LMFDB - Embedded newform 1323.2.o.e.440.5

Newspace parameters

Level:	$ N $	$=$	$ 1323 = 3^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1323.o (of order $6$, degree $2$, not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$10.5642081874$
Analytic rank:	$0$
Dimension:	$48$
Relative dimension:	$24$ over $\Q(\zeta_{6})$
Twist minimal:	no (minimal twist has level 441)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			440.5
Character	$\chi$	$=$	1323.440
Dual form			1323.2.o.e.881.5

$q$-expansion

$f(q)$	$=$	$q+(-1.61855 - 0.934468i) q^{2} +(0.746462 + 1.29291i) q^{4} +(-1.25287 - 2.17003i) q^{5} +0.947692i q^{8} +O(q^{10})$	\(q+(-1.61855 - 0.934468i) q^{2} +(0.746462 + 1.29291i) q^{4} +(-1.25287 - 2.17003i) q^{5} +0.947692i q^{8} +4.68306i q^{10} +(4.85803 + 2.80479i) q^{11} +(0.384312 - 0.221883i) q^{13} +(2.37851 - 4.11970i) q^{16} -3.07770 q^{17} -2.57419i q^{19} +(1.87044 - 3.23969i) q^{20} +(-5.24197 - 9.07935i) q^{22} +(-6.83476 + 3.94605i) q^{23} +(-0.639351 + 1.10739i) q^{25} -0.829370 q^{26} +(-2.71041 - 1.56485i) q^{29} +(-9.06457 + 5.23343i) q^{31} +(-6.05802 + 3.49760i) q^{32} +(4.98140 + 2.87601i) q^{34} -1.41634 q^{37} +(-2.40550 + 4.16645i) q^{38} +(2.05652 - 1.18733i) q^{40} +(-1.64665 - 2.85208i) q^{41} +(-4.75676 + 8.23894i) q^{43} +8.37467i q^{44} +14.7498 q^{46} +(1.07190 - 1.85659i) q^{47} +(2.06964 - 1.19491i) q^{50} +(0.573749 + 0.331254i) q^{52} +4.85412i q^{53} -14.0561i q^{55} +(2.92461 + 5.06558i) q^{58} +(3.65496 + 6.33057i) q^{59} +(7.40950 + 4.27788i) q^{61} +19.5619 q^{62} +3.55953 q^{64} +(-0.962984 - 0.555979i) q^{65} +(0.934442 + 1.61850i) q^{67} +(-2.29739 - 3.97919i) q^{68} +2.95338i q^{71} -8.51943i q^{73} +(2.29241 + 1.32352i) q^{74} +(3.32820 - 1.92154i) q^{76} +(0.287130 - 0.497324i) q^{79} -11.9198 q^{80} +6.15497i q^{82} +(-4.23521 + 7.33560i) q^{83} +(3.85595 + 6.67870i) q^{85} +(15.3981 - 8.89008i) q^{86} +(-2.65807 + 4.60392i) q^{88} +7.57858 q^{89} +(-10.2038 - 5.89115i) q^{92} +(-3.46984 + 2.00332i) q^{94} +(-5.58607 + 3.22512i) q^{95} +(-3.22662 - 1.86289i) q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 48q + 24q^{4} + O(q^{10}) $	$ 48q + 24q^{4} + 24q^{11} - 24q^{16} + 48q^{23} - 24q^{25} - 120q^{32} - 48q^{50} - 48q^{64} - 120q^{65} + 168q^{74} - 24q^{79} - 24q^{85} - 24q^{86} + 144q^{92} - 96q^{95} + O(q^{100}) $

Display 100 coefficients 10 coefficients

Character values

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

$n$	$785$	$1081$
$\chi(n)$	$e\left(\frac{1}{6}\right)$	$-1$

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 $

$2$	−1.61855	−	0.934468i	−1.14449	−	0.660769i	−0.196948	−	0.980414i	$-0.563103\pi$
$2$	−1.61855	−	0.934468i	−1.14449	−	0.660769i	−0.947537	+	0.319645i	$0.896436\pi$
$3$		0			0
$3$		0			0
$4$	0.746462	+	1.29291i	0.373231	+	0.646455i
$5$	−1.25287	−	2.17003i	−0.560299	−	0.970467i	−0.997470	−	0.0710881i	$-0.977353\pi$
$5$	−1.25287	−	2.17003i	−0.560299	−	0.970467i	0.437171	−	0.899378i	$-0.355980\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$			0.947692i			0.335060i
$9$		0			0
$10$			4.68306i			1.48091i
$11$	4.85803	+	2.80479i	1.46475	+	0.845675i	0.999225	−	0.0393590i	$-0.0125316\pi$
$11$	4.85803	+	2.80479i	1.46475	+	0.845675i	0.465527	+	0.885034i	$0.345865\pi$
$12$		0			0
$13$	0.384312	−	0.221883i	0.106589	−	0.0615392i	−0.445758	−	0.895154i	$-0.647066\pi$
$13$	0.384312	−	0.221883i	0.106589	−	0.0615392i	0.552347	+	0.833614i	$0.313732\pi$
$14$		0			0
$15$		0			0
$16$	2.37851	−	4.11970i	0.594628	−	1.02993i
$17$	−3.07770			−0.746451			−0.373226	−	0.927741i	$-0.621748\pi$
$17$	−3.07770			−0.746451			−0.373226	+	0.927741i	$0.621748\pi$
$18$		0			0
$19$		−	2.57419i		−	0.590560i	−0.955411	−	0.295280i	$-0.904587\pi$
$19$		−	2.57419i		−	0.590560i	0.955411	−	0.295280i	$-0.0954130\pi$
$20$	1.87044	−	3.23969i	0.418242	−	0.724417i
$21$		0			0
$22$	−5.24197	−	9.07935i	−1.11759	−	1.93572i
$23$	−6.83476	+	3.94605i	−1.42515	+	0.822808i	−0.996732	−	0.0807749i	$-0.974261\pi$
$23$	−6.83476	+	3.94605i	−1.42515	+	0.822808i	−0.428413	+	0.903583i	$0.640927\pi$
$24$		0			0
$25$	−0.639351	+	1.10739i	−0.127870	+	0.221478i
$26$	−0.829370			−0.162653
$27$		0			0
$28$		0			0
$29$	−2.71041	−	1.56485i	−0.503310	−	0.290586i	0.226770	−	0.973948i	$-0.427184\pi$
$29$	−2.71041	−	1.56485i	−0.503310	−	0.290586i	−0.730079	+	0.683362i	$0.760517\pi$
$30$		0			0
$31$	−9.06457	+	5.23343i	−1.62805	+	0.939952i	−0.643370	+	0.765556i	$0.722464\pi$
$31$	−9.06457	+	5.23343i	−1.62805	+	0.939952i	−0.984675	+	0.174397i	$0.944202\pi$
$32$	−6.05802	+	3.49760i	−1.07092	+	0.618294i
$33$		0			0
$34$	4.98140	+	2.87601i	0.854303	+	0.493232i
$35$		0			0
$36$		0			0
$37$	−1.41634			−0.232844			−0.116422	−	0.993200i	$-0.537143\pi$
$37$	−1.41634			−0.232844			−0.116422	+	0.993200i	$0.537143\pi$
$38$	−2.40550	+	4.16645i	−0.390224	+	0.675887i
$39$		0			0
$40$	2.05652	−	1.18733i	0.325164	−	0.187734i
$41$	−1.64665	−	2.85208i	−0.257163	−	0.445420i	0.708318	−	0.705894i	$-0.249455\pi$
$41$	−1.64665	−	2.85208i	−0.257163	−	0.445420i	−0.965481	+	0.260474i	$0.916121\pi$
$42$		0			0
$43$	−4.75676	+	8.23894i	−0.725398	+	1.25643i	0.233411	+	0.972378i	$0.425011\pi$
$43$	−4.75676	+	8.23894i	−0.725398	+	1.25643i	−0.958810	+	0.284049i	$0.908322\pi$
$44$			8.37467i			1.26253i
$45$		0			0
$46$	14.7498			2.17474
$47$	1.07190	−	1.85659i	0.156353	−	0.270811i	−0.777198	−	0.629256i	$-0.783360\pi$
$47$	1.07190	−	1.85659i	0.156353	−	0.270811i	0.933551	+	0.358445i	$0.116693\pi$
