LMFDB - Embedded newform 1386.2.r.b.1277.4

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:	$ 2^{2}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			1277.4
Root			$-0.258819 - 0.965926i$ of defining polynomial
Character	$\chi$	$=$	1386.1277
Dual form			1386.2.r.b.89.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.866025 - 0.500000i) q^{2} +(0.500000 - 0.866025i) q^{4} +(1.22474 + 2.12132i) q^{5} +(-1.00000 + 2.44949i) q^{7} -1.00000i q^{8} +O(q^{10})$	\(q+(0.866025 - 0.500000i) q^{2} +(0.500000 - 0.866025i) q^{4} +(1.22474 + 2.12132i) q^{5} +(-1.00000 + 2.44949i) q^{7} -1.00000i q^{8} +(2.12132 + 1.22474i) q^{10} +(0.866025 + 0.500000i) q^{11} +2.44949i q^{13} +(0.358719 + 2.62132i) q^{14} +(-0.500000 - 0.866025i) q^{16} +(-0.866025 + 1.50000i) q^{17} +(-1.50000 + 0.866025i) q^{19} +2.44949 q^{20} +1.00000 q^{22} +(-6.27231 + 3.62132i) q^{23} +(-0.500000 + 0.866025i) q^{25} +(1.22474 + 2.12132i) q^{26} +(1.62132 + 2.09077i) q^{28} -1.24264i q^{29} +(1.24264 + 0.717439i) q^{31} +(-0.866025 - 0.500000i) q^{32} +1.73205i q^{34} +(-6.42090 + 0.878680i) q^{35} +(1.62132 + 2.80821i) q^{37} +(-0.866025 + 1.50000i) q^{38} +(2.12132 - 1.22474i) q^{40} +1.43488 q^{41} +1.00000 q^{43} +(0.866025 - 0.500000i) q^{44} +(-3.62132 + 6.27231i) q^{46} +(3.82282 + 6.62132i) q^{47} +(-5.00000 - 4.89898i) q^{49} +1.00000i q^{50} +(2.12132 + 1.22474i) q^{52} +2.44949i q^{55} +(2.44949 + 1.00000i) q^{56} +(-0.621320 - 1.07616i) q^{58} +(-7.49706 + 12.9853i) q^{59} +(9.00000 - 5.19615i) q^{61} +1.43488 q^{62} -1.00000 q^{64} +(-5.19615 + 3.00000i) q^{65} +(6.24264 - 10.8126i) q^{67} +(0.866025 + 1.50000i) q^{68} +(-5.12132 + 3.97141i) q^{70} +7.24264i q^{71} +(4.24264 + 2.44949i) q^{73} +(2.80821 + 1.62132i) q^{74} +1.73205i q^{76} +(-2.09077 + 1.62132i) q^{77} +(-4.00000 - 6.92820i) q^{79} +(1.22474 - 2.12132i) q^{80} +(1.24264 - 0.717439i) q^{82} +12.8418 q^{83} -4.24264 q^{85} +(0.866025 - 0.500000i) q^{86} +(0.500000 - 0.866025i) q^{88} +(-0.507306 - 0.878680i) q^{89} +(-6.00000 - 2.44949i) q^{91} +7.24264i q^{92} +(6.62132 + 3.82282i) q^{94} +(-3.67423 - 2.12132i) q^{95} +3.16693i q^{97} +(-6.77962 - 1.74264i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{4} - 8 q^{7}+O(q^{10}) $ 8 * q + 4 * q^4 - 8 * q^7	$ 8 q + 4 q^{4} - 8 q^{7} - 4 q^{16} - 12 q^{19} + 8 q^{22} - 4 q^{25} - 4 q^{28} - 24 q^{31} - 4 q^{37} + 8 q^{43} - 12 q^{46} - 40 q^{49} + 12 q^{58} + 72 q^{61} - 8 q^{64} + 16 q^{67} - 24 q^{70} - 32 q^{79} - 24 q^{82} + 4 q^{88} - 48 q^{91} + 36 q^{94}+O(q^{100}) $ 8 * q + 4 * q^4 - 8 * q^7 - 4 * q^16 - 12 * q^19 + 8 * q^22 - 4 * q^25 - 4 * q^28 - 24 * q^31 - 4 * q^37 + 8 * q^43 - 12 * q^46 - 40 * q^49 + 12 * q^58 + 72 * q^61 - 8 * q^64 + 16 * q^67 - 24 * q^70 - 32 * q^79 - 24 * q^82 + 4 * q^88 - 48 * q^91 + 36 * q^94

Display 100 coefficients 10 coefficients

Character values

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

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

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
$5$	1.22474	+	2.12132i	0.547723	+	0.948683i	0.998430	+	0.0560116i	$0.0178384\pi$
$5$	1.22474	+	2.12132i	0.547723	+	0.948683i	−0.450708	+	0.892672i	$0.648828\pi$
$6$		0			0
$7$	−1.00000	+	2.44949i	−0.377964	+	0.925820i
$7$	−1.00000	+	2.44949i	−0.377964	+	0.925820i
$8$		−	1.00000i		−	0.353553i
$9$		0			0
$10$	2.12132	+	1.22474i	0.670820	+	0.387298i
$11$	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	$0.110320\pi$
$13$			2.44949i			0.679366i	−0.940540	+	0.339683i	$0.889680\pi$
$14$	0.358719	+	2.62132i	0.0958718	+	0.700577i
$15$		0			0
$16$	−0.500000	−	0.866025i	−0.125000	−	0.216506i
$17$	−0.866025	+	1.50000i	−0.210042	+	0.363803i	−0.951727	−	0.306944i	$-0.900693\pi$
$17$	−0.866025	+	1.50000i	−0.210042	+	0.363803i	0.741685	+	0.670748i	$0.234027\pi$
$18$		0			0
$19$	−1.50000	+	0.866025i	−0.344124	+	0.198680i	−0.662094	−	0.749421i	$-0.730332\pi$
$19$	−1.50000	+	0.866025i	−0.344124	+	0.198680i	0.317970	+	0.948101i	$0.396999\pi$
$20$	2.44949			0.547723
$21$		0			0
$22$	1.00000			0.213201
$23$	−6.27231	+	3.62132i	−1.30787	+	0.755097i	−0.981740	−	0.190228i	$-0.939077\pi$
$23$	−6.27231	+	3.62132i	−1.30787	+	0.755097i	−0.326127	+	0.945326i	$0.605744\pi$
$24$		0			0
$25$	−0.500000	+	0.866025i	−0.100000	+	0.173205i
$26$	1.22474	+	2.12132i	0.240192	+	0.416025i
$27$		0			0
$28$	1.62132	+	2.09077i	0.306401	+	0.395118i
$29$		−	1.24264i		−	0.230753i	−0.993322	−	0.115376i	$-0.963193\pi$
$29$		−	1.24264i		−	0.230753i	0.993322	−	0.115376i	$-0.0368074\pi$
$30$		0			0
$31$	1.24264	+	0.717439i	0.223185	+	0.128856i	0.607424	−	0.794378i	$-0.292203\pi$
$31$	1.24264	+	0.717439i	0.223185	+	0.128856i	−0.384239	+	0.923234i	$0.625536\pi$
$32$	−0.866025	−	0.500000i	−0.153093	−	0.0883883i
$33$		0			0
$34$			1.73205i			0.297044i
$35$	−6.42090	+	0.878680i	−1.08533	+	0.148524i
$36$		0			0
$37$	1.62132	+	2.80821i	0.266543	+	0.461667i	0.967967	−	0.251078i	$-0.0807851\pi$
$37$	1.62132	+	2.80821i	0.266543	+	0.461667i	−0.701423	+	0.712745i	$0.747452\pi$
$38$	−0.866025	+	1.50000i	−0.140488	+	0.243332i
$39$		0			0
$40$	2.12132	−	1.22474i	0.335410	−	0.193649i
$41$	1.43488			0.224090			0.112045	−	0.993703i	$-0.464260\pi$
$41$	1.43488			0.224090			0.112045	+	0.993703i	$0.464260\pi$
$42$		0			0
$43$	1.00000			0.152499			0.0762493	−	0.997089i	$-0.475706\pi$
$43$	1.00000			0.152499			0.0762493	+	0.997089i	$0.475706\pi$
$44$	0.866025	−	0.500000i	0.130558	−	0.0753778i
$45$		0			0
$46$	−3.62132	+	6.27231i	−0.533935	+	0.924802i
$47$	3.82282	+	6.62132i	0.557616	+	0.965819i	0.997695	+	0.0678598i	$0.0216171\pi$
$47$	3.82282	+	6.62132i	0.557616	+	0.965819i	−0.440079	+	0.897959i	$0.645050\pi$
$48$		0			0
$49$	−5.00000	−	4.89898i	−0.714286	−	0.699854i
$50$			1.00000i			0.141421i
$51$		0			0
$52$	2.12132	+	1.22474i	0.294174	+	0.169842i
$53$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$53$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$54$		0			0
$55$			2.44949i			0.330289i
$56$	2.44949	+	1.00000i	0.327327	+	0.133631i
$57$		0			0
$58$	−0.621320	−	1.07616i	−0.0815834	−	0.141307i
$59$	−7.49706	+	12.9853i	−0.976034	+	1.69054i	−0.299552	+	0.954080i	$0.596837\pi$
$59$	−7.49706	+	12.9853i	−0.976034	+	1.69054i	−0.676481	+	0.736460i	$0.736496\pi$
$60$		0			0
$61$	9.00000	−	5.19615i	1.15233	−	0.665299i	0.202878	−	0.979204i	$-0.434971\pi$
$61$	9.00000	−	5.19615i	1.15233	−	0.665299i	0.949454	+	0.313905i	$0.101637\pi$
$62$	1.43488			0.182230
$63$		0			0
$64$	−1.00000			−0.125000
$65$	−5.19615	+	3.00000i	−0.644503	+	0.372104i
$66$		0			0
$67$	6.24264	−	10.8126i	0.762660	−	1.32097i	−0.178815	−	0.983883i	$-0.557226\pi$
$67$	6.24264	−	10.8126i	0.762660	−	1.32097i	0.941475	−	0.337083i	$-0.109440\pi$
$68$	0.866025	+	1.50000i	0.105021	+	0.181902i
$69$		0			0
$70$	−5.12132	+	3.97141i	−0.612115	+	0.474674i
$71$			7.24264i			0.859543i	0.902938	+	0.429772i	$0.141406\pi$
$71$			7.24264i			0.859543i	−0.902938	+	0.429772i	$0.858594\pi$
$72$		0			0
$73$	4.24264	+	2.44949i	0.496564	+	0.286691i	0.727293	−	0.686327i	$-0.240778\pi$
$73$	4.24264	+	2.44949i	0.496564	+	0.286691i	−0.230730	+	0.973018i	$0.574111\pi$
