LMFDB - Embedded newform 1386.2.k.a.991.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1386,2,Mod(793,1386)]

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([0, 2, 0]))

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

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

chi := DirichletCharacter("1386.793");

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.k (of order $3$, 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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 154)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			991.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	1386.991
Dual form			1386.2.k.a.793.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 - 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(-2.00000 - 3.46410i) q^{5} +(-2.50000 - 0.866025i) q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+(-0.500000 - 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(-2.00000 - 3.46410i) q^{5} +(-2.50000 - 0.866025i) q^{7} +1.00000 q^{8} +(-2.00000 + 3.46410i) q^{10} +(-0.500000 + 0.866025i) q^{11} -1.00000 q^{13} +(0.500000 + 2.59808i) q^{14} +(-0.500000 - 0.866025i) q^{16} +(1.00000 - 1.73205i) q^{17} +(-3.00000 - 5.19615i) q^{19} +4.00000 q^{20} +1.00000 q^{22} +(-1.00000 - 1.73205i) q^{23} +(-5.50000 + 9.52628i) q^{25} +(0.500000 + 0.866025i) q^{26} +(2.00000 - 1.73205i) q^{28} -1.00000 q^{29} +(-2.00000 + 3.46410i) q^{31} +(-0.500000 + 0.866025i) q^{32} -2.00000 q^{34} +(2.00000 + 10.3923i) q^{35} +(1.00000 + 1.73205i) q^{37} +(-3.00000 + 5.19615i) q^{38} +(-2.00000 - 3.46410i) q^{40} +2.00000 q^{41} +4.00000 q^{43} +(-0.500000 - 0.866025i) q^{44} +(-1.00000 + 1.73205i) q^{46} +(1.00000 + 1.73205i) q^{47} +(5.50000 + 4.33013i) q^{49} +11.0000 q^{50} +(0.500000 - 0.866025i) q^{52} +(-6.00000 + 10.3923i) q^{53} +4.00000 q^{55} +(-2.50000 - 0.866025i) q^{56} +(0.500000 + 0.866025i) q^{58} +(4.50000 - 7.79423i) q^{59} +(2.50000 + 4.33013i) q^{61} +4.00000 q^{62} +1.00000 q^{64} +(2.00000 + 3.46410i) q^{65} +(4.50000 - 7.79423i) q^{67} +(1.00000 + 1.73205i) q^{68} +(8.00000 - 6.92820i) q^{70} -4.00000 q^{71} +(1.00000 - 1.73205i) q^{73} +(1.00000 - 1.73205i) q^{74} +6.00000 q^{76} +(2.00000 - 1.73205i) q^{77} +(7.50000 + 12.9904i) q^{79} +(-2.00000 + 3.46410i) q^{80} +(-1.00000 - 1.73205i) q^{82} +6.00000 q^{83} -8.00000 q^{85} +(-2.00000 - 3.46410i) q^{86} +(-0.500000 + 0.866025i) q^{88} +(3.00000 + 5.19615i) q^{89} +(2.50000 + 0.866025i) q^{91} +2.00000 q^{92} +(1.00000 - 1.73205i) q^{94} +(-12.0000 + 20.7846i) q^{95} -5.00000 q^{97} +(1.00000 - 6.92820i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} - q^{4} - 4 q^{5} - 5 q^{7} + 2 q^{8}+O(q^{10}) $ 2 * q - q^2 - q^4 - 4 * q^5 - 5 * q^7 + 2 * q^8	$ 2 q - q^{2} - q^{4} - 4 q^{5} - 5 q^{7} + 2 q^{8} - 4 q^{10} - q^{11} - 2 q^{13} + q^{14} - q^{16} + 2 q^{17} - 6 q^{19} + 8 q^{20} + 2 q^{22} - 2 q^{23} - 11 q^{25} + q^{26} + 4 q^{28} - 2 q^{29} - 4 q^{31} - q^{32} - 4 q^{34} + 4 q^{35} + 2 q^{37} - 6 q^{38} - 4 q^{40} + 4 q^{41} + 8 q^{43} - q^{44} - 2 q^{46} + 2 q^{47} + 11 q^{49} + 22 q^{50} + q^{52} - 12 q^{53} + 8 q^{55} - 5 q^{56} + q^{58} + 9 q^{59} + 5 q^{61} + 8 q^{62} + 2 q^{64} + 4 q^{65} + 9 q^{67} + 2 q^{68} + 16 q^{70} - 8 q^{71} + 2 q^{73} + 2 q^{74} + 12 q^{76} + 4 q^{77} + 15 q^{79} - 4 q^{80} - 2 q^{82} + 12 q^{83} - 16 q^{85} - 4 q^{86} - q^{88} + 6 q^{89} + 5 q^{91} + 4 q^{92} + 2 q^{94} - 24 q^{95} - 10 q^{97} + 2 q^{98}+O(q^{100}) $ 2 * q - q^2 - q^4 - 4 * q^5 - 5 * q^7 + 2 * q^8 - 4 * q^10 - q^11 - 2 * q^13 + q^14 - q^16 + 2 * q^17 - 6 * q^19 + 8 * q^20 + 2 * q^22 - 2 * q^23 - 11 * q^25 + q^26 + 4 * q^28 - 2 * q^29 - 4 * q^31 - q^32 - 4 * q^34 + 4 * q^35 + 2 * q^37 - 6 * q^38 - 4 * q^40 + 4 * q^41 + 8 * q^43 - q^44 - 2 * q^46 + 2 * q^47 + 11 * q^49 + 22 * q^50 + q^52 - 12 * q^53 + 8 * q^55 - 5 * q^56 + q^58 + 9 * q^59 + 5 * q^61 + 8 * q^62 + 2 * q^64 + 4 * q^65 + 9 * q^67 + 2 * q^68 + 16 * q^70 - 8 * q^71 + 2 * q^73 + 2 * q^74 + 12 * q^76 + 4 * q^77 + 15 * q^79 - 4 * q^80 - 2 * q^82 + 12 * q^83 - 16 * q^85 - 4 * q^86 - q^88 + 6 * q^89 + 5 * q^91 + 4 * q^92 + 2 * q^94 - 24 * q^95 - 10 * q^97 + 2 * q^98

Display 100 coefficients 10 coefficients

Character values

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

$n$	$155$	$199$	$1135$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	−0.500000	−	0.866025i	−0.353553	−	0.612372i
$2$	−0.500000	−	0.866025i	−0.353553	−	0.612372i
$3$		0			0
$3$		0			0
$4$	−0.500000	+	0.866025i	−0.250000	+	0.433013i
$5$	−2.00000	−	3.46410i	−0.894427	−	1.54919i	−0.834512	−	0.550990i	$-0.814250\pi$
$5$	−2.00000	−	3.46410i	−0.894427	−	1.54919i	−0.0599153	−	0.998203i	$-0.519083\pi$
$6$		0			0
$7$	−2.50000	−	0.866025i	−0.944911	−	0.327327i
$7$	−2.50000	−	0.866025i	−0.944911	−	0.327327i
$8$	1.00000			0.353553
$9$		0			0
$10$	−2.00000	+	3.46410i	−0.632456	+	1.09545i
$11$	−0.500000	+	0.866025i	−0.150756	+	0.261116i
$11$	−0.500000	+	0.866025i	−0.150756	+	0.261116i
$12$		0			0
$13$	−1.00000			−0.277350			−0.138675	−	0.990338i	$-0.544284\pi$
$13$	−1.00000			−0.277350			−0.138675	+	0.990338i	$0.544284\pi$
$14$	0.500000	+	2.59808i	0.133631	+	0.694365i
$15$		0			0
$16$	−0.500000	−	0.866025i	−0.125000	−	0.216506i
$17$	1.00000	−	1.73205i	0.242536	−	0.420084i	−0.718900	−	0.695113i	$-0.755354\pi$
$17$	1.00000	−	1.73205i	0.242536	−	0.420084i	0.961436	+	0.275029i	$0.0886875\pi$
$18$		0			0
$19$	−3.00000	−	5.19615i	−0.688247	−	1.19208i	−0.972404	−	0.233301i	$-0.925047\pi$
$19$	−3.00000	−	5.19615i	−0.688247	−	1.19208i	0.284157	−	0.958778i	$-0.408286\pi$
$20$	4.00000			0.894427
$21$		0			0
$22$	1.00000			0.213201
$23$	−1.00000	−	1.73205i	−0.208514	−	0.361158i	0.742732	−	0.669588i	$-0.233529\pi$
$23$	−1.00000	−	1.73205i	−0.208514	−	0.361158i	−0.951247	+	0.308431i	$0.900196\pi$
$24$		0			0
$25$	−5.50000	+	9.52628i	−1.10000	+	1.90526i
$26$	0.500000	+	0.866025i	0.0980581	+	0.169842i
$27$		0			0
$28$	2.00000	−	1.73205i	0.377964	−	0.327327i
$29$	−1.00000			−0.185695			−0.0928477	−	0.995680i	$-0.529597\pi$
$29$	−1.00000			−0.185695			−0.0928477	+	0.995680i	$0.529597\pi$
$30$		0			0
$31$	−2.00000	+	3.46410i	−0.359211	+	0.622171i	−0.987829	−	0.155543i	$-0.950287\pi$
$31$	−2.00000	+	3.46410i	−0.359211	+	0.622171i	0.628619	+	0.777714i	$0.283621\pi$
$32$	−0.500000	+	0.866025i	−0.0883883	+	0.153093i
$33$		0			0
$34$	−2.00000			−0.342997
$35$	2.00000	+	10.3923i	0.338062	+	1.75662i
$36$		0			0
$37$	1.00000	+	1.73205i	0.164399	+	0.284747i	0.936442	−	0.350823i	$-0.114098\pi$
$37$	1.00000	+	1.73205i	0.164399	+	0.284747i	−0.772043	+	0.635571i	$0.780765\pi$
$38$	−3.00000	+	5.19615i	−0.486664	+	0.842927i
$39$		0			0
$40$	−2.00000	−	3.46410i	−0.316228	−	0.547723i
$41$	2.00000			0.312348			0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000			0.312348			0.156174	+	0.987730i	$0.450084\pi$
$42$		0			0
$43$	4.00000			0.609994			0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000			0.609994			0.304997	+	0.952353i	$0.401344\pi$
$44$	−0.500000	−	0.866025i	−0.0753778	−	0.130558i
$45$		0			0
$46$	−1.00000	+	1.73205i	−0.147442	+	0.255377i
$47$	1.00000	+	1.73205i	0.145865	+	0.252646i	0.929695	−	0.368329i	$-0.120070\pi$
$47$	1.00000	+	1.73205i	0.145865	+	0.252646i	−0.783830	+	0.620975i	$0.786737\pi$
$48$		0			0
$49$	5.50000	+	4.33013i	0.785714	+	0.618590i
$50$	11.0000			1.55563
$51$		0			0
$52$	0.500000	−	0.866025i	0.0693375	−	0.120096i
$53$	−6.00000	+	10.3923i	−0.824163	+	1.42749i	0.0783936	+	0.996922i	$0.475021\pi$
