LMFDB - Embedded newform 675.2.e.b.451.2

Newspace parameters

Level:	$ N $	$=$	$ 675 = 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	675.e (of order $3$, degree $2$, not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$5.38990213644$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(\zeta_{3})$
Coefficient field:	6.0.954288.1
Defining polynomial:	$ x^{6} - x^{5} - 2x^{4} + 3x^{3} - 6x^{2} - 9x + 27 $ x^6 - x^5 - 2x^4 + 3x^3 - 6x^2 - 9x + 27
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 45)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			451.2
Root			$-1.62241 + 0.606458i$ of defining polynomial
Character	$\chi$	$=$	675.451
Dual form			675.2.e.b.226.2

$q$-expansion

$f(q)$	$=$	$q+(-0.285997 + 0.495361i) q^{2} +(0.836412 + 1.44871i) q^{4} +(0.714003 - 1.23669i) q^{7} -2.10083 q^{8} +O(q^{10})$	\(q+(-0.285997 + 0.495361i) q^{2} +(0.836412 + 1.44871i) q^{4} +(0.714003 - 1.23669i) q^{7} -2.10083 q^{8} +(1.33641 - 2.31473i) q^{11} +(2.33641 + 4.04678i) q^{13} +(0.408405 + 0.707378i) q^{14} +(-1.07199 + 1.85675i) q^{16} +2.67282 q^{17} +4.67282 q^{19} +(0.764419 + 1.32401i) q^{22} +(2.95882 + 5.12483i) q^{23} -2.67282 q^{26} +2.38880 q^{28} +(-4.74482 + 8.21826i) q^{29} +(-3.48040 - 6.02823i) q^{31} +(-2.71400 - 4.70079i) q^{32} +(-0.764419 + 1.32401i) q^{34} +1.81681 q^{37} +(-1.33641 + 2.31473i) q^{38} +(-0.735581 - 1.27406i) q^{41} +(0.235581 - 0.408039i) q^{43} +4.47116 q^{44} -3.38485 q^{46} +(-3.47842 + 6.02480i) q^{47} +(2.48040 + 4.29618i) q^{49} +(-3.90841 + 6.76956i) q^{52} -1.14399 q^{53} +(-1.50000 + 2.59808i) q^{56} +(-2.71400 - 4.70079i) q^{58} +(-0.571993 - 0.990721i) q^{59} +(1.26442 - 2.19004i) q^{61} +3.98153 q^{62} -1.18319 q^{64} +(-3.29523 - 5.70751i) q^{67} +(2.23558 + 3.87214i) q^{68} +12.8745 q^{71} +1.71203 q^{73} +(-0.519602 + 0.899976i) q^{74} +(3.90841 + 6.76956i) q^{76} +(-1.90841 - 3.30545i) q^{77} +(0.143987 - 0.249392i) q^{79} +0.841495 q^{82} +(2.14201 - 3.71007i) q^{83} +(0.134751 + 0.233396i) q^{86} +(-2.80757 + 4.86286i) q^{88} +3.00000 q^{89} +6.67282 q^{91} +(-4.94958 + 8.57293i) q^{92} +(-1.98963 - 3.44615i) q^{94} +(3.91764 - 6.78555i) q^{97} -2.83754 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - q^{2} - 5 q^{4} + 5 q^{7} + 6 q^{8}+O(q^{10}) $ 6 * q - q^2 - 5 * q^4 + 5 * q^7 + 6 * q^8	$ 6 q - q^{2} - 5 q^{4} + 5 q^{7} + 6 q^{8} - 2 q^{11} + 4 q^{13} - 9 q^{14} - 5 q^{16} - 4 q^{17} + 8 q^{19} - 4 q^{22} - 3 q^{23} + 4 q^{26} - 10 q^{28} - 7 q^{29} - 8 q^{31} - 17 q^{32} + 4 q^{34} - 12 q^{37} + 2 q^{38} - 13 q^{41} + 10 q^{43} + 44 q^{44} - 6 q^{46} - 13 q^{47} + 2 q^{49} - 12 q^{52} - 4 q^{53} - 9 q^{56} - 17 q^{58} - 2 q^{59} - q^{61} + 84 q^{62} - 30 q^{64} + 11 q^{67} + 22 q^{68} + 20 q^{71} + 16 q^{73} - 16 q^{74} + 12 q^{76} - 2 q^{79} + 58 q^{82} + 15 q^{83} + 28 q^{86} - 24 q^{88} + 18 q^{89} + 20 q^{91} - 39 q^{92} + 31 q^{94} - 18 q^{97} - 80 q^{98}+O(q^{100}) $ 6 * q - q^2 - 5 * q^4 + 5 * q^7 + 6 * q^8 - 2 * q^11 + 4 * q^13 - 9 * q^14 - 5 * q^16 - 4 * q^17 + 8 * q^19 - 4 * q^22 - 3 * q^23 + 4 * q^26 - 10 * q^28 - 7 * q^29 - 8 * q^31 - 17 * q^32 + 4 * q^34 - 12 * q^37 + 2 * q^38 - 13 * q^41 + 10 * q^43 + 44 * q^44 - 6 * q^46 - 13 * q^47 + 2 * q^49 - 12 * q^52 - 4 * q^53 - 9 * q^56 - 17 * q^58 - 2 * q^59 - q^61 + 84 * q^62 - 30 * q^64 + 11 * q^67 + 22 * q^68 + 20 * q^71 + 16 * q^73 - 16 * q^74 + 12 * q^76 - 2 * q^79 + 58 * q^82 + 15 * q^83 + 28 * q^86 - 24 * q^88 + 18 * q^89 + 20 * q^91 - 39 * q^92 + 31 * q^94 - 18 * q^97 - 80 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/675\mathbb{Z}\right)^\times$.

$n$	$326$	$352$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$

Coefficient data

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

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

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

$2$	−0.285997	+	0.495361i	−0.202230	+	0.350273i	−0.949247	−	0.314533i	$-0.898152\pi$
$2$	−0.285997	+	0.495361i	−0.202230	+	0.350273i	0.747017	+	0.664805i	$0.231486\pi$
$3$		0			0
$3$		0			0
$4$	0.836412	+	1.44871i	0.418206	+	0.724354i
$5$		0			0
$5$		0			0
$6$		0			0
$7$	0.714003	−	1.23669i	0.269868	−	0.467425i	−0.698960	−	0.715161i	$-0.746353\pi$
$7$	0.714003	−	1.23669i	0.269868	−	0.467425i	0.968828	+	0.247736i	$0.0796866\pi$
$8$	−2.10083			−0.742756
$9$		0			0
$10$		0			0
$11$	1.33641	−	2.31473i	0.402943	−	0.697918i	−0.591136	−	0.806572i	$-0.701321\pi$
$11$	1.33641	−	2.31473i	0.402943	−	0.697918i	0.994080	+	0.108653i	$0.0346538\pi$
$12$		0			0
$13$	2.33641	+	4.04678i	0.648004	+	1.12238i	0.983599	+	0.180370i	$0.0577294\pi$
$13$	2.33641	+	4.04678i	0.648004	+	1.12238i	−0.335595	+	0.942006i	$0.608937\pi$
$14$	0.408405	+	0.707378i	0.109151	+	0.189055i
$15$		0			0
$16$	−1.07199	+	1.85675i	−0.267998	+	0.464187i
$17$	2.67282			0.648255			0.324127	−	0.946013i	$-0.394929\pi$
$17$	2.67282			0.648255			0.324127	+	0.946013i	$0.394929\pi$
$18$		0			0
$19$	4.67282			1.07202			0.536010	−	0.844212i	$-0.319931\pi$
$19$	4.67282			1.07202			0.536010	+	0.844212i	$0.319931\pi$
$20$		0			0
$21$		0			0
$22$	0.764419	+	1.32401i	0.162975	+	0.282280i
$23$	2.95882	+	5.12483i	0.616957	+	1.06860i	0.990038	+	0.140802i	$0.0449680\pi$
$23$	2.95882	+	5.12483i	0.616957	+	1.06860i	−0.373081	+	0.927799i	$0.621699\pi$
$24$		0			0
$25$		0			0
$26$	−2.67282			−0.524184
$27$		0			0
$28$	2.38880			0.451441
$29$	−4.74482	+	8.21826i	−0.881090	+	1.52609i	−0.0309603	+	0.999521i	$0.509857\pi$
$29$	−4.74482	+	8.21826i	−0.881090	+	1.52609i	−0.850130	+	0.526573i	$0.823477\pi$
$30$		0			0
