LMFDB - Embedded newform 1638.2.bj.i.127.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1638,2,Mod(127,1638)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1638, 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("1638.127");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1638.bj (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:	$13.0794958511$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 2 x^{15} + 2 x^{14} + 18 x^{13} + 143 x^{12} - 148 x^{11} + 172 x^{10} + 1612 x^{9} + \cdots + 97344 $ x^16 - 2x^15 + 2x^14 + 18x^13 + 143x^12 - 148x^11 + 172x^10 + 1612x^9 + 6655x^8 + 478x^7 + 1106x^6 + 11266x^5 + 55249x^4 + 8856x^3 + 288x^2 + 7488*x + 97344
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			127.2
Root			$-1.29491 + 1.29491i$ of defining polynomial
Character	$\chi$	$=$	1638.127
Dual form			1638.2.bj.i.1135.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.866025 + 0.500000i) q^{2} +(0.500000 - 0.866025i) q^{4} -1.34861i q^{5} +(-0.866025 - 0.500000i) q^{7} +1.00000i q^{8} +O(q^{10})$	\(q+(-0.866025 + 0.500000i) q^{2} +(0.500000 - 0.866025i) q^{4} -1.34861i q^{5} +(-0.866025 - 0.500000i) q^{7} +1.00000i q^{8} +(0.674306 + 1.16793i) q^{10} +(-1.11892 + 0.646009i) q^{11} +(-0.343043 + 3.58920i) q^{13} +1.00000 q^{14} +(-0.500000 - 0.866025i) q^{16} +(-1.76555 + 3.05803i) q^{17} +(1.98887 + 1.14828i) q^{19} +(-1.16793 - 0.674306i) q^{20} +(0.646009 - 1.11892i) q^{22} +(-3.08058 - 5.33572i) q^{23} +3.18125 q^{25} +(-1.49751 - 3.27986i) q^{26} +(-0.866025 + 0.500000i) q^{28} +(-4.11174 - 7.12175i) q^{29} -9.90596i q^{31} +(0.866025 + 0.500000i) q^{32} -3.53111i q^{34} +(-0.674306 + 1.16793i) q^{35} +(3.71625 - 2.14558i) q^{37} -2.29655 q^{38} +1.34861 q^{40} +(-6.84283 + 3.95071i) q^{41} +(-6.26649 + 10.8539i) q^{43} +1.29202i q^{44} +(5.33572 + 3.08058i) q^{46} +10.7063i q^{47} +(0.500000 + 0.866025i) q^{49} +(-2.75504 + 1.59062i) q^{50} +(2.93681 + 2.09168i) q^{52} -3.30912 q^{53} +(0.871215 + 1.50899i) q^{55} +(0.500000 - 0.866025i) q^{56} +(7.12175 + 4.11174i) q^{58} +(-9.40363 - 5.42919i) q^{59} +(-2.64171 + 4.57558i) q^{61} +(4.95298 + 8.57881i) q^{62} -1.00000 q^{64} +(4.84043 + 0.462632i) q^{65} +(-7.85978 + 4.53784i) q^{67} +(1.76555 + 3.05803i) q^{68} -1.34861i q^{70} +(-6.89175 - 3.97895i) q^{71} +3.28169i q^{73} +(-2.14558 + 3.71625i) q^{74} +(1.98887 - 1.14828i) q^{76} +1.29202 q^{77} -3.32636 q^{79} +(-1.16793 + 0.674306i) q^{80} +(3.95071 - 6.84283i) q^{82} -0.731184i q^{83} +(4.12410 + 2.38105i) q^{85} -12.5330i q^{86} +(-0.646009 - 1.11892i) q^{88} +(-11.5236 + 6.65313i) q^{89} +(2.09168 - 2.93681i) q^{91} -6.16116 q^{92} +(-5.35313 - 9.27189i) q^{94} +(1.54858 - 2.68222i) q^{95} +(-10.5806 - 6.10870i) q^{97} +(-0.866025 - 0.500000i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 8 q^{4}+O(q^{10}) $ 16 * q + 8 * q^4	$ 16 q + 8 q^{4} + 2 q^{10} + 12 q^{11} + 10 q^{13} + 16 q^{14} - 8 q^{16} + 6 q^{17} - 4 q^{22} + 12 q^{23} - 20 q^{25} + 2 q^{26} - 16 q^{29} - 2 q^{35} - 6 q^{37} + 4 q^{40} - 12 q^{41} - 6 q^{43} + 6 q^{46} + 8 q^{49} + 24 q^{50} - 4 q^{52} + 40 q^{53} + 20 q^{55} + 8 q^{56} + 6 q^{58} - 6 q^{59} - 2 q^{61} + 14 q^{62} - 16 q^{64} + 52 q^{65} - 30 q^{67} - 6 q^{68} - 12 q^{71} - 24 q^{74} - 8 q^{77} - 16 q^{79} + 2 q^{82} + 6 q^{85} + 4 q^{88} - 30 q^{89} + 4 q^{91} + 24 q^{92} - 8 q^{94} + 40 q^{95} - 24 q^{97}+O(q^{100}) $ 16 * q + 8 * q^4 + 2 * q^10 + 12 * q^11 + 10 * q^13 + 16 * q^14 - 8 * q^16 + 6 * q^17 - 4 * q^22 + 12 * q^23 - 20 * q^25 + 2 * q^26 - 16 * q^29 - 2 * q^35 - 6 * q^37 + 4 * q^40 - 12 * q^41 - 6 * q^43 + 6 * q^46 + 8 * q^49 + 24 * q^50 - 4 * q^52 + 40 * q^53 + 20 * q^55 + 8 * q^56 + 6 * q^58 - 6 * q^59 - 2 * q^61 + 14 * q^62 - 16 * q^64 + 52 * q^65 - 30 * q^67 - 6 * q^68 - 12 * q^71 - 24 * q^74 - 8 * q^77 - 16 * q^79 + 2 * q^82 + 6 * q^85 + 4 * q^88 - 30 * q^89 + 4 * q^91 + 24 * q^92 - 8 * q^94 + 40 * q^95 - 24 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1638\mathbb{Z}\right)^\times$.

$n$	$379$	$703$	$911$
$\chi(n)$	$e\left(\frac{5}{6}\right)$	$1$	$1$

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 $

$2$	−0.866025	+	0.500000i	−0.612372	+	0.353553i
$2$	−0.866025	+	0.500000i	−0.612372	+	0.353553i
$3$		0			0
$3$		0			0
$4$	0.500000	−	0.866025i	0.250000	−	0.433013i
$5$		−	1.34861i		−	0.603118i	−0.953448	−	0.301559i	$-0.902493\pi$
$5$		−	1.34861i		−	0.603118i	0.953448	−	0.301559i	$-0.0975070\pi$
$6$		0			0
$7$	−0.866025	−	0.500000i	−0.327327	−	0.188982i
$7$	−0.866025	−	0.500000i	−0.327327	−	0.188982i
$8$			1.00000i			0.353553i
$9$		0			0
$10$	0.674306	+	1.16793i	0.213234	+	0.369333i
$11$	−1.11892	+	0.646009i	−0.337367	+	0.194779i	−0.659107	−	0.752049i	$-0.729066\pi$
$11$	−1.11892	+	0.646009i	−0.337367	+	0.194779i	0.321740	+	0.946828i	$0.395732\pi$
$12$		0			0
$13$	−0.343043	+	3.58920i	−0.0951431	+	0.995464i
$13$	−0.343043	+	3.58920i	−0.0951431	+	0.995464i
$14$	1.00000			0.267261
$15$		0			0
$16$	−0.500000	−	0.866025i	−0.125000	−	0.216506i
$17$	−1.76555	+	3.05803i	−0.428210	+	0.741681i	−0.996714	−	0.0809991i	$-0.974189\pi$
$17$	−1.76555	+	3.05803i	−0.428210	+	0.741681i	0.568504	+	0.822680i	$0.307522\pi$
$18$		0			0
$19$	1.98887	+	1.14828i	0.456279	+	0.263433i	0.710478	−	0.703719i	$-0.248479\pi$
$19$	1.98887	+	1.14828i	0.456279	+	0.263433i	−0.254200	+	0.967152i	$0.581812\pi$
$20$	−1.16793	−	0.674306i	−0.261158	−	0.150779i
$21$		0			0
$22$	0.646009	−	1.11892i	0.137730	−	0.238555i
$23$	−3.08058	−	5.33572i	−0.642345	−	1.11257i	−0.984908	−	0.173079i	$-0.944628\pi$
$23$	−3.08058	−	5.33572i	−0.642345	−	1.11257i	0.342563	−	0.939495i	$-0.388705\pi$
$24$		0			0
$25$	3.18125			0.636249
$26$	−1.49751	−	3.27986i	−0.293687	−	0.643233i
$27$		0			0
$28$	−0.866025	+	0.500000i	−0.163663	+	0.0944911i
$29$	−4.11174	−	7.12175i	−0.763532	−	1.32248i	−0.941019	−	0.338353i	$-0.890130\pi$
$29$	−4.11174	−	7.12175i	−0.763532	−	1.32248i	0.177488	−	0.984123i	$-0.443203\pi$
$30$		0			0
$31$		−	9.90596i		−	1.77916i	−0.456777	−	0.889581i	$-0.650996\pi$
