LMFDB - Embedded newform 260.2.x.a.121.1

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			121.1
Root			$1.20036 - 0.747754i$ of defining polynomial
Character	$\chi$	$=$	260.121
Dual form			260.2.x.a.101.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.41342 + 2.44811i) q^{3} +1.00000i q^{5} +(-1.81414 + 1.04739i) q^{7} +(-2.49551 - 4.32235i) q^{9} +O(q^{10})$	\(q+(-1.41342 + 2.44811i) q^{3} +1.00000i q^{5} +(-1.81414 + 1.04739i) q^{7} +(-2.49551 - 4.32235i) q^{9} +(1.50000 + 0.866025i) q^{11} +(-3.59030 - 0.331331i) q^{13} +(-2.44811 - 1.41342i) q^{15} +(-1.81414 - 3.14218i) q^{17} +(0.926118 - 0.534695i) q^{19} -5.92163i q^{21} +(-3.90893 + 6.77046i) q^{23} -1.00000 q^{25} +5.62828 q^{27} +(0.263457 - 0.456321i) q^{29} +5.84325i q^{31} +(-4.24026 + 2.44811i) q^{33} +(-1.04739 - 1.81414i) q^{35} +(8.44242 + 4.87423i) q^{37} +(5.88573 - 8.32114i) q^{39} +(-3.69615 - 2.13397i) q^{41} +(4.67238 + 8.09281i) q^{43} +(4.32235 - 2.49551i) q^{45} -3.46410i q^{47} +(-1.30593 + 2.26194i) q^{49} +10.2566 q^{51} +12.5939 q^{53} +(-0.866025 + 1.50000i) q^{55} +3.02299i q^{57} +(-1.21564 + 0.701848i) q^{59} +(5.55440 + 9.62050i) q^{61} +(9.05440 + 5.22756i) q^{63} +(0.331331 - 3.59030i) q^{65} +(-9.38201 - 5.41671i) q^{67} +(-11.0499 - 19.1390i) q^{69} +(-12.2709 + 7.08460i) q^{71} +2.64469i q^{73} +(1.41342 - 2.44811i) q^{75} -3.62828 q^{77} +13.5729 q^{79} +(-0.468594 + 0.811629i) q^{81} -15.7925i q^{83} +(3.14218 - 1.81414i) q^{85} +(0.744750 + 1.28994i) q^{87} +(-4.78436 - 2.76225i) q^{89} +(6.86033 - 3.15937i) q^{91} +(-14.3049 - 8.25896i) q^{93} +(0.534695 + 0.926118i) q^{95} +(-13.1589 + 7.59730i) q^{97} -8.64469i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−1.41342	+	2.44811i	−0.816038	+	1.41342i	0.0925423	+	0.995709i	$0.470501\pi$
$3$	−1.41342	+	2.44811i	−0.816038	+	1.41342i	−0.908580	+	0.417710i	$0.862833\pi$
$4$		0			0
$5$			1.00000i			0.447214i
$5$			1.00000i			0.447214i
$6$		0			0
$7$	−1.81414	+	1.04739i	−0.685680	+	0.395878i	−0.801992	−	0.597335i	$-0.796226\pi$
$7$	−1.81414	+	1.04739i	−0.685680	+	0.395878i	0.116312	+	0.993213i	$0.462893\pi$
$8$		0			0
$9$	−2.49551	−	4.32235i	−0.831836	−	1.44078i
$10$		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
$13$	−3.59030	−	0.331331i	−0.995769	−	0.0918946i
$13$	−3.59030	−	0.331331i	−0.995769	−	0.0918946i
$14$		0			0
$15$	−2.44811	−	1.41342i	−0.632100	−	0.364943i
$16$		0			0
$17$	−1.81414	−	3.14218i	−0.439993	−	0.762091i	0.557695	−	0.830046i	$-0.311686\pi$
$17$	−1.81414	−	3.14218i	−0.439993	−	0.762091i	−0.997688	+	0.0679550i	$0.978353\pi$
$18$		0			0
$19$	0.926118	−	0.534695i	0.212466	−	0.122667i	−0.389991	−	0.920819i	$-0.627522\pi$
$19$	0.926118	−	0.534695i	0.212466	−	0.122667i	0.602457	+	0.798151i	$0.294189\pi$
$20$		0			0
$21$		−	5.92163i		−	1.29220i
$22$		0			0
$23$	−3.90893	+	6.77046i	−0.815068	+	1.41174i	0.0942118	+	0.995552i	$0.469967\pi$
$23$	−3.90893	+	6.77046i	−0.815068	+	1.41174i	−0.909279	+	0.416186i	$0.863366\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$	5.62828			1.08316
$28$		0			0
$29$	0.263457	−	0.456321i	0.0489227	−	0.0847366i	−0.840527	−	0.541770i	$-0.817755\pi$
$29$	0.263457	−	0.456321i	0.0489227	−	0.0847366i	0.889450	+	0.457033i	$0.151088\pi$
$30$		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
$33$	−4.24026	+	2.44811i	−0.738134	+	0.426162i
$34$		0			0
$35$	−1.04739	−	1.81414i	−0.177042	−	0.306646i
$36$		0			0
$37$	8.44242	+	4.87423i	1.38792	+	0.801319i	0.993081	−	0.117429i	$-0.0374654\pi$
$37$	8.44242	+	4.87423i	1.38792	+	0.801319i	0.394844	+	0.918748i	$0.370799\pi$
$38$		0			0
$39$	5.88573	−	8.32114i	0.942471	−	1.33245i
$40$		0			0
$41$	−3.69615	−	2.13397i	−0.577242	−	0.333271i	0.182795	−	0.983151i	$-0.441486\pi$
$41$	−3.69615	−	2.13397i	−0.577242	−	0.333271i	−0.760037	+	0.649880i	$0.774819\pi$
$42$		0			0
$43$	4.67238	+	8.09281i	0.712532	+	1.23414i	0.963904	+	0.266251i	$0.0857849\pi$
$43$	4.67238	+	8.09281i	0.712532	+	1.23414i	−0.251372	+	0.967891i	$0.580882\pi$
$44$		0			0
$45$	4.32235	−	2.49551i	0.644337	−	0.372008i
$46$		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
$49$	−1.30593	+	2.26194i	−0.186562	+	0.323134i
$50$		0			0
$51$	10.2566			1.43621
$52$		0			0
$53$	12.5939			1.72990			0.864952	−	0.501854i	$-0.167349\pi$
$53$	12.5939			1.72990			0.864952	+	0.501854i	$0.167349\pi$
$54$		0			0
$55$	−0.866025	+	1.50000i	−0.116775	+	0.202260i
$56$		0			0
$57$			3.02299i			0.400405i
$58$		0			0
$59$	−1.21564	+	0.701848i	−0.158262	+	0.0913729i	−0.577040	−	0.816716i	$-0.695792\pi$
$59$	−1.21564	+	0.701848i	−0.158262	+	0.0913729i	0.418777	+	0.908089i	$0.362459\pi$
$60$		0			0
$61$	5.55440	+	9.62050i	0.711168	+	1.23178i	0.964419	+	0.264378i	$0.0851667\pi$
$61$	5.55440	+	9.62050i	0.711168	+	1.23178i	−0.253251	+	0.967400i	$0.581500\pi$
$62$		0			0
$63$	9.05440	+	5.22756i	1.14075	+	0.658610i
$64$		0			0
$65$	0.331331	−	3.59030i	0.0410965	−	0.445321i
$66$		0			0
$67$	−9.38201	−	5.41671i	−1.14620	−	0.661756i	−0.198238	−	0.980154i	$-0.563522\pi$
