LMFDB - Embedded newform 104.2.e.a.77.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [104,2,Mod(77,104)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("104.77"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(104, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 104 = 2^{3} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	104.e (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$0.830444181021$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			77.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	104.77
Dual form			104.2.e.a.77.1

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-1.00000 + 1.00000i) q^{2} -1.00000i q^{3} -2.00000i q^{4} +1.00000 q^{5} +(1.00000 + 1.00000i) q^{6} -3.00000i q^{7} +(2.00000 + 2.00000i) q^{8} +2.00000 q^{9} +(-1.00000 + 1.00000i) q^{10} +2.00000 q^{11} -2.00000 q^{12} +(-3.00000 + 2.00000i) q^{13} +(3.00000 + 3.00000i) q^{14} -1.00000i q^{15} -4.00000 q^{16} +3.00000 q^{17} +(-2.00000 + 2.00000i) q^{18} -2.00000i q^{20} -3.00000 q^{21} +(-2.00000 + 2.00000i) q^{22} -6.00000 q^{23} +(2.00000 - 2.00000i) q^{24} -4.00000 q^{25} +(1.00000 - 5.00000i) q^{26} -5.00000i q^{27} -6.00000 q^{28} +6.00000i q^{29} +(1.00000 + 1.00000i) q^{30} +(4.00000 - 4.00000i) q^{32} -2.00000i q^{33} +(-3.00000 + 3.00000i) q^{34} -3.00000i q^{35} -4.00000i q^{36} +3.00000 q^{37} +(2.00000 + 3.00000i) q^{39} +(2.00000 + 2.00000i) q^{40} +10.0000i q^{41} +(3.00000 - 3.00000i) q^{42} +9.00000i q^{43} -4.00000i q^{44} +2.00000 q^{45} +(6.00000 - 6.00000i) q^{46} +7.00000i q^{47} +4.00000i q^{48} -2.00000 q^{49} +(4.00000 - 4.00000i) q^{50} -3.00000i q^{51} +(4.00000 + 6.00000i) q^{52} -6.00000i q^{53} +(5.00000 + 5.00000i) q^{54} +2.00000 q^{55} +(6.00000 - 6.00000i) q^{56} +(-6.00000 - 6.00000i) q^{58} +10.0000 q^{59} -2.00000 q^{60} -10.0000i q^{61} -6.00000i q^{63} +8.00000i q^{64} +(-3.00000 + 2.00000i) q^{65} +(2.00000 + 2.00000i) q^{66} -12.0000 q^{67} -6.00000i q^{68} +6.00000i q^{69} +(3.00000 + 3.00000i) q^{70} +5.00000i q^{71} +(4.00000 + 4.00000i) q^{72} -6.00000i q^{73} +(-3.00000 + 3.00000i) q^{74} +4.00000i q^{75} -6.00000i q^{77} +(-5.00000 - 1.00000i) q^{78} -4.00000 q^{80} +1.00000 q^{81} +(-10.0000 - 10.0000i) q^{82} -16.0000 q^{83} +6.00000i q^{84} +3.00000 q^{85} +(-9.00000 - 9.00000i) q^{86} +6.00000 q^{87} +(4.00000 + 4.00000i) q^{88} -4.00000i q^{89} +(-2.00000 + 2.00000i) q^{90} +(6.00000 + 9.00000i) q^{91} +12.0000i q^{92} +(-7.00000 - 7.00000i) q^{94} +(-4.00000 - 4.00000i) q^{96} -18.0000i q^{97} +(2.00000 - 2.00000i) q^{98} +4.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{5} + 2 q^{6} + 4 q^{8} + 4 q^{9} - 2 q^{10} + 4 q^{11} - 4 q^{12} - 6 q^{13} + 6 q^{14} - 8 q^{16} + 6 q^{17} - 4 q^{18} - 6 q^{21} - 4 q^{22} - 12 q^{23} + 4 q^{24} - 8 q^{25} + 2 q^{26}+ \cdots + 8 q^{99}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^5 + 2 * q^6 + 4 * q^8 + 4 * q^9 - 2 * q^10 + 4 * q^11 - 4 * q^12 - 6 * q^13 + 6 * q^14 - 8 * q^16 + 6 * q^17 - 4 * q^18 - 6 * q^21 - 4 * q^22 - 12 * q^23 + 4 * q^24 - 8 * q^25 + 2 * q^26 - 12 * q^28 + 2 * q^30 + 8 * q^32 - 6 * q^34 + 6 * q^37 + 4 * q^39 + 4 * q^40 + 6 * q^42 + 4 * q^45 + 12 * q^46 - 4 * q^49 + 8 * q^50 + 8 * q^52 + 10 * q^54 + 4 * q^55 + 12 * q^56 - 12 * q^58 + 20 * q^59 - 4 * q^60 - 6 * q^65 + 4 * q^66 - 24 * q^67 + 6 * q^70 + 8 * q^72 - 6 * q^74 - 10 * q^78 - 8 * q^80 + 2 * q^81 - 20 * q^82 - 32 * q^83 + 6 * q^85 - 18 * q^86 + 12 * q^87 + 8 * q^88 - 4 * q^90 + 12 * q^91 - 14 * q^94 - 8 * q^96 + 4 * q^98 + 8 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/104\mathbb{Z}\right)^\times$.

$n$	$41$	$53$	$79$
$\chi(n)$	$-1$	$-1$	$1$

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	+	1.00000i	−0.707107	+	0.707107i
$2$	−1.00000	+	1.00000i	−0.707107	+	0.707107i
$3$		−	1.00000i		−	0.577350i	−0.957427	−	0.288675i	$-0.906785\pi$
$3$		−	1.00000i		−	0.577350i	0.957427	−	0.288675i	$-0.0932147\pi$
$4$		−	2.00000i		−	1.00000i
$5$	1.00000			0.447214			0.223607	−	0.974679i	$-0.428217\pi$
$5$	1.00000			0.447214			0.223607	+	0.974679i	$0.428217\pi$
$6$	1.00000	+	1.00000i	0.408248	+	0.408248i
$7$		−	3.00000i		−	1.13389i	−0.823754	−	0.566947i	$-0.808125\pi$
$7$		−	3.00000i		−	1.13389i	0.823754	−	0.566947i	$-0.191875\pi$
$8$	2.00000	+	2.00000i	0.707107	+	0.707107i
$9$	2.00000			0.666667
$10$	−1.00000	+	1.00000i	−0.316228	+	0.316228i
$11$	2.00000			0.603023			0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000			0.603023			0.301511	+	0.953463i	$0.402509\pi$
$12$	−2.00000			−0.577350
$13$	−3.00000	+	2.00000i	−0.832050	+	0.554700i
$13$	−3.00000	+	2.00000i	−0.832050	+	0.554700i
$14$	3.00000	+	3.00000i	0.801784	+	0.801784i
$15$		−	1.00000i		−	0.258199i
$16$	−4.00000			−1.00000
$17$	3.00000			0.727607			0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000			0.727607			0.363803	+	0.931476i	$0.381478\pi$
$18$	−2.00000	+	2.00000i	−0.471405	+	0.471405i
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$		−	2.00000i		−	0.447214i
$21$	−3.00000			−0.654654
$22$	−2.00000	+	2.00000i	−0.426401	+	0.426401i
$23$	−6.00000			−1.25109			−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000			−1.25109			−0.625543	+	0.780189i	$0.715123\pi$
$24$	2.00000	−	2.00000i	0.408248	−	0.408248i
$25$	−4.00000			−0.800000
$26$	1.00000	−	5.00000i	0.196116	−	0.980581i
$27$		−	5.00000i		−	0.962250i
$28$	−6.00000			−1.13389
$29$			6.00000i			1.11417i	0.830455	+	0.557086i	$0.188081\pi$
$29$			6.00000i			1.11417i	−0.830455	+	0.557086i	$0.811919\pi$
$30$	1.00000	+	1.00000i	0.182574	+	0.182574i
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$	4.00000	−	4.00000i	0.707107	−	0.707107i
$33$		−	2.00000i		−	0.348155i
$34$	−3.00000	+	3.00000i	−0.514496	+	0.514496i
$35$		−	3.00000i		−	0.507093i
$36$		−	4.00000i		−	0.666667i
$37$	3.00000			0.493197			0.246598	−	0.969118i	$-0.420687\pi$
$37$	3.00000			0.493197			0.246598	+	0.969118i	$0.420687\pi$
$38$		0			0
$39$	2.00000	+	3.00000i	0.320256	+	0.480384i
$40$	2.00000	+	2.00000i	0.316228	+	0.316228i
$41$			10.0000i			1.56174i	0.624695	+	0.780869i	$0.285223\pi$
$41$			10.0000i			1.56174i	−0.624695	+	0.780869i	$0.714777\pi$
$42$	3.00000	−	3.00000i	0.462910	−	0.462910i
$43$			9.00000i			1.37249i	0.727372	+	0.686244i	$0.240742\pi$
$43$			9.00000i			1.37249i	−0.727372	+	0.686244i	$0.759258\pi$
$44$		−	4.00000i		−	0.603023i
$45$	2.00000			0.298142
$46$	6.00000	−	6.00000i	0.884652	−	0.884652i
$47$			7.00000i			1.02105i	0.859861	+	0.510527i	$0.170550\pi$
$47$			7.00000i			1.02105i	−0.859861	+	0.510527i	$0.829450\pi$
$48$			4.00000i			0.577350i
$49$	−2.00000			−0.285714
$50$	4.00000	−	4.00000i	0.565685	−	0.565685i
$51$		−	3.00000i		−	0.420084i
$52$	4.00000	+	6.00000i	0.554700	+	0.832050i
$53$		−	6.00000i		−	0.824163i	−0.911147	−	0.412082i	$-0.864802\pi$
$53$		−	6.00000i		−	0.824163i	0.911147	−	0.412082i	$-0.135198\pi$
