LMFDB - Embedded newform 605.2.m.a.112.2

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			112.2
Root			$1.63550 - 0.570223i$ of defining polynomial
Character	$\chi$	$=$	605.112
Dual form			605.2.m.a.578.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.477487 - 3.01474i) q^{3} +(1.17557 - 1.61803i) q^{4} +(1.93903 + 1.11362i) q^{5} +(-6.00747 - 1.95194i) q^{9} +(-4.31662 - 4.31662i) q^{12} +(4.28314 - 5.31393i) q^{15} +(-1.23607 - 3.80423i) q^{16} +(4.08135 - 1.82828i) q^{20} +(2.84169 - 2.84169i) q^{23} +(2.51969 + 4.31870i) q^{25} +(-4.59592 + 9.02000i) q^{27} +(-3.07468 + 9.46289i) q^{31} +(-10.2205 + 7.42564i) q^{36} +(-0.326303 - 2.06020i) q^{37} +(-9.47494 - 10.4749i) q^{45} +(13.0135 + 2.06113i) q^{47} +(-12.0589 + 1.90995i) q^{48} +(-6.65740 + 2.16312i) q^{49} +(-2.33269 - 4.57816i) q^{53} +(-1.94946 + 2.68321i) q^{59} +(-3.56298 - 13.1772i) q^{60} +(-7.60845 - 2.47214i) q^{64} +(11.4749 + 11.4749i) q^{67} +(-7.21007 - 9.92381i) q^{69} +(-0.927051 - 2.85317i) q^{71} +(14.2229 - 5.53406i) q^{75} +(1.83970 - 8.75303i) q^{80} +(9.66765 + 7.02396i) q^{81} +9.00000i q^{89} +(-1.25734 - 7.93855i) q^{92} +(27.0600 + 13.7878i) q^{93} +(-4.44184 + 2.26323i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/605\mathbb{Z}\right)^\times$.

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

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		0.891007	−	0.453990i	$-0.150000\pi$
$2$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$3$	0.477487	−	3.01474i	0.275677	−	1.74056i	−0.329218	−	0.944254i	$-0.606785\pi$
$3$	0.477487	−	3.01474i	0.275677	−	1.74056i	0.604896	−	0.796305i	$-0.293215\pi$
$4$	1.17557	−	1.61803i	0.587785	−	0.809017i
$5$	1.93903	+	1.11362i	0.867161	+	0.498027i
$5$	1.93903	+	1.11362i	0.867161	+	0.498027i
$6$		0			0
$7$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$7$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$8$		0			0
$9$	−6.00747	−	1.95194i	−2.00249	−	0.650648i
$10$		0			0
$11$		0			0
$11$		0			0
$12$	−4.31662	−	4.31662i	−1.24610	−	1.24610i
$13$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$13$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$14$		0			0
$15$	4.28314	−	5.31393i	1.10590	−	1.37205i
$16$	−1.23607	−	3.80423i	−0.309017	−	0.951057i
$17$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$17$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$18$		0			0
$19$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$19$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$20$	4.08135	−	1.82828i	0.912617	−	0.408815i
$21$		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
$25$	2.51969	+	4.31870i	0.503937	+	0.863740i
$26$		0			0
$27$	−4.59592	+	9.02000i	−0.884485	+	1.73590i
$28$		0			0
$29$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$29$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$30$		0			0
$31$	−3.07468	+	9.46289i	−0.552229	+	1.69959i	0.150923	+	0.988546i	$0.451776\pi$
$31$	−3.07468	+	9.46289i	−0.552229	+	1.69959i	−0.703151	+	0.711040i	$0.748224\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−10.2205	+	7.42564i	−1.70342	+	1.23761i
$37$	−0.326303	−	2.06020i	−0.0536439	−	0.338694i	−0.999884	−	0.0152291i	$-0.995152\pi$
$37$	−0.326303	−	2.06020i	−0.0536439	−	0.338694i	0.946240	−	0.323465i	$-0.104848\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$41$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$42$		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
$47$	13.0135	+	2.06113i	1.89821	+	0.300647i	0.992402	−	0.123038i	$-0.0392637\pi$
$47$	13.0135	+	2.06113i	1.89821	+	0.300647i	0.905810	+	0.423685i	$0.139264\pi$
$48$	−12.0589	+	1.90995i	−1.74056	+	0.275677i
$49$	−6.65740	+	2.16312i	−0.951057	+	0.309017i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−2.33269	−	4.57816i	−0.320420	−	0.628859i	0.673473	−	0.739212i	$-0.264802\pi$
$53$	−2.33269	−	4.57816i	−0.320420	−	0.628859i	−0.993892	+	0.110353i	$0.964802\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−1.94946	+	2.68321i	−0.253798	+	0.349324i	−0.916837	−	0.399262i	$-0.869267\pi$
$59$	−1.94946	+	2.68321i	−0.253798	+	0.349324i	0.663039	+	0.748585i	$0.269267\pi$
$60$	−3.56298	−	13.1772i	−0.459978	−	1.70116i
$61$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$61$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$62$		0			0
$63$		0			0
$64$	−7.60845	−	2.47214i	−0.951057	−	0.309017i
$65$		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
$69$	−7.21007	−	9.92381i	−0.867990	−	1.19469i
$70$		0			0
$71$	−0.927051	−	2.85317i	−0.110021	−	0.338609i	0.880855	−	0.473386i	$-0.156968\pi$
$71$	−0.927051	−	2.85317i	−0.110021	−	0.338609i	−0.990876	+	0.134777i	$0.956968\pi$
$72$		0			0
