LMFDB - Embedded newform 1170.2.e.b.469.1

Newspace parameters

Level:	$ N $	$=$	$ 1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1170.e (of order $2$, degree $1$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$9.34249703649$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 390)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			469.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1170.469
Dual form			1170.2.e.b.469.2

$q$-expansion

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} +(-2.00000 + 1.00000i) q^{5} +4.00000i q^{7} +1.00000i q^{8} +O(q^{10})$	\(q-1.00000i q^{2} -1.00000 q^{4} +(-2.00000 + 1.00000i) q^{5} +4.00000i q^{7} +1.00000i q^{8} +(1.00000 + 2.00000i) q^{10} -2.00000 q^{11} -1.00000i q^{13} +4.00000 q^{14} +1.00000 q^{16} -4.00000i q^{17} -2.00000 q^{19} +(2.00000 - 1.00000i) q^{20} +2.00000i q^{22} -6.00000i q^{23} +(3.00000 - 4.00000i) q^{25} -1.00000 q^{26} -4.00000i q^{28} -2.00000 q^{29} -4.00000 q^{31} -1.00000i q^{32} -4.00000 q^{34} +(-4.00000 - 8.00000i) q^{35} -6.00000i q^{37} +2.00000i q^{38} +(-1.00000 - 2.00000i) q^{40} +6.00000 q^{41} -8.00000i q^{43} +2.00000 q^{44} -6.00000 q^{46} -8.00000i q^{47} -9.00000 q^{49} +(-4.00000 - 3.00000i) q^{50} +1.00000i q^{52} +10.0000i q^{53} +(4.00000 - 2.00000i) q^{55} -4.00000 q^{56} +2.00000i q^{58} -14.0000 q^{59} +10.0000 q^{61} +4.00000i q^{62} -1.00000 q^{64} +(1.00000 + 2.00000i) q^{65} +4.00000i q^{67} +4.00000i q^{68} +(-8.00000 + 4.00000i) q^{70} -8.00000 q^{71} -10.0000i q^{73} -6.00000 q^{74} +2.00000 q^{76} -8.00000i q^{77} +8.00000 q^{79} +(-2.00000 + 1.00000i) q^{80} -6.00000i q^{82} +12.0000i q^{83} +(4.00000 + 8.00000i) q^{85} -8.00000 q^{86} -2.00000i q^{88} -18.0000 q^{89} +4.00000 q^{91} +6.00000i q^{92} -8.00000 q^{94} +(4.00000 - 2.00000i) q^{95} +6.00000i q^{97} +9.00000i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} - 4 q^{5}+O(q^{10}) $ 2 * q - 2 * q^4 - 4 * q^5	$ 2 q - 2 q^{4} - 4 q^{5} + 2 q^{10} - 4 q^{11} + 8 q^{14} + 2 q^{16} - 4 q^{19} + 4 q^{20} + 6 q^{25} - 2 q^{26} - 4 q^{29} - 8 q^{31} - 8 q^{34} - 8 q^{35} - 2 q^{40} + 12 q^{41} + 4 q^{44} - 12 q^{46} - 18 q^{49} - 8 q^{50} + 8 q^{55} - 8 q^{56} - 28 q^{59} + 20 q^{61} - 2 q^{64} + 2 q^{65} - 16 q^{70} - 16 q^{71} - 12 q^{74} + 4 q^{76} + 16 q^{79} - 4 q^{80} + 8 q^{85} - 16 q^{86} - 36 q^{89} + 8 q^{91} - 16 q^{94} + 8 q^{95}+O(q^{100}) $ 2 * q - 2 * q^4 - 4 * q^5 + 2 * q^10 - 4 * q^11 + 8 * q^14 + 2 * q^16 - 4 * q^19 + 4 * q^20 + 6 * q^25 - 2 * q^26 - 4 * q^29 - 8 * q^31 - 8 * q^34 - 8 * q^35 - 2 * q^40 + 12 * q^41 + 4 * q^44 - 12 * q^46 - 18 * q^49 - 8 * q^50 + 8 * q^55 - 8 * q^56 - 28 * q^59 + 20 * q^61 - 2 * q^64 + 2 * q^65 - 16 * q^70 - 16 * q^71 - 12 * q^74 + 4 * q^76 + 16 * q^79 - 4 * q^80 + 8 * q^85 - 16 * q^86 - 36 * q^89 + 8 * q^91 - 16 * q^94 + 8 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1170\mathbb{Z}\right)^\times$.

$n$	$911$	$937$	$1081$
$\chi(n)$	$1$	$-1$	$1$

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
$3$		0			0
$3$		0			0
$4$	−1.00000			−0.500000
$5$	−2.00000	+	1.00000i	−0.894427	+	0.447214i
$5$	−2.00000	+	1.00000i	−0.894427	+	0.447214i
$6$		0			0
$7$			4.00000i			1.51186i	0.654654	+	0.755929i	$0.272814\pi$
$7$			4.00000i			1.51186i	−0.654654	+	0.755929i	$0.727186\pi$
$8$			1.00000i			0.353553i
$9$		0			0
$10$	1.00000	+	2.00000i	0.316228	+	0.632456i
$11$	−2.00000			−0.603023			−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000			−0.603023			−0.301511	+	0.953463i	$0.597491\pi$
$12$		0			0
$13$		−	1.00000i		−	0.277350i
$13$		−	1.00000i		−	0.277350i
$14$	4.00000			1.06904
$15$		0			0
$16$	1.00000			0.250000
$17$		−	4.00000i		−	0.970143i	−0.874475	−	0.485071i	$-0.838794\pi$
$17$		−	4.00000i		−	0.970143i	0.874475	−	0.485071i	$-0.161206\pi$
$18$		0			0
$19$	−2.00000			−0.458831			−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000			−0.458831			−0.229416	+	0.973329i	$0.573682\pi$
$20$	2.00000	−	1.00000i	0.447214	−	0.223607i
$21$		0			0
$22$			2.00000i			0.426401i
$23$		−	6.00000i		−	1.25109i	−0.780189	−	0.625543i	$-0.784877\pi$
$23$		−	6.00000i		−	1.25109i	0.780189	−	0.625543i	$-0.215123\pi$
$24$		0			0
$25$	3.00000	−	4.00000i	0.600000	−	0.800000i
$26$	−1.00000			−0.196116
$27$		0			0
$28$		−	4.00000i		−	0.755929i
$29$	−2.00000			−0.371391			−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000			−0.371391			−0.185695	+	0.982607i	$0.559454\pi$
$30$		0			0
$31$	−4.00000			−0.718421			−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000			−0.718421			−0.359211	+	0.933257i	$0.616954\pi$
$32$		−	1.00000i		−	0.176777i
$33$		0			0
$34$	−4.00000			−0.685994
$35$	−4.00000	−	8.00000i	−0.676123	−	1.35225i
$36$		0			0
$37$		−	6.00000i		−	0.986394i	−0.869918	−	0.493197i	$-0.835828\pi$
$37$		−	6.00000i		−	0.986394i	0.869918	−	0.493197i	$-0.164172\pi$
$38$			2.00000i			0.324443i
$39$		0			0
