LMFDB - Embedded newform 117.2.ba.a.98.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [117,2,Mod(71,117)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(117, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([6, 5]))

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

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

chi := DirichletCharacter("117.71");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 117 = 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	117.ba (of order $12$, degree $4$, 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:	$0.934249703649$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$2$ over $\Q(\zeta_{12})$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			98.2
Root			$-0.965926 + 0.258819i$ of defining polynomial
Character	$\chi$	$=$	117.98
Dual form			117.2.ba.a.80.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.517638 + 1.93185i) q^{2} +(-1.73205 + 1.00000i) q^{4} +(-1.93185 + 1.93185i) q^{5} +(-0.500000 - 0.133975i) q^{7} +O(q^{10})$	\(q+(0.517638 + 1.93185i) q^{2} +(-1.73205 + 1.00000i) q^{4} +(-1.93185 + 1.93185i) q^{5} +(-0.500000 - 0.133975i) q^{7} +(-4.73205 - 2.73205i) q^{10} +(4.05317 - 1.08604i) q^{11} +(3.59808 + 0.232051i) q^{13} -1.03528i q^{14} +(-2.00000 + 3.46410i) q^{16} +(-3.34607 - 5.79555i) q^{17} +(1.63397 - 6.09808i) q^{19} +(1.41421 - 5.27792i) q^{20} +(4.19615 + 7.26795i) q^{22} +(-1.22474 + 2.12132i) q^{23} -2.46410i q^{25} +(1.41421 + 7.07107i) q^{26} +(1.00000 - 0.267949i) q^{28} +(-1.22474 - 0.707107i) q^{29} +(4.63397 + 4.63397i) q^{31} +(-7.72741 - 2.07055i) q^{32} +(9.46410 - 9.46410i) q^{34} +(1.22474 - 0.707107i) q^{35} +(0.830127 + 3.09808i) q^{37} +12.6264 q^{38} +(-0.378937 - 1.41421i) q^{41} +(-6.69615 + 3.86603i) q^{43} +(-5.93426 + 5.93426i) q^{44} +(-4.73205 - 1.26795i) q^{46} +(-2.31079 - 2.31079i) q^{47} +(-5.83013 - 3.36603i) q^{49} +(4.76028 - 1.27551i) q^{50} +(-6.46410 + 3.19615i) q^{52} -5.93426i q^{53} +(-5.73205 + 9.92820i) q^{55} +(0.732051 - 2.73205i) q^{58} +(0.0507680 - 0.189469i) q^{59} +(-6.59808 - 11.4282i) q^{61} +(-6.55343 + 11.3509i) q^{62} -8.00000i q^{64} +(-7.39924 + 6.50266i) q^{65} +(-7.59808 + 2.03590i) q^{67} +(11.5911 + 6.69213i) q^{68} +(2.00000 + 2.00000i) q^{70} +(15.1266 + 4.05317i) q^{71} +(2.90192 - 2.90192i) q^{73} +(-5.55532 + 3.20736i) q^{74} +(3.26795 + 12.1962i) q^{76} -2.17209 q^{77} +7.19615 q^{79} +(-2.82843 - 10.5558i) q^{80} +(2.53590 - 1.46410i) q^{82} +(-5.27792 + 5.27792i) q^{83} +(17.6603 + 4.73205i) q^{85} +(-10.9348 - 10.9348i) q^{86} +(4.38134 - 1.17398i) q^{89} +(-1.76795 - 0.598076i) q^{91} -4.89898i q^{92} +(3.26795 - 5.66025i) q^{94} +(8.62398 + 14.9372i) q^{95} +(-1.30385 + 4.86603i) q^{97} +(3.48477 - 13.0053i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{7}+O(q^{10}) $ 8 * q - 4 * q^7	$ 8 q - 4 q^{7} - 24 q^{10} + 8 q^{13} - 16 q^{16} + 20 q^{19} - 8 q^{22} + 8 q^{28} + 44 q^{31} + 48 q^{34} - 28 q^{37} - 12 q^{43} - 24 q^{46} - 12 q^{49} - 24 q^{52} - 32 q^{55} - 8 q^{58} - 32 q^{61} - 40 q^{67} + 16 q^{70} + 44 q^{73} + 40 q^{76} + 16 q^{79} + 48 q^{82} + 72 q^{85} - 28 q^{91} + 40 q^{94} - 52 q^{97}+O(q^{100}) $ 8 * q - 4 * q^7 - 24 * q^10 + 8 * q^13 - 16 * q^16 + 20 * q^19 - 8 * q^22 + 8 * q^28 + 44 * q^31 + 48 * q^34 - 28 * q^37 - 12 * q^43 - 24 * q^46 - 12 * q^49 - 24 * q^52 - 32 * q^55 - 8 * q^58 - 32 * q^61 - 40 * q^67 + 16 * q^70 + 44 * q^73 + 40 * q^76 + 16 * q^79 + 48 * q^82 + 72 * q^85 - 28 * q^91 + 40 * q^94 - 52 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/117\mathbb{Z}\right)^\times$.

$n$	$28$	$92$
$\chi(n)$	$e\left(\frac{11}{12}\right)$	$-1$

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 $

$2$	0.517638	+	1.93185i	0.366025	+	1.36603i	0.866025	+	0.500000i	$0.166667\pi$
$2$	0.517638	+	1.93185i	0.366025	+	1.36603i	−0.500000	+	0.866025i	$0.666667\pi$
$3$		0			0
$3$		0			0
$4$	−1.73205	+	1.00000i	−0.866025	+	0.500000i
$5$	−1.93185	+	1.93185i	−0.863950	+	0.863950i	−0.991794	−	0.127844i	$-0.959194\pi$
$5$	−1.93185	+	1.93185i	−0.863950	+	0.863950i	0.127844	+	0.991794i	$0.459194\pi$
$6$		0			0
$7$	−0.500000	−	0.133975i	−0.188982	−	0.0506376i	0.163087	−	0.986612i	$-0.447855\pi$
$7$	−0.500000	−	0.133975i	−0.188982	−	0.0506376i	−0.352069	+	0.935974i	$0.614522\pi$
$8$		0			0
$9$		0			0
$10$	−4.73205	−	2.73205i	−1.49641	−	0.863950i
$11$	4.05317	−	1.08604i	1.22208	−	0.327455i	0.410588	−	0.911821i	$-0.365324\pi$
$11$	4.05317	−	1.08604i	1.22208	−	0.327455i	0.811490	+	0.584367i	$0.198657\pi$
$12$		0			0
$13$	3.59808	+	0.232051i	0.997927	+	0.0643593i
$13$	3.59808	+	0.232051i	0.997927	+	0.0643593i
$14$		−	1.03528i		−	0.276689i
$15$		0			0
$16$	−2.00000	+	3.46410i	−0.500000	+	0.866025i
$17$	−3.34607	−	5.79555i	−0.811540	−	1.40563i	−0.911786	−	0.410666i	$-0.865296\pi$
$17$	−3.34607	−	5.79555i	−0.811540	−	1.40563i	0.100246	−	0.994963i	$-0.468037\pi$
$18$		0			0
$19$	1.63397	−	6.09808i	0.374859	−	1.39899i	−0.478691	−	0.877984i	$-0.658888\pi$
$19$	1.63397	−	6.09808i	0.374859	−	1.39899i	0.853550	−	0.521011i	$-0.174445\pi$
$20$	1.41421	−	5.27792i	0.316228	−	1.18018i
$21$		0			0
$22$	4.19615	+	7.26795i	0.894623	+	1.54953i
$23$	−1.22474	+	2.12132i	−0.255377	+	0.442326i	−0.964998	−	0.262258i	$-0.915533\pi$
$23$	−1.22474	+	2.12132i	−0.255377	+	0.442326i	0.709621	+	0.704584i	$0.248866\pi$
$24$		0			0
$25$		−	2.46410i		−	0.492820i
$26$	1.41421	+	7.07107i	0.277350	+	1.38675i
$27$		0			0
$28$	1.00000	−	0.267949i	0.188982	−	0.0506376i
$29$	−1.22474	−	0.707107i	−0.227429	−	0.131306i	0.381956	−	0.924180i	$-0.375251\pi$
$29$	−1.22474	−	0.707107i	−0.227429	−	0.131306i	−0.609386	+	0.792874i	$0.708584\pi$
$30$		0			0
$31$	4.63397	+	4.63397i	0.832286	+	0.832286i	0.987829	−	0.155543i	$-0.0497126\pi$
$31$	4.63397	+	4.63397i	0.832286	+	0.832286i	−0.155543	+	0.987829i	$0.549713\pi$
$32$	−7.72741	−	2.07055i	−1.36603	−	0.366025i
$33$		0			0
$34$	9.46410	−	9.46410i	1.62308	−	1.62308i
$35$	1.22474	−	0.707107i	0.207020	−	0.119523i
$36$		0			0
$37$	0.830127	+	3.09808i	0.136472	+	0.509321i	0.999988	+	0.00499824i	$0.00159099\pi$
$37$	0.830127	+	3.09808i	0.136472	+	0.509321i	−0.863515	+	0.504322i	$0.831742\pi$
$38$	12.6264			2.04827
$39$		0			0
$40$		0			0
$41$	−0.378937	−	1.41421i	−0.0591801	−	0.220863i	0.930002	−	0.367554i	$-0.119805\pi$
$41$	−0.378937	−	1.41421i	−0.0591801	−	0.220863i	−0.989182	+	0.146690i	$0.953138\pi$
$42$		0			0
$43$	−6.69615	+	3.86603i	−1.02115	+	0.589563i	−0.914438	−	0.404726i	$-0.867367\pi$
$43$	−6.69615	+	3.86603i	−1.02115	+	0.589563i	−0.106716	+	0.994290i	$0.534033\pi$
$44$	−5.93426	+	5.93426i	−0.894623	+	0.894623i
$45$		0			0
$46$	−4.73205	−	1.26795i	−0.697703	−	0.186949i
$47$	−2.31079	−	2.31079i	−0.337063	−	0.337063i	0.518198	−	0.855261i	$-0.326603\pi$
$47$	−2.31079	−	2.31079i	−0.337063	−	0.337063i	−0.855261	+	0.518198i	$0.826603\pi$
$48$		0			0
$49$	−5.83013	−	3.36603i	−0.832875	−	0.480861i
$50$	4.76028	−	1.27551i	0.673205	−	0.180385i
$51$		0			0
$52$	−6.46410	+	3.19615i	−0.896410	+	0.443227i
$53$		−	5.93426i		−	0.815133i	−0.913176	−	0.407566i	$-0.866378\pi$
$53$		−	5.93426i		−	0.815133i	0.913176	−	0.407566i	$-0.133622\pi$
$54$		0			0
$55$	−5.73205	+	9.92820i	−0.772910	+	1.33872i
$56$		0			0
$57$		0			0
$58$	0.732051	−	2.73205i	0.0961230	−	0.358736i
$59$	0.0507680	−	0.189469i	0.00660943	−	0.0246667i	−0.962542	−	0.271131i	$-0.912602\pi$
$59$	0.0507680	−	0.189469i	0.00660943	−	0.0246667i	0.969152	+	0.246465i	$0.0792689\pi$
$60$		0			0