$48$		0			0
$49$		0			0
$50$	2.06964	−	1.19491i	0.292691	−	0.168985i
$51$		0			0
$52$	0.573749	+	0.331254i	0.0795647	+	0.0459367i
$53$			4.85412i			0.666764i	0.942792	+	0.333382i	$0.108190\pi$
$53$			4.85412i			0.666764i	−0.942792	+	0.333382i	$0.891810\pi$
$54$		0			0
$55$		−	14.0561i		−	1.89532i
$56$		0			0
$57$		0			0
$58$	2.92461	+	5.06558i	0.384020	+	0.665143i
$59$	3.65496	+	6.33057i	0.475835	+	0.824170i	0.999617	−	0.0276824i	$-0.00881270\pi$
$59$	3.65496	+	6.33057i	0.475835	+	0.824170i	−0.523782	+	0.851852i	$0.675479\pi$
$60$		0			0
$61$	7.40950	+	4.27788i	0.948690	+	0.547726i	0.892674	−	0.450704i	$-0.148827\pi$
$61$	7.40950	+	4.27788i	0.948690	+	0.547726i	0.0560160	+	0.998430i	$0.482160\pi$
$62$	19.5619			2.48437
$63$		0			0
$64$	3.55953			0.444941
$65$	−0.962984	−	0.555979i	−0.119443	−	0.0689607i
$66$		0			0
$67$	0.934442	+	1.61850i	0.114160	+	0.197731i	0.917444	−	0.397865i	$-0.130249\pi$
$67$	0.934442	+	1.61850i	0.114160	+	0.197731i	−0.803284	+	0.595597i	$0.796916\pi$
$68$	−2.29739	−	3.97919i	−0.278599	−	0.482548i
$69$		0			0
$70$		0			0
$71$			2.95338i			0.350501i	0.984524	+	0.175251i	$0.0560736\pi$
$71$			2.95338i			0.350501i	−0.984524	+	0.175251i	$0.943926\pi$
$72$		0			0
$73$		−	8.51943i		−	0.997124i	−0.866854	−	0.498562i	$-0.833862\pi$
$73$		−	8.51943i		−	0.997124i	0.866854	−	0.498562i	$-0.166138\pi$
$74$	2.29241	+	1.32352i	0.266487	+	0.153856i
$75$		0			0
$76$	3.32820	−	1.92154i	0.381771	−	0.220415i
$77$		0			0
$78$		0			0
$79$	0.287130	−	0.497324i	0.0323047	−	0.0559533i	−0.849421	−	0.527716i	$-0.823049\pi$
$79$	0.287130	−	0.497324i	0.0323047	−	0.0559533i	0.881726	+	0.471762i	$0.156382\pi$
$80$	−11.9198			−1.33268
$81$		0			0
$82$			6.15497i			0.679702i
$83$	−4.23521	+	7.33560i	−0.464875	+	0.805186i	−0.999196	−	0.0400951i	$-0.987234\pi$
$83$	−4.23521	+	7.33560i	−0.464875	+	0.805186i	0.534321	+	0.845281i	$0.320567\pi$
$84$		0			0
$85$	3.85595	+	6.67870i	0.418236	+	0.724406i
$86$	15.3981	−	8.89008i	1.66042	−	0.958642i
$87$		0			0
$88$	−2.65807	+	4.60392i	−0.283352	+	0.490779i
$89$	7.57858			0.803328			0.401664	−	0.915787i	$-0.368432\pi$
$89$	7.57858			0.803328			0.401664	+	0.915787i	$0.368432\pi$
$90$		0			0
$91$		0			0
$92$	−10.2038	−	5.89115i	−1.06382	−	0.614195i
$93$		0			0
$94$	−3.46984	+	2.00332i	−0.357887	+	0.206626i
$95$	−5.58607	+	3.22512i	−0.573119	+	0.330890i
$96$		0			0
$97$	−3.22662	−	1.86289i	−0.327614	−	0.189148i	0.327167	−	0.944966i	$-0.393906\pi$
$97$	−3.22662	−	1.86289i	−0.327614	−	0.189148i	−0.654781	+	0.755818i	$0.727239\pi$
$98$		0			0
$99$		0			0
$100$	−1.90901			−0.190901
$101$	3.76725	−	6.52506i	0.374855	−	0.649268i	−0.615450	−	0.788176i	$-0.711026\pi$
$101$	3.76725	−	6.52506i	0.374855	−	0.649268i	0.990305	+	0.138908i	$0.0443592\pi$
$102$		0			0
$103$	−12.4045	+	7.16173i	−1.22225	+	0.705666i	−0.965397	−	0.260784i	$-0.916019\pi$
$103$	−12.4045	+	7.16173i	−1.22225	+	0.705666i	−0.256853	+	0.966451i	$0.582685\pi$
$104$	0.210276	+	0.364210i	0.0206193	+	0.0357137i
$105$		0			0
$106$	4.53602	−	7.85662i	0.440577	−	0.763102i
$107$			13.4866i			1.30380i	0.758305	+	0.651900i	$0.226028\pi$
$107$			13.4866i			1.30380i	−0.758305	+	0.651900i	$0.773972\pi$
$108$		0			0
$109$	−0.918695			−0.0879950			−0.0439975	−	0.999032i	$-0.514009\pi$
$109$	−0.918695			−0.0879950			−0.0439975	+	0.999032i	$0.514009\pi$
$110$	−13.1350	+	22.7504i	−1.25237	+	2.16917i
$111$		0			0
$112$		0			0
$113$	−4.10412	+	2.36952i	−0.386083	+	0.222905i	−0.680462	−	0.732784i	$-0.738221\pi$
$113$	−4.10412	+	2.36952i	−0.386083	+	0.222905i	0.294378	+	0.955689i	$0.404887\pi$
$114$		0			0
$115$	17.1261	+	9.88775i	1.59702	+	0.922037i
$116$		−	4.67242i		−	0.433823i
$117$		0			0
$118$		−	13.6618i		−	1.25767i
$119$		0			0
$120$		0			0
$121$	10.2337	+	17.7252i	0.930332	+	1.61138i
$122$	−7.99508	−	13.8479i	−0.723841	−	1.25373i
$123$		0			0
$124$	−13.5327	−	7.81312i	−1.21527	−	0.701639i
$125$	−9.32458			−0.834016
$126$		0			0
$127$	7.37245			0.654200			0.327100	−	0.944990i	$-0.393929\pi$
$127$	7.37245			0.654200			0.327100	+	0.944990i	$0.393929\pi$
$128$	6.35477	+	3.66893i	0.561688	+	0.324291i
$129$		0			0
$130$	1.03909	+	1.79976i	0.0911342	+	0.157849i
$131$	−3.93150	−	6.80955i	−0.343497	−	0.594953i	0.641583	−	0.767054i	$-0.278278\pi$
$131$	−3.93150	−	6.80955i	−0.343497	−	0.594953i	−0.985079	+	0.172100i	$0.944945\pi$
$132$		0			0
$133$		0			0
$134$		−	3.49283i		−	0.301734i
$135$		0			0
$136$		−	2.91671i		−	0.250106i
$137$	10.0198	+	5.78491i	0.856046	+	0.494238i	0.862686	−	0.505740i	$-0.168780\pi$
$137$	10.0198	+	5.78491i	0.856046	+	0.494238i	−0.00664016	+	0.999978i	$0.502114\pi$
$138$		0			0
$139$	−16.9741	+	9.79999i	−1.43972	+	0.831224i	−0.997830	−	0.0658437i	$-0.979026\pi$
$139$	−16.9741	+	9.79999i	−1.43972	+	0.831224i	−0.441893	+	0.897068i	$0.645693\pi$
$140$		0			0
$141$		0			0
$142$	2.75984	−	4.78018i	0.231600	−	0.401144i
$143$	2.48933			0.208169
$144$		0			0
$145$			7.84221i			0.651260i
$146$	−7.96114	+	13.7891i	−0.658868	+	1.14119i
$147$		0			0
$148$	−1.05724	−	1.83120i	−0.0869047	−	0.150523i
$149$	13.6315	−	7.87012i	1.11673	−	0.644746i	0.176167	−	0.984360i	$-0.443630\pi$
$149$	13.6315	−	7.87012i	1.11673	−	0.644746i	0.940565	+	0.339615i	$0.110297\pi$
$150$		0			0
$151$	0.991353	−	1.71707i	0.0806752	−	0.139734i	−0.822865	−	0.568237i	$-0.807626\pi$
$151$	0.991353	−	1.71707i	0.0806752	−	0.139734i	0.903540	+	0.428503i	$0.140959\pi$
$152$	2.43954			0.197873