$31$	−3.48040	−	6.02823i	−0.625098	−	1.08270i	−0.988522	−	0.151078i	$-0.951726\pi$
$31$	−3.48040	−	6.02823i	−0.625098	−	1.08270i	0.363424	−	0.931624i	$-0.381608\pi$
$32$	−2.71400	−	4.70079i	−0.479773	−	0.830990i
$33$		0			0
$34$	−0.764419	+	1.32401i	−0.131097	+	0.227066i
$35$		0			0
$36$		0			0
$37$	1.81681			0.298682			0.149341	−	0.988786i	$-0.452285\pi$
$37$	1.81681			0.298682			0.149341	+	0.988786i	$0.452285\pi$
$38$	−1.33641	+	2.31473i	−0.216795	+	0.375499i
$39$		0			0
$40$		0			0
$41$	−0.735581	−	1.27406i	−0.114879	−	0.198975i	0.802853	−	0.596177i	$-0.203315\pi$
$41$	−0.735581	−	1.27406i	−0.114879	−	0.198975i	−0.917731	+	0.397202i	$0.869981\pi$
$42$		0			0
$43$	0.235581	−	0.408039i	0.0359258	−	0.0622254i	−0.847503	−	0.530790i	$-0.821895\pi$
$43$	0.235581	−	0.408039i	0.0359258	−	0.0622254i	0.883429	+	0.468565i	$0.155229\pi$
$44$	4.47116			0.674053
$45$		0			0
$46$	−3.38485			−0.499069
$47$	−3.47842	+	6.02480i	−0.507380	+	0.878808i	0.492584	+	0.870265i	$0.336053\pi$
$47$	−3.47842	+	6.02480i	−0.507380	+	0.878808i	−0.999964	+	0.00854274i	$0.997281\pi$
$48$		0			0
$49$	2.48040	+	4.29618i	0.354343	+	0.613739i
$50$		0			0
$51$		0			0
$52$	−3.90841	+	6.76956i	−0.541998	+	0.938769i
$53$	−1.14399			−0.157139			−0.0785693	−	0.996909i	$-0.525035\pi$
$53$	−1.14399			−0.157139			−0.0785693	+	0.996909i	$0.525035\pi$
$54$		0			0
$55$		0			0
$56$	−1.50000	+	2.59808i	−0.200446	+	0.347183i
$57$		0			0
$58$	−2.71400	−	4.70079i	−0.356366	−	0.617244i
$59$	−0.571993	−	0.990721i	−0.0744672	−	0.128981i	0.826387	−	0.563102i	$-0.190392\pi$
$59$	−0.571993	−	0.990721i	−0.0744672	−	0.128981i	−0.900854	+	0.434121i	$0.857059\pi$
$60$		0			0
$61$	1.26442	−	2.19004i	0.161892	−	0.280406i	−0.773655	−	0.633607i	$-0.781574\pi$
$61$	1.26442	−	2.19004i	0.161892	−	0.280406i	0.935547	+	0.353201i	$0.114907\pi$
$62$	3.98153			0.505655
$63$		0			0
$64$	−1.18319			−0.147899
$65$		0			0
$66$		0			0
$67$	−3.29523	−	5.70751i	−0.402577	−	0.697283i	0.591459	−	0.806335i	$-0.298552\pi$
$67$	−3.29523	−	5.70751i	−0.402577	−	0.697283i	−0.994036	+	0.109051i	$0.965219\pi$
$68$	2.23558	+	3.87214i	0.271104	+	0.469566i
$69$		0			0
$70$		0			0
$71$	12.8745			1.52792			0.763960	−	0.645263i	$-0.223252\pi$
$71$	12.8745			1.52792			0.763960	+	0.645263i	$0.223252\pi$
$72$		0			0
$73$	1.71203			0.200378			0.100189	−	0.994968i	$-0.468055\pi$
$73$	1.71203			0.200378			0.100189	+	0.994968i	$0.468055\pi$
$74$	−0.519602	+	0.899976i	−0.0604025	+	0.104620i
$75$		0			0
$76$	3.90841	+	6.76956i	0.448325	+	0.776521i
$77$	−1.90841	−	3.30545i	−0.217483	−	0.376692i
$78$		0			0
$79$	0.143987	−	0.249392i	0.0161998	−	0.0280588i	−0.857812	−	0.513964i	$-0.828177\pi$
$79$	0.143987	−	0.249392i	0.0161998	−	0.0280588i	0.874012	+	0.485905i	$0.161510\pi$
$80$		0			0
$81$		0			0
$82$	0.841495			0.0929276
$83$	2.14201	−	3.71007i	0.235116	−	0.407233i	−0.724190	−	0.689600i	$-0.757786\pi$
$83$	2.14201	−	3.71007i	0.235116	−	0.407233i	0.959306	+	0.282367i	$0.0911196\pi$
$84$		0			0
$85$		0			0
$86$	0.134751	+	0.233396i	0.0145306	+	0.0251677i
$87$		0			0
$88$	−2.80757	+	4.86286i	−0.299288	+	0.518383i
$89$	3.00000			0.317999			0.159000	−	0.987279i	$-0.449173\pi$
$89$	3.00000			0.317999			0.159000	+	0.987279i	$0.449173\pi$
$90$		0			0
$91$	6.67282			0.699502
$92$	−4.94958	+	8.57293i	−0.516030	+	0.893790i
$93$		0			0
$94$	−1.98963	−	3.44615i	−0.205215	−	0.355443i
$95$		0			0
$96$		0			0
$97$	3.91764	−	6.78555i	0.397776	−	0.688968i	−0.595675	−	0.803225i	$-0.703115\pi$
$97$	3.91764	−	6.78555i	0.397776	−	0.688968i	0.993451	+	0.114257i	$0.0364487\pi$
$98$	−2.83754			−0.286635
$99$		0			0
$100$		0			0
$101$	−2.10083	+	3.63875i	−0.209040	+	0.362069i	−0.951413	−	0.307919i	$-0.900367\pi$
$101$	−2.10083	+	3.63875i	−0.209040	+	0.362069i	0.742372	+	0.669988i	$0.233701\pi$
$102$		0			0
$103$	−0.908405	−	1.57340i	−0.0895078	−	0.155032i	0.817795	−	0.575509i	$-0.195196\pi$
$103$	−0.908405	−	1.57340i	−0.0895078	−	0.155032i	−0.907303	+	0.420477i	$0.861863\pi$
$104$	−4.90841	−	8.50161i	−0.481309	−	0.833651i
$105$		0			0
$106$	0.327176	−	0.566686i	0.0317782	−	0.0550414i
$107$	11.9176			1.15212			0.576061	−	0.817407i	$-0.304589\pi$
$107$	11.9176			1.15212			0.576061	+	0.817407i	$0.304589\pi$
$108$		0			0
$109$	−16.6521			−1.59498			−0.797491	−	0.603331i	$-0.793840\pi$
$109$	−16.6521			−1.59498			−0.797491	+	0.603331i	$0.793840\pi$
$110$		0			0
$111$		0			0
$112$	1.53081	+	2.65145i	0.144648	+	0.250538i
$113$	−10.0616	−	17.4272i	−0.946518	−	1.63942i	−0.752682	−	0.658384i	$-0.771240\pi$
$113$	−10.0616	−	17.4272i	−0.946518	−	1.63942i	−0.193836	−	0.981034i	$-0.562093\pi$
$114$		0			0
$115$		0			0
$116$	−15.8745			−1.47391
$117$		0			0
$118$	0.654353			0.0602380
$119$	1.90841	−	3.30545i	0.174943	−	0.303011i
$120$		0			0
$121$	1.92801	+	3.33941i	0.175273	+	0.303582i
$122$	0.723239	+	1.25269i	0.0654790	+	0.113413i
$123$		0			0
$124$	5.82209	−	10.0842i	0.522839	−	0.905584i
$125$		0			0
$126$		0			0
$127$	−2.18714			−0.194078			−0.0970388	−	0.995281i	$-0.530937\pi$
$127$	−2.18714			−0.194078			−0.0970388	+	0.995281i	$0.530937\pi$
$128$	5.76640	−	9.98769i	0.509682	−	0.882795i
$129$		0			0
$130$		0			0
$131$	−3.00000	−	5.19615i	−0.262111	−	0.453990i	0.704692	−	0.709514i	$-0.251085\pi$
$131$	−3.00000	−	5.19615i	−0.262111	−	0.453990i	−0.966803	+	0.255524i	$0.917752\pi$
$132$		0			0
$133$	3.33641	−	5.77883i	0.289304	−	0.501089i
$134$	3.76970			0.325653
$135$		0			0
$136$	−5.61515			−0.481495
$137$	5.10083	−	8.83490i	0.435793	−	0.754816i	−0.561567	−	0.827431i	$-0.689801\pi$
$137$	5.10083	−	8.83490i	0.435793	−	0.754816i	0.997360	+	0.0726153i	$0.0231345\pi$