$40$	−1.00000	−	2.00000i	−0.158114	−	0.316228i
$41$	6.00000			0.937043			0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000			0.937043			0.468521	+	0.883452i	$0.344787\pi$
$42$		0			0
$43$		−	8.00000i		−	1.21999i	−0.792406	−	0.609994i	$-0.791172\pi$
$43$		−	8.00000i		−	1.21999i	0.792406	−	0.609994i	$-0.208828\pi$
$44$	2.00000			0.301511
$45$		0			0
$46$	−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	$-0.198301\pi$
$48$		0			0
$49$	−9.00000			−1.28571
$50$	−4.00000	−	3.00000i	−0.565685	−	0.424264i
$51$		0			0
$52$			1.00000i			0.138675i
$53$			10.0000i			1.37361i	0.726844	+	0.686803i	$0.240986\pi$
$53$			10.0000i			1.37361i	−0.726844	+	0.686803i	$0.759014\pi$
$54$		0			0
$55$	4.00000	−	2.00000i	0.539360	−	0.269680i
$56$	−4.00000			−0.534522
$57$		0			0
$58$			2.00000i			0.262613i
$59$	−14.0000			−1.82264			−0.911322	−	0.411693i	$-0.864937\pi$
$59$	−14.0000			−1.82264			−0.911322	+	0.411693i	$0.864937\pi$
$60$		0			0
$61$	10.0000			1.28037			0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000			1.28037			0.640184	+	0.768221i	$0.278858\pi$
$62$			4.00000i			0.508001i
$63$		0			0
$64$	−1.00000			−0.125000
$65$	1.00000	+	2.00000i	0.124035	+	0.248069i
$66$		0			0
$67$			4.00000i			0.488678i	0.969690	+	0.244339i	$0.0785709\pi$
$67$			4.00000i			0.488678i	−0.969690	+	0.244339i	$0.921429\pi$
$68$			4.00000i			0.485071i
$69$		0			0
$70$	−8.00000	+	4.00000i	−0.956183	+	0.478091i
$71$	−8.00000			−0.949425			−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000			−0.949425			−0.474713	+	0.880141i	$0.657448\pi$
$72$		0			0
$73$		−	10.0000i		−	1.17041i	−0.810885	−	0.585206i	$-0.801014\pi$
$73$		−	10.0000i		−	1.17041i	0.810885	−	0.585206i	$-0.198986\pi$
$74$	−6.00000			−0.697486
$75$		0			0
$76$	2.00000			0.229416
$77$		−	8.00000i		−	0.911685i
$78$		0			0
$79$	8.00000			0.900070			0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000			0.900070			0.450035	+	0.893011i	$0.351411\pi$
$80$	−2.00000	+	1.00000i	−0.223607	+	0.111803i
$81$		0			0
$82$		−	6.00000i		−	0.662589i
$83$			12.0000i			1.31717i	0.752506	+	0.658586i	$0.228845\pi$
$83$			12.0000i			1.31717i	−0.752506	+	0.658586i	$0.771155\pi$
$84$		0			0
$85$	4.00000	+	8.00000i	0.433861	+	0.867722i
$86$	−8.00000			−0.862662
$87$		0			0
$88$		−	2.00000i		−	0.213201i
$89$	−18.0000			−1.90800			−0.953998	−	0.299813i	$-0.903076\pi$
$89$	−18.0000			−1.90800			−0.953998	+	0.299813i	$0.903076\pi$
$90$		0			0
$91$	4.00000			0.419314
$92$			6.00000i			0.625543i
$93$		0			0
$94$	−8.00000			−0.825137
$95$	4.00000	−	2.00000i	0.410391	−	0.205196i
$96$		0			0
$97$			6.00000i			0.609208i	0.952479	+	0.304604i	$0.0985241\pi$
$97$			6.00000i			0.609208i	−0.952479	+	0.304604i	$0.901476\pi$
$98$			9.00000i			0.909137i
$99$		0			0
$100$	−3.00000	+	4.00000i	−0.300000	+	0.400000i
$101$	14.0000			1.39305			0.696526	−	0.717532i	$-0.254728\pi$
$101$	14.0000			1.39305			0.696526	+	0.717532i	$0.254728\pi$
$102$		0			0
$103$		−	6.00000i		−	0.591198i	−0.955312	−	0.295599i	$-0.904481\pi$
$103$		−	6.00000i		−	0.591198i	0.955312	−	0.295599i	$-0.0955191\pi$
$104$	1.00000			0.0980581
$105$		0			0
$106$	10.0000			0.971286
$107$			8.00000i			0.773389i	0.922208	+	0.386695i	$0.126383\pi$
$107$			8.00000i			0.773389i	−0.922208	+	0.386695i	$0.873617\pi$
$108$		0			0
$109$	−12.0000			−1.14939			−0.574696	−	0.818367i	$-0.694880\pi$
$109$	−12.0000			−1.14939			−0.574696	+	0.818367i	$0.694880\pi$
$110$	−2.00000	−	4.00000i	−0.190693	−	0.381385i
$111$		0			0
$112$			4.00000i			0.377964i
$113$		−	12.0000i		−	1.12887i	−0.825479	−	0.564433i	$-0.809095\pi$
$113$		−	12.0000i		−	1.12887i	0.825479	−	0.564433i	$-0.190905\pi$
$114$		0			0
$115$	6.00000	+	12.0000i	0.559503	+	1.11901i
$116$	2.00000			0.185695
$117$		0			0
$118$			14.0000i			1.28880i
$119$	16.0000			1.46672
$120$		0			0
$121$	−7.00000			−0.636364
$122$		−	10.0000i		−	0.905357i
$123$		0			0
$124$	4.00000			0.359211
$125$	−2.00000	+	11.0000i	−0.178885	+	0.983870i
$126$		0			0
$127$		−	18.0000i		−	1.59724i	−0.601834	−	0.798621i	$-0.705563\pi$
$127$		−	18.0000i		−	1.59724i	0.601834	−	0.798621i	$-0.294437\pi$
$128$			1.00000i			0.0883883i
$129$		0			0
$130$	2.00000	−	1.00000i	0.175412	−	0.0877058i
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$		0			0
$133$		−	8.00000i		−	0.693688i
$134$	4.00000			0.345547
$135$		0			0
$136$	4.00000			0.342997
$137$		−	2.00000i		−	0.170872i	−0.996344	−	0.0854358i	$-0.972772\pi$
$137$		−	2.00000i		−	0.170872i	0.996344	−	0.0854358i	$-0.0272282\pi$
$138$		0			0
$139$	−16.0000			−1.35710			−0.678551	−	0.734553i	$-0.737392\pi$
$139$	−16.0000			−1.35710			−0.678551	+	0.734553i	$0.737392\pi$
$140$	4.00000	+	8.00000i	0.338062	+	0.676123i
$141$		0			0
$142$			8.00000i			0.671345i
$143$			2.00000i			0.167248i
$144$		0			0
$145$	4.00000	−	2.00000i	0.332182	−	0.166091i
$146$	−10.0000			−0.827606
$147$		0			0
$148$			6.00000i			0.493197i
$149$		0			0			−	1.00000i	$-0.5\pi$
