LMFDB - Embedded newform 1152.2.w.b.143.2

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			143.2
Character	$\chi$	$=$	1152.143
Dual form			1152.2.w.b.1007.2

$q$-expansion

$f(q)$	$=$	$q+(-1.12344 + 2.71222i) q^{5} +(-3.03150 + 3.03150i) q^{7} +O(q^{10})$	$q+(-1.12344 + 2.71222i) q^{5} +(-3.03150 + 3.03150i) q^{7} +(-0.616847 + 1.48920i) q^{11} +(3.35133 - 1.38816i) q^{13} +4.76709 q^{17} +(-1.13264 - 2.73444i) q^{19} +(-4.11192 + 4.11192i) q^{23} +(-2.55851 - 2.55851i) q^{25} +(-8.16664 + 3.38273i) q^{29} -6.16305i q^{31} +(-4.81640 - 11.6278i) q^{35} +(-9.38963 - 3.88931i) q^{37} +(-0.169917 - 0.169917i) q^{41} +(-7.57687 - 3.13844i) q^{43} +2.44049i q^{47} -11.3800i q^{49} +(1.99073 + 0.824586i) q^{53} +(-3.34605 - 3.34605i) q^{55} +(-1.21488 - 0.503218i) q^{59} +(1.04489 + 2.52258i) q^{61} +10.6491i q^{65} +(-3.91551 + 1.62186i) q^{67} +(5.28919 + 5.28919i) q^{71} +(-1.57482 + 1.57482i) q^{73} +(-2.64454 - 6.38449i) q^{77} +13.7711 q^{79} +(-3.31934 + 1.37492i) q^{83} +(-5.35554 + 12.9294i) q^{85} +(6.99010 - 6.99010i) q^{89} +(-5.95133 + 14.3678i) q^{91} +8.68886 q^{95} -13.5584 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times$.

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

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
$5$	−1.12344	+	2.71222i	−0.502418	+	1.21294i	0.445745	+	0.895160i	$0.352939\pi$
$5$	−1.12344	+	2.71222i	−0.502418	+	1.21294i	−0.948163	+	0.317784i	$0.897061\pi$
$6$		0			0
$7$	−3.03150	+	3.03150i	−1.14580	+	1.14580i	−0.158430	+	0.987370i	$0.550643\pi$
$7$	−3.03150	+	3.03150i	−1.14580	+	1.14580i	−0.987370	+	0.158430i	$0.949357\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−0.616847	+	1.48920i	−0.185986	+	0.449011i	−0.989180	−	0.146707i	$-0.953132\pi$
$11$	−0.616847	+	1.48920i	−0.185986	+	0.449011i	0.803194	+	0.595718i	$0.203132\pi$
$12$		0			0
$13$	3.35133	−	1.38816i	0.929490	−	0.385008i	0.134005	−	0.990981i	$-0.457216\pi$
$13$	3.35133	−	1.38816i	0.929490	−	0.385008i	0.795485	+	0.605973i	$0.207216\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	4.76709			1.15619			0.578094	−	0.815970i	$-0.303797\pi$
$17$	4.76709			1.15619			0.578094	+	0.815970i	$0.303797\pi$
$18$		0			0
$19$	−1.13264	−	2.73444i	−0.259846	−	0.627323i	0.739082	−	0.673615i	$-0.235259\pi$
$19$	−1.13264	−	2.73444i	−0.259846	−	0.627323i	−0.998928	+	0.0462921i	$0.985259\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−4.11192	+	4.11192i	−0.857395	+	0.857395i	−0.991031	−	0.133636i	$-0.957335\pi$
$23$	−4.11192	+	4.11192i	−0.857395	+	0.857395i	0.133636	+	0.991031i	$0.457335\pi$
$24$		0			0
$25$	−2.55851	−	2.55851i	−0.511702	−	0.511702i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−8.16664	+	3.38273i	−1.51651	+	0.628158i	−0.976888	−	0.213751i	$-0.931432\pi$
$29$	−8.16664	+	3.38273i	−1.51651	+	0.628158i	−0.539620	+	0.841909i	$0.681432\pi$
$30$		0			0
$31$		−	6.16305i		−	1.10692i	−0.832877	−	0.553458i	$-0.813308\pi$
$31$		−	6.16305i		−	1.10692i	0.832877	−	0.553458i	$-0.186692\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−4.81640	−	11.6278i	−0.814121	−	1.96546i
$36$		0			0
$37$	−9.38963	−	3.88931i	−1.54365	−	0.639399i	−0.561493	−	0.827482i	$-0.689773\pi$
$37$	−9.38963	−	3.88931i	−1.54365	−	0.639399i	−0.982153	+	0.188083i	$0.939773\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−0.169917	−	0.169917i	−0.0265365	−	0.0265365i	0.693714	−	0.720251i	$-0.255973\pi$
$41$	−0.169917	−	0.169917i	−0.0265365	−	0.0265365i	−0.720251	+	0.693714i	$0.755973\pi$
$42$		0			0
$43$	−7.57687	−	3.13844i	−1.15546	−	0.478608i	−0.279100	−	0.960262i	$-0.590036\pi$
$43$	−7.57687	−	3.13844i	−1.15546	−	0.478608i	−0.876361	+	0.481654i	$0.840036\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			2.44049i			0.355981i	0.984032	+	0.177991i	$0.0569597\pi$
$47$			2.44049i			0.355981i	−0.984032	+	0.177991i	$0.943040\pi$
$48$		0			0
$49$		−	11.3800i		−	1.62572i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	1.99073	+	0.824586i	0.273447	+	0.113266i	0.515193	−	0.857074i	$-0.327720\pi$
$53$	1.99073	+	0.824586i	0.273447	+	0.113266i	−0.241746	+	0.970340i	$0.577720\pi$
$54$		0			0
$55$	−3.34605	−	3.34605i	−0.451182	−	0.451182i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−1.21488	−	0.503218i	−0.158163	−	0.0655134i	0.302197	−	0.953245i	$-0.402280\pi$
$59$	−1.21488	−	0.503218i	−0.158163	−	0.0655134i	−0.460361	+	0.887732i	$0.652280\pi$
$60$		0			0
$61$	1.04489	+	2.52258i	0.133784	+	0.322983i	0.976548	−	0.215300i	$-0.0690731\pi$
$61$	1.04489	+	2.52258i	0.133784	+	0.322983i	−0.842764	+	0.538283i	$0.819073\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			10.6491i			1.32085i
$66$		0			0
$67$	−3.91551	+	1.62186i	−0.478355	+	0.198141i	−0.608815	−	0.793312i	$-0.708355\pi$
