LMFDB - Embedded newform 1386.2.r.b.89.3

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			89.3
Root			$0.258819 - 0.965926i$ of defining polynomial
Character	$\chi$	$=$	1386.89
Dual form			1386.2.r.b.1277.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.866025 + 0.500000i) q^{2} +(0.500000 + 0.866025i) q^{4} +(-1.22474 + 2.12132i) q^{5} +(-1.00000 + 2.44949i) q^{7} +1.00000i q^{8} +O(q^{10})$	\(q+(0.866025 + 0.500000i) q^{2} +(0.500000 + 0.866025i) q^{4} +(-1.22474 + 2.12132i) q^{5} +(-1.00000 + 2.44949i) q^{7} +1.00000i q^{8} +(-2.12132 + 1.22474i) q^{10} +(0.866025 - 0.500000i) q^{11} +2.44949i q^{13} +(-2.09077 + 1.62132i) q^{14} +(-0.500000 + 0.866025i) q^{16} +(-0.866025 - 1.50000i) q^{17} +(-1.50000 - 0.866025i) q^{19} -2.44949 q^{20} +1.00000 q^{22} +(1.07616 + 0.621320i) q^{23} +(-0.500000 - 0.866025i) q^{25} +(-1.22474 + 2.12132i) q^{26} +(-2.62132 + 0.358719i) q^{28} -7.24264i q^{29} +(-7.24264 + 4.18154i) q^{31} +(-0.866025 + 0.500000i) q^{32} -1.73205i q^{34} +(-3.97141 - 5.12132i) q^{35} +(-2.62132 + 4.54026i) q^{37} +(-0.866025 - 1.50000i) q^{38} +(-2.12132 - 1.22474i) q^{40} -8.36308 q^{41} +1.00000 q^{43} +(0.866025 + 0.500000i) q^{44} +(0.621320 + 1.07616i) q^{46} +(1.37333 - 2.37868i) q^{47} +(-5.00000 - 4.89898i) q^{49} -1.00000i q^{50} +(-2.12132 + 1.22474i) q^{52} +2.44949i q^{55} +(-2.44949 - 1.00000i) q^{56} +(3.62132 - 6.27231i) q^{58} +(2.30090 + 3.98528i) q^{59} +(9.00000 + 5.19615i) q^{61} -8.36308 q^{62} -1.00000 q^{64} +(-5.19615 - 3.00000i) q^{65} +(-2.24264 - 3.88437i) q^{67} +(0.866025 - 1.50000i) q^{68} +(-0.878680 - 6.42090i) q^{70} +1.24264i q^{71} +(-4.24264 + 2.44949i) q^{73} +(-4.54026 + 2.62132i) q^{74} -1.73205i q^{76} +(0.358719 + 2.62132i) q^{77} +(-4.00000 + 6.92820i) q^{79} +(-1.22474 - 2.12132i) q^{80} +(-7.24264 - 4.18154i) q^{82} +7.94282 q^{83} +4.24264 q^{85} +(0.866025 + 0.500000i) q^{86} +(0.500000 + 0.866025i) q^{88} +(-2.95680 + 5.12132i) q^{89} +(-6.00000 - 2.44949i) q^{91} +1.24264i q^{92} +(2.37868 - 1.37333i) q^{94} +(3.67423 - 2.12132i) q^{95} +6.63103i q^{97} +(-1.88064 - 6.74264i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/1386\mathbb{Z}\right)^\times$.

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

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
$5$	−1.22474	+	2.12132i	−0.547723	+	0.948683i	0.450708	+	0.892672i	$0.351172\pi$
$5$	−1.22474	+	2.12132i	−0.547723	+	0.948683i	−0.998430	+	0.0560116i	$0.982162\pi$
$6$		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	$0.889680\pi$
$14$	−2.09077	+	1.62132i	−0.558782	+	0.433316i
$15$		0			0
$16$	−0.500000	+	0.866025i	−0.125000	+	0.216506i
$17$	−0.866025	−	1.50000i	−0.210042	−	0.363803i	0.741685	−	0.670748i	$-0.234027\pi$
$17$	−0.866025	−	1.50000i	−0.210042	−	0.363803i	−0.951727	+	0.306944i	$0.900693\pi$
$18$		0			0
$19$	−1.50000	−	0.866025i	−0.344124	−	0.198680i	0.317970	−	0.948101i	$-0.396999\pi$
$19$	−1.50000	−	0.866025i	−0.344124	−	0.198680i	−0.662094	+	0.749421i	$0.730332\pi$
$20$	−2.44949			−0.547723
$21$		0			0
$22$	1.00000			0.213201
$23$	1.07616	+	0.621320i	0.224395	+	0.129554i	0.607983	−	0.793950i	$-0.291979\pi$
$23$	1.07616	+	0.621320i	0.224395	+	0.129554i	−0.383589	+	0.923504i	$0.625312\pi$
$24$		0			0
$25$	−0.500000	−	0.866025i	−0.100000	−	0.173205i
$26$	−1.22474	+	2.12132i	−0.240192	+	0.416025i
$27$		0			0
$28$	−2.62132	+	0.358719i	−0.495383	+	0.0677916i
$29$		−	7.24264i		−	1.34492i	−0.740131	−	0.672462i	$-0.765237\pi$
$29$		−	7.24264i		−	1.34492i	0.740131	−	0.672462i	$-0.234763\pi$
$30$		0			0
$31$	−7.24264	+	4.18154i	−1.30082	+	0.751027i	−0.980544	−	0.196299i	$-0.937108\pi$
$31$	−7.24264	+	4.18154i	−1.30082	+	0.751027i	−0.320273	+	0.947325i	$0.603774\pi$
$32$	−0.866025	+	0.500000i	−0.153093	+	0.0883883i
$33$		0			0
$34$		−	1.73205i		−	0.297044i
$35$	−3.97141	−	5.12132i	−0.671290	−	0.865661i
$36$		0			0
$37$	−2.62132	+	4.54026i	−0.430942	+	0.746414i	−0.996955	−	0.0779826i	$-0.975152\pi$
$37$	−2.62132	+	4.54026i	−0.430942	+	0.746414i	0.566012	+	0.824397i	$0.308485\pi$
$38$	−0.866025	−	1.50000i	−0.140488	−	0.243332i
$39$		0			0
$40$	−2.12132	−	1.22474i	−0.335410	−	0.193649i
$41$	−8.36308			−1.30609			−0.653047	−	0.757317i	$-0.726510\pi$
$41$	−8.36308			−1.30609			−0.653047	+	0.757317i	$0.726510\pi$
$42$		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
$46$	0.621320	+	1.07616i	0.0916087	+	0.158671i
$47$	1.37333	−	2.37868i	0.200321	−	0.346966i	−0.748311	−	0.663348i	$-0.769135\pi$
$47$	1.37333	−	2.37868i	0.200321	−	0.346966i	0.948632	+	0.316382i	$0.102468\pi$
$48$		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	$-0.666667\pi$
$53$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$54$		0			0
$55$			2.44949i			0.330289i
$56$	−2.44949	−	1.00000i	−0.327327	−	0.133631i
$57$		0			0
$58$	3.62132	−	6.27231i	0.475503	−	0.823595i
$59$	2.30090	+	3.98528i	0.299552	+	0.518839i	0.976034	−	0.217620i	$-0.0698294\pi$
$59$	2.30090	+	3.98528i	0.299552	+	0.518839i	−0.676481	+	0.736460i	$0.736496\pi$
$60$		0			0
$61$	9.00000	+	5.19615i	1.15233	+	0.665299i	0.949454	−	0.313905i	$-0.101637\pi$
$61$	9.00000	+	5.19615i	1.15233	+	0.665299i	0.202878	+	0.979204i	$0.434971\pi$
$62$	−8.36308			−1.06211
$63$		0			0
$64$	−1.00000			−0.125000
$65$	−5.19615	−	3.00000i	−0.644503	−	0.372104i
$66$		0			0
$67$	−2.24264	−	3.88437i	−0.273982	−	0.474551i	0.695896	−	0.718143i	$-0.255008\pi$
$67$	−2.24264	−	3.88437i	−0.273982	−	0.474551i	−0.969878	+	0.243592i	$0.921674\pi$
$68$	0.866025	−	1.50000i	0.105021	−	0.181902i
$69$		0			0
$70$	−0.878680	−	6.42090i	−0.105022	−	0.767444i
$71$			1.24264i			0.147474i	0.997278	+	0.0737372i	$0.0234926\pi$
$71$			1.24264i			0.147474i	−0.997278	+	0.0737372i	$0.976507\pi$
$72$		0			0
$73$	−4.24264	+	2.44949i	−0.496564	+	0.286691i	−0.727293	−	0.686327i	$-0.759222\pi$
$73$	−4.24264	+	2.44949i	−0.496564	+	0.286691i	0.230730	+	0.973018i	$0.425889\pi$
$74$	−4.54026	+	2.62132i	−0.527795	+	0.304722i
$75$		0			0
$76$		−	1.73205i		−	0.198680i
$77$	0.358719	+	2.62132i	0.0408799	+	0.298727i
$78$		0			0
$79$	−4.00000	+	6.92820i	−0.450035	+	0.779484i	−0.998388	−	0.0567635i	$-0.981922\pi$
$79$	−4.00000	+	6.92820i	−0.450035	+	0.779484i	0.548352	+	0.836247i	$0.315255\pi$
$80$	−1.22474	−	2.12132i	−0.136931	−	0.237171i
$81$		0			0
$82$	−7.24264	−	4.18154i	−0.799816	−	0.461774i
$83$	7.94282			0.871837			0.435919	−	0.899986i	$-0.356424\pi$
$83$	7.94282			0.871837			0.435919	+	0.899986i	$0.356424\pi$
$84$		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
$89$	−2.95680	+	5.12132i	−0.313420	+	0.542859i	−0.979100	−	0.203378i	$-0.934808\pi$
$89$	−2.95680	+	5.12132i	−0.313420	+	0.542859i	0.665681	+	0.746237i	$0.268141\pi$
$90$		0			0
$91$	−6.00000	−	2.44949i	−0.628971	−	0.256776i
$92$			1.24264i			0.129554i
