LMFDB - Embedded newform 1323.2.g.h.667.2

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:	$24$
Relative dimension:	$12$ over $\Q(\zeta_{3})$
Twist minimal:	no (minimal twist has level 441)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			667.2
Character	$\chi$	$=$	1323.667
Dual form			1323.2.g.h.361.2

$q$-expansion

$f(q)$	$=$	$q+(-1.35757 - 2.35137i) q^{2} +(-2.68597 + 4.65224i) q^{4} -1.58639 q^{5} +9.15528 q^{8} +O(q^{10})$	\(q+(-1.35757 - 2.35137i) q^{2} +(-2.68597 + 4.65224i) q^{4} -1.58639 q^{5} +9.15528 q^{8} +(2.15363 + 3.73020i) q^{10} +1.34875 q^{11} +(-1.58916 - 2.75251i) q^{13} +(-7.05696 - 12.2230i) q^{16} +(-1.40027 - 2.42534i) q^{17} +(0.312846 - 0.541866i) q^{19} +(4.26101 - 7.38028i) q^{20} +(-1.83102 - 3.17142i) q^{22} +0.284867 q^{23} -2.48336 q^{25} +(-4.31479 + 7.47343i) q^{26} +(-2.27396 + 3.93861i) q^{29} +(3.71502 - 6.43461i) q^{31} +(-10.0053 + 17.3297i) q^{32} +(-3.80191 + 6.58511i) q^{34} +(-4.01126 + 6.94770i) q^{37} -1.69884 q^{38} -14.5239 q^{40} +(5.01329 + 8.68327i) q^{41} +(-3.12937 + 5.42022i) q^{43} +(-3.62271 + 6.27472i) q^{44} +(-0.386726 - 0.669829i) q^{46} +(-5.57383 - 9.65415i) q^{47} +(3.37132 + 5.83930i) q^{50} +17.0738 q^{52} +(1.39349 + 2.41359i) q^{53} -2.13965 q^{55} +12.3482 q^{58} +(-2.28734 + 3.96180i) q^{59} +(-0.192507 - 0.333432i) q^{61} -20.1736 q^{62} +26.1036 q^{64} +(2.52104 + 4.36656i) q^{65} +(1.26958 - 2.19898i) q^{67} +15.0443 q^{68} +1.45208 q^{71} +(-0.234067 - 0.405416i) q^{73} +21.7822 q^{74} +(1.68059 + 2.91087i) q^{76} +(7.85620 + 13.6073i) q^{79} +(11.1951 + 19.3905i) q^{80} +(13.6117 - 23.5762i) q^{82} +(-6.99338 + 12.1129i) q^{83} +(2.22138 + 3.84754i) q^{85} +16.9933 q^{86} +12.3482 q^{88} +(1.29353 - 2.24046i) q^{89} +(-0.765146 + 1.32527i) q^{92} +(-15.1337 + 26.2123i) q^{94} +(-0.496297 + 0.859612i) q^{95} +(-7.22962 + 12.5221i) q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24q - 4q^{2} - 12q^{4} + 24q^{8} + O(q^{10}) $	$ 24q - 4q^{2} - 12q^{4} + 24q^{8} + 40q^{11} - 12q^{16} + 64q^{23} + 24q^{25} - 16q^{29} - 48q^{32} - 12q^{37} - 56q^{44} + 24q^{46} + 4q^{50} - 32q^{53} + 96q^{64} - 60q^{65} - 12q^{67} + 112q^{71} + 136q^{74} + 12q^{79} + 12q^{85} + 152q^{86} - 16q^{92} - 64q^{95} + 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 $

$2$	−1.35757	−	2.35137i	−0.959944	−	1.66267i	−0.722624	−	0.691241i	$-0.757064\pi$
$2$	−1.35757	−	2.35137i	−0.959944	−	1.66267i	−0.237320	−	0.971432i	$-0.576269\pi$
$3$		0			0
$3$		0			0
$4$	−2.68597	+	4.65224i	−1.34299	+	2.32612i
$5$	−1.58639			−0.709457			−0.354728	−	0.934969i	$-0.615427\pi$
$5$	−1.58639			−0.709457			−0.354728	+	0.934969i	$0.615427\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$	9.15528			3.23688
$9$		0			0
$10$	2.15363	+	3.73020i	0.681039	+	1.17959i
$11$	1.34875			0.406664			0.203332	−	0.979110i	$-0.434823\pi$
$11$	1.34875			0.406664			0.203332	+	0.979110i	$0.434823\pi$
$12$		0			0
$13$	−1.58916	−	2.75251i	−0.440754	−	0.763409i	0.556991	−	0.830518i	$-0.311956\pi$
$13$	−1.58916	−	2.75251i	−0.440754	−	0.763409i	−0.997746	+	0.0671096i	$0.978622\pi$
$14$		0			0
$15$		0			0
$16$	−7.05696	−	12.2230i	−1.76424	−	3.05575i
$17$	−1.40027	−	2.42534i	−0.339615	−	0.588230i	0.644745	−	0.764397i	$-0.276963\pi$
$17$	−1.40027	−	2.42534i	−0.339615	−	0.588230i	−0.984360	+	0.176167i	$0.943630\pi$
$18$		0			0
$19$	0.312846	−	0.541866i	0.0717719	−	0.124313i	−0.827906	−	0.560867i	$-0.810468\pi$
$19$	0.312846	−	0.541866i	0.0717719	−	0.124313i	0.899678	+	0.436554i	$0.143801\pi$
$20$	4.26101	−	7.38028i	0.952791	−	1.65028i
$21$		0			0
$22$	−1.83102	−	3.17142i	−0.390375	−	0.676149i
$23$	0.284867			0.0593989			0.0296995	−	0.999559i	$-0.490545\pi$
$23$	0.284867			0.0593989			0.0296995	+	0.999559i	$0.490545\pi$
$24$		0			0
$25$	−2.48336			−0.496671
$26$	−4.31479	+	7.47343i	−0.846199	+	1.46566i
$27$		0			0
$28$		0			0
$29$	−2.27396	+	3.93861i	−0.422264	+	0.731382i	−0.996161	−	0.0875454i	$-0.972098\pi$
$29$	−2.27396	+	3.93861i	−0.422264	+	0.731382i	0.573897	+	0.818928i	$0.305431\pi$
$30$		0			0
$31$	3.71502	−	6.43461i	0.667238	−	1.15569i	−0.311435	−	0.950267i	$-0.600810\pi$
$31$	3.71502	−	6.43461i	0.667238	−	1.15569i	0.978673	−	0.205423i	$-0.0658569\pi$
$32$	−10.0053	+	17.3297i	−1.76870	+	3.06348i
$33$		0			0
$34$	−3.80191	+	6.58511i	−0.652023	+	1.12934i
$35$		0			0
$36$		0			0
$37$	−4.01126	+	6.94770i	−0.659447	+	1.14220i	0.321312	+	0.946973i	$0.395876\pi$
$37$	−4.01126	+	6.94770i	−0.659447	+	1.14220i	−0.980759	+	0.195222i	$0.937457\pi$
$38$	−1.69884			−0.275588
$39$		0			0
$40$	−14.5239			−2.29643
$41$	5.01329	+	8.68327i	0.782944	+	1.35610i	0.930219	+	0.367004i	$0.119616\pi$
$41$	5.01329	+	8.68327i	0.782944	+	1.35610i	−0.147275	+	0.989096i	$0.547050\pi$
$42$		0			0
$43$	−3.12937	+	5.42022i	−0.477224	+	0.826576i	−0.999659	−	0.0261027i	$-0.991690\pi$
$43$	−3.12937	+	5.42022i	−0.477224	+	0.826576i	0.522435	+	0.852679i	$0.325024\pi$
$44$	−3.62271	+	6.27472i	−0.546144	+	0.945950i
$45$		0			0
$46$	−0.386726	−	0.669829i	−0.0570197	−	0.0987609i
$47$	−5.57383	−	9.65415i	−0.813026	−	1.40820i	−0.910737	−	0.412988i	$-0.864485\pi$
$47$	−5.57383	−	9.65415i	−0.813026	−	1.40820i	0.0977106	−	0.995215i	$-0.468848\pi$
$48$		0			0
$49$		0			0
$50$	3.37132	+	5.83930i	0.476777	+	0.825802i
$51$		0			0
$52$	17.0738			2.36771
$53$	1.39349	+	2.41359i	0.191410	+	0.331532i	0.945718	−	0.324989i	$-0.105361\pi$