$67$	−3.91551	+	1.62186i	−0.478355	+	0.198141i	0.130459	+	0.991454i	$0.458355\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	5.28919	+	5.28919i	0.627711	+	0.627711i	0.947492	−	0.319781i	$-0.103609\pi$
$71$	5.28919	+	5.28919i	0.627711	+	0.627711i	−0.319781	+	0.947492i	$0.603609\pi$
$72$		0			0
$73$	−1.57482	+	1.57482i	−0.184319	+	0.184319i	−0.793235	−	0.608916i	$-0.791605\pi$
$73$	−1.57482	+	1.57482i	−0.184319	+	0.184319i	0.608916	+	0.793235i	$0.291605\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−2.64454	−	6.38449i	−0.301373	−	0.727580i
$78$		0			0
$79$	13.7711			1.54937			0.774684	−	0.632348i	$-0.217909\pi$
$79$	13.7711			1.54937			0.774684	+	0.632348i	$0.217909\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−3.31934	+	1.37492i	−0.364345	+	0.150917i	−0.557343	−	0.830282i	$-0.688179\pi$
$83$	−3.31934	+	1.37492i	−0.364345	+	0.150917i	0.192997	+	0.981199i	$0.438179\pi$
$84$		0			0
$85$	−5.35554	+	12.9294i	−0.580890	+	1.40239i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	6.99010	−	6.99010i	0.740949	−	0.740949i	−0.231811	−	0.972761i	$-0.574465\pi$
$89$	6.99010	−	6.99010i	0.740949	−	0.740949i	0.972761	+	0.231811i	$0.0744652\pi$
$90$		0			0
$91$	−5.95133	+	14.3678i	−0.623869	+	1.50615i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	8.68886			0.891459
$96$		0			0
$97$	−13.5584			−1.37665			−0.688323	−	0.725404i	$-0.741653\pi$
$97$	−13.5584			−1.37665			−0.688323	+	0.725404i	$0.741653\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−1.10479	+	2.66720i	−0.109931	+	0.265396i	−0.969265	−	0.246018i	$-0.920878\pi$
$101$	−1.10479	+	2.66720i	−0.109931	+	0.265396i	0.859335	+	0.511413i	$0.170878\pi$
$102$		0			0
$103$	12.8009	−	12.8009i	1.26131	−	1.26131i	0.310851	−	0.950459i	$-0.399386\pi$
$103$	12.8009	−	12.8009i	1.26131	−	1.26131i	0.950459	−	0.310851i	$-0.100614\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	0.720677	−	1.73987i	0.0696704	−	0.168199i	−0.885209	−	0.465194i	$-0.845984\pi$
$107$	0.720677	−	1.73987i	0.0696704	−	0.168199i	0.954879	+	0.296995i	$0.0959845\pi$
$108$		0			0
$109$	4.45017	−	1.84332i	0.426249	−	0.176558i	−0.159238	−	0.987240i	$-0.550904\pi$
$109$	4.45017	−	1.84332i	0.426249	−	0.176558i	0.585486	+	0.810682i	$0.300904\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−13.6612			−1.28514			−0.642568	−	0.766229i	$-0.722131\pi$
$113$	−13.6612			−1.28514			−0.642568	+	0.766229i	$0.722131\pi$
$114$		0			0
$115$	−6.53296	−	15.7720i	−0.609201	−	1.47074i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−14.4514	+	14.4514i	−1.32476	+	1.32476i
$120$		0			0
$121$	5.94096	+	5.94096i	0.540087	+	0.540087i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−3.74754	+	1.55228i	−0.335190	+	0.138840i
$126$		0			0
$127$			13.2014i			1.17143i	0.810515	+	0.585717i	$0.199187\pi$
$127$			13.2014i			1.17143i	−0.810515	+	0.585717i	$0.800813\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	2.56553	+	6.19375i	0.224152	+	0.541150i	0.995446	−	0.0953287i	$-0.0303902\pi$
$131$	2.56553	+	6.19375i	0.224152	+	0.541150i	−0.771294	+	0.636479i	$0.780390\pi$
$132$		0			0
$133$	11.7231	+	4.85585i	1.01652	+	0.421056i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	9.77333	+	9.77333i	0.834992	+	0.834992i	0.988195	−	0.153203i	$-0.0489587\pi$
$137$	9.77333	+	9.77333i	0.834992	+	0.834992i	−0.153203	+	0.988195i	$0.548959\pi$
$138$		0			0
$139$	8.26616	+	3.42396i	0.701127	+	0.290416i	0.704627	−	0.709578i	$-0.251114\pi$
$139$	8.26616	+	3.42396i	0.701127	+	0.290416i	−0.00350050	+	0.999994i	$0.501114\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			5.84708i			0.488957i
$144$		0			0
$145$		−	25.9501i		−	2.15504i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−18.9803	−	7.86190i	−1.55493	−	0.644072i	−0.570729	−	0.821139i	$-0.693339\pi$
$149$	−18.9803	−	7.86190i	−1.55493	−	0.644072i	−0.984199	+	0.177066i	$0.943339\pi$
$150$		0			0
$151$	−7.83326	−	7.83326i	−0.637462	−	0.637462i	0.312467	−	0.949929i	$-0.398845\pi$
$151$	−7.83326	−	7.83326i	−0.637462	−	0.637462i	−0.949929	+	0.312467i	$0.898845\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	16.7156	+	6.92381i	1.34263	+	0.556134i
$156$		0			0
$157$	3.58014	+	8.64323i	0.285727	+	0.689805i	0.999949	−	0.0101140i	$-0.00321944\pi$
$157$	3.58014	+	8.64323i	0.285727	+	0.689805i	−0.714222	+	0.699919i	$0.753219\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	24.9306i		−	1.96481i
$162$		0			0
$163$	4.73693	−	1.96210i	0.371025	−	0.153684i	−0.189377	−	0.981904i	$-0.560647\pi$
$163$	4.73693	−	1.96210i	0.371025	−	0.153684i	0.560402	+	0.828221i	$0.310647\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	4.55199	+	4.55199i	0.352244	+	0.352244i	0.860944	−	0.508700i	$-0.169874\pi$