$153$		0			0
$154$		0			0
$155$	22.7134	+	13.1136i	1.82438	+	1.05331i
$156$		0			0
$157$	7.26790	−	4.19612i	0.580041	−	0.334887i	−0.181108	−	0.983463i	$-0.557969\pi$
$157$	7.26790	−	4.19612i	0.580041	−	0.334887i	0.761150	+	0.648576i	$0.224635\pi$
$158$	−0.929468	+	0.536628i	−0.0739445	+	0.0426919i
$159$		0			0
$160$	15.1798	+	8.76405i	1.20007	+	0.692859i
$161$		0			0
$162$		0			0
$163$	−1.07411			−0.0841306			−0.0420653	−	0.999115i	$-0.513394\pi$
$163$	−1.07411			−0.0841306			−0.0420653	+	0.999115i	$0.513394\pi$
$164$	2.45832	−	4.25794i	0.191963	−	0.332489i
$165$		0			0
$166$	13.7098	−	7.91534i	1.06408	−	0.614349i
$167$	3.99731	+	6.92354i	0.309321	+	0.535760i	0.978214	−	0.207599i	$-0.0665649\pi$
$167$	3.99731	+	6.92354i	0.309321	+	0.535760i	−0.668893	+	0.743359i	$0.733232\pi$
$168$		0			0
$169$	−6.40154	+	11.0878i	−0.492426	+	0.852907i
$170$		−	14.4130i		−	1.10543i
$171$		0			0
$172$	−14.2030			−1.08297
$173$	0.501744	−	0.869046i	0.0381469	−	0.0660723i	−0.846322	−	0.532672i	$-0.821188\pi$
$173$	0.501744	−	0.869046i	0.0381469	−	0.0660723i	0.884468	+	0.466600i	$0.154521\pi$
$174$		0			0
$175$		0			0
$176$	23.1098	−	13.3424i	1.74197	−	1.00572i
$177$		0			0
$178$	−12.2663	−	7.08195i	−0.919398	−	0.530814i
$179$		−	1.47539i		−	0.110276i	−0.998479	−	0.0551379i	$-0.982440\pi$
$179$		−	1.47539i		−	0.110276i	0.998479	−	0.0551379i	$-0.0175599\pi$
$180$		0			0
$181$			15.0440i			1.11821i	0.829096	+	0.559106i	$0.188855\pi$
$181$			15.0440i			1.11821i	−0.829096	+	0.559106i	$0.811145\pi$
$182$		0			0
$183$		0			0
$184$	−3.73964	−	6.47724i	−0.275690	−	0.477509i
$185$	1.77448	+	3.07349i	0.130462	+	0.225967i
$186$		0			0
$187$	−14.9516	−	8.63229i	−1.09337	−	0.631255i
$188$	3.20054			0.233423
$189$		0			0
$190$	12.0551			0.874568
$191$	−11.4521	−	6.61187i	−0.828644	−	0.478418i	0.0247439	−	0.999694i	$-0.492123\pi$
$191$	−11.4521	−	6.61187i	−0.828644	−	0.478418i	−0.853388	+	0.521276i	$0.825456\pi$
$192$		0			0
$193$	0.777855	+	1.34728i	0.0559912	+	0.0969796i	0.892662	−	0.450726i	$-0.148835\pi$
$193$	0.777855	+	1.34728i	0.0559912	+	0.0969796i	−0.836671	+	0.547705i	$0.815501\pi$
$194$	3.48163	+	6.03035i	0.249966	+	0.432954i
$195$		0			0
$196$		0			0
$197$			4.96185i			0.353517i	0.984254	+	0.176759i	$0.0565612\pi$
$197$			4.96185i			0.353517i	−0.984254	+	0.176759i	$0.943439\pi$
$198$		0			0
$199$		−	11.1922i		−	0.793394i	−0.917950	−	0.396697i	$-0.870156\pi$
$199$		−	11.1922i		−	0.793394i	0.917950	−	0.396697i	$-0.129844\pi$
$200$	−1.04946	−	0.605908i	−0.0742083	−	0.0428442i
$201$		0			0
$202$	−12.1949	+	7.04074i	−0.858032	+	0.495385i
$203$		0			0
$204$		0			0
$205$	−4.12606	+	7.14655i	−0.288177	+	0.499137i
$206$	26.7696			1.86513
$207$		0			0
$208$		−	2.11100i		−	0.146372i
$209$	7.22006	−	12.5055i	0.499422	−	0.865024i
$210$		0			0
$211$	7.68026	+	13.3026i	0.528731	+	0.915789i	0.999439	+	0.0334999i	$0.0106654\pi$
$211$	7.68026	+	13.3026i	0.528731	+	0.915789i	−0.470708	+	0.882289i	$0.656001\pi$
$212$	−6.27594	+	3.62342i	−0.431033	+	0.248857i
$213$		0			0
$214$	12.6028	−	21.8287i	0.861511	−	1.49218i
$215$	23.8383			1.62576
$216$		0			0
$217$		0			0
$218$	1.48695	+	0.858492i	0.100709	+	0.0581444i
$219$		0			0
$220$	18.1733	−	10.4923i	1.22524	−	0.707394i
$221$	−1.18280	+	0.682888i	−0.0795635	+	0.0459360i
$222$		0			0
$223$	2.76845	+	1.59837i	0.185389	+	0.107034i	0.589822	−	0.807533i	$-0.299198\pi$
$223$	2.76845	+	1.59837i	0.185389	+	0.107034i	−0.404433	+	0.914568i	$0.632531\pi$
$224$		0			0
$225$		0			0
$226$	8.85695			0.589156
$227$	−7.33494	+	12.7045i	−0.486837	+	0.843227i	−0.999885	−	0.0151329i	$-0.995183\pi$
$227$	−7.33494	+	12.7045i	−0.486837	+	0.843227i	0.513048	+	0.858360i	$0.328516\pi$
$228$		0			0
$229$	−2.92550	+	1.68904i	−0.193322	+	0.111615i	−0.593537	−	0.804807i	$-0.702269\pi$
$229$	−2.92550	+	1.68904i	−0.193322	+	0.111615i	0.400215	+	0.916421i	$0.368936\pi$
$230$	−18.4796	−	32.0076i	−1.21851	−	2.11052i
$231$		0			0
$232$	1.48300	−	2.56863i	0.0973637	−	0.168639i
$233$			4.88009i			0.319706i	0.987141	+	0.159853i	$0.0511020\pi$
$233$			4.88009i			0.319706i	−0.987141	+	0.159853i	$0.948898\pi$
$234$		0			0
$235$	−5.37180			−0.350418
$236$	−5.45657	+	9.45106i	−0.355193	+	0.615212i
$237$		0			0
$238$		0			0
$239$	−13.6253	+	7.86657i	−0.881347	+	0.508846i	−0.871102	−	0.491101i	$-0.836594\pi$
$239$	−13.6253	+	7.86657i	−0.881347	+	0.508846i	−0.0102448	+	0.999948i	$0.503261\pi$
$240$		0			0
$241$	0.666305	+	0.384691i	0.0429205	+	0.0247801i	0.521307	−	0.853369i	$-0.325445\pi$
$241$	0.666305	+	0.384691i	0.0429205	+	0.0247801i	−0.478386	+	0.878150i	$0.658778\pi$
$242$		−	38.2521i		−	2.45894i
$243$		0			0
$244$			12.7731i			0.817714i
$245$		0			0
$246$		0			0
$247$	−0.571169	−	0.989293i	−0.0363426	−	0.0629472i
$248$	−4.95968	−	8.59042i	−0.314940	−	0.545492i
$249$		0			0
$250$	15.0923	+	8.71353i	0.954519	+	0.551092i
$251$	1.14544			0.0722996			0.0361498	−	0.999346i	$-0.488491\pi$
$251$	1.14544			0.0722996			0.0361498	+	0.999346i	$0.488491\pi$
$252$		0			0
$253$	−44.2713			−2.78331
$254$	−11.9327	−	6.88933i	−0.748722	−	0.432275i
$255$		0			0
$256$	−10.4165	−	18.0420i	−0.651033	−	1.12762i
$257$	−14.4917	−	25.1004i	−0.903969	−	1.56572i	−0.822295	−	0.569061i	$-0.807307\pi$
$257$	−14.4917	−	25.1004i	−0.903969	−	1.56572i	−0.0816738	−	0.996659i	$-0.526027\pi$
$258$		0			0
$259$		0			0
$260$		−	1.66007i		−	0.102953i
$261$		0			0
$262$			14.6954i			0.907887i
$263$	−11.8643	−	6.84988i	−0.731586	−	0.422381i	0.0874160	−	0.996172i	$-0.472139\pi$
