LMFDB - Embedded newform 260.2.x.a.101.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [260,2,Mod(101,260)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(260, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 5]))

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

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

chi := DirichletCharacter("260.101");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 260 = 2^{2} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	260.x (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:	$2.07611045255$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	8.0.22581504.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} + 5x^{6} + 2x^{5} - 11x^{4} + 4x^{3} + 20x^{2} - 32x + 16 $ x^8 - 4x^7 + 5x^6 + 2x^5 - 11x^4 + 4x^3 + 20x^2 - 32*x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			101.3
Root			$0.665665 - 1.24775i$ of defining polynomial
Character	$\chi$	$=$	260.101
Dual form			260.2.x.a.121.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.0473938 + 0.0820885i) q^{3} -1.00000i q^{5} +(0.716063 + 0.413419i) q^{7} +(1.49551 - 2.59030i) q^{9} +O(q^{10})$	\(q+(0.0473938 + 0.0820885i) q^{3} -1.00000i q^{5} +(0.716063 + 0.413419i) q^{7} +(1.49551 - 2.59030i) q^{9} +(1.50000 - 0.866025i) q^{11} +(3.32235 + 1.40072i) q^{13} +(0.0820885 - 0.0473938i) q^{15} +(0.716063 - 1.24026i) q^{17} +(-0.926118 - 0.534695i) q^{19} +0.0783740i q^{21} +(1.54290 + 2.67238i) q^{23} -1.00000 q^{25} +0.567874 q^{27} +(-3.72756 - 6.45632i) q^{29} +5.84325i q^{31} +(0.142181 + 0.0820885i) q^{33} +(0.413419 - 0.716063i) q^{35} +(0.851811 - 0.491793i) q^{37} +(0.0424756 + 0.339112i) q^{39} +(-3.69615 + 2.13397i) q^{41} +(-4.77046 + 8.26268i) q^{43} +(-2.59030 - 1.49551i) q^{45} +3.46410i q^{47} +(-3.15817 - 5.47011i) q^{49} +0.135748 q^{51} +0.334308 q^{53} +(-0.866025 - 1.50000i) q^{55} -0.101365i q^{57} +(-9.98052 - 5.76225i) q^{59} +(-1.35824 + 2.35255i) q^{61} +(2.14176 - 1.23654i) q^{63} +(1.40072 - 3.32235i) q^{65} +(-11.9122 + 6.87752i) q^{67} +(-0.146248 + 0.253309i) q^{69} +(8.46704 + 4.88845i) q^{71} +11.1806i q^{73} +(-0.0473938 - 0.0820885i) q^{75} +1.43213 q^{77} -0.252387 q^{79} +(-4.45961 - 7.72427i) q^{81} +5.67165i q^{83} +(-1.24026 - 0.716063i) q^{85} +(0.353326 - 0.611979i) q^{87} +(3.98052 - 2.29815i) q^{89} +(1.79992 + 2.37653i) q^{91} +(-0.479664 + 0.276934i) q^{93} +(-0.534695 + 0.926118i) q^{95} +(8.25698 + 4.76717i) q^{97} -5.18059i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 2 q^{3} + 6 q^{7} - 4 q^{9}+O(q^{10}) $ 8 * q - 2 * q^3 + 6 * q^7 - 4 * q^9	$ 8 q - 2 q^{3} + 6 q^{7} - 4 q^{9} + 12 q^{11} - 8 q^{13} - 6 q^{15} + 6 q^{17} - 6 q^{23} - 8 q^{25} + 4 q^{27} - 6 q^{33} - 6 q^{35} + 6 q^{37} - 4 q^{39} + 12 q^{41} + 10 q^{43} - 4 q^{49} + 24 q^{53} - 24 q^{59} - 4 q^{61} + 24 q^{63} - 54 q^{67} - 24 q^{69} - 36 q^{71} + 2 q^{75} + 12 q^{77} - 16 q^{79} + 8 q^{81} + 18 q^{85} - 6 q^{87} - 24 q^{89} + 24 q^{93} - 30 q^{97}+O(q^{100}) $ 8 * q - 2 * q^3 + 6 * q^7 - 4 * q^9 + 12 * q^11 - 8 * q^13 - 6 * q^15 + 6 * q^17 - 6 * q^23 - 8 * q^25 + 4 * q^27 - 6 * q^33 - 6 * q^35 + 6 * q^37 - 4 * q^39 + 12 * q^41 + 10 * q^43 - 4 * q^49 + 24 * q^53 - 24 * q^59 - 4 * q^61 + 24 * q^63 - 54 * q^67 - 24 * q^69 - 36 * q^71 + 2 * q^75 + 12 * q^77 - 16 * q^79 + 8 * q^81 + 18 * q^85 - 6 * q^87 - 24 * q^89 + 24 * q^93 - 30 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/260\mathbb{Z}\right)^\times$.

$n$	$41$	$131$	$157$
$\chi(n)$	$e\left(\frac{5}{6}\right)$	$1$	$1$

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
$3$	0.0473938	+	0.0820885i	0.0273628	+	0.0473938i	0.879383	−	0.476116i	$-0.157956\pi$
$3$	0.0473938	+	0.0820885i	0.0273628	+	0.0473938i	−0.852020	+	0.523510i	$0.824622\pi$
$4$		0			0
$5$		−	1.00000i		−	0.447214i
$5$		−	1.00000i		−	0.447214i
$6$		0			0
$7$	0.716063	+	0.413419i	0.270646	+	0.156258i	0.629181	−	0.777259i	$-0.283390\pi$
$7$	0.716063	+	0.413419i	0.270646	+	0.156258i	−0.358535	+	0.933516i	$0.616724\pi$
$8$		0			0
$9$	1.49551	−	2.59030i	0.498503	−	0.863432i
$10$		0			0
$11$	1.50000	−	0.866025i	0.452267	−	0.261116i	−0.256520	−	0.966539i	$-0.582576\pi$
$11$	1.50000	−	0.866025i	0.452267	−	0.261116i	0.708787	+	0.705422i	$0.249243\pi$
$12$		0			0
$13$	3.32235	+	1.40072i	0.921453	+	0.388490i
$13$	3.32235	+	1.40072i	0.921453	+	0.388490i
$14$		0			0
$15$	0.0820885	−	0.0473938i	0.0211951	−	0.0122370i
$16$		0			0
$17$	0.716063	−	1.24026i	0.173671	−	0.300807i	−0.766030	−	0.642805i	$-0.777770\pi$
$17$	0.716063	−	1.24026i	0.173671	−	0.300807i	0.939700	+	0.341999i	$0.111104\pi$
$18$		0			0
$19$	−0.926118	−	0.534695i	−0.212466	−	0.122667i	0.389991	−	0.920819i	$-0.372478\pi$
$19$	−0.926118	−	0.534695i	−0.212466	−	0.122667i	−0.602457	+	0.798151i	$0.705811\pi$
$20$		0			0
$21$			0.0783740i			0.0171026i
$22$		0			0
$23$	1.54290	+	2.67238i	0.321717	+	0.557231i	0.980842	−	0.194803i	$-0.0624066\pi$
$23$	1.54290	+	2.67238i	0.321717	+	0.557231i	−0.659125	+	0.752033i	$0.729073\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$	0.567874			0.109287
$28$		0			0
$29$	−3.72756	−	6.45632i	−0.692190	−	1.19891i	−0.971119	−	0.238597i	$-0.923313\pi$
$29$	−3.72756	−	6.45632i	−0.692190	−	1.19891i	0.278928	−	0.960312i	$-0.410021\pi$
$30$		0			0
$31$			5.84325i			1.04948i	0.851263	+	0.524740i	$0.175837\pi$
$31$			5.84325i			1.04948i	−0.851263	+	0.524740i	$0.824163\pi$
$32$		0			0
$33$	0.142181	+	0.0820885i	0.0247506	+	0.0142898i
$34$		0			0
$35$	0.413419	−	0.716063i	0.0698806	−	0.121037i
