LMFDB - Embedded newform 1638.2.bj.a.127.1

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:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 546)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			127.1
Root			$0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	1638.127
Dual form			1638.2.bj.a.1135.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.866025 + 0.500000i) q^{2} +(0.500000 - 0.866025i) q^{4} -0.732051i q^{5} +(0.866025 + 0.500000i) q^{7} +1.00000i q^{8} +O(q^{10})$	\(q+(-0.866025 + 0.500000i) q^{2} +(0.500000 - 0.866025i) q^{4} -0.732051i q^{5} +(0.866025 + 0.500000i) q^{7} +1.00000i q^{8} +(0.366025 + 0.633975i) q^{10} +(-4.50000 + 2.59808i) q^{11} +(0.866025 - 3.50000i) q^{13} -1.00000 q^{14} +(-0.500000 - 0.866025i) q^{16} +(1.13397 - 1.96410i) q^{17} +(-0.401924 - 0.232051i) q^{19} +(-0.633975 - 0.366025i) q^{20} +(2.59808 - 4.50000i) q^{22} +(3.73205 + 6.46410i) q^{23} +4.46410 q^{25} +(1.00000 + 3.46410i) q^{26} +(0.866025 - 0.500000i) q^{28} +(-1.76795 - 3.06218i) q^{29} -3.26795i q^{31} +(0.866025 + 0.500000i) q^{32} +2.26795i q^{34} +(0.366025 - 0.633975i) q^{35} +(5.83013 - 3.36603i) q^{37} +0.464102 q^{38} +0.732051 q^{40} +(7.33013 - 4.23205i) q^{41} +(-4.36603 + 7.56218i) q^{43} +5.19615i q^{44} +(-6.46410 - 3.73205i) q^{46} -3.92820i q^{47} +(0.500000 + 0.866025i) q^{49} +(-3.86603 + 2.23205i) q^{50} +(-2.59808 - 2.50000i) q^{52} +9.92820 q^{53} +(1.90192 + 3.29423i) q^{55} +(-0.500000 + 0.866025i) q^{56} +(3.06218 + 1.76795i) q^{58} +(-7.73205 - 4.46410i) q^{59} +(-1.86603 + 3.23205i) q^{61} +(1.63397 + 2.83013i) q^{62} -1.00000 q^{64} +(-2.56218 - 0.633975i) q^{65} +(8.66025 - 5.00000i) q^{67} +(-1.13397 - 1.96410i) q^{68} +0.732051i q^{70} +(-6.29423 - 3.63397i) q^{71} -1.46410i q^{73} +(-3.36603 + 5.83013i) q^{74} +(-0.401924 + 0.232051i) q^{76} -5.19615 q^{77} +9.00000 q^{79} +(-0.633975 + 0.366025i) q^{80} +(-4.23205 + 7.33013i) q^{82} -9.26795i q^{83} +(-1.43782 - 0.830127i) q^{85} -8.73205i q^{86} +(-2.59808 - 4.50000i) q^{88} +(14.5981 - 8.42820i) q^{89} +(2.50000 - 2.59808i) q^{91} +7.46410 q^{92} +(1.96410 + 3.40192i) q^{94} +(-0.169873 + 0.294229i) q^{95} +(-12.2942 - 7.09808i) q^{97} +(-0.866025 - 0.500000i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{4}+O(q^{10}) $ 4 * q + 2 * q^4	$ 4 q + 2 q^{4} - 2 q^{10} - 18 q^{11} - 4 q^{14} - 2 q^{16} + 8 q^{17} - 12 q^{19} - 6 q^{20} + 8 q^{23} + 4 q^{25} + 4 q^{26} - 14 q^{29} - 2 q^{35} + 6 q^{37} - 12 q^{38} - 4 q^{40} + 12 q^{41} - 14 q^{43} - 12 q^{46} + 2 q^{49} - 12 q^{50} + 12 q^{53} + 18 q^{55} - 2 q^{56} - 12 q^{58} - 24 q^{59} - 4 q^{61} + 10 q^{62} - 4 q^{64} + 14 q^{65} - 8 q^{68} + 6 q^{71} - 10 q^{74} - 12 q^{76} + 36 q^{79} - 6 q^{80} - 10 q^{82} - 30 q^{85} + 48 q^{89} + 10 q^{91} + 16 q^{92} - 6 q^{94} - 18 q^{95} - 18 q^{97}+O(q^{100}) $ 4 * q + 2 * q^4 - 2 * q^10 - 18 * q^11 - 4 * q^14 - 2 * q^16 + 8 * q^17 - 12 * q^19 - 6 * q^20 + 8 * q^23 + 4 * q^25 + 4 * q^26 - 14 * q^29 - 2 * q^35 + 6 * q^37 - 12 * q^38 - 4 * q^40 + 12 * q^41 - 14 * q^43 - 12 * q^46 + 2 * q^49 - 12 * q^50 + 12 * q^53 + 18 * q^55 - 2 * q^56 - 12 * q^58 - 24 * q^59 - 4 * q^61 + 10 * q^62 - 4 * q^64 + 14 * q^65 - 8 * q^68 + 6 * q^71 - 10 * q^74 - 12 * q^76 + 36 * q^79 - 6 * q^80 - 10 * q^82 - 30 * q^85 + 48 * q^89 + 10 * q^91 + 16 * q^92 - 6 * q^94 - 18 * q^95 - 18 * 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
$5$		−	0.732051i		−	0.327383i	−0.986512	−	0.163692i	$-0.947660\pi$
$5$		−	0.732051i		−	0.327383i	0.986512	−	0.163692i	$-0.0523402\pi$
$6$		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
$10$	0.366025	+	0.633975i	0.115747	+	0.200480i
$11$	−4.50000	+	2.59808i	−1.35680	+	0.783349i	−0.989191	−	0.146631i	$-0.953157\pi$
$11$	−4.50000	+	2.59808i	−1.35680	+	0.783349i	−0.367610	+	0.929980i	$0.619824\pi$
$12$		0			0
$13$	0.866025	−	3.50000i	0.240192	−	0.970725i
$13$	0.866025	−	3.50000i	0.240192	−	0.970725i
$14$	−1.00000			−0.267261
$15$		0			0
$16$	−0.500000	−	0.866025i	−0.125000	−	0.216506i
$17$	1.13397	−	1.96410i	0.275029	−	0.476365i	−0.695113	−	0.718900i	$-0.744646\pi$
$17$	1.13397	−	1.96410i	0.275029	−	0.476365i	0.970143	+	0.242536i	$0.0779791\pi$
$18$		0			0
$19$	−0.401924	−	0.232051i	−0.0922076	−	0.0532361i	0.453187	−	0.891415i	$-0.350287\pi$
$19$	−0.401924	−	0.232051i	−0.0922076	−	0.0532361i	−0.545395	+	0.838179i	$0.683620\pi$
$20$	−0.633975	−	0.366025i	−0.141761	−	0.0818458i
$21$		0			0
$22$	2.59808	−	4.50000i	0.553912	−	0.959403i
$23$	3.73205	+	6.46410i	0.778186	+	1.34786i	0.932986	+	0.359912i	$0.117193\pi$
$23$	3.73205	+	6.46410i	0.778186	+	1.34786i	−0.154800	+	0.987946i	$0.549473\pi$
$24$		0			0
$25$	4.46410			0.892820
$26$	1.00000	+	3.46410i	0.196116	+	0.679366i
$27$		0			0
$28$	0.866025	−	0.500000i	0.163663	−	0.0944911i
$29$	−1.76795	−	3.06218i	−0.328300	−	0.568632i	0.653875	−	0.756603i	$-0.273142\pi$
$29$	−1.76795	−	3.06218i	−0.328300	−	0.568632i	−0.982175	+	0.187971i	$0.939809\pi$
$30$		0			0
$31$		−	3.26795i		−	0.586941i	−0.955968	−	0.293471i	$-0.905190\pi$
$31$		−	3.26795i		−	0.586941i	0.955968	−	0.293471i	$-0.0948102\pi$
$32$	0.866025	+	0.500000i	0.153093	+	0.0883883i
$33$		0			0
$34$			2.26795i			0.388950i
$35$	0.366025	−	0.633975i	0.0618696	−	0.107161i
$36$		0			0
$37$	5.83013	−	3.36603i	0.958467	−	0.553371i	0.0627661	−	0.998028i	$-0.480008\pi$
$37$	5.83013	−	3.36603i	0.958467	−	0.553371i	0.895701	+	0.444657i	$0.146674\pi$
$38$	0.464102			0.0752872
$39$		0			0
$40$	0.732051			0.115747
$41$	7.33013	−	4.23205i	1.14477	−	0.660935i	0.197165	−	0.980370i	$-0.436826\pi$
$41$	7.33013	−	4.23205i	1.14477	−	0.660935i	0.947608	+	0.319435i	$0.103493\pi$
$42$		0			0
$43$	−4.36603	+	7.56218i	−0.665813	+	1.15322i	0.313252	+	0.949670i	$0.398582\pi$
$43$	−4.36603	+	7.56218i	−0.665813	+	1.15322i	−0.979064	+	0.203551i	$0.934752\pi$
$44$			5.19615i			0.783349i
$45$		0			0
