LMFDB - Embedded newform 112.2.i.c.81.1

Newspace parameters

Level:	$ N $	$=$	$ 112 = 2^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	112.i (of order $3$, degree $2$, not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$0.894324502638$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
Defining polynomial:	$x^{2} - x + 1$
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 56)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			81.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	112.81
Dual form			112.2.i.c.65.1

$q$-expansion

$f(q)$	$=$	$q+(1.50000 - 2.59808i) q^{3} +(0.500000 + 0.866025i) q^{5} +(-2.00000 + 1.73205i) q^{7} +(-3.00000 - 5.19615i) q^{9} +O(q^{10})$	\(q+(1.50000 - 2.59808i) q^{3} +(0.500000 + 0.866025i) q^{5} +(-2.00000 + 1.73205i) q^{7} +(-3.00000 - 5.19615i) q^{9} +(-0.500000 + 0.866025i) q^{11} +2.00000 q^{13} +3.00000 q^{15} +(-1.50000 + 2.59808i) q^{17} +(2.50000 + 4.33013i) q^{19} +(1.50000 + 7.79423i) q^{21} +(-1.50000 - 2.59808i) q^{23} +(2.00000 - 3.46410i) q^{25} -9.00000 q^{27} -6.00000 q^{29} +(-0.500000 + 0.866025i) q^{31} +(1.50000 + 2.59808i) q^{33} +(-2.50000 - 0.866025i) q^{35} +(2.50000 + 4.33013i) q^{37} +(3.00000 - 5.19615i) q^{39} -10.0000 q^{41} +4.00000 q^{43} +(3.00000 - 5.19615i) q^{45} +(0.500000 + 0.866025i) q^{47} +(1.00000 - 6.92820i) q^{49} +(4.50000 + 7.79423i) q^{51} +(4.50000 - 7.79423i) q^{53} -1.00000 q^{55} +15.0000 q^{57} +(1.50000 - 2.59808i) q^{59} +(-1.50000 - 2.59808i) q^{61} +(15.0000 + 5.19615i) q^{63} +(1.00000 + 1.73205i) q^{65} +(5.50000 - 9.52628i) q^{67} -9.00000 q^{69} -16.0000 q^{71} +(-3.50000 + 6.06218i) q^{73} +(-6.00000 - 10.3923i) q^{75} +(-0.500000 - 2.59808i) q^{77} +(-5.50000 - 9.52628i) q^{79} +(-4.50000 + 7.79423i) q^{81} +4.00000 q^{83} -3.00000 q^{85} +(-9.00000 + 15.5885i) q^{87} +(4.50000 + 7.79423i) q^{89} +(-4.00000 + 3.46410i) q^{91} +(1.50000 + 2.59808i) q^{93} +(-2.50000 + 4.33013i) q^{95} +6.00000 q^{97} +6.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2q + 3q^{3} + q^{5} - 4q^{7} - 6q^{9} + O(q^{10}) $	$ 2q + 3q^{3} + q^{5} - 4q^{7} - 6q^{9} - q^{11} + 4q^{13} + 6q^{15} - 3q^{17} + 5q^{19} + 3q^{21} - 3q^{23} + 4q^{25} - 18q^{27} - 12q^{29} - q^{31} + 3q^{33} - 5q^{35} + 5q^{37} + 6q^{39} - 20q^{41} + 8q^{43} + 6q^{45} + q^{47} + 2q^{49} + 9q^{51} + 9q^{53} - 2q^{55} + 30q^{57} + 3q^{59} - 3q^{61} + 30q^{63} + 2q^{65} + 11q^{67} - 18q^{69} - 32q^{71} - 7q^{73} - 12q^{75} - q^{77} - 11q^{79} - 9q^{81} + 8q^{83} - 6q^{85} - 18q^{87} + 9q^{89} - 8q^{91} + 3q^{93} - 5q^{95} + 12q^{97} + 12q^{99} + O(q^{100}) $

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/112\mathbb{Z}\right)^\times$.

$n$	$15$	$17$	$85$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$	$1$

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			0
$2$		0			0
$3$	1.50000	−	2.59808i	0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$3$	1.50000	−	2.59808i	0.866025	−	1.50000i	0.866025	−	0.500000i	$-0.166667\pi$
$4$		0			0
$5$	0.500000	+	0.866025i	0.223607	+	0.387298i	0.955901	−	0.293691i	$-0.0948835\pi$
$5$	0.500000	+	0.866025i	0.223607	+	0.387298i	−0.732294	+	0.680989i	$0.761550\pi$
$6$		0			0
$7$	−2.00000	+	1.73205i	−0.755929	+	0.654654i
$7$	−2.00000	+	1.73205i	−0.755929	+	0.654654i
$8$		0			0
$9$	−3.00000	−	5.19615i	−1.00000	−	1.73205i
$10$		0			0
$11$	−0.500000	+	0.866025i	−0.150756	+	0.261116i	−0.931505	−	0.363727i	$-0.881504\pi$
$11$	−0.500000	+	0.866025i	−0.150756	+	0.261116i	0.780750	+	0.624844i	$0.214837\pi$
$12$		0			0
$13$	2.00000			0.554700			0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000			0.554700			0.277350	+	0.960769i	$0.410544\pi$
$14$		0			0
$15$	3.00000			0.774597
$16$		0			0
$17$	−1.50000	+	2.59808i	−0.363803	+	0.630126i	−0.988583	−	0.150675i	$-0.951855\pi$
$17$	−1.50000	+	2.59808i	−0.363803	+	0.630126i	0.624780	+	0.780801i	$0.285189\pi$
$18$		0			0
$19$	2.50000	+	4.33013i	0.573539	+	0.993399i	0.996199	+	0.0871106i	$0.0277634\pi$
$19$	2.50000	+	4.33013i	0.573539	+	0.993399i	−0.422659	+	0.906289i	$0.638903\pi$
$20$		0			0
$21$	1.50000	+	7.79423i	0.327327	+	1.70084i
$22$		0			0
$23$	−1.50000	−	2.59808i	−0.312772	−	0.541736i	0.666190	−	0.745782i	$-0.267924\pi$
$23$	−1.50000	−	2.59808i	−0.312772	−	0.541736i	−0.978961	+	0.204046i	$0.934591\pi$
$24$		0			0
$25$	2.00000	−	3.46410i	0.400000	−	0.692820i
$26$		0			0
$27$	−9.00000			−1.73205
$28$		0			0
$29$	−6.00000			−1.11417			−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000			−1.11417			−0.557086	+	0.830455i	$0.688081\pi$
$30$		0			0
$31$	−0.500000	+	0.866025i	−0.0898027	+	0.155543i	−0.907428	−	0.420208i	$-0.861957\pi$
$31$	−0.500000	+	0.866025i	−0.0898027	+	0.155543i	0.817625	+	0.575751i	$0.195290\pi$
$32$		0			0
$33$	1.50000	+	2.59808i	0.261116	+	0.452267i
$34$		0			0
$35$	−2.50000	−	0.866025i	−0.422577	−	0.146385i
$36$		0			0
$37$	2.50000	+	4.33013i	0.410997	+	0.711868i	0.994999	−	0.0998840i	$-0.0318472\pi$
$37$	2.50000	+	4.33013i	0.410997	+	0.711868i	−0.584002	+	0.811752i	$0.698514\pi$
$38$		0			0
$39$	3.00000	−	5.19615i	0.480384	−	0.832050i
$40$		0			0
$41$	−10.0000			−1.56174			−0.780869	−	0.624695i	$-0.785223\pi$
$41$	−10.0000			−1.56174			−0.780869	+	0.624695i	$0.785223\pi$
$42$		0			0
