LMFDB - Embedded newform 2646.2.h.d.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:	$x^{2} - x + 1$
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.d.361.1

$q$-expansion

$f(q)$	$=$	$q+(-0.500000 - 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +3.00000 q^{5} +1.00000 q^{8} +O(q^{10})$	\(q+(-0.500000 - 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +3.00000 q^{5} +1.00000 q^{8} +(-1.50000 - 2.59808i) q^{10} +3.00000 q^{11} +(-0.500000 - 0.866025i) q^{13} +(-0.500000 - 0.866025i) q^{16} +(-1.50000 - 2.59808i) q^{17} +(-3.50000 + 6.06218i) q^{19} +(-1.50000 + 2.59808i) q^{20} +(-1.50000 - 2.59808i) q^{22} +9.00000 q^{23} +4.00000 q^{25} +(-0.500000 + 0.866025i) q^{26} +(1.50000 - 2.59808i) q^{29} +(4.00000 - 6.92820i) q^{31} +(-0.500000 + 0.866025i) q^{32} +(-1.50000 + 2.59808i) q^{34} +(0.500000 - 0.866025i) q^{37} +7.00000 q^{38} +3.00000 q^{40} +(-1.50000 - 2.59808i) q^{41} +(0.500000 - 0.866025i) q^{43} +(-1.50000 + 2.59808i) q^{44} +(-4.50000 - 7.79423i) q^{46} +(-2.00000 - 3.46410i) q^{50} +1.00000 q^{52} +(1.50000 + 2.59808i) q^{53} +9.00000 q^{55} -3.00000 q^{58} +(1.00000 + 1.73205i) q^{61} -8.00000 q^{62} +1.00000 q^{64} +(-1.50000 - 2.59808i) q^{65} +(2.00000 - 3.46410i) q^{67} +3.00000 q^{68} -12.0000 q^{71} +(5.50000 + 9.52628i) q^{73} -1.00000 q^{74} +(-3.50000 - 6.06218i) q^{76} +(8.00000 + 13.8564i) q^{79} +(-1.50000 - 2.59808i) q^{80} +(-1.50000 + 2.59808i) q^{82} +(4.50000 - 7.79423i) q^{83} +(-4.50000 - 7.79423i) q^{85} -1.00000 q^{86} +3.00000 q^{88} +(-1.50000 + 2.59808i) q^{89} +(-4.50000 + 7.79423i) q^{92} +(-10.5000 + 18.1865i) q^{95} +(-0.500000 + 0.866025i) q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} - q^{4} + 6 q^{5} + 2 q^{8} + O(q^{10}) $	$ 2 q - q^{2} - q^{4} + 6 q^{5} + 2 q^{8} - 3 q^{10} + 6 q^{11} - q^{13} - q^{16} - 3 q^{17} - 7 q^{19} - 3 q^{20} - 3 q^{22} + 18 q^{23} + 8 q^{25} - q^{26} + 3 q^{29} + 8 q^{31} - q^{32} - 3 q^{34} + q^{37} + 14 q^{38} + 6 q^{40} - 3 q^{41} + q^{43} - 3 q^{44} - 9 q^{46} - 4 q^{50} + 2 q^{52} + 3 q^{53} + 18 q^{55} - 6 q^{58} + 2 q^{61} - 16 q^{62} + 2 q^{64} - 3 q^{65} + 4 q^{67} + 6 q^{68} - 24 q^{71} + 11 q^{73} - 2 q^{74} - 7 q^{76} + 16 q^{79} - 3 q^{80} - 3 q^{82} + 9 q^{83} - 9 q^{85} - 2 q^{86} + 6 q^{88} - 3 q^{89} - 9 q^{92} - 21 q^{95} - q^{97} + O(q^{100}) $

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
$5$	3.00000			1.34164			0.670820	−	0.741620i	$-0.265942\pi$
$5$	3.00000			1.34164			0.670820	+	0.741620i	$0.265942\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$	1.00000			0.353553
$9$		0			0
$10$	−1.50000	−	2.59808i	−0.474342	−	0.821584i
$11$	3.00000			0.904534			0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000			0.904534			0.452267	+	0.891883i	$0.350615\pi$
$12$		0			0
$13$	−0.500000	−	0.866025i	−0.138675	−	0.240192i	0.788320	−	0.615265i	$-0.210951\pi$
$13$	−0.500000	−	0.866025i	−0.138675	−	0.240192i	−0.926995	+	0.375073i	$0.877618\pi$
$14$		0			0
$15$		0			0
$16$	−0.500000	−	0.866025i	−0.125000	−	0.216506i
$17$	−1.50000	−	2.59808i	−0.363803	−	0.630126i	0.624780	−	0.780801i	$-0.285189\pi$
$17$	−1.50000	−	2.59808i	−0.363803	−	0.630126i	−0.988583	+	0.150675i	$0.951855\pi$
$18$		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	$0.869927\pi$
$20$	−1.50000	+	2.59808i	−0.335410	+	0.580948i
$21$		0			0
$22$	−1.50000	−	2.59808i	−0.319801	−	0.553912i
$23$	9.00000			1.87663			0.938315	−	0.345782i	$-0.112386\pi$
$23$	9.00000			1.87663			0.938315	+	0.345782i	$0.112386\pi$
$24$		0			0
$25$	4.00000			0.800000
$26$	−0.500000	+	0.866025i	−0.0980581	+	0.169842i
$27$		0			0
$28$		0			0
$29$	1.50000	−	2.59808i	0.278543	−	0.482451i	−0.692480	−	0.721437i	$-0.743482\pi$
$29$	1.50000	−	2.59808i	0.278543	−	0.482451i	0.971023	+	0.238987i	$0.0768152\pi$
$30$		0			0
$31$	4.00000	−	6.92820i	0.718421	−	1.24434i	−0.243204	−	0.969975i	$-0.578198\pi$
$31$	4.00000	−	6.92820i	0.718421	−	1.24434i	0.961625	−	0.274367i	$-0.0884683\pi$
$32$	−0.500000	+	0.866025i	−0.0883883	+	0.153093i
$33$		0			0
$34$	−1.50000	+	2.59808i	−0.257248	+	0.445566i
$35$		0			0
$36$		0			0
$37$	0.500000	−	0.866025i	0.0821995	−	0.142374i	−0.821995	−	0.569495i	$-0.807139\pi$
$37$	0.500000	−	0.866025i	0.0821995	−	0.142374i	0.904194	+	0.427121i	$0.140472\pi$
$38$	7.00000			1.13555
$39$		0			0
$40$	3.00000			0.474342
$41$	−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	$0.142372\pi$
$44$	−1.50000	+	2.59808i	−0.226134	+	0.391675i
$45$		0			0
$46$	−4.50000	−	7.79423i	−0.663489	−	1.14920i
$47$		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
$50$	−2.00000	−	3.46410i	−0.282843	−	0.489898i
$51$		0			0
$52$	1.00000			0.138675
$53$	1.50000	+	2.59808i	0.206041	+	0.356873i	0.950464	−	0.310835i	$-0.100609\pi$
$53$	1.50000	+	2.59808i	0.206041	+	0.356873i	−0.744423	+	0.667708i	$0.767275\pi$
$54$		0			0
$55$	9.00000			1.21356
$56$		0			0
$57$		0			0
$58$	−3.00000			−0.393919
$59$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$59$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$60$		0			0
$61$	1.00000	+	1.73205i	0.128037	+	0.221766i	0.922916	−	0.385002i	$-0.125799\pi$
$61$	1.00000	+	1.73205i	0.128037	+	0.221766i	−0.794879	+	0.606768i	$0.792466\pi$
