LMFDB - Embedded newform 378.2.f.d.127.2

Newspace parameters

Level:	$ N $	$=$	$ 378 = 2 \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	378.f (of order $3$, degree $2$, not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$3.01834519640$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-3})$
Defining polynomial:	$x^{4} - 2 x^{2} + 4$
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 126)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			127.2
Root			$-1.22474 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	378.127
Dual form			378.2.f.d.253.2

$q$-expansion

$f(q)$	$=$	$q+(0.500000 - 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(1.72474 + 2.98735i) q^{5} +(0.500000 - 0.866025i) q^{7} -1.00000 q^{8} +O(q^{10})$	\(q+(0.500000 - 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(1.72474 + 2.98735i) q^{5} +(0.500000 - 0.866025i) q^{7} -1.00000 q^{8} +3.44949 q^{10} +(1.00000 - 1.73205i) q^{11} +(2.44949 + 4.24264i) q^{13} +(-0.500000 - 0.866025i) q^{14} +(-0.500000 + 0.866025i) q^{16} -2.00000 q^{17} +7.44949 q^{19} +(1.72474 - 2.98735i) q^{20} +(-1.00000 - 1.73205i) q^{22} +(-0.500000 - 0.866025i) q^{23} +(-3.44949 + 5.97469i) q^{25} +4.89898 q^{26} -1.00000 q^{28} +(1.44949 - 2.51059i) q^{29} +(-3.00000 - 5.19615i) q^{31} +(0.500000 + 0.866025i) q^{32} +(-1.00000 + 1.73205i) q^{34} +3.44949 q^{35} -7.79796 q^{37} +(3.72474 - 6.45145i) q^{38} +(-1.72474 - 2.98735i) q^{40} +(-4.89898 - 8.48528i) q^{41} +(1.44949 - 2.51059i) q^{43} -2.00000 q^{44} -1.00000 q^{46} +(-4.89898 + 8.48528i) q^{47} +(-0.500000 - 0.866025i) q^{49} +(3.44949 + 5.97469i) q^{50} +(2.44949 - 4.24264i) q^{52} +1.10102 q^{53} +6.89898 q^{55} +(-0.500000 + 0.866025i) q^{56} +(-1.44949 - 2.51059i) q^{58} +(-1.00000 - 1.73205i) q^{59} +(-5.72474 + 9.91555i) q^{61} -6.00000 q^{62} +1.00000 q^{64} +(-8.44949 + 14.6349i) q^{65} +(1.55051 + 2.68556i) q^{67} +(1.00000 + 1.73205i) q^{68} +(1.72474 - 2.98735i) q^{70} -9.89898 q^{71} +2.89898 q^{73} +(-3.89898 + 6.75323i) q^{74} +(-3.72474 - 6.45145i) q^{76} +(-1.00000 - 1.73205i) q^{77} +(-3.94949 + 6.84072i) q^{79} -3.44949 q^{80} -9.79796 q^{82} +(1.00000 - 1.73205i) q^{83} +(-3.44949 - 5.97469i) q^{85} +(-1.44949 - 2.51059i) q^{86} +(-1.00000 + 1.73205i) q^{88} +7.10102 q^{89} +4.89898 q^{91} +(-0.500000 + 0.866025i) q^{92} +(4.89898 + 8.48528i) q^{94} +(12.8485 + 22.2542i) q^{95} +(3.44949 - 5.97469i) q^{97} -1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} - 2 q^{4} + 2 q^{5} + 2 q^{7} - 4 q^{8} + O(q^{10}) $	$ 4 q + 2 q^{2} - 2 q^{4} + 2 q^{5} + 2 q^{7} - 4 q^{8} + 4 q^{10} + 4 q^{11} - 2 q^{14} - 2 q^{16} - 8 q^{17} + 20 q^{19} + 2 q^{20} - 4 q^{22} - 2 q^{23} - 4 q^{25} - 4 q^{28} - 4 q^{29} - 12 q^{31} + 2 q^{32} - 4 q^{34} + 4 q^{35} + 8 q^{37} + 10 q^{38} - 2 q^{40} - 4 q^{43} - 8 q^{44} - 4 q^{46} - 2 q^{49} + 4 q^{50} + 24 q^{53} + 8 q^{55} - 2 q^{56} + 4 q^{58} - 4 q^{59} - 18 q^{61} - 24 q^{62} + 4 q^{64} - 24 q^{65} + 16 q^{67} + 4 q^{68} + 2 q^{70} - 20 q^{71} - 8 q^{73} + 4 q^{74} - 10 q^{76} - 4 q^{77} - 6 q^{79} - 4 q^{80} + 4 q^{83} - 4 q^{85} + 4 q^{86} - 4 q^{88} + 48 q^{89} - 2 q^{92} + 22 q^{95} + 4 q^{97} - 4 q^{98} + O(q^{100}) $

Display 100 coefficients 10 coefficients

Character values

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

$n$	$29$	$325$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$

Coefficient data

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

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

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

$2$	0.500000	−	0.866025i	0.353553	−	0.612372i
$2$	0.500000	−	0.866025i	0.353553	−	0.612372i
$3$		0			0
$3$		0			0
$4$	−0.500000	−	0.866025i	−0.250000	−	0.433013i
$5$	1.72474	+	2.98735i	0.771329	+	1.33598i	0.936835	+	0.349773i	$0.113741\pi$
$5$	1.72474	+	2.98735i	0.771329	+	1.33598i	−0.165505	+	0.986209i	$0.552925\pi$
$6$		0			0
$7$	0.500000	−	0.866025i	0.188982	−	0.327327i
$7$	0.500000	−	0.866025i	0.188982	−	0.327327i
$8$	−1.00000			−0.353553
$9$		0			0
$10$	3.44949			1.09082
$11$	1.00000	−	1.73205i	0.301511	−	0.522233i	−0.674967	−	0.737848i	$-0.735842\pi$
$11$	1.00000	−	1.73205i	0.301511	−	0.522233i	0.976478	+	0.215615i	$0.0691756\pi$
$12$		0			0
$13$	2.44949	+	4.24264i	0.679366	+	1.17670i	0.975172	+	0.221449i	$0.0710785\pi$
$13$	2.44949	+	4.24264i	0.679366	+	1.17670i	−0.295806	+	0.955248i	$0.595588\pi$
$14$	−0.500000	−	0.866025i	−0.133631	−	0.231455i
$15$		0			0
$16$	−0.500000	+	0.866025i	−0.125000	+	0.216506i
$17$	−2.00000			−0.485071			−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000			−0.485071			−0.242536	+	0.970143i	$0.577979\pi$
$18$		0			0
$19$	7.44949			1.70903			0.854515	−	0.519427i	$-0.173854\pi$
$19$	7.44949			1.70903			0.854515	+	0.519427i	$0.173854\pi$
$20$	1.72474	−	2.98735i	0.385665	−	0.667991i
$21$		0			0
$22$	−1.00000	−	1.73205i	−0.213201	−	0.369274i
$23$	−0.500000	−	0.866025i	−0.104257	−	0.180579i	0.809177	−	0.587565i	$-0.199913\pi$
$23$	−0.500000	−	0.866025i	−0.104257	−	0.180579i	−0.913434	+	0.406986i	$0.866580\pi$
$24$		0			0
$25$	−3.44949	+	5.97469i	−0.689898	+	1.19494i
$26$	4.89898			0.960769
$27$		0			0
$28$	−1.00000			−0.188982
$29$	1.44949	−	2.51059i	0.269163	−	0.466205i	−0.699483	−	0.714650i	$-0.746586\pi$
$29$	1.44949	−	2.51059i	0.269163	−	0.466205i	0.968646	+	0.248445i	$0.0799195\pi$
$30$		0			0
$31$	−3.00000	−	5.19615i	−0.538816	−	0.933257i	−0.998968	−	0.0454165i	$-0.985539\pi$
$31$	−3.00000	−	5.19615i	−0.538816	−	0.933257i	0.460152	−	0.887840i	$-0.347795\pi$
$32$	0.500000	+	0.866025i	0.0883883	+	0.153093i
$33$		0			0
$34$	−1.00000	+	1.73205i	−0.171499	+	0.297044i
$35$	3.44949			0.583070
$36$		0			0
$37$	−7.79796			−1.28198			−0.640988	−	0.767551i	$-0.721475\pi$
$37$	−7.79796			−1.28198			−0.640988	+	0.767551i	$0.721475\pi$
$38$	3.72474	−	6.45145i	0.604233	−	1.04656i
