LMFDB - Embedded newform 1152.2.w.a.431.5

Newspace parameters

Level:	$ N $	$=$	$ 1152 = 2^{7} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1152.w (of order $8$, degree $4$, not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$9.19876631285$
Analytic rank:	$0$
Dimension:	$32$
Relative dimension:	$8$ over $\Q(\zeta_{8})$
Twist minimal:	no (minimal twist has level 288)
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label			431.5
Character	$\chi$	$=$	1152.431
Dual form			1152.2.w.a.719.5

$q$-expansion

$f(q)$	$=$	$q+(0.0913223 - 0.0378270i) q^{5} +(3.05457 + 3.05457i) q^{7} +O(q^{10})$	$q+(0.0913223 - 0.0378270i) q^{5} +(3.05457 + 3.05457i) q^{7} +(-5.25649 + 2.17731i) q^{11} +(-1.57202 + 3.79519i) q^{13} +2.56471 q^{17} +(-2.64058 - 1.09376i) q^{19} +(-4.03079 - 4.03079i) q^{23} +(-3.52863 + 3.52863i) q^{25} +(-2.06984 + 4.99704i) q^{29} -1.44203i q^{31} +(0.394496 + 0.163406i) q^{35} +(-2.07758 - 5.01573i) q^{37} +(-0.296726 + 0.296726i) q^{41} +(2.72382 + 6.57588i) q^{43} -7.42367i q^{47} +11.6608i q^{49} +(-1.53235 - 3.69941i) q^{53} +(-0.397674 + 0.397674i) q^{55} +(0.988255 + 2.38586i) q^{59} +(10.4346 + 4.32215i) q^{61} +0.406051i q^{65} +(0.690522 - 1.66707i) q^{67} +(-2.97957 + 2.97957i) q^{71} +(9.22401 + 9.22401i) q^{73} +(-22.7071 - 9.40558i) q^{77} +1.12181 q^{79} +(-4.13529 + 9.98348i) q^{83} +(0.234216 - 0.0970152i) q^{85} +(12.0590 + 12.0590i) q^{89} +(-16.3945 + 6.79084i) q^{91} -0.282518 q^{95} -18.5545 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q+O(q^{10}) $ 32 * q	$ 32 q - 8 q^{11} + 16 q^{29} + 24 q^{35} + 16 q^{53} + 32 q^{55} + 32 q^{59} + 32 q^{61} + 16 q^{67} + 16 q^{71} + 16 q^{77} + 32 q^{79} - 40 q^{83} + 48 q^{91} - 80 q^{95}+O(q^{100}) $ 32 * q - 8 * q^11 + 16 * q^29 + 24 * q^35 + 16 * q^53 + 32 * q^55 + 32 * q^59 + 32 * q^61 + 16 * q^67 + 16 * q^71 + 16 * q^77 + 32 * q^79 - 40 * q^83 + 48 * q^91 - 80 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times$.

$n$	$127$	$641$	$901$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{3}{8}\right)$

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
$5$	0.0913223	−	0.0378270i	0.0408406	−	0.0169167i	−0.362170	−	0.932112i	$-0.617964\pi$
$5$	0.0913223	−	0.0378270i	0.0408406	−	0.0169167i	0.403010	+	0.915195i	$0.367964\pi$
$6$		0			0
$7$	3.05457	+	3.05457i	1.15452	+	1.15452i	0.985636	+	0.168884i	$0.0540163\pi$
$7$	3.05457	+	3.05457i	1.15452	+	1.15452i	0.168884	+	0.985636i	$0.445984\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−5.25649	+	2.17731i	−1.58489	+	0.656483i	−0.989179	−	0.146715i	$-0.953130\pi$
$11$	−5.25649	+	2.17731i	−1.58489	+	0.656483i	−0.595712	+	0.803198i	$0.703130\pi$
$12$		0			0
$13$	−1.57202	+	3.79519i	−0.436000	+	1.05260i	0.541317	+	0.840818i	$0.317926\pi$
$13$	−1.57202	+	3.79519i	−0.436000	+	1.05260i	−0.977318	+	0.211779i	$0.932074\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	2.56471			0.622034			0.311017	−	0.950404i	$-0.399330\pi$
$17$	2.56471			0.622034			0.311017	+	0.950404i	$0.399330\pi$
$18$		0			0
$19$	−2.64058	−	1.09376i	−0.605791	−	0.250927i	0.0586367	−	0.998279i	$-0.481325\pi$
$19$	−2.64058	−	1.09376i	−0.605791	−	0.250927i	−0.664428	+	0.747353i	$0.731325\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−4.03079	−	4.03079i	−0.840477	−	0.840477i	0.148443	−	0.988921i	$-0.452574\pi$
$23$	−4.03079	−	4.03079i	−0.840477	−	0.840477i	−0.988921	+	0.148443i	$0.952574\pi$
$24$		0			0
$25$	−3.52863	+	3.52863i	−0.705725	+	0.705725i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−2.06984	+	4.99704i	−0.384360	+	0.927927i	0.606752	+	0.794892i	$0.292472\pi$
$29$	−2.06984	+	4.99704i	−0.384360	+	0.927927i	−0.991111	+	0.133035i	$0.957528\pi$
$30$		0			0
$31$		−	1.44203i		−	0.258996i	−0.991580	−	0.129498i	$-0.958663\pi$
$31$		−	1.44203i		−	0.258996i	0.991580	−	0.129498i	$-0.0413366\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	0.394496	+	0.163406i	0.0666820	+	0.0276206i
$36$		0			0
$37$	−2.07758	−	5.01573i	−0.341552	−	0.824581i	−0.997559	−	0.0698256i	$-0.977756\pi$
$37$	−2.07758	−	5.01573i	−0.341552	−	0.824581i	0.656007	−	0.754755i	$-0.272244\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−0.296726	+	0.296726i	−0.0463409	+	0.0463409i	−0.729897	−	0.683557i	$-0.760432\pi$
$41$	−0.296726	+	0.296726i	−0.0463409	+	0.0463409i	0.683557	+	0.729897i	$0.260432\pi$
$42$		0			0
$43$	2.72382	+	6.57588i	0.415378	+	1.00281i	0.983669	+	0.179984i	$0.0576047\pi$
$43$	2.72382	+	6.57588i	0.415378	+	1.00281i	−0.568291	+	0.822828i	$0.692395\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	7.42367i		−	1.08285i	−0.840748	−	0.541427i	$-0.817884\pi$
$47$		−	7.42367i		−	1.08285i	0.840748	−	0.541427i	$-0.182116\pi$
$48$		0			0
$49$			11.6608i			1.66583i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−1.53235	−	3.69941i	−0.210484	−	0.508153i	0.783014	−	0.622004i	$-0.213681\pi$