$67$	−9.38201	−	5.41671i	−1.14620	−	0.661756i	−0.947957	+	0.318398i	$0.896855\pi$
$68$		0			0
$69$	−11.0499	−	19.1390i	−1.33025	−	2.30406i
$70$		0			0
$71$	−12.2709	+	7.08460i	−1.45629	+	0.840787i	−0.998826	−	0.0484428i	$-0.984574\pi$
$71$	−12.2709	+	7.08460i	−1.45629	+	0.840787i	−0.457460	+	0.889230i	$0.651241\pi$
$72$		0			0
$73$			2.64469i			0.309538i	0.987951	+	0.154769i	$0.0494633\pi$
$73$			2.64469i			0.309538i	−0.987951	+	0.154769i	$0.950537\pi$
$74$		0			0
$75$	1.41342	−	2.44811i	0.163208	−	0.282684i
$76$		0			0
$77$	−3.62828			−0.413481
$78$		0			0
$79$	13.5729			1.52707			0.763535	−	0.645766i	$-0.223462\pi$
$79$	13.5729			1.52707			0.763535	+	0.645766i	$0.223462\pi$
$80$		0			0
$81$	−0.468594	+	0.811629i	−0.0520660	+	0.0901809i
$82$		0			0
$83$		−	15.7925i		−	1.73345i	−0.498789	−	0.866724i	$-0.666222\pi$
$83$		−	15.7925i		−	1.73345i	0.498789	−	0.866724i	$-0.333778\pi$
$84$		0			0
$85$	3.14218	−	1.81414i	0.340817	−	0.196771i
$86$		0			0
$87$	0.744750	+	1.28994i	0.0798456	+	0.138297i
$88$		0			0
$89$	−4.78436	−	2.76225i	−0.507141	−	0.292798i	0.224516	−	0.974470i	$-0.427920\pi$
$89$	−4.78436	−	2.76225i	−0.507141	−	0.292798i	−0.731658	+	0.681672i	$0.761253\pi$
$90$		0			0
$91$	6.86033	−	3.15937i	0.719158	−	0.331192i
$92$		0			0
$93$	−14.3049	−	8.25896i	−1.48335	−	0.856415i
$94$		0			0
$95$	0.534695	+	0.926118i	0.0548585	+	0.0950177i
$96$		0			0
$97$	−13.1589	+	7.59730i	−1.33608	+	0.771389i	−0.986224	−	0.165413i	$-0.947104\pi$
$97$	−13.1589	+	7.59730i	−1.33608	+	0.771389i	−0.349860	+	0.936802i	$0.613771\pi$
$98$		0			0
$99$		−	8.64469i		−	0.868824i
$100$		0			0
$101$	−1.83133	+	3.17196i	−0.182224	+	0.315622i	−0.942638	−	0.333818i	$-0.891663\pi$
$101$	−1.83133	+	3.17196i	−0.182224	+	0.315622i	0.760413	+	0.649439i	$0.224996\pi$
$102$		0			0
$103$	13.7804			1.35783			0.678914	−	0.734218i	$-0.262451\pi$
$103$	13.7804			1.35783			0.678914	+	0.734218i	$0.262451\pi$
$104$		0			0
$105$	5.92163			0.577892
$106$		0			0
$107$	−1.61856	+	2.80342i	−0.156472	+	0.271017i	−0.933594	−	0.358333i	$-0.883345\pi$
$107$	−1.61856	+	2.80342i	−0.156472	+	0.271017i	0.777122	+	0.629350i	$0.216679\pi$
$108$		0			0
$109$			9.12979i			0.874476i	0.899346	+	0.437238i	$0.144043\pi$
$109$			9.12979i			0.874476i	−0.899346	+	0.437238i	$0.855957\pi$
$110$		0			0
$111$	−23.8654	+	13.7787i	−2.26520	+	1.30781i
$112$		0			0
$113$	5.47680	+	9.48610i	0.515214	+	0.892377i	0.999844	+	0.0176577i	$0.00562092\pi$
$113$	5.47680	+	9.48610i	0.515214	+	0.892377i	−0.484630	+	0.874719i	$0.661046\pi$
$114$		0			0
$115$	−6.77046	−	3.90893i	−0.631349	−	0.364509i
$116$		0			0
$117$	7.52748	+	16.3453i	0.695916	+	1.51113i
$118$		0			0
$119$	6.58220	+	3.80024i	0.603390	+	0.348367i
$120$		0			0
$121$	−4.00000	−	6.92820i	−0.363636	−	0.629837i
$122$		0			0
$123$	10.4484	−	6.03240i	0.942103	−	0.543923i
$124$		0			0
$125$		−	1.00000i		−	0.0894427i
$126$		0			0
$127$	0.453810	−	0.786022i	0.0402691	−	0.0697482i	−0.845188	−	0.534469i	$-0.820512\pi$
$127$	0.453810	−	0.786022i	0.0402691	−	0.0697482i	0.885458	+	0.464720i	$0.153845\pi$
$128$		0			0
$129$	−26.4161			−2.32581
$130$		0			0
$131$	−13.1626			−1.15002			−0.575012	−	0.818145i	$-0.695003\pi$
$131$	−13.1626			−1.15002			−0.575012	+	0.818145i	$0.695003\pi$
$132$		0			0
$133$	−1.12007	+	1.94002i	−0.0971225	+	0.168221i
$134$		0			0
$135$			5.62828i			0.484405i
$136$		0			0
$137$	10.4484	−	6.03240i	0.892669	−	0.515383i	0.0178546	−	0.999841i	$-0.494316\pi$
$137$	10.4484	−	6.03240i	0.892669	−	0.515383i	0.874815	+	0.484458i	$0.160983\pi$
$138$		0			0
$139$	2.80593	+	4.86002i	0.237996	+	0.412221i	0.960139	−	0.279522i	$-0.0901761\pi$
$139$	2.80593	+	4.86002i	0.237996	+	0.412221i	−0.722143	+	0.691744i	$0.756843\pi$
$140$		0			0
$141$	8.48052	+	4.89623i	0.714188	+	0.412337i
$142$		0			0
$143$	−5.09850	−	3.60628i	−0.426358	−	0.301573i
$144$		0			0
$145$	0.456321	+	0.263457i	0.0378954	+	0.0218789i
$146$		0			0
$147$	−3.69166	−	6.39414i	−0.304483	−	0.527380i
$148$		0			0
$149$	18.1767	−	10.4943i	1.48909	−	0.859727i	0.489168	−	0.872189i	$-0.337300\pi$
$149$	18.1767	−	10.4943i	1.48909	−	0.859727i	0.999922	+	0.0124625i	$0.00396705\pi$
$150$		0			0
$151$		−	6.99102i		−	0.568921i	−0.958688	−	0.284460i	$-0.908186\pi$
$151$		−	6.99102i		−	0.568921i	0.958688	−	0.284460i	$-0.0918144\pi$
$152$		0			0
$153$	−9.05440	+	15.6827i	−0.732005	+	1.26787i
$154$		0			0
$155$	−5.84325			−0.469341
$156$		0			0
$157$	3.74761			0.299092			0.149546	−	0.988755i	$-0.452219\pi$
$157$	3.74761			0.299092			0.149546	+	0.988755i	$0.452219\pi$
$158$		0			0
$159$	−17.8005	+	30.8313i	−1.41167	+	2.44508i
$160$		0			0
$161$		−	16.3767i		−	1.29067i
$162$		0			0
$163$	−5.52377	+	3.18915i	−0.432655	+	0.249793i	−0.700477	−	0.713675i	$-0.747029\pi$
$163$	−5.52377	+	3.18915i	−0.432655	+	0.249793i	0.267822	+	0.963468i	$0.413696\pi$
$164$		0			0
$165$	−2.44811	−	4.24026i	−0.190585	−	0.330104i
$166$		0			0
$167$	−13.4484	−	7.76445i	−1.04067	−	0.600831i	−0.120647	−	0.992695i	$-0.538497\pi$
$167$	−13.4484	−	7.76445i	−1.04067	−	0.600831i	−0.920023	+	0.391864i	$0.871830\pi$
