LMFDB - Embedded newform 2646.2.h.c.667.1

Newspace parameters

Level:	$ N $	$=$	$ 2646 = 2 \cdot 3^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2646.h (of order $3$, degree $2$, not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$21.1284163748$
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 126)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			667.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	2646.667
Dual form			2646.2.h.c.361.1

$q$-expansion

$f(q)$	$=$	$q+(-0.500000 - 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +2.00000 q^{5} +1.00000 q^{8} +O(q^{10})$	\(q+(-0.500000 - 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +2.00000 q^{5} +1.00000 q^{8} +(-1.00000 - 1.73205i) q^{10} -1.00000 q^{11} +(-3.00000 - 5.19615i) q^{13} +(-0.500000 - 0.866025i) q^{16} +(2.50000 + 4.33013i) q^{17} +(-3.50000 + 6.06218i) q^{19} +(-1.00000 + 1.73205i) q^{20} +(0.500000 + 0.866025i) q^{22} -4.00000 q^{23} -1.00000 q^{25} +(-3.00000 + 5.19615i) q^{26} +(-2.00000 + 3.46410i) q^{29} +(-3.00000 + 5.19615i) q^{31} +(-0.500000 + 0.866025i) q^{32} +(2.50000 - 4.33013i) q^{34} +(-1.00000 + 1.73205i) q^{37} +7.00000 q^{38} +2.00000 q^{40} +(-1.50000 - 2.59808i) q^{41} +(0.500000 - 0.866025i) q^{43} +(0.500000 - 0.866025i) q^{44} +(2.00000 + 3.46410i) q^{46} +(0.500000 + 0.866025i) q^{50} +6.00000 q^{52} +(6.00000 + 10.3923i) q^{53} -2.00000 q^{55} +4.00000 q^{58} +(3.50000 - 6.06218i) q^{59} +(-6.00000 - 10.3923i) q^{61} +6.00000 q^{62} +1.00000 q^{64} +(-6.00000 - 10.3923i) q^{65} +(-6.50000 + 11.2583i) q^{67} -5.00000 q^{68} +8.00000 q^{71} +(0.500000 + 0.866025i) q^{73} +2.00000 q^{74} +(-3.50000 - 6.06218i) q^{76} +(3.00000 + 5.19615i) q^{79} +(-1.00000 - 1.73205i) q^{80} +(-1.50000 + 2.59808i) q^{82} +(-8.00000 + 13.8564i) q^{83} +(5.00000 + 8.66025i) q^{85} -1.00000 q^{86} -1.00000 q^{88} +(3.00000 - 5.19615i) q^{89} +(2.00000 - 3.46410i) q^{92} +(-7.00000 + 12.1244i) q^{95} +(-2.50000 + 4.33013i) q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} - q^{4} + 4 q^{5} + 2 q^{8}+O(q^{10}) $ 2 * q - q^2 - q^4 + 4 * q^5 + 2 * q^8	$ 2 q - q^{2} - q^{4} + 4 q^{5} + 2 q^{8} - 2 q^{10} - 2 q^{11} - 6 q^{13} - q^{16} + 5 q^{17} - 7 q^{19} - 2 q^{20} + q^{22} - 8 q^{23} - 2 q^{25} - 6 q^{26} - 4 q^{29} - 6 q^{31} - q^{32} + 5 q^{34} - 2 q^{37} + 14 q^{38} + 4 q^{40} - 3 q^{41} + q^{43} + q^{44} + 4 q^{46} + q^{50} + 12 q^{52} + 12 q^{53} - 4 q^{55} + 8 q^{58} + 7 q^{59} - 12 q^{61} + 12 q^{62} + 2 q^{64} - 12 q^{65} - 13 q^{67} - 10 q^{68} + 16 q^{71} + q^{73} + 4 q^{74} - 7 q^{76} + 6 q^{79} - 2 q^{80} - 3 q^{82} - 16 q^{83} + 10 q^{85} - 2 q^{86} - 2 q^{88} + 6 q^{89} + 4 q^{92} - 14 q^{95} - 5 q^{97}+O(q^{100}) $ 2 * q - q^2 - q^4 + 4 * q^5 + 2 * q^8 - 2 * q^10 - 2 * q^11 - 6 * q^13 - q^16 + 5 * q^17 - 7 * q^19 - 2 * q^20 + q^22 - 8 * q^23 - 2 * q^25 - 6 * q^26 - 4 * q^29 - 6 * q^31 - q^32 + 5 * q^34 - 2 * q^37 + 14 * q^38 + 4 * q^40 - 3 * q^41 + q^43 + q^44 + 4 * q^46 + q^50 + 12 * q^52 + 12 * q^53 - 4 * q^55 + 8 * q^58 + 7 * q^59 - 12 * q^61 + 12 * q^62 + 2 * q^64 - 12 * q^65 - 13 * q^67 - 10 * q^68 + 16 * q^71 + q^73 + 4 * q^74 - 7 * q^76 + 6 * q^79 - 2 * q^80 - 3 * q^82 - 16 * q^83 + 10 * q^85 - 2 * q^86 - 2 * q^88 + 6 * q^89 + 4 * q^92 - 14 * q^95 - 5 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2646\mathbb{Z}\right)^\times$.

$n$	$785$	$1081$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{3}\right)$

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			0.894427			0.447214	−	0.894427i	$-0.352416\pi$
$5$	2.00000			0.894427			0.447214	+	0.894427i	$0.352416\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$	1.00000			0.353553
$9$		0			0
$10$	−1.00000	−	1.73205i	−0.316228	−	0.547723i
$11$	−1.00000			−0.301511			−0.150756	−	0.988571i	$-0.548171\pi$
$11$	−1.00000			−0.301511			−0.150756	+	0.988571i	$0.548171\pi$
$12$		0			0
$13$	−3.00000	−	5.19615i	−0.832050	−	1.44115i	−0.896410	−	0.443227i	$-0.853834\pi$
$13$	−3.00000	−	5.19615i	−0.832050	−	1.44115i	0.0643593	−	0.997927i	$-0.479500\pi$
$14$		0			0
$15$		0			0
$16$	−0.500000	−	0.866025i	−0.125000	−	0.216506i
$17$	2.50000	+	4.33013i	0.606339	+	1.05021i	0.991838	+	0.127502i	$0.0406959\pi$
$17$	2.50000	+	4.33013i	0.606339	+	1.05021i	−0.385499	+	0.922708i	$0.625971\pi$
$18$		0			0
$19$	−3.50000	+	6.06218i	−0.802955	+	1.39076i	0.114708	+	0.993399i	$0.463407\pi$
$19$	−3.50000	+	6.06218i	−0.802955	+	1.39076i	−0.917663	+	0.397360i	$0.869927\pi$
$20$	−1.00000	+	1.73205i	−0.223607	+	0.387298i
$21$		0			0
$22$	0.500000	+	0.866025i	0.106600	+	0.184637i
$23$	−4.00000			−0.834058			−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000			−0.834058			−0.417029	+	0.908893i	$0.636929\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$	−3.00000	+	5.19615i	−0.588348	+	1.01905i
$27$		0			0
$28$		0			0
$29$	−2.00000	+	3.46410i	−0.371391	+	0.643268i	−0.989780	−	0.142605i	$-0.954452\pi$
$29$	−2.00000	+	3.46410i	−0.371391	+	0.643268i	0.618389	+	0.785872i	$0.287786\pi$
$30$		0			0
$31$	−3.00000	+	5.19615i	−0.538816	+	0.933257i	0.460152	+	0.887840i	$0.347795\pi$
$31$	−3.00000	+	5.19615i	−0.538816	+	0.933257i	−0.998968	+	0.0454165i	$0.985539\pi$
$32$	−0.500000	+	0.866025i	−0.0883883	+	0.153093i
$33$		0			0
$34$	2.50000	−	4.33013i	0.428746	−	0.742611i
$35$		0			0
$36$		0			0
$37$	−1.00000	+	1.73205i	−0.164399	+	0.284747i	−0.936442	−	0.350823i	$-0.885902\pi$
$37$	−1.00000	+	1.73205i	−0.164399	+	0.284747i	0.772043	+	0.635571i	$0.219235\pi$
$38$	7.00000			1.13555