$149$		0			0				1.00000i	$0.5\pi$
$150$		0			0
$151$	8.00000			0.651031			0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000			0.651031			0.325515	+	0.945537i	$0.394462\pi$
$152$		−	2.00000i		−	0.162221i
$153$		0			0
$154$	−8.00000			−0.644658
$155$	8.00000	−	4.00000i	0.642575	−	0.321288i
$156$		0			0
$157$		−	10.0000i		−	0.798087i	−0.916932	−	0.399043i	$-0.869342\pi$
$157$		−	10.0000i		−	0.798087i	0.916932	−	0.399043i	$-0.130658\pi$
$158$		−	8.00000i		−	0.636446i
$159$		0			0
$160$	1.00000	+	2.00000i	0.0790569	+	0.158114i
$161$	24.0000			1.89146
$162$		0			0
$163$			4.00000i			0.313304i	0.987654	+	0.156652i	$0.0500701\pi$
$163$			4.00000i			0.313304i	−0.987654	+	0.156652i	$0.949930\pi$
$164$	−6.00000			−0.468521
$165$		0			0
$166$	12.0000			0.931381
$167$		0			0		1.00000			$0$
$167$		0			0		−1.00000			$\pi$
$168$		0			0
$169$	−1.00000			−0.0769231
$170$	8.00000	−	4.00000i	0.613572	−	0.306786i
$171$		0			0
$172$			8.00000i			0.609994i
$173$			10.0000i			0.760286i	0.924928	+	0.380143i	$0.124125\pi$
$173$			10.0000i			0.760286i	−0.924928	+	0.380143i	$0.875875\pi$
$174$		0			0
$175$	16.0000	+	12.0000i	1.20949	+	0.907115i
$176$	−2.00000			−0.150756
$177$		0			0
$178$			18.0000i			1.34916i
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$	6.00000			0.445976			0.222988	−	0.974821i	$-0.428419\pi$
$181$	6.00000			0.445976			0.222988	+	0.974821i	$0.428419\pi$
$182$		−	4.00000i		−	0.296500i
$183$		0			0
$184$	6.00000			0.442326
$185$	6.00000	+	12.0000i	0.441129	+	0.882258i
$186$		0			0
$187$			8.00000i			0.585018i
$188$			8.00000i			0.583460i
$189$		0			0
$190$	−2.00000	−	4.00000i	−0.145095	−	0.290191i
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$		0			0
$193$		−	6.00000i		−	0.431889i	−0.976406	−	0.215945i	$-0.930717\pi$
$193$		−	6.00000i		−	0.431889i	0.976406	−	0.215945i	$-0.0692831\pi$
$194$	6.00000			0.430775
$195$		0			0
$196$	9.00000			0.642857
$197$		−	26.0000i		−	1.85242i	−0.377004	−	0.926212i	$-0.623046\pi$
$197$		−	26.0000i		−	1.85242i	0.377004	−	0.926212i	$-0.376954\pi$
$198$		0			0
$199$	−24.0000			−1.70131			−0.850657	−	0.525720i	$-0.823796\pi$
$199$	−24.0000			−1.70131			−0.850657	+	0.525720i	$0.823796\pi$
$200$	4.00000	+	3.00000i	0.282843	+	0.212132i
$201$		0			0
$202$		−	14.0000i		−	0.985037i
$203$		−	8.00000i		−	0.561490i
$204$		0			0
$205$	−12.0000	+	6.00000i	−0.838116	+	0.419058i
$206$	−6.00000			−0.418040
$207$		0			0
$208$		−	1.00000i		−	0.0693375i
$209$	4.00000			0.276686
$210$		0			0
$211$	−28.0000			−1.92760			−0.963800	−	0.266627i	$-0.914091\pi$
$211$	−28.0000			−1.92760			−0.963800	+	0.266627i	$0.914091\pi$
$212$		−	10.0000i		−	0.686803i
$213$		0			0
$214$	8.00000			0.546869
$215$	8.00000	+	16.0000i	0.545595	+	1.09119i
$216$		0			0
$217$		−	16.0000i		−	1.08615i
$218$			12.0000i			0.812743i
$219$		0			0
$220$	−4.00000	+	2.00000i	−0.269680	+	0.134840i
$221$	−4.00000			−0.269069
$222$		0			0
$223$			16.0000i			1.07144i	0.844396	+	0.535720i	$0.179960\pi$
$223$			16.0000i			1.07144i	−0.844396	+	0.535720i	$0.820040\pi$
$224$	4.00000			0.267261
$225$		0			0
$226$	−12.0000			−0.798228
$227$		−	12.0000i		−	0.796468i	−0.917284	−	0.398234i	$-0.869623\pi$
$227$		−	12.0000i		−	0.796468i	0.917284	−	0.398234i	$-0.130377\pi$
$228$		0			0
$229$	16.0000			1.05731			0.528655	−	0.848837i	$-0.322697\pi$
$229$	16.0000			1.05731			0.528655	+	0.848837i	$0.322697\pi$
$230$	12.0000	−	6.00000i	0.791257	−	0.395628i
$231$		0			0
$232$		−	2.00000i		−	0.131306i
$233$		−	4.00000i		−	0.262049i	−0.991379	−	0.131024i	$-0.958173\pi$
$233$		−	4.00000i		−	0.262049i	0.991379	−	0.131024i	$-0.0418266\pi$
$234$		0			0
$235$	8.00000	+	16.0000i	0.521862	+	1.04372i
$236$	14.0000			0.911322
$237$		0			0
$238$		−	16.0000i		−	1.03713i
$239$	−12.0000			−0.776215			−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000			−0.776215			−0.388108	+	0.921614i	$0.626871\pi$
$240$		0			0
$241$	−30.0000			−1.93247			−0.966235	−	0.257663i	$-0.917048\pi$
$241$	−30.0000			−1.93247			−0.966235	+	0.257663i	$0.917048\pi$
$242$			7.00000i			0.449977i
$243$		0			0
$244$	−10.0000			−0.640184
$245$	18.0000	−	9.00000i	1.14998	−	0.574989i
$246$		0			0
$247$			2.00000i			0.127257i
$248$		−	4.00000i		−	0.254000i
$249$		0			0
$250$	11.0000	+	2.00000i	0.695701	+	0.126491i
$251$	12.0000			0.757433			0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000			0.757433			0.378717	+	0.925513i	$0.376365\pi$
$252$		0			0
$253$			12.0000i			0.754434i
$254$	−18.0000			−1.12942
$255$		0			0
$256$	1.00000			0.0625000
$257$			12.0000i			0.748539i	0.927320	+	0.374270i	$0.122107\pi$
$257$			12.0000i			0.748539i	−0.927320	+	0.374270i	$0.877893\pi$
$258$		0			0
$259$	24.0000			1.49129
$260$	−1.00000	−	2.00000i	−0.0620174	−	0.124035i
$261$		0			0
$262$		0			0
$263$		−	10.0000i		−	0.616626i	−0.951285	−	0.308313i	$-0.900236\pi$
$263$		−	10.0000i		−	0.616626i	0.951285	−	0.308313i	$-0.0997645\pi$