$36$		0			0
$37$	0.851811	−	0.491793i	0.140037	−	0.0808503i	−0.428345	−	0.903615i	$-0.640903\pi$
$37$	0.851811	−	0.491793i	0.140037	−	0.0808503i	0.568382	+	0.822765i	$0.307570\pi$
$38$		0			0
$39$	0.0424756	+	0.339112i	0.00680154	+	0.0543013i
$40$		0			0
$41$	−3.69615	+	2.13397i	−0.577242	+	0.333271i	−0.760037	−	0.649880i	$-0.774819\pi$
$41$	−3.69615	+	2.13397i	−0.577242	+	0.333271i	0.182795	+	0.983151i	$0.441486\pi$
$42$		0			0
$43$	−4.77046	+	8.26268i	−0.727488	+	1.26005i	0.230453	+	0.973083i	$0.425979\pi$
$43$	−4.77046	+	8.26268i	−0.727488	+	1.26005i	−0.957942	+	0.286963i	$0.907354\pi$
$44$		0			0
$45$	−2.59030	−	1.49551i	−0.386138	−	0.222937i
$46$		0			0
$47$			3.46410i			0.505291i	0.967559	+	0.252646i	$0.0813007\pi$
$47$			3.46410i			0.505291i	−0.967559	+	0.252646i	$0.918699\pi$
$48$		0			0
$49$	−3.15817	−	5.47011i	−0.451167	−	0.781444i
$50$		0			0
$51$	0.135748			0.0190085
$52$		0			0
$53$	0.334308			0.0459207			0.0229603	−	0.999736i	$-0.492691\pi$
$53$	0.334308			0.0459207			0.0229603	+	0.999736i	$0.492691\pi$
$54$		0			0
$55$	−0.866025	−	1.50000i	−0.116775	−	0.202260i
$56$		0			0
$57$		−	0.101365i		−	0.0134261i
$58$		0			0
$59$	−9.98052	−	5.76225i	−1.29935	−	0.750181i	−0.319060	−	0.947734i	$-0.603367\pi$
$59$	−9.98052	−	5.76225i	−1.29935	−	0.750181i	−0.980292	+	0.197553i	$0.936701\pi$
$60$		0			0
$61$	−1.35824	+	2.35255i	−0.173905	+	0.301213i	−0.939782	−	0.341775i	$-0.888972\pi$
$61$	−1.35824	+	2.35255i	−0.173905	+	0.301213i	0.765877	+	0.642988i	$0.222305\pi$
$62$		0			0
$63$	2.14176	−	1.23654i	0.269836	−	0.155790i
$64$		0			0
$65$	1.40072	−	3.32235i	0.173738	−	0.412086i
$66$		0			0
$67$	−11.9122	+	6.87752i	−1.45531	+	0.840223i	−0.998775	−	0.0494832i	$-0.984243\pi$
$67$	−11.9122	+	6.87752i	−1.45531	+	0.840223i	−0.456534	+	0.889706i	$0.650909\pi$
$68$		0			0
$69$	−0.146248	+	0.253309i	−0.0176062	+	0.0304948i
$70$		0			0
$71$	8.46704	+	4.88845i	1.00485	+	0.580152i	0.909680	−	0.415309i	$-0.136327\pi$
$71$	8.46704	+	4.88845i	1.00485	+	0.580152i	0.0951721	+	0.995461i	$0.469660\pi$
$72$		0			0
$73$			11.1806i			1.30859i	0.756240	+	0.654295i	$0.227034\pi$
$73$			11.1806i			1.30859i	−0.756240	+	0.654295i	$0.772966\pi$
$74$		0			0
$75$	−0.0473938	−	0.0820885i	−0.00547256	−	0.00947876i
$76$		0			0
$77$	1.43213			0.163206
$78$		0			0
$79$	−0.252387			−0.0283958			−0.0141979	−	0.999899i	$-0.504519\pi$
$79$	−0.252387			−0.0283958			−0.0141979	+	0.999899i	$0.504519\pi$
$80$		0			0
$81$	−4.45961	−	7.72427i	−0.495512	−	0.858252i
$82$		0			0
$83$			5.67165i			0.622544i	0.950321	+	0.311272i	$0.100755\pi$
$83$			5.67165i			0.622544i	−0.950321	+	0.311272i	$0.899245\pi$
$84$		0			0
$85$	−1.24026	−	0.716063i	−0.134525	−	0.0776679i
$86$		0			0
$87$	0.353326	−	0.611979i	0.0378806	−	0.0656110i
$88$		0			0
$89$	3.98052	−	2.29815i	0.421934	−	0.243604i	−0.273971	−	0.961738i	$-0.588337\pi$
$89$	3.98052	−	2.29815i	0.421934	−	0.243604i	0.695904	+	0.718135i	$0.255004\pi$
$90$		0			0
$91$	1.79992	+	2.37653i	0.188683	+	0.249128i
$92$		0			0
$93$	−0.479664	+	0.276934i	−0.0497388	+	0.0287167i
$94$		0			0
$95$	−0.534695	+	0.926118i	−0.0548585	+	0.0950177i
$96$		0			0
$97$	8.25698	+	4.76717i	0.838370	+	0.484033i	0.856710	−	0.515799i	$-0.172505\pi$
$97$	8.25698	+	4.76717i	0.838370	+	0.484033i	−0.0183401	+	0.999832i	$0.505838\pi$
$98$		0			0
$99$		−	5.18059i		−	0.520669i
$100$		0			0
$101$	−2.90072	−	5.02419i	−0.288632	−	0.499926i	0.684851	−	0.728683i	$-0.259867\pi$
$101$	−2.90072	−	5.02419i	−0.288632	−	0.499926i	−0.973484	+	0.228757i	$0.926534\pi$
$102$		0			0
$103$	10.0760			0.992814			0.496407	−	0.868090i	$-0.334652\pi$
$103$	10.0760			0.992814			0.496407	+	0.868090i	$0.334652\pi$
$104$		0			0
$105$	0.0783740			0.00764852
$106$		0			0
$107$	−8.13977	−	14.0985i	−0.786902	−	1.36295i	−0.927856	−	0.372938i	$-0.878351\pi$
$107$	−8.13977	−	14.0985i	−0.786902	−	1.36295i	0.140955	−	0.990016i	$-0.454983\pi$
$108$		0			0
$109$			3.12979i			0.299780i	0.988703	+	0.149890i	$0.0478919\pi$
$109$			3.12979i			0.299780i	−0.988703	+	0.149890i	$0.952108\pi$
$110$		0			0
$111$	0.0807411	+	0.0466159i	0.00766361	+	0.00442458i
$112$		0			0
$113$	5.08538	−	8.80813i	0.478392	−	0.828599i	−0.521301	−	0.853373i	$-0.674553\pi$
$113$	5.08538	−	8.80813i	0.478392	−	0.828599i	0.999693	+	0.0247735i	$0.00788647\pi$
$114$		0			0
$115$	2.67238	−	1.54290i	0.249201	−	0.143876i
$116$		0			0
$117$	8.59687	−	6.51107i	0.794781	−	0.601949i
$118$		0			0
$119$	1.02549	−	0.592068i	0.0940068	−	0.0542748i
$120$		0			0
$121$	−4.00000	+	6.92820i	−0.363636	+	0.629837i
$122$		0			0
$123$	−0.350349	−	0.202274i	−0.0315899	−	0.0182385i
$124$		0			0
$125$			1.00000i			0.0894427i
$126$		0			0
$127$	2.98401	+	5.16846i	0.264788	+	0.458627i	0.967508	−	0.252840i	$-0.0813646\pi$
$127$	2.98401	+	5.16846i	0.264788	+	0.458627i	−0.702720	+	0.711467i	$0.748031\pi$
$128$		0			0
$129$	−0.904361			−0.0796245
$130$		0			0
$131$	16.6267			1.45268			0.726342	−	0.687334i	$-0.241219\pi$
$131$	16.6267			1.45268			0.726342	+	0.687334i	$0.241219\pi$
$132$		0			0
$133$	−0.442106	−	0.765750i	−0.0383355	−	0.0663990i
$134$		0			0
$135$		−	0.567874i		−	0.0488748i
$136$		0			0
$137$	−0.350349	−	0.202274i	−0.0299324	−	0.0172815i	0.484959	−	0.874537i	$-0.338834\pi$
$137$	−0.350349	−	0.202274i	−0.0299324	−	0.0172815i	−0.514892	+	0.857255i	$0.672168\pi$
$138$		0			0
$139$	4.65817	−	8.06819i	0.395101	−	0.684335i	−0.598013	−	0.801486i	$-0.704043\pi$
$139$	4.65817	−	8.06819i	0.395101	−	0.684335i	0.993114	+	0.117152i	$0.0373763\pi$