$53$	1.39349	+	2.41359i	0.191410	+	0.331532i	−0.754308	+	0.656521i	$0.772027\pi$
$54$		0			0
$55$	−2.13965			−0.288510
$56$		0			0
$57$		0			0
$58$	12.3482			1.62140
$59$	−2.28734	+	3.96180i	−0.297787	+	0.515782i	−0.975629	−	0.219425i	$-0.929582\pi$
$59$	−2.28734	+	3.96180i	−0.297787	+	0.515782i	0.677842	+	0.735207i	$0.262915\pi$
$60$		0			0
$61$	−0.192507	−	0.333432i	−0.0246480	−	0.0426916i	0.853438	−	0.521194i	$-0.174513\pi$
$61$	−0.192507	−	0.333432i	−0.0246480	−	0.0426916i	−0.878086	+	0.478502i	$0.841180\pi$
$62$	−20.1736			−2.56205
$63$		0			0
$64$	26.1036			3.26295
$65$	2.52104	+	4.36656i	0.312696	+	0.541605i
$66$		0			0
$67$	1.26958	−	2.19898i	0.155104	−	0.268648i	−0.777993	−	0.628273i	$-0.783762\pi$
$67$	1.26958	−	2.19898i	0.155104	−	0.268648i	0.933097	+	0.359625i	$0.117095\pi$
$68$	15.0443			1.82439
$69$		0			0
$70$		0			0
$71$	1.45208			0.172330			0.0861651	−	0.996281i	$-0.472539\pi$
$71$	1.45208			0.172330			0.0861651	+	0.996281i	$0.472539\pi$
$72$		0			0
$73$	−0.234067	−	0.405416i	−0.0273955	−	0.0474503i	0.852003	−	0.523538i	$-0.175388\pi$
$73$	−0.234067	−	0.405416i	−0.0273955	−	0.0474503i	−0.879398	+	0.476087i	$0.842055\pi$
$74$	21.7822			2.53213
$75$		0			0
$76$	1.68059	+	2.91087i	0.192777	+	0.333900i
$77$		0			0
$78$		0			0
$79$	7.85620	+	13.6073i	0.883892	+	1.53095i	0.846978	+	0.531627i	$0.178419\pi$
$79$	7.85620	+	13.6073i	0.883892	+	1.53095i	0.0369135	+	0.999318i	$0.488247\pi$
$80$	11.1951	+	19.3905i	1.25165	+	2.16792i
$81$		0			0
$82$	13.6117	−	23.5762i	1.50317	−	2.60356i
$83$	−6.99338	+	12.1129i	−0.767623	+	1.32956i	0.171225	+	0.985232i	$0.445228\pi$
$83$	−6.99338	+	12.1129i	−0.767623	+	1.32956i	−0.938848	+	0.344331i	$0.888106\pi$
$84$		0			0
$85$	2.22138	+	3.84754i	0.240942	+	0.417324i
$86$	16.9933			1.83243
$87$		0			0
$88$	12.3482			1.31632
$89$	1.29353	−	2.24046i	0.137114	−	0.237488i	−0.789289	−	0.614022i	$-0.789551\pi$
$89$	1.29353	−	2.24046i	0.137114	−	0.237488i	0.926403	+	0.376534i	$0.122884\pi$
$90$		0			0
$91$		0			0
$92$	−0.765146	+	1.32527i	−0.0797719	+	0.138169i
$93$		0			0
$94$	−15.1337	+	26.2123i	−1.56092	+	2.70359i
$95$	−0.496297	+	0.859612i	−0.0509190	+	0.0881944i
$96$		0			0
$97$	−7.22962	+	12.5221i	−0.734057	+	1.27142i	0.221079	+	0.975256i	$0.429042\pi$
$97$	−7.22962	+	12.5221i	−0.734057	+	1.27142i	−0.955136	+	0.296168i	$0.904291\pi$
$98$		0			0
$99$		0			0
$100$	6.67023	−	11.5532i	0.667023	−	1.15532i
$101$	9.83776			0.978894			0.489447	−	0.872033i	$-0.337199\pi$
$101$	9.83776			0.978894			0.489447	+	0.872033i	$0.337199\pi$
$102$		0			0
$103$	−11.0579			−1.08957			−0.544786	−	0.838575i	$-0.683389\pi$
$103$	−11.0579			−1.08957			−0.544786	+	0.838575i	$0.683389\pi$
$104$	−14.5492	−	25.2000i	−1.42667	−	2.47106i
$105$		0			0
$106$	3.78350	−	6.55322i	0.367486	−	0.636505i
$107$	−0.962153	+	1.66650i	−0.0930149	+	0.161106i	−0.908778	−	0.417279i	$-0.862984\pi$
$107$	−0.962153	+	1.66650i	−0.0930149	+	0.161106i	0.815764	+	0.578386i	$0.196317\pi$
$108$		0			0
$109$	9.30341	+	16.1140i	0.891105	+	1.54344i	0.838553	+	0.544821i	$0.183402\pi$
$109$	9.30341	+	16.1140i	0.891105	+	1.54344i	0.0525523	+	0.998618i	$0.483264\pi$
$110$	2.90472	+	5.03112i	0.276954	+	0.479698i
$111$		0			0
$112$		0			0
$113$	−1.59338	−	2.75982i	−0.149893	−	0.259622i	0.781295	−	0.624162i	$-0.214560\pi$
$113$	−1.59338	−	2.75982i	−0.149893	−	0.259622i	−0.931188	+	0.364540i	$0.881226\pi$
$114$		0			0
$115$	−0.451911			−0.0421410
$116$	−12.2156	−	21.1580i	−1.13419	−	1.96447i
$117$		0			0
$118$	12.4209			1.14344
$119$		0			0
$120$		0			0
$121$	−9.18087			−0.834624
$122$	−0.522682	+	0.905312i	−0.0473214	+	0.0819631i
$123$		0			0
$124$	19.9569	+	34.5664i	1.79218	+	3.10415i
$125$	11.8715			1.06182
$126$		0			0
$127$	−8.37387			−0.743061			−0.371530	−	0.928421i	$-0.621167\pi$
$127$	−8.37387			−0.743061			−0.371530	+	0.928421i	$0.621167\pi$
$128$	−15.4267	−	26.7199i	−1.36354	−	2.36173i
$129$		0			0
$130$	6.84495	−	11.8558i	0.600341	−	1.03982i
$131$	−11.9726			−1.04605			−0.523024	−	0.852318i	$-0.675196\pi$
$131$	−11.9726			−1.04605			−0.523024	+	0.852318i	$0.675196\pi$
$132$		0			0
$133$		0			0
$134$	−6.89415			−0.595564
$135$		0			0
$136$	−12.8199	−	22.2046i	−1.09929	−	1.90403i
$137$	−16.5505			−1.41401			−0.707003	−	0.707211i	$-0.749953\pi$
$137$	−16.5505			−1.41401			−0.707003	+	0.707211i	$0.749953\pi$
$138$		0			0
$139$	−3.95119	−	6.84367i	−0.335136	−	0.580472i	0.648375	−	0.761321i	$-0.275449\pi$
$139$	−3.95119	−	6.84367i	−0.335136	−	0.580472i	−0.983511	+	0.180849i	$0.942116\pi$
$140$		0			0
$141$		0			0
$142$	−1.97130	−	3.41438i	−0.165427	−	0.286529i
$143$	−2.14339	−	3.71245i	−0.179239	−	0.310451i
$144$		0			0
$145$	3.60739	−	6.24819i	0.299578	−	0.518884i
$146$	−0.635523	+	1.10076i	−0.0525962	+	0.0910994i
$147$		0			0
$148$	−21.5483	−	37.3227i	−1.77126	−	3.06791i
$149$	13.6685			1.11977			0.559885	−	0.828570i	$-0.310845\pi$
$149$	13.6685			1.11977			0.559885	+	0.828570i	$0.310845\pi$
$150$		0			0
$151$	3.89963			0.317348			0.158674	−	0.987331i	$-0.449278\pi$
$151$	3.89963			0.317348			0.158674	+	0.987331i	$0.449278\pi$
$152$	2.86420	−	4.96093i	0.232317	−	0.402385i
$153$		0			0
$154$		0			0
$155$	−5.89349	+	10.2078i	−0.473376	+	0.819912i
$156$		0			0
$157$	−0.147176	+	0.254917i	−0.0117459	+	0.0203446i	−0.871839	−	0.489793i	$-0.837072\pi$