$74$	2.80821	+	1.62132i	0.326448	+	0.188475i
$75$		0			0
$76$			1.73205i			0.198680i
$77$	−2.09077	+	1.62132i	−0.238265	+	0.184767i
$78$		0			0
$79$	−4.00000	−	6.92820i	−0.450035	−	0.779484i	0.548352	−	0.836247i	$-0.315255\pi$
$79$	−4.00000	−	6.92820i	−0.450035	−	0.779484i	−0.998388	+	0.0567635i	$0.981922\pi$
$80$	1.22474	−	2.12132i	0.136931	−	0.237171i
$81$		0			0
$82$	1.24264	−	0.717439i	0.137227	−	0.0792279i
$83$	12.8418			1.40957			0.704785	−	0.709421i	$-0.251043\pi$
$83$	12.8418			1.40957			0.704785	+	0.709421i	$0.251043\pi$
$84$		0			0
$85$	−4.24264			−0.460179
$86$	0.866025	−	0.500000i	0.0933859	−	0.0539164i
$87$		0			0
$88$	0.500000	−	0.866025i	0.0533002	−	0.0923186i
$89$	−0.507306	−	0.878680i	−0.0537743	−	0.0931399i	0.837885	−	0.545846i	$-0.183792\pi$
$89$	−0.507306	−	0.878680i	−0.0537743	−	0.0931399i	−0.891660	+	0.452707i	$0.850458\pi$
$90$		0			0
$91$	−6.00000	−	2.44949i	−0.628971	−	0.256776i
$92$			7.24264i			0.755097i
$93$		0			0
$94$	6.62132	+	3.82282i	0.682937	+	0.394294i
$95$	−3.67423	−	2.12132i	−0.376969	−	0.217643i
$96$		0			0
$97$			3.16693i			0.321553i	0.986991	+	0.160776i	$0.0513998\pi$
$97$			3.16693i			0.321553i	−0.986991	+	0.160776i	$0.948600\pi$
$98$	−6.77962	−	1.74264i	−0.684845	−	0.176033i
$99$		0			0
$100$	0.500000	+	0.866025i	0.0500000	+	0.0866025i
$101$	5.25770	−	9.10660i	0.523161	−	0.906141i	−0.476476	−	0.879187i	$-0.658086\pi$
$101$	5.25770	−	9.10660i	0.523161	−	0.906141i	0.999637	−	0.0269533i	$-0.00858054\pi$
$102$		0			0
$103$	14.1213	−	8.15295i	1.39142	−	0.803334i	0.397943	−	0.917410i	$-0.369724\pi$
$103$	14.1213	−	8.15295i	1.39142	−	0.803334i	0.993472	+	0.114076i	$0.0363908\pi$
$104$	2.44949			0.240192
$105$		0			0
$106$		0			0
$107$	−12.5446	+	7.24264i	−1.21273	+	0.700173i	−0.963354	−	0.268233i	$-0.913560\pi$
$107$	−12.5446	+	7.24264i	−1.21273	+	0.700173i	−0.249380	+	0.968406i	$0.580227\pi$
$108$		0			0
$109$	4.87868	−	8.45012i	0.467293	−	0.809375i	−0.532009	−	0.846739i	$-0.678563\pi$
$109$	4.87868	−	8.45012i	0.467293	−	0.809375i	0.999302	+	0.0373639i	$0.0118961\pi$
$110$	1.22474	+	2.12132i	0.116775	+	0.202260i
$111$		0			0
$112$	2.62132	−	0.358719i	0.247691	−	0.0338958i
$113$			1.75736i			0.165318i	0.996578	+	0.0826592i	$0.0263413\pi$
$113$			1.75736i			0.165318i	−0.996578	+	0.0826592i	$0.973659\pi$
$114$		0			0
$115$	−15.3640	−	8.87039i	−1.43270	−	0.827168i
$116$	−1.07616	−	0.621320i	−0.0999188	−	0.0576881i
$117$		0			0
$118$			14.9941i			1.38032i
$119$	−2.80821	−	3.62132i	−0.257428	−	0.331966i
$120$		0			0
$121$	0.500000	+	0.866025i	0.0454545	+	0.0787296i
$122$	5.19615	−	9.00000i	0.470438	−	0.814822i
$123$		0			0
$124$	1.24264	−	0.717439i	0.111592	−	0.0644279i
$125$	9.79796			0.876356
$126$		0			0
$127$	12.7574			1.13203			0.566016	−	0.824394i	$-0.308484\pi$
$127$	12.7574			1.13203			0.566016	+	0.824394i	$0.308484\pi$
$128$	−0.866025	+	0.500000i	−0.0765466	+	0.0441942i
$129$		0			0
$130$	−3.00000	+	5.19615i	−0.263117	+	0.455733i
$131$	−5.70346	−	9.87868i	−0.498313	−	0.863104i	0.501685	−	0.865051i	$-0.332714\pi$
$131$	−5.70346	−	9.87868i	−0.498313	−	0.863104i	−0.999998	+	0.00194634i	$0.999380\pi$
$132$		0			0
$133$	−0.621320	−	4.54026i	−0.0538753	−	0.393690i
$134$		−	12.4853i		−	1.07856i
$135$		0			0
$136$	1.50000	+	0.866025i	0.128624	+	0.0742611i
$137$	1.52192	+	0.878680i	0.130026	+	0.0750707i	0.563602	−	0.826046i	$-0.309415\pi$
$137$	1.52192	+	0.878680i	0.130026	+	0.0750707i	−0.433576	+	0.901117i	$0.642748\pi$
$138$		0			0
$139$		−	0.297173i		−	0.0252059i	−0.999921	−	0.0126029i	$-0.995988\pi$
$139$		−	0.297173i		−	0.0252059i	0.999921	−	0.0126029i	$-0.00401175\pi$
$140$	−2.44949	+	6.00000i	−0.207020	+	0.507093i
$141$		0			0
$142$	3.62132	+	6.27231i	0.303894	+	0.526361i
$143$	−1.22474	+	2.12132i	−0.102418	+	0.177394i
$144$		0			0
$145$	2.63604	−	1.52192i	0.218911	−	0.126388i
$146$	4.89898			0.405442
$147$		0			0
$148$	3.24264			0.266543
$149$	4.11999	−	2.37868i	0.337523	−	0.194869i	−0.321653	−	0.946858i	$-0.604238\pi$
$149$	4.11999	−	2.37868i	0.337523	−	0.194869i	0.659176	+	0.751989i	$0.270905\pi$
$150$		0			0
$151$	−1.37868	+	2.38794i	−0.112195	+	0.194328i	−0.916655	−	0.399679i	$-0.869122\pi$
$151$	−1.37868	+	2.38794i	−0.112195	+	0.194328i	0.804460	+	0.594007i	$0.202455\pi$
$152$	0.866025	+	1.50000i	0.0702439	+	0.121666i
$153$		0			0
$154$	−1.00000	+	2.44949i	−0.0805823	+	0.197386i
$155$			3.51472i			0.282309i
$156$		0			0
$157$	3.62132	+	2.09077i	0.289013	+	0.166862i	0.637497	−	0.770453i	$-0.279970\pi$
$157$	3.62132	+	2.09077i	0.289013	+	0.166862i	−0.348484	+	0.937315i	$0.613303\pi$
$158$	−6.92820	−	4.00000i	−0.551178	−	0.318223i
$159$		0			0
$160$		−	2.44949i		−	0.193649i
$161$	−2.59808	−	18.9853i	−0.204757	−	1.49625i
$162$		0			0
$163$	7.48528	+	12.9649i	0.586292	+	1.01549i	0.994713	+	0.102694i	$0.0327464\pi$
$163$	7.48528	+	12.9649i	0.586292	+	1.01549i	−0.408420	+	0.912794i	$0.633920\pi$
$164$	0.717439	−	1.24264i	0.0560226	−	0.0970339i
$165$		0			0
$166$	11.1213	−	6.42090i	0.863182	−	0.498358i
$167$	−0.420266			−0.0325212			−0.0162606	−	0.999868i	$-0.505176\pi$
$167$	−0.420266			−0.0325212			−0.0162606	+	0.999868i	$0.505176\pi$
$168$		0			0
$169$	7.00000			0.538462
$170$	−3.67423	+	2.12132i	−0.281801	+	0.162698i
$171$		0			0
$172$	0.500000	−	0.866025i	0.0381246	−	0.0660338i
$173$	−1.01461	−	1.75736i	−0.0771395	−	0.133610i	0.824875	−	0.565315i	$-0.191245\pi$
$173$	−1.01461	−	1.75736i	−0.0771395	−	0.133610i	−0.902015	+	0.431705i	$0.857912\pi$
$174$		0			0
$175$	−1.62132	−	2.09077i	−0.122560	−	0.158047i
$176$		−	1.00000i		−	0.0753778i
$177$		0			0
$178$	−0.878680	−	0.507306i	−0.0658598	−	0.0380242i
$179$	−7.79423	−	4.50000i	−0.582568	−	0.336346i	0.179585	−	0.983742i	$-0.442524\pi$
$179$	−7.79423	−	4.50000i	−0.582568	−	0.336346i	−0.762153	+	0.647397i	$0.775858\pi$
$180$		0			0
$181$			4.89898i			0.364138i	0.983286	+	0.182069i	$0.0582795\pi$
$181$			4.89898i			0.364138i	−0.983286	+	0.182069i	$0.941721\pi$
$182$	−6.42090	+	0.878680i	−0.475949	+	0.0651321i
$183$		0			0
$184$	3.62132	+	6.27231i	0.266967	+	0.462401i
$185$	−3.97141	+	6.87868i	−0.291984	+	0.505731i
$186$		0			0
$187$	−1.50000	+	0.866025i	−0.109691	+	0.0633300i
$188$	7.64564			0.557616
$189$		0			0
$190$	−4.24264			−0.307794
$191$	−14.6969	+	8.48528i	−1.06343	+	0.613973i	−0.926380	−	0.376590i	$-0.877096\pi$
$191$	−14.6969	+	8.48528i	−1.06343	+	0.613973i	−0.137053	+	0.990564i	$0.543763\pi$
$192$		0			0
$193$	10.3640	−	17.9509i	0.746014	−	1.29213i	−0.203705	−	0.979032i	$-0.565298\pi$
$193$	10.3640	−	17.9509i	0.746014	−	1.29213i	0.949719	−	0.313102i	$-0.101368\pi$
$194$	1.58346	+	2.74264i	0.113686	+	0.196910i
$195$		0			0
$196$	−6.74264	+	1.88064i	−0.481617	+	0.134331i
$197$		−	3.72792i		−	0.265603i	−0.991143	−	0.132802i	$-0.957603\pi$
$197$		−	3.72792i		−	0.265603i	0.991143	−	0.132802i	$-0.0423974\pi$
$198$		0			0
$199$	−6.36396	−	3.67423i	−0.451129	−	0.260460i	0.257178	−	0.966364i	$-0.417207\pi$
$199$	−6.36396	−	3.67423i	−0.451129	−	0.260460i	−0.708307	+	0.705905i	$0.750541\pi$
$200$	0.866025	+	0.500000i	0.0612372	+	0.0353553i
$201$		0			0
$202$		−	10.5154i		−	0.739861i
$203$	3.04384	+	1.24264i	0.213635	+	0.0872163i
