LMFDB - Embedded newform 1620.3.o.f.701.4

Newspace parameters

Level:	$ N $	$=$	$ 1620 = 2^{2} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1620.o (of order $6$, degree $2$, not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$44.1418028264$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	8.0.3317760000.8
Defining polynomial:	$ x^{8} + 4x^{6} + 7x^{4} + 36x^{2} + 81 $ x^8 + 4x^6 + 7x^4 + 36*x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 180)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			701.4
Root			$0.178197 - 1.72286i$ of defining polynomial
Character	$\chi$	$=$	1620.701
Dual form			1620.3.o.f.1241.4

$q$-expansion

$f(q)$	$=$	$q+(1.93649 + 1.11803i) q^{5} +(6.74342 + 11.6799i) q^{7} +O(q^{10})$	$q+(1.93649 + 1.11803i) q^{5} +(6.74342 + 11.6799i) q^{7} +(-15.2932 + 8.82952i) q^{11} +(3.74342 - 6.48379i) q^{13} +16.9706i q^{17} -10.9737 q^{19} +(18.9674 + 10.9508i) q^{23} +(2.50000 + 4.33013i) q^{25} +(-41.0128 + 23.6788i) q^{29} +(8.48683 - 14.6996i) q^{31} +30.1575i q^{35} -5.53950 q^{37} +(-57.4985 - 33.1968i) q^{41} +(-19.4868 - 33.7522i) q^{43} +(28.2014 - 16.2821i) q^{47} +(-66.4473 + 115.090i) q^{49} +11.2392i q^{53} -39.4868 q^{55} +(27.6051 + 15.9378i) q^{59} +(-23.4605 - 40.6348i) q^{61} +(14.4982 - 8.37053i) q^{65} +(38.0000 - 65.8179i) q^{67} -77.7445i q^{71} +94.9210 q^{73} +(-206.257 - 119.082i) q^{77} +(3.46050 + 5.99376i) q^{79} +(-53.8243 + 31.0755i) q^{83} +(-18.9737 + 32.8634i) q^{85} -62.2626i q^{89} +100.974 q^{91} +(-21.2504 - 12.2689i) q^{95} +(62.4868 + 108.230i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 16 q^{7}+O(q^{10}) $ 8 * q + 16 * q^7	$ 8 q + 16 q^{7} - 8 q^{13} + 64 q^{19} + 20 q^{25} - 8 q^{31} - 272 q^{37} - 80 q^{43} - 228 q^{49} - 240 q^{55} + 40 q^{61} + 304 q^{67} + 304 q^{73} - 200 q^{79} + 656 q^{91} + 424 q^{97}+O(q^{100}) $ 8 * q + 16 * q^7 - 8 * q^13 + 64 * q^19 + 20 * q^25 - 8 * q^31 - 272 * q^37 - 80 * q^43 - 228 * q^49 - 240 * q^55 + 40 * q^61 + 304 * q^67 + 304 * q^73 - 200 * q^79 + 656 * q^91 + 424 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times$.

$n$	$811$	$1297$	$1541$
$\chi(n)$	$1$	$1$	$e\left(\frac{5}{6}\right)$

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 $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	1.93649	+	1.11803i	0.387298	+	0.223607i
$5$	1.93649	+	1.11803i	0.387298	+	0.223607i
$6$		0			0
$7$	6.74342	+	11.6799i	0.963345	+	1.66856i	0.713997	+	0.700149i	$0.246883\pi$
$7$	6.74342	+	11.6799i	0.963345	+	1.66856i	0.249348	+	0.968414i	$0.419784\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−15.2932	+	8.82952i	−1.39029	+	0.802684i	−0.993347	−	0.115159i	$-0.963262\pi$
$11$	−15.2932	+	8.82952i	−1.39029	+	0.802684i	−0.396942	+	0.917843i	$0.629929\pi$
$12$		0			0
$13$	3.74342	−	6.48379i	0.287955	−	0.498753i	−0.685366	−	0.728198i	$-0.740358\pi$
$13$	3.74342	−	6.48379i	0.287955	−	0.498753i	0.973322	+	0.229446i	$0.0736913\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			16.9706i			0.998268i	0.866525	+	0.499134i	$0.166349\pi$
$17$			16.9706i			0.998268i	−0.866525	+	0.499134i	$0.833651\pi$
$18$		0			0
$19$	−10.9737			−0.577561			−0.288781	−	0.957395i	$-0.593250\pi$
$19$	−10.9737			−0.577561			−0.288781	+	0.957395i	$0.593250\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	18.9674	+	10.9508i	0.824670	+	0.476124i	0.852024	−	0.523502i	$-0.175375\pi$
$23$	18.9674	+	10.9508i	0.824670	+	0.476124i	−0.0273540	+	0.999626i	$0.508708\pi$
$24$		0			0
$25$	2.50000	+	4.33013i	0.100000	+	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−41.0128	+	23.6788i	−1.41424	+	0.816509i	−0.995784	−	0.0917300i	$-0.970760\pi$
$29$	−41.0128	+	23.6788i	−1.41424	+	0.816509i	−0.418451	+	0.908239i	$0.637427\pi$
$30$		0			0
$31$	8.48683	−	14.6996i	0.273769	−	0.474181i	−0.696055	−	0.717989i	$-0.745063\pi$
$31$	8.48683	−	14.6996i	0.273769	−	0.474181i	0.969824	+	0.243807i	$0.0783964\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$			30.1575i			0.861642i
$36$		0			0
$37$	−5.53950			−0.149716			−0.0748581	−	0.997194i	$-0.523850\pi$
$37$	−5.53950			−0.149716			−0.0748581	+	0.997194i	$0.523850\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−57.4985	−	33.1968i	−1.40240	−	0.809677i	−0.407764	−	0.913087i	$-0.633691\pi$
$41$	−57.4985	−	33.1968i	−1.40240	−	0.809677i	−0.994639	+	0.103410i	$0.967025\pi$
$42$		0			0
$43$	−19.4868	−	33.7522i	−0.453182	−	0.784935i	0.545399	−	0.838176i	$-0.316378\pi$
$43$	−19.4868	−	33.7522i	−0.453182	−	0.784935i	−0.998582	+	0.0532417i	$0.983045\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	28.2014	−	16.2821i	0.600029	−	0.346427i	−0.169024	−	0.985612i	$-0.554061\pi$
$47$	28.2014	−	16.2821i	0.600029	−	0.346427i	0.769053	+	0.639185i	$0.220728\pi$
$48$		0			0