$54$	5.00000	+	5.00000i	0.680414	+	0.680414i
$55$	2.00000			0.269680
$56$	6.00000	−	6.00000i	0.801784	−	0.801784i
$57$		0			0
$58$	−6.00000	−	6.00000i	−0.787839	−	0.787839i
$59$	10.0000			1.30189			0.650945	−	0.759125i	$-0.274373\pi$
$59$	10.0000			1.30189			0.650945	+	0.759125i	$0.274373\pi$
$60$	−2.00000			−0.258199
$61$		−	10.0000i		−	1.28037i	−0.768221	−	0.640184i	$-0.778858\pi$
$61$		−	10.0000i		−	1.28037i	0.768221	−	0.640184i	$-0.221142\pi$
$62$		0			0
$63$		−	6.00000i		−	0.755929i
$64$			8.00000i			1.00000i
$65$	−3.00000	+	2.00000i	−0.372104	+	0.248069i
$66$	2.00000	+	2.00000i	0.246183	+	0.246183i
$67$	−12.0000			−1.46603			−0.733017	−	0.680211i	$-0.761888\pi$
$67$	−12.0000			−1.46603			−0.733017	+	0.680211i	$0.761888\pi$
$68$		−	6.00000i		−	0.727607i
$69$			6.00000i			0.722315i
$70$	3.00000	+	3.00000i	0.358569	+	0.358569i
$71$			5.00000i			0.593391i	0.954972	+	0.296695i	$0.0958846\pi$
$71$			5.00000i			0.593391i	−0.954972	+	0.296695i	$0.904115\pi$
$72$	4.00000	+	4.00000i	0.471405	+	0.471405i
$73$		−	6.00000i		−	0.702247i	−0.936329	−	0.351123i	$-0.885800\pi$
$73$		−	6.00000i		−	0.702247i	0.936329	−	0.351123i	$-0.114200\pi$
$74$	−3.00000	+	3.00000i	−0.348743	+	0.348743i
$75$			4.00000i			0.461880i
$76$		0			0
$77$		−	6.00000i		−	0.683763i
$78$	−5.00000	−	1.00000i	−0.566139	−	0.113228i
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$	−4.00000			−0.447214
$81$	1.00000			0.111111
$82$	−10.0000	−	10.0000i	−1.10432	−	1.10432i
$83$	−16.0000			−1.75623			−0.878114	−	0.478451i	$-0.841198\pi$
$83$	−16.0000			−1.75623			−0.878114	+	0.478451i	$0.841198\pi$
$84$			6.00000i			0.654654i
$85$	3.00000			0.325396
$86$	−9.00000	−	9.00000i	−0.970495	−	0.970495i
$87$	6.00000			0.643268
$88$	4.00000	+	4.00000i	0.426401	+	0.426401i
$89$		−	4.00000i		−	0.423999i	−0.977270	−	0.212000i	$-0.932002\pi$
$89$		−	4.00000i		−	0.423999i	0.977270	−	0.212000i	$-0.0679975\pi$
$90$	−2.00000	+	2.00000i	−0.210819	+	0.210819i
$91$	6.00000	+	9.00000i	0.628971	+	0.943456i
$92$			12.0000i			1.25109i
$93$		0			0
$94$	−7.00000	−	7.00000i	−0.721995	−	0.721995i
$95$		0			0
$96$	−4.00000	−	4.00000i	−0.408248	−	0.408248i
$97$		−	18.0000i		−	1.82762i	−0.406138	−	0.913812i	$-0.633125\pi$
$97$		−	18.0000i		−	1.82762i	0.406138	−	0.913812i	$-0.366875\pi$
$98$	2.00000	−	2.00000i	0.202031	−	0.202031i
$99$	4.00000			0.402015
$100$			8.00000i			0.800000i
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$	3.00000	+	3.00000i	0.297044	+	0.297044i
$103$	4.00000			0.394132			0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000			0.394132			0.197066	+	0.980390i	$0.436859\pi$
$104$	−10.0000	−	2.00000i	−0.980581	−	0.196116i
$105$	−3.00000			−0.292770
$106$	6.00000	+	6.00000i	0.582772	+	0.582772i
$107$			12.0000i			1.16008i	0.814587	+	0.580042i	$0.196964\pi$
$107$			12.0000i			1.16008i	−0.814587	+	0.580042i	$0.803036\pi$
$108$	−10.0000			−0.962250
$109$	15.0000			1.43674			0.718370	−	0.695662i	$-0.244889\pi$
$109$	15.0000			1.43674			0.718370	+	0.695662i	$0.244889\pi$
$110$	−2.00000	+	2.00000i	−0.190693	+	0.190693i
$111$		−	3.00000i		−	0.284747i
$112$			12.0000i			1.13389i
$113$	−6.00000			−0.564433			−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000			−0.564433			−0.282216	+	0.959351i	$0.591070\pi$
$114$		0			0
$115$	−6.00000			−0.559503
$116$	12.0000			1.11417
$117$	−6.00000	+	4.00000i	−0.554700	+	0.369800i
$118$	−10.0000	+	10.0000i	−0.920575	+	0.920575i
$119$		−	9.00000i		−	0.825029i
$120$	2.00000	−	2.00000i	0.182574	−	0.182574i
$121$	−7.00000			−0.636364
$122$	10.0000	+	10.0000i	0.905357	+	0.905357i
$123$	10.0000			0.901670
$124$		0			0
$125$	−9.00000			−0.804984
$126$	6.00000	+	6.00000i	0.534522	+	0.534522i
$127$	18.0000			1.59724			0.798621	−	0.601834i	$-0.205563\pi$
$127$	18.0000			1.59724			0.798621	+	0.601834i	$0.205563\pi$
$128$	−8.00000	−	8.00000i	−0.707107	−	0.707107i
$129$	9.00000			0.792406
$130$	1.00000	−	5.00000i	0.0877058	−	0.438529i
$131$		−	15.0000i		−	1.31056i	−0.755388	−	0.655278i	$-0.772551\pi$
$131$		−	15.0000i		−	1.31056i	0.755388	−	0.655278i	$-0.227449\pi$
$132$	−4.00000			−0.348155
$133$		0			0
$134$	12.0000	−	12.0000i	1.03664	−	1.03664i
$135$		−	5.00000i		−	0.430331i
$136$	6.00000	+	6.00000i	0.514496	+	0.514496i
$137$			2.00000i			0.170872i	0.996344	+	0.0854358i	$0.0272282\pi$
$137$			2.00000i			0.170872i	−0.996344	+	0.0854358i	$0.972772\pi$
$138$	−6.00000	−	6.00000i	−0.510754	−	0.510754i
$139$		−	9.00000i		−	0.763370i	−0.924292	−	0.381685i	$-0.875344\pi$
$139$		−	9.00000i		−	0.763370i	0.924292	−	0.381685i	$-0.124656\pi$
$140$	−6.00000			−0.507093
$141$	7.00000			0.589506
$142$	−5.00000	−	5.00000i	−0.419591	−	0.419591i
$143$	−6.00000	+	4.00000i	−0.501745	+	0.334497i
$144$	−8.00000			−0.666667
$145$			6.00000i			0.498273i
$146$	6.00000	+	6.00000i	0.496564	+	0.496564i
$147$			2.00000i			0.164957i
$148$		−	6.00000i		−	0.493197i
$149$	−10.0000			−0.819232			−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000			−0.819232			−0.409616	+	0.912258i	$0.634337\pi$
$150$	−4.00000	−	4.00000i	−0.326599	−	0.326599i
$151$			15.0000i			1.22068i	0.792139	+	0.610341i	$0.208968\pi$
$151$			15.0000i			1.22068i	−0.792139	+	0.610341i	$0.791032\pi$
$152$		0			0
$153$	6.00000			0.485071
$154$	6.00000	+	6.00000i	0.483494	+	0.483494i
$155$		0			0
$156$	6.00000	−	4.00000i	0.480384	−	0.320256i
$157$		−	18.0000i		−	1.43656i	−0.695756	−	0.718278i	$-0.744931\pi$
$157$		−	18.0000i		−	1.43656i	0.695756	−	0.718278i	$-0.255069\pi$
$158$		0			0
$159$	−6.00000			−0.475831
$160$	4.00000	−	4.00000i	0.316228	−	0.316228i
$161$			18.0000i			1.41860i
$162$	−1.00000	+	1.00000i	−0.0785674	+	0.0785674i
$163$	−6.00000			−0.469956			−0.234978	−	0.972001i	$-0.575502\pi$
$163$	−6.00000			−0.469956			−0.234978	+	0.972001i	$0.575502\pi$
$164$	20.0000			1.56174
$165$		−	2.00000i		−	0.155700i
$166$	16.0000	−	16.0000i	1.24184	−	1.24184i
$167$		−	8.00000i		−	0.619059i	−0.950890	−	0.309529i	$-0.899829\pi$
$167$		−	8.00000i		−	0.619059i	0.950890	−	0.309529i	$-0.100171\pi$
$168$	−6.00000	−	6.00000i	−0.462910	−	0.462910i
$169$	5.00000	−	12.0000i	0.384615	−	0.923077i
$170$	−3.00000	+	3.00000i	−0.230089	+	0.230089i
$171$		0			0
$172$	18.0000			1.37249
$173$			24.0000i			1.82469i	0.409426	+	0.912343i	$0.365729\pi$
$173$			24.0000i			1.82469i	−0.409426	+	0.912343i	$0.634271\pi$
$174$	−6.00000	+	6.00000i	−0.454859	+	0.454859i
$175$			12.0000i			0.907115i
$176$	−8.00000			−0.603023
$177$		−	10.0000i		−	0.751646i
$178$	4.00000	+	4.00000i	0.299813	+	0.299813i
$179$		−	9.00000i		−	0.672692i	−0.941739	−	0.336346i	$-0.890809\pi$
$179$		−	9.00000i		−	0.672692i	0.941739	−	0.336346i	$-0.109191\pi$
$180$		−	4.00000i		−	0.298142i
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$	−15.0000	−	3.00000i	−1.11187	−	0.222375i
$183$	−10.0000			−0.739221