$204$		0			0
$205$	1.75736	+	3.04384i	0.122739	+	0.212591i
$206$	8.15295	−	14.1213i	0.568043	−	0.983879i
$207$		0			0
$208$	2.12132	−	1.22474i	0.147087	−	0.0849208i
$209$	−1.73205			−0.119808
$210$		0			0
$211$	−22.4853			−1.54795			−0.773975	−	0.633216i	$-0.781735\pi$
$211$	−22.4853			−1.54795			−0.773975	+	0.633216i	$0.781735\pi$
$212$		0			0
$213$		0			0
$214$	−7.24264	+	12.5446i	−0.495097	+	0.857533i
$215$	1.22474	+	2.12132i	0.0835269	+	0.144673i
$216$		0			0
$217$	−3.00000	+	2.32640i	−0.203653	+	0.157926i
$218$		−	9.75736i		−	0.660852i
$219$		0			0
$220$	2.12132	+	1.22474i	0.143019	+	0.0825723i
$221$	−3.67423	−	2.12132i	−0.247156	−	0.142695i
$222$		0			0
$223$		−	9.37769i		−	0.627977i	−0.949427	−	0.313988i	$-0.898335\pi$
$223$		−	9.37769i		−	0.627977i	0.949427	−	0.313988i	$-0.101665\pi$
$224$	2.09077	−	1.62132i	0.139695	−	0.108329i
$225$		0			0
$226$	0.878680	+	1.52192i	0.0584489	+	0.101236i
$227$	3.16693	−	5.48528i	0.210196	−	0.364071i	−0.741579	−	0.670865i	$-0.765923\pi$
$227$	3.16693	−	5.48528i	0.210196	−	0.364071i	0.951776	+	0.306794i	$0.0992563\pi$
$228$		0			0
$229$	−12.0000	+	6.92820i	−0.792982	+	0.457829i	−0.841011	−	0.541017i	$-0.818039\pi$
$229$	−12.0000	+	6.92820i	−0.792982	+	0.457829i	0.0480291	+	0.998846i	$0.484706\pi$
$230$	−17.7408			−1.16979
$231$		0			0
$232$	−1.24264			−0.0815834
$233$	−15.1427	+	8.74264i	−0.992031	+	0.572749i	−0.905881	−	0.423533i	$-0.860790\pi$
$233$	−15.1427	+	8.74264i	−0.992031	+	0.572749i	−0.0861503	+	0.996282i	$0.527457\pi$
$234$		0			0
$235$	−9.36396	+	16.2189i	−0.610837	+	1.05800i
$236$	7.49706	+	12.9853i	0.488017	+	0.845270i
$237$		0			0
$238$	−4.24264	−	1.73205i	−0.275010	−	0.112272i
$239$			12.7279i			0.823301i	0.911342	+	0.411650i	$0.135048\pi$
$239$			12.7279i			0.823301i	−0.911342	+	0.411650i	$0.864952\pi$
$240$		0			0
$241$	−10.9706	−	6.33386i	−0.706676	−	0.408000i	0.103153	−	0.994665i	$-0.467107\pi$
$241$	−10.9706	−	6.33386i	−0.706676	−	0.408000i	−0.809829	+	0.586666i	$0.800440\pi$
$242$	0.866025	+	0.500000i	0.0556702	+	0.0321412i
$243$		0			0
$244$		−	10.3923i		−	0.665299i
$245$	4.26858	−	16.6066i	0.272710	−	1.06096i
$246$		0			0
$247$	−2.12132	−	3.67423i	−0.134976	−	0.233786i
$248$	0.717439	−	1.24264i	0.0455574	−	0.0789078i
$249$		0			0
$250$	8.48528	−	4.89898i	0.536656	−	0.309839i
$251$	18.4582			1.16507			0.582536	−	0.812805i	$-0.302060\pi$
$251$	18.4582			1.16507			0.582536	+	0.812805i	$0.302060\pi$
$252$		0			0
$253$	−7.24264			−0.455341
$254$	11.0482	−	6.37868i	0.693226	−	0.400234i
$255$		0			0
$256$	−0.500000	+	0.866025i	−0.0312500	+	0.0541266i
$257$	7.85578	+	13.6066i	0.490030	+	0.848756i	0.999934	−	0.0114746i	$-0.00365256\pi$
$257$	7.85578	+	13.6066i	0.490030	+	0.848756i	−0.509904	+	0.860231i	$0.670319\pi$
$258$		0			0
$259$	−8.50000	+	1.16320i	−0.528164	+	0.0722776i
$260$			6.00000i			0.372104i
$261$		0			0
$262$	−9.87868	−	5.70346i	−0.610307	−	0.352361i
$263$	−22.9369	−	13.2426i	−1.41435	−	0.816576i	−0.418557	−	0.908191i	$-0.637464\pi$
$263$	−22.9369	−	13.2426i	−1.41435	−	0.816576i	−0.995795	+	0.0916144i	$0.970797\pi$
$264$		0			0
$265$		0			0
$266$	−2.80821	−	3.62132i	−0.172182	−	0.222037i
$267$		0			0
$268$	−6.24264	−	10.8126i	−0.381330	−	0.660483i
$269$	13.3491	−	23.1213i	0.813909	−	1.40973i	−0.0961996	−	0.995362i	$-0.530669\pi$
$269$	13.3491	−	23.1213i	0.813909	−	1.40973i	0.910109	−	0.414370i	$-0.135998\pi$
$270$		0			0
$271$	1.75736	−	1.01461i	0.106752	−	0.0616333i	−0.445673	−	0.895196i	$-0.647036\pi$
$271$	1.75736	−	1.01461i	0.106752	−	0.0616333i	0.552425	+	0.833562i	$0.313702\pi$
$272$	1.73205			0.105021
$273$		0			0
$274$	1.75736			0.106166
$275$	−0.866025	+	0.500000i	−0.0522233	+	0.0301511i
$276$		0			0
$277$	14.7279	−	25.5095i	0.884915	−	1.53272i	0.0391041	−	0.999235i	$-0.487550\pi$
$277$	14.7279	−	25.5095i	0.884915	−	1.53272i	0.845811	−	0.533483i	$-0.179117\pi$
$278$	−0.148586	−	0.257359i	−0.00891162	−	0.0154354i
$279$		0			0
$280$	0.878680	+	6.42090i	0.0525112	+	0.383722i
$281$		−	23.4853i		−	1.40101i	−0.713645	−	0.700507i	$-0.752957\pi$
$281$		−	23.4853i		−	1.40101i	0.713645	−	0.700507i	$-0.247043\pi$
$282$		0			0
$283$	−26.4853	−	15.2913i	−1.57439	−	0.908973i	−0.995621	−	0.0934783i	$-0.970201\pi$
$283$	−26.4853	−	15.2913i	−1.57439	−	0.908973i	−0.578765	−	0.815494i	$-0.696465\pi$
$284$	6.27231	+	3.62132i	0.372193	+	0.214886i
$285$		0			0
$286$			2.44949i			0.144841i
$287$	−1.43488	+	3.51472i	−0.0846982	+	0.207467i
$288$		0			0
$289$	7.00000	+	12.1244i	0.411765	+	0.713197i
$290$	1.52192	−	2.63604i	0.0893701	−	0.154794i
$291$		0			0
$292$	4.24264	−	2.44949i	0.248282	−	0.143346i
$293$	−7.05130			−0.411941			−0.205971	−	0.978558i	$-0.566035\pi$
$293$	−7.05130			−0.411941			−0.205971	+	0.978558i	$0.566035\pi$
$294$		0			0
$295$	−36.7279			−2.13838
$296$	2.80821	−	1.62132i	0.163224	−	0.0942373i
$297$		0			0
$298$	2.37868	−	4.11999i	0.137793	−	0.238665i
$299$	−8.87039	−	15.3640i	−0.512988	−	0.888521i
$300$		0			0
$301$	−1.00000	+	2.44949i	−0.0576390	+	0.141186i
$302$			2.75736i			0.158668i
$303$		0			0
$304$	1.50000	+	0.866025i	0.0860309	+	0.0496700i
$305$	22.0454	+	12.7279i	1.26232	+	0.728799i
$306$		0			0
$307$		−	7.52255i		−	0.429335i	−0.976687	−	0.214667i	$-0.931133\pi$
$307$		−	7.52255i		−	0.429335i	0.976687	−	0.214667i	$-0.0688667\pi$
$308$	0.358719	+	2.62132i	0.0204399	+	0.149364i
$309$		0			0
$310$	1.75736	+	3.04384i	0.0998113	+	0.172878i
$311$	13.2005	−	22.8640i	0.748532	−	1.29650i	−0.199994	−	0.979797i	$-0.564092\pi$
$311$	13.2005	−	22.8640i	0.748532	−	1.29650i	0.948526	−	0.316699i	$-0.102575\pi$
$312$		0			0
$313$	6.98528	−	4.03295i	0.394831	−	0.227956i	−0.289420	−	0.957202i	$-0.593462\pi$
$313$	6.98528	−	4.03295i	0.394831	−	0.227956i	0.684251	+	0.729246i	$0.260129\pi$
$314$	4.18154			0.235978
$315$		0			0
$316$	−8.00000			−0.450035
$317$	6.71807	−	3.87868i	0.377324	−	0.217848i	−0.299329	−	0.954150i	$-0.596763\pi$
$317$	6.71807	−	3.87868i	0.377324	−	0.217848i	0.676654	+	0.736302i	$0.263430\pi$
$318$		0			0
$319$	0.621320	−	1.07616i	0.0347873	−	0.0602533i
$320$	−1.22474	−	2.12132i	−0.0684653	−	0.118585i
$321$		0			0
$322$	−11.7426	−	15.1427i	−0.654392	−	0.843870i
$323$		−	3.00000i		−	0.166924i
$324$		0			0
$325$	−2.12132	−	1.22474i	−0.117670	−	0.0679366i
$326$	12.9649	+	7.48528i	0.718059	+	0.414571i
$327$		0			0
$328$		−	1.43488i		−	0.0792279i
$329$	−20.0417	+	2.74264i	−1.10493	+	0.151207i
$330$		0			0
$331$	7.00000	+	12.1244i	0.384755	+	0.666415i	0.991735	−	0.128302i	$-0.0409527\pi$
$331$	7.00000	+	12.1244i	0.384755	+	0.666415i	−0.606980	+	0.794717i	$0.707619\pi$
$332$	6.42090	−	11.1213i	0.352393	−	0.610362i
$333$		0			0
$334$	−0.363961	+	0.210133i	−0.0199151	+	0.0114980i
$335$	30.5826			1.67090
$336$		0			0
$337$	−24.2426			−1.32058			−0.660290	−	0.751010i	$-0.729567\pi$
$337$	−24.2426			−1.32058			−0.660290	+	0.751010i	$0.729567\pi$
$338$	6.06218	−	3.50000i	0.329739	−	0.190375i
$339$		0			0
$340$	−2.12132	+	3.67423i	−0.115045	+	0.199263i
$341$	0.717439	+	1.24264i	0.0388515	+	0.0672928i
$342$		0			0
$343$	17.0000	−	7.34847i	0.917914	−	0.396780i
$344$		−	1.00000i		−	0.0539164i
$345$		0			0
$346$	−1.75736	−	1.01461i	−0.0944762	−	0.0545459i