$62$	−8.00000			−1.01600
$63$		0			0
$64$	1.00000			0.125000
$65$	−1.50000	−	2.59808i	−0.186052	−	0.322252i
$66$		0			0
$67$	2.00000	−	3.46410i	0.244339	−	0.423207i	−0.717607	−	0.696449i	$-0.754762\pi$
$67$	2.00000	−	3.46410i	0.244339	−	0.423207i	0.961946	+	0.273241i	$0.0880957\pi$
$68$	3.00000			0.363803
$69$		0			0
$70$		0			0
$71$	−12.0000			−1.42414			−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000			−1.42414			−0.712069	+	0.702109i	$0.752242\pi$
$72$		0			0
$73$	5.50000	+	9.52628i	0.643726	+	1.11497i	0.984594	+	0.174855i	$0.0559458\pi$
$73$	5.50000	+	9.52628i	0.643726	+	1.11497i	−0.340868	+	0.940111i	$0.610721\pi$
$74$	−1.00000			−0.116248
$75$		0			0
$76$	−3.50000	−	6.06218i	−0.401478	−	0.695379i
$77$		0			0
$78$		0			0
$79$	8.00000	+	13.8564i	0.900070	+	1.55897i	0.827401	+	0.561611i	$0.189818\pi$
$79$	8.00000	+	13.8564i	0.900070	+	1.55897i	0.0726692	+	0.997356i	$0.476848\pi$
$80$	−1.50000	−	2.59808i	−0.167705	−	0.290474i
$81$		0			0
$82$	−1.50000	+	2.59808i	−0.165647	+	0.286910i
$83$	4.50000	−	7.79423i	0.493939	−	0.855528i	−0.506036	−	0.862512i	$-0.668890\pi$
$83$	4.50000	−	7.79423i	0.493939	−	0.855528i	0.999976	+	0.00698436i	$0.00222321\pi$
$84$		0			0
$85$	−4.50000	−	7.79423i	−0.488094	−	0.845403i
$86$	−1.00000			−0.107833
$87$		0			0
$88$	3.00000			0.319801
$89$	−1.50000	+	2.59808i	−0.159000	+	0.275396i	−0.934508	−	0.355942i	$-0.884160\pi$
$89$	−1.50000	+	2.59808i	−0.159000	+	0.275396i	0.775509	+	0.631337i	$0.217494\pi$
$90$		0			0
$91$		0			0
$92$	−4.50000	+	7.79423i	−0.469157	+	0.812605i
$93$		0			0
$94$		0			0
$95$	−10.5000	+	18.1865i	−1.07728	+	1.86590i
$96$		0			0
$97$	−0.500000	+	0.866025i	−0.0507673	+	0.0879316i	−0.890292	−	0.455389i	$-0.849500\pi$
$97$	−0.500000	+	0.866025i	−0.0507673	+	0.0879316i	0.839525	+	0.543321i	$0.182833\pi$
$98$		0			0
$99$		0			0
$100$	−2.00000	+	3.46410i	−0.200000	+	0.346410i
$101$	3.00000			0.298511			0.149256	−	0.988799i	$-0.452312\pi$
$101$	3.00000			0.298511			0.149256	+	0.988799i	$0.452312\pi$
$102$		0			0
$103$	13.0000			1.28093			0.640464	−	0.767988i	$-0.278742\pi$
$103$	13.0000			1.28093			0.640464	+	0.767988i	$0.278742\pi$
$104$	−0.500000	−	0.866025i	−0.0490290	−	0.0849208i
$105$		0			0
$106$	1.50000	−	2.59808i	0.145693	−	0.252347i
$107$	4.50000	−	7.79423i	0.435031	−	0.753497i	−0.562267	−	0.826956i	$-0.690071\pi$
$107$	4.50000	−	7.79423i	0.435031	−	0.753497i	0.997298	+	0.0734594i	$0.0234039\pi$
$108$		0			0
$109$	6.50000	+	11.2583i	0.622587	+	1.07835i	0.989002	+	0.147901i	$0.0472517\pi$
$109$	6.50000	+	11.2583i	0.622587	+	1.07835i	−0.366415	+	0.930451i	$0.619415\pi$
$110$	−4.50000	−	7.79423i	−0.429058	−	0.743151i
$111$		0			0
$112$		0			0
$113$	−4.50000	−	7.79423i	−0.423324	−	0.733219i	0.572938	−	0.819599i	$-0.305804\pi$
$113$	−4.50000	−	7.79423i	−0.423324	−	0.733219i	−0.996262	+	0.0863794i	$0.972470\pi$
$114$		0			0
$115$	27.0000			2.51776
$116$	1.50000	+	2.59808i	0.139272	+	0.241225i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−2.00000			−0.181818
$122$	1.00000	−	1.73205i	0.0905357	−	0.156813i
$123$		0			0
$124$	4.00000	+	6.92820i	0.359211	+	0.622171i
$125$	−3.00000			−0.268328
$126$		0			0
$127$	−4.00000			−0.354943			−0.177471	−	0.984126i	$-0.556792\pi$
$127$	−4.00000			−0.354943			−0.177471	+	0.984126i	$0.556792\pi$
$128$	−0.500000	−	0.866025i	−0.0441942	−	0.0765466i
$129$		0			0
$130$	−1.50000	+	2.59808i	−0.131559	+	0.227866i
$131$	15.0000			1.31056			0.655278	−	0.755388i	$-0.272551\pi$
$131$	15.0000			1.31056			0.655278	+	0.755388i	$0.272551\pi$
$132$		0			0
$133$		0			0
$134$	−4.00000			−0.345547
$135$		0			0
$136$	−1.50000	−	2.59808i	−0.128624	−	0.222783i
$137$	9.00000			0.768922			0.384461	−	0.923141i	$-0.374387\pi$
$137$	9.00000			0.768922			0.384461	+	0.923141i	$0.374387\pi$
$138$		0			0
$139$	−3.50000	−	6.06218i	−0.296866	−	0.514187i	0.678551	−	0.734553i	$-0.262608\pi$
$139$	−3.50000	−	6.06218i	−0.296866	−	0.514187i	−0.975417	+	0.220366i	$0.929275\pi$
$140$		0			0
$141$		0			0
$142$	6.00000	+	10.3923i	0.503509	+	0.872103i
$143$	−1.50000	−	2.59808i	−0.125436	−	0.217262i
$144$		0			0
$145$	4.50000	−	7.79423i	0.373705	−	0.647275i
$146$	5.50000	−	9.52628i	0.455183	−	0.788400i
$147$		0			0
$148$	0.500000	+	0.866025i	0.0410997	+	0.0711868i
$149$	9.00000			0.737309			0.368654	−	0.929567i	$-0.379819\pi$
$149$	9.00000			0.737309			0.368654	+	0.929567i	$0.379819\pi$
$150$		0			0
$151$	−7.00000			−0.569652			−0.284826	−	0.958579i	$-0.591936\pi$
$151$	−7.00000			−0.569652			−0.284826	+	0.958579i	$0.591936\pi$
$152$	−3.50000	+	6.06218i	−0.283887	+	0.491708i
$153$		0			0
$154$		0			0
$155$	12.0000	−	20.7846i	0.963863	−	1.66946i
$156$		0			0
$157$	−11.0000	+	19.0526i	−0.877896	+	1.52056i	−0.0242497	+	0.999706i	$0.507720\pi$
$157$	−11.0000	+	19.0526i	−0.877896	+	1.52056i	−0.853646	+	0.520854i	$0.825614\pi$
$158$	8.00000	−	13.8564i	0.636446	−	1.10236i
$159$		0			0
$160$	−1.50000	+	2.59808i	−0.118585	+	0.205396i
$161$		0			0
$162$		0			0
$163$	9.50000	−	16.4545i	0.744097	−	1.28881i	−0.206518	−	0.978443i	$-0.566213\pi$
$163$	9.50000	−	16.4545i	0.744097	−	1.28881i	0.950615	−	0.310372i	$-0.100454\pi$
$164$	3.00000			0.234261
$165$		0			0
$166$	−9.00000			−0.698535
