LMFDB - Embedded newform 845.2.l.a.654.1

Newspace parameters

Level:	$ N $	$=$	$ 845 = 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	845.l (of order $6$, degree $2$, not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$6.74735897080$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{12})$
Defining polynomial:	$x^{4} - x^{2} + 1$
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 65)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			654.1
Root			$0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	845.654
Dual form			845.2.l.a.699.1

$q$-expansion

$f(q)$	$=$	$q+(-0.500000 + 0.866025i) q^{2} +(-1.73205 - 1.00000i) q^{3} +(0.500000 + 0.866025i) q^{4} +(1.00000 - 2.00000i) q^{5} +(1.73205 - 1.00000i) q^{6} -3.00000 q^{8} +(0.500000 + 0.866025i) q^{9} +O(q^{10})$	\(q+(-0.500000 + 0.866025i) q^{2} +(-1.73205 - 1.00000i) q^{3} +(0.500000 + 0.866025i) q^{4} +(1.00000 - 2.00000i) q^{5} +(1.73205 - 1.00000i) q^{6} -3.00000 q^{8} +(0.500000 + 0.866025i) q^{9} +(1.23205 + 1.86603i) q^{10} +(1.73205 + 1.00000i) q^{11} -2.00000i q^{12} +(-3.73205 + 2.46410i) q^{15} +(0.500000 - 0.866025i) q^{16} -1.00000 q^{18} +(5.19615 - 3.00000i) q^{19} +(2.23205 - 0.133975i) q^{20} +(-1.73205 + 1.00000i) q^{22} +(-5.19615 - 3.00000i) q^{23} +(5.19615 + 3.00000i) q^{24} +(-3.00000 - 4.00000i) q^{25} +4.00000i q^{27} +(-3.00000 + 5.19615i) q^{29} +(-0.267949 - 4.46410i) q^{30} -6.00000i q^{31} +(-2.50000 - 4.33013i) q^{32} +(-2.00000 - 3.46410i) q^{33} +(-0.500000 + 0.866025i) q^{36} +(3.00000 - 5.19615i) q^{37} +6.00000i q^{38} +(-3.00000 + 6.00000i) q^{40} +(-6.92820 - 4.00000i) q^{41} +(-5.19615 + 3.00000i) q^{43} +2.00000i q^{44} +(2.23205 - 0.133975i) q^{45} +(5.19615 - 3.00000i) q^{46} +8.00000 q^{47} +(-1.73205 + 1.00000i) q^{48} +(3.50000 - 6.06218i) q^{49} +(4.96410 - 0.598076i) q^{50} -12.0000i q^{53} +(-3.46410 - 2.00000i) q^{54} +(3.73205 - 2.46410i) q^{55} -12.0000 q^{57} +(-3.00000 - 5.19615i) q^{58} +(-1.73205 + 1.00000i) q^{59} +(-4.00000 - 2.00000i) q^{60} +(-3.00000 - 5.19615i) q^{61} +(5.19615 + 3.00000i) q^{62} +7.00000 q^{64} +4.00000 q^{66} +(6.00000 - 10.3923i) q^{67} +(6.00000 + 10.3923i) q^{69} +(1.73205 - 1.00000i) q^{71} +(-1.50000 - 2.59808i) q^{72} +6.00000 q^{73} +(3.00000 + 5.19615i) q^{74} +(1.19615 + 9.92820i) q^{75} +(5.19615 + 3.00000i) q^{76} +(-1.23205 - 1.86603i) q^{80} +(5.50000 - 9.52628i) q^{81} +(6.92820 - 4.00000i) q^{82} +4.00000 q^{83} -6.00000i q^{86} +(10.3923 - 6.00000i) q^{87} +(-5.19615 - 3.00000i) q^{88} +(-6.92820 - 4.00000i) q^{89} +(-1.00000 + 2.00000i) q^{90} -6.00000i q^{92} +(-6.00000 + 10.3923i) q^{93} +(-4.00000 + 6.92820i) q^{94} +(-0.803848 - 13.3923i) q^{95} +10.0000i q^{96} +(-3.00000 - 5.19615i) q^{97} +(3.50000 + 6.06218i) q^{98} +2.00000i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{2} + 2 q^{4} + 4 q^{5} - 12 q^{8} + 2 q^{9} + O(q^{10}) $	$ 4 q - 2 q^{2} + 2 q^{4} + 4 q^{5} - 12 q^{8} + 2 q^{9} - 2 q^{10} - 8 q^{15} + 2 q^{16} - 4 q^{18} + 2 q^{20} - 12 q^{25} - 12 q^{29} - 8 q^{30} - 10 q^{32} - 8 q^{33} - 2 q^{36} + 12 q^{37} - 12 q^{40} + 2 q^{45} + 32 q^{47} + 14 q^{49} + 6 q^{50} + 8 q^{55} - 48 q^{57} - 12 q^{58} - 16 q^{60} - 12 q^{61} + 28 q^{64} + 16 q^{66} + 24 q^{67} + 24 q^{69} - 6 q^{72} + 24 q^{73} + 12 q^{74} - 16 q^{75} + 2 q^{80} + 22 q^{81} + 16 q^{83} - 4 q^{90} - 24 q^{93} - 16 q^{94} - 24 q^{95} - 12 q^{97} + 14 q^{98} + O(q^{100}) $

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/845\mathbb{Z}\right)^\times$.

$n$	$171$	$677$
$\chi(n)$	$e\left(\frac{1}{6}\right)$	$-1$

Coefficient data

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

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

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

$2$	−0.500000	+	0.866025i	−0.353553	+	0.612372i	−0.986869	−	0.161521i	$-0.948360\pi$
$2$	−0.500000	+	0.866025i	−0.353553	+	0.612372i	0.633316	+	0.773893i	$0.281693\pi$
$3$	−1.73205	−	1.00000i	−1.00000	−	0.577350i	−0.0917517	−	0.995782i	$-0.529247\pi$
$3$	−1.73205	−	1.00000i	−1.00000	−	0.577350i	−0.908248	+	0.418432i	$0.862580\pi$
$4$	0.500000	+	0.866025i	0.250000	+	0.433013i
$5$	1.00000	−	2.00000i	0.447214	−	0.894427i
$5$	1.00000	−	2.00000i	0.447214	−	0.894427i
$6$	1.73205	−	1.00000i	0.707107	−	0.408248i
$7$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$7$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$8$	−3.00000			−1.06066
$9$	0.500000	+	0.866025i	0.166667	+	0.288675i
$10$	1.23205	+	1.86603i	0.389609	+	0.590089i
$11$	1.73205	+	1.00000i	0.522233	+	0.301511i	0.737848	−	0.674967i	$-0.235842\pi$
$11$	1.73205	+	1.00000i	0.522233	+	0.301511i	−0.215615	+	0.976478i	$0.569176\pi$
$12$		−	2.00000i		−	0.577350i
$13$		0			0
$13$		0			0
$14$		0			0
$15$	−3.73205	+	2.46410i	−0.963611	+	0.636228i
$16$	0.500000	−	0.866025i	0.125000	−	0.216506i
$17$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$17$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$18$	−1.00000			−0.235702
$19$	5.19615	−	3.00000i	1.19208	−	0.688247i	0.233301	−	0.972404i	$-0.425047\pi$
$19$	5.19615	−	3.00000i	1.19208	−	0.688247i	0.958778	+	0.284157i	$0.0917138\pi$
$20$	2.23205	−	0.133975i	0.499102	−	0.0299576i
$21$		0			0
$22$	−1.73205	+	1.00000i	−0.369274	+	0.213201i
$23$	−5.19615	−	3.00000i	−1.08347	−	0.625543i	−0.151642	−	0.988436i	$-0.548456\pi$
$23$	−5.19615	−	3.00000i	−1.08347	−	0.625543i	−0.931831	+	0.362892i	$0.881789\pi$
$24$	5.19615	+	3.00000i	1.06066	+	0.612372i
$25$	−3.00000	−	4.00000i	−0.600000	−	0.800000i
$26$		0			0
$27$			4.00000i			0.769800i
$28$		0			0
$29$	−3.00000	+	5.19615i	−0.557086	+	0.964901i	0.440652	+	0.897678i	$0.354747\pi$
$29$	−3.00000	+	5.19615i	−0.557086	+	0.964901i	−0.997738	+	0.0672232i	$0.978586\pi$
$30$	−0.267949	−	4.46410i	−0.0489206	−	0.815030i
$31$		−	6.00000i		−	1.07763i	−0.842424	−	0.538816i	$-0.818872\pi$
$31$		−	6.00000i		−	1.07763i	0.842424	−	0.538816i	$-0.181128\pi$
$32$	−2.50000	−	4.33013i	−0.441942	−	0.765466i
$33$	−2.00000	−	3.46410i	−0.348155	−	0.603023i
$34$		0			0
$35$		0			0
$36$	−0.500000	+	0.866025i	−0.0833333	+	0.144338i