$184$	−12.0000	−	12.0000i	−0.884652	−	0.884652i
$185$	3.00000			0.220564
$186$		0			0
$187$	6.00000			0.438763
$188$	14.0000			1.02105
$189$	−15.0000			−1.09109
$190$		0			0
$191$	−18.0000			−1.30243			−0.651217	−	0.758891i	$-0.725741\pi$
$191$	−18.0000			−1.30243			−0.651217	+	0.758891i	$0.725741\pi$
$192$	8.00000			0.577350
$193$			24.0000i			1.72756i	0.503871	+	0.863779i	$0.331909\pi$
$193$			24.0000i			1.72756i	−0.503871	+	0.863779i	$0.668091\pi$
$194$	18.0000	+	18.0000i	1.29232	+	1.29232i
$195$	2.00000	+	3.00000i	0.143223	+	0.214834i
$196$			4.00000i			0.285714i
$197$	23.0000			1.63868			0.819341	−	0.573306i	$-0.194340\pi$
$197$	23.0000			1.63868			0.819341	+	0.573306i	$0.194340\pi$
$198$	−4.00000	+	4.00000i	−0.284268	+	0.284268i
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$	−8.00000	−	8.00000i	−0.565685	−	0.565685i
$201$			12.0000i			0.846415i
$202$		0			0
$203$	18.0000			1.26335
$204$	−6.00000			−0.420084
$205$			10.0000i			0.698430i
$206$	−4.00000	+	4.00000i	−0.278693	+	0.278693i
$207$	−12.0000			−0.834058
$208$	12.0000	−	8.00000i	0.832050	−	0.554700i
$209$		0			0
$210$	3.00000	−	3.00000i	0.207020	−	0.207020i
$211$			5.00000i			0.344214i	0.985078	+	0.172107i	$0.0550575\pi$
$211$			5.00000i			0.344214i	−0.985078	+	0.172107i	$0.944942\pi$
$212$	−12.0000			−0.824163
$213$	5.00000			0.342594
$214$	−12.0000	−	12.0000i	−0.820303	−	0.820303i
$215$			9.00000i			0.613795i
$216$	10.0000	−	10.0000i	0.680414	−	0.680414i
$217$		0			0
$218$	−15.0000	+	15.0000i	−1.01593	+	1.01593i
$219$	−6.00000			−0.405442
$220$		−	4.00000i		−	0.269680i
$221$	−9.00000	+	6.00000i	−0.605406	+	0.403604i
$222$	3.00000	+	3.00000i	0.201347	+	0.201347i
$223$		−	21.0000i		−	1.40626i	−0.711059	−	0.703132i	$-0.751784\pi$
$223$		−	21.0000i		−	1.40626i	0.711059	−	0.703132i	$-0.248216\pi$
$224$	−12.0000	−	12.0000i	−0.801784	−	0.801784i
$225$	−8.00000			−0.533333
$226$	6.00000	−	6.00000i	0.399114	−	0.399114i
$227$	−2.00000			−0.132745			−0.0663723	−	0.997795i	$-0.521143\pi$
$227$	−2.00000			−0.132745			−0.0663723	+	0.997795i	$0.521143\pi$
$228$		0			0
$229$	−15.0000			−0.991228			−0.495614	−	0.868543i	$-0.665057\pi$
$229$	−15.0000			−0.991228			−0.495614	+	0.868543i	$0.665057\pi$
$230$	6.00000	−	6.00000i	0.395628	−	0.395628i
$231$	−6.00000			−0.394771
$232$	−12.0000	+	12.0000i	−0.787839	+	0.787839i
$233$	9.00000			0.589610			0.294805	−	0.955557i	$-0.404745\pi$
$233$	9.00000			0.589610			0.294805	+	0.955557i	$0.404745\pi$
$234$	2.00000	−	10.0000i	0.130744	−	0.653720i
$235$			7.00000i			0.456630i
$236$		−	20.0000i		−	1.30189i
$237$		0			0
$238$	9.00000	+	9.00000i	0.583383	+	0.583383i
$239$		−	19.0000i		−	1.22901i	−0.788914	−	0.614504i	$-0.789356\pi$
$239$		−	19.0000i		−	1.22901i	0.788914	−	0.614504i	$-0.210644\pi$
$240$			4.00000i			0.258199i
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$	7.00000	−	7.00000i	0.449977	−	0.449977i
$243$		−	16.0000i		−	1.02640i
$244$	−20.0000			−1.28037
$245$	−2.00000			−0.127775
$246$	−10.0000	+	10.0000i	−0.637577	+	0.637577i
$247$		0			0
$248$		0			0
$249$			16.0000i			1.01396i
$250$	9.00000	−	9.00000i	0.569210	−	0.569210i
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$	−12.0000			−0.755929
$253$	−12.0000			−0.754434
$254$	−18.0000	+	18.0000i	−1.12942	+	1.12942i
$255$		−	3.00000i		−	0.187867i
$256$	16.0000			1.00000
$257$	3.00000			0.187135			0.0935674	−	0.995613i	$-0.470173\pi$
$257$	3.00000			0.187135			0.0935674	+	0.995613i	$0.470173\pi$
$258$	−9.00000	+	9.00000i	−0.560316	+	0.560316i
$259$		−	9.00000i		−	0.559233i
$260$	4.00000	+	6.00000i	0.248069	+	0.372104i
$261$			12.0000i			0.742781i
$262$	15.0000	+	15.0000i	0.926703	+	0.926703i
$263$	−6.00000			−0.369976			−0.184988	−	0.982741i	$-0.559225\pi$
$263$	−6.00000			−0.369976			−0.184988	+	0.982741i	$0.559225\pi$
$264$	4.00000	−	4.00000i	0.246183	−	0.246183i
$265$		−	6.00000i		−	0.368577i
$266$		0			0
$267$	−4.00000			−0.244796
$268$			24.0000i			1.46603i
$269$			6.00000i			0.365826i	0.983129	+	0.182913i	$0.0585527\pi$
$269$			6.00000i			0.365826i	−0.983129	+	0.182913i	$0.941447\pi$
$270$	5.00000	+	5.00000i	0.304290	+	0.304290i
$271$			15.0000i			0.911185i	0.890188	+	0.455593i	$0.150573\pi$
$271$			15.0000i			0.911185i	−0.890188	+	0.455593i	$0.849427\pi$
$272$	−12.0000			−0.727607
$273$	9.00000	−	6.00000i	0.544705	−	0.363137i
$274$	−2.00000	−	2.00000i	−0.120824	−	0.120824i
$275$	−8.00000			−0.482418
$276$	12.0000			0.722315
$277$		−	18.0000i		−	1.08152i	−0.841178	−	0.540758i	$-0.818138\pi$
$277$		−	18.0000i		−	1.08152i	0.841178	−	0.540758i	$-0.181862\pi$
$278$	9.00000	+	9.00000i	0.539784	+	0.539784i
$279$		0			0
$280$	6.00000	−	6.00000i	0.358569	−	0.358569i
$281$		−	20.0000i		−	1.19310i	−0.802576	−	0.596550i	$-0.796538\pi$
$281$		−	20.0000i		−	1.19310i	0.802576	−	0.596550i	$-0.203462\pi$
$282$	−7.00000	+	7.00000i	−0.416844	+	0.416844i
$283$			4.00000i			0.237775i	0.992908	+	0.118888i	$0.0379328\pi$
$283$			4.00000i			0.237775i	−0.992908	+	0.118888i	$0.962067\pi$
$284$	10.0000			0.593391
$285$		0			0
$286$	2.00000	−	10.0000i	0.118262	−	0.591312i
$287$	30.0000			1.77084
$288$	8.00000	−	8.00000i	0.471405	−	0.471405i
$289$	−8.00000			−0.470588
$290$	−6.00000	−	6.00000i	−0.352332	−	0.352332i
$291$	−18.0000			−1.05518
$292$	−12.0000			−0.702247
$293$	−1.00000			−0.0584206			−0.0292103	−	0.999573i	$-0.509299\pi$
$293$	−1.00000			−0.0584206			−0.0292103	+	0.999573i	$0.509299\pi$
$294$	−2.00000	−	2.00000i	−0.116642	−	0.116642i
$295$	10.0000			0.582223
$296$	6.00000	+	6.00000i	0.348743	+	0.348743i
$297$		−	10.0000i		−	0.580259i
$298$	10.0000	−	10.0000i	0.579284	−	0.579284i
$299$	18.0000	−	12.0000i	1.04097	−	0.693978i
$300$	8.00000			0.461880
$301$	27.0000			1.55625
$302$	−15.0000	−	15.0000i	−0.863153	−	0.863153i
$303$		0			0
$304$		0			0
$305$		−	10.0000i		−	0.572598i
$306$	−6.00000	+	6.00000i	−0.342997	+	0.342997i
$307$	18.0000			1.02731			0.513657	−	0.857996i	$-0.328290\pi$
$307$	18.0000			1.02731			0.513657	+	0.857996i	$0.328290\pi$
$308$	−12.0000			−0.683763
$309$		−	4.00000i		−	0.227552i
$310$		0			0
$311$	12.0000			0.680458			0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000			0.680458			0.340229	+	0.940343i	$0.389495\pi$
$312$	−2.00000	+	10.0000i	−0.113228	+	0.566139i
$313$	9.00000			0.508710			0.254355	−	0.967111i	$-0.418137\pi$
$313$	9.00000			0.508710			0.254355	+	0.967111i	$0.418137\pi$
$314$	18.0000	+	18.0000i	1.01580	+	1.01580i
$315$		−	6.00000i		−	0.338062i
$316$		0			0
$317$	−2.00000			−0.112331			−0.0561656	−	0.998421i	$-0.517887\pi$
$317$	−2.00000			−0.112331			−0.0561656	+	0.998421i	$0.517887\pi$
$318$	6.00000	−	6.00000i	0.336463	−	0.336463i
$319$			12.0000i			0.671871i
$320$			8.00000i			0.447214i
$321$	12.0000			0.669775
$322$	−18.0000	−	18.0000i	−1.00310	−	1.00310i
$323$		0			0
$324$		−	2.00000i		−	0.111111i
$325$	12.0000	−	8.00000i	0.665640	−	0.443760i
