LMFDB - Embedded newform 1110.2.h.f.961.1

Newspace parameters

Level:	$ N $	$=$	$ 1110 = 2 \cdot 3 \cdot 5 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1110.h (of order $2$, degree $1$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$8.86339462436$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.279290944.1
Defining polynomial:	$x^{6} + 26 x^{4} + 169 x^{2} + 16$
Coefficient ring:	$\Z[a_1, \ldots, a_{37}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			961.1
Root			$-3.44055i$ of defining polynomial
Character	$\chi$	$=$	1110.961
Dual form			1110.2.h.f.961.4

$q$-expansion

$f(q)$	$=$	$q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{6} -4.44055 q^{7} +1.00000i q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{6} -4.44055 q^{7} +1.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} -2.44055 q^{11} -1.00000 q^{12} -0.440552i q^{13} +4.44055i q^{14} -1.00000i q^{15} +1.00000 q^{16} +4.83740i q^{17} -1.00000i q^{18} +0.440552i q^{19} +1.00000i q^{20} -4.44055 q^{21} +2.44055i q^{22} +2.83740i q^{23} +1.00000i q^{24} -1.00000 q^{25} -0.440552 q^{26} +1.00000 q^{27} +4.44055 q^{28} +2.00000i q^{29} -1.00000 q^{30} +5.27795i q^{31} -1.00000i q^{32} -2.44055 q^{33} +4.83740 q^{34} +4.44055i q^{35} -1.00000 q^{36} +(-0.198422 + 6.07953i) q^{37} +0.440552 q^{38} -0.440552i q^{39} +1.00000 q^{40} -2.00000 q^{41} +4.44055i q^{42} -0.396844i q^{43} +2.44055 q^{44} -1.00000i q^{45} +2.83740 q^{46} -10.8811 q^{47} +1.00000 q^{48} +12.7185 q^{49} +1.00000i q^{50} +4.83740i q^{51} +0.440552i q^{52} -2.44055 q^{53} -1.00000i q^{54} +2.44055i q^{55} -4.44055i q^{56} +0.440552i q^{57} +2.00000 q^{58} -2.39684i q^{59} +1.00000i q^{60} +2.00000i q^{61} +5.27795 q^{62} -4.44055 q^{63} -1.00000 q^{64} -0.440552 q^{65} +2.44055i q^{66} -13.7622 q^{67} -4.83740i q^{68} +2.83740i q^{69} +4.44055 q^{70} -9.67479 q^{71} +1.00000i q^{72} +5.71850 q^{73} +(6.07953 + 0.198422i) q^{74} -1.00000 q^{75} -0.440552i q^{76} +10.8374 q^{77} -0.440552 q^{78} -13.2779i q^{79} -1.00000i q^{80} +1.00000 q^{81} +2.00000i q^{82} -7.71850 q^{83} +4.44055 q^{84} +4.83740 q^{85} -0.396844 q^{86} +2.00000i q^{87} -2.44055i q^{88} -15.3217i q^{89} -1.00000 q^{90} +1.95629i q^{91} -2.83740i q^{92} +5.27795i q^{93} +10.8811i q^{94} +0.440552 q^{95} -1.00000i q^{96} +2.88110i q^{97} -12.7185i q^{98} -2.44055 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6q + 6q^{3} - 6q^{4} - 6q^{7} + 6q^{9} + O(q^{10}) $	$ 6q + 6q^{3} - 6q^{4} - 6q^{7} + 6q^{9} - 6q^{10} + 6q^{11} - 6q^{12} + 6q^{16} - 6q^{21} - 6q^{25} + 18q^{26} + 6q^{27} + 6q^{28} - 6q^{30} + 6q^{33} + 10q^{34} - 6q^{36} - 2q^{37} - 18q^{38} + 6q^{40} - 12q^{41} - 6q^{44} - 2q^{46} - 24q^{47} + 6q^{48} + 16q^{49} + 6q^{53} + 12q^{58} - 8q^{62} - 6q^{63} - 6q^{64} + 18q^{65} + 6q^{70} - 20q^{71} - 26q^{73} - 4q^{74} - 6q^{75} + 46q^{77} + 18q^{78} + 6q^{81} + 14q^{83} + 6q^{84} + 10q^{85} - 4q^{86} - 6q^{90} - 18q^{95} + 6q^{99} + O(q^{100}) $

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1110\mathbb{Z}\right)^\times$.

$n$	$371$	$631$	$667$
$\chi(n)$	$1$	$-1$	$1$

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 $

$2$		−	1.00000i		−	0.707107i
$2$		−	1.00000i		−	0.707107i
$3$	1.00000			0.577350
$3$	1.00000			0.577350
$4$	−1.00000			−0.500000
$5$		−	1.00000i		−	0.447214i
$5$		−	1.00000i		−	0.447214i
$6$		−	1.00000i		−	0.408248i
$7$	−4.44055			−1.67837			−0.839185	−	0.543845i	$-0.816968\pi$
$7$	−4.44055			−1.67837			−0.839185	+	0.543845i	$0.816968\pi$
$8$			1.00000i			0.353553i
$9$	1.00000			0.333333
$10$	−1.00000			−0.316228
$11$	−2.44055			−0.735854			−0.367927	−	0.929855i	$-0.619932\pi$
$11$	−2.44055			−0.735854			−0.367927	+	0.929855i	$0.619932\pi$
$12$	−1.00000			−0.288675
$13$		−	0.440552i		−	0.122187i	−0.998132	−	0.0610935i	$-0.980541\pi$
$13$		−	0.440552i		−	0.122187i	0.998132	−	0.0610935i	$-0.0194588\pi$
$14$			4.44055i			1.18679i
$15$		−	1.00000i		−	0.258199i
$16$	1.00000			0.250000
$17$			4.83740i			1.17324i	0.809862	+	0.586620i	$0.199542\pi$
$17$			4.83740i			1.17324i	−0.809862	+	0.586620i	$0.800458\pi$
$18$		−	1.00000i		−	0.235702i
$19$			0.440552i			0.101069i	0.998722	+	0.0505347i	$0.0160926\pi$
$19$			0.440552i			0.101069i	−0.998722	+	0.0505347i	$0.983907\pi$
$20$			1.00000i			0.223607i
$21$	−4.44055			−0.969008
$22$			2.44055i			0.520327i
$23$			2.83740i			0.591638i	0.955244	+	0.295819i	$0.0955925\pi$
$23$			2.83740i			0.591638i	−0.955244	+	0.295819i	$0.904407\pi$
$24$			1.00000i			0.204124i
$25$	−1.00000			−0.200000
$26$	−0.440552			−0.0863993
$27$	1.00000			0.192450
$28$	4.44055			0.839185
$29$			2.00000i			0.371391i	0.982607	+	0.185695i	$0.0594537\pi$
$29$			2.00000i			0.371391i	−0.982607	+	0.185695i	$0.940546\pi$
$30$	−1.00000			−0.182574
$31$			5.27795i			0.947947i	0.880539	+	0.473974i	$0.157181\pi$
$31$			5.27795i			0.947947i	−0.880539	+	0.473974i	$0.842819\pi$
$32$		−	1.00000i		−	0.176777i
$33$	−2.44055			−0.424846
$34$	4.83740			0.829607
$35$			4.44055i			0.750590i
$36$	−1.00000			−0.166667
$37$	−0.198422	+	6.07953i	−0.0326204	+	0.999468i
$37$	−0.198422	+	6.07953i	−0.0326204	+	0.999468i
$38$	0.440552			0.0714669
$39$		−	0.440552i		−	0.0705447i
$40$	1.00000			0.158114
$41$	−2.00000			−0.312348			−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000			−0.312348			−0.156174	+	0.987730i	$0.549916\pi$