$37$	3.00000	−	5.19615i	0.493197	−	0.854242i	−0.506772	−	0.862080i	$-0.669162\pi$
$37$	3.00000	−	5.19615i	0.493197	−	0.854242i	0.999969	+	0.00783774i	$0.00249486\pi$
$38$			6.00000i			0.973329i
$39$		0			0
$40$	−3.00000	+	6.00000i	−0.474342	+	0.948683i
$41$	−6.92820	−	4.00000i	−1.08200	−	0.624695i	−0.150567	−	0.988600i	$-0.548110\pi$
$41$	−6.92820	−	4.00000i	−1.08200	−	0.624695i	−0.931436	+	0.363905i	$0.881443\pi$
$42$		0			0
$43$	−5.19615	+	3.00000i	−0.792406	+	0.457496i	−0.840809	−	0.541332i	$-0.817920\pi$
$43$	−5.19615	+	3.00000i	−0.792406	+	0.457496i	0.0484030	+	0.998828i	$0.484587\pi$
$44$			2.00000i			0.301511i
$45$	2.23205	−	0.133975i	0.332734	−	0.0199718i
$46$	5.19615	−	3.00000i	0.766131	−	0.442326i
$47$	8.00000			1.16692			0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000			1.16692			0.583460	+	0.812142i	$0.301699\pi$
$48$	−1.73205	+	1.00000i	−0.250000	+	0.144338i
$49$	3.50000	−	6.06218i	0.500000	−	0.866025i
$50$	4.96410	−	0.598076i	0.702030	−	0.0845807i
$51$		0			0
$52$		0			0
$53$		−	12.0000i		−	1.64833i	−0.566352	−	0.824163i	$-0.691646\pi$
$53$		−	12.0000i		−	1.64833i	0.566352	−	0.824163i	$-0.308354\pi$
$54$	−3.46410	−	2.00000i	−0.471405	−	0.272166i
$55$	3.73205	−	2.46410i	0.503230	−	0.332259i
$56$		0			0
$57$	−12.0000			−1.58944
$58$	−3.00000	−	5.19615i	−0.393919	−	0.682288i
$59$	−1.73205	+	1.00000i	−0.225494	+	0.130189i	−0.608492	−	0.793560i	$-0.708225\pi$
$59$	−1.73205	+	1.00000i	−0.225494	+	0.130189i	0.382998	+	0.923749i	$0.374892\pi$
$60$	−4.00000	−	2.00000i	−0.516398	−	0.258199i
$61$	−3.00000	−	5.19615i	−0.384111	−	0.665299i	0.607535	−	0.794293i	$-0.292159\pi$
$61$	−3.00000	−	5.19615i	−0.384111	−	0.665299i	−0.991645	+	0.128994i	$0.958825\pi$
$62$	5.19615	+	3.00000i	0.659912	+	0.381000i
$63$		0			0
$64$	7.00000			0.875000
$65$		0			0
$66$	4.00000			0.492366
$67$	6.00000	−	10.3923i	0.733017	−	1.26962i	−0.222571	−	0.974916i	$-0.571445\pi$
$67$	6.00000	−	10.3923i	0.733017	−	1.26962i	0.955588	−	0.294706i	$-0.0952216\pi$
$68$		0			0
$69$	6.00000	+	10.3923i	0.722315	+	1.25109i
$70$		0			0
$71$	1.73205	−	1.00000i	0.205557	−	0.118678i	−0.393688	−	0.919244i	$-0.628801\pi$
$71$	1.73205	−	1.00000i	0.205557	−	0.118678i	0.599245	+	0.800566i	$0.295468\pi$
$72$	−1.50000	−	2.59808i	−0.176777	−	0.306186i
$73$	6.00000			0.702247			0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000			0.702247			0.351123	+	0.936329i	$0.385800\pi$
$74$	3.00000	+	5.19615i	0.348743	+	0.604040i
$75$	1.19615	+	9.92820i	0.138120	+	1.14641i
$76$	5.19615	+	3.00000i	0.596040	+	0.344124i
$77$		0			0
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$	−1.23205	−	1.86603i	−0.137747	−	0.208628i
$81$	5.50000	−	9.52628i	0.611111	−	1.05848i
$82$	6.92820	−	4.00000i	0.765092	−	0.441726i
$83$	4.00000			0.439057			0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000			0.439057			0.219529	+	0.975606i	$0.429548\pi$
$84$		0			0
$85$		0			0
$86$		−	6.00000i		−	0.646997i
$87$	10.3923	−	6.00000i	1.11417	−	0.643268i
$88$	−5.19615	−	3.00000i	−0.553912	−	0.319801i
$89$	−6.92820	−	4.00000i	−0.734388	−	0.423999i	0.0856373	−	0.996326i	$-0.472707\pi$
$89$	−6.92820	−	4.00000i	−0.734388	−	0.423999i	−0.820025	+	0.572327i	$0.806041\pi$
$90$	−1.00000	+	2.00000i	−0.105409	+	0.210819i
$91$		0			0
$92$		−	6.00000i		−	0.625543i
$93$	−6.00000	+	10.3923i	−0.622171	+	1.07763i
$94$	−4.00000	+	6.92820i	−0.412568	+	0.714590i
$95$	−0.803848	−	13.3923i	−0.0824730	−	1.37402i
$96$			10.0000i			1.02062i
$97$	−3.00000	−	5.19615i	−0.304604	−	0.527589i	0.672569	−	0.740034i	$-0.265191\pi$
$97$	−3.00000	−	5.19615i	−0.304604	−	0.527589i	−0.977173	+	0.212445i	$0.931857\pi$
$98$	3.50000	+	6.06218i	0.353553	+	0.612372i
$99$			2.00000i			0.201008i
$100$	1.96410	−	4.59808i	0.196410	−	0.459808i
$101$	−3.00000	+	5.19615i	−0.298511	+	0.517036i	−0.975796	−	0.218685i	$-0.929823\pi$
$101$	−3.00000	+	5.19615i	−0.298511	+	0.517036i	0.677284	+	0.735721i	$0.263157\pi$
$102$		0			0
$103$			6.00000i			0.591198i	0.955312	+	0.295599i	$0.0955191\pi$
$103$			6.00000i			0.591198i	−0.955312	+	0.295599i	$0.904481\pi$
$104$		0			0
$105$		0			0
$106$	10.3923	+	6.00000i	1.00939	+	0.582772i
$107$	5.19615	+	3.00000i	0.502331	+	0.290021i	0.729676	−	0.683793i	$-0.239671\pi$
$107$	5.19615	+	3.00000i	0.502331	+	0.290021i	−0.227345	+	0.973814i	$0.573004\pi$
$108$	−3.46410	+	2.00000i	−0.333333	+	0.192450i
$109$		−	12.0000i		−	1.14939i	−0.818367	−	0.574696i	$-0.805120\pi$
$109$		−	12.0000i		−	1.14939i	0.818367	−	0.574696i	$-0.194880\pi$
$110$	0.267949	+	4.46410i	0.0255480	+	0.425635i
$111$	−10.3923	+	6.00000i	−0.986394	+	0.569495i
$112$		0			0
$113$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$113$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$114$	6.00000	−	10.3923i	0.561951	−	0.973329i
$115$	−11.1962	+	7.39230i	−1.04405	+	0.689336i
$116$	−6.00000			−0.557086
$117$		0			0
$118$		−	2.00000i		−	0.184115i
$119$		0			0
$120$	11.1962	−	7.39230i	1.02206	−	0.674822i
$121$	−3.50000	−	6.06218i	−0.318182	−	0.551107i
$122$	6.00000			0.543214
$123$	8.00000	+	13.8564i	0.721336	+	1.24939i
$124$	5.19615	−	3.00000i	0.466628	−	0.269408i
$125$	−11.0000	+	2.00000i	−0.983870	+	0.178885i
$126$		0			0
$127$	1.73205	+	1.00000i	0.153695	+	0.0887357i	0.574875	−	0.818241i	$-0.305051\pi$
$127$	1.73205	+	1.00000i	0.153695	+	0.0887357i	−0.421180	+	0.906977i	$0.638384\pi$
$128$	1.50000	−	2.59808i	0.132583	−	0.229640i
$129$	12.0000			1.05654
$130$		0			0
$131$	−12.0000			−1.04844			−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000			−1.04844			−0.524222	+	0.851581i	$0.675644\pi$
$132$	2.00000	−	3.46410i	0.174078	−	0.301511i
$133$		0			0
$134$	6.00000	+	10.3923i	0.518321	+	0.897758i
$135$	8.00000	+	4.00000i	0.688530	+	0.344265i
$136$		0			0
$137$	1.00000	+	1.73205i	0.0854358	+	0.147979i	0.905577	−	0.424182i	$-0.139438\pi$
$137$	1.00000	+	1.73205i	0.0854358	+	0.147979i	−0.820141	+	0.572161i	$0.806105\pi$
$138$	−12.0000			−1.02151
$139$	2.00000	+	3.46410i	0.169638	+	0.293821i	0.938293	−	0.345843i	$-0.112407\pi$
$139$	2.00000	+	3.46410i	0.169638	+	0.293821i	−0.768655	+	0.639664i	$0.779074\pi$
$140$		0			0