$43$	4.00000			0.609994			0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000			0.609994			0.304997	+	0.952353i	$0.401344\pi$
$44$		0			0
$45$	3.00000	−	5.19615i	0.447214	−	0.774597i
$46$		0			0
$47$	0.500000	+	0.866025i	0.0729325	+	0.126323i	0.900185	−	0.435507i	$-0.143431\pi$
$47$	0.500000	+	0.866025i	0.0729325	+	0.126323i	−0.827253	+	0.561830i	$0.810098\pi$
$48$		0			0
$49$	1.00000	−	6.92820i	0.142857	−	0.989743i
$50$		0			0
$51$	4.50000	+	7.79423i	0.630126	+	1.09141i
$52$		0			0
$53$	4.50000	−	7.79423i	0.618123	−	1.07062i	−0.371706	−	0.928351i	$-0.621227\pi$
$53$	4.50000	−	7.79423i	0.618123	−	1.07062i	0.989828	−	0.142269i	$-0.0454398\pi$
$54$		0			0
$55$	−1.00000			−0.134840
$56$		0			0
$57$	15.0000			1.98680
$58$		0			0
$59$	1.50000	−	2.59808i	0.195283	−	0.338241i	−0.751710	−	0.659494i	$-0.770771\pi$
$59$	1.50000	−	2.59808i	0.195283	−	0.338241i	0.946993	+	0.321253i	$0.104104\pi$
$60$		0			0
$61$	−1.50000	−	2.59808i	−0.192055	−	0.332650i	0.753876	−	0.657017i	$-0.228182\pi$
$61$	−1.50000	−	2.59808i	−0.192055	−	0.332650i	−0.945931	+	0.324367i	$0.894849\pi$
$62$		0			0
$63$	15.0000	+	5.19615i	1.88982	+	0.654654i
$64$		0			0
$65$	1.00000	+	1.73205i	0.124035	+	0.214834i
$66$		0			0
$67$	5.50000	−	9.52628i	0.671932	−	1.16382i	−0.305424	−	0.952217i	$-0.598798\pi$
$67$	5.50000	−	9.52628i	0.671932	−	1.16382i	0.977356	−	0.211604i	$-0.0678686\pi$
$68$		0			0
$69$	−9.00000			−1.08347
$70$		0			0
$71$	−16.0000			−1.89885			−0.949425	−	0.313993i	$-0.898333\pi$
$71$	−16.0000			−1.89885			−0.949425	+	0.313993i	$0.898333\pi$
$72$		0			0
$73$	−3.50000	+	6.06218i	−0.409644	+	0.709524i	−0.994850	−	0.101361i	$-0.967680\pi$
$73$	−3.50000	+	6.06218i	−0.409644	+	0.709524i	0.585206	+	0.810885i	$0.301014\pi$
$74$		0			0
$75$	−6.00000	−	10.3923i	−0.692820	−	1.20000i
$76$		0			0
$77$	−0.500000	−	2.59808i	−0.0569803	−	0.296078i
$78$		0			0
$79$	−5.50000	−	9.52628i	−0.618798	−	1.07179i	−0.989705	−	0.143120i	$-0.954286\pi$
$79$	−5.50000	−	9.52628i	−0.618798	−	1.07179i	0.370907	−	0.928670i	$-0.379047\pi$
$80$		0			0
$81$	−4.50000	+	7.79423i	−0.500000	+	0.866025i
$82$		0			0
$83$	4.00000			0.439057			0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000			0.439057			0.219529	+	0.975606i	$0.429548\pi$
$84$		0			0
$85$	−3.00000			−0.325396
$86$		0			0
$87$	−9.00000	+	15.5885i	−0.964901	+	1.67126i
$88$		0			0
$89$	4.50000	+	7.79423i	0.476999	+	0.826187i	0.999653	−	0.0263586i	$-0.00839118\pi$
$89$	4.50000	+	7.79423i	0.476999	+	0.826187i	−0.522654	+	0.852545i	$0.675058\pi$
$90$		0			0
$91$	−4.00000	+	3.46410i	−0.419314	+	0.363137i
$92$		0			0
$93$	1.50000	+	2.59808i	0.155543	+	0.269408i
$94$		0			0
$95$	−2.50000	+	4.33013i	−0.256495	+	0.444262i
$96$		0			0
$97$	6.00000			0.609208			0.304604	−	0.952479i	$-0.401476\pi$
$97$	6.00000			0.609208			0.304604	+	0.952479i	$0.401476\pi$
$98$		0			0
$99$	6.00000			0.603023
$100$		0			0
$101$	6.50000	−	11.2583i	0.646774	−	1.12025i	−0.337115	−	0.941464i	$-0.609451\pi$
$101$	6.50000	−	11.2583i	0.646774	−	1.12025i	0.983889	−	0.178782i	$-0.0572157\pi$
$102$		0			0
$103$	2.50000	+	4.33013i	0.246332	+	0.426660i	0.962505	−	0.271263i	$-0.0874412\pi$
$103$	2.50000	+	4.33013i	0.246332	+	0.426660i	−0.716173	+	0.697923i	$0.754108\pi$
$104$		0			0
$105$	−6.00000	+	5.19615i	−0.585540	+	0.507093i
$106$		0			0
$107$	−1.50000	−	2.59808i	−0.145010	−	0.251166i	0.784366	−	0.620298i	$-0.212988\pi$
$107$	−1.50000	−	2.59808i	−0.145010	−	0.251166i	−0.929377	+	0.369132i	$0.879655\pi$
$108$		0			0
$109$	−5.50000	+	9.52628i	−0.526804	+	0.912452i	0.472708	+	0.881219i	$0.343277\pi$
$109$	−5.50000	+	9.52628i	−0.526804	+	0.912452i	−0.999512	+	0.0312328i	$0.990057\pi$
$110$		0			0
$111$	15.0000			1.42374
$112$		0			0
$113$	−10.0000			−0.940721			−0.470360	−	0.882474i	$-0.655876\pi$
$113$	−10.0000			−0.940721			−0.470360	+	0.882474i	$0.655876\pi$
$114$		0			0
$115$	1.50000	−	2.59808i	0.139876	−	0.242272i
$116$		0			0
$117$	−6.00000	−	10.3923i	−0.554700	−	0.960769i
$118$		0			0
$119$	−1.50000	−	7.79423i	−0.137505	−	0.714496i
$120$		0			0
$121$	5.00000	+	8.66025i	0.454545	+	0.787296i
$122$		0			0
$123$	−15.0000	+	25.9808i	−1.35250	+	2.34261i
$124$		0			0
$125$	9.00000			0.804984
$126$		0			0
$127$	8.00000			0.709885			0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000			0.709885			0.354943	+	0.934888i	$0.384500\pi$
$128$		0			0
$129$	6.00000	−	10.3923i	0.528271	−	0.914991i
$130$		0			0
$131$	8.50000	+	14.7224i	0.742648	+	1.28630i	0.951285	+	0.308312i	$0.0997640\pi$
$131$	8.50000	+	14.7224i	0.742648	+	1.28630i	−0.208637	+	0.977993i	$0.566903\pi$
$132$		0			0
$133$	−12.5000	−	4.33013i	−1.08389	−	0.375470i
$134$		0			0
$135$	−4.50000	−	7.79423i	−0.387298	−	0.670820i
$136$		0			0
$137$	−1.50000	+	2.59808i	−0.128154	+	0.221969i	−0.922961	−	0.384893i	$-0.874238\pi$
$137$	−1.50000	+	2.59808i	−0.128154	+	0.221969i	0.794808	+	0.606861i	$0.207572\pi$
$138$		0			0
$139$	−4.00000			−0.339276			−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000			−0.339276			−0.169638	+	0.985506i	$0.554260\pi$
$140$		0			0
$141$	3.00000			0.252646
$142$		0			0
$143$	−1.00000	+	1.73205i	−0.0836242	+	0.144841i
$144$		0			0
$145$	−3.00000	−	5.19615i	−0.249136	−	0.431517i
$146$		0			0
$147$	−16.5000	−	12.9904i	−1.36090	−	1.07143i
$148$		0			0
$149$	−7.50000	−	12.9904i	−0.614424	−	1.06421i	−0.990485	−	0.137619i	$-0.956055\pi$