$264$		0			0
$265$	−10.0000	−	20.0000i	−0.614295	−	1.22859i
$266$	−8.00000			−0.490511
$267$		0			0
$268$		−	4.00000i		−	0.244339i
$269$	−14.0000			−0.853595			−0.426798	−	0.904347i	$-0.640358\pi$
$269$	−14.0000			−0.853595			−0.426798	+	0.904347i	$0.640358\pi$
$270$		0			0
$271$	16.0000			0.971931			0.485965	−	0.873978i	$-0.338468\pi$
$271$	16.0000			0.971931			0.485965	+	0.873978i	$0.338468\pi$
$272$		−	4.00000i		−	0.242536i
$273$		0			0
$274$	−2.00000			−0.120824
$275$	−6.00000	+	8.00000i	−0.361814	+	0.482418i
$276$		0			0
$277$			2.00000i			0.120168i	0.998193	+	0.0600842i	$0.0191369\pi$
$277$			2.00000i			0.120168i	−0.998193	+	0.0600842i	$0.980863\pi$
$278$			16.0000i			0.959616i
$279$		0			0
$280$	8.00000	−	4.00000i	0.478091	−	0.239046i
$281$	18.0000			1.07379			0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000			1.07379			0.536895	+	0.843649i	$0.319597\pi$
$282$		0			0
$283$			4.00000i			0.237775i	0.992908	+	0.118888i	$0.0379328\pi$
$283$			4.00000i			0.237775i	−0.992908	+	0.118888i	$0.962067\pi$
$284$	8.00000			0.474713
$285$		0			0
$286$	2.00000			0.118262
$287$			24.0000i			1.41668i
$288$		0			0
$289$	1.00000			0.0588235
$290$	−2.00000	−	4.00000i	−0.117444	−	0.234888i
$291$		0			0
$292$			10.0000i			0.585206i
$293$		−	14.0000i		−	0.817889i	−0.912559	−	0.408944i	$-0.865897\pi$
$293$		−	14.0000i		−	0.817889i	0.912559	−	0.408944i	$-0.134103\pi$
$294$		0			0
$295$	28.0000	−	14.0000i	1.63022	−	0.815112i
$296$	6.00000			0.348743
$297$		0			0
$298$		0			0
$299$	−6.00000			−0.346989
$300$		0			0
$301$	32.0000			1.84445
$302$		−	8.00000i		−	0.460348i
$303$		0			0
$304$	−2.00000			−0.114708
$305$	−20.0000	+	10.0000i	−1.14520	+	0.572598i
$306$		0			0
$307$			28.0000i			1.59804i	0.601302	+	0.799022i	$0.294649\pi$
$307$			28.0000i			1.59804i	−0.601302	+	0.799022i	$0.705351\pi$
$308$			8.00000i			0.455842i
$309$		0			0
$310$	−4.00000	−	8.00000i	−0.227185	−	0.454369i
$311$	−8.00000			−0.453638			−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000			−0.453638			−0.226819	+	0.973937i	$0.572833\pi$
$312$		0			0
$313$		−	8.00000i		−	0.452187i	−0.974106	−	0.226093i	$-0.927405\pi$
$313$		−	8.00000i		−	0.452187i	0.974106	−	0.226093i	$-0.0725954\pi$
$314$	−10.0000			−0.564333
$315$		0			0
$316$	−8.00000			−0.450035
$317$			30.0000i			1.68497i	0.538721	+	0.842484i	$0.318908\pi$
$317$			30.0000i			1.68497i	−0.538721	+	0.842484i	$0.681092\pi$
$318$		0			0
$319$	4.00000			0.223957
$320$	2.00000	−	1.00000i	0.111803	−	0.0559017i
$321$		0			0
$322$		−	24.0000i		−	1.33747i
$323$			8.00000i			0.445132i
$324$		0			0
$325$	−4.00000	−	3.00000i	−0.221880	−	0.166410i
$326$	4.00000			0.221540
$327$		0			0
$328$			6.00000i			0.331295i
$329$	32.0000			1.76422
$330$		0			0
$331$	18.0000			0.989369			0.494685	−	0.869072i	$-0.335284\pi$
$331$	18.0000			0.989369			0.494685	+	0.869072i	$0.335284\pi$
$332$		−	12.0000i		−	0.658586i
$333$		0			0
$334$		0			0
$335$	−4.00000	−	8.00000i	−0.218543	−	0.437087i
$336$		0			0
$337$		−	32.0000i		−	1.74315i	−0.490261	−	0.871576i	$-0.663099\pi$
$337$		−	32.0000i		−	1.74315i	0.490261	−	0.871576i	$-0.336901\pi$
$338$			1.00000i			0.0543928i
$339$		0			0
$340$	−4.00000	−	8.00000i	−0.216930	−	0.433861i
$341$	8.00000			0.433224
$342$		0			0
$343$		−	8.00000i		−	0.431959i
$344$	8.00000			0.431331
$345$		0			0
$346$	10.0000			0.537603
$347$			24.0000i			1.28839i	0.764862	+	0.644194i	$0.222807\pi$
$347$			24.0000i			1.28839i	−0.764862	+	0.644194i	$0.777193\pi$
$348$		0			0
$349$	28.0000			1.49881			0.749403	−	0.662114i	$-0.230341\pi$
$349$	28.0000			1.49881			0.749403	+	0.662114i	$0.230341\pi$
$350$	12.0000	−	16.0000i	0.641427	−	0.855236i
$351$		0			0
$352$			2.00000i			0.106600i
$353$		−	26.0000i		−	1.38384i	−0.721974	−	0.691920i	$-0.756765\pi$
$353$		−	26.0000i		−	1.38384i	0.721974	−	0.691920i	$-0.243235\pi$
$354$		0			0
$355$	16.0000	−	8.00000i	0.849192	−	0.424596i
$356$	18.0000			0.953998
$357$		0			0
$358$		0			0
$359$	20.0000			1.05556			0.527780	−	0.849381i	$-0.323025\pi$
$359$	20.0000			1.05556			0.527780	+	0.849381i	$0.323025\pi$
$360$		0			0
$361$	−15.0000			−0.789474
$362$		−	6.00000i		−	0.315353i
$363$		0			0
$364$	−4.00000			−0.209657
$365$	10.0000	+	20.0000i	0.523424	+	1.04685i
$366$		0			0
$367$			22.0000i			1.14839i	0.818718	+	0.574195i	$0.194685\pi$
$367$			22.0000i			1.14839i	−0.818718	+	0.574195i	$0.805315\pi$
$368$		−	6.00000i		−	0.312772i
$369$		0			0
$370$	12.0000	−	6.00000i	0.623850	−	0.311925i
$371$	−40.0000			−2.07670
$372$		0			0
$373$			2.00000i			0.103556i	0.998659	+	0.0517780i	$0.0164888\pi$
$373$			2.00000i			0.103556i	−0.998659	+	0.0517780i	$0.983511\pi$
$374$	8.00000			0.413670
$375$		0			0
$376$	8.00000			0.412568
$377$			2.00000i			0.103005i
$378$		0			0
$379$	−26.0000			−1.33553			−0.667765	−	0.744372i	$-0.732749\pi$
$379$	−26.0000			−1.33553			−0.667765	+	0.744372i	$0.732749\pi$
$380$	−4.00000	+	2.00000i	−0.205196	+	0.102598i
