LMFDB - Embedded newform 1638.2.dm.c.415.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1638,2,Mod(415,1638)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1638, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2, 3])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1638.415"); 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.dm (of order $6$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,0,6,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(10)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$13.0794958511$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(\zeta_{6})$
Coefficient field:	12.0.752609431977984.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 15x^{10} + 90x^{8} - 247x^{6} + 270x^{4} + 21x^{2} + 49 $ x^12 - 15x^10 + 90x^8 - 247x^6 + 270x^4 + 21*x^2 + 49
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 546)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			415.3
Root			$-2.23871 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	1638.415
Dual form			1638.2.dm.c.1117.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-0.866025 - 0.500000i) q^{2} +(0.500000 + 0.866025i) q^{4} +(1.95878 + 1.13090i) q^{5} +(-2.41839 + 1.07303i) q^{7} -1.00000i q^{8} +(-1.13090 - 1.95878i) q^{10} +(1.09275 - 0.630901i) q^{11} +(-1.26180 - 3.37755i) q^{13} +(2.63090 + 0.279927i) q^{14} +(-0.500000 + 0.866025i) q^{16} +(0.188776 + 0.326969i) q^{17} +(-6.71612 - 3.87755i) q^{19} +2.26180i q^{20} -1.26180 q^{22} +(2.24665 - 3.89131i) q^{23} +(0.0578747 + 0.100242i) q^{25} +(-0.596023 + 3.55595i) q^{26} +(-2.13846 - 1.55787i) q^{28} +2.40786 q^{29} +(-3.46410 + 2.00000i) q^{31} +(0.866025 - 0.500000i) q^{32} -0.377552i q^{34} +(-5.95058 - 0.633140i) q^{35} +(-2.92505 - 1.68878i) q^{37} +(3.87755 + 6.71612i) q^{38} +(1.13090 - 1.95878i) q^{40} -9.90116i q^{41} +8.75510 q^{43} +(1.09275 + 0.630901i) q^{44} +(-3.89131 + 2.24665i) q^{46} +(-2.85105 - 1.64605i) q^{47} +(4.69723 - 5.18999i) q^{49} -0.115749i q^{50} +(2.29414 - 2.78153i) q^{52} +(1.50000 + 2.59808i) q^{53} +2.85395 q^{55} +(1.07303 + 2.41839i) q^{56} +(-2.08526 - 1.20393i) q^{58} +(-4.04404 + 2.33483i) q^{59} +(4.45058 - 7.70863i) q^{61} +4.00000 q^{62} -1.00000 q^{64} +(1.34809 - 8.04285i) q^{65} +(-1.11900 + 0.646053i) q^{67} +(-0.188776 + 0.326969i) q^{68} +(4.83678 + 3.52360i) q^{70} +4.11575i q^{71} +(8.26232 - 4.77026i) q^{73} +(1.68878 + 2.92505i) q^{74} -7.75510i q^{76} +(-1.96573 + 2.69832i) q^{77} +(3.14605 - 5.44912i) q^{79} +(-1.95878 + 1.13090i) q^{80} +(-4.95058 + 8.57465i) q^{82} -6.39446i q^{83} +0.853947i q^{85} +(-7.58214 - 4.37755i) q^{86} +(-0.630901 - 1.09275i) q^{88} +(9.85325 + 5.68878i) q^{89} +(6.67573 + 6.81429i) q^{91} +4.49330 q^{92} +(1.64605 + 2.85105i) q^{94} +(-8.77026 - 15.1905i) q^{95} -8.03030i q^{97} +(-6.66292 + 2.14605i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 6 q^{4} + 12 q^{13} + 18 q^{14} - 6 q^{16} - 18 q^{17} + 12 q^{22} - 6 q^{25} - 12 q^{29} - 24 q^{35} + 6 q^{38} + 24 q^{43} - 18 q^{49} + 6 q^{52} + 18 q^{53} + 48 q^{55} + 6 q^{56} + 6 q^{61} + 48 q^{62}+ \cdots - 24 q^{95}+O(q^{100}) $ 12 * q + 6 * q^4 + 12 * q^13 + 18 * q^14 - 6 * q^16 - 18 * q^17 + 12 * q^22 - 6 * q^25 - 12 * q^29 - 24 * q^35 + 6 * q^38 + 24 * q^43 - 18 * q^49 + 6 * q^52 + 18 * q^53 + 48 * q^55 + 6 * q^56 + 6 * q^61 + 48 * q^62 - 12 * q^64 + 12 * q^65 + 18 * q^68 + 24 * q^77 + 24 * q^79 - 12 * q^82 + 6 * q^88 - 24 * q^91 + 6 * q^94 - 24 * q^95

Character values

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

$n$	$379$	$703$	$911$
$\chi(n)$	$-1$	$e\left(\frac{1}{3}\right)$	$1$

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 $

$2$	−0.866025	−	0.500000i	−0.612372	−	0.353553i
$2$	−0.866025	−	0.500000i	−0.612372	−	0.353553i
$3$		0			0
$3$		0			0
$4$	0.500000	+	0.866025i	0.250000	+	0.433013i
$5$	1.95878	+	1.13090i	0.875992	+	0.505754i	0.869335	−	0.494223i	$-0.164548\pi$
$5$	1.95878	+	1.13090i	0.875992	+	0.505754i	0.00665735	+	0.999978i	$0.497881\pi$
$6$		0			0
$7$	−2.41839	+	1.07303i	−0.914066	+	0.405566i
$7$	−2.41839	+	1.07303i	−0.914066	+	0.405566i
$8$		−	1.00000i		−	0.353553i
$9$		0			0
$10$	−1.13090	−	1.95878i	−0.357622	−	0.619420i
$11$	1.09275	−	0.630901i	0.329477	−	0.190224i	−0.326132	−	0.945324i	$-0.605745\pi$
$11$	1.09275	−	0.630901i	0.329477	−	0.190224i	0.655609	+	0.755101i	$0.272412\pi$
$12$		0			0
$13$	−1.26180	−	3.37755i	−0.349961	−	0.936764i
$13$	−1.26180	−	3.37755i	−0.349961	−	0.936764i
$14$	2.63090	+	0.279927i	0.703138	+	0.0748137i
$15$		0			0
$16$	−0.500000	+	0.866025i	−0.125000	+	0.216506i
$17$	0.188776	+	0.326969i	0.0457849	+	0.0793017i	0.888010	−	0.459825i	$-0.152088\pi$
$17$	0.188776	+	0.326969i	0.0457849	+	0.0793017i	−0.842225	+	0.539127i	$0.818754\pi$
$18$		0			0
$19$	−6.71612	−	3.87755i	−1.54078	−	0.889571i	−0.998790	−	0.0491885i	$-0.984336\pi$
$19$	−6.71612	−	3.87755i	−1.54078	−	0.889571i	−0.541993	−	0.840383i	$-0.682330\pi$
$20$			2.26180i			0.505754i
$21$		0			0
$22$	−1.26180			−0.269017
$23$	2.24665	−	3.89131i	0.468459	−	0.811395i	−0.530891	−	0.847440i	$-0.678143\pi$
$23$	2.24665	−	3.89131i	0.468459	−	0.811395i	0.999350	+	0.0360452i	$0.0114760\pi$
$24$		0			0
$25$	0.0578747	+	0.100242i	0.0115749	+	0.0200484i
$26$	−0.596023	+	3.55595i	−0.116890	+	0.697379i
$27$		0			0
$28$	−2.13846	−	1.55787i	−0.404132	−	0.294411i
$29$	2.40786			0.447127			0.223564	−	0.974689i	$-0.428231\pi$
$29$	2.40786			0.447127			0.223564	+	0.974689i	$0.428231\pi$
$30$		0			0
$31$	−3.46410	+	2.00000i	−0.622171	+	0.359211i	−0.777714	−	0.628619i	$-0.783621\pi$
$31$	−3.46410	+	2.00000i	−0.622171	+	0.359211i	0.155543	+	0.987829i	$0.450287\pi$
$32$	0.866025	−	0.500000i	0.153093	−	0.0883883i
$33$		0			0
$34$		−	0.377552i		−	0.0647496i
$35$	−5.95058	−	0.633140i	−1.00583	−	0.107020i
$36$		0			0
$37$	−2.92505	−	1.68878i	−0.480875	−	0.277633i	0.239906	−	0.970796i	$-0.422883\pi$
$37$	−2.92505	−	1.68878i	−0.480875	−	0.277633i	−0.720781	+	0.693163i	$0.756217\pi$
$38$	3.87755	+	6.71612i	0.629022	+	1.08950i
$39$		0			0
$40$	1.13090	−	1.95878i	0.178811	−	0.309710i
$41$		−	9.90116i		−	1.54630i	−0.634223	−	0.773150i	$-0.718680\pi$
$41$		−	9.90116i		−	1.54630i	0.634223	−	0.773150i	$-0.281320\pi$
$42$		0			0
$43$	8.75510			1.33514			0.667570	−	0.744547i	$-0.267334\pi$
$43$	8.75510			1.33514			0.667570	+	0.744547i	$0.267334\pi$
$44$	1.09275	+	0.630901i	0.164739	+	0.0951119i
$45$		0			0
$46$	−3.89131	+	2.24665i	−0.573743	+	0.331251i
$47$	−2.85105	−	1.64605i	−0.415868	−	0.240101i	0.277440	−	0.960743i	$-0.410514\pi$
$47$	−2.85105	−	1.64605i	−0.415868	−	0.240101i	−0.693308	+	0.720642i	$0.743847\pi$
$48$		0			0
$49$	4.69723	−	5.18999i	0.671033	−	0.741428i
$50$		−	0.115749i		−	0.0163694i
$51$		0			0
$52$	2.29414	−	2.78153i	0.318141	−	0.385729i
$53$	1.50000	+	2.59808i	0.206041	+	0.356873i	0.950464	−	0.310835i	$-0.100609\pi$
$53$	1.50000	+	2.59808i	0.206041	+	0.356873i	−0.744423	+	0.667708i	$0.767275\pi$
$54$		0			0
$55$	2.85395			0.384826
$56$	1.07303	+	2.41839i	0.143389	+	0.323171i
$57$		0			0
$58$	−2.08526	−	1.20393i	−0.273809	−	0.158083i
$59$	−4.04404	+	2.33483i	−0.526489	+	0.303969i	−0.739586	−	0.673062i	$-0.764979\pi$
$59$	−4.04404	+	2.33483i	−0.526489	+	0.303969i	0.213096	+	0.977031i	$0.431645\pi$
$60$		0			0
$61$	4.45058	−	7.70863i	0.569838	−	0.986989i	−0.426743	−	0.904373i	$-0.640339\pi$
$61$	4.45058	−	7.70863i	0.569838	−	0.986989i	0.996581	−	0.0826158i	$-0.0263274\pi$