$73$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$73$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$74$		0			0
$75$	14.2229	−	5.53406i	1.64231	−	0.639019i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$79$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$80$	1.83970	−	8.75303i	0.205685	−	0.978618i
$81$	9.66765	+	7.02396i	1.07418	+	0.780440i
$82$		0			0
$83$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$83$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$84$		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
$92$	−1.25734	−	7.93855i	−0.131087	−	0.827651i
$93$	27.0600	+	13.7878i	2.80599	+	1.42972i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−4.44184	+	2.26323i	−0.451000	+	0.229796i	−0.664711	−	0.747101i	$-0.731445\pi$
$97$	−4.44184	+	2.26323i	−0.451000	+	0.229796i	0.213710	+	0.976897i	$0.431445\pi$
$98$		0			0
$99$		0			0
$100$	9.94987	+	1.00000i	0.994987	+	0.100000i
$101$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$101$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$102$		0			0
$103$	11.1044	−	1.75876i	1.09415	−	0.173296i	0.416829	−	0.908985i	$-0.363141\pi$
$103$	11.1044	−	1.75876i	1.09415	−	0.173296i	0.677320	+	0.735689i	$0.263141\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$107$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$108$	9.19184	+	18.0400i	0.884485	+	1.73590i
$109$		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
$113$	2.68980	−	16.9828i	0.253036	−	1.59760i	−0.454382	−	0.890807i	$-0.650140\pi$
$113$	2.68980	−	16.9828i	0.253036	−	1.59760i	0.707417	−	0.706796i	$-0.249860\pi$
$114$		0			0
$115$	8.67469	−	2.34555i	0.808919	−	0.218724i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$		0			0
$122$		0			0
$123$		0			0
$124$	11.6968	+	16.0992i	1.05040	+	1.44575i
$125$	0.0763444	+	11.1801i	0.00682845	+	0.999977i
$126$		0			0
$127$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$127$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$128$		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
$135$	−18.9565	+	12.3719i	−1.63152	+	1.06481i
$136$		0			0
$137$	6.48986	−	12.7371i	0.554466	−	1.08820i	−0.428350	−	0.903613i	$-0.640905\pi$
$137$	6.48986	−	12.7371i	0.554466	−	1.08820i	0.982816	−	0.184588i	$-0.0590949\pi$
$138$		0			0
$139$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$139$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$140$		0			0
$141$	12.4275	−	38.2480i	1.04659	−	3.22107i
$142$		0			0
$143$		0			0
$144$			25.2665i			2.10554i
$145$		0			0
$146$		0			0
$147$	3.34241	+	21.1031i	0.275677	+	1.74056i
$148$	−3.71706	−	1.89394i	−0.305540	−	0.155681i
$149$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$149$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$150$		0			0
$151$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$151$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−16.5000	+	14.9248i	−1.32531	+	1.19879i
$156$		0			0
$157$	−23.0122	−	3.64478i	−1.83658	−	0.290885i	−0.860684	−	0.509140i	$-0.829964\pi$
$157$	−23.0122	−	3.64478i	−1.83658	−	0.290885i	−0.975892	+	0.218255i	$0.929964\pi$
$158$		0			0
$159$	−14.9158	+	4.84643i	−1.18290	+	0.384347i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	11.5245	+	22.6182i	0.902671	+	1.77159i	0.547558	+	0.836768i	$0.315557\pi$
$163$	11.5245	+	22.6182i	0.902671	+	1.77159i	0.355112	+	0.934824i	$0.384443\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$167$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$168$		0			0
$169$	−7.64121	+	10.5172i	−0.587785	+	0.809017i
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$173$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	7.15831	+	7.15831i	0.538052	+	0.538052i
$178$		0			0
$179$	12.3435	+	16.9894i	0.922596	+	1.26984i	0.962678	+	0.270648i	$0.0872379\pi$
$179$	12.3435	+	16.9894i	0.922596	+	1.26984i	−0.0400827	+	0.999196i	$0.512762\pi$
$180$	−28.0873	+	3.01674i	−2.09350	+	0.224855i
$181$	−3.07468	−	9.46289i	−0.228539	−	0.703371i	−0.997913	−	0.0645725i	$-0.979432\pi$
$181$	−3.07468	−	9.46289i	−0.228539	−	0.703371i	0.769374	−	0.638799i	$-0.220568\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	1.66157	−	4.35816i	0.122161	−	0.320418i
$186$		0			0
$187$		0			0
$188$	18.6332	−	18.6332i	1.35897	−	1.35897i
$189$		0			0
$190$		0			0
$191$	−18.7824	−	13.6462i	−1.35905	−	0.987407i	−0.998505	−	0.0546656i	$-0.982591\pi$
$191$	−18.7824	−	13.6462i	−1.35905	−	0.987407i	−0.360545	−	0.932742i	$-0.617409\pi$
$192$	−11.0858	+	21.7571i	−0.800047	+	1.57018i
$193$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$193$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$194$		0			0
$195$		0			0
$196$	−4.32624	+	13.3148i	−0.309017	+	0.951057i
