LMFDB - Embedded newform 980.2.g.a.391.25

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [980,2,Mod(391,980)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(980, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1]))

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

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

chi := DirichletCharacter("980.391");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 980 = 2^{2} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	980.g (of order $2$, degree $1$, 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:	$7.82533939809$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	no (minimal twist has level 140)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			391.25
Character	$\chi$	$=$	980.391
Dual form			980.2.g.a.391.27

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(1.17639 - 0.784927i) q^{2} -2.99814 q^{3} +(0.767779 - 1.84676i) q^{4} +1.00000i q^{5} +(-3.52698 + 2.35332i) q^{6} +(-0.546365 - 2.77516i) q^{8} +5.98886 q^{9} +(0.784927 + 1.17639i) q^{10} +2.23657i q^{11} +(-2.30191 + 5.53685i) q^{12} -3.17109i q^{13} -2.99814i q^{15} +(-2.82103 - 2.83580i) q^{16} -3.44551i q^{17} +(7.04523 - 4.70082i) q^{18} +2.05235 q^{19} +(1.84676 + 0.767779i) q^{20} +(1.75554 + 2.63107i) q^{22} -2.66137i q^{23} +(1.63808 + 8.32031i) q^{24} -1.00000 q^{25} +(-2.48908 - 3.73043i) q^{26} -8.96105 q^{27} -7.38092 q^{29} +(-2.35332 - 3.52698i) q^{30} -4.89199 q^{31} +(-5.54453 - 1.12170i) q^{32} -6.70555i q^{33} +(-2.70447 - 4.05325i) q^{34} +(4.59812 - 11.0600i) q^{36} -11.1938 q^{37} +(2.41436 - 1.61095i) q^{38} +9.50739i q^{39} +(2.77516 - 0.546365i) q^{40} -1.46011i q^{41} -9.95752i q^{43} +(4.13040 + 1.71719i) q^{44} +5.98886i q^{45} +(-2.08898 - 3.13080i) q^{46} +6.12686 q^{47} +(8.45786 + 8.50215i) q^{48} +(-1.17639 + 0.784927i) q^{50} +10.3301i q^{51} +(-5.85624 - 2.43470i) q^{52} +4.65777 q^{53} +(-10.5417 + 7.03377i) q^{54} -2.23657 q^{55} -6.15325 q^{57} +(-8.68283 + 5.79348i) q^{58} -7.11876 q^{59} +(-5.53685 - 2.30191i) q^{60} +2.53665i q^{61} +(-5.75488 + 3.83985i) q^{62} +(-7.40297 + 3.03249i) q^{64} +3.17109 q^{65} +(-5.26337 - 7.88833i) q^{66} +0.0527695i q^{67} +(-6.36302 - 2.64539i) q^{68} +7.97917i q^{69} -0.212347i q^{71} +(-3.27210 - 16.6200i) q^{72} -14.8744i q^{73} +(-13.1682 + 8.78631i) q^{74} +2.99814 q^{75} +(1.57575 - 3.79020i) q^{76} +(7.46261 + 11.1844i) q^{78} +0.461203i q^{79} +(2.83580 - 2.82103i) q^{80} +8.89991 q^{81} +(-1.14608 - 1.71766i) q^{82} -10.9174 q^{83} +3.44551 q^{85} +(-7.81593 - 11.7139i) q^{86} +22.1291 q^{87} +(6.20682 - 1.22198i) q^{88} -7.02049i q^{89} +(4.70082 + 7.04523i) q^{90} +(-4.91490 - 2.04334i) q^{92} +14.6669 q^{93} +(7.20756 - 4.80914i) q^{94} +2.05235i q^{95} +(16.6233 + 3.36302i) q^{96} +0.185459i q^{97} +13.3945i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 4 q^{2} + 4 q^{4} - 4 q^{8} + 32 q^{9} + 28 q^{16} - 8 q^{22} - 32 q^{25} - 40 q^{29} - 4 q^{32} + 60 q^{36} - 16 q^{37} + 36 q^{44} - 4 q^{46} + 4 q^{50} + 16 q^{53} + 48 q^{57} - 4 q^{58} - 28 q^{60}+ \cdots + 16 q^{93}+O(q^{100}) $ 32 * q - 4 * q^2 + 4 * q^4 - 4 * q^8 + 32 * q^9 + 28 * q^16 - 8 * q^22 - 32 * q^25 - 40 * q^29 - 4 * q^32 + 60 * q^36 - 16 * q^37 + 36 * q^44 - 4 * q^46 + 4 * q^50 + 16 * q^53 + 48 * q^57 - 4 * q^58 - 28 * q^60 + 4 * q^64 - 8 * q^65 - 8 * q^72 - 76 * q^74 + 120 * q^78 + 72 * q^81 - 56 * q^86 - 8 * q^88 - 4 * q^92 + 16 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/980\mathbb{Z}\right)^\times$.

$n$	$101$	$197$	$491$
$\chi(n)$	$-1$	$1$	$-1$

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 $

$2$	1.17639	−	0.784927i	0.831832	−	0.555027i
$2$	1.17639	−	0.784927i	0.831832	−	0.555027i
$3$	−2.99814			−1.73098			−0.865490	−	0.500927i	$-0.832992\pi$
$3$	−2.99814			−1.73098			−0.865490	+	0.500927i	$0.832992\pi$
$4$	0.767779	−	1.84676i	0.383889	−	0.923379i
$5$			1.00000i			0.447214i
$5$			1.00000i			0.447214i
$6$	−3.52698	+	2.35332i	−1.43988	+	0.960741i
$7$		0			0
$7$		0			0
$8$	−0.546365	−	2.77516i	−0.193169	−	0.981165i
$9$	5.98886			1.99629
$10$	0.784927	+	1.17639i	0.248216	+	0.372007i
$11$			2.23657i			0.674351i	0.941442	+	0.337175i	$0.109472\pi$
$11$			2.23657i			0.674351i	−0.941442	+	0.337175i	$0.890528\pi$
$12$	−2.30191	+	5.53685i	−0.664504	+	1.59835i
$13$		−	3.17109i		−	0.879502i	−0.898120	−	0.439751i	$-0.855067\pi$
$13$		−	3.17109i		−	0.879502i	0.898120	−	0.439751i	$-0.144933\pi$
$14$		0			0
$15$		−	2.99814i		−	0.774117i
$16$	−2.82103	−	2.83580i	−0.705258	−	0.708951i
$17$		−	3.44551i		−	0.835658i	−0.908526	−	0.417829i	$-0.862791\pi$
$17$		−	3.44551i		−	0.835658i	0.908526	−	0.417829i	$-0.137209\pi$
$18$	7.04523	−	4.70082i	1.66058	−	1.10799i
$19$	2.05235			0.470842			0.235421	−	0.971893i	$-0.424353\pi$
$19$	2.05235			0.470842			0.235421	+	0.971893i	$0.424353\pi$
$20$	1.84676	+	0.767779i	0.412948	+	0.171681i
$21$		0			0
$22$	1.75554	+	2.63107i	0.374283	+	0.560947i
$23$		−	2.66137i		−	0.554934i	−0.960735	−	0.277467i	$-0.910505\pi$
$23$		−	2.66137i		−	0.554934i	0.960735	−	0.277467i	$-0.0894949\pi$
$24$	1.63808	+	8.32031i	0.334372	+	1.69838i
$25$	−1.00000			−0.200000
$26$	−2.48908	−	3.73043i	−0.488148	−	0.731598i
$27$	−8.96105			−1.72455
$28$		0			0
$29$	−7.38092			−1.37060			−0.685301	−	0.728260i	$-0.740329\pi$
$29$	−7.38092			−1.37060			−0.685301	+	0.728260i	$0.740329\pi$
$30$	−2.35332	−	3.52698i	−0.429656	−	0.643936i
$31$	−4.89199			−0.878627			−0.439313	−	0.898334i	$-0.644778\pi$
$31$	−4.89199			−0.878627			−0.439313	+	0.898334i	$0.644778\pi$
$32$	−5.54453	−	1.12170i	−0.980143	−	0.198291i
$33$		−	6.70555i		−	1.16729i
$34$	−2.70447	−	4.05325i	−0.463813	−	0.695127i
$35$		0			0
$36$	4.59812	−	11.0600i	0.766354	−	1.84333i
$37$	−11.1938			−1.84025			−0.920123	−	0.391628i	$-0.871912\pi$
$37$	−11.1938			−1.84025			−0.920123	+	0.391628i	$0.871912\pi$
$38$	2.41436	−	1.61095i	0.391662	−	0.261330i
$39$			9.50739i			1.52240i
