LMFDB - Embedded newform 260.2.x.a.121.2

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			121.2
Root			$-1.27597 - 0.609843i$ of defining polynomial
Character	$\chi$	$=$	260.121
Dual form			260.2.x.a.101.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.800098 + 1.38581i) q^{3} -1.00000i q^{5} +(3.75184 - 2.16612i) q^{7} +(0.219687 + 0.380509i) q^{9} +O(q^{10})$	\(q+(-0.800098 + 1.38581i) q^{3} -1.00000i q^{5} +(3.75184 - 2.16612i) q^{7} +(0.219687 + 0.380509i) q^{9} +(1.50000 + 0.866025i) q^{11} +(-3.11256 + 1.81988i) q^{13} +(1.38581 + 0.800098i) q^{15} +(3.75184 + 6.49837i) q^{17} +(4.65213 - 2.68591i) q^{19} +6.93244i q^{21} +(-0.580411 + 1.00530i) q^{23} -1.00000 q^{25} -5.50367 q^{27} +(1.01236 - 1.75347i) q^{29} -7.86488i q^{31} +(-2.40029 + 1.38581i) q^{33} +(-2.16612 - 3.75184i) q^{35} +(-8.25551 - 4.76632i) q^{37} +(-0.0316594 - 5.76950i) q^{39} +(6.69615 + 3.86603i) q^{41} +(2.09277 + 3.62479i) q^{43} +(0.380509 - 0.219687i) q^{45} -3.46410i q^{47} +(5.88418 - 10.1917i) q^{49} -12.0073 q^{51} -12.6471 q^{53} +(0.866025 - 1.50000i) q^{55} +8.59596i q^{57} +(5.49674 - 3.17354i) q^{59} +(-1.85154 - 3.20696i) q^{61} +(1.64846 + 0.951738i) q^{63} +(1.81988 + 3.11256i) q^{65} +(-4.55568 - 2.63022i) q^{67} +(-0.928771 - 1.60868i) q^{69} +(-10.8377 + 6.25714i) q^{71} +5.23898i q^{73} +(0.800098 - 1.38581i) q^{75} +7.50367 q^{77} -8.16719 q^{79} +(3.74441 - 6.48552i) q^{81} +0.456760i q^{83} +(6.49837 - 3.75184i) q^{85} +(1.61998 + 2.80589i) q^{87} +(-11.4967 - 6.63765i) q^{89} +(-7.73572 + 13.5701i) q^{91} +(10.8992 + 6.29268i) q^{93} +(-2.68591 - 4.65213i) q^{95} +(-2.43371 + 1.40511i) q^{97} +0.761018i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/260\mathbb{Z}\right)^\times$.

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

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
$3$	−0.800098	+	1.38581i	−0.461937	+	0.800098i	−0.999057	−	0.0434075i	$-0.986179\pi$
$3$	−0.800098	+	1.38581i	−0.461937	+	0.800098i	0.537121	+	0.843505i	$0.319512\pi$
$4$		0			0
$5$		−	1.00000i		−	0.447214i
$5$		−	1.00000i		−	0.447214i
$6$		0			0
$7$	3.75184	−	2.16612i	1.41806	−	0.818718i	0.421932	−	0.906627i	$-0.361352\pi$
$7$	3.75184	−	2.16612i	1.41806	−	0.818718i	0.996128	+	0.0879098i	$0.0280187\pi$
$8$		0			0
$9$	0.219687	+	0.380509i	0.0732290	+	0.126836i
$10$		0			0
$11$	1.50000	+	0.866025i	0.452267	+	0.261116i	0.708787	−	0.705422i	$-0.249243\pi$
$11$	1.50000	+	0.866025i	0.452267	+	0.261116i	−0.256520	+	0.966539i	$0.582576\pi$
$12$		0			0
$13$	−3.11256	+	1.81988i	−0.863269	+	0.504745i
$13$	−3.11256	+	1.81988i	−0.863269	+	0.504745i
$14$		0			0
$15$	1.38581	+	0.800098i	0.357815	+	0.206584i
$16$		0			0
$17$	3.75184	+	6.49837i	0.909954	+	1.57609i	0.814126	+	0.580688i	$0.197216\pi$
$17$	3.75184	+	6.49837i	0.909954	+	1.57609i	0.0958278	+	0.995398i	$0.469450\pi$
$18$		0			0
$19$	4.65213	−	2.68591i	1.06727	−	0.616190i	0.139837	−	0.990175i	$-0.455342\pi$
$19$	4.65213	−	2.68591i	1.06727	−	0.616190i	0.927435	+	0.373985i	$0.122009\pi$
$20$		0			0
$21$			6.93244i			1.51278i
$22$		0			0
$23$	−0.580411	+	1.00530i	−0.121024	+	0.209620i	−0.920172	−	0.391515i	$-0.871951\pi$
$23$	−0.580411	+	1.00530i	−0.121024	+	0.209620i	0.799148	+	0.601135i	$0.205284\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$	−5.50367			−1.05918
$28$		0			0
$29$	1.01236	−	1.75347i	0.187991	−	0.325610i	−0.756589	−	0.653891i	$-0.773136\pi$
$29$	1.01236	−	1.75347i	0.187991	−	0.325610i	0.944580	+	0.328280i	$0.106469\pi$
$30$		0			0
$31$		−	7.86488i		−	1.41257i	−0.707925	−	0.706287i	$-0.750369\pi$
$31$		−	7.86488i		−	1.41257i	0.707925	−	0.706287i	$-0.249631\pi$
$32$		0			0
$33$	−2.40029	+	1.38581i	−0.417837	+	0.241239i
$34$		0			0
$35$	−2.16612	−	3.75184i	−0.366142	−	0.634176i
$36$		0			0
$37$	−8.25551	−	4.76632i	−1.35720	−	0.783578i	−0.367952	−	0.929845i	$-0.619941\pi$
$37$	−8.25551	−	4.76632i	−1.35720	−	0.783578i	−0.989245	+	0.146267i	$0.953274\pi$
$38$		0			0
$39$	−0.0316594	−	5.76950i	−0.00506956	−	0.923860i
$40$		0			0
$41$	6.69615	+	3.86603i	1.04576	+	0.603772i	0.921460	−	0.388473i	$-0.126997\pi$
$41$	6.69615	+	3.86603i	1.04576	+	0.603772i	0.124303	+	0.992244i	$0.460331\pi$
$42$		0			0
$43$	2.09277	+	3.62479i	0.319145	+	0.552776i	0.980310	−	0.197465i	$-0.0632709\pi$
$43$	2.09277	+	3.62479i	0.319145	+	0.552776i	−0.661165	+	0.750241i	$0.729938\pi$
$44$		0			0
$45$	0.380509	−	0.219687i	0.0567229	−	0.0327490i
$46$		0			0
$47$		−	3.46410i		−	0.505291i	−0.967559	−	0.252646i	$-0.918699\pi$
$47$		−	3.46410i		−	0.505291i	0.967559	−	0.252646i	$-0.0813007\pi$
$48$		0			0
$49$	5.88418	−	10.1917i	0.840597	−	1.45596i
$50$		0			0
$51$	−12.0073			−1.68136
$52$		0			0
$53$	−12.6471			−1.73721			−0.868607	−	0.495502i	$-0.834984\pi$
$53$	−12.6471			−1.73721			−0.868607	+	0.495502i	$0.834984\pi$
$54$		0			0
$55$	0.866025	−	1.50000i	0.116775	−	0.202260i
$56$		0			0
$57$			8.59596i			1.13856i
$58$		0			0
$59$	5.49674	−	3.17354i	0.715615	−	0.413160i	−0.0975219	−	0.995233i	$-0.531092\pi$
$59$	5.49674	−	3.17354i	0.715615	−	0.413160i	0.813136	+	0.582073i	$0.197758\pi$
$60$		0			0
