LMFDB - Embedded newform 378.2.g.g.163.1

Newspace parameters

Level:	$ N $	$=$	$ 378 = 2 \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	378.g (of order $3$, degree $2$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$3.01834519640$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{7})$
Defining polynomial:	$ x^{4} + 7x^{2} + 49 $ x^4 + 7*x^2 + 49
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			163.1
Root			$1.32288 - 2.29129i$ of defining polynomial
Character	$\chi$	$=$	378.163
Dual form			378.2.g.g.109.1

$q$-expansion

$f(q)$	$=$	$q+(-0.500000 + 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(-0.822876 + 1.42526i) q^{5} +2.64575 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+(-0.500000 + 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(-0.822876 + 1.42526i) q^{5} +2.64575 q^{7} +1.00000 q^{8} +(-0.822876 - 1.42526i) q^{10} +(0.822876 + 1.42526i) q^{11} +0.645751 q^{13} +(-1.32288 + 2.29129i) q^{14} +(-0.500000 + 0.866025i) q^{16} +(0.822876 + 1.42526i) q^{17} +(-1.00000 + 1.73205i) q^{19} +1.64575 q^{20} -1.64575 q^{22} +(-4.64575 + 8.04668i) q^{23} +(1.14575 + 1.98450i) q^{25} +(-0.322876 + 0.559237i) q^{26} +(-1.32288 - 2.29129i) q^{28} +7.64575 q^{29} +(-0.322876 - 0.559237i) q^{31} +(-0.500000 - 0.866025i) q^{32} -1.64575 q^{34} +(-2.17712 + 3.77089i) q^{35} +(-1.96863 + 3.40976i) q^{37} +(-1.00000 - 1.73205i) q^{38} +(-0.822876 + 1.42526i) q^{40} -4.93725 q^{41} +5.00000 q^{43} +(0.822876 - 1.42526i) q^{44} +(-4.64575 - 8.04668i) q^{46} +(5.46863 - 9.47194i) q^{47} +7.00000 q^{49} -2.29150 q^{50} +(-0.322876 - 0.559237i) q^{52} +(-3.00000 - 5.19615i) q^{53} -2.70850 q^{55} +2.64575 q^{56} +(-3.82288 + 6.62141i) q^{58} +(-6.82288 - 11.8176i) q^{59} +(-6.32288 + 10.9515i) q^{61} +0.645751 q^{62} +1.00000 q^{64} +(-0.531373 + 0.920365i) q^{65} +(-4.14575 - 7.18065i) q^{67} +(0.822876 - 1.42526i) q^{68} +(-2.17712 - 3.77089i) q^{70} +10.3542 q^{71} +(5.29150 + 9.16515i) q^{73} +(-1.96863 - 3.40976i) q^{74} +2.00000 q^{76} +(2.17712 + 3.77089i) q^{77} +(7.61438 - 13.1885i) q^{79} +(-0.822876 - 1.42526i) q^{80} +(2.46863 - 4.27579i) q^{82} -2.70850 q^{83} -2.70850 q^{85} +(-2.50000 + 4.33013i) q^{86} +(0.822876 + 1.42526i) q^{88} +(5.46863 - 9.47194i) q^{89} +1.70850 q^{91} +9.29150 q^{92} +(5.46863 + 9.47194i) q^{94} +(-1.64575 - 2.85052i) q^{95} -7.58301 q^{97} +(-3.50000 + 6.06218i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{2} - 2 q^{4} + 2 q^{5} + 4 q^{8}+O(q^{10}) $ 4 * q - 2 * q^2 - 2 * q^4 + 2 * q^5 + 4 * q^8	$ 4 q - 2 q^{2} - 2 q^{4} + 2 q^{5} + 4 q^{8} + 2 q^{10} - 2 q^{11} - 8 q^{13} - 2 q^{16} - 2 q^{17} - 4 q^{19} - 4 q^{20} + 4 q^{22} - 8 q^{23} - 6 q^{25} + 4 q^{26} + 20 q^{29} + 4 q^{31} - 2 q^{32} + 4 q^{34} - 14 q^{35} + 8 q^{37} - 4 q^{38} + 2 q^{40} + 12 q^{41} + 20 q^{43} - 2 q^{44} - 8 q^{46} + 6 q^{47} + 28 q^{49} + 12 q^{50} + 4 q^{52} - 12 q^{53} - 32 q^{55} - 10 q^{58} - 22 q^{59} - 20 q^{61} - 8 q^{62} + 4 q^{64} - 18 q^{65} - 6 q^{67} - 2 q^{68} - 14 q^{70} + 52 q^{71} + 8 q^{74} + 8 q^{76} + 14 q^{77} + 4 q^{79} + 2 q^{80} - 6 q^{82} - 32 q^{83} - 32 q^{85} - 10 q^{86} - 2 q^{88} + 6 q^{89} + 28 q^{91} + 16 q^{92} + 6 q^{94} + 4 q^{95} + 12 q^{97} - 14 q^{98}+O(q^{100}) $ 4 * q - 2 * q^2 - 2 * q^4 + 2 * q^5 + 4 * q^8 + 2 * q^10 - 2 * q^11 - 8 * q^13 - 2 * q^16 - 2 * q^17 - 4 * q^19 - 4 * q^20 + 4 * q^22 - 8 * q^23 - 6 * q^25 + 4 * q^26 + 20 * q^29 + 4 * q^31 - 2 * q^32 + 4 * q^34 - 14 * q^35 + 8 * q^37 - 4 * q^38 + 2 * q^40 + 12 * q^41 + 20 * q^43 - 2 * q^44 - 8 * q^46 + 6 * q^47 + 28 * q^49 + 12 * q^50 + 4 * q^52 - 12 * q^53 - 32 * q^55 - 10 * q^58 - 22 * q^59 - 20 * q^61 - 8 * q^62 + 4 * q^64 - 18 * q^65 - 6 * q^67 - 2 * q^68 - 14 * q^70 + 52 * q^71 + 8 * q^74 + 8 * q^76 + 14 * q^77 + 4 * q^79 + 2 * q^80 - 6 * q^82 - 32 * q^83 - 32 * q^85 - 10 * q^86 - 2 * q^88 + 6 * q^89 + 28 * q^91 + 16 * q^92 + 6 * q^94 + 4 * q^95 + 12 * q^97 - 14 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/378\mathbb{Z}\right)^\times$.

$n$	$29$	$325$
$\chi(n)$	$1$	$e\left(\frac{1}{3}\right)$

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
$5$	−0.822876	+	1.42526i	−0.368001	+	0.637397i	−0.989253	−	0.146214i	$-0.953291\pi$
$5$	−0.822876	+	1.42526i	−0.368001	+	0.637397i	0.621252	+	0.783611i	$0.286624\pi$
$6$		0			0
$7$	2.64575			1.00000
$7$	2.64575			1.00000
$8$	1.00000			0.353553
$9$		0			0
$10$	−0.822876	−	1.42526i	−0.260216	−	0.450708i
$11$	0.822876	+	1.42526i	0.248106	+	0.429733i	0.963000	−	0.269500i	$-0.0868584\pi$
$11$	0.822876	+	1.42526i	0.248106	+	0.429733i	−0.714894	+	0.699233i	$0.753525\pi$
$12$		0			0
$13$	0.645751			0.179099			0.0895496	−	0.995982i	$-0.471457\pi$
$13$	0.645751			0.179099			0.0895496	+	0.995982i	$0.471457\pi$
$14$	−1.32288	+	2.29129i	−0.353553	+	0.612372i
$15$		0			0
$16$	−0.500000	+	0.866025i	−0.125000	+	0.216506i
$17$	0.822876	+	1.42526i	0.199577	+	0.345677i	0.948391	−	0.317103i	$-0.102710\pi$
$17$	0.822876	+	1.42526i	0.199577	+	0.345677i	−0.748815	+	0.662780i	$0.769377\pi$
$18$		0			0
$19$	−1.00000	+	1.73205i	−0.229416	+	0.397360i	−0.957635	−	0.287984i	$-0.907015\pi$
$19$	−1.00000	+	1.73205i	−0.229416	+	0.397360i	0.728219	+	0.685344i	$0.240348\pi$
$20$	1.64575			0.368001
$21$		0			0
$22$	−1.64575			−0.350875
$23$	−4.64575	+	8.04668i	−0.968706	+	1.67785i	−0.269396	+	0.963029i	$0.586824\pi$
$23$	−4.64575	+	8.04668i	−0.968706	+	1.67785i	−0.699310	+	0.714819i	$0.746509\pi$
$24$		0			0
$25$	1.14575	+	1.98450i	0.229150	+	0.396900i
$26$	−0.322876	+	0.559237i	−0.0633211	+	0.109675i
$27$		0			0
$28$	−1.32288	−	2.29129i	−0.250000	−	0.433013i
$29$	7.64575			1.41978			0.709890	−	0.704312i	$-0.248745\pi$
$29$	7.64575			1.41978			0.709890	+	0.704312i	$0.248745\pi$
$30$		0			0
$31$	−0.322876	−	0.559237i	−0.0579902	−	0.100442i	0.835573	−	0.549380i	$-0.185136\pi$
$31$	−0.322876	−	0.559237i	−0.0579902	−	0.100442i	−0.893563	+	0.448938i	$0.851803\pi$
$32$	−0.500000	−	0.866025i	−0.0883883	−	0.153093i
$33$		0			0
$34$	−1.64575			−0.282244
