LMFDB - Embedded newform 1386.2.r.c.89.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1386,2,Mod(89,1386)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1386, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 5, 0]))

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

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

chi := DirichletCharacter("1386.89");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1386.r (of order $6$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$11.0672657201$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			89.2
Root			$0.965926 + 0.258819i$ of defining polynomial
Character	$\chi$	$=$	1386.89
Dual form			1386.2.r.c.1277.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.866025 - 0.500000i) q^{2} +(0.500000 + 0.866025i) q^{4} +(1.96593 - 3.40508i) q^{5} +(2.19067 + 1.48356i) q^{7} -1.00000i q^{8} +O(q^{10})$	\(q+(-0.866025 - 0.500000i) q^{2} +(0.500000 + 0.866025i) q^{4} +(1.96593 - 3.40508i) q^{5} +(2.19067 + 1.48356i) q^{7} -1.00000i q^{8} +(-3.40508 + 1.96593i) q^{10} +(-0.866025 + 0.500000i) q^{11} -2.44949i q^{13} +(-1.15539 - 2.38014i) q^{14} +(-0.500000 + 0.866025i) q^{16} +(-3.64626 - 6.31552i) q^{17} +(3.17914 + 1.83548i) q^{19} +3.93185 q^{20} +1.00000 q^{22} +(2.94098 + 1.69798i) q^{23} +(-5.22973 - 9.05816i) q^{25} +(-1.22474 + 2.12132i) q^{26} +(-0.189469 + 2.63896i) q^{28} -5.16088i q^{29} +(-1.99465 + 1.15161i) q^{31} +(0.866025 - 0.500000i) q^{32} +7.29253i q^{34} +(9.35835 - 4.54284i) q^{35} +(1.98709 - 3.44174i) q^{37} +(-1.83548 - 3.17914i) q^{38} +(-3.40508 - 1.96593i) q^{40} +5.92480 q^{41} -10.7128 q^{43} +(-0.866025 - 0.500000i) q^{44} +(-1.69798 - 2.94098i) q^{46} +(-4.31199 + 7.46859i) q^{47} +(2.59808 + 6.50000i) q^{49} +10.4595i q^{50} +(2.12132 - 1.22474i) q^{52} +(11.5854 - 6.68885i) q^{53} +3.93185i q^{55} +(1.48356 - 2.19067i) q^{56} +(-2.58044 + 4.46945i) q^{58} +(0.975056 + 1.68885i) q^{59} +(5.16813 + 2.98382i) q^{61} +2.30323 q^{62} -1.00000 q^{64} +(-8.34072 - 4.81552i) q^{65} +(-2.33378 - 4.04222i) q^{67} +(3.64626 - 6.31552i) q^{68} +(-10.3760 - 0.744963i) q^{70} +8.90754i q^{71} +(-2.37466 + 1.37101i) q^{73} +(-3.44174 + 1.98709i) q^{74} +3.67095i q^{76} +(-2.63896 - 0.189469i) q^{77} +(7.69798 - 13.3333i) q^{79} +(1.96593 + 3.40508i) q^{80} +(-5.13103 - 2.96240i) q^{82} -6.05521 q^{83} -28.6731 q^{85} +(9.27755 + 5.35640i) q^{86} +(0.500000 + 0.866025i) q^{88} +(-5.87404 + 10.1741i) q^{89} +(3.63397 - 5.36603i) q^{91} +3.39595i q^{92} +(7.46859 - 4.31199i) q^{94} +(12.4999 - 7.21682i) q^{95} +5.85765i q^{97} +(1.00000 - 6.92820i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{4} + 8 q^{5}+O(q^{10}) $ 8 * q + 4 * q^4 + 8 * q^5	$ 8 q + 4 q^{4} + 8 q^{5} - 4 q^{16} - 4 q^{17} + 24 q^{19} + 16 q^{20} + 8 q^{22} + 24 q^{23} - 4 q^{25} + 12 q^{31} + 16 q^{35} + 12 q^{37} - 8 q^{38} + 16 q^{41} - 32 q^{43} + 8 q^{46} + 48 q^{53} - 4 q^{58} + 16 q^{59} + 24 q^{62} - 8 q^{64} - 12 q^{65} - 24 q^{67} + 4 q^{68} - 20 q^{70} - 24 q^{73} - 12 q^{74} + 40 q^{79} + 8 q^{80} + 12 q^{82} + 72 q^{83} - 32 q^{85} - 24 q^{86} + 4 q^{88} - 16 q^{89} + 36 q^{91} + 24 q^{95} + 8 q^{98}+O(q^{100}) $ 8 * q + 4 * q^4 + 8 * q^5 - 4 * q^16 - 4 * q^17 + 24 * q^19 + 16 * q^20 + 8 * q^22 + 24 * q^23 - 4 * q^25 + 12 * q^31 + 16 * q^35 + 12 * q^37 - 8 * q^38 + 16 * q^41 - 32 * q^43 + 8 * q^46 + 48 * q^53 - 4 * q^58 + 16 * q^59 + 24 * q^62 - 8 * q^64 - 12 * q^65 - 24 * q^67 + 4 * q^68 - 20 * q^70 - 24 * q^73 - 12 * q^74 + 40 * q^79 + 8 * q^80 + 12 * q^82 + 72 * q^83 - 32 * q^85 - 24 * q^86 + 4 * q^88 - 16 * q^89 + 36 * q^91 + 24 * q^95 + 8 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1386\mathbb{Z}\right)^\times$.

$n$	$155$	$199$	$1135$
$\chi(n)$	$-1$	$e\left(\frac{5}{6}\right)$	$1$

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
$5$	1.96593	−	3.40508i	0.879189	−	1.52280i	0.0269561	−	0.999637i	$-0.491419\pi$
$5$	1.96593	−	3.40508i	0.879189	−	1.52280i	0.852233	−	0.523163i	$-0.175248\pi$
$6$		0			0
$7$	2.19067	+	1.48356i	0.827996	+	0.560734i
$7$	2.19067	+	1.48356i	0.827996	+	0.560734i
$8$		−	1.00000i		−	0.353553i
$9$		0			0
$10$	−3.40508	+	1.96593i	−1.07678	+	0.621680i
$11$	−0.866025	+	0.500000i	−0.261116	+	0.150756i
$11$	−0.866025	+	0.500000i	−0.261116	+	0.150756i
$12$		0			0
$13$		−	2.44949i		−	0.679366i	−0.940540	−	0.339683i	$-0.889680\pi$
$13$		−	2.44949i		−	0.679366i	0.940540	−	0.339683i	$-0.110320\pi$
$14$	−1.15539	−	2.38014i	−0.308792	−	0.636119i
$15$		0			0
$16$	−0.500000	+	0.866025i	−0.125000	+	0.216506i
$17$	−3.64626	−	6.31552i	−0.884349	−	1.53174i	−0.846458	−	0.532456i	$-0.821269\pi$
$17$	−3.64626	−	6.31552i	−0.884349	−	1.53174i	−0.0378910	−	0.999282i	$-0.512064\pi$
$18$		0			0
$19$	3.17914	+	1.83548i	0.729344	+	0.421087i	0.818182	−	0.574959i	$-0.194982\pi$
$19$	3.17914	+	1.83548i	0.729344	+	0.421087i	−0.0888382	+	0.996046i	$0.528315\pi$
$20$	3.93185			0.879189
$21$		0			0
$22$	1.00000			0.213201
$23$	2.94098	+	1.69798i	0.613237	+	0.354053i	0.774231	−	0.632903i	$-0.218137\pi$
$23$	2.94098	+	1.69798i	0.613237	+	0.354053i	−0.160994	+	0.986955i	$0.551470\pi$
$24$		0			0
$25$	−5.22973	−	9.05816i	−1.04595	−	1.81163i
$26$	−1.22474	+	2.12132i	−0.240192	+	0.416025i
$27$		0			0
$28$	−0.189469	+	2.63896i	−0.0358062	+	0.498716i
$29$		−	5.16088i		−	0.958351i	−0.877719	−	0.479175i	$-0.840936\pi$
$29$		−	5.16088i		−	0.958351i	0.877719	−	0.479175i	$-0.159064\pi$
$30$		0			0
$31$	−1.99465	+	1.15161i	−0.358250	+	0.206836i	−0.668313	−	0.743880i	$-0.732983\pi$
$31$	−1.99465	+	1.15161i	−0.358250	+	0.206836i	0.310063	+	0.950716i	$0.399650\pi$
$32$	0.866025	−	0.500000i	0.153093	−	0.0883883i
$33$		0			0
$34$			7.29253i			1.25066i
$35$	9.35835	−	4.54284i	1.58185	−	0.767880i
$36$		0			0
$37$	1.98709	−	3.44174i	0.326675	−	0.565818i	−0.655175	−	0.755477i	$-0.727405\pi$
$37$	1.98709	−	3.44174i	0.326675	−	0.565818i	0.981850	+	0.189659i	$0.0607383\pi$
$38$	−1.83548	−	3.17914i	−0.297753	−	0.515724i
$39$		0			0
$40$	−3.40508	−	1.96593i	−0.538391	−	0.310840i
$41$	5.92480			0.925299			0.462649	−	0.886541i	$-0.346899\pi$
$41$	5.92480			0.925299			0.462649	+	0.886541i	$0.346899\pi$
$42$		0			0
$43$	−10.7128			−1.63369			−0.816843	−	0.576860i	$-0.804278\pi$
$43$	−10.7128			−1.63369			−0.816843	+	0.576860i	$0.804278\pi$
$44$	−0.866025	−	0.500000i	−0.130558	−	0.0753778i
$45$		0			0
$46$	−1.69798	−	2.94098i	−0.250353	−	0.433624i
$47$	−4.31199	+	7.46859i	−0.628969	+	1.08941i	0.358791	+	0.933418i	$0.383189\pi$
$47$	−4.31199	+	7.46859i	−0.628969	+	1.08941i	−0.987759	+	0.155987i	$0.950144\pi$
$48$		0			0
$49$	2.59808	+	6.50000i	0.371154	+	0.928571i
$50$			10.4595i			1.47919i
$51$		0			0
$52$	2.12132	−	1.22474i	0.294174	−	0.169842i
$53$	11.5854	−	6.68885i	1.59138	−	0.918784i	0.598309	−	0.801265i	$-0.295839\pi$
$53$	11.5854	−	6.68885i	1.59138	−	0.918784i	0.993071	−	0.117519i	$-0.0374940\pi$
$54$		0			0
$55$			3.93185i			0.530171i
$56$	1.48356	−	2.19067i	0.198250	−	0.292741i
$57$		0			0
$58$	−2.58044	+	4.46945i	−0.338828	+	0.586868i
$59$	0.975056	+	1.68885i	0.126941	+	0.219869i	0.922490	−	0.386021i	$-0.126151\pi$
$59$	0.975056	+	1.68885i	0.126941	+	0.219869i	−0.795549	+	0.605890i	$0.792817\pi$
$60$		0			0
$61$	5.16813	+	2.98382i	0.661711	+	0.382039i	0.792929	−	0.609314i	$-0.208555\pi$
$61$	5.16813	+	2.98382i	0.661711	+	0.382039i	−0.131217	+	0.991354i	$0.541889\pi$