$46$	−6.46410	−	3.73205i	−0.953080	−	0.550261i
$47$		−	3.92820i		−	0.572987i	−0.958082	−	0.286494i	$-0.907510\pi$
$47$		−	3.92820i		−	0.572987i	0.958082	−	0.286494i	$-0.0924897\pi$
$48$		0			0
$49$	0.500000	+	0.866025i	0.0714286	+	0.123718i
$50$	−3.86603	+	2.23205i	−0.546739	+	0.315660i
$51$		0			0
$52$	−2.59808	−	2.50000i	−0.360288	−	0.346688i
$53$	9.92820			1.36374			0.681872	−	0.731472i	$-0.261166\pi$
$53$	9.92820			1.36374			0.681872	+	0.731472i	$0.261166\pi$
$54$		0			0
$55$	1.90192	+	3.29423i	0.256455	+	0.444194i
$56$	−0.500000	+	0.866025i	−0.0668153	+	0.115728i
$57$		0			0
$58$	3.06218	+	1.76795i	0.402084	+	0.232143i
$59$	−7.73205	−	4.46410i	−1.00663	−	0.581177i	−0.0964249	−	0.995340i	$-0.530741\pi$
$59$	−7.73205	−	4.46410i	−1.00663	−	0.581177i	−0.910202	+	0.414164i	$0.864074\pi$
$60$		0			0
$61$	−1.86603	+	3.23205i	−0.238920	+	0.413822i	−0.960405	−	0.278609i	$-0.910127\pi$
$61$	−1.86603	+	3.23205i	−0.238920	+	0.413822i	0.721485	+	0.692430i	$0.243460\pi$
$62$	1.63397	+	2.83013i	0.207515	+	0.359426i
$63$		0			0
$64$	−1.00000			−0.125000
$65$	−2.56218	−	0.633975i	−0.317799	−	0.0786349i
$66$		0			0
$67$	8.66025	−	5.00000i	1.05802	−	0.610847i	0.133135	−	0.991098i	$-0.457496\pi$
$67$	8.66025	−	5.00000i	1.05802	−	0.610847i	0.924883	+	0.380251i	$0.124162\pi$
$68$	−1.13397	−	1.96410i	−0.137515	−	0.238182i
$69$		0			0
$70$			0.732051i			0.0874968i
$71$	−6.29423	−	3.63397i	−0.746988	−	0.431273i	0.0776169	−	0.996983i	$-0.475269\pi$
$71$	−6.29423	−	3.63397i	−0.746988	−	0.431273i	−0.824604	+	0.565710i	$0.808602\pi$
$72$		0			0
$73$		−	1.46410i		−	0.171360i	−0.996323	−	0.0856801i	$-0.972694\pi$
$73$		−	1.46410i		−	0.171360i	0.996323	−	0.0856801i	$-0.0273063\pi$
$74$	−3.36603	+	5.83013i	−0.391293	+	0.677738i
$75$		0			0
$76$	−0.401924	+	0.232051i	−0.0461038	+	0.0266181i
$77$	−5.19615			−0.592157
$78$		0			0
$79$	9.00000			1.01258			0.506290	−	0.862364i	$-0.331017\pi$
$79$	9.00000			1.01258			0.506290	+	0.862364i	$0.331017\pi$
$80$	−0.633975	+	0.366025i	−0.0708805	+	0.0409229i
$81$		0			0
$82$	−4.23205	+	7.33013i	−0.467352	+	0.809477i
$83$		−	9.26795i		−	1.01729i	−0.860976	−	0.508645i	$-0.830147\pi$
$83$		−	9.26795i		−	1.01729i	0.860976	−	0.508645i	$-0.169853\pi$
$84$		0			0
$85$	−1.43782	−	0.830127i	−0.155954	−	0.0900399i
$86$		−	8.73205i		−	0.941601i
$87$		0			0
$88$	−2.59808	−	4.50000i	−0.276956	−	0.479702i
$89$	14.5981	−	8.42820i	1.54739	−	0.893388i	0.549053	−	0.835787i	$-0.314988\pi$
$89$	14.5981	−	8.42820i	1.54739	−	0.893388i	0.998340	−	0.0576004i	$-0.0183449\pi$
$90$		0			0
$91$	2.50000	−	2.59808i	0.262071	−	0.272352i
$92$	7.46410			0.778186
$93$		0			0
$94$	1.96410	+	3.40192i	0.202582	+	0.350882i
$95$	−0.169873	+	0.294229i	−0.0174286	+	0.0301872i
$96$		0			0
$97$	−12.2942	−	7.09808i	−1.24829	−	0.720700i	−0.277522	−	0.960719i	$-0.589513\pi$
$97$	−12.2942	−	7.09808i	−1.24829	−	0.720700i	−0.970768	+	0.240019i	$0.922846\pi$
$98$	−0.866025	−	0.500000i	−0.0874818	−	0.0505076i
$99$		0			0
$100$	2.23205	−	3.86603i	0.223205	−	0.386603i
$101$	−2.46410	−	4.26795i	−0.245187	−	0.424677i	0.716997	−	0.697076i	$-0.245516\pi$
$101$	−2.46410	−	4.26795i	−0.245187	−	0.424677i	−0.962184	+	0.272399i	$0.912183\pi$
$102$		0			0
$103$	16.1962			1.59585			0.797927	−	0.602754i	$-0.205930\pi$
$103$	16.1962			1.59585			0.797927	+	0.602754i	$0.205930\pi$
$104$	3.50000	+	0.866025i	0.343203	+	0.0849208i
$105$		0			0
$106$	−8.59808	+	4.96410i	−0.835119	+	0.482156i
$107$	−3.03590	−	5.25833i	−0.293491	−	0.508342i	0.681141	−	0.732152i	$-0.261484\pi$
$107$	−3.03590	−	5.25833i	−0.293491	−	0.508342i	−0.974633	+	0.223810i	$0.928151\pi$
$108$		0			0
$109$			11.2679i			1.07927i	0.841898	+	0.539637i	$0.181438\pi$
$109$			11.2679i			1.07927i	−0.841898	+	0.539637i	$0.818562\pi$
$110$	−3.29423	−	1.90192i	−0.314092	−	0.181341i
$111$		0			0
$112$		−	1.00000i		−	0.0944911i
$113$	3.83013	−	6.63397i	0.360308	−	0.624072i	−0.627703	−	0.778453i	$-0.716005\pi$
$113$	3.83013	−	6.63397i	0.360308	−	0.624072i	0.988011	+	0.154381i	$0.0493382\pi$
$114$		0			0
$115$	4.73205	−	2.73205i	0.441266	−	0.254765i
$116$	−3.53590			−0.328300
$117$		0			0
$118$	8.92820			0.821908
$119$	1.96410	−	1.13397i	0.180049	−	0.103951i
$120$		0			0
$121$	8.00000	−	13.8564i	0.727273	−	1.25967i
$122$		−	3.73205i		−	0.337884i
$123$		0			0
$124$	−2.83013	−	1.63397i	−0.254153	−	0.146735i
$125$		−	6.92820i		−	0.619677i
$126$		0			0
$127$	7.19615	+	12.4641i	0.638555	+	1.10601i	0.985750	+	0.168217i	$0.0538010\pi$
$127$	7.19615	+	12.4641i	0.638555	+	1.10601i	−0.347195	+	0.937793i	$0.612866\pi$
$128$	0.866025	−	0.500000i	0.0765466	−	0.0441942i
$129$		0			0
$130$	2.53590	−	0.732051i	0.222413	−	0.0642051i
$131$	14.9282			1.30428			0.652142	−	0.758097i	$-0.273871\pi$
$131$	14.9282			1.30428			0.652142	+	0.758097i	$0.273871\pi$
$132$		0			0
$133$	−0.232051	−	0.401924i	−0.0201214	−	0.0348512i
$134$	−5.00000	+	8.66025i	−0.431934	+	0.748132i
$135$		0			0
$136$	1.96410	+	1.13397i	0.168420	+	0.0972375i
$137$	16.8564	+	9.73205i	1.44014	+	0.831465i	0.997858	−	0.0654110i	$-0.0208359\pi$
$137$	16.8564	+	9.73205i	1.44014	+	0.831465i	0.442282	+	0.896876i	$0.354169\pi$
$138$		0			0
$139$	−8.52628	+	14.7679i	−0.723190	+	1.25260i	0.236525	+	0.971625i	$0.423991\pi$
$139$	−8.52628	+	14.7679i	−0.723190	+	1.25260i	−0.959715	+	0.280976i	$0.909342\pi$
$140$	−0.366025	−	0.633975i	−0.0309348	−	0.0535806i
$141$		0			0
$142$	7.26795			0.609913
$143$	5.19615	+	18.0000i	0.434524	+	1.50524i
$144$		0			0
$145$	−2.24167	+	1.29423i	−0.186161	+	0.107480i
$146$	0.732051	+	1.26795i	0.0605850	+	0.104936i
$147$		0			0
$148$		−	6.73205i		−	0.553371i
$149$	6.92820	+	4.00000i	0.567581	+	0.327693i	0.756182	−	0.654361i	$-0.227062\pi$
$149$	6.92820	+	4.00000i	0.567581	+	0.327693i	−0.188602	+	0.982054i	$0.560396\pi$
$150$		0			0
$151$		−	1.19615i		−	0.0973415i	−0.998815	−	0.0486708i	$-0.984501\pi$
$151$		−	1.19615i		−	0.0973415i	0.998815	−	0.0486708i	$-0.0154985\pi$