$263$	−11.8643	−	6.84988i	−0.731586	−	0.422381i	−0.819002	+	0.573790i	$0.805472\pi$
$264$		0			0
$265$	10.5336	−	6.08156i	0.647072	−	0.373587i
$266$		0			0
$267$		0			0
$268$	−1.39505	+	2.41630i	−0.0852163	+	0.147599i
$269$	10.4795			0.638944			0.319472	−	0.947596i	$-0.396494\pi$
$269$	10.4795			0.638944			0.319472	+	0.947596i	$0.396494\pi$
$270$		0			0
$271$		−	6.54607i		−	0.397646i	−0.980035	−	0.198823i	$-0.936288\pi$
$271$		−	6.54607i		−	0.397646i	0.980035	−	0.198823i	$-0.0637118\pi$
$272$	−7.32034	+	12.6792i	−0.443861	+	0.768790i
$273$		0			0
$274$	−10.8116	−	18.7263i	−0.653155	−	1.13130i
$275$	−6.21198	+	3.58649i	−0.374596	+	0.216273i
$276$		0			0
$277$	−11.2156	+	19.4261i	−0.673883	+	1.16720i	0.302911	+	0.953019i	$0.402041\pi$
$277$	−11.2156	+	19.4261i	−0.673883	+	1.16720i	−0.976794	+	0.214181i	$0.931292\pi$
$278$	36.6311			2.19699
$279$		0			0
$280$		0			0
$281$	−19.3552	−	11.1747i	−1.15463	−	0.666627i	−0.204621	−	0.978841i	$-0.565596\pi$
$281$	−19.3552	−	11.1747i	−1.15463	−	0.666627i	−0.950012	+	0.312214i	$0.898929\pi$
$282$		0			0
$283$	−16.4296	+	9.48563i	−0.976638	+	0.563862i	−0.901254	−	0.433292i	$-0.857352\pi$
$283$	−16.4296	+	9.48563i	−0.976638	+	0.563862i	−0.0753848	+	0.997155i	$0.524019\pi$
$284$	−3.81845	+	2.20459i	−0.226584	+	0.130818i
$285$		0			0
$286$	−4.02910	−	2.32620i	−0.238246	−	0.137551i
$287$		0			0
$288$		0			0
$289$	−7.52777			−0.442810
$290$	7.32830	−	12.6930i	0.430333	−	0.745358i
$291$		0			0
$292$	11.0149	−	6.35943i	0.644596	−	0.372158i
$293$	−4.41136	−	7.64069i	−0.257714	−	0.446374i	0.707915	−	0.706298i	$-0.249636\pi$
$293$	−4.41136	−	7.64069i	−0.257714	−	0.446374i	−0.965629	+	0.259924i	$0.916303\pi$
$294$		0			0
$295$	9.15835	−	15.8627i	0.533220	−	0.923563i
$296$		−	1.34225i		−	0.0780167i
$297$		0			0
$298$	−29.4175			−1.70411
$299$	−1.75112	+	3.03303i	−0.101270	+	0.175405i
$300$		0			0
$301$		0			0
$302$	−3.20910	+	1.85278i	−0.184663	+	0.106615i
$303$		0			0
$304$	−10.6049	−	6.12275i	−0.608233	−	0.351164i
$305$		−	21.4385i		−	1.22756i
$306$		0			0
$307$			28.7533i			1.64104i	0.571620	+	0.820519i	$0.306315\pi$
$307$			28.7533i			1.64104i	−0.571620	+	0.820519i	$0.693685\pi$
$308$		0			0
$309$		0			0
$310$	−24.5085	−	42.4499i	−1.39199	−	2.41099i
$311$	6.64294	+	11.5059i	0.376687	+	0.652441i	0.990578	−	0.136950i	$-0.0437300\pi$
$311$	6.64294	+	11.5059i	0.376687	+	0.652441i	−0.613891	+	0.789391i	$0.710397\pi$
$312$		0			0
$313$	27.3227	+	15.7748i	1.54437	+	0.891643i	0.998555	+	0.0537372i	$0.0171133\pi$
$313$	27.3227	+	15.7748i	1.54437	+	0.891643i	0.545815	+	0.837905i	$0.316220\pi$
$314$	−15.6846			−0.885132
$315$		0			0
$316$	0.857328			0.0482285
$317$	−17.3819	−	10.0354i	−0.976264	−	0.563646i	−0.0751236	−	0.997174i	$-0.523935\pi$
$317$	−17.3819	−	10.0354i	−0.976264	−	0.563646i	−0.901140	+	0.433528i	$0.857268\pi$
$318$		0			0
$319$	−8.77816	−	15.2042i	−0.491483	−	0.851273i
$320$	−4.45961	−	7.72428i	−0.249300	−	0.431800i
$321$		0			0
$322$		0			0
$323$			7.92259i			0.440824i
$324$		0			0
$325$			0.567444i			0.0314761i
$326$	1.73849	+	1.00372i	0.0962862	+	0.0555909i
$327$		0			0
$328$	2.70289	−	1.56052i	0.149242	−	0.0861651i
$329$		0			0
$330$		0			0
$331$	−0.623957	+	1.08073i	−0.0342958	+	0.0594020i	−0.882664	−	0.470005i	$-0.844252\pi$
$331$	−0.623957	+	1.08073i	−0.0342958	+	0.0594020i	0.848368	+	0.529407i	$0.177585\pi$
$332$	−12.6457			−0.694023
$333$		0			0
$334$		−	14.9414i		−	0.817559i
$335$	2.34146	−	4.05553i	0.127928	−	0.221577i
$336$		0			0
$337$	6.58745	+	11.4098i	0.358842	+	0.621532i	0.987768	−	0.155933i	$-0.0498385\pi$
$337$	6.58745	+	11.4098i	0.358842	+	0.621532i	−0.628926	+	0.777465i	$0.716505\pi$
$338$	20.7224	−	11.9641i	1.12715	−	0.650759i
$339$		0			0
$340$	−5.75664	+	9.97079i	−0.312198	+	0.540742i
$341$	−58.7146			−3.17958
$342$		0			0
$343$		0			0
$344$	−7.80798	−	4.50794i	−0.420978	−	0.243052i
$345$		0			0
$346$	−1.62419	+	0.937727i	−0.0873171	+	0.0504125i
$347$	23.4411	−	13.5337i	1.25838	−	0.726529i	0.285624	−	0.958342i	$-0.407799\pi$
$347$	23.4411	−	13.5337i	1.25838	−	0.726529i	0.972760	+	0.231813i	$0.0744658\pi$
$348$		0			0
$349$	−30.3413	−	17.5176i	−1.62413	−	0.937694i	−0.985798	−	0.167936i	$-0.946290\pi$
$349$	−30.3413	−	17.5176i	−1.62413	−	0.937694i	−0.638336	−	0.769758i	$-0.720377\pi$
$350$		0			0
$351$		0			0
$352$	−39.2401			−2.09150
$353$	1.26256	−	2.18682i	0.0671992	−	0.116392i	−0.830468	−	0.557066i	$-0.811927\pi$
$353$	1.26256	−	2.18682i	0.0671992	−	0.116392i	0.897667	+	0.440674i	$0.145260\pi$
$354$		0			0
$355$	6.40892	−	3.70019i	0.340150	−	0.196386i
$356$	5.65713	+	9.79843i	0.299827	+	0.519316i
$357$		0			0
$358$	−1.37871	+	2.38799i	−0.0728669	+	0.126209i
$359$			7.26762i			0.383570i	0.981437	+	0.191785i	$0.0614277\pi$
$359$			7.26762i			0.383570i	−0.981437	+	0.191785i	$0.938572\pi$
$360$		0			0
$361$	12.3735			0.651239
$362$	14.0581	−	24.3494i	0.738880	−	1.27978i
$363$		0			0
$364$		0			0
$365$	−18.4874	+	10.6737i	−0.967675	+	0.558688i
$366$		0			0
$367$	11.6714	+	6.73848i	0.609242	+	0.351746i	0.772669	−	0.634809i	$-0.218921\pi$
$367$	11.6714	+	6.73848i	0.609242	+	0.351746i	−0.163427	+	0.986555i	$0.552255\pi$
$368$			37.5429i			1.95706i
$369$		0			0
$370$		−	6.63278i		−	0.344822i
$371$		0			0
$372$		0			0
$373$	−11.8820	−	20.5801i	−0.615224	−	1.06560i	−0.990345	−	0.138624i	$-0.955732\pi$
$373$	−11.8820	−	20.5801i	−0.615224	−	1.06560i	0.375121	−	0.926976i	$-0.377601\pi$