$197$		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
$201$	40.0730	−	29.1148i	2.82653	−	2.05360i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−22.6182	+	11.5245i	−1.57207	+	0.801010i
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$211$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$212$	−10.1499	−	1.60758i	−0.697095	−	0.110409i
$213$	−9.04421	+	1.43246i	−0.619699	+	0.0981507i
$214$		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
$223$	−3.19123	+	20.1486i	−0.213700	+	1.34925i	0.614544	+	0.788883i	$0.289340\pi$
$223$	−3.19123	+	20.1486i	−0.213700	+	1.34925i	−0.828244	+	0.560368i	$0.810660\pi$
$224$		0			0
$225$	−6.70707	−	30.8627i	−0.447138	−	2.05752i
$226$		0			0
$227$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$227$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$228$		0			0
$229$	28.3887	+	9.22404i	1.87598	+	0.609542i	0.989039	+	0.147652i	$0.0471715\pi$
$229$	28.3887	+	9.22404i	1.87598	+	0.609542i	0.886937	+	0.461890i	$0.152828\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$233$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$234$		0			0
$235$	22.9382	+	18.4887i	1.49632	+	1.20607i
$236$	2.04979	+	6.30860i	0.133430	+	0.410655i
$237$		0			0
$238$		0			0
$239$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$239$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$240$	−25.5096	−	9.72567i	−1.64664	−	0.627789i
$241$		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
$245$	−15.3178	−	3.21948i	−0.978618	−	0.205685i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	8.34346	−	25.6785i	0.526634	−	1.62081i	−0.234427	−	0.972134i	$-0.575321\pi$
$251$	8.34346	−	25.6785i	0.526634	−	1.62081i	0.761061	−	0.648680i	$-0.224679\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−12.9443	+	9.40456i	−0.809017	+	0.587785i
$257$	−4.92606	−	31.1019i	−0.307279	−	1.94008i	−0.339510	−	0.940603i	$-0.610261\pi$
$257$	−4.92606	−	31.1019i	−0.307279	−	1.94008i	0.0322308	−	0.999480i	$-0.489739\pi$
$258$		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
$267$	27.1326	+	4.29738i	1.66049	+	0.262996i
$268$	32.0564	−	5.07724i	1.95816	−	0.310142i
$269$	−12.6172	+	4.09957i	−0.769284	+	0.249955i	−0.667258	−	0.744826i	$-0.732532\pi$
$269$	−12.6172	+	4.09957i	−0.769284	+	0.249955i	−0.102025	+	0.994782i	$0.532532\pi$
$270$		0			0
$271$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$271$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$	−24.5330			−1.47671
$277$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$277$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$278$		0			0
$279$	36.9421	−	50.8464i	2.21166	−	3.04410i
$280$		0			0
$281$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$281$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$282$		0			0
$283$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$283$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$284$	−5.70634	−	1.85410i	−0.338609	−	0.110021i
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−9.99235	−	13.7533i	−0.587785	−	0.809017i
$290$		0			0
$291$	4.70212	+	14.4716i	0.275643	+	0.848342i
$292$		0			0
$293$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$293$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$294$		0			0
$295$	−6.76815	+	3.03185i	−0.394057	+	0.176521i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	7.76567	−	29.5188i	0.448351	−	1.70427i
$301$		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
$311$	−9.70820	+	7.05342i	−0.550502	+	0.399963i	−0.827970	−	0.560772i	$-0.810505\pi$
$311$	−9.70820	+	7.05342i	−0.550502	+	0.399963i	0.277469	+	0.960735i	$0.410505\pi$
$312$		0			0
$313$	30.7771	+	15.6817i	1.73962	+	0.886382i	0.968245	+	0.250004i	$0.0804318\pi$
$313$	30.7771	+	15.6817i	1.73962	+	0.886382i	0.771377	+	0.636378i	$0.219568\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−31.6382	+	16.1204i	−1.77698	+	0.905415i	−0.854378	+	0.519652i	$0.826062\pi$
$317$	−31.6382	+	16.1204i	−1.77698	+	0.905415i	−0.922598	+	0.385763i	$0.873938\pi$
$318$		0			0
$319$		0			0
$320$	−12.0000	−	13.2665i	−0.670820	−	0.741620i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	22.7300	−	7.38543i	1.26278	−	0.410302i
$325$		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
$333$	−2.06113	+	13.0135i	−0.112949	+	0.713134i
$334$		0			0
$335$	9.47151	+	35.0290i	0.517484	+	1.91384i
$336$		0			0
$337$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$337$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$338$		0			0
$339$	−49.9142	−	16.2181i	−2.71097	−	0.880846i
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$	−2.92917	−	27.2719i	−0.157701	−	1.46827i
$346$		0			0
