LMFDB - Embedded newform 1110.2.e.a.739.2

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	$8.86339462436$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
Defining polynomial:	$x^{2} + 1$
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			739.2
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	1110.739
Dual form			1110.2.e.a.739.1

$q$-expansion

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000i q^{3} +1.00000 q^{4} +(2.00000 + 1.00000i) q^{5} -1.00000i q^{6} -3.00000i q^{7} -1.00000 q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000i q^{3} +1.00000 q^{4} +(2.00000 + 1.00000i) q^{5} -1.00000i q^{6} -3.00000i q^{7} -1.00000 q^{8} -1.00000 q^{9} +(-2.00000 - 1.00000i) q^{10} -3.00000 q^{11} +1.00000i q^{12} -4.00000 q^{13} +3.00000i q^{14} +(-1.00000 + 2.00000i) q^{15} +1.00000 q^{16} +7.00000 q^{17} +1.00000 q^{18} +4.00000i q^{19} +(2.00000 + 1.00000i) q^{20} +3.00000 q^{21} +3.00000 q^{22} +6.00000 q^{23} -1.00000i q^{24} +(3.00000 + 4.00000i) q^{25} +4.00000 q^{26} -1.00000i q^{27} -3.00000i q^{28} +9.00000i q^{29} +(1.00000 - 2.00000i) q^{30} +5.00000i q^{31} -1.00000 q^{32} -3.00000i q^{33} -7.00000 q^{34} +(3.00000 - 6.00000i) q^{35} -1.00000 q^{36} +(6.00000 + 1.00000i) q^{37} -4.00000i q^{38} -4.00000i q^{39} +(-2.00000 - 1.00000i) q^{40} +7.00000 q^{41} -3.00000 q^{42} +1.00000 q^{43} -3.00000 q^{44} +(-2.00000 - 1.00000i) q^{45} -6.00000 q^{46} -8.00000i q^{47} +1.00000i q^{48} -2.00000 q^{49} +(-3.00000 - 4.00000i) q^{50} +7.00000i q^{51} -4.00000 q^{52} -1.00000i q^{53} +1.00000i q^{54} +(-6.00000 - 3.00000i) q^{55} +3.00000i q^{56} -4.00000 q^{57} -9.00000i q^{58} -6.00000i q^{59} +(-1.00000 + 2.00000i) q^{60} +5.00000i q^{61} -5.00000i q^{62} +3.00000i q^{63} +1.00000 q^{64} +(-8.00000 - 4.00000i) q^{65} +3.00000i q^{66} +2.00000i q^{67} +7.00000 q^{68} +6.00000i q^{69} +(-3.00000 + 6.00000i) q^{70} +12.0000 q^{71} +1.00000 q^{72} +4.00000i q^{73} +(-6.00000 - 1.00000i) q^{74} +(-4.00000 + 3.00000i) q^{75} +4.00000i q^{76} +9.00000i q^{77} +4.00000i q^{78} +4.00000i q^{79} +(2.00000 + 1.00000i) q^{80} +1.00000 q^{81} -7.00000 q^{82} +14.0000i q^{83} +3.00000 q^{84} +(14.0000 + 7.00000i) q^{85} -1.00000 q^{86} -9.00000 q^{87} +3.00000 q^{88} -6.00000i q^{89} +(2.00000 + 1.00000i) q^{90} +12.0000i q^{91} +6.00000 q^{92} -5.00000 q^{93} +8.00000i q^{94} +(-4.00000 + 8.00000i) q^{95} -1.00000i q^{96} +7.00000 q^{97} +2.00000 q^{98} +3.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2q - 2q^{2} + 2q^{4} + 4q^{5} - 2q^{8} - 2q^{9} + O(q^{10}) $	$ 2q - 2q^{2} + 2q^{4} + 4q^{5} - 2q^{8} - 2q^{9} - 4q^{10} - 6q^{11} - 8q^{13} - 2q^{15} + 2q^{16} + 14q^{17} + 2q^{18} + 4q^{20} + 6q^{21} + 6q^{22} + 12q^{23} + 6q^{25} + 8q^{26} + 2q^{30} - 2q^{32} - 14q^{34} + 6q^{35} - 2q^{36} + 12q^{37} - 4q^{40} + 14q^{41} - 6q^{42} + 2q^{43} - 6q^{44} - 4q^{45} - 12q^{46} - 4q^{49} - 6q^{50} - 8q^{52} - 12q^{55} - 8q^{57} - 2q^{60} + 2q^{64} - 16q^{65} + 14q^{68} - 6q^{70} + 24q^{71} + 2q^{72} - 12q^{74} - 8q^{75} + 4q^{80} + 2q^{81} - 14q^{82} + 6q^{84} + 28q^{85} - 2q^{86} - 18q^{87} + 6q^{88} + 4q^{90} + 12q^{92} - 10q^{93} - 8q^{95} + 14q^{97} + 4q^{98} + 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 $

$2$	−1.00000			−0.707107
$2$	−1.00000			−0.707107
$3$			1.00000i			0.577350i
$3$			1.00000i			0.577350i
$4$	1.00000			0.500000
$5$	2.00000	+	1.00000i	0.894427	+	0.447214i
$5$	2.00000	+	1.00000i	0.894427	+	0.447214i
$6$		−	1.00000i		−	0.408248i
$7$		−	3.00000i		−	1.13389i	−0.823754	−	0.566947i	$-0.808125\pi$
$7$		−	3.00000i		−	1.13389i	0.823754	−	0.566947i	$-0.191875\pi$
$8$	−1.00000			−0.353553
$9$	−1.00000			−0.333333
$10$	−2.00000	−	1.00000i	−0.632456	−	0.316228i
$11$	−3.00000			−0.904534			−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000			−0.904534			−0.452267	+	0.891883i	$0.649385\pi$
$12$			1.00000i			0.288675i
$13$	−4.00000			−1.10940			−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000			−1.10940			−0.554700	+	0.832050i	$0.687167\pi$
$14$			3.00000i			0.801784i
$15$	−1.00000	+	2.00000i	−0.258199	+	0.516398i
$16$	1.00000			0.250000
$17$	7.00000			1.69775			0.848875	−	0.528594i	$-0.177281\pi$
$17$	7.00000			1.69775			0.848875	+	0.528594i	$0.177281\pi$
$18$	1.00000			0.235702
$19$			4.00000i			0.917663i	0.888523	+	0.458831i	$0.151732\pi$
$19$			4.00000i			0.917663i	−0.888523	+	0.458831i	$0.848268\pi$
$20$	2.00000	+	1.00000i	0.447214	+	0.223607i
$21$	3.00000			0.654654
$22$	3.00000			0.639602
$23$	6.00000			1.25109			0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000			1.25109			0.625543	+	0.780189i	$0.284877\pi$
$24$		−	1.00000i		−	0.204124i
$25$	3.00000	+	4.00000i	0.600000	+	0.800000i
$26$	4.00000			0.784465
$27$		−	1.00000i		−	0.192450i
$28$		−	3.00000i		−	0.566947i
$29$			9.00000i			1.67126i	0.549294	+	0.835629i	$0.314897\pi$
$29$			9.00000i			1.67126i	−0.549294	+	0.835629i	$0.685103\pi$
$30$	1.00000	−	2.00000i	0.182574	−	0.365148i
$31$			5.00000i			0.898027i	0.893525	+	0.449013i	$0.148224\pi$
$31$			5.00000i			0.898027i	−0.893525	+	0.449013i	$0.851776\pi$
$32$	−1.00000			−0.176777
$33$		−	3.00000i		−	0.522233i
$34$	−7.00000			−1.20049
$35$	3.00000	−	6.00000i	0.507093	−	1.01419i
$36$	−1.00000			−0.166667
$37$	6.00000	+	1.00000i	0.986394	+	0.164399i
$37$	6.00000	+	1.00000i	0.986394	+	0.164399i
$38$		−	4.00000i		−	0.648886i
$39$		−	4.00000i		−	0.640513i
$40$	−2.00000	−	1.00000i	−0.316228	−	0.158114i
$41$	7.00000			1.09322			0.546608	−	0.837389i	$-0.315919\pi$
$41$	7.00000			1.09322			0.546608	+	0.837389i	$0.315919\pi$
$42$	−3.00000			−0.462910
$43$	1.00000			0.152499			0.0762493	−	0.997089i	$-0.475706\pi$