$140$		0			0
$141$	−0.284363	+	0.164177i	−0.0239477	+	0.0138262i
$142$		0			0
$143$	6.19658	−	0.776156i	0.518184	−	0.0649054i
$144$		0			0
$145$	−6.45632	+	3.72756i	−0.536168	+	0.309557i
$146$		0			0
$147$	0.299355	−	0.518498i	0.0246904	−	0.0427650i
$148$		0			0
$149$	9.41179	+	5.43390i	0.771044	+	0.445162i	0.833247	−	0.552901i	$-0.186479\pi$
$149$	9.41179	+	5.43390i	0.771044	+	0.445162i	−0.0622030	+	0.998064i	$0.519813\pi$
$150$		0			0
$151$		−	0.991015i		−	0.0806477i	−0.999187	−	0.0403238i	$-0.987161\pi$
$151$		−	0.991015i		−	0.0806477i	0.999187	−	0.0403238i	$-0.0128390\pi$
$152$		0			0
$153$	−2.14176	−	3.70963i	−0.173151	−	0.299906i
$154$		0			0
$155$	5.84325			0.469341
$156$		0			0
$157$	17.5729			1.40247			0.701235	−	0.712930i	$-0.252632\pi$
$157$	17.5729			1.40247			0.701235	+	0.712930i	$0.252632\pi$
$158$		0			0
$159$	0.0158441	+	0.0274428i	0.00125652	+	0.00217636i
$160$		0			0
$161$			2.55146i			0.201083i
$162$		0			0
$163$	−14.9666	−	8.64098i	−1.17228	−	0.676814i	−0.218061	−	0.975935i	$-0.569973\pi$
$163$	−14.9666	−	8.64098i	−1.17228	−	0.676814i	−0.954215	+	0.299122i	$0.903306\pi$
$164$		0			0
$165$	0.0820885	−	0.142181i	0.00639058	−	0.0110688i
$166$		0			0
$167$	−2.64965	+	1.52978i	−0.205036	+	0.118378i	−0.599002	−	0.800747i	$-0.704436\pi$
$167$	−2.64965	+	1.52978i	−0.205036	+	0.118378i	0.393966	+	0.919125i	$0.371103\pi$
$168$		0			0
$169$	9.07597	+	9.30735i	0.698151	+	0.715950i
$170$		0			0
$171$	−2.77003	+	1.59928i	−0.211830	+	0.122300i
$172$		0			0
$173$	−1.71006	+	2.96190i	−0.130013	+	0.225189i	−0.923681	−	0.383161i	$-0.874835\pi$
$173$	−1.71006	+	2.96190i	−0.130013	+	0.225189i	0.793668	+	0.608351i	$0.208169\pi$
$174$		0			0
$175$	−0.716063	−	0.413419i	−0.0541293	−	0.0312516i
$176$		0			0
$177$		−	1.09238i		−	0.0821083i
$178$		0			0
$179$	−5.19109	−	8.99123i	−0.388000	−	0.672036i	0.604180	−	0.796848i	$-0.293501\pi$
$179$	−5.19109	−	8.99123i	−0.388000	−	0.672036i	−0.992180	+	0.124811i	$0.960167\pi$
$180$		0			0
$181$	−10.3492			−0.769247			−0.384624	−	0.923073i	$-0.625669\pi$
$181$	−10.3492			−0.769247			−0.384624	+	0.923073i	$0.625669\pi$
$182$		0			0
$183$	−0.257489			−0.0190342
$184$		0			0
$185$	−0.491793	−	0.851811i	−0.0361574	−	0.0626264i
$186$		0			0
$187$		−	2.48052i		−	0.181393i
$188$		0			0
$189$	0.406634	+	0.234770i	0.0295782	+	0.0170770i
$190$		0			0
$191$	7.75296	−	13.4285i	0.560984	−	0.971653i	−0.436427	−	0.899740i	$-0.643756\pi$
$191$	7.75296	−	13.4285i	0.560984	−	0.971653i	0.997411	−	0.0719134i	$-0.0229105\pi$
$192$		0			0
$193$	4.82401	−	2.78514i	0.347239	−	0.200479i	−0.316229	−	0.948683i	$-0.602417\pi$
$193$	4.82401	−	2.78514i	0.347239	−	0.200479i	0.663469	+	0.748204i	$0.269084\pi$
$194$		0			0
$195$	0.339112	−	0.0424756i	0.0242843	−	0.00304174i
$196$		0			0
$197$	−21.5405	+	12.4364i	−1.53470	+	0.886058i	−0.535561	+	0.844497i	$0.679900\pi$
$197$	−21.5405	+	12.4364i	−1.53470	+	0.886058i	−0.999136	+	0.0415608i	$0.986767\pi$
$198$		0			0
$199$	−9.32443	+	16.1504i	−0.660991	+	1.14487i	0.319364	+	0.947632i	$0.396531\pi$
$199$	−9.32443	+	16.1504i	−0.660991	+	1.14487i	−0.980356	+	0.197239i	$0.936803\pi$
$200$		0			0
$201$	−1.12913	−	0.651904i	−0.0796427	−	0.0459817i
$202$		0			0
$203$		−	6.16418i		−	0.432640i
$204$		0			0
$205$	2.13397	+	3.69615i	0.149043	+	0.258150i
$206$		0			0
$207$	9.22968			0.641507
$208$		0			0
$209$	−1.85224			−0.128122
$210$		0			0
$211$	−4.82235	−	8.35255i	−0.331984	−	0.575013i	0.650917	−	0.759149i	$-0.274385\pi$
$211$	−4.82235	−	8.35255i	−0.331984	−	0.575013i	−0.982901	+	0.184136i	$0.941051\pi$
$212$		0			0
$213$			0.926728i			0.0634984i
$214$		0			0
$215$	8.26268	+	4.77046i	0.563510	+	0.325343i
$216$		0			0
$217$	−2.41571	+	4.18414i	−0.163989	+	0.284038i
$218$		0			0
$219$	−0.917797	+	0.529891i	−0.0620190	+	0.0358067i
$220$		0			0
$221$	4.11626	−	3.11756i	0.276890	−	0.209710i
$222$		0			0
$223$	−1.00558	+	0.580573i	−0.0673387	+	0.0388780i	−0.533291	−	0.845932i	$-0.679045\pi$
$223$	−1.00558	+	0.580573i	−0.0673387	+	0.0388780i	0.465953	+	0.884810i	$0.345712\pi$
$224$		0			0
$225$	−1.49551	+	2.59030i	−0.0997005	+	0.172686i
$226$		0			0
$227$	−23.5957	−	13.6230i	−1.56610	−	0.904191i	−0.996617	−	0.0821911i	$-0.973808\pi$
$227$	−23.5957	−	13.6230i	−1.56610	−	0.904191i	−0.569488	−	0.822000i	$-0.692858\pi$
$228$		0			0
$229$			24.3432i			1.60864i	0.594193	+	0.804322i	$0.297471\pi$
$229$			24.3432i			1.60864i	−0.594193	+	0.804322i	$0.702529\pi$
$230$		0			0
$231$	0.0678739	+	0.117561i	0.00446577	+	0.00773495i
$232$		0			0
$233$	−23.0238			−1.50834			−0.754171	−	0.656678i	$-0.771961\pi$
$233$	−23.0238			−1.50834			−0.754171	+	0.656678i	$0.771961\pi$
$234$		0			0
$235$	3.46410			0.225973
$236$		0			0
$237$	−0.0119616	−	0.0207181i	−0.000776989	−	0.00134578i
$238$		0			0
$239$			23.7057i			1.53340i	0.642008	+	0.766698i	$0.278102\pi$
$239$			23.7057i			1.53340i	−0.642008	+	0.766698i	$0.721898\pi$
$240$		0			0
$241$	−9.37968	−	5.41536i	−0.604198	−	0.348834i	0.166493	−	0.986043i	$-0.446756\pi$
$241$	−9.37968	−	5.41536i	−0.604198	−	0.348834i	−0.770691	+	0.637209i	$0.780089\pi$
$242$		0			0
$243$	1.27453	−	2.20754i	0.0817609	−	0.141614i
$244$		0			0
$245$	−5.47011	+	3.15817i	−0.349472	+	0.201768i
$246$		0			0
$247$	−2.32793	−	3.07367i	−0.148123	−	0.195573i
$248$		0			0
$249$	−0.465577	+	0.268801i	−0.0295047	+	0.0170346i
$250$		0			0
$251$	−0.560405	+	0.970649i	−0.0353724	+	0.0612668i	−0.883170	−	0.469054i	$-0.844595\pi$
$251$	−0.560405	+	0.970649i	−0.0353724	+	0.0612668i	0.847797	+	0.530321i	$0.177928\pi$