$157$	−0.147176	+	0.254917i	−0.0117459	+	0.0203446i	0.860093	+	0.510138i	$0.170406\pi$
$158$	21.3306	−	36.9457i	1.69697	−	2.93925i
$159$		0			0
$160$	15.8723	−	27.4917i	1.25482	−	2.17341i
$161$		0			0
$162$		0			0
$163$	−5.35455	+	9.27436i	−0.419401	+	0.726424i	−0.995879	−	0.0906886i	$-0.971093\pi$
$163$	−5.35455	+	9.27436i	−0.419401	+	0.726424i	0.576478	+	0.817112i	$0.304427\pi$
$164$	−53.8622			−4.20593
$165$		0			0
$166$	37.9759			2.94750
$167$	1.59872	+	2.76907i	0.123713	+	0.214277i	0.921229	−	0.389020i	$-0.127186\pi$
$167$	1.59872	+	2.76907i	0.123713	+	0.214277i	−0.797516	+	0.603298i	$0.793853\pi$
$168$		0			0
$169$	1.44913	−	2.50997i	0.111472	−	0.193074i
$170$	6.03133	−	10.4466i	0.462582	−	0.801215i
$171$		0			0
$172$	−16.8108	−	29.1171i	−1.28181	−	2.22016i
$173$	5.71875	+	9.90517i	0.434789	+	0.753076i	0.997278	−	0.0737284i	$-0.0234898\pi$
$173$	5.71875	+	9.90517i	0.434789	+	0.753076i	−0.562490	+	0.826804i	$0.690156\pi$
$174$		0			0
$175$		0			0
$176$	−9.51809	−	16.4858i	−0.717453	−	1.24266i
$177$		0			0
$178$	−7.02421			−0.526487
$179$	0.549275	+	0.951372i	0.0410547	+	0.0711089i	0.885823	−	0.464024i	$-0.153595\pi$
$179$	0.549275	+	0.951372i	0.0410547	+	0.0711089i	−0.844768	+	0.535133i	$0.820262\pi$
$180$		0			0
$181$	−3.19013			−0.237120			−0.118560	−	0.992947i	$-0.537828\pi$
$181$	−3.19013			−0.237120			−0.118560	+	0.992947i	$0.537828\pi$
$182$		0			0
$183$		0			0
$184$	2.60804			0.192267
$185$	6.36343	−	11.0218i	0.467849	−	0.810338i
$186$		0			0
$187$	−1.88861	−	3.27118i	−0.138109	−	0.239212i
$188$	59.8846			4.36753
$189$		0			0
$190$	2.69503			0.195518
$191$	1.93407	+	3.34992i	0.139945	+	0.242391i	0.927475	−	0.373884i	$-0.121974\pi$
$191$	1.93407	+	3.34992i	0.139945	+	0.242391i	−0.787531	+	0.616275i	$0.788641\pi$
$192$		0			0
$193$	2.06793	−	3.58175i	0.148853	−	0.257820i	−0.781951	−	0.623340i	$-0.785775\pi$
$193$	2.06793	−	3.58175i	0.148853	−	0.257820i	0.930804	+	0.365520i	$0.119109\pi$
$194$	39.2588			2.81862
$195$		0			0
$196$		0			0
$197$	0.889267			0.0633576			0.0316788	−	0.999498i	$-0.489915\pi$
$197$	0.889267			0.0633576			0.0316788	+	0.999498i	$0.489915\pi$
$198$		0			0
$199$	3.16193	+	5.47663i	0.224143	+	0.388228i	0.956062	−	0.293164i	$-0.0947083\pi$
$199$	3.16193	+	5.47663i	0.224143	+	0.388228i	−0.731919	+	0.681392i	$0.761375\pi$
$200$	−22.7358			−1.60767
$201$		0			0
$202$	−13.3554	−	23.1323i	−0.939684	−	1.62758i
$203$		0			0
$204$		0			0
$205$	−7.95305	−	13.7751i	−0.555465	−	0.962093i
$206$	15.0119	+	26.0014i	1.04593	+	1.81160i
$207$		0			0
$208$	−22.4293	+	38.8487i	−1.55519	+	2.69367i
$209$	0.421952	−	0.730843i	0.0291870	−	0.0505535i
$210$		0			0
$211$	5.71291	+	9.89505i	0.393293	+	0.681204i	0.992882	−	0.119105i	$-0.0380025\pi$
$211$	5.71291	+	9.89505i	0.393293	+	0.681204i	−0.599589	+	0.800308i	$0.704669\pi$
$212$	−14.9715			−1.02825
$213$		0			0
$214$	5.22475			0.357156
$215$	4.96441	−	8.59860i	0.338570	−	0.586420i
$216$		0			0
$217$		0			0
$218$	25.2600	−	43.7516i	1.71082	−	2.96323i
$219$		0			0
$220$	5.74705	−	9.95417i	0.387466	−	0.671110i
$221$	−4.45051	+	7.70850i	−0.299373	+	0.518530i
$222$		0			0
$223$	−8.35953	+	14.4791i	−0.559796	+	0.969595i	0.437717	+	0.899113i	$0.355787\pi$
$223$	−8.35953	+	14.4791i	−0.559796	+	0.969595i	−0.997513	+	0.0704822i	$0.977546\pi$
$224$		0			0
$225$		0			0
$226$	−4.32625	+	7.49328i	−0.287778	+	0.498446i
$227$	17.0700			1.13298			0.566489	−	0.824070i	$-0.308302\pi$
$227$	17.0700			1.13298			0.566489	+	0.824070i	$0.308302\pi$
$228$		0			0
$229$	−19.7894			−1.30772			−0.653861	−	0.756615i	$-0.726852\pi$
$229$	−19.7894			−1.30772			−0.653861	+	0.756615i	$0.726852\pi$
$230$	0.613500	+	1.06261i	0.0404530	+	0.0700666i
$231$		0			0
$232$	−20.8187	+	36.0591i	−1.36682	+	2.36740i
$233$	2.96579	−	5.13691i	0.194296	−	0.336530i	−0.752374	−	0.658736i	$-0.771091\pi$
$233$	2.96579	−	5.13691i	0.194296	−	0.336530i	0.946669	+	0.322207i	$0.104425\pi$
$234$		0			0
$235$	8.84228	+	15.3153i	0.576807	+	0.999058i
$236$	−12.2875	−	21.2826i	−0.799847	−	1.38538i
$237$		0			0
$238$		0			0
$239$	10.0277	+	17.3685i	0.648637	+	1.12347i	0.983449	+	0.181187i	$0.0579939\pi$
$239$	10.0277	+	17.3685i	0.648637	+	1.12347i	−0.334812	+	0.942285i	$0.608673\pi$
$240$		0			0
$241$	−29.2887			−1.88665			−0.943326	−	0.331869i	$-0.892321\pi$
$241$	−29.2887			−1.88665			−0.943326	+	0.331869i	$0.892321\pi$
$242$	12.4636	+	21.5877i	0.801193	+	1.38771i
$243$		0			0
$244$	2.06827			0.132408
$245$		0			0
$246$		0			0
$247$	−1.98865			−0.126535
$248$	34.0121	−	58.9107i	2.15977	−	3.74083i
$249$		0			0
$250$	−16.1164	−	27.9144i	−1.01929	−	1.76546i
$251$	−22.7856			−1.43821			−0.719106	−	0.694901i	$-0.755448\pi$
$251$	−22.7856			−1.43821			−0.719106	+	0.694901i	$0.755448\pi$
$252$		0			0
$253$	0.384215			0.0241554
$254$	11.3681	+	19.6901i	0.713297	+	1.23547i
$255$		0			0
$256$	−15.7821	+	27.3354i	−0.986381	+	1.70846i
$257$	−24.2889			−1.51510			−0.757550	−	0.652778i	$-0.773604\pi$
$257$	−24.2889			−1.51510			−0.757550	+	0.652778i	$0.773604\pi$
$258$		0			0
$259$		0			0
$260$	−27.0857			−1.67979
$261$		0			0
$262$	16.2536	+	28.1520i	1.00415	+	1.73924i
$263$	8.61155			0.531011			0.265506	−	0.964109i	$-0.414461\pi$
$263$	8.61155			0.531011			0.265506	+	0.964109i	$0.414461\pi$
$264$		0			0
$265$	−2.21062	−	3.82890i	−0.135797	−	0.235208i
$266$		0			0
$267$		0			0
$268$	6.82011	+	11.8128i	0.416605	+	0.721581i