$62$	4.00000			0.508001
$63$		0			0
$64$	−1.00000			−0.125000
$65$	1.34809	−	8.04285i	0.167210	−	0.997593i
$66$		0			0
$67$	−1.11900	+	0.646053i	−0.136707	+	0.0789279i	−0.566794	−	0.823860i	$-0.691816\pi$
$67$	−1.11900	+	0.646053i	−0.136707	+	0.0789279i	0.430087	+	0.902788i	$0.358483\pi$
$68$	−0.188776	+	0.326969i	−0.0228924	+	0.0396509i
$69$		0			0
$70$	4.83678	+	3.52360i	0.578106	+	0.421151i
$71$			4.11575i			0.488450i	0.969719	+	0.244225i	$0.0785335\pi$
$71$			4.11575i			0.488450i	−0.969719	+	0.244225i	$0.921467\pi$
$72$		0			0
$73$	8.26232	−	4.77026i	0.967032	−	0.558316i	0.0687018	−	0.997637i	$-0.478114\pi$
$73$	8.26232	−	4.77026i	0.967032	−	0.558316i	0.898330	+	0.439321i	$0.144781\pi$
$74$	1.68878	+	2.92505i	0.196316	+	0.340030i
$75$		0			0
$76$		−	7.75510i		−	0.889571i
$77$	−1.96573	+	2.69832i	−0.224016	+	0.307502i
$78$		0			0
$79$	3.14605	−	5.44912i	0.353959	−	0.613074i	−0.632981	−	0.774168i	$-0.718169\pi$
$79$	3.14605	−	5.44912i	0.353959	−	0.613074i	0.986939	+	0.161093i	$0.0515020\pi$
$80$	−1.95878	+	1.13090i	−0.218998	+	0.126439i
$81$		0			0
$82$	−4.95058	+	8.57465i	−0.546700	+	0.946912i
$83$		−	6.39446i		−	0.701883i	−0.936397	−	0.350941i	$-0.885862\pi$
$83$		−	6.39446i		−	0.701883i	0.936397	−	0.350941i	$-0.114138\pi$
$84$		0			0
$85$			0.853947i			0.0926236i
$86$	−7.58214	−	4.37755i	−0.817603	−	0.472044i
$87$		0			0
$88$	−0.630901	−	1.09275i	−0.0672543	−	0.116488i
$89$	9.85325	+	5.68878i	1.04444	+	0.603009i	0.921088	−	0.389354i	$-0.127302\pi$
$89$	9.85325	+	5.68878i	1.04444	+	0.603009i	0.123354	+	0.992363i	$0.460635\pi$
$90$		0			0
$91$	6.67573	+	6.81429i	0.699807	+	0.714332i
$92$	4.49330			0.468459
$93$		0			0
$94$	1.64605	+	2.85105i	0.169777	+	0.294063i
$95$	−8.77026	−	15.1905i	−0.899809	−	1.55852i
$96$		0			0
$97$		−	8.03030i		−	0.815354i	−0.913126	−	0.407677i	$-0.866339\pi$
$97$		−	8.03030i		−	0.815354i	0.913126	−	0.407677i	$-0.133661\pi$
$98$	−6.66292	+	2.14605i	−0.673056	+	0.216784i
$99$		0			0
$100$	−0.0578747	+	0.100242i	−0.00578747	+	0.0100242i
$101$	5.81968	+	10.0800i	0.579079	+	1.00300i	0.995585	+	0.0938620i	$0.0299213\pi$
$101$	5.81968	+	10.0800i	0.579079	+	1.00300i	−0.416506	+	0.909133i	$0.636745\pi$
$102$		0			0
$103$	−6.23150	+	10.7933i	−0.614008	+	1.06349i	0.376550	+	0.926396i	$0.377110\pi$
$103$	−6.23150	+	10.7933i	−0.614008	+	1.06349i	−0.990558	+	0.137096i	$0.956223\pi$
$104$	−3.37755	+	1.26180i	−0.331196	+	0.123730i
$105$		0			0
$106$		−	3.00000i		−	0.291386i
$107$	8.90116	−	15.4173i	0.860507	−	1.49044i	−0.0109328	−	0.999940i	$-0.503480\pi$
$107$	8.90116	−	15.4173i	0.860507	−	1.49044i	0.871440	−	0.490502i	$-0.163187\pi$
$108$		0			0
$109$	2.83944	−	1.63935i	0.271969	−	0.157022i	−0.357813	−	0.933793i	$-0.616477\pi$
$109$	2.83944	−	1.63935i	0.271969	−	0.157022i	0.629782	+	0.776772i	$0.283144\pi$
$110$	−2.47159	−	1.42697i	−0.235657	−	0.136057i
$111$		0			0
$112$	0.279927	−	2.63090i	0.0264506	−	0.248597i
$113$	−1.32241			−0.124402			−0.0622009	−	0.998064i	$-0.519812\pi$
$113$	−1.32241			−0.124402			−0.0622009	+	0.998064i	$0.519812\pi$
$114$		0			0
$115$	8.80138	−	5.08148i	0.820733	−	0.473850i
$116$	1.20393	+	2.08526i	0.111782	+	0.193612i
$117$		0			0
$118$	4.66966			0.429877
$119$	−0.807380	−	0.588178i	−0.0740124	−	0.0539182i
$120$		0			0
$121$	−4.70393	+	8.14744i	−0.427630	+	0.740677i
$122$	−7.70863	+	4.45058i	−0.697906	+	0.402936i
$123$		0			0
$124$	−3.46410	−	2.00000i	−0.311086	−	0.179605i
$125$		−	11.0472i		−	0.988092i
$126$		0			0
$127$	−16.3642			−1.45208			−0.726042	−	0.687650i	$-0.758642\pi$
$127$	−16.3642			−1.45208			−0.726042	+	0.687650i	$0.758642\pi$
$128$	0.866025	+	0.500000i	0.0765466	+	0.0441942i
$129$		0			0
$130$	−5.18890	+	6.29127i	−0.455097	+	0.551781i
$131$	−5.71908	+	9.90574i	−0.499678	+	0.865468i	−1.00000		0.000371455i	$-0.999882\pi$
$131$	−5.71908	+	9.90574i	−0.499678	+	0.865468i	0.500322	+	0.865840i	$0.333215\pi$
$132$		0			0
$133$	20.4029	+	2.17086i	1.76916	+	0.188238i
$134$	1.29211			0.111621
$135$		0			0
$136$	0.326969	−	0.188776i	0.0280374	−	0.0161874i
$137$	−9.65276	+	5.57303i	−0.824691	+	0.476136i	−0.852032	−	0.523490i	$-0.824630\pi$
$137$	−9.65276	+	5.57303i	−0.824691	+	0.476136i	0.0273402	+	0.999626i	$0.491296\pi$
$138$		0			0
$139$	−2.49330			−0.211479			−0.105740	−	0.994394i	$-0.533721\pi$
$139$	−2.49330			−0.211479			−0.105740	+	0.994394i	$0.533721\pi$
$140$	−2.42697	−	5.46992i	−0.205117	−	0.462293i
$141$		0			0
$142$	2.05787	−	3.56434i	0.172693	−	0.299113i
$143$	−3.50974	−	2.89476i	−0.293499	−	0.242072i
$144$		0			0
$145$	4.71645	+	2.72305i	0.391680	+	0.226137i
$146$	−9.54051			−0.789578
$147$		0			0
$148$		−	3.37755i		−	0.277633i
$149$	0.0262436	+	0.0151517i	0.00214996	+	0.00124128i	0.501075	−	0.865404i	$-0.332938\pi$
$149$	0.0262436	+	0.0151517i	0.00214996	+	0.00124128i	−0.498925	+	0.866645i	$0.666272\pi$
$150$		0			0
$151$	−3.73171	+	2.15451i	−0.303683	+	0.175331i	−0.644096	−	0.764945i	$-0.722766\pi$
$151$	−3.73171	+	2.15451i	−0.303683	+	0.175331i	0.340413	+	0.940276i	$0.389433\pi$
$152$	−3.87755	+	6.71612i	−0.314511	+	0.544749i
$153$		0			0
$154$	3.05153	−	1.35395i	0.245899	−	0.109104i
$155$	−9.04721			−0.726689
$156$		0			0
$157$	−11.3557	−	19.6687i	−0.906284	−	1.56973i	−0.819185	−	0.573530i	$-0.805574\pi$
$157$	−11.3557	−	19.6687i	−0.906284	−	1.56973i	−0.0870987	−	0.996200i	$-0.527760\pi$
$158$	−5.44912	+	3.14605i	−0.433509	+	0.250287i
$159$		0			0
$160$	2.26180			0.178811
$161$	−1.25780	+	11.8214i	−0.0991283	+	0.931659i
$162$		0			0
$163$	−10.7455	−	6.20393i	−0.841654	−	0.485929i	0.0161722	−	0.999869i	$-0.494852\pi$
$163$	−10.7455	−	6.20393i	−0.841654	−	0.485929i	−0.857826	+	0.513940i	$0.828185\pi$
$164$	8.57465	−	4.95058i	0.669568	−	0.386575i
$165$		0			0
$166$	−3.19723	+	5.53776i	−0.248153	+	0.429814i
$167$		−	24.6732i		−	1.90927i	−0.297784	−	0.954633i	$-0.596247\pi$
$167$		−	24.6732i		−	1.90927i	0.297784	−	0.954633i	$-0.403753\pi$
$168$		0			0
$169$	−9.81571	+	8.52360i	−0.755055	+	0.655662i
$170$	0.426974	−	0.739540i	0.0327474	−	0.0567201i
$171$		0			0
$172$	4.37755	+	7.58214i	0.333785	+	0.578133i
$173$	11.9893	−	20.7661i	0.911532	−	1.57882i	0.0996316	−	0.995024i	$-0.468234\pi$
$173$	11.9893	−	20.7661i	0.911532	−	1.57882i	0.811901	−	0.583796i	$-0.198433\pi$
$174$		0			0
$175$	−0.247526	−	0.180323i	−0.0187112	−	0.0136311i
$176$			1.26180i			0.0951119i
$177$		0			0
$178$	−5.68878	−	9.85325i	−0.426392	−	0.738532i
$179$	−2.85395	−	4.94318i	−0.213314	−	0.369471i	0.739436	−	0.673227i	$-0.235092\pi$
$179$	−2.85395	−	4.94318i	−0.213314	−	0.369471i	−0.952750	+	0.303756i	$0.901759\pi$
$180$		0			0
$181$	−16.6697			−1.23905			−0.619523	−	0.784979i	$-0.712674\pi$
$181$	−16.6697			−1.23905			−0.619523	+	0.784979i	$0.712674\pi$
$182$	−2.37421	−	9.23922i	−0.175988	−	0.684856i
$183$		0			0
$184$	−3.89131	−	2.24665i	−0.286871	−	0.165625i
$185$	−3.81968	−	6.61587i	−0.280828	−	0.486409i
$186$		0			0
$187$	0.412571	+	0.238198i	0.0301702	+	0.0174187i
$188$		−	3.29211i		−	0.240101i
$189$		0			0
$190$			17.5405i			1.27252i
$191$	3.63935	−	6.30355i	0.263334	−	0.456109i	−0.703791	−	0.710407i	$-0.748511\pi$
$191$	3.63935	−	6.30355i	0.263334	−	0.456109i	0.967126	+	0.254298i	$0.0818444\pi$
$192$		0			0
$193$	14.9900	−	8.65451i	1.07901	−	0.622965i	0.148379	−	0.988931i	$-0.452595\pi$
$193$	14.9900	−	8.65451i	1.07901	−	0.622965i	0.930628	+	0.365966i	$0.119261\pi$
