.931614i
$433$	0.272746	−	0.535294i	0.0131073	−	0.0257246i	−0.884360	−	0.466805i	$-0.845405\pi$
$433$	0.272746	−	0.535294i	0.0131073	−	0.0257246i	0.897467	+	0.441081i	$0.145405\pi$
$434$		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
$443$	9.75478	−	4.97031i	0.463464	−	0.236146i	−0.206636	−	0.978418i	$-0.566252\pi$
$443$	9.75478	−	4.97031i	0.463464	−	0.236146i	0.670099	+	0.742271i	$0.266252\pi$
$444$	−37.3447	+	12.1340i	−1.77230	+	0.575855i
$445$	−19.6943	−	4.13933i	−0.933600	−	0.196223i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−22.9236	+	31.5517i	−1.08183	+	1.48902i	−0.224346	+	0.974510i	$0.572024\pi$
$449$	−22.9236	+	31.5517i	−1.08183	+	1.48902i	−0.857487	+	0.514505i	$0.827976\pi$
$450$		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		0.156434	−	0.987688i	$-0.450000\pi$
$457$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$458$		0			0
$459$		0			0
$460$	24.4422	−	30.3245i	1.13962	−	1.41388i
$461$		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
$465$	−12.9834	+	34.0544i	−0.602091	+	1.57923i
$466$		0			0
$467$	4.43748	−	28.0171i	0.205342	−	1.29648i	−0.642523	−	0.766267i	$-0.722112\pi$
$467$	4.43748	−	28.0171i	0.205342	−	1.29648i	0.847865	−	0.530212i	$-0.177888\pi$
$468$		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
$477$	−3.84338	−	1.95830i	−0.175976	−	0.0896645i
$478$		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
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$	2.13325	−	42.5581i	0.0968659	−	1.93246i
$486$		0			0
$487$	−10.6097	−	20.8228i	−0.480773	−	0.943571i	−0.996238	−	0.0866600i	$-0.972381\pi$
$487$	−10.6097	−	20.8228i	−0.480773	−	0.943571i	0.515465	−	0.856911i	$-0.327619\pi$
$488$		0			0
$489$	−2.65510	−	3.65443i	−0.120068	−	0.165259i
$490$		0			0
$491$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$491$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$492$		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	$-0.453055\pi$
$499$	18.9258	−	6.14936i	0.847235	−	0.275283i	0.700287	+	0.713861i	$0.253055\pi$
$500$	−6.76445	+	21.3130i	−0.302516	+	0.953144i
$501$		0			0
$502$		0			0
$503$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$503$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$504$		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	$-0.423418\pi$
$509$	−3.15430	−	1.02489i	−0.139812	−	0.0454276i	−0.378087	+	0.925770i	$0.623418\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	29.4215	+	23.7144i	1.29647	+	1.04498i
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	13.3236	+	41.0059i	0.583718	+	1.79650i	0.604358	+	0.796713i	$0.293430\pi$
$521$	13.3236	+	41.0059i	0.583718	+	1.79650i	−0.0206400	+	0.999787i	$0.506570\pi$
$522$		0			0
$523$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$523$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$524$		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
$537$	33.9766	−	5.38136i	1.46620	−	0.232223i
$538$		0			0
$539$		0			0
$540$	24.2665	+	1.21637i	1.04426	+	0.0523444i
$541$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$541$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$542$		0			0
$543$	−7.39955	+	14.5224i	−0.317545	+	0.623217i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$547$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$548$	5.79989	+	36.6191i	0.247759	+	1.56429i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−9.02982	+	42.9626i	−0.383295	+	1.82366i
$556$		0			0
$557$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$557$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$563$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$564$	11.8242	+	3.84193i	0.497891	+	0.161774i
$565$	7.29800	+	26.9906i	0.307029	+	1.13550i
$566$		0			0
$567$		0			0
$568$		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
