LMFDB - Embedded newform 1323.2.g.c.667.1

Newspace parameters

Level:	$ N $	$=$	$ 1323 = 3^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1323.g (of order $3$, degree $2$, not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$10.5642081874$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(\zeta_{3})$
Coefficient field:	6.0.309123.1
Defining polynomial:	$x^{6} - 3 x^{5} + 10 x^{4} - 15 x^{3} + 19 x^{2} - 12 x + 3$
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 63)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			667.1
Root			$0.500000 - 2.05195i$ of defining polynomial
Character	$\chi$	$=$	1323.667
Dual form			1323.2.g.c.361.1

$q$-expansion

$f(q)$	$=$	$q+(-1.23025 - 2.13086i) q^{2} +(-2.02704 + 3.51094i) q^{4} +2.59358 q^{5} +5.05408 q^{8} +O(q^{10})$	\(q+(-1.23025 - 2.13086i) q^{2} +(-2.02704 + 3.51094i) q^{4} +2.59358 q^{5} +5.05408 q^{8} +(-3.19076 - 5.52655i) q^{10} -4.51459 q^{11} +(-0.500000 - 0.866025i) q^{13} +(-2.16372 - 3.74766i) q^{16} +(-0.472958 - 0.819187i) q^{17} +(2.02704 - 3.51094i) q^{19} +(-5.25729 + 9.10590i) q^{20} +(5.55408 + 9.61996i) q^{22} +0.273346 q^{23} +1.72665 q^{25} +(-1.23025 + 2.13086i) q^{26} +(1.23025 - 2.13086i) q^{29} +(-1.16372 + 2.01561i) q^{31} +(-0.269748 + 0.467216i) q^{32} +(-1.16372 + 2.01561i) q^{34} +(-0.890369 + 1.54216i) q^{37} -9.97509 q^{38} +13.1082 q^{40} +(-3.20321 - 5.54812i) q^{41} +(5.21780 - 9.03749i) q^{43} +(9.15126 - 15.8505i) q^{44} +(-0.336285 - 0.582462i) q^{46} +(-6.08113 - 10.5328i) q^{47} +(-2.12422 - 3.67926i) q^{50} +4.05408 q^{52} +(-3.13667 - 5.43288i) q^{53} -11.7089 q^{55} -6.05408 q^{58} +(-1.36333 + 2.36135i) q^{59} +(1.13667 + 1.96878i) q^{61} +5.72665 q^{62} -7.32743 q^{64} +(-1.29679 - 2.24611i) q^{65} +(7.90856 - 13.6980i) q^{67} +3.83482 q^{68} -3.27335 q^{71} +(0.753696 + 1.30544i) q^{73} +4.38151 q^{74} +(8.21780 + 14.2336i) q^{76} +(-7.35447 - 12.7383i) q^{79} +(-5.61177 - 9.71987i) q^{80} +(-7.88151 + 13.6512i) q^{82} +(-0.472958 + 0.819187i) q^{83} +(-1.22665 - 2.12463i) q^{85} -25.6768 q^{86} -22.8171 q^{88} +(-7.17830 + 12.4332i) q^{89} +(-0.554084 + 0.959702i) q^{92} +(-14.9626 + 25.9161i) q^{94} +(5.25729 - 9.10590i) q^{95} +(5.74484 - 9.95036i) q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6q - q^{2} - 3q^{4} + 10q^{5} + 12q^{8} + O(q^{10}) $	$ 6q - q^{2} - 3q^{4} + 10q^{5} + 12q^{8} + 4q^{11} - 3q^{13} - 3q^{16} - 12q^{17} + 3q^{19} - 16q^{20} + 15q^{22} + 12q^{25} - q^{26} + q^{29} + 3q^{31} - 8q^{32} + 3q^{34} + 3q^{37} - 16q^{38} + 42q^{40} - 22q^{41} + 3q^{43} + 23q^{44} - 12q^{46} - 9q^{47} + 10q^{50} + 6q^{52} - 18q^{53} - 12q^{55} - 18q^{58} - 9q^{59} + 6q^{61} + 36q^{62} - 24q^{64} - 5q^{65} - 12q^{68} - 18q^{71} - 3q^{73} - 12q^{74} + 21q^{76} - 15q^{79} + 11q^{80} - 9q^{82} - 12q^{83} - 9q^{85} - 68q^{86} - 42q^{88} - 2q^{89} + 15q^{92} - 24q^{94} + 16q^{95} - 3q^{97} + O(q^{100}) $

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1323\mathbb{Z}\right)^\times$.

$n$	$785$	$1081$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{3}\right)$

Coefficient data

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

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

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

$2$	−1.23025	−	2.13086i	−0.869920	−	1.50675i	−0.862078	−	0.506776i	$-0.830837\pi$
$2$	−1.23025	−	2.13086i	−0.869920	−	1.50675i	−0.00784213	−	0.999969i	$-0.502496\pi$
$3$		0			0
$3$		0			0
$4$	−2.02704	+	3.51094i	−1.01352	+	1.75547i
$5$	2.59358			1.15988			0.579942	−	0.814658i	$-0.303075\pi$
$5$	2.59358			1.15988			0.579942	+	0.814658i	$0.303075\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$	5.05408			1.78689
$9$		0			0
$10$	−3.19076	−	5.52655i	−1.00901	−	1.74765i
$11$	−4.51459			−1.36120			−0.680600	−	0.732655i	$-0.738281\pi$
$11$	−4.51459			−1.36120			−0.680600	+	0.732655i	$0.738281\pi$
$12$		0			0
$13$	−0.500000	−	0.866025i	−0.138675	−	0.240192i	0.788320	−	0.615265i	$-0.210951\pi$
$13$	−0.500000	−	0.866025i	−0.138675	−	0.240192i	−0.926995	+	0.375073i	$0.877618\pi$
$14$		0			0
$15$		0			0
$16$	−2.16372	−	3.74766i	−0.540929	−	0.936916i
$17$	−0.472958	−	0.819187i	−0.114709	−	0.198682i	0.802954	−	0.596041i	$-0.203260\pi$
$17$	−0.472958	−	0.819187i	−0.114709	−	0.198682i	−0.917663	+	0.397359i	$0.869927\pi$
$18$		0			0
$19$	2.02704	−	3.51094i	0.465035	−	0.805465i	−0.534168	−	0.845378i	$-0.679375\pi$
$19$	2.02704	−	3.51094i	0.465035	−	0.805465i	0.999203	+	0.0399136i	$0.0127083\pi$
$20$	−5.25729	+	9.10590i	−1.17557	+	2.03614i
$21$		0			0
$22$	5.55408	+	9.61996i	1.18413	+	2.05098i
$23$	0.273346			0.0569966			0.0284983	−	0.999594i	$-0.490927\pi$
$23$	0.273346			0.0569966			0.0284983	+	0.999594i	$0.490927\pi$
$24$		0			0
$25$	1.72665			0.345331
$26$	−1.23025	+	2.13086i	−0.241272	+	0.417896i
$27$		0			0
$28$		0			0
$29$	1.23025	−	2.13086i	0.228452	−	0.395691i	−0.728897	−	0.684623i	$-0.759967\pi$
$29$	1.23025	−	2.13086i	0.228452	−	0.395691i	0.957350	+	0.288932i	$0.0933002\pi$
$30$		0			0
$31$	−1.16372	+	2.01561i	−0.209009	+	0.362015i	−0.951403	−	0.307949i	$-0.900357\pi$
$31$	−1.16372	+	2.01561i	−0.209009	+	0.362015i	0.742393	+	0.669964i	$0.233691\pi$
$32$	−0.269748	+	0.467216i	−0.0476851	+	0.0825930i
$33$		0			0
$34$	−1.16372	+	2.01561i	−0.199576	+	0.345675i
$35$		0			0
$36$		0			0
$37$	−0.890369	+	1.54216i	−0.146376	+	0.253530i	−0.929885	−	0.367849i	$-0.880094\pi$
$37$	−0.890369	+	1.54216i	−0.146376	+	0.253530i	0.783510	+	0.621380i	$0.213428\pi$
$38$	−9.97509			−1.61817
$39$		0			0
