LMFDB - Embedded newform 819.2.y.h.811.6

Newspace parameters

Level:	$ N $	$=$	$ 819 = 3^{2} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	819.y (of order $4$, degree $2$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$6.53974792554$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
Defining polynomial:	$ x^{12} + 35x^{8} + 295x^{4} + 169 $ x^12 + 35x^8 + 295x^4 + 169
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 91)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			811.6
Root			$-0.626770 + 0.626770i$ of defining polynomial
Character	$\chi$	$=$	819.811
Dual form			819.2.y.h.307.6

$q$-expansion

$f(q)$	$=$	$q+(1.45161 - 1.45161i) q^{2} -2.21432i q^{4} +(2.01464 + 2.01464i) q^{5} +(-1.13594 - 2.38948i) q^{7} +(-0.311108 - 0.311108i) q^{8} +O(q^{10})$	\(q+(1.45161 - 1.45161i) q^{2} -2.21432i q^{4} +(2.01464 + 2.01464i) q^{5} +(-1.13594 - 2.38948i) q^{7} +(-0.311108 - 0.311108i) q^{8} +5.84892 q^{10} +(-0.451606 - 0.451606i) q^{11} +(3.40251 + 1.19288i) q^{13} +(-5.11753 - 1.81964i) q^{14} +3.52543 q^{16} +4.32672 q^{17} +(-3.40251 - 3.40251i) q^{19} +(4.46105 - 4.46105i) q^{20} -1.31111 q^{22} -0.933323i q^{23} +3.11753i q^{25} +(6.67068 - 3.20751i) q^{26} +(-5.29108 + 2.51534i) q^{28} -6.33185 q^{29} +(5.47781 + 5.47781i) q^{31} +(5.73975 - 5.73975i) q^{32} +(6.28070 - 6.28070i) q^{34} +(2.52543 - 7.10246i) q^{35} +(2.14050 + 2.14050i) q^{37} -9.87820 q^{38} -1.25354i q^{40} +(-1.81964 - 1.81964i) q^{41} -10.4795i q^{43} +(-1.00000 + 1.00000i) q^{44} +(-1.35482 - 1.35482i) q^{46} +(-5.90958 + 5.90958i) q^{47} +(-4.41926 + 5.42864i) q^{49} +(4.52543 + 4.52543i) q^{50} +(2.64141 - 7.53424i) q^{52} -3.36196 q^{53} -1.81964i q^{55} +(-0.389986 + 1.09679i) q^{56} +(-9.19135 + 9.19135i) q^{58} +(0.255657 - 0.255657i) q^{59} +7.78989i q^{61} +15.9032 q^{62} -9.61285i q^{64} +(4.45161 + 9.25803i) q^{65} +(7.28100 - 7.28100i) q^{67} -9.58075i q^{68} +(-6.64405 - 13.9759i) q^{70} +(-5.56914 + 5.56914i) q^{71} +(-8.86144 + 8.86144i) q^{73} +6.21432 q^{74} +(-7.53424 + 7.53424i) q^{76} +(-0.566106 + 1.59210i) q^{77} -13.7971 q^{79} +(7.10246 + 7.10246i) q^{80} -5.28281 q^{82} +(-4.30785 - 4.30785i) q^{83} +(8.71678 + 8.71678i) q^{85} +(-15.2121 - 15.2121i) q^{86} +0.280996i q^{88} +(-5.61214 + 5.61214i) q^{89} +(-1.01470 - 9.48527i) q^{91} -2.06668 q^{92} +17.1568i q^{94} -13.7096i q^{95} +(-0.236784 - 0.236784i) q^{97} +(1.46522 + 14.2953i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 4 q^{2} - 8 q^{7} - 4 q^{8}+O(q^{10}) $ 12 * q + 4 * q^2 - 8 * q^7 - 4 * q^8	$ 12 q + 4 q^{2} - 8 q^{7} - 4 q^{8} + 8 q^{11} - 8 q^{14} + 16 q^{16} - 16 q^{22} - 20 q^{28} + 4 q^{29} + 16 q^{32} + 4 q^{35} + 12 q^{37} - 12 q^{44} + 24 q^{46} + 28 q^{50} + 12 q^{53} - 44 q^{58} + 40 q^{65} + 60 q^{67} + 4 q^{70} + 48 q^{74} - 4 q^{79} + 12 q^{85} - 36 q^{86} - 32 q^{91} - 24 q^{92} + 28 q^{98}+O(q^{100}) $ 12 * q + 4 * q^2 - 8 * q^7 - 4 * q^8 + 8 * q^11 - 8 * q^14 + 16 * q^16 - 16 * q^22 - 20 * q^28 + 4 * q^29 + 16 * q^32 + 4 * q^35 + 12 * q^37 - 12 * q^44 + 24 * q^46 + 28 * q^50 + 12 * q^53 - 44 * q^58 + 40 * q^65 + 60 * q^67 + 4 * q^70 + 48 * q^74 - 4 * q^79 + 12 * q^85 - 36 * q^86 - 32 * q^91 - 24 * q^92 + 28 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/819\mathbb{Z}\right)^\times$.

$n$	$92$	$379$	$703$
$\chi(n)$	$1$	$e\left(\frac{3}{4}\right)$	$-1$

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 $

$2$	1.45161	−	1.45161i	1.02644	−	1.02644i	0.0267996	−	0.999641i	$-0.491468\pi$
$2$	1.45161	−	1.45161i	1.02644	−	1.02644i	0.999641	−	0.0267996i	$-0.00853160\pi$
$3$		0			0
$3$		0			0
$4$		−	2.21432i		−	1.10716i
$5$	2.01464	+	2.01464i	0.900973	+	0.900973i	0.995520	−	0.0945469i	$-0.0301402\pi$
$5$	2.01464	+	2.01464i	0.900973	+	0.900973i	−0.0945469	+	0.995520i	$0.530140\pi$
$6$		0			0
$7$	−1.13594	−	2.38948i	−0.429347	−	0.903140i
$7$	−1.13594	−	2.38948i	−0.429347	−	0.903140i
$8$	−0.311108	−	0.311108i	−0.109993	−	0.109993i
$9$		0			0
$10$	5.84892			1.84959
$11$	−0.451606	−	0.451606i	−0.136164	−	0.136164i	0.635739	−	0.771904i	$-0.280695\pi$
$11$	−0.451606	−	0.451606i	−0.136164	−	0.136164i	−0.771904	+	0.635739i	$0.780695\pi$
$12$		0			0
$13$	3.40251	+	1.19288i	0.943685	+	0.330844i
$13$	3.40251	+	1.19288i	0.943685	+	0.330844i
$14$	−5.11753	−	1.81964i	−1.36772	−	0.486321i
$15$		0			0
$16$	3.52543			0.881357
$17$	4.32672			1.04938			0.524692	−	0.851292i	$-0.324180\pi$
$17$	4.32672			1.04938			0.524692	+	0.851292i	$0.324180\pi$
$18$		0			0
$19$	−3.40251	−	3.40251i	−0.780588	−	0.780588i	0.199342	−	0.979930i	$-0.436120\pi$
$19$	−3.40251	−	3.40251i	−0.780588	−	0.780588i	−0.979930	+	0.199342i	$0.936120\pi$
$20$	4.46105	−	4.46105i	0.997522	−	0.997522i
$21$		0			0
$22$	−1.31111			−0.279529
$23$		−	0.933323i		−	0.194611i	−0.995255	−	0.0973057i	$-0.968978\pi$
$23$		−	0.933323i		−	0.194611i	0.995255	−	0.0973057i	$-0.0310225\pi$
$24$		0			0
$25$			3.11753i			0.623506i
$26$	6.67068	−	3.20751i	1.30823	−	0.629045i
$27$		0			0
$28$	−5.29108	+	2.51534i	−0.999920	+	0.475355i
$29$	−6.33185			−1.17580			−0.587898	−	0.808935i	$-0.700044\pi$
$29$	−6.33185			−1.17580			−0.587898	+	0.808935i	$0.700044\pi$
$30$		0			0
$31$	5.47781	+	5.47781i	0.983843	+	0.983843i	0.999872	−	0.0160282i	$-0.00510215\pi$
$31$	5.47781	+	5.47781i	0.983843	+	0.983843i	−0.0160282	+	0.999872i	$0.505102\pi$
$32$	5.73975	−	5.73975i	1.01465	−	1.01465i
$33$		0			0
$34$	6.28070	−	6.28070i	1.07713	−	1.07713i
$35$	2.52543	−	7.10246i	0.426875	−	1.20053i
$36$		0			0
$37$	2.14050	+	2.14050i	0.351896	+	0.351896i	0.860815	−	0.508919i	$-0.169955\pi$
$37$	2.14050	+	2.14050i	0.351896	+	0.351896i	−0.508919	+	0.860815i	$0.669955\pi$
$38$	−9.87820			−1.60246
$39$		0			0
$40$		−	1.25354i		−	0.198202i
