LMFDB - Embedded newform 378.2.m.a.251.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [378,2,Mod(125,378)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(378, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([5, 3]))

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

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

chi := DirichletCharacter("378.125");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 378 = 2 \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	378.m (of order $6$, degree $2$, not 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:	$3.01834519640$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 6x^{14} + 9x^{12} + 54x^{10} - 288x^{8} + 486x^{6} + 729x^{4} - 4374x^{2} + 6561 $ x^16 - 6x^14 + 9x^12 + 54x^10 - 288x^8 + 486x^6 + 729x^4 - 4374*x^2 + 6561
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3^{4} $
Twist minimal:	no (minimal twist has level 126)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			251.5
Root			$-1.62181 - 0.608059i$ of defining polynomial
Character	$\chi$	$=$	378.251
Dual form			378.2.m.a.125.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.866025 + 0.500000i) q^{2} +(0.500000 + 0.866025i) q^{4} +(-1.94556 - 3.36980i) q^{5} +(0.343982 - 2.62329i) q^{7} +1.00000i q^{8} +O(q^{10})$	\(q+(0.866025 + 0.500000i) q^{2} +(0.500000 + 0.866025i) q^{4} +(-1.94556 - 3.36980i) q^{5} +(0.343982 - 2.62329i) q^{7} +1.00000i q^{8} -3.89111i q^{10} +(-3.41614 - 1.97231i) q^{11} +(2.46687 - 1.42425i) q^{13} +(1.60954 - 2.09985i) q^{14} +(-0.500000 + 0.866025i) q^{16} -0.742117 q^{17} +1.78474i q^{19} +(1.94556 - 3.36980i) q^{20} +(-1.97231 - 3.41614i) q^{22} +(5.41535 - 3.12656i) q^{23} +(-5.07039 + 8.78217i) q^{25} +2.84849 q^{26} +(2.44383 - 1.01375i) q^{28} +(2.50079 + 1.44383i) q^{29} +(3.04125 - 1.75587i) q^{31} +(-0.866025 + 0.500000i) q^{32} +(-0.642692 - 0.371058i) q^{34} +(-9.50923 + 3.94462i) q^{35} +3.00158 q^{37} +(-0.892369 + 1.54563i) q^{38} +(3.36980 - 1.94556i) q^{40} +(5.24705 + 9.08816i) q^{41} +(0.471521 - 0.816699i) q^{43} -3.94462i q^{44} +6.25311 q^{46} +(-1.09263 + 1.89248i) q^{47} +(-6.76335 - 1.80473i) q^{49} +(-8.78217 + 5.07039i) q^{50} +(2.46687 + 1.42425i) q^{52} +15.3490i q^{55} +(2.62329 + 0.343982i) q^{56} +(1.44383 + 2.50079i) q^{58} +(-0.0105673 - 0.0183031i) q^{59} +(-2.13832 - 1.23456i) q^{61} +3.51174 q^{62} -1.00000 q^{64} +(-9.59886 - 5.54191i) q^{65} +(-6.72463 - 11.6474i) q^{67} +(-0.371058 - 0.642692i) q^{68} +(-10.2075 - 1.33847i) q^{70} -1.94304i q^{71} -4.85486i q^{73} +(2.59944 + 1.50079i) q^{74} +(-1.54563 + 0.892369i) q^{76} +(-6.34904 + 8.28311i) q^{77} +(-1.81806 + 3.14898i) q^{79} +3.89111 q^{80} +10.4941i q^{82} +(4.02998 - 6.98012i) q^{83} +(1.44383 + 2.50079i) q^{85} +(0.816699 - 0.471521i) q^{86} +(1.97231 - 3.41614i) q^{88} +9.26646 q^{89} +(-2.88766 - 6.96124i) q^{91} +(5.41535 + 3.12656i) q^{92} +(-1.89248 + 1.09263i) q^{94} +(6.01422 - 3.47231i) q^{95} +(16.2983 + 9.40980i) q^{97} +(-4.95487 - 4.94462i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 8 q^{4} + 2 q^{7}+O(q^{10}) $ 16 * q + 8 * q^4 + 2 * q^7	$ 16 q + 8 q^{4} + 2 q^{7} + 12 q^{11} + 6 q^{14} - 8 q^{16} + 48 q^{23} - 8 q^{25} + 4 q^{28} + 12 q^{29} - 8 q^{37} + 4 q^{43} + 24 q^{46} - 8 q^{49} - 60 q^{50} + 6 q^{56} - 12 q^{58} - 16 q^{64} - 84 q^{65} - 28 q^{67} - 36 q^{74} - 78 q^{77} - 4 q^{79} - 12 q^{85} + 24 q^{86} + 24 q^{91} + 48 q^{92} - 12 q^{95}+O(q^{100}) $ 16 * q + 8 * q^4 + 2 * q^7 + 12 * q^11 + 6 * q^14 - 8 * q^16 + 48 * q^23 - 8 * q^25 + 4 * q^28 + 12 * q^29 - 8 * q^37 + 4 * q^43 + 24 * q^46 - 8 * q^49 - 60 * q^50 + 6 * q^56 - 12 * q^58 - 16 * q^64 - 84 * q^65 - 28 * q^67 - 36 * q^74 - 78 * q^77 - 4 * q^79 - 12 * q^85 + 24 * q^86 + 24 * q^91 + 48 * q^92 - 12 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/378\mathbb{Z}\right)^\times$.

$n$	$29$	$325$
$\chi(n)$	$e\left(\frac{1}{6}\right)$	$-1$

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
$5$	−1.94556	−	3.36980i	−0.870080	−	1.50702i	−0.861913	−	0.507056i	$-0.830734\pi$
$5$	−1.94556	−	3.36980i	−0.870080	−	1.50702i	−0.00816625	−	0.999967i	$-0.502599\pi$
$6$		0			0
$7$	0.343982	−	2.62329i	0.130013	−	0.991512i
$7$	0.343982	−	2.62329i	0.130013	−	0.991512i
$8$			1.00000i			0.353553i
$9$		0			0
$10$		−	3.89111i		−	1.23048i
$11$	−3.41614	−	1.97231i	−1.03001	−	0.594674i	−0.113019	−	0.993593i	$-0.536052\pi$
$11$	−3.41614	−	1.97231i	−1.03001	−	0.594674i	−0.916986	+	0.398919i	$0.869385\pi$
$12$		0			0
$13$	2.46687	−	1.42425i	0.684186	−	0.395015i	−0.117244	−	0.993103i	$-0.537406\pi$
$13$	2.46687	−	1.42425i	0.684186	−	0.395015i	0.801430	+	0.598088i	$0.204073\pi$
$14$	1.60954	−	2.09985i	0.430169	−	0.561208i
$15$		0			0
$16$	−0.500000	+	0.866025i	−0.125000	+	0.216506i
$17$	−0.742117			−0.179990			−0.0899949	−	0.995942i	$-0.528685\pi$
$17$	−0.742117			−0.179990			−0.0899949	+	0.995942i	$0.528685\pi$
$18$		0			0
$19$			1.78474i			0.409447i	0.978820	+	0.204723i	$0.0656295\pi$
$19$			1.78474i			0.409447i	−0.978820	+	0.204723i	$0.934370\pi$
$20$	1.94556	−	3.36980i	0.435040	−	0.753511i
$21$		0			0
$22$	−1.97231	−	3.41614i	−0.420498	−	0.728324i
$23$	5.41535	−	3.12656i	1.12918	−	0.651932i	0.185451	−	0.982654i	$-0.440626\pi$
$23$	5.41535	−	3.12656i	1.12918	−	0.651932i	0.943728	+	0.330722i	$0.107292\pi$
$24$		0			0
$25$	−5.07039	+	8.78217i	−1.01408	+	1.75643i
$26$	2.84849			0.558636
$27$		0			0
$28$	2.44383	−	1.01375i	0.461841	−	0.191581i
$29$	2.50079	+	1.44383i	0.464385	+	0.268113i	0.713886	−	0.700262i	$-0.246933\pi$
$29$	2.50079	+	1.44383i	0.464385	+	0.268113i	−0.249501	+	0.968374i	$0.580267\pi$
$30$		0			0
$31$	3.04125	−	1.75587i	0.546225	−	0.315363i	−0.201373	−	0.979515i	$-0.564540\pi$
$31$	3.04125	−	1.75587i	0.546225	−	0.315363i	0.747598	+	0.664152i	$0.231207\pi$
$32$	−0.866025	+	0.500000i	−0.153093	+	0.0883883i
$33$		0			0
$34$	−0.642692	−	0.371058i	−0.110221	−	0.0636360i
$35$	−9.50923	+	3.94462i	−1.60735	+	0.666762i
$36$		0			0
$37$	3.00158			0.493456			0.246728	−	0.969085i	$-0.420645\pi$
$37$	3.00158			0.493456			0.246728	+	0.969085i	$0.420645\pi$
$38$	−0.892369	+	1.54563i	−0.144761	+	0.250734i
$39$		0			0
$40$	3.36980	−	1.94556i	0.532813	−	0.307620i
$41$	5.24705	+	9.08816i	0.819452	+	1.41933i	0.906087	+	0.423092i	$0.139055\pi$
$41$	5.24705	+	9.08816i	0.819452	+	1.41933i	−0.0866345	+	0.996240i	$0.527611\pi$
$42$		0			0
$43$	0.471521	−	0.816699i	0.0719063	−	0.124545i	−0.827830	−	0.560978i	$-0.810425\pi$
$43$	0.471521	−	0.816699i	0.0719063	−	0.124545i	0.899737	+	0.436433i	$0.143758\pi$
$44$		−	3.94462i		−	0.594674i
$45$		0			0
$46$	6.25311			0.921971
$47$	−1.09263	+	1.89248i	−0.159376	+	0.276047i	−0.934644	−	0.355585i	$-0.884282\pi$
$47$	−1.09263	+	1.89248i	−0.159376	+	0.276047i	0.775268	+	0.631633i	$0.217615\pi$
$48$		0			0
$49$	−6.76335	−	1.80473i	−0.966193	−	0.257819i
$50$	−8.78217	+	5.07039i	−1.24199	+	0.717061i
$51$		0			0
$52$	2.46687	+	1.42425i	0.342093	+	0.197507i
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$		0			0
$55$			15.3490i			2.06965i
$56$	2.62329	+	0.343982i	0.350553	+	0.0459665i
$57$		0			0
$58$	1.44383	+	2.50079i	0.189584	+	0.328370i
$59$	−0.0105673	−	0.0183031i	−0.00137575	−	0.00238286i	0.865337	−	0.501191i	$-0.167105\pi$
$59$	−0.0105673	−	0.0183031i	−0.00137575	−	0.00238286i	−0.866712	+	0.498808i	$0.833771\pi$
$60$		0			0
$61$	−2.13832	−	1.23456i	−0.273783	−	0.158069i	0.356822	−	0.934172i	$-0.383860\pi$
$61$	−2.13832	−	1.23456i	−0.273783	−	0.158069i	−0.630606	+	0.776103i	$0.717193\pi$
$62$	3.51174			0.445991
$63$		0			0