$141$	−13.8564	−	8.00000i	−1.16692	−	0.673722i
$142$			2.00000i			0.167836i
$143$		0			0
$144$	1.00000			0.0833333
$145$	7.39230	+	11.1962i	0.613898	+	0.929790i
$146$	−3.00000	+	5.19615i	−0.248282	+	0.430037i
$147$	−12.1244	+	7.00000i	−1.00000	+	0.577350i
$148$	6.00000			0.493197
$149$	−17.3205	+	10.0000i	−1.41895	+	0.819232i	−0.996207	−	0.0870170i	$-0.972267\pi$
$149$	−17.3205	+	10.0000i	−1.41895	+	0.819232i	−0.422744	+	0.906249i	$0.638933\pi$
$150$	−9.19615	−	3.92820i	−0.750863	−	0.320736i
$151$			18.0000i			1.46482i	0.680864	+	0.732410i	$0.261604\pi$
$151$			18.0000i			1.46482i	−0.680864	+	0.732410i	$0.738396\pi$
$152$	−15.5885	+	9.00000i	−1.26439	+	0.729996i
$153$		0			0
$154$		0			0
$155$	−12.0000	−	6.00000i	−0.963863	−	0.481932i
$156$		0			0
$157$			12.0000i			0.957704i	0.877896	+	0.478852i	$0.158947\pi$
$157$			12.0000i			0.957704i	−0.877896	+	0.478852i	$0.841053\pi$
$158$		0			0
$159$	−12.0000	+	20.7846i	−0.951662	+	1.64833i
$160$	−11.1603	+	0.669873i	−0.882296	+	0.0529581i
$161$		0			0
$162$	5.50000	+	9.52628i	0.432121	+	0.748455i
$163$	6.00000	+	10.3923i	0.469956	+	0.813988i	0.999410	−	0.0343508i	$-0.0109363\pi$
$163$	6.00000	+	10.3923i	0.469956	+	0.813988i	−0.529454	+	0.848339i	$0.677603\pi$
$164$		−	8.00000i		−	0.624695i
$165$	−8.92820	+	0.535898i	−0.695060	+	0.0417196i
$166$	−2.00000	+	3.46410i	−0.155230	+	0.268866i
$167$	−8.00000	+	13.8564i	−0.619059	+	1.07224i	0.370599	+	0.928793i	$0.379152\pi$
$167$	−8.00000	+	13.8564i	−0.619059	+	1.07224i	−0.989658	+	0.143448i	$0.954181\pi$
$168$		0			0
$169$		0			0
$170$		0			0
$171$	5.19615	+	3.00000i	0.397360	+	0.229416i
$172$	−5.19615	−	3.00000i	−0.396203	−	0.228748i
$173$	10.3923	−	6.00000i	0.790112	−	0.456172i	−0.0498898	−	0.998755i	$-0.515887\pi$
$173$	10.3923	−	6.00000i	0.790112	−	0.456172i	0.840002	+	0.542583i	$0.182554\pi$
$174$			12.0000i			0.909718i
$175$		0			0
$176$	1.73205	−	1.00000i	0.130558	−	0.0753778i
$177$	4.00000			0.300658
$178$	6.92820	−	4.00000i	0.519291	−	0.299813i
$179$	6.00000	−	10.3923i	0.448461	−	0.776757i	−0.549825	−	0.835280i	$-0.685306\pi$
$179$	6.00000	−	10.3923i	0.448461	−	0.776757i	0.998286	+	0.0585225i	$0.0186389\pi$
$180$	1.23205	+	1.86603i	0.0918316	+	0.139085i
$181$	−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
$183$			12.0000i			0.887066i
$184$	15.5885	+	9.00000i	1.14920	+	0.663489i
$185$	−7.39230	−	11.1962i	−0.543493	−	0.823157i
$186$	−6.00000	−	10.3923i	−0.439941	−	0.762001i
$187$		0			0
$188$	4.00000	+	6.92820i	0.291730	+	0.505291i
$189$		0			0
$190$	12.0000	+	6.00000i	0.870572	+	0.435286i
$191$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$191$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$192$	−12.1244	−	7.00000i	−0.875000	−	0.505181i
$193$	−3.00000	+	5.19615i	−0.215945	+	0.374027i	−0.953564	−	0.301189i	$-0.902616\pi$
$193$	−3.00000	+	5.19615i	−0.215945	+	0.374027i	0.737620	+	0.675216i	$0.235950\pi$
$194$	6.00000			0.430775
$195$		0			0
$196$	7.00000			0.500000
$197$	−1.00000	+	1.73205i	−0.0712470	+	0.123404i	−0.899448	−	0.437028i	$-0.856031\pi$
$197$	−1.00000	+	1.73205i	−0.0712470	+	0.123404i	0.828201	+	0.560431i	$0.189365\pi$
$198$	−1.73205	−	1.00000i	−0.123091	−	0.0710669i
$199$	12.0000	+	20.7846i	0.850657	+	1.47338i	0.880616	+	0.473831i	$0.157129\pi$
$199$	12.0000	+	20.7846i	0.850657	+	1.47338i	−0.0299585	+	0.999551i	$0.509538\pi$
$200$	9.00000	+	12.0000i	0.636396	+	0.848528i
$201$	−20.7846	+	12.0000i	−1.46603	+	0.846415i
$202$	−3.00000	−	5.19615i	−0.211079	−	0.365600i
$203$		0			0
$204$		0			0
$205$	−14.9282	+	9.85641i	−1.04263	+	0.688401i
$206$	−5.19615	−	3.00000i	−0.362033	−	0.209020i
$207$		−	6.00000i		−	0.417029i
$208$		0			0
$209$	12.0000			0.830057
$210$		0			0
$211$	6.00000	−	10.3923i	0.413057	−	0.715436i	−0.582165	−	0.813070i	$-0.697794\pi$
$211$	6.00000	−	10.3923i	0.413057	−	0.715436i	0.995222	+	0.0976347i	$0.0311277\pi$
$212$	10.3923	−	6.00000i	0.713746	−	0.412082i
$213$	−4.00000			−0.274075
$214$	−5.19615	+	3.00000i	−0.355202	+	0.205076i
$215$	0.803848	+	13.3923i	0.0548219	+	0.913348i
$216$		−	12.0000i		−	0.816497i
$217$		0			0
$218$	10.3923	+	6.00000i	0.703856	+	0.406371i
$219$	−10.3923	−	6.00000i	−0.702247	−	0.405442i
$220$	4.00000	+	2.00000i	0.269680	+	0.134840i
$221$		0			0
$222$		−	12.0000i		−	0.805387i
$223$	12.0000	−	20.7846i	0.803579	−	1.39184i	−0.113666	−	0.993519i	$-0.536260\pi$
$223$	12.0000	−	20.7846i	0.803579	−	1.39184i	0.917246	−	0.398321i	$-0.130407\pi$
$224$		0			0
$225$	1.96410	−	4.59808i	0.130940	−	0.306538i
$226$		0			0
$227$	−2.00000	−	3.46410i	−0.132745	−	0.229920i	0.791989	−	0.610535i	$-0.209046\pi$
$227$	−2.00000	−	3.46410i	−0.132745	−	0.229920i	−0.924734	+	0.380615i	$0.875712\pi$
$228$	−6.00000	−	10.3923i	−0.397360	−	0.688247i
$229$			12.0000i			0.792982i	0.918039	+	0.396491i	$0.129772\pi$
$229$			12.0000i			0.792982i	−0.918039	+	0.396491i	$0.870228\pi$
$230$	−0.803848	−	13.3923i	−0.0530041	−	0.883062i
$231$		0			0
$232$	9.00000	−	15.5885i	0.590879	−	1.02343i
$233$		−	24.0000i		−	1.57229i	−0.618041	−	0.786146i	$-0.712073\pi$
$233$		−	24.0000i		−	1.57229i	0.618041	−	0.786146i	$-0.287927\pi$
$234$		0			0
$235$	8.00000	−	16.0000i	0.521862	−	1.04372i
$236$	−1.73205	−	1.00000i	−0.112747	−	0.0650945i
$237$		0			0
$238$		0			0
$239$			10.0000i			0.646846i	0.946254	+	0.323423i	$0.104834\pi$
$239$			10.0000i			0.646846i	−0.946254	+	0.323423i	$0.895166\pi$
$240$	0.267949	+	4.46410i	0.0172960	+	0.288157i
$241$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$241$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$242$	7.00000			0.449977
$243$	−8.66025	+	5.00000i	−0.555556	+	0.320750i
$244$	3.00000	−	5.19615i	0.192055	−	0.332650i
$245$	−8.62436	−	13.0622i	−0.550990	−	0.834512i
$246$	−16.0000			−1.02012
$247$		0			0
$248$			18.0000i			1.14300i
$249$	−6.92820	−	4.00000i	−0.439057	−	0.253490i
$250$	3.76795	−	10.5263i	0.238306	−	0.665740i
$251$	−6.00000	−	10.3923i	−0.378717	−	0.655956i	0.612159	−	0.790735i	$-0.290301\pi$
$251$	−6.00000	−	10.3923i	−0.378717	−	0.655956i	−0.990876	+	0.134778i	$0.956968\pi$
$252$		0			0
$253$	−6.00000	−	10.3923i	−0.377217	−	0.653359i