$347$	13.4361	+	7.75736i	0.721290	+	0.416437i	0.815227	−	0.579141i	$-0.196612\pi$
$347$	13.4361	+	7.75736i	0.721290	+	0.416437i	−0.0939374	+	0.995578i	$0.529945\pi$
$348$		0			0
$349$			14.6969i			0.786709i	0.919387	+	0.393355i	$0.128686\pi$
$349$			14.6969i			0.786709i	−0.919387	+	0.393355i	$0.871314\pi$
$350$	−2.44949	−	1.00000i	−0.130931	−	0.0534522i
$351$		0			0
$352$	−0.500000	−	0.866025i	−0.0266501	−	0.0461593i
$353$	4.39167	−	7.60660i	0.233745	−	0.404859i	−0.725162	−	0.688578i	$-0.758235\pi$
$353$	4.39167	−	7.60660i	0.233745	−	0.404859i	0.958907	+	0.283720i	$0.0915685\pi$
$354$		0			0
$355$	−15.3640	+	8.87039i	−0.815434	+	0.470791i
$356$	−1.01461			−0.0537743
$357$		0			0
$358$	−9.00000			−0.475665
$359$	0.891519	−	0.514719i	0.0470526	−	0.0271658i	−0.476289	−	0.879289i	$-0.658018\pi$
$359$	0.891519	−	0.514719i	0.0470526	−	0.0271658i	0.523342	+	0.852123i	$0.324685\pi$
$360$		0			0
$361$	−8.00000	+	13.8564i	−0.421053	+	0.729285i
$362$	2.44949	+	4.24264i	0.128742	+	0.222988i
$363$		0			0
$364$	−5.12132	+	3.97141i	−0.268430	+	0.208158i
$365$			12.0000i			0.628109i
$366$		0			0
$367$	10.2426	+	5.91359i	0.534661	+	0.308687i	0.742913	−	0.669388i	$-0.233444\pi$
$367$	10.2426	+	5.91359i	0.534661	+	0.308687i	−0.208251	+	0.978075i	$0.566777\pi$
$368$	6.27231	+	3.62132i	0.326967	+	0.188774i
$369$		0			0
$370$			7.94282i			0.412927i
$371$		0			0
$372$		0			0
$373$	17.0000	+	29.4449i	0.880227	+	1.52460i	0.851089	+	0.525022i	$0.175943\pi$
$373$	17.0000	+	29.4449i	0.880227	+	1.52460i	0.0291379	+	0.999575i	$0.490724\pi$
$374$	−0.866025	+	1.50000i	−0.0447811	+	0.0775632i
$375$		0			0
$376$	6.62132	−	3.82282i	0.341469	−	0.197147i
$377$	3.04384			0.156766
$378$		0			0
$379$	−32.7279			−1.68112			−0.840560	−	0.541718i	$-0.817774\pi$
$379$	−32.7279			−1.68112			−0.840560	+	0.541718i	$0.817774\pi$
$380$	−3.67423	+	2.12132i	−0.188484	+	0.108821i
$381$		0			0
$382$	−8.48528	+	14.6969i	−0.434145	+	0.751961i
$383$	19.2372	+	33.3198i	0.982975	+	1.70256i	0.650611	+	0.759411i	$0.274513\pi$
$383$	19.2372	+	33.3198i	0.982975	+	1.70256i	0.332364	+	0.943151i	$0.392154\pi$
$384$		0			0
$385$	−6.00000	−	2.44949i	−0.305788	−	0.124838i
$386$		−	20.7279i		−	1.05502i
$387$		0			0
$388$	2.74264	+	1.58346i	0.139236	+	0.0803882i
$389$	−31.8073	−	18.3640i	−1.61269	−	0.931090i	−0.988742	−	0.149627i	$-0.952193\pi$
$389$	−31.8073	−	18.3640i	−1.61269	−	0.931090i	−0.623952	−	0.781462i	$-0.714474\pi$
$390$		0			0
$391$		−	12.5446i		−	0.634409i
$392$	−4.89898	+	5.00000i	−0.247436	+	0.252538i
$393$		0			0
$394$	−1.86396	−	3.22848i	−0.0939050	−	0.162648i
$395$	9.79796	−	16.9706i	0.492989	−	0.853882i
$396$		0			0
$397$	28.3492	−	16.3674i	1.42281	−	0.821458i	0.426270	−	0.904596i	$-0.359827\pi$
$397$	28.3492	−	16.3674i	1.42281	−	0.821458i	0.996538	+	0.0831378i	$0.0264941\pi$
$398$	−7.34847			−0.368345
$399$		0			0
$400$	1.00000			0.0500000
$401$	24.4588	−	14.1213i	1.22142	−	0.705185i	0.256197	−	0.966625i	$-0.417531\pi$
$401$	24.4588	−	14.1213i	1.22142	−	0.705185i	0.965220	+	0.261440i	$0.0841972\pi$
$402$		0			0
$403$	−1.75736	+	3.04384i	−0.0875403	+	0.151624i
$404$	−5.25770	−	9.10660i	−0.261580	−	0.453070i
$405$		0			0
$406$	3.25736	−	0.445759i	0.161660	−	0.0221227i
$407$			3.24264i			0.160732i
$408$		0			0
$409$	0.363961	+	0.210133i	0.0179967	+	0.0103904i	0.508971	−	0.860783i	$-0.330026\pi$
$409$	0.363961	+	0.210133i	0.0179967	+	0.0103904i	−0.490975	+	0.871174i	$0.663359\pi$
$410$	3.04384	+	1.75736i	0.150324	+	0.0867898i
$411$		0			0
$412$		−	16.3059i		−	0.803334i
$413$	−24.3103	−	31.3492i	−1.19623	−	1.54260i
$414$		0			0
$415$	15.7279	+	27.2416i	0.772053	+	1.33724i
$416$	1.22474	−	2.12132i	0.0600481	−	0.104006i
$417$		0			0
$418$	−1.50000	+	0.866025i	−0.0733674	+	0.0423587i
$419$	−4.60181			−0.224813			−0.112406	−	0.993662i	$-0.535856\pi$
$419$	−4.60181			−0.224813			−0.112406	+	0.993662i	$0.535856\pi$
$420$		0			0
$421$	28.6985			1.39868			0.699339	−	0.714790i	$-0.253478\pi$
$421$	28.6985			1.39868			0.699339	+	0.714790i	$0.253478\pi$
$422$	−19.4728	+	11.2426i	−0.947922	+	0.547283i
$423$		0			0
$424$		0			0
$425$	−0.866025	−	1.50000i	−0.0420084	−	0.0727607i
$426$		0			0
$427$	3.72792	+	27.2416i	0.180407	+	1.31831i
$428$			14.4853i			0.700173i
$429$		0			0
$430$	2.12132	+	1.22474i	0.102299	+	0.0590624i
$431$	14.6969	+	8.48528i	0.707927	+	0.408722i	0.810293	−	0.586025i	$-0.199308\pi$
$431$	14.6969	+	8.48528i	0.707927	+	0.408722i	−0.102366	+	0.994747i	$0.532641\pi$
$432$		0			0
$433$			21.3280i			1.02496i	0.858700	+	0.512478i	$0.171273\pi$
$433$			21.3280i			1.02496i	−0.858700	+	0.512478i	$0.828727\pi$
$434$	−1.43488	+	3.51472i	−0.0688763	+	0.168712i
$435$		0			0
$436$	−4.87868	−	8.45012i	−0.233646	−	0.404687i
$437$	6.27231	−	10.8640i	0.300045	−	0.519694i
$438$		0			0
$439$	21.6213	−	12.4831i	1.03193	−	0.595785i	0.114393	−	0.993436i	$-0.463508\pi$
$439$	21.6213	−	12.4831i	1.03193	−	0.595785i	0.917537	+	0.397651i	$0.130175\pi$
$440$	2.44949			0.116775
$441$		0			0
$442$	−4.24264			−0.201802
$443$	−2.59808	+	1.50000i	−0.123438	+	0.0712672i	−0.560448	−	0.828190i	$-0.689371\pi$
$443$	−2.59808	+	1.50000i	−0.123438	+	0.0712672i	0.437009	+	0.899457i	$0.356038\pi$
$444$		0			0
$445$	1.24264	−	2.15232i	0.0589068	−	0.102030i
$446$	−4.68885	−	8.12132i	−0.222023	−	0.384556i
$447$		0			0
$448$	1.00000	−	2.44949i	0.0472456	−	0.115728i
$449$			13.4558i			0.635021i	0.948255	+	0.317510i	$0.102847\pi$
$449$			13.4558i			0.635021i	−0.948255	+	0.317510i	$0.897153\pi$
$450$		0			0
$451$	1.24264	+	0.717439i	0.0585137	+	0.0337829i
$452$	1.52192	+	0.878680i	0.0715850	+	0.0413296i
$453$		0			0
$454$		−	6.33386i		−	0.297263i
$455$	−2.15232	−	15.7279i	−0.100902	−	0.737336i
$456$		0			0
$457$	−16.8492	−	29.1837i	−0.788174	−	1.36516i	−0.927084	−	0.374853i	$-0.877693\pi$
$457$	−16.8492	−	29.1837i	−0.788174	−	1.36516i	0.138910	−	0.990305i	$-0.455640\pi$
$458$	−6.92820	+	12.0000i	−0.323734	+	0.560723i
$459$		0			0
$460$	−15.3640	+	8.87039i	−0.716348	+	0.413584i
$461$	−19.4728			−0.906940			−0.453470	−	0.891272i	$-0.649814\pi$
$461$	−19.4728			−0.906940			−0.453470	+	0.891272i	$0.649814\pi$
$462$		0			0
$463$	−29.4558			−1.36893			−0.684465	−	0.729046i	$-0.739964\pi$
$463$	−29.4558			−1.36893			−0.684465	+	0.729046i	$0.739964\pi$
$464$	−1.07616	+	0.621320i	−0.0499594	+	0.0288441i
$465$		0			0
$466$	−8.74264	+	15.1427i	−0.404995	+	0.701472i
$467$	2.59808	+	4.50000i	0.120225	+	0.208235i	0.919856	−	0.392256i	$-0.128305\pi$
$467$	2.59808	+	4.50000i	0.120225	+	0.208235i	−0.799632	+	0.600491i	$0.794972\pi$
$468$		0			0
$469$	20.2426	+	26.1039i	0.934718	+	1.20536i
$470$			18.7279i			0.863855i
$471$		0			0
$472$	12.9853	+	7.49706i	0.597696	+	0.345080i
$473$	0.866025	+	0.500000i	0.0398199	+	0.0229900i
$474$		0			0
$475$		−	1.73205i		−	0.0794719i
$476$	−4.54026	+	0.621320i	−0.208102	+	0.0284782i
$477$		0			0
$478$	6.36396	+	11.0227i	0.291081	+	0.504167i
$479$	−13.1390	+	22.7574i	−0.600335	+	1.03981i	0.392435	+	0.919780i	$0.371633\pi$
$479$	−13.1390	+	22.7574i	−0.600335	+	1.03981i	−0.992770	+	0.120031i	$0.961701\pi$
$480$		0			0
$481$	−6.87868	+	3.97141i	−0.313641	+	0.181081i