$269$	7.61561	+	13.1906i	0.464332	+	0.804247i	0.999171	−	0.0407073i	$-0.0129611\pi$
$269$	7.61561	+	13.1906i	0.464332	+	0.804247i	−0.534839	+	0.844954i	$0.679628\pi$
$270$		0			0
$271$	−2.33910	+	4.05144i	−0.142090	+	0.246108i	−0.928284	−	0.371873i	$-0.878716\pi$
$271$	−2.33910	+	4.05144i	−0.142090	+	0.246108i	0.786193	+	0.617981i	$0.212049\pi$
$272$	−19.7633	+	34.2310i	−1.19832	+	2.07556i
$273$		0			0
$274$	22.4684	+	38.9164i	1.35737	+	2.35103i
$275$	−3.34943			−0.201978
$276$		0			0
$277$	−16.3907			−0.984824			−0.492412	−	0.870362i	$-0.663885\pi$
$277$	−16.3907			−0.984824			−0.492412	+	0.870362i	$0.663885\pi$
$278$	−10.7280	+	18.5815i	−0.643423	+	1.11444i
$279$		0			0
$280$		0			0
$281$	−1.75702	+	3.04325i	−0.104815	+	0.181545i	−0.913663	−	0.406473i	$-0.866758\pi$
$281$	−1.75702	+	3.04325i	−0.104815	+	0.181545i	0.808848	+	0.588018i	$0.200092\pi$
$282$		0			0
$283$	13.0354	−	22.5780i	0.774874	−	1.34212i	−0.159992	−	0.987118i	$-0.551147\pi$
$283$	13.0354	−	22.5780i	0.774874	−	1.34212i	0.934865	−	0.355002i	$-0.115520\pi$
$284$	−3.90025	+	6.75543i	−0.231437	+	0.400861i
$285$		0			0
$286$	−5.81958	+	10.0798i	−0.344119	+	0.596031i
$287$		0			0
$288$		0			0
$289$	4.57850	−	7.93019i	0.269323	−	0.466482i
$290$	−19.5891			−1.15031
$291$		0			0
$292$	2.51479			0.147167
$293$	−9.44192	−	16.3539i	−0.551603	−	0.955404i	−0.998159	−	0.0606487i	$-0.980683\pi$
$293$	−9.44192	−	16.3539i	−0.551603	−	0.955404i	0.446556	−	0.894756i	$-0.352650\pi$
$294$		0			0
$295$	3.62863	−	6.28497i	0.211267	−	0.365925i
$296$	−36.7242	+	63.6082i	−2.13455	+	3.69715i
$297$		0			0
$298$	−18.5559	−	32.1398i	−1.07492	−	1.86181i
$299$	−0.452700	−	0.784099i	−0.0261803	−	0.0453456i
$300$		0			0
$301$		0			0
$302$	−5.29401	−	9.16950i	−0.304636	−	0.527645i
$303$		0			0
$304$	−8.83097			−0.506491
$305$	0.305392	+	0.528954i	0.0174867	+	0.0302878i
$306$		0			0
$307$	21.6407			1.23510			0.617551	−	0.786531i	$-0.288125\pi$
$307$	21.6407			1.23510			0.617551	+	0.786531i	$0.288125\pi$
$308$		0			0
$309$		0			0
$310$	32.0032			1.81766
$311$	−2.24724	+	3.89234i	−0.127429	+	0.220714i	−0.922680	−	0.385567i	$-0.874006\pi$
$311$	−2.24724	+	3.89234i	−0.127429	+	0.220714i	0.795251	+	0.606281i	$0.207339\pi$
$312$		0			0
$313$	4.30102	+	7.44958i	0.243108	+	0.421075i	0.961598	−	0.274462i	$-0.0884997\pi$
$313$	4.30102	+	7.44958i	0.243108	+	0.421075i	−0.718490	+	0.695537i	$0.755166\pi$
$314$	0.799206			0.0451018
$315$		0			0
$316$	−84.4062			−4.74822
$317$	−4.03128	−	6.98237i	−0.226419	−	0.392169i	0.730325	−	0.683100i	$-0.239369\pi$
$317$	−4.03128	−	6.98237i	−0.226419	−	0.392169i	−0.956744	+	0.290930i	$0.906035\pi$
$318$		0			0
$319$	−3.06701	+	5.31221i	−0.171719	+	0.297427i
$320$	−41.4105			−2.31492
$321$		0			0
$322$		0			0
$323$	−1.75228			−0.0974992
$324$		0			0
$325$	3.94646	+	6.83546i	0.218910	+	0.379163i
$326$	29.0766			1.61041
$327$		0			0
$328$	45.8981	+	79.4978i	2.53430	+	4.38953i
$329$		0			0
$330$		0			0
$331$	11.4513	+	19.8342i	0.629419	+	1.09019i	0.987668	+	0.156560i	$0.0500405\pi$
$331$	11.4513	+	19.8342i	0.629419	+	1.09019i	−0.358249	+	0.933626i	$0.616626\pi$
$332$	−37.5681	−	65.0698i	−2.06182	−	3.57117i
$333$		0			0
$334$	4.34075	−	7.51840i	0.237515	−	0.411388i
$335$	−2.01405	+	3.48844i	−0.110039	+	0.190594i
$336$		0			0
$337$	−6.81891	−	11.8107i	−0.371450	−	0.643369i	0.618339	−	0.785911i	$-0.287806\pi$
$337$	−6.81891	−	11.8107i	−0.371450	−	0.643369i	−0.989789	+	0.142542i	$0.954472\pi$
$338$	−7.86916			−0.428026
$339$		0			0
$340$	−23.8662			−1.29433
$341$	5.01065	−	8.67869i	0.271342	−	0.469978i
$342$		0			0
$343$		0			0
$344$	−28.6502	+	49.6237i	−1.54472	+	2.67553i
$345$		0			0
$346$	15.5272	−	26.8938i	0.834746	−	1.44582i
$347$	−1.41282	+	2.44707i	−0.0758440	+	0.131366i	−0.901453	−	0.432877i	$-0.857498\pi$
$347$	−1.41282	+	2.44707i	−0.0758440	+	0.131366i	0.825609	+	0.564243i	$0.190832\pi$
$348$		0			0
$349$	1.81202	−	3.13851i	0.0969951	−	0.168000i	−0.813444	−	0.581643i	$-0.802410\pi$
$349$	1.81202	−	3.13851i	0.0969951	−	0.168000i	0.910440	+	0.413642i	$0.135744\pi$
$350$		0			0
$351$		0			0
$352$	−13.4947	+	23.3734i	−0.719268	+	1.24581i
$353$	−2.75401			−0.146581			−0.0732907	−	0.997311i	$-0.523350\pi$
$353$	−2.75401			−0.146581			−0.0732907	+	0.997311i	$0.523350\pi$
$354$		0			0
$355$	−2.30357			−0.122261
$356$	6.94877	+	12.0356i	0.368284	+	0.637887i
$357$		0			0
$358$	1.49135	−	2.58310i	0.0788205	−	0.136521i
$359$	−8.40076	+	14.5505i	−0.443375	+	0.767948i	−0.997937	−	0.0641941i	$-0.979552\pi$
$359$	−8.40076	+	14.5505i	−0.443375	+	0.767948i	0.554562	+	0.832142i	$0.312886\pi$
$360$		0			0
$361$	9.30425	+	16.1154i	0.489698	+	0.848181i
$362$	4.33081	+	7.50119i	0.227622	+	0.394254i
$363$		0			0
$364$		0			0
$365$	0.371322	+	0.643149i	0.0194359	+	0.0336640i
$366$		0			0
$367$	23.9339			1.24934			0.624670	−	0.780889i	$-0.285233\pi$
$367$	23.9339			1.24934			0.624670	+	0.780889i	$0.285233\pi$
$368$	−2.01030	−	3.48193i	−0.104794	−	0.181508i
$369$		0			0
$370$	−34.5551			−1.79644
$371$		0			0
$372$		0			0
$373$	−19.1606			−0.992098			−0.496049	−	0.868295i	$-0.665216\pi$
$373$	−19.1606			−0.992098			−0.496049	+	0.868295i	$0.665216\pi$
$374$	−5.12784	+	8.88168i	−0.265154	+	0.459261i
$375$		0			0
$376$	−51.0299	−	88.3865i	−2.63167	−	4.55818i
$377$	14.4548			0.744458
$378$		0			0
$379$	10.0770			0.517622			0.258811	−	0.965928i	$-0.416669\pi$