$194$	−4.01515	+	6.95445i	−0.288271	+	0.499300i
$195$		0			0
$196$	6.84328	+	1.47292i	0.488806	+	0.105209i
$197$		−	3.21459i		−	0.229030i	−0.993422	−	0.114515i	$-0.963469\pi$
$197$		−	3.21459i		−	0.229030i	0.993422	−	0.114515i	$-0.0365314\pi$
$198$		0			0
$199$	−5.85173	−	10.1355i	−0.414818	−	0.718487i	0.580591	−	0.814195i	$-0.302822\pi$
$199$	−5.85173	−	10.1355i	−0.414818	−	0.718487i	−0.995409	+	0.0957088i	$0.969488\pi$
$200$	0.100242	−	0.0578747i	0.00708817	−	0.00409236i
$201$		0			0
$202$		−	11.6394i		−	0.818942i
$203$	−5.82313	+	2.58369i	−0.408704	+	0.181340i
$204$		0			0
$205$	11.1972	−	19.3942i	0.782048	−	1.35455i
$206$	10.7933	−	6.23150i	0.752003	−	0.434169i
$207$		0			0
$208$	3.55595	+	0.596023i	0.246561	+	0.0413268i
$209$	−9.78541			−0.676871
$210$		0			0
$211$	−8.06411			−0.555157			−0.277578	−	0.960703i	$-0.589532\pi$
$211$	−8.06411			−0.555157			−0.277578	+	0.960703i	$0.589532\pi$
$212$	−1.50000	+	2.59808i	−0.103020	+	0.178437i
$213$		0			0
$214$	−15.4173	+	8.90116i	−1.05390	+	0.608471i
$215$	17.1493	+	9.90116i	1.16957	+	0.675253i
$216$		0			0
$217$	6.23150	−	8.55385i	0.423022	−	0.580673i
$218$	−3.27871			−0.222062
$219$		0			0
$220$	1.42697	+	2.47159i	0.0962065	+	0.166635i
$221$	0.866158	−	1.05017i	0.0582641	−	0.0706421i
$222$		0			0
$223$			23.7854i			1.59279i	0.604778	+	0.796394i	$0.293262\pi$
$223$			23.7854i			1.59279i	−0.604778	+	0.796394i	$0.706738\pi$
$224$	−1.55787	+	2.13846i	−0.104090	+	0.142882i
$225$		0			0
$226$	1.14524	+	0.661205i	0.0761802	+	0.0439827i
$227$	8.86074	−	5.11575i	0.588108	−	0.339544i	−0.176241	−	0.984347i	$-0.556394\pi$
$227$	8.86074	−	5.11575i	0.588108	−	0.339544i	0.764349	+	0.644803i	$0.223061\pi$
$228$		0			0
$229$	−0.906910	−	0.523604i	−0.0599303	−	0.0346008i	0.469735	−	0.882807i	$-0.344349\pi$
$229$	−0.906910	−	0.523604i	−0.0599303	−	0.0346008i	−0.529666	+	0.848206i	$0.677683\pi$
$230$	−10.1630			−0.670126
$231$		0			0
$232$		−	2.40786i		−	0.158083i
$233$	−1.87580	+	3.24898i	−0.122888	+	0.212848i	−0.920905	−	0.389787i	$-0.872549\pi$
$233$	−1.87580	+	3.24898i	−0.122888	+	0.212848i	0.798018	+	0.602634i	$0.205882\pi$
$234$		0			0
$235$	−3.72305	−	6.44850i	−0.242865	−	0.420654i
$236$	−4.04404	−	2.33483i	−0.263245	−	0.151984i
$237$		0			0
$238$	0.405123	+	0.913067i	0.0262602	+	0.0591854i
$239$			29.1968i			1.88858i	0.329112	+	0.944291i	$0.393251\pi$
$239$			29.1968i			1.88858i	−0.329112	+	0.944291i	$0.606749\pi$
$240$		0			0
$241$	−1.47908	+	0.853947i	−0.0952759	+	0.0550076i	−0.546881	−	0.837210i	$-0.684185\pi$
$241$	−1.47908	+	0.853947i	−0.0952759	+	0.0550076i	0.451605	+	0.892218i	$0.350852\pi$
$242$	8.14744	−	4.70393i	0.523737	−	0.302380i
$243$		0			0
$244$	8.90116			0.569838
$245$	15.0702	−	4.85395i	0.962800	−	0.310107i
$246$		0			0
$247$	−4.62222	+	27.5767i	−0.294105	+	1.75467i
$248$	2.00000	+	3.46410i	0.127000	+	0.219971i
$249$		0			0
$250$	−5.52360	+	9.56716i	−0.349343	+	0.605081i
$251$	−19.4496			−1.22765			−0.613824	−	0.789443i	$-0.710370\pi$
$251$	−19.4496			−1.22765			−0.613824	+	0.789443i	$0.710370\pi$
$252$		0			0
$253$		−	5.66966i		−	0.356448i
$254$	14.1718	+	8.18208i	0.889216	+	0.513389i
$255$		0			0
$256$	−0.500000	−	0.866025i	−0.0312500	−	0.0541266i
$257$	6.14605	−	10.6453i	0.383380	−	0.664034i	−0.608163	−	0.793812i	$-0.708093\pi$
$257$	6.14605	−	10.6453i	0.383380	−	0.664034i	0.991543	+	0.129778i	$0.0414266\pi$
$258$		0			0
$259$	8.88600	+	0.945469i	0.552149	+	0.0587486i
$260$	7.63935	−	2.85395i	0.473773	−	0.176994i
$261$		0			0
$262$	9.90574	−	5.71908i	0.611978	−	0.353326i
$263$	1.65451	+	2.86569i	0.102021	+	0.176706i	0.912517	−	0.409038i	$-0.134136\pi$
$263$	1.65451	+	2.86569i	0.102021	+	0.176706i	−0.810496	+	0.585744i	$0.800802\pi$
$264$		0			0
$265$			6.78541i			0.416824i
$266$	−16.5840	−	12.0815i	−1.01683	−	0.740763i
$267$		0			0
$268$	−1.11900	−	0.646053i	−0.0683536	−	0.0394640i
$269$	16.0062	+	27.7236i	0.975918	+	1.69034i	0.676873	+	0.736100i	$0.263335\pi$
$269$	16.0062	+	27.7236i	0.975918	+	1.69034i	0.299045	+	0.954239i	$0.403332\pi$
$270$		0			0
$271$	−19.3202	−	11.1545i	−1.17362	−	0.677588i	−0.219088	−	0.975705i	$-0.570308\pi$
$271$	−19.3202	−	11.1545i	−1.17362	−	0.677588i	−0.954529	+	0.298117i	$0.903641\pi$
$272$	−0.377552			−0.0228924
$273$		0			0
$274$	11.1461			0.673358
$275$	0.126485	+	0.0730264i	0.00762736	+	0.00440366i
$276$		0			0
$277$	5.20568	+	9.01650i	0.312779	+	0.541749i	0.978963	−	0.204038i	$-0.0654067\pi$
$277$	5.20568	+	9.01650i	0.312779	+	0.541749i	−0.666184	+	0.745788i	$0.732073\pi$
$278$	2.15926	+	1.24665i	0.129504	+	0.0747691i
$279$		0			0
$280$	−0.633140	+	5.95058i	−0.0378374	+	0.355615i
$281$			29.5102i			1.76043i	0.474574	+	0.880216i	$0.342602\pi$
$281$			29.5102i			1.76043i	−0.474574	+	0.880216i	$0.657398\pi$
$282$		0			0
$283$	−1.60730	−	2.78392i	−0.0955439	−	0.165487i	0.814292	−	0.580456i	$-0.197126\pi$
$283$	−1.60730	−	2.78392i	−0.0955439	−	0.165487i	−0.909835	+	0.414969i	$0.863792\pi$
$284$	−3.56434	+	2.05787i	−0.211505	+	0.122112i
$285$		0			0
$286$	1.59214	+	4.26180i	0.0941455	+	0.252006i
$287$	10.6242	+	23.9449i	0.627127	+	1.41342i
$288$		0			0
$289$	8.42873	−	14.5990i	0.495807	−	0.858764i
$290$	−2.72305	−	4.71645i	−0.159903	−	0.276960i
$291$		0			0
$292$	8.26232	+	4.77026i	0.483516	+	0.279158i
$293$			24.7248i			1.44444i	0.691664	+	0.722219i	$0.256878\pi$
$293$			24.7248i			1.44444i	−0.691664	+	0.722219i	$0.743122\pi$
$294$		0			0
$295$	−10.5618			−0.614934
$296$	−1.68878	+	2.92505i	−0.0981581	+	0.170015i
$297$		0			0
$298$	−0.0151517	−	0.0262436i	−0.000877716	−	0.00152025i
$299$	−15.9779	−	2.67811i	−0.924028	−	0.154879i
$300$		0			0
$301$	−21.1733	+	9.39446i	−1.22041	+	0.541488i
$302$	4.30901			0.247956
$303$		0			0
$304$	6.71612	−	3.87755i	0.385196	−	0.222393i
$305$	17.4354	−	10.0663i	0.998348	−	0.576396i
$306$		0			0
$307$			6.93939i			0.396052i	0.980197	+	0.198026i	$0.0634531\pi$
$307$			6.93939i			0.396052i	−0.980197	+	0.198026i	$0.936547\pi$
$308$	−3.31968	−	0.353213i	−0.189156	−	0.0201262i
$309$		0			0
$310$	7.83511	+	4.52360i	0.445005	+	0.256923i
$311$	−5.53876	−	9.59341i	−0.314074	−	0.543992i	0.665166	−	0.746695i	$-0.268361\pi$
$311$	−5.53876	−	9.59341i	−0.314074	−	0.543992i	−0.979240	+	0.202703i	$0.935027\pi$
$312$		0			0
$313$	10.7039	−	18.5397i	0.605022	−	1.04793i	−0.387026	−	0.922069i	$-0.626498\pi$
$313$	10.7039	−	18.5397i	0.605022	−	1.04793i	0.992048	−	0.125860i	$-0.0401689\pi$
$314$			22.7114i			1.28168i
$315$		0			0
$316$	6.29211			0.353959
$317$	−18.3754	−	10.6091i	−1.03207	−	0.595864i	−0.114490	−	0.993424i	$-0.536523\pi$
$317$	−18.3754	−	10.6091i	−1.03207	−	0.595864i	−0.917576	+	0.397561i	$0.869857\pi$
$318$		0			0
$319$	2.63119	−	1.51912i	0.147318	−	0.0850543i
$320$	−1.95878	−	1.13090i	−0.109499	−	0.0632193i
$321$		0			0
$322$	7.00000	−	9.60876i	0.390095	−	0.535475i
$323$		−	2.92795i		−	0.162916i
$324$		0			0
$325$	0.265546	−	0.321960i	0.0147298	−	0.0178591i
$326$	6.20393	+	10.7455i	0.343604	+	0.595139i
$327$		0			0
$328$	−9.90116			−0.546700
$329$	8.66120	+	0.921550i	0.477508	+	0.0508067i
$330$		0			0
$331$	18.2205	+	10.5196i	1.00149	+	0.578212i	0.908689	−	0.417473i	$-0.137084\pi$
$331$	18.2205	+	10.5196i	1.00149	+	0.578212i	0.0928025	+	0.995685i	$0.470417\pi$
$332$	5.53776	−	3.19723i	0.303924	−	0.175471i
$333$		0			0
$334$	−12.3366	+	21.3676i	−0.675028	+	1.16918i
$335$	−2.92249			−0.159673
$336$		0			0
$337$	−25.5574			−1.39220			−0.696101	−	0.717944i	$-0.745083\pi$