$64$	−1.00000			−0.125000
$65$	−9.59886	−	5.54191i	−1.19059	−	0.687389i
$66$		0			0
$67$	−6.72463	−	11.6474i	−0.821544	−	1.42296i	−0.904532	−	0.426406i	$-0.859779\pi$
$67$	−6.72463	−	11.6474i	−0.821544	−	1.42296i	0.0829874	−	0.996551i	$-0.473554\pi$
$68$	−0.371058	−	0.642692i	−0.0449974	−	0.0779379i
$69$		0			0
$70$	−10.2075	−	1.33847i	−1.22003	−	0.159978i
$71$		−	1.94304i		−	0.230597i	−0.993331	−	0.115298i	$-0.963218\pi$
$71$		−	1.94304i		−	0.230597i	0.993331	−	0.115298i	$-0.0367824\pi$
$72$		0			0
$73$		−	4.85486i		−	0.568218i	−0.958792	−	0.284109i	$-0.908302\pi$
$73$		−	4.85486i		−	0.568218i	0.958792	−	0.284109i	$-0.0916978\pi$
$74$	2.59944	+	1.50079i	0.302179	+	0.174463i
$75$		0			0
$76$	−1.54563	+	0.892369i	−0.177296	+	0.102362i
$77$	−6.34904	+	8.28311i	−0.723540	+	0.943948i
$78$		0			0
$79$	−1.81806	+	3.14898i	−0.204548	+	0.354288i	−0.949989	−	0.312284i	$-0.898906\pi$
$79$	−1.81806	+	3.14898i	−0.204548	+	0.354288i	0.745440	+	0.666572i	$0.232239\pi$
$80$	3.89111			0.435040
$81$		0			0
$82$			10.4941i			1.15888i
$83$	4.02998	−	6.98012i	0.442347	−	0.766168i	−0.555516	−	0.831506i	$-0.687479\pi$
$83$	4.02998	−	6.98012i	0.442347	−	0.766168i	0.997863	+	0.0653378i	$0.0208125\pi$
$84$		0			0
$85$	1.44383	+	2.50079i	0.156605	+	0.271249i
$86$	0.816699	−	0.471521i	0.0880669	−	0.0508454i
$87$		0			0
$88$	1.97231	−	3.41614i	0.210249	−	0.364162i
$89$	9.26646			0.982243			0.491122	−	0.871091i	$-0.336587\pi$
$89$	9.26646			0.982243			0.491122	+	0.871091i	$0.336587\pi$
$90$		0			0
$91$	−2.88766	−	6.96124i	−0.302709	−	0.729736i
$92$	5.41535	+	3.12656i	0.564589	+	0.325966i
$93$		0			0
$94$	−1.89248	+	1.09263i	−0.195195	+	0.112696i
$95$	6.01422	−	3.47231i	0.617046	−	0.356251i
$96$		0			0
$97$	16.2983	+	9.40980i	1.65484	+	0.955421i	0.975043	+	0.222018i	$0.0712643\pi$
$97$	16.2983	+	9.40980i	1.65484	+	0.955421i	0.679794	+	0.733403i	$0.262069\pi$
$98$	−4.95487	−	4.94462i	−0.500517	−	0.499482i
$99$		0			0
$100$	−10.1408			−1.01408
$101$	−4.14079	+	7.17206i	−0.412024	+	0.713647i	−0.995111	−	0.0987631i	$-0.968511\pi$
$101$	−4.14079	+	7.17206i	−0.412024	+	0.713647i	0.583087	+	0.812410i	$0.301845\pi$
$102$		0			0
$103$	−14.7646	+	8.52435i	−1.45480	+	0.839929i	−0.998748	−	0.0500247i	$-0.984070\pi$
$103$	−14.7646	+	8.52435i	−1.45480	+	0.839929i	−0.456051	+	0.889953i	$0.650737\pi$
$104$	1.42425	+	2.46687i	0.139659	+	0.241896i
$105$		0			0
$106$		0			0
$107$			14.3369i			1.38600i	0.720936	+	0.693001i	$0.243712\pi$
$107$			14.3369i			1.38600i	−0.720936	+	0.693001i	$0.756288\pi$
$108$		0			0
$109$	11.2800			1.08042			0.540212	−	0.841529i	$-0.318344\pi$
$109$	11.2800			1.08042			0.540212	+	0.841529i	$0.318344\pi$
$110$	−7.67448	+	13.2926i	−0.731733	+	1.26740i
$111$		0			0
$112$	2.09985	+	1.60954i	0.198417	+	0.152088i
$113$	8.51501	−	4.91614i	0.801024	−	0.462472i	−0.0428049	−	0.999083i	$-0.513629\pi$
$113$	8.51501	−	4.91614i	0.801024	−	0.462472i	0.843829	+	0.536612i	$0.180296\pi$
$114$		0			0
$115$	−21.0718	−	12.1658i	−1.96495	−	1.13447i
$116$			2.88766i			0.268113i
$117$		0			0
$118$		−	0.0211346i		−	0.00194560i
$119$	−0.255275	+	1.94679i	−0.0234010	+	0.178462i
$120$		0			0
$121$	2.28001	+	3.94910i	0.207274	+	0.359009i
$122$	−1.23456	−	2.13832i	−0.111772	−	0.193594i
$123$		0			0
$124$	3.04125	+	1.75587i	0.273112	+	0.157682i
$125$	20.0033			1.78915
$126$		0			0
$127$	2.94462			0.261293			0.130646	−	0.991429i	$-0.458295\pi$
$127$	2.94462			0.261293			0.130646	+	0.991429i	$0.458295\pi$
$128$	−0.866025	−	0.500000i	−0.0765466	−	0.0441942i
$129$		0			0
$130$	−5.54191	−	9.59886i	−0.486057	−	0.841876i
$131$	−7.53255	−	13.0468i	−0.658122	−	1.13990i	−0.981101	−	0.193495i	$-0.938018\pi$
$131$	−7.53255	−	13.0468i	−0.658122	−	1.13990i	0.322979	−	0.946406i	$-0.395316\pi$
$132$		0			0
$133$	4.68189	+	0.613917i	0.405972	+	0.0532334i
$134$		−	13.4493i		−	1.16184i
$135$		0			0
$136$		−	0.742117i		−	0.0636360i
$137$	13.6139	+	7.85997i	1.16311	+	0.671523i	0.952048	−	0.305950i	$-0.0989739\pi$
$137$	13.6139	+	7.85997i	1.16311	+	0.671523i	0.211064	+	0.977472i	$0.432307\pi$
$138$		0			0
$139$	2.86373	−	1.65337i	0.242898	−	0.140237i	−0.373610	−	0.927586i	$-0.621880\pi$
$139$	2.86373	−	1.65337i	0.242898	−	0.140237i	0.616508	+	0.787349i	$0.288547\pi$
$140$	−8.17075	−	6.26292i	−0.690555	−	0.529313i
$141$		0			0
$142$	0.971521	−	1.68272i	0.0815282	−	0.141211i
$143$	−11.2362			−0.939620
$144$		0			0
$145$		−	11.2362i		−	0.933118i
$146$	2.42743	−	4.20443i	0.200896	−	0.347961i
$147$		0			0
$148$	1.50079	+	2.59944i	0.123364	+	0.213673i
$149$	−9.52765	+	5.50079i	−0.780535	+	0.450642i	−0.836620	−	0.547784i	$-0.815472\pi$
$149$	−9.52765	+	5.50079i	−0.780535	+	0.450642i	0.0560848	+	0.998426i	$0.482138\pi$
$150$		0			0
$151$	0.719988	−	1.24706i	0.0585918	−	0.101484i	−0.835242	−	0.549883i	$-0.814672\pi$
$151$	0.719988	−	1.24706i	0.0585918	−	0.101484i	0.893834	+	0.448399i	$0.148006\pi$
$152$	−1.78474			−0.144761
$153$		0			0
$154$	−9.63998	+	3.99886i	−0.776812	+	0.322237i
$155$	−11.8339	−	6.83228i	−0.950518	−	0.548782i
$156$		0			0
$157$	14.3822	−	8.30354i	1.14782	−	0.662695i	0.199465	−	0.979905i	$-0.436079\pi$
$157$	14.3822	−	8.30354i	1.14782	−	0.662695i	0.948355	+	0.317210i	$0.102746\pi$
$158$	−3.14898	+	1.81806i	−0.250519	+	0.144637i
$159$		0			0
$160$	3.36980	+	1.94556i	0.266406	+	0.153810i
$161$	−6.33909	−	15.2815i	−0.499591	−	1.20435i
$162$		0			0
$163$	−12.3955			−0.970887			−0.485444	−	0.874268i	$-0.661342\pi$
$163$	−12.3955			−0.970887			−0.485444	+	0.874268i	$0.661342\pi$
$164$	−5.24705	+	9.08816i	−0.409726	+	0.709666i
$165$		0			0
$166$	6.98012	−	4.02998i	0.541763	−	0.312787i
$167$	5.86087	+	10.1513i	0.453528	+	0.785534i	0.998602	−	0.0528541i	$-0.0168318\pi$
$167$	5.86087	+	10.1513i	0.453528	+	0.785534i	−0.545074	+	0.838388i	$0.683498\pi$
$168$		0			0
$169$	−2.44304	+	4.23147i	−0.187926	+	0.325498i
$170$			2.88766i			0.221474i
$171$		0			0
$172$	0.943042			0.0719063
$173$	8.38548	−	14.5241i	0.637536	−	1.10425i	−0.348435	−	0.937333i	$-0.613287\pi$
$173$	8.38548	−	14.5241i	0.637536	−	1.10425i	0.985972	−	0.166913i	$-0.0533798\pi$
$174$		0			0
$175$	21.2941	+	16.3220i	1.60968	+	1.23383i
$176$	3.41614	−	1.97231i	0.257501	−	0.148668i
$177$		0			0
$178$	8.02499	+	4.63323i	0.601499	+	0.347275i
$179$		−	5.77532i		−	0.431668i	−0.976430	−	0.215834i	$-0.930753\pi$
$179$		−	5.77532i		−	0.431668i	0.976430	−	0.215834i	$-0.0692470\pi$
$180$		0			0
$181$		−	5.53310i		−	0.411272i	−0.978629	−	0.205636i	$-0.934074\pi$
$181$		−	5.53310i		−	0.411272i	0.978629	−	0.205636i	$-0.0659263\pi$
$182$	0.979830	−	7.47244i	0.0726298	−	0.553894i
$183$		0			0
$184$	3.12656	+	5.41535i	0.230493	+	0.399225i
$185$	−5.83974	−	10.1147i	−0.429346	−	0.743649i
$186$		0			0
$187$	2.53518	+	1.46368i	0.185390	+	0.107035i
$188$	−2.18525			−0.159376
$189$		0			0
$190$	6.94462			0.503816
$191$	5.38124	+	3.10686i	0.389373	+	0.224805i	0.681888	−	0.731456i	$-0.261159\pi$
$191$	5.38124	+	3.10686i	0.389373	+	0.224805i	−0.292515	+	0.956261i	$0.594492\pi$
$192$		0			0
$193$	3.90271	+	6.75970i	0.280923	+	0.486574i	0.971612	−	0.236578i	$-0.0760260\pi$
$193$	3.90271	+	6.75970i	0.280923	+	0.486574i	−0.690689	+	0.723152i	$0.742693\pi$
$194$	9.40980	+	16.2983i	0.675584	+	1.17015i
$195$		0			0
$196$	−1.81873	−	6.75960i	−0.129910	−	0.482829i
$197$			12.7737i			0.910092i	0.890468	+	0.455046i	$0.150377\pi$
$197$			12.7737i			0.910092i	−0.890468	+	0.455046i	$0.849623\pi$
$198$		0			0
$199$		−	1.81201i		−	0.128450i	−0.997935	−	0.0642250i	$-0.979542\pi$
$199$		−	1.81201i		−	0.128450i	0.997935	−	0.0642250i	$-0.0204575\pi$