$374$	16.1332	+	27.9435i	0.834228	+	1.44492i
$375$		0			0
$376$	1.75947	+	1.01583i	0.0907379	+	0.0523875i
$377$	−1.38886			−0.0715297
$378$		0			0
$379$	21.2283			1.09042			0.545211	−	0.838299i	$-0.316449\pi$
$379$	21.2283			1.09042			0.545211	+	0.838299i	$0.316449\pi$
$380$	−8.33958	−	4.81486i	−0.427812	−	0.246997i
$381$		0			0
$382$	12.3572	+	21.4032i	0.632248	+	1.09509i
$383$	6.47930	+	11.2225i	0.331077	+	0.573442i	0.982723	−	0.185081i	$-0.0592547\pi$
$383$	6.47930	+	11.2225i	0.331077	+	0.573442i	−0.651646	+	0.758523i	$0.725921\pi$
$384$		0			0
$385$		0			0
$386$		−	2.90752i		−	0.147989i
$387$		0			0
$388$		−	5.56231i		−	0.282384i
$389$	−9.48037	−	5.47350i	−0.480674	−	0.277517i	0.240023	−	0.970767i	$-0.422845\pi$
$389$	−9.48037	−	5.47350i	−0.480674	−	0.277517i	−0.720697	+	0.693250i	$0.756178\pi$
$390$		0			0
$391$	21.0353	−	12.1447i	1.06380	−	0.614186i
$392$		0			0
$393$		0			0
$394$	4.63669	−	8.03098i	0.233593	−	0.404595i
$395$	−1.43894			−0.0724011
$396$		0			0
$397$		−	29.8901i		−	1.50014i	−0.661358	−	0.750071i	$-0.730019\pi$
$397$		−	29.8901i		−	1.50014i	0.661358	−	0.750071i	$-0.269981\pi$
$398$	−10.4588	+	18.1151i	−0.524250	+	0.908028i
$399$		0			0
$400$	3.04141	+	5.26788i	0.152071	+	0.263394i
$401$	−16.6233	+	9.59744i	−0.830126	+	0.479273i	−0.853896	−	0.520444i	$-0.825766\pi$
$401$	−16.6233	+	9.59744i	−0.830126	+	0.479273i	0.0237698	+	0.999717i	$0.492433\pi$
$402$		0			0
$403$	−2.32242	+	4.02254i	−0.115688	+	0.200377i
$404$	11.2484			0.559630
$405$		0			0
$406$		0			0
$407$	−6.88060	−	3.97252i	−0.341059	−	0.196910i
$408$		0			0
$409$	24.9737	−	14.4186i	1.23487	−	0.712953i	0.266830	−	0.963744i	$-0.414024\pi$
$409$	24.9737	−	14.4186i	1.23487	−	0.712953i	0.968041	+	0.250790i	$0.0806905\pi$
$410$	13.3565	−	7.71135i	0.659628	−	0.380837i
$411$		0			0
$412$	−18.5190	−	10.6919i	−0.912363	−	0.526753i
$413$		0			0
$414$		0			0
$415$	21.2246			1.04188
$416$	−1.55211	+	2.68834i	−0.0760986	+	0.131807i
$417$		0			0
$418$	−23.3720	+	13.4938i	−1.14316	+	0.660005i
$419$	5.06390	+	8.77094i	0.247388	+	0.428488i	0.962800	−	0.270214i	$-0.0870945\pi$
$419$	5.06390	+	8.77094i	0.247388	+	0.428488i	−0.715412	+	0.698702i	$0.753761\pi$
$420$		0			0
$421$	12.7094	−	22.0134i	0.619419	−	1.07287i	−0.370173	−	0.928963i	$-0.620702\pi$
$421$	12.7094	−	22.0134i	0.619419	−	1.07287i	0.989592	−	0.143902i	$-0.0459651\pi$
$422$		−	28.7079i		−	1.39748i
$423$		0			0
$424$	−4.60021			−0.223406
$425$	1.96773	−	3.40821i	0.0954490	−	0.165322i
$426$		0			0
$427$		0			0
$428$	−17.4370	+	10.0673i	−0.842849	+	0.486619i
$429$		0			0
$430$	−38.5834	−	22.2762i	−1.86066	−	1.07425i
$431$		−	10.8434i		−	0.522308i	−0.965297	−	0.261154i	$-0.915897\pi$
$431$		−	10.8434i		−	0.522308i	0.965297	−	0.261154i	$-0.0841031\pi$
$432$		0			0
$433$		−	13.0519i		−	0.627233i	−0.949550	−	0.313616i	$-0.898459\pi$
$433$		−	13.0519i		−	0.627233i	0.949550	−	0.313616i	$-0.101541\pi$
$434$		0			0
$435$		0			0
$436$	−0.685771	−	1.18779i	−0.0328425	−	0.0568849i
$437$	10.1579	+	17.5940i	0.485918	+	0.841634i
$438$		0			0
$439$	−27.6736	−	15.9773i	−1.32079	−	0.762557i	−0.336933	−	0.941529i	$-0.609389\pi$
$439$	−27.6736	−	15.9773i	−1.32079	−	0.762557i	−0.983854	+	0.178972i	$0.942723\pi$
$440$	13.3208			0.635046
$441$		0			0
$442$	2.55255			0.121412
$443$	−21.4748	−	12.3985i	−1.02030	−	0.589068i	−0.106107	−	0.994355i	$-0.533838\pi$
$443$	−21.4748	−	12.3985i	−1.02030	−	0.589068i	−0.914190	+	0.405286i	$0.867172\pi$
$444$		0			0
$445$	−9.49496	−	16.4458i	−0.450104	−	0.779603i
$446$	−2.98724	−	5.17406i	−0.141450	−	0.244999i
$447$		0			0
$448$		0			0
$449$		−	13.7710i		−	0.649892i	−0.945733	−	0.324946i	$-0.894654\pi$
$449$		−	13.7710i		−	0.649892i	0.945733	−	0.324946i	$-0.105346\pi$
$450$		0			0
$451$		−	18.4740i		−	0.869906i
$452$	−6.12715	−	3.53751i	−0.288197	−	0.166390i
$453$		0			0
$454$	23.7439	−	13.7085i	1.11436	−	0.643374i
$455$		0			0
$456$		0			0
$457$	−13.6554	+	23.6518i	−0.638771	+	1.10638i	0.346931	+	0.937890i	$0.387224\pi$
$457$	−13.6554	+	23.6518i	−0.638771	+	1.10638i	−0.985703	+	0.168494i	$0.946110\pi$
$458$	6.31340			0.295006
$459$		0			0
$460$			29.5233i			1.37653i
$461$	−5.51822	+	9.55784i	−0.257009	+	0.445153i	−0.965439	−	0.260628i	$-0.916070\pi$
$461$	−5.51822	+	9.55784i	−0.257009	+	0.445153i	0.708430	+	0.705781i	$0.249404\pi$
$462$		0			0
$463$	−12.2346	−	21.1910i	−0.568591	−	0.984829i	−0.996706	−	0.0811042i	$-0.974155\pi$
$463$	−12.2346	−	21.1910i	−0.568591	−	0.984829i	0.428115	−	0.903724i	$-0.359178\pi$
$464$	−12.8935	+	7.44405i	−0.598564	+	0.345581i
$465$		0			0
$466$	4.56029	−	7.89866i	0.211251	−	0.365898i
$467$	15.9048			0.735988			0.367994	−	0.929828i	$-0.380045\pi$
$467$	15.9048			0.735988			0.367994	+	0.929828i	$0.380045\pi$
$468$		0			0
$469$		0			0
$470$	8.69451	+	5.01978i	0.401048	+	0.231545i
$471$		0			0
$472$	−5.99943	+	3.46377i	−0.276146	+	0.159433i
$473$	−46.2169	+	26.6834i	−2.12506	+	1.22690i
$474$		0			0
$475$	2.85063	+	1.64581i	0.130796	+	0.0755151i
$476$		0			0
$477$		0			0
$478$	29.4042			1.34492
$479$	−6.92685	+	11.9976i	−0.316496	+	0.548187i	−0.979754	−	0.200204i	$-0.935840\pi$
$479$	−6.92685	+	11.9976i	−0.316496	+	0.548187i	0.663259	+	0.748390i	$0.269173\pi$
$480$		0			0
$481$	−0.544315	+	0.314261i	−0.0248186	+	0.0143290i
$482$	−0.718964	−	1.24528i	−0.0327479	−	0.0567210i
$483$		0			0
$484$	−15.2781	+	26.4624i	−0.694458	+	1.20284i
$485$			9.33582i			0.423918i
$486$		0			0
