LMFDB - Embedded newform 380.2.r.a.49.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [380,2,Mod(49,380)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(380, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 4]))

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

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

chi := DirichletCharacter("380.49");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 380 = 2^{2} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	380.r (of order $6$, degree $2$, 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:	$3.03431527681$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$10$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 20 x^{18} + 261 x^{16} - 1994 x^{14} + 11074 x^{12} - 39211 x^{10} + 99376 x^{8} - 134299 x^{6} + 124617 x^{4} - 24768 x^{2} + 4096 $ x^20 - 20x^18 + 261x^16 - 1994x^14 + 11074x^12 - 39211x^10 + 99376x^8 - 134299x^6 + 124617x^4 - 24768*x^2 + 4096
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			49.3
Root			$-1.74361 + 1.00667i$ of defining polynomial
Character	$\chi$	$=$	380.49
Dual form			380.2.r.a.349.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.74361 - 1.00667i) q^{3} +(-1.95659 + 1.08248i) q^{5} +1.34403i q^{7} +(0.526784 + 0.912416i) q^{9} +O(q^{10})$	\(q+(-1.74361 - 1.00667i) q^{3} +(-1.95659 + 1.08248i) q^{5} +1.34403i q^{7} +(0.526784 + 0.912416i) q^{9} +5.25594 q^{11} +(2.10918 - 1.21773i) q^{13} +(4.50123 + 0.0822214i) q^{15} +(-1.17765 - 0.679914i) q^{17} +(2.89815 + 3.25587i) q^{19} +(1.35300 - 2.34346i) q^{21} +(7.05514 - 4.07329i) q^{23} +(2.65647 - 4.23594i) q^{25} +3.91884i q^{27} +(-1.03597 - 1.79435i) q^{29} -0.513207 q^{31} +(-9.16431 - 5.29102i) q^{33} +(-1.45488 - 2.62971i) q^{35} +5.57175i q^{37} -4.90344 q^{39} +(2.70353 - 4.68265i) q^{41} +(11.0197 + 6.36221i) q^{43} +(-2.01837 - 1.21499i) q^{45} +(-2.82785 + 1.63266i) q^{47} +5.19359 q^{49} +(1.36890 + 2.37101i) q^{51} +(-10.1892 + 5.88276i) q^{53} +(-10.2837 + 5.68946i) q^{55} +(-1.77564 - 8.59447i) q^{57} +(0.0175979 - 0.0304805i) q^{59} +(0.518372 + 0.897846i) q^{61} +(-1.22631 + 0.708011i) q^{63} +(-2.80861 + 4.66574i) q^{65} +(-0.664028 + 0.383377i) q^{67} -16.4019 q^{69} +(5.68450 - 9.84583i) q^{71} +(1.86429 + 1.07635i) q^{73} +(-8.89606 + 4.71162i) q^{75} +7.06413i q^{77} +(-6.48576 + 11.2337i) q^{79} +(5.52535 - 9.57019i) q^{81} +4.20304i q^{83} +(3.04016 + 0.0555329i) q^{85} +4.17153i q^{87} +(-3.65426 - 6.32937i) q^{89} +(1.63667 + 2.83479i) q^{91} +(0.894833 + 0.516632i) q^{93} +(-9.19491 - 3.23321i) q^{95} +(-0.721716 - 0.416683i) q^{97} +(2.76875 + 4.79561i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + q^{5} + 10 q^{9}+O(q^{10}) $ 20 * q + q^5 + 10 * q^9	$ 20 q + q^{5} + 10 q^{9} - 5 q^{15} + 14 q^{19} - 8 q^{21} + 9 q^{25} - 16 q^{29} + 8 q^{31} - 2 q^{35} - 8 q^{39} + 26 q^{41} - 32 q^{45} - 44 q^{49} + 26 q^{51} - 12 q^{55} + 4 q^{59} + 2 q^{61} - 18 q^{65} + 48 q^{69} - 2 q^{71} + 46 q^{75} - 16 q^{79} + 26 q^{81} - 39 q^{85} - 40 q^{89} - 4 q^{91} - 43 q^{95} - 20 q^{99}+O(q^{100}) $ 20 * q + q^5 + 10 * q^9 - 5 * q^15 + 14 * q^19 - 8 * q^21 + 9 * q^25 - 16 * q^29 + 8 * q^31 - 2 * q^35 - 8 * q^39 + 26 * q^41 - 32 * q^45 - 44 * q^49 + 26 * q^51 - 12 * q^55 + 4 * q^59 + 2 * q^61 - 18 * q^65 + 48 * q^69 - 2 * q^71 + 46 * q^75 - 16 * q^79 + 26 * q^81 - 39 * q^85 - 40 * q^89 - 4 * q^91 - 43 * q^95 - 20 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/380\mathbb{Z}\right)^\times$.

$n$	$21$	$77$	$191$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$-1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$	−1.74361	−	1.00667i	−1.00667	−	0.581203i	−0.0964577	−	0.995337i	$-0.530751\pi$
$3$	−1.74361	−	1.00667i	−1.00667	−	0.581203i	−0.910216	+	0.414134i	$0.864085\pi$
$4$		0			0
$5$	−1.95659	+	1.08248i	−0.875013	+	0.484100i
$5$	−1.95659	+	1.08248i	−0.875013	+	0.484100i
$6$		0			0
$7$			1.34403i			0.507994i	0.967205	+	0.253997i	$0.0817454\pi$
$7$			1.34403i			0.507994i	−0.967205	+	0.253997i	$0.918255\pi$
$8$		0			0
$9$	0.526784	+	0.912416i	0.175595	+	0.304139i
$10$		0			0
$11$	5.25594			1.58473			0.792363	−	0.610050i	$-0.208851\pi$
$11$	5.25594			1.58473			0.792363	+	0.610050i	$0.208851\pi$
$12$		0			0
$13$	2.10918	−	1.21773i	0.584980	−	0.337738i	−0.178130	−	0.984007i	$-0.557005\pi$
$13$	2.10918	−	1.21773i	0.584980	−	0.337738i	0.763110	+	0.646269i	$0.223671\pi$
$14$		0			0
$15$	4.50123	+	0.0822214i	1.16221	+	0.0212295i
$16$		0			0
$17$	−1.17765	−	0.679914i	−0.285621	−	0.164903i	0.350344	−	0.936621i	$-0.386065\pi$
$17$	−1.17765	−	0.679914i	−0.285621	−	0.164903i	−0.635965	+	0.771718i	$0.719398\pi$
$18$		0			0
$19$	2.89815	+	3.25587i	0.664882	+	0.746949i
$19$	2.89815	+	3.25587i	0.664882	+	0.746949i
$20$		0			0
$21$	1.35300	−	2.34346i	0.295248	−	0.511384i
$22$		0			0
$23$	7.05514	−	4.07329i	1.47110	−	0.849339i	0.471625	−	0.881799i	$-0.343668\pi$
$23$	7.05514	−	4.07329i	1.47110	−	0.849339i	0.999473	+	0.0324603i	$0.0103342\pi$
$24$		0			0
$25$	2.65647	−	4.23594i	0.531294	−	0.847187i
$26$		0			0
$27$			3.91884i			0.754182i
$28$		0			0
$29$	−1.03597	−	1.79435i	−0.192375	−	0.333203i	0.753662	−	0.657262i	$-0.228286\pi$
$29$	−1.03597	−	1.79435i	−0.192375	−	0.333203i	−0.946037	+	0.324059i	$0.894952\pi$
$30$		0			0
$31$	−0.513207			−0.0921747			−0.0460873	−	0.998937i	$-0.514675\pi$
$31$	−0.513207			−0.0921747			−0.0460873	+	0.998937i	$0.514675\pi$
$32$		0			0
$33$	−9.16431	−	5.29102i	−1.59530	−	0.921048i