$200$	−8.78217	−	5.07039i	−0.620993	−	0.358530i
$201$		0			0
$202$	−7.17206	+	4.14079i	−0.504624	+	0.291345i
$203$	4.64782	−	6.06365i	0.326213	−	0.425585i
$204$		0			0
$205$	20.4169	−	35.3631i	1.42598	−	2.46986i
$206$	−17.0487			−1.18784
$207$		0			0
$208$			2.84849i			0.197507i
$209$	3.52006	−	6.09692i	0.243487	−	0.421732i
$210$		0			0
$211$	−1.88766	−	3.26953i	−0.129952	−	0.225083i	0.793706	−	0.608302i	$-0.208149\pi$
$211$	−1.88766	−	3.26953i	−0.129952	−	0.225083i	−0.923658	+	0.383218i	$0.874816\pi$
$212$		0			0
$213$		0			0
$214$	−7.16846	+	12.4161i	−0.490026	+	0.848750i
$215$	−3.66949			−0.250257
$216$		0			0
$217$	−3.56002	−	8.58209i	−0.241670	−	0.582590i
$218$	9.76874	+	5.63998i	0.661622	+	0.381988i
$219$		0			0
$220$	−13.2926	+	7.67448i	−0.896187	+	0.517414i
$221$	−1.83070	+	1.05696i	−0.123146	+	0.0710987i
$222$		0			0
$223$	11.0662	+	6.38910i	0.741051	+	0.427846i	0.822451	−	0.568836i	$-0.192606\pi$
$223$	11.0662	+	6.38910i	0.741051	+	0.427846i	−0.0814006	+	0.996681i	$0.525939\pi$
$224$	1.01375	+	2.44383i	0.0677341	+	0.163285i
$225$		0			0
$226$	9.83228			0.654034
$227$	9.99110	−	17.3051i	0.663133	−	1.14858i	−0.316655	−	0.948541i	$-0.602560\pi$
$227$	9.99110	−	17.3051i	0.663133	−	1.14858i	0.979788	−	0.200039i	$-0.0641068\pi$
$228$		0			0
$229$	8.77402	−	5.06568i	0.579804	−	0.334750i	−0.181252	−	0.983437i	$-0.558015\pi$
$229$	8.77402	−	5.06568i	0.579804	−	0.334750i	0.761055	+	0.648687i	$0.224682\pi$
$230$	−12.1658	−	21.0718i	−0.802188	−	1.38943i
$231$		0			0
$232$	−1.44383	+	2.50079i	−0.0947921	+	0.164185i
$233$			7.31007i			0.478898i	0.970909	+	0.239449i	$0.0769669\pi$
$233$			7.31007i			0.478898i	−0.970909	+	0.239449i	$0.923033\pi$
$234$		0			0
$235$	8.50307			0.554679
$236$	0.0105673	−	0.0183031i	0.000687873	−	0.00119143i
$237$		0			0
$238$	−1.19447	+	1.55833i	−0.0774260	+	0.101012i
$239$	−7.28317	+	4.20494i	−0.471109	+	0.271995i	−0.716704	−	0.697378i	$-0.754350\pi$
$239$	−7.28317	+	4.20494i	−0.471109	+	0.271995i	0.245595	+	0.969373i	$0.421017\pi$
$240$		0			0
$241$	−7.75277	−	4.47607i	−0.499400	−	0.288329i	0.229066	−	0.973411i	$-0.426433\pi$
$241$	−7.75277	−	4.47607i	−0.499400	−	0.288329i	−0.728466	+	0.685082i	$0.759766\pi$
$242$			4.56002i			0.293129i
$243$		0			0
$244$		−	2.46911i		−	0.158069i
$245$	7.07690	+	26.3024i	0.452127	+	1.68040i
$246$		0			0
$247$	2.54191	+	4.40271i	0.161738	+	0.280138i
$248$	1.75587	+	3.04125i	0.111498	+	0.193120i
$249$		0			0
$250$	17.3234	+	10.0017i	1.09563	+	0.632561i
$251$	−12.6432			−0.798033			−0.399017	−	0.916944i	$-0.630648\pi$
$251$	−12.6432			−0.798033			−0.399017	+	0.916944i	$0.630648\pi$
$252$		0			0
$253$	−24.6661			−1.55075
$254$	2.55012	+	1.47231i	0.160008	+	0.0923809i
$255$		0			0
$256$	−0.500000	−	0.866025i	−0.0312500	−	0.0541266i
$257$	8.15329	+	14.1219i	0.508588	+	0.880900i	0.999951	+	0.00994523i	$0.00316572\pi$
$257$	8.15329	+	14.1219i	0.508588	+	0.880900i	−0.491362	+	0.870955i	$0.663501\pi$
$258$		0			0
$259$	1.03249	−	7.87402i	0.0641557	−	0.489268i
$260$		−	11.0838i		−	0.687389i
$261$		0			0
$262$		−	15.0651i		−	0.930725i
$263$	−20.5434	−	11.8608i	−1.26676	−	0.731366i	−0.292389	−	0.956300i	$-0.594450\pi$
$263$	−20.5434	−	11.8608i	−1.26676	−	0.731366i	−0.974374	+	0.224934i	$0.927783\pi$
$264$		0			0
$265$		0			0
$266$	3.74768	+	2.87261i	0.229785	+	0.176131i
$267$		0			0
$268$	6.72463	−	11.6474i	0.410772	−	0.711478i
$269$	−7.28288			−0.444045			−0.222022	−	0.975042i	$-0.571266\pi$
$269$	−7.28288			−0.444045			−0.222022	+	0.975042i	$0.571266\pi$
$270$		0			0
$271$		−	22.6879i		−	1.37819i	−0.724669	−	0.689097i	$-0.758007\pi$
$271$		−	22.6879i		−	1.37819i	0.724669	−	0.689097i	$-0.241993\pi$
$272$	0.371058	−	0.642692i	0.0224987	−	0.0389689i
$273$		0			0
$274$	7.85997	+	13.6139i	0.474838	+	0.822444i
$275$	34.6423	−	20.0007i	2.08901	−	1.20609i
$276$		0			0
$277$	−12.0838	+	20.9298i	−0.726046	+	1.25755i	0.232496	+	0.972597i	$0.425311\pi$
$277$	−12.0838	+	20.9298i	−0.726046	+	1.25755i	−0.958542	+	0.284951i	$0.908023\pi$
$278$	3.30675			0.198326
$279$		0			0
$280$	−3.94462	−	9.50923i	−0.235736	−	0.568285i
$281$	4.11229	+	2.37423i	0.245319	+	0.141635i	0.617619	−	0.786478i	$-0.288097\pi$
$281$	4.11229	+	2.37423i	0.245319	+	0.141635i	−0.372300	+	0.928112i	$0.621431\pi$
$282$		0			0
$283$	−25.4484	+	14.6926i	−1.51275	+	0.873387i	−0.512861	+	0.858471i	$0.671415\pi$
$283$	−25.4484	+	14.6926i	−1.51275	+	0.873387i	−0.999889	+	0.0149153i	$0.995252\pi$
$284$	1.68272	−	0.971521i	0.0998513	−	0.0576492i
$285$		0			0
$286$	−9.73085	−	5.61811i	−0.575398	−	0.332206i
$287$	25.6458	−	10.6384i	1.51382	−	0.627965i
$288$		0			0
$289$	−16.4493			−0.967604
$290$	5.61811	−	9.73085i	0.329907	−	0.571415i
$291$		0			0
$292$	4.20443	−	2.42743i	0.246046	−	0.142055i
$293$	−3.31206	−	5.73666i	−0.193493	−	0.335139i	0.752913	−	0.658121i	$-0.228648\pi$
$293$	−3.31206	−	5.73666i	−0.193493	−	0.335139i	−0.946405	+	0.322981i	$0.895315\pi$
$294$		0			0
$295$	−0.0411186	+	0.0712195i	−0.00239402	+	0.00414656i
$296$			3.00158i			0.174463i
$297$		0			0
$298$	−11.0016			−0.637304
$299$	8.90597	−	15.4256i	0.515046	−	0.892085i
$300$		0			0
$301$	−1.98025	−	1.51787i	−0.114140	−	0.0874885i
$302$	1.24706	−	0.719988i	0.0717600	−	0.0414307i
$303$		0			0
$304$	−1.54563	−	0.892369i	−0.0886479	−	0.0511809i
$305$			9.60761i			0.550130i
$306$		0			0
$307$			21.7242i			1.23987i	0.784655	+	0.619933i	$0.212840\pi$
$307$			21.7242i			1.23987i	−0.784655	+	0.619933i	$0.787160\pi$
$308$	−10.3479	−	1.35688i	−0.589626	−	0.0773152i
$309$		0			0
$310$	−6.83228	−	11.8339i	−0.388048	−	0.672118i
$311$	−3.14900	−	5.45422i	−0.178563	−	0.309281i	0.762825	−	0.646605i	$-0.223812\pi$
$311$	−3.14900	−	5.45422i	−0.178563	−	0.309281i	−0.941389	+	0.337324i	$0.890478\pi$
$312$		0			0
$313$	−19.2423	−	11.1095i	−1.08764	−	0.627948i	−0.154691	−	0.987963i	$-0.549438\pi$
$313$	−19.2423	−	11.1095i	−1.08764	−	0.627948i	−0.932946	+	0.360015i	$0.882771\pi$
$314$	16.6071			0.937192
$315$		0			0
$316$	−3.63613			−0.204548
$317$	−13.5632	−	7.83070i	−0.761784	−	0.439816i	0.0681519	−	0.997675i	$-0.478290\pi$
$317$	−13.5632	−	7.83070i	−0.761784	−	0.439816i	−0.829936	+	0.557859i	$0.811623\pi$
$318$		0			0
$319$	−5.69536	−	9.86466i	−0.318879	−	0.552315i
$320$	1.94556	+	3.36980i	0.108760	+	0.188378i
$321$		0			0
$322$	2.15096	−	16.4038i	0.119868	−	0.914145i
$323$		−	1.32448i		−	0.0736963i
$324$		0			0
$325$			28.8859i			1.60230i
$326$	−10.7348	−	6.19773i	−0.594545	−	0.343260i
$327$		0			0
$328$	−9.08816	+	5.24705i	−0.501810	+	0.289720i
$329$	4.58870	+	3.51726i	0.252983	+	0.193913i
$330$		0			0
$331$	−0.636129	+	1.10181i	−0.0349648	+	0.0605608i	−0.882978	−	0.469414i	$-0.844465\pi$
$331$	−0.636129	+	1.10181i	−0.0349648	+	0.0605608i	0.848013	+	0.529975i	$0.177799\pi$
$332$	8.05995			0.442347
$333$		0			0
$334$			11.7217i			0.641386i
$335$	−26.1663	+	45.3214i	−1.42962	+	2.47617i
$336$		0			0
$337$	−3.78001	−	6.54717i	−0.205910	−	0.356647i	0.744512	−	0.667609i	$-0.232682\pi$
$337$	−3.78001	−	6.54717i	−0.205910	−	0.356647i	−0.950422	+	0.310962i	$0.899349\pi$
$338$	−4.23147	+	2.44304i	−0.230162	+	0.132884i
$339$		0			0
$340$	−1.44383	+	2.50079i	−0.0783027	+	0.135624i
$341$	−13.8525			−0.750153
$342$		0			0
$343$	−7.06081	+	17.1215i	−0.381248	+	0.924473i
$344$	0.816699	+	0.471521i	0.0440334	+	0.0254227i
$345$		0			0
$346$	14.5241	−	8.38548i	0.780820	−	0.450806i
$347$	−19.1470	+	11.0545i	−1.02787	+	0.593439i	−0.916373	−	0.400326i	$-0.868897\pi$
$347$	−19.1470	+	11.0545i	−1.02787	+	0.593439i	−0.111494	+	0.993765i	$0.535564\pi$
$348$		0			0
$349$	−12.7682	−	7.37173i	−0.683467	−	0.394600i	0.117693	−	0.993050i	$-0.462450\pi$