$40$	2.77516	−	0.546365i	0.438791	−	0.0863878i
$41$		−	1.46011i		−	0.228031i	−0.993479	−	0.114016i	$-0.963629\pi$
$41$		−	1.46011i		−	0.228031i	0.993479	−	0.114016i	$-0.0363714\pi$
$42$		0			0
$43$		−	9.95752i		−	1.51851i	−0.650794	−	0.759254i	$-0.725564\pi$
$43$		−	9.95752i		−	1.51851i	0.650794	−	0.759254i	$-0.274436\pi$
$44$	4.13040	+	1.71719i	0.622681	+	0.258876i
$45$			5.98886i			0.892767i
$46$	−2.08898	−	3.13080i	−0.308003	−	0.461612i
$47$	6.12686			0.893694			0.446847	−	0.894610i	$-0.352547\pi$
$47$	6.12686			0.893694			0.446847	+	0.894610i	$0.352547\pi$
$48$	8.45786	+	8.50215i	1.22079	+	1.22718i
$49$		0			0
$50$	−1.17639	+	0.784927i	−0.166366	+	0.111005i
$51$			10.3301i			1.44651i
$52$	−5.85624	−	2.43470i	−0.812114	−	0.337632i
$53$	4.65777			0.639793			0.319897	−	0.947452i	$-0.396352\pi$
$53$	4.65777			0.639793			0.319897	+	0.947452i	$0.396352\pi$
$54$	−10.5417	+	7.03377i	−1.43454	+	0.957175i
$55$	−2.23657			−0.301579
$56$		0			0
$57$	−6.15325			−0.815018
$58$	−8.68283	+	5.79348i	−1.14011	+	0.760722i
$59$	−7.11876			−0.926784			−0.463392	−	0.886153i	$-0.653368\pi$
$59$	−7.11876			−0.926784			−0.463392	+	0.886153i	$0.653368\pi$
$60$	−5.53685	−	2.30191i	−0.714804	−	0.297175i
$61$			2.53665i			0.324785i	0.986726	+	0.162393i	$0.0519211\pi$
$61$			2.53665i			0.324785i	−0.986726	+	0.162393i	$0.948079\pi$
$62$	−5.75488	+	3.83985i	−0.730870	+	0.487662i
$63$		0			0
$64$	−7.40297	+	3.03249i	−0.925371	+	0.379062i
$65$	3.17109			0.393325
$66$	−5.26337	−	7.88833i	−0.647876	−	0.970987i
$67$			0.0527695i			0.00644683i	0.999995	+	0.00322341i	$0.00102605\pi$
$67$			0.0527695i			0.00644683i	−0.999995	+	0.00322341i	$0.998974\pi$
$68$	−6.36302	−	2.64539i	−0.771629	−	0.320800i
$69$			7.97917i			0.960579i
$70$		0			0
$71$		−	0.212347i		−	0.0252009i	−0.999921	−	0.0126005i	$-0.995989\pi$
$71$		−	0.212347i		−	0.0252009i	0.999921	−	0.0126005i	$-0.00401095\pi$
$72$	−3.27210	−	16.6200i	−0.385621	−	1.95869i
$73$		−	14.8744i		−	1.74092i	−0.492241	−	0.870459i	$-0.663822\pi$
$73$		−	14.8744i		−	1.74092i	0.492241	−	0.870459i	$-0.336178\pi$
$74$	−13.1682	+	8.78631i	−1.53078	+	1.02139i
$75$	2.99814			0.346196
$76$	1.57575	−	3.79020i	0.180751	−	0.434766i
$77$		0			0
$78$	7.46261	+	11.1844i	0.844974	+	1.26638i
$79$			0.461203i			0.0518894i	0.999663	+	0.0259447i	$0.00825938\pi$
$79$			0.461203i			0.0518894i	−0.999663	+	0.0259447i	$0.991741\pi$
$80$	2.83580	−	2.82103i	0.317052	−	0.315401i
$81$	8.89991			0.988879
$82$	−1.14608	−	1.71766i	−0.126564	−	0.189684i
$83$	−10.9174			−1.19834			−0.599168	−	0.800624i	$-0.704502\pi$
$83$	−10.9174			−1.19834			−0.599168	+	0.800624i	$0.704502\pi$
$84$		0			0
$85$	3.44551			0.373718
$86$	−7.81593	−	11.7139i	−0.842813	−	1.26314i
$87$	22.1291			2.37248
$88$	6.20682	−	1.22198i	0.661650	−	0.130264i
$89$		−	7.02049i		−	0.744170i	−0.928199	−	0.372085i	$-0.878643\pi$
$89$		−	7.02049i		−	0.744170i	0.928199	−	0.372085i	$-0.121357\pi$
$90$	4.70082	+	7.04523i	0.495510	+	0.742633i
$91$		0			0
$92$	−4.91490	−	2.04334i	−0.512414	−	0.213033i
$93$	14.6669			1.52088
$94$	7.20756	−	4.80914i	0.743403	−	0.496025i
$95$			2.05235i			0.210567i
$96$	16.6233	+	3.36302i	1.69661	+	0.343237i
$97$			0.185459i			0.0188305i	0.999956	+	0.00941523i	$0.00299701\pi$
$97$			0.185459i			0.0188305i	−0.999956	+	0.00941523i	$0.997003\pi$
$98$		0			0
$99$			13.3945i			1.34620i
$100$	−0.767779	+	1.84676i	−0.0767779	+	0.184676i
$101$		−	6.24891i		−	0.621790i	−0.950444	−	0.310895i	$-0.899371\pi$
$101$		−	6.24891i		−	0.621790i	0.950444	−	0.310895i	$-0.100629\pi$
$102$	8.10839	+	12.1522i	0.802850	+	1.20325i
$103$	−11.4184			−1.12508			−0.562542	−	0.826769i	$-0.690176\pi$
$103$	−11.4184			−1.12508			−0.562542	+	0.826769i	$0.690176\pi$
$104$	−8.80027	+	1.73257i	−0.862937	+	0.169893i
$105$		0			0
$106$	5.47934	−	3.65601i	0.532201	−	0.355103i
$107$			2.60387i			0.251726i	0.992048	+	0.125863i	$0.0401700\pi$
$107$			2.60387i			0.251726i	−0.992048	+	0.125863i	$0.959830\pi$
$108$	−6.88010	+	16.5489i	−0.662038	+	1.59242i
$109$	1.00189			0.0959638			0.0479819	−	0.998848i	$-0.484721\pi$
$109$	1.00189			0.0959638			0.0479819	+	0.998848i	$0.484721\pi$
$110$	−2.63107	+	1.75554i	−0.250863	+	0.167384i
$111$	33.5606			3.18543
$112$		0			0
$113$	14.8588			1.39780			0.698899	−	0.715220i	$-0.253674\pi$
$113$	14.8588			1.39780			0.698899	+	0.715220i	$0.253674\pi$
$114$	−7.23861	+	4.82985i	−0.677958	+	0.452357i
$115$	2.66137			0.248174
$116$	−5.66691	+	13.6308i	−0.526160	+	1.26559i
$117$		−	18.9912i		−	1.75574i
$118$	−8.37443	+	5.58771i	−0.770929	+	0.514391i
$119$		0			0
$120$	−8.32031	+	1.63808i	−0.759537	+	0.149536i
$121$	5.99776			0.545251
$122$	1.99109	+	2.98409i	0.180265	+	0.270167i
$123$			4.37763i			0.394718i
$124$	−3.75596	+	9.03432i	−0.337295	+	0.811306i
$125$		−	1.00000i		−	0.0894427i
$126$		0			0
$127$			3.02360i			0.268301i	0.990961	+	0.134151i	$0.0428306\pi$
$127$			3.02360i			0.268301i	−0.990961	+	0.134151i	$0.957169\pi$
$128$	−6.32848	+	9.37818i	−0.559364	+	0.828922i
$129$			29.8541i			2.62851i
$130$	3.73043	−	2.48908i	0.327181	−	0.218306i
$131$	15.7053			1.37218			0.686091	−	0.727516i	$-0.259325\pi$
$131$	15.7053			1.37218			0.686091	+	0.727516i	$0.259325\pi$
$132$	−12.3835	−	5.14838i	−1.07785	−	0.448109i
$133$		0			0
$134$	0.0414202	+	0.0620775i	0.00357816	+	0.00536268i
$135$		−	8.96105i		−	0.771244i
$136$	−9.56181	+	1.88250i	−0.819919	+	0.161423i
$137$	9.61436			0.821410			0.410705	−	0.911768i	$-0.365283\pi$
$137$	9.61436			0.821410			0.410705	+	0.911768i	$0.365283\pi$
$138$	6.26306	+	9.38660i	0.533147	+	0.799040i
$139$	7.49745			0.635925			0.317963	−	0.948103i	$-0.397001\pi$
$139$	7.49745			0.635925			0.317963	+	0.948103i	$0.397001\pi$
$140$		0			0
$141$	−18.3692			−1.54697
$142$	−0.166677	−	0.249802i	−0.0139872	−	0.0209629i
$143$	7.09236			0.593093
$144$	−16.8948	−	16.9832i	−1.40790	−	1.41527i
$145$		−	7.38092i		−	0.612952i
$146$	−11.6753	−	17.4981i	−0.966257	−	1.44815i