$152$	0.232051	−	0.401924i	0.0188218	−	0.0326003i
$153$		0			0
$154$	4.50000	−	2.59808i	0.362620	−	0.209359i
$155$	−2.39230			−0.192155
$156$		0			0
$157$	−11.4641			−0.914935			−0.457467	−	0.889226i	$-0.651243\pi$
$157$	−11.4641			−0.914935			−0.457467	+	0.889226i	$0.651243\pi$
$158$	−7.79423	+	4.50000i	−0.620076	+	0.358001i
$159$		0			0
$160$	0.366025	−	0.633975i	0.0289368	−	0.0501201i
$161$			7.46410i			0.588254i
$162$		0			0
$163$	−6.16987	−	3.56218i	−0.483262	−	0.279011i	0.238513	−	0.971139i	$-0.423340\pi$
$163$	−6.16987	−	3.56218i	−0.483262	−	0.279011i	−0.721775	+	0.692128i	$0.756673\pi$
$164$		−	8.46410i		−	0.660935i
$165$		0			0
$166$	4.63397	+	8.02628i	0.359666	+	0.622960i
$167$	4.39230	−	2.53590i	0.339887	−	0.196234i	−0.320335	−	0.947304i	$-0.603796\pi$
$167$	4.39230	−	2.53590i	0.339887	−	0.196234i	0.660222	+	0.751071i	$0.270462\pi$
$168$		0			0
$169$	−11.5000	−	6.06218i	−0.884615	−	0.466321i
$170$	1.66025			0.127336
$171$		0			0
$172$	4.36603	+	7.56218i	0.332906	+	0.576611i
$173$	−1.09808	+	1.90192i	−0.0834852	+	0.144601i	−0.904745	−	0.425954i	$-0.859938\pi$
$173$	−1.09808	+	1.90192i	−0.0834852	+	0.144601i	0.821260	+	0.570555i	$0.193272\pi$
$174$		0			0
$175$	3.86603	+	2.23205i	0.292244	+	0.168727i
$176$	4.50000	+	2.59808i	0.339200	+	0.195837i
$177$		0			0
$178$	−8.42820	+	14.5981i	−0.631721	+	1.09417i
$179$	1.73205	+	3.00000i	0.129460	+	0.224231i	0.923467	−	0.383677i	$-0.125342\pi$
$179$	1.73205	+	3.00000i	0.129460	+	0.224231i	−0.794008	+	0.607908i	$0.792009\pi$
$180$		0			0
$181$	14.2679			1.06053			0.530264	−	0.847832i	$-0.322093\pi$
$181$	14.2679			1.06053			0.530264	+	0.847832i	$0.322093\pi$
$182$	−0.866025	+	3.50000i	−0.0641941	+	0.259437i
$183$		0			0
$184$	−6.46410	+	3.73205i	−0.476540	+	0.275130i
$185$	−2.46410	−	4.26795i	−0.181164	−	0.313786i
$186$		0			0
$187$			11.7846i			0.861776i
$188$	−3.40192	−	1.96410i	−0.248111	−	0.143247i
$189$		0			0
$190$		−	0.339746i		−	0.0246478i
$191$	−4.09808	+	7.09808i	−0.296526	+	0.513599i	−0.975339	−	0.220713i	$-0.929162\pi$
$191$	−4.09808	+	7.09808i	−0.296526	+	0.513599i	0.678812	+	0.734312i	$0.262495\pi$
$192$		0			0
$193$	6.57180	−	3.79423i	0.473048	−	0.273115i	−0.244467	−	0.969658i	$-0.578613\pi$
$193$	6.57180	−	3.79423i	0.473048	−	0.273115i	0.717515	+	0.696543i	$0.245280\pi$
$194$	14.1962			1.01922
$195$		0			0
$196$	1.00000			0.0714286
$197$	−0.232051	+	0.133975i	−0.0165329	+	0.00954529i	−0.508244	−	0.861213i	$-0.669705\pi$
$197$	−0.232051	+	0.133975i	−0.0165329	+	0.00954529i	0.491711	+	0.870759i	$0.336372\pi$
$198$		0			0
$199$	12.2942	−	21.2942i	0.871515	−	1.50951i	0.0110851	−	0.999939i	$-0.496471\pi$
$199$	12.2942	−	21.2942i	0.871515	−	1.50951i	0.860430	−	0.509569i	$-0.170195\pi$
$200$			4.46410i			0.315660i
$201$		0			0
$202$	4.26795	+	2.46410i	0.300292	+	0.173374i
$203$		−	3.53590i		−	0.248171i
$204$		0			0
$205$	−3.09808	−	5.36603i	−0.216379	−	0.374779i
$206$	−14.0263	+	8.09808i	−0.977257	+	0.564220i
$207$		0			0
$208$	−3.46410	+	1.00000i	−0.240192	+	0.0693375i
$209$	2.41154			0.166810
$210$		0			0
$211$	−12.4641	−	21.5885i	−0.858064	−	1.48621i	−0.873773	−	0.486334i	$-0.838334\pi$
$211$	−12.4641	−	21.5885i	−0.858064	−	1.48621i	0.0157088	−	0.999877i	$-0.495000\pi$
$212$	4.96410	−	8.59808i	0.340936	−	0.590518i
$213$		0			0
$214$	5.25833	+	3.03590i	0.359452	+	0.207530i
$215$	5.53590	+	3.19615i	0.377545	+	0.217976i
$216$		0			0
$217$	1.63397	−	2.83013i	0.110921	−	0.192122i
$218$	−5.63397	−	9.75833i	−0.381581	−	0.660918i
$219$		0			0
$220$	3.80385			0.256455
$221$	−5.89230	−	5.66987i	−0.396359	−	0.381397i
$222$		0			0
$223$	−4.73205	+	2.73205i	−0.316882	+	0.182952i	−0.650002	−	0.759933i	$-0.725232\pi$
$223$	−4.73205	+	2.73205i	−0.316882	+	0.182952i	0.333120	+	0.942884i	$0.391899\pi$
$224$	0.500000	+	0.866025i	0.0334077	+	0.0578638i
$225$		0			0
$226$			7.66025i			0.509553i
$227$	11.5359	+	6.66025i	0.765664	+	0.442057i	0.831326	−	0.555785i	$-0.187582\pi$
$227$	11.5359	+	6.66025i	0.765664	+	0.442057i	−0.0656613	+	0.997842i	$0.520916\pi$
$228$		0			0
$229$			17.3923i			1.14932i	0.818394	+	0.574658i	$0.194865\pi$
$229$			17.3923i			1.14932i	−0.818394	+	0.574658i	$0.805135\pi$
$230$	−2.73205	+	4.73205i	−0.180146	+	0.312022i
$231$		0			0
$232$	3.06218	−	1.76795i	0.201042	−	0.116072i
$233$	−18.5885			−1.21777			−0.608885	−	0.793258i	$-0.708383\pi$
$233$	−18.5885			−1.21777			−0.608885	+	0.793258i	$0.708383\pi$
$234$		0			0
$235$	−2.87564			−0.187586
$236$	−7.73205	+	4.46410i	−0.503314	+	0.290588i
$237$		0			0
$238$	−1.13397	+	1.96410i	−0.0735047	+	0.127314i
$239$		−	26.5885i		−	1.71986i	−0.510408	−	0.859932i	$-0.670506\pi$
$239$		−	26.5885i		−	1.71986i	0.510408	−	0.859932i	$-0.329494\pi$
$240$		0			0
$241$	9.46410	+	5.46410i	0.609636	+	0.351974i	0.772823	−	0.634621i	$-0.218844\pi$
$241$	9.46410	+	5.46410i	0.609636	+	0.351974i	−0.163187	+	0.986595i	$0.552177\pi$
$242$			16.0000i			1.02852i
$243$		0			0
$244$	1.86603	+	3.23205i	0.119460	+	0.206911i
$245$	0.633975	−	0.366025i	0.0405032	−	0.0233845i
$246$		0			0
$247$	−1.16025	+	1.20577i	−0.0738252	+	0.0767214i
$248$	3.26795			0.207515
$249$		0			0
$250$	3.46410	+	6.00000i	0.219089	+	0.379473i
$251$	−8.75833	+	15.1699i	−0.552821	+	0.957514i	0.445249	+	0.895407i	$0.353115\pi$
$251$	−8.75833	+	15.1699i	−0.552821	+	0.957514i	−0.998070	+	0.0621069i	$0.980218\pi$
$252$		0			0
$253$	−33.5885	−	19.3923i	−2.11169	−	1.21918i
$254$	−12.4641	−	7.19615i	−0.782067	−	0.451527i
$255$		0			0
$256$	−0.500000	+	0.866025i	−0.0312500	+	0.0541266i
$257$	12.5263	+	21.6962i	0.781368	+	1.35337i	0.931145	+	0.364649i	$0.118811\pi$
$257$	12.5263	+	21.6962i	0.781368	+	1.35337i	−0.149777	+	0.988720i	$0.547856\pi$
$258$		0			0
$259$	6.73205			0.418309
$260$	−1.83013	+	1.90192i	−0.113500	+	0.117952i
$261$		0			0
$262$	−12.9282	+	7.46410i	−0.798707	+	0.461134i