$349$	−12.7682	−	7.37173i	−0.683467	−	0.394600i	−0.801160	+	0.598450i	$0.795783\pi$
$350$	10.2802	+	24.7823i	0.549501	+	1.32467i
$351$		0			0
$352$	3.94462			0.210249
$353$	−8.63881	+	14.9629i	−0.459798	+	0.796393i	−0.998950	−	0.0458154i	$-0.985411\pi$
$353$	−8.63881	+	14.9629i	−0.459798	+	0.796393i	0.539152	+	0.842208i	$0.318745\pi$
$354$		0			0
$355$	−6.54767	+	3.78030i	−0.347514	+	0.200638i
$356$	4.63323	+	8.02499i	0.245561	+	0.425324i
$357$		0			0
$358$	2.88766	−	5.00158i	0.152618	−	0.264342i
$359$			10.9129i			0.575963i	0.957636	+	0.287982i	$0.0929842\pi$
$359$			10.9129i			0.575963i	−0.957636	+	0.287982i	$0.907016\pi$
$360$		0			0
$361$	15.8147			0.832353
$362$	2.76655	−	4.79180i	0.145407	−	0.251852i
$363$		0			0
$364$	4.58478	−	5.98141i	0.240308	−	0.313511i
$365$	−16.3599	+	9.44541i	−0.856318	+	0.494395i
$366$		0			0
$367$	30.9407	+	17.8636i	1.61509	+	0.932472i	0.988166	+	0.153391i	$0.0490194\pi$
$367$	30.9407	+	17.8636i	1.61509	+	0.932472i	0.626923	+	0.779081i	$0.284314\pi$
$368$			6.25311i			0.325966i
$369$		0			0
$370$		−	11.6795i		−	0.607187i
$371$		0			0
$372$		0			0
$373$	16.0300	+	27.7648i	0.830003	+	1.43761i	0.898035	+	0.439923i	$0.144994\pi$
$373$	16.0300	+	27.7648i	0.830003	+	1.43761i	−0.0680328	+	0.997683i	$0.521672\pi$
$374$	1.46368	+	2.53518i	0.0756853	+	0.131091i
$375$		0			0
$376$	−1.89248	−	1.09263i	−0.0975974	−	0.0563479i
$377$	8.22549			0.423634
$378$		0			0
$379$	34.8891			1.79214			0.896068	−	0.443918i	$-0.146412\pi$
$379$	34.8891			1.79214			0.896068	+	0.443918i	$0.146412\pi$
$380$	6.01422	+	3.47231i	0.308523	+	0.178126i
$381$		0			0
$382$	3.10686	+	5.38124i	0.158961	+	0.275328i
$383$	8.76711	+	15.1851i	0.447978	+	0.775921i	0.998254	−	0.0590616i	$-0.0188108\pi$
$383$	8.76711	+	15.1851i	0.447978	+	0.775921i	−0.550276	+	0.834983i	$0.685477\pi$
$384$		0			0
$385$	40.2649	+	5.27976i	2.05209	+	0.269082i
$386$			7.80542i			0.397286i
$387$		0			0
$388$			18.8196i			0.955421i
$389$	6.60060	+	3.81086i	0.334664	+	0.193218i	0.657910	−	0.753097i	$-0.271441\pi$
$389$	6.60060	+	3.81086i	0.334664	+	0.193218i	−0.323246	+	0.946315i	$0.604774\pi$
$390$		0			0
$391$	−4.01882	+	2.32027i	−0.203241	+	0.117341i
$392$	1.80473	−	6.76335i	0.0911527	−	0.341601i
$393$		0			0
$394$	−6.38687	+	11.0624i	−0.321766	+	0.557315i
$395$	14.1486			0.711893
$396$		0			0
$397$			37.6469i			1.88944i	0.327873	+	0.944722i	$0.393668\pi$
$397$			37.6469i			1.88944i	−0.327873	+	0.944722i	$0.606332\pi$
$398$	0.906005	−	1.56925i	0.0454139	−	0.0786592i
$399$		0			0
$400$	−5.07039	−	8.78217i	−0.253519	−	0.439108i
$401$	−18.5689	+	10.7207i	−0.927284	+	0.535368i	−0.885952	−	0.463778i	$-0.846494\pi$
$401$	−18.5689	+	10.7207i	−0.927284	+	0.535368i	−0.0413326	+	0.999145i	$0.513160\pi$
$402$		0			0
$403$	5.00158	−	8.66299i	0.249146	−	0.431534i
$404$	−8.28158			−0.412024
$405$		0			0
$406$	7.05696	−	2.92737i	0.350231	−	0.145283i
$407$	−10.2538	−	5.92004i	−0.508262	−	0.293445i
$408$		0			0
$409$	−25.6086	+	14.7851i	−1.26627	+	0.731079i	−0.974279	−	0.225344i	$-0.927649\pi$
$409$	−25.6086	+	14.7851i	−1.26627	+	0.731079i	−0.291986	+	0.956423i	$0.594316\pi$
$410$	35.3631	−	20.4169i	1.74646	−	1.00832i
$411$		0			0
$412$	−14.7646	−	8.52435i	−0.727400	−	0.419964i
$413$	−0.0516494	+	0.0214252i	−0.00254150	+	0.00105427i
$414$		0			0
$415$	−31.3622			−1.53951
$416$	−1.42425	+	2.46687i	−0.0698294	+	0.120948i
$417$		0			0
$418$	6.09692	−	3.52006i	0.298210	−	0.172172i
$419$	−3.56481	−	6.17443i	−0.174152	−	0.301641i	0.765715	−	0.643180i	$-0.222385\pi$
$419$	−3.56481	−	6.17443i	−0.174152	−	0.301641i	−0.939868	+	0.341539i	$0.889052\pi$
$420$		0			0
$421$	−2.31007	+	4.00115i	−0.112586	+	0.195004i	−0.916812	−	0.399319i	$-0.869247\pi$
$421$	−2.31007	+	4.00115i	−0.112586	+	0.195004i	0.804226	+	0.594323i	$0.202580\pi$
$422$		−	3.77532i		−	0.183780i
$423$		0			0
$424$		0			0
$425$	3.76282	−	6.51739i	0.182524	−	0.316140i
$426$		0			0
$427$	−3.97415	+	5.18477i	−0.192323	+	0.250908i
$428$	−12.4161	+	7.16846i	−0.600157	+	0.346501i
$429$		0			0
$430$	−3.17787	−	1.83474i	−0.153250	−	0.0884792i
$431$		−	4.00771i		−	0.193045i	−0.995331	−	0.0965223i	$-0.969228\pi$
$431$		−	4.00771i		−	0.193045i	0.995331	−	0.0965223i	$-0.0307719\pi$
$432$		0			0
$433$		−	29.4125i		−	1.41348i	−0.707475	−	0.706738i	$-0.750166\pi$
$433$		−	29.4125i		−	1.41348i	0.707475	−	0.706738i	$-0.249834\pi$
$434$	1.20797	−	9.21232i	0.0579845	−	0.442205i
$435$		0			0
$436$	5.63998	+	9.76874i	0.270106	+	0.467838i
$437$	5.58008	+	9.66498i	0.266931	+	0.462339i
$438$		0			0
$439$	18.5130	+	10.6885i	0.883575	+	0.510133i	0.871836	−	0.489799i	$-0.162930\pi$
$439$	18.5130	+	10.6885i	0.883575	+	0.510133i	0.0117398	+	0.999931i	$0.496263\pi$
$440$	−15.3490			−0.731733
$441$		0			0
$442$	−2.11392			−0.100549
$443$	−5.05227	−	2.91693i	−0.240041	−	0.138587i	0.375155	−	0.926962i	$-0.377590\pi$
$443$	−5.05227	−	2.91693i	−0.240041	−	0.138587i	−0.615195	+	0.788375i	$0.710923\pi$
$444$		0			0
$445$	−18.0284	−	31.2262i	−0.854630	−	1.48026i
$446$	6.38910	+	11.0662i	0.302533	+	0.524002i
$447$		0			0
$448$	−0.343982	+	2.62329i	−0.0162516	+	0.123939i
$449$		−	22.5823i		−	1.06573i	−0.846202	−	0.532863i	$-0.821116\pi$
$449$		−	22.5823i		−	1.06573i	0.846202	−	0.532863i	$-0.178884\pi$
$450$		0			0
$451$		−	41.3953i		−	1.94923i
$452$	8.51501	+	4.91614i	0.400512	+	0.231236i
$453$		0			0
$454$	17.3051	−	9.99110i	0.812168	−	0.468906i
$455$	−17.8399	+	23.2743i	−0.836347	+	1.09112i
$456$		0			0
$457$	−19.9311	+	34.5218i	−0.932340	+	1.61486i	−0.153029	+	0.988222i	$0.548903\pi$
$457$	−19.9311	+	34.5218i	−0.932340	+	1.61486i	−0.779310	+	0.626638i	$0.784430\pi$
$458$	10.1314			0.473408
$459$		0			0
$460$		−	24.3316i		−	1.13447i
$461$	−3.68254	+	6.37834i	−0.171513	+	0.297069i	−0.938949	−	0.344056i	$-0.888199\pi$
$461$	−3.68254	+	6.37834i	−0.171513	+	0.297069i	0.767436	+	0.641125i	$0.221532\pi$
$462$		0			0
$463$	−14.3457	−	24.8475i	−0.666702	−	1.15476i	−0.978821	−	0.204718i	$-0.934372\pi$
$463$	−14.3457	−	24.8475i	−0.666702	−	1.15476i	0.312119	−	0.950043i	$-0.398961\pi$
$464$	−2.50079	+	1.44383i	−0.116096	+	0.0670282i
$465$		0			0
$466$	−3.65503	+	6.33070i	−0.169316	+	0.293264i
$467$	−13.6704			−0.632590			−0.316295	−	0.948661i	$-0.602439\pi$
$467$	−13.6704			−0.632590			−0.316295	+	0.948661i	$0.602439\pi$
$468$		0			0
$469$	−32.8677	+	13.6342i	−1.51769	+	0.629569i
$470$	7.36387	+	4.25153i	0.339670	+	0.196109i
$471$		0			0
$472$	0.0183031	−	0.0105673i	0.000842469	−	0.000486400i
$473$	−3.22157	+	1.85997i	−0.148128	+	0.0855216i
$474$		0			0
$475$	−15.6739	−	9.04931i	−0.719166	−	0.415211i
$476$	−1.81361	+	0.752321i	−0.0831266	+	0.0344826i
$477$		0			0
$478$	−8.40988			−0.384659
$479$	5.20537	−	9.01596i	0.237839	−	0.411950i	−0.722255	−	0.691627i	$-0.756894\pi$
$479$	5.20537	−	9.01596i	0.237839	−	0.411950i	0.960094	+	0.279677i	$0.0902275\pi$
$480$		0			0
$481$	7.40449	−	4.27499i	0.337616	−	0.194923i
$482$	−4.47607	−	7.75277i	−0.203879	−	0.353129i
$483$		0			0
$484$	−2.28001	+	3.94910i	−0.103637	+	0.179504i
$485$		−	73.2292i		−	3.32517i
$486$		0			0
$487$	2.33850			0.105968			0.0529838	−	0.998595i	$-0.483127\pi$
$487$	2.33850			0.105968			0.0529838	+	0.998595i	$0.483127\pi$
$488$	1.23456	−	2.13832i	0.0558858	−	0.0967970i
$489$		0			0
$490$	−7.02242	+	26.3170i	−0.317240	+	1.18888i
$491$	29.3448	−	16.9422i	1.32431	−	0.764591i	0.339898	−	0.940462i	$-0.389608\pi$
$491$	29.3448	−	16.9422i	1.32431	−	0.764591i	0.984413	+	0.175871i	$0.0562742\pi$
$492$		0			0
$493$	−1.85588	−	1.07149i	−0.0835845	−	0.0482575i