$42$			4.44055i			0.685192i
$43$		−	0.396844i		−	0.0605182i	−0.999542	−	0.0302591i	$-0.990367\pi$
$43$		−	0.396844i		−	0.0605182i	0.999542	−	0.0302591i	$-0.00963323\pi$
$44$	2.44055			0.367927
$45$		−	1.00000i		−	0.149071i
$46$	2.83740			0.418351
$47$	−10.8811			−1.58717			−0.793586	−	0.608458i	$-0.791788\pi$
$47$	−10.8811			−1.58717			−0.793586	+	0.608458i	$0.791788\pi$
$48$	1.00000			0.144338
$49$	12.7185			1.81693
$50$			1.00000i			0.141421i
$51$			4.83740i			0.677371i
$52$			0.440552i			0.0610935i
$53$	−2.44055			−0.335236			−0.167618	−	0.985852i	$-0.553607\pi$
$53$	−2.44055			−0.335236			−0.167618	+	0.985852i	$0.553607\pi$
$54$		−	1.00000i		−	0.136083i
$55$			2.44055i			0.329084i
$56$		−	4.44055i		−	0.593394i
$57$			0.440552i			0.0583525i
$58$	2.00000			0.262613
$59$		−	2.39684i		−	0.312043i	−0.987754	−	0.156021i	$-0.950133\pi$
$59$		−	2.39684i		−	0.312043i	0.987754	−	0.156021i	$-0.0498668\pi$
$60$			1.00000i			0.129099i
$61$			2.00000i			0.256074i	0.991769	+	0.128037i	$0.0408676\pi$
$61$			2.00000i			0.256074i	−0.991769	+	0.128037i	$0.959132\pi$
$62$	5.27795			0.670300
$63$	−4.44055			−0.559457
$64$	−1.00000			−0.125000
$65$	−0.440552			−0.0546437
$66$			2.44055i			0.300411i
$67$	−13.7622			−1.68132			−0.840661	−	0.541562i	$-0.817833\pi$
$67$	−13.7622			−1.68132			−0.840661	+	0.541562i	$0.817833\pi$
$68$		−	4.83740i		−	0.586620i
$69$			2.83740i			0.341582i
$70$	4.44055			0.530747
$71$	−9.67479			−1.14819			−0.574093	−	0.818790i	$-0.694645\pi$
$71$	−9.67479			−1.14819			−0.574093	+	0.818790i	$0.694645\pi$
$72$			1.00000i			0.117851i
$73$	5.71850			0.669300			0.334650	−	0.942343i	$-0.391382\pi$
$73$	5.71850			0.669300			0.334650	+	0.942343i	$0.391382\pi$
$74$	6.07953	+	0.198422i	0.706730	+	0.0230661i
$75$	−1.00000			−0.115470
$76$		−	0.440552i		−	0.0505347i
$77$	10.8374			1.23504
$78$	−0.440552			−0.0498827
$79$		−	13.2779i		−	1.49389i	−0.664888	−	0.746943i	$-0.731521\pi$
$79$		−	13.2779i		−	1.49389i	0.664888	−	0.746943i	$-0.268479\pi$
$80$		−	1.00000i		−	0.111803i
$81$	1.00000			0.111111
$82$			2.00000i			0.220863i
$83$	−7.71850			−0.847215			−0.423608	−	0.905846i	$-0.639236\pi$
$83$	−7.71850			−0.847215			−0.423608	+	0.905846i	$0.639236\pi$
$84$	4.44055			0.484504
$85$	4.83740			0.524689
$86$	−0.396844			−0.0427928
$87$			2.00000i			0.214423i
$88$		−	2.44055i		−	0.260164i
$89$		−	15.3217i		−	1.62409i	−0.583593	−	0.812046i	$-0.698354\pi$
$89$		−	15.3217i		−	1.62409i	0.583593	−	0.812046i	$-0.301646\pi$
$90$	−1.00000			−0.105409
$91$			1.95629i			0.205075i
$92$		−	2.83740i		−	0.295819i
$93$			5.27795i			0.547298i
$94$			10.8811i			1.12230i
$95$	0.440552			0.0451996
$96$		−	1.00000i		−	0.102062i
$97$			2.88110i			0.292532i	0.989245	+	0.146266i	$0.0467255\pi$
$97$			2.88110i			0.292532i	−0.989245	+	0.146266i	$0.953275\pi$
$98$		−	12.7185i		−	1.28476i
$99$	−2.44055			−0.245285
$100$	1.00000			0.100000
$101$	−12.1591			−1.20987			−0.604935	−	0.796275i	$-0.706801\pi$
$101$	−12.1591			−1.20987			−0.604935	+	0.796275i	$0.706801\pi$
$102$	4.83740			0.478974
$103$			10.5559i			1.04010i	0.854135	+	0.520052i	$0.174087\pi$
$103$			10.5559i			1.04010i	−0.854135	+	0.520052i	$0.825913\pi$
$104$	0.440552			0.0431996
$105$			4.44055i			0.433353i
$106$			2.44055i			0.237047i
$107$	18.9248			1.82953			0.914765	−	0.403986i	$-0.132375\pi$
$107$	18.9248			1.82953			0.914765	+	0.403986i	$0.132375\pi$
$108$	−1.00000			−0.0962250
$109$			12.0437i			1.15358i	0.816893	+	0.576789i	$0.195695\pi$
$109$			12.0437i			1.15358i	−0.816893	+	0.576789i	$0.804305\pi$
$110$	2.44055			0.232697
$111$	−0.198422	+	6.07953i	−0.0188334	+	0.577043i
$112$	−4.44055			−0.419593
$113$		−	2.00000i		−	0.188144i	−0.995565	−	0.0940721i	$-0.970012\pi$
$113$		−	2.00000i		−	0.188144i	0.995565	−	0.0940721i	$-0.0299884\pi$
$114$	0.440552			0.0412614
$115$	2.83740			0.264589
$116$		−	2.00000i		−	0.185695i
$117$		−	0.440552i		−	0.0407290i
$118$	−2.39684			−0.220647
$119$		−	21.4807i		−	1.96913i
$120$	1.00000			0.0912871
$121$	−5.04371			−0.458519
$122$	2.00000			0.181071
$123$	−2.00000			−0.180334
$124$		−	5.27795i		−	0.473974i
$125$			1.00000i			0.0894427i
$126$			4.44055i			0.395596i
$127$	−7.55945			−0.670793			−0.335396	−	0.942077i	$-0.608870\pi$
$127$	−7.55945			−0.670793			−0.335396	+	0.942077i	$0.608870\pi$
$128$			1.00000i			0.0883883i
$129$		−	0.396844i		−	0.0349402i
$130$			0.440552i			0.0386389i
$131$		−	12.1591i		−	1.06234i	−0.847265	−	0.531171i	$-0.821752\pi$
$131$		−	12.1591i		−	1.06234i	0.847265	−	0.531171i	$-0.178248\pi$
$132$	2.44055			0.212423
$133$		−	1.95629i		−	0.169632i
$134$			13.7622i			1.18887i
$135$		−	1.00000i		−	0.0860663i
$136$	−4.83740			−0.414803
$137$	3.27795			0.280054			0.140027	−	0.990148i	$-0.455281\pi$
$137$	3.27795			0.280054			0.140027	+	0.990148i	$0.455281\pi$
$138$	2.83740			0.241535
$139$	5.76221			0.488744			0.244372	−	0.969682i	$-0.421418\pi$
$139$	5.76221			0.488744			0.244372	+	0.969682i	$0.421418\pi$
$140$		−	4.44055i		−	0.375295i
$141$	−10.8811			−0.916354
$142$			9.67479i			0.811890i
$143$			1.07519i			0.0899118i
$144$	1.00000			0.0833333
$145$	2.00000			0.166091
$146$		−	5.71850i		−	0.473266i
$147$	12.7185			1.04900
$148$	0.198422	−	6.07953i	0.0163102	−	0.499734i
$149$	1.60316			0.131336			0.0656678	−	0.997842i	$-0.479082\pi$
$149$	1.60316			0.131336			0.0656678	+	0.997842i	$0.479082\pi$