$53$	−6.00000	+	10.3923i	−0.824163	+	1.42749i	−0.902557	+	0.430570i	$0.858312\pi$
$54$		0			0
$55$	4.00000			0.539360
$56$	−2.50000	−	0.866025i	−0.334077	−	0.115728i
$57$		0			0
$58$	0.500000	+	0.866025i	0.0656532	+	0.113715i
$59$	4.50000	−	7.79423i	0.585850	−	1.01472i	−0.408919	−	0.912571i	$-0.634094\pi$
$59$	4.50000	−	7.79423i	0.585850	−	1.01472i	0.994769	−	0.102151i	$-0.0325726\pi$
$60$		0			0
$61$	2.50000	+	4.33013i	0.320092	+	0.554416i	0.980507	−	0.196485i	$-0.0629528\pi$
$61$	2.50000	+	4.33013i	0.320092	+	0.554416i	−0.660415	+	0.750901i	$0.729619\pi$
$62$	4.00000			0.508001
$63$		0			0
$64$	1.00000			0.125000
$65$	2.00000	+	3.46410i	0.248069	+	0.429669i
$66$		0			0
$67$	4.50000	−	7.79423i	0.549762	−	0.952217i	−0.448528	−	0.893769i	$-0.648052\pi$
$67$	4.50000	−	7.79423i	0.549762	−	0.952217i	0.998290	−	0.0584478i	$-0.0186151\pi$
$68$	1.00000	+	1.73205i	0.121268	+	0.210042i
$69$		0			0
$70$	8.00000	−	6.92820i	0.956183	−	0.828079i
$71$	−4.00000			−0.474713			−0.237356	−	0.971423i	$-0.576281\pi$
$71$	−4.00000			−0.474713			−0.237356	+	0.971423i	$0.576281\pi$
$72$		0			0
$73$	1.00000	−	1.73205i	0.117041	−	0.202721i	−0.801553	−	0.597924i	$-0.795992\pi$
$73$	1.00000	−	1.73205i	0.117041	−	0.202721i	0.918594	+	0.395203i	$0.129326\pi$
$74$	1.00000	−	1.73205i	0.116248	−	0.201347i
$75$		0			0
$76$	6.00000			0.688247
$77$	2.00000	−	1.73205i	0.227921	−	0.197386i
$78$		0			0
$79$	7.50000	+	12.9904i	0.843816	+	1.46153i	0.886646	+	0.462450i	$0.153029\pi$
$79$	7.50000	+	12.9904i	0.843816	+	1.46153i	−0.0428296	+	0.999082i	$0.513637\pi$
$80$	−2.00000	+	3.46410i	−0.223607	+	0.387298i
$81$		0			0
$82$	−1.00000	−	1.73205i	−0.110432	−	0.191273i
$83$	6.00000			0.658586			0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000			0.658586			0.329293	+	0.944228i	$0.393190\pi$
$84$		0			0
$85$	−8.00000			−0.867722
$86$	−2.00000	−	3.46410i	−0.215666	−	0.373544i
$87$		0			0
$88$	−0.500000	+	0.866025i	−0.0533002	+	0.0923186i
$89$	3.00000	+	5.19615i	0.317999	+	0.550791i	0.980071	−	0.198650i	$-0.0636557\pi$
$89$	3.00000	+	5.19615i	0.317999	+	0.550791i	−0.662071	+	0.749441i	$0.730322\pi$
$90$		0			0
$91$	2.50000	+	0.866025i	0.262071	+	0.0907841i
$92$	2.00000			0.208514
$93$		0			0
$94$	1.00000	−	1.73205i	0.103142	−	0.178647i
$95$	−12.0000	+	20.7846i	−1.23117	+	2.13246i
$96$		0			0
$97$	−5.00000			−0.507673			−0.253837	−	0.967247i	$-0.581693\pi$
$97$	−5.00000			−0.507673			−0.253837	+	0.967247i	$0.581693\pi$
$98$	1.00000	−	6.92820i	0.101015	−	0.699854i
$99$		0			0
$100$	−5.50000	−	9.52628i	−0.550000	−	0.952628i
$101$	−7.50000	+	12.9904i	−0.746278	+	1.29259i	0.203317	+	0.979113i	$0.434828\pi$
$101$	−7.50000	+	12.9904i	−0.746278	+	1.29259i	−0.949595	+	0.313478i	$0.898506\pi$
$102$		0			0
$103$	−6.00000	−	10.3923i	−0.591198	−	1.02398i	−0.994071	−	0.108729i	$-0.965322\pi$
$103$	−6.00000	−	10.3923i	−0.591198	−	1.02398i	0.402874	−	0.915255i	$-0.368011\pi$
$104$	−1.00000			−0.0980581
$105$		0			0
$106$	12.0000			1.16554
$107$	4.00000	+	6.92820i	0.386695	+	0.669775i	0.992003	−	0.126217i	$-0.0402834\pi$
$107$	4.00000	+	6.92820i	0.386695	+	0.669775i	−0.605308	+	0.795991i	$0.706950\pi$
$108$		0			0
$109$	−5.00000	+	8.66025i	−0.478913	+	0.829502i	−0.999708	−	0.0241802i	$-0.992302\pi$
$109$	−5.00000	+	8.66025i	−0.478913	+	0.829502i	0.520794	+	0.853682i	$0.325636\pi$
$110$	−2.00000	−	3.46410i	−0.190693	−	0.330289i
$111$		0			0
$112$	0.500000	+	2.59808i	0.0472456	+	0.245495i
$113$	−17.0000			−1.59923			−0.799613	−	0.600516i	$-0.794962\pi$
$113$	−17.0000			−1.59923			−0.799613	+	0.600516i	$0.794962\pi$
$114$		0			0
$115$	−4.00000	+	6.92820i	−0.373002	+	0.646058i
$116$	0.500000	−	0.866025i	0.0464238	−	0.0804084i
$117$		0			0
$118$	−9.00000			−0.828517
$119$	−4.00000	+	3.46410i	−0.366679	+	0.317554i
$120$		0			0
$121$	−0.500000	−	0.866025i	−0.0454545	−	0.0787296i
$122$	2.50000	−	4.33013i	0.226339	−	0.392031i
$123$		0			0
$124$	−2.00000	−	3.46410i	−0.179605	−	0.311086i
$125$	24.0000			2.14663
$126$		0			0
$127$	5.00000			0.443678			0.221839	−	0.975083i	$-0.428794\pi$
$127$	5.00000			0.443678			0.221839	+	0.975083i	$0.428794\pi$
$128$	−0.500000	−	0.866025i	−0.0441942	−	0.0765466i
$129$		0			0
$130$	2.00000	−	3.46410i	0.175412	−	0.303822i
$131$	−9.00000	−	15.5885i	−0.786334	−	1.36197i	−0.928199	−	0.372084i	$-0.878643\pi$
$131$	−9.00000	−	15.5885i	−0.786334	−	1.36197i	0.141865	−	0.989886i	$-0.454690\pi$
$132$		0			0
$133$	3.00000	+	15.5885i	0.260133	+	1.35169i
$134$	−9.00000			−0.777482
$135$		0			0
$136$	1.00000	−	1.73205i	0.0857493	−	0.148522i
$137$	−4.50000	+	7.79423i	−0.384461	+	0.665906i	−0.991694	−	0.128618i	$-0.958946\pi$
$137$	−4.50000	+	7.79423i	−0.384461	+	0.665906i	0.607233	+	0.794524i	$0.292279\pi$
$138$		0			0
$139$	8.00000			0.678551			0.339276	−	0.940687i	$-0.389818\pi$
$139$	8.00000			0.678551			0.339276	+	0.940687i	$0.389818\pi$
$140$	−10.0000	−	3.46410i	−0.845154	−	0.292770i
$141$		0			0
$142$	2.00000	+	3.46410i	0.167836	+	0.290701i
$143$	0.500000	−	0.866025i	0.0418121	−	0.0724207i
$144$		0			0
$145$	2.00000	+	3.46410i	0.166091	+	0.287678i
$146$	−2.00000			−0.165521
$147$		0			0
$148$	−2.00000			−0.164399
$149$	−5.00000	−	8.66025i	−0.409616	−	0.709476i	0.585231	−	0.810867i	$-0.301004\pi$
$149$	−5.00000	−	8.66025i	−0.409616	−	0.709476i	−0.994847	+	0.101391i	$0.967671\pi$
$150$		0			0
$151$	−4.50000	+	7.79423i	−0.366205	+	0.634285i	−0.988969	−	0.148124i	$-0.952676\pi$
$151$	−4.50000	+	7.79423i	−0.366205	+	0.634285i	0.622764	+	0.782410i	$0.286010\pi$
$152$	−3.00000	−	5.19615i	−0.243332	−	0.421464i
$153$		0			0
$154$	−2.50000	−	0.866025i	−0.201456	−	0.0697863i
$155$	16.0000			1.28515
$156$		0			0
$157$	−2.00000	+	3.46410i	−0.159617	+	0.276465i	−0.934731	−	0.355357i	$-0.884359\pi$
$157$	−2.00000	+	3.46410i	−0.159617	+	0.276465i	0.775113	+	0.631822i	$0.217693\pi$
$158$	7.50000	−	12.9904i	0.596668	−	1.03346i
$159$		0			0
$160$	4.00000			0.316228
$161$	1.00000	+	5.19615i	0.0788110	+	0.409514i
$162$		0			0
$163$	−6.50000	−	11.2583i	−0.509119	−	0.881820i	−0.999944	−	0.0105623i	$-0.996638\pi$
$163$	−6.50000	−	11.2583i	−0.509119	−	0.881820i	0.490825	−	0.871258i	$-0.336695\pi$
$164$	−1.00000	+	1.73205i	−0.0780869	+	0.135250i
$165$		0			0
$166$	−3.00000	−	5.19615i	−0.232845	−	0.403300i
$167$	17.0000			1.31550			0.657750	−	0.753237i	$-0.271508\pi$
$167$	17.0000			1.31550			0.657750	+	0.753237i	$0.271508\pi$
$168$		0			0
$169$	−12.0000			−0.923077
$170$	4.00000	+	6.92820i	0.306786	+	0.531369i
$171$		0			0
$172$	−2.00000	+	3.46410i	−0.152499	+	0.264135i
$173$	−2.50000	−	4.33013i	−0.190071	−	0.329213i	0.755202	−	0.655492i	$-0.227539\pi$
$173$	−2.50000	−	4.33013i	−0.190071	−	0.329213i	−0.945274	+	0.326278i	$0.894205\pi$
$174$		0			0
$175$	22.0000	−	19.0526i	1.66304	−	1.44024i
$176$	1.00000			0.0753778
$177$		0			0
$178$	3.00000	−	5.19615i	0.224860	−	0.389468i
$179$	6.50000	−	11.2583i	0.485833	−	0.841487i	−0.514035	−	0.857769i	$-0.671850\pi$
$179$	6.50000	−	11.2583i	0.485833	−	0.841487i	0.999867	+	0.0162823i	$0.00518305\pi$
$180$		0			0
$181$	−22.0000			−1.63525			−0.817624	−	0.575753i	$-0.804709\pi$
$181$	−22.0000			−1.63525			−0.817624	+	0.575753i	$0.804709\pi$
$182$	−0.500000	−	2.59808i	−0.0370625	−	0.192582i
$183$		0			0
$184$	−1.00000	−	1.73205i	−0.0737210	−	0.127688i
$185$	4.00000	−	6.92820i	0.294086	−	0.509372i
$186$		0			0
$187$	1.00000	+	1.73205i	0.0731272	+	0.126660i
$188$	−2.00000			−0.145865
$189$		0			0
$190$	24.0000			1.74114
$191$	7.00000	+	12.1244i	0.506502	+	0.877288i	0.999972	+	0.00752447i	$0.00239513\pi$