$379$	10.0770			0.517622			0.258811	+	0.965928i	$0.416669\pi$
$380$	−2.66608	−	4.61779i	−0.136767	−	0.236888i
$381$		0			0
$382$	5.25127	−	9.09546i	0.268678	−	0.465364i
$383$	−20.1435			−1.02929			−0.514643	−	0.857405i	$-0.672075\pi$
$383$	−20.1435			−1.02929			−0.514643	+	0.857405i	$0.672075\pi$
$384$		0			0
$385$		0			0
$386$	−11.2294			−0.571561
$387$		0			0
$388$	−38.8372	−	67.2679i	−1.97166	−	3.41501i
$389$	−13.3947			−0.679139			−0.339570	−	0.940581i	$-0.610281\pi$
$389$	−13.3947			−0.679139			−0.339570	+	0.940581i	$0.610281\pi$
$390$		0			0
$391$	−0.398891	−	0.690899i	−0.0201728	−	0.0349402i
$392$		0			0
$393$		0			0
$394$	−1.20724	−	2.09100i	−0.0608198	−	0.105343i
$395$	−12.4630	−	21.5866i	−0.627083	−	1.08614i
$396$		0			0
$397$	−9.00664	+	15.6000i	−0.452031	+	0.782940i	−0.998512	−	0.0545313i	$-0.982634\pi$
$397$	−9.00664	+	15.6000i	−0.452031	+	0.782940i	0.546482	+	0.837471i	$0.315967\pi$
$398$	8.58506	−	14.8698i	0.430330	−	0.745354i
$399$		0			0
$400$	17.5249	+	30.3541i	0.876247	+	1.51770i
$401$	−28.8675			−1.44157			−0.720787	−	0.693157i	$-0.756219\pi$
$401$	−28.8675			−1.44157			−0.720787	+	0.693157i	$0.756219\pi$
$402$		0			0
$403$	−23.6151			−1.17635
$404$	−26.4240	+	45.7676i	−1.31464	+	2.27703i
$405$		0			0
$406$		0			0
$407$	−5.41019	+	9.37073i	−0.268173	+	0.464490i
$408$		0			0
$409$	−5.42937	+	9.40395i	−0.268465	+	0.464995i	−0.968466	−	0.249147i	$-0.919850\pi$
$409$	−5.42937	+	9.40395i	−0.268465	+	0.464995i	0.700000	+	0.714142i	$0.253183\pi$
$410$	−21.5936	+	37.4012i	−1.06643	+	1.84711i
$411$		0			0
$412$	29.7014	−	51.4443i	1.46328	−	2.53448i
$413$		0			0
$414$		0			0
$415$	11.0943	−	19.2158i	0.544595	−	0.943267i
$416$	63.6001			3.11825
$417$		0			0
$418$	−2.29131			−0.112072
$419$	−0.247572	−	0.428807i	−0.0120947	−	0.0209486i	0.859915	−	0.510438i	$-0.170517\pi$
$419$	−0.247572	−	0.428807i	−0.0120947	−	0.0209486i	−0.872009	+	0.489489i	$0.837183\pi$
$420$		0			0
$421$	9.50320	−	16.4600i	0.463158	−	0.802212i	−0.535959	−	0.844244i	$-0.680050\pi$
$421$	9.50320	−	16.4600i	0.463158	−	0.802212i	0.999116	+	0.0420318i	$0.0133831\pi$
$422$	15.5113	−	26.8664i	0.755079	−	1.30784i
$423$		0			0
$424$	12.7578	+	22.0971i	0.619572	+	1.07313i
$425$	3.47737	+	6.02298i	0.168677	+	0.292157i
$426$		0			0
$427$		0			0
$428$	−5.16864	−	8.95234i	−0.249835	−	0.432728i
$429$		0			0
$430$	−26.9580			−1.30003
$431$	−8.46073	−	14.6544i	−0.407539	−	0.705878i	0.587074	−	0.809533i	$-0.300280\pi$
$431$	−8.46073	−	14.6544i	−0.407539	−	0.705878i	−0.994613	+	0.103655i	$0.966946\pi$
$432$		0			0
$433$	33.4740			1.60866			0.804330	−	0.594183i	$-0.202524\pi$
$433$	33.4740			1.60866			0.804330	+	0.594183i	$0.202524\pi$
$434$		0			0
$435$		0			0
$436$	−99.9548			−4.78697
$437$	0.0891197	−	0.154360i	0.00426317	−	0.00738403i
$438$		0			0
$439$	10.4657	+	18.1272i	0.499502	+	0.865163i	1.00000		0.000574559i	$-0.000182888\pi$
$439$	10.4657	+	18.1272i	0.499502	+	0.865163i	−0.500498	+	0.865738i	$0.666850\pi$
$440$	−19.5891			−0.933874
$441$		0			0
$442$	24.1674			1.14953
$443$	−15.4290	−	26.7238i	−0.733054	−	1.26969i	−0.955572	−	0.294759i	$-0.904761\pi$
$443$	−15.4290	−	26.7238i	−0.733054	−	1.26969i	0.222517	−	0.974929i	$-0.428573\pi$
$444$		0			0
$445$	−2.05205	+	3.55425i	−0.0972763	+	0.168487i
$446$	45.3945			2.14949
$447$		0			0
$448$		0			0
$449$	33.2789			1.57053			0.785263	−	0.619162i	$-0.212528\pi$
$449$	33.2789			1.57053			0.785263	+	0.619162i	$0.212528\pi$
$450$		0			0
$451$	6.76168	+	11.7116i	0.318395	+	0.551477i
$452$	17.1191			0.805217
$453$		0			0
$454$	−23.1737	−	40.1380i	−1.08760	−	1.88377i
$455$		0			0
$456$		0			0
$457$	−11.8952	−	20.6031i	−0.556434	−	0.963772i	−0.997790	−	0.0664402i	$-0.978836\pi$
$457$	−11.8952	−	20.6031i	−0.556434	−	0.963772i	0.441356	−	0.897332i	$-0.354498\pi$
$458$	26.8654	+	46.5323i	1.25534	+	2.17431i
$459$		0			0
$460$	1.21382	−	2.10240i	0.0565947	−	0.0980249i
$461$	8.53122	−	14.7765i	0.397339	−	0.688211i	−0.596058	−	0.802941i	$-0.703267\pi$
$461$	8.53122	−	14.7765i	0.397339	−	0.688211i	0.993397	+	0.114731i	$0.0366005\pi$
$462$		0			0
$463$	18.1243	+	31.3922i	0.842306	+	1.45892i	0.887940	+	0.459959i	$0.152136\pi$
$463$	18.1243	+	31.3922i	0.842306	+	1.45892i	−0.0456338	+	0.998958i	$0.514531\pi$
$464$	64.1889			2.97990
$465$		0			0
$466$	−16.1051			−0.746052
$467$	4.09580	−	7.09413i	0.189531	−	0.328277i	−0.755563	−	0.655076i	$-0.772637\pi$
$467$	4.09580	−	7.09413i	0.189531	−	0.328277i	0.945094	+	0.326799i	$0.105970\pi$
$468$		0			0
$469$		0			0
$470$	24.0080	−	41.5830i	1.10740	−	1.91808i
$471$		0			0
$472$	−20.9413	+	36.2714i	−0.963900	+	1.66952i
$473$	−4.22074	+	7.31054i	−0.194070	+	0.336139i
$474$		0			0
$475$	−0.776909	+	1.34565i	−0.0356470	+	0.0617425i
$476$		0			0
$477$		0			0
$478$	27.2265	−	47.1577i	1.24531	−	2.15694i
$479$	−25.5549			−1.16763			−0.583817	−	0.811885i	$-0.698441\pi$
$479$	−25.5549			−1.16763			−0.583817	+	0.811885i	$0.698441\pi$
$480$		0			0
$481$	25.4982			1.16262
$482$	39.7614	+	68.8687i	1.81108	+	3.13688i
$483$		0			0
$484$	24.6596	−	42.7116i	1.12089	−	1.94144i
$485$	11.4690	−	19.8649i	0.520782	−	0.902020i
$486$		0			0
$487$	3.46140	+	5.99533i	0.156851	+	0.271674i	0.933732	−	0.357974i	$-0.116532\pi$
$487$	3.46140	+	5.99533i	0.156851	+	0.271674i	−0.776880	+	0.629648i	$0.783199\pi$
$488$	−1.76246	−	3.05266i	−0.0797826	−	0.138188i
$489$		0			0
$490$		0			0