$62$	2.30323			0.292510
$63$		0			0
$64$	−1.00000			−0.125000
$65$	−8.34072	−	4.81552i	−1.03454	−	0.597291i
$66$		0			0
$67$	−2.33378	−	4.04222i	−0.285116	−	0.493835i	0.687521	−	0.726164i	$-0.258699\pi$
$67$	−2.33378	−	4.04222i	−0.285116	−	0.493835i	−0.972637	+	0.232329i	$0.925365\pi$
$68$	3.64626	−	6.31552i	0.442175	−	0.765869i
$69$		0			0
$70$	−10.3760	−	0.744963i	−1.24017	−	0.0890401i
$71$			8.90754i			1.05713i	0.848893	+	0.528565i	$0.177270\pi$
$71$			8.90754i			1.05713i	−0.848893	+	0.528565i	$0.822730\pi$
$72$		0			0
$73$	−2.37466	+	1.37101i	−0.277933	+	0.160464i	−0.632487	−	0.774571i	$-0.717966\pi$
$73$	−2.37466	+	1.37101i	−0.277933	+	0.160464i	0.354554	+	0.935035i	$0.384633\pi$
$74$	−3.44174	+	1.98709i	−0.400094	+	0.230994i
$75$		0			0
$76$			3.67095i			0.421087i
$77$	−2.63896	−	0.189469i	−0.300737	−	0.0215920i
$78$		0			0
$79$	7.69798	−	13.3333i	0.866090	−	1.50011i	0.000129281	−	1.00000i	$-0.499959\pi$
$79$	7.69798	−	13.3333i	0.866090	−	1.50011i	0.865961	−	0.500112i	$-0.166708\pi$
$80$	1.96593	+	3.40508i	0.219797	+	0.380700i
$81$		0			0
$82$	−5.13103	−	2.96240i	−0.566628	−	0.327143i
$83$	−6.05521			−0.664646			−0.332323	−	0.943166i	$-0.607832\pi$
$83$	−6.05521			−0.664646			−0.332323	+	0.943166i	$0.607832\pi$
$84$		0			0
$85$	−28.6731			−3.11004
$86$	9.27755	+	5.35640i	1.00042	+	0.577595i
$87$		0			0
$88$	0.500000	+	0.866025i	0.0533002	+	0.0923186i
$89$	−5.87404	+	10.1741i	−0.622647	+	1.07846i	0.366344	+	0.930479i	$0.380609\pi$
$89$	−5.87404	+	10.1741i	−0.622647	+	1.07846i	−0.988991	+	0.147976i	$0.952724\pi$
$90$		0			0
$91$	3.63397	−	5.36603i	0.380944	−	0.562512i
$92$			3.39595i			0.354053i
$93$		0			0
$94$	7.46859	−	4.31199i	0.770326	−	0.444748i
$95$	12.4999	−	7.21682i	1.28246	−	0.740430i
$96$		0			0
$97$			5.85765i			0.594754i	0.954760	+	0.297377i	$0.0961119\pi$
$97$			5.85765i			0.594754i	−0.954760	+	0.297377i	$0.903888\pi$
$98$	1.00000	−	6.92820i	0.101015	−	0.699854i
$99$		0			0
$100$	5.22973	−	9.05816i	0.522973	−	0.905816i
$101$	−7.76028	−	13.4412i	−0.772177	−	1.33745i	−0.936368	−	0.351021i	$-0.885835\pi$
$101$	−7.76028	−	13.4412i	−0.772177	−	1.33745i	0.164191	−	0.986429i	$-0.447499\pi$
$102$		0			0
$103$	10.7413	+	6.20150i	1.05837	+	0.611052i	0.924982	−	0.380011i	$-0.124080\pi$
$103$	10.7413	+	6.20150i	1.05837	+	0.611052i	0.133391	+	0.991063i	$0.457413\pi$
$104$	−2.44949			−0.240192
$105$		0			0
$106$	−13.3777			−1.29936
$107$	−11.9441	−	6.89595i	−1.15468	−	0.666657i	−0.204660	−	0.978833i	$-0.565609\pi$
$107$	−11.9441	−	6.89595i	−1.15468	−	0.666657i	−0.950024	+	0.312176i	$0.898942\pi$
$108$		0			0
$109$	−2.45534	−	4.25277i	−0.235179	−	0.407341i	0.724146	−	0.689647i	$-0.242234\pi$
$109$	−2.45534	−	4.25277i	−0.235179	−	0.407341i	−0.959325	+	0.282306i	$0.908901\pi$
$110$	1.96593	−	3.40508i	0.187444	−	0.324662i
$111$		0			0
$112$	−2.38014	+	1.15539i	−0.224902	+	0.109175i
$113$		−	16.1270i		−	1.51710i	−0.651614	−	0.758551i	$-0.725908\pi$
$113$		−	16.1270i		−	1.51710i	0.651614	−	0.758551i	$-0.274092\pi$
$114$		0			0
$115$	11.5635	−	6.67619i	1.07830	−	0.622558i
$116$	4.46945	−	2.58044i	0.414978	−	0.239588i
$117$		0			0
$118$		−	1.95011i		−	0.179522i
$119$	1.38171	−	19.2447i	0.126661	−	1.76416i
$120$		0			0
$121$	0.500000	−	0.866025i	0.0454545	−	0.0787296i
$122$	−2.98382	−	5.16813i	−0.270143	−	0.467901i
$123$		0			0
$124$	−1.99465	−	1.15161i	−0.179125	−	0.103418i
$125$	−21.4658			−1.91996
$126$		0			0
$127$	9.94522			0.882495			0.441248	−	0.897385i	$-0.354536\pi$
$127$	9.94522			0.882495			0.441248	+	0.897385i	$0.354536\pi$
$128$	0.866025	+	0.500000i	0.0765466	+	0.0441942i
$129$		0			0
$130$	4.81552	+	8.34072i	0.422349	+	0.731529i
$131$	9.24264	−	16.0087i	0.807533	−	1.39869i	−0.107034	−	0.994255i	$-0.534135\pi$
$131$	9.24264	−	16.0087i	0.807533	−	1.39869i	0.914567	−	0.404433i	$-0.132531\pi$
$132$		0			0
$133$	4.24140	+	8.73737i	0.367776	+	0.757626i
$134$			4.66755i			0.403215i
$135$		0			0
$136$	−6.31552	+	3.64626i	−0.541551	+	0.312665i
$137$	−2.41786	+	1.39595i	−0.206572	+	0.119264i	−0.599717	−	0.800212i	$-0.704720\pi$
$137$	−2.41786	+	1.39595i	−0.206572	+	0.119264i	0.393145	+	0.919476i	$0.371387\pi$
$138$		0			0
$139$		−	5.10583i		−	0.433071i	−0.976275	−	0.216535i	$-0.930524\pi$
$139$		−	5.10583i		−	0.433071i	0.976275	−	0.216535i	$-0.0694757\pi$
$140$	8.61339	+	5.83315i	0.727965	+	0.492991i
$141$		0			0
$142$	4.45377	−	7.71415i	0.373752	−	0.647358i
$143$	1.22474	+	2.12132i	0.102418	+	0.177394i
$144$		0			0
$145$	−17.5732	−	10.1459i	−1.45938	−	0.842571i
$146$	2.74202			0.226931
$147$		0			0
$148$	3.97418			0.326675
$149$	5.93596	+	3.42713i	0.486292	+	0.280761i	0.723035	−	0.690811i	$-0.242746\pi$
$149$	5.93596	+	3.42713i	0.486292	+	0.280761i	−0.236743	+	0.971572i	$0.576080\pi$
$150$		0			0
$151$	7.23351	+	12.5288i	0.588655	+	1.01958i	0.994409	+	0.105598i	$0.0336756\pi$
$151$	7.23351	+	12.5288i	0.588655	+	1.01958i	−0.405754	+	0.913982i	$0.632991\pi$
$152$	1.83548	−	3.17914i	0.148877	−	0.257862i
$153$		0			0
$154$	2.19067	+	1.48356i	0.176529	+	0.119549i
$155$			9.05594i			0.727391i
$156$		0			0
$157$	14.4305	−	8.33145i	1.15168	−	0.664922i	0.202383	−	0.979306i	$-0.435131\pi$
$157$	14.4305	−	8.33145i	1.15168	−	0.664922i	0.949296	+	0.314384i	$0.101798\pi$
$158$	−13.3333	+	7.69798i	−1.06074	+	0.612418i
$159$		0			0
$160$		−	3.93185i		−	0.310840i
$161$	3.92367	+	8.08284i	0.309228	+	0.637017i
$162$		0			0
$163$	−0.481740	+	0.834398i	−0.0377328	+	0.0653551i	−0.884275	−	0.466966i	$-0.845347\pi$
$163$	−0.481740	+	0.834398i	−0.0377328	+	0.0653551i	0.846542	+	0.532321i	$0.178680\pi$
$164$	2.96240	+	5.13103i	0.231325	+	0.400666i
$165$		0			0
$166$	5.24397	+	3.02761i	0.407011	+	0.234988i
$167$	−1.93426			−0.149677			−0.0748386	−	0.997196i	$-0.523844\pi$
$167$	−1.93426			−0.149677			−0.0748386	+	0.997196i	$0.523844\pi$
$168$		0			0
$169$	7.00000			0.538462
$170$	24.8317	+	14.3366i	1.90450	+	1.09956i
$171$		0			0
$172$	−5.35640	−	9.27755i	−0.408421	−	0.707407i
$173$	−9.87902	+	17.1110i	−0.751088	+	1.30092i	0.196208	+	0.980562i	$0.437137\pi$
$173$	−9.87902	+	17.1110i	−0.751088	+	1.30092i	−0.947296	+	0.320360i	$0.896196\pi$
$174$		0			0
$175$	1.98174	−	27.6021i	0.149805	−	2.08652i
$176$		−	1.00000i		−	0.0753778i
$177$		0			0
$178$	10.1741	−	5.87404i	0.762583	−	0.440278i
$179$	−9.13593	+	5.27463i	−0.682852	+	0.394245i	−0.800929	−	0.598760i	$-0.795660\pi$
$179$	−9.13593	+	5.27463i	−0.682852	+	0.394245i	0.118077	+	0.993004i	$0.462327\pi$
$180$		0			0
$181$			4.12560i			0.306653i	0.988176	+	0.153327i	$0.0489987\pi$
$181$			4.12560i			0.306653i	−0.988176	+	0.153327i	$0.951001\pi$
$182$	−5.83013	+	2.83013i	−0.432158	+	0.209783i
$183$		0			0
$184$	1.69798	−	2.94098i	0.125176	−	0.216812i
$185$	−7.81294	−	13.5324i	−0.574418	−	0.994922i
$186$		0			0
$187$	6.31552	+	3.64626i	0.461836	+	0.266641i
$188$	−8.62398			−0.628969
$189$		0			0
$190$	−14.4336			−1.04713
$191$	−19.9354	−	11.5097i	−1.44248	−	0.832813i	−0.444461	−	0.895798i	$-0.646605\pi$
$191$	−19.9354	−	11.5097i	−1.44248	−	0.832813i	−0.998014	+	0.0629850i	$0.979938\pi$
$192$		0			0
$193$	−4.60708	−	7.97970i	−0.331625	−	0.574391i	0.651206	−	0.758901i	$-0.274264\pi$
$193$	−4.60708	−	7.97970i	−0.331625	−	0.574391i	−0.982831	+	0.184510i	$0.940930\pi$
$194$	2.92883	−	5.07287i	0.210277	−	0.364211i
$195$		0			0