$93$		0			0
$94$	2.37868	−	1.37333i	0.245342	−	0.141648i
$95$	3.67423	−	2.12132i	0.376969	−	0.217643i
$96$		0			0
$97$			6.63103i			0.673279i	0.941634	+	0.336640i	$0.109290\pi$
$97$			6.63103i			0.673279i	−0.941634	+	0.336640i	$0.890710\pi$
$98$	−1.88064	−	6.74264i	−0.189973	−	0.681110i
$99$		0			0
$100$	0.500000	−	0.866025i	0.0500000	−	0.0866025i
$101$	−6.98975	−	12.1066i	−0.695506	−	1.20465i	−0.970010	−	0.243066i	$-0.921847\pi$
$101$	−6.98975	−	12.1066i	−0.695506	−	1.20465i	0.274504	−	0.961586i	$-0.411486\pi$
$102$		0			0
$103$	9.87868	+	5.70346i	0.973375	+	0.561978i	0.900264	−	0.435345i	$-0.143374\pi$
$103$	9.87868	+	5.70346i	0.973375	+	0.561978i	0.0731117	+	0.997324i	$0.476707\pi$
$104$	−2.44949			−0.240192
$105$		0			0
$106$		0			0
$107$	2.15232	+	1.24264i	0.208072	+	0.120131i	0.600415	−	0.799688i	$-0.295002\pi$
$107$	2.15232	+	1.24264i	0.208072	+	0.120131i	−0.392343	+	0.919819i	$0.628335\pi$
$108$		0			0
$109$	9.12132	+	15.7986i	0.873664	+	1.51323i	0.858179	+	0.513350i	$0.171596\pi$
$109$	9.12132	+	15.7986i	0.873664	+	1.51323i	0.0154849	+	0.999880i	$0.495071\pi$
$110$	−1.22474	+	2.12132i	−0.116775	+	0.202260i
$111$		0			0
$112$	−1.62132	−	2.09077i	−0.153200	−	0.197559i
$113$		−	10.2426i		−	0.963547i	−0.876296	−	0.481773i	$-0.839993\pi$
$113$		−	10.2426i		−	0.963547i	0.876296	−	0.481773i	$-0.160007\pi$
$114$		0			0
$115$	−2.63604	+	1.52192i	−0.245812	+	0.141920i
$116$	6.27231	−	3.62132i	0.582369	−	0.336231i
$117$		0			0
$118$			4.60181i			0.423631i
$119$	4.54026	−	0.621320i	0.416205	−	0.0569563i
$120$		0			0
$121$	0.500000	−	0.866025i	0.0454545	−	0.0787296i
$122$	5.19615	+	9.00000i	0.470438	+	0.814822i
$123$		0			0
$124$	−7.24264	−	4.18154i	−0.650408	−	0.375513i
$125$	−9.79796			−0.876356
$126$		0			0
$127$	21.2426			1.88498			0.942490	−	0.334235i	$-0.108478\pi$
$127$	21.2426			1.88498			0.942490	+	0.334235i	$0.108478\pi$
$128$	−0.866025	−	0.500000i	−0.0765466	−	0.0441942i
$129$		0			0
$130$	−3.00000	−	5.19615i	−0.263117	−	0.455733i
$131$	−8.15295	+	14.1213i	−0.712326	+	1.23379i	0.251655	+	0.967817i	$0.419025\pi$
$131$	−8.15295	+	14.1213i	−0.712326	+	1.23379i	−0.963982	+	0.265969i	$0.914308\pi$
$132$		0			0
$133$	3.62132	−	2.80821i	0.314008	−	0.243503i
$134$		−	4.48528i		−	0.387469i
$135$		0			0
$136$	1.50000	−	0.866025i	0.128624	−	0.0742611i
$137$	8.87039	−	5.12132i	0.757848	−	0.437544i	−0.0706744	−	0.997499i	$-0.522515\pi$
$137$	8.87039	−	5.12132i	0.757848	−	0.437544i	0.828523	+	0.559956i	$0.189182\pi$
$138$		0			0
$139$			10.0951i			0.856258i	0.903718	+	0.428129i	$0.140827\pi$
$139$			10.0951i			0.856258i	−0.903718	+	0.428129i	$0.859173\pi$
$140$	2.44949	−	6.00000i	0.207020	−	0.507093i
$141$		0			0
$142$	−0.621320	+	1.07616i	−0.0521400	+	0.0903092i
$143$	1.22474	+	2.12132i	0.102418	+	0.177394i
$144$		0			0
$145$	15.3640	+	8.87039i	1.27591	+	0.736646i
$146$	−4.89898			−0.405442
$147$		0			0
$148$	−5.24264			−0.430942
$149$	11.4685	+	6.62132i	0.939533	+	0.542440i	0.889814	−	0.456324i	$-0.150834\pi$
$149$	11.4685	+	6.62132i	0.939533	+	0.542440i	0.0497192	+	0.998763i	$0.484167\pi$
$150$		0			0
$151$	−5.62132	−	9.73641i	−0.457457	−	0.792338i	0.541369	−	0.840785i	$-0.317906\pi$
$151$	−5.62132	−	9.73641i	−0.457457	−	0.792338i	−0.998826	+	0.0484470i	$0.984573\pi$
$152$	0.866025	−	1.50000i	0.0702439	−	0.121666i
$153$		0			0
$154$	−1.00000	+	2.44949i	−0.0805823	+	0.197386i
$155$		−	20.4853i		−	1.64542i
$156$		0			0
$157$	−0.621320	+	0.358719i	−0.0495868	+	0.0286289i	−0.524588	−	0.851356i	$-0.675781\pi$
$157$	−0.621320	+	0.358719i	−0.0495868	+	0.0286289i	0.475002	+	0.879985i	$0.342447\pi$
$158$	−6.92820	+	4.00000i	−0.551178	+	0.318223i
$159$		0			0
$160$		−	2.44949i		−	0.193649i
$161$	−2.59808	+	2.01472i	−0.204757	+	0.158782i
$162$		0			0
$163$	−9.48528	+	16.4290i	−0.742945	+	1.28682i	0.208204	+	0.978085i	$0.433238\pi$
$163$	−9.48528	+	16.4290i	−0.742945	+	1.28682i	−0.951149	+	0.308732i	$0.900095\pi$
$164$	−4.18154	−	7.24264i	−0.326523	−	0.565555i
$165$		0			0
$166$	6.87868	+	3.97141i	0.533889	+	0.308241i
$167$	14.2767			1.10476			0.552381	−	0.833592i	$-0.313719\pi$
$167$	14.2767			1.10476			0.552381	+	0.833592i	$0.313719\pi$
$168$		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
$173$	−5.91359	+	10.2426i	−0.449602	+	0.778734i	−0.998360	−	0.0572477i	$-0.981768\pi$
$173$	−5.91359	+	10.2426i	−0.449602	+	0.778734i	0.548758	+	0.835981i	$0.315101\pi$
$174$		0			0
$175$	2.62132	−	0.358719i	0.198153	−	0.0271166i
$176$			1.00000i			0.0753778i
$177$		0			0
$178$	−5.12132	+	2.95680i	−0.383859	+	0.221621i
$179$	−7.79423	+	4.50000i	−0.582568	+	0.336346i	−0.762153	−	0.647397i	$-0.775858\pi$
$179$	−7.79423	+	4.50000i	−0.582568	+	0.336346i	0.179585	+	0.983742i	$0.442524\pi$
$180$		0			0
$181$			4.89898i			0.364138i	0.983286	+	0.182069i	$0.0582795\pi$
$181$			4.89898i			0.364138i	−0.983286	+	0.182069i	$0.941721\pi$
$182$	−3.97141	−	5.12132i	−0.294380	−	0.379618i
$183$		0			0
$184$	−0.621320	+	1.07616i	−0.0458043	+	0.0793355i
$185$	−6.42090	−	11.1213i	−0.472074	−	0.817656i
$186$		0			0
$187$	−1.50000	−	0.866025i	−0.109691	−	0.0633300i
$188$	2.74666			0.200321
$189$		0			0
$190$	4.24264			0.307794
$191$	14.6969	+	8.48528i	1.06343	+	0.613973i	0.926380	−	0.376590i	$-0.122904\pi$
$191$	14.6969	+	8.48528i	1.06343	+	0.613973i	0.137053	+	0.990564i	$0.456237\pi$
$192$		0			0
$193$	−2.36396	−	4.09450i	−0.170162	−	0.294729i	0.768315	−	0.640072i	$-0.221096\pi$
$193$	−2.36396	−	4.09450i	−0.170162	−	0.294729i	−0.938476	+	0.345344i	$0.887762\pi$
$194$	−3.31552	+	5.74264i	−0.238040	+	0.412298i
$195$		0			0
$196$	1.74264	−	6.77962i	0.124474	−	0.484258i
$197$		−	21.7279i		−	1.54805i	−0.633155	−	0.774025i	$-0.718240\pi$
$197$		−	21.7279i		−	1.54805i	0.633155	−	0.774025i	$-0.281760\pi$
$198$		0			0
$199$	6.36396	−	3.67423i	0.451129	−	0.260460i	−0.257178	−	0.966364i	$-0.582793\pi$
$199$	6.36396	−	3.67423i	0.451129	−	0.260460i	0.708307	+	0.705905i	$0.249459\pi$
$200$	0.866025	−	0.500000i	0.0612372	−	0.0353553i
$201$		0			0
$202$		−	13.9795i		−	0.983594i
$203$	17.7408	+	7.24264i	1.24516	+	0.508334i
$204$		0			0
$205$	10.2426	−	17.7408i	0.715377	−	1.23907i
$206$	5.70346	+	9.87868i	0.397379	+	0.688280i
$207$		0			0
$208$	−2.12132	−	1.22474i	−0.147087	−	0.0849208i
$209$	−1.73205			−0.119808
$210$		0			0
$211$	−5.51472			−0.379649			−0.189824	−	0.981818i	$-0.560792\pi$
$211$	−5.51472			−0.379649			−0.189824	+	0.981818i	$0.560792\pi$
$212$		0			0
$213$		0			0
$214$	1.24264	+	2.15232i	0.0849452	+	0.147129i
$215$	−1.22474	+	2.12132i	−0.0835269	+	0.144673i
$216$		0			0
$217$	−3.00000	−	21.9223i	−0.203653	−	1.48818i
$218$			18.2426i			1.23555i
$219$		0			0
$220$	−2.12132	+	1.22474i	−0.143019	+	0.0825723i
$221$	3.67423	−	2.12132i	0.247156	−	0.142695i
$222$		0			0