$263$	1.56218	+	2.70577i	0.0963280	+	0.166845i	0.910162	−	0.414252i	$-0.135957\pi$
$263$	1.56218	+	2.70577i	0.0963280	+	0.166845i	−0.813834	+	0.581097i	$0.802624\pi$
$264$		0			0
$265$		−	7.26795i		−	0.446467i
$266$	0.401924	+	0.232051i	0.0246435	+	0.0142279i
$267$		0			0
$268$		−	10.0000i		−	0.610847i
$269$	−2.00000	+	3.46410i	−0.121942	+	0.211210i	−0.920534	−	0.390664i	$-0.872246\pi$
$269$	−2.00000	+	3.46410i	−0.121942	+	0.211210i	0.798591	+	0.601874i	$0.205579\pi$
$270$		0			0
$271$	6.63397	−	3.83013i	0.402985	−	0.232664i	−0.284786	−	0.958591i	$-0.591923\pi$
$271$	6.63397	−	3.83013i	0.402985	−	0.232664i	0.687771	+	0.725927i	$0.258589\pi$
$272$	−2.26795			−0.137515
$273$		0			0
$274$	−19.4641			−1.17587
$275$	−20.0885	+	11.5981i	−1.21138	+	0.699390i
$276$		0			0
$277$	−7.19615	+	12.4641i	−0.432375	+	0.748895i	−0.997077	−	0.0763993i	$-0.975658\pi$
$277$	−7.19615	+	12.4641i	−0.432375	+	0.748895i	0.564702	+	0.825295i	$0.308991\pi$
$278$		−	17.0526i		−	1.02274i
$279$		0			0
$280$	0.633975	+	0.366025i	0.0378872	+	0.0218742i
$281$		−	12.7321i		−	0.759530i	−0.925083	−	0.379765i	$-0.876005\pi$
$281$		−	12.7321i		−	0.759530i	0.925083	−	0.379765i	$-0.123995\pi$
$282$		0			0
$283$	−8.92820	−	15.4641i	−0.530727	−	0.919245i	−0.999357	−	0.0358512i	$-0.988586\pi$
$283$	−8.92820	−	15.4641i	−0.530727	−	0.919245i	0.468631	−	0.883394i	$-0.344748\pi$
$284$	−6.29423	+	3.63397i	−0.373494	+	0.215637i
$285$		0			0
$286$	−13.5000	−	12.9904i	−0.798272	−	0.768137i
$287$	8.46410			0.499620
$288$		0			0
$289$	5.92820	+	10.2679i	0.348718	+	0.603997i
$290$	1.29423	−	2.24167i	0.0759997	−	0.131635i
$291$		0			0
$292$	−1.26795	−	0.732051i	−0.0742011	−	0.0428400i
$293$	−0.928203	−	0.535898i	−0.0542262	−	0.0313075i	0.472642	−	0.881255i	$-0.343300\pi$
$293$	−0.928203	−	0.535898i	−0.0542262	−	0.0313075i	−0.526868	+	0.849947i	$0.676634\pi$
$294$		0			0
$295$	−3.26795	+	5.66025i	−0.190267	+	0.329553i
$296$	3.36603	+	5.83013i	0.195646	+	0.338869i
$297$		0			0
$298$	−8.00000			−0.463428
$299$	25.8564	−	7.46410i	1.49531	−	0.431660i
$300$		0			0
$301$	−7.56218	+	4.36603i	−0.435877	+	0.251654i
$302$	0.598076	+	1.03590i	0.0344154	+	0.0596093i
$303$		0			0
$304$			0.464102i			0.0266181i
$305$	2.36603	+	1.36603i	0.135478	+	0.0782184i
$306$		0			0
$307$		−	5.00000i		−	0.285365i	−0.989769	−	0.142683i	$-0.954427\pi$
$307$		−	5.00000i		−	0.285365i	0.989769	−	0.142683i	$-0.0455728\pi$
$308$	−2.59808	+	4.50000i	−0.148039	+	0.256411i
$309$		0			0
$310$	2.07180	−	1.19615i	0.117670	−	0.0679369i
$311$	−14.8038			−0.839449			−0.419725	−	0.907652i	$-0.637873\pi$
$311$	−14.8038			−0.839449			−0.419725	+	0.907652i	$0.637873\pi$
$312$		0			0
$313$	6.87564			0.388634			0.194317	−	0.980939i	$-0.437751\pi$
$313$	6.87564			0.388634			0.194317	+	0.980939i	$0.437751\pi$
$314$	9.92820	−	5.73205i	0.560281	−	0.323478i
$315$		0			0
$316$	4.50000	−	7.79423i	0.253145	−	0.438460i
$317$		−	14.7846i		−	0.830386i	−0.909733	−	0.415193i	$-0.863714\pi$
$317$		−	14.7846i		−	0.830386i	0.909733	−	0.415193i	$-0.136286\pi$
$318$		0			0
$319$	15.9115	+	9.18653i	0.890875	+	0.514347i
$320$			0.732051i			0.0409229i
$321$		0			0
$322$	−3.73205	−	6.46410i	−0.207979	−	0.360230i
$323$	−0.911543	+	0.526279i	−0.0507196	+	0.0292830i
$324$		0			0
$325$	3.86603	−	15.6244i	0.214449	−	0.866683i
$326$	7.12436			0.394582
$327$		0			0
$328$	4.23205	+	7.33013i	0.233676	+	0.404739i
$329$	1.96410	−	3.40192i	0.108284	−	0.187554i
$330$		0			0
$331$	5.87564	+	3.39230i	0.322955	+	0.186458i	0.652709	−	0.757609i	$-0.273633\pi$
$331$	5.87564	+	3.39230i	0.322955	+	0.186458i	−0.329754	+	0.944067i	$0.606966\pi$
$332$	−8.02628	−	4.63397i	−0.440499	−	0.254322i
$333$		0			0
$334$	−2.53590	+	4.39230i	−0.138758	+	0.240336i
$335$	−3.66025	−	6.33975i	−0.199981	−	0.346377i
$336$		0			0
$337$	−11.0000			−0.599208			−0.299604	−	0.954064i	$-0.596855\pi$
$337$	−11.0000			−0.599208			−0.299604	+	0.954064i	$0.596855\pi$
$338$	12.9904	−	0.500000i	0.706584	−	0.0271964i
$339$		0			0
$340$	−1.43782	+	0.830127i	−0.0779769	+	0.0450200i
$341$	8.49038	+	14.7058i	0.459780	+	0.796362i
$342$		0			0
$343$			1.00000i			0.0539949i
$344$	−7.56218	−	4.36603i	−0.407725	−	0.235400i
$345$		0			0
$346$		−	2.19615i		−	0.118066i
$347$	−3.23205	+	5.59808i	−0.173506	+	0.300520i	−0.939643	−	0.342156i	$-0.888843\pi$
$347$	−3.23205	+	5.59808i	−0.173506	+	0.300520i	0.766138	+	0.642677i	$0.222176\pi$
$348$		0			0
$349$	5.32051	−	3.07180i	0.284800	−	0.164430i	−0.350794	−	0.936453i	$-0.614088\pi$
$349$	5.32051	−	3.07180i	0.284800	−	0.164430i	0.635595	+	0.772023i	$0.280755\pi$
$350$	−4.46410			−0.238616
$351$		0			0
$352$	−5.19615			−0.276956
$353$	−27.1244	+	15.6603i	−1.44368	+	0.833511i	−0.998093	−	0.0617256i	$-0.980340\pi$
$353$	−27.1244	+	15.6603i	−1.44368	+	0.833511i	−0.445591	+	0.895237i	$0.647006\pi$
$354$		0			0
$355$	−2.66025	+	4.60770i	−0.141192	+	0.244551i
$356$		−	16.8564i		−	0.893388i
$357$		0			0
$358$	−3.00000	−	1.73205i	−0.158555	−	0.0915417i
$359$			34.4449i			1.81793i	0.416872	+	0.908965i	$0.363126\pi$
$359$			34.4449i			1.81793i	−0.416872	+	0.908965i	$0.636874\pi$
$360$		0			0
$361$	−9.39230	−	16.2679i	−0.494332	−	0.856208i
$362$	−12.3564	+	7.13397i	−0.649438	+	0.374953i
$363$		0			0
$364$	−1.00000	−	3.46410i	−0.0524142	−	0.181568i
$365$	−1.07180			−0.0561004
$366$		0			0
$367$	−6.66025	−	11.5359i	−0.347662	−	0.602169i	0.638171	−	0.769894i	$-0.279691\pi$
$367$	−6.66025	−	11.5359i	−0.347662	−	0.602169i	−0.985834	+	0.167725i	$0.946358\pi$
$368$	3.73205	−	6.46410i	0.194547	−	0.336965i
$369$		0			0
$370$	4.26795	+	2.46410i	0.221880	+	0.128103i
$371$	8.59808	+	4.96410i	0.446390	+	0.257723i
$372$		0			0
$373$	7.56218	−	13.0981i	0.391555	−	0.678193i	−0.601100	−	0.799174i	$-0.705271\pi$
$373$	7.56218	−	13.0981i	0.391555	−	0.678193i	0.992655	+	0.120981i	$0.0386040\pi$
$374$	−5.89230	−	10.2058i	−0.304684	−	0.527728i
$375$		0			0
$376$	3.92820			0.202582
$377$	−12.2487	+	3.53590i	−0.630841	+	0.182108i