$326$	6.00000	−	6.00000i	0.332309	−	0.332309i
$327$		−	15.0000i		−	0.829502i
$328$	−20.0000	+	20.0000i	−1.10432	+	1.10432i
$329$	21.0000			1.15777
$330$	2.00000	+	2.00000i	0.110096	+	0.110096i
$331$	12.0000			0.659580			0.329790	−	0.944054i	$-0.393022\pi$
$331$	12.0000			0.659580			0.329790	+	0.944054i	$0.393022\pi$
$332$			32.0000i			1.75623i
$333$	6.00000			0.328798
$334$	8.00000	+	8.00000i	0.437741	+	0.437741i
$335$	−12.0000			−0.655630
$336$	12.0000			0.654654
$337$	−27.0000			−1.47078			−0.735392	−	0.677642i	$-0.763002\pi$
$337$	−27.0000			−1.47078			−0.735392	+	0.677642i	$0.763002\pi$
$338$	7.00000	+	17.0000i	0.380750	+	0.924678i
$339$			6.00000i			0.325875i
$340$		−	6.00000i		−	0.325396i
$341$		0			0
$342$		0			0
$343$		−	15.0000i		−	0.809924i
$344$	−18.0000	+	18.0000i	−0.970495	+	0.970495i
$345$			6.00000i			0.323029i
$346$	−24.0000	−	24.0000i	−1.29025	−	1.29025i
$347$			27.0000i			1.44944i	0.689046	+	0.724718i	$0.258030\pi$
$347$			27.0000i			1.44944i	−0.689046	+	0.724718i	$0.741970\pi$
$348$		−	12.0000i		−	0.643268i
$349$	−15.0000			−0.802932			−0.401466	−	0.915874i	$-0.631499\pi$
$349$	−15.0000			−0.802932			−0.401466	+	0.915874i	$0.631499\pi$
$350$	−12.0000	−	12.0000i	−0.641427	−	0.641427i
$351$	10.0000	+	15.0000i	0.533761	+	0.800641i
$352$	8.00000	−	8.00000i	0.426401	−	0.426401i
$353$		−	16.0000i		−	0.851594i	−0.904819	−	0.425797i	$-0.859994\pi$
$353$		−	16.0000i		−	0.851594i	0.904819	−	0.425797i	$-0.140006\pi$
$354$	10.0000	+	10.0000i	0.531494	+	0.531494i
$355$			5.00000i			0.265372i
$356$	−8.00000			−0.423999
$357$	−9.00000			−0.476331
$358$	9.00000	+	9.00000i	0.475665	+	0.475665i
$359$		−	4.00000i		−	0.211112i	−0.994413	−	0.105556i	$-0.966338\pi$
$359$		−	4.00000i		−	0.211112i	0.994413	−	0.105556i	$-0.0336622\pi$
$360$	4.00000	+	4.00000i	0.210819	+	0.210819i
$361$	−19.0000			−1.00000
$362$		0			0
$363$			7.00000i			0.367405i
$364$	18.0000	−	12.0000i	0.943456	−	0.628971i
$365$		−	6.00000i		−	0.314054i
$366$	10.0000	−	10.0000i	0.522708	−	0.522708i
$367$	28.0000			1.46159			0.730794	−	0.682598i	$-0.239150\pi$
$367$	28.0000			1.46159			0.730794	+	0.682598i	$0.239150\pi$
$368$	24.0000			1.25109
$369$			20.0000i			1.04116i
$370$	−3.00000	+	3.00000i	−0.155963	+	0.155963i
$371$	−18.0000			−0.934513
$372$		0			0
$373$			4.00000i			0.207112i	0.994624	+	0.103556i	$0.0330221\pi$
$373$			4.00000i			0.207112i	−0.994624	+	0.103556i	$0.966978\pi$
$374$	−6.00000	+	6.00000i	−0.310253	+	0.310253i
$375$			9.00000i			0.464758i
$376$	−14.0000	+	14.0000i	−0.721995	+	0.721995i
$377$	−12.0000	−	18.0000i	−0.618031	−	0.927047i
$378$	15.0000	−	15.0000i	0.771517	−	0.771517i
$379$	−30.0000			−1.54100			−0.770498	−	0.637442i	$-0.779993\pi$
$379$	−30.0000			−1.54100			−0.770498	+	0.637442i	$0.779993\pi$
$380$		0			0
$381$		−	18.0000i		−	0.922168i
$382$	18.0000	−	18.0000i	0.920960	−	0.920960i
$383$		−	1.00000i		−	0.0510976i	−0.999674	−	0.0255488i	$-0.991867\pi$
$383$		−	1.00000i		−	0.0510976i	0.999674	−	0.0255488i	$-0.00813332\pi$
$384$	−8.00000	+	8.00000i	−0.408248	+	0.408248i
$385$		−	6.00000i		−	0.305788i
$386$	−24.0000	−	24.0000i	−1.22157	−	1.22157i
$387$			18.0000i			0.914991i
$388$	−36.0000			−1.82762
$389$			6.00000i			0.304212i	0.988364	+	0.152106i	$0.0486055\pi$
$389$			6.00000i			0.304212i	−0.988364	+	0.152106i	$0.951394\pi$
$390$	−5.00000	−	1.00000i	−0.253185	−	0.0506370i
$391$	−18.0000			−0.910299
$392$	−4.00000	−	4.00000i	−0.202031	−	0.202031i
$393$	−15.0000			−0.756650
$394$	−23.0000	+	23.0000i	−1.15872	+	1.15872i
$395$		0			0
$396$		−	8.00000i		−	0.402015i
$397$	18.0000			0.903394			0.451697	−	0.892171i	$-0.350819\pi$
$397$	18.0000			0.903394			0.451697	+	0.892171i	$0.350819\pi$
$398$		0			0
$399$		0			0
$400$	16.0000			0.800000
$401$			10.0000i			0.499376i	0.968326	+	0.249688i	$0.0803281\pi$
$401$			10.0000i			0.499376i	−0.968326	+	0.249688i	$0.919672\pi$
$402$	−12.0000	−	12.0000i	−0.598506	−	0.598506i
$403$		0			0
$404$		0			0
$405$	1.00000			0.0496904
$406$	−18.0000	+	18.0000i	−0.893325	+	0.893325i
$407$	6.00000			0.297409
$408$	6.00000	−	6.00000i	0.297044	−	0.297044i
$409$			6.00000i			0.296681i	0.988936	+	0.148340i	$0.0473931\pi$
$409$			6.00000i			0.296681i	−0.988936	+	0.148340i	$0.952607\pi$
$410$	−10.0000	−	10.0000i	−0.493865	−	0.493865i
$411$	2.00000			0.0986527
$412$		−	8.00000i		−	0.394132i
$413$		−	30.0000i		−	1.47620i
$414$	12.0000	−	12.0000i	0.589768	−	0.589768i
$415$	−16.0000			−0.785409
$416$	−4.00000	+	20.0000i	−0.196116	+	0.980581i
$417$	−9.00000			−0.440732
$418$		0			0
$419$		−	9.00000i		−	0.439679i	−0.975536	−	0.219839i	$-0.929447\pi$
$419$		−	9.00000i		−	0.439679i	0.975536	−	0.219839i	$-0.0705533\pi$
$420$			6.00000i			0.292770i
$421$	−3.00000			−0.146211			−0.0731055	−	0.997324i	$-0.523291\pi$
$421$	−3.00000			−0.146211			−0.0731055	+	0.997324i	$0.523291\pi$
$422$	−5.00000	−	5.00000i	−0.243396	−	0.243396i
$423$			14.0000i			0.680703i
$424$	12.0000	−	12.0000i	0.582772	−	0.582772i
$425$	−12.0000			−0.582086
$426$	−5.00000	+	5.00000i	−0.242251	+	0.242251i
$427$	−30.0000			−1.45180
$428$	24.0000			1.16008
$429$	4.00000	+	6.00000i	0.193122	+	0.289683i
$430$	−9.00000	−	9.00000i	−0.434019	−	0.434019i
$431$			5.00000i			0.240842i	0.992723	+	0.120421i	$0.0384244\pi$
$431$			5.00000i			0.240842i	−0.992723	+	0.120421i	$0.961576\pi$
$432$			20.0000i			0.962250i
$433$	9.00000			0.432512			0.216256	−	0.976337i	$-0.430615\pi$
$433$	9.00000			0.432512			0.216256	+	0.976337i	$0.430615\pi$
$434$		0			0
$435$	6.00000			0.287678
$436$		−	30.0000i		−	1.43674i
$437$		0			0
$438$	6.00000	−	6.00000i	0.286691	−	0.286691i
$439$	−10.0000			−0.477274			−0.238637	−	0.971109i	$-0.576701\pi$
$439$	−10.0000			−0.477274			−0.238637	+	0.971109i	$0.576701\pi$
$440$	4.00000	+	4.00000i	0.190693	+	0.190693i
$441$	−4.00000			−0.190476
$442$	3.00000	−	15.0000i	0.142695	−	0.713477i
$443$			9.00000i			0.427603i	0.976877	+	0.213801i	$0.0685846\pi$
$443$			9.00000i			0.427603i	−0.976877	+	0.213801i	$0.931415\pi$
$444$	−6.00000			−0.284747
$445$		−	4.00000i		−	0.189618i
$446$	21.0000	+	21.0000i	0.994379	+	0.994379i
$447$			10.0000i			0.472984i
$448$	24.0000			1.13389
$449$		−	14.0000i		−	0.660701i	−0.943858	−	0.330350i	$-0.892833\pi$
$449$		−	14.0000i		−	0.660701i	0.943858	−	0.330350i	$-0.107167\pi$
$450$	8.00000	−	8.00000i	0.377124	−	0.377124i
$451$			20.0000i			0.941763i
$452$			12.0000i			0.564433i
$453$	15.0000			0.704761
$454$	2.00000	−	2.00000i	0.0938647	−	0.0938647i
$455$	6.00000	+	9.00000i	0.281284	+	0.421927i
$456$		0			0
$457$			12.0000i			0.561336i	0.959805	+	0.280668i	$0.0905560\pi$
$457$			12.0000i			0.561336i	−0.959805	+	0.280668i	$0.909444\pi$
$458$	15.0000	−	15.0000i	0.700904	−	0.700904i
$459$		−	15.0000i		−	0.700140i
$460$			12.0000i			0.559503i
$461$	7.00000			0.326023			0.163011	−	0.986624i	$-0.447879\pi$