$61$	−6.59808	−	11.4282i	−0.844797	−	1.46323i	−0.885797	−	0.464072i	$-0.846388\pi$
$61$	−6.59808	−	11.4282i	−0.844797	−	1.46323i	0.0410002	−	0.999159i	$-0.486946\pi$
$62$	−6.55343	+	11.3509i	−0.832286	+	1.44156i
$63$		0			0
$64$		−	8.00000i		−	1.00000i
$65$	−7.39924	+	6.50266i	−0.917762	+	0.806556i
$66$		0			0
$67$	−7.59808	+	2.03590i	−0.928253	+	0.248725i	−0.691109	−	0.722750i	$-0.742878\pi$
$67$	−7.59808	+	2.03590i	−0.928253	+	0.248725i	−0.237143	+	0.971475i	$0.576211\pi$
$68$	11.5911	+	6.69213i	1.40563	+	0.811540i
$69$		0			0
$70$	2.00000	+	2.00000i	0.239046	+	0.239046i
$71$	15.1266	+	4.05317i	1.79520	+	0.481023i	0.993212	−	0.116315i	$-0.0371081\pi$
$71$	15.1266	+	4.05317i	1.79520	+	0.481023i	0.801990	+	0.597338i	$0.203775\pi$
$72$		0			0
$73$	2.90192	−	2.90192i	0.339644	−	0.339644i	−0.516589	−	0.856233i	$-0.672798\pi$
$73$	2.90192	−	2.90192i	0.339644	−	0.339644i	0.856233	+	0.516589i	$0.172798\pi$
$74$	−5.55532	+	3.20736i	−0.645793	+	0.372849i
$75$		0			0
$76$	3.26795	+	12.1962i	0.374859	+	1.39899i
$77$	−2.17209			−0.247532
$78$		0			0
$79$	7.19615			0.809630			0.404815	−	0.914399i	$-0.367336\pi$
$79$	7.19615			0.809630			0.404815	+	0.914399i	$0.367336\pi$
$80$	−2.82843	−	10.5558i	−0.316228	−	1.18018i
$81$		0			0
$82$	2.53590	−	1.46410i	0.280043	−	0.161683i
$83$	−5.27792	+	5.27792i	−0.579327	+	0.579327i	−0.934718	−	0.355391i	$-0.884348\pi$
$83$	−5.27792	+	5.27792i	−0.579327	+	0.579327i	0.355391	+	0.934718i	$0.384348\pi$
$84$		0			0
$85$	17.6603	+	4.73205i	1.91552	+	0.513263i
$86$	−10.9348	−	10.9348i	−1.17913	−	1.17913i
$87$		0			0
$88$		0			0
$89$	4.38134	−	1.17398i	0.464421	−	0.124441i	−0.0190181	−	0.999819i	$-0.506054\pi$
$89$	4.38134	−	1.17398i	0.464421	−	0.124441i	0.483439	+	0.875378i	$0.339387\pi$
$90$		0			0
$91$	−1.76795	−	0.598076i	−0.185331	−	0.0626954i
$92$		−	4.89898i		−	0.510754i
$93$		0			0
$94$	3.26795	−	5.66025i	0.337063	−	0.583811i
$95$	8.62398	+	14.9372i	0.884802	+	1.53252i
$96$		0			0
$97$	−1.30385	+	4.86603i	−0.132386	+	0.494070i	−0.999995	−	0.00317651i	$-0.998989\pi$
$97$	−1.30385	+	4.86603i	−0.132386	+	0.494070i	0.867609	+	0.497247i	$0.165656\pi$
$98$	3.48477	−	13.0053i	0.352015	−	1.31374i
$99$		0			0
$100$	2.46410	+	4.26795i	0.246410	+	0.426795i
$101$	0.568406	−	0.984508i	0.0565585	−	0.0979622i	−0.836360	−	0.548181i	$-0.815321\pi$
$101$	0.568406	−	0.984508i	0.0565585	−	0.0979622i	0.892918	+	0.450219i	$0.148654\pi$
$102$		0			0
$103$			8.66025i			0.853320i	0.904412	+	0.426660i	$0.140310\pi$
$103$			8.66025i			0.853320i	−0.904412	+	0.426660i	$0.859690\pi$
$104$		0			0
$105$		0			0
$106$	11.4641	−	3.07180i	1.11349	−	0.298359i
$107$	1.46498	+	0.845807i	0.141625	+	0.0817673i	0.569138	−	0.822242i	$-0.307277\pi$
$107$	1.46498	+	0.845807i	0.141625	+	0.0817673i	−0.427513	+	0.904009i	$0.640610\pi$
$108$		0			0
$109$	−11.2942	−	11.2942i	−1.08179	−	1.08179i	−0.996343	−	0.0854483i	$-0.972768\pi$
$109$	−11.2942	−	11.2942i	−1.08179	−	1.08179i	−0.0854483	−	0.996343i	$-0.527232\pi$
$110$	−22.1469	−	5.93426i	−2.11163	−	0.565809i
$111$		0			0
$112$	1.46410	−	1.46410i	0.138345	−	0.138345i
$113$	3.91447	−	2.26002i	0.368242	−	0.212605i	−0.304448	−	0.952529i	$-0.598472\pi$
$113$	3.91447	−	2.26002i	0.368242	−	0.212605i	0.672690	+	0.739924i	$0.265139\pi$
$114$		0			0
$115$	−1.73205	−	6.46410i	−0.161515	−	0.602781i
$116$	2.82843			0.262613
$117$		0			0
$118$	0.392305			0.0361146
$119$	0.896575	+	3.34607i	0.0821889	+	0.306733i
$120$		0			0
$121$	5.72243	−	3.30385i	0.520221	−	0.300350i
$122$	18.6622	−	18.6622i	1.68959	−	1.68959i
$123$		0			0
$124$	−12.6603	−	3.39230i	−1.13692	−	0.304638i
$125$	−4.89898	−	4.89898i	−0.438178	−	0.438178i
$126$		0			0
$127$	8.59808	+	4.96410i	0.762956	+	0.440493i	0.830356	−	0.557233i	$-0.188137\pi$
$127$	8.59808	+	4.96410i	0.762956	+	0.440493i	−0.0674001	+	0.997726i	$0.521470\pi$
$128$		0			0
$129$		0			0
$130$	−16.3923	−	10.9282i	−1.43770	−	0.958467i
$131$		−	7.07107i		−	0.617802i	−0.951094	−	0.308901i	$-0.900039\pi$
$131$		−	7.07107i		−	0.617802i	0.951094	−	0.308901i	$-0.0999612\pi$
$132$		0			0
$133$	−1.63397	+	2.83013i	−0.141684	+	0.245403i
$134$	−7.86611	−	13.6245i	−0.679528	−	1.17698i
$135$		0			0
$136$		0			0
$137$	−3.86370	+	14.4195i	−0.330098	+	1.23194i	0.578988	+	0.815336i	$0.303448\pi$
$137$	−3.86370	+	14.4195i	−0.330098	+	1.23194i	−0.909086	+	0.416608i	$0.863219\pi$
$138$		0			0
$139$	−1.69615	−	2.93782i	−0.143866	−	0.249183i	0.785083	−	0.619390i	$-0.212620\pi$
$139$	−1.69615	−	2.93782i	−0.143866	−	0.249183i	−0.928949	+	0.370207i	$0.879287\pi$
$140$	−1.41421	+	2.44949i	−0.119523	+	0.207020i
$141$		0			0
$142$			31.3205i			2.62836i
$143$	14.8356	−	2.96713i	1.24062	−	0.248124i
$144$		0			0
$145$	3.73205	−	1.00000i	0.309930	−	0.0830455i
$146$	7.10823	+	4.10394i	0.588282	+	0.339644i
$147$		0			0
$148$	−4.53590	−	4.53590i	−0.372849	−	0.372849i
$149$	−6.17449	−	1.65445i	−0.505834	−	0.135538i	−0.00312781	−	0.999995i	$-0.500996\pi$
$149$	−6.17449	−	1.65445i	−0.505834	−	0.135538i	−0.502706	+	0.864457i	$0.667662\pi$
$150$		0			0
$151$	−8.46410	+	8.46410i	−0.688799	+	0.688799i	−0.961966	−	0.273168i	$-0.911929\pi$
$151$	−8.46410	+	8.46410i	−0.688799	+	0.688799i	0.273168	+	0.961966i	$0.411929\pi$
$152$		0			0
$153$		0			0
$154$	−1.12436	−	4.19615i	−0.0906032	−	0.338136i
$155$	−17.9043			−1.43811
$156$		0			0
$157$	−11.3923			−0.909205			−0.454602	−	0.890694i	$-0.650219\pi$
$157$	−11.3923			−0.909205			−0.454602	+	0.890694i	$0.650219\pi$
$158$	3.72500	+	13.9019i	0.296345	+	1.10598i
$159$		0			0
$160$	18.9282	−	10.9282i	1.49641	−	0.863950i
$161$	0.896575	−	0.896575i	0.0706600	−	0.0706600i
$162$		0			0
$163$	16.0622	+	4.30385i	1.25809	+	0.337103i	0.825455	−	0.564467i	$-0.190918\pi$
$163$	16.0622	+	4.30385i	1.25809	+	0.337103i	0.432632	+	0.901571i	$0.357585\pi$
$164$	2.07055	+	2.07055i	0.161683	+	0.161683i
$165$		0			0
$166$	−12.9282	−	7.46410i	−1.00342	−	0.579327i
$167$	5.27792	−	1.41421i	0.408417	−	0.109435i	−0.0487602	−	0.998811i	$-0.515527\pi$
$167$	5.27792	−	1.41421i	0.408417	−	0.109435i	0.457178	+	0.889375i	$0.348860\pi$
$168$		0			0
$169$	12.8923	+	1.66987i	0.991716	+	0.128452i
$170$			36.5665i			2.80452i
$171$		0			0
$172$	7.73205	−	13.3923i	0.589563	−	1.02115i
$173$	7.02030	+	12.1595i	0.533744	+	0.924471i	0.999223	+	0.0394122i	$0.0125486\pi$
$173$	7.02030	+	12.1595i	0.533744	+	0.924471i	−0.465480	+	0.885059i	$0.654118\pi$
$174$		0			0
$175$	−0.330127	+	1.23205i	−0.0249553	+	0.0931343i
$176$	−4.34418	+	16.2127i	−0.327455	+	1.22208i
$177$		0			0
$178$	4.53590	+	7.85641i	0.339980	+	0.588863i
$179$	−8.24504	+	14.2808i	−0.616264	+	1.06740i	0.373898	+	0.927470i	$0.378021\pi$
$179$	−8.24504	+	14.2808i	−0.616264	+	1.06740i	−0.990161	+	0.139930i	$0.955312\pi$
$180$		0			0
$181$			6.00000i			0.445976i	0.974821	+	0.222988i	$0.0715812\pi$
$181$			6.00000i			0.445976i	−0.974821	+	0.222988i	$0.928419\pi$
$182$	0.240237	−	3.72500i	0.0178075	−	0.276116i
$183$		0			0
$184$		0			0
$185$	−7.58871	−	4.38134i	−0.557933	−	0.322123i
$186$		0			0
$187$	−19.8564	−	19.8564i	−1.45204	−	1.45204i
$188$	6.31319	+	1.69161i	0.460437	+	0.123374i
$189$		0			0
$190$	−24.3923	+	24.3923i	−1.76960	+	1.76960i
$191$	11.2629	−	6.50266i	0.814958	−	0.470516i	−0.0337168	−	0.999431i	$-0.510734\pi$
$191$	11.2629	−	6.50266i	0.814958	−	0.470516i	0.848675	+	0.528915i	$0.177401\pi$
$192$		0			0
$193$	0.303848	+	1.13397i	0.0218714	+	0.0816253i	0.975999	−	0.217775i	$-0.0698800\pi$
$193$	0.303848	+	1.13397i	0.0218714	+	0.0816253i	−0.954128	+	0.299401i	$0.903213\pi$
$194$	−10.0754			−0.723369