$378$		0			0
$379$	−31.2224	+	18.0263i	−1.60379	+	0.925948i	−0.613069	+	0.790030i	$0.710065\pi$
$379$	−31.2224	+	18.0263i	−1.60379	+	0.925948i	−0.990720	+	0.135918i	$0.956602\pi$
$380$	0.169873	+	0.294229i	0.00871430	+	0.0150936i
$381$		0			0
$382$		−	8.19615i		−	0.419352i
$383$	−26.8468	−	15.5000i	−1.37181	−	0.792013i	−0.380651	−	0.924719i	$-0.624300\pi$
$383$	−26.8468	−	15.5000i	−1.37181	−	0.792013i	−0.991155	+	0.132706i	$0.957633\pi$
$384$		0			0
$385$			3.80385i			0.193862i
$386$	−3.79423	+	6.57180i	−0.193121	+	0.334496i
$387$		0			0
$388$	−12.2942	+	7.09808i	−0.624145	+	0.360350i
$389$	−21.4641			−1.08827			−0.544137	−	0.838997i	$-0.683143\pi$
$389$	−21.4641			−1.08827			−0.544137	+	0.838997i	$0.683143\pi$
$390$		0			0
$391$	16.9282			0.856096
$392$	−0.866025	+	0.500000i	−0.0437409	+	0.0252538i
$393$		0			0
$394$	0.133975	−	0.232051i	0.00674954	−	0.0116906i
$395$		−	6.58846i		−	0.331501i
$396$		0			0
$397$	−18.5263	−	10.6962i	−0.929807	−	0.536825i	−0.0430567	−	0.999073i	$-0.513710\pi$
$397$	−18.5263	−	10.6962i	−0.929807	−	0.536825i	−0.886751	+	0.462248i	$0.847043\pi$
$398$			24.5885i			1.23251i
$399$		0			0
$400$	−2.23205	−	3.86603i	−0.111603	−	0.193301i
$401$	9.58846	−	5.53590i	0.478825	−	0.276450i	−0.241102	−	0.970500i	$-0.577509\pi$
$401$	9.58846	−	5.53590i	0.478825	−	0.276450i	0.719927	+	0.694050i	$0.244175\pi$
$402$		0			0
$403$	−11.4378	−	2.83013i	−0.569759	−	0.140979i
$404$	−4.92820			−0.245187
$405$		0			0
$406$	1.76795	+	3.06218i	0.0877418	+	0.151973i
$407$	−17.4904	+	30.2942i	−0.866966	+	1.50163i
$408$		0			0
$409$	−14.3660	−	8.29423i	−0.710354	−	0.410123i	0.100838	−	0.994903i	$-0.467848\pi$
$409$	−14.3660	−	8.29423i	−0.710354	−	0.410123i	−0.811192	+	0.584780i	$0.801181\pi$
$410$	5.36603	+	3.09808i	0.265009	+	0.153003i
$411$		0			0
$412$	8.09808	−	14.0263i	0.398964	−	0.691025i
$413$	−4.46410	−	7.73205i	−0.219664	−	0.380469i
$414$		0			0
$415$	−6.78461			−0.333043
$416$	2.50000	−	2.59808i	0.122573	−	0.127381i
$417$		0			0
$418$	−2.08846	+	1.20577i	−0.102150	+	0.0589762i
$419$	−7.29423	−	12.6340i	−0.356346	−	0.617210i	0.631001	−	0.775782i	$-0.282644\pi$
$419$	−7.29423	−	12.6340i	−0.356346	−	0.617210i	−0.987347	+	0.158572i	$0.949311\pi$
$420$		0			0
$421$			25.3205i			1.23405i	0.786945	+	0.617023i	$0.211661\pi$
$421$			25.3205i			1.23405i	−0.786945	+	0.617023i	$0.788339\pi$
$422$	21.5885	+	12.4641i	1.05091	+	0.606743i
$423$		0			0
$424$			9.92820i			0.482156i
$425$	5.06218	−	8.76795i	0.245552	−	0.425308i
$426$		0			0
$427$	−3.23205	+	1.86603i	−0.156410	+	0.0903033i
$428$	−6.07180			−0.293491
$429$		0			0
$430$	−6.39230			−0.308264
$431$	−11.3660	+	6.56218i	−0.547482	+	0.316089i	−0.748106	−	0.663579i	$-0.769036\pi$
$431$	−11.3660	+	6.56218i	−0.547482	+	0.316089i	0.200624	+	0.979668i	$0.435703\pi$
$432$		0			0
$433$	6.07180	−	10.5167i	0.291792	−	0.505398i	−0.682442	−	0.730940i	$-0.739082\pi$
$433$	6.07180	−	10.5167i	0.291792	−	0.505398i	0.974234	+	0.225542i	$0.0724152\pi$
$434$			3.26795i			0.156867i
$435$		0			0
$436$	9.75833	+	5.63397i	0.467339	+	0.269818i
$437$		−	3.46410i		−	0.165710i
$438$		0			0
$439$	11.6603	+	20.1962i	0.556514	+	0.963910i	0.997784	+	0.0665356i	$0.0211946\pi$
$439$	11.6603	+	20.1962i	0.556514	+	0.963910i	−0.441270	+	0.897374i	$0.645472\pi$
$440$	−3.29423	+	1.90192i	−0.157046	+	0.0906707i
$441$		0			0
$442$	7.93782	+	1.96410i	0.377564	+	0.0934228i
$443$	25.0000			1.18779			0.593893	−	0.804544i	$-0.297590\pi$
$443$	25.0000			1.18779			0.593893	+	0.804544i	$0.297590\pi$
$444$		0			0
$445$	−6.16987	−	10.6865i	−0.292480	−	0.506590i
$446$	2.73205	−	4.73205i	0.129366	−	0.224069i
$447$		0			0
$448$	−0.866025	−	0.500000i	−0.0409159	−	0.0236228i
$449$	−25.0981	−	14.4904i	−1.18445	−	0.683843i	−0.227411	−	0.973799i	$-0.573026\pi$
$449$	−25.0981	−	14.4904i	−1.18445	−	0.683843i	−0.957040	+	0.289955i	$0.906360\pi$
$450$		0			0
$451$	−21.9904	+	38.0885i	−1.03549	+	1.79352i
$452$	−3.83013	−	6.63397i	−0.180154	−	0.312036i
$453$		0			0
$454$	−13.3205			−0.625162
$455$	−1.90192	−	1.83013i	−0.0891636	−	0.0857977i
$456$		0			0
$457$	−25.7321	+	14.8564i	−1.20369	+	0.694953i	−0.961375	−	0.275243i	$-0.911242\pi$
$457$	−25.7321	+	14.8564i	−1.20369	+	0.694953i	−0.242320	+	0.970196i	$0.577908\pi$
$458$	−8.69615	−	15.0622i	−0.406345	−	0.703809i
$459$		0			0
$460$		−	5.46410i		−	0.254765i
$461$	−6.00000	−	3.46410i	−0.279448	−	0.161339i	0.353726	−	0.935349i	$-0.384915\pi$
$461$	−6.00000	−	3.46410i	−0.279448	−	0.161339i	−0.633173	+	0.774010i	$0.718248\pi$
$462$		0			0
$463$			15.3397i			0.712898i	0.934315	+	0.356449i	$0.116013\pi$
$463$			15.3397i			0.712898i	−0.934315	+	0.356449i	$0.883987\pi$
$464$	−1.76795	+	3.06218i	−0.0820750	+	0.142158i
$465$		0			0
$466$	16.0981	−	9.29423i	0.745729	−	0.430547i
$467$	−18.9282			−0.875893			−0.437946	−	0.899001i	$-0.644294\pi$
$467$	−18.9282			−0.875893			−0.437946	+	0.899001i	$0.644294\pi$
$468$		0			0
$469$	10.0000			0.461757
$470$	2.49038	−	1.43782i	0.114873	−	0.0663218i
$471$		0			0
$472$	4.46410	−	7.73205i	0.205477	−	0.355896i
$473$		−	45.3731i		−	2.08626i
$474$		0			0
$475$	−1.79423	−	1.03590i	−0.0823249	−	0.0475303i
$476$		−	2.26795i		−	0.103951i
$477$		0			0
$478$	13.2942	+	23.0263i	0.608064	+	1.05320i
$479$	−4.20577	+	2.42820i	−0.192167	+	0.110947i	−0.592996	−	0.805205i	$-0.702055\pi$
$479$	−4.20577	+	2.42820i	−0.192167	+	0.110947i	0.400830	+	0.916153i	$0.368722\pi$
$480$		0			0
$481$	−6.73205	−	23.3205i	−0.306955	−	1.06332i
$482$	−10.9282			−0.497766
$483$		0			0
$484$	−8.00000	−	13.8564i	−0.363636	−	0.629837i
$485$	−5.19615	+	9.00000i	−0.235945	+	0.408669i
$486$		0			0
$487$	−25.5000	−	14.7224i	−1.15552	−	0.667137i	−0.205290	−	0.978701i	$-0.565814\pi$
$487$	−25.5000	−	14.7224i	−1.15552	−	0.667137i	−0.950225	+	0.311564i	$0.899147\pi$
$488$	−3.23205	−	1.86603i	−0.146308	−	0.0844710i
$489$		0			0
$490$	−0.366025	+	0.633975i	−0.0165353	+	0.0286401i