$35$	−2.17712	+	3.77089i	−0.368001	+	0.637397i
$36$		0			0
$37$	−1.96863	+	3.40976i	−0.323640	+	0.560561i	−0.981236	−	0.192809i	$-0.938240\pi$
$37$	−1.96863	+	3.40976i	−0.323640	+	0.560561i	0.657596	+	0.753371i	$0.271573\pi$
$38$	−1.00000	−	1.73205i	−0.162221	−	0.280976i
$39$		0			0
$40$	−0.822876	+	1.42526i	−0.130108	+	0.225354i
$41$	−4.93725			−0.771070			−0.385535	−	0.922693i	$-0.625983\pi$
$41$	−4.93725			−0.771070			−0.385535	+	0.922693i	$0.625983\pi$
$42$		0			0
$43$	5.00000			0.762493			0.381246	−	0.924473i	$-0.375495\pi$
$43$	5.00000			0.762493			0.381246	+	0.924473i	$0.375495\pi$
$44$	0.822876	−	1.42526i	0.124053	−	0.214866i
$45$		0			0
$46$	−4.64575	−	8.04668i	−0.684979	−	1.18642i
$47$	5.46863	−	9.47194i	0.797681	−	1.38162i	−0.123441	−	0.992352i	$-0.539393\pi$
$47$	5.46863	−	9.47194i	0.797681	−	1.38162i	0.921123	−	0.389273i	$-0.127274\pi$
$48$		0			0
$49$	7.00000			1.00000
$50$	−2.29150			−0.324067
$51$		0			0
$52$	−0.322876	−	0.559237i	−0.0447748	−	0.0775522i
$53$	−3.00000	−	5.19615i	−0.412082	−	0.713746i	0.583036	−	0.812447i	$-0.301865\pi$
$53$	−3.00000	−	5.19615i	−0.412082	−	0.713746i	−0.995117	+	0.0987002i	$0.968532\pi$
$54$		0			0
$55$	−2.70850			−0.365214
$56$	2.64575			0.353553
$57$		0			0
$58$	−3.82288	+	6.62141i	−0.501968	+	0.869434i
$59$	−6.82288	−	11.8176i	−0.888263	−	1.53852i	−0.841928	−	0.539590i	$-0.818579\pi$
$59$	−6.82288	−	11.8176i	−0.888263	−	1.53852i	−0.0463350	−	0.998926i	$-0.514754\pi$
$60$		0			0
$61$	−6.32288	+	10.9515i	−0.809561	+	1.40220i	0.103607	+	0.994618i	$0.466962\pi$
$61$	−6.32288	+	10.9515i	−0.809561	+	1.40220i	−0.913168	+	0.407583i	$0.866372\pi$
$62$	0.645751			0.0820105
$63$		0			0
$64$	1.00000			0.125000
$65$	−0.531373	+	0.920365i	−0.0659087	+	0.114157i
$66$		0			0
$67$	−4.14575	−	7.18065i	−0.506484	−	0.877256i	−0.999972	−	0.00750349i	$-0.997612\pi$
$67$	−4.14575	−	7.18065i	−0.506484	−	0.877256i	0.493488	−	0.869753i	$-0.335722\pi$
$68$	0.822876	−	1.42526i	0.0997883	−	0.172838i
$69$		0			0
$70$	−2.17712	−	3.77089i	−0.260216	−	0.450708i
$71$	10.3542			1.22882			0.614412	−	0.788986i	$-0.289393\pi$
$71$	10.3542			1.22882			0.614412	+	0.788986i	$0.289393\pi$
$72$		0			0
$73$	5.29150	+	9.16515i	0.619324	+	1.07270i	0.989609	+	0.143782i	$0.0459264\pi$
$73$	5.29150	+	9.16515i	0.619324	+	1.07270i	−0.370286	+	0.928918i	$0.620740\pi$
$74$	−1.96863	−	3.40976i	−0.228848	−	0.396377i
$75$		0			0
$76$	2.00000			0.229416
$77$	2.17712	+	3.77089i	0.248106	+	0.429733i
$78$		0			0
$79$	7.61438	−	13.1885i	0.856684	−	1.48382i	−0.0183890	−	0.999831i	$-0.505854\pi$
$79$	7.61438	−	13.1885i	0.856684	−	1.48382i	0.875073	−	0.483990i	$-0.160813\pi$
$80$	−0.822876	−	1.42526i	−0.0920003	−	0.159349i
$81$		0			0
$82$	2.46863	−	4.27579i	0.272614	−	0.472182i
$83$	−2.70850			−0.297296			−0.148648	−	0.988890i	$-0.547492\pi$
$83$	−2.70850			−0.297296			−0.148648	+	0.988890i	$0.547492\pi$
$84$		0			0
$85$	−2.70850			−0.293778
$86$	−2.50000	+	4.33013i	−0.269582	+	0.466930i
$87$		0			0
$88$	0.822876	+	1.42526i	0.0877188	+	0.151933i
$89$	5.46863	−	9.47194i	0.579673	−	1.00402i	−0.415843	−	0.909436i	$-0.636514\pi$
$89$	5.46863	−	9.47194i	0.579673	−	1.00402i	0.995517	−	0.0945873i	$-0.0301532\pi$
$90$		0			0
$91$	1.70850			0.179099
$92$	9.29150			0.968706
$93$		0			0
$94$	5.46863	+	9.47194i	0.564046	+	0.976956i
$95$	−1.64575	−	2.85052i	−0.168851	−	0.292458i
$96$		0			0
$97$	−7.58301			−0.769938			−0.384969	−	0.922930i	$-0.625788\pi$
$97$	−7.58301			−0.769938			−0.384969	+	0.922930i	$0.625788\pi$
$98$	−3.50000	+	6.06218i	−0.353553	+	0.612372i
$99$		0			0
$100$	1.14575	−	1.98450i	0.114575	−	0.198450i
$101$	−6.82288	−	11.8176i	−0.678902	−	1.17589i	−0.975312	−	0.220831i	$-0.929123\pi$
$101$	−6.82288	−	11.8176i	−0.678902	−	1.17589i	0.296411	−	0.955061i	$-0.404210\pi$
$102$		0			0
$103$	5.96863	−	10.3380i	0.588106	−	1.01863i	−0.406374	−	0.913707i	$-0.633207\pi$
$103$	5.96863	−	10.3380i	0.588106	−	1.01863i	0.994480	−	0.104923i	$-0.0334597\pi$
$104$	0.645751			0.0633211
$105$		0			0
$106$	6.00000			0.582772
$107$	3.00000	−	5.19615i	0.290021	−	0.502331i	−0.683793	−	0.729676i	$-0.739671\pi$
$107$	3.00000	−	5.19615i	0.290021	−	0.502331i	0.973814	+	0.227345i	$0.0730044\pi$
$108$		0			0
$109$	4.32288	+	7.48744i	0.414056	+	0.717167i	0.995329	−	0.0965423i	$-0.0307783\pi$
$109$	4.32288	+	7.48744i	0.414056	+	0.717167i	−0.581272	+	0.813709i	$0.697445\pi$
$110$	1.35425	−	2.34563i	0.129123	−	0.223647i
$111$		0			0
$112$	−1.32288	+	2.29129i	−0.125000	+	0.216506i
$113$	7.64575			0.719252			0.359626	−	0.933097i	$-0.382904\pi$
$113$	7.64575			0.719252			0.359626	+	0.933097i	$0.382904\pi$
$114$		0			0
$115$	−7.64575	−	13.2428i	−0.712970	−	1.23490i
$116$	−3.82288	−	6.62141i	−0.354945	−	0.614783i
$117$		0			0
$118$	13.6458			1.25619
$119$	2.17712	+	3.77089i	0.199577	+	0.345677i
$120$		0			0
$121$	4.14575	−	7.18065i	0.376886	−	0.652787i
$122$	−6.32288	−	10.9515i	−0.572446	−	0.991506i
$123$		0			0
$124$	−0.322876	+	0.559237i	−0.0289951	+	0.0502210i
$125$	−12.0000			−1.07331
$126$		0			0
$127$	6.64575			0.589715			0.294858	−	0.955541i	$-0.404728\pi$
$127$	6.64575			0.589715			0.294858	+	0.955541i	$0.404728\pi$
$128$	−0.500000	+	0.866025i	−0.0441942	+	0.0765466i
$129$		0			0
$130$	−0.531373	−	0.920365i	−0.0466045	−	0.0807214i
$131$	−3.29150	+	5.70105i	−0.287580	+	0.498103i	−0.973232	−	0.229827i	$-0.926184\pi$
$131$	−3.29150	+	5.70105i	−0.287580	+	0.498103i	0.685652	+	0.727930i	$0.259517\pi$
$132$		0			0
$133$	−2.64575	+	4.58258i	−0.229416	+	0.397360i
$134$	8.29150			0.716277
$135$		0			0
$136$	0.822876	+	1.42526i	0.0705610	+	0.122215i
$137$	4.64575	+	8.04668i	0.396913	+	0.687474i	0.993343	−	0.115191i	$-0.0367480\pi$
$137$	4.64575	+	8.04668i	0.396913	+	0.687474i	−0.596430	+	0.802665i	$0.703415\pi$
$138$		0			0
$139$	−19.5830			−1.66101			−0.830504	−	0.557012i	$-0.811948\pi$
$139$	−19.5830			−1.66101			−0.830504	+	0.557012i	$0.811948\pi$
$140$	4.35425			0.368001
$141$		0			0
$142$	−5.17712	+	8.96704i	−0.434455	+	0.752497i
$143$	0.531373	+	0.920365i	0.0444356	+	0.0769648i