$61$	−1.85154	−	3.20696i	−0.237066	−	0.410610i	0.722805	−	0.691052i	$-0.242852\pi$
$61$	−1.85154	−	3.20696i	−0.237066	−	0.410610i	−0.959871	+	0.280442i	$0.909519\pi$
$62$		0			0
$63$	1.64846	+	0.951738i	0.207686	+	0.119908i
$64$		0			0
$65$	1.81988	+	3.11256i	0.225729	+	0.386066i
$66$		0			0
$67$	−4.55568	−	2.63022i	−0.556565	−	0.321333i	0.195200	−	0.980763i	$-0.437464\pi$
$67$	−4.55568	−	2.63022i	−0.556565	−	0.321333i	−0.751766	+	0.659430i	$0.770798\pi$
$68$		0			0
$69$	−0.928771	−	1.60868i	−0.111811	−	0.193662i
$70$		0			0
$71$	−10.8377	+	6.25714i	−1.28620	+	0.742586i	−0.977974	−	0.208729i	$-0.933067\pi$
$71$	−10.8377	+	6.25714i	−1.28620	+	0.742586i	−0.308222	+	0.951314i	$0.599734\pi$
$72$		0			0
$73$			5.23898i			0.613177i	0.951842	+	0.306588i	$0.0991875\pi$
$73$			5.23898i			0.613177i	−0.951842	+	0.306588i	$0.900813\pi$
$74$		0			0
$75$	0.800098	−	1.38581i	0.0923873	−	0.160020i
$76$		0			0
$77$	7.50367			0.855123
$78$		0			0
$79$	−8.16719			−0.918880			−0.459440	−	0.888209i	$-0.651950\pi$
$79$	−8.16719			−0.918880			−0.459440	+	0.888209i	$0.651950\pi$
$80$		0			0
$81$	3.74441	−	6.48552i	0.416046	−	0.720613i
$82$		0			0
$83$			0.456760i			0.0501359i	0.999686	+	0.0250679i	$0.00798021\pi$
$83$			0.456760i			0.0501359i	−0.999686	+	0.0250679i	$0.992020\pi$
$84$		0			0
$85$	6.49837	−	3.75184i	0.704847	−	0.406944i
$86$		0			0
$87$	1.61998	+	2.80589i	0.173680	+	0.300823i
$88$		0			0
$89$	−11.4967	−	6.63765i	−1.21865	−	0.703589i	−0.254022	−	0.967198i	$-0.581754\pi$
$89$	−11.4967	−	6.63765i	−1.21865	−	0.703589i	−0.964630	+	0.263609i	$0.915087\pi$
$90$		0			0
$91$	−7.73572	+	13.5701i	−0.810924	+	1.42253i
$92$		0			0
$93$	10.8992	+	6.29268i	1.13020	+	0.652520i
$94$		0			0
$95$	−2.68591	−	4.65213i	−0.275568	−	0.477298i
$96$		0			0
$97$	−2.43371	+	1.40511i	−0.247106	+	0.142667i	−0.618439	−	0.785833i	$-0.712234\pi$
$97$	−2.43371	+	1.40511i	−0.247106	+	0.142667i	0.371332	+	0.928500i	$0.378901\pi$
$98$		0			0
$99$			0.761018i			0.0764852i
$100$		0			0
$101$	−3.31988	+	5.75021i	−0.330341	+	0.572167i	−0.982579	−	0.185847i	$-0.940497\pi$
$101$	−3.31988	+	5.75021i	−0.330341	+	0.572167i	0.652238	+	0.758014i	$0.273830\pi$
$102$		0			0
$103$	7.37605			0.726784			0.363392	−	0.931636i	$-0.381619\pi$
$103$	7.37605			0.726784			0.363392	+	0.931636i	$0.381619\pi$
$104$		0			0
$105$	6.93244			0.676537
$106$		0			0
$107$	3.95668	−	6.85317i	0.382507	−	0.662521i	−0.608913	−	0.793237i	$-0.708394\pi$
$107$	3.95668	−	6.85317i	0.382507	−	0.662521i	0.991420	+	0.130716i	$0.0417275\pi$
$108$		0			0
$109$			9.18301i			0.879572i	0.898102	+	0.439786i	$0.144946\pi$
$109$			9.18301i			0.879572i	−0.898102	+	0.439786i	$0.855054\pi$
$110$		0			0
$111$	13.2104	−	7.62704i	1.25388	−	0.723927i
$112$		0			0
$113$	2.88793	+	5.00204i	0.271674	+	0.470552i	0.969291	−	0.245919i	$-0.0790896\pi$
$113$	2.88793	+	5.00204i	0.271674	+	0.470552i	−0.697617	+	0.716471i	$0.745756\pi$
$114$		0			0
$115$	1.00530	+	0.580411i	0.0937448	+	0.0541236i
$116$		0			0
$117$	−1.37627	−	0.784552i	−0.127236	−	0.0725319i
$118$		0			0
$119$	28.1525	+	16.2539i	2.58074	+	1.48999i
$120$		0			0
$121$	−4.00000	−	6.92820i	−0.363636	−	0.629837i
$122$		0			0
$123$	−10.7152	+	6.18640i	−0.966153	+	0.557809i
$124$		0			0
$125$			1.00000i			0.0894427i
$126$		0			0
$127$	9.48389	−	16.4266i	0.841559	−	1.45762i	−0.0470176	−	0.998894i	$-0.514972\pi$
$127$	9.48389	−	16.4266i	0.841559	−	1.45762i	0.888576	−	0.458729i	$-0.151695\pi$
$128$		0			0
$129$	−6.69770			−0.589699
$130$		0			0
$131$	−1.34637			−0.117633			−0.0588165	−	0.998269i	$-0.518733\pi$
$131$	−1.34637			−0.117633			−0.0588165	+	0.998269i	$0.518733\pi$
$132$		0			0
$133$	11.6360	−	20.1542i	1.00897	−	1.74759i
$134$		0			0
$135$			5.50367i			0.473681i
$136$		0			0
$137$	−10.7152	+	6.18640i	−0.915457	+	0.528540i	−0.882183	−	0.470907i	$-0.843927\pi$
$137$	−10.7152	+	6.18640i	−0.915457	+	0.528540i	−0.0332744	+	0.999446i	$0.510594\pi$
$138$		0			0
$139$	−4.38418	−	7.59362i	−0.371861	−	0.644083i	0.617991	−	0.786185i	$-0.287947\pi$
$139$	−4.38418	−	7.59362i	−0.371861	−	0.644083i	−0.989852	+	0.142103i	$0.954614\pi$
$140$		0			0
$141$	4.80059	+	2.77162i	0.404282	+	0.233413i
$142$		0			0
$143$	−6.24490	+	0.0342681i	−0.522225	+	0.00286565i
$144$		0			0
$145$	−1.75347	−	1.01236i	−0.145617	−	0.0840722i
$146$		0			0
$147$	9.41584	+	16.3087i	0.776605	+	1.34512i
$148$		0			0
$149$	4.10443	−	2.36970i	0.336248	−	0.194133i	−0.322363	−	0.946616i	$-0.604477\pi$
$149$	4.10443	−	2.36970i	0.336248	−	0.194133i	0.658612	+	0.752483i	$0.271144\pi$
$150$		0			0
$151$			1.56063i			0.127002i	0.997982	+	0.0635010i	$0.0202266\pi$
$151$			1.56063i			0.127002i	−0.997982	+	0.0635010i	$0.979773\pi$
$152$		0			0
$153$	−1.64846	+	2.85521i	−0.133270	+	0.230830i
$154$		0			0
$155$	−7.86488			−0.631723
$156$		0			0
$157$	−9.15332			−0.730515			−0.365257	−	0.930907i	$-0.619019\pi$
$157$	−9.15332			−0.730515			−0.365257	+	0.930907i	$0.619019\pi$
$158$		0			0
$159$	10.1189	−	17.5265i	0.802483	−	1.38994i
$160$		0			0
$161$			5.02897i			0.396338i
$162$		0			0
$163$	2.28893	−	1.32151i	0.179283	−	0.103509i	−0.407673	−	0.913128i	$-0.633660\pi$