$571$		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
$575$	−15.7903	−	40.5821i	−0.658503	−	1.69239i
$576$	0.782740	+	2.40902i	0.0326141	+	0.100376i
$577$	6.29956	−	39.7739i	0.262254	−	1.65581i	−0.407486	−	0.913212i	$-0.633594\pi$
$577$	6.29956	−	39.7739i	0.262254	−	1.65581i	0.669740	−	0.742596i	$-0.266406\pi$
$578$		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
$587$	34.4210	+	17.5384i	1.42070	+	0.723885i	0.984405	−	0.175916i	$-0.0562887\pi$
$587$	34.4210	+	17.5384i	1.42070	+	0.723885i	0.436300	+	0.899801i	$0.356289\pi$
$588$	22.6511	+	3.58758i	0.934114	+	0.147949i
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−47.3512	+	7.49970i	−1.94612	+	0.308236i
$593$		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
$597$	−14.7991	−	29.0449i	−0.605686	−	1.18873i
$598$		0			0
$599$	−21.1603	−	29.1246i	−0.864585	−	1.19000i	−0.980457	−	0.196735i	$-0.936966\pi$
$599$	−21.1603	−	29.1246i	−0.864585	−	1.19000i	0.115872	−	0.993264i	$-0.463034\pi$
$600$		0			0
$601$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$601$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$602$		0			0
$603$	0.106827	+	0.674478i	0.00435032	+	0.0274669i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$607$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$613$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$614$		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	$-0.393603\pi$
$619$	0.951057	+	0.309017i	0.0382262	+	0.0124204i	−0.289841	+	0.957075i	$0.593603\pi$
$620$	−22.1608	+	38.5862i	−0.890000	+	1.54966i
$621$	−38.2800	+	27.8120i	−1.53612	+	1.11606i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	16.8967	+	18.4255i	0.675869	+	0.737022i
$626$		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	0.984836	−	0.173489i	$-0.0555042\pi$
$631$	2.16312	+	6.65740i	0.0861124	+	0.265027i	−0.898723	+	0.438516i	$0.855504\pi$
$632$		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	0.703518	+	0.710678i	$0.251612\pi$
$641$	−7.17425	+	22.0801i	−0.283366	+	0.872111i	−0.986884	+	0.161433i	$0.948388\pi$
$642$		0			0
$643$	49.4815	+	7.83709i	1.95136	+	0.309065i	1.00000		0.000979141i	$0.000311670\pi$
$643$	49.4815	+	7.83709i	1.95136	+	0.309065i	0.951359	+	0.308086i	$0.0996883\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	48.9692	−	7.75596i	1.92518	−	0.304918i	0.927546	−	0.373709i	$-0.121914\pi$
$647$	48.9692	−	7.75596i	1.92518	−	0.304918i	0.997631	+	0.0687910i	$0.0219142\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$	−2.50379	−	4.91397i	−0.0980561	−	0.192446i
$653$	15.3073	−	30.0423i	0.599021	−	1.17565i	−0.370084	−	0.928998i	$-0.620671\pi$
$653$	15.3073	−	30.0423i	0.599021	−	1.17565i	0.969105	−	0.246647i	$-0.0793288\pi$
$654$		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		−0.156434	−	0.987688i	$-0.550000\pi$
$673$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$674$		0			0
$675$	14.6912	−	22.8495i	0.565463	−	0.879476i
$676$	−21.0344	+	15.2824i	−0.809017	+	0.587785i
$677$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$677$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$678$		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
$685$	38.7322	+	14.7668i	1.47988	+	0.564212i
$686$		0			0
$687$	7.64913	−	48.2947i	0.291832	−	1.84256i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−13.7533	−	9.99235i	−0.523200	−	0.380127i	0.294608	−	0.955618i	$-0.404811\pi$
$691$	−13.7533	−	9.99235i	−0.523200	−	0.380127i	−0.817808	+	0.575491i	$0.804811\pi$
$692$		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	$-0.400000\pi$
$701$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$	10.3087	−	9.32456i	0.388248	−	0.351183i
$706$		0			0
$707$		0			0
$708$	4.93303	−	9.68162i	0.185395	−	0.363857i
$709$	11.1679	+	15.3713i	0.419420	+	0.577282i	0.965484	−	0.260461i	$-0.0838746\pi$