$347$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$347$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$348$		0			0
$349$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$349$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$350$		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
$355$	1.37978	−	6.56477i	0.0732309	−	0.348422i
$356$	14.5623	+	10.5801i	0.771801	+	0.560746i
$357$		0			0
$358$		0			0
$359$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$359$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$360$		0			0
$361$	−5.87132	+	18.0701i	−0.309017	+	0.951057i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	2.99217	+	18.8918i	0.156190	+	0.986146i	0.933903	+	0.357526i	$0.116380\pi$
$367$	2.99217	+	18.8918i	0.156190	+	0.986146i	−0.777713	+	0.628620i	$0.783620\pi$
$368$	−14.3229	−	7.29790i	−0.746635	−	0.380429i
$369$		0			0
$370$		0			0
$371$		0			0
$372$	54.1200	−	27.5755i	2.80599	−	1.42972i
$373$		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
$379$	−28.3887	+	9.22404i	−1.45823	+	0.473807i	−0.927528	−	0.373753i	$-0.878071\pi$
$379$	−28.3887	+	9.22404i	−1.45823	+	0.473807i	−0.530700	+	0.847560i	$0.678071\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−11.4551	−	22.4818i	−0.585327	−	1.14877i	−0.973821	−	0.227317i	$-0.927005\pi$
$383$	−11.4551	−	22.4818i	−0.585327	−	1.14877i	0.388494	−	0.921451i	$-0.372995\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	−1.55971	+	9.84763i	−0.0791824	+	0.499938i
$389$	−21.4441	+	29.5153i	−1.08726	+	1.49648i	−0.235988	+	0.971756i	$0.575833\pi$
$389$	−21.4441	+	29.5153i	−1.08726	+	1.49648i	−0.851270	+	0.524727i	$0.824167\pi$
$390$		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
$400$	13.3148	−	14.9237i	0.665741	−	0.746183i
$401$	−8.19915	−	25.2344i	−0.409446	−	1.26014i	−0.917125	−	0.398599i	$-0.869497\pi$
$401$	−8.19915	−	25.2344i	−0.409446	−	1.26014i	0.507679	−	0.861546i	$-0.330503\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	10.9238	+	24.3858i	0.542810	+	1.21174i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$409$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$410$		0			0
$411$	−35.3000	−	25.6470i	−1.74122	−	1.26507i
$412$	10.2083	−	20.0349i	0.502925	−	0.987046i
$413$		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
$421$	−32.1985	+	23.3936i	−1.56926	+	1.14013i	−0.641394	+	0.767211i	$0.721644\pi$
$421$	−32.1985	+	23.3936i	−1.56926	+	1.14013i	−0.927863	+	0.372921i	$0.878356\pi$
$422$		0			0
$423$	−74.1549	−	37.7838i	−3.60553	−	1.83711i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$431$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$432$	39.9950	+	6.33458i	1.92426	+	0.304773i
$433$	−41.1006	+	6.50970i	−1.97517	+	0.312836i	−0.985327	+	0.170676i	$0.945405\pi$
$433$	−41.1006	+	6.50970i	−1.97517	+	0.312836i	−0.989844	+	0.142160i	$0.954595\pi$
$434$		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
$443$	6.35852	−	40.1461i	0.302102	−	1.90740i	−0.105825	−	0.994385i	$-0.533748\pi$
$443$	6.35852	−	40.1461i	0.302102	−	1.90740i	0.407928	−	0.913014i	$-0.366252\pi$
$444$	−7.48456	+	10.3016i	−0.355202	+	0.488893i
$445$	−10.0226	+	17.4513i	−0.475117	+	0.827270i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	37.0912	+	12.0517i	1.75044	+	0.568753i	0.996140	−	0.0877747i	$-0.0279756\pi$
$449$	37.0912	+	12.0517i	1.75044	+	0.568753i	0.754302	+	0.656528i	$0.227976\pi$
$450$		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		0.453990	−	0.891007i	$-0.350000\pi$
$457$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$458$		0			0
$459$		0			0
$460$	6.40253	−	16.7933i	0.298519	−	0.782992i
$461$		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
$465$	37.1158	+	56.8695i	1.72121	+	2.63726i
$466$		0			0
$467$	−14.8042	+	29.0549i	−0.685056	+	1.34450i	0.242257	+	0.970212i	$0.422112\pi$
$467$	−14.8042	+	29.0549i	−0.685056	+	1.34450i	−0.927313	+	0.374286i	$0.877888\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−21.9761	+	67.6355i	−1.01260	+	3.11648i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	5.07724	+	32.0564i	0.232471	+	1.46776i
$478$		0			0
$479$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$479$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$480$		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
$487$	−36.9803	−	5.85710i	−1.67574	−	0.265410i	−0.755036	−	0.655683i	$-0.772381\pi$
$487$	−36.9803	−	5.85710i	−1.67574	−	0.265410i	−0.920699	+	0.390273i	$0.872381\pi$
$488$		0			0
$489$	73.6906	−	23.9435i	3.33240	−	1.08276i
$490$		0			0
$491$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$491$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	39.7995			1.78705
$497$		0			0
$498$		0			0