$138$		0			0
$139$	−4.00000	−	6.92820i	−0.339276	−	0.587643i	0.645021	−	0.764165i	$-0.276849\pi$
$139$	−4.00000	−	6.92820i	−0.339276	−	0.587643i	−0.984297	+	0.176522i	$0.943515\pi$
$140$		0			0
$141$		0			0
$142$	−3.68206	+	6.37751i	−0.308992	+	0.535189i
$143$	12.4896			1.04444
$144$		0			0
$145$		0			0
$146$	−0.489634	+	0.848071i	−0.0405224	+	0.0701868i
$147$		0			0
$148$	1.51960	+	2.63203i	0.124910	+	0.216351i
$149$	−10.0381	−	17.3865i	−0.822351	−	1.42435i	−0.903927	−	0.427687i	$-0.859329\pi$
$149$	−10.0381	−	17.3865i	−0.822351	−	1.42435i	0.0815762	−	0.996667i	$-0.474005\pi$
$150$		0			0
$151$	−1.51960	+	2.63203i	−0.123663	+	0.214191i	−0.921210	−	0.389066i	$-0.872798\pi$
$151$	−1.51960	+	2.63203i	−0.123663	+	0.214191i	0.797546	+	0.603258i	$0.206131\pi$
$152$	−9.81681			−0.796248
$153$		0			0
$154$	2.18319			0.175926
$155$		0			0
$156$		0			0
$157$	0.100830	+	0.174643i	0.00804714	+	0.0139381i	0.870021	−	0.493015i	$-0.164105\pi$
$157$	0.100830	+	0.174643i	0.00804714	+	0.0139381i	−0.861974	+	0.506953i	$0.830772\pi$
$158$	0.0823593	+	0.142651i	0.00655216	+	0.0113487i
$159$		0			0
$160$		0			0
$161$	8.45043			0.665987
$162$		0			0
$163$	−17.8168			−1.39552			−0.697760	−	0.716331i	$-0.745820\pi$
$163$	−17.8168			−1.39552			−0.697760	+	0.716331i	$0.745820\pi$
$164$	1.23050	−	2.13129i	0.0960858	−	0.166425i
$165$		0			0
$166$	1.22522	+	2.12214i	0.0950952	+	0.164710i
$167$	−7.05042	−	12.2117i	−0.545578	−	0.944968i	−0.998570	−	0.0534538i	$-0.982977\pi$
$167$	−7.05042	−	12.2117i	−0.545578	−	0.944968i	0.452993	−	0.891514i	$-0.350356\pi$
$168$		0			0
$169$	−4.41764	+	7.65158i	−0.339819	+	0.588583i
$170$		0			0
$171$		0			0
$172$	0.788172			0.0600976
$173$	−2.18319	+	3.78140i	−0.165985	+	0.287494i	−0.937005	−	0.349317i	$-0.886414\pi$
$173$	−2.18319	+	3.78140i	−0.165985	+	0.287494i	0.771020	+	0.636811i	$0.219747\pi$
$174$		0			0
$175$		0			0
$176$	2.86525	+	4.96276i	0.215976	+	0.374082i
$177$		0			0
$178$	−0.857990	+	1.48608i	−0.0643091	+	0.111387i
$179$	15.1625			1.13330			0.566648	−	0.823960i	$-0.308240\pi$
$179$	15.1625			1.13330			0.566648	+	0.823960i	$0.308240\pi$
$180$		0			0
$181$	3.20166			0.237978			0.118989	−	0.992896i	$-0.462035\pi$
$181$	3.20166			0.237978			0.118989	+	0.992896i	$0.462035\pi$
$182$	−1.90841	+	3.30545i	−0.141460	+	0.245017i
$183$		0			0
$184$	−6.21598	−	10.7664i	−0.458248	−	0.793709i
$185$		0			0
$186$		0			0
$187$	3.57199	−	6.18687i	0.261210	−	0.452429i
$188$	−11.6376			−0.848757
$189$		0			0
$190$		0			0
$191$	−1.41877	+	2.45738i	−0.102659	+	0.177810i	−0.912779	−	0.408453i	$-0.866068\pi$
$191$	−1.41877	+	2.45738i	−0.102659	+	0.177810i	0.810121	+	0.586263i	$0.199402\pi$
$192$		0			0
$193$	−9.39409	−	16.2710i	−0.676201	−	1.17121i	−0.976116	−	0.217249i	$-0.930292\pi$
$193$	−9.39409	−	16.2710i	−0.676201	−	1.17121i	0.299915	−	0.953966i	$-0.403042\pi$
$194$	2.24086	+	3.88129i	0.160885	+	0.278660i
$195$		0			0
$196$	−4.14927	+	7.18675i	−0.296376	+	0.513339i
$197$	5.83528			0.415747			0.207873	−	0.978156i	$-0.433346\pi$
$197$	5.83528			0.415747			0.207873	+	0.978156i	$0.433346\pi$
$198$		0			0
$199$	13.0761			0.926943			0.463472	−	0.886112i	$-0.346604\pi$
$199$	13.0761			0.926943			0.463472	+	0.886112i	$0.346604\pi$
$200$		0			0
$201$		0			0
$202$	−1.20166	−	2.08134i	−0.0845486	−	0.146442i
$203$	6.77563	+	11.7357i	0.475556	+	0.823687i
$204$		0			0
$205$		0			0
$206$	1.03920			0.0724047
$207$		0			0
$208$	−10.0185			−0.694656
$209$	6.24482	−	10.8163i	0.431963	−	0.748182i
$210$		0			0
$211$	−4.19243	−	7.26149i	−0.288618	−	0.499902i	0.684862	−	0.728673i	$-0.259863\pi$
$211$	−4.19243	−	7.26149i	−0.288618	−	0.499902i	−0.973480	+	0.228771i	$0.926529\pi$
$212$	−0.956844	−	1.65730i	−0.0657163	−	0.113824i
$213$		0			0
$214$	−3.40841	+	5.90353i	−0.232994	+	0.403557i
$215$		0			0
$216$		0			0
$217$	−9.94006			−0.674776
$218$	4.76244	−	8.24879i	0.322553	−	0.558679i
$219$		0			0
$220$		0			0
$221$	6.24482	+	10.8163i	0.420072	+	0.727586i
$222$		0			0
$223$	4.58321	−	7.93834i	0.306914	−	0.531591i	−0.670772	−	0.741664i	$-0.734037\pi$
$223$	4.58321	−	7.93834i	0.306914	−	0.531591i	0.977686	+	0.210073i	$0.0673702\pi$
$224$	−7.75123			−0.517901
$225$		0			0
$226$	11.5104			0.765658
$227$	1.33641	−	2.31473i	0.0887008	−	0.153634i	−0.818261	−	0.574846i	$-0.805062\pi$
$227$	1.33641	−	2.31473i	0.0887008	−	0.153634i	0.906962	+	0.421212i	$0.138395\pi$
$228$		0			0
$229$	−1.27365	−	2.20603i	−0.0841654	−	0.145779i	0.820870	−	0.571115i	$-0.193489\pi$
$229$	−1.27365	−	2.20603i	−0.0841654	−	0.145779i	−0.905035	+	0.425336i	$0.860156\pi$
$230$		0			0
$231$		0			0
$232$	9.96806	−	17.2652i	0.654435	−	1.13351i
$233$	−6.22013			−0.407494			−0.203747	−	0.979024i	$-0.565312\pi$
$233$	−6.22013			−0.407494			−0.203747	+	0.979024i	$0.565312\pi$
$234$		0			0
$235$		0			0
$236$	0.956844	−	1.65730i	0.0622852	−	0.107881i
$237$		0			0
$238$	1.09159	+	1.89070i	0.0707576	+	0.122556i
$239$	4.06163	+	7.03494i	0.262725	+	0.455053i	0.966965	−	0.254909i	$-0.0820455\pi$
$239$	4.06163	+	7.03494i	0.262725	+	0.455053i	−0.704240	+	0.709962i	$0.748712\pi$
$240$		0			0
$241$	13.1821	−	22.8320i	0.849131	−	1.47074i	−0.0328536	−	0.999460i	$-0.510460\pi$
$241$	13.1821	−	22.8320i	0.849131	−	1.47074i	0.881985	−	0.471278i	$-0.156207\pi$
$242$	−2.20561			−0.141782
$243$		0			0
$244$	4.23030			0.270817
$245$		0			0
$246$		0			0
$247$	10.9176	+	18.9099i	0.694673	+	1.20321i
$248$	7.31173	+	12.6643i	0.464295	+	0.804183i
$249$		0			0
$250$		0			0
$251$	0.549569			0.0346885			0.0173443	−	0.999850i	$-0.494479\pi$
$251$	0.549569			0.0346885			0.0173443	+	0.999850i	$0.494479\pi$