$150$			1.00000i			0.0816497i
$151$	4.28150			0.348424			0.174212	−	0.984708i	$-0.444262\pi$
$151$	4.28150			0.348424			0.174212	+	0.984708i	$0.444262\pi$
$152$	−0.440552			−0.0357335
$153$			4.83740i			0.391080i
$154$		−	10.8374i		−	0.873302i
$155$	5.27795			0.423935
$156$			0.440552i			0.0352724i
$157$	3.60316			0.287563			0.143782	−	0.989609i	$-0.454074\pi$
$157$	3.60316			0.287563			0.143782	+	0.989609i	$0.454074\pi$
$158$	−13.2779			−1.05634
$159$	−2.44055			−0.193548
$160$	−1.00000			−0.0790569
$161$		−	12.5996i		−	0.992988i
$162$		−	1.00000i		−	0.0785674i
$163$			3.32165i			0.260172i	0.991503	+	0.130086i	$0.0415254\pi$
$163$			3.32165i			0.260172i	−0.991503	+	0.130086i	$0.958475\pi$
$164$	2.00000			0.156174
$165$			2.44055i			0.189997i
$166$			7.71850i			0.599072i
$167$		−	2.83740i		−	0.219564i	−0.993956	−	0.109782i	$-0.964985\pi$
$167$		−	2.83740i		−	0.219564i	0.993956	−	0.109782i	$-0.0350153\pi$
$168$		−	4.44055i		−	0.342596i
$169$	12.8059			0.985070
$170$		−	4.83740i		−	0.371011i
$171$			0.440552i			0.0336898i
$172$			0.396844i			0.0302591i
$173$	−12.9964			−0.988102			−0.494051	−	0.869433i	$-0.664484\pi$
$173$	−12.9964			−0.988102			−0.494051	+	0.869433i	$0.664484\pi$
$174$	2.00000			0.151620
$175$	4.44055			0.335674
$176$	−2.44055			−0.183964
$177$		−	2.39684i		−	0.180158i
$178$	−15.3217			−1.14841
$179$			4.72205i			0.352943i	0.984306	+	0.176471i	$0.0564683\pi$
$179$			4.72205i			0.352943i	−0.984306	+	0.176471i	$0.943532\pi$
$180$			1.00000i			0.0745356i
$181$	20.5559			1.52791			0.763954	−	0.645271i	$-0.223256\pi$
$181$	20.5559			1.52791			0.763954	+	0.645271i	$0.223256\pi$
$182$	1.95629			0.145010
$183$			2.00000i			0.147844i
$184$	−2.83740			−0.209176
$185$	6.07953	+	0.198422i	0.446976	+	0.0145883i
$186$	5.27795			0.386998
$187$		−	11.8059i		−	0.863334i
$188$	10.8811			0.793586
$189$	−4.44055			−0.323003
$190$		−	0.440552i		−	0.0319610i
$191$		−	9.16260i		−	0.662983i	−0.943458	−	0.331491i	$-0.892448\pi$
$191$		−	9.16260i		−	0.662983i	0.943458	−	0.331491i	$-0.107552\pi$
$192$	−1.00000			−0.0721688
$193$			14.8811i			1.07117i	0.844483	+	0.535583i	$0.179908\pi$
$193$			14.8811i			1.07117i	−0.844483	+	0.535583i	$0.820092\pi$
$194$	2.88110			0.206851
$195$	−0.440552			−0.0315486
$196$	−12.7185			−0.908464
$197$	−9.55945			−0.681082			−0.340541	−	0.940230i	$-0.610610\pi$
$197$	−9.55945			−0.681082			−0.340541	+	0.940230i	$0.610610\pi$
$198$			2.44055i			0.173442i
$199$			20.4843i			1.45209i	0.687647	+	0.726045i	$0.258644\pi$
$199$			20.4843i			1.45209i	−0.687647	+	0.726045i	$0.741356\pi$
$200$		−	1.00000i		−	0.0707107i
$201$	−13.7622			−0.970711
$202$			12.1591i			0.855508i
$203$		−	8.88110i		−	0.623331i
$204$		−	4.83740i		−	0.338685i
$205$			2.00000i			0.139686i
$206$	10.5559			0.735464
$207$			2.83740i			0.197213i
$208$		−	0.440552i		−	0.0305468i
$209$		−	1.07519i		−	0.0743724i
$210$	4.44055			0.306427
$211$	0.881103			0.0606577			0.0303288	−	0.999540i	$-0.490345\pi$
$211$	0.881103			0.0606577			0.0303288	+	0.999540i	$0.490345\pi$
$212$	2.44055			0.167618
$213$	−9.67479			−0.662906
$214$		−	18.9248i		−	1.29367i
$215$	−0.396844			−0.0270645
$216$			1.00000i			0.0680414i
$217$		−	23.4370i		−	1.59101i
$218$	12.0437			0.815703
$219$	5.71850			0.386420
$220$		−	2.44055i		−	0.164542i
$221$	2.13112			0.143355
$222$	6.07953	+	0.198422i	0.408031	+	0.0133172i
$223$	1.51574			0.101502			0.0507508	−	0.998711i	$-0.483839\pi$
$223$	1.51574			0.101502			0.0507508	+	0.998711i	$0.483839\pi$
$224$			4.44055i			0.296697i
$225$	−1.00000			−0.0666667
$226$	−2.00000			−0.133038
$227$			14.5559i			0.966109i	0.875590	+	0.483054i	$0.160473\pi$
$227$			14.5559i			0.966109i	−0.875590	+	0.483054i	$0.839527\pi$
$228$		−	0.440552i		−	0.0291762i
$229$	15.6748			1.03582			0.517910	−	0.855435i	$-0.326710\pi$
$229$	15.6748			1.03582			0.517910	+	0.855435i	$0.326710\pi$
$230$		−	2.83740i		−	0.187092i
$231$	10.8374			0.713048
$232$	−2.00000			−0.131306
$233$	−11.2779			−0.738843			−0.369421	−	0.929262i	$-0.620444\pi$
$233$	−11.2779			−0.738843			−0.369421	+	0.929262i	$0.620444\pi$
$234$	−0.440552			−0.0287998
$235$			10.8811i			0.709805i
$236$			2.39684i			0.156021i
$237$		−	13.2779i		−	0.862495i
$238$	−21.4807			−1.39239
$239$		−	22.5559i		−	1.45902i	−0.683970	−	0.729510i	$-0.739748\pi$
$239$		−	22.5559i		−	1.45902i	0.683970	−	0.729510i	$-0.260252\pi$
$240$		−	1.00000i		−	0.0645497i
$241$		−	26.5559i		−	1.71062i	−0.518120	−	0.855308i	$-0.673368\pi$
$241$		−	26.5559i		−	1.71062i	0.518120	−	0.855308i	$-0.326632\pi$
$242$			5.04371i			0.324222i
$243$	1.00000			0.0641500
$244$		−	2.00000i		−	0.128037i
$245$		−	12.7185i		−	0.812555i
$246$			2.00000i			0.127515i
$247$	0.194086			0.0123494
$248$	−5.27795			−0.335150
$249$	−7.71850			−0.489140
$250$	1.00000			0.0632456
$251$		−	3.27795i		−	0.206902i	−0.994635	−	0.103451i	$-0.967011\pi$
$251$		−	3.27795i		−	0.206902i	0.994635	−	0.103451i	$-0.0329885\pi$
$252$	4.44055			0.279728
$253$		−	6.92481i		−	0.435359i
$254$			7.55945i			0.474322i
$255$	4.83740			0.302929
$256$	1.00000			0.0625000
$257$			20.0437i			1.25029i	0.780508	+	0.625146i	$0.214961\pi$
$257$			20.0437i			1.25029i	−0.780508	+	0.625146i	$0.785039\pi$
$258$	−0.396844			−0.0247064
$259$	0.881103	−	26.9964i	0.0547491	−	1.67748i
$260$	0.440552			0.0273219
$261$			2.00000i			0.123797i