$49$	−66.4473	+	115.090i	−1.35607	+	2.34878i
$50$		0			0
$51$		0			0
$52$		0			0
$53$			11.2392i			0.212061i	0.994363	+	0.106030i	$0.0338141\pi$
$53$			11.2392i			0.212061i	−0.994363	+	0.106030i	$0.966186\pi$
$54$		0			0
$55$	−39.4868			−0.717942
$56$		0			0
$57$		0			0
$58$		0			0
$59$	27.6051	+	15.9378i	0.467884	+	0.270133i	0.715353	−	0.698763i	$-0.246266\pi$
$59$	27.6051	+	15.9378i	0.467884	+	0.270133i	−0.247470	+	0.968896i	$0.579599\pi$
$60$		0			0
$61$	−23.4605	−	40.6348i	−0.384598	−	0.666144i	0.607115	−	0.794614i	$-0.292327\pi$
$61$	−23.4605	−	40.6348i	−0.384598	−	0.666144i	−0.991713	+	0.128470i	$0.958993\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	14.4982	−	8.37053i	0.223049	−	0.128777i
$66$		0			0
$67$	38.0000	−	65.8179i	0.567164	−	0.982357i	−0.429681	−	0.902981i	$-0.641374\pi$
$67$	38.0000	−	65.8179i	0.567164	−	0.982357i	0.996845	−	0.0793762i	$-0.0252928\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	77.7445i		−	1.09499i	−0.836808	−	0.547497i	$-0.815581\pi$
$71$		−	77.7445i		−	1.09499i	0.836808	−	0.547497i	$-0.184419\pi$
$72$		0			0
$73$	94.9210			1.30029			0.650144	−	0.759811i	$-0.274709\pi$
$73$	94.9210			1.30029			0.650144	+	0.759811i	$0.274709\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−206.257	−	119.082i	−2.67866	−	1.54652i
$78$		0			0
$79$	3.46050	+	5.99376i	0.0438038	+	0.0758704i	0.887096	−	0.461585i	$-0.152719\pi$
$79$	3.46050	+	5.99376i	0.0438038	+	0.0758704i	−0.843292	+	0.537455i	$0.819386\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−53.8243	+	31.0755i	−0.648485	+	0.374403i	−0.787876	−	0.615834i	$-0.788819\pi$
$83$	−53.8243	+	31.0755i	−0.648485	+	0.374403i	0.139390	+	0.990237i	$0.455486\pi$
$84$		0			0
$85$	−18.9737	+	32.8634i	−0.223220	+	0.386628i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	62.2626i		−	0.699580i	−0.936828	−	0.349790i	$-0.886253\pi$
$89$		−	62.2626i		−	0.699580i	0.936828	−	0.349790i	$-0.113747\pi$
$90$		0			0
$91$	100.974			1.10960
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−21.2504	−	12.2689i	−0.223689	−	0.129147i
$96$		0			0
$97$	62.4868	+	108.230i	0.644194	+	1.11578i	0.984487	+	0.175457i	$0.0561405\pi$
$97$	62.4868	+	108.230i	0.644194	+	1.11578i	−0.340293	+	0.940320i	$0.610526\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−97.2221	+	56.1312i	−0.962595	+	0.555754i	−0.896971	−	0.442090i	$-0.854237\pi$
$101$	−97.2221	+	56.1312i	−0.962595	+	0.555754i	−0.0656242	+	0.997844i	$0.520904\pi$
$102$		0			0
$103$	85.1512	−	147.486i	0.826711	−	1.43191i	−0.0738934	−	0.997266i	$-0.523542\pi$
$103$	85.1512	−	147.486i	0.826711	−	1.43191i	0.900604	−	0.434640i	$-0.143124\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			93.3381i			0.872319i	0.899869	+	0.436159i	$0.143662\pi$
$107$			93.3381i			0.872319i	−0.899869	+	0.436159i	$0.856338\pi$
$108$		0			0
$109$	−68.8683			−0.631820			−0.315910	−	0.948789i	$-0.602310\pi$
$109$	−68.8683			−0.631820			−0.315910	+	0.948789i	$0.602310\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	95.5301	+	55.1543i	0.845399	+	0.488091i	0.859096	−	0.511815i	$-0.171027\pi$
$113$	95.5301	+	55.1543i	0.845399	+	0.488091i	−0.0136967	+	0.999906i	$0.504360\pi$
$114$		0			0
$115$	24.4868	+	42.4124i	0.212929	+	0.368804i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−198.215	+	114.440i	−1.66567	+	0.961677i
$120$		0			0
$121$	95.4210	−	165.274i	0.788603	−	1.36590i
$122$		0			0
$123$		0			0
$124$		0			0
$125$			11.1803i			0.0894427i
$126$		0			0
$127$	−65.3815			−0.514815			−0.257407	−	0.966303i	$-0.582868\pi$
$127$	−65.3815			−0.514815			−0.257407	+	0.966303i	$0.582868\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−40.9161	−	23.6229i	−0.312336	−	0.180328i	0.335635	−	0.941992i	$-0.391049\pi$
$131$	−40.9161	−	23.6229i	−0.312336	−	0.180328i	−0.647972	+	0.761665i	$0.724382\pi$
$132$		0			0
$133$	−74.0000	−	128.172i	−0.556391	−	0.963697i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−187.096	+	108.020i	−1.36566	+	0.788465i	−0.990371	−	0.138442i	$-0.955791\pi$
$137$	−187.096	+	108.020i	−1.36566	+	0.788465i	−0.375291	+	0.926907i	$0.622457\pi$
$138$		0			0
$139$	−105.921	+	183.461i	−0.762022	+	1.31986i	0.179785	+	0.983706i	$0.442460\pi$
$139$	−105.921	+	183.461i	−0.762022	+	1.31986i	−0.941807	+	0.336154i	$0.890874\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			132.210i			0.924548i
$144$		0			0
$145$	−105.895			−0.730308
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−80.1402	−	46.2689i	−0.537853	−	0.310530i	0.206355	−	0.978477i	$-0.433840\pi$
$149$	−80.1402	−	46.2689i	−0.537853	−	0.310530i	−0.744209	+	0.667947i	$0.767173\pi$
$150$		0			0
$151$	61.9210	+	107.250i	0.410073	+	0.710267i	0.994897	−	0.100893i	$-0.0321699\pi$