$461$	7.00000			0.326023			0.163011	+	0.986624i	$0.447879\pi$
$462$	6.00000	−	6.00000i	0.279145	−	0.279145i
$463$			24.0000i			1.11537i	0.830051	+	0.557687i	$0.188311\pi$
$463$			24.0000i			1.11537i	−0.830051	+	0.557687i	$0.811689\pi$
$464$		−	24.0000i		−	1.11417i
$465$		0			0
$466$	−9.00000	+	9.00000i	−0.416917	+	0.416917i
$467$			12.0000i			0.555294i	0.960683	+	0.277647i	$0.0895545\pi$
$467$			12.0000i			0.555294i	−0.960683	+	0.277647i	$0.910445\pi$
$468$	8.00000	+	12.0000i	0.369800	+	0.554700i
$469$			36.0000i			1.66233i
$470$	−7.00000	−	7.00000i	−0.322886	−	0.322886i
$471$	−18.0000			−0.829396
$472$	20.0000	+	20.0000i	0.920575	+	0.920575i
$473$			18.0000i			0.827641i
$474$		0			0
$475$		0			0
$476$	−18.0000			−0.825029
$477$		−	12.0000i		−	0.549442i
$478$	19.0000	+	19.0000i	0.869040	+	0.869040i
$479$		−	29.0000i		−	1.32504i	−0.749043	−	0.662522i	$-0.769486\pi$
$479$		−	29.0000i		−	1.32504i	0.749043	−	0.662522i	$-0.230514\pi$
$480$	−4.00000	−	4.00000i	−0.182574	−	0.182574i
$481$	−9.00000	+	6.00000i	−0.410365	+	0.273576i
$482$		0			0
$483$	18.0000			0.819028
$484$			14.0000i			0.636364i
$485$		−	18.0000i		−	0.817338i
$486$	16.0000	+	16.0000i	0.725775	+	0.725775i
$487$			12.0000i			0.543772i	0.962329	+	0.271886i	$0.0876473\pi$
$487$			12.0000i			0.543772i	−0.962329	+	0.271886i	$0.912353\pi$
$488$	20.0000	−	20.0000i	0.905357	−	0.905357i
$489$			6.00000i			0.271329i
$490$	2.00000	−	2.00000i	0.0903508	−	0.0903508i
$491$		−	15.0000i		−	0.676941i	−0.940977	−	0.338470i	$-0.890091\pi$
$491$		−	15.0000i		−	0.676941i	0.940977	−	0.338470i	$-0.109909\pi$
$492$		−	20.0000i		−	0.901670i
$493$			18.0000i			0.810679i
$494$		0			0
$495$	4.00000			0.179787
$496$		0			0
$497$	15.0000			0.672842
$498$	−16.0000	−	16.0000i	−0.716977	−	0.716977i
$499$	30.0000			1.34298			0.671492	−	0.741012i	$-0.265654\pi$
$499$	30.0000			1.34298			0.671492	+	0.741012i	$0.265654\pi$
$500$			18.0000i			0.804984i
$501$	−8.00000			−0.357414
$502$		0			0
$503$	24.0000			1.07011			0.535054	−	0.844818i	$-0.320291\pi$
$503$	24.0000			1.07011			0.535054	+	0.844818i	$0.320291\pi$
$504$	12.0000	−	12.0000i	0.534522	−	0.534522i
$505$		0			0
$506$	12.0000	−	12.0000i	0.533465	−	0.533465i
$507$	−12.0000	−	5.00000i	−0.532939	−	0.222058i
$508$		−	36.0000i		−	1.59724i
$509$	−10.0000			−0.443242			−0.221621	−	0.975133i	$-0.571135\pi$
$509$	−10.0000			−0.443242			−0.221621	+	0.975133i	$0.571135\pi$
$510$	3.00000	+	3.00000i	0.132842	+	0.132842i
$511$	−18.0000			−0.796273
$512$	−16.0000	+	16.0000i	−0.707107	+	0.707107i
$513$		0			0
$514$	−3.00000	+	3.00000i	−0.132324	+	0.132324i
$515$	4.00000			0.176261
$516$		−	18.0000i		−	0.792406i
$517$			14.0000i			0.615719i
$518$	9.00000	+	9.00000i	0.395437	+	0.395437i
$519$	24.0000			1.05348
$520$	−10.0000	−	2.00000i	−0.438529	−	0.0877058i
$521$	−3.00000			−0.131432			−0.0657162	−	0.997838i	$-0.520933\pi$
$521$	−3.00000			−0.131432			−0.0657162	+	0.997838i	$0.520933\pi$
$522$	−12.0000	−	12.0000i	−0.525226	−	0.525226i
$523$		−	36.0000i		−	1.57417i	−0.616844	−	0.787085i	$-0.711589\pi$
$523$		−	36.0000i		−	1.57417i	0.616844	−	0.787085i	$-0.288411\pi$
$524$	−30.0000			−1.31056
$525$	12.0000			0.523723
$526$	6.00000	−	6.00000i	0.261612	−	0.261612i
$527$		0			0
$528$			8.00000i			0.348155i
$529$	13.0000			0.565217
$530$	6.00000	+	6.00000i	0.260623	+	0.260623i
$531$	20.0000			0.867926
$532$		0			0
$533$	−20.0000	−	30.0000i	−0.866296	−	1.29944i
$534$	4.00000	−	4.00000i	0.173097	−	0.173097i
$535$			12.0000i			0.518805i
$536$	−24.0000	−	24.0000i	−1.03664	−	1.03664i
$537$	−9.00000			−0.388379
$538$	−6.00000	−	6.00000i	−0.258678	−	0.258678i
$539$	−4.00000			−0.172292
$540$	−10.0000			−0.430331
$541$	−33.0000			−1.41878			−0.709390	−	0.704816i	$-0.751030\pi$
$541$	−33.0000			−1.41878			−0.709390	+	0.704816i	$0.751030\pi$
$542$	−15.0000	−	15.0000i	−0.644305	−	0.644305i
$543$		0			0
$544$	12.0000	−	12.0000i	0.514496	−	0.514496i
$545$	15.0000			0.642529
$546$	−3.00000	+	15.0000i	−0.128388	+	0.641941i
$547$			27.0000i			1.15444i	0.816590	+	0.577218i	$0.195862\pi$
$547$			27.0000i			1.15444i	−0.816590	+	0.577218i	$0.804138\pi$
$548$	4.00000			0.170872
$549$		−	20.0000i		−	0.853579i
$550$	8.00000	−	8.00000i	0.341121	−	0.341121i
$551$		0			0
$552$	−12.0000	+	12.0000i	−0.510754	+	0.510754i
$553$		0			0
$554$	18.0000	+	18.0000i	0.764747	+	0.764747i
$555$		−	3.00000i		−	0.127343i
$556$	−18.0000			−0.763370
$557$	23.0000			0.974541			0.487271	−	0.873251i	$-0.337993\pi$
$557$	23.0000			0.974541			0.487271	+	0.873251i	$0.337993\pi$
$558$		0			0
$559$	−18.0000	−	27.0000i	−0.761319	−	1.14198i
$560$			12.0000i			0.507093i
$561$		−	6.00000i		−	0.253320i
$562$	20.0000	+	20.0000i	0.843649	+	0.843649i
$563$		−	21.0000i		−	0.885044i	−0.896758	−	0.442522i	$-0.854084\pi$
$563$		−	21.0000i		−	0.885044i	0.896758	−	0.442522i	$-0.145916\pi$
$564$		−	14.0000i		−	0.589506i
$565$	−6.00000			−0.252422
$566$	−4.00000	−	4.00000i	−0.168133	−	0.168133i
$567$		−	3.00000i		−	0.125988i
$568$	−10.0000	+	10.0000i	−0.419591	+	0.419591i
$569$	15.0000			0.628833			0.314416	−	0.949285i	$-0.398191\pi$
$569$	15.0000			0.628833			0.314416	+	0.949285i	$0.398191\pi$
$570$		0			0
$571$		−	5.00000i		−	0.209243i	−0.994512	−	0.104622i	$-0.966637\pi$
$571$		−	5.00000i		−	0.209243i	0.994512	−	0.104622i	$-0.0333632\pi$
$572$	8.00000	+	12.0000i	0.334497	+	0.501745i
$573$			18.0000i			0.751961i
$574$	−30.0000	+	30.0000i	−1.25218	+	1.25218i
$575$	24.0000			1.00087
$576$			16.0000i			0.666667i
$577$		−	18.0000i		−	0.749350i	−0.927156	−	0.374675i	$-0.877754\pi$
$577$		−	18.0000i		−	0.749350i	0.927156	−	0.374675i	$-0.122246\pi$
$578$	8.00000	−	8.00000i	0.332756	−	0.332756i
$579$	24.0000			0.997406
$580$	12.0000			0.498273
$581$			48.0000i			1.99138i
$582$	18.0000	−	18.0000i	0.746124	−	0.746124i
$583$		−	12.0000i		−	0.496989i
$584$	12.0000	−	12.0000i	0.496564	−	0.496564i
$585$	−6.00000	+	4.00000i	−0.248069	+	0.165380i
$586$	1.00000	−	1.00000i	0.0413096	−	0.0413096i
$587$	−2.00000			−0.0825488			−0.0412744	−	0.999148i	$-0.513142\pi$
$587$	−2.00000			−0.0825488			−0.0412744	+	0.999148i	$0.513142\pi$
$588$	4.00000			0.164957
$589$		0			0
$590$	−10.0000	+	10.0000i	−0.411693	+	0.411693i
$591$		−	23.0000i		−	0.946094i
$592$	−12.0000			−0.493197
$593$			14.0000i			0.574911i	0.957794	+	0.287456i	$0.0928094\pi$
$593$			14.0000i			0.574911i	−0.957794	+	0.287456i	$0.907191\pi$
$594$	10.0000	+	10.0000i	0.410305	+	0.410305i
$595$		−	9.00000i		−	0.368964i
$596$			20.0000i			0.819232i
$597$		0			0
$598$	−6.00000	+	30.0000i	−0.245358	+	1.22679i
$599$	−30.0000			−1.22577			−0.612883	−	0.790173i	$-0.709990\pi$
$599$	−30.0000			−1.22577			−0.612883	+	0.790173i	$0.709990\pi$
$600$	−8.00000	+	8.00000i	−0.326599	+	0.326599i
$601$	27.0000			1.10135			0.550676	−	0.834719i	$-0.314370\pi$
$601$	27.0000			1.10135			0.550676	+	0.834719i	$0.314370\pi$