$195$		0			0
$196$	13.4641			0.961722
$197$	1.83032	+	6.83083i	0.130405	+	0.486677i	0.999975	−	0.00713319i	$-0.00227058\pi$
$197$	1.83032	+	6.83083i	0.130405	+	0.486677i	−0.869570	+	0.493810i	$0.835604\pi$
$198$		0			0
$199$	−0.401924	+	0.232051i	−0.0284916	+	0.0164496i	−0.514178	−	0.857683i	$-0.671903\pi$
$199$	−0.401924	+	0.232051i	−0.0284916	+	0.0164496i	0.485687	+	0.874133i	$0.338570\pi$
$200$		0			0
$201$		0			0
$202$	2.19615	+	0.588457i	0.154521	+	0.0414037i
$203$	0.517638	+	0.517638i	0.0363311	+	0.0363311i
$204$		0			0
$205$	3.46410	+	2.00000i	0.241943	+	0.139686i
$206$	−16.7303	+	4.48288i	−1.16566	+	0.312337i
$207$		0			0
$208$	−8.00000	+	12.0000i	−0.554700	+	0.832050i
$209$		−	26.4911i		−	1.83243i
$210$		0			0
$211$	7.59808	−	13.1603i	0.523073	−	0.905989i	−0.476566	−	0.879139i	$-0.658119\pi$
$211$	7.59808	−	13.1603i	0.523073	−	0.905989i	0.999639	−	0.0268507i	$-0.00854788\pi$
$212$	5.93426	+	10.2784i	0.407566	+	0.705926i
$213$		0			0
$214$	−0.875644	+	3.26795i	−0.0598578	+	0.223392i
$215$	5.46739	−	20.4046i	0.372873	−	1.39158i
$216$		0			0
$217$	−1.69615	−	2.93782i	−0.115142	−	0.199432i
$218$	15.9725	−	27.6651i	1.08179	−	1.87372i
$219$		0			0
$220$		−	22.9282i		−	1.54582i
$221$	−10.6945	−	21.6293i	−0.719392	−	1.45494i
$222$		0			0
$223$	−23.7583	+	6.36603i	−1.59098	+	0.426301i	−0.942301	−	0.334766i	$-0.891343\pi$
$223$	−23.7583	+	6.36603i	−1.59098	+	0.426301i	−0.648674	+	0.761066i	$0.724676\pi$
$224$	3.58630	+	2.07055i	0.239620	+	0.138345i
$225$		0			0
$226$	6.39230	+	6.39230i	0.425210	+	0.425210i
$227$	6.31319	+	1.69161i	0.419021	+	0.112276i	0.462168	−	0.886792i	$-0.347072\pi$
$227$	6.31319	+	1.69161i	0.419021	+	0.112276i	−0.0431468	+	0.999069i	$0.513738\pi$
$228$		0			0
$229$	−3.73205	+	3.73205i	−0.246621	+	0.246621i	−0.819582	−	0.572961i	$-0.805794\pi$
$229$	−3.73205	+	3.73205i	−0.246621	+	0.246621i	0.572961	+	0.819582i	$0.305794\pi$
$230$	11.5911	−	6.69213i	0.764295	−	0.441266i
$231$		0			0
$232$		0			0
$233$	−18.9396			−1.24077			−0.620387	−	0.784296i	$-0.713024\pi$
$233$	−18.9396			−1.24077			−0.620387	+	0.784296i	$0.713024\pi$
$234$		0			0
$235$	8.92820			0.582412
$236$	0.101536	+	0.378937i	0.00660943	+	0.0246667i
$237$		0			0
$238$	−6.00000	+	3.46410i	−0.388922	+	0.224544i
$239$	−4.62158	+	4.62158i	−0.298945	+	0.298945i	−0.840601	−	0.541656i	$-0.817798\pi$
$239$	−4.62158	+	4.62158i	−0.298945	+	0.298945i	0.541656	+	0.840601i	$0.317798\pi$
$240$		0			0
$241$	−11.2942	−	3.02628i	−0.727525	−	0.194940i	−0.123998	−	0.992282i	$-0.539572\pi$
$241$	−11.2942	−	3.02628i	−0.727525	−	0.194940i	−0.603527	+	0.797343i	$0.706238\pi$
$242$	9.34469	+	9.34469i	0.600700	+	0.600700i
$243$		0			0
$244$	22.8564	+	13.1962i	1.46323	+	0.844797i
$245$	17.7656	−	4.76028i	1.13500	−	0.304123i
$246$		0			0
$247$	7.29423	−	21.5622i	0.464121	−	1.37197i
$248$		0			0
$249$		0			0
$250$	6.92820	−	12.0000i	0.438178	−	0.758947i
$251$	−10.6945	−	18.5235i	−0.675033	−	1.16919i	−0.976459	−	0.215702i	$-0.930796\pi$
$251$	−10.6945	−	18.5235i	−0.675033	−	1.16919i	0.301426	−	0.953490i	$-0.402537\pi$
$252$		0			0
$253$	−2.66025	+	9.92820i	−0.167249	+	0.624181i
$254$	−5.13922	+	19.1798i	−0.322463	+	1.20345i
$255$		0			0
$256$	−8.00000	−	13.8564i	−0.500000	−	0.866025i
$257$	−0.656339	+	1.13681i	−0.0409413	+	0.0709124i	−0.885770	−	0.464125i	$-0.846369\pi$
$257$	−0.656339	+	1.13681i	−0.0409413	+	0.0709124i	0.844829	+	0.535037i	$0.179702\pi$
$258$		0			0
$259$		−	1.66025i		−	0.103163i
$260$	6.31319	−	18.6622i	0.391528	−	1.15738i
$261$		0			0
$262$	13.6603	−	3.66025i	0.843933	−	0.226131i
$263$	−20.3166	−	11.7298i	−1.25278	−	0.723291i	−0.281117	−	0.959674i	$-0.590705\pi$
$263$	−20.3166	−	11.7298i	−1.25278	−	0.723291i	−0.971660	+	0.236382i	$0.924038\pi$
$264$		0			0
$265$	11.4641	+	11.4641i	0.704234	+	0.704234i
$266$	−6.31319	−	1.69161i	−0.387087	−	0.103720i
$267$		0			0
$268$	11.1244	−	11.1244i	0.679528	−	0.679528i
$269$	−17.8671	+	10.3156i	−1.08938	+	0.628953i	−0.933411	−	0.358809i	$-0.883183\pi$
$269$	−17.8671	+	10.3156i	−1.08938	+	0.628953i	−0.155968	+	0.987762i	$0.549850\pi$
$270$		0			0
$271$	2.62436	+	9.79423i	0.159418	+	0.594957i	0.998686	+	0.0512393i	$0.0163171\pi$
$271$	2.62436	+	9.79423i	0.159418	+	0.594957i	−0.839268	+	0.543718i	$0.817016\pi$
$272$	26.7685			1.62308
$273$		0			0
$274$	−29.8564			−1.80369
$275$	−2.67612	−	9.98743i	−0.161376	−	0.602265i
$276$		0			0
$277$	17.1962	−	9.92820i	1.03322	−	0.596528i	0.115312	−	0.993329i	$-0.463213\pi$
$277$	17.1962	−	9.92820i	1.03322	−	0.596528i	0.917904	+	0.396801i	$0.129880\pi$
$278$	4.79744	−	4.79744i	0.287732	−	0.287732i
$279$		0			0
$280$		0			0
$281$	16.6288	+	16.6288i	0.991990	+	0.991990i	0.999968	−	0.00797773i	$-0.00253942\pi$
$281$	16.6288	+	16.6288i	0.991990	+	0.991990i	−0.00797773	+	0.999968i	$0.502539\pi$
$282$		0			0
$283$	19.7942	+	11.4282i	1.17664	+	0.679336i	0.955236	−	0.295845i	$-0.0956012\pi$
$283$	19.7942	+	11.4282i	1.17664	+	0.679336i	0.221409	+	0.975181i	$0.428934\pi$
$284$	−30.2533	+	8.10634i	−1.79520	+	0.481023i
$285$		0			0
$286$	13.4115	+	27.1244i	0.793041	+	1.60390i
$287$			0.757875i			0.0447359i
$288$		0			0
$289$	−13.8923	+	24.0622i	−0.817194	+	1.41542i
$290$	3.86370	+	6.69213i	0.226884	+	0.392975i
$291$		0			0
$292$	−2.12436	+	7.92820i	−0.124319	+	0.463963i
$293$	−3.05506	+	11.4016i	−0.178479	+	0.666091i	0.817454	+	0.575993i	$0.195385\pi$
$293$	−3.05506	+	11.4016i	−0.178479	+	0.666091i	−0.995933	+	0.0900977i	$0.971282\pi$
$294$		0			0
$295$	0.267949	+	0.464102i	0.0156006	+	0.0270210i
$296$		0			0
$297$		0			0
$298$		−	12.7846i		−	0.740593i
$299$	−4.89898	+	7.34847i	−0.283315	+	0.424973i
$300$		0			0
$301$	3.86603	−	1.03590i	0.222834	−	0.0597082i
$302$	−20.7327	−	11.9700i	−1.19303	−	0.688799i
$303$		0			0
$304$	17.8564	+	17.8564i	1.02414	+	1.02414i
$305$	34.8241	+	9.33109i	1.99402	+	0.534297i
$306$		0			0
$307$	19.2942	−	19.2942i	1.10118	−	1.10118i	0.106911	−	0.994269i	$-0.465904\pi$
$307$	19.2942	−	19.2942i	1.10118	−	1.10118i	0.994269	−	0.106911i	$-0.0340961\pi$
$308$	3.76217	−	2.17209i	0.214369	−	0.123766i
$309$		0			0
$310$	−9.26795	−	34.5885i	−0.526384	−	1.96449i
$311$	11.1106			0.630026			0.315013	−	0.949087i	$-0.397991\pi$
$311$	11.1106			0.630026			0.315013	+	0.949087i	$0.397991\pi$
$312$		0			0
$313$	−3.19615			−0.180657			−0.0903286	−	0.995912i	$-0.528792\pi$
$313$	−3.19615			−0.180657			−0.0903286	+	0.995912i	$0.528792\pi$
$314$	−5.89709	−	22.0082i	−0.332792	−	1.24200i
$315$		0			0
$316$	−12.4641	+	7.19615i	−0.701160	+	0.404815i
$317$	10.5558	−	10.5558i	0.592875	−	0.592875i	−0.345532	−	0.938407i	$-0.612302\pi$
$317$	10.5558	−	10.5558i	0.592875	−	0.592875i	0.938407	+	0.345532i	$0.112302\pi$
$318$		0			0
$319$	−5.73205	−	1.53590i	−0.320933	−	0.0859938i
$320$	15.4548	+	15.4548i	0.863950	+	0.863950i
$321$		0			0
$322$	2.19615	+	1.26795i	0.122387	+	0.0706600i
$323$	−40.8091	+	10.9348i	−2.27068	+	0.608427i
$324$		0			0
$325$	0.571797	−	8.86603i	0.0317176	−	0.491799i
$326$			33.2576i			1.84197i
$327$		0			0
$328$		0			0
$329$	0.845807	+	1.46498i	0.0466309	+	0.0807670i
$330$		0			0
$331$	1.40192	−	5.23205i	0.0770567	−	0.287580i	−0.916635	−	0.399725i	$-0.869106\pi$
$331$	1.40192	−	5.23205i	0.0770567	−	0.287580i	0.993692	+	0.112146i	$0.0357723\pi$
$332$	3.86370	−	14.4195i	0.212048	−	0.791375i
$333$		0			0
$334$	5.46410	+	9.46410i	0.298982	+	0.517853i
$335$	10.7453	−	18.6114i	0.587079	−	1.01685i
$336$		0			0
$337$		−	27.9282i		−	1.52135i	−0.649135	−	0.760673i	$-0.724869\pi$
$337$		−	27.9282i		−	1.52135i	0.649135	−	0.760673i	$-0.275131\pi$
