LMFDB - Embedded newform 605.2.m.a.233.2

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			233.2
Root			$1.04771 - 1.37924i$ of defining polynomial
Character	$\chi$	$=$	605.233
Dual form			605.2.m.a.457.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(3.01474 + 0.477487i) q^{3} +(-1.17557 + 1.61803i) q^{4} +(-0.914138 + 2.04067i) q^{5} +(6.00747 + 1.95194i) q^{9} +(-4.31662 + 4.31662i) q^{12} +(-3.73028 + 5.71560i) q^{15} +(-1.23607 - 3.80423i) q^{16} +(-2.22725 - 3.87806i) q^{20} +(2.84169 + 2.84169i) q^{23} +(-3.32870 - 3.73092i) q^{25} +(9.02000 + 4.59592i) q^{27} +(-3.07468 + 9.46289i) q^{31} +(-10.2205 + 7.42564i) q^{36} +(-2.06020 + 0.326303i) q^{37} +(-9.47494 + 10.4749i) q^{45} +(2.06113 - 13.0135i) q^{47} +(-1.90995 - 12.0589i) q^{48} +(6.65740 - 2.16312i) q^{49} +(4.57816 - 2.33269i) q^{53} +(1.94946 - 2.68321i) q^{59} +(-4.86283 - 12.7548i) q^{60} +(7.60845 + 2.47214i) q^{64} +(11.4749 - 11.4749i) q^{67} +(7.21007 + 9.92381i) q^{69} +(-0.927051 - 2.85317i) q^{71} +(-8.25370 - 12.8371i) q^{75} +(8.89312 + 0.955176i) q^{80} +(9.66765 + 7.02396i) q^{81} -9.00000i q^{89} +(-7.93855 + 1.25734i) q^{92} +(-13.7878 + 27.0600i) q^{93} +(2.26323 + 4.44184i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 2 q^{3} - 16 q^{12} + 8 q^{15} + 16 q^{16} - 12 q^{20} + 72 q^{23} - 2 q^{25} - 22 q^{27} - 24 q^{36} + 14 q^{37} - 72 q^{45} + 24 q^{47} - 8 q^{48} + 12 q^{53} + 28 q^{60} + 104 q^{67} + 12 q^{71}+ \cdots + 34 q^{97}+O(q^{100}) $ 16 * q + 2 * q^3 - 16 * q^12 + 8 * q^15 + 16 * q^16 - 12 * q^20 + 72 * q^23 - 2 * q^25 - 22 * q^27 - 24 * q^36 + 14 * q^37 - 72 * q^45 + 24 * q^47 - 8 * q^48 + 12 * q^53 + 28 * q^60 + 104 * q^67 + 12 * q^71 - 32 * q^75 + 8 * q^81 - 36 * q^92 - 66 * q^93 + 34 * q^97

Character values

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

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

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		−0.453990	−	0.891007i	$-0.650000\pi$
$2$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$3$	3.01474	+	0.477487i	1.74056	+	0.275677i	0.944254	−	0.329218i	$-0.106785\pi$
$3$	3.01474	+	0.477487i	1.74056	+	0.275677i	0.796305	+	0.604896i	$0.206785\pi$
$4$	−1.17557	+	1.61803i	−0.587785	+	0.809017i
$5$	−0.914138	+	2.04067i	−0.408815	+	0.912617i
$5$	−0.914138	+	2.04067i	−0.408815	+	0.912617i
$6$		0			0
$7$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$7$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$8$		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		0.891007	−	0.453990i	$-0.150000\pi$
$13$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$14$		0			0
$15$	−3.73028	+	5.71560i	−0.963154	+	1.47576i
$16$	−1.23607	−	3.80423i	−0.309017	−	0.951057i
$17$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$17$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$18$		0			0
$19$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$19$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$20$	−2.22725	−	3.87806i	−0.498027	−	0.867161i
$21$		0			0
$22$		0			0
$23$	2.84169	+	2.84169i	0.592533	+	0.592533i	0.938315	−	0.345782i	$-0.112386\pi$
$23$	2.84169	+	2.84169i	0.592533	+	0.592533i	−0.345782	+	0.938315i	$0.612386\pi$
$24$		0			0
$25$	−3.32870	−	3.73092i	−0.665741	−	0.746183i
$26$		0			0
$27$	9.02000	+	4.59592i	1.73590	+	0.884485i
$28$		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
$37$	−2.06020	+	0.326303i	−0.338694	+	0.0536439i	−0.323465	−	0.946240i	$-0.604848\pi$
$37$	−2.06020	+	0.326303i	−0.338694	+	0.0536439i	−0.0152291	+	0.999884i	$0.504848\pi$
$38$		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	$-0.750000\pi$
$43$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$44$		0			0
$45$	−9.47494	+	10.4749i	−1.41244	+	1.56151i
$46$		0			0
$47$	2.06113	−	13.0135i	0.300647	−	1.89821i	−0.123038	−	0.992402i	$-0.539264\pi$
$47$	2.06113	−	13.0135i	0.300647	−	1.89821i	0.423685	−	0.905810i	$-0.360736\pi$
$48$	−1.90995	−	12.0589i	−0.275677	−	1.74056i