$602$	−27.0000	+	27.0000i	−1.10044	+	1.10044i
$603$	−24.0000			−0.977356
$604$	30.0000			1.22068
$605$	−7.00000			−0.284590
$606$		0			0
$607$	−22.0000			−0.892952			−0.446476	−	0.894795i	$-0.647321\pi$
$607$	−22.0000			−0.892952			−0.446476	+	0.894795i	$0.647321\pi$
$608$		0			0
$609$		−	18.0000i		−	0.729397i
$610$	10.0000	+	10.0000i	0.404888	+	0.404888i
$611$	−14.0000	−	21.0000i	−0.566379	−	0.849569i
$612$		−	12.0000i		−	0.485071i
$613$	−6.00000			−0.242338			−0.121169	−	0.992632i	$-0.538664\pi$
$613$	−6.00000			−0.242338			−0.121169	+	0.992632i	$0.538664\pi$
$614$	−18.0000	+	18.0000i	−0.726421	+	0.726421i
$615$	10.0000			0.403239
$616$	12.0000	−	12.0000i	0.483494	−	0.483494i
$617$		−	8.00000i		−	0.322068i	−0.986949	−	0.161034i	$-0.948517\pi$
$617$		−	8.00000i		−	0.322068i	0.986949	−	0.161034i	$-0.0514829\pi$
$618$	4.00000	+	4.00000i	0.160904	+	0.160904i
$619$	−30.0000			−1.20580			−0.602901	−	0.797816i	$-0.705989\pi$
$619$	−30.0000			−1.20580			−0.602901	+	0.797816i	$0.705989\pi$
$620$		0			0
$621$			30.0000i			1.20386i
$622$	−12.0000	+	12.0000i	−0.481156	+	0.481156i
$623$	−12.0000			−0.480770
$624$	−8.00000	−	12.0000i	−0.320256	−	0.480384i
$625$	11.0000			0.440000
$626$	−9.00000	+	9.00000i	−0.359712	+	0.359712i
$627$		0			0
$628$	−36.0000			−1.43656
$629$	9.00000			0.358854
$630$	6.00000	+	6.00000i	0.239046	+	0.239046i
$631$		−	45.0000i		−	1.79142i	−0.444637	−	0.895711i	$-0.646667\pi$
$631$		−	45.0000i		−	1.79142i	0.444637	−	0.895711i	$-0.353333\pi$
$632$		0			0
$633$	5.00000			0.198732
$634$	2.00000	−	2.00000i	0.0794301	−	0.0794301i
$635$	18.0000			0.714308
$636$			12.0000i			0.475831i
$637$	6.00000	−	4.00000i	0.237729	−	0.158486i
$638$	−12.0000	−	12.0000i	−0.475085	−	0.475085i
$639$			10.0000i			0.395594i
$640$	−8.00000	−	8.00000i	−0.316228	−	0.316228i
$641$	−18.0000			−0.710957			−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000			−0.710957			−0.355479	+	0.934684i	$0.615682\pi$
$642$	−12.0000	+	12.0000i	−0.473602	+	0.473602i
$643$	24.0000			0.946468			0.473234	−	0.880937i	$-0.343087\pi$
$643$	24.0000			0.946468			0.473234	+	0.880937i	$0.343087\pi$
$644$	36.0000			1.41860
$645$	9.00000			0.354375
$646$		0			0
$647$	18.0000			0.707653			0.353827	−	0.935311i	$-0.384880\pi$
$647$	18.0000			0.707653			0.353827	+	0.935311i	$0.384880\pi$
$648$	2.00000	+	2.00000i	0.0785674	+	0.0785674i
$649$	20.0000			0.785069
$650$	−4.00000	+	20.0000i	−0.156893	+	0.784465i
$651$		0			0
$652$			12.0000i			0.469956i
$653$			24.0000i			0.939193i	0.882881	+	0.469596i	$0.155601\pi$
$653$			24.0000i			0.939193i	−0.882881	+	0.469596i	$0.844399\pi$
$654$	15.0000	+	15.0000i	0.586546	+	0.586546i
$655$		−	15.0000i		−	0.586098i
$656$		−	40.0000i		−	1.56174i
$657$		−	12.0000i		−	0.468165i
$658$	−21.0000	+	21.0000i	−0.818665	+	0.818665i
$659$			36.0000i			1.40236i	0.712984	+	0.701180i	$0.247343\pi$
$659$			36.0000i			1.40236i	−0.712984	+	0.701180i	$0.752657\pi$
$660$	−4.00000			−0.155700
$661$	−18.0000			−0.700119			−0.350059	−	0.936727i	$-0.613839\pi$
$661$	−18.0000			−0.700119			−0.350059	+	0.936727i	$0.613839\pi$
$662$	−12.0000	+	12.0000i	−0.466393	+	0.466393i
$663$	6.00000	+	9.00000i	0.233021	+	0.349531i
$664$	−32.0000	−	32.0000i	−1.24184	−	1.24184i
$665$		0			0
$666$	−6.00000	+	6.00000i	−0.232495	+	0.232495i
$667$		−	36.0000i		−	1.39393i
$668$	−16.0000			−0.619059
$669$	−21.0000			−0.811907
$670$	12.0000	−	12.0000i	0.463600	−	0.463600i
$671$		−	20.0000i		−	0.772091i
$672$	−12.0000	+	12.0000i	−0.462910	+	0.462910i
$673$	19.0000			0.732396			0.366198	−	0.930537i	$-0.380659\pi$
$673$	19.0000			0.732396			0.366198	+	0.930537i	$0.380659\pi$
$674$	27.0000	−	27.0000i	1.04000	−	1.04000i
$675$			20.0000i			0.769800i
$676$	−24.0000	−	10.0000i	−0.923077	−	0.384615i
$677$			12.0000i			0.461197i	0.973049	+	0.230599i	$0.0740685\pi$
$677$			12.0000i			0.461197i	−0.973049	+	0.230599i	$0.925932\pi$
$678$	−6.00000	−	6.00000i	−0.230429	−	0.230429i
$679$	−54.0000			−2.07233
$680$	6.00000	+	6.00000i	0.230089	+	0.230089i
$681$			2.00000i			0.0766402i
$682$		0			0
$683$	4.00000			0.153056			0.0765279	−	0.997067i	$-0.475617\pi$
$683$	4.00000			0.153056			0.0765279	+	0.997067i	$0.475617\pi$
$684$		0			0
$685$			2.00000i			0.0764161i
$686$	15.0000	+	15.0000i	0.572703	+	0.572703i
$687$			15.0000i			0.572286i
$688$		−	36.0000i		−	1.37249i
$689$	12.0000	+	18.0000i	0.457164	+	0.685745i
$690$	−6.00000	−	6.00000i	−0.228416	−	0.228416i
$691$	12.0000			0.456502			0.228251	−	0.973602i	$-0.426699\pi$
$691$	12.0000			0.456502			0.228251	+	0.973602i	$0.426699\pi$
$692$	48.0000			1.82469
$693$		−	12.0000i		−	0.455842i
$694$	−27.0000	−	27.0000i	−1.02491	−	1.02491i
$695$		−	9.00000i		−	0.341389i
$696$	12.0000	+	12.0000i	0.454859	+	0.454859i
$697$			30.0000i			1.13633i
$698$	15.0000	−	15.0000i	0.567758	−	0.567758i
$699$		−	9.00000i		−	0.340411i
$700$	24.0000			0.907115
$701$		−	30.0000i		−	1.13308i	−0.824033	−	0.566542i	$-0.808281\pi$
$701$		−	30.0000i		−	1.13308i	0.824033	−	0.566542i	$-0.191719\pi$
$702$	−25.0000	−	5.00000i	−0.943564	−	0.188713i
$703$		0			0
$704$			16.0000i			0.603023i
$705$	7.00000			0.263635
$706$	16.0000	+	16.0000i	0.602168	+	0.602168i
$707$		0			0
$708$	−20.0000			−0.751646
$709$	30.0000			1.12667			0.563337	−	0.826227i	$-0.309517\pi$
$709$	30.0000			1.12667			0.563337	+	0.826227i	$0.309517\pi$
$710$	−5.00000	−	5.00000i	−0.187647	−	0.187647i
$711$		0			0
$712$	8.00000	−	8.00000i	0.299813	−	0.299813i
$713$		0			0
$714$	9.00000	−	9.00000i	0.336817	−	0.336817i
$715$	−6.00000	+	4.00000i	−0.224387	+	0.149592i
$716$	−18.0000			−0.672692
$717$	−19.0000			−0.709568
$718$	4.00000	+	4.00000i	0.149279	+	0.149279i
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$	−8.00000			−0.298142
$721$		−	12.0000i		−	0.446903i
$722$	19.0000	−	19.0000i	0.707107	−	0.707107i
$723$		0			0
$724$		0			0
$725$		−	24.0000i		−	0.891338i
$726$	−7.00000	−	7.00000i	−0.259794	−	0.259794i
$727$	18.0000			0.667583			0.333792	−	0.942647i	$-0.391672\pi$
$727$	18.0000			0.667583			0.333792	+	0.942647i	$0.391672\pi$
$728$	−6.00000	+	30.0000i	−0.222375	+	1.11187i
$729$	−13.0000			−0.481481
$730$	6.00000	+	6.00000i	0.222070	+	0.222070i
$731$			27.0000i			0.998631i
$732$			20.0000i			0.739221i
$733$	−51.0000			−1.88373			−0.941864	−	0.335994i	$-0.890928\pi$
$733$	−51.0000			−1.88373			−0.941864	+	0.335994i	$0.890928\pi$
$734$	−28.0000	+	28.0000i	−1.03350	+	1.03350i
$735$			2.00000i			0.0737711i
$736$	−24.0000	+	24.0000i	−0.884652	+	0.884652i
$737$	−24.0000			−0.884051
$738$	−20.0000	−	20.0000i	−0.736210	−	0.736210i
$739$	30.0000			1.10357			0.551784	−	0.833987i	$-0.313947\pi$
$739$	30.0000			1.10357			0.551784	+	0.833987i	$0.313947\pi$
$740$		−	6.00000i		−	0.220564i
$741$		0			0
$742$	18.0000	−	18.0000i	0.660801	−	0.660801i
$743$			19.0000i			0.697042i	0.937301	+	0.348521i	$0.113316\pi$