$338$	3.44760	+	25.7704i	0.187525	+	1.40173i
$339$		0			0
$340$	−35.3205	+	9.46410i	−1.91552	+	0.513263i
$341$	23.8150	+	13.7496i	1.28965	+	0.744582i
$342$		0			0
$343$	5.02628	+	5.02628i	0.271394	+	0.271394i
$344$		0			0
$345$		0			0
$346$	−19.8564	+	19.8564i	−1.06749	+	1.06749i
$347$	−25.2156	+	14.5582i	−1.35364	+	0.781527i	−0.988758	−	0.149525i	$-0.952226\pi$
$347$	−25.2156	+	14.5582i	−1.35364	+	0.781527i	−0.364887	+	0.931052i	$0.618892\pi$
$348$		0			0
$349$	−0.545517	−	2.03590i	−0.0292009	−	0.108979i	0.949787	−	0.312896i	$-0.101299\pi$
$349$	−0.545517	−	2.03590i	−0.0292009	−	0.108979i	−0.978988	+	0.203917i	$0.934633\pi$
$350$	−2.55103			−0.136358
$351$		0			0
$352$	−33.5692			−1.78925
$353$	−0.466870	−	1.74238i	−0.0248490	−	0.0927377i	0.952388	−	0.304890i	$-0.0986195\pi$
$353$	−0.466870	−	1.74238i	−0.0248490	−	0.0927377i	−0.977237	+	0.212152i	$0.931953\pi$
$354$		0			0
$355$	−37.0526	+	21.3923i	−1.96655	+	1.13539i
$356$	−6.41473	+	6.41473i	−0.339980	+	0.339980i
$357$		0			0
$358$	−31.8564	−	8.53590i	−1.68366	−	0.451136i
$359$	7.07107	+	7.07107i	0.373197	+	0.373197i	0.868640	−	0.495443i	$-0.164994\pi$
$359$	7.07107	+	7.07107i	0.373197	+	0.373197i	−0.495443	+	0.868640i	$0.664994\pi$
$360$		0			0
$361$	−18.0622	−	10.4282i	−0.950641	−	0.548853i
$362$	−11.5911	+	3.10583i	−0.609215	+	0.163239i
$363$		0			0
$364$	3.66025	−	0.732051i	0.191849	−	0.0383699i
$365$			11.2122i			0.586872i
$366$		0			0
$367$	2.69615	−	4.66987i	0.140738	−	0.243765i	−0.787037	−	0.616906i	$-0.788386\pi$
$367$	2.69615	−	4.66987i	0.140738	−	0.243765i	0.927775	+	0.373141i	$0.121719\pi$
$368$	−4.89898	−	8.48528i	−0.255377	−	0.442326i
$369$		0			0
$370$	4.53590	−	16.9282i	0.235810	−	0.880055i
$371$	−0.795040	+	2.96713i	−0.0412764	+	0.154046i
$372$		0			0
$373$	3.79423	+	6.57180i	0.196458	+	0.340275i	0.947377	−	0.320119i	$-0.103723\pi$
$373$	3.79423	+	6.57180i	0.196458	+	0.340275i	−0.750920	+	0.660394i	$0.770389\pi$
$374$	28.0812	−	48.6381i	1.45204	−	2.51501i
$375$		0			0
$376$		0			0
$377$	−4.24264	−	2.82843i	−0.218507	−	0.145671i
$378$		0			0
$379$	22.3564	−	5.99038i	1.14837	−	0.307705i	0.366059	−	0.930592i	$-0.380707\pi$
$379$	22.3564	−	5.99038i	1.14837	−	0.307705i	0.782312	+	0.622886i	$0.214040\pi$
$380$	−29.8744	−	17.2480i	−1.53252	−	0.884802i
$381$		0			0
$382$	18.3923	+	18.3923i	0.941032	+	0.941032i
$383$	9.65926	+	2.58819i	0.493565	+	0.132250i	0.497012	−	0.867744i	$-0.334431\pi$
$383$	9.65926	+	2.58819i	0.493565	+	0.132250i	−0.00344689	+	0.999994i	$0.501097\pi$
$384$		0			0
$385$	4.19615	−	4.19615i	0.213856	−	0.213856i
$386$	−2.03339	+	1.17398i	−0.103497	+	0.0597539i
$387$		0			0
$388$	−2.60770	−	9.73205i	−0.132386	−	0.494070i
$389$	6.69213			0.339304			0.169652	−	0.985504i	$-0.445736\pi$
$389$	6.69213			0.339304			0.169652	+	0.985504i	$0.445736\pi$
$390$		0			0
$391$	16.3923			0.828994
$392$		0			0
$393$		0			0
$394$	−12.2487	+	7.07180i	−0.617081	+	0.356272i
$395$	−13.9019	+	13.9019i	−0.699480	+	0.699480i
$396$		0			0
$397$	−15.3301	−	4.10770i	−0.769397	−	0.206159i	−0.147292	−	0.989093i	$-0.547056\pi$
$397$	−15.3301	−	4.10770i	−0.769397	−	0.206159i	−0.622105	+	0.782934i	$0.713722\pi$
$398$	−0.656339	−	0.656339i	−0.0328993	−	0.0328993i
$399$		0			0
$400$	8.53590	+	4.92820i	0.426795	+	0.246410i
$401$	38.4983	−	10.3156i	1.92251	−	0.515136i	0.935842	−	0.352419i	$-0.114641\pi$
$401$	38.4983	−	10.3156i	1.92251	−	0.515136i	0.986673	−	0.162717i	$-0.0520258\pi$
$402$		0			0
$403$	15.5981	+	17.7487i	0.776996	+	0.884126i
$404$			2.27362i			0.113117i
$405$		0			0
$406$	−0.732051	+	1.26795i	−0.0363311	+	0.0629273i
$407$	6.72930	+	11.6555i	0.333559	+	0.577741i
$408$		0			0
$409$	−4.42820	+	16.5263i	−0.218961	+	0.817172i	0.765774	+	0.643110i	$0.222356\pi$
$409$	−4.42820	+	16.5263i	−0.218961	+	0.817172i	−0.984735	+	0.174062i	$0.944311\pi$
$410$	−2.07055	+	7.72741i	−0.102257	+	0.381629i
$411$		0			0
$412$	−8.66025	−	15.0000i	−0.426660	−	0.738997i
$413$	−0.0507680	+	0.0879327i	−0.00249813	+	0.00432689i
$414$		0			0
$415$		−	20.3923i		−	1.00102i
$416$	−27.3233	−	9.24316i	−1.33964	−	0.453183i
$417$		0			0
$418$	51.1769	−	13.7128i	2.50314	−	0.670716i
$419$	−2.92996	−	1.69161i	−0.143138	−	0.0826408i	0.426721	−	0.904383i	$-0.359669\pi$
$419$	−2.92996	−	1.69161i	−0.143138	−	0.0826408i	−0.569859	+	0.821743i	$0.693002\pi$
$420$		0			0
$421$	15.3660	+	15.3660i	0.748894	+	0.748894i	0.974272	−	0.225377i	$-0.0723615\pi$
$421$	15.3660	+	15.3660i	0.748894	+	0.748894i	−0.225377	+	0.974272i	$0.572361\pi$
$422$	29.3567	+	7.86611i	1.42906	+	0.382916i
$423$		0			0
$424$		0			0
$425$	−14.2808	+	8.24504i	−0.692722	+	0.399943i
$426$		0			0
$427$	1.76795	+	6.59808i	0.0855571	+	0.319303i
$428$	−3.38323			−0.163535
$429$		0			0
$430$	42.2487			2.03741
$431$	4.10394	+	15.3161i	0.197680	+	0.737751i	0.991557	+	0.129673i	$0.0413927\pi$
$431$	4.10394	+	15.3161i	0.197680	+	0.737751i	−0.793877	+	0.608078i	$0.791941\pi$
$432$		0			0
$433$	−15.9904	+	9.23205i	−0.768449	+	0.443664i	−0.832321	−	0.554294i	$-0.812988\pi$
$433$	−15.9904	+	9.23205i	−0.768449	+	0.443664i	0.0638723	+	0.997958i	$0.479655\pi$
$434$	4.79744	−	4.79744i	0.230285	−	0.230285i
$435$		0			0
$436$	30.8564	+	8.26795i	1.47775	+	0.395963i
$437$	10.9348	+	10.9348i	0.523081	+	0.523081i
$438$		0			0
$439$	−8.30385	−	4.79423i	−0.396321	−	0.228816i	0.288574	−	0.957457i	$-0.406819\pi$
$439$	−8.30385	−	4.79423i	−0.396321	−	0.228816i	−0.684895	+	0.728641i	$0.740152\pi$
$440$		0			0
$441$		0			0
$442$	36.2487	−	31.8564i	1.72418	−	1.51525i
$443$		−	23.7370i		−	1.12778i	−0.825850	−	0.563890i	$-0.809304\pi$
$443$		−	23.7370i		−	1.12778i	0.825850	−	0.563890i	$-0.190696\pi$
$444$		0			0
$445$	−6.19615	+	10.7321i	−0.293726	+	0.508748i
$446$	−24.5964	−	42.6023i	−1.16467	−	2.01728i
$447$		0			0
$448$	−1.07180	+	4.00000i	−0.0506376	+	0.188982i
$449$	2.26002	−	8.43451i	0.106657	−	0.398049i	−0.891871	−	0.452290i	$-0.850607\pi$
$449$	2.26002	−	8.43451i	0.106657	−	0.398049i	0.998528	+	0.0542408i	$0.0172738\pi$
$450$		0			0
$451$	−3.07180	−	5.32051i	−0.144645	−	0.250533i
$452$	−4.52004	+	7.82894i	−0.212605	+	0.368242i
$453$		0			0
$454$			13.0718i			0.613490i
$455$	4.57081	−	2.26002i	0.214283	−	0.105951i
$456$		0			0
$457$	−11.0622	+	2.96410i	−0.517467	+	0.138655i	−0.508095	−	0.861301i	$-0.669650\pi$
$457$	−11.0622	+	2.96410i	−0.517467	+	0.138655i	−0.00937223	+	0.999956i	$0.502983\pi$
$458$	−9.14162	−	5.27792i	−0.427160	−	0.246621i
$459$		0			0
$460$	9.46410	+	9.46410i	0.441266	+	0.441266i
$461$	−2.58819	−	0.693504i	−0.120544	−	0.0322997i	0.198043	−	0.980193i	$-0.436542\pi$
$461$	−2.58819	−	0.693504i	−0.120544	−	0.0322997i	−0.318587	+	0.947894i	$0.603208\pi$
$462$		0			0
$463$	−4.83013	+	4.83013i	−0.224475	+	0.224475i	−0.810380	−	0.585905i	$-0.800739\pi$
$463$	−4.83013	+	4.83013i	−0.224475	+	0.224475i	0.585905	+	0.810380i	$0.300739\pi$
$464$	4.89898	−	2.82843i	0.227429	−	0.131306i
$465$		0			0
$466$	−9.80385	−	36.5885i	−0.454154	−	1.69493i
$467$	−22.0454			−1.02014			−0.510070	−	0.860133i	$-0.670380\pi$
$467$	−22.0454			−1.02014			−0.510070	+	0.860133i	$0.670380\pi$
$468$		0			0
$469$	4.07180			0.188018
$470$	4.62158	+	17.2480i	0.213177	+	0.795589i
$471$		0			0
$472$		0			0
$473$	−22.9420	+	22.9420i	−1.05487	+	1.05487i
$474$		0			0
$475$	−15.0263	−	4.02628i	−0.689453	−	0.184738i
$476$	−4.89898	−	4.89898i	−0.224544	−	0.224544i
$477$		0			0
$478$	−11.3205	−	6.53590i	−0.517788	−	0.298945i
$479$	−3.86370	+	1.03528i	−0.176537	+	0.0473030i	−0.346005	−	0.938233i	$-0.612462\pi$