$196$	−4.33013	+	5.50000i	−0.309295	+	0.392857i
$197$			6.52180i			0.464659i	0.972637	+	0.232330i	$0.0746348\pi$
$197$			6.52180i			0.464659i	−0.972637	+	0.232330i	$0.925365\pi$
$198$		0			0
$199$	−12.9813	+	7.49473i	−0.920217	+	0.531288i	−0.883704	−	0.468046i	$-0.844958\pi$
$199$	−12.9813	+	7.49473i	−0.920217	+	0.531288i	−0.0365128	+	0.999333i	$0.511625\pi$
$200$	−9.05816	+	5.22973i	−0.640508	+	0.369798i
$201$		0			0
$202$			15.5206i			1.09202i
$203$	7.65649	−	11.3058i	0.537380	−	0.793510i
$204$		0			0
$205$	11.6477	−	20.1744i	0.813512	−	1.40904i
$206$	−6.20150	−	10.7413i	−0.432079	−	0.748383i
$207$		0			0
$208$	2.12132	+	1.22474i	0.147087	+	0.0849208i
$209$	−3.67095			−0.253925
$210$		0			0
$211$	−9.10759			−0.626992			−0.313496	−	0.949589i	$-0.601500\pi$
$211$	−9.10759			−0.626992			−0.313496	+	0.949589i	$0.601500\pi$
$212$	11.5854	+	6.68885i	0.795690	+	0.459392i
$213$		0			0
$214$	6.89595	+	11.9441i	0.471398	+	0.816485i
$215$	−21.0606	+	36.4780i	−1.43632	+	2.48778i
$216$		0			0
$217$	−6.07812	−	0.436389i	−0.412609	−	0.0296240i
$218$			4.91067i			0.332593i
$219$		0			0
$220$	−3.40508	+	1.96593i	−0.229571	+	0.132543i
$221$	−15.4698	+	8.93149i	−1.04061	+	0.600797i
$222$		0			0
$223$		−	7.28788i		−	0.488033i	−0.969771	−	0.244016i	$-0.921535\pi$
$223$		−	7.28788i		−	0.488033i	0.969771	−	0.244016i	$-0.0784651\pi$
$224$	2.63896	+	0.189469i	0.176323	+	0.0126594i
$225$		0			0
$226$	−8.06350	+	13.9664i	−0.536376	+	0.929031i
$227$	−0.465428	−	0.806145i	−0.0308915	−	0.0535057i	0.850166	−	0.526514i	$-0.176501\pi$
$227$	−0.465428	−	0.806145i	−0.0308915	−	0.0535057i	−0.881058	+	0.473009i	$0.843168\pi$
$228$		0			0
$229$	20.8048	+	12.0117i	1.37482	+	0.793753i	0.991530	−	0.129874i	$-0.0414574\pi$
$229$	20.8048	+	12.0117i	1.37482	+	0.793753i	0.383291	+	0.923628i	$0.374791\pi$
$230$	−13.3524			−0.880430
$231$		0			0
$232$	−5.16088			−0.338828
$233$	15.1427	+	8.74264i	0.992031	+	0.572749i	0.905881	−	0.423533i	$-0.139210\pi$
$233$	15.1427	+	8.74264i	0.992031	+	0.572749i	0.0861503	+	0.996282i	$0.472543\pi$
$234$		0			0
$235$	16.9541	+	29.3654i	1.10596	+	1.91559i
$236$	−0.975056	+	1.68885i	−0.0634707	+	0.109935i
$237$		0			0
$238$	−10.8189	+	15.9755i	−0.701287	+	1.03554i
$239$			0.712115i			0.0460629i	0.999735	+	0.0230315i	$0.00733179\pi$
$239$			0.712115i			0.0460629i	−0.999735	+	0.0230315i	$0.992668\pi$
$240$		0			0
$241$	−13.5097	+	7.79983i	−0.870237	+	0.502432i	−0.867427	−	0.497565i	$-0.834228\pi$
$241$	−13.5097	+	7.79983i	−0.870237	+	0.502432i	−0.00280996	+	0.999996i	$0.500894\pi$
$242$	−0.866025	+	0.500000i	−0.0556702	+	0.0321412i
$243$		0			0
$244$			5.96764i			0.382039i
$245$	27.2407	+	3.93185i	1.74034	+	0.251197i
$246$		0			0
$247$	4.49598	−	7.78726i	0.286072	−	0.495492i
$248$	1.15161	+	1.99465i	0.0731275	+	0.126661i
$249$		0			0
$250$	18.5899	+	10.7329i	1.17573	+	0.678807i
$251$	22.2326			1.40331			0.701655	−	0.712516i	$-0.252445\pi$
$251$	22.2326			1.40331			0.701655	+	0.712516i	$0.252445\pi$
$252$		0			0
$253$	−3.39595			−0.213502
$254$	−8.61281	−	4.97261i	−0.540416	−	0.312009i
$255$		0			0
$256$	−0.500000	−	0.866025i	−0.0312500	−	0.0541266i
$257$	−1.47443	+	2.55379i	−0.0919727	+	0.159301i	−0.908341	−	0.418230i	$-0.862651\pi$
$257$	−1.47443	+	2.55379i	−0.0919727	+	0.159301i	0.816368	+	0.577531i	$0.195984\pi$
$258$		0			0
$259$	9.45909	−	4.59174i	0.587759	−	0.285317i
$260$		−	9.63103i		−	0.597291i
$261$		0			0
$262$	−16.0087	+	9.24264i	−0.989022	+	0.571012i
$263$	−16.4066	+	9.47237i	−1.01168	+	0.584091i	−0.911682	−	0.410897i	$-0.865216\pi$
$263$	−16.4066	+	9.47237i	−1.01168	+	0.584091i	−0.0999939	+	0.994988i	$0.531882\pi$
$264$		0			0
$265$		−	52.5991i		−	3.23114i
$266$	0.695530	−	9.68749i	0.0426457	−	0.593978i
$267$		0			0
$268$	2.33378	−	4.04222i	0.142558	−	0.246918i
$269$	12.1165	+	20.9863i	0.738754	+	1.27956i	0.953057	+	0.302792i	$0.0979189\pi$
$269$	12.1165	+	20.9863i	0.738754	+	1.27956i	−0.214302	+	0.976767i	$0.568748\pi$
$270$		0			0
$271$	19.4275	+	11.2165i	1.18013	+	0.681351i	0.956046	−	0.293218i	$-0.0947261\pi$
$271$	19.4275	+	11.2165i	1.18013	+	0.681351i	0.224089	+	0.974569i	$0.428059\pi$
$272$	7.29253			0.442175
$273$		0			0
$274$	2.79191			0.168665
$275$	9.05816	+	5.22973i	0.546227	+	0.315365i
$276$		0			0
$277$	4.92392	+	8.52849i	0.295850	+	0.512427i	0.975182	−	0.221404i	$-0.0710638\pi$
$277$	4.92392	+	8.52849i	0.295850	+	0.512427i	−0.679332	+	0.733831i	$0.737731\pi$
$278$	−2.55291	+	4.42178i	−0.153114	+	0.265201i
$279$		0			0
$280$	−4.54284	−	9.35835i	−0.271487	−	0.559269i
$281$			10.8623i			0.647990i	0.946059	+	0.323995i	$0.105026\pi$
$281$			10.8623i			0.647990i	−0.946059	+	0.323995i	$0.894974\pi$
$282$		0			0
$283$	−8.32792	+	4.80813i	−0.495044	+	0.285814i	−0.726664	−	0.686993i	$-0.758930\pi$
$283$	−8.32792	+	4.80813i	−0.495044	+	0.285814i	0.231621	+	0.972806i	$0.425597\pi$
$284$	−7.71415	+	4.45377i	−0.457751	+	0.264283i
$285$		0			0
$286$		−	2.44949i		−	0.144841i
$287$	12.9793	+	8.78982i	0.766143	+	0.518847i
$288$		0			0
$289$	−18.0905	+	31.3336i	−1.06415	+	1.84316i
$290$	10.1459	+	17.5732i	0.595788	+	1.03193i
$291$		0			0
$292$	−2.37466	−	1.37101i	−0.138966	−	0.0802322i
$293$	−14.4546			−0.844450			−0.422225	−	0.906491i	$-0.638751\pi$
$293$	−14.4546			−0.844450			−0.422225	+	0.906491i	$0.638751\pi$
$294$		0			0
$295$	7.66755			0.446422
$296$	−3.44174	−	1.98709i	−0.200047	−	0.115497i
$297$		0			0
$298$	−3.42713	−	5.93596i	−0.198528	−	0.343861i
$299$	4.15918	−	7.20390i	0.240531	−	0.416613i
$300$		0			0
$301$	−23.4682	−	15.8931i	−1.35268	−	0.916064i
$302$		−	14.4670i		−	0.832484i
$303$		0			0
$304$	−3.17914	+	1.83548i	−0.182336	+	0.105272i
$305$	20.3203	−	11.7319i	1.16354	−	0.671769i
$306$		0			0
$307$		−	0.928932i		−	0.0530170i	−0.999649	−	0.0265085i	$-0.991561\pi$
$307$		−	0.928932i		−	0.0530170i	0.999649	−	0.0265085i	$-0.00843890\pi$
$308$	−1.15539	−	2.38014i	−0.0658347	−	0.135621i
$309$		0			0
$310$	4.52797	−	7.84267i	0.257171	−	0.445434i
$311$	13.3841	+	23.1819i	0.758940	+	1.31452i	0.943392	+	0.331681i	$0.107616\pi$
$311$	13.3841	+	23.1819i	0.758940	+	1.31452i	−0.184451	+	0.982842i	$0.559051\pi$
$312$		0			0
$313$	6.88437	+	3.97469i	0.389127	+	0.224663i	0.681782	−	0.731555i	$-0.261205\pi$
$313$	6.88437	+	3.97469i	0.389127	+	0.224663i	−0.292655	+	0.956218i	$0.594539\pi$
$314$	−16.6629			−0.940342
$315$		0			0
$316$	15.3960			0.866090
$317$	14.4394	+	8.33657i	0.810995	+	0.468228i	0.847301	−	0.531112i	$-0.178226\pi$
$317$	14.4394	+	8.33657i	0.810995	+	0.468228i	−0.0363060	+	0.999341i	$0.511559\pi$
$318$		0			0
$319$	2.58044	+	4.46945i	0.144477	+	0.250241i
$320$	−1.96593	+	3.40508i	−0.109899	+	0.190350i
$321$		0			0
$322$	0.643427	−	8.96178i	0.0358568	−	0.499420i
$323$		−	26.7705i		−	1.48955i
$324$		0			0
$325$	−22.1879	+	12.8102i	−1.23076	+	0.710580i
$326$	0.834398	−	0.481740i	0.0462130	−	0.0266811i
$327$		0			0
$328$		−	5.92480i		−	0.327143i
$329$	−20.5263	+	9.96410i	−1.13165	+	0.549339i
$330$		0			0
$331$	1.58933	−	2.75280i	0.0873574	−	0.151307i	−0.819036	−	0.573742i	$-0.805491\pi$
$331$	1.58933	−	2.75280i	0.0873574	−	0.151307i	0.906393	+	0.422435i	$0.138824\pi$
$332$	−3.02761	−	5.24397i	−0.166161	−	0.287800i
$333$		0			0
$334$	1.67511	+	0.967128i	0.0916582	+	0.0529189i
$335$	−18.3521			−1.00268
$336$		0			0
$337$	15.7473			0.857812			0.428906	−	0.903349i	$-0.358899\pi$