$491$	3.19615	+	5.53590i	0.144240	+	0.249832i	0.929089	−	0.369856i	$-0.120593\pi$
$491$	3.19615	+	5.53590i	0.144240	+	0.249832i	−0.784849	+	0.619687i	$0.787259\pi$
$492$		0			0
$493$	−8.01924			−0.361168
$494$	0.401924	−	1.62436i	0.0180834	−	0.0730832i
$495$		0			0
$496$	−2.83013	+	1.63397i	−0.127076	+	0.0733676i
$497$	−3.63397	−	6.29423i	−0.163006	−	0.282335i
$498$		0			0
$499$			20.5885i			0.921666i	0.887487	+	0.460833i	$0.152449\pi$
$499$			20.5885i			0.921666i	−0.887487	+	0.460833i	$0.847551\pi$
$500$	−6.00000	−	3.46410i	−0.268328	−	0.154919i
$501$		0			0
$502$		−	17.5167i		−	0.781807i
$503$	−15.4641	+	26.7846i	−0.689510	+	1.19427i	0.282486	+	0.959271i	$0.408841\pi$
$503$	−15.4641	+	26.7846i	−0.689510	+	1.19427i	−0.971996	+	0.234995i	$0.924492\pi$
$504$		0			0
$505$	−3.12436	+	1.80385i	−0.139032	+	0.0802702i
$506$	38.7846			1.72419
$507$		0			0
$508$	14.3923			0.638555
$509$	25.6865	−	14.8301i	1.13854	−	0.657334i	0.192468	−	0.981303i	$-0.438351\pi$
$509$	25.6865	−	14.8301i	1.13854	−	0.657334i	0.946068	+	0.323969i	$0.105017\pi$
$510$		0			0
$511$	0.732051	−	1.26795i	0.0323840	−	0.0560908i
$512$		−	1.00000i		−	0.0441942i
$513$		0			0
$514$	−21.6962	−	12.5263i	−0.956976	−	0.552511i
$515$		−	11.8564i		−	0.522456i
$516$		0			0
$517$	10.2058	+	17.6769i	0.448849	+	0.777430i
$518$	−5.83013	+	3.36603i	−0.256161	+	0.147895i
$519$		0			0
$520$	0.633975	−	2.56218i	0.0278016	−	0.112359i
$521$	21.5885			0.945807			0.472904	−	0.881114i	$-0.343206\pi$
$521$	21.5885			0.945807			0.472904	+	0.881114i	$0.343206\pi$
$522$		0			0
$523$	12.5981	+	21.8205i	0.550875	+	0.954144i	0.998212	+	0.0597784i	$0.0190394\pi$
$523$	12.5981	+	21.8205i	0.550875	+	0.954144i	−0.447336	+	0.894366i	$0.647627\pi$
$524$	7.46410	−	12.9282i	0.326071	−	0.564771i
$525$		0			0
$526$	−2.70577	−	1.56218i	−0.117977	−	0.0681142i
$527$	−6.41858	−	3.70577i	−0.279598	−	0.161426i
$528$		0			0
$529$	−16.3564	+	28.3301i	−0.711148	+	1.23174i
$530$	3.63397	+	6.29423i	0.157850	+	0.273404i
$531$		0			0
$532$	−0.464102			−0.0201214
$533$	−8.46410	−	29.3205i	−0.366621	−	1.27001i
$534$		0			0
$535$	−3.84936	+	2.22243i	−0.166423	+	0.0960841i
$536$	5.00000	+	8.66025i	0.215967	+	0.374066i
$537$		0			0
$538$		−	4.00000i		−	0.172452i
$539$	−4.50000	−	2.59808i	−0.193829	−	0.111907i
$540$		0			0
$541$			8.05256i			0.346207i	0.984904	+	0.173103i	$0.0553794\pi$
$541$			8.05256i			0.346207i	−0.984904	+	0.173103i	$0.944621\pi$
$542$	−3.83013	+	6.63397i	−0.164518	+	0.284954i
$543$		0			0
$544$	1.96410	−	1.13397i	0.0842102	−	0.0486188i
$545$	8.24871			0.353336
$546$		0			0
$547$	−4.19615			−0.179415			−0.0897073	−	0.995968i	$-0.528593\pi$
$547$	−4.19615			−0.179415			−0.0897073	+	0.995968i	$0.528593\pi$
$548$	16.8564	−	9.73205i	0.720070	−	0.415733i
$549$		0			0
$550$	11.5981	−	20.0885i	0.494544	−	0.856575i
$551$			1.64102i			0.0699096i
$552$		0			0
$553$	7.79423	+	4.50000i	0.331444	+	0.191359i
$554$		−	14.3923i		−	0.611470i
$555$		0			0
$556$	8.52628	+	14.7679i	0.361595	+	0.626301i
$557$	−17.0885	+	9.86603i	−0.724061	+	0.418037i	−0.816246	−	0.577705i	$-0.803948\pi$
$557$	−17.0885	+	9.86603i	−0.724061	+	0.418037i	0.0921844	+	0.995742i	$0.470615\pi$
$558$		0			0
$559$	22.6865	+	21.8301i	0.959538	+	0.923316i
$560$	−0.732051			−0.0309348
$561$		0			0
$562$	6.36603	+	11.0263i	0.268535	+	0.465116i
$563$	6.83013	−	11.8301i	0.287856	−	0.498580i	−0.685442	−	0.728127i	$-0.740391\pi$
$563$	6.83013	−	11.8301i	0.287856	−	0.498580i	0.973298	+	0.229547i	$0.0737244\pi$
$564$		0			0
$565$	−4.85641	−	2.80385i	−0.204311	−	0.117959i
$566$	15.4641	+	8.92820i	0.650005	+	0.375280i
$567$		0			0
$568$	3.63397	−	6.29423i	0.152478	−	0.264100i
$569$	−0.830127	−	1.43782i	−0.0348007	−	0.0602766i	0.848100	−	0.529835i	$-0.177746\pi$
$569$	−0.830127	−	1.43782i	−0.0348007	−	0.0602766i	−0.882901	+	0.469559i	$0.844413\pi$
$570$		0			0
$571$	39.5167			1.65372			0.826860	−	0.562407i	$-0.190125\pi$
$571$	39.5167			1.65372			0.826860	+	0.562407i	$0.190125\pi$
$572$	18.1865	+	4.50000i	0.760417	+	0.188154i
$573$		0			0
$574$	−7.33013	+	4.23205i	−0.305954	+	0.176642i
$575$	16.6603	+	28.8564i	0.694781	+	1.20340i
$576$		0			0
$577$		−	28.5885i		−	1.19015i	−0.803669	−	0.595077i	$-0.797122\pi$
$577$		−	28.5885i		−	1.19015i	0.803669	−	0.595077i	$-0.202878\pi$
$578$	−10.2679	−	5.92820i	−0.427090	−	0.246581i
$579$		0			0
$580$			2.58846i			0.107480i
$581$	4.63397	−	8.02628i	0.192250	−	0.332986i
$582$		0			0
$583$	−44.6769	+	25.7942i	−1.85033	+	1.06829i
$584$	1.46410			0.0605850
$585$		0			0
$586$	1.07180			0.0442755
$587$	4.43782	−	2.56218i	0.183169	−	0.105752i	−0.405612	−	0.914045i	$-0.632942\pi$
$587$	4.43782	−	2.56218i	0.183169	−	0.105752i	0.588781	+	0.808293i	$0.299608\pi$
$588$		0			0
$589$	−0.758330	+	1.31347i	−0.0312465	+	0.0541204i
$590$		−	6.53590i		−	0.269079i
$591$		0			0
$592$	−5.83013	−	3.36603i	−0.239617	−	0.138343i
$593$			37.0000i			1.51941i	0.650269	+	0.759704i	$0.274656\pi$
$593$			37.0000i			1.51941i	−0.650269	+	0.759704i	$0.725344\pi$
$594$		0			0
$595$	−0.830127	−	1.43782i	−0.0340319	−	0.0589450i
$596$	6.92820	−	4.00000i	0.283790	−	0.163846i
$597$		0			0
$598$	−18.6603	+	19.3923i	−0.763075	+	0.793010i
$599$	47.5692			1.94363			0.971813	−	0.235754i	$-0.0757559\pi$
$599$	47.5692			1.94363			0.971813	+	0.235754i	$0.0757559\pi$
$600$		0			0
$601$	24.1506	+	41.8301i	0.985125	+	1.70629i	0.641380	+	0.767223i	$0.278362\pi$
$601$	24.1506	+	41.8301i	0.985125	+	1.70629i	0.343745	+	0.939063i	$0.388304\pi$
$602$	4.36603	−	7.56218i	0.177946	−	0.308211i
$603$		0			0
$604$	−1.03590	−	0.598076i	−0.0421501	−	0.0243354i
$605$	−10.1436	−	5.85641i	−0.412396	−	0.238097i
$606$		0			0
$607$	12.8301	−	22.2224i	0.520759	−	0.901981i	−0.478950	−	0.877842i	$-0.658982\pi$
$607$	12.8301	−	22.2224i	0.520759	−	0.901981i	0.999709	−	0.0241384i	$-0.00768424\pi$
$608$	−0.232051	−	0.401924i	−0.00941090	−	0.0163002i
