LMFDB - Embedded newform 1078.2.e.c.177.1

Newspace parameters

Level:	$ N $	$=$	$ 1078 = 2 \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1078.e (of order $3$, degree $2$, not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$8.60787333789$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
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			177.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	1078.177
Dual form			1078.2.e.c.67.1

$q$-expansion

$f(q)$	$=$	$q+(-0.500000 + 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(1.00000 - 1.73205i) q^{5} +1.00000 q^{8} +(1.50000 - 2.59808i) q^{9} +O(q^{10})$	\(q+(-0.500000 + 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(1.00000 - 1.73205i) q^{5} +1.00000 q^{8} +(1.50000 - 2.59808i) q^{9} +(1.00000 + 1.73205i) q^{10} +(0.500000 + 0.866025i) q^{11} -2.00000 q^{13} +(-0.500000 + 0.866025i) q^{16} +(1.00000 + 1.73205i) q^{17} +(1.50000 + 2.59808i) q^{18} -2.00000 q^{20} -1.00000 q^{22} +(4.00000 - 6.92820i) q^{23} +(0.500000 + 0.866025i) q^{25} +(1.00000 - 1.73205i) q^{26} -2.00000 q^{29} +(-4.00000 - 6.92820i) q^{31} +(-0.500000 - 0.866025i) q^{32} -2.00000 q^{34} -3.00000 q^{36} +(1.00000 - 1.73205i) q^{37} +(1.00000 - 1.73205i) q^{40} -10.0000 q^{41} +4.00000 q^{43} +(0.500000 - 0.866025i) q^{44} +(-3.00000 - 5.19615i) q^{45} +(4.00000 + 6.92820i) q^{46} +(4.00000 - 6.92820i) q^{47} -1.00000 q^{50} +(1.00000 + 1.73205i) q^{52} +(-3.00000 - 5.19615i) q^{53} +2.00000 q^{55} +(1.00000 - 1.73205i) q^{58} +(5.00000 - 8.66025i) q^{61} +8.00000 q^{62} +1.00000 q^{64} +(-2.00000 + 3.46410i) q^{65} +(6.00000 + 10.3923i) q^{67} +(1.00000 - 1.73205i) q^{68} +16.0000 q^{71} +(1.50000 - 2.59808i) q^{72} +(-7.00000 - 12.1244i) q^{73} +(1.00000 + 1.73205i) q^{74} +(1.00000 + 1.73205i) q^{80} +(-4.50000 - 7.79423i) q^{81} +(5.00000 - 8.66025i) q^{82} +4.00000 q^{85} +(-2.00000 + 3.46410i) q^{86} +(0.500000 + 0.866025i) q^{88} +(-3.00000 + 5.19615i) q^{89} +6.00000 q^{90} -8.00000 q^{92} +(4.00000 + 6.92820i) q^{94} -10.0000 q^{97} +3.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} - q^{4} + 2 q^{5} + 2 q^{8} + 3 q^{9}+O(q^{10}) $ 2 * q - q^2 - q^4 + 2 * q^5 + 2 * q^8 + 3 * q^9	$ 2 q - q^{2} - q^{4} + 2 q^{5} + 2 q^{8} + 3 q^{9} + 2 q^{10} + q^{11} - 4 q^{13} - q^{16} + 2 q^{17} + 3 q^{18} - 4 q^{20} - 2 q^{22} + 8 q^{23} + q^{25} + 2 q^{26} - 4 q^{29} - 8 q^{31} - q^{32} - 4 q^{34} - 6 q^{36} + 2 q^{37} + 2 q^{40} - 20 q^{41} + 8 q^{43} + q^{44} - 6 q^{45} + 8 q^{46} + 8 q^{47} - 2 q^{50} + 2 q^{52} - 6 q^{53} + 4 q^{55} + 2 q^{58} + 10 q^{61} + 16 q^{62} + 2 q^{64} - 4 q^{65} + 12 q^{67} + 2 q^{68} + 32 q^{71} + 3 q^{72} - 14 q^{73} + 2 q^{74} + 2 q^{80} - 9 q^{81} + 10 q^{82} + 8 q^{85} - 4 q^{86} + q^{88} - 6 q^{89} + 12 q^{90} - 16 q^{92} + 8 q^{94} - 20 q^{97} + 6 q^{99}+O(q^{100}) $ 2 * q - q^2 - q^4 + 2 * q^5 + 2 * q^8 + 3 * q^9 + 2 * q^10 + q^11 - 4 * q^13 - q^16 + 2 * q^17 + 3 * q^18 - 4 * q^20 - 2 * q^22 + 8 * q^23 + q^25 + 2 * q^26 - 4 * q^29 - 8 * q^31 - q^32 - 4 * q^34 - 6 * q^36 + 2 * q^37 + 2 * q^40 - 20 * q^41 + 8 * q^43 + q^44 - 6 * q^45 + 8 * q^46 + 8 * q^47 - 2 * q^50 + 2 * q^52 - 6 * q^53 + 4 * q^55 + 2 * q^58 + 10 * q^61 + 16 * q^62 + 2 * q^64 - 4 * q^65 + 12 * q^67 + 2 * q^68 + 32 * q^71 + 3 * q^72 - 14 * q^73 + 2 * q^74 + 2 * q^80 - 9 * q^81 + 10 * q^82 + 8 * q^85 - 4 * q^86 + q^88 - 6 * q^89 + 12 * q^90 - 16 * q^92 + 8 * q^94 - 20 * q^97 + 6 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1078\mathbb{Z}\right)^\times$.

$n$	$199$	$981$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$1$

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		0.866025	−	0.500000i	$-0.166667\pi$
$3$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$4$	−0.500000	−	0.866025i	−0.250000	−	0.433013i
$5$	1.00000	−	1.73205i	0.447214	−	0.774597i	−0.550990	−	0.834512i	$-0.685750\pi$
$5$	1.00000	−	1.73205i	0.447214	−	0.774597i	0.998203	+	0.0599153i	$0.0190830\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$	1.00000			0.353553
$9$	1.50000	−	2.59808i	0.500000	−	0.866025i
$10$	1.00000	+	1.73205i	0.316228	+	0.547723i
$11$	0.500000	+	0.866025i	0.150756	+	0.261116i
$11$	0.500000	+	0.866025i	0.150756	+	0.261116i
$12$		0			0
$13$	−2.00000			−0.554700			−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000			−0.554700			−0.277350	+	0.960769i	$0.589456\pi$
$14$		0			0
$15$		0			0
$16$	−0.500000	+	0.866025i	−0.125000	+	0.216506i
$17$	1.00000	+	1.73205i	0.242536	+	0.420084i	0.961436	−	0.275029i	$-0.0886875\pi$
$17$	1.00000	+	1.73205i	0.242536	+	0.420084i	−0.718900	+	0.695113i	$0.755354\pi$
$18$	1.50000	+	2.59808i	0.353553	+	0.612372i
$19$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$19$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$20$	−2.00000			−0.447214
$21$		0			0
$22$	−1.00000			−0.213201
$23$	4.00000	−	6.92820i	0.834058	−	1.44463i	−0.0607377	−	0.998154i	$-0.519345\pi$
$23$	4.00000	−	6.92820i	0.834058	−	1.44463i	0.894795	−	0.446476i	$-0.147321\pi$
$24$		0			0
$25$	0.500000	+	0.866025i	0.100000	+	0.173205i
$26$	1.00000	−	1.73205i	0.196116	−	0.339683i
$27$		0			0
$28$		0			0
$29$	−2.00000			−0.371391			−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000			−0.371391			−0.185695	+	0.982607i	$0.559454\pi$
$30$		0			0
$31$	−4.00000	−	6.92820i	−0.718421	−	1.24434i	−0.961625	−	0.274367i	$-0.911532\pi$
$31$	−4.00000	−	6.92820i	−0.718421	−	1.24434i	0.243204	−	0.969975i	$-0.421802\pi$
$32$	−0.500000	−	0.866025i	−0.0883883	−	0.153093i
$33$		0			0
$34$	−2.00000			−0.342997
$35$		0			0