$337$	−25.5574			−1.39220			−0.696101	+	0.717944i	$0.745083\pi$
$338$	12.7625	−	2.47380i	0.694186	−	0.134557i
$339$		0			0
$340$	−0.739540	+	0.426974i	−0.0401072	+	0.0231559i
$341$	−2.52360	+	4.37101i	−0.136661	+	0.236704i
$342$		0			0
$343$	−5.79073	+	17.5917i	−0.312670	+	0.949862i
$344$		−	8.75510i		−	0.472044i
$345$		0			0
$346$	−20.7661	+	11.9893i	−1.11639	+	0.644551i
$347$	9.16517	+	15.8745i	0.492012	+	0.852190i	0.999958	−	0.00919912i	$-0.00292821\pi$
$347$	9.16517	+	15.8745i	0.492012	+	0.852190i	−0.507946	+	0.861389i	$0.669595\pi$
$348$		0			0
$349$		−	4.19326i		−	0.224460i	−0.993682	−	0.112230i	$-0.964201\pi$
$349$		−	4.19326i		−	0.224460i	0.993682	−	0.112230i	$-0.0357994\pi$
$350$	0.124202	+	0.279927i	0.00663888	+	0.0149627i
$351$		0			0
$352$	0.630901	−	1.09275i	0.0336271	−	0.0582439i
$353$	18.9338	−	10.9315i	1.00775	−	0.581823i	0.0972156	−	0.995263i	$-0.469006\pi$
$353$	18.9338	−	10.9315i	1.00775	−	0.581823i	0.910531	+	0.413440i	$0.135673\pi$
$354$		0			0
$355$	−4.65451	+	8.06184i	−0.247036	+	0.427878i
$356$			11.3776i			0.603009i
$357$		0			0
$358$			5.70789i			0.301672i
$359$	22.4341	+	12.9523i	1.18403	+	0.683598i	0.956943	−	0.290277i	$-0.0937475\pi$
$359$	22.4341	+	12.9523i	1.18403	+	0.683598i	0.227084	+	0.973875i	$0.427081\pi$
$360$		0			0
$361$	20.5708	+	35.6297i	1.08267	+	1.87525i
$362$	14.4363	+	8.33483i	0.758758	+	0.438069i
$363$		0			0
$364$	−2.56348	+	9.18850i	−0.134363	+	0.481608i
$365$	21.5787			1.12948
$366$		0			0
$367$	8.93146	+	15.4697i	0.466218	+	0.807514i	0.999256	−	0.0385779i	$-0.0122828\pi$
$367$	8.93146	+	15.4697i	0.466218	+	0.807514i	−0.533037	+	0.846092i	$0.678949\pi$
$368$	2.24665	+	3.89131i	0.117115	+	0.202849i
$369$		0			0
$370$			7.63935i			0.397151i
$371$	−6.41539	−	4.67362i	−0.333070	−	0.242642i
$372$		0			0
$373$	7.56236	−	13.0984i	0.391564	−	0.678209i	−0.601092	−	0.799180i	$-0.705267\pi$
$373$	7.56236	−	13.0984i	0.391564	−	0.678209i	0.992656	+	0.120971i	$0.0386008\pi$
$374$	−0.238198	−	0.412571i	−0.0123169	−	0.0213335i
$375$		0			0
$376$	−1.64605	+	2.85105i	−0.0848887	+	0.147032i
$377$	−3.03824	−	8.13265i	−0.156477	−	0.418853i
$378$		0			0
$379$		−	12.5236i		−	0.643294i	−0.946860	−	0.321647i	$-0.895764\pi$
$379$		−	12.5236i		−	0.643294i	0.946860	−	0.321647i	$-0.104236\pi$
$380$	8.77026	−	15.1905i	0.449905	−	0.779258i
$381$		0			0
$382$	−6.30355	+	3.63935i	−0.322517	+	0.186206i
$383$	−1.36723	−	0.789373i	−0.0698624	−	0.0403351i	0.464662	−	0.885488i	$-0.346176\pi$
$383$	−1.36723	−	0.789373i	−0.0698624	−	0.0403351i	−0.534524	+	0.845153i	$0.679509\pi$
$384$		0			0
$385$	−6.90196	+	3.06236i	−0.351756	+	0.156072i
$386$	−17.3090			−0.881005
$387$		0			0
$388$	6.95445	−	4.01515i	0.353059	−	0.203838i
$389$	2.40512	+	4.16580i	0.121945	+	0.211214i	0.920534	−	0.390661i	$-0.127754\pi$
$389$	2.40512	+	4.16580i	0.121945	+	0.211214i	−0.798590	+	0.601876i	$0.794420\pi$
$390$		0			0
$391$	1.69645			0.0857933
$392$	−5.18999	−	4.69723i	−0.262134	−	0.237246i
$393$		0			0
$394$	−1.60730	+	2.78392i	−0.0809744	+	0.140252i
$395$	12.3248	−	7.11575i	0.620130	−	0.358032i
$396$		0			0
$397$	23.3380	+	13.4742i	1.17130	+	0.676250i	0.953986	−	0.299852i	$-0.0969373\pi$
$397$	23.3380	+	13.4742i	1.17130	+	0.676250i	0.217313	+	0.976102i	$0.430271\pi$
$398$			11.7035i			0.586642i
$399$		0			0
$400$	−0.115749			−0.00578747
$401$	−17.8364	−	10.2978i	−0.890705	−	0.514249i	−0.0165321	−	0.999863i	$-0.505263\pi$
$401$	−17.8364	−	10.2978i	−0.890705	−	0.514249i	−0.874173	+	0.485614i	$0.838596\pi$
$402$		0			0
$403$	11.1261	+	9.17658i	0.554231	+	0.457118i
$404$	−5.81968	+	10.0800i	−0.289540	+	0.501498i
$405$		0			0
$406$	6.33483	+	0.674024i	0.314392	+	0.0334513i
$407$	−4.26180			−0.211250
$408$		0			0
$409$	4.59774	−	2.65451i	0.227344	−	0.131257i	−0.382002	−	0.924161i	$-0.624765\pi$
$409$	4.59774	−	2.65451i	0.227344	−	0.131257i	0.609346	+	0.792904i	$0.291432\pi$
$410$	−19.3942	+	11.1972i	−0.957810	+	0.552992i
$411$		0			0
$412$	−12.4630			−0.614008
$413$	7.27474	−	9.98589i	0.357967	−	0.491374i
$414$		0			0
$415$	7.23150	−	12.5253i	0.354980	−	0.614844i
$416$	−2.78153	−	2.29414i	−0.136376	−	0.112480i
$417$		0			0
$418$	8.47441	+	4.89270i	0.414497	+	0.239310i
$419$	20.5842			1.00560			0.502802	−	0.864401i	$-0.332302\pi$
$419$	20.5842			1.00560			0.502802	+	0.864401i	$0.332302\pi$
$420$		0			0
$421$			12.8878i			0.628111i	0.949405	+	0.314055i	$0.101688\pi$
$421$			12.8878i			0.628111i	−0.949405	+	0.314055i	$0.898312\pi$
$422$	6.98373	+	4.03206i	0.339963	+	0.196277i
$423$		0			0
$424$	2.59808	−	1.50000i	0.126174	−	0.0728464i
$425$	−0.0218507	+	0.0378465i	−0.00105991	+	0.00183582i
$426$		0			0
$427$	−2.49168	+	23.4181i	−0.120581	+	1.13328i
$428$	17.8023			0.860507
$429$		0			0
$430$	−9.90116	−	17.1493i	−0.477476	−	0.827013i
$431$	23.0588	−	13.3130i	1.11070	−	0.641264i	0.171690	−	0.985151i	$-0.445077\pi$
$431$	23.0588	−	13.3130i	1.11070	−	0.641264i	0.939011	+	0.343887i	$0.111744\pi$
$432$		0			0
$433$	−28.0810			−1.34949			−0.674744	−	0.738052i	$-0.735746\pi$
$433$	−28.0810			−1.34949			−0.674744	+	0.738052i	$0.735746\pi$
$434$	−9.67356	+	4.29211i	−0.464346	+	0.206028i
$435$		0			0
$436$	2.83944	+	1.63935i	0.135985	+	0.0785108i
$437$	−30.1775	+	17.4230i	−1.44359	+	0.833455i
$438$		0			0
$439$	−13.8820	+	24.0444i	−0.662554	+	1.14758i	0.317389	+	0.948295i	$0.397194\pi$
$439$	−13.8820	+	24.0444i	−0.662554	+	1.14758i	−0.979942	+	0.199281i	$0.936139\pi$
$440$		−	2.85395i		−	0.136057i
$441$		0			0
$442$	−1.27520	+	0.476396i	−0.0606551	+	0.0226598i
$443$	2.28092	−	3.95067i	0.108370	−	0.187702i	−0.806740	−	0.590906i	$-0.798770\pi$
$443$	2.28092	−	3.95067i	0.108370	−	0.187702i	0.915110	+	0.403204i	$0.132104\pi$
$444$		0			0
$445$	12.8669	+	22.2861i	0.609949	+	1.05646i
$446$	11.8927	−	20.5988i	0.563136	−	0.975380i
$447$		0			0
$448$	2.41839	−	1.07303i	0.114258	−	0.0506957i
$449$		−	2.78190i		−	0.131286i	−0.997843	−	0.0656430i	$-0.979090\pi$
$449$		−	2.78190i		−	0.131286i	0.997843	−	0.0656430i	$-0.0209098\pi$
$450$		0			0
$451$	−6.24665	−	10.8195i	−0.294143	−	0.509471i
$452$	−0.661205	−	1.14524i	−0.0311004	−	0.0538676i
$453$		0			0
$454$	−10.2315			−0.480188
$455$	5.36999	+	20.8973i	0.251749	+	0.979680i
$456$		0			0
$457$	25.8126	+	14.9029i	1.20746	+	0.697129i	0.962204	−	0.272329i	$-0.0877939\pi$
$457$	25.8126	+	14.9029i	1.20746	+	0.697129i	0.245258	+	0.969458i	$0.421127\pi$
$458$	0.523604	+	0.906910i	0.0244664	+	0.0423771i
$459$		0			0
$460$	8.80138	+	5.08148i	0.410366	+	0.236925i
$461$			18.6180i			0.867128i	0.901123	+	0.433564i	$0.142744\pi$
$461$			18.6180i			0.867128i	−0.901123	+	0.433564i	$0.857256\pi$
$462$		0			0
$463$		−	42.0338i		−	1.95348i	−0.214434	−	0.976738i	$-0.568791\pi$
$463$		−	42.0338i		−	1.95348i	0.214434	−	0.976738i	$-0.431209\pi$
$464$	−1.20393	+	2.08526i	−0.0558909	+	0.0968059i
$465$		0			0
$466$	3.24898	−	1.87580i	0.150506	−	0.0868947i
$467$	−19.4608	+	33.7071i	−0.900538	+	1.55978i	−0.0737402	+	0.997277i	$0.523494\pi$
$467$	−19.4608	+	33.7071i	−0.900538	+	1.55978i	−0.826798	+	0.562500i	$0.809840\pi$
$468$		0			0
$469$	2.01294	−	2.76312i	0.0929489	−	0.127589i
$470$			7.44609i			0.343463i
$471$		0			0
$472$	2.33483	+	4.04404i	0.107469	+	0.186142i
$473$	9.56716	−	5.52360i	0.439899	−	0.253976i
$474$		0			0
$475$		−	0.897649i		−	0.0411869i
$476$	0.105687	−	0.993301i	0.00484416	−	0.0455279i
$477$		0			0
$478$	14.5984	−	25.2851i	0.667715	−	1.15652i