$168$		0			0
$169$	12.7804	+	2.37915i	0.983111	+	0.183012i
$170$		0			0
$171$	−4.62227	−	2.66867i	−0.353474	−	0.204078i
$172$		0			0
$173$	−2.38802	−	4.13617i	−0.181558	−	0.314467i	0.760853	−	0.648924i	$-0.224781\pi$
$173$	−2.38802	−	4.13617i	−0.181558	−	0.314467i	−0.942411	+	0.334456i	$0.891447\pi$
$174$		0			0
$175$	1.81414	−	1.04739i	0.137136	−	0.0791755i
$176$		0			0
$177$		−	3.96802i		−	0.298255i
$178$		0			0
$179$	7.85134	−	13.5989i	0.586837	−	1.01643i	−0.407807	−	0.913068i	$-0.633706\pi$
$179$	7.85134	−	13.5989i	0.586837	−	1.01643i	0.994644	−	0.103363i	$-0.0329604\pi$
$180$		0			0
$181$	10.8851			0.809080			0.404540	−	0.914520i	$-0.367432\pi$
$181$	10.8851			0.809080			0.404540	+	0.914520i	$0.367432\pi$
$182$		0			0
$183$	−31.4028			−2.32136
$184$		0			0
$185$	−4.87423	+	8.44242i	−0.358361	+	0.620699i
$186$		0			0
$187$		−	6.28436i		−	0.459558i
$188$		0			0
$189$	−10.2105	+	5.89502i	−0.742703	+	0.428800i
$190$		0			0
$191$	2.97909	+	5.15994i	0.215560	+	0.373360i	0.953446	−	0.301565i	$-0.0975091\pi$
$191$	2.97909	+	5.15994i	0.215560	+	0.373360i	−0.737886	+	0.674925i	$0.764176\pi$
$192$		0			0
$193$	11.0587	+	6.38473i	0.796021	+	0.459583i	0.842078	−	0.539356i	$-0.181332\pi$
$193$	11.0587	+	6.38473i	0.796021	+	0.459583i	−0.0460568	+	0.998939i	$0.514666\pi$
$194$		0			0
$195$	8.32114	+	5.88573i	0.595889	+	0.421486i
$196$		0			0
$197$	−13.9499	−	8.05397i	−0.993888	−	0.573822i	−0.0874540	−	0.996169i	$-0.527873\pi$
$197$	−13.9499	−	8.05397i	−0.993888	−	0.573822i	−0.906434	+	0.422347i	$0.861206\pi$
$198$		0			0
$199$	−4.26403	−	7.38551i	−0.302269	−	0.523545i	0.674381	−	0.738384i	$-0.264411\pi$
$199$	−4.26403	−	7.38551i	−0.302269	−	0.523545i	−0.976650	+	0.214839i	$0.931077\pi$
$200$		0			0
$201$	26.5214	−	15.3122i	1.87068	−	1.08004i
$202$		0			0
$203$			1.10377i			0.0774696i
$204$		0			0
$205$	2.13397	−	3.69615i	0.149043	−	0.258150i
$206$		0			0
$207$	39.0190			2.71201
$208$		0			0
$209$	1.85224			0.128122
$210$		0			0
$211$	2.09030	−	3.62050i	0.143902	−	0.249245i	−0.785061	−	0.619419i	$-0.787368\pi$
$211$	2.09030	−	3.62050i	0.143902	−	0.249245i	0.928963	+	0.370173i	$0.120702\pi$
$212$		0			0
$213$		−	40.0540i		−	2.74446i
$214$		0			0
$215$	−8.09281	+	4.67238i	−0.551925	+	0.318654i
$216$		0			0
$217$	−6.12019	−	10.6005i	−0.415465	−	0.719607i
$218$		0			0
$219$	−6.47451	−	3.73806i	−0.437507	−	0.252595i
$220$		0			0
$221$	5.47219	+	11.8824i	0.368100	+	0.799299i
$222$		0			0
$223$	−9.09249	−	5.24955i	−0.608878	−	0.351536i	0.163648	−	0.986519i	$-0.447674\pi$
$223$	−9.09249	−	5.24955i	−0.608878	−	0.351536i	−0.772526	+	0.634983i	$0.781007\pi$
$224$		0			0
$225$	2.49551	+	4.32235i	0.166367	+	0.288156i
$226$		0			0
$227$	13.4977	−	7.79288i	0.895872	−	0.517232i	0.0200131	−	0.999800i	$-0.493629\pi$
$227$	13.4977	−	7.79288i	0.895872	−	0.517232i	0.875858	+	0.482568i	$0.160296\pi$
$228$		0			0
$229$			19.2714i			1.27349i	0.771074	+	0.636745i	$0.219720\pi$
$229$			19.2714i			1.27349i	−0.771074	+	0.636745i	$0.780280\pi$
$230$		0			0
$231$	5.12828	−	8.88244i	0.337416	−	0.584422i
$232$		0			0
$233$	2.48794			0.162991			0.0814953	−	0.996674i	$-0.474030\pi$
$233$	2.48794			0.162991			0.0814953	+	0.996674i	$0.474030\pi$
$234$		0			0
$235$	3.46410			0.225973
$236$		0			0
$237$	−19.1842	+	33.2280i	−1.24615	+	2.15839i
$238$		0			0
$239$			16.7775i			1.08525i	0.839976	+	0.542624i	$0.182569\pi$
$239$			16.7775i			1.08525i	−0.839976	+	0.542624i	$0.817431\pi$
$240$		0			0
$241$	25.1835	−	14.5397i	1.62221	−	0.936585i	0.635887	−	0.771782i	$-0.280634\pi$
$241$	25.1835	−	14.5397i	1.62221	−	0.936585i	0.986326	−	0.164803i	$-0.0526990\pi$
$242$		0			0
$243$	7.11778	+	12.3284i	0.456606	+	0.790864i
$244$		0			0
$245$	−2.26194	−	1.30593i	−0.144510	−	0.0834330i
$246$		0			0
$247$	−3.50220	+	1.61286i	−0.222840	+	0.102624i
$248$		0			0
$249$	38.6617	+	22.3214i	2.45009	+	1.41456i
$250$		0			0
$251$	9.56040	+	16.5591i	0.603447	+	1.04520i	0.992295	+	0.123899i	$0.0395400\pi$
$251$	9.56040	+	16.5591i	0.603447	+	1.04520i	−0.388847	+	0.921302i	$0.627127\pi$
$252$		0			0
$253$	−11.7268	+	6.77046i	−0.737256	+	0.425655i
$254$		0			0
$255$			10.2566i			0.642291i
$256$		0			0
$257$	6.85453	−	11.8724i	0.427574	−	0.740580i	−0.569083	−	0.822280i	$-0.692702\pi$
$257$	6.85453	−	11.8724i	0.427574	−	0.740580i	0.996657	+	0.0817004i	$0.0260351\pi$
$258$		0			0
$259$	−20.4210			−1.26890
$260$		0			0
$261$	−2.62983			−0.162783
$262$		0			0
$263$	5.95675	−	10.3174i	0.367309	−	0.636197i	−0.621835	−	0.783148i	$-0.713613\pi$
$263$	5.95675	−	10.3174i	0.367309	−	0.636197i	0.989144	+	0.146951i	$0.0469459\pi$
$264$		0			0
$265$			12.5939i			0.773637i
$266$		0			0
$267$	13.5246	−	7.80844i	0.827693	−	0.477869i
$268$		0			0
$269$	10.4656	+	18.1270i	0.638100	+	1.10522i	0.985849	+	0.167634i	$0.0536127\pi$
$269$	10.4656	+	18.1270i	0.638100	+	1.10522i	−0.347749	+	0.937588i	$0.613054\pi$
$270$		0			0
$271$	1.69014	+	0.975805i	0.102669	+	0.0592760i	0.550455	−	0.834865i	$-0.314454\pi$
$271$	1.69014	+	0.975805i	0.102669	+	0.0592760i	−0.447786	+	0.894141i	$0.647787\pi$