$40$	13.1082			2.07258
$41$	−3.20321	−	5.54812i	−0.500257	−	0.866471i	−1.00000		0.000297253i	$-0.999905\pi$
$41$	−3.20321	−	5.54812i	−0.500257	−	0.866471i	0.499743	−	0.866174i	$-0.333428\pi$
$42$		0			0
$43$	5.21780	−	9.03749i	0.795707	−	1.37820i	−0.126682	−	0.991943i	$-0.540433\pi$
$43$	5.21780	−	9.03749i	0.795707	−	1.37820i	0.922389	−	0.386262i	$-0.126234\pi$
$44$	9.15126	−	15.8505i	1.37960	−	2.38955i
$45$		0			0
$46$	−0.336285	−	0.582462i	−0.0495825	−	0.0858794i
$47$	−6.08113	−	10.5328i	−0.887023	−	1.53637i	−0.843377	−	0.537323i	$-0.819436\pi$
$47$	−6.08113	−	10.5328i	−0.887023	−	1.53637i	−0.0436467	−	0.999047i	$-0.513898\pi$
$48$		0			0
$49$		0			0
$50$	−2.12422	−	3.67926i	−0.300410	−	0.520326i
$51$		0			0
$52$	4.05408			0.562200
$53$	−3.13667	−	5.43288i	−0.430855	−	0.746263i	0.566092	−	0.824342i	$-0.308455\pi$
$53$	−3.13667	−	5.43288i	−0.430855	−	0.746263i	−0.996947	+	0.0780790i	$0.975121\pi$
$54$		0			0
$55$	−11.7089			−1.57883
$56$		0			0
$57$		0			0
$58$	−6.05408			−0.794940
$59$	−1.36333	+	2.36135i	−0.177490	+	0.307422i	−0.941020	−	0.338350i	$-0.890131\pi$
$59$	−1.36333	+	2.36135i	−0.177490	+	0.307422i	0.763530	+	0.645772i	$0.223464\pi$
$60$		0			0
$61$	1.13667	+	1.96878i	0.145536	+	0.252076i	0.929573	−	0.368639i	$-0.120176\pi$
$61$	1.13667	+	1.96878i	0.145536	+	0.252076i	−0.784037	+	0.620714i	$0.786843\pi$
$62$	5.72665			0.727286
$63$		0			0
$64$	−7.32743			−0.915929
$65$	−1.29679	−	2.24611i	−0.160847	−	0.278595i
$66$		0			0
$67$	7.90856	−	13.6980i	0.966184	−	1.67348i	0.259784	−	0.965667i	$-0.416349\pi$
$67$	7.90856	−	13.6980i	0.966184	−	1.67348i	0.706400	−	0.707813i	$-0.250318\pi$
$68$	3.83482			0.465041
$69$		0			0
$70$		0			0
$71$	−3.27335			−0.388475			−0.194237	−	0.980955i	$-0.562223\pi$
$71$	−3.27335			−0.388475			−0.194237	+	0.980955i	$0.562223\pi$
$72$		0			0
$73$	0.753696	+	1.30544i	0.0882134	+	0.152790i	0.906756	−	0.421656i	$-0.138551\pi$
$73$	0.753696	+	1.30544i	0.0882134	+	0.152790i	−0.818543	+	0.574446i	$0.805218\pi$
$74$	4.38151			0.509341
$75$		0			0
$76$	8.21780	+	14.2336i	0.942646	+	1.63271i
$77$		0			0
$78$		0			0
$79$	−7.35447	−	12.7383i	−0.827443	−	1.43317i	−0.900038	−	0.435811i	$-0.856461\pi$
$79$	−7.35447	−	12.7383i	−0.827443	−	1.43317i	0.0725952	−	0.997361i	$-0.476872\pi$
$80$	−5.61177	−	9.71987i	−0.627415	−	1.08671i
$81$		0			0
$82$	−7.88151	+	13.6512i	−0.870368	+	1.50752i
$83$	−0.472958	+	0.819187i	−0.0519139	+	0.0899175i	−0.890815	−	0.454367i	$-0.849865\pi$
$83$	−0.472958	+	0.819187i	−0.0519139	+	0.0899175i	0.838901	+	0.544285i	$0.183199\pi$
$84$		0			0
$85$	−1.22665	−	2.12463i	−0.133049	−	0.230448i
$86$	−25.6768			−2.76881
$87$		0			0
$88$	−22.8171			−2.43231
$89$	−7.17830	+	12.4332i	−0.760899	+	1.31792i	0.181489	+	0.983393i	$0.441908\pi$
$89$	−7.17830	+	12.4332i	−0.760899	+	1.31792i	−0.942388	+	0.334522i	$0.891425\pi$
$90$		0			0
$91$		0			0
$92$	−0.554084	+	0.959702i	−0.0577673	+	0.100056i
$93$		0			0
$94$	−14.9626	+	25.9161i	−1.54328	+	2.67304i
$95$	5.25729	−	9.10590i	0.539387	−	0.934246i
$96$		0			0
$97$	5.74484	−	9.95036i	0.583300	−	1.01031i	−0.411785	−	0.911281i	$-0.635094\pi$
$97$	5.74484	−	9.95036i	0.583300	−	1.01031i	0.995085	−	0.0990246i	$-0.0315722\pi$
$98$		0			0
$99$		0			0
$100$	−3.50000	+	6.06218i	−0.350000	+	0.606218i
$101$	3.67977			0.366150			0.183075	−	0.983099i	$-0.441395\pi$
$101$	3.67977			0.366150			0.183075	+	0.983099i	$0.441395\pi$
$102$		0			0
$103$	−9.72665			−0.958396			−0.479198	−	0.877707i	$-0.659072\pi$
$103$	−9.72665			−0.958396			−0.479198	+	0.877707i	$0.659072\pi$
$104$	−2.52704	−	4.37697i	−0.247797	−	0.429197i
$105$		0			0
$106$	−7.71780	+	13.3676i	−0.749619	+	1.29838i
$107$	−0.687159	+	1.19019i	−0.0664301	+	0.115060i	−0.897327	−	0.441365i	$-0.854494\pi$
$107$	−0.687159	+	1.19019i	−0.0664301	+	0.115060i	0.830897	+	0.556426i	$0.187828\pi$
$108$		0			0
$109$	1.69961	+	2.94381i	0.162793	+	0.281966i	0.935869	−	0.352347i	$-0.114616\pi$
$109$	1.69961	+	2.94381i	0.162793	+	0.281966i	−0.773076	+	0.634313i	$0.781283\pi$
$110$	14.4050	+	24.9501i	1.37346	+	2.37890i
$111$		0			0
$112$		0			0
$113$	5.19436	+	8.99689i	0.488644	+	0.846356i	0.999915	−	0.0130636i	$-0.00415840\pi$
$113$	5.19436	+	8.99689i	0.488644	+	0.846356i	−0.511271	+	0.859420i	$0.670825\pi$
$114$		0			0
$115$	0.708945			0.0661095
$116$	4.98755	+	8.63868i	0.463082	+	0.802082i
$117$		0			0
$118$	6.70895			0.617608
$119$		0			0
$120$		0			0
$121$	9.38151			0.852865
$122$	2.79679	−	4.84418i	0.253209	−	0.438572i
$123$		0			0
$124$	−4.71780	−	8.17147i	−0.423671	−	0.733820i
$125$	−8.48968			−0.759340
$126$		0			0
$127$	0.672570			0.0596809			0.0298405	−	0.999555i	$-0.490500\pi$
$127$	0.672570			0.0596809			0.0298405	+	0.999555i	$0.490500\pi$
$128$	9.55408	+	16.5482i	0.844470	+	1.46266i
$129$		0			0
$130$	−3.19076	+	5.52655i	−0.279848	+	0.484711i
$131$	−7.91381			−0.691433			−0.345717	−	0.938339i	$-0.612364\pi$
$131$	−7.91381			−0.691433			−0.345717	+	0.938339i	$0.612364\pi$
$132$		0			0
$133$		0			0
$134$	−38.9181			−3.36201
$135$		0			0
$136$	−2.39037	−	4.14024i	−0.204972	−	0.355023i
$137$	3.67257			0.313769			0.156884	−	0.987617i	$-0.449855\pi$
$137$	3.67257			0.313769			0.156884	+	0.987617i	$0.449855\pi$
$138$		0			0
$139$	1.02704	+	1.77889i	0.0871126	+	0.150883i	0.906289	−	0.422658i	$-0.138903\pi$
$139$	1.02704	+	1.77889i	0.0871126	+	0.150883i	−0.819177	+	0.573541i	$0.805569\pi$
$140$		0			0
$141$		0			0
$142$	4.02704	+	6.97504i	0.337942	+	0.585332i
$143$	2.25729	+	3.90975i	0.188764	+	0.326950i
$144$		0			0
$145$	3.19076	−	5.52655i	0.264978	−	0.458955i