$41$	−1.81964	−	1.81964i	−0.284181	−	0.284181i	0.550593	−	0.834774i	$-0.314402\pi$
$41$	−1.81964	−	1.81964i	−0.284181	−	0.284181i	−0.834774	+	0.550593i	$0.814402\pi$
$42$		0			0
$43$		−	10.4795i		−	1.59811i	−0.601259	−	0.799054i	$-0.705334\pi$
$43$		−	10.4795i		−	1.59811i	0.601259	−	0.799054i	$-0.294666\pi$
$44$	−1.00000	+	1.00000i	−0.150756	+	0.150756i
$45$		0			0
$46$	−1.35482	−	1.35482i	−0.199757	−	0.199757i
$47$	−5.90958	+	5.90958i	−0.862002	+	0.862002i	−0.991570	−	0.129569i	$-0.958641\pi$
$47$	−5.90958	+	5.90958i	−0.862002	+	0.862002i	0.129569	+	0.991570i	$0.458641\pi$
$48$		0			0
$49$	−4.41926	+	5.42864i	−0.631323	+	0.775520i
$50$	4.52543	+	4.52543i	0.639992	+	0.639992i
$51$		0			0
$52$	2.64141	−	7.53424i	0.366297	−	1.04481i
$53$	−3.36196			−0.461801			−0.230901	−	0.972977i	$-0.574167\pi$
$53$	−3.36196			−0.461801			−0.230901	+	0.972977i	$0.574167\pi$
$54$		0			0
$55$		−	1.81964i		−	0.245361i
$56$	−0.389986	+	1.09679i	−0.0521141	+	0.146564i
$57$		0			0
$58$	−9.19135	+	9.19135i	−1.20688	+	1.20688i
$59$	0.255657	−	0.255657i	0.0332837	−	0.0332837i	−0.690269	−	0.723553i	$-0.742508\pi$
$59$	0.255657	−	0.255657i	0.0332837	−	0.0332837i	0.723553	+	0.690269i	$0.242508\pi$
$60$		0			0
$61$			7.78989i			0.997394i	0.866776	+	0.498697i	$0.166188\pi$
$61$			7.78989i			0.997394i	−0.866776	+	0.498697i	$0.833812\pi$
$62$	15.9032			2.01971
$63$		0			0
$64$		−	9.61285i		−	1.20161i
$65$	4.45161	+	9.25803i	0.552154	+	1.14832i
$66$		0			0
$67$	7.28100	−	7.28100i	0.889515	−	0.889515i	−0.104961	−	0.994476i	$-0.533472\pi$
$67$	7.28100	−	7.28100i	0.889515	−	0.889515i	0.994476	+	0.104961i	$0.0334718\pi$
$68$		−	9.58075i		−	1.16184i
$69$		0			0
$70$	−6.64405	−	13.9759i	−0.794116	−	1.67044i
$71$	−5.56914	+	5.56914i	−0.660935	+	0.660935i	−0.955600	−	0.294665i	$-0.904792\pi$
$71$	−5.56914	+	5.56914i	−0.660935	+	0.660935i	0.294665	+	0.955600i	$0.404792\pi$
$72$		0			0
$73$	−8.86144	+	8.86144i	−1.03715	+	1.03715i	−0.0378706	+	0.999283i	$0.512057\pi$
$73$	−8.86144	+	8.86144i	−1.03715	+	1.03715i	−0.999283	+	0.0378706i	$0.987943\pi$
$74$	6.21432			0.722400
$75$		0			0
$76$	−7.53424	+	7.53424i	−0.864236	+	0.864236i
$77$	−0.566106	+	1.59210i	−0.0645137	+	0.181437i
$78$		0			0
$79$	−13.7971			−1.55229			−0.776145	−	0.630554i	$-0.782828\pi$
$79$	−13.7971			−1.55229			−0.776145	+	0.630554i	$0.782828\pi$
$80$	7.10246	+	7.10246i	0.794079	+	0.794079i
$81$		0			0
$82$	−5.28281			−0.583389
$83$	−4.30785	−	4.30785i	−0.472848	−	0.472848i	0.429987	−	0.902835i	$-0.358518\pi$
$83$	−4.30785	−	4.30785i	−0.472848	−	0.472848i	−0.902835	+	0.429987i	$0.858518\pi$
$84$		0			0
$85$	8.71678	+	8.71678i	0.945468	+	0.945468i
$86$	−15.2121	−	15.2121i	−1.64036	−	1.64036i
$87$		0			0
$88$			0.280996i			0.0299543i
$89$	−5.61214	+	5.61214i	−0.594885	+	0.594885i	−0.938947	−	0.344062i	$-0.888197\pi$
$89$	−5.61214	+	5.61214i	−0.594885	+	0.594885i	0.344062	+	0.938947i	$0.388197\pi$
$90$		0			0
$91$	−1.01470	−	9.48527i	−0.106370	−	0.994327i
$92$	−2.06668			−0.215466
$93$		0			0
$94$			17.1568i			1.76959i
$95$		−	13.7096i		−	1.40658i
$96$		0			0
$97$	−0.236784	−	0.236784i	−0.0240417	−	0.0240417i	0.694984	−	0.719025i	$-0.255412\pi$
$97$	−0.236784	−	0.236784i	−0.0240417	−	0.0240417i	−0.719025	+	0.694984i	$0.755412\pi$
$98$	1.46522	+	14.2953i	0.148009	+	1.44404i
$99$		0			0
$100$	6.90321			0.690321
$101$	−9.21955			−0.917380			−0.458690	−	0.888596i	$-0.651681\pi$
$101$	−9.21955			−0.917380			−0.458690	+	0.888596i	$0.651681\pi$
$102$		0			0
$103$	−2.50708			−0.247030			−0.123515	−	0.992343i	$-0.539417\pi$
$103$	−2.50708			−0.247030			−0.123515	+	0.992343i	$0.539417\pi$
$104$	−0.687433	−	1.42966i	−0.0674084	−	0.140190i
$105$		0			0
$106$	−4.88025	+	4.88025i	−0.474011	+	0.474011i
$107$	−2.88247			−0.278659			−0.139329	−	0.990246i	$-0.544495\pi$
$107$	−2.88247			−0.278659			−0.139329	+	0.990246i	$0.544495\pi$
$108$		0			0
$109$	−3.54839	+	3.54839i	−0.339875	+	0.339875i	−0.856320	−	0.516446i	$-0.827255\pi$
$109$	−3.54839	+	3.54839i	−0.339875	+	0.339875i	0.516446	+	0.856320i	$0.327255\pi$
$110$	−2.64141	−	2.64141i	−0.251848	−	0.251848i
$111$		0			0
$112$	−4.00469	−	8.42395i	−0.378408	−	0.795989i
$113$	16.3526			1.53832			0.769161	−	0.639055i	$-0.220674\pi$
$113$	16.3526			1.53832			0.769161	+	0.639055i	$0.220674\pi$
$114$		0			0
$115$	1.88031	−	1.88031i	0.175340	−	0.175340i
$116$			14.0207i			1.30179i
$117$		0			0
$118$		−	0.742226i		−	0.0683274i
$119$	−4.91492	−	10.3386i	−0.450550	−	0.947741i
$120$		0			0
$121$		−	10.5921i		−	0.962919i
$122$	11.3079	+	11.3079i	1.02377	+	1.02377i
$123$		0			0
$124$	12.1296	−	12.1296i	1.08927	−	1.08927i
$125$	3.79249	−	3.79249i	0.339211	−	0.339211i
$126$		0			0
$127$			13.8272i			1.22696i	0.789709	+	0.613481i	$0.210231\pi$
$127$			13.8272i			1.22696i	−0.789709	+	0.613481i	$0.789769\pi$
$128$	−2.47457	−	2.47457i	−0.218723	−	0.218723i
$129$		0			0
$130$	19.9010	+	6.97703i	1.74543	+	0.611926i
$131$			12.8301i			1.12097i	0.828166	+	0.560483i	$0.189385\pi$
$131$			12.8301i			1.12097i	−0.828166	+	0.560483i	$0.810615\pi$
$132$		0			0
$133$	−4.26517	+	11.9953i	−0.369838	+	1.04012i
$134$		−	21.1383i		−	1.82607i
$135$		0			0
$136$	−1.34608	−	1.34608i	−0.115425	−	0.115425i
$137$	−2.40075	−	2.40075i	−0.205110	−	0.205110i	0.597075	−	0.802185i	$-0.296329\pi$
$137$	−2.40075	−	2.40075i	−0.205110	−	0.205110i	−0.802185	+	0.597075i	$0.796329\pi$
$138$		0			0
$139$			3.34184i			0.283451i	0.989906	+	0.141726i	$0.0452651\pi$
$139$			3.34184i			0.283451i	−0.989906	+	0.141726i	$0.954735\pi$
$140$	−15.7271	−	5.59210i	−1.32918	−	0.472619i
$141$		0			0
$142$			16.1684i			1.35682i
$143$	−0.997882	−	2.07530i	−0.0834471	−	0.173545i
$144$		0			0
$145$	−12.7564	−	12.7564i	−1.05936	−	1.05936i
$146$			25.7266i			2.12915i
$147$		0			0
$148$	4.73975	−	4.73975i	0.389605	−	0.389605i