$53$	−1.53235	−	3.69941i	−0.210484	−	0.508153i	−0.993498	+	0.113851i	$0.963681\pi$
$54$		0			0
$55$	−0.397674	+	0.397674i	−0.0536223	+	0.0536223i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	0.988255	+	2.38586i	0.128660	+	0.310612i	0.975062	−	0.221931i	$-0.0712361\pi$
$59$	0.988255	+	2.38586i	0.128660	+	0.310612i	−0.846402	+	0.532544i	$0.821236\pi$
$60$		0			0
$61$	10.4346	+	4.32215i	1.33601	+	0.553395i	0.932365	−	0.361519i	$-0.117742\pi$
$61$	10.4346	+	4.32215i	1.33601	+	0.553395i	0.403649	+	0.914914i	$0.367742\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			0.406051i			0.0503644i
$66$		0			0
$67$	0.690522	−	1.66707i	0.0843606	−	0.203665i	−0.876070	−	0.482184i	$-0.839844\pi$
$67$	0.690522	−	1.66707i	0.0843606	−	0.203665i	0.960431	+	0.278519i	$0.0898437\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−2.97957	+	2.97957i	−0.353610	+	0.353610i	−0.861451	−	0.507841i	$-0.830444\pi$
$71$	−2.97957	+	2.97957i	−0.353610	+	0.353610i	0.507841	+	0.861451i	$0.330444\pi$
$72$		0			0
$73$	9.22401	+	9.22401i	1.07959	+	1.07959i	0.996546	+	0.0830429i	$0.0264638\pi$
$73$	9.22401	+	9.22401i	1.07959	+	1.07959i	0.0830429	+	0.996546i	$0.473536\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−22.7071	−	9.40558i	−2.58771	−	1.07187i
$78$		0			0
$79$	1.12181			0.126213			0.0631065	−	0.998007i	$-0.479899\pi$
$79$	1.12181			0.126213			0.0631065	+	0.998007i	$0.479899\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−4.13529	+	9.98348i	−0.453907	+	1.09583i	0.516916	+	0.856036i	$0.327080\pi$
$83$	−4.13529	+	9.98348i	−0.453907	+	1.09583i	−0.970824	+	0.239794i	$0.922920\pi$
$84$		0			0
$85$	0.234216	−	0.0970152i	0.0254042	−	0.0105228i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	12.0590	+	12.0590i	1.27825	+	1.27825i	0.941644	+	0.336609i	$0.109280\pi$
$89$	12.0590	+	12.0590i	1.27825	+	1.27825i	0.336609	+	0.941644i	$0.390720\pi$
$90$		0			0
$91$	−16.3945	+	6.79084i	−1.71862	+	0.711874i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−0.282518			−0.0289857
$96$		0			0
$97$	−18.5545			−1.88393			−0.941963	−	0.335717i	$-0.891022\pi$
$97$	−18.5545			−1.88393			−0.941963	+	0.335717i	$0.891022\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−5.24603	+	2.17298i	−0.521999	+	0.216219i	−0.628095	−	0.778137i	$-0.716165\pi$
$101$	−5.24603	+	2.17298i	−0.521999	+	0.216219i	0.106096	+	0.994356i	$0.466165\pi$
$102$		0			0
$103$	11.0095	+	11.0095i	1.08479	+	1.08479i	0.996055	+	0.0887398i	$0.0282839\pi$
$103$	11.0095	+	11.0095i	1.08479	+	1.08479i	0.0887398	+	0.996055i	$0.471716\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	0.182628	−	0.0756468i	0.0176553	−	0.00731305i	−0.373838	−	0.927494i	$-0.621959\pi$
$107$	0.182628	−	0.0756468i	0.0176553	−	0.00731305i	0.391494	+	0.920181i	$0.371959\pi$
$108$		0			0
$109$	2.10265	−	5.07626i	0.201398	−	0.486217i	−0.790621	−	0.612305i	$-0.790242\pi$
$109$	2.10265	−	5.07626i	0.201398	−	0.486217i	0.992019	+	0.126088i	$0.0402423\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−5.91777			−0.556697			−0.278348	−	0.960480i	$-0.589787\pi$
$113$	−5.91777			−0.556697			−0.278348	+	0.960480i	$0.589787\pi$
$114$		0			0
$115$	−0.520573	−	0.215629i	−0.0485437	−	0.0201075i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	7.83410	+	7.83410i	0.718151	+	0.718151i
$120$		0			0
$121$	15.1118	−	15.1118i	1.37380	−	1.37380i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−0.377900	+	0.912331i	−0.0338004	+	0.0816014i
$126$		0			0
$127$			9.70061i			0.860790i	0.902641	+	0.430395i	$0.141626\pi$
$127$			9.70061i			0.860790i	−0.902641	+	0.430395i	$0.858374\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	12.7327	+	5.27407i	1.11246	+	0.460798i	0.861786	−	0.507273i	$-0.169346\pi$
$131$	12.7327	+	5.27407i	1.11246	+	0.460798i	0.250678	+	0.968070i	$0.419346\pi$
$132$		0			0
$133$	−4.72486	−	11.4068i	−0.409698	−	0.989098i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	5.62861	−	5.62861i	0.480884	−	0.480884i	−0.424530	−	0.905414i	$-0.639561\pi$
$137$	5.62861	−	5.62861i	0.480884	−	0.480884i	0.905414	+	0.424530i	$0.139561\pi$
$138$		0			0
$139$	−0.202389	−	0.488610i	−0.0171664	−	0.0414434i	0.915064	−	0.403309i	$-0.132140\pi$
$139$	−0.202389	−	0.488610i	−0.0171664	−	0.0414434i	−0.932230	+	0.361865i	$0.882140\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	23.3722i		−	1.95448i
$144$		0			0
$145$			0.534637i			0.0443992i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−1.35052	−	3.26046i	−0.110639	−	0.267107i	0.858856	−	0.512217i	$-0.171176\pi$
$149$	−1.35052	−	3.26046i	−0.110639	−	0.267107i	−0.969495	+	0.245110i	$0.921176\pi$
$150$		0			0
$151$	13.6492	−	13.6492i	1.11076	−	1.11076i	0.117709	−	0.993048i	$-0.462445\pi$
$151$	13.6492	−	13.6492i	1.11076	−	1.11076i	0.993048	−	0.117709i	$-0.0375552\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−0.0545476	−	0.131690i	−0.00438137	−	0.0105776i
$156$		0			0