$36$	−3.00000			−0.500000
$37$	1.00000	−	1.73205i	0.164399	−	0.284747i	−0.772043	−	0.635571i	$-0.780765\pi$
$37$	1.00000	−	1.73205i	0.164399	−	0.284747i	0.936442	+	0.350823i	$0.114098\pi$
$38$		0			0
$39$		0			0
$40$	1.00000	−	1.73205i	0.158114	−	0.273861i
$41$	−10.0000			−1.56174			−0.780869	−	0.624695i	$-0.785223\pi$
$41$	−10.0000			−1.56174			−0.780869	+	0.624695i	$0.785223\pi$
$42$		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
$45$	−3.00000	−	5.19615i	−0.447214	−	0.774597i
$46$	4.00000	+	6.92820i	0.589768	+	1.02151i
$47$	4.00000	−	6.92820i	0.583460	−	1.01058i	−0.411606	−	0.911362i	$-0.635032\pi$
$47$	4.00000	−	6.92820i	0.583460	−	1.01058i	0.995066	−	0.0992202i	$-0.0316348\pi$
$48$		0			0
$49$		0			0
$50$	−1.00000			−0.141421
$51$		0			0
$52$	1.00000	+	1.73205i	0.138675	+	0.240192i
$53$	−3.00000	−	5.19615i	−0.412082	−	0.713746i	0.583036	−	0.812447i	$-0.301865\pi$
$53$	−3.00000	−	5.19615i	−0.412082	−	0.713746i	−0.995117	+	0.0987002i	$0.968532\pi$
$54$		0			0
$55$	2.00000			0.269680
$56$		0			0
$57$		0			0
$58$	1.00000	−	1.73205i	0.131306	−	0.227429i
$59$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$59$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$60$		0			0
$61$	5.00000	−	8.66025i	0.640184	−	1.10883i	−0.345207	−	0.938527i	$-0.612191\pi$
$61$	5.00000	−	8.66025i	0.640184	−	1.10883i	0.985391	−	0.170305i	$-0.0544754\pi$
$62$	8.00000			1.01600
$63$		0			0
$64$	1.00000			0.125000
$65$	−2.00000	+	3.46410i	−0.248069	+	0.429669i
$66$		0			0
$67$	6.00000	+	10.3923i	0.733017	+	1.26962i	0.955588	+	0.294706i	$0.0952216\pi$
$67$	6.00000	+	10.3923i	0.733017	+	1.26962i	−0.222571	+	0.974916i	$0.571445\pi$
$68$	1.00000	−	1.73205i	0.121268	−	0.210042i
$69$		0			0
$70$		0			0
$71$	16.0000			1.89885			0.949425	−	0.313993i	$-0.101667\pi$
$71$	16.0000			1.89885			0.949425	+	0.313993i	$0.101667\pi$
$72$	1.50000	−	2.59808i	0.176777	−	0.306186i
$73$	−7.00000	−	12.1244i	−0.819288	−	1.41905i	−0.906208	−	0.422833i	$-0.861036\pi$
$73$	−7.00000	−	12.1244i	−0.819288	−	1.41905i	0.0869195	−	0.996215i	$-0.472298\pi$
$74$	1.00000	+	1.73205i	0.116248	+	0.201347i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$79$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$80$	1.00000	+	1.73205i	0.111803	+	0.193649i
$81$	−4.50000	−	7.79423i	−0.500000	−	0.866025i
$82$	5.00000	−	8.66025i	0.552158	−	0.956365i
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$	4.00000			0.433861
$86$	−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	$-0.936344\pi$
$89$	−3.00000	+	5.19615i	−0.317999	+	0.550791i	0.662071	+	0.749441i	$0.269678\pi$
$90$	6.00000			0.632456
$91$		0			0
$92$	−8.00000			−0.834058
$93$		0			0
$94$	4.00000	+	6.92820i	0.412568	+	0.714590i
$95$		0			0
$96$		0			0
$97$	−10.0000			−1.01535			−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000			−1.01535			−0.507673	+	0.861550i	$0.669494\pi$
$98$		0			0
$99$	3.00000			0.301511
$100$	0.500000	−	0.866025i	0.0500000	−	0.0866025i
$101$	9.00000	+	15.5885i	0.895533	+	1.55111i	0.833143	+	0.553058i	$0.186539\pi$
$101$	9.00000	+	15.5885i	0.895533	+	1.55111i	0.0623905	+	0.998052i	$0.480128\pi$
$102$		0			0
$103$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$103$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$104$	−2.00000			−0.196116
$105$		0			0
$106$	6.00000			0.582772
$107$	2.00000	−	3.46410i	0.193347	−	0.334887i	−0.753010	−	0.658009i	$-0.771399\pi$
$107$	2.00000	−	3.46410i	0.193347	−	0.334887i	0.946357	+	0.323122i	$0.104732\pi$
$108$		0			0
$109$	1.00000	+	1.73205i	0.0957826	+	0.165900i	0.909935	−	0.414751i	$-0.136131\pi$
$109$	1.00000	+	1.73205i	0.0957826	+	0.165900i	−0.814152	+	0.580651i	$0.802798\pi$
$110$	−1.00000	+	1.73205i	−0.0953463	+	0.165145i
$111$		0			0
$112$		0			0
$113$	2.00000			0.188144			0.0940721	−	0.995565i	$-0.470012\pi$
$113$	2.00000			0.188144			0.0940721	+	0.995565i	$0.470012\pi$
$114$		0			0
$115$	−8.00000	−	13.8564i	−0.746004	−	1.29212i
$116$	1.00000	+	1.73205i	0.0928477	+	0.160817i
$117$	−3.00000	+	5.19615i	−0.277350	+	0.480384i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.500000	+	0.866025i	−0.0454545	+	0.0787296i
$122$	5.00000	+	8.66025i	0.452679	+	0.784063i
$123$		0			0
$124$	−4.00000	+	6.92820i	−0.359211	+	0.622171i
$125$	12.0000			1.07331
$126$		0			0
$127$	8.00000			0.709885			0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000			0.709885			0.354943	+	0.934888i	$0.384500\pi$
$128$	−0.500000	+	0.866025i	−0.0441942	+	0.0765466i
$129$		0			0
$130$	−2.00000	−	3.46410i	−0.175412	−	0.303822i
$131$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$131$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$132$		0			0
$133$		0			0
$134$	−12.0000			−1.03664
$135$		0			0
$136$	1.00000	+	1.73205i	0.0857493	+	0.148522i
$137$	3.00000	+	5.19615i	0.256307	+	0.443937i	0.965250	−	0.261329i	$-0.0841608\pi$
$137$	3.00000	+	5.19615i	0.256307	+	0.443937i	−0.708942	+	0.705266i	$0.750827\pi$
$138$		0			0
$139$	16.0000			1.35710			0.678551	−	0.734553i	$-0.262608\pi$
$139$	16.0000			1.35710			0.678551	+	0.734553i	$0.262608\pi$
$140$		0			0
$141$		0			0
$142$	−8.00000	+	13.8564i	−0.671345	+	1.16280i
$143$	−1.00000	−	1.73205i	−0.0836242	−	0.144841i
$144$	1.50000	+	2.59808i	0.125000	+	0.216506i
$145$	−2.00000	+	3.46410i	−0.166091	+	0.287678i
$146$	14.0000			1.15865
$147$		0			0
$148$	−2.00000			−0.164399
$149$	−7.00000	+	12.1244i	−0.573462	+	0.993266i	0.422744	+	0.906249i	$0.361067\pi$
$149$	−7.00000	+	12.1244i	−0.573462	+	0.993266i	−0.996207	+	0.0870170i	$0.972267\pi$
$150$		0			0