$479$	−3.86370	+	1.03528i	−0.176537	+	0.0473030i	0.169468	+	0.985536i	$0.445795\pi$
$480$		0			0
$481$	2.26795	+	11.3397i	0.103410	+	0.517048i
$482$		−	23.3853i		−	1.06517i
$483$		0			0
$484$	−6.60770	+	11.4449i	−0.300350	+	0.520221i
$485$	−6.88160	−	11.9193i	−0.312477	−	0.541227i
$486$		0			0
$487$	−5.16987	+	19.2942i	−0.234269	+	0.874305i	0.744208	+	0.667948i	$0.232827\pi$
$487$	−5.16987	+	19.2942i	−0.234269	+	0.874305i	−0.978477	+	0.206357i	$0.933839\pi$
$488$		0			0
$489$		0			0
$490$	18.3923	+	31.8564i	0.830880	+	1.43913i
$491$	13.0561	−	22.6138i	0.589213	−	1.02055i	−0.405123	−	0.914262i	$-0.632771\pi$
$491$	13.0561	−	22.6138i	0.589213	−	1.02055i	0.994336	−	0.106285i	$-0.0338955\pi$
$492$		0			0
$493$			9.46410i			0.426242i
$494$	45.4307	+	2.92996i	2.04402	+	0.131825i
$495$		0			0
$496$	−25.3205	+	6.78461i	−1.13692	+	0.304638i
$497$	−7.02030	−	4.05317i	−0.314903	−	0.181810i
$498$		0			0
$499$	−24.2679	−	24.2679i	−1.08638	−	1.08638i	−0.995898	−	0.0904848i	$-0.971158\pi$
$499$	−24.2679	−	24.2679i	−1.08638	−	1.08638i	−0.0904848	−	0.995898i	$-0.528842\pi$
$500$	13.3843	+	3.58630i	0.598562	+	0.160384i
$501$		0			0
$502$	30.2487	−	30.2487i	1.35007	−	1.35007i
$503$	23.2702	−	13.4350i	1.03756	−	0.599038i	0.118422	−	0.992963i	$-0.462216\pi$
$503$	23.2702	−	13.4350i	1.03756	−	0.599038i	0.919143	+	0.393925i	$0.128883\pi$
$504$		0			0
$505$	0.803848	+	3.00000i	0.0357707	+	0.133498i
$506$	−20.5569			−0.913864
$507$		0			0
$508$	−19.8564			−0.880986
$509$	2.15849	+	8.05558i	0.0956732	+	0.357057i	0.997121	−	0.0758313i	$-0.0241611\pi$
$509$	2.15849	+	8.05558i	0.0956732	+	0.357057i	−0.901447	+	0.432888i	$0.857494\pi$
$510$		0			0
$511$	−1.83975	+	1.06218i	−0.0813856	+	0.0469880i
$512$	22.6274	−	22.6274i	1.00000	−	1.00000i
$513$		0			0
$514$	−2.53590	−	0.679492i	−0.111854	−	0.0299711i
$515$	−16.7303	−	16.7303i	−0.737226	−	0.737226i
$516$		0			0
$517$	−11.8756	−	6.85641i	−0.522290	−	0.301544i
$518$	3.20736	−	0.859411i	0.140924	−	0.0377603i
$519$		0			0
$520$		0			0
$521$			26.5654i			1.16385i	0.813241	+	0.581927i	$0.197701\pi$
$521$			26.5654i			1.16385i	−0.813241	+	0.581927i	$0.802299\pi$
$522$		0			0
$523$	21.3923	−	37.0526i	0.935420	−	1.62020i	0.161537	−	0.986867i	$-0.448355\pi$
$523$	21.3923	−	37.0526i	0.935420	−	1.62020i	0.773883	−	0.633329i	$-0.218312\pi$
$524$	7.07107	+	12.2474i	0.308901	+	0.535032i
$525$		0			0
$526$	12.1436	−	45.3205i	0.529486	−	1.97607i
$527$	11.3509	−	42.3620i	0.494452	−	1.84532i
$528$		0			0
$529$	8.50000	+	14.7224i	0.369565	+	0.640106i
$530$	−16.2127	+	28.0812i	−0.704234	+	1.21977i
$531$		0			0
$532$		−	6.53590i		−	0.283367i
$533$	−1.03528	−	5.17638i	−0.0448428	−	0.224214i
$534$		0			0
$535$	−4.46410	+	1.19615i	−0.193000	+	0.0517142i
$536$		0			0
$537$		0			0
$538$	−29.1769	−	29.1769i	−1.25791	−	1.25791i
$539$	−27.2862	−	7.31130i	−1.17530	−	0.314920i
$540$		0			0
$541$	−30.6865	+	30.6865i	−1.31932	+	1.31932i	−0.405001	+	0.914316i	$0.632729\pi$
$541$	−30.6865	+	30.6865i	−1.31932	+	1.31932i	−0.914316	+	0.405001i	$0.867271\pi$
$542$	−17.5625	+	10.1397i	−0.754375	+	0.435539i
$543$		0			0
$544$	13.8564	+	51.7128i	0.594089	+	2.21717i
$545$	43.6375			1.86923
$546$		0			0
$547$	−17.3923			−0.743641			−0.371821	−	0.928305i	$-0.621266\pi$
$547$	−17.3923			−0.743641			−0.371821	+	0.928305i	$0.621266\pi$
$548$	−7.72741	−	28.8391i	−0.330098	−	1.23194i
$549$		0			0
$550$	17.9090	−	10.3397i	0.763641	−	0.440888i
$551$	−6.31319	+	6.31319i	−0.268951	+	0.268951i
$552$		0			0
$553$	−3.59808	−	0.964102i	−0.153006	−	0.0409978i
$554$	28.0812	+	28.0812i	1.19306	+	1.19306i
$555$		0			0
$556$	5.87564	+	3.39230i	0.249183	+	0.143866i
$557$	4.38134	−	1.17398i	0.185643	−	0.0497430i	−0.164800	−	0.986327i	$-0.552698\pi$
$557$	4.38134	−	1.17398i	0.185643	−	0.0497430i	0.350443	+	0.936584i	$0.386031\pi$
$558$		0			0
$559$	−24.9904	+	12.3564i	−1.05698	+	0.522620i
$560$			5.65685i			0.239046i
$561$		0			0
$562$	−23.5167	+	40.7321i	−0.991990	+	1.71818i
$563$	14.8492	+	25.7196i	0.625821	+	1.08395i	0.988381	+	0.151993i	$0.0485693\pi$
$563$	14.8492	+	25.7196i	0.625821	+	1.08395i	−0.362561	+	0.931960i	$0.618097\pi$
$564$		0			0
$565$	−3.19615	+	11.9282i	−0.134463	+	0.501823i
$566$	−11.8313	+	44.1552i	−0.497309	+	1.85598i
$567$		0			0
$568$		0			0
$569$	−3.67423	+	6.36396i	−0.154032	+	0.266791i	−0.932706	−	0.360637i	$-0.882559\pi$
$569$	−3.67423	+	6.36396i	−0.154032	+	0.266791i	0.778674	+	0.627428i	$0.215893\pi$
$570$		0			0
$571$		−	21.7128i		−	0.908653i	−0.890835	−	0.454326i	$-0.849880\pi$
$571$		−	21.7128i		−	0.908653i	0.890835	−	0.454326i	$-0.150120\pi$
$572$	−22.7290	+	19.9749i	−0.950345	+	0.835191i
$573$		0			0
$574$	−1.46410	+	0.392305i	−0.0611104	+	0.0163745i
$575$	5.22715	+	3.01790i	0.217987	+	0.125855i
$576$		0			0
$577$	9.19615	+	9.19615i	0.382841	+	0.382841i	0.872125	−	0.489284i	$-0.162742\pi$
$577$	9.19615	+	9.19615i	0.382841	+	0.382841i	−0.489284	+	0.872125i	$0.662742\pi$
$578$	−53.6757	−	14.3824i	−2.23262	−	0.598228i
$579$		0			0
$580$	−5.46410	+	5.46410i	−0.226884	+	0.226884i
$581$	3.34607	−	1.93185i	0.138818	−	0.0801467i
$582$		0			0
$583$	−6.44486	−	24.0526i	−0.266919	−	0.996155i
$584$		0			0
$585$		0			0
$586$	−23.6077			−0.975225
$587$	9.98743	+	37.2736i	0.412225	+	1.53845i	0.790329	+	0.612682i	$0.209910\pi$
$587$	9.98743	+	37.2736i	0.412225	+	1.53845i	−0.378104	+	0.925763i	$0.623424\pi$
$588$		0			0
$589$	35.8301	−	20.6865i	1.47635	−	0.852374i
$590$	−0.757875	+	0.757875i	−0.0312012	+	0.0312012i
$591$		0			0
$592$	−12.3923	−	3.32051i	−0.509321	−	0.136472i
$593$	−4.10394	−	4.10394i	−0.168529	−	0.168529i	0.617804	−	0.786332i	$-0.288023\pi$
$593$	−4.10394	−	4.10394i	−0.168529	−	0.168529i	−0.786332	+	0.617804i	$0.788023\pi$
$594$		0			0
$595$	−8.19615	−	4.73205i	−0.336009	−	0.193995i
$596$	12.3490	−	3.30890i	0.505834	−	0.135538i
$597$		0			0
$598$	−16.7321	−	5.66025i	−0.684224	−	0.231465i
$599$			15.2789i			0.624281i	0.950036	+	0.312140i	$0.101046\pi$
$599$			15.2789i			0.624281i	−0.950036	+	0.312140i	$0.898954\pi$
$600$		0			0
$601$	18.3923	−	31.8564i	0.750238	−	1.29945i	−0.197470	−	0.980309i	$-0.563272\pi$
$601$	18.3923	−	31.8564i	0.750238	−	1.29945i	0.947707	−	0.319141i	$-0.103394\pi$
$602$	4.00240	+	6.93237i	0.163126	+	0.282542i
$603$		0			0
$604$	6.19615	−	23.1244i	0.252118	−	0.940917i
$605$	−4.67235	+	17.4374i	−0.189958	+	0.708932i
$606$		0			0
$607$	19.1962	+	33.2487i	0.779148	+	1.34952i	0.932433	+	0.361342i	$0.117681\pi$
$607$	19.1962	+	33.2487i	0.779148	+	1.34952i	−0.153286	+	0.988182i	$0.548985\pi$
$608$	−25.2528	+	43.7391i	−1.02414	+	1.77385i
$609$		0			0
$610$			72.1051i			2.91945i
$611$	−7.77817	−	8.85062i	−0.314671	−	0.358058i
$612$		0			0
$613$	−6.33013	+	1.69615i	−0.255671	+	0.0685070i	−0.384378	−	0.923176i	$-0.625584\pi$
$613$	−6.33013	+	1.69615i	−0.255671	+	0.0685070i	0.128707	+	0.991683i	$0.458917\pi$
$614$	47.2610	+	27.2862i	1.90730	+	1.10118i
$615$		0			0
$616$		0			0
$617$	−29.9251	−	8.01841i	−1.20474	−	0.322809i	−0.400044	−	0.916496i	$-0.631005\pi$
$617$	−29.9251	−	8.01841i	−1.20474	−	0.322809i	−0.804696	+	0.593687i	$0.797672\pi$
$618$		0			0
$619$	12.8301	−	12.8301i	0.515686	−	0.515686i	−0.400577	−	0.916263i	$-0.631190\pi$
$619$	12.8301	−	12.8301i	0.515686	−	0.515686i	0.916263	+	0.400577i	$0.131190\pi$
$620$	31.0112	−	17.9043i	1.24544	−	0.719054i
$621$		0			0
$622$	5.75129	+	21.4641i	0.230606	+	0.860632i
$623$	−2.34795			−0.0940688
$624$		0			0
$625$	31.2487			1.24995
$626$	−1.65445	−	6.17449i	−0.0661251	−	0.246782i