$191$	7.00000	+	12.1244i	0.506502	+	0.877288i	−0.493469	+	0.869763i	$0.664272\pi$
$192$		0			0
$193$	11.0000	−	19.0526i	0.791797	−	1.37143i	−0.133056	−	0.991109i	$-0.542479\pi$
$193$	11.0000	−	19.0526i	0.791797	−	1.37143i	0.924853	−	0.380325i	$-0.124188\pi$
$194$	2.50000	+	4.33013i	0.179490	+	0.310885i
$195$		0			0
$196$	−6.50000	+	2.59808i	−0.464286	+	0.185577i
$197$	3.00000			0.213741			0.106871	−	0.994273i	$-0.465917\pi$
$197$	3.00000			0.213741			0.106871	+	0.994273i	$0.465917\pi$
$198$		0			0
$199$	−5.00000	+	8.66025i	−0.354441	+	0.613909i	−0.987022	−	0.160585i	$-0.948662\pi$
$199$	−5.00000	+	8.66025i	−0.354441	+	0.613909i	0.632581	+	0.774494i	$0.281995\pi$
$200$	−5.50000	+	9.52628i	−0.388909	+	0.673610i
$201$		0			0
$202$	15.0000			1.05540
$203$	2.50000	+	0.866025i	0.175466	+	0.0607831i
$204$		0			0
$205$	−4.00000	−	6.92820i	−0.279372	−	0.483887i
$206$	−6.00000	+	10.3923i	−0.418040	+	0.724066i
$207$		0			0
$208$	0.500000	+	0.866025i	0.0346688	+	0.0600481i
$209$	6.00000			0.415029
$210$		0			0
$211$	−14.0000			−0.963800			−0.481900	−	0.876226i	$-0.660053\pi$
$211$	−14.0000			−0.963800			−0.481900	+	0.876226i	$0.660053\pi$
$212$	−6.00000	−	10.3923i	−0.412082	−	0.713746i
$213$		0			0
$214$	4.00000	−	6.92820i	0.273434	−	0.473602i
$215$	−8.00000	−	13.8564i	−0.545595	−	0.944999i
$216$		0			0
$217$	8.00000	−	6.92820i	0.543075	−	0.470317i
$218$	10.0000			0.677285
$219$		0			0
$220$	−2.00000	+	3.46410i	−0.134840	+	0.233550i
$221$	−1.00000	+	1.73205i	−0.0672673	+	0.116510i
$222$		0			0
$223$	−26.0000			−1.74109			−0.870544	−	0.492090i	$-0.836233\pi$
$223$	−26.0000			−1.74109			−0.870544	+	0.492090i	$0.836233\pi$
$224$	2.00000	−	1.73205i	0.133631	−	0.115728i
$225$		0			0
$226$	8.50000	+	14.7224i	0.565412	+	0.979322i
$227$	5.00000	−	8.66025i	0.331862	−	0.574801i	−0.651015	−	0.759065i	$-0.725657\pi$
$227$	5.00000	−	8.66025i	0.331862	−	0.574801i	0.982877	+	0.184263i	$0.0589899\pi$
$228$		0			0
$229$	8.00000	+	13.8564i	0.528655	+	0.915657i	0.999442	+	0.0334101i	$0.0106368\pi$
$229$	8.00000	+	13.8564i	0.528655	+	0.915657i	−0.470787	+	0.882247i	$0.656030\pi$
$230$	8.00000			0.527504
$231$		0			0
$232$	−1.00000			−0.0656532
$233$	3.00000	+	5.19615i	0.196537	+	0.340411i	0.947403	−	0.320043i	$-0.103697\pi$
$233$	3.00000	+	5.19615i	0.196537	+	0.340411i	−0.750867	+	0.660454i	$0.770364\pi$
$234$		0			0
$235$	4.00000	−	6.92820i	0.260931	−	0.451946i
$236$	4.50000	+	7.79423i	0.292925	+	0.507361i
$237$		0			0
$238$	5.00000	+	1.73205i	0.324102	+	0.112272i
$239$	−19.0000			−1.22901			−0.614504	−	0.788914i	$-0.710644\pi$
$239$	−19.0000			−1.22901			−0.614504	+	0.788914i	$0.710644\pi$
$240$		0			0
$241$	−15.0000	+	25.9808i	−0.966235	+	1.67357i	−0.259975	+	0.965615i	$0.583714\pi$
$241$	−15.0000	+	25.9808i	−0.966235	+	1.67357i	−0.706260	+	0.707953i	$0.749619\pi$
$242$	−0.500000	+	0.866025i	−0.0321412	+	0.0556702i
$243$		0			0
$244$	−5.00000			−0.320092
$245$	4.00000	−	27.7128i	0.255551	−	1.77051i
$246$		0			0
$247$	3.00000	+	5.19615i	0.190885	+	0.330623i
$248$	−2.00000	+	3.46410i	−0.127000	+	0.219971i
$249$		0			0
$250$	−12.0000	−	20.7846i	−0.758947	−	1.31453i
$251$	−12.0000			−0.757433			−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000			−0.757433			−0.378717	+	0.925513i	$0.623635\pi$
$252$		0			0
$253$	2.00000			0.125739
$254$	−2.50000	−	4.33013i	−0.156864	−	0.271696i
$255$		0			0
$256$	−0.500000	+	0.866025i	−0.0312500	+	0.0541266i
$257$	−10.5000	−	18.1865i	−0.654972	−	1.13444i	−0.981901	−	0.189396i	$-0.939347\pi$
$257$	−10.5000	−	18.1865i	−0.654972	−	1.13444i	0.326929	−	0.945049i	$-0.393986\pi$
$258$		0			0
$259$	−1.00000	−	5.19615i	−0.0621370	−	0.322873i
$260$	−4.00000			−0.248069
$261$		0			0
$262$	−9.00000	+	15.5885i	−0.556022	+	0.963058i
$263$	1.50000	−	2.59808i	0.0924940	−	0.160204i	−0.816066	−	0.577959i	$-0.803849\pi$
$263$	1.50000	−	2.59808i	0.0924940	−	0.160204i	0.908560	+	0.417755i	$0.137183\pi$
$264$		0			0
$265$	48.0000			2.94862
$266$	12.0000	−	10.3923i	0.735767	−	0.637193i
$267$		0			0
$268$	4.50000	+	7.79423i	0.274881	+	0.476108i
$269$	6.00000	−	10.3923i	0.365826	−	0.633630i	−0.623082	−	0.782157i	$-0.714120\pi$
$269$	6.00000	−	10.3923i	0.365826	−	0.633630i	0.988908	+	0.148527i	$0.0474530\pi$
$270$		0			0
$271$	−12.5000	−	21.6506i	−0.759321	−	1.31518i	−0.943197	−	0.332233i	$-0.892198\pi$
$271$	−12.5000	−	21.6506i	−0.759321	−	1.31518i	0.183876	−	0.982949i	$-0.441135\pi$
$272$	−2.00000			−0.121268
$273$		0			0
$274$	9.00000			0.543710
$275$	−5.50000	−	9.52628i	−0.331662	−	0.574456i
$276$		0			0
$277$	1.50000	−	2.59808i	0.0901263	−	0.156103i	−0.817438	−	0.576017i	$-0.804606\pi$
$277$	1.50000	−	2.59808i	0.0901263	−	0.156103i	0.907564	+	0.419914i	$0.137940\pi$
$278$	−4.00000	−	6.92820i	−0.239904	−	0.415526i
$279$		0			0
$280$	2.00000	+	10.3923i	0.119523	+	0.621059i
$281$	10.0000			0.596550			0.298275	−	0.954480i	$-0.403589\pi$
$281$	10.0000			0.596550			0.298275	+	0.954480i	$0.403589\pi$
$282$		0			0
$283$	3.00000	−	5.19615i	0.178331	−	0.308879i	−0.762978	−	0.646425i	$-0.776263\pi$
$283$	3.00000	−	5.19615i	0.178331	−	0.308879i	0.941309	+	0.337546i	$0.109597\pi$
$284$	2.00000	−	3.46410i	0.118678	−	0.205557i
$285$		0			0
$286$	−1.00000			−0.0591312
$287$	−5.00000	−	1.73205i	−0.295141	−	0.102240i
$288$		0			0
$289$	6.50000	+	11.2583i	0.382353	+	0.662255i
$290$	2.00000	−	3.46410i	0.117444	−	0.203419i
$291$		0			0
$292$	1.00000	+	1.73205i	0.0585206	+	0.101361i
$293$	6.00000			0.350524			0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000			0.350524			0.175262	+	0.984522i	$0.443923\pi$
$294$		0			0
$295$	−36.0000			−2.09600
$296$	1.00000	+	1.73205i	0.0581238	+	0.100673i
$297$		0			0
$298$	−5.00000	+	8.66025i	−0.289642	+	0.501675i
$299$	1.00000	+	1.73205i	0.0578315	+	0.100167i
$300$		0			0
$301$	−10.0000	−	3.46410i	−0.576390	−	0.199667i
$302$	9.00000			0.517892
$303$		0			0
$304$	−3.00000	+	5.19615i	−0.172062	+	0.298020i
$305$	10.0000	−	17.3205i	0.572598	−	0.991769i
$306$		0			0
$307$	32.0000			1.82634			0.913168	−	0.407583i	$-0.133628\pi$
$307$	32.0000			1.82634			0.913168	+	0.407583i	$0.133628\pi$
$308$	0.500000	+	2.59808i	0.0284901	+	0.148039i
$309$		0			0
$310$	−8.00000	−	13.8564i	−0.454369	−	0.786991i
$311$	−14.0000	+	24.2487i	−0.793867	+	1.37502i	0.129689	+	0.991555i	$0.458602\pi$
$311$	−14.0000	+	24.2487i	−0.793867	+	1.37502i	−0.923556	+	0.383464i	$0.874731\pi$
$312$		0			0
$313$	−0.500000	−	0.866025i	−0.0282617	−	0.0489506i	0.851549	−	0.524276i	$-0.175664\pi$
$313$	−0.500000	−	0.866025i	−0.0282617	−	0.0489506i	−0.879810	+	0.475325i	$0.842331\pi$
$314$	4.00000			0.225733
$315$		0			0
$316$	−15.0000			−0.843816
$317$	−6.00000	−	10.3923i	−0.336994	−	0.583690i	0.646872	−	0.762598i	$-0.276077\pi$
$317$	−6.00000	−	10.3923i	−0.336994	−	0.583690i	−0.983866	+	0.178908i	$0.942743\pi$
$318$		0			0
$319$	0.500000	−	0.866025i	0.0279946	−	0.0484881i
$320$	−2.00000	−	3.46410i	−0.111803	−	0.193649i
$321$		0			0
$322$	4.00000	−	3.46410i	0.222911	−	0.193047i
$323$	−12.0000			−0.667698
$324$		0			0
$325$	5.50000	−	9.52628i	0.305085	−	0.528423i
$326$	−6.50000	+	11.2583i	−0.360002	+	0.623541i
$327$		0			0
$328$	2.00000			0.110432
$329$	−1.00000	−	5.19615i	−0.0551318	−	0.286473i
$330$		0			0
$331$	−3.50000	−	6.06218i	−0.192377	−	0.333207i	0.753660	−	0.657264i	$-0.228286\pi$
$331$	−3.50000	−	6.06218i	−0.192377	−	0.333207i	−0.946038	+	0.324057i	$0.894953\pi$
$332$	−3.00000	+	5.19615i	−0.164646	+	0.285176i
$333$		0			0
$334$	−8.50000	−	14.7224i	−0.465099	−	0.805576i