$479$	−1.98029	+	1.14332i	−0.0904817	+	0.0522397i	−0.544558	−	0.838723i	$-0.683303\pi$
$479$	−1.98029	+	1.14332i	−0.0904817	+	0.0522397i	0.454076	+	0.890963i	$0.349969\pi$
$480$		0			0
$481$	−2.01310	+	12.0104i	−0.0917894	+	0.547627i
$482$	1.70789			0.0777925
$483$		0			0
$484$	−9.40786			−0.427630
$485$	9.08148	−	15.7296i	0.412369	−	0.714244i
$486$		0			0
$487$	26.7087	−	15.4203i	1.21029	−	0.698759i	0.247465	−	0.968897i	$-0.420402\pi$
$487$	26.7087	−	15.4203i	1.21029	−	0.698759i	0.962822	+	0.270138i	$0.0870692\pi$
$488$	−7.70863	−	4.45058i	−0.348953	−	0.201468i
$489$		0			0
$490$	−15.4781	−	3.33146i	−0.699232	−	0.150500i
$491$	−41.1754			−1.85822			−0.929111	−	0.369802i	$-0.879426\pi$
$491$	−41.1754			−1.85822			−0.929111	+	0.369802i	$0.879426\pi$
$492$		0			0
$493$	0.454545	+	0.787295i	0.0204717	+	0.0354580i
$494$	17.7913	−	21.5710i	0.800470	−	0.970527i
$495$		0			0
$496$		−	4.00000i		−	0.179605i
$497$	−4.41631	−	9.95349i	−0.198098	−	0.446475i
$498$		0			0
$499$	−13.3961	−	7.73423i	−0.599691	−	0.346232i	0.169229	−	0.985577i	$-0.445872\pi$
$499$	−13.3961	−	7.73423i	−0.599691	−	0.346232i	−0.768920	+	0.639345i	$0.779206\pi$
$500$	9.56716	−	5.52360i	0.427857	−	0.247023i
$501$		0			0
$502$	16.8438	+	9.72480i	0.751778	+	0.434039i
$503$	0.986602			0.0439904			0.0219952	−	0.999758i	$-0.492998\pi$
$503$	0.986602			0.0439904			0.0219952	+	0.999758i	$0.492998\pi$
$504$		0			0
$505$			26.3259i			1.17149i
$506$	−2.83483	+	4.91007i	−0.126024	+	0.218279i
$507$		0			0
$508$	−8.18208	−	14.1718i	−0.363021	−	0.628771i
$509$	−14.8588	−	8.57875i	−0.658606	−	0.380246i	0.133140	−	0.991097i	$-0.457494\pi$
$509$	−14.8588	−	8.57875i	−0.658606	−	0.380246i	−0.791746	+	0.610851i	$0.790827\pi$
$510$		0			0
$511$	−14.8629	+	20.4020i	−0.657497	+	0.902533i
$512$			1.00000i			0.0441942i
$513$		0			0
$514$	−10.6453	+	6.14605i	−0.469543	+	0.271091i
$515$	−24.4122	+	14.0944i	−1.07573	+	0.621074i
$516$		0			0
$517$	−4.15399			−0.182692
$518$	−7.22277	−	5.26180i	−0.317350	−	0.231190i
$519$		0			0
$520$	−8.04285	−	1.34809i	−0.352702	−	0.0591175i
$521$	−16.2787	−	28.1955i	−0.713183	−	1.23527i	−0.963656	−	0.267146i	$-0.913919\pi$
$521$	−16.2787	−	28.1955i	−0.713183	−	1.23527i	0.250473	−	0.968124i	$-0.419414\pi$
$522$		0			0
$523$	17.4248	−	30.1806i	0.761932	−	1.31970i	−0.179922	−	0.983681i	$-0.557585\pi$
$523$	17.4248	−	30.1806i	0.761932	−	1.31970i	0.941854	−	0.336023i	$-0.109082\pi$
$524$	−11.4382			−0.499678
$525$		0			0
$526$		−	3.30901i		−	0.144280i
$527$	−1.30788	−	0.755103i	−0.0569720	−	0.0328928i
$528$		0			0
$529$	1.40512	+	2.43374i	0.0610923	+	0.105815i
$530$	3.39270	−	5.87633i	0.147370	−	0.255252i
$531$		0			0
$532$	8.32143	+	18.7549i	0.360780	+	0.813127i
$533$	−33.4417	+	12.4933i	−1.44852	+	0.541145i
$534$		0			0
$535$	34.8708	−	20.1327i	1.50760	−	0.870411i
$536$	0.646053	+	1.11900i	0.0279052	+	0.0483333i
$537$		0			0
$538$		−	32.0125i		−	1.38016i
$539$	1.85854	−	8.63487i	0.0800528	−	0.371930i
$540$		0			0
$541$	23.3905	+	13.5045i	1.00563	+	0.580603i	0.909910	−	0.414805i	$-0.136150\pi$
$541$	23.3905	+	13.5045i	1.00563	+	0.580603i	0.0957238	+	0.995408i	$0.469483\pi$
$542$	11.1545	+	19.3202i	0.479127	+	0.829872i
$543$		0			0
$544$	0.326969	+	0.188776i	0.0140187	+	0.00809370i
$545$	7.41579			0.317657
$546$		0			0
$547$	33.1719			1.41833			0.709165	−	0.705043i	$-0.249072\pi$
$547$	33.1719			1.41833			0.709165	+	0.705043i	$0.249072\pi$
$548$	−9.65276	−	5.57303i	−0.412346	−	0.238068i
$549$		0			0
$550$	−0.0730264	−	0.126485i	−0.00311386	−	0.00539336i
$551$	−16.1714	−	9.33658i	−0.688926	−	0.397752i
$552$		0			0
$553$	−1.76133	+	16.5539i	−0.0748995	+	0.703944i
$554$		−	10.4114i		−	0.442336i
$555$		0			0
$556$	−1.24665	−	2.15926i	−0.0528698	−	0.0915731i
$557$	−16.4136	+	9.47640i	−0.695466	+	0.401528i	−0.805657	−	0.592383i	$-0.798187\pi$
$557$	−16.4136	+	9.47640i	−0.695466	+	0.401528i	0.110190	+	0.993911i	$0.464854\pi$
$558$		0			0
$559$	−11.0472	−	29.5708i	−0.467247	−	1.25071i
$560$	3.52360	−	4.83678i	0.148899	−	0.204391i
$561$		0			0
$562$	14.7551	−	25.5566i	0.622406	−	1.07804i
$563$	−15.7720	−	27.3179i	−0.664711	−	1.15131i	−0.979364	−	0.202106i	$-0.935221\pi$
$563$	−15.7720	−	27.3179i	−0.664711	−	1.15131i	0.314653	−	0.949207i	$-0.398112\pi$
$564$		0			0
$565$	−2.59031	−	1.49551i	−0.108975	−	0.0629167i
$566$			3.21459i			0.135119i
$567$		0			0
$568$	4.11575			0.172693
$569$	1.53206	−	2.65360i	0.0642272	−	0.111245i	−0.832124	−	0.554590i	$-0.812875\pi$
$569$	1.53206	−	2.65360i	0.0642272	−	0.111245i	0.896351	+	0.443345i	$0.146208\pi$
$570$		0			0
$571$	2.05514	+	3.55961i	0.0860050	+	0.148965i	0.905819	−	0.423665i	$-0.139257\pi$
$571$	2.05514	+	3.55961i	0.0860050	+	0.148965i	−0.819814	+	0.572630i	$0.805923\pi$
$572$	0.752063	−	4.48690i	0.0314453	−	0.187607i
$573$		0			0
$574$	2.77160	−	26.0490i	0.115685	−	1.08726i
$575$	0.520097			0.0216895
$576$		0			0
$577$	−16.4954	+	9.52360i	−0.686711	+	0.396473i	−0.802379	−	0.596815i	$-0.796433\pi$
$577$	−16.4954	+	9.52360i	−0.686711	+	0.396473i	0.115668	+	0.993288i	$0.463099\pi$
$578$	−14.5990	+	8.42873i	−0.607238	+	0.350589i
$579$		0			0
$580$			5.44609i			0.226137i
$581$	6.86142	+	15.4643i	0.284660	+	0.641567i
$582$		0			0
$583$	3.27826	+	1.89270i	0.135772	+	0.0783878i
$584$	−4.77026	−	8.26232i	−0.197395	−	0.341897i
$585$		0			0
$586$	12.3624	−	21.4123i	0.510686	−	0.884534i
$587$		−	36.2216i		−	1.49503i	−0.664247	−	0.747513i	$-0.731248\pi$
$587$		−	36.2216i		−	1.49503i	0.664247	−	0.747513i	$-0.268752\pi$
$588$		0			0
$589$	31.0204			1.27817
$590$	9.14682	+	5.28092i	0.376569	+	0.217412i
$591$		0			0
$592$	2.92505	−	1.68878i	0.120219	−	0.0694083i
$593$	4.35164	+	2.51242i	0.178700	+	0.103173i	0.586682	−	0.809817i	$-0.300434\pi$
$593$	4.35164	+	2.51242i	0.178700	+	0.103173i	−0.407982	+	0.912990i	$0.633767\pi$
$594$		0			0
$595$	−0.916308	−	2.06518i	−0.0375650	−	0.0846640i
$596$			0.0303035i			0.00124128i
$597$		0			0
$598$	12.4982	+	10.3083i	0.511091	+	0.421537i
$599$	15.4096	+	26.6902i	0.629620	+	1.09053i	0.987628	+	0.156815i	$0.0501226\pi$
$599$	15.4096	+	26.6902i	0.629620	+	1.09053i	−0.358008	+	0.933718i	$0.616544\pi$
$600$		0			0
$601$	9.12121			0.372062			0.186031	−	0.982544i	$-0.440438\pi$
$601$	9.12121			0.372062			0.186031	+	0.982544i	$0.440438\pi$
$602$	23.0338	+	2.45079i	0.938788	+	0.0998868i
$603$		0			0
$604$	−3.73171	−	2.15451i	−0.151841	−	0.0876656i
$605$	−18.4279	+	10.6394i	−0.749201	+	0.432551i
$606$		0			0
$607$	−11.1821	+	19.3679i	−0.453866	+	0.786120i	−0.998622	−	0.0524749i	$-0.983289\pi$
$607$	−11.1821	+	19.3679i	−0.453866	+	0.786120i	0.544756	+	0.838595i	$0.316622\pi$
$608$	−7.75510			−0.314511
$609$		0			0
$610$	−20.1327			−0.815147
$611$	−1.96217	+	11.7066i	−0.0793809	+	0.473596i
$612$		0			0
$613$	−29.9508	+	17.2921i	−1.20970	+	0.698422i	−0.962694	−	0.270592i	$-0.912781\pi$
$613$	−29.9508	+	17.2921i	−1.20970	+	0.698422i	−0.247008	+	0.969014i	$0.579447\pi$
$614$	3.46970	−	6.00969i	0.140026	−	0.242531i
$615$		0			0
$616$	2.69832	+	1.96573i	0.108718	+	0.0792015i
$617$		−	1.21810i		−	0.0490389i	−0.999699	−	0.0245194i	$-0.992194\pi$
$617$		−	1.21810i		−	0.0490389i	0.999699	−	0.0245194i	$-0.00780556\pi$
$618$		0			0
$619$	−6.93507	+	4.00397i	−0.278744	+	0.160933i	−0.632855	−	0.774271i	$-0.718117\pi$
$619$	−6.93507	+	4.00397i	−0.278744	+	0.160933i	0.354111	+	0.935204i	$0.384784\pi$
$620$	−4.52360	−	7.83511i	−0.181672	−	0.314666i