$252$		0			0
$253$	4.62870	+	2.67238i	0.291004	+	0.168011i
$254$		0			0
$255$		−	0.135748i		−	0.00850086i
$256$		0			0
$257$	8.31534	+	14.4026i	0.518697	+	0.898409i	0.999764	+	0.0217255i	$0.00691599\pi$
$257$	8.31534	+	14.4026i	0.518697	+	0.898409i	−0.481067	+	0.876684i	$0.659751\pi$
$258$		0			0
$259$	0.813267			0.0505340
$260$		0			0
$261$	−22.2984			−1.38023
$262$		0			0
$263$	−12.2510	−	21.2193i	−0.755427	−	1.30844i	−0.945162	−	0.326603i	$-0.894096\pi$
$263$	−12.2510	−	21.2193i	−0.755427	−	1.30844i	0.189734	−	0.981836i	$-0.439237\pi$
$264$		0			0
$265$		−	0.334308i		−	0.0205364i
$266$		0			0
$267$	0.377303	+	0.217836i	0.0230906	+	0.0133314i
$268$		0			0
$269$	3.26643	−	5.65763i	0.199158	−	0.344952i	−0.749098	−	0.662460i	$-0.769513\pi$
$269$	3.26643	−	5.65763i	0.199158	−	0.344952i	0.948256	+	0.317508i	$0.102846\pi$
$270$		0			0
$271$	4.89831	−	2.82804i	0.297551	−	0.171791i	−0.343791	−	0.939046i	$-0.611711\pi$
$271$	4.89831	−	2.82804i	0.297551	−	0.171791i	0.641342	+	0.767255i	$0.278378\pi$
$272$		0			0
$273$	−0.109780	+	0.260386i	−0.00664419	+	0.0157593i
$274$		0			0
$275$	−1.50000	+	0.866025i	−0.0904534	+	0.0522233i
$276$		0			0
$277$	1.85782	−	3.21784i	0.111626	−	0.193341i	−0.804800	−	0.593546i	$-0.797728\pi$
$277$	1.85782	−	3.21784i	0.111626	−	0.193341i	0.916426	+	0.400205i	$0.131061\pi$
$278$		0			0
$279$	15.1357	+	8.73863i	0.906154	+	0.523168i
$280$		0			0
$281$			9.70447i			0.578920i	0.957190	+	0.289460i	$0.0934758\pi$
$281$			9.70447i			0.578920i	−0.957190	+	0.289460i	$0.906524\pi$
$282$		0			0
$283$	−12.0988	−	20.9558i	−0.719200	−	1.24569i	−0.961317	−	0.275444i	$-0.911175\pi$
$283$	−12.0988	−	20.9558i	−0.719200	−	1.24569i	0.242117	−	0.970247i	$-0.422158\pi$
$284$		0			0
$285$	−0.101365			−0.00600433
$286$		0			0
$287$	−3.52890			−0.208305
$288$		0			0
$289$	7.47451	+	12.9462i	0.439677	+	0.761543i
$290$		0			0
$291$			0.903737i			0.0529780i
$292$		0			0
$293$	−3.14218	−	1.81414i	−0.183568	−	0.105983i	0.405400	−	0.914139i	$-0.367132\pi$
$293$	−3.14218	−	1.81414i	−0.183568	−	0.105983i	−0.588968	+	0.808156i	$0.700466\pi$
$294$		0			0
$295$	−5.76225	+	9.98052i	−0.335491	+	0.581088i
$296$		0			0
$297$	0.851811	−	0.491793i	0.0494271	−	0.0285367i
$298$		0			0
$299$	1.38279	+	11.0398i	0.0799689	+	0.638446i
$300$		0			0
$301$	−6.83190	+	3.94440i	−0.393784	+	0.227351i
$302$		0			0
$303$	0.274952	−	0.476231i	0.0157956	−	0.0273588i
$304$		0			0
$305$	2.35255	+	1.35824i	0.134707	+	0.0777729i
$306$		0			0
$307$		−	9.40129i		−	0.536560i	−0.963341	−	0.268280i	$-0.913545\pi$
$307$		−	9.40129i		−	0.536560i	0.963341	−	0.268280i	$-0.0864552\pi$
$308$		0			0
$309$	0.477538	+	0.827121i	0.0271662	+	0.0470532i
$310$		0			0
$311$	25.5370			1.44807			0.724034	−	0.689764i	$-0.242286\pi$
$311$	25.5370			1.44807			0.724034	+	0.689764i	$0.242286\pi$
$312$		0			0
$313$	5.25656			0.297118			0.148559	−	0.988904i	$-0.452536\pi$
$313$	5.25656			0.297118			0.148559	+	0.988904i	$0.452536\pi$
$314$		0			0
$315$	−1.23654	−	2.14176i	−0.0696713	−	0.120674i
$316$		0			0
$317$		−	14.1536i		−	0.794947i	−0.917614	−	0.397474i	$-0.869887\pi$
$317$		−	14.1536i		−	0.794947i	0.917614	−	0.397474i	$-0.130113\pi$
$318$		0			0
$319$	−11.1827	−	6.45632i	−0.626110	−	0.361485i
$320$		0			0
$321$	0.771550	−	1.33636i	0.0430637	−	0.0745885i
$322$		0			0
$323$	−1.32632	+	0.765750i	−0.0737983	+	0.0426075i
$324$		0			0
$325$	−3.32235	−	1.40072i	−0.184291	−	0.0776980i
$326$		0			0
$327$	−0.256920	+	0.148333i	−0.0142077	+	0.00820282i
$328$		0			0
$329$	−1.43213	+	2.48052i	−0.0789557	+	0.136755i
$330$		0			0
$331$	−16.0945	−	9.29214i	−0.884632	−	0.510742i	−0.0124490	−	0.999923i	$-0.503963\pi$
$331$	−16.0945	−	9.29214i	−0.884632	−	0.510742i	−0.872183	+	0.489180i	$0.837296\pi$
$332$		0			0
$333$		−	2.94192i		−	0.161216i
$334$		0			0
$335$	6.87752	+	11.9122i	0.375759	+	0.650834i
$336$		0			0
$337$	22.4060			1.22053			0.610267	−	0.792196i	$-0.291062\pi$
$337$	22.4060			1.22053			0.610267	+	0.792196i	$0.291062\pi$
$338$		0			0
$339$	0.964061			0.0523606
$340$		0			0
$341$	5.06040	+	8.76488i	0.274036	+	0.474645i
$342$		0			0
$343$		−	11.0105i		−	0.594509i
$344$		0			0
$345$	0.253309	+	0.146248i	0.0136377	+	0.00787372i
$346$		0			0
$347$	−10.0862	+	17.4699i	−0.541457	+	0.937831i	0.457364	+	0.889280i	$0.348794\pi$
$347$	−10.0862	+	17.4699i	−0.541457	+	0.937831i	−0.998821	+	0.0485514i	$0.984540\pi$
$348$		0			0
$349$	24.7634	−	14.2972i	1.32556	−	0.765310i	0.340947	−	0.940083i	$-0.389252\pi$
$349$	24.7634	−	14.2972i	1.32556	−	0.765310i	0.984609	+	0.174773i	$0.0559192\pi$
$350$		0			0
$351$	1.88667	+	0.795432i	0.100703	+	0.0424570i
$352$		0			0
$353$	9.66167	−	5.57817i	0.514239	−	0.296896i	−0.220336	−	0.975424i	$-0.570715\pi$
$353$	9.66167	−	5.57817i	0.514239	−	0.296896i	0.734574	+	0.678528i	$0.237382\pi$
$354$		0			0
$355$	4.88845	−	8.46704i	0.259452	−	0.449384i
$356$		0			0
$357$	0.0972040	+	0.0561207i	0.00514458	+	0.00297022i
$358$		0			0
$359$			11.0490i			0.583145i	0.956549	+	0.291572i	$0.0941784\pi$
$359$			11.0490i			0.583145i	−0.956549	+	0.291572i	$0.905822\pi$
$360$		0			0
$361$	−8.92820	−	15.4641i	−0.469905	−	0.813900i
$362$		0			0
$363$	−0.758301			−0.0398005
$364$		0			0
$365$	11.1806			0.585219
$366$		0			0
$367$	−12.2026	−	21.1355i	−0.636970	−	1.10326i	−0.986094	−	0.166188i	$-0.946854\pi$
$367$	−12.2026	−	21.1355i	−0.636970	−	1.10326i	0.349124	−	0.937076i	$-0.386479\pi$
$368$		0			0
$369$			12.7655i			0.664545i
$370$		0			0
$371$	0.239385	+	0.138209i	0.0124283	+	0.00717546i