$167$	7.50000	+	12.9904i	0.580367	+	1.00523i	0.995436	+	0.0954356i	$0.0304244\pi$
$167$	7.50000	+	12.9904i	0.580367	+	1.00523i	−0.415068	+	0.909790i	$0.636242\pi$
$168$		0			0
$169$	6.00000	−	10.3923i	0.461538	−	0.799408i
$170$	−4.50000	+	7.79423i	−0.345134	+	0.597790i
$171$		0			0
$172$	0.500000	+	0.866025i	0.0381246	+	0.0660338i
$173$	3.00000	+	5.19615i	0.228086	+	0.395056i	0.957241	−	0.289292i	$-0.0934200\pi$
$173$	3.00000	+	5.19615i	0.228086	+	0.395056i	−0.729155	+	0.684349i	$0.760087\pi$
$174$		0			0
$175$		0			0
$176$	−1.50000	−	2.59808i	−0.113067	−	0.195837i
$177$		0			0
$178$	3.00000			0.224860
$179$	−10.5000	−	18.1865i	−0.784807	−	1.35933i	−0.929114	−	0.369792i	$-0.879429\pi$
$179$	−10.5000	−	18.1865i	−0.784807	−	1.35933i	0.144308	−	0.989533i	$-0.453905\pi$
$180$		0			0
$181$	−2.00000			−0.148659			−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000			−0.148659			−0.0743294	+	0.997234i	$0.523682\pi$
$182$		0			0
$183$		0			0
$184$	9.00000			0.663489
$185$	1.50000	−	2.59808i	0.110282	−	0.191014i
$186$		0			0
$187$	−4.50000	−	7.79423i	−0.329073	−	0.569970i
$188$		0			0
$189$		0			0
$190$	21.0000			1.52350
$191$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$191$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$192$		0			0
$193$	−7.00000	+	12.1244i	−0.503871	+	0.872730i	0.496119	+	0.868255i	$0.334758\pi$
$193$	−7.00000	+	12.1244i	−0.503871	+	0.872730i	−0.999990	+	0.00447566i	$0.998575\pi$
$194$	1.00000			0.0717958
$195$		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	$0.721573\pi$
$198$		0			0
$199$	−12.5000	−	21.6506i	−0.886102	−	1.53477i	−0.844446	−	0.535641i	$-0.820070\pi$
$199$	−12.5000	−	21.6506i	−0.886102	−	1.53477i	−0.0416556	−	0.999132i	$-0.513263\pi$
$200$	4.00000			0.282843
$201$		0			0
$202$	−1.50000	−	2.59808i	−0.105540	−	0.182800i
$203$		0			0
$204$		0			0
$205$	−4.50000	−	7.79423i	−0.314294	−	0.544373i
$206$	−6.50000	−	11.2583i	−0.452876	−	0.784405i
$207$		0			0
$208$	−0.500000	+	0.866025i	−0.0346688	+	0.0600481i
$209$	−10.5000	+	18.1865i	−0.726300	+	1.25799i
$210$		0			0
$211$	−2.50000	−	4.33013i	−0.172107	−	0.298098i	0.767049	−	0.641588i	$-0.221724\pi$
$211$	−2.50000	−	4.33013i	−0.172107	−	0.298098i	−0.939156	+	0.343490i	$0.888391\pi$
$212$	−3.00000			−0.206041
$213$		0			0
$214$	−9.00000			−0.615227
$215$	1.50000	−	2.59808i	0.102299	−	0.177187i
$216$		0			0
$217$		0			0
$218$	6.50000	−	11.2583i	0.440236	−	0.762510i
$219$		0			0
$220$	−4.50000	+	7.79423i	−0.303390	+	0.525487i
$221$	−1.50000	+	2.59808i	−0.100901	+	0.174766i
$222$		0			0
$223$	−0.500000	+	0.866025i	−0.0334825	+	0.0579934i	−0.882281	−	0.470723i	$-0.843993\pi$
$223$	−0.500000	+	0.866025i	−0.0334825	+	0.0579934i	0.848799	+	0.528716i	$0.177326\pi$
$224$		0			0
$225$		0			0
$226$	−4.50000	+	7.79423i	−0.299336	+	0.518464i
$227$	−3.00000			−0.199117			−0.0995585	−	0.995032i	$-0.531743\pi$
$227$	−3.00000			−0.199117			−0.0995585	+	0.995032i	$0.531743\pi$
$228$		0			0
$229$	13.0000			0.859064			0.429532	−	0.903052i	$-0.358679\pi$
$229$	13.0000			0.859064			0.429532	+	0.903052i	$0.358679\pi$
$230$	−13.5000	−	23.3827i	−0.890164	−	1.54181i
$231$		0			0
$232$	1.50000	−	2.59808i	0.0984798	−	0.170572i
$233$	1.50000	−	2.59808i	0.0982683	−	0.170206i	−0.812700	−	0.582683i	$-0.802003\pi$
$233$	1.50000	−	2.59808i	0.0982683	−	0.170206i	0.910968	+	0.412477i	$0.135336\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−1.50000	−	2.59808i	−0.0970269	−	0.168056i	0.813426	−	0.581669i	$-0.197600\pi$
$239$	−1.50000	−	2.59808i	−0.0970269	−	0.168056i	−0.910453	+	0.413613i	$0.864267\pi$
$240$		0			0
$241$	13.0000			0.837404			0.418702	−	0.908124i	$-0.362485\pi$
$241$	13.0000			0.837404			0.418702	+	0.908124i	$0.362485\pi$
$242$	1.00000	+	1.73205i	0.0642824	+	0.111340i
$243$		0			0
$244$	−2.00000			−0.128037
$245$		0			0
$246$		0			0
$247$	7.00000			0.445399
$248$	4.00000	−	6.92820i	0.254000	−	0.439941i
$249$		0			0
$250$	1.50000	+	2.59808i	0.0948683	+	0.164317i
$251$	12.0000			0.757433			0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000			0.757433			0.378717	+	0.925513i	$0.376365\pi$
$252$		0			0
$253$	27.0000			1.69748
$254$	2.00000	+	3.46410i	0.125491	+	0.217357i
$255$		0			0
$256$	−0.500000	+	0.866025i	−0.0312500	+	0.0541266i
$257$	−21.0000			−1.30994			−0.654972	−	0.755653i	$-0.727320\pi$
$257$	−21.0000			−1.30994			−0.654972	+	0.755653i	$0.727320\pi$
$258$		0			0
$259$		0			0
$260$	3.00000			0.186052
$261$		0			0
$262$	−7.50000	−	12.9904i	−0.463352	−	0.802548i
$263$	−9.00000			−0.554964			−0.277482	−	0.960731i	$-0.589500\pi$
$263$	−9.00000			−0.554964			−0.277482	+	0.960731i	$0.589500\pi$
$264$		0			0
$265$	4.50000	+	7.79423i	0.276433	+	0.478796i
$266$		0			0
$267$		0			0
$268$	2.00000	+	3.46410i	0.122169	+	0.211604i
$269$	−7.50000	−	12.9904i	−0.457283	−	0.792038i	0.541533	−	0.840679i	$-0.317844\pi$
$269$	−7.50000	−	12.9904i	−0.457283	−	0.792038i	−0.998816	+	0.0486418i	$0.984511\pi$
$270$		0			0
$271$	2.50000	−	4.33013i	0.151864	−	0.263036i	−0.780049	−	0.625719i	$-0.784806\pi$
$271$	2.50000	−	4.33013i	0.151864	−	0.263036i	0.931913	+	0.362682i	$0.118139\pi$
$272$	−1.50000	+	2.59808i	−0.0909509	+	0.157532i
$273$		0			0