$146$	1.85447	−	3.21204i	0.153477	−	0.265830i
$147$		0			0
$148$	−3.60963	−	6.25206i	−0.296710	−	0.513917i
$149$	13.5438			1.10955			0.554774	−	0.832001i	$-0.312805\pi$
$149$	13.5438			1.10955			0.554774	+	0.832001i	$0.312805\pi$
$150$		0			0
$151$	9.92821			0.807946			0.403973	−	0.914771i	$-0.367629\pi$
$151$	9.92821			0.807946			0.403973	+	0.914771i	$0.367629\pi$
$152$	10.2448	−	17.7446i	0.830966	−	1.43928i
$153$		0			0
$154$		0			0
$155$	−3.01819	+	5.22765i	−0.242427	+	0.419895i
$156$		0			0
$157$	−3.02704	+	5.24299i	−0.241584	+	0.418436i	−0.961166	−	0.275972i	$-0.911000\pi$
$157$	−3.02704	+	5.24299i	−0.241584	+	0.418436i	0.719581	+	0.694408i	$0.244334\pi$
$158$	−18.0957	+	31.3427i	−1.43962	+	2.49349i
$159$		0			0
$160$	−0.699612	+	1.21176i	−0.0553092	+	0.0957983i
$161$		0			0
$162$		0			0
$163$	−8.90856	+	15.4301i	−0.697772	+	1.20858i	0.271465	+	0.962448i	$0.412492\pi$
$163$	−8.90856	+	15.4301i	−0.697772	+	1.20858i	−0.969237	+	0.246128i	$0.920842\pi$
$164$	25.9722			2.02809
$165$		0			0
$166$	2.32743			0.180644
$167$	−4.23385	−	7.33325i	−0.327625	−	0.567464i	0.654415	−	0.756136i	$-0.272915\pi$
$167$	−4.23385	−	7.33325i	−0.327625	−	0.567464i	−0.982040	+	0.188672i	$0.939582\pi$
$168$		0			0
$169$	6.00000	−	10.3923i	0.461538	−	0.799408i
$170$	−3.01819	+	5.22765i	−0.231484	+	0.400943i
$171$		0			0
$172$	21.1534	+	36.6388i	1.61293	+	2.79368i
$173$	8.67830	+	15.0313i	0.659799	+	1.14281i	0.980667	+	0.195682i	$0.0626920\pi$
$173$	8.67830	+	15.0313i	0.659799	+	1.14281i	−0.320868	+	0.947124i	$0.603975\pi$
$174$		0			0
$175$		0			0
$176$	9.76829	+	16.9192i	0.736312	+	1.27533i
$177$		0			0
$178$	35.3245			2.64768
$179$	−5.67471	−	9.82888i	−0.424147	−	0.734645i	0.572193	−	0.820119i	$-0.306093\pi$
$179$	−5.67471	−	9.82888i	−0.424147	−	0.734645i	−0.996340	+	0.0854741i	$0.972759\pi$
$180$		0			0
$181$	21.8889			1.62699			0.813495	−	0.581572i	$-0.197562\pi$
$181$	21.8889			1.62699			0.813495	+	0.581572i	$0.197562\pi$
$182$		0			0
$183$		0			0
$184$	1.38151			0.101847
$185$	−2.30924	+	3.99973i	−0.169779	+	0.294066i
$186$		0			0
$187$	2.13521	+	3.69829i	0.156142	+	0.270446i
$188$	49.3068			3.59607
$189$		0			0
$190$	−25.8712			−1.87689
$191$	−0.350874	−	0.607731i	−0.0253883	−	0.0439739i	0.853052	−	0.521826i	$-0.174749\pi$
$191$	−0.350874	−	0.607731i	−0.0253883	−	0.0439739i	−0.878440	+	0.477852i	$0.841416\pi$
$192$		0			0
$193$	−6.07227	+	10.5175i	−0.437092	+	0.757065i	−0.997464	−	0.0711760i	$-0.977325\pi$
$193$	−6.07227	+	10.5175i	−0.437092	+	0.757065i	0.560372	+	0.828241i	$0.310658\pi$
$194$	−28.2704			−2.02970
$195$		0			0
$196$		0			0
$197$	16.4107			1.16921			0.584607	−	0.811317i	$-0.301249\pi$
$197$	16.4107			1.16921			0.584607	+	0.811317i	$0.301249\pi$
$198$		0			0
$199$	−11.3530	−	19.6640i	−0.804794	−	1.39394i	−0.916430	−	0.400194i	$-0.868943\pi$
$199$	−11.3530	−	19.6640i	−0.804794	−	1.39394i	0.111637	−	0.993749i	$-0.464391\pi$
$200$	8.72665			0.617068
$201$		0			0
$202$	−4.52704	−	7.84107i	−0.318522	−	0.551696i
$203$		0			0
$204$		0			0
$205$	−8.30778	−	14.3895i	−0.580241	−	1.00501i
$206$	11.9662	+	20.7261i	0.833727	+	1.44406i
$207$		0			0
$208$	−2.16372	+	3.74766i	−0.150027	+	0.259854i
$209$	−9.15126	+	15.8505i	−0.633006	+	1.09640i
$210$		0			0
$211$	−2.28074	−	3.95035i	−0.157012	−	0.271954i	0.776778	−	0.629775i	$-0.216853\pi$
$211$	−2.28074	−	3.95035i	−0.157012	−	0.271954i	−0.933790	+	0.357822i	$0.883520\pi$
$212$	25.4327			1.74672
$213$		0			0
$214$	3.38151			0.231156
$215$	13.5328	−	23.4395i	0.922928	−	1.59856i
$216$		0			0
$217$		0			0
$218$	4.18190	−	7.24327i	0.283234	−	0.490576i
$219$		0			0
$220$	23.7345	−	41.1094i	1.60018	−	2.77160i
$221$	−0.472958	+	0.819187i	−0.0318146	+	0.0551045i
$222$		0			0
$223$	−6.66225	+	11.5394i	−0.446137	+	0.772733i	−0.998131	−	0.0611159i	$-0.980534\pi$
$223$	−6.66225	+	11.5394i	−0.446137	+	0.772733i	0.551993	+	0.833849i	$0.313867\pi$
$224$		0			0
$225$		0			0
$226$	12.7807	−	22.1369i	0.850162	−	1.47252i
$227$	−1.38151			−0.0916943			−0.0458472	−	0.998948i	$-0.514599\pi$
$227$	−1.38151			−0.0916943			−0.0458472	+	0.998948i	$0.514599\pi$
$228$		0			0
$229$	−17.9794			−1.18811			−0.594055	−	0.804424i	$-0.702474\pi$
$229$	−17.9794			−1.18811			−0.594055	+	0.804424i	$0.702474\pi$
$230$	−0.872181	−	1.51066i	−0.0575099	−	0.0996101i
$231$		0			0
$232$	6.21780	−	10.7695i	0.408219	−	0.707055i
$233$	−9.49115	+	16.4391i	−0.621786	+	1.07696i	0.367367	+	0.930076i	$0.380259\pi$
$233$	−9.49115	+	16.4391i	−0.621786	+	1.07696i	−0.989153	+	0.146888i	$0.953074\pi$
$234$		0			0
$235$	−15.7719	−	27.3177i	−1.02884	−	1.78201i
$236$	−5.52704	−	9.57312i	−0.359780	−	0.623157i
$237$		0			0
$238$		0			0
$239$	2.44592	+	4.23645i	0.158213	+	0.274033i	0.934224	−	0.356686i	$-0.116093\pi$
$239$	2.44592	+	4.23645i	0.158213	+	0.274033i	−0.776011	+	0.630719i	$0.782760\pi$
$240$		0			0
$241$	−26.1593			−1.68507			−0.842535	−	0.538641i	$-0.818938\pi$
$241$	−26.1593			−1.68507			−0.842535	+	0.538641i	$0.818938\pi$
$242$	−11.5416	−	19.9907i	−0.741924	−	1.28505i
$243$		0			0
$244$	−9.21634			−0.590016
$245$		0			0
$246$		0			0
$247$	−4.05408			−0.257955
$248$	−5.88151	+	10.1871i	−0.373477	+	0.646880i
$249$		0			0
$250$	10.4445	+	18.0903i	0.660565	+	1.14413i
$251$	−18.4576			−1.16503			−0.582516	−	0.812819i	$-0.697932\pi$
$251$	−18.4576			−1.16503			−0.582516	+	0.812819i	$0.697932\pi$
$252$		0			0
$253$	−1.23405			−0.0775838
$254$	−0.827430	−	1.43315i	−0.0519176	−	0.0899239i
$255$		0			0
$256$	16.1804	−	28.0253i	1.01128	−	1.75158i