$254$	−1.73205	+	1.00000i	−0.108679	+	0.0627456i
$255$		0			0
$256$	8.50000	+	14.7224i	0.531250	+	0.920152i
$257$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$257$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$258$	−6.00000	+	10.3923i	−0.373544	+	0.646997i
$259$		0			0
$260$		0			0
$261$	−6.00000			−0.371391
$262$	6.00000	−	10.3923i	0.370681	−	0.642039i
$263$	−5.19615	−	3.00000i	−0.320408	−	0.184988i	0.331166	−	0.943572i	$-0.392558\pi$
$263$	−5.19615	−	3.00000i	−0.320408	−	0.184988i	−0.651575	+	0.758585i	$0.725891\pi$
$264$	6.00000	+	10.3923i	0.369274	+	0.639602i
$265$	−24.0000	−	12.0000i	−1.47431	−	0.737154i
$266$		0			0
$267$	8.00000	+	13.8564i	0.489592	+	0.847998i
$268$	12.0000			0.733017
$269$	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	$0.648441\pi$
$270$	−7.46410	+	4.92820i	−0.454251	+	0.299921i
$271$	5.19615	+	3.00000i	0.315644	+	0.182237i	0.649449	−	0.760405i	$-0.275000\pi$
$271$	5.19615	+	3.00000i	0.315644	+	0.182237i	−0.333805	+	0.942642i	$0.608333\pi$
$272$		0			0
$273$		0			0
$274$	−2.00000			−0.120824
$275$	−1.19615	−	9.92820i	−0.0721307	−	0.598693i
$276$	−6.00000	+	10.3923i	−0.361158	+	0.625543i
$277$	−10.3923	+	6.00000i	−0.624413	+	0.360505i	−0.778585	−	0.627539i	$-0.784062\pi$
$277$	−10.3923	+	6.00000i	−0.624413	+	0.360505i	0.154172	+	0.988044i	$0.450729\pi$
$278$	−4.00000			−0.239904
$279$	5.19615	−	3.00000i	0.311086	−	0.179605i
$280$		0			0
$281$		−	8.00000i		−	0.477240i	−0.971113	−	0.238620i	$-0.923305\pi$
$281$		−	8.00000i		−	0.477240i	0.971113	−	0.238620i	$-0.0766950\pi$
$282$	13.8564	−	8.00000i	0.825137	−	0.476393i
$283$	19.0526	+	11.0000i	1.13256	+	0.653882i	0.944577	−	0.328291i	$-0.106473\pi$
$283$	19.0526	+	11.0000i	1.13256	+	0.653882i	0.187980	+	0.982173i	$0.439806\pi$
$284$	1.73205	+	1.00000i	0.102778	+	0.0593391i
$285$	−12.0000	+	24.0000i	−0.710819	+	1.42164i
$286$		0			0
$287$		0			0
$288$	2.50000	−	4.33013i	0.147314	−	0.255155i
$289$	−8.50000	+	14.7224i	−0.500000	+	0.866025i
$290$	−13.3923	+	0.803848i	−0.786423	+	0.0472036i
$291$			12.0000i			0.703452i
$292$	3.00000	+	5.19615i	0.175562	+	0.304082i
$293$	−13.0000	−	22.5167i	−0.759468	−	1.31544i	−0.943122	−	0.332446i	$-0.892126\pi$
$293$	−13.0000	−	22.5167i	−0.759468	−	1.31544i	0.183654	−	0.982991i	$-0.441207\pi$
$294$		−	14.0000i		−	0.816497i
$295$	0.267949	+	4.46410i	0.0156006	+	0.259910i
$296$	−9.00000	+	15.5885i	−0.523114	+	0.906061i
$297$	−4.00000	+	6.92820i	−0.232104	+	0.402015i
$298$		−	20.0000i		−	1.15857i
$299$		0			0
$300$	−8.00000	+	6.00000i	−0.461880	+	0.346410i
$301$		0			0
$302$	−15.5885	−	9.00000i	−0.897015	−	0.517892i
$303$	10.3923	−	6.00000i	0.597022	−	0.344691i
$304$		−	6.00000i		−	0.344124i
$305$	−13.3923	+	0.803848i	−0.766841	+	0.0460282i
$306$		0			0
$307$	−12.0000			−0.684876			−0.342438	−	0.939540i	$-0.611253\pi$
$307$	−12.0000			−0.684876			−0.342438	+	0.939540i	$0.611253\pi$
$308$		0			0
$309$	6.00000	−	10.3923i	0.341328	−	0.591198i
$310$	11.1962	−	7.39230i	0.635899	−	0.419855i
$311$	24.0000			1.36092			0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000			1.36092			0.680458	+	0.732787i	$0.261781\pi$
$312$		0			0
$313$			8.00000i			0.452187i	0.974106	+	0.226093i	$0.0725954\pi$
$313$			8.00000i			0.452187i	−0.974106	+	0.226093i	$0.927405\pi$
$314$	−10.3923	−	6.00000i	−0.586472	−	0.338600i
$315$		0			0
$316$		0			0
$317$	2.00000			0.112331			0.0561656	−	0.998421i	$-0.482113\pi$
$317$	2.00000			0.112331			0.0561656	+	0.998421i	$0.482113\pi$
$318$	−12.0000	−	20.7846i	−0.672927	−	1.16554i
$319$	−10.3923	+	6.00000i	−0.581857	+	0.335936i
$320$	7.00000	−	14.0000i	0.391312	−	0.782624i
$321$	−6.00000	−	10.3923i	−0.334887	−	0.580042i
$322$		0			0
$323$		0			0
$324$	11.0000			0.611111
$325$		0			0
$326$	−12.0000			−0.664619
$327$	−12.0000	+	20.7846i	−0.663602	+	1.14939i
$328$	20.7846	+	12.0000i	1.14764	+	0.662589i
$329$		0			0
$330$	4.00000	−	8.00000i	0.220193	−	0.440386i
$331$	25.9808	−	15.0000i	1.42803	−	0.824475i	0.431066	−	0.902320i	$-0.358137\pi$
$331$	25.9808	−	15.0000i	1.42803	−	0.824475i	0.996965	+	0.0778456i	$0.0248041\pi$
$332$	2.00000	+	3.46410i	0.109764	+	0.190117i
$333$	6.00000			0.328798
$334$	−8.00000	−	13.8564i	−0.437741	−	0.758189i
$335$	−14.7846	−	22.3923i	−0.807770	−	1.22342i
$336$		0			0
$337$			32.0000i			1.74315i	0.490261	+	0.871576i	$0.336901\pi$
$337$			32.0000i			1.74315i	−0.490261	+	0.871576i	$0.663099\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	6.00000	−	10.3923i	0.324918	−	0.562775i
$342$	−5.19615	+	3.00000i	−0.280976	+	0.162221i
$343$		0			0
$344$	15.5885	−	9.00000i	0.840473	−	0.485247i
$345$	26.7846	−	1.60770i	1.44203	−	0.0865554i
$346$			12.0000i			0.645124i
$347$	−5.19615	+	3.00000i	−0.278944	+	0.161048i	−0.632945	−	0.774197i	$-0.718154\pi$
$347$	−5.19615	+	3.00000i	−0.278944	+	0.161048i	0.354001	+	0.935245i	$0.384821\pi$
$348$	10.3923	+	6.00000i	0.557086	+	0.321634i
$349$	10.3923	+	6.00000i	0.556287	+	0.321173i	0.751654	−	0.659558i	$-0.229256\pi$
$349$	10.3923	+	6.00000i	0.556287	+	0.321173i	−0.195367	+	0.980730i	$0.562590\pi$
$350$		0			0
$351$		0			0
$352$		−	10.0000i		−	0.533002i
$353$	−7.00000	+	12.1244i	−0.372572	+	0.645314i	−0.989960	−	0.141344i	$-0.954858\pi$
$353$	−7.00000	+	12.1244i	−0.372572	+	0.645314i	0.617388	+	0.786659i	$0.288191\pi$
$354$	−2.00000	+	3.46410i	−0.106299	+	0.184115i
$355$	−0.267949	−	4.46410i	−0.0142213	−	0.236930i
$356$		−	8.00000i		−	0.423999i
$357$		0			0
$358$	6.00000	+	10.3923i	0.317110	+	0.549250i
$359$			2.00000i			0.105556i	0.998606	+	0.0527780i	$0.0168076\pi$
$359$			2.00000i			0.105556i	−0.998606	+	0.0527780i	$0.983192\pi$
$360$	−6.69615	+	0.401924i	−0.352918	+	0.0211832i
$361$	8.50000	−	14.7224i	0.447368	−	0.774865i
$362$	1.00000	−	1.73205i	0.0525588	−	0.0910346i
$363$			14.0000i			0.734809i
$364$		0			0
$365$	6.00000	−	12.0000i	0.314054	−	0.628109i
$366$	−10.3923	−	6.00000i	−0.543214	−	0.313625i
$367$	15.5885	+	9.00000i	0.813711	+	0.469796i	0.848243	−	0.529607i	$-0.177661\pi$
$367$	15.5885	+	9.00000i	0.813711	+	0.469796i	−0.0345320	+	0.999404i	$0.510994\pi$
$368$	−5.19615	+	3.00000i	−0.270868	+	0.156386i