$167$	4.55199	+	4.55199i	0.352244	+	0.352244i	−0.508700	+	0.860944i	$0.669874\pi$
$168$		0			0
$169$	0.111994	−	0.111994i	0.00861490	−	0.00861490i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−4.28444	−	10.3436i	−0.325740	−	0.786406i	−0.998899	−	0.0469083i	$-0.985063\pi$
$173$	−4.28444	−	10.3436i	−0.325740	−	0.786406i	0.673159	−	0.739498i	$-0.264937\pi$
$174$		0			0
$175$	15.5123			1.17262
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−10.2246	+	4.23519i	−0.764226	+	0.316553i	−0.730531	−	0.682880i	$-0.760727\pi$
$179$	−10.2246	+	4.23519i	−0.764226	+	0.316553i	−0.0336946	+	0.999432i	$0.510727\pi$
$180$		0			0
$181$	−1.33358	+	3.21956i	−0.0991246	+	0.239308i	−0.965661	−	0.259806i	$-0.916341\pi$
$181$	−1.33358	+	3.21956i	−0.0991246	+	0.239308i	0.866536	+	0.499114i	$0.166341\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	21.0974	−	21.0974i	1.55111	−	1.55111i
$186$		0			0
$187$	−2.94056	+	7.09915i	−0.215035	+	0.519141i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−4.82446			−0.349085			−0.174543	−	0.984650i	$-0.555845\pi$
$191$	−4.82446			−0.349085			−0.174543	+	0.984650i	$0.555845\pi$
$192$		0			0
$193$	0.222878			0.0160431			0.00802155	−	0.999968i	$-0.497447\pi$
$193$	0.222878			0.0160431			0.00802155	+	0.999968i	$0.497447\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−9.44160	+	22.7940i	−0.672686	+	1.62401i	0.104341	+	0.994542i	$0.466727\pi$
$197$	−9.44160	+	22.7940i	−0.672686	+	1.62401i	−0.777028	+	0.629467i	$0.783273\pi$
$198$		0			0
$199$	−14.8710	+	14.8710i	−1.05418	+	1.05418i	−0.0557310	+	0.998446i	$0.517749\pi$
$199$	−14.8710	+	14.8710i	−1.05418	+	1.05418i	−0.998446	+	0.0557310i	$0.982251\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	14.5024	−	35.0120i	1.01787	−	2.45736i
$204$		0			0
$205$	0.651743	−	0.269961i	0.0455197	−	0.0188549i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	4.77079			0.330003
$210$		0			0
$211$	0.644000	+	1.55475i	0.0443348	+	0.107034i	0.944496	−	0.328524i	$-0.106551\pi$
$211$	0.644000	+	1.55475i	0.0443348	+	0.107034i	−0.900161	+	0.435558i	$0.856551\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	17.0243	−	17.0243i	1.16105	−	1.16105i
$216$		0			0
$217$	18.6833	+	18.6833i	1.26830	+	1.26830i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	15.9761	−	6.61750i	1.07467	−	0.445141i
$222$		0			0
$223$			23.4280i			1.56885i	0.620221	+	0.784427i	$0.287043\pi$
$223$			23.4280i			1.56885i	−0.620221	+	0.784427i	$0.712957\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−5.10690	−	12.3292i	−0.338957	−	0.818315i	−0.997816	−	0.0660486i	$-0.978961\pi$
$227$	−5.10690	−	12.3292i	−0.338957	−	0.818315i	0.658859	−	0.752266i	$-0.271039\pi$
$228$		0			0
$229$	−7.82666	−	3.24191i	−0.517200	−	0.214231i	0.108786	−	0.994065i	$-0.465304\pi$
$229$	−7.82666	−	3.24191i	−0.517200	−	0.214231i	−0.625987	+	0.779834i	$0.715304\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	9.66805	+	9.66805i	0.633375	+	0.633375i	0.948913	−	0.315538i	$-0.102185\pi$
$233$	9.66805	+	9.66805i	0.633375	+	0.633375i	−0.315538	+	0.948913i	$0.602185\pi$
$234$		0			0
$235$	−6.61915	−	2.74174i	−0.431785	−	0.178851i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	25.8579i		−	1.67261i	−0.548268	−	0.836303i	$-0.684713\pi$
$239$		−	25.8579i		−	1.67261i	0.548268	−	0.836303i	$-0.315287\pi$
$240$		0			0
$241$			24.9358i			1.60626i	0.595805	+	0.803129i	$0.296833\pi$
$241$			24.9358i			1.60626i	−0.595805	+	0.803129i	$0.703167\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	30.8652	+	12.7848i	1.97190	+	0.816789i
$246$		0			0
$247$	−7.59170	−	7.59170i	−0.483048	−	0.483048i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−11.5486	−	4.78358i	−0.728939	−	0.301937i	−0.0128233	−	0.999918i	$-0.504082\pi$
$251$	−11.5486	−	4.78358i	−0.728939	−	0.301937i	−0.716116	+	0.697981i	$0.754082\pi$
$252$		0			0
$253$	−3.58705	−	8.65990i	−0.225516	−	0.544443i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			17.0780i			1.06530i	0.846337	+	0.532648i	$0.178803\pi$
$257$			17.0780i			1.06530i	−0.846337	+	0.532648i	$0.821197\pi$
$258$		0			0
$259$	40.2552	−	16.6742i	2.50133	−	1.03609i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	10.4595	+	10.4595i	0.644963	+	0.644963i	0.951771	−	0.306809i	$-0.0992612\pi$
$263$	10.4595	+	10.4595i	0.644963	+	0.644963i	−0.306809	+	0.951771i	$0.599261\pi$
$264$		0			0
$265$	−4.47292	+	4.47292i	−0.274770	+	0.274770i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−3.47587	−	8.39150i	−0.211928	−	0.511639i	0.781792	−	0.623540i	$-0.214306\pi$
$269$	−3.47587	−	8.39150i	−0.211928	−	0.511639i	−0.993719	+	0.111901i	$0.964306\pi$
$270$		0			0
$271$	5.66173			0.343925			0.171963	−	0.985103i	$-0.444989\pi$