$257$	11.7339			0.731938			0.365969	−	0.930627i	$-0.380738\pi$
$257$	11.7339			0.731938			0.365969	+	0.930627i	$0.380738\pi$
$258$		0			0
$259$		0			0
$260$	10.5146			0.652087
$261$		0			0
$262$	9.73599	+	16.8632i	0.601491	+	1.04181i
$263$	7.52179			0.463813			0.231907	−	0.972738i	$-0.425504\pi$
$263$	7.52179			0.463813			0.231907	+	0.972738i	$0.425504\pi$
$264$		0			0
$265$	−8.13521	−	14.0906i	−0.499742	−	0.865579i
$266$		0			0
$267$		0			0
$268$	32.0620	+	55.5329i	1.95850	+	3.39221i
$269$	−9.41741	−	16.3114i	−0.574190	−	0.994526i	−0.996129	−	0.0879017i	$-0.971984\pi$
$269$	−9.41741	−	16.3114i	−0.574190	−	0.994526i	0.421939	−	0.906624i	$-0.361349\pi$
$270$		0			0
$271$	11.9911	−	20.7693i	0.728410	−	1.26164i	−0.229145	−	0.973392i	$-0.573593\pi$
$271$	11.9911	−	20.7693i	0.728410	−	1.26164i	0.957555	−	0.288251i	$-0.0930738\pi$
$272$	−2.04669	+	3.54498i	−0.124099	+	0.214946i
$273$		0			0
$274$	−4.51819	−	7.82573i	−0.272954	−	0.472770i
$275$	−7.79513			−0.470064
$276$		0			0
$277$	7.16225			0.430338			0.215169	−	0.976577i	$-0.430970\pi$
$277$	7.16225			0.430338			0.215169	+	0.976577i	$0.430970\pi$
$278$	2.52704	−	4.37697i	0.151562	−	0.262513i
$279$		0			0
$280$		0			0
$281$	7.44085	−	12.8879i	0.443884	−	0.768830i	−0.554090	−	0.832457i	$-0.686933\pi$
$281$	7.44085	−	12.8879i	0.443884	−	0.768830i	0.997974	+	0.0636271i	$0.0202668\pi$
$282$		0			0
$283$	−9.99854	+	17.3180i	−0.594351	+	1.02945i	0.399287	+	0.916826i	$0.369258\pi$
$283$	−9.99854	+	17.3180i	−0.594351	+	1.02945i	−0.993638	+	0.112621i	$0.964076\pi$
$284$	6.63521	−	11.4925i	0.393727	−	0.681956i
$285$		0			0
$286$	5.55408	−	9.61996i	0.328420	−	0.568840i
$287$		0			0
$288$		0			0
$289$	8.05262	−	13.9475i	0.473684	−	0.820444i
$290$	−15.7017			−0.922038
$291$		0			0
$292$	−6.11109			−0.357625
$293$	7.53278	+	13.0472i	0.440070	+	0.762223i	0.997694	−	0.0678705i	$-0.0216205\pi$
$293$	7.53278	+	13.0472i	0.440070	+	0.762223i	−0.557625	+	0.830093i	$0.688287\pi$
$294$		0			0
$295$	−3.53590	+	6.12435i	−0.205868	+	0.356574i
$296$	−4.50000	+	7.79423i	−0.261557	+	0.453030i
$297$		0			0
$298$	−16.6623	−	28.8599i	−0.965218	−	1.67181i
$299$	−0.136673	−	0.236725i	−0.00790401	−	0.0136901i
$300$		0			0
$301$		0			0
$302$	−12.2142	−	21.1556i	−0.702848	−	1.21737i
$303$		0			0
$304$	−17.5438			−1.00620
$305$	2.94805	+	5.10618i	0.168805	+	0.292379i
$306$		0			0
$307$	−27.2704			−1.55641			−0.778203	−	0.628013i	$-0.783868\pi$
$307$	−27.2704			−1.55641			−0.778203	+	0.628013i	$0.783868\pi$
$308$		0			0
$309$		0			0
$310$	14.8525			0.843567
$311$	−7.99115	+	13.8411i	−0.453136	+	0.784855i	−0.998579	−	0.0532931i	$-0.983028\pi$
$311$	−7.99115	+	13.8411i	−0.453136	+	0.784855i	0.545443	+	0.838148i	$0.316362\pi$
$312$		0			0
$313$	−5.79893	−	10.0440i	−0.327775	−	0.567722i	0.654295	−	0.756239i	$-0.272965\pi$
$313$	−5.79893	−	10.0440i	−0.327775	−	0.567722i	−0.982070	+	0.188517i	$0.939632\pi$
$314$	14.8961			0.840636
$315$		0			0
$316$	59.6313			3.35452
$317$	−1.00885	−	1.74739i	−0.0566629	−	0.0981430i	0.836303	−	0.548268i	$-0.184713\pi$
$317$	−1.00885	−	1.74739i	−0.0566629	−	0.0981430i	−0.892965	+	0.450125i	$0.851379\pi$
$318$		0			0
$319$	−5.55408	+	9.61996i	−0.310969	+	0.538614i
$320$	−19.0043			−1.06237
$321$		0			0
$322$		0			0
$323$	−3.83482			−0.213375
$324$		0			0
$325$	−0.863327	−	1.49533i	−0.0478888	−	0.0829458i
$326$	43.8391			2.42802
$327$		0			0
$328$	−16.1893	−	28.0407i	−0.893904	−	1.54829i
$329$		0			0
$330$		0			0
$331$	9.85447	+	17.0684i	0.541651	+	0.938167i	0.998809	+	0.0487815i	$0.0155338\pi$
$331$	9.85447	+	17.0684i	0.541651	+	0.938167i	−0.457159	+	0.889385i	$0.651133\pi$
$332$	−1.91741	−	3.32105i	−0.105232	−	0.182266i
$333$		0			0
$334$	−10.4174	+	18.0435i	−0.570015	+	0.987296i
$335$	20.5115	−	35.5269i	1.12066	−	1.94104i
$336$		0			0
$337$	14.5256	+	25.1590i	0.791259	+	1.37050i	0.925188	+	0.379509i	$0.123907\pi$
$337$	14.5256	+	25.1590i	0.791259	+	1.37050i	−0.133929	+	0.990991i	$0.542759\pi$
$338$	−29.5261			−1.60601
$339$		0			0
$340$	9.94592			0.539393
$341$	5.25370	−	9.09967i	0.284504	−	0.492775i
$342$		0			0
$343$		0			0
$344$	26.3712	−	45.6763i	1.42184	−	2.46270i
$345$		0			0
$346$	21.3530	−	36.9845i	1.14794	−	1.98830i
$347$	14.5416	−	25.1868i	0.780636	−	1.35210i	−0.150936	−	0.988544i	$-0.548229\pi$
$347$	14.5416	−	25.1868i	0.780636	−	1.35210i	0.931572	−	0.363557i	$-0.118438\pi$
$348$		0			0
$349$	−12.3815	+	21.4454i	−0.662767	+	1.14795i	0.317118	+	0.948386i	$0.397285\pi$
$349$	−12.3815	+	21.4454i	−0.662767	+	1.14795i	−0.979885	+	0.199561i	$0.936049\pi$
$350$		0			0
$351$		0			0
$352$	1.21780	−	2.10929i	0.0649089	−	0.112426i
$353$	33.3025			1.77251			0.886257	−	0.463193i	$-0.153296\pi$
$353$	33.3025			1.77251			0.886257	+	0.463193i	$0.153296\pi$
$354$		0			0
$355$	−8.48968			−0.450586
$356$	−29.1015	−	50.4052i	−1.54237	−	2.67147i
$357$		0			0
$358$	−13.9626	+	24.1840i	−0.737949	+	1.27816i
$359$	12.7683	−	22.1153i	0.673884	−	1.16720i	−0.302909	−	0.953019i	$-0.597958\pi$
$359$	12.7683	−	22.1153i	0.673884	−	1.16720i	0.976794	−	0.214182i	$-0.0687087\pi$
$360$		0			0
$361$	1.28220	+	2.22084i	0.0674842	+	0.116886i
$362$	−26.9289	−	46.6422i	−1.41535	−	2.45146i
$363$		0			0
$364$		0			0
$365$	1.95477	+	3.38576i	0.102317	+	0.177219i
$366$		0			0
$367$	27.4504			1.43290			0.716449	−	0.697639i	$-0.245766\pi$
$367$	27.4504			1.43290			0.716449	+	0.697639i	$0.245766\pi$
$368$	−0.591443	−	1.02441i	−0.0308311	−	0.0534011i
$369$		0			0
$370$	11.3638			0.590776
$371$		0			0
$372$		0			0
$373$	16.3274			0.845402			0.422701	−	0.906269i	$-0.361082\pi$