$151$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$151$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$152$		0			0
$153$	6.00000			0.485071
$154$		0			0
$155$	−16.0000			−1.28515
$156$		0			0
$157$	−7.00000	−	12.1244i	−0.558661	−	0.967629i	−0.997609	−	0.0691164i	$-0.977982\pi$
$157$	−7.00000	−	12.1244i	−0.558661	−	0.967629i	0.438948	−	0.898513i	$-0.355351\pi$
$158$		0			0
$159$		0			0
$160$	−2.00000			−0.158114
$161$		0			0
$162$	9.00000			0.707107
$163$	10.0000	−	17.3205i	0.783260	−	1.35665i	−0.146772	−	0.989170i	$-0.546888\pi$
$163$	10.0000	−	17.3205i	0.783260	−	1.35665i	0.930033	−	0.367477i	$-0.119778\pi$
$164$	5.00000	+	8.66025i	0.390434	+	0.676252i
$165$		0			0
$166$		0			0
$167$	−16.0000			−1.23812			−0.619059	−	0.785345i	$-0.712486\pi$
$167$	−16.0000			−1.23812			−0.619059	+	0.785345i	$0.712486\pi$
$168$		0			0
$169$	−9.00000			−0.692308
$170$	−2.00000	+	3.46410i	−0.153393	+	0.265684i
$171$		0			0
$172$	−2.00000	−	3.46410i	−0.152499	−	0.264135i
$173$	−7.00000	+	12.1244i	−0.532200	+	0.921798i	0.467093	+	0.884208i	$0.345301\pi$
$173$	−7.00000	+	12.1244i	−0.532200	+	0.921798i	−0.999293	+	0.0375896i	$0.988032\pi$
$174$		0			0
$175$		0			0
$176$	−1.00000			−0.0753778
$177$		0			0
$178$	−3.00000	−	5.19615i	−0.224860	−	0.389468i
$179$	−2.00000	−	3.46410i	−0.149487	−	0.258919i	0.781551	−	0.623841i	$-0.214429\pi$
$179$	−2.00000	−	3.46410i	−0.149487	−	0.258919i	−0.931038	+	0.364922i	$0.881096\pi$
$180$	−3.00000	+	5.19615i	−0.223607	+	0.387298i
$181$	22.0000			1.63525			0.817624	−	0.575753i	$-0.195291\pi$
$181$	22.0000			1.63525			0.817624	+	0.575753i	$0.195291\pi$
$182$		0			0
$183$		0			0
$184$	4.00000	−	6.92820i	0.294884	−	0.510754i
$185$	−2.00000	−	3.46410i	−0.147043	−	0.254686i
$186$		0			0
$187$	−1.00000	+	1.73205i	−0.0731272	+	0.126660i
$188$	−8.00000			−0.583460
$189$		0			0
$190$		0			0
$191$	−4.00000	+	6.92820i	−0.289430	+	0.501307i	−0.973674	−	0.227946i	$-0.926799\pi$
$191$	−4.00000	+	6.92820i	−0.289430	+	0.501307i	0.684244	+	0.729253i	$0.260132\pi$
$192$		0			0
$193$	−1.00000	−	1.73205i	−0.0719816	−	0.124676i	0.827788	−	0.561041i	$-0.189599\pi$
$193$	−1.00000	−	1.73205i	−0.0719816	−	0.124676i	−0.899770	+	0.436365i	$0.856266\pi$
$194$	5.00000	−	8.66025i	0.358979	−	0.621770i
$195$		0			0
$196$		0			0
$197$	−18.0000			−1.28245			−0.641223	−	0.767354i	$-0.721573\pi$
$197$	−18.0000			−1.28245			−0.641223	+	0.767354i	$0.721573\pi$
$198$	−1.50000	+	2.59808i	−0.106600	+	0.184637i
$199$	8.00000	+	13.8564i	0.567105	+	0.982255i	0.996850	+	0.0793045i	$0.0252700\pi$
$199$	8.00000	+	13.8564i	0.567105	+	0.982255i	−0.429745	+	0.902950i	$0.641397\pi$
$200$	0.500000	+	0.866025i	0.0353553	+	0.0612372i
$201$		0			0
$202$	−18.0000			−1.26648
$203$		0			0
$204$		0			0
$205$	−10.0000	+	17.3205i	−0.698430	+	1.20972i
$206$		0			0
$207$	−12.0000	−	20.7846i	−0.834058	−	1.44463i
$208$	1.00000	−	1.73205i	0.0693375	−	0.120096i
$209$		0			0
$210$		0			0
$211$	4.00000			0.275371			0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000			0.275371			0.137686	+	0.990476i	$0.456034\pi$
$212$	−3.00000	+	5.19615i	−0.206041	+	0.356873i
$213$		0			0
$214$	2.00000	+	3.46410i	0.136717	+	0.236801i
$215$	4.00000	−	6.92820i	0.272798	−	0.472500i
$216$		0			0
$217$		0			0
$218$	−2.00000			−0.135457
$219$		0			0
$220$	−1.00000	−	1.73205i	−0.0674200	−	0.116775i
$221$	−2.00000	−	3.46410i	−0.134535	−	0.233021i
$222$		0			0
$223$	−16.0000			−1.07144			−0.535720	−	0.844396i	$-0.679960\pi$
$223$	−16.0000			−1.07144			−0.535720	+	0.844396i	$0.679960\pi$
$224$		0			0
$225$	3.00000			0.200000
$226$	−1.00000	+	1.73205i	−0.0665190	+	0.115214i
$227$	8.00000	+	13.8564i	0.530979	+	0.919682i	0.999346	+	0.0361484i	$0.0115089\pi$
$227$	8.00000	+	13.8564i	0.530979	+	0.919682i	−0.468368	+	0.883534i	$0.655158\pi$
$228$		0			0
$229$	−11.0000	+	19.0526i	−0.726900	+	1.25903i	0.231287	+	0.972886i	$0.425707\pi$
$229$	−11.0000	+	19.0526i	−0.726900	+	1.25903i	−0.958187	+	0.286143i	$0.907627\pi$
$230$	16.0000			1.05501
$231$		0			0
$232$	−2.00000			−0.131306
$233$	3.00000	−	5.19615i	0.196537	−	0.340411i	−0.750867	−	0.660454i	$-0.770364\pi$
$233$	3.00000	−	5.19615i	0.196537	−	0.340411i	0.947403	+	0.320043i	$0.103697\pi$
$234$	−3.00000	−	5.19615i	−0.196116	−	0.339683i
$235$	−8.00000	−	13.8564i	−0.521862	−	0.903892i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−8.00000			−0.517477			−0.258738	−	0.965947i	$-0.583307\pi$
$239$	−8.00000			−0.517477			−0.258738	+	0.965947i	$0.583307\pi$
$240$		0			0
$241$	9.00000	+	15.5885i	0.579741	+	1.00414i	0.995509	+	0.0946700i	$0.0301796\pi$
$241$	9.00000	+	15.5885i	0.579741	+	1.00414i	−0.415768	+	0.909471i	$0.636487\pi$
$242$	−0.500000	−	0.866025i	−0.0321412	−	0.0556702i
$243$		0			0
$244$	−10.0000			−0.640184
$245$		0			0
$246$		0			0
$247$		0			0
$248$	−4.00000	−	6.92820i	−0.254000	−	0.439941i
$249$		0			0
$250$	−6.00000	+	10.3923i	−0.379473	+	0.657267i
$251$	−24.0000			−1.51487			−0.757433	−	0.652913i	$-0.773547\pi$
$251$	−24.0000			−1.51487			−0.757433	+	0.652913i	$0.773547\pi$
$252$		0			0
$253$	8.00000			0.502956
$254$	−4.00000	+	6.92820i	−0.250982	+	0.434714i
$255$		0			0
$256$	−0.500000	−	0.866025i	−0.0312500	−	0.0541266i
$257$	−15.0000	+	25.9808i	−0.935674	+	1.62064i	−0.162247	+	0.986750i	$0.551874\pi$
$257$	−15.0000	+	25.9808i	−0.935674	+	1.62064i	−0.773427	+	0.633885i	$0.781459\pi$
$258$		0			0
$259$		0			0
$260$	4.00000			0.248069
$261$	−3.00000	+	5.19615i	−0.185695	+	0.321634i
$262$		0			0
$263$	12.0000	+	20.7846i	0.739952	+	1.28163i	0.952517	+	0.304487i	$0.0984850\pi$