$271$	5.66173			0.343925			0.171963	+	0.985103i	$0.444989\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	5.38834	−	2.23192i	0.324929	−	0.134590i
$276$		0			0
$277$	−5.27625	+	12.7380i	−0.317019	+	0.765352i	0.682390	+	0.730988i	$0.260940\pi$
$277$	−5.27625	+	12.7380i	−0.317019	+	0.765352i	−0.999409	+	0.0343637i	$0.989060\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−5.75892	+	5.75892i	−0.343548	+	0.343548i	−0.857700	−	0.514151i	$-0.828107\pi$
$281$	−5.75892	+	5.75892i	−0.343548	+	0.343548i	0.514151	+	0.857700i	$0.328107\pi$
$282$		0			0
$283$	−10.4136	+	25.1407i	−0.619025	+	1.49446i	0.233812	+	0.972282i	$0.424880\pi$
$283$	−10.4136	+	25.1407i	−0.619025	+	1.49446i	−0.852837	+	0.522177i	$0.825120\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	1.03021			0.0608111
$288$		0			0
$289$	5.72512			0.336772
$290$		0			0
$291$		0			0
$292$		0			0
$293$	5.21345	−	12.5864i	0.304573	−	0.735304i	−0.695290	−	0.718729i	$-0.744724\pi$
$293$	5.21345	−	12.5864i	0.304573	−	0.735304i	0.999863	−	0.0165743i	$-0.00527600\pi$
$294$		0			0
$295$	2.72968	−	2.72968i	0.158928	−	0.158928i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−8.07236	+	19.4884i	−0.466837	+	1.12704i
$300$		0			0
$301$	32.4835	−	13.4551i	1.87232	−	0.775539i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−8.01566			−0.458975
$306$		0			0
$307$	8.98052	+	21.6809i	0.512545	+	1.23739i	0.942398	+	0.334495i	$0.108566\pi$
$307$	8.98052	+	21.6809i	0.512545	+	1.23739i	−0.429852	+	0.902899i	$0.641434\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−12.7177	+	12.7177i	−0.721157	+	0.721157i	−0.968841	−	0.247684i	$-0.920330\pi$
$311$	−12.7177	+	12.7177i	−0.721157	+	0.721157i	0.247684	+	0.968841i	$0.420330\pi$
$312$		0			0
$313$	10.1388	+	10.1388i	0.573081	+	0.573081i	0.932988	−	0.359907i	$-0.117192\pi$
$313$	10.1388	+	10.1388i	0.573081	+	0.573081i	−0.359907	+	0.932988i	$0.617192\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	2.52668	−	1.04659i	0.141913	−	0.0587822i	−0.310597	−	0.950542i	$-0.600529\pi$
$317$	2.52668	−	1.04659i	0.141913	−	0.0587822i	0.452509	+	0.891760i	$0.350529\pi$
$318$		0			0
$319$		−	14.2484i		−	0.797757i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−5.39940	−	13.0353i	−0.300431	−	0.725304i
$324$		0			0
$325$	−12.1260	−	5.02277i	−0.672631	−	0.278613i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−7.39834	−	7.39834i	−0.407884	−	0.407884i
$330$		0			0
$331$	5.49794	+	2.27732i	0.302194	+	0.125173i	0.528628	−	0.848854i	$-0.322707\pi$
$331$	5.49794	+	2.27732i	0.302194	+	0.125173i	−0.226434	+	0.974027i	$0.572707\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	12.4418i		−	0.679768i
$336$		0			0
$337$		−	5.85273i		−	0.318818i	−0.987213	−	0.159409i	$-0.949041\pi$
$337$		−	5.85273i		−	0.318818i	0.987213	−	0.159409i	$-0.0509589\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	9.17801	+	3.80166i	0.497017	+	0.205871i
$342$		0			0
$343$	13.2780	+	13.2780i	0.716946	+	0.716946i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	7.23994	+	2.99888i	0.388660	+	0.160988i	0.568452	−	0.822716i	$-0.307542\pi$
$347$	7.23994	+	2.99888i	0.388660	+	0.160988i	−0.179792	+	0.983705i	$0.557542\pi$
$348$		0			0
$349$	−4.01973	−	9.70449i	−0.215171	−	0.519469i	0.779032	−	0.626984i	$-0.215711\pi$
$349$	−4.01973	−	9.70449i	−0.215171	−	0.519469i	−0.994204	+	0.107515i	$0.965711\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			16.6657i			0.887026i	0.896268	+	0.443513i	$0.146268\pi$
$353$			16.6657i			0.887026i	−0.896268	+	0.443513i	$0.853732\pi$
$354$		0			0
$355$	−20.2875	+	8.40338i	−1.07675	+	0.446005i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−9.54649	−	9.54649i	−0.503845	−	0.503845i	0.408786	−	0.912630i	$-0.365952\pi$
$359$	−9.54649	−	9.54649i	−0.503845	−	0.503845i	−0.912630	+	0.408786i	$0.865952\pi$
$360$		0			0
$361$	7.24075	−	7.24075i	0.381092	−	0.381092i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−2.50205	−	6.04049i	−0.130963	−	0.316173i
$366$		0			0
$367$	6.31072			0.329417			0.164708	−	0.986342i	$-0.447332\pi$
$367$	6.31072			0.329417			0.164708	+	0.986342i	$0.447332\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−8.53463	+	3.53516i	−0.443096	+	0.183536i
$372$		0			0
$373$	−3.06259	+	7.39375i	−0.158575	+	0.382834i	−0.983120	−	0.182963i	$-0.941431\pi$
$373$	−3.06259	+	7.39375i	−0.158575	+	0.382834i	0.824545	+	0.565797i	$0.191431\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−22.6733	+	22.6733i	−1.16773	+	1.16773i
$378$		0			0
$379$	8.31838	−	20.0823i	0.427286	−	1.03156i	−0.552858	−	0.833275i	$-0.686463\pi$
$379$	8.31838	−	20.0823i	0.427286	−	1.03156i	0.980144	−	0.198285i	$-0.0635372\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	7.95308			0.406383			0.203192	−	0.979139i	$-0.434869\pi$