$373$	16.3274			0.845402			0.422701	+	0.906269i	$0.361082\pi$
$374$	5.25370	−	9.09967i	0.271662	−	0.470533i
$375$		0			0
$376$	−30.7345	−	53.2338i	−1.58501	−	2.74532i
$377$	−2.46050			−0.126722
$378$		0			0
$379$	12.0364			0.618267			0.309134	−	0.951019i	$-0.399961\pi$
$379$	12.0364			0.618267			0.309134	+	0.951019i	$0.399961\pi$
$380$	21.3135	+	36.9161i	1.09336	+	1.89376i
$381$		0			0
$382$	−0.863327	+	1.49533i	−0.0441716	+	0.0765075i
$383$	12.4356			0.635429			0.317715	−	0.948186i	$-0.397085\pi$
$383$	12.4356			0.635429			0.317715	+	0.948186i	$0.397085\pi$
$384$		0			0
$385$		0			0
$386$	29.8817			1.52094
$387$		0			0
$388$	23.2901	+	40.3396i	1.18237	+	2.04793i
$389$	−20.6008			−1.04450			−0.522250	−	0.852792i	$-0.674907\pi$
$389$	−20.6008			−1.04450			−0.522250	+	0.852792i	$0.674907\pi$
$390$		0			0
$391$	−0.129281	−	0.223922i	−0.00653803	−	0.0113242i
$392$		0			0
$393$		0			0
$394$	−20.1893	−	34.9689i	−1.01712	−	1.76171i
$395$	−19.0744	−	33.0378i	−0.959738	−	1.66231i
$396$		0			0
$397$	11.8186	−	20.4704i	0.593157	−	1.02738i	−0.400647	−	0.916233i	$-0.631215\pi$
$397$	11.8186	−	20.4704i	0.593157	−	1.02738i	0.993804	−	0.111146i	$-0.0354521\pi$
$398$	−27.9341	+	48.3833i	−1.40021	+	2.42524i
$399$		0			0
$400$	−3.73599	−	6.47092i	−0.186799	−	0.323546i
$401$	2.56440			0.128060			0.0640300	−	0.997948i	$-0.479605\pi$
$401$	2.56440			0.128060			0.0640300	+	0.997948i	$0.479605\pi$
$402$		0			0
$403$	2.32743			0.115938
$404$	−7.45904	+	12.9194i	−0.371101	+	0.642766i
$405$		0			0
$406$		0			0
$407$	4.01965	−	6.96224i	0.199247	−	0.345105i
$408$		0			0
$409$	17.1623	−	29.7259i	0.848619	−	1.46985i	−0.0338223	−	0.999428i	$-0.510768\pi$
$409$	17.1623	−	29.7259i	0.848619	−	1.46985i	0.882441	−	0.470423i	$-0.155899\pi$
$410$	−20.4413	+	35.4054i	−1.00953	+	1.74855i
$411$		0			0
$412$	19.7163	−	34.1497i	0.971354	−	1.68243i
$413$		0			0
$414$		0			0
$415$	−1.22665	+	2.12463i	−0.0602141	+	0.104294i
$416$	0.539495			0.0264509
$417$		0			0
$418$	45.0335			2.20266
$419$	2.02850	+	3.51347i	0.0990989	+	0.171644i	0.911312	−	0.411717i	$-0.135071\pi$
$419$	2.02850	+	3.51347i	0.0990989	+	0.171644i	−0.812213	+	0.583361i	$0.801737\pi$
$420$		0			0
$421$	10.5344	−	18.2462i	0.513417	−	0.889264i	−0.486462	−	0.873702i	$-0.661713\pi$
$421$	10.5344	−	18.2462i	0.513417	−	0.889264i	0.999879	−	0.0155624i	$-0.00495387\pi$
$422$	−5.61177	+	9.71987i	−0.273177	+	0.473156i
$423$		0			0
$424$	−15.8530	−	27.4582i	−0.769890	−	1.33349i
$425$	−0.816635	−	1.41445i	−0.0396126	−	0.0686110i
$426$		0			0
$427$		0			0
$428$	−2.78580	−	4.82515i	−0.134657	−	0.233232i
$429$		0			0
$430$	−66.5949			−3.21149
$431$	11.3092	+	19.5882i	0.544747	+	0.943530i	0.998623	+	0.0524646i	$0.0167077\pi$
$431$	11.3092	+	19.5882i	0.544747	+	0.943530i	−0.453876	+	0.891065i	$0.649959\pi$
$432$		0			0
$433$	2.41789			0.116196			0.0580982	−	0.998311i	$-0.481496\pi$
$433$	2.41789			0.116196			0.0580982	+	0.998311i	$0.481496\pi$
$434$		0			0
$435$		0			0
$436$	−13.7807			−0.659978
$437$	0.554084	−	0.959702i	0.0265054	−	0.0459088i
$438$		0			0
$439$	11.7448	+	20.3427i	0.560551	+	0.970902i	0.997448	+	0.0713911i	$0.0227438\pi$
$439$	11.7448	+	20.3427i	0.560551	+	0.970902i	−0.436898	+	0.899511i	$0.643923\pi$
$440$	−59.1780			−2.82120
$441$		0			0
$442$	2.32743			0.110705
$443$	−6.70895	−	11.6202i	−0.318752	−	0.552094i	0.661476	−	0.749966i	$-0.269930\pi$
$443$	−6.70895	−	11.6202i	−0.318752	−	0.552094i	−0.980228	+	0.197872i	$0.936597\pi$
$444$		0			0
$445$	−18.6175	+	32.2465i	−0.882554	+	1.52863i
$446$	32.7850			1.55242
$447$		0			0
$448$		0			0
$449$	9.16225			0.432393			0.216197	−	0.976350i	$-0.430635\pi$
$449$	9.16225			0.432393			0.216197	+	0.976350i	$0.430635\pi$
$450$		0			0
$451$	14.4612	+	25.0475i	0.680950	+	1.17944i
$452$	−42.1167			−1.98100
$453$		0			0
$454$	1.69961	+	2.94381i	0.0797667	+	0.138160i
$455$		0			0
$456$		0			0
$457$	−4.40856	−	7.63584i	−0.206224	−	0.357190i	0.744298	−	0.667847i	$-0.232784\pi$
$457$	−4.40856	−	7.63584i	−0.206224	−	0.357190i	−0.950522	+	0.310658i	$0.899451\pi$
$458$	22.1192	+	38.3115i	1.03356	+	1.79018i
$459$		0			0
$460$	−1.43706	+	2.48906i	−0.0670033	+	0.116053i
$461$	−2.82957	+	4.90095i	−0.131786	+	0.228260i	−0.924365	−	0.381509i	$-0.875405\pi$
$461$	−2.82957	+	4.90095i	−0.131786	+	0.228260i	0.792579	+	0.609769i	$0.208738\pi$
$462$		0			0
$463$	−7.86333	−	13.6197i	−0.365440	−	0.632960i	0.623407	−	0.781898i	$-0.285748\pi$
$463$	−7.86333	−	13.6197i	−0.365440	−	0.632960i	−0.988847	+	0.148937i	$0.952415\pi$
$464$	−10.6477			−0.494305
$465$		0			0
$466$	46.7060			2.16361
$467$	−10.9985	+	19.0500i	−0.508952	+	0.881530i	0.490995	+	0.871163i	$0.336633\pi$
$467$	−10.9985	+	19.0500i	−0.508952	+	0.881530i	−0.999946	+	0.0103675i	$0.996700\pi$
$468$		0			0
$469$		0			0
$470$	−38.8068	+	67.2153i	−1.79002	+	3.10041i
$471$		0			0
$472$	−6.89037	+	11.9345i	−0.317155	+	0.549328i
$473$	−23.5562	+	40.8006i	−1.08312	+	1.87601i
$474$		0			0
$475$	3.50000	−	6.06218i	0.160591	−	0.278152i
$476$		0			0
$477$		0			0
$478$	6.01819	−	10.4238i	0.275265	−	0.476774i
$479$	−24.9751			−1.14114			−0.570571	−	0.821249i	$-0.693278\pi$
$479$	−24.9751			−1.14114			−0.570571	+	0.821249i	$0.693278\pi$
$480$		0			0
$481$	1.78074			0.0811947
$482$	32.1826	+	55.7419i	1.46588	+	2.53897i
$483$		0			0
$484$	−19.0167	+	32.9379i	−0.864397	+	1.49718i
$485$	14.8997	−	25.8070i	0.676561	−	1.17184i
$486$		0			0
$487$	8.79893	+	15.2402i	0.398717	+	0.690599i	0.993568	−	0.113238i	$-0.0361221\pi$