$263$	12.0000	+	20.7846i	0.739952	+	1.28163i	−0.212565	+	0.977147i	$0.568182\pi$
$264$		0			0
$265$	−12.0000			−0.737154
$266$		0			0
$267$		0			0
$268$	6.00000	−	10.3923i	0.366508	−	0.634811i
$269$	9.00000	+	15.5885i	0.548740	+	0.950445i	0.998361	+	0.0572259i	$0.0182255\pi$
$269$	9.00000	+	15.5885i	0.548740	+	0.950445i	−0.449622	+	0.893219i	$0.648441\pi$
$270$		0			0
$271$	8.00000	−	13.8564i	0.485965	−	0.841717i	−0.513905	−	0.857847i	$-0.671801\pi$
$271$	8.00000	−	13.8564i	0.485965	−	0.841717i	0.999870	+	0.0161307i	$0.00513477\pi$
$272$	−2.00000			−0.121268
$273$		0			0
$274$	−6.00000			−0.362473
$275$	−0.500000	+	0.866025i	−0.0301511	+	0.0522233i
$276$		0			0
$277$	−15.0000	−	25.9808i	−0.901263	−	1.56103i	−0.825857	−	0.563880i	$-0.809308\pi$
$277$	−15.0000	−	25.9808i	−0.901263	−	1.56103i	−0.0754058	−	0.997153i	$-0.524025\pi$
$278$	−8.00000	+	13.8564i	−0.479808	+	0.831052i
$279$	−24.0000			−1.43684
$280$		0			0
$281$	−22.0000			−1.31241			−0.656205	−	0.754583i	$-0.727839\pi$
$281$	−22.0000			−1.31241			−0.656205	+	0.754583i	$0.727839\pi$
$282$		0			0
$283$	−12.0000	−	20.7846i	−0.713326	−	1.23552i	−0.963602	−	0.267342i	$-0.913855\pi$
$283$	−12.0000	−	20.7846i	−0.713326	−	1.23552i	0.250276	−	0.968175i	$-0.419479\pi$
$284$	−8.00000	−	13.8564i	−0.474713	−	0.822226i
$285$		0			0
$286$	2.00000			0.118262
$287$		0			0
$288$	−3.00000			−0.176777
$289$	6.50000	−	11.2583i	0.382353	−	0.662255i
$290$	−2.00000	−	3.46410i	−0.117444	−	0.203419i
$291$		0			0
$292$	−7.00000	+	12.1244i	−0.409644	+	0.709524i
$293$	30.0000			1.75262			0.876309	−	0.481749i	$-0.159998\pi$
$293$	30.0000			1.75262			0.876309	+	0.481749i	$0.159998\pi$
$294$		0			0
$295$		0			0
$296$	1.00000	−	1.73205i	0.0581238	−	0.100673i
$297$		0			0
$298$	−7.00000	−	12.1244i	−0.405499	−	0.702345i
$299$	−8.00000	+	13.8564i	−0.462652	+	0.801337i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−10.0000	−	17.3205i	−0.572598	−	0.991769i
$306$	−3.00000	+	5.19615i	−0.171499	+	0.297044i
$307$	16.0000			0.913168			0.456584	−	0.889680i	$-0.349073\pi$
$307$	16.0000			0.913168			0.456584	+	0.889680i	$0.349073\pi$
$308$		0			0
$309$		0			0
$310$	8.00000	−	13.8564i	0.454369	−	0.786991i
$311$	4.00000	+	6.92820i	0.226819	+	0.392862i	0.956864	−	0.290537i	$-0.0938340\pi$
$311$	4.00000	+	6.92820i	0.226819	+	0.392862i	−0.730044	+	0.683400i	$0.760501\pi$
$312$		0			0
$313$	−7.00000	+	12.1244i	−0.395663	+	0.685309i	−0.993186	−	0.116543i	$-0.962819\pi$
$313$	−7.00000	+	12.1244i	−0.395663	+	0.685309i	0.597522	+	0.801852i	$0.296152\pi$
$314$	14.0000			0.790066
$315$		0			0
$316$		0			0
$317$	9.00000	−	15.5885i	0.505490	−	0.875535i	−0.494489	−	0.869184i	$-0.664645\pi$
$317$	9.00000	−	15.5885i	0.505490	−	0.875535i	0.999980	−	0.00635137i	$-0.00202172\pi$
$318$		0			0
$319$	−1.00000	−	1.73205i	−0.0559893	−	0.0969762i
$320$	1.00000	−	1.73205i	0.0559017	−	0.0968246i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−4.50000	+	7.79423i	−0.250000	+	0.433013i
$325$	−1.00000	−	1.73205i	−0.0554700	−	0.0960769i
$326$	10.0000	+	17.3205i	0.553849	+	0.959294i
$327$		0			0
$328$	−10.0000			−0.552158
$329$		0			0
$330$		0			0
$331$	−2.00000	+	3.46410i	−0.109930	+	0.190404i	−0.915742	−	0.401768i	$-0.868396\pi$
$331$	−2.00000	+	3.46410i	−0.109930	+	0.190404i	0.805812	+	0.592172i	$0.201729\pi$
$332$		0			0
$333$	−3.00000	−	5.19615i	−0.164399	−	0.284747i
$334$	8.00000	−	13.8564i	0.437741	−	0.758189i
$335$	24.0000			1.31126
$336$		0			0
$337$	18.0000			0.980522			0.490261	−	0.871576i	$-0.336901\pi$
$337$	18.0000			0.980522			0.490261	+	0.871576i	$0.336901\pi$
$338$	4.50000	−	7.79423i	0.244768	−	0.423950i
$339$		0			0
$340$	−2.00000	−	3.46410i	−0.108465	−	0.187867i
$341$	4.00000	−	6.92820i	0.216612	−	0.375183i
$342$		0			0
$343$		0			0
$344$	4.00000			0.215666
$345$		0			0
$346$	−7.00000	−	12.1244i	−0.376322	−	0.651809i
$347$	14.0000	+	24.2487i	0.751559	+	1.30174i	0.947067	+	0.321037i	$0.104031\pi$
$347$	14.0000	+	24.2487i	0.751559	+	1.30174i	−0.195507	+	0.980702i	$0.562635\pi$
$348$		0			0
$349$	−2.00000			−0.107058			−0.0535288	−	0.998566i	$-0.517047\pi$
$349$	−2.00000			−0.107058			−0.0535288	+	0.998566i	$0.517047\pi$
$350$		0			0
$351$		0			0
$352$	0.500000	−	0.866025i	0.0266501	−	0.0461593i
$353$	9.00000	+	15.5885i	0.479022	+	0.829690i	0.999711	−	0.0240566i	$-0.00765819\pi$
$353$	9.00000	+	15.5885i	0.479022	+	0.829690i	−0.520689	+	0.853746i	$0.674325\pi$
$354$		0			0
$355$	16.0000	−	27.7128i	0.849192	−	1.47084i
$356$	6.00000			0.317999
$357$		0			0
$358$	4.00000			0.211407
$359$	−16.0000	+	27.7128i	−0.844448	+	1.46263i	0.0416523	+	0.999132i	$0.486738\pi$
$359$	−16.0000	+	27.7128i	−0.844448	+	1.46263i	−0.886100	+	0.463494i	$0.846596\pi$
$360$	−3.00000	−	5.19615i	−0.158114	−	0.273861i
$361$	9.50000	+	16.4545i	0.500000	+	0.866025i
$362$	−11.0000	+	19.0526i	−0.578147	+	1.00138i
$363$		0			0
$364$		0			0
$365$	−28.0000			−1.46559
$366$		0			0
$367$	8.00000	+	13.8564i	0.417597	+	0.723299i	0.995697	−	0.0926670i	$-0.0295392\pi$
$367$	8.00000	+	13.8564i	0.417597	+	0.723299i	−0.578101	+	0.815966i	$0.696206\pi$
$368$	4.00000	+	6.92820i	0.208514	+	0.361158i
$369$	−15.0000	+	25.9808i	−0.780869	+	1.35250i
$370$	4.00000			0.207950
$371$		0			0
$372$		0			0
$373$	−7.00000	+	12.1244i	−0.362446	+	0.627775i	−0.988363	−	0.152115i	$-0.951392\pi$
$373$	−7.00000	+	12.1244i	−0.362446	+	0.627775i	0.625917	+	0.779890i	$0.284725\pi$
$374$	−1.00000	−	1.73205i	−0.0517088	−	0.0895622i
$375$		0			0
$376$	4.00000	−	6.92820i	0.206284	−	0.357295i
$377$	4.00000			0.206010
$378$		0			0
