LMFDB - Embedded newform 605.2.m.a.602.1

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			602.1
Root			$-0.0369185 - 1.73166i$ of defining polynomial
Character	$\chi$	$=$	605.602
Dual form			605.2.m.a.403.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.71963 - 1.38572i) q^{3} +(1.90211 + 0.618034i) q^{4} +(-2.22328 + 0.238794i) q^{5} +(3.71282 + 5.11026i) q^{9} +(-4.31662 - 4.31662i) q^{12} +(6.37741 + 2.43142i) q^{15} +(3.23607 + 2.35114i) q^{16} +(-4.37651 - 0.919850i) q^{20} +(2.84169 - 2.84169i) q^{23} +(4.88595 - 1.06181i) q^{25} +(-1.58365 - 9.99875i) q^{27} +(8.04962 - 5.84839i) q^{31} +(3.90389 + 12.0149i) q^{36} +(1.85853 - 0.946968i) q^{37} +(-9.47494 - 10.4749i) q^{45} +(5.98164 - 11.7396i) q^{47} +(-5.54289 - 10.8785i) q^{48} +(4.11450 - 5.66312i) q^{49} +(-0.803790 + 5.07493i) q^{53} +(-3.15430 - 1.02489i) q^{59} +(10.6279 + 8.56629i) q^{60} +(4.70228 + 6.47214i) q^{64} +(11.4749 + 11.4749i) q^{67} +(-11.6661 + 3.79056i) q^{69} +(2.42705 + 1.76336i) q^{71} +(-14.7594 - 3.88284i) q^{75} +(-7.75613 - 4.45449i) q^{80} +(-3.69271 + 11.3650i) q^{81} +9.00000i q^{89} +(7.16147 - 3.64895i) q^{92} +(-29.9962 + 4.75094i) q^{93} +(4.92382 + 0.779856i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/605\mathbb{Z}\right)^\times$.

$n$	$122$	$486$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$e\left(\frac{3}{10}\right)$

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		−0.987688	−	0.156434i	$-0.950000\pi$
$2$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$3$	−2.71963	−	1.38572i	−1.57018	−	0.800047i	−0.570408	−	0.821362i	$-0.693215\pi$
$3$	−2.71963	−	1.38572i	−1.57018	−	0.800047i	−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		0.891007	−	0.453990i	$-0.150000\pi$
$7$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$8$		0			0
$9$	3.71282	+	5.11026i	1.23761	+	1.70342i
$10$		0			0
$11$		0			0
$11$		0			0
$12$	−4.31662	−	4.31662i	−1.24610	−	1.24610i
$13$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$13$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$14$		0			0
$15$	6.37741	+	2.43142i	1.64664	+	0.627789i
$16$	3.23607	+	2.35114i	0.809017	+	0.587785i
$17$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$17$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$18$		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
$23$	2.84169	−	2.84169i	0.592533	−	0.592533i	−0.345782	−	0.938315i	$-0.612386\pi$
$23$	2.84169	−	2.84169i	0.592533	−	0.592533i	0.938315	+	0.345782i	$0.112386\pi$
$24$		0			0
$25$	4.88595	−	1.06181i	0.977191	−	0.212362i
$26$		0			0
$27$	−1.58365	−	9.99875i	−0.304773	−	1.92426i
$28$		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	$-0.348224\pi$
$31$	8.04962	−	5.84839i	1.44575	−	1.05040i	0.986800	−	0.161942i	$-0.0517756\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	3.90389	+	12.0149i	0.650648	+	2.00249i
$37$	1.85853	−	0.946968i	0.305540	−	0.155681i	−0.294497	−	0.955652i	$-0.595152\pi$
$37$	1.85853	−	0.946968i	0.305540	−	0.155681i	0.600038	+	0.799972i	$0.295152\pi$
$38$		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	$-0.250000\pi$
$43$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$44$		0			0
$45$	−9.47494	−	10.4749i	−1.41244	−	1.56151i
$46$		0			0
$47$	5.98164	−	11.7396i	0.872512	−	1.71240i	0.189653	−	0.981851i	$-0.439264\pi$
$47$	5.98164	−	11.7396i	0.872512	−	1.71240i	0.682859	−	0.730550i	$-0.260736\pi$
$48$	−5.54289	−	10.8785i	−0.800047	−	1.57018i
$49$	4.11450	−	5.66312i	0.587785	−	0.809017i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−0.803790	+	5.07493i	−0.110409	+	0.697095i	0.868940	+	0.494918i	$0.164802\pi$
$53$	−0.803790	+	5.07493i	−0.110409	+	0.697095i	−0.979349	+	0.202178i	$0.935198\pi$
$54$		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	$0.669267\pi$
$60$	10.6279	+	8.56629i	1.37205	+	1.10590i
$61$		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
$67$	11.4749	+	11.4749i	1.40189	+	1.40189i	0.794101	+	0.607785i	$0.207942\pi$
$67$	11.4749	+	11.4749i	1.40189	+	1.40189i	0.607785	+	0.794101i	$0.292058\pi$
$68$		0			0
$69$	−11.6661	+	3.79056i	−1.40444	+	0.456329i
$70$		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		−0.453990	−	0.891007i	$-0.650000\pi$