$163$	2.28893	−	1.32151i	0.179283	−	0.103509i	0.586956	+	0.809619i	$0.300326\pi$
$164$		0			0
$165$	1.38581	+	2.40029i	0.107885	+	0.186863i
$166$		0			0
$167$	7.71515	+	4.45435i	0.597017	+	0.344688i	0.767867	−	0.640609i	$-0.221318\pi$
$167$	7.71515	+	4.45435i	0.597017	+	0.344688i	−0.170850	+	0.985297i	$0.554651\pi$
$168$		0			0
$169$	6.37605	−	11.3290i	0.490466	−	0.871460i
$170$		0			0
$171$	2.04402	+	1.18012i	0.156310	+	0.0902458i
$172$		0			0
$173$	6.90396	+	11.9580i	0.524899	+	0.909151i	0.999580	+	0.0289933i	$0.00923014\pi$
$173$	6.90396	+	11.9580i	0.524899	+	0.909151i	−0.474681	+	0.880158i	$0.657437\pi$
$174$		0			0
$175$	−3.75184	+	2.16612i	−0.283612	+	0.163744i
$176$		0			0
$177$			10.1566i			0.763416i
$178$		0			0
$179$	−12.1751	+	21.0879i	−0.910009	+	1.57618i	−0.0959612	+	0.995385i	$0.530592\pi$
$179$	−12.1751	+	21.0879i	−0.910009	+	1.57618i	−0.814048	+	0.580797i	$0.802741\pi$
$180$		0			0
$181$	24.8336			1.84587			0.922935	−	0.384956i	$-0.125783\pi$
$181$	24.8336			1.84587			0.922935	+	0.384956i	$0.125783\pi$
$182$		0			0
$183$	5.92566			0.438037
$184$		0			0
$185$	−4.76632	+	8.25551i	−0.350427	+	0.606957i
$186$		0			0
$187$			12.9967i			0.950416i
$188$		0			0
$189$	−20.6489	+	11.9216i	−1.50198	+	0.867171i
$190$		0			0
$191$	−2.98438	−	5.16909i	−0.215942	−	0.374022i	0.737622	−	0.675214i	$-0.235949\pi$
$191$	−2.98438	−	5.16909i	−0.215942	−	0.374022i	−0.953564	+	0.301192i	$0.902615\pi$
$192$		0			0
$193$	−7.83988	−	4.52636i	−0.564327	−	0.325814i	0.190553	−	0.981677i	$-0.438972\pi$
$193$	−7.83988	−	4.52636i	−0.564327	−	0.325814i	−0.754880	+	0.655862i	$0.772305\pi$
$194$		0			0
$195$	−5.76950	+	0.0316594i	−0.413163	+	0.00226718i
$196$		0			0
$197$	−9.86320	−	5.69452i	−0.702724	−	0.405718i	0.105637	−	0.994405i	$-0.466312\pi$
$197$	−9.86320	−	5.69452i	−0.702724	−	0.405718i	−0.808361	+	0.588687i	$0.799645\pi$
$198$		0			0
$199$	5.38863	+	9.33339i	0.381990	+	0.661626i	0.991347	−	0.131270i	$-0.0419055\pi$
$199$	5.38863	+	9.33339i	0.381990	+	0.661626i	−0.609357	+	0.792896i	$0.708572\pi$
$200$		0			0
$201$	7.28998	−	4.20887i	0.514196	−	0.296871i
$202$		0			0
$203$		−	8.77162i		−	0.615647i
$204$		0			0
$205$	3.86603	−	6.69615i	0.270015	−	0.467680i
$206$		0			0
$207$	−0.510035			−0.0354499
$208$		0			0
$209$	9.30426			0.643589
$210$		0			0
$211$	1.61256	−	2.79304i	0.111013	−	0.192280i	−0.805166	−	0.593050i	$-0.797924\pi$
$211$	1.61256	−	2.79304i	0.111013	−	0.192280i	0.916179	+	0.400769i	$0.131257\pi$
$212$		0			0
$213$		−	20.0253i		−	1.37211i
$214$		0			0
$215$	3.62479	−	2.09277i	0.247209	−	0.142726i
$216$		0			0
$217$	−17.0363	−	29.5078i	−1.15650	−	2.00312i
$218$		0			0
$219$	−7.26023	−	4.19170i	−0.490601	−	0.283249i
$220$		0			0
$221$	−23.5041	−	13.3987i	−1.58106	−	0.901292i
$222$		0			0
$223$	−14.7046	−	8.48968i	−0.984690	−	0.568511i	−0.0810069	−	0.996714i	$-0.525814\pi$
$223$	−14.7046	−	8.48968i	−0.984690	−	0.568511i	−0.903683	+	0.428203i	$0.859147\pi$
$224$		0			0
$225$	−0.219687	−	0.380509i	−0.0146458	−	0.0253673i
$226$		0			0
$227$	2.07891	−	1.20026i	0.137982	−	0.0796641i	−0.429420	−	0.903105i	$-0.641282\pi$
$227$	2.07891	−	1.20026i	0.137982	−	0.0796641i	0.567402	+	0.823441i	$0.307949\pi$
$228$		0			0
$229$			7.35671i			0.486145i	0.970008	+	0.243073i	$0.0781553\pi$
$229$			7.35671i			0.486145i	−0.970008	+	0.243073i	$0.921845\pi$
$230$		0			0
$231$	−6.00367	+	10.3987i	−0.395013	+	0.684182i
$232$		0			0
$233$	−3.37410			−0.221045			−0.110522	−	0.993874i	$-0.535252\pi$
$233$	−3.37410			−0.221045			−0.110522	+	0.993874i	$0.535252\pi$
$234$		0			0
$235$	−3.46410			−0.225973
$236$		0			0
$237$	6.53455	−	11.3182i	0.424464	−	0.735194i
$238$		0			0
$239$			23.7807i			1.53824i	0.639103	+	0.769121i	$0.279306\pi$
$239$			23.7807i			1.53824i	−0.639103	+	0.769121i	$0.720694\pi$
$240$		0			0
$241$	19.3307	−	11.1606i	1.24520	−	0.718918i	0.275054	−	0.961429i	$-0.411304\pi$
$241$	19.3307	−	11.1606i	1.24520	−	0.718918i	0.970149	+	0.242511i	$0.0779709\pi$
$242$		0			0
$243$	−2.26371	−	3.92086i	−0.145217	−	0.251523i
$244$		0			0
$245$	−10.1917	−	5.88418i	−0.651124	−	0.375926i
$246$		0			0
$247$	−9.59199	+	16.8264i	−0.610324	+	1.07064i
$248$		0			0
$249$	−0.632982	−	0.365452i	−0.0401136	−	0.0231596i
$250$		0			0
$251$	−2.31119	−	4.00310i	−0.145881	−	0.252673i	0.783820	−	0.620988i	$-0.213268\pi$
$251$	−2.31119	−	4.00310i	−0.145881	−	0.252673i	−0.929701	+	0.368314i	$0.879935\pi$
$252$		0			0
$253$	−1.74123	+	1.00530i	−0.109470	+	0.0632028i
$254$		0			0
$255$			12.0073i			0.751929i
$256$		0			0
$257$	10.9320	−	18.9347i	0.681916	−	1.18111i	−0.292479	−	0.956272i	$-0.594480\pi$
$257$	10.9320	−	18.9347i	0.681916	−	1.18111i	0.974395	−	0.224842i	$-0.0721866\pi$
$258$		0			0
$259$	−41.2977			−2.56612
$260$		0			0
$261$	0.889612			0.0550656
$262$		0			0
$263$	10.0895	−	17.4756i	0.622146	−	1.07759i	−0.366939	−	0.930245i	$-0.619594\pi$
$263$	10.0895	−	17.4756i	0.622146	−	1.07759i	0.989085	−	0.147344i	$-0.0470724\pi$
$264$		0			0
$265$			12.6471i			0.776906i
$266$		0			0
$267$	18.3970	−	10.6215i	1.12588	−	0.650027i
$268$		0			0
$269$	−3.64344	−	6.31062i	−0.222144	−	0.384765i	0.733315	−	0.679889i	$-0.237972\pi$