$337$	15.7473			0.857812			0.428906	+	0.903349i	$0.358899\pi$
$338$	−6.06218	−	3.50000i	−0.329739	−	0.190375i
$339$		0			0
$340$	−14.3366	−	24.8317i	−0.777510	−	1.34669i
$341$	1.15161	−	1.99465i	0.0623633	−	0.108016i
$342$		0			0
$343$	−3.95164	+	18.0938i	−0.213368	+	0.976972i
$344$			10.7128i			0.577595i
$345$		0			0
$346$	17.1110	−	9.87902i	0.919891	−	0.531099i
$347$	22.7976	−	13.1622i	1.22384	−	0.706584i	0.258105	−	0.966117i	$-0.416902\pi$
$347$	22.7976	−	13.1622i	1.22384	−	0.706584i	0.965734	+	0.259533i	$0.0835686\pi$
$348$		0			0
$349$			32.9044i			1.76133i	0.473737	+	0.880666i	$0.342905\pi$
$349$			32.9044i			1.76133i	−0.473737	+	0.880666i	$0.657095\pi$
$350$	−15.5173	+	22.9132i	−0.829433	+	1.22476i
$351$		0			0
$352$	−0.500000	+	0.866025i	−0.0266501	+	0.0461593i
$353$	4.74943	+	8.22626i	0.252787	+	0.437839i	0.964292	−	0.264841i	$-0.0853196\pi$
$353$	4.74943	+	8.22626i	0.252787	+	0.437839i	−0.711505	+	0.702681i	$0.751986\pi$
$354$		0			0
$355$	30.3309	+	17.5116i	1.60980	+	0.929417i
$356$	−11.7481			−0.622647
$357$		0			0
$358$	10.5493			0.557546
$359$	−2.59578	−	1.49867i	−0.137000	−	0.0790970i	0.429933	−	0.902861i	$-0.358537\pi$
$359$	−2.59578	−	1.49867i	−0.137000	−	0.0790970i	−0.566933	+	0.823764i	$0.691870\pi$
$360$		0			0
$361$	−2.76206	−	4.78403i	−0.145372	−	0.251791i
$362$	2.06280	−	3.57287i	0.108418	−	0.187786i
$363$		0			0
$364$	6.46410	+	0.464102i	0.338811	+	0.0243255i
$365$			10.7812i			0.564314i
$366$		0			0
$367$	14.3336	−	8.27551i	0.748208	−	0.431978i	−0.0768379	−	0.997044i	$-0.524482\pi$
$367$	14.3336	−	8.27551i	0.748208	−	0.431978i	0.825046	+	0.565065i	$0.191149\pi$
$368$	−2.94098	+	1.69798i	−0.153309	+	0.0885132i
$369$		0			0
$370$			15.6259i			0.812350i
$371$	35.3032	+	2.53465i	1.83285	+	0.131593i
$372$		0			0
$373$	9.73753	−	16.8659i	0.504190	−	0.873283i	−0.495798	−	0.868438i	$-0.665124\pi$
$373$	9.73753	−	16.8659i	0.504190	−	0.873283i	0.999988	−	0.00484535i	$-0.00154233\pi$
$374$	−3.64626	−	6.31552i	−0.188544	−	0.326568i
$375$		0			0
$376$	7.46859	+	4.31199i	0.385163	+	0.222374i
$377$	−12.6415			−0.651071
$378$		0			0
$379$	2.20809			0.113422			0.0567111	−	0.998391i	$-0.481939\pi$
$379$	2.20809			0.113422			0.0567111	+	0.998391i	$0.481939\pi$
$380$	12.4999	+	7.21682i	0.641231	+	0.370215i
$381$		0			0
$382$	11.5097	+	19.9354i	0.588888	+	1.01998i
$383$	−1.41849	+	2.45690i	−0.0724816	+	0.125542i	−0.899988	−	0.435914i	$-0.856425\pi$
$383$	−1.41849	+	2.45690i	−0.0724816	+	0.125542i	0.827507	+	0.561456i	$0.189759\pi$
$384$		0			0
$385$	−5.83315	+	8.61339i	−0.297285	+	0.438979i
$386$			9.21416i			0.468989i
$387$		0			0
$388$	−5.07287	+	2.92883i	−0.257536	+	0.148689i
$389$	16.4686	−	9.50814i	0.834991	−	0.482082i	−0.0205678	−	0.999788i	$-0.506547\pi$
$389$	16.4686	−	9.50814i	0.834991	−	0.482082i	0.855558	+	0.517706i	$0.173214\pi$
$390$		0			0
$391$		−	24.7651i		−	1.25242i
$392$	6.50000	−	2.59808i	0.328300	−	0.131223i
$393$		0			0
$394$	3.26090	−	5.64805i	0.164282	−	0.284544i
$395$	−30.2673	−	52.4245i	−1.52291	−	2.63776i
$396$		0			0
$397$	9.23729	+	5.33315i	0.463606	+	0.267663i	0.713559	−	0.700595i	$-0.247082\pi$
$397$	9.23729	+	5.33315i	0.463606	+	0.267663i	−0.249953	+	0.968258i	$0.580415\pi$
$398$	14.9895			0.751354
$399$		0			0
$400$	10.4595			0.522973
$401$	28.4216	+	16.4092i	1.41931	+	0.819437i	0.996238	−	0.0866614i	$-0.0276198\pi$
$401$	28.4216	+	16.4092i	1.41931	+	0.819437i	0.423068	+	0.906098i	$0.360953\pi$
$402$		0			0
$403$	2.82086	+	4.88588i	0.140517	+	0.243383i
$404$	7.76028	−	13.4412i	0.386088	−	0.668725i
$405$		0			0
$406$	−12.2836	+	5.96285i	−0.609625	+	0.295931i
$407$			3.97418i			0.196993i
$408$		0			0
$409$	−6.25334	+	3.61037i	−0.309208	+	0.178521i	−0.646572	−	0.762853i	$-0.723798\pi$
$409$	−6.25334	+	3.61037i	−0.309208	+	0.178521i	0.337364	+	0.941374i	$0.390465\pi$
$410$	−20.1744	+	11.6477i	−0.996345	+	0.575240i
$411$		0			0
$412$			12.4030i			0.611052i
$413$	−0.369485	+	5.14626i	−0.0181812	+	0.253231i
$414$		0			0
$415$	−11.9041	+	20.6185i	−0.584349	+	1.01212i
$416$	−1.22474	−	2.12132i	−0.0600481	−	0.104006i
$417$		0			0
$418$	3.17914	+	1.83548i	0.155497	+	0.0897760i
$419$	−15.9184			−0.777665			−0.388832	−	0.921309i	$-0.627121\pi$
$419$	−15.9184			−0.777665			−0.388832	+	0.921309i	$0.627121\pi$
$420$		0			0
$421$	4.23889			0.206591			0.103295	−	0.994651i	$-0.467061\pi$
$421$	4.23889			0.206591			0.103295	+	0.994651i	$0.467061\pi$
$422$	7.88740	+	4.55379i	0.383953	+	0.221675i
$423$		0			0
$424$	−6.68885	−	11.5854i	−0.324839	−	0.562638i
$425$	−38.1379	+	66.0569i	−1.84996	+	3.20423i
$426$		0			0
$427$	6.89498	+	14.2038i	0.333672	+	0.687371i
$428$		−	13.7919i		−	0.666657i
$429$		0			0
$430$	36.4780	−	21.0606i	1.75912	−	1.01563i
$431$	−17.4567	+	10.0786i	−0.840860	+	0.485471i	−0.857556	−	0.514390i	$-0.828018\pi$
$431$	−17.4567	+	10.0786i	−0.840860	+	0.485471i	0.0166966	+	0.999861i	$0.494685\pi$
$432$		0			0
$433$		−	5.81292i		−	0.279351i	−0.990197	−	0.139676i	$-0.955394\pi$
$433$		−	5.81292i		−	0.279351i	0.990197	−	0.139676i	$-0.0446060\pi$
$434$	5.04561	+	3.41698i	0.242197	+	0.164020i
$435$		0			0
$436$	2.45534	−	4.25277i	0.117589	−	0.203671i
$437$	6.23319	+	10.7962i	0.298174	+	0.516452i
$438$		0			0
$439$	24.0204	+	13.8682i	1.14643	+	0.661893i	0.948015	−	0.318225i	$-0.103087\pi$
$439$	24.0204	+	13.8682i	1.14643	+	0.661893i	0.198416	+	0.980118i	$0.436420\pi$
$440$	3.93185			0.187444
$441$		0			0
$442$	17.8630			0.849655
$443$	11.4746	+	6.62486i	0.545175	+	0.314757i	0.747173	−	0.664629i	$-0.231410\pi$
$443$	11.4746	+	6.62486i	0.545175	+	0.314757i	−0.201999	+	0.979386i	$0.564744\pi$
$444$		0			0
$445$	23.0958	+	40.0032i	1.09485	+	1.89633i
$446$	−3.64394	+	6.31149i	−0.172546	+	0.298858i
$447$		0			0
$448$	−2.19067	−	1.48356i	−0.103499	−	0.0700918i
$449$			21.9232i			1.03462i	0.855798	+	0.517310i	$0.173067\pi$
$449$			21.9232i			1.03462i	−0.855798	+	0.517310i	$0.826933\pi$
$450$		0			0
$451$	−5.13103	+	2.96240i	−0.241611	+	0.139494i
$452$	13.9664	−	8.06350i	0.656924	−	0.379275i
$453$		0			0
$454$			0.930856i			0.0436872i
$455$	−11.1276	−	22.9232i	−0.521672	−	1.07466i
$456$		0			0
$457$	16.3092	−	28.2483i	0.762911	−	1.32140i	−0.178433	−	0.983952i	$-0.557103\pi$
$457$	16.3092	−	28.2483i	0.762911	−	1.32140i	0.941344	−	0.337449i	$-0.109564\pi$
$458$	−12.0117	−	20.8048i	−0.561268	−	0.972145i
$459$		0			0
$460$	11.5635	+	6.67619i	0.539151	+	0.311279i
$461$	−31.6262			−1.47298			−0.736489	−	0.676450i	$-0.763518\pi$
$461$	−31.6262			−1.47298			−0.736489	+	0.676450i	$0.763518\pi$
$462$		0			0
$463$	11.5687			0.537642			0.268821	−	0.963190i	$-0.413366\pi$
$463$	11.5687			0.537642			0.268821	+	0.963190i	$0.413366\pi$
$464$	4.46945	+	2.58044i	0.207489	+	0.119794i
$465$		0			0
$466$	−8.74264	−	15.1427i	−0.404995	−	0.701472i
$467$	3.92392	−	6.79643i	0.181578	−	0.314501i	−0.760840	−	0.648939i	$-0.775213\pi$
$467$	3.92392	−	6.79643i	0.181578	−	0.314501i	0.942418	+	0.334438i	$0.108546\pi$
$468$		0			0
$469$	0.884355	−	12.3175i	0.0408357	−	0.568768i
$470$		−	33.9082i		−	1.56407i
$471$		0			0
$472$	1.68885	−	0.975056i	0.0777355	−	0.0448806i
$473$	9.27755	−	5.35640i	0.426582	−	0.246287i
$474$		0			0
$475$		−	38.3962i		−	1.76174i
$476$	17.3572	−	8.42575i	0.795568	−	0.386194i
$477$		0			0
$478$	0.356058	−	0.616710i	0.0162857	−	0.0282077i
$479$	−20.3161	−	35.1885i	−0.928267	−	1.60781i	−0.786221	−	0.617945i	$-0.787965\pi$
