LMFDB - Embedded newform 252.2.b.d.55.1

Newspace parameters

Level:	$ N $	$=$	$ 252 = 2^{2} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	252.b (of order $2$, degree $1$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$2.01223013094$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.0.2312.1
Defining polynomial:	$x^{4} - x^{3} - 2 x + 4$
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 84)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			55.1
Root			$-0.780776 - 1.17915i$ of defining polynomial
Character	$\chi$	$=$	252.55
Dual form			252.2.b.d.55.2

$q$-expansion

$f(q)$	$=$	$q+(-0.780776 - 1.17915i) q^{2} +(-0.780776 + 1.84130i) q^{4} +1.69614i q^{5} +(-2.56155 - 0.662153i) q^{7} +(2.78078 - 0.516994i) q^{8} +O(q^{10})$	\(q+(-0.780776 - 1.17915i) q^{2} +(-0.780776 + 1.84130i) q^{4} +1.69614i q^{5} +(-2.56155 - 0.662153i) q^{7} +(2.78078 - 0.516994i) q^{8} +(2.00000 - 1.32431i) q^{10} +3.02045i q^{11} +6.04090i q^{13} +(1.21922 + 3.53744i) q^{14} +(-2.78078 - 2.87529i) q^{16} +4.34475i q^{17} +1.12311 q^{19} +(-3.12311 - 1.32431i) q^{20} +(3.56155 - 2.35829i) q^{22} -3.02045i q^{23} +2.12311 q^{25} +(7.12311 - 4.71659i) q^{26} +(3.21922 - 4.19960i) q^{28} +2.00000 q^{29} +(-1.21922 + 5.52390i) q^{32} +(5.12311 - 3.39228i) q^{34} +(1.12311 - 4.34475i) q^{35} -7.12311 q^{37} +(-0.876894 - 1.32431i) q^{38} +(0.876894 + 4.71659i) q^{40} -7.73704i q^{41} +8.10887i q^{43} +(-5.56155 - 2.35829i) q^{44} +(-3.56155 + 2.35829i) q^{46} -10.2462 q^{47} +(6.12311 + 3.39228i) q^{49} +(-1.65767 - 2.50345i) q^{50} +(-11.1231 - 4.71659i) q^{52} +4.24621 q^{53} -5.12311 q^{55} +(-7.46543 - 0.516994i) q^{56} +(-1.56155 - 2.35829i) q^{58} +4.00000 q^{59} -9.43318i q^{61} +(7.46543 - 2.87529i) q^{64} -10.2462 q^{65} -2.06798i q^{67} +(-8.00000 - 3.39228i) q^{68} +(-6.00000 + 2.06798i) q^{70} +12.4536i q^{71} -3.39228i q^{73} +(5.56155 + 8.39919i) q^{74} +(-0.876894 + 2.06798i) q^{76} +(2.00000 - 7.73704i) q^{77} -4.71659i q^{79} +(4.87689 - 4.71659i) q^{80} +(-9.12311 + 6.04090i) q^{82} +6.24621 q^{83} -7.36932 q^{85} +(9.56155 - 6.33122i) q^{86} +(1.56155 + 8.39919i) q^{88} +7.73704i q^{89} +(4.00000 - 15.4741i) q^{91} +(5.56155 + 2.35829i) q^{92} +(8.00000 + 12.0818i) q^{94} +1.90495i q^{95} -8.68951i q^{97} +(-0.780776 - 9.86866i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{2} + q^{4} - 2 q^{7} + 7 q^{8} + O(q^{10}) $	$ 4 q + q^{2} + q^{4} - 2 q^{7} + 7 q^{8} + 8 q^{10} + 9 q^{14} - 7 q^{16} - 12 q^{19} + 4 q^{20} + 6 q^{22} - 8 q^{25} + 12 q^{26} + 17 q^{28} + 8 q^{29} - 9 q^{32} + 4 q^{34} - 12 q^{35} - 12 q^{37} - 20 q^{38} + 20 q^{40} - 14 q^{44} - 6 q^{46} - 8 q^{47} + 8 q^{49} - 19 q^{50} - 28 q^{52} - 16 q^{53} - 4 q^{55} - q^{56} + 2 q^{58} + 16 q^{59} + q^{64} - 8 q^{65} - 32 q^{68} - 24 q^{70} + 14 q^{74} - 20 q^{76} + 8 q^{77} + 36 q^{80} - 20 q^{82} - 8 q^{83} + 20 q^{85} + 30 q^{86} - 2 q^{88} + 16 q^{91} + 14 q^{92} + 32 q^{94} + q^{98} + O(q^{100}) $

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/252\mathbb{Z}\right)^\times$.