$43$	1.00000			0.152499			0.0762493	+	0.997089i	$0.475706\pi$
$44$	−3.00000			−0.452267
$45$	−2.00000	−	1.00000i	−0.298142	−	0.149071i
$46$	−6.00000			−0.884652
$47$		−	8.00000i		−	1.16692i	−0.812142	−	0.583460i	$-0.801699\pi$
$47$		−	8.00000i		−	1.16692i	0.812142	−	0.583460i	$-0.198301\pi$
$48$			1.00000i			0.144338i
$49$	−2.00000			−0.285714
$50$	−3.00000	−	4.00000i	−0.424264	−	0.565685i
$51$			7.00000i			0.980196i
$52$	−4.00000			−0.554700
$53$		−	1.00000i		−	0.137361i	−0.997639	−	0.0686803i	$-0.978121\pi$
$53$		−	1.00000i		−	0.137361i	0.997639	−	0.0686803i	$-0.0218788\pi$
$54$			1.00000i			0.136083i
$55$	−6.00000	−	3.00000i	−0.809040	−	0.404520i
$56$			3.00000i			0.400892i
$57$	−4.00000			−0.529813
$58$		−	9.00000i		−	1.18176i
$59$		−	6.00000i		−	0.781133i	−0.920575	−	0.390567i	$-0.872279\pi$
$59$		−	6.00000i		−	0.781133i	0.920575	−	0.390567i	$-0.127721\pi$
$60$	−1.00000	+	2.00000i	−0.129099	+	0.258199i
$61$			5.00000i			0.640184i	0.947386	+	0.320092i	$0.103714\pi$
$61$			5.00000i			0.640184i	−0.947386	+	0.320092i	$0.896286\pi$
$62$		−	5.00000i		−	0.635001i
$63$			3.00000i			0.377964i
$64$	1.00000			0.125000
$65$	−8.00000	−	4.00000i	−0.992278	−	0.496139i
$66$			3.00000i			0.369274i
$67$			2.00000i			0.244339i	0.992509	+	0.122169i	$0.0389851\pi$
$67$			2.00000i			0.244339i	−0.992509	+	0.122169i	$0.961015\pi$
$68$	7.00000			0.848875
$69$			6.00000i			0.722315i
$70$	−3.00000	+	6.00000i	−0.358569	+	0.717137i
$71$	12.0000			1.42414			0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000			1.42414			0.712069	+	0.702109i	$0.247758\pi$
$72$	1.00000			0.117851
$73$			4.00000i			0.468165i	0.972217	+	0.234082i	$0.0752085\pi$
$73$			4.00000i			0.468165i	−0.972217	+	0.234082i	$0.924791\pi$
$74$	−6.00000	−	1.00000i	−0.697486	−	0.116248i
$75$	−4.00000	+	3.00000i	−0.461880	+	0.346410i
$76$			4.00000i			0.458831i
$77$			9.00000i			1.02565i
$78$			4.00000i			0.452911i
$79$			4.00000i			0.450035i	0.974355	+	0.225018i	$0.0722440\pi$
$79$			4.00000i			0.450035i	−0.974355	+	0.225018i	$0.927756\pi$
$80$	2.00000	+	1.00000i	0.223607	+	0.111803i
$81$	1.00000			0.111111
$82$	−7.00000			−0.773021
$83$			14.0000i			1.53670i	0.640030	+	0.768350i	$0.278922\pi$
$83$			14.0000i			1.53670i	−0.640030	+	0.768350i	$0.721078\pi$
$84$	3.00000			0.327327
$85$	14.0000	+	7.00000i	1.51851	+	0.759257i
$86$	−1.00000			−0.107833
$87$	−9.00000			−0.964901
$88$	3.00000			0.319801
$89$		−	6.00000i		−	0.635999i	−0.948091	−	0.317999i	$-0.896989\pi$
$89$		−	6.00000i		−	0.635999i	0.948091	−	0.317999i	$-0.103011\pi$
$90$	2.00000	+	1.00000i	0.210819	+	0.105409i
$91$			12.0000i			1.25794i
$92$	6.00000			0.625543
$93$	−5.00000			−0.518476
$94$			8.00000i			0.825137i
$95$	−4.00000	+	8.00000i	−0.410391	+	0.820783i
$96$		−	1.00000i		−	0.102062i
$97$	7.00000			0.710742			0.355371	−	0.934725i	$-0.384354\pi$
$97$	7.00000			0.710742			0.355371	+	0.934725i	$0.384354\pi$
$98$	2.00000			0.202031
$99$	3.00000			0.301511
$100$	3.00000	+	4.00000i	0.300000	+	0.400000i
$101$	2.00000			0.199007			0.0995037	−	0.995037i	$-0.468274\pi$
$101$	2.00000			0.199007			0.0995037	+	0.995037i	$0.468274\pi$
$102$		−	7.00000i		−	0.693103i
$103$	−14.0000			−1.37946			−0.689730	−	0.724066i	$-0.742271\pi$
$103$	−14.0000			−1.37946			−0.689730	+	0.724066i	$0.742271\pi$
$104$	4.00000			0.392232
$105$	6.00000	+	3.00000i	0.585540	+	0.292770i
$106$			1.00000i			0.0971286i
$107$		−	18.0000i		−	1.74013i	−0.492941	−	0.870063i	$-0.664078\pi$
$107$		−	18.0000i		−	1.74013i	0.492941	−	0.870063i	$-0.335922\pi$
$108$		−	1.00000i		−	0.0962250i
$109$		−	11.0000i		−	1.05361i	−0.849987	−	0.526804i	$-0.823390\pi$
$109$		−	11.0000i		−	1.05361i	0.849987	−	0.526804i	$-0.176610\pi$
$110$	6.00000	+	3.00000i	0.572078	+	0.286039i
$111$	−1.00000	+	6.00000i	−0.0949158	+	0.569495i
$112$		−	3.00000i		−	0.283473i
$113$	1.00000			0.0940721			0.0470360	−	0.998893i	$-0.485022\pi$
$113$	1.00000			0.0940721			0.0470360	+	0.998893i	$0.485022\pi$
$114$	4.00000			0.374634
$115$	12.0000	+	6.00000i	1.11901	+	0.559503i
$116$			9.00000i			0.835629i
$117$	4.00000			0.369800
$118$			6.00000i			0.552345i
$119$		−	21.0000i		−	1.92507i
$120$	1.00000	−	2.00000i	0.0912871	−	0.182574i
$121$	−2.00000			−0.181818
$122$		−	5.00000i		−	0.452679i
$123$			7.00000i			0.631169i
$124$			5.00000i			0.449013i
$125$	2.00000	+	11.0000i	0.178885	+	0.983870i
$126$		−	3.00000i		−	0.267261i
$127$		−	8.00000i		−	0.709885i	−0.934888	−	0.354943i	$-0.884500\pi$
$127$		−	8.00000i		−	0.709885i	0.934888	−	0.354943i	$-0.115500\pi$
$128$	−1.00000			−0.0883883
$129$			1.00000i			0.0880451i
$130$	8.00000	+	4.00000i	0.701646	+	0.350823i
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		−	3.00000i		−	0.261116i
$133$	12.0000			1.04053
$134$		−	2.00000i		−	0.172774i
$135$	1.00000	−	2.00000i	0.0860663	−	0.172133i
$136$	−7.00000			−0.600245
$137$			12.0000i			1.02523i	0.858619	+	0.512615i	$0.171323\pi$
$137$			12.0000i			1.02523i	−0.858619	+	0.512615i	$0.828677\pi$
$138$		−	6.00000i		−	0.510754i
$139$	−15.0000			−1.27228			−0.636142	−	0.771572i	$-0.719471\pi$
$139$	−15.0000			−1.27228			−0.636142	+	0.771572i	$0.719471\pi$
$140$	3.00000	−	6.00000i	0.253546	−	0.507093i
$141$	8.00000			0.673722
$142$	−12.0000			−1.00702
$143$	12.0000			1.00349
$144$	−1.00000			−0.0833333
$145$	−9.00000	+	18.0000i	−0.747409	+	1.49482i
$146$		−	4.00000i		−	0.331042i
$147$		−	2.00000i		−	0.164957i
$148$	6.00000	+	1.00000i	0.493197	+	0.0821995i
$149$		0			0			−	1.00000i	$-0.5\pi$
$149$		0			0				1.00000i	$0.5\pi$
$150$	4.00000	−	3.00000i	0.326599	−	0.244949i
$151$	2.00000			0.162758			0.0813788	−	0.996683i	$-0.474068\pi$