$379$	4.00000			0.205466			0.102733	−	0.994709i	$-0.467241\pi$
$379$	4.00000			0.205466			0.102733	+	0.994709i	$0.467241\pi$
$380$		0			0
$381$		0			0
$382$	−4.00000	−	6.92820i	−0.204658	−	0.354478i
$383$	4.00000	−	6.92820i	0.204390	−	0.354015i	−0.745548	−	0.666452i	$-0.767812\pi$
$383$	4.00000	−	6.92820i	0.204390	−	0.354015i	0.949938	+	0.312437i	$0.101145\pi$
$384$		0			0
$385$		0			0
$386$	2.00000			0.101797
$387$	6.00000	−	10.3923i	0.304997	−	0.528271i
$388$	5.00000	+	8.66025i	0.253837	+	0.439658i
$389$	9.00000	+	15.5885i	0.456318	+	0.790366i	0.998763	−	0.0497253i	$-0.0158346\pi$
$389$	9.00000	+	15.5885i	0.456318	+	0.790366i	−0.542445	+	0.840091i	$0.682501\pi$
$390$		0			0
$391$	16.0000			0.809155
$392$		0			0
$393$		0			0
$394$	9.00000	−	15.5885i	0.453413	−	0.785335i
$395$		0			0
$396$	−1.50000	−	2.59808i	−0.0753778	−	0.130558i
$397$	9.00000	−	15.5885i	0.451697	−	0.782362i	−0.546795	−	0.837267i	$-0.684152\pi$
$397$	9.00000	−	15.5885i	0.451697	−	0.782362i	0.998492	+	0.0549046i	$0.0174855\pi$
$398$	−16.0000			−0.802008
$399$		0			0
$400$	−1.00000			−0.0500000
$401$	−9.00000	+	15.5885i	−0.449439	+	0.778450i	−0.998350	−	0.0574304i	$-0.981709\pi$
$401$	−9.00000	+	15.5885i	−0.449439	+	0.778450i	0.548911	+	0.835881i	$0.315043\pi$
$402$		0			0
$403$	8.00000	+	13.8564i	0.398508	+	0.690237i
$404$	9.00000	−	15.5885i	0.447767	−	0.775555i
$405$	−18.0000			−0.894427
$406$		0			0
$407$	2.00000			0.0991363
$408$		0			0
$409$	5.00000	+	8.66025i	0.247234	+	0.428222i	0.962757	−	0.270367i	$-0.0871450\pi$
$409$	5.00000	+	8.66025i	0.247234	+	0.428222i	−0.715523	+	0.698589i	$0.753812\pi$
$410$	−10.0000	−	17.3205i	−0.493865	−	0.855399i
$411$		0			0
$412$		0			0
$413$		0			0
$414$	24.0000			1.17954
$415$		0			0
$416$	1.00000	+	1.73205i	0.0490290	+	0.0849208i
$417$		0			0
$418$		0			0
$419$	−16.0000			−0.781651			−0.390826	−	0.920465i	$-0.627810\pi$
$419$	−16.0000			−0.781651			−0.390826	+	0.920465i	$0.627810\pi$
$420$		0			0
$421$	22.0000			1.07221			0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000			1.07221			0.536107	+	0.844150i	$0.319894\pi$
$422$	−2.00000	+	3.46410i	−0.0973585	+	0.168630i
$423$	−12.0000	−	20.7846i	−0.583460	−	1.01058i
$424$	−3.00000	−	5.19615i	−0.145693	−	0.252347i
$425$	−1.00000	+	1.73205i	−0.0485071	+	0.0840168i
$426$		0			0
$427$		0			0
$428$	−4.00000			−0.193347
$429$		0			0
$430$	4.00000	+	6.92820i	0.192897	+	0.334108i
$431$	4.00000	+	6.92820i	0.192673	+	0.333720i	0.946135	−	0.323772i	$-0.104951\pi$
$431$	4.00000	+	6.92820i	0.192673	+	0.333720i	−0.753462	+	0.657491i	$0.771618\pi$
$432$		0			0
$433$	−26.0000			−1.24948			−0.624740	−	0.780833i	$-0.714795\pi$
$433$	−26.0000			−1.24948			−0.624740	+	0.780833i	$0.714795\pi$
$434$		0			0
$435$		0			0
$436$	1.00000	−	1.73205i	0.0478913	−	0.0829502i
$437$		0			0
$438$		0			0
$439$	20.0000	−	34.6410i	0.954548	−	1.65333i	0.219149	−	0.975691i	$-0.429672\pi$
$439$	20.0000	−	34.6410i	0.954548	−	1.65333i	0.735399	−	0.677634i	$-0.236995\pi$
$440$	2.00000			0.0953463
$441$		0			0
$442$	4.00000			0.190261
$443$	−6.00000	+	10.3923i	−0.285069	+	0.493753i	−0.972626	−	0.232377i	$-0.925350\pi$
$443$	−6.00000	+	10.3923i	−0.285069	+	0.493753i	0.687557	+	0.726130i	$0.258683\pi$
$444$		0			0
$445$	6.00000	+	10.3923i	0.284427	+	0.492642i
$446$	8.00000	−	13.8564i	0.378811	−	0.656120i
$447$		0			0
$448$		0			0
$449$	−30.0000			−1.41579			−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000			−1.41579			−0.707894	+	0.706319i	$0.750354\pi$
$450$	−1.50000	+	2.59808i	−0.0707107	+	0.122474i
$451$	−5.00000	−	8.66025i	−0.235441	−	0.407795i
$452$	−1.00000	−	1.73205i	−0.0470360	−	0.0814688i
$453$		0			0
$454$	−16.0000			−0.750917
$455$		0			0
$456$		0			0
$457$	11.0000	−	19.0526i	0.514558	−	0.891241i	−0.485299	−	0.874348i	$-0.661289\pi$
$457$	11.0000	−	19.0526i	0.514558	−	0.891241i	0.999857	−	0.0168929i	$-0.00537742\pi$
$458$	−11.0000	−	19.0526i	−0.513996	−	0.890268i
$459$		0			0
$460$	−8.00000	+	13.8564i	−0.373002	+	0.646058i
$461$	30.0000			1.39724			0.698620	−	0.715493i	$-0.253798\pi$
$461$	30.0000			1.39724			0.698620	+	0.715493i	$0.253798\pi$
$462$		0			0
$463$	−8.00000			−0.371792			−0.185896	−	0.982569i	$-0.559519\pi$
$463$	−8.00000			−0.371792			−0.185896	+	0.982569i	$0.559519\pi$
$464$	1.00000	−	1.73205i	0.0464238	−	0.0804084i
$465$		0			0
$466$	3.00000	+	5.19615i	0.138972	+	0.240707i
$467$	12.0000	−	20.7846i	0.555294	−	0.961797i	−0.442587	−	0.896726i	$-0.645939\pi$
$467$	12.0000	−	20.7846i	0.555294	−	0.961797i	0.997881	−	0.0650714i	$-0.0207275\pi$
$468$	6.00000			0.277350
$469$		0			0
$470$	16.0000			0.738025
$471$		0			0
$472$		0			0
$473$	2.00000	+	3.46410i	0.0919601	+	0.159280i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−18.0000			−0.824163
$478$	4.00000	−	6.92820i	0.182956	−	0.316889i
$479$	4.00000	+	6.92820i	0.182765	+	0.316558i	0.942821	−	0.333300i	$-0.108162\pi$
$479$	4.00000	+	6.92820i	0.182765	+	0.316558i	−0.760056	+	0.649857i	$0.774829\pi$
$480$		0			0
$481$	−2.00000	+	3.46410i	−0.0911922	+	0.157949i
$482$	−18.0000			−0.819878
$483$		0			0
$484$	1.00000			0.0454545
$485$	−10.0000	+	17.3205i	−0.454077	+	0.786484i
$486$		0			0
$487$	20.0000	+	34.6410i	0.906287	+	1.56973i	0.819181	+	0.573535i	$0.194428\pi$
$487$	20.0000	+	34.6410i	0.906287	+	1.56973i	0.0871056	+	0.996199i	$0.472238\pi$
$488$	5.00000	−	8.66025i	0.226339	−	0.392031i
$489$		0			0
$490$		0			0
$491$	12.0000			0.541552			0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000			0.541552			0.270776	+	0.962642i	$0.412720\pi$
$492$		0			0
$493$	−2.00000	−	3.46410i	−0.0900755	−	0.156015i