$709$	11.1679	+	15.3713i	0.419420	+	0.577282i	−0.546064	+	0.837743i	$0.683875\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−13.5558	−	85.5883i	−0.507671	−	3.20531i
$714$		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	$-0.999931\pi$
$719$	−48.5039	+	15.7599i	−1.80889	+	0.587744i	−0.808890	+	0.587961i	$0.799931\pi$
$720$	2.77143	+	0.582495i	0.103285	+	0.0217083i
$721$		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		0.453990	−	0.891007i	$-0.350000\pi$
$733$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$734$		0			0
$735$	16.0906	−	19.9630i	0.593510	−	0.736345i
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$739$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$740$	−19.0946	+	50.0837i	−0.701933	+	1.84111i
$741$		0			0
$742$		0			0
$743$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$743$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$744$		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	0.733588	+	0.679594i	$0.237844\pi$
$751$	−7.10739	+	21.8743i	−0.259352	+	0.798205i	−0.992941	+	0.118611i	$0.962156\pi$
$752$	13.5250	+	6.89133i	0.493206	+	0.251301i
$753$	−43.6842	−	6.91890i	−1.59194	−	0.252139i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−54.3353	+	8.60586i	−1.97485	+	0.312785i	−0.983807	+	0.179232i	$0.942639\pi$
$757$	−54.3353	+	8.60586i	−1.97485	+	0.312785i	−0.991042	+	0.133554i	$0.957361\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$761$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$762$		0			0
$763$		0			0
$764$	−27.2925	−	37.5649i	−0.987407	−	1.35905i
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−4.10009	−	25.8869i	−0.147949	−	0.934114i
$769$		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
$773$	25.6636	−	13.0763i	0.923056	−	0.470320i	0.0731890	−	0.997318i	$-0.476682\pi$
$773$	25.6636	−	13.0763i	0.923056	−	0.470320i	0.849867	+	0.526998i	$0.176682\pi$
$774$		0			0
$775$	25.0706	+	42.9705i	0.900561	+	1.54355i
$776$		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
$785$	−19.9188	+	5.38584i	−0.710931	+	0.192229i
$786$		0			0
$787$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$787$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$	45.5410	−	20.4005i	1.61517	−	0.723530i
$796$	−12.2987	−	37.8516i	−0.435917	−	1.34161i
$797$	−6.56866	+	41.4729i	−0.232674	+	1.46905i	0.543970	+	0.839105i	$0.316921\pi$
$797$	−6.56866	+	41.4729i	−0.232674	+	1.46905i	−0.776644	+	0.629940i	$0.783079\pi$
$798$		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
$807$	−19.3632	−	9.86606i	−0.681618	−	0.347302i
$808$		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
$811$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$811$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$812$		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	$-0.600000\pi$
$821$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$822$		0			0
$823$	−2.11834	−	13.3746i	−0.0738405	−	0.466211i	−0.996707	−	0.0810902i	$-0.974160\pi$
$823$	−2.11834	−	13.3746i	−0.0738405	−	0.466211i	0.922866	−	0.385121i	$-0.125840\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$827$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$828$	−4.91397	+	2.50379i	−0.170772	+	0.0870128i
$829$	27.5806	−	8.96149i	0.957915	−	0.311246i	0.211987	−	0.977272i	$-0.432006\pi$
$829$	27.5806	−	8.96149i	0.957915	−	0.311246i	0.745928	+	0.666027i	$0.232006\pi$
$830$		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	$-0.616849\pi$
$839$	−34.6973	−	11.2738i	−1.19788	−	0.389216i	−0.838982	+	0.544160i	$0.816849\pi$
$840$		0			0
$841$	23.4615	−	17.0458i	0.809017	−	0.587785i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	3.10432	+	28.9027i	0.106792	+	0.994281i
$846$		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
$852$	1.53753	−	9.70760i	0.0526750	−	0.332577i