$49$	6.65740	−	2.16312i	0.951057	−	0.309017i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	4.57816	−	2.33269i	0.628859	−	0.320420i	−0.110353	−	0.993892i	$-0.535198\pi$
$53$	4.57816	−	2.33269i	0.628859	−	0.320420i	0.739212	+	0.673473i	$0.235198\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	1.94946	−	2.68321i	0.253798	−	0.349324i	−0.663039	−	0.748585i	$-0.730733\pi$
$59$	1.94946	−	2.68321i	0.253798	−	0.349324i	0.916837	+	0.399262i	$0.130733\pi$
$60$	−4.86283	−	12.7548i	−0.627789	−	1.64664i
$61$		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	0.607785	−	0.794101i	$-0.292058\pi$
$67$	11.4749	−	11.4749i	1.40189	−	1.40189i	0.794101	−	0.607785i	$-0.207942\pi$
$68$		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		−0.156434	−	0.987688i	$-0.550000\pi$
$73$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$74$		0			0
$75$	−8.25370	−	12.8371i	−0.953055	−	1.48231i
$76$		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	$0.100000\pi$
$80$	8.89312	+	0.955176i	0.994281	+	0.106792i
$81$	9.66765	+	7.02396i	1.07418	+	0.780440i
$82$		0			0
$83$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$83$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	9.00000i		−	0.953998i	−0.878904	−	0.476999i	$-0.841725\pi$
$89$		−	9.00000i		−	0.953998i	0.878904	−	0.476999i	$-0.158275\pi$
$90$		0			0
$91$		0			0
$92$	−7.93855	+	1.25734i	−0.827651	+	0.131087i
$93$	−13.7878	+	27.0600i	−1.42972	+	2.80599i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	2.26323	+	4.44184i	0.229796	+	0.451000i	0.976897	−	0.213710i	$-0.0685548\pi$
$97$	2.26323	+	4.44184i	0.229796	+	0.451000i	−0.747101	+	0.664711i	$0.768555\pi$
$98$		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
$103$	1.75876	+	11.1044i	0.173296	+	1.09415i	0.908985	+	0.416829i	$0.136859\pi$
$103$	1.75876	+	11.1044i	0.173296	+	1.09415i	−0.735689	+	0.677320i	$0.763141\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$107$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$108$	−18.0400	+	9.19184i	−1.73590	+	0.884485i
$109$		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
$113$	16.9828	+	2.68980i	1.59760	+	0.253036i	0.890807	−	0.454382i	$-0.150140\pi$
$113$	16.9828	+	2.68980i	1.59760	+	0.253036i	0.706796	+	0.707417i	$0.250140\pi$
$114$		0			0
$115$	−8.39665	+	3.20126i	−0.782992	+	0.298519i
$116$		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
$125$	10.6565	−	3.38223i	0.953144	−	0.302516i
$126$		0			0
$127$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$127$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$128$		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
$135$	−17.6243	+	14.2056i	−1.51686	+	1.22262i
$136$		0			0
$137$	−12.7371	−	6.48986i	−1.08820	−	0.554466i	−0.184588	−	0.982816i	$-0.559095\pi$
$137$	−12.7371	−	6.48986i	−1.08820	−	0.554466i	−0.903613	+	0.428350i	$0.859095\pi$
$138$		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
$147$	21.1031	−	3.34241i	1.74056	−	0.275677i
$148$	1.89394	−	3.71706i	0.155681	−	0.305540i
$149$		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
$157$	−3.64478	+	23.0122i	−0.290885	+	1.83658i	0.218255	+	0.975892i	$0.429964\pi$
$157$	−3.64478	+	23.0122i	−0.290885	+	1.83658i	−0.509140	+	0.860684i	$0.670036\pi$
$158$		0			0
$159$	14.9158	−	4.84643i	1.18290	−	0.384347i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−22.6182	+	11.5245i	−1.77159	+	0.902671i	−0.836768	+	0.547558i	$0.815557\pi$
$163$	−22.6182	+	11.5245i	−1.77159	+	0.902671i	−0.934824	+	0.355112i	$0.884443\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$167$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$168$		0			0
$169$	7.64121	−	10.5172i	0.587785	−	0.809017i
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$173$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$174$		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	$-0.912762\pi$
$179$	−12.3435	−	16.9894i	−0.922596	−	1.26984i	0.0400827	−	0.999196i	$-0.487238\pi$
$180$	−5.81035	−	27.6448i	−0.433078	−	2.06052i
$181$	−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
$185$	1.21743	−	4.50247i	0.0895069	−	0.331028i