$383$	7.95308			0.406383			0.203192	+	0.979139i	$0.434869\pi$
$384$		0			0
$385$	20.2871			1.03393
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−2.38215	+	5.75102i	−0.120780	+	0.291588i	−0.972693	−	0.232094i	$-0.925442\pi$
$389$	−2.38215	+	5.75102i	−0.120780	+	0.291588i	0.851913	+	0.523683i	$0.175442\pi$
$390$		0			0
$391$	−19.6019	+	19.6019i	−0.991310	+	0.991310i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−15.4710	+	37.3503i	−0.778430	+	1.87930i
$396$		0			0
$397$	25.3118	−	10.4845i	1.27036	−	0.526202i	0.357288	−	0.933994i	$-0.383701\pi$
$397$	25.3118	−	10.4845i	1.27036	−	0.526202i	0.913075	+	0.407792i	$0.133701\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−1.81017			−0.0903954			−0.0451977	−	0.998978i	$-0.514392\pi$
$401$	−1.81017			−0.0903954			−0.0451977	+	0.998978i	$0.514392\pi$
$402$		0			0
$403$	−8.55532	−	20.6544i	−0.426171	−	1.02887i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	11.5839	−	11.5839i	0.574194	−	0.574194i
$408$		0			0
$409$	−14.4331	−	14.4331i	−0.713672	−	0.713672i	0.253630	−	0.967301i	$-0.418376\pi$
$409$	−14.4331	−	14.4331i	−0.713672	−	0.713672i	−0.967301	+	0.253630i	$0.918376\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	5.20841	−	2.15739i	0.256289	−	0.106158i
$414$		0			0
$415$		−	10.5474i		−	0.517754i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	1.33902	+	3.23269i	0.0654155	+	0.157927i	0.953207	−	0.302320i	$-0.0977610\pi$
$419$	1.33902	+	3.23269i	0.0654155	+	0.157927i	−0.887791	+	0.460247i	$0.847761\pi$
$420$		0			0
$421$	−10.5557	−	4.37232i	−0.514454	−	0.213094i	0.110325	−	0.993896i	$-0.464811\pi$
$421$	−10.5557	−	4.37232i	−0.514454	−	0.213094i	−0.624779	+	0.780802i	$0.714811\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−12.1966	−	12.1966i	−0.591624	−	0.591624i
$426$		0			0
$427$	−10.8148	−	4.47963i	−0.523364	−	0.216784i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	3.31967i		−	0.159903i	−0.996799	−	0.0799513i	$-0.974524\pi$
$431$		−	3.31967i		−	0.159903i	0.996799	−	0.0799513i	$-0.0254765\pi$
$432$		0			0
$433$			4.53298i			0.217841i	0.994050	+	0.108921i	$0.0347394\pi$
$433$			4.53298i			0.217841i	−0.994050	+	0.108921i	$0.965261\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	15.9011	+	6.58646i	0.760654	+	0.315073i
$438$		0			0
$439$	−20.9703	−	20.9703i	−1.00086	−	1.00086i	−1.00000		0.000858061i	$-0.999727\pi$
$439$	−20.9703	−	20.9703i	−1.00086	−	1.00086i	−0.000858061	−	1.00000i	$-0.500273\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	31.9708	+	13.2427i	1.51898	+	0.629181i	0.977386	−	0.211462i	$-0.0678225\pi$
$443$	31.9708	+	13.2427i	1.51898	+	0.629181i	0.541590	+	0.840643i	$0.317822\pi$
$444$		0			0
$445$	11.1058	+	26.8117i	0.526464	+	1.27100i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	21.0485i		−	0.993340i	−0.867940	−	0.496670i	$-0.834556\pi$
$449$		−	21.0485i		−	0.993340i	0.867940	−	0.496670i	$-0.165444\pi$
$450$		0			0
$451$	0.357852	−	0.148227i	0.0168506	−	0.00697975i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−32.2827	−	32.2827i	−1.51344	−	1.51344i
$456$		0			0
$457$	−4.62656	+	4.62656i	−0.216421	+	0.216421i	−0.806988	−	0.590567i	$-0.798904\pi$
$457$	−4.62656	+	4.62656i	−0.216421	+	0.216421i	0.590567	+	0.806988i	$0.298904\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	4.84388	+	11.6942i	0.225602	+	0.544652i	0.995633	−	0.0933550i	$-0.0297592\pi$
$461$	4.84388	+	11.6942i	0.225602	+	0.544652i	−0.770031	+	0.638007i	$0.779759\pi$
$462$		0			0
$463$	−14.2219			−0.660950			−0.330475	−	0.943815i	$-0.607209\pi$
$463$	−14.2219			−0.660950			−0.330475	+	0.943815i	$0.607209\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	34.5241	−	14.3003i	1.59758	−	0.661741i	0.606512	−	0.795074i	$-0.292568\pi$
$467$	34.5241	−	14.3003i	1.59758	−	0.661741i	0.991071	+	0.133333i	$0.0425681\pi$
$468$		0			0
$469$	6.95321	−	16.7865i	0.321069	−	0.775130i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	9.34754	−	9.34754i	0.429800	−	0.429800i
$474$		0			0
$475$	−4.09821	+	9.89396i	−0.188039	+	0.453966i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−13.4016			−0.612333			−0.306167	−	0.951978i	$-0.599046\pi$
$479$	−13.4016			−0.612333			−0.306167	+	0.951978i	$0.599046\pi$
$480$		0			0
$481$	−36.8667			−1.68098
$482$		0			0
$483$		0			0
$484$		0			0
$485$	15.2320	−	36.7734i	0.691652	−	1.66979i
$486$		0			0
$487$	−1.26750	+	1.26750i	−0.0574358	+	0.0574358i	−0.735241	−	0.677805i	$-0.762931\pi$
$487$	−1.26750	+	1.26750i	−0.0574358	+	0.0574358i	0.677805	+	0.735241i	$0.262931\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−8.81902	+	21.2910i	−0.397997	+	0.960849i	0.590144	+	0.807298i	$0.299071\pi$