$494$		0			0
$495$	3.00000	−	5.19615i	0.134840	−	0.233550i
$496$	8.00000			0.359211
$497$		0			0
$498$		0			0
$499$	14.0000	−	24.2487i	0.626726	−	1.08552i	−0.361478	−	0.932381i	$-0.617728\pi$
$499$	14.0000	−	24.2487i	0.626726	−	1.08552i	0.988204	−	0.153141i	$-0.0489388\pi$
$500$	−6.00000	−	10.3923i	−0.268328	−	0.464758i
$501$		0			0
$502$	12.0000	−	20.7846i	0.535586	−	0.927663i
$503$	24.0000			1.07011			0.535054	−	0.844818i	$-0.320291\pi$
$503$	24.0000			1.07011			0.535054	+	0.844818i	$0.320291\pi$
$504$		0			0
$505$	36.0000			1.60198
$506$	−4.00000	+	6.92820i	−0.177822	+	0.307996i
$507$		0			0
$508$	−4.00000	−	6.92820i	−0.177471	−	0.307389i
$509$	1.00000	−	1.73205i	0.0443242	−	0.0767718i	−0.843012	−	0.537895i	$-0.819220\pi$
$509$	1.00000	−	1.73205i	0.0443242	−	0.0767718i	0.887336	+	0.461123i	$0.152553\pi$
$510$		0			0
$511$		0			0
$512$	1.00000			0.0441942
$513$		0			0
$514$	−15.0000	−	25.9808i	−0.661622	−	1.14596i
$515$		0			0
$516$		0			0
$517$	8.00000			0.351840
$518$		0			0
$519$		0			0
$520$	−2.00000	+	3.46410i	−0.0877058	+	0.151911i
$521$	17.0000	+	29.4449i	0.744784	+	1.29000i	0.950296	+	0.311348i	$0.100781\pi$
$521$	17.0000	+	29.4449i	0.744784	+	1.29000i	−0.205512	+	0.978655i	$0.565886\pi$
$522$	−3.00000	−	5.19615i	−0.131306	−	0.227429i
$523$	4.00000	−	6.92820i	0.174908	−	0.302949i	−0.765222	−	0.643767i	$-0.777371\pi$
$523$	4.00000	−	6.92820i	0.174908	−	0.302949i	0.940129	+	0.340818i	$0.110704\pi$
$524$		0			0
$525$		0			0
$526$	−24.0000			−1.04645
$527$	8.00000	−	13.8564i	0.348485	−	0.603595i
$528$		0			0
$529$	−20.5000	−	35.5070i	−0.891304	−	1.54378i
$530$	6.00000	−	10.3923i	0.260623	−	0.451413i
$531$		0			0
$532$		0			0
$533$	20.0000			0.866296
$534$		0			0
$535$	−4.00000	−	6.92820i	−0.172935	−	0.299532i
$536$	6.00000	+	10.3923i	0.259161	+	0.448879i
$537$		0			0
$538$	−18.0000			−0.776035
$539$		0			0
$540$		0			0
$541$	−19.0000	+	32.9090i	−0.816874	+	1.41487i	0.0911008	+	0.995842i	$0.470961\pi$
$541$	−19.0000	+	32.9090i	−0.816874	+	1.41487i	−0.907975	+	0.419025i	$0.862372\pi$
$542$	8.00000	+	13.8564i	0.343629	+	0.595184i
$543$		0			0
$544$	1.00000	−	1.73205i	0.0428746	−	0.0742611i
$545$	4.00000			0.171341
$546$		0			0
$547$	−36.0000			−1.53925			−0.769624	−	0.638497i	$-0.779557\pi$
$547$	−36.0000			−1.53925			−0.769624	+	0.638497i	$0.779557\pi$
$548$	3.00000	−	5.19615i	0.128154	−	0.221969i
$549$	−15.0000	−	25.9808i	−0.640184	−	1.10883i
$550$	−0.500000	−	0.866025i	−0.0213201	−	0.0369274i
$551$		0			0
$552$		0			0
$553$		0			0
$554$	30.0000			1.27458
$555$		0			0
$556$	−8.00000	−	13.8564i	−0.339276	−	0.587643i
$557$	5.00000	+	8.66025i	0.211857	+	0.366947i	0.952296	−	0.305177i	$-0.0987156\pi$
$557$	5.00000	+	8.66025i	0.211857	+	0.366947i	−0.740439	+	0.672124i	$0.765382\pi$
$558$	12.0000	−	20.7846i	0.508001	−	0.879883i
$559$	−8.00000			−0.338364
$560$		0			0
$561$		0			0
$562$	11.0000	−	19.0526i	0.464007	−	0.803684i
$563$	4.00000	+	6.92820i	0.168580	+	0.291989i	0.937921	−	0.346850i	$-0.112749\pi$
$563$	4.00000	+	6.92820i	0.168580	+	0.291989i	−0.769341	+	0.638838i	$0.779415\pi$
$564$		0			0
$565$	2.00000	−	3.46410i	0.0841406	−	0.145736i
$566$	24.0000			1.00880
$567$		0			0
$568$	16.0000			0.671345
$569$	3.00000	−	5.19615i	0.125767	−	0.217834i	−0.796266	−	0.604947i	$-0.793194\pi$
$569$	3.00000	−	5.19615i	0.125767	−	0.217834i	0.922032	+	0.387113i	$0.126528\pi$
$570$		0			0
$571$	22.0000	+	38.1051i	0.920671	+	1.59465i	0.798379	+	0.602155i	$0.205691\pi$
$571$	22.0000	+	38.1051i	0.920671	+	1.59465i	0.122292	+	0.992494i	$0.460975\pi$
$572$	−1.00000	+	1.73205i	−0.0418121	+	0.0724207i
$573$		0			0
$574$		0			0
$575$	8.00000			0.333623
$576$	1.50000	−	2.59808i	0.0625000	−	0.108253i
$577$	1.00000	+	1.73205i	0.0416305	+	0.0721062i	0.886090	−	0.463513i	$-0.153411\pi$
$577$	1.00000	+	1.73205i	0.0416305	+	0.0721062i	−0.844459	+	0.535620i	$0.820078\pi$
$578$	6.50000	+	11.2583i	0.270364	+	0.468285i
$579$		0			0
$580$	4.00000			0.166091
$581$		0			0
$582$		0			0
$583$	3.00000	−	5.19615i	0.124247	−	0.215203i
$584$	−7.00000	−	12.1244i	−0.289662	−	0.501709i
$585$	6.00000	+	10.3923i	0.248069	+	0.429669i
$586$	−15.0000	+	25.9808i	−0.619644	+	1.07326i
$587$	−24.0000			−0.990586			−0.495293	−	0.868726i	$-0.664939\pi$
$587$	−24.0000			−0.990586			−0.495293	+	0.868726i	$0.664939\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	1.00000	+	1.73205i	0.0410997	+	0.0711868i
$593$	21.0000	−	36.3731i	0.862367	−	1.49366i	−0.00727173	−	0.999974i	$-0.502315\pi$
$593$	21.0000	−	36.3731i	0.862367	−	1.49366i	0.869638	−	0.493689i	$-0.164352\pi$
$594$		0			0
$595$		0			0
$596$	14.0000			0.573462
$597$		0			0
$598$	−8.00000	−	13.8564i	−0.327144	−	0.566631i
$599$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$599$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$600$		0			0
$601$	46.0000			1.87638			0.938190	−	0.346122i	$-0.112502\pi$
$601$	46.0000			1.87638			0.938190	+	0.346122i	$0.112502\pi$
$602$		0			0
$603$	36.0000			1.46603
$604$		0			0
$605$	1.00000	+	1.73205i	0.0406558	+	0.0704179i
$606$		0			0
$607$	−8.00000	+	13.8564i	−0.324710	+	0.562414i	−0.981454	−	0.191700i	$-0.938600\pi$
$607$	−8.00000	+	13.8564i	−0.324710	+	0.562414i	0.656744	+	0.754114i	$0.271933\pi$
$608$		0			0
$609$		0			0
$610$	20.0000			0.809776
$611$	−8.00000	+	13.8564i	−0.323645	+	0.560570i
$612$	−3.00000	−	5.19615i	−0.121268	−	0.210042i
$613$	−11.0000	−	19.0526i	−0.444286	−	0.769526i	0.553716	−	0.832705i	$-0.313209\pi$
$613$	−11.0000	−	19.0526i	−0.444286	−	0.769526i	−0.998002	+	0.0631797i	$0.979876\pi$