$252$		0			0
$253$	15.8168			0.994394
$254$	0.625515	−	1.08342i	0.0392483	−	0.0679801i
$255$		0			0
$256$	2.11515	+	3.66355i	0.132197	+	0.228972i
$257$	9.00000	+	15.5885i	0.561405	+	0.972381i	0.997374	+	0.0724199i	$0.0230722\pi$
$257$	9.00000	+	15.5885i	0.561405	+	0.972381i	−0.435970	+	0.899961i	$0.643595\pi$
$258$		0			0
$259$	1.29721	−	2.24683i	0.0806046	−	0.139611i
$260$		0			0
$261$		0			0
$262$	3.43196			0.212027
$263$	−5.94761	+	10.3016i	−0.366745	+	0.635221i	−0.989055	−	0.147550i	$-0.952861\pi$
$263$	−5.94761	+	10.3016i	−0.366745	+	0.635221i	0.622309	+	0.782771i	$0.286195\pi$
$264$		0			0
$265$		0			0
$266$	1.90841	+	3.30545i	0.117012	+	0.202670i
$267$		0			0
$268$	5.51234	−	9.54766i	0.336720	−	0.583216i
$269$	−28.5737			−1.74217			−0.871084	−	0.491134i	$-0.836583\pi$
$269$	−28.5737			−1.74217			−0.871084	+	0.491134i	$0.836583\pi$
$270$		0			0
$271$	−23.3641			−1.41927			−0.709635	−	0.704570i	$-0.751140\pi$
$271$	−23.3641			−1.41927			−0.709635	+	0.704570i	$0.751140\pi$
$272$	−2.86525	+	4.96276i	−0.173731	+	0.300911i
$273$		0			0
$274$	2.91764	+	5.05350i	0.176261	+	0.305293i
$275$		0			0
$276$		0			0
$277$	−7.53807	+	13.0563i	−0.452919	+	0.784479i	−0.998566	−	0.0535366i	$-0.982951\pi$
$277$	−7.53807	+	13.0563i	−0.452919	+	0.784479i	0.545647	+	0.838015i	$0.316284\pi$
$278$	4.57595			0.274447
$279$		0			0
$280$		0			0
$281$	3.32605	−	5.76088i	0.198415	−	0.343665i	−0.749599	−	0.661892i	$-0.769754\pi$
$281$	3.32605	−	5.76088i	0.198415	−	0.343665i	0.948015	+	0.318226i	$0.103087\pi$
$282$		0			0
$283$	13.4485	+	23.2934i	0.799428	+	1.38465i	0.919989	+	0.391943i	$0.128197\pi$
$283$	13.4485	+	23.2934i	0.799428	+	1.38465i	−0.120562	+	0.992706i	$0.538470\pi$
$284$	10.7684	+	18.6514i	0.638985	+	1.10675i
$285$		0			0
$286$	−3.57199	+	6.18687i	−0.211216	+	0.365838i
$287$	−2.10083			−0.124008
$288$		0			0
$289$	−9.85601			−0.579765
$290$		0			0
$291$		0			0
$292$	1.43196	+	2.48023i	0.0837991	+	0.145144i
$293$	−6.19243	−	10.7256i	−0.361765	−	0.626596i	0.626486	−	0.779433i	$-0.284493\pi$
$293$	−6.19243	−	10.7256i	−0.361765	−	0.626596i	−0.988251	+	0.152837i	$0.951159\pi$
$294$		0			0
$295$		0			0
$296$	−3.81681			−0.221848
$297$		0			0
$298$	11.4834			0.665217
$299$	−13.8260	+	23.9474i	−0.799581	+	1.38491i
$300$		0			0
$301$	−0.336412	−	0.582682i	−0.0193905	−	0.0335853i
$302$	−0.869202	−	1.50550i	−0.0500169	−	0.0866319i
$303$		0			0
$304$	−5.00924	+	8.67625i	−0.287299	+	0.497617i
$305$		0			0
$306$		0			0
$307$	2.49359			0.142317			0.0711583	−	0.997465i	$-0.477330\pi$
$307$	2.49359			0.142317			0.0711583	+	0.997465i	$0.477330\pi$
$308$	3.19243	−	5.52944i	0.181905	−	0.315069i
$309$		0			0
$310$		0			0
$311$	−12.1101	−	20.9752i	−0.686699	−	1.18940i	−0.972900	−	0.231228i	$-0.925726\pi$
$311$	−12.1101	−	20.9752i	−0.686699	−	1.18940i	0.286201	−	0.958170i	$-0.407608\pi$
$312$		0			0
$313$	−17.5420	+	30.3837i	−0.991534	+	1.71739i	−0.383315	+	0.923618i	$0.625218\pi$
$313$	−17.5420	+	30.3837i	−0.991534	+	1.71739i	−0.608219	+	0.793770i	$0.708116\pi$
$314$	−0.115349			−0.00650950
$315$		0			0
$316$	0.481728			0.0270993
$317$	5.23558	−	9.06829i	0.294060	−	0.509326i	−0.680706	−	0.732557i	$-0.738327\pi$
$317$	5.23558	−	9.06829i	0.294060	−	0.509326i	0.974766	+	0.223231i	$0.0716603\pi$
$318$		0			0
$319$	12.6821	+	21.9660i	0.710059	+	1.22986i
$320$		0			0
$321$		0			0
$322$	−2.41679	+	4.18601i	−0.134683	+	0.233277i
$323$	12.4896			0.694942
$324$		0			0
$325$		0			0
$326$	5.09555	−	8.82575i	0.282216	−	0.488813i
$327$		0			0
$328$	1.54533	+	2.67659i	0.0853267	+	0.147790i
$329$	4.96721	+	8.60346i	0.273851	+	0.474324i
$330$		0			0
$331$	−8.38880	+	14.5298i	−0.461090	+	0.798632i	−0.999016	−	0.0443606i	$-0.985875\pi$
$331$	−8.38880	+	14.5298i	−0.461090	+	0.798632i	0.537925	+	0.842993i	$0.319208\pi$
$332$	7.16641			0.393308
$333$		0			0
$334$	8.06558			0.441329
$335$		0			0
$336$		0			0
$337$	−13.5905	−	23.5394i	−0.740320	−	1.28227i	−0.952350	−	0.305008i	$-0.901341\pi$
$337$	−13.5905	−	23.5394i	−0.740320	−	1.28227i	0.212030	−	0.977263i	$-0.431993\pi$
$338$	−2.52686	−	4.37665i	−0.137443	−	0.238058i
$339$		0			0
$340$		0			0
$341$	−18.6050			−1.00752
$342$		0			0
$343$	17.0801			0.922239
$344$	−0.494917	+	0.857221i	−0.0266841	+	0.0462182i
$345$		0			0
$346$	−1.24877	−	2.16293i	−0.0671343	−	0.116280i
$347$	11.7829	+	20.4086i	0.632539	+	1.09559i	0.987031	+	0.160530i	$0.0513204\pi$
$347$	11.7829	+	20.4086i	0.632539	+	1.09559i	−0.354492	+	0.935059i	$0.615346\pi$
$348$		0			0
$349$	5.35601	−	9.27689i	0.286701	−	0.496580i	−0.686319	−	0.727300i	$-0.740775\pi$
$349$	5.35601	−	9.27689i	0.286701	−	0.496580i	0.973020	+	0.230720i	$0.0741081\pi$
$350$		0			0
$351$		0			0
$352$	−14.5081			−0.773285
$353$	−13.6336	+	23.6141i	−0.725644	+	1.25685i	0.233064	+	0.972461i	$0.425125\pi$
$353$	−13.6336	+	23.6141i	−0.725644	+	1.25685i	−0.958708	+	0.284392i	$0.908208\pi$
$354$		0			0
$355$		0			0
$356$	2.50924	+	4.34612i	0.132989	+	0.230344i
$357$		0			0
$358$	−4.33641	+	7.51089i	−0.229186	+	0.396963i
$359$	10.6807			0.563707			0.281854	−	0.959457i	$-0.409051\pi$
$359$	10.6807			0.563707			0.281854	+	0.959457i	$0.409051\pi$
$360$		0			0
$361$	2.83528			0.149225
$362$	−0.915664	+	1.58598i	−0.0481262	+	0.0833571i
$363$		0			0
$364$	5.58123	+	9.66697i	0.292536	+	0.506687i
$365$		0			0
$366$		0			0
$367$	−4.23558	+	7.33624i	−0.221096	+	0.382949i	−0.955141	−	0.296152i	$-0.904297\pi$
$367$	−4.23558	+	7.33624i	−0.221096	+	0.382949i	0.734045	+	0.679100i	$0.237630\pi$
$368$	−12.6873			−0.661373
$369$		0			0
$370$		0			0
$371$	−0.816810	+	1.41476i	−0.0424067	+	0.0734505i