$621$		0			0
$622$			11.0775i			0.444168i
$623$	−29.9332	−	3.18489i	−1.19925	−	0.127600i
$624$		0			0
$625$	12.7827	−	22.1402i	0.511307	−	0.885610i
$626$	−18.5397	+	10.7039i	−0.740997	+	0.427815i
$627$		0			0
$628$	11.3557	−	19.6687i	0.453142	−	0.784865i
$629$		−	1.27520i		−	0.0508456i
$630$		0			0
$631$			36.8843i			1.46834i	0.678966	+	0.734169i	$0.262428\pi$
$631$			36.8843i			1.46834i	−0.678966	+	0.734169i	$0.737572\pi$
$632$	−5.44912	−	3.14605i	−0.216754	−	0.125143i
$633$		0			0
$634$	10.6091	+	18.3754i	0.421339	+	0.729781i
$635$	−32.0537	−	18.5062i	−1.27201	−	0.734398i
$636$		0			0
$637$	−23.4564	−	9.31639i	−0.929378	−	0.369129i
$638$	−3.03824			−0.120285
$639$		0			0
$640$	1.13090	+	1.95878i	0.0447028	+	0.0774275i
$641$	21.5748	+	37.3686i	0.852153	+	1.47597i	0.879262	+	0.476339i	$0.158037\pi$
$641$	21.5748	+	37.3686i	0.852153	+	1.47597i	−0.0271091	+	0.999632i	$0.508630\pi$
$642$		0			0
$643$		−	5.10235i		−	0.201217i	−0.994926	−	0.100609i	$-0.967921\pi$
$643$		−	5.10235i		−	0.201217i	0.994926	−	0.100609i	$-0.0320790\pi$
$644$	−10.8666	+	4.82143i	−0.428202	+	0.189991i
$645$		0			0
$646$	−1.46398	+	2.53568i	−0.0575994	+	0.0997650i
$647$	18.4079	+	31.8833i	0.723687	+	1.25346i	0.959512	+	0.281668i	$0.0908876\pi$
$647$	18.4079	+	31.8833i	0.723687	+	1.25346i	−0.235825	+	0.971796i	$0.575779\pi$
$648$		0			0
$649$	−2.94609	+	5.10278i	−0.115644	+	0.200302i
$650$	−0.390950	+	0.146053i	−0.0153343	+	0.00572866i
$651$		0			0
$652$		−	12.4079i		−	0.485929i
$653$	14.8669	−	25.7502i	0.581786	−	1.00768i	−0.413482	−	0.910513i	$-0.635687\pi$
$653$	14.8669	−	25.7502i	0.581786	−	1.00768i	0.995268	−	0.0971708i	$-0.0309793\pi$
$654$		0			0
$655$	−22.4048	+	12.9354i	−0.875429	+	0.505429i
$656$	8.57465	+	4.95058i	0.334784	+	0.193288i
$657$		0			0
$658$	−7.04005	−	5.12869i	−0.274450	−	0.199937i
$659$	−16.9216			−0.659171			−0.329585	−	0.944126i	$-0.606909\pi$
$659$	−16.9216			−0.659171			−0.329585	+	0.944126i	$0.606909\pi$
$660$		0			0
$661$	−35.9451	+	20.7529i	−1.39810	+	0.807194i	−0.994194	−	0.107606i	$-0.965681\pi$
$661$	−35.9451	+	20.7529i	−1.39810	+	0.807194i	−0.403907	+	0.914800i	$0.632348\pi$
$662$	−10.5196	−	18.2205i	−0.408857	−	0.708162i
$663$		0			0
$664$	−6.39446			−0.248153
$665$	37.5097	+	27.3259i	1.45457	+	1.05965i
$666$		0			0
$667$	5.40961	−	9.36972i	0.209461	−	0.362797i
$668$	21.3676	−	12.3366i	0.826737	−	0.477317i
$669$		0			0
$670$	2.53095	+	1.46124i	0.0977791	+	0.0564528i
$671$		−	11.2315i		−	0.433587i
$672$		0			0
$673$	15.6394			0.602853			0.301426	−	0.953489i	$-0.402537\pi$
$673$	15.6394			0.602853			0.301426	+	0.953489i	$0.402537\pi$
$674$	22.1334	+	12.7787i	0.852546	+	0.492217i
$675$		0			0
$676$	−12.2895	−	4.23885i	−0.472674	−	0.163033i
$677$	6.61178	−	11.4519i	0.254111	−	0.440134i	−0.710542	−	0.703654i	$-0.751550\pi$
$677$	6.61178	−	11.4519i	0.254111	−	0.440134i	0.964654	+	0.263521i	$0.0848837\pi$
$678$		0			0
$679$	8.61673	+	19.4204i	0.330680	+	0.745287i
$680$	0.853947			0.0327474
$681$		0			0
$682$	4.37101	−	2.52360i	0.167375	−	0.0966338i
$683$	22.4935	−	12.9866i	0.860688	−	0.496919i	−0.00355454	−	0.999994i	$-0.501131\pi$
$683$	22.4935	−	12.9866i	0.860688	−	0.496919i	0.864243	+	0.503075i	$0.167798\pi$
$684$		0			0
$685$	−25.2102			−0.963231
$686$	13.8108	−	12.3395i	0.527297	−	0.471124i
$687$		0			0
$688$	−4.37755	+	7.58214i	−0.166893	+	0.289066i
$689$	6.88243	−	8.34459i	0.262200	−	0.317903i
$690$		0			0
$691$	−14.6467	−	8.45630i	−0.557188	−	0.321693i	0.194828	−	0.980837i	$-0.437585\pi$
$691$	−14.6467	−	8.45630i	−0.557188	−	0.321693i	−0.752016	+	0.659145i	$0.770918\pi$
$692$	23.9787			0.911532
$693$		0			0
$694$		−	18.3303i		−	0.695810i
$695$	−4.88382	−	2.81968i	−0.185254	−	0.106956i
$696$		0			0
$697$	3.23737	−	1.86910i	0.122624	−	0.0707972i
$698$	−2.09663	+	3.63147i	−0.0793587	+	0.137453i
$699$		0			0
$700$	0.0324014	−	0.304525i	0.00122466	−	0.0115100i
$701$	33.0731			1.24915			0.624577	−	0.780964i	$-0.285272\pi$
$701$	33.0731			1.24915			0.624577	+	0.780964i	$0.285272\pi$
$702$		0			0
$703$	13.0966	+	22.6840i	0.493949	+	0.855544i
$704$	−1.09275	+	0.630901i	−0.0411847	+	0.0237780i
$705$		0			0
$706$	−21.8629			−0.822822
$707$	−24.8903	−	18.1327i	−0.936097	−	0.681949i
$708$		0			0
$709$	−10.4254	−	6.01912i	−0.391535	−	0.226053i	0.291290	−	0.956635i	$-0.405916\pi$
$709$	−10.4254	−	6.01912i	−0.391535	−	0.226053i	−0.682825	+	0.730582i	$0.739249\pi$
$710$	8.06184	−	4.65451i	0.302555	−	0.174680i
$711$		0			0
$712$	5.68878	−	9.85325i	0.213196	−	0.369266i
$713$			17.9732i			0.673102i
$714$		0			0
$715$	−3.60112	−	9.63935i	−0.134674	−	0.360491i
$716$	2.85395	−	4.94318i	0.106657	−	0.184735i
$717$		0			0
$718$	−12.9523	−	22.4341i	−0.483377	−	0.837233i
$719$	−3.59214	+	6.22178i	−0.133964	+	0.232033i	−0.925201	−	0.379476i	$-0.876104\pi$
$719$	−3.59214	+	6.22178i	−0.133964	+	0.232033i	0.791237	+	0.611510i	$0.209437\pi$
$720$		0			0
$721$	3.48873	−	32.7889i	0.129927	−	1.22112i
$722$		−	41.1416i		−	1.53113i
$723$		0			0
$724$	−8.33483	−	14.4363i	−0.309761	−	0.536523i
$725$	0.139354	+	0.241368i	0.00517547	+	0.00896418i
$726$		0			0
$727$	−41.8967			−1.55386			−0.776932	−	0.629585i	$-0.783225\pi$
$727$	−41.8967			−1.55386			−0.776932	+	0.629585i	$0.783225\pi$
$728$	6.81429	−	6.67573i	0.252554	−	0.247419i
$729$		0			0
$730$	−18.6877	−	10.7894i	−0.691664	−	0.399333i
$731$	1.65275	+	2.86265i	0.0611292	+	0.105879i
$732$		0			0
$733$	−19.5022	−	11.2596i	−0.720329	−	0.415882i	0.0945445	−	0.995521i	$-0.469861\pi$
$733$	−19.5022	−	11.2596i	−0.720329	−	0.415882i	−0.814874	+	0.579638i	$0.803194\pi$
$734$		−	17.8629i		−	0.659332i
$735$		0			0
$736$		−	4.49330i		−	0.165625i
$737$	−0.815191	+	1.41195i	−0.0300279	+	0.0520099i
$738$		0			0
$739$	10.3992	−	6.00397i	0.382540	−	0.220859i	−0.296383	−	0.955069i	$-0.595780\pi$
$739$	10.3992	−	6.00397i	0.382540	−	0.220859i	0.678923	+	0.734210i	$0.262447\pi$
$740$	3.81968	−	6.61587i	0.140414	−	0.243204i
$741$		0			0
$742$	3.21908	+	7.25517i	0.118176	+	0.266346i
$743$			9.52360i			0.349387i	0.984623	+	0.174694i	$0.0558935\pi$
$743$			9.52360i			0.349387i	−0.984623	+	0.174694i	$0.944107\pi$
$744$		0			0
$745$	0.0342702	+	0.0593577i	0.00125556	+	0.00217470i
$746$	−13.0984	+	7.56236i	−0.479566	+	0.276878i
$747$		0			0
$748$			0.476396i			0.0174187i
$749$	−4.98335	+	46.8361i	−0.182088	+	1.71135i
$750$		0			0
$751$	19.7248	−	34.1644i	0.719768	−	1.24668i	−0.241323	−	0.970445i	$-0.577581\pi$
$751$	19.7248	−	34.1644i	0.719768	−	1.24668i	0.961091	−	0.276230i	$-0.0890852\pi$
$752$	2.85105	−	1.64605i	0.103967	−	0.0600254i
$753$		0			0
$754$	−1.43514	+	8.56220i	−0.0522646	+	0.311817i
$755$	−9.74613			−0.354698
$756$		0			0
$757$	19.4854			0.708208			0.354104	−	0.935206i	$-0.384786\pi$
$757$	19.4854			0.708208			0.354104	+	0.935206i	$0.384786\pi$
$758$	−6.26180	+	10.8458i	−0.227439	+	0.393936i
$759$		0			0
$760$	−15.1905	+	8.77026i	−0.551018	+	0.318131i
$761$	−5.73138	−	3.30901i	−0.207762	−	0.119952i	0.392509	−	0.919748i	$-0.371607\pi$
$761$	−5.73138	−	3.30901i	−0.207762	−	0.119952i	−0.600271	+	0.799797i	$0.704941\pi$
$762$		0			0
$763$	−5.10782	+	7.01140i	−0.184915	+	0.253830i
$764$	7.27871			0.263334
$765$		0			0
$766$	0.789373	+	1.36723i	0.0285212	+	0.0494002i
$767$	12.9888	+	10.7129i	0.468998	+	0.386819i
$768$		0			0
$769$		−	14.3900i		−	0.518918i	−0.965754	−	0.259459i	$-0.916456\pi$
$769$		−	14.3900i		−	0.518918i	0.965754	−	0.259459i	$-0.0835442\pi$