$73$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$74$		0			0
$75$	−14.7594	−	3.88284i	−1.70427	−	0.448351i
$76$		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
$81$	−3.69271	+	11.3650i	−0.410302	+	1.26278i
$82$		0			0
$83$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$83$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$			9.00000i			0.953998i	0.878904	+	0.476999i	$0.158275\pi$
$89$			9.00000i			0.953998i	−0.878904	+	0.476999i	$0.841725\pi$
$90$		0			0
$91$		0			0
$92$	7.16147	−	3.64895i	0.746635	−	0.380429i
$93$	−29.9962	+	4.75094i	−3.11047	+	0.492649i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	4.92382	+	0.779856i	0.499938	+	0.0791824i	0.401310	−	0.915942i	$-0.368555\pi$
$97$	4.92382	+	0.779856i	0.499938	+	0.0791824i	0.0986273	+	0.995124i	$0.468555\pi$
$98$		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
$103$	5.10413	+	10.0174i	0.502925	+	0.987046i	0.993303	+	0.115536i	$0.0368587\pi$
$103$	5.10413	+	10.0174i	0.502925	+	0.987046i	−0.490378	+	0.871510i	$0.663141\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$107$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$108$	3.16729	−	19.9975i	0.304773	−	1.92426i
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$		0			0
$111$	−6.36675			−0.604305
$112$		0			0
$113$	−15.3204	−	7.80612i	−1.44122	−	0.734338i	−0.453599	−	0.891206i	$-0.649860\pi$
$113$	−15.3204	−	7.80612i	−1.44122	−	0.734338i	−0.987620	+	0.156868i	$0.949860\pi$
$114$		0			0
$115$	−5.63929	+	6.99645i	−0.525867	+	0.652422i
$116$		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		−0.156434	−	0.987688i	$-0.550000\pi$
$127$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$128$		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
$135$	5.90853	+	21.8519i	0.508525	+	1.88071i
$136$		0			0
$137$	2.23625	+	14.1191i	0.191056	+	1.20628i	0.877673	+	0.479260i	$0.159095\pi$
$137$	2.23625	+	14.1191i	0.191056	+	1.20628i	−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
$141$	−32.5357	+	23.6386i	−2.74000	+	1.99073i
$142$		0			0
$143$		0			0
$144$			25.2665i			2.10554i
$145$		0			0
$146$		0			0
$147$	−19.0374	+	9.70005i	−1.57018	+	0.800047i
$148$	4.12039	−	0.652606i	0.338694	−	0.0536439i
$149$		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
$157$	−10.5776	+	20.7596i	−0.844181	+	1.65680i	−0.0939948	+	0.995573i	$0.529964\pi$
$157$	−10.5776	+	20.7596i	−0.844181	+	1.65680i	−0.750186	+	0.661226i	$0.770036\pi$
$158$		0			0
$159$	9.21846	−	12.6881i	0.731071	−	1.00623i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	3.97108	−	25.0724i	0.311039	−	1.96382i	0.0488556	−	0.998806i	$-0.484443\pi$
$163$	3.97108	−	25.0724i	0.311039	−	1.96382i	0.262184	−	0.965018i	$-0.415557\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$167$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$168$		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	$-0.850000\pi$
$173$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	7.15831	+	7.15831i	0.538052	+	0.538052i
$178$		0			0
$179$	19.9722	−	6.48936i	1.49279	−	0.485037i	0.554885	−	0.831927i	$-0.312762\pi$
$179$	19.9722	−	6.48936i	1.49279	−	0.485037i	0.937906	+	0.346890i	$0.112762\pi$
$180$	−11.5485	−	25.7803i	−0.860777	−	1.92155i
$181$	8.04962	+	5.84839i	0.598323	+	0.434707i	0.845283	−	0.534318i	$-0.179432\pi$
$181$	8.04962	+	5.84839i	0.598323	+	0.434707i	−0.246960	+	0.969026i	$0.579432\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−3.90590	+	2.54918i	−0.287168	+	0.187420i
$186$		0			0
$187$		0			0
$188$	18.6332	−	18.6332i	1.35897	−	1.35897i
$189$		0			0
$190$		0			0
$191$	7.17425	−	22.0801i	0.519111	−	1.59766i	−0.256565	−	0.966527i	$-0.582591\pi$
$191$	7.17425	−	22.0801i	0.519111	−	1.59766i	0.775676	−	0.631132i	$-0.217409\pi$
$192$	−3.81990	−	24.1179i	−0.275677	−	1.74056i
$193$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$193$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$194$		0			0
$195$		0			0
$196$	11.3262	−	8.22899i	0.809017	−	0.587785i
$197$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$197$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$198$		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