$149$	−7.50000	−	12.9904i	−0.614424	−	1.06421i	0.376061	−	0.926595i	$-0.377278\pi$
$150$		0			0
$151$	7.50000	−	12.9904i	0.610341	−	1.05714i	−0.380841	−	0.924640i	$-0.624366\pi$
$151$	7.50000	−	12.9904i	0.610341	−	1.05714i	0.991183	−	0.132502i	$-0.0423010\pi$
$152$		0			0
$153$	18.0000			1.45521
$154$		0			0
$155$	−1.00000			−0.0803219
$156$		0			0
$157$	−7.50000	+	12.9904i	−0.598565	+	1.03675i	0.394468	+	0.918910i	$0.370929\pi$
$157$	−7.50000	+	12.9904i	−0.598565	+	1.03675i	−0.993033	+	0.117836i	$0.962404\pi$
$158$		0			0
$159$	−13.5000	−	23.3827i	−1.07062	−	1.85437i
$160$		0			0
$161$	7.50000	+	2.59808i	0.591083	+	0.204757i
$162$		0			0
$163$	4.50000	+	7.79423i	0.352467	+	0.610491i	0.986681	−	0.162667i	$-0.0520095\pi$
$163$	4.50000	+	7.79423i	0.352467	+	0.610491i	−0.634214	+	0.773158i	$0.718676\pi$
$164$		0			0
$165$	−1.50000	+	2.59808i	−0.116775	+	0.202260i
$166$		0			0
$167$	20.0000			1.54765			0.773823	−	0.633402i	$-0.218342\pi$
$167$	20.0000			1.54765			0.773823	+	0.633402i	$0.218342\pi$
$168$		0			0
$169$	−9.00000			−0.692308
$170$		0			0
$171$	15.0000	−	25.9808i	1.14708	−	1.98680i
$172$		0			0
$173$	10.5000	+	18.1865i	0.798300	+	1.38270i	0.920722	+	0.390218i	$0.127601\pi$
$173$	10.5000	+	18.1865i	0.798300	+	1.38270i	−0.122422	+	0.992478i	$0.539066\pi$
$174$		0			0
$175$	2.00000	+	10.3923i	0.151186	+	0.785584i
$176$		0			0
$177$	−4.50000	−	7.79423i	−0.338241	−	0.585850i
$178$		0			0
$179$	−0.500000	+	0.866025i	−0.0373718	+	0.0647298i	−0.884106	−	0.467286i	$-0.845232\pi$
$179$	−0.500000	+	0.866025i	−0.0373718	+	0.0647298i	0.846735	+	0.532016i	$0.178565\pi$
$180$		0			0
$181$	22.0000			1.63525			0.817624	−	0.575753i	$-0.195291\pi$
$181$	22.0000			1.63525			0.817624	+	0.575753i	$0.195291\pi$
$182$		0			0
$183$	−9.00000			−0.665299
$184$		0			0
$185$	−2.50000	+	4.33013i	−0.183804	+	0.318357i
$186$		0			0
$187$	−1.50000	−	2.59808i	−0.109691	−	0.189990i
$188$		0			0
$189$	18.0000	−	15.5885i	1.30931	−	1.13389i
$190$		0			0
$191$	8.50000	+	14.7224i	0.615038	+	1.06528i	0.990378	+	0.138390i	$0.0441928\pi$
$191$	8.50000	+	14.7224i	0.615038	+	1.06528i	−0.375339	+	0.926887i	$0.622474\pi$
$192$		0			0
$193$	2.50000	−	4.33013i	0.179954	−	0.311689i	−0.761911	−	0.647682i	$-0.775738\pi$
$193$	2.50000	−	4.33013i	0.179954	−	0.311689i	0.941865	+	0.335993i	$0.109072\pi$
$194$		0			0
$195$	6.00000			0.429669
$196$		0			0
$197$	18.0000			1.28245			0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000			1.28245			0.641223	+	0.767354i	$0.278427\pi$
$198$		0			0
$199$	−4.50000	+	7.79423i	−0.318997	+	0.552518i	−0.980279	−	0.197619i	$-0.936679\pi$
$199$	−4.50000	+	7.79423i	−0.318997	+	0.552518i	0.661282	+	0.750137i	$0.270013\pi$
$200$		0			0
$201$	−16.5000	−	28.5788i	−1.16382	−	2.01580i
$202$		0			0
$203$	12.0000	−	10.3923i	0.842235	−	0.729397i
$204$		0			0
$205$	−5.00000	−	8.66025i	−0.349215	−	0.604858i
$206$		0			0
$207$	−9.00000	+	15.5885i	−0.625543	+	1.08347i
$208$		0			0
$209$	−5.00000			−0.345857
$210$		0			0
$211$	−12.0000			−0.826114			−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000			−0.826114			−0.413057	+	0.910705i	$0.635539\pi$
$212$		0			0
$213$	−24.0000	+	41.5692i	−1.64445	+	2.84828i
$214$		0			0
$215$	2.00000	+	3.46410i	0.136399	+	0.236250i
$216$		0			0
$217$	−0.500000	−	2.59808i	−0.0339422	−	0.176369i
$218$		0			0
$219$	10.5000	+	18.1865i	0.709524	+	1.22893i
$220$		0			0
$221$	−3.00000	+	5.19615i	−0.201802	+	0.349531i
$222$		0			0
$223$	−24.0000			−1.60716			−0.803579	−	0.595198i	$-0.797074\pi$
$223$	−24.0000			−1.60716			−0.803579	+	0.595198i	$0.797074\pi$
$224$		0			0
$225$	−24.0000			−1.60000
$226$		0			0
$227$	3.50000	−	6.06218i	0.232303	−	0.402361i	−0.726182	−	0.687502i	$-0.758707\pi$
$227$	3.50000	−	6.06218i	0.232303	−	0.402361i	0.958485	+	0.285141i	$0.0920405\pi$
$228$		0			0
$229$	−3.50000	−	6.06218i	−0.231287	−	0.400600i	0.726900	−	0.686743i	$-0.240960\pi$
$229$	−3.50000	−	6.06218i	−0.231287	−	0.400600i	−0.958187	+	0.286143i	$0.907627\pi$
$230$		0			0
$231$	−7.50000	−	2.59808i	−0.493464	−	0.170941i
$232$		0			0
$233$	6.50000	+	11.2583i	0.425829	+	0.737558i	0.996497	−	0.0836229i	$-0.0266491\pi$
$233$	6.50000	+	11.2583i	0.425829	+	0.737558i	−0.570668	+	0.821181i	$0.693316\pi$
$234$		0			0
$235$	−0.500000	+	0.866025i	−0.0326164	+	0.0564933i
$236$		0			0
$237$	−33.0000			−2.14358
$238$		0			0
$239$	4.00000			0.258738			0.129369	−	0.991596i	$-0.458705\pi$
$239$	4.00000			0.258738			0.129369	+	0.991596i	$0.458705\pi$
$240$		0			0
$241$	8.50000	−	14.7224i	0.547533	−	0.948355i	−0.450910	−	0.892570i	$-0.648900\pi$
$241$	8.50000	−	14.7224i	0.547533	−	0.948355i	0.998443	−	0.0557856i	$-0.0177663\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	6.50000	−	2.59808i	0.415270	−	0.165985i
$246$		0			0
$247$	5.00000	+	8.66025i	0.318142	+	0.551039i
$248$		0			0
$249$	6.00000	−	10.3923i	0.380235	−	0.658586i
$250$		0			0
$251$	−24.0000			−1.51487			−0.757433	−	0.652913i	$-0.773547\pi$
$251$	−24.0000			−1.51487			−0.757433	+	0.652913i	$0.773547\pi$
$252$		0			0
$253$	3.00000			0.188608
$254$		0			0
$255$	−4.50000	+	7.79423i	−0.281801	+	0.488094i
$256$		0			0
$257$	6.50000	+	11.2583i	0.405459	+	0.702275i	0.994375	−	0.105919i	$-0.0337784\pi$
$257$	6.50000	+	11.2583i	0.405459	+	0.702275i	−0.588916	+	0.808194i	$0.700445\pi$