$269$	−3.64344	−	6.31062i	−0.222144	−	0.384765i	−0.955459	+	0.295124i	$0.904639\pi$
$270$		0			0
$271$	−4.23650	−	2.44595i	−0.257349	−	0.148581i	0.365775	−	0.930703i	$-0.380804\pi$
$271$	−4.23650	−	2.44595i	−0.257349	−	0.148581i	−0.623125	+	0.782122i	$0.714137\pi$
$272$		0			0
$273$	−12.6162	−	21.5776i	−0.763569	−	1.30594i
$274$		0			0
$275$	−1.50000	−	0.866025i	−0.0904534	−	0.0522233i
$276$		0			0
$277$	4.40029	+	7.62153i	0.264388	+	0.457933i	0.967403	−	0.253241i	$-0.0814967\pi$
$277$	4.40029	+	7.62153i	0.264388	+	0.457933i	−0.703015	+	0.711175i	$0.748163\pi$
$278$		0			0
$279$	2.99266	−	1.72781i	0.179166	−	0.103441i
$280$		0			0
$281$		−	12.6085i		−	0.752161i	−0.926587	−	0.376081i	$-0.877272\pi$
$281$		−	12.6085i		−	0.752161i	0.926587	−	0.376081i	$-0.122728\pi$
$282$		0			0
$283$	−10.4505	+	18.1007i	−0.621216	+	1.07598i	0.368044	+	0.929808i	$0.380028\pi$
$283$	−10.4505	+	18.1007i	−0.621216	+	1.07598i	−0.989260	+	0.146169i	$0.953306\pi$
$284$		0			0
$285$	8.59596			0.509181
$286$		0			0
$287$	33.4972			1.97727
$288$		0			0
$289$	−19.6525	+	34.0392i	−1.15603	+	2.00231i
$290$		0			0
$291$		−	4.49689i		−	0.263612i
$292$		0			0
$293$	−0.599706	+	0.346241i	−0.0350352	+	0.0202276i	−0.517415	−	0.855734i	$-0.673106\pi$
$293$	−0.599706	+	0.346241i	−0.0350352	+	0.0202276i	0.482380	+	0.875962i	$0.339772\pi$
$294$		0			0
$295$	−3.17354	−	5.49674i	−0.184771	−	0.320033i
$296$		0			0
$297$	−8.25551	−	4.76632i	−0.479033	−	0.276570i
$298$		0			0
$299$	−0.0229665	−	4.18534i	−0.00132819	−	0.242044i
$300$		0			0
$301$	15.7035	+	9.06642i	0.905134	+	0.522580i
$302$		0			0
$303$	−5.31246	−	9.20145i	−0.305193	−	0.528610i
$304$		0			0
$305$	−3.20696	+	1.85154i	−0.183630	+	0.106019i
$306$		0			0
$307$			8.83168i			0.504051i	0.967721	+	0.252025i	$0.0810966\pi$
$307$			8.83168i			0.504051i	−0.967721	+	0.252025i	$0.918903\pi$
$308$		0			0
$309$	−5.90157	+	10.2218i	−0.335728	+	0.581499i
$310$		0			0
$311$	−11.3958			−0.646198			−0.323099	−	0.946365i	$-0.604725\pi$
$311$	−11.3958			−0.646198			−0.323099	+	0.946365i	$0.604725\pi$
$312$		0			0
$313$	−3.38496			−0.191329			−0.0956647	−	0.995414i	$-0.530498\pi$
$313$	−3.38496			−0.191329			−0.0956647	+	0.995414i	$0.530498\pi$
$314$		0			0
$315$	0.951738	−	1.64846i	0.0536243	−	0.0928801i
$316$		0			0
$317$		−	0.557104i		−	0.0312901i	−0.999878	−	0.0156450i	$-0.995020\pi$
$317$		−	0.557104i		−	0.0312901i	0.999878	−	0.0156450i	$-0.00498017\pi$
$318$		0			0
$319$	3.03709	−	1.75347i	0.170044	−	0.0981752i
$320$		0			0
$321$	6.33146	+	10.9664i	0.353388	+	0.612086i
$322$		0			0
$323$	34.9080	+	20.1542i	1.94234	+	1.12141i
$324$		0			0
$325$	3.11256	−	1.81988i	0.172654	−	0.100949i
$326$		0			0
$327$	−12.7259	−	7.34730i	−0.703744	−	0.406307i
$328$		0			0
$329$	−7.50367	−	12.9967i	−0.413691	−	0.716533i
$330$		0			0
$331$	3.43266	−	1.98185i	0.188676	−	0.108932i	−0.402687	−	0.915338i	$-0.631924\pi$
$331$	3.43266	−	1.98185i	0.188676	−	0.108932i	0.591363	+	0.806406i	$0.298590\pi$
$332$		0			0
$333$		−	4.18839i		−	0.229522i
$334$		0			0
$335$	−2.63022	+	4.55568i	−0.143705	+	0.248904i
$336$		0			0
$337$	16.7243			0.911030			0.455515	−	0.890228i	$-0.349455\pi$
$337$	16.7243			0.911030			0.455515	+	0.890228i	$0.349455\pi$
$338$		0			0
$339$	−9.24251			−0.501984
$340$		0			0
$341$	6.81119	−	11.7973i	0.368847	−	0.638861i
$342$		0			0
$343$		−	20.6577i		−	1.11541i
$344$		0			0
$345$	−1.60868	+	0.928771i	−0.0866084	+	0.0500034i
$346$		0			0
$347$	−3.79043	−	6.56521i	−0.203481	−	0.352439i	0.746167	−	0.665759i	$-0.231892\pi$
$347$	−3.79043	−	6.56521i	−0.203481	−	0.352439i	−0.949648	+	0.313320i	$0.898559\pi$
$348$		0			0
$349$	23.6733	+	13.6678i	1.26720	+	0.731621i	0.974458	−	0.224569i	$-0.0720975\pi$
$349$	23.6733	+	13.6678i	1.26720	+	0.731621i	0.292746	+	0.956190i	$0.405431\pi$
$350$		0			0
$351$	17.1305	−	10.0160i	0.914359	−	0.534616i
$352$		0			0
$353$	22.5964	+	13.0461i	1.20269	+	0.694372i	0.961152	−	0.276020i	$-0.0890156\pi$
$353$	22.5964	+	13.0461i	1.20269	+	0.694372i	0.241536	+	0.970392i	$0.422349\pi$
$354$		0			0
$355$	6.25714	+	10.8377i	0.332094	+	0.575204i
$356$		0			0
$357$	−45.0496	+	26.0094i	−2.38428	+	1.37656i
$358$		0			0
$359$			0.694176i			0.0366372i	0.999832	+	0.0183186i	$0.00583132\pi$
$359$			0.694176i			0.0366372i	−0.999832	+	0.0183186i	$0.994169\pi$
$360$		0			0
$361$	4.92820	−	8.53590i	0.259379	−	0.449258i
$362$		0			0
$363$	12.8016			0.671908
$364$		0			0
$365$	5.23898			0.274221
$366$		0			0
$367$	−11.4109	+	19.7643i	−0.595644	+	1.03169i	0.397812	+	0.917467i	$0.369770\pi$
$367$	−11.4109	+	19.7643i	−0.595644	+	1.03169i	−0.993456	+	0.114218i	$0.963564\pi$
$368$		0			0
$369$			3.39726i			0.176854i
$370$		0			0
$371$	−47.4499	+	27.3952i	−2.46347	+	1.42229i
$372$		0			0
$373$	3.94065	+	6.82540i	0.204039	+	0.353406i	0.949826	−	0.312779i	$-0.101260\pi$
$373$	3.94065	+	6.82540i	0.204039	+	0.353406i	−0.745787	+	0.666184i	$0.767926\pi$
$374$		0			0
$375$	−1.38581	−	0.800098i	−0.0715629	−	0.0413169i
$376$		0			0
$377$	0.0400587	+	7.30015i	0.00206313	+	0.375977i
$378$		0			0