$144$		0			0
$145$	−6.29150	+	10.8972i	−0.522481	+	0.904963i
$146$	−10.5830			−0.875856
$147$		0			0
$148$	3.93725			0.323640
$149$	−3.53137	+	6.11652i	−0.289301	+	0.501085i	−0.973643	−	0.228077i	$-0.926756\pi$
$149$	−3.53137	+	6.11652i	−0.289301	+	0.501085i	0.684342	+	0.729161i	$0.260090\pi$
$150$		0			0
$151$	−3.61438	−	6.26029i	−0.294134	−	0.509455i	0.680649	−	0.732610i	$-0.261698\pi$
$151$	−3.61438	−	6.26029i	−0.294134	−	0.509455i	−0.974783	+	0.223155i	$0.928365\pi$
$152$	−1.00000	+	1.73205i	−0.0811107	+	0.140488i
$153$		0			0
$154$	−4.35425			−0.350875
$155$	1.06275			0.0853618
$156$		0			0
$157$	2.00000	+	3.46410i	0.159617	+	0.276465i	0.934731	−	0.355357i	$-0.115641\pi$
$157$	2.00000	+	3.46410i	0.159617	+	0.276465i	−0.775113	+	0.631822i	$0.782307\pi$
$158$	7.61438	+	13.1885i	0.605767	+	1.04922i
$159$		0			0
$160$	1.64575			0.130108
$161$	−12.2915	+	21.2895i	−0.968706	+	1.67785i
$162$		0			0
$163$	0.500000	−	0.866025i	0.0391630	−	0.0678323i	−0.845780	−	0.533533i	$-0.820864\pi$
$163$	0.500000	−	0.866025i	0.0391630	−	0.0678323i	0.884943	+	0.465700i	$0.154198\pi$
$164$	2.46863	+	4.27579i	0.192767	+	0.333883i
$165$		0			0
$166$	1.35425	−	2.34563i	0.105110	−	0.182056i
$167$	12.5830			0.973702			0.486851	−	0.873485i	$-0.338145\pi$
$167$	12.5830			0.973702			0.486851	+	0.873485i	$0.338145\pi$
$168$		0			0
$169$	−12.5830			−0.967923
$170$	1.35425	−	2.34563i	0.103866	−	0.179901i
$171$		0			0
$172$	−2.50000	−	4.33013i	−0.190623	−	0.330169i
$173$	−3.29150	+	5.70105i	−0.250248	+	0.433443i	−0.963594	−	0.267369i	$-0.913846\pi$
$173$	−3.29150	+	5.70105i	−0.250248	+	0.433443i	0.713346	+	0.700812i	$0.247179\pi$
$174$		0			0
$175$	3.03137	+	5.25049i	0.229150	+	0.396900i
$176$	−1.64575			−0.124053
$177$		0			0
$178$	5.46863	+	9.47194i	0.409891	+	0.709952i
$179$	−0.531373	−	0.920365i	−0.0397167	−	0.0687913i	0.845484	−	0.534001i	$-0.179312\pi$
$179$	−0.531373	−	0.920365i	−0.0397167	−	0.0687913i	−0.885200	+	0.465210i	$0.845979\pi$
$180$		0			0
$181$	−13.2915			−0.987950			−0.493975	−	0.869476i	$-0.664457\pi$
$181$	−13.2915			−0.987950			−0.493975	+	0.869476i	$0.664457\pi$
$182$	−0.854249	+	1.47960i	−0.0633211	+	0.109675i
$183$		0			0
$184$	−4.64575	+	8.04668i	−0.342489	+	0.593209i
$185$	−3.23987	−	5.61162i	−0.238200	−	0.412575i
$186$		0			0
$187$	−1.35425	+	2.34563i	−0.0990325	+	0.171529i
$188$	−10.9373			−0.797681
$189$		0			0
$190$	3.29150			0.238791
$191$	−13.4059	+	23.2197i	−0.970015	+	1.68012i	−0.274526	+	0.961580i	$0.588521\pi$
$191$	−13.4059	+	23.2197i	−0.970015	+	1.68012i	−0.695489	+	0.718537i	$0.744812\pi$
$192$		0			0
$193$	3.50000	+	6.06218i	0.251936	+	0.436365i	0.964059	−	0.265689i	$-0.0855996\pi$
$193$	3.50000	+	6.06218i	0.251936	+	0.436365i	−0.712123	+	0.702055i	$0.752266\pi$
$194$	3.79150	−	6.56708i	0.272214	−	0.471489i
$195$		0			0
$196$	−3.50000	−	6.06218i	−0.250000	−	0.433013i
$197$	15.2915			1.08947			0.544737	−	0.838607i	$-0.316629\pi$
$197$	15.2915			1.08947			0.544737	+	0.838607i	$0.316629\pi$
$198$		0			0
$199$	−10.9686	−	18.9982i	−0.777545	−	1.34675i	−0.933353	−	0.358961i	$-0.883131\pi$
$199$	−10.9686	−	18.9982i	−0.777545	−	1.34675i	0.155807	−	0.987787i	$-0.450202\pi$
$200$	1.14575	+	1.98450i	0.0810169	+	0.140325i
$201$		0			0
$202$	13.6458			0.960112
$203$	20.2288			1.41978
$204$		0			0
$205$	4.06275	−	7.03688i	0.283754	−	0.491477i
$206$	5.96863	+	10.3380i	0.415854	+	0.720280i
$207$		0			0
$208$	−0.322876	+	0.559237i	−0.0223874	+	0.0387761i
$209$	−3.29150			−0.227678
$210$		0			0
$211$	−16.8745			−1.16169			−0.580845	−	0.814015i	$-0.697278\pi$
$211$	−16.8745			−1.16169			−0.580845	+	0.814015i	$0.697278\pi$
$212$	−3.00000	+	5.19615i	−0.206041	+	0.356873i
$213$		0			0
$214$	3.00000	+	5.19615i	0.205076	+	0.355202i
$215$	−4.11438	+	7.12631i	−0.280598	+	0.486010i
$216$		0			0
$217$	−0.854249	−	1.47960i	−0.0579902	−	0.100442i
$218$	−8.64575			−0.585564
$219$		0			0
$220$	1.35425	+	2.34563i	0.0913034	+	0.158142i
$221$	0.531373	+	0.920365i	0.0357440	+	0.0619105i
$222$		0			0
$223$	17.8745			1.19697			0.598483	−	0.801136i	$-0.295770\pi$
$223$	17.8745			1.19697			0.598483	+	0.801136i	$0.295770\pi$
$224$	−1.32288	−	2.29129i	−0.0883883	−	0.153093i
$225$		0			0
$226$	−3.82288	+	6.62141i	−0.254294	+	0.440450i
$227$	−3.00000	−	5.19615i	−0.199117	−	0.344881i	0.749125	−	0.662428i	$-0.230474\pi$
$227$	−3.00000	−	5.19615i	−0.199117	−	0.344881i	−0.948242	+	0.317547i	$0.897141\pi$
$228$		0			0
$229$	7.32288	−	12.6836i	0.483909	−	0.838155i	−0.515920	−	0.856637i	$-0.672550\pi$
$229$	7.32288	−	12.6836i	0.483909	−	0.838155i	0.999829	+	0.0184814i	$0.00588315\pi$
$230$	15.2915			1.00829
$231$		0			0
$232$	7.64575			0.501968
$233$	4.35425	−	7.54178i	0.285256	−	0.494078i	−0.687415	−	0.726265i	$-0.741255\pi$
$233$	4.35425	−	7.54178i	0.285256	−	0.494078i	0.972671	+	0.232186i	$0.0745879\pi$
$234$		0			0
$235$	9.00000	+	15.5885i	0.587095	+	1.01688i
$236$	−6.82288	+	11.8176i	−0.444131	+	0.769258i
$237$		0			0
$238$	−4.35425			−0.282244
$239$	4.93725			0.319364			0.159682	−	0.987168i	$-0.448953\pi$
$239$	4.93725			0.319364			0.159682	+	0.987168i	$0.448953\pi$
$240$		0			0
$241$	−2.50000	−	4.33013i	−0.161039	−	0.278928i	0.774202	−	0.632938i	$-0.218151\pi$
$241$	−2.50000	−	4.33013i	−0.161039	−	0.278928i	−0.935242	+	0.354010i	$0.884818\pi$
$242$	4.14575	+	7.18065i	0.266499	+	0.461590i
$243$		0			0
$244$	12.6458			0.809561
$245$	−5.76013	+	9.97684i	−0.368001	+	0.637397i
$246$		0			0
$247$	−0.645751	+	1.11847i	−0.0410882	+	0.0711668i
$248$	−0.322876	−	0.559237i	−0.0205026	−	0.0355116i
$249$		0			0
$250$	6.00000	−	10.3923i	0.379473	−	0.657267i
$251$	−18.0000			−1.13615			−0.568075	−	0.822977i	$-0.692312\pi$
$251$	−18.0000			−1.13615			−0.568075	+	0.822977i	$0.692312\pi$
$252$		0			0
$253$	−15.2915			−0.961369
$254$	−3.32288	+	5.75539i	−0.208496	+	0.361125i
$255$		0			0
$256$	−0.500000	−	0.866025i	−0.0312500	−	0.0541266i
$257$	7.93725	−	13.7477i	0.495112	−	0.857560i	−0.504872	−	0.863194i	$-0.668460\pi$