$258$		0			0
$259$	−12.5000	−	4.33013i	−0.776712	−	0.269061i
$260$		0			0
$261$	18.0000	+	31.1769i	1.11417	+	1.92980i
$262$		0			0
$263$	1.50000	−	2.59808i	0.0924940	−	0.160204i	−0.816066	−	0.577959i	$-0.803849\pi$
$263$	1.50000	−	2.59808i	0.0924940	−	0.160204i	0.908560	+	0.417755i	$0.137183\pi$
$264$		0			0
$265$	9.00000			0.552866
$266$		0			0
$267$	27.0000			1.65237
$268$		0			0
$269$	8.50000	−	14.7224i	0.518254	−	0.897643i	−0.481521	−	0.876435i	$-0.659915\pi$
$269$	8.50000	−	14.7224i	0.518254	−	0.897643i	0.999775	−	0.0212079i	$-0.00675120\pi$
$270$		0			0
$271$	−1.50000	−	2.59808i	−0.0911185	−	0.157822i	0.816864	−	0.576831i	$-0.195711\pi$
$271$	−1.50000	−	2.59808i	−0.0911185	−	0.157822i	−0.907982	+	0.419009i	$0.862378\pi$
$272$		0			0
$273$	3.00000	+	15.5885i	0.181568	+	0.943456i
$274$		0			0
$275$	2.00000	+	3.46410i	0.120605	+	0.208893i
$276$		0			0
$277$	−3.50000	+	6.06218i	−0.210295	+	0.364241i	−0.951807	−	0.306699i	$-0.900776\pi$
$277$	−3.50000	+	6.06218i	−0.210295	+	0.364241i	0.741512	+	0.670940i	$0.234109\pi$
$278$		0			0
$279$	6.00000			0.359211
$280$		0			0
$281$	−18.0000			−1.07379			−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000			−1.07379			−0.536895	+	0.843649i	$0.680403\pi$
$282$		0			0
$283$	−8.50000	+	14.7224i	−0.505273	+	0.875158i	0.494709	+	0.869059i	$0.335275\pi$
$283$	−8.50000	+	14.7224i	−0.505273	+	0.875158i	−0.999981	+	0.00609896i	$0.998059\pi$
$284$		0			0
$285$	7.50000	+	12.9904i	0.444262	+	0.769484i
$286$		0			0
$287$	20.0000	−	17.3205i	1.18056	−	1.02240i
$288$		0			0
$289$	4.00000	+	6.92820i	0.235294	+	0.407541i
$290$		0			0
$291$	9.00000	−	15.5885i	0.527589	−	0.913812i
$292$		0			0
$293$	6.00000			0.350524			0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000			0.350524			0.175262	+	0.984522i	$0.443923\pi$
$294$		0			0
$295$	3.00000			0.174667
$296$		0			0
$297$	4.50000	−	7.79423i	0.261116	−	0.452267i
$298$		0			0
$299$	−3.00000	−	5.19615i	−0.173494	−	0.300501i
$300$		0			0
$301$	−8.00000	+	6.92820i	−0.461112	+	0.399335i
$302$		0			0
$303$	−19.5000	−	33.7750i	−1.12025	−	1.94032i
$304$		0			0
$305$	1.50000	−	2.59808i	0.0858898	−	0.148765i
$306$		0			0
$307$	−4.00000			−0.228292			−0.114146	−	0.993464i	$-0.536413\pi$
$307$	−4.00000			−0.228292			−0.114146	+	0.993464i	$0.536413\pi$
$308$		0			0
$309$	15.0000			0.853320
$310$		0			0
$311$	5.50000	−	9.52628i	0.311876	−	0.540186i	−0.666892	−	0.745154i	$-0.732376\pi$
$311$	5.50000	−	9.52628i	0.311876	−	0.540186i	0.978769	+	0.204968i	$0.0657092\pi$
$312$		0			0
$313$	−15.5000	−	26.8468i	−0.876112	−	1.51747i	−0.855574	−	0.517681i	$-0.826795\pi$
$313$	−15.5000	−	26.8468i	−0.876112	−	1.51747i	−0.0205381	−	0.999789i	$-0.506538\pi$
$314$		0			0
$315$	3.00000	+	15.5885i	0.169031	+	0.878310i
$316$		0			0
$317$	−13.5000	−	23.3827i	−0.758236	−	1.31330i	−0.943750	−	0.330661i	$-0.892728\pi$
$317$	−13.5000	−	23.3827i	−0.758236	−	1.31330i	0.185514	−	0.982642i	$-0.440605\pi$
$318$		0			0
$319$	3.00000	−	5.19615i	0.167968	−	0.290929i
$320$		0			0
$321$	−9.00000			−0.502331
$322$		0			0
$323$	−15.0000			−0.834622
$324$		0			0
$325$	4.00000	−	6.92820i	0.221880	−	0.384308i
$326$		0			0
$327$	16.5000	+	28.5788i	0.912452	+	1.58041i
$328$		0			0
$329$	−2.50000	−	0.866025i	−0.137829	−	0.0477455i
$330$		0			0
$331$	−3.50000	−	6.06218i	−0.192377	−	0.333207i	0.753660	−	0.657264i	$-0.228286\pi$
$331$	−3.50000	−	6.06218i	−0.192377	−	0.333207i	−0.946038	+	0.324057i	$0.894953\pi$
$332$		0			0
$333$	15.0000	−	25.9808i	0.821995	−	1.42374i
$334$		0			0
$335$	11.0000			0.600994
$336$		0			0
$337$	14.0000			0.762629			0.381314	−	0.924445i	$-0.375472\pi$
$337$	14.0000			0.762629			0.381314	+	0.924445i	$0.375472\pi$
$338$		0			0
$339$	−15.0000	+	25.9808i	−0.814688	+	1.41108i
$340$		0			0
$341$	−0.500000	−	0.866025i	−0.0270765	−	0.0468979i
$342$		0			0
$343$	10.0000	+	15.5885i	0.539949	+	0.841698i
$344$		0			0
$345$	−4.50000	−	7.79423i	−0.242272	−	0.419627i
$346$		0			0
$347$	1.50000	−	2.59808i	0.0805242	−	0.139472i	−0.822951	−	0.568112i	$-0.807674\pi$
$347$	1.50000	−	2.59808i	0.0805242	−	0.139472i	0.903475	+	0.428640i	$0.141007\pi$
$348$		0			0
$349$	−6.00000			−0.321173			−0.160586	−	0.987022i	$-0.551338\pi$
$349$	−6.00000			−0.321173			−0.160586	+	0.987022i	$0.551338\pi$
$350$		0			0
$351$	−18.0000			−0.960769
$352$		0			0
$353$	2.50000	−	4.33013i	0.133062	−	0.230469i	−0.791794	−	0.610789i	$-0.790853\pi$
$353$	2.50000	−	4.33013i	0.133062	−	0.230469i	0.924855	+	0.380319i	$0.124186\pi$
$354$		0			0
$355$	−8.00000	−	13.8564i	−0.424596	−	0.735422i
$356$		0			0
$357$	−22.5000	−	7.79423i	−1.19083	−	0.412514i
$358$		0			0
$359$	−7.50000	−	12.9904i	−0.395835	−	0.685606i	0.597372	−	0.801964i	$-0.296211\pi$
$359$	−7.50000	−	12.9904i	−0.395835	−	0.685606i	−0.993207	+	0.116358i	$0.962878\pi$
$360$		0			0
$361$	−3.00000	+	5.19615i	−0.157895	+	0.273482i
$362$		0			0
$363$	30.0000			1.57459
$364$		0			0
$365$	−7.00000			−0.366397
$366$		0			0
$367$	9.50000	−	16.4545i	0.495896	−	0.858917i	−0.504093	−	0.863649i	$-0.668173\pi$
$367$	9.50000	−	16.4545i	0.495896	−	0.858917i	0.999989	+	0.00473247i	$0.00150640\pi$
$368$		0			0
$369$	30.0000	+	51.9615i	1.56174	+	2.70501i
$370$		0			0
$371$	4.50000	+	23.3827i	0.233628	+	1.21397i
$372$		0			0
$373$	−9.50000	−	16.4545i	−0.491891	−	0.851981i	0.508065	−	0.861319i	$-0.330361\pi$