$186$		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	$-0.617409\pi$
$192$	21.7571	+	11.0858i	1.57018	+	0.800047i
$193$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$193$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$194$		0			0
$195$		0			0
$196$	−4.32624	+	13.3148i	−0.309017	+	0.951057i
$197$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$197$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$198$		0			0
$199$			19.8997i			1.41066i	0.708881	+	0.705328i	$0.249200\pi$
$199$			19.8997i			1.41066i	−0.708881	+	0.705328i	$0.750800\pi$
$200$		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
$207$	11.5245	+	22.6182i	0.801010	+	1.57207i
$208$		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	$0.600000\pi$
$212$	−1.60758	+	10.1499i	−0.110409	+	0.697095i
$213$	−1.43246	−	9.04421i	−0.0981507	−	0.619699i
$214$		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
$223$	−20.1486	−	3.19123i	−1.34925	−	0.213700i	−0.560368	−	0.828244i	$-0.689340\pi$
$223$	−20.1486	−	3.19123i	−1.34925	−	0.213700i	−0.788883	+	0.614544i	$0.789340\pi$
$224$		0			0
$225$	−12.7145	−	28.9108i	−0.847636	−	1.92739i
$226$		0			0
$227$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$227$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$228$		0			0
$229$	−28.3887	−	9.22404i	−1.87598	−	0.609542i	−0.989039	−	0.147652i	$-0.952828\pi$
$229$	−28.3887	−	9.22404i	−1.87598	−	0.609542i	−0.886937	−	0.461890i	$-0.847172\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$233$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$234$		0			0
$235$	24.6721	+	16.1022i	1.60943	+	1.05039i
$236$	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	$0.300000\pi$
$240$	26.3543	+	7.12595i	1.70116	+	0.459978i
$241$		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
$245$	−1.67156	+	15.5630i	−0.106792	+	0.994281i
$246$		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
$257$	−31.1019	+	4.92606i	−1.94008	+	0.307279i	−0.999480	−	0.0322308i	$-0.989739\pi$
$257$	−31.1019	+	4.92606i	−1.94008	+	0.307279i	−0.940603	+	0.339510i	$0.889739\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$263$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$264$		0			0
$265$	0.575188	+	11.4749i	0.0353335	+	0.704900i
$266$		0			0
$267$	4.29738	−	27.1326i	0.262996	−	1.66049i
$268$	5.07724	+	32.0564i	0.310142	+	1.95816i
$269$	12.6172	−	4.09957i	0.769284	−	0.249955i	0.102025	−	0.994782i	$-0.467468\pi$
$269$	12.6172	−	4.09957i	0.769284	−	0.249955i	0.667258	+	0.744826i	$0.267468\pi$
$270$		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		−0.453990	−	0.891007i	$-0.650000\pi$
$277$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$278$		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		−0.987688	−	0.156434i	$-0.950000\pi$
$283$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$284$	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		−0.156434	−	0.987688i	$-0.550000\pi$
$293$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$294$		0			0
$295$	3.69347	+	6.43104i	0.215042	+	0.374430i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	30.4737	+	1.73620i	1.75940	+	0.100240i
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$307$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$308$		0			0
$309$			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
$313$	−15.6817	+	30.7771i	−0.886382	+	1.73962i	−0.250004	+	0.968245i	$0.580432\pi$
$313$	−15.6817	+	30.7771i	−0.886382	+	1.73962i	−0.636378	+	0.771377i	$0.719568\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	16.1204	+	31.6382i	0.905415	+	1.77698i	0.519652	+	0.854378i	$0.326062\pi$
$317$	16.1204	+	31.6382i	0.905415	+	1.77698i	0.385763	+	0.922598i	$0.373938\pi$
$318$		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
$333$	−13.0135	−	2.06113i	−0.713134	−	0.112949i
$334$		0			0
$335$	12.9269	+	33.9063i	0.706274	+	1.85250i
$336$		0			0
$337$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$337$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$338$		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