$39$		0			0
$40$	−1.72474	−	2.98735i	−0.272706	−	0.472341i
$41$	−4.89898	−	8.48528i	−0.765092	−	1.32518i	−0.940198	−	0.340629i	$-0.889360\pi$
$41$	−4.89898	−	8.48528i	−0.765092	−	1.32518i	0.175106	−	0.984550i	$-0.443973\pi$
$42$		0			0
$43$	1.44949	−	2.51059i	0.221045	−	0.382861i	−0.734080	−	0.679062i	$-0.762387\pi$
$43$	1.44949	−	2.51059i	0.221045	−	0.382861i	0.955126	+	0.296201i	$0.0957199\pi$
$44$	−2.00000			−0.301511
$45$		0			0
$46$	−1.00000			−0.147442
$47$	−4.89898	+	8.48528i	−0.714590	+	1.23771i	0.248528	+	0.968625i	$0.420053\pi$
$47$	−4.89898	+	8.48528i	−0.714590	+	1.23771i	−0.963118	+	0.269081i	$0.913280\pi$
$48$		0			0
$49$	−0.500000	−	0.866025i	−0.0714286	−	0.123718i
$50$	3.44949	+	5.97469i	0.487832	+	0.844949i
$51$		0			0
$52$	2.44949	−	4.24264i	0.339683	−	0.588348i
$53$	1.10102			0.151237			0.0756184	−	0.997137i	$-0.475907\pi$
$53$	1.10102			0.151237			0.0756184	+	0.997137i	$0.475907\pi$
$54$		0			0
$55$	6.89898			0.930258
$56$	−0.500000	+	0.866025i	−0.0668153	+	0.115728i
$57$		0			0
$58$	−1.44949	−	2.51059i	−0.190327	−	0.329657i
$59$	−1.00000	−	1.73205i	−0.130189	−	0.225494i	0.793560	−	0.608492i	$-0.208225\pi$
$59$	−1.00000	−	1.73205i	−0.130189	−	0.225494i	−0.923749	+	0.382998i	$0.874892\pi$
$60$		0			0
$61$	−5.72474	+	9.91555i	−0.732978	+	1.26956i	0.222626	+	0.974904i	$0.428537\pi$
$61$	−5.72474	+	9.91555i	−0.732978	+	1.26956i	−0.955605	+	0.294652i	$0.904796\pi$
$62$	−6.00000			−0.762001
$63$		0			0
$64$	1.00000			0.125000
$65$	−8.44949	+	14.6349i	−1.04803	+	1.81524i
$66$		0			0
$67$	1.55051	+	2.68556i	0.189425	+	0.328094i	0.945059	−	0.326901i	$-0.106004\pi$
$67$	1.55051	+	2.68556i	0.189425	+	0.328094i	−0.755634	+	0.654994i	$0.772671\pi$
$68$	1.00000	+	1.73205i	0.121268	+	0.210042i
$69$		0			0
$70$	1.72474	−	2.98735i	0.206146	−	0.357056i
$71$	−9.89898			−1.17479			−0.587396	−	0.809299i	$-0.699847\pi$
$71$	−9.89898			−1.17479			−0.587396	+	0.809299i	$0.699847\pi$
$72$		0			0
$73$	2.89898			0.339300			0.169650	−	0.985504i	$-0.445736\pi$
$73$	2.89898			0.339300			0.169650	+	0.985504i	$0.445736\pi$
$74$	−3.89898	+	6.75323i	−0.453247	+	0.785047i
$75$		0			0
$76$	−3.72474	−	6.45145i	−0.427258	−	0.740032i
$77$	−1.00000	−	1.73205i	−0.113961	−	0.197386i
$78$		0			0
$79$	−3.94949	+	6.84072i	−0.444352	+	0.769641i	−0.998007	−	0.0631057i	$-0.979899\pi$
$79$	−3.94949	+	6.84072i	−0.444352	+	0.769641i	0.553655	+	0.832746i	$0.313233\pi$
$80$	−3.44949			−0.385665
$81$		0			0
$82$	−9.79796			−1.08200
$83$	1.00000	−	1.73205i	0.109764	−	0.190117i	−0.805910	−	0.592037i	$-0.798324\pi$
$83$	1.00000	−	1.73205i	0.109764	−	0.190117i	0.915675	+	0.401920i	$0.131657\pi$
$84$		0			0
$85$	−3.44949	−	5.97469i	−0.374150	−	0.648046i
$86$	−1.44949	−	2.51059i	−0.156302	−	0.270724i
$87$		0			0
$88$	−1.00000	+	1.73205i	−0.106600	+	0.184637i
$89$	7.10102			0.752707			0.376353	−	0.926476i	$-0.377178\pi$
$89$	7.10102			0.752707			0.376353	+	0.926476i	$0.377178\pi$
$90$		0			0
$91$	4.89898			0.513553
$92$	−0.500000	+	0.866025i	−0.0521286	+	0.0902894i
$93$		0			0
$94$	4.89898	+	8.48528i	0.505291	+	0.875190i
$95$	12.8485	+	22.2542i	1.31823	+	2.28323i
$96$		0			0
$97$	3.44949	−	5.97469i	0.350243	−	0.606638i	−0.636049	−	0.771649i	$-0.719432\pi$
$97$	3.44949	−	5.97469i	0.350243	−	0.606638i	0.986292	+	0.165011i	$0.0527658\pi$
$98$	−1.00000			−0.101015
$99$		0			0
$100$	6.89898			0.689898
$101$	3.62372	−	6.27647i	0.360574	−	0.624533i	−0.627481	−	0.778632i	$-0.715914\pi$
$101$	3.62372	−	6.27647i	0.360574	−	0.624533i	0.988055	+	0.154099i	$0.0492475\pi$
$102$		0			0
$103$	−7.00000	−	12.1244i	−0.689730	−	1.19465i	−0.971925	−	0.235291i	$-0.924396\pi$
$103$	−7.00000	−	12.1244i	−0.689730	−	1.19465i	0.282194	−	0.959357i	$-0.408938\pi$
$104$	−2.44949	−	4.24264i	−0.240192	−	0.416025i
$105$		0			0
$106$	0.550510	−	0.953512i	0.0534703	−	0.0926132i
$107$	12.0000			1.16008			0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000			1.16008			0.580042	+	0.814587i	$0.303036\pi$
$108$		0			0
$109$	−16.6969			−1.59928			−0.799638	−	0.600482i	$-0.794975\pi$
$109$	−16.6969			−1.59928			−0.799638	+	0.600482i	$0.794975\pi$
$110$	3.44949	−	5.97469i	0.328896	−	0.569664i
$111$		0			0
$112$	0.500000	+	0.866025i	0.0472456	+	0.0818317i
$113$	−7.94949	−	13.7689i	−0.747825	−	1.29527i	−0.948863	−	0.315688i	$-0.897765\pi$
$113$	−7.94949	−	13.7689i	−0.747825	−	1.29527i	0.201038	−	0.979583i	$-0.435569\pi$
$114$		0			0
$115$	1.72474	−	2.98735i	0.160833	−	0.278571i
$116$	−2.89898			−0.269163
$117$		0			0
$118$	−2.00000			−0.184115
$119$	−1.00000	+	1.73205i	−0.0916698	+	0.158777i
$120$		0			0
$121$	3.50000	+	6.06218i	0.318182	+	0.551107i
$122$	5.72474	+	9.91555i	0.518294	+	0.897712i
$123$		0			0
$124$	−3.00000	+	5.19615i	−0.269408	+	0.466628i
$125$	−6.55051			−0.585895
$126$		0			0
$127$	−3.00000			−0.266207			−0.133103	−	0.991102i	$-0.542494\pi$
$127$	−3.00000			−0.266207			−0.133103	+	0.991102i	$0.542494\pi$
$128$	0.500000	−	0.866025i	0.0441942	−	0.0765466i
$129$		0			0
$130$	8.44949	+	14.6349i	0.741069	+	1.28357i
$131$	−6.72474	−	11.6476i	−0.587544	−	1.01766i	−0.994553	−	0.104232i	$-0.966762\pi$
$131$	−6.72474	−	11.6476i	−0.587544	−	1.01766i	0.407009	−	0.913424i	$-0.366572\pi$
$132$		0			0
$133$	3.72474	−	6.45145i	0.322976	−	0.559411i
$134$	3.10102			0.267887
$135$		0			0
$136$	2.00000			0.171499
$137$	−5.89898	+	10.2173i	−0.503984	+	0.872926i	0.496006	+	0.868319i	$0.334800\pi$
$137$	−5.89898	+	10.2173i	−0.503984	+	0.872926i	−0.999989	+	0.00460626i	$0.998534\pi$
$138$		0			0
$139$	−4.72474	−	8.18350i	−0.400748	−	0.694115i	0.593069	−	0.805152i	$-0.297916\pi$
$139$	−4.72474	−	8.18350i	−0.400748	−	0.694115i	−0.993816	+	0.111037i	$0.964583\pi$
$140$	−1.72474	−	2.98735i	−0.145768	−	0.252477i
$141$		0			0
$142$	−4.94949	+	8.57277i	−0.415352	+	0.719411i
$143$	9.79796			0.819346
$144$		0			0
$145$	10.0000			0.830455