$39$		0			0
$40$	2.00000			0.316228
$41$	−1.50000	−	2.59808i	−0.234261	−	0.405751i	0.724797	−	0.688963i	$-0.241934\pi$
$41$	−1.50000	−	2.59808i	−0.234261	−	0.405751i	−0.959058	+	0.283211i	$0.908600\pi$
$42$		0			0
$43$	0.500000	−	0.866025i	0.0762493	−	0.132068i	−0.825380	−	0.564578i	$-0.809039\pi$
$43$	0.500000	−	0.866025i	0.0762493	−	0.132068i	0.901629	+	0.432511i	$0.142372\pi$
$44$	0.500000	−	0.866025i	0.0753778	−	0.130558i
$45$		0			0
$46$	2.00000	+	3.46410i	0.294884	+	0.510754i
$47$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$47$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$48$		0			0
$49$		0			0
$50$	0.500000	+	0.866025i	0.0707107	+	0.122474i
$51$		0			0
$52$	6.00000			0.832050
$53$	6.00000	+	10.3923i	0.824163	+	1.42749i	0.902557	+	0.430570i	$0.141688\pi$
$53$	6.00000	+	10.3923i	0.824163	+	1.42749i	−0.0783936	+	0.996922i	$0.524979\pi$
$54$		0			0
$55$	−2.00000			−0.269680
$56$		0			0
$57$		0			0
$58$	4.00000			0.525226
$59$	3.50000	−	6.06218i	0.455661	−	0.789228i	−0.543065	−	0.839691i	$-0.682736\pi$
$59$	3.50000	−	6.06218i	0.455661	−	0.789228i	0.998726	+	0.0504625i	$0.0160695\pi$
$60$		0			0
$61$	−6.00000	−	10.3923i	−0.768221	−	1.33060i	−0.938527	−	0.345207i	$-0.887809\pi$
$61$	−6.00000	−	10.3923i	−0.768221	−	1.33060i	0.170305	−	0.985391i	$-0.445525\pi$
$62$	6.00000			0.762001
$63$		0			0
$64$	1.00000			0.125000
$65$	−6.00000	−	10.3923i	−0.744208	−	1.28901i
$66$		0			0
$67$	−6.50000	+	11.2583i	−0.794101	+	1.37542i	0.129307	+	0.991605i	$0.458725\pi$
$67$	−6.50000	+	11.2583i	−0.794101	+	1.37542i	−0.923408	+	0.383819i	$0.874609\pi$
$68$	−5.00000			−0.606339
$69$		0			0
$70$		0			0
$71$	8.00000			0.949425			0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000			0.949425			0.474713	+	0.880141i	$0.342552\pi$
$72$		0			0
$73$	0.500000	+	0.866025i	0.0585206	+	0.101361i	0.893801	−	0.448463i	$-0.148028\pi$
$73$	0.500000	+	0.866025i	0.0585206	+	0.101361i	−0.835281	+	0.549823i	$0.814695\pi$
$74$	2.00000			0.232495
$75$		0			0
$76$	−3.50000	−	6.06218i	−0.401478	−	0.695379i
$77$		0			0
$78$		0			0
$79$	3.00000	+	5.19615i	0.337526	+	0.584613i	0.983967	−	0.178352i	$-0.0570765\pi$
$79$	3.00000	+	5.19615i	0.337526	+	0.584613i	−0.646440	+	0.762964i	$0.723743\pi$
$80$	−1.00000	−	1.73205i	−0.111803	−	0.193649i
$81$		0			0
$82$	−1.50000	+	2.59808i	−0.165647	+	0.286910i
$83$	−8.00000	+	13.8564i	−0.878114	+	1.52094i	−0.0247060	+	0.999695i	$0.507865\pi$
$83$	−8.00000	+	13.8564i	−0.878114	+	1.52094i	−0.853408	+	0.521243i	$0.825468\pi$
$84$		0			0
$85$	5.00000	+	8.66025i	0.542326	+	0.939336i
$86$	−1.00000			−0.107833
$87$		0			0
$88$	−1.00000			−0.106600
$89$	3.00000	−	5.19615i	0.317999	−	0.550791i	−0.662071	−	0.749441i	$-0.730322\pi$
$89$	3.00000	−	5.19615i	0.317999	−	0.550791i	0.980071	+	0.198650i	$0.0636557\pi$
$90$		0			0
$91$		0			0
$92$	2.00000	−	3.46410i	0.208514	−	0.361158i
$93$		0			0
$94$		0			0
$95$	−7.00000	+	12.1244i	−0.718185	+	1.24393i
$96$		0			0
$97$	−2.50000	+	4.33013i	−0.253837	+	0.439658i	−0.964579	−	0.263795i	$-0.915026\pi$
$97$	−2.50000	+	4.33013i	−0.253837	+	0.439658i	0.710742	+	0.703452i	$0.248359\pi$
$98$		0			0
$99$		0			0
$100$	0.500000	−	0.866025i	0.0500000	−	0.0866025i
$101$	−4.00000			−0.398015			−0.199007	−	0.979998i	$-0.563772\pi$
$101$	−4.00000			−0.398015			−0.199007	+	0.979998i	$0.563772\pi$
$102$		0			0
$103$	−14.0000			−1.37946			−0.689730	−	0.724066i	$-0.742271\pi$
$103$	−14.0000			−1.37946			−0.689730	+	0.724066i	$0.742271\pi$
$104$	−3.00000	−	5.19615i	−0.294174	−	0.509525i
$105$		0			0
$106$	6.00000	−	10.3923i	0.582772	−	1.00939i
$107$	1.50000	−	2.59808i	0.145010	−	0.251166i	−0.784366	−	0.620298i	$-0.787012\pi$
$107$	1.50000	−	2.59808i	0.145010	−	0.251166i	0.929377	+	0.369132i	$0.120345\pi$
$108$		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
$113$	−5.00000	−	8.66025i	−0.470360	−	0.814688i	0.529065	−	0.848581i	$-0.322543\pi$
$113$	−5.00000	−	8.66025i	−0.470360	−	0.814688i	−0.999425	+	0.0338931i	$0.989209\pi$
$114$		0			0
$115$	−8.00000			−0.746004
$116$	−2.00000	−	3.46410i	−0.185695	−	0.321634i
$117$		0			0
$118$	−7.00000			−0.644402
$119$		0			0
$120$		0			0
$121$	−10.0000			−0.909091
$122$	−6.00000	+	10.3923i	−0.543214	+	0.940875i
$123$		0			0
$124$	−3.00000	−	5.19615i	−0.269408	−	0.466628i
$125$	−12.0000			−1.07331
$126$		0			0
$127$	−12.0000			−1.06483			−0.532414	−	0.846484i	$-0.678715\pi$
$127$	−12.0000			−1.06483			−0.532414	+	0.846484i	$0.678715\pi$
$128$	−0.500000	−	0.866025i	−0.0441942	−	0.0765466i
$129$		0			0
$130$	−6.00000	+	10.3923i	−0.526235	+	0.911465i
$131$	−4.00000			−0.349482			−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000			−0.349482			−0.174741	+	0.984614i	$0.555909\pi$
$132$		0			0
$133$		0			0
$134$	13.0000			1.12303
$135$		0			0
$136$	2.50000	+	4.33013i	0.214373	+	0.371305i
$137$	19.0000			1.62328			0.811640	−	0.584158i	$-0.198575\pi$
$137$	19.0000			1.62328			0.811640	+	0.584158i	$0.198575\pi$
$138$		0			0
$139$	−2.50000	−	4.33013i	−0.212047	−	0.367277i	0.740308	−	0.672268i	$-0.234680\pi$
$139$	−2.50000	−	4.33013i	−0.212047	−	0.367277i	−0.952355	+	0.304991i	$0.901346\pi$
$140$		0			0
$141$		0			0
$142$	−4.00000	−	6.92820i	−0.335673	−	0.581402i
$143$	3.00000	+	5.19615i	0.250873	+	0.434524i
$144$		0			0
$145$	−4.00000	+	6.92820i	−0.332182	+	0.575356i
$146$	0.500000	−	0.866025i	0.0413803	−	0.0716728i
$147$		0			0
$148$	−1.00000	−	1.73205i	−0.0821995	−	0.142374i
$149$	24.0000			1.96616			0.983078	−	0.183186i	$-0.0586410\pi$
$149$	24.0000			1.96616			0.983078	+	0.183186i	$0.0586410\pi$