$345$	−26.8423	+	5.64167i	−1.44514	+	0.303737i
$346$		0			0
$347$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$347$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$348$		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	0.239511	−	0.970894i	$-0.423013\pi$
$353$	−13.7414	−	13.7414i	−0.731383	−	0.731383i	−0.970894	+	0.239511i	$0.923013\pi$
$354$		0			0
$355$	6.66984	+	0.716382i	0.353998	+	0.0380216i
$356$	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
$367$	18.8918	−	2.99217i	0.986146	−	0.156190i	0.357526	−	0.933903i	$-0.383620\pi$
$367$	18.8918	−	2.99217i	0.986146	−	0.156190i	0.628620	+	0.777713i	$0.283620\pi$
$368$	7.29790	−	14.3229i	0.380429	−	0.746635i
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−27.5755	−	54.1200i	−1.42972	−	2.80599i
$373$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$373$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$374$		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	0.530700	−	0.847560i	$-0.321929\pi$
$379$	28.3887	−	9.22404i	1.45823	−	0.473807i	0.927528	+	0.373753i	$0.121929\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	22.4818	−	11.4551i	1.14877	−	0.585327i	0.227317	−	0.973821i	$-0.427005\pi$
$383$	22.4818	−	11.4551i	1.14877	−	0.585327i	0.921451	+	0.388494i	$0.127005\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	−9.84763	−	1.55971i	−0.499938	−	0.0791824i
$389$	21.4441	−	29.5153i	1.08726	−	1.49648i	0.235988	−	0.971756i	$-0.424167\pi$
$389$	21.4441	−	29.5153i	1.08726	−	1.49648i	0.851270	−	0.524727i	$-0.175833\pi$
$390$		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	−0.0501886	+	0.998740i	$0.515982\pi$
$397$	−20.8997	+	20.8997i	−1.04893	+	1.04893i	−0.998740	+	0.0501886i	$0.984018\pi$
$398$		0			0
$399$		0			0
$400$	−10.0787	+	17.2748i	−0.503937	+	0.863740i
$401$	−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
$405$	−23.1712	+	13.3077i	−1.15139	+	0.661263i
$406$		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
$412$	−20.0349	−	10.2083i	−0.987046	−	0.502925i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	24.0000i		−	1.17248i	−0.810139	−	0.586238i	$-0.800608\pi$
$419$		−	24.0000i		−	1.17248i	0.810139	−	0.586238i	$-0.199392\pi$
$420$		0			0
$421$	−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
$423$	37.7838	−	74.1549i	1.83711	−	3.60553i
$424$		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	$0.600000\pi$
$432$	6.33458	−	39.9950i	0.304773	−	1.92426i
$433$	−6.50970	−	41.1006i	−0.312836	−	1.97517i	−0.170676	−	0.985327i	$-0.554595\pi$
$433$	−6.50970	−	41.1006i	−0.312836	−	1.97517i	−0.142160	−	0.989844i	$-0.545405\pi$
$434$		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
$443$	40.1461	+	6.35852i	1.90740	+	0.302102i	0.994385	−	0.105825i	$-0.0337485\pi$
$443$	40.1461	+	6.35852i	1.90740	+	0.302102i	0.913014	+	0.407928i	$0.133748\pi$
$444$	7.48456	−	10.3016i	0.355202	−	0.488893i
$445$	18.3661	+	8.22724i	0.870635	+	0.390009i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−37.0912	−	12.0517i	−1.75044	−	0.568753i	−0.754302	−	0.656528i	$-0.772024\pi$
$449$	−37.0912	−	12.0517i	−1.75044	−	0.568753i	−0.996140	+	0.0877747i	$0.972024\pi$
$450$		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		−0.891007	−	0.453990i	$-0.850000\pi$
$457$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$458$		0			0
$459$		0			0
$460$	4.69110	−	17.3494i	0.218724	−	0.808919i
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$	−0.575188	−	0.575188i	−0.0267313	−	0.0267313i	0.693615	−	0.720346i	$-0.256017\pi$
$463$	−0.575188	−	0.575188i	−0.0267313	−	0.0267313i	−0.720346	+	0.693615i	$0.756017\pi$
$464$		0			0
$465$	−42.6167	−	52.8729i	−1.97630	−	2.45192i
$466$		0			0
$467$	29.0549	+	14.8042i	1.34450	+	0.685056i	0.970212	−	0.242257i	$-0.0778878\pi$
$467$	29.0549	+	14.8042i	1.34450	+	0.685056i	0.374286	+	0.927313i	$0.377888\pi$
$468$		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
$477$	32.0564	−	5.07724i	1.46776	−	0.232471i
$478$		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