$770$	7.50845	+	0.798898i	0.270586	+	0.0287903i
$771$		0			0
$772$	14.9900	+	8.65451i	0.539503	+	0.311482i
$773$	47.8358	−	27.6180i	1.72053	−	0.993351i	0.802706	−	0.596375i	$-0.203393\pi$
$773$	47.8358	−	27.6180i	1.72053	−	0.993351i	0.917829	−	0.396976i	$-0.129940\pi$
$774$		0			0
$775$	−0.400968	−	0.231499i	−0.0144032	−	0.00831568i
$776$	−8.03030			−0.288271
$777$		0			0
$778$		−	4.81025i		−	0.172456i
$779$	−38.3922	+	66.4973i	−1.37554	+	2.38251i
$780$		0			0
$781$	2.59663	+	4.49750i	0.0929148	+	0.160933i
$782$	−1.46917	−	0.848227i	−0.0525375	−	0.0303325i
$783$		0			0
$784$	2.14605	+	6.66292i	0.0766447	+	0.237961i
$785$		−	51.3687i		−	1.83343i
$786$		0			0
$787$	0.642336	−	0.370853i	0.0228968	−	0.0132195i	−0.488508	−	0.872559i	$-0.662459\pi$
$787$	0.642336	−	0.370853i	0.0228968	−	0.0132195i	0.511405	+	0.859340i	$0.329125\pi$
$788$	2.78392	−	1.60730i	0.0991730	−	0.0572576i
$789$		0			0
$790$	−14.2315			−0.506334
$791$	3.19810	−	1.41898i	0.113711	−	0.0504531i
$792$		0			0
$793$	−31.6520	−	5.30529i	−1.12400	−	0.188397i
$794$	−13.4742	−	23.3380i	−0.478181	−	0.828234i
$795$		0			0
$796$	5.85173	−	10.1355i	0.207409	−	0.359243i
$797$	−21.4685			−0.760452			−0.380226	−	0.924894i	$-0.624154\pi$
$797$	−21.4685			−0.760452			−0.380226	+	0.924894i	$0.624154\pi$
$798$		0			0
$799$		−	1.24294i		−	0.0439721i
$800$	0.100242	+	0.0578747i	0.00354409	+	0.00204618i
$801$		0			0
$802$	10.2978	+	17.8364i	0.363629	+	0.629824i
$803$	6.01912	−	10.4254i	0.212410	−	0.367905i
$804$		0			0
$805$	−15.8326	+	21.7331i	−0.558026	+	0.765992i
$806$	−5.04721	−	13.5102i	−0.177780	−	0.475877i
$807$		0			0
$808$	10.0800	−	5.81968i	0.354612	−	0.204736i
$809$	−6.63487	−	11.4919i	−0.233270	−	0.404035i	0.725499	−	0.688223i	$-0.241609\pi$
$809$	−6.63487	−	11.4919i	−0.233270	−	0.404035i	−0.958768	+	0.284189i	$0.908276\pi$
$810$		0			0
$811$			14.8236i			0.520529i	0.965537	+	0.260264i	$0.0838097\pi$
$811$			14.8236i			0.520529i	−0.965537	+	0.260264i	$0.916190\pi$
$812$	−5.14911	−	3.75114i	−0.180698	−	0.131639i
$813$		0			0
$814$	3.69083	+	2.13090i	0.129363	+	0.0746880i
$815$	−14.0321	−	24.3042i	−0.491522	−	0.851340i
$816$		0			0
$817$	−58.8003	−	33.9484i	−2.05716	−	1.18770i
$818$	−5.30901			−0.185625
$819$		0			0
$820$	22.3945			0.782048
$821$	28.2743	+	16.3242i	0.986779	+	0.569717i	0.904310	−	0.426876i	$-0.140386\pi$
$821$	28.2743	+	16.3242i	0.986779	+	0.569717i	0.0824692	+	0.996594i	$0.473719\pi$
$822$		0			0
$823$	5.67187	+	9.82397i	0.197709	+	0.342442i	0.947785	−	0.318909i	$-0.103317\pi$
$823$	5.67187	+	9.82397i	0.197709	+	0.342442i	−0.750076	+	0.661351i	$0.769983\pi$
$824$	10.7933	+	6.23150i	0.376001	+	0.217085i
$825$		0			0
$826$	−11.2931	+	5.01067i	−0.392936	+	0.174343i
$827$			14.7988i			0.514605i	0.966331	+	0.257302i	$0.0828337\pi$
$827$			14.7988i			0.514605i	−0.966331	+	0.257302i	$0.917166\pi$
$828$		0			0
$829$	−6.23598	−	10.8010i	−0.216585	−	0.375136i	0.737177	−	0.675700i	$-0.236158\pi$
$829$	−6.23598	−	10.8010i	−0.216585	−	0.375136i	−0.953762	+	0.300564i	$0.902825\pi$
$830$	−12.5253	+	7.23150i	−0.434760	+	0.251009i
$831$		0			0
$832$	1.26180	+	3.37755i	0.0437451	+	0.117096i
$833$	2.58369	+	0.556104i	0.0895196	+	0.0192679i
$834$		0			0
$835$	27.9029	−	48.3293i	0.965620	−	1.67250i
$836$	−4.89270	−	8.47441i	−0.169218	−	0.293094i
$837$		0			0
$838$	−17.8264	−	10.2921i	−0.615804	−	0.355535i
$839$		−	10.0890i		−	0.348309i	−0.984718	−	0.174155i	$-0.944281\pi$
$839$		−	10.0890i		−	0.348309i	0.984718	−	0.174155i	$-0.0557193\pi$
$840$		0			0
$841$	−23.2022			−0.800077
$842$	6.44388	−	11.1611i	0.222071	−	0.384638i
$843$		0			0
$844$	−4.03206	−	6.98373i	−0.138789	−	0.240390i
$845$	−28.8662	+	5.59525i	−0.993026	+	0.192483i
$846$		0			0
$847$	2.63352	−	24.7511i	0.0904887	−	0.850459i
$848$	−3.00000			−0.103020
$849$		0			0
$850$	0.0378465	−	0.0218507i	0.00129812	−	0.000749472i
$851$	−13.1431	+	7.58818i	−0.450540	+	0.260119i
$852$		0			0
$853$		−	1.48784i		−	0.0509426i	−0.999676	−	0.0254713i	$-0.991891\pi$
$853$		−	1.48784i		−	0.0509426i	0.999676	−	0.0254713i	$-0.00810864\pi$
$854$	13.8669	−	19.0348i	0.474515	−	0.651357i
$855$		0			0
$856$	−15.4173	−	8.90116i	−0.526951	−	0.304235i
$857$	−22.2832	−	38.5956i	−0.761179	−	1.31840i	−0.942243	−	0.334930i	$-0.891287\pi$
$857$	−22.2832	−	38.5956i	−0.761179	−	1.31840i	0.181064	−	0.983471i	$-0.442046\pi$
$858$		0			0
$859$	9.06411	−	15.6995i	0.309264	−	0.535660i	−0.668938	−	0.743318i	$-0.733251\pi$
$859$	9.06411	−	15.6995i	0.309264	−	0.535660i	0.978201	+	0.207658i	$0.0665841\pi$
$860$			19.8023i			0.675253i
$861$		0			0
$862$	−26.6260			−0.906884
$863$	19.8838	+	11.4799i	0.676852	+	0.390780i	0.798668	−	0.601772i	$-0.205539\pi$
$863$	19.8838	+	11.4799i	0.676852	+	0.390780i	−0.121816	+	0.992553i	$0.538872\pi$
$864$		0			0
$865$	46.9689	−	27.1175i	1.59699	−	0.922023i
$866$	24.3189	+	14.0405i	0.826389	+	0.477116i
$867$		0			0
$868$	10.5236	+	1.11971i	0.357194	+	0.0380054i
$869$		−	7.93939i		−	0.269325i
$870$		0			0
$871$	3.59403	+	2.96428i	0.121779	+	0.100441i
$872$	−1.63935	−	2.83944i	−0.0555155	−	0.0961557i
$873$		0			0
$874$	34.8460			1.17868
$875$	11.8539	+	26.7165i	0.400737	+	0.903182i
$876$		0			0
$877$	26.6309	+	15.3753i	0.899261	+	0.519188i	0.876960	−	0.480563i	$-0.159568\pi$
$877$	26.6309	+	15.3753i	0.899261	+	0.519188i	0.0223003	+	0.999751i	$0.492901\pi$
$878$	24.0444	−	13.8820i	0.811459	−	0.468496i
$879$		0			0
$880$	−1.42697	+	2.47159i	−0.0481033	+	0.0833173i
$881$	28.7551			0.968784			0.484392	−	0.874851i	$-0.339041\pi$
$881$	28.7551			0.968784			0.484392	+	0.874851i	$0.339041\pi$
$882$		0			0
$883$	−31.3696			−1.05567			−0.527836	−	0.849346i	$-0.676996\pi$
$883$	−31.3696			−1.05567			−0.527836	+	0.849346i	$0.676996\pi$
$884$	1.34255	+	0.225029i	0.0451550	+	0.00756856i
$885$		0			0
$886$	−3.95067	+	2.28092i	−0.132725	+	0.0766290i
$887$	4.40786	−	7.63463i	0.148001	−	0.256346i	−0.782487	−	0.622666i	$-0.786049\pi$
$887$	4.40786	−	7.63463i	0.148001	−	0.256346i	0.930489	+	0.366321i	$0.119383\pi$
$888$		0			0
$889$	39.5749	−	17.5592i	1.32730	−	0.588916i
$890$		−	25.7338i		−	0.862598i
$891$		0			0
$892$	−20.5988	+	11.8927i	−0.689698	+	0.398197i
$893$	12.7653	+	22.1102i	0.427175	+	0.739888i
$894$		0			0
$895$		−	12.9101i		−	0.431538i
$896$	−2.63090	−	0.279927i	−0.0878922	−	0.00935171i
$897$		0			0
$898$	−1.39095	+	2.40920i	−0.0464166	+	0.0803959i
$899$	−8.34105	+	4.81571i	−0.278190	+	0.160613i
$900$		0			0
$901$	−0.566327	+	0.980908i	−0.0188671	+	0.0326788i
$902$			12.4933i			0.415981i
$903$		0			0
$904$			1.32241i			0.0439827i
$905$	−32.6522	−	18.8517i	−1.08539	−	0.626653i
$906$		0			0
$907$	2.01515	+	3.49035i	0.0669120	+	0.115895i	0.897541	−	0.440932i	$-0.145352\pi$
$907$	2.01515	+	3.49035i	0.0669120	+	0.115895i	−0.830629	+	0.556827i	$0.812019\pi$
$908$	8.86074	+	5.11575i	0.294054	+	0.169772i
$909$		0			0
$910$	5.79809	−	20.7826i	0.192205	−	0.688936i
$911$	14.1212			0.467857			0.233928	−	0.972254i	$-0.424842\pi$
$911$	14.1212			0.467857			0.233928	+	0.972254i	$0.424842\pi$
$912$		0			0
$913$	−4.03427	−	6.98756i	−0.133515	−	0.231255i
$914$	−14.9029	−	25.8126i	−0.492944	−	0.853805i
$915$		0			0
$916$		−	1.04721i		−	0.0346008i
$917$	3.20185	−	30.0927i	0.105734	−	0.993747i
$918$		0			0
$919$	5.06411	−	8.77130i	0.167050	−	0.289339i	−0.770332	−	0.637644i	$-0.779909\pi$
$919$	5.06411	−	8.77130i	0.167050	−	0.289339i	0.937381	+	0.348305i	$0.113243\pi$