$31$		−	9.90596i		−	1.77916i	0.456777	−	0.889581i	$-0.349004\pi$
$32$	0.866025	+	0.500000i	0.153093	+	0.0883883i
$33$		0			0
$34$		−	3.53111i		−	0.605580i
$35$	−0.674306	+	1.16793i	−0.113979	+	0.197417i
$36$		0			0
$37$	3.71625	−	2.14558i	0.610947	−	0.352730i	−0.162389	−	0.986727i	$-0.551920\pi$
$37$	3.71625	−	2.14558i	0.610947	−	0.352730i	0.773336	+	0.633996i	$0.218587\pi$
$38$	−2.29655			−0.372550
$39$		0			0
$40$	1.34861			0.213234
$41$	−6.84283	+	3.95071i	−1.06867	+	0.616997i	−0.927818	−	0.373033i	$-0.878318\pi$
$41$	−6.84283	+	3.95071i	−1.06867	+	0.616997i	−0.140853	+	0.990031i	$0.544984\pi$
$42$		0			0
$43$	−6.26649	+	10.8539i	−0.955630	+	1.65520i	−0.222710	+	0.974885i	$0.571490\pi$
$43$	−6.26649	+	10.8539i	−0.955630	+	1.65520i	−0.732920	+	0.680315i	$0.761843\pi$
$44$			1.29202i			0.194779i
$45$		0			0
$46$	5.33572	+	3.08058i	0.786709	+	0.454207i
$47$			10.7063i			1.56167i	0.624738	+	0.780834i	$0.285206\pi$
$47$			10.7063i			1.56167i	−0.624738	+	0.780834i	$0.714794\pi$
$48$		0			0
$49$	0.500000	+	0.866025i	0.0714286	+	0.123718i
$50$	−2.75504	+	1.59062i	−0.389621	+	0.224948i
$51$		0			0
$52$	2.93681	+	2.09168i	0.407263	+	0.290064i
$53$	−3.30912			−0.454543			−0.227272	−	0.973831i	$-0.572980\pi$
$53$	−3.30912			−0.454543			−0.227272	+	0.973831i	$0.572980\pi$
$54$		0			0
$55$	0.871215	+	1.50899i	0.117475	+	0.203472i
$56$	0.500000	−	0.866025i	0.0668153	−	0.115728i
$57$		0			0
$58$	7.12175	+	4.11174i	0.935132	+	0.539898i
$59$	−9.40363	−	5.42919i	−1.22425	−	0.706820i	−0.258428	−	0.966031i	$-0.583204\pi$
$59$	−9.40363	−	5.42919i	−1.22425	−	0.706820i	−0.965821	+	0.259210i	$0.916538\pi$
$60$		0			0
$61$	−2.64171	+	4.57558i	−0.338237	+	0.585843i	−0.984101	−	0.177609i	$-0.943164\pi$
$61$	−2.64171	+	4.57558i	−0.338237	+	0.585843i	0.645864	+	0.763452i	$0.276497\pi$
$62$	4.95298	+	8.57881i	0.629029	+	1.08951i
$63$		0			0
$64$	−1.00000			−0.125000
$65$	4.84043	+	0.462632i	0.600382	+	0.0573825i
$66$		0			0
$67$	−7.85978	+	4.53784i	−0.960225	+	0.554386i	−0.896242	−	0.443565i	$-0.853714\pi$
$67$	−7.85978	+	4.53784i	−0.960225	+	0.554386i	−0.0639825	+	0.997951i	$0.520380\pi$
$68$	1.76555	+	3.05803i	0.214105	+	0.370841i
$69$		0			0
$70$		−	1.34861i		−	0.161190i
$71$	−6.89175	−	3.97895i	−0.817900	−	0.472215i	0.0317917	−	0.999495i	$-0.489879\pi$
$71$	−6.89175	−	3.97895i	−0.817900	−	0.472215i	−0.849692	+	0.527280i	$0.823212\pi$
$72$		0			0
$73$			3.28169i			0.384092i	0.981386	+	0.192046i	$0.0615123\pi$
$73$			3.28169i			0.384092i	−0.981386	+	0.192046i	$0.938488\pi$
$74$	−2.14558	+	3.71625i	−0.249418	+	0.432005i
$75$		0			0
$76$	1.98887	−	1.14828i	0.228139	−	0.131716i
$77$	1.29202			0.147239
$78$		0			0
$79$	−3.32636			−0.374245			−0.187123	−	0.982337i	$-0.559916\pi$
$79$	−3.32636			−0.374245			−0.187123	+	0.982337i	$0.559916\pi$
$80$	−1.16793	+	0.674306i	−0.130579	+	0.0753897i
$81$		0			0
$82$	3.95071	−	6.84283i	0.436283	−	0.755664i
$83$		−	0.731184i		−	0.0802579i	−0.999195	−	0.0401289i	$-0.987223\pi$
$83$		−	0.731184i		−	0.0802579i	0.999195	−	0.0401289i	$-0.0127769\pi$
$84$		0			0
$85$	4.12410	+	2.38105i	0.447321	+	0.258261i
$86$		−	12.5330i		−	1.35146i
$87$		0			0
$88$	−0.646009	−	1.11892i	−0.0688648	−	0.119277i
$89$	−11.5236	+	6.65313i	−1.22150	+	0.705231i	−0.965237	−	0.261377i	$-0.915823\pi$
$89$	−11.5236	+	6.65313i	−1.22150	+	0.705231i	−0.256259	+	0.966608i	$0.582490\pi$
$90$		0			0
$91$	2.09168	−	2.93681i	0.219268	−	0.307862i
$92$	−6.16116			−0.642345
$93$		0			0
$94$	−5.35313	−	9.27189i	−0.552133	−	0.956323i
$95$	1.54858	−	2.68222i	0.158881	−	0.275190i
$96$		0			0
$97$	−10.5806	−	6.10870i	−1.07429	−	0.620244i	−0.144943	−	0.989440i	$-0.546300\pi$
$97$	−10.5806	−	6.10870i	−1.07429	−	0.620244i	−0.929352	+	0.369196i	$0.879633\pi$
$98$	−0.866025	−	0.500000i	−0.0874818	−	0.0505076i
$99$		0			0
$100$	1.59062	−	2.75504i	0.159062	−	0.275504i
$101$	6.32905	+	10.9622i	0.629764	+	1.09078i	0.987599	+	0.156998i	$0.0501816\pi$
$101$	6.32905	+	10.9622i	0.629764	+	1.09078i	−0.357835	+	0.933785i	$0.616485\pi$
$102$		0			0
$103$	11.8042			1.16310			0.581551	−	0.813510i	$-0.302446\pi$
$103$	11.8042			1.16310			0.581551	+	0.813510i	$0.302446\pi$
$104$	−3.58920	−	0.343043i	−0.351950	−	0.0336382i
$105$		0			0
$106$	2.86579	−	1.65456i	0.278350	−	0.160705i
$107$	5.14877	+	8.91794i	0.497751	+	0.862130i	0.999997	−	0.00259527i	$-0.000826100\pi$
$107$	5.14877	+	8.91794i	0.497751	+	0.862130i	−0.502246	+	0.864725i	$0.667493\pi$
$108$		0			0
$109$			18.8876i			1.80910i	0.426368	+	0.904550i	$0.359793\pi$
$109$			18.8876i			1.80910i	−0.426368	+	0.904550i	$0.640207\pi$
$110$	−1.50899	−	0.871215i	−0.143876	−	0.0830671i
$111$		0			0
$112$			1.00000i			0.0944911i
$113$	−6.42036	+	11.1204i	−0.603977	+	1.04612i	0.388236	+	0.921560i	$0.373085\pi$
$113$	−6.42036	+	11.1204i	−0.603977	+	1.04612i	−0.992212	+	0.124558i	$0.960249\pi$
$114$		0			0
$115$	−7.19581	+	4.15451i	−0.671013	+	0.387410i
$116$	−8.22349			−0.763532
$117$		0			0
$118$	10.8584			0.999595
$119$	3.05803	−	1.76555i	0.280329	−	0.161848i
$120$		0			0
$121$	−4.66535	+	8.08062i	−0.424122	+	0.734601i
$122$		−	5.28343i		−	0.478339i
$123$		0			0
$124$	−8.57881	−	4.95298i	−0.770400	−	0.444791i
$125$		−	11.0333i		−	0.986851i
$126$		0			0
$127$	−5.67048	−	9.82157i	−0.503174	−	0.871523i	−0.999993	−	0.00366893i	$-0.998832\pi$
$127$	−5.67048	−	9.82157i	−0.503174	−	0.871523i	0.496819	−	0.867854i	$-0.334501\pi$
$128$	0.866025	−	0.500000i	0.0765466	−	0.0441942i
$129$		0			0
$130$	−4.42325	+	2.01956i	−0.387945	+	0.177128i
$131$	12.8560			1.12324			0.561618	−	0.827397i	$-0.310179\pi$
$131$	12.8560			1.12324			0.561618	+	0.827397i	$0.310179\pi$
$132$		0			0
$133$	−1.14828	−	1.98887i	−0.0995682	−	0.172457i
$134$	4.53784	−	7.85978i	0.392010	−	0.678981i
$135$		0			0
$136$	−3.05803	−	1.76555i	−0.262224	−	0.151395i
$137$	−10.0049	−	5.77635i	−0.854779	−	0.493507i	0.00748143	−	0.999972i	$-0.497619\pi$
$137$	−10.0049	−	5.77635i	−0.854779	−	0.493507i	−0.862260	+	0.506465i	$0.830952\pi$
$138$		0			0
$139$	4.29208	−	7.43409i	0.364049	−	0.630552i	−0.624574	−	0.780966i	$-0.714727\pi$