$369$		−	8.00000i		−	0.416463i
$370$	13.3923	−	0.803848i	0.696233	−	0.0417900i
$371$		0			0
$372$	−12.0000			−0.622171
$373$	3.46410	−	2.00000i	0.179364	−	0.103556i	−0.407630	−	0.913147i	$-0.633645\pi$
$373$	3.46410	−	2.00000i	0.179364	−	0.103556i	0.586994	+	0.809591i	$0.300311\pi$
$374$		0			0
$375$	21.0526	+	7.53590i	1.08715	+	0.389152i
$376$	−24.0000			−1.23771
$377$		0			0
$378$		0			0
$379$	15.5885	+	9.00000i	0.800725	+	0.462299i	0.843725	−	0.536776i	$-0.180358\pi$
$379$	15.5885	+	9.00000i	0.800725	+	0.462299i	−0.0429994	+	0.999075i	$0.513691\pi$
$380$	11.1962	−	7.39230i	0.574351	−	0.379217i
$381$	−2.00000	−	3.46410i	−0.102463	−	0.177471i
$382$		0			0
$383$	4.00000	+	6.92820i	0.204390	+	0.354015i	0.949938	−	0.312437i	$-0.101145\pi$
$383$	4.00000	+	6.92820i	0.204390	+	0.354015i	−0.745548	+	0.666452i	$0.767812\pi$
$384$	−5.19615	+	3.00000i	−0.265165	+	0.153093i
$385$		0			0
$386$	−3.00000	−	5.19615i	−0.152696	−	0.264477i
$387$	−5.19615	−	3.00000i	−0.264135	−	0.152499i
$388$	3.00000	−	5.19615i	0.152302	−	0.263795i
$389$	6.00000			0.304212			0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000			0.304212			0.152106	+	0.988364i	$0.451394\pi$
$390$		0			0
$391$		0			0
$392$	−10.5000	+	18.1865i	−0.530330	+	0.918559i
$393$	20.7846	+	12.0000i	1.04844	+	0.605320i
$394$	−1.00000	−	1.73205i	−0.0503793	−	0.0872595i
$395$		0			0
$396$	−1.73205	+	1.00000i	−0.0870388	+	0.0502519i
$397$	−9.00000	−	15.5885i	−0.451697	−	0.782362i	0.546795	−	0.837267i	$-0.315848\pi$
$397$	−9.00000	−	15.5885i	−0.451697	−	0.782362i	−0.998492	+	0.0549046i	$0.982515\pi$
$398$	−24.0000			−1.20301
$399$		0			0
$400$	−4.96410	+	0.598076i	−0.248205	+	0.0299038i
$401$	13.8564	+	8.00000i	0.691956	+	0.399501i	0.804344	−	0.594163i	$-0.202517\pi$
$401$	13.8564	+	8.00000i	0.691956	+	0.399501i	−0.112388	+	0.993664i	$0.535850\pi$
$402$		−	24.0000i		−	1.19701i
$403$		0			0
$404$	−6.00000			−0.298511
$405$	−13.5526	−	20.5263i	−0.673432	−	1.01996i
$406$		0			0
$407$	10.3923	−	6.00000i	0.515127	−	0.297409i
$408$		0			0
$409$	20.7846	−	12.0000i	1.02773	−	0.593362i	0.111398	−	0.993776i	$-0.464467\pi$
$409$	20.7846	−	12.0000i	1.02773	−	0.593362i	0.916334	+	0.400414i	$0.131134\pi$
$410$	−1.07180	−	17.8564i	−0.0529323	−	0.881865i
$411$		−	4.00000i		−	0.197305i
$412$	−5.19615	+	3.00000i	−0.255996	+	0.147799i
$413$		0			0
$414$	5.19615	+	3.00000i	0.255377	+	0.147442i
$415$	4.00000	−	8.00000i	0.196352	−	0.392705i
$416$		0			0
$417$		−	8.00000i		−	0.391762i
$418$	−6.00000	+	10.3923i	−0.293470	+	0.508304i
$419$	6.00000	−	10.3923i	0.293119	−	0.507697i	−0.681426	−	0.731887i	$-0.738640\pi$
$419$	6.00000	−	10.3923i	0.293119	−	0.507697i	0.974546	+	0.224189i	$0.0719734\pi$
$420$		0			0
$421$		−	36.0000i		−	1.75453i	−0.480004	−	0.877266i	$-0.659365\pi$
$421$		−	36.0000i		−	1.75453i	0.480004	−	0.877266i	$-0.340635\pi$
$422$	6.00000	+	10.3923i	0.292075	+	0.505889i
$423$	4.00000	+	6.92820i	0.194487	+	0.336861i
$424$			36.0000i			1.74831i
$425$		0			0
$426$	2.00000	−	3.46410i	0.0969003	−	0.167836i
$427$		0			0
$428$			6.00000i			0.290021i
$429$		0			0
$430$	−12.0000	−	6.00000i	−0.578691	−	0.289346i
$431$	−8.66025	−	5.00000i	−0.417150	−	0.240842i	0.276707	−	0.960954i	$-0.410757\pi$
$431$	−8.66025	−	5.00000i	−0.417150	−	0.240842i	−0.693857	+	0.720113i	$0.744090\pi$
$432$	3.46410	+	2.00000i	0.166667	+	0.0962250i
$433$	13.8564	−	8.00000i	0.665896	−	0.384455i	−0.128624	−	0.991693i	$-0.541056\pi$
$433$	13.8564	−	8.00000i	0.665896	−	0.384455i	0.794520	+	0.607238i	$0.207723\pi$
$434$		0			0
$435$	−1.60770	−	26.7846i	−0.0770831	−	1.28422i
$436$	10.3923	−	6.00000i	0.497701	−	0.287348i
$437$	−36.0000			−1.72211
$438$	10.3923	−	6.00000i	0.496564	−	0.286691i
$439$	−4.00000	+	6.92820i	−0.190910	+	0.330665i	−0.945552	−	0.325471i	$-0.894477\pi$
$439$	−4.00000	+	6.92820i	−0.190910	+	0.330665i	0.754642	+	0.656136i	$0.227810\pi$
$440$	−11.1962	+	7.39230i	−0.533756	+	0.352414i
$441$	7.00000			0.333333
$442$		0			0
$443$		−	6.00000i		−	0.285069i	−0.989790	−	0.142534i	$-0.954475\pi$
$443$		−	6.00000i		−	0.285069i	0.989790	−	0.142534i	$-0.0455251\pi$
$444$	−10.3923	−	6.00000i	−0.493197	−	0.284747i
$445$	−14.9282	+	9.85641i	−0.707665	+	0.467238i
$446$	12.0000	+	20.7846i	0.568216	+	0.984180i
$447$	40.0000			1.89194
$448$		0			0
$449$	13.8564	−	8.00000i	0.653924	−	0.377543i	−0.136034	−	0.990704i	$-0.543436\pi$
$449$	13.8564	−	8.00000i	0.653924	−	0.377543i	0.789958	+	0.613161i	$0.210102\pi$
$450$	3.00000	+	4.00000i	0.141421	+	0.188562i
$451$	−8.00000	−	13.8564i	−0.376705	−	0.652473i
$452$		0			0
$453$	18.0000	−	31.1769i	0.845714	−	1.46482i
$454$	4.00000			0.187729
$455$		0			0
$456$	36.0000			1.68585
$457$	−15.0000	+	25.9808i	−0.701670	+	1.21533i	0.266209	+	0.963915i	$0.414229\pi$
$457$	−15.0000	+	25.9808i	−0.701670	+	1.21533i	−0.967880	+	0.251414i	$0.919105\pi$
$458$	−10.3923	−	6.00000i	−0.485601	−	0.280362i
$459$		0			0
$460$	−12.0000	−	6.00000i	−0.559503	−	0.279751i
$461$	3.46410	−	2.00000i	0.161339	−	0.0931493i	−0.417156	−	0.908835i	$-0.636973\pi$
$461$	3.46410	−	2.00000i	0.161339	−	0.0931493i	0.578496	+	0.815685i	$0.303640\pi$
$462$		0			0
$463$	24.0000			1.11537			0.557687	−	0.830051i	$-0.311689\pi$
$463$	24.0000			1.11537			0.557687	+	0.830051i	$0.311689\pi$
$464$	3.00000	+	5.19615i	0.139272	+	0.241225i
$465$	14.7846	+	22.3923i	0.685620	+	1.03842i
$466$	20.7846	+	12.0000i	0.962828	+	0.555889i
$467$			18.0000i			0.832941i	0.909149	+	0.416470i	$0.136733\pi$
$467$			18.0000i			0.832941i	−0.909149	+	0.416470i	$0.863267\pi$
$468$		0			0
$469$		0			0
$470$	9.85641	+	14.9282i	0.454642	+	0.688587i
$471$	12.0000	−	20.7846i	0.552931	−	0.957704i
$472$	5.19615	−	3.00000i	0.239172	−	0.138086i
$473$	−12.0000			−0.551761
$474$		0			0
$475$	−27.5885	−	11.7846i	−1.26585	−	0.540715i
$476$		0			0
$477$	10.3923	−	6.00000i	0.475831	−	0.274721i
$478$	−8.66025	−	5.00000i	−0.396111	−	0.228695i
$479$	19.0526	+	11.0000i	0.870534	+	0.502603i	0.867526	−	0.497393i	$-0.165709\pi$
$479$	19.0526	+	11.0000i	0.870534	+	0.502603i	0.00300810	+	0.999995i	$0.499042\pi$
$480$	20.0000	+	10.0000i	0.912871	+	0.456435i
$481$		0			0
$482$		0			0