$151$	61.9210	+	107.250i	0.410073	+	0.710267i	−0.584824	+	0.811160i	$0.698837\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	32.8694	−	18.9771i	0.212060	−	0.122433i
$156$		0			0
$157$	15.7434	−	27.2684i	0.100277	−	0.173684i	−0.811522	−	0.584322i	$-0.801361\pi$
$157$	15.7434	−	27.2684i	0.100277	−	0.173684i	0.911799	+	0.410638i	$0.134694\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			295.384i			1.83469i
$162$		0			0
$163$	−101.132			−0.620440			−0.310220	−	0.950665i	$-0.600403\pi$
$163$	−101.132			−0.620440			−0.310220	+	0.950665i	$0.600403\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	128.002	+	73.9020i	0.766479	+	0.442527i	0.831617	−	0.555349i	$-0.187416\pi$
$167$	128.002	+	73.9020i	0.766479	+	0.442527i	−0.0651382	+	0.997876i	$0.520749\pi$
$168$		0			0
$169$	56.4737	+	97.8153i	0.334164	+	0.578789i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−287.589	+	166.040i	−1.66237	+	0.959767i	−0.690785	+	0.723061i	$0.742735\pi$
$173$	−287.589	+	166.040i	−1.66237	+	0.959767i	−0.971581	+	0.236707i	$0.923932\pi$
$174$		0			0
$175$	−33.7171	+	58.3997i	−0.192669	+	0.333713i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			45.8688i			0.256250i	0.991758	+	0.128125i	$0.0408959\pi$
$179$			45.8688i			0.256250i	−0.991758	+	0.128125i	$0.959104\pi$
$180$		0			0
$181$	132.868			0.734079			0.367040	−	0.930205i	$-0.380371\pi$
$181$	132.868			0.734079			0.367040	+	0.930205i	$0.380371\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−10.7272	−	6.19335i	−0.0579849	−	0.0334776i
$186$		0			0
$187$	−149.842	−	259.534i	−0.801294	−	1.38788i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−142.006	+	81.9871i	−0.743486	+	0.429252i	−0.823336	−	0.567555i	$-0.807889\pi$
$191$	−142.006	+	81.9871i	−0.743486	+	0.429252i	0.0798492	+	0.996807i	$0.474556\pi$
$192$		0			0
$193$	−55.0000	+	95.2628i	−0.284974	+	0.493590i	−0.972603	−	0.232473i	$-0.925318\pi$
$193$	−55.0000	+	95.2628i	−0.284974	+	0.493590i	0.687629	+	0.726062i	$0.258652\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			211.108i			1.07162i	0.844340	+	0.535808i	$0.179993\pi$
$197$			211.108i			1.07162i	−0.844340	+	0.535808i	$0.820007\pi$
$198$		0			0
$199$	−18.1053			−0.0909816			−0.0454908	−	0.998965i	$-0.514485\pi$
$199$	−18.1053			−0.0909816			−0.0454908	+	0.998965i	$0.514485\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−553.133	−	319.352i	−2.72479	−	1.57316i
$204$		0			0
$205$	−74.2302	−	128.571i	−0.362099	−	0.627173i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	167.822	−	96.8923i	0.802978	−	0.463599i
$210$		0			0
$211$	−170.789	+	295.816i	−0.809428	+	1.40197i	0.103833	+	0.994595i	$0.466889\pi$
$211$	−170.789	+	295.816i	−0.809428	+	1.40197i	−0.913261	+	0.407376i	$0.866444\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	87.1478i		−	0.405338i
$216$		0			0
$217$	228.921			1.05494
$218$		0			0
$219$		0			0
$220$		0			0
$221$	110.034	+	63.5279i	0.497889	+	0.287456i
$222$		0			0
$223$	−29.6644	−	51.3803i	−0.133024	−	0.230405i	0.791817	−	0.610759i	$-0.209135\pi$
$223$	−29.6644	−	51.3803i	−0.133024	−	0.230405i	−0.924841	+	0.380354i	$0.875802\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	100.107	−	57.7965i	0.440998	−	0.254610i	−0.263023	−	0.964790i	$-0.584719\pi$
$227$	100.107	−	57.7965i	0.440998	−	0.254610i	0.704021	+	0.710179i	$0.251386\pi$
$228$		0			0
$229$	126.947	−	219.879i	0.554355	−	0.960171i	−0.443598	−	0.896226i	$-0.646298\pi$
$229$	126.947	−	219.879i	0.554355	−	0.960171i	0.997953	−	0.0639455i	$-0.0203684\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	206.401i		−	0.885840i	−0.896561	−	0.442920i	$-0.853943\pi$
$233$		−	206.401i		−	0.885840i	0.896561	−	0.442920i	$-0.146057\pi$
$234$		0			0
$235$	72.8157			0.309854
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−299.901	−	173.148i	−1.25482	−	0.724469i	−0.282755	−	0.959192i	$-0.591248\pi$
$239$	−299.901	−	173.148i	−1.25482	−	0.724469i	−0.972062	+	0.234723i	$0.924582\pi$
$240$		0			0
$241$	91.0263	+	157.662i	0.377703	+	0.654200i	0.990728	−	0.135863i	$-0.0433808\pi$
$241$	91.0263	+	157.662i	0.377703	+	0.654200i	−0.613025	+	0.790064i	$0.710047\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−257.349	+	148.581i	−1.05041	+	0.606452i
$246$		0			0
$247$	−41.0790	+	71.1509i	−0.166312	+	0.288060i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	140.807i		−	0.560985i	−0.959856	−	0.280493i	$-0.909502\pi$
$251$		−	140.807i		−	0.560985i	0.959856	−	0.280493i	$-0.0904979\pi$
$252$		0			0
$253$	−386.763			−1.52871
$254$		0			0
$255$		0			0
$256$		0			0
$257$	253.425	+	146.315i	0.986091	+	0.569320i	0.904104	−	0.427313i	$-0.140540\pi$
$257$	253.425	+	146.315i	0.986091	+	0.569320i	0.0819876	+	0.996633i	$0.473873\pi$
$258$		0			0