$920$	−5.08148	−	8.80138i	−0.167531	−	0.290173i
$921$		0			0
$922$	9.30901	−	16.1237i	0.306576	−	0.531005i
$923$	13.9012	−	5.19326i	0.457562	−	0.170938i
$924$		0			0
$925$		−	0.390950i		−	0.0128543i
$926$	−21.0169	+	36.4023i	−0.690658	+	1.19626i
$927$		0			0
$928$	2.08526	−	1.20393i	0.0684521	−	0.0395209i
$929$	30.3811	+	17.5405i	0.996770	+	0.575485i	0.907291	−	0.420503i	$-0.138146\pi$
$929$	30.3811	+	17.5405i	0.996770	+	0.575485i	0.0894790	+	0.995989i	$0.471480\pi$
$930$		0			0
$931$	−51.6716	+	16.6429i	−1.69347	+	0.545448i
$932$	−3.75160			−0.122888
$933$		0			0
$934$	33.7071	−	19.4608i	1.10293	−	0.636776i
$935$	0.538756	+	0.933153i	0.0176192	+	0.0305174i
$936$		0			0
$937$	22.6055			0.738491			0.369245	−	0.929332i	$-0.379616\pi$
$937$	22.6055			0.738491			0.369245	+	0.929332i	$0.379616\pi$
$938$	−3.12482	+	1.38646i	−0.102029	+	0.0452696i
$939$		0			0
$940$	3.72305	−	6.44850i	0.121432	−	0.210327i
$941$	−23.8013	+	13.7417i	−0.775901	+	0.447967i	−0.834976	−	0.550287i	$-0.814518\pi$
$941$	−23.8013	+	13.7417i	−0.775901	+	0.447967i	0.0590745	+	0.998254i	$0.481185\pi$
$942$		0			0
$943$	−38.5285	−	22.2444i	−1.25466	−	0.724379i
$944$		−	4.66966i		−	0.151984i
$945$		0			0
$946$	−11.0472			−0.359176
$947$	−24.5757	−	14.1888i	−0.798602	−	0.461073i	0.0443799	−	0.999015i	$-0.485869\pi$
$947$	−24.5757	−	14.1888i	−0.798602	−	0.461073i	−0.842982	+	0.537941i	$0.819202\pi$
$948$		0			0
$949$	−26.5372	−	21.8873i	−0.861434	−	0.710492i
$950$	−0.448824	+	0.777386i	−0.0145618	+	0.0252217i
$951$		0			0
$952$	−0.588178	+	0.807380i	−0.0190630	+	0.0261674i
$953$	−9.58864			−0.310606			−0.155303	−	0.987867i	$-0.549635\pi$
$953$	−9.58864			−0.310606			−0.155303	+	0.987867i	$0.549635\pi$
$954$		0			0
$955$	14.2574	−	8.23150i	0.461358	−	0.266365i
$956$	−25.2851	+	14.5984i	−0.817780	+	0.472146i
$957$		0			0
$958$	2.28664			0.0738780
$959$	17.3642	−	23.8354i	0.560718	−	0.769686i
$960$		0			0
$961$	−7.50000	+	12.9904i	−0.241935	+	0.419045i
$962$	7.74859	−	9.39476i	0.249825	−	0.302899i
$963$		0			0
$964$	−1.47908	−	0.853947i	−0.0476380	−	0.0275038i
$965$	39.1496			1.26027
$966$		0			0
$967$		−	31.2271i		−	1.00419i	−0.864811	−	0.502097i	$-0.832562\pi$
$967$		−	31.2271i		−	1.00419i	0.864811	−	0.502097i	$-0.167438\pi$
$968$	8.14744	+	4.70393i	0.261869	+	0.151190i
$969$		0			0
$970$	−15.7296	+	9.08148i	−0.505046	+	0.291589i
$971$	−18.3833	+	31.8408i	−0.589947	+	1.02182i	0.404291	+	0.914630i	$0.367518\pi$
$971$	−18.3833	+	31.8408i	−0.589947	+	1.02182i	−0.994239	+	0.107188i	$0.965815\pi$
$972$		0			0
$973$	6.02978	−	2.67538i	0.193306	−	0.0857687i
$974$	−30.8405			−0.988195
$975$		0			0
$976$	4.45058	+	7.70863i	0.142460	+	0.246747i
$977$	14.4811	−	8.36065i	0.463290	−	0.267481i	−0.250136	−	0.968211i	$-0.580475\pi$
$977$	14.4811	−	8.36065i	0.463290	−	0.267481i	0.713427	+	0.700730i	$0.247142\pi$
$978$		0			0
$979$	14.3562			0.458827
$980$	11.7387	+	10.6242i	0.374980	+	0.339378i
$981$		0			0
$982$	35.6590	+	20.5877i	1.13792	+	0.656981i
$983$	−39.2534	+	22.6630i	−1.25199	+	0.722836i	−0.971504	−	0.237022i	$-0.923829\pi$
$983$	−39.2534	+	22.6630i	−1.25199	+	0.722836i	−0.280485	+	0.959859i	$0.590495\pi$
$984$		0			0
$985$	3.63539	−	6.29668i	0.115833	−	0.200629i
$986$		−	0.909090i		−	0.0289513i
$987$		0			0
$988$	−26.1933	+	9.78541i	−0.833319	+	0.311315i
$989$	19.6697	−	34.0688i	0.625459	−	1.08333i
$990$		0			0
$991$	−25.7417	−	44.5859i	−0.817712	−	1.41632i	−0.907364	−	0.420346i	$-0.861909\pi$
$991$	−25.7417	−	44.5859i	−0.817712	−	1.41632i	0.0896517	−	0.995973i	$-0.471425\pi$
$992$	−2.00000	+	3.46410i	−0.0635001	+	0.109985i
$993$		0			0
$994$	−1.15211	+	10.8281i	−0.0365427	+	0.343447i
$995$		−	26.4709i		−	0.839185i
$996$		0			0
$997$	26.7333	+	46.3034i	0.846651	+	1.46644i	0.884180	+	0.467146i	$0.154718\pi$
$997$	26.7333	+	46.3034i	0.846651	+	1.46644i	−0.0375294	+	0.999296i	$0.511949\pi$
$998$	7.73423	+	13.3961i	0.244823	+	0.424046i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1638.2.dm.c.415.3		12
3.2	odd	2		546.2.bk.b.415.4	yes	12
7.4	even	3	inner	1638.2.dm.c.1117.4		12
13.12	even	2	inner	1638.2.dm.c.415.4		12
21.2	odd	6		3822.2.c.k.883.3		6
21.5	even	6		3822.2.c.j.883.1		6
21.11	odd	6		546.2.bk.b.25.3	✓	12
39.38	odd	2		546.2.bk.b.415.3	yes	12
91.25	even	6	inner	1638.2.dm.c.1117.3		12
273.116	odd	6		546.2.bk.b.25.4	yes	12
273.194	even	6		3822.2.c.j.883.6		6
273.233	odd	6		3822.2.c.k.883.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
546.2.bk.b.25.3	✓	12	21.11	odd	6
546.2.bk.b.25.4	yes	12	273.116	odd	6
546.2.bk.b.415.3	yes	12	39.38	odd	2
546.2.bk.b.415.4	yes	12	3.2	odd	2
1638.2.dm.c.415.3		12	1.1	even	1	trivial
1638.2.dm.c.415.4		12	13.12	even	2	inner
1638.2.dm.c.1117.3		12	91.25	even	6	inner
1638.2.dm.c.1117.4		12	7.4	even	3	inner
3822.2.c.j.883.1		6	21.5	even	6
3822.2.c.j.883.6		6	273.194	even	6
3822.2.c.k.883.3		6	21.2	odd	6
3822.2.c.k.883.4		6	273.233	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+(-0.866025 - 0.500000i) q^{2} +(0.500000 + 0.866025i) q^{4} +(1.95878 + 1.13090i) q^{5} +(-2.41839 + 1.07303i) q^{7} -1.00000i q^{8} +(-1.13090 - 1.95878i) q^{10} +(1.09275 - 0.630901i) q^{11} +(-1.26180 - 3.37755i) q^{13} +(2.63090 + 0.279927i) q^{14} +(-0.500000 + 0.866025i) q^{16} +(0.188776 + 0.326969i) q^{17} +(-6.71612 - 3.87755i) q^{19} +2.26180i q^{20} -1.26180 q^{22} +(2.24665 - 3.89131i) q^{23} +(0.0578747 + 0.100242i) q^{25} +(-0.596023 + 3.55595i) q^{26} +(-2.13846 - 1.55787i) q^{28} +2.40786 q^{29} +(-3.46410 + 2.00000i) q^{31} +(0.866025 - 0.500000i) q^{32} -0.377552i q^{34} +(-5.95058 - 0.633140i) q^{35} +(-2.92505 - 1.68878i) q^{37} +(3.87755 + 6.71612i) q^{38} +(1.13090 - 1.95878i) q^{40} -9.90116i q^{41} +8.75510 q^{43} +(1.09275 + 0.630901i) q^{44} +(-3.89131 + 2.24665i) q^{46} +(-2.85105 - 1.64605i) q^{47} +(4.69723 - 5.18999i) q^{49} -0.115749i q^{50} +(2.29414 - 2.78153i) q^{52} +(1.50000 + 2.59808i) q^{53} +2.85395 q^{55} +(1.07303 + 2.41839i) q^{56} +(-2.08526 - 1.20393i) q^{58} +(-4.04404 + 2.33483i) q^{59} +(4.45058 - 7.70863i) q^{61} +4.00000 q^{62} -1.00000 q^{64} +(1.34809 - 8.04285i) q^{65} +(-1.11900 + 0.646053i) q^{67} +(-0.188776 + 0.326969i) q^{68} +(4.83678 + 3.52360i) q^{70} +4.11575i q^{71} +(8.26232 - 4.77026i) q^{73} +(1.68878 + 2.92505i) q^{74} -7.75510i q^{76} +(-1.96573 + 2.69832i) q^{77} +(3.14605 - 5.44912i) q^{79} +(-1.95878 + 1.13090i) q^{80} +(-4.95058 + 8.57465i) q^{82} -6.39446i q^{83} +0.853947i q^{85} +(-7.58214 - 4.37755i) q^{86} +(-0.630901 - 1.09275i) q^{88} +(9.85325 + 5.68878i) q^{89} +(6.67573 + 6.81429i) q^{91} +4.49330 q^{92} +(1.64605 + 2.85105i) q^{94} +(-8.77026 - 15.1905i) q^{95} -8.03030i q^{97} +(-6.66292 + 2.14605i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 6 q^{4} + 12 q^{13} + 18 q^{14} - 6 q^{16} - 18 q^{17} + 12 q^{22} - 6 q^{25} - 12 q^{29} - 24 q^{35} + 6 q^{38} + 24 q^{43} - 18 q^{49} + 6 q^{52} + 18 q^{53} + 48 q^{55} + 6 q^{56} + 6 q^{61} + 48 q^{62}+ \cdots - 24 q^{95}+O(q^{100}) \) 12 * q + 6 * q^4 + 12 * q^13 + 18 * q^14 - 6 * q^16 - 18 * q^17 + 12 * q^22 - 6 * q^25 - 12 * q^29 - 24 * q^35 + 6 * q^38 + 24 * q^43 - 18 * q^49 + 6 * q^52 + 18 * q^53 + 48 * q^55 + 6 * q^56 + 6 * q^61 + 48 * q^62 - 12 * q^64 + 12 * q^65 + 18 * q^68 + 24 * q^77 + 24 * q^79 - 12 * q^82 + 6 * q^88 - 24 * q^91 + 6 * q^94 - 24 * q^95

Introduction

L-functions

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