$n$	$29$	$73$	$127$
$\chi(n)$	$1$	$-1$	$-1$

Coefficient data

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

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

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

$2$	−0.780776	−	1.17915i	−0.552092	−	0.833783i
$2$	−0.780776	−	1.17915i	−0.552092	−	0.833783i
$3$		0			0
$3$		0			0
$4$	−0.780776	+	1.84130i	−0.390388	+	0.920650i
$5$			1.69614i			0.758537i	0.925287	+	0.379269i	$0.123824\pi$
$5$			1.69614i			0.758537i	−0.925287	+	0.379269i	$0.876176\pi$
$6$		0			0
$7$	−2.56155	−	0.662153i	−0.968176	−	0.250270i
$7$	−2.56155	−	0.662153i	−0.968176	−	0.250270i
$8$	2.78078	−	0.516994i	0.983153	−	0.182785i
$9$		0			0
$10$	2.00000	−	1.32431i	0.632456	−	0.418783i
$11$			3.02045i			0.910699i	0.890313	+	0.455350i	$0.150486\pi$
$11$			3.02045i			0.910699i	−0.890313	+	0.455350i	$0.849514\pi$
$12$		0			0
$13$			6.04090i			1.67544i	0.546098	+	0.837722i	$0.316113\pi$
$13$			6.04090i			1.67544i	−0.546098	+	0.837722i	$0.683887\pi$
$14$	1.21922	+	3.53744i	0.325851	+	0.945421i
$15$		0			0
$16$	−2.78078	−	2.87529i	−0.695194	−	0.718822i
$17$			4.34475i			1.05376i	0.849940	+	0.526879i	$0.176638\pi$
$17$			4.34475i			1.05376i	−0.849940	+	0.526879i	$0.823362\pi$
$18$		0			0
$19$	1.12311			0.257658			0.128829	−	0.991667i	$-0.458878\pi$
$19$	1.12311			0.257658			0.128829	+	0.991667i	$0.458878\pi$
$20$	−3.12311	−	1.32431i	−0.698348	−	0.296124i
$21$		0			0
$22$	3.56155	−	2.35829i	0.759326	−	0.502790i
$23$		−	3.02045i		−	0.629807i	−0.949124	−	0.314903i	$-0.898028\pi$
$23$		−	3.02045i		−	0.629807i	0.949124	−	0.314903i	$-0.101972\pi$
$24$		0			0
$25$	2.12311			0.424621
$26$	7.12311	−	4.71659i	1.39696	−	0.924999i
$27$		0			0
$28$	3.21922	−	4.19960i	0.608376	−	0.793649i
$29$	2.00000			0.371391			0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000			0.371391			0.185695	+	0.982607i	$0.440546\pi$
$30$		0			0
$31$		0			0			−	1.00000i	$-0.5\pi$
$31$		0			0				1.00000i	$0.5\pi$
$32$	−1.21922	+	5.52390i	−0.215530	+	0.976497i
$33$		0			0
$34$	5.12311	−	3.39228i	0.878605	−	0.581772i
$35$	1.12311	−	4.34475i	0.189839	−	0.734398i
$36$		0			0
$37$	−7.12311			−1.17103			−0.585516	−	0.810661i	$-0.699108\pi$
$37$	−7.12311			−1.17103			−0.585516	+	0.810661i	$0.699108\pi$
$38$	−0.876894	−	1.32431i	−0.142251	−	0.214831i
$39$		0			0
$40$	0.876894	+	4.71659i	0.138649	+	0.745758i
$41$		−	7.73704i		−	1.20832i	−0.796862	−	0.604161i	$-0.793508\pi$
$41$		−	7.73704i		−	1.20832i	0.796862	−	0.604161i	$-0.206492\pi$
$42$		0			0
$43$			8.10887i			1.23659i	0.785946	+	0.618296i	$0.212177\pi$
$43$			8.10887i			1.23659i	−0.785946	+	0.618296i	$0.787823\pi$
$44$	−5.56155	−	2.35829i	−0.838436	−	0.355526i
$45$		0			0
$46$	−3.56155	+	2.35829i	−0.525122	+	0.347712i
$47$	−10.2462			−1.49456			−0.747282	−	0.664507i	$-0.768641\pi$
$47$	−10.2462			−1.49456			−0.747282	+	0.664507i	$0.768641\pi$
$48$		0			0
$49$	6.12311	+	3.39228i	0.874729	+	0.484612i
$50$	−1.65767	−	2.50345i	−0.234430	−	0.354042i
$51$		0			0
$52$	−11.1231	−	4.71659i	−1.54250	−	0.654073i
$53$	4.24621			0.583262			0.291631	−	0.956531i	$-0.405802\pi$
$53$	4.24621			0.583262			0.291631	+	0.956531i	$0.405802\pi$
$54$		0			0
$55$	−5.12311			−0.690799
$56$	−7.46543	−	0.516994i	−0.997611	−	0.0690862i
$57$		0			0
$58$	−1.56155	−	2.35829i	−0.205042	−	0.309659i
$59$	4.00000			0.520756			0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000			0.520756			0.260378	+	0.965507i	$0.416153\pi$
$60$		0			0
$61$		−	9.43318i		−	1.20779i	−0.797062	−	0.603897i	$-0.793614\pi$
$61$		−	9.43318i		−	1.20779i	0.797062	−	0.603897i	$-0.206386\pi$
$62$		0			0
$63$		0			0
$64$	7.46543	−	2.87529i	0.933179	−	0.359411i
$65$	−10.2462			−1.27089
$66$		0			0
$67$		−	2.06798i		−	0.252643i	−0.991989	−	0.126322i	$-0.959683\pi$
$67$		−	2.06798i		−	0.252643i	0.991989	−	0.126322i	$-0.0403172\pi$
$68$	−8.00000	−	3.39228i	−0.970143	−	0.411375i
$69$		0			0
$70$	−6.00000	+	2.06798i	−0.717137	+	0.247170i
$71$			12.4536i			1.47797i	0.673720	+	0.738987i	$0.264695\pi$
$71$			12.4536i			1.47797i	−0.673720	+	0.738987i	$0.735305\pi$
$72$		0			0
$73$		−	3.39228i		−	0.397037i	−0.980097	−	0.198518i	$-0.936387\pi$
$73$		−	3.39228i		−	0.397037i	0.980097	−	0.198518i	$-0.0636129\pi$
$74$	5.56155	+	8.39919i	0.646517	+	0.976386i
$75$		0			0
$76$	−0.876894	+	2.06798i	−0.100587	+	0.237213i
$77$	2.00000	−	7.73704i	0.227921	−	0.881717i
$78$		0			0
$79$		−	4.71659i		−	0.530658i	−0.964158	−	0.265329i	$-0.914519\pi$
$79$		−	4.71659i		−	0.530658i	0.964158	−	0.265329i	$-0.0854805\pi$
$80$	4.87689	−	4.71659i	0.545253	−	0.527331i
$81$		0			0
$82$	−9.12311	+	6.04090i	−1.00748	+	0.667105i
$83$	6.24621			0.685611			0.342805	−	0.939406i	$-0.388623\pi$
$83$	6.24621			0.685611			0.342805	+	0.939406i	$0.388623\pi$
$84$		0			0
$85$	−7.36932			−0.799315
$86$	9.56155	−	6.33122i	1.03105	−	0.682712i
$87$		0			0
$88$	1.56155	+	8.39919i	0.166462	+	0.895357i
$89$			7.73704i			0.820124i	0.912058	+	0.410062i	$0.134493\pi$
$89$			7.73704i			0.820124i	−0.912058	+	0.410062i	$0.865507\pi$
$90$		0			0
$91$	4.00000	−	15.4741i	0.419314	−	1.62212i
$92$	5.56155	+	2.35829i	0.579832	+	0.245869i
$93$		0			0
$94$	8.00000	+	12.0818i	0.825137	+	1.24614i
$95$			1.90495i			0.195443i
$96$		0			0
$97$		−	8.68951i		−	0.882286i	−0.897437	−	0.441143i	$-0.854573\pi$
$97$		−	8.68951i		−	0.882286i	0.897437	−	0.441143i	$-0.145427\pi$
$98$	−0.780776	−	9.86866i	−0.0788703	−	0.996885i
$99$		0			0
$100$	−1.65767	+	3.90928i	−0.165767	+	0.390928i
$101$		−	6.99337i		−	0.695866i	−0.937519	−	0.347933i	$-0.886884\pi$
$101$		−	6.99337i		−	0.695866i	0.937519	−	0.347933i	$-0.113116\pi$
$102$		0			0
$103$	8.00000			0.788263			0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000			0.788263			0.394132	+	0.919054i	$0.371045\pi$
$104$	3.12311	+	16.7984i	0.306246	+	1.64722i
$105$		0			0
$106$	−3.31534	−	5.00691i	−0.322014	−	0.486314i
$107$		−	5.66906i		−	0.548049i	−0.961723	−	0.274024i	$-0.911645\pi$
$107$		−	5.66906i		−	0.548049i	0.961723	−	0.274024i	$-0.0883549\pi$
$108$		0			0
$109$	8.24621			0.789844			0.394922	−	0.918715i	$-0.370772\pi$
$109$	8.24621			0.789844			0.394922	+	0.918715i	$0.370772\pi$
$110$	4.00000	+	6.04090i	0.381385	+	0.575977i
$111$		0			0
$112$	5.21922	+	9.20650i	0.493170	+	0.869933i
$113$	−12.2462			−1.15203			−0.576013	−	0.817440i	$-0.695392\pi$