$149$	−2.62936	+	2.62936i	−0.215406	+	0.215406i	−0.806559	−	0.591153i	$-0.798673\pi$
$149$	−2.62936	+	2.62936i	−0.215406	+	0.215406i	0.591153	+	0.806559i	$0.298673\pi$
$150$		0			0
$151$	−4.78346	−	4.78346i	−0.389272	−	0.389272i	0.485156	−	0.874428i	$-0.338763\pi$
$151$	−4.78346	−	4.78346i	−0.389272	−	0.389272i	−0.874428	+	0.485156i	$0.838763\pi$
$152$			2.11709i			0.171719i
$153$		0			0
$154$	1.48935	+	3.13287i	0.120015	+	0.252454i
$155$			22.0716i			1.77283i
$156$		0			0
$157$			3.42542i			0.273379i	0.990614	+	0.136689i	$0.0436462\pi$
$157$			3.42542i			0.273379i	−0.990614	+	0.136689i	$0.956354\pi$
$158$	−20.0279	+	20.0279i	−1.59333	+	1.59333i
$159$		0			0
$160$	23.1270			1.82835
$161$	−2.23016	+	1.06020i	−0.175761	+	0.0835557i
$162$		0			0
$163$	10.4494	+	10.4494i	0.818459	+	0.818459i	0.985885	−	0.167426i	$-0.0535455\pi$
$163$	10.4494	+	10.4494i	0.818459	+	0.818459i	−0.167426	+	0.985885i	$0.553545\pi$
$164$	−4.02928	+	4.02928i	−0.314634	+	0.314634i
$165$		0			0
$166$	−12.5066			−0.970701
$167$	16.2326	−	16.2326i	1.25611	−	1.25611i	0.303180	−	0.952933i	$-0.401952\pi$
$167$	16.2326	−	16.2326i	1.25611	−	1.25611i	0.952933	−	0.303180i	$-0.0980483\pi$
$168$		0			0
$169$	10.1541	+	8.11753i	0.781084	+	0.624426i
$170$	25.3067			1.94093
$171$		0			0
$172$	−23.2050			−1.76936
$173$	−14.3810			−1.09337			−0.546685	−	0.837338i	$-0.684111\pi$
$173$	−14.3810			−1.09337			−0.546685	+	0.837338i	$0.684111\pi$
$174$		0			0
$175$	7.44929	−	3.54134i	0.563113	−	0.267700i
$176$	−1.59210	−	1.59210i	−0.120009	−	0.120009i
$177$		0			0
$178$			16.2932i			1.22123i
$179$		−	4.11108i		−	0.307276i	−0.988127	−	0.153638i	$-0.950901\pi$
$179$		−	4.11108i		−	0.307276i	0.988127	−	0.153638i	$-0.0490990\pi$
$180$		0			0
$181$	21.5760			1.60373			0.801867	−	0.597502i	$-0.203840\pi$
$181$	21.5760			1.60373			0.801867	+	0.597502i	$0.203840\pi$
$182$	−15.2418	−	12.2959i	−1.12980	−	0.911435i
$183$		0			0
$184$	−0.290364	+	0.290364i	−0.0214059	+	0.0214059i
$185$			8.62466i			0.634097i
$186$		0			0
$187$	−1.95397	−	1.95397i	−0.142889	−	0.142889i
$188$	13.0857	+	13.0857i	0.954373	+	0.954373i
$189$		0			0
$190$	−19.9010	−	19.9010i	−1.44377	−	1.44377i
$191$	−10.3017			−0.745408			−0.372704	−	0.927950i	$-0.621569\pi$
$191$	−10.3017			−0.745408			−0.372704	+	0.927950i	$0.621569\pi$
$192$		0			0
$193$	6.28592	+	6.28592i	0.452470	+	0.452470i	0.896174	−	0.443703i	$-0.146336\pi$
$193$	6.28592	+	6.28592i	0.452470	+	0.452470i	−0.443703	+	0.896174i	$0.646336\pi$
$194$	−0.687433			−0.0493548
$195$		0			0
$196$	12.0207	+	9.78566i	0.858625	+	0.698976i
$197$	3.25088	−	3.25088i	0.231616	−	0.231616i	−0.581751	−	0.813367i	$-0.697632\pi$
$197$	3.25088	−	3.25088i	0.231616	−	0.231616i	0.813367	+	0.581751i	$0.197632\pi$
$198$		0			0
$199$	−6.20116			−0.439589			−0.219794	−	0.975546i	$-0.570539\pi$
$199$	−6.20116			−0.439589			−0.219794	+	0.975546i	$0.570539\pi$
$200$	0.969888	−	0.969888i	0.0685815	−	0.0685815i
$201$		0			0
$202$	−13.3832	+	13.3832i	−0.941636	+	0.941636i
$203$	7.19263	+	15.1299i	0.504824	+	1.06191i
$204$		0			0
$205$		−	7.33185i		−	0.512079i
$206$	−3.63929	+	3.63929i	−0.253561	+	0.253561i
$207$		0			0
$208$	11.9953	+	4.20540i	0.831724	+	0.291592i
$209$			3.07318i			0.212577i
$210$		0			0
$211$	−9.55554			−0.657830			−0.328915	−	0.944359i	$-0.606683\pi$
$211$	−9.55554			−0.657830			−0.328915	+	0.944359i	$0.606683\pi$
$212$			7.44446i			0.511288i
$213$		0			0
$214$	−4.18421	+	4.18421i	−0.286027	+	0.286027i
$215$	21.1124	−	21.1124i	1.43985	−	1.43985i
$216$		0			0
$217$	6.86665	−	19.3116i	0.466138	−	1.31096i
$218$			10.3017i			0.697722i
$219$		0			0
$220$	−4.02928			−0.271654
$221$	14.7217	+	5.16124i	0.990289	+	0.347183i
$222$		0			0
$223$	−2.58074	−	2.58074i	−0.172819	−	0.172819i	0.615398	−	0.788217i	$-0.288995\pi$
$223$	−2.58074	−	2.58074i	−0.172819	−	0.172819i	−0.788217	+	0.615398i	$0.788995\pi$
$224$	−20.2351	−	7.19500i	−1.35201	−	0.480736i
$225$		0			0
$226$	23.7375	−	23.7375i	1.57900	−	1.57900i
$227$	−5.25403	−	5.25403i	−0.348722	−	0.348722i	0.510911	−	0.859633i	$-0.329308\pi$
$227$	−5.25403	−	5.25403i	−0.348722	−	0.348722i	−0.859633	+	0.510911i	$0.829308\pi$
$228$		0			0
$229$	−0.729224	+	0.729224i	−0.0481885	+	0.0481885i	−0.730790	−	0.682602i	$-0.760848\pi$
$229$	−0.729224	+	0.729224i	−0.0481885	+	0.0481885i	0.682602	+	0.730790i	$0.260848\pi$
$230$		−	5.45893i		−	0.359952i
$231$		0			0
$232$	1.96989	+	1.96989i	0.129330	+	0.129330i
$233$		−	24.6128i		−	1.61244i	−0.591615	−	0.806221i	$-0.701509\pi$
$233$		−	24.6128i		−	1.61244i	0.591615	−	0.806221i	$-0.298491\pi$
$234$		0			0
$235$	−23.8113			−1.55328
$236$	−0.566106	−	0.566106i	−0.0368503	−	0.0368503i
$237$		0			0
$238$	−22.1421	−	7.87310i	−1.43526	−	0.510337i
$239$	6.52543	−	6.52543i	0.422095	−	0.422095i	−0.463830	−	0.885924i	$-0.653525\pi$
$239$	6.52543	−	6.52543i	0.422095	−	0.422095i	0.885924	+	0.463830i	$0.153525\pi$
$240$		0			0
$241$	20.1563	−	20.1563i	1.29838	−	1.29838i	0.368920	−	0.929461i	$-0.379728\pi$
$241$	20.1563	−	20.1563i	1.29838	−	1.29838i	0.929461	−	0.368920i	$-0.120272\pi$
$242$	−15.3756	−	15.3756i	−0.988379	−	0.988379i
$243$		0			0
$244$	17.2493			1.10427
$245$	−19.8400	+	2.03353i	−1.26753	+	0.129918i
$246$		0			0
$247$	−7.51828	−	15.6358i	−0.478377	−	0.994883i
$248$		−	3.40838i		−	0.216432i
$249$		0			0
$250$		−	11.0104i		−	0.696359i
$251$	14.4448			0.911747			0.455873	−	0.890045i	$-0.349327\pi$
$251$	14.4448			0.911747			0.455873	+	0.890045i	$0.349327\pi$
$252$		0			0
$253$	−0.421494	+	0.421494i	−0.0264991	+	0.0264991i
$254$	20.0716	+	20.0716i	1.25940	+	1.25940i
$255$		0			0
$256$	12.0415			0.752593
$257$	−14.6237			−0.912201			−0.456101	−	0.889928i	$-0.650754\pi$
$257$	−14.6237			−0.912201			−0.456101	+	0.889928i	$0.650754\pi$