$146$	1.44949	−	2.51059i	0.119961	−	0.207778i
$147$		0			0
$148$	3.89898	+	6.75323i	0.320494	+	0.555112i
$149$	−3.00000	−	5.19615i	−0.245770	−	0.425685i	0.716578	−	0.697507i	$-0.245707\pi$
$149$	−3.00000	−	5.19615i	−0.245770	−	0.425685i	−0.962348	+	0.271821i	$0.912374\pi$
$150$		0			0
$151$	2.50000	−	4.33013i	0.203447	−	0.352381i	−0.746190	−	0.665733i	$-0.768119\pi$
$151$	2.50000	−	4.33013i	0.203447	−	0.352381i	0.949637	+	0.313353i	$0.101452\pi$
$152$	−7.44949			−0.604233
$153$		0			0
$154$	−2.00000			−0.161165
$155$	10.3485	−	17.9241i	0.831209	−	1.43970i
$156$		0			0
$157$	−3.17423	−	5.49794i	−0.253332	−	0.438783i	0.711109	−	0.703081i	$-0.248193\pi$
$157$	−3.17423	−	5.49794i	−0.253332	−	0.438783i	−0.964441	+	0.264298i	$0.914860\pi$
$158$	3.94949	+	6.84072i	0.314205	+	0.544218i
$159$		0			0
$160$	−1.72474	+	2.98735i	−0.136353	+	0.236170i
$161$	−1.00000			−0.0788110
$162$		0			0
$163$	−0.202041			−0.0158251			−0.00791254	−	0.999969i	$-0.502519\pi$
$163$	−0.202041			−0.0158251			−0.00791254	+	0.999969i	$0.502519\pi$
$164$	−4.89898	+	8.48528i	−0.382546	+	0.662589i
$165$		0			0
$166$	−1.00000	−	1.73205i	−0.0776151	−	0.134433i
$167$	9.34847	+	16.1920i	0.723406	+	1.25298i	0.959627	+	0.281277i	$0.0907579\pi$
$167$	9.34847	+	16.1920i	0.723406	+	1.25298i	−0.236220	+	0.971700i	$0.575909\pi$
$168$		0			0
$169$	−5.50000	+	9.52628i	−0.423077	+	0.732791i
$170$	−6.89898			−0.529128
$171$		0			0
$172$	−2.89898			−0.221045
$173$	−6.44949	+	11.1708i	−0.490346	+	0.849304i	−0.999938	−	0.0111123i	$-0.996463\pi$
$173$	−6.44949	+	11.1708i	−0.490346	+	0.849304i	0.509593	+	0.860416i	$0.329796\pi$
$174$		0			0
$175$	3.44949	+	5.97469i	0.260757	+	0.451644i
$176$	1.00000	+	1.73205i	0.0753778	+	0.130558i
$177$		0			0
$178$	3.55051	−	6.14966i	0.266122	−	0.460937i
$179$	8.69694			0.650040			0.325020	−	0.945707i	$-0.394629\pi$
$179$	8.69694			0.650040			0.325020	+	0.945707i	$0.394629\pi$
$180$		0			0
$181$	4.34847			0.323219			0.161610	−	0.986855i	$-0.448331\pi$
$181$	4.34847			0.323219			0.161610	+	0.986855i	$0.448331\pi$
$182$	2.44949	−	4.24264i	0.181568	−	0.314485i
$183$		0			0
$184$	0.500000	+	0.866025i	0.0368605	+	0.0638442i
$185$	−13.4495	−	23.2952i	−0.988826	−	1.71270i
$186$		0			0
$187$	−2.00000	+	3.46410i	−0.146254	+	0.253320i
$188$	9.79796			0.714590
$189$		0			0
$190$	25.6969			1.86425
$191$	6.94949	−	12.0369i	0.502847	−	0.870957i	−0.497147	−	0.867666i	$-0.665619\pi$
$191$	6.94949	−	12.0369i	0.502847	−	0.870957i	0.999995	−	0.00329106i	$-0.00104758\pi$
$192$		0			0
$193$	4.05051	+	7.01569i	0.291562	+	0.505000i	0.974179	−	0.225776i	$-0.0724917\pi$
$193$	4.05051	+	7.01569i	0.291562	+	0.505000i	−0.682617	+	0.730776i	$0.739158\pi$
$194$	−3.44949	−	5.97469i	−0.247659	−	0.428958i
$195$		0			0
$196$	−0.500000	+	0.866025i	−0.0357143	+	0.0618590i
$197$	12.6969			0.904619			0.452310	−	0.891861i	$-0.350600\pi$
$197$	12.6969			0.904619			0.452310	+	0.891861i	$0.350600\pi$
$198$		0			0
$199$	6.89898			0.489056			0.244528	−	0.969642i	$-0.421367\pi$
$199$	6.89898			0.489056			0.244528	+	0.969642i	$0.421367\pi$
$200$	3.44949	−	5.97469i	0.243916	−	0.422474i
$201$		0			0
$202$	−3.62372	−	6.27647i	−0.254964	−	0.441611i
$203$	−1.44949	−	2.51059i	−0.101734	−	0.176209i
$204$		0			0
$205$	16.8990	−	29.2699i	1.18028	−	2.04430i
$206$	−14.0000			−0.975426
$207$		0			0
$208$	−4.89898			−0.339683
$209$	7.44949	−	12.9029i	0.515292	−	0.892512i
$210$		0			0
$211$	−1.55051	−	2.68556i	−0.106742	−	0.184882i	0.807707	−	0.589584i	$-0.200708\pi$
$211$	−1.55051	−	2.68556i	−0.106742	−	0.184882i	−0.914448	+	0.404703i	$0.867375\pi$
$212$	−0.550510	−	0.953512i	−0.0378092	−	0.0654875i
$213$		0			0
$214$	6.00000	−	10.3923i	0.410152	−	0.710403i
$215$	10.0000			0.681994
$216$		0			0
$217$	−6.00000			−0.407307
$218$	−8.34847	+	14.4600i	−0.565430	+	0.979353i
$219$		0			0
$220$	−3.44949	−	5.97469i	−0.232565	−	0.402814i
$221$	−4.89898	−	8.48528i	−0.329541	−	0.570782i
$222$		0			0
$223$	10.4495	−	18.0990i	0.699750	−	1.21200i	−0.268804	−	0.963195i	$-0.586628\pi$
$223$	10.4495	−	18.0990i	0.699750	−	1.21200i	0.968553	−	0.248807i	$-0.0800384\pi$
$224$	1.00000			0.0668153
$225$		0			0
$226$	−15.8990			−1.05758
$227$	−0.275255	+	0.476756i	−0.0182693	+	0.0316434i	−0.875016	−	0.484095i	$-0.839149\pi$
$227$	−0.275255	+	0.476756i	−0.0182693	+	0.0316434i	0.856746	+	0.515738i	$0.172482\pi$
$228$		0			0
$229$	11.6237	+	20.1329i	0.768117	+	1.33042i	0.938583	+	0.345055i	$0.112140\pi$
$229$	11.6237	+	20.1329i	0.768117	+	1.33042i	−0.170465	+	0.985364i	$0.554527\pi$
$230$	−1.72474	−	2.98735i	−0.113726	−	0.196980i
$231$		0			0
$232$	−1.44949	+	2.51059i	−0.0951637	+	0.164828i
$233$	7.00000			0.458585			0.229293	−	0.973358i	$-0.426359\pi$
$233$	7.00000			0.458585			0.229293	+	0.973358i	$0.426359\pi$
$234$		0			0
$235$	−33.7980			−2.20474
$236$	−1.00000	+	1.73205i	−0.0650945	+	0.112747i
$237$		0			0
$238$	1.00000	+	1.73205i	0.0648204	+	0.112272i
$239$	−6.39898	−	11.0834i	−0.413916	−	0.716923i	0.581398	−	0.813619i	$-0.302506\pi$
$239$	−6.39898	−	11.0834i	−0.413916	−	0.716923i	−0.995314	+	0.0966962i	$0.969172\pi$
$240$		0			0
$241$	4.44949	−	7.70674i	0.286617	−	0.496435i	−0.686383	−	0.727240i	$-0.740803\pi$
$241$	4.44949	−	7.70674i	0.286617	−	0.496435i	0.973000	+	0.230805i	$0.0741360\pi$
$242$	7.00000			0.449977
$243$		0			0
$244$	11.4495			0.732978
$245$	1.72474	−	2.98735i	0.110190	−	0.190855i
$246$		0			0
$247$	18.2474	+	31.6055i	1.16106	+	2.01101i
$248$	3.00000	+	5.19615i	0.190500	+	0.329956i
$249$		0			0
$250$	−3.27526	+	5.67291i	−0.207145	+	0.358786i
$251$	−12.5505			−0.792181			−0.396091	−	0.918211i	$-0.629633\pi$
$251$	−12.5505			−0.792181			−0.396091	+	0.918211i	$0.629633\pi$
$252$		0			0
$253$	−2.00000			−0.125739
$254$	−1.50000	+	2.59808i	−0.0941184	+	0.163018i
$255$		0			0
$256$	−0.500000	−	0.866025i	−0.0312500	−	0.0541266i