$274$	−4.50000	−	7.79423i	−0.271855	−	0.470867i
$275$	12.0000			0.723627
$276$		0			0
$277$	−1.00000			−0.0600842			−0.0300421	−	0.999549i	$-0.509564\pi$
$277$	−1.00000			−0.0600842			−0.0300421	+	0.999549i	$0.509564\pi$
$278$	−3.50000	+	6.06218i	−0.209916	+	0.363585i
$279$		0			0
$280$		0			0
$281$	−10.5000	+	18.1865i	−0.626377	+	1.08492i	0.361895	+	0.932219i	$0.382130\pi$
$281$	−10.5000	+	18.1865i	−0.626377	+	1.08492i	−0.988273	+	0.152699i	$0.951204\pi$
$282$		0			0
$283$	−2.00000	+	3.46410i	−0.118888	+	0.205919i	−0.919327	−	0.393494i	$-0.871266\pi$
$283$	−2.00000	+	3.46410i	−0.118888	+	0.205919i	0.800439	+	0.599414i	$0.204600\pi$
$284$	6.00000	−	10.3923i	0.356034	−	0.616670i
$285$		0			0
$286$	−1.50000	+	2.59808i	−0.0886969	+	0.153627i
$287$		0			0
$288$		0			0
$289$	4.00000	−	6.92820i	0.235294	−	0.407541i
$290$	−9.00000			−0.528498
$291$		0			0
$292$	−11.0000			−0.643726
$293$	4.50000	+	7.79423i	0.262893	+	0.455344i	0.967009	−	0.254741i	$-0.0819901\pi$
$293$	4.50000	+	7.79423i	0.262893	+	0.455344i	−0.704117	+	0.710084i	$0.748657\pi$
$294$		0			0
$295$		0			0
$296$	0.500000	−	0.866025i	0.0290619	−	0.0503367i
$297$		0			0
$298$	−4.50000	−	7.79423i	−0.260678	−	0.451508i
$299$	−4.50000	−	7.79423i	−0.260242	−	0.450752i
$300$		0			0
$301$		0			0
$302$	3.50000	+	6.06218i	0.201402	+	0.348839i
$303$		0			0
$304$	7.00000			0.401478
$305$	3.00000	+	5.19615i	0.171780	+	0.297531i
$306$		0			0
$307$	28.0000			1.59804			0.799022	−	0.601302i	$-0.205351\pi$
$307$	28.0000			1.59804			0.799022	+	0.601302i	$0.205351\pi$
$308$		0			0
$309$		0			0
$310$	−24.0000			−1.36311
$311$	12.0000	−	20.7846i	0.680458	−	1.17859i	−0.294384	−	0.955687i	$-0.595114\pi$
$311$	12.0000	−	20.7846i	0.680458	−	1.17859i	0.974841	−	0.222900i	$-0.0715523\pi$
$312$		0			0
$313$	−5.00000	−	8.66025i	−0.282617	−	0.489506i	0.689412	−	0.724370i	$-0.257869\pi$
$313$	−5.00000	−	8.66025i	−0.282617	−	0.489506i	−0.972028	+	0.234863i	$0.924536\pi$
$314$	22.0000			1.24153
$315$		0			0
$316$	−16.0000			−0.900070
$317$	9.00000	+	15.5885i	0.505490	+	0.875535i	0.999980	+	0.00635137i	$0.00202172\pi$
$317$	9.00000	+	15.5885i	0.505490	+	0.875535i	−0.494489	+	0.869184i	$0.664645\pi$
$318$		0			0
$319$	4.50000	−	7.79423i	0.251952	−	0.436393i
$320$	3.00000			0.167705
$321$		0			0
$322$		0			0
$323$	21.0000			1.16847
$324$		0			0
$325$	−2.00000	−	3.46410i	−0.110940	−	0.192154i
$326$	−19.0000			−1.05231
$327$		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	$0.903893\pi$
$332$	4.50000	+	7.79423i	0.246970	+	0.427764i
$333$		0			0
$334$	7.50000	−	12.9904i	0.410382	−	0.710802i
$335$	6.00000	−	10.3923i	0.327815	−	0.567792i
$336$		0			0
$337$	6.50000	+	11.2583i	0.354078	+	0.613280i	0.986960	−	0.160968i	$-0.0514616\pi$
$337$	6.50000	+	11.2583i	0.354078	+	0.613280i	−0.632882	+	0.774248i	$0.718128\pi$
$338$	−12.0000			−0.652714
$339$		0			0
$340$	9.00000			0.488094
$341$	12.0000	−	20.7846i	0.649836	−	1.12555i
$342$		0			0
$343$		0			0
$344$	0.500000	−	0.866025i	0.0269582	−	0.0466930i
$345$		0			0
$346$	3.00000	−	5.19615i	0.161281	−	0.279347i
$347$	6.00000	−	10.3923i	0.322097	−	0.557888i	−0.658824	−	0.752297i	$-0.728946\pi$
$347$	6.00000	−	10.3923i	0.322097	−	0.557888i	0.980921	+	0.194409i	$0.0622790\pi$
$348$		0			0
$349$	11.5000	−	19.9186i	0.615581	−	1.06622i	−0.374701	−	0.927146i	$-0.622255\pi$
$349$	11.5000	−	19.9186i	0.615581	−	1.06622i	0.990282	−	0.139072i	$-0.0444119\pi$
$350$		0			0
$351$		0			0
$352$	−1.50000	+	2.59808i	−0.0799503	+	0.138478i
$353$	3.00000			0.159674			0.0798369	−	0.996808i	$-0.474560\pi$
$353$	3.00000			0.159674			0.0798369	+	0.996808i	$0.474560\pi$
$354$		0			0
$355$	−36.0000			−1.91068
$356$	−1.50000	−	2.59808i	−0.0794998	−	0.137698i
$357$		0			0
$358$	−10.5000	+	18.1865i	−0.554942	+	0.961188i
$359$	4.50000	−	7.79423i	0.237501	−	0.411364i	−0.722496	−	0.691375i	$-0.757005\pi$
$359$	4.50000	−	7.79423i	0.237501	−	0.411364i	0.959997	+	0.280012i	$0.0903384\pi$
$360$		0			0
$361$	−15.0000	−	25.9808i	−0.789474	−	1.36741i
$362$	1.00000	+	1.73205i	0.0525588	+	0.0910346i
$363$		0			0
$364$		0			0
$365$	16.5000	+	28.5788i	0.863649	+	1.49588i
$366$		0			0
$367$	−17.0000			−0.887393			−0.443696	−	0.896177i	$-0.646333\pi$
$367$	−17.0000			−0.887393			−0.443696	+	0.896177i	$0.646333\pi$
$368$	−4.50000	−	7.79423i	−0.234579	−	0.406302i
$369$		0			0
$370$	−3.00000			−0.155963
$371$		0			0
$372$		0			0
$373$	−13.0000			−0.673114			−0.336557	−	0.941663i	$-0.609263\pi$
$373$	−13.0000			−0.673114			−0.336557	+	0.941663i	$0.609263\pi$
$374$	−4.50000	+	7.79423i	−0.232689	+	0.403030i
$375$		0			0
$376$		0			0
$377$	−3.00000			−0.154508
$378$		0			0
$379$	−28.0000			−1.43826			−0.719132	−	0.694874i	$-0.755460\pi$
$379$	−28.0000			−1.43826			−0.719132	+	0.694874i	$0.755460\pi$
$380$	−10.5000	−	18.1865i	−0.538639	−	0.932949i
$381$		0			0
$382$		0			0
$383$	15.0000			0.766464			0.383232	−	0.923652i	$-0.374811\pi$
$383$	15.0000			0.766464			0.383232	+	0.923652i	$0.374811\pi$
$384$		0			0
$385$		0			0
$386$	14.0000			0.712581
$387$		0			0
$388$	−0.500000	−	0.866025i	−0.0253837	−	0.0439658i
$389$	−27.0000			−1.36895			−0.684477	−	0.729034i	$-0.739969\pi$
$389$	−27.0000			−1.36895			−0.684477	+	0.729034i	$0.739969\pi$