$258$		0			0
$259$	2.68320	−	7.54617i	0.166726	−	0.468896i
$260$	20.5002	−	9.85728i	1.27137	−	0.611322i
$261$		0			0
$262$	18.6242	+	18.6242i	1.15061	+	1.15061i
$263$	2.61729			0.161389			0.0806946	−	0.996739i	$-0.474286\pi$
$263$	2.61729			0.161389			0.0806946	+	0.996739i	$0.474286\pi$
$264$		0			0
$265$	−6.77314	−	6.77314i	−0.416071	−	0.416071i
$266$	11.2211	+	23.6038i	0.688009	+	1.44724i
$267$		0			0
$268$	−16.1225	−	16.1225i	−0.984836	−	0.984836i
$269$		−	9.82340i		−	0.598944i	−0.954105	−	0.299472i	$-0.903190\pi$
$269$		−	9.82340i		−	0.598944i	0.954105	−	0.299472i	$-0.0968104\pi$
$270$		0			0
$271$	−3.43438	+	3.43438i	−0.208624	+	0.208624i	−0.803682	−	0.595059i	$-0.797129\pi$
$271$	−3.43438	+	3.43438i	−0.208624	+	0.208624i	0.595059	+	0.803682i	$0.297129\pi$
$272$	15.2535			0.924882
$273$		0			0
$274$	−6.96989			−0.421066
$275$	1.40790	−	1.40790i	0.0848993	−	0.0848993i
$276$		0			0
$277$		−	5.89877i		−	0.354423i	−0.984173	−	0.177211i	$-0.943292\pi$
$277$		−	5.89877i		−	0.354423i	0.984173	−	0.177211i	$-0.0567076\pi$
$278$	4.85104	+	4.85104i	0.290946	+	0.290946i
$279$		0			0
$280$	−2.99531	+	1.42395i	−0.179004	+	0.0850973i
$281$	−9.23729	−	9.23729i	−0.551050	−	0.551050i	0.375694	−	0.926744i	$-0.377405\pi$
$281$	−9.23729	−	9.23729i	−0.551050	−	0.551050i	−0.926744	+	0.375694i	$0.877405\pi$
$282$		0			0
$283$	8.32721			0.495001			0.247501	−	0.968888i	$-0.420391\pi$
$283$	8.32721			0.495001			0.247501	+	0.968888i	$0.420391\pi$
$284$	12.3319	+	12.3319i	0.731761	+	0.731761i
$285$		0			0
$286$	−4.46105	−	1.56399i	−0.263788	−	0.0924806i
$287$	−2.28100	+	6.41503i	−0.134643	+	0.378667i
$288$		0			0
$289$	1.72054			0.101208
$290$	−37.0345			−2.17474
$291$		0			0
$292$	19.6221	+	19.6221i	1.14829	+	1.14829i
$293$	−15.4903	+	15.4903i	−0.904955	+	0.904955i	−0.995860	−	0.0909047i	$-0.971024\pi$
$293$	−15.4903	+	15.4903i	−0.904955	+	0.904955i	0.0909047	+	0.995860i	$0.471024\pi$
$294$		0			0
$295$	1.03011			0.0599754
$296$		−	1.33185i		−	0.0774123i
$297$		0			0
$298$			7.63359i			0.442202i
$299$	1.11334	−	3.17564i	0.0643860	−	0.183652i
$300$		0			0
$301$	−25.0406	+	11.9041i	−1.44331	+	0.686142i
$302$	−13.8874			−0.799130
$303$		0			0
$304$	−11.9953	−	11.9953i	−0.687977	−	0.687977i
$305$	−15.6938	+	15.6938i	−0.898625	+	0.898625i
$306$		0			0
$307$	−5.77526	+	5.77526i	−0.329611	+	0.329611i	−0.852439	−	0.522827i	$-0.824877\pi$
$307$	−5.77526	+	5.77526i	−0.329611	+	0.329611i	0.522827	+	0.852439i	$0.324877\pi$
$308$	3.52543	+	1.25354i	0.200880	+	0.0714270i
$309$		0			0
$310$	32.0393	+	32.0393i	1.81971	+	1.81971i
$311$	−2.62562			−0.148885			−0.0744426	−	0.997225i	$-0.523718\pi$
$311$	−2.62562			−0.148885			−0.0744426	+	0.997225i	$0.523718\pi$
$312$		0			0
$313$			19.8489i			1.12193i	0.827840	+	0.560964i	$0.189569\pi$
$313$			19.8489i			1.12193i	−0.827840	+	0.560964i	$0.810431\pi$
$314$	4.97237	+	4.97237i	0.280607	+	0.280607i
$315$		0			0
$316$			30.5511i			1.71863i
$317$	−1.14764	+	1.14764i	−0.0644581	+	0.0644581i	−0.738601	−	0.674143i	$-0.764513\pi$
$317$	−1.14764	+	1.14764i	−0.0644581	+	0.0644581i	0.674143	+	0.738601i	$0.264513\pi$
$318$		0			0
$319$	2.85950	+	2.85950i	0.160101	+	0.160101i
$320$	19.3664	−	19.3664i	1.08262	−	1.08262i
$321$		0			0
$322$	−1.69832	+	4.77631i	−0.0946435	+	0.266173i
$323$	−14.7217	−	14.7217i	−0.819137	−	0.819137i
$324$		0			0
$325$	−3.71883	+	10.6074i	−0.206283	+	0.588394i
$326$	30.3368			1.68020
$327$		0			0
$328$			1.13221i			0.0625159i
$329$	20.8338	+	7.40790i	1.14861	+	0.408411i
$330$		0			0
$331$	−2.90321	+	2.90321i	−0.159575	+	0.159575i	−0.782378	−	0.622803i	$-0.785994\pi$
$331$	−2.90321	+	2.90321i	−0.159575	+	0.159575i	0.622803	+	0.782378i	$0.285994\pi$
$332$	−9.53896	+	9.53896i	−0.523518	+	0.523518i
$333$		0			0
$334$		−	47.1266i		−	2.57865i
$335$	29.3371			1.60286
$336$		0			0
$337$			4.47304i			0.243662i	0.992551	+	0.121831i	$0.0388766\pi$
$337$			4.47304i			0.243662i	−0.992551	+	0.121831i	$0.961123\pi$
$338$	26.5232	−	2.95629i	1.44267	−	0.160801i
$339$		0			0
$340$	19.3017	−	19.3017i	1.04678	−	1.04678i
$341$		−	4.94762i		−	0.267929i
$342$		0			0
$343$	17.9917	+	4.39312i	0.971459	+	0.237206i
$344$	−3.26025	+	3.26025i	−0.175781	+	0.175781i
$345$		0			0
$346$	−20.8756	+	20.8756i	−1.12228	+	1.12228i
$347$	−8.81135			−0.473018			−0.236509	−	0.971629i	$-0.576003\pi$
$347$	−8.81135			−0.473018			−0.236509	+	0.971629i	$0.576003\pi$
$348$		0			0
$349$	11.8133	−	11.8133i	0.632351	−	0.632351i	−0.316306	−	0.948657i	$-0.602443\pi$
$349$	11.8133	−	11.8133i	0.632351	−	0.632351i	0.948657	+	0.316306i	$0.102443\pi$
$350$	5.67280	−	15.9541i	0.303224	−	0.852781i
$351$		0			0
$352$	−5.18421			−0.276319
$353$	−2.91007	−	2.91007i	−0.154887	−	0.154887i	0.625410	−	0.780297i	$-0.284932\pi$
$353$	−2.91007	−	2.91007i	−0.154887	−	0.154887i	−0.780297	+	0.625410i	$0.784932\pi$
$354$		0			0
$355$	−22.4396			−1.19097
$356$	12.4271	+	12.4271i	0.658633	+	0.658633i
$357$		0			0
$358$	−5.96767	−	5.96767i	−0.315401	−	0.315401i
$359$	−3.52320	−	3.52320i	−0.185948	−	0.185948i	0.607994	−	0.793942i	$-0.291974\pi$
$359$	−3.52320	−	3.52320i	−0.185948	−	0.185948i	−0.793942	+	0.607994i	$0.791974\pi$
$360$		0			0
$361$			4.15410i			0.218637i
$362$	31.3199	−	31.3199i	1.64614	−	1.64614i
$363$		0			0
$364$	−21.0034	+	2.24687i	−1.10088	+	0.117768i
$365$	−35.7052			−1.86890
$366$		0			0
$367$		−	19.8112i		−	1.03414i	−0.855944	−	0.517068i	$-0.827024\pi$
$367$		−	19.8112i		−	1.03414i	0.855944	−	0.517068i	$-0.172976\pi$
$368$		−	3.29036i		−	0.171522i
$369$		0			0
$370$	12.5196	+	12.5196i	0.650863	+	0.650863i
$371$	3.81900	+	8.03335i	0.198273	+	0.417071i
$372$		0			0
$373$	−24.5368			−1.27047			−0.635234	−	0.772320i	$-0.719096\pi$
$373$	−24.5368			−1.27047			−0.635234	+	0.772320i	$0.719096\pi$