$151$	2.00000			0.162758			0.0813788	+	0.996683i	$0.474068\pi$
$152$		−	4.00000i		−	0.324443i
$153$	−7.00000			−0.565916
$154$		−	9.00000i		−	0.725241i
$155$	−5.00000	+	10.0000i	−0.401610	+	0.803219i
$156$		−	4.00000i		−	0.320256i
$157$			7.00000i			0.558661i	0.960195	+	0.279330i	$0.0901125\pi$
$157$			7.00000i			0.558661i	−0.960195	+	0.279330i	$0.909888\pi$
$158$		−	4.00000i		−	0.318223i
$159$	1.00000			0.0793052
$160$	−2.00000	−	1.00000i	−0.158114	−	0.0790569i
$161$		−	18.0000i		−	1.41860i
$162$	−1.00000			−0.0785674
$163$	−9.00000			−0.704934			−0.352467	−	0.935824i	$-0.614657\pi$
$163$	−9.00000			−0.704934			−0.352467	+	0.935824i	$0.614657\pi$
$164$	7.00000			0.546608
$165$	3.00000	−	6.00000i	0.233550	−	0.467099i
$166$		−	14.0000i		−	1.08661i
$167$	12.0000			0.928588			0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000			0.928588			0.464294	+	0.885681i	$0.346308\pi$
$168$	−3.00000			−0.231455
$169$	3.00000			0.230769
$170$	−14.0000	−	7.00000i	−1.07375	−	0.536875i
$171$		−	4.00000i		−	0.305888i
$172$	1.00000			0.0762493
$173$			9.00000i			0.684257i	0.939653	+	0.342129i	$0.111148\pi$
$173$			9.00000i			0.684257i	−0.939653	+	0.342129i	$0.888852\pi$
$174$	9.00000			0.682288
$175$	12.0000	−	9.00000i	0.907115	−	0.680336i
$176$	−3.00000			−0.226134
$177$	6.00000			0.450988
$178$			6.00000i			0.449719i
$179$			4.00000i			0.298974i	0.988764	+	0.149487i	$0.0477622\pi$
$179$			4.00000i			0.298974i	−0.988764	+	0.149487i	$0.952238\pi$
$180$	−2.00000	−	1.00000i	−0.149071	−	0.0745356i
$181$	−18.0000			−1.33793			−0.668965	−	0.743294i	$-0.733262\pi$
$181$	−18.0000			−1.33793			−0.668965	+	0.743294i	$0.733262\pi$
$182$		−	12.0000i		−	0.889499i
$183$	−5.00000			−0.369611
$184$	−6.00000			−0.442326
$185$	11.0000	+	8.00000i	0.808736	+	0.588172i
$186$	5.00000			0.366618
$187$	−21.0000			−1.53567
$188$		−	8.00000i		−	0.583460i
$189$	−3.00000			−0.218218
$190$	4.00000	−	8.00000i	0.290191	−	0.580381i
$191$			15.0000i			1.08536i	0.839939	+	0.542681i	$0.182591\pi$
$191$			15.0000i			1.08536i	−0.839939	+	0.542681i	$0.817409\pi$
$192$			1.00000i			0.0721688i
$193$	−14.0000			−1.00774			−0.503871	−	0.863779i	$-0.668091\pi$
$193$	−14.0000			−1.00774			−0.503871	+	0.863779i	$0.668091\pi$
$194$	−7.00000			−0.502571
$195$	4.00000	−	8.00000i	0.286446	−	0.572892i
$196$	−2.00000			−0.142857
$197$		−	18.0000i		−	1.28245i	−0.767354	−	0.641223i	$-0.778427\pi$
$197$		−	18.0000i		−	1.28245i	0.767354	−	0.641223i	$-0.221573\pi$
$198$	−3.00000			−0.213201
$199$		−	16.0000i		−	1.13421i	−0.823646	−	0.567105i	$-0.808063\pi$
$199$		−	16.0000i		−	1.13421i	0.823646	−	0.567105i	$-0.191937\pi$
$200$	−3.00000	−	4.00000i	−0.212132	−	0.282843i
$201$	−2.00000			−0.141069
$202$	−2.00000			−0.140720
$203$	27.0000			1.89503
$204$			7.00000i			0.490098i
$205$	14.0000	+	7.00000i	0.977802	+	0.488901i
$206$	14.0000			0.975426
$207$	−6.00000			−0.417029
$208$	−4.00000			−0.277350
$209$		−	12.0000i		−	0.830057i
$210$	−6.00000	−	3.00000i	−0.414039	−	0.207020i
$211$	−13.0000			−0.894957			−0.447478	−	0.894295i	$-0.647678\pi$
$211$	−13.0000			−0.894957			−0.447478	+	0.894295i	$0.647678\pi$
$212$		−	1.00000i		−	0.0686803i
$213$			12.0000i			0.822226i
$214$			18.0000i			1.23045i
$215$	2.00000	+	1.00000i	0.136399	+	0.0681994i
$216$			1.00000i			0.0680414i
$217$	15.0000			1.01827
$218$			11.0000i			0.745014i
$219$	−4.00000			−0.270295
$220$	−6.00000	−	3.00000i	−0.404520	−	0.202260i
$221$	−28.0000			−1.88348
$222$	1.00000	−	6.00000i	0.0671156	−	0.402694i
$223$		−	11.0000i		−	0.736614i	−0.929704	−	0.368307i	$-0.879937\pi$
$223$		−	11.0000i		−	0.736614i	0.929704	−	0.368307i	$-0.120063\pi$
$224$			3.00000i			0.200446i
$225$	−3.00000	−	4.00000i	−0.200000	−	0.266667i
$226$	−1.00000			−0.0665190
$227$	−13.0000			−0.862840			−0.431420	−	0.902151i	$-0.641987\pi$
$227$	−13.0000			−0.862840			−0.431420	+	0.902151i	$0.641987\pi$
$228$	−4.00000			−0.264906
$229$	10.0000			0.660819			0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000			0.660819			0.330409	+	0.943838i	$0.392813\pi$
$230$	−12.0000	−	6.00000i	−0.791257	−	0.395628i
$231$	−9.00000			−0.592157
$232$		−	9.00000i		−	0.590879i
$233$		−	6.00000i		−	0.393073i	−0.980497	−	0.196537i	$-0.937031\pi$
$233$		−	6.00000i		−	0.393073i	0.980497	−	0.196537i	$-0.0629694\pi$
$234$	−4.00000			−0.261488
$235$	8.00000	−	16.0000i	0.521862	−	1.04372i
$236$		−	6.00000i		−	0.390567i
$237$	−4.00000			−0.259828
$238$			21.0000i			1.36123i
$239$			9.00000i			0.582162i	0.956698	+	0.291081i	$0.0940149\pi$
$239$			9.00000i			0.582162i	−0.956698	+	0.291081i	$0.905985\pi$
$240$	−1.00000	+	2.00000i	−0.0645497	+	0.129099i
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$	2.00000			0.128565
$243$			1.00000i			0.0641500i
$244$			5.00000i			0.320092i
$245$	−4.00000	−	2.00000i	−0.255551	−	0.127775i
$246$		−	7.00000i		−	0.446304i
$247$		−	16.0000i		−	1.01806i
$248$		−	5.00000i		−	0.317500i
$249$	−14.0000			−0.887214
$250$	−2.00000	−	11.0000i	−0.126491	−	0.695701i
$251$		−	20.0000i		−	1.26239i	−0.775625	−	0.631194i	$-0.782565\pi$
$251$		−	20.0000i		−	1.26239i	0.775625	−	0.631194i	$-0.217435\pi$
$252$			3.00000i			0.188982i
$253$	−18.0000			−1.13165
$254$			8.00000i			0.501965i
$255$	−7.00000	+	14.0000i	−0.438357	+	0.876714i
$256$	1.00000			0.0625000
$257$	−18.0000			−1.12281			−0.561405	−	0.827541i	$-0.689739\pi$
$257$	−18.0000			−1.12281			−0.561405	+	0.827541i	$0.689739\pi$
$258$		−	1.00000i		−	0.0622573i
$259$	3.00000	−	18.0000i	0.186411	−	1.11847i
$260$	−8.00000	−	4.00000i	−0.496139	−	0.248069i
$261$		−	9.00000i		−	0.557086i
$262$		0			0
$263$		−	31.0000i		−	1.91154i	−0.294112	−	0.955771i	$-0.595024\pi$