$139$	4.29208	−	7.43409i	0.364049	−	0.630552i	0.988623	+	0.150414i	$0.0480607\pi$
$140$	0.674306	+	1.16793i	0.0569893	+	0.0987083i
$141$		0			0
$142$	7.95791			0.667813
$143$	−1.93481	−	4.23763i	−0.161797	−	0.354369i
$144$		0			0
$145$	−9.60448	+	5.54515i	−0.797608	+	0.460499i
$146$	−1.64084	−	2.84202i	−0.135797	−	0.235208i
$147$		0			0
$148$		−	4.29115i		−	0.352730i
$149$	−8.76683	−	5.06153i	−0.718207	−	0.414657i	0.0958856	−	0.995392i	$-0.469432\pi$
$149$	−8.76683	−	5.06153i	−0.718207	−	0.414657i	−0.814092	+	0.580736i	$0.802765\pi$
$150$		0			0
$151$			3.86590i			0.314603i	0.987551	+	0.157301i	$0.0502794\pi$
$151$			3.86590i			0.314603i	−0.987551	+	0.157301i	$0.949721\pi$
$152$	−1.14828	+	1.98887i	−0.0931375	+	0.161319i
$153$		0			0
$154$	−1.11892	+	0.646009i	−0.0901651	+	0.0520569i
$155$	−13.3593			−1.07304
$156$		0			0
$157$	4.71896			0.376614			0.188307	−	0.982110i	$-0.439700\pi$
$157$	4.71896			0.376614			0.188307	+	0.982110i	$0.439700\pi$
$158$	2.88072	−	1.66318i	0.229177	−	0.132316i
$159$		0			0
$160$	0.674306	−	1.16793i	0.0533086	−	0.0923331i
$161$			6.16116i			0.485567i
$162$		0			0
$163$	−20.2815	−	11.7095i	−1.58857	−	0.917161i	−0.993543	−	0.113454i	$-0.963809\pi$
$163$	−20.2815	−	11.7095i	−1.58857	−	0.917161i	−0.595026	−	0.803707i	$-0.702858\pi$
$164$			7.90142i			0.616997i
$165$		0			0
$166$	0.365592	+	0.633224i	0.0283754	+	0.0491477i
$167$	21.0853	−	12.1736i	1.63163	−	0.942021i	0.648038	−	0.761608i	$-0.275590\pi$
$167$	21.0853	−	12.1736i	1.63163	−	0.942021i	0.983591	−	0.180414i	$-0.0577436\pi$
$168$		0			0
$169$	−12.7646	−	2.46250i	−0.981896	−	0.189423i
$170$	−4.76210			−0.365236
$171$		0			0
$172$	6.26649	+	10.8539i	0.477815	+	0.827600i
$173$	9.06378	−	15.6989i	0.689107	−	1.19357i	−0.283021	−	0.959114i	$-0.591337\pi$
$173$	9.06378	−	15.6989i	0.689107	−	1.19357i	0.972127	−	0.234454i	$-0.0753302\pi$
$174$		0			0
$175$	−2.75504	−	1.59062i	−0.208261	−	0.120240i
$176$	1.11892	+	0.646009i	0.0843418	+	0.0486947i
$177$		0			0
$178$	6.65313	−	11.5236i	0.498673	−	0.863728i
$179$	8.75612	+	15.1660i	0.654463	+	1.13356i	0.982028	+	0.188735i	$0.0604386\pi$
$179$	8.75612	+	15.1660i	0.654463	+	1.13356i	−0.327565	+	0.944829i	$0.606228\pi$
$180$		0			0
$181$	−24.5640			−1.82583			−0.912914	−	0.408152i	$-0.866173\pi$
$181$	−24.5640			−1.82583			−0.912914	+	0.408152i	$0.866173\pi$
$182$	−0.343043	+	3.58920i	−0.0254281	+	0.266049i
$183$		0			0
$184$	5.33572	−	3.08058i	0.393354	−	0.227103i
$185$	−2.89355	−	5.01177i	−0.212738	−	0.368473i
$186$		0			0
$187$		−	4.56225i		−	0.333625i
$188$	9.27189	+	5.35313i	0.676222	+	0.390417i
$189$		0			0
$190$			3.09716i			0.224691i
$191$	−6.64065	+	11.5020i	−0.480501	+	0.832252i	−0.999750	−	0.0223712i	$-0.992878\pi$
$191$	−6.64065	+	11.5020i	−0.480501	+	0.832252i	0.519249	+	0.854623i	$0.326212\pi$
$192$		0			0
$193$	18.9996	−	10.9694i	1.36762	−	0.789596i	0.376997	−	0.926214i	$-0.376957\pi$
$193$	18.9996	−	10.9694i	1.36762	−	0.789596i	0.990624	+	0.136618i	$0.0436234\pi$
$194$	12.2174			0.877158
$195$		0			0
$196$	1.00000			0.0714286
$197$	−13.1392	+	7.58594i	−0.936132	+	0.540476i	−0.888746	−	0.458401i	$-0.848422\pi$
$197$	−13.1392	+	7.58594i	−0.936132	+	0.540476i	−0.0473862	+	0.998877i	$0.515089\pi$
$198$		0			0
$199$	7.01383	−	12.1483i	0.497197	−	0.861171i	−0.502798	−	0.864404i	$-0.667696\pi$
$199$	7.01383	−	12.1483i	0.497197	−	0.861171i	0.999995	+	0.00323355i	$0.00102927\pi$
$200$			3.18125i			0.224948i
$201$		0			0
$202$	−10.9622	−	6.32905i	−0.771300	−	0.445310i
$203$			8.22349i			0.577176i
$204$		0			0
$205$	5.32798	+	9.22832i	0.372122	+	0.644534i
$206$	−10.2227	+	5.90210i	−0.712252	+	0.411219i
$207$		0			0
$208$	3.27986	−	1.49751i	0.227417	−	0.103834i
$209$	−2.96719			−0.205245
$210$		0			0
$211$	−7.16564	−	12.4113i	−0.493303	−	0.854426i	0.506667	−	0.862142i	$-0.330877\pi$
$211$	−7.16564	−	12.4113i	−0.493303	−	0.854426i	−0.999970	+	0.00771563i	$0.997544\pi$
$212$	−1.65456	+	2.86579i	−0.113636	+	0.196823i
$213$		0			0
$214$	−8.91794	−	5.14877i	−0.609618	−	0.351963i
$215$	14.6377	+	8.45106i	0.998280	+	0.576357i
$216$		0			0
$217$	−4.95298	+	8.57881i	−0.336230	+	0.582368i
$218$	−9.44378	−	16.3571i	−0.639613	−	1.10784i
$219$		0			0
$220$	1.74243			0.117475
$221$	−10.3702	−	7.38596i	−0.697575	−	0.496833i
$222$		0			0
$223$	−10.6241	+	6.13382i	−0.711442	+	0.410751i	−0.811595	−	0.584221i	$-0.801400\pi$
$223$	−10.6241	+	6.13382i	−0.711442	+	0.410751i	0.100153	+	0.994972i	$0.468067\pi$
$224$	−0.500000	−	0.866025i	−0.0334077	−	0.0578638i
$225$		0			0
$226$		−	12.8407i		−	0.854152i
$227$	−14.5897	−	8.42336i	−0.968352	−	0.559078i	−0.0696185	−	0.997574i	$-0.522178\pi$
$227$	−14.5897	−	8.42336i	−0.968352	−	0.559078i	−0.898733	+	0.438495i	$0.855512\pi$
$228$		0			0
$229$			9.18953i			0.607261i	0.952790	+	0.303631i	$0.0981989\pi$
$229$			9.18953i			0.607261i	−0.952790	+	0.303631i	$0.901801\pi$
$230$	4.15451	−	7.19581i	0.273940	−	0.474478i
$231$		0			0
$232$	7.12175	−	4.11174i	0.467566	−	0.269949i
$233$	25.1142			1.64529			0.822643	−	0.568558i	$-0.192498\pi$
$233$	25.1142			1.64529			0.822643	+	0.568558i	$0.192498\pi$
$234$		0			0
$235$	14.4386			0.941870
$236$	−9.40363	+	5.42919i	−0.612124	+	0.353410i
$237$		0			0
$238$	−1.76555	+	3.05803i	−0.114444	+	0.198223i
$239$		−	22.1206i		−	1.43086i	−0.698682	−	0.715432i	$-0.746230\pi$
$239$		−	22.1206i		−	1.43086i	0.698682	−	0.715432i	$-0.253770\pi$
$240$		0			0
$241$	−5.87308	−	3.39083i	−0.378318	−	0.218422i	0.298768	−	0.954326i	$-0.403424\pi$
$241$	−5.87308	−	3.39083i	−0.378318	−	0.218422i	−0.677086	+	0.735904i	$0.736758\pi$
$242$		−	9.33069i		−	0.599800i
$243$		0			0
$244$	2.64171	+	4.57558i	0.169118	+	0.292922i
$245$	1.16793	−	0.674306i	0.0746165	−	0.0430798i
$246$		0			0
$247$	−4.80366	+	6.74454i	−0.305649	+	0.429145i
$248$	9.90596			0.629029
$249$		0			0
$250$	5.51666	+	9.55514i	0.348904	+	0.604320i
$251$	−0.613386	+	1.06242i	−0.0387166	+	0.0670591i	−0.884734	−	0.466096i	$-0.845660\pi$
$251$	−0.613386	+	1.06242i	−0.0387166	+	0.0670591i	0.846018	+	0.533155i	$0.178994\pi$