$374$	−5.67280			−0.293334
$375$		0			0
$376$	3.67704			0.189629
$377$	−21.5442	−	7.55311i	−1.10958	−	0.389005i
$378$		0			0
$379$	−8.78346	+	8.78346i	−0.451176	+	0.451176i	−0.895745	−	0.444569i	$-0.853357\pi$
$379$	−8.78346	+	8.78346i	−0.451176	+	0.451176i	0.444569	+	0.895745i	$0.353357\pi$
$380$	−30.3575			−1.55731
$381$		0			0
$382$	−14.9541	+	14.9541i	−0.765117	+	0.765117i
$383$	−17.8990	−	17.8990i	−0.914596	−	0.914596i	0.0820333	−	0.996630i	$-0.473859\pi$
$383$	−17.8990	−	17.8990i	−0.914596	−	0.914596i	−0.996630	+	0.0820333i	$0.973859\pi$
$384$		0			0
$385$	−4.34801	+	2.06702i	−0.221595	+	0.105345i
$386$	18.2494			0.928868
$387$		0			0
$388$	−0.524315	+	0.524315i	−0.0266181	+	0.0266181i
$389$			12.0667i			0.611805i	0.952063	+	0.305902i	$0.0989581\pi$
$389$			12.0667i			0.611805i	−0.952063	+	0.305902i	$0.901042\pi$
$390$		0			0
$391$		−	4.03823i		−	0.204222i
$392$	3.06376	−	0.314025i	0.154743	−	0.0158607i
$393$		0			0
$394$		−	9.43801i		−	0.475480i
$395$	−27.7961	−	27.7961i	−1.39857	−	1.39857i
$396$		0			0
$397$	−7.82177	+	7.82177i	−0.392563	+	0.392563i	−0.875600	−	0.483037i	$-0.839534\pi$
$397$	−7.82177	+	7.82177i	−0.392563	+	0.392563i	0.483037	+	0.875600i	$0.339534\pi$
$398$	−9.00164	+	9.00164i	−0.451212	+	0.451212i
$399$		0			0
$400$			10.9906i			0.549532i
$401$	17.6178	+	17.6178i	0.879789	+	0.879789i	0.993512	−	0.113723i	$-0.0362777\pi$
$401$	17.6178	+	17.6178i	0.879789	+	0.879789i	−0.113723	+	0.993512i	$0.536278\pi$
$402$		0			0
$403$	12.1039	+	25.1726i	0.602940	+	1.25394i
$404$			20.4150i			1.01569i
$405$		0			0
$406$	32.4035	+	11.5217i	1.60816	+	0.571813i
$407$		−	1.93332i		−	0.0958313i
$408$		0			0
$409$	−24.0225	−	24.0225i	−1.18783	−	1.18783i	−0.977665	−	0.210169i	$-0.932599\pi$
$409$	−24.0225	−	24.0225i	−1.18783	−	1.18783i	−0.210169	−	0.977665i	$-0.567401\pi$
$410$	−10.6430	−	10.6430i	−0.525618	−	0.525618i
$411$		0			0
$412$			5.55147i			0.273501i
$413$	−0.901299	−	0.320476i	−0.0443500	−	0.0157696i
$414$		0			0
$415$		−	17.3575i		−	0.852047i
$416$	26.3763	−	12.6827i	1.29321	−	0.621822i
$417$		0			0
$418$	4.46105	+	4.46105i	0.218197	+	0.218197i
$419$		−	19.6899i		−	0.961912i	−0.876745	−	0.480956i	$-0.840290\pi$
$419$		−	19.6899i		−	0.961912i	0.876745	−	0.480956i	$-0.159710\pi$
$420$		0			0
$421$	26.6042	−	26.6042i	1.29661	−	1.29661i	0.365989	−	0.930619i	$-0.380731\pi$
$421$	26.6042	−	26.6042i	1.29661	−	1.29661i	0.930619	−	0.365989i	$-0.119269\pi$
$422$	−13.8709	+	13.8709i	−0.675224	+	0.675224i
$423$		0			0
$424$	1.04593	+	1.04593i	0.0507950	+	0.0507950i
$425$			13.4887i			0.654298i
$426$		0			0
$427$	18.6138	−	8.84888i	0.900786	−	0.428228i
$428$			6.38271i			0.308520i
$429$		0			0
$430$		−	61.2937i		−	2.95585i
$431$	4.81135	−	4.81135i	0.231754	−	0.231754i	−0.581670	−	0.813425i	$-0.697601\pi$
$431$	4.81135	−	4.81135i	0.231754	−	0.231754i	0.813425	+	0.581670i	$0.197601\pi$
$432$		0			0
$433$	−8.34704			−0.401133			−0.200567	−	0.979680i	$-0.564278\pi$
$433$	−8.34704			−0.401133			−0.200567	+	0.979680i	$0.564278\pi$
$434$	−18.0652	−	38.0005i	−0.867157	−	1.82408i
$435$		0			0
$436$	7.85728	+	7.85728i	0.376295	+	0.376295i
$437$	−3.17564	+	3.17564i	−0.151911	+	0.151911i
$438$		0			0
$439$	−2.42350			−0.115667			−0.0578336	−	0.998326i	$-0.518419\pi$
$439$	−2.42350			−0.115667			−0.0578336	+	0.998326i	$0.518419\pi$
$440$	−0.566106	+	0.566106i	−0.0269880	+	0.0269880i
$441$		0			0
$442$	28.8622	−	13.8780i	1.37283	−	0.660110i
$443$	−13.7763			−0.654532			−0.327266	−	0.944932i	$-0.606127\pi$
$443$	−13.7763			−0.654532			−0.327266	+	0.944932i	$0.606127\pi$
$444$		0			0
$445$	−22.6128			−1.07195
$446$	−7.49245			−0.354778
$447$		0			0
$448$	−22.9697	+	10.9197i	−1.08522	+	0.515905i
$449$	24.7447	+	24.7447i	1.16777	+	1.16777i	0.982730	+	0.185043i	$0.0592423\pi$
$449$	24.7447	+	24.7447i	1.16777	+	1.16777i	0.185043	+	0.982730i	$0.440758\pi$
$450$		0			0
$451$			1.64353i			0.0773906i
$452$		−	36.2099i		−	1.70317i
$453$		0			0
$454$	−15.2535			−0.715885
$455$	17.0651	−	21.1536i	0.800026	−	0.991698i
$456$		0			0
$457$	−2.84521	+	2.84521i	−0.133093	+	0.133093i	−0.770515	−	0.637422i	$-0.780001\pi$
$457$	−2.84521	+	2.84521i	−0.133093	+	0.133093i	0.637422	+	0.770515i	$0.280001\pi$
$458$			2.11709i			0.0989252i
$459$		0			0
$460$	−4.16360	−	4.16360i	−0.194129	−	0.194129i
$461$	2.38575	+	2.38575i	0.111115	+	0.111115i	0.760479	−	0.649363i	$-0.224964\pi$
$461$	2.38575	+	2.38575i	0.111115	+	0.111115i	−0.649363	+	0.760479i	$0.724964\pi$
$462$		0			0
$463$	17.5899	+	17.5899i	0.817471	+	0.817471i	0.985741	−	0.168270i	$-0.0538180\pi$
$463$	17.5899	+	17.5899i	0.817471	+	0.817471i	−0.168270	+	0.985741i	$0.553818\pi$
$464$	−22.3225			−1.03630
$465$		0			0
$466$	−35.7282	−	35.7282i	−1.65507	−	1.65507i
$467$	21.4287			0.991602			0.495801	−	0.868436i	$-0.334874\pi$
$467$	21.4287			0.991602			0.495801	+	0.868436i	$0.334874\pi$
$468$		0			0
$469$	−25.6686	−	9.12701i	−1.18527	−	0.421446i
$470$	−34.5647	+	34.5647i	−1.59435	+	1.59435i
$471$		0			0
$472$	−0.159074			−0.00732196
$473$	−4.73260	+	4.73260i	−0.217605	+	0.217605i
$474$		0			0
$475$	10.6074	−	10.6074i	0.486702	−	0.486702i
$476$	−22.8930	+	10.8832i	−1.04930	+	0.498830i
$477$		0			0
$478$		−	18.9447i		−	0.866510i
$479$	−4.61425	+	4.61425i	−0.210831	+	0.210831i	−0.804620	−	0.593790i	$-0.797631\pi$
$479$	−4.61425	+	4.61425i	−0.210831	+	0.210831i	0.593790	+	0.804620i	$0.297631\pi$
$480$		0			0
$481$	4.72971	+	9.83641i	0.215656	+	0.448501i
$482$		−	58.5180i		−	2.66542i
$483$		0			0
$484$	−23.4543			−1.06610
$485$		−	0.954067i		−	0.0433220i
$486$		0			0
$487$	9.62867	−	9.62867i	0.436317	−	0.436317i	−0.454454	−	0.890770i	$-0.650166\pi$
$487$	9.62867	−	9.62867i	0.436317	−	0.436317i	0.890770	+	0.454454i	$0.150166\pi$
$488$	2.42350	−	2.42350i	0.109707	−	0.109707i