$147$		0			0
$148$	−8.59435	+	20.6722i	−0.706451	+	1.69925i
$149$	6.50132			0.532609			0.266305	−	0.963889i	$-0.414197\pi$
$149$	6.50132			0.532609			0.266305	+	0.963889i	$0.414197\pi$
$150$	3.52698	−	2.35332i	0.287977	−	0.192148i
$151$		−	23.6747i		−	1.92662i	−0.268396	−	0.963309i	$-0.586494\pi$
$151$		−	23.6747i		−	1.92662i	0.268396	−	0.963309i	$-0.413506\pi$
$152$	−1.12133	−	5.69560i	−0.0909521	−	0.461974i
$153$		−	20.6347i		−	1.66821i
$154$		0			0
$155$		−	4.89199i		−	0.392934i
$156$	17.5578	+	7.29957i	1.40575	+	0.584433i
$157$			7.83249i			0.625100i	0.949901	+	0.312550i	$0.101183\pi$
$157$			7.83249i			0.625100i	−0.949901	+	0.312550i	$0.898817\pi$
$158$	0.362011	+	0.542554i	0.0288000	+	0.0431632i
$159$	−13.9647			−1.10747
$160$	1.12170	−	5.54453i	0.0886783	−	0.438333i
$161$		0			0
$162$	10.4697	−	6.98578i	0.822581	−	0.548855i
$163$		−	10.8516i		−	0.849961i	−0.905203	−	0.424980i	$-0.860281\pi$
$163$		−	10.8516i		−	0.849961i	0.905203	−	0.424980i	$-0.139719\pi$
$164$	−2.69648	−	1.12104i	−0.210560	−	0.0875389i
$165$	6.70555			0.522027
$166$	−12.8430	+	8.56933i	−0.996814	+	0.665109i
$167$	11.7476			0.909058			0.454529	−	0.890732i	$-0.349808\pi$
$167$	11.7476			0.909058			0.454529	+	0.890732i	$0.349808\pi$
$168$		0			0
$169$	2.94418			0.226476
$170$	4.05325	−	2.70447i	0.310870	−	0.207423i
$171$	12.2913			0.939937
$172$	−18.3891	−	7.64517i	−1.40216	−	0.582939i
$173$			16.4874i			1.25352i	0.779214	+	0.626758i	$0.215618\pi$
$173$			16.4874i			1.25352i	−0.779214	+	0.626758i	$0.784382\pi$
$174$	26.0324	−	17.3697i	1.97351	−	1.31679i
$175$		0			0
$176$	6.34247	−	6.30943i	0.478082	−	0.475591i
$177$	21.3431			1.60424
$178$	−5.51057	−	8.25882i	−0.413035	−	0.619025i
$179$			9.10959i			0.680883i	0.940266	+	0.340441i	$0.110577\pi$
$179$			9.10959i			0.680883i	−0.940266	+	0.340441i	$0.889423\pi$
$180$	11.0600	+	4.59812i	0.824363	+	0.342724i
$181$			16.5755i			1.23205i	0.787728	+	0.616023i	$0.211257\pi$
$181$			16.5755i			1.23205i	−0.787728	+	0.616023i	$0.788743\pi$
$182$		0			0
$183$		−	7.60525i		−	0.562196i
$184$	−7.38571	+	1.45408i	−0.544482	+	0.107196i
$185$		−	11.1938i		−	0.822983i
$186$	17.2539	−	11.5124i	1.26512	−	0.844132i
$187$	7.70611			0.563527
$188$	4.70407	−	11.3148i	0.343080	−	0.825218i
$189$		0			0
$190$	1.61095	+	2.41436i	0.116870	+	0.175156i
$191$		−	2.99295i		−	0.216562i	−0.994120	−	0.108281i	$-0.965465\pi$
$191$		−	2.99295i		−	0.216562i	0.994120	−	0.108281i	$-0.0345347\pi$
$192$	22.1952	−	9.09185i	1.60180	−	0.656148i
$193$	−14.7088			−1.05877			−0.529383	−	0.848383i	$-0.677577\pi$
$193$	−14.7088			−1.05877			−0.529383	+	0.848383i	$0.677577\pi$
$194$	0.145571	+	0.218171i	0.0104514	+	0.0156638i
$195$	−9.50739			−0.680838
$196$		0			0
$197$	4.81748			0.343231			0.171616	−	0.985164i	$-0.445101\pi$
$197$	4.81748			0.343231			0.171616	+	0.985164i	$0.445101\pi$
$198$	10.5137	+	15.7571i	0.747177	+	1.11981i
$199$	−1.27436			−0.0903370			−0.0451685	−	0.998979i	$-0.514382\pi$
$199$	−1.27436			−0.0903370			−0.0451685	+	0.998979i	$0.514382\pi$
$200$	0.546365	+	2.77516i	0.0386338	+	0.196233i
$201$		−	0.158211i		−	0.0111593i
$202$	−4.90494	−	7.35115i	−0.345110	−	0.517225i
$203$		0			0
$204$	19.0772	+	7.93125i	1.33567	+	0.555298i
$205$	1.46011			0.101979
$206$	−13.4324	+	8.96257i	−0.935881	+	0.624452i
$207$		−	15.9386i		−	1.10781i
$208$	−8.99259	+	8.94575i	−0.623524	+	0.620276i
$209$			4.59023i			0.317513i
$210$		0			0
$211$			3.70986i			0.255397i	0.991813	+	0.127698i	$0.0407590\pi$
$211$			3.70986i			0.255397i	−0.991813	+	0.127698i	$0.959241\pi$
$212$	3.57613	−	8.60177i	0.245610	−	0.590772i
$213$			0.636646i			0.0436222i
$214$	2.04385	+	3.06317i	0.139715	+	0.209394i
$215$	9.95752			0.679097
$216$	4.89600	+	24.8683i	0.333131	+	1.69207i
$217$		0			0
$218$	1.17861	−	0.786412i	0.0798258	−	0.0532625i
$219$			44.5956i			3.01349i
$220$	−1.71719	+	4.13040i	−0.115773	+	0.278472i
$221$	−10.9260			−0.734963
$222$	39.4803	−	26.3426i	2.64974	−	1.76800i
$223$	12.9581			0.867737			0.433869	−	0.900976i	$-0.357148\pi$
$223$	12.9581			0.867737			0.433869	+	0.900976i	$0.357148\pi$
$224$		0			0
$225$	−5.98886			−0.399258
$226$	17.4797	−	11.6631i	1.16273	−	0.775816i
$227$	−8.89253			−0.590218			−0.295109	−	0.955464i	$-0.595356\pi$
$227$	−8.89253			−0.590218			−0.295109	+	0.955464i	$0.595356\pi$
$228$	−4.72433	+	11.3636i	−0.312877	+	0.752570i
$229$			12.9457i			0.855478i	0.903902	+	0.427739i	$0.140690\pi$
$229$			12.9457i			0.855478i	−0.903902	+	0.427739i	$0.859310\pi$
$230$	3.13080	−	2.08898i	0.206439	−	0.137743i
$231$		0			0
$232$	4.03267	+	20.4832i	0.264758	+	1.34479i
$233$	−15.2162			−0.996846			−0.498423	−	0.866934i	$-0.666087\pi$
$233$	−15.2162			−0.996846			−0.498423	+	0.866934i	$0.666087\pi$
$234$	−14.9067	−	22.3411i	−0.974484	−	1.46048i
$235$			6.12686i			0.399672i
$236$	−5.46564	+	13.1466i	−0.355783	+	0.855773i
$237$		−	1.38275i		−	0.0898194i
$238$		0			0
$239$		−	0.0438513i		−	0.00283650i	−0.999999	−	0.00141825i	$-0.999549\pi$
$239$		−	0.0438513i		−	0.00283650i	0.999999	−	0.00141825i	$-0.000451444\pi$
$240$	−8.50215	+	8.45786i	−0.548811	+	0.545952i
$241$			2.30058i			0.148194i	0.997251	+	0.0740968i	$0.0236074\pi$
$241$			2.30058i			0.148194i	−0.997251	+	0.0740968i	$0.976393\pi$
$242$	7.05570	−	4.70781i	0.453557	−	0.302629i
$243$	0.199935			0.0128258
$244$	4.68458	+	1.94759i	0.299900	+	0.124682i
$245$		0			0
$246$	3.43612	+	5.14979i	0.219079	+	0.328339i
$247$		−	6.50820i		−	0.414107i
$248$	2.67281	+	13.5760i	0.169724	+	0.862078i
$249$	32.7318			2.07429
$250$	−0.784927	−	1.17639i	−0.0496432	−	0.0744013i
$251$	−6.32409			−0.399173			−0.199587	−	0.979880i	$-0.563960\pi$
$251$	−6.32409			−0.399173			−0.199587	+	0.979880i	$0.563960\pi$
$252$		0			0
$253$	5.95233			0.374220
$254$	2.37331	+	3.55693i	0.148914	+	0.223181i
$255$	−10.3301			−0.646897
$256$	−0.0835627	+	15.9998i	−0.00522267	+	0.999986i
$257$			14.2414i			0.888351i	0.895940	+	0.444176i	$0.146503\pi$