$372$		0			0
$373$	5.06163	+	8.76700i	0.262081	+	0.453938i	0.966795	−	0.255554i	$-0.0822579\pi$
$373$	5.06163	+	8.76700i	0.262081	+	0.453938i	−0.704714	+	0.709492i	$0.748925\pi$
$374$	2.04316	+	3.53885i	0.105649	+	0.182990i
$375$		0			0
$376$	7.30757	−	12.6571i	0.376859	−	0.652740i
$377$	−44.3434			−2.28380
$378$		0			0
$379$	11.9216			0.612371			0.306186	−	0.951972i	$-0.400947\pi$
$379$	11.9216			0.612371			0.306186	+	0.951972i	$0.400947\pi$
$380$		0			0
$381$		0			0
$382$	−0.811528	−	1.40561i	−0.0415214	−	0.0719171i
$383$	−4.90841	−	8.50161i	−0.250808	−	0.434412i	0.712941	−	0.701224i	$-0.247363\pi$
$383$	−4.90841	−	8.50161i	−0.250808	−	0.434412i	−0.963748	+	0.266813i	$0.914030\pi$
$384$		0			0
$385$		0			0
$386$	10.7467			0.546993
$387$		0			0
$388$	13.1070			0.665409
$389$	4.61007	−	7.98487i	0.233740	−	0.404849i	−0.725166	−	0.688574i	$-0.758237\pi$
$389$	4.61007	−	7.98487i	0.233740	−	0.404849i	0.958906	+	0.283725i	$0.0915703\pi$
$390$		0			0
$391$	7.90841	+	13.6978i	0.399945	+	0.692725i
$392$	−5.21090	−	9.02554i	−0.263190	−	0.455858i
$393$		0			0
$394$	−1.66887	+	2.89057i	−0.0840765	+	0.145625i
$395$		0			0
$396$		0			0
$397$	22.9793			1.15330			0.576648	−	0.816993i	$-0.304360\pi$
$397$	22.9793			1.15330			0.576648	+	0.816993i	$0.304360\pi$
$398$	−3.73973	+	6.47741i	−0.187456	+	0.324683i
$399$		0			0
$400$		0			0
$401$	−5.53279	−	9.58307i	−0.276294	−	0.478556i	0.694167	−	0.719814i	$-0.255773\pi$
$401$	−5.53279	−	9.58307i	−0.276294	−	0.478556i	−0.970461	+	0.241259i	$0.922440\pi$
$402$		0			0
$403$	16.2633	−	28.1688i	0.810132	−	1.40319i
$404$	−7.02864			−0.349688
$405$		0			0
$406$	−7.75123			−0.384687
$407$	2.42801	−	4.20543i	0.120352	−	0.208455i
$408$		0			0
$409$	8.81681	+	15.2712i	0.435963	+	0.755110i	0.997374	−	0.0724270i	$-0.0230744\pi$
$409$	8.81681	+	15.2712i	0.435963	+	0.755110i	−0.561411	+	0.827537i	$0.689741\pi$
$410$		0			0
$411$		0			0
$412$	1.51960	−	2.63203i	0.0748654	−	0.129671i
$413$	−1.63362			−0.0803852
$414$		0			0
$415$		0			0
$416$	12.6821	−	21.9660i	0.621789	−	1.07697i
$417$		0			0
$418$	3.57199	+	6.18687i	0.174712	+	0.302610i
$419$	18.5173	+	32.0730i	0.904631	+	1.56687i	0.821411	+	0.570336i	$0.193187\pi$
$419$	18.5173	+	32.0730i	0.904631	+	1.56687i	0.0832199	+	0.996531i	$0.473480\pi$
$420$		0			0
$421$	−2.52884	+	4.38007i	−0.123248	+	0.213472i	−0.921047	−	0.389452i	$-0.872664\pi$
$421$	−2.52884	+	4.38007i	−0.123248	+	0.213472i	0.797799	+	0.602924i	$0.205998\pi$
$422$	4.79608			0.233469
$423$		0			0
$424$	2.40332			0.116716
$425$		0			0
$426$		0			0
$427$	−1.80560	−	3.12739i	−0.0873790	−	0.151345i
$428$	9.96806	+	17.2652i	0.481824	+	0.834544i
$429$		0			0
$430$		0			0
$431$	5.23030			0.251935			0.125967	−	0.992034i	$-0.459797\pi$
$431$	5.23030			0.251935			0.125967	+	0.992034i	$0.459797\pi$
$432$		0			0
$433$	−34.3434			−1.65044			−0.825219	−	0.564813i	$-0.808948\pi$
$433$	−34.3434			−1.65044			−0.825219	+	0.564813i	$0.808948\pi$
$434$	2.84283	−	4.92392i	0.136460	−	0.236356i
$435$		0			0
$436$	−13.9280	−	24.1240i	−0.667031	−	1.15533i
$437$	13.8260	+	23.9474i	0.661389	+	1.14556i
$438$		0			0
$439$	9.77365	−	16.9285i	0.466471	−	0.807952i	−0.532796	−	0.846244i	$-0.678859\pi$
$439$	9.77365	−	16.9285i	0.466471	−	0.807952i	0.999267	+	0.0382924i	$0.0121918\pi$
$440$		0			0
$441$		0			0
$442$	−7.14399			−0.339805
$443$	−5.25208	+	9.09686i	−0.249534	+	0.432205i	−0.963396	−	0.268081i	$-0.913611\pi$
$443$	−5.25208	+	9.09686i	−0.249534	+	0.432205i	0.713863	+	0.700286i	$0.246944\pi$
$444$		0			0
$445$		0			0
$446$	2.62156	+	4.54068i	0.124135	+	0.215007i
$447$		0			0
$448$	−0.844801	+	1.46324i	−0.0399131	+	0.0691315i
$449$	−22.8560			−1.07864			−0.539321	−	0.842100i	$-0.681319\pi$
$449$	−22.8560			−1.07864			−0.539321	+	0.842100i	$0.681319\pi$
$450$		0			0
$451$	−3.93216			−0.185158
$452$	16.8313	−	29.1527i	0.791679	−	1.37123i
$453$		0			0
$454$	0.764419	+	1.32401i	0.0358759	+	0.0621390i
$455$		0			0
$456$		0			0
$457$	−7.01319	+	12.1472i	−0.328063	+	0.568222i	−0.982127	−	0.188217i	$-0.939729\pi$
$457$	−7.01319	+	12.1472i	−0.328063	+	0.568222i	0.654064	+	0.756439i	$0.273063\pi$
$458$	1.45704			0.0680832
$459$		0			0
$460$		0			0
$461$	−12.0513	+	20.8734i	−0.561283	+	0.972171i	0.436102	+	0.899897i	$0.356359\pi$
$461$	−12.0513	+	20.8734i	−0.561283	+	0.972171i	−0.997385	+	0.0722736i	$0.976975\pi$
$462$		0			0
$463$	−16.9700	−	29.3930i	−0.788664	−	1.36601i	−0.926785	−	0.375591i	$-0.877440\pi$
$463$	−16.9700	−	29.3930i	−0.788664	−	1.36601i	0.138121	−	0.990415i	$-0.455894\pi$
$464$	−10.1728	−	17.6198i	−0.472261	−	0.817981i
$465$		0			0
$466$	1.77894	−	3.08121i	0.0824077	−	0.142734i
$467$	−27.3720			−1.26663			−0.633313	−	0.773896i	$-0.718305\pi$
$467$	−27.3720			−1.26663			−0.633313	+	0.773896i	$0.718305\pi$
$468$		0			0
$469$	−9.41123			−0.434570
$470$		0			0
$471$		0			0
$472$	1.20166	+	2.08134i	0.0553109	+	0.0958013i
$473$	−0.629668	−	1.09062i	−0.0289521	−	0.0501466i
$474$		0			0
$475$		0			0
$476$	6.38485			0.292649
$477$		0			0
$478$	−4.64645			−0.212524
$479$	2.61515	−	4.52957i	0.119489	−	0.206961i	−0.800076	−	0.599898i	$-0.795208\pi$
$479$	2.61515	−	4.52957i	0.119489	−	0.206961i	0.919565	+	0.392937i	$0.128541\pi$
$480$		0			0
$481$	4.24482	+	7.35224i	0.193547	+	0.335233i
$482$	7.54005	+	13.0597i	0.343440	+	0.594855i
$483$		0			0
$484$	−3.22522	+	5.58624i	−0.146601	+	0.253920i
$485$		0			0
$486$		0			0
$487$	24.0185			1.08838			0.544190	−	0.838962i	$-0.316837\pi$
$487$	24.0185			1.08838			0.544190	+	0.838962i	$0.316837\pi$
$488$	−2.65633	+	4.60090i	−0.120246	+	0.208273i
$489$		0			0