$487$	28.7987			1.30499			0.652496	−	0.757792i	$-0.273722\pi$
$487$	28.7987			1.30499			0.652496	+	0.757792i	$0.273722\pi$
$488$	−4.05411	+	7.02193i	−0.183521	+	0.317868i
$489$		0			0
$490$		0			0
$491$	33.5627	−	19.3774i	1.51466	−	0.874492i	0.514812	−	0.857303i	$-0.327862\pi$
$491$	33.5627	−	19.3774i	1.51466	−	0.874492i	0.999852	−	0.0171884i	$-0.00547151\pi$
$492$		0			0
$493$	8.34181	+	4.81615i	0.375696	+	0.216908i
$494$			2.13496i			0.0960562i
$495$		0			0
$496$			49.7911i			2.23569i
$497$		0			0
$498$		0			0
$499$	1.73333	+	3.00222i	0.0775946	+	0.134398i	0.902212	−	0.431294i	$-0.141943\pi$
$499$	1.73333	+	3.00222i	0.0775946	+	0.134398i	−0.824617	+	0.565691i	$0.808609\pi$
$500$	−6.96045	−	12.0558i	−0.311281	−	0.539154i
$501$		0			0
$502$	−1.85395	−	1.07038i	−0.0827458	−	0.0477733i
$503$	−28.2202			−1.25828			−0.629138	−	0.777293i	$-0.716592\pi$
$503$	−28.2202			−1.25828			−0.629138	+	0.777293i	$0.716592\pi$
$504$		0			0
$505$	−18.8794			−0.840124
$506$	71.6552	+	41.3701i	3.18546	+	1.83913i
$507$		0			0
$508$	5.50326	+	9.53193i	0.244168	+	0.422911i
$509$	17.9062	+	31.0144i	0.793678	+	1.37469i	0.923675	+	0.383176i	$0.125170\pi$
$509$	17.9062	+	31.0144i	0.793678	+	1.37469i	−0.129997	+	0.991514i	$0.541497\pi$
$510$		0			0
$511$		0			0
$512$			24.2599i			1.07215i
$513$		0			0
$514$			54.1682i			2.38926i
$515$	31.0823	+	17.9454i	1.36965	+	0.790768i
$516$		0			0
$517$	10.4147	−	6.01291i	0.458036	−	0.264447i
$518$		0			0
$519$		0			0
$520$	0.526897	−	0.912612i	0.0231060	−	0.0400207i
$521$	−26.9216			−1.17946			−0.589729	−	0.807601i	$-0.700766\pi$
$521$	−26.9216			−1.17946			−0.589729	+	0.807601i	$0.700766\pi$
$522$		0			0
$523$		−	9.03185i		−	0.394935i	−0.980309	−	0.197468i	$-0.936728\pi$
$523$		−	9.03185i		−	0.394935i	0.980309	−	0.197468i	$-0.0632717\pi$
$524$	5.86943	−	10.1662i	0.256407	−	0.444110i
$525$		0			0
$526$	12.8020	+	22.1737i	0.558193	+	0.966819i
$527$	27.8980	−	16.1069i	1.21526	−	0.701629i
$528$		0			0
$529$	19.6426	−	34.0220i	0.854026	−	1.47922i
$530$	−22.7321			−0.987420
$531$		0			0
$532$		0			0
$533$	−1.26565	−	0.730726i	−0.0548216	−	0.0316513i
$534$		0			0
$535$	29.2664	−	16.8969i	1.26530	−	0.730518i
$536$	−1.53384	+	0.885563i	−0.0662518	+	0.0382505i
$537$		0			0
$538$	−16.9615	−	9.79273i	−0.731262	−	0.422195i
$539$		0			0
$540$		0			0
$541$	11.3358			0.487366			0.243683	−	0.969855i	$-0.421644\pi$
$541$	11.3358			0.487366			0.243683	+	0.969855i	$0.421644\pi$
$542$	−6.11710	+	10.5951i	−0.262752	+	0.455100i
$543$		0			0
$544$	18.6447	−	10.7646i	0.799387	−	0.461526i
$545$	1.15100	+	1.99360i	0.0493035	+	0.0853962i
$546$		0			0
$547$	19.4246	−	33.6444i	0.830537	−	1.43853i	−0.0670762	−	0.997748i	$-0.521367\pi$
$547$	19.4246	−	33.6444i	0.830537	−	1.43853i	0.897613	−	0.440784i	$-0.145300\pi$
$548$			17.2729i			0.737861i
$549$		0			0
$550$	13.4058			0.571627
$551$	−4.02823	+	6.97711i	−0.171609	+	0.297235i
$552$		0			0
$553$		0			0
$554$	36.3061	−	20.9613i	1.54250	−	0.890562i
$555$		0			0
$556$	−25.3410	−	14.6306i	−1.07470	−	0.620478i
$557$		−	7.26499i		−	0.307827i	−0.988084	−	0.153914i	$-0.950812\pi$
$557$		−	7.26499i		−	0.307827i	0.988084	−	0.153914i	$-0.0491878\pi$
$558$		0			0
$559$			4.22177i			0.178562i
$560$		0			0
$561$		0			0
$562$	20.8848	+	36.1736i	0.880973	+	1.52589i
$563$	−11.5409	−	19.9894i	−0.486390	−	0.842453i	0.513487	−	0.858097i	$-0.328353\pi$
$563$	−11.5409	−	19.9894i	−0.486390	−	0.842453i	−0.999878	+	0.0156446i	$0.995020\pi$
$564$		0			0
$565$	10.2838	+	5.93738i	0.432644	+	0.249787i
$566$	35.4561			1.49033
$567$		0			0
$568$	−2.79889			−0.117439
$569$	15.5482	+	8.97677i	0.651815	+	0.376326i	0.789151	−	0.614199i	$-0.210521\pi$
$569$	15.5482	+	8.97677i	0.651815	+	0.376326i	−0.137336	+	0.990525i	$0.543854\pi$
$570$		0			0
$571$	7.04234	+	12.1977i	0.294713	+	0.510457i	0.974918	−	0.222564i	$-0.0714427\pi$
$571$	7.04234	+	12.1977i	0.294713	+	0.510457i	−0.680205	+	0.733022i	$0.738109\pi$
$572$	1.85819	+	3.21849i	0.0776950	+	0.134572i
$573$		0			0
$574$		0			0
$575$		−	10.0916i		−	0.420851i
$576$		0			0
$577$			30.0675i			1.25172i	0.779934	+	0.625862i	$0.215253\pi$
$577$			30.0675i			1.25172i	−0.779934	+	0.625862i	$0.784747\pi$
$578$	12.1841	+	7.03447i	0.506790	+	0.292595i
$579$		0			0
$580$	−10.1393	+	5.85392i	−0.421011	+	0.243071i
$581$		0			0
$582$		0			0
$583$	−13.6148	+	23.5815i	−0.563866	+	0.976644i
$584$	8.07379			0.334096
$585$		0			0
$586$			16.4891i			0.681158i
$587$	18.0979	−	31.3465i	0.746981	−	1.29381i	−0.202283	−	0.979327i	$-0.564836\pi$
$587$	18.0979	−	31.3465i	0.746981	−	1.29381i	0.949264	−	0.314481i	$-0.101831\pi$
$588$		0			0
$589$	13.4719	+	23.3339i	0.555098	+	0.961459i
$590$	−29.6464	+	17.1164i	−1.22052	+	0.704670i
$591$		0			0
$592$	−3.36877	+	5.83489i	−0.138456	+	0.239812i
$593$	−2.04317			−0.0839028			−0.0419514	−	0.999120i	$-0.513357\pi$
$593$	−2.04317			−0.0839028			−0.0419514	+	0.999120i	$0.513357\pi$
$594$		0			0
$595$		0			0
$596$	20.3507	+	11.7495i	0.833598	+	0.481278i
$597$		0			0
$598$	5.66854	−	3.27273i	0.231804	−	0.133832i
$599$	15.7873	−	9.11478i	0.645050	−	0.372420i	−0.141507	−	0.989937i	$-0.545195\pi$
$599$	15.7873	−	9.11478i	0.645050	−	0.372420i	0.786557	+	0.617517i	$0.211861\pi$
$600$		0			0
$601$	32.1713	+	18.5741i	1.31230	+	0.757654i	0.982476	−	0.186390i	$-0.0596787\pi$
$601$	32.1713	+	18.5741i	1.31230	+	0.757654i	0.329820	+	0.944044i	$0.393012\pi$
$602$		0			0
$603$		0			0
$604$	2.96003			0.120442
$605$	25.6428	−	44.4146i	1.04253	−	1.80571i