$483$		0			0
$484$	3.50000	−	6.06218i	0.159091	−	0.275554i
$485$	−13.3923	+	0.803848i	−0.608113	+	0.0365008i
$486$		−	10.0000i		−	0.453609i
$487$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$487$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$488$	9.00000	+	15.5885i	0.407411	+	0.705656i
$489$		−	24.0000i		−	1.08532i
$490$	15.6244	−	0.937822i	0.705836	−	0.0423665i
$491$	−6.00000	+	10.3923i	−0.270776	+	0.468998i	−0.969061	−	0.246822i	$-0.920614\pi$
$491$	−6.00000	+	10.3923i	−0.270776	+	0.468998i	0.698285	+	0.715820i	$0.253947\pi$
$492$	−8.00000	+	13.8564i	−0.360668	+	0.624695i
$493$		0			0
$494$		0			0
$495$	4.00000	+	2.00000i	0.179787	+	0.0898933i
$496$	−5.19615	−	3.00000i	−0.233314	−	0.134704i
$497$		0			0
$498$	6.92820	−	4.00000i	0.310460	−	0.179244i
$499$			6.00000i			0.268597i	0.990941	+	0.134298i	$0.0428781\pi$
$499$			6.00000i			0.268597i	−0.990941	+	0.134298i	$0.957122\pi$
$500$	−7.23205	−	8.52628i	−0.323427	−	0.381307i
$501$	27.7128	−	16.0000i	1.23812	−	0.714827i
$502$	12.0000			0.535586
$503$	5.19615	−	3.00000i	0.231685	−	0.133763i	−0.379664	−	0.925124i	$-0.623960\pi$
$503$	5.19615	−	3.00000i	0.231685	−	0.133763i	0.611349	+	0.791361i	$0.290627\pi$
$504$		0			0
$505$	7.39230	+	11.1962i	0.328953	+	0.498222i
$506$	12.0000			0.533465
$507$		0			0
$508$			2.00000i			0.0887357i
$509$	−17.3205	−	10.0000i	−0.767718	−	0.443242i	0.0643419	−	0.997928i	$-0.479505\pi$
$509$	−17.3205	−	10.0000i	−0.767718	−	0.443242i	−0.832060	+	0.554686i	$0.812839\pi$
$510$		0			0
$511$		0			0
$512$	−11.0000			−0.486136
$513$	12.0000	+	20.7846i	0.529813	+	0.917663i
$514$		0			0
$515$	12.0000	+	6.00000i	0.528783	+	0.264392i
$516$	6.00000	+	10.3923i	0.264135	+	0.457496i
$517$	13.8564	+	8.00000i	0.609404	+	0.351840i
$518$		0			0
$519$	−24.0000			−1.05348
$520$		0			0
$521$	−30.0000			−1.31432			−0.657162	−	0.753749i	$-0.728243\pi$
$521$	−30.0000			−1.31432			−0.657162	+	0.753749i	$0.728243\pi$
$522$	3.00000	−	5.19615i	0.131306	−	0.227429i
$523$	−36.3731	−	21.0000i	−1.59048	−	0.918266i	−0.993224	−	0.116218i	$-0.962923\pi$
$523$	−36.3731	−	21.0000i	−1.59048	−	0.918266i	−0.597259	−	0.802048i	$-0.703744\pi$
$524$	−6.00000	−	10.3923i	−0.262111	−	0.453990i
$525$		0			0
$526$	5.19615	−	3.00000i	0.226563	−	0.130806i
$527$		0			0
$528$	−4.00000			−0.174078
$529$	6.50000	+	11.2583i	0.282609	+	0.489493i
$530$	22.3923	−	14.7846i	0.972660	−	0.642202i
$531$	−1.73205	−	1.00000i	−0.0751646	−	0.0433963i
$532$		0			0
$533$		0			0
$534$	−16.0000			−0.692388
$535$	11.1962	−	7.39230i	0.484052	−	0.319597i
$536$	−18.0000	+	31.1769i	−0.777482	+	1.34664i
$537$	−20.7846	+	12.0000i	−0.896922	+	0.517838i
$538$	−18.0000			−0.776035
$539$	12.1244	−	7.00000i	0.522233	−	0.301511i
$540$	0.535898	+	8.92820i	0.0230614	+	0.384209i
$541$		−	12.0000i		−	0.515920i	−0.966156	−	0.257960i	$-0.916950\pi$
$541$		−	12.0000i		−	0.515920i	0.966156	−	0.257960i	$-0.0830503\pi$
$542$	−5.19615	+	3.00000i	−0.223194	+	0.128861i
$543$	3.46410	+	2.00000i	0.148659	+	0.0858282i
$544$		0			0
$545$	−24.0000	−	12.0000i	−1.02805	−	0.514024i
$546$		0			0
$547$			18.0000i			0.769624i	0.922995	+	0.384812i	$0.125734\pi$
$547$			18.0000i			0.769624i	−0.922995	+	0.384812i	$0.874266\pi$
$548$	−1.00000	+	1.73205i	−0.0427179	+	0.0739895i
$549$	3.00000	−	5.19615i	0.128037	−	0.221766i
$550$	9.19615	+	3.92820i	0.392125	+	0.167499i
$551$			36.0000i			1.53365i
$552$	−18.0000	−	31.1769i	−0.766131	−	1.32698i
$553$		0			0
$554$		−	12.0000i		−	0.509831i
$555$	1.60770	+	26.7846i	0.0682429	+	1.13694i
$556$	−2.00000	+	3.46410i	−0.0848189	+	0.146911i
$557$	7.00000	−	12.1244i	0.296600	−	0.513725i	−0.678756	−	0.734364i	$-0.737481\pi$
$557$	7.00000	−	12.1244i	0.296600	−	0.513725i	0.975356	+	0.220638i	$0.0708140\pi$
$558$			6.00000i			0.254000i
$559$		0			0
$560$		0			0
$561$		0			0
$562$	6.92820	+	4.00000i	0.292249	+	0.168730i
$563$	−25.9808	+	15.0000i	−1.09496	+	0.632175i	−0.934892	−	0.354932i	$-0.884504\pi$
$563$	−25.9808	+	15.0000i	−1.09496	+	0.632175i	−0.160066	+	0.987106i	$0.551171\pi$
$564$		−	16.0000i		−	0.673722i
$565$		0			0
$566$	−19.0526	+	11.0000i	−0.800839	+	0.462364i
$567$		0			0
$568$	−5.19615	+	3.00000i	−0.218026	+	0.125877i
$569$	−9.00000	+	15.5885i	−0.377300	+	0.653502i	−0.990668	−	0.136295i	$-0.956481\pi$
$569$	−9.00000	+	15.5885i	−0.377300	+	0.653502i	0.613369	+	0.789797i	$0.289814\pi$
$570$	−14.7846	−	22.3923i	−0.619259	−	0.937910i
$571$	12.0000			0.502184			0.251092	−	0.967963i	$-0.419210\pi$
$571$	12.0000			0.502184			0.251092	+	0.967963i	$0.419210\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	3.58846	+	29.7846i	0.149649	+	1.24210i
$576$	3.50000	+	6.06218i	0.145833	+	0.252591i
$577$	−18.0000			−0.749350			−0.374675	−	0.927156i	$-0.622246\pi$
$577$	−18.0000			−0.749350			−0.374675	+	0.927156i	$0.622246\pi$
$578$	−8.50000	−	14.7224i	−0.353553	−	0.612372i
$579$	10.3923	−	6.00000i	0.431889	−	0.249351i
$580$	−6.00000	+	12.0000i	−0.249136	+	0.498273i
$581$		0			0
$582$	−10.3923	−	6.00000i	−0.430775	−	0.248708i
$583$	12.0000	−	20.7846i	0.496989	−	0.860811i
$584$	−18.0000			−0.744845
$585$		0			0
$586$	26.0000			1.07405
$587$	10.0000	−	17.3205i	0.412744	−	0.714894i	−0.582445	−	0.812870i	$-0.697904\pi$
$587$	10.0000	−	17.3205i	0.412744	−	0.714894i	0.995189	+	0.0979766i	$0.0312370\pi$
$588$	−12.1244	−	7.00000i	−0.500000	−	0.288675i
$589$	−18.0000	−	31.1769i	−0.741677	−	1.28462i
$590$	−4.00000	−	2.00000i	−0.164677	−	0.0823387i
$591$	3.46410	−	2.00000i	0.142494	−	0.0822690i
$592$	−3.00000	−	5.19615i	−0.123299	−	0.213561i
$593$	22.0000			0.903432			0.451716	−	0.892162i	$-0.350812\pi$
$593$	22.0000			0.903432			0.451716	+	0.892162i	$0.350812\pi$
$594$	−4.00000	−	6.92820i	−0.164122	−	0.284268i
$595$		0			0
$596$	−17.3205	−	10.0000i	−0.709476	−	0.409616i
$597$		−	48.0000i		−	1.96451i
$598$		0			0
$599$	24.0000			0.980613			0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000			0.980613			0.490307	+	0.871550i	$0.336885\pi$
$600$	−3.58846	−	29.7846i	−0.146498	−	1.21595i
$601$	3.00000	−	5.19615i	0.122373	−	0.211955i	−0.798330	−	0.602220i	$-0.794283\pi$