$257$			14.2414i			0.888351i	−0.895940	+	0.444176i	$0.853497\pi$
$258$	23.4333	+	35.1200i	1.45889	+	2.18648i
$259$		0			0
$260$	2.43470	−	5.85624i	0.150993	−	0.363188i
$261$	−44.2033			−2.73612
$262$	18.4756	−	12.3275i	1.14142	−	0.761598i
$263$		−	11.7484i		−	0.724438i	−0.932093	−	0.362219i	$-0.882019\pi$
$263$		−	11.7484i		−	0.724438i	0.932093	−	0.362219i	$-0.117981\pi$
$264$	−18.6089	+	3.66368i	−1.14530	+	0.225484i
$265$			4.65777i			0.286124i
$266$		0			0
$267$			21.0484i			1.28814i
$268$	0.0974526	+	0.0405153i	0.00595286	+	0.00247487i
$269$		−	8.48931i		−	0.517603i	−0.965931	−	0.258801i	$-0.916673\pi$
$269$		−	8.48931i		−	0.517603i	0.965931	−	0.258801i	$-0.0833275\pi$
$270$	−7.03377	−	10.5417i	−0.428062	−	0.641546i
$271$	−7.97372			−0.484369			−0.242184	−	0.970230i	$-0.577864\pi$
$271$	−7.97372			−0.484369			−0.242184	+	0.970230i	$0.577864\pi$
$272$	−9.77078	+	9.71988i	−0.592440	+	0.589354i
$273$		0			0
$274$	11.3102	−	7.54657i	0.683275	−	0.455905i
$275$		−	2.23657i		−	0.134870i
$276$	14.7356	+	6.12623i	0.886978	+	0.368756i
$277$	−13.5951			−0.816849			−0.408425	−	0.912792i	$-0.633922\pi$
$277$	−13.5951			−0.816849			−0.408425	+	0.912792i	$0.633922\pi$
$278$	8.81991	−	5.88495i	0.528983	−	0.352956i
$279$	−29.2974			−1.75399
$280$		0			0
$281$	9.48286			0.565700			0.282850	−	0.959164i	$-0.408720\pi$
$281$	9.48286			0.565700			0.282850	+	0.959164i	$0.408720\pi$
$282$	−21.6093	+	14.4185i	−1.28682	+	0.858608i
$283$	−21.5492			−1.28097			−0.640483	−	0.767972i	$-0.721266\pi$
$283$	−21.5492			−1.28097			−0.640483	+	0.767972i	$0.721266\pi$
$284$	−0.392153	−	0.163035i	−0.0232700	−	0.00967436i
$285$		−	6.15325i		−	0.364487i
$286$	8.34337	−	5.56699i	0.493354	−	0.329183i
$287$		0			0
$288$	−33.2054	−	6.71772i	−1.95665	−	0.395845i
$289$	5.12849			0.301676
$290$	−5.79348	−	8.68283i	−0.340205	−	0.509873i
$291$		−	0.556031i		−	0.0325951i
$292$	−27.4694	−	11.4203i	−1.60753	−	0.668320i
$293$		−	28.9496i		−	1.69125i	−0.533776	−	0.845626i	$-0.679227\pi$
$293$		−	28.9496i		−	1.69125i	0.533776	−	0.845626i	$-0.320773\pi$
$294$		0			0
$295$		−	7.11876i		−	0.414470i
$296$	6.11589	+	31.0645i	0.355479	+	1.80559i
$297$		−	20.0420i		−	1.16295i
$298$	7.64808	−	5.10306i	0.443041	−	0.295613i
$299$	−8.43944			−0.488066
$300$	2.30191	−	5.53685i	0.132901	−	0.319670i
$301$		0			0
$302$	−18.5829	−	27.8506i	−1.06933	−	1.60262i
$303$			18.7351i			1.07631i
$304$	−5.78975	−	5.82007i	−0.332065	−	0.333804i
$305$	−2.53665			−0.145248
$306$	−16.1967	−	24.2744i	−0.925904	−	1.38767i
$307$	8.00589			0.456920			0.228460	−	0.973553i	$-0.426631\pi$
$307$	8.00589			0.456920			0.228460	+	0.973553i	$0.426631\pi$
$308$		0			0
$309$	34.2339			1.94750
$310$	−3.83985	−	5.75488i	−0.218089	−	0.326855i
$311$	13.7527			0.779841			0.389921	−	0.920848i	$-0.372502\pi$
$311$	13.7527			0.779841			0.389921	+	0.920848i	$0.372502\pi$
$312$	26.3845	−	5.19450i	1.49373	−	0.294081i
$313$		−	10.6358i		−	0.601173i	−0.953754	−	0.300587i	$-0.902818\pi$
$313$		−	10.6358i		−	0.601173i	0.953754	−	0.300587i	$-0.0971825\pi$
$314$	6.14793	+	9.21404i	0.346948	+	0.519979i
$315$		0			0
$316$	0.851730	+	0.354102i	0.0479136	+	0.0199198i
$317$	−16.9835			−0.953888			−0.476944	−	0.878934i	$-0.658256\pi$
$317$	−16.9835			−0.953888			−0.476944	+	0.878934i	$0.658256\pi$
$318$	−16.4279	+	10.9612i	−0.921228	+	0.614676i
$319$		−	16.5079i		−	0.924267i
$320$	−3.03249	−	7.40297i	−0.169522	−	0.413839i
$321$		−	7.80679i		−	0.435732i
$322$		0			0
$323$		−	7.07140i		−	0.393463i
$324$	6.83316	−	16.4360i	0.379620	−	0.913110i
$325$			3.17109i			0.175900i
$326$	−8.51770	−	12.7657i	−0.471752	−	0.707025i
$327$	−3.00381			−0.166111
$328$	−4.05204	+	0.797755i	−0.223737	+	0.0440486i
$329$		0			0
$330$	7.88833	−	5.26337i	0.434239	−	0.289739i
$331$			8.33018i			0.457868i	0.973442	+	0.228934i	$0.0735240\pi$
$331$			8.33018i			0.457868i	−0.973442	+	0.228934i	$0.926476\pi$
$332$	−8.38211	+	20.1617i	−0.460028	+	1.10652i
$333$	−67.0381			−3.67366
$334$	13.8198	−	9.22102i	0.756183	−	0.504552i
$335$	−0.0527695			−0.00288311
$336$		0			0
$337$	−27.0772			−1.47499			−0.737495	−	0.675353i	$-0.763991\pi$
$337$	−27.0772			−1.47499			−0.737495	+	0.675353i	$0.763991\pi$
$338$	3.46350	−	2.31097i	0.188390	−	0.125700i
$339$	−44.5488			−2.41956
$340$	2.64539	−	6.36302i	0.143466	−	0.345083i
$341$		−	10.9413i		−	0.592503i
$342$	14.4593	−	9.64775i	0.781870	−	0.521690i
$343$		0			0
$344$	−27.6337	+	5.44044i	−1.48991	+	0.293329i
$345$	−7.97917			−0.429584
$346$	12.9414	+	19.3956i	0.695735	+	1.04271i
$347$			27.7721i			1.49089i	0.666569	+	0.745443i	$0.267762\pi$
$347$			27.7721i			1.49089i	−0.666569	+	0.745443i	$0.732238\pi$
$348$	16.9902	−	40.8670i	0.910771	−	2.19070i
$349$		−	9.64063i		−	0.516051i	−0.966138	−	0.258026i	$-0.916928\pi$
$349$		−	9.64063i		−	0.516051i	0.966138	−	0.258026i	$-0.0830719\pi$
$350$		0			0
$351$			28.4163i			1.51675i
$352$	2.50876	−	12.4007i	0.133718	−	0.660960i
$353$		−	16.3139i		−	0.868304i	−0.900840	−	0.434152i	$-0.857048\pi$
$353$		−	16.3139i		−	0.868304i	0.900840	−	0.434152i	$-0.142952\pi$
$354$	25.1077	−	16.7528i	1.33446	−	0.890399i
$355$	0.212347			0.0112702
$356$	−12.9651	−	5.39018i	−0.687151	−	0.285679i
$357$		0			0
$358$	7.15037	+	10.7164i	0.377909	+	0.566380i
$359$			1.60208i			0.0845545i	0.999106	+	0.0422773i	$0.0134613\pi$
$359$			1.60208i			0.0845545i	−0.999106	+	0.0422773i	$0.986539\pi$
$360$	16.6200	−	3.27210i	0.875952	−	0.172455i
$361$	−14.7878			−0.778308
$362$	13.0106	+	19.4992i	0.683819	+	1.02486i
$363$	−17.9822			−0.943818
$364$		0			0
$365$	14.8744			0.778562
$366$	−5.96957	−	8.94673i	−0.312034	−	0.467653i
$367$	−1.26052			−0.0657985			−0.0328993	−	0.999459i	$-0.510474\pi$
$367$	−1.26052			−0.0657985			−0.0328993	+	0.999459i	$0.510474\pi$
$368$	−7.54712	+	7.50781i	−0.393421	+	0.391371i
$369$		−	8.74443i		−	0.455217i
$370$	−8.78631	−	13.1682i	−0.456778	−	0.684584i
$371$		0			0
$372$	11.2609	−	27.0862i	0.583851	−	1.40435i