$606$		0			0
$607$	−17.1730	+	9.91482i	−0.697030	+	0.402430i	−0.806240	−	0.591588i	$-0.798501\pi$
$607$	−17.1730	+	9.91482i	−0.697030	+	0.402430i	0.109211	+	0.994019i	$0.465168\pi$
$608$	9.00349	+	15.5945i	0.365140	+	0.632440i
$609$		0			0
$610$	−20.0336	+	34.6991i	−0.811135	+	1.40493i
$611$		−	0.951346i		−	0.0384873i
$612$		0			0
$613$	−12.6882			−0.512473			−0.256237	−	0.966614i	$-0.582483\pi$
$613$	−12.6882			−0.512473			−0.256237	+	0.966614i	$0.582483\pi$
$614$	26.8690	−	46.5385i	1.08435	−	1.87814i
$615$		0			0
$616$		0			0
$617$	−4.37247	+	2.52445i	−0.176029	+	0.101630i	−0.585426	−	0.810726i	$-0.699073\pi$
$617$	−4.37247	+	2.52445i	−0.176029	+	0.101630i	0.409397	+	0.912357i	$0.365739\pi$
$618$		0			0
$619$	−0.231999	−	0.133945i	−0.00932485	−	0.00538370i	0.495330	−	0.868705i	$-0.335047\pi$
$619$	−0.231999	−	0.133945i	−0.00932485	−	0.00538370i	−0.504655	+	0.863321i	$0.668380\pi$
$620$			39.1552i			1.57251i
$621$		0			0
$622$		−	24.8305i		−	0.995611i
$623$		0			0
$624$		0			0
$625$	14.8792	+	25.7716i	0.595169	+	1.03086i
$626$	−29.4820	−	51.0644i	−1.17834	−	2.04094i
$627$		0			0
$628$	10.8504	+	6.26449i	0.432979	+	0.249981i
$629$	4.35905			0.173807
$630$		0			0
$631$	37.7899			1.50439			0.752197	−	0.658938i	$-0.228994\pi$
$631$	37.7899			1.50439			0.752197	+	0.658938i	$0.228994\pi$
$632$	0.471310	+	0.272111i	0.0187477	+	0.0108240i
$633$		0			0
$634$	18.7556	+	32.4856i	0.744880	+	1.29017i
$635$	−9.23671	−	15.9984i	−0.366547	−	0.634879i
$636$		0			0
$637$		0			0
$638$			32.8117i			1.29903i
$639$		0			0
$640$		−	18.3867i		−	0.726799i
$641$	−29.7991	−	17.2045i	−1.17699	−	0.679537i	−0.221676	−	0.975120i	$-0.571153\pi$
$641$	−29.7991	−	17.2045i	−1.17699	−	0.679537i	−0.955317	+	0.295583i	$0.904486\pi$
$642$		0			0
$643$	−0.676278	+	0.390449i	−0.0266698	+	0.0153978i	−0.513276	−	0.858224i	$-0.671568\pi$
$643$	−0.676278	+	0.390449i	−0.0266698	+	0.0153978i	0.486606	+	0.873622i	$0.338235\pi$
$644$		0			0
$645$		0			0
$646$	7.40341	−	12.8231i	0.291283	−	0.504517i
$647$	−19.6436			−0.772271			−0.386136	−	0.922442i	$-0.626190\pi$
$647$	−19.6436			−0.772271			−0.386136	+	0.922442i	$0.626190\pi$
$648$		0			0
$649$			41.0055i			1.60961i
$650$	0.530259	−	0.918435i	0.0207985	−	0.0360240i
$651$		0			0
$652$	−0.801781	−	1.38872i	−0.0314001	−	0.0543867i
$653$	−2.77600	+	1.60272i	−0.108633	+	0.0627194i	−0.553332	−	0.832961i	$-0.686644\pi$
$653$	−2.77600	+	1.60272i	−0.108633	+	0.0627194i	0.444699	+	0.895680i	$0.353311\pi$
$654$		0			0
$655$	−9.85129	+	17.0629i	−0.384922	+	0.666704i
$656$	−15.6663			−0.611666
$657$		0			0
$658$		0			0
$659$	24.2959	+	14.0273i	0.946435	+	0.546425i	0.891972	−	0.452091i	$-0.149322\pi$
$659$	24.2959	+	14.0273i	0.946435	+	0.546425i	0.0544636	+	0.998516i	$0.482655\pi$
$660$		0			0
$661$	−28.0490	+	16.1941i	−1.09098	+	0.629878i	−0.933837	−	0.357698i	$-0.883562\pi$
$661$	−28.0490	+	16.1941i	−1.09098	+	0.629878i	−0.157143	+	0.987576i	$0.550228\pi$
$662$	2.01981	−	1.16614i	0.0785020	−	0.0453232i
$663$		0			0
$664$	−6.95189	−	4.01367i	−0.269785	−	0.155761i
$665$		0			0
$666$		0			0
$667$	24.7000			0.956386
$668$	−5.96768	+	10.3363i	−0.230897	+	0.399925i
$669$		0			0
$670$	−7.57953	+	4.37605i	−0.292823	+	0.169061i
$671$	23.9971	+	41.5641i	0.926397	+	1.60457i
$672$		0			0
$673$	−10.3088	+	17.8554i	−0.397375	+	0.688273i	−0.993401	−	0.114692i	$-0.963412\pi$
$673$	−10.3088	+	17.8554i	−0.397375	+	0.688273i	0.596026	+	0.802965i	$0.296745\pi$
$674$		−	24.6231i		−	0.948445i
$675$		0			0
$676$	−19.1140			−0.735155
$677$	−25.1655	+	43.5880i	−0.967190	+	1.67522i	−0.263578	+	0.964638i	$0.584903\pi$
$677$	−25.1655	+	43.5880i	−0.967190	+	1.67522i	−0.703612	+	0.710585i	$0.748431\pi$
$678$		0			0
$679$		0			0
$680$	−6.32935	+	3.65425i	−0.242719	+	0.140134i
$681$		0			0
$682$	95.0324	+	54.8670i	3.63898	+	2.10097i
$683$			13.7445i			0.525920i	0.964807	+	0.262960i	$0.0846987\pi$
$683$			13.7445i			0.525920i	−0.964807	+	0.262960i	$0.915301\pi$
$684$		0			0
$685$		−	28.9909i		−	1.10769i
$686$		0			0
$687$		0			0
$688$	22.6280	+	39.1929i	0.862685	+	1.49421i
$689$	1.07705	+	1.86550i	0.0410321	+	0.0710698i
$690$		0			0
$691$	−25.4328	−	14.6837i	−0.967511	−	0.558593i	−0.0690343	−	0.997614i	$-0.521992\pi$
$691$	−25.4328	−	14.6837i	−0.967511	−	0.558593i	−0.898476	+	0.439022i	$0.855325\pi$
$692$	1.49813			0.0569504
$693$		0			0
$694$	−50.5874			−1.92027
$695$	42.5325	+	24.5562i	1.61335	+	0.931468i
$696$		0			0
$697$	5.06789	+	8.77784i	0.191960	+	0.332484i
$698$	32.7392	+	56.7060i	1.23920	+	2.14635i
$699$		0			0
$700$		0			0
$701$		−	44.2011i		−	1.66945i	−0.550666	−	0.834726i	$-0.685626\pi$
$701$		−	44.2011i		−	1.66945i	0.550666	−	0.834726i	$-0.314374\pi$
$702$		0			0
$703$			3.64592i			0.137508i
$704$	17.2923	+	9.98371i	0.651728	+	0.376275i
$705$		0			0
$706$	−4.08702	+	2.35964i	−0.153817	+	0.0888063i
$707$		0			0
$708$		0			0
$709$	5.66629	−	9.81430i	0.212802	−	0.368584i	−0.739788	−	0.672840i	$-0.765074\pi$
$709$	5.66629	−	9.81430i	0.212802	−	0.368584i	0.952590	+	0.304256i	$0.0984077\pi$
$710$	−13.8308			−0.519062
$711$		0			0
$712$			7.18216i			0.269163i
$713$	41.3028	−	71.5385i	1.54680	−	2.67914i
$714$		0			0
$715$	−3.11881	−	5.40193i	−0.116637	−	0.202021i
$716$	1.90755	−	1.10132i	0.0712884	−	0.0411584i
$717$		0			0
$718$	6.79136	−	11.7630i	0.253451	−	0.438991i
$719$	−36.1294			−1.34740			−0.673700	−	0.739005i	$-0.735296\pi$
$719$	−36.1294			−1.34740			−0.673700	+	0.739005i	$0.735296\pi$