$487$	8.79893	+	15.2402i	0.398717	+	0.690599i	−0.594851	+	0.803836i	$0.702789\pi$
$488$	5.74484	+	9.95036i	0.260057	+	0.450432i
$489$		0			0
$490$		0			0
$491$	6.89757	+	11.9469i	0.311283	+	0.539158i	0.978640	−	0.205580i	$-0.0659080\pi$
$491$	6.89757	+	11.9469i	0.311283	+	0.539158i	−0.667358	+	0.744737i	$0.732575\pi$
$492$		0			0
$493$	−2.32743			−0.104822
$494$	4.98755	+	8.63868i	0.224400	+	0.388673i
$495$		0			0
$496$	10.0718			0.452237
$497$		0			0
$498$		0			0
$499$	13.0875			0.585879			0.292939	−	0.956131i	$-0.405367\pi$
$499$	13.0875			0.585879			0.292939	+	0.956131i	$0.405367\pi$
$500$	17.2089	−	29.8068i	0.769607	−	1.33300i
$501$		0			0
$502$	22.7075	+	39.3305i	1.01348	+	1.75541i
$503$	22.3068			0.994611			0.497305	−	0.867576i	$-0.334323\pi$
$503$	22.3068			0.994611			0.497305	+	0.867576i	$0.334323\pi$
$504$		0			0
$505$	9.54377			0.424692
$506$	1.51819	+	2.62958i	0.0674917	+	0.116899i
$507$		0			0
$508$	−1.36333	+	2.36135i	−0.0604879	+	0.104768i
$509$	15.8932			0.704453			0.352226	−	0.935915i	$-0.385425\pi$
$509$	15.8932			0.704453			0.352226	+	0.935915i	$0.385425\pi$
$510$		0			0
$511$		0			0
$512$	−41.4078			−1.82998
$513$		0			0
$514$	−14.4356	−	25.0032i	−0.636727	−	1.10284i
$515$	−25.2268			−1.11163
$516$		0			0
$517$	27.4538	+	47.5514i	1.20742	+	2.09131i
$518$		0			0
$519$		0			0
$520$	−6.55408	−	11.3520i	−0.287416	−	0.497818i
$521$	2.20895	+	3.82600i	0.0967756	+	0.167620i	0.910348	−	0.413843i	$-0.135814\pi$
$521$	2.20895	+	3.82600i	0.0967756	+	0.167620i	−0.813573	+	0.581463i	$0.802480\pi$
$522$		0			0
$523$	−12.6367	+	21.8874i	−0.552563	+	0.957067i	0.445526	+	0.895269i	$0.353017\pi$
$523$	−12.6367	+	21.8874i	−0.552563	+	0.957067i	−0.998089	+	0.0617982i	$0.980316\pi$
$524$	16.0416	−	27.7849i	0.700782	−	1.21379i
$525$		0			0
$526$	−9.25370	−	16.0279i	−0.403480	−	0.698848i
$527$	2.20155			0.0959012
$528$		0			0
$529$	−22.9253			−0.996751
$530$	−20.0167	+	34.6700i	−0.869471	+	1.50597i
$531$		0			0
$532$		0			0
$533$	−3.20321	+	5.54812i	−0.138746	+	0.240316i
$534$		0			0
$535$	−1.78220	+	3.08686i	−0.0770513	+	0.133457i
$536$	39.9705	−	69.2310i	1.72646	−	2.99032i
$537$		0			0
$538$	−23.1716	+	40.1344i	−0.998998	+	1.73032i
$539$		0			0
$540$		0			0
$541$	1.71926	−	2.97785i	0.0739168	−	0.128028i	−0.826698	−	0.562646i	$-0.809783\pi$
$541$	1.71926	−	2.97785i	0.0739168	−	0.128028i	0.900615	+	0.434618i	$0.143117\pi$
$542$	−59.0085			−2.53463
$543$		0			0
$544$	0.510317			0.0218797
$545$	4.40808	+	7.63501i	0.188821	+	0.327048i
$546$		0			0
$547$	3.46410	−	6.00000i	0.148114	−	0.256542i	−0.782416	−	0.622756i	$-0.786013\pi$
$547$	3.46410	−	6.00000i	0.148114	−	0.256542i	0.930531	+	0.366214i	$0.119346\pi$
$548$	−7.44445	+	12.8942i	−0.318011	+	0.550812i
$549$		0			0
$550$	9.58998	+	16.6103i	0.408918	+	0.708267i
$551$	−4.98755	−	8.63868i	−0.212477	−	0.368020i
$552$		0			0
$553$		0			0
$554$	−8.81138	−	15.2618i	−0.374360	−	0.648410i
$555$		0			0
$556$	−8.32743			−0.353162
$557$	16.7917	+	29.0841i	0.711488	+	1.23233i	0.964298	+	0.264818i	$0.0853119\pi$
$557$	16.7917	+	29.0841i	0.711488	+	1.23233i	−0.252810	+	0.967516i	$0.581355\pi$
$558$		0			0
$559$	−10.4356			−0.441379
$560$		0			0
$561$		0			0
$562$	−36.6165			−1.54457
$563$	21.2396	−	36.7880i	0.895142	−	1.55043i	0.0615128	−	0.998106i	$-0.480407\pi$
$563$	21.2396	−	36.7880i	0.895142	−	1.55043i	0.833629	−	0.552325i	$-0.186259\pi$
$564$		0			0
$565$	13.4720	+	23.3341i	0.566770	+	0.981675i
$566$	49.2029			2.06815
$567$		0			0
$568$	−16.5438			−0.694161
$569$	5.20175	+	9.00969i	0.218069	+	0.377706i	0.954217	−	0.299114i	$-0.0966910\pi$
$569$	5.20175	+	9.00969i	0.218069	+	0.377706i	−0.736149	+	0.676820i	$0.763358\pi$
$570$		0			0
$571$	−8.92480	+	15.4582i	−0.373491	+	0.646906i	−0.990100	−	0.140364i	$-0.955173\pi$
$571$	−8.92480	+	15.4582i	−0.373491	+	0.646906i	0.616609	+	0.787270i	$0.288506\pi$
$572$	−18.3025			−0.765267
$573$		0			0
$574$		0			0
$575$	0.471974			0.0196827
$576$		0			0
$577$	−5.97150	−	10.3429i	−0.248597	−	0.430582i	0.714540	−	0.699595i	$-0.246636\pi$
$577$	−5.97150	−	10.3429i	−0.248597	−	0.430582i	−0.963137	+	0.269013i	$0.913303\pi$
$578$	−39.6270			−1.64827
$579$		0			0
$580$	12.9356	+	22.4051i	0.537122	+	0.930322i
$581$		0			0
$582$		0			0
$583$	14.1608	+	24.5272i	0.586480	+	1.01581i
$584$	3.80924	+	6.59780i	0.157628	+	0.273019i
$585$		0			0
$586$	18.5344	−	32.1026i	0.765650	−	1.32615i
$587$	11.9299	−	20.6631i	0.492398	−	0.852859i	−0.507563	−	0.861614i	$-0.669454\pi$
$587$	11.9299	−	20.6631i	0.492398	−	0.852859i	0.999962	+	0.00875568i	$0.00278706\pi$
$588$		0			0
$589$	4.71780	+	8.17147i	0.194394	+	0.336699i
$590$	17.4002			0.716354
$591$		0			0
$592$	7.70602			0.316715
$593$	9.79007	−	16.9569i	0.402030	−	0.696336i	−0.591941	−	0.805981i	$-0.701638\pi$
$593$	9.79007	−	16.9569i	0.402030	−	0.696336i	0.993971	+	0.109645i	$0.0349714\pi$
$594$		0			0
$595$		0			0
$596$	−27.4538	+	47.5514i	−1.12455	+	1.94778i
$597$		0			0
$598$	−0.336285	+	0.582462i	−0.0137517	+	0.0238187i
$599$	9.27335	−	16.0619i	0.378899	−	0.656272i	−0.612004	−	0.790855i	$-0.709636\pi$
$599$	9.27335	−	16.0619i	0.378899	−	0.656272i	0.990902	+	0.134583i	$0.0429696\pi$
$600$		0			0
$601$	9.09931	−	15.7605i	0.371169	−	0.642883i	−0.618577	−	0.785724i	$-0.712290\pi$
$601$	9.09931	−	15.7605i	0.371169	−	0.642883i	0.989746	+	0.142841i	$0.0456238\pi$
$602$		0			0
$603$		0			0
$604$	−20.1249	+	34.8573i	−0.818870	+	1.41832i
$605$	24.3317			0.989224
$606$		0			0
$607$	−22.3097			−0.905524			−0.452762	−	0.891631i	$-0.649561\pi$