$499$	−11.6968	+	16.0992i	−0.523620	+	0.720701i	−0.986141	−	0.165907i	$-0.946945\pi$
$499$	−11.6968	+	16.0992i	−0.523620	+	0.720701i	0.462522	+	0.886608i	$0.346945\pi$
$500$	18.1795	+	13.0194i	0.813012	+	0.582247i
$501$		0			0
$502$		0			0
$503$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$503$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	28.0581	+	28.0581i	1.24610	+	1.24610i
$508$		0			0
$509$	1.94946	+	2.68321i	0.0864084	+	0.118931i	0.850033	−	0.526730i	$-0.176582\pi$
$509$	1.94946	+	2.68321i	0.0864084	+	0.118931i	−0.763624	+	0.645661i	$0.776582\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	23.4904	+	8.95582i	1.03511	+	0.394641i
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	34.8817	+	25.3430i	1.52819	+	1.11030i	0.957232	+	0.289321i	$0.0934296\pi$
$521$	34.8817	+	25.3430i	1.52819	+	1.11030i	0.570962	+	0.820977i	$0.306570\pi$
$522$		0			0
$523$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$523$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$			6.84962i			0.297810i
$530$		0			0
$531$	16.9488	−	12.3140i	0.735515	−	0.534383i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	57.1123	−	29.1002i	2.46458	−	1.25576i
$538$		0			0
$539$		0			0
$540$	−2.26650	+	45.2164i	−0.0975346	+	1.94580i
$541$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$541$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$542$		0			0
$543$	−29.9962	+	4.75094i	−1.28726	+	0.203882i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$547$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$548$	−12.9797	−	25.4741i	−0.554466	−	1.08820i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−12.3453	−	7.09016i	−0.524030	−	0.300961i
$556$		0			0
$557$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$557$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$563$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$564$	−47.2772	−	65.0715i	−1.99073	−	2.74000i
$565$	24.1280	−	29.9347i	1.01507	−	1.25936i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$569$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$570$		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
$575$	19.4326	+	5.11224i	0.810394	+	0.213195i
$576$	40.8821	+	29.7026i	1.70342	+	1.23761i
$577$	11.8938	−	23.3429i	0.495146	−	0.971779i	−0.499290	−	0.866435i	$-0.666406\pi$
$577$	11.8938	−	23.3429i	0.495146	−	0.971779i	0.994436	−	0.105344i	$-0.0335944\pi$
$578$		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
$587$	−4.57582	−	28.8906i	−0.188864	−	1.19244i	−0.881864	−	0.471504i	$-0.843711\pi$
$587$	−4.57582	−	28.8906i	−0.188864	−	1.19244i	0.693000	−	0.720938i	$-0.256289\pi$
$588$	38.0749	+	19.4001i	1.57018	+	0.800047i
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−7.43412	+	3.78787i	−0.305540	+	0.155681i
$593$		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
$597$	−59.9925	−	9.50188i	−2.45533	−	0.388886i
$598$		0			0
$599$	34.2380	−	11.1246i	1.39893	−	0.454539i	0.490084	−	0.871675i	$-0.336966\pi$
$599$	34.2380	−	11.1246i	1.39893	−	0.454539i	0.908844	+	0.417136i	$0.136966\pi$
$600$		0			0
$601$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$601$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$602$		0			0
$603$	−46.5369	−	91.3337i	−1.89513	−	3.71940i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$607$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$613$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$614$		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
$619$	0.587785	+	0.809017i	0.0236251	+	0.0325171i	0.820666	−	0.571408i	$-0.193603\pi$
$619$	0.587785	+	0.809017i	0.0236251	+	0.0325171i	−0.797041	+	0.603925i	$0.793603\pi$
$620$	4.75194	+	44.2427i	0.190842	+	1.77683i
$621$	12.5719	+	38.6922i	0.504491	+	1.55266i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−12.3024	+	21.7635i	−0.492095	+	0.870542i
$626$		0			0
$627$		0			0
$628$	−32.9499	+	32.9499i	−1.31484	+	1.31484i
$629$		0			0
$630$		0			0
$631$	−5.66312	−	4.11450i	−0.225445	−	0.163796i	0.469329	−	0.883023i	$-0.344496\pi$
$631$	−5.66312	−	4.11450i	−0.225445	−	0.163796i	−0.694774	+	0.719228i	$0.744496\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$	−9.69286	+	29.8316i	−0.384347	+	1.18290i
$637$		0			0
$638$		0			0
$639$			18.9499i			0.749645i
$640$		0			0
$641$	−18.7824	+	13.6462i	−0.741862	+	0.538994i	−0.893294	−	0.449474i	$-0.851612\pi$
$641$	−18.7824	+	13.6462i	−0.741862	+	0.538994i	0.151432	+	0.988468i	$0.451612\pi$
$642$		0			0
$643$	−7.02515	−	3.57949i	−0.277045	−	0.141161i	0.309948	−	0.950753i	$-0.399688\pi$
$643$	−7.02515	−	3.57949i	−0.277045	−	0.141161i	−0.586993	+	0.809592i	$0.699688\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	10.1537	−	5.17359i	0.399185	−	0.203395i	−0.242861	−	0.970061i	$-0.578086\pi$