$390$		0			0
$391$	−13.5000	−	23.3827i	−0.682724	−	1.18251i
$392$		0			0
$393$		0			0
$394$	9.00000	+	15.5885i	0.453413	+	0.785335i
$395$	24.0000	+	41.5692i	1.20757	+	2.09157i
$396$		0			0
$397$	−6.50000	+	11.2583i	−0.326226	+	0.565039i	−0.981760	−	0.190126i	$-0.939110\pi$
$397$	−6.50000	+	11.2583i	−0.326226	+	0.565039i	0.655534	+	0.755166i	$0.272444\pi$
$398$	−12.5000	+	21.6506i	−0.626568	+	1.08525i
$399$		0			0
$400$	−2.00000	−	3.46410i	−0.100000	−	0.173205i
$401$	−27.0000			−1.34832			−0.674158	−	0.738587i	$-0.735493\pi$
$401$	−27.0000			−1.34832			−0.674158	+	0.738587i	$0.735493\pi$
$402$		0			0
$403$	−8.00000			−0.398508
$404$	−1.50000	+	2.59808i	−0.0746278	+	0.129259i
$405$		0			0
$406$		0			0
$407$	1.50000	−	2.59808i	0.0743522	−	0.128782i
$408$		0			0
$409$	−17.0000	+	29.4449i	−0.840596	+	1.45595i	0.0487958	+	0.998809i	$0.484462\pi$
$409$	−17.0000	+	29.4449i	−0.840596	+	1.45595i	−0.889392	+	0.457146i	$0.848872\pi$
$410$	−4.50000	+	7.79423i	−0.222239	+	0.384930i
$411$		0			0
$412$	−6.50000	+	11.2583i	−0.320232	+	0.554658i
$413$		0			0
$414$		0			0
$415$	13.5000	−	23.3827i	0.662689	−	1.14781i
$416$	1.00000			0.0490290
$417$		0			0
$418$	21.0000			1.02714
$419$	−4.50000	−	7.79423i	−0.219839	−	0.380773i	0.734919	−	0.678155i	$-0.237220\pi$
$419$	−4.50000	−	7.79423i	−0.219839	−	0.380773i	−0.954759	+	0.297382i	$0.903887\pi$
$420$		0			0
$421$	−17.5000	+	30.3109i	−0.852898	+	1.47726i	0.0256838	+	0.999670i	$0.491824\pi$
$421$	−17.5000	+	30.3109i	−0.852898	+	1.47726i	−0.878582	+	0.477592i	$0.841510\pi$
$422$	−2.50000	+	4.33013i	−0.121698	+	0.210787i
$423$		0			0
$424$	1.50000	+	2.59808i	0.0728464	+	0.126174i
$425$	−6.00000	−	10.3923i	−0.291043	−	0.504101i
$426$		0			0
$427$		0			0
$428$	4.50000	+	7.79423i	0.217516	+	0.376748i
$429$		0			0
$430$	−3.00000			−0.144673
$431$	13.5000	+	23.3827i	0.650272	+	1.12630i	0.983057	+	0.183301i	$0.0586785\pi$
$431$	13.5000	+	23.3827i	0.650272	+	1.12630i	−0.332785	+	0.943003i	$0.607988\pi$
$432$		0			0
$433$	−2.00000			−0.0961139			−0.0480569	−	0.998845i	$-0.515303\pi$
$433$	−2.00000			−0.0961139			−0.0480569	+	0.998845i	$0.515303\pi$
$434$		0			0
$435$		0			0
$436$	−13.0000			−0.622587
$437$	−31.5000	+	54.5596i	−1.50685	+	2.60994i
$438$		0			0
$439$	4.00000	+	6.92820i	0.190910	+	0.330665i	0.945552	−	0.325471i	$-0.105523\pi$
$439$	4.00000	+	6.92820i	0.190910	+	0.330665i	−0.754642	+	0.656136i	$0.772190\pi$
$440$	9.00000			0.429058
$441$		0			0
$442$	3.00000			0.142695
$443$	18.0000	+	31.1769i	0.855206	+	1.48126i	0.876454	+	0.481486i	$0.159903\pi$
$443$	18.0000	+	31.1769i	0.855206	+	1.48126i	−0.0212481	+	0.999774i	$0.506764\pi$
$444$		0			0
$445$	−4.50000	+	7.79423i	−0.213320	+	0.369482i
$446$	1.00000			0.0473514
$447$		0			0
$448$		0			0
$449$	−6.00000			−0.283158			−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000			−0.283158			−0.141579	+	0.989927i	$0.545218\pi$
$450$		0			0
$451$	−4.50000	−	7.79423i	−0.211897	−	0.367016i
$452$	9.00000			0.423324
$453$		0			0
$454$	1.50000	+	2.59808i	0.0703985	+	0.121934i
$455$		0			0
$456$		0			0
$457$	5.00000	+	8.66025i	0.233890	+	0.405110i	0.958950	−	0.283577i	$-0.0915211\pi$
$457$	5.00000	+	8.66025i	0.233890	+	0.405110i	−0.725059	+	0.688686i	$0.758188\pi$
$458$	−6.50000	−	11.2583i	−0.303725	−	0.526067i
$459$		0			0
$460$	−13.5000	+	23.3827i	−0.629441	+	1.09022i
$461$	4.50000	−	7.79423i	0.209586	−	0.363013i	−0.741998	−	0.670402i	$-0.766122\pi$
$461$	4.50000	−	7.79423i	0.209586	−	0.363013i	0.951584	+	0.307388i	$0.0994551\pi$
$462$		0			0
$463$	−20.5000	−	35.5070i	−0.952716	−	1.65015i	−0.739511	−	0.673145i	$-0.764943\pi$
$463$	−20.5000	−	35.5070i	−0.952716	−	1.65015i	−0.213205	−	0.977007i	$-0.568390\pi$
$464$	−3.00000			−0.139272
$465$		0			0
$466$	−3.00000			−0.138972
$467$	1.50000	−	2.59808i	0.0694117	−	0.120225i	−0.829231	−	0.558906i	$-0.811221\pi$
$467$	1.50000	−	2.59808i	0.0694117	−	0.120225i	0.898642	+	0.438682i	$0.144554\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	1.50000	−	2.59808i	0.0689701	−	0.119460i
$474$		0			0
$475$	−14.0000	+	24.2487i	−0.642364	+	1.11261i
$476$		0			0
$477$		0			0
$478$	−1.50000	+	2.59808i	−0.0686084	+	0.118833i
$479$	−3.00000			−0.137073			−0.0685367	−	0.997649i	$-0.521833\pi$
$479$	−3.00000			−0.137073			−0.0685367	+	0.997649i	$0.521833\pi$
$480$		0			0
$481$	−1.00000			−0.0455961
$482$	−6.50000	−	11.2583i	−0.296067	−	0.512803i
$483$		0			0
$484$	1.00000	−	1.73205i	0.0454545	−	0.0787296i
$485$	−1.50000	+	2.59808i	−0.0681115	+	0.117973i
$486$		0			0
$487$	12.5000	+	21.6506i	0.566429	+	0.981084i	0.996915	+	0.0784867i	$0.0250088\pi$
$487$	12.5000	+	21.6506i	0.566429	+	0.981084i	−0.430486	+	0.902597i	$0.641658\pi$
$488$	1.00000	+	1.73205i	0.0452679	+	0.0784063i
$489$		0			0
$490$		0			0
$491$	10.5000	+	18.1865i	0.473858	+	0.820747i	0.999552	−	0.0299272i	$-0.00952753\pi$
$491$	10.5000	+	18.1865i	0.473858	+	0.820747i	−0.525694	+	0.850674i	$0.676194\pi$
$492$		0			0
$493$	−9.00000			−0.405340
$494$	−3.50000	−	6.06218i	−0.157472	−	0.272750i
$495$		0			0
$496$	−8.00000			−0.359211
$497$		0			0
$498$		0			0
$499$	−25.0000			−1.11915			−0.559577	−	0.828778i	$-0.689036\pi$
$499$	−25.0000			−1.11915			−0.559577	+	0.828778i	$0.689036\pi$