$627$		0			0
$628$	19.7321	−	11.3923i	0.787395	−	0.454602i
$629$	15.1774	−	15.1774i	0.605163	−	0.605163i
$630$		0			0
$631$	38.4545	+	10.3038i	1.53085	+	0.410190i	0.923296	−	0.384090i	$-0.125485\pi$
$631$	38.4545	+	10.3038i	1.53085	+	0.410190i	0.607553	+	0.794279i	$0.292151\pi$
$632$		0			0
$633$		0			0
$634$	25.8564	+	14.9282i	1.02689	+	0.592875i
$635$	−26.2001	+	7.02030i	−1.03972	+	0.278592i
$636$		0			0
$637$	−20.1962	−	13.4641i	−0.800201	−	0.533467i
$638$		−	11.8685i		−	0.469879i
$639$		0			0
$640$		0			0
$641$	−13.4722	−	23.3345i	−0.532120	−	0.921658i	−0.999297	−	0.0374946i	$-0.988062\pi$
$641$	−13.4722	−	23.3345i	−0.532120	−	0.921658i	0.467177	−	0.884164i	$-0.345271\pi$
$642$		0			0
$643$	5.96410	−	22.2583i	0.235201	−	0.877783i	−0.742857	−	0.669450i	$-0.766530\pi$
$643$	5.96410	−	22.2583i	0.235201	−	0.877783i	0.978058	−	0.208333i	$-0.0668036\pi$
$644$	−0.656339	+	2.44949i	−0.0258634	+	0.0965234i
$645$		0			0
$646$	−42.2487	−	73.1769i	−1.66225	−	2.87911i
$647$	−14.1929	+	24.5828i	−0.557981	+	0.966451i	0.439684	+	0.898152i	$0.355090\pi$
$647$	−14.1929	+	24.5828i	−0.557981	+	0.966451i	−0.997665	+	0.0682984i	$0.978243\pi$
$648$		0			0
$649$		−	0.823085i		−	0.0323089i
$650$	17.4238	−	3.48477i	0.683419	−	0.136684i
$651$		0			0
$652$	−32.1244	+	8.60770i	−1.25809	+	0.337103i
$653$	9.79796	+	5.65685i	0.383424	+	0.221370i	0.679307	−	0.733854i	$-0.262281\pi$
$653$	9.79796	+	5.65685i	0.383424	+	0.221370i	−0.295883	+	0.955224i	$0.595614\pi$
$654$		0			0
$655$	13.6603	+	13.6603i	0.533750	+	0.533750i
$656$	5.65685	+	1.51575i	0.220863	+	0.0591801i
$657$		0			0
$658$	−2.39230	+	2.39230i	−0.0932618	+	0.0932618i
$659$	40.3930	−	23.3209i	1.57349	−	0.908454i	0.577752	−	0.816212i	$-0.303930\pi$
$659$	40.3930	−	23.3209i	1.57349	−	0.908454i	0.995737	−	0.0922417i	$-0.0294032\pi$
$660$		0			0
$661$	9.59808	+	35.8205i	0.373322	+	1.39326i	0.855781	+	0.517338i	$0.173077\pi$
$661$	9.59808	+	35.8205i	0.373322	+	1.39326i	−0.482459	+	0.875918i	$0.660256\pi$
$662$	10.8332			0.421046
$663$		0			0
$664$		0			0
$665$	−2.31079	−	8.62398i	−0.0896086	−	0.334424i
$666$		0			0
$667$	3.00000	−	1.73205i	0.116160	−	0.0670653i
$668$	−7.72741	+	7.72741i	−0.298982	+	0.298982i
$669$		0			0
$670$	41.5167	+	11.1244i	1.60393	+	0.429771i
$671$	−39.1547	−	39.1547i	−1.51155	−	1.51155i
$672$		0			0
$673$	7.28461	+	4.20577i	0.280801	+	0.162121i	0.633786	−	0.773508i	$-0.281500\pi$
$673$	7.28461	+	4.20577i	0.280801	+	0.162121i	−0.352985	+	0.935629i	$0.614833\pi$
$674$	53.9531	−	14.4567i	2.07820	−	0.556851i
$675$		0			0
$676$	−24.0000	+	10.0000i	−0.923077	+	0.384615i
$677$		−	13.5873i		−	0.522204i	−0.965311	−	0.261102i	$-0.915914\pi$
$677$		−	13.5873i		−	0.522204i	0.965311	−	0.261102i	$-0.0840858\pi$
$678$		0			0
$679$	1.30385	−	2.25833i	0.0500371	−	0.0866668i
$680$		0			0
$681$		0			0
$682$	−14.2346	+	53.1244i	−0.545072	+	2.03424i
$683$	4.53365	−	16.9198i	0.173475	−	0.647418i	−0.823331	−	0.567561i	$-0.807887\pi$
$683$	4.53365	−	16.9198i	0.173475	−	0.647418i	0.996806	−	0.0798568i	$-0.0254463\pi$
$684$		0			0
$685$	−20.3923	−	35.3205i	−0.779150	−	1.34953i
$686$	−7.10823	+	12.3118i	−0.271394	+	0.470067i
$687$		0			0
$688$		−	30.9282i		−	1.17913i
$689$	1.37705	−	21.3519i	0.0524614	−	0.813443i
$690$		0			0
$691$	24.4282	−	6.54552i	0.929293	−	0.249003i	0.237740	−	0.971329i	$-0.423594\pi$
$691$	24.4282	−	6.54552i	0.929293	−	0.249003i	0.691553	+	0.722326i	$0.256927\pi$
$692$	−24.3190	−	14.0406i	−0.924471	−	0.533744i
$693$		0			0
$694$	−41.1769	−	41.1769i	−1.56305	−	1.56305i
$695$	8.95215	+	2.39872i	0.339574	+	0.0909887i
$696$		0			0
$697$	−6.92820	+	6.92820i	−0.262424	+	0.262424i
$698$	3.65067	−	2.10772i	0.138180	−	0.0797783i
$699$		0			0
$700$	−0.660254	−	2.46410i	−0.0249553	−	0.0931343i
$701$	−23.1822			−0.875580			−0.437790	−	0.899077i	$-0.644239\pi$
$701$	−23.1822			−0.875580			−0.437790	+	0.899077i	$0.644239\pi$
$702$		0			0
$703$	20.2487			0.763695
$704$	−8.68835	−	32.4254i	−0.327455	−	1.22208i
$705$		0			0
$706$	3.12436	−	1.80385i	0.117587	−	0.0678887i
$707$	−0.416102	+	0.416102i	−0.0156491	+	0.0156491i
$708$		0			0
$709$	−11.6962	−	3.13397i	−0.439258	−	0.117699i	0.0324106	−	0.999475i	$-0.489682\pi$
$709$	−11.6962	−	3.13397i	−0.439258	−	0.117699i	−0.471669	+	0.881776i	$0.656348\pi$
$710$	−60.5066	−	60.5066i	−2.27077	−	2.27077i
$711$		0			0
$712$		0			0
$713$	−15.5056	+	4.15471i	−0.580689	+	0.155595i
$714$		0			0
$715$	−22.9282	+	34.3923i	−0.857466	+	1.28620i
$716$		−	32.9802i		−	1.23253i
$717$		0			0
$718$	−10.0000	+	17.3205i	−0.373197	+	0.646396i
$719$	21.9575	+	38.0315i	0.818876	+	1.41833i	0.906511	+	0.422182i	$0.138736\pi$
$719$	21.9575	+	38.0315i	0.818876	+	1.41833i	−0.0876356	+	0.996153i	$0.527931\pi$
$720$		0			0
$721$	1.16025	−	4.33013i	0.0432101	−	0.161262i
$722$	10.7961	−	40.2915i	0.401788	−	1.49949i
$723$		0			0
$724$	−6.00000	−	10.3923i	−0.222988	−	0.386227i
$725$	−1.74238	+	3.01790i	−0.0647105	+	0.112082i
$726$		0			0
$727$			35.1051i			1.30198i	0.759088	+	0.650988i	$0.225645\pi$
$727$			35.1051i			1.30198i	−0.759088	+	0.650988i	$0.774355\pi$
$728$		0			0
$729$		0			0
$730$	−21.6603	+	5.80385i	−0.801682	+	0.214810i
$731$	44.8115	+	25.8719i	1.65741	+	0.956908i
$732$		0			0
$733$	−32.7583	−	32.7583i	−1.20996	−	1.20996i	−0.971040	−	0.238916i	$-0.923208\pi$
$733$	−32.7583	−	32.7583i	−1.20996	−	1.20996i	−0.238916	−	0.971040i	$-0.576792\pi$
$734$	10.4171	+	2.79126i	0.384503	+	0.103027i
$735$		0			0
$736$	13.8564	−	13.8564i	0.510754	−	0.510754i
$737$	−28.5852	+	16.5037i	−1.05295	+	0.607921i
$738$		0			0
$739$	−0.562178	−	2.09808i	−0.0206800	−	0.0771790i	0.954815	−	0.297202i	$-0.0960534\pi$
$739$	−0.562178	−	2.09808i	−0.0206800	−	0.0771790i	−0.975495	+	0.220023i	$0.929387\pi$
$740$	17.5254			0.644245
$741$		0			0
$742$	−6.14359			−0.225538
$743$	−1.77955	−	6.64136i	−0.0652853	−	0.243648i	0.925570	−	0.378576i	$-0.123586\pi$
$743$	−1.77955	−	6.64136i	−0.0652853	−	0.243648i	−0.990855	+	0.134928i	$0.956920\pi$
$744$		0			0
$745$	15.1244	−	8.73205i	0.554114	−	0.319918i
$746$	−10.7317	+	10.7317i	−0.392915	+	0.392915i
$747$		0			0
$748$	54.2487	+	14.5359i	1.98353	+	0.531485i
$749$	−0.619174	−	0.619174i	−0.0226241	−	0.0226241i
$750$		0			0
$751$	−8.78461	−	5.07180i	−0.320555	−	0.185072i	0.331085	−	0.943601i	$-0.392585\pi$
$751$	−8.78461	−	5.07180i	−0.320555	−	0.185072i	−0.651640	+	0.758528i	$0.725919\pi$
$752$	12.6264	−	3.38323i	0.460437	−	0.123374i
$753$		0			0
$754$	3.26795	−	9.66025i	0.119012	−	0.351806i
$755$		−	32.7028i		−	1.19018i
$756$		0			0
$757$	14.8038	−	25.6410i	0.538055	−	0.931939i	−0.460954	−	0.887424i	$-0.652493\pi$
$757$	14.8038	−	25.6410i	0.538055	−	0.931939i	0.999009	−	0.0445144i	$-0.0141741\pi$
$758$	23.1451	+	40.0884i	0.840666	+	1.45608i
$759$		0			0
$760$		0			0
$761$	0.314566	−	1.17398i	0.0114030	−	0.0425566i	−0.959990	−	0.280035i	$-0.909654\pi$
$761$	0.314566	−	1.17398i	0.0114030	−	0.0425566i	0.971393	+	0.237478i	$0.0763207\pi$
$762$		0			0
$763$	4.13397	+	7.16025i	0.149660	+	0.259219i
$764$	−13.0053	+	22.5259i	−0.470516	+	0.814958i
$765$		0			0
$766$			20.0000i			0.722629i
$767$	0.226633	−	0.669942i	0.00818326	−	0.0241902i
$768$		0			0
$769$	33.4904	−	8.97372i	1.20769	−	0.323601i	0.401837	−	0.915711i	$-0.368372\pi$
$769$	33.4904	−	8.97372i	1.20769	−	0.323601i	0.805857	+	0.592110i	$0.201705\pi$
$770$	10.2784	+	5.93426i	0.370409	+	0.213856i
$771$		0			0
$772$	−1.66025	−	1.66025i	−0.0597539	−	0.0597539i
$773$	−3.81294	−	1.02167i	−0.137142	−	0.0367470i	0.189595	−	0.981862i	$-0.439282\pi$