$252$		0			0
$253$	6.89384	+	3.98016i	0.433412	+	0.250231i
$254$	9.82157	+	5.67048i	0.616260	+	0.355798i
$255$		0			0
$256$	−0.500000	+	0.866025i	−0.0312500	+	0.0541266i
$257$	8.94177	+	15.4876i	0.557772	+	0.966090i	0.997682	+	0.0680481i	$0.0216771\pi$
$257$	8.94177	+	15.4876i	0.557772	+	0.966090i	−0.439910	+	0.898042i	$0.644990\pi$
$258$		0			0
$259$	−4.29115			−0.266639
$260$	2.82087	−	3.96062i	0.174943	−	0.245627i
$261$		0			0
$262$	−11.1336	+	6.42801i	−0.687838	+	0.397124i
$263$	−3.47009	−	6.01037i	−0.213975	−	0.370615i	0.738980	−	0.673727i	$-0.235308\pi$
$263$	−3.47009	−	6.01037i	−0.213975	−	0.370615i	−0.952955	+	0.303112i	$0.901974\pi$
$264$		0			0
$265$			4.46272i			0.274143i
$266$	1.98887	+	1.14828i	0.121946	+	0.0704053i
$267$		0			0
$268$			9.07569i			0.554386i
$269$	3.84257	−	6.65552i	0.234285	−	0.405794i	−0.724779	−	0.688981i	$-0.758058\pi$
$269$	3.84257	−	6.65552i	0.234285	−	0.405794i	0.959065	+	0.283187i	$0.0913916\pi$
$270$		0			0
$271$	4.06286	−	2.34569i	0.246801	−	0.142491i	−0.371498	−	0.928434i	$-0.621156\pi$
$271$	4.06286	−	2.34569i	0.246801	−	0.142491i	0.618299	+	0.785943i	$0.287822\pi$
$272$	3.53111			0.214105
$273$		0			0
$274$	11.5527			0.697924
$275$	−3.55956	+	2.05511i	−0.214650	+	0.123928i
$276$		0			0
$277$	−11.7976	+	20.4340i	−0.708847	+	1.22776i	0.256438	+	0.966561i	$0.417451\pi$
$277$	−11.7976	+	20.4340i	−0.708847	+	1.22776i	−0.965285	+	0.261198i	$0.915882\pi$
$278$			8.58415i			0.514843i
$279$		0			0
$280$	−1.16793	−	0.674306i	−0.0697973	−	0.0402975i
$281$		−	9.32765i		−	0.556441i	−0.960517	−	0.278221i	$-0.910255\pi$
$281$		−	9.32765i		−	0.556441i	0.960517	−	0.278221i	$-0.0897446\pi$
$282$		0			0
$283$	−8.08625	−	14.0058i	−0.480678	−	0.832558i	0.519077	−	0.854728i	$-0.326276\pi$
$283$	−8.08625	−	14.0058i	−0.480678	−	0.832558i	−0.999754	+	0.0221698i	$0.992943\pi$
$284$	−6.89175	+	3.97895i	−0.408950	+	0.236107i
$285$		0			0
$286$	3.79441	+	2.70249i	0.224368	+	0.159802i
$287$	7.90142			0.466406
$288$		0			0
$289$	2.26564	+	3.92420i	0.133273	+	0.230835i
$290$	5.54515	−	9.60448i	0.325622	−	0.563994i
$291$		0			0
$292$	2.84202	+	1.64084i	0.166317	+	0.0960231i
$293$	23.9023	+	13.8000i	1.39639	+	0.806203i	0.994012	−	0.109272i	$-0.0348520\pi$
$293$	23.9023	+	13.8000i	1.39639	+	0.806203i	0.402373	+	0.915476i	$0.368185\pi$
$294$		0			0
$295$	−7.32187	+	12.6819i	−0.426296	+	0.738366i
$296$	2.14558	+	3.71625i	0.124709	+	0.216002i
$297$		0			0
$298$	10.1231			0.586413
$299$	20.2077	−	9.22642i	1.16864	−	0.533577i
$300$		0			0
$301$	10.8539	−	6.26649i	0.625607	−	0.361194i
$302$	−1.93295	−	3.34797i	−0.111229	−	0.192654i
$303$		0			0
$304$		−	2.29655i		−	0.131716i
$305$	6.17068	+	3.56265i	0.353332	+	0.203996i
$306$		0			0
$307$		−	15.9372i		−	0.909584i	−0.890598	−	0.454792i	$-0.849714\pi$
$307$		−	15.9372i		−	0.909584i	0.890598	−	0.454792i	$-0.150286\pi$
$308$	0.646009	−	1.11892i	0.0368098	−	0.0637564i
$309$		0			0
$310$	11.5695	−	6.67965i	0.657103	−	0.379378i
$311$	−8.56954			−0.485934			−0.242967	−	0.970035i	$-0.578121\pi$
$311$	−8.56954			−0.485934			−0.242967	+	0.970035i	$0.578121\pi$
$312$		0			0
$313$	19.1599			1.08298			0.541490	−	0.840707i	$-0.317860\pi$
$313$	19.1599			1.08298			0.541490	+	0.840707i	$0.317860\pi$
$314$	−4.08674	+	2.35948i	−0.230628	+	0.133153i
$315$		0			0
$316$	−1.66318	+	2.88072i	−0.0935613	+	0.162053i
$317$		−	4.23963i		−	0.238121i	−0.992887	−	0.119061i	$-0.962012\pi$
$317$		−	4.23963i		−	0.238121i	0.992887	−	0.119061i	$-0.0379883\pi$
$318$		0			0
$319$	9.20143	+	5.31245i	0.515181	+	0.297440i
$320$			1.34861i			0.0753897i
$321$		0			0
$322$	−3.08058	−	5.33572i	−0.171674	−	0.297348i
$323$	−7.02292	+	4.05469i	−0.390766	+	0.225609i
$324$		0			0
$325$	−1.09130	+	11.4181i	−0.0605347	+	0.633363i
$326$	23.4190			1.29706
$327$		0			0
$328$	−3.95071	−	6.84283i	−0.218141	−	0.377832i
$329$	5.35313	−	9.27189i	0.295128	−	0.511176i
$330$		0			0
$331$	−0.0541701	−	0.0312751i	−0.00297746	−	0.00171904i	0.498511	−	0.866884i	$-0.333881\pi$
$331$	−0.0541701	−	0.0312751i	−0.00297746	−	0.00171904i	−0.501488	+	0.865165i	$0.667214\pi$
$332$	−0.633224	−	0.365592i	−0.0347527	−	0.0200645i
$333$		0			0
$334$	−12.1736	+	21.0853i	−0.666110	+	1.15374i
$335$	6.11979	+	10.5998i	0.334360	+	0.579128i
$336$		0			0
$337$	−10.3047			−0.561334			−0.280667	−	0.959805i	$-0.590556\pi$
$337$	−10.3047			−0.561334			−0.280667	+	0.959805i	$0.590556\pi$
$338$	12.2858	−	4.24974i	0.668257	−	0.231155i
$339$		0			0
$340$	4.12410	−	2.38105i	0.223660	−	0.129130i
$341$	6.39933	+	11.0840i	0.346543	+	0.600231i
$342$		0			0
$343$		−	1.00000i		−	0.0539949i
$344$	−10.8539	−	6.26649i	−0.585202	−	0.337866i
$345$		0			0
$346$			18.1276i			0.974544i
$347$	9.05710	−	15.6874i	0.486210	−	0.842141i	−0.513664	−	0.857991i	$-0.671712\pi$
$347$	9.05710	−	15.6874i	0.486210	−	0.842141i	0.999874	+	0.0158503i	$0.00504551\pi$
$348$		0			0
$349$	2.21514	−	1.27891i	0.118574	−	0.0684587i	−0.439540	−	0.898223i	$-0.644859\pi$
$349$	2.21514	−	1.27891i	0.118574	−	0.0684587i	0.558114	+	0.829764i	$0.311525\pi$
$350$	3.18125			0.170045
$351$		0			0
$352$	−1.29202			−0.0688648
$353$	−5.37541	+	3.10350i	−0.286104	+	0.165182i	−0.636184	−	0.771538i	$-0.719488\pi$
$353$	−5.37541	+	3.10350i	−0.286104	+	0.165182i	0.350079	+	0.936720i	$0.386155\pi$
$354$		0			0
$355$	−5.36606	+	9.29430i	−0.284801	+	0.493290i
$356$			13.3063i			0.705231i
$357$		0			0
$358$	−15.1660	−	8.75612i	−0.801550	−	0.462775i
$359$		−	7.92186i		−	0.418100i	−0.977905	−	0.209050i	$-0.932963\pi$
$359$		−	7.92186i		−	0.418100i	0.977905	−	0.209050i	$-0.0670371\pi$
$360$		0			0
$361$	−6.86292	−	11.8869i	−0.361207	−	0.625628i
$362$	21.2731	−	12.2820i	1.11809	−	0.645528i
$363$		0			0
$364$	−1.49751	−	3.27986i	−0.0784910	−	0.171911i
$365$	4.42572			0.231653
$366$		0			0
$367$	7.99513	+	13.8480i	0.417342	+	0.722858i	0.995671	−	0.0929458i	$-0.0296283\pi$
$367$	7.99513	+	13.8480i	0.417342	+	0.722858i	−0.578329	+	0.815804i	$0.696295\pi$
$368$	−3.08058	+	5.33572i	−0.160586	+	0.278144i
$369$		0			0