$482$	−12.6677			−0.576999
$483$		0			0
$484$	1.00000			0.0454545
$485$	−6.71807	+	3.87868i	−0.305052	+	0.176122i
$486$		0			0
$487$	−0.606602	+	1.05066i	−0.0274877	+	0.0476102i	−0.879442	−	0.476006i	$-0.842084\pi$
$487$	−0.606602	+	1.05066i	−0.0274877	+	0.0476102i	0.851954	+	0.523616i	$0.175417\pi$
$488$	−5.19615	−	9.00000i	−0.235219	−	0.407411i
$489$		0			0
$490$	−4.60660	−	16.5160i	−0.208105	−	0.746118i
$491$		−	17.2721i		−	0.779478i	−0.920925	−	0.389739i	$-0.872565\pi$
$491$		−	17.2721i		−	0.779478i	0.920925	−	0.389739i	$-0.127435\pi$
$492$		0			0
$493$	1.86396	+	1.07616i	0.0839486	+	0.0484677i
$494$	−3.67423	−	2.12132i	−0.165312	−	0.0954427i
$495$		0			0
$496$		−	1.43488i		−	0.0644279i
$497$	−17.7408	−	7.24264i	−0.795782	−	0.324877i
$498$		0			0
$499$	2.60660	+	4.51477i	0.116688	+	0.202109i	0.918453	−	0.395530i	$-0.129439\pi$
$499$	2.60660	+	4.51477i	0.116688	+	0.202109i	−0.801766	+	0.597639i	$0.796106\pi$
$500$	4.89898	−	8.48528i	0.219089	−	0.379473i
$501$		0			0
$502$	15.9853	−	9.22911i	0.713458	−	0.411915i
$503$	20.1903			0.900239			0.450120	−	0.892968i	$-0.351381\pi$
$503$	20.1903			0.900239			0.450120	+	0.892968i	$0.351381\pi$
$504$		0			0
$505$	25.7574			1.14619
$506$	−6.27231	+	3.62132i	−0.278838	+	0.160987i
$507$		0			0
$508$	6.37868	−	11.0482i	0.283008	−	0.490184i
$509$	20.1903	+	34.9706i	0.894918	+	1.55004i	0.833906	+	0.551906i	$0.186099\pi$
$509$	20.1903	+	34.9706i	0.894918	+	1.55004i	0.0610113	+	0.998137i	$0.480567\pi$
$510$		0			0
$511$	−10.2426	+	7.94282i	−0.453108	+	0.351369i
$512$			1.00000i			0.0441942i
$513$		0			0
$514$	13.6066	+	7.85578i	0.600161	+	0.346503i
$515$	34.5900	+	19.9706i	1.52422	+	0.880008i
$516$		0			0
$517$			7.64564i			0.336255i
$518$	−6.77962	+	5.25736i	−0.297879	+	0.230995i
$519$		0			0
$520$	3.00000	+	5.19615i	0.131559	+	0.227866i
$521$	12.9649	−	22.4558i	0.568002	−	0.983808i	−0.428761	−	0.903418i	$-0.641050\pi$
$521$	12.9649	−	22.4558i	0.568002	−	0.983808i	0.996763	−	0.0803906i	$-0.0256168\pi$
$522$		0			0
$523$	−18.7279	+	10.8126i	−0.818915	+	0.472801i	−0.850042	−	0.526715i	$-0.823424\pi$
$523$	−18.7279	+	10.8126i	−0.818915	+	0.472801i	0.0311273	+	0.999515i	$0.490090\pi$
$524$	−11.4069			−0.498313
$525$		0			0
$526$	−26.4853			−1.15481
$527$	−2.15232	+	1.24264i	−0.0937564	+	0.0541303i
$528$		0			0
$529$	14.7279	−	25.5095i	0.640344	−	1.10911i
$530$		0			0
$531$		0			0
$532$	−4.24264	−	1.73205i	−0.183942	−	0.0750939i
$533$			3.51472i			0.152239i
$534$		0			0
$535$	−30.7279	−	17.7408i	−1.32848	−	0.767001i
$536$	−10.8126	−	6.24264i	−0.467032	−	0.269641i
$537$		0			0
$538$		−	26.6982i		−	1.15104i
$539$	−1.88064	−	6.74264i	−0.0810048	−	0.290426i
$540$		0			0
$541$	9.75736	+	16.9002i	0.419502	+	0.726598i	0.995889	−	0.0905782i	$-0.0288715\pi$
$541$	9.75736	+	16.9002i	0.419502	+	0.726598i	−0.576388	+	0.817176i	$0.695538\pi$
$542$	1.01461	−	1.75736i	0.0435813	−	0.0754850i
$543$		0			0
$544$	1.50000	−	0.866025i	0.0643120	−	0.0371305i
$545$	23.9006			1.02379
$546$		0			0
$547$	−10.5147			−0.449577			−0.224788	−	0.974408i	$-0.572169\pi$
$547$	−10.5147			−0.449577			−0.224788	+	0.974408i	$0.572169\pi$
$548$	1.52192	−	0.878680i	0.0650131	−	0.0375353i
$549$		0			0
$550$	−0.500000	+	0.866025i	−0.0213201	+	0.0369274i
$551$	1.07616	+	1.86396i	0.0458459	+	0.0794074i
$552$		0			0
$553$	20.9706	−	2.86976i	0.891759	−	0.122034i
$554$		−	29.4558i		−	1.25146i
$555$		0			0
$556$	−0.257359	−	0.148586i	−0.0109145	−	0.00630147i
$557$	33.5139	+	19.3492i	1.42003	+	0.819854i	0.996301	−	0.0859359i	$-0.0273880\pi$
$557$	33.5139	+	19.3492i	1.42003	+	0.819854i	0.423728	+	0.905790i	$0.360721\pi$
$558$		0			0
$559$			2.44949i			0.103602i
$560$	3.97141	+	5.12132i	0.167823	+	0.216415i
$561$		0			0
$562$	−11.7426	−	20.3389i	−0.495333	−	0.857943i
$563$	−2.53653	+	4.39340i	−0.106902	+	0.185160i	−0.914514	−	0.404555i	$-0.867426\pi$
$563$	−2.53653	+	4.39340i	−0.106902	+	0.185160i	0.807612	+	0.589715i	$0.200760\pi$
$564$		0			0
$565$	−3.72792	+	2.15232i	−0.156835	+	0.0905486i
$566$	−30.5826			−1.28548
$567$		0			0
$568$	7.24264			0.303894
$569$	−24.6435	+	14.2279i	−1.03311	+	0.596466i	−0.917874	−	0.396873i	$-0.870095\pi$
$569$	−24.6435	+	14.2279i	−1.03311	+	0.596466i	−0.115235	+	0.993338i	$0.536762\pi$
$570$		0			0
$571$	13.7426	−	23.8030i	0.575112	−	0.996123i	−0.420918	−	0.907099i	$-0.638292\pi$
$571$	13.7426	−	23.8030i	0.575112	−	0.996123i	0.996030	−	0.0890238i	$-0.0283747\pi$
$572$	1.22474	+	2.12132i	0.0512092	+	0.0886969i
$573$		0			0
$574$	0.514719	+	3.76127i	0.0214839	+	0.156993i
$575$		−	7.24264i		−	0.302039i
$576$		0			0
$577$	3.00000	+	1.73205i	0.124892	+	0.0721062i	0.561144	−	0.827718i	$-0.310361\pi$
$577$	3.00000	+	1.73205i	0.124892	+	0.0721062i	−0.436253	+	0.899824i	$0.643695\pi$
$578$	12.1244	+	7.00000i	0.504307	+	0.291162i
$579$		0			0
$580$		−	3.04384i		−	0.126388i
$581$	−12.8418	+	31.4558i	−0.532767	+	1.30501i
$582$		0			0
$583$		0			0
$584$	2.44949	−	4.24264i	0.101361	−	0.175562i
$585$		0			0
$586$	−6.10660	+	3.52565i	−0.252261	+	0.145643i
$587$	−14.4508			−0.596446			−0.298223	−	0.954496i	$-0.596394\pi$
$587$	−14.4508			−0.596446			−0.298223	+	0.954496i	$0.596394\pi$
$588$		0			0
$589$	−2.48528			−0.102404
$590$	−31.8073	+	18.3640i	−1.30949	+	0.756032i
$591$		0			0
$592$	1.62132	−	2.80821i	0.0666359	−	0.115417i
$593$	2.89525	+	5.01472i	0.118894	+	0.205930i	0.919329	−	0.393489i	$-0.128732\pi$
$593$	2.89525	+	5.01472i	0.118894	+	0.205930i	−0.800436	+	0.599418i	$0.795399\pi$
$594$		0			0
$595$	4.24264	−	10.3923i	0.173931	−	0.426043i
$596$		−	4.75736i		−	0.194869i
$597$		0			0
$598$	−15.3640	−	8.87039i	−0.628279	−	0.362737i
$599$	−15.5885	−	9.00000i	−0.636927	−	0.367730i	0.146503	−	0.989210i	$-0.453198\pi$
$599$	−15.5885	−	9.00000i	−0.636927	−	0.367730i	−0.783430	+	0.621480i	$0.786532\pi$
$600$		0			0
$601$		−	17.1464i		−	0.699417i	−0.936858	−	0.349709i	$-0.886281\pi$
$601$		−	17.1464i		−	0.699417i	0.936858	−	0.349709i	$-0.113719\pi$
$602$	0.358719	+	2.62132i	0.0146203	+	0.106837i
$603$		0			0
$604$	1.37868	+	2.38794i	0.0560977	+	0.0971640i
$605$	−1.22474	+	2.12132i	−0.0497930	+	0.0862439i
$606$		0			0
$607$	29.4853	−	17.0233i	1.19677	−	0.690956i	0.236937	−	0.971525i	$-0.423857\pi$
$607$	29.4853	−	17.0233i	1.19677	−	0.690956i	0.959834	+	0.280569i	$0.0905232\pi$
$608$	1.73205			0.0702439
$609$		0			0
$610$	25.4558			1.03068
$611$	−16.2189	+	9.36396i	−0.656145	+	0.378825i
$612$		0			0
$613$	−16.3640	+	28.3432i	−0.660934	+	1.14477i	0.319436	+	0.947608i	$0.396506\pi$
$613$	−16.3640	+	28.3432i	−0.660934	+	1.14477i	−0.980371	+	0.197164i	$0.936827\pi$
$614$	−3.76127	−	6.51472i	−0.151793	−	0.262913i
$615$		0			0
$616$	1.62132	+	2.09077i	0.0653249	+	0.0842395i
$617$		−	9.21320i		−	0.370910i	−0.982653	−	0.185455i	$-0.940624\pi$
$617$		−	9.21320i		−	0.370910i	0.982653	−	0.185455i	$-0.0593758\pi$
$618$		0			0
$619$	−30.2132	−	17.4436i	−1.21437	−	0.701118i	−0.250663	−	0.968074i	$-0.580649\pi$
$619$	−30.2132	−	17.4436i	−1.21437	−	0.701118i	−0.963708	+	0.266957i	$0.913982\pi$
$620$	3.04384	+	1.75736i	0.122243	+	0.0705772i
$621$		0			0
$622$		−	26.4010i		−	1.05858i
$623$	2.65962	−	0.363961i	0.106556	−	0.0145818i