$490$		0			0
$491$	7.38880	+	12.7978i	0.333452	+	0.577556i	0.983186	−	0.182606i	$-0.0584531\pi$
$491$	7.38880	+	12.7978i	0.333452	+	0.577556i	−0.649734	+	0.760161i	$0.725120\pi$
$492$		0			0
$493$	−12.6821	+	21.9660i	−0.571171	+	0.989298i
$494$	−12.4896			−0.561935
$495$		0			0
$496$	14.9239			0.670101
$497$	9.19243	−	15.9217i	0.412337	−	0.714188i
$498$		0			0
$499$	−12.4280	−	21.5259i	−0.556354	−	0.963633i	−0.997797	−	0.0663440i	$-0.978867\pi$
$499$	−12.4280	−	21.5259i	−0.556354	−	0.963633i	0.441443	−	0.897289i	$-0.354467\pi$
$500$		0			0
$501$		0			0
$502$	−0.157175	+	0.272235i	−0.00701506	+	0.0121504i
$503$	38.9154			1.73515			0.867576	−	0.497305i	$-0.165677\pi$
$503$	38.9154			1.73515			0.867576	+	0.497305i	$0.165677\pi$
$504$		0			0
$505$		0			0
$506$	−4.52355	+	7.83503i	−0.201097	+	0.348309i
$507$		0			0
$508$	−1.82935	−	3.16853i	−0.0811644	−	0.140581i
$509$	1.01037	+	1.75001i	0.0447837	+	0.0775676i	0.887548	−	0.460715i	$-0.152407\pi$
$509$	1.01037	+	1.75001i	0.0447837	+	0.0775676i	−0.842765	+	0.538282i	$0.819073\pi$
$510$		0			0
$511$	1.22239	−	2.11725i	0.0540755	−	0.0936615i
$512$	20.6459			0.912428
$513$		0			0
$514$	−10.2959			−0.454132
$515$		0			0
$516$		0			0
$517$	9.29721	+	16.1032i	0.408891	+	0.708220i
$518$	0.741995	+	1.28517i	0.0326014	+	0.0564672i
$519$		0			0
$520$		0			0
$521$	23.0290			1.00892			0.504460	−	0.863435i	$-0.331692\pi$
$521$	23.0290			1.00892			0.504460	+	0.863435i	$0.331692\pi$
$522$		0			0
$523$	41.1170			1.79792			0.898961	−	0.438028i	$-0.144323\pi$
$523$	41.1170			1.79792			0.898961	+	0.438028i	$0.144323\pi$
$524$	5.01847	−	8.69225i	0.219233	−	0.379723i
$525$		0			0
$526$	−3.40199	−	5.89242i	−0.148334	−	0.256922i
$527$	−9.30249	−	16.1124i	−0.405223	−	0.701867i
$528$		0			0
$529$	−6.00924	+	10.4083i	−0.261271	+	0.452535i
$530$		0			0
$531$		0			0
$532$	11.1625			0.483954
$533$	3.43724	−	5.95348i	0.148883	−	0.257874i
$534$		0			0
$535$		0			0
$536$	6.92272	+	11.9905i	0.299016	+	0.517911i
$537$		0			0
$538$	8.17198	−	14.1543i	0.352319	−	0.610234i
$539$	13.2593			0.571120
$540$		0			0
$541$	5.20957			0.223977			0.111988	−	0.993710i	$-0.464278\pi$
$541$	5.20957			0.223977			0.111988	+	0.993710i	$0.464278\pi$
$542$	6.68206	−	11.5737i	0.287019	−	0.497132i
$543$		0			0
$544$	−7.25405	−	12.5644i	−0.311015	−	0.538694i
$545$		0			0
$546$		0			0
$547$	20.0204	−	34.6764i	0.856013	−	1.48266i	−0.0196900	−	0.999806i	$-0.506268\pi$
$547$	20.0204	−	34.6764i	0.856013	−	1.48266i	0.875702	−	0.482851i	$-0.160399\pi$
$548$	17.0656			0.729005
$549$		0			0
$550$		0			0
$551$	−22.1717	+	38.4025i	−0.944546	+	1.63600i
$552$		0			0
$553$	−0.205614	−	0.356133i	−0.00874359	−	0.0151443i
$554$	−4.31173	−	7.46813i	−0.183188	−	0.317290i
$555$		0			0
$556$	6.69129	−	11.5897i	0.283774	−	0.491511i
$557$	−14.4033			−0.610288			−0.305144	−	0.952306i	$-0.598705\pi$
$557$	−14.4033			−0.610288			−0.305144	+	0.952306i	$0.598705\pi$
$558$		0			0
$559$	2.20166			0.0931203
$560$		0			0
$561$		0			0
$562$	1.90248	+	3.29518i	0.0802511	+	0.138999i
$563$	−14.6840	−	25.4335i	−0.618858	−	1.07189i	−0.989694	−	0.143196i	$-0.954262\pi$
$563$	−14.6840	−	25.4335i	−0.618858	−	1.07189i	0.370836	−	0.928698i	$-0.379071\pi$
$564$		0			0
$565$		0			0
$566$	−15.3849			−0.646674
$567$		0			0
$568$	−27.0471			−1.13487
$569$	23.4033	−	40.5357i	0.981118	−	1.69935i	0.323062	−	0.946378i	$-0.395288\pi$
$569$	23.4033	−	40.5357i	0.981118	−	1.69935i	0.658056	−	0.752969i	$-0.271379\pi$
$570$		0			0
$571$	−10.0000	−	17.3205i	−0.418487	−	0.724841i	0.577301	−	0.816532i	$-0.304106\pi$
$571$	−10.0000	−	17.3205i	−0.418487	−	0.724841i	−0.995788	+	0.0916910i	$0.970773\pi$
$572$	10.4465	+	18.0938i	0.436789	+	0.756541i
$573$		0			0
$574$	0.600830	−	1.04067i	0.0250782	−	0.0434367i
$575$		0			0
$576$		0			0
$577$	−28.2386			−1.17559			−0.587794	−	0.809010i	$-0.700004\pi$
$577$	−28.2386			−1.17559			−0.587794	+	0.809010i	$0.700004\pi$
$578$	2.81879	−	4.88228i	0.117246	−	0.203076i
$579$		0			0
$580$		0			0
$581$	−3.05880	−	5.29801i	−0.126901	−	0.219798i
$582$		0			0
$583$	−1.52884	+	2.64802i	−0.0633180	+	0.109670i
$584$	−3.59668			−0.148832
$585$		0			0
$586$	7.08405			0.292639
$587$	−9.04118	+	15.6598i	−0.373169	+	0.646348i	−0.990051	−	0.140707i	$-0.955062\pi$
$587$	−9.04118	+	15.6598i	−0.373169	+	0.646348i	0.616882	+	0.787056i	$0.288396\pi$
$588$		0			0
$589$	−16.2633	−	28.1688i	−0.670117	−	1.16068i
$590$		0			0
$591$		0			0
$592$	−1.94761	+	3.37336i	−0.0800462	+	0.138644i
$593$	−7.73840			−0.317778			−0.158889	−	0.987296i	$-0.550791\pi$
$593$	−7.73840			−0.317778			−0.158889	+	0.987296i	$0.550791\pi$
$594$		0			0
$595$		0			0
$596$	16.7919	−	29.0845i	0.687824	−	1.19135i
$597$		0			0
$598$	−7.90841	−	13.6978i	−0.323399	−	0.560143i
$599$	−13.9608	−	24.1808i	−0.570423	−	0.988001i	−0.996522	−	0.0833249i	$-0.973446\pi$
$599$	−13.9608	−	24.1808i	−0.570423	−	0.988001i	0.426100	−	0.904676i	$-0.359887\pi$
$600$		0			0
$601$	−19.2201	+	33.2902i	−0.784006	+	1.35794i	0.145586	+	0.989346i	$0.453493\pi$
$601$	−19.2201	+	33.2902i	−0.784006	+	1.35794i	−0.929591	+	0.368592i	$0.879840\pi$
$602$	0.384851			0.0156853
$603$		0			0
$604$	−5.08405			−0.206867
$605$		0			0
$606$		0			0
$607$	−0.319917	−	0.554113i	−0.0129850	−	0.0224907i	0.859460	−	0.511203i	$-0.170800\pi$
$607$	−0.319917	−	0.554113i	−0.0129850	−	0.0224907i	−0.872445	+	0.488712i	$0.837467\pi$
$608$	−12.6821	−	21.9660i	−0.514325	−	0.890838i
$609$		0			0
$610$		0			0
$611$	−32.5081			−1.31514
$612$		0			0
$613$	−42.7467			−1.72652			−0.863262	−	0.504757i	$-0.831582\pi$
$613$	−42.7467			−1.72652			−0.863262	+	0.504757i	$0.831582\pi$