$150$		0			0
$151$	10.0000			0.813788			0.406894	−	0.913475i	$-0.366612\pi$
$151$	10.0000			0.813788			0.406894	+	0.913475i	$0.366612\pi$
$152$	−3.50000	+	6.06218i	−0.283887	+	0.491708i
$153$		0			0
$154$		0			0
$155$	−6.00000	+	10.3923i	−0.481932	+	0.834730i
$156$		0			0
$157$	1.00000	−	1.73205i	0.0798087	−	0.138233i	−0.823359	−	0.567521i	$-0.807902\pi$
$157$	1.00000	−	1.73205i	0.0798087	−	0.138233i	0.903167	+	0.429289i	$0.141236\pi$
$158$	3.00000	−	5.19615i	0.238667	−	0.413384i
$159$		0			0
$160$	−1.00000	+	1.73205i	−0.0790569	+	0.136931i
$161$		0			0
$162$		0			0
$163$	2.00000	−	3.46410i	0.156652	−	0.271329i	−0.777007	−	0.629492i	$-0.783263\pi$
$163$	2.00000	−	3.46410i	0.156652	−	0.271329i	0.933659	+	0.358162i	$0.116597\pi$
$164$	3.00000			0.234261
$165$		0			0
$166$	16.0000			1.24184
$167$	−10.0000	−	17.3205i	−0.773823	−	1.34030i	−0.935454	−	0.353450i	$-0.885009\pi$
$167$	−10.0000	−	17.3205i	−0.773823	−	1.34030i	0.161630	−	0.986851i	$-0.448325\pi$
$168$		0			0
$169$	−11.5000	+	19.9186i	−0.884615	+	1.53220i
$170$	5.00000	−	8.66025i	0.383482	−	0.664211i
$171$		0			0
$172$	0.500000	+	0.866025i	0.0381246	+	0.0660338i
$173$	−1.00000	−	1.73205i	−0.0760286	−	0.131685i	0.825505	−	0.564396i	$-0.190891\pi$
$173$	−1.00000	−	1.73205i	−0.0760286	−	0.131685i	−0.901533	+	0.432710i	$0.857557\pi$
$174$		0			0
$175$		0			0
$176$	0.500000	+	0.866025i	0.0376889	+	0.0652791i
$177$		0			0
$178$	−6.00000			−0.449719
$179$	12.0000	+	20.7846i	0.896922	+	1.55351i	0.831408	+	0.555663i	$0.187536\pi$
$179$	12.0000	+	20.7846i	0.896922	+	1.55351i	0.0655145	+	0.997852i	$0.479131\pi$
$180$		0			0
$181$		0			0			−	1.00000i	$-0.5\pi$
$181$		0			0				1.00000i	$0.5\pi$
$182$		0			0
$183$		0			0
$184$	−4.00000			−0.294884
$185$	−2.00000	+	3.46410i	−0.147043	+	0.254686i
$186$		0			0
$187$	−2.50000	−	4.33013i	−0.182818	−	0.316650i
$188$		0			0
$189$		0			0
$190$	14.0000			1.01567
$191$	6.00000	+	10.3923i	0.434145	+	0.751961i	0.997225	−	0.0744412i	$-0.0237173\pi$
$191$	6.00000	+	10.3923i	0.434145	+	0.751961i	−0.563081	+	0.826402i	$0.690384\pi$
$192$		0			0
$193$	−8.50000	+	14.7224i	−0.611843	+	1.05974i	0.379086	+	0.925361i	$0.376238\pi$
$193$	−8.50000	+	14.7224i	−0.611843	+	1.05974i	−0.990930	+	0.134382i	$0.957095\pi$
$194$	5.00000			0.358979
$195$		0			0
$196$		0			0
$197$	−10.0000			−0.712470			−0.356235	−	0.934396i	$-0.615940\pi$
$197$	−10.0000			−0.712470			−0.356235	+	0.934396i	$0.615940\pi$
$198$		0			0
$199$	7.00000	+	12.1244i	0.496217	+	0.859473i	0.999990	−	0.00436292i	$-0.00138876\pi$
$199$	7.00000	+	12.1244i	0.496217	+	0.859473i	−0.503774	+	0.863836i	$0.668055\pi$
$200$	−1.00000			−0.0707107
$201$		0			0
$202$	2.00000	+	3.46410i	0.140720	+	0.243733i
$203$		0			0
$204$		0			0
$205$	−3.00000	−	5.19615i	−0.209529	−	0.362915i
$206$	7.00000	+	12.1244i	0.487713	+	0.844744i
$207$		0			0
$208$	−3.00000	+	5.19615i	−0.208013	+	0.360288i
$209$	3.50000	−	6.06218i	0.242100	−	0.419330i
$210$		0			0
$211$	8.00000	+	13.8564i	0.550743	+	0.953914i	0.998221	+	0.0596196i	$0.0189888\pi$
$211$	8.00000	+	13.8564i	0.550743	+	0.953914i	−0.447478	+	0.894295i	$0.647678\pi$
$212$	−12.0000			−0.824163
$213$		0			0
$214$	−3.00000			−0.205076
$215$	1.00000	−	1.73205i	0.0681994	−	0.118125i
$216$		0			0
$217$		0			0
$218$	1.00000	−	1.73205i	0.0677285	−	0.117309i
$219$		0			0
$220$	1.00000	−	1.73205i	0.0674200	−	0.116775i
$221$	15.0000	−	25.9808i	1.00901	−	1.74766i
$222$		0			0
$223$	−2.00000	+	3.46410i	−0.133930	+	0.231973i	−0.925188	−	0.379509i	$-0.876093\pi$
$223$	−2.00000	+	3.46410i	−0.133930	+	0.231973i	0.791258	+	0.611482i	$0.209426\pi$
$224$		0			0
$225$		0			0
$226$	−5.00000	+	8.66025i	−0.332595	+	0.576072i
$227$	3.00000			0.199117			0.0995585	−	0.995032i	$-0.468257\pi$
$227$	3.00000			0.199117			0.0995585	+	0.995032i	$0.468257\pi$
$228$		0			0
$229$	26.0000			1.71813			0.859064	−	0.511868i	$-0.171046\pi$
$229$	26.0000			1.71813			0.859064	+	0.511868i	$0.171046\pi$
$230$	4.00000	+	6.92820i	0.263752	+	0.456832i
$231$		0			0
$232$	−2.00000	+	3.46410i	−0.131306	+	0.227429i
$233$	−14.5000	+	25.1147i	−0.949927	+	1.64532i	−0.204354	+	0.978897i	$0.565509\pi$
$233$	−14.5000	+	25.1147i	−0.949927	+	1.64532i	−0.745573	+	0.666424i	$0.767824\pi$
$234$		0			0
$235$		0			0
$236$	3.50000	+	6.06218i	0.227831	+	0.394614i
$237$		0			0
$238$		0			0
$239$	3.00000	+	5.19615i	0.194054	+	0.336111i	0.946590	−	0.322440i	$-0.104503\pi$
$239$	3.00000	+	5.19615i	0.194054	+	0.336111i	−0.752536	+	0.658551i	$0.771170\pi$
$240$		0			0
$241$	−23.0000			−1.48156			−0.740780	−	0.671748i	$-0.765544\pi$
$241$	−23.0000			−1.48156			−0.740780	+	0.671748i	$0.765544\pi$
$242$	5.00000	+	8.66025i	0.321412	+	0.556702i
$243$		0			0
$244$	12.0000			0.768221
$245$		0			0
$246$		0			0
$247$	42.0000			2.67240
$248$	−3.00000	+	5.19615i	−0.190500	+	0.329956i
$249$		0			0
$250$	6.00000	+	10.3923i	0.379473	+	0.657267i
$251$	3.00000			0.189358			0.0946792	−	0.995508i	$-0.469817\pi$
$251$	3.00000			0.189358			0.0946792	+	0.995508i	$0.469817\pi$
$252$		0			0
$253$	4.00000			0.251478
$254$	6.00000	+	10.3923i	0.376473	+	0.652071i
$255$		0			0
$256$	−0.500000	+	0.866025i	−0.0312500	+	0.0541266i
$257$	−15.0000			−0.935674			−0.467837	−	0.883815i	$-0.654967\pi$
$257$	−15.0000			−0.935674			−0.467837	+	0.883815i	$0.654967\pi$
$258$		0			0
$259$		0			0
$260$	12.0000			0.744208
$261$		0			0
$262$	2.00000	+	3.46410i	0.123560	+	0.214013i
$263$	−18.0000			−1.10993			−0.554964	−	0.831875i	$-0.687268\pi$
$263$	−18.0000			−1.10993			−0.554964	+	0.831875i	$0.687268\pi$