$34$		0			0
$35$	−1.45488	−	2.62971i	−0.245920	−	0.444501i
$36$		0			0
$37$			5.57175i			0.915991i	0.888955	+	0.457995i	$0.151432\pi$
$37$			5.57175i			0.915991i	−0.888955	+	0.457995i	$0.848568\pi$
$38$		0			0
$39$	−4.90344			−0.785179
$40$		0			0
$41$	2.70353	−	4.68265i	0.422220	−	0.731307i	−0.573936	−	0.818900i	$-0.694584\pi$
$41$	2.70353	−	4.68265i	0.422220	−	0.731307i	0.996156	+	0.0875933i	$0.0279176\pi$
$42$		0			0
$43$	11.0197	+	6.36221i	1.68048	+	0.970228i	0.961339	+	0.275369i	$0.0888000\pi$
$43$	11.0197	+	6.36221i	1.68048	+	0.970228i	0.719146	+	0.694859i	$0.244533\pi$
$44$		0			0
$45$	−2.01837	−	1.21499i	−0.300881	−	0.181120i
$46$		0			0
$47$	−2.82785	+	1.63266i	−0.412485	+	0.238148i	−0.691857	−	0.722035i	$-0.743207\pi$
$47$	−2.82785	+	1.63266i	−0.412485	+	0.238148i	0.279372	+	0.960183i	$0.409874\pi$
$48$		0			0
$49$	5.19359			0.741942
$50$		0			0
$51$	1.36890	+	2.37101i	0.191685	+	0.332008i
$52$		0			0
$53$	−10.1892	+	5.88276i	−1.39960	+	0.808060i	−0.994351	−	0.106146i	$-0.966149\pi$
$53$	−10.1892	+	5.88276i	−1.39960	+	0.808060i	−0.405250	+	0.914206i	$0.632815\pi$
$54$		0			0
$55$	−10.2837	+	5.68946i	−1.38666	+	0.767166i
$56$		0			0
$57$	−1.77564	−	8.59447i	−0.235190	−	1.13837i
$58$		0			0
$59$	0.0175979	−	0.0304805i	0.00229105	−	0.00396822i	−0.864878	−	0.501983i	$-0.832604\pi$
$59$	0.0175979	−	0.0304805i	0.00229105	−	0.00396822i	0.867169	+	0.498015i	$0.165937\pi$
$60$		0			0
$61$	0.518372	+	0.897846i	0.0663707	+	0.114957i	0.897301	−	0.441419i	$-0.145525\pi$
$61$	0.518372	+	0.897846i	0.0663707	+	0.114957i	−0.830930	+	0.556376i	$0.812191\pi$
$62$		0			0
$63$	−1.22631	+	0.708011i	−0.154501	+	0.0892010i
$64$		0			0
$65$	−2.80861	+	4.66574i	−0.348366	+	0.578714i
$66$		0			0
$67$	−0.664028	+	0.383377i	−0.0811239	+	0.0468369i	−0.540013	−	0.841657i	$-0.681581\pi$
$67$	−0.664028	+	0.383377i	−0.0811239	+	0.0468369i	0.458889	+	0.888493i	$0.348247\pi$
$68$		0			0
$69$	−16.4019			−1.97455
$70$		0			0
$71$	5.68450	−	9.84583i	0.674625	−	1.16849i	−0.301953	−	0.953323i	$-0.597638\pi$
$71$	5.68450	−	9.84583i	0.674625	−	1.16849i	0.976578	−	0.215163i	$-0.0690282\pi$
$72$		0			0
$73$	1.86429	+	1.07635i	0.218199	+	0.125977i	0.605116	−	0.796137i	$-0.293127\pi$
$73$	1.86429	+	1.07635i	0.218199	+	0.125977i	−0.386917	+	0.922115i	$0.626460\pi$
$74$		0			0
$75$	−8.89606	+	4.71162i	−1.02723	+	0.544051i
$76$		0			0
$77$			7.06413i			0.805032i
$78$		0			0
$79$	−6.48576	+	11.2337i	−0.729705	+	1.26389i	0.227302	+	0.973824i	$0.427009\pi$
$79$	−6.48576	+	11.2337i	−0.729705	+	1.26389i	−0.957008	+	0.290062i	$0.906324\pi$
$80$		0			0
$81$	5.52535	−	9.57019i	0.613928	−	1.06335i
$82$		0			0
$83$			4.20304i			0.461343i	0.973032	+	0.230672i	$0.0740923\pi$
$83$			4.20304i			0.461343i	−0.973032	+	0.230672i	$0.925908\pi$
$84$		0			0
$85$	3.04016	+	0.0555329i	0.329752	+	0.00602339i
$86$		0			0
$87$			4.17153i			0.447235i
$88$		0			0
$89$	−3.65426	−	6.32937i	−0.387351	−	0.670912i	0.604741	−	0.796422i	$-0.293277\pi$
$89$	−3.65426	−	6.32937i	−0.387351	−	0.670912i	−0.992092	+	0.125510i	$0.959943\pi$
$90$		0			0
$91$	1.63667	+	2.83479i	0.171569	+	0.297166i
$92$		0			0
$93$	0.894833	+	0.516632i	0.0927898	+	0.0535722i
$94$		0			0
$95$	−9.19491	−	3.23321i	−0.943378	−	0.331720i
$96$		0			0
$97$	−0.721716	−	0.416683i	−0.0732791	−	0.0423077i	0.462913	−	0.886404i	$-0.346804\pi$
$97$	−0.721716	−	0.416683i	−0.0732791	−	0.0423077i	−0.536192	+	0.844096i	$0.680138\pi$
$98$		0			0
$99$	2.76875	+	4.79561i	0.278269	+	0.481977i
$100$		0			0
$101$	−7.40992	−	12.8344i	−0.737315	−	1.27707i	−0.953700	−	0.300759i	$-0.902760\pi$
$101$	−7.40992	−	12.8344i	−0.737315	−	1.27707i	0.216385	−	0.976308i	$-0.430573\pi$
$102$		0			0
$103$		−	9.40773i		−	0.926971i	−0.886104	−	0.463486i	$-0.846599\pi$
$103$		−	9.40773i		−	0.926971i	0.886104	−	0.463486i	$-0.153401\pi$
$104$		0			0
$105$	−0.110508	+	6.04977i	−0.0107845	+	0.590397i
$106$		0			0
$107$		−	12.8130i		−	1.23868i	−0.785124	−	0.619338i	$-0.787401\pi$
$107$		−	12.8130i		−	1.23868i	0.785124	−	0.619338i	$-0.212599\pi$
$108$		0			0
$109$	0.996875	−	1.72664i	0.0954833	−	0.165382i	−0.814327	−	0.580406i	$-0.802894\pi$
$109$	0.996875	−	1.72664i	0.0954833	−	0.165382i	0.909810	+	0.415025i	$0.136227\pi$
$110$		0			0
$111$	5.60894	−	9.71497i	0.532377	−	0.922104i
$112$		0			0
$113$			8.34647i			0.785170i	0.919716	+	0.392585i	$0.128419\pi$
$113$			8.34647i			0.785170i	−0.919716	+	0.392585i	$0.871581\pi$
$114$		0			0
$115$	−9.39474	+	15.6068i	−0.876064	+	1.45534i
$116$		0			0
$117$	2.22216	+	1.28296i	0.205439	+	0.118610i
$118$		0			0
$119$	0.913823	−	1.58279i	0.0837700	−	0.145094i
$120$		0			0
$121$	16.6249			1.51136
$122$		0			0
$123$	−9.42780	+	5.44314i	−0.850076	+	0.490791i
$124$		0			0
$125$	−0.612300	+	11.1636i	−0.0547658	+	0.998499i
$126$		0			0
$127$	17.8240	−	10.2907i	1.58163	−	0.913153i	0.587005	−	0.809583i	$-0.300307\pi$
$127$	17.8240	−	10.2907i	1.58163	−	0.913153i	0.994622	−	0.103569i	$-0.0330264\pi$
$128$		0			0
$129$	−12.8093	−	22.1864i	−1.12780	−	1.95341i
$130$		0			0
$131$	−5.36554	+	9.29339i	−0.468790	+	0.811967i	−0.999364	−	0.0356712i	$-0.988643\pi$
$131$	−5.36554	+	9.29339i	−0.468790	+	0.811967i	0.530574	+	0.847639i	$0.321976\pi$
$132$		0			0
$133$	−4.37598	+	3.89519i	−0.379446	+	0.337756i
$134$		0			0
$135$	−4.24207	−	7.66756i	−0.365100	−	0.659919i
$136$		0			0
$137$	14.8771	−	8.58931i	1.27104	−	0.733834i	0.295855	−	0.955233i	$-0.404396\pi$
$137$	14.8771	−	8.58931i	1.27104	−	0.733834i	0.975183	+	0.221399i	$0.0710622\pi$