$479$	−20.3161	−	35.1885i	−0.928267	−	1.60781i	−0.142046	−	0.989860i	$-0.545368\pi$
$480$		0			0
$481$	−8.43050	−	4.86735i	−0.384398	−	0.221932i
$482$	15.5997			0.710545
$483$		0			0
$484$	1.00000			0.0454545
$485$	19.9458	+	11.5157i	0.905692	+	0.522901i
$486$		0			0
$487$	12.6074	+	21.8367i	0.571297	+	0.989515i	0.996433	+	0.0843858i	$0.0268928\pi$
$487$	12.6074	+	21.8367i	0.571297	+	0.989515i	−0.425136	+	0.905129i	$0.639774\pi$
$488$	2.98382	−	5.16813i	0.135071	−	0.233950i
$489$		0			0
$490$	−21.6252	−	17.0254i	−0.976926	−	0.769130i
$491$		−	2.83551i		−	0.127965i	−0.997951	−	0.0639824i	$-0.979620\pi$
$491$		−	2.83551i		−	0.127965i	0.997951	−	0.0639824i	$-0.0203802\pi$
$492$		0			0
$493$	−32.5936	+	18.8179i	−1.46794	+	0.847516i
$494$	−7.78726	+	4.49598i	−0.350365	+	0.202284i
$495$		0			0
$496$		−	2.30323i		−	0.103418i
$497$	−13.2149	+	19.5135i	−0.592769	+	0.875299i
$498$		0			0
$499$	−7.29858	+	12.6415i	−0.326729	+	0.565912i	−0.981861	−	0.189603i	$-0.939280\pi$
$499$	−7.29858	+	12.6415i	−0.326729	+	0.565912i	0.655131	+	0.755515i	$0.272613\pi$
$500$	−10.7329	−	18.5899i	−0.479989	−	0.831366i
$501$		0			0
$502$	−19.2540	−	11.1163i	−0.859349	−	0.496145i
$503$	−14.4955			−0.646322			−0.323161	−	0.946344i	$-0.604745\pi$
$503$	−14.4955			−0.646322			−0.323161	+	0.946344i	$0.604745\pi$
$504$		0			0
$505$	−61.0245			−2.71556
$506$	2.94098	+	1.69798i	0.130743	+	0.0754843i
$507$		0			0
$508$	4.97261	+	8.61281i	0.220624	+	0.382132i
$509$	14.4421	−	25.0144i	0.640134	−	1.10874i	−0.345269	−	0.938504i	$-0.612212\pi$
$509$	14.4421	−	25.0144i	0.640134	−	1.10874i	0.985403	−	0.170240i	$-0.0544544\pi$
$510$		0			0
$511$	−7.23607	−	0.519527i	−0.320105	−	0.0229825i
$512$			1.00000i			0.0441942i
$513$		0			0
$514$	2.55379	−	1.47443i	0.112643	−	0.0650345i
$515$	42.2333	−	24.3834i	1.86102	−	1.07446i
$516$		0			0
$517$		−	8.62398i		−	0.379282i
$518$	−10.4877	−	0.752982i	−0.460802	−	0.0330841i
$519$		0			0
$520$	−4.81552	+	8.34072i	−0.211174	+	0.365765i
$521$	−2.96937	−	5.14310i	−0.130090	−	0.225323i	0.793621	−	0.608413i	$-0.208193\pi$
$521$	−2.96937	−	5.14310i	−0.130090	−	0.225323i	−0.923711	+	0.383090i	$0.874860\pi$
$522$		0			0
$523$	−15.6424	−	9.03115i	−0.683995	−	0.394905i	0.117364	−	0.993089i	$-0.462556\pi$
$523$	−15.6424	−	9.03115i	−0.683995	−	0.394905i	−0.801358	+	0.598184i	$0.795889\pi$
$524$	18.4853			0.807533
$525$		0			0
$526$	18.9447			0.826030
$527$	14.5461	+	8.39817i	0.633636	+	0.365830i
$528$		0			0
$529$	−5.73375	−	9.93115i	−0.249294	−	0.431789i
$530$	−26.2996	+	45.5522i	−1.14238	+	1.97866i
$531$		0			0
$532$	−5.44609	+	8.04184i	−0.236118	+	0.348658i
$533$		−	14.5127i		−	0.628617i
$534$		0			0
$535$	−46.9626	+	27.1139i	−2.03037	+	1.17223i
$536$	−4.04222	+	2.33378i	−0.174597	+	0.100804i
$537$		0			0
$538$		−	24.2329i		−	1.04476i
$539$	−5.50000	−	4.33013i	−0.236902	−	0.186512i
$540$		0			0
$541$	7.59039	−	13.1469i	0.326336	−	0.565231i	−0.655446	−	0.755242i	$-0.727519\pi$
$541$	7.59039	−	13.1469i	0.326336	−	0.565231i	0.981782	+	0.190011i	$0.0608525\pi$
$542$	−11.2165	−	19.4275i	−0.481788	−	0.834481i
$543$		0			0
$544$	−6.31552	−	3.64626i	−0.270775	−	0.156332i
$545$	−19.3080			−0.827065
$546$		0			0
$547$	12.6498			0.540865			0.270433	−	0.962739i	$-0.412833\pi$
$547$	12.6498			0.540865			0.270433	+	0.962739i	$0.412833\pi$
$548$	−2.41786	−	1.39595i	−0.103286	−	0.0596322i
$549$		0			0
$550$	−5.22973	−	9.05816i	−0.222996	−	0.386241i
$551$	9.47266	−	16.4071i	0.403549	−	0.698967i
$552$		0			0
$553$	36.6445	−	17.7884i	1.55828	−	0.756440i
$554$		−	9.84785i		−	0.418395i
$555$		0			0
$556$	4.42178	−	2.55291i	0.187525	−	0.108268i
$557$	2.43639	−	1.40665i	0.103233	−	0.0596017i	−0.447495	−	0.894287i	$-0.647684\pi$
$557$	2.43639	−	1.40665i	0.103233	−	0.0596017i	0.550728	+	0.834685i	$0.314350\pi$
$558$		0			0
$559$			26.2409i			1.10987i
$560$	−0.744963	+	10.3760i	−0.0314804	+	0.438466i
$561$		0			0
$562$	5.43115	−	9.40702i	0.229099	−	0.396811i
$563$	−2.94887	−	5.10759i	−0.124280	−	0.215259i	0.797171	−	0.603753i	$-0.206329\pi$
$563$	−2.94887	−	5.10759i	−0.124280	−	0.215259i	−0.921451	+	0.388494i	$0.872995\pi$
$564$		0			0
$565$	−54.9138	−	31.7045i	−2.31024	−	1.33382i
$566$	9.61625			0.404201
$567$		0			0
$568$	8.90754			0.373752
$569$	24.4687	+	14.1270i	1.02578	+	0.592235i	0.915773	−	0.401696i	$-0.131579\pi$
$569$	24.4687	+	14.1270i	1.02578	+	0.592235i	0.110008	+	0.993931i	$0.464912\pi$
$570$		0			0
$571$	16.6142	+	28.7766i	0.695281	+	1.20426i	0.970086	+	0.242762i	$0.0780535\pi$
$571$	16.6142	+	28.7766i	0.695281	+	1.20426i	−0.274805	+	0.961500i	$0.588613\pi$
$572$	−1.22474	+	2.12132i	−0.0512092	+	0.0886969i
$573$		0			0
$574$	−6.84549	−	14.1019i	−0.285725	−	0.588600i
$575$		−	35.5198i		−	1.48128i
$576$		0			0
$577$	−7.80012	+	4.50340i	−0.324723	+	0.187479i	−0.653496	−	0.756930i	$-0.726698\pi$
$577$	−7.80012	+	4.50340i	−0.324723	+	0.187479i	0.328773	+	0.944409i	$0.393365\pi$
$578$	31.3336	−	18.0905i	1.30331	−	0.752465i
$579$		0			0
$580$		−	20.2918i		−	0.842571i
$581$	−13.2650	−	8.98329i	−0.550324	−	0.372690i
$582$		0			0
$583$	−6.68885	+	11.5854i	−0.277024	+	0.479819i
$584$	1.37101	+	2.37466i	0.0567328	+	0.0982640i
$585$		0			0
$586$	12.5181	+	7.22732i	0.517118	+	0.298558i
$587$	39.5783			1.63357			0.816785	−	0.576942i	$-0.195754\pi$
$587$	39.5783			1.63357			0.816785	+	0.576942i	$0.195754\pi$
$588$		0			0
$589$	−8.45503			−0.348383
$590$	−6.64029	−	3.83378i	−0.273377	−	0.157834i
$591$		0			0
$592$	1.98709	+	3.44174i	0.0816688	+	0.141455i
$593$	−17.9542	+	31.0976i	−0.737290	+	1.27702i	0.216421	+	0.976300i	$0.430562\pi$
$593$	−17.9542	+	31.0976i	−0.737290	+	1.27702i	−0.953711	+	0.300724i	$0.902772\pi$
$594$		0			0
$595$	−62.8134	−	42.5384i	−2.57510	−	1.74391i
$596$			6.85425i			0.280761i
$597$		0			0
$598$	−7.20390	+	4.15918i	−0.294590	+	0.170081i
$599$	−28.9870	+	16.7357i	−1.18438	+	0.683800i	−0.957023	−	0.290012i	$-0.906341\pi$
$599$	−28.9870	+	16.7357i	−1.18438	+	0.683800i	−0.227354	+	0.973812i	$0.573007\pi$
$600$		0			0
$601$			27.5546i			1.12398i	0.827146	+	0.561988i	$0.189963\pi$
$601$			27.5546i			1.12398i	−0.827146	+	0.561988i	$0.810037\pi$
$602$	12.3775	+	25.4979i	0.504469	+	1.03922i
$603$		0			0
$604$	−7.23351	+	12.5288i	−0.294327	+	0.509790i
$605$	−1.96593	−	3.40508i	−0.0799263	−	0.138436i
$606$		0			0
$607$	−39.1116	−	22.5811i	−1.58749	−	0.916539i	−0.993719	−	0.111906i	$-0.964305\pi$
$607$	−39.1116	−	22.5811i	−1.58749	−	0.916539i	−0.593773	−	0.804633i	$-0.702362\pi$
$608$	3.67095			0.148877
$609$		0			0
$610$	−23.4639			−0.950025
$611$	18.2942	+	10.5622i	0.740105	+	0.427300i
$612$		0			0
$613$	13.4202	+	23.2445i	0.542037	+	0.938836i	0.998787	+	0.0492407i	$0.0156802\pi$
$613$	13.4202	+	23.2445i	0.542037	+	0.938836i	−0.456750	+	0.889595i	$0.650987\pi$
$614$	−0.464466	+	0.804479i	−0.0187443	+	0.0324661i
$615$		0			0
$616$	−0.189469	+	2.63896i	−0.00763391	+	0.106327i
$617$		−	37.4317i		−	1.50694i	−0.657481	−	0.753471i	$-0.728378\pi$
$617$		−	37.4317i		−	1.50694i	0.657481	−	0.753471i	$-0.271622\pi$
$618$		0			0
$619$	−10.7567	+	6.21039i	−0.432349	+	0.249617i	−0.700347	−	0.713803i	$-0.746971\pi$
$619$	−10.7567	+	6.21039i	−0.432349	+	0.249617i	0.267998	+	0.963419i	$0.413638\pi$
$620$	−7.84267	+	4.52797i	−0.314969	+	0.181848i
$621$		0			0
$622$		−	26.7681i		−	1.07330i