$264$		0			0
$265$	12.0000	+	20.7846i	0.737154	+	1.27679i
$266$		0			0
$267$		0			0
$268$	−6.50000	−	11.2583i	−0.397051	−	0.687712i
$269$	−10.0000	−	17.3205i	−0.609711	−	1.05605i	−0.991288	−	0.131713i	$-0.957952\pi$
$269$	−10.0000	−	17.3205i	−0.609711	−	1.05605i	0.381577	−	0.924337i	$-0.375381\pi$
$270$		0			0
$271$	−3.00000	+	5.19615i	−0.182237	+	0.315644i	−0.942642	−	0.333805i	$-0.891667\pi$
$271$	−3.00000	+	5.19615i	−0.182237	+	0.315644i	0.760405	+	0.649449i	$0.225000\pi$
$272$	2.50000	−	4.33013i	0.151585	−	0.262553i
$273$		0			0
$274$	−9.50000	−	16.4545i	−0.573916	−	0.994052i
$275$	1.00000			0.0603023
$276$		0			0
$277$	2.00000			0.120168			0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000			0.120168			0.0600842	+	0.998193i	$0.480863\pi$
$278$	−2.50000	+	4.33013i	−0.149940	+	0.259704i
$279$		0			0
$280$		0			0
$281$	11.0000	−	19.0526i	0.656205	−	1.13658i	−0.325385	−	0.945582i	$-0.605494\pi$
$281$	11.0000	−	19.0526i	0.656205	−	1.13658i	0.981590	−	0.190999i	$-0.0611727\pi$
$282$		0			0
$283$	2.00000	−	3.46410i	0.118888	−	0.205919i	−0.800439	−	0.599414i	$-0.795400\pi$
$283$	2.00000	−	3.46410i	0.118888	−	0.205919i	0.919327	+	0.393494i	$0.128734\pi$
$284$	−4.00000	+	6.92820i	−0.237356	+	0.411113i
$285$		0			0
$286$	3.00000	−	5.19615i	0.177394	−	0.307255i
$287$		0			0
$288$		0			0
$289$	−4.00000	+	6.92820i	−0.235294	+	0.407541i
$290$	8.00000			0.469776
$291$		0			0
$292$	−1.00000			−0.0585206
$293$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$293$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$294$		0			0
$295$	7.00000	−	12.1244i	0.407556	−	0.705907i
$296$	−1.00000	+	1.73205i	−0.0581238	+	0.100673i
$297$		0			0
$298$	−12.0000	−	20.7846i	−0.695141	−	1.20402i
$299$	12.0000	+	20.7846i	0.693978	+	1.20201i
$300$		0			0
$301$		0			0
$302$	−5.00000	−	8.66025i	−0.287718	−	0.498342i
$303$		0			0
$304$	7.00000			0.401478
$305$	−12.0000	−	20.7846i	−0.687118	−	1.19012i
$306$		0			0
$307$	−7.00000			−0.399511			−0.199756	−	0.979846i	$-0.564015\pi$
$307$	−7.00000			−0.399511			−0.199756	+	0.979846i	$0.564015\pi$
$308$		0			0
$309$		0			0
$310$	12.0000			0.681554
$311$	−1.00000	+	1.73205i	−0.0567048	+	0.0982156i	−0.892984	−	0.450088i	$-0.851393\pi$
$311$	−1.00000	+	1.73205i	−0.0567048	+	0.0982156i	0.836280	+	0.548303i	$0.184726\pi$
$312$		0			0
$313$	−8.50000	−	14.7224i	−0.480448	−	0.832161i	0.519300	−	0.854592i	$-0.326193\pi$
$313$	−8.50000	−	14.7224i	−0.480448	−	0.832161i	−0.999748	+	0.0224310i	$0.992859\pi$
$314$	−2.00000			−0.112867
$315$		0			0
$316$	−6.00000			−0.337526
$317$	−3.00000	−	5.19615i	−0.168497	−	0.291845i	0.769395	−	0.638774i	$-0.220558\pi$
$317$	−3.00000	−	5.19615i	−0.168497	−	0.291845i	−0.937892	+	0.346929i	$0.887225\pi$
$318$		0			0
$319$	2.00000	−	3.46410i	0.111979	−	0.193952i
$320$	2.00000			0.111803
$321$		0			0
$322$		0			0
$323$	−35.0000			−1.94745
$324$		0			0
$325$	3.00000	+	5.19615i	0.166410	+	0.288231i
$326$	−4.00000			−0.221540
$327$		0			0
$328$	−1.50000	−	2.59808i	−0.0828236	−	0.143455i
$329$		0			0
$330$		0			0
$331$	−4.00000	−	6.92820i	−0.219860	−	0.380808i	0.734905	−	0.678170i	$-0.237227\pi$
$331$	−4.00000	−	6.92820i	−0.219860	−	0.380808i	−0.954765	+	0.297361i	$0.903893\pi$
$332$	−8.00000	−	13.8564i	−0.439057	−	0.760469i
$333$		0			0
$334$	−10.0000	+	17.3205i	−0.547176	+	0.947736i
$335$	−13.0000	+	22.5167i	−0.710266	+	1.23022i
$336$		0			0
$337$	4.50000	+	7.79423i	0.245131	+	0.424579i	0.962168	−	0.272456i	$-0.0878358\pi$
$337$	4.50000	+	7.79423i	0.245131	+	0.424579i	−0.717038	+	0.697034i	$0.754502\pi$
$338$	23.0000			1.25104
$339$		0			0
$340$	−10.0000			−0.542326
$341$	3.00000	−	5.19615i	0.162459	−	0.281387i
$342$		0			0
$343$		0			0
$344$	0.500000	−	0.866025i	0.0269582	−	0.0466930i
$345$		0			0
$346$	−1.00000	+	1.73205i	−0.0537603	+	0.0931156i
$347$	−1.50000	+	2.59808i	−0.0805242	+	0.139472i	−0.903475	−	0.428640i	$-0.858993\pi$
$347$	−1.50000	+	2.59808i	−0.0805242	+	0.139472i	0.822951	+	0.568112i	$0.192326\pi$
$348$		0			0
$349$	−7.00000	+	12.1244i	−0.374701	+	0.649002i	−0.990282	−	0.139072i	$-0.955588\pi$
$349$	−7.00000	+	12.1244i	−0.374701	+	0.649002i	0.615581	+	0.788074i	$0.288921\pi$
$350$		0			0
$351$		0			0
$352$	0.500000	−	0.866025i	0.0266501	−	0.0461593i
$353$	−15.0000			−0.798369			−0.399185	−	0.916871i	$-0.630707\pi$
$353$	−15.0000			−0.798369			−0.399185	+	0.916871i	$0.630707\pi$
$354$		0			0
$355$	16.0000			0.849192
$356$	3.00000	+	5.19615i	0.159000	+	0.275396i
$357$		0			0
$358$	12.0000	−	20.7846i	0.634220	−	1.09850i
$359$	1.00000	−	1.73205i	0.0527780	−	0.0914141i	−0.838429	−	0.545010i	$-0.816526\pi$
$359$	1.00000	−	1.73205i	0.0527780	−	0.0914141i	0.891207	+	0.453596i	$0.149859\pi$
$360$		0			0
$361$	−15.0000	−	25.9808i	−0.789474	−	1.36741i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	1.00000	+	1.73205i	0.0523424	+	0.0906597i
$366$		0			0
$367$	22.0000			1.14839			0.574195	−	0.818718i	$-0.305315\pi$
$367$	22.0000			1.14839			0.574195	+	0.818718i	$0.305315\pi$
$368$	2.00000	+	3.46410i	0.104257	+	0.180579i
$369$		0			0
$370$	4.00000			0.207950
$371$		0			0
$372$		0			0
$373$	22.0000			1.13912			0.569558	−	0.821951i	$-0.307114\pi$
$373$	22.0000			1.13912			0.569558	+	0.821951i	$0.307114\pi$
$374$	−2.50000	+	4.33013i	−0.129272	+	0.223906i
$375$		0			0
$376$		0			0
$377$	24.0000			1.23606
$378$		0			0
$379$	−17.0000			−0.873231			−0.436616	−	0.899648i	$-0.643823\pi$
$379$	−17.0000			−0.873231			−0.436616	+	0.899648i	$0.643823\pi$