$370$	5.01177	+	2.89355i	0.260550	+	0.150428i
$371$	2.86579	+	1.65456i	0.148784	+	0.0859006i
$372$		0			0
$373$	−7.59496	+	13.1549i	−0.393252	+	0.681132i	−0.992876	−	0.119149i	$-0.961983\pi$
$373$	−7.59496	+	13.1549i	−0.393252	+	0.681132i	0.599624	+	0.800282i	$0.295317\pi$
$374$	2.28113	+	3.95103i	0.117954	+	0.204303i
$375$		0			0
$376$	−10.7063			−0.552133
$377$	26.9719	−	12.3148i	1.38912	−	0.634244i
$378$		0			0
$379$	11.8872	−	6.86310i	0.610607	−	0.352534i	−0.162596	−	0.986693i	$-0.551987\pi$
$379$	11.8872	−	6.86310i	0.610607	−	0.352534i	0.773203	+	0.634159i	$0.218653\pi$
$380$	−1.54858	−	2.68222i	−0.0794404	−	0.137595i
$381$		0			0
$382$		−	13.2813i		−	0.679531i
$383$	−0.366939	−	0.211852i	−0.0187497	−	0.0108251i	0.490596	−	0.871387i	$-0.336779\pi$
$383$	−0.366939	−	0.211852i	−0.0187497	−	0.0108251i	−0.509346	+	0.860562i	$0.670112\pi$
$384$		0			0
$385$		−	1.74243i		−	0.0888025i
$386$	−10.9694	+	18.9996i	−0.558329	+	0.967054i
$387$		0			0
$388$	−10.5806	+	6.10870i	−0.537147	+	0.310122i
$389$	23.3287			1.18281			0.591406	−	0.806374i	$-0.298573\pi$
$389$	23.3287			1.18281			0.591406	+	0.806374i	$0.298573\pi$
$390$		0			0
$391$	21.7557			1.10023
$392$	−0.866025	+	0.500000i	−0.0437409	+	0.0252538i
$393$		0			0
$394$	7.58594	−	13.1392i	0.382174	−	0.661945i
$395$			4.48597i			0.225714i
$396$		0			0
$397$	22.9083	+	13.2261i	1.14973	+	0.663800i	0.948823	−	0.315810i	$-0.102276\pi$
$397$	22.9083	+	13.2261i	1.14973	+	0.663800i	0.200912	+	0.979609i	$0.435609\pi$
$398$			14.0277i			0.703143i
$399$		0			0
$400$	−1.59062	−	2.75504i	−0.0795311	−	0.137752i
$401$	21.0649	−	12.1618i	1.05193	−	0.607333i	0.128742	−	0.991678i	$-0.458906\pi$
$401$	21.0649	−	12.1618i	1.05193	−	0.607333i	0.923189	+	0.384346i	$0.125573\pi$
$402$		0			0
$403$	35.5544	+	3.39817i	1.77109	+	0.169275i
$404$	12.6581			0.629764
$405$		0			0
$406$	−4.11174	−	7.12175i	−0.204062	−	0.353446i
$407$	−2.77212	+	4.80145i	−0.137409	+	0.237999i
$408$		0			0
$409$	−4.16586	−	2.40516i	−0.205989	−	0.118928i	0.393457	−	0.919343i	$-0.371279\pi$
$409$	−4.16586	−	2.40516i	−0.205989	−	0.118928i	−0.599446	+	0.800415i	$0.704612\pi$
$410$	−9.22832	−	5.32798i	−0.455754	−	0.263130i
$411$		0			0
$412$	5.90210	−	10.2227i	0.290776	−	0.503638i
$413$	5.42919	+	9.40363i	0.267153	+	0.462723i
$414$		0			0
$415$	−0.986084			−0.0484049
$416$	−2.09168	+	2.93681i	−0.102553	+	0.143989i
$417$		0			0
$418$	2.56966	−	1.48359i	0.125686	−	0.0725649i
$419$	3.32439	+	5.75801i	0.162407	+	0.281297i	0.935731	−	0.352713i	$-0.114741\pi$
$419$	3.32439	+	5.75801i	0.162407	+	0.281297i	−0.773324	+	0.634011i	$0.781408\pi$
$420$		0			0
$421$		−	15.2834i		−	0.744869i	−0.928059	−	0.372434i	$-0.878523\pi$
$421$		−	15.2834i		−	0.744869i	0.928059	−	0.372434i	$-0.121477\pi$
$422$	12.4113	+	7.16564i	0.604171	+	0.348818i
$423$		0			0
$424$		−	3.30912i		−	0.160705i
$425$	−5.61666	+	9.72834i	−0.272448	+	0.471894i
$426$		0			0
$427$	4.57558	−	2.64171i	0.221428	−	0.127841i
$428$	10.2975			0.497751
$429$		0			0
$430$	−16.9021			−0.815092
$431$	−31.5708	+	18.2274i	−1.52071	+	0.877983i	−0.521009	+	0.853551i	$0.674444\pi$
$431$	−31.5708	+	18.2274i	−1.52071	+	0.877983i	−0.999701	+	0.0244320i	$0.992222\pi$
$432$		0			0
$433$	−0.457951	+	0.793194i	−0.0220077	+	0.0381185i	−0.876820	−	0.480820i	$-0.840339\pi$
$433$	−0.457951	+	0.793194i	−0.0220077	+	0.0381185i	0.854812	+	0.518938i	$0.173673\pi$
$434$		−	9.90596i		−	0.475501i
$435$		0			0
$436$	16.3571	+	9.44378i	0.783363	+	0.452275i
$437$		−	14.1494i		−	0.676858i
$438$		0			0
$439$	−13.3296	−	23.0876i	−0.636188	−	1.10191i	−0.986262	−	0.165189i	$-0.947177\pi$
$439$	−13.3296	−	23.0876i	−0.636188	−	1.10191i	0.350074	−	0.936722i	$-0.386157\pi$
$440$	−1.50899	+	0.871215i	−0.0719382	+	0.0415336i
$441$		0			0
$442$	12.6738	+	1.21132i	0.602833	+	0.0576167i
$443$	28.1700			1.33840			0.669199	−	0.743083i	$-0.266637\pi$
$443$	28.1700			1.33840			0.669199	+	0.743083i	$0.266637\pi$
$444$		0			0
$445$	8.97250	+	15.5408i	0.425337	+	0.736705i
$446$	6.13382	−	10.6241i	0.290445	−	0.503065i
$447$		0			0
$448$	0.866025	+	0.500000i	0.0409159	+	0.0236228i
$449$	−22.1050	−	12.7623i	−1.04320	−	0.602290i	−0.122460	−	0.992473i	$-0.539078\pi$
$449$	−22.1050	−	12.7623i	−1.04320	−	0.602290i	−0.920737	+	0.390183i	$0.872412\pi$
$450$		0			0
$451$	5.10439	−	8.84106i	0.240356	−	0.416309i
$452$	6.42036	+	11.1204i	0.301988	+	0.523059i
$453$		0			0
$454$	16.8467			0.790656
$455$	−3.96062	−	2.82087i	−0.185677	−	0.132244i
$456$		0			0
$457$	−30.7007	+	17.7251i	−1.43612	+	0.829143i	−0.997577	−	0.0695681i	$-0.977838\pi$
$457$	−30.7007	+	17.7251i	−1.43612	+	0.829143i	−0.438541	+	0.898711i	$0.644505\pi$
$458$	−4.59477	−	7.95837i	−0.214699	−	0.371870i
$459$		0			0
$460$			8.30901i			0.387410i
$461$	3.16691	+	1.82842i	0.147498	+	0.0851579i	0.571933	−	0.820301i	$-0.306194\pi$
$461$	3.16691	+	1.82842i	0.147498	+	0.0851579i	−0.424435	+	0.905458i	$0.639527\pi$
$462$		0			0
$463$		−	2.86203i		−	0.133010i	−0.997786	−	0.0665048i	$-0.978815\pi$
$463$		−	2.86203i		−	0.133010i	0.997786	−	0.0665048i	$-0.0211848\pi$
$464$	−4.11174	+	7.12175i	−0.190883	+	0.330619i
$465$		0			0
$466$	−21.7495	+	12.5571i	−1.00753	+	0.581697i
$467$	12.4480			0.576024			0.288012	−	0.957627i	$-0.407006\pi$
$467$	12.4480			0.576024			0.288012	+	0.957627i	$0.407006\pi$
$468$		0			0
$469$	9.07569			0.419076
$470$	−12.5042	+	7.21929i	−0.576775	+	0.333001i
$471$		0			0
$472$	5.42919	−	9.40363i	0.249899	−	0.432837i
$473$		−	16.1928i		−	0.744547i
$474$		0			0
$475$	6.32709	+	3.65295i	0.290307	+	0.167609i
$476$		−	3.53111i		−	0.161848i
$477$		0			0
$478$	11.0603	+	19.1570i	0.505887	+	0.876222i
$479$	2.49108	−	1.43822i	0.113820	−	0.0657141i	−0.442009	−	0.897011i	$-0.645734\pi$
$479$	2.49108	−	1.43822i	0.113820	−	0.0657141i	0.555829	+	0.831296i	$0.312401\pi$
$480$		0			0
$481$	6.42606	+	14.0744i	0.293003	+	0.641735i
$482$	6.78165			0.308896
$483$		0			0
$484$	4.66535	+	8.08062i	0.212061	+	0.367301i
$485$	−8.23826	+	14.2691i	−0.374080	+	0.647926i
$486$		0			0