$272$		0			0
$273$	−1.96202	+	21.2604i	−0.118747	+	1.28674i
$274$		0			0
$275$	−1.50000	−	0.866025i	−0.0904534	−	0.0522233i
$276$		0			0
$277$	6.24026	+	10.8084i	0.374941	+	0.649416i	0.990318	−	0.138816i	$-0.0443297\pi$
$277$	6.24026	+	10.8084i	0.374941	+	0.649416i	−0.615377	+	0.788233i	$0.710996\pi$
$278$		0			0
$279$	25.2566	−	14.5819i	1.51207	−	0.872994i
$280$		0			0
$281$		−	2.29553i		−	0.136940i	−0.997653	−	0.0684698i	$-0.978188\pi$
$281$		−	2.29553i		−	0.136940i	0.997653	−	0.0684698i	$-0.0218117\pi$
$282$		0			0
$283$	7.46484	−	12.9295i	0.443739	−	0.768578i	−0.554225	−	0.832367i	$-0.686985\pi$
$283$	7.46484	−	12.9295i	0.443739	−	0.768578i	0.997963	+	0.0637892i	$0.0203185\pi$
$284$		0			0
$285$	−3.02299			−0.179067
$286$		0			0
$287$	8.94045			0.527738
$288$		0			0
$289$	1.91780	−	3.32172i	0.112812	−	0.195395i
$290$		0			0
$291$		−	42.9527i		−	2.51793i
$292$		0			0
$293$	1.24026	−	0.716063i	0.0724566	−	0.0418329i	−0.463334	−	0.886184i	$-0.653347\pi$
$293$	1.24026	−	0.716063i	0.0724566	−	0.0418329i	0.535791	+	0.844351i	$0.320014\pi$
$294$		0			0
$295$	−0.701848	−	1.21564i	−0.0408632	−	0.0707771i
$296$		0			0
$297$	8.44242	+	4.87423i	0.489879	+	0.282832i
$298$		0			0
$299$	16.2775	−	23.0128i	0.941350	−	1.33086i
$300$		0			0
$301$	−16.9527	−	9.78765i	−0.977138	−	0.564151i
$302$		0			0
$303$	−5.17688	−	8.96661i	−0.297404	−	0.515118i
$304$		0			0
$305$	−9.62050	+	5.55440i	−0.550868	+	0.318044i
$306$		0			0
$307$			17.3833i			0.992118i	0.868289	+	0.496059i	$0.165220\pi$
$307$			17.3833i			0.992118i	−0.868289	+	0.496059i	$0.834780\pi$
$308$		0			0
$309$	−19.4775	+	33.7361i	−1.10804	+	1.91918i
$310$		0			0
$311$	−20.2164			−1.14637			−0.573185	−	0.819426i	$-0.694292\pi$
$311$	−20.2164			−1.14637			−0.573185	+	0.819426i	$0.694292\pi$
$312$		0			0
$313$	−4.86425			−0.274944			−0.137472	−	0.990506i	$-0.543898\pi$
$313$	−4.86425			−0.274944			−0.137472	+	0.990506i	$0.543898\pi$
$314$		0			0
$315$	−5.22756	+	9.05440i	−0.294540	+	0.510157i
$316$		0			0
$317$		−	23.6177i		−	1.32650i	−0.748396	−	0.663252i	$-0.769176\pi$
$317$		−	23.6177i		−	1.32650i	0.748396	−	0.663252i	$-0.230824\pi$
$318$		0			0
$319$	0.790371	−	0.456321i	0.0442523	−	0.0255491i
$320$		0			0
$321$	−4.57540	−	7.92482i	−0.255374	−	0.442320i
$322$		0			0
$323$	−3.36022	−	1.94002i	−0.186967	−	0.107946i
$324$		0			0
$325$	3.59030	+	0.331331i	0.199154	+	0.0183789i
$326$		0			0
$327$	−22.3508	−	12.9042i	−1.23600	−	0.713605i
$328$		0			0
$329$	3.62828	+	6.28436i	0.200033	+	0.346468i
$330$		0			0
$331$	−12.8863	+	7.43991i	−0.708295	+	0.408934i	−0.810429	−	0.585836i	$-0.800766\pi$
$331$	−12.8863	+	7.43991i	−0.708295	+	0.408934i	0.102134	+	0.994771i	$0.467433\pi$
$332$		0			0
$333$		−	48.6547i		−	2.66626i
$334$		0			0
$335$	5.41671	−	9.38201i	0.295946	−	0.512594i
$336$		0			0
$337$	−29.1906			−1.59012			−0.795058	−	0.606534i	$-0.792559\pi$
$337$	−29.1906			−1.59012			−0.795058	+	0.606534i	$0.792559\pi$
$338$		0			0
$339$	−30.9641			−1.68174
$340$		0			0
$341$	−5.06040	+	8.76488i	−0.274036	+	0.474645i
$342$		0			0
$343$		−	20.1348i		−	1.08718i
$344$		0			0
$345$	19.1390	−	11.0499i	1.03041	−	0.594907i
$346$		0			0
$347$	12.1125	+	20.9795i	0.650234	+	1.12624i	0.983066	+	0.183252i	$0.0586623\pi$
$347$	12.1125	+	20.9795i	0.650234	+	1.12624i	−0.332833	+	0.942986i	$0.608004\pi$
$348$		0			0
$349$	−11.1557	−	6.44076i	−0.597152	−	0.344766i	0.170768	−	0.985311i	$-0.445375\pi$
$349$	−11.1557	−	6.44076i	−0.597152	−	0.344766i	−0.767920	+	0.640545i	$0.778708\pi$
$350$		0			0
$351$	−20.2072	−	1.86482i	−1.07858	−	0.0995368i
$352$		0			0
$353$	14.0441	+	8.10837i	0.747492	+	0.431565i	0.824787	−	0.565443i	$-0.191295\pi$
$353$	14.0441	+	8.10837i	0.747492	+	0.431565i	−0.0772948	+	0.997008i	$0.524628\pi$
$354$		0			0
$355$	−7.08460	−	12.2709i	−0.376011	−	0.651271i
$356$		0			0
$357$	−18.6068	+	10.7427i	−0.984777	+	0.568562i
$358$		0			0
$359$			9.19261i			0.485167i	0.970131	+	0.242584i	$0.0779949\pi$
$359$			9.19261i			0.485167i	−0.970131	+	0.242584i	$0.922005\pi$
$360$		0			0
$361$	−8.92820	+	15.4641i	−0.469905	+	0.813900i
$362$		0			0
$363$	22.6147			1.18696
$364$		0			0
$365$	−2.64469			−0.138430
$366$		0			0
$367$	2.30066	−	3.98486i	0.120094	−	0.208008i	−0.799711	−	0.600385i	$-0.795014\pi$
$367$	2.30066	−	3.98486i	0.120094	−	0.208008i	0.919804	+	0.392377i	$0.128347\pi$
$368$		0			0
$369$			21.3014i			1.10891i
$370$		0			0
$371$	−22.8471	+	13.1908i	−1.18616	+	0.684831i
$372$		0			0
$373$	10.2463	+	17.7471i	0.530532	+	0.918908i	0.999365	+	0.0356212i	$0.0113410\pi$
$373$	10.2463	+	17.7471i	0.530532	+	0.918908i	−0.468834	+	0.883286i	$0.655326\pi$
$374$		0			0
$375$	2.44811	+	1.41342i	0.126420	+	0.0729887i
$376$		0			0
$377$	−1.09708	+	1.55103i	−0.0565025	+	0.0798823i
$378$		0			0
$379$	6.95307	+	4.01436i	0.357155	+	0.206204i	0.667832	−	0.744312i	$-0.267222\pi$
$379$	6.95307	+	4.01436i	0.357155	+	0.206204i	−0.310677	+	0.950516i	$0.600556\pi$
$380$		0			0
$381$	1.28285	+	2.22196i	0.0657223	+	0.113834i
$382$		0			0