$157$	−15.4587	−	6.40321i	−1.23374	−	0.511032i	−0.331987	−	0.943284i	$-0.607719\pi$
$157$	−15.4587	−	6.40321i	−1.23374	−	0.511032i	−0.901753	+	0.432252i	$0.857719\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	24.6247i		−	1.94070i
$162$		0			0
$163$	6.04288	−	14.5888i	0.473315	−	1.14268i	−0.489374	−	0.872074i	$-0.662775\pi$
$163$	6.04288	−	14.5888i	0.473315	−	1.14268i	0.962689	−	0.270609i	$-0.0872252\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	6.58872	−	6.58872i	0.509850	−	0.509850i	−0.404630	−	0.914480i	$-0.632600\pi$
$167$	6.58872	−	6.58872i	0.509850	−	0.509850i	0.914480	+	0.404630i	$0.132600\pi$
$168$		0			0
$169$	−2.73986	−	2.73986i	−0.210759	−	0.210759i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	17.8204	+	7.38143i	1.35486	+	0.561200i	0.937640	−	0.347608i	$-0.113006\pi$
$173$	17.8204	+	7.38143i	1.35486	+	0.561200i	0.417216	+	0.908807i	$0.363006\pi$
$174$		0			0
$175$	−21.5569			−1.62955
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−4.07369	+	9.83475i	−0.304482	+	0.735084i	0.695383	+	0.718639i	$0.255235\pi$
$179$	−4.07369	+	9.83475i	−0.304482	+	0.735084i	−0.999865	+	0.0164446i	$0.994765\pi$
$180$		0			0
$181$	8.06280	−	3.33972i	0.599303	−	0.248239i	−0.0623442	−	0.998055i	$-0.519858\pi$
$181$	8.06280	−	3.33972i	0.599303	−	0.248239i	0.661647	+	0.749815i	$0.269858\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−0.379459	−	0.379459i	−0.0278984	−	0.0278984i
$186$		0			0
$187$	−13.4814	+	5.58417i	−0.985856	+	0.408355i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	13.0520			0.944411			0.472205	−	0.881489i	$-0.343458\pi$
$191$	13.0520			0.944411			0.472205	+	0.881489i	$0.343458\pi$
$192$		0			0
$193$	−3.05926			−0.220210			−0.110105	−	0.993920i	$-0.535119\pi$
$193$	−3.05926			−0.220210			−0.110105	+	0.993920i	$0.535119\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	5.46774	−	2.26481i	0.389560	−	0.161361i	−0.179302	−	0.983794i	$-0.557384\pi$
$197$	5.46774	−	2.26481i	0.389560	−	0.161361i	0.568862	+	0.822433i	$0.307384\pi$
$198$		0			0
$199$	4.48835	+	4.48835i	0.318170	+	0.318170i	0.848064	−	0.529894i	$-0.177768\pi$
$199$	4.48835	+	4.48835i	0.318170	+	0.318170i	−0.529894	+	0.848064i	$0.677768\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−21.5863	+	8.94133i	−1.51506	+	0.627559i
$204$		0			0
$205$	−0.0158735	+	0.0383220i	−0.00110865	+	0.00267652i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	16.2617			1.12484
$210$		0			0
$211$	−0.322439	−	0.133558i	−0.0221976	−	0.00919454i	0.371557	−	0.928410i	$-0.378824\pi$
$211$	−0.322439	−	0.133558i	−0.0221976	−	0.00919454i	−0.393755	+	0.919216i	$0.628824\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	0.497491	+	0.497491i	0.0339286	+	0.0339286i
$216$		0			0
$217$	4.40478	−	4.40478i	0.299016	−	0.299016i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−4.03178	+	9.73358i	−0.271207	+	0.654752i
$222$		0			0
$223$		−	19.1612i		−	1.28313i	−0.767068	−	0.641566i	$-0.778285\pi$
$223$		−	19.1612i		−	1.28313i	0.767068	−	0.641566i	$-0.221715\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	19.4451	+	8.05441i	1.29061	+	0.534590i	0.919169	−	0.393864i	$-0.128862\pi$
$227$	19.4451	+	8.05441i	1.29061	+	0.534590i	0.371446	+	0.928455i	$0.378862\pi$
$228$		0			0
$229$	−6.97193	−	16.8317i	−0.460718	−	1.11227i	−0.968103	−	0.250552i	$-0.919388\pi$
$229$	−6.97193	−	16.8317i	−0.460718	−	1.11227i	0.507385	−	0.861719i	$-0.330612\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−16.7768	+	16.7768i	−1.09909	+	1.09909i	−0.104569	+	0.994518i	$0.533346\pi$
$233$	−16.7768	+	16.7768i	−1.09909	+	1.09909i	−0.994518	+	0.104569i	$0.966654\pi$
$234$		0			0
$235$	−0.280815	−	0.677947i	−0.0183183	−	0.0442244i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			9.10976i			0.589261i	0.955611	+	0.294631i	$0.0951966\pi$
$239$			9.10976i			0.589261i	−0.955611	+	0.294631i	$0.904803\pi$
$240$		0			0
$241$			7.39995i			0.476673i	0.971183	+	0.238336i	$0.0766020\pi$
$241$			7.39995i			0.476673i	−0.971183	+	0.238336i	$0.923398\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	0.441093	+	1.06489i	0.0281804	+	0.0680336i
$246$		0			0
$247$	8.30210	−	8.30210i	0.528250	−	0.528250i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−4.84250	−	11.6908i	−0.305656	−	0.737919i	−0.999836	−	0.0181174i	$-0.994233\pi$
$251$	−4.84250	−	11.6908i	−0.305656	−	0.737919i	0.694180	−	0.719802i	$-0.255767\pi$
$252$		0			0
$253$	29.9641	+	12.4115i	1.88382	+	0.780306i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			6.02571i			0.375873i	0.982181	+	0.187937i	$0.0601800\pi$
$257$			6.02571i			0.375873i	−0.982181	+	0.187937i	$0.939820\pi$
$258$		0			0
$259$	8.97478	−	21.6670i	0.557666	−	1.34632i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−6.79218	+	6.79218i	−0.418824	+	0.418824i	−0.884798	−	0.465974i	$-0.845704\pi$
$263$	−6.79218	+	6.79218i	−0.418824	+	0.418824i	0.465974	+	0.884798i	$0.345704\pi$
$264$		0			0