$113$	−12.2462			−1.15203			−0.576013	+	0.817440i	$0.695392\pi$
$114$		0			0
$115$	5.12311			0.477732
$116$	−1.56155	+	3.68260i	−0.144987	+	0.341921i
$117$		0			0
$118$	−3.12311	−	4.71659i	−0.287505	−	0.434197i
$119$	2.87689	−	11.1293i	0.263724	−	1.02022i
$120$		0			0
$121$	1.87689			0.170627
$122$	−11.1231	+	7.36520i	−1.00704	+	0.666814i
$123$		0			0
$124$		0			0
$125$			12.0818i			1.08063i
$126$		0			0
$127$			19.4470i			1.72564i	0.505510	+	0.862821i	$0.331304\pi$
$127$			19.4470i			1.72564i	−0.505510	+	0.862821i	$0.668696\pi$
$128$	−9.21922	−	6.55789i	−0.814872	−	0.579641i
$129$		0			0
$130$	8.00000	+	12.0818i	0.701646	+	1.05964i
$131$	22.2462			1.94366			0.971830	−	0.235682i	$-0.0757323\pi$
$131$	22.2462			1.94366			0.971830	+	0.235682i	$0.0757323\pi$
$132$		0			0
$133$	−2.87689	−	0.743668i	−0.249458	−	0.0644842i
$134$	−2.43845	+	1.61463i	−0.210650	+	0.139482i
$135$		0			0
$136$	2.24621	+	12.0818i	0.192611	+	1.03601i
$137$	16.2462			1.38801			0.694004	−	0.719971i	$-0.255845\pi$
$137$	16.2462			1.38801			0.694004	+	0.719971i	$0.255845\pi$
$138$		0			0
$139$	12.0000			1.01783			0.508913	−	0.860818i	$-0.330047\pi$
$139$	12.0000			1.01783			0.508913	+	0.860818i	$0.330047\pi$
$140$	7.12311	+	5.46026i	0.602012	+	0.461476i
$141$		0			0
$142$	14.6847	−	9.72350i	1.23231	−	0.815978i
$143$	−18.2462			−1.52582
$144$		0			0
$145$			3.39228i			0.281714i
$146$	−4.00000	+	2.64861i	−0.331042	+	0.219201i
$147$		0			0
$148$	5.56155	−	13.1158i	0.457157	−	1.07811i
$149$	10.0000			0.819232			0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000			0.819232			0.409616	+	0.912258i	$0.365663\pi$
$150$		0			0
$151$		−	8.10887i		−	0.659891i	−0.944000	−	0.329945i	$-0.892970\pi$
$151$		−	8.10887i		−	0.659891i	0.944000	−	0.329945i	$-0.107030\pi$
$152$	3.12311	−	0.580639i	0.253317	−	0.0470960i
$153$		0			0
$154$	−10.6847	+	3.68260i	−0.860994	+	0.296752i
$155$		0			0
$156$		0			0
$157$			4.13595i			0.330085i	0.986286	+	0.165042i	$0.0527761\pi$
$157$			4.13595i			0.330085i	−0.986286	+	0.165042i	$0.947224\pi$
$158$	−5.56155	+	3.68260i	−0.442453	+	0.292972i
$159$		0			0
$160$	−9.36932	−	2.06798i	−0.740710	−	0.163488i
$161$	−2.00000	+	7.73704i	−0.157622	+	0.609764i
$162$		0			0
$163$			11.5012i			0.900840i	0.892817	+	0.450420i	$0.148726\pi$
$163$			11.5012i			0.900840i	−0.892817	+	0.450420i	$0.851274\pi$
$164$	14.2462	+	6.04090i	1.11244	+	0.471715i
$165$		0			0
$166$	−4.87689	−	7.36520i	−0.378520	−	0.571651i
$167$	−2.24621			−0.173817			−0.0869085	−	0.996216i	$-0.527699\pi$
$167$	−2.24621			−0.173817			−0.0869085	+	0.996216i	$0.527699\pi$
$168$		0			0
$169$	−23.4924			−1.80711
$170$	5.75379	+	8.68951i	0.441295	+	0.666455i
$171$		0			0
$172$	−14.9309	−	6.33122i	−1.13847	−	0.482751i
$173$		−	8.48071i		−	0.644776i	−0.946608	−	0.322388i	$-0.895514\pi$
$173$		−	8.48071i		−	0.644776i	0.946608	−	0.322388i	$-0.104486\pi$
$174$		0			0
$175$	−5.43845	−	1.40582i	−0.411108	−	0.106270i
$176$	8.68466	−	8.39919i	0.654631	−	0.633113i
$177$		0			0
$178$	9.12311	−	6.04090i	0.683806	−	0.452784i
$179$		−	2.27678i		−	0.170175i	−0.996374	−	0.0850873i	$-0.972883\pi$
$179$		−	2.27678i		−	0.170175i	0.996374	−	0.0850873i	$-0.0271169\pi$
$180$		0			0
$181$		−	6.04090i		−	0.449016i	−0.974472	−	0.224508i	$-0.927922\pi$
$181$		−	6.04090i		−	0.449016i	0.974472	−	0.224508i	$-0.0720775\pi$
$182$	−21.3693	+	7.36520i	−1.58400	+	0.545945i
$183$		0			0
$184$	−1.56155	−	8.39919i	−0.115119	−	0.619197i
$185$		−	12.0818i		−	0.888271i
$186$		0			0
$187$	−13.1231			−0.959657
$188$	8.00000	−	18.8664i	0.583460	−	1.37597i
$189$		0			0
$190$	2.24621	−	1.48734i	0.162957	−	0.107903i
$191$			9.06134i			0.655656i	0.944737	+	0.327828i	$0.106317\pi$
$191$			9.06134i			0.655656i	−0.944737	+	0.327828i	$0.893683\pi$
$192$		0			0
$193$	9.36932			0.674418			0.337209	−	0.941430i	$-0.390517\pi$
$193$	9.36932			0.674418			0.337209	+	0.941430i	$0.390517\pi$
$194$	−10.2462	+	6.78456i	−0.735635	+	0.487103i
$195$		0			0
$196$	−11.0270	+	8.62586i	−0.787642	+	0.616133i
$197$	−0.246211			−0.0175418			−0.00877091	−	0.999962i	$-0.502792\pi$
$197$	−0.246211			−0.0175418			−0.00877091	+	0.999962i	$0.502792\pi$
$198$		0			0
$199$	−5.12311			−0.363167			−0.181584	−	0.983375i	$-0.558122\pi$
$199$	−5.12311			−0.363167			−0.181584	+	0.983375i	$0.558122\pi$
$200$	5.90388	−	1.09763i	0.417468	−	0.0776143i
$201$		0			0
$202$	−8.24621	+	5.46026i	−0.580201	+	0.384182i
$203$	−5.12311	−	1.32431i	−0.359572	−	0.0929481i
$204$		0			0
$205$	13.1231			0.916557
$206$	−6.24621	−	9.43318i	−0.435194	−	0.657241i
$207$		0			0
$208$	17.3693	−	16.7984i	1.20435	−	1.16476i
$209$			3.39228i			0.234649i
$210$		0			0
$211$		−	3.97292i		−	0.273507i	−0.990605	−	0.136754i	$-0.956333\pi$
$211$		−	3.97292i		−	0.273507i	0.990605	−	0.136754i	$-0.0436668\pi$
$212$	−3.31534	+	7.81855i	−0.227699	+	0.536980i
$213$		0			0
$214$	−6.68466	+	4.42627i	−0.456954	+	0.302574i
$215$	−13.7538			−0.938001
$216$		0			0
$217$		0			0
$218$	−6.43845	−	9.72350i	−0.436067	−	0.658558i
$219$		0			0
$220$	4.00000	−	9.43318i	0.269680	−	0.635985i
$221$	−26.2462			−1.76551
$222$		0			0
$223$	−18.8769			−1.26409			−0.632045	−	0.774932i	$-0.717784\pi$
$223$	−18.8769			−1.26409			−0.632045	+	0.774932i	$0.717784\pi$
$224$	6.78078	−	13.3425i	0.453060	−	0.891480i
$225$		0			0
$226$	9.56155	+	14.4401i	0.636025	+	0.960540i
$227$	16.4924			1.09464			0.547320	−	0.836923i	$-0.315648\pi$
$227$	16.4924			1.09464			0.547320	+	0.836923i	$0.315648\pi$
$228$		0			0
$229$			18.1227i			1.19758i	0.800906	+	0.598790i	$0.204352\pi$
$229$			18.1227i			1.19758i	−0.800906	+	0.598790i	$0.795648\pi$
$230$	−4.00000	−	6.04090i	−0.263752	−	0.398325i
$231$		0			0
$232$	5.56155	−	1.03399i	0.365134	−	0.0678846i
$233$	10.4924			0.687381			0.343691	−	0.939083i	$-0.388323\pi$
$233$	10.4924			0.687381			0.343691	+	0.939083i	$0.388323\pi$
$234$		0			0
$235$		−	17.3790i		−	1.13368i
$236$	−3.12311	+	7.36520i	−0.203297	+	0.479434i
$237$		0			0
$238$	−15.3693	+	5.29723i	−0.996245	+	0.343368i
$239$			2.27678i			0.147273i	0.997285	+	0.0736363i	$0.0234604\pi$
$239$			2.27678i			0.147273i	−0.997285	+	0.0736363i	$0.976540\pi$
$240$		0			0
$241$			1.90495i			0.122708i	0.998116	+	0.0613542i	$0.0195419\pi$
$241$			1.90495i			0.122708i	−0.998116	+	0.0613542i	$0.980458\pi$
$242$	−1.46543	−	2.21313i	−0.0942017	−	0.142266i