$223$			4.47871i			0.299917i	0.988692	+	0.149958i	$0.0479140\pi$
$223$			4.47871i			0.299917i	−0.988692	+	0.149958i	$0.952086\pi$
$224$	−0.358719	−	2.62132i	−0.0239680	−	0.175144i
$225$		0			0
$226$	5.12132	−	8.87039i	0.340665	−	0.590049i
$227$	−6.63103	−	11.4853i	−0.440117	−	0.762305i	0.557581	−	0.830123i	$-0.311730\pi$
$227$	−6.63103	−	11.4853i	−0.440117	−	0.762305i	−0.997698	+	0.0678178i	$0.978396\pi$
$228$		0			0
$229$	−12.0000	−	6.92820i	−0.792982	−	0.457829i	0.0480291	−	0.998846i	$-0.484706\pi$
$229$	−12.0000	−	6.92820i	−0.792982	−	0.457829i	−0.841011	+	0.541017i	$0.818039\pi$
$230$	−3.04384			−0.200705
$231$		0			0
$232$	7.24264			0.475503
$233$	−0.445759	−	0.257359i	−0.0292027	−	0.0168602i	0.485328	−	0.874332i	$-0.338700\pi$
$233$	−0.445759	−	0.257359i	−0.0292027	−	0.0168602i	−0.514530	+	0.857472i	$0.672034\pi$
$234$		0			0
$235$	3.36396	+	5.82655i	0.219441	+	0.380082i
$236$	−2.30090	+	3.98528i	−0.149776	+	0.259420i
$237$		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
$241$	22.9706	−	13.2621i	1.47966	−	0.854284i	0.479929	−	0.877307i	$-0.340662\pi$
$241$	22.9706	−	13.2621i	1.47966	−	0.854284i	0.999735	+	0.0230229i	$0.00732905\pi$
$242$	0.866025	−	0.500000i	0.0556702	−	0.0321412i
$243$		0			0
$244$			10.3923i			0.665299i
$245$	16.5160	−	4.60660i	1.05517	−	0.294305i
$246$		0			0
$247$	2.12132	−	3.67423i	0.134976	−	0.233786i
$248$	−4.18154	−	7.24264i	−0.265528	−	0.459908i
$249$		0			0
$250$	−8.48528	−	4.89898i	−0.536656	−	0.309839i
$251$	−1.13770			−0.0718113			−0.0359056	−	0.999355i	$-0.511432\pi$
$251$	−1.13770			−0.0718113			−0.0359056	+	0.999355i	$0.511432\pi$
$252$		0			0
$253$	1.24264			0.0781242
$254$	18.3967	+	10.6213i	1.15431	+	0.666441i
$255$		0			0
$256$	−0.500000	−	0.866025i	−0.0312500	−	0.0541266i
$257$	−4.39167	+	7.60660i	−0.273945	+	0.474487i	−0.969868	−	0.243629i	$-0.921662\pi$
$257$	−4.39167	+	7.60660i	−0.273945	+	0.474487i	0.695923	+	0.718116i	$0.254995\pi$
$258$		0			0
$259$	−8.50000	−	10.9612i	−0.528164	−	0.681093i
$260$		−	6.00000i		−	0.372104i
$261$		0			0
$262$	−14.1213	+	8.15295i	−0.872418	+	0.503691i
$263$	−8.23999	+	4.75736i	−0.508099	+	0.293351i	−0.732052	−	0.681249i	$-0.761437\pi$
$263$	−8.23999	+	4.75736i	−0.508099	+	0.293351i	0.223953	+	0.974600i	$0.428104\pi$
$264$		0			0
$265$		0			0
$266$	4.54026	−	0.621320i	0.278381	−	0.0380956i
$267$		0			0
$268$	2.24264	−	3.88437i	0.136991	−	0.237276i
$269$	10.8996	+	18.8787i	0.664561	+	1.15105i	0.979404	+	0.201910i	$0.0647148\pi$
$269$	10.8996	+	18.8787i	0.664561	+	1.15105i	−0.314843	+	0.949144i	$0.601952\pi$
$270$		0			0
$271$	10.2426	+	5.91359i	0.622196	+	0.359225i	0.777724	−	0.628606i	$-0.216374\pi$
$271$	10.2426	+	5.91359i	0.622196	+	0.359225i	−0.155527	+	0.987832i	$0.549708\pi$
$272$	1.73205			0.105021
$273$		0			0
$274$	10.2426			0.618781
$275$	−0.866025	−	0.500000i	−0.0522233	−	0.0301511i
$276$		0			0
$277$	−10.7279	−	18.5813i	−0.644578	−	1.11644i	−0.984399	−	0.175952i	$-0.943700\pi$
$277$	−10.7279	−	18.5813i	−0.644578	−	1.11644i	0.339820	−	0.940490i	$-0.389634\pi$
$278$	−5.04757	+	8.74264i	−0.302733	+	0.524349i
$279$		0			0
$280$	5.12132	−	3.97141i	0.306057	−	0.237337i
$281$			6.51472i			0.388636i	0.980939	+	0.194318i	$0.0622493\pi$
$281$			6.51472i			0.388636i	−0.980939	+	0.194318i	$0.937751\pi$
$282$		0			0
$283$	−9.51472	+	5.49333i	−0.565591	+	0.326544i	−0.755387	−	0.655279i	$-0.772551\pi$
$283$	−9.51472	+	5.49333i	−0.565591	+	0.326544i	0.189795	+	0.981824i	$0.439218\pi$
$284$	−1.07616	+	0.621320i	−0.0638583	+	0.0368686i
$285$		0			0
$286$			2.44949i			0.144841i
$287$	8.36308	−	20.4853i	0.493657	−	1.20921i
$288$		0			0
$289$	7.00000	−	12.1244i	0.411765	−	0.713197i
$290$	8.87039	+	15.3640i	0.520887	+	0.902203i
$291$		0			0
$292$	−4.24264	−	2.44949i	−0.248282	−	0.143346i
$293$	17.4436			1.01907			0.509533	−	0.860451i	$-0.329818\pi$
$293$	17.4436			1.01907			0.509533	+	0.860451i	$0.329818\pi$
$294$		0			0
$295$	−11.2721			−0.656286
$296$	−4.54026	−	2.62132i	−0.263897	−	0.152361i
$297$		0			0
$298$	6.62132	+	11.4685i	0.383563	+	0.664350i
$299$	−1.52192	+	2.63604i	−0.0880148	+	0.152446i
$300$		0			0
$301$	−1.00000	+	2.44949i	−0.0576390	+	0.141186i
$302$		−	11.2426i		−	0.646941i
$303$		0			0
$304$	1.50000	−	0.866025i	0.0860309	−	0.0496700i
$305$	−22.0454	+	12.7279i	−1.26232	+	0.728799i
$306$		0			0
$307$			27.1185i			1.54773i	0.633349	+	0.773866i	$0.281680\pi$
$307$			27.1185i			1.54773i	−0.633349	+	0.773866i	$0.718320\pi$
$308$	−2.09077	+	1.62132i	−0.119133	+	0.0923833i
$309$		0			0
$310$	10.2426	−	17.7408i	0.581743	−	1.00761i
$311$	5.85204	+	10.1360i	0.331839	+	0.574762i	0.982872	−	0.184287i	$-0.0589976\pi$
$311$	5.85204	+	10.1360i	0.331839	+	0.574762i	−0.651033	+	0.759049i	$0.725664\pi$
$312$		0			0
$313$	−9.98528	−	5.76500i	−0.564401	−	0.325857i	0.190509	−	0.981685i	$-0.438986\pi$
$313$	−9.98528	−	5.76500i	−0.564401	−	0.325857i	−0.754910	+	0.655828i	$0.772320\pi$
$314$	−0.717439			−0.0404874
$315$		0			0
$316$	−8.00000			−0.450035
$317$	14.0665	+	8.12132i	0.790056	+	0.456139i	0.839982	−	0.542614i	$-0.182565\pi$
$317$	14.0665	+	8.12132i	0.790056	+	0.456139i	−0.0499265	+	0.998753i	$0.515899\pi$
$318$		0			0
$319$	−3.62132	−	6.27231i	−0.202755	−	0.351182i
$320$	1.22474	−	2.12132i	0.0684653	−	0.118585i
$321$		0			0
$322$	−3.25736	+	0.445759i	−0.181526	+	0.0248412i
$323$			3.00000i			0.166924i
$324$		0			0
$325$	2.12132	−	1.22474i	0.117670	−	0.0679366i
$326$	−16.4290	+	9.48528i	−0.909918	+	0.525341i
$327$		0			0
$328$		−	8.36308i		−	0.461774i
$329$	4.45322	+	5.74264i	0.245514	+	0.316602i
$330$		0			0
$331$	7.00000	−	12.1244i	0.384755	−	0.666415i	−0.606980	−	0.794717i	$-0.707619\pi$
$331$	7.00000	−	12.1244i	0.384755	−	0.666415i	0.991735	+	0.128302i	$0.0409527\pi$
$332$	3.97141	+	6.87868i	0.217959	+	0.377517i
$333$		0			0
$334$	12.3640	+	7.13834i	0.676526	+	0.390592i
$335$	10.9867			0.600265
$336$		0			0
$337$	−15.7574			−0.858358			−0.429179	−	0.903219i	$-0.641197\pi$
$337$	−15.7574			−0.858358			−0.429179	+	0.903219i	$0.641197\pi$
$338$	6.06218	+	3.50000i	0.329739	+	0.190375i
$339$		0			0
$340$	2.12132	+	3.67423i	0.115045	+	0.199263i
$341$	−4.18154	+	7.24264i	−0.226443	+	0.392211i
$342$		0			0
$343$	17.0000	−	7.34847i	0.917914	−	0.396780i
$344$			1.00000i			0.0539164i
$345$		0			0
$346$	−10.2426	+	5.91359i	−0.550648	+	0.317917i
$347$	28.1331	−	16.2426i	1.51026	−	0.871951i	0.510335	−	0.859976i	$-0.329522\pi$
$347$	28.1331	−	16.2426i	1.51026	−	0.871951i	0.999928	−	0.0119747i	$-0.00381174\pi$
$348$		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
$353$	−7.85578	−	13.6066i	−0.418121	−	0.724206i	0.577630	−	0.816299i	$-0.303978\pi$
$353$	−7.85578	−	13.6066i	−0.418121	−	0.724206i	−0.995750	+	0.0920926i	$0.970644\pi$
$354$		0			0
$355$	−2.63604	−	1.52192i	−0.139906	−	0.0807750i