$265$	−0.279875	−	0.279875i	−0.0171926	−	0.0171926i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−13.8548	−	5.73883i	−0.844740	−	0.349903i	−0.0820193	−	0.996631i	$-0.526137\pi$
$269$	−13.8548	−	5.73883i	−0.844740	−	0.349903i	−0.762721	+	0.646728i	$0.776137\pi$
$270$		0			0
$271$	−5.62497			−0.341692			−0.170846	−	0.985298i	$-0.554650\pi$
$271$	−5.62497			−0.341692			−0.170846	+	0.985298i	$0.554650\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	10.8653	−	26.2311i	0.655201	−	1.58179i
$276$		0			0
$277$	−1.00843	+	0.417706i	−0.0605908	+	0.0250975i	−0.412773	−	0.910834i	$-0.635440\pi$
$277$	−1.00843	+	0.417706i	−0.0605908	+	0.0250975i	0.352182	+	0.935931i	$0.385440\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	21.7589	+	21.7589i	1.29803	+	1.29803i	0.929693	+	0.368335i	$0.120072\pi$
$281$	21.7589	+	21.7589i	1.29803	+	1.29803i	0.368335	+	0.929693i	$0.379928\pi$
$282$		0			0
$283$	22.3614	−	9.26239i	1.32925	−	0.550592i	0.398807	−	0.917035i	$-0.369424\pi$
$283$	22.3614	−	9.26239i	1.32925	−	0.550592i	0.930441	+	0.366443i	$0.119424\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−1.81274			−0.107003
$288$		0			0
$289$	−10.4223			−0.613074
$290$		0			0
$291$		0			0
$292$		0			0
$293$	29.2423	−	12.1126i	1.70835	−	0.707623i	0.708355	−	0.705856i	$-0.249438\pi$
$293$	29.2423	−	12.1126i	1.70835	−	0.707623i	0.999998	−	0.00176683i	$-0.000562400\pi$
$294$		0			0
$295$	0.180500	+	0.180500i	0.0105091	+	0.0105091i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	21.6341	−	8.96114i	1.25113	−	0.518236i
$300$		0			0
$301$	−11.7664	+	28.4066i	−0.678204	+	1.63733i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	1.11641			0.0639252
$306$		0			0
$307$	−13.7854	−	5.71009i	−0.786773	−	0.325892i	−0.0471275	−	0.998889i	$-0.515007\pi$
$307$	−13.7854	−	5.71009i	−0.786773	−	0.325892i	−0.739645	+	0.672997i	$0.765007\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−11.4220	−	11.4220i	−0.647684	−	0.647684i	0.304749	−	0.952433i	$-0.401427\pi$
$311$	−11.4220	−	11.4220i	−0.647684	−	0.647684i	−0.952433	+	0.304749i	$0.901427\pi$
$312$		0			0
$313$	2.24707	−	2.24707i	0.127012	−	0.127012i	−0.640743	−	0.767755i	$-0.721374\pi$
$313$	2.24707	−	2.24707i	0.127012	−	0.127012i	0.767755	+	0.640743i	$0.221374\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	1.05906	−	2.55679i	0.0594826	−	0.143604i	−0.891344	−	0.453328i	$-0.850237\pi$
$317$	1.05906	−	2.55679i	0.0594826	−	0.143604i	0.950826	+	0.309724i	$0.100237\pi$
$318$		0			0
$319$		−	30.7736i		−	1.72299i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−6.77233	−	2.80519i	−0.376823	−	0.156085i
$324$		0			0
$325$	−7.84475	−	18.9389i	−0.435148	−	1.05054i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	22.6761	−	22.6761i	1.25018	−	1.25018i
$330$		0			0
$331$	0.123439	+	0.298008i	0.00678481	+	0.0163800i	0.927236	−	0.374478i	$-0.122178\pi$
$331$	0.123439	+	0.298008i	0.00678481	+	0.0163800i	−0.920451	+	0.390858i	$0.872178\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	0.178361i		−	0.00974489i
$336$		0			0
$337$		−	16.5067i		−	0.899179i	−0.893235	−	0.449590i	$-0.851570\pi$
$337$		−	16.5067i		−	0.899179i	0.893235	−	0.449590i	$-0.148430\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	3.13974	+	7.58001i	0.170027	+	0.410481i
$342$		0			0
$343$	−14.2368	+	14.2368i	−0.768716	+	0.768716i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−5.96248	−	14.3947i	−0.320083	−	0.772748i	−0.999248	−	0.0387633i	$-0.987658\pi$
$347$	−5.96248	−	14.3947i	−0.320083	−	0.772748i	0.679166	−	0.733985i	$-0.262342\pi$
$348$		0			0
$349$	1.57618	+	0.652873i	0.0843707	+	0.0349475i	0.424470	−	0.905442i	$-0.360461\pi$
$349$	1.57618	+	0.652873i	0.0843707	+	0.0349475i	−0.340099	+	0.940390i	$0.610461\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			18.7091i			0.995786i	0.867239	+	0.497893i	$0.165893\pi$
$353$			18.7091i			0.995786i	−0.867239	+	0.497893i	$0.834107\pi$
$354$		0			0
$355$	−0.159393	+	0.384810i	−0.00845972	+	0.0204236i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−23.5065	+	23.5065i	−1.24062	+	1.24062i	−0.280882	+	0.959742i	$0.590627\pi$
$359$	−23.5065	+	23.5065i	−1.24062	+	1.24062i	−0.959742	+	0.280882i	$0.909373\pi$
$360$		0			0
$361$	−7.65868	−	7.65868i	−0.403088	−	0.403088i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	1.19127	+	0.493442i	0.0623542	+	0.0258279i
$366$		0			0
$367$	2.95190			0.154088			0.0770440	−	0.997028i	$-0.475452\pi$
$367$	2.95190			0.154088			0.0770440	+	0.997028i	$0.475452\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	6.61945	−	15.9808i	0.343665	−	0.829680i
$372$		0			0
$373$	−19.7552	+	8.18287i	−1.02289	+	0.423693i	−0.830140	−	0.557556i	$-0.811739\pi$
$373$	−19.7552	+	8.18287i	−1.02289	+	0.423693i	−0.192746	+	0.981249i	$0.561739\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−15.7109	−	15.7109i	−0.809152	−	0.809152i
$378$		0			0