$487$	−5.85710	+	36.9803i	−0.265410	+	1.67574i	0.390273	+	0.920699i	$0.372381\pi$
$487$	−5.85710	+	36.9803i	−0.265410	+	1.67574i	−0.655683	+	0.755036i	$0.727619\pi$
$488$		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	−0.462522	−	0.886608i	$-0.653055\pi$
$499$	11.6968	−	16.0992i	0.523620	−	0.720701i	0.986141	+	0.165907i	$0.0530552\pi$
$500$	−7.05488	+	21.2186i	−0.315504	+	0.948924i
$501$		0			0
$502$		0			0
$503$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$503$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$504$		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	0.763624	−	0.645661i	$-0.223418\pi$
$509$	−1.94946	−	2.68321i	−0.0864084	−	0.118931i	−0.850033	+	0.526730i	$0.823418\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−24.2682	−	6.56189i	−1.06939	−	0.289151i
$516$		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		0.453990	−	0.891007i	$-0.350000\pi$
$523$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$524$		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
$537$	−29.1002	−	57.1123i	−1.25576	−	2.46458i
$538$		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
$543$	−4.75094	−	29.9962i	−0.203882	−	1.28726i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$547$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$548$	25.4741	−	12.9797i	1.08820	−	0.554466i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$	5.82009	−	12.9925i	0.247049	−	0.551499i
$556$		0			0
$557$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$557$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$563$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$564$	47.2772	+	65.0715i	1.99073	+	2.74000i
$565$	−21.0136	+	32.1974i	−0.884049	+	1.35456i
$566$		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
$575$	1.14296	−	20.0612i	0.0476649	−	0.836611i
$576$	40.8821	+	29.7026i	1.70342	+	1.23761i
$577$	−23.3429	−	11.8938i	−0.971779	−	0.495146i	−0.105344	−	0.994436i	$-0.533594\pi$
$577$	−23.3429	−	11.8938i	−0.971779	−	0.495146i	−0.866435	+	0.499290i	$0.833594\pi$
$578$		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
$587$	−28.8906	+	4.57582i	−1.19244	+	0.188864i	−0.720938	−	0.693000i	$-0.756289\pi$
$587$	−28.8906	+	4.57582i	−1.19244	+	0.188864i	−0.471504	+	0.881864i	$0.656289\pi$
$588$	−19.4001	+	38.0749i	−0.800047	+	1.57018i
$589$		0			0
$590$		0			0
$591$		0			0
$592$	3.78787	+	7.43412i	0.155681	+	0.305540i
$593$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$593$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−9.50188	+	59.9925i	−0.388886	+	2.45533i
$598$		0			0
$599$	−34.2380	+	11.1246i	−1.39893	+	0.454539i	−0.908844	−	0.417136i	$-0.863034\pi$
$599$	−34.2380	+	11.1246i	−1.39893	+	0.454539i	−0.490084	+	0.871675i	$0.663034\pi$
$600$		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
$603$	91.3337	−	46.5369i	3.71940	−	1.89513i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$607$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$613$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−7.73350	+	7.73350i	−0.311339	+	0.311339i	−0.845428	−	0.534089i	$-0.820655\pi$
$617$	−7.73350	+	7.73350i	−0.311339	+	0.311339i	0.534089	+	0.845428i	$0.320655\pi$
$618$		0			0
$619$	−0.587785	−	0.809017i	−0.0236251	−	0.0325171i	0.797041	−	0.603925i	$-0.206397\pi$
$619$	−0.587785	−	0.809017i	−0.0236251	−	0.0325171i	−0.820666	+	0.571408i	$0.806397\pi$
$620$	43.5458	−	9.15239i	1.74884	−	0.367569i
$621$	12.5719	+	38.6922i	0.504491	+	1.55266i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−2.83947	+	24.8382i	−0.113579	+	0.993529i
$626$		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
$643$	3.57949	−	7.02515i	0.141161	−	0.277045i	−0.809592	−	0.586993i	$-0.800312\pi$
$643$	3.57949	−	7.02515i	0.141161	−	0.277045i	0.950753	+	0.309948i	$0.100312\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−5.17359	−	10.1537i	−0.203395	−	0.399185i	0.766666	−	0.642046i	$-0.221914\pi$
$647$	−5.17359	−	10.1537i	−0.203395	−	0.399185i	−0.970061	+	0.242861i	$0.921914\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$	7.94216	−	50.1448i	0.311039	−	1.96382i