$138$		0			0
$139$	3.66394	+	6.34613i	0.310771	+	0.538272i	0.978530	−	0.206106i	$-0.0660793\pi$
$139$	3.66394	+	6.34613i	0.310771	+	0.538272i	−0.667758	+	0.744378i	$0.732746\pi$
$140$		0			0
$141$	6.57423			0.553650
$142$		0			0
$143$	11.0857	−	6.40033i	0.927033	−	0.535223i
$144$		0			0
$145$	3.96932	+	2.38939i	0.329634	+	0.198428i
$146$		0			0
$147$	−9.05560	−	5.22825i	−0.746893	−	0.431219i
$148$		0			0
$149$	−6.12292	+	10.6052i	−0.501609	+	0.868812i	0.498389	+	0.866953i	$0.333925\pi$
$149$	−6.12292	+	10.6052i	−0.501609	+	0.868812i	−0.999998	+	0.00185904i	$0.999408\pi$
$150$		0			0
$151$	−11.5577			−0.940549			−0.470274	−	0.882520i	$-0.655845\pi$
$151$	−11.5577			−0.940549			−0.470274	+	0.882520i	$0.655845\pi$
$152$		0			0
$153$		−	1.43267i		−	0.115825i
$154$		0			0
$155$	1.00413	−	0.555537i	0.0806540	−	0.0446218i
$156$		0			0
$157$	−3.58528	−	2.06996i	−0.286137	−	0.165201i	0.350062	−	0.936727i	$-0.386161\pi$
$157$	−3.58528	−	2.06996i	−0.286137	−	0.165201i	−0.636198	+	0.771526i	$0.719494\pi$
$158$		0			0
$159$	23.6881			1.87859
$160$		0			0
$161$	5.47460	+	9.48229i	0.431459	+	0.747309i
$162$		0			0
$163$		−	9.41672i		−	0.737575i	−0.929514	−	0.368787i	$-0.879773\pi$
$163$		−	9.41672i		−	0.737575i	0.929514	−	0.368787i	$-0.120227\pi$
$164$		0			0
$165$	23.6582	+	0.432151i	1.84179	+	0.0336429i
$166$		0			0
$167$	−11.8395	+	6.83551i	−0.916164	+	0.528948i	−0.882409	−	0.470482i	$-0.844080\pi$
$167$	−11.8395	+	6.83551i	−0.916164	+	0.528948i	−0.0337550	+	0.999430i	$0.510747\pi$
$168$		0			0
$169$	−3.53425	+	6.12151i	−0.271866	+	0.470885i
$170$		0			0
$171$	−1.44401	+	4.35946i	−0.110426	+	0.333376i
$172$		0			0
$173$	10.1999	+	5.88891i	0.775484	+	0.447726i	0.834827	−	0.550512i	$-0.185567\pi$
$173$	10.1999	+	5.88891i	0.775484	+	0.447726i	−0.0593437	+	0.998238i	$0.518901\pi$
$174$		0			0
$175$	5.69321	+	3.57037i	0.430366	+	0.269894i
$176$		0			0
$177$	−0.0613678	+	0.0354307i	−0.00461269	+	0.00266314i
$178$		0			0
$179$	16.5727			1.23870			0.619350	−	0.785115i	$-0.287396\pi$
$179$	16.5727			1.23870			0.619350	+	0.785115i	$0.287396\pi$
$180$		0			0
$181$	7.19552	+	12.4630i	0.534839	+	0.926368i	0.999171	+	0.0407069i	$0.0129610\pi$
$181$	7.19552	+	12.4630i	0.534839	+	0.926368i	−0.464332	+	0.885661i	$0.653706\pi$
$182$		0			0
$183$		−	2.08733i		−	0.154300i
$184$		0			0
$185$	−6.03132	−	10.9016i	−0.443431	−	0.801504i
$186$		0			0
$187$	−6.18964	−	3.57359i	−0.452631	−	0.261327i
$188$		0			0
$189$	−5.26703			−0.383120
$190$		0			0
$191$	−5.97170			−0.432097			−0.216049	−	0.976383i	$-0.569317\pi$
$191$	−5.97170			−0.432097			−0.216049	+	0.976383i	$0.569317\pi$
$192$		0			0
$193$	−14.1046	−	8.14331i	−1.01527	−	0.586168i	−0.102542	−	0.994729i	$-0.532698\pi$
$193$	−14.1046	−	8.14331i	−1.01527	−	0.586168i	−0.912731	+	0.408560i	$0.866031\pi$
$194$		0			0
$195$	9.59401	−	5.30788i	0.687041	−	0.380105i
$196$		0			0
$197$		−	11.5233i		−	0.820998i	−0.911861	−	0.410499i	$-0.865355\pi$
$197$		−	11.5233i		−	0.820998i	0.911861	−	0.410499i	$-0.134645\pi$
$198$		0			0
$199$	4.79943	+	8.31285i	0.340222	+	0.589283i	0.984474	−	0.175531i	$-0.0561643\pi$
$199$	4.79943	+	8.31285i	0.340222	+	0.589283i	−0.644251	+	0.764814i	$0.722831\pi$
$200$		0			0
$201$	1.54374			0.108887
$202$		0			0
$203$	2.41166	−	1.39237i	0.169265	−	0.0977253i
$204$		0			0
$205$	−0.220814	+	12.0885i	−0.0154223	+	0.844299i
$206$		0			0
$207$	7.43307	+	4.29148i	0.516634	+	0.298279i
$208$		0			0
$209$	15.2325	+	17.1127i	1.05366	+	1.18371i
$210$		0			0
$211$	7.28207	−	12.6129i	0.501318	−	0.868308i	−0.498681	−	0.866786i	$-0.666182\pi$
$211$	7.28207	−	12.6129i	0.501318	−	0.868308i	0.999999	−	0.00152265i	$-0.000484675\pi$
$212$		0			0
$213$	−19.8231	+	11.4449i	−1.35826	+	0.784189i
$214$		0			0
$215$	−28.4479	−	0.519642i	−1.94013	−	0.0354393i
$216$		0			0
$217$		−	0.689764i		−	0.0468242i
$218$		0			0
$219$	−2.16707	−	3.75347i	−0.146437	−	0.253636i
$220$		0			0
$221$	−3.31182			−0.222777
$222$		0			0
$223$	6.55816	+	3.78635i	0.439167	+	0.253553i	0.703244	−	0.710949i	$-0.251734\pi$
$223$	6.55816	+	3.78635i	0.439167	+	0.253553i	−0.264077	+	0.964501i	$0.585067\pi$
$224$		0			0
$225$	5.26432	+	0.192385i	0.350955	+	0.0128257i
$226$		0			0
$227$		−	6.86640i		−	0.455739i	−0.973692	−	0.227869i	$-0.926824\pi$
$227$		−	6.86640i		−	0.455739i	0.973692	−	0.227869i	$-0.0731759\pi$
$228$		0			0
$229$	−22.1011			−1.46048			−0.730240	−	0.683191i	$-0.760592\pi$
$229$	−22.1011			−1.46048			−0.730240	+	0.683191i	$0.760592\pi$
$230$		0			0
$231$	7.11127	−	12.3171i	0.467887	−	0.810404i
$232$		0			0
$233$	−2.57806	−	1.48844i	−0.168894	−	0.0975111i	0.413170	−	0.910654i	$-0.364421\pi$
$233$	−2.57806	−	1.48844i	−0.168894	−	0.0975111i	−0.582064	+	0.813143i	$0.697755\pi$
$234$		0			0
$235$	3.76562	−	6.25555i	0.245642	−	0.408067i
$236$		0			0
$237$	22.6173	−	13.0581i	1.46915	−	0.848214i
$238$		0			0
$239$	9.71289			0.628275			0.314137	−	0.949378i	$-0.398285\pi$
$239$	9.71289			0.628275			0.314137	+	0.949378i	$0.398285\pi$
$240$		0			0
$241$	9.34287	+	16.1823i	0.601827	+	1.04239i	0.992544	+	0.121884i	$0.0388937\pi$
$241$	9.34287	+	16.1823i	0.601827	+	1.04239i	−0.390717	+	0.920511i	$0.627773\pi$
$242$		0			0
$243$	−9.08665	+	5.24618i	−0.582909	+	0.336543i
$244$		0			0
$245$	−10.1617	+	5.62196i	−0.649209	+	0.359174i
$246$		0			0
$247$	10.0775	+	3.33803i	0.641216	+	0.212394i
$248$		0			0
$249$	4.23109	−	7.32846i	0.268134	−	0.464422i
$250$		0			0
$251$	2.10091	+	3.63888i	0.132608	+	0.229684i	0.924681	−	0.380742i	$-0.124332\pi$