$257$	7.93725	−	13.7477i	0.495112	−	0.857560i	0.999984	+	0.00563467i	$0.00179358\pi$
$258$		0			0
$259$	−5.20850	+	9.02138i	−0.323640	+	0.560561i
$260$	1.06275			0.0659087
$261$		0			0
$262$	−3.29150	−	5.70105i	−0.203350	−	0.352212i
$263$	−5.46863	−	9.47194i	−0.337210	−	0.584065i	0.646697	−	0.762747i	$-0.276150\pi$
$263$	−5.46863	−	9.47194i	−0.337210	−	0.584065i	−0.983907	+	0.178682i	$0.942817\pi$
$264$		0			0
$265$	9.87451			0.606586
$266$	−2.64575	−	4.58258i	−0.162221	−	0.280976i
$267$		0			0
$268$	−4.14575	+	7.18065i	−0.253242	+	0.438628i
$269$	13.6458	+	23.6351i	0.831996	+	1.44106i	0.896453	+	0.443139i	$0.146135\pi$
$269$	13.6458	+	23.6351i	0.831996	+	1.44106i	−0.0644567	+	0.997921i	$0.520531\pi$
$270$		0			0
$271$	10.6144	−	18.3846i	0.644778	−	1.11679i	−0.339575	−	0.940579i	$-0.610283\pi$
$271$	10.6144	−	18.3846i	0.644778	−	1.11679i	0.984353	−	0.176209i	$-0.0563833\pi$
$272$	−1.64575			−0.0997883
$273$		0			0
$274$	−9.29150			−0.561320
$275$	−1.88562	+	3.26599i	−0.113707	+	0.196947i
$276$		0			0
$277$	8.96863	+	15.5341i	0.538873	+	0.933355i	0.998965	+	0.0454837i	$0.0144829\pi$
$277$	8.96863	+	15.5341i	0.538873	+	0.933355i	−0.460093	+	0.887871i	$0.652184\pi$
$278$	9.79150	−	16.9594i	0.587255	−	1.01716i
$279$		0			0
$280$	−2.17712	+	3.77089i	−0.130108	+	0.225354i
$281$	−17.5203			−1.04517			−0.522586	−	0.852587i	$-0.675032\pi$
$281$	−17.5203			−1.04517			−0.522586	+	0.852587i	$0.675032\pi$
$282$		0			0
$283$	−4.14575	−	7.18065i	−0.246439	−	0.426845i	0.716096	−	0.698002i	$-0.245927\pi$
$283$	−4.14575	−	7.18065i	−0.246439	−	0.426845i	−0.962535	+	0.271156i	$0.912594\pi$
$284$	−5.17712	−	8.96704i	−0.307206	−	0.532096i
$285$		0			0
$286$	−1.06275			−0.0628415
$287$	−13.0627			−0.771070
$288$		0			0
$289$	7.14575	−	12.3768i	0.420338	−	0.728047i
$290$	−6.29150	−	10.8972i	−0.369450	−	0.639906i
$291$		0			0
$292$	5.29150	−	9.16515i	0.309662	−	0.536350i
$293$	22.9373			1.34001			0.670004	−	0.742357i	$-0.266292\pi$
$293$	22.9373			1.34001			0.670004	+	0.742357i	$0.266292\pi$
$294$		0			0
$295$	22.4575			1.30753
$296$	−1.96863	+	3.40976i	−0.114424	+	0.198188i
$297$		0			0
$298$	−3.53137	−	6.11652i	−0.204567	−	0.354320i
$299$	−3.00000	+	5.19615i	−0.173494	+	0.300501i
$300$		0			0
$301$	13.2288			0.762493
$302$	7.22876			0.415968
$303$		0			0
$304$	−1.00000	−	1.73205i	−0.0573539	−	0.0993399i
$305$	−10.4059	−	18.0235i	−0.595839	−	1.03202i
$306$		0			0
$307$	−13.5830			−0.775223			−0.387612	−	0.921823i	$-0.626700\pi$
$307$	−13.5830			−0.775223			−0.387612	+	0.921823i	$0.626700\pi$
$308$	2.17712	−	3.77089i	0.124053	−	0.214866i
$309$		0			0
$310$	−0.531373	+	0.920365i	−0.0301800	+	0.0522732i
$311$	−11.7601	−	20.3691i	−0.666856	−	1.15503i	−0.978779	−	0.204921i	$-0.934306\pi$
$311$	−11.7601	−	20.3691i	−0.666856	−	1.15503i	0.311923	−	0.950107i	$-0.399027\pi$
$312$		0			0
$313$	−11.6458	+	20.1710i	−0.658257	+	1.14013i	0.322810	+	0.946464i	$0.395373\pi$
$313$	−11.6458	+	20.1710i	−0.658257	+	1.14013i	−0.981067	+	0.193670i	$0.937961\pi$
$314$	−4.00000			−0.225733
$315$		0			0
$316$	−15.2288			−0.856684
$317$	10.4059	−	18.0235i	0.584452	−	1.01230i	−0.410491	−	0.911865i	$-0.634643\pi$
$317$	10.4059	−	18.0235i	0.584452	−	1.01230i	0.994943	−	0.100437i	$-0.0320240\pi$
$318$		0			0
$319$	6.29150	+	10.8972i	0.352257	+	0.610126i
$320$	−0.822876	+	1.42526i	−0.0460001	+	0.0796746i
$321$		0			0
$322$	−12.2915	−	21.2895i	−0.684979	−	1.18642i
$323$	−3.29150			−0.183144
$324$		0			0
$325$	0.739870	+	1.28149i	0.0410406	+	0.0710844i
$326$	0.500000	+	0.866025i	0.0276924	+	0.0479647i
$327$		0			0
$328$	−4.93725			−0.272614
$329$	14.4686	−	25.0604i	0.797681	−	1.38162i
$330$		0			0
$331$	0.0627461	−	0.108679i	0.00344884	−	0.00597356i	−0.864296	−	0.502984i	$-0.832236\pi$
$331$	0.0627461	−	0.108679i	0.00344884	−	0.00597356i	0.867745	+	0.497010i	$0.165569\pi$
$332$	1.35425	+	2.34563i	0.0743241	+	0.128733i
$333$		0			0
$334$	−6.29150	+	10.8972i	−0.344256	+	0.596268i
$335$	13.6458			0.745547
$336$		0			0
$337$	−9.41699			−0.512976			−0.256488	−	0.966547i	$-0.582565\pi$
$337$	−9.41699			−0.512976			−0.256488	+	0.966547i	$0.582565\pi$
$338$	6.29150	−	10.8972i	0.342213	−	0.592730i
$339$		0			0
$340$	1.35425	+	2.34563i	0.0734444	+	0.127210i
$341$	0.531373	−	0.920365i	0.0287755	−	0.0498406i
$342$		0			0
$343$	18.5203			1.00000
$344$	5.00000			0.269582
$345$		0			0
$346$	−3.29150	−	5.70105i	−0.176952	−	0.306490i
$347$	6.00000	+	10.3923i	0.322097	+	0.557888i	0.980921	−	0.194409i	$-0.0622790\pi$
$347$	6.00000	+	10.3923i	0.322097	+	0.557888i	−0.658824	+	0.752297i	$0.728946\pi$
$348$		0			0
$349$	25.2288			1.35046			0.675232	−	0.737605i	$-0.264043\pi$
$349$	25.2288			1.35046			0.675232	+	0.737605i	$0.264043\pi$
$350$	−6.06275			−0.324067
$351$		0			0
$352$	0.822876	−	1.42526i	0.0438594	−	0.0759667i
$353$	6.00000	+	10.3923i	0.319348	+	0.553127i	0.980352	−	0.197256i	$-0.0632029\pi$
$353$	6.00000	+	10.3923i	0.319348	+	0.553127i	−0.661004	+	0.750382i	$0.729870\pi$
$354$		0			0
$355$	−8.52026	+	14.7575i	−0.452208	+	0.783248i
$356$	−10.9373			−0.579673
$357$		0			0
$358$	1.06275			0.0561679
$359$	15.5830	−	26.9906i	0.822440	−	1.42451i	−0.0814209	−	0.996680i	$-0.525946\pi$
$359$	15.5830	−	26.9906i	0.822440	−	1.42451i	0.903860	−	0.427827i	$-0.140721\pi$
$360$		0			0
$361$	7.50000	+	12.9904i	0.394737	+	0.683704i
$362$	6.64575	−	11.5108i	0.349293	−	0.604993i
$363$		0			0
$364$	−0.854249	−	1.47960i	−0.0447748	−	0.0775522i
$365$	−17.4170			−0.911647
$366$		0			0
$367$	0.937254	+	1.62337i	0.0489243	+	0.0847393i	0.889450	−	0.457032i	$-0.151087\pi$
$367$	0.937254	+	1.62337i	0.0489243	+	0.0847393i	−0.840526	+	0.541771i	$0.817754\pi$
$368$	−4.64575	−	8.04668i	−0.242177	−	0.419462i
$369$		0			0
$370$	6.47974			0.336866
$371$	−7.93725	−	13.7477i	−0.412082	−	0.713746i
$372$		0			0
$373$	8.29150	−	14.3613i	0.429318	−	0.743600i	−0.567495	−	0.823377i	$-0.692087\pi$
$373$	8.29150	−	14.3613i	0.429318	−	0.743600i	0.996813	+	0.0797767i	$0.0254207\pi$
$374$	−1.35425	−	2.34563i	−0.0700265	−	0.121290i
$375$		0			0