$489$		0			0
$490$	−25.8479	+	31.7517i	−1.16769	+	1.43439i
$491$		−	9.21924i		−	0.416059i	−0.978123	−	0.208029i	$-0.933295\pi$
$491$		−	9.21924i		−	0.416059i	0.978123	−	0.208029i	$-0.0667049\pi$
$492$		0			0
$493$	−27.3962			−1.23386
$494$	−33.6106	−	11.7835i	−1.51221	−	0.530163i
$495$		0			0
$496$	19.3116	+	19.3116i	0.867117	+	0.867117i
$497$	19.6336	+	6.98113i	0.880687	+	0.313147i
$498$		0			0
$499$	−6.65947	+	6.65947i	−0.298119	+	0.298119i	−0.840277	−	0.542158i	$-0.817608\pi$
$499$	−6.65947	+	6.65947i	−0.298119	+	0.298119i	0.542158	+	0.840277i	$0.317608\pi$
$500$	−8.39779	−	8.39779i	−0.375561	−	0.375561i
$501$		0			0
$502$	20.9681	−	20.9681i	0.935854	−	0.935854i
$503$		−	1.81069i		−	0.0807346i	−0.999185	−	0.0403673i	$-0.987147\pi$
$503$		−	1.81069i		−	0.0807346i	0.999185	−	0.0403673i	$-0.0128528\pi$
$504$		0			0
$505$	−18.5741	−	18.5741i	−0.826535	−	0.826535i
$506$			1.22369i			0.0543996i
$507$		0			0
$508$	30.6178			1.35844
$509$	22.2864	+	22.2864i	0.987827	+	0.987827i	0.999927	−	0.0121000i	$-0.00385163\pi$
$509$	22.2864	+	22.2864i	0.987827	+	0.987827i	−0.0121000	+	0.999927i	$0.503852\pi$
$510$		0			0
$511$	31.2404	+	11.1082i	1.38199	+	0.491396i
$512$	22.4286	−	22.4286i	0.991215	−	0.991215i
$513$		0			0
$514$	−21.2278	+	21.2278i	−0.936320	+	0.936320i
$515$	−5.05086	−	5.05086i	−0.222567	−	0.222567i
$516$		0			0
$517$	5.33761			0.234748
$518$	−7.05912	−	14.8490i	−0.310160	−	0.652428i
$519$		0			0
$520$	1.49532	−	4.26517i	0.0655739	−	0.187040i
$521$		−	15.8744i		−	0.695472i	−0.937592	−	0.347736i	$-0.886951\pi$
$521$		−	15.8744i		−	0.695472i	0.937592	−	0.347736i	$-0.113049\pi$
$522$		0			0
$523$			3.90795i			0.170883i	0.996343	+	0.0854413i	$0.0272300\pi$
$523$			3.90795i			0.170883i	−0.996343	+	0.0854413i	$0.972770\pi$
$524$	28.4098			1.24109
$525$		0			0
$526$	3.79928	−	3.79928i	0.165656	−	0.165656i
$527$	23.7010	+	23.7010i	1.03243	+	1.03243i
$528$		0			0
$529$	22.1289			0.962126
$530$	−19.6639			−0.854143
$531$		0			0
$532$	26.5614	+	9.44446i	1.15158	+	0.409469i
$533$	−4.02074	−	8.36196i	−0.174158	−	0.362197i
$534$		0			0
$535$	−5.80713	−	5.80713i	−0.251064	−	0.251064i
$536$	−4.53035			−0.195681
$537$		0			0
$538$	−14.2597	−	14.2597i	−0.614780	−	0.614780i
$539$	4.44737	−	0.455841i	0.191562	−	0.0196345i
$540$		0			0
$541$	8.57406	+	8.57406i	0.368628	+	0.368628i	0.866977	−	0.498349i	$-0.166060\pi$
$541$	8.57406	+	8.57406i	0.368628	+	0.368628i	−0.498349	+	0.866977i	$0.666060\pi$
$542$			9.97073i			0.428280i
$543$		0			0
$544$	24.8343	−	24.8343i	1.06476	−	1.06476i
$545$	−14.2975			−0.612436
$546$		0			0
$547$	−4.72546			−0.202046			−0.101023	−	0.994884i	$-0.532212\pi$
$547$	−4.72546			−0.202046			−0.101023	+	0.994884i	$0.532212\pi$
$548$	−5.31603	+	5.31603i	−0.227090	+	0.227090i
$549$		0			0
$550$		−	4.08742i		−	0.174288i
$551$	21.5442	+	21.5442i	0.917812	+	0.917812i
$552$		0			0
$553$	15.6727	+	32.9678i	0.666470	+	1.40193i
$554$	−8.56268	−	8.56268i	−0.363794	−	0.363794i
$555$		0			0
$556$	7.39991			0.313826
$557$	23.8415	+	23.8415i	1.01019	+	1.01019i	0.999947	+	0.0102475i	$0.00326193\pi$
$557$	23.8415	+	23.8415i	1.01019	+	1.01019i	0.0102475	+	0.999947i	$0.496738\pi$
$558$		0			0
$559$	12.5007	−	35.6565i	0.528725	−	1.50811i
$560$	8.90321	−	25.0392i	0.376229	−	1.05810i
$561$		0			0
$562$	−26.8178			−1.13124
$563$	41.2776			1.73965			0.869823	−	0.493365i	$-0.164233\pi$
$563$	41.2776			1.73965			0.869823	+	0.493365i	$0.164233\pi$
$564$		0			0
$565$	32.9446	+	32.9446i	1.38599	+	1.38599i
$566$	12.0878	−	12.0878i	0.508089	−	0.508089i
$567$		0			0
$568$	3.46520			0.145397
$569$		−	43.5402i		−	1.82530i	−0.408742	−	0.912650i	$-0.634032\pi$
$569$		−	43.5402i		−	1.82530i	0.408742	−	0.912650i	$-0.365968\pi$
$570$		0			0
$571$		−	21.2701i		−	0.890126i	−0.895499	−	0.445063i	$-0.853181\pi$
$571$		−	21.2701i		−	0.890126i	0.895499	−	0.445063i	$-0.146819\pi$
$572$	−4.59538	+	2.20963i	−0.192143	+	0.0923893i
$573$		0			0
$574$	6.00098	+	12.6232i	0.250476	+	0.526882i
$575$	2.90967			0.121341
$576$		0			0
$577$	8.73703	+	8.73703i	0.363727	+	0.363727i	0.865183	−	0.501456i	$-0.167202\pi$
$577$	8.73703	+	8.73703i	0.363727	+	0.363727i	−0.501456	+	0.865183i	$0.667202\pi$
$578$	2.49754	−	2.49754i	0.103884	−	0.103884i
$579$		0			0
$580$	−28.2467	+	28.2467i	−1.17288	+	1.17288i
$581$	−5.40006	+	15.1870i	−0.224032	+	0.630064i
$582$		0			0
$583$	1.51828	+	1.51828i	0.0628808	+	0.0628808i
$584$	5.51373			0.228160
$585$		0			0
$586$			44.9717i			1.85776i
$587$	30.6931	+	30.6931i	1.26684	+	1.26684i	0.947711	+	0.319131i	$0.103391\pi$
$587$	30.6931	+	30.6931i	1.26684	+	1.26684i	0.319131	+	0.947711i	$0.396609\pi$
$588$		0			0
$589$		−	37.2766i		−	1.53595i
$590$	1.49532	−	1.49532i	0.0615612	−	0.0615612i
$591$		0			0
$592$	7.54617	+	7.54617i	0.310146	+	0.310146i
$593$	5.18036	−	5.18036i	0.212732	−	0.212732i	−0.592695	−	0.805427i	$-0.701936\pi$
$593$	5.18036	−	5.18036i	0.212732	−	0.212732i	0.805427	+	0.592695i	$0.201936\pi$
$594$		0			0
$595$	10.9268	−	30.7304i	0.447956	−	1.25982i
$596$	5.82225	+	5.82225i	0.238488	+	0.238488i
$597$		0			0
$598$	−2.99365	−	6.22591i	−0.122419	−	0.254596i
$599$	18.5254			0.756928			0.378464	−	0.925616i	$-0.376452\pi$
$599$	18.5254			0.756928			0.378464	+	0.925616i	$0.376452\pi$
$600$		0			0
$601$		−	30.9807i		−	1.26373i	−0.775079	−	0.631864i	$-0.782290\pi$
$601$		−	30.9807i		−	1.26373i	0.775079	−	0.631864i	$-0.217710\pi$
$602$	−19.0690	+	53.6291i	−0.777193	+	2.18576i
$603$		0			0
$604$	−10.5921	+	10.5921i	−0.430987	+	0.430987i
$605$	21.3393	−	21.3393i	0.867564	−	0.867564i
$606$		0			0
$607$			29.4674i			1.19605i	0.801479	+	0.598023i	$0.204047\pi$
$607$			29.4674i			1.19605i	−0.801479	+	0.598023i	$0.795953\pi$
$608$	−39.0591			−1.58405
$609$		0			0
$610$			45.5625i			1.84477i