$379$	26.9703	+	15.5713i	1.38537	+	0.799843i	0.992789	−	0.119875i	$-0.0382494\pi$
$379$	26.9703	+	15.5713i	1.38537	+	0.799843i	0.392580	+	0.919718i	$0.371583\pi$
$380$		0			0
$381$	15.1761	+	26.2857i	0.777494	+	1.34666i
$382$		0			0
$383$	29.8885	−	17.2561i	1.52723	−	0.881747i	0.527754	−	0.849397i	$-0.323034\pi$
$383$	29.8885	−	17.2561i	1.52723	−	0.881747i	0.999477	−	0.0323495i	$-0.0102990\pi$
$384$		0			0
$385$		−	7.50367i		−	0.382422i
$386$		0			0
$387$	−0.919510	+	1.59264i	−0.0467413	+	0.0809584i
$388$		0			0
$389$	−6.18414			−0.313548			−0.156774	−	0.987634i	$-0.550109\pi$
$389$	−6.18414			−0.313548			−0.156774	+	0.987634i	$0.550109\pi$
$390$		0			0
$391$	−8.71043			−0.440505
$392$		0			0
$393$	1.07723	−	1.86582i	0.0543390	−	0.0941179i
$394$		0			0
$395$			8.16719i			0.410936i
$396$		0			0
$397$	19.6209	−	11.3281i	0.984745	−	0.568543i	0.0810459	−	0.996710i	$-0.474174\pi$
$397$	19.6209	−	11.3281i	0.984745	−	0.568543i	0.903699	+	0.428167i	$0.140841\pi$
$398$		0			0
$399$	18.6199	+	32.2506i	0.932161	+	1.61455i
$400$		0			0
$401$	3.35847	+	1.93902i	0.167714	+	0.0968298i	0.581508	−	0.813541i	$-0.302463\pi$
$401$	3.35847	+	1.93902i	0.167714	+	0.0968298i	−0.413793	+	0.910371i	$0.635796\pi$
$402$		0			0
$403$	14.3132	+	24.4799i	0.712990	+	1.21943i
$404$		0			0
$405$	−6.48552	−	3.74441i	−0.322268	−	0.186061i
$406$		0			0
$407$	−8.25551	−	14.2990i	−0.409210	−	0.708773i
$408$		0			0
$409$	12.0037	−	6.93032i	0.593543	−	0.342682i	−0.172954	−	0.984930i	$-0.555331\pi$
$409$	12.0037	−	6.93032i	0.593543	−	0.342682i	0.766497	+	0.642248i	$0.221998\pi$
$410$		0			0
$411$		−	19.7989i		−	0.976607i
$412$		0			0
$413$	13.7486	−	23.8132i	0.676523	−	1.17177i
$414$		0			0
$415$	0.456760			0.0224214
$416$		0			0
$417$	14.0311			0.687106
$418$		0			0
$419$	15.1751	−	26.2840i	0.741352	−	1.28406i	−0.210528	−	0.977588i	$-0.567518\pi$
$419$	15.1751	−	26.2840i	0.741352	−	1.28406i	0.951880	−	0.306471i	$-0.0991484\pi$
$420$		0			0
$421$			0.608516i			0.0296573i	0.999890	+	0.0148286i	$0.00472027\pi$
$421$			0.608516i			0.0296573i	−0.999890	+	0.0148286i	$0.995280\pi$
$422$		0			0
$423$	1.31812	−	0.761018i	0.0640892	−	0.0370019i
$424$		0			0
$425$	−3.75184	−	6.49837i	−0.181991	−	0.315217i
$426$		0			0
$427$	−13.8934	−	8.02134i	−0.672347	−	0.388180i
$428$		0			0
$429$	4.94905	−	8.68167i	0.238942	−	0.419155i
$430$		0			0
$431$	−34.0819	−	19.6772i	−1.64167	−	0.947818i	−0.980240	−	0.197810i	$-0.936617\pi$
$431$	−34.0819	−	19.6772i	−1.64167	−	0.947818i	−0.661429	−	0.750008i	$-0.730050\pi$
$432$		0			0
$433$	3.21474	+	5.56810i	0.154491	+	0.267586i	0.932873	−	0.360204i	$-0.117293\pi$
$433$	3.21474	+	5.56810i	0.154491	+	0.267586i	−0.778383	+	0.627790i	$0.783960\pi$
$434$		0			0
$435$	2.80589	−	1.61998i	0.134532	−	0.0776721i
$436$		0			0
$437$			6.23572i			0.298295i
$438$		0			0
$439$	0.735722	−	1.27431i	0.0351141	−	0.0608194i	−0.847934	−	0.530101i	$-0.822154\pi$
$439$	0.735722	−	1.27431i	0.0351141	−	0.0608194i	0.883048	+	0.469282i	$0.155487\pi$
$440$		0			0
$441$	5.17071			0.246224
$442$		0			0
$443$	−16.9193			−0.803860			−0.401930	−	0.915670i	$-0.631660\pi$
$443$	−16.9193			−0.803860			−0.401930	+	0.915670i	$0.631660\pi$
$444$		0			0
$445$	−6.63765	+	11.4967i	−0.314655	+	0.544998i
$446$		0			0
$447$			7.58396i			0.358709i
$448$		0			0
$449$	−7.89557	+	4.55851i	−0.372615	+	0.215129i	−0.674600	−	0.738183i	$-0.735684\pi$
$449$	−7.89557	+	4.55851i	−0.372615	+	0.215129i	0.301985	+	0.953313i	$0.402351\pi$
$450$		0			0
$451$	6.69615	+	11.5981i	0.315310	+	0.546132i
$452$		0			0
$453$	−2.16273	−	1.24865i	−0.101614	−	0.0586669i
$454$		0			0
$455$	13.5701	+	7.73572i	0.636176	+	0.362656i
$456$		0			0
$457$	−15.2041	−	8.77811i	−0.711220	−	0.410623i	0.100293	−	0.994958i	$-0.468022\pi$
$457$	−15.2041	−	8.77811i	−0.711220	−	0.410623i	−0.811512	+	0.584335i	$0.801355\pi$
$458$		0			0
$459$	−20.6489	−	35.7649i	−0.963807	−	1.66936i
$460$		0			0
$461$	−13.2973	+	7.67722i	−0.619318	+	0.357564i	−0.776604	−	0.629990i	$-0.783059\pi$
$461$	−13.2973	+	7.67722i	−0.619318	+	0.357564i	0.157285	+	0.987553i	$0.449726\pi$
$462$		0			0
$463$		−	18.8614i		−	0.876562i	−0.898838	−	0.438281i	$-0.855588\pi$
$463$		−	18.8614i		−	0.876562i	0.898838	−	0.438281i	$-0.144412\pi$
$464$		0			0
$465$	6.29268	−	10.8992i	0.291816	−	0.505440i
$466$		0			0
$467$	32.9302			1.52383			0.761913	−	0.647679i	$-0.224260\pi$
$467$	32.9302			1.52383			0.761913	+	0.647679i	$0.224260\pi$
$468$		0			0
$469$	−22.7896			−1.05232
$470$		0			0
$471$	7.32355	−	12.6848i	0.337452	−	0.584483i
$472$		0			0
$473$			7.24958i			0.333336i
$474$		0			0
$475$	−4.65213	+	2.68591i	−0.213454	+	0.123238i
$476$		0			0
$477$	−2.77840	−	4.81234i	−0.127214	−	0.220342i
$478$		0			0
$479$	−14.9368	−	8.62378i	−0.682481	−	0.394031i	0.118308	−	0.992977i	$-0.462253\pi$
$479$	−14.9368	−	8.62378i	−0.682481	−	0.394031i	−0.800789	+	0.598946i	$0.795586\pi$
$480$		0			0
$481$	34.3699	−	0.188601i	1.56713	−	0.00859944i
$482$		0			0
$483$	−6.96919	−	4.02367i	−0.317109	−	0.183083i
$484$		0			0
$485$	1.40511	+	2.43371i	0.0638026	+	0.110509i
$486$		0			0
$487$	−15.5663	+	8.98720i	−0.705376	+	0.407249i	−0.809346	−	0.587331i	$-0.800178\pi$