$379$	−4.26476	+	1.76652i	−0.219066	+	0.0907402i	−0.489518	−	0.871993i	$-0.662827\pi$
$379$	−4.26476	+	1.76652i	−0.219066	+	0.0907402i	0.270451	+	0.962734i	$0.412827\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	9.45328			0.483040			0.241520	−	0.970396i	$-0.422354\pi$
$383$	9.45328			0.483040			0.241520	+	0.970396i	$0.422354\pi$
$384$		0			0
$385$	−2.42945			−0.123816
$386$		0			0
$387$		0			0
$388$		0			0
$389$	20.8554	−	8.63860i	1.05741	−	0.437994i	0.214880	−	0.976640i	$-0.431064\pi$
$389$	20.8554	−	8.63860i	1.05741	−	0.437994i	0.842532	+	0.538646i	$0.181064\pi$
$390$		0			0
$391$	−10.3378	−	10.3378i	−0.522806	−	0.522806i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	0.102446	−	0.0424345i	0.00515461	−	0.00213511i
$396$		0			0
$397$	3.29268	−	7.94922i	0.165255	−	0.398960i	−0.819460	−	0.573137i	$-0.805726\pi$
$397$	3.29268	−	7.94922i	0.165255	−	0.398960i	0.984714	+	0.174177i	$0.0557264\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−32.5903			−1.62748			−0.813740	−	0.581229i	$-0.802572\pi$
$401$	−32.5903			−1.62748			−0.813740	+	0.581229i	$0.802572\pi$
$402$		0			0
$403$	5.47278	+	2.26690i	0.272619	+	0.112922i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	21.8416	+	21.8416i	1.08265	+	1.08265i
$408$		0			0
$409$	5.41708	−	5.41708i	0.267858	−	0.267858i	−0.560379	−	0.828236i	$-0.689344\pi$
$409$	5.41708	−	5.41708i	0.267858	−	0.267858i	0.828236	+	0.560379i	$0.189344\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−4.26908	+	10.3065i	−0.210068	+	0.507149i
$414$		0			0
$415$			1.06814i			0.0524329i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−22.0298	−	9.12504i	−1.07623	−	0.445787i	−0.227042	−	0.973885i	$-0.572906\pi$
$419$	−22.0298	−	9.12504i	−1.07623	−	0.445787i	−0.849184	+	0.528097i	$0.822906\pi$
$420$		0			0
$421$	2.82553	+	6.82144i	0.137708	+	0.332457i	0.977656	−	0.210210i	$-0.0674147\pi$
$421$	2.82553	+	6.82144i	0.137708	+	0.332457i	−0.839948	+	0.542667i	$0.817415\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−9.04991	+	9.04991i	−0.438985	+	0.438985i
$426$		0			0
$427$	18.6709	+	45.0756i	0.903549	+	2.18136i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			6.73112i			0.324226i	0.986772	+	0.162113i	$0.0518310\pi$
$431$			6.73112i			0.324226i	−0.986772	+	0.162113i	$0.948169\pi$
$432$		0			0
$433$		−	12.1160i		−	0.582257i	−0.956684	−	0.291129i	$-0.905969\pi$
$433$		−	12.1160i		−	0.582257i	0.956684	−	0.291129i	$-0.0940308\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	6.23489	+	15.0524i	0.298255	+	0.720052i
$438$		0			0
$439$	−14.0767	+	14.0767i	−0.671843	+	0.671843i	−0.958141	−	0.286298i	$-0.907575\pi$
$439$	−14.0767	+	14.0767i	−0.671843	+	0.671843i	0.286298	+	0.958141i	$0.407575\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	14.5313	+	35.0817i	0.690404	+	1.66678i	0.743965	+	0.668218i	$0.232943\pi$
$443$	14.5313	+	35.0817i	0.690404	+	1.66678i	−0.0535615	+	0.998565i	$0.517057\pi$
$444$		0			0
$445$	1.55741	+	0.645102i	0.0738285	+	0.0305808i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	25.9809i		−	1.22611i	−0.790039	−	0.613056i	$-0.789940\pi$
$449$		−	25.9809i		−	1.22611i	0.790039	−	0.613056i	$-0.210060\pi$
$450$		0			0
$451$	0.913674	−	2.20580i	0.0430232	−	0.103867i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−1.24031	+	1.24031i	−0.0581467	+	0.0581467i
$456$		0			0
$457$	6.23785	+	6.23785i	0.291794	+	0.291794i	0.837789	−	0.545994i	$-0.183848\pi$
$457$	6.23785	+	6.23785i	0.291794	+	0.291794i	−0.545994	+	0.837789i	$0.683848\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	23.3278	+	9.66269i	1.08648	+	0.450036i	0.852779	−	0.522271i	$-0.174915\pi$
$461$	23.3278	+	9.66269i	1.08648	+	0.450036i	0.233704	+	0.972308i	$0.424915\pi$
$462$		0			0
$463$	−3.99595			−0.185707			−0.0928537	−	0.995680i	$-0.529599\pi$
$463$	−3.99595			−0.185707			−0.0928537	+	0.995680i	$0.529599\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−7.26272	+	17.5338i	−0.336079	+	0.811366i	0.662006	+	0.749499i	$0.269706\pi$
$467$	−7.26272	+	17.5338i	−0.336079	+	0.811366i	−0.998084	+	0.0618671i	$0.980294\pi$
$468$		0			0
$469$	7.20142	−	2.98293i	0.332531	−	0.137739i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−28.6354	−	28.6354i	−1.31666	−	1.31666i
$474$		0			0
$475$	13.1771	−	5.45814i	0.604607	−	0.250437i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−21.7665			−0.994537			−0.497268	−	0.867597i	$-0.665664\pi$
$479$	−21.7665			−0.994537			−0.497268	+	0.867597i	$0.665664\pi$
$480$		0			0
$481$	22.3017			1.01687
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−1.69444	+	0.701861i	−0.0769407	+	0.0318699i
$486$		0			0
$487$	9.10128	+	9.10128i	0.412419	+	0.412419i	0.882580	−	0.470162i	$-0.155804\pi$
$487$	9.10128	+	9.10128i	0.412419	+	0.412419i	−0.470162	+	0.882580i	$0.655804\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−15.2645	+	6.32276i	−0.688876	+	0.285342i	−0.699532	−	0.714601i	$-0.746608\pi$