$491$	−8.81902	+	21.2910i	−0.397997	+	0.960849i	−0.988141	+	0.153551i	$0.950929\pi$
$492$		0			0
$493$	−38.9311	+	16.1258i	−1.75337	+	0.726269i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−32.0684			−1.43846
$498$		0			0
$499$	2.76574	+	6.67709i	0.123812	+	0.298908i	0.973617	−	0.228189i	$-0.0732805\pi$
$499$	2.76574	+	6.67709i	0.123812	+	0.298908i	−0.849805	+	0.527097i	$0.823281\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−17.1789	+	17.1789i	−0.765968	+	0.765968i	−0.977394	−	0.211426i	$-0.932189\pi$
$503$	−17.1789	+	17.1789i	−0.765968	+	0.765968i	0.211426	+	0.977394i	$0.432189\pi$
$504$		0			0
$505$	−5.99287	−	5.99287i	−0.266679	−	0.266679i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	0.432418	−	0.179113i	0.0191666	−	0.00793905i	−0.373080	−	0.927799i	$-0.621698\pi$
$509$	0.432418	−	0.179113i	0.0191666	−	0.00793905i	0.392246	+	0.919860i	$0.371698\pi$
$510$		0			0
$511$		−	9.54815i		−	0.422385i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	20.3379	+	49.0999i	0.896193	+	2.16360i
$516$		0			0
$517$	−3.63437	−	1.50541i	−0.159839	−	0.0662077i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−20.6495	−	20.6495i	−0.904670	−	0.904670i	0.0911655	−	0.995836i	$-0.470941\pi$
$521$	−20.6495	−	20.6495i	−0.904670	−	0.904670i	−0.995836	+	0.0911655i	$0.970941\pi$
$522$		0			0
$523$	40.6453	+	16.8358i	1.77729	+	0.736180i	0.993321	+	0.115386i	$0.0368104\pi$
$523$	40.6453	+	16.8358i	1.77729	+	0.736180i	0.783974	+	0.620794i	$0.213190\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	29.3798i		−	1.27980i
$528$		0			0
$529$		−	10.8158i		−	0.470252i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−0.805318	−	0.333574i	−0.0348822	−	0.0144487i
$534$		0			0
$535$	3.90927	+	3.90927i	0.169013	+	0.169013i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	16.9471	+	7.01973i	0.729964	+	0.302361i
$540$		0			0
$541$	−7.26862	−	17.5480i	−0.312502	−	0.754447i	−0.999611	−	0.0278925i	$-0.991120\pi$
$541$	−7.26862	−	17.5480i	−0.312502	−	0.754447i	0.687109	−	0.726555i	$-0.258880\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			14.1407i			0.605721i
$546$		0			0
$547$	5.22407	−	2.16388i	0.223365	−	0.0925208i	−0.268195	−	0.963365i	$-0.586427\pi$
$547$	5.22407	−	2.16388i	0.223365	−	0.0925208i	0.491560	+	0.870844i	$0.336427\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	18.4998	+	18.4998i	0.788116	+	0.788116i
$552$		0			0
$553$	−41.7471	+	41.7471i	−1.77527	+	1.77527i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	7.20051	+	17.3836i	0.305095	+	0.736565i	0.999850	+	0.0173184i	$0.00551288\pi$
$557$	7.20051	+	17.3836i	0.305095	+	0.736565i	−0.694755	+	0.719247i	$0.744487\pi$
$558$		0			0
$559$	−29.7492			−1.25826
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−43.4445	+	17.9953i	−1.83097	+	0.758411i	−0.864007	+	0.503479i	$0.832053\pi$
$563$	−43.4445	+	17.9953i	−1.83097	+	0.758411i	−0.966959	+	0.254932i	$0.917947\pi$
$564$		0			0
$565$	15.3475	−	37.0522i	0.645675	−	1.55880i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−5.66061	+	5.66061i	−0.237305	+	0.237305i	−0.815733	−	0.578428i	$-0.803666\pi$
$569$	−5.66061	+	5.66061i	−0.237305	+	0.237305i	0.578428	+	0.815733i	$0.303666\pi$
$570$		0			0
$571$	13.7365	−	33.1629i	0.574856	−	1.38783i	−0.322522	−	0.946562i	$-0.604531\pi$
$571$	13.7365	−	33.1629i	0.574856	−	1.38783i	0.897378	−	0.441263i	$-0.145469\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	21.0408			0.877461
$576$		0			0
$577$	4.38583			0.182584			0.0912922	−	0.995824i	$-0.470900\pi$
$577$	4.38583			0.182584			0.0912922	+	0.995824i	$0.470900\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	5.89454	−	14.2307i	0.244547	−	0.590388i
$582$		0			0
$583$	−2.45595	+	2.45595i	−0.101715	+	0.101715i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	4.24918	−	10.2584i	0.175382	−	0.423411i	−0.811605	−	0.584206i	$-0.801406\pi$
$587$	4.24918	−	10.2584i	0.175382	−	0.423411i	0.986988	+	0.160795i	$0.0514060\pi$
$588$		0			0
$589$	−16.8525	+	6.98052i	−0.694394	+	0.287627i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	5.76763			0.236848			0.118424	−	0.992963i	$-0.462216\pi$
$593$	5.76763			0.236848			0.118424	+	0.992963i	$0.462216\pi$
$594$		0			0
$595$	−22.9602	−	55.4309i	−0.941277	−	2.27244i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	13.4401	−	13.4401i	0.549149	−	0.549149i	−0.377045	−	0.926195i	$-0.623060\pi$
$599$	13.4401	−	13.4401i	0.549149	−	0.549149i	0.926195	+	0.377045i	$0.123060\pi$
$600$		0			0
$601$	8.58677	+	8.58677i	0.350262	+	0.350262i	0.860207	−	0.509945i	$-0.170334\pi$
$601$	8.58677	+	8.58677i	0.350262	+	0.350262i	−0.509945	+	0.860207i	$0.670334\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−22.7875	+	9.43890i	−0.926444	+	0.383746i