$383$	−26.1041	+	15.0712i	−1.33386	+	0.770104i	−0.985889	−	0.167401i	$-0.946462\pi$
$383$	−26.1041	+	15.0712i	−1.33386	+	0.770104i	−0.347971	+	0.937505i	$0.613129\pi$
$384$		0			0
$385$		−	3.62828i		−	0.184914i
$386$		0			0
$387$	23.3199	−	40.3913i	1.18542	−	2.05321i
$388$		0			0
$389$	−35.8264			−1.81647			−0.908236	−	0.418459i	$-0.862570\pi$
$389$	−35.8264			−1.81647			−0.908236	+	0.418459i	$0.862570\pi$
$390$		0			0
$391$	28.3654			1.43450
$392$		0			0
$393$	18.6043	−	32.2236i	0.938463	−	1.62547i
$394$		0			0
$395$			13.5729i			0.682926i
$396$		0			0
$397$	−13.3221	+	7.69152i	−0.668617	+	0.386026i	−0.795552	−	0.605885i	$-0.792819\pi$
$397$	−13.3221	+	7.69152i	−0.668617	+	0.386026i	0.126935	+	0.991911i	$0.459486\pi$
$398$		0			0
$399$	−3.16626	−	5.48413i	−0.158511	−	0.274550i
$400$		0			0
$401$	−8.46704	−	4.88845i	−0.422824	−	0.244117i	0.273461	−	0.961883i	$-0.411832\pi$
$401$	−8.46704	−	4.88845i	−0.422824	−	0.244117i	−0.696285	+	0.717766i	$0.745165\pi$
$402$		0			0
$403$	1.93605	−	20.9790i	0.0964415	−	1.04504i
$404$		0			0
$405$	−0.811629	−	0.468594i	−0.0403301	−	0.0232846i
$406$		0			0
$407$	8.44242	+	14.6227i	0.418475	+	0.724820i
$408$		0			0
$409$	0.871721	−	0.503289i	0.0431039	−	0.0248860i	−0.478293	−	0.878200i	$-0.658744\pi$
$409$	0.871721	−	0.503289i	0.0431039	−	0.0248860i	0.521397	+	0.853314i	$0.325411\pi$
$410$		0			0
$411$			34.1052i			1.68229i
$412$		0			0
$413$	1.47022	−	2.54650i	0.0723450	−	0.125305i
$414$		0			0
$415$	15.7925			0.775221
$416$		0			0
$417$	−15.8638			−0.776855
$418$		0			0
$419$	−4.85134	+	8.40278i	−0.237004	+	0.410502i	−0.959853	−	0.280503i	$-0.909499\pi$
$419$	−4.85134	+	8.40278i	−0.237004	+	0.410502i	0.722849	+	0.691006i	$0.242832\pi$
$420$		0			0
$421$			14.2955i			0.696721i	0.937361	+	0.348361i	$0.113262\pi$
$421$			14.2955i			0.696721i	−0.937361	+	0.348361i	$0.886738\pi$
$422$		0			0
$423$	−14.9730	+	8.64469i	−0.728014	+	0.420319i
$424$		0			0
$425$	1.81414	+	3.14218i	0.0879987	+	0.152418i
$426$		0			0
$427$	−20.1529	−	11.6353i	−0.975267	−	0.563071i
$428$		0			0
$429$	16.0349	−	7.38452i	0.774173	−	0.356528i
$430$		0			0
$431$	10.5197	+	6.07357i	0.506718	+	0.292554i	0.731483	−	0.681859i	$-0.238828\pi$
$431$	10.5197	+	6.07357i	0.506718	+	0.292554i	−0.224766	+	0.974413i	$0.572162\pi$
$432$		0			0
$433$	−0.104510	−	0.181016i	−0.00502242	−	0.00869909i	0.863503	−	0.504343i	$-0.168265\pi$
$433$	−0.104510	−	0.181016i	−0.00502242	−	0.00869909i	−0.868526	+	0.495644i	$0.834932\pi$
$434$		0			0
$435$	−1.28994	+	0.744750i	−0.0618481	+	0.0357080i
$436$		0			0
$437$			8.36033i			0.399929i
$438$		0			0
$439$	−13.8603	+	24.0068i	−0.661517	+	1.14578i	0.318700	+	0.947856i	$0.396754\pi$
$439$	−13.8603	+	24.0068i	−0.661517	+	1.14578i	−0.980217	+	0.197926i	$0.936579\pi$
$440$		0			0
$441$	13.0359			0.620755
$442$		0			0
$443$	−7.98798			−0.379521			−0.189760	−	0.981830i	$-0.560771\pi$
$443$	−7.98798			−0.379521			−0.189760	+	0.981830i	$0.560771\pi$
$444$		0			0
$445$	2.76225	−	4.78436i	0.130943	−	0.226801i
$446$		0			0
$447$			59.3314i			2.80628i
$448$		0			0
$449$	6.17667	−	3.56610i	0.291495	−	0.168295i	−0.347121	−	0.937820i	$-0.612841\pi$
$449$	6.17667	−	3.56610i	0.291495	−	0.168295i	0.638616	+	0.769526i	$0.279507\pi$
$450$		0			0
$451$	−3.69615	−	6.40192i	−0.174045	−	0.301455i
$452$		0			0
$453$	17.1148	+	9.88124i	0.804124	+	0.464261i
$454$		0			0
$455$	3.15937	+	6.86033i	0.148114	+	0.321617i
$456$		0			0
$457$	−27.4364	−	15.8404i	−1.28342	−	0.740984i	−0.305949	−	0.952048i	$-0.598974\pi$
$457$	−27.4364	−	15.8404i	−1.28342	−	0.740984i	−0.977472	+	0.211064i	$0.932307\pi$
$458$		0			0
$459$	−10.2105	−	17.6851i	−0.476584	−	0.825468i
$460$		0			0
$461$	−10.2649	+	5.92643i	−0.478083	+	0.276021i	−0.719617	−	0.694371i	$-0.755683\pi$
$461$	−10.2649	+	5.92643i	−0.478083	+	0.276021i	0.241534	+	0.970392i	$0.422349\pi$
$462$		0			0
$463$		−	12.7655i		−	0.593263i	−0.954992	−	0.296632i	$-0.904137\pi$
$463$		−	12.7655i		−	0.593263i	0.954992	−	0.296632i	$-0.0958633\pi$
$464$		0			0
$465$	8.25896	−	14.3049i	0.383000	−	0.663376i
$466$		0			0
$467$	31.3402			1.45025			0.725125	−	0.688617i	$-0.241782\pi$
$467$	31.3402			1.45025			0.725125	+	0.688617i	$0.241782\pi$
$468$		0			0
$469$	22.6937			1.04790
$470$		0			0
$471$	−5.29695	+	9.17458i	−0.244070	+	0.422742i
$472$		0			0
$473$			16.1856i			0.744215i
$474$		0			0
$475$	−0.926118	+	0.534695i	−0.0424932	+	0.0245335i
$476$		0			0
$477$	−31.4282	−	54.4352i	−1.43900	−	2.49242i
$478$		0			0
$479$	13.9656	+	8.06303i	0.638103	+	0.368409i	0.783884	−	0.620908i	$-0.213236\pi$
$479$	13.9656	+	8.06303i	0.638103	+	0.368409i	−0.145780	+	0.989317i	$0.546569\pi$
$480$		0			0
$481$	−28.6958	−	20.2972i	−1.30842	−	0.925471i
$482$		0			0
$483$	40.0921	+	23.1472i	1.82426	+	1.05323i
$484$		0			0
$485$	−7.59730	−	13.1589i	−0.344976	−	0.597515i
$486$		0			0
$487$	−4.84109	+	2.79501i	−0.219371	+	0.126654i	−0.605659	−	0.795724i	$-0.707090\pi$
$487$	−4.84109	+	2.79501i	−0.219371	+	0.126654i	0.386288	+	0.922378i	$0.373757\pi$
$488$		0			0
$489$		−	18.0304i		−	0.815364i
$490$		0			0