$614$	−8.00000	+	13.8564i	−0.322854	+	0.559199i
$615$		0			0
$616$		0			0
$617$	−6.00000			−0.241551			−0.120775	−	0.992680i	$-0.538538\pi$
$617$	−6.00000			−0.241551			−0.120775	+	0.992680i	$0.538538\pi$
$618$		0			0
$619$	−8.00000	−	13.8564i	−0.321547	−	0.556936i	0.659260	−	0.751915i	$-0.270870\pi$
$619$	−8.00000	−	13.8564i	−0.321547	−	0.556936i	−0.980807	+	0.194979i	$0.937536\pi$
$620$	8.00000	+	13.8564i	0.321288	+	0.556487i
$621$		0			0
$622$	−8.00000			−0.320771
$623$		0			0
$624$		0			0
$625$	9.50000	−	16.4545i	0.380000	−	0.658179i
$626$	−7.00000	−	12.1244i	−0.279776	−	0.484587i
$627$		0			0
$628$	−7.00000	+	12.1244i	−0.279330	+	0.483814i
$629$	4.00000			0.159490
$630$		0			0
$631$		0			0			−	1.00000i	$-0.5\pi$
$631$		0			0				1.00000i	$0.5\pi$
$632$		0			0
$633$		0			0
$634$	9.00000	+	15.5885i	0.357436	+	0.619097i
$635$	8.00000	−	13.8564i	0.317470	−	0.549875i
$636$		0			0
$637$		0			0
$638$	2.00000			0.0791808
$639$	24.0000	−	41.5692i	0.949425	−	1.64445i
$640$	1.00000	+	1.73205i	0.0395285	+	0.0684653i
$641$	15.0000	+	25.9808i	0.592464	+	1.02618i	0.993899	+	0.110291i	$0.0351782\pi$
$641$	15.0000	+	25.9808i	0.592464	+	1.02618i	−0.401435	+	0.915888i	$0.631488\pi$
$642$		0			0
$643$	32.0000			1.26196			0.630978	−	0.775800i	$-0.282654\pi$
$643$	32.0000			1.26196			0.630978	+	0.775800i	$0.282654\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$647$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$648$	−4.50000	−	7.79423i	−0.176777	−	0.306186i
$649$		0			0
$650$	2.00000			0.0784465
$651$		0			0
$652$	−20.0000			−0.783260
$653$	13.0000	−	22.5167i	0.508729	−	0.881145i	−0.491220	−	0.871036i	$-0.663449\pi$
$653$	13.0000	−	22.5167i	0.508729	−	0.881145i	0.999949	−	0.0101092i	$-0.00321793\pi$
$654$		0			0
$655$		0			0
$656$	5.00000	−	8.66025i	0.195217	−	0.338126i
$657$	−42.0000			−1.63858
$658$		0			0
$659$	28.0000			1.09073			0.545363	−	0.838200i	$-0.316392\pi$
$659$	28.0000			1.09073			0.545363	+	0.838200i	$0.316392\pi$
$660$		0			0
$661$	1.00000	+	1.73205i	0.0388955	+	0.0673690i	0.884818	−	0.465937i	$-0.154283\pi$
$661$	1.00000	+	1.73205i	0.0388955	+	0.0673690i	−0.845922	+	0.533306i	$0.820949\pi$
$662$	−2.00000	−	3.46410i	−0.0777322	−	0.134636i
$663$		0			0
$664$		0			0
$665$		0			0
$666$	6.00000			0.232495
$667$	−8.00000	+	13.8564i	−0.309761	+	0.536522i
$668$	8.00000	+	13.8564i	0.309529	+	0.536120i
$669$		0			0
$670$	−12.0000	+	20.7846i	−0.463600	+	0.802980i
$671$	10.0000			0.386046
$672$		0			0
$673$	−14.0000			−0.539660			−0.269830	−	0.962908i	$-0.586968\pi$
$673$	−14.0000			−0.539660			−0.269830	+	0.962908i	$0.586968\pi$
$674$	−9.00000	+	15.5885i	−0.346667	+	0.600445i
$675$		0			0
$676$	4.50000	+	7.79423i	0.173077	+	0.299778i
$677$	13.0000	−	22.5167i	0.499631	−	0.865386i	−0.500369	−	0.865812i	$-0.666802\pi$
$677$	13.0000	−	22.5167i	0.499631	−	0.865386i	1.00000		0.000426509i	$0.000135762\pi$
$678$		0			0
$679$		0			0
$680$	4.00000			0.153393
$681$		0			0
$682$	4.00000	+	6.92820i	0.153168	+	0.265295i
$683$	−6.00000	−	10.3923i	−0.229584	−	0.397650i	0.728101	−	0.685470i	$-0.240403\pi$
$683$	−6.00000	−	10.3923i	−0.229584	−	0.397650i	−0.957685	+	0.287819i	$0.907070\pi$
$684$		0			0
$685$	12.0000			0.458496
$686$		0			0
$687$		0			0
$688$	−2.00000	+	3.46410i	−0.0762493	+	0.132068i
$689$	6.00000	+	10.3923i	0.228582	+	0.395915i
$690$		0			0
$691$	12.0000	−	20.7846i	0.456502	−	0.790684i	−0.542272	−	0.840203i	$-0.682436\pi$
$691$	12.0000	−	20.7846i	0.456502	−	0.790684i	0.998773	+	0.0495194i	$0.0157690\pi$
$692$	14.0000			0.532200
$693$		0			0
$694$	−28.0000			−1.06287
$695$	16.0000	−	27.7128i	0.606915	−	1.05121i
$696$		0			0
$697$	−10.0000	−	17.3205i	−0.378777	−	0.656061i
$698$	1.00000	−	1.73205i	0.0378506	−	0.0655591i
$699$		0			0
$700$		0			0
$701$	30.0000			1.13308			0.566542	−	0.824033i	$-0.308281\pi$
$701$	30.0000			1.13308			0.566542	+	0.824033i	$0.308281\pi$
$702$		0			0
$703$		0			0
$704$	0.500000	+	0.866025i	0.0188445	+	0.0326396i
$705$		0			0
$706$	−18.0000			−0.677439
$707$		0			0
$708$		0			0
$709$	−15.0000	+	25.9808i	−0.563337	+	0.975728i	0.433865	+	0.900978i	$0.357149\pi$
$709$	−15.0000	+	25.9808i	−0.563337	+	0.975728i	−0.997202	+	0.0747503i	$0.976184\pi$
$710$	16.0000	+	27.7128i	0.600469	+	1.04004i
$711$		0			0
$712$	−3.00000	+	5.19615i	−0.112430	+	0.194734i
$713$	−64.0000			−2.39682
$714$		0			0
$715$	−4.00000			−0.149592
$716$	−2.00000	+	3.46410i	−0.0747435	+	0.129460i
$717$		0			0
$718$	−16.0000	−	27.7128i	−0.597115	−	1.03423i
$719$	−4.00000	+	6.92820i	−0.149175	+	0.258378i	−0.930923	−	0.365216i	$-0.880995\pi$
$719$	−4.00000	+	6.92820i	−0.149175	+	0.258378i	0.781748	+	0.623595i	$0.214328\pi$
$720$	6.00000			0.223607
$721$		0			0
$722$	−19.0000			−0.707107
$723$		0			0
$724$	−11.0000	−	19.0526i	−0.408812	−	0.708083i
$725$	−1.00000	−	1.73205i	−0.0371391	−	0.0643268i
$726$		0			0
$727$	−32.0000			−1.18681			−0.593407	−	0.804902i	$-0.702218\pi$
$727$	−32.0000			−1.18681			−0.593407	+	0.804902i	$0.702218\pi$
$728$		0			0
$729$	−27.0000			−1.00000
$730$	14.0000	−	24.2487i	0.518163	−	0.897485i
$731$	4.00000	+	6.92820i	0.147945	+	0.256249i
$732$		0			0
$733$	−11.0000	+	19.0526i	−0.406294	+	0.703722i	−0.994471	−	0.105010i	$-0.966513\pi$
$733$	−11.0000	+	19.0526i	−0.406294	+	0.703722i	0.588177	+	0.808732i	$0.299846\pi$
$734$	−16.0000			−0.590571
$735$		0			0
$736$	−8.00000			−0.294884
$737$	−6.00000	+	10.3923i	−0.221013	+	0.382805i
$738$	−15.0000	−	25.9808i	−0.552158	−	0.956365i
$739$	−6.00000	−	10.3923i	−0.220714	−	0.382287i	0.734311	−	0.678813i	$-0.237505\pi$