$263$		−	31.0000i		−	1.91154i	0.294112	−	0.955771i	$-0.404976\pi$
$264$			3.00000i			0.184637i
$265$	1.00000	−	2.00000i	0.0614295	−	0.122859i
$266$	−12.0000			−0.735767
$267$	6.00000			0.367194
$268$			2.00000i			0.122169i
$269$	20.0000			1.21942			0.609711	−	0.792624i	$-0.291286\pi$
$269$	20.0000			1.21942			0.609711	+	0.792624i	$0.291286\pi$
$270$	−1.00000	+	2.00000i	−0.0608581	+	0.121716i
$271$	22.0000			1.33640			0.668202	−	0.743980i	$-0.267064\pi$
$271$	22.0000			1.33640			0.668202	+	0.743980i	$0.267064\pi$
$272$	7.00000			0.424437
$273$	−12.0000			−0.726273
$274$		−	12.0000i		−	0.724947i
$275$	−9.00000	−	12.0000i	−0.542720	−	0.723627i
$276$			6.00000i			0.361158i
$277$	−28.0000			−1.68236			−0.841178	−	0.540758i	$-0.818138\pi$
$277$	−28.0000			−1.68236			−0.841178	+	0.540758i	$0.818138\pi$
$278$	15.0000			0.899640
$279$		−	5.00000i		−	0.299342i
$280$	−3.00000	+	6.00000i	−0.179284	+	0.358569i
$281$			30.0000i			1.78965i	0.446417	+	0.894825i	$0.352700\pi$
$281$			30.0000i			1.78965i	−0.446417	+	0.894825i	$0.647300\pi$
$282$	−8.00000			−0.476393
$283$	−4.00000			−0.237775			−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000			−0.237775			−0.118888	+	0.992908i	$0.537933\pi$
$284$	12.0000			0.712069
$285$	−8.00000	−	4.00000i	−0.473879	−	0.236940i
$286$	−12.0000			−0.709575
$287$		−	21.0000i		−	1.23959i
$288$	1.00000			0.0589256
$289$	32.0000			1.88235
$290$	9.00000	−	18.0000i	0.528498	−	1.05700i
$291$			7.00000i			0.410347i
$292$			4.00000i			0.234082i
$293$			29.0000i			1.69420i	0.531435	+	0.847099i	$0.321653\pi$
$293$			29.0000i			1.69420i	−0.531435	+	0.847099i	$0.678347\pi$
$294$			2.00000i			0.116642i
$295$	6.00000	−	12.0000i	0.349334	−	0.698667i
$296$	−6.00000	−	1.00000i	−0.348743	−	0.0581238i
$297$			3.00000i			0.174078i
$298$		0			0
$299$	−24.0000			−1.38796
$300$	−4.00000	+	3.00000i	−0.230940	+	0.173205i
$301$		−	3.00000i		−	0.172917i
$302$	−2.00000			−0.115087
$303$			2.00000i			0.114897i
$304$			4.00000i			0.229416i
$305$	−5.00000	+	10.0000i	−0.286299	+	0.572598i
$306$	7.00000			0.400163
$307$		−	18.0000i		−	1.02731i	−0.857996	−	0.513657i	$-0.828290\pi$
$307$		−	18.0000i		−	1.02731i	0.857996	−	0.513657i	$-0.171710\pi$
$308$			9.00000i			0.512823i
$309$		−	14.0000i		−	0.796432i
$310$	5.00000	−	10.0000i	0.283981	−	0.567962i
$311$		−	15.0000i		−	0.850572i	−0.905059	−	0.425286i	$-0.860174\pi$
$311$		−	15.0000i		−	0.850572i	0.905059	−	0.425286i	$-0.139826\pi$
$312$			4.00000i			0.226455i
$313$	6.00000			0.339140			0.169570	−	0.985518i	$-0.445762\pi$
$313$	6.00000			0.339140			0.169570	+	0.985518i	$0.445762\pi$
$314$		−	7.00000i		−	0.395033i
$315$	−3.00000	+	6.00000i	−0.169031	+	0.338062i
$316$			4.00000i			0.225018i
$317$			27.0000i			1.51647i	0.651981	+	0.758236i	$0.273938\pi$
$317$			27.0000i			1.51647i	−0.651981	+	0.758236i	$0.726062\pi$
$318$	−1.00000			−0.0560772
$319$		−	27.0000i		−	1.51171i
$320$	2.00000	+	1.00000i	0.111803	+	0.0559017i
$321$	18.0000			1.00466
$322$			18.0000i			1.00310i
$323$			28.0000i			1.55796i
$324$	1.00000			0.0555556
$325$	−12.0000	−	16.0000i	−0.665640	−	0.887520i
$326$	9.00000			0.498464
$327$	11.0000			0.608301
$328$	−7.00000			−0.386510
$329$	−24.0000			−1.32316
$330$	−3.00000	+	6.00000i	−0.165145	+	0.330289i
$331$			20.0000i			1.09930i	0.835395	+	0.549650i	$0.185239\pi$
$331$			20.0000i			1.09930i	−0.835395	+	0.549650i	$0.814761\pi$
$332$			14.0000i			0.768350i
$333$	−6.00000	−	1.00000i	−0.328798	−	0.0547997i
$334$	−12.0000			−0.656611
$335$	−2.00000	+	4.00000i	−0.109272	+	0.218543i
$336$	3.00000			0.163663
$337$		−	18.0000i		−	0.980522i	−0.871576	−	0.490261i	$-0.836901\pi$
$337$		−	18.0000i		−	0.980522i	0.871576	−	0.490261i	$-0.163099\pi$
$338$	−3.00000			−0.163178
$339$			1.00000i			0.0543125i
$340$	14.0000	+	7.00000i	0.759257	+	0.379628i
$341$		−	15.0000i		−	0.812296i
$342$			4.00000i			0.216295i
$343$		−	15.0000i		−	0.809924i
$344$	−1.00000			−0.0539164
$345$	−6.00000	+	12.0000i	−0.323029	+	0.646058i
$346$		−	9.00000i		−	0.483843i
$347$	−28.0000			−1.50312			−0.751559	−	0.659665i	$-0.770698\pi$
$347$	−28.0000			−1.50312			−0.751559	+	0.659665i	$0.770698\pi$
$348$	−9.00000			−0.482451
$349$		0			0			−	1.00000i	$-0.5\pi$
$349$		0			0				1.00000i	$0.5\pi$
$350$	−12.0000	+	9.00000i	−0.641427	+	0.481070i
$351$			4.00000i			0.213504i
$352$	3.00000			0.159901
$353$	31.0000			1.64996			0.824982	−	0.565159i	$-0.191185\pi$
$353$	31.0000			1.64996			0.824982	+	0.565159i	$0.191185\pi$
$354$	−6.00000			−0.318896
$355$	24.0000	+	12.0000i	1.27379	+	0.636894i
$356$		−	6.00000i		−	0.317999i
$357$	21.0000			1.11144
$358$		−	4.00000i		−	0.211407i
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$	2.00000	+	1.00000i	0.105409	+	0.0527046i
$361$	3.00000			0.157895
$362$	18.0000			0.946059
$363$		−	2.00000i		−	0.104973i
$364$			12.0000i			0.628971i
$365$	−4.00000	+	8.00000i	−0.209370	+	0.418739i
$366$	5.00000			0.261354
$367$		−	23.0000i		−	1.20059i	−0.799779	−	0.600295i	$-0.795050\pi$
$367$		−	23.0000i		−	1.20059i	0.799779	−	0.600295i	$-0.204950\pi$
$368$	6.00000			0.312772
$369$	−7.00000			−0.364405
$370$	−11.0000	−	8.00000i	−0.571863	−	0.415900i
$371$	−3.00000			−0.155752
$372$	−5.00000			−0.259238
$373$			14.0000i			0.724893i	0.932005	+	0.362446i	$0.118058\pi$
$373$			14.0000i			0.724893i	−0.932005	+	0.362446i	$0.881942\pi$
$374$	21.0000			1.08588
$375$	−11.0000	+	2.00000i	−0.568038	+	0.103280i
$376$			8.00000i			0.412568i
$377$		−	36.0000i		−	1.85409i
$378$	3.00000			0.154303
$379$	20.0000			1.02733			0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000			1.02733			0.513665	+	0.857991i	$0.328287\pi$