$611$	−27.1568	+	13.0580i	−1.09865	+	0.528270i
$612$		0			0
$613$	13.9398	−	13.9398i	0.563022	−	0.563022i	−0.367142	−	0.930165i	$-0.619664\pi$
$613$	13.9398	−	13.9398i	0.563022	−	0.563022i	0.930165	+	0.367142i	$0.119664\pi$
$614$			16.7668i			0.676653i
$615$		0			0
$616$	0.671436	−	0.319196i	0.0270529	−	0.0128608i
$617$	27.7052	−	27.7052i	1.11537	−	1.11537i	0.122957	−	0.992412i	$-0.460762\pi$
$617$	27.7052	−	27.7052i	1.11537	−	1.11537i	0.992412	−	0.122957i	$-0.0392377\pi$
$618$		0			0
$619$	4.58642	−	4.58642i	0.184344	−	0.184344i	−0.608902	−	0.793246i	$-0.708390\pi$
$619$	4.58642	−	4.58642i	0.184344	−	0.184344i	0.793246	+	0.608902i	$0.208390\pi$
$620$	48.8736			1.96281
$621$		0			0
$622$	−3.81137	+	3.81137i	−0.152822	+	0.152822i
$623$	19.7852	+	7.03503i	0.792677	+	0.281853i
$624$		0			0
$625$	30.8687			1.23475
$626$	28.8128	+	28.8128i	1.15159	+	1.15159i
$627$		0			0
$628$	7.58498			0.302674
$629$	9.26134	+	9.26134i	0.369274	+	0.369274i
$630$		0			0
$631$	11.1175	+	11.1175i	0.442582	+	0.442582i	0.892879	−	0.450297i	$-0.148682\pi$
$631$	11.1175	+	11.1175i	0.442582	+	0.442582i	−0.450297	+	0.892879i	$0.648682\pi$
$632$	4.29237	+	4.29237i	0.170741	+	0.170741i
$633$		0			0
$634$			3.33185i			0.132325i
$635$	−27.8567	+	27.8567i	−1.10546	+	1.10546i
$636$		0			0
$637$	−21.5123	+	13.1994i	−0.852347	+	0.522977i
$638$	8.30174			0.328669
$639$		0			0
$640$		−	9.97073i		−	0.394128i
$641$		−	22.3590i		−	0.883129i	−0.897229	−	0.441565i	$-0.854424\pi$
$641$		−	22.3590i		−	0.883129i	0.897229	−	0.441565i	$-0.145576\pi$
$642$		0			0
$643$	−5.69880	−	5.69880i	−0.224739	−	0.224739i	0.585752	−	0.810491i	$-0.300799\pi$
$643$	−5.69880	−	5.69880i	−0.224739	−	0.224739i	−0.810491	+	0.585752i	$0.800799\pi$
$644$	2.34763	+	4.93829i	0.0925096	+	0.194596i
$645$		0			0
$646$	−42.7402			−1.68159
$647$	−7.73510			−0.304098			−0.152049	−	0.988373i	$-0.548587\pi$
$647$	−7.73510			−0.304098			−0.152049	+	0.988373i	$0.548587\pi$
$648$		0			0
$649$	−0.230912			−0.00906410
$650$	9.99952	+	20.7961i	0.392214	+	0.815689i
$651$		0			0
$652$	23.1383	−	23.1383i	0.906165	−	0.906165i
$653$	−21.3921			−0.837137			−0.418568	−	0.908185i	$-0.637468\pi$
$653$	−21.3921			−0.837137			−0.418568	+	0.908185i	$0.637468\pi$
$654$		0			0
$655$	−25.8479	+	25.8479i	−1.00996	+	1.00996i
$656$	−6.41503	−	6.41503i	−0.250465	−	0.250465i
$657$		0			0
$658$	40.9958	−	19.4891i	1.59818	−	0.759766i
$659$	1.68445			0.0656167			0.0328084	−	0.999462i	$-0.489555\pi$
$659$	1.68445			0.0656167			0.0328084	+	0.999462i	$0.489555\pi$
$660$		0			0
$661$	−23.4056	+	23.4056i	−0.910372	+	0.910372i	−0.996301	−	0.0859290i	$-0.972614\pi$
$661$	−23.4056	+	23.4056i	−0.910372	+	0.910372i	0.0859290	+	0.996301i	$0.472614\pi$
$662$			8.42864i			0.327588i
$663$		0			0
$664$			2.68041i			0.104020i
$665$	−32.7589	+	15.5734i	−1.27034	+	0.603910i
$666$		0			0
$667$			5.90967i			0.228823i
$668$	−35.9441	−	35.9441i	−1.39072	−	1.39072i
$669$		0			0
$670$	42.5860	−	42.5860i	1.64524	−	1.64524i
$671$	3.51796	−	3.51796i	0.135809	−	0.135809i
$672$		0			0
$673$		−	26.2464i		−	1.01173i	−0.862614	−	0.505863i	$-0.831174\pi$
$673$		−	26.2464i		−	1.01173i	0.862614	−	0.505863i	$-0.168826\pi$
$674$	6.49309	+	6.49309i	0.250105	+	0.250105i
$675$		0			0
$676$	17.9748	−	22.4844i	0.691339	−	0.864785i
$677$			36.7658i			1.41303i	0.707700	+	0.706513i	$0.249733\pi$
$677$			36.7658i			1.41303i	−0.707700	+	0.706513i	$0.750267\pi$
$678$		0			0
$679$	−0.296818	+	0.834764i	−0.0113908	+	0.0320353i
$680$		−	5.42372i		−	0.207990i
$681$		0			0
$682$	−7.18200	−	7.18200i	−0.275013	−	0.275013i
$683$	14.5812	+	14.5812i	0.557934	+	0.557934i	0.928719	−	0.370785i	$-0.120911\pi$
$683$	14.5812	+	14.5812i	0.557934	+	0.557934i	−0.370785	+	0.928719i	$0.620911\pi$
$684$		0			0
$685$		−	9.67329i		−	0.369597i
$686$	32.4939	−	19.7397i	1.24062	−	0.753667i
$687$		0			0
$688$		−	36.9447i		−	1.40850i
$689$	−11.4391	−	4.01040i	−0.435795	−	0.152784i
$690$		0			0
$691$	30.4663	+	30.4663i	1.15899	+	1.15899i	0.984693	+	0.174299i	$0.0557658\pi$
$691$	30.4663	+	30.4663i	1.15899	+	1.15899i	0.174299	+	0.984693i	$0.444234\pi$
$692$			31.8442i			1.21054i
$693$		0			0
$694$	−12.7906	+	12.7906i	−0.485525	+	0.485525i
$695$	−6.73260	+	6.73260i	−0.255382	+	0.255382i
$696$		0			0
$697$	−7.87310	−	7.87310i	−0.298215	−	0.298215i
$698$		−	34.2965i		−	1.29814i
$699$		0			0
$700$	−7.84166	−	16.4951i	−0.296387	−	0.623457i
$701$		−	17.7368i		−	0.669911i	−0.942234	−	0.334955i	$-0.891279\pi$
$701$		−	17.7368i		−	0.669911i	0.942234	−	0.334955i	$-0.108721\pi$
$702$		0			0
$703$		−	14.5661i		−	0.549371i
$704$	−4.34122	+	4.34122i	−0.163616	+	0.163616i
$705$		0			0
$706$	−8.44854			−0.317965
$707$	10.4729	+	22.0300i	0.393874	+	0.828522i
$708$		0			0
$709$	19.2422	+	19.2422i	0.722656	+	0.722656i	0.969146	−	0.246489i	$-0.0792770\pi$
$709$	19.2422	+	19.2422i	0.722656	+	0.722656i	−0.246489	+	0.969146i	$0.579277\pi$
$710$	−32.5734	+	32.5734i	−1.22246	+	1.22246i
$711$		0			0
$712$	3.49196			0.130867
$713$	5.11257	−	5.11257i	0.191467	−	0.191467i
$714$		0			0
$715$	2.17061	−	6.19135i	0.0811762	−	0.231543i
$716$	−9.10324			−0.340204
$717$		0			0
$718$	−10.2286			−0.381728
$719$	−42.3551			−1.57958			−0.789789	−	0.613379i	$-0.789810\pi$
$719$	−42.3551			−1.57958			−0.789789	+	0.613379i	$0.789810\pi$
$720$		0			0
$721$	2.84790	+	5.99062i	0.106061	+	0.223102i
$722$	6.03011	+	6.03011i	0.224418	+	0.224418i
$723$		0			0
$724$		−	47.7762i		−	1.77559i
$725$		−	19.7397i		−	0.733116i
$726$		0			0
$727$	23.2484			0.862234			0.431117	−	0.902296i	$-0.358120\pi$
$727$	23.2484			0.862234			0.431117	+	0.902296i	$0.358120\pi$
$728$	−2.63526	+	3.26662i	−0.0976693	+	0.121069i
$729$		0			0
$730$	−51.8299	+	51.8299i	−1.91831	+	1.91831i
$731$		−	45.3419i		−	1.67703i