$372$		0			0
$373$	2.65566	−	4.59974i	0.137505	−	0.238165i	−0.789047	−	0.614333i	$-0.789425\pi$
$373$	2.65566	−	4.59974i	0.137505	−	0.238165i	0.926552	+	0.376168i	$0.122758\pi$
$374$		0			0
$375$	−0.0820885	+	0.0473938i	−0.00423903	+	0.00244740i
$376$		0			0
$377$	−3.34074	−	26.6714i	−0.172057	−	1.37365i
$378$		0			0
$379$	29.0469	−	16.7703i	1.49204	−	0.861430i	0.492082	−	0.870549i	$-0.336236\pi$
$379$	29.0469	−	16.7703i	1.49204	−	0.861430i	0.999958	+	0.00911888i	$0.00290267\pi$
$380$		0			0
$381$	−0.282847	+	0.489906i	−0.0144907	+	0.0250986i
$382$		0			0
$383$	20.6138	+	11.9014i	1.05331	+	0.608131i	0.923575	−	0.383417i	$-0.125253\pi$
$383$	20.6138	+	11.9014i	1.05331	+	0.608131i	0.129739	+	0.991548i	$0.458586\pi$
$384$		0			0
$385$		−	1.43213i		−	0.0729879i
$386$		0			0
$387$	14.2685	+	24.7138i	0.725310	+	1.25627i
$388$		0			0
$389$	−26.2787			−1.33238			−0.666191	−	0.745781i	$-0.732077\pi$
$389$	−26.2787			−1.33238			−0.666191	+	0.745781i	$0.732077\pi$
$390$		0			0
$391$	4.41926			0.223492
$392$		0			0
$393$	0.788003	+	1.36486i	0.0397495	+	0.0688482i
$394$		0			0
$395$			0.252387i			0.0126990i
$396$		0			0
$397$	28.8317	+	16.6460i	1.44702	+	0.835439i	0.998303	−	0.0582340i	$-0.0185469\pi$
$397$	28.8317	+	16.6460i	1.44702	+	0.835439i	0.448719	+	0.893673i	$0.351880\pi$
$398$		0			0
$399$	0.0419062	−	0.0725836i	0.00209793	−	0.00363373i
$400$		0			0
$401$	12.2709	−	7.08460i	0.612779	−	0.353788i	−0.161273	−	0.986910i	$-0.551560\pi$
$401$	12.2709	−	7.08460i	0.612779	−	0.353788i	0.774052	+	0.633122i	$0.218227\pi$
$402$		0			0
$403$	−8.18476	+	19.4133i	−0.407712	+	0.967046i
$404$		0			0
$405$	−7.72427	+	4.45961i	−0.383822	+	0.221600i
$406$		0			0
$407$	0.851811	−	1.47538i	0.0422227	−	0.0731319i
$408$		0			0
$409$	5.93213	+	3.42491i	0.293325	+	0.169351i	0.639440	−	0.768841i	$-0.279166\pi$
$409$	5.93213	+	3.42491i	0.293325	+	0.169351i	−0.346116	+	0.938192i	$0.612499\pi$
$410$		0			0
$411$		−	0.0383462i		−	0.00189148i
$412$		0			0
$413$	−4.76445	−	8.25227i	−0.234443	−	0.406068i
$414$		0			0
$415$	5.67165			0.278410
$416$		0			0
$417$	0.883073			0.0432443
$418$		0			0
$419$	8.19109	+	14.1874i	0.400161	+	0.693099i	0.993745	−	0.111673i	$-0.0356209\pi$
$419$	8.19109	+	14.1874i	0.400161	+	0.693099i	−0.593584	+	0.804772i	$0.702288\pi$
$420$		0			0
$421$		−	21.7045i		−	1.05781i	−0.848681	−	0.528906i	$-0.822603\pi$
$421$		−	21.7045i		−	1.05781i	0.848681	−	0.528906i	$-0.177397\pi$
$422$		0			0
$423$	8.97305	+	5.18059i	0.436284	+	0.251889i
$424$		0			0
$425$	−0.716063	+	1.24026i	−0.0347342	+	0.0601613i
$426$		0			0
$427$	−1.94518	+	1.12305i	−0.0941337	+	0.0543481i
$428$		0			0
$429$	0.357393	+	0.471883i	0.0172551	+	0.0227827i
$430$		0			0
$431$	28.0495	−	16.1944i	1.35110	−	0.780056i	0.362693	−	0.931909i	$-0.381857\pi$
$431$	28.0495	−	16.1944i	1.35110	−	0.780056i	0.988403	+	0.151853i	$0.0485240\pi$
$432$		0			0
$433$	14.3987	−	24.9393i	0.691959	−	1.19851i	−0.279236	−	0.960223i	$-0.590081\pi$
$433$	14.3987	−	24.9393i	0.691959	−	1.19851i	0.971195	−	0.238286i	$-0.0765855\pi$
$434$		0			0
$435$	−0.611979	−	0.353326i	−0.0293422	−	0.0169407i
$436$		0			0
$437$		−	3.29992i		−	0.157857i
$438$		0			0
$439$	−8.79992	−	15.2419i	−0.419997	−	0.727457i	0.575941	−	0.817491i	$-0.304636\pi$
$439$	−8.79992	−	15.2419i	−0.419997	−	0.727457i	−0.995939	+	0.0900341i	$0.971302\pi$
$440$		0			0
$441$	−18.8923			−0.899632
$442$		0			0
$443$	−14.4043			−0.684370			−0.342185	−	0.939633i	$-0.611167\pi$
$443$	−14.4043			−0.684370			−0.342185	+	0.939633i	$0.611167\pi$
$444$		0			0
$445$	−2.29815	−	3.98052i	−0.108943	−	0.188695i
$446$		0			0
$447$			1.03013i			0.0487236i
$448$		0			0
$449$	−2.58821	−	1.49430i	−0.122145	−	0.0705206i	0.437683	−	0.899130i	$-0.355799\pi$
$449$	−2.58821	−	1.49430i	−0.122145	−	0.0705206i	−0.559828	+	0.828609i	$0.689133\pi$
$450$		0			0
$451$	−3.69615	+	6.40192i	−0.174045	+	0.301455i
$452$		0			0
$453$	0.0813509	−	0.0469680i	0.00382220	−	0.00220675i
$454$		0			0
$455$	2.37653	−	1.79992i	0.111413	−	0.0843818i
$456$		0			0
$457$	−23.0540	+	13.3102i	−1.07842	+	0.622626i	−0.930470	−	0.366369i	$-0.880601\pi$
$457$	−23.0540	+	13.3102i	−1.07842	+	0.622626i	−0.147950	+	0.988995i	$0.547267\pi$
$458$		0			0
$459$	0.406634	−	0.704310i	0.0189800	−	0.0328744i
$460$		0			0
$461$	7.26488	+	4.19438i	0.338359	+	0.195352i	0.659546	−	0.751664i	$-0.270748\pi$
$461$	7.26488	+	4.19438i	0.338359	+	0.195352i	−0.321187	+	0.947016i	$0.604082\pi$
$462$		0			0
$463$		−	21.3014i		−	0.989960i	−0.868904	−	0.494980i	$-0.835175\pi$
$463$		−	21.3014i		−	0.989960i	0.868904	−	0.494980i	$-0.164825\pi$
$464$		0			0
$465$	0.276934	+	0.479664i	0.0128425	+	0.0222439i
$466$		0			0
$467$	2.12392			0.0982833			0.0491417	−	0.998792i	$-0.484351\pi$
$467$	2.12392			0.0982833			0.0491417	+	0.998792i	$0.484351\pi$
$468$		0			0
$469$	−11.3732			−0.525165
$470$		0			0
$471$	0.832846	+	1.44253i	0.0383755	+	0.0664684i
$472$		0			0
$473$			16.5254i			0.759837i
$474$		0			0
$475$	0.926118	+	0.534695i	0.0424932	+	0.0245335i
$476$		0			0
$477$	0.499960	−	0.865955i	0.0228916	−	0.0396494i
$478$		0			0
$479$	−25.1617	+	14.5271i	−1.14967	+	0.663762i	−0.948806	−	0.315858i	$-0.897708\pi$
$479$	−25.1617	+	14.5271i	−1.14967	+	0.663762i	−0.200862	+	0.979619i	$0.564374\pi$
$480$		0			0
$481$	3.51887	−	0.440759i	0.160447	−	0.0200969i
$482$		0			0
$483$	−0.209445	+	0.120923i	−0.00953010	+	0.00550220i
$484$		0			0
$485$	4.76717	−	8.25698i	0.216466	−	0.374930i
$486$		0			0