$380$	−4.00000	+	8.00000i	−0.205196	+	0.410391i
$381$	8.00000			0.409852
$382$		−	15.0000i		−	0.767467i
$383$	6.00000			0.306586			0.153293	−	0.988181i	$-0.451012\pi$
$383$	6.00000			0.306586			0.153293	+	0.988181i	$0.451012\pi$
$384$		−	1.00000i		−	0.0510310i
$385$	−9.00000	+	18.0000i	−0.458682	+	0.917365i
$386$	14.0000			0.712581
$387$	−1.00000			−0.0508329
$388$	7.00000			0.355371
$389$		−	31.0000i		−	1.57176i	−0.618378	−	0.785881i	$-0.712210\pi$
$389$		−	31.0000i		−	1.57176i	0.618378	−	0.785881i	$-0.287790\pi$
$390$	−4.00000	+	8.00000i	−0.202548	+	0.405096i
$391$	42.0000			2.12403
$392$	2.00000			0.101015
$393$		0			0
$394$			18.0000i			0.906827i
$395$	−4.00000	+	8.00000i	−0.201262	+	0.402524i
$396$	3.00000			0.150756
$397$		−	38.0000i		−	1.90717i	−0.301131	−	0.953583i	$-0.597364\pi$
$397$		−	38.0000i		−	1.90717i	0.301131	−	0.953583i	$-0.402636\pi$
$398$			16.0000i			0.802008i
$399$			12.0000i			0.600751i
$400$	3.00000	+	4.00000i	0.150000	+	0.200000i
$401$			10.0000i			0.499376i	0.968326	+	0.249688i	$0.0803281\pi$
$401$			10.0000i			0.499376i	−0.968326	+	0.249688i	$0.919672\pi$
$402$	2.00000			0.0997509
$403$		−	20.0000i		−	0.996271i
$404$	2.00000			0.0995037
$405$	2.00000	+	1.00000i	0.0993808	+	0.0496904i
$406$	−27.0000			−1.33999
$407$	−18.0000	−	3.00000i	−0.892227	−	0.148704i
$408$		−	7.00000i		−	0.346552i
$409$			14.0000i			0.692255i	0.938187	+	0.346128i	$0.112504\pi$
$409$			14.0000i			0.692255i	−0.938187	+	0.346128i	$0.887496\pi$
$410$	−14.0000	−	7.00000i	−0.691411	−	0.345705i
$411$	−12.0000			−0.591916
$412$	−14.0000			−0.689730
$413$	−18.0000			−0.885722
$414$	6.00000			0.294884
$415$	−14.0000	+	28.0000i	−0.687233	+	1.37447i
$416$	4.00000			0.196116
$417$		−	15.0000i		−	0.734553i
$418$			12.0000i			0.586939i
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$	6.00000	+	3.00000i	0.292770	+	0.146385i
$421$		−	30.0000i		−	1.46211i	−0.682318	−	0.731055i	$-0.739028\pi$
$421$		−	30.0000i		−	1.46211i	0.682318	−	0.731055i	$-0.260972\pi$
$422$	13.0000			0.632830
$423$			8.00000i			0.388973i
$424$			1.00000i			0.0485643i
$425$	21.0000	+	28.0000i	1.01865	+	1.35820i
$426$		−	12.0000i		−	0.581402i
$427$	15.0000			0.725901
$428$		−	18.0000i		−	0.870063i
$429$			12.0000i			0.579365i
$430$	−2.00000	−	1.00000i	−0.0964486	−	0.0482243i
$431$			25.0000i			1.20421i	0.798418	+	0.602104i	$0.205671\pi$
$431$			25.0000i			1.20421i	−0.798418	+	0.602104i	$0.794329\pi$
$432$		−	1.00000i		−	0.0481125i
$433$			34.0000i			1.63394i	0.576683	+	0.816968i	$0.304347\pi$
$433$			34.0000i			1.63394i	−0.576683	+	0.816968i	$0.695653\pi$
$434$	−15.0000			−0.720023
$435$	−18.0000	−	9.00000i	−0.863034	−	0.431517i
$436$		−	11.0000i		−	0.526804i
$437$			24.0000i			1.14808i
$438$	4.00000			0.191127
$439$			9.00000i			0.429547i	0.976664	+	0.214773i	$0.0689013\pi$
$439$			9.00000i			0.429547i	−0.976664	+	0.214773i	$0.931099\pi$
$440$	6.00000	+	3.00000i	0.286039	+	0.143019i
$441$	2.00000			0.0952381
$442$	28.0000			1.33182
$443$		−	36.0000i		−	1.71041i	−0.518289	−	0.855206i	$-0.673431\pi$
$443$		−	36.0000i		−	1.71041i	0.518289	−	0.855206i	$-0.326569\pi$
$444$	−1.00000	+	6.00000i	−0.0474579	+	0.284747i
$445$	6.00000	−	12.0000i	0.284427	−	0.568855i
$446$			11.0000i			0.520865i
$447$		0			0
$448$		−	3.00000i		−	0.141737i
$449$		−	6.00000i		−	0.283158i	−0.989927	−	0.141579i	$-0.954782\pi$
$449$		−	6.00000i		−	0.283158i	0.989927	−	0.141579i	$-0.0452178\pi$
$450$	3.00000	+	4.00000i	0.141421	+	0.188562i
$451$	−21.0000			−0.988851
$452$	1.00000			0.0470360
$453$			2.00000i			0.0939682i
$454$	13.0000			0.610120
$455$	−12.0000	+	24.0000i	−0.562569	+	1.12514i
$456$	4.00000			0.187317
$457$	−13.0000			−0.608114			−0.304057	−	0.952654i	$-0.598341\pi$
$457$	−13.0000			−0.608114			−0.304057	+	0.952654i	$0.598341\pi$
$458$	−10.0000			−0.467269
$459$		−	7.00000i		−	0.326732i
$460$	12.0000	+	6.00000i	0.559503	+	0.279751i
$461$		−	25.0000i		−	1.16437i	−0.813058	−	0.582183i	$-0.802199\pi$
$461$		−	25.0000i		−	1.16437i	0.813058	−	0.582183i	$-0.197801\pi$
$462$	9.00000			0.418718
$463$	26.0000			1.20832			0.604161	−	0.796862i	$-0.293508\pi$
$463$	26.0000			1.20832			0.604161	+	0.796862i	$0.293508\pi$
$464$			9.00000i			0.417815i
$465$	−10.0000	−	5.00000i	−0.463739	−	0.231869i
$466$			6.00000i			0.277945i
$467$	−23.0000			−1.06431			−0.532157	−	0.846646i	$-0.678618\pi$
$467$	−23.0000			−1.06431			−0.532157	+	0.846646i	$0.678618\pi$
$468$	4.00000			0.184900
$469$	6.00000			0.277054
$470$	−8.00000	+	16.0000i	−0.369012	+	0.738025i
$471$	−7.00000			−0.322543
$472$			6.00000i			0.276172i
$473$	−3.00000			−0.137940
$474$	4.00000			0.183726
$475$	−16.0000	+	12.0000i	−0.734130	+	0.550598i
$476$		−	21.0000i		−	0.962533i
$477$			1.00000i			0.0457869i
$478$		−	9.00000i		−	0.411650i
$479$			24.0000i			1.09659i	0.836286	+	0.548294i	$0.184723\pi$
$479$			24.0000i			1.09659i	−0.836286	+	0.548294i	$0.815277\pi$
$480$	1.00000	−	2.00000i	0.0456435	−	0.0912871i
$481$	−24.0000	−	4.00000i	−1.09431	−	0.182384i
$482$		0			0
$483$	18.0000			0.819028
$484$	−2.00000			−0.0909091
$485$	14.0000	+	7.00000i	0.635707	+	0.317854i
$486$		−	1.00000i		−	0.0453609i
$487$	22.0000			0.996915			0.498458	−	0.866914i	$-0.333900\pi$
$487$	22.0000			0.996915			0.498458	+	0.866914i	$0.333900\pi$
$488$		−	5.00000i		−	0.226339i
$489$		−	9.00000i		−	0.406994i
$490$	4.00000	+	2.00000i	0.180702	+	0.0903508i
$491$	−8.00000			−0.361035			−0.180517	−	0.983572i	$-0.557777\pi$
$491$	−8.00000			−0.361035			−0.180517	+	0.983572i	$0.557777\pi$
$492$			7.00000i			0.315584i
$493$			63.0000i			2.83738i
$494$			16.0000i			0.719874i