$251$	2.10091	+	3.63888i	0.132608	+	0.229684i	−0.792073	+	0.610426i	$0.790998\pi$
$252$		0			0
$253$	37.0814	−	21.4090i	2.33129	−	1.34597i
$254$		0			0
$255$	−5.24495	−	3.15728i	−0.328452	−	0.197716i
$256$		0			0
$257$	−24.3946	+	14.0842i	−1.52169	+	0.878548i	−0.522018	+	0.852934i	$0.674821\pi$
$257$	−24.3946	+	14.0842i	−1.52169	+	0.878548i	−0.999672	+	0.0256140i	$0.991846\pi$
$258$		0			0
$259$	−7.48859			−0.465318
$260$		0			0
$261$	1.09146	−	1.89047i	0.0675599	−	0.117017i
$262$		0			0
$263$	−2.34563	−	1.35425i	−0.144637	−	0.0835065i	0.425935	−	0.904754i	$-0.359945\pi$
$263$	−2.34563	−	1.35425i	−0.144637	−	0.0835065i	−0.570572	+	0.821247i	$0.693279\pi$
$264$		0			0
$265$	13.5682	−	22.5398i	0.833486	−	1.38461i
$266$		0			0
$267$			14.7146i			0.900519i
$268$		0			0
$269$	5.64101	−	9.77052i	0.343938	−	0.595719i	−0.641222	−	0.767356i	$-0.721572\pi$
$269$	5.64101	−	9.77052i	0.343938	−	0.595719i	0.985160	+	0.171637i	$0.0549055\pi$
$270$		0			0
$271$	−3.16690	+	5.48523i	−0.192375	+	0.333204i	−0.946037	−	0.324059i	$-0.894952\pi$
$271$	−3.16690	+	5.48523i	−0.192375	+	0.333204i	0.753662	+	0.657263i	$0.228286\pi$
$272$		0			0
$273$		−	6.59035i		−	0.398866i
$274$		0			0
$275$	13.9623	−	22.2638i	0.841956	−	1.34256i
$276$		0			0
$277$			11.1435i			0.669549i	0.942298	+	0.334774i	$0.108660\pi$
$277$			11.1435i			0.669549i	−0.942298	+	0.334774i	$0.891340\pi$
$278$		0			0
$279$	−0.270349	−	0.468258i	−0.0161854	−	0.0280339i
$280$		0			0
$281$	−12.9061	−	22.3541i	−0.769916	−	1.33353i	−0.937608	−	0.347694i	$-0.886965\pi$
$281$	−12.9061	−	22.3541i	−0.769916	−	1.33353i	0.167693	−	0.985839i	$-0.446368\pi$
$282$		0			0
$283$	24.9942	+	14.4304i	1.48575	+	0.857799i	0.999868	−	0.0162249i	$-0.00516476\pi$
$283$	24.9942	+	14.4304i	1.48575	+	0.857799i	0.485883	+	0.874024i	$0.338498\pi$
$284$		0			0
$285$	12.7775	+	14.8937i	0.756877	+	0.882228i
$286$		0			0
$287$	6.29360	+	3.63361i	0.371500	+	0.214485i
$288$		0			0
$289$	−7.57543	−	13.1210i	−0.445614	−	0.771826i
$290$		0			0
$291$	0.838927	+	1.45306i	0.0491788	+	0.0851801i
$292$		0			0
$293$		−	22.7742i		−	1.33048i	−0.746628	−	0.665242i	$-0.768328\pi$
$293$		−	22.7742i		−	1.33048i	0.746628	−	0.665242i	$-0.231672\pi$
$294$		0			0
$295$	−0.00143733	+	0.0786871i	−8.36848e−5	+	0.00458134i
$296$		0			0
$297$			20.5972i			1.19517i
$298$		0			0
$299$	9.92035	−	17.1825i	0.573709	−	0.993692i
$300$		0			0
$301$	−8.55098	+	14.8107i	−0.492870	+	0.853676i
$302$		0			0
$303$			29.8375i			1.71412i
$304$		0			0
$305$	−1.98614	−	1.19559i	−0.113726	−	0.0684592i
$306$		0			0
$307$	−14.0275	−	8.09880i	−0.800593	−	0.462223i	0.0430854	−	0.999071i	$-0.486281\pi$
$307$	−14.0275	−	8.09880i	−0.800593	−	0.462223i	−0.843679	+	0.536849i	$0.819615\pi$
$308$		0			0
$309$	−9.47052	+	16.4034i	−0.538759	+	0.933158i
$310$		0			0
$311$	−28.3483			−1.60749			−0.803743	−	0.594977i	$-0.797161\pi$
$311$	−28.3483			−1.60749			−0.803743	+	0.594977i	$0.797161\pi$
$312$		0			0
$313$	−2.67539	+	1.54464i	−0.151222	+	0.0873081i	−0.573702	−	0.819064i	$-0.694493\pi$
$313$	−2.67539	+	1.54464i	−0.151222	+	0.0873081i	0.422480	+	0.906372i	$0.361160\pi$
$314$		0			0
$315$	1.63298	−	2.71275i	0.0920079	−	0.152846i
$316$		0			0
$317$	−20.8236	+	12.0225i	−1.16957	+	0.675251i	−0.953579	−	0.301144i	$-0.902632\pi$
$317$	−20.8236	+	12.0225i	−1.16957	+	0.675251i	−0.215991	+	0.976395i	$0.569298\pi$
$318$		0			0
$319$	−5.44500	−	9.43101i	−0.304861	−	0.528035i
$320$		0			0
$321$	−12.8985	+	22.3408i	−0.719923	+	1.24694i
$322$		0			0
$323$	−1.19928	−	5.80476i	−0.0667298	−	0.322986i
$324$		0			0
$325$	0.444724	−	12.1692i	0.0246689	−	0.675026i
$326$		0			0
$327$	−3.47632	+	2.00706i	−0.192241	+	0.110990i
$328$		0			0
$329$	−2.19434	−	3.80071i	−0.120978	−	0.209540i
$330$		0			0
$331$	−20.7717			−1.14171			−0.570857	−	0.821049i	$-0.693389\pi$
$331$	−20.7717			−1.14171			−0.570857	+	0.821049i	$0.693389\pi$
$332$		0			0
$333$	−5.08376	+	2.93511i	−0.278588	+	0.160843i
$334$		0			0
$335$	0.884231	−	1.46891i	0.0483107	−	0.0802550i
$336$		0			0
$337$	2.10139	+	1.21324i	0.114470	+	0.0660892i	0.556142	−	0.831087i	$-0.312281\pi$
$337$	2.10139	+	1.21324i	0.114470	+	0.0660892i	−0.441672	+	0.897177i	$0.645614\pi$
$338$		0			0
$339$	8.40217	−	14.5530i	0.456343	−	0.790410i
$340$		0			0
$341$	−2.69739			−0.146072
$342$		0			0
$343$			16.3885i			0.884896i
$344$		0			0
$345$	32.0917	−	17.7547i	1.72776	−	0.955882i
$346$		0			0
$347$	−2.48834	−	1.43664i	−0.133581	−	0.0771230i	0.431720	−	0.902008i	$-0.357907\pi$
$347$	−2.48834	−	1.43664i	−0.133581	−	0.0771230i	−0.565301	+	0.824885i	$0.691240\pi$
$348$		0			0
$349$	5.89385			0.315490			0.157745	−	0.987480i	$-0.449578\pi$
$349$	5.89385			0.315490			0.157745	+	0.987480i	$0.449578\pi$
$350$		0			0
$351$	4.77211	+	8.26553i	0.254716	+	0.441181i
$352$		0			0
$353$		−	12.0238i		−	0.639962i	−0.947424	−	0.319981i	$-0.896324\pi$
$353$		−	12.0238i		−	0.639962i	0.947424	−	0.319981i	$-0.103676\pi$
$354$		0			0
$355$	−0.464289	+	25.4176i	−0.0246419	+	1.34903i
$356$		0			0
$357$	−3.18670	+	1.83984i	−0.168658	+	0.0973748i
$358$		0			0
$359$	−2.26590	+	3.92466i	−0.119590	+	0.207136i	−0.919605	−	0.392844i	$-0.871491\pi$
$359$	−2.26590	+	3.92466i	−0.119590	+	0.207136i	0.800015	+	0.599979i	$0.204825\pi$
$360$		0			0
$361$	−2.20143	+	18.8720i	−0.115865	+	0.993265i
$362$		0			0
$363$	−28.9874	−	16.7359i	−1.52144	−	0.878406i
$364$		0			0
$365$	−4.81278	−	0.0879123i	−0.251912	−	0.00460154i
$366$		0			0