$381$		0			0
$382$		0			0
$383$		−	12.0000i		−	0.613171i	−0.951843	−	0.306586i	$-0.900813\pi$
$383$		−	12.0000i		−	0.613171i	0.951843	−	0.306586i	$-0.0991866\pi$
$384$		0			0
$385$	8.00000	+	16.0000i	0.407718	+	0.815436i
$386$	−6.00000			−0.305392
$387$		0			0
$388$		−	6.00000i		−	0.304604i
$389$	−26.0000			−1.31825			−0.659126	−	0.752032i	$-0.729074\pi$
$389$	−26.0000			−1.31825			−0.659126	+	0.752032i	$0.729074\pi$
$390$		0			0
$391$	−24.0000			−1.21373
$392$		−	9.00000i		−	0.454569i
$393$		0			0
$394$	−26.0000			−1.30986
$395$	−16.0000	+	8.00000i	−0.805047	+	0.402524i
$396$		0			0
$397$			6.00000i			0.301131i	0.988600	+	0.150566i	$0.0481095\pi$
$397$			6.00000i			0.301131i	−0.988600	+	0.150566i	$0.951890\pi$
$398$			24.0000i			1.20301i
$399$		0			0
$400$	3.00000	−	4.00000i	0.150000	−	0.200000i
$401$	18.0000			0.898877			0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000			0.898877			0.449439	+	0.893311i	$0.351624\pi$
$402$		0			0
$403$			4.00000i			0.199254i
$404$	−14.0000			−0.696526
$405$		0			0
$406$	−8.00000			−0.397033
$407$			12.0000i			0.594818i
$408$		0			0
$409$	2.00000			0.0988936			0.0494468	−	0.998777i	$-0.484254\pi$
$409$	2.00000			0.0988936			0.0494468	+	0.998777i	$0.484254\pi$
$410$	6.00000	+	12.0000i	0.296319	+	0.592638i
$411$		0			0
$412$			6.00000i			0.295599i
$413$		−	56.0000i		−	2.75558i
$414$		0			0
$415$	−12.0000	−	24.0000i	−0.589057	−	1.17811i
$416$	−1.00000			−0.0490290
$417$		0			0
$418$		−	4.00000i		−	0.195646i
$419$	16.0000			0.781651			0.390826	−	0.920465i	$-0.372190\pi$
$419$	16.0000			0.781651			0.390826	+	0.920465i	$0.372190\pi$
$420$		0			0
$421$	28.0000			1.36464			0.682318	−	0.731055i	$-0.260972\pi$
$421$	28.0000			1.36464			0.682318	+	0.731055i	$0.260972\pi$
$422$			28.0000i			1.36302i
$423$		0			0
$424$	−10.0000			−0.485643
$425$	−16.0000	−	12.0000i	−0.776114	−	0.582086i
$426$		0			0
$427$			40.0000i			1.93574i
$428$		−	8.00000i		−	0.386695i
$429$		0			0
$430$	16.0000	−	8.00000i	0.771589	−	0.385794i
$431$	8.00000			0.385346			0.192673	−	0.981263i	$-0.438284\pi$
$431$	8.00000			0.385346			0.192673	+	0.981263i	$0.438284\pi$
$432$		0			0
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$	−16.0000			−0.768025
$435$		0			0
$436$	12.0000			0.574696
$437$			12.0000i			0.574038i
$438$		0			0
$439$	−32.0000			−1.52728			−0.763638	−	0.645644i	$-0.776589\pi$
$439$	−32.0000			−1.52728			−0.763638	+	0.645644i	$0.776589\pi$
$440$	2.00000	+	4.00000i	0.0953463	+	0.190693i
$441$		0			0
$442$			4.00000i			0.190261i
$443$			8.00000i			0.380091i	0.981775	+	0.190046i	$0.0608636\pi$
$443$			8.00000i			0.380091i	−0.981775	+	0.190046i	$0.939136\pi$
$444$		0			0
$445$	36.0000	−	18.0000i	1.70656	−	0.853282i
$446$	16.0000			0.757622
$447$		0			0
$448$		−	4.00000i		−	0.188982i
$449$	6.00000			0.283158			0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000			0.283158			0.141579	+	0.989927i	$0.454782\pi$
$450$		0			0
$451$	−12.0000			−0.565058
$452$			12.0000i			0.564433i
$453$		0			0
$454$	−12.0000			−0.563188
$455$	−8.00000	+	4.00000i	−0.375046	+	0.187523i
$456$		0			0
$457$		−	26.0000i		−	1.21623i	−0.793849	−	0.608114i	$-0.791926\pi$
$457$		−	26.0000i		−	1.21623i	0.793849	−	0.608114i	$-0.208074\pi$
$458$		−	16.0000i		−	0.747631i
$459$		0			0
$460$	−6.00000	−	12.0000i	−0.279751	−	0.559503i
$461$	4.00000			0.186299			0.0931493	−	0.995652i	$-0.470307\pi$
$461$	4.00000			0.186299			0.0931493	+	0.995652i	$0.470307\pi$
$462$		0			0
$463$			36.0000i			1.67306i	0.547920	+	0.836531i	$0.315420\pi$
$463$			36.0000i			1.67306i	−0.547920	+	0.836531i	$0.684580\pi$
$464$	−2.00000			−0.0928477
$465$		0			0
$466$	−4.00000			−0.185296
$467$			16.0000i			0.740392i	0.928954	+	0.370196i	$0.120709\pi$
$467$			16.0000i			0.740392i	−0.928954	+	0.370196i	$0.879291\pi$
$468$		0			0
$469$	−16.0000			−0.738811
$470$	16.0000	−	8.00000i	0.738025	−	0.369012i
$471$		0			0
$472$		−	14.0000i		−	0.644402i
$473$			16.0000i			0.735681i
$474$		0			0
$475$	−6.00000	+	8.00000i	−0.275299	+	0.367065i
$476$	−16.0000			−0.733359
$477$		0			0
$478$			12.0000i			0.548867i
$479$	−12.0000			−0.548294			−0.274147	−	0.961688i	$-0.588395\pi$
$479$	−12.0000			−0.548294			−0.274147	+	0.961688i	$0.588395\pi$
$480$		0			0
$481$	−6.00000			−0.273576
$482$			30.0000i			1.36646i
$483$		0			0
$484$	7.00000			0.318182
$485$	−6.00000	−	12.0000i	−0.272446	−	0.544892i
$486$		0			0
$487$		−	40.0000i		−	1.81257i	−0.422664	−	0.906287i	$-0.638905\pi$
$487$		−	40.0000i		−	1.81257i	0.422664	−	0.906287i	$-0.361095\pi$
$488$			10.0000i			0.452679i
$489$		0			0
$490$	−9.00000	−	18.0000i	−0.406579	−	0.813157i
$491$	−8.00000			−0.361035			−0.180517	−	0.983572i	$-0.557777\pi$
$491$	−8.00000			−0.361035			−0.180517	+	0.983572i	$0.557777\pi$
$492$		0			0
$493$			8.00000i			0.360302i
$494$	2.00000			0.0899843
$495$		0			0
$496$	−4.00000			−0.179605
$497$		−	32.0000i		−	1.43540i