$500$	1.50000	−	2.59808i	0.0670820	−	0.116190i
$501$		0			0
$502$	−6.00000	−	10.3923i	−0.267793	−	0.463831i
$503$	−24.0000			−1.07011			−0.535054	−	0.844818i	$-0.679709\pi$
$503$	−24.0000			−1.07011			−0.535054	+	0.844818i	$0.679709\pi$
$504$		0			0
$505$	9.00000			0.400495
$506$	−13.5000	−	23.3827i	−0.600148	−	1.03949i
$507$		0			0
$508$	2.00000	−	3.46410i	0.0887357	−	0.153695i
$509$	−9.00000			−0.398918			−0.199459	−	0.979906i	$-0.563918\pi$
$509$	−9.00000			−0.398918			−0.199459	+	0.979906i	$0.563918\pi$
$510$		0			0
$511$		0			0
$512$	1.00000			0.0441942
$513$		0			0
$514$	10.5000	+	18.1865i	0.463135	+	0.802174i
$515$	39.0000			1.71855
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	−1.50000	−	2.59808i	−0.0657794	−	0.113933i
$521$	−1.50000	−	2.59808i	−0.0657162	−	0.113824i	0.831295	−	0.555831i	$-0.187600\pi$
$521$	−1.50000	−	2.59808i	−0.0657162	−	0.113824i	−0.897011	+	0.442007i	$0.854267\pi$
$522$		0			0
$523$	−3.50000	+	6.06218i	−0.153044	+	0.265081i	−0.932345	−	0.361569i	$-0.882241\pi$
$523$	−3.50000	+	6.06218i	−0.153044	+	0.265081i	0.779301	+	0.626650i	$0.215574\pi$
$524$	−7.50000	+	12.9904i	−0.327639	+	0.567487i
$525$		0			0
$526$	4.50000	+	7.79423i	0.196209	+	0.339845i
$527$	−24.0000			−1.04546
$528$		0			0
$529$	58.0000			2.52174
$530$	4.50000	−	7.79423i	0.195468	−	0.338560i
$531$		0			0
$532$		0			0
$533$	−1.50000	+	2.59808i	−0.0649722	+	0.112535i
$534$		0			0
$535$	13.5000	−	23.3827i	0.583656	−	1.01092i
$536$	2.00000	−	3.46410i	0.0863868	−	0.149626i
$537$		0			0
$538$	−7.50000	+	12.9904i	−0.323348	+	0.560055i
$539$		0			0
$540$		0			0
$541$	−5.50000	+	9.52628i	−0.236463	+	0.409567i	−0.959697	−	0.281037i	$-0.909322\pi$
$541$	−5.50000	+	9.52628i	−0.236463	+	0.409567i	0.723234	+	0.690604i	$0.242655\pi$
$542$	−5.00000			−0.214768
$543$		0			0
$544$	3.00000			0.128624
$545$	19.5000	+	33.7750i	0.835288	+	1.44676i
$546$		0			0
$547$	−5.50000	+	9.52628i	−0.235163	+	0.407314i	−0.959320	−	0.282321i	$-0.908896\pi$
$547$	−5.50000	+	9.52628i	−0.235163	+	0.407314i	0.724157	+	0.689635i	$0.242229\pi$
$548$	−4.50000	+	7.79423i	−0.192230	+	0.332953i
$549$		0			0
$550$	−6.00000	−	10.3923i	−0.255841	−	0.443129i
$551$	10.5000	+	18.1865i	0.447315	+	0.774772i
$552$		0			0
$553$		0			0
$554$	0.500000	+	0.866025i	0.0212430	+	0.0367939i
$555$		0			0
$556$	7.00000			0.296866
$557$	−4.50000	−	7.79423i	−0.190671	−	0.330252i	0.754802	−	0.655953i	$-0.227733\pi$
$557$	−4.50000	−	7.79423i	−0.190671	−	0.330252i	−0.945473	+	0.325701i	$0.894400\pi$
$558$		0			0
$559$	−1.00000			−0.0422955
$560$		0			0
$561$		0			0
$562$	21.0000			0.885832
$563$	−6.00000	+	10.3923i	−0.252870	+	0.437983i	−0.964315	−	0.264758i	$-0.914708\pi$
$563$	−6.00000	+	10.3923i	−0.252870	+	0.437983i	0.711445	+	0.702742i	$0.248041\pi$
$564$		0			0
$565$	−13.5000	−	23.3827i	−0.567949	−	0.983717i
$566$	4.00000			0.168133
$567$		0			0
$568$	−12.0000			−0.503509
$569$	−9.00000	−	15.5885i	−0.377300	−	0.653502i	0.613369	−	0.789797i	$-0.289814\pi$
$569$	−9.00000	−	15.5885i	−0.377300	−	0.653502i	−0.990668	+	0.136295i	$0.956481\pi$
$570$		0			0
$571$	−16.0000	+	27.7128i	−0.669579	+	1.15975i	0.308443	+	0.951243i	$0.400192\pi$
$571$	−16.0000	+	27.7128i	−0.669579	+	1.15975i	−0.978022	+	0.208502i	$0.933141\pi$
$572$	3.00000			0.125436
$573$		0			0
$574$		0			0
$575$	36.0000			1.50130
$576$		0			0
$577$	−12.5000	−	21.6506i	−0.520382	−	0.901328i	−0.999719	−	0.0236970i	$-0.992456\pi$
$577$	−12.5000	−	21.6506i	−0.520382	−	0.901328i	0.479337	−	0.877631i	$-0.340877\pi$
$578$	−8.00000			−0.332756
$579$		0			0
$580$	4.50000	+	7.79423i	0.186852	+	0.323638i
$581$		0			0
$582$		0			0
$583$	4.50000	+	7.79423i	0.186371	+	0.322804i
$584$	5.50000	+	9.52628i	0.227592	+	0.394200i
$585$		0			0
$586$	4.50000	−	7.79423i	0.185893	−	0.321977i
$587$	−1.50000	+	2.59808i	−0.0619116	+	0.107234i	−0.895320	−	0.445424i	$-0.853053\pi$
$587$	−1.50000	+	2.59808i	−0.0619116	+	0.107234i	0.833408	+	0.552658i	$0.186386\pi$
$588$		0			0
$589$	28.0000	+	48.4974i	1.15372	+	1.99830i
$590$		0			0
$591$		0			0
$592$	−1.00000			−0.0410997
$593$	−19.5000	+	33.7750i	−0.800769	+	1.38697i	0.118342	+	0.992973i	$0.462242\pi$
$593$	−19.5000	+	33.7750i	−0.800769	+	1.38697i	−0.919111	+	0.394000i	$0.871091\pi$
$594$		0			0
$595$		0			0
$596$	−4.50000	+	7.79423i	−0.184327	+	0.319264i
$597$		0			0
$598$	−4.50000	+	7.79423i	−0.184019	+	0.318730i
$599$	−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
$601$	−12.5000	+	21.6506i	−0.509886	+	0.883148i	0.490049	+	0.871695i	$0.336979\pi$
$601$	−12.5000	+	21.6506i	−0.509886	+	0.883148i	−0.999934	+	0.0114528i	$0.996354\pi$
$602$		0			0
$603$		0			0
$604$	3.50000	−	6.06218i	0.142413	−	0.246667i
$605$	−6.00000			−0.243935
$606$		0			0
$607$	13.0000			0.527654			0.263827	−	0.964570i	$-0.415015\pi$
$607$	13.0000			0.527654			0.263827	+	0.964570i	$0.415015\pi$
$608$	−3.50000	−	6.06218i	−0.141944	−	0.245854i
$609$		0			0
$610$	3.00000	−	5.19615i	0.121466	−	0.210386i
$611$		0			0
$612$		0			0
$613$	−11.5000	−	19.9186i	−0.464481	−	0.804504i	0.534697	−	0.845044i	$-0.320426\pi$
$613$	−11.5000	−	19.9186i	−0.464481	−	0.804504i	−0.999178	+	0.0405396i	$0.987092\pi$
$614$	−14.0000	−	24.2487i	−0.564994	−	0.978598i