$380$	−7.00000	−	12.1244i	−0.359092	−	0.621966i
$381$		0			0
$382$	6.00000	−	10.3923i	0.306987	−	0.531717i
$383$	4.00000			0.204390			0.102195	−	0.994764i	$-0.467413\pi$
$383$	4.00000			0.204390			0.102195	+	0.994764i	$0.467413\pi$
$384$		0			0
$385$		0			0
$386$	17.0000			0.865277
$387$		0			0
$388$	−2.50000	−	4.33013i	−0.126918	−	0.219829i
$389$	−8.00000			−0.405616			−0.202808	−	0.979219i	$-0.565007\pi$
$389$	−8.00000			−0.405616			−0.202808	+	0.979219i	$0.565007\pi$
$390$		0			0
$391$	−10.0000	−	17.3205i	−0.505722	−	0.875936i
$392$		0			0
$393$		0			0
$394$	5.00000	+	8.66025i	0.251896	+	0.436297i
$395$	6.00000	+	10.3923i	0.301893	+	0.522894i
$396$		0			0
$397$	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	$0.0174855\pi$
$398$	7.00000	−	12.1244i	0.350878	−	0.607739i
$399$		0			0
$400$	0.500000	+	0.866025i	0.0250000	+	0.0433013i
$401$	−9.00000			−0.449439			−0.224719	−	0.974424i	$-0.572147\pi$
$401$	−9.00000			−0.449439			−0.224719	+	0.974424i	$0.572147\pi$
$402$		0			0
$403$	36.0000			1.79329
$404$	2.00000	−	3.46410i	0.0995037	−	0.172345i
$405$		0			0
$406$		0			0
$407$	1.00000	−	1.73205i	0.0495682	−	0.0858546i
$408$		0			0
$409$	5.50000	−	9.52628i	0.271957	−	0.471044i	−0.697406	−	0.716677i	$-0.745662\pi$
$409$	5.50000	−	9.52628i	0.271957	−	0.471044i	0.969363	+	0.245633i	$0.0789957\pi$
$410$	−3.00000	+	5.19615i	−0.148159	+	0.256620i
$411$		0			0
$412$	7.00000	−	12.1244i	0.344865	−	0.597324i
$413$		0			0
$414$		0			0
$415$	−16.0000	+	27.7128i	−0.785409	+	1.36037i
$416$	6.00000			0.294174
$417$		0			0
$418$	−7.00000			−0.342381
$419$	6.00000	+	10.3923i	0.293119	+	0.507697i	0.974546	−	0.224189i	$-0.0719734\pi$
$419$	6.00000	+	10.3923i	0.293119	+	0.507697i	−0.681426	+	0.731887i	$0.738640\pi$
$420$		0			0
$421$	6.00000	−	10.3923i	0.292422	−	0.506490i	−0.681960	−	0.731390i	$-0.738872\pi$
$421$	6.00000	−	10.3923i	0.292422	−	0.506490i	0.974382	+	0.224900i	$0.0722054\pi$
$422$	8.00000	−	13.8564i	0.389434	−	0.674519i
$423$		0			0
$424$	6.00000	+	10.3923i	0.291386	+	0.504695i
$425$	−2.50000	−	4.33013i	−0.121268	−	0.210042i
$426$		0			0
$427$		0			0
$428$	1.50000	+	2.59808i	0.0725052	+	0.125583i
$429$		0			0
$430$	−2.00000			−0.0964486
$431$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$431$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$432$		0			0
$433$	25.0000			1.20142			0.600712	−	0.799466i	$-0.294884\pi$
$433$	25.0000			1.20142			0.600712	+	0.799466i	$0.294884\pi$
$434$		0			0
$435$		0			0
$436$	−2.00000			−0.0957826
$437$	14.0000	−	24.2487i	0.669711	−	1.15997i
$438$		0			0
$439$	−12.0000	−	20.7846i	−0.572729	−	0.991995i	−0.996284	−	0.0861252i	$-0.972552\pi$
$439$	−12.0000	−	20.7846i	−0.572729	−	0.991995i	0.423556	−	0.905870i	$-0.360782\pi$
$440$	−2.00000			−0.0953463
$441$		0			0
$442$	−30.0000			−1.42695
$443$	3.50000	+	6.06218i	0.166290	+	0.288023i	0.937113	−	0.349027i	$-0.113488\pi$
$443$	3.50000	+	6.06218i	0.166290	+	0.288023i	−0.770823	+	0.637050i	$0.780155\pi$
$444$		0			0
$445$	6.00000	−	10.3923i	0.284427	−	0.492642i
$446$	4.00000			0.189405
$447$		0			0
$448$		0			0
$449$	−17.0000			−0.802280			−0.401140	−	0.916017i	$-0.631386\pi$
$449$	−17.0000			−0.802280			−0.401140	+	0.916017i	$0.631386\pi$
$450$		0			0
$451$	1.50000	+	2.59808i	0.0706322	+	0.122339i
$452$	10.0000			0.470360
$453$		0			0
$454$	−1.50000	−	2.59808i	−0.0703985	−	0.121934i
$455$		0			0
$456$		0			0
$457$	−0.500000	−	0.866025i	−0.0233890	−	0.0405110i	0.854094	−	0.520119i	$-0.174112\pi$
$457$	−0.500000	−	0.866025i	−0.0233890	−	0.0405110i	−0.877483	+	0.479608i	$0.840779\pi$
$458$	−13.0000	−	22.5167i	−0.607450	−	1.05213i
$459$		0			0
$460$	4.00000	−	6.92820i	0.186501	−	0.323029i
$461$	−7.00000	+	12.1244i	−0.326023	+	0.564688i	−0.981719	−	0.190337i	$-0.939042\pi$
$461$	−7.00000	+	12.1244i	−0.326023	+	0.564688i	0.655696	+	0.755025i	$0.272375\pi$
$462$		0			0
$463$	4.00000	+	6.92820i	0.185896	+	0.321981i	0.943878	−	0.330294i	$-0.107148\pi$
$463$	4.00000	+	6.92820i	0.185896	+	0.321981i	−0.757982	+	0.652275i	$0.773815\pi$
$464$	4.00000			0.185695
$465$		0			0
$466$	29.0000			1.34340
$467$	6.50000	−	11.2583i	0.300784	−	0.520973i	−0.675530	−	0.737333i	$-0.736085\pi$
$467$	6.50000	−	11.2583i	0.300784	−	0.520973i	0.976314	+	0.216359i	$0.0694183\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$	3.50000	−	6.06218i	0.161101	−	0.279034i
$473$	−0.500000	+	0.866025i	−0.0229900	+	0.0398199i
$474$		0			0
$475$	3.50000	−	6.06218i	0.160591	−	0.278152i
$476$		0			0
$477$		0			0
$478$	3.00000	−	5.19615i	0.137217	−	0.237666i
$479$	−20.0000			−0.913823			−0.456912	−	0.889512i	$-0.651044\pi$
$479$	−20.0000			−0.913823			−0.456912	+	0.889512i	$0.651044\pi$
$480$		0			0
$481$	12.0000			0.547153
$482$	11.5000	+	19.9186i	0.523811	+	0.907267i
$483$		0			0
$484$	5.00000	−	8.66025i	0.227273	−	0.393648i
$485$	−5.00000	+	8.66025i	−0.227038	+	0.393242i
$486$		0			0
$487$	5.00000	+	8.66025i	0.226572	+	0.392434i	0.956790	−	0.290780i	$-0.0939149\pi$
$487$	5.00000	+	8.66025i	0.226572	+	0.392434i	−0.730218	+	0.683214i	$0.760582\pi$
$488$	−6.00000	−	10.3923i	−0.271607	−	0.470438i
$489$		0			0
$490$		0			0
$491$	16.5000	+	28.5788i	0.744635	+	1.28974i	0.950365	+	0.311136i	$0.100710\pi$
$491$	16.5000	+	28.5788i	0.744635	+	1.28974i	−0.205731	+	0.978609i	$0.565957\pi$
$492$		0			0
$493$	−20.0000			−0.900755
$494$	−21.0000	−	36.3731i	−0.944835	−	1.63650i
$495$		0			0
$496$	6.00000			0.269408
$497$		0			0
$498$		0			0
$499$	−29.0000			−1.29822			−0.649109	−	0.760695i	$-0.724858\pi$