$498$		0			0
$499$	−30.0000			−1.34298			−0.671492	−	0.741012i	$-0.734346\pi$
$499$	−30.0000			−1.34298			−0.671492	+	0.741012i	$0.734346\pi$
$500$	2.00000	−	11.0000i	0.0894427	−	0.491935i
$501$		0			0
$502$		−	12.0000i		−	0.535586i
$503$			18.0000i			0.802580i	0.915951	+	0.401290i	$0.131438\pi$
$503$			18.0000i			0.802580i	−0.915951	+	0.401290i	$0.868562\pi$
$504$		0			0
$505$	−28.0000	+	14.0000i	−1.24598	+	0.622992i
$506$	12.0000			0.533465
$507$		0			0
$508$			18.0000i			0.798621i
$509$	−16.0000			−0.709188			−0.354594	−	0.935020i	$-0.615381\pi$
$509$	−16.0000			−0.709188			−0.354594	+	0.935020i	$0.615381\pi$
$510$		0			0
$511$	40.0000			1.76950
$512$		−	1.00000i		−	0.0441942i
$513$		0			0
$514$	12.0000			0.529297
$515$	6.00000	+	12.0000i	0.264392	+	0.528783i
$516$		0			0
$517$			16.0000i			0.703679i
$518$		−	24.0000i		−	1.05450i
$519$		0			0
$520$	−2.00000	+	1.00000i	−0.0877058	+	0.0438529i
$521$	−38.0000			−1.66481			−0.832405	−	0.554168i	$-0.813037\pi$
$521$	−38.0000			−1.66481			−0.832405	+	0.554168i	$0.813037\pi$
$522$		0			0
$523$		−	8.00000i		−	0.349816i	−0.984585	−	0.174908i	$-0.944037\pi$
$523$		−	8.00000i		−	0.349816i	0.984585	−	0.174908i	$-0.0559627\pi$
$524$		0			0
$525$		0			0
$526$	−10.0000			−0.436021
$527$			16.0000i			0.696971i
$528$		0			0
$529$	−13.0000			−0.565217
$530$	−20.0000	+	10.0000i	−0.868744	+	0.434372i
$531$		0			0
$532$			8.00000i			0.346844i
$533$		−	6.00000i		−	0.259889i
$534$		0			0
$535$	−8.00000	−	16.0000i	−0.345870	−	0.691740i
$536$	−4.00000			−0.172774
$537$		0			0
$538$			14.0000i			0.603583i
$539$	18.0000			0.775315
$540$		0			0
$541$	−8.00000			−0.343947			−0.171973	−	0.985102i	$-0.555014\pi$
$541$	−8.00000			−0.343947			−0.171973	+	0.985102i	$0.555014\pi$
$542$		−	16.0000i		−	0.687259i
$543$		0			0
$544$	−4.00000			−0.171499
$545$	24.0000	−	12.0000i	1.02805	−	0.514024i
$546$		0			0
$547$			44.0000i			1.88130i	0.339372	+	0.940652i	$0.389785\pi$
$547$			44.0000i			1.88130i	−0.339372	+	0.940652i	$0.610215\pi$
$548$			2.00000i			0.0854358i
$549$		0			0
$550$	8.00000	+	6.00000i	0.341121	+	0.255841i
$551$	4.00000			0.170406
$552$		0			0
$553$			32.0000i			1.36078i
$554$	2.00000			0.0849719
$555$		0			0
$556$	16.0000			0.678551
$557$		−	14.0000i		−	0.593199i	−0.955002	−	0.296600i	$-0.904147\pi$
$557$		−	14.0000i		−	0.593199i	0.955002	−	0.296600i	$-0.0958526\pi$
$558$		0			0
$559$	−8.00000			−0.338364
$560$	−4.00000	−	8.00000i	−0.169031	−	0.338062i
$561$		0			0
$562$		−	18.0000i		−	0.759284i
$563$		−	16.0000i		−	0.674320i	−0.941447	−	0.337160i	$-0.890534\pi$
$563$		−	16.0000i		−	0.674320i	0.941447	−	0.337160i	$-0.109466\pi$
$564$		0			0
$565$	12.0000	+	24.0000i	0.504844	+	1.00969i
$566$	4.00000			0.168133
$567$		0			0
$568$		−	8.00000i		−	0.335673i
$569$	−42.0000			−1.76073			−0.880366	−	0.474295i	$-0.842703\pi$
$569$	−42.0000			−1.76073			−0.880366	+	0.474295i	$0.842703\pi$
$570$		0			0
$571$	−36.0000			−1.50655			−0.753277	−	0.657704i	$-0.771528\pi$
$571$	−36.0000			−1.50655			−0.753277	+	0.657704i	$0.771528\pi$
$572$		−	2.00000i		−	0.0836242i
$573$		0			0
$574$	24.0000			1.00174
$575$	−24.0000	−	18.0000i	−1.00087	−	0.750652i
$576$		0			0
$577$			26.0000i			1.08239i	0.840896	+	0.541197i	$0.182029\pi$
$577$			26.0000i			1.08239i	−0.840896	+	0.541197i	$0.817971\pi$
$578$		−	1.00000i		−	0.0415945i
$579$		0			0
$580$	−4.00000	+	2.00000i	−0.166091	+	0.0830455i
$581$	−48.0000			−1.99138
$582$		0			0
$583$		−	20.0000i		−	0.828315i
$584$	10.0000			0.413803
$585$		0			0
$586$	−14.0000			−0.578335
$587$		−	36.0000i		−	1.48588i	−0.669359	−	0.742940i	$-0.733431\pi$
$587$		−	36.0000i		−	1.48588i	0.669359	−	0.742940i	$-0.266569\pi$
$588$		0			0
$589$	8.00000			0.329634
$590$	−14.0000	−	28.0000i	−0.576371	−	1.15274i
$591$		0			0
$592$		−	6.00000i		−	0.246598i
$593$			14.0000i			0.574911i	0.957794	+	0.287456i	$0.0928094\pi$
$593$			14.0000i			0.574911i	−0.957794	+	0.287456i	$0.907191\pi$
$594$		0			0
$595$	−32.0000	+	16.0000i	−1.31187	+	0.655936i
$596$		0			0
$597$		0			0
$598$			6.00000i			0.245358i
$599$	−16.0000			−0.653742			−0.326871	−	0.945069i	$-0.605994\pi$
$599$	−16.0000			−0.653742			−0.326871	+	0.945069i	$0.605994\pi$
$600$		0			0
$601$	−10.0000			−0.407909			−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000			−0.407909			−0.203954	+	0.978980i	$0.565379\pi$
$602$		−	32.0000i		−	1.30422i
$603$		0			0
$604$	−8.00000			−0.325515
$605$	14.0000	−	7.00000i	0.569181	−	0.284590i
$606$		0			0
$607$			18.0000i			0.730597i	0.930890	+	0.365299i	$0.119033\pi$
$607$			18.0000i			0.730597i	−0.930890	+	0.365299i	$0.880967\pi$
$608$			2.00000i			0.0811107i
$609$		0			0
$610$	10.0000	+	20.0000i	0.404888	+	0.809776i
$611$	−8.00000			−0.323645
$612$		0			0
$613$		−	26.0000i		−	1.05013i	−0.851062	−	0.525065i	$-0.824041\pi$
$613$		−	26.0000i		−	1.05013i	0.851062	−	0.525065i	$-0.175959\pi$
$614$	28.0000			1.12999
$615$		0			0