$376$	5.46863	−	9.47194i	0.282023	−	0.488478i
$377$	4.93725			0.254282
$378$		0			0
$379$	4.41699			0.226886			0.113443	−	0.993545i	$-0.463812\pi$
$379$	4.41699			0.226886			0.113443	+	0.993545i	$0.463812\pi$
$380$	−1.64575	+	2.85052i	−0.0844253	+	0.146229i
$381$		0			0
$382$	−13.4059	−	23.2197i	−0.685905	−	1.18802i
$383$	0.291503	−	0.504897i	0.0148951	−	0.0257990i	−0.858482	−	0.512844i	$-0.828592\pi$
$383$	0.291503	−	0.504897i	0.0148951	−	0.0257990i	0.873377	+	0.487045i	$0.161925\pi$
$384$		0			0
$385$	−7.16601			−0.365214
$386$	−7.00000			−0.356291
$387$		0			0
$388$	3.79150	+	6.56708i	0.192484	+	0.333393i
$389$	−4.35425	−	7.54178i	−0.220769	−	0.382383i	0.734273	−	0.678855i	$-0.237523\pi$
$389$	−4.35425	−	7.54178i	−0.220769	−	0.382383i	−0.955042	+	0.296471i	$0.904190\pi$
$390$		0			0
$391$	−15.2915			−0.773325
$392$	7.00000			0.353553
$393$		0			0
$394$	−7.64575	+	13.2428i	−0.385187	+	0.667164i
$395$	12.5314	+	21.7050i	0.630522	+	1.09210i
$396$		0			0
$397$	5.67712	−	9.83307i	0.284927	−	0.493508i	−0.687665	−	0.726028i	$-0.741364\pi$
$397$	5.67712	−	9.83307i	0.284927	−	0.493508i	0.972591	+	0.232521i	$0.0746974\pi$
$398$	21.9373			1.09962
$399$		0			0
$400$	−2.29150			−0.114575
$401$	−13.4059	+	23.2197i	−0.669458	+	1.15953i	0.308598	+	0.951192i	$0.400140\pi$
$401$	−13.4059	+	23.2197i	−0.669458	+	1.15953i	−0.978056	+	0.208342i	$0.933193\pi$
$402$		0			0
$403$	−0.208497	−	0.361128i	−0.0103860	−	0.0179891i
$404$	−6.82288	+	11.8176i	−0.339451	+	0.587946i
$405$		0			0
$406$	−10.1144	+	17.5186i	−0.501968	+	0.869434i
$407$	−6.47974			−0.321189
$408$		0			0
$409$	8.43725	+	14.6138i	0.417195	+	0.722604i	0.995656	−	0.0931066i	$-0.0296798\pi$
$409$	8.43725	+	14.6138i	0.417195	+	0.722604i	−0.578461	+	0.815710i	$0.696346\pi$
$410$	4.06275	+	7.03688i	0.200645	+	0.347527i
$411$		0			0
$412$	−11.9373			−0.588106
$413$	−18.0516	−	31.2663i	−0.888263	−	1.53852i
$414$		0			0
$415$	2.22876	−	3.86032i	0.109405	−	0.189496i
$416$	−0.322876	−	0.559237i	−0.0158303	−	0.0274189i
$417$		0			0
$418$	1.64575	−	2.85052i	0.0804963	−	0.139424i
$419$	13.0627			0.638157			0.319078	−	0.947728i	$-0.396627\pi$
$419$	13.0627			0.638157			0.319078	+	0.947728i	$0.396627\pi$
$420$		0			0
$421$	−25.2915			−1.23263			−0.616316	−	0.787499i	$-0.711376\pi$
$421$	−25.2915			−1.23263			−0.616316	+	0.787499i	$0.711376\pi$
$422$	8.43725	−	14.6138i	0.410719	−	0.711386i
$423$		0			0
$424$	−3.00000	−	5.19615i	−0.145693	−	0.252347i
$425$	−1.88562	+	3.26599i	−0.0914661	+	0.158424i
$426$		0			0
$427$	−16.7288	+	28.9751i	−0.809561	+	1.40220i
$428$	−6.00000			−0.290021
$429$		0			0
$430$	−4.11438	−	7.12631i	−0.198413	−	0.343661i
$431$	4.40588	+	7.63121i	0.212224	+	0.367582i	0.952410	−	0.304819i	$-0.0985961\pi$
$431$	4.40588	+	7.63121i	0.212224	+	0.367582i	−0.740186	+	0.672402i	$0.765263\pi$
$432$		0			0
$433$	−19.0000			−0.913082			−0.456541	−	0.889702i	$-0.650912\pi$
$433$	−19.0000			−0.913082			−0.456541	+	0.889702i	$0.650912\pi$
$434$	1.70850			0.0820105
$435$		0			0
$436$	4.32288	−	7.48744i	0.207028	−	0.358583i
$437$	−9.29150	−	16.0934i	−0.444473	−	0.769850i
$438$		0			0
$439$	8.58301	−	14.8662i	0.409644	−	0.709525i	−0.585205	−	0.810885i	$-0.698986\pi$
$439$	8.58301	−	14.8662i	0.409644	−	0.709525i	0.994850	+	0.101360i	$0.0323194\pi$
$440$	−2.70850			−0.129123
$441$		0			0
$442$	−1.06275			−0.0505497
$443$	4.35425	−	7.54178i	0.206877	−	0.358321i	−0.743852	−	0.668344i	$-0.767003\pi$
$443$	4.35425	−	7.54178i	0.206877	−	0.358321i	0.950729	+	0.310023i	$0.100337\pi$
$444$		0			0
$445$	9.00000	+	15.5885i	0.426641	+	0.738964i
$446$	−8.93725	+	15.4798i	−0.423191	+	0.732989i
$447$		0			0
$448$	2.64575			0.125000
$449$	7.64575			0.360825			0.180413	−	0.983591i	$-0.442257\pi$
$449$	7.64575			0.360825			0.180413	+	0.983591i	$0.442257\pi$
$450$		0			0
$451$	−4.06275	−	7.03688i	−0.191307	−	0.331354i
$452$	−3.82288	−	6.62141i	−0.179813	−	0.311445i
$453$		0			0
$454$	6.00000			0.281594
$455$	−1.40588	+	2.43506i	−0.0659087	+	0.114157i
$456$		0			0
$457$	−1.14575	+	1.98450i	−0.0535960	+	0.0928310i	−0.891579	−	0.452866i	$-0.850402\pi$
$457$	−1.14575	+	1.98450i	−0.0535960	+	0.0928310i	0.837983	+	0.545697i	$0.183735\pi$
$458$	7.32288	+	12.6836i	0.342176	+	0.592665i
$459$		0			0
$460$	−7.64575	+	13.2428i	−0.356485	+	0.617450i
$461$	−2.22876			−0.103804			−0.0519018	−	0.998652i	$-0.516528\pi$
$461$	−2.22876			−0.103804			−0.0519018	+	0.998652i	$0.516528\pi$
$462$		0			0
$463$	29.2915			1.36129			0.680646	−	0.732613i	$-0.261699\pi$
$463$	29.2915			1.36129			0.680646	+	0.732613i	$0.261699\pi$
$464$	−3.82288	+	6.62141i	−0.177473	+	0.307391i
$465$		0			0
$466$	4.35425	+	7.54178i	0.201707	+	0.349366i
$467$	13.1144	−	22.7148i	0.606861	−	1.05111i	−0.384893	−	0.922961i	$-0.625762\pi$
$467$	13.1144	−	22.7148i	0.606861	−	1.05111i	0.991754	−	0.128153i	$-0.0409049\pi$
$468$		0			0
$469$	−10.9686	−	18.9982i	−0.506484	−	0.877256i
$470$	−18.0000			−0.830278
$471$		0			0
$472$	−6.82288	−	11.8176i	−0.314048	−	0.543948i
$473$	4.11438	+	7.12631i	0.189179	+	0.327668i
$474$		0			0
$475$	−4.58301			−0.210283
$476$	2.17712	−	3.77089i	0.0997883	−	0.172838i
$477$		0			0
$478$	−2.46863	+	4.27579i	−0.112912	+	0.195570i
$479$	0.822876	+	1.42526i	0.0375981	+	0.0651219i	0.884212	−	0.467086i	$-0.154696\pi$
$479$	0.822876	+	1.42526i	0.0375981	+	0.0651219i	−0.846614	+	0.532207i	$0.821363\pi$
$480$		0			0
$481$	−1.27124	+	2.20186i	−0.0579637	+	0.100396i
$482$	5.00000			0.227744
$483$		0			0
$484$	−8.29150			−0.376886
$485$	6.23987	−	10.8078i	0.283338	−	0.490756i
$486$		0			0
$487$	3.93725	+	6.81952i	0.178414	+	0.309022i	0.941337	−	0.337467i	$-0.109570\pi$
$487$	3.93725	+	6.81952i	0.178414	+	0.309022i	−0.762923	+	0.646489i	$0.776237\pi$
$488$	−6.32288	+	10.9515i	−0.286223	+	0.495753i
$489$		0			0
$490$	−5.76013	−	9.97684i	−0.260216	−	0.450708i
$491$	−37.7490			−1.70359			−0.851795	−	0.523876i	$-0.824486\pi$
$491$	−37.7490			−1.70359			−0.851795	+	0.523876i	$0.824486\pi$
$492$		0			0
$493$	6.29150	+	10.8972i	0.283355	+	0.490785i