$262$	−12.1591			−0.751189
$263$	−17.4370			−1.07521			−0.537606	−	0.843196i	$-0.680671\pi$
$263$	−17.4370			−1.07521			−0.537606	+	0.843196i	$0.680671\pi$
$264$		−	2.44055i		−	0.150206i
$265$			2.44055i			0.149922i
$266$	−1.95629			−0.119948
$267$		−	15.3217i		−	0.937670i
$268$	13.7622			0.840661
$269$	−16.4406			−1.00240			−0.501199	−	0.865332i	$-0.667108\pi$
$269$	−16.4406			−1.00240			−0.501199	+	0.865332i	$0.667108\pi$
$270$	−1.00000			−0.0608581
$271$	−24.3181			−1.47722			−0.738610	−	0.674133i	$-0.764517\pi$
$271$	−24.3181			−1.47722			−0.738610	+	0.674133i	$0.764517\pi$
$272$			4.83740i			0.293310i
$273$			1.95629i			0.118400i
$274$		−	3.27795i		−	0.198028i
$275$	2.44055			0.147171
$276$		−	2.83740i		−	0.170791i
$277$			0.353137i			0.0212179i	0.999944	+	0.0106090i	$0.00337700\pi$
$277$			0.353137i			0.0212179i	−0.999944	+	0.0106090i	$0.996623\pi$
$278$		−	5.76221i		−	0.345594i
$279$			5.27795i			0.315982i
$280$	−4.44055			−0.265374
$281$		−	20.2028i		−	1.20520i	−0.798045	−	0.602598i	$-0.794132\pi$
$281$		−	20.2028i		−	1.20520i	0.798045	−	0.602598i	$-0.205868\pi$
$282$			10.8811i			0.647960i
$283$		−	13.6469i		−	0.811222i	−0.914046	−	0.405611i	$-0.867059\pi$
$283$		−	13.6469i		−	0.811222i	0.914046	−	0.405611i	$-0.132941\pi$
$284$	9.67479			0.574093
$285$	0.440552			0.0260960
$286$	1.07519			0.0635773
$287$	8.88110			0.524235
$288$		−	1.00000i		−	0.0589256i
$289$	−6.40040			−0.376494
$290$		−	2.00000i		−	0.117444i
$291$			2.88110i			0.168893i
$292$	−5.71850			−0.334650
$293$	−6.44055			−0.376261			−0.188131	−	0.982144i	$-0.560243\pi$
$293$	−6.44055			−0.376261			−0.188131	+	0.982144i	$0.560243\pi$
$294$		−	12.7185i		−	0.741758i
$295$	−2.39684			−0.139550
$296$	−6.07953	−	0.198422i	−0.353365	−	0.0115330i
$297$	−2.44055			−0.141615
$298$		−	1.60316i		−	0.0928683i
$299$	1.25002			0.0722905
$300$	1.00000			0.0577350
$301$			1.76221i			0.101572i
$302$		−	4.28150i		−	0.246373i
$303$	−12.1591			−0.698519
$304$			0.440552i			0.0252674i
$305$	2.00000			0.114520
$306$	4.83740			0.276536
$307$	−20.3181			−1.15962			−0.579808	−	0.814753i	$-0.696872\pi$
$307$	−20.3181			−1.15962			−0.579808	+	0.814753i	$0.696872\pi$
$308$	−10.8374			−0.617518
$309$			10.5559i			0.600504i
$310$		−	5.27795i		−	0.299767i
$311$			6.23779i			0.353713i	0.984237	+	0.176856i	$0.0565928\pi$
$311$			6.23779i			0.353713i	−0.984237	+	0.176856i	$0.943407\pi$
$312$	0.440552			0.0249413
$313$		−	8.55589i		−	0.483608i	−0.970325	−	0.241804i	$-0.922261\pi$
$313$		−	8.55589i		−	0.483608i	0.970325	−	0.241804i	$-0.0777391\pi$
$314$		−	3.60316i		−	0.203338i
$315$			4.44055i			0.250197i
$316$			13.2779i			0.746943i
$317$	−33.5960			−1.88694			−0.943471	−	0.331455i	$-0.892461\pi$
$317$	−33.5960			−1.88694			−0.943471	+	0.331455i	$0.892461\pi$
$318$			2.44055i			0.136859i
$319$		−	4.88110i		−	0.273289i
$320$			1.00000i			0.0559017i
$321$	18.9248			1.05628
$322$	−12.5996			−0.702148
$323$	−2.13112			−0.118579
$324$	−1.00000			−0.0555556
$325$			0.440552i			0.0244374i
$326$	3.32165			0.183969
$327$			12.0437i			0.666019i
$328$		−	2.00000i		−	0.110432i
$329$	48.3181			2.66386
$330$	2.44055			0.134348
$331$			31.2779i			1.71919i	0.510975	+	0.859596i	$0.329285\pi$
$331$			31.2779i			1.71919i	−0.510975	+	0.859596i	$0.670715\pi$
$332$	7.71850			0.423608
$333$	−0.198422	+	6.07953i	−0.0108735	+	0.333156i
$334$	−2.83740			−0.155255
$335$			13.7622i			0.751910i
$336$	−4.44055			−0.242252
$337$	−5.63108			−0.306745			−0.153372	−	0.988168i	$-0.549013\pi$
$337$	−5.63108			−0.306745			−0.153372	+	0.988168i	$0.549013\pi$
$338$		−	12.8059i		−	0.696550i
$339$		−	2.00000i		−	0.108625i
$340$	−4.83740			−0.262345
$341$		−	12.8811i		−	0.697551i
$342$	0.440552			0.0238223
$343$	−25.3933			−1.37111
$344$	0.396844			0.0213964
$345$	2.83740			0.152760
$346$			12.9964i			0.698693i
$347$			0.881103i			0.0473001i	0.999720	+	0.0236501i	$0.00752875\pi$
$347$			0.881103i			0.0473001i	−0.999720	+	0.0236501i	$0.992471\pi$
$348$		−	2.00000i		−	0.107211i
$349$	0.555895			0.0297564			0.0148782	−	0.999889i	$-0.495264\pi$
$349$	0.555895			0.0297564			0.0148782	+	0.999889i	$0.495264\pi$
$350$		−	4.44055i		−	0.237357i
$351$		−	0.440552i		−	0.0235149i
$352$			2.44055i			0.130082i
$353$		−	5.20631i		−	0.277104i	−0.990355	−	0.138552i	$-0.955755\pi$
$353$		−	5.20631i		−	0.277104i	0.990355	−	0.138552i	$-0.0442448\pi$
$354$	−2.39684			−0.127391
$355$			9.67479i			0.513485i
$356$			15.3217i			0.812046i
$357$		−	21.4807i		−	1.13688i
$358$	4.72205			0.249568
$359$	−15.4370			−0.814734			−0.407367	−	0.913265i	$-0.633553\pi$
$359$	−15.4370			−0.814734			−0.407367	+	0.913265i	$0.633553\pi$
$360$	1.00000			0.0527046
$361$	18.8059			0.989785
$362$		−	20.5559i		−	1.08039i
$363$	−5.04371			−0.264726
$364$		−	1.95629i		−	0.102538i
$365$		−	5.71850i		−	0.299320i
$366$	2.00000			0.104542
$367$	−14.9964			−0.782808			−0.391404	−	0.920219i	$-0.628011\pi$
$367$	−14.9964			−0.782808			−0.391404	+	0.920219i	$0.628011\pi$
$368$			2.83740i			0.147909i
$369$	−2.00000			−0.104116
$370$	0.198422	−	6.07953i	0.0103155	−	0.316059i
$371$	10.8374			0.562650
$372$		−	5.27795i		−	0.273649i
$373$	33.5086			1.73501			0.867506	−	0.497427i	$-0.165722\pi$
$373$	33.5086			1.73501			0.867506	+	0.497427i	$0.165722\pi$
$374$	−11.8059			−0.610469
$375$			1.00000i			0.0516398i
$376$		−	10.8811i		−	0.561150i