$623$	−27.9620	+	13.5737i	−1.12028	+	0.543817i
$624$		0			0
$625$	−16.0515	+	27.8020i	−0.642059	+	1.11208i
$626$	−3.97469	−	6.88437i	−0.158861	−	0.275155i
$627$		0			0
$628$	14.4305	+	8.33145i	0.575840	+	0.332461i
$629$	−28.9818			−1.15558
$630$		0			0
$631$	−5.57938			−0.222112			−0.111056	−	0.993814i	$-0.535423\pi$
$631$	−5.57938			−0.222112			−0.111056	+	0.993814i	$0.535423\pi$
$632$	−13.3333	−	7.69798i	−0.530370	−	0.306209i
$633$		0			0
$634$	−8.33657	−	14.4394i	−0.331088	−	0.573460i
$635$	19.5516	−	33.8643i	0.775880	−	1.34386i
$636$		0			0
$637$	15.9217	−	6.36396i	0.630840	−	0.252149i
$638$		−	5.16088i		−	0.204321i
$639$		0			0
$640$	3.40508	−	1.96593i	0.134598	−	0.0777100i
$641$	−29.7432	+	17.1723i	−1.17479	+	0.678263i	−0.954803	−	0.297240i	$-0.903934\pi$
$641$	−29.7432	+	17.1723i	−1.17479	+	0.678263i	−0.219984	+	0.975504i	$0.570600\pi$
$642$		0			0
$643$			21.2432i			0.837748i	0.908044	+	0.418874i	$0.137575\pi$
$643$			21.2432i			0.837748i	−0.908044	+	0.418874i	$0.862425\pi$
$644$	−5.03811	+	7.43942i	−0.198529	+	0.293154i
$645$		0			0
$646$	−13.3853	+	23.1839i	−0.526636	+	0.912160i
$647$	−12.0354	−	20.8459i	−0.473160	−	0.819537i	0.526368	−	0.850257i	$-0.323553\pi$
$647$	−12.0354	−	20.8459i	−0.473160	−	0.819537i	−0.999528	+	0.0307199i	$0.990220\pi$
$648$		0			0
$649$	−1.68885	−	0.975056i	−0.0662930	−	0.0382743i
$650$	25.6203			1.00491
$651$		0			0
$652$	−0.963479			−0.0377328
$653$	25.2388	+	14.5716i	0.987672	+	0.570233i	0.904578	−	0.426309i	$-0.140186\pi$
$653$	25.2388	+	14.5716i	0.987672	+	0.570233i	0.0830943	+	0.996542i	$0.473520\pi$
$654$		0			0
$655$	−36.3407	−	62.9439i	−1.41995	−	2.45942i
$656$	−2.96240	+	5.13103i	−0.115662	+	0.200333i
$657$		0			0
$658$	22.7583	+	1.63397i	0.887212	+	0.0636990i
$659$			39.4790i			1.53788i	0.639318	+	0.768942i	$0.279217\pi$
$659$			39.4790i			1.53788i	−0.639318	+	0.768942i	$0.720783\pi$
$660$		0			0
$661$	−33.0582	+	19.0861i	−1.28581	+	0.742365i	−0.977905	−	0.209051i	$-0.932962\pi$
$661$	−33.0582	+	19.0861i	−1.28581	+	0.742365i	−0.307909	+	0.951416i	$0.599629\pi$
$662$	−2.75280	+	1.58933i	−0.106990	+	0.0617710i
$663$		0			0
$664$			6.05521i			0.234988i
$665$	38.0898	+	2.73472i	1.47706	+	0.106048i
$666$		0			0
$667$	8.76305	−	15.1780i	0.339307	−	0.587696i
$668$	−0.967128	−	1.67511i	−0.0374193	−	0.0648121i
$669$		0			0
$670$	15.8934	+	9.17606i	0.614016	+	0.354502i
$671$	−5.96764			−0.230378
$672$		0			0
$673$	−10.7014			−0.412509			−0.206255	−	0.978498i	$-0.566128\pi$
$673$	−10.7014			−0.412509			−0.206255	+	0.978498i	$0.566128\pi$
$674$	−13.6376	−	7.87367i	−0.525301	−	0.303282i
$675$		0			0
$676$	3.50000	+	6.06218i	0.134615	+	0.233161i
$677$	8.12925	−	14.0803i	0.312432	−	0.541149i	−0.666456	−	0.745544i	$-0.732190\pi$
$677$	8.12925	−	14.0803i	0.312432	−	0.541149i	0.978888	+	0.204396i	$0.0655229\pi$
$678$		0			0
$679$	−8.69020	+	12.8322i	−0.333499	+	0.492454i
$680$			28.6731i			1.09956i
$681$		0			0
$682$	−1.99465	+	1.15161i	−0.0763792	+	0.0440975i
$683$	−0.845958	+	0.488414i	−0.0323697	+	0.0186887i	−0.516098	−	0.856530i	$-0.672616\pi$
$683$	−0.845958	+	0.488414i	−0.0323697	+	0.0186887i	0.483728	+	0.875218i	$0.339282\pi$
$684$		0			0
$685$			10.9774i			0.419423i
$686$	12.4691	−	13.6938i	0.476073	−	0.522834i
$687$		0			0
$688$	5.35640	−	9.27755i	0.204211	−	0.353703i
$689$	−16.3843	−	28.3784i	−0.624191	−	1.08113i
$690$		0			0
$691$	15.3783	+	8.87869i	0.585020	+	0.337761i	0.763126	−	0.646250i	$-0.223664\pi$
$691$	15.3783	+	8.87869i	0.585020	+	0.337761i	−0.178106	+	0.984011i	$0.556997\pi$
$692$	−19.7580			−0.751088
$693$		0			0
$694$	−26.3244			−0.999261
$695$	−17.3858	−	10.0377i	−0.659480	−	0.380751i
$696$		0			0
$697$	−21.6034	−	37.4182i	−0.818287	−	1.41731i
$698$	16.4522	−	28.4961i	0.622725	−	1.07859i
$699$		0			0
$700$	24.8950	−	12.0848i	0.940941	−	0.456762i
$701$		−	9.85160i		−	0.372090i	−0.982541	−	0.186045i	$-0.940433\pi$
$701$		−	9.85160i		−	0.372090i	0.982541	−	0.186045i	$-0.0595670\pi$
$702$		0			0
$703$	12.6344	−	7.29450i	0.476517	−	0.275117i
$704$	0.866025	−	0.500000i	0.0326396	−	0.0188445i
$705$		0			0
$706$		−	9.49886i		−	0.357494i
$707$	2.94066	−	40.9581i	0.110595	−	1.54039i
$708$		0			0
$709$	19.4994	−	33.7739i	0.732314	−	1.26840i	−0.223578	−	0.974686i	$-0.571774\pi$
$709$	19.4994	−	33.7739i	0.732314	−	1.26840i	0.955892	−	0.293718i	$-0.0948928\pi$
$710$	−17.5116	−	30.3309i	−0.657197	−	1.13830i
$711$		0			0
$712$	10.1741	+	5.87404i	0.381292	+	0.220139i
$713$	−7.82165			−0.292923
$714$		0			0
$715$	9.63103			0.360180
$716$	−9.13593	−	5.27463i	−0.341426	−	0.197122i
$717$		0			0
$718$	1.49867	+	2.59578i	0.0559300	+	0.0968736i
$719$	−6.32864	+	10.9615i	−0.236019	+	0.408796i	−0.959568	−	0.281476i	$-0.909176\pi$
$719$	−6.32864	+	10.9615i	−0.236019	+	0.408796i	0.723550	+	0.690272i	$0.242509\pi$
$720$		0			0
$721$	14.3304	+	29.5209i	0.533690	+	1.09941i
$722$			5.52412i			0.205587i
$723$		0			0
$724$	−3.57287	+	2.06280i	−0.132785	+	0.0766634i
$725$	−46.7480	+	26.9900i	−1.73618	+	1.00238i
$726$		0			0
$727$		−	15.8671i		−	0.588478i	−0.955732	−	0.294239i	$-0.904934\pi$
$727$		−	15.8671i		−	0.588478i	0.955732	−	0.294239i	$-0.0950662\pi$
$728$	−5.36603	−	3.63397i	−0.198878	−	0.134684i
$729$		0			0
$730$	5.39060	−	9.33680i	0.199515	−	0.345571i
$731$	39.0617	+	67.6568i	1.44475	+	2.50238i
$732$		0			0
$733$	−21.8797	−	12.6322i	−0.808144	−	0.466582i	0.0381669	−	0.999271i	$-0.487848\pi$
$733$	−21.8797	−	12.6322i	−0.808144	−	0.466582i	−0.846311	+	0.532689i	$0.821181\pi$
$734$	−16.5510			−0.610910
$735$		0			0
$736$	3.39595			0.125176
$737$	4.04222	+	2.33378i	0.148897	+	0.0859657i
$738$		0			0
$739$	2.35023	+	4.07072i	0.0864546	+	0.149744i	0.906010	−	0.423256i	$-0.139113\pi$
$739$	2.35023	+	4.07072i	0.0864546	+	0.149744i	−0.819556	+	0.573000i	$0.805780\pi$
$740$	7.81294	−	13.5324i	0.287209	−	0.497461i
$741$		0			0
$742$	−29.3061	−	19.8467i	−1.07586	−	0.728594i
$743$			37.5807i			1.37870i	0.724427	+	0.689352i	$0.242105\pi$
$743$			37.5807i			1.37870i	−0.724427	+	0.689352i	$0.757895\pi$
$744$		0			0
$745$	23.3393	−	13.4749i	0.855086	−	0.493684i
$746$	−16.8659	+	9.73753i	−0.617505	+	0.356516i
$747$		0			0
$748$			7.29253i			0.266641i
$749$	−15.9351	−	32.8267i	−0.582256	−	1.19946i
$750$		0			0
$751$	13.6493	−	23.6413i	0.498070	−	0.862682i	−0.501928	−	0.864910i	$-0.667376\pi$
$751$	13.6493	−	23.6413i	0.498070	−	0.862682i	0.999998	+	0.00222750i	$0.000709036\pi$
$752$	−4.31199	−	7.46859i	−0.157242	−	0.272351i
$753$		0			0
$754$	10.9479	+	6.32076i	0.398698	+	0.230188i
$755$	56.8822			2.07015
$756$		0			0
$757$	−1.76874			−0.0642858			−0.0321429	−	0.999483i	$-0.510233\pi$
$757$	−1.76874			−0.0642858			−0.0321429	+	0.999483i	$0.510233\pi$
$758$	−1.91227	−	1.10405i	−0.0694566	−	0.0401008i
$759$		0			0
$760$	−7.21682	−	12.4999i	−0.261781	−	0.453419i
$761$	−15.4241	+	26.7153i	−0.559122	+	0.968428i	0.438448	+	0.898757i	$0.355528\pi$
$761$	−15.4241	+	26.7153i	−0.559122	+	0.968428i	−0.997570	+	0.0696715i	$0.977805\pi$
$762$		0			0
$763$	0.930419	−	12.9591i	0.0336834	−	0.469149i
$764$		−	23.0194i		−	0.832813i
$765$		0			0
$766$	2.45690	−	1.41849i	0.0887715	−	0.0512522i
$767$	4.13681	−	2.38839i	0.149372	−	0.0862398i
$768$		0			0
$769$		−	39.1842i		−	1.41302i	−0.707704	−	0.706509i	$-0.750269\pi$
$769$		−	39.1842i		−	1.41302i	0.707704	−	0.706509i	$-0.249731\pi$