$495$	6.00000	+	3.00000i	0.269680	+	0.134840i
$496$			5.00000i			0.224507i
$497$		−	36.0000i		−	1.61482i
$498$	14.0000			0.627355
$499$			24.0000i			1.07439i	0.843459	+	0.537194i	$0.180516\pi$
$499$			24.0000i			1.07439i	−0.843459	+	0.537194i	$0.819484\pi$
$500$	2.00000	+	11.0000i	0.0894427	+	0.491935i
$501$			12.0000i			0.536120i
$502$			20.0000i			0.892644i
$503$	−14.0000			−0.624229			−0.312115	−	0.950044i	$-0.601037\pi$
$503$	−14.0000			−0.624229			−0.312115	+	0.950044i	$0.601037\pi$
$504$		−	3.00000i		−	0.133631i
$505$	4.00000	+	2.00000i	0.177998	+	0.0889988i
$506$	18.0000			0.800198
$507$			3.00000i			0.133235i
$508$		−	8.00000i		−	0.354943i
$509$	−10.0000			−0.443242			−0.221621	−	0.975133i	$-0.571135\pi$
$509$	−10.0000			−0.443242			−0.221621	+	0.975133i	$0.571135\pi$
$510$	7.00000	−	14.0000i	0.309965	−	0.619930i
$511$	12.0000			0.530849
$512$	−1.00000			−0.0441942
$513$	4.00000			0.176604
$514$	18.0000			0.793946
$515$	−28.0000	−	14.0000i	−1.23383	−	0.616914i
$516$			1.00000i			0.0440225i
$517$			24.0000i			1.05552i
$518$	−3.00000	+	18.0000i	−0.131812	+	0.790875i
$519$	−9.00000			−0.395056
$520$	8.00000	+	4.00000i	0.350823	+	0.175412i
$521$	37.0000			1.62100			0.810500	−	0.585739i	$-0.199196\pi$
$521$	37.0000			1.62100			0.810500	+	0.585739i	$0.199196\pi$
$522$			9.00000i			0.393919i
$523$	−4.00000			−0.174908			−0.0874539	−	0.996169i	$-0.527873\pi$
$523$	−4.00000			−0.174908			−0.0874539	+	0.996169i	$0.527873\pi$
$524$		0			0
$525$	9.00000	+	12.0000i	0.392792	+	0.523723i
$526$			31.0000i			1.35166i
$527$			35.0000i			1.52462i
$528$		−	3.00000i		−	0.130558i
$529$	13.0000			0.565217
$530$	−1.00000	+	2.00000i	−0.0434372	+	0.0868744i
$531$			6.00000i			0.260378i
$532$	12.0000			0.520266
$533$	−28.0000			−1.21281
$534$	−6.00000			−0.259645
$535$	18.0000	−	36.0000i	0.778208	−	1.55642i
$536$		−	2.00000i		−	0.0863868i
$537$	−4.00000			−0.172613
$538$	−20.0000			−0.862261
$539$	6.00000			0.258438
$540$	1.00000	−	2.00000i	0.0430331	−	0.0860663i
$541$		−	30.0000i		−	1.28980i	−0.764267	−	0.644900i	$-0.776899\pi$
$541$		−	30.0000i		−	1.28980i	0.764267	−	0.644900i	$-0.223101\pi$
$542$	−22.0000			−0.944981
$543$		−	18.0000i		−	0.772454i
$544$	−7.00000			−0.300123
$545$	11.0000	−	22.0000i	0.471188	−	0.942376i
$546$	12.0000			0.513553
$547$	−23.0000			−0.983409			−0.491704	−	0.870762i	$-0.663626\pi$
$547$	−23.0000			−0.983409			−0.491704	+	0.870762i	$0.663626\pi$
$548$			12.0000i			0.512615i
$549$		−	5.00000i		−	0.213395i
$550$	9.00000	+	12.0000i	0.383761	+	0.511682i
$551$	−36.0000			−1.53365
$552$		−	6.00000i		−	0.255377i
$553$	12.0000			0.510292
$554$	28.0000			1.18961
$555$	−8.00000	+	11.0000i	−0.339581	+	0.466924i
$556$	−15.0000			−0.636142
$557$	12.0000			0.508456			0.254228	−	0.967144i	$-0.418179\pi$
$557$	12.0000			0.508456			0.254228	+	0.967144i	$0.418179\pi$
$558$			5.00000i			0.211667i
$559$	−4.00000			−0.169182
$560$	3.00000	−	6.00000i	0.126773	−	0.253546i
$561$		−	21.0000i		−	0.886621i
$562$		−	30.0000i		−	1.26547i
$563$	21.0000			0.885044			0.442522	−	0.896758i	$-0.354084\pi$
$563$	21.0000			0.885044			0.442522	+	0.896758i	$0.354084\pi$
$564$	8.00000			0.336861
$565$	2.00000	+	1.00000i	0.0841406	+	0.0420703i
$566$	4.00000			0.168133
$567$		−	3.00000i		−	0.125988i
$568$	−12.0000			−0.503509
$569$			24.0000i			1.00613i	0.864248	+	0.503066i	$0.167795\pi$
$569$			24.0000i			1.00613i	−0.864248	+	0.503066i	$0.832205\pi$
$570$	8.00000	+	4.00000i	0.335083	+	0.167542i
$571$	−33.0000			−1.38101			−0.690504	−	0.723329i	$-0.742611\pi$
$571$	−33.0000			−1.38101			−0.690504	+	0.723329i	$0.742611\pi$
$572$	12.0000			0.501745
$573$	−15.0000			−0.626634
$574$			21.0000i			0.876523i
$575$	18.0000	+	24.0000i	0.750652	+	1.00087i
$576$	−1.00000			−0.0416667
$577$	22.0000			0.915872			0.457936	−	0.888985i	$-0.348589\pi$
$577$	22.0000			0.915872			0.457936	+	0.888985i	$0.348589\pi$
$578$	−32.0000			−1.33102
$579$		−	14.0000i		−	0.581820i
$580$	−9.00000	+	18.0000i	−0.373705	+	0.747409i
$581$	42.0000			1.74245
$582$		−	7.00000i		−	0.290159i
$583$			3.00000i			0.124247i
$584$		−	4.00000i		−	0.165521i
$585$	8.00000	+	4.00000i	0.330759	+	0.165380i
$586$		−	29.0000i		−	1.19798i
$587$	47.0000			1.93990			0.969949	−	0.243309i	$-0.0782329\pi$
$587$	47.0000			1.93990			0.969949	+	0.243309i	$0.0782329\pi$
$588$		−	2.00000i		−	0.0824786i
$589$	−20.0000			−0.824086
$590$	−6.00000	+	12.0000i	−0.247016	+	0.494032i
$591$	18.0000			0.740421
$592$	6.00000	+	1.00000i	0.246598	+	0.0410997i
$593$		−	46.0000i		−	1.88899i	−0.328521	−	0.944497i	$-0.606550\pi$
$593$		−	46.0000i		−	1.88899i	0.328521	−	0.944497i	$-0.393450\pi$
$594$		−	3.00000i		−	0.123091i
$595$	21.0000	−	42.0000i	0.860916	−	1.72183i
$596$		0			0
$597$	16.0000			0.654836
$598$	24.0000			0.981433
$599$	−30.0000			−1.22577			−0.612883	−	0.790173i	$-0.709990\pi$
$599$	−30.0000			−1.22577			−0.612883	+	0.790173i	$0.709990\pi$
$600$	4.00000	−	3.00000i	0.163299	−	0.122474i
$601$	−23.0000			−0.938190			−0.469095	−	0.883148i	$-0.655420\pi$
$601$	−23.0000			−0.938190			−0.469095	+	0.883148i	$0.655420\pi$
$602$			3.00000i			0.122271i
$603$		−	2.00000i		−	0.0814463i
$604$	2.00000			0.0813788
$605$	−4.00000	−	2.00000i	−0.162623	−	0.0813116i
$606$		−	2.00000i		−	0.0812444i
$607$	32.0000			1.29884			0.649420	−	0.760430i	$-0.275012\pi$
$607$	32.0000			1.29884			0.649420	+	0.760430i	$0.275012\pi$
$608$		−	4.00000i		−	0.162221i
$609$			27.0000i			1.09410i
$610$	5.00000	−	10.0000i	0.202444	−	0.404888i
$611$			32.0000i			1.29458i
$612$	−7.00000			−0.282958
$613$		−	31.0000i		−	1.25208i	−0.779792	−	0.626039i	$-0.784675\pi$
$613$		−	31.0000i		−	1.25208i	0.779792	−	0.626039i	$-0.215325\pi$