$616$	8.00000			0.322329
$617$			34.0000i			1.36879i	0.729112	+	0.684394i	$0.239933\pi$
$617$			34.0000i			1.36879i	−0.729112	+	0.684394i	$0.760067\pi$
$618$		0			0
$619$	−42.0000			−1.68812			−0.844061	−	0.536247i	$-0.819842\pi$
$619$	−42.0000			−1.68812			−0.844061	+	0.536247i	$0.819842\pi$
$620$	−8.00000	+	4.00000i	−0.321288	+	0.160644i
$621$		0			0
$622$			8.00000i			0.320771i
$623$		−	72.0000i		−	2.88462i
$624$		0			0
$625$	−7.00000	−	24.0000i	−0.280000	−	0.960000i
$626$	−8.00000			−0.319744
$627$		0			0
$628$			10.0000i			0.399043i
$629$	−24.0000			−0.956943
$630$		0			0
$631$	−32.0000			−1.27390			−0.636950	−	0.770905i	$-0.719804\pi$
$631$	−32.0000			−1.27390			−0.636950	+	0.770905i	$0.719804\pi$
$632$			8.00000i			0.318223i
$633$		0			0
$634$	30.0000			1.19145
$635$	18.0000	+	36.0000i	0.714308	+	1.42862i
$636$		0			0
$637$			9.00000i			0.356593i
$638$		−	4.00000i		−	0.158362i
$639$		0			0
$640$	−1.00000	−	2.00000i	−0.0395285	−	0.0790569i
$641$	2.00000			0.0789953			0.0394976	−	0.999220i	$-0.487424\pi$
$641$	2.00000			0.0789953			0.0394976	+	0.999220i	$0.487424\pi$
$642$		0			0
$643$			20.0000i			0.788723i	0.918955	+	0.394362i	$0.129034\pi$
$643$			20.0000i			0.788723i	−0.918955	+	0.394362i	$0.870966\pi$
$644$	−24.0000			−0.945732
$645$		0			0
$646$	8.00000			0.314756
$647$			6.00000i			0.235884i	0.993020	+	0.117942i	$0.0376297\pi$
$647$			6.00000i			0.235884i	−0.993020	+	0.117942i	$0.962370\pi$
$648$		0			0
$649$	28.0000			1.09910
$650$	−3.00000	+	4.00000i	−0.117670	+	0.156893i
$651$		0			0
$652$		−	4.00000i		−	0.156652i
$653$		−	26.0000i		−	1.01746i	−0.860927	−	0.508729i	$-0.830115\pi$
$653$		−	26.0000i		−	1.01746i	0.860927	−	0.508729i	$-0.169885\pi$
$654$		0			0
$655$		0			0
$656$	6.00000			0.234261
$657$		0			0
$658$		−	32.0000i		−	1.24749i
$659$	40.0000			1.55818			0.779089	−	0.626913i	$-0.215682\pi$
$659$	40.0000			1.55818			0.779089	+	0.626913i	$0.215682\pi$
$660$		0			0
$661$	24.0000			0.933492			0.466746	−	0.884391i	$-0.345426\pi$
$661$	24.0000			0.933492			0.466746	+	0.884391i	$0.345426\pi$
$662$		−	18.0000i		−	0.699590i
$663$		0			0
$664$	−12.0000			−0.465690
$665$	8.00000	+	16.0000i	0.310227	+	0.620453i
$666$		0			0
$667$			12.0000i			0.464642i
$668$		0			0
$669$		0			0
$670$	−8.00000	+	4.00000i	−0.309067	+	0.154533i
$671$	−20.0000			−0.772091
$672$		0			0
$673$		−	12.0000i		−	0.462566i	−0.972887	−	0.231283i	$-0.925708\pi$
$673$		−	12.0000i		−	0.462566i	0.972887	−	0.231283i	$-0.0742923\pi$
$674$	−32.0000			−1.23259
$675$		0			0
$676$	1.00000			0.0384615
$677$		−	22.0000i		−	0.845529i	−0.906240	−	0.422764i	$-0.861060\pi$
$677$		−	22.0000i		−	0.845529i	0.906240	−	0.422764i	$-0.138940\pi$
$678$		0			0
$679$	−24.0000			−0.921035
$680$	−8.00000	+	4.00000i	−0.306786	+	0.153393i
$681$		0			0
$682$		−	8.00000i		−	0.306336i
$683$			12.0000i			0.459167i	0.973289	+	0.229584i	$0.0737364\pi$
$683$			12.0000i			0.459167i	−0.973289	+	0.229584i	$0.926264\pi$
$684$		0			0
$685$	2.00000	+	4.00000i	0.0764161	+	0.152832i
$686$	−8.00000			−0.305441
$687$		0			0
$688$		−	8.00000i		−	0.304997i
$689$	10.0000			0.380970
$690$		0			0
$691$	2.00000			0.0760836			0.0380418	−	0.999276i	$-0.487888\pi$
$691$	2.00000			0.0760836			0.0380418	+	0.999276i	$0.487888\pi$
$692$		−	10.0000i		−	0.380143i
$693$		0			0
$694$	24.0000			0.911028
$695$	32.0000	−	16.0000i	1.21383	−	0.606915i
$696$		0			0
$697$		−	24.0000i		−	0.909065i
$698$		−	28.0000i		−	1.05982i
$699$		0			0
$700$	−16.0000	−	12.0000i	−0.604743	−	0.453557i
$701$	30.0000			1.13308			0.566542	−	0.824033i	$-0.308281\pi$
$701$	30.0000			1.13308			0.566542	+	0.824033i	$0.308281\pi$
$702$		0			0
$703$			12.0000i			0.452589i
$704$	2.00000			0.0753778
$705$		0			0
$706$	−26.0000			−0.978523
$707$			56.0000i			2.10610i
$708$		0			0
$709$	32.0000			1.20179			0.600893	−	0.799330i	$-0.294812\pi$
$709$	32.0000			1.20179			0.600893	+	0.799330i	$0.294812\pi$
$710$	−8.00000	−	16.0000i	−0.300235	−	0.600469i
$711$		0			0
$712$		−	18.0000i		−	0.674579i
$713$			24.0000i			0.898807i
$714$		0			0
$715$	−2.00000	−	4.00000i	−0.0747958	−	0.149592i
$716$		0			0
$717$		0			0
$718$		−	20.0000i		−	0.746393i
$719$	24.0000			0.895049			0.447524	−	0.894272i	$-0.352306\pi$
$719$	24.0000			0.895049			0.447524	+	0.894272i	$0.352306\pi$
$720$		0			0
$721$	24.0000			0.893807
$722$			15.0000i			0.558242i
$723$		0			0
$724$	−6.00000			−0.222988
$725$	−6.00000	+	8.00000i	−0.222834	+	0.297113i
$726$		0			0
$727$			14.0000i			0.519231i	0.965712	+	0.259616i	$0.0835959\pi$
$727$			14.0000i			0.519231i	−0.965712	+	0.259616i	$0.916404\pi$
$728$			4.00000i			0.148250i
$729$		0			0
$730$	20.0000	−	10.0000i	0.740233	−	0.370117i
$731$	−32.0000			−1.18356
$732$		0			0
$733$			2.00000i			0.0738717i	0.999318	+	0.0369358i	$0.0117597\pi$
$733$			2.00000i			0.0738717i	−0.999318	+	0.0369358i	$0.988240\pi$
$734$	22.0000			0.812035
$735$		0			0
$736$	−6.00000			−0.221163