$487$	−15.5663	+	8.98720i	−0.705376	+	0.407249i	0.103971	+	0.994580i	$0.466845\pi$
$488$		0			0
$489$			4.22936i			0.191258i
$490$		0			0
$491$	13.0977	−	22.6858i	0.591089	−	1.02380i	−0.402997	−	0.915201i	$-0.632032\pi$
$491$	13.0977	−	22.6858i	0.591089	−	1.02380i	0.994086	−	0.108595i	$-0.0346350\pi$
$492$		0			0
$493$	15.1929			0.684253
$494$		0			0
$495$	0.761018			0.0342052
$496$		0			0
$497$	−27.1075	+	46.9515i	−1.21594	+	2.10606i
$498$		0			0
$499$		−	18.3428i		−	0.821139i	−0.911829	−	0.410569i	$-0.865330\pi$
$499$		−	18.3428i		−	0.821139i	0.911829	−	0.410569i	$-0.134670\pi$
$500$		0			0
$501$	−12.3458	+	7.12783i	−0.551568	+	0.318448i
$502$		0			0
$503$	7.12811	+	12.3462i	0.317827	+	0.550492i	0.980034	−	0.198829i	$-0.0637137\pi$
$503$	7.12811	+	12.3462i	0.317827	+	0.550492i	−0.662208	+	0.749320i	$0.730380\pi$
$504$		0			0
$505$	5.75021	+	3.31988i	0.255881	+	0.147733i
$506$		0			0
$507$	10.5984	+	17.9003i	0.470690	+	0.794980i
$508$		0			0
$509$	27.5380	+	15.8991i	1.22060	+	0.704714i	0.965046	−	0.262081i	$-0.0844086\pi$
$509$	27.5380	+	15.8991i	1.22060	+	0.704714i	0.255555	+	0.966795i	$0.417742\pi$
$510$		0			0
$511$	11.3483	+	19.6558i	0.502018	+	0.869521i
$512$		0			0
$513$	−25.6038	+	14.7824i	−1.13043	+	0.652657i
$514$		0			0
$515$		−	7.37605i		−	0.325028i
$516$		0			0
$517$	3.00000	−	5.19615i	0.131940	−	0.228527i
$518$		0			0
$519$	−22.0954			−0.969880
$520$		0			0
$521$	−45.1676			−1.97883			−0.989414	−	0.145122i	$-0.953642\pi$
$521$	−45.1676			−1.97883			−0.989414	+	0.145122i	$0.953642\pi$
$522$		0			0
$523$	−13.0573	+	22.6159i	−0.570955	+	0.988924i	0.425513	+	0.904952i	$0.360094\pi$
$523$	−13.0573	+	22.6159i	−0.570955	+	0.988924i	−0.996468	+	0.0839712i	$0.973240\pi$
$524$		0			0
$525$		−	6.93244i		−	0.302557i
$526$		0			0
$527$	51.1089	−	29.5078i	2.22634	−	1.28538i
$528$		0			0
$529$	10.8262	+	18.7516i	0.470706	+	0.815287i
$530$		0			0
$531$	2.41512	+	1.39437i	0.104807	+	0.0605106i
$532$		0			0
$533$	−27.8779	+	0.152976i	−1.20753	+	0.00662615i
$534$		0			0
$535$	−6.85317	−	3.95668i	−0.296288	−	0.171062i
$536$		0			0
$537$	−19.4825	−	33.7447i	−0.840733	−	1.45619i
$538$		0			0
$539$	17.6525	−	10.1917i	0.760349	−	0.438988i
$540$		0			0
$541$			31.4750i			1.35321i	0.736344	+	0.676607i	$0.236550\pi$
$541$			31.4750i			1.35321i	−0.736344	+	0.676607i	$0.763450\pi$
$542$		0			0
$543$	−19.8693	+	34.4147i	−0.852675	+	1.47688i
$544$		0			0
$545$	9.18301			0.393357
$546$		0			0
$547$	−5.70391			−0.243881			−0.121941	−	0.992537i	$-0.538912\pi$
$547$	−5.70391			−0.243881			−0.121941	+	0.992537i	$0.538912\pi$
$548$		0			0
$549$	0.813519	−	1.40906i	0.0347201	−	0.0601370i
$550$		0			0
$551$		−	10.8765i		−	0.463353i
$552$		0			0
$553$	−30.6419	+	17.6911i	−1.30303	+	0.752303i
$554$		0			0
$555$	−7.62704	−	13.2104i	−0.323750	−	0.560751i
$556$		0			0
$557$	18.0757	+	10.4360i	0.765890	+	0.442187i	0.831406	−	0.555665i	$-0.187536\pi$
$557$	18.0757	+	10.4360i	0.765890	+	0.442187i	−0.0655165	+	0.997851i	$0.520870\pi$
$558$		0			0
$559$	−13.1106	−	7.47378i	−0.554519	−	0.316107i
$560$		0			0
$561$	−18.0110	−	10.3987i	−0.760426	−	0.439032i
$562$		0			0
$563$	−16.6157	−	28.7793i	−0.700270	−	1.21290i	−0.968372	−	0.249513i	$-0.919730\pi$
$563$	−16.6157	−	28.7793i	−0.700270	−	1.21290i	0.268102	−	0.963391i	$-0.413604\pi$
$564$		0			0
$565$	5.00204	−	2.88793i	0.210437	−	0.121496i
$566$		0			0
$567$		−	32.4435i		−	1.36250i
$568$		0			0
$569$	−20.0336	+	34.6992i	−0.839851	+	1.45466i	0.0501680	+	0.998741i	$0.484024\pi$
$569$	−20.0336	+	34.6992i	−0.839851	+	1.45466i	−0.890019	+	0.455924i	$0.849309\pi$
$570$		0			0
$571$	−4.53590			−0.189821			−0.0949107	−	0.995486i	$-0.530257\pi$
$571$	−4.53590			−0.189821			−0.0949107	+	0.995486i	$0.530257\pi$
$572$		0			0
$573$	9.55117			0.399006
$574$		0			0
$575$	0.580411	−	1.00530i	0.0242048	−	0.0419240i
$576$		0			0
$577$		−	35.2706i		−	1.46834i	−0.678968	−	0.734168i	$-0.737573\pi$
$577$		−	35.2706i		−	1.46834i	0.678968	−	0.734168i	$-0.262427\pi$
$578$		0			0
$579$	12.5453	−	7.24306i	0.521367	−	0.301011i
$580$		0			0
$581$	0.989397	+	1.71369i	0.0410471	+	0.0710957i
$582$		0			0
$583$	−18.9707	−	10.9527i	−0.785684	−	0.453615i
$584$		0			0
$585$	−0.784552	+	1.37627i	−0.0324372	+	0.0569018i
$586$		0			0
$587$	−35.6511	−	20.5832i	−1.47148	−	0.849558i	−0.471991	−	0.881603i	$-0.656464\pi$
$587$	−35.6511	−	20.5832i	−1.47148	−	0.849558i	−0.999486	+	0.0320457i	$0.989798\pi$
$588$		0			0
$589$	−21.1244	−	36.5885i	−0.870414	−	1.50760i
$590$		0			0
$591$	15.7831	−	9.11235i	0.649228	−	0.374832i
$592$		0			0
$593$			9.39726i			0.385899i	0.981209	+	0.192950i	$0.0618054\pi$
$593$			9.39726i			0.385899i	−0.981209	+	0.192950i	$0.938195\pi$
$594$		0			0
$595$	16.2539	−	28.1525i	0.666344	−	1.15414i
$596$		0			0
$597$	−17.2457			−0.705821
$598$		0			0
$599$	21.7881			0.890239			0.445119	−	0.895471i	$-0.353161\pi$
$599$	21.7881			0.890239			0.445119	+	0.895471i	$0.353161\pi$
$600$		0			0
$601$	−12.5481	+	21.7340i	−0.511848	+	0.886547i	0.488058	+	0.872811i	$0.337706\pi$
$601$	−12.5481	+	21.7340i	−0.511848	+	0.886547i	−0.999906	+	0.0137352i	$0.995628\pi$