$494$	−0.645751	−	1.11847i	−0.0290537	−	0.0503225i
$495$		0			0
$496$	0.645751			0.0289951
$497$	27.3948			1.22882
$498$		0			0
$499$	−18.0830	+	31.3207i	−0.809506	+	1.40211i	0.103700	+	0.994609i	$0.466932\pi$
$499$	−18.0830	+	31.3207i	−0.809506	+	1.40211i	−0.913206	+	0.407498i	$0.866401\pi$
$500$	6.00000	+	10.3923i	0.268328	+	0.464758i
$501$		0			0
$502$	9.00000	−	15.5885i	0.401690	−	0.695747i
$503$	−27.8745			−1.24286			−0.621431	−	0.783469i	$-0.713449\pi$
$503$	−27.8745			−1.24286			−0.621431	+	0.783469i	$0.713449\pi$
$504$		0			0
$505$	22.4575			0.999346
$506$	7.64575	−	13.2428i	0.339895	−	0.588716i
$507$		0			0
$508$	−3.32288	−	5.75539i	−0.147429	−	0.255354i
$509$	−19.9373	+	34.5323i	−0.883703	+	1.53062i	−0.0365105	+	0.999333i	$0.511624\pi$
$509$	−19.9373	+	34.5323i	−0.883703	+	1.53062i	−0.847193	+	0.531286i	$0.821709\pi$
$510$		0			0
$511$	14.0000	+	24.2487i	0.619324	+	1.07270i
$512$	1.00000			0.0441942
$513$		0			0
$514$	7.93725	+	13.7477i	0.350097	+	0.606386i
$515$	9.82288	+	17.0137i	0.432848	+	0.749714i
$516$		0			0
$517$	18.0000			0.791639
$518$	−5.20850	−	9.02138i	−0.228848	−	0.396377i
$519$		0			0
$520$	−0.531373	+	0.920365i	−0.0233022	+	0.0403607i
$521$	19.9373	+	34.5323i	0.873467	+	1.51289i	0.858387	+	0.513003i	$0.171467\pi$
$521$	19.9373	+	34.5323i	0.873467	+	1.51289i	0.0150801	+	0.999886i	$0.495200\pi$
$522$		0			0
$523$	0.500000	−	0.866025i	0.0218635	−	0.0378686i	−0.854887	−	0.518815i	$-0.826373\pi$
$523$	0.500000	−	0.866025i	0.0218635	−	0.0378686i	0.876750	+	0.480946i	$0.159707\pi$
$524$	6.58301			0.287580
$525$		0			0
$526$	10.9373			0.476887
$527$	0.531373	−	0.920365i	0.0231470	−	0.0400917i
$528$		0			0
$529$	−31.6660	−	54.8471i	−1.37678	−	2.38466i
$530$	−4.93725	+	8.55157i	−0.214461	+	0.371457i
$531$		0			0
$532$	5.29150			0.229416
$533$	−3.18824			−0.138098
$534$		0			0
$535$	4.93725	+	8.55157i	0.213456	+	0.369717i
$536$	−4.14575	−	7.18065i	−0.179069	−	0.310157i
$537$		0			0
$538$	−27.2915			−1.17662
$539$	5.76013	+	9.97684i	0.248106	+	0.429733i
$540$		0			0
$541$	20.5830	−	35.6508i	0.884933	−	1.53275i	0.0391415	−	0.999234i	$-0.487538\pi$
$541$	20.5830	−	35.6508i	0.884933	−	1.53275i	0.845791	−	0.533514i	$-0.179129\pi$
$542$	10.6144	+	18.3846i	0.455927	+	0.789688i
$543$		0			0
$544$	0.822876	−	1.42526i	0.0352805	−	0.0611076i
$545$	−14.2288			−0.609493
$546$		0			0
$547$	7.70850			0.329592			0.164796	−	0.986328i	$-0.447303\pi$
$547$	7.70850			0.329592			0.164796	+	0.986328i	$0.447303\pi$
$548$	4.64575	−	8.04668i	0.198457	−	0.343737i
$549$		0			0
$550$	−1.88562	−	3.26599i	−0.0804032	−	0.139262i
$551$	−7.64575	+	13.2428i	−0.325720	+	0.564164i
$552$		0			0
$553$	20.1458	−	34.8935i	0.856684	−	1.48382i
$554$	−17.9373			−0.762081
$555$		0			0
$556$	9.79150	+	16.9594i	0.415252	+	0.719238i
$557$	16.1144	+	27.9109i	0.682788	+	1.18262i	0.974126	+	0.226004i	$0.0725661\pi$
$557$	16.1144	+	27.9109i	0.682788	+	1.18262i	−0.291338	+	0.956620i	$0.594101\pi$
$558$		0			0
$559$	3.22876			0.136562
$560$	−2.17712	−	3.77089i	−0.0920003	−	0.159349i
$561$		0			0
$562$	8.76013	−	15.1730i	0.369524	−	0.640034i
$563$	17.2288	+	29.8411i	0.726106	+	1.25765i	0.958517	+	0.285034i	$0.0920050\pi$
$563$	17.2288	+	29.8411i	0.726106	+	1.25765i	−0.232412	+	0.972617i	$0.574662\pi$
$564$		0			0
$565$	−6.29150	+	10.8972i	−0.264686	+	0.458449i
$566$	8.29150			0.348518
$567$		0			0
$568$	10.3542			0.434455
$569$	0.531373	−	0.920365i	0.0222763	−	0.0385837i	−0.854672	−	0.519168i	$-0.826242\pi$
$569$	0.531373	−	0.920365i	0.0222763	−	0.0385837i	0.876949	+	0.480584i	$0.159575\pi$
$570$		0			0
$571$	0.645751	+	1.11847i	0.0270239	+	0.0468067i	0.879221	−	0.476414i	$-0.158064\pi$
$571$	0.645751	+	1.11847i	0.0270239	+	0.0468067i	−0.852197	+	0.523221i	$0.824730\pi$
$572$	0.531373	−	0.920365i	0.0222178	−	0.0384824i
$573$		0			0
$574$	6.53137	−	11.3127i	0.272614	−	0.472182i
$575$	−21.2915			−0.887917
$576$		0			0
$577$	10.8542	+	18.8001i	0.451868	+	0.782659i	0.998502	−	0.0547129i	$-0.0174244\pi$
$577$	10.8542	+	18.8001i	0.451868	+	0.782659i	−0.546634	+	0.837372i	$0.684091\pi$
$578$	7.14575	+	12.3768i	0.297224	+	0.514807i
$579$		0			0
$580$	12.5830			0.522481
$581$	−7.16601			−0.297296
$582$		0			0
$583$	4.93725	−	8.55157i	0.204480	−	0.354170i
$584$	5.29150	+	9.16515i	0.218964	+	0.379257i
$585$		0			0
$586$	−11.4686	+	19.8642i	−0.473765	+	0.820584i
$587$	38.2288			1.57787			0.788935	−	0.614477i	$-0.210633\pi$
$587$	38.2288			1.57787			0.788935	+	0.614477i	$0.210633\pi$
$588$		0			0
$589$	1.29150			0.0532154
$590$	−11.2288	+	19.4488i	−0.462281	+	0.800693i
$591$		0			0
$592$	−1.96863	−	3.40976i	−0.0809101	−	0.140140i
$593$	−12.5314	+	21.7050i	−0.514602	+	0.891316i	0.485255	+	0.874373i	$0.338727\pi$
$593$	−12.5314	+	21.7050i	−0.514602	+	0.891316i	−0.999856	+	0.0169436i	$0.994606\pi$
$594$		0			0
$595$	−7.16601			−0.293778
$596$	7.06275			0.289301
$597$		0			0
$598$	−3.00000	−	5.19615i	−0.122679	−	0.212486i
$599$	10.9373	+	18.9439i	0.446884	+	0.774026i	0.998181	−	0.0602830i	$-0.0192003\pi$
$599$	10.9373	+	18.9439i	0.446884	+	0.774026i	−0.551297	+	0.834309i	$0.685867\pi$
$600$		0			0
$601$	−4.87451			−0.198835			−0.0994177	−	0.995046i	$-0.531698\pi$
$601$	−4.87451			−0.198835			−0.0994177	+	0.995046i	$0.531698\pi$
$602$	−6.61438	+	11.4564i	−0.269582	+	0.466930i
$603$		0			0
$604$	−3.61438	+	6.26029i	−0.147067	+	0.254727i
$605$	6.82288	+	11.8176i	0.277389	+	0.480452i
$606$		0			0
$607$	−13.2915	+	23.0216i	−0.539485	+	0.934416i	0.459446	+	0.888206i	$0.348048\pi$
$607$	−13.2915	+	23.0216i	−0.539485	+	0.934416i	−0.998932	+	0.0462106i	$0.985285\pi$
$608$	2.00000			0.0811107
$609$		0			0
$610$	20.8118			0.842644
$611$	3.53137	−	6.11652i	0.142864	−	0.247448i
$612$		0			0
$613$	−13.1974	−	22.8585i	−0.533037	−	0.923248i	−0.999256	−	0.0385780i	$-0.987717\pi$
$613$	−13.1974	−	22.8585i	−0.533037	−	0.923248i	0.466218	−	0.884670i	$-0.345616\pi$
$614$	6.79150	−	11.7632i	0.274083	−	0.474725i
$615$		0			0
$616$	2.17712	+	3.77089i	0.0877188	+	0.151933i
$617$	−35.5203			−1.42999			−0.714996	−	0.699129i	$-0.753571\pi$