$367$	−29.9143	+	17.2710i	−1.56151	+	0.901539i	−0.564407	+	0.825496i	$0.690895\pi$
$367$	−29.9143	+	17.2710i	−1.56151	+	0.901539i	−0.997105	+	0.0760429i	$0.975771\pi$
$368$		0			0
$369$	5.69670			0.296558
$370$		0			0
$371$	−7.90659	−	13.6946i	−0.410490	−	0.710989i
$372$		0			0
$373$			24.0801i			1.24682i	0.781894	+	0.623411i	$0.214254\pi$
$373$			24.0801i			1.24682i	−0.781894	+	0.623411i	$0.785746\pi$
$374$		0			0
$375$	12.3057	−	18.8485i	0.635462	−	0.973333i
$376$		0			0
$377$	−4.37008	−	2.52307i	−0.225071	−	0.129945i
$378$		0			0
$379$	−30.1565			−1.54904			−0.774518	−	0.632552i	$-0.782008\pi$
$379$	−30.1565			−1.54904			−0.774518	+	0.632552i	$0.782008\pi$
$380$		0			0
$381$	−41.4375			−2.12291
$382$		0			0
$383$	−33.6192	−	19.4101i	−1.71786	−	0.991809i	−0.922797	−	0.385286i	$-0.874103\pi$
$383$	−33.6192	−	19.4101i	−1.71786	−	0.991809i	−0.795066	−	0.606522i	$-0.792564\pi$
$384$		0			0
$385$	−7.64678	−	13.8216i	−0.389716	−	0.704413i
$386$		0			0
$387$			13.4060i			0.681467i
$388$		0			0
$389$	18.6935	+	32.3781i	0.947799	+	1.64164i	0.750048	+	0.661383i	$0.230030\pi$
$389$	18.6935	+	32.3781i	0.947799	+	1.64164i	0.197750	+	0.980252i	$0.436636\pi$
$390$		0			0
$391$	−11.0779			−0.560236
$392$		0			0
$393$	18.7108	−	10.8027i	0.943836	−	0.544924i
$394$		0			0
$395$	0.529733	−	29.0004i	0.0266538	−	1.45917i
$396$		0			0
$397$	13.4790	+	7.78211i	0.676492	+	0.390573i	0.798532	−	0.601952i	$-0.205610\pi$
$397$	13.4790	+	7.78211i	0.676492	+	0.390573i	−0.122040	+	0.992525i	$0.538944\pi$
$398$		0			0
$399$	11.5512	−	2.38651i	0.578283	−	0.119475i
$400$		0			0
$401$	−11.3113	+	19.5918i	−0.564860	+	0.978366i	0.432203	+	0.901777i	$0.357737\pi$
$401$	−11.3113	+	19.5918i	−0.564860	+	0.978366i	−0.997063	+	0.0765898i	$0.975597\pi$
$402$		0			0
$403$	−1.08244	+	0.624949i	−0.0539203	+	0.0311309i
$404$		0			0
$405$	−0.451290	+	24.7060i	−0.0224248	+	1.22765i
$406$		0			0
$407$			29.2848i			1.45159i
$408$		0			0
$409$	−18.1239	−	31.3915i	−0.896169	−	1.55221i	−0.832351	−	0.554249i	$-0.813005\pi$
$409$	−18.1239	−	31.3915i	−0.896169	−	1.55221i	−0.0638187	−	0.997962i	$-0.520328\pi$
$410$		0			0
$411$	−34.5865			−1.70603
$412$		0			0
$413$	0.0409666	+	0.0236521i	0.00201583	+	0.00116384i
$414$		0			0
$415$	−4.54970	−	8.22361i	−0.223336	−	0.403681i
$416$		0			0
$417$		−	14.7536i		−	0.722486i
$418$		0			0
$419$	14.5598			0.711293			0.355647	−	0.934621i	$-0.384261\pi$
$419$	14.5598			0.711293			0.355647	+	0.934621i	$0.384261\pi$
$420$		0			0
$421$	−0.784161	+	1.35821i	−0.0382177	+	0.0661950i	−0.884502	−	0.466537i	$-0.845501\pi$
$421$	−0.784161	+	1.35821i	−0.0382177	+	0.0661950i	0.846284	+	0.532732i	$0.178835\pi$
$422$		0			0
$423$	−2.97934	−	1.72012i	−0.144860	−	0.0836351i
$424$		0			0
$425$	−6.00846	+	3.18226i	−0.291453	+	0.154362i
$426$		0			0
$427$	−1.20673	+	0.696705i	−0.0583977	+	0.0337159i
$428$		0			0
$429$	−25.7722			−1.24429
$430$		0			0
$431$	12.7303	+	22.0495i	0.613197	+	1.06209i	0.990698	+	0.136080i	$0.0434503\pi$
$431$	12.7303	+	22.0495i	0.613197	+	1.06209i	−0.377500	+	0.926009i	$0.623216\pi$
$432$		0			0
$433$	−14.6212	+	8.44155i	−0.702650	+	0.405675i	−0.808334	−	0.588725i	$-0.799630\pi$
$433$	−14.6212	+	8.44155i	−0.702650	+	0.405675i	0.105684	+	0.994400i	$0.466297\pi$
$434$		0			0
$435$	−4.51560	−	8.16197i	−0.216507	−	0.391337i
$436$		0			0
$437$	33.7090	+	11.1656i	1.61252	+	0.534125i
$438$		0			0
$439$	9.93240	−	17.2034i	0.474048	−	0.821075i	−0.525511	−	0.850787i	$-0.676126\pi$
$439$	9.93240	−	17.2034i	0.474048	−	0.821075i	0.999558	+	0.0297121i	$0.00945905\pi$
$440$		0			0
$441$	2.73590	+	4.73872i	0.130281	+	0.225653i
$442$		0			0
$443$	−4.46422	+	2.57742i	−0.212101	+	0.122457i	−0.602288	−	0.798279i	$-0.705744\pi$
$443$	−4.46422	+	2.57742i	−0.212101	+	0.122457i	0.390186	+	0.920736i	$0.372411\pi$
$444$		0			0
$445$	14.0013	+	8.42830i	0.663726	+	0.399540i
$446$		0			0
$447$	21.3520	−	12.3276i	1.00991	−	0.583074i
$448$		0			0
$449$	33.2207			1.56778			0.783892	−	0.620897i	$-0.213232\pi$
$449$	33.2207			1.56778			0.783892	+	0.620897i	$0.213232\pi$
$450$		0			0
$451$	14.2096	−	24.6117i	0.669103	−	1.15892i
$452$		0			0
$453$	20.1520	+	11.6348i	0.946826	+	0.546650i
$454$		0			0
$455$	−6.27088	−	3.77485i	−0.293983	−	0.176968i
$456$		0			0
$457$		−	11.7126i		−	0.547894i	−0.961745	−	0.273947i	$-0.911671\pi$
$457$		−	11.7126i		−	0.547894i	0.961745	−	0.273947i	$-0.0883293\pi$
$458$		0			0
$459$	2.66448	−	4.61501i	0.124367	−	0.215410i
$460$		0			0
$461$	3.68501	−	6.38263i	0.171628	−	0.297269i	−0.767361	−	0.641215i	$-0.778431\pi$
$461$	3.68501	−	6.38263i	0.171628	−	0.297269i	0.938989	+	0.343947i	$0.111764\pi$
$462$		0			0
$463$		−	28.8020i		−	1.33854i	−0.743019	−	0.669271i	$-0.766607\pi$
$463$		−	28.8020i		−	1.33854i	0.743019	−	0.669271i	$-0.233393\pi$
$464$		0			0
$465$	−2.31006	−	0.0421966i	−0.107127	−	0.00195682i
$466$		0			0
$467$		−	34.1251i		−	1.57912i	−0.613673	−	0.789561i	$-0.710309\pi$
$467$		−	34.1251i		−	1.57912i	0.613673	−	0.789561i	$-0.289691\pi$
$468$		0			0
$469$	−0.515268	−	0.892471i	−0.0237929	−	0.0412105i
$470$		0			0
$471$	4.16756	+	7.21842i	0.192031	+	0.332607i
$472$		0			0
$473$	57.9188	+	33.4394i	2.66311	+	1.53755i
$474$		0			0
$475$	21.4905	−	3.62725i	0.986053	−	0.166430i
$476$		0			0
$477$	−10.7351	−	6.19789i	−0.491525	−	0.283782i
$478$		0			0
$479$	14.4130	+	24.9640i	0.658546	+	1.14064i	0.980992	+	0.194048i	$0.0621617\pi$
$479$	14.4130	+	24.9640i	0.658546	+	1.14064i	−0.322446	+	0.946588i	$0.604505\pi$
$480$		0			0