$377$	0.881103			0.0453791
$378$			4.44055i			0.228397i
$379$	34.6433			1.77951			0.889754	−	0.456441i	$-0.150876\pi$
$379$	34.6433			1.77951			0.889754	+	0.456441i	$0.150876\pi$
$380$	−0.440552			−0.0225998
$381$	−7.55945			−0.387282
$382$	−9.16260			−0.468800
$383$			8.51219i			0.434952i	0.976066	+	0.217476i	$0.0697825\pi$
$383$			8.51219i			0.434952i	−0.976066	+	0.217476i	$0.930218\pi$
$384$			1.00000i			0.0510310i
$385$		−	10.8374i		−	0.552325i
$386$	14.8811			0.757428
$387$		−	0.396844i		−	0.0201727i
$388$		−	2.88110i		−	0.146266i
$389$			27.6748i			1.40317i	0.712587	+	0.701584i	$0.247523\pi$
$389$			27.6748i			1.40317i	−0.712587	+	0.701584i	$0.752477\pi$
$390$			0.440552i			0.0223082i
$391$	−13.7256			−0.694134
$392$			12.7185i			0.642381i
$393$		−	12.1591i		−	0.613343i
$394$			9.55945i			0.481598i
$395$	−13.2779			−0.668086
$396$	2.44055			0.122642
$397$	9.36536			0.470034			0.235017	−	0.971991i	$-0.424485\pi$
$397$	9.36536			0.470034			0.235017	+	0.971991i	$0.424485\pi$
$398$	20.4843			1.02678
$399$		−	1.95629i		−	0.0979371i
$400$	−1.00000			−0.0500000
$401$			7.88466i			0.393741i	0.980430	+	0.196870i	$0.0630778\pi$
$401$			7.88466i			0.393741i	−0.980430	+	0.196870i	$0.936922\pi$
$402$			13.7622i			0.686396i
$403$	2.32521			0.115827
$404$	12.1591			0.604935
$405$		−	1.00000i		−	0.0496904i
$406$	−8.88110			−0.440762
$407$	0.484259	−	14.8374i	0.0240038	−	0.735462i
$408$	−4.83740			−0.239487
$409$			38.7307i			1.91511i	0.288249	+	0.957556i	$0.406927\pi$
$409$			38.7307i			1.91511i	−0.288249	+	0.957556i	$0.593073\pi$
$410$	2.00000			0.0987730
$411$	3.27795			0.161689
$412$		−	10.5559i		−	0.520052i
$413$			10.6433i			0.523723i
$414$	2.83740			0.139450
$415$			7.71850i			0.378886i
$416$	−0.440552			−0.0215998
$417$	5.76221			0.282176
$418$	−1.07519			−0.0525892
$419$	34.6712			1.69380			0.846900	−	0.531752i	$-0.178466\pi$
$419$	34.6712			1.69380			0.846900	+	0.531752i	$0.178466\pi$
$420$		−	4.44055i		−	0.216677i
$421$		−	23.1118i		−	1.12640i	−0.826321	−	0.563200i	$-0.809570\pi$
$421$		−	23.1118i		−	1.12640i	0.826321	−	0.563200i	$-0.190430\pi$
$422$		−	0.881103i		−	0.0428914i
$423$	−10.8811			−0.529057
$424$		−	2.44055i		−	0.118524i
$425$		−	4.83740i		−	0.234648i
$426$			9.67479i			0.468745i
$427$		−	8.88110i		−	0.429787i
$428$	−18.9248			−0.914765
$429$			1.07519i			0.0519106i
$430$			0.396844i			0.0191375i
$431$		−	9.95629i		−	0.479578i	−0.970825	−	0.239789i	$-0.922922\pi$
$431$		−	9.95629i		−	0.479578i	0.970825	−	0.239789i	$-0.0770782\pi$
$432$	1.00000			0.0481125
$433$	−8.04371			−0.386556			−0.193278	−	0.981144i	$-0.561912\pi$
$433$	−8.04371			−0.386556			−0.193278	+	0.981144i	$0.561912\pi$
$434$	−23.4370			−1.12501
$435$	2.00000			0.0958927
$436$		−	12.0437i		−	0.576789i
$437$	−1.25002			−0.0597965
$438$		−	5.71850i		−	0.273240i
$439$		−	13.2779i		−	0.633722i	−0.948472	−	0.316861i	$-0.897371\pi$
$439$		−	13.2779i		−	0.633722i	0.948472	−	0.316861i	$-0.102629\pi$
$440$	−2.44055			−0.116349
$441$	12.7185			0.605643
$442$		−	2.13112i		−	0.101367i
$443$	−2.41262			−0.114627			−0.0573136	−	0.998356i	$-0.518253\pi$
$443$	−2.41262			−0.114627			−0.0573136	+	0.998356i	$0.518253\pi$
$444$	0.198422	−	6.07953i	0.00941669	−	0.288522i
$445$	−15.3217			−0.726316
$446$		−	1.51574i		−	0.0717724i
$447$	1.60316			0.0758267
$448$	4.44055			0.209796
$449$			22.2465i			1.04988i	0.851141	+	0.524938i	$0.175911\pi$
$449$			22.2465i			1.04988i	−0.851141	+	0.524938i	$0.824089\pi$
$450$			1.00000i			0.0471405i
$451$	4.88110			0.229842
$452$			2.00000i			0.0940721i
$453$	4.28150			0.201162
$454$	14.5559			0.683142
$455$	1.95629			0.0917124
$456$	−0.440552			−0.0206307
$457$		−	20.5559i		−	0.961564i	−0.876840	−	0.480782i	$-0.840353\pi$
$457$		−	20.5559i		−	0.961564i	0.876840	−	0.480782i	$-0.159647\pi$
$458$		−	15.6748i		−	0.732435i
$459$			4.83740i			0.225790i
$460$	−2.83740			−0.132294
$461$			12.6433i			0.588858i	0.955673	+	0.294429i	$0.0951294\pi$
$461$			12.6433i			0.588858i	−0.955673	+	0.294429i	$0.904871\pi$
$462$		−	10.8374i		−	0.504201i
$463$			32.4055i			1.50601i	0.658014	+	0.753006i	$0.271397\pi$
$463$			32.4055i			1.50601i	−0.658014	+	0.753006i	$0.728603\pi$
$464$			2.00000i			0.0928477i
$465$	5.27795			0.244759
$466$			11.2779i			0.522441i
$467$		−	26.5559i		−	1.22886i	−0.788971	−	0.614430i	$-0.789386\pi$
$467$		−	26.5559i		−	1.22886i	0.788971	−	0.614430i	$-0.210614\pi$
$468$			0.440552i			0.0203645i
$469$	61.1118			2.82188
$470$	10.8811			0.501908
$471$	3.60316			0.166025
$472$	2.39684			0.110324
$473$			0.968518i			0.0445325i
$474$	−13.2779			−0.609876
$475$		−	0.440552i		−	0.0202139i
$476$			21.4807i			0.984567i
$477$	−2.44055			−0.111745
$478$	−22.5559			−1.03168
$479$		−	31.0122i		−	1.41698i	−0.705718	−	0.708492i	$-0.749376\pi$
$479$		−	31.0122i		−	1.41698i	0.705718	−	0.708492i	$-0.250624\pi$
$480$	−1.00000			−0.0456435
$481$	2.67835	+	0.0874152i	0.122122	+	0.00398579i
$482$	−26.5559			−1.20959
$483$		−	12.5996i		−	0.573302i
$484$	5.04371			0.229259
$485$	2.88110			0.130824
$486$		−	1.00000i		−	0.0453609i
$487$		−	7.20631i		−	0.326549i	−0.986581	−	0.163275i	$-0.947794\pi$
$487$		−	7.20631i		−	0.326549i	0.986581	−	0.163275i	$-0.0522056\pi$
$488$	−2.00000			−0.0905357
$489$			3.32165i			0.150210i
$490$	−12.7185			−0.574563
$491$	2.44055			0.110141			0.0550703	−	0.998482i	$-0.482462\pi$