$356$	−5.91359			−0.313420
$357$		0			0
$358$	−9.00000			−0.475665
$359$	30.2854	+	17.4853i	1.59840	+	0.922838i	0.991795	+	0.127836i	$0.0408030\pi$
$359$	30.2854	+	17.4853i	1.59840	+	0.922838i	0.606607	+	0.795002i	$0.292530\pi$
$360$		0			0
$361$	−8.00000	−	13.8564i	−0.421053	−	0.729285i
$362$	−2.44949	+	4.24264i	−0.128742	+	0.222988i
$363$		0			0
$364$	−0.878680	−	6.42090i	−0.0460553	−	0.336546i
$365$		−	12.0000i		−	0.628109i
$366$		0			0
$367$	1.75736	−	1.01461i	0.0917334	−	0.0529623i	−0.453432	−	0.891291i	$-0.649800\pi$
$367$	1.75736	−	1.01461i	0.0917334	−	0.0529623i	0.545165	+	0.838329i	$0.316467\pi$
$368$	−1.07616	+	0.621320i	−0.0560986	+	0.0323886i
$369$		0			0
$370$		−	12.8418i		−	0.667613i
$371$		0			0
$372$		0			0
$373$	17.0000	−	29.4449i	0.880227	−	1.52460i	0.0291379	−	0.999575i	$-0.490724\pi$
$373$	17.0000	−	29.4449i	0.880227	−	1.52460i	0.851089	−	0.525022i	$-0.175943\pi$
$374$	−0.866025	−	1.50000i	−0.0447811	−	0.0775632i
$375$		0			0
$376$	2.37868	+	1.37333i	0.122671	+	0.0708242i
$377$	17.7408			0.913696
$378$		0			0
$379$	−7.27208			−0.373542			−0.186771	−	0.982404i	$-0.559802\pi$
$379$	−7.27208			−0.373542			−0.186771	+	0.982404i	$0.559802\pi$
$380$	3.67423	+	2.12132i	0.188484	+	0.108821i
$381$		0			0
$382$	8.48528	+	14.6969i	0.434145	+	0.751961i
$383$	−17.5051	+	30.3198i	−0.894471	+	1.54927i	−0.0600136	+	0.998198i	$0.519114\pi$
$383$	−17.5051	+	30.3198i	−0.894471	+	1.54927i	−0.834458	+	0.551072i	$0.814219\pi$
$384$		0			0
$385$	−6.00000	−	2.44949i	−0.305788	−	0.124838i
$386$		−	4.72792i		−	0.240645i
$387$		0			0
$388$	−5.74264	+	3.31552i	−0.291538	+	0.168320i
$389$	−9.76191	+	5.63604i	−0.494948	+	0.285759i	−0.726625	−	0.687034i	$-0.758912\pi$
$389$	−9.76191	+	5.63604i	−0.494948	+	0.285759i	0.231677	+	0.972793i	$0.425579\pi$
$390$		0			0
$391$		−	2.15232i		−	0.108847i
$392$	4.89898	−	5.00000i	0.247436	−	0.252538i
$393$		0			0
$394$	10.8640	−	18.8169i	0.547318	−	0.947983i
$395$	−9.79796	−	16.9706i	−0.492989	−	0.853882i
$396$		0			0
$397$	−1.34924	−	0.778985i	−0.0677165	−	0.0390962i	0.465759	−	0.884911i	$-0.345781\pi$
$397$	−1.34924	−	0.778985i	−0.0677165	−	0.0390962i	−0.533476	+	0.845815i	$0.679115\pi$
$398$	7.34847			0.368345
$399$		0			0
$400$	1.00000			0.0500000
$401$	17.1104	+	9.87868i	0.854451	+	0.493318i	0.862150	−	0.506653i	$-0.169117\pi$
$401$	17.1104	+	9.87868i	0.854451	+	0.493318i	−0.00769892	+	0.999970i	$0.502451\pi$
$402$		0			0
$403$	−10.2426	−	17.7408i	−0.510222	−	0.883731i
$404$	6.98975	−	12.1066i	0.347753	−	0.602326i
$405$		0			0
$406$	11.7426	+	15.1427i	0.582777	+	0.751519i
$407$			5.24264i			0.259868i
$408$		0			0
$409$	−12.3640	+	7.13834i	−0.611359	+	0.352968i	−0.773497	−	0.633800i	$-0.781494\pi$
$409$	−12.3640	+	7.13834i	−0.611359	+	0.352968i	0.162138	+	0.986768i	$0.448161\pi$
$410$	17.7408	−	10.2426i	0.876154	−	0.505848i
$411$		0			0
$412$			11.4069i			0.561978i
$413$	−12.0628	+	1.65076i	−0.593572	+	0.0812285i
$414$		0			0
$415$	−9.72792	+	16.8493i	−0.477525	+	0.827097i
$416$	−1.22474	−	2.12132i	−0.0600481	−	0.104006i
$417$		0			0
$418$	−1.50000	−	0.866025i	−0.0733674	−	0.0423587i
$419$	14.9941			0.732510			0.366255	−	0.930514i	$-0.380640\pi$
$419$	14.9941			0.732510			0.366255	+	0.930514i	$0.380640\pi$
$420$		0			0
$421$	−30.6985			−1.49615			−0.748076	−	0.663613i	$-0.769022\pi$
$421$	−30.6985			−1.49615			−0.748076	+	0.663613i	$0.769022\pi$
$422$	−4.77589	−	2.75736i	−0.232487	−	0.134226i
$423$		0			0
$424$		0			0
$425$	−0.866025	+	1.50000i	−0.0420084	+	0.0727607i
$426$		0			0
$427$	−21.7279	+	16.8493i	−1.05149	+	0.815393i
$428$			2.48528i			0.120131i
$429$		0			0
$430$	−2.12132	+	1.22474i	−0.102299	+	0.0590624i
$431$	−14.6969	+	8.48528i	−0.707927	+	0.408722i	−0.810293	−	0.586025i	$-0.800692\pi$
$431$	−14.6969	+	8.48528i	−0.707927	+	0.408722i	0.102366	+	0.994747i	$0.467359\pi$
$432$		0			0
$433$			17.8639i			0.858483i	0.903190	+	0.429241i	$0.141219\pi$
$433$			17.8639i			0.858483i	−0.903190	+	0.429241i	$0.858781\pi$
$434$	8.36308	−	20.4853i	0.401441	−	0.983325i
$435$		0			0
$436$	−9.12132	+	15.7986i	−0.436832	+	0.756615i
$437$	−1.07616	−	1.86396i	−0.0514796	−	0.0891653i
$438$		0			0
$439$	17.3787	+	10.0336i	0.829439	+	0.478877i	0.853661	−	0.520830i	$-0.174377\pi$
$439$	17.3787	+	10.0336i	0.829439	+	0.478877i	−0.0242215	+	0.999707i	$0.507711\pi$
$440$	−2.44949			−0.116775
$441$		0			0
$442$	4.24264			0.201802
$443$	−2.59808	−	1.50000i	−0.123438	−	0.0712672i	0.437009	−	0.899457i	$-0.356038\pi$
$443$	−2.59808	−	1.50000i	−0.123438	−	0.0712672i	−0.560448	+	0.828190i	$0.689371\pi$
$444$		0			0
$445$	−7.24264	−	12.5446i	−0.343334	−	0.594672i
$446$	−2.23936	+	3.87868i	−0.106037	+	0.183661i
$447$		0			0
$448$	1.00000	−	2.44949i	0.0472456	−	0.115728i
$449$			37.4558i			1.76765i	0.467817	+	0.883825i	$0.345041\pi$
$449$			37.4558i			1.76765i	−0.467817	+	0.883825i	$0.654959\pi$
$450$		0			0
$451$	−7.24264	+	4.18154i	−0.341043	+	0.196901i
$452$	8.87039	−	5.12132i	0.417228	−	0.240887i
$453$		0			0
$454$		−	13.2621i		−	0.622419i
$455$	12.5446	−	9.72792i	0.588101	−	0.456052i
$456$		0			0
$457$	12.8492	−	22.2555i	0.601062	−	1.04107i	−0.391598	−	0.920136i	$-0.628078\pi$
$457$	12.8492	−	22.2555i	0.601062	−	1.04107i	0.992661	−	0.120934i	$-0.0385889\pi$
$458$	−6.92820	−	12.0000i	−0.323734	−	0.560723i
$459$		0			0
$460$	−2.63604	−	1.52192i	−0.122906	−	0.0709598i
$461$	−4.77589			−0.222435			−0.111218	−	0.993796i	$-0.535475\pi$
$461$	−4.77589			−0.222435			−0.111218	+	0.993796i	$0.535475\pi$
$462$		0			0
$463$	21.4558			0.997138			0.498569	−	0.866850i	$-0.333859\pi$
$463$	21.4558			0.997138			0.498569	+	0.866850i	$0.333859\pi$
$464$	6.27231	+	3.62132i	0.291185	+	0.168116i
$465$		0			0
$466$	−0.257359	−	0.445759i	−0.0119219	−	0.0206494i
$467$	2.59808	−	4.50000i	0.120225	−	0.208235i	−0.799632	−	0.600491i	$-0.794972\pi$
$467$	2.59808	−	4.50000i	0.120225	−	0.208235i	0.919856	+	0.392256i	$0.128305\pi$
$468$		0			0
$469$	11.7574	−	1.60896i	0.542904	−	0.0742948i
$470$			6.72792i			0.310336i
$471$		0			0
$472$	−3.98528	+	2.30090i	−0.183437	+	0.105908i
$473$	0.866025	−	0.500000i	0.0398199	−	0.0229900i
$474$		0			0
$475$			1.73205i			0.0794719i
$476$	2.80821	+	3.62132i	0.128714	+	0.165983i
$477$		0			0
$478$	−6.36396	+	11.0227i	−0.291081	+	0.504167i
$479$	−18.0379	−	31.2426i	−0.824175	−	1.42751i	−0.902548	−	0.430589i	$-0.858306\pi$
$479$	−18.0379	−	31.2426i	−0.824175	−	1.42751i	0.0783735	−	0.996924i	$-0.475027\pi$
$480$		0			0
$481$	−11.1213	−	6.42090i	−0.507089	−	0.292768i
$482$	26.5241			1.20814
$483$		0			0
$484$	1.00000			0.0454545
$485$	−14.0665	−	8.12132i	−0.638729	−	0.368770i
$486$		0			0
$487$	20.6066	+	35.6917i	0.933774	+	1.61734i	0.776805	+	0.629741i	$0.216839\pi$
$487$	20.6066	+	35.6917i	0.933774	+	1.61734i	0.156969	+	0.987604i	$0.449828\pi$