$491$	−9.14772	+	15.8443i	−0.412831	+	0.715044i	−0.995198	−	0.0978817i	$-0.968793\pi$
$491$	−9.14772	+	15.8443i	−0.412831	+	0.715044i	0.582367	+	0.812926i	$0.302127\pi$
$492$		0			0
$493$	−1.91179			−0.0861027
$494$		0			0
$495$	8.64469			0.388550
$496$		0			0
$497$	14.8407	−	25.7049i	0.665698	−	1.15302i
$498$		0			0
$499$			0.553868i			0.0247945i	0.999923	+	0.0123973i	$0.00394627\pi$
$499$			0.553868i			0.0247945i	−0.999923	+	0.0123973i	$0.996054\pi$
$500$		0			0
$501$	38.0165	−	21.9489i	1.69845	−	0.980602i
$502$		0			0
$503$	−7.34162	−	12.7161i	−0.327347	−	0.566981i	0.654638	−	0.755943i	$-0.272821\pi$
$503$	−7.34162	−	12.7161i	−0.327347	−	0.566981i	−0.981984	+	0.188962i	$0.939488\pi$
$504$		0			0
$505$	−3.17196	−	1.83133i	−0.141150	−	0.0814931i
$506$		0			0
$507$	−23.8886	+	27.9252i	−1.06093	+	1.24020i
$508$		0			0
$509$	5.99545	+	3.46148i	0.265744	+	0.153427i	0.626952	−	0.779058i	$-0.284302\pi$
$509$	5.99545	+	3.46148i	0.265744	+	0.153427i	−0.361208	+	0.932485i	$0.617636\pi$
$510$		0			0
$511$	−2.77003	−	4.79784i	−0.122539	−	0.212244i
$512$		0			0
$513$	5.21245	−	3.00941i	0.230135	−	0.132869i
$514$		0			0
$515$			13.7804i			0.607239i
$516$		0			0
$517$	3.00000	−	5.19615i	0.131940	−	0.228527i
$518$		0			0
$519$	13.5011			0.592632
$520$		0			0
$521$	−18.3551			−0.804152			−0.402076	−	0.915606i	$-0.631711\pi$
$521$	−18.3551			−0.804152			−0.402076	+	0.915606i	$0.631711\pi$
$522$		0			0
$523$	9.13563	−	15.8234i	0.399473	−	0.691908i	−0.594188	−	0.804326i	$-0.702526\pi$
$523$	9.13563	−	15.8234i	0.399473	−	0.691908i	0.993661	+	0.112418i	$0.0358597\pi$
$524$		0			0
$525$			5.92163i			0.258441i
$526$		0			0
$527$	18.3606	−	10.6005i	0.799798	−	0.461764i
$528$		0			0
$529$	−19.0594	−	33.0119i	−0.828670	−	1.43530i
$530$		0			0
$531$	6.06726	+	3.50294i	0.263297	+	0.152014i
$532$		0			0
$533$	12.5632	+	8.88625i	0.544174	+	0.384906i
$534$		0			0
$535$	−2.80342	−	1.61856i	−0.121202	−	0.0699763i
$536$		0			0
$537$	22.1945	+	38.4420i	0.957763	+	1.65889i
$538$		0			0
$539$	−3.91780	+	2.26194i	−0.168751	+	0.0974287i
$540$		0			0
$541$		−	31.8881i		−	1.37098i	−0.728084	−	0.685488i	$-0.759589\pi$
$541$		−	31.8881i		−	1.37098i	0.728084	−	0.685488i	$-0.240411\pi$
$542$		0			0
$543$	−15.3852	+	26.6479i	−0.660240	+	1.14357i
$544$		0			0
$545$	−9.12979			−0.391077
$546$		0			0
$547$	44.7966			1.91537			0.957683	−	0.287826i	$-0.0929325\pi$
$547$	44.7966			1.91537			0.957683	+	0.287826i	$0.0929325\pi$
$548$		0			0
$549$	27.7221	−	48.0161i	1.18315	−	2.04928i
$550$		0			0
$551$		−	0.563476i		−	0.0240049i
$552$		0			0
$553$	−24.6231	+	14.2162i	−1.04708	+	0.604533i
$554$		0			0
$555$	−13.7787	−	23.8654i	−0.584872	−	1.01303i
$556$		0			0
$557$	22.7820	+	13.1532i	0.965306	+	0.557319i	0.897802	−	0.440400i	$-0.145163\pi$
$557$	22.7820	+	13.1532i	0.965306	+	0.557319i	0.0675037	+	0.997719i	$0.478497\pi$
$558$		0			0
$559$	−14.0938	−	30.6037i	−0.596106	−	1.29440i
$560$		0			0
$561$	15.3848	+	8.88244i	0.649548	+	0.375017i
$562$		0			0
$563$	−2.89492	−	5.01415i	−0.122006	−	0.211321i	0.798552	−	0.601925i	$-0.205599\pi$
$563$	−2.89492	−	5.01415i	−0.122006	−	0.211321i	−0.920559	+	0.390604i	$0.872266\pi$
$564$		0			0
$565$	−9.48610	+	5.47680i	−0.399083	+	0.230411i
$566$		0			0
$567$		−	1.96321i		−	0.0824471i
$568$		0			0
$569$	11.8184	−	20.4700i	0.495452	−	0.858149i	−0.504534	−	0.863392i	$-0.668336\pi$
$569$	11.8184	−	20.4700i	0.495452	−	0.858149i	0.999986	+	0.00524320i	$0.00166897\pi$
$570$		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
$573$	−16.8428			−0.703620
$574$		0			0
$575$	3.90893	−	6.77046i	0.163014	−	0.282348i
$576$		0			0
$577$			34.2415i			1.42549i	0.701422	+	0.712746i	$0.252549\pi$
$577$			34.2415i			1.42549i	−0.701422	+	0.712746i	$0.747451\pi$
$578$		0			0
$579$	−31.2611	+	18.0486i	−1.29917	+	0.750074i
$580$		0			0
$581$	16.5409	+	28.6497i	0.686233	+	1.18859i
$582$		0			0
$583$	18.8908	+	10.9066i	0.782379	+	0.451707i
$584$		0			0
$585$	−16.3453	+	7.52748i	−0.675797	+	0.311223i
$586$		0			0
$587$	−4.88091	−	2.81800i	−0.201457	−	0.116311i	0.395878	−	0.918303i	$-0.370440\pi$
$587$	−4.88091	−	2.81800i	−0.201457	−	0.116311i	−0.597335	+	0.801992i	$0.703774\pi$
$588$		0			0
$589$	3.12436	+	5.41154i	0.128737	+	0.222979i
$590$		0			0
$591$	39.4341	−	22.7673i	1.62210	−	0.936521i
$592$		0			0
$593$			15.3014i			0.628353i	0.949365	+	0.314177i	$0.101728\pi$
$593$			15.3014i			0.628353i	−0.949365	+	0.314177i	$0.898272\pi$
$594$		0			0
$595$	−3.80024	+	6.58220i	−0.155795	+	0.269844i
$596$		0			0
$597$	24.1074			0.986651
$598$		0			0
$599$	9.82414			0.401404			0.200702	−	0.979652i	$-0.435678\pi$
$599$	9.82414			0.401404			0.200702	+	0.979652i	$0.435678\pi$
$600$		0			0
$601$	23.0945	−	40.0008i	0.942043	−	1.63167i	0.180476	−	0.983579i	$-0.442236\pi$
$601$	23.0945	−	40.0008i	0.942043	−	1.63167i	0.761566	−	0.648087i	$-0.224431\pi$
$602$		0			0
$603$			54.0697i			2.20189i
$604$		0			0
$605$	6.92820	−	4.00000i	0.281672	−	0.162623i
$606$		0			0
$607$	−12.3482	−	21.3877i	−0.501198	−	0.868100i	−0.999999	−	0.00138384i	$-0.999560\pi$