$647$	10.1537	−	5.17359i	0.399185	−	0.203395i	0.642046	+	0.766666i	$0.278086\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$	50.1448	+	7.94216i	1.96382	+	0.311039i
$653$	37.9348	−	6.00828i	1.48450	−	0.235122i	0.639047	−	0.769167i	$-0.279329\pi$
$653$	37.9348	−	6.00828i	1.48450	−	0.235122i	0.845456	+	0.534045i	$0.179329\pi$
$654$		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
$669$	59.2190	+	19.2414i	2.28954	+	0.743916i
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$673$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$674$		0			0
$675$	−50.5350	+	2.87917i	−1.94509	+	0.110819i
$676$	8.03444	+	24.7275i	0.309017	+	0.951057i
$677$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$677$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$678$		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
$685$	26.7683	−	17.4703i	1.02276	−	0.667506i
$686$		0			0
$687$	41.3633	−	81.1800i	1.57811	−	3.09721i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	5.25329	−	16.1680i	0.199845	−	0.615058i	−0.800041	−	0.599945i	$-0.795189\pi$
$691$	5.25329	−	16.1680i	0.199845	−	0.615058i	0.999886	−	0.0151132i	$-0.00481087\pi$
$692$		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		0.809017	−	0.587785i	$-0.200000\pi$
$701$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$	66.6913	−	60.3246i	2.51174	−	2.27195i
$706$		0			0
$707$		0			0
$708$	19.9975	−	3.16729i	0.751552	−	0.119034i
$709$	−18.0701	+	5.87132i	−0.678636	+	0.220502i	−0.627998	−	0.778215i	$-0.716125\pi$
$709$	−18.0701	+	5.87132i	−0.678636	+	0.220502i	−0.0506378	+	0.998717i	$0.516125\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	18.1533	+	35.6279i	0.679847	+	1.33427i
$714$		0			0
$715$		0			0
$716$	42.0000			1.56961
$717$		0			0
$718$		0			0
$719$	−29.9770	+	41.2599i	−1.11796	+	1.53873i	−0.308811	+	0.951123i	$0.599931\pi$
$719$	−29.9770	+	41.2599i	−1.11796	+	1.53873i	−0.809144	+	0.587610i	$0.800069\pi$
$720$	−28.1374	+	48.9925i	−1.04862	+	1.82584i
$721$		0			0
$722$		0			0
$723$		0			0
$724$	−18.9258	−	6.14936i	−0.703371	−	0.228539i
$725$		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
$729$	10.1195	+	13.9283i	0.374797	+	0.515864i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$733$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$734$		0			0
$735$	−17.0199	+	44.6419i	−0.627789	+	1.64664i
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$739$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$740$	−5.09836	−	7.81180i	−0.187420	−	0.287168i
$741$		0			0
$742$		0			0
$743$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$743$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	18.6074	−	13.5191i	0.678993	−	0.493318i	−0.194030	−	0.980996i	$-0.562156\pi$
$751$	18.6074	−	13.5191i	0.678993	−	0.493318i	0.873024	+	0.487678i	$0.162156\pi$
$752$	−8.24453	−	52.0539i	−0.300647	−	1.89821i
$753$	−73.4301	−	37.4145i	−2.67594	−	1.36346i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−1.13375	+	0.577674i	−0.0412068	+	0.0209959i	−0.474473	−	0.880270i	$-0.657361\pi$
$757$	−1.13375	+	0.577674i	−0.0412068	+	0.0209959i	0.433266	+	0.901266i	$0.357361\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$761$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$762$		0			0
$763$		0			0
$764$	−44.1602	+	14.3485i	−1.59766	+	0.519111i
$765$		0			0
$766$		0			0
$767$		0			0
$768$	22.1715	+	43.5141i	0.800047	+	1.57018i
$769$		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
$773$	−7.44074	+	46.9790i	−0.267625	+	1.68972i	0.377795	+	0.925889i	$0.376682\pi$
$773$	−7.44074	+	46.9790i	−0.267625	+	1.68972i	−0.645420	+	0.763828i	$0.723318\pi$
$774$		0			0
$775$	−48.6146	+	10.5649i	−1.74629	+	0.379502i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	16.4580	+	22.6525i	0.587785	+	0.809017i
$785$	−40.5625	−	32.6943i	−1.44774	−	1.16691i
$786$		0			0
$787$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$787$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$	−34.3193	−	7.21318i	−1.21718	−	0.255825i
$796$	−32.1985	−	23.3936i	−1.14124	−	0.829163i
$797$	17.1369	−	33.6330i	0.607019	−	1.19134i	−0.359113	−	0.933294i	$-0.616921\pi$
$797$	17.1369	−	33.6330i	0.607019	−	1.19134i	0.966132	−	0.258048i	$-0.0830794\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	17.5675	−	54.0672i	0.620717	−	1.91037i
$802$		0			0
$803$		0			0
$804$		−	99.0660i		−	3.49379i
$805$		0			0
$806$		0			0
$807$	6.33458	+	39.9950i	0.222988	+	1.40789i
$808$		0			0
$809$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$809$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$810$		0			0