$853$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$853$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$854$		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
$863$	57.5711	+	9.11837i	1.95974	+	0.310393i	0.999514	+	0.0311832i	$0.00992752\pi$
$863$	57.5711	+	9.11837i	1.95974	+	0.310393i	0.960230	+	0.279210i	$0.0900725\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−27.5049	+	4.35634i	−0.934114	+	0.147949i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−2.73926	+	5.37610i	−0.0927099	+	0.181954i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$877$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$878$		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
$883$	−22.7445	+	11.5889i	−0.765413	+	0.389997i	−0.792686	−	0.609631i	$-0.791318\pi$
$883$	−22.7445	+	11.5889i	−0.765413	+	0.389997i	0.0272727	+	0.999628i	$0.491318\pi$
$884$		0			0
$885$	−6.63971	−	10.1735i	−0.223191	−	0.341978i
$886$		0			0
$887$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$887$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$888$		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
$895$	23.3861	−	40.7197i	0.781711	−	1.36111i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	2.10790	−	2.36260i	0.0702633	−	0.0787534i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	9.09556	+	20.3045i	0.302347	+	0.674943i
$906$		0			0
$907$	5.71876	−	36.1068i	0.189888	−	1.19891i	−0.690030	−	0.723781i	$-0.742403\pi$
$907$	5.71876	−	36.1068i	0.189888	−	1.19891i	0.879918	−	0.475126i	$-0.157597\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	5.36641	+	3.89893i	0.177797	+	0.129177i	0.673124	−	0.739529i	$-0.264952\pi$
$911$	5.36641	+	3.89893i	0.177797	+	0.129177i	−0.495327	+	0.868706i	$0.664952\pi$
$912$		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	$-0.700000\pi$
$919$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$920$		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
$927$	−2.42924	−	4.76765i	−0.0797866	−	0.156590i
$928$		0			0
$929$	31.1914	+	42.9313i	1.02336	+	1.40853i	0.909823	+	0.414996i	$0.136217\pi$
$929$	31.1914	+	42.9313i	1.02336	+	1.40853i	0.113534	+	0.993534i	$0.463783\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	−3.07507	−	19.4152i	−0.100673	−	0.635625i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$937$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$938$		0			0
$939$	11.9522	−	3.88349i	0.390044	−	0.126733i
$940$	14.2121	−	9.27553i	0.463549	−	0.302534i
$941$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$941$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$942$		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		0.453990	−	0.891007i	$-0.350000\pi$
$953$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$954$		0			0
$955$	−51.6165	+	5.54393i	−1.67027	+	0.179397i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$	−27.3808	−	10.4390i	−0.883710	−	0.336919i
$961$	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	−0.472932	−	0.881099i	$-0.656804\pi$
$971$	13.3236	−	41.0059i	0.427575	−	1.31594i	0.900507	−	0.434842i	$-0.143196\pi$
$972$	−5.83824	−	2.97473i	−0.187261	−	0.0954145i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−20.5209	+	3.25018i	−0.656520	+	0.103983i	−0.475802	−	0.879552i	$-0.657842\pi$
$977$	−20.5209	+	3.25018i	−0.656520	+	0.103983i	−0.180718	+	0.983535i	$0.557842\pi$
$978$		0			0
$979$		0			0
$980$	23.2164	−	21.0000i	0.741620	−	0.670820i
$981$		0			0
$982$		0			0
$983$	4.66028	−	9.14632i	0.148640	−	0.291722i	−0.804666	−	0.593727i	$-0.797656\pi$
$983$	4.66028	−	9.14632i	0.148640	−	0.291722i	0.953306	+	0.302005i	$0.0976558\pi$
$984$		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
$993$	14.5224	−	7.39955i	0.460855	−	0.234817i
$994$		0			0
$995$	−43.5458	−	9.15239i	−1.38049	−	0.290150i
$996$		0			0