$491$	−18.7262	−	32.4348i	−0.845103	−	1.46376i	−0.885532	−	0.464578i	$-0.846206\pi$
$491$	−18.7262	−	32.4348i	−0.845103	−	1.46376i	0.0404294	−	0.999182i	$-0.487127\pi$
$492$		0			0
$493$	12.7366			0.573628
$494$	2.69973	+	4.67607i	0.121467	+	0.210386i
$495$		0			0
$496$	−104.867			−4.70867
$497$		0			0
$498$		0			0
$499$	25.6250			1.14713			0.573566	−	0.819159i	$-0.305560\pi$
$499$	25.6250			1.14713			0.573566	+	0.819159i	$0.305560\pi$
$500$	−31.8867	+	55.2293i	−1.42601	+	2.46993i
$501$		0			0
$502$	30.9329	+	53.5774i	1.38060	+	2.39127i
$503$	−5.79692			−0.258472			−0.129236	−	0.991614i	$-0.541252\pi$
$503$	−5.79692			−0.258472			−0.129236	+	0.991614i	$0.541252\pi$
$504$		0			0
$505$	−15.6066			−0.694483
$506$	−0.521598	−	0.903434i	−0.0231878	−	0.0401625i
$507$		0			0
$508$	22.4920	−	38.9573i	0.997921	−	1.72845i
$509$	25.1395			1.11429			0.557144	−	0.830416i	$-0.311897\pi$
$509$	25.1395			1.11429			0.557144	+	0.830416i	$0.311897\pi$
$510$		0			0
$511$		0			0
$512$	23.9940			1.06039
$513$		0			0
$514$	32.9738	+	57.1123i	1.45441	+	2.51911i
$515$	17.5423			0.773004
$516$		0			0
$517$	−7.51771	−	13.0211i	−0.330629	−	0.572665i
$518$		0			0
$519$		0			0
$520$	23.0808	+	39.9771i	1.01216	+	1.75311i
$521$	−3.64828	−	6.31900i	−0.159834	−	0.276841i	0.774975	−	0.631992i	$-0.217763\pi$
$521$	−3.64828	−	6.31900i	−0.159834	−	0.276841i	−0.934809	+	0.355152i	$0.884429\pi$
$522$		0			0
$523$	8.38637	−	14.5256i	0.366710	−	0.635161i	−0.622339	−	0.782748i	$-0.713817\pi$
$523$	8.38637	−	14.5256i	0.366710	−	0.635161i	0.989049	+	0.147587i	$0.0471506\pi$
$524$	32.1580	−	55.6993i	1.40483	−	2.43324i
$525$		0			0
$526$	−11.6908	−	20.2490i	−0.509741	−	0.882898i
$527$	−20.8081			−0.906416
$528$		0			0
$529$	−22.9189			−0.996472
$530$	−6.00212	+	10.3960i	−0.260716	+	0.451573i
$531$		0			0
$532$		0			0
$533$	15.9339	−	27.5982i	0.690172	−	1.19541i
$534$		0			0
$535$	1.52635	−	2.64372i	0.0659900	−	0.114298i
$536$	11.6234	−	20.1322i	0.502053	−	0.869581i
$537$		0			0
$538$	20.6774	−	35.8143i	0.891466	−	1.54406i
$539$		0			0
$540$		0			0
$541$	2.64908	−	4.58834i	0.113893	−	0.197268i	−0.803444	−	0.595381i	$-0.797001\pi$
$541$	2.64908	−	4.58834i	0.113893	−	0.197268i	0.917337	+	0.398112i	$0.130335\pi$
$542$	12.7019			0.545595
$543$		0			0
$544$	56.0404			2.40271
$545$	−14.7589	−	25.5631i	−0.632200	−	1.09500i
$546$		0			0
$547$	16.4325	−	28.4619i	0.702603	−	1.21694i	−0.264947	−	0.964263i	$-0.585354\pi$
$547$	16.4325	−	28.4619i	0.702603	−	1.21694i	0.967550	−	0.252681i	$-0.0813123\pi$
$548$	44.4542	−	76.9970i	1.89899	−	3.28915i
$549$		0			0
$550$	4.54708	+	7.87577i	0.193888	+	0.335824i
$551$	1.42280	+	2.46436i	0.0606133	+	0.104985i
$552$		0			0
$553$		0			0
$554$	22.2515	+	38.5408i	0.945376	+	1.63744i
$555$		0			0
$556$	42.4512			1.80033
$557$	−9.40798	−	16.2951i	−0.398629	−	0.690446i	0.594928	−	0.803779i	$-0.297181\pi$
$557$	−9.40798	−	16.2951i	−0.398629	−	0.690446i	−0.993557	+	0.113333i	$0.963847\pi$
$558$		0			0
$559$	19.8923			0.841354
$560$		0			0
$561$		0			0
$562$	9.54108			0.402466
$563$	−13.8325	+	23.9586i	−0.582970	+	1.00973i	0.412155	+	0.911114i	$0.364776\pi$
$563$	−13.8325	+	23.9586i	−0.582970	+	1.00973i	−0.995125	+	0.0986197i	$0.968557\pi$
$564$		0			0
$565$	2.52773	+	4.37816i	0.106343	+	0.184191i
$566$	−70.7856			−2.97534
$567$		0			0
$568$	13.2942			0.557812
$569$	−20.0916	−	34.7996i	−0.842282	−	1.45888i	−0.887961	−	0.459920i	$-0.847878\pi$
$569$	−20.0916	−	34.7996i	−0.842282	−	1.45888i	0.0456782	−	0.998956i	$-0.485455\pi$
$570$		0			0
$571$	3.40565	−	5.89875i	0.142522	−	0.246855i	−0.785924	−	0.618323i	$-0.787812\pi$
$571$	3.40565	−	5.89875i	0.142522	−	0.246855i	0.928446	+	0.371468i	$0.121146\pi$
$572$	23.0283			0.962862
$573$		0			0
$574$		0			0
$575$	−0.707427			−0.0295017
$576$		0			0
$577$	−18.2111	−	31.5425i	−0.758138	−	1.31313i	−0.943799	−	0.330519i	$-0.892776\pi$
$577$	−18.2111	−	31.5425i	−0.758138	−	1.31313i	0.185661	−	0.982614i	$-0.440557\pi$
$578$	−24.8625			−1.03414
$579$		0			0
$580$	19.3787	+	33.5649i	0.804658	+	1.39371i
$581$		0			0
$582$		0			0
$583$	1.87947	+	3.25534i	0.0778397	+	0.134822i
$584$	−2.14295	−	3.71170i	−0.0886759	−	0.153591i
$585$		0			0
$586$	−25.6361	+	44.4030i	−1.05902	+	1.83427i
$587$	5.57943	−	9.66385i	0.230288	−	0.398870i	−0.727605	−	0.685996i	$-0.759367\pi$
$587$	5.57943	−	9.66385i	0.230288	−	0.398870i	0.957893	+	0.287126i	$0.0927000\pi$
$588$		0			0
$589$	−2.32446	−	4.02609i	−0.0957779	−	0.165892i
$590$	−19.7044			−0.811218
$591$		0			0
$592$	113.229			4.65369
$593$	−9.90427	+	17.1547i	−0.406720	+	0.704459i	−0.994520	−	0.104547i	$-0.966661\pi$
$593$	−9.90427	+	17.1547i	−0.406720	+	0.704459i	0.587800	+	0.809006i	$0.299994\pi$
$594$		0			0
$595$		0			0
$596$	−36.7133	+	63.5893i	−1.50384	+	2.60472i
$597$		0			0
$598$	−1.22914	+	2.12893i	−0.0502633	+	0.0870586i
$599$	−9.06600	+	15.7028i	−0.370427	+	0.641598i	−0.989631	−	0.143632i	$-0.954122\pi$
$599$	−9.06600	+	15.7028i	−0.370427	+	0.641598i	0.619204	+	0.785230i	$0.287455\pi$
$600$		0			0
$601$	12.3285	−	21.3536i	0.502889	−	0.871030i	−0.497105	−	0.867690i	$-0.665604\pi$
$601$	12.3285	−	21.3536i	0.502889	−	0.871030i	0.999994	−	0.00333942i	$-0.00106297\pi$
$602$		0			0
$603$		0			0
$604$	−10.4743	+	18.1420i	−0.426194	+	0.738189i
$605$	14.5645			0.592130
$606$		0			0
$607$	−17.2775			−0.701273			−0.350637	−	0.936512i	$-0.614035\pi$