$487$	−26.2570	−	15.1595i	−1.18982	−	0.686941i	−0.231552	−	0.972822i	$-0.574380\pi$
$487$	−26.2570	−	15.1595i	−1.18982	−	0.686941i	−0.958265	+	0.285881i	$0.907714\pi$
$488$		0			0
$489$		−	1.63811i		−	0.0740781i
$490$		0			0
$491$	19.0759	+	33.0405i	0.860884	+	1.49110i	0.871076	+	0.491148i	$0.163422\pi$
$491$	19.0759	+	33.0405i	0.860884	+	1.49110i	−0.0101919	+	0.999948i	$0.503244\pi$
$492$		0			0
$493$	−10.6767			−0.480853
$494$		0			0
$495$	−5.18059			−0.232850
$496$		0			0
$497$	4.04196	+	7.00087i	0.181306	+	0.314032i
$498$		0			0
$499$		−	16.5179i		−	0.739444i	−0.929142	−	0.369722i	$-0.879453\pi$
$499$		−	16.5179i		−	0.739444i	0.929142	−	0.369722i	$-0.120547\pi$
$500$		0			0
$501$	−0.251154	−	0.145004i	−0.0112207	−	0.00647829i
$502$		0			0
$503$	−5.88081	+	10.1859i	−0.262212	+	0.454165i	−0.966830	−	0.255423i	$-0.917785\pi$
$503$	−5.88081	+	10.1859i	−0.262212	+	0.454165i	0.704617	+	0.709588i	$0.251119\pi$
$504$		0			0
$505$	−5.02419	+	2.90072i	−0.223574	+	0.129080i
$506$		0			0
$507$	−0.333882	+	1.18614i	−0.0148282	+	0.0526785i
$508$		0			0
$509$	27.5930	−	15.9308i	1.22304	−	0.706122i	0.257474	−	0.966285i	$-0.417110\pi$
$509$	27.5930	−	15.9308i	1.22304	−	0.706122i	0.965565	+	0.260164i	$0.0837765\pi$
$510$		0			0
$511$	−4.62227	+	8.00601i	−0.204477	+	0.354165i
$512$		0			0
$513$	−0.525918	−	0.303639i	−0.0232199	−	0.0134060i
$514$		0			0
$515$		−	10.0760i		−	0.444000i
$516$		0			0
$517$	3.00000	+	5.19615i	0.131940	+	0.228527i
$518$		0			0
$519$	−0.324184			−0.0142301
$520$		0			0
$521$	−19.5013			−0.854367			−0.427183	−	0.904165i	$-0.640494\pi$
$521$	−19.5013			−0.854367			−0.427183	+	0.904165i	$0.640494\pi$
$522$		0			0
$523$	22.2830	+	38.5952i	0.974365	+	1.68765i	0.682014	+	0.731340i	$0.261105\pi$
$523$	22.2830	+	38.5952i	0.974365	+	1.68765i	0.292352	+	0.956311i	$0.405562\pi$
$524$		0			0
$525$		−	0.0783740i		−	0.00342052i
$526$		0			0
$527$	7.24714	+	4.18414i	0.315690	+	0.182264i
$528$		0			0
$529$	6.73891	−	11.6721i	0.292996	−	0.507484i
$530$		0			0
$531$	−29.8519	+	17.2350i	−1.29546	+	0.747935i
$532$		0			0
$533$	−15.2690	+	1.91253i	−0.661374	+	0.0828407i
$534$		0			0
$535$	−14.0985	+	8.13977i	−0.609531	+	0.351913i
$536$		0			0
$537$	0.492051	−	0.852257i	0.0212336	−	0.0367776i
$538$		0			0
$539$	−9.47451	−	5.47011i	−0.408096	−	0.235614i
$540$		0			0
$541$		−	3.74450i		−	0.160989i	−0.996755	−	0.0804943i	$-0.974350\pi$
$541$		−	3.74450i		−	0.160989i	0.996755	−	0.0804943i	$-0.0256499\pi$
$542$		0			0
$543$	−0.490486	−	0.849547i	−0.0210488	−	0.0364576i
$544$		0			0
$545$	3.12979			0.134066
$546$		0			0
$547$	38.3803			1.64102			0.820511	−	0.571630i	$-0.193689\pi$
$547$	38.3803			1.64102			0.820511	+	0.571630i	$0.193689\pi$
$548$		0			0
$549$	4.06253	+	7.03651i	0.173385	+	0.300311i
$550$		0			0
$551$			7.97242i			0.339637i
$552$		0			0
$553$	−0.180725	−	0.104342i	−0.00768522	−	0.00443706i
$554$		0			0
$555$	0.0466159	−	0.0807411i	0.00197873	−	0.00342727i
$556$		0			0
$557$	−23.0763	+	13.3231i	−0.977772	+	0.564517i	−0.901597	−	0.432577i	$-0.857604\pi$
$557$	−23.0763	+	13.3231i	−0.977772	+	0.564517i	−0.0761755	+	0.997094i	$0.524271\pi$
$558$		0			0
$559$	−27.4228	+	20.7694i	−1.15986	+	0.878452i
$560$		0			0
$561$	0.203622	−	0.117561i	0.00859691	−	0.00496343i
$562$		0			0
$563$	−8.34675	+	14.4570i	−0.351774	+	0.609290i	−0.986560	−	0.163398i	$-0.947755\pi$
$563$	−8.34675	+	14.4570i	−0.351774	+	0.609290i	0.634787	+	0.772687i	$0.281088\pi$
$564$		0			0
$565$	−8.80813	−	5.08538i	−0.370561	−	0.213943i
$566$		0			0
$567$		−	7.37475i		−	0.309710i
$568$		0			0
$569$	−21.9620	−	38.0393i	−0.920694	−	1.59469i	−0.798344	−	0.602202i	$-0.794290\pi$
$569$	−21.9620	−	38.0393i	−0.920694	−	1.59469i	−0.122350	−	0.992487i	$-0.539043\pi$
$570$		0			0
$571$	−11.4641			−0.479758			−0.239879	−	0.970803i	$-0.577108\pi$
$571$	−11.4641			−0.479758			−0.239879	+	0.970803i	$0.577108\pi$
$572$		0			0
$573$	1.46977			0.0614004
$574$		0			0
$575$	−1.54290	−	2.67238i	−0.0643434	−	0.111446i
$576$		0			0
$577$		−	44.9354i		−	1.87069i	−0.353743	−	0.935343i	$-0.615091\pi$
$577$		−	44.9354i		−	1.87069i	0.353743	−	0.935343i	$-0.384909\pi$
$578$		0			0
$579$	0.457256	+	0.263997i	0.0190029	+	0.0109713i
$580$		0			0
$581$	−2.34477	+	4.06126i	−0.0972773	+	0.168489i
$582$		0			0
$583$	0.501461	−	0.289519i	0.0207684	−	0.0119906i
$584$		0			0
$585$	−6.51107	−	8.59687i	−0.269200	−	0.355437i
$586$		0			0
$587$	−21.2364	+	12.2608i	−0.876520	+	0.506059i	−0.869509	−	0.493916i	$-0.835565\pi$
$587$	−21.2364	+	12.2608i	−0.876520	+	0.506059i	−0.00701059	+	0.999975i	$0.502232\pi$
$588$		0			0
$589$	3.12436	−	5.41154i	0.128737	−	0.222979i
$590$		0			0
$591$	−2.04177	−	1.17882i	−0.0839873	−	0.0484901i
$592$		0			0
$593$			18.7655i			0.770607i	0.922790	+	0.385303i	$0.125903\pi$
$593$			18.7655i			0.770607i	−0.922790	+	0.385303i	$0.874097\pi$
$594$		0			0
$595$	−0.592068	−	1.02549i	−0.0242724	−	0.0420411i
$596$		0			0
$597$	−1.76768			−0.0723464
$598$		0			0
$599$	−35.9293			−1.46803			−0.734015	−	0.679133i	$-0.762356\pi$
$599$	−35.9293			−1.46803			−0.734015	+	0.679133i	$0.762356\pi$
$600$		0			0
$601$	19.8863	+	34.4441i	0.811179	+	1.40500i	0.912039	+	0.410103i	$0.134507\pi$
$601$	19.8863	+	34.4441i	0.811179	+	1.40500i	−0.100860	+	0.994901i	$0.532160\pi$
$602$		0			0
$603$			41.1415i			1.67541i
$604$		0			0
$605$	6.92820	+	4.00000i	0.281672	+	0.162623i
$606$		0			0
$607$	−16.7306	+	28.9783i	−0.679076	+	1.17619i	0.296184	+	0.955131i	$0.404286\pi$
$607$	−16.7306	+	28.9783i	−0.679076	+	1.17619i	−0.975260	+	0.221063i	$0.929047\pi$