$373$	9.22744			0.477779			0.238889	−	0.971047i	$-0.423217\pi$
$373$	9.22744			0.477779			0.238889	+	0.971047i	$0.423217\pi$
$374$	9.06538	−	6.04873i	0.468759	−	0.312773i
$375$			2.99814i			0.154823i
$376$	−3.34750	−	17.0030i	−0.172634	−	0.876862i
$377$			23.4056i			1.20545i
$378$		0			0
$379$			2.53516i			0.130223i	0.997878	+	0.0651113i	$0.0207403\pi$
$379$			2.53516i			0.130223i	−0.997878	+	0.0651113i	$0.979260\pi$
$380$	3.79020	+	1.57575i	0.194433	+	0.0808344i
$381$		−	9.06518i		−	0.464423i
$382$	−2.34925	−	3.52087i	−0.120198	−	0.180144i
$383$	−1.32487			−0.0676977			−0.0338489	−	0.999427i	$-0.510776\pi$
$383$	−1.32487			−0.0676977			−0.0338489	+	0.999427i	$0.510776\pi$
$384$	18.9737	−	28.1171i	0.968248	−	1.43485i
$385$		0			0
$386$	−17.3033	+	11.5454i	−0.880715	+	0.587644i
$387$		−	59.6343i		−	3.03138i
$388$	0.342497	+	0.142391i	0.0173877	+	0.00722882i
$389$	32.2269			1.63397			0.816983	−	0.576662i	$-0.195645\pi$
$389$	32.2269			1.63397			0.816983	+	0.576662i	$0.195645\pi$
$390$	−11.1844	+	7.46261i	−0.566343	+	0.377884i
$391$	−9.16976			−0.463735
$392$		0			0
$393$	−47.0869			−2.37522
$394$	5.66723	−	3.78137i	0.285511	−	0.190503i
$395$	−0.461203			−0.0232056
$396$	24.7364	+	10.2840i	1.24305	+	0.516791i
$397$			35.1742i			1.76534i	0.469993	+	0.882670i	$0.344257\pi$
$397$			35.1742i			1.76534i	−0.469993	+	0.882670i	$0.655743\pi$
$398$	−1.49914	+	1.00028i	−0.0751452	+	0.0501395i
$399$		0			0
$400$	2.82103	+	2.83580i	0.141052	+	0.141790i
$401$	31.9246			1.59424			0.797120	−	0.603820i	$-0.206356\pi$
$401$	31.9246			1.59424			0.797120	+	0.603820i	$0.206356\pi$
$402$	−0.124184	−	0.186117i	−0.00619373	−	0.00928268i
$403$			15.5129i			0.772754i
$404$	−11.5402	−	4.79778i	−0.574148	−	0.238699i
$405$			8.89991i			0.442240i
$406$		0			0
$407$		−	25.0357i		−	1.24097i
$408$	28.6677	−	5.64401i	1.41926	−	0.279420i
$409$			16.4142i			0.811632i	0.913955	+	0.405816i	$0.133013\pi$
$409$			16.4142i			0.811632i	−0.913955	+	0.405816i	$0.866987\pi$
$410$	1.71766	−	1.14608i	0.0848292	−	0.0566010i
$411$	−28.8252			−1.42184
$412$	−8.76677	+	21.0869i	−0.431908	+	1.03888i
$413$		0			0
$414$	−12.5106	−	18.7500i	−0.614864	−	0.921510i
$415$		−	10.9174i		−	0.535912i
$416$	−3.55702	+	17.5822i	−0.174397	+	0.862038i
$417$	−22.4784			−1.10077
$418$	3.60299	+	5.39989i	0.176228	+	0.264117i
$419$	11.8654			0.579665			0.289832	−	0.957077i	$-0.406400\pi$
$419$	11.8654			0.579665			0.289832	+	0.957077i	$0.406400\pi$
$420$		0			0
$421$	10.3433			0.504101			0.252051	−	0.967714i	$-0.418895\pi$
$421$	10.3433			0.504101			0.252051	+	0.967714i	$0.418895\pi$
$422$	2.91197	+	4.36423i	0.141752	+	0.212447i
$423$	36.6929			1.78407
$424$	−2.54484	−	12.9260i	−0.123588	−	0.627743i
$425$			3.44551i			0.167132i
$426$	0.499720	+	0.748942i	0.0242115	+	0.0362864i
$427$		0			0
$428$	4.80873	+	1.99920i	0.232439	+	0.0966349i
$429$	−21.2639			−1.02663
$430$	11.7139	−	7.81593i	0.564895	−	0.376918i
$431$			29.6571i			1.42853i	0.699874	+	0.714267i	$0.253240\pi$
$431$			29.6571i			1.42853i	−0.699874	+	0.714267i	$0.746760\pi$
$432$	25.2794	+	25.4118i	1.21626	+	1.22262i
$433$			29.4107i			1.41339i	0.707520	+	0.706693i	$0.249814\pi$
$433$			29.4107i			1.41339i	−0.707520	+	0.706693i	$0.750186\pi$
$434$		0			0
$435$			22.1291i			1.06101i
$436$	0.769231	−	1.85025i	0.0368395	−	0.0886110i
$437$		−	5.46207i		−	0.261286i
$438$	35.0043	+	52.4617i	1.67257	+	2.50672i
$439$	8.82381			0.421138			0.210569	−	0.977579i	$-0.432468\pi$
$439$	8.82381			0.421138			0.210569	+	0.977579i	$0.432468\pi$
$440$	1.22198	+	6.20682i	0.0582557	+	0.295899i
$441$		0			0
$442$	−12.8532	+	8.57612i	−0.611366	+	0.407925i
$443$		−	19.5668i		−	0.929649i	−0.885403	−	0.464824i	$-0.846117\pi$
$443$		−	19.5668i		−	0.929649i	0.885403	−	0.464824i	$-0.153883\pi$
$444$	25.7671	−	61.9783i	1.22285	−	2.94136i
$445$	7.02049			0.332803
$446$	15.2437	−	10.1711i	0.721812	−	0.481618i
$447$	−19.4919			−0.921935
$448$		0			0
$449$	5.02309			0.237054			0.118527	−	0.992951i	$-0.462183\pi$
$449$	5.02309			0.237054			0.118527	+	0.992951i	$0.462183\pi$
$450$	−7.04523	+	4.70082i	−0.332115	+	0.221599i
$451$	3.26564			0.153773
$452$	11.4083	−	27.4406i	0.536600	−	1.29070i
$453$			70.9801i			3.33493i
$454$	−10.4611	+	6.97998i	−0.490962	+	0.327587i
$455$		0			0
$456$	3.36192	+	17.0762i	0.157436	+	0.799667i
$457$	−27.8449			−1.30253			−0.651265	−	0.758851i	$-0.725761\pi$
$457$	−27.8449			−1.30253			−0.651265	+	0.758851i	$0.725761\pi$
$458$	10.1615	+	15.2292i	0.474814	+	0.711614i
$459$			30.8753i			1.44114i
$460$	2.04334	−	4.91490i	0.0952713	−	0.229159i
$461$		−	19.8494i		−	0.924481i	−0.886755	−	0.462240i	$-0.847046\pi$
$461$		−	19.8494i		−	0.924481i	0.886755	−	0.462240i	$-0.152954\pi$
$462$		0			0
$463$			35.7118i			1.65967i	0.558012	+	0.829833i	$0.311564\pi$
$463$			35.7118i			1.65967i	−0.558012	+	0.829833i	$0.688436\pi$
$464$	20.8218	+	20.9308i	0.966628	+	0.971690i
$465$			14.6669i			0.680160i
$466$	−17.9001	+	11.9436i	−0.829208	+	0.553277i
$467$	22.6111			1.04632			0.523158	−	0.852236i	$-0.324754\pi$
$467$	22.6111			1.04632			0.523158	+	0.852236i	$0.324754\pi$
$468$	−35.0722	−	14.5811i	−1.62121	−	0.674010i
$469$		0			0
$470$	4.80914	+	7.20756i	0.221829	+	0.332460i
$471$		−	23.4829i		−	1.08204i
$472$	3.88944	+	19.7557i	0.179026	+	0.909329i
$473$	22.2707			1.02401
$474$	−1.08536	−	1.62665i	−0.0498522	−	0.0747147i
$475$	−2.05235			−0.0941684
$476$		0			0
$477$	27.8947			1.27721
$478$	−0.0344201	−	0.0515861i	−0.00157434	−	0.00235949i
$479$	−8.56399			−0.391299			−0.195649	−	0.980674i	$-0.562681\pi$
$479$	−8.56399			−0.391299			−0.195649	+	0.980674i	$0.562681\pi$
$480$	−3.36302	+	16.6233i	−0.153500	+	0.758746i
$481$			35.4965i			1.61850i
$482$	1.80579	+	2.70638i	0.0822515	+	0.123272i
$483$		0			0
$484$	4.60495	−	11.0764i	0.209316	−	0.503473i
$485$	−0.185459			−0.00842124
$486$	0.235201	−	0.156934i	0.0106689	−	0.00711869i
$487$		−	31.6451i		−	1.43397i	−0.697086	−	0.716987i	$-0.745521\pi$