$732$		0			0
$733$	24.1280	+	24.1280i	0.891188	+	0.891188i	0.994635	−	0.103447i	$-0.0329873\pi$
$733$	24.1280	+	24.1280i	0.891188	+	0.891188i	−0.103447	+	0.994635i	$0.532987\pi$
$734$	−28.7580	−	28.7580i	−1.06148	−	1.06148i
$735$		0			0
$736$	−5.35704	−	5.35704i	−0.197463	−	0.197463i
$737$	−6.57628			−0.242240
$738$		0			0
$739$	4.46590	+	4.46590i	0.164281	+	0.164281i	0.784460	−	0.620179i	$-0.212940\pi$
$739$	4.46590	+	4.46590i	0.164281	+	0.164281i	−0.620179	+	0.784460i	$0.712940\pi$
$740$	19.0977			0.702047
$741$		0			0
$742$	17.2050	+	6.11758i	0.631614	+	0.224583i
$743$	−10.0114	+	10.0114i	−0.367282	+	0.367282i	−0.866485	−	0.499203i	$-0.833626\pi$
$743$	−10.0114	+	10.0114i	−0.367282	+	0.367282i	0.499203	+	0.866485i	$0.333626\pi$
$744$		0			0
$745$	−10.5944			−0.388149
$746$	−35.6178	+	35.6178i	−1.30406	+	1.30406i
$747$		0			0
$748$	−4.32672	+	4.32672i	−0.158201	+	0.158201i
$749$	3.27432	+	6.88761i	0.119641	+	0.251668i
$750$		0			0
$751$		−	8.16686i		−	0.298013i	−0.988836	−	0.149006i	$-0.952392\pi$
$751$		−	8.16686i		−	0.298013i	0.988836	−	0.149006i	$-0.0476075\pi$
$752$	−20.8338	+	20.8338i	−0.759731	+	0.759731i
$753$		0			0
$754$	−42.2378	+	20.3095i	−1.53821	+	0.739628i
$755$		−	19.2739i		−	0.701448i
$756$		0			0
$757$	10.5210			0.382392			0.191196	−	0.981552i	$-0.438763\pi$
$757$	10.5210			0.382392			0.191196	+	0.981552i	$0.438763\pi$
$758$			25.5002i			0.926210i
$759$		0			0
$760$	−4.26517	+	4.26517i	−0.154714	+	0.154714i
$761$	−5.95138	+	5.95138i	−0.215737	+	0.215737i	−0.806699	−	0.590962i	$-0.798748\pi$
$761$	−5.95138	+	5.95138i	−0.215737	+	0.215737i	0.590962	+	0.806699i	$0.298748\pi$
$762$		0			0
$763$	12.5096	+	4.44805i	0.452878	+	0.161030i
$764$			22.8113i			0.825286i
$765$		0			0
$766$	−51.9646			−1.87756
$767$	1.17484	−	0.564907i	0.0424210	−	0.0203976i
$768$		0			0
$769$	9.73800	+	9.73800i	0.351161	+	0.351161i	0.860541	−	0.509380i	$-0.170125\pi$
$769$	9.73800	+	9.73800i	0.351161	+	0.351161i	−0.509380	+	0.860541i	$0.670125\pi$
$770$	−3.31111	+	9.31209i	−0.119324	+	0.335584i
$771$		0			0
$772$	13.9190	−	13.9190i	0.500957	−	0.500957i
$773$	16.0246	+	16.0246i	0.576364	+	0.576364i	0.933899	−	0.357536i	$-0.116383\pi$
$773$	16.0246	+	16.0246i	0.576364	+	0.576364i	−0.357536	+	0.933899i	$0.616383\pi$
$774$		0			0
$775$	−17.0772	+	17.0772i	−0.613433	+	0.613433i
$776$			0.147331i			0.00528886i
$777$		0			0
$778$	17.5161	+	17.5161i	0.627981	+	0.627981i
$779$			12.3827i			0.443657i
$780$		0			0
$781$	5.03011			0.179992
$782$	−5.86192	−	5.86192i	−0.209622	−	0.209622i
$783$		0			0
$784$	−15.5798	+	19.1383i	−0.556421	+	0.683510i
$785$	−6.90099	+	6.90099i	−0.246307	+	0.246307i
$786$		0			0
$787$	−17.9844	+	17.9844i	−0.641075	+	0.641075i	−0.950820	−	0.309745i	$-0.899756\pi$
$787$	−17.9844	+	17.9844i	−0.641075	+	0.641075i	0.309745	+	0.950820i	$0.399756\pi$
$788$	−7.19850	−	7.19850i	−0.256436	−	0.256436i
$789$		0

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:	\( N \)	\(=\)	\( 819 = 3^{2} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	819.y (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.45161 - 1.45161i) q^{2} -2.21432i q^{4} +(2.01464 + 2.01464i) q^{5} +(-1.13594 - 2.38948i) q^{7} +(-0.311108 - 0.311108i) q^{8} +O(q^{10})\)	\(q+(1.45161 - 1.45161i) q^{2} -2.21432i q^{4} +(2.01464 + 2.01464i) q^{5} +(-1.13594 - 2.38948i) q^{7} +(-0.311108 - 0.311108i) q^{8} +5.84892 q^{10} +(-0.451606 - 0.451606i) q^{11} +(3.40251 + 1.19288i) q^{13} +(-5.11753 - 1.81964i) q^{14} +3.52543 q^{16} +4.32672 q^{17} +(-3.40251 - 3.40251i) q^{19} +(4.46105 - 4.46105i) q^{20} -1.31111 q^{22} -0.933323i q^{23} +3.11753i q^{25} +(6.67068 - 3.20751i) q^{26} +(-5.29108 + 2.51534i) q^{28} -6.33185 q^{29} +(5.47781 + 5.47781i) q^{31} +(5.73975 - 5.73975i) q^{32} +(6.28070 - 6.28070i) q^{34} +(2.52543 - 7.10246i) q^{35} +(2.14050 + 2.14050i) q^{37} -9.87820 q^{38} -1.25354i q^{40} +(-1.81964 - 1.81964i) q^{41} -10.4795i q^{43} +(-1.00000 + 1.00000i) q^{44} +(-1.35482 - 1.35482i) q^{46} +(-5.90958 + 5.90958i) q^{47} +(-4.41926 + 5.42864i) q^{49} +(4.52543 + 4.52543i) q^{50} +(2.64141 - 7.53424i) q^{52} -3.36196 q^{53} -1.81964i q^{55} +(-0.389986 + 1.09679i) q^{56} +(-9.19135 + 9.19135i) q^{58} +(0.255657 - 0.255657i) q^{59} +7.78989i q^{61} +15.9032 q^{62} -9.61285i q^{64} +(4.45161 + 9.25803i) q^{65} +(7.28100 - 7.28100i) q^{67} -9.58075i q^{68} +(-6.64405 - 13.9759i) q^{70} +(-5.56914 + 5.56914i) q^{71} +(-8.86144 + 8.86144i) q^{73} +6.21432 q^{74} +(-7.53424 + 7.53424i) q^{76} +(-0.566106 + 1.59210i) q^{77} -13.7971 q^{79} +(7.10246 + 7.10246i) q^{80} -5.28281 q^{82} +(-4.30785 - 4.30785i) q^{83} +(8.71678 + 8.71678i) q^{85} +(-15.2121 - 15.2121i) q^{86} +0.280996i q^{88} +(-5.61214 + 5.61214i) q^{89} +(-1.01470 - 9.48527i) q^{91} -2.06668 q^{92} +17.1568i q^{94} -13.7096i q^{95} +(-0.236784 - 0.236784i) q^{97} +(1.46522 + 14.2953i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 4 q^{2} - 8 q^{7} - 4 q^{8}+O(q^{10}) \) 12 * q + 4 * q^2 - 8 * q^7 - 4 * q^8	\( 12 q + 4 q^{2} - 8 q^{7} - 4 q^{8} + 8 q^{11} - 8 q^{14} + 16 q^{16} - 16 q^{22} - 20 q^{28} + 4 q^{29} + 16 q^{32} + 4 q^{35} + 12 q^{37} - 12 q^{44} + 24 q^{46} + 28 q^{50} + 12 q^{53} - 44 q^{58} + 40 q^{65} + 60 q^{67} + 4 q^{70} + 48 q^{74} - 4 q^{79} + 12 q^{85} - 36 q^{86} - 32 q^{91} - 24 q^{92} + 28 q^{98}+O(q^{100}) \) 12 * q + 4 * q^2 - 8 * q^7 - 4 * q^8 + 8 * q^11 - 8 * q^14 + 16 * q^16 - 16 * q^22 - 20 * q^28 + 4 * q^29 + 16 * q^32 + 4 * q^35 + 12 * q^37 - 12 * q^44 + 24 * q^46 + 28 * q^50 + 12 * q^53 - 44 * q^58 + 40 * q^65 + 60 * q^67 + 4 * q^70 + 48 * q^74 - 4 * q^79 + 12 * q^85 - 36 * q^86 - 32 * q^91 - 24 * q^92 + 28 * q^98

Introduction

L-functions

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