$614$	−0.713157	+	1.23522i	−0.0287807	+	0.0498496i
$615$		0			0
$616$	4.00924	+	6.94420i	0.161537	+	0.279790i
$617$	−10.5513	−	18.2753i	−0.424778	−	0.735737i	0.571622	−	0.820517i	$-0.306314\pi$
$617$	−10.5513	−	18.2753i	−0.424778	−	0.735737i	−0.996400	+	0.0847805i	$0.972981\pi$
$618$		0			0
$619$	6.82605	−	11.8231i	0.274362	−	0.475209i	−0.695612	−	0.718418i	$-0.744867\pi$
$619$	6.82605	−	11.8231i	0.274362	−	0.475209i	0.969974	+	0.243209i	$0.0782000\pi$
$620$		0			0
$621$		0			0
$622$	13.8538			0.555485
$623$	2.14201	−	3.71007i	0.0858178	−	0.148641i
$624$		0			0
$625$		0			0
$626$	−10.0339	−	17.3793i	−0.401036	−	0.694615i
$627$		0			0
$628$	−0.168672	+	0.292148i	−0.00673073	+	0.0116580i
$629$	4.85601			0.193622
$630$		0			0
$631$	33.2593			1.32403			0.662017	−	0.749489i	$-0.269701\pi$
$631$	33.2593			1.32403			0.662017	+	0.749489i	$0.269701\pi$
$632$	−0.302491	+	0.523930i	−0.0120325	+	0.0208408i
$633$		0			0
$634$	2.99472	+	5.18700i	0.118935	+	0.206002i
$635$		0			0
$636$		0			0
$637$	−11.5905	+	20.0753i	−0.459231	+	0.795411i
$638$	−14.5081			−0.574381
$639$		0			0
$640$		0			0
$641$	13.1429	−	22.7641i	0.519112	−	0.899128i	−0.480642	−	0.876917i	$-0.659596\pi$
$641$	13.1429	−	22.7641i	0.519112	−	0.899128i	0.999753	−	0.0222106i	$-0.00707044\pi$
$642$		0			0
$643$	10.2913	+	17.8250i	0.405848	+	0.702950i	0.994420	−	0.105495i	$-0.0336427\pi$
$643$	10.2913	+	17.8250i	0.405848	+	0.702950i	−0.588571	+	0.808445i	$0.700309\pi$
$644$	7.06804	+	12.2422i	0.278520	+	0.482410i
$645$		0			0
$646$	−3.57199	+	6.18687i	−0.140538	+	0.243419i
$647$	23.2527			0.914159			0.457079	−	0.889426i	$-0.348896\pi$
$647$	23.2527			0.914159			0.457079	+	0.889426i	$0.348896\pi$
$648$		0			0
$649$	−3.05767			−0.120024
$650$		0			0
$651$		0			0
$652$	−14.9022	−	25.8114i	−0.583615	−	1.01085i
$653$	−9.37957	−	16.2459i	−0.367051	−	0.635751i	0.622052	−	0.782976i	$-0.286299\pi$
$653$	−9.37957	−	16.2459i	−0.367051	−	0.635751i	−0.989103	+	0.147225i	$0.952966\pi$
$654$		0			0
$655$		0			0
$656$	3.15415			0.123149
$657$		0			0
$658$	−5.68242			−0.221524
$659$	−0.140034	+	0.242545i	−0.00545494	+	0.00944823i	−0.868740	−	0.495268i	$-0.835070\pi$
$659$	−0.140034	+	0.242545i	−0.00545494	+	0.00944823i	0.863285	+	0.504717i	$0.168403\pi$
$660$		0			0
$661$	19.8930	+	34.4556i	0.773746	+	1.34017i	0.935496	+	0.353336i	$0.114953\pi$
$661$	19.8930	+	34.4556i	0.773746	+	1.34017i	−0.161750	+	0.986832i	$0.551714\pi$
$662$	−4.79834	−	8.31097i	−0.186493	−	0.323015i
$663$		0			0
$664$	−4.50000	+	7.79423i	−0.174634	+	0.302475i
$665$		0			0
$666$		0			0
$667$	−56.1562			−2.17438
$668$	11.7941	−	20.4280i	0.456327	−	0.790382i
$669$		0			0
$670$		0			0
$671$	−3.37957	−	5.85358i	−0.130467	−	0.225975i
$672$		0			0
$673$	16.7644	−	29.0368i	0.646221	−	1.11929i	−0.337797	−	0.941219i	$-0.609682\pi$
$673$	16.7644	−	29.0368i	0.646221	−	1.11929i	0.984018	−	0.178068i	$-0.0569848\pi$
$674$	15.5473			0.598860
$675$		0			0
$676$	−14.7799			−0.568456
$677$	−13.7437	+	23.8048i	−0.528213	+	0.914891i	0.471246	+	0.882002i	$0.343804\pi$
$677$	−13.7437	+	23.8048i	−0.528213	+	0.914891i	−0.999459	+	0.0328897i	$0.989529\pi$
$678$		0			0
$679$	−5.59442	−	9.68981i	−0.214694	−	0.371861i
$680$		0			0
$681$		0			0
$682$	5.32096	−	9.21618i	0.203750	−	0.352906i
$683$	−34.5865			−1.32342			−0.661708	−	0.749762i	$-0.730168\pi$
$683$	−34.5865			−1.32342			−0.661708	+	0.749762i	$0.730168\pi$
$684$		0			0
$685$		0			0
$686$	−4.88485	+	8.46081i	−0.186504	+	0.323035i
$687$		0			0
$688$	0.505083	+	0.874830i	0.0192561	+	0.0333526i
$689$	−2.67282	−	4.62947i	−0.101826	−	0.176369i
$690$		0			0
$691$	−20.3641	+	35.2717i	−0.774688	+	1.34180i	0.160282	+	0.987071i	$0.448760\pi$
$691$	−20.3641	+	35.2717i	−0.774688	+	1.34180i	−0.934970	+	0.354727i	$0.884574\pi$
$692$	−7.30418			−0.277663
$693$		0			0
$694$	−13.4795			−0.511674
$695$		0			0
$696$		0			0
$697$	−1.96608	−	3.40535i	−0.0744706	−	0.128987i
$698$	3.06360	+	5.30632i	0.115959	+	0.200847i
$699$		0			0
$700$		0			0
$701$	19.4712			0.735416			0.367708	−	0.929941i	$-0.380143\pi$
$701$	19.4712			0.735416			0.367708	+	0.929941i	$0.380143\pi$
$702$		0			0
$703$	8.48963			0.320193
$704$	−1.58123	+	2.73877i	−0.0595948	+	0.103221i
$705$		0			0
$706$	−7.79834	−	13.5071i	−0.293494	−	0.508347i
$707$	3.00000	+	5.19615i	0.112827	+	0.195421i
$708$		0			0
$709$	−7.54316	+	13.0651i	−0.283289	+	0.490671i	−0.972193	−	0.234182i	$-0.924759\pi$
$709$	−7.54316	+	13.0651i	−0.283289	+	0.490671i	0.688904	+	0.724853i	$0.258092\pi$
$710$		0			0
$711$		0			0
$712$	−6.30249			−0.236196
$713$	20.5957	−	35.6729i	0.771317	−	1.33596i
$714$		0			0
$715$		0			0
$716$	12.6821	+	21.9660i	0.473951	+	0.820907i
$717$		0			0
$718$	−3.05465	+	5.29081i	−0.113999	+	0.197451i
$719$	3.43196			0.127990			0.0639952	−	0.997950i	$-0.479616\pi$
$719$	3.43196			0.127990			0.0639952	+	0.997950i	$0.479616\pi$
$720$		0			0
$721$	−2.59442			−0.0966211
$722$	−0.810881	+	1.40449i	−0.0301779	+	0.0522696i
$723$		0			0
$724$	2.67791	+	4.63827i	0.0995236	+	0.172380i
$725$		0			0
$726$		0			0
$727$	−17.8857	+	30.9789i	−0.663344	+	1.14895i	0.316388	+	0.948630i	$0.397530\pi$
$727$	−17.8857	+	30.9789i	−0.663344	+	1.14895i	−0.979732	+	0.200315i	$0.935803\pi$
$728$	−14.0185			−0.519559
$729$		0			0
$730$		0			0
$731$	0.629668	−	1.09062i	0.0232891	−	0.0403379i
$732$		0			0
$733$	−11.0000	−	19.0526i	−0.406294	−	0.703722i	0.588177	−	0.808732i	$-0.299846\pi$
$733$	−11.0000	−	19.0526i	−0.406294	−	0.703722i	−0.994471	+	0.105010i	$0.966513\pi$
$734$	−2.42272	−	4.19628i	−0.0894244	−	0.154888i
$735$		0			0