$257$	13.8990	+	24.0737i	0.866995	+	1.50168i	0.865053	+	0.501680i	$0.167285\pi$
$257$	13.8990	+	24.0737i	0.866995	+	1.50168i	0.00194150	+	0.999998i	$0.499382\pi$
$258$		0			0
$259$	−3.89898	+	6.75323i	−0.242271	+	0.419625i
$260$	16.8990			1.04803
$261$		0			0
$262$	−13.4495			−0.830912
$263$	8.05051	−	13.9439i	0.496416	−	0.859817i	−0.503576	−	0.863951i	$-0.667983\pi$
$263$	8.05051	−	13.9439i	0.496416	−	0.859817i	0.999991	+	0.00413383i	$0.00131584\pi$
$264$		0			0
$265$	1.89898	+	3.28913i	0.116653	+	0.202050i
$266$	−3.72474	−	6.45145i	−0.228379	−	0.395564i
$267$		0			0
$268$	1.55051	−	2.68556i	0.0947125	−	0.164047i
$269$	3.65153			0.222638			0.111319	−	0.993785i	$-0.464493\pi$
$269$	3.65153			0.222638			0.111319	+	0.993785i	$0.464493\pi$
$270$		0			0
$271$	16.8990			1.02654			0.513270	−	0.858227i	$-0.328434\pi$
$271$	16.8990			1.02654			0.513270	+	0.858227i	$0.328434\pi$
$272$	1.00000	−	1.73205i	0.0606339	−	0.105021i
$273$		0			0
$274$	5.89898	+	10.2173i	0.356370	+	0.617252i
$275$	6.89898	+	11.9494i	0.416024	+	0.720575i
$276$		0			0
$277$	−5.34847	+	9.26382i	−0.321358	+	0.556609i	−0.980769	−	0.195174i	$-0.937473\pi$
$277$	−5.34847	+	9.26382i	−0.321358	+	0.556609i	0.659410	+	0.751783i	$0.270806\pi$
$278$	−9.44949			−0.566743
$279$		0			0
$280$	−3.44949			−0.206146
$281$	−9.50000	+	16.4545i	−0.566722	+	0.981592i	0.430165	+	0.902750i	$0.358455\pi$
$281$	−9.50000	+	16.4545i	−0.566722	+	0.981592i	−0.996887	+	0.0788417i	$0.974878\pi$
$282$		0			0
$283$	10.2753	+	17.7973i	0.610801	+	1.05794i	0.991106	+	0.133077i	$0.0424856\pi$
$283$	10.2753	+	17.7973i	0.610801	+	1.05794i	−0.380305	+	0.924861i	$0.624181\pi$
$284$	4.94949	+	8.57277i	0.293698	+	0.508700i
$285$		0			0
$286$	4.89898	−	8.48528i	0.289683	−	0.501745i
$287$	−9.79796			−0.578355
$288$		0			0
$289$	−13.0000			−0.764706
$290$	5.00000	−	8.66025i	0.293610	−	0.508548i
$291$		0			0
$292$	−1.44949	−	2.51059i	−0.0848250	−	0.146921i
$293$	13.6237	+	23.5970i	0.795906	+	1.37855i	0.922262	+	0.386565i	$0.126339\pi$
$293$	13.6237	+	23.5970i	0.795906	+	1.37855i	−0.126356	+	0.991985i	$0.540328\pi$
$294$		0			0
$295$	3.44949	−	5.97469i	0.200837	−	0.347860i
$296$	7.79796			0.453247
$297$		0			0
$298$	−6.00000			−0.347571
$299$	2.44949	−	4.24264i	0.141658	−	0.245358i
$300$		0			0
$301$	−1.44949	−	2.51059i	−0.0835472	−	0.144708i
$302$	−2.50000	−	4.33013i	−0.143859	−	0.249171i
$303$		0			0
$304$	−3.72474	+	6.45145i	−0.213629	+	0.370016i
$305$	−39.4949			−2.26147
$306$		0			0
$307$	0.752551			0.0429504			0.0214752	−	0.999769i	$-0.493164\pi$
$307$	0.752551			0.0429504			0.0214752	+	0.999769i	$0.493164\pi$
$308$	−1.00000	+	1.73205i	−0.0569803	+	0.0986928i
$309$		0			0
$310$	−10.3485	−	17.9241i	−0.587754	−	1.01802i
$311$	0.651531	+	1.12848i	0.0369449	+	0.0639905i	0.883907	−	0.467663i	$-0.154904\pi$
$311$	0.651531	+	1.12848i	0.0369449	+	0.0639905i	−0.846962	+	0.531654i	$0.821571\pi$
$312$		0			0
$313$	−12.3485	+	21.3882i	−0.697977	+	1.20893i	0.271190	+	0.962526i	$0.412583\pi$
$313$	−12.3485	+	21.3882i	−0.697977	+	1.20893i	−0.969167	+	0.246405i	$0.920751\pi$
$314$	−6.34847			−0.358265
$315$		0			0
$316$	7.89898			0.444352
$317$	−4.34847	+	7.53177i	−0.244234	+	0.423026i	−0.961916	−	0.273345i	$-0.911870\pi$
$317$	−4.34847	+	7.53177i	−0.244234	+	0.423026i	0.717682	+	0.696371i	$0.245203\pi$
$318$		0			0
$319$	−2.89898	−	5.02118i	−0.162312	−	0.281132i
$320$	1.72474	+	2.98735i	0.0964162	+	0.166998i
$321$		0			0
$322$	−0.500000	+	0.866025i	−0.0278639	+	0.0482617i
$323$	−14.8990			−0.829001
$324$		0			0
$325$	−33.7980			−1.87477
$326$	−0.101021	+	0.174973i	−0.00559501	+	0.00969084i
$327$		0			0
$328$	4.89898	+	8.48528i	0.270501	+	0.468521i
$329$	4.89898	+	8.48528i	0.270089	+	0.467809i
$330$		0			0
$331$	12.3485	−	21.3882i	0.678733	−	1.17560i	−0.296629	−	0.954993i	$-0.595863\pi$
$331$	12.3485	−	21.3882i	0.678733	−	1.17560i	0.975363	−	0.220608i	$-0.0708041\pi$
$332$	−2.00000			−0.109764
$333$		0			0
$334$	18.6969			1.02305
$335$	−5.34847	+	9.26382i	−0.292218	+	0.506137i
$336$		0			0
$337$	−17.6969	−	30.6520i	−0.964014	−	1.66972i	−0.712242	−	0.701934i	$-0.752320\pi$
$337$	−17.6969	−	30.6520i	−0.964014	−	1.66972i	−0.251772	−	0.967787i	$-0.581013\pi$
$338$	5.50000	+	9.52628i	0.299161	+	0.518161i
$339$		0			0
$340$	−3.44949	+	5.97469i	−0.187075	+	0.324023i
$341$	−12.0000			−0.649836
$342$		0			0
$343$	−1.00000			−0.0539949
$344$	−1.44949	+	2.51059i	−0.0781512	+	0.135362i
$345$		0			0
$346$	6.44949	+	11.1708i	0.346727	+	0.600548i
$347$	−9.79796	−	16.9706i	−0.525982	−	0.911028i	−0.999542	−	0.0302659i	$-0.990365\pi$
$347$	−9.79796	−	16.9706i	−0.525982	−	0.911028i	0.473560	−	0.880762i	$-0.342969\pi$
$348$		0			0
$349$	−10.4495	+	18.0990i	−0.559348	+	0.968820i	0.438203	+	0.898876i	$0.355615\pi$
$349$	−10.4495	+	18.0990i	−0.559348	+	0.968820i	−0.997551	+	0.0699435i	$0.977718\pi$
$350$	6.89898			0.368766
$351$		0			0
$352$	2.00000			0.106600
$353$	−3.00000	+	5.19615i	−0.159674	+	0.276563i	−0.934751	−	0.355303i	$-0.884378\pi$
$353$	−3.00000	+	5.19615i	−0.159674	+	0.276563i	0.775077	+	0.631867i	$0.217711\pi$
$354$		0			0
$355$	−17.0732	−	29.5717i	−0.906152	−	1.56950i
$356$	−3.55051	−	6.14966i	−0.188177	−	0.325932i
$357$		0			0
$358$	4.34847	−	7.53177i	0.229824	−	0.398066i
$359$	−10.7980			−0.569894			−0.284947	−	0.958543i	$-0.591976\pi$
$359$	−10.7980			−0.569894			−0.284947	+	0.958543i	$0.591976\pi$
$360$		0			0
$361$	36.4949			1.92078
$362$	2.17423	−	3.76588i	0.114275	−	0.197931i
$363$		0			0
$364$	−2.44949	−	4.24264i	−0.128388	−	0.222375i
$365$	5.00000	+	8.66025i	0.261712	+	0.453298i
$366$		0			0
$367$	−2.89898	+	5.02118i	−0.151325	+	0.262103i	−0.931715	−	0.363190i	$-0.881687\pi$
$367$	−2.89898	+	5.02118i	−0.151325	+	0.262103i	0.780389	+	0.625294i	$0.215021\pi$
$368$	1.00000			0.0521286
$369$		0			0
$370$	−26.8990			−1.39841
$371$	0.550510	−	0.953512i	0.0285811	−	0.0495039i
$372$		0			0