$653$	6.00828	+	37.9348i	0.235122	+	1.48450i	0.769167	+	0.639047i	$0.220671\pi$
$653$	6.00828	+	37.9348i	0.235122	+	1.48450i	−0.534045	+	0.845456i	$0.679329\pi$
$654$		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		0.891007	−	0.453990i	$-0.150000\pi$
$673$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$674$		0			0
$675$	−12.8779	−	48.9513i	−0.495671	−	1.88414i
$676$	8.03444	+	24.7275i	0.309017	+	0.951057i
$677$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$677$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	35.2164	+	35.2164i	1.34752	+	1.34752i	0.888350	+	0.459167i	$0.151852\pi$
$683$	35.2164	+	35.2164i	1.34752	+	1.34752i	0.459167	+	0.888350i	$0.348148\pi$
$684$		0			0
$685$	24.8871	−	20.0596i	0.950888	−	0.766437i
$686$		0			0
$687$	−81.1800	−	41.3633i	−3.09721	−	1.57811i
$688$		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
$708$	3.16729	+	19.9975i	0.119034	+	0.751552i
$709$	18.0701	−	5.87132i	0.678636	−	0.220502i	0.0506378	−	0.998717i	$-0.483875\pi$
$709$	18.0701	−	5.87132i	0.678636	−	0.220502i	0.627998	+	0.778215i	$0.283875\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−35.6279	+	18.1533i	−1.33427	+	0.679847i
$714$		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	$-0.400069\pi$
$719$	29.9770	−	41.2599i	1.11796	−	1.53873i	0.809144	−	0.587610i	$-0.199931\pi$
$720$	51.5607	+	23.0971i	1.92155	+	0.860777i
$721$		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	0.184510	−	0.982831i	$-0.440930\pi$
$727$	31.4749	−	31.4749i	1.16734	−	1.16734i	0.982831	−	0.184510i	$-0.0590699\pi$
$728$		0			0
$729$	−10.1195	−	13.9283i	−0.374797	−	0.515864i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$733$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$734$		0			0
$735$	−12.4704	+	46.1201i	−0.459978	+	1.70116i
$736$		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	$0.100000\pi$
$740$	5.85399	+	7.26281i	0.215197	+	0.266986i
$741$		0			0
$742$		0			0
$743$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$743$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$744$		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	$0.162156\pi$
$752$	−52.0539	+	8.24453i	−1.89821	+	0.300647i
$753$	37.4145	−	73.4301i	1.36346	−	2.67594i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	0.577674	+	1.13375i	0.0209959	+	0.0412068i	0.901266	−	0.433266i	$-0.142639\pi$
$757$	0.577674	+	1.13375i	0.0209959	+	0.0412068i	−0.880270	+	0.474473i	$0.842639\pi$
$758$		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
$768$	−43.5141	+	22.1715i	−1.57018	+	0.800047i
$769$		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
$773$	−46.9790	−	7.44074i	−1.68972	−	0.267625i	−0.763828	−	0.645420i	$-0.776682\pi$
$773$	−46.9790	−	7.44074i	−1.68972	−	0.267625i	−0.925889	+	0.377795i	$0.876682\pi$
$774$		0			0
$775$	45.5400	−	20.0278i	1.63584	−	0.719419i
$776$		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
$785$	−43.6286	−	28.4742i	−1.55717	−	1.01629i
$786$		0			0
$787$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$787$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$	−3.74509	+	34.8685i	−0.132825	+	1.23666i
$796$	−32.1985	−	23.3936i	−1.14124	−	0.829163i
$797$	−33.6330	−	17.1369i	−1.19134	−	0.607019i	−0.258048	−	0.966132i	$-0.583079\pi$
$797$	−33.6330	−	17.1369i	−1.19134	−	0.607019i	−0.933294	+	0.359113i	$0.883079\pi$
$798$		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
$807$	39.9950	−	6.33458i	1.40789	−	0.222988i
$808$		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
$823$	49.6782	−	25.3123i	1.73167	−	0.882331i	0.758689	−	0.651453i	$-0.225840\pi$
$823$	49.6782	−	25.3123i	1.73167	−	0.882331i	0.972983	−	0.230878i	$-0.0741598\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$827$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$828$	−50.1448	−	7.94216i	−1.74265	−	0.276009i
$829$	−17.0458	+	23.4615i	−0.592024	+	0.814851i	−0.994949	−	0.100382i	$-0.967994\pi$
$829$	−17.0458	+	23.4615i	−0.592024	+	0.814851i	0.402925	+	0.915233i	$0.367994\pi$