$259$	−37.3552	−	64.7010i	−0.144228	−	0.249811i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	234.652	−	135.476i	0.892211	−	0.515118i	0.0175460	−	0.999846i	$-0.494415\pi$
$263$	234.652	−	135.476i	0.892211	−	0.515118i	0.874665	+	0.484728i	$0.161081\pi$
$264$		0			0
$265$	−12.5658	+	21.7647i	−0.0474182	+	0.0821308i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			275.083i			1.02261i	0.859398	+	0.511307i	$0.170838\pi$
$269$			275.083i			1.02261i	−0.859398	+	0.511307i	$0.829162\pi$
$270$		0			0
$271$	248.158			0.915712			0.457856	−	0.889026i	$-0.348617\pi$
$271$	248.158			0.915712			0.457856	+	0.889026i	$0.348617\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−76.4659	−	44.1476i	−0.278058	−	0.160537i
$276$		0			0
$277$	171.020	+	296.215i	0.617399	+	1.06937i	0.989958	+	0.141358i	$0.0451470\pi$
$277$	171.020	+	296.215i	0.617399	+	1.06937i	−0.372559	+	0.928008i	$0.621520\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	304.961	−	176.070i	1.08527	−	0.626582i	0.152958	−	0.988233i	$-0.451120\pi$
$281$	304.961	−	176.070i	1.08527	−	0.626582i	0.932314	+	0.361650i	$0.117787\pi$
$282$		0			0
$283$	−124.816	+	216.187i	−0.441045	+	0.763912i	−0.997767	−	0.0667866i	$-0.978725\pi$
$283$	−124.816	+	216.187i	−0.441045	+	0.763912i	0.556723	+	0.830699i	$0.312059\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	895.439i		−	3.12000i
$288$		0			0
$289$	1.00000			0.00346021
$290$		0			0
$291$		0			0
$292$		0			0
$293$	32.4719	+	18.7476i	0.110825	+	0.0639851i	0.554388	−	0.832258i	$-0.312952\pi$
$293$	32.4719	+	18.7476i	0.110825	+	0.0639851i	−0.443563	+	0.896243i	$0.646286\pi$
$294$		0			0
$295$	35.6381	+	61.7270i	0.120807	+	0.209244i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	142.006	−	81.9871i	0.474936	−	0.274204i
$300$		0			0
$301$	262.816	−	455.210i	0.873142	−	1.51233i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	104.919i		−	0.343995i
$306$		0			0
$307$	457.842			1.49134			0.745671	−	0.666314i	$-0.232129\pi$
$307$	457.842			1.49134			0.745671	+	0.666314i	$0.232129\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	7.34847	+	4.24264i	0.0236285	+	0.0136419i	0.511768	−	0.859124i	$-0.328991\pi$
$311$	7.34847	+	4.24264i	0.0236285	+	0.0136419i	−0.488139	+	0.872766i	$0.662324\pi$
$312$		0			0
$313$	0.789328	+	1.36716i	0.00252181	+	0.00436791i	0.867284	−	0.497814i	$-0.165864\pi$
$313$	0.789328	+	1.36716i	0.00252181	+	0.00436791i	−0.864762	+	0.502182i	$0.832531\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−458.296	+	264.597i	−1.44573	+	0.834692i	−0.998223	−	0.0595890i	$-0.981021\pi$
$317$	−458.296	+	264.597i	−1.44573	+	0.834692i	−0.447506	+	0.894281i	$0.647688\pi$
$318$		0			0
$319$	418.144	−	724.248i	1.31080	−	2.27037i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	186.229i		−	0.576561i
$324$		0			0
$325$	37.4342			0.115182
$326$		0			0
$327$		0			0
$328$		0			0
$329$	380.347	+	219.594i	1.15607	+	0.667458i
$330$		0			0
$331$	195.381	+	338.411i	0.590276	+	1.02239i	0.994195	+	0.107594i	$0.0343146\pi$
$331$	195.381	+	338.411i	0.590276	+	1.02239i	−0.403919	+	0.914795i	$0.632352\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	147.173	−	84.9706i	0.439323	−	0.253644i
$336$		0			0
$337$	−207.355	+	359.150i	−0.615297	+	1.06573i	0.375035	+	0.927011i	$0.377631\pi$
$337$	−207.355	+	359.150i	−0.615297	+	1.06573i	−0.990332	+	0.138715i	$0.955703\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			299.739i			0.878999i
$342$		0			0
$343$	−1131.47			−3.29876
$344$		0			0
$345$		0			0
$346$		0			0
$347$	111.420	+	64.3281i	0.321094	+	0.185384i	0.651880	−	0.758322i	$-0.273981\pi$
$347$	111.420	+	64.3281i	0.321094	+	0.185384i	−0.330786	+	0.943706i	$0.607314\pi$
$348$		0			0
$349$	18.5395	+	32.1114i	0.0531218	+	0.0920096i	0.891364	−	0.453289i	$-0.149750\pi$
$349$	18.5395	+	32.1114i	0.0531218	+	0.0920096i	−0.838242	+	0.545299i	$0.816416\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	165.050	−	95.2918i	0.467565	−	0.269949i	−0.247655	−	0.968848i	$-0.579660\pi$
$353$	165.050	−	95.2918i	0.467565	−	0.269949i	0.715220	+	0.698900i	$0.246327\pi$
$354$		0			0
$355$	86.9210	−	150.552i	0.244848	−	0.424089i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			254.782i			0.709699i	0.934923	+	0.354849i	$0.115468\pi$
$359$			254.782i			0.709699i	−0.934923	+	0.354849i	$0.884532\pi$
$360$		0			0
$361$	−240.579			−0.666423
$362$		0			0
$363$		0			0
$364$		0			0
$365$	183.814	+	106.125i	0.503599	+	0.290753i
$366$		0			0
$367$	−150.230	−	260.206i	−0.409347	−	0.709009i	0.585470	−	0.810694i	$-0.300910\pi$
$367$	−150.230	−	260.206i	−0.409347	−	0.709009i	−0.994817	+	0.101685i	$0.967577\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−131.273	+	75.7908i	−0.353837	+	0.204288i
$372$		0			0
$373$	−352.046	+	609.761i	−0.943823	+	1.63475i	−0.185732	+	0.982601i	$0.559466\pi$