$614$			18.0000i			0.726421i
$615$	−7.00000	+	14.0000i	−0.282267	+	0.564534i
$616$		−	9.00000i		−	0.362620i
$617$		−	8.00000i		−	0.322068i	−0.986949	−	0.161034i	$-0.948517\pi$
$617$		−	8.00000i		−	0.322068i	0.986949	−	0.161034i	$-0.0514829\pi$
$618$			14.0000i			0.563163i
$619$	25.0000			1.00483			0.502417	−	0.864625i	$-0.332444\pi$
$619$	25.0000			1.00483			0.502417	+	0.864625i	$0.332444\pi$
$620$	−5.00000	+	10.0000i	−0.200805	+	0.401610i
$621$		−	6.00000i		−	0.240772i
$622$			15.0000i			0.601445i
$623$	−18.0000			−0.721155
$624$		−	4.00000i		−	0.160128i
$625$	−7.00000	+	24.0000i	−0.280000	+	0.960000i
$626$	−6.00000			−0.239808
$627$	12.0000			0.479234
$628$			7.00000i			0.279330i
$629$	42.0000	+	7.00000i	1.67465	+	0.279108i
$630$	3.00000	−	6.00000i	0.119523	−	0.239046i
$631$			25.0000i			0.995234i	0.867397	+	0.497617i	$0.165792\pi$
$631$			25.0000i			0.995234i	−0.867397	+	0.497617i	$0.834208\pi$
$632$		−	4.00000i		−	0.159111i
$633$		−	13.0000i		−	0.516704i
$634$		−	27.0000i		−	1.07231i
$635$	8.00000	−	16.0000i	0.317470	−	0.634941i
$636$	1.00000			0.0396526
$637$	8.00000			0.316972
$638$			27.0000i			1.06894i
$639$	−12.0000			−0.474713
$640$	−2.00000	−	1.00000i	−0.0790569	−	0.0395285i
$641$	17.0000			0.671460			0.335730	−	0.941958i	$-0.391017\pi$
$641$	17.0000			0.671460			0.335730	+	0.941958i	$0.391017\pi$
$642$	−18.0000			−0.710403
$643$	31.0000			1.22252			0.611260	−	0.791430i	$-0.290663\pi$
$643$	31.0000			1.22252			0.611260	+	0.791430i	$0.290663\pi$
$644$		−	18.0000i		−	0.709299i
$645$	−1.00000	+	2.00000i	−0.0393750	+	0.0787499i
$646$		−	28.0000i		−	1.10165i
$647$	−48.0000			−1.88707			−0.943537	−	0.331266i	$-0.892524\pi$
$647$	−48.0000			−1.88707			−0.943537	+	0.331266i	$0.892524\pi$
$648$	−1.00000			−0.0392837
$649$			18.0000i			0.706562i
$650$	12.0000	+	16.0000i	0.470679	+	0.627572i
$651$			15.0000i			0.587896i
$652$	−9.00000			−0.352467
$653$	36.0000			1.40879			0.704394	−	0.709809i	$-0.251219\pi$
$653$	36.0000			1.40879			0.704394	+	0.709809i	$0.251219\pi$
$654$	−11.0000			−0.430134
$655$		0			0
$656$	7.00000			0.273304
$657$		−	4.00000i		−	0.156055i
$658$	24.0000			0.935617
$659$	−20.0000			−0.779089			−0.389545	−	0.921008i	$-0.627368\pi$
$659$	−20.0000			−0.779089			−0.389545	+	0.921008i	$0.627368\pi$
$660$	3.00000	−	6.00000i	0.116775	−	0.233550i
$661$			35.0000i			1.36134i	0.732589	+	0.680671i	$0.238312\pi$
$661$			35.0000i			1.36134i	−0.732589	+	0.680671i	$0.761688\pi$
$662$		−	20.0000i		−	0.777322i
$663$		−	28.0000i		−	1.08743i
$664$		−	14.0000i		−	0.543305i
$665$	24.0000	+	12.0000i	0.930680	+	0.465340i
$666$	6.00000	+	1.00000i	0.232495	+	0.0387492i
$667$			54.0000i			2.09089i
$668$	12.0000			0.464294
$669$	11.0000			0.425285
$670$	2.00000	−	4.00000i	0.0772667	−	0.154533i
$671$		−	15.0000i		−	0.579069i
$672$	−3.00000			−0.115728
$673$		−	26.0000i		−	1.00223i	−0.865382	−	0.501113i	$-0.832924\pi$
$673$		−	26.0000i		−	1.00223i	0.865382	−	0.501113i	$-0.167076\pi$
$674$			18.0000i			0.693334i
$675$	4.00000	−	3.00000i	0.153960	−	0.115470i
$676$	3.00000			0.115385
$677$			42.0000i			1.61419i	0.590421	+	0.807096i	$0.298962\pi$
$677$			42.0000i			1.61419i	−0.590421	+	0.807096i	$0.701038\pi$
$678$		−	1.00000i		−	0.0384048i
$679$		−	21.0000i		−	0.805906i
$680$	−14.0000	−	7.00000i	−0.536875	−	0.268438i
$681$		−	13.0000i		−	0.498161i
$682$			15.0000i			0.574380i
$683$	1.00000			0.0382639			0.0191320	−	0.999817i	$-0.493910\pi$
$683$	1.00000			0.0382639			0.0191320	+	0.999817i	$0.493910\pi$
$684$		−	4.00000i		−	0.152944i
$685$	−12.0000	+	24.0000i	−0.458496	+	0.916993i
$686$			15.0000i			0.572703i
$687$			10.0000i			0.381524i
$688$	1.00000			0.0381246
$689$			4.00000i			0.152388i
$690$	6.00000	−	12.0000i	0.228416	−	0.456832i
$691$	−43.0000			−1.63580			−0.817899	−	0.575362i	$-0.804861\pi$
$691$	−43.0000			−1.63580			−0.817899	+	0.575362i	$0.804861\pi$
$692$			9.00000i			0.342129i
$693$		−	9.00000i		−	0.341882i
$694$	28.0000			1.06287
$695$	−30.0000	−	15.0000i	−1.13796	−	0.568982i
$696$	9.00000			0.341144
$697$	49.0000			1.85601
$698$		0			0
$699$	6.00000			0.226941
$700$	12.0000	−	9.00000i	0.453557	−	0.340168i
$701$			10.0000i			0.377695i	0.982006	+	0.188847i	$0.0604752\pi$
$701$			10.0000i			0.377695i	−0.982006	+	0.188847i	$0.939525\pi$
$702$		−	4.00000i		−	0.150970i
$703$	−4.00000	+	24.0000i	−0.150863	+	0.905177i
$704$	−3.00000			−0.113067
$705$	16.0000	+	8.00000i	0.602595	+	0.301297i
$706$	−31.0000			−1.16670
$707$		−	6.00000i		−	0.225653i
$708$	6.00000			0.225494
$709$		−	11.0000i		−	0.413114i	−0.978435	−	0.206557i	$-0.933774\pi$
$709$		−	11.0000i		−	0.413114i	0.978435	−	0.206557i	$-0.0662258\pi$
$710$	−24.0000	−	12.0000i	−0.900704	−	0.450352i
$711$		−	4.00000i		−	0.150012i
$712$			6.00000i			0.224860i
$713$			30.0000i			1.12351i
$714$	−21.0000			−0.785905
$715$	24.0000	+	12.0000i	0.897549	+	0.448775i
$716$			4.00000i			0.149487i
$717$	−9.00000			−0.336111
$718$		0			0
$719$	50.0000			1.86469			0.932343	−	0.361576i	$-0.117761\pi$
$719$	50.0000			1.86469			0.932343	+	0.361576i	$0.117761\pi$
$720$	−2.00000	−	1.00000i	−0.0745356	−	0.0372678i
$721$			42.0000i			1.56416i
$722$	−3.00000			−0.111648
$723$		0			0
$724$	−18.0000			−0.668965
$725$	−36.0000	+	27.0000i	−1.33701	+	1.00275i
$726$			2.00000i			0.0742270i
$727$	−18.0000			−0.667583			−0.333792	−	0.942647i	$-0.608328\pi$
$727$	−18.0000			−0.667583			−0.333792	+	0.942647i	$0.608328\pi$
$728$		−	12.0000i		−	0.444750i
$729$	−1.00000			−0.0370370
$730$	4.00000	−	8.00000i	0.148047	−	0.296093i
$731$	7.00000			0.258904
$732$	−5.00000			−0.184805