$830$		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	$-0.983151\pi$
$839$	−21.4441	−	29.5153i	−0.740332	−	1.01898i	0.258267	−	0.966074i	$-0.416849\pi$
$840$		0			0
$841$	−8.96149	−	27.5806i	−0.309017	−	0.951057i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	14.4771	+	25.2074i	0.498027	+	0.867161i
$846$		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
$852$	16.3178	+	8.31433i	0.559038	+	0.284844i
$853$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$853$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$857$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$858$		0			0
$859$			31.0000i			1.05771i	0.848713	+	0.528853i	$0.177378\pi$
$859$			31.0000i			1.05771i	−0.848713	+	0.528853i	$0.822622\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−3.34912	+	6.57301i	−0.114005	+	0.223748i	−0.940958	−	0.338524i	$-0.890072\pi$
$863$	−3.34912	+	6.57301i	−0.114005	+	0.223748i	0.826953	+	0.562272i	$0.190072\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	23.5573	+	46.2337i	0.800047	+	1.57018i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	4.92606	+	31.1019i	0.166722	+	1.05264i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$877$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$878$		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
$883$	53.0085	+	8.39572i	1.78388	+	0.282538i	0.959130	−	0.282964i	$-0.0913178\pi$
$883$	53.0085	+	8.39572i	1.78388	+	0.282538i	0.824747	+	0.565503i	$0.191318\pi$
$884$		0			0
$885$	8.06410	+	21.1515i	0.271072	+	0.710999i
$886$		0			0
$887$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$887$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$888$		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
$895$	45.9534	−	9.65843i	1.53605	−	0.322846i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	61.7255	+	13.4141i	2.05752	+	0.447138i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	22.1214	+	2.37597i	0.735339	+	0.0789799i
$906$		0			0
$907$	42.6530	+	21.7328i	1.41627	+	0.721626i	0.983674	−	0.179962i	$-0.0575975\pi$
$907$	42.6530	+	21.7328i	1.41627	+	0.721626i	0.432597	+	0.901588i	$0.357597\pi$
$908$		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
$927$	−11.1095	+	70.1423i	−0.364882	+	2.30378i
$928$		0			0
$929$	−50.4688	+	16.3983i	−1.65583	+	0.538011i	−0.979991	−	0.199042i	$-0.936217\pi$
$929$	−50.4688	+	16.3983i	−1.65583	+	0.538011i	−0.675835	+	0.737053i	$0.736217\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	−32.6356	+	16.6287i	−1.06844	+	0.544398i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$937$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$938$		0			0
$939$	−61.9718	+	85.2969i	−2.02237	+	2.78356i
$940$	−55.0578	+	20.9910i	−1.79579	+	0.684652i
$941$		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	−0.377215	+	0.926126i	$0.623118\pi$
$947$	−40.1082	+	40.1082i	−1.30334	+	1.30334i	−0.926126	+	0.377215i	$0.876882\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	33.4921	+	103.078i	1.08605	+	3.34253i
$952$		0			0
$953$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$953$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$954$		0			0
$955$	45.0173	−	25.8543i	1.45672	−	0.836625i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$	−42.5114	+	34.2651i	−1.37205	+	1.10590i
$961$	−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	$-0.250000\pi$
$967$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$968$		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	$0.0568043\pi$
$972$	−12.0589	+	1.90995i	−0.386791	+	0.0612616i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	26.7675	+	52.5341i	0.856367	+	1.68072i	0.724311	+	0.689473i	$0.242158\pi$
$977$	26.7675	+	52.5341i	0.856367	+	1.68072i	0.132056	+	0.991242i	$0.457842\pi$
$978$		0			0
$979$		0			0
$980$	−23.2164	−	21.0000i	−0.741620	−	0.670820i
$981$		0			0
$982$		0			0
$983$	9.67699	+	61.0981i	0.308648	+	1.94873i	0.316012	+	0.948755i	$0.397656\pi$
$983$	9.67699	+	61.0981i	0.308648	+	1.94873i	−0.00736431	+	0.999973i	$0.502344\pi$