$335$	−36.0000			−1.96689
$336$		0			0
$337$	−30.0000			−1.63420			−0.817102	−	0.576493i	$-0.804421\pi$
$337$	−30.0000			−1.63420			−0.817102	+	0.576493i	$0.804421\pi$
$338$	6.00000	+	10.3923i	0.326357	+	0.565267i
$339$		0			0
$340$	4.00000	−	6.92820i	0.216930	−	0.375735i
$341$	−2.00000	−	3.46410i	−0.108306	−	0.187592i
$342$		0			0
$343$	−10.0000	−	15.5885i	−0.539949	−	0.841698i
$344$	4.00000			0.215666
$345$		0			0
$346$	−2.50000	+	4.33013i	−0.134401	+	0.232789i
$347$	−14.0000	+	24.2487i	−0.751559	+	1.30174i	0.195507	+	0.980702i	$0.437365\pi$
$347$	−14.0000	+	24.2487i	−0.751559	+	1.30174i	−0.947067	+	0.321037i	$0.895969\pi$
$348$		0			0
$349$	2.00000			0.107058			0.0535288	−	0.998566i	$-0.482953\pi$
$349$	2.00000			0.107058			0.0535288	+	0.998566i	$0.482953\pi$
$350$	−27.5000	−	9.52628i	−1.46994	−	0.509201i
$351$		0			0
$352$	−0.500000	−	0.866025i	−0.0266501	−	0.0461593i
$353$	−9.00000	+	15.5885i	−0.479022	+	0.829690i	−0.999711	−	0.0240566i	$-0.992342\pi$
$353$	−9.00000	+	15.5885i	−0.479022	+	0.829690i	0.520689	+	0.853746i	$0.325675\pi$
$354$		0			0
$355$	8.00000	+	13.8564i	0.424596	+	0.735422i
$356$	−6.00000			−0.317999
$357$		0			0
$358$	−13.0000			−0.687071
$359$	−15.5000	−	26.8468i	−0.818059	−	1.41692i	−0.907111	−	0.420892i	$-0.861717\pi$
$359$	−15.5000	−	26.8468i	−0.818059	−	1.41692i	0.0890519	−	0.996027i	$-0.471616\pi$
$360$		0			0
$361$	−8.50000	+	14.7224i	−0.447368	+	0.774865i
$362$	11.0000	+	19.0526i	0.578147	+	1.00138i
$363$		0			0
$364$	−2.00000	+	1.73205i	−0.104828	+	0.0907841i
$365$	−8.00000			−0.418739
$366$		0			0
$367$	7.00000	−	12.1244i	0.365397	−	0.632886i	−0.623443	−	0.781869i	$-0.714267\pi$
$367$	7.00000	−	12.1244i	0.365397	−	0.632886i	0.988840	+	0.148983i	$0.0475999\pi$
$368$	−1.00000	+	1.73205i	−0.0521286	+	0.0902894i
$369$		0			0
$370$	−8.00000			−0.415900
$371$	24.0000	−	20.7846i	1.24602	−	1.07908i
$372$		0			0
$373$	3.50000	+	6.06218i	0.181223	+	0.313888i	0.942297	−	0.334777i	$-0.108661\pi$
$373$	3.50000	+	6.06218i	0.181223	+	0.313888i	−0.761074	+	0.648665i	$0.775328\pi$
$374$	1.00000	−	1.73205i	0.0517088	−	0.0895622i
$375$		0			0
$376$	1.00000	+	1.73205i	0.0515711	+	0.0893237i
$377$	1.00000			0.0515026
$378$		0			0
$379$	−29.0000			−1.48963			−0.744815	−	0.667271i	$-0.767462\pi$
$379$	−29.0000			−1.48963			−0.744815	+	0.667271i	$0.767462\pi$
$380$	−12.0000	−	20.7846i	−0.615587	−	1.06623i
$381$		0			0
$382$	7.00000	−	12.1244i	0.358151	−	0.620336i
$383$	4.00000	+	6.92820i	0.204390	+	0.354015i	0.949938	−	0.312437i	$-0.101145\pi$
$383$	4.00000	+	6.92820i	0.204390	+	0.354015i	−0.745548	+	0.666452i	$0.767812\pi$
$384$		0			0
$385$	−10.0000	−	3.46410i	−0.509647	−	0.176547i
$386$	−22.0000			−1.11977
$387$		0			0
$388$	2.50000	−	4.33013i	0.126918	−	0.219829i
$389$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$389$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$390$		0			0
$391$	−4.00000			−0.202289
$392$	5.50000	+	4.33013i	0.277792	+	0.218704i
$393$		0			0
$394$	−1.50000	−	2.59808i	−0.0755689	−	0.130889i
$395$	30.0000	−	51.9615i	1.50946	−	2.61447i
$396$		0			0
$397$	15.0000	+	25.9808i	0.752828	+	1.30394i	0.946447	+	0.322860i	$0.104644\pi$
$397$	15.0000	+	25.9808i	0.752828	+	1.30394i	−0.193618	+	0.981077i	$0.562022\pi$
$398$	10.0000			0.501255
$399$		0			0
$400$	11.0000			0.550000
$401$	7.50000	+	12.9904i	0.374532	+	0.648709i	0.990257	−	0.139253i	$-0.0444700\pi$
$401$	7.50000	+	12.9904i	0.374532	+	0.648709i	−0.615725	+	0.787961i	$0.711137\pi$
$402$		0			0
$403$	2.00000	−	3.46410i	0.0996271	−	0.172559i
$404$	−7.50000	−	12.9904i	−0.373139	−	0.646296i
$405$		0			0
$406$	−0.500000	−	2.59808i	−0.0248146	−	0.128940i
$407$	−2.00000			−0.0991363
$408$		0			0
$409$	16.0000	−	27.7128i	0.791149	−	1.37031i	−0.134107	−	0.990967i	$-0.542817\pi$
$409$	16.0000	−	27.7128i	0.791149	−	1.37031i	0.925256	−	0.379344i	$-0.123850\pi$
$410$	−4.00000	+	6.92820i	−0.197546	+	0.342160i
$411$		0			0
$412$	12.0000			0.591198
$413$	−18.0000	+	15.5885i	−0.885722	+	0.767058i
$414$		0			0
$415$	−12.0000	−	20.7846i	−0.589057	−	1.02028i
$416$	0.500000	−	0.866025i	0.0245145	−	0.0424604i
$417$		0			0
$418$	−3.00000	−	5.19615i	−0.146735	−	0.254152i
$419$	20.0000			0.977064			0.488532	−	0.872546i	$-0.337533\pi$
$419$	20.0000			0.977064			0.488532	+	0.872546i	$0.337533\pi$
$420$		0			0
$421$	−20.0000			−0.974740			−0.487370	−	0.873195i	$-0.662044\pi$
$421$	−20.0000			−0.974740			−0.487370	+	0.873195i	$0.662044\pi$
$422$	7.00000	+	12.1244i	0.340755	+	0.590204i
$423$		0			0
$424$	−6.00000	+	10.3923i	−0.291386	+	0.504695i
$425$	11.0000	+	19.0526i	0.533578	+	0.924185i
$426$		0			0
$427$	−2.50000	−	12.9904i	−0.120983	−	0.628649i
$428$	−8.00000			−0.386695
$429$		0			0
$430$	−8.00000	+	13.8564i	−0.385794	+	0.668215i
$431$	0.500000	−	0.866025i	0.0240842	−	0.0417150i	−0.853732	−	0.520712i	$-0.825666\pi$
$431$	0.500000	−	0.866025i	0.0240842	−	0.0417150i	0.877816	+	0.478997i	$0.159000\pi$
$432$		0			0
$433$	−10.0000			−0.480569			−0.240285	−	0.970702i	$-0.577241\pi$
$433$	−10.0000			−0.480569			−0.240285	+	0.970702i	$0.577241\pi$
$434$	−10.0000	−	3.46410i	−0.480015	−	0.166282i
$435$		0			0
$436$	−5.00000	−	8.66025i	−0.239457	−	0.414751i
$437$	−6.00000	+	10.3923i	−0.287019	+	0.497131i
$438$		0			0
$439$	2.50000	+	4.33013i	0.119318	+	0.206666i	0.919498	−	0.393095i	$-0.128596\pi$
$439$	2.50000	+	4.33013i	0.119318	+	0.206666i	−0.800179	+	0.599761i	$0.795262\pi$
$440$	4.00000			0.190693
$441$		0			0
$442$	2.00000			0.0951303
$443$	−6.00000	−	10.3923i	−0.285069	−	0.493753i	0.687557	−	0.726130i	$-0.258683\pi$
$443$	−6.00000	−	10.3923i	−0.285069	−	0.493753i	−0.972626	+	0.232377i	$0.925350\pi$
$444$		0			0
$445$	12.0000	−	20.7846i	0.568855	−	0.985285i
$446$	13.0000	+	22.5167i	0.615568	+	1.06619i
$447$		0			0
$448$	−2.50000	−	0.866025i	−0.118114	−	0.0409159i
$449$	−30.0000			−1.41579			−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000			−1.41579			−0.707894	+	0.706319i	$0.750354\pi$
$450$		0			0
$451$	−1.00000	+	1.73205i	−0.0470882	+	0.0815591i
$452$	8.50000	−	14.7224i	0.399806	−	0.692485i
$453$		0			0
$454$	−10.0000			−0.469323
$455$	−2.00000	−	10.3923i	−0.0937614	−	0.487199i
$456$		0			0
$457$	−1.00000	−	1.73205i	−0.0467780	−	0.0810219i	0.841688	−	0.539964i	$-0.181562\pi$
$457$	−1.00000	−	1.73205i	−0.0467780	−	0.0810219i	−0.888466	+	0.458942i	$0.848229\pi$
$458$	8.00000	−	13.8564i	0.373815	−	0.647467i
$459$		0			0
$460$	−4.00000	−	6.92820i	−0.186501	−	0.323029i
$461$	3.00000			0.139724			0.0698620	−	0.997557i	$-0.477744\pi$
$461$	3.00000			0.139724			0.0698620	+	0.997557i	$0.477744\pi$
$462$		0			0
$463$	−14.0000			−0.650635			−0.325318	−	0.945605i	$-0.605471\pi$
$463$	−14.0000			−0.650635			−0.325318	+	0.945605i	$0.605471\pi$
$464$	0.500000	+	0.866025i	0.0232119	+	0.0402042i
$465$		0			0
$466$	3.00000	−	5.19615i	0.138972	−	0.240707i
$467$	−6.00000	−	10.3923i	−0.277647	−	0.480899i	0.693153	−	0.720791i	$-0.256221\pi$
$467$	−6.00000	−	10.3923i	−0.277647	−	0.480899i	−0.970799	+	0.239892i	$0.922888\pi$
$468$		0			0
$469$	−18.0000	+	15.5885i	−0.831163	+	0.719808i
$470$	−8.00000			−0.369012
$471$		0			0
$472$	4.50000	−	7.79423i	0.207129	−	0.358758i
$473$	−2.00000	+	3.46410i	−0.0919601	+	0.159280i
$474$		0			0
$475$	66.0000			3.02829
$476$	−1.00000	−	5.19615i	−0.0458349	−	0.238165i
$477$		0			0
$478$	9.50000	+	16.4545i	0.434520	+	0.752611i
$479$	−18.5000	+	32.0429i	−0.845287	+	1.46408i	0.0400855	+	0.999196i	$0.487237\pi$