$201$	−15.3065	−	47.1087i	−1.07964	−	3.32279i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	25.0724	+	3.97108i	1.74265	+	0.276009i
$208$		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	$0.800000\pi$
$212$	−4.66538	+	9.15632i	−0.320420	+	0.628859i
$213$	−4.15717	−	8.15890i	−0.284844	−	0.559038i
$214$		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
$223$	18.1763	+	9.26130i	1.21718	+	0.620182i	0.940177	−	0.340687i	$-0.110660\pi$
$223$	18.1763	+	9.26130i	1.21718	+	0.620182i	0.277000	+	0.960870i	$0.410660\pi$
$224$		0			0
$225$	23.5668	+	21.0262i	1.57112	+	1.40174i
$226$		0			0
$227$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$227$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$228$		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		0.156434	−	0.987688i	$-0.450000\pi$
$233$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$234$		0			0
$235$	−10.4955	+	27.5289i	−0.684652	+	1.79579i
$236$	−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	$0.900000\pi$
$240$	14.9211	+	22.8624i	0.963154	+	1.47576i
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$		0			0
$243$	4.31662	−	4.31662i	0.276912	−	0.276912i
$244$		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
$257$	28.0574	−	14.2960i	1.75017	−	0.891758i	0.789652	−	0.613555i	$-0.210261\pi$
$257$	28.0574	−	14.2960i	1.75017	−	0.891758i	0.960522	−	0.278203i	$-0.0897388\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$263$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$264$		0			0
$265$	0.575188	−	11.4749i	0.0353335	−	0.704900i
$266$		0			0
$267$	12.4715	−	24.4767i	0.763243	−	1.49795i
$268$	14.7347	+	28.9185i	0.900067	+	1.76648i
$269$	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
$276$	−24.5330			−1.47671
$277$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$277$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$278$		0			0
$279$	59.7735	+	19.4216i	3.57855	+	1.16274i
$280$		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		−0.891007	−	0.453990i	$-0.850000\pi$
$283$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$284$	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
$291$	−12.3103	−	8.94396i	−0.721643	−	0.524304i
$292$		0			0
$293$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$293$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$294$		0			0
$295$	7.25763	+	1.52540i	0.422555	+	0.0888121i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	−25.6743	−	16.5074i	−1.48231	−	0.953055i
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$307$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$308$		0			0
$309$		−	34.3166i		−	1.95220i
$310$		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
$313$	−34.1166	+	5.40354i	−1.92839	+	0.305426i	−0.998111	−	0.0614365i	$-0.980432\pi$
$313$	−34.1166	+	5.40354i	−1.92839	+	0.305426i	−0.930275	+	0.366863i	$0.880432\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	35.0712	+	5.55473i	1.96979	+	0.311985i	0.996650	+	0.0817838i	$0.0260617\pi$
$317$	35.0712	+	5.55473i	1.96979	+	0.311985i	0.973143	+	0.230201i	$0.0739383\pi$
$318$		0			0
$319$		0			0
$320$	−12.0000	−	13.2665i	−0.670820	−	0.741620i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−14.0479	+	19.3353i	−0.780440	+	1.07418i
$325$		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	$-0.588164\pi$
$331$	−9.94987			−0.546895			−0.273447	+	0.961887i	$0.588164\pi$
$332$		0			0
$333$	11.7396	+	5.98164i	0.643328	+	0.327792i
$334$		0			0
$335$	−28.2522	−	22.7719i	−1.54358	−	1.24416i
$336$		0			0
$337$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$337$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$338$		0			0
$339$	30.8487	+	42.4595i	1.67547	+	2.30609i
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$	25.0319	−	11.2133i	1.34767	−	0.603702i
$346$		0			0
$347$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$347$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$348$		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
$353$	−13.7414	+	13.7414i	−0.731383	+	0.731383i	−0.970894	−	0.239511i	$-0.923013\pi$
$353$	−13.7414	+	13.7414i	−0.731383	+	0.731383i	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
$367$	−17.0426	+	8.68362i	−0.889615	+	0.453282i	−0.838179	−	0.545395i	$-0.816380\pi$