$607$	−17.2775			−0.701273			−0.350637	+	0.936512i	$0.614035\pi$
$608$	6.26024	+	10.8431i	0.253886	+	0.439744i
$609$		0			0
$610$	0.829179	−	1.43618i	0.0335725	−	0.0581492i
$611$	−17.7154	+	30.6840i	−0.716689	+	1.24134i
$612$		0			0
$613$	−9.77828	−	16.9365i	−0.394941	−	0.684058i	0.598153	−	0.801382i	$-0.295902\pi$
$613$	−9.77828	−	16.9365i	−0.394941	−	0.684058i	−0.993094	+	0.117324i	$0.962568\pi$
$614$	−29.3787	−	50.8855i	−1.18563	−	2.05357i
$615$		0			0
$616$		0			0
$617$	−10.8723	−	18.8314i	−0.437702	−	0.758122i	0.559810	−	0.828621i	$-0.310874\pi$
$617$	−10.8723	−	18.8314i	−0.437702	−	0.758122i	−0.997512	+	0.0704988i	$0.977541\pi$
$618$		0			0
$619$	−33.8048			−1.35873			−0.679366	−	0.733800i	$-0.737745\pi$
$619$	−33.8048			−1.35873			−0.679366	+	0.733800i	$0.737745\pi$
$620$	−31.6595	−	54.8359i	−1.27148	−	2.20226i
$621$		0			0
$622$	12.2031			0.489300
$623$		0			0
$624$		0			0
$625$	−6.41615			−0.256646
$626$	11.6778	−	20.2266i	0.466740	−	0.808418i
$627$		0			0
$628$	−0.790623	−	1.36940i	−0.0315493	−	0.0546450i
$629$	22.4674			0.895832
$630$		0			0
$631$	−23.6410			−0.941134			−0.470567	−	0.882364i	$-0.655951\pi$
$631$	−23.6410			−0.941134			−0.470567	+	0.882364i	$0.655951\pi$
$632$	71.9258	+	124.579i	2.86105	+	4.95549i
$633$		0			0
$634$	−10.9454	+	18.9581i	−0.434699	+	0.752921i
$635$	13.2843			0.527169
$636$		0			0
$637$		0			0
$638$	16.6547			0.659365
$639$		0			0
$640$	24.4729	+	42.3883i	0.967375	+	1.67554i
$641$	−15.9180			−0.628724			−0.314362	−	0.949303i	$-0.601791\pi$
$641$	−15.9180			−0.628724			−0.314362	+	0.949303i	$0.601791\pi$
$642$		0			0
$643$	13.2527	+	22.9544i	0.522636	+	0.905231i	0.999653	+	0.0263376i	$0.00838450\pi$
$643$	13.2527	+	22.9544i	0.522636	+	0.905231i	−0.477017	+	0.878894i	$0.658282\pi$
$644$		0			0
$645$		0			0
$646$	2.37883	+	4.12026i	0.0935938	+	0.162109i
$647$	−0.00801958	−	0.0138903i	−0.000315282	−	0.000546085i	0.865868	−	0.500273i	$-0.166767\pi$
$647$	−0.00801958	−	0.0138903i	−0.000315282	−	0.000546085i	−0.866183	+	0.499727i	$0.833434\pi$
$648$		0			0
$649$	−3.08506	+	5.34348i	−0.121099	+	0.209750i
$650$	10.7152	−	18.5592i	0.420283	−	0.727951i
$651$		0			0
$652$	−28.7644	−	49.8214i	−1.12650	−	1.95115i
$653$	33.2879			1.30266			0.651328	−	0.758796i	$-0.274212\pi$
$653$	33.2879			1.30266			0.651328	+	0.758796i	$0.274212\pi$
$654$		0			0
$655$	18.9932			0.742126
$656$	70.7571	−	122.555i	2.76260	−	4.78497i
$657$		0			0
$658$		0			0
$659$	−19.4156	+	33.6288i	−0.756324	+	1.30999i	0.188389	+	0.982094i	$0.439673\pi$
$659$	−19.4156	+	33.6288i	−0.756324	+	1.30999i	−0.944713	+	0.327897i	$0.893660\pi$
$660$		0			0
$661$	2.65322	−	4.59551i	0.103198	−	0.178745i	−0.809802	−	0.586703i	$-0.800426\pi$
$661$	2.65322	−	4.59551i	0.103198	−	0.178745i	0.913001	+	0.407958i	$0.133759\pi$
$662$	31.0917	−	53.8525i	1.20842	−	2.09304i
$663$		0			0
$664$	−64.0264	+	110.897i	−2.48471	+	4.30364i
$665$		0			0
$666$		0			0
$667$	−0.647777	+	1.12198i	−0.0250820	+	0.0434433i
$668$	−17.1765			−0.664579
$669$		0			0
$670$	10.9368			0.422527
$671$	−0.259644	−	0.449717i	−0.0100235	−	0.0173611i
$672$		0			0
$673$	−3.03565	+	5.25789i	−0.117016	+	0.202677i	−0.918584	−	0.395227i	$-0.870666\pi$
$673$	−3.03565	+	5.25789i	−0.117016	+	0.202677i	0.801568	+	0.597903i	$0.203999\pi$
$674$	−18.5142	+	32.0676i	−0.713142	+	1.23520i
$675$		0			0
$676$	7.78465	+	13.4834i	0.299410	+	0.518593i
$677$	17.3925	+	30.1247i	0.668449	+	1.15779i	0.978338	+	0.207014i	$0.0663747\pi$
$677$	17.3925	+	30.1247i	0.668449	+	1.15779i	−0.309889	+	0.950773i	$0.600292\pi$
$678$		0			0
$679$		0			0
$680$	20.3373	+	35.2253i	0.779901	+	1.35083i
$681$		0			0
$682$	−27.2091			−1.04189
$683$	9.71206	+	16.8218i	0.371622	+	0.643667i	0.989815	−	0.142358i	$-0.0454686\pi$
$683$	9.71206	+	16.8218i	0.371622	+	0.643667i	−0.618194	+	0.786026i	$0.712135\pi$
$684$		0			0
$685$	26.2556			1.00318
$686$		0			0
$687$		0			0
$688$	88.3352			3.36775
$689$	4.42895	−	7.67117i	0.168730	−	0.292248i
$690$		0			0
$691$	−3.31837	−	5.74759i	−0.126237	−	0.218649i	0.795979	−	0.605324i	$-0.206957\pi$
$691$	−3.31837	−	5.74759i	−0.126237	−	0.218649i	−0.922216	+	0.386676i	$0.873623\pi$
$692$	−61.4416			−2.33566
$693$		0			0
$694$	7.67197			0.291224
$695$	6.26814	+	10.8567i	0.237764	+	0.411820i
$696$		0			0
$697$	14.0399	−	24.3178i	0.531799	−	0.921103i
$698$	−9.83974			−0.372440
$699$		0			0
$700$		0			0
$701$	13.9153			0.525574			0.262787	−	0.964854i	$-0.415358\pi$
$701$	13.9153			0.525574			0.262787	+	0.964854i	$0.415358\pi$
$702$		0			0
$703$	2.50982	+	4.34713i	0.0946595	+	0.163955i
$704$	35.2072			1.32692
$705$		0			0
$706$	3.73876	+	6.47571i	0.140710	+	0.243717i
$707$		0			0
$708$		0			0
$709$	−17.0778	−	29.5796i	−0.641370	−	1.11089i	−0.985127	−	0.171827i	$-0.945033\pi$
$709$	−17.0778	−	29.5796i	−0.641370	−	1.11089i	0.343757	−	0.939059i	$-0.388300\pi$
$710$	3.12725	+	5.41655i	0.117364	+	0.203280i
$711$		0			0
$712$	11.8426	−	20.5120i	0.443821	−	0.768721i
$713$	1.05829	−	1.83301i	0.0396332	−	0.0686467i
$714$		0			0
$715$	3.40025	+	5.88941i	0.127162	+	0.220251i
$716$	−5.90135			−0.220544
$717$		0			0
$718$	45.6183			1.70246
$719$	22.1450	−	38.3563i	0.825870	−	1.43045i	−0.0753825	−	0.997155i	$-0.524018\pi$
$719$	22.1450	−	38.3563i	0.825870	−	1.43045i	0.901253	−	0.433294i	$-0.142649\pi$
$720$		0			0
$721$		0			0
$722$	25.2623	−	43.7556i	0.940165	−	1.62841i