$491$	−15.2645	+	6.32276i	−0.688876	+	0.285342i	0.0106556	+	0.999943i	$0.496608\pi$
$492$		0			0
$493$	−5.30855	+	12.8160i	−0.239085	+	0.577202i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−18.2026			−0.816500
$498$		0			0
$499$	32.2595	+	13.3623i	1.44413	+	0.598180i	0.960796	−	0.277255i	$-0.0894246\pi$
$499$	32.2595	+	13.3623i	1.44413	+	0.598180i	0.483337	+	0.875434i	$0.339425\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	7.34111	+	7.34111i	0.327324	+	0.327324i	0.851568	−	0.524244i	$-0.175652\pi$
$503$	7.34111	+	7.34111i	0.327324	+	0.327324i	−0.524244	+	0.851568i	$0.675652\pi$
$504$		0			0
$505$	−0.396882	+	0.396882i	−0.0176610	+	0.0176610i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	7.55299	−	18.2345i	0.334780	−	0.808231i	−0.663419	−	0.748248i	$-0.730895\pi$
$509$	7.55299	−	18.2345i	0.334780	−	0.808231i	0.998199	−	0.0599832i	$-0.0191047\pi$
$510$		0			0
$511$			56.3508i			2.49281i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	1.42186	+	0.588955i	0.0626548	+	0.0259525i
$516$		0			0
$517$	16.1636	+	39.0225i	0.710876	+	1.71621i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−26.3209	+	26.3209i	−1.15314	+	1.15314i	−0.167218	+	0.985920i	$0.553478\pi$
$521$	−26.3209	+	26.3209i	−1.15314	+	1.15314i	−0.985920	+	0.167218i	$0.946522\pi$
$522$		0			0
$523$	−5.50275	−	13.2848i	−0.240619	−	0.580905i	0.756726	−	0.653732i	$-0.226798\pi$
$523$	−5.50275	−	13.2848i	−0.240619	−	0.580905i	−0.997345	+	0.0728276i	$0.976798\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	3.69839i		−	0.161104i
$528$		0			0
$529$			9.49450i			0.412805i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−0.659674	−	1.59259i	−0.0285737	−	0.0689829i
$534$		0			0
$535$	0.0138165	−	0.0138165i	0.000597339	−	0.000597339i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−25.3892	−	61.2950i	−1.09359	−	2.64016i
$540$		0			0
$541$	1.34772	+	0.558245i	0.0579431	+	0.0240008i	0.411467	−	0.911425i	$-0.365017\pi$
$541$	1.34772	+	0.558245i	0.0579431	+	0.0240008i	−0.353524	+	0.935426i	$0.615017\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	0.543113i		−	0.0232644i
$546$		0			0
$547$	−3.11417	+	7.51828i	−0.133152	+	0.321458i	−0.976367	−	0.216119i	$-0.930660\pi$
$547$	−3.11417	+	7.51828i	−0.133152	+	0.321458i	0.843215	+	0.537577i	$0.180660\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	10.9312	−	10.9312i	0.465683	−	0.465683i
$552$		0			0
$553$	3.42664	+	3.42664i	0.145715	+	0.145715i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−10.3810	−	4.29993i	−0.439855	−	0.182194i	0.151755	−	0.988418i	$-0.451508\pi$
$557$	−10.3810	−	4.29993i	−0.439855	−	0.182194i	−0.591610	+	0.806224i	$0.701508\pi$
$558$		0			0
$559$	−29.2386			−1.23666
$560$		0			0
$561$		0			0
$562$		0			0
$563$	1.95005	−	4.70784i	0.0821848	−	0.198412i	−0.877445	−	0.479677i	$-0.840754\pi$
$563$	1.95005	−	4.70784i	0.0821848	−	0.198412i	0.959630	+	0.281265i	$0.0907540\pi$
$564$		0			0
$565$	−0.540424	+	0.223851i	−0.0227358	+	0.00941749i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−3.12423	−	3.12423i	−0.130974	−	0.130974i	0.638581	−	0.769555i	$-0.279522\pi$
$569$	−3.12423	−	3.12423i	−0.130974	−	0.130974i	−0.769555	+	0.638581i	$0.779522\pi$
$570$		0			0
$571$	−33.4029	+	13.8360i	−1.39787	+	0.579017i	−0.949198	−	0.314680i	$-0.898103\pi$
$571$	−33.4029	+	13.8360i	−1.39787	+	0.579017i	−0.448672	+	0.893697i	$0.648103\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	28.4463			1.18629
$576$		0			0
$577$	−14.0315			−0.584141			−0.292070	−	0.956397i	$-0.594344\pi$
$577$	−14.0315			−0.584141			−0.292070	+	0.956397i	$0.594344\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−43.1268	+	17.8637i	−1.78920	+	0.741112i
$582$		0			0
$583$	16.1095	+	16.1095i	0.667188	+	0.667188i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	12.0165	−	4.97740i	0.495975	−	0.205439i	−0.120652	−	0.992695i	$-0.538499\pi$
$587$	12.0165	−	4.97740i	0.495975	−	0.205439i	0.616627	+	0.787255i	$0.288499\pi$
$588$		0			0
$589$	−1.57724	+	3.80780i	−0.0649891	+	0.156898i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−17.6528			−0.724914			−0.362457	−	0.932000i	$-0.618062\pi$
$593$	−17.6528			−0.724914			−0.362457	+	0.932000i	$0.618062\pi$
$594$		0			0
$595$	1.01177	+	0.419088i	0.0414785	+	0.0171809i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−5.25995	−	5.25995i	−0.214916	−	0.214916i	0.591436	−	0.806352i	$-0.298561\pi$
$599$	−5.25995	−	5.25995i	−0.214916	−	0.214916i	−0.806352	+	0.591436i	$0.798561\pi$
$600$		0			0
$601$	6.54574	−	6.54574i	0.267006	−	0.267006i	−0.560886	−	0.827893i	$-0.689540\pi$
$601$	6.54574	−	6.54574i	0.267006	−	0.267006i	0.827893	+	0.560886i	$0.189540\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	0.808413	−	1.95168i	0.0328667	−	0.0793472i
$606$		0			0
$607$			1.72276i			0.0699248i	0.999389	+	0.0349624i	$0.0111311\pi$