$997$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$997$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$998$		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.403.2		16
5.2	odd	4	inner	605.2.m.a.282.1		16
11.2	odd	10	inner	605.2.m.a.578.1		16
11.3	even	5	inner	605.2.m.a.118.1		16
11.4	even	5	inner	605.2.m.a.233.1		16
11.5	even	5		55.2.e.b.43.2	yes	4
11.6	odd	10		55.2.e.b.43.2	yes	4
11.7	odd	10	inner	605.2.m.a.233.1		16
11.8	odd	10	inner	605.2.m.a.118.1		16
11.9	even	5	inner	605.2.m.a.578.1		16
11.10	odd	2	CM	605.2.m.a.403.2		16
33.5	odd	10		495.2.k.a.208.2		4
33.17	even	10		495.2.k.a.208.2		4
44.27	odd	10		880.2.bd.d.593.1		4
44.39	even	10		880.2.bd.d.593.1		4
55.2	even	20	inner	605.2.m.a.457.1		16
55.7	even	20	inner	605.2.m.a.112.1		16
55.17	even	20		55.2.e.b.32.2	✓	4
55.27	odd	20		55.2.e.b.32.2	✓	4
55.28	even	20		275.2.e.a.32.1		4
55.32	even	4	inner	605.2.m.a.282.1		16
55.37	odd	20	inner	605.2.m.a.112.1		16
55.38	odd	20		275.2.e.a.32.1		4
55.39	odd	10		275.2.e.a.43.1		4
55.42	odd	20	inner	605.2.m.a.457.1		16
55.47	odd	20	inner	605.2.m.a.602.2		16
55.49	even	10		275.2.e.a.43.1		4
55.52	even	20	inner	605.2.m.a.602.2		16
165.17	odd	20		495.2.k.a.307.2		4
165.137	even	20		495.2.k.a.307.2		4
220.27	even	20		880.2.bd.d.417.1		4
220.127	odd	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	55.17	even	20
55.2.e.b.32.2	✓	4	55.27	odd	20
55.2.e.b.43.2	yes	4	11.5	even	5
55.2.e.b.43.2	yes	4	11.6	odd	10
275.2.e.a.32.1		4	55.28	even	20
275.2.e.a.32.1		4	55.38	odd	20
275.2.e.a.43.1		4	55.39	odd	10
275.2.e.a.43.1		4	55.49	even	10
495.2.k.a.208.2		4	33.5	odd	10
495.2.k.a.208.2		4	33.17	even	10
495.2.k.a.307.2		4	165.17	odd	20
495.2.k.a.307.2		4	165.137	even	20
605.2.m.a.112.1		16	55.7	even	20	inner
605.2.m.a.112.1		16	55.37	odd	20	inner
605.2.m.a.118.1		16	11.3	even	5	inner
605.2.m.a.118.1		16	11.8	odd	10	inner
605.2.m.a.233.1		16	11.4	even	5	inner
605.2.m.a.233.1		16	11.7	odd	10	inner
605.2.m.a.282.1		16	5.2	odd	4	inner
605.2.m.a.282.1		16	55.32	even	4	inner
605.2.m.a.403.2		16	1.1	even	1	trivial
605.2.m.a.403.2		16	11.10	odd	2	CM
605.2.m.a.457.1		16	55.2	even	20	inner
605.2.m.a.457.1		16	55.42	odd	20	inner
605.2.m.a.578.1		16	11.2	odd	10	inner
605.2.m.a.578.1		16	11.9	even	5	inner
605.2.m.a.602.2		16	55.47	odd	20	inner
605.2.m.a.602.2		16	55.52	even	20	inner
880.2.bd.d.417.1		4	220.27	even	20
880.2.bd.d.417.1		4	220.127	odd	20
880.2.bd.d.593.1		4	44.27	odd	10
880.2.bd.d.593.1		4	44.39	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)

\(f(q)\)	\(=\)	\(q+(1.45956 - 0.743682i) q^{3} +(1.90211 - 0.618034i) q^{4} +(0.459925 - 2.18826i) q^{5} +(-0.186107 + 0.256155i) q^{9} +(2.31662 - 2.31662i) q^{12} +(-0.956081 - 3.53593i) q^{15} +(3.23607 - 2.35114i) q^{16} +(-0.477588 - 4.44656i) q^{20} +(6.15831 + 6.15831i) q^{23} +(-4.57694 - 2.01287i) q^{25} +(-0.849903 + 5.36608i) q^{27} +(-8.04962 - 5.84839i) q^{31} +(-0.195685 + 0.602256i) q^{36} +(-10.6790 - 5.44124i) q^{37} +(0.474937 + 0.525063i) q^{45} +(1.72283 + 3.38125i) q^{47} +(2.97473 - 5.83824i) q^{48} +(4.11450 + 5.66312i) q^{49} +(2.13118 + 13.4557i) q^{53} +(3.15430 - 1.02489i) q^{59} +(-4.00390 - 6.13484i) q^{60} +(4.70228 - 6.47214i) q^{64} +(1.52506 - 1.52506i) q^{67} +(13.5682 + 4.40859i) q^{69} +(2.42705 - 1.76336i) q^{71} +(-8.17724 + 0.465888i) q^{75} +(-3.65655 - 8.16270i) q^{80} +(2.45665 + 7.56078i) q^{81} -9.00000i q^{89} +(15.5199 + 7.90776i) q^{92} +(-16.0982 - 2.54971i) q^{93} +(18.8218 - 2.98108i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 2 q^{3} - 16 q^{12} + 8 q^{15} + 16 q^{16} - 12 q^{20} + 72 q^{23} - 2 q^{25} - 22 q^{27} - 24 q^{36} + 14 q^{37} - 72 q^{45} + 24 q^{47} - 8 q^{48} + 12 q^{53} + 28 q^{60} + 104 q^{67} + 12 q^{71}+ \cdots + 34 q^{97}+O(q^{100}) \) 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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