$723$		0			0
$724$	8.56860	−	14.8413i	0.318450	−	0.551571i
$725$	5.64705	−	9.78099i	0.209726	−	0.363257i
$726$		0			0
$727$	−14.1247	+	24.4647i	−0.523857	+	0.907346i	0.475758	+	0.879576i	$0.342174\pi$
$727$	−14.1247	+	24.4647i	−0.523857	+	0.907346i	−0.999614	+	0.0277700i	$0.991159\pi$
$728$		0			0
$729$		0			0
$730$	1.00819	−	1.74623i	0.0373147	−	0.0646310i
$731$	17.5278			0.648290
$732$		0			0
$733$	25.0169			0.924020			0.462010	−	0.886875i	$-0.347128\pi$
$733$	25.0169			0.924020			0.462010	+	0.886875i	$0.347128\pi$
$734$	−32.4919	−	56.2776i	−1.19930	−	2.07724i
$735$		0			0
$736$	−2.85018	+	4.93666i	−0.105059	+	0.181968i
$737$	1.71235	−	2.96587i	0.0630752	−	0.109249i
$738$		0			0
$739$	−16.0115	−	27.7327i	−0.588992	−	1.02016i	−0.994365	−	0.106013i	$-0.966192\pi$
$739$	−16.0115	−	27.7327i	−0.588992	−	1.02016i	0.405373	−	0.914151i	$-0.367142\pi$
$740$	34.1840	+	59.2084i	1.25663	+	2.17655i
$741$		0			0
$742$		0			0
$743$	−19.4031	−	33.6072i	−0.711833	−	1.23293i	−0.964169	−	0.265290i	$-0.914532\pi$
$743$	−19.4031	−	33.6072i	−0.711833	−	1.23293i	0.252336	−	0.967640i	$-0.418801\pi$
$744$		0			0
$745$	−21.6837			−0.794428
$746$	26.0118	+	45.0537i	0.952359	+	1.64953i
$747$		0			0
$748$	20.2911			0.741915
$749$		0			0
$750$		0			0
$751$	21.6991			0.791811			0.395905	−	0.918291i	$-0.370431\pi$
$751$	21.6991			0.791811			0.395905	+	0.918291i	$0.370431\pi$
$752$	−78.6685	+	136.258i	−2.86874	+	4.96881i
$753$		0			0
$754$	−19.6233	−	33.9885i	−0.714638	−	1.23779i
$755$	−6.18635			−0.225144
$756$		0			0
$757$	33.5242			1.21846			0.609229	−	0.792995i	$-0.291479\pi$
$757$	33.5242			1.21846			0.609229	+	0.792995i	$0.291479\pi$
$758$	−13.6802	−	23.6949i	−0.496889	−	0.860637i
$759$		0			0
$760$	−4.54374	+	7.86999i	−0.164819	+	0.285475i
$761$	−13.3210			−0.482884			−0.241442	−	0.970415i	$-0.577620\pi$
$761$	−13.3210			−0.482884			−0.241442	+	0.970415i	$0.577620\pi$
$762$		0			0
$763$		0			0
$764$	−20.7795			−0.751775
$765$		0			0
$766$	27.3462	+	47.3649i	0.988057	+	1.71137i
$767$	14.5398			0.525003
$768$		0			0
$769$	−27.3568	−	47.3833i	−0.986510	−	1.70869i	−0.635022	−	0.772494i	$-0.719009\pi$
$769$	−27.3568	−	47.3833i	−0.986510	−	1.70869i	−0.351488	−	0.936192i	$-0.614324\pi$
$770$		0			0
$771$		0			0
$772$	11.1088	+	19.2410i	0.399814	+	0.692498i
$773$	−1.18021	−	2.04418i	−0.0424491	−	0.0735240i	0.844020	−	0.536311i	$-0.180183\pi$
$773$	−1.18021	−	2.04418i	−0.0424491	−	0.0735240i	−0.886469	+	0.462787i	$0.846849\pi$
$774$		0			0
$775$	−9.22573	+	15.9794i	−0.331398	+	0.573998i
$776$	−66.1892	+	114.643i	−2.37606	+	4.11545i
$777$		0			0
$778$	18.1842	+	31.4960i	0.651936	+	1.12919i
$779$	6.27356			0.224774
$780$		0			0
$781$	1.95850			0.0700805
$782$	−1.08304	+	1.87588i	−0.0387295	+	0.0670814i
$783$		0			0
$784$		0			0
$785$	0.233479	−	0.404398i	0.00833323	−	0.0144336i
$786$		0			0
$787$	−0.833971	+	1.44448i	−0.0297278	+	0.0514901i	−0.880507	−	0.474034i	$-0.842797\pi$
$787$	−0.833971	+	1.44448i	−0.0297278	+	0.0514901i	0.850779	+	0.525524i	$0.176131\pi$
$788$	−2.38855	+	4.13708i	−0.0850884	+	0.147377i
$789$		0			0
$790$	−33.8388	+	58.6105i	−1.20393	+	2.08527i
$791$		0			0
$792$		0			0
$793$	−0.611849	+	1.05975i	−0.0217274	+	0.0376330i
$794$	48.9085			1.73570
$795$		0			0
$796$	−33.9715			−1.20409

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)

\(f(q)\)	\(=\)	\(q+(-1.35757 - 2.35137i) q^{2} +(-2.68597 + 4.65224i) q^{4} -1.58639 q^{5} +9.15528 q^{8} +O(q^{10})\)	\(q+(-1.35757 - 2.35137i) q^{2} +(-2.68597 + 4.65224i) q^{4} -1.58639 q^{5} +9.15528 q^{8} +(2.15363 + 3.73020i) q^{10} +1.34875 q^{11} +(-1.58916 - 2.75251i) q^{13} +(-7.05696 - 12.2230i) q^{16} +(-1.40027 - 2.42534i) q^{17} +(0.312846 - 0.541866i) q^{19} +(4.26101 - 7.38028i) q^{20} +(-1.83102 - 3.17142i) q^{22} +0.284867 q^{23} -2.48336 q^{25} +(-4.31479 + 7.47343i) q^{26} +(-2.27396 + 3.93861i) q^{29} +(3.71502 - 6.43461i) q^{31} +(-10.0053 + 17.3297i) q^{32} +(-3.80191 + 6.58511i) q^{34} +(-4.01126 + 6.94770i) q^{37} -1.69884 q^{38} -14.5239 q^{40} +(5.01329 + 8.68327i) q^{41} +(-3.12937 + 5.42022i) q^{43} +(-3.62271 + 6.27472i) q^{44} +(-0.386726 - 0.669829i) q^{46} +(-5.57383 - 9.65415i) q^{47} +(3.37132 + 5.83930i) q^{50} +17.0738 q^{52} +(1.39349 + 2.41359i) q^{53} -2.13965 q^{55} +12.3482 q^{58} +(-2.28734 + 3.96180i) q^{59} +(-0.192507 - 0.333432i) q^{61} -20.1736 q^{62} +26.1036 q^{64} +(2.52104 + 4.36656i) q^{65} +(1.26958 - 2.19898i) q^{67} +15.0443 q^{68} +1.45208 q^{71} +(-0.234067 - 0.405416i) q^{73} +21.7822 q^{74} +(1.68059 + 2.91087i) q^{76} +(7.85620 + 13.6073i) q^{79} +(11.1951 + 19.3905i) q^{80} +(13.6117 - 23.5762i) q^{82} +(-6.99338 + 12.1129i) q^{83} +(2.22138 + 3.84754i) q^{85} +16.9933 q^{86} +12.3482 q^{88} +(1.29353 - 2.24046i) q^{89} +(-0.765146 + 1.32527i) q^{92} +(-15.1337 + 26.2123i) q^{94} +(-0.496297 + 0.859612i) q^{95} +(-7.22962 + 12.5221i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24q - 4q^{2} - 12q^{4} + 24q^{8} + O(q^{10}) \)	\( 24q - 4q^{2} - 12q^{4} + 24q^{8} + 40q^{11} - 12q^{16} + 64q^{23} + 24q^{25} - 16q^{29} - 48q^{32} - 12q^{37} - 56q^{44} + 24q^{46} + 4q^{50} - 32q^{53} + 96q^{64} - 60q^{65} - 12q^{67} + 112q^{71} + 136q^{74} + 12q^{79} + 12q^{85} + 152q^{86} - 16q^{92} - 64q^{95} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

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