$609$		0			0
$610$	−2.73205			−0.110618
$611$	−13.7487	−	3.40192i	−0.556213	−	0.137627i
$612$		0			0
$613$	−41.3205	+	23.8564i	−1.66892	+	0.963551i	−0.700698	+	0.713458i	$0.747128\pi$
$613$	−41.3205	+	23.8564i	−1.66892	+	0.963551i	−0.968222	+	0.250093i	$0.919539\pi$
$614$	2.50000	+	4.33013i	0.100892	+	0.174750i
$615$		0			0
$616$		−	5.19615i		−	0.209359i
$617$	−11.9545	−	6.90192i	−0.481269	−	0.277861i	0.239676	−	0.970853i	$-0.422959\pi$
$617$	−11.9545	−	6.90192i	−0.481269	−	0.277861i	−0.720945	+	0.692992i	$0.756292\pi$
$618$		0			0
$619$			11.3923i			0.457895i	0.973439	+	0.228948i	$0.0735285\pi$
$619$			11.3923i			0.457895i	−0.973439	+	0.228948i	$0.926472\pi$
$620$	−1.19615	+	2.07180i	−0.0480386	+	0.0832054i
$621$		0			0
$622$	12.8205	−	7.40192i	0.514056	−	0.296790i
$623$	16.8564			0.675338
$624$		0			0
$625$	17.2487			0.689948
$626$	−5.95448	+	3.43782i	−0.237989	+	0.137403i
$627$		0			0
$628$	−5.73205	+	9.92820i	−0.228734	+	0.396178i
$629$		−	15.2679i		−	0.608773i
$630$		0			0
$631$	1.62436	+	0.937822i	0.0646646	+	0.0373341i	0.531984	−	0.846755i	$-0.321447\pi$
$631$	1.62436	+	0.937822i	0.0646646	+	0.0373341i	−0.467319	+	0.884089i	$0.654780\pi$
$632$			9.00000i			0.358001i
$633$		0			0
$634$	7.39230	+	12.8038i	0.293586	+	0.508506i
$635$	9.12436	−	5.26795i	0.362089	−	0.209052i
$636$		0			0
$637$	3.46410	−	1.00000i	0.137253	−	0.0396214i
$638$	−18.3731			−0.727397
$639$		0			0
$640$	−0.366025	−	0.633975i	−0.0144684	−	0.0250600i
$641$	8.66025	−	15.0000i	0.342059	−	0.592464i	−0.642756	−	0.766071i	$-0.722209\pi$
$641$	8.66025	−	15.0000i	0.342059	−	0.592464i	0.984815	+	0.173607i	$0.0555422\pi$
$642$		0			0
$643$	2.00962	+	1.16025i	0.0792516	+	0.0457560i	0.539102	−	0.842240i	$-0.318764\pi$
$643$	2.00962	+	1.16025i	0.0792516	+	0.0457560i	−0.459851	+	0.887996i	$0.652097\pi$
$644$	6.46410	+	3.73205i	0.254721	+	0.147063i
$645$		0			0
$646$	0.526279	−	0.911543i	0.0207062	−	0.0358642i
$647$	11.8660	+	20.5526i	0.466502	+	0.808004i	0.999268	−	0.0382579i	$-0.0121809\pi$
$647$	11.8660	+	20.5526i	0.466502	+	0.808004i	−0.532766	+	0.846262i	$0.678848\pi$
$648$		0			0
$649$	46.3923			1.82106
$650$	4.46410	+	15.4641i	0.175096	+	0.606552i
$651$		0			0
$652$	−6.16987	+	3.56218i	−0.241631	+	0.139506i
$653$	−7.35641	−	12.7417i	−0.287878	−	0.498620i	0.685425	−	0.728144i	$-0.259617\pi$
$653$	−7.35641	−	12.7417i	−0.287878	−	0.498620i	−0.973303	+	0.229523i	$0.926283\pi$
$654$		0			0
$655$		−	10.9282i		−	0.427000i
$656$	−7.33013	−	4.23205i	−0.286193	−	0.165234i
$657$		0			0
$658$			3.92820i			0.153137i
$659$	−6.62436	+	11.4737i	−0.258048	+	0.446953i	−0.965719	−	0.259590i	$-0.916413\pi$
$659$	−6.62436	+	11.4737i	−0.258048	+	0.446953i	0.707671	+	0.706542i	$0.249746\pi$
$660$		0			0
$661$	19.5167	−	11.2679i	0.759110	−	0.438272i	−0.0698660	−	0.997556i	$-0.522257\pi$
$661$	19.5167	−	11.2679i	0.759110	−	0.438272i	0.828976	+	0.559284i	$0.188924\pi$
$662$	−6.78461			−0.263691
$663$		0			0
$664$	9.26795			0.359666
$665$	−0.294229	+	0.169873i	−0.0114097	+	0.00658739i
$666$		0			0
$667$	13.1962	−	22.8564i	0.510957	−	0.885004i
$668$		−	5.07180i		−	0.196234i
$669$		0			0
$670$	6.33975	+	3.66025i	0.244926	+	0.141408i
$671$		−	19.3923i		−	0.748632i
$672$		0			0
$673$	−21.8205	−	37.7942i	−0.841119	−	1.45686i	−0.888950	−	0.458005i	$-0.848564\pi$
$673$	−21.8205	−	37.7942i	−0.841119	−	1.45686i	0.0478308	−	0.998855i	$-0.484769\pi$
$674$	9.52628	−	5.50000i	0.366939	−	0.211852i
$675$		0			0
$676$	−11.0000	+	6.92820i	−0.423077	+	0.266469i
$677$	−43.2679			−1.66292			−0.831461	−	0.555583i	$-0.812495\pi$
$677$	−43.2679			−1.66292			−0.831461	+	0.555583i	$0.812495\pi$
$678$		0			0
$679$	−7.09808	−	12.2942i	−0.272399	−	0.471809i
$680$	0.830127	−	1.43782i	0.0318339	−	0.0551380i
$681$		0			0
$682$	−14.7058	−	8.49038i	−0.563113	−	0.325113i
$683$	15.0000	+	8.66025i	0.573959	+	0.331375i	0.758729	−	0.651406i	$-0.225821\pi$
$683$	15.0000	+	8.66025i	0.573959	+	0.331375i	−0.184770	+	0.982782i	$0.559154\pi$
$684$		0			0
$685$	7.12436	−	12.3397i	0.272208	−	0.471477i
$686$	−0.500000	−	0.866025i	−0.0190901	−	0.0330650i
$687$		0			0
$688$	8.73205			0.332906
$689$	8.59808	−	34.7487i	0.327561	−	1.32382i
$690$		0			0
$691$	32.1051	−	18.5359i	1.22134	−	0.705139i	0.256133	−	0.966641i	$-0.417551\pi$
$691$	32.1051	−	18.5359i	1.22134	−	0.705139i	0.965203	+	0.261503i	$0.0842180\pi$
$692$	1.09808	+	1.90192i	0.0417426	+	0.0723003i
$693$		0			0
$694$		−	6.46410i		−	0.245374i
$695$	10.8109	+	6.24167i	0.410080	+	0.236760i
$696$		0			0
$697$		−	19.1962i		−	0.727106i
$698$	−3.07180	+	5.32051i	−0.116269	+	0.201384i
$699$		0			0
$700$	3.86603	−	2.23205i	0.146122	−	0.0843636i
$701$	25.7846			0.973871			0.486936	−	0.873438i	$-0.338115\pi$
$701$	25.7846			0.973871			0.486936	+	0.873438i	$0.338115\pi$
$702$		0			0
$703$	−3.12436			−0.117837
$704$	4.50000	−	2.59808i	0.169600	−	0.0979187i
$705$		0			0
$706$	15.6603	−	27.1244i	0.589381	−	1.02084i
$707$		−	4.92820i		−	0.185344i
$708$		0			0
$709$	44.4904	+	25.6865i	1.67087	+	0.964678i	0.967152	+	0.254200i	$0.0818123\pi$
$709$	44.4904	+	25.6865i	1.67087	+	0.964678i	0.703720	+	0.710478i	$0.251521\pi$
$710$		−	5.32051i		−	0.199675i
$711$		0			0
$712$	8.42820	+	14.5981i	0.315860	+	0.547086i
$713$	21.1244	−	12.1962i	0.791113	−	0.456749i
$714$		0			0
$715$	13.1769	−	3.80385i	0.492789	−	0.142256i
$716$	3.46410			0.129460
$717$		0			0
$718$	−17.2224	−	29.8301i	−0.642735	−	1.11325i
$719$	−19.7224	+	34.1603i	−0.735523	+	1.27396i	0.218971	+	0.975731i	$0.429730\pi$
$719$	−19.7224	+	34.1603i	−0.735523	+	1.27396i	−0.954494	+	0.298231i	$0.903603\pi$
$720$		0			0
$721$	14.0263	+	8.09808i	0.522366	+	0.301588i
$722$	16.2679	+	9.39230i	0.605430	+	0.349545i
$723$		0			0
$724$	7.13397	−	12.3564i	0.265132	−	0.459222i
$725$	−7.89230	−	13.6699i	−0.293113	−	0.507686i
$726$		0			0
$727$	3.60770			0.133802			0.0669010	−	0.997760i	$-0.478689\pi$