$606$		0			0
$607$		−	13.3743i		−	0.542847i	−0.962460	−	0.271424i	$-0.912506\pi$
$607$		−	13.3743i		−	0.542847i	0.962460	−	0.271424i	$-0.0874944\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	3.38780	+	8.17886i	0.137056	+	0.330881i
$612$		0			0
$613$	−10.6749	−	4.42168i	−0.431154	−	0.178590i	0.156542	−	0.987671i	$-0.449965\pi$
$613$	−10.6749	−	4.42168i	−0.431154	−	0.178590i	−0.587697	+	0.809081i	$0.699965\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−14.8713	−	14.8713i	−0.598695	−	0.598695i	0.341270	−	0.939965i	$-0.389143\pi$
$617$	−14.8713	−	14.8713i	−0.598695	−	0.598695i	−0.939965	+	0.341270i	$0.889143\pi$
$618$		0			0
$619$	6.25062	+	2.58909i	0.251234	+	0.104064i	0.504746	−	0.863268i	$-0.331586\pi$
$619$	6.25062	+	2.58909i	0.251234	+	0.104064i	−0.253513	+	0.967332i	$0.581586\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			42.3810i			1.69796i
$624$		0			0
$625$		−	29.9995i		−	1.19998i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−44.7612	−	18.5407i	−1.78475	−	0.739266i
$630$		0			0
$631$	5.78346	+	5.78346i	0.230236	+	0.230236i	0.812791	−	0.582555i	$-0.197947\pi$
$631$	5.78346	+	5.78346i	0.230236	+	0.230236i	−0.582555	+	0.812791i	$0.697947\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−35.8052	−	14.8310i	−1.42088	−	0.588550i
$636$		0			0
$637$	−15.7973	−	38.1381i	−0.625913	−	1.51109i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			29.5203i			1.16598i	0.812479	+	0.582990i	$0.198117\pi$
$641$			29.5203i			1.16598i	−0.812479	+	0.582990i	$0.801883\pi$
$642$		0			0
$643$	−10.0660	+	4.16947i	−0.396964	+	0.164428i	−0.572230	−	0.820093i	$-0.693921\pi$
$643$	−10.0660	+	4.16947i	−0.396964	+	0.164428i	0.175266	+	0.984521i	$0.443921\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−31.9256	−	31.9256i	−1.25512	−	1.25512i	−0.953393	−	0.301730i	$-0.902436\pi$
$647$	−31.9256	−	31.9256i	−1.25512	−	1.25512i	−0.301730	−	0.953393i	$-0.597564\pi$
$648$		0			0
$649$	1.49878	−	1.49878i	0.0588324	−	0.0588324i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	6.05626	+	14.6211i	0.237000	+	0.572168i	0.996970	−	0.0777900i	$-0.0247864\pi$
$653$	6.05626	+	14.6211i	0.237000	+	0.572168i	−0.759970	+	0.649958i	$0.774786\pi$
$654$		0			0
$655$	−19.6811			−0.769002
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−28.2612	+	11.7062i	−1.10090	+	0.456008i	−0.857795	−	0.513992i	$-0.828166\pi$
$659$	−28.2612	+	11.7062i	−1.10090	+	0.456008i	−0.243106	+	0.970000i	$0.578166\pi$
$660$		0			0
$661$	6.64573	−	16.0442i	0.258489	−	0.624047i	−0.740350	−	0.672221i	$-0.765340\pi$
$661$	6.64573	−	16.0442i	0.258489	−	0.624047i	0.998839	+	0.0481743i	$0.0153403\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−26.3403	+	26.3403i	−1.02143	+	1.02143i
$666$		0			0
$667$	19.6711	−	47.4901i	0.761666	−	1.83883i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−4.40116			−0.169905
$672$		0			0
$673$	34.2884			1.32172			0.660860	−	0.750509i	$-0.270192\pi$
$673$	34.2884			1.32172			0.660860	+	0.750509i	$0.270192\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−16.6560	+	40.2112i	−0.640142	+	1.54544i	0.186345	+	0.982484i	$0.440336\pi$
$677$	−16.6560	+	40.2112i	−0.640142	+	1.54544i	−0.826487	+	0.562956i	$0.809664\pi$
$678$		0			0
$679$	41.1023	−	41.1023i	1.57736	−	1.57736i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	5.35596	−	12.9304i	0.204940	−	0.494769i	−0.787673	−	0.616094i	$-0.788714\pi$
$683$	5.35596	−	12.9304i	0.204940	−	0.494769i	0.992613	+	0.121325i	$0.0387142\pi$
$684$		0			0
$685$	−37.4872	+	15.5277i	−1.43231	+	0.593284i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	7.81623			0.297775
$690$		0			0
$691$	19.9998	+	48.2837i	0.760826	+	1.83680i	0.480898	+	0.876777i	$0.340311\pi$
$691$	19.9998	+	48.2837i	0.760826	+	1.83680i	0.279929	+	0.960021i	$0.409689\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−18.5731	+	18.5731i	−0.704517	+	0.704517i
$696$		0			0
$697$	−0.810008	−	0.810008i	−0.0306812	−	0.0306812i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	22.6661	−	9.38859i	0.856085	−	0.354602i	0.0889098	−	0.996040i	$-0.471662\pi$
$701$	22.6661	−	9.38859i	0.856085	−	0.354602i	0.767175	+	0.641438i	$0.221662\pi$
$702$		0			0
$703$			30.0806i			1.13451i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−4.73644	−	11.4348i	−0.178132	−	0.430049i
$708$		0			0
$709$	−6.91323	−	2.86355i	−0.259632	−	0.107543i	0.249071	−	0.968485i	$-0.419875\pi$
$709$	−6.91323	−	2.86355i	−0.259632	−	0.107543i	−0.508703	+	0.860942i	$0.669875\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	25.3420	+	25.3420i	0.949064	+	0.949064i
$714$		0			0