$479$	−18.5000	+	32.0429i	−0.845287	+	1.46408i	−0.885372	+	0.464883i	$0.846096\pi$
$480$		0			0
$481$	−1.00000	−	1.73205i	−0.0455961	−	0.0789747i
$482$	30.0000			1.36646
$483$		0			0
$484$	1.00000			0.0454545
$485$	10.0000	+	17.3205i	0.454077	+	0.786484i
$486$		0			0
$487$	2.00000	−	3.46410i	0.0906287	−	0.156973i	−0.817147	−	0.576429i	$-0.804446\pi$
$487$	2.00000	−	3.46410i	0.0906287	−	0.156973i	0.907776	+	0.419456i	$0.137779\pi$
$488$	2.50000	+	4.33013i	0.113170	+	0.196016i
$489$		0			0
$490$	−26.0000	+	10.3923i	−1.17456	+	0.469476i
$491$	−18.0000			−0.812329			−0.406164	−	0.913800i	$-0.633134\pi$
$491$	−18.0000			−0.812329			−0.406164	+	0.913800i	$0.633134\pi$
$492$		0			0
$493$	−1.00000	+	1.73205i	−0.0450377	+	0.0780076i
$494$	3.00000	−	5.19615i	0.134976	−	0.233786i
$495$		0			0
$496$	4.00000			0.179605
$497$	10.0000	+	3.46410i	0.448561	+	0.155386i
$498$		0			0
$499$	8.00000	+	13.8564i	0.358129	+	0.620298i	0.987648	−	0.156687i	$-0.0500814\pi$
$499$	8.00000	+	13.8564i	0.358129	+	0.620298i	−0.629519	+	0.776985i	$0.716748\pi$
$500$	−12.0000	+	20.7846i	−0.536656	+	0.929516i
$501$		0			0
$502$	6.00000	+	10.3923i	0.267793	+	0.463831i
$503$	21.0000			0.936344			0.468172	−	0.883637i	$-0.344913\pi$
$503$	21.0000			0.936344			0.468172	+	0.883637i	$0.344913\pi$
$504$		0			0
$505$	60.0000			2.66996
$506$	−1.00000	−	1.73205i	−0.0444554	−	0.0769991i
$507$		0			0
$508$	−2.50000	+	4.33013i	−0.110920	+	0.192118i
$509$	1.00000	+	1.73205i	0.0443242	+	0.0767718i	0.887336	−	0.461123i	$-0.152553\pi$
$509$	1.00000	+	1.73205i	0.0443242	+	0.0767718i	−0.843012	+	0.537895i	$0.819220\pi$
$510$		0			0
$511$	−4.00000	+	3.46410i	−0.176950	+	0.153243i
$512$	1.00000			0.0441942
$513$		0			0
$514$	−10.5000	+	18.1865i	−0.463135	+	0.802174i
$515$	−24.0000	+	41.5692i	−1.05757	+	1.83176i
$516$		0			0
$517$	−2.00000			−0.0879599
$518$	−4.00000	+	3.46410i	−0.175750	+	0.152204i
$519$		0			0
$520$	2.00000	+	3.46410i	0.0877058	+	0.151911i
$521$	−13.0000	+	22.5167i	−0.569540	+	0.986473i	0.427071	+	0.904218i	$0.359545\pi$
$521$	−13.0000	+	22.5167i	−0.569540	+	0.986473i	−0.996611	+	0.0822547i	$0.973788\pi$
$522$		0			0
$523$	−10.0000	−	17.3205i	−0.437269	−	0.757373i	0.560208	−	0.828352i	$-0.310721\pi$
$523$	−10.0000	−	17.3205i	−0.437269	−	0.757373i	−0.997478	+	0.0709788i	$0.977388\pi$
$524$	18.0000			0.786334
$525$		0			0
$526$	−3.00000			−0.130806
$527$	4.00000	+	6.92820i	0.174243	+	0.301797i
$528$		0			0
$529$	9.50000	−	16.4545i	0.413043	−	0.715412i
$530$	−24.0000	−	41.5692i	−1.04249	−	1.80565i
$531$		0			0
$532$	−15.0000	−	5.19615i	−0.650332	−	0.225282i
$533$	−2.00000			−0.0866296
$534$		0			0
$535$	16.0000	−	27.7128i	0.691740	−	1.19813i
$536$	4.50000	−	7.79423i	0.194370	−	0.336659i
$537$		0			0
$538$	−12.0000			−0.517357
$539$	−6.50000	+	2.59808i	−0.279975	+	0.111907i
$540$		0			0
$541$	12.5000	+	21.6506i	0.537417	+	0.930834i	0.999042	+	0.0437584i	$0.0139332\pi$
$541$	12.5000	+	21.6506i	0.537417	+	0.930834i	−0.461625	+	0.887075i	$0.652733\pi$
$542$	−12.5000	+	21.6506i	−0.536921	+	0.929974i
$543$		0			0
$544$	1.00000	+	1.73205i	0.0428746	+	0.0742611i
$545$	40.0000			1.71341
$546$		0			0
$547$		0			0			−	1.00000i	$-0.5\pi$
$547$		0			0				1.00000i	$0.5\pi$
$548$	−4.50000	−	7.79423i	−0.192230	−	0.332953i
$549$		0			0
$550$	−5.50000	+	9.52628i	−0.234521	+	0.406202i
$551$	3.00000	+	5.19615i	0.127804	+	0.221364i
$552$		0			0
$553$	−7.50000	−	38.9711i	−0.318932	−	1.65722i
$554$	−3.00000			−0.127458
$555$		0			0
$556$	−4.00000	+	6.92820i	−0.169638	+	0.293821i
$557$	19.0000	−	32.9090i	0.805056	−	1.39440i	−0.111198	−	0.993798i	$-0.535469\pi$
$557$	19.0000	−	32.9090i	0.805056	−	1.39440i	0.916253	−	0.400599i	$-0.131198\pi$
$558$		0			0
$559$	−4.00000			−0.169182
$560$	8.00000	−	6.92820i	0.338062	−	0.292770i
$561$		0			0
$562$	−5.00000	−	8.66025i	−0.210912	−	0.365311i
$563$	13.0000	−	22.5167i	0.547885	−	0.948964i	−0.450535	−	0.892759i	$-0.648767\pi$
$563$	13.0000	−	22.5167i	0.547885	−	0.948964i	0.998419	−	0.0562051i	$-0.0179001\pi$
$564$		0			0
$565$	34.0000	+	58.8897i	1.43039	+	2.47751i
$566$	−6.00000			−0.252199
$567$		0			0
$568$	−4.00000			−0.167836
$569$	18.0000	+	31.1769i	0.754599	+	1.30700i	0.945573	+	0.325409i	$0.105502\pi$
$569$	18.0000	+	31.1769i	0.754599	+	1.30700i	−0.190974	+	0.981595i	$0.561165\pi$
$570$		0			0
$571$	10.0000	−	17.3205i	0.418487	−	0.724841i	−0.577301	−	0.816532i	$-0.695894\pi$
$571$	10.0000	−	17.3205i	0.418487	−	0.724841i	0.995788	+	0.0916910i	$0.0292272\pi$
$572$	0.500000	+	0.866025i	0.0209061	+	0.0362103i
$573$		0			0
$574$	1.00000	+	5.19615i	0.0417392	+	0.216883i
$575$	22.0000			0.917463
$576$		0			0
$577$	3.50000	−	6.06218i	0.145707	−	0.252372i	−0.783930	−	0.620850i	$-0.786788\pi$
$577$	3.50000	−	6.06218i	0.145707	−	0.252372i	0.929636	+	0.368478i	$0.120121\pi$
$578$	6.50000	−	11.2583i	0.270364	−	0.468285i
$579$		0			0
$580$	−4.00000			−0.166091
$581$	−15.0000	−	5.19615i	−0.622305	−	0.215573i
$582$		0			0
$583$	−6.00000	−	10.3923i	−0.248495	−	0.430405i
$584$	1.00000	−	1.73205i	0.0413803	−	0.0716728i
$585$		0			0
$586$	−3.00000	−	5.19615i	−0.123929	−	0.214651i
$587$	3.00000			0.123823			0.0619116	−	0.998082i	$-0.480280\pi$
$587$	3.00000			0.123823			0.0619116	+	0.998082i	$0.480280\pi$
$588$		0			0
$589$	24.0000			0.988903
$590$	18.0000	+	31.1769i	0.741048	+	1.28353i
$591$		0			0
$592$	1.00000	−	1.73205i	0.0410997	−	0.0711868i
$593$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$593$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$594$		0			0
$595$	20.0000	+	6.92820i	0.819920	+	0.284029i
$596$	10.0000			0.409616
$597$		0			0
$598$	1.00000	−	1.73205i	0.0408930	−	0.0708288i
$599$	−12.0000	+	20.7846i	−0.490307	+	0.849236i	−0.999938	−	0.0111569i	$-0.996449\pi$
$599$	−12.0000	+	20.7846i	−0.490307	+	0.849236i	0.509631	+	0.860393i	$0.329782\pi$
$600$		0			0
$601$	26.0000			1.06056			0.530281	−	0.847822i	$-0.322086\pi$
$601$	26.0000			1.06056			0.530281	+	0.847822i	$0.322086\pi$
$602$	2.00000	+	10.3923i	0.0815139	+	0.423559i
$603$		0			0
$604$	−4.50000	−	7.79423i	−0.183102	−	0.317143i
$605$	−2.00000	+	3.46410i	−0.0813116	+	0.140836i
$606$		0			0
$607$	8.00000	+	13.8564i	0.324710	+	0.562414i	0.981454	−	0.191700i	$-0.0614000\pi$
$607$	8.00000	+	13.8564i	0.324710	+	0.562414i	−0.656744	+	0.754114i	$0.728067\pi$
$608$	6.00000			0.243332
$609$		0			0
$610$	−20.0000			−0.809776
$611$	−1.00000	−	1.73205i	−0.0404557	−	0.0700713i
$612$		0			0
$613$	−11.0000	+	19.0526i	−0.444286	+	0.769526i	−0.998002	−	0.0631797i	$-0.979876\pi$
$613$	−11.0000	+	19.0526i	−0.444286	+	0.769526i	0.553716	+	0.832705i	$0.313209\pi$
$614$	−16.0000	−	27.7128i	−0.645707	−	1.11840i
$615$		0			0
$616$	2.00000	−	1.73205i	0.0805823	−	0.0697863i
$617$	27.0000			1.08698			0.543490	−	0.839416i	$-0.317103\pi$
$617$	27.0000			1.08698			0.543490	+	0.839416i	$0.317103\pi$
$618$		0			0
$619$	−10.0000	+	17.3205i	−0.401934	+	0.696170i	−0.993959	−	0.109749i	$-0.964995\pi$
$619$	−10.0000	+	17.3205i	−0.401934	+	0.696170i	0.592025	+	0.805919i	$0.298329\pi$
$620$	−8.00000	+	13.8564i	−0.321288	+	0.556487i
$621$		0			0
$622$	28.0000			1.12270
$623$	−3.00000	−	15.5885i	−0.120192	−	0.624538i
$624$		0			0
$625$	−20.5000	−	35.5070i	−0.820000	−	1.42028i
$626$	−0.500000	+	0.866025i	−0.0199840	+	0.0346133i
$627$		0			0
$628$	−2.00000	−	3.46410i	−0.0798087	−	0.138233i
$629$	4.00000			0.159490
$630$		0			0
$631$		0			0			−	1.00000i	$-0.5\pi$
$631$		0			0				1.00000i	$0.5\pi$
$632$	7.50000	+	12.9904i	0.298334	+	0.516730i
$633$		0			0
$634$	−6.00000	+	10.3923i	−0.238290	+	0.412731i