$499$	−29.0000			−1.29822			−0.649109	+	0.760695i	$0.724858\pi$
$500$	6.00000	−	10.3923i	0.268328	−	0.464758i
$501$		0			0
$502$	−1.50000	−	2.59808i	−0.0669483	−	0.115958i
$503$	6.00000			0.267527			0.133763	−	0.991013i	$-0.457294\pi$
$503$	6.00000			0.267527			0.133763	+	0.991013i	$0.457294\pi$
$504$		0			0
$505$	−8.00000			−0.355995
$506$	−2.00000	−	3.46410i	−0.0889108	−	0.153998i
$507$		0			0
$508$	6.00000	−	10.3923i	0.266207	−	0.461084i
$509$	30.0000			1.32973			0.664863	−	0.746965i	$-0.268490\pi$
$509$	30.0000			1.32973			0.664863	+	0.746965i	$0.268490\pi$
$510$		0			0
$511$		0			0
$512$	1.00000			0.0441942
$513$		0			0
$514$	7.50000	+	12.9904i	0.330811	+	0.572981i
$515$	−28.0000			−1.23383
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	−6.00000	−	10.3923i	−0.263117	−	0.455733i
$521$	4.50000	+	7.79423i	0.197149	+	0.341471i	0.947603	−	0.319451i	$-0.103499\pi$
$521$	4.50000	+	7.79423i	0.197149	+	0.341471i	−0.750454	+	0.660922i	$0.770165\pi$
$522$		0			0
$523$	−14.0000	+	24.2487i	−0.612177	+	1.06032i	0.378695	+	0.925521i	$0.376373\pi$
$523$	−14.0000	+	24.2487i	−0.612177	+	1.06032i	−0.990873	+	0.134801i	$0.956961\pi$
$524$	2.00000	−	3.46410i	0.0873704	−	0.151330i
$525$		0			0
$526$	9.00000	+	15.5885i	0.392419	+	0.679689i
$527$	−30.0000			−1.30682
$528$		0			0
$529$	−7.00000			−0.304348
$530$	12.0000	−	20.7846i	0.521247	−	0.902826i
$531$		0			0
$532$		0			0
$533$	−9.00000	+	15.5885i	−0.389833	+	0.675211i
$534$		0			0
$535$	3.00000	−	5.19615i	0.129701	−	0.224649i
$536$	−6.50000	+	11.2583i	−0.280757	+	0.486286i
$537$		0			0
$538$	−10.0000	+	17.3205i	−0.431131	+	0.746740i
$539$		0			0
$540$		0			0
$541$	12.0000	−	20.7846i	0.515920	−	0.893600i	−0.483909	−	0.875118i	$-0.660783\pi$
$541$	12.0000	−	20.7846i	0.515920	−	0.893600i	0.999829	−	0.0184818i	$-0.00588327\pi$
$542$	6.00000			0.257722
$543$		0			0
$544$	−5.00000			−0.214373
$545$	2.00000	+	3.46410i	0.0856706	+	0.148386i
$546$		0			0
$547$	10.5000	−	18.1865i	0.448948	−	0.777600i	−0.549370	−	0.835579i	$-0.685132\pi$
$547$	10.5000	−	18.1865i	0.448948	−	0.777600i	0.998318	+	0.0579790i	$0.0184657\pi$
$548$	−9.50000	+	16.4545i	−0.405820	+	0.702901i
$549$		0			0
$550$	−0.500000	−	0.866025i	−0.0213201	−	0.0369274i
$551$	−14.0000	−	24.2487i	−0.596420	−	1.03303i
$552$		0			0
$553$		0			0
$554$	−1.00000	−	1.73205i	−0.0424859	−	0.0735878i
$555$		0			0
$556$	5.00000			0.212047
$557$	−14.0000	−	24.2487i	−0.593199	−	1.02745i	−0.993798	−	0.111198i	$-0.964531\pi$
$557$	−14.0000	−	24.2487i	−0.593199	−	1.02745i	0.400599	−	0.916253i	$-0.368802\pi$
$558$		0			0
$559$	−6.00000			−0.253773
$560$		0			0
$561$		0			0
$562$	−22.0000			−0.928014
$563$	−15.5000	+	26.8468i	−0.653247	+	1.13146i	0.329083	+	0.944301i	$0.393260\pi$
$563$	−15.5000	+	26.8468i	−0.653247	+	1.13146i	−0.982330	+	0.187156i	$0.940073\pi$
$564$		0			0
$565$	−10.0000	−	17.3205i	−0.420703	−	0.728679i
$566$	−4.00000			−0.168133
$567$		0			0
$568$	8.00000			0.335673
$569$	−7.50000	−	12.9904i	−0.314416	−	0.544585i	0.664897	−	0.746935i	$-0.268475\pi$
$569$	−7.50000	−	12.9904i	−0.314416	−	0.544585i	−0.979313	+	0.202350i	$0.935142\pi$
$570$		0			0
$571$	16.5000	−	28.5788i	0.690504	−	1.19599i	−0.281170	−	0.959658i	$-0.590722\pi$
$571$	16.5000	−	28.5788i	0.690504	−	1.19599i	0.971673	−	0.236329i	$-0.0759443\pi$
$572$	−6.00000			−0.250873
$573$		0			0
$574$		0			0
$575$	4.00000			0.166812
$576$		0			0
$577$	−17.5000	−	30.3109i	−0.728535	−	1.26186i	−0.957503	−	0.288425i	$-0.906868\pi$
$577$	−17.5000	−	30.3109i	−0.728535	−	1.26186i	0.228968	−	0.973434i	$-0.426465\pi$
$578$	8.00000			0.332756
$579$		0			0
$580$	−4.00000	−	6.92820i	−0.166091	−	0.287678i
$581$		0			0
$582$		0			0
$583$	−6.00000	−	10.3923i	−0.248495	−	0.430405i
$584$	0.500000	+	0.866025i	0.0206901	+	0.0358364i
$585$		0			0
$586$		0			0
$587$	23.5000	−	40.7032i	0.969949	−	1.68000i	0.274263	−	0.961655i	$-0.411566\pi$
$587$	23.5000	−	40.7032i	0.969949	−	1.68000i	0.695686	−	0.718346i	$-0.255100\pi$
$588$		0			0
$589$	−21.0000	−	36.3731i	−0.865290	−	1.49873i
$590$	−14.0000			−0.576371
$591$		0			0
$592$	2.00000			0.0821995
$593$	3.00000	−	5.19615i	0.123195	−	0.213380i	−0.797831	−	0.602881i	$-0.794019\pi$
$593$	3.00000	−	5.19615i	0.123195	−	0.213380i	0.921026	+	0.389501i	$0.127353\pi$
$594$		0			0
$595$		0			0
$596$	−12.0000	+	20.7846i	−0.491539	+	0.851371i
$597$		0			0
$598$	12.0000	−	20.7846i	0.490716	−	0.849946i
$599$	12.0000	−	20.7846i	0.490307	−	0.849236i	−0.509631	−	0.860393i	$-0.670218\pi$
$599$	12.0000	−	20.7846i	0.490307	−	0.849236i	0.999938	+	0.0111569i	$0.00355143\pi$
$600$		0			0
$601$	−9.50000	+	16.4545i	−0.387513	+	0.671192i	−0.992114	−	0.125336i	$-0.959999\pi$
$601$	−9.50000	+	16.4545i	−0.387513	+	0.671192i	0.604601	+	0.796528i	$0.293332\pi$
$602$		0			0
$603$		0			0
$604$	−5.00000	+	8.66025i	−0.203447	+	0.352381i
$605$	−20.0000			−0.813116
$606$		0			0
$607$	24.0000			0.974130			0.487065	−	0.873366i	$-0.338067\pi$
$607$	24.0000			0.974130			0.487065	+	0.873366i	$0.338067\pi$
$608$	−3.50000	−	6.06218i	−0.141944	−	0.245854i
$609$		0			0
$610$	−12.0000	+	20.7846i	−0.485866	+	0.841544i
$611$		0			0
$612$		0			0
$613$	−21.0000	−	36.3731i	−0.848182	−	1.46909i	−0.882829	−	0.469695i	$-0.844364\pi$
$613$	−21.0000	−	36.3731i	−0.848182	−	1.46909i	0.0346469	−	0.999400i	$-0.488969\pi$
$614$	3.50000	+	6.06218i	0.141249	+	0.244650i
$615$		0			0
$616$		0			0
$617$	−8.50000	−	14.7224i	−0.342197	−	0.592703i	0.642643	−	0.766165i	$-0.277838\pi$
$617$	−8.50000	−	14.7224i	−0.342197	−	0.592703i	−0.984840	+	0.173463i	$0.944504\pi$