$494$			5.08381i			0.228732i
$495$		0			0
$496$			3.51174i			0.157682i
$497$	−5.09717	−	0.668371i	−0.228639	−	0.0299805i
$498$		0			0
$499$	8.30223	+	14.3799i	0.371659	+	0.643732i	0.989821	−	0.142319i	$-0.0454558\pi$
$499$	8.30223	+	14.3799i	0.371659	+	0.643732i	−0.618162	+	0.786051i	$0.712123\pi$
$500$	10.0017	+	17.3234i	0.447288	+	0.774726i
$501$		0			0
$502$	−10.9494	−	6.32161i	−0.488694	−	0.282147i
$503$	−35.3661			−1.57690			−0.788449	−	0.615100i	$-0.789115\pi$
$503$	−35.3661			−1.57690			−0.788449	+	0.615100i	$0.789115\pi$
$504$		0			0
$505$	32.2246			1.43398
$506$	−21.3615	−	12.3331i	−0.949635	−	0.548272i
$507$		0			0
$508$	1.47231	+	2.55012i	0.0653232	+	0.113143i
$509$	−18.5291	−	32.0933i	−0.821287	−	1.42251i	−0.904724	−	0.425998i	$-0.859923\pi$
$509$	−18.5291	−	32.0933i	−0.821287	−	1.42251i	0.0834371	−	0.996513i	$-0.473410\pi$
$510$		0			0
$511$	−12.7357	−	1.66998i	−0.563396	−	0.0738757i
$512$		−	1.00000i		−	0.0441942i
$513$		0			0
$514$			16.3066i			0.719252i
$515$	57.4507	+	33.1692i	2.53158	+	1.46161i
$516$		0			0
$517$	7.46513	−	4.30999i	0.328316	−	0.189553i
$518$	4.83117	−	6.30286i	0.212270	−	0.276932i
$519$		0			0
$520$	5.54191	−	9.59886i	0.243029	−	0.420938i
$521$	1.78309			0.0781187			0.0390594	−	0.999237i	$-0.487564\pi$
$521$	1.78309			0.0781187			0.0390594	+	0.999237i	$0.487564\pi$
$522$		0			0
$523$		−	24.0538i		−	1.05180i	−0.850546	−	0.525901i	$-0.823728\pi$
$523$		−	24.0538i		−	1.05180i	0.850546	−	0.525901i	$-0.176272\pi$
$524$	7.53255	−	13.0468i	0.329061	−	0.569950i
$525$		0			0
$526$	−11.8608	−	20.5434i	−0.517154	−	0.895737i
$527$	−2.25696	+	1.30306i	−0.0983149	+	0.0567621i
$528$		0			0
$529$	8.05069	−	13.9442i	0.350030	−	0.606270i
$530$		0			0
$531$		0			0
$532$	1.80928	+	4.36160i	0.0784422	+	0.189099i
$533$	25.8876	+	14.9462i	1.12132	+	0.647392i
$534$		0			0
$535$	48.3126	−	27.8933i	2.08874	−	1.20593i
$536$	11.6474	−	6.72463i	0.503091	−	0.290460i
$537$		0			0
$538$	−6.30716	−	3.64144i	−0.271921	−	0.156994i
$539$	19.5451	+	19.5046i	0.841866	+	0.840124i
$540$		0			0
$541$	30.0032			1.28994			0.644968	−	0.764209i	$-0.276871\pi$
$541$	30.0032			1.28994			0.644968	+	0.764209i	$0.276871\pi$
$542$	11.3440	−	19.6483i	0.487265	−	0.843968i
$543$		0			0
$544$	0.642692	−	0.371058i	0.0275552	−	0.0159090i
$545$	−21.9458	−	38.0113i	−0.940056	−	1.62822i
$546$		0			0
$547$	−10.7816	+	18.6743i	−0.460987	+	0.798454i	−0.999010	−	0.0444765i	$-0.985838\pi$
$547$	−10.7816	+	18.6743i	−0.460987	+	0.798454i	0.538023	+	0.842930i	$0.319171\pi$
$548$			15.7199i			0.671523i
$549$		0			0
$550$	40.0015			1.70567
$551$	−2.57686	+	4.46325i	−0.109778	+	0.190141i
$552$		0			0
$553$	7.63532	+	5.85251i	0.324687	+	0.248874i
$554$	−20.9298	+	12.0838i	−0.889221	+	0.513392i
$555$		0			0
$556$	2.86373	+	1.65337i	0.121449	+	0.0701187i
$557$		−	36.9477i		−	1.56552i	−0.622321	−	0.782762i	$-0.713810\pi$
$557$		−	36.9477i		−	1.56552i	0.622321	−	0.782762i	$-0.286190\pi$
$558$		0			0
$559$		−	2.68625i		−	0.113616i
$560$	1.33847	−	10.2075i	0.0565608	−	0.431347i
$561$		0			0
$562$	2.37423	+	4.11229i	0.100151	+	0.173467i
$563$	7.58422	+	13.1363i	0.319637	+	0.553627i	0.980412	−	0.196957i	$-0.0631058\pi$
$563$	7.58422	+	13.1363i	0.319637	+	0.553627i	−0.660776	+	0.750584i	$0.729772\pi$
$564$		0			0
$565$	−33.1329	−	19.1293i	−1.39391	−	0.804774i
$566$	−29.3853			−1.23516
$567$		0			0
$568$	1.94304			0.0815282
$569$	31.8084	+	18.3646i	1.33348	+	0.769885i	0.985831	−	0.167740i	$-0.0536470\pi$
$569$	31.8084	+	18.3646i	1.33348	+	0.769885i	0.347648	+	0.937625i	$0.386980\pi$
$570$		0			0
$571$	−5.61387	−	9.72351i	−0.234933	−	0.406916i	0.724320	−	0.689464i	$-0.242154\pi$
$571$	−5.61387	−	9.72351i	−0.234933	−	0.406916i	−0.959253	+	0.282548i	$0.908820\pi$
$572$	−5.61811	−	9.73085i	−0.234905	−	0.406867i
$573$		0			0
$574$	27.5291	+	3.60978i	1.14904	+	0.150669i
$575$			63.4114i			2.64444i
$576$		0			0
$577$			36.5515i			1.52166i	0.648952	+	0.760829i	$0.275208\pi$
$577$			36.5515i			1.52166i	−0.648952	+	0.760829i	$0.724792\pi$
$578$	−14.2455	−	8.22463i	−0.592534	−	0.342100i
$579$		0			0
$580$	9.73085	−	5.61811i	0.404052	−	0.233279i
$581$	−16.9247	−	12.9729i	−0.702154	−	0.538205i
$582$		0			0
$583$		0			0
$584$	4.85486			0.200896
$585$		0			0
$586$		−	6.62413i		−	0.273640i
$587$	4.99738	−	8.65571i	0.206264	−	0.357259i	−0.744271	−	0.667878i	$-0.767203\pi$
$587$	4.99738	−	8.65571i	0.206264	−	0.357259i	0.950535	+	0.310619i	$0.100536\pi$
$588$		0			0
$589$	3.13376	+	5.42784i	0.129124	+	0.223650i
$590$	−0.0712195	+	0.0411186i	−0.00293206	+	0.00169283i
$591$		0			0
$592$	−1.50079	+	2.59944i	−0.0616820	+	0.106836i
$593$	7.78223			0.319578			0.159789	−	0.987151i	$-0.448919\pi$
$593$	7.78223			0.319578			0.159789	+	0.987151i	$0.448919\pi$
$594$		0			0
$595$	7.05696	−	2.92737i	0.289307	−	0.120010i
$596$	−9.52765	−	5.50079i	−0.390268	−	0.225321i
$597$		0			0
$598$	15.4256	−	8.90597i	0.630800	−	0.364192i
$599$	−21.6614	+	12.5062i	−0.885061	+	0.510990i	−0.872324	−	0.488929i	$-0.837388\pi$
$599$	−21.6614	+	12.5062i	−0.885061	+	0.510990i	−0.0127373	+	0.999919i	$0.504055\pi$
$600$		0			0
$601$	−25.9925	−	15.0068i	−1.06026	−	0.612139i	−0.134753	−	0.990879i	$-0.543024\pi$
$601$	−25.9925	−	15.0068i	−1.06026	−	0.612139i	−0.925503	+	0.378740i	$0.876357\pi$
$602$	−0.956010	−	2.30464i	−0.0389640	−	0.0939300i
$603$		0			0
$604$	1.43998			0.0585918
$605$	8.87179	−	15.3664i	0.360689	−	0.624732i
$606$		0			0
$607$	−3.96882	+	2.29140i	−0.161089	+	0.0930050i	−0.578378	−	0.815769i	$-0.696314\pi$
$607$	−3.96882	+	2.29140i	−0.161089	+	0.0930050i	0.417288	+	0.908774i	$0.362981\pi$
$608$	−0.892369	−	1.54563i	−0.0361903	−	0.0626835i
$609$		0			0
$610$	−4.80380	+	8.32043i	−0.194500	+	0.336884i
$611$			6.22468i			0.251823i
$612$		0			0
$613$	30.5522			1.23399			0.616996	−	0.786966i	$-0.288349\pi$
$613$	30.5522			1.23399			0.616996	+	0.786966i	$0.288349\pi$
$614$	−10.8621	+	18.8137i	−0.438359	+	0.759259i
$615$		0			0
$616$	−8.28311	−	6.34904i	−0.333736	−	0.255810i
$617$	28.2484	−	16.3092i	1.13724	−	0.656585i	0.191493	−	0.981494i	$-0.438667\pi$
$617$	28.2484	−	16.3092i	1.13724	−	0.656585i	0.945745	+	0.324909i	$0.105334\pi$
$618$		0			0
$619$	17.3244	+	10.0023i	0.696327	+	0.402024i	0.805978	−	0.591946i	$-0.201640\pi$
$619$	17.3244	+	10.0023i	0.696327	+	0.402024i	−0.109651	+	0.993970i	$0.534973\pi$
$620$		−	13.6646i		−	0.548782i
$621$		0			0
$622$		−	6.29800i		−	0.252527i
$623$	3.18749	−	24.3087i	0.127704	−	0.973906i
$624$		0			0
$625$	−13.5657	−	23.4965i	−0.542628	−	0.939859i
$626$	−11.1095	−	19.2423i	−0.444026	−	0.769076i
$627$		0			0
$628$	14.3822	+	8.30354i	0.573910	+	0.331347i
$629$	−2.22752			−0.0888171
$630$		0			0
$631$	6.09634			0.242692			0.121346	−	0.992610i	$-0.461279\pi$
$631$	6.09634			0.242692			0.121346	+	0.992610i	$0.461279\pi$
$632$	−3.14898	−	1.81806i	−0.125260	−	0.0723187i
$633$		0			0
$634$	−7.83070	−	13.5632i	−0.310997	−	0.538663i
$635$	−5.72893	−	9.92279i	−0.227345	−	0.393774i
$636$		0			0
$637$	−19.2547	+	5.18065i	−0.762898	+	0.205265i
$638$		−	11.3907i		−	0.450963i
$639$		0			0
$640$			3.89111i			0.153810i
$641$	−28.9612	−	16.7207i	−1.14390	−	0.660429i	−0.196504	−	0.980503i	$-0.562959\pi$
$641$	−28.9612	−	16.7207i	−1.14390	−	0.660429i	−0.947393	+	0.320074i	$0.896292\pi$
$642$		0			0
$643$	16.6022	−	9.58527i	0.654726	−	0.378006i	−0.135539	−	0.990772i	$-0.543276\pi$
$643$	16.6022	−	9.58527i	0.654726	−	0.378006i	0.790264	+	0.612766i	$0.209943\pi$
$644$	10.0647	−	13.1306i	0.396603	−	0.517418i
$645$		0			0
$646$	0.662242	−	1.14704i	0.0260556	−	0.0451296i
$647$	44.6049			1.75360			0.876800	−	0.480854i	$-0.159673\pi$