$243$		0			0
$244$	17.3693	+	7.36520i	1.11196	+	0.471509i
$245$	−5.75379	+	10.3857i	−0.367596	+	0.663515i
$246$		0			0
$247$			6.78456i			0.431691i
$248$		0			0
$249$		0			0
$250$	14.2462	−	9.43318i	0.901010	−	0.596607i
$251$	−22.2462			−1.40417			−0.702084	−	0.712094i	$-0.747747\pi$
$251$	−22.2462			−1.40417			−0.702084	+	0.712094i	$0.747747\pi$
$252$		0			0
$253$	9.12311			0.573565
$254$	22.9309	−	15.1838i	1.43881	−	0.952713i
$255$		0			0
$256$	−0.534565	+	15.9911i	−0.0334103	+	0.999442i
$257$		−	19.8188i		−	1.23626i	−0.786074	−	0.618132i	$-0.787890\pi$
$257$		−	19.8188i		−	1.23626i	0.786074	−	0.618132i	$-0.212110\pi$
$258$		0			0
$259$	18.2462	+	4.71659i	1.13376	+	0.293075i
$260$	8.00000	−	18.8664i	0.496139	−	1.17004i
$261$		0			0
$262$	−17.3693	−	26.2316i	−1.07308	−	1.62059i
$263$		−	4.92539i		−	0.303713i	−0.988403	−	0.151856i	$-0.951475\pi$
$263$		−	4.92539i		−	0.303713i	0.988403	−	0.151856i	$-0.0485251\pi$
$264$		0			0
$265$			7.20217i			0.442426i
$266$	1.36932	+	3.97292i	0.0839582	+	0.243595i
$267$		0			0
$268$	3.80776	+	1.61463i	0.232596	+	0.0986290i
$269$			22.4674i			1.36986i	0.728607	+	0.684932i	$0.240168\pi$
$269$			22.4674i			1.36986i	−0.728607	+	0.684932i	$0.759832\pi$
$270$		0			0
$271$	−4.49242			−0.272895			−0.136448	−	0.990647i	$-0.543569\pi$
$271$	−4.49242			−0.272895			−0.136448	+	0.990647i	$0.543569\pi$
$272$	12.4924	−	12.0818i	0.757464	−	0.732566i
$273$		0			0
$274$	−12.6847	−	19.1567i	−0.766308	−	1.15730i
$275$			6.41273i			0.386702i
$276$		0			0
$277$	3.12311			0.187649			0.0938246	−	0.995589i	$-0.470091\pi$
$277$	3.12311			0.187649			0.0938246	+	0.995589i	$0.470091\pi$
$278$	−9.36932	−	14.1498i	−0.561934	−	0.848647i
$279$		0			0
$280$	0.876894	−	12.6624i	0.0524045	−	0.756725i
$281$	0.246211			0.0146877			0.00734387	−	0.999973i	$-0.497662\pi$
$281$	0.246211			0.0146877			0.00734387	+	0.999973i	$0.497662\pi$
$282$		0			0
$283$	−17.1231			−1.01786			−0.508931	−	0.860807i	$-0.669959\pi$
$283$	−17.1231			−1.01786			−0.508931	+	0.860807i	$0.669959\pi$
$284$	−22.9309	−	9.72350i	−1.36070	−	0.576983i
$285$		0			0
$286$	14.2462	+	21.5150i	0.842396	+	1.27221i
$287$	−5.12311	+	19.8188i	−0.302407	+	1.16987i
$288$		0			0
$289$	−1.87689			−0.110406
$290$	4.00000	−	2.64861i	0.234888	−	0.155532i
$291$		0			0
$292$	6.24621	+	2.64861i	0.365532	+	0.154998i
$293$		−	6.99337i		−	0.408557i	−0.978913	−	0.204278i	$-0.934515\pi$
$293$		−	6.99337i		−	0.408557i	0.978913	−	0.204278i	$-0.0654848\pi$
$294$		0			0
$295$			6.78456i			0.395013i
$296$	−19.8078	+	3.68260i	−1.15130	+	0.214047i
$297$		0			0
$298$	−7.80776	−	11.7915i	−0.452292	−	0.683062i
$299$	18.2462			1.05521
$300$		0			0
$301$	5.36932	−	20.7713i	0.309482	−	1.19724i
$302$	−9.56155	+	6.33122i	−0.550206	+	0.364320i
$303$		0			0
$304$	−3.12311	−	3.22925i	−0.179122	−	0.185210i
$305$	16.0000			0.916157
$306$		0			0
$307$	21.6155			1.23366			0.616832	−	0.787095i	$-0.288416\pi$
$307$	21.6155			1.23366			0.616832	+	0.787095i	$0.288416\pi$
$308$	12.6847	+	9.72350i	0.722775	+	0.554048i
$309$		0			0
$310$		0			0
$311$	8.00000			0.453638			0.226819	−	0.973937i	$-0.427167\pi$
$311$	8.00000			0.453638			0.226819	+	0.973937i	$0.427167\pi$
$312$		0			0
$313$			25.6509i			1.44988i	0.688814	+	0.724938i	$0.258132\pi$
$313$			25.6509i			1.44988i	−0.688814	+	0.724938i	$0.741868\pi$
$314$	4.87689	−	3.22925i	0.275219	−	0.182237i
$315$		0			0
$316$	8.68466	+	3.68260i	0.488550	+	0.207163i
$317$	−18.4924			−1.03864			−0.519319	−	0.854580i	$-0.673814\pi$
$317$	−18.4924			−1.03864			−0.519319	+	0.854580i	$0.673814\pi$
$318$		0			0
$319$			6.04090i			0.338225i
$320$	4.87689	+	12.6624i	0.272627	+	0.707851i
$321$		0			0
$322$	10.6847	−	3.68260i	0.595433	−	0.205223i
$323$			4.87962i			0.271509i
$324$		0			0
$325$			12.8255i			0.711429i
$326$	13.5616	−	8.97983i	0.751105	−	0.497347i
$327$		0			0
$328$	−4.00000	−	21.5150i	−0.220863	−	1.18797i
$329$	26.2462	+	6.78456i	1.44700	+	0.374045i
$330$		0			0
$331$		−	5.46026i		−	0.300123i	−0.988677	−	0.150061i	$-0.952053\pi$
$331$		−	5.46026i		−	0.300123i	0.988677	−	0.150061i	$-0.0479472\pi$
$332$	−4.87689	+	11.5012i	−0.267654	+	0.631208i
$333$		0			0
$334$	1.75379	+	2.64861i	0.0959631	+	0.144926i
$335$	3.50758			0.191639
$336$		0			0
$337$	−8.24621			−0.449200			−0.224600	−	0.974451i	$-0.572108\pi$
$337$	−8.24621			−0.449200			−0.224600	+	0.974451i	$0.572108\pi$
$338$	18.3423	+	27.7010i	0.997691	+	1.50674i
$339$		0			0
$340$	5.75379	−	13.5691i	0.312043	−	0.735889i
$341$		0			0
$342$		0			0
$343$	−13.4384	−	12.7439i	−0.725608	−	0.688108i
$344$	4.19224	+	22.5490i	0.226030	+	1.21576i
$345$		0			0
$346$	−10.0000	+	6.62153i	−0.537603	+	0.355976i
$347$		−	34.7123i		−	1.86345i	−0.363162	−	0.931726i	$-0.618303\pi$
$347$		−	34.7123i		−	1.86345i	0.363162	−	0.931726i	$-0.381697\pi$
$348$		0			0
$349$			4.13595i			0.221392i	0.993854	+	0.110696i	$0.0353080\pi$
$349$			4.13595i			0.221392i	−0.993854	+	0.110696i	$0.964692\pi$
$350$	2.58854	+	7.51036i	0.138363	+	0.401446i
$351$		0			0
$352$	−16.6847	−	3.68260i	−0.889295	−	0.196283i
$353$			6.24970i			0.332638i	0.986072	+	0.166319i	$0.0531881\pi$
$353$			6.24970i			0.332638i	−0.986072	+	0.166319i	$0.946812\pi$
$354$		0			0
$355$	−21.1231			−1.12110
$356$	−14.2462	−	6.04090i	−0.755048	−	0.320167i
$357$		0			0
$358$	−2.68466	+	1.77766i	−0.141889	+	0.0939520i
$359$		−	1.11550i		−	0.0588740i	−0.999567	−	0.0294370i	$-0.990629\pi$
$359$		−	1.11550i		−	0.0588740i	0.999567	−	0.0294370i	$-0.00937144\pi$
$360$		0			0
$361$	−17.7386			−0.933612
$362$	−7.12311	+	4.71659i	−0.374382	+	0.247898i
$363$		0			0
$364$	25.3693	+	19.4470i	1.32971	+	1.01930i
$365$	5.75379			0.301167
$366$		0			0
$367$	8.63068			0.450518			0.225259	−	0.974299i	$-0.427677\pi$
$367$	8.63068			0.450518			0.225259	+	0.974299i	$0.427677\pi$
$368$	−8.68466	+	8.39919i	−0.452719	+	0.437838i
$369$		0			0
$370$	−14.2462	+	9.43318i	−0.740625	+	0.490408i
$371$	−10.8769	−	2.81164i	−0.564700	−	0.145973i
$372$		0			0
$373$	−10.0000			−0.517780			−0.258890	−	0.965907i	$-0.583357\pi$
$373$	−10.0000			−0.517780			−0.258890	+	0.965907i	$0.583357\pi$
$374$	10.2462	+	15.4741i	0.529819	+	0.800145i
$375$		0			0
$376$	−28.4924	+	5.29723i	−1.46938	+	0.273184i
$377$			12.0818i			0.622244i
$378$		0			0
$379$			18.7033i			0.960725i	0.877070	+	0.480363i	$0.159495\pi$