$487$	−6.74495	−	3.89420i	−0.305643	−	0.176463i	0.339332	−	0.940667i	$-0.389799\pi$
$487$	−6.74495	−	3.89420i	−0.305643	−	0.176463i	−0.644975	+	0.764204i	$0.723132\pi$
$488$	−4.57558	−	2.64171i	−0.207127	−	0.119585i
$489$		0			0
$490$	−0.674306	+	1.16793i	−0.0304620	+	0.0527618i
$491$	−1.14360	−	1.98078i	−0.0516102	−	0.0893914i	0.839066	−	0.544029i	$-0.183102\pi$
$491$	−1.14360	−	1.98078i	−0.0516102	−	0.0893914i	−0.890676	+	0.454638i	$0.849769\pi$
$492$		0			0
$493$	29.0380			1.30781
$494$	0.787817	−	8.24277i	0.0354455	−	0.370860i
$495$		0			0
$496$	−8.57881	+	4.95298i	−0.385200	+	0.222395i
$497$	3.97895	+	6.89175i	0.178480	+	0.309137i
$498$		0			0
$499$		−	28.0079i		−	1.25381i	−0.779097	−	0.626903i	$-0.784322\pi$
$499$		−	28.0079i		−	1.25381i	0.779097	−	0.626903i	$-0.215678\pi$
$500$	−9.55514	−	5.51666i	−0.427319	−	0.246713i
$501$		0			0
$502$		−	1.22677i		−	0.0547535i
$503$	−22.0810	+	38.2454i	−0.984543	+	1.70528i	−0.340594	+	0.940210i	$0.610628\pi$
$503$	−22.0810	+	38.2454i	−0.984543	+	1.70528i	−0.643949	+	0.765069i	$0.722705\pi$
$504$		0			0
$505$	14.7838	−	8.53543i	0.657870	−	0.379822i
$506$	−7.96032			−0.353880
$507$		0			0
$508$	−11.3410			−0.503174
$509$	13.3612	−	7.71409i	0.592225	−	0.341921i	−0.173752	−	0.984789i	$-0.555589\pi$
$509$	13.3612	−	7.71409i	0.592225	−	0.341921i	0.765977	+	0.642868i	$0.222256\pi$
$510$		0			0
$511$	1.64084	−	2.84202i	0.0725866	−	0.125724i
$512$		−	1.00000i		−	0.0441942i
$513$		0			0
$514$	−15.4876	−	8.94177i	−0.683129	−	0.394405i
$515$		−	15.9193i		−	0.701488i
$516$		0			0
$517$	−6.91634	−	11.9794i	−0.304180	−	0.526856i
$518$	3.71625	−	2.14558i	0.163282	−	0.0942712i
$519$		0			0
$520$	−0.462632	+	4.84043i	−0.0202878	+	0.212267i
$521$	−24.0874			−1.05529			−0.527643	−	0.849466i	$-0.676924\pi$
$521$	−24.0874			−1.05529			−0.527643	+	0.849466i	$0.676924\pi$
$522$		0			0
$523$	−10.2046	−	17.6748i	−0.446215	−	0.772866i	0.551921	−	0.833896i	$-0.313895\pi$
$523$	−10.2046	−	17.6748i	−0.446215	−	0.772866i	−0.998136	+	0.0610299i	$0.980561\pi$
$524$	6.42801	−	11.1336i	0.280809	−	0.486375i
$525$		0			0
$526$	6.01037	+	3.47009i	0.262065	+	0.151303i
$527$	30.2927	+	17.4895i	1.31957	+	0.761855i
$528$		0			0
$529$	−7.47993	+	12.9556i	−0.325214	+	0.563288i
$530$	−2.23136	−	3.86483i	−0.0969242	−	0.167878i
$531$		0			0
$532$	−2.29655			−0.0995682
$533$	−11.8325	−	25.9155i	−0.512522	−	1.12253i
$534$		0			0
$535$	12.0268	−	6.94370i	0.519966	−	0.300202i
$536$	−4.53784	−	7.85978i	−0.196005	−	0.339491i
$537$		0			0
$538$			7.68513i			0.331330i
$539$	−1.11892	−	0.646009i	−0.0481953	−	0.0278256i
$540$		0			0
$541$		−	0.995612i		−	0.0428047i	−0.999771	−	0.0214024i	$-0.993187\pi$
$541$		−	0.995612i		−	0.0428047i	0.999771	−	0.0214024i	$-0.00681310\pi$
$542$	−2.34569	+	4.06286i	−0.100756	+	0.174515i
$543$		0			0
$544$	−3.05803	+	1.76555i	−0.131112	+	0.0756975i
$545$	25.4720			1.09110
$546$		0			0
$547$	10.8590			0.464296			0.232148	−	0.972680i	$-0.425425\pi$
$547$	10.8590			0.464296			0.232148	+	0.972680i	$0.425425\pi$
$548$	−10.0049	+	5.77635i	−0.427390	+	0.246753i
$549$		0			0
$550$	2.05511	−	3.55956i	0.0876303	−	0.151780i
$551$		−	18.8857i		−	0.804557i
$552$		0			0
$553$	2.88072	+	1.66318i	0.122501	+	0.0707257i
$554$		−	23.5951i		−	1.00246i
$555$		0			0
$556$	−4.29208	−	7.43409i	−0.182025	−	0.315276i
$557$	−14.4876	+	8.36441i	−0.613859	+	0.354411i	−0.774474	−	0.632606i	$-0.781985\pi$
$557$	−14.4876	+	8.36441i	−0.613859	+	0.354411i	0.160616	+	0.987017i	$0.448652\pi$
$558$		0			0
$559$	−36.8070	−	26.2150i	−1.55677	−	1.10878i
$560$	1.34861			0.0569893
$561$		0			0
$562$	4.66383	+	8.07798i	0.196732	+	0.340749i
$563$	12.0094	−	20.8010i	0.506138	−	0.876656i	−0.493837	−	0.869555i	$-0.664406\pi$
$563$	12.0094	−	20.8010i	0.506138	−	0.876656i	0.999975	−	0.00710189i	$-0.00226062\pi$
$564$		0			0
$565$	14.9971	+	8.65857i	0.630932	+	0.364269i
$566$	14.0058	+	8.08625i	0.588707	+	0.339890i
$567$		0			0
$568$	3.97895	−	6.89175i	0.166953	−	0.289171i
$569$	21.4551	+	37.1613i	0.899445	+	1.55788i	0.828205	+	0.560425i	$0.189362\pi$
$569$	21.4551	+	37.1613i	0.899445	+	1.55788i	0.0712394	+	0.997459i	$0.477305\pi$
$570$		0			0
$571$	−10.7484			−0.449807			−0.224904	−	0.974381i	$-0.572207\pi$
$571$	−10.7484			−0.449807			−0.224904	+	0.974381i	$0.572207\pi$
$572$	−4.63730	−	0.443218i	−0.193895	−	0.0185319i
$573$		0			0
$574$	−6.84283	+	3.95071i	−0.285614	+	0.164899i
$575$	−9.80008	−	16.9742i	−0.408691	−	0.707874i
$576$		0			0
$577$		−	3.70579i		−	0.154274i	−0.997020	−	0.0771371i	$-0.975422\pi$
$577$		−	3.70579i		−	0.154274i	0.997020	−	0.0771371i	$-0.0245779\pi$
$578$	−3.92420	−	2.26564i	−0.163225	−	0.0942380i
$579$		0			0
$580$			11.0903i			0.460499i
$581$	−0.365592	+	0.633224i	−0.0151673	+	0.0262706i
$582$		0			0
$583$	3.70265	−	2.13772i	0.153348	−	0.0885354i
$584$	−3.28169			−0.135797
$585$		0			0
$586$	−27.6000			−1.14014
$587$	14.3886	−	8.30727i	0.593881	−	0.342878i	−0.172749	−	0.984966i	$-0.555265\pi$
$587$	14.3886	−	8.30727i	0.593881	−	0.342878i	0.766631	+	0.642088i	$0.221932\pi$
$588$		0			0
$589$	11.3748	−	19.7017i	0.468689	−	0.811794i
$590$		−	14.6437i		−	0.602873i
$591$		0			0
$592$	−3.71625	−	2.14558i	−0.152737	−	0.0881826i
$593$			12.2152i			0.501620i	0.968036	+	0.250810i	$0.0806969\pi$
$593$			12.2152i			0.501620i	−0.968036	+	0.250810i	$0.919303\pi$
$594$		0			0
$595$	−2.38105	−	4.12410i	−0.0976134	−	0.169071i
$596$	−8.76683	+	5.06153i	−0.359103	+	0.207328i
$597$		0			0
$598$	−12.8872	+	18.0942i	−0.526996	+	0.739925i
$599$	−32.8665			−1.34289			−0.671443	−	0.741056i	$-0.734325\pi$
$599$	−32.8665			−1.34289			−0.671443	+	0.741056i	$0.734325\pi$
$600$		0			0
$601$	14.4802	+	25.0805i	0.590661	+	1.02306i	0.994144	+	0.108068i	$0.0344664\pi$
$601$	14.4802	+	25.0805i	0.590661	+	1.02306i	−0.403482	+	0.914988i	$0.632200\pi$
$602$	−6.26649	+	10.8539i	−0.255403	+	0.442371i
$603$		0			0
$604$	3.34797	+	1.93295i	0.136227	+	0.0786507i
$605$	10.8976	+	6.29174i	0.443051	+	0.255796i
$606$		0			0
$607$	10.6364	−	18.4227i	0.431716	−	0.747755i	−0.565305	−	0.824882i	$-0.691241\pi$