$635$	−10.0000	−	17.3205i	−0.396838	−	0.687343i
$636$		0			0
$637$	−5.50000	−	4.33013i	−0.217918	−	0.171566i
$638$	−1.00000			−0.0395904
$639$		0			0
$640$	−2.00000	+	3.46410i	−0.0790569	+	0.136931i
$641$	4.50000	−	7.79423i	0.177739	−	0.307854i	−0.763367	−	0.645966i	$-0.776455\pi$
$641$	4.50000	−	7.79423i	0.177739	−	0.307854i	0.941106	+	0.338112i	$0.109788\pi$
$642$		0			0
$643$	1.00000			0.0394362			0.0197181	−	0.999806i	$-0.493723\pi$
$643$	1.00000			0.0394362			0.0197181	+	0.999806i	$0.493723\pi$
$644$	−5.00000	−	1.73205i	−0.197028	−	0.0682524i
$645$		0			0
$646$	6.00000	+	10.3923i	0.236067	+	0.408880i
$647$	12.0000	−	20.7846i	0.471769	−	0.817127i	−0.527710	−	0.849425i	$-0.676949\pi$
$647$	12.0000	−	20.7846i	0.471769	−	0.817127i	0.999478	+	0.0322975i	$0.0102824\pi$
$648$		0			0
$649$	4.50000	+	7.79423i	0.176640	+	0.305950i
$650$	−11.0000			−0.431455
$651$		0			0
$652$	13.0000			0.509119
$653$	11.0000	+	19.0526i	0.430463	+	0.745584i	0.996913	−	0.0785119i	$-0.0250169\pi$
$653$	11.0000	+	19.0526i	0.430463	+	0.745584i	−0.566450	+	0.824096i	$0.691684\pi$
$654$		0			0
$655$	−36.0000	+	62.3538i	−1.40664	+	2.43637i
$656$	−1.00000	−	1.73205i	−0.0390434	−	0.0676252i
$657$		0			0
$658$	−4.00000	+	3.46410i	−0.155936	+	0.135045i
$659$	32.0000			1.24654			0.623272	−	0.782006i	$-0.285803\pi$
$659$	32.0000			1.24654			0.623272	+	0.782006i	$0.285803\pi$
$660$		0			0
$661$	5.00000	−	8.66025i	0.194477	−	0.336845i	−0.752252	−	0.658876i	$-0.771032\pi$
$661$	5.00000	−	8.66025i	0.194477	−	0.336845i	0.946729	+	0.322031i	$0.104366\pi$
$662$	−3.50000	+	6.06218i	−0.136031	+	0.235613i
$663$		0			0
$664$	6.00000			0.232845
$665$	48.0000	−	41.5692i	1.86136	−	1.61199i
$666$		0			0
$667$	1.00000	+	1.73205i	0.0387202	+	0.0670653i
$668$	−8.50000	+	14.7224i	−0.328875	+	0.569628i
$669$		0			0
$670$	18.0000	+	31.1769i	0.695401	+	1.20447i
$671$	−5.00000			−0.193023
$672$		0			0
$673$	16.0000			0.616755			0.308377	−	0.951264i	$-0.400214\pi$
$673$	16.0000			0.616755			0.308377	+	0.951264i	$0.400214\pi$
$674$	15.0000	+	25.9808i	0.577778	+	1.00074i
$675$		0			0
$676$	6.00000	−	10.3923i	0.230769	−	0.399704i
$677$	19.0000	+	32.9090i	0.730229	+	1.26479i	0.956785	+	0.290796i	$0.0939201\pi$
$677$	19.0000	+	32.9090i	0.730229	+	1.26479i	−0.226556	+	0.973998i	$0.572747\pi$
$678$		0			0
$679$	12.5000	+	4.33013i	0.479706	+	0.166175i
$680$	−8.00000			−0.306786
$681$		0			0
$682$	−2.00000	+	3.46410i	−0.0765840	+	0.132647i
$683$	−16.5000	+	28.5788i	−0.631355	+	1.09354i	0.355920	+	0.934516i	$0.384168\pi$
$683$	−16.5000	+	28.5788i	−0.631355	+	1.09354i	−0.987275	+	0.159022i	$0.949166\pi$
$684$		0			0
$685$	36.0000			1.37549
$686$	−8.50000	+	16.4545i	−0.324532	+	0.628235i
$687$		0			0
$688$	−2.00000	−	3.46410i	−0.0762493	−	0.132068i
$689$	6.00000	−	10.3923i	0.228582	−	0.395915i
$690$		0			0
$691$	−7.50000	−	12.9904i	−0.285313	−	0.494177i	0.687372	−	0.726306i	$-0.258764\pi$
$691$	−7.50000	−	12.9904i	−0.285313	−	0.494177i	−0.972685	+	0.232128i	$0.925431\pi$
$692$	5.00000			0.190071
$693$		0			0
$694$	28.0000			1.06287
$695$	−16.0000	−	27.7128i	−0.606915	−	1.05121i
$696$		0			0
$697$	2.00000	−	3.46410i	0.0757554	−	0.131212i
$698$	−1.00000	−	1.73205i	−0.0378506	−	0.0655591i
$699$		0			0
$700$	5.50000	+	28.5788i	0.207880	+	1.08018i
$701$	−39.0000			−1.47301			−0.736505	−	0.676432i	$-0.763525\pi$
$701$	−39.0000			−1.47301			−0.736505	+	0.676432i	$0.763525\pi$
$702$		0			0
$703$	6.00000	−	10.3923i	0.226294	−	0.391953i
$704$	−0.500000	+	0.866025i	−0.0188445	+	0.0326396i
$705$		0			0
$706$	18.0000			0.677439
$707$	30.0000	−	25.9808i	1.12827	−	0.977107i
$708$		0			0
$709$	−3.00000	−	5.19615i	−0.112667	−	0.195146i	0.804178	−	0.594389i	$-0.202606\pi$
$709$	−3.00000	−	5.19615i	−0.112667	−	0.195146i	−0.916845	+	0.399244i	$0.869273\pi$
$710$	8.00000	−	13.8564i	0.300235	−	0.520022i
$711$		0			0
$712$	3.00000	+	5.19615i	0.112430	+	0.194734i
$713$	8.00000			0.299602
$714$		0			0
$715$	−4.00000			−0.149592
$716$	6.50000	+	11.2583i	0.242916	+	0.420744i
$717$		0			0
$718$	−15.5000	+	26.8468i	−0.578455	+	1.00191i
$719$	−13.0000	−	22.5167i	−0.484818	−	0.839730i	0.515030	−	0.857172i	$-0.327781\pi$
$719$	−13.0000	−	22.5167i	−0.484818	−	0.839730i	−0.999848	+	0.0174426i	$0.994448\pi$
$720$		0			0
$721$	6.00000	+	31.1769i	0.223452	+	1.16109i
$722$	17.0000			0.632674
$723$		0			0
$724$	11.0000	−	19.0526i	0.408812	−	0.708083i
$725$	5.50000	−	9.52628i	0.204265	−	0.353797i
$726$		0			0
$727$	−34.0000			−1.26099			−0.630495	−	0.776193i	$-0.717148\pi$
$727$	−34.0000			−1.26099			−0.630495	+	0.776193i	$0.717148\pi$
$728$	2.50000	+	0.866025i	0.0926562	+	0.0320970i
$729$		0			0
$730$	4.00000	+	6.92820i	0.148047	+	0.256424i
$731$	4.00000	−	6.92820i	0.147945	−	0.256249i
$732$		0			0
$733$	0.500000	+	0.866025i	0.0184679	+	0.0319874i	0.875112	−	0.483921i	$-0.160788\pi$
$733$	0.500000	+	0.866025i	0.0184679	+	0.0319874i	−0.856644	+	0.515908i	$0.827454\pi$
$734$	−14.0000			−0.516749
$735$		0			0
$736$	2.00000			0.0737210
$737$	4.50000	+	7.79423i	0.165760	+	0.287104i
$738$		0			0
$739$	−9.00000	+	15.5885i	−0.331070	+	0.573431i	−0.982722	−	0.185088i	$-0.940743\pi$
$739$	−9.00000	+	15.5885i	−0.331070	+	0.573431i	0.651652	+	0.758518i	$0.274076\pi$
$740$	4.00000	+	6.92820i	0.147043	+	0.254686i
$741$		0			0
$742$	−30.0000	−	10.3923i	−1.10133	−	0.381514i
$743$	−36.0000			−1.32071			−0.660356	−	0.750953i	$-0.729595\pi$
$743$	−36.0000			−1.32071			−0.660356	+	0.750953i	$0.729595\pi$
$744$		0			0
$745$	−20.0000	+	34.6410i	−0.732743	+	1.26915i
$746$	3.50000	−	6.06218i	0.128144	−	0.221952i
$747$		0			0
$748$	−2.00000			−0.0731272
$749$	−4.00000	−	20.7846i	−0.146157	−	0.759453i
$750$		0			0
$751$	22.0000	+	38.1051i	0.802791	+	1.39048i	0.917772	+	0.397108i	$0.129986\pi$
$751$	22.0000	+	38.1051i	0.802791	+	1.39048i	−0.114981	+	0.993368i	$0.536681\pi$
$752$	1.00000	−	1.73205i	0.0364662	−	0.0631614i
$753$		0			0
$754$	−0.500000	−	0.866025i	−0.0182089	−	0.0315388i
$755$	36.0000			1.31017
$756$		0			0
$757$	26.0000			0.944986			0.472493	−	0.881334i	$-0.343354\pi$
$757$	26.0000			0.944986			0.472493	+	0.881334i	$0.343354\pi$
$758$	14.5000	+	25.1147i	0.526664	+	0.912208i
$759$		0			0
$760$	−12.0000	+	20.7846i	−0.435286	+	0.753937i
$761$	27.0000	+	46.7654i	0.978749	+	1.69524i	0.666962	+	0.745091i	$0.267594\pi$
$761$	27.0000	+	46.7654i	0.978749	+	1.69524i	0.311787	+	0.950152i	$0.399073\pi$
$762$		0			0
$763$	20.0000	−	17.3205i	0.724049	−	0.627044i
$764$	−14.0000			−0.506502
$765$		0			0
$766$	4.00000	−	6.92820i	0.144526	−	0.250326i
$767$	−4.50000	+	7.79423i	−0.162486	+	0.281433i
$768$		0			0
$769$	−38.0000			−1.37032			−0.685158	−	0.728395i	$-0.740267\pi$
$769$	−38.0000			−1.37032			−0.685158	+	0.728395i	$0.740267\pi$
$770$	2.00000	+	10.3923i	0.0720750	+	0.374513i
$771$		0			0
$772$	11.0000	+	19.0526i	0.395899	+	0.685717i
$773$	−12.0000	+	20.7846i	−0.431610	+	0.747570i	−0.997012	−	0.0772449i	$-0.975388\pi$
$773$	−12.0000	+	20.7846i	−0.431610	+	0.747570i	0.565402	+	0.824815i	$0.308721\pi$
$774$		0			0
$775$	−22.0000	−	38.1051i	−0.790263	−	1.36878i
$776$	−5.00000			−0.179490
$777$		0			0
$778$		0			0
$779$	−6.00000	−	10.3923i	−0.214972	−	0.372343i
$780$		0			0
$781$	2.00000	−	3.46410i	0.0715656	−	0.123955i
$782$	2.00000	+	3.46410i	0.0715199	+	0.123876i
$783$		0			0
$784$	1.00000	−	6.92820i	0.0357143	−	0.247436i
$785$	16.0000			0.571064
$786$		0			0
$787$	11.0000	−	19.0526i	0.392108	−	0.679150i	−0.600620	−	0.799535i	$-0.705079\pi$