$373$	−1.44949	−	2.51059i	−0.0750517	−	0.129993i	0.826057	−	0.563587i	$-0.190579\pi$
$373$	−1.44949	−	2.51059i	−0.0750517	−	0.129993i	−0.901109	+	0.433593i	$0.857246\pi$
$374$	2.00000	+	3.46410i	0.103418	+	0.179124i
$375$		0			0
$376$	4.89898	−	8.48528i	0.252646	−	0.437595i
$377$	14.2020			0.731442
$378$		0			0
$379$	−26.4949			−1.36095			−0.680476	−	0.732771i	$-0.738227\pi$
$379$	−26.4949			−1.36095			−0.680476	+	0.732771i	$0.738227\pi$
$380$	12.8485	−	22.2542i	0.659113	−	1.14162i
$381$		0			0
$382$	−6.94949	−	12.0369i	−0.355567	−	0.615860i
$383$	3.44949	+	5.97469i	0.176261	+	0.305292i	0.940597	−	0.339526i	$-0.110266\pi$
$383$	3.44949	+	5.97469i	0.176261	+	0.305292i	−0.764336	+	0.644818i	$0.776933\pi$
$384$		0			0
$385$	3.44949	−	5.97469i	0.175802	−	0.304498i
$386$	8.10102			0.412331
$387$		0			0
$388$	−6.89898			−0.350243
$389$	−7.55051	+	13.0779i	−0.382826	+	0.663074i	−0.991465	−	0.130373i	$-0.958382\pi$
$389$	−7.55051	+	13.0779i	−0.382826	+	0.663074i	0.608639	+	0.793447i	$0.291716\pi$
$390$		0			0
$391$	1.00000	+	1.73205i	0.0505722	+	0.0875936i
$392$	0.500000	+	0.866025i	0.0252538	+	0.0437409i
$393$		0			0
$394$	6.34847	−	10.9959i	0.319831	−	0.553964i
$395$	−27.2474			−1.37097
$396$		0			0
$397$	9.30306			0.466907			0.233454	−	0.972368i	$-0.424997\pi$
$397$	9.30306			0.466907			0.233454	+	0.972368i	$0.424997\pi$
$398$	3.44949	−	5.97469i	0.172907	−	0.299484i
$399$		0			0
$400$	−3.44949	−	5.97469i	−0.172474	−	0.298735i
$401$	−5.05051	−	8.74774i	−0.252210	−	0.436841i	0.711924	−	0.702257i	$-0.247824\pi$
$401$	−5.05051	−	8.74774i	−0.252210	−	0.436841i	−0.964134	+	0.265416i	$0.914491\pi$
$402$		0			0
$403$	14.6969	−	25.4558i	0.732107	−	1.26805i
$404$	−7.24745			−0.360574
$405$		0			0
$406$	−2.89898			−0.143874
$407$	−7.79796	+	13.5065i	−0.386530	+	0.669490i
$408$		0			0
$409$	−2.89898	−	5.02118i	−0.143345	−	0.248281i	0.785409	−	0.618977i	$-0.212453\pi$
$409$	−2.89898	−	5.02118i	−0.143345	−	0.248281i	−0.928754	+	0.370696i	$0.879119\pi$
$410$	−16.8990	−	29.2699i	−0.834581	−	1.44554i
$411$		0			0
$412$	−7.00000	+	12.1244i	−0.344865	+	0.597324i
$413$	−2.00000			−0.0984136
$414$		0			0
$415$	6.89898			0.338658
$416$	−2.44949	+	4.24264i	−0.120096	+	0.208013i
$417$		0			0
$418$	−7.44949	−	12.9029i	−0.364366	−	0.631101i
$419$	12.2753	+	21.2614i	0.599685	+	1.03869i	0.992867	+	0.119225i	$0.0380410\pi$
$419$	12.2753	+	21.2614i	0.599685	+	1.03869i	−0.393182	+	0.919461i	$0.628626\pi$
$420$		0			0
$421$	−6.55051	+	11.3458i	−0.319252	+	0.552961i	−0.980332	−	0.197354i	$-0.936765\pi$
$421$	−6.55051	+	11.3458i	−0.319252	+	0.552961i	0.661080	+	0.750316i	$0.270098\pi$
$422$	−3.10102			−0.150955
$423$		0			0
$424$	−1.10102			−0.0534703
$425$	6.89898	−	11.9494i	0.334650	−	0.579630i
$426$		0			0
$427$	5.72474	+	9.91555i	0.277040	+	0.479847i
$428$	−6.00000	−	10.3923i	−0.290021	−	0.502331i
$429$		0			0
$430$	5.00000	−	8.66025i	0.241121	−	0.417635i
$431$	7.59592			0.365882			0.182941	−	0.983124i	$-0.441438\pi$
$431$	7.59592			0.365882			0.182941	+	0.983124i	$0.441438\pi$
$432$		0			0
$433$	11.7980			0.566974			0.283487	−	0.958976i	$-0.408509\pi$
$433$	11.7980			0.566974			0.283487	+	0.958976i	$0.408509\pi$
$434$	−3.00000	+	5.19615i	−0.144005	+	0.249423i
$435$		0			0
$436$	8.34847	+	14.4600i	0.399819	+	0.692507i
$437$	−3.72474	−	6.45145i	−0.178179	−	0.308615i
$438$		0			0
$439$	−10.8990	+	18.8776i	−0.520180	+	0.900978i	0.479545	+	0.877517i	$0.340802\pi$
$439$	−10.8990	+	18.8776i	−0.520180	+	0.900978i	−0.999725	+	0.0234607i	$0.992532\pi$
$440$	−6.89898			−0.328896
$441$		0			0
$442$	−9.79796			−0.466041
$443$	−2.55051	+	4.41761i	−0.121178	+	0.209887i	−0.920233	−	0.391372i	$-0.872001\pi$
$443$	−2.55051	+	4.41761i	−0.121178	+	0.209887i	0.799054	+	0.601259i	$0.205334\pi$
$444$		0			0
$445$	12.2474	+	21.2132i	0.580585	+	1.00560i
$446$	−10.4495	−	18.0990i	−0.494798	−	0.857015i
$447$		0			0
$448$	0.500000	−	0.866025i	0.0236228	−	0.0409159i
$449$	18.5959			0.877596			0.438798	−	0.898586i	$-0.355404\pi$
$449$	18.5959			0.877596			0.438798	+	0.898586i	$0.355404\pi$
$450$		0			0
$451$	−19.5959			−0.922736
$452$	−7.94949	+	13.7689i	−0.373913	+	0.647636i
$453$		0			0
$454$	0.275255	+	0.476756i	0.0129184	+	0.0223753i
$455$	8.44949	+	14.6349i	0.396118	+	0.686097i
$456$		0			0
$457$	−15.7474	+	27.2754i	−0.736635	+	1.27589i	0.217368	+	0.976090i	$0.430253\pi$
$457$	−15.7474	+	27.2754i	−0.736635	+	1.27589i	−0.954002	+	0.299799i	$0.903080\pi$
$458$	23.2474			1.08628
$459$		0			0
$460$	−3.44949			−0.160833
$461$	10.1742	−	17.6223i	0.473861	−	0.820752i	−0.525691	−	0.850676i	$-0.676193\pi$
$461$	10.1742	−	17.6223i	0.473861	−	0.820752i	0.999552	+	0.0299238i	$0.00952645\pi$
$462$		0			0
$463$	12.8485	+	22.2542i	0.597119	+	1.03424i	0.993244	+	0.116044i	$0.0370213\pi$
$463$	12.8485	+	22.2542i	0.597119	+	1.03424i	−0.396125	+	0.918197i	$0.629645\pi$
$464$	1.44949	+	2.51059i	0.0672909	+	0.116551i
$465$		0			0
$466$	3.50000	−	6.06218i	0.162134	−	0.280825i
$467$	−10.0000			−0.462745			−0.231372	−	0.972865i	$-0.574322\pi$
$467$	−10.0000			−0.462745			−0.231372	+	0.972865i	$0.574322\pi$
$468$		0			0
$469$	3.10102			0.143192
$470$	−16.8990	+	29.2699i	−0.779492	+	1.35012i
$471$		0			0
$472$	1.00000	+	1.73205i	0.0460287	+	0.0797241i
$473$	−2.89898	−	5.02118i	−0.133295	−	0.230874i
$474$		0			0
$475$	−25.6969	+	44.5084i	−1.17906	+	2.04219i
$476$	2.00000			0.0916698
$477$		0			0
$478$	−12.7980			−0.585365
$479$	−14.7980	+	25.6308i	−0.676136	+	1.17110i	0.299999	+	0.953939i	$0.403013\pi$
$479$	−14.7980	+	25.6308i	−0.676136	+	1.17110i	−0.976135	+	0.217163i	$0.930320\pi$
$480$		0			0
$481$	−19.1010	−	33.0839i	−0.870932	−	1.50850i
$482$	−4.44949	−	7.70674i	−0.202669	−	0.351032i
$483$		0			0
$484$	3.50000	−	6.06218i	0.159091	−	0.275554i
$485$	23.7980			1.08061
$486$		0			0
$487$	22.3939			1.01476			0.507382	−	0.861721i	$-0.330613\pi$