$743$			19.0000i			0.697042i	−0.937301	+	0.348521i	$0.886684\pi$
$744$		0			0
$745$	−10.0000			−0.366372
$746$	−4.00000	−	4.00000i	−0.146450	−	0.146450i
$747$	−32.0000			−1.17082
$748$		−	12.0000i		−	0.438763i
$749$	36.0000			1.31541
$750$	−9.00000	−	9.00000i	−0.328634	−	0.328634i
$751$	22.0000			0.802791			0.401396	−	0.915905i	$-0.368525\pi$
$751$	22.0000			0.802791			0.401396	+	0.915905i	$0.368525\pi$
$752$		−	28.0000i		−	1.02105i
$753$		0			0
$754$	30.0000	+	6.00000i	1.09254	+	0.218507i
$755$			15.0000i			0.545906i
$756$			30.0000i			1.09109i
$757$		−	38.0000i		−	1.38113i	−0.723269	−	0.690567i	$-0.757361\pi$
$757$		−	38.0000i		−	1.38113i	0.723269	−	0.690567i	$-0.242639\pi$
$758$	30.0000	−	30.0000i	1.08965	−	1.08965i
$759$			12.0000i			0.435572i
$760$		0			0
$761$		−	10.0000i		−	0.362500i	−0.983437	−	0.181250i	$-0.941986\pi$
$761$		−	10.0000i		−	0.362500i	0.983437	−	0.181250i	$-0.0580143\pi$
$762$	18.0000	+	18.0000i	0.652071	+	0.652071i
$763$		−	45.0000i		−	1.62911i
$764$			36.0000i			1.30243i
$765$	6.00000			0.216930
$766$	1.00000	+	1.00000i	0.0361315	+	0.0361315i
$767$	−30.0000	+	20.0000i	−1.08324	+	0.722158i
$768$		−	16.0000i		−	0.577350i
$769$			36.0000i			1.29819i	0.760706	+	0.649097i	$0.224853\pi$
$769$			36.0000i			1.29819i	−0.760706	+	0.649097i	$0.775147\pi$
$770$	6.00000	+	6.00000i	0.216225	+	0.216225i
$771$		−	3.00000i		−	0.108042i
$772$	48.0000			1.72756
$773$	49.0000			1.76241			0.881204	−	0.472737i	$-0.156734\pi$
$773$	49.0000			1.76241			0.881204	+	0.472737i	$0.156734\pi$
$774$	−18.0000	−	18.0000i	−0.646997	−	0.646997i
$775$		0			0
$776$	36.0000	−	36.0000i	1.29232	−	1.29232i
$777$	−9.00000			−0.322873
$778$	−6.00000	−	6.00000i	−0.215110	−	0.215110i
$779$		0			0
$780$	6.00000	−	4.00000i	0.214834	−	0.143223i
$781$			10.0000i			0.357828i
$782$	18.0000	−	18.0000i	0.643679	−	0.643679i
$783$	30.0000			1.07211
$784$	8.00000			0.285714
$785$		−	18.0000i		−	0.642448i
$786$	15.0000	−	15.0000i	0.535032	−	0.535032i
$787$	18.0000			0.641631			0.320815	−	0.947142i	$-0.396043\pi$
$787$	18.0000			0.641631			0.320815	+	0.947142i	$0.396043\pi$
$788$		−	46.0000i		−	1.63868i
$789$			6.00000i			0.213606i
$790$		0			0
$791$			18.0000i			0.640006i
$792$	8.00000	+	8.00000i	0.284268	+	0.284268i
$793$	20.0000	+	30.0000i	0.710221	+	1.06533i
$794$	−18.0000	+	18.0000i	−0.638796	+	0.638796i
$795$	−6.00000			−0.212798
$796$		0			0
$797$			12.0000i			0.425062i	0.977154	+	0.212531i	$0.0681706\pi$
$797$			12.0000i			0.425062i	−0.977154	+	0.212531i	$0.931829\pi$
$798$		0			0
$799$			21.0000i			0.742927i
$800$	−16.0000	+	16.0000i	−0.565685	+	0.565685i
$801$		−	8.00000i		−	0.282666i
$802$	−10.0000	−	10.0000i	−0.353112	−	0.353112i
$803$		−	12.0000i		−	0.423471i
$804$	24.0000			0.846415
$805$			18.0000i			0.634417i
$806$		0			0
$807$	6.00000			0.211210
$808$		0			0
$809$	−15.0000			−0.527372			−0.263686	−	0.964609i	$-0.584938\pi$
$809$	−15.0000			−0.527372			−0.263686	+	0.964609i	$0.584938\pi$
$810$	−1.00000	+	1.00000i	−0.0351364	+	0.0351364i
$811$	12.0000			0.421377			0.210688	−	0.977553i	$-0.432429\pi$
$811$	12.0000			0.421377			0.210688	+	0.977553i	$0.432429\pi$
$812$		−	36.0000i		−	1.26335i
$813$	15.0000			0.526073
$814$	−6.00000	+	6.00000i	−0.210300	+	0.210300i
$815$	−6.00000			−0.210171
$816$			12.0000i			0.420084i
$817$		0			0
$818$	−6.00000	−	6.00000i	−0.209785	−	0.209785i
$819$	12.0000	+	18.0000i	0.419314	+	0.628971i
$820$	20.0000			0.698430
$821$	−43.0000			−1.50071			−0.750355	−	0.661035i	$-0.770118\pi$
$821$	−43.0000			−1.50071			−0.750355	+	0.661035i	$0.770118\pi$
$822$	−2.00000	+	2.00000i	−0.0697580	+	0.0697580i
$823$	14.0000			0.488009			0.244005	−	0.969774i	$-0.421539\pi$
$823$	14.0000			0.488009			0.244005	+	0.969774i	$0.421539\pi$
$824$	8.00000	+	8.00000i	0.278693	+	0.278693i
$825$			8.00000i			0.278524i
$826$	30.0000	+	30.0000i	1.04383	+	1.04383i
$827$	−32.0000			−1.11275			−0.556375	−	0.830932i	$-0.687808\pi$
$827$	−32.0000			−1.11275			−0.556375	+	0.830932i	$0.687808\pi$
$828$			24.0000i			0.834058i
$829$			16.0000i			0.555703i	0.960624	+	0.277851i	$0.0896223\pi$
$829$			16.0000i			0.555703i	−0.960624	+	0.277851i	$0.910378\pi$
$830$	16.0000	−	16.0000i	0.555368	−	0.555368i
$831$	−18.0000			−0.624413
$832$	−16.0000	−	24.0000i	−0.554700	−	0.832050i
$833$	−6.00000			−0.207888
$834$	9.00000	−	9.00000i	0.311645	−	0.311645i
$835$		−	8.00000i		−	0.276851i
$836$		0			0
$837$		0			0
$838$	9.00000	+	9.00000i	0.310900	+	0.310900i
$839$			16.0000i			0.552381i	0.961103	+	0.276191i	$0.0890721\pi$
$839$			16.0000i			0.552381i	−0.961103	+	0.276191i	$0.910928\pi$
$840$	−6.00000	−	6.00000i	−0.207020	−	0.207020i
$841$	−7.00000			−0.241379
$842$	3.00000	−	3.00000i	0.103387	−	0.103387i
$843$	−20.0000			−0.688837
$844$	10.0000			0.344214
$845$	5.00000	−	12.0000i	0.172005	−	0.412813i
$846$	−14.0000	−	14.0000i	−0.481330	−	0.481330i
$847$			21.0000i			0.721569i
$848$			24.0000i			0.824163i
$849$	4.00000			0.137280
$850$	12.0000	−	12.0000i	0.411597	−	0.411597i
$851$	−18.0000			−0.617032
$852$		−	10.0000i		−	0.342594i
$853$	9.00000			0.308154			0.154077	−	0.988059i	$-0.450760\pi$
$853$	9.00000			0.308154			0.154077	+	0.988059i	$0.450760\pi$
$854$	30.0000	−	30.0000i	1.02658	−	1.02658i
$855$		0			0
$856$	−24.0000	+	24.0000i	−0.820303	+	0.820303i
$857$	−42.0000			−1.43469			−0.717346	−	0.696717i	$-0.754643\pi$
$857$	−42.0000			−1.43469			−0.717346	+	0.696717i	$0.754643\pi$
$858$	−10.0000	−	2.00000i	−0.341394	−	0.0682789i
$859$		−	4.00000i		−	0.136478i	−0.997669	−	0.0682391i	$-0.978262\pi$
$859$		−	4.00000i		−	0.136478i	0.997669	−	0.0682391i	$-0.0217381\pi$
$860$	18.0000			0.613795
$861$		−	30.0000i		−	1.02240i
$862$	−5.00000	−	5.00000i	−0.170301	−	0.170301i
$863$			49.0000i			1.66798i	0.551780	+	0.833990i	$0.313949\pi$
$863$			49.0000i			1.66798i	−0.551780	+	0.833990i	$0.686051\pi$
$864$	−20.0000	−	20.0000i	−0.680414	−	0.680414i
$865$			24.0000i			0.816024i
$866$	−9.00000	+	9.00000i	−0.305832	+	0.305832i
$867$			8.00000i			0.271694i
$868$		0			0
$869$		0			0
$870$	−6.00000	+	6.00000i	−0.203419	+	0.203419i
$871$	36.0000	−	24.0000i	1.21981	−	0.813209i
$872$	30.0000	+	30.0000i	1.01593	+	1.01593i
$873$		−	36.0000i		−	1.21842i
$874$		0			0
$875$			27.0000i			0.912767i
$876$			12.0000i			0.405442i
$877$	33.0000			1.11433			0.557165	−	0.830402i	$-0.311889\pi$
$877$	33.0000			1.11433			0.557165	+	0.830402i	$0.311889\pi$
$878$	10.0000	−	10.0000i	0.337484	−	0.337484i
$879$			1.00000i			0.0337292i
$880$	−8.00000			−0.269680
$881$	−33.0000			−1.11180			−0.555899	−	0.831250i	$-0.687626\pi$
$881$	−33.0000			−1.11180			−0.555899	+	0.831250i	$0.687626\pi$
$882$	4.00000	−	4.00000i	0.134687	−	0.134687i
$883$			29.0000i			0.975928i	0.872864	+	0.487964i	$0.162260\pi$
$883$			29.0000i			0.975928i	−0.872864	+	0.487964i	$0.837740\pi$
$884$	12.0000	+	18.0000i	0.403604	+	0.605406i