$373$	−9.50000	−	16.4545i	−0.491891	−	0.851981i	−0.999956	+	0.00933789i	$0.997028\pi$
$374$		0			0
$375$	13.5000	−	23.3827i	0.697137	−	1.20748i
$376$		0			0
$377$	−12.0000			−0.618031
$378$		0			0
$379$	16.0000			0.821865			0.410932	−	0.911666i	$-0.365203\pi$
$379$	16.0000			0.821865			0.410932	+	0.911666i	$0.365203\pi$
$380$		0			0
$381$	12.0000	−	20.7846i	0.614779	−	1.06483i
$382$		0			0
$383$	4.50000	+	7.79423i	0.229939	+	0.398266i	0.957790	−	0.287469i	$-0.0928139\pi$
$383$	4.50000	+	7.79423i	0.229939	+	0.398266i	−0.727851	+	0.685736i	$0.759481\pi$
$384$		0			0
$385$	2.00000	−	1.73205i	0.101929	−	0.0882735i
$386$		0			0
$387$	−12.0000	−	20.7846i	−0.609994	−	1.05654i
$388$		0			0
$389$	−9.50000	+	16.4545i	−0.481669	+	0.834275i	−0.999779	−	0.0210389i	$-0.993303\pi$
$389$	−9.50000	+	16.4545i	−0.481669	+	0.834275i	0.518110	+	0.855314i	$0.326636\pi$
$390$		0			0
$391$	9.00000			0.455150
$392$		0			0
$393$	51.0000			2.57261
$394$		0			0
$395$	5.50000	−	9.52628i	0.276735	−	0.479319i
$396$		0			0
$397$	8.50000	+	14.7224i	0.426603	+	0.738898i	0.996569	−	0.0827707i	$-0.0263769\pi$
$397$	8.50000	+	14.7224i	0.426603	+	0.738898i	−0.569966	+	0.821668i	$0.693044\pi$
$398$		0			0
$399$	−30.0000	+	25.9808i	−1.50188	+	1.30066i
$400$		0			0
$401$	−1.50000	−	2.59808i	−0.0749064	−	0.129742i	0.826139	−	0.563466i	$-0.190532\pi$
$401$	−1.50000	−	2.59808i	−0.0749064	−	0.129742i	−0.901046	+	0.433724i	$0.857199\pi$
$402$		0			0
$403$	−1.00000	+	1.73205i	−0.0498135	+	0.0862796i
$404$		0			0
$405$	−9.00000			−0.447214
$406$		0			0
$407$	−5.00000			−0.247841
$408$		0			0
$409$	−9.50000	+	16.4545i	−0.469745	+	0.813622i	−0.999402	−	0.0345902i	$-0.988987\pi$
$409$	−9.50000	+	16.4545i	−0.469745	+	0.813622i	0.529657	+	0.848212i	$0.322321\pi$
$410$		0			0
$411$	4.50000	+	7.79423i	0.221969	+	0.384461i
$412$		0			0
$413$	1.50000	+	7.79423i	0.0738102	+	0.383529i
$414$		0			0
$415$	2.00000	+	3.46410i	0.0981761	+	0.170046i
$416$		0			0
$417$	−6.00000	+	10.3923i	−0.293821	+	0.508913i
$418$		0			0
$419$	−20.0000			−0.977064			−0.488532	−	0.872546i	$-0.662467\pi$
$419$	−20.0000			−0.977064			−0.488532	+	0.872546i	$0.662467\pi$
$420$		0			0
$421$	10.0000			0.487370			0.243685	−	0.969854i	$-0.421644\pi$
$421$	10.0000			0.487370			0.243685	+	0.969854i	$0.421644\pi$
$422$		0			0
$423$	3.00000	−	5.19615i	0.145865	−	0.252646i
$424$		0			0
$425$	6.00000	+	10.3923i	0.291043	+	0.504101i
$426$		0			0
$427$	7.50000	+	2.59808i	0.362950	+	0.125730i
$428$		0			0
$429$	3.00000	+	5.19615i	0.144841	+	0.250873i
$430$		0			0
$431$	−20.5000	+	35.5070i	−0.987450	+	1.71031i	−0.356953	+	0.934122i	$0.616185\pi$
$431$	−20.5000	+	35.5070i	−0.987450	+	1.71031i	−0.630497	+	0.776192i	$0.717149\pi$
$432$		0			0
$433$	−26.0000			−1.24948			−0.624740	−	0.780833i	$-0.714795\pi$
$433$	−26.0000			−1.24948			−0.624740	+	0.780833i	$0.714795\pi$
$434$		0			0
$435$	−18.0000			−0.863034
$436$		0			0
$437$	7.50000	−	12.9904i	0.358774	−	0.621414i
$438$		0			0
$439$	−7.50000	−	12.9904i	−0.357955	−	0.619997i	0.629664	−	0.776868i	$-0.283193\pi$
$439$	−7.50000	−	12.9904i	−0.357955	−	0.619997i	−0.987619	+	0.156871i	$0.949859\pi$
$440$		0			0
$441$	−39.0000	+	15.5885i	−1.85714	+	0.742307i
$442$		0			0
$443$	−13.5000	−	23.3827i	−0.641404	−	1.11094i	−0.985119	−	0.171871i	$-0.945019\pi$
$443$	−13.5000	−	23.3827i	−0.641404	−	1.11094i	0.343715	−	0.939074i	$-0.388315\pi$
$444$		0			0
$445$	−4.50000	+	7.79423i	−0.213320	+	0.369482i
$446$		0			0
$447$	−45.0000			−2.12843
$448$		0			0
$449$	−2.00000			−0.0943858			−0.0471929	−	0.998886i	$-0.515028\pi$
$449$	−2.00000			−0.0943858			−0.0471929	+	0.998886i	$0.515028\pi$
$450$		0			0
$451$	5.00000	−	8.66025i	0.235441	−	0.407795i
$452$		0			0
$453$	−22.5000	−	38.9711i	−1.05714	−	1.83102i
$454$		0			0
$455$	−5.00000	−	1.73205i	−0.234404	−	0.0811998i
$456$		0			0
$457$	8.50000	+	14.7224i	0.397613	+	0.688686i	0.993431	−	0.114433i	$-0.0365053\pi$
$457$	8.50000	+	14.7224i	0.397613	+	0.688686i	−0.595818	+	0.803120i	$0.703172\pi$
$458$		0			0
$459$	13.5000	−	23.3827i	0.630126	−	1.09141i
$460$		0			0
$461$	−6.00000			−0.279448			−0.139724	−	0.990190i	$-0.544622\pi$
$461$	−6.00000			−0.279448			−0.139724	+	0.990190i	$0.544622\pi$
$462$		0			0
$463$	16.0000			0.743583			0.371792	−	0.928316i	$-0.378744\pi$
$463$	16.0000			0.743583			0.371792	+	0.928316i	$0.378744\pi$
$464$		0			0
$465$	−1.50000	+	2.59808i	−0.0695608	+	0.120483i
$466$		0			0
$467$	12.5000	+	21.6506i	0.578431	+	1.00187i	0.995660	+	0.0930703i	$0.0296681\pi$
$467$	12.5000	+	21.6506i	0.578431	+	1.00187i	−0.417229	+	0.908802i	$0.636999\pi$
$468$		0			0
$469$	5.50000	+	28.5788i	0.253966	+	1.31965i
$470$		0			0
$471$	22.5000	+	38.9711i	1.03675	+	1.79570i
$472$		0			0
$473$	−2.00000	+	3.46410i	−0.0919601	+	0.159280i
$474$		0			0
$475$	20.0000			0.917663
$476$		0			0
$477$	−54.0000			−2.47249
$478$		0			0
$479$	−10.5000	+	18.1865i	−0.479757	+	0.830964i	−0.999730	−	0.0232187i	$-0.992609\pi$
$479$	−10.5000	+	18.1865i	−0.479757	+	0.830964i	0.519973	+	0.854183i	$0.325942\pi$
$480$		0			0
$481$	5.00000	+	8.66025i	0.227980	+	0.394874i
$482$		0			0
$483$	18.0000	−	15.5885i	0.819028	−	0.709299i
$484$		0			0
$485$	3.00000	+	5.19615i	0.136223	+	0.235945i
$486$		0			0