$379$			18.7033i			0.960725i	−0.877070	+	0.480363i	$0.840505\pi$
$380$	−3.50758	−	1.48734i	−0.179935	−	0.0762988i
$381$		0			0
$382$	10.6847	−	7.07488i	0.546675	−	0.361983i
$383$	−26.2462			−1.34112			−0.670559	−	0.741856i	$-0.733946\pi$
$383$	−26.2462			−1.34112			−0.670559	+	0.741856i	$0.733946\pi$
$384$		0			0
$385$	13.1231	+	3.39228i	0.668815	+	0.172887i
$386$	−7.31534	−	11.0478i	−0.372341	−	0.562318i
$387$		0			0
$388$	16.0000	+	6.78456i	0.812277	+	0.344434i
$389$	−0.246211			−0.0124834			−0.00624170	−	0.999981i	$-0.501987\pi$
$389$	−0.246211			−0.0124834			−0.00624170	+	0.999981i	$0.501987\pi$
$390$		0			0
$391$	13.1231			0.663664
$392$	18.7808	+	6.26757i	0.948572	+	0.316560i
$393$		0			0
$394$	0.192236	+	0.290319i	0.00968471	+	0.0146261i
$395$	8.00000			0.402524
$396$		0			0
$397$		−	16.2177i		−	0.813945i	−0.913440	−	0.406973i	$-0.866584\pi$
$397$		−	16.2177i		−	0.813945i	0.913440	−	0.406973i	$-0.133416\pi$
$398$	4.00000	+	6.04090i	0.200502	+	0.302803i
$399$		0			0
$400$	−5.90388	−	6.10454i	−0.295194	−	0.305227i
$401$	8.24621			0.411796			0.205898	−	0.978573i	$-0.433988\pi$
$401$	8.24621			0.411796			0.205898	+	0.978573i	$0.433988\pi$
$402$		0			0
$403$		0			0
$404$	12.8769	+	5.46026i	0.640649	+	0.271658i
$405$		0			0
$406$	2.43845	+	7.07488i	0.121018	+	0.351121i
$407$		−	21.5150i		−	1.06646i
$408$		0			0
$409$		−	27.5559i		−	1.36255i	−0.732028	−	0.681275i	$-0.761426\pi$
$409$		−	27.5559i		−	1.36255i	0.732028	−	0.681275i	$-0.238574\pi$
$410$	−10.2462	−	15.4741i	−0.506024	−	0.764210i
$411$		0			0
$412$	−6.24621	+	14.7304i	−0.307729	+	0.725715i
$413$	−10.2462	−	2.64861i	−0.504183	−	0.130330i
$414$		0			0
$415$			10.5945i			0.520061i
$416$	−33.3693	−	7.36520i	−1.63607	−	0.361109i
$417$		0			0
$418$	4.00000	−	2.64861i	0.195646	−	0.129548i
$419$	16.4924			0.805708			0.402854	−	0.915264i	$-0.368018\pi$
$419$	16.4924			0.805708			0.402854	+	0.915264i	$0.368018\pi$
$420$		0			0
$421$	19.1231			0.932003			0.466002	−	0.884784i	$-0.345694\pi$
$421$	19.1231			0.932003			0.466002	+	0.884784i	$0.345694\pi$
$422$	−4.68466	+	3.10196i	−0.228046	+	0.151001i
$423$		0			0
$424$	11.8078	−	2.19526i	0.573436	−	0.106611i
$425$			9.22437i			0.447448i
$426$		0			0
$427$	−6.24621	+	24.1636i	−0.302275	+	1.16936i
$428$	10.4384	+	4.42627i	0.504561	+	0.213952i
$429$		0			0
$430$	10.7386	+	16.2177i	0.517863	+	0.782089i
$431$		−	15.1022i		−	0.727449i	−0.931507	−	0.363725i	$-0.881505\pi$
$431$		−	15.1022i		−	0.727449i	0.931507	−	0.363725i	$-0.118495\pi$
$432$		0			0
$433$			6.78456i			0.326045i	0.986622	+	0.163023i	$0.0521244\pi$
$433$			6.78456i			0.326045i	−0.986622	+	0.163023i	$0.947876\pi$
$434$		0			0
$435$		0			0
$436$	−6.43845	+	15.1838i	−0.308346	+	0.727170i
$437$		−	3.39228i		−	0.162275i
$438$		0			0
$439$	31.3693			1.49718			0.748588	−	0.663036i	$-0.230732\pi$
$439$	31.3693			1.49718			0.748588	+	0.663036i	$0.230732\pi$
$440$	−14.2462	+	2.64861i	−0.679161	+	0.126268i
$441$		0			0
$442$	20.4924	+	30.9481i	0.974725	+	1.47205i
$443$			35.8735i			1.70440i	0.523213	+	0.852202i	$0.324733\pi$
$443$			35.8735i			1.70440i	−0.523213	+	0.852202i	$0.675267\pi$
$444$		0			0
$445$	−13.1231			−0.622095
$446$	14.7386	+	22.2586i	0.697895	+	1.05398i
$447$		0			0
$448$	−21.0270	+	2.42194i	−0.993432	+	0.114426i
$449$	−28.2462			−1.33302			−0.666511	−	0.745496i	$-0.732213\pi$
$449$	−28.2462			−1.33302			−0.666511	+	0.745496i	$0.732213\pi$
$450$		0			0
$451$	23.3693			1.10042
$452$	9.56155	−	22.5490i	0.449738	−	1.06061i
$453$		0			0
$454$	−12.8769	−	19.4470i	−0.604343	−	0.912693i
$455$	26.2462	+	6.78456i	1.23044	+	0.318065i
$456$		0			0
$457$	−16.2462			−0.759966			−0.379983	−	0.924994i	$-0.624070\pi$
$457$	−16.2462			−0.759966			−0.379983	+	0.924994i	$0.624070\pi$
$458$	21.3693	−	14.1498i	0.998523	−	0.661175i
$459$		0			0
$460$	−4.00000	+	9.43318i	−0.186501	+	0.439824i
$461$		−	17.1702i		−	0.799697i	−0.916581	−	0.399848i	$-0.869063\pi$
$461$		−	17.1702i		−	0.799697i	0.916581	−	0.399848i	$-0.130937\pi$
$462$		0			0
$463$		−	39.4746i		−	1.83454i	−0.398264	−	0.917271i	$-0.630387\pi$
$463$		−	39.4746i		−	1.83454i	0.398264	−	0.917271i	$-0.369613\pi$
$464$	−5.56155	−	5.75058i	−0.258189	−	0.266964i
$465$		0			0
$466$	−8.19224	−	12.3721i	−0.379498	−	0.573127i
$467$	17.7538			0.821547			0.410774	−	0.911737i	$-0.365259\pi$
$467$	17.7538			0.821547			0.410774	+	0.911737i	$0.365259\pi$
$468$		0			0
$469$	−1.36932	+	5.29723i	−0.0632292	+	0.244603i
$470$	−20.4924	+	13.5691i	−0.945245	+	0.625897i
$471$		0			0
$472$	11.1231	−	2.06798i	0.511982	−	0.0951863i
$473$	−24.4924			−1.12616
$474$		0			0
$475$	2.38447			0.109407
$476$	18.2462	+	13.9867i	0.836314	+	0.641081i
$477$		0			0
$478$	2.68466	−	1.77766i	0.122793	−	0.0813081i
$479$	−20.4924			−0.936323			−0.468161	−	0.883643i	$-0.655083\pi$
$479$	−20.4924			−0.936323			−0.468161	+	0.883643i	$0.655083\pi$
$480$		0			0
$481$		−	43.0299i		−	1.96200i
$482$	2.24621	−	1.48734i	0.102312	−	0.0677463i
$483$		0			0
$484$	−1.46543	+	3.45593i	−0.0666107	+	0.157088i
$485$	14.7386			0.669247
$486$		0			0
$487$		−	32.2725i		−	1.46240i	−0.682161	−	0.731202i	$-0.738960\pi$
$487$		−	32.2725i		−	1.46240i	0.682161	−	0.731202i	$-0.261040\pi$
$488$	−4.87689	−	26.2316i	−0.220767	−	1.18745i
$489$		0			0
$490$	16.7386	−	1.32431i	0.756174	−	0.0598261i
$491$			11.7100i			0.528463i	0.964459	+	0.264231i	$0.0851183\pi$
$491$			11.7100i			0.528463i	−0.964459	+	0.264231i	$0.914882\pi$
$492$		0			0
$493$			8.68951i			0.391356i
$494$	8.00000	−	5.29723i	0.359937	−	0.238334i
$495$		0			0
$496$		0			0
$497$	8.24621	−	31.9006i	0.369893	−	1.43094i
$498$		0			0
$499$		−	17.5420i		−	0.785290i	−0.919690	−	0.392645i	$-0.871560\pi$
$499$		−	17.5420i		−	0.785290i	0.919690	−	0.392645i	$-0.128440\pi$
$500$	−22.2462	−	9.43318i	−0.994881	−	0.421865i
$501$		0			0
$502$	17.3693	+	26.2316i	0.775231	+	1.17077i
$503$	22.7386			1.01387			0.506933	−	0.861986i	$-0.330779\pi$
$503$	22.7386			1.01387			0.506933	+	0.861986i	$0.330779\pi$
$504$		0			0
$505$	11.8617			0.527840
$506$	−7.12311	−	10.7575i	−0.316661	−	0.478229i
$507$		0			0
$508$	−35.8078	−	15.1838i	−1.58871	−	0.673670i
$509$		−	1.69614i		−	0.0751801i	−0.999293	−	0.0375901i	$-0.988032\pi$
$509$		−	1.69614i		−	0.0751801i	0.999293	−	0.0375901i	$-0.0119681\pi$
$510$		0			0
$511$	−2.24621	+	8.68951i	−0.0993665	+	0.384401i