$624$		0			0
$625$	14.5000	+	25.1147i	0.580000	+	1.00459i
$626$	4.03295	−	6.98528i	0.161189	−	0.279188i
$627$		0			0
$628$	3.62132	−	2.09077i	0.144506	−	0.0834308i
$629$	−5.61642			−0.223941
$630$		0			0
$631$	37.6985			1.50075			0.750376	−	0.661011i	$-0.229872\pi$
$631$	37.6985			1.50075			0.750376	+	0.661011i	$0.229872\pi$
$632$	−6.92820	+	4.00000i	−0.275589	+	0.159111i
$633$		0			0
$634$	3.87868	−	6.71807i	0.154042	−	0.266809i
$635$	15.6245	+	27.0624i	0.620040	+	1.07394i
$636$		0			0
$637$	12.0000	−	12.2474i	0.475457	−	0.485262i
$638$		−	1.24264i		−	0.0491966i
$639$		0			0
$640$	−2.12132	−	1.22474i	−0.0838525	−	0.0484123i
$641$	0.630399	+	0.363961i	0.0248993	+	0.0143756i	0.512398	−	0.858748i	$-0.328757\pi$
$641$	0.630399	+	0.363961i	0.0248993	+	0.0143756i	−0.487499	+	0.873124i	$0.662091\pi$
$642$		0			0
$643$			25.2633i			0.996288i	0.867094	+	0.498144i	$0.165985\pi$
$643$			25.2633i			0.996288i	−0.867094	+	0.498144i	$0.834015\pi$
$644$	−17.7408	−	7.24264i	−0.699084	−	0.285400i
$645$		0			0
$646$	−1.50000	−	2.59808i	−0.0590167	−	0.102220i
$647$	−19.7700	+	34.2426i	−0.777239	+	1.34622i	0.156289	+	0.987711i	$0.450047\pi$
$647$	−19.7700	+	34.2426i	−0.777239	+	1.34622i	−0.933528	+	0.358506i	$0.883286\pi$
$648$		0			0
$649$	−12.9853	+	7.49706i	−0.509717	+	0.294285i
$650$	−2.44949			−0.0960769
$651$		0			0
$652$	14.9706			0.586292
$653$	−2.41344	+	1.39340i	−0.0944451	+	0.0545279i	−0.546479	−	0.837473i	$-0.684032\pi$
$653$	−2.41344	+	1.39340i	−0.0944451	+	0.0545279i	0.452034	+	0.892001i	$0.350699\pi$
$654$		0			0
$655$	13.9706	−	24.1977i	0.545875	−	0.945483i
$656$	−0.717439	−	1.24264i	−0.0280113	−	0.0485170i
$657$		0			0
$658$	−15.9853	+	12.3960i	−0.623171	+	0.483248i
$659$		−	16.2426i		−	0.632723i	−0.948639	−	0.316362i	$-0.897539\pi$
$659$		−	16.2426i		−	0.632723i	0.948639	−	0.316362i	$-0.102461\pi$
$660$		0			0
$661$	15.6213	+	9.01897i	0.607599	+	0.350797i	0.772025	−	0.635592i	$-0.219244\pi$
$661$	15.6213	+	9.01897i	0.607599	+	0.350797i	−0.164426	+	0.986389i	$0.552577\pi$
$662$	12.1244	+	7.00000i	0.471226	+	0.272063i
$663$		0			0
$664$		−	12.8418i		−	0.498358i
$665$	8.87039	−	6.87868i	0.343979	−	0.266744i
$666$		0			0
$667$	4.50000	+	7.79423i	0.174241	+	0.301794i
$668$	−0.210133	+	0.363961i	−0.00813029	+	0.0140821i
$669$		0			0
$670$	26.4853	−	15.2913i	1.02322	−	0.590754i
$671$	10.3923			0.401190
$672$		0			0
$673$	38.2426			1.47415			0.737073	−	0.675813i	$-0.236207\pi$
$673$	38.2426			1.47415			0.737073	+	0.675813i	$0.236207\pi$
$674$	−20.9947	+	12.1213i	−0.808687	+	0.466896i
$675$		0			0
$676$	3.50000	−	6.06218i	0.134615	−	0.233161i
$677$	−4.24309	−	7.34924i	−0.163075	−	0.282454i	0.772895	−	0.634534i	$-0.218808\pi$
$677$	−4.24309	−	7.34924i	−0.163075	−	0.282454i	−0.935970	+	0.352080i	$0.885475\pi$
$678$		0			0
$679$	−7.75736	−	3.16693i	−0.297700	−	0.121536i
$680$			4.24264i			0.162698i
$681$		0			0
$682$	1.24264	+	0.717439i	0.0475832	+	0.0274722i
$683$	−5.64191	−	3.25736i	−0.215882	−	0.124639i	0.388160	−	0.921592i	$-0.373111\pi$
$683$	−5.64191	−	3.25736i	−0.215882	−	0.124639i	−0.604042	+	0.796953i	$0.706444\pi$
$684$		0			0
$685$			4.30463i			0.164472i
$686$	11.0482	−	14.8640i	0.421822	−	0.567509i
$687$		0			0
$688$	−0.500000	−	0.866025i	−0.0190623	−	0.0330169i
$689$		0			0
$690$		0			0
$691$	−18.5147	+	10.6895i	−0.704333	+	0.406647i	−0.808959	−	0.587865i	$-0.799969\pi$
$691$	−18.5147	+	10.6895i	−0.704333	+	0.406647i	0.104626	+	0.994512i	$0.466635\pi$
$692$	−2.02922			−0.0771395
$693$		0			0
$694$	15.5147			0.588931
$695$	0.630399	−	0.363961i	0.0239124	−	0.0138058i
$696$		0			0
$697$	−1.24264	+	2.15232i	−0.0470684	+	0.0815248i
$698$	7.34847	+	12.7279i	0.278144	+	0.481759i
$699$		0			0
$700$	−2.62132	+	0.358719i	−0.0990766	+	0.0135583i
$701$			40.7574i			1.53938i	0.638415	+	0.769692i	$0.279590\pi$
$701$			40.7574i			1.53938i	−0.638415	+	0.769692i	$0.720410\pi$
$702$		0			0
$703$	−4.86396	−	2.80821i	−0.183448	−	0.105914i
$704$	−0.866025	−	0.500000i	−0.0326396	−	0.0188445i
$705$		0			0
$706$		−	8.78335i		−	0.330566i
$707$	17.0488	+	21.9853i	0.641187	+	0.826842i
$708$		0			0
$709$	−1.37868	−	2.38794i	−0.0517774	−	0.0896811i	0.838975	−	0.544170i	$-0.183155\pi$
$709$	−1.37868	−	2.38794i	−0.0517774	−	0.0896811i	−0.890752	+	0.454489i	$0.849822\pi$
$710$	−8.87039	+	15.3640i	−0.332900	+	0.576599i
$711$		0			0
$712$	−0.878680	+	0.507306i	−0.0329299	+	0.0190121i
$713$	−10.3923			−0.389195
$714$		0			0
$715$	−6.00000			−0.224387
$716$	−7.79423	+	4.50000i	−0.291284	+	0.168173i
$717$		0			0
$718$	0.514719	−	0.891519i	0.0192091	−	0.0332712i
$719$	3.40256	+	5.89340i	0.126894	+	0.219787i	0.922472	−	0.386065i	$-0.126166\pi$
$719$	3.40256	+	5.89340i	0.126894	+	0.219787i	−0.795578	+	0.605851i	$0.792833\pi$
$720$		0			0
$721$	5.84924	+	42.7430i	0.217837	+	1.59183i
$722$			16.0000i			0.595458i
$723$		0			0
$724$	4.24264	+	2.44949i	0.157676	+	0.0910346i
$725$	1.07616	+	0.621320i	0.0399675	+	0.0230753i
$726$		0			0
$727$		−	38.1051i		−	1.41324i	−0.707593	−	0.706620i	$-0.750219\pi$
$727$		−	38.1051i		−	1.41324i	0.707593	−	0.706620i	$-0.249781\pi$
$728$	−2.44949	+	6.00000i	−0.0907841	+	0.222375i
$729$		0			0
$730$	6.00000	+	10.3923i	0.222070	+	0.384636i
$731$	−0.866025	+	1.50000i	−0.0320311	+	0.0554795i
$732$		0			0
$733$	−4.45584	+	2.57258i	−0.164580	+	0.0950205i	−0.580028	−	0.814597i	$-0.696958\pi$
$733$	−4.45584	+	2.57258i	−0.164580	+	0.0950205i	0.415447	+	0.909617i	$0.363625\pi$
$734$	11.8272			0.436549
$735$		0			0
$736$	7.24264			0.266967
$737$	10.8126	−	6.24264i	0.398286	−	0.229951i
$738$		0			0
$739$	−1.72792	+	2.99285i	−0.0635626	+	0.110094i	−0.896055	−	0.443942i	$-0.853580\pi$
$739$	−1.72792	+	2.99285i	−0.0635626	+	0.110094i	0.832493	+	0.554036i	$0.186913\pi$
$740$	3.97141	+	6.87868i	0.145992	+	0.252865i
$741$		0			0
$742$		0			0
$743$		−	35.6985i		−	1.30965i	−0.755780	−	0.654825i	$-0.772742\pi$
$743$		−	35.6985i		−	1.30965i	0.755780	−	0.654825i	$-0.227258\pi$
$744$		0			0
$745$	10.0919	+	5.82655i	0.369738	+	0.213468i
$746$	29.4449	+	17.0000i	1.07805	+	0.622414i
$747$		0			0
$748$			1.73205i			0.0633300i
$749$	−5.19615	−	37.9706i	−0.189863	−	1.38741i
$750$		0			0
$751$	10.8787	+	18.8424i	0.396969	+	0.687570i	0.993350	−	0.115131i	$-0.0367288\pi$
$751$	10.8787	+	18.8424i	0.396969	+	0.687570i	−0.596382	+	0.802701i	$0.703395\pi$
$752$	3.82282	−	6.62132i	0.139404	−	0.241455i
$753$		0			0
$754$	2.63604	−	1.52192i	0.0959989	−	0.0554250i
$755$	−6.75412			−0.245808
$756$		0			0
$757$	41.6690			1.51449			0.757244	−	0.653132i	$-0.226546\pi$
$757$	41.6690			1.51449			0.757244	+	0.653132i	$0.226546\pi$
$758$	−28.3432	+	16.3640i	−1.02947	+	0.594366i
$759$		0			0
$760$	−2.12132	+	3.67423i	−0.0769484	+	0.133278i
$761$	18.0379	+	31.2426i	0.653875	+	1.13254i	0.982175	+	0.187971i	$0.0601911\pi$
$761$	18.0379	+	31.2426i	0.653875	+	1.13254i	−0.328299	+	0.944574i	$0.606476\pi$
$762$		0			0
$763$	15.8198	+	20.4004i	0.572715	+	0.738544i
$764$			16.9706i			0.613973i
$765$		0			0
$766$	33.3198	+	19.2372i	1.20389	+	0.695068i
$767$	−31.8073	−	18.3640i	−1.14850	−	0.663084i
$768$		0			0
$769$			8.53716i			0.307858i	0.988082	+	0.153929i	$0.0491927\pi$