$607$	−12.3482	−	21.3877i	−0.501198	−	0.868100i	0.498801	−	0.866716i	$-0.333774\pi$
$608$		0			0
$609$	−2.70216	−	1.56009i	−0.109497	−	0.0632182i
$610$		0			0
$611$	−1.14776	+	12.4371i	−0.0464335	+	0.503153i
$612$		0			0
$613$	−14.4155	−	8.32277i	−0.582235	−	0.336154i	0.179786	−	0.983706i	$-0.442459\pi$
$613$	−14.4155	−	8.32277i	−0.582235	−	0.336154i	−0.762021	+	0.647552i	$0.775793\pi$
$614$		0			0
$615$	6.03240	+	10.4484i	0.243250	+	0.421321i
$616$		0			0
$617$	−1.39918	+	0.807820i	−0.0563291	+	0.0325216i	−0.527900	−	0.849307i	$-0.677020\pi$
$617$	−1.39918	+	0.807820i	−0.0563291	+	0.0325216i	0.471571	+	0.881828i	$0.343687\pi$
$618$		0			0
$619$		−	23.3922i		−	0.940213i	−0.882610	−	0.470106i	$-0.844216\pi$
$619$		−	23.3922i		−	0.940213i	0.882610	−	0.470106i	$-0.155784\pi$
$620$		0			0
$621$	−22.0005	+	38.1060i	−0.882851	+	1.52914i
$622$		0			0
$623$	11.5727			0.463649
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	−2.61799	+	4.53449i	−0.104552	+	0.181090i
$628$		0			0
$629$		−	35.3701i		−	1.41030i
$630$		0			0
$631$	2.77458	−	1.60190i	0.110454	−	0.0637708i	−0.443755	−	0.896148i	$-0.646354\pi$
$631$	2.77458	−	1.60190i	0.110454	−	0.0637708i	0.554209	+	0.832377i	$0.313021\pi$
$632$		0			0
$633$	5.90893	+	10.2346i	0.234859	+	0.406787i
$634$		0			0
$635$	0.786022	+	0.453810i	0.0311923	+	0.0180089i
$636$		0			0
$637$	5.43813	−	7.68834i	0.215467	−	0.304623i
$638$		0			0
$639$	61.2442	+	35.3593i	2.42278	+	1.39879i
$640$		0			0
$641$	−9.98794	−	17.2996i	−0.394500	−	0.683294i	0.598537	−	0.801095i	$-0.295749\pi$
$641$	−9.98794	−	17.2996i	−0.394500	−	0.683294i	−0.993037	+	0.117801i	$0.962416\pi$
$642$		0			0
$643$	−24.7136	+	14.2684i	−0.974607	+	0.562690i	−0.900638	−	0.434571i	$-0.856900\pi$
$643$	−24.7136	+	14.2684i	−0.974607	+	0.562690i	−0.0739696	+	0.997260i	$0.523567\pi$
$644$		0			0
$645$		−	26.4161i		−	1.04013i
$646$		0			0
$647$	3.04325	−	5.27107i	0.119643	−	0.207227i	−0.799983	−	0.600022i	$-0.795158\pi$
$647$	3.04325	−	5.27107i	0.119643	−	0.207227i	0.919626	+	0.392795i	$0.128492\pi$
$648$		0			0
$649$	−2.43127			−0.0954359
$650$		0			0
$651$	34.6016			1.35614
$652$		0			0
$653$	−9.81499	+	17.0001i	−0.384090	+	0.665264i	−0.991643	−	0.129016i	$-0.958818\pi$
$653$	−9.81499	+	17.0001i	−0.384090	+	0.665264i	0.607552	+	0.794280i	$0.292152\pi$
$654$		0			0
$655$		−	13.1626i		−	0.514306i
$656$		0			0
$657$	11.4313	−	6.59985i	0.445976	−	0.257485i
$658$		0			0
$659$	9.79752	+	16.9698i	0.381657	+	0.661049i	0.991299	−	0.131627i	$-0.0420202\pi$
$659$	9.79752	+	16.9698i	0.381657	+	0.661049i	−0.609642	+	0.792677i	$0.708687\pi$
$660$		0			0
$661$	30.0735	+	17.3630i	1.16972	+	0.675341i	0.953615	−	0.301029i	$-0.0973300\pi$
$661$	30.0735	+	17.3630i	1.16972	+	0.675341i	0.216109	+	0.976369i	$0.430663\pi$
$662$		0			0
$663$	−36.8241	−	3.39831i	−1.43013	−	0.131980i
$664$		0			0
$665$	−1.94002	−	1.12007i	−0.0752308	−	0.0434345i
$666$		0			0
$667$	2.05967	+	3.56745i	0.0797506	+	0.138132i
$668$		0			0
$669$	25.7030	−	14.8396i	0.993736	−	0.573734i
$670$		0			0
$671$			19.2410i			0.742790i
$672$		0			0
$673$	−17.0051	+	29.4538i	−0.655500	+	1.13536i	0.326268	+	0.945277i	$0.394209\pi$
$673$	−17.0051	+	29.4538i	−0.655500	+	1.13536i	−0.981768	+	0.190082i	$0.939125\pi$
$674$		0			0
$675$	−5.62828			−0.216633
$676$		0			0
$677$	29.9209			1.14995			0.574977	−	0.818169i	$-0.305011\pi$
$677$	29.9209			1.14995			0.574977	+	0.818169i	$0.305011\pi$
$678$		0			0
$679$	15.9147	−	27.5651i	0.610751	−	1.05785i
$680$		0			0
$681$			44.0584i			1.68832i
$682$		0			0
$683$	24.1251	−	13.9286i	0.923121	−	0.532964i	0.0384916	−	0.999259i	$-0.487745\pi$
$683$	24.1251	−	13.9286i	0.923121	−	0.532964i	0.884629	+	0.466295i	$0.154411\pi$
$684$		0			0
$685$	6.03240	+	10.4484i	0.230486	+	0.399214i
$686$		0			0
$687$	−47.1786	−	27.2386i	−1.79998	−	1.03922i
$688$		0			0
$689$	−45.2158	−	4.17274i	−1.72258	−	0.158969i
$690$		0			0
$691$	31.4550	+	18.1606i	1.19661	+	0.690860i	0.959797	−	0.280696i	$-0.0905652\pi$
$691$	31.4550	+	18.1606i	1.19661	+	0.690860i	0.236809	+	0.971556i	$0.423899\pi$
$692$		0			0
$693$	9.05440	+	15.6827i	0.343948	+	0.595736i
$694$		0			0
$695$	−4.86002	+	2.80593i	−0.184351	+	0.106435i
$696$		0			0
$697$			15.4853i			0.586548i
$698$		0			0
$699$	−3.51651	+	6.09077i	−0.133007	+	0.230374i
$700$		0			0
$701$	2.16156			0.0816411			0.0408206	−	0.999166i	$-0.487003\pi$
$701$	2.16156			0.0816411			0.0408206	+	0.999166i	$0.487003\pi$
$702$		0			0
$703$	10.4249			0.393183
$704$		0			0
$705$	−4.89623	+	8.48052i	−0.184403	+	0.319395i
$706$		0			0
$707$		−	7.67250i		−	0.288554i
$708$		0			0
$709$	−41.2371	+	23.8082i	−1.54869	+	0.894137i	−0.550449	+	0.834869i	$0.685543\pi$
$709$	−41.2371	+	23.8082i	−1.54869	+	0.894137i	−0.998242	+	0.0592680i	$0.981123\pi$
$710$		0			0
$711$	−33.8713	−	58.6668i	−1.27027	−	2.20018i
$712$		0			0
$713$	−39.5615	−	22.8408i	−1.48159	−	0.855396i
$714$		0			0
$715$	3.60628	−	5.09850i	0.134867	−	0.190673i
$716$		0			0
$717$	−41.0733	−	23.7137i	−1.53391	−	0.885603i
$718$		0			0