$491$	2.44055			0.110141			0.0550703	+	0.998482i	$0.482462\pi$
$492$	2.00000			0.0901670
$493$	−9.67479			−0.435731
$494$		−	0.194086i		−	0.00873233i
$495$			2.44055i			0.109695i
$496$			5.27795i			0.236987i
$497$	42.9614			1.92708
$498$			7.71850i			0.345874i
$499$			36.7587i			1.64554i	0.568372	+	0.822772i	$0.307573\pi$
$499$			36.7587i			1.64554i	−0.568372	+	0.822772i	$0.692427\pi$
$500$		−	1.00000i		−	0.0447214i
$501$		−	2.83740i		−	0.126765i
$502$	−3.27795			−0.146302
$503$			29.1118i			1.29803i	0.760775	+	0.649015i	$0.224819\pi$
$503$			29.1118i			1.29803i	−0.760775	+	0.649015i	$0.775181\pi$
$504$		−	4.44055i		−	0.197798i
$505$			12.1591i			0.541071i
$506$	−6.92481			−0.307845
$507$	12.8059			0.568731
$508$	7.55945			0.335396
$509$	6.90903			0.306237			0.153119	−	0.988208i	$-0.451068\pi$
$509$	6.90903			0.306237			0.153119	+	0.988208i	$0.451068\pi$
$510$		−	4.83740i		−	0.214203i
$511$	−25.3933			−1.12333
$512$		−	1.00000i		−	0.0441942i
$513$			0.440552i			0.0194508i
$514$	20.0437			0.884090
$515$	10.5559			0.465148
$516$			0.396844i			0.0174701i
$517$	26.5559			1.16793
$518$	−26.9964	−	0.881103i	−1.18616	−	0.0387135i
$519$	−12.9964			−0.570481
$520$		−	0.440552i		−	0.0193195i
$521$	40.5559			1.77679			0.888393	−	0.459084i	$-0.151822\pi$
$521$	40.5559			1.77679			0.888393	+	0.459084i	$0.151822\pi$
$522$	2.00000			0.0875376
$523$		−	42.4772i		−	1.85740i	−0.370837	−	0.928698i	$-0.620929\pi$
$523$		−	42.4772i		−	1.85740i	0.370837	−	0.928698i	$-0.379071\pi$
$524$			12.1591i			0.531171i
$525$	4.44055			0.193802
$526$			17.4370i			0.760289i
$527$	−25.5315			−1.11217
$528$	−2.44055			−0.106211
$529$	14.9492			0.649965
$530$	2.44055			0.106011
$531$		−	2.39684i		−	0.104014i
$532$			1.95629i			0.0848160i
$533$			0.881103i			0.0381648i
$534$	−15.3217			−0.663033
$535$		−	18.9248i		−	0.818191i
$536$		−	13.7622i		−	0.594437i
$537$			4.72205i			0.203772i
$538$			16.4406i			0.708803i
$539$	−31.0402			−1.33699
$540$			1.00000i			0.0430331i
$541$		−	20.0437i		−	0.861746i	−0.902413	−	0.430873i	$-0.858206\pi$
$541$		−	20.0437i		−	0.861746i	0.902413	−	0.430873i	$-0.141794\pi$
$542$			24.3181i			1.04455i
$543$	20.5559			0.882138
$544$	4.83740			0.207402
$545$	12.0437			0.515896
$546$	1.95629			0.0837216
$547$			20.1153i			0.860070i	0.902812	+	0.430035i	$0.141499\pi$
$547$			20.1153i			0.860070i	−0.902812	+	0.430035i	$0.858501\pi$
$548$	−3.27795			−0.140027
$549$			2.00000i			0.0853579i
$550$		−	2.44055i		−	0.104065i
$551$	−0.881103			−0.0375363
$552$	−2.83740			−0.120768
$553$			58.9614i			2.50729i
$554$	0.353137			0.0150033
$555$	6.07953	+	0.198422i	0.258061	+	0.00842255i
$556$	−5.76221			−0.244372
$557$			11.7622i			0.498381i	0.968455	+	0.249190i	$0.0801645\pi$
$557$			11.7622i			0.498381i	−0.968455	+	0.249190i	$0.919836\pi$
$558$	5.27795			0.223433
$559$	−0.174830			−0.00739453
$560$			4.44055i			0.187648i
$561$		−	11.8059i		−	0.498446i
$562$	−20.2028			−0.852202
$563$			42.9614i			1.81061i	0.424765	+	0.905304i	$0.360357\pi$
$563$			42.9614i			1.81061i	−0.424765	+	0.905304i	$0.639643\pi$
$564$	10.8811			0.458177
$565$	−2.00000			−0.0841406
$566$	−13.6469			−0.573620
$567$	−4.44055			−0.186486
$568$		−	9.67479i		−	0.405945i
$569$			41.8775i			1.75560i	0.479029	+	0.877799i	$0.340989\pi$
$569$			41.8775i			1.75560i	−0.479029	+	0.877799i	$0.659011\pi$
$570$		−	0.440552i		−	0.0184527i
$571$	−37.7622			−1.58030			−0.790150	−	0.612914i	$-0.789997\pi$
$571$	−37.7622			−1.58030			−0.790150	+	0.612914i	$0.789997\pi$
$572$		−	1.07519i		−	0.0449559i
$573$		−	9.16260i		−	0.382773i
$574$		−	8.88110i		−	0.370690i
$575$		−	2.83740i		−	0.118328i
$576$	−1.00000			−0.0416667
$577$			2.56300i			0.106699i	0.998576	+	0.0533496i	$0.0169898\pi$
$577$			2.56300i			0.106699i	−0.998576	+	0.0533496i	$0.983010\pi$
$578$			6.40040i			0.266221i
$579$			14.8811i			0.618438i
$580$	−2.00000			−0.0830455
$581$	34.2744			1.42194
$582$	2.88110			0.119426
$583$	5.95629			0.246684
$584$			5.71850i			0.236633i
$585$	−0.440552			−0.0182146
$586$			6.44055i			0.266057i
$587$			45.1992i			1.86557i	0.360432	+	0.932785i	$0.382629\pi$
$587$			45.1992i			1.86557i	−0.360432	+	0.932785i	$0.617371\pi$
$588$	−12.7185			−0.524502
$589$	−2.32521			−0.0958085
$590$			2.39684i			0.0986765i
$591$	−9.55945			−0.393223
$592$	−0.198422	+	6.07953i	−0.00815510	+	0.249867i
$593$	43.5086			1.78669			0.893343	−	0.449376i	$-0.148354\pi$
$593$	43.5086			1.78669			0.893343	+	0.449376i	$0.148354\pi$
$594$			2.44055i			0.100137i
$595$	−21.4807			−0.880623
$596$	−1.60316			−0.0656678
$597$			20.4843i			0.838365i
$598$		−	1.25002i		−	0.0511171i
$599$	26.6433			1.08862			0.544308	−	0.838885i	$-0.316792\pi$
$599$	26.6433			1.08862			0.544308	+	0.838885i	$0.316792\pi$
$600$		−	1.00000i		−	0.0408248i
$601$	0.924811			0.0377238			0.0188619	−	0.999822i	$-0.493996\pi$
$601$	0.924811			0.0377238			0.0188619	+	0.999822i	$0.493996\pi$
$602$	1.76221			0.0718222
$603$	−13.7622			−0.560440
$604$	−4.28150			−0.174212
$605$			5.04371i			0.205056i
$606$			12.1591i			0.493928i
$607$			2.55589i			0.103741i	0.998654	+	0.0518703i	$0.0165182\pi$
$607$			2.55589i			0.103741i	−0.998654	+	0.0518703i	$0.983482\pi$
$608$	0.440552			0.0178667
$609$		−	8.88110i		−	0.359880i
$610$		−	2.00000i		−	0.0809776i
$611$			4.79369i			0.193932i
$612$		−	4.83740i		−	0.195540i