$487$	−6.50000	+	11.2583i	−0.294543	+	0.510164i	−0.974879	−	0.222737i	$-0.928501\pi$
$487$	−6.50000	+	11.2583i	−0.294543	+	0.510164i	0.680335	+	0.732901i	$0.261834\pi$
$488$		0			0
$489$	27.0000			1.22098
$490$		0			0
$491$		0			0			−	1.00000i	$-0.5\pi$
$491$		0			0				1.00000i	$0.5\pi$
$492$		0			0
$493$	9.00000	−	15.5885i	0.405340	−	0.702069i
$494$		0			0
$495$	3.00000	+	5.19615i	0.134840	+	0.233550i
$496$		0			0
$497$	32.0000	−	27.7128i	1.43540	−	1.24309i
$498$		0			0
$499$	−3.50000	−	6.06218i	−0.156682	−	0.271380i	0.776989	−	0.629515i	$-0.216746\pi$
$499$	−3.50000	−	6.06218i	−0.156682	−	0.271380i	−0.933670	+	0.358134i	$0.883413\pi$
$500$		0			0
$501$	30.0000	−	51.9615i	1.34030	−	2.32147i
$502$		0			0
$503$	−16.0000			−0.713405			−0.356702	−	0.934218i	$-0.616099\pi$
$503$	−16.0000			−0.713405			−0.356702	+	0.934218i	$0.616099\pi$
$504$		0			0
$505$	13.0000			0.578492
$506$		0			0
$507$	−13.5000	+	23.3827i	−0.599556	+	1.03846i
$508$		0			0
$509$	−3.50000	−	6.06218i	−0.155135	−	0.268701i	0.777973	−	0.628297i	$-0.216248\pi$
$509$	−3.50000	−	6.06218i	−0.155135	−	0.268701i	−0.933108	+	0.359596i	$0.882915\pi$
$510$		0			0
$511$	−3.50000	−	18.1865i	−0.154831	−	0.804525i
$512$		0			0
$513$	−22.5000	−	38.9711i	−0.993399	−	1.72062i
$514$		0			0
$515$	−2.50000	+	4.33013i	−0.110163	+	0.190808i
$516$		0			0
$517$	−1.00000			−0.0439799
$518$		0			0
$519$	63.0000			2.76539
$520$		0			0
$521$	−7.50000	+	12.9904i	−0.328581	+	0.569119i	−0.982231	−	0.187678i	$-0.939904\pi$
$521$	−7.50000	+	12.9904i	−0.328581	+	0.569119i	0.653650	+	0.756797i	$0.273237\pi$
$522$		0			0
$523$	6.50000	+	11.2583i	0.284225	+	0.492292i	0.972421	−	0.233233i	$-0.0749303\pi$
$523$	6.50000	+	11.2583i	0.284225	+	0.492292i	−0.688196	+	0.725525i	$0.741597\pi$
$524$		0			0
$525$	30.0000	+	10.3923i	1.30931	+	0.453557i
$526$		0			0
$527$	−1.50000	−	2.59808i	−0.0653410	−	0.113174i
$528$		0			0
$529$	7.00000	−	12.1244i	0.304348	−	0.527146i
$530$		0			0
$531$	−18.0000			−0.781133
$532$		0			0
$533$	−20.0000			−0.866296
$534$		0			0
$535$	1.50000	−	2.59808i	0.0648507	−	0.112325i
$536$		0			0
$537$	1.50000	+	2.59808i	0.0647298	+	0.112115i
$538$		0			0
$539$	5.50000	+	4.33013i	0.236902	+	0.186512i
$540$		0			0
$541$	12.5000	+	21.6506i	0.537417	+	0.930834i	0.999042	+	0.0437584i	$0.0139332\pi$
$541$	12.5000	+	21.6506i	0.537417	+	0.930834i	−0.461625	+	0.887075i	$0.652733\pi$
$542$		0			0
$543$	33.0000	−	57.1577i	1.41617	−	2.45287i
$544$		0			0
$545$	−11.0000			−0.471188
$546$		0			0
$547$		0			0			−	1.00000i	$-0.5\pi$
$547$		0			0				1.00000i	$0.5\pi$
$548$		0			0
$549$	−9.00000	+	15.5885i	−0.384111	+	0.665299i
$550$		0			0
$551$	−15.0000	−	25.9808i	−0.639021	−	1.10682i
$552$		0			0
$553$	27.5000	+	9.52628i	1.16942	+	0.405099i
$554$		0			0
$555$	7.50000	+	12.9904i	0.318357	+	0.551411i
$556$		0			0
$557$	−5.50000	+	9.52628i	−0.233042	+	0.403641i	−0.958702	−	0.284413i	$-0.908201\pi$
$557$	−5.50000	+	9.52628i	−0.233042	+	0.403641i	0.725660	+	0.688054i	$0.241535\pi$
$558$		0			0
$559$	8.00000			0.338364
$560$		0			0
$561$	−9.00000			−0.379980
$562$		0			0
$563$	5.50000	−	9.52628i	0.231797	−	0.401485i	−0.726540	−	0.687124i	$-0.758873\pi$
$563$	5.50000	−	9.52628i	0.231797	−	0.401485i	0.958337	+	0.285640i	$0.0922060\pi$
$564$		0			0
$565$	−5.00000	−	8.66025i	−0.210352	−	0.364340i
$566$		0			0
$567$	−4.50000	−	23.3827i	−0.188982	−	0.981981i
$568$		0			0
$569$	0.500000	+	0.866025i	0.0209611	+	0.0363057i	0.876316	−	0.481737i	$-0.159994\pi$
$569$	0.500000	+	0.866025i	0.0209611	+	0.0363057i	−0.855355	+	0.518043i	$0.826661\pi$
$570$		0			0
$571$	−8.50000	+	14.7224i	−0.355714	+	0.616115i	−0.987240	−	0.159240i	$-0.949096\pi$
$571$	−8.50000	+	14.7224i	−0.355714	+	0.616115i	0.631526	+	0.775355i	$0.282429\pi$
$572$		0			0
$573$	51.0000			2.13056
$574$		0			0
$575$	−12.0000			−0.500435
$576$		0			0
$577$	−15.5000	+	26.8468i	−0.645273	+	1.11765i	0.338965	+	0.940799i	$0.389923\pi$
$577$	−15.5000	+	26.8468i	−0.645273	+	1.11765i	−0.984238	+	0.176847i	$0.943410\pi$
$578$		0			0
$579$	−7.50000	−	12.9904i	−0.311689	−	0.539862i
$580$		0			0
$581$	−8.00000	+	6.92820i	−0.331896	+	0.287430i
$582$		0			0
$583$	4.50000	+	7.79423i	0.186371	+	0.322804i
$584$		0			0
$585$	6.00000	−	10.3923i	0.248069	−	0.429669i
$586$		0			0
$587$	−12.0000			−0.495293			−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000			−0.495293			−0.247647	+	0.968850i	$0.579657\pi$
$588$		0			0
$589$	−5.00000			−0.206021
$590$		0			0
$591$	27.0000	−	46.7654i	1.11063	−	1.92367i
$592$		0			0
$593$	−21.5000	−	37.2391i	−0.882899	−	1.52923i	−0.848103	−	0.529832i	$-0.822255\pi$
$593$	−21.5000	−	37.2391i	−0.882899	−	1.52923i	−0.0347964	−	0.999394i	$-0.511078\pi$
$594$		0			0
$595$	6.00000	−	5.19615i	0.245976	−	0.213021i
$596$		0			0
$597$	13.5000	+	23.3827i	0.552518	+	0.956990i
$598$		0			0
$599$	−10.5000	+	18.1865i	−0.429018	+	0.743082i	−0.996786	−	0.0801071i	$-0.974474\pi$
$599$	−10.5000	+	18.1865i	−0.429018	+	0.743082i	0.567768	+	0.823189i	$0.307807\pi$
$600$		0			0
$601$	−34.0000			−1.38689			−0.693444	−	0.720510i	$-0.743908\pi$
$601$	−34.0000			−1.38689			−0.693444	+	0.720510i	$0.743908\pi$
$602$		0			0
$603$	−66.0000			−2.68773
$604$		0			0
$605$	−5.00000	+	8.66025i	−0.203279	+	0.352089i
$606$		0			0
$607$	−3.50000	−	6.06218i	−0.142061	−	0.246056i	0.786212	−	0.617957i	$-0.212039\pi$