$608$		0			0
$609$	0.506008	−	0.292144i	0.0205045	−	0.0118383i
$610$		0			0
$611$	−4.85224	+	11.5089i	−0.196300	+	0.465602i
$612$		0			0
$613$	17.1212	−	9.88495i	0.691520	−	0.399249i	−0.112661	−	0.993633i	$-0.535937\pi$
$613$	17.1212	−	9.88495i	0.691520	−	0.399249i	0.804181	+	0.594384i	$0.202604\pi$
$614$		0			0
$615$	−0.202274	+	0.350349i	−0.00815649	+	0.0141275i
$616$		0			0
$617$	−16.8950	−	9.75436i	−0.680169	−	0.392696i	0.119750	−	0.992804i	$-0.461791\pi$
$617$	−16.8950	−	9.75436i	−0.680169	−	0.392696i	−0.799919	+	0.600108i	$0.795124\pi$
$618$		0			0
$619$		−	40.4640i		−	1.62639i	−0.581994	−	0.813193i	$-0.697727\pi$
$619$		−	40.4640i		−	1.62639i	0.581994	−	0.813193i	$-0.302273\pi$
$620$		0			0
$621$	0.876173	+	1.51758i	0.0351596	+	0.0608983i
$622$		0			0
$623$	3.80040			0.152260
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	−0.0877845	−	0.152047i	−0.00350578	−	0.00607218i
$628$		0			0
$629$		−	1.40862i		−	0.0561653i
$630$		0			0
$631$	−16.9707	−	9.79806i	−0.675594	−	0.390054i	0.122599	−	0.992456i	$-0.460877\pi$
$631$	−16.9707	−	9.79806i	−0.675594	−	0.390054i	−0.798193	+	0.602402i	$0.794211\pi$
$632$		0			0
$633$	0.457099	−	0.791718i	0.0181680	−	0.0314680i
$634$		0			0
$635$	5.16846	−	2.98401i	0.205104	−	0.118417i
$636$		0			0
$637$	−2.83044	−	22.5973i	−0.112146	−	0.895338i
$638$		0			0
$639$	25.3250	−	14.6214i	1.00184	−	0.578414i
$640$		0			0
$641$	15.5238	−	26.8881i	0.613155	−	1.06202i	−0.377550	−	0.925989i	$-0.623233\pi$
$641$	15.5238	−	26.8881i	0.613155	−	1.06202i	0.990705	−	0.136026i	$-0.0434332\pi$
$642$		0			0
$643$	−5.14990	−	2.97329i	−0.203092	−	0.117255i	0.395005	−	0.918679i	$-0.370743\pi$
$643$	−5.14990	−	2.97329i	−0.203092	−	0.117255i	−0.598097	+	0.801424i	$0.704076\pi$
$644$		0			0
$645$			0.904361i			0.0356092i
$646$		0			0
$647$	21.2510	+	36.8078i	0.835462	+	1.44706i	0.893654	+	0.448757i	$0.148133\pi$
$647$	21.2510	+	36.8078i	0.835462	+	1.44706i	−0.0581916	+	0.998305i	$0.518533\pi$
$648$		0			0
$649$	−19.9610			−0.783539
$650$		0			0
$651$	−0.457959			−0.0179488
$652$		0			0
$653$	15.3054	+	26.5097i	0.598945	+	1.03740i	0.992977	+	0.118307i	$0.0377467\pi$
$653$	15.3054	+	26.5097i	0.598945	+	1.03740i	−0.394032	+	0.919097i	$0.628920\pi$
$654$		0			0
$655$		−	16.6267i		−	0.649660i
$656$		0			0
$657$	28.9610	+	16.7207i	1.12988	+	0.652335i
$658$		0			0
$659$	12.7191	−	22.0302i	0.495467	−	0.858175i	−0.504519	−	0.863401i	$-0.668330\pi$
$659$	12.7191	−	22.0302i	0.495467	−	0.858175i	0.999986	+	0.00522582i	$0.00166344\pi$
$660$		0			0
$661$	−0.288909	+	0.166802i	−0.0112373	+	0.00648784i	−0.505608	−	0.862763i	$-0.668732\pi$
$661$	−0.288909	+	0.166802i	−0.0112373	+	0.00648784i	0.494371	+	0.869251i	$0.335398\pi$
$662$		0			0
$663$	0.451001	+	0.190145i	0.0175154	+	0.00738461i
$664$		0			0
$665$	−0.765750	+	0.442106i	−0.0296945	+	0.0171441i
$666$		0			0
$667$	11.5025	−	19.9229i	0.445379	−	0.771419i
$668$		0			0
$669$	−0.0953167	−	0.0550311i	−0.00368516	−	0.00212763i
$670$		0			0
$671$			4.70510i			0.181638i
$672$		0			0
$673$	4.90706	+	8.49928i	0.189153	+	0.327623i	0.944968	−	0.327162i	$-0.106092\pi$
$673$	4.90706	+	8.49928i	0.189153	+	0.327623i	−0.755815	+	0.654785i	$0.772759\pi$
$674$		0			0
$675$	−0.567874			−0.0218575
$676$		0			0
$677$	−23.2414			−0.893241			−0.446620	−	0.894724i	$-0.647373\pi$
$677$	−23.2414			−0.893241			−0.446620	+	0.894724i	$0.647373\pi$
$678$		0			0
$679$	3.94168	+	6.82719i	0.151268	+	0.262004i
$680$		0			0
$681$		−	2.58258i		−	0.0989648i
$682$		0			0
$683$	4.56144	+	2.63355i	0.174539	+	0.100770i	0.584724	−	0.811232i	$-0.301203\pi$
$683$	4.56144	+	2.63355i	0.174539	+	0.100770i	−0.410186	+	0.912002i	$0.634536\pi$
$684$		0			0
$685$	−0.202274	+	0.350349i	−0.00772850	+	0.0133862i
$686$		0			0
$687$	−1.99830	+	1.15372i	−0.0762398	+	0.0440171i
$688$		0			0
$689$	1.11069	+	0.468271i	0.0423138	+	0.0178397i
$690$		0			0
$691$	17.1334	−	9.89199i	0.651787	−	0.376309i	−0.137354	−	0.990522i	$-0.543860\pi$
$691$	17.1334	−	9.89199i	0.651787	−	0.376309i	0.789140	+	0.614213i	$0.210526\pi$
$692$		0			0
$693$	2.14176	−	3.70963i	0.0813586	−	0.140917i
$694$		0			0
$695$	−8.06819	−	4.65817i	−0.306044	−	0.176694i
$696$		0			0
$697$			6.11224i			0.231518i
$698$		0			0
$699$	−1.09119	−	1.88999i	−0.0412725	−	0.0714861i
$700$		0			0
$701$	18.1256			0.684595			0.342298	−	0.939592i	$-0.388795\pi$
$701$	18.1256			0.684595			0.342298	+	0.939592i	$0.388795\pi$
$702$		0			0
$703$	−1.05184			−0.0396708
$704$		0			0
$705$	0.164177	+	0.284363i	0.00618326	+	0.0107097i
$706$		0			0
$707$		−	4.79685i		−	0.180404i
$708$		0			0
$709$	−22.3514	−	12.9046i	−0.839424	−	0.484642i	0.0176445	−	0.999844i	$-0.494383\pi$
$709$	−22.3514	−	12.9046i	−0.839424	−	0.484642i	−0.857068	+	0.515203i	$0.827717\pi$
$710$		0			0
$711$	−0.377447	+	0.653758i	−0.0141554	+	0.0245178i
$712$		0			0
$713$	−15.6154	+	9.01556i	−0.584802	+	0.337635i
$714$		0			0
$715$	−0.776156	−	6.19658i	−0.0290266	−	0.231739i
$716$		0			0
$717$	−1.94597	+	1.12350i	−0.0726734	+	0.0419580i
$718$		0			0
$719$	−4.51338	+	7.81741i	−0.168321	+	0.291540i	−0.937830	−	0.347096i	$-0.887168\pi$
$719$	−4.51338	+	7.81741i	−0.168321	+	0.291540i	0.769509	+	0.638636i	$0.220501\pi$
$720$		0			0
$721$	7.21503	+	4.16560i	0.268702	+	0.155135i
$722$		0			0
$723$		−	1.02662i		−	0.0381803i
$724$		0			0
$725$	3.72756	+	6.45632i	0.138438	+	0.239782i
$726$		0			0
$727$	24.8934			0.923245			0.461623	−	0.887076i	$-0.347267\pi$
$727$	24.8934			0.923245			0.461623	+	0.887076i	$0.347267\pi$
$728$		0			0