$601$	3.00000	−	5.19615i	0.122373	−	0.211955i	0.920703	+	0.390264i	$0.127616\pi$
$602$		0			0
$603$	12.0000			0.488678
$604$	−15.5885	+	9.00000i	−0.634285	+	0.366205i
$605$	−15.6244	+	0.937822i	−0.635220	+	0.0381279i
$606$			12.0000i			0.487467i
$607$	−15.5885	+	9.00000i	−0.632716	+	0.365299i	−0.781803	−	0.623525i	$-0.785700\pi$
$607$	−15.5885	+	9.00000i	−0.632716	+	0.365299i	0.149087	+	0.988824i	$0.452366\pi$
$608$	−25.9808	−	15.0000i	−1.05366	−	0.608330i
$609$		0			0
$610$	6.00000	−	12.0000i	0.242933	−	0.485866i
$611$		0			0
$612$		0			0
$613$	15.0000	−	25.9808i	0.605844	−	1.04935i	−0.386073	−	0.922468i	$-0.626169\pi$
$613$	15.0000	−	25.9808i	0.605844	−	1.04935i	0.991917	−	0.126885i	$-0.0404979\pi$
$614$	6.00000	−	10.3923i	0.242140	−	0.419399i
$615$	35.7128	−	2.14359i	1.44008	−	0.0864380i
$616$		0			0
$617$	17.0000	+	29.4449i	0.684394	+	1.18541i	0.973627	+	0.228147i	$0.0732666\pi$
$617$	17.0000	+	29.4449i	0.684394	+	1.18541i	−0.289233	+	0.957259i	$0.593400\pi$
$618$	6.00000	+	10.3923i	0.241355	+	0.418040i
$619$		−	18.0000i		−	0.723481i	−0.932279	−	0.361741i	$-0.882183\pi$
$619$		−	18.0000i		−	0.723481i	0.932279	−	0.361741i	$-0.117817\pi$
$620$	−0.803848	−	13.3923i	−0.0322833	−	0.537848i
$621$	12.0000	−	20.7846i	0.481543	−	0.834058i
$622$	−12.0000	+	20.7846i	−0.481156	+	0.833387i
$623$		0			0
$624$		0			0
$625$	−7.00000	+	24.0000i	−0.280000	+	0.960000i
$626$	−6.92820	−	4.00000i	−0.276907	−	0.159872i
$627$	−20.7846	−	12.0000i	−0.830057	−	0.479234i
$628$	−10.3923	+	6.00000i	−0.414698	+	0.239426i
$629$		0			0
$630$		0			0
$631$	−25.9808	+	15.0000i	−1.03428	+	0.597141i	−0.918207	−	0.396100i	$-0.870363\pi$
$631$	−25.9808	+	15.0000i	−1.03428	+	0.597141i	−0.116071	+	0.993241i	$0.537030\pi$
$632$		0			0
$633$	−20.7846	+	12.0000i	−0.826114	+	0.476957i
$634$	−1.00000	+	1.73205i	−0.0397151	+	0.0687885i
$635$	3.73205	−	2.46410i	0.148102	−	0.0977849i
$636$	−24.0000			−0.951662
$637$		0			0
$638$		−	12.0000i		−	0.475085i
$639$	1.73205	+	1.00000i	0.0685189	+	0.0395594i
$640$	−3.69615	−	5.59808i	−0.146103	−	0.221283i
$641$	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	$0.631488\pi$
$642$	12.0000			0.473602
$643$	−18.0000	−	31.1769i	−0.709851	−	1.22950i	−0.964912	−	0.262573i	$-0.915429\pi$
$643$	−18.0000	−	31.1769i	−0.709851	−	1.22950i	0.255062	−	0.966925i	$-0.417904\pi$
$644$		0			0
$645$	12.0000	−	24.0000i	0.472500	−	0.944999i
$646$		0			0
$647$	−5.19615	−	3.00000i	−0.204282	−	0.117942i	0.394369	−	0.918952i	$-0.370963\pi$
$647$	−5.19615	−	3.00000i	−0.204282	−	0.117942i	−0.598651	+	0.801010i	$0.704296\pi$
$648$	−16.5000	+	28.5788i	−0.648181	+	1.12268i
$649$	−4.00000			−0.157014
$650$		0			0
$651$		0			0
$652$	−6.00000	+	10.3923i	−0.234978	+	0.406994i
$653$	31.1769	+	18.0000i	1.22005	+	0.704394i	0.964928	−	0.262515i	$-0.0845520\pi$
$653$	31.1769	+	18.0000i	1.22005	+	0.704394i	0.255119	+	0.966910i	$0.417885\pi$
$654$	−12.0000	−	20.7846i	−0.469237	−	0.812743i
$655$	−12.0000	+	24.0000i	−0.468879	+	0.937758i
$656$	−6.92820	+	4.00000i	−0.270501	+	0.156174i
$657$	3.00000	+	5.19615i	0.117041	+	0.202721i
$658$		0			0
$659$	−18.0000	−	31.1769i	−0.701180	−	1.21448i	−0.968052	−	0.250748i	$-0.919323\pi$
$659$	−18.0000	−	31.1769i	−0.701180	−	1.21448i	0.266872	−	0.963732i	$-0.414010\pi$
$660$	−4.92820	−	7.46410i	−0.191830	−	0.290540i
$661$	−10.3923	−	6.00000i	−0.404214	−	0.233373i	0.284087	−	0.958799i	$-0.408310\pi$
$661$	−10.3923	−	6.00000i	−0.404214	−	0.233373i	−0.688301	+	0.725426i	$0.741643\pi$
$662$			30.0000i			1.16598i
$663$		0			0
$664$	−12.0000			−0.465690
$665$		0			0
$666$	−3.00000	+	5.19615i	−0.116248	+	0.201347i
$667$	31.1769	−	18.0000i	1.20717	−	0.696963i
$668$	−16.0000			−0.619059
$669$	−41.5692	+	24.0000i	−1.60716	+	0.927894i
$670$	26.7846	−	1.60770i	1.03478	−	0.0621107i
$671$		−	12.0000i		−	0.463255i
$672$		0			0
$673$	−41.5692	−	24.0000i	−1.60238	−	0.925132i	−0.991011	−	0.133783i	$-0.957287\pi$
$673$	−41.5692	−	24.0000i	−1.60238	−	0.925132i	−0.611365	−	0.791349i	$-0.709379\pi$
$674$	−27.7128	−	16.0000i	−1.06746	−	0.616297i
$675$	16.0000	−	12.0000i	0.615840	−	0.461880i
$676$		0			0
$677$			36.0000i			1.38359i	0.722093	+	0.691796i	$0.243180\pi$
$677$			36.0000i			1.38359i	−0.722093	+	0.691796i	$0.756820\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$			8.00000i			0.306561i
$682$	6.00000	+	10.3923i	0.229752	+	0.397942i
$683$	−22.0000	−	38.1051i	−0.841807	−	1.45805i	−0.888366	−	0.459136i	$-0.848159\pi$
$683$	−22.0000	−	38.1051i	−0.841807	−	1.45805i	0.0465592	−	0.998916i	$-0.485174\pi$
$684$			6.00000i			0.229416i
$685$	4.46410	−	0.267949i	0.170565	−	0.0102378i
$686$		0			0
$687$	12.0000	−	20.7846i	0.457829	−	0.792982i
$688$			6.00000i			0.228748i
$689$		0			0
$690$	−12.0000	+	24.0000i	−0.456832	+	0.913664i
$691$	36.3731	+	21.0000i	1.38370	+	0.798878i	0.992595	−	0.121470i	$-0.0387608\pi$
$691$	36.3731	+	21.0000i	1.38370	+	0.798878i	0.391102	+	0.920348i	$0.372094\pi$
$692$	10.3923	+	6.00000i	0.395056	+	0.228086i
$693$		0			0
$694$		−	6.00000i		−	0.227757i
$695$	8.92820	−	0.535898i	0.338666	−	0.0203278i
$696$	−31.1769	+	18.0000i	−1.18176	+	0.682288i
$697$		0			0
$698$	−10.3923	+	6.00000i	−0.393355	+	0.227103i
$699$	−24.0000	+	41.5692i	−0.907763	+	1.57229i
$700$		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
$703$		−	36.0000i		−	1.35777i
$704$	12.1244	+	7.00000i	0.456954	+	0.263822i
$705$	−29.8564	+	19.7128i	−1.12446	+	0.742427i
$706$	−7.00000	−	12.1244i	−0.263448	−	0.456306i
$707$		0			0
$708$	2.00000	+	3.46410i	0.0751646	+	0.130189i
$709$	10.3923	−	6.00000i	0.390291	−	0.225335i	−0.291995	−	0.956420i	$-0.594319\pi$
$709$	10.3923	−	6.00000i	0.390291	−	0.225335i	0.682286	+	0.731085i	$0.260986\pi$
$710$	4.00000	+	2.00000i	0.150117	+	0.0750587i
$711$		0			0
$712$	20.7846	+	12.0000i	0.778936	+	0.449719i
$713$	−18.0000	+	31.1769i	−0.674105	+	1.16758i
$714$		0			0
$715$		0			0
$716$	12.0000			0.448461
$717$	10.0000	−	17.3205i	0.373457	−	0.646846i
$718$	−1.73205	−	1.00000i	−0.0646396	−	0.0373197i
$719$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$719$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$720$	1.00000	−	2.00000i	0.0372678	−	0.0745356i