$984$		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
$993$	−29.9962	−	4.75094i	−0.951902	−	0.150766i
$994$		0			0
$995$	−40.6089	−	18.1911i	−1.28739	−	0.576697i
$996$		0			0
$997$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$997$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$998$		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.233.2		16
5.2	odd	4	inner	605.2.m.a.112.2		16
11.2	odd	10	inner	605.2.m.a.118.2		16
11.3	even	5	inner	605.2.m.a.403.1		16
11.4	even	5		55.2.e.b.43.1	yes	4
11.5	even	5	inner	605.2.m.a.578.2		16
11.6	odd	10	inner	605.2.m.a.578.2		16
11.7	odd	10		55.2.e.b.43.1	yes	4
11.8	odd	10	inner	605.2.m.a.403.1		16
11.9	even	5	inner	605.2.m.a.118.2		16
11.10	odd	2	CM	605.2.m.a.233.2		16
33.26	odd	10		495.2.k.a.208.1		4
33.29	even	10		495.2.k.a.208.1		4
44.7	even	10		880.2.bd.d.593.2		4
44.15	odd	10		880.2.bd.d.593.2		4
55.2	even	20	inner	605.2.m.a.602.1		16
55.4	even	10		275.2.e.a.43.2		4
55.7	even	20		55.2.e.b.32.1	✓	4
55.17	even	20	inner	605.2.m.a.457.2		16
55.18	even	20		275.2.e.a.32.2		4
55.27	odd	20	inner	605.2.m.a.457.2		16
55.29	odd	10		275.2.e.a.43.2		4
55.32	even	4	inner	605.2.m.a.112.2		16
55.37	odd	20		55.2.e.b.32.1	✓	4
55.42	odd	20	inner	605.2.m.a.602.1		16
55.47	odd	20	inner	605.2.m.a.282.2		16
55.48	odd	20		275.2.e.a.32.2		4
55.52	even	20	inner	605.2.m.a.282.2		16
165.62	odd	20		495.2.k.a.307.1		4
165.92	even	20		495.2.k.a.307.1		4
220.7	odd	20		880.2.bd.d.417.2		4
220.147	even	20		880.2.bd.d.417.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
55.2.e.b.32.1	✓	4	55.7	even	20
55.2.e.b.32.1	✓	4	55.37	odd	20
55.2.e.b.43.1	yes	4	11.4	even	5
55.2.e.b.43.1	yes	4	11.7	odd	10
275.2.e.a.32.2		4	55.18	even	20
275.2.e.a.32.2		4	55.48	odd	20
275.2.e.a.43.2		4	55.4	even	10
275.2.e.a.43.2		4	55.29	odd	10
495.2.k.a.208.1		4	33.26	odd	10
495.2.k.a.208.1		4	33.29	even	10
495.2.k.a.307.1		4	165.62	odd	20
495.2.k.a.307.1		4	165.92	even	20
605.2.m.a.112.2		16	5.2	odd	4	inner
605.2.m.a.112.2		16	55.32	even	4	inner
605.2.m.a.118.2		16	11.2	odd	10	inner
605.2.m.a.118.2		16	11.9	even	5	inner
605.2.m.a.233.2		16	1.1	even	1	trivial
605.2.m.a.233.2		16	11.10	odd	2	CM
605.2.m.a.282.2		16	55.47	odd	20	inner
605.2.m.a.282.2		16	55.52	even	20	inner
605.2.m.a.403.1		16	11.3	even	5	inner
605.2.m.a.403.1		16	11.8	odd	10	inner
605.2.m.a.457.2		16	55.17	even	20	inner
605.2.m.a.457.2		16	55.27	odd	20	inner
605.2.m.a.578.2		16	11.5	even	5	inner
605.2.m.a.578.2		16	11.6	odd	10	inner
605.2.m.a.602.1		16	55.2	even	20	inner
605.2.m.a.602.1		16	55.42	odd	20	inner
880.2.bd.d.417.2		4	220.7	odd	20
880.2.bd.d.417.2		4	220.147	even	20
880.2.bd.d.593.2		4	44.7	even	10
880.2.bd.d.593.2		4	44.15	odd	10

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+(3.01474 + 0.477487i) q^{3} +(-1.17557 + 1.61803i) q^{4} +(-0.914138 + 2.04067i) q^{5} +(6.00747 + 1.95194i) q^{9} +(-4.31662 + 4.31662i) q^{12} +(-3.73028 + 5.71560i) q^{15} +(-1.23607 - 3.80423i) q^{16} +(-2.22725 - 3.87806i) q^{20} +(2.84169 + 2.84169i) q^{23} +(-3.32870 - 3.73092i) q^{25} +(9.02000 + 4.59592i) q^{27} +(-3.07468 + 9.46289i) q^{31} +(-10.2205 + 7.42564i) q^{36} +(-2.06020 + 0.326303i) q^{37} +(-9.47494 + 10.4749i) q^{45} +(2.06113 - 13.0135i) q^{47} +(-1.90995 - 12.0589i) q^{48} +(6.65740 - 2.16312i) q^{49} +(4.57816 - 2.33269i) q^{53} +(1.94946 - 2.68321i) q^{59} +(-4.86283 - 12.7548i) q^{60} +(7.60845 + 2.47214i) q^{64} +(11.4749 - 11.4749i) q^{67} +(7.21007 + 9.92381i) q^{69} +(-0.927051 - 2.85317i) q^{71} +(-8.25370 - 12.8371i) q^{75} +(8.89312 + 0.955176i) q^{80} +(9.66765 + 7.02396i) q^{81} -9.00000i q^{89} +(-7.93855 + 1.25734i) q^{92} +(-13.7878 + 27.0600i) q^{93} +(2.26323 + 4.44184i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 2 q^{3} - 16 q^{12} + 8 q^{15} + 16 q^{16} - 12 q^{20} + 72 q^{23} - 2 q^{25} - 22 q^{27} - 24 q^{36} + 14 q^{37} - 72 q^{45} + 24 q^{47} - 8 q^{48} + 12 q^{53} + 28 q^{60} + 104 q^{67} + 12 q^{71}+ \cdots + 34 q^{97}+O(q^{100}) \) 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