$488$	−5.19615	+	9.00000i	−0.235219	+	0.407411i
$489$		0			0
$490$	16.6066	+	4.26858i	0.750210	+	0.192835i
$491$			42.7279i			1.92828i	0.265387	+	0.964142i	$0.414500\pi$
$491$			42.7279i			1.92828i	−0.265387	+	0.964142i	$0.585500\pi$
$492$		0			0
$493$	−10.8640	+	6.27231i	−0.489288	+	0.282491i
$494$	3.67423	−	2.12132i	0.165312	−	0.0954427i
$495$		0			0
$496$		−	8.36308i		−	0.375513i
$497$	−3.04384	−	1.24264i	−0.136535	−	0.0557401i
$498$		0			0
$499$	−18.6066	+	32.2276i	−0.832946	+	1.44270i	0.0627461	+	0.998030i	$0.480014\pi$
$499$	−18.6066	+	32.2276i	−0.832946	+	1.44270i	−0.895692	+	0.444675i	$0.853319\pi$
$500$	−4.89898	−	8.48528i	−0.219089	−	0.379473i
$501$		0			0
$502$	−0.985281	−	0.568852i	−0.0439753	−	0.0253891i
$503$	0.594346			0.0265006			0.0132503	−	0.999912i	$-0.495782\pi$
$503$	0.594346			0.0265006			0.0132503	+	0.999912i	$0.495782\pi$
$504$		0			0
$505$	34.2426			1.52378
$506$	1.07616	+	0.621320i	0.0478411	+	0.0276211i
$507$		0			0
$508$	10.6213	+	18.3967i	0.471245	+	0.816220i
$509$	0.594346	−	1.02944i	0.0263439	−	0.0456290i	−0.852553	−	0.522641i	$-0.824947\pi$
$509$	0.594346	−	1.02944i	0.0263439	−	0.0456290i	0.878897	+	0.477012i	$0.158280\pi$
$510$		0			0
$511$	−1.75736	−	12.8418i	−0.0777410	−	0.568088i
$512$		−	1.00000i		−	0.0441942i
$513$		0			0
$514$	−7.60660	+	4.39167i	−0.335513	+	0.193708i
$515$	−24.1977	+	13.9706i	−1.06628	+	0.615617i
$516$		0			0
$517$		−	2.74666i		−	0.120798i
$518$	−1.88064	−	13.7426i	−0.0826305	−	0.603817i
$519$		0			0
$520$	3.00000	−	5.19615i	0.131559	−	0.227866i
$521$	−16.4290	−	28.4558i	−0.719767	−	1.24667i	−0.961092	−	0.276229i	$-0.910915\pi$
$521$	−16.4290	−	28.4558i	−0.719767	−	1.24667i	0.241325	−	0.970444i	$-0.422418\pi$
$522$		0			0
$523$	6.72792	+	3.88437i	0.294191	+	0.169852i	0.639831	−	0.768516i	$-0.279005\pi$
$523$	6.72792	+	3.88437i	0.294191	+	0.169852i	−0.345639	+	0.938368i	$0.612338\pi$
$524$	−16.3059			−0.712326
$525$		0			0
$526$	−9.51472			−0.414861
$527$	12.5446	+	7.24264i	0.546452	+	0.315494i
$528$		0			0
$529$	−10.7279	−	18.5813i	−0.466431	−	0.807883i
$530$		0			0
$531$		0			0
$532$	4.24264	+	1.73205i	0.183942	+	0.0750939i
$533$		−	20.4853i		−	0.887316i
$534$		0			0
$535$	−5.27208	+	3.04384i	−0.227932	+	0.131596i
$536$	3.88437	−	2.24264i	0.167779	−	0.0968673i
$537$		0			0
$538$			21.7992i			0.939831i
$539$	−6.77962	−	1.74264i	−0.292019	−	0.0750608i
$540$		0			0
$541$	18.2426	−	31.5972i	0.784312	−	1.35847i	−0.145097	−	0.989417i	$-0.546349\pi$
$541$	18.2426	−	31.5972i	0.784312	−	1.35847i	0.929409	−	0.369051i	$-0.120317\pi$
$542$	5.91359	+	10.2426i	0.254010	+	0.439959i
$543$		0			0
$544$	1.50000	+	0.866025i	0.0643120	+	0.0371305i
$545$	−44.6852			−1.91410
$546$		0			0
$547$	−27.4853			−1.17519			−0.587593	−	0.809157i	$-0.699924\pi$
$547$	−27.4853			−1.17519			−0.587593	+	0.809157i	$0.699924\pi$
$548$	8.87039	+	5.12132i	0.378924	+	0.218772i
$549$		0			0
$550$	−0.500000	−	0.866025i	−0.0213201	−	0.0369274i
$551$	−6.27231	+	10.8640i	−0.267209	+	0.462820i
$552$		0			0
$553$	−12.9706	−	16.7262i	−0.551564	−	0.711269i
$554$		−	21.4558i		−	0.911571i
$555$		0			0
$556$	−8.74264	+	5.04757i	−0.370771	+	0.214064i
$557$	−17.9254	+	10.3492i	−0.759524	+	0.438511i	−0.829125	−	0.559063i	$-0.811161\pi$
$557$	−17.9254	+	10.3492i	−0.759524	+	0.438511i	0.0696007	+	0.997575i	$0.477827\pi$
$558$		0			0
$559$			2.44949i			0.103602i
$560$	6.42090	−	0.878680i	0.271332	−	0.0371310i
$561$		0			0
$562$	−3.25736	+	5.64191i	−0.137403	+	0.237990i
$563$	−14.7840	−	25.6066i	−0.623070	−	1.07919i	−0.988911	−	0.148512i	$-0.952552\pi$
$563$	−14.7840	−	25.6066i	−0.623070	−	1.07919i	0.365840	−	0.930678i	$-0.380782\pi$
$564$		0			0
$565$	21.7279	+	12.5446i	0.914101	+	0.527756i
$566$	−10.9867			−0.461803
$567$		0			0
$568$	−1.24264			−0.0521400
$569$	19.4473	+	11.2279i	0.815275	+	0.470699i	0.848784	−	0.528739i	$-0.177335\pi$
$569$	19.4473	+	11.2279i	0.815275	+	0.470699i	−0.0335097	+	0.999438i	$0.510668\pi$
$570$		0			0
$571$	5.25736	+	9.10601i	0.220014	+	0.381075i	0.954812	−	0.297211i	$-0.0960565\pi$
$571$	5.25736	+	9.10601i	0.220014	+	0.381075i	−0.734798	+	0.678286i	$0.762723\pi$
$572$	−1.22474	+	2.12132i	−0.0512092	+	0.0886969i
$573$		0			0
$574$	17.4853	−	13.5592i	0.729822	−	0.565951i
$575$		−	1.24264i		−	0.0518217i
$576$		0			0
$577$	3.00000	−	1.73205i	0.124892	−	0.0721062i	−0.436253	−	0.899824i	$-0.643695\pi$
$577$	3.00000	−	1.73205i	0.124892	−	0.0721062i	0.561144	+	0.827718i	$0.310361\pi$
$578$	12.1244	−	7.00000i	0.504307	−	0.291162i
$579$		0			0
$580$			17.7408i			0.736646i
$581$	−7.94282	+	19.4558i	−0.329523	+	0.807164i
$582$		0			0
$583$		0			0
$584$	−2.44949	−	4.24264i	−0.101361	−	0.175562i
$585$		0			0
$586$	15.1066	+	8.72180i	0.624048	+	0.360294i
$587$	−34.0467			−1.40526			−0.702628	−	0.711557i	$-0.747990\pi$
$587$	−34.0467			−1.40526			−0.702628	+	0.711557i	$0.747990\pi$
$588$		0			0
$589$	14.4853			0.596856
$590$	−9.76191	−	5.63604i	−0.401891	−	0.232032i
$591$		0			0
$592$	−2.62132	−	4.54026i	−0.107736	−	0.186604i
$593$	12.6932	−	21.9853i	0.521248	−	0.902827i	−0.478447	−	0.878116i	$-0.658800\pi$
$593$	12.6932	−	21.9853i	0.521248	−	0.902827i	0.999695	−	0.0247109i	$-0.00786652\pi$
$594$		0			0
$595$	−4.24264	+	10.3923i	−0.173931	+	0.426043i
$596$			13.2426i			0.542440i
$597$		0			0
$598$	−2.63604	+	1.52192i	−0.107796	+	0.0622358i
$599$	−15.5885	+	9.00000i	−0.636927	+	0.367730i	−0.783430	−	0.621480i	$-0.786532\pi$
$599$	−15.5885	+	9.00000i	−0.636927	+	0.367730i	0.146503	+	0.989210i	$0.453198\pi$
$600$		0			0
$601$		−	17.1464i		−	0.699417i	−0.936858	−	0.349709i	$-0.886281\pi$
$601$		−	17.1464i		−	0.699417i	0.936858	−	0.349709i	$-0.113719\pi$
$602$	−2.09077	+	1.62132i	−0.0852134	+	0.0660801i
$603$		0			0
$604$	5.62132	−	9.73641i	0.228728	−	0.396169i
$605$	1.22474	+	2.12132i	0.0497930	+	0.0862439i
$606$		0			0
$607$	12.5147	+	7.22538i	0.507957	+	0.293269i	0.731993	−	0.681312i	$-0.238590\pi$
$607$	12.5147	+	7.22538i	0.507957	+	0.293269i	−0.224037	+	0.974581i	$0.571924\pi$
$608$	1.73205			0.0702439
$609$		0			0
$610$	−25.4558			−1.03068
$611$	5.82655	+	3.36396i	0.235717	+	0.136091i
$612$		0			0
$613$	−3.63604	−	6.29780i	−0.146858	−	0.254366i	0.783206	−	0.621762i	$-0.213583\pi$
$613$	−3.63604	−	6.29780i	−0.146858	−	0.254366i	−0.930065	+	0.367396i	$0.880249\pi$
$614$	−13.5592	+	23.4853i	−0.547206	+	0.947789i
$615$		0			0
$616$	−2.62132	+	0.358719i	−0.105616	+	0.0144532i
$617$		−	33.2132i		−	1.33711i	−0.743661	−	0.668557i	$-0.766912\pi$
$617$		−	33.2132i		−	1.33711i	0.743661	−	0.668557i	$-0.233088\pi$
$618$		0			0
$619$	12.2132	−	7.05130i	0.490890	−	0.283416i	−0.234054	−	0.972224i	$-0.575199\pi$
$619$	12.2132	−	7.05130i	0.490890	−	0.283416i	0.724944	+	0.688808i	$0.241866\pi$
$620$	17.7408	−	10.2426i	0.712487	−	0.411354i