$617$	−35.5203			−1.42999			−0.714996	+	0.699129i	$0.753571\pi$
$618$		0			0
$619$	−1.14575	−	1.98450i	−0.0460516	−	0.0797638i	0.842081	−	0.539351i	$-0.181331\pi$
$619$	−1.14575	−	1.98450i	−0.0460516	−	0.0797638i	−0.888132	+	0.459588i	$0.847997\pi$
$620$	−0.531373	−	0.920365i	−0.0213405	−	0.0369628i
$621$		0			0
$622$	23.5203			0.943076
$623$	14.4686	−	25.0604i	0.579673	−	1.00402i
$624$		0			0
$625$	4.14575	−	7.18065i	0.165830	−	0.287226i
$626$	−11.6458	−	20.1710i	−0.465458	−	0.806197i
$627$		0			0
$628$	2.00000	−	3.46410i	0.0798087	−	0.138233i
$629$	−6.47974			−0.258364
$630$		0			0
$631$	21.9373			0.873308			0.436654	−	0.899629i	$-0.356163\pi$
$631$	21.9373			0.873308			0.436654	+	0.899629i	$0.356163\pi$
$632$	7.61438	−	13.1885i	0.302884	−	0.524610i
$633$		0			0
$634$	10.4059	+	18.0235i	0.413270	+	0.715805i
$635$	−5.46863	+	9.47194i	−0.217016	+	0.375882i
$636$		0			0
$637$	4.52026			0.179099
$638$	−12.5830			−0.498166
$639$		0			0
$640$	−0.822876	−	1.42526i	−0.0325270	−	0.0563384i
$641$	−4.11438	−	7.12631i	−0.162508	−	0.281472i	0.773259	−	0.634090i	$-0.218625\pi$
$641$	−4.11438	−	7.12631i	−0.162508	−	0.281472i	−0.935768	+	0.352617i	$0.885292\pi$
$642$		0			0
$643$	−34.8745			−1.37532			−0.687658	−	0.726035i	$-0.741361\pi$
$643$	−34.8745			−1.37532			−0.687658	+	0.726035i	$0.741361\pi$
$644$	24.5830			0.968706
$645$		0			0
$646$	1.64575	−	2.85052i	0.0647512	−	0.112152i
$647$	10.9373	+	18.9439i	0.429988	+	0.744761i	0.996872	−	0.0790370i	$-0.0251845\pi$
$647$	10.9373	+	18.9439i	0.429988	+	0.744761i	−0.566884	+	0.823798i	$0.691851\pi$
$648$		0			0
$649$	11.2288	−	19.4488i	0.440767	−	0.763431i
$650$	−1.47974			−0.0580402
$651$		0			0
$652$	−1.00000			−0.0391630
$653$	3.00000	−	5.19615i	0.117399	−	0.203341i	−0.801337	−	0.598213i	$-0.795878\pi$
$653$	3.00000	−	5.19615i	0.117399	−	0.203341i	0.918736	+	0.394872i	$0.129211\pi$
$654$		0			0
$655$	−5.41699	−	9.38251i	−0.211659	−	0.366605i
$656$	2.46863	−	4.27579i	0.0963837	−	0.166941i
$657$		0			0
$658$	14.4686	+	25.0604i	0.564046	+	0.976956i
$659$	2.22876			0.0868200			0.0434100	−	0.999057i	$-0.486178\pi$
$659$	2.22876			0.0868200			0.0434100	+	0.999057i	$0.486178\pi$
$660$		0			0
$661$	−19.5830	−	33.9188i	−0.761691	−	1.31929i	−0.941979	−	0.335673i	$-0.891036\pi$
$661$	−19.5830	−	33.9188i	−0.761691	−	1.31929i	0.180288	−	0.983614i	$-0.442297\pi$
$662$	0.0627461	+	0.108679i	0.00243870	+	0.00422394i
$663$		0			0
$664$	−2.70850			−0.105110
$665$	−4.35425	−	7.54178i	−0.168851	−	0.292458i
$666$		0			0
$667$	−35.5203	+	61.5229i	−1.37535	+	2.38218i
$668$	−6.29150	−	10.8972i	−0.243426	−	0.421625i
$669$		0			0
$670$	−6.82288	+	11.8176i	−0.263591	+	0.456552i
$671$	−20.8118			−0.803429
$672$		0			0
$673$	−47.7490			−1.84059			−0.920295	−	0.391226i	$-0.872051\pi$
$673$	−47.7490			−1.84059			−0.920295	+	0.391226i	$0.872051\pi$
$674$	4.70850	−	8.15536i	0.181365	−	0.314133i
$675$		0			0
$676$	6.29150	+	10.8972i	0.241981	+	0.419123i
$677$	7.06275	−	12.2330i	0.271443	−	0.470154i	−0.697788	−	0.716304i	$-0.745832\pi$
$677$	7.06275	−	12.2330i	0.271443	−	0.470154i	0.969232	+	0.246150i	$0.0791657\pi$
$678$		0			0
$679$	−20.0627			−0.769938
$680$	−2.70850			−0.103866
$681$		0			0
$682$	0.531373	+	0.920365i	0.0203473	+	0.0352426i
$683$	−10.4059	−	18.0235i	−0.398170	−	0.689651i	0.595330	−	0.803481i	$-0.297021\pi$
$683$	−10.4059	−	18.0235i	−0.398170	−	0.689651i	−0.993500	+	0.113831i	$0.963688\pi$
$684$		0			0
$685$	−15.2915			−0.584258
$686$	−9.26013	+	16.0390i	−0.353553	+	0.612372i
$687$		0			0
$688$	−2.50000	+	4.33013i	−0.0953116	+	0.165085i
$689$	−1.93725	−	3.35542i	−0.0738035	−	0.127831i
$690$		0			0
$691$	−12.3745	+	21.4333i	−0.470748	+	0.815360i	−0.999440	−	0.0334536i	$-0.989349\pi$
$691$	−12.3745	+	21.4333i	−0.470748	+	0.815360i	0.528692	+	0.848814i	$0.322683\pi$
$692$	6.58301			0.250248
$693$		0			0
$694$	−12.0000			−0.455514
$695$	16.1144	−	27.9109i	0.611253	−	1.05872i
$696$		0			0
$697$	−4.06275	−	7.03688i	−0.153887	−	0.266541i
$698$	−12.6144	+	21.8487i	−0.477461	+	0.826987i
$699$		0			0
$700$	3.03137	−	5.25049i	0.114575	−	0.198450i
$701$	−5.41699			−0.204597			−0.102299	−	0.994754i	$-0.532620\pi$
$701$	−5.41699			−0.204597			−0.102299	+	0.994754i	$0.532620\pi$
$702$		0			0
$703$	−3.93725	−	6.81952i	−0.148496	−	0.257203i
$704$	0.822876	+	1.42526i	0.0310133	+	0.0537166i
$705$		0			0
$706$	−12.0000			−0.451626
$707$	−18.0516	−	31.2663i	−0.678902	−	1.17589i
$708$		0			0
$709$	4.90588	−	8.49723i	0.184244	−	0.319120i	−0.759077	−	0.651000i	$-0.774350\pi$
$709$	4.90588	−	8.49723i	0.184244	−	0.319120i	0.943322	+	0.331880i	$0.107683\pi$
$710$	−8.52026	−	14.7575i	−0.319760	−	0.553840i
$711$		0			0
$712$	5.46863	−	9.47194i	0.204945	−	0.354976i
$713$	6.00000			0.224702
$714$		0			0
$715$	−1.74902			−0.0654095
$716$	−0.531373	+	0.920365i	−0.0198583	+	0.0343957i
$717$		0			0
$718$	15.5830	+	26.9906i	0.581553	+	1.00728i
$719$	−3.53137	+	6.11652i	−0.131698	+	0.228108i	−0.924331	−	0.381591i	$-0.875376\pi$
$719$	−3.53137	+	6.11652i	−0.131698	+	0.228108i	0.792633	+	0.609699i	$0.208710\pi$
$720$		0			0
$721$	15.7915	−	27.3517i	0.588106	−	1.01863i
$722$	−15.0000			−0.558242
$723$		0			0
$724$	6.64575	+	11.5108i	0.246987	+	0.427795i
$725$	8.76013	+	15.1730i	0.325343	+	0.563511i
$726$		0			0
$727$	25.2288			0.935683			0.467841	−	0.883812i	$-0.345032\pi$
$727$	25.2288			0.935683			0.467841	+	0.883812i	$0.345032\pi$
$728$	1.70850			0.0633211
$729$		0			0
$730$	8.70850	−	15.0836i	0.322316	−	0.558268i
$731$	4.11438	+	7.12631i	0.152176	+	0.263576i
$732$		0			0
$733$	−4.38562	+	7.59612i	−0.161987	+	0.280569i	−0.935581	−	0.353112i	$-0.885123\pi$
$733$	−4.38562	+	7.59612i	−0.161987	+	0.280569i	0.773594	+	0.633681i	$0.218457\pi$
$734$	−1.87451			−0.0691893
$735$		0			0
$736$	9.29150			0.342489
$737$	6.82288	−	11.8176i	0.251324	−	0.435306i
$738$		0			0
$739$	−10.7288	−	18.5828i	−0.394664	−	0.683578i	0.598395	−	0.801202i	$-0.295806\pi$
$739$	−10.7288	−	18.5828i	−0.394664	−	0.683578i	−0.993058	+	0.117624i	$0.962472\pi$