$811$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$811$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$812$		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		−0.809017	−	0.587785i	$-0.800000\pi$
$821$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$822$		0			0
$823$	−25.3123	−	49.6782i	−0.882331	−	1.73167i	−0.651453	−	0.758689i	$-0.725840\pi$
$823$	−25.3123	−	49.6782i	−0.882331	−	1.73167i	−0.230878	−	0.972983i	$-0.574160\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$827$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$828$	−7.94216	+	50.1448i	−0.276009	+	1.74265i
$829$	17.0458	−	23.4615i	0.592024	−	0.814851i	−0.402925	−	0.915233i	$-0.632006\pi$
$829$	17.0458	−	23.4615i	0.592024	−	0.814851i	0.994949	+	0.100382i	$0.0320064\pi$
$830$		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
$839$	21.4441	+	29.5153i	0.740332	+	1.01898i	0.998599	+	0.0529065i	$0.0168485\pi$
$839$	21.4441	+	29.5153i	0.740332	+	1.01898i	−0.258267	+	0.966074i	$0.583151\pi$
$840$		0			0
$841$	−8.96149	−	27.5806i	−0.309017	−	0.951057i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−26.5288	+	11.8838i	−0.912617	+	0.408815i
$846$		0			0
$847$		0			0
$848$	−14.5330	+	14.5330i	−0.499065	+	0.499065i
$849$		0			0
$850$		0			0
$851$	−6.78168	−	4.92718i	−0.232473	−	0.168902i
$852$	−8.31433	+	16.3178i	−0.284844	+	0.559038i
$853$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$853$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$854$		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
$863$	6.57301	+	3.34912i	0.223748	+	0.114005i	0.562272	−	0.826953i	$-0.309928\pi$
$863$	6.57301	+	3.34912i	0.223748	+	0.114005i	−0.338524	+	0.940958i	$0.609928\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−46.2337	+	23.5573i	−1.57018	+	0.800047i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	31.1019	−	4.92606i	1.05264	−	0.166722i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$877$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$878$		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
$883$	8.39572	−	53.0085i	0.282538	−	1.78388i	−0.282964	−	0.959130i	$-0.591318\pi$
$883$	8.39572	−	53.0085i	0.282538	−	1.78388i	0.565503	−	0.824747i	$-0.308682\pi$
$884$		0			0
$885$	5.90853	+	21.8519i	0.198613	+	0.734542i
$886$		0			0
$887$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$887$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$888$		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
$895$	5.01467	+	46.6889i	0.167622	+	1.56064i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−57.8216	−	25.4291i	−1.92739	−	0.847636i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	4.57620	−	21.7729i	0.152118	−	0.723755i
$906$		0			0
$907$	−21.7328	+	42.6530i	−0.721626	+	1.41627i	0.179962	+	0.983674i	$0.442403\pi$
$907$	−21.7328	+	42.6530i	−0.721626	+	1.41627i	−0.901588	+	0.432597i	$0.857597\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	2.04979	−	6.30860i	0.0679125	−	0.209013i	−0.911341	−	0.411652i	$-0.864952\pi$
$911$	2.04979	−	6.30860i	0.0679125	−	0.209013i	0.979253	+	0.202639i	$0.0649518\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	48.2977	−	35.0903i	1.59580	−	1.15942i
$917$		0			0
$918$		0			0
$919$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$919$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$920$		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
$927$	−70.1423	−	11.1095i	−2.30378	−	0.364882i
$928$		0			0
$929$	50.4688	−	16.3983i	1.65583	−	0.538011i	0.675835	−	0.737053i	$-0.263783\pi$
$929$	50.4688	−	16.3983i	1.65583	−	0.538011i	0.979991	+	0.199042i	$0.0637830\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	16.6287	+	32.6356i	0.544398	+	1.06844i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$937$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$938$		0			0
$939$	61.9718	−	85.2969i	2.02237	−	2.78356i
$940$	56.8809	−	15.3800i	1.85525	−	0.501641i
$941$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$941$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$942$		0			0
$943$		0			0
$944$	12.6172	+	4.09957i	0.410655	+	0.133430i
$945$		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
$951$	33.4921	+	103.078i	1.08605	+	3.34253i
$952$		0			0
$953$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$953$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$954$		0			0
$955$	−21.2230	−	47.3771i	−0.686759	−	1.53309i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$	−45.7248	+	29.8422i	−1.47576	+	0.963154i
$961$	−55.0132	−	39.9694i	−1.77462	−	1.28934i
$962$		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
$971$	34.8817	−	25.3430i	1.11941	−	0.813296i	0.135287	−	0.990806i	$-0.456804\pi$