$373$	−352.046	+	609.761i	−0.943823	+	1.63475i	−0.758091	+	0.652149i	$0.773868\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			354.558i			0.940472i
$378$		0			0
$379$	333.789			0.880711			0.440355	−	0.897824i	$-0.354852\pi$
$379$	333.789			0.880711			0.440355	+	0.897824i	$0.354852\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	226.030	+	130.498i	0.590155	+	0.340726i	0.765159	−	0.643841i	$-0.222660\pi$
$383$	226.030	+	130.498i	0.590155	+	0.340726i	−0.175004	+	0.984568i	$0.555994\pi$
$384$		0			0
$385$	−266.276	−	461.204i	−0.691626	−	1.19793i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	31.0859	−	17.9474i	0.0799122	−	0.0461373i	−0.459511	−	0.888172i	$-0.651975\pi$
$389$	31.0859	−	17.9474i	0.0799122	−	0.0461373i	0.539424	+	0.842035i	$0.318642\pi$
$390$		0			0
$391$	−185.842	+	321.888i	−0.475299	+	0.823242i
$392$		0			0
$393$		0			0
$394$		0			0
$395$			15.4758i			0.0391793i
$396$		0			0
$397$	−90.3552			−0.227595			−0.113797	−	0.993504i	$-0.536301\pi$
$397$	−90.3552			−0.227595			−0.113797	+	0.993504i	$0.536301\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	37.8381	+	21.8458i	0.0943593	+	0.0544784i	0.546437	−	0.837500i	$-0.315984\pi$
$401$	37.8381	+	21.8458i	0.0943593	+	0.0544784i	−0.452078	+	0.891978i	$0.649317\pi$
$402$		0			0
$403$	−63.5395	−	110.054i	−0.157666	−	0.273086i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	84.7166	−	48.9112i	0.208149	−	0.120175i
$408$		0			0
$409$	130.026	−	225.212i	0.317913	−	0.550641i	−0.662140	−	0.749381i	$-0.730351\pi$
$409$	130.026	−	225.212i	0.317913	−	0.550641i	0.980052	+	0.198739i	$0.0636847\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			429.902i			1.04092i
$414$		0			0
$415$	−138.974			−0.334876
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−228.012	−	131.643i	−0.544181	−	0.314183i	0.202591	−	0.979263i	$-0.435064\pi$
$419$	−228.012	−	131.643i	−0.544181	−	0.314183i	−0.746772	+	0.665081i	$0.768397\pi$
$420$		0			0
$421$	−183.605	−	318.013i	−0.436116	−	0.755376i	0.561270	−	0.827633i	$-0.310313\pi$
$421$	−183.605	−	318.013i	−0.436116	−	0.755376i	−0.997386	+	0.0722573i	$0.976980\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−73.4847	+	42.4264i	−0.172905	+	0.0998268i
$426$		0			0
$427$	316.408	−	548.034i	0.741002	−	1.28345i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	690.768i		−	1.60271i	−0.598189	−	0.801355i	$-0.704113\pi$
$431$		−	690.768i		−	1.60271i	0.598189	−	0.801355i	$-0.295887\pi$
$432$		0			0
$433$	−117.500			−0.271363			−0.135682	−	0.990752i	$-0.543322\pi$
$433$	−117.500			−0.271363			−0.135682	+	0.990752i	$0.543322\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−208.142	−	120.171i	−0.476298	−	0.274991i
$438$		0			0
$439$	−25.0000	−	43.3013i	−0.0569476	−	0.0986362i	0.836146	−	0.548507i	$-0.184803\pi$
$439$	−25.0000	−	43.3013i	−0.0569476	−	0.0986362i	−0.893094	+	0.449871i	$0.851470\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	456.217	−	263.397i	1.02984	−	0.594576i	0.112898	−	0.993607i	$-0.463987\pi$
$443$	456.217	−	263.397i	1.02984	−	0.594576i	0.916937	+	0.399031i	$0.130653\pi$
$444$		0			0
$445$	69.6117	−	120.571i	0.156431	−	0.270946i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			321.882i			0.716887i	0.933552	+	0.358443i	$0.116692\pi$
$449$			321.882i			0.716887i	−0.933552	+	0.358443i	$0.883308\pi$
$450$		0			0
$451$	1172.45			2.59966
$452$		0			0
$453$		0			0
$454$		0			0
$455$	195.535	+	112.892i	0.429747	+	0.248114i
$456$		0			0
$457$	216.540	+	375.057i	0.473828	+	0.820695i	0.999551	−	0.0299614i	$-0.00953842\pi$
$457$	216.540	+	375.057i	0.473828	+	0.820695i	−0.525723	+	0.850656i	$0.676205\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	421.554	−	243.384i	0.914433	−	0.527948i	0.0325783	−	0.999469i	$-0.489628\pi$
$461$	421.554	−	243.384i	0.914433	−	0.527948i	0.881855	+	0.471521i	$0.156295\pi$
$462$		0			0
$463$	−211.612	+	366.522i	−0.457045	+	0.791625i	−0.998803	−	0.0489094i	$-0.984425\pi$
$463$	−211.612	+	366.522i	−0.457045	+	0.791625i	0.541758	+	0.840534i	$0.317759\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			295.831i			0.633472i	0.948514	+	0.316736i	$0.102587\pi$
$467$			295.831i			0.633472i	−0.948514	+	0.316736i	$0.897413\pi$
$468$		0			0
$469$	1025.00			2.18550
$470$		0			0
$471$		0			0
$472$		0			0
$473$	596.031	+	344.119i	1.26011	+	0.727524i
$474$		0			0
$475$	−27.4342	−	47.5174i	−0.0577561	−	0.100037i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	230.993	−	133.364i	0.482240	−	0.278421i	−0.239109	−	0.970993i	$-0.576855\pi$
$479$	230.993	−	133.364i	0.482240	−	0.278421i	0.721350	+	0.692571i	$0.243522\pi$
$480$		0			0
$481$	−20.7367	+	35.9169i	−0.0431116	+	0.0746714i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			279.450i			0.576185i