$618$		0			0
$619$	−37.0000			−1.48716			−0.743578	−	0.668649i	$-0.766873\pi$
$619$	−37.0000			−1.48716			−0.743578	+	0.668649i	$0.766873\pi$
$620$	−6.00000	−	10.3923i	−0.240966	−	0.417365i
$621$		0			0
$622$	2.00000			0.0801927
$623$		0			0
$624$		0			0
$625$	−19.0000			−0.760000
$626$	−8.50000	+	14.7224i	−0.339728	+	0.588427i
$627$		0			0
$628$	1.00000	+	1.73205i	0.0399043	+	0.0691164i
$629$	−10.0000			−0.398726
$630$		0			0
$631$	8.00000			0.318475			0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000			0.318475			0.159237	+	0.987240i	$0.449096\pi$
$632$	3.00000	+	5.19615i	0.119334	+	0.206692i
$633$		0			0
$634$	−3.00000	+	5.19615i	−0.119145	+	0.206366i
$635$	−24.0000			−0.952411
$636$		0			0
$637$		0			0
$638$	−4.00000			−0.158362
$639$		0			0
$640$	−1.00000	−	1.73205i	−0.0395285	−	0.0684653i
$641$	−1.00000			−0.0394976			−0.0197488	−	0.999805i	$-0.506287\pi$
$641$	−1.00000			−0.0394976			−0.0197488	+	0.999805i	$0.506287\pi$
$642$		0			0
$643$	−3.50000	−	6.06218i	−0.138027	−	0.239069i	0.788723	−	0.614749i	$-0.210743\pi$
$643$	−3.50000	−	6.06218i	−0.138027	−	0.239069i	−0.926750	+	0.375680i	$0.877409\pi$
$644$		0			0
$645$		0			0
$646$	17.5000	+	30.3109i	0.688528	+	1.19257i
$647$	−6.00000	−	10.3923i	−0.235884	−	0.408564i	0.723645	−	0.690172i	$-0.242465\pi$
$647$	−6.00000	−	10.3923i	−0.235884	−	0.408564i	−0.959529	+	0.281609i	$0.909132\pi$
$648$		0			0
$649$	−3.50000	+	6.06218i	−0.137387	+	0.237961i
$650$	3.00000	−	5.19615i	0.117670	−	0.203810i
$651$		0			0
$652$	2.00000	+	3.46410i	0.0783260	+	0.135665i
$653$	6.00000			0.234798			0.117399	−	0.993085i	$-0.462544\pi$
$653$	6.00000			0.234798			0.117399	+	0.993085i	$0.462544\pi$
$654$		0			0
$655$	−8.00000			−0.312586
$656$	−1.50000	+	2.59808i	−0.0585652	+	0.101438i
$657$		0			0
$658$		0			0
$659$	8.00000	−	13.8564i	0.311636	−	0.539769i	−0.667081	−	0.744985i	$-0.732456\pi$
$659$	8.00000	−	13.8564i	0.311636	−	0.539769i	0.978717	+	0.205216i	$0.0657898\pi$
$660$		0			0
$661$	14.0000	−	24.2487i	0.544537	−	0.943166i	−0.454099	−	0.890951i	$-0.650039\pi$
$661$	14.0000	−	24.2487i	0.544537	−	0.943166i	0.998636	−	0.0522143i	$-0.0166279\pi$
$662$	−4.00000	+	6.92820i	−0.155464	+	0.269272i
$663$		0			0
$664$	−8.00000	+	13.8564i	−0.310460	+	0.537733i
$665$		0			0
$666$		0			0
$667$	8.00000	−	13.8564i	0.309761	−	0.536522i
$668$	20.0000			0.773823
$669$		0			0
$670$	26.0000			1.00447
$671$	6.00000	+	10.3923i	0.231627	+	0.401190i
$672$		0			0
$673$	−7.00000	+	12.1244i	−0.269830	+	0.467360i	−0.968818	−	0.247774i	$-0.920301\pi$
$673$	−7.00000	+	12.1244i	−0.269830	+	0.467360i	0.698988	+	0.715134i	$0.253634\pi$
$674$	4.50000	−	7.79423i	0.173334	−	0.300222i
$675$		0			0
$676$	−11.5000	−	19.9186i	−0.442308	−	0.766099i
$677$	−15.0000	−	25.9808i	−0.576497	−	0.998522i	−0.995877	−	0.0907112i	$-0.971086\pi$
$677$	−15.0000	−	25.9808i	−0.576497	−	0.998522i	0.419380	−	0.907811i	$-0.362247\pi$
$678$		0			0
$679$		0			0
$680$	5.00000	+	8.66025i	0.191741	+	0.332106i
$681$		0			0
$682$	−6.00000			−0.229752
$683$	19.5000	+	33.7750i	0.746147	+	1.29236i	0.949657	+	0.313291i	$0.101432\pi$
$683$	19.5000	+	33.7750i	0.746147	+	1.29236i	−0.203510	+	0.979073i	$0.565235\pi$
$684$		0			0
$685$	38.0000			1.45191
$686$		0			0
$687$		0			0
$688$	−1.00000			−0.0381246
$689$	36.0000	−	62.3538i	1.37149	−	2.37549i
$690$		0			0
$691$	16.0000	+	27.7128i	0.608669	+	1.05425i	0.991460	+	0.130410i	$0.0416295\pi$
$691$	16.0000	+	27.7128i	0.608669	+	1.05425i	−0.382791	+	0.923835i	$0.625037\pi$
$692$	2.00000			0.0760286
$693$		0			0
$694$	3.00000			0.113878
$695$	−5.00000	−	8.66025i	−0.189661	−	0.328502i
$696$		0			0
$697$	7.50000	−	12.9904i	0.284083	−	0.492046i
$698$	14.0000			0.529908
$699$		0			0
$700$		0			0
$701$	−8.00000			−0.302156			−0.151078	−	0.988522i	$-0.548274\pi$
$701$	−8.00000			−0.302156			−0.151078	+	0.988522i	$0.548274\pi$
$702$		0			0
$703$	−7.00000	−	12.1244i	−0.264010	−	0.457279i
$704$	−1.00000			−0.0376889
$705$		0			0
$706$	7.50000	+	12.9904i	0.282266	+	0.488899i
$707$		0			0
$708$		0			0
$709$	2.00000	+	3.46410i	0.0751116	+	0.130097i	0.901135	−	0.433539i	$-0.142735\pi$
$709$	2.00000	+	3.46410i	0.0751116	+	0.130097i	−0.826023	+	0.563636i	$0.809402\pi$
$710$	−8.00000	−	13.8564i	−0.300235	−	0.520022i
$711$		0			0
$712$	3.00000	−	5.19615i	0.112430	−	0.194734i
$713$	12.0000	−	20.7846i	0.449404	−	0.778390i
$714$		0			0
$715$	6.00000	+	10.3923i	0.224387	+	0.388650i
$716$	−24.0000			−0.896922
$717$		0			0
$718$	−2.00000			−0.0746393
$719$	−3.00000	+	5.19615i	−0.111881	+	0.193784i	−0.916529	−	0.399969i	$-0.869021\pi$
$719$	−3.00000	+	5.19615i	−0.111881	+	0.193784i	0.804648	+	0.593753i	$0.202354\pi$
$720$		0			0
$721$		0			0
$722$	−15.0000	+	25.9808i	−0.558242	+	0.966904i
$723$		0			0
$724$		0			0
$725$	2.00000	−	3.46410i	0.0742781	−	0.128654i
$726$		0			0
$727$	−7.00000	+	12.1244i	−0.259616	+	0.449667i	−0.966139	−	0.258022i	$-0.916929\pi$
$727$	−7.00000	+	12.1244i	−0.259616	+	0.449667i	0.706523	+	0.707690i	$0.250263\pi$
$728$		0			0
$729$		0			0
$730$	1.00000	−	1.73205i	0.0370117	−	0.0641061i
$731$	5.00000			0.184932
$732$		0			0
$733$	18.0000			0.664845			0.332423	−	0.943131i	$-0.392134\pi$
$733$	18.0000			0.664845			0.332423	+	0.943131i	$0.392134\pi$
$734$	−11.0000	−	19.0526i	−0.406017	−	0.703243i
$735$		0			0
$736$	2.00000	−	3.46410i	0.0737210	−	0.127688i
$737$	6.50000	−	11.2583i	0.239431	−	0.414706i
$738$		0			0
$739$	16.5000	+	28.5788i	0.606962	+	1.05129i	0.991738	+	0.128279i	$0.0409454\pi$