$367$	−17.0426	+	8.68362i	−0.889615	+	0.453282i	−0.0514358	+	0.998676i	$0.516380\pi$
$368$	15.8771	−	2.51469i	0.827651	−	0.131087i
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−59.9925	−	9.50188i	−3.11047	−	0.492649i
$373$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$373$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$374$		0			0
$375$	33.7414	+	5.10819i	1.74240	+	0.263786i
$376$		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
$383$	−3.94715	+	24.9213i	−0.201690	+	1.27342i	0.654224	+	0.756301i	$0.272995\pi$
$383$	−3.94715	+	24.9213i	−0.201690	+	1.27342i	−0.855914	+	0.517119i	$0.827005\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	8.88368	+	4.52646i	0.451000	+	0.229796i
$389$	−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
$397$	−20.8997	−	20.8997i	−1.04893	−	1.04893i	−0.998740	−	0.0501886i	$-0.984018\pi$
$397$	−20.8997	−	20.8997i	−1.04893	−	1.04893i	−0.0501886	−	0.998740i	$-0.515982\pi$
$398$		0			0
$399$		0			0
$400$	18.3078	+	8.05147i	0.915388	+	0.402574i
$401$	21.4656	+	15.5957i	1.07194	+	0.778812i	0.976261	−	0.216600i	$-0.0694966\pi$
$401$	21.4656	+	15.5957i	1.07194	+	0.778812i	0.0956827	+	0.995412i	$0.469497\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	5.49605	−	26.1494i	0.273101	−	1.29937i
$406$		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
$411$	13.4834	−	41.4977i	0.665088	−	2.04693i
$412$	3.51753	+	22.2088i	0.173296	+	1.09415i
$413$		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	$0.199392\pi$
$419$			24.0000i			1.17248i	−0.810139	+	0.586238i	$0.800608\pi$
$420$		0			0
$421$	12.2987	+	37.8516i	0.599403	+	1.84477i	0.531460	+	0.847084i	$0.321644\pi$
$421$	12.2987	+	37.8516i	0.599403	+	1.84477i	0.0679432	+	0.997689i	$0.478356\pi$
$422$		0			0
$423$	82.2013	−	13.0194i	3.99676	−	0.633025i
$424$		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	$0.800000\pi$
$432$	18.3837	−	36.0800i	0.884485	−	1.73590i
$433$	−18.8919	−	37.0774i	−0.907886	−	1.78183i	−0.466805	−	0.884360i	$-0.654595\pi$
$433$	−18.8919	−	37.0774i	−0.907886	−	1.78183i	−0.441081	−	0.897467i	$-0.645405\pi$
$434$		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
$441$	44.2164			2.10554
$442$		0			0
$443$	−36.2163	−	18.4531i	−1.72069	−	0.876735i	−0.978418	−	0.206636i	$-0.933748\pi$
$443$	−36.2163	−	18.4531i	−1.72069	−	0.876735i	−0.742271	−	0.670099i	$-0.766252\pi$
$444$	−12.1103	−	3.93487i	−0.574728	−	0.186741i
$445$	−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	$-0.827976\pi$
$449$	−22.9236	−	31.5517i	−1.08183	−	1.48902i	−0.224346	−	0.974510i	$-0.572024\pi$
$450$		0			0
$451$		0			0
$452$	−24.3166	−	24.3166i	−1.14376	−	1.14376i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$457$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$458$		0			0
$459$		0			0
$460$	−15.0506	+	9.82276i	−0.701738	+	0.457989i
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$	−0.575188	+	0.575188i	−0.0267313	+	0.0267313i	−0.720346	−	0.693615i	$-0.756017\pi$
$463$	−0.575188	+	0.575188i	−0.0267313	+	0.0267313i	0.693615	+	0.720346i	$0.256017\pi$
$464$		0			0
$465$	65.5556	−	17.7256i	3.04007	−	0.822004i
$466$		0			0
$467$	−5.10118	−	32.2076i	−0.236054	−	1.49039i	−0.766267	−	0.642523i	$-0.777888\pi$
$467$	−5.10118	−	32.2076i	−0.236054	−	1.49039i	0.530212	−	0.847865i	$-0.322112\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	57.5342	−	41.8010i	2.65103	−	1.92609i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−28.9185	+	14.7347i	−1.32409	+	0.674657i
$478$		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
$485$	−11.1332	−	0.558061i	−0.505535	−	0.0253403i
$486$		0			0
$487$	−16.9980	+	33.3604i	−0.770251	+	1.51170i	0.0866600	+	0.996238i	$0.472381\pi$
$487$	−16.9980	+	33.3604i	−0.770251	+	1.51170i	−0.856911	+	0.515465i	$0.827619\pi$
$488$		0			0
$489$	−45.5433	+	62.6850i	−2.05954	+	2.83471i
$490$		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		−0.891007	−	0.453990i	$-0.850000\pi$
$503$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	28.0581	+	28.0581i	1.24610	+	1.24610i
$508$		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
$515$	−13.7400	−	21.0527i	−0.605458	−	0.927693i