$615$		0			0
$616$		0			0
$617$	−22.5000	−	38.9711i	−0.905816	−	1.56892i	−0.819818	−	0.572624i	$-0.805926\pi$
$617$	−22.5000	−	38.9711i	−0.905816	−	1.56892i	−0.0859976	−	0.996295i	$-0.527408\pi$
$618$		0			0
$619$	−17.0000			−0.683288			−0.341644	−	0.939829i	$-0.610984\pi$
$619$	−17.0000			−0.683288			−0.341644	+	0.939829i	$0.610984\pi$
$620$	12.0000	+	20.7846i	0.481932	+	0.834730i
$621$		0			0
$622$	−24.0000			−0.962312
$623$		0			0
$624$		0			0
$625$	−29.0000			−1.16000
$626$	−5.00000	+	8.66025i	−0.199840	+	0.346133i
$627$		0			0
$628$	−11.0000	−	19.0526i	−0.438948	−	0.760280i
$629$	−3.00000			−0.119618
$630$		0			0
$631$	8.00000			0.318475			0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000			0.318475			0.159237	+	0.987240i	$0.449096\pi$
$632$	8.00000	+	13.8564i	0.318223	+	0.551178i
$633$		0			0
$634$	9.00000	−	15.5885i	0.357436	−	0.619097i
$635$	−12.0000			−0.476205
$636$		0			0
$637$		0			0
$638$	−9.00000			−0.356313
$639$		0			0
$640$	−1.50000	−	2.59808i	−0.0592927	−	0.102698i
$641$	33.0000			1.30342			0.651711	−	0.758468i	$-0.274052\pi$
$641$	33.0000			1.30342			0.651711	+	0.758468i	$0.274052\pi$
$642$		0			0
$643$	14.5000	+	25.1147i	0.571824	+	0.990429i	0.996379	+	0.0850262i	$0.0270974\pi$
$643$	14.5000	+	25.1147i	0.571824	+	0.990429i	−0.424555	+	0.905402i	$0.639569\pi$
$644$		0			0
$645$		0			0
$646$	−10.5000	−	18.1865i	−0.413117	−	0.715540i
$647$	10.5000	+	18.1865i	0.412798	+	0.714986i	0.995194	−	0.0979182i	$-0.0312184\pi$
$647$	10.5000	+	18.1865i	0.412798	+	0.714986i	−0.582397	+	0.812905i	$0.697885\pi$
$648$		0			0
$649$		0			0
$650$	−2.00000	+	3.46410i	−0.0784465	+	0.135873i
$651$		0			0
$652$	9.50000	+	16.4545i	0.372049	+	0.644407i
$653$	−15.0000			−0.586995			−0.293498	−	0.955960i	$-0.594819\pi$
$653$	−15.0000			−0.586995			−0.293498	+	0.955960i	$0.594819\pi$
$654$		0			0
$655$	45.0000			1.75830
$656$	−1.50000	+	2.59808i	−0.0585652	+	0.101438i
$657$		0			0
$658$		0			0
$659$	1.50000	−	2.59808i	0.0584317	−	0.101207i	−0.835330	−	0.549749i	$-0.814723\pi$
$659$	1.50000	−	2.59808i	0.0584317	−	0.101207i	0.893762	+	0.448542i	$0.148057\pi$
$660$		0			0
$661$	−11.0000	+	19.0526i	−0.427850	+	0.741059i	−0.996682	−	0.0813955i	$-0.974062\pi$
$661$	−11.0000	+	19.0526i	−0.427850	+	0.741059i	0.568831	+	0.822454i	$0.307396\pi$
$662$	−4.00000	+	6.92820i	−0.155464	+	0.269272i
$663$		0			0
$664$	4.50000	−	7.79423i	0.174634	−	0.302475i
$665$		0			0
$666$		0			0
$667$	13.5000	−	23.3827i	0.522722	−	0.905381i
$668$	−15.0000			−0.580367
$669$		0			0
$670$	−12.0000			−0.463600
$671$	3.00000	+	5.19615i	0.115814	+	0.200595i
$672$		0			0
$673$	−17.5000	+	30.3109i	−0.674575	+	1.16840i	0.302017	+	0.953302i	$0.402340\pi$
$673$	−17.5000	+	30.3109i	−0.674575	+	1.16840i	−0.976593	+	0.215096i	$0.930993\pi$
$674$	6.50000	−	11.2583i	0.250371	−	0.433655i
$675$		0			0
$676$	6.00000	+	10.3923i	0.230769	+	0.399704i
$677$	15.0000	+	25.9808i	0.576497	+	0.998522i	0.995877	+	0.0907112i	$0.0289140\pi$
$677$	15.0000	+	25.9808i	0.576497	+	0.998522i	−0.419380	+	0.907811i	$0.637753\pi$
$678$		0			0
$679$		0			0
$680$	−4.50000	−	7.79423i	−0.172567	−	0.298895i
$681$		0			0
$682$	−24.0000			−0.919007
$683$	−4.50000	−	7.79423i	−0.172188	−	0.298238i	0.766997	−	0.641651i	$-0.221750\pi$
$683$	−4.50000	−	7.79423i	−0.172188	−	0.298238i	−0.939184	+	0.343413i	$0.888417\pi$
$684$		0			0
$685$	27.0000			1.03162
$686$		0			0
$687$		0			0
$688$	−1.00000			−0.0381246
$689$	1.50000	−	2.59808i	0.0571454	−	0.0989788i
$690$		0			0
$691$	22.0000	+	38.1051i	0.836919	+	1.44959i	0.892458	+	0.451130i	$0.148979\pi$
$691$	22.0000	+	38.1051i	0.836919	+	1.44959i	−0.0555386	+	0.998457i	$0.517688\pi$
$692$	−6.00000			−0.228086
$693$		0			0
$694$	−12.0000			−0.455514
$695$	−10.5000	−	18.1865i	−0.398288	−	0.689855i
$696$		0			0
$697$	−4.50000	+	7.79423i	−0.170450	+	0.295227i
$698$	−23.0000			−0.870563
$699$		0			0
$700$		0			0
$701$	18.0000			0.679851			0.339925	−	0.940452i	$-0.389598\pi$
$701$	18.0000			0.679851			0.339925	+	0.940452i	$0.389598\pi$
$702$		0			0
$703$	3.50000	+	6.06218i	0.132005	+	0.228639i
$704$	3.00000			0.113067
$705$		0			0
$706$	−1.50000	−	2.59808i	−0.0564532	−	0.0977799i
$707$		0			0
$708$		0			0
$709$	−13.0000	−	22.5167i	−0.488225	−	0.845631i	0.511683	−	0.859174i	$-0.329022\pi$
$709$	−13.0000	−	22.5167i	−0.488225	−	0.845631i	−0.999908	+	0.0135434i	$0.995689\pi$
$710$	18.0000	+	31.1769i	0.675528	+	1.17005i
$711$		0			0
$712$	−1.50000	+	2.59808i	−0.0562149	+	0.0973670i
$713$	36.0000	−	62.3538i	1.34821	−	2.33517i
$714$		0			0
$715$	−4.50000	−	7.79423i	−0.168290	−	0.291488i
$716$	21.0000			0.784807
$717$		0			0
$718$	−9.00000			−0.335877
$719$	7.50000	−	12.9904i	0.279703	−	0.484459i	−0.691608	−	0.722273i	$-0.743097\pi$
$719$	7.50000	−	12.9904i	0.279703	−	0.484459i	0.971311	+	0.237814i	$0.0764307\pi$
$720$		0			0
$721$		0			0
$722$	−15.0000	+	25.9808i	−0.558242	+	0.966904i
$723$		0			0
$724$	1.00000	−	1.73205i	0.0371647	−	0.0643712i
$725$	6.00000	−	10.3923i	0.222834	−	0.385961i
$726$		0			0
$727$	−6.50000	+	11.2583i	−0.241072	+	0.417548i	−0.961020	−	0.276479i	$-0.910832\pi$
$727$	−6.50000	+	11.2583i	−0.241072	+	0.417548i	0.719948	+	0.694028i	$0.244166\pi$