$602$		0			0
$603$		−	2.31130i		−	0.0941236i
$604$		0			0
$605$	−6.92820	+	4.00000i	−0.281672	+	0.162623i
$606$		0			0
$607$	16.9888	+	29.4254i	0.689552	+	1.19434i	0.971983	+	0.235052i	$0.0755259\pi$
$607$	16.9888	+	29.4254i	0.689552	+	1.19434i	−0.282431	+	0.959288i	$0.591141\pi$
$608$		0			0
$609$	12.1558	+	7.01815i	0.492578	+	0.284390i
$610$		0			0
$611$	6.30426	+	10.7822i	0.255043	+	0.436202i
$612$		0			0
$613$	18.5736	+	10.7235i	0.750182	+	0.433118i	0.825760	−	0.564022i	$-0.190747\pi$
$613$	18.5736	+	10.7235i	0.750182	+	0.433118i	−0.0755778	+	0.997140i	$0.524080\pi$
$614$		0			0
$615$	6.18640	+	10.7152i	0.249460	+	0.432077i
$616$		0			0
$617$	29.5092	−	17.0372i	1.18800	−	0.685890i	0.230146	−	0.973156i	$-0.426080\pi$
$617$	29.5092	−	17.0372i	1.18800	−	0.685890i	0.957851	+	0.287266i	$0.0927464\pi$
$618$		0			0
$619$		−	26.9791i		−	1.08438i	−0.840256	−	0.542191i	$-0.817595\pi$
$619$		−	26.9791i		−	1.08438i	0.840256	−	0.542191i	$-0.182405\pi$
$620$		0			0
$621$	3.19439	−	5.53285i	0.128186	−	0.222026i
$622$		0			0
$623$	−57.5118			−2.30416
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	−7.44432	+	12.8939i	−0.297297	+	0.514934i
$628$		0			0
$629$		−	71.5298i		−	2.85208i
$630$		0			0
$631$	−32.8863	+	18.9869i	−1.30918	+	0.755857i	−0.981960	−	0.189090i	$-0.939446\pi$
$631$	−32.8863	+	18.9869i	−1.30918	+	0.755857i	−0.327223	+	0.944947i	$0.606113\pi$
$632$		0			0
$633$	2.58041	+	4.46940i	0.102562	+	0.177643i
$634$		0			0
$635$	−16.4266	−	9.48389i	−0.651869	−	0.376357i
$636$		0			0
$637$	0.232834	+	42.4308i	0.00922521	+	1.68117i
$638$		0			0
$639$	−4.76179	−	2.74922i	−0.188374	−	0.108758i
$640$		0			0
$641$	−4.12590	−	7.14627i	−0.162963	−	0.282261i	0.772967	−	0.634446i	$-0.218772\pi$
$641$	−4.12590	−	7.14627i	−0.162963	−	0.282261i	−0.935930	+	0.352186i	$0.885439\pi$
$642$		0			0
$643$	41.6992	−	24.0750i	1.64446	−	0.949427i	0.665233	−	0.746636i	$-0.268332\pi$
$643$	41.6992	−	24.0750i	1.64446	−	0.949427i	0.979222	−	0.202791i	$-0.0650011\pi$
$644$		0			0
$645$			6.69770i			0.263722i
$646$		0			0
$647$	−1.08951	+	1.88709i	−0.0428332	+	0.0741893i	−0.886647	−	0.462446i	$-0.846972\pi$
$647$	−1.08951	+	1.88709i	−0.0428332	+	0.0741893i	0.843814	+	0.536636i	$0.180305\pi$
$648$		0			0
$649$	10.9935			0.431532
$650$		0			0
$651$	54.5229			2.13692
$652$		0			0
$653$	−16.9341	+	29.3308i	−0.662684	+	1.14780i	0.317224	+	0.948351i	$0.397249\pi$
$653$	−16.9341	+	29.3308i	−0.662684	+	1.14780i	−0.979908	+	0.199451i	$0.936084\pi$
$654$		0			0
$655$			1.34637i			0.0526071i
$656$		0			0
$657$	−1.99348	+	1.15094i	−0.0777730	+	0.0449023i
$658$		0			0
$659$	−13.2246	−	22.9056i	−0.515155	−	0.892275i	−0.999845	−	0.0175892i	$-0.994401\pi$
$659$	−13.2246	−	22.9056i	−0.515155	−	0.892275i	0.484690	−	0.874686i	$-0.338932\pi$
$660$		0			0
$661$	−26.3259	−	15.1993i	−1.02396	−	0.591182i	−0.108710	−	0.994074i	$-0.534672\pi$
$661$	−26.3259	−	15.1993i	−1.02396	−	0.591182i	−0.915248	+	0.402891i	$0.868005\pi$
$662$		0			0
$663$	37.3736	−	21.8520i	1.45147	−	0.848660i
$664$		0			0
$665$	−20.1542	−	11.6360i	−0.781545	−	0.451225i
$666$		0			0
$667$	1.17517	+	2.03546i	0.0455029	+	0.0788134i
$668$		0			0
$669$	23.5302	−	13.5851i	0.909729	−	0.525232i
$670$		0			0
$671$		−	6.41393i		−	0.247607i
$672$		0			0
$673$	−18.1976	+	31.5192i	−0.701467	+	1.21498i	0.266485	+	0.963839i	$0.414138\pi$
$673$	−18.1976	+	31.5192i	−0.701467	+	1.21498i	−0.967952	+	0.251137i	$0.919196\pi$
$674$		0			0
$675$	5.50367			0.211836
$676$		0			0
$677$	36.0043			1.38376			0.691880	−	0.722013i	$-0.256783\pi$
$677$	36.0043			1.38376			0.691880	+	0.722013i	$0.256783\pi$
$678$		0			0
$679$	−6.08726	+	10.5435i	−0.233608	+	0.404620i
$680$		0			0
$681$			3.84130i			0.147199i
$682$		0			0
$683$	−11.1107	+	6.41478i	−0.425140	+	0.245455i	−0.697274	−	0.716805i	$-0.745604\pi$
$683$	−11.1107	+	6.41478i	−0.425140	+	0.245455i	0.272134	+	0.962259i	$0.412271\pi$
$684$		0			0
$685$	6.18640	+	10.7152i	0.236370	+	0.409405i
$686$		0			0
$687$	−10.1950	−	5.88609i	−0.388964	−	0.224568i
$688$		0			0
$689$	39.3649	−	23.0162i	1.49968	−	0.876849i
$690$		0			0
$691$	28.5608	+	16.4896i	1.08650	+	0.627294i	0.932644	−	0.360799i	$-0.117496\pi$
$691$	28.5608	+	16.4896i	1.08650	+	0.627294i	0.153861	+	0.988092i	$0.450829\pi$
$692$		0			0
$693$	1.64846	+	2.85521i	0.0626197	+	0.108461i
$694$		0			0
$695$	−7.59362	+	4.38418i	−0.288043	+	0.166301i
$696$		0			0
$697$			58.0188i			2.19762i
$698$		0			0
$699$	2.69961	−	4.67586i	0.102109	−	0.176857i
$700$		0			0
$701$	40.7352			1.53855			0.769273	−	0.638921i	$-0.220619\pi$
$701$	40.7352			1.53855			0.769273	+	0.638921i	$0.220619\pi$
$702$		0			0
$703$	−51.2076			−1.93133
$704$		0			0
$705$	2.77162	−	4.80059i	0.104385	−	0.180801i
$706$		0			0
$707$			28.7651i			1.08182i
$708$		0			0
$709$	−15.2932	+	8.82956i	−0.574350	+	0.331601i	−0.758885	−	0.651225i	$-0.774255\pi$
$709$	−15.2932	+	8.82956i	−0.574350	+	0.331601i	0.184535	+	0.982826i	$0.440922\pi$
$710$		0			0
$711$	−1.79422	−	3.10769i	−0.0672886	−	0.116547i
$712$		0			0
$713$	7.90658	+	4.56487i	0.296104	+	0.170956i
$714$		0			0