$787$	11.0000	−	19.0526i	0.392108	−	0.679150i	0.992727	+	0.120384i	$0.0384127\pi$
$788$	−1.50000	+	2.59808i	−0.0534353	+	0.0925526i
$789$		0			0
$790$	−60.0000			−2.13470
$791$	42.5000	+	14.7224i	1.51113	+	0.523469i
$792$		0			0
$793$	−2.50000	−	4.33013i	−0.0887776	−	0.153767i
$794$	15.0000	−	25.9808i	0.532330	−	0.922023i
$795$		0			0
$796$	−5.00000	−	8.66025i	−0.177220	−	0.306955i
$797$	−38.0000			−1.34603			−0.673015	−	0.739629i	$-0.735001\pi$
$797$	−38.0000			−1.34603			−0.673015	+	0.739629i	$0.735001\pi$
$798$		0			0
$799$	4.00000			0.141510
$800$	−5.50000	−	9.52628i	−0.194454	−	0.336805i
$801$		0			0
$802$	7.50000	−	12.9904i	0.264834	−	0.458706i
$803$	1.00000	+	1.73205i	0.0352892	+	0.0611227i
$804$		0			0
$805$	16.0000	−	13.8564i	0.563926	−	0.488374i
$806$	−4.00000			−0.140894
$807$		0			0
$808$	−7.50000	+	12.9904i	−0.263849	+	0.457000i
$809$	−24.0000	+	41.5692i	−0.843795	+	1.46150i	0.0428684	+	0.999081i	$0.486350\pi$
$809$	−24.0000	+	41.5692i	−0.843795	+	1.46150i	−0.886664	+	0.462415i	$0.846983\pi$
$810$		0			0
$811$	−28.0000			−0.983213			−0.491606	−	0.870817i	$-0.663590\pi$
$811$	−28.0000			−0.983213			−0.491606	+	0.870817i	$0.663590\pi$
$812$	−2.00000	+	1.73205i	−0.0701862	+	0.0607831i
$813$		0			0
$814$	1.00000	+	1.73205i	0.0350500	+	0.0607083i
$815$	−26.0000	+	45.0333i	−0.910740	+	1.57745i
$816$		0			0
$817$	−12.0000	−	20.7846i	−0.419827	−	0.727161i
$818$	−32.0000			−1.11885
$819$		0			0
$820$	8.00000			0.279372
$821$	−7.50000	−	12.9904i	−0.261752	−	0.453367i	0.704956	−	0.709251i	$-0.250967\pi$
$821$	−7.50000	−	12.9904i	−0.261752	−	0.453367i	−0.966708	+	0.255884i	$0.917634\pi$
$822$		0			0
$823$	−19.0000	+	32.9090i	−0.662298	+	1.14713i	0.317712	+	0.948187i	$0.397086\pi$
$823$	−19.0000	+	32.9090i	−0.662298	+	1.14713i	−0.980010	+	0.198947i	$0.936248\pi$
$824$	−6.00000	−	10.3923i	−0.209020	−	0.362033i
$825$		0			0
$826$	22.5000	+	7.79423i	0.782875	+	0.271196i
$827$	22.0000			0.765015			0.382507	−	0.923952i	$-0.375061\pi$
$827$	22.0000			0.765015			0.382507	+	0.923952i	$0.375061\pi$
$828$		0			0
$829$	−8.00000	+	13.8564i	−0.277851	+	0.481253i	−0.970851	−	0.239686i	$-0.922956\pi$
$829$	−8.00000	+	13.8564i	−0.277851	+	0.481253i	0.692999	+	0.720938i	$0.256289\pi$
$830$	−12.0000	+	20.7846i	−0.416526	+	0.721444i
$831$		0			0
$832$	−1.00000			−0.0346688
$833$	13.0000	−	5.19615i	0.450423	−	0.180036i
$834$		0			0
$835$	−34.0000	−	58.8897i	−1.17662	−	2.03796i
$836$	−3.00000	+	5.19615i	−0.103757	+	0.179713i
$837$		0			0
$838$	−10.0000	−	17.3205i	−0.345444	−	0.598327i
$839$	−54.0000			−1.86429			−0.932144	−	0.362089i	$-0.882064\pi$
$839$	−54.0000			−1.86429			−0.932144	+	0.362089i	$0.882064\pi$
$840$		0			0
$841$	−28.0000			−0.965517
$842$	10.0000	+	17.3205i	0.344623	+	0.596904i
$843$		0			0
$844$	7.00000	−	12.1244i	0.240950	−	0.417338i
$845$	24.0000	+	41.5692i	0.825625	+	1.43002i
$846$		0			0
$847$	0.500000	+	2.59808i	0.0171802	+	0.0892710i
$848$	12.0000			0.412082
$849$		0			0
$850$	11.0000	−	19.0526i	0.377297	−	0.653497i
$851$	2.00000	−	3.46410i	0.0685591	−	0.118748i
$852$		0			0
$853$	50.0000			1.71197			0.855984	−	0.517003i	$-0.172952\pi$
$853$	50.0000			1.71197			0.855984	+	0.517003i	$0.172952\pi$
$854$	−10.0000	+	8.66025i	−0.342193	+	0.296348i
$855$		0			0
$856$	4.00000	+	6.92820i	0.136717	+	0.236801i
$857$	−2.00000	+	3.46410i	−0.0683187	+	0.118331i	−0.898161	−	0.439666i	$-0.855097\pi$
$857$	−2.00000	+	3.46410i	−0.0683187	+	0.118331i	0.829843	+	0.557998i	$0.188430\pi$
$858$		0			0
$859$	8.50000	+	14.7224i	0.290016	+	0.502323i	0.973813	−	0.227349i	$-0.0730059\pi$
$859$	8.50000	+	14.7224i	0.290016	+	0.502323i	−0.683797	+	0.729672i	$0.739673\pi$
$860$	16.0000			0.545595
$861$		0			0
$862$	−1.00000			−0.0340601
$863$	5.00000	+	8.66025i	0.170202	+	0.294798i	0.938490	−	0.345305i	$-0.112225\pi$
$863$	5.00000	+	8.66025i	0.170202	+	0.294798i	−0.768288	+	0.640104i	$0.778891\pi$
$864$		0			0
$865$	−10.0000	+	17.3205i	−0.340010	+	0.588915i
$866$	5.00000	+	8.66025i	0.169907	+	0.294287i
$867$		0			0
$868$	2.00000	+	10.3923i	0.0678844	+	0.352738i
$869$	−15.0000			−0.508840
$870$		0			0
$871$	−4.50000	+	7.79423i	−0.152477	+	0.264097i
$872$	−5.00000	+	8.66025i	−0.169321	+	0.293273i
$873$		0			0
$874$	12.0000			0.405906
$875$	−60.0000	−	20.7846i	−2.02837	−	0.702648i
$876$		0			0
$877$	−12.5000	−	21.6506i	−0.422095	−	0.731090i	0.574049	−	0.818821i	$-0.305372\pi$
$877$	−12.5000	−	21.6506i	−0.422095	−	0.731090i	−0.996144	+	0.0877308i	$0.972038\pi$
$878$	2.50000	−	4.33013i	0.0843709	−	0.146135i
$879$		0			0
$880$	−2.00000	−	3.46410i	−0.0674200	−	0.116775i
$881$	−9.00000			−0.303218			−0.151609	−	0.988441i	$-0.548445\pi$
$881$	−9.00000			−0.303218			−0.151609	+	0.988441i	$0.548445\pi$
$882$		0			0
$883$	1.00000			0.0336527			0.0168263	−	0.999858i	$-0.494644\pi$
$883$	1.00000			0.0336527			0.0168263	+	0.999858i	$0.494644\pi$
$884$	−1.00000	−	1.73205i	−0.0336336	−	0.0582552i
$885$		0			0
$886$	−6.00000	+	10.3923i	−0.201574	+	0.349136i
$887$	−8.50000	−	14.7224i	−0.285402	−	0.494331i	0.687305	−	0.726369i	$-0.258794\pi$
$887$	−8.50000	−	14.7224i	−0.285402	−	0.494331i	−0.972707	+	0.232038i	$0.925460\pi$
$888$		0			0
$889$	−12.5000	−	4.33013i	−0.419237	−	0.145228i
$890$	−24.0000			−0.804482
$891$		0			0
$892$	13.0000	−	22.5167i	0.435272	−	0.753914i
$893$	6.00000	−	10.3923i	0.200782	−	0.347765i
$894$		0			0
$895$	−52.0000			−1.73817
$896$	0.500000	+	2.59808i	0.0167038	+	0.0867956i
$897$		0			0
$898$	15.0000	+	25.9808i	0.500556	+	0.866989i
$899$	2.00000	−	3.46410i	0.0667037	−	0.115534i
$900$		0			0
$901$	12.0000	+	20.7846i	0.399778	+	0.692436i
$902$	2.00000			0.0665927
$903$		0			0
$904$	−17.0000			−0.565412
$905$	44.0000	+	76.2102i	1.46261	+	2.53331i
$906$		0			0
$907$	−22.0000	+	38.1051i	−0.730498	+	1.26526i	0.226173	+	0.974087i	$0.427379\pi$
$907$	−22.0000	+	38.1051i	−0.730498	+	1.26526i	−0.956671	+	0.291172i	$0.905955\pi$
$908$	5.00000	+	8.66025i	0.165931	+	0.287401i
$909$		0			0
$910$	−8.00000	+	6.92820i	−0.265197	+	0.229668i
$911$	54.0000			1.78910			0.894550	−	0.446968i	$-0.147496\pi$
$911$	54.0000			1.78910			0.894550	+	0.446968i	$0.147496\pi$
$912$		0			0
$913$	−3.00000	+	5.19615i	−0.0992855	+	0.171968i
$914$	−1.00000	+	1.73205i	−0.0330771	+	0.0572911i
$915$		0			0
$916$	−16.0000			−0.528655
$917$	9.00000	+	46.7654i	0.297206	+	1.54433i
$918$		0			0
$919$	12.0000	+	20.7846i	0.395843	+	0.685621i	0.993208	−	0.116348i	$-0.0371189\pi$
$919$	12.0000	+	20.7846i	0.395843	+	0.685621i	−0.597365	+	0.801970i	$0.703786\pi$
$920$	−4.00000	+	6.92820i	−0.131876	+	0.228416i
$921$		0			0
$922$	−1.50000	−	2.59808i	−0.0493999	−	0.0855631i
$923$	4.00000			0.131662
$924$		0			0
$925$	−22.0000			−0.723356
$926$	7.00000	+	12.1244i	0.230034	+	0.398431i
$927$		0			0
$928$	0.500000	−	0.866025i	0.0164133	−	0.0284287i
$929$	−16.5000	−	28.5788i	−0.541347	−	0.937641i	−0.998827	−	0.0484211i	$-0.984581\pi$
$929$	−16.5000	−	28.5788i	−0.541347	−	0.937641i	0.457480	−	0.889220i	$-0.348752\pi$
$930$		0			0
$931$	6.00000	−	41.5692i	0.196642	−	1.36238i
$932$	−6.00000			−0.196537
$933$		0			0
$934$	−6.00000	+	10.3923i	−0.196326	+	0.340047i
$935$	4.00000	−	6.92820i	0.130814	−	0.226576i
$936$		0			0
$937$	−12.0000			−0.392023			−0.196011	−	0.980602i	$-0.562799\pi$
$937$	−12.0000			−0.392023			−0.196011	+	0.980602i	$0.562799\pi$
$938$	22.5000	+	7.79423i	0.734651	+	0.254491i
$939$		0			0
$940$	4.00000	+	6.92820i	0.130466	+	0.225973i
$941$	27.5000	−	47.6314i	0.896474	−	1.55274i	0.0645052	−	0.997917i	$-0.479453\pi$