$516$		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	$0.493430\pi$
$521$	−13.3236	+	41.0059i	−0.583718	+	1.79650i	−0.604358	+	0.796713i	$0.706570\pi$
$522$		0			0
$523$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$523$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$			6.84962i			0.297810i
$530$		0			0
$531$	−6.47387	−	19.9245i	−0.280942	−	0.864650i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−63.3094	−	10.0272i	−2.73200	−	0.432707i
$538$		0			0
$539$		0			0
$540$	−2.26650	+	45.2164i	−0.0975346	+	1.94580i
$541$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$541$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$542$		0			0
$543$	−13.7878	−	27.0600i	−0.591689	−	1.16126i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$547$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$548$	−4.47250	+	28.2383i	−0.191056	+	1.20628i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$	14.1551	−	1.52034i	0.600849	−	0.0645349i
$556$		0			0
$557$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$557$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$563$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$564$	−76.4961	+	24.8551i	−3.22107	+	1.04659i
$565$	35.9255	+	13.6968i	1.51140	+	0.576228i
$566$		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
$573$	−50.1082	+	50.1082i	−2.09330	+	2.09330i
$574$		0			0
$575$	10.8670	−	16.9017i	0.453186	−	0.704849i
$576$	−15.6156	+	48.0597i	−0.650648	+	2.00249i
$577$	4.09833	+	25.8758i	0.170616	+	1.07723i	0.913212	+	0.407486i	$0.133594\pi$
$577$	4.09833	+	25.8758i	0.170616	+	1.07723i	−0.742596	+	0.669740i	$0.766406\pi$
$578$		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
$587$	26.0626	−	13.2795i	1.07572	−	0.548105i	0.175916	−	0.984405i	$-0.443711\pi$
$587$	26.0626	−	13.2795i	1.07572	−	0.548105i	0.899801	+	0.436300i	$0.143711\pi$
$588$	−42.2063	+	6.68482i	−1.74056	+	0.275677i
$589$		0			0
$590$		0			0
$591$		0			0
$592$	8.24078	+	1.30521i	0.338694	+	0.0536439i
$593$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$593$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−27.5755	+	54.1200i	−1.12859	+	2.21498i
$598$		0			0
$599$	−21.1603	+	29.1246i	−0.864585	+	1.19000i	0.115872	+	0.993264i	$0.463034\pi$
$599$	−21.1603	+	29.1246i	−0.864585	+	1.19000i	−0.980457	+	0.196735i	$0.936966\pi$
$600$		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
$603$	−16.0355	+	101.244i	−0.653017	+	4.12298i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$607$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$613$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−7.73350	−	7.73350i	−0.311339	−	0.311339i	0.534089	−	0.845428i	$-0.320655\pi$
$617$	−7.73350	−	7.73350i	−0.311339	−	0.311339i	−0.845428	+	0.534089i	$0.820655\pi$
$618$		0			0
$619$	0.951057	−	0.309017i	0.0382262	−	0.0124204i	−0.289841	−	0.957075i	$-0.593603\pi$
$619$	0.951057	−	0.309017i	0.0382262	−	0.0124204i	0.328068	+	0.944654i	$0.393603\pi$
$620$	−40.6089	+	18.1911i	−1.63089	+	0.730573i
$621$	−32.9135	−	23.9131i	−1.32077	−	0.959599i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	22.7451	−	10.3759i	0.909804	−	0.415037i
$626$		0			0
$627$		0			0
$628$	−32.9499	+	32.9499i	−1.31484	+	1.31484i
$629$		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
$636$	25.3762	−	18.4369i	1.00623	−	0.731071i
$637$		0			0
$638$		0			0
$639$			18.9499i			0.749645i
$640$		0			0
$641$	7.17425	+	22.0801i	0.283366	+	0.872111i	0.986884	+	0.161433i	$0.0516116\pi$
$641$	7.17425	+	22.0801i	0.283366	+	0.872111i	−0.703518	+	0.710678i	$0.748388\pi$
$642$		0			0
$643$	7.78744	−	1.23341i	0.307106	−	0.0486409i	−0.000979141	−	1.00000i	$-0.500312\pi$
$643$	7.78744	−	1.23341i	0.307106	−	0.0486409i	0.308086	+	0.951359i	$0.400312\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−11.2555	−	1.78270i	−0.442500	−	0.0700851i	−0.0687910	−	0.997631i	$-0.521914\pi$
$647$	−11.2555	−	1.78270i	−0.442500	−	0.0700851i	−0.373709	+	0.927546i	$0.621914\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$	23.0491	−	45.2363i	0.902671	−	1.77159i
$653$	17.4367	+	34.2215i	0.682351	+	1.33919i	0.928998	+	0.370084i	$0.120671\pi$
$653$	17.4367	+	34.2215i	0.682351	+	1.33919i	−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