$613$	10.1591			0.410320			0.205160	−	0.978728i	$-0.434228\pi$
$613$	10.1591			0.410320			0.205160	+	0.978728i	$0.434228\pi$
$614$			20.3181i			0.819972i
$615$			2.00000i			0.0806478i
$616$			10.8374i			0.436651i
$617$	−12.1591			−0.489505			−0.244752	−	0.969586i	$-0.578707\pi$
$617$	−12.1591			−0.489505			−0.244752	+	0.969586i	$0.578707\pi$
$618$	10.5559			0.424620
$619$	23.5244			0.945526			0.472763	−	0.881190i	$-0.343257\pi$
$619$	23.5244			0.945526			0.472763	+	0.881190i	$0.343257\pi$
$620$	−5.27795			−0.211967
$621$			2.83740i			0.113861i
$622$	6.23779			0.250113
$623$			68.0366i			2.72583i
$624$		−	0.440552i		−	0.0176362i
$625$	1.00000			0.0400000
$626$	−8.55589			−0.341962
$627$		−	1.07519i		−	0.0429389i
$628$	−3.60316			−0.143782
$629$	−29.4091	−	0.959846i	−1.17262	−	0.0382716i
$630$	4.44055			0.176916
$631$		−	3.60316i		−	0.143439i	−0.997425	−	0.0717197i	$-0.977151\pi$
$631$		−	3.60316i		−	0.143439i	0.997425	−	0.0717197i	$-0.0228487\pi$
$632$	13.2779			0.528168
$633$	0.881103			0.0350207
$634$			33.5960i			1.33427i
$635$			7.55945i			0.299988i
$636$	2.44055			0.0967742
$637$		−	5.60316i		−	0.222005i
$638$	−4.88110			−0.193245
$639$	−9.67479			−0.382729
$640$	1.00000			0.0395285
$641$	−37.4370			−1.47867			−0.739336	−	0.673336i	$-0.764861\pi$
$641$	−37.4370			−1.47867			−0.739336	+	0.673336i	$0.764861\pi$
$642$		−	18.9248i		−	0.746903i
$643$			0.678345i			0.0267513i	0.999911	+	0.0133757i	$0.00425773\pi$
$643$			0.678345i			0.0267513i	−0.999911	+	0.0133757i	$0.995742\pi$
$644$			12.5996i			0.496494i
$645$	−0.396844			−0.0156257
$646$			2.13112i			0.0838479i
$647$			6.92481i			0.272242i	0.990692	+	0.136121i	$0.0434637\pi$
$647$			6.92481i			0.272242i	−0.990692	+	0.136121i	$0.956536\pi$
$648$			1.00000i			0.0392837i
$649$			5.84962i			0.229618i
$650$	0.440552			0.0172799
$651$		−	23.4370i		−	0.918568i
$652$		−	3.32165i		−	0.130086i
$653$		−	15.4441i		−	0.604375i	−0.953249	−	0.302187i	$-0.902283\pi$
$653$		−	15.4441i		−	0.604375i	0.953249	−	0.302187i	$-0.0977168\pi$
$654$	12.0437			0.470946
$655$	−12.1591			−0.475093
$656$	−2.00000			−0.0780869
$657$	5.71850			0.223100
$658$		−	48.3181i		−	1.88364i
$659$	−18.9527			−0.738294			−0.369147	−	0.929371i	$-0.620350\pi$
$659$	−18.9527			−0.738294			−0.369147	+	0.929371i	$0.620350\pi$
$660$		−	2.44055i		−	0.0949983i
$661$			36.3618i			1.41431i	0.707058	+	0.707155i	$0.250022\pi$
$661$			36.3618i			1.41431i	−0.707058	+	0.707155i	$0.749978\pi$
$662$	31.2779			1.21565
$663$	2.13112			0.0827659
$664$		−	7.71850i		−	0.299536i
$665$	−1.95629			−0.0758618
$666$	6.07953	+	0.198422i	0.235577	+	0.00768870i
$667$	−5.67479			−0.219729
$668$			2.83740i			0.109782i
$669$	1.51574			0.0586019
$670$	13.7622			0.531680
$671$		−	4.88110i		−	0.188433i
$672$			4.44055i			0.171298i
$673$	9.80591			0.377990			0.188995	−	0.981978i	$-0.439477\pi$
$673$	9.80591			0.377990			0.188995	+	0.981978i	$0.439477\pi$
$674$			5.63108i			0.216901i
$675$	−1.00000			−0.0384900
$676$	−12.8059			−0.492535
$677$	28.2902			1.08728			0.543640	−	0.839319i	$-0.317046\pi$
$677$	28.2902			1.08728			0.543640	+	0.839319i	$0.317046\pi$
$678$	−2.00000			−0.0768095
$679$		−	12.7937i		−	0.490977i
$680$			4.83740i			0.185506i
$681$			14.5559i			0.557783i
$682$	−12.8811			−0.493243
$683$			24.3181i			0.930506i	0.885178	+	0.465253i	$0.154037\pi$
$683$			24.3181i			0.930506i	−0.885178	+	0.465253i	$0.845963\pi$
$684$		−	0.440552i		−	0.0168449i
$685$		−	3.27795i		−	0.125244i
$686$			25.3933i			0.969520i
$687$	15.6748			0.598031
$688$		−	0.396844i		−	0.0151295i
$689$			1.07519i			0.0409614i
$690$		−	2.83740i		−	0.108018i
$691$	5.53152			0.210429			0.105214	−	0.994450i	$-0.466447\pi$
$691$	5.53152			0.210429			0.105214	+	0.994450i	$0.466447\pi$
$692$	12.9964			0.494051
$693$	10.8374			0.411679
$694$	0.881103			0.0334462
$695$		−	5.76221i		−	0.218573i
$696$	−2.00000			−0.0758098
$697$		−	9.67479i		−	0.366459i
$698$		−	0.555895i		−	0.0210409i
$699$	−11.2779			−0.426571
$700$	−4.44055			−0.167837
$701$			23.6748i			0.894185i	0.894488	+	0.447092i	$0.147540\pi$
$701$			23.6748i			0.894185i	−0.894488	+	0.447092i	$0.852460\pi$
$702$	−0.440552			−0.0166276
$703$	−2.67835	−	0.0874152i	−0.101016	−	0.00329693i
$704$	2.44055			0.0919818
$705$			10.8811i			0.409806i
$706$	−5.20631			−0.195942
$707$	53.9929			2.03061
$708$			2.39684i			0.0900789i
$709$		−	3.07519i		−	0.115491i	−0.998331	−	0.0577456i	$-0.981609\pi$
$709$		−	3.07519i		−	0.115491i	0.998331	−	0.0577456i	$-0.0183912\pi$
$710$	9.67479			0.363088
$711$		−	13.2779i		−	0.497962i
$712$	15.3217			0.574203
$713$	−14.9756			−0.560842
$714$	−21.4807			−0.803895
$715$	1.07519			0.0402098
$716$		−	4.72205i		−	0.176471i
$717$		−	22.5559i		−	0.842365i
$718$			15.4370i			0.576104i
$719$	−31.3496			−1.16914			−0.584571	−	0.811342i	$-0.698737\pi$
$719$	−31.3496			−1.16914			−0.584571	+	0.811342i	$0.698737\pi$
$720$		−	1.00000i		−	0.0372678i
$721$		−	46.8740i		−	1.74568i
$722$		−	18.8059i		−	0.699884i
$723$		−	26.5559i		−	0.987624i
$724$	−20.5559			−0.763954
$725$		−	2.00000i		−	0.0742781i
$726$			5.04371i			0.187190i
$727$			20.0874i			0.745001i	0.928032	+	0.372500i	$0.121500\pi$
$727$			20.0874i			0.745001i	−0.928032	+	0.372500i	$0.878500\pi$
$728$	−1.95629			−0.0725050
$729$	1.00000			0.0370370
$730$	−5.71850			−0.211651
$731$	1.91969			0.0710024