$720$		0			0
$721$		0			0
$722$	−20.0271	−	11.5627i	−0.745333	−	0.430318i
$723$		0			0
$724$	−19.4505	+	11.2298i	−0.722874	+	0.417351i
$725$	3.46580	−	2.00098i	0.128717	−	0.0743146i
$726$		0			0
$727$	6.20547	+	3.58273i	0.230148	+	0.132876i	0.610640	−	0.791908i	$-0.290912\pi$
$727$	6.20547	+	3.58273i	0.230148	+	0.132876i	−0.380492	+	0.924784i	$0.624245\pi$
$728$		0			0
$729$		0			0
$730$	39.8970			1.47665
$731$	14.6399	−	25.3570i	0.541475	−	0.937862i
$732$		0			0
$733$	−41.4391	+	23.9249i	−1.53059	+	0.883685i	−0.531253	+	0.847213i	$0.678279\pi$
$733$	−41.4391	+	23.9249i	−1.53059	+	0.883685i	−0.999335	+	0.0364726i	$0.988388\pi$
$734$	−12.5938	−	21.8131i	−0.464846	−	0.805136i
$735$		0			0
$736$	27.6034	−	47.8105i	1.01747	−	1.76232i
$737$			10.4836i			0.386170i
$738$		0			0
$739$	15.3483			0.564597			0.282299	−	0.959327i	$-0.408903\pi$
$739$	15.3483			0.564597			0.282299	+	0.959327i	$0.408903\pi$
$740$	−2.64917	+	4.58849i	−0.0973853	+	0.168676i
$741$		0			0
$742$		0			0
$743$	−34.9422	+	20.1739i	−1.28191	+	0.740109i	−0.977197	−	0.212337i	$-0.931893\pi$
$743$	−34.9422	+	20.1739i	−1.28191	+	0.740109i	−0.304709	+	0.952445i	$0.598559\pi$
$744$		0			0
$745$	−34.1568	−	19.7204i	−1.25141	−	0.722501i
$746$			44.4132i			1.62608i
$747$		0			0
$748$		−	25.7747i		−	0.942417i
$749$		0			0
$750$		0			0
$751$	−16.9449	−	29.3494i	−0.618327	−	1.07097i	−0.989791	−	0.142527i	$-0.954477\pi$
$751$	−16.9449	−	29.3494i	−0.618327	−	1.07097i	0.371463	−	0.928448i	$-0.378856\pi$
$752$	−5.09906	−	8.83183i	−0.185944	−	0.322064i
$753$		0			0
$754$	2.24793	+	1.29784i	0.0818647	+	0.0472646i
$755$	−4.96814			−0.180809
$756$		0			0
$757$	−5.73237			−0.208346			−0.104173	−	0.994559i	$-0.533220\pi$
$757$	−5.73237			−0.208346			−0.104173	+	0.994559i	$0.533220\pi$
$758$	−34.3589	−	19.8371i	−1.24797	−	0.720517i
$759$		0			0
$760$	−3.05642	−	5.29387i	−0.110868	−	0.192029i
$761$	13.2666	+	22.9784i	0.480914	+	0.832968i	0.999760	−	0.0218999i	$-0.00697151\pi$
$761$	13.2666	+	22.9784i	0.480914	+	0.832968i	−0.518846	+	0.854868i	$0.673638\pi$
$762$		0			0
$763$		0			0
$764$		−	19.7420i		−	0.714242i
$765$		0			0
$766$		−	24.2188i		−	0.875061i
$767$	2.80929	+	1.62194i	0.101438	+	0.0585650i
$768$		0			0
$769$	−23.3944	+	13.5068i	−0.843623	+	0.487066i	−0.858494	−	0.512823i	$-0.828600\pi$
$769$	−23.3944	+	13.5068i	−0.843623	+	0.487066i	0.0148711	+	0.999889i	$0.495266\pi$
$770$		0			0
$771$		0			0
$772$	−1.16128	+	2.01139i	−0.0417953	+	0.0723916i
$773$	−22.6017			−0.812927			−0.406464	−	0.913667i	$-0.633238\pi$
$773$	−22.6017			−0.812927			−0.406464	+	0.913667i	$0.633238\pi$
$774$		0			0
$775$		−	13.3840i		−	0.480768i
$776$	1.76545	−	3.05784i	0.0633758	−	0.109770i
$777$		0			0
$778$	10.2296	+	17.7182i	0.366750	+	0.635229i
$779$	−7.34180	+	4.23879i	−0.263047	+	0.151870i
$780$		0			0
$781$	−8.28359	+	14.3476i	−0.296410	+	0.513398i
$782$	−45.3955			−1.62334
$783$		0			0
$784$		0			0
$785$	−18.2114	−	10.5144i	−0.649993	−	0.375274i
$786$		0			0
$787$	−22.6864	+	13.0980i	−0.808683	+	0.466893i	−0.846498	−	0.532391i	$-0.821293\pi$
$787$	−22.6864	+	13.0980i	−0.808683	+	0.466893i	0.0378153	+	0.999285i	$0.487960\pi$
$788$	−6.41523	+	3.70383i	−0.228533	+	0.131944i
$789$		0			0
$790$	2.32900	+	1.34465i	0.0828620	+	0.0478404i
$791$		0			0
$792$		0			0
$793$	3.79675			0.134827
$794$	−27.9314	+	48.3785i	−0.991247	+	1.71689i
$795$		0			0
$796$	14.4705	−	8.35456i	0.512894	−	0.296120i

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.o (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.61855 - 0.934468i) q^{2} +(0.746462 + 1.29291i) q^{4} +(-1.25287 - 2.17003i) q^{5} +0.947692i q^{8} +O(q^{10})\)	\(q+(-1.61855 - 0.934468i) q^{2} +(0.746462 + 1.29291i) q^{4} +(-1.25287 - 2.17003i) q^{5} +0.947692i q^{8} +4.68306i q^{10} +(4.85803 + 2.80479i) q^{11} +(0.384312 - 0.221883i) q^{13} +(2.37851 - 4.11970i) q^{16} -3.07770 q^{17} -2.57419i q^{19} +(1.87044 - 3.23969i) q^{20} +(-5.24197 - 9.07935i) q^{22} +(-6.83476 + 3.94605i) q^{23} +(-0.639351 + 1.10739i) q^{25} -0.829370 q^{26} +(-2.71041 - 1.56485i) q^{29} +(-9.06457 + 5.23343i) q^{31} +(-6.05802 + 3.49760i) q^{32} +(4.98140 + 2.87601i) q^{34} -1.41634 q^{37} +(-2.40550 + 4.16645i) q^{38} +(2.05652 - 1.18733i) q^{40} +(-1.64665 - 2.85208i) q^{41} +(-4.75676 + 8.23894i) q^{43} +8.37467i q^{44} +14.7498 q^{46} +(1.07190 - 1.85659i) q^{47} +(2.06964 - 1.19491i) q^{50} +(0.573749 + 0.331254i) q^{52} +4.85412i q^{53} -14.0561i q^{55} +(2.92461 + 5.06558i) q^{58} +(3.65496 + 6.33057i) q^{59} +(7.40950 + 4.27788i) q^{61} +19.5619 q^{62} +3.55953 q^{64} +(-0.962984 - 0.555979i) q^{65} +(0.934442 + 1.61850i) q^{67} +(-2.29739 - 3.97919i) q^{68} +2.95338i q^{71} -8.51943i q^{73} +(2.29241 + 1.32352i) q^{74} +(3.32820 - 1.92154i) q^{76} +(0.287130 - 0.497324i) q^{79} -11.9198 q^{80} +6.15497i q^{82} +(-4.23521 + 7.33560i) q^{83} +(3.85595 + 6.67870i) q^{85} +(15.3981 - 8.89008i) q^{86} +(-2.65807 + 4.60392i) q^{88} +7.57858 q^{89} +(-10.2038 - 5.89115i) q^{92} +(-3.46984 + 2.00332i) q^{94} +(-5.58607 + 3.22512i) q^{95} +(-3.22662 - 1.86289i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48q + 24q^{4} + O(q^{10}) \)	\( 48q + 24q^{4} + 24q^{11} - 24q^{16} + 48q^{23} - 24q^{25} - 120q^{32} - 48q^{50} - 48q^{64} - 120q^{65} + 168q^{74} - 24q^{79} - 24q^{85} - 24q^{86} + 144q^{92} - 96q^{95} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

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