$481$	6.78491	+	11.7518i	0.309365	+	0.535836i
$482$		0			0
$483$		−	22.0446i		−	1.00306i
$484$		0			0
$485$	1.86315	+	0.0340331i	0.0846013	+	0.00154536i
$486$		0			0
$487$		−	16.5796i		−	0.751294i	−0.926763	−	0.375647i	$-0.877421\pi$
$487$		−	16.5796i		−	0.751294i	0.926763	−	0.375647i	$-0.122579\pi$
$488$		0			0
$489$	−9.47957	+	16.4191i	−0.428681	+	0.742497i
$490$		0			0
$491$	−4.94615	+	8.56698i	−0.223217	+	0.386623i	−0.955783	−	0.294073i	$-0.904989\pi$
$491$	−4.94615	+	8.56698i	−0.223217	+	0.386623i	0.732566	+	0.680696i	$0.238322\pi$
$492$		0			0
$493$			2.81748i			0.126893i
$494$		0			0
$495$	−10.6084	−	6.38591i	−0.476814	−	0.287026i
$496$		0			0
$497$	13.2331	+	7.64011i	0.593584	+	0.342706i
$498$		0			0
$499$	−15.4949	+	26.8380i	−0.693649	+	1.20144i	0.276985	+	0.960874i	$0.410665\pi$
$499$	−15.4949	+	26.8380i	−0.693649	+	1.20144i	−0.970634	+	0.240561i	$0.922669\pi$
$500$		0			0
$501$	27.5245			1.22970
$502$		0			0
$503$	−11.1406	+	6.43203i	−0.496735	+	0.286790i	−0.727364	−	0.686252i	$-0.759255\pi$
$503$	−11.1406	+	6.43203i	−0.496735	+	0.286790i	0.230629	+	0.973042i	$0.425922\pi$
$504$		0			0
$505$	28.3911	+	17.0905i	1.26339	+	0.760516i
$506$		0			0
$507$	12.3247	−	7.11568i	0.547360	−	0.316018i
$508$		0			0
$509$	−7.35312	−	12.7360i	−0.325921	−	0.564512i	0.655777	−	0.754955i	$-0.272341\pi$
$509$	−7.35312	−	12.7360i	−0.325921	−	0.564512i	−0.981698	+	0.190442i	$0.939008\pi$
$510$		0			0
$511$	−1.44664	+	2.50566i	−0.0639957	+	0.110844i
$512$		0			0
$513$	−12.7593	+	11.3574i	−0.563335	+	0.501442i
$514$		0			0
$515$	10.1837	+	18.4071i	0.448747	+	0.811112i
$516$		0			0
$517$	−14.8630	+	8.58118i	−0.653676	+	0.377400i
$518$		0			0
$519$	−11.8564	−	20.5359i	−0.520439	−	0.901427i
$520$		0			0
$521$	5.35528			0.234619			0.117310	−	0.993095i	$-0.462573\pi$
$521$	5.35528			0.234619			0.117310	+	0.993095i	$0.462573\pi$
$522$		0			0
$523$	13.8388	−	7.98981i	0.605127	−	0.349370i	−0.165929	−	0.986138i	$-0.553062\pi$
$523$	13.8388	−	7.98981i	0.605127	−	0.349370i	0.771056	+	0.636768i	$0.219729\pi$
$524$		0			0
$525$	−6.33254	−	11.9565i	−0.276375	−	0.521826i
$526$		0			0
$527$	0.604376	+	0.348937i	0.0263270	+	0.0151999i
$528$		0			0
$529$	21.6833	−	37.5566i	0.942753	−	1.63290i
$530$		0			0
$531$	0.0370812			0.00160919
$532$		0			0
$533$		−	13.1687i		−	0.570400i
$534$		0			0
$535$	13.8698	+	25.0697i	0.599643	+	1.08386i
$536$		0			0
$537$	−28.8963	−	16.6833i	−1.24697	−	0.719937i
$538$		0			0
$539$	27.2972			1.17577
$540$		0			0
$541$	−17.9500	−	31.0904i	−0.771732	−	1.33668i	−0.936613	−	0.350366i	$-0.886057\pi$
$541$	−17.9500	−	31.0904i	−0.771732	−	1.33668i	0.164881	−	0.986313i	$-0.447276\pi$
$542$		0			0
$543$		−	28.9742i		−	1.24340i
$544$		0			0
$545$	−0.0814211	+	4.45742i	−0.00348770	+	0.190935i
$546$		0			0
$547$	22.6473	−	13.0754i	0.968327	−	0.559064i	0.0696011	−	0.997575i	$-0.477827\pi$
$547$	22.6473	−	13.0754i	0.968327	−	0.559064i	0.898726	+	0.438511i	$0.144494\pi$
$548$		0			0
$549$	−0.546140	+	0.945942i	−0.0233087	+	0.0403718i
$550$		0			0
$551$	2.83979	−	8.57329i	0.120979	−	0.365235i
$552$		0			0
$553$	−15.0983	−	8.71704i	−0.642047	−	0.370686i
$554$		0			0
$555$	−0.458118	+	25.0798i	−0.0194460	+	1.06458i
$556$		0			0
$557$	3.64444	−	2.10412i	0.154420	−	0.0891543i	−0.420799	−	0.907154i	$-0.638250\pi$
$557$	3.64444	−	2.10412i	0.154420	−	0.0891543i	0.575219	+	0.818000i	$0.304917\pi$
$558$		0			0
$559$	30.9899			1.31073
$560$		0			0
$561$	7.19488	+	12.4619i	0.303768	+	0.526142i
$562$		0			0
$563$			40.5225i			1.70782i	0.520422	+	0.853909i	$0.325775\pi$
$563$			40.5225i			1.70782i	−0.520422	+	0.853909i	$0.674225\pi$
$564$		0			0
$565$	−9.03489	−	16.3306i	−0.380101	−	0.687034i
$566$		0			0
$567$	12.8626	+	7.42622i	0.540178	+	0.311872i
$568$		0			0
$569$	−23.9522			−1.00413			−0.502064	−	0.864831i	$-0.667426\pi$
$569$	−23.9522			−1.00413			−0.502064	+	0.864831i	$0.667426\pi$
$570$		0			0
$571$	−7.78949			−0.325980			−0.162990	−	0.986628i	$-0.552114\pi$
$571$	−7.78949			−0.325980			−0.162990	+	0.986628i	$0.552114\pi$
$572$		0			0
$573$	10.4123	+	6.01155i	0.434981	+	0.251136i
$574$		0			0
$575$	1.48759	−	40.7057i	0.0620369	−	1.69754i
$576$		0			0
$577$		−	34.1385i		−	1.42121i	−0.703593	−	0.710603i	$-0.748422\pi$
$577$		−	34.1385i		−	1.42121i	0.703593	−	0.710603i	$-0.251578\pi$
$578$		0			0
$579$	16.3953	+	28.3975i	0.681366	+	1.18016i
$580$		0			0
$581$	−5.64899			−0.234360
$582$		0			0
$583$	−53.5541	+	30.9195i	−2.21798	+	1.28055i
$584$		0			0
$585$	−5.73663	−	0.104788i	−0.237181	−	0.00433244i
$586$		0			0
$587$	−19.1737	−	11.0700i	−0.791384	−	0.456906i	0.0490654	−	0.998796i	$-0.484376\pi$
$587$	−19.1737	−	11.0700i	−0.791384	−	0.456906i	−0.840450	+	0.541890i	$0.817709\pi$
$588$		0			0
$589$	−1.48735	−	1.67094i	−0.0612853	−	0.0688498i
$590$		0			0
$591$	−11.6002	+	20.0921i	−0.477167	+	0.826477i
$592$		0			0
$593$	35.8131	−	20.6767i	1.47067	−	0.849089i	0.471208	−	0.882022i	$-0.343818\pi$
$593$	35.8131	−	20.6767i	1.47067	−	0.849089i	0.999458	+	0.0329325i	$0.0104846\pi$
$594$		0			0
$595$	−0.0746377	+	4.08606i	−0.00305985	+	0.167512i
$596$		0			0
$597$		−	19.3258i		−	0.790954i
$598$		0			0
$599$	16.8243	+	29.1406i	0.687423	+	1.19065i	0.972669	+	0.232197i	$0.0745912\pi$
$599$	16.8243	+	29.1406i	0.687423	+	1.19065i	−0.285246	+	0.958454i	$0.592075\pi$
$600$		0			0
$601$	38.4939			1.57020			0.785100	−	0.619369i	$-0.212611\pi$
$601$	38.4939			1.57020			0.785100	+	0.619369i	$0.212611\pi$
$602$		0			0
$603$	−0.699598	−	0.403913i	−0.0284899	−	0.0164486i