$729$	−26.5160			−0.982075
$730$		0			0
$731$	6.83190	+	11.8332i	0.252687	+	0.437667i
$732$		0			0
$733$		−	13.2793i		−	0.490484i	−0.969462	−	0.245242i	$-0.921133\pi$
$733$		−	13.2793i		−	0.490484i	0.969462	−	0.245242i	$-0.0788674\pi$
$734$		0			0
$735$	−0.518498	−	0.299355i	−0.0191251	−	0.0110419i
$736$		0			0
$737$	−11.9122	+	20.6326i	−0.438792	+	0.760010i
$738$		0			0
$739$	16.9656	−	9.79508i	0.624089	−	0.360318i	−0.154370	−	0.988013i	$-0.549335\pi$
$739$	16.9656	−	9.79508i	0.624089	−	0.360318i	0.778459	+	0.627695i	$0.216002\pi$
$740$		0			0
$741$	0.141984	−	0.336769i	0.00521590	−	0.0123715i
$742$		0			0
$743$	44.3804	−	25.6230i	1.62816	−	0.940017i	0.643515	−	0.765434i	$-0.277475\pi$
$743$	44.3804	−	25.6230i	1.62816	−	0.940017i	0.984642	−	0.174583i	$-0.0558578\pi$
$744$		0			0
$745$	5.43390	−	9.41179i	0.199083	−	0.344821i
$746$		0			0
$747$	14.6912	+	8.48199i	0.537524	+	0.310340i
$748$		0			0
$749$		−	13.4606i		−	0.491838i
$750$		0			0
$751$	−10.5992	−	18.3584i	−0.386772	−	0.669908i	0.605242	−	0.796042i	$-0.293077\pi$
$751$	−10.5992	−	18.3584i	−0.386772	−	0.669908i	−0.992013	+	0.126134i	$0.959743\pi$
$752$		0			0
$753$	−0.106239			−0.00387156
$754$		0			0
$755$	−0.991015			−0.0360667
$756$		0			0
$757$	16.4747	+	28.5350i	0.598783	+	1.03712i	0.993001	+	0.118106i	$0.0376823\pi$
$757$	16.4747	+	28.5350i	0.598783	+	1.03712i	−0.394218	+	0.919017i	$0.628984\pi$
$758$		0			0
$759$			0.506618i			0.0183891i
$760$		0			0
$761$	45.2367	+	26.1174i	1.63983	+	0.946756i	0.980891	+	0.194559i	$0.0623277\pi$
$761$	45.2367	+	26.1174i	1.63983	+	0.946756i	0.658939	+	0.752197i	$0.271006\pi$
$762$		0			0
$763$	−1.29392	+	2.24113i	−0.0468429	+	0.0811343i
$764$		0			0
$765$	−3.70963	+	2.14176i	−0.134122	+	0.0774353i
$766$		0			0
$767$	−25.0874	−	33.1241i	−0.905854	−	1.19604i
$768$		0			0
$769$	23.2717	−	13.4359i	0.839200	−	0.484513i	−0.0177920	−	0.999842i	$-0.505664\pi$
$769$	23.2717	−	13.4359i	0.839200	−	0.484513i	0.856992	+	0.515329i	$0.172330\pi$
$770$		0			0
$771$	−0.788191	+	1.36519i	−0.0283860	+	0.0491660i
$772$		0			0
$773$	10.3533	+	5.97746i	0.372381	+	0.214994i	0.674498	−	0.738276i	$-0.264360\pi$
$773$	10.3533	+	5.97746i	0.372381	+	0.214994i	−0.302117	+	0.953271i	$0.597693\pi$
$774$		0			0
$775$		−	5.84325i		−	0.209896i
$776$		0			0
$777$	0.0385438	+	0.0667598i	0.00138275	+	0.00239500i
$778$		0			0
$779$	4.56410			0.163526
$780$		0			0
$781$	16.9341			0.605949
$782$		0			0
$783$	−2.11678	−	3.66638i	−0.0756477	−	0.131026i
$784$		0			0
$785$		−	17.5729i		−	0.627204i
$786$		0			0
$787$	29.8724	+	17.2468i	1.06484	+	0.614783i	0.926766	−	0.375639i	$-0.122577\pi$
$787$	29.8724	+	17.2468i	1.06484	+	0.614783i	0.138070	+	0.990422i	$0.455910\pi$
$788$		0			0
$789$	1.16124	−	2.01133i	0.0413413	−	0.0716051i
$790$		0			0
$791$

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:	\( N \)	\(=\)	\( 260 = 2^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	260.x (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.0473938 + 0.0820885i) q^{3} -1.00000i q^{5} +(0.716063 + 0.413419i) q^{7} +(1.49551 - 2.59030i) q^{9} +O(q^{10})\)	\(q+(0.0473938 + 0.0820885i) q^{3} -1.00000i q^{5} +(0.716063 + 0.413419i) q^{7} +(1.49551 - 2.59030i) q^{9} +(1.50000 - 0.866025i) q^{11} +(3.32235 + 1.40072i) q^{13} +(0.0820885 - 0.0473938i) q^{15} +(0.716063 - 1.24026i) q^{17} +(-0.926118 - 0.534695i) q^{19} +0.0783740i q^{21} +(1.54290 + 2.67238i) q^{23} -1.00000 q^{25} +0.567874 q^{27} +(-3.72756 - 6.45632i) q^{29} +5.84325i q^{31} +(0.142181 + 0.0820885i) q^{33} +(0.413419 - 0.716063i) q^{35} +(0.851811 - 0.491793i) q^{37} +(0.0424756 + 0.339112i) q^{39} +(-3.69615 + 2.13397i) q^{41} +(-4.77046 + 8.26268i) q^{43} +(-2.59030 - 1.49551i) q^{45} +3.46410i q^{47} +(-3.15817 - 5.47011i) q^{49} +0.135748 q^{51} +0.334308 q^{53} +(-0.866025 - 1.50000i) q^{55} -0.101365i q^{57} +(-9.98052 - 5.76225i) q^{59} +(-1.35824 + 2.35255i) q^{61} +(2.14176 - 1.23654i) q^{63} +(1.40072 - 3.32235i) q^{65} +(-11.9122 + 6.87752i) q^{67} +(-0.146248 + 0.253309i) q^{69} +(8.46704 + 4.88845i) q^{71} +11.1806i q^{73} +(-0.0473938 - 0.0820885i) q^{75} +1.43213 q^{77} -0.252387 q^{79} +(-4.45961 - 7.72427i) q^{81} +5.67165i q^{83} +(-1.24026 - 0.716063i) q^{85} +(0.353326 - 0.611979i) q^{87} +(3.98052 - 2.29815i) q^{89} +(1.79992 + 2.37653i) q^{91} +(-0.479664 + 0.276934i) q^{93} +(-0.534695 + 0.926118i) q^{95} +(8.25698 + 4.76717i) q^{97} -5.18059i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{3} + 6 q^{7} - 4 q^{9}+O(q^{10}) \) 8 * q - 2 * q^3 + 6 * q^7 - 4 * q^9	\( 8 q - 2 q^{3} + 6 q^{7} - 4 q^{9} + 12 q^{11} - 8 q^{13} - 6 q^{15} + 6 q^{17} - 6 q^{23} - 8 q^{25} + 4 q^{27} - 6 q^{33} - 6 q^{35} + 6 q^{37} - 4 q^{39} + 12 q^{41} + 10 q^{43} - 4 q^{49} + 24 q^{53} - 24 q^{59} - 4 q^{61} + 24 q^{63} - 54 q^{67} - 24 q^{69} - 36 q^{71} + 2 q^{75} + 12 q^{77} - 16 q^{79} + 8 q^{81} + 18 q^{85} - 6 q^{87} - 24 q^{89} + 24 q^{93} - 30 q^{97}+O(q^{100}) \) 8 * q - 2 * q^3 + 6 * q^7 - 4 * q^9 + 12 * q^11 - 8 * q^13 - 6 * q^15 + 6 * q^17 - 6 * q^23 - 8 * q^25 + 4 * q^27 - 6 * q^33 - 6 * q^35 + 6 * q^37 - 4 * q^39 + 12 * q^41 + 10 * q^43 - 4 * q^49 + 24 * q^53 - 24 * q^59 - 4 * q^61 + 24 * q^63 - 54 * q^67 - 24 * q^69 - 36 * q^71 + 2 * q^75 + 12 * q^77 - 16 * q^79 + 8 * q^81 + 18 * q^85 - 6 * q^87 - 24 * q^89 + 24 * q^93 - 30 * q^97

Introduction

L-functions

Modular forms

Varieties

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Database

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