$487$	22.3939			1.01476			0.507382	+	0.861721i	$0.330613\pi$
$488$	5.72474	−	9.91555i	0.259147	−	0.448856i
$489$		0			0
$490$	−1.72474	−	2.98735i	−0.0779160	−	0.134955i
$491$	−1.89898	−	3.28913i	−0.0856997	−	0.148436i	0.819989	−	0.572379i	$-0.193979\pi$
$491$	−1.89898	−	3.28913i	−0.0856997	−	0.148436i	−0.905689	+	0.423942i	$0.860646\pi$
$492$		0			0
$493$	−2.89898	+	5.02118i	−0.130563	+	0.226143i
$494$	36.4949			1.64198
$495$		0			0
$496$	6.00000			0.269408
$497$	−4.94949	+	8.57277i	−0.222015	+	0.384541i
$498$		0			0
$499$	−16.6969	−	28.9199i	−0.747458	−	1.29463i	−0.949038	−	0.315163i	$-0.897941\pi$
$499$	−16.6969	−	28.9199i	−0.747458	−	1.29463i	0.201580	−	0.979472i	$-0.435392\pi$
$500$	3.27526	+	5.67291i	0.146474	+	0.253700i
$501$		0			0
$502$	−6.27526	+	10.8691i	−0.280078	+	0.485110i
$503$	−24.4949			−1.09217			−0.546087	−	0.837729i	$-0.683883\pi$
$503$	−24.4949			−1.09217			−0.546087	+	0.837729i	$0.683883\pi$
$504$		0			0
$505$	25.0000			1.11249
$506$	−1.00000	+	1.73205i	−0.0444554	+	0.0769991i
$507$		0			0
$508$	1.50000	+	2.59808i	0.0665517	+	0.115271i
$509$	−8.44949	−	14.6349i	−0.374517	−	0.648683i	0.615738	−	0.787951i	$-0.288858\pi$
$509$	−8.44949	−	14.6349i	−0.374517	−	0.648683i	−0.990255	+	0.139269i	$0.955525\pi$
$510$		0			0
$511$	1.44949	−	2.51059i	0.0641217	−	0.111062i
$512$	−1.00000			−0.0441942
$513$		0			0
$514$	27.7980			1.22612
$515$	24.1464	−	41.8228i	1.06402	−	1.84293i
$516$		0			0
$517$	9.79796	+	16.9706i	0.430914	+	0.746364i
$518$	3.89898	+	6.75323i	0.171311	+	0.296720i
$519$		0			0
$520$	8.44949	−	14.6349i	0.370535	−	0.641785i
$521$	38.6969			1.69534			0.847672	−	0.530521i	$-0.178004\pi$
$521$	38.6969			1.69534			0.847672	+	0.530521i	$0.178004\pi$
$522$		0			0
$523$	0.348469			0.0152375			0.00761875	−	0.999971i	$-0.497575\pi$
$523$	0.348469			0.0152375			0.00761875	+	0.999971i	$0.497575\pi$
$524$	−6.72474	+	11.6476i	−0.293772	+	0.508828i
$525$		0			0
$526$	−8.05051	−	13.9439i	−0.351019	−	0.607983i
$527$	6.00000	+	10.3923i	0.261364	+	0.452696i
$528$		0			0
$529$	11.0000	−	19.0526i	0.478261	−	0.828372i
$530$	3.79796			0.164973
$531$		0			0
$532$	−7.44949			−0.322976
$533$	24.0000	−	41.5692i	1.03956	−	1.80056i
$534$		0			0
$535$	20.6969	+	35.8481i	0.894807	+	1.54985i
$536$	−1.55051	−	2.68556i	−0.0669718	−	0.115999i
$537$		0			0
$538$	1.82577	−	3.16232i	0.0787143	−	0.136337i
$539$	−2.00000			−0.0861461
$540$		0			0
$541$	30.4949			1.31108			0.655539	−	0.755161i	$-0.272441\pi$
$541$	30.4949			1.31108			0.655539	+	0.755161i	$0.272441\pi$
$542$	8.44949	−	14.6349i	0.362937	−	0.628625i
$543$		0			0
$544$	−1.00000	−	1.73205i	−0.0428746	−	0.0742611i
$545$	−28.7980	−	49.8795i	−1.23357	−	2.13660i
$546$		0			0
$547$	−15.7980	+	27.3629i	−0.675472	+	1.16995i	0.300859	+	0.953669i	$0.402727\pi$
$547$	−15.7980	+	27.3629i	−0.675472	+	1.16995i	−0.976331	+	0.216283i	$0.930607\pi$
$548$	11.7980			0.503984
$549$		0			0
$550$	13.7980			0.588347
$551$	10.7980	−	18.7026i	0.460009	−	0.796758i
$552$		0			0
$553$	3.94949	+	6.84072i	0.167949	+	0.290897i
$554$	5.34847	+	9.26382i	0.227235	+	0.393582i
$555$		0			0
$556$	−4.72474	+	8.18350i	−0.200374	+	0.347058i
$557$	3.10102			0.131394			0.0656972	−	0.997840i	$-0.479073\pi$
$557$	3.10102			0.131394			0.0656972	+	0.997840i	$0.479073\pi$
$558$		0			0
$559$	14.2020			0.600682
$560$	−1.72474	+	2.98735i	−0.0728838	+	0.126238i
$561$		0			0
$562$	9.50000	+	16.4545i	0.400733	+	0.694090i
$563$	6.97219	+	12.0762i	0.293843	+	0.508951i	0.974715	−	0.223451i	$-0.0717324\pi$
$563$	6.97219	+	12.0762i	0.293843	+	0.508951i	−0.680872	+	0.732402i	$0.738399\pi$
$564$		0			0
$565$	27.4217	−	47.4957i	1.15364	−	1.99816i
$566$	20.5505			0.863802
$567$		0			0
$568$	9.89898			0.415352
$569$	−15.0000	+	25.9808i	−0.628833	+	1.08917i	0.358954	+	0.933355i	$0.383134\pi$
$569$	−15.0000	+	25.9808i	−0.628833	+	1.08917i	−0.987786	+	0.155815i	$0.950200\pi$
$570$		0			0
$571$	−7.10102	−	12.2993i	−0.297168	−	0.514711i	0.678319	−	0.734768i	$-0.262709\pi$
$571$	−7.10102	−	12.2993i	−0.297168	−	0.514711i	−0.975487	+	0.220057i	$0.929376\pi$
$572$	−4.89898	−	8.48528i	−0.204837	−	0.354787i
$573$		0			0
$574$	−4.89898	+	8.48528i	−0.204479	+	0.354169i
$575$	6.89898			0.287707
$576$		0			0
$577$	23.5959			0.982311			0.491155	−	0.871072i	$-0.336575\pi$
$577$	23.5959			0.982311			0.491155	+	0.871072i	$0.336575\pi$
$578$	−6.50000	+	11.2583i	−0.270364	+	0.468285i
$579$		0			0
$580$	−5.00000	−	8.66025i	−0.207614	−	0.359597i
$581$	−1.00000	−	1.73205i	−0.0414870	−	0.0718576i
$582$		0			0
$583$	1.10102	−	1.90702i	0.0455996	−	0.0789808i
$584$	−2.89898			−0.119961
$585$		0			0
$586$	27.2474			1.12558
$587$	−9.07321	+	15.7153i	−0.374492	+	0.648639i	−0.990251	−	0.139296i	$-0.955516\pi$
$587$	−9.07321	+	15.7153i	−0.374492	+	0.648639i	0.615759	+	0.787934i	$0.288849\pi$
$588$		0			0
$589$	−22.3485	−	38.7087i	−0.920853	−	1.59496i
$590$	−3.44949	−	5.97469i	−0.142013	−	0.245974i
$591$		0			0
$592$	3.89898	−	6.75323i	0.160247	−	0.277556i
$593$	14.6969			0.603531			0.301765	−	0.953382i	$-0.402424\pi$
$593$	14.6969			0.603531			0.301765	+	0.953382i	$0.402424\pi$
$594$		0			0
$595$	−6.89898			−0.282831
$596$	−3.00000	+	5.19615i	−0.122885	+	0.212843i
$597$		0			0
$598$	−2.44949	−	4.24264i	−0.100167	−	0.173494i
$599$	−7.10102	−	12.2993i	−0.290140	−	0.502537i	0.683703	−	0.729761i	$-0.260368\pi$
$599$	−7.10102	−	12.2993i	−0.290140	−	0.502537i	−0.973843	+	0.227224i	$0.927035\pi$
$600$		0			0
$601$	6.34847	−	10.9959i	0.258959	−	0.448531i	−0.707004	−	0.707210i	$-0.749954\pi$
$601$	6.34847	−	10.9959i	0.258959	−	0.448531i	0.965963	+	0.258679i	$0.0832871\pi$
$602$	−2.89898			−0.118154
$603$		0			0
$604$	−5.00000			−0.203447
$605$	−12.0732	+	20.9114i	−0.490846	+	0.850170i
$606$		0			0
$607$	4.34847	+	7.53177i	0.176499	+	0.305705i	0.940679	−	0.339298i	$-0.110189\pi$
$607$	4.34847	+	7.53177i	0.176499	+	0.305705i	−0.764180	+	0.645003i	$0.776856\pi$