$607$	−22.3097			−0.905524			−0.452762	+	0.891631i	$0.649561\pi$
$608$	1.09358	+	1.89413i	0.0443505	+	0.0768173i
$609$		0			0
$610$	7.25370	−	12.5638i	0.293694	−	0.508692i
$611$	−6.08113	+	10.5328i	−0.246016	+	0.426112i
$612$		0			0
$613$	−5.11849	−	8.86548i	−0.206734	−	0.358073i	0.743950	−	0.668235i	$-0.232950\pi$
$613$	−5.11849	−	8.86548i	−0.206734	−	0.358073i	−0.950684	+	0.310162i	$0.899617\pi$
$614$	33.5495	+	58.1094i	1.35395	+	2.34511i
$615$		0			0
$616$		0			0
$617$	−5.66372	−	9.80984i	−0.228013	−	0.394929i	0.729206	−	0.684294i	$-0.239889\pi$
$617$	−5.66372	−	9.80984i	−0.228013	−	0.394929i	−0.957219	+	0.289364i	$0.906556\pi$
$618$		0			0
$619$	8.63327			0.347000			0.173500	−	0.984834i	$-0.444492\pi$
$619$	8.63327			0.347000			0.173500	+	0.984834i	$0.444492\pi$
$620$	−12.2360	−	21.1934i	−0.491409	−	0.851145i
$621$		0			0
$622$	39.3245			1.57677
$623$		0			0
$624$		0			0
$625$	−30.6519			−1.22608
$626$	−14.2683	+	24.7134i	−0.570275	+	0.987746i
$627$		0			0
$628$	−12.2719	−	21.2555i	−0.489701	−	0.848188i
$629$	1.68443			0.0671626
$630$		0			0
$631$	−14.8535			−0.591308			−0.295654	−	0.955295i	$-0.595538\pi$
$631$	−14.8535			−0.591308			−0.295654	+	0.955295i	$0.595538\pi$
$632$	−37.1701	−	64.3805i	−1.47855	−	2.56092i
$633$		0			0
$634$	−2.48229	+	4.29945i	−0.0985844	+	0.170753i
$635$	1.74436			0.0692229
$636$		0			0
$637$		0			0
$638$	27.3317			1.08207
$639$		0			0
$640$	24.7793	+	42.9190i	0.979487	+	1.69652i
$641$	34.1593			1.34921			0.674606	−	0.738178i	$-0.264313\pi$
$641$	34.1593			1.34921			0.674606	+	0.738178i	$0.264313\pi$
$642$		0			0
$643$	5.41741	+	9.38323i	0.213642	+	0.370039i	0.952852	−	0.303437i	$-0.0981341\pi$
$643$	5.41741	+	9.38323i	0.213642	+	0.370039i	−0.739210	+	0.673475i	$0.764801\pi$
$644$		0			0
$645$		0			0
$646$	4.71780	+	8.17147i	0.185619	+	0.321502i
$647$	16.4846	+	28.5522i	0.648077	+	1.12250i	0.983582	+	0.180464i	$0.0577600\pi$
$647$	16.4846	+	28.5522i	0.648077	+	1.12250i	−0.335504	+	0.942039i	$0.608907\pi$
$648$		0			0
$649$	6.15486	−	10.6605i	0.241599	−	0.418462i
$650$	−2.12422	+	3.67926i	−0.0833188	+	0.144312i
$651$		0			0
$652$	−36.1160	−	62.5548i	−1.41441	−	2.44984i
$653$	3.93113			0.153837			0.0769185	−	0.997037i	$-0.475492\pi$
$653$	3.93113			0.153837			0.0769185	+	0.997037i	$0.475492\pi$
$654$		0			0
$655$	−20.5251			−0.801982
$656$	−13.8617	+	24.0091i	−0.541207	+	0.937399i
$657$		0			0
$658$		0			0
$659$	8.40856	−	14.5640i	0.327551	−	0.567335i	−0.654474	−	0.756084i	$-0.727110\pi$
$659$	8.40856	−	14.5640i	0.327551	−	0.567335i	0.982025	+	0.188749i	$0.0604434\pi$
$660$		0			0
$661$	8.51080	−	14.7411i	0.331032	−	0.573364i	−0.651683	−	0.758492i	$-0.725937\pi$
$661$	8.51080	−	14.7411i	0.331032	−	0.573364i	0.982714	+	0.185128i	$0.0592700\pi$
$662$	24.2470	−	41.9970i	0.942386	−	1.63226i
$663$		0			0
$664$	−2.39037	+	4.14024i	−0.0927643	+	0.160672i
$665$		0			0
$666$		0			0
$667$	0.336285	−	0.582462i	0.0130210	−	0.0225530i
$668$	34.3288			1.32822
$669$		0			0
$670$	−100.937			−3.89954
$671$	−5.13161	−	8.88821i	−0.198104	−	0.343126i
$672$		0			0
$673$	−14.3727	+	24.8942i	−0.554025	+	0.959600i	0.443953	+	0.896050i	$0.353576\pi$
$673$	−14.3727	+	24.8942i	−0.554025	+	0.959600i	−0.997979	+	0.0635501i	$0.979758\pi$
$674$	35.7403	−	61.9039i	1.37666	−	2.38445i
$675$		0			0
$676$	24.3245	+	42.1313i	0.935558	+	1.62043i
$677$	−3.01819	−	5.22765i	−0.115998	−	0.200915i	0.802180	−	0.597082i	$-0.203673\pi$
$677$	−3.01819	−	5.22765i	−0.115998	−	0.200915i	−0.918178	+	0.396167i	$0.870340\pi$
$678$		0			0
$679$		0			0
$680$	−6.19961	−	10.7380i	−0.237744	−	0.411785i
$681$		0			0
$682$	−25.8535			−0.989981
$683$	10.2556	+	17.7633i	0.392421	+	0.679693i	0.992768	−	0.120046i	$-0.0383043\pi$
$683$	10.2556	+	17.7633i	0.392421	+	0.679693i	−0.600347	+	0.799739i	$0.704971\pi$
$684$		0			0
$685$	9.52510			0.363935
$686$		0			0
$687$		0			0
$688$	−45.1593			−1.72168
$689$	−3.13667	+	5.43288i	−0.119498	+	0.206976i
$690$		0			0
$691$	7.50146	+	12.9929i	0.285369	+	0.494274i	0.972699	−	0.232072i	$-0.0745505\pi$
$691$	7.50146	+	12.9929i	0.285369	+	0.494274i	−0.687330	+	0.726346i	$0.741217\pi$
$692$	−70.3652			−2.67488
$693$		0			0
$694$	−71.5595			−2.71636
$695$	2.66372	+	4.61369i	0.101040	+	0.175007i
$696$		0			0
$697$	−3.02997	+	5.24806i	−0.114768	+	0.198784i
$698$	60.9296			2.30622
$699$		0			0
$700$		0			0
$701$	−38.5113			−1.45455			−0.727275	−	0.686346i	$-0.759214\pi$
$701$	−38.5113			−1.45455			−0.727275	+	0.686346i	$0.759214\pi$
$702$		0			0
$703$	3.60963	+	6.25206i	0.136140	+	0.235801i
$704$	33.0803			1.24676
$705$		0			0
$706$	−40.9705	−	70.9630i	−1.54195	−	2.67073i
$707$		0			0
$708$		0			0
$709$	−3.82004	−	6.61650i	−0.143465	−	0.248488i	0.785334	−	0.619072i	$-0.212491\pi$
$709$	−3.82004	−	6.61650i	−0.143465	−	0.248488i	−0.928799	+	0.370584i	$0.879158\pi$
$710$	10.4445	+	18.0903i	0.391973	+	0.678918i
$711$		0			0
$712$	−36.2798	+	62.8384i	−1.35964	+	2.35497i
$713$	−0.318097	+	0.550960i	−0.0119128	+	0.0206336i
$714$		0			0
$715$	5.85447	+	10.1402i	0.218945	+	0.379224i
$716$	46.0115			1.71953
$717$		0			0
$718$	−62.8329			−2.34490
$719$	−15.0182	+	26.0123i	−0.560084	+	0.970094i	0.437405	+	0.899265i	$0.355898\pi$
$719$	−15.0182	+	26.0123i	−0.560084	+	0.970094i	−0.997488	+	0.0708289i	$0.977436\pi$
$720$		0			0
$721$		0			0
$722$	3.15486	−	5.46438i	0.117412	−	0.203363i
$723$		0			0
$724$	−44.3697	+	76.8506i	−1.64899	+	2.85613i
$725$	2.12422	−	3.67926i	0.0788916	−	0.136644i
$726$		0			0
$727$	−1.72812	+	2.99319i	−0.0640923	+	0.111011i	−0.896291	−	0.443466i	$-0.853749\pi$