$721$		0			0
$722$	8.50000	+	14.7224i	0.316337	+	0.547912i
$723$		0			0
$724$	−1.00000	−	1.73205i	−0.0371647	−	0.0643712i
$725$	29.7846	−	3.58846i	1.10617	−	0.133272i
$726$	−12.1244	−	7.00000i	−0.449977	−	0.259794i
$727$		−	26.0000i		−	0.964287i	−0.876092	−	0.482143i	$-0.839858\pi$
$727$		−	26.0000i		−	0.964287i	0.876092	−	0.482143i	$-0.160142\pi$
$728$		0			0
$729$	−13.0000			−0.481481
$730$	7.39230	+	11.1962i	0.273601	+	0.414388i
$731$		0			0
$732$	−10.3923	+	6.00000i	−0.384111	+	0.221766i
$733$	42.0000			1.55131			0.775653	−	0.631160i	$-0.217421\pi$
$733$	42.0000			1.55131			0.775653	+	0.631160i	$0.217421\pi$
$734$	−15.5885	+	9.00000i	−0.575380	+	0.332196i
$735$	1.87564	+	31.2487i	0.0691842	+	1.15263i
$736$			30.0000i			1.10581i
$737$	20.7846	−	12.0000i	0.765611	−	0.442026i
$738$	6.92820	+	4.00000i	0.255031	+	0.147242i
$739$	−5.19615	−	3.00000i	−0.191144	−	0.110357i	0.401374	−	0.915914i	$-0.368533\pi$
$739$	−5.19615	−	3.00000i	−0.191144	−	0.110357i	−0.592518	+	0.805557i	$0.701866\pi$
$740$	6.00000	−	12.0000i	0.220564	−	0.441129i
$741$		0			0
$742$		0			0
$743$	−8.00000	+	13.8564i	−0.293492	+	0.508342i	−0.974633	−	0.223810i	$-0.928151\pi$
$743$	−8.00000	+	13.8564i	−0.293492	+	0.508342i	0.681141	+	0.732152i	$0.261484\pi$
$744$	18.0000	−	31.1769i	0.659912	−	1.14300i
$745$	2.67949	+	44.6410i	0.0981690	+	1.63552i
$746$			4.00000i			0.146450i
$747$	2.00000	+	3.46410i	0.0731762	+	0.126745i
$748$		0			0
$749$		0			0
$750$	−17.0526	+	14.4641i	−0.622671	+	0.528154i
$751$	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
$753$			24.0000i			0.874609i
$754$		0			0
$755$	36.0000	+	18.0000i	1.31017	+	0.655087i
$756$		0			0
$757$	−17.3205	−	10.0000i	−0.629525	−	0.363456i	0.151043	−	0.988527i	$-0.451737\pi$
$757$	−17.3205	−	10.0000i	−0.629525	−	0.363456i	−0.780568	+	0.625071i	$0.785070\pi$
$758$	−15.5885	+	9.00000i	−0.566198	+	0.326895i
$759$			24.0000i			0.871145i
$760$	2.41154	+	40.1769i	0.0874758	+	1.45737i
$761$	34.6410	−	20.0000i	1.25574	−	0.724999i	0.283493	−	0.958974i	$-0.408507\pi$
$761$	34.6410	−	20.0000i	1.25574	−	0.724999i	0.972243	+	0.233975i	$0.0751733\pi$
$762$	4.00000			0.144905
$763$		0			0
$764$		0			0
$765$		0			0
$766$	−8.00000			−0.289052
$767$		0			0
$768$		−	34.0000i		−	1.22687i
$769$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$769$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$770$		0			0
$771$		0			0
$772$	−6.00000			−0.215945
$773$	19.0000	+	32.9090i	0.683383	+	1.18365i	0.973942	+	0.226796i	$0.0728252\pi$
$773$	19.0000	+	32.9090i	0.683383	+	1.18365i	−0.290560	+	0.956857i	$0.593841\pi$
$774$	5.19615	−	3.00000i	0.186772	−	0.107833i
$775$	−24.0000	+	18.0000i	−0.862105	+	0.646579i
$776$	9.00000	+	15.5885i	0.323081	+	0.559593i
$777$		0			0
$778$	−3.00000	+	5.19615i	−0.107555	+	0.186291i
$779$	−48.0000			−1.71978
$780$		0			0
$781$	4.00000			0.143131
$782$		0			0
$783$	−20.7846	−	12.0000i	−0.742781	−	0.428845i
$784$	−3.50000	−	6.06218i	−0.125000	−	0.216506i
$785$	24.0000	+	12.0000i	0.856597	+	0.428298i

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:	\( N \)	\(=\)	\( 845 = 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	845.l (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 + 0.866025i) q^{2} +(-1.73205 - 1.00000i) q^{3} +(0.500000 + 0.866025i) q^{4} +(1.00000 - 2.00000i) q^{5} +(1.73205 - 1.00000i) q^{6} -3.00000 q^{8} +(0.500000 + 0.866025i) q^{9} +O(q^{10})\)	\(q+(-0.500000 + 0.866025i) q^{2} +(-1.73205 - 1.00000i) q^{3} +(0.500000 + 0.866025i) q^{4} +(1.00000 - 2.00000i) q^{5} +(1.73205 - 1.00000i) q^{6} -3.00000 q^{8} +(0.500000 + 0.866025i) q^{9} +(1.23205 + 1.86603i) q^{10} +(1.73205 + 1.00000i) q^{11} -2.00000i q^{12} +(-3.73205 + 2.46410i) q^{15} +(0.500000 - 0.866025i) q^{16} -1.00000 q^{18} +(5.19615 - 3.00000i) q^{19} +(2.23205 - 0.133975i) q^{20} +(-1.73205 + 1.00000i) q^{22} +(-5.19615 - 3.00000i) q^{23} +(5.19615 + 3.00000i) q^{24} +(-3.00000 - 4.00000i) q^{25} +4.00000i q^{27} +(-3.00000 + 5.19615i) q^{29} +(-0.267949 - 4.46410i) q^{30} -6.00000i q^{31} +(-2.50000 - 4.33013i) q^{32} +(-2.00000 - 3.46410i) q^{33} +(-0.500000 + 0.866025i) q^{36} +(3.00000 - 5.19615i) q^{37} +6.00000i q^{38} +(-3.00000 + 6.00000i) q^{40} +(-6.92820 - 4.00000i) q^{41} +(-5.19615 + 3.00000i) q^{43} +2.00000i q^{44} +(2.23205 - 0.133975i) q^{45} +(5.19615 - 3.00000i) q^{46} +8.00000 q^{47} +(-1.73205 + 1.00000i) q^{48} +(3.50000 - 6.06218i) q^{49} +(4.96410 - 0.598076i) q^{50} -12.0000i q^{53} +(-3.46410 - 2.00000i) q^{54} +(3.73205 - 2.46410i) q^{55} -12.0000 q^{57} +(-3.00000 - 5.19615i) q^{58} +(-1.73205 + 1.00000i) q^{59} +(-4.00000 - 2.00000i) q^{60} +(-3.00000 - 5.19615i) q^{61} +(5.19615 + 3.00000i) q^{62} +7.00000 q^{64} +4.00000 q^{66} +(6.00000 - 10.3923i) q^{67} +(6.00000 + 10.3923i) q^{69} +(1.73205 - 1.00000i) q^{71} +(-1.50000 - 2.59808i) q^{72} +6.00000 q^{73} +(3.00000 + 5.19615i) q^{74} +(1.19615 + 9.92820i) q^{75} +(5.19615 + 3.00000i) q^{76} +(-1.23205 - 1.86603i) q^{80} +(5.50000 - 9.52628i) q^{81} +(6.92820 - 4.00000i) q^{82} +4.00000 q^{83} -6.00000i q^{86} +(10.3923 - 6.00000i) q^{87} +(-5.19615 - 3.00000i) q^{88} +(-6.92820 - 4.00000i) q^{89} +(-1.00000 + 2.00000i) q^{90} -6.00000i q^{92} +(-6.00000 + 10.3923i) q^{93} +(-4.00000 + 6.92820i) q^{94} +(-0.803848 - 13.3923i) q^{95} +10.0000i q^{96} +(-3.00000 - 5.19615i) q^{97} +(3.50000 + 6.06218i) q^{98} +2.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{2} + 2 q^{4} + 4 q^{5} - 12 q^{8} + 2 q^{9} + O(q^{10}) \)	\( 4 q - 2 q^{2} + 2 q^{4} + 4 q^{5} - 12 q^{8} + 2 q^{9} - 2 q^{10} - 8 q^{15} + 2 q^{16} - 4 q^{18} + 2 q^{20} - 12 q^{25} - 12 q^{29} - 8 q^{30} - 10 q^{32} - 8 q^{33} - 2 q^{36} + 12 q^{37} - 12 q^{40} + 2 q^{45} + 32 q^{47} + 14 q^{49} + 6 q^{50} + 8 q^{55} - 48 q^{57} - 12 q^{58} - 16 q^{60} - 12 q^{61} + 28 q^{64} + 16 q^{66} + 24 q^{67} + 24 q^{69} - 6 q^{72} + 24 q^{73} + 12 q^{74} - 16 q^{75} + 2 q^{80} + 22 q^{81} + 16 q^{83} - 4 q^{90} - 24 q^{93} - 16 q^{94} - 24 q^{95} - 12 q^{97} + 14 q^{98} + O(q^{100}) \)

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