$608$	3.72474	+	6.45145i	0.151058	+	0.261641i
$609$		0			0
$610$	−19.7474	+	34.2036i	−0.799551	+	1.38486i
$611$	−48.0000			−1.94187
$612$		0			0
$613$	14.6969			0.593604			0.296802	−	0.954939i	$-0.404080\pi$
$613$	14.6969			0.593604			0.296802	+	0.954939i	$0.404080\pi$
$614$	0.376276	−	0.651729i	0.0151852	−	0.0263016i
$615$		0			0
$616$	1.00000	+	1.73205i	0.0402911	+	0.0697863i
$617$	21.6969	+	37.5802i	0.873486	+	1.51292i	0.858367	+	0.513036i	$0.171479\pi$
$617$	21.6969	+	37.5802i	0.873486	+	1.51292i	0.0151189	+	0.999886i	$0.495187\pi$
$618$		0			0
$619$	2.07321	−	3.59091i	0.0833295	−	0.144331i	−0.821349	−	0.570426i	$-0.806778\pi$
$619$	2.07321	−	3.59091i	0.0833295	−	0.144331i	0.904678	+	0.426096i	$0.140111\pi$
$620$	−20.6969			−0.831209
$621$		0			0
$622$	1.30306			0.0522480
$623$	3.55051	−	6.14966i	0.142248	−	0.246381i
$624$		0			0
$625$	5.94949	+	10.3048i	0.237980	+	0.412193i
$626$	12.3485	+	21.3882i	0.493544	+	0.854843i
$627$		0			0
$628$	−3.17423	+	5.49794i	−0.126666	+	0.219392i
$629$	15.5959			0.621850
$630$		0			0
$631$	18.1010			0.720590			0.360295	−	0.932838i	$-0.382676\pi$
$631$	18.1010			0.720590			0.360295	+	0.932838i	$0.382676\pi$
$632$	3.94949	−	6.84072i	0.157102	−	0.272109i
$633$		0			0
$634$	4.34847	+	7.53177i	0.172700	+	0.299125i
$635$	−5.17423	−	8.96204i	−0.205333	−	0.355648i
$636$		0			0
$637$	2.44949	−	4.24264i	0.0970523	−	0.168100i
$638$	−5.79796			−0.229543
$639$		0			0
$640$	3.44949			0.136353
$641$	20.7474	−	35.9356i	0.819475	−	1.41937i	−0.0865947	−	0.996244i	$-0.527599\pi$
$641$	20.7474	−	35.9356i	0.819475	−	1.41937i	0.906070	−	0.423129i	$-0.139068\pi$
$642$		0			0
$643$	−9.69694	−	16.7956i	−0.382410	−	0.662353i	0.608996	−	0.793173i	$-0.291572\pi$
$643$	−9.69694	−	16.7956i	−0.382410	−	0.662353i	−0.991406	+	0.130820i	$0.958239\pi$
$644$	0.500000	+	0.866025i	0.0197028	+	0.0341262i
$645$		0			0
$646$	−7.44949	+	12.9029i	−0.293096	+	0.507658i
$647$	21.3031			0.837510			0.418755	−	0.908099i	$-0.362467\pi$
$647$	21.3031			0.837510			0.418755	+	0.908099i	$0.362467\pi$
$648$		0			0
$649$	−4.00000			−0.157014
$650$	−16.8990	+	29.2699i	−0.662833	+	1.14806i
$651$		0			0
$652$	0.101021	+	0.174973i	0.00395627	+	0.00685246i
$653$	−4.89898	−	8.48528i	−0.191712	−	0.332055i	0.754106	−	0.656753i	$-0.228071\pi$
$653$	−4.89898	−	8.48528i	−0.191712	−	0.332055i	−0.945818	+	0.324698i	$0.894737\pi$
$654$		0			0
$655$	23.1969	−	40.1783i	0.906379	−	1.56990i
$656$	9.79796			0.382546
$657$		0			0
$658$	9.79796			0.381964
$659$	2.34847	−	4.06767i	0.0914834	−	0.158454i	−0.816652	−	0.577130i	$-0.804172\pi$
$659$	2.34847	−	4.06767i	0.0914834	−	0.158454i	0.908136	+	0.418676i	$0.137506\pi$
$660$		0			0
$661$	−4.72474	−	8.18350i	−0.183771	−	0.318301i	0.759391	−	0.650635i	$-0.225497\pi$
$661$	−4.72474	−	8.18350i	−0.183771	−	0.318301i	−0.943162	+	0.332334i	$0.892164\pi$
$662$	−12.3485	−	21.3882i	−0.479937	−	0.831275i
$663$		0			0
$664$	−1.00000	+	1.73205i	−0.0388075	+	0.0672166i
$665$	25.6969			0.996485
$666$		0			0
$667$	−2.89898			−0.112249
$668$	9.34847	−	16.1920i	0.361703	−	0.626488i
$669$		0			0
$670$	5.34847	+	9.26382i	0.206629	+	0.357893i
$671$	11.4495	+	19.8311i	0.442003	+	0.765571i
$672$		0			0
$673$	−15.2980	+	26.4968i	−0.589693	+	1.02138i	0.404579	+	0.914503i	$0.367418\pi$
$673$	−15.2980	+	26.4968i	−0.589693	+	1.02138i	−0.994272	+	0.106875i	$0.965915\pi$
$674$	−35.3939			−1.36332
$675$		0			0
$676$	11.0000			0.423077
$677$	−7.34847	+	12.7279i	−0.282425	+	0.489174i	−0.971981	−	0.235058i	$-0.924472\pi$
$677$	−7.34847	+	12.7279i	−0.282425	+	0.489174i	0.689557	+	0.724232i	$0.257805\pi$
$678$		0			0
$679$	−3.44949	−	5.97469i	−0.132379	−	0.229288i
$680$	3.44949	+	5.97469i	0.132282	+	0.229119i
$681$		0			0
$682$	−6.00000	+	10.3923i	−0.229752	+	0.397942i
$683$	−32.2020			−1.23218			−0.616088	−	0.787677i	$-0.711284\pi$
$683$	−32.2020			−1.23218			−0.616088	+	0.787677i	$0.711284\pi$
$684$		0			0
$685$	−40.6969			−1.55495
$686$	−0.500000	+	0.866025i	−0.0190901	+	0.0330650i
$687$		0			0
$688$	1.44949	+	2.51059i	0.0552613	+	0.0957153i
$689$	2.69694	+	4.67123i	0.102745	+	0.177960i
$690$		0			0
$691$	−3.47730	+	6.02285i	−0.132283	+	0.229120i	−0.924556	−	0.381046i	$-0.875564\pi$
$691$	−3.47730	+	6.02285i	−0.132283	+	0.229120i	0.792274	+	0.610166i	$0.208897\pi$
$692$	12.8990			0.490346
$693$		0			0
$694$	−19.5959			−0.743851
$695$	16.2980	−	28.2289i	0.618217	−	1.07078i
$696$		0			0
$697$	9.79796	+	16.9706i	0.371124	+	0.642806i
$698$	10.4495	+	18.0990i	0.395519	+	0.685059i
$699$		0			0
$700$	3.44949	−	5.97469i	0.130378	−	0.225822i
$701$	−51.3939			−1.94112			−0.970560	−	0.240860i	$-0.922571\pi$
$701$	−51.3939			−1.94112			−0.970560	+	0.240860i	$0.922571\pi$
$702$		0			0
$703$	−58.0908			−2.19094
$704$	1.00000	−	1.73205i	0.0376889	−	0.0652791i
$705$		0			0
$706$	3.00000	+	5.19615i	0.112906	+	0.195560i
$707$	−3.62372	−	6.27647i	−0.136284	−	0.236051i
$708$		0			0
$709$	5.79796	−	10.0424i	0.217747	−	0.377149i	−0.736372	−	0.676577i	$-0.763463\pi$
$709$	5.79796	−	10.0424i	0.217747	−	0.377149i	0.954119	+	0.299428i	$0.0967959\pi$
$710$	−34.1464			−1.28149
$711$		0			0
$712$	−7.10102			−0.266122
$713$	−3.00000	+	5.19615i	−0.112351	+	0.194597i
$714$		0			0
$715$	16.8990	+	29.2699i	0.631986	+	1.09463i
$716$	−4.34847	−	7.53177i	−0.162510	−	0.281475i
$717$		0			0
$718$	−5.39898	+	9.35131i	−0.201488	+	0.348988i
$719$	9.79796			0.365402			0.182701	−	0.983169i	$-0.441516\pi$
$719$	9.79796			0.365402			0.182701	+	0.983169i	$0.441516\pi$
$720$		0			0
$721$	−14.0000			−0.521387
$722$	18.2474	−	31.6055i	0.679100	−	1.17624i
$723$		0			0
$724$	−2.17423	−	3.76588i	−0.0808048	−	0.139958i
$725$	10.0000	+	17.3205i	0.371391	+	0.643268i
$726$		0			0
$727$	−20.2474	+	35.0696i	−0.750936	+	1.30066i	0.196433	+	0.980517i	$0.437064\pi$
$727$	−20.2474	+	35.0696i	−0.750936	+	1.30066i	−0.947369	+	0.320143i	$0.896269\pi$