$669$	−36.5993	−	50.3747i	−1.41501	−	1.94760i
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$673$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$674$		0			0
$675$	−18.3544	−	47.1719i	−0.706462	−	1.81565i
$676$	−21.0344	−	15.2824i	−0.809017	−	0.587785i
$677$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$677$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	35.2164	−	35.2164i	1.34752	−	1.34752i	0.459167	−	0.888350i	$-0.348148\pi$
$683$	35.2164	−	35.2164i	1.34752	−	1.34752i	0.888350	−	0.459167i	$-0.151852\pi$
$684$		0			0
$685$	−8.34338	−	30.8568i	−0.318784	−	1.17898i
$686$		0			0
$687$	14.2528	+	89.9887i	0.543779	+	3.43328i
$688$		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
$705$	66.6913	−	60.3246i	2.51174	−	2.27195i
$706$		0			0
$707$		0			0
$708$	9.19184	+	18.0400i	0.345450	+	0.677985i
$709$	11.1679	−	15.3713i	0.419420	−	0.577282i	−0.546064	−	0.837743i	$-0.683875\pi$
$709$	11.1679	−	15.3713i	0.419420	−	0.577282i	0.965484	+	0.260461i	$0.0838746\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	6.25520	−	39.4938i	0.234259	−	1.47905i
$714$		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	−0.808890	−	0.587961i	$-0.799931\pi$
$719$	−48.5039	−	15.7599i	−1.80889	−	0.587744i	−1.00000		0.000216702i	$0.999931\pi$
$720$	−6.03349	−	56.1745i	−0.224855	−	2.09350i
$721$		0			0
$722$		0			0
$723$		0			0
$724$	11.6968	+	16.0992i	0.434707	+	0.598323i
$725$		0			0
$726$		0			0
$727$	31.4749	+	31.4749i	1.16734	+	1.16734i	0.982831	+	0.184510i	$0.0590699\pi$
$727$	31.4749	+	31.4749i	1.16734	+	1.16734i	0.184510	+	0.982831i	$0.440930\pi$
$728$		0			0
$729$	16.3737	−	5.32015i	0.606435	−	0.197043i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$733$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$734$		0			0
$735$	40.0092	−	26.1120i	1.47576	−	0.963154i
$736$		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	$0.300000\pi$
$740$	−9.00495	+	2.43485i	−0.331028	+	0.0895069i
$741$		0			0
$742$		0			0
$743$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$743$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$744$		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	$-0.237844\pi$
$752$	46.9585	−	23.9266i	1.71240	−	0.872512i
$753$	81.3979	−	12.8922i	2.96630	−	0.469816i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	1.25677	+	0.199053i	0.0456781	+	0.00723470i	0.179232	−	0.983807i	$-0.442639\pi$
$757$	1.25677	+	0.199053i	0.0456781	+	0.00723470i	−0.133554	+	0.991042i	$0.542639\pi$
$758$		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
$768$	7.63980	−	48.2358i	0.275677	−	1.74056i
$769$		0			0			−	1.00000i	$-0.5\pi$
$769$		0			0				1.00000i	$0.5\pi$
$770$		0			0
$771$	−96.1161			−3.46154
$772$		0			0
$773$	42.3804	+	21.5939i	1.52432	+	0.776678i	0.997318	−	0.0731890i	$-0.0233176\pi$
$773$	42.3804	+	21.5939i	1.52432	+	0.776678i	0.526998	+	0.849867i	$0.323318\pi$
$774$		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
$785$	18.5596	−	48.6803i	0.662421	−	1.73748i
$786$		0			0
$787$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$787$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$	−17.4654	+	30.4106i	−0.619433	+	1.07855i
$796$	12.2987	−	37.8516i	0.435917	−	1.34161i
$797$	5.90497	+	37.2825i	0.209165	+	1.32061i	0.839105	+	0.543970i	$0.183079\pi$
$797$	5.90497	+	37.2825i	0.209165	+	1.32061i	−0.629940	+	0.776644i	$0.716921\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	−45.9923	+	33.4154i	−1.62506	+	1.18067i
$802$		0			0
$803$		0			0
$804$		−	99.0660i		−	3.49379i
$805$		0			0
$806$		0			0
$807$	−36.0800	+	18.3837i	−1.27008	+	0.647136i
$808$		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
$815$	−2.84169	+	56.6913i	−0.0995400	+	1.98581i
$816$		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
$823$	−8.72202	+	55.0687i	−0.304030	+	1.91957i	0.0810902	+	0.996707i	$0.474160\pi$
$823$	−8.72202	+	55.0687i	−0.304030	+	1.91957i	−0.385121	+	0.922866i	$0.625840\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$827$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$828$	45.2363	+	23.0491i	1.57207	+	0.801010i
$829$	27.5806	+	8.96149i	0.957915	+	0.311246i	0.745928	−	0.666027i	$-0.232006\pi$