$733$		−	11.0000i		−	0.406294i	−0.979148	−	0.203147i	$-0.934883\pi$
$733$		−	11.0000i		−	0.406294i	0.979148	−	0.203147i	$-0.0651170\pi$
$734$			23.0000i			0.848945i
$735$	2.00000	−	4.00000i	0.0737711	−	0.147542i
$736$	−6.00000			−0.221163
$737$		−	6.00000i		−	0.221013i
$738$	7.00000			0.257674
$739$	5.00000			0.183928			0.0919640	−	0.995762i	$-0.470686\pi$
$739$	5.00000			0.183928			0.0919640	+	0.995762i	$0.470686\pi$
$740$	11.0000	+	8.00000i	0.404368	+	0.294086i
$741$	16.0000			0.587775
$742$	3.00000			0.110133
$743$		−	51.0000i		−	1.87101i	−0.353315	−	0.935504i	$-0.614946\pi$
$743$		−	51.0000i		−	1.87101i	0.353315	−	0.935504i	$-0.385054\pi$
$744$	5.00000			0.183309
$745$		0			0
$746$		−	14.0000i		−	0.512576i
$747$		−	14.0000i		−	0.512233i
$748$	−21.0000			−0.767836
$749$	−54.0000			−1.97312
$750$	11.0000	−	2.00000i	0.401663	−	0.0730297i
$751$	−18.0000			−0.656829			−0.328415	−	0.944534i	$-0.606514\pi$
$751$	−18.0000			−0.656829			−0.328415	+	0.944534i	$0.606514\pi$
$752$		−	8.00000i		−	0.291730i
$753$	20.0000			0.728841
$754$			36.0000i			1.31104i
$755$	4.00000	+	2.00000i	0.145575	+	0.0727875i
$756$	−3.00000			−0.109109
$757$	2.00000			0.0726912			0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000			0.0726912			0.0363456	+	0.999339i	$0.488428\pi$
$758$	−20.0000			−0.726433
$759$		−	18.0000i		−	0.653359i
$760$	4.00000	−	8.00000i	0.145095	−	0.290191i
$761$	−13.0000			−0.471250			−0.235625	−	0.971844i	$-0.575714\pi$
$761$	−13.0000			−0.471250			−0.235625	+	0.971844i	$0.575714\pi$
$762$	−8.00000			−0.289809
$763$	−33.0000			−1.19468
$764$			15.0000i			0.542681i
$765$	−14.0000	−	7.00000i	−0.506171	−	0.253086i
$766$	−6.00000			−0.216789
$767$			24.0000i			0.866590i
$768$			1.00000i			0.0360844i
$769$		−	26.0000i		−	0.937584i	−0.883309	−	0.468792i	$-0.844689\pi$
$769$		−	26.0000i		−	0.937584i	0.883309	−	0.468792i	$-0.155311\pi$
$770$	9.00000	−	18.0000i	0.324337	−	0.648675i
$771$		−	18.0000i		−	0.648254i
$772$	−14.0000			−0.503871
$773$		−	11.0000i		−	0.395643i	−0.980238	−	0.197821i	$-0.936613\pi$
$773$		−	11.0000i		−	0.395643i	0.980238	−	0.197821i	$-0.0633866\pi$
$774$	1.00000			0.0359443
$775$	−20.0000	+	15.0000i	−0.718421	+	0.538816i
$776$	−7.00000			−0.251285
$777$	18.0000	+	3.00000i	0.645746	+	0.107624i
$778$			31.0000i			1.11140i
$779$			28.0000i			1.00320i
$780$	4.00000	−	8.00000i	0.143223	−	0.286446i
$781$	−36.0000			−1.28818
$782$	−42.0000			−1.50192
$783$	9.00000			0.321634
$784$	−2.00000			−0.0714286
$785$	−7.00000	+	14.0000i	−0.249841	+	0.499681i
$786$		0			0
$787$		−	28.0000i		−	0.998092i	−0.866575	−	0.499046i	$-0.833684\pi$
$787$		−	28.0000i		−	0.998092i	0.866575	−	0.499046

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.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000i q^{3} +1.00000 q^{4} +(2.00000 + 1.00000i) q^{5} -1.00000i q^{6} -3.00000i q^{7} -1.00000 q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000i q^{3} +1.00000 q^{4} +(2.00000 + 1.00000i) q^{5} -1.00000i q^{6} -3.00000i q^{7} -1.00000 q^{8} -1.00000 q^{9} +(-2.00000 - 1.00000i) q^{10} -3.00000 q^{11} +1.00000i q^{12} -4.00000 q^{13} +3.00000i q^{14} +(-1.00000 + 2.00000i) q^{15} +1.00000 q^{16} +7.00000 q^{17} +1.00000 q^{18} +4.00000i q^{19} +(2.00000 + 1.00000i) q^{20} +3.00000 q^{21} +3.00000 q^{22} +6.00000 q^{23} -1.00000i q^{24} +(3.00000 + 4.00000i) q^{25} +4.00000 q^{26} -1.00000i q^{27} -3.00000i q^{28} +9.00000i q^{29} +(1.00000 - 2.00000i) q^{30} +5.00000i q^{31} -1.00000 q^{32} -3.00000i q^{33} -7.00000 q^{34} +(3.00000 - 6.00000i) q^{35} -1.00000 q^{36} +(6.00000 + 1.00000i) q^{37} -4.00000i q^{38} -4.00000i q^{39} +(-2.00000 - 1.00000i) q^{40} +7.00000 q^{41} -3.00000 q^{42} +1.00000 q^{43} -3.00000 q^{44} +(-2.00000 - 1.00000i) q^{45} -6.00000 q^{46} -8.00000i q^{47} +1.00000i q^{48} -2.00000 q^{49} +(-3.00000 - 4.00000i) q^{50} +7.00000i q^{51} -4.00000 q^{52} -1.00000i q^{53} +1.00000i q^{54} +(-6.00000 - 3.00000i) q^{55} +3.00000i q^{56} -4.00000 q^{57} -9.00000i q^{58} -6.00000i q^{59} +(-1.00000 + 2.00000i) q^{60} +5.00000i q^{61} -5.00000i q^{62} +3.00000i q^{63} +1.00000 q^{64} +(-8.00000 - 4.00000i) q^{65} +3.00000i q^{66} +2.00000i q^{67} +7.00000 q^{68} +6.00000i q^{69} +(-3.00000 + 6.00000i) q^{70} +12.0000 q^{71} +1.00000 q^{72} +4.00000i q^{73} +(-6.00000 - 1.00000i) q^{74} +(-4.00000 + 3.00000i) q^{75} +4.00000i q^{76} +9.00000i q^{77} +4.00000i q^{78} +4.00000i q^{79} +(2.00000 + 1.00000i) q^{80} +1.00000 q^{81} -7.00000 q^{82} +14.0000i q^{83} +3.00000 q^{84} +(14.0000 + 7.00000i) q^{85} -1.00000 q^{86} -9.00000 q^{87} +3.00000 q^{88} -6.00000i q^{89} +(2.00000 + 1.00000i) q^{90} +12.0000i q^{91} +6.00000 q^{92} -5.00000 q^{93} +8.00000i q^{94} +(-4.00000 + 8.00000i) q^{95} -1.00000i q^{96} +7.00000 q^{97} +2.00000 q^{98} +3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2q - 2q^{2} + 2q^{4} + 4q^{5} - 2q^{8} - 2q^{9} + O(q^{10}) \)	\( 2q - 2q^{2} + 2q^{4} + 4q^{5} - 2q^{8} - 2q^{9} - 4q^{10} - 6q^{11} - 8q^{13} - 2q^{15} + 2q^{16} + 14q^{17} + 2q^{18} + 4q^{20} + 6q^{21} + 6q^{22} + 12q^{23} + 6q^{25} + 8q^{26} + 2q^{30} - 2q^{32} - 14q^{34} + 6q^{35} - 2q^{36} + 12q^{37} - 4q^{40} + 14q^{41} - 6q^{42} + 2q^{43} - 6q^{44} - 4q^{45} - 12q^{46} - 4q^{49} - 6q^{50} - 8q^{52} - 12q^{55} - 8q^{57} - 2q^{60} + 2q^{64} - 16q^{65} + 14q^{68} - 6q^{70} + 24q^{71} + 2q^{72} - 12q^{74} - 8q^{75} + 4q^{80} + 2q^{81} - 14q^{82} + 6q^{84} + 28q^{85} - 2q^{86} - 18q^{87} + 6q^{88} + 4q^{90} + 12q^{92} - 10q^{93} - 8q^{95} + 14q^{97} + 4q^{98} + 6q^{99} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

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