$487$		−	31.6451i		−	1.43397i	0.697086	−	0.716987i	$-0.254479\pi$
$488$	7.03961	−	1.38594i	0.318668	−	0.0627384i
$489$			32.5346i			1.47126i
$490$		0			0
$491$		−	35.7781i		−	1.61464i	−0.590113	−	0.807321i	$-0.700917\pi$
$491$		−	35.7781i		−	1.61464i	0.590113	−	0.807321i	$-0.299083\pi$
$492$	8.08443	+	3.36105i	0.364474	+	0.151528i
$493$			25.4310i			1.14535i
$494$	−5.10846	−	7.65617i	−0.229841	−	0.344467i
$495$	−13.3945			−0.602038
$496$	13.8004	+	13.8727i	0.619658	+	0.622903i
$497$		0			0
$498$	38.5053	−	25.6921i	1.72546	−	1.15129i
$499$		−	41.3148i		−	1.84950i	−0.380569	−	0.924752i	$-0.624272\pi$
$499$		−	41.3148i		−	1.84950i	0.380569	−	0.924752i	$-0.375728\pi$
$500$	−1.84676	−	0.767779i	−0.0825895	−	0.0343361i
$501$	−35.2210			−1.57356
$502$	−7.43959	+	4.96395i	−0.332045	+	0.221552i
$503$	29.0170			1.29381			0.646903	−	0.762572i	$-0.276064\pi$
$503$	29.0170			1.29381			0.646903	+	0.762572i	$0.276064\pi$
$504$		0			0
$505$	6.24891			0.278073
$506$	7.00226	−	4.67215i	0.311288	−	0.207702i
$507$	−8.82708			−0.392024
$508$	5.58386	+	2.32146i	0.247744	+	0.102998i
$509$		−	20.0311i		−	0.887861i	−0.896061	−	0.443931i	$-0.853584\pi$
$509$		−	20.0311i		−	0.887861i	0.896061	−	0.443931i	$-0.146416\pi$
$510$	−12.1522	+	8.10839i	−0.538110	+	0.359046i
$511$		0			0
$512$	12.4604	+	18.8875i	0.550675	+	0.834720i
$513$	−18.3912			−0.811993
$514$	11.1784	+	16.7534i	0.493059	+	0.738959i
$515$		−	11.4184i		−	0.503153i
$516$	55.1333	+	22.9213i	2.42711	+	1.00906i
$517$			13.7031i			0.602663i
$518$		0			0
$519$		−	49.4317i		−	2.16981i
$520$	−1.73257	−	8.80027i	−0.0759783	−	0.385917i
$521$			21.4442i			0.939487i	0.882803	+	0.469743i	$0.155654\pi$
$521$			21.4442i			0.939487i	−0.882803	+	0.469743i	$0.844346\pi$
$522$	−52.0003	+	34.6964i	−2.27599	+	1.51862i
$523$	−41.0040			−1.79298			−0.896491	−	0.443063i	$-0.853892\pi$
$523$	−41.0040			−1.79298			−0.896491	+	0.443063i	$0.853892\pi$
$524$	12.0582	−	29.0040i	0.526766	−	1.26704i
$525$		0			0
$526$	−9.22165	−	13.8207i	−0.402083	−	0.602611i
$527$			16.8554i			0.734231i
$528$	−19.0156	+	18.9166i	−0.827549	+	0.823238i
$529$	15.9171			0.692049
$530$	3.65601	+	5.47934i	0.158807	+	0.238007i
$531$	−42.6333			−1.85013
$532$		0			0
$533$	−4.63015			−0.200554
$534$	16.5215	+	24.7611i	0.714954	+	1.07152i
$535$	−2.60387			−0.112575
$536$	0.146444	−	0.0288314i	0.00632540	−	0.00124533i
$537$		−	27.3119i		−	1.17859i
$538$	−6.66349	−	9.98673i	−0.287284	−	0.430559i
$539$		0			0
$540$	−16.5489	−	6.88010i	−0.712151	−	0.296072i
$541$	3.45282			0.148448			0.0742242	−	0.997242i	$-0.476352\pi$
$541$	3.45282			0.148448			0.0742242	+	0.997242i	$0.476352\pi$
$542$	−9.38019	+	6.25879i	−0.402914	+	0.268838i
$543$		−	49.6957i		−	2.13265i
$544$	−3.86483	+	19.1037i	−0.165703	+	0.819064i
$545$			1.00189i			0.0429163i
$546$		0			0
$547$		−	28.2607i		−	1.20834i	−0.796855	−	0.604170i	$-0.793505\pi$
$547$		−	28.2607i		−	1.20834i	0.796855	−	0.604170i	$-0.206495\pi$
$548$	7.38170	−	17.7554i	0.315331	−	0.758473i
$549$			15.1917i			0.648365i
$550$	−1.75554	−	2.63107i	−0.0748566	−	0.112189i
$551$	−15.1483			−0.645337
$552$	22.1434	−	4.35953i	0.942487	−	0.185554i
$553$		0			0
$554$	−15.9931	+	10.6711i	−0.679481	+	0.453374i
$555$			33.5606i			1.42457i
$556$	5.75638	−	13.8460i	0.244125	−	0.587200i
$557$	2.07653			0.0879854			0.0439927	−	0.999032i	$-0.485992\pi$
$557$	2.07653			0.0879854			0.0439927	+	0.999032i	$0.485992\pi$
$558$	−34.4652	+	22.9964i	−1.45903	+	0.973514i
$559$	−31.5762			−1.33553
$560$		0			0
$561$	−23.1040			−0.975453
$562$	11.1555	−	7.44335i	0.470567	−	0.313979i
$563$	45.9095			1.93485			0.967426	−	0.253152i	$-0.0814673\pi$
$563$	45.9095			1.93485			0.967426	+	0.253152i	$0.0814673\pi$
$564$	−14.1035	+	33.9235i	−0.593864	+	1.42844i
$565$			14.8588i			0.625115i
$566$	−25.3502	+	16.9145i	−1.06555	+	0.710971i
$567$		0			0
$568$	−0.589295	+	0.116019i	−0.0247263	+	0.00486803i
$569$	7.92825			0.332370			0.166185	−	0.986095i	$-0.446855\pi$
$569$	7.92825			0.332370			0.166185	+	0.986095i	$0.446855\pi$
$570$	−4.82985	−	7.23861i	−0.202300	−	0.303192i
$571$		−	21.1673i		−	0.885825i	−0.896565	−	0.442912i	$-0.853945\pi$
$571$		−	21.1673i		−	0.885825i	0.896565	−	0.442912i	$-0.146055\pi$
$572$	5.44536	−	13.0979i	0.227682	−	0.547650i
$573$			8.97330i			0.374865i
$574$		0			0
$575$			2.66137i			0.110987i
$576$	−44.3354	+	18.1612i	−1.84731	+	0.756716i
$577$			10.3315i			0.430104i	0.976603	+	0.215052i	$0.0689922\pi$
$577$			10.3315i			0.430104i	−0.976603	+	0.215052i	$0.931008\pi$
$578$	6.03309	−	4.02549i	0.250944	−	0.167438i
$579$	44.0992			1.83270
$580$	−13.6308	−	5.66691i	−0.565987	−	0.235306i
$581$		0			0
$582$	−0.436444	−	0.654109i	−0.0180912	−	0.0271137i
$583$			10.4174i			0.431445i
$584$	−41.2788	+	8.12685i	−1.70813	+	0.336291i
$585$	18.9912			0.785191
$586$	−22.7233	−	34.0559i	−0.938691	−	1.40684i
$587$	20.4660			0.844722			0.422361	−	0.906428i	$-0.361201\pi$
$587$	20.4660			0.844722			0.422361	+	0.906428i	$0.361201\pi$
$588$		0			0
$589$	−10.0401			−0.413694
$590$	−5.58771	−	8.37443i	−0.230042	−	0.344770i
$591$	−14.4435			−0.594126
$592$	31.5780	+	31.7434i	1.29785	+	1.30464i
$593$		−	44.8473i		−	1.84166i	−0.389968	−	0.920828i	$-0.627514\pi$
$593$		−	44.8473i		−	1.84166i	0.389968	−	0.920828i	$-0.372486\pi$
$594$	−15.7315	−	23.5772i	−0.645471	−	0.967383i
$595$		0			0
$596$	4.99158	−	12.0064i	0.204463	−	0.491800i
$597$	3.82072			0.156371
$598$	−9.92806	+	6.62435i	−0.405989	+	0.270890i
$599$			16.3388i			0.667587i	0.942646	+	0.333793i	$0.108329\pi$
$599$			16.3388i			0.667587i	−0.942646	+	0.333793i	$0.891671\pi$
$600$	−1.63808	−	8.32031i	−0.0668743	−	0.339675i
$601$		−	39.8029i		−	1.62359i	−0.583941	−	0.811796i	$-0.698490\pi$
$601$		−	39.8029i		−	1.62359i	0.583941	−	0.811796i	$-0.301510\pi$
$602$		0			0
$603$			0.316030i			0.0128697i
$604$	−43.7214	−	18.1769i	−1.77900	−	0.739608i
$605$			5.99776i			0.243844i
$606$	14.7057	+	22.0398i	0.597379	+	0.895305i