$829$	27.5806	+	8.96149i	0.957915	+	0.311246i	0.211987	+	0.977272i	$0.432006\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−71.2243	−	71.2243i	−2.46187	−	2.46187i
$838$		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
$848$	−14.5330	+	14.5330i	−0.499065	+	0.499065i
$849$		0			0
$850$		0			0
$851$	2.59037	−	7.97235i	0.0887968	−	0.273289i
$852$	−2.86492	−	18.0884i	−0.0981507	−	0.619699i
$853$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$853$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$857$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$858$		0			0
$859$		−	31.0000i		−	1.05771i	−0.848713	−	0.528853i	$-0.822622\pi$
$859$		−	31.0000i		−	1.05771i	0.848713	−	0.528853i	$-0.177378\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−7.28624	+	1.15403i	−0.248027	+	0.0392835i	−0.279210	−	0.960230i	$-0.590072\pi$
$863$	−7.28624	+	1.15403i	−0.248027	+	0.0392835i	0.0311832	+	0.999514i	$0.490072\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	51.2505	+	8.11728i	1.74056	+	0.275677i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	14.2960	+	28.0574i	0.483845	+	0.949600i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$877$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$878$		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
$883$	−47.8196	−	24.3653i	−1.60926	−	0.819958i	−0.999628	−	0.0272727i	$-0.991318\pi$
$883$	−47.8196	−	24.3653i	−1.60926	−	0.819958i	−0.609631	−	0.792686i	$-0.708682\pi$
$884$		0			0
$885$	−17.6243	−	14.2056i	−0.592434	−	0.477515i
$886$		0			0
$887$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$887$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$	28.8496	+	28.8496i	0.965957	+	0.965957i
$893$		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
$900$	31.8318	+	54.5592i	1.06106	+	1.81864i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−19.2931	−	11.0804i	−0.641325	−	0.368325i
$906$		0			0
$907$	−7.48861	−	47.2812i	−0.248655	−	1.56995i	−0.723781	−	0.690030i	$-0.757597\pi$
$907$	−7.48861	−	47.2812i	−0.248655	−	1.56995i	0.475126	−	0.879918i	$-0.342403\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−5.36641	+	3.89893i	−0.177797	+	0.129177i	−0.673124	−	0.739529i	$-0.735048\pi$
$911$	−5.36641	+	3.89893i	−0.177797	+	0.129177i	0.495327	+	0.868706i	$0.335048\pi$
$912$		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
$925$	8.07519	−	6.60025i	0.265511	−	0.217015i
$926$		0			0
$927$	−32.2409	+	63.2763i	−1.05893	+	2.07827i
$928$		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
$933$	5.72985	−	36.1768i	0.187587	−	1.18438i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$937$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$938$		0			0
$939$	100.273	+	32.5805i	3.27227	+	1.06322i
$940$	−36.9774	+	45.8765i	−1.20607	+	1.49632i
$941$		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
$947$	−40.1082	−	40.1082i	−1.30334	−	1.30334i	−0.926126	−	0.377215i	$-0.876882\pi$
$947$	−40.1082	−	40.1082i	−1.30334	−	1.30334i	−0.377215	−	0.926126i	$-0.623118\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	−87.6834	−	63.7057i	−2.84333	−	2.06580i
$952$		0			0
$953$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$953$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$954$		0			0
$955$	−10.6778	+	50.8034i	−0.345525	+	1.64396i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$	14.2519	+	52.7087i	0.459978	+	1.70116i
$961$	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	$-0.750000\pi$
$967$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−13.3236	−	41.0059i	−0.427575	−	1.31594i	−0.900507	−	0.434842i	$-0.856804\pi$
$971$	−13.3236	−	41.0059i	−0.427575	−	1.31594i	0.472932	−	0.881099i	$-0.343196\pi$
$972$	10.8785	−	5.54289i	0.348929	−	0.177788i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	58.2345	+	9.22344i	1.86309	+	0.295084i	0.983535	−	0.180718i	$-0.0578422\pi$
$977$	58.2345	+	9.22344i	1.86309	+	0.295084i	0.879552	+	0.475802i	$0.157842\pi$
$978$		0			0
$979$		0			0
$980$	−23.2164	+	21.0000i	−0.741620	+	0.670820i
$981$		0			0
$982$		0			0
$983$	28.0837	+	55.1174i	0.895732	+	1.75797i	0.593727	+	0.804666i	$0.297656\pi$
$983$	28.0837	+	55.1174i	0.895732	+	1.75797i	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	$-0.897103\pi$