$770$	9.35835	−	4.54284i	0.337252	−	0.163713i
$771$		0			0
$772$	4.60708	−	7.97970i	0.165813	−	0.287196i
$773$	27.0296	+	46.8166i	0.972186	+	1.68388i	0.688924	+	0.724834i	$0.258084\pi$
$773$	27.0296	+	46.8166i	0.972186	+	1.68388i	0.283263	+	0.959042i	$0.408583\pi$
$774$		0			0
$775$	20.8630	+	12.0452i	0.749420	+	0.432678i
$776$	5.85765			0.210277
$777$		0			0
$778$	−19.0163			−0.681767
$779$	18.8358	+	10.8748i	0.674861	+	0.389631i
$780$		0			0
$781$	−4.45377	−	7.71415i	−0.159368	−	0.276034i
$782$	−12.3825	+	21.4472i	−0.442799	+	0.766950i
$783$		0			0
$784$	−6.92820	−	1.00000i	−0.247436	−	0.0357143i
$785$		−	65.5161i		−	2.33837i
$786$		0			0
$787$	−8.76131	+	5.05834i	−0.312307	+	0.180311i	−0.647958	−	0.761676i	$-0.724377\pi$
$787$	−8.76131	+	5.05834i	−0.312307	+	0.180311i	0.335651	+	0.941986i	$0.391044\pi$
$788$	−5.64805	+	3.26090i	−0.201203	+	0.116165i
$789$		0			0
$790$			60.5346i			2.15372i
$791$	23.9254	−	35.3290i	0.850691	−	1.25615i
$792$		0			0
$793$	7.30884	−	12.6593i	0.259545	−	0.449544i
$794$	−5.33315	−	9.23729i	−0.189267	−	0.327819i
$795$		0			0
$796$	−12.9813	−	7.49473i	−0.460109	−	0.265644i
$797$	44.4401			1.57415			0.787075	−	0.616858i	$-0.211595\pi$
$797$	44.4401			1.57415			0.787075	+	0.616858i	$0.211595\pi$
$798$		0			0
$799$	62.8906			2.22491
$800$	−9.05816	−	5.22973i	−0.320254	−	0.184899i
$801$		0			0
$802$	−16.4092	−	28.4216i	−0.579429	−	1.00360i
$803$	1.37101	−	2.37466i	0.0483819	−	0.0837998i
$804$		0			0
$805$	35.2364	+	2.52986i	1.24192	+	0.0891658i
$806$		−	5.64173i		−	0.198721i
$807$		0			0
$808$	−13.4412	+	7.76028i	−0.472860	+	0.273006i
$809$	42.6703	−	24.6357i	1.50021	−	0.866146i	0.500209	−	0.865905i	$-0.333256\pi$
$809$	42.6703	−	24.6357i	1.50021	−	0.866146i	1.00000		0.000241636i	$-7.69150e-5\pi$
$810$		0			0
$811$			21.4369i			0.752750i	0.926467	+	0.376375i	$0.122830\pi$
$811$			21.4369i			0.752750i	−0.926467	+	0.376375i	$0.877170\pi$
$812$	13.6193	+	0.977824i	0.477945	+	0.0343149i
$813$		0			0
$814$	1.98709	−	3.44174i	0.0696474	−	0.120633i
$815$	1.89413	+	3.28073i	0.0663484	+	0.114919i
$816$		0			0
$817$	−34.0574	−	19.6631i	−1.19152	−	0.687924i
$818$	7.22073			0.252467
$819$		0			0
$820$	23.2954			0.813512
$821$	18.7461	+	10.8231i	0.654244	+	0.377728i	0.790080	−	0.613003i	$-0.210039\pi$
$821$	18.7461	+	10.8231i	0.654244	+	0.377728i	−0.135836	+	0.990731i	$0.543372\pi$
$822$		0			0
$823$	3.93650	+	6.81821i	0.137218	+	0.237668i	0.926442	−	0.376437i	$-0.122851\pi$
$823$	3.93650	+	6.81821i	0.137218	+	0.237668i	−0.789225	+	0.614104i	$0.789517\pi$
$824$	6.20150	−	10.7413i	0.216040	−	0.374191i
$825$		0			0
$826$	2.89312	−	4.27205i	0.100664	−	0.148644i
$827$		−	21.5865i		−	0.750635i	−0.926896	−	0.375317i	$-0.877534\pi$
$827$		−	21.5865i		−	0.750635i	0.926896	−	0.375317i	$-0.122466\pi$
$828$		0			0
$829$	6.04678	−	3.49111i	0.210013	−	0.121251i	−0.391304	−	0.920261i	$-0.627976\pi$
$829$	6.04678	−	3.49111i	0.210013	−	0.121251i	0.601318	+	0.799010i	$0.294643\pi$
$830$	20.6185	−	11.9041i	0.715679	−	0.413197i
$831$		0			0
$832$			2.44949i			0.0849208i
$833$	31.5776	−	40.1089i	1.09410	−	1.38969i
$834$		0			0
$835$	−3.80260	+	6.58630i	−0.131595	+	0.227928i
$836$	−1.83548	−	3.17914i	−0.0634812	−	0.109953i
$837$		0			0
$838$	13.7857	+	7.95920i	0.476220	+	0.274946i
$839$	−9.03111			−0.311789			−0.155894	−	0.987774i	$-0.549826\pi$
$839$	−9.03111			−0.311789			−0.155894	+	0.987774i	$0.549826\pi$
$840$		0			0
$841$	2.36536			0.0815640
$842$	−3.67099	−	2.11944i	−0.126511	−	0.0730409i
$843$		0			0
$844$	−4.55379	−	7.88740i	−0.156748	−	0.271496i
$845$	13.7615	−	23.8356i	0.473409	−	0.819969i
$846$		0			0
$847$	2.38014	−	1.15539i	0.0817826	−	0.0396998i
$848$			13.3777i			0.459392i
$849$		0			0
$850$	66.0569	−	38.1379i	2.26573	−	1.30812i
$851$	11.6880	−	6.74806i	0.400659	−	0.231320i
$852$		0			0
$853$			25.1105i			0.859768i	0.902884	+	0.429884i	$0.141446\pi$
$853$			25.1105i			0.859768i	−0.902884	+	0.429884i	$0.858554\pi$
$854$	1.13068	−	15.7484i	0.0386911	−	0.538898i
$855$		0			0
$856$	−6.89595	+	11.9441i	−0.235699	+	0.408242i
$857$	25.9758	+	44.9914i	0.887315	+	1.53688i	0.843037	+	0.537856i	$0.180766\pi$
$857$	25.9758	+	44.9914i	0.887315	+	1.53688i	0.0442787	+	0.999019i	$0.485901\pi$
$858$		0			0
$859$	16.7954	+	9.69686i	0.573053	+	0.330852i	0.758368	−	0.651827i	$-0.225997\pi$
$859$	16.7954	+	9.69686i	0.573053	+	0.330852i	−0.185315	+	0.982679i	$0.559330\pi$
$860$	−42.1211			−1.43632
$861$		0			0
$862$	20.1573			0.686559
$863$	−49.7575	−	28.7275i	−1.69377	−	0.977896i	−0.951429	−	0.307869i	$-0.900384\pi$
$863$	−49.7575	−	28.7275i	−1.69377	−	0.977896i	−0.742337	−	0.670027i	$-0.766283\pi$
$864$		0			0
$865$	38.8428	+	67.2778i	1.32070	+	2.28751i
$866$	−2.90646	+	5.03414i	−0.0987656	+	0.171067i
$867$		0			0
$868$	−2.66113	−	5.48200i	−0.0903248	−	0.186071i
$869$			15.3960i			0.522272i
$870$		0			0
$871$	−9.90137	+	5.71656i	−0.335495	+	0.193698i
$872$	−4.25277	+	2.45534i	−0.144017	+	0.0831482i
$873$		0			0
$874$		−	12.4664i		−	0.421681i
$875$	−47.0244	−	31.8459i	−1.58972	−	1.07659i
$876$		0			0
$877$	12.9828	−	22.4869i	0.438399	−	0.759329i	−0.559167	−	0.829055i	$-0.688879\pi$
$877$	12.9828	−	22.4869i	0.438399	−	0.759329i	0.997566	+	0.0697257i	$0.0222124\pi$
$878$	−13.8682	−	24.0204i	−0.468029	−	0.810650i
$879$		0			0
$880$	−3.40508	−	1.96593i	−0.114785	−	0.0662713i
$881$	−11.9446			−0.402423			−0.201211	−	0.979548i	$-0.564488\pi$
$881$	−11.9446			−0.402423			−0.201211	+	0.979548i	$0.564488\pi$
$882$		0			0
$883$	−52.7238			−1.77430			−0.887149	−	0.461483i	$-0.847318\pi$
$883$	−52.7238			−1.77430			−0.887149	+	0.461483i	$0.847318\pi$
$884$	−15.4698	−	8.93149i	−0.520305	−	0.300398i
$885$		0			0
$886$	−6.62486	−	11.4746i	−0.222567	−	0.385497i
$887$	13.8518	−	23.9921i	0.465099	−	0.805575i	−0.534107	−	0.845417i	$-0.679352\pi$
$887$	13.8518	−	23.9921i	0.465099	−	0.805575i	0.999206	+	0.0398422i	$0.0126855\pi$
$888$		0			0
$889$	21.7867	+	14.7544i	0.730702	+	0.494846i
$890$		−	46.1917i		−	1.54835i
$891$		0			0
$892$	6.31149	−	3.64394i	0.211324	−	0.122008i
$893$	−27.4168	+	15.8291i	−0.917469	+	0.529701i
$894$		0			0
$895$			41.4781i			1.38646i
$896$	1.15539	+	2.38014i	0.0385990	+	0.0795149i
$897$		0			0
$898$	10.9616	−	18.9861i	0.365793	−	0.633573i
$899$	5.94333	+	10.2942i	0.198221	+	0.343329i
$900$		0			0
$901$	−84.4870	−	48.7786i	−2.81467	−	1.62505i
$902$	5.92480			0.197274
$903$		0			0
$904$	−16.1270			−0.536376
$905$	14.0480	+	8.11062i	0.466972	+	0.269606i
$906$		0			0
$907$	−15.7674	−	27.3100i	−0.523548	−	0.906812i	−0.999624	−	0.0274080i	$-0.991275\pi$
$907$	−15.7674	−	27.3100i	−0.523548	−	0.906812i	0.476076	−	0.879404i	$-0.342059\pi$
$908$	0.465428	−	0.806145i	0.0154458	−	0.0267529i
$909$		0			0
$910$	−1.82478	+	25.4159i	−0.0604908	+	0.842529i
$911$		−	0.137644i		−	0.00456035i	−0.999997	−	0.00228018i	$-0.999274\pi$
$911$		−	0.137644i		−	0.00456035i	0.999997	−	0.00228018i	$-0.000725803\pi$
$912$		0			0
$913$	5.24397	−	3.02761i	0.173550	−	0.100199i
$914$	−28.2483	+	16.3092i	−0.934372	+	0.539460i
$915$		0			0
$916$			24.0233i			0.793753i
$917$	43.9975	−	21.3578i	1.45293	−	0.705296i
$918$		0			0
$919$	23.7828	−	41.1930i	0.784521	−	1.35883i	−0.144763	−	0.989466i	$-0.546242\pi$
$919$	23.7828	−	41.1930i	0.784521	−	1.35883i	0.929285	−	0.369365i	$-0.120425\pi$
$920$	−6.67619	−	11.5635i	−0.220108	−	0.381237i