$740$	−3.23987	+	5.61162i	−0.119100	+	0.206287i
$741$		0			0
$742$	15.8745			0.582772
$743$	−13.0627			−0.479226			−0.239613	−	0.970869i	$-0.577021\pi$
$743$	−13.0627			−0.479226			−0.239613	+	0.970869i	$0.577021\pi$
$744$		0			0
$745$	−5.81176	−	10.0663i	−0.212926	−	0.368799i
$746$	8.29150	+	14.3613i	0.303573	+	0.525805i
$747$		0			0
$748$	2.70850			0.0990325
$749$	7.93725	−	13.7477i	0.290021	−	0.502331i
$750$		0			0
$751$	−0.228757	+	0.396218i	−0.00834745	+	0.0144582i	−0.870169	−	0.492753i	$-0.835990\pi$
$751$	−0.228757	+	0.396218i	−0.00834745	+	0.0144582i	0.861822	+	0.507212i	$0.169324\pi$
$752$	5.46863	+	9.47194i	0.199420	+	0.345406i
$753$		0			0
$754$	−2.46863	+	4.27579i	−0.0899021	+	0.155715i
$755$	11.8967			0.432967
$756$		0			0
$757$	−40.9778			−1.48936			−0.744681	−	0.667420i	$-0.767398\pi$
$757$	−40.9778			−1.48936			−0.744681	+	0.667420i	$0.767398\pi$
$758$	−2.20850	+	3.82523i	−0.0802162	+	0.138939i
$759$		0			0
$760$	−1.64575	−	2.85052i	−0.0596977	−	0.103399i
$761$	9.29150	−	16.0934i	0.336817	−	0.583384i	−0.647015	−	0.762477i	$-0.723983\pi$
$761$	9.29150	−	16.0934i	0.336817	−	0.583384i	0.983832	+	0.179093i	$0.0573164\pi$
$762$		0			0
$763$	11.4373	+	19.8099i	0.414056	+	0.717167i
$764$	26.8118			0.970015
$765$		0			0
$766$	0.291503	+	0.504897i	0.0105324	+	0.0182427i
$767$	−4.40588	−	7.63121i	−0.159087	−	0.275547i
$768$		0			0
$769$	19.4170			0.700195			0.350097	−	0.936713i	$-0.386148\pi$
$769$	19.4170			0.700195			0.350097	+	0.936713i	$0.386148\pi$
$770$	3.58301	−	6.20595i	0.129123	−	0.223647i
$771$		0			0
$772$	3.50000	−	6.06218i	0.125968	−	0.218183i
$773$	−19.1660	−	33.1965i	−0.689353	−	1.19400i	−0.972047	−	0.234785i	$-0.924561\pi$
$773$	−19.1660	−	33.1965i	−0.689353	−	1.19400i	0.282694	−	0.959210i	$-0.408772\pi$
$774$		0			0
$775$	0.739870	−	1.28149i	0.0265769	−	0.0460326i
$776$	−7.58301			−0.272214
$777$		0			0
$778$	8.70850			0.312215
$779$	4.93725	−	8.55157i	0.176895	−	0.306392i
$780$		0			0
$781$	8.52026	+	14.7575i	0.304879	+	0.528066i
$782$	7.64575	−	13.2428i	0.273412	−	0.473563i
$783$		0			0
$784$	−3.50000	+	6.06218i	−0.125000	+	0.216506i
$785$	−6.58301			−0.234958
$786$		0			0
$787$	11.1458	+	19.3050i	0.397303	+	0.688149i	0.993392	−	0.114769i	$-0.0366128\pi$
$787$	11.1458	+	19.3050i	0.397303	+	0.688149i	−0.596089	+	0.802918i	$0.703279\pi$
$788$	−7.64575	−	13.2428i	−0.272369	−	0.471756i
$789$

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:	\( N \)	\(=\)	\( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	378.g (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 + 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(-0.822876 + 1.42526i) q^{5} +2.64575 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+(-0.500000 + 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(-0.822876 + 1.42526i) q^{5} +2.64575 q^{7} +1.00000 q^{8} +(-0.822876 - 1.42526i) q^{10} +(0.822876 + 1.42526i) q^{11} +0.645751 q^{13} +(-1.32288 + 2.29129i) q^{14} +(-0.500000 + 0.866025i) q^{16} +(0.822876 + 1.42526i) q^{17} +(-1.00000 + 1.73205i) q^{19} +1.64575 q^{20} -1.64575 q^{22} +(-4.64575 + 8.04668i) q^{23} +(1.14575 + 1.98450i) q^{25} +(-0.322876 + 0.559237i) q^{26} +(-1.32288 - 2.29129i) q^{28} +7.64575 q^{29} +(-0.322876 - 0.559237i) q^{31} +(-0.500000 - 0.866025i) q^{32} -1.64575 q^{34} +(-2.17712 + 3.77089i) q^{35} +(-1.96863 + 3.40976i) q^{37} +(-1.00000 - 1.73205i) q^{38} +(-0.822876 + 1.42526i) q^{40} -4.93725 q^{41} +5.00000 q^{43} +(0.822876 - 1.42526i) q^{44} +(-4.64575 - 8.04668i) q^{46} +(5.46863 - 9.47194i) q^{47} +7.00000 q^{49} -2.29150 q^{50} +(-0.322876 - 0.559237i) q^{52} +(-3.00000 - 5.19615i) q^{53} -2.70850 q^{55} +2.64575 q^{56} +(-3.82288 + 6.62141i) q^{58} +(-6.82288 - 11.8176i) q^{59} +(-6.32288 + 10.9515i) q^{61} +0.645751 q^{62} +1.00000 q^{64} +(-0.531373 + 0.920365i) q^{65} +(-4.14575 - 7.18065i) q^{67} +(0.822876 - 1.42526i) q^{68} +(-2.17712 - 3.77089i) q^{70} +10.3542 q^{71} +(5.29150 + 9.16515i) q^{73} +(-1.96863 - 3.40976i) q^{74} +2.00000 q^{76} +(2.17712 + 3.77089i) q^{77} +(7.61438 - 13.1885i) q^{79} +(-0.822876 - 1.42526i) q^{80} +(2.46863 - 4.27579i) q^{82} -2.70850 q^{83} -2.70850 q^{85} +(-2.50000 + 4.33013i) q^{86} +(0.822876 + 1.42526i) q^{88} +(5.46863 - 9.47194i) q^{89} +1.70850 q^{91} +9.29150 q^{92} +(5.46863 + 9.47194i) q^{94} +(-1.64575 - 2.85052i) q^{95} -7.58301 q^{97} +(-3.50000 + 6.06218i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{2} - 2 q^{4} + 2 q^{5} + 4 q^{8}+O(q^{10}) \) 4 * q - 2 * q^2 - 2 * q^4 + 2 * q^5 + 4 * q^8	\( 4 q - 2 q^{2} - 2 q^{4} + 2 q^{5} + 4 q^{8} + 2 q^{10} - 2 q^{11} - 8 q^{13} - 2 q^{16} - 2 q^{17} - 4 q^{19} - 4 q^{20} + 4 q^{22} - 8 q^{23} - 6 q^{25} + 4 q^{26} + 20 q^{29} + 4 q^{31} - 2 q^{32} + 4 q^{34} - 14 q^{35} + 8 q^{37} - 4 q^{38} + 2 q^{40} + 12 q^{41} + 20 q^{43} - 2 q^{44} - 8 q^{46} + 6 q^{47} + 28 q^{49} + 12 q^{50} + 4 q^{52} - 12 q^{53} - 32 q^{55} - 10 q^{58} - 22 q^{59} - 20 q^{61} - 8 q^{62} + 4 q^{64} - 18 q^{65} - 6 q^{67} - 2 q^{68} - 14 q^{70} + 52 q^{71} + 8 q^{74} + 8 q^{76} + 14 q^{77} + 4 q^{79} + 2 q^{80} - 6 q^{82} - 32 q^{83} - 32 q^{85} - 10 q^{86} - 2 q^{88} + 6 q^{89} + 28 q^{91} + 16 q^{92} + 6 q^{94} + 4 q^{95} + 12 q^{97} - 14 q^{98}+O(q^{100}) \) 4 * q - 2 * q^2 - 2 * q^4 + 2 * q^5 + 4 * q^8 + 2 * q^10 - 2 * q^11 - 8 * q^13 - 2 * q^16 - 2 * q^17 - 4 * q^19 - 4 * q^20 + 4 * q^22 - 8 * q^23 - 6 * q^25 + 4 * q^26 + 20 * q^29 + 4 * q^31 - 2 * q^32 + 4 * q^34 - 14 * q^35 + 8 * q^37 - 4 * q^38 + 2 * q^40 + 12 * q^41 + 20 * q^43 - 2 * q^44 - 8 * q^46 + 6 * q^47 + 28 * q^49 + 12 * q^50 + 4 * q^52 - 12 * q^53 - 32 * q^55 - 10 * q^58 - 22 * q^59 - 20 * q^61 - 8 * q^62 + 4 * q^64 - 18 * q^65 - 6 * q^67 - 2 * q^68 - 14 * q^70 + 52 * q^71 + 8 * q^74 + 8 * q^76 + 14 * q^77 + 4 * q^79 + 2 * q^80 - 6 * q^82 - 32 * q^83 - 32 * q^85 - 10 * q^86 - 2 * q^88 + 6 * q^89 + 28 * q^91 + 16 * q^92 + 6 * q^94 + 4 * q^95 + 12 * q^97 - 14 * q^98

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