$607$			1.72276i			0.0699248i	−0.999389	+	0.0349624i	$0.988869\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	28.1743	+	11.6702i	1.13981	+	0.472125i
$612$		0			0
$613$	16.6267	+	40.1405i	0.671547	+	1.62126i	0.778982	+	0.627046i	$0.215736\pi$
$613$	16.6267	+	40.1405i	0.671547	+	1.62126i	−0.107435	+	0.994212i	$0.534264\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−7.78933	+	7.78933i	−0.313587	+	0.313587i	−0.846297	−	0.532711i	$-0.821173\pi$
$617$	−7.78933	+	7.78933i	−0.313587	+	0.313587i	0.532711	+	0.846297i	$0.321173\pi$
$618$		0			0
$619$	−11.0171	−	26.5976i	−0.442814	−	1.06905i	−0.974957	−	0.222394i	$-0.928613\pi$
$619$	−11.0171	−	26.5976i	−0.442814	−	1.06905i	0.532143	−	0.846655i	$-0.321387\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			73.6703i			2.95154i
$624$		0			0
$625$		−	24.8535i		−	0.994141i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−5.32840	−	12.8639i	−0.212457	−	0.512917i
$630$		0			0
$631$	21.0548	−	21.0548i	0.838177	−	0.838177i	−0.150442	−	0.988619i	$-0.548070\pi$
$631$	21.0548	−	21.0548i	0.838177	−	0.838177i	0.988619	+	0.150442i	$0.0480696\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	0.366944	+	0.885882i	0.0145617	+	0.0351552i
$636$		0			0
$637$	−44.2551	−	18.3311i	−1.75345	−	0.726303i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	5.65123i		−	0.223210i	−0.993753	−	0.111605i	$-0.964401\pi$
$641$		−	5.65123i		−	0.223210i	0.993753	−	0.111605i	$-0.0355992\pi$
$642$		0			0
$643$	−17.0022	+	41.0470i	−0.670502	+	1.61874i	0.110256	+	0.993903i	$0.464833\pi$
$643$	−17.0022	+	41.0470i	−0.670502	+	1.61874i	−0.780759	+	0.624833i	$0.785167\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	0.0151270	−	0.0151270i	0.000594703	−	0.000594703i	−0.706809	−	0.707404i	$-0.749866\pi$
$647$	0.0151270	−	0.0151270i	0.000594703	−	0.000594703i	0.707404	+	0.706809i	$0.249866\pi$
$648$		0			0
$649$	−10.3895	−	10.3895i	−0.407824	−	0.407824i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	28.8575	+	11.9531i	1.12928	+	0.467763i	0.867536	−	0.497375i	$-0.165703\pi$
$653$	28.8575	+	11.9531i	1.12928	+	0.467763i	0.261744	+	0.965137i	$0.415703\pi$
$654$		0			0
$655$	1.36228			0.0532289
$656$		0			0
$657$		0			0
$658$		0			0
$659$	7.25678	−	17.5194i	0.282684	−	0.682459i	−0.717212	−	0.696855i	$-0.754582\pi$
$659$	7.25678	−	17.5194i	0.282684	−	0.682459i	0.999896	+	0.0143952i	$0.00458229\pi$
$660$		0			0
$661$	−15.6310	+	6.47456i	−0.607974	+	0.251831i	−0.665362	−	0.746521i	$-0.731723\pi$
$661$	−15.6310	+	6.47456i	−0.607974	+	0.251831i	0.0573877	+	0.998352i	$0.481723\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−0.862971	−	0.862971i	−0.0334646	−	0.0334646i
$666$		0			0
$667$	28.4851	−	11.7989i	1.10295	−	0.456856i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−64.2600			−2.48073
$672$		0			0
$673$	26.5987			1.02530			0.512652	−	0.858596i	$-0.328663\pi$
$673$	26.5987			1.02530			0.512652	+	0.858596i	$0.328663\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−26.3000	+	10.8938i	−1.01079	+	0.418683i	−0.825743	−	0.564047i	$-0.809244\pi$
$677$	−26.3000	+	10.8938i	−1.01079	+	0.418683i	−0.185047	+	0.982730i	$0.559244\pi$
$678$		0			0
$679$	−56.6761	−	56.6761i	−2.17503	−	2.17503i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	29.9528	−	12.4069i	1.14611	−	0.474735i	0.272884	−	0.962047i	$-0.412022\pi$
$683$	29.9528	−	12.4069i	1.14611	−	0.474735i	0.873228	+	0.487312i	$0.162022\pi$
$684$		0			0
$685$	0.301105	−	0.726931i	0.0115046	−	0.0277746i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	16.4489			0.626652
$690$		0			0
$691$	9.94272	+	4.11841i	0.378239	+	0.156672i	0.563699	−	0.825980i	$-0.309378\pi$
$691$	9.94272	+	4.11841i	0.378239	+	0.156672i	−0.185460	+	0.982652i	$0.559378\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−0.0369653	−	0.0369653i	−0.00140217	−	0.00140217i
$696$		0			0
$697$	−0.761018	+	0.761018i	−0.0288256	+	0.0288256i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	5.59430	−	13.5058i	0.211294	−	0.510108i	−0.782329	−	0.622866i	$-0.785968\pi$
$701$	5.59430	−	13.5058i	0.211294	−	0.510108i	0.993623	+	0.112757i	$0.0359683\pi$
$702$		0			0
$703$			15.5168i			0.585228i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−22.6619	−	9.38686i	−0.852288	−	0.353029i
$708$		0			0
$709$	4.92207	+	11.8829i	0.184852	+	0.446273i	0.988955	−	0.148218i	$-0.0473537\pi$
$709$	4.92207	+	11.8829i	0.184852	+	0.446273i	−0.804102	+	0.594491i	$0.797354\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−5.81252	+	5.81252i	−0.217680	+	0.217680i
$714$		0			0
$715$	−0.884098	−	2.13440i	−0.0330634	−	0.0798221i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	16.1029i		−	0.600537i	−0.953855	−	0.300268i	$-0.902924\pi$
$719$		−	16.1029i		−	0.600537i	0.953855	−	0.300268i	$-0.0970762\pi$
$720$		0			0
$721$			67.2584i			2.50483i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−10.3290	−	24.9364i	−0.383609	−	0.926113i