$486$		0			0
$487$	247.750			0.508727			0.254364	−	0.967109i	$-0.418134\pi$
$487$	247.750			0.508727			0.254364	+	0.967109i	$0.418134\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	147.566	+	85.1971i	0.300541	+	0.173517i	0.642686	−	0.766130i	$-0.277820\pi$
$491$	147.566	+	85.1971i	0.300541	+	0.173517i	−0.342145	+	0.939647i	$0.611153\pi$
$492$		0			0
$493$	−401.842	−	696.011i	−0.815095	−	1.41179i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	908.051	−	524.264i	1.82706	−	1.05486i
$498$		0			0
$499$	−23.6975	+	41.0453i	−0.0474900	+	0.0822551i	−0.888793	−	0.458308i	$-0.848456\pi$
$499$	−23.6975	+	41.0453i	−0.0474900	+	0.0822551i	0.841303	+	0.540563i	$0.181789\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			321.064i			0.638298i	0.947705	+	0.319149i	$0.103397\pi$
$503$			321.064i			0.638298i	−0.947705	+	0.319149i	$0.896603\pi$
$504$		0			0
$505$	−251.026			−0.497082
$506$		0			0
$507$		0			0
$508$		0			0
$509$	105.763	+	61.0623i	0.207786	+	0.119965i	0.600282	−	0.799788i	$-0.295055\pi$
$509$	105.763	+	61.0623i	0.207786	+	0.119965i	−0.392496	+	0.919754i	$0.628388\pi$
$510$		0			0
$511$	640.092	+	1108.67i	1.25263	+	2.16961i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	329.789	−	190.404i	0.640368	−	0.369716i
$516$		0			0
$517$	−287.526	+	498.010i	−0.556143	+	0.963268i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	79.4567i		−	0.152508i	−0.997088	−	0.0762540i	$-0.975704\pi$
$521$		−	79.4567i		−	0.152508i	0.997088	−	0.0762540i	$-0.0242960\pi$
$522$		0			0
$523$	−864.605			−1.65316			−0.826582	−	0.562816i	$-0.809718\pi$
$523$	−864.605			−1.65316			−0.826582	+	0.562816i	$0.809718\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	249.461	+	144.026i	0.473360	+	0.273295i
$528$		0			0
$529$	−24.6580	−	42.7089i	−0.0466125	−	0.0807352i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−430.482	+	248.539i	−0.807658	+	0.466302i
$534$		0			0
$535$	−104.355	+	180.748i	−0.195056	+	0.337848i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	2346.79i		−	4.35398i
$540$		0			0
$541$	33.6057			0.0621177			0.0310589	−	0.999518i	$-0.490112\pi$
$541$	33.6057			0.0621177			0.0310589	+	0.999518i	$0.490112\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−133.363	−	76.9971i	−0.244703	−	0.141279i
$546$		0			0
$547$	279.540	+	484.177i	0.511041	+	0.885149i	0.999918	+	0.0127965i	$0.00407336\pi$
$547$	279.540	+	484.177i	0.511041	+	0.885149i	−0.488877	+	0.872353i	$0.662593\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	450.061	−	259.843i	0.816808	−	0.471584i
$552$		0			0
$553$	−46.6712	+	80.8368i	−0.0843963	+	0.146179i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	310.308i		−	0.557105i	−0.960421	−	0.278553i	$-0.910145\pi$
$557$		−	310.308i		−	0.557105i	0.960421	−	0.278553i	$-0.0898547\pi$
$558$		0			0
$559$	−291.789			−0.521984
$560$		0			0
$561$		0			0
$562$		0			0
$563$	490.768	+	283.345i	0.871701	+	0.503277i	0.867913	−	0.496716i	$-0.165461\pi$
$563$	490.768	+	283.345i	0.871701	+	0.503277i	0.00378817	+	0.999993i	$0.498794\pi$
$564$		0			0
$565$	123.329	+	213.612i	0.218281	+	0.378074i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	116.093	−	67.0262i	0.204029	−	0.117796i	−0.394504	−	0.918894i	$-0.629084\pi$
$569$	116.093	−	67.0262i	0.204029	−	0.117796i	0.598534	+	0.801098i	$0.295750\pi$
$570$		0			0
$571$	330.960	−	573.240i	0.579615	−	1.00392i	−0.415908	−	0.909407i	$-0.636536\pi$
$571$	330.960	−	573.240i	0.579615	−	1.00392i	0.995523	−	0.0945161i	$-0.0301304\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			109.508i			0.190449i
$576$		0			0
$577$	76.7630			0.133038			0.0665191	−	0.997785i	$-0.478811\pi$
$577$	76.7630			0.133038			0.0665191	+	0.997785i	$0.478811\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−725.919	−	419.109i	−1.24943	−	0.721359i
$582$		0			0
$583$	−99.2370	−	171.884i	−0.170218	−	0.294826i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	592.841	−	342.277i	1.00995	−	0.583095i	0.0987734	−	0.995110i	$-0.468508\pi$
$587$	592.841	−	342.277i	1.00995	−	0.583095i	0.911177	+	0.412015i	$0.135175\pi$
$588$		0			0
$589$	−93.1317	+	161.309i	−0.158118	+	0.273869i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	592.146i		−	0.998560i	−0.866441	−	0.499280i	$-0.833598\pi$
$593$		−	592.146i		−	0.998560i	0.866441	−	0.499280i	$-0.166402\pi$
$594$		0			0
$595$	−511.789			−0.860150
$596$		0			0
$597$		0			0
$598$		0			0
$599$	877.078	+	506.381i	1.46424	+	0.845377i	0.999203	−	0.0399156i	$-0.0127089\pi$
$599$	877.078	+	506.381i	1.46424	+	0.845377i	0.465034	+	0.885293i	$0.346042\pi$
$600$		0			0
$601$	−161.789	−	280.227i	−0.269200	−	0.466268i	0.699455	−	0.714676i	$-0.253426\pi$