$715$	0.0342681	+	6.24490i	0.00128156	+	0.233546i
$716$		0			0
$717$	−32.9555	−	19.0269i	−1.23074	−	0.710571i
$718$		0			0
$719$	24.7836	+	42.9265i	0.924273	+	1.60089i	0.792727	+	0.609577i	$0.208661\pi$
$719$	24.7836	+	42.9265i	0.924273	+	1.60089i	0.131546	+	0.991310i	$0.458006\pi$
$720$		0			0
$721$	27.6737	−	15.9774i	1.03062	−	0.595031i
$722$		0			0
$723$			35.7183i			1.32838i
$724$		0			0
$725$	−1.01236	+	1.75347i	−0.0375983	+	0.0651221i
$726$		0			0
$727$	−36.4738			−1.35274			−0.676370	−	0.736562i	$-0.736448\pi$
$727$	−36.4738			−1.35274			−0.676370	+	0.736562i	$0.736448\pi$
$728$		0			0
$729$	29.7112			1.10042
$730$		0			0
$731$	−15.7035	+	27.1993i	−0.580815	+	1.00600i
$732$		0			0
$733$		−	28.8491i		−	1.06556i	−0.846252	−	0.532782i	$-0.821146\pi$
$733$		−	28.8491i		−	1.06556i	0.846252	−	0.532782i	$-0.178854\pi$
$734$		0			0
$735$	16.3087	−	9.41584i	0.601556	−	0.347308i
$736$		0			0
$737$	−4.55568	−	7.89067i	−0.167811	−	0.290657i
$738$		0			0
$739$	17.1330	+	9.89173i	0.630247	+	0.363873i	0.780848	−	0.624721i	$-0.214787\pi$
$739$	17.1330	+	9.89173i	0.630247	+	0.363873i	−0.150601	+	0.988595i	$0.548121\pi$
$740$		0			0
$741$	−15.6436	−	26.7554i	−0.574683	−	0.982885i
$742$		0			0
$743$	−22.8635	−	13.2003i	−0.838781	−	0.484271i	0.0180685	−	0.999837i	$-0.494248\pi$
$743$	−22.8635	−	13.2003i	−0.838781	−	0.484271i	−0.856850	+	0.515566i	$0.827582\pi$
$744$		0			0
$745$	−2.36970	−	4.10443i	−0.0868190	−	0.150375i
$746$		0			0
$747$	−0.173801	+	0.100344i	−0.00635905	+	0.00367140i
$748$		0			0
$749$		−	34.2826i		−	1.25266i
$750$		0			0
$751$	−6.90171	+	11.9541i	−0.251847	+	0.436212i	−0.964034	−	0.265778i	$-0.914371\pi$
$751$	−6.90171	+	11.9541i	−0.251847	+	0.436212i	0.712187	+	0.701990i	$0.247705\pi$
$752$		0			0
$753$	7.39671			0.269551
$754$		0			0
$755$	1.56063			0.0567970
$756$		0			0
$757$	2.59080	−	4.48739i	0.0941642	−	0.163097i	−0.815095	−	0.579327i	$-0.803315\pi$
$757$	2.59080	−	4.48739i	0.0941642	−	0.163097i	0.909259	+	0.416230i	$0.136649\pi$
$758$		0			0
$759$		−	3.21736i		−	0.116783i
$760$		0			0
$761$	0.00743288	−	0.00429137i	0.000269442	−	0.000155562i	−0.499865	−	0.866103i	$-0.666617\pi$
$761$	0.00743288	−	0.00429137i	0.000269442	−	0.000155562i	0.500135	+	0.865948i	$0.333284\pi$
$762$		0			0
$763$	19.8915	+	34.4531i	0.720121	+	1.24729i
$764$		0			0
$765$	2.85521	+	1.64846i	0.103230	+	0.0596001i
$766$		0			0
$767$	−11.3335	+	19.8813i	−0.409227	+	0.717871i
$768$		0			0
$769$	10.2610	+	5.92417i	0.370020	+	0.213631i	0.673467	−	0.739217i	$-0.264804\pi$
$769$	10.2610	+	5.92417i	0.370020	+	0.213631i	−0.303447	+	0.952848i	$0.598138\pi$
$770$		0			0
$771$	17.4933	+	30.2992i	0.630005	+	1.09120i
$772$		0			0
$773$	−18.2262	+	10.5229i	−0.655550	+	0.378482i	−0.790579	−	0.612360i	$-0.790220\pi$
$773$	−18.2262	+	10.5229i	−0.655550	+	0.378482i	0.135030	+	0.990842i	$0.456887\pi$
$774$		0			0
$775$			7.86488i			0.282515i
$776$		0			0
$777$	33.0422	−	57.2308i	1.18538	−	2.05314i
$778$		0			0
$779$	41.5352			1.48815
$780$		0			0
$781$	−21.6754			−0.775605
$782$		0			0
$783$	−5.57172	+	9.65050i	−0.199117	+	0.344881i
$784$		0			0
$785$			9.15332i			0.326696i
$786$		0			0
$787$	−4.34029	+	2.50587i	−0.154715	+	0.0893246i	−0.575359	−	0.817901i	$-0.695138\pi$
$787$	−4.34029	+	2.50587i	−0.154715	+	0.0893246i	0.420644	+	0.907226i	$0.361804\pi$
$788$		0			0
$789$	16.1452	+	27.9643i	0.574784	+	0.995556i
$790$		0			0
$791$	21.6701	+	12.5112i	0.770499

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)

\(f(q)\)	\(=\)	\(q+(-0.800098 + 1.38581i) q^{3} -1.00000i q^{5} +(3.75184 - 2.16612i) q^{7} +(0.219687 + 0.380509i) q^{9} +O(q^{10})\)	\(q+(-0.800098 + 1.38581i) q^{3} -1.00000i q^{5} +(3.75184 - 2.16612i) q^{7} +(0.219687 + 0.380509i) q^{9} +(1.50000 + 0.866025i) q^{11} +(-3.11256 + 1.81988i) q^{13} +(1.38581 + 0.800098i) q^{15} +(3.75184 + 6.49837i) q^{17} +(4.65213 - 2.68591i) q^{19} +6.93244i q^{21} +(-0.580411 + 1.00530i) q^{23} -1.00000 q^{25} -5.50367 q^{27} +(1.01236 - 1.75347i) q^{29} -7.86488i q^{31} +(-2.40029 + 1.38581i) q^{33} +(-2.16612 - 3.75184i) q^{35} +(-8.25551 - 4.76632i) q^{37} +(-0.0316594 - 5.76950i) q^{39} +(6.69615 + 3.86603i) q^{41} +(2.09277 + 3.62479i) q^{43} +(0.380509 - 0.219687i) q^{45} -3.46410i q^{47} +(5.88418 - 10.1917i) q^{49} -12.0073 q^{51} -12.6471 q^{53} +(0.866025 - 1.50000i) q^{55} +8.59596i q^{57} +(5.49674 - 3.17354i) q^{59} +(-1.85154 - 3.20696i) q^{61} +(1.64846 + 0.951738i) q^{63} +(1.81988 + 3.11256i) q^{65} +(-4.55568 - 2.63022i) q^{67} +(-0.928771 - 1.60868i) q^{69} +(-10.8377 + 6.25714i) q^{71} +5.23898i q^{73} +(0.800098 - 1.38581i) q^{75} +7.50367 q^{77} -8.16719 q^{79} +(3.74441 - 6.48552i) q^{81} +0.456760i q^{83} +(6.49837 - 3.75184i) q^{85} +(1.61998 + 2.80589i) q^{87} +(-11.4967 - 6.63765i) q^{89} +(-7.73572 + 13.5701i) q^{91} +(10.8992 + 6.29268i) q^{93} +(-2.68591 - 4.65213i) q^{95} +(-2.43371 + 1.40511i) q^{97} +0.761018i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{3} + 6 q^{7} - 4 q^{9}+O(q^{10}) \) 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

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more