$728$		0			0
$729$		0			0
$730$	16.5000	−	28.5788i	0.610692	−	1.05775i
$731$	−3.00000			−0.110959
$732$		0			0
$733$	1.00000			0.0369358			0.0184679	−	0.999829i	$-0.494121\pi$
$733$	1.00000			0.0369358			0.0184679	+	0.999829i	$0.494121\pi$
$734$	8.50000	+	14.7224i	0.313741	+	0.543415i
$735$		0			0
$736$	−4.50000	+	7.79423i	−0.165872	+	0.287299i
$737$	6.00000	−	10.3923i	0.221013	−	0.382805i
$738$		0			0
$739$	−11.5000	−	19.9186i	−0.423034	−	0.732717i	0.573200	−	0.819415i	$-0.305702\pi$
$739$	−11.5000	−	19.9186i	−0.423034	−	0.732717i	−0.996235	+	0.0866983i	$0.972368\pi$
$740$	1.50000	+	2.59808i	0.0551411	+	0.0955072i
$741$		0			0
$742$		0			0
$743$	10.5000	+	18.1865i	0.385208	+	0.667199i	0.991798	−	0.127815i	$-0.0407965\pi$
$743$	10.5000	+	18.1865i	0.385208	+	0.667199i	−0.606590	+	0.795015i	$0.707463\pi$
$744$		0			0
$745$	27.0000			0.989203
$746$	6.50000	+	11.2583i	0.237982	+	0.412197i
$747$		0			0
$748$	9.00000			0.329073
$749$		0			0
$750$		0			0
$751$	−13.0000			−0.474377			−0.237188	−	0.971464i	$-0.576226\pi$
$751$	−13.0000			−0.474377			−0.237188	+	0.971464i	$0.576226\pi$
$752$		0			0
$753$		0			0
$754$	1.50000	+	2.59808i	0.0546268	+	0.0946164i
$755$	−21.0000			−0.764268
$756$		0			0
$757$	−22.0000			−0.799604			−0.399802	−	0.916602i	$-0.630921\pi$
$757$	−22.0000			−0.799604			−0.399802	+	0.916602i	$0.630921\pi$
$758$	14.0000	+	24.2487i	0.508503	+	0.880753i
$759$		0			0
$760$	−10.5000	+	18.1865i	−0.380875	+	0.659695i
$761$	−45.0000			−1.63125			−0.815624	−	0.578582i	$-0.803606\pi$
$761$	−45.0000			−1.63125			−0.815624	+	0.578582i	$0.803606\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$	−7.50000	−	12.9904i	−0.270986	−	0.469362i
$767$		0			0
$768$		0			0
$769$	11.5000	+	19.9186i	0.414701	+	0.718283i	0.995397	−	0.0958377i	$-0.0305530\pi$
$769$	11.5000	+	19.9186i	0.414701	+	0.718283i	−0.580696	+	0.814120i	$0.697220\pi$
$770$		0			0
$771$		0			0
$772$	−7.00000	−	12.1244i	−0.251936	−	0.436365i
$773$	−13.5000	−	23.3827i	−0.485561	−	0.841017i	0.514301	−	0.857610i	$-0.328051\pi$
$773$	−13.5000	−	23.3827i	−0.485561	−	0.841017i	−0.999862	+	0.0165929i	$0.994718\pi$
$774$		0			0
$775$	16.0000	−	27.7128i	0.574737	−	0.995474i
$776$	−0.500000	+	0.866025i	−0.0179490	+	0.0310885i
$777$		0			0
$778$	13.5000	+	23.3827i	0.483998	+	0.838310i
$779$	21.0000			0.752403
$780$		0			0
$781$	−36.0000			−1.28818
$782$	−13.5000	+	23.3827i	−0.482759	+	0.836163i
$783$		0			0
$784$		0			0
$785$	−33.0000	+	57.1577i	−1.17782	+	2.04004i
$786$		0			0
$787$	−14.0000	+	24.2487i	−0.499046	+	0.864373i	−0.999999	−	0.00110111i	$-0.999650\pi$
$787$	−14.0000	+	24.2487i	−0.499046	+	0.864373i	0.500953	+	0.865474i	$0.332983\pi$
$788$	9.00000	−	15.5885i	0.320612	−	0.555316i
$789$		0			0
$790$	24.0000	−	41.5692i	0.853882	−	1.47897i
$791$		0			0
$792$		0			0
$793$	1.00000	−	1.73205i	0.0355110	−	0.0615069i
$794$	13.0000			0.461353

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)

\(f(q)\)	\(=\)	\(q+(-0.500000 - 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +3.00000 q^{5} +1.00000 q^{8} +O(q^{10})\)	\(q+(-0.500000 - 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +3.00000 q^{5} +1.00000 q^{8} +(-1.50000 - 2.59808i) q^{10} +3.00000 q^{11} +(-0.500000 - 0.866025i) q^{13} +(-0.500000 - 0.866025i) q^{16} +(-1.50000 - 2.59808i) q^{17} +(-3.50000 + 6.06218i) q^{19} +(-1.50000 + 2.59808i) q^{20} +(-1.50000 - 2.59808i) q^{22} +9.00000 q^{23} +4.00000 q^{25} +(-0.500000 + 0.866025i) q^{26} +(1.50000 - 2.59808i) q^{29} +(4.00000 - 6.92820i) q^{31} +(-0.500000 + 0.866025i) q^{32} +(-1.50000 + 2.59808i) q^{34} +(0.500000 - 0.866025i) q^{37} +7.00000 q^{38} +3.00000 q^{40} +(-1.50000 - 2.59808i) q^{41} +(0.500000 - 0.866025i) q^{43} +(-1.50000 + 2.59808i) q^{44} +(-4.50000 - 7.79423i) q^{46} +(-2.00000 - 3.46410i) q^{50} +1.00000 q^{52} +(1.50000 + 2.59808i) q^{53} +9.00000 q^{55} -3.00000 q^{58} +(1.00000 + 1.73205i) q^{61} -8.00000 q^{62} +1.00000 q^{64} +(-1.50000 - 2.59808i) q^{65} +(2.00000 - 3.46410i) q^{67} +3.00000 q^{68} -12.0000 q^{71} +(5.50000 + 9.52628i) q^{73} -1.00000 q^{74} +(-3.50000 - 6.06218i) q^{76} +(8.00000 + 13.8564i) q^{79} +(-1.50000 - 2.59808i) q^{80} +(-1.50000 + 2.59808i) q^{82} +(4.50000 - 7.79423i) q^{83} +(-4.50000 - 7.79423i) q^{85} -1.00000 q^{86} +3.00000 q^{88} +(-1.50000 + 2.59808i) q^{89} +(-4.50000 + 7.79423i) q^{92} +(-10.5000 + 18.1865i) q^{95} +(-0.500000 + 0.866025i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} - q^{4} + 6 q^{5} + 2 q^{8} + O(q^{10}) \)	\( 2 q - q^{2} - q^{4} + 6 q^{5} + 2 q^{8} - 3 q^{10} + 6 q^{11} - q^{13} - q^{16} - 3 q^{17} - 7 q^{19} - 3 q^{20} - 3 q^{22} + 18 q^{23} + 8 q^{25} - q^{26} + 3 q^{29} + 8 q^{31} - q^{32} - 3 q^{34} + q^{37} + 14 q^{38} + 6 q^{40} - 3 q^{41} + q^{43} - 3 q^{44} - 9 q^{46} - 4 q^{50} + 2 q^{52} + 3 q^{53} + 18 q^{55} - 6 q^{58} + 2 q^{61} - 16 q^{62} + 2 q^{64} - 3 q^{65} + 4 q^{67} + 6 q^{68} - 24 q^{71} + 11 q^{73} - 2 q^{74} - 7 q^{76} + 16 q^{79} - 3 q^{80} - 3 q^{82} + 9 q^{83} - 9 q^{85} - 2 q^{86} + 6 q^{88} - 3 q^{89} - 9 q^{92} - 21 q^{95} - q^{97} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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