$921$		0			0
$922$	27.3891	+	15.8131i	0.902011	+	0.520776i
$923$	21.8189			0.718179
$924$		0			0
$925$	−41.5677			−1.36674
$926$	−10.0188	−	5.78434i	−0.329237	−	0.190085i
$927$		0			0
$928$	−2.58044	−	4.46945i	−0.0847070	−	0.146717i
$929$	3.76320	−	6.51805i	0.123467	−	0.213850i	−0.797666	−	0.603100i	$-0.793932\pi$
$929$	3.76320	−	6.51805i	0.123467	−	0.213850i	0.921132	+	0.389249i	$0.127266\pi$
$930$		0			0
$931$	−3.67095	+	25.4331i	−0.120311	+	0.833536i
$932$			17.4853i			0.572749i
$933$		0			0
$934$	−6.79643	+	3.92392i	−0.222386	+	0.128395i
$935$	24.8317	−	14.3366i	0.812082	−	0.468856i
$936$		0			0
$937$		−	7.64372i		−	0.249709i	−0.992175	−	0.124855i	$-0.960154\pi$
$937$		−	7.64372i		−	0.249709i	0.992175	−	0.124855i	$-0.0398465\pi$
$938$	−6.92461	+	10.2251i	−0.226097	+	0.333860i
$939$		0			0
$940$	−16.9541	+	29.3654i	−0.552982	+	0.957793i
$941$	3.92910	+	6.80539i	0.128085	+	0.221850i	0.922935	−	0.384957i	$-0.125784\pi$
$941$	3.92910	+	6.80539i	0.128085	+	0.221850i	−0.794850	+	0.606806i	$0.792450\pi$
$942$		0			0
$943$	17.4247	+	10.0602i	0.567428	+	0.327604i
$944$	−1.95011			−0.0634707
$945$		0			0
$946$	−10.7128			−0.348303
$947$	−33.1290	−	19.1270i	−1.07655	−	0.621544i	−0.146584	−	0.989198i	$-0.546828\pi$
$947$	−33.1290	−	19.1270i	−1.07655	−	0.621544i	−0.929963	+	0.367654i	$0.880161\pi$
$948$		0			0
$949$	3.35827	+	5.81670i	0.109014	+	0.188818i
$950$	−19.1981	+	33.2520i	−0.622868	+	1.07884i
$951$		0			0
$952$	−19.2447	−	1.38171i	−0.623724	−	0.0447813i
$953$			12.6569i			0.409996i	0.978762	+	0.204998i	$0.0657187\pi$
$953$			12.6569i			0.409996i	−0.978762	+	0.204998i	$0.934281\pi$
$954$		0			0
$955$	−78.3830	+	45.2545i	−2.53642	+	1.46440i
$956$	−0.616710	+	0.356058i	−0.0199458	+	0.0115157i
$957$		0			0
$958$			40.6322i			1.31277i
$959$	−7.36773	−	0.528979i	−0.237916	−	0.0170816i
$960$		0			0
$961$	−12.8476	+	22.2527i	−0.414438	+	0.717828i
$962$	4.86735	+	8.43050i	0.156930	+	0.271810i
$963$		0			0
$964$	−13.5097	−	7.79983i	−0.435118	−	0.251216i
$965$	−36.2287			−1.16624
$966$		0			0
$967$	37.2489			1.19784			0.598922	−	0.800807i	$-0.295596\pi$
$967$	37.2489			1.19784			0.598922	+	0.800807i	$0.295596\pi$
$968$	−0.866025	−	0.500000i	−0.0278351	−	0.0160706i
$969$		0			0
$970$	−11.5157	−	19.9458i	−0.369747	−	0.640421i
$971$	−28.9586	+	50.1577i	−0.929325	+	1.60964i	−0.144872	+	0.989450i	$0.546277\pi$
$971$	−28.9586	+	50.1577i	−0.929325	+	1.60964i	−0.784453	+	0.620188i	$0.787056\pi$
$972$		0			0
$973$	7.57482	−	11.1852i	0.242838	−	0.358581i
$974$		−	25.2148i		−	0.807936i
$975$		0			0
$976$	−5.16813	+	2.98382i	−0.165428	+	0.0955098i
$977$	11.2435	−	6.49145i	0.359712	−	0.207680i	−0.309242	−	0.950983i	$-0.600075\pi$
$977$	11.2435	−	6.49145i	0.359712	−	0.207680i	0.668954	+	0.743303i	$0.266742\pi$
$978$		0			0
$979$		−	11.7481i		−	0.375470i
$980$	10.2153	+	25.5570i	0.326314	+	0.816390i
$981$		0			0
$982$	−1.41776	+	2.45562i	−0.0452424	+	0.0783621i
$983$	9.60828	+	16.6420i	0.306457	+	0.530799i	0.977585	−	0.210543i	$-0.0675232\pi$
$983$	9.60828	+	16.6420i	0.306457	+	0.530799i	−0.671128	+	0.741342i	$0.734190\pi$
$984$		0			0
$985$	22.2073	+	12.8214i	0.707583	+	0.408523i
$986$	37.6358			1.19857
$987$		0			0
$988$	8.99196			0.286072
$989$	−31.5061	−	18.1901i	−1.00184	−	0.578411i
$990$		0			0
$991$	15.7408	+	27.2639i	0.500024	+	0.866067i	1.00000		2.77966e-5i	$8.84792e-6\pi$
$991$	15.7408	+	27.2639i	0.500024	+	0.866067i	−0.499976	+	0.866039i	$0.666658\pi$
$992$	−1.15161	+	1.99465i	−0.0365637	+	0.0633303i
$993$		0			0
$994$	21.2012	−	10.2917i	0.672461	−	0.326434i
$995$			58.9364i			1.86841i
$996$		0			0
$997$	36.6287	−	21.1476i	1.16004	−	0.669750i	0.208728	−	0.977974i	$-0.433068\pi$
$997$	36.6287	−	21.1476i	1.16004	−	0.669750i	0.951314	+	0.308223i	$0.0997344\pi$
$998$	12.6415	−	7.29858i	0.400160	−	0.231033i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1386.2.r.c.89.2	yes	8
3.2	odd	2		1386.2.r.a.89.3	✓	8
7.3	odd	6		1386.2.r.a.1277.3	yes	8
21.17	even	6	inner	1386.2.r.c.1277.2	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1386.2.r.a.89.3	✓	8	3.2	odd	2
1386.2.r.a.1277.3	yes	8	7.3	odd	6
1386.2.r.c.89.2	yes	8	1.1	even	1	trivial
1386.2.r.c.1277.2	yes	8	21.17	even	6	inner

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:	\( N \)	\(=\)	\( 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1386.r (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.866025 - 0.500000i) q^{2} +(0.500000 + 0.866025i) q^{4} +(1.96593 - 3.40508i) q^{5} +(2.19067 + 1.48356i) q^{7} -1.00000i q^{8} +O(q^{10})\)	\(q+(-0.866025 - 0.500000i) q^{2} +(0.500000 + 0.866025i) q^{4} +(1.96593 - 3.40508i) q^{5} +(2.19067 + 1.48356i) q^{7} -1.00000i q^{8} +(-3.40508 + 1.96593i) q^{10} +(-0.866025 + 0.500000i) q^{11} -2.44949i q^{13} +(-1.15539 - 2.38014i) q^{14} +(-0.500000 + 0.866025i) q^{16} +(-3.64626 - 6.31552i) q^{17} +(3.17914 + 1.83548i) q^{19} +3.93185 q^{20} +1.00000 q^{22} +(2.94098 + 1.69798i) q^{23} +(-5.22973 - 9.05816i) q^{25} +(-1.22474 + 2.12132i) q^{26} +(-0.189469 + 2.63896i) q^{28} -5.16088i q^{29} +(-1.99465 + 1.15161i) q^{31} +(0.866025 - 0.500000i) q^{32} +7.29253i q^{34} +(9.35835 - 4.54284i) q^{35} +(1.98709 - 3.44174i) q^{37} +(-1.83548 - 3.17914i) q^{38} +(-3.40508 - 1.96593i) q^{40} +5.92480 q^{41} -10.7128 q^{43} +(-0.866025 - 0.500000i) q^{44} +(-1.69798 - 2.94098i) q^{46} +(-4.31199 + 7.46859i) q^{47} +(2.59808 + 6.50000i) q^{49} +10.4595i q^{50} +(2.12132 - 1.22474i) q^{52} +(11.5854 - 6.68885i) q^{53} +3.93185i q^{55} +(1.48356 - 2.19067i) q^{56} +(-2.58044 + 4.46945i) q^{58} +(0.975056 + 1.68885i) q^{59} +(5.16813 + 2.98382i) q^{61} +2.30323 q^{62} -1.00000 q^{64} +(-8.34072 - 4.81552i) q^{65} +(-2.33378 - 4.04222i) q^{67} +(3.64626 - 6.31552i) q^{68} +(-10.3760 - 0.744963i) q^{70} +8.90754i q^{71} +(-2.37466 + 1.37101i) q^{73} +(-3.44174 + 1.98709i) q^{74} +3.67095i q^{76} +(-2.63896 - 0.189469i) q^{77} +(7.69798 - 13.3333i) q^{79} +(1.96593 + 3.40508i) q^{80} +(-5.13103 - 2.96240i) q^{82} -6.05521 q^{83} -28.6731 q^{85} +(9.27755 + 5.35640i) q^{86} +(0.500000 + 0.866025i) q^{88} +(-5.87404 + 10.1741i) q^{89} +(3.63397 - 5.36603i) q^{91} +3.39595i q^{92} +(7.46859 - 4.31199i) q^{94} +(12.4999 - 7.21682i) q^{95} +5.85765i q^{97} +(1.00000 - 6.92820i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{4} + 8 q^{5}+O(q^{10}) \) 8 * q + 4 * q^4 + 8 * q^5	\( 8 q + 4 q^{4} + 8 q^{5} - 4 q^{16} - 4 q^{17} + 24 q^{19} + 16 q^{20} + 8 q^{22} + 24 q^{23} - 4 q^{25} + 12 q^{31} + 16 q^{35} + 12 q^{37} - 8 q^{38} + 16 q^{41} - 32 q^{43} + 8 q^{46} + 48 q^{53} - 4 q^{58} + 16 q^{59} + 24 q^{62} - 8 q^{64} - 12 q^{65} - 24 q^{67} + 4 q^{68} - 20 q^{70} - 24 q^{73} - 12 q^{74} + 40 q^{79} + 8 q^{80} + 12 q^{82} + 72 q^{83} - 32 q^{85} - 24 q^{86} + 4 q^{88} - 16 q^{89} + 36 q^{91} + 24 q^{95} + 8 q^{98}+O(q^{100}) \) 8 * q + 4 * q^4 + 8 * q^5 - 4 * q^16 - 4 * q^17 + 24 * q^19 + 16 * q^20 + 8 * q^22 + 24 * q^23 - 4 * q^25 + 12 * q^31 + 16 * q^35 + 12 * q^37 - 8 * q^38 + 16 * q^41 - 32 * q^43 + 8 * q^46 + 48 * q^53 - 4 * q^58 + 16 * q^59 + 24 * q^62 - 8 * q^64 - 12 * q^65 - 24 * q^67 + 4 * q^68 - 20 * q^70 - 24 * q^73 - 12 * q^74 + 40 * q^79 + 8 * q^80 + 12 * q^82 + 72 * q^83 - 32 * q^85 - 24 * q^86 + 4 * q^88 - 16 * q^89 + 36 * q^91 + 24 * q^95 + 8 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more