$739$	−6.00000	−	10.3923i	−0.220714	−	0.382287i	−0.955025	+	0.296526i	$0.904172\pi$
$740$	−2.00000	+	3.46410i	−0.0735215	+	0.127343i
$741$		0			0
$742$		0			0
$743$		0			0			−	1.00000i	$-0.5\pi$
$743$		0			0				1.00000i	$0.5\pi$
$744$		0			0
$745$	14.0000	+	24.2487i	0.512920	+	0.888404i
$746$	−7.00000	−	12.1244i	−0.256288	−	0.443904i
$747$		0			0
$748$	2.00000			0.0731272
$749$		0			0
$750$		0			0
$751$	16.0000	−	27.7128i	0.583848	−	1.01125i	−0.411170	−	0.911559i	$-0.634880\pi$
$751$	16.0000	−	27.7128i	0.583848	−	1.01125i	0.995018	−	0.0996961i	$-0.0317870\pi$
$752$	4.00000	+	6.92820i	0.145865	+	0.252646i
$753$		0			0
$754$	−2.00000	+	3.46410i	−0.0728357	+	0.126155i
$755$		0			0
$756$		0			0
$757$	14.0000			0.508839			0.254419	−	0.967094i	$-0.418116\pi$
$757$	14.0000			0.508839			0.254419	+	0.967094i	$0.418116\pi$
$758$	−2.00000	+	3.46410i	−0.0726433	+	0.125822i
$759$		0			0
$760$		0			0
$761$	−3.00000	+	5.19615i	−0.108750	+	0.188360i	−0.915264	−	0.402854i	$-0.868018\pi$
$761$	−3.00000	+	5.19615i	−0.108750	+	0.188360i	0.806514	+	0.591215i	$0.201351\pi$
$762$		0			0
$763$		0			0
$764$	8.00000			0.289430
$765$	6.00000	−	10.3923i	0.216930	−	0.375735i
$766$	4.00000	+	6.92820i	0.144526	+	0.250326i
$767$		0			0
$768$		0			0
$769$	−34.0000			−1.22607			−0.613036	−	0.790055i	$-0.710052\pi$
$769$	−34.0000			−1.22607			−0.613036	+	0.790055i	$0.710052\pi$
$770$		0			0
$771$		0			0
$772$	−1.00000	+	1.73205i	−0.0359908	+	0.0623379i
$773$	9.00000	+	15.5885i	0.323708	+	0.560678i	0.981250	−	0.192740i	$-0.0617373\pi$
$773$	9.00000	+	15.5885i	0.323708	+	0.560678i	−0.657542	+	0.753418i	$0.728404\pi$
$774$	6.00000	+	10.3923i	0.215666	+	0.373544i
$775$	4.00000	−	6.92820i	0.143684	−	0.248868i
$776$	−10.0000			−0.358979
$777$		0			0
$778$	−18.0000			−0.645331
$779$		0			0
$780$		0			0
$781$	8.00000	+	13.8564i	0.286263	+	0.495821i
$782$	−8.00000	+	13.8564i	−0.286079	+	0.495504i
$783$		0			0
$784$		0			0
$785$	−28.0000			−0.999363
$786$		0			0
$787$	16.0000	+	27.7128i	0.570338	+	0.987855i	0.996531	+	0.0832226i	$0.0265213\pi$
$787$	16.0000	+	27.7128i	0.570338	+	0.987855i	−0.426193	+	0.904632i	$0.640145\pi$
$788$	9.00000	+	15.5885i	0.320612	+	0.555316i
$789$		0			0
$790$		0			0
$791$		0			0
$792$	3.00000			0.106600
$793$	−10.0000	+	17.3205i	−0.355110	+

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:	\( N \)	\(=\)	\( 1078 = 2 \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1078.e (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 + 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(1.00000 - 1.73205i) q^{5} +1.00000 q^{8} +(1.50000 - 2.59808i) q^{9} +O(q^{10})\)	\(q+(-0.500000 + 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(1.00000 - 1.73205i) q^{5} +1.00000 q^{8} +(1.50000 - 2.59808i) q^{9} +(1.00000 + 1.73205i) q^{10} +(0.500000 + 0.866025i) q^{11} -2.00000 q^{13} +(-0.500000 + 0.866025i) q^{16} +(1.00000 + 1.73205i) q^{17} +(1.50000 + 2.59808i) q^{18} -2.00000 q^{20} -1.00000 q^{22} +(4.00000 - 6.92820i) q^{23} +(0.500000 + 0.866025i) q^{25} +(1.00000 - 1.73205i) q^{26} -2.00000 q^{29} +(-4.00000 - 6.92820i) q^{31} +(-0.500000 - 0.866025i) q^{32} -2.00000 q^{34} -3.00000 q^{36} +(1.00000 - 1.73205i) q^{37} +(1.00000 - 1.73205i) q^{40} -10.0000 q^{41} +4.00000 q^{43} +(0.500000 - 0.866025i) q^{44} +(-3.00000 - 5.19615i) q^{45} +(4.00000 + 6.92820i) q^{46} +(4.00000 - 6.92820i) q^{47} -1.00000 q^{50} +(1.00000 + 1.73205i) q^{52} +(-3.00000 - 5.19615i) q^{53} +2.00000 q^{55} +(1.00000 - 1.73205i) q^{58} +(5.00000 - 8.66025i) q^{61} +8.00000 q^{62} +1.00000 q^{64} +(-2.00000 + 3.46410i) q^{65} +(6.00000 + 10.3923i) q^{67} +(1.00000 - 1.73205i) q^{68} +16.0000 q^{71} +(1.50000 - 2.59808i) q^{72} +(-7.00000 - 12.1244i) q^{73} +(1.00000 + 1.73205i) q^{74} +(1.00000 + 1.73205i) q^{80} +(-4.50000 - 7.79423i) q^{81} +(5.00000 - 8.66025i) q^{82} +4.00000 q^{85} +(-2.00000 + 3.46410i) q^{86} +(0.500000 + 0.866025i) q^{88} +(-3.00000 + 5.19615i) q^{89} +6.00000 q^{90} -8.00000 q^{92} +(4.00000 + 6.92820i) q^{94} -10.0000 q^{97} +3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} - q^{4} + 2 q^{5} + 2 q^{8} + 3 q^{9}+O(q^{10}) \) 2 * q - q^2 - q^4 + 2 * q^5 + 2 * q^8 + 3 * q^9	\( 2 q - q^{2} - q^{4} + 2 q^{5} + 2 q^{8} + 3 q^{9} + 2 q^{10} + q^{11} - 4 q^{13} - q^{16} + 2 q^{17} + 3 q^{18} - 4 q^{20} - 2 q^{22} + 8 q^{23} + q^{25} + 2 q^{26} - 4 q^{29} - 8 q^{31} - q^{32} - 4 q^{34} - 6 q^{36} + 2 q^{37} + 2 q^{40} - 20 q^{41} + 8 q^{43} + q^{44} - 6 q^{45} + 8 q^{46} + 8 q^{47} - 2 q^{50} + 2 q^{52} - 6 q^{53} + 4 q^{55} + 2 q^{58} + 10 q^{61} + 16 q^{62} + 2 q^{64} - 4 q^{65} + 12 q^{67} + 2 q^{68} + 32 q^{71} + 3 q^{72} - 14 q^{73} + 2 q^{74} + 2 q^{80} - 9 q^{81} + 10 q^{82} + 8 q^{85} - 4 q^{86} + q^{88} - 6 q^{89} + 12 q^{90} - 16 q^{92} + 8 q^{94} - 20 q^{97} + 6 q^{99}+O(q^{100}) \) 2 * q - q^2 - q^4 + 2 * q^5 + 2 * q^8 + 3 * q^9 + 2 * q^10 + q^11 - 4 * q^13 - q^16 + 2 * q^17 + 3 * q^18 - 4 * q^20 - 2 * q^22 + 8 * q^23 + q^25 + 2 * q^26 - 4 * q^29 - 8 * q^31 - q^32 - 4 * q^34 - 6 * q^36 + 2 * q^37 + 2 * q^40 - 20 * q^41 + 8 * q^43 + q^44 - 6 * q^45 + 8 * q^46 + 8 * q^47 - 2 * q^50 + 2 * q^52 - 6 * q^53 + 4 * q^55 + 2 * q^58 + 10 * q^61 + 16 * q^62 + 2 * q^64 - 4 * q^65 + 12 * q^67 + 2 * q^68 + 32 * q^71 + 3 * q^72 - 14 * q^73 + 2 * q^74 + 2 * q^80 - 9 * q^81 + 10 * q^82 + 8 * q^85 - 4 * q^86 + q^88 - 6 * q^89 + 12 * q^90 - 16 * q^92 + 8 * q^94 - 20 * q^97 + 6 * q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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