$715$	−15.8586	−	6.56884i	−0.593078	−	0.245661i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			32.1395i			1.19860i	0.800525	+	0.599300i	$0.204554\pi$
$719$			32.1395i			1.19860i	−0.800525	+	0.599300i	$0.795446\pi$
$720$		0			0
$721$			77.6119i			2.89042i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	29.5492	+	12.2397i	1.09743	+	0.454570i
$726$		0			0
$727$	−24.0313	−	24.0313i	−0.891271	−	0.891271i	0.103372	−	0.994643i	$-0.467037\pi$
$727$	−24.0313	−	24.0313i	−0.891271	−	0.891271i	−0.994643	+	0.103372i	$0.967037\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−36.1196	−	14.9612i	−1.33593	−	0.553361i
$732$		0			0
$733$	−3.28110	−	7.92127i	−0.121190	−	0.292579i	0.851629	−	0.524145i	$-0.175615\pi$
$733$	−3.28110	−	7.92127i	−0.121190	−	0.292579i	−0.972819	+	0.231566i	$0.925615\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	6.83141i		−	0.251638i
$738$		0			0
$739$	−4.01435	+	1.66280i	−0.147670	+	0.0611671i	−0.455295	−	0.890341i	$-0.650466\pi$
$739$	−4.01435	+	1.66280i	−0.147670	+	0.0611671i	0.307624	+	0.951508i	$0.400466\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	20.6253	+	20.6253i	0.756669	+	0.756669i	0.975715	−	0.219046i	$-0.0702944\pi$
$743$	20.6253	+	20.6253i	0.756669	+	0.756669i	−0.219046	+	0.975715i	$0.570294\pi$
$744$		0			0
$745$	42.6465	−	42.6465i	1.56245	−	1.56245i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	3.08968	+	7.45915i	0.112894	+	0.272551i
$750$		0			0
$751$	−13.6475			−0.498004			−0.249002	−	0.968503i	$-0.580103\pi$
$751$	−13.6475			−0.498004			−0.249002	+	0.968503i	$0.580103\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	30.0458	−	12.4454i	1.09348	−	0.452933i
$756$		0			0
$757$	1.50145	−	3.62481i	0.0545710	−	0.131746i	−0.894243	−	0.447582i	$-0.852285\pi$
$757$	1.50145	−	3.62481i	0.0545710	−	0.131746i	0.948814	+	0.315836i	$0.102285\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−10.5824	+	10.5824i	−0.383612	+	0.383612i	−0.872402	−	0.488790i	$-0.837438\pi$
$761$	−10.5824	+	10.5824i	−0.383612	+	0.383612i	0.488790	+	0.872402i	$0.337438\pi$
$762$		0			0
$763$	−7.90266	+	19.0787i	−0.286096	+	0.690696i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−4.76999			−0.172234
$768$		0			0
$769$	−42.6343			−1.53743			−0.768715	−	0.639591i	$-0.779104\pi$
$769$	−42.6343			−1.53743			−0.768715	+	0.639591i	$0.779104\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	13.5374	−	32.6822i	0.486906	−	1.17550i	−0.469362	−	0.883006i	$-0.655516\pi$
$773$	13.5374	−	32.6822i	0.486906	−	1.17550i	0.956268	−	0.292490i	$-0.0944839\pi$
$774$		0			0
$775$	−15.7682	+	15.7682i	−0.566411	+	0.566411i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−0.272172	+	0.657081i	−0.00975157	+	0.0235424i
$780$		0			0
$781$	−11.1393	+	4.61404i	−0.398595	+	0.165103i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−27.4645			−0.980249
$786$		0			0
$787$	11.1547	+	26.9298i	0.397622	+	0.959945i	0.988228	+	0.152985i	$0.0488887\pi$
$787$	11.1547	+	26.9298i	0.397622	+	0.959945i	−0.590606	+	0.806960i	$0.701111\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	41.4139	−	41.4139i	1.47251	−	1.47251i
$792$		0			0
$793$	7.00350	+	7.00350i	0.248702	+	0.248702i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	3.44606	−	1.42740i	0.122066	−	0.0505612i	−0.320815	−	0.947142i	$-0.603957\pi$
$797$	3.44606	−	1.42740i	0.122066	−	0.0505612i	0.442880	+	0.896581i	$0.353957\pi$
$798$		0			0
$799$			11.6340i			0.411582i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−1.37380	−	3.31665i	−0.0484804	−	0.117042i
$804$		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)

\(f(q)\)	\(=\)	\(q+(-1.12344 + 2.71222i) q^{5} +(-3.03150 + 3.03150i) q^{7} +O(q^{10})\)	\(q+(-1.12344 + 2.71222i) q^{5} +(-3.03150 + 3.03150i) q^{7} +(-0.616847 + 1.48920i) q^{11} +(3.35133 - 1.38816i) q^{13} +4.76709 q^{17} +(-1.13264 - 2.73444i) q^{19} +(-4.11192 + 4.11192i) q^{23} +(-2.55851 - 2.55851i) q^{25} +(-8.16664 + 3.38273i) q^{29} -6.16305i q^{31} +(-4.81640 - 11.6278i) q^{35} +(-9.38963 - 3.88931i) q^{37} +(-0.169917 - 0.169917i) q^{41} +(-7.57687 - 3.13844i) q^{43} +2.44049i q^{47} -11.3800i q^{49} +(1.99073 + 0.824586i) q^{53} +(-3.34605 - 3.34605i) q^{55} +(-1.21488 - 0.503218i) q^{59} +(1.04489 + 2.52258i) q^{61} +10.6491i q^{65} +(-3.91551 + 1.62186i) q^{67} +(5.28919 + 5.28919i) q^{71} +(-1.57482 + 1.57482i) q^{73} +(-2.64454 - 6.38449i) q^{77} +13.7711 q^{79} +(-3.31934 + 1.37492i) q^{83} +(-5.35554 + 12.9294i) q^{85} +(6.99010 - 6.99010i) q^{89} +(-5.95133 + 14.3678i) q^{91} +8.68886 q^{95} -13.5584 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q+O(q^{10}) \) 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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