$732$		−	2.00000i		−	0.0739221i
$733$	7.83384			0.289350			0.144675	−	0.989479i	$-0.453786\pi$
$733$	7.83384			0.289350			0.144675	+	0.989479i	$0.453786\pi$
$734$			14.9964i			0.553529i
$735$		−	12.7185i		−	0.469129i
$736$	2.83740			0.104588
$737$	33.5874			1.23721
$738$			2.00000i			0.0736210i
$739$	−52.6362			−1.93625			−0.968127	−	0.250460i	$-0.919418\pi$
$739$	−52.6362			−1.93625			−0.968127	+	0.250460i	$0.919418\pi$
$740$	−6.07953	−	0.198422i	−0.223488	−	0.00729414i
$741$	0.194086			0.00712992
$742$		−	10.8374i		−	0.397853i
$743$	−8.55589			−0.313885			−0.156943	−	0.987608i	$-0.550164\pi$
$743$	−8.55589			−0.313885			−0.156943	+	0.987608i	$0.550164\pi$
$744$	−5.27795			−0.193499
$745$		−	1.60316i		−	0.0587351i
$746$		−	33.5086i		−	1.22684i
$747$	−7.71850			−0.282405
$748$			11.8059i			0.431667i
$749$	−84.0366			−3.07063
$750$	1.00000			0.0365148
$751$	16.0000			0.583848			0.291924	−	0.956441i	$-0.405705\pi$
$751$	16.0000			0.583848			0.291924	+	0.956441i	$0.405705\pi$
$752$	−10.8811			−0.396793
$753$		−	3.27795i		−	0.119455i
$754$		−	0.881103i		−	0.0320879i
$755$		−	4.28150i		−	0.155820i
$756$	4.44055			0.161501
$757$		−	39.4020i		−	1.43209i	−0.698055	−	0.716044i	$-0.745951\pi$
$757$		−	39.4020i		−	1.43209i	0.698055	−	0.716044i	$-0.254049\pi$
$758$		−	34.6433i		−	1.25830i
$759$		−	6.92481i		−	0.251355i
$760$			0.440552i			0.0159805i
$761$	−40.5559			−1.47015			−0.735075	−	0.677986i	$-0.762853\pi$
$761$	−40.5559			−1.47015			−0.735075	+	0.677986i	$0.762853\pi$
$762$			7.55945i			0.273850i
$763$		−	53.4807i		−	1.93613i
$764$			9.16260i			0.331491i
$765$	4.83740			0.174896
$766$	8.51219			0.307558
$767$	−1.05593			−0.0381276
$768$	1.00000			0.0360844
$769$			4.08742i			0.147396i	0.997281	+	0.0736980i	$0.0234801\pi$
$769$			4.08742i			0.147396i	−0.997281	+	0.0736980i	$0.976520\pi$
$770$	−10.8374			−0.390553
$771$			20.0437i			0.721856i
$772$		−	14.8811i		−	0.535583i
$773$	24.6783			0.887618			0.443809	−	0.896121i	$-0.353627\pi$
$773$	24.6783			0.887618			0.443809	+	0.896121i	$0.353627\pi$
$774$	−0.396844			−0.0142643
$775$		−	5.27795i		−	0.189589i
$776$	−2.88110			−0.103426
$777$	0.881103	−	26.9964i	0.0316094	−	0.968492i
$778$	27.6748			0.992189
$779$		−	0.881103i		−	0.0315688i
$780$	0.440552			0.0157743
$781$	23.6118			0.844898
$782$			13.7256i			0.490827i
$783$			2.00000i			0.0714742i
$784$	12.7185			0.454232
$785$		−	3.60316i		−	0.128602i
$786$	−12.1591			−0.433699
$787$	−37.1992			−1.32601			−0.663004	−	0.748616i	$-0.730719\pi$
$787$	−37.1992			−1.32601			−0.663004	+	0.748616i	$0.730719\pi$
$788$	9.55945			0.340541
$789$	−17.4370			−0.620774
$790$			13.2779i			0.472408i
$791$			8.88110i			0.315776i
$792$		−	2.44055i		−	0.0867212i
$793$	0.881103			0.0312889

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:	\( N \)	\(=\)	\( 1110 = 2 \cdot 3 \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1110.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{6} -4.44055 q^{7} +1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{6} -4.44055 q^{7} +1.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} -2.44055 q^{11} -1.00000 q^{12} -0.440552i q^{13} +4.44055i q^{14} -1.00000i q^{15} +1.00000 q^{16} +4.83740i q^{17} -1.00000i q^{18} +0.440552i q^{19} +1.00000i q^{20} -4.44055 q^{21} +2.44055i q^{22} +2.83740i q^{23} +1.00000i q^{24} -1.00000 q^{25} -0.440552 q^{26} +1.00000 q^{27} +4.44055 q^{28} +2.00000i q^{29} -1.00000 q^{30} +5.27795i q^{31} -1.00000i q^{32} -2.44055 q^{33} +4.83740 q^{34} +4.44055i q^{35} -1.00000 q^{36} +(-0.198422 + 6.07953i) q^{37} +0.440552 q^{38} -0.440552i q^{39} +1.00000 q^{40} -2.00000 q^{41} +4.44055i q^{42} -0.396844i q^{43} +2.44055 q^{44} -1.00000i q^{45} +2.83740 q^{46} -10.8811 q^{47} +1.00000 q^{48} +12.7185 q^{49} +1.00000i q^{50} +4.83740i q^{51} +0.440552i q^{52} -2.44055 q^{53} -1.00000i q^{54} +2.44055i q^{55} -4.44055i q^{56} +0.440552i q^{57} +2.00000 q^{58} -2.39684i q^{59} +1.00000i q^{60} +2.00000i q^{61} +5.27795 q^{62} -4.44055 q^{63} -1.00000 q^{64} -0.440552 q^{65} +2.44055i q^{66} -13.7622 q^{67} -4.83740i q^{68} +2.83740i q^{69} +4.44055 q^{70} -9.67479 q^{71} +1.00000i q^{72} +5.71850 q^{73} +(6.07953 + 0.198422i) q^{74} -1.00000 q^{75} -0.440552i q^{76} +10.8374 q^{77} -0.440552 q^{78} -13.2779i q^{79} -1.00000i q^{80} +1.00000 q^{81} +2.00000i q^{82} -7.71850 q^{83} +4.44055 q^{84} +4.83740 q^{85} -0.396844 q^{86} +2.00000i q^{87} -2.44055i q^{88} -15.3217i q^{89} -1.00000 q^{90} +1.95629i q^{91} -2.83740i q^{92} +5.27795i q^{93} +10.8811i q^{94} +0.440552 q^{95} -1.00000i q^{96} +2.88110i q^{97} -12.7185i q^{98} -2.44055 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6q + 6q^{3} - 6q^{4} - 6q^{7} + 6q^{9} + O(q^{10}) \)	\( 6q + 6q^{3} - 6q^{4} - 6q^{7} + 6q^{9} - 6q^{10} + 6q^{11} - 6q^{12} + 6q^{16} - 6q^{21} - 6q^{25} + 18q^{26} + 6q^{27} + 6q^{28} - 6q^{30} + 6q^{33} + 10q^{34} - 6q^{36} - 2q^{37} - 18q^{38} + 6q^{40} - 12q^{41} - 6q^{44} - 2q^{46} - 24q^{47} + 6q^{48} + 16q^{49} + 6q^{53} + 12q^{58} - 8q^{62} - 6q^{63} - 6q^{64} + 18q^{65} + 6q^{70} - 20q^{71} - 26q^{73} - 4q^{74} - 6q^{75} + 46q^{77} + 18q^{78} + 6q^{81} + 14q^{83} + 6q^{84} + 10q^{85} - 4q^{86} - 6q^{90} - 18q^{95} + 6q^{99} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

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