$647$	44.6049			1.75360			0.876800	+	0.480854i	$0.159673\pi$
$648$		0			0
$649$			0.0833680i			0.00327248i
$650$	−14.4430	+	25.0159i	−0.566500	+	0.981206i
$651$		0			0
$652$	−6.19773	−	10.7348i	−0.242722	−	0.420407i
$653$	0.564755	−	0.326061i	0.0221006	−	0.0127598i	−0.488909	−	0.872335i	$-0.662605\pi$
$653$	0.564755	−	0.326061i	0.0221006	−	0.0127598i	0.511010	+	0.859575i	$0.329272\pi$
$654$		0			0
$655$	−29.3100	+	50.7664i	−1.14524	+	1.98361i
$656$	−10.4941			−0.409726
$657$		0			0
$658$	2.21530	+	5.34039i	0.0863614	+	0.208190i
$659$	−26.2738	−	15.1692i	−1.02348	−	0.590908i	−0.108372	−	0.994110i	$-0.534564\pi$
$659$	−26.2738	−	15.1692i	−1.02348	−	0.590908i	−0.915111	+	0.403202i	$0.867897\pi$
$660$		0			0
$661$	−11.1004	+	6.40881i	−0.431755	+	0.249274i	−0.700094	−	0.714051i	$-0.746859\pi$
$661$	−11.1004	+	6.40881i	−0.431755	+	0.249274i	0.268339	+	0.963325i	$0.413525\pi$
$662$	−1.10181	+	0.636129i	−0.0428230	+	0.0247239i
$663$		0			0
$664$	6.98012	+	4.02998i	0.270881	+	0.156393i
$665$	−7.04011	−	16.9715i	−0.273004	−	0.658126i
$666$		0			0
$667$	18.0569			0.699165
$668$	−5.86087	+	10.1513i	−0.226764	+	0.392767i
$669$		0			0
$670$	−45.3214	+	26.1663i	−1.75092	+	1.01089i
$671$	4.86986	+	8.43484i	0.187999	+	0.325623i
$672$		0			0
$673$	11.2246	−	19.4416i	0.432678	−	0.749420i	−0.564425	−	0.825484i	$-0.690902\pi$
$673$	11.2246	−	19.4416i	0.432678	−	0.749420i	0.997103	+	0.0760644i	$0.0242355\pi$
$674$		−	7.56002i		−	0.291201i
$675$		0			0
$676$	−4.88608			−0.187926
$677$	−25.5903	+	44.3237i	−0.983516	+	1.70350i	−0.335163	+	0.942160i	$0.608791\pi$
$677$	−25.5903	+	44.3237i	−0.983516	+	1.70350i	−0.648353	+	0.761340i	$0.724542\pi$
$678$		0			0
$679$	30.2910	−	39.5183i	1.16246	−	1.51657i
$680$	−2.50079	+	1.44383i	−0.0959009	+	0.0553684i
$681$		0			0
$682$	−11.9966	−	6.92623i	−0.459373	−	0.265219i
$683$		−	14.5616i		−	0.557184i	−0.960410	−	0.278592i	$-0.910132\pi$
$683$		−	14.5616i		−	0.557184i	0.960410	−	0.278592i	$-0.0898677\pi$
$684$		0			0
$685$		−	61.1681i		−	2.33711i
$686$	−14.6756	+	11.2972i	−0.560316	+	0.431330i
$687$		0			0
$688$	0.471521	+	0.816699i	0.0179766	+	0.0311363i
$689$		0			0
$690$		0			0
$691$	−21.1757	−	12.2258i	−0.805560	−	0.465090i	0.0398517	−	0.999206i	$-0.487311\pi$
$691$	−21.1757	−	12.2258i	−0.805560	−	0.465090i	−0.845412	+	0.534115i	$0.820645\pi$
$692$	16.7710			0.637536
$693$		0			0
$694$	−22.1091			−0.839250
$695$	−11.1431	−	6.43347i	−0.422682	−	0.244035i
$696$		0			0
$697$	−3.89393	−	6.74448i	−0.147493	−	0.255465i
$698$	−7.37173	−	12.7682i	−0.279024	−	0.483284i
$699$		0			0
$700$	−3.48824	+	26.6022i	−0.131843	+	1.00547i
$701$		−	2.21697i		−	0.0837337i	−0.999123	−	0.0418669i	$-0.986669\pi$
$701$		−	2.21697i		−	0.0837337i	0.999123	−	0.0418669i	$-0.0133305\pi$
$702$		0			0
$703$			5.35703i			0.202044i
$704$	3.41614	+	1.97231i	0.128751	+	0.0743342i
$705$		0			0
$706$	−14.9629	+	8.63881i	−0.563135	+	0.325126i
$707$	17.3901	+	13.3296i	0.654021	+	0.501310i
$708$		0			0
$709$	12.1962	−	21.1244i	0.458036	−	0.793342i	−0.540821	−	0.841138i	$-0.681886\pi$
$709$	12.1962	−	21.1244i	0.458036	−	0.793342i	0.998857	+	0.0477959i	$0.0152197\pi$
$710$	−7.56060			−0.283744
$711$		0			0
$712$			9.26646i			0.347275i
$713$	10.9796	−	19.0173i	0.411190	−	0.712203i
$714$		0			0
$715$	21.8607	+	37.8639i	0.817544	+	1.41603i
$716$	5.00158	−	2.88766i	0.186918	−	0.107917i
$717$		0			0
$718$	−5.45647	+	9.45088i	−0.203634	+	0.352704i
$719$	2.22752			0.0830725			0.0415363	−	0.999137i	$-0.486775\pi$
$719$	2.22752			0.0830725			0.0415363	+	0.999137i	$0.486775\pi$
$720$		0			0
$721$	17.2831	+	41.6641i	0.643657	+	1.55165i
$722$	13.6959	+	7.90736i	0.509710	+	0.294281i
$723$		0			0
$724$	4.79180	−	2.76655i	0.178086	−	0.102818i
$725$	−25.3599	+	14.6416i	−0.941844	+	0.543774i
$726$		0			0
$727$	10.4880	+	6.05523i	0.388977	+	0.224576i	0.681717	−	0.731616i	$-0.261234\pi$
$727$	10.4880	+	6.05523i	0.388977	+	0.224576i	−0.292740	+	0.956192i	$0.594567\pi$
$728$	6.96124	−	2.88766i	0.258001	−	0.107024i
$729$		0			0
$730$	−18.8908			−0.699180
$731$	−0.349924	+	0.606086i	−0.0129424	+	0.0224169i
$732$		0			0
$733$	13.5673	−	7.83306i	0.501118	−	0.289321i	−0.228057	−	0.973648i	$-0.573237\pi$
$733$	13.5673	−	7.83306i	0.501118	−	0.289321i	0.729175	+	0.684327i	$0.239904\pi$
$734$	17.8636	+	30.9407i	0.659357	+	1.14204i
$735$		0			0
$736$	−3.12656	+	5.41535i	−0.115246	+	0.199613i
$737$			53.0522i			1.95420i
$738$		0			0
$739$	−8.10454			−0.298130			−0.149065	−	0.988827i	$-0.547626\pi$
$739$	−8.10454			−0.298130			−0.149065	+	0.988827i	$0.547626\pi$
$740$	5.83974	−	10.1147i	0.214673	−	0.371825i
$741$		0			0
$742$		0			0
$743$	10.5429	−	6.08697i	0.386783	−	0.223309i	−0.293982	−	0.955811i	$-0.594981\pi$
$743$	10.5429	−	6.08697i	0.386783	−	0.223309i	0.680765	+	0.732502i	$0.261647\pi$
$744$		0			0
$745$	37.0732	+	21.4042i	1.35826	+	0.784189i
$746$			32.0600i			1.17380i
$747$		0			0
$748$			2.92737i			0.107035i
$749$	37.6100	+	4.93164i	1.37424	+	0.180198i
$750$		0			0
$751$	−17.3062	−	29.9752i	−0.631511	−	1.09381i	−0.987243	−	0.159221i	$-0.949102\pi$
$751$	−17.3062	−	29.9752i	−0.631511	−	1.09381i	0.355732	−	0.934588i	$-0.384232\pi$
$752$	−1.09263	−	1.89248i	−0.0398440	−	0.0690118i
$753$		0			0
$754$	7.12348	+	4.11274i	0.259422	+	0.149777i
$755$	−5.60311			−0.203918
$756$		0			0
$757$	−39.0553			−1.41949			−0.709744	−	0.704459i	$-0.751190\pi$
$757$	−39.0553			−1.41949			−0.709744	+	0.704459i	$0.751190\pi$
$758$	30.2149	+	17.4446i	1.09745	+	0.633615i
$759$		0			0
$760$	3.47231	+	6.01422i	0.125954	+	0.218159i
$761$	−5.11262	−	8.85532i	−0.185332	−	0.321005i	0.758356	−	0.651840i	$-0.226003\pi$
$761$	−5.11262	−	8.85532i	−0.185332	−	0.321005i	−0.943688	+	0.330835i	$0.892670\pi$
$762$		0			0
$763$	3.88010	−	29.5907i	0.140469	−	1.07125i
$764$			6.21372i			0.224805i
$765$		0			0
$766$			17.5342i			0.633537i
$767$	−0.0521363	−	0.0301009i	−0.00188253	−	0.00108688i
$768$		0			0
$769$	−26.6746	+	15.4006i	−0.961910	+	0.555359i	−0.896760	−	0.442517i	$-0.854086\pi$
$769$	−26.6746	+	15.4006i	−0.961910	+	0.555359i	−0.0651494	+	0.997876i	$0.520752\pi$
$770$	32.2305	+	24.7048i	1.16151	+	0.890301i
$771$		0			0
$772$	−3.90271	+	6.75970i	−0.140462	+	0.243287i
$773$	35.7833			1.28704			0.643518	−	0.765431i	$-0.277474\pi$
$773$	35.7833			1.28704			0.643518	+	0.765431i	$0.277474\pi$
$774$		0			0
$775$			35.6117i			1.27921i
$776$	−9.40980	+	16.2983i	−0.337792	+	0.585073i
$777$		0			0
$778$	3.81086	+	6.60060i	0.136626	+	0.236643i
$779$	−16.2200	+	9.36461i	−0.581141	+	0.335522i
$780$		0			0
$781$	−3.83228	+	6.63771i	−0.137130	+	0.237516i
$782$	−4.64054			−0.165945
$783$		0			0
$784$	4.94462	−	4.95487i	0.176594	−	0.176960i
$785$	−55.9626	−	32.3100i	−1.99739	−	1.15319i
$786$		0			0
$787$	13.2859	−	7.67064i	0.473592	−	0.273429i	−0.244150	−	0.969737i	$-0.578509\pi$
$787$	13.2859	−	7.67064i	0.473592	−	0.273429i	0.717742	+	0.696309i	$0.245176\pi$
$788$	−11.0624	+	6.38687i	−0.394081	+	0.227523i
$789$		0			0
$790$	12.2530	+	7.07430i	0.435944	+	0.251692i
$791$	−9.96748	−	24.0284i	−0.354403	−	0.854353i
$792$		0			0
$793$	−7.03326			−0.249758
$794$	−18.8234	+	32.6032i	−0.668019	+	1.15704i
$795$		0			0
$796$	1.56925	−	0.906005i	0.0556205	−	0.0321125i
$797$	17.5200	+	30.3455i	0.620590	+	1.07489i	0.989376	+	0.145379i	$0.0464402\pi$
$797$	17.5200	+	30.3455i	0.620590	+	1.07489i	−0.368786	+	0.929514i	$0.620226\pi$
$798$		0			0
$799$	0.810856	−	1.40444i	0.0286860	−	0.0496857i
$800$		−	10.1408i		−	0.358530i
$801$		0			0
$802$	−21.4415			−0.757124
$803$	−9.57529	+	16.5849i	−0.337905	+	0.585268i