$512$	19.2732	−	11.8551i	0.851763	−	0.523927i
$513$		0			0
$514$	−23.3693	+	15.4741i	−1.03078	+	0.682532i
$515$			13.5691i			0.597927i
$516$		0			0
$517$		−	30.9481i		−	1.36110i
$518$	−8.68466	−	25.1976i	−0.381582	−	1.10712i
$519$		0			0
$520$	−28.4924	+	5.29723i	−1.24948	+	0.232299i
$521$		−	33.8056i		−	1.48105i	−0.672030	−	0.740524i	$-0.734577\pi$
$521$		−	33.8056i		−	1.48105i	0.672030	−	0.740524i	$-0.265423\pi$
$522$		0			0
$523$	0.492423			0.0215321			0.0107661	−	0.999942i	$-0.496573\pi$
$523$	0.492423			0.0215321			0.0107661	+	0.999942i	$0.496573\pi$
$524$	−17.3693	+	40.9620i	−0.758782	+	1.78943i
$525$		0			0
$526$	−5.80776	+	3.84563i	−0.253231	+	0.167677i
$527$		0			0
$528$		0			0
$529$	13.8769			0.603343
$530$	8.49242	−	5.62329i	0.368887	−	0.244260i
$531$		0			0
$532$	3.61553	−	4.71659i	0.156753	−	0.204490i
$533$	46.7386			2.02447
$534$		0			0
$535$	9.61553			0.415716
$536$	−1.06913	−	5.75058i	−0.0461794	−	0.248387i
$537$		0			0
$538$	26.4924	−	17.5420i	1.14217	−	0.756291i
$539$	−10.2462	+	18.4945i	−0.441336	+	0.796615i
$540$		0			0
$541$	11.1231			0.478220			0.239110	−	0.970993i	$-0.423144\pi$
$541$	11.1231			0.478220			0.239110	+	0.970993i	$0.423144\pi$
$542$	3.50758	+	5.29723i	0.150663	+	0.227535i
$543$		0			0
$544$	−24.0000	−	5.29723i	−1.02899	−	0.227117i
$545$			13.9867i			0.599126i
$546$		0			0
$547$		−	33.0161i		−	1.41167i	−0.708378	−	0.705834i	$-0.750573\pi$
$547$		−	33.0161i		−	1.41167i	0.708378	−	0.705834i	$-0.249427\pi$
$548$	−12.6847	+	29.9142i	−0.541862	+	1.27787i
$549$		0			0
$550$	7.56155	−	5.00691i	0.322426	−	0.213495i
$551$	2.24621			0.0956918
$552$		0			0
$553$	−3.12311	+	12.0818i	−0.132808	+	0.513770i
$554$	−2.43845	−	3.68260i	−0.103600	−	0.156459i
$555$		0			0
$556$	−9.36932	+	22.0956i	−0.397348	+	0.937063i
$557$	−14.0000			−0.593199			−0.296600	−	0.955002i	$-0.595853\pi$
$557$	−14.0000			−0.593199			−0.296600	+	0.955002i	$0.595853\pi$
$558$		0			0
$559$	−48.9848			−2.07184
$560$	−15.6155	+	8.85254i	−0.659877	+	0.374088i
$561$		0			0
$562$	−0.192236	−	0.290319i	−0.00810898	−	0.0122464i
$563$	−14.2462			−0.600406			−0.300203	−	0.953875i	$-0.597054\pi$
$563$	−14.2462			−0.600406			−0.300203	+	0.953875i	$0.597054\pi$
$564$		0			0
$565$		−	20.7713i		−	0.873855i
$566$	13.3693	+	20.1907i	0.561954	+	0.848677i
$567$		0			0
$568$	6.43845	+	34.6307i	0.270151	+	1.45307i
$569$	−34.9848			−1.46664			−0.733320	−	0.679883i	$-0.762031\pi$
$569$	−34.9848			−1.46664			−0.733320	+	0.679883i	$0.762031\pi$
$570$		0			0
$571$			40.9620i			1.71420i	0.515146	+	0.857102i	$0.327738\pi$
$571$			40.9620i			1.71420i	−0.515146	+	0.857102i	$0.672262\pi$
$572$	14.2462	−	33.5968i	0.595664	−	1.40475i
$573$		0			0
$574$	27.3693	−	9.43318i	1.14237	−	0.393733i
$575$		−	6.41273i		−	0.267429i
$576$		0			0
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	1.46543	+	2.21313i	0.0609541	+	0.0920543i
$579$		0			0
$580$	−6.24621	−	2.64861i	−0.259360	−	0.109978i
$581$	−16.0000	−	4.13595i	−0.663792	−	0.171588i
$582$		0			0
$583$			12.8255i			0.531176i
$584$	−1.75379	−	9.43318i	−0.0725723	−	0.390348i
$585$		0			0
$586$	−8.24621	+	5.46026i	−0.340648	+	0.225561i
$587$	−38.2462			−1.57859			−0.789295	−	0.614014i	$-0.789554\pi$
$587$	−38.2462			−1.57859			−0.789295	+	0.614014i	$0.789554\pi$
$588$		0			0
$589$		0			0
$590$	8.00000	−	5.29723i	0.329355	−	0.218083i
$591$		0			0
$592$	19.8078	+	20.4810i	0.814094	+	0.841763i
$593$			21.7238i			0.892088i	0.895011	+	0.446044i	$0.147168\pi$
$593$			21.7238i			0.892088i	−0.895011	+	0.446044i	$0.852832\pi$
$594$		0			0
$595$	18.8769	+	4.87962i	0.773877	+	0.200045i
$596$	−7.80776	+	18.4130i	−0.319818	+	0.754226i
$597$		0			0
$598$	−14.2462	−	21.5150i	−0.582571	−	0.879813i
$599$			19.2382i			0.786051i	0.919528	+	0.393026i	$0.128572\pi$
$599$			19.2382i			0.786051i	−0.919528	+	0.393026i	$0.871428\pi$
$600$		0			0
$601$		−	5.29723i		−	0.216078i	−0.994147	−	0.108039i	$-0.965543\pi$
$601$		−	5.29723i		−	0.216078i	0.994147	−	0.108039i	$-0.0344572\pi$
$602$	−28.6847	+	9.88653i	−1.16910	+	0.402945i
$603$		0			0
$604$	14.9309	+	6.33122i	0.607528	+	0.257613i
$605$			3.18348i			0.129427i
$606$		0			0
$607$	33.6155			1.36441			0.682206	−	0.731160i	$-0.261021\pi$
$607$	33.6155			1.36441			0.682206	+	0.731160i	$0.261021\pi$
$608$	−1.36932	+	6.20393i	−0.0555331	+	0.251602i
$609$		0			0
$610$	−12.4924	−	18.8664i	−0.505803	−	0.763876i
$611$		−	61.8963i		−	2.50406i
$612$		0			0
$613$	−40.7386			−1.64542			−0.822709	−	0.568463i	$-0.807538\pi$
$613$	−40.7386			−1.64542			−0.822709	+	0.568463i	$0.807538\pi$
$614$	−16.8769	−	25.4879i	−0.681096	−	1.02861i
$615$		0			0
$616$	1.56155	−	22.5490i	0.0629168	−	0.908523i
$617$	−15.7538			−0.634224			−0.317112	−	0.948388i	$-0.602713\pi$
$617$	−15.7538			−0.634224			−0.317112	+	0.948388i	$0.602713\pi$
$618$		0			0
$619$	−20.0000			−0.803868			−0.401934	−	0.915669i	$-0.631662\pi$
$619$	−20.0000			−0.803868			−0.401934	+	0.915669i	$0.631662\pi$
$620$		0			0
$621$		0			0
$622$	−6.24621	−	9.43318i	−0.250450	−	0.378236i
$623$	5.12311	−	19.8188i	0.205253	−	0.794025i
$624$		0			0
$625$	−9.87689			−0.395076
$626$	30.2462	−	20.0276i	1.20888	−	0.800465i
$627$		0			0
$628$	−7.61553	−	3.22925i	−0.303893	−	0.128861i
$629$		−	30.9481i		−	1.23398i
$630$		0			0
$631$			17.9597i			0.714963i	0.933920	+	0.357481i	$0.116364\pi$
$631$			17.9597i			0.714963i	−0.933920	+	0.357481i	$0.883636\pi$
$632$	−2.43845	−	13.1158i	−0.0969962	−	0.521718i
$633$		0			0
$634$	14.4384	+	21.8053i	0.573424	+	0.865999i
$635$	−32.9848			−1.30896
$636$		0			0
$637$	−20.4924	+	36.9890i	−0.811939	+	1.46556i
$638$	7.12311	−	4.71659i	0.282006	−	0.186732i
$639$		0			0
$640$	11.1231	−	15.6371i	0.439679	−	0.618111i
$641$	9.50758			0.375527			0.187763	−	0.982214i	$-0.439876\pi$
$641$	9.50758			0.375527			0.187763	+	0.982214i	$0.439876\pi$
$642$		0			0
$643$	−29.6155			−1.16792			−0.583961	−	0.811782i	$-0.698498\pi$
$643$	−29.6155			−1.16792			−0.583961	+	0.811782i	$0.698498\pi$
$644$	−12.6847	−	9.72350i	−0.499846	−	0.383159i
$645$		0			0
$646$	5.75379	−	3.80989i	0.226380	−	0.149898i
$647$	32.9848			1.29677			0.648384	−	0.761313i	$-0.275445\pi$
$647$	32.9848			1.29677			0.648384	+	0.761313i	$0.275445\pi$
$648$		0			0
$649$			12.0818i			0.474252i
$650$	15.1231	−	10.0138i	0.593177	−	0.392774i