$607$	10.6364	−	18.4227i	0.431716	−	0.747755i	0.997021	+	0.0771273i	$0.0245748\pi$
$608$	1.14828	+	1.98887i	0.0465687	+	0.0806594i
$609$		0			0
$610$	−7.12529			−0.288495
$611$	−38.4269	−	3.67271i	−1.55458	−	0.148582i
$612$		0			0
$613$	16.4098	−	9.47422i	0.662787	−	0.382660i	−0.130551	−	0.991442i	$-0.541675\pi$
$613$	16.4098	−	9.47422i	0.662787	−	0.382660i	0.793338	+	0.608781i	$0.208341\pi$
$614$	7.96860	+	13.8020i	0.321586	+	0.557004i
$615$		0			0
$616$			1.29202i			0.0520569i
$617$	−36.0364	−	20.8056i	−1.45077	−	0.837602i	−0.452244	−	0.891894i	$-0.649377\pi$
$617$	−36.0364	−	20.8056i	−1.45077	−	0.837602i	−0.998525	+	0.0542919i	$0.982710\pi$
$618$		0			0
$619$			33.3758i			1.34149i	0.741690	+	0.670743i	$0.234025\pi$
$619$			33.3758i			1.34149i	−0.741690	+	0.670743i	$0.765975\pi$
$620$	−6.67965	+	11.5695i	−0.268261	+	0.464642i
$621$		0			0
$622$	7.42144	−	4.28477i	0.297572	−	0.171804i
$623$	13.3063			0.533104
$624$		0			0
$625$	1.02655			0.0410621
$626$	−16.5929	+	9.57994i	−0.663188	+	0.382892i
$627$		0			0
$628$	2.35948	−	4.08674i	0.0941535	−	0.163079i
$629$			15.1525i			0.604171i
$630$		0			0
$631$	−15.6685	−	9.04620i	−0.623752	−	0.360123i	0.154576	−	0.987981i	$-0.450599\pi$
$631$	−15.6685	−	9.04620i	−0.623752	−	0.360123i	−0.778328	+	0.627857i	$0.783932\pi$
$632$		−	3.32636i		−	0.132316i
$633$		0			0
$634$	2.11982	+	3.67163i	0.0841886	+	0.145819i
$635$	−13.2455	+	7.64728i	−0.525631	+	0.303473i
$636$		0			0
$637$	−3.27986	+	1.49751i	−0.129953	+	0.0593336i
$638$	−10.6249			−0.420643
$639$		0			0
$640$	−0.674306	−	1.16793i	−0.0266543	−	0.0461666i
$641$	−4.70086	+	8.14213i	−0.185673	+	0.321595i	−0.943803	−	0.330508i	$-0.892780\pi$
$641$	−4.70086	+	8.14213i	−0.185673	+	0.321595i	0.758130	+	0.652103i	$0.226113\pi$
$642$		0			0
$643$	−35.7323	−	20.6300i	−1.40914	−	0.813569i	−0.413837	−	0.910351i	$-0.635812\pi$
$643$	−35.7323	−	20.6300i	−1.40914	−	0.813569i	−0.995306	+	0.0967822i	$0.969145\pi$
$644$	5.33572	+	3.08058i	0.210257	+	0.121392i
$645$		0			0
$646$	4.05469	−	7.02292i	0.159530	−	0.276313i
$647$	9.47607	+	16.4130i	0.372543	+	0.645263i	0.989956	−	0.141376i	$-0.0451527\pi$
$647$	9.47607	+	16.4130i	0.372543	+	0.645263i	−0.617413	+	0.786639i	$0.711819\pi$
$648$		0			0
$649$	14.0292			0.550695
$650$	−4.76396	−	10.4340i	−0.186858	−	0.409256i
$651$		0			0
$652$	−20.2815	+	11.7095i	−0.794284	+	0.458580i
$653$	2.84907	+	4.93473i	0.111493	+	0.193111i	0.916372	−	0.400327i	$-0.131104\pi$
$653$	2.84907	+	4.93473i	0.111493	+	0.193111i	−0.804880	+	0.593438i	$0.797770\pi$
$654$		0			0
$655$		−	17.3378i		−	0.677443i
$656$	6.84283	+	3.95071i	0.267168	+	0.154249i
$657$		0			0
$658$			10.7063i			0.417373i
$659$	−2.06833	+	3.58246i	−0.0805709	+	0.139553i	−0.903495	−	0.428598i	$-0.859008\pi$
$659$	−2.06833	+	3.58246i	−0.0805709	+	0.139553i	0.822924	+	0.568151i	$0.192341\pi$
$660$		0			0
$661$	−0.636113	+	0.367260i	−0.0247419	+	0.0142848i	−0.512320	−	0.858795i	$-0.671214\pi$
$661$	−0.636113	+	0.367260i	−0.0247419	+	0.0142848i	0.487578	+	0.873079i	$0.337880\pi$
$662$	0.0625502			0.00243108
$663$		0			0
$664$	0.731184			0.0283754
$665$	−2.68222	+	1.54858i	−0.104012	+	0.0600513i
$666$		0			0
$667$	−25.3331	+	43.8782i	−0.980902	+	1.69897i
$668$		−	24.3472i		−	0.942021i
$669$		0			0
$670$	−10.5998	−	6.11979i	−0.409506	−	0.236428i
$671$		−	6.82628i		−	0.263526i
$672$		0			0
$673$	9.17498	+	15.8915i	0.353669	+	0.612573i	0.986889	−	0.161399i	$-0.0516005\pi$
$673$	9.17498	+	15.8915i	0.353669	+	0.612573i	−0.633220	+	0.773972i	$0.718267\pi$
$674$	8.92415	−	5.15236i	0.343745	−	0.198461i
$675$		0			0
$676$	−8.51491	+	9.82326i	−0.327496	+	0.377818i
$677$	26.0206			1.00005			0.500026	−	0.866010i	$-0.333324\pi$
$677$	26.0206			1.00005			0.500026	+	0.866010i	$0.333324\pi$
$678$		0			0
$679$	6.10870	+	10.5806i	0.234430	+	0.406045i
$680$	−2.38105	+	4.12410i	−0.0913090	+	0.158152i
$681$		0			0
$682$	−11.0840	−	6.39933i	−0.424427	−	0.245043i
$683$	33.2640	+	19.2050i	1.27281	+	0.734859i	0.975516	−	0.219927i	$-0.0705818\pi$
$683$	33.2640	+	19.2050i	1.27281	+	0.734859i	0.297296	+	0.954785i	$0.403915\pi$
$684$		0			0
$685$	−7.79005	+	13.4928i	−0.297643	+	0.515532i
$686$	0.500000	+	0.866025i	0.0190901	+	0.0330650i
$687$		0			0
$688$	12.5330			0.477815
$689$	1.13517	−	11.8771i	0.0432466	−	0.452481i
$690$		0			0
$691$	4.37089	−	2.52354i	0.166277	−	0.0959998i	−0.414552	−	0.910025i	$-0.636062\pi$
$691$	4.37089	−	2.52354i	0.166277	−	0.0959998i	0.580829	+	0.814026i	$0.302729\pi$
$692$	−9.06378	−	15.6989i	−0.344553	−	0.596784i
$693$		0			0
$694$			18.1142i			0.687605i
$695$	−10.0257	−	5.78835i	−0.380297	−	0.219564i
$696$		0			0
$697$		−	27.9008i		−	1.05682i
$698$	−1.27891	+	2.21514i	−0.0484076	+	0.0838444i
$699$		0			0
$700$	−2.75504	+	1.59062i	−0.104131	+	0.0601199i
$701$	−41.3523			−1.56185			−0.780927	−	0.624623i	$-0.785253\pi$
$701$	−41.3523			−1.56185			−0.780927	+	0.624623i	$0.785253\pi$
$702$		0			0
$703$	9.85485			0.371683
$704$	1.11892	−	0.646009i	0.0421709	−	0.0243474i
$705$		0			0
$706$	3.10350	−	5.37541i	0.116802	−	0.202306i
$707$		−	12.6581i		−	0.476057i
$708$		0			0
$709$	28.7863	+	16.6198i	1.08109	+	0.624170i	0.931191	−	0.364531i	$-0.118771\pi$
$709$	28.7863	+	16.6198i	1.08109	+	0.624170i	0.149902	+	0.988701i	$0.452104\pi$
$710$		−	10.7321i		−	0.402770i
$711$		0			0
$712$	−6.65313	−	11.5236i	−0.249337	−	0.431864i
$713$	−52.8554	+	30.5161i	−1.97945	+	1.14284i
$714$		0			0
$715$	−5.71492	+	2.60931i	−0.213726	+	0.0975828i
$716$	17.5122			0.654463
$717$		0			0
$718$	3.96093	+	6.86053i	0.147821	+	0.256033i
$719$	−7.34020	+	12.7136i	−0.273743	+	0.474137i	−0.969817	−	0.243833i	$-0.921595\pi$
$719$	−7.34020	+	12.7136i	−0.273743	+	0.474137i	0.696074	+	0.717970i	$0.254929\pi$
$720$		0			0
$721$	−10.2227	−	5.90210i	−0.380715	−	0.219806i
$722$	11.8869	+	6.86292i	0.442386	+	0.255412i
$723$		0			0
$724$	−12.2820	+	21.2731i	−0.456457	+	0.790607i
$725$	−13.0805	−	22.6560i	−0.485796	−	0.841424i
$726$		0			0
$727$	25.4749			0.944812			0.472406	−	0.881381i	$-0.343386\pi$