$736$	16.0605	−	27.8176i	0.591998	−	1.02537i
$737$	−17.6151			−0.648862
$738$		0			0
$739$	6.08631			0.223889			0.111944	−	0.993714i	$-0.464292\pi$
$739$	6.08631			0.223889			0.111944	+	0.993714i	$0.464292\pi$
$740$		0			0
$741$		0			0
$742$	−0.467210	−	0.809231i	−0.0171518	−	0.0297078i
$743$	12.7509	+	22.0853i	0.467787	+	0.810231i	0.999322	−	0.0368054i	$-0.0117182\pi$
$743$	12.7509	+	22.0853i	0.467787	+	0.810231i	−0.531536	+	0.847036i	$0.678385\pi$
$744$		0			0
$745$		0			0
$746$	−5.79043			−0.212003
$747$		0			0
$748$	11.9506			0.436958
$749$	8.50924	−	14.7384i	0.310921	−	0.538530i
$750$		0			0
$751$	9.19638	+	15.9286i	0.335581	+	0.581243i	0.983596	−	0.180384i	$-0.0577342\pi$
$751$	9.19638	+	15.9286i	0.335581	+	0.581243i	−0.648016	+	0.761627i	$0.724401\pi$
$752$	−7.45769	−	12.9171i	−0.271954	−	0.471038i
$753$		0			0
$754$	12.6821	−	21.9660i	0.461853	−	0.799953i
$755$		0			0
$756$		0			0
$757$	41.8986			1.52283			0.761415	−	0.648264i	$-0.224505\pi$
$757$	41.8986			1.52283			0.761415	+	0.648264i	$0.224505\pi$
$758$	−3.40954	+	5.90549i	−0.123840	+	0.214497i
$759$		0			0
$760$		0			0
$761$	−3.98568	−	6.90340i	−0.144481	−	0.250248i	0.784698	−	0.619878i	$-0.212818\pi$
$761$	−3.98568	−	6.90340i	−0.144481	−	0.250248i	−0.929179	+	0.369630i	$0.879485\pi$
$762$		0			0
$763$	−11.8896	+	20.5935i	−0.430434	+	0.745534i
$764$	−4.74671			−0.171730
$765$		0			0
$766$	5.61515			0.202884
$767$	2.67282	−	4.62947i	0.0965101	−	0.167160i
$768$		0			0
$769$	−3.01432	−	5.22095i	−0.108699	−	0.188272i	0.806544	−	0.591174i	$-0.201335\pi$
$769$	−3.01432	−	5.22095i	−0.108699	−	0.188272i	−0.915244	+	0.402901i	$0.868002\pi$
$770$		0			0
$771$		0			0
$772$	15.7147	−	27.2186i	0.565583	−	0.979618i
$773$	44.4033			1.59708			0.798538	−	0.601944i	$-0.205607\pi$
$773$	44.4033			1.59708			0.798538	+	0.601944i	$0.205607\pi$
$774$		0			0
$775$		0			0
$776$	−8.23030	+	14.2553i	−0.295451	+	0.511735i
$777$		0			0
$778$	2.63693	+	4.56729i	0.0945384	+	0.163745i
$779$	−3.43724	−	5.95348i	−0.123152	−	0.213305i
$780$		0			0
$781$	17.2056	−	29.8010i	0.615665	−	1.06636i
$782$	−9.04711			−0.323524
$783$		0			0
$784$	−10.6359			−0.379853
$785$		0			0
$786$		0			0
$787$	8.41877	+	14.5817i	0.300097	+	0.519783i	0.976158	−	0.217063i	$-0.0696477\pi$
$787$	8.41877	+	14.5817i	0.300097	+	0.519783i	−0.676061	+	0.736846i	$0.736314\pi$
$788$	4.88070	+	8.45362i	0.173868	+	0.301148i
$789$		0			0
$790$		0			0
$791$	−28.7361			−1.02174
$792$		0			0
$793$	11.8168			0.419627
$794$	−6.57199	+	11.3830i	−0.233231	+	0.403968i

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:	\( N \)	\(=\)	\( 675 = 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	675.e (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.285997 + 0.495361i) q^{2} +(0.836412 + 1.44871i) q^{4} +(0.714003 - 1.23669i) q^{7} -2.10083 q^{8} +O(q^{10})\)	\(q+(-0.285997 + 0.495361i) q^{2} +(0.836412 + 1.44871i) q^{4} +(0.714003 - 1.23669i) q^{7} -2.10083 q^{8} +(1.33641 - 2.31473i) q^{11} +(2.33641 + 4.04678i) q^{13} +(0.408405 + 0.707378i) q^{14} +(-1.07199 + 1.85675i) q^{16} +2.67282 q^{17} +4.67282 q^{19} +(0.764419 + 1.32401i) q^{22} +(2.95882 + 5.12483i) q^{23} -2.67282 q^{26} +2.38880 q^{28} +(-4.74482 + 8.21826i) q^{29} +(-3.48040 - 6.02823i) q^{31} +(-2.71400 - 4.70079i) q^{32} +(-0.764419 + 1.32401i) q^{34} +1.81681 q^{37} +(-1.33641 + 2.31473i) q^{38} +(-0.735581 - 1.27406i) q^{41} +(0.235581 - 0.408039i) q^{43} +4.47116 q^{44} -3.38485 q^{46} +(-3.47842 + 6.02480i) q^{47} +(2.48040 + 4.29618i) q^{49} +(-3.90841 + 6.76956i) q^{52} -1.14399 q^{53} +(-1.50000 + 2.59808i) q^{56} +(-2.71400 - 4.70079i) q^{58} +(-0.571993 - 0.990721i) q^{59} +(1.26442 - 2.19004i) q^{61} +3.98153 q^{62} -1.18319 q^{64} +(-3.29523 - 5.70751i) q^{67} +(2.23558 + 3.87214i) q^{68} +12.8745 q^{71} +1.71203 q^{73} +(-0.519602 + 0.899976i) q^{74} +(3.90841 + 6.76956i) q^{76} +(-1.90841 - 3.30545i) q^{77} +(0.143987 - 0.249392i) q^{79} +0.841495 q^{82} +(2.14201 - 3.71007i) q^{83} +(0.134751 + 0.233396i) q^{86} +(-2.80757 + 4.86286i) q^{88} +3.00000 q^{89} +6.67282 q^{91} +(-4.94958 + 8.57293i) q^{92} +(-1.98963 - 3.44615i) q^{94} +(3.91764 - 6.78555i) q^{97} -2.83754 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - q^{2} - 5 q^{4} + 5 q^{7} + 6 q^{8}+O(q^{10}) \) 6 * q - q^2 - 5 * q^4 + 5 * q^7 + 6 * q^8	\( 6 q - q^{2} - 5 q^{4} + 5 q^{7} + 6 q^{8} - 2 q^{11} + 4 q^{13} - 9 q^{14} - 5 q^{16} - 4 q^{17} + 8 q^{19} - 4 q^{22} - 3 q^{23} + 4 q^{26} - 10 q^{28} - 7 q^{29} - 8 q^{31} - 17 q^{32} + 4 q^{34} - 12 q^{37} + 2 q^{38} - 13 q^{41} + 10 q^{43} + 44 q^{44} - 6 q^{46} - 13 q^{47} + 2 q^{49} - 12 q^{52} - 4 q^{53} - 9 q^{56} - 17 q^{58} - 2 q^{59} - q^{61} + 84 q^{62} - 30 q^{64} + 11 q^{67} + 22 q^{68} + 20 q^{71} + 16 q^{73} - 16 q^{74} + 12 q^{76} - 2 q^{79} + 58 q^{82} + 15 q^{83} + 28 q^{86} - 24 q^{88} + 18 q^{89} + 20 q^{91} - 39 q^{92} + 31 q^{94} - 18 q^{97} - 80 q^{98}+O(q^{100}) \) 6 * q - q^2 - 5 * q^4 + 5 * q^7 + 6 * q^8 - 2 * q^11 + 4 * q^13 - 9 * q^14 - 5 * q^16 - 4 * q^17 + 8 * q^19 - 4 * q^22 - 3 * q^23 + 4 * q^26 - 10 * q^28 - 7 * q^29 - 8 * q^31 - 17 * q^32 + 4 * q^34 - 12 * q^37 + 2 * q^38 - 13 * q^41 + 10 * q^43 + 44 * q^44 - 6 * q^46 - 13 * q^47 + 2 * q^49 - 12 * q^52 - 4 * q^53 - 9 * q^56 - 17 * q^58 - 2 * q^59 - q^61 + 84 * q^62 - 30 * q^64 + 11 * q^67 + 22 * q^68 + 20 * q^71 + 16 * q^73 - 16 * q^74 + 12 * q^76 - 2 * q^79 + 58 * q^82 + 15 * q^83 + 28 * q^86 - 24 * q^88 + 18 * q^89 + 20 * q^91 - 39 * q^92 + 31 * q^94 - 18 * q^97 - 80 * q^98

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