$607$	9.40746			0.381837			0.190919	−	0.981606i	$-0.438853\pi$
$607$	9.40746			0.381837			0.190919	+	0.981606i	$0.438853\pi$
$608$	−11.3793	−	2.30213i	−0.461493	−	0.0933636i
$609$		0			0
$610$	−2.98409	+	1.99109i	−0.120822	+	0.0806168i
$611$		−	19.4288i		−	0.786006i
$612$	−38.1072	−	15.8429i	−1.54039	−	0.640410i
$613$	13.6359			0.550750			0.275375	−	0.961337i	$-0.411198\pi$
$613$	13.6359			0.550750			0.275375	+	0.961337i	$0.411198\pi$
$614$	9.41803	−	6.28404i	0.380081	−	0.253603i
$615$	−4.37763			−0.176523
$616$		0			0
$617$	39.1144			1.57469			0.787343	−	0.616515i	$-0.211456\pi$
$617$	39.1144			1.57469			0.787343	+	0.616515i	$0.211456\pi$
$618$	40.2723	−	26.8711i	1.61999	−	1.08091i
$619$	19.0190			0.764438			0.382219	−	0.924072i	$-0.375160\pi$
$619$	19.0190			0.764438			0.382219	+	0.924072i	$0.375160\pi$
$620$	−9.03432	−	3.75596i	−0.362827	−	0.150843i
$621$			23.8487i			0.957013i
$622$	16.1785	−	10.7948i	0.648697	−	0.432833i
$623$		0			0
$624$	26.9611	−	26.8206i	1.07931	−	1.07368i
$625$	1.00000			0.0400000
$626$	−8.34836	−	12.5119i	−0.333668	−	0.500075i
$627$		−	13.7622i		−	0.549608i
$628$	14.4647	+	6.01362i	0.577205	+	0.239969i
$629$			38.5683i			1.53782i
$630$		0			0
$631$			16.4987i			0.656802i	0.944538	+	0.328401i	$0.106510\pi$
$631$			16.4987i			0.656802i	−0.944538	+	0.328401i	$0.893490\pi$
$632$	1.27991	−	0.251985i	0.0509121	−	0.0100234i
$633$		−	11.1227i		−	0.442087i
$634$	−19.9792	+	13.3308i	−0.793475	+	0.529434i
$635$	−3.02360			−0.119988
$636$	−10.7218	+	25.7893i	−0.425146	+	1.02261i
$637$		0			0
$638$	−12.9575	−	19.4197i	−0.512993	−	0.768835i
$639$		−	1.27171i		−	0.0503083i
$640$	−9.37818	−	6.32848i	−0.370705	−	0.250155i
$641$	26.9446			1.06425			0.532124	−	0.846667i	$-0.321394\pi$
$641$	26.9446			1.06425			0.532124	+	0.846667i	$0.321394\pi$
$642$	−6.12776	−	9.18381i	−0.241843	−	0.362456i
$643$	43.6730			1.72229			0.861147	−	0.508355i	$-0.169746\pi$
$643$	43.6730			1.72229			0.861147	+	0.508355i	$0.169746\pi$
$644$		0			0
$645$	−29.8541			−1.17550
$646$	−5.55053	−	8.31871i	−0.218383	−	0.327295i
$647$	28.1079			1.10504			0.552518	−	0.833501i	$-0.313667\pi$
$647$	28.1079			1.10504			0.552518	+	0.833501i	$0.313667\pi$
$648$	−4.86260	−	24.6986i	−0.191021	−	0.970254i
$649$		−	15.9216i		−	0.624978i
$650$	2.48908	+	3.73043i	0.0976296	+	0.146320i
$651$		0			0
$652$	−20.0402	−	8.33161i	−0.784836	−	0.326291i
$653$	−41.6734			−1.63081			−0.815403	−	0.578893i	$-0.803485\pi$
$653$	−41.6734			−1.63081			−0.815403	+	0.578893i	$0.803485\pi$
$654$	−3.53365	+	2.35778i	−0.138177	+	0.0921963i
$655$			15.7053i			0.613658i
$656$	−4.14060	+	4.11903i	−0.161663	+	0.160821i
$657$		−	89.0808i		−	3.47537i
$658$		0			0
$659$			47.0951i			1.83457i	0.398236	+	0.917283i	$0.369622\pi$
$659$			47.0951i			1.83457i	−0.398236	+	0.917283i	$0.630378\pi$
$660$	5.14838	−	12.3835i	0.200400	−	0.482028i
$661$		−	11.6385i		−	0.452685i	−0.974048	−	0.226342i	$-0.927323\pi$
$661$		−	11.6385i		−	0.452685i	0.974048	−	0.226342i	$-0.0726768\pi$
$662$	6.53858	+	9.79952i	0.254129	+	0.380869i
$663$	32.7578			1.27221
$664$	5.96486	+	30.2973i	0.231481	+	1.17577i
$665$		0			0
$666$	−78.8628	+	52.6200i	−3.05587	+	2.03898i
$667$			19.6434i			0.760594i
$668$	9.01957	−	21.6950i	0.348978	−	0.839405i
$669$	−38.8502			−1.50203
$670$	−0.0620775	+	0.0414202i	−0.00239826	+	0.00160020i
$671$	−5.67340			−0.219019
$672$		0			0
$673$	14.9849			0.577626			0.288813	−	0.957385i	$-0.406739\pi$
$673$	14.9849			0.577626			0.288813	+	0.957385i	$0.406739\pi$
$674$	−31.8533	+	21.2536i	−1.22694	+	0.818659i
$675$	8.96105			0.344911
$676$	2.26048	−	5.43719i	0.0869416	−	0.209123i
$677$		−	23.0186i		−	0.884678i	−0.896848	−	0.442339i	$-0.854149\pi$
$677$		−	23.0186i		−	0.884678i	0.896848	−	0.442339i	$-0.145851\pi$
$678$	−52.4067	+	34.9676i	−2.01267	+	1.34292i
$679$		0			0
$680$	−1.88250	−	9.56181i	−0.0721907	−	0.366679i
$681$	26.6611			1.02165
$682$	−8.58809	−	12.8712i	−0.328855	−	0.492863i
$683$		−	37.2694i		−	1.42607i	−0.701126	−	0.713037i	$-0.747319\pi$
$683$		−	37.2694i		−	1.42607i	0.701126	−	0.713037i	$-0.252681\pi$
$684$	9.43697	−	22.6990i	0.360832	−	0.867918i
$685$			9.61436i			0.367346i
$686$		0			0
$687$		−	38.8132i		−	1.48082i
$688$	−28.2376	+	28.0905i	−1.07655	+	1.07094i
$689$		−	14.7702i		−	0.562700i
$690$	−9.38660	+	6.26306i	−0.357342	+	0.238431i
$691$	−27.9938			−1.06493			−0.532467	−	0.846451i	$-0.678735\pi$
$691$	−27.9938			−1.06493			−0.532467	+	0.846451i	$0.678735\pi$
$692$	30.4483	+	12.6587i	1.15747	+	0.481211i
$693$		0			0
$694$	21.7991	+	32.6708i	0.827483	+	1.24017i
$695$			7.49745i			0.284394i
$696$	−12.0905	−	61.4116i	−0.458291	−	2.32780i
$697$	−5.03083			−0.190556
$698$	−7.56719	−	11.3411i	−0.286423	−	0.429268i
$699$	45.6203			1.72552
$700$		0			0
$701$	29.2334			1.10413			0.552065	−	0.833801i	$-0.313840\pi$
$701$	29.2334			1.10413			0.552065	+	0.833801i	$0.313840\pi$
$702$	22.3047	+	33.4286i	0.841837	+	1.26168i
$703$	−22.9736			−0.866466
$704$	−6.78238	−	16.5573i	−0.255621	−	0.624025i
$705$		−	18.3692i		−	0.691824i
$706$	−12.8053	−	19.1915i	−0.481932	−	0.722283i
$707$		0			0
$708$	16.3868	−	39.4155i	0.615852	−	1.48133i
$709$	−4.16149			−0.156288			−0.0781440	−	0.996942i	$-0.524899\pi$
$709$	−4.16149			−0.156288			−0.0781440	+	0.996942i	$0.524899\pi$
$710$	0.249802	−	0.166677i	0.00937490	−	0.00625526i
$711$			2.76208i			0.103586i
$712$	−19.4829	+	3.83575i	−0.730154	+	0.143751i
$713$			13.0194i			0.487580i
$714$		0			0
$715$			7.09236i			0.265239i
$716$	16.8232	+	6.99415i	0.628713	+	0.261384i
$717$			0.131472i			0.00490993i
$718$	1.25752	+	1.88467i	0.0469301	+	0.0703352i
$719$	−42.2226			−1.57464			−0.787318	−	0.616547i	$-0.788531\pi$
$719$	−42.2226			−1.57464			−0.787318	+	0.616547i	$0.788531\pi$
$720$	16.9832	−	16.8948i	0.632928	−	0.629631i
$721$		0			0
$722$	−17.3962	+	11.6074i	−0.647421	+	0.431982i
$723$		−	6.89748i		−	0.256520i
$724$	30.6109	+	12.7263i	1.13765	+	0.472970i