$726$		0			0
$727$	−21.0560	+	21.0560i	−0.780923	+	0.780923i	−0.979987	−	0.199064i	$-0.936210\pi$
$727$	−21.0560	+	21.0560i	−0.780923	+	0.780923i	0.199064	+	0.979987i	$0.436210\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	6.98581	+	16.8652i	0.258379	+	0.623783i
$732$		0			0
$733$	41.2426	+	17.0832i	1.52333	+	0.630984i	0.978255	−	0.207404i	$-0.0665015\pi$
$733$	41.2426	+	17.0832i	1.52333	+	0.630984i	0.545074	+	0.838388i	$0.316501\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			10.2664i			0.378168i
$738$		0			0
$739$	−11.2262	+	27.1024i	−0.412962	+	0.996977i	0.571377	+	0.820688i	$0.306409\pi$
$739$	−11.2262	+	27.1024i	−0.412962	+	0.996977i	−0.984338	+	0.176290i	$0.943591\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−15.8059	+	15.8059i	−0.579863	+	0.579863i	−0.934865	−	0.355003i	$-0.884480\pi$
$743$	−15.8059	+	15.8059i	−0.579863	+	0.579863i	0.355003	+	0.934865i	$0.384480\pi$
$744$		0			0
$745$	−0.246666	−	0.246666i	−0.00903715	−	0.00903715i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	0.788918	+	0.326780i	0.0288264	+	0.0119403i
$750$		0			0
$751$	12.4955			0.455969			0.227984	−	0.973665i	$-0.426787\pi$
$751$	12.4955			0.455969			0.227984	+	0.973665i	$0.426787\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	0.730170	−	1.76279i	0.0265736	−	0.0641544i
$756$		0			0
$757$	30.0398	−	12.4429i	1.09182	−	0.452245i	0.237176	−	0.971467i	$-0.423778\pi$
$757$	30.0398	−	12.4429i	1.09182	−	0.452245i	0.854639	+	0.519222i	$0.173778\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	15.5032	+	15.5032i	0.561992	+	0.561992i	0.929873	−	0.367881i	$-0.119917\pi$
$761$	15.5032	+	15.5032i	0.561992	+	0.561992i	−0.367881	+	0.929873i	$0.619917\pi$
$762$		0			0
$763$	21.9285	−	9.08308i	0.793865	−	0.328830i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−10.6084			−0.383046
$768$		0			0
$769$	−7.08253			−0.255403			−0.127701	−	0.991813i	$-0.540760\pi$
$769$	−7.08253			−0.255403			−0.127701	+	0.991813i	$0.540760\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	30.3534	−	12.5728i	1.09173	−	0.452211i	0.237123	−	0.971480i	$-0.423796\pi$
$773$	30.3534	−	12.5728i	1.09173	−	0.452211i	0.854611	+	0.519268i	$0.173796\pi$
$774$		0			0
$775$	5.08838	+	5.08838i	0.182780	+	0.182780i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	1.10808	−	0.458981i	0.0397011	−	0.0164447i
$780$		0			0
$781$	9.17464	−	22.1495i	0.328294	−	0.792573i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−1.65394			−0.0590316
$786$		0			0
$787$	37.1276	+	15.3788i	1.32346	+	0.548194i	0.928782	−	0.370627i	$-0.120857\pi$
$787$	37.1276	+	15.3788i	1.32346	+	0.548194i	0.394675	+	0.918821i	$0.370857\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−18.0763	−	18.0763i	−0.642718	−	0.642718i
$792$		0			0
$793$	−32.8068	+	32.8068i	−1.16500	+	1.16500i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	1.87919	−	4.53677i	0.0665644	−	0.160701i	−0.887097	−	0.461584i	$-0.847281\pi$
$797$	1.87919	−	4.53677i	0.0665644	−	0.160701i	0.953661	+	0.300883i	$0.0972814\pi$
$798$		0			0
$799$		−	19.0396i		−	0.673572i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−68.5694	−	28.4024i	−2.41976	−	1.00230i
$804$		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:	\( N \)	\(=\)	\( 1152 = 2^{7} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1152.w (of order \(8\), degree \(4\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.0913223 - 0.0378270i) q^{5} +(3.05457 + 3.05457i) q^{7} +O(q^{10})\)	\(q+(0.0913223 - 0.0378270i) q^{5} +(3.05457 + 3.05457i) q^{7} +(-5.25649 + 2.17731i) q^{11} +(-1.57202 + 3.79519i) q^{13} +2.56471 q^{17} +(-2.64058 - 1.09376i) q^{19} +(-4.03079 - 4.03079i) q^{23} +(-3.52863 + 3.52863i) q^{25} +(-2.06984 + 4.99704i) q^{29} -1.44203i q^{31} +(0.394496 + 0.163406i) q^{35} +(-2.07758 - 5.01573i) q^{37} +(-0.296726 + 0.296726i) q^{41} +(2.72382 + 6.57588i) q^{43} -7.42367i q^{47} +11.6608i q^{49} +(-1.53235 - 3.69941i) q^{53} +(-0.397674 + 0.397674i) q^{55} +(0.988255 + 2.38586i) q^{59} +(10.4346 + 4.32215i) q^{61} +0.406051i q^{65} +(0.690522 - 1.66707i) q^{67} +(-2.97957 + 2.97957i) q^{71} +(9.22401 + 9.22401i) q^{73} +(-22.7071 - 9.40558i) q^{77} +1.12181 q^{79} +(-4.13529 + 9.98348i) q^{83} +(0.234216 - 0.0970152i) q^{85} +(12.0590 + 12.0590i) q^{89} +(-16.3945 + 6.79084i) q^{91} -0.282518 q^{95} -18.5545 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q+O(q^{10}) \) 32 * q	\( 32 q - 8 q^{11} + 16 q^{29} + 24 q^{35} + 16 q^{53} + 32 q^{55} + 32 q^{59} + 32 q^{61} + 16 q^{67} + 16 q^{71} + 16 q^{77} + 32 q^{79} - 40 q^{83} + 48 q^{91} - 80 q^{95}+O(q^{100}) \) 32 * q - 8 * q^11 + 16 * q^29 + 24 * q^35 + 16 * q^53 + 32 * q^55 + 32 * q^59 + 32 * q^61 + 16 * q^67 + 16 * q^71 + 16 * q^77 + 32 * q^79 - 40 * q^83 + 48 * q^91 - 80 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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