$769$			8.53716i			0.307858i	−0.988082	+	0.153929i	$0.950807\pi$
$770$	−6.42090	+	0.878680i	−0.231393	+	0.0316654i
$771$		0			0
$772$	−10.3640	−	17.9509i	−0.373007	−	0.646067i
$773$	15.7986	−	27.3640i	0.568236	−	0.984213i	−0.428505	−	0.903540i	$-0.640959\pi$
$773$	15.7986	−	27.3640i	0.568236	−	0.984213i	0.996741	−	0.0806737i	$-0.0257072\pi$
$774$		0			0
$775$	−1.24264	+	0.717439i	−0.0446370	+	0.0257712i
$776$	3.16693			0.113686
$777$		0			0
$778$	−36.7279			−1.31676
$779$	−2.15232	+	1.24264i	−0.0771148	+	0.0445222i
$780$		0			0
$781$	−3.62132	+	6.27231i	−0.129581	+	0.224441i
$782$	−6.27231	−	10.8640i	−0.224297	−	0.388494i
$783$		0			0
$784$	−1.74264	+	6.77962i	−0.0622372	+	0.242129i
$785$			10.2426i			0.365576i
$786$		0			0
$787$	6.47056	+	3.73578i	0.230651	+	0.133166i	0.610872	−	0.791729i	$-0.290819\pi$
$787$	6.47056	+	3.73578i	0.230651	+	0.133166i	−0.380222	+	0.924895i	$0.624152\pi$
$788$	−3.22848	−	1.86396i	−0.115010	−	0.0664009i
$789$		0			0
$790$		−	19.5959i		−	0.697191i
$791$	−4.30463	−	1.75736i	−0.153055	−	0.0624845i
$792$		0			0
$793$	12.7279	+	22.0454i	0.451982	+	0.782855i
$794$	16.3674	−	28.3492i	0.580859	−	1.00608i
$795$		0			0
$796$	−6.36396	+	3.67423i	−0.225565	+	0.130230i
$797$	21.0308			0.744949			0.372475	−	0.928042i	$-0.378509\pi$
$797$	21.0308			0.744949			0.372475	+	0.928042i	$0.378509\pi$
$798$		0			0
$799$	−13.2426			−0.468491
$800$	0.866025	−	0.500000i	0.0306186	−	0.0176777i
$801$		0			0
$802$	14.1213	−	24.4588i	0.498641	−	0.863672i
$803$	2.44949	+	4.24264i	0.0864406	+	0.149720i
$804$		0			0
$805$	37.0919	−	28.7635i	1.30732	−	1.01378i
$806$			3.51472i			0.123801i
$807$		0			0
$808$	−9.10660	−	5.25770i	−0.320369	−	0.184965i
$809$	−39.7862	−	22.9706i	−1.39881	−	0.807602i	−0.404540	−	0.914520i	$-0.632568\pi$
$809$	−39.7862	−	22.9706i	−1.39881	−	0.807602i	−0.994268	+	0.106918i	$0.965902\pi$
$810$		0			0
$811$			56.5123i			1.98442i	0.124589	+	0.992208i	$0.460239\pi$
$811$			56.5123i			1.98442i	−0.124589	+	0.992208i	$0.539761\pi$
$812$	2.59808	−	2.01472i	0.0911746	−	0.0707028i
$813$		0			0
$814$	1.62132	+	2.80821i	0.0568272	+	0.0984277i
$815$	−18.3351	+	31.7574i	−0.642251	+	1.11241i
$816$		0			0
$817$	−1.50000	+	0.866025i	−0.0524784	+	0.0302984i
$818$	0.420266			0.0146943
$819$		0			0
$820$	3.51472			0.122739
$821$	8.23999	−	4.75736i	0.287578	−	0.166033i	−0.349271	−	0.937022i	$-0.613571\pi$
$821$	8.23999	−	4.75736i	0.287578	−	0.166033i	0.636849	+	0.770989i	$0.280238\pi$
$822$		0			0
$823$	21.6066	−	37.4237i	0.753159	−	1.30451i	−0.193126	−	0.981174i	$-0.561863\pi$
$823$	21.6066	−	37.4237i	0.753159	−	1.30451i	0.946284	−	0.323335i	$-0.104804\pi$
$824$	−8.15295	−	14.1213i	−0.284021	−	0.491940i
$825$		0			0
$826$	−36.7279	−	14.9941i	−1.27793	−	0.521712i
$827$			14.4853i			0.503703i	0.967766	+	0.251851i	$0.0810394\pi$
$827$			14.4853i			0.503703i	−0.967766	+	0.251851i	$0.918961\pi$
$828$		0			0
$829$	−24.3198	−	14.0410i	−0.844662	−	0.487666i	0.0141844	−	0.999899i	$-0.495485\pi$
$829$	−24.3198	−	14.0410i	−0.844662	−	0.487666i	−0.858846	+	0.512234i	$0.828818\pi$
$830$	27.2416	+	15.7279i	0.945568	+	0.545924i
$831$		0			0
$832$		−	2.44949i		−	0.0849208i
$833$	11.6786	−	3.25736i	0.404639	−	0.112861i
$834$		0			0
$835$	−0.514719	−	0.891519i	−0.0178126	−	0.0308523i
$836$	−0.866025	+	1.50000i	−0.0299521	+	0.0518786i
$837$		0			0
$838$	−3.98528	+	2.30090i	−0.137669	+	0.0794834i
$839$	−24.2487			−0.837158			−0.418579	−	0.908180i	$-0.637472\pi$
$839$	−24.2487			−0.837158			−0.418579	+	0.908180i	$0.637472\pi$
$840$		0			0
$841$	27.4558			0.946753
$842$	24.8536	−	14.3492i	0.856512	−	0.494508i
$843$		0			0
$844$	−11.2426	+	19.4728i	−0.386988	+	0.670282i
$845$	8.57321	+	14.8492i	0.294928	+	0.510829i
$846$		0			0
$847$	−2.62132	+	0.358719i	−0.0900696	+	0.0123257i
$848$		0			0
$849$		0			0
$850$	−1.50000	−	0.866025i	−0.0514496	−	0.0297044i
$851$	−20.3389	−	11.7426i	−0.697207	−	0.402533i
$852$		0			0
$853$			12.2474i			0.419345i	0.977772	+	0.209672i	$0.0672397\pi$
$853$			12.2474i			0.419345i	−0.977772	+	0.209672i	$0.932760\pi$
$854$	16.8493	+	21.7279i	0.576570	+	0.743514i
$855$		0			0
$856$	7.24264	+	12.5446i	0.247548	+	0.428766i
$857$	−8.09140	+	14.0147i	−0.276397	+	0.478734i	−0.970487	−	0.241155i	$-0.922474\pi$
$857$	−8.09140	+	14.0147i	−0.276397	+	0.478734i	0.694090	+	0.719889i	$0.255807\pi$
$858$		0			0
$859$	28.0919	−	16.2189i	0.958483	−	0.553380i	0.0627770	−	0.998028i	$-0.480004\pi$
$859$	28.0919	−	16.2189i	0.958483	−	0.553380i	0.895706	+	0.444647i	$0.146671\pi$
$860$	2.44949			0.0835269
$861$		0			0
$862$	16.9706			0.578020
$863$	−22.0454	+	12.7279i	−0.750434	+	0.433264i	−0.825851	−	0.563889i	$-0.809305\pi$
$863$	−22.0454	+	12.7279i	−0.750434	+	0.433264i	0.0754164	+	0.997152i	$0.475971\pi$
$864$		0			0
$865$	2.48528	−	4.30463i	0.0845021	−	0.146362i
$866$	10.6640	+	18.4706i	0.362377	+	0.627655i
$867$		0			0
$868$	0.514719	+	3.76127i	0.0174707	+	0.127666i
$869$		−	8.00000i		−	0.271381i
$870$		0			0
$871$	26.4853	+	15.2913i	0.897420	+	0.518125i
$872$	−8.45012	−	4.87868i	−0.286157	−	0.165213i
$873$		0			0
$874$		−	12.5446i		−	0.424328i
$875$	−9.79796	+	24.0000i	−0.331231	+	0.811348i
$876$		0			0
$877$	−18.1213	−	31.3870i	−0.611914	−	1.05987i	−0.990918	−	0.134470i	$-0.957067\pi$
$877$	−18.1213	−	31.3870i	−0.611914	−	1.05987i	0.379004	−	0.925395i	$-0.376267\pi$
$878$	12.4831	−	21.6213i	0.421283	−	0.729684i
$879$		0			0
$880$	2.12132	−	1.22474i	0.0715097	−	0.0412861i
$881$	14.6969			0.495152			0.247576	−	0.968868i	$-0.420366\pi$
$881$	14.6969			0.495152			0.247576	+	0.968868i	$0.420366\pi$
$882$		0			0
$883$	−32.7279			−1.10138			−0.550691	−	0.834709i	$-0.685636\pi$
$883$	−32.7279			−1.10138			−0.550691	+	0.834709i	$0.685636\pi$
$884$	−3.67423	+	2.12132i	−0.123578	+	0.0713477i
$885$		0			0
$886$	−1.50000	+	2.59808i	−0.0503935	+	0.0872841i
$887$	6.92820	+	12.0000i	0.232626	+	0.402921i	0.958580	−	0.284823i	$-0.0919348\pi$
$887$	6.92820	+	12.0000i	0.232626	+	0.402921i	−0.725954	+	0.687743i	$0.758601\pi$
$888$		0			0
$889$	−12.7574	+	31.2490i	−0.427868	+	1.04806i
$890$		−	2.48528i		−	0.0833068i
$891$		0			0
$892$	−8.12132	−	4.68885i	−0.271922	−	0.156994i
$893$	−11.4685	−	6.62132i	−0.383777	−	0.221574i
$894$		0			0
$895$		−	22.0454i		−	0.736897i
$896$	−0.358719	−	2.62132i	−0.0119840	−	0.0875722i
$897$		0			0
$898$	6.72792	+	11.6531i	0.224514	+	0.388869i
$899$	0.891519	−	1.54416i	0.0297338	−	0.0515005i
$900$		0			0
$901$		0			0
$902$	1.43488			0.0477762
$903$		0			0
$904$	1.75736			0.0584489
$905$	−10.3923	+	6.00000i	−0.345452	+	0.199447i
$906$		0			0
$907$	−24.2426	+	41.9895i	−0.804964	+	1.39424i	0.111352	+	0.993781i	$0.464482\pi$
$907$	−24.2426	+	41.9895i	−0.804964	+	1.39424i	−0.916316	+	0.400457i	$0.868851\pi$
$908$	−3.16693	−	5.48528i	−0.105098	−	0.182035i
$909$		0			0
$910$	−9.72792	−	12.5446i	−0.322477	−	0.415850i
$911$		−	9.72792i		−	0.322300i	−0.986930	−	0.161150i	$-0.948480\pi$
$911$		−	9.72792i		−	0.322300i	0.986930	−	0.161150i	$-0.0515204\pi$
$912$		0			0
$913$	11.1213	+	6.42090i	0.368062	+	0.212501i
$914$	−29.1837	−	16.8492i	−0.965312	−	0.557323i
$915$		0			0
$916$			13.8564i			0.457829i