$651$		0			0
$652$	−21.1771	−	8.97983i	−0.829358	−	0.351677i
$653$	32.7386			1.28116			0.640581	−	0.767891i	$-0.278694\pi$
$653$	32.7386			1.28116			0.640581	+	0.767891i	$0.278694\pi$
$654$		0			0
$655$			37.7327i			1.47434i
$656$	−22.2462	+	21.5150i	−0.868569	+	0.840018i
$657$		0			0
$658$	−12.4924	−	36.2454i	−0.487005	−	1.41299i
$659$		−	38.1045i		−	1.48434i	−0.670210	−	0.742171i	$-0.733796\pi$
$659$		−	38.1045i		−	1.48434i	0.670210	−	0.742171i	$-0.266204\pi$
$660$		0			0
$661$			2.64861i			0.103019i	0.998673	+	0.0515096i	$0.0164033\pi$
$661$			2.64861i			0.103019i	−0.998673	+	0.0515096i	$0.983597\pi$
$662$	−6.43845	+	4.26324i	−0.250237	+	0.165696i
$663$		0			0
$664$	17.3693	−	3.22925i	0.674060	−	0.125319i
$665$	1.26137	−	4.87962i	0.0489137	−	0.189223i
$666$		0			0
$667$		−	6.04090i		−	0.233904i
$668$	1.75379	−	4.13595i	0.0678561	−	0.160025i
$669$		0			0
$670$	−2.73863	−	4.13595i	−0.105803	−	0.159786i
$671$	28.4924			1.09994
$672$		0			0
$673$	29.8617			1.15109			0.575543	−	0.817772i	$-0.304791\pi$
$673$	29.8617			1.15109			0.575543	+	0.817772i	$0.304791\pi$
$674$	6.43845	+	9.72350i	0.248000	+	0.374535i
$675$		0			0
$676$	18.3423	−	43.2566i	0.705474	−	1.66372i
$677$			32.6443i			1.25462i	0.778769	+	0.627311i	$0.215844\pi$
$677$			32.6443i			1.25462i	−0.778769	+	0.627311i	$0.784156\pi$
$678$		0			0
$679$	−5.75379	+	22.2586i	−0.220810	+	0.854208i
$680$	−20.4924	+	3.80989i	−0.785849	+	0.146103i
$681$		0			0
$682$		0			0
$683$			29.0890i			1.11306i	0.830828	+	0.556529i	$0.187867\pi$
$683$			29.0890i			1.11306i	−0.830828	+	0.556529i	$0.812133\pi$
$684$		0			0
$685$			27.5559i			1.05286i
$686$	−4.53457	+	25.7961i	−0.173131	+	0.984899i
$687$		0			0
$688$	23.3153	−	22.5490i	0.888889	−	0.859671i
$689$			25.6509i			0.977222i
$690$		0			0
$691$	−12.0000			−0.456502			−0.228251	−	0.973602i	$-0.573301\pi$
$691$	−12.0000			−0.456502			−0.228251	+	0.973602i	$0.573301\pi$
$692$	15.6155	+	6.62153i	0.593613	+	0.251713i
$693$		0			0
$694$	−40.9309	+	27.1025i	−1.55371	+	1.02880i
$695$			20.3537i			0.772060i
$696$		0			0
$697$	33.6155			1.27328
$698$	4.87689	−	3.22925i	0.184593	−	0.122229i
$699$		0			0
$700$	6.83475	−	8.91618i	0.258329	−	0.337000i
$701$	34.0000			1.28416			0.642081	−	0.766637i	$-0.278071\pi$
$701$	34.0000			1.28416			0.642081	+	0.766637i	$0.278071\pi$
$702$		0			0
$703$	−8.00000			−0.301726
$704$	8.68466	+	22.5490i	0.327315	+	0.849846i
$705$		0			0
$706$	7.36932	−	4.87962i	0.277348	−	0.183647i
$707$	−4.63068	+	17.9139i	−0.174155	+	0.673721i
$708$		0			0
$709$	6.00000			0.225335			0.112667	−	0.993633i	$-0.464061\pi$
$709$	6.00000			0.225335			0.112667	+	0.993633i	$0.464061\pi$
$710$	16.4924	+	24.9073i	0.618950	+	0.934752i
$711$		0			0
$712$	4.00000	+	21.5150i	0.149906	+	0.806308i
$713$		0			0
$714$		0			0
$715$		−	30.9481i		−	1.15740i
$716$	4.19224	+	1.77766i	0.156671	+	0.0664341i
$717$		0			0
$718$	−1.31534	+	0.870958i	−0.0490881	+	0.0325039i
$719$	−4.49242			−0.167539			−0.0837695	−	0.996485i	$-0.526696\pi$
$719$	−4.49242			−0.167539			−0.0837695	+	0.996485i	$0.526696\pi$
$720$		0			0
$721$	−20.4924	−	5.29723i	−0.763178	−	0.197279i
$722$	13.8499	+	20.9165i	0.515440	+	0.778430i
$723$		0			0
$724$	11.1231	+	4.71659i	0.413387	+	0.175291i
$725$	4.24621			0.157700
$726$		0			0
$727$	32.9848			1.22334			0.611670	−	0.791113i	$-0.290498\pi$
$727$	32.9848			1.22334			0.611670	+	0.791113i	$0.290498\pi$
$728$	3.12311	−	45.0979i	0.115750	−	1.67144i
$729$		0			0
$730$	−4.49242	−	6.78456i	−0.166272	−	0.251108i
$731$	−35.2311			−1.30307
$732$		0			0
$733$			16.6354i			0.614441i	0.951638	+	0.307220i	$0.0993989\pi$
$733$			16.6354i			0.614441i	−0.951638	+	0.307220i	$0.900601\pi$
$734$	−6.73863	−	10.1768i	−0.248728	−	0.375634i
$735$		0			0
$736$	16.6847	+	3.68260i	0.615005	+	0.135742i
$737$	6.24621			0.230082
$738$		0			0
$739$		−	5.87787i		−	0.216221i	−0.994139	−	0.108110i	$-0.965520\pi$
$739$		−	5.87787i		−	0.216221i	0.994139	−	0.108110i	$-0.0344800\pi$
$740$	22.2462	+	9.43318i	0.817787	+	0.346771i
$741$		0			0
$742$	5.17708	+	15.0207i	0.190057	+	0.551428i
$743$		−	13.6149i		−	0.499482i	−0.968313	−	0.249741i	$-0.919654\pi$
$743$		−	13.6149i		−	0.499482i	0.968313	−	0.249741i	$-0.0803455\pi$
$744$		0			0
$745$			16.9614i			0.621418i
$746$	7.80776	+	11.7915i	0.285863	+	0.431716i
$747$		0			0
$748$	10.2462	−	24.1636i	0.374639	−	0.883508i
$749$	−3.75379	+	14.5216i	−0.137160	+	0.530608i
$750$		0			0
$751$		−	30.7851i		−	1.12336i	−0.827353	−	0.561682i	$-0.810154\pi$
$751$		−	30.7851i		−	1.12336i	0.827353	−	0.561682i	$-0.189846\pi$
$752$	28.4924	+	29.4608i	1.03901	+	1.07433i
$753$		0			0
$754$	14.2462	−	9.43318i	0.518816	−	0.343536i
$755$	13.7538			0.500552
$756$		0			0
$757$	30.9848			1.12616			0.563082	−	0.826401i	$-0.309616\pi$
$757$	30.9848			1.12616			0.563082	+	0.826401i	$0.309616\pi$
$758$	22.0540	−	14.6031i	0.801036	−	0.530409i
$759$		0			0
$760$	0.984845	+	5.29723i	0.0357241	+	0.192151i
$761$			33.8056i			1.22545i	0.790296	+	0.612725i	$0.209927\pi$
$761$			33.8056i			1.22545i	−0.790296	+	0.612725i	$0.790073\pi$
$762$		0			0
$763$	−21.1231	−	5.46026i	−0.764708	−	0.197675i
$764$	−16.6847	−	7.07488i	−0.603630	−	0.255960i
$765$		0			0
$766$	20.4924	+	30.9481i	0.740421	+	1.11820i
$767$			24.1636i			0.872496i
$768$		0			0
$769$			44.5173i			1.60533i	0.596427	+	0.802667i	$0.296586\pi$
$769$			44.5173i			1.60533i	−0.596427	+	0.802667i	$0.703414\pi$
$770$	−6.24621	−	18.1227i	−0.225098	−	0.653096i
$771$		0			0
$772$	−7.31534	+	17.2517i	−0.263285	+	0.620903i
$773$			1.69614i			0.0610060i	0.999535	+	0.0305030i	$0.00971091\pi$
$773$			1.69614i			0.0610060i	−0.999535	+	0.0305030i	$0.990289\pi$
$774$		0			0
$775$		0			0
$776$	−4.49242	−	24.1636i	−0.161269	−	0.867422i
$777$		0			0
$778$	0.192236	+	0.290319i	0.00689199	+	0.0104085i
$779$		−	8.68951i		−	0.311334i
$780$		0			0
$781$	−37.6155			−1.34599
$782$	−10.2462	−	15.4741i	−0.366404	−	0.553352i
$783$		0			0
$784$	−7.27320	−	27.0389i	−0.259757	−	0.965674i
$785$	−7.01515			−0.250382
$786$		0			0
$787$	−20.9848			−0.748029			−0.374014	−	0.927423i	$-0.622019\pi$
$787$	−20.9848			−0.748029			−0.374014	+	0.927423i	$0.622019\pi$
$788$	0.192236	−	0.453349i	0.00684812	−	0.0161499i
$789$		0			0
$790$	−6.24621	−	9.43318i	−0.222230	−	0.335617i
$791$	31.3693	+	8.10887i	1.11536	+	0.288318i