$601$	−161.789	−	280.227i	−0.269200	−	0.466268i	−0.968656	+	0.248408i	$0.920093\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	369.564	−	213.368i	0.610849	−	0.352674i
$606$		0			0
$607$	399.914	−	692.672i	0.658837	−	1.14114i	−0.322080	−	0.946713i	$-0.604382\pi$
$607$	399.914	−	692.672i	0.658837	−	1.14114i	0.980917	−	0.194427i	$-0.0622848\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	243.802i		−	0.399022i
$612$		0			0
$613$	−680.302			−1.10979			−0.554896	−	0.831920i	$-0.687242\pi$
$613$	−680.302			−1.10979			−0.554896	+	0.831920i	$0.687242\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−229.382	−	132.434i	−0.371770	−	0.214641i	0.302461	−	0.953162i	$-0.402192\pi$
$617$	−229.382	−	132.434i	−0.371770	−	0.214641i	−0.674231	+	0.738520i	$0.735525\pi$
$618$		0			0
$619$	−267.921	−	464.053i	−0.432829	−	0.749681i	0.564287	−	0.825579i	$-0.309151\pi$
$619$	−267.921	−	464.053i	−0.432829	−	0.749681i	−0.997116	+	0.0758974i	$0.975818\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	727.224	−	419.863i	1.16729	−	0.673937i
$624$		0			0
$625$	−12.5000	+	21.6506i	−0.0200000	+	0.0346410i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	94.0085i		−	0.149457i
$630$		0			0
$631$	307.026			0.486571			0.243286	−	0.969955i	$-0.421775\pi$
$631$	307.026			0.486571			0.243286	+	0.969955i	$0.421775\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−126.611	−	73.0987i	−0.199387	−	0.115116i
$636$		0			0
$637$	497.480	+	861.661i	0.780973	+	1.35269i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−1079.35	+	623.165i	−1.68386	+	0.972177i	−0.724806	+	0.688953i	$0.758071\pi$
$641$	−1079.35	+	623.165i	−1.68386	+	0.972177i	−0.959054	+	0.283224i	$0.908596\pi$
$642$		0			0
$643$	−77.6975	+	134.576i	−0.120836	+	0.209294i	−0.920098	−	0.391689i	$-0.871891\pi$
$643$	−77.6975	+	134.576i	−0.120836	+	0.209294i	0.799262	+	0.600983i	$0.205224\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	142.649i		−	0.220478i	−0.993905	−	0.110239i	$-0.964838\pi$
$647$		−	142.649i		−	0.220478i	0.993905	−	0.110239i	$-0.0351617\pi$
$648$		0			0
$649$	−562.894			−0.867325
$650$		0			0
$651$		0			0
$652$		0			0
$653$	800.709	+	462.289i	1.22620	+	0.707947i	0.966233	−	0.257671i	$-0.0829549\pi$
$653$	800.709	+	462.289i	1.22620	+	0.707947i	0.259967	+	0.965617i	$0.416288\pi$
$654$		0			0
$655$	−52.8224	−	91.4911i	−0.0806449	−	0.139681i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	304.880	−	176.023i	0.462641	−	0.267106i	−0.250513	−	0.968113i	$-0.580599\pi$
$659$	304.880	−	176.023i	0.462641	−	0.267106i	0.713154	+	0.701007i	$0.247266\pi$
$660$		0			0
$661$	321.460	−	556.786i	0.486325	−	0.842339i	−0.513552	−	0.858059i	$-0.671671\pi$
$661$	321.460	−	556.786i	0.486325	−	0.842339i	0.999876	+	0.0157198i	$0.00500396\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	330.938i		−	0.497651i
$666$		0			0
$667$	−1037.21			−1.55504
$668$		0			0
$669$		0			0
$670$		0			0
$671$	717.571	+	414.290i	1.06941	+	0.617422i
$672$		0			0
$673$	−491.302	−	850.961i	−0.730019	−	1.26443i	−0.956875	−	0.290501i	$-0.906178\pi$
$673$	−491.302	−	850.961i	−0.730019	−	1.26443i	0.226856	−	0.973928i	$-0.427155\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	334.952	−	193.384i	0.494759	−	0.285649i	−0.231788	−	0.972766i	$-0.574458\pi$
$677$	334.952	−	193.384i	0.494759	−	0.285649i	0.726546	+	0.687117i	$0.241124\pi$
$678$		0			0
$679$	−842.749	+	1459.68i	−1.24116	+	2.14976i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	504.539i		−	0.738710i	−0.929288	−	0.369355i	$-0.879579\pi$
$683$		−	504.539i		−	0.738710i	0.929288	−	0.369355i	$-0.120421\pi$
$684$		0			0
$685$	−483.079			−0.705225
$686$		0			0
$687$		0			0
$688$		0			0
$689$	72.8727	+	42.0731i	0.105766	+	0.0610640i
$690$		0			0
$691$	135.540	+	234.761i	0.196150	+	0.339741i	0.947277	−	0.320416i	$-0.103823\pi$
$691$	135.540	+	234.761i	0.196150	+	0.339741i	−0.751127	+	0.660158i	$0.770489\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−410.230	+	236.847i	−0.590259	+	0.340786i
$696$		0			0
$697$	563.368	−	975.782i	0.808275	−	1.39997i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	358.559i		−	0.511496i	−0.966743	−	0.255748i	$-0.917678\pi$
$701$		−	358.559i		−	0.511496i	0.966743	−	0.255748i	$-0.0823218\pi$
$702$		0			0
$703$	60.7886			0.0864703
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−1311.22	−	757.032i	−1.85462	−	1.07077i
$708$		0			0
$709$	366.789	+	635.298i	0.517333	+	0.896048i	0.999797	+	0.0201318i	$0.00640859\pi$
$709$	366.789	+	635.298i	0.517333	+	0.896048i	−0.482464	+	0.875916i	$0.660258\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	321.947	−	185.876i	0.451538	−	0.260696i
$714$		0			0
$715$	−147.816	+	256.024i	−0.206735	+	0.358076i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			1108.37i			1.54155i	0.637110	+	0.770773i	$0.280130\pi$