$739$	16.5000	+	28.5788i	0.606962	+	1.05129i	−0.384776	+	0.923010i	$0.625721\pi$
$740$	−2.00000	−	3.46410i	−0.0735215	−	0.127343i
$741$		0			0
$742$		0			0
$743$	−3.00000	−	5.19615i	−0.110059	−	0.190628i	0.805735	−	0.592277i	$-0.201771\pi$
$743$	−3.00000	−	5.19615i	−0.110059	−	0.190628i	−0.915794	+	0.401648i	$0.868437\pi$
$744$		0			0
$745$	48.0000			1.75858
$746$	−11.0000	−	19.0526i	−0.402739	−	0.697564i
$747$		0			0
$748$	5.00000			0.182818
$749$		0			0
$750$		0			0
$751$	−18.0000			−0.656829			−0.328415	−	0.944534i	$-0.606514\pi$
$751$	−18.0000			−0.656829			−0.328415	+	0.944534i	$0.606514\pi$
$752$		0			0
$753$		0			0
$754$	−12.0000	−	20.7846i	−0.437014	−	0.756931i
$755$	20.0000			0.727875
$756$		0			0
$757$	−48.0000			−1.74459			−0.872295	−	0.488980i	$-0.837369\pi$
$757$	−48.0000			−1.74459			−0.872295	+	0.488980i	$0.837369\pi$
$758$	8.50000	+	14.7224i	0.308734	+	0.534743i
$759$		0			0
$760$	−7.00000	+	12.1244i	−0.253917	+	0.439797i
$761$	10.0000			0.362500			0.181250	−	0.983437i	$-0.441986\pi$
$761$	10.0000			0.362500			0.181250	+	0.983437i	$0.441986\pi$
$762$		0			0
$763$		0			0
$764$	−12.0000			−0.434145
$765$		0			0
$766$	−2.00000	−	3.46410i	−0.0722629	−	0.125163i
$767$	−42.0000			−1.51653
$768$		0			0
$769$	11.0000	+	19.0526i	0.396670	+	0.687053i	0.993313	−	0.115454i	$-0.0368323\pi$
$769$	11.0000	+	19.0526i	0.396670	+	0.687053i	−0.596643	+	0.802507i	$0.703499\pi$
$770$		0			0
$771$		0			0
$772$	−8.50000	−	14.7224i	−0.305922	−	0.529872i
$773$	26.0000	+	45.0333i	0.935155	+	1.61974i	0.774357	+	0.632749i	$0.218073\pi$
$773$	26.0000	+	45.0333i	0.935155	+	1.61974i	0.160798	+	0.986987i	$0.448593\pi$
$774$		0			0
$775$	3.00000	−	5.19615i	0.107763	−	0.186651i
$776$	−2.50000	+	4.33013i	−0.0897448	+	0.155443i
$777$		0			0
$778$	4.00000	+	6.92820i	0.143407	+	0.248388i
$779$	21.0000			0.752403
$780$		0			0
$781$	−8.00000			−0.286263
$782$	−10.0000	+	17.3205i	−0.357599	+	0.619380i
$783$		0			0
$784$		0			0
$785$	2.00000	−	3.46410i	0.0713831	−	0.123639i
$786$		0			0
$787$	−6.00000	+	10.3923i	−0.213877	+	0.370446i	−0.952925	−	0.303207i	$-0.901942\pi$
$787$	−6.00000	+	10.3923i	−0.213877	+	0.370446i	0.739048	+	0.673653i	$0.235276\pi$
$788$	5.00000	−	8.66025i	0.178118	−	0.308509i
$789$		0			0
$790$	6.00000	−	10.3923i	0.213470	−	0.369742i
$791$		0			0
$792$		0			0
$793$	−36.0000	+	62.3538i

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:	\( N \)	\(=\)	\( 2646 = 2 \cdot 3^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2646.h (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 - 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +2.00000 q^{5} +1.00000 q^{8} +O(q^{10})\)	\(q+(-0.500000 - 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +2.00000 q^{5} +1.00000 q^{8} +(-1.00000 - 1.73205i) q^{10} -1.00000 q^{11} +(-3.00000 - 5.19615i) q^{13} +(-0.500000 - 0.866025i) q^{16} +(2.50000 + 4.33013i) q^{17} +(-3.50000 + 6.06218i) q^{19} +(-1.00000 + 1.73205i) q^{20} +(0.500000 + 0.866025i) q^{22} -4.00000 q^{23} -1.00000 q^{25} +(-3.00000 + 5.19615i) q^{26} +(-2.00000 + 3.46410i) q^{29} +(-3.00000 + 5.19615i) q^{31} +(-0.500000 + 0.866025i) q^{32} +(2.50000 - 4.33013i) q^{34} +(-1.00000 + 1.73205i) q^{37} +7.00000 q^{38} +2.00000 q^{40} +(-1.50000 - 2.59808i) q^{41} +(0.500000 - 0.866025i) q^{43} +(0.500000 - 0.866025i) q^{44} +(2.00000 + 3.46410i) q^{46} +(0.500000 + 0.866025i) q^{50} +6.00000 q^{52} +(6.00000 + 10.3923i) q^{53} -2.00000 q^{55} +4.00000 q^{58} +(3.50000 - 6.06218i) q^{59} +(-6.00000 - 10.3923i) q^{61} +6.00000 q^{62} +1.00000 q^{64} +(-6.00000 - 10.3923i) q^{65} +(-6.50000 + 11.2583i) q^{67} -5.00000 q^{68} +8.00000 q^{71} +(0.500000 + 0.866025i) q^{73} +2.00000 q^{74} +(-3.50000 - 6.06218i) q^{76} +(3.00000 + 5.19615i) q^{79} +(-1.00000 - 1.73205i) q^{80} +(-1.50000 + 2.59808i) q^{82} +(-8.00000 + 13.8564i) q^{83} +(5.00000 + 8.66025i) q^{85} -1.00000 q^{86} -1.00000 q^{88} +(3.00000 - 5.19615i) q^{89} +(2.00000 - 3.46410i) q^{92} +(-7.00000 + 12.1244i) q^{95} +(-2.50000 + 4.33013i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} - q^{4} + 4 q^{5} + 2 q^{8}+O(q^{10}) \) 2 * q - q^2 - q^4 + 4 * q^5 + 2 * q^8	\( 2 q - q^{2} - q^{4} + 4 q^{5} + 2 q^{8} - 2 q^{10} - 2 q^{11} - 6 q^{13} - q^{16} + 5 q^{17} - 7 q^{19} - 2 q^{20} + q^{22} - 8 q^{23} - 2 q^{25} - 6 q^{26} - 4 q^{29} - 6 q^{31} - q^{32} + 5 q^{34} - 2 q^{37} + 14 q^{38} + 4 q^{40} - 3 q^{41} + q^{43} + q^{44} + 4 q^{46} + q^{50} + 12 q^{52} + 12 q^{53} - 4 q^{55} + 8 q^{58} + 7 q^{59} - 12 q^{61} + 12 q^{62} + 2 q^{64} - 12 q^{65} - 13 q^{67} - 10 q^{68} + 16 q^{71} + q^{73} + 4 q^{74} - 7 q^{76} + 6 q^{79} - 2 q^{80} - 3 q^{82} - 16 q^{83} + 10 q^{85} - 2 q^{86} - 2 q^{88} + 6 q^{89} + 4 q^{92} - 14 q^{95} - 5 q^{97}+O(q^{100}) \) 2 * q - q^2 - q^4 + 4 * q^5 + 2 * q^8 - 2 * q^10 - 2 * q^11 - 6 * q^13 - q^16 + 5 * q^17 - 7 * q^19 - 2 * q^20 + q^22 - 8 * q^23 - 2 * q^25 - 6 * q^26 - 4 * q^29 - 6 * q^31 - q^32 + 5 * q^34 - 2 * q^37 + 14 * q^38 + 4 * q^40 - 3 * q^41 + q^43 + q^44 + 4 * q^46 + q^50 + 12 * q^52 + 12 * q^53 - 4 * q^55 + 8 * q^58 + 7 * q^59 - 12 * q^61 + 12 * q^62 + 2 * q^64 - 12 * q^65 - 13 * q^67 - 10 * q^68 + 16 * q^71 + q^73 + 4 * q^74 - 7 * q^76 + 6 * q^79 - 2 * q^80 - 3 * q^82 - 16 * q^83 + 10 * q^85 - 2 * q^86 - 2 * q^88 + 6 * q^89 + 4 * q^92 - 14 * q^95 - 5 * q^97

Introduction

L-functions

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Varieties

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Groups

Database

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