$804$		0			0
$805$	−39.1627	+	51.0926i	−1.38031	+	1.80078i
$806$	8.66299	−	5.00158i	0.305141	−	0.176173i
$807$		0			0
$808$	−7.17206	−	4.14079i	−0.252312	−	0.145673i
$809$			27.2925i			0.959553i	0.877391	+	0.479777i	$0.159282\pi$
$809$			27.2925i			0.959553i	−0.877391	+	0.479777i	$0.840718\pi$
$810$		0			0
$811$			27.7628i			0.974883i	0.873156	+	0.487442i	$0.162070\pi$
$811$			27.7628i			0.974883i	−0.873156	+	0.487442i	$0.837930\pi$
$812$	7.57519	+	0.993303i	0.265837	+	0.0348581i
$813$		0			0
$814$	−5.92004	−	10.2538i	−0.207497	−	0.359396i
$815$	24.1161	+	41.7703i	0.844749	+	1.46315i
$816$		0			0
$817$	1.45759	+	0.841542i	0.0509947	+	0.0294418i
$818$	−29.5703			−1.03390
$819$		0			0
$820$	40.8338			1.42598
$821$	38.4968	+	22.2262i	1.34355	+	0.775698i	0.987326	−	0.158703i	$-0.0507311\pi$
$821$	38.4968	+	22.2262i	1.34355	+	0.775698i	0.356223	+	0.934401i	$0.384064\pi$
$822$		0			0
$823$	25.5577	+	44.2672i	0.890884	+	1.54306i	0.838818	+	0.544413i	$0.183247\pi$
$823$	25.5577	+	44.2672i	0.890884	+	1.54306i	0.0520663	+	0.998644i	$0.483419\pi$
$824$	−8.52435	−	14.7646i	−0.296960	−	0.514349i
$825$		0			0
$826$	−0.0554423	−	0.00726992i	−0.00192908	−	0.000252953i
$827$		−	14.5414i		−	0.505653i	−0.967512	−	0.252826i	$-0.918640\pi$
$827$		−	14.5414i		−	0.505653i	0.967512	−	0.252826i	$-0.0813601\pi$
$828$		0			0
$829$		−	27.9681i		−	0.971373i	−0.874133	−	0.485686i	$-0.838570\pi$
$829$		−	27.9681i		−	0.971373i	0.874133	−	0.485686i	$-0.161430\pi$
$830$	−27.1605	−	15.6811i	−0.942754	−	0.544299i
$831$		0			0
$832$	−2.46687	+	1.42425i	−0.0855233	+	0.0493769i
$833$	5.01920	+	1.33932i	0.173905	+	0.0464047i
$834$		0			0
$835$	22.8053	−	39.5000i	0.789211	−	1.36695i
$836$	7.04011			0.243487
$837$		0			0
$838$		−	7.12962i		−	0.246289i
$839$	0.499354	−	0.864906i	0.0172396	−	0.0298599i	−0.857277	−	0.514856i	$-0.827846\pi$
$839$	0.499354	−	0.864906i	0.0172396	−	0.0298599i	0.874517	+	0.484996i	$0.161179\pi$
$840$		0			0
$841$	−10.3307	−	17.8933i	−0.356231	−	0.617011i
$842$	−4.00115	+	2.31007i	−0.137889	+	0.0796102i
$843$		0			0
$844$	1.88766	−	3.26953i	0.0649760	−	0.112542i
$845$	19.0123			0.654044
$846$		0			0
$847$	11.1439	−	4.62273i	0.382910	−	0.158839i
$848$		0			0
$849$		0			0
$850$	6.51739	−	3.76282i	0.223545	−	0.129064i
$851$	16.2546	−	9.38460i	0.557200	−	0.321700i
$852$		0			0
$853$	8.48739	+	4.90020i	0.290603	+	0.167780i	0.638214	−	0.769859i	$-0.279674\pi$
$853$	8.48739	+	4.90020i	0.290603	+	0.167780i	−0.347611	+	0.937639i	$0.613007\pi$
$854$	−6.03410	+	2.50307i	−0.206483	+	0.0856531i
$855$		0			0
$856$	−14.3369			−0.490026
$857$	−3.85002	+	6.66842i	−0.131514	+	0.227789i	−0.924260	−	0.381763i	$-0.875317\pi$
$857$	−3.85002	+	6.66842i	−0.131514	+	0.227789i	0.792746	+	0.609552i	$0.208651\pi$
$858$		0			0
$859$	−16.4022	+	9.46979i	−0.559634	+	0.323105i	−0.752999	−	0.658022i	$-0.771393\pi$
$859$	−16.4022	+	9.46979i	−0.559634	+	0.323105i	0.193364	+	0.981127i	$0.438060\pi$
$860$	−1.83474	−	3.17787i	−0.0625642	−	0.108364i
$861$		0			0
$862$	2.00385	−	3.47078i	0.0682515	−	0.118215i
$863$			17.4540i			0.594141i	0.954856	+	0.297070i	$0.0960096\pi$
$863$			17.4540i			0.594141i	−0.954856	+	0.297070i	$0.903990\pi$
$864$		0			0
$865$	−65.2578			−2.21883
$866$	14.7063	−	25.4720i	0.499740	−	0.865574i
$867$		0			0
$868$	5.65229	−	7.37411i	0.191851	−	0.250294i
$869$	12.4215	−	7.17157i	0.421371	−	0.243279i
$870$		0			0
$871$	−33.1776	−	19.1551i	−1.12418	−	0.649045i
$872$			11.2800i			0.381988i
$873$		0			0
$874$			11.1602i			0.377498i
$875$	6.88078	−	52.4746i	0.232613	−	1.77397i
$876$		0			0
$877$	−0.196152	−	0.339746i	−0.00662360	−	0.0114724i	0.862695	−	0.505725i	$-0.168775\pi$
$877$	−0.196152	−	0.339746i	−0.00662360	−	0.0114724i	−0.869318	+	0.494253i	$0.835442\pi$
$878$	10.6885	+	18.5130i	0.360718	+	0.624782i
$879$		0			0
$880$	−13.2926	−	7.67448i	−0.448093	−	0.258707i
$881$	−37.0259			−1.24744			−0.623718	−	0.781650i	$-0.714378\pi$
$881$	−37.0259			−1.24744			−0.623718	+	0.781650i	$0.714378\pi$
$882$		0			0
$883$	−29.9586			−1.00819			−0.504094	−	0.863649i	$-0.668174\pi$
$883$	−29.9586			−1.00819			−0.504094	+	0.863649i	$0.668174\pi$
$884$	−1.83070	−	1.05696i	−0.0615732	−	0.0355493i
$885$		0			0
$886$	−2.91693	−	5.05227i	−0.0979962	−	0.169734i
$887$	14.4930	+	25.1026i	0.486626	+	0.842861i	0.999882	−	0.0153745i	$-0.00489405\pi$
$887$	14.4930	+	25.1026i	0.486626	+	0.842861i	−0.513256	+	0.858236i	$0.671561\pi$
$888$		0			0
$889$	1.01290	−	7.72461i	0.0339714	−	0.259075i
$890$		−	36.0569i		−	1.20863i
$891$		0			0
$892$			12.7782i			0.427846i
$893$	−3.37759	−	1.95005i	−0.113027	−	0.0652560i
$894$		0			0
$895$	−19.4617	+	11.2362i	−0.650533	+	0.375586i
$896$	−1.60954	+	2.09985i	−0.0537711	+	0.0701510i
$897$		0			0
$898$	11.2912	−	19.5569i	0.376791	−	0.652621i
$899$	10.1407			0.338211
$900$		0			0
$901$		0			0
$902$	20.6976	−	35.8493i	0.689156	−	1.19365i
$903$		0			0
$904$	4.91614	+	8.51501i	0.163508	+	0.283205i
$905$	−18.6455	+	10.7650i	−0.619796	+	0.357839i
$906$		0			0
$907$	1.94773	−	3.37357i	0.0646733	−	0.112017i	−0.831876	−	0.554962i	$-0.812733\pi$
$907$	1.94773	−	3.37357i	0.0646733	−	0.112017i	0.896549	+	0.442945i	$0.146066\pi$
$908$	19.9822			0.663133
$909$		0			0
$910$	−27.0870	+	11.2362i	−0.897924	+	0.372477i
$911$	−1.32768	−	0.766538i	−0.0439881	−	0.0253966i	0.477845	−	0.878444i	$-0.341418\pi$
$911$	−1.32768	−	0.766538i	−0.0439881	−	0.0253966i	−0.521833	+	0.853048i	$0.674752\pi$
$912$		0			0
$913$	−27.5339	+	15.8967i	−0.911240	+	0.526105i
$914$	−34.5218	+	19.9311i	−1.14188	+	0.659264i
$915$		0			0
$916$	8.77402	+	5.06568i	0.289902	+	0.167375i
$917$	−36.8166	+	15.2723i	−1.21579	+	0.504334i
$918$		0			0
$919$	28.2531			0.931984			0.465992	−	0.884789i	$-0.345698\pi$
$919$	28.2531			0.931984			0.465992	+	0.884789i	$0.345698\pi$
$920$	12.1658	−	21.0718i	0.401094	−	0.694715i
$921$		0			0
$922$	−6.37834	+	3.68254i	−0.210059	+	0.121278i
$923$	−2.76737	−	4.79323i	−0.0910892	−	0.157771i
$924$		0			0
$925$	−15.2192	+	26.3603i	−0.500403	+	0.866723i
$926$		−	28.6914i		−	0.942859i
$927$		0			0
$928$	−2.88766			−0.0947921
$929$	−1.64363	+	2.84685i	−0.0539257	+	0.0934021i	−0.891728	−	0.452571i	$-0.850507\pi$
$929$	−1.64363	+	2.84685i	−0.0539257	+	0.0934021i	0.837802	+	0.545974i	$0.183840\pi$
$930$		0			0
$931$	3.22097	−	12.0708i	0.105563	−	0.395605i
$932$	−6.33070	+	3.65503i	−0.207369	+	0.119725i
$933$		0			0
$934$	−11.8389	−	6.83519i	−0.387380	−	0.223654i
$935$		−	11.3907i		−	0.372517i
$936$		0			0
$937$		−	35.5084i		−	1.16001i	−0.814613	−	0.580005i	$-0.803051\pi$
$937$		−	35.5084i		−	1.16001i	0.814613	−	0.580005i	$-0.196949\pi$
$938$	−35.2814	−	4.62630i	−1.15198	−	0.151054i
$939$		0			0
$940$	4.25153	+	7.36387i	0.138670	+	0.240183i
$941$	−6.24941	−	10.8243i	−0.203725	−	0.352862i	0.746001	−	0.665945i	$-0.231972\pi$
$941$	−6.24941	−	10.8243i	−0.203725	−	0.352862i	−0.949726	+	0.313083i	$0.898638\pi$
$942$		0			0
$943$	56.8293	+	32.8104i	1.85062	+	1.06845i
$944$	0.0211346			0.000687873
$945$		0			0
$946$	−3.71994			−0.120946
$947$	−31.2769	−	18.0577i	−1.01636	−	0.586796i	−0.103313	−	0.994649i	$-0.532944\pi$
$947$	−31.2769	−	18.0577i	−1.01636	−	0.586796i	−0.913048	+	0.407852i	$0.866278\pi$
$948$		0			0
$949$	−6.91452	−	11.9763i	−0.224455	−	0.388767i
$950$	−9.04931	−	15.6739i	−0.293598	−	0.508527i
$951$		0			0
$952$	−1.94679	−	0.255275i	−0.0630959	−	0.00827350i
$953$			45.2925i			1.46717i	0.679599	+	0.733584i	$0.262154\pi$
$953$			45.2925i			1.46717i	−0.679599	+	0.733584i	$0.737846\pi$
$954$		0			0
$955$		−	24.1783i		−	0.782392i