$719$	−10.1469	−	17.5749i	−0.378414	−	0.655433i	0.612417	−	0.790535i	$-0.290197\pi$
$719$	−10.1469	−	17.5749i	−0.378414	−	0.655433i	−0.990832	+	0.135102i	$0.956864\pi$
$720$		0			0
$721$	−24.9996	+	14.4335i	−0.931035	+	0.537533i
$722$		0			0
$723$			82.2029i			3.05716i
$724$		0			0
$725$	−0.263457	+	0.456321i	−0.00978454	+	0.0169473i
$726$		0			0
$727$	11.0681			0.410494			0.205247	−	0.978710i	$-0.434200\pi$
$727$	11.0681			0.410494			0.205247	+	0.978710i	$0.434200\pi$
$728$		0			0
$729$	−43.0532			−1.59456
$730$		0			0
$731$	16.9527	−	29.3630i	0.627019	−	1.08603i
$732$		0			0
$733$			23.4002i			0.864304i	0.901801	+	0.432152i	$0.142246\pi$
$733$			23.4002i			0.864304i	−0.901801	+	0.432152i	$0.857754\pi$
$734$		0			0
$735$	6.39414	−	3.69166i	0.235852	−	0.136169i
$736$		0			0
$737$	−9.38201	−	16.2501i	−0.345591	−	0.598581i
$738$		0			0
$739$	−22.1617	−	12.7951i	−0.815232	−	0.470675i	0.0335372	−	0.999437i	$-0.489323\pi$
$739$	−22.1617	−	12.7951i	−0.815232	−	0.470675i	−0.848770	+	0.528763i	$0.822656\pi$
$740$		0			0
$741$	1.00161	−	10.8534i	0.0367951	−	0.398711i
$742$		0			0
$743$	7.28694	+	4.20712i	0.267332	+	0.154344i	0.627675	−	0.778476i	$-0.284007\pi$
$743$	7.28694	+	4.20712i	0.267332	+	0.154344i	−0.360343	+	0.932820i	$0.617340\pi$
$744$		0			0
$745$	10.4943	+	18.1767i	0.384482	+	0.665942i
$746$		0			0
$747$	−68.2605	+	39.4102i	−2.49752	+	1.44194i
$748$		0			0
$749$		−	6.78106i		−	0.247775i
$750$		0			0
$751$	20.2595	−	35.0905i	0.739279	−	1.28047i	−0.213541	−	0.976934i	$-0.568500\pi$
$751$	20.2595	−	35.0905i	0.739279	−	1.28047i	0.952820	−	0.303535i	$-0.0981671\pi$
$752$		0			0
$753$	−54.0514			−1.96974
$754$		0			0
$755$	6.99102			0.254429
$756$		0			0
$757$	5.67593	−	9.83100i	0.206295	−	0.357314i	−0.744249	−	0.667902i	$-0.767193\pi$
$757$	5.67593	−	9.83100i	0.206295	−	0.357314i	0.950545	+	0.310588i	$0.100526\pi$
$758$		0			0
$759$		−	38.2780i		−	1.38940i
$760$		0			0
$761$	−5.86364	+	3.38538i	−0.212557	+	0.122720i	−0.602499	−	0.798120i	$-0.705828\pi$
$761$	−5.86364	+	3.38538i	−0.212557	+	0.122720i	0.389942	+	0.920839i	$0.372495\pi$
$762$		0			0
$763$	−9.56249	−	16.5627i	−0.346185	−	0.599611i
$764$		0			0
$765$	−15.6827	−	9.05440i	−0.567008	−	0.327362i
$766$		0			0
$767$	4.59704	−	2.11707i	0.165990	−	0.0764428i
$768$		0			0
$769$	−20.0563	−	11.5795i	−0.723250	−	0.417569i	0.0926975	−	0.995694i	$-0.470451\pi$
$769$	−20.0563	−	11.5795i	−0.723250	−	0.417569i	−0.815948	+	0.578126i	$0.803784\pi$
$770$		0			0
$771$	19.3766	+	33.5613i	0.697833	+	1.20868i
$772$		0			0
$773$	36.3333	−	20.9770i	1.30682	−	0.754491i	0.325253	−	0.945627i	$-0.394551\pi$
$773$	36.3333	−	20.9770i	1.30682	−	0.754491i	0.981564	+	0.191136i	$0.0612172\pi$
$774$		0			0
$775$		−	5.84325i		−	0.209896i
$776$		0			0
$777$	28.8634	−	49.9928i	1.03547	−	1.79348i
$778$		0			0
$779$	−4.56410			−0.163526
$780$		0			0
$781$	−24.5418			−0.878174
$782$		0			0
$783$	1.48281	−	2.56830i	0.0529913	−	0.0917835i
$784$		0			0
$785$			3.74761i			0.133758i
$786$		0			0
$787$	32.4026	−	18.7076i	1.15503	−	0.666856i	0.204920	−	0.978779i	$-0.434306\pi$
$787$	32.4026	−	18.7076i	1.15503	−	0.666856i	0.950107	+	0.311923i	$0.100973\pi$
$788$		0			0
$789$	16.8388	+	29.1656i	0.599476	+	1.03832i
$790$		0			0
$791$	−19.8714	−	11.4727i	−0.706544	&m

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)

\(f(q)\)	\(=\)	\(q+(-1.41342 + 2.44811i) q^{3} +1.00000i q^{5} +(-1.81414 + 1.04739i) q^{7} +(-2.49551 - 4.32235i) q^{9} +O(q^{10})\)	\(q+(-1.41342 + 2.44811i) q^{3} +1.00000i q^{5} +(-1.81414 + 1.04739i) q^{7} +(-2.49551 - 4.32235i) q^{9} +(1.50000 + 0.866025i) q^{11} +(-3.59030 - 0.331331i) q^{13} +(-2.44811 - 1.41342i) q^{15} +(-1.81414 - 3.14218i) q^{17} +(0.926118 - 0.534695i) q^{19} -5.92163i q^{21} +(-3.90893 + 6.77046i) q^{23} -1.00000 q^{25} +5.62828 q^{27} +(0.263457 - 0.456321i) q^{29} +5.84325i q^{31} +(-4.24026 + 2.44811i) q^{33} +(-1.04739 - 1.81414i) q^{35} +(8.44242 + 4.87423i) q^{37} +(5.88573 - 8.32114i) q^{39} +(-3.69615 - 2.13397i) q^{41} +(4.67238 + 8.09281i) q^{43} +(4.32235 - 2.49551i) q^{45} -3.46410i q^{47} +(-1.30593 + 2.26194i) q^{49} +10.2566 q^{51} +12.5939 q^{53} +(-0.866025 + 1.50000i) q^{55} +3.02299i q^{57} +(-1.21564 + 0.701848i) q^{59} +(5.55440 + 9.62050i) q^{61} +(9.05440 + 5.22756i) q^{63} +(0.331331 - 3.59030i) q^{65} +(-9.38201 - 5.41671i) q^{67} +(-11.0499 - 19.1390i) q^{69} +(-12.2709 + 7.08460i) q^{71} +2.64469i q^{73} +(1.41342 - 2.44811i) q^{75} -3.62828 q^{77} +13.5729 q^{79} +(-0.468594 + 0.811629i) q^{81} -15.7925i q^{83} +(3.14218 - 1.81414i) q^{85} +(0.744750 + 1.28994i) q^{87} +(-4.78436 - 2.76225i) q^{89} +(6.86033 - 3.15937i) q^{91} +(-14.3049 - 8.25896i) q^{93} +(0.534695 + 0.926118i) q^{95} +(-13.1589 + 7.59730i) q^{97} -8.64469i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{3} + 6 q^{7} - 4 q^{9}+O(q^{10}) \) 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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Representations

Groups

Database

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