$719$			1108.37i			1.54155i	−0.637110	+	0.770773i	$0.719870\pi$
$720$		0			0
$721$	2296.84			3.18563
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−205.064	−	118.394i	−0.282847	−	0.163302i
$726$		0			0
$727$	−632.519	−	1095.56i	−0.870040	−	1.50695i	−0.861954	−	0.506987i	$-0.830759\pi$
$727$	−632.519	−	1095.56i	−0.870040	−	1.50695i	−0.00808623	−	0.999967i	$-0.502574\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	572.794	−	330.703i	0.783575	−	0.452397i
$732$		0			0
$733$	293.375	−	508.140i	0.400238	−	0.693233i	−0.593516	−	0.804822i	$-0.702261\pi$
$733$	293.375	−	508.140i	0.400238	−	0.693233i	0.993754	+	0.111589i	$0.0355940\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			1342.09i			1.82101i
$738$		0			0
$739$	−215.973			−0.292250			−0.146125	−	0.989266i	$-0.546680\pi$
$739$	−215.973			−0.292250			−0.146125	+	0.989266i	$0.546680\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−519.581	−	299.980i	−0.699302	−	0.403742i	0.107786	−	0.994174i	$-0.465624\pi$
$743$	−519.581	−	299.980i	−0.699302	−	0.403742i	−0.807087	+	0.590432i	$0.798957\pi$
$744$		0			0
$745$	−103.460	−	179.199i	−0.138873	−	0.240535i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−1090.18	+	629.418i	−1.45552	+	0.840344i
$750$		0			0
$751$	−406.828	+	704.648i	−0.541716	+	0.938279i	0.457090	+	0.889420i	$0.348892\pi$
$751$	−406.828	+	704.648i	−0.541716	+	0.938279i	−0.998806	+	0.0488587i	$0.984442\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			276.919i			0.366780i
$756$		0			0
$757$	1070.43			1.41405			0.707023	−	0.707190i	$-0.250038\pi$
$757$	1070.43			1.41405			0.707023	+	0.707190i	$0.250038\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	567.153	+	327.446i	0.745273	+	0.430284i	0.823983	−	0.566614i	$-0.191747\pi$
$761$	567.153	+	327.446i	0.745273	+	0.430284i	−0.0787104	+	0.996898i	$0.525080\pi$
$762$		0			0
$763$	−464.408	−	804.378i	−0.608660	−	1.05423i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	206.675	−	119.324i	0.269459	−	0.155572i
$768$		0			0
$769$	−341.315	+	591.175i	−0.443843	+	0.768759i	−0.997971	−	0.0636729i	$-0.979719\pi$
$769$	−341.315	+	591.175i	−0.443843	+	0.768759i	0.554128	+	0.832432i	$0.313052\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			770.690i			0.997012i	0.866886	+	0.498506i	$0.166118\pi$
$773$			770.690i			0.997012i	−0.866886	+	0.498506i	$0.833882\pi$
$774$		0			0
$775$	84.8683			0.109508
$776$		0			0
$777$		0			0
$778$		0			0
$779$	630.969	+	364.290i	0.809974	+	0.467638i
$780$		0			0
$781$	686.447	+	1188.96i	0.878933	+	1.52236i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	60.9740	−	35.2033i	0.0776739	−	0.0448450i
$786$		0			0
$787$	126.935	−	219.857i	0.161289	−	0.279361i	−0.774042	−	0.633134i	$-0.781768\pi$
$787$	126.935	−	219.857i	0.161289	−	0.279361i	0.935331	+	0.353773i	$0.115102\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			1487.71i			1.88080i
$792$		0			0
$793$	−351.290			−0.442988
$794$		0			0
$795$		0			0
$796$		0			0
$797$	914.401	+	527.929i	1.14730	+	0.662396i	0.948229	−	0.317589i	$-0.102873\pi$
$797$	914.401	+	527.929i	1.14730	+	0.662396i	0.199075	+	0.979984i	$0.436206\pi$
$798$		0			0
$799$	276.316	+	478.593i	0.345827	+	0.598990i
$800$		0			0

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1620.o (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.93649 + 1.11803i) q^{5} +(6.74342 + 11.6799i) q^{7} +O(q^{10})\)	\(q+(1.93649 + 1.11803i) q^{5} +(6.74342 + 11.6799i) q^{7} +(-15.2932 + 8.82952i) q^{11} +(3.74342 - 6.48379i) q^{13} +16.9706i q^{17} -10.9737 q^{19} +(18.9674 + 10.9508i) q^{23} +(2.50000 + 4.33013i) q^{25} +(-41.0128 + 23.6788i) q^{29} +(8.48683 - 14.6996i) q^{31} +30.1575i q^{35} -5.53950 q^{37} +(-57.4985 - 33.1968i) q^{41} +(-19.4868 - 33.7522i) q^{43} +(28.2014 - 16.2821i) q^{47} +(-66.4473 + 115.090i) q^{49} +11.2392i q^{53} -39.4868 q^{55} +(27.6051 + 15.9378i) q^{59} +(-23.4605 - 40.6348i) q^{61} +(14.4982 - 8.37053i) q^{65} +(38.0000 - 65.8179i) q^{67} -77.7445i q^{71} +94.9210 q^{73} +(-206.257 - 119.082i) q^{77} +(3.46050 + 5.99376i) q^{79} +(-53.8243 + 31.0755i) q^{83} +(-18.9737 + 32.8634i) q^{85} -62.2626i q^{89} +100.974 q^{91} +(-21.2504 - 12.2689i) q^{95} +(62.4868 + 108.230i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 16 q^{7}+O(q^{10}) \) 8 * q + 16 * q^7	\( 8 q + 16 q^{7} - 8 q^{13} + 64 q^{19} + 20 q^{25} - 8 q^{31} - 272 q^{37} - 80 q^{43} - 228 q^{49} - 240 q^{55} + 40 q^{61} + 304 q^{67} + 304 q^{73} - 200 q^{79} + 656 q^{91} + 424 q^{97}+O(q^{100}) \) 8 * q + 16 * q^7 - 8 * q^13 + 64 * q^19 + 20 * q^25 - 8 * q^31 - 272 * q^37 - 80 * q^43 - 228 * q^49 - 240 * q^55 + 40 * q^61 + 304 * q^67 + 304 * q^73 - 200 * q^79 + 656 * q^91 + 424 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more