$773$	−3.81294	−	1.02167i	−0.137142	−	0.0367470i	−0.326737	+	0.945115i	$0.605949\pi$
$774$		0			0
$775$	11.4186	−	11.4186i	0.410168	−	0.410168i
$776$		0			0
$777$		0			0
$778$	3.46410	+	12.9282i	0.124194	+	0.463499i
$779$	−9.24316			−0.331170
$780$		0			0
$781$	65.7128			2.35139
$782$	8.48528	+	31.6675i	0.303433	+	1.13243i
$783$		0			0
$784$	23.3205	−	13.4641i	0.832875	−	0.480861i
$785$	22.0082	−	22.0082i	0.785508	−	0.785508i
$786$		0			0
$787$	−12.1603	−	3.25833i	−0.433466	−	0.116147i	0.0354866	−	0.999370i	$-0.488702\pi$
$787$	−12.1603	−	3.25833i	−0.433466	−	0.116147i	−0.468953	+	0.883223i	$0.655369\pi$
$788$	−10.0010	−	10.0010i	−0.356272	−	0.356272i
$789$		0			0
$790$	−34.0526	−	19.6603i	−1.21154	−	0.699480i
$791$	−2.26002	+	0.605571i	−0.0803571	+	0.0215316i
$792$		0			0
$793$	−21.0885	−	42.6506i	−0.748873	−	1.51457i
$794$		−	31.7418i		−	1.12648i
$795$		0			0
$796$	0.464102	−	0.803848i	0.0164496	−	0.0284916i
$797$	−14.2808	−	24.7351i	−0.505853	−	0.876163i	−0.999977	−	0.00677190i	$-0.997844\pi$
$797$	−14.2808	−	24.7351i	−0.505853	−	0.876163i	0.494124	−	0.869391i	$-0.335489\pi$
$798$		0			0
$799$	−5.66025	+	21.1244i	−0.200245	+	0.747326i
$800$	−5.10205	+	19.0411i	−0.180385	+	0.673205i
$801$		0			0
$802$	39.8564	+	69.0333i	1.40738	+	2.43765i
$803$	8.61038	−	14.9136i	0.303854	−	0.526290i
$804$		0			0
$805$			3.46410i			0.122094i
$806$	−26.2137	+	39.3206i	−0.923339	+	1.38501i
$807$		0			0
$808$		0			0
$809$	41.8816	+	24.1803i	1.47248	+	0.850135i	0.999521	−	0.0309507i	$-0.00985347\pi$
$809$	41.8816	+	24.1803i	1.47248	+	0.850135i	0.472956	+	0.881086i	$0.343187\pi$
$810$		0			0
$811$	8.09808	+	8.09808i	0.284362	+	0.284362i	0.834846	−	0.550484i	$-0.185557\pi$
$811$	8.09808	+	8.09808i	0.284362	+	0.284362i	−0.550484	+	0.834846i	$0.685557\pi$
$812$	−1.41421	−	0.378937i	−0.0496292	−	0.0132981i
$813$		0			0
$814$	−19.0333	+	19.0333i	−0.667118	+	0.667118i
$815$	−39.3441	+	22.7153i	−1.37817	+	0.795684i
$816$		0			0
$817$	12.6340	+	47.1506i	0.442007	+	1.64959i
$818$	−34.2185			−1.19642
$819$		0			0
$820$	−8.00000			−0.279372
$821$	−13.6109	−	50.7965i	−0.475023	−	1.77281i	−0.621305	−	0.783569i	$-0.713397\pi$
$821$	−13.6109	−	50.7965i	−0.475023	−	1.77281i	0.146282	−	0.989243i	$-0.453269\pi$
$822$		0			0
$823$	3.80385	−	2.19615i	0.132594	−	0.0765531i	−0.432236	−	0.901761i	$-0.642275\pi$
$823$	3.80385	−	2.19615i	0.132594	−	0.0765531i	0.564830	+	0.825208i	$0.308942\pi$
$824$		0			0
$825$		0			0
$826$	−0.196152	−	0.0525589i	−0.00682502	−	0.00182876i
$827$	17.5254	+	17.5254i	0.609417	+	0.609417i	0.942794	−	0.333377i	$-0.108188\pi$
$827$	17.5254	+	17.5254i	0.609417	+	0.609417i	−0.333377	+	0.942794i	$0.608188\pi$
$828$		0			0
$829$	16.2058	+	9.35641i	0.562850	+	0.324961i	0.754288	−	0.656543i	$-0.227982\pi$
$829$	16.2058	+	9.35641i	0.562850	+	0.324961i	−0.191439	+	0.981505i	$0.561315\pi$
$830$	39.3949	−	10.5558i	1.36742	−	0.366398i
$831$		0			0
$832$	1.85641	−	28.7846i	0.0643593	−	0.997927i
$833$			45.0518i			1.56095i
$834$		0			0
$835$	−7.46410	+	12.9282i	−0.258306	+	0.447399i
$836$	26.4911	+	45.8840i	0.916215	+	1.58693i
$837$		0			0
$838$	1.75129	−	6.53590i	0.0604973	−	0.225779i
$839$	−3.71140	+	13.8511i	−0.128132	+	0.478194i	−0.999932	−	0.0116675i	$-0.996286\pi$
$839$	−3.71140	+	13.8511i	−0.128132	+	0.478194i	0.871800	+	0.489862i	$0.162953\pi$
$840$		0			0
$841$	−13.5000	−	23.3827i	−0.465517	−	0.806300i
$842$	−21.7308	+	37.6389i	−0.748894	+	1.29712i
$843$		0			0
$844$			30.3923i			1.04615i
$845$	−28.1320	+	21.6801i	−0.967769	+	0.745817i
$846$		0			0
$847$	−3.30385	+	0.885263i	−0.113522	+	0.0304180i
$848$	20.5569	+	11.8685i	0.705926	+	0.407566i
$849$		0			0
$850$	−23.3205	−	23.3205i	−0.799887	−	0.799887i
$851$	−7.58871	−	2.03339i	−0.260137	−	0.0697036i
$852$		0			0
$853$	11.2224	−	11.2224i	0.384249	−	0.384249i	−0.488381	−	0.872630i	$-0.662413\pi$
$853$	11.2224	−	11.2224i	0.384249	−	0.384249i	0.872630	+	0.488381i	$0.162413\pi$
$854$	−11.8313	+	6.83083i	−0.404860	+	0.233746i
$855$		0			0
$856$		0			0
$857$	3.28169			0.112101			0.0560503	−	0.998428i	$-0.482149\pi$
$857$	3.28169			0.112101			0.0560503	+	0.998428i	$0.482149\pi$
$858$		0			0
$859$	−51.1962			−1.74679			−0.873395	−	0.487012i	$-0.838087\pi$
$859$	−51.1962			−1.74679			−0.873395	+	0.487012i	$0.838087\pi$
$860$	10.9348	+	40.8091i	0.372873	+	1.39158i
$861$		0			0
$862$	−27.4641	+	15.8564i	−0.935431	+	0.540071i
$863$	10.3156	−	10.3156i	0.351147	−	0.351147i	−0.509389	−	0.860536i	$-0.670129\pi$
$863$	10.3156	−	10.3156i	0.351147	−	0.351147i	0.860536	+	0.509389i	$0.170129\pi$
$864$		0			0
$865$	−37.0526	−	9.92820i	−1.25982	−	0.337569i
$866$	−26.1122	−	26.1122i	−0.887328	−	0.887328i
$867$		0			0
$868$	5.87564	+	3.39230i	0.199432	+	0.115142i
$869$	29.1672	−	7.81534i	0.989431	−	0.265117i
$870$		0			0
$871$	−27.8109	+	5.56218i	−0.942336	+	0.188467i
$872$		0			0
$873$		0			0
$874$	−15.4641	+	26.7846i	−0.523081	+	0.906003i
$875$	1.79315	+	3.10583i	0.0606196	+	0.104996i
$876$		0			0
$877$	6.15064	−	22.9545i	0.207692	−	0.775118i	−0.780920	−	0.624631i	$-0.785249\pi$
$877$	6.15064	−	22.9545i	0.207692	−	0.775118i	0.988612	−	0.150487i	$-0.0480840\pi$
$878$	4.96335	−	18.5235i	0.167505	−	0.625137i
$879$		0			0
$880$	−22.9282	−	39.7128i	−0.772910	−	1.33872i
$881$	0.240237	−	0.416102i	0.00809378	−	0.0140188i	−0.861950	−	0.506993i	$-0.830757\pi$
$881$	0.240237	−	0.416102i	0.00809378	−	0.0140188i	0.870044	+	0.492974i	$0.164090\pi$
$882$		0			0
$883$			23.7846i			0.800416i	0.916424	+	0.400208i	$0.131062\pi$
$883$			23.7846i			0.800416i	−0.916424	+	0.400208i	$0.868938\pi$
$884$	40.1528	+	26.7685i	1.35048	+	0.900323i
$885$		0			0
$886$	45.8564	−	12.2872i	1.54058	−	0.412796i
$887$	−38.6878	−	22.3364i	−1.29901	−	0.749983i	−0.318776	−	0.947830i	$-0.603272\pi$
$887$	−38.6878	−	22.3364i	−1.29901	−	0.749983i	−0.980233	+	0.197847i	$0.936605\pi$
$888$		0			0
$889$	−3.63397	−	3.63397i	−0.121880	−	0.121880i
$890$	−23.9401	−	6.41473i	−0.802474	−	0.215022i
$891$		0			0
$892$	34.7846	−	34.7846i	1.16467	−	1.16467i
$893$	−17.8671	+	10.3156i	−0.597901	+	0.345198i
$894$		0			0
$895$	−11.6603	−	43.5167i	−0.389759	−	1.45460i
$896$		0			0
$897$		0			0
$898$	17.4641			0.582785
$899$	−2.39872	−	8.95215i	−0.0800018	−	0.298571i
$900$		0			0
$901$	−34.3923	+	19.8564i	−1.14577	+	0.661513i
$902$	8.68835	−	8.68835i	0.289291	−	0.289291i
$903$		0			0
$904$		0			0
$905$	−11.5911	−	11.5911i	−0.385302	−	0.385302i
$906$		0			0
$907$	−15.5885	−	9.00000i	−0.517606	−	0.298840i	0.218348	−	0.975871i	$-0.429933\pi$
$907$	−15.5885	−	9.00000i	−0.517606	−	0.298840i	−0.735955	+	0.677031i	$0.763266\pi$
$908$	−12.6264	+	3.38323i	−0.419021	+	0.112276i
$909$		0			0
$910$	6.73205	+	7.66025i	0.223165	+	0.253935i
$911$			52.0213i			1.72354i	0.507297	+	0.861771i	$0.330645\pi$
$911$			52.0213i			1.72354i	−0.507297	+	0.861771i	$0.669355\pi$
$912$		0			0
$913$	−15.6603	+	27.1244i	−0.518279	+	0.897685i
$914$	−11.4524	−	19.8362i	−0.378812	−	0.656122i
$915$		0			0
$916$	2.73205	−	10.1962i	0.0902695	−	0.336890i
$917$	−0.947343	+	3.53553i	−0.0312840	+	0.116754i
$918$		0			0
$919$	14.5885	+	25.2679i	0.481229	+	0.833513i	0.999768	−	0.0215411i	$-0.00685729\pi$
$919$	14.5885	+	25.2679i	0.481229	+	0.833513i	−0.518539	+	0.855054i	$0.673524\pi$
$920$		0			0
$921$		0			0
$922$		−	5.35898i		−	0.176489i
$923$	53.4863	+	18.0938i	1.76052	+	0.595564i
$924$		0			0
$925$	7.63397	−	2.04552i	0.251004	−	0.0672562i
$926$	−11.8313	−	6.83083i	−0.388802	−	0.224475i
$927$		0			0
$928$	8.00000	+	8.00000i	0.262613	+	0.262613i