$956$	−7.28317	−	4.20494i	−0.235554	−	0.135997i
$957$		0			0
$958$	9.01596	−	5.20537i	0.291293	−	0.168178i
$959$	25.3019	−	33.0095i	0.817042	−	1.06593i
$960$		0			0
$961$	−9.33386	+	16.1667i	−0.301092	+	0.521507i
$962$	8.54997			0.275662
$963$		0			0
$964$		−	8.95213i		−	0.288329i
$965$	15.1859	−	26.3028i	0.488851	−	0.846716i
$966$		0			0
$967$	12.0000	+	20.7845i	0.385893	+	0.668385i	0.991893	−	0.127079i	$-0.0405602\pi$
$967$	12.0000	+	20.7845i	0.385893	+	0.668385i	−0.606000	+	0.795465i	$0.707227\pi$
$968$	−3.94910	+	2.28001i	−0.126929	+	0.0732824i
$969$		0			0
$970$	36.6146	−	63.4184i	1.17562	−	2.03624i
$971$	33.3626			1.07066			0.535328	−	0.844644i	$-0.320188\pi$
$971$	33.3626			1.07066			0.535328	+	0.844644i	$0.320188\pi$
$972$		0			0
$973$	−3.35222	−	8.08113i	−0.107467	−	0.259069i
$974$	2.02520	+	1.16925i	0.0648917	+	0.0374652i
$975$		0			0
$976$	2.13832	−	1.23456i	0.0684458	−	0.0395172i
$977$	29.8846	−	17.2539i	0.956091	−	0.552000i	0.0611236	−	0.998130i	$-0.480532\pi$
$977$	29.8846	−	17.2539i	0.956091	−	0.552000i	0.894968	+	0.446131i	$0.147198\pi$
$978$		0			0
$979$	−31.6555	−	18.2763i	−1.01172	−	0.584114i
$980$	−19.2401	+	19.2800i	−0.614602	+	0.615876i
$981$		0			0
$982$	33.8844			1.08130
$983$	1.20651	−	2.08973i	0.0384817	−	0.0666522i	−0.846143	−	0.532956i	$-0.821081\pi$
$983$	1.20651	−	2.08973i	0.0384817	−	0.0666522i	0.884625	+	0.466304i	$0.154415\pi$
$984$		0			0
$985$	43.0450	−	24.8521i	1.37153	−	0.791852i
$986$	−1.07149	−	1.85588i	−0.0341232	−	0.0591032i
$987$		0			0
$988$	−2.54191	+	4.40271i	−0.0808688	+	0.140069i
$989$		−	5.89695i		−	0.187512i
$990$		0			0
$991$	48.5982			1.54377			0.771887	−	0.635760i	$-0.219313\pi$
$991$	48.5982			1.54377			0.771887	+	0.635760i	$0.219313\pi$
$992$	−1.75587	+	3.04125i	−0.0557488	+	0.0965598i
$993$		0			0
$994$	−4.08010	−	3.12741i	−0.129413	−	0.0991955i
$995$	−6.10612	+	3.52537i	−0.193577	+	0.111762i
$996$		0			0
$997$	38.8449	+	22.4271i	1.23023	+	0.710274i	0.967078	−	0.254481i	$-0.0819045\pi$
$997$	38.8449	+	22.4271i	1.23023	+	0.710274i	0.263152	+	0.964754i	$0.415238\pi$
$998$			16.6045i			0.525605i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	378.2.m.a.251.5		16
3.2	odd	2		126.2.m.a.83.1	yes	16
4.3	odd	2		3024.2.cc.b.2897.1		16
7.2	even	3		2646.2.l.b.521.1		16
7.3	odd	6		2646.2.t.a.1979.1		16
7.4	even	3		2646.2.t.a.1979.4		16
7.5	odd	6		2646.2.l.b.521.4		16
7.6	odd	2	inner	378.2.m.a.251.8		16
9.2	odd	6		1134.2.d.a.1133.9		16
9.4	even	3		126.2.m.a.41.4	yes	16
9.5	odd	6	inner	378.2.m.a.125.8		16
9.7	even	3		1134.2.d.a.1133.8		16
12.11	even	2		1008.2.cc.b.209.7		16
21.2	odd	6		882.2.l.a.227.8		16
21.5	even	6		882.2.l.a.227.5		16
21.11	odd	6		882.2.t.b.803.7		16
21.17	even	6		882.2.t.b.803.6		16
21.20	even	2		126.2.m.a.83.4	yes	16
28.27	even	2		3024.2.cc.b.2897.8		16
36.23	even	6		3024.2.cc.b.881.8		16
36.31	odd	6		1008.2.cc.b.545.2		16
63.4	even	3		882.2.l.a.509.1		16
63.5	even	6		2646.2.t.a.2285.4		16
63.13	odd	6		126.2.m.a.41.1	✓	16
63.20	even	6		1134.2.d.a.1133.16		16
63.23	odd	6		2646.2.t.a.2285.1		16
63.31	odd	6		882.2.l.a.509.4		16
63.32	odd	6		2646.2.l.b.1097.8		16
63.34	odd	6		1134.2.d.a.1133.1		16
63.40	odd	6		882.2.t.b.815.7		16
63.41	even	6	inner	378.2.m.a.125.5		16
63.58	even	3		882.2.t.b.815.6		16
63.59	even	6		2646.2.l.b.1097.5		16
84.83	odd	2		1008.2.cc.b.209.2		16
252.139	even	6		1008.2.cc.b.545.7		16
252.167	odd	6		3024.2.cc.b.881.1		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
126.2.m.a.41.1	✓	16	63.13	odd	6
126.2.m.a.41.4	yes	16	9.4	even	3
126.2.m.a.83.1	yes	16	3.2	odd	2
126.2.m.a.83.4	yes	16	21.20	even	2
378.2.m.a.125.5		16	63.41	even	6	inner
378.2.m.a.125.8		16	9.5	odd	6	inner
378.2.m.a.251.5		16	1.1	even	1	trivial
378.2.m.a.251.8		16	7.6	odd	2	inner
882.2.l.a.227.5		16	21.5	even	6
882.2.l.a.227.8		16	21.2	odd	6
882.2.l.a.509.1		16	63.4	even	3
882.2.l.a.509.4		16	63.31	odd	6
882.2.t.b.803.6		16	21.17	even	6
882.2.t.b.803.7		16	21.11	odd	6
882.2.t.b.815.6		16	63.58	even	3
882.2.t.b.815.7		16	63.40	odd	6
1008.2.cc.b.209.2		16	84.83	odd	2
1008.2.cc.b.209.7		16	12.11	even	2
1008.2.cc.b.545.2		16	36.31	odd	6
1008.2.cc.b.545.7		16	252.139	even	6
1134.2.d.a.1133.1		16	63.34	odd	6
1134.2.d.a.1133.8		16	9.7	even	3
1134.2.d.a.1133.9		16	9.2	odd	6
1134.2.d.a.1133.16		16	63.20	even	6
2646.2.l.b.521.1		16	7.2	even	3
2646.2.l.b.521.4		16	7.5	odd	6
2646.2.l.b.1097.5		16	63.59	even	6
2646.2.l.b.1097.8		16	63.32	odd	6
2646.2.t.a.1979.1		16	7.3	odd	6
2646.2.t.a.1979.4		16	7.4	even	3
2646.2.t.a.2285.1		16	63.23	odd	6
2646.2.t.a.2285.4		16	63.5	even	6
3024.2.cc.b.881.1		16	252.167	odd	6
3024.2.cc.b.881.8		16	36.23	even	6
3024.2.cc.b.2897.1		16	4.3	odd	2
3024.2.cc.b.2897.8		16	28.27	even	2

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:	\( N \)	\(=\)	\( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	378.m (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.866025 + 0.500000i) q^{2} +(0.500000 + 0.866025i) q^{4} +(-1.94556 - 3.36980i) q^{5} +(0.343982 - 2.62329i) q^{7} +1.00000i q^{8} +O(q^{10})\)	\(q+(0.866025 + 0.500000i) q^{2} +(0.500000 + 0.866025i) q^{4} +(-1.94556 - 3.36980i) q^{5} +(0.343982 - 2.62329i) q^{7} +1.00000i q^{8} -3.89111i q^{10} +(-3.41614 - 1.97231i) q^{11} +(2.46687 - 1.42425i) q^{13} +(1.60954 - 2.09985i) q^{14} +(-0.500000 + 0.866025i) q^{16} -0.742117 q^{17} +1.78474i q^{19} +(1.94556 - 3.36980i) q^{20} +(-1.97231 - 3.41614i) q^{22} +(5.41535 - 3.12656i) q^{23} +(-5.07039 + 8.78217i) q^{25} +2.84849 q^{26} +(2.44383 - 1.01375i) q^{28} +(2.50079 + 1.44383i) q^{29} +(3.04125 - 1.75587i) q^{31} +(-0.866025 + 0.500000i) q^{32} +(-0.642692 - 0.371058i) q^{34} +(-9.50923 + 3.94462i) q^{35} +3.00158 q^{37} +(-0.892369 + 1.54563i) q^{38} +(3.36980 - 1.94556i) q^{40} +(5.24705 + 9.08816i) q^{41} +(0.471521 - 0.816699i) q^{43} -3.94462i q^{44} +6.25311 q^{46} +(-1.09263 + 1.89248i) q^{47} +(-6.76335 - 1.80473i) q^{49} +(-8.78217 + 5.07039i) q^{50} +(2.46687 + 1.42425i) q^{52} +15.3490i q^{55} +(2.62329 + 0.343982i) q^{56} +(1.44383 + 2.50079i) q^{58} +(-0.0105673 - 0.0183031i) q^{59} +(-2.13832 - 1.23456i) q^{61} +3.51174 q^{62} -1.00000 q^{64} +(-9.59886 - 5.54191i) q^{65} +(-6.72463 - 11.6474i) q^{67} +(-0.371058 - 0.642692i) q^{68} +(-10.2075 - 1.33847i) q^{70} -1.94304i q^{71} -4.85486i q^{73} +(2.59944 + 1.50079i) q^{74} +(-1.54563 + 0.892369i) q^{76} +(-6.34904 + 8.28311i) q^{77} +(-1.81806 + 3.14898i) q^{79} +3.89111 q^{80} +10.4941i q^{82} +(4.02998 - 6.98012i) q^{83} +(1.44383 + 2.50079i) q^{85} +(0.816699 - 0.471521i) q^{86} +(1.97231 - 3.41614i) q^{88} +9.26646 q^{89} +(-2.88766 - 6.96124i) q^{91} +(5.41535 + 3.12656i) q^{92} +(-1.89248 + 1.09263i) q^{94} +(6.01422 - 3.47231i) q^{95} +(16.2983 + 9.40980i) q^{97} +(-4.95487 - 4.94462i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 8 q^{4} + 2 q^{7}+O(q^{10}) \) 16 * q + 8 * q^4 + 2 * q^7	\( 16 q + 8 q^{4} + 2 q^{7} + 12 q^{11} + 6 q^{14} - 8 q^{16} + 48 q^{23} - 8 q^{25} + 4 q^{28} + 12 q^{29} - 8 q^{37} + 4 q^{43} + 24 q^{46} - 8 q^{49} - 60 q^{50} + 6 q^{56} - 12 q^{58} - 16 q^{64} - 84 q^{65} - 28 q^{67} - 36 q^{74} - 78 q^{77} - 4 q^{79} - 12 q^{85} + 24 q^{86} + 24 q^{91} + 48 q^{92} - 12 q^{95}+O(q^{100}) \) 16 * q + 8 * q^4 + 2 * q^7 + 12 * q^11 + 6 * q^14 - 8 * q^16 + 48 * q^23 - 8 * q^25 + 4 * q^28 + 12 * q^29 - 8 * q^37 + 4 * q^43 + 24 * q^46 - 8 * q^49 - 60 * q^50 + 6 * q^56 - 12 * q^58 - 16 * q^64 - 84 * q^65 - 28 * q^67 - 36 * q^74 - 78 * q^77 - 4 * q^79 - 12 * q^85 + 24 * q^86 + 24 * q^91 + 48 * q^92 - 12 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more