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:	\( N \)	\(=\)	\( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	252.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.780776 - 1.17915i) q^{2} +(-0.780776 + 1.84130i) q^{4} +1.69614i q^{5} +(-2.56155 - 0.662153i) q^{7} +(2.78078 - 0.516994i) q^{8} +O(q^{10})\)	\(q+(-0.780776 - 1.17915i) q^{2} +(-0.780776 + 1.84130i) q^{4} +1.69614i q^{5} +(-2.56155 - 0.662153i) q^{7} +(2.78078 - 0.516994i) q^{8} +(2.00000 - 1.32431i) q^{10} +3.02045i q^{11} +6.04090i q^{13} +(1.21922 + 3.53744i) q^{14} +(-2.78078 - 2.87529i) q^{16} +4.34475i q^{17} +1.12311 q^{19} +(-3.12311 - 1.32431i) q^{20} +(3.56155 - 2.35829i) q^{22} -3.02045i q^{23} +2.12311 q^{25} +(7.12311 - 4.71659i) q^{26} +(3.21922 - 4.19960i) q^{28} +2.00000 q^{29} +(-1.21922 + 5.52390i) q^{32} +(5.12311 - 3.39228i) q^{34} +(1.12311 - 4.34475i) q^{35} -7.12311 q^{37} +(-0.876894 - 1.32431i) q^{38} +(0.876894 + 4.71659i) q^{40} -7.73704i q^{41} +8.10887i q^{43} +(-5.56155 - 2.35829i) q^{44} +(-3.56155 + 2.35829i) q^{46} -10.2462 q^{47} +(6.12311 + 3.39228i) q^{49} +(-1.65767 - 2.50345i) q^{50} +(-11.1231 - 4.71659i) q^{52} +4.24621 q^{53} -5.12311 q^{55} +(-7.46543 - 0.516994i) q^{56} +(-1.56155 - 2.35829i) q^{58} +4.00000 q^{59} -9.43318i q^{61} +(7.46543 - 2.87529i) q^{64} -10.2462 q^{65} -2.06798i q^{67} +(-8.00000 - 3.39228i) q^{68} +(-6.00000 + 2.06798i) q^{70} +12.4536i q^{71} -3.39228i q^{73} +(5.56155 + 8.39919i) q^{74} +(-0.876894 + 2.06798i) q^{76} +(2.00000 - 7.73704i) q^{77} -4.71659i q^{79} +(4.87689 - 4.71659i) q^{80} +(-9.12311 + 6.04090i) q^{82} +6.24621 q^{83} -7.36932 q^{85} +(9.56155 - 6.33122i) q^{86} +(1.56155 + 8.39919i) q^{88} +7.73704i q^{89} +(4.00000 - 15.4741i) q^{91} +(5.56155 + 2.35829i) q^{92} +(8.00000 + 12.0818i) q^{94} +1.90495i q^{95} -8.68951i q^{97} +(-0.780776 - 9.86866i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{2} + q^{4} - 2 q^{7} + 7 q^{8} + O(q^{10}) \)	\( 4 q + q^{2} + q^{4} - 2 q^{7} + 7 q^{8} + 8 q^{10} + 9 q^{14} - 7 q^{16} - 12 q^{19} + 4 q^{20} + 6 q^{22} - 8 q^{25} + 12 q^{26} + 17 q^{28} + 8 q^{29} - 9 q^{32} + 4 q^{34} - 12 q^{35} - 12 q^{37} - 20 q^{38} + 20 q^{40} - 14 q^{44} - 6 q^{46} - 8 q^{47} + 8 q^{49} - 19 q^{50} - 28 q^{52} - 16 q^{53} - 4 q^{55} - q^{56} + 2 q^{58} + 16 q^{59} + q^{64} - 8 q^{65} - 32 q^{68} - 24 q^{70} + 14 q^{74} - 20 q^{76} + 8 q^{77} + 36 q^{80} - 20 q^{82} - 8 q^{83} + 20 q^{85} + 30 q^{86} - 2 q^{88} + 16 q^{91} + 14 q^{92} + 32 q^{94} + q^{98} + O(q^{100}) \)

Introduction

L-functions

Modular forms

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