$621$		0			0
$622$			11.7041i			0.469291i
$623$	−9.58783	−	12.3640i	−0.384128	−	0.495352i
$624$		0			0
$625$	14.5000	−	25.1147i	0.580000	−	1.00459i
$626$	−5.76500	−	9.98528i	−0.230416	−	0.399092i
$627$		0			0
$628$	−0.621320	−	0.358719i	−0.0247934	−	0.0143145i
$629$	9.08052			0.362064
$630$		0			0
$631$	−21.6985			−0.863803			−0.431902	−	0.901921i	$-0.642157\pi$
$631$	−21.6985			−0.863803			−0.431902	+	0.901921i	$0.642157\pi$
$632$	−6.92820	−	4.00000i	−0.275589	−	0.159111i
$633$		0			0
$634$	8.12132	+	14.0665i	0.322539	+	0.558654i
$635$	−26.0168	+	45.0624i	−1.03245	+	1.78825i
$636$		0			0
$637$	12.0000	−	12.2474i	0.475457	−	0.485262i
$638$		−	7.24264i		−	0.286739i
$639$		0			0
$640$	2.12132	−	1.22474i	0.0838525	−	0.0484123i
$641$	−21.4150	+	12.3640i	−0.845842	+	0.488347i	−0.859246	−	0.511563i	$-0.829067\pi$
$641$	−21.4150	+	12.3640i	−0.845842	+	0.488347i	0.0134037	+	0.999910i	$0.495733\pi$
$642$		0			0
$643$		−	30.1623i		−	1.18949i	−0.803916	−	0.594743i	$-0.797254\pi$
$643$		−	30.1623i		−	1.18949i	0.803916	−	0.594743i	$-0.202746\pi$
$644$	−3.04384	−	1.24264i	−0.119944	−	0.0489669i
$645$		0			0
$646$	−1.50000	+	2.59808i	−0.0590167	+	0.102220i
$647$	−14.8710	−	25.7574i	−0.584640	−	1.01263i	−0.994920	−	0.100667i	$-0.967902\pi$
$647$	−14.8710	−	25.7574i	−0.584640	−	1.01263i	0.410280	−	0.911960i	$-0.365431\pi$
$648$		0			0
$649$	3.98528	+	2.30090i	0.156436	+	0.0903184i
$650$	2.44949			0.0960769
$651$		0			0
$652$	−18.9706			−0.742945
$653$	−39.1558	−	22.6066i	−1.53228	−	0.884665i	−0.999256	−	0.0385672i	$-0.987721\pi$
$653$	−39.1558	−	22.6066i	−1.53228	−	0.884665i	−0.533028	−	0.846097i	$-0.678946\pi$
$654$		0			0
$655$	−19.9706	−	34.5900i	−0.780314	−	1.35154i
$656$	4.18154	−	7.24264i	0.163262	−	0.282778i
$657$		0			0
$658$	0.985281	+	7.19988i	0.0384103	+	0.280681i
$659$			7.75736i			0.302184i	0.988520	+	0.151092i	$0.0482789\pi$
$659$			7.75736i			0.302184i	−0.988520	+	0.151092i	$0.951721\pi$
$660$		0			0
$661$	11.3787	−	6.56948i	0.442579	−	0.255523i	−0.262112	−	0.965038i	$-0.584419\pi$
$661$	11.3787	−	6.56948i	0.442579	−	0.255523i	0.704691	+	0.709514i	$0.251086\pi$
$662$	12.1244	−	7.00000i	0.471226	−	0.272063i
$663$		0			0
$664$			7.94282i			0.308241i
$665$	1.52192	+	11.1213i	0.0590174	+	0.431266i
$666$		0			0
$667$	4.50000	−	7.79423i	0.174241	−	0.301794i
$668$	7.13834	+	12.3640i	0.276191	+	0.478376i
$669$		0			0
$670$	9.51472	+	5.49333i	0.367586	+	0.212226i
$671$	10.3923			0.401190
$672$		0			0
$673$	29.7574			1.14706			0.573531	−	0.819184i	$-0.305573\pi$
$673$	29.7574			1.14706			0.573531	+	0.819184i	$0.305573\pi$
$674$	−13.6463	−	7.87868i	−0.525635	−	0.303475i
$675$		0			0
$676$	3.50000	+	6.06218i	0.134615	+	0.233161i
$677$	12.9033	−	22.3492i	0.495916	−	0.858951i	−0.504073	−	0.863661i	$-0.668166\pi$
$677$	12.9033	−	22.3492i	0.495916	−	0.858951i	0.999989	+	0.00470976i	$0.00149917\pi$
$678$		0			0
$679$	−16.2426	−	6.63103i	−0.623335	−	0.254476i
$680$			4.24264i			0.162698i
$681$		0			0
$682$	−7.24264	+	4.18154i	−0.277335	+	0.160119i
$683$	−20.3389	+	11.7426i	−0.778244	+	0.449320i	−0.835808	−	0.549022i	$-0.815000\pi$
$683$	−20.3389	+	11.7426i	−0.778244	+	0.449320i	0.0575633	+	0.998342i	$0.481667\pi$
$684$		0			0
$685$			25.0892i			0.958611i
$686$	18.3967	+	2.13604i	0.702388	+	0.0815543i
$687$		0			0
$688$	−0.500000	+	0.866025i	−0.0190623	+	0.0330169i
$689$		0			0
$690$		0			0
$691$	−35.4853	−	20.4874i	−1.34992	−	0.779379i	−0.361685	−	0.932300i	$-0.617799\pi$
$691$	−35.4853	−	20.4874i	−1.34992	−	0.779379i	−0.988238	+	0.152921i	$0.951132\pi$
$692$	−11.8272			−0.449602
$693$		0			0
$694$	32.4853			1.23312
$695$	−21.4150	−	12.3640i	−0.812318	−	0.468992i
$696$		0			0
$697$	7.24264	+	12.5446i	0.274335	+	0.475161i
$698$	−7.34847	+	12.7279i	−0.278144	+	0.481759i
$699$		0			0
$700$	1.62132	+	2.09077i	0.0612801	+	0.0790237i
$701$		−	49.2426i		−	1.85987i	−0.367725	−	0.929934i	$-0.619863\pi$
$701$		−	49.2426i		−	1.85987i	0.367725	−	0.929934i	$-0.380137\pi$
$702$		0			0
$703$	7.86396	−	4.54026i	0.296595	−	0.171239i
$704$	−0.866025	+	0.500000i	−0.0326396	+	0.0188445i
$705$		0			0
$706$		−	15.7116i		−	0.591312i
$707$	36.6447	−	5.01472i	1.37817	−	0.188598i
$708$		0			0
$709$	−5.62132	+	9.73641i	−0.211113	+	0.365659i	−0.952063	−	0.305901i	$-0.901042\pi$
$709$	−5.62132	+	9.73641i	−0.211113	+	0.365659i	0.740950	+	0.671560i	$0.234376\pi$
$710$	−1.52192	−	2.63604i	−0.0571166	−	0.0989288i
$711$		0			0
$712$	−5.12132	−	2.95680i	−0.191930	−	0.110811i
$713$	−10.3923			−0.389195
$714$		0			0
$715$	−6.00000			−0.224387
$716$	−7.79423	−	4.50000i	−0.291284	−	0.168173i
$717$		0			0
$718$	17.4853	+	30.2854i	0.652545	+	1.13024i
$719$	15.6500	−	27.1066i	0.583647	−	1.01091i	−0.411396	−	0.911457i	$-0.634959\pi$
$719$	15.6500	−	27.1066i	0.583647	−	1.01091i	0.995043	−	0.0994490i	$-0.0317080\pi$
$720$		0			0
$721$	−23.8492	+	18.4943i	−0.888192	+	0.688762i
$722$		−	16.0000i		−	0.595458i
$723$		0			0
$724$	−4.24264	+	2.44949i	−0.157676	+	0.0910346i
$725$	−6.27231	+	3.62132i	−0.232948	+	0.134492i
$726$		0			0
$727$			38.1051i			1.41324i	0.707593	+	0.706620i	$0.249781\pi$
$727$			38.1051i			1.41324i	−0.707593	+	0.706620i	$0.750219\pi$
$728$	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
$733$	46.4558	+	26.8213i	1.71589	+	0.990667i	0.926095	+	0.377291i	$0.123144\pi$
$733$	46.4558	+	26.8213i	1.71589	+	0.990667i	0.789791	+	0.613376i	$0.210189\pi$
$734$	2.02922			0.0749000
$735$		0			0
$736$	−1.24264			−0.0458043
$737$	−3.88437	−	2.24264i	−0.143083	−	0.0826087i
$738$		0			0
$739$	23.7279	+	41.0980i	0.872846	+	1.51181i	0.859040	+	0.511908i	$0.171061\pi$
$739$	23.7279	+	41.0980i	0.872846	+	1.51181i	0.0138057	+	0.999905i	$0.495605\pi$
$740$	6.42090	−	11.1213i	0.236037	−	0.408828i
$741$		0			0
$742$		0			0
$743$		−	23.6985i		−	0.869413i	−0.900572	−	0.434707i	$-0.856852\pi$
$743$		−	23.6985i		−	0.869413i	0.900572	−	0.434707i	$-0.143148\pi$
$744$		0			0
$745$	−28.0919	+	16.2189i	−1.02921	+	0.594213i
$746$	29.4449	−	17.0000i	1.07805	−	0.622414i
$747$		0			0
$748$		−	1.73205i		−	0.0633300i
$749$	−5.19615	+	4.02944i	−0.189863	+	0.147232i
$750$		0			0
$751$	15.1213	−	26.1909i	0.551785	−	0.955719i	−0.446361	−	0.894853i	$-0.647280\pi$
$751$	15.1213	−	26.1909i	0.551785	−	0.955719i	0.998146	−	0.0608664i	$-0.0193864\pi$
$752$	1.37333	+	2.37868i	0.0500802	+	0.0867415i
$753$		0			0
$754$	15.3640	+	8.87039i	0.559522	+	0.323040i
$755$	27.5387			1.00224
$756$		0			0
$757$	−51.6690			−1.87794			−0.938972	−	0.343994i	$-0.888220\pi$
$757$	−51.6690			−1.87794			−0.938972	+	0.343994i	$0.888220\pi$
$758$	−6.29780	−	3.63604i	−0.228747	−	0.132067i
$759$		0			0
$760$	2.12132	+	3.67423i	0.0769484	+	0.133278i
$761$	13.1390	−	22.7574i	0.476287	−	0.824954i	−0.523344	−	0.852122i	$-0.675316\pi$