$607$	−3.50000	−	6.06218i	−0.142061	−	0.246056i	−0.928272	+	0.371901i	$0.878706\pi$
$608$		0			0
$609$	−9.00000	−	46.7654i	−0.364698	−	1.89503i
$610$		0			0
$611$	1.00000	+	1.73205i	0.0404557	+	0.0700713i
$612$		0			0
$613$	10.5000	−	18.1865i	0.424091	−	0.734547i	−0.572244	−	0.820083i	$-0.693927\pi$
$613$	10.5000	−	18.1865i	0.424091	−	0.734547i	0.996335	+	0.0855362i	$0.0272603\pi$
$614$		0			0
$615$	−30.0000			−1.20972
$616$		0			0
$617$	22.0000			0.885687			0.442843	−	0.896599i	$-0.353970\pi$
$617$	22.0000			0.885687			0.442843	+	0.896599i	$0.353970\pi$
$618$		0			0
$619$	−2.50000	+	4.33013i	−0.100483	+	0.174042i	−0.911884	−	0.410448i	$-0.865372\pi$
$619$	−2.50000	+	4.33013i	−0.100483	+	0.174042i	0.811400	+	0.584491i	$0.198706\pi$
$620$		0			0
$621$	13.5000	+	23.3827i	0.541736	+	0.938315i
$622$		0			0
$623$	−22.5000	−	7.79423i	−0.901443	−	0.312269i
$624$		0			0
$625$	−5.50000	−	9.52628i	−0.220000	−	0.381051i
$626$		0			0
$627$	−7.50000	+	12.9904i	−0.299521	+	0.518786i
$628$		0			0
$629$	−15.0000			−0.598089
$630$		0			0
$631$		0			0			−	1.00000i	$-0.5\pi$
$631$		0			0				1.00000i	$0.5\pi$
$632$		0			0
$633$	−18.0000	+	31.1769i	−0.715436	+	1.23917i
$634$		0			0
$635$	4.00000	+	6.92820i	0.158735	+	0.274937i
$636$		0			0
$637$	2.00000	−	13.8564i	0.0792429	−	0.549011i
$638$		0			0
$639$	48.0000	+	83.1384i	1.89885	+	3.28891i
$640$		0			0
$641$	−7.50000	+	12.9904i	−0.296232	+	0.513089i	−0.975271	−	0.221013i	$-0.929064\pi$
$641$	−7.50000	+	12.9904i	−0.296232	+	0.513089i	0.679039	+	0.734103i	$0.262397\pi$
$642$		0			0
$643$	44.0000			1.73519			0.867595	−	0.497271i	$-0.165665\pi$
$643$	44.0000			1.73519			0.867595	+	0.497271i	$0.165665\pi$
$644$		0			0
$645$	12.0000			0.472500
$646$		0			0
$647$	21.5000	−	37.2391i	0.845252	−	1.46402i	−0.0401498	−	0.999194i	$-0.512784\pi$
$647$	21.5000	−	37.2391i	0.845252	−	1.46402i	0.885402	−	0.464826i	$-0.153883\pi$
$648$		0			0
$649$	1.50000	+	2.59808i	0.0588802	+	0.101983i
$650$		0			0
$651$	−7.50000	−	2.59808i	−0.293948	−	0.101827i
$652$		0			0
$653$	2.50000	+	4.33013i	0.0978326	+	0.169451i	0.910787	−	0.412876i	$-0.135476\pi$
$653$	2.50000	+	4.33013i	0.0978326	+	0.169451i	−0.812955	+	0.582327i	$0.802142\pi$
$654$		0			0
$655$	−8.50000	+	14.7224i	−0.332122	+	0.575253i
$656$		0			0
$657$	42.0000			1.63858
$658$		0			0
$659$	36.0000			1.40236			0.701180	−	0.712984i	$-0.252657\pi$
$659$	36.0000			1.40236			0.701180	+	0.712984i	$0.252657\pi$
$660$		0			0
$661$	0.500000	−	0.866025i	0.0194477	−	0.0336845i	−0.856138	−	0.516748i	$-0.827143\pi$
$661$	0.500000	−	0.866025i	0.0194477	−	0.0336845i	0.875585	+	0.483063i	$0.160476\pi$
$662$		0			0
$663$	9.00000	+	15.5885i	0.349531	+	0.605406i
$664$		0			0
$665$	−2.50000	−	12.9904i	−0.0969458	−	0.503745i
$666$		0			0
$667$	9.00000	+	15.5885i	0.348481	+	0.603587i
$668$		0			0
$669$	−36.0000	+	62.3538i	−1.39184	+	2.41074i
$670$		0			0
$671$	3.00000			0.115814
$672$		0			0
$673$	−2.00000			−0.0770943			−0.0385472	−	0.999257i	$-0.512273\pi$
$673$	−2.00000			−0.0770943			−0.0385472	+	0.999257i	$0.512273\pi$
$674$		0			0
$675$	−18.0000	+	31.1769i	−0.692820	+	1.20000i
$676$		0			0
$677$	4.50000	+	7.79423i	0.172949	+	0.299557i	0.939450	−	0.342687i	$-0.111337\pi$
$677$	4.50000	+	7.79423i	0.172949	+	0.299557i	−0.766501	+	0.642244i	$0.778004\pi$
$678$		0			0
$679$	−12.0000	+	10.3923i	−0.460518	+	0.398820i
$680$		0			0
$681$	−10.5000	−	18.1865i	−0.402361	−	0.696909i
$682$		0			0
$683$	19.5000	−	33.7750i	0.746147	−	1.29236i	−0.203510	−	0.979073i	$-0.565235\pi$
$683$	19.5000	−	33.7750i	0.746147	−	1.29236i	0.949657	−	0.313291i	$-0.101432\pi$
$684$		0			0
$685$	−3.00000			−0.114624
$686$		0			0
$687$	−21.0000			−0.801200
$688$		0			0
$689$	9.00000	−	15.5885i	0.342873	−	0.593873i
$690$		0			0
$691$	−23.5000	−	40.7032i	−0.893982	−	1.54842i	−0.835059	−	0.550160i	$-0.814567\pi$
$691$	−23.5000	−	40.7032i	−0.893982	−	1.54842i	−0.0589228	−	0.998263i	$-0.518767\pi$
$692$		0			0
$693$	−12.0000	+	10.3923i	−0.455842	+	0.394771i
$694$		0			0
$695$	−2.00000	−	3.46410i	−0.0758643	−	0.131401i
$696$		0			0
$697$	15.0000	−	25.9808i	0.568166	−	0.984092i
$698$		0			0
$699$	39.0000			1.47512
$700$		0			0
$701$	−18.0000			−0.679851			−0.339925	−	0.940452i	$-0.610402\pi$
$701$	−18.0000			−0.679851			−0.339925	+	0.940452i	$0.610402\pi$
$702$		0			0
$703$	−12.5000	+	21.6506i	−0.471446	+	0.816569i
$704$		0			0
$705$	1.50000	+	2.59808i	0.0564933	+	0.0978492i
$706$		0			0
$707$	6.50000	+	33.7750i	0.244458	+	1.27024i
$708$		0			0
$709$	−13.5000	−	23.3827i	−0.507003	−	0.878155i	−0.999967	−	0.00810550i	$-0.997420\pi$
$709$	−13.5000	−	23.3827i	−0.507003	−	0.878155i	0.492964	−	0.870050i	$-0.335913\pi$
$710$		0			0
$711$	−33.0000	+	57.1577i	−1.23760	+	2.14358i
$712$		0			0
$713$	3.00000			0.112351
$714$		0			0
$715$	−2.00000			−0.0747958
$716$		0			0
$717$	6.00000	−	10.3923i	0.224074	−	0.388108i
$718$		0			0
$719$	14.5000	+	25.1147i	0.540759	+	0.936622i	0.998861	+	0.0477220i	$0.0151961\pi$
$719$	14.5000	+	25.1147i	0.540759	+	0.936622i	−0.458102	+	0.888900i	$0.651471\pi$
$720$		0			0
$721$	−12.5000	−	4.33013i	−0.465524	−	0.161262i
$722$		0			0
$723$	−25.5000	−	44.1673i	−0.948355	−	1.64260i
$724$		0			0
$725$	−12.0000	+	20.7846i	−0.445669	+	0.771921i
$726$		0			0
$727$	−16.0000			−0.593407			−0.296704	−	0.954970i	$-0.595887\pi$