$727$	3.60770			0.133802			0.0669010	+	0.997760i	$0.478689\pi$
$728$	2.59808	+	2.50000i	0.0962911	+	0.0926562i
$729$		0			0
$730$	0.928203	−	0.535898i	0.0343543	−	0.0198345i
$731$	9.90192	+	17.1506i	0.366236	+	0.634339i
$732$		0			0
$733$		−	25.7846i		−	0.952376i	−0.879343	−	0.476188i	$-0.842018\pi$
$733$		−	25.7846i		−	0.952376i	0.879343	−	0.476188i	$-0.157982\pi$
$734$	11.5359	+	6.66025i	0.425798	+	0.245834i
$735$		0			0
$736$			7.46410i			0.275130i
$737$	−25.9808	+	45.0000i	−0.957014	+	1.65760i
$738$		0			0
$739$	−33.9282	+	19.5885i	−1.24807	+	0.720573i	−0.970724	−	0.240197i	$-0.922788\pi$
$739$	−33.9282	+	19.5885i	−1.24807	+	0.720573i	−0.277345	+	0.960770i	$0.589455\pi$
$740$	−4.92820			−0.181164
$741$		0			0
$742$	−9.92820			−0.364476
$743$	−12.1699	+	7.02628i	−0.446469	+	0.257769i	−0.706338	−	0.707875i	$-0.749654\pi$
$743$	−12.1699	+	7.02628i	−0.446469	+	0.257769i	0.259869	+	0.965644i	$0.416321\pi$
$744$		0			0
$745$	2.92820	−	5.07180i	0.107281	−	0.185816i
$746$			15.1244i			0.553742i
$747$		0			0
$748$	10.2058	+	5.89230i	0.373160	+	0.215444i
$749$		−	6.07180i		−	0.221859i
$750$		0			0
$751$	2.57180	+	4.45448i	0.0938462	+	0.162546i	0.909126	−	0.416520i	$-0.136750\pi$
$751$	2.57180	+	4.45448i	0.0938462	+	0.162546i	−0.815280	+	0.579067i	$0.803417\pi$
$752$	−3.40192	+	1.96410i	−0.124055	+	0.0716234i
$753$		0			0
$754$	8.83975	−	9.18653i	0.321925	−	0.334554i
$755$	−0.875644			−0.0318680
$756$		0			0
$757$	−8.09808	−	14.0263i	−0.294330	−	0.509794i	0.680499	−	0.732749i	$-0.261763\pi$
$757$	−8.09808	−	14.0263i	−0.294330	−	0.509794i	−0.974829	+	0.222955i	$0.928430\pi$
$758$	18.0263	−	31.2224i	0.654744	−	1.13405i
$759$		0			0
$760$	−0.294229	−	0.169873i	−0.0106728	−	0.00616194i
$761$	15.8038	+	9.12436i	0.572889	+	0.330758i	0.758302	−	0.651903i	$-0.226029\pi$
$761$	15.8038	+	9.12436i	0.572889	+	0.330758i	−0.185413	+	0.982661i	$0.559362\pi$
$762$		0			0
$763$	−5.63397	+	9.75833i	−0.203964	+	0.353275i
$764$	4.09808	+	7.09808i	0.148263	+	0.256799i
$765$		0			0
$766$	31.0000			1.12008
$767$	−22.3205	+	23.1962i	−0.805947	+	0.837565i
$768$		0			0
$769$	−37.4711	+	21.6340i	−1.35124	+	0.780141i	−0.988424	−	0.151718i	$-0.951519\pi$
$769$	−37.4711	+	21.6340i	−1.35124	+	0.780141i	−0.362820	+	0.931859i	$0.618186\pi$
$770$	−1.90192	−	3.29423i	−0.0685406	−	0.118716i
$771$		0			0
$772$		−	7.58846i		−	0.273115i
$773$	−16.6077	−	9.58846i	−0.597337	−	0.344873i	0.170656	−	0.985331i	$-0.445411\pi$
$773$	−16.6077	−	9.58846i	−0.597337	−	0.344873i	−0.767993	+	0.640458i	$0.778745\pi$
$774$		0			0
$775$		−	14.5885i		−	0.524033i
$776$	7.09808	−	12.2942i	0.254806	−	0.441337i
$777$		0			0
$778$	18.5885	−	10.7321i	0.666428	−	0.384763i
$779$	−3.92820			−0.140742
$780$		0			0
$781$	37.7654			1.35135
$782$	−14.6603	+

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)

\(f(q)\)	\(=\)	\(q+(-0.866025 + 0.500000i) q^{2} +(0.500000 - 0.866025i) q^{4} -0.732051i q^{5} +(0.866025 + 0.500000i) q^{7} +1.00000i q^{8} +O(q^{10})\)	\(q+(-0.866025 + 0.500000i) q^{2} +(0.500000 - 0.866025i) q^{4} -0.732051i q^{5} +(0.866025 + 0.500000i) q^{7} +1.00000i q^{8} +(0.366025 + 0.633975i) q^{10} +(-4.50000 + 2.59808i) q^{11} +(0.866025 - 3.50000i) q^{13} -1.00000 q^{14} +(-0.500000 - 0.866025i) q^{16} +(1.13397 - 1.96410i) q^{17} +(-0.401924 - 0.232051i) q^{19} +(-0.633975 - 0.366025i) q^{20} +(2.59808 - 4.50000i) q^{22} +(3.73205 + 6.46410i) q^{23} +4.46410 q^{25} +(1.00000 + 3.46410i) q^{26} +(0.866025 - 0.500000i) q^{28} +(-1.76795 - 3.06218i) q^{29} -3.26795i q^{31} +(0.866025 + 0.500000i) q^{32} +2.26795i q^{34} +(0.366025 - 0.633975i) q^{35} +(5.83013 - 3.36603i) q^{37} +0.464102 q^{38} +0.732051 q^{40} +(7.33013 - 4.23205i) q^{41} +(-4.36603 + 7.56218i) q^{43} +5.19615i q^{44} +(-6.46410 - 3.73205i) q^{46} -3.92820i q^{47} +(0.500000 + 0.866025i) q^{49} +(-3.86603 + 2.23205i) q^{50} +(-2.59808 - 2.50000i) q^{52} +9.92820 q^{53} +(1.90192 + 3.29423i) q^{55} +(-0.500000 + 0.866025i) q^{56} +(3.06218 + 1.76795i) q^{58} +(-7.73205 - 4.46410i) q^{59} +(-1.86603 + 3.23205i) q^{61} +(1.63397 + 2.83013i) q^{62} -1.00000 q^{64} +(-2.56218 - 0.633975i) q^{65} +(8.66025 - 5.00000i) q^{67} +(-1.13397 - 1.96410i) q^{68} +0.732051i q^{70} +(-6.29423 - 3.63397i) q^{71} -1.46410i q^{73} +(-3.36603 + 5.83013i) q^{74} +(-0.401924 + 0.232051i) q^{76} -5.19615 q^{77} +9.00000 q^{79} +(-0.633975 + 0.366025i) q^{80} +(-4.23205 + 7.33013i) q^{82} -9.26795i q^{83} +(-1.43782 - 0.830127i) q^{85} -8.73205i q^{86} +(-2.59808 - 4.50000i) q^{88} +(14.5981 - 8.42820i) q^{89} +(2.50000 - 2.59808i) q^{91} +7.46410 q^{92} +(1.96410 + 3.40192i) q^{94} +(-0.169873 + 0.294229i) q^{95} +(-12.2942 - 7.09808i) q^{97} +(-0.866025 - 0.500000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{4}+O(q^{10}) \) 4 * q + 2 * q^4	\( 4 q + 2 q^{4} - 2 q^{10} - 18 q^{11} - 4 q^{14} - 2 q^{16} + 8 q^{17} - 12 q^{19} - 6 q^{20} + 8 q^{23} + 4 q^{25} + 4 q^{26} - 14 q^{29} - 2 q^{35} + 6 q^{37} - 12 q^{38} - 4 q^{40} + 12 q^{41} - 14 q^{43} - 12 q^{46} + 2 q^{49} - 12 q^{50} + 12 q^{53} + 18 q^{55} - 2 q^{56} - 12 q^{58} - 24 q^{59} - 4 q^{61} + 10 q^{62} - 4 q^{64} + 14 q^{65} - 8 q^{68} + 6 q^{71} - 10 q^{74} - 12 q^{76} + 36 q^{79} - 6 q^{80} - 10 q^{82} - 30 q^{85} + 48 q^{89} + 10 q^{91} + 16 q^{92} - 6 q^{94} - 18 q^{95} - 18 q^{97}+O(q^{100}) \) 4 * q + 2 * q^4 - 2 * q^10 - 18 * q^11 - 4 * q^14 - 2 * q^16 + 8 * q^17 - 12 * q^19 - 6 * q^20 + 8 * q^23 + 4 * q^25 + 4 * q^26 - 14 * q^29 - 2 * q^35 + 6 * q^37 - 12 * q^38 - 4 * q^40 + 12 * q^41 - 14 * q^43 - 12 * q^46 + 2 * q^49 - 12 * q^50 + 12 * q^53 + 18 * q^55 - 2 * q^56 - 12 * q^58 - 24 * q^59 - 4 * q^61 + 10 * q^62 - 4 * q^64 + 14 * q^65 - 8 * q^68 + 6 * q^71 - 10 * q^74 - 12 * q^76 + 36 * q^79 - 6 * q^80 - 10 * q^82 - 30 * q^85 + 48 * q^89 + 10 * q^91 + 16 * q^92 - 6 * q^94 - 18 * q^95 - 18 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more