$604$		0			0
$605$	−32.5281	+	17.9962i	−1.32246	+	0.731648i
$606$		0			0
$607$			22.1827i			0.900367i	0.892936	+	0.450183i	$0.148641\pi$
$607$			22.1827i			0.900367i	−0.892936	+	0.450183i	$0.851359\pi$
$608$		0			0
$609$	−5.60665			−0.227193
$610$		0			0
$611$	−3.97629	+	6.88714i	−0.160864	+	0.278624i
$612$		0			0
$613$	−4.13445	−	2.38703i	−0.166989	−	0.0964111i	0.414176	−	0.910197i	$-0.364070\pi$
$613$	−4.13445	−	2.38703i	−0.166989	−	0.0964111i	−0.581165	+	0.813786i	$0.697403\pi$
$614$		0			0
$615$	12.5542	−	20.8554i	0.506235	−	0.840970i
$616$		0			0
$617$	14.0178	−	8.09317i	0.564335	−	0.325819i	−0.190549	−	0.981678i	$-0.561027\pi$
$617$	14.0178	−	8.09317i	0.564335	−	0.325819i	0.754883	+	0.655859i	$0.227693\pi$
$618$		0			0
$619$	−13.4892			−0.542176			−0.271088	−	0.962555i	$-0.587383\pi$
$619$	−13.4892			−0.542176			−0.271088	+	0.962555i	$0.587383\pi$
$620$		0			0
$621$	15.9626	+	27.6480i	0.640556	+	1.10948i
$622$		0			0
$623$	8.50684	−	4.91143i	0.340819	−	0.196772i
$624$		0			0
$625$	−10.8863	−	22.5053i	−0.435453	−	0.900212i
$626$		0			0
$627$	−9.33268	−	45.1720i	−0.372711	−	1.80400i
$628$		0			0
$629$	3.78832	−	6.56156i	0.151050	−	0.261626i
$630$		0			0
$631$	13.2207	+	22.8989i	0.526308	+	0.911592i	0.999530	+	0.0306488i	$0.00975735\pi$
$631$	13.2207	+	22.8989i	0.526308	+	0.911592i	−0.473222	+	0.880943i	$0.656909\pi$
$632$		0			0
$633$	−25.3942	+	14.6613i	−1.00933	+	0.582735i
$634$		0			0
$635$	−23.7348	+	39.4288i	−0.941886	+	1.56469i
$636$		0			0
$637$	10.9542	−	6.32441i	0.434021	−	0.250582i
$638$		0			0
$639$	11.9780			0.473842
$640$		0			0
$641$	−4.27817	+	7.41000i	−0.168977	+	0.292677i	−0.938061	−	0.346471i	$-0.887380\pi$
$641$	−4.27817	+	7.41000i	−0.168977	+	0.292677i	0.769083	+	0.639149i	$0.220713\pi$
$642$		0			0
$643$	24.2681	+	14.0112i	0.957042	+	0.552548i	0.895261	−	0.445541i	$-0.146989\pi$
$643$	24.2681	+	14.0112i	0.957042	+	0.552548i	0.0617804	+	0.998090i	$0.480322\pi$
$644$		0			0
$645$	49.0790	+	29.5438i	1.93248	+	1.16329i
$646$		0			0
$647$			9.57376i			0.376383i	0.982132	+	0.188192i	$0.0602626\pi$
$647$			9.57376i			0.376383i	−0.982132	+	0.188192i	$0.939737\pi$
$648$		0			0
$649$	0.0924936	−	0.160204i	0.00363069	−	0.00628854i
$650$		0			0
$651$	−0.694367	+	1.20268i	−0.0272144	+	0.0471367i
$652$		0			0
$653$			16.4168i			0.642439i	0.947005	+	0.321219i	$0.104093\pi$
$653$			16.4168i			0.642439i	−0.947005	+	0.321219i	$0.895907\pi$
$654$		0			0
$655$	0.438238	−	23.9914i	0.0171234	−	0.937423i
$656$		0			0
$657$			2.26801i			0.0884837i
$658$		0			0
$659$	7.08162	+	12.2657i	0.275861	+	0.477805i	0.970352	−	0.241697i	$-0.0777039\pi$
$659$	7.08162	+	12.2657i	0.275861	+	0.477805i	−0.694491	+	0.719501i	$0.744371\pi$
$660$		0			0
$661$	−18.5170	−	32.0724i	−0.720229	−	1.24747i	−0.960908	−	0.276868i	$-0.910704\pi$
$661$	−18.5170	−	32.0724i	−0.720229	−	1.24747i	0.240679	−	0.970605i	$-0.422630\pi$
$662$		0			0
$663$	5.77452	+	3.33392i	0.224264	+	0.129479i
$664$		0			0
$665$	4.34552	−	12.3582i	0.168512	−	0.479230i
$666$		0			0
$667$	−14.6178	−	8.43960i	−0.566004	−	0.326783i
$668$		0			0
$669$	−7.62324	−	13.2038i	−0.294732	−	0.510490i
$670$		0			0
$671$	2.72453	+	4.71903i	0.105179	+	0.182176i
$672$		0			0
$673$		−	42.3293i		−	1.63167i	−0.578282	−	0.815837i	$-0.696277\pi$
$673$		−	42.3293i		−	1.63167i	0.578282	−	0.815837i	$-0.303723\pi$
$674$		0			0
$675$	16.6000	+	10.4103i	0.638933	+	0.400693i
$676$		0			0
$677$			13.9856i			0.537510i	0.963209	+	0.268755i	$0.0866121\pi$
$677$			13.9856i			0.537510i	−0.963209	+	0.268755i	$0.913388\pi$
$678$		0			0
$679$	0.560033	−	0.970005i	0.0214921	−	0.0372254i
$680$		0			0
$681$	−6.91222	+	11.9723i	−0.264877	+	0.458780i
$682$		0			0
$683$			11.6668i			0.446416i	0.974771	+	0.223208i	$0.0716529\pi$
$683$			11.6668i			0.446416i	−0.974771	+	0.223208i	$0.928347\pi$
$684$		0			0
$685$	−19.8106	+	32.9099i	−0.756925	+	1.25742i
$686$		0			0
$687$	38.5356	+	22.2486i	1.47023	+	0.848835i
$688$		0			0
$689$	−14.3273	+	24.8156i	−0.545825	+	0.945397i
$690$		0			0
$691$	−15.7886			−0.600627			−0.300313	−	0.953841i	$-0.597091\pi$
$691$	−15.7886			−0.600627			−0.300313	+	0.953841i	$0.597091\pi$
$692$		0			0
$693$	−6.44542	+	3.72127i	−0.244841	+	0.141359i
$694$		0			0
$695$	−14.0384	−	8.45062i	−0.532506	−	0.320550i
$696$		0			0
$697$	−6.36760	+	3.67633i	−0.241190	+	0.139251i
$698$		0			0
$699$	2.99675	+	5.19053i	0.113348	+	0.196324i
$700$		0			0
$701$	2.64450	−	4.58042i	0.0998816	−	0.173000i	−0.811754	−	0.584000i	$-0.801487\pi$
$701$	2.64450	−	4.58042i	0.0998816	−	0.173000i	0.911635	+	0.411000i	$0.134820\pi$
$702$		0			0
$703$	−18.1409	+	16.1478i	−0.684198	+	0.609025i
$704$		0			0
$705$	−12.8631	+	7.11648i	−0.484451	+	0.268022i
$706$		0			0
$707$	17.2497	−	9.95913i	0.648743	−	0.374552i
$708$		0			0
$709$	−12.2529	−	21.2226i	−0.460166	−	0.797031i	0.538803	−	0.842432i	$-0.318877\pi$
$709$	−12.2529	−	21.2226i	−0.460166	−	0.797031i	−0.998969	+	0.0454011i	$0.985543\pi$
$710$		0			0
$711$	−13.6664			−0.512529
$712$		0			0
$713$	−3.62075	+	2.09044i	−0.135598	+	0.0782875i
$714$		0			0
$715$	−14.7619	+	24.5229i	−0.552064	+	0.917104i
$716$		0			0
$717$	−16.9355	−	9.77771i	−0.632467	−	0.365155i
$718$		0			0
$719$	22.4239	−	38.8393i	0.836269	−	1.44846i	−0.0567236	−	0.998390i	$-0.518065\pi$
$719$	22.4239	−	38.8393i	0.836269	−	1.44846i	0.892993	−	0.450071i	$-0.148601\pi$
$720$		0			0
$721$	12.6442			0.470896
$722$		0			0
$723$		−	37.6209i		−	1.39914i
$724$		0			0
$725$	−10.3528	−	0.378343i	−0.384493	−	0.0140513i