$727$	−16.0000			−0.593407			−0.296704	+	0.954970i	$0.595887\pi$
$728$		0			0
$729$	−27.0000			−1.00000
$730$		0			0
$731$	−6.00000	+	10.3923i	−0.221918	+	0.384373i
$732$		0			0
$733$	−5.50000	−	9.52628i	−0.203147	−	0.351861i	0.746394	−	0.665505i	$-0.231784\pi$
$733$	−5.50000	−	9.52628i	−0.203147	−	0.351861i	−0.949541	+	0.313644i	$0.898450\pi$
$734$		0			0
$735$	3.00000	−	20.7846i	0.110657	−	0.766652i
$736$		0			0
$737$	5.50000	+	9.52628i	0.202595	+	0.350905i
$738$		0			0
$739$	−20.5000	+	35.5070i	−0.754105	+	1.30615i	0.191714	+	0.981451i	$0.438596\pi$
$739$	−20.5000	+	35.5070i	−0.754105	+	1.30615i	−0.945818	+	0.324697i	$0.894738\pi$
$740$		0			0
$741$	30.0000			1.10208
$742$		0			0
$743$	−32.0000			−1.17397			−0.586983	−	0.809599i	$-0.699684\pi$
$743$	−32.0000			−1.17397			−0.586983	+	0.809599i	$0.699684\pi$
$744$		0			0
$745$	7.50000	−	12.9904i	0.274779	−	0.475931i
$746$		0			0
$747$	−12.0000	−	20.7846i	−0.439057	−	0.760469i
$748$		0			0
$749$	7.50000	+	2.59808i	0.274044	+	0.0949316i
$750$		0			0
$751$	−23.5000	−	40.7032i	−0.857527	−	1.48528i	−0.874281	−	0.485421i	$-0.838666\pi$
$751$	−23.5000	−	40.7032i	−0.857527	−	1.48528i	0.0167534	−	0.999860i	$-0.494667\pi$
$752$		0			0
$753$	−36.0000	+	62.3538i	−1.31191	+	2.27230i
$754$		0			0
$755$	15.0000			0.545906
$756$		0			0
$757$	34.0000			1.23575			0.617876	−	0.786276i	$-0.287994\pi$
$757$	34.0000			1.23575			0.617876	+	0.786276i	$0.287994\pi$
$758$		0			0
$759$	4.50000	−	7.79423i	0.163340	−	0.282913i
$760$		0			0
$761$	−13.5000	−	23.3827i	−0.489375	−	0.847622i	0.510551	−	0.859848i	$-0.329442\pi$
$761$	−13.5000	−	23.3827i	−0.489375	−	0.847622i	−0.999925	+	0.0122260i	$0.996108\pi$
$762$		0			0
$763$	−5.50000	−	28.5788i	−0.199113	−	1.03462i
$764$		0			0
$765$	9.00000	+	15.5885i	0.325396	+	0.563602i
$766$		0			0
$767$	3.00000	−	5.19615i	0.108324	−	0.187622i
$768$		0			0
$769$	14.0000			0.504853			0.252426	−	0.967616i	$-0.418771\pi$
$769$	14.0000			0.504853			0.252426	+	0.967616i	$0.418771\pi$
$770$		0			0
$771$	39.0000			1.40455
$772$		0			0
$773$	−17.5000	+	30.3109i	−0.629431	+	1.09021i	0.358235	+	0.933632i	$0.383379\pi$
$773$	−17.5000	+	30.3109i	−0.629431	+	1.09021i	−0.987666	+	0.156575i	$0.949955\pi$
$774$		0			0
$775$	2.00000	+	3.46410i	0.0718421	+	0.124434i
$776$		0			0
$777$	−30.0000	+	25.9808i	−1.07624	+	0.932055i
$778$		0			0
$779$	−25.0000	−	43.3013i	−0.895718	−	1.55143i
$780$		0			0
$781$	8.00000	−	13.8564i	0.286263	−	0.495821i
$782$		0			0
$783$	54.0000			1.92980
$784$		0			0
$785$	−15.0000			−0.535373
$786$		0			0
$787$	−6.50000	+	11.2583i	−0.231700	+	0.401316i	−0.958308	−	0.285736i	$-0.907762\pi$
$787$	−6.50000	+	11.2583i	−0.231700	+	0.401316i	0.726609	+	0.687052i	$0.241095\pi$
$788$		0			0
$789$	−4.50000	−	7.79423i	−0.160204	−	0.277482i
$790$		0			0
$791$	20.0000	−	17.3205i	0.711118	−	0.615846i
$792$		0			0
$793$	−3.00000	−	5.19615i	−0.106533	−	0.184521i
$794$		0			0
$795$	13.5000	−	23.3827i	0.478796	−	0.829298i
$796$		0			0

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 112 = 2^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	112.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.50000 - 2.59808i) q^{3} +(0.500000 + 0.866025i) q^{5} +(-2.00000 + 1.73205i) q^{7} +(-3.00000 - 5.19615i) q^{9} +O(q^{10})\)	\(q+(1.50000 - 2.59808i) q^{3} +(0.500000 + 0.866025i) q^{5} +(-2.00000 + 1.73205i) q^{7} +(-3.00000 - 5.19615i) q^{9} +(-0.500000 + 0.866025i) q^{11} +2.00000 q^{13} +3.00000 q^{15} +(-1.50000 + 2.59808i) q^{17} +(2.50000 + 4.33013i) q^{19} +(1.50000 + 7.79423i) q^{21} +(-1.50000 - 2.59808i) q^{23} +(2.00000 - 3.46410i) q^{25} -9.00000 q^{27} -6.00000 q^{29} +(-0.500000 + 0.866025i) q^{31} +(1.50000 + 2.59808i) q^{33} +(-2.50000 - 0.866025i) q^{35} +(2.50000 + 4.33013i) q^{37} +(3.00000 - 5.19615i) q^{39} -10.0000 q^{41} +4.00000 q^{43} +(3.00000 - 5.19615i) q^{45} +(0.500000 + 0.866025i) q^{47} +(1.00000 - 6.92820i) q^{49} +(4.50000 + 7.79423i) q^{51} +(4.50000 - 7.79423i) q^{53} -1.00000 q^{55} +15.0000 q^{57} +(1.50000 - 2.59808i) q^{59} +(-1.50000 - 2.59808i) q^{61} +(15.0000 + 5.19615i) q^{63} +(1.00000 + 1.73205i) q^{65} +(5.50000 - 9.52628i) q^{67} -9.00000 q^{69} -16.0000 q^{71} +(-3.50000 + 6.06218i) q^{73} +(-6.00000 - 10.3923i) q^{75} +(-0.500000 - 2.59808i) q^{77} +(-5.50000 - 9.52628i) q^{79} +(-4.50000 + 7.79423i) q^{81} +4.00000 q^{83} -3.00000 q^{85} +(-9.00000 + 15.5885i) q^{87} +(4.50000 + 7.79423i) q^{89} +(-4.00000 + 3.46410i) q^{91} +(1.50000 + 2.59808i) q^{93} +(-2.50000 + 4.33013i) q^{95} +6.00000 q^{97} +6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2q + 3q^{3} + q^{5} - 4q^{7} - 6q^{9} + O(q^{10}) \)	\( 2q + 3q^{3} + q^{5} - 4q^{7} - 6q^{9} - q^{11} + 4q^{13} + 6q^{15} - 3q^{17} + 5q^{19} + 3q^{21} - 3q^{23} + 4q^{25} - 18q^{27} - 12q^{29} - q^{31} + 3q^{33} - 5q^{35} + 5q^{37} + 6q^{39} - 20q^{41} + 8q^{43} + 6q^{45} + q^{47} + 2q^{49} + 9q^{51} + 9q^{53} - 2q^{55} + 30q^{57} + 3q^{59} - 3q^{61} + 30q^{63} + 2q^{65} + 11q^{67} - 18q^{69} - 32q^{71} - 7q^{73} - 12q^{75} - q^{77} - 11q^{79} - 9q^{81} + 8q^{83} - 6q^{85} - 18q^{87} + 9q^{89} - 8q^{91} + 3q^{93} - 5q^{95} + 12q^{97} + 12q^{99} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

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