$728$	−4.89898			−0.181568
$729$		0			0
$730$	10.0000			0.370117
$731$	−2.89898	+	5.02118i	−0.107223	+	0.185715i
$732$		0			0
$733$	6.27526	+	10.8691i	0.231782	+	0.401458i	0.958333	−	0.285655i	$-0.0922111\pi$
$733$	6.27526	+	10.8691i	0.231782	+	0.401458i	−0.726551	+	0.687113i	$0.758878\pi$
$734$	2.89898	+	5.02118i	0.107003	+	0.185335i
$735$		0			0
$736$	0.500000	−	0.866025i	0.0184302	−	0.0319221i
$737$	6.20204			0.228455
$738$		0			0
$739$	−25.5959			−0.941561			−0.470781	−	0.882250i	$-0.656028\pi$
$739$	−25.5959			−0.941561			−0.470781	+	0.882250i	$0.656028\pi$
$740$	−13.4495	+	23.2952i	−0.494413	+	0.856349i
$741$		0			0
$742$	−0.550510	−	0.953512i	−0.0202099	−	0.0350045i
$743$	18.0000	+	31.1769i	0.660356	+	1.14377i	0.980522	+	0.196409i	$0.0629279\pi$
$743$	18.0000	+	31.1769i	0.660356	+	1.14377i	−0.320166	+	0.947361i	$0.603739\pi$
$744$		0			0
$745$	10.3485	−	17.9241i	0.379139	−	0.656687i
$746$	−2.89898			−0.106139
$747$		0			0
$748$	4.00000			0.146254
$749$	6.00000	−	10.3923i	0.219235	−	0.379727i
$750$		0			0
$751$	−20.2980	−	35.1571i	−0.740683	−	1.28290i	−0.952185	−	0.305523i	$-0.901169\pi$
$751$	−20.2980	−	35.1571i	−0.740683	−	1.28290i	0.211502	−	0.977378i	$-0.432165\pi$
$752$	−4.89898	−	8.48528i	−0.178647	−	0.309426i
$753$		0			0
$754$	7.10102	−	12.2993i	0.258604	−	0.447915i
$755$	17.2474			0.627699
$756$		0			0
$757$	23.3939			0.850265			0.425132	−	0.905131i	$-0.360228\pi$
$757$	23.3939			0.850265			0.425132	+	0.905131i	$0.360228\pi$
$758$	−13.2474	+	22.9453i	−0.481169	+	0.833409i
$759$		0			0
$760$	−12.8485	−	22.2542i	−0.466063	−	0.807245i
$761$	1.00000	+	1.73205i	0.0362500	+	0.0627868i	0.883581	−	0.468278i	$-0.155125\pi$
$761$	1.00000	+	1.73205i	0.0362500	+	0.0627868i	−0.847331	+	0.531065i	$0.821792\pi$
$762$		0			0
$763$	−8.34847	+	14.4600i	−0.302235	+	0.523486i
$764$	−13.8990			−0.502847
$765$		0			0
$766$	6.89898			0.249270
$767$	4.89898	−	8.48528i	0.176892	−	0.306386i
$768$		0			0
$769$	−27.0454	−	46.8440i	−0.975282	−	1.68924i	−0.679000	−	0.734138i	$-0.737586\pi$
$769$	−27.0454	−	46.8440i	−0.975282	−	1.68924i	−0.296282	−	0.955100i	$-0.595747\pi$
$770$	−3.44949	−	5.97469i	−0.124311	−	0.215313i
$771$		0			0
$772$	4.05051	−	7.01569i	0.145781	−	0.252500i
$773$	19.9444			0.717350			0.358675	−	0.933463i	$-0.383229\pi$
$773$	19.9444			0.717350			0.358675	+	0.933463i	$0.383229\pi$
$774$		0			0
$775$	41.3939			1.48691
$776$	−3.44949	+	5.97469i	−0.123829	+	0.214479i
$777$		0			0
$778$	7.55051	+	13.0779i	0.270699	+	0.468864i
$779$	−36.4949	−	63.2110i	−1.30757	−	2.26477i
$780$		0			0
$781$	−9.89898	+	17.1455i	−0.354213	+	0.613515i
$782$	2.00000			0.0715199
$783$		0			0
$784$	1.00000			0.0357143
$785$	10.9495	−	18.9651i	0.390804	−	0.676892i
$786$		0			0
$787$	−23.6969	−	41.0443i	−0.844705	−	1.46307i	−0.885877	−	0.463919i	$-0.846443\pi$
$787$	−23.6969	−	41.0443i	−0.844705	−	1.46307i	0.0411728	−	0.999152i	$-0.486891\pi$
$788$	−6.34847	−	10.9959i	−0.226155	−	0.391712i
$789$		0			0
$790$	−13.6237	+	23.5970i	−0.484710	+	0.839543i
$791$

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.f (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.500000 - 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(1.72474 + 2.98735i) q^{5} +(0.500000 - 0.866025i) q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q+(0.500000 - 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(1.72474 + 2.98735i) q^{5} +(0.500000 - 0.866025i) q^{7} -1.00000 q^{8} +3.44949 q^{10} +(1.00000 - 1.73205i) q^{11} +(2.44949 + 4.24264i) q^{13} +(-0.500000 - 0.866025i) q^{14} +(-0.500000 + 0.866025i) q^{16} -2.00000 q^{17} +7.44949 q^{19} +(1.72474 - 2.98735i) q^{20} +(-1.00000 - 1.73205i) q^{22} +(-0.500000 - 0.866025i) q^{23} +(-3.44949 + 5.97469i) q^{25} +4.89898 q^{26} -1.00000 q^{28} +(1.44949 - 2.51059i) q^{29} +(-3.00000 - 5.19615i) q^{31} +(0.500000 + 0.866025i) q^{32} +(-1.00000 + 1.73205i) q^{34} +3.44949 q^{35} -7.79796 q^{37} +(3.72474 - 6.45145i) q^{38} +(-1.72474 - 2.98735i) q^{40} +(-4.89898 - 8.48528i) q^{41} +(1.44949 - 2.51059i) q^{43} -2.00000 q^{44} -1.00000 q^{46} +(-4.89898 + 8.48528i) q^{47} +(-0.500000 - 0.866025i) q^{49} +(3.44949 + 5.97469i) q^{50} +(2.44949 - 4.24264i) q^{52} +1.10102 q^{53} +6.89898 q^{55} +(-0.500000 + 0.866025i) q^{56} +(-1.44949 - 2.51059i) q^{58} +(-1.00000 - 1.73205i) q^{59} +(-5.72474 + 9.91555i) q^{61} -6.00000 q^{62} +1.00000 q^{64} +(-8.44949 + 14.6349i) q^{65} +(1.55051 + 2.68556i) q^{67} +(1.00000 + 1.73205i) q^{68} +(1.72474 - 2.98735i) q^{70} -9.89898 q^{71} +2.89898 q^{73} +(-3.89898 + 6.75323i) q^{74} +(-3.72474 - 6.45145i) q^{76} +(-1.00000 - 1.73205i) q^{77} +(-3.94949 + 6.84072i) q^{79} -3.44949 q^{80} -9.79796 q^{82} +(1.00000 - 1.73205i) q^{83} +(-3.44949 - 5.97469i) q^{85} +(-1.44949 - 2.51059i) q^{86} +(-1.00000 + 1.73205i) q^{88} +7.10102 q^{89} +4.89898 q^{91} +(-0.500000 + 0.866025i) q^{92} +(4.89898 + 8.48528i) q^{94} +(12.8485 + 22.2542i) q^{95} +(3.44949 - 5.97469i) q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} - 2 q^{4} + 2 q^{5} + 2 q^{7} - 4 q^{8} + O(q^{10}) \)	\( 4 q + 2 q^{2} - 2 q^{4} + 2 q^{5} + 2 q^{7} - 4 q^{8} + 4 q^{10} + 4 q^{11} - 2 q^{14} - 2 q^{16} - 8 q^{17} + 20 q^{19} + 2 q^{20} - 4 q^{22} - 2 q^{23} - 4 q^{25} - 4 q^{28} - 4 q^{29} - 12 q^{31} + 2 q^{32} - 4 q^{34} + 4 q^{35} + 8 q^{37} + 10 q^{38} - 2 q^{40} - 4 q^{43} - 8 q^{44} - 4 q^{46} - 2 q^{49} + 4 q^{50} + 24 q^{53} + 8 q^{55} - 2 q^{56} + 4 q^{58} - 4 q^{59} - 18 q^{61} - 24 q^{62} + 4 q^{64} - 24 q^{65} + 16 q^{67} + 4 q^{68} + 2 q^{70} - 20 q^{71} - 8 q^{73} + 4 q^{74} - 10 q^{76} - 4 q^{77} - 6 q^{79} - 4 q^{80} + 4 q^{83} - 4 q^{85} + 4 q^{86} - 4 q^{88} + 48 q^{89} - 2 q^{92} + 22 q^{95} + 4 q^{97} - 4 q^{98} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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