$725$	7.38092			0.274120
$726$	−21.1540	+	14.1147i	−0.785098	+	0.523845i
$727$	−27.2605			−1.01104			−0.505519	−	0.862816i	$-0.668699\pi$
$727$	−27.2605			−1.01104			−0.505519	+	0.862816i	$0.668699\pi$
$728$		0			0
$729$	−27.2992			−1.01108
$730$	17.4981	−	11.6753i	0.647633	−	0.432123i
$731$	−34.3087			−1.26895
$732$	−14.0451	−	5.83915i	−0.519120	−	0.215821i
$733$		−	15.3060i		−	0.565340i	−0.959217	−	0.282670i	$-0.908780\pi$
$733$		−	15.3060i		−	0.565340i	0.959217	−	0.282670i	$-0.0912201\pi$
$734$	−1.48286	+	0.989415i	−0.0547333	+	0.0365200i
$735$		0			0
$736$	−2.98526	+	14.7560i	−0.110038	+	0.543915i
$737$	−0.118023			−0.00434742
$738$	−6.86374	−	10.2868i	−0.252658	−	0.378664i
$739$		−	49.1549i		−	1.80819i	−0.427330	−	0.904096i	$-0.640546\pi$
$739$		−	49.1549i		−	1.80819i	0.427330	−	0.904096i	$-0.359454\pi$
$740$	−20.6722	−	8.59435i	−0.759926	−	0.315935i
$741$			19.5125i			0.716810i
$742$		0			0
$743$			35.2067i			1.29161i	0.763503	+	0.645805i	$0.223478\pi$
$743$			35.2067i			1.29161i	−0.763503	+	0.645805i	$0.776522\pi$
$744$	−8.01346	−	40.7029i	−0.293788	−	1.49224i
$745$			6.50132i			0.238190i
$746$	10.8551	−	7.24287i	0.397432	−	0.265180i
$747$	−65.3825			−2.39222
$748$	5.91659	−	14.2313i	0.216332	−	0.520349i
$749$		0			0
$750$	2.35332	+	3.52698i	0.0859313	+	0.128787i
$751$			0.674682i			0.0246195i	0.999924	+	0.0123098i	$0.00391841\pi$
$751$			0.674682i			0.0246195i	−0.999924	+	0.0123098i	$0.996082\pi$
$752$	−17.2841	−	17.3746i	−0.630285	−	0.633585i
$753$	18.9605			0.690960
$754$	18.3717	+	27.5340i	0.669057	+	1.00273i
$755$	23.6747			0.861609
$756$		0			0
$757$	−45.8640			−1.66695			−0.833477	−	0.552553i	$-0.813654\pi$
$757$	−45.8640			−1.66695			−0.833477	+	0.552553i	$0.813654\pi$
$758$	1.98992	+	2.98234i	0.0722771	+	0.108323i
$759$	−17.8460			−0.647767
$760$	5.69560	−	1.12133i	0.206601	−	0.0406750i
$761$		−	21.8503i		−	0.792074i	−0.918235	−	0.396037i	$-0.870385\pi$
$761$		−	21.8503i		−	0.792074i	0.918235	−	0.396037i	$-0.129615\pi$
$762$	−7.11551	−	10.6642i	−0.257768	−	0.386322i
$763$		0			0
$764$	−5.52726	−	2.29793i	−0.199969	−	0.0831360i
$765$	20.6347			0.746048
$766$	−1.55856	+	1.03993i	−0.0563131	+	0.0375741i
$767$			22.5742i			0.815109i
$768$	0.250533	−	47.9696i	0.00904033	−	1.73096i
$769$			16.0214i			0.577745i	0.957368	+	0.288872i	$0.0932803\pi$
$769$			16.0214i			0.577745i	−0.957368	+	0.288872i	$0.906720\pi$
$770$		0			0
$771$		−	42.6976i		−	1.53772i
$772$	−11.2931	+	27.1637i	−0.406449	+	0.977642i
$773$		−	42.3415i		−	1.52292i	−0.648214	−	0.761458i	$-0.724484\pi$
$773$		−	42.3415i		−	1.52292i	0.648214	−	0.761458i	$-0.275516\pi$
$774$	−46.8085	−	70.1530i	−1.68250	−	2.52160i
$775$	4.89199			0.175725
$776$	0.514676	−	0.101328i	0.0184758	−	0.00363746i
$777$		0			0
$778$	37.9113	−	25.2957i	1.35919	−	0.906896i
$779$		−	2.99667i		−	0.107367i
$780$	−7.29957	+	17.5578i	−0.261367	+	0.628672i
$781$	0.474928			0.0169942
$782$	−10.7872	+	7.19760i	−0.385750	+	0.257385i
$783$	66.1408			2.36368
$784$		0

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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 980 = 2^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	980.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(1.17639 - 0.784927i) q^{2} -2.99814 q^{3} +(0.767779 - 1.84676i) q^{4} +1.00000i q^{5} +(-3.52698 + 2.35332i) q^{6} +(-0.546365 - 2.77516i) q^{8} +5.98886 q^{9} +(0.784927 + 1.17639i) q^{10} +2.23657i q^{11} +(-2.30191 + 5.53685i) q^{12} -3.17109i q^{13} -2.99814i q^{15} +(-2.82103 - 2.83580i) q^{16} -3.44551i q^{17} +(7.04523 - 4.70082i) q^{18} +2.05235 q^{19} +(1.84676 + 0.767779i) q^{20} +(1.75554 + 2.63107i) q^{22} -2.66137i q^{23} +(1.63808 + 8.32031i) q^{24} -1.00000 q^{25} +(-2.48908 - 3.73043i) q^{26} -8.96105 q^{27} -7.38092 q^{29} +(-2.35332 - 3.52698i) q^{30} -4.89199 q^{31} +(-5.54453 - 1.12170i) q^{32} -6.70555i q^{33} +(-2.70447 - 4.05325i) q^{34} +(4.59812 - 11.0600i) q^{36} -11.1938 q^{37} +(2.41436 - 1.61095i) q^{38} +9.50739i q^{39} +(2.77516 - 0.546365i) q^{40} -1.46011i q^{41} -9.95752i q^{43} +(4.13040 + 1.71719i) q^{44} +5.98886i q^{45} +(-2.08898 - 3.13080i) q^{46} +6.12686 q^{47} +(8.45786 + 8.50215i) q^{48} +(-1.17639 + 0.784927i) q^{50} +10.3301i q^{51} +(-5.85624 - 2.43470i) q^{52} +4.65777 q^{53} +(-10.5417 + 7.03377i) q^{54} -2.23657 q^{55} -6.15325 q^{57} +(-8.68283 + 5.79348i) q^{58} -7.11876 q^{59} +(-5.53685 - 2.30191i) q^{60} +2.53665i q^{61} +(-5.75488 + 3.83985i) q^{62} +(-7.40297 + 3.03249i) q^{64} +3.17109 q^{65} +(-5.26337 - 7.88833i) q^{66} +0.0527695i q^{67} +(-6.36302 - 2.64539i) q^{68} +7.97917i q^{69} -0.212347i q^{71} +(-3.27210 - 16.6200i) q^{72} -14.8744i q^{73} +(-13.1682 + 8.78631i) q^{74} +2.99814 q^{75} +(1.57575 - 3.79020i) q^{76} +(7.46261 + 11.1844i) q^{78} +0.461203i q^{79} +(2.83580 - 2.82103i) q^{80} +8.89991 q^{81} +(-1.14608 - 1.71766i) q^{82} -10.9174 q^{83} +3.44551 q^{85} +(-7.81593 - 11.7139i) q^{86} +22.1291 q^{87} +(6.20682 - 1.22198i) q^{88} -7.02049i q^{89} +(4.70082 + 7.04523i) q^{90} +(-4.91490 - 2.04334i) q^{92} +14.6669 q^{93} +(7.20756 - 4.80914i) q^{94} +2.05235i q^{95} +(16.6233 + 3.36302i) q^{96} +0.185459i q^{97} +13.3945i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 4 q^{2} + 4 q^{4} - 4 q^{8} + 32 q^{9} + 28 q^{16} - 8 q^{22} - 32 q^{25} - 40 q^{29} - 4 q^{32} + 60 q^{36} - 16 q^{37} + 36 q^{44} - 4 q^{46} + 4 q^{50} + 16 q^{53} + 48 q^{57} - 4 q^{58} - 28 q^{60}+ \cdots + 16 q^{93}+O(q^{100}) \) 32 * q - 4 * q^2 + 4 * q^4 - 4 * q^8 + 32 * q^9 + 28 * q^16 - 8 * q^22 - 32 * q^25 - 40 * q^29 - 4 * q^32 + 60 * q^36 - 16 * q^37 + 36 * q^44 - 4 * q^46 + 4 * q^50 + 16 * q^53 + 48 * q^57 - 4 * q^58 - 28 * q^60 + 4 * q^64 - 8 * q^65 - 8 * q^72 - 76 * q^74 + 120 * q^78 + 72 * q^81 - 56 * q^86 - 8 * q^88 - 4 * q^92 + 16 * q^93

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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