$727$	25.4749			0.944812			0.472406	+	0.881381i	$0.343386\pi$
$728$	2.93681	+	2.09168i	0.108846	+	0.0775229i
$729$		0			0
$730$	−3.83279	+	2.21286i	−0.141858	+	0.0819017i
$731$	−22.1276	−	38.3262i	−0.818420	−	1.41755i
$732$		0			0
$733$		−	13.7814i		−	0.509027i	−0.967069	−	0.254514i	$-0.918085\pi$
$733$		−	13.7814i		−	0.509027i	0.967069	−	0.254514i	$-0.0819154\pi$
$734$	−13.8480	−	7.99513i	−0.511138	−	0.295105i
$735$		0			0
$736$		−	6.16116i		−	0.227103i
$737$	5.86298	−	10.1550i	0.215965	−	0.374063i
$738$		0			0
$739$	29.1960	−	16.8563i	1.07399	−	0.620069i	0.144722	−	0.989472i	$-0.453771\pi$
$739$	29.1960	−	16.8563i	1.07399	−	0.620069i	0.929269	+	0.369403i	$0.120438\pi$
$740$	−5.78710			−0.212738
$741$		0			0
$742$	−3.30912			−0.121482
$743$	−25.1238	+	14.5052i	−0.921701	+	0.532144i	−0.884177	−	0.467151i	$-0.845280\pi$
$743$	−25.1238	+	14.5052i	−0.921701	+	0.532144i	−0.0375238	+	0.999296i	$0.511947\pi$
$744$		0			0
$745$	−6.82604	+	11.8231i	−0.250087	+	0.433163i
$746$		−	15.1899i		−	0.556142i
$747$		0			0
$748$	−3.95103	−	2.28113i	−0.144464	−	0.0834063i
$749$		−	10.2975i		−	0.376264i
$750$		0			0
$751$	−13.9330	−	24.1326i	−0.508422	−	0.880613i	−0.999952	−	0.00975221i	$-0.996896\pi$
$751$	−13.9330	−	24.1326i	−0.508422	−	0.880613i	0.491531	−	0.870860i	$-0.336438\pi$
$752$	9.27189	−	5.35313i	0.338111	−	0.195209i
$753$		0			0
$754$	−17.2009	+	24.1508i	−0.626421	+	0.879522i
$755$	5.21360			0.189742
$756$		0			0
$757$	7.24779	+	12.5535i	0.263426	+	0.456266i	0.967150	−	0.254207i	$-0.0818143\pi$
$757$	7.24779	+	12.5535i	0.263426	+	0.456266i	−0.703724	+	0.710473i	$0.748481\pi$
$758$	−6.86310	+	11.8872i	−0.249279	+	0.431764i
$759$		0			0
$760$	2.68222	+	1.54858i	0.0972942	+	0.0561729i
$761$	8.33828	+	4.81411i	0.302262	+	0.174511i	0.643459	−	0.765481i	$-0.277499\pi$
$761$	8.33828	+	4.81411i	0.302262	+	0.174511i	−0.341196	+	0.939992i	$0.610832\pi$
$762$		0			0
$763$	9.44378	−	16.3571i	0.341888	−	0.592167i
$764$	6.64065	+	11.5020i	0.240250	+	0.416126i
$765$		0			0
$766$	0.423704			0.0153091
$767$	22.7123	−	31.8890i	0.820093	−	1.15145i
$768$		0			0
$769$	15.7328	−	9.08333i	0.567339	−	0.327553i	−0.188747	−	0.982026i	$-0.560443\pi$
$769$	15.7328	−	9.08333i	0.567339	−	0.327553i	0.756086	+	0.654473i	$0.227109\pi$
$770$	0.871215	+	1.50899i	0.0313964	+	0.0543802i
$771$		0			0
$772$		−	21.9388i		−	0.789596i
$773$	−18.5429	−	10.7058i	−0.666944	−	0.385060i	0.127974	−	0.991778i	$-0.459153\pi$
$773$	−18.5429	−	10.7058i	−0.666944	−	0.385060i	−0.794918	+	0.606718i	$0.792486\pi$
$774$		0			0
$775$		−	31.5133i		−	1.13199i
$776$	6.10870	−	10.5806i	0.219289	−	0.379820i
$777$		0			0
$778$	−20.2032	+	11.6643i	−0.724321	+	0.418187i
$779$	−18.1460			−0.650149
$780$		0			0
$781$	10.2818			0.367910
$782$	−18.8410	+	10.8779i	−0.673753	+

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:	\( N \)	\(=\)	\( 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1638.bj (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.866025 + 0.500000i) q^{2} +(0.500000 - 0.866025i) q^{4} -1.34861i q^{5} +(-0.866025 - 0.500000i) q^{7} +1.00000i q^{8} +O(q^{10})\)	\(q+(-0.866025 + 0.500000i) q^{2} +(0.500000 - 0.866025i) q^{4} -1.34861i q^{5} +(-0.866025 - 0.500000i) q^{7} +1.00000i q^{8} +(0.674306 + 1.16793i) q^{10} +(-1.11892 + 0.646009i) q^{11} +(-0.343043 + 3.58920i) q^{13} +1.00000 q^{14} +(-0.500000 - 0.866025i) q^{16} +(-1.76555 + 3.05803i) q^{17} +(1.98887 + 1.14828i) q^{19} +(-1.16793 - 0.674306i) q^{20} +(0.646009 - 1.11892i) q^{22} +(-3.08058 - 5.33572i) q^{23} +3.18125 q^{25} +(-1.49751 - 3.27986i) q^{26} +(-0.866025 + 0.500000i) q^{28} +(-4.11174 - 7.12175i) q^{29} -9.90596i q^{31} +(0.866025 + 0.500000i) q^{32} -3.53111i q^{34} +(-0.674306 + 1.16793i) q^{35} +(3.71625 - 2.14558i) q^{37} -2.29655 q^{38} +1.34861 q^{40} +(-6.84283 + 3.95071i) q^{41} +(-6.26649 + 10.8539i) q^{43} +1.29202i q^{44} +(5.33572 + 3.08058i) q^{46} +10.7063i q^{47} +(0.500000 + 0.866025i) q^{49} +(-2.75504 + 1.59062i) q^{50} +(2.93681 + 2.09168i) q^{52} -3.30912 q^{53} +(0.871215 + 1.50899i) q^{55} +(0.500000 - 0.866025i) q^{56} +(7.12175 + 4.11174i) q^{58} +(-9.40363 - 5.42919i) q^{59} +(-2.64171 + 4.57558i) q^{61} +(4.95298 + 8.57881i) q^{62} -1.00000 q^{64} +(4.84043 + 0.462632i) q^{65} +(-7.85978 + 4.53784i) q^{67} +(1.76555 + 3.05803i) q^{68} -1.34861i q^{70} +(-6.89175 - 3.97895i) q^{71} +3.28169i q^{73} +(-2.14558 + 3.71625i) q^{74} +(1.98887 - 1.14828i) q^{76} +1.29202 q^{77} -3.32636 q^{79} +(-1.16793 + 0.674306i) q^{80} +(3.95071 - 6.84283i) q^{82} -0.731184i q^{83} +(4.12410 + 2.38105i) q^{85} -12.5330i q^{86} +(-0.646009 - 1.11892i) q^{88} +(-11.5236 + 6.65313i) q^{89} +(2.09168 - 2.93681i) q^{91} -6.16116 q^{92} +(-5.35313 - 9.27189i) q^{94} +(1.54858 - 2.68222i) q^{95} +(-10.5806 - 6.10870i) q^{97} +(-0.866025 - 0.500000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 8 q^{4}+O(q^{10}) \) 16 * q + 8 * q^4	\( 16 q + 8 q^{4} + 2 q^{10} + 12 q^{11} + 10 q^{13} + 16 q^{14} - 8 q^{16} + 6 q^{17} - 4 q^{22} + 12 q^{23} - 20 q^{25} + 2 q^{26} - 16 q^{29} - 2 q^{35} - 6 q^{37} + 4 q^{40} - 12 q^{41} - 6 q^{43} + 6 q^{46} + 8 q^{49} + 24 q^{50} - 4 q^{52} + 40 q^{53} + 20 q^{55} + 8 q^{56} + 6 q^{58} - 6 q^{59} - 2 q^{61} + 14 q^{62} - 16 q^{64} + 52 q^{65} - 30 q^{67} - 6 q^{68} - 12 q^{71} - 24 q^{74} - 8 q^{77} - 16 q^{79} + 2 q^{82} + 6 q^{85} + 4 q^{88} - 30 q^{89} + 4 q^{91} + 24 q^{92} - 8 q^{94} + 40 q^{95} - 24 q^{97}+O(q^{100}) \) 16 * q + 8 * q^4 + 2 * q^10 + 12 * q^11 + 10 * q^13 + 16 * q^14 - 8 * q^16 + 6 * q^17 - 4 * q^22 + 12 * q^23 - 20 * q^25 + 2 * q^26 - 16 * q^29 - 2 * q^35 - 6 * q^37 + 4 * q^40 - 12 * q^41 - 6 * q^43 + 6 * q^46 + 8 * q^49 + 24 * q^50 - 4 * q^52 + 40 * q^53 + 20 * q^55 + 8 * q^56 + 6 * q^58 - 6 * q^59 - 2 * q^61 + 14 * q^62 - 16 * q^64 + 52 * q^65 - 30 * q^67 - 6 * q^68 - 12 * q^71 - 24 * q^74 - 8 * q^77 - 16 * q^79 + 2 * q^82 + 6 * q^85 + 4 * q^88 - 30 * q^89 + 4 * q^91 + 24 * q^92 - 8 * q^94 + 40 * q^95 - 24 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more