$761$	13.1390	−	22.7574i	0.476287	−	0.824954i	0.999631	+	0.0271681i	$0.00864893\pi$
$762$		0			0
$763$	−47.8198	+	6.54399i	−1.73119	+	0.236908i
$764$			16.9706i			0.613973i
$765$		0			0
$766$	−30.3198	+	17.5051i	−1.09550	+	0.632487i
$767$	−9.76191	+	5.63604i	−0.352482	+	0.203506i
$768$		0			0
$769$		−	33.0321i		−	1.19117i	−0.803294	−	0.595583i	$-0.796921\pi$
$769$		−	33.0321i		−	1.19117i	0.803294	−	0.595583i	$-0.203079\pi$
$770$	−3.97141	−	5.12132i	−0.143120	−	0.184560i
$771$		0			0
$772$	2.36396	−	4.09450i	0.0850808	−	0.147364i
$773$	8.45012	+	14.6360i	0.303930	+	0.526422i	0.977022	−	0.213136i	$-0.0683679\pi$
$773$	8.45012	+	14.6360i	0.303930	+	0.526422i	−0.673093	+	0.739558i	$0.735035\pi$
$774$		0			0
$775$	7.24264	+	4.18154i	0.260163	+	0.150205i
$776$	−6.63103			−0.238040
$777$		0			0
$778$	−11.2721			−0.404124
$779$	12.5446	+	7.24264i	0.449458	+	0.259495i
$780$		0			0
$781$	0.621320	+	1.07616i	0.0222326	+	0.0385080i
$782$	1.07616	−	1.86396i	0.0384833	−	0.0666551i
$783$		0			0
$784$	6.74264	−	1.88064i	0.240809	−	0.0671656i
$785$		−	1.75736i		−	0.0627228i
$786$		0			0
$787$	−27.4706	+	15.8601i	−0.979220	+	0.565353i	−0.902034	−	0.431664i	$-0.857927\pi$
$787$	−27.4706	+	15.8601i	−0.979220	+	0.565353i	−0.0771853	+	0.997017i	$0.524593\pi$
$788$	18.8169	−	10.8640i	0.670325	−	0.387013i
$789$		0			0
$790$		−	19.5959i		−	0.697191i
$791$	25.0892	+	10.2426i	0.892071	+	0.364186i
$792$		0			0
$793$	−12.7279	+	22.0454i	−0.451982	+	0.782855i
$794$	−0.778985	−	1.34924i	−0.0276452	−	0.0478828i
$795$		0			0
$796$	6.36396	+	3.67423i	0.225565	+	0.130230i
$797$	−27.9590			−0.990359			−0.495179	−	0.868791i	$-0.664898\pi$
$797$	−27.9590			−0.990359			−0.495179	+	0.868791i	$0.664898\pi$
$798$		0			0
$799$	−4.75736			−0.168303
$800$	0.866025	+	0.500000i	0.0306186	+	0.0176777i
$801$		0			0
$802$	9.87868	+	17.1104i	0.348828	+	0.604188i
$803$	−2.44949	+	4.24264i	−0.0864406	+	0.149720i
$804$		0			0
$805$	−1.09188	−	7.97887i	−0.0384838	−	0.281218i
$806$		−	20.4853i		−	0.721563i
$807$		0			0
$808$	12.1066	−	6.98975i	0.425909	−	0.245899i
$809$	19.0016	−	10.9706i	0.668060	−	0.385704i	−0.127281	−	0.991867i	$-0.540625\pi$
$809$	19.0016	−	10.9706i	0.668060	−	0.385704i	0.795341	+	0.606162i	$0.207292\pi$
$810$		0			0
$811$			21.8713i			0.768006i	0.923332	+	0.384003i	$0.125455\pi$
$811$			21.8713i			0.768006i	−0.923332	+	0.384003i	$0.874545\pi$
$812$	2.59808	+	18.9853i	0.0911746	+	0.666253i
$813$		0			0
$814$	−2.62132	+	4.54026i	−0.0918772	+	0.159136i
$815$	−23.2341	−	40.2426i	−0.813855	−	1.40964i
$816$		0			0
$817$	−1.50000	−	0.866025i	−0.0524784	−	0.0302984i
$818$	−14.2767			−0.499172
$819$		0			0
$820$	20.4853			0.715377
$821$	22.9369	+	13.2426i	0.800504	+	0.462171i	0.843647	−	0.536898i	$-0.180404\pi$
$821$	22.9369	+	13.2426i	0.800504	+	0.462171i	−0.0431432	+	0.999069i	$0.513737\pi$
$822$		0			0
$823$	0.393398	+	0.681386i	0.0137130	+	0.0237516i	0.872800	−	0.488077i	$-0.162302\pi$
$823$	0.393398	+	0.681386i	0.0137130	+	0.0237516i	−0.859087	+	0.511829i	$0.828968\pi$
$824$	−5.70346	+	9.87868i	−0.198689	+	0.344140i
$825$		0			0
$826$	−11.2721	−	4.60181i	−0.392206	−	0.160117i
$827$			2.48528i			0.0864217i	0.999066	+	0.0432109i	$0.0137587\pi$
$827$			2.48528i			0.0864217i	−0.999066	+	0.0432109i	$0.986241\pi$
$828$		0			0
$829$	39.3198	−	22.7013i	1.36563	−	0.788449i	0.375266	−	0.926917i	$-0.377551\pi$
$829$	39.3198	−	22.7013i	1.36563	−	0.788449i	0.990367	+	0.138468i	$0.0442179\pi$
$830$	−16.8493	+	9.72792i	−0.584846	+	0.337661i
$831$		0			0
$832$		−	2.44949i		−	0.0849208i
$833$	−3.01834	+	11.7426i	−0.104579	+	0.406858i
$834$		0			0
$835$	−17.4853	+	30.2854i	−0.605103	+	1.04807i
$836$	−0.866025	−	1.50000i	−0.0299521	−	0.0518786i
$837$		0			0
$838$	12.9853	+	7.49706i	0.448569	+	0.258981i
$839$	−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
$841$	−23.4558			−0.808822
$842$	−26.5857	−	15.3492i	−0.916203	−	0.528970i
$843$		0			0
$844$	−2.75736	−	4.77589i	−0.0949122	−	0.164393i
$845$	−8.57321	+	14.8492i	−0.294928	+	0.510829i
$846$		0			0
$847$	1.62132	+	2.09077i	0.0557092	+	0.0718397i
$848$		0			0
$849$		0			0
$850$	−1.50000	+	0.866025i	−0.0514496	+	0.0297044i
$851$	−5.64191	+	3.25736i	−0.193402	+	0.111661i
$852$		0			0
$853$			12.2474i			0.419345i	0.977772	+	0.209672i	$0.0672397\pi$
$853$			12.2474i			0.419345i	−0.977772	+	0.209672i	$0.932760\pi$
$854$	−27.2416	+	3.72792i	−0.932187	+	0.127567i
$855$		0			0
$856$	−1.24264	+	2.15232i	−0.0424726	+	0.0735647i
$857$	−17.8894	−	30.9853i	−0.611089	−	1.05844i	−0.991057	−	0.133438i	$-0.957398\pi$
$857$	−17.8894	−	30.9853i	−0.611089	−	1.05844i	0.379968	−	0.925000i	$-0.375935\pi$
$858$		0			0
$859$	−10.0919	−	5.82655i	−0.344331	−	0.198799i	0.317855	−	0.948139i	$-0.397038\pi$
$859$	−10.0919	−	5.82655i	−0.344331	−	0.198799i	−0.662185	+	0.749340i	$0.730371\pi$
$860$	−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	$-0.190695\pi$
$863$	22.0454	+	12.7279i	0.750434	+	0.433264i	−0.0754164	+	0.997152i	$0.524029\pi$
$864$		0			0
$865$	−14.4853	−	25.0892i	−0.492514	−	0.853060i
$866$	−8.93193	+	15.4706i	−0.303519	+	0.525711i
$867$		0			0
$868$	17.4853	−	13.5592i	0.593489	−	0.460230i
$869$			8.00000i			0.271381i
$870$		0			0
$871$	9.51472	−	5.49333i	0.322394	−	0.186134i
$872$	−15.7986	+	9.12132i	−0.535008	+	0.308887i
$873$		0			0
$874$		−	2.15232i		−	0.0728032i
$875$	9.79796	−	24.0000i	0.331231	−	0.811348i
$876$		0			0
$877$	−13.8787	+	24.0386i	−0.468650	+	0.811725i	−0.999358	−	0.0358295i	$-0.988593\pi$
$877$	−13.8787	+	24.0386i	−0.468650	+	0.811725i	0.530708	+	0.847555i	$0.321926\pi$
$878$	10.0336	+	17.3787i	0.338617	+	0.586502i
$879$		0			0
$880$	−2.12132	−	1.22474i	−0.0715097	−	0.0412861i
$881$	−14.6969			−0.495152			−0.247576	−	0.968868i	$-0.579634\pi$
$881$	−14.6969			−0.495152			−0.247576	+	0.968868i	$0.579634\pi$
$882$		0			0
$883$	−7.27208			−0.244725			−0.122362	−	0.992485i	$-0.539047\pi$
$883$	−7.27208			−0.244725			−0.122362	+	0.992485i	$0.539047\pi$
$884$	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	−0.725954	−	0.687743i	$-0.758601\pi$
$887$	6.92820	−	12.0000i	0.232626	−	0.402921i	0.958580	+	0.284823i	$0.0919348\pi$
$888$		0			0
$889$	−21.2426	+	52.0336i	−0.712455	+	1.74515i
$890$		−	14.4853i		−	0.485548i
$891$		0			0
$892$	−3.87868	+	2.23936i	−0.129868	+	0.0749792i
$893$	−4.11999	+	2.37868i	−0.137870	+	0.0795995i
$894$		0			0
$895$		−	22.0454i		−	0.736897i
$896$	2.09077	−	1.62132i	0.0698477	−	0.0541645i
$897$		0			0
$898$	−18.7279	+	32.4377i	−0.624959	+	1.08246i
$899$	30.2854	+	52.4558i	1.01007	+	1.74950i
$900$		0			0
$901$		0			0
$902$	−8.36308			−0.278460
$903$		0			0
$904$	10.2426			0.340665
$905$	−10.3923	−	6.00000i	−0.345452	−	0.199447i
$906$		0			0
$907$	−15.7574	−	27.2925i	−0.523215	−	0.906234i	−0.999635	−	0.0270166i	$-0.991399\pi$