$726$		0			0
$727$	28.1376	+	16.2453i	1.04357	+	0.602504i	0.920842	−	0.389937i	$-0.127503\pi$
$727$	28.1376	+	16.2453i	1.04357	+	0.602504i	0.122726	+	0.992441i	$0.460836\pi$
$728$		0			0
$729$	−12.0273			−0.445457
$730$		0			0
$731$	−8.65152	−	14.9849i	−0.319988	−	0.554235i
$732$		0			0
$733$		−	11.1969i		−	0.413568i	−0.978387	−	0.206784i	$-0.933700\pi$
$733$		−	11.1969i		−	0.413568i	0.978387	−	0.206784i	$-0.0662998\pi$
$734$		0			0
$735$	23.3776	+	0.427025i	0.862294	+	0.0157510i
$736$		0			0
$737$	−3.49009	+	2.01501i	−0.128559	+	0.0742237i
$738$		0			0
$739$	0.466361	−	0.807761i	0.0171554	−	0.0297140i	−0.857320	−	0.514783i	$-0.827872\pi$
$739$	0.466361	−	0.807761i	0.0171554	−	0.0297140i	0.874476	+	0.485069i	$0.161206\pi$
$740$		0			0
$741$	−14.2109	−	15.9650i	−0.522051	−	0.586488i
$742$		0			0
$743$	23.3370	+	13.4736i	0.856153	+	0.494300i	0.862722	−	0.505678i	$-0.168758\pi$
$743$	23.3370	+	13.4736i	0.856153	+	0.494300i	−0.00656939	+	0.999978i	$0.502091\pi$
$744$		0			0
$745$	0.500098	−	27.3780i	0.0183222	−	1.00305i
$746$		0			0
$747$	−3.83492	+	2.21409i	−0.140312	+	0.0810094i
$748$		0			0
$749$	17.2210			0.629240
$750$		0			0
$751$	2.33645	+	4.04686i	0.0852584	+	0.147672i	0.905501	−	0.424343i	$-0.139495\pi$
$751$	2.33645	+	4.04686i	0.0852584	+	0.147672i	−0.820243	+	0.572015i	$0.806162\pi$
$752$		0			0
$753$		−	8.45971i		−	0.308289i
$754$		0			0
$755$	22.6136	−	12.5109i	0.822992	−	0.455320i
$756$		0			0
$757$	−37.0902	−	21.4140i	−1.34807	−	0.778306i	−0.360090	−	0.932918i	$-0.617254\pi$
$757$	−37.0902	−	21.4140i	−1.34807	−	0.778306i	−0.987975	+	0.154612i	$0.950587\pi$
$758$		0			0
$759$	−86.2073			−3.12913
$760$		0			0
$761$	−16.7169			−0.605987			−0.302994	−	0.952993i	$-0.597986\pi$
$761$	−16.7169			−0.605987			−0.302994	+	0.952993i	$0.597986\pi$
$762$		0			0
$763$	2.32065	+	1.33983i	0.0840131	+	0.0485050i
$764$		0			0
$765$	1.55084	+	2.80315i	0.0560707	+	0.101348i
$766$		0			0
$767$		−	0.0857182i		−	0.00309511i
$768$		0			0
$769$	−7.70852	−	13.3516i	−0.277976	−	0.481469i	0.692905	−	0.721029i	$-0.256330\pi$
$769$	−7.70852	−	13.3516i	−0.277976	−	0.481469i	−0.970882	+	0.239559i	$0.922997\pi$
$770$		0			0
$771$	56.7128			2.04246
$772$		0			0
$773$	−28.7343	+	16.5897i	−1.03350	+	0.596691i	−0.917985	−	0.396615i	$-0.870185\pi$
$773$	−28.7343	+	16.5897i	−1.03350	+	0.596691i	−0.115514	+	0.993306i	$0.536852\pi$
$774$		0			0
$775$	−1.36332	+	2.17391i	−0.0489719	+	0.0780892i
$776$		0			0
$777$	13.0572	+	7.53856i	0.468423	+	0.270444i
$778$		0			0
$779$	23.0813	−	4.76868i	0.826975	−	0.170856i
$780$		0			0
$781$	29.8774	−	51.7491i	1.06910	−	1.85173i
$782$		0			0
$783$	7.03179	−	4.05980i	0.251296	−	0.145086i
$784$		0			0
$785$	9.25562	+	0.169067i	0.330347	+	0.00603427i
$786$		0			0
$787$			12.8318i			0.457405i	0.973496	+	0.228702i	$0.0734483\pi$
$787$			12.8318i			0.457405i	−0.973496	+	0.228702i	$0.926552\pi$
$788$		0			0
$789$	2.72657	+	4.72256i	0.0970685	+	0.168128i

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:	\( N \)	\(=\)	\( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	380.r (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.74361 - 1.00667i) q^{3} +(-1.95659 + 1.08248i) q^{5} +1.34403i q^{7} +(0.526784 + 0.912416i) q^{9} +O(q^{10})\)	\(q+(-1.74361 - 1.00667i) q^{3} +(-1.95659 + 1.08248i) q^{5} +1.34403i q^{7} +(0.526784 + 0.912416i) q^{9} +5.25594 q^{11} +(2.10918 - 1.21773i) q^{13} +(4.50123 + 0.0822214i) q^{15} +(-1.17765 - 0.679914i) q^{17} +(2.89815 + 3.25587i) q^{19} +(1.35300 - 2.34346i) q^{21} +(7.05514 - 4.07329i) q^{23} +(2.65647 - 4.23594i) q^{25} +3.91884i q^{27} +(-1.03597 - 1.79435i) q^{29} -0.513207 q^{31} +(-9.16431 - 5.29102i) q^{33} +(-1.45488 - 2.62971i) q^{35} +5.57175i q^{37} -4.90344 q^{39} +(2.70353 - 4.68265i) q^{41} +(11.0197 + 6.36221i) q^{43} +(-2.01837 - 1.21499i) q^{45} +(-2.82785 + 1.63266i) q^{47} +5.19359 q^{49} +(1.36890 + 2.37101i) q^{51} +(-10.1892 + 5.88276i) q^{53} +(-10.2837 + 5.68946i) q^{55} +(-1.77564 - 8.59447i) q^{57} +(0.0175979 - 0.0304805i) q^{59} +(0.518372 + 0.897846i) q^{61} +(-1.22631 + 0.708011i) q^{63} +(-2.80861 + 4.66574i) q^{65} +(-0.664028 + 0.383377i) q^{67} -16.4019 q^{69} +(5.68450 - 9.84583i) q^{71} +(1.86429 + 1.07635i) q^{73} +(-8.89606 + 4.71162i) q^{75} +7.06413i q^{77} +(-6.48576 + 11.2337i) q^{79} +(5.52535 - 9.57019i) q^{81} +4.20304i q^{83} +(3.04016 + 0.0555329i) q^{85} +4.17153i q^{87} +(-3.65426 - 6.32937i) q^{89} +(1.63667 + 2.83479i) q^{91} +(0.894833 + 0.516632i) q^{93} +(-9.19491 - 3.23321i) q^{95} +(-0.721716 - 0.416683i) q^{97} +(2.76875 + 4.79561i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + q^{5} + 10 q^{9}+O(q^{10}) \) 20 * q + q^5 + 10 * q^9	\( 20 q + q^{5} + 10 q^{9} - 5 q^{15} + 14 q^{19} - 8 q^{21} + 9 q^{25} - 16 q^{29} + 8 q^{31} - 2 q^{35} - 8 q^{39} + 26 q^{41} - 32 q^{45} - 44 q^{49} + 26 q^{51} - 12 q^{55} + 4 q^{59} + 2 q^{61} - 18 q^{65} + 48 q^{69} - 2 q^{71} + 46 q^{75} - 16 q^{79} + 26 q^{81} - 39 q^{85} - 40 q^{89} - 4 q^{91} - 43 q^{95} - 20 q^{99}+O(q^{100}) \) 20 * q + q^5 + 10 * q^9 - 5 * q^15 + 14 * q^19 - 8 * q^21 + 9 * q^25 - 16 * q^29 + 8 * q^31 - 2 * q^35 - 8 * q^39 + 26 * q^41 - 32 * q^45 - 44 * q^49 + 26 * q^51 - 12 * q^55 + 4 * q^59 + 2 * q^61 - 18 * q^65 + 48 * q^69 - 2 * q^71 + 46 * q^75 - 16 * q^79 + 26 * q^81 - 39 * q^85 - 40 * q^89 - 4 * q^91 - 43 * q^95 - 20 * q^99

Introduction

L-functions

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