$941$	27.5000	−	47.6314i	0.896474	−	1.55274i	0.831969	−	0.554822i	$-0.187214\pi$
$942$		0			0
$943$	−2.00000	−	3.46410i	−0.0651290	−	0.112807i
$944$	−9.00000			−0.292925
$945$		0			0
$946$	4.00000			0.130051
$947$	−8.00000	−	13.8564i	−0.259965	−	0.450273i	0.706267	−	0.707945i	$-0.250378\pi$
$947$	−8.00000	−	13.8564i	−0.259965	−	0.450273i	−0.966232	+	0.257673i	$0.917044\pi$
$948$		0			0
$949$	−1.00000	+	1.73205i	−0.0324614	+	0.0562247i
$950$	−33.0000	−	57.1577i	−1.07066	−	1.85444i
$951$		0			0
$952$	−4.00000	+	3.46410i	−0.129641	+	0.112272i
$953$	−56.0000			−1.81402			−0.907009	−	0.421111i	$-0.861640\pi$
$953$	−56.0000			−1.81402			−0.907009	+	0.421111i	$0.861640\pi$
$954$		0			0
$955$	28.0000	−	48.4974i	0.906059	−	1.56934i
$956$	9.50000	−	16.4545i	0.307252	−	0.532176i
$957$		0			0
$958$	37.0000			1.19542
$959$	18.0000	−	15.5885i	0.581250	−	0.503378i
$960$		0			0
$961$	7.50000	+	12.9904i	0.241935	+	0.419045i
$962$	−1.00000	+	1.73205i	−0.0322413	+	0.0558436i
$963$		0			0
$964$	−15.0000	−	25.9808i	−0.483117	−	0.836784i
$965$	−88.0000			−2.83282
$966$		0			0
$967$	56.0000			1.80084			0.900419	−	0.435023i	$-0.143260\pi$
$967$	56.0000			1.80084			0.900419	+	0.435023i	$0.143260\pi$
$968$	−0.500000	−	0.866025i	−0.0160706	−	0.0278351i
$969$		0			0
$970$	10.0000	−	17.3205i	0.321081	−	0.556128i
$971$	−6.50000	−	11.2583i	−0.208595	−	0.361297i	0.742677	−	0.669650i	$-0.233556\pi$
$971$	−6.50000	−	11.2583i	−0.208595	−	0.361297i	−0.951272	+	0.308353i	$0.900222\pi$
$972$		0			0
$973$	−20.0000	−	6.92820i	−0.641171	−	0.222108i
$974$	−4.00000			−0.128168
$975$		0			0
$976$	2.50000	−	4.33013i	0.0800230	−	0.138604i
$977$	7.00000	−	12.1244i	0.223950	−	0.387893i	−0.732054	−	0.681247i	$-0.761438\pi$
$977$	7.00000	−	12.1244i	0.223950	−	0.387893i	0.956004	+	0.293354i	$0.0947715\pi$
$978$		0			0
$979$	−6.00000			−0.191761
$980$	22.0000	+	17.3205i	0.702764	+	0.553283i
$981$		0			0
$982$	9.00000	+	15.5885i	0.287202	+	0.497448i
$983$	9.00000	−	15.5885i	0.287055	−	0.497195i	−0.686050	−	0.727554i	$-0.740657\pi$
$983$	9.00000	−	15.5885i	0.287055	−	0.497195i	0.973106	+	0.230360i	$0.0739903\pi$
$984$		0			0
$985$	−6.00000	−	10.3923i	−0.191176	−	0.331126i
$986$	2.00000			0.0636930
$987$		0			0
$988$	−6.00000			−0.190885
$989$	−4.00000	−	6.92820i	−0.127193	−	0.220304i
$990$		0			0
$991$	10.0000	−	17.3205i	0.317660	−	0.550204i	−0.662339	−	0.749204i	$-0.730436\pi$
$991$	10.0000	−	17.3205i	0.317660	−	0.550204i	0.979999	+	0.199000i	$0.0637695\pi$
$992$	−2.00000	−	3.46410i	−0.0635001	−	0.109985i
$993$		0			0
$994$	−2.00000	−	10.3923i	−0.0634361	−	0.329624i
$995$	40.0000			1.26809
$996$		0			0
$997$	21.0000	−	36.3731i	0.665077	−	1.15195i	−0.314188	−	0.949361i	$-0.601732\pi$
$997$	21.0000	−	36.3731i	0.665077	−	1.15195i	0.979265	−	0.202586i	$-0.0649345\pi$
$998$	8.00000	−	13.8564i	0.253236	−	0.438617i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1386.2.k.a.991.1		2
3.2	odd	2		154.2.e.d.67.1	yes	2
7.2	even	3	inner	1386.2.k.a.793.1		2
7.3	odd	6		9702.2.a.bb.1.1		1
7.4	even	3		9702.2.a.cg.1.1		1
12.11	even	2		1232.2.q.a.529.1		2
21.2	odd	6		154.2.e.d.23.1	✓	2
21.5	even	6		1078.2.e.g.177.1		2
21.11	odd	6		1078.2.a.a.1.1		1
21.17	even	6		1078.2.a.f.1.1		1
21.20	even	2		1078.2.e.g.67.1		2
84.11	even	6		8624.2.a.bd.1.1		1
84.23	even	6		1232.2.q.a.177.1		2
84.59	odd	6		8624.2.a.d.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
154.2.e.d.23.1	✓	2	21.2	odd	6
154.2.e.d.67.1	yes	2	3.2	odd	2
1078.2.a.a.1.1		1	21.11	odd	6
1078.2.a.f.1.1		1	21.17	even	6
1078.2.e.g.67.1		2	21.20	even	2
1078.2.e.g.177.1		2	21.5	even	6
1232.2.q.a.177.1		2	84.23	even	6
1232.2.q.a.529.1		2	12.11	even	2
1386.2.k.a.793.1		2	7.2	even	3	inner
1386.2.k.a.991.1		2	1.1	even	1	trivial
8624.2.a.d.1.1		1	84.59	odd	6
8624.2.a.bd.1.1		1	84.11	even	6
9702.2.a.bb.1.1		1	7.3	odd	6
9702.2.a.cg.1.1		1	7.4	even	3

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.k (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 - 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(-2.00000 - 3.46410i) q^{5} +(-2.50000 - 0.866025i) q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+(-0.500000 - 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(-2.00000 - 3.46410i) q^{5} +(-2.50000 - 0.866025i) q^{7} +1.00000 q^{8} +(-2.00000 + 3.46410i) q^{10} +(-0.500000 + 0.866025i) q^{11} -1.00000 q^{13} +(0.500000 + 2.59808i) q^{14} +(-0.500000 - 0.866025i) q^{16} +(1.00000 - 1.73205i) q^{17} +(-3.00000 - 5.19615i) q^{19} +4.00000 q^{20} +1.00000 q^{22} +(-1.00000 - 1.73205i) q^{23} +(-5.50000 + 9.52628i) q^{25} +(0.500000 + 0.866025i) q^{26} +(2.00000 - 1.73205i) q^{28} -1.00000 q^{29} +(-2.00000 + 3.46410i) q^{31} +(-0.500000 + 0.866025i) q^{32} -2.00000 q^{34} +(2.00000 + 10.3923i) q^{35} +(1.00000 + 1.73205i) q^{37} +(-3.00000 + 5.19615i) q^{38} +(-2.00000 - 3.46410i) q^{40} +2.00000 q^{41} +4.00000 q^{43} +(-0.500000 - 0.866025i) q^{44} +(-1.00000 + 1.73205i) q^{46} +(1.00000 + 1.73205i) q^{47} +(5.50000 + 4.33013i) q^{49} +11.0000 q^{50} +(0.500000 - 0.866025i) q^{52} +(-6.00000 + 10.3923i) q^{53} +4.00000 q^{55} +(-2.50000 - 0.866025i) q^{56} +(0.500000 + 0.866025i) q^{58} +(4.50000 - 7.79423i) q^{59} +(2.50000 + 4.33013i) q^{61} +4.00000 q^{62} +1.00000 q^{64} +(2.00000 + 3.46410i) q^{65} +(4.50000 - 7.79423i) q^{67} +(1.00000 + 1.73205i) q^{68} +(8.00000 - 6.92820i) q^{70} -4.00000 q^{71} +(1.00000 - 1.73205i) q^{73} +(1.00000 - 1.73205i) q^{74} +6.00000 q^{76} +(2.00000 - 1.73205i) q^{77} +(7.50000 + 12.9904i) q^{79} +(-2.00000 + 3.46410i) q^{80} +(-1.00000 - 1.73205i) q^{82} +6.00000 q^{83} -8.00000 q^{85} +(-2.00000 - 3.46410i) q^{86} +(-0.500000 + 0.866025i) q^{88} +(3.00000 + 5.19615i) q^{89} +(2.50000 + 0.866025i) q^{91} +2.00000 q^{92} +(1.00000 - 1.73205i) q^{94} +(-12.0000 + 20.7846i) q^{95} -5.00000 q^{97} +(1.00000 - 6.92820i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} - q^{4} - 4 q^{5} - 5 q^{7} + 2 q^{8}+O(q^{10}) \) 2 * q - q^2 - q^4 - 4 * q^5 - 5 * q^7 + 2 * q^8	\( 2 q - q^{2} - q^{4} - 4 q^{5} - 5 q^{7} + 2 q^{8} - 4 q^{10} - q^{11} - 2 q^{13} + q^{14} - q^{16} + 2 q^{17} - 6 q^{19} + 8 q^{20} + 2 q^{22} - 2 q^{23} - 11 q^{25} + q^{26} + 4 q^{28} - 2 q^{29} - 4 q^{31} - q^{32} - 4 q^{34} + 4 q^{35} + 2 q^{37} - 6 q^{38} - 4 q^{40} + 4 q^{41} + 8 q^{43} - q^{44} - 2 q^{46} + 2 q^{47} + 11 q^{49} + 22 q^{50} + q^{52} - 12 q^{53} + 8 q^{55} - 5 q^{56} + q^{58} + 9 q^{59} + 5 q^{61} + 8 q^{62} + 2 q^{64} + 4 q^{65} + 9 q^{67} + 2 q^{68} + 16 q^{70} - 8 q^{71} + 2 q^{73} + 2 q^{74} + 12 q^{76} + 4 q^{77} + 15 q^{79} - 4 q^{80} - 2 q^{82} + 12 q^{83} - 16 q^{85} - 4 q^{86} - q^{88} + 6 q^{89} + 5 q^{91} + 4 q^{92} + 2 q^{94} - 24 q^{95} - 10 q^{97} + 2 q^{98}+O(q^{100}) \) 2 * q - q^2 - q^4 - 4 * q^5 - 5 * q^7 + 2 * q^8 - 4 * q^10 - q^11 - 2 * q^13 + q^14 - q^16 + 2 * q^17 - 6 * q^19 + 8 * q^20 + 2 * q^22 - 2 * q^23 - 11 * q^25 + q^26 + 4 * q^28 - 2 * q^29 - 4 * q^31 - q^32 - 4 * q^34 + 4 * q^35 + 2 * q^37 - 6 * q^38 - 4 * q^40 + 4 * q^41 + 8 * q^43 - q^44 - 2 * q^46 + 2 * q^47 + 11 * q^49 + 22 * q^50 + q^52 - 12 * q^53 + 8 * q^55 - 5 * q^56 + q^58 + 9 * q^59 + 5 * q^61 + 8 * q^62 + 2 * q^64 + 4 * q^65 + 9 * q^67 + 2 * q^68 + 16 * q^70 - 8 * q^71 + 2 * q^73 + 2 * q^74 + 12 * q^76 + 4 * q^77 + 15 * q^79 - 4 * q^80 - 2 * q^82 + 12 * q^83 - 16 * q^85 - 4 * q^86 - q^88 + 6 * q^89 + 5 * q^91 + 4 * q^92 + 2 * q^94 - 24 * q^95 - 10 * q^97 + 2 * q^98

Introduction

L-functions

Modular forms

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