$971$	34.8817	−	25.3430i	1.11941	−	0.813296i	0.984119	+	0.177510i	$0.0568043\pi$
$972$	−1.90995	−	12.0589i	−0.0612616	−	0.386791i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−52.5341	+	26.7675i	−1.68072	+	0.856367i	−0.689473	+	0.724311i	$0.742158\pi$
$977$	−52.5341	+	26.7675i	−1.68072	+	0.856367i	−0.991242	+	0.132056i	$0.957842\pi$
$978$		0			0
$979$		0			0
$980$	−23.2164	+	21.0000i	−0.741620	+	0.670820i
$981$		0			0
$982$		0			0
$983$	61.0981	−	9.67699i	1.94873	−	0.308648i	0.948755	−	0.316012i	$-0.102344\pi$
$983$	61.0981	−	9.67699i	1.94873	−	0.308648i	0.999973	+	0.00736431i	$0.00234415\pi$
$984$		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
$993$	−4.75094	+	29.9962i	−0.150766	+	0.951902i
$994$		0			0
$995$	22.1608	−	38.5862i	0.702545	−	1.22327i
$996$		0			0
$997$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$997$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$998$		0			0
$999$	20.0826	+	6.52524i	0.635386	+	0.206449i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	605.2.m.a.112.2		16
5.3	odd	4	inner	605.2.m.a.233.2		16
11.2	odd	10	inner	605.2.m.a.602.1		16
11.3	even	5	inner	605.2.m.a.282.2		16
11.4	even	5		55.2.e.b.32.1	✓	4
11.5	even	5	inner	605.2.m.a.457.2		16
11.6	odd	10	inner	605.2.m.a.457.2		16
11.7	odd	10		55.2.e.b.32.1	✓	4
11.8	odd	10	inner	605.2.m.a.282.2		16
11.9	even	5	inner	605.2.m.a.602.1		16
11.10	odd	2	CM	605.2.m.a.112.2		16
33.26	odd	10		495.2.k.a.307.1		4
33.29	even	10		495.2.k.a.307.1		4
44.7	even	10		880.2.bd.d.417.2		4
44.15	odd	10		880.2.bd.d.417.2		4
55.3	odd	20	inner	605.2.m.a.403.1		16
55.4	even	10		275.2.e.a.32.2		4
55.7	even	20		275.2.e.a.43.2		4
55.8	even	20	inner	605.2.m.a.403.1		16
55.13	even	20	inner	605.2.m.a.118.2		16
55.18	even	20		55.2.e.b.43.1	yes	4
55.28	even	20	inner	605.2.m.a.578.2		16
55.29	odd	10		275.2.e.a.32.2		4
55.37	odd	20		275.2.e.a.43.2		4
55.38	odd	20	inner	605.2.m.a.578.2		16
55.43	even	4	inner	605.2.m.a.233.2		16
55.48	odd	20		55.2.e.b.43.1	yes	4
55.53	odd	20	inner	605.2.m.a.118.2		16
165.128	odd	20		495.2.k.a.208.1		4
165.158	even	20		495.2.k.a.208.1		4
220.103	even	20		880.2.bd.d.593.2		4
220.183	odd	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	11.4	even	5
55.2.e.b.32.1	✓	4	11.7	odd	10
55.2.e.b.43.1	yes	4	55.18	even	20
55.2.e.b.43.1	yes	4	55.48	odd	20
275.2.e.a.32.2		4	55.4	even	10
275.2.e.a.32.2		4	55.29	odd	10
275.2.e.a.43.2		4	55.7	even	20
275.2.e.a.43.2		4	55.37	odd	20
495.2.k.a.208.1		4	165.128	odd	20
495.2.k.a.208.1		4	165.158	even	20
495.2.k.a.307.1		4	33.26	odd	10
495.2.k.a.307.1		4	33.29	even	10
605.2.m.a.112.2		16	1.1	even	1	trivial
605.2.m.a.112.2		16	11.10	odd	2	CM
605.2.m.a.118.2		16	55.13	even	20	inner
605.2.m.a.118.2		16	55.53	odd	20	inner
605.2.m.a.233.2		16	5.3	odd	4	inner
605.2.m.a.233.2		16	55.43	even	4	inner
605.2.m.a.282.2		16	11.3	even	5	inner
605.2.m.a.282.2		16	11.8	odd	10	inner
605.2.m.a.403.1		16	55.3	odd	20	inner
605.2.m.a.403.1		16	55.8	even	20	inner
605.2.m.a.457.2		16	11.5	even	5	inner
605.2.m.a.457.2		16	11.6	odd	10	inner
605.2.m.a.578.2		16	55.28	even	20	inner
605.2.m.a.578.2		16	55.38	odd	20	inner
605.2.m.a.602.1		16	11.2	odd	10	inner
605.2.m.a.602.1		16	11.9	even	5	inner
880.2.bd.d.417.2		4	44.7	even	10
880.2.bd.d.417.2		4	44.15	odd	10
880.2.bd.d.593.2		4	220.103	even	20
880.2.bd.d.593.2		4	220.183	odd	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)

\(f(q)\)	\(=\)	\(q+(0.477487 - 3.01474i) q^{3} +(1.17557 - 1.61803i) q^{4} +(1.93903 + 1.11362i) q^{5} +(-6.00747 - 1.95194i) q^{9} +(-4.31662 - 4.31662i) q^{12} +(4.28314 - 5.31393i) q^{15} +(-1.23607 - 3.80423i) q^{16} +(4.08135 - 1.82828i) q^{20} +(2.84169 - 2.84169i) q^{23} +(2.51969 + 4.31870i) q^{25} +(-4.59592 + 9.02000i) q^{27} +(-3.07468 + 9.46289i) q^{31} +(-10.2205 + 7.42564i) q^{36} +(-0.326303 - 2.06020i) q^{37} +(-9.47494 - 10.4749i) q^{45} +(13.0135 + 2.06113i) q^{47} +(-12.0589 + 1.90995i) q^{48} +(-6.65740 + 2.16312i) q^{49} +(-2.33269 - 4.57816i) q^{53} +(-1.94946 + 2.68321i) q^{59} +(-3.56298 - 13.1772i) q^{60} +(-7.60845 - 2.47214i) q^{64} +(11.4749 + 11.4749i) q^{67} +(-7.21007 - 9.92381i) q^{69} +(-0.927051 - 2.85317i) q^{71} +(14.2229 - 5.53406i) q^{75} +(1.83970 - 8.75303i) q^{80} +(9.66765 + 7.02396i) q^{81} +9.00000i q^{89} +(-1.25734 - 7.93855i) q^{92} +(27.0600 + 13.7878i) q^{93} +(-4.44184 + 2.26323i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 2 q^{3} - 16 q^{12} + 8 q^{15} + 16 q^{16} - 12 q^{20} + 72 q^{23} - 2 q^{25} - 22 q^{27} - 24 q^{36} + 14 q^{37} - 72 q^{45} + 24 q^{47} - 8 q^{48} + 12 q^{53} + 28 q^{60} + 104 q^{67} + 12 q^{71}+ \cdots + 34 q^{97}+O(q^{100}) \) 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