$727$	−1.72812	+	2.99319i	−0.0640923	+	0.111011i	0.832199	+	0.554478i	$0.187082\pi$
$728$		0			0
$729$		0			0
$730$	4.80972	−	8.33068i	0.178016	−	0.308332i
$731$	−9.87120			−0.365099
$732$		0			0
$733$	38.5261			1.42299			0.711496	−	0.702690i	$-0.248018\pi$
$733$	38.5261			1.42299			0.711496	+	0.702690i	$0.248018\pi$
$734$	−33.7709	−	58.4929i	−1.24651	−	2.15901i
$735$		0			0
$736$	−0.0737345	+	0.127712i	−0.00271789	+	0.00470752i
$737$	−35.7039	+	61.8409i	−1.31517	+	2.27794i
$738$		0			0
$739$	−22.5620	−	39.0785i	−0.829955	−	1.43752i	−0.898073	−	0.439847i	$-0.855033\pi$
$739$	−22.5620	−	39.0785i	−0.829955	−	1.43752i	0.0681179	−	0.997677i	$-0.478301\pi$
$740$	−9.36186	−	16.2152i	−0.344149	−	0.596084i
$741$		0			0
$742$		0			0
$743$	4.74338	+	8.21577i	0.174018	+	0.301407i	0.939821	−	0.341668i	$-0.110992\pi$
$743$	4.74338	+	8.21577i	0.174018	+	0.301407i	−0.765803	+	0.643075i	$0.777658\pi$
$744$		0			0
$745$	35.1268			1.28695
$746$	−20.0869	−	34.7915i	−0.735432	−	1.27381i
$747$		0			0
$748$	−17.3126			−0.633013
$749$		0			0
$750$		0			0
$751$	−9.83190			−0.358771			−0.179386	−	0.983779i	$-0.557411\pi$
$751$	−9.83190			−0.358771			−0.179386	+	0.983779i	$0.557411\pi$
$752$	−26.3157	+	45.5800i	−0.959633	+	1.66213i
$753$		0			0
$754$	3.02704	+	5.24299i	0.110238	+	0.190938i
$755$	25.7496			0.937124
$756$		0			0
$757$	−41.8171			−1.51987			−0.759934	−	0.650000i	$-0.774769\pi$
$757$	−41.8171			−1.51987			−0.759934	+	0.650000i	$0.774769\pi$
$758$	−14.8078	−	25.6478i	−0.537843	−	0.931571i
$759$		0			0
$760$	26.5708	−	46.0220i	0.963825	−	1.66939i
$761$	−22.9794			−0.833001			−0.416501	−	0.909135i	$-0.636744\pi$
$761$	−22.9794			−0.833001			−0.416501	+	0.909135i	$0.636744\pi$
$762$		0			0
$763$		0			0
$764$	2.84494			0.102926
$765$		0			0
$766$	−15.2989	−	26.4985i	−0.552773	−	0.957430i
$767$	2.72665			0.0984538
$768$		0			0
$769$	−3.04329	−	5.27113i	−0.109744	−	0.190082i	0.805923	−	0.592021i	$-0.201670\pi$
$769$	−3.04329	−	5.27113i	−0.109744	−	0.190082i	−0.915666	+	0.401939i	$0.868336\pi$
$770$		0			0
$771$		0			0
$772$	−24.6175	−	42.6388i	−0.886003	−	1.53460i
$773$	−20.9107	−	36.2184i	−0.752105	−	1.30268i	−0.946801	−	0.321821i	$-0.895705\pi$
$773$	−20.9107	−	36.2184i	−0.752105	−	1.30268i	0.194695	−	0.980864i	$-0.437628\pi$
$774$		0			0
$775$	−2.00933	+	3.48027i	−0.0721774	+	0.125015i
$776$	29.0349	−	50.2899i	1.04229	−	1.80530i
$777$		0			0
$778$	25.3442	+	43.8974i	0.908632	+	1.57380i
$779$	−25.9722			−0.930550
$780$		0			0
$781$	14.7778			0.528792
$782$	−0.318097	+	0.550960i	−0.0113751	+	0.0197023i
$783$		0			0
$784$		0			0
$785$	−7.85087	+	13.5981i	−0.280210	+	0.485337i
$786$		0			0
$787$	−16.1460	+	27.9657i	−0.575543	+	0.996870i	0.420439	+	0.907321i	$0.361876\pi$
$787$	−16.1460	+	27.9657i	−0.575543	+	0.996870i	−0.995982	+	0.0895491i	$0.971457\pi$
$788$	−33.2652	+	57.6170i	−1.18502	+	2.05252i
$789$		0			0
$790$	−46.9327	+	81.2898i	−1.66979	+	2.89216i
$791$		0			0
$792$		0			0
$793$	1.13667	−	1.96878i	0.0403644	−	0.0699133i
$794$	−58.1593			−2.06400

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:	\( N \)	\(=\)	\( 1323 = 3^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1323.g (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.23025 - 2.13086i) q^{2} +(-2.02704 + 3.51094i) q^{4} +2.59358 q^{5} +5.05408 q^{8} +O(q^{10})\)	\(q+(-1.23025 - 2.13086i) q^{2} +(-2.02704 + 3.51094i) q^{4} +2.59358 q^{5} +5.05408 q^{8} +(-3.19076 - 5.52655i) q^{10} -4.51459 q^{11} +(-0.500000 - 0.866025i) q^{13} +(-2.16372 - 3.74766i) q^{16} +(-0.472958 - 0.819187i) q^{17} +(2.02704 - 3.51094i) q^{19} +(-5.25729 + 9.10590i) q^{20} +(5.55408 + 9.61996i) q^{22} +0.273346 q^{23} +1.72665 q^{25} +(-1.23025 + 2.13086i) q^{26} +(1.23025 - 2.13086i) q^{29} +(-1.16372 + 2.01561i) q^{31} +(-0.269748 + 0.467216i) q^{32} +(-1.16372 + 2.01561i) q^{34} +(-0.890369 + 1.54216i) q^{37} -9.97509 q^{38} +13.1082 q^{40} +(-3.20321 - 5.54812i) q^{41} +(5.21780 - 9.03749i) q^{43} +(9.15126 - 15.8505i) q^{44} +(-0.336285 - 0.582462i) q^{46} +(-6.08113 - 10.5328i) q^{47} +(-2.12422 - 3.67926i) q^{50} +4.05408 q^{52} +(-3.13667 - 5.43288i) q^{53} -11.7089 q^{55} -6.05408 q^{58} +(-1.36333 + 2.36135i) q^{59} +(1.13667 + 1.96878i) q^{61} +5.72665 q^{62} -7.32743 q^{64} +(-1.29679 - 2.24611i) q^{65} +(7.90856 - 13.6980i) q^{67} +3.83482 q^{68} -3.27335 q^{71} +(0.753696 + 1.30544i) q^{73} +4.38151 q^{74} +(8.21780 + 14.2336i) q^{76} +(-7.35447 - 12.7383i) q^{79} +(-5.61177 - 9.71987i) q^{80} +(-7.88151 + 13.6512i) q^{82} +(-0.472958 + 0.819187i) q^{83} +(-1.22665 - 2.12463i) q^{85} -25.6768 q^{86} -22.8171 q^{88} +(-7.17830 + 12.4332i) q^{89} +(-0.554084 + 0.959702i) q^{92} +(-14.9626 + 25.9161i) q^{94} +(5.25729 - 9.10590i) q^{95} +(5.74484 - 9.95036i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6q - q^{2} - 3q^{4} + 10q^{5} + 12q^{8} + O(q^{10}) \)	\( 6q - q^{2} - 3q^{4} + 10q^{5} + 12q^{8} + 4q^{11} - 3q^{13} - 3q^{16} - 12q^{17} + 3q^{19} - 16q^{20} + 15q^{22} + 12q^{25} - q^{26} + q^{29} + 3q^{31} - 8q^{32} + 3q^{34} + 3q^{37} - 16q^{38} + 42q^{40} - 22q^{41} + 3q^{43} + 23q^{44} - 12q^{46} - 9q^{47} + 10q^{50} + 6q^{52} - 18q^{53} - 12q^{55} - 18q^{58} - 9q^{59} + 6q^{61} + 36q^{62} - 24q^{64} - 5q^{65} - 12q^{68} - 18q^{71} - 3q^{73} - 12q^{74} + 21q^{76} - 15q^{79} + 11q^{80} - 9q^{82} - 12q^{83} - 9q^{85} - 68q^{86} - 42q^{88} - 2q^{89} + 15q^{92} - 24q^{94} + 16q^{95} - 3q^{97} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

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