$917$	29.9012	−	4.09188i	0.987424	−	0.135126i
$918$		0			0
$919$	−4.37868	−	7.58410i	−0.144439	−	0.250176i	0.784724	−	0.619845i	$-0.212805\pi$
$919$	−4.37868	−	7.58410i	−0.144439	−	0.250176i	−0.929164	+	0.369669i	$0.879471\pi$
$920$	−8.87039	+	15.3640i	−0.292448	+	0.506535i
$921$		0			0
$922$	−16.8640	+	9.73641i	−0.555385	+	0.320652i
$923$	−17.7408			−0.583945
$924$		0			0
$925$	−3.24264			−0.106617
$926$	−25.5095	+	14.7279i	−0.838294	+	0.483990i
$927$		0			0
$928$	−0.621320	+	1.07616i	−0.0203958	+	0.0353266i
$929$	−14.4868	−	25.0919i	−0.475297	−	0.823238i	0.524303	−	0.851532i	$-0.324326\pi$
$929$	−14.4868	−	25.0919i	−0.475297	−	0.823238i	−0.999600	+	0.0282938i	$0.990993\pi$
$930$		0			0
$931$	11.7426	+	3.01834i	0.384850	+	0.0989222i
$932$			17.4853i			0.572749i
$933$		0			0
$934$	4.50000	+	2.59808i	0.147244	+	0.0850117i
$935$	−3.67423	−	2.12132i	−0.120160	−	0.0693746i
$936$		0			0
$937$			56.0921i			1.83245i	0.400666	+	0.916224i	$0.368779\pi$
$937$			56.0921i			1.83245i	−0.400666	+	0.916224i	$0.631221\pi$
$938$	30.5826	+	12.4853i	0.998556	+	0.407659i
$939$		0			0
$940$	9.36396	+	16.2189i	0.305419	+	0.529001i
$941$	7.70719	−	13.3492i	0.251247	−	0.435173i	−0.712622	−	0.701548i	$-0.752493\pi$
$941$	7.70719	−	13.3492i	0.251247	−	0.435173i	0.963869	+	0.266375i	$0.0858260\pi$
$942$		0			0
$943$	−9.00000	+	5.19615i	−0.293080	+	0.169210i
$944$	14.9941			0.488017
$945$		0			0
$946$	1.00000			0.0325128
$947$	18.1865	−	10.5000i	0.590983	−	0.341204i	−0.174503	−	0.984657i	$-0.555832\pi$
$947$	18.1865	−	10.5000i	0.590983	−	0.341204i	0.765486	+	0.643452i	$0.222499\pi$
$948$		0			0
$949$	−6.00000	+	10.3923i	−0.194768	+	0.337348i
$950$	−0.866025	−	1.50000i	−0.0280976	−	0.0486664i
$951$		0			0
$952$	−3.62132	+	2.80821i	−0.117368	+	0.0910146i
$953$		−	48.0000i		−	1.55487i	−0.628962	−	0.777436i	$-0.716520\pi$
$953$		−	48.0000i		−	1.55487i	0.628962	−	0.777436i	$-0.283480\pi$
$954$		0			0
$955$	−36.0000	−	20.7846i	−1.16493	−	0.672574i
$956$	11.0227	+	6.36396i	0.356500	+	0.205825i
$957$		0			0
$958$			26.2779i			0.849002i
$959$	−3.67423	+	2.84924i	−0.118647	+	0.0920068i
$960$		0			0
$961$	−14.4706	−	25.0637i	−0.466792	−	0.808508i
$962$	−3.97141	+	6.87868i	−0.128043	+	0.221778i
$963$		0			0
$964$	−10.9706	+	6.33386i	−0.353338	+	0.204000i
$965$	50.7728			1.63444
$966$		0			0
$967$	24.7574			0.796143			0.398072	−	0.917354i	$-0.369680\pi$
$967$	24.7574			0.796143			0.398072	+	0.917354i	$0.369680\pi$
$968$	0.866025	−	0.500000i	0.0278351	−	0.0160706i
$969$		0			0
$970$	−3.87868	+	6.71807i	−0.124537	+	0.215704i
$971$	28.5533	+	49.4558i	0.916320	+	1.58711i	0.804956	+	0.593334i	$0.202189\pi$
$971$	28.5533	+	49.4558i	0.916320	+	1.58711i	0.111364	+	0.993780i	$0.464478\pi$
$972$		0			0
$973$	0.727922	+	0.297173i	0.0233361	+	0.00952693i
$974$			1.21320i			0.0388735i
$975$		0			0
$976$	−9.00000	−	5.19615i	−0.288083	−	0.166325i
$977$	53.2223	+	30.7279i	1.70273	+	0.983073i	0.942973	+	0.332870i	$0.108017\pi$
$977$	53.2223	+	30.7279i	1.70273	+	0.983073i	0.759760	+	0.650203i	$0.225316\pi$
$978$		0			0
$979$		−	1.01461i		−	0.0324271i
$980$	−12.2474	−	12.0000i	−0.391230	−	0.383326i
$981$		0			0
$982$	−8.63604	−	14.9581i	−0.275587	−	0.477331i
$983$	−29.0352	+	50.2904i	−0.926078	+	1.60401i	−0.136259	+	0.990673i	$0.543508\pi$
$983$	−29.0352	+	50.2904i	−0.926078	+	1.60401i	−0.789819	+	0.613340i	$0.789825\pi$
$984$		0			0
$985$	7.90812	−	4.56575i	0.251974	−	0.145477i
$986$	2.15232			0.0685437
$987$		0			0
$988$	−4.24264			−0.134976
$989$	−6.27231	+	3.62132i	−0.199448	+	0.115151i
$990$		0			0
$991$	−6.24264	+	10.8126i	−0.198304	+	0.343473i	−0.947979	−	0.318334i	$-0.896877\pi$
$991$	−6.24264	+	10.8126i	−0.198304	+	0.343473i	0.749675	+	0.661807i	$0.230210\pi$
$992$	−0.717439	−	1.24264i	−0.0227787	−	0.0394539i
$993$		0			0
$994$	−18.9853	+	2.59808i	−0.602177	+	0.0824060i
$995$		−	18.0000i		−	0.570638i
$996$		0			0
$997$	−27.5772	−	15.9217i	−0.873378	−	0.504245i	−0.00490839	−	0.999988i	$-0.501562\pi$
$997$	−27.5772	−	15.9217i	−0.873378	−	0.504245i	−0.868469	+	0.495743i	$0.834896\pi$
$998$	4.51477	+	2.60660i	0.142912	+	0.0825105i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1386.2.r.b.1277.4	yes	8
3.2	odd	2	inner	1386.2.r.b.1277.1	yes	8
7.5	odd	6	inner	1386.2.r.b.89.1	✓	8
21.5	even	6	inner	1386.2.r.b.89.4	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1386.2.r.b.89.1	✓	8	7.5	odd	6	inner
1386.2.r.b.89.4	yes	8	21.5	even	6	inner
1386.2.r.b.1277.1	yes	8	3.2	odd	2	inner
1386.2.r.b.1277.4	yes	8	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+(0.866025 - 0.500000i) q^{2} +(0.500000 - 0.866025i) q^{4} +(1.22474 + 2.12132i) q^{5} +(-1.00000 + 2.44949i) q^{7} -1.00000i q^{8} +O(q^{10})\)	\(q+(0.866025 - 0.500000i) q^{2} +(0.500000 - 0.866025i) q^{4} +(1.22474 + 2.12132i) q^{5} +(-1.00000 + 2.44949i) q^{7} -1.00000i q^{8} +(2.12132 + 1.22474i) q^{10} +(0.866025 + 0.500000i) q^{11} +2.44949i q^{13} +(0.358719 + 2.62132i) q^{14} +(-0.500000 - 0.866025i) q^{16} +(-0.866025 + 1.50000i) q^{17} +(-1.50000 + 0.866025i) q^{19} +2.44949 q^{20} +1.00000 q^{22} +(-6.27231 + 3.62132i) q^{23} +(-0.500000 + 0.866025i) q^{25} +(1.22474 + 2.12132i) q^{26} +(1.62132 + 2.09077i) q^{28} -1.24264i q^{29} +(1.24264 + 0.717439i) q^{31} +(-0.866025 - 0.500000i) q^{32} +1.73205i q^{34} +(-6.42090 + 0.878680i) q^{35} +(1.62132 + 2.80821i) q^{37} +(-0.866025 + 1.50000i) q^{38} +(2.12132 - 1.22474i) q^{40} +1.43488 q^{41} +1.00000 q^{43} +(0.866025 - 0.500000i) q^{44} +(-3.62132 + 6.27231i) q^{46} +(3.82282 + 6.62132i) q^{47} +(-5.00000 - 4.89898i) q^{49} +1.00000i q^{50} +(2.12132 + 1.22474i) q^{52} +2.44949i q^{55} +(2.44949 + 1.00000i) q^{56} +(-0.621320 - 1.07616i) q^{58} +(-7.49706 + 12.9853i) q^{59} +(9.00000 - 5.19615i) q^{61} +1.43488 q^{62} -1.00000 q^{64} +(-5.19615 + 3.00000i) q^{65} +(6.24264 - 10.8126i) q^{67} +(0.866025 + 1.50000i) q^{68} +(-5.12132 + 3.97141i) q^{70} +7.24264i q^{71} +(4.24264 + 2.44949i) q^{73} +(2.80821 + 1.62132i) q^{74} +1.73205i q^{76} +(-2.09077 + 1.62132i) q^{77} +(-4.00000 - 6.92820i) q^{79} +(1.22474 - 2.12132i) q^{80} +(1.24264 - 0.717439i) q^{82} +12.8418 q^{83} -4.24264 q^{85} +(0.866025 - 0.500000i) q^{86} +(0.500000 - 0.866025i) q^{88} +(-0.507306 - 0.878680i) q^{89} +(-6.00000 - 2.44949i) q^{91} +7.24264i q^{92} +(6.62132 + 3.82282i) q^{94} +(-3.67423 - 2.12132i) q^{95} +3.16693i q^{97} +(-6.77962 - 1.74264i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{4} - 8 q^{7}+O(q^{10}) \) 8 * q + 4 * q^4 - 8 * q^7	\( 8 q + 4 q^{4} - 8 q^{7} - 4 q^{16} - 12 q^{19} + 8 q^{22} - 4 q^{25} - 4 q^{28} - 24 q^{31} - 4 q^{37} + 8 q^{43} - 12 q^{46} - 40 q^{49} + 12 q^{58} + 72 q^{61} - 8 q^{64} + 16 q^{67} - 24 q^{70} - 32 q^{79} - 24 q^{82} + 4 q^{88} - 48 q^{91} + 36 q^{94}+O(q^{100}) \) 8 * q + 4 * q^4 - 8 * q^7 - 4 * q^16 - 12 * q^19 + 8 * q^22 - 4 * q^25 - 4 * q^28 - 24 * q^31 - 4 * q^37 + 8 * q^43 - 12 * q^46 - 40 * q^49 + 12 * q^58 + 72 * q^61 - 8 * q^64 + 16 * q^67 - 24 * q^70 - 32 * q^79 - 24 * q^82 + 4 * q^88 - 48 * q^91 + 36 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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