$737$		−	8.00000i		−	0.294684i
$738$		0			0
$739$	2.00000			0.0735712			0.0367856	−	0.999323i	$-0.488288\pi$
$739$	2.00000			0.0735712			0.0367856	+	0.999323i	$0.488288\pi$
$740$	−6.00000	−	12.0000i	−0.220564	−	0.441129i
$741$		0			0
$742$			40.0000i			1.46845i
$743$		−	4.00000i		−	0.146746i	−0.997305	−	0.0733729i	$-0.976624\pi$
$743$		−	4.00000i		−	0.146746i	0.997305	−	0.0733729i	$-0.0233763\pi$
$744$		0			0
$745$		0			0
$746$	2.00000			0.0732252
$747$		0			0
$748$		−	8.00000i		−	0.292509i
$749$	−32.0000			−1.16925
$750$		0			0
$751$	48.0000			1.75154			0.875772	−	0.482724i	$-0.160353\pi$
$751$	48.0000			1.75154			0.875772	+	0.482724i	$0.160353\pi$
$752$		−	8.00000i		−	0.291730i
$753$		0			0
$754$	2.00000			0.0728357
$755$	−16.0000	+	8.00000i	−0.582300	+	0.291150i
$756$		0			0
$757$		−	22.0000i		−	0.799604i	−0.916602	−	0.399802i	$-0.869079\pi$
$757$		−	22.0000i		−	0.799604i	0.916602	−	0.399802i	$-0.130921\pi$
$758$			26.0000i			0.944363i
$759$		0			0
$760$	2.00000	+	4.00000i	0.0725476	+	0.145095i
$761$	−14.0000			−0.507500			−0.253750	−	0.967270i	$-0.581664\pi$
$761$	−14.0000			−0.507500			−0.253750	+	0.967270i	$0.581664\pi$
$762$		0			0
$763$		−	48.0000i		−	1.73772i
$764$		0			0
$765$		0			0
$766$	−12.0000			−0.433578
$767$			14.0000i			0.505511i
$768$		0			0
$769$	22.0000			0.793340			0.396670	−	0.917961i	$-0.370166\pi$
$769$	22.0000			0.793340			0.396670	+	0.917961i	$0.370166\pi$
$770$	16.0000	−	8.00000i	0.576600	−	0.288300i
$771$		0			0
$772$			6.00000i			0.215945i
$773$			14.0000i			0.503545i	0.967786	+	0.251773i	$0.0810135\pi$
$773$			14.0000i			0.503545i	−0.967786	+	0.251773i	$0.918987\pi$
$774$		0			0
$775$	−12.0000	+	16.0000i	−0.431053	+	0.574737i
$776$	−6.00000			−0.215387
$777$		0			0
$778$			26.0000i			0.932145i
$779$	−12.0000			−0.429945
$780$		0			0
$781$	16.0000			0.572525
$782$			24.0000i			0.858238i
$783$		0			0
$784$	−9.00000			−0.321429
$785$	10.0000	+	20.0000i	0.356915	+	0.713831i
$786$		0			0
$787$		−	28.0000i		−	0.998092i	−0.866575	−	0.499046i	$-0.833684\pi$
$787$		−	28.0000i		−	0.998092i	0.866575	−	0.499046i	$-0.166316\pi$
$788$			26.0000i			0.926212i
$789$		0			0
$790$	8.00000	+	16.0000i	0.284627	+	0.569254i
$791$	48.0000			1.70668
$792$		0			0
$793$		−	10.0000i		−	0.355110i
$794$	6.00000			0.212932
$795$		0			0
$796$

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:	\( N \)	\(=\)	\( 1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1170.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} +(-2.00000 + 1.00000i) q^{5} +4.00000i q^{7} +1.00000i q^{8} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000 q^{4} +(-2.00000 + 1.00000i) q^{5} +4.00000i q^{7} +1.00000i q^{8} +(1.00000 + 2.00000i) q^{10} -2.00000 q^{11} -1.00000i q^{13} +4.00000 q^{14} +1.00000 q^{16} -4.00000i q^{17} -2.00000 q^{19} +(2.00000 - 1.00000i) q^{20} +2.00000i q^{22} -6.00000i q^{23} +(3.00000 - 4.00000i) q^{25} -1.00000 q^{26} -4.00000i q^{28} -2.00000 q^{29} -4.00000 q^{31} -1.00000i q^{32} -4.00000 q^{34} +(-4.00000 - 8.00000i) q^{35} -6.00000i q^{37} +2.00000i q^{38} +(-1.00000 - 2.00000i) q^{40} +6.00000 q^{41} -8.00000i q^{43} +2.00000 q^{44} -6.00000 q^{46} -8.00000i q^{47} -9.00000 q^{49} +(-4.00000 - 3.00000i) q^{50} +1.00000i q^{52} +10.0000i q^{53} +(4.00000 - 2.00000i) q^{55} -4.00000 q^{56} +2.00000i q^{58} -14.0000 q^{59} +10.0000 q^{61} +4.00000i q^{62} -1.00000 q^{64} +(1.00000 + 2.00000i) q^{65} +4.00000i q^{67} +4.00000i q^{68} +(-8.00000 + 4.00000i) q^{70} -8.00000 q^{71} -10.0000i q^{73} -6.00000 q^{74} +2.00000 q^{76} -8.00000i q^{77} +8.00000 q^{79} +(-2.00000 + 1.00000i) q^{80} -6.00000i q^{82} +12.0000i q^{83} +(4.00000 + 8.00000i) q^{85} -8.00000 q^{86} -2.00000i q^{88} -18.0000 q^{89} +4.00000 q^{91} +6.00000i q^{92} -8.00000 q^{94} +(4.00000 - 2.00000i) q^{95} +6.00000i q^{97} +9.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} - 4 q^{5}+O(q^{10}) \) 2 * q - 2 * q^4 - 4 * q^5	\( 2 q - 2 q^{4} - 4 q^{5} + 2 q^{10} - 4 q^{11} + 8 q^{14} + 2 q^{16} - 4 q^{19} + 4 q^{20} + 6 q^{25} - 2 q^{26} - 4 q^{29} - 8 q^{31} - 8 q^{34} - 8 q^{35} - 2 q^{40} + 12 q^{41} + 4 q^{44} - 12 q^{46} - 18 q^{49} - 8 q^{50} + 8 q^{55} - 8 q^{56} - 28 q^{59} + 20 q^{61} - 2 q^{64} + 2 q^{65} - 16 q^{70} - 16 q^{71} - 12 q^{74} + 4 q^{76} + 16 q^{79} - 4 q^{80} + 8 q^{85} - 16 q^{86} - 36 q^{89} + 8 q^{91} - 16 q^{94} + 8 q^{95}+O(q^{100}) \) 2 * q - 2 * q^4 - 4 * q^5 + 2 * q^10 - 4 * q^11 + 8 * q^14 + 2 * q^16 - 4 * q^19 + 4 * q^20 + 6 * q^25 - 2 * q^26 - 4 * q^29 - 8 * q^31 - 8 * q^34 - 8 * q^35 - 2 * q^40 + 12 * q^41 + 4 * q^44 - 12 * q^46 - 18 * q^49 - 8 * q^50 + 8 * q^55 - 8 * q^56 - 28 * q^59 + 20 * q^61 - 2 * q^64 + 2 * q^65 - 16 * q^70 - 16 * q^71 - 12 * q^74 + 4 * q^76 + 16 * q^79 - 4 * q^80 + 8 * q^85 - 16 * q^86 - 36 * q^89 + 8 * q^91 - 16 * q^94 + 8 * q^95

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