$907$	−15.7574	−	27.2925i	−0.523215	−	0.906234i	0.476420	−	0.879218i	$-0.341934\pi$
$908$	6.63103	−	11.4853i	0.220058	−	0.381152i
$909$		0			0
$910$	15.7279	−	2.15232i	0.521376	−	0.0713486i
$911$		−	15.7279i		−	0.521089i	−0.965462	−	0.260545i	$-0.916098\pi$
$911$		−	15.7279i		−	0.521089i	0.965462	−	0.260545i	$-0.0839021\pi$
$912$		0			0
$913$	6.87868	−	3.97141i	0.227651	−	0.131434i
$914$	22.2555	−	12.8492i	0.736148	−	0.425015i
$915$		0			0
$916$		−	13.8564i		−	0.457829i
$917$	−26.4371	−	34.0919i	−0.873029	−	1.12581i
$918$		0			0
$919$	−8.62132	+	14.9326i	−0.284391	+	0.492580i	−0.972461	−	0.233064i	$-0.925125\pi$
$919$	−8.62132	+	14.9326i	−0.284391	+	0.492580i	0.688070	+	0.725644i	$0.258458\pi$
$920$	−1.52192	−	2.63604i	−0.0501761	−	0.0869076i
$921$		0			0
$922$	−4.13604	−	2.38794i	−0.136213	−	0.0786427i
$923$	−3.04384			−0.100189
$924$		0			0
$925$	5.24264			0.172377
$926$	18.5813	+	10.7279i	0.610620	+	0.352541i
$927$		0			0
$928$	3.62132	+	6.27231i	0.118876	+	0.205899i
$929$	7.55860	−	13.0919i	0.247990	−	0.429531i	−0.714978	−	0.699147i	$-0.753563\pi$
$929$	7.55860	−	13.0919i	0.247990	−	0.429531i	0.962968	+	0.269616i	$0.0868968\pi$
$930$		0			0
$931$	3.25736	+	11.6786i	0.106756	+	0.382751i
$932$		−	0.514719i		−	0.0168602i
$933$		0			0
$934$	4.50000	−	2.59808i	0.147244	−	0.0850117i
$935$	3.67423	−	2.12132i	0.120160	−	0.0693746i
$936$		0			0
$937$			7.59466i			0.248107i	0.992276	+	0.124053i	$0.0395894\pi$
$937$			7.59466i			0.248107i	−0.992276	+	0.124053i	$0.960411\pi$
$938$	10.9867	+	4.48528i	0.358727	+	0.146450i
$939$		0			0
$940$	−3.36396	+	5.82655i	−0.109720	+	0.190041i
$941$	−9.43924	−	16.3492i	−0.307710	−	0.532970i	0.670151	−	0.742225i	$-0.266229\pi$
$941$	−9.43924	−	16.3492i	−0.307710	−	0.532970i	−0.977861	+	0.209255i	$0.932896\pi$
$942$		0			0
$943$	−9.00000	−	5.19615i	−0.293080	−	0.169210i
$944$	−4.60181			−0.149776
$945$		0			0
$946$	1.00000			0.0325128
$947$	18.1865	+	10.5000i	0.590983	+	0.341204i	0.765486	−	0.643452i	$-0.222499\pi$
$947$	18.1865	+	10.5000i	0.590983	+	0.341204i	−0.174503	+	0.984657i	$0.555832\pi$
$948$		0			0
$949$	−6.00000	−	10.3923i	−0.194768	−	0.337348i
$950$	−0.866025	+	1.50000i	−0.0280976	+	0.0486664i
$951$		0			0
$952$	0.621320	+	4.54026i	0.0201371	+	0.147151i
$953$			48.0000i			1.55487i	0.628962	+	0.777436i	$0.283480\pi$
$953$			48.0000i			1.55487i	−0.628962	+	0.777436i	$0.716520\pi$
$954$		0			0
$955$	−36.0000	+	20.7846i	−1.16493	+	0.672574i
$956$	−11.0227	+	6.36396i	−0.356500	+	0.205825i
$957$		0			0
$958$		−	36.0759i		−	1.16556i
$959$	3.67423	+	26.8492i	0.118647	+	0.867007i
$960$		0			0
$961$	19.4706	−	33.7240i	0.628083	−	1.08787i
$962$	−6.42090	−	11.1213i	−0.207018	−	0.358566i
$963$		0			0
$964$	22.9706	+	13.2621i	0.739832	+	0.427142i
$965$	11.5810			0.372805
$966$		0			0
$967$	33.2426			1.06901			0.534506	−	0.845165i	$-0.320498\pi$
$967$	33.2426			1.06901			0.534506	+	0.845165i	$0.320498\pi$
$968$	0.866025	+	0.500000i	0.0278351	+	0.0160706i
$969$		0			0
$970$	−8.12132	−	14.0665i	−0.260760	−	0.451649i
$971$	−0.840532	+	1.45584i	−0.0269740	+	0.0467203i	−0.879197	−	0.476458i	$-0.841920\pi$
$971$	−0.840532	+	1.45584i	−0.0269740	+	0.0467203i	0.852223	+	0.523178i	$0.175254\pi$
$972$		0			0
$973$	−24.7279	−	10.0951i	−0.792741	−	0.323635i
$974$			41.2132i			1.32056i
$975$		0			0
$976$	−9.00000	+	5.19615i	−0.288083	+	0.166325i
$977$	9.13151	−	5.27208i	0.292143	−	0.168669i	−0.346765	−	0.937952i	$-0.612720\pi$
$977$	9.13151	−	5.27208i	0.292143	−	0.168669i	0.638908	+	0.769283i	$0.279387\pi$
$978$		0			0
$979$			5.91359i			0.188999i
$980$	12.2474	+	12.0000i	0.391230	+	0.383326i
$981$		0			0
$982$	−21.3640	+	37.0035i	−0.681751	+	1.18083i
$983$	27.3031	+	47.2904i	0.870834	+	1.50833i	0.861135	+	0.508376i	$0.169754\pi$
$983$	27.3031	+	47.2904i	0.870834	+	1.50833i	0.00969837	+	0.999953i	$0.496913\pi$
$984$		0			0
$985$	46.0919	+	26.6112i	1.46861	+	0.847902i
$986$	−12.5446			−0.399502
$987$		0			0
$988$	4.24264			0.134976
$989$	1.07616	+	0.621320i	0.0342198	+	0.0197568i
$990$		0			0
$991$	2.24264	+	3.88437i	0.0712398	+	0.123391i	0.899445	−	0.437034i	$-0.143971\pi$
$991$	2.24264	+	3.88437i	0.0712398	+	0.123391i	−0.828205	+	0.560425i	$0.810638\pi$
$992$	4.18154	−	7.24264i	0.132764	−	0.229954i
$993$		0			0
$994$	−2.01472	−	2.59808i	−0.0639030	−	0.0824060i
$995$			18.0000i			0.570638i
$996$		0			0
$997$	27.5772	−	15.9217i	0.873378	−	0.504245i	0.00490839	−	0.999988i	$-0.498438\pi$
$997$	27.5772	−	15.9217i	0.873378	−	0.504245i	0.868469	+	0.495743i	$0.165104\pi$
$998$	−32.2276	+	18.6066i	−1.02015	+	0.588982i
$999$		0			0

Twists

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

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

\(f(q)\)	\(=\)	\(q+(0.866025 + 0.500000i) q^{2} +(0.500000 + 0.866025i) q^{4} +(-1.22474 + 2.12132i) q^{5} +(-1.00000 + 2.44949i) q^{7} +1.00000i q^{8} +O(q^{10})\)	\(q+(0.866025 + 0.500000i) q^{2} +(0.500000 + 0.866025i) q^{4} +(-1.22474 + 2.12132i) q^{5} +(-1.00000 + 2.44949i) q^{7} +1.00000i q^{8} +(-2.12132 + 1.22474i) q^{10} +(0.866025 - 0.500000i) q^{11} +2.44949i q^{13} +(-2.09077 + 1.62132i) q^{14} +(-0.500000 + 0.866025i) q^{16} +(-0.866025 - 1.50000i) q^{17} +(-1.50000 - 0.866025i) q^{19} -2.44949 q^{20} +1.00000 q^{22} +(1.07616 + 0.621320i) q^{23} +(-0.500000 - 0.866025i) q^{25} +(-1.22474 + 2.12132i) q^{26} +(-2.62132 + 0.358719i) q^{28} -7.24264i q^{29} +(-7.24264 + 4.18154i) q^{31} +(-0.866025 + 0.500000i) q^{32} -1.73205i q^{34} +(-3.97141 - 5.12132i) q^{35} +(-2.62132 + 4.54026i) q^{37} +(-0.866025 - 1.50000i) q^{38} +(-2.12132 - 1.22474i) q^{40} -8.36308 q^{41} +1.00000 q^{43} +(0.866025 + 0.500000i) q^{44} +(0.621320 + 1.07616i) q^{46} +(1.37333 - 2.37868i) q^{47} +(-5.00000 - 4.89898i) q^{49} -1.00000i q^{50} +(-2.12132 + 1.22474i) q^{52} +2.44949i q^{55} +(-2.44949 - 1.00000i) q^{56} +(3.62132 - 6.27231i) q^{58} +(2.30090 + 3.98528i) q^{59} +(9.00000 + 5.19615i) q^{61} -8.36308 q^{62} -1.00000 q^{64} +(-5.19615 - 3.00000i) q^{65} +(-2.24264 - 3.88437i) q^{67} +(0.866025 - 1.50000i) q^{68} +(-0.878680 - 6.42090i) q^{70} +1.24264i q^{71} +(-4.24264 + 2.44949i) q^{73} +(-4.54026 + 2.62132i) q^{74} -1.73205i q^{76} +(0.358719 + 2.62132i) q^{77} +(-4.00000 + 6.92820i) q^{79} +(-1.22474 - 2.12132i) q^{80} +(-7.24264 - 4.18154i) q^{82} +7.94282 q^{83} +4.24264 q^{85} +(0.866025 + 0.500000i) q^{86} +(0.500000 + 0.866025i) q^{88} +(-2.95680 + 5.12132i) q^{89} +(-6.00000 - 2.44949i) q^{91} +1.24264i q^{92} +(2.37868 - 1.37333i) q^{94} +(3.67423 - 2.12132i) q^{95} +6.63103i q^{97} +(-1.88064 - 6.74264i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{4} - 8 q^{7}+O(q^{10}) \) 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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