$929$	−8.38375	−	2.24642i	−0.275062	−	0.0737026i	0.118651	−	0.992936i	$-0.462143\pi$
$929$	−8.38375	−	2.24642i	−0.275062	−	0.0737026i	−0.393713	+	0.919233i	$0.628810\pi$
$930$		0			0
$931$	−30.0526	+	30.0526i	−0.984933	+	0.984933i
$932$	32.8043	−	18.9396i	1.07454	−	0.620387i
$933$		0			0
$934$	−11.4115	−	42.5885i	−0.373397	−	1.39354i
$935$	76.7193			2.50899
$936$		0			0
$937$	−51.5692			−1.68469			−0.842346	−	0.538936i	$-0.818826\pi$
$937$	−51.5692			−1.68469			−0.842346	+	0.538936i	$0.818826\pi$
$938$	2.10772	+	7.86611i	0.0688194	+	0.256838i
$939$		0			0
$940$	−15.4641	+	8.92820i	−0.504383	+	0.291206i
$941$	6.55343	−	6.55343i	0.213636	−	0.213636i	−0.592174	−	0.805810i	$-0.701730\pi$
$941$	6.55343	−	6.55343i	0.213636	−	0.213636i	0.805810	+	0.592174i	$0.201730\pi$
$942$		0			0
$943$	3.46410	+	0.928203i	0.112807	+	0.0302265i
$944$	0.554803	+	0.554803i	0.0180573	+	0.0180573i
$945$		0			0
$946$	−56.1962	−	32.4449i	−1.82709	−	1.05487i
$947$	−3.86370	+	1.03528i	−0.125553	+	0.0336420i	−0.321049	−	0.947063i	$-0.604035\pi$
$947$	−3.86370	+	1.03528i	−0.125553	+	0.0336420i	0.195495	+	0.980705i	$0.437369\pi$
$948$		0			0
$949$	11.1147	−	9.76795i	0.360800	−	0.317081i
$950$		−	31.1127i		−	1.00943i
$951$		0			0
$952$		0			0
$953$	−29.7864	−	51.5916i	−0.964877	−	1.67122i	−0.709946	−	0.704256i	$-0.751281\pi$
$953$	−29.7864	−	51.5916i	−0.964877	−	1.67122i	−0.254931	−	0.966959i	$-0.582053\pi$
$954$		0			0
$955$	−9.19615	+	34.3205i	−0.297581	+	1.11059i
$956$	3.38323	−	12.6264i	0.109421	−	0.408367i
$957$		0			0
$958$	−4.00000	−	6.92820i	−0.129234	−	0.223840i
$959$	3.86370	−	6.69213i	0.124765	−	0.216100i
$960$		0			0
$961$			11.9474i			0.385401i
$962$	−20.7327	+	10.2512i	−0.668450	+	0.330513i
$963$		0			0
$964$	22.5885	−	6.05256i	0.727525	−	0.194940i
$965$	−2.77766	−	1.60368i	−0.0894160	−	0.0516244i
$966$		0			0
$967$	5.73205	+	5.73205i	0.184330	+	0.184330i	0.793240	−	0.608909i	$-0.208393\pi$
$967$	5.73205	+	5.73205i	0.184330	+	0.184330i	−0.608909	+	0.793240i	$0.708393\pi$
$968$		0			0
$969$		0			0
$970$	19.4641	−	19.4641i	0.624955	−	0.624955i
$971$	45.0518	−	26.0106i	1.44578	−	0.834721i	0.447554	−	0.894257i	$-0.352295\pi$
$971$	45.0518	−	26.0106i	1.44578	−	0.834721i	0.998226	+	0.0595357i	$0.0189620\pi$
$972$		0			0
$973$	0.454483	+	1.69615i	0.0145700	+	0.0543762i
$974$	−39.9497			−1.28007
$975$		0			0
$976$	52.7846			1.68959
$977$	−2.65256	−	9.89949i	−0.0848630	−	0.316713i	0.910425	−	0.413674i	$-0.135755\pi$
$977$	−2.65256	−	9.89949i	−0.0848630	−	0.316713i	−0.995288	+	0.0969607i	$0.969088\pi$
$978$		0			0
$979$	16.4833	−	9.51666i	0.526810	−	0.304154i
$980$	−26.0106	+	26.0106i	−0.830880	+	0.830880i
$981$		0			0
$982$	50.4449	+	13.5167i	1.60976	+	0.431334i
$983$	−4.27981	−	4.27981i	−0.136505	−	0.136505i	0.635553	−	0.772057i	$-0.280772\pi$
$983$	−4.27981	−	4.27981i	−0.136505	−	0.136505i	−0.772057	+	0.635553i	$0.780772\pi$
$984$		0			0
$985$	−16.7321	−	9.66025i	−0.533127	−	0.307801i
$986$	−18.2832	+	4.89898i	−0.582257	+	0.156015i
$987$		0			0
$988$	8.92820	+	44.6410i	0.284044	+	1.42022i
$989$		−	18.9396i		−	0.602244i
$990$		0			0
$991$	−4.80385	+	8.32051i	−0.152599	+	0.264310i	−0.932182	−	0.361989i	$-0.882098\pi$
$991$	−4.80385	+	8.32051i	−0.152599	+	0.264310i	0.779583	+	0.626299i	$0.215431\pi$
$992$	−26.2137	−	45.4035i	−0.832286	−	1.44156i
$993$		0			0
$994$	4.19615	−	15.6603i	0.133094	−	0.496713i
$995$	0.328169	−	1.22474i	0.0104037	−	0.0388270i
$996$		0			0
$997$	19.8923	+	34.4545i	0.629996	+	1.09118i	0.987552	+	0.157293i	$0.0502767\pi$
$997$	19.8923	+	34.4545i	0.629996	+	1.09118i	−0.357556	+	0.933892i	$0.616390\pi$
$998$	34.3201	−	59.4441i	1.08638	−	1.88167i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	117.2.ba.a.98.2	yes	8
3.2	odd	2	inner	117.2.ba.a.98.1	yes	8
13.2	odd	12	inner	117.2.ba.a.80.1	✓	8
13.4	even	6		1521.2.i.e.944.2		8
13.6	odd	12		1521.2.i.d.746.2		8
13.7	odd	12		1521.2.i.e.746.3		8
13.9	even	3		1521.2.i.d.944.3		8
39.2	even	12	inner	117.2.ba.a.80.2	yes	8
39.17	odd	6		1521.2.i.e.944.3		8
39.20	even	12		1521.2.i.e.746.2		8
39.32	even	12		1521.2.i.d.746.3		8
39.35	odd	6		1521.2.i.d.944.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
117.2.ba.a.80.1	✓	8	13.2	odd	12	inner
117.2.ba.a.80.2	yes	8	39.2	even	12	inner
117.2.ba.a.98.1	yes	8	3.2	odd	2	inner
117.2.ba.a.98.2	yes	8	1.1	even	1	trivial
1521.2.i.d.746.2		8	13.6	odd	12
1521.2.i.d.746.3		8	39.32	even	12
1521.2.i.d.944.2		8	39.35	odd	6
1521.2.i.d.944.3		8	13.9	even	3
1521.2.i.e.746.2		8	39.20	even	12
1521.2.i.e.746.3		8	13.7	odd	12
1521.2.i.e.944.2		8	13.4	even	6
1521.2.i.e.944.3		8	39.17	odd	6

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:	\( N \)	\(=\)	\( 117 = 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	117.ba (of order \(12\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\(q+(0.517638 + 1.93185i) q^{2} +(-1.73205 + 1.00000i) q^{4} +(-1.93185 + 1.93185i) q^{5} +(-0.500000 - 0.133975i) q^{7} +O(q^{10})\)	\(q+(0.517638 + 1.93185i) q^{2} +(-1.73205 + 1.00000i) q^{4} +(-1.93185 + 1.93185i) q^{5} +(-0.500000 - 0.133975i) q^{7} +(-4.73205 - 2.73205i) q^{10} +(4.05317 - 1.08604i) q^{11} +(3.59808 + 0.232051i) q^{13} -1.03528i q^{14} +(-2.00000 + 3.46410i) q^{16} +(-3.34607 - 5.79555i) q^{17} +(1.63397 - 6.09808i) q^{19} +(1.41421 - 5.27792i) q^{20} +(4.19615 + 7.26795i) q^{22} +(-1.22474 + 2.12132i) q^{23} -2.46410i q^{25} +(1.41421 + 7.07107i) q^{26} +(1.00000 - 0.267949i) q^{28} +(-1.22474 - 0.707107i) q^{29} +(4.63397 + 4.63397i) q^{31} +(-7.72741 - 2.07055i) q^{32} +(9.46410 - 9.46410i) q^{34} +(1.22474 - 0.707107i) q^{35} +(0.830127 + 3.09808i) q^{37} +12.6264 q^{38} +(-0.378937 - 1.41421i) q^{41} +(-6.69615 + 3.86603i) q^{43} +(-5.93426 + 5.93426i) q^{44} +(-4.73205 - 1.26795i) q^{46} +(-2.31079 - 2.31079i) q^{47} +(-5.83013 - 3.36603i) q^{49} +(4.76028 - 1.27551i) q^{50} +(-6.46410 + 3.19615i) q^{52} -5.93426i q^{53} +(-5.73205 + 9.92820i) q^{55} +(0.732051 - 2.73205i) q^{58} +(0.0507680 - 0.189469i) q^{59} +(-6.59808 - 11.4282i) q^{61} +(-6.55343 + 11.3509i) q^{62} -8.00000i q^{64} +(-7.39924 + 6.50266i) q^{65} +(-7.59808 + 2.03590i) q^{67} +(11.5911 + 6.69213i) q^{68} +(2.00000 + 2.00000i) q^{70} +(15.1266 + 4.05317i) q^{71} +(2.90192 - 2.90192i) q^{73} +(-5.55532 + 3.20736i) q^{74} +(3.26795 + 12.1962i) q^{76} -2.17209 q^{77} +7.19615 q^{79} +(-2.82843 - 10.5558i) q^{80} +(2.53590 - 1.46410i) q^{82} +(-5.27792 + 5.27792i) q^{83} +(17.6603 + 4.73205i) q^{85} +(-10.9348 - 10.9348i) q^{86} +(4.38134 - 1.17398i) q^{89} +(-1.76795 - 0.598076i) q^{91} -4.89898i q^{92} +(3.26795 - 5.66025i) q^{94} +(8.62398 + 14.9372i) q^{95} +(-1.30385 + 4.86603i) q^{97} +(3.48477 - 13.0053i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{7}+O(q^{10}) \) 8 * q - 4 * q^7	\( 8 q - 4 q^{7} - 24 q^{10} + 8 q^{13} - 16 q^{16} + 20 q^{19} - 8 q^{22} + 8 q^{28} + 44 q^{31} + 48 q^{34} - 28 q^{37} - 12 q^{43} - 24 q^{46} - 12 q^{49} - 24 q^{52} - 32 q^{55} - 8 q^{58} - 32 q^{61} - 40 q^{67} + 16 q^{70} + 44 q^{73} + 40 q^{76} + 16 q^{79} + 48 q^{82} + 72 q^{85} - 28 q^{91} + 40 q^{94} - 52 q^{97}+O(q^{100}) \) 8 * q - 4 * q^7 - 24 * q^10 + 8 * q^13 - 16 * q^16 + 20 * q^19 - 8 * q^22 + 8 * q^28 + 44 * q^31 + 48 * q^34 - 28 * q^37 - 12 * q^43 - 24 * q^46 - 12 * q^49 - 24 * q^52 - 32 * q^55 - 8 * q^58 - 32 * q^61 - 40 * q^67 + 16 * q^70 + 44 * q^73 + 40 * q^76 + 16 * q^79 + 48 * q^82 + 72 * q^85 - 28 * q^91 + 40 * q^94 - 52 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more