$991$	−59.6992			−1.89641			−0.948205	+	0.317660i	$0.897103\pi$
$992$		0			0
$993$	27.0600	+	13.7878i	0.858723	+	0.437541i
$994$		0			0
$995$	4.75194	+	44.2427i	0.150647	+	1.40259i
$996$		0			0
$997$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$997$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$998$		0			0
$999$	−12.4117	−	17.0833i	−0.392690	−	0.540492i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	605.2.m.a.602.1		16
5.3	odd	4	inner	605.2.m.a.118.2		16
11.2	odd	10		55.2.e.b.32.1	✓	4
11.3	even	5	inner	605.2.m.a.457.2		16
11.4	even	5	inner	605.2.m.a.282.2		16
11.5	even	5	inner	605.2.m.a.112.2		16
11.6	odd	10	inner	605.2.m.a.112.2		16
11.7	odd	10	inner	605.2.m.a.282.2		16
11.8	odd	10	inner	605.2.m.a.457.2		16
11.9	even	5		55.2.e.b.32.1	✓	4
11.10	odd	2	CM	605.2.m.a.602.1		16
33.2	even	10		495.2.k.a.307.1		4
33.20	odd	10		495.2.k.a.307.1		4
44.31	odd	10		880.2.bd.d.417.2		4
44.35	even	10		880.2.bd.d.417.2		4
55.2	even	20		275.2.e.a.43.2		4
55.3	odd	20	inner	605.2.m.a.578.2		16
55.8	even	20	inner	605.2.m.a.578.2		16
55.9	even	10		275.2.e.a.32.2		4
55.13	even	20		55.2.e.b.43.1	yes	4
55.18	even	20	inner	605.2.m.a.403.1		16
55.24	odd	10		275.2.e.a.32.2		4
55.28	even	20	inner	605.2.m.a.233.2		16
55.38	odd	20	inner	605.2.m.a.233.2		16
55.42	odd	20		275.2.e.a.43.2		4
55.43	even	4	inner	605.2.m.a.118.2		16
55.48	odd	20	inner	605.2.m.a.403.1		16
55.53	odd	20		55.2.e.b.43.1	yes	4
165.53	even	20		495.2.k.a.208.1		4
165.68	odd	20		495.2.k.a.208.1		4
220.123	odd	20		880.2.bd.d.593.2		4
220.163	even	20		880.2.bd.d.593.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
55.2.e.b.32.1	✓	4	11.2	odd	10
55.2.e.b.32.1	✓	4	11.9	even	5
55.2.e.b.43.1	yes	4	55.13	even	20
55.2.e.b.43.1	yes	4	55.53	odd	20
275.2.e.a.32.2		4	55.9	even	10
275.2.e.a.32.2		4	55.24	odd	10
275.2.e.a.43.2		4	55.2	even	20
275.2.e.a.43.2		4	55.42	odd	20
495.2.k.a.208.1		4	165.53	even	20
495.2.k.a.208.1		4	165.68	odd	20
495.2.k.a.307.1		4	33.2	even	10
495.2.k.a.307.1		4	33.20	odd	10
605.2.m.a.112.2		16	11.5	even	5	inner
605.2.m.a.112.2		16	11.6	odd	10	inner
605.2.m.a.118.2		16	5.3	odd	4	inner
605.2.m.a.118.2		16	55.43	even	4	inner
605.2.m.a.233.2		16	55.28	even	20	inner
605.2.m.a.233.2		16	55.38	odd	20	inner
605.2.m.a.282.2		16	11.4	even	5	inner
605.2.m.a.282.2		16	11.7	odd	10	inner
605.2.m.a.403.1		16	55.18	even	20	inner
605.2.m.a.403.1		16	55.48	odd	20	inner
605.2.m.a.457.2		16	11.3	even	5	inner
605.2.m.a.457.2		16	11.8	odd	10	inner
605.2.m.a.578.2		16	55.3	odd	20	inner
605.2.m.a.578.2		16	55.8	even	20	inner
605.2.m.a.602.1		16	1.1	even	1	trivial
605.2.m.a.602.1		16	11.10	odd	2	CM
880.2.bd.d.417.2		4	44.31	odd	10
880.2.bd.d.417.2		4	44.35	even	10
880.2.bd.d.593.2		4	220.123	odd	20
880.2.bd.d.593.2		4	220.163	even	20

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)

\(f(q)\)	\(=\)	\(q+(-2.71963 - 1.38572i) q^{3} +(1.90211 + 0.618034i) q^{4} +(-2.22328 + 0.238794i) q^{5} +(3.71282 + 5.11026i) q^{9} +(-4.31662 - 4.31662i) q^{12} +(6.37741 + 2.43142i) q^{15} +(3.23607 + 2.35114i) q^{16} +(-4.37651 - 0.919850i) q^{20} +(2.84169 - 2.84169i) q^{23} +(4.88595 - 1.06181i) q^{25} +(-1.58365 - 9.99875i) q^{27} +(8.04962 - 5.84839i) q^{31} +(3.90389 + 12.0149i) q^{36} +(1.85853 - 0.946968i) q^{37} +(-9.47494 - 10.4749i) q^{45} +(5.98164 - 11.7396i) q^{47} +(-5.54289 - 10.8785i) q^{48} +(4.11450 - 5.66312i) q^{49} +(-0.803790 + 5.07493i) q^{53} +(-3.15430 - 1.02489i) q^{59} +(10.6279 + 8.56629i) q^{60} +(4.70228 + 6.47214i) q^{64} +(11.4749 + 11.4749i) q^{67} +(-11.6661 + 3.79056i) q^{69} +(2.42705 + 1.76336i) q^{71} +(-14.7594 - 3.88284i) q^{75} +(-7.75613 - 4.45449i) q^{80} +(-3.69271 + 11.3650i) q^{81} +9.00000i q^{89} +(7.16147 - 3.64895i) q^{92} +(-29.9962 + 4.75094i) q^{93} +(4.92382 + 0.779856i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 2 q^{3} - 16 q^{12} + 8 q^{15} + 16 q^{16} - 12 q^{20} + 72 q^{23} - 2 q^{25} - 22 q^{27} - 24 q^{36} + 14 q^{37} - 72 q^{45} + 24 q^{47} - 8 q^{48} + 12 q^{53} + 28 q^{60} + 104 q^{67} + 12 q^{71}+ \cdots + 34 q^{97}+O(q^{100}) \) 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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more