$885$		−	10.0000i		−	0.336146i
$886$	−9.00000	−	9.00000i	−0.302361	−	0.302361i
$887$	−42.0000			−1.41022			−0.705111	−	0.709097i	$-0.749103\pi$
$887$	−42.0000			−1.41022			−0.705111	+	0.709097i	$0.749103\pi$
$888$	6.00000	−	6.00000i	0.201347	−	0.201347i
$889$		−	54.0000i		−	1.81110i
$890$	4.00000	+	4.00000i	0.134080	+	0.134080i
$891$	2.00000			0.0670025
$892$	−42.0000			−1.40626
$893$		0			0
$894$	−10.0000	−	10.0000i	−0.334450	−	0.334450i
$895$		−	9.00000i		−	0.300837i
$896$	−24.0000	+	24.0000i	−0.801784	+	0.801784i
$897$	−12.0000	−	18.0000i	−0.400668	−	0.601003i
$898$	14.0000	+	14.0000i	0.467186	+	0.467186i
$899$		0			0
$900$			16.0000i			0.533333i
$901$		−	18.0000i		−	0.599667i
$902$	−20.0000	−	20.0000i	−0.665927	−	0.665927i
$903$		−	27.0000i		−	0.898504i
$904$	−12.0000	−	12.0000i	−0.399114	−	0.399114i
$905$		0			0
$906$	−15.0000	+	15.0000i	−0.498342	+	0.498342i
$907$			17.0000i			0.564476i	0.959344	+	0.282238i	$0.0910767\pi$
$907$			17.0000i			0.564476i	−0.959344	+	0.282238i	$0.908923\pi$
$908$			4.00000i			0.132745i
$909$		0			0
$910$	−15.0000	−	3.00000i	−0.497245	−	0.0994490i
$911$	42.0000			1.39152			0.695761	−	0.718273i	$-0.255067\pi$
$911$	42.0000			1.39152			0.695761	+	0.718273i	$0.255067\pi$
$912$		0			0
$913$	−32.0000			−1.05905
$914$	−12.0000	−	12.0000i	−0.396925	−	0.396925i
$915$	−10.0000			−0.330590
$916$			30.0000i			0.991228i
$917$	−45.0000			−1.48603
$918$	15.0000	+	15.0000i	0.495074	+	0.495074i
$919$		0			0			−	1.00000i	$-0.5\pi$
$919$		0			0				1.00000i	$0.5\pi$
$920$	−12.0000	−	12.0000i	−0.395628	−	0.395628i
$921$		−	18.0000i		−	0.593120i
$922$	−7.00000	+	7.00000i	−0.230533	+	0.230533i
$923$	−10.0000	−	15.0000i	−0.329154	−	0.493731i
$924$			12.0000i			0.394771i
$925$	−12.0000			−0.394558
$926$	−24.0000	−	24.0000i	−0.788689	−	0.788689i
$927$	8.00000			0.262754
$928$	24.0000	+	24.0000i	0.787839	+	0.787839i
$929$			56.0000i			1.83730i	0.395072	+	0.918650i	$0.370720\pi$
$929$			56.0000i			1.83730i	−0.395072	+	0.918650i	$0.629280\pi$
$930$		0			0
$931$		0			0
$932$		−	18.0000i		−	0.589610i
$933$		−	12.0000i		−	0.392862i
$934$	−12.0000	−	12.0000i	−0.392652	−	0.392652i
$935$	6.00000			0.196221
$936$	−20.0000	−	4.00000i	−0.653720	−	0.130744i
$937$	18.0000			0.588034			0.294017	−	0.955800i	$-0.405008\pi$
$937$	18.0000			0.588034			0.294017	+	0.955800i	$0.405008\pi$
$938$	−36.0000	−	36.0000i	−1.17544	−	1.17544i
$939$		−	9.00000i		−	0.293704i
$940$	14.0000			0.456630
$941$	17.0000			0.554184			0.277092	−	0.960843i	$-0.410629\pi$
$941$	17.0000			0.554184			0.277092	+	0.960843i	$0.410629\pi$
$942$	18.0000	−	18.0000i	0.586472	−	0.586472i
$943$		−	60.0000i		−	1.95387i
$944$	−40.0000			−1.30189
$945$	−15.0000			−0.487950
$946$	−18.0000	−	18.0000i	−0.585230	−	0.585230i
$947$	−2.00000			−0.0649913			−0.0324956	−	0.999472i	$-0.510346\pi$
$947$	−2.00000			−0.0649913			−0.0324956	+	0.999472i	$0.510346\pi$
$948$		0			0
$949$	12.0000	+	18.0000i	0.389536	+	0.584305i
$950$		0			0
$951$			2.00000i			0.0648544i
$952$	18.0000	−	18.0000i	0.583383	−	0.583383i
$953$	−21.0000			−0.680257			−0.340128	−	0.940379i	$-0.610471\pi$
$953$	−21.0000			−0.680257			−0.340128	+	0.940379i	$0.610471\pi$
$954$	12.0000	+	12.0000i	0.388514	+	0.388514i
$955$	−18.0000			−0.582466
$956$	−38.0000			−1.22901
$957$	12.0000			0.387905
$958$	29.0000	+	29.0000i	0.936947	+	0.936947i
$959$	6.00000			0.193750
$960$	8.00000			0.258199
$961$	31.0000			1.00000
$962$	3.00000	−	15.0000i	0.0967239	−	0.483619i
$963$			24.0000i			0.773389i
$964$		0			0
$965$			24.0000i			0.772587i
$966$	−18.0000	+	18.0000i	−0.579141	+	0.579141i
$967$			57.0000i			1.83300i	0.400039	+	0.916498i	$0.368997\pi$
$967$			57.0000i			1.83300i	−0.400039	+	0.916498i	$0.631003\pi$
$968$	−14.0000	−	14.0000i	−0.449977	−	0.449977i
$969$		0			0
$970$	18.0000	+	18.0000i	0.577945	+	0.577945i
$971$			45.0000i			1.44412i	0.691831	+	0.722059i	$0.256804\pi$
$971$			45.0000i			1.44412i	−0.691831	+	0.722059i	$0.743196\pi$
$972$	−32.0000			−1.02640
$973$	−27.0000			−0.865580
$974$	−12.0000	−	12.0000i	−0.384505	−	0.384505i
$975$	−8.00000	−	12.0000i	−0.256205	−	0.384308i
$976$			40.0000i			1.28037i
$977$		−	28.0000i		−	0.895799i	−0.894084	−	0.447900i	$-0.852172\pi$
$977$		−	28.0000i		−	0.895799i	0.894084	−	0.447900i	$-0.147828\pi$
$978$	−6.00000	−	6.00000i	−0.191859	−	0.191859i
$979$		−	8.00000i		−	0.255681i
$980$			4.00000i			0.127775i
$981$	30.0000			0.957826
$982$	15.0000	+	15.0000i	0.478669	+	0.478669i
$983$		−	11.0000i		−	0.350846i	−0.984493	−	0.175423i	$-0.943871\pi$
$983$		−	11.0000i		−	0.350846i	0.984493	−	0.175423i	$-0.0561292\pi$
$984$	20.0000	+	20.0000i	0.637577	+	0.637577i
$985$	23.0000			0.732841
$986$	−18.0000	−	18.0000i	−0.573237	−	0.573237i
$987$		−	21.0000i		−	0.668437i
$988$		0			0
$989$		−	54.0000i		−	1.71710i
$990$	−4.00000	+	4.00000i	−0.127128	+	0.127128i
$991$	52.0000			1.65183			0.825917	−	0.563791i	$-0.190658\pi$
$991$	52.0000			1.65183			0.825917	+	0.563791i	$0.190658\pi$
$992$		0			0
$993$		−	12.0000i		−	0.380808i
$994$	−15.0000	+	15.0000i	−0.475771	+	0.475771i
$995$		0			0
$996$	32.0000			1.01396
$997$		−	18.0000i		−	0.570066i	−0.958518	−	0.285033i	$-0.907995\pi$
$997$		−	18.0000i		−	0.570066i	0.958518	−	0.285033i	$-0.0920045\pi$
$998$	−30.0000	+	30.0000i	−0.949633	+	0.949633i
$999$		−	15.0000i		−	0.474579i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	104.2.e.a.77.2	yes	2
3.2	odd	2		936.2.m.c.181.1		2
4.3	odd	2		416.2.e.b.337.2		2
8.3	odd	2		416.2.e.a.337.1		2
8.5	even	2		104.2.e.b.77.2	yes	2
12.11	even	2		3744.2.m.b.1585.2		2
13.12	even	2		104.2.e.b.77.1	yes	2
24.5	odd	2		936.2.m.b.181.1		2
24.11	even	2		3744.2.m.c.1585.2		2
39.38	odd	2		936.2.m.b.181.2		2
52.51	odd	2		416.2.e.a.337.2		2
104.51	odd	2		416.2.e.b.337.1		2
104.77	even	2	inner	104.2.e.a.77.1	✓	2
156.155	even	2		3744.2.m.c.1585.1		2
312.77	odd	2		936.2.m.c.181.2		2
312.155	even	2		3744.2.m.b.1585.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
104.2.e.a.77.1	✓	2	104.77	even	2	inner
104.2.e.a.77.2	yes	2	1.1	even	1	trivial
104.2.e.b.77.1	yes	2	13.12	even	2
104.2.e.b.77.2	yes	2	8.5	even	2
416.2.e.a.337.1		2	8.3	odd	2
416.2.e.a.337.2		2	52.51	odd	2
416.2.e.b.337.1		2	104.51	odd	2
416.2.e.b.337.2		2	4.3	odd	2
936.2.m.b.181.1		2	24.5	odd	2
936.2.m.b.181.2		2	39.38	odd	2
936.2.m.c.181.1		2	3.2	odd	2
936.2.m.c.181.2		2	312.77	odd	2
3744.2.m.b.1585.1		2	312.155	even	2
3744.2.m.b.1585.2		2	12.11	even	2
3744.2.m.c.1585.1		2	156.155	even	2
3744.2.m.c.1585.2		2	24.11	even	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 104 = 2^{3} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	104.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.00000 + 1.00000i) q^{2} -1.00000i q^{3} -2.00000i q^{4} +1.00000 q^{5} +(1.00000 + 1.00000i) q^{6} -3.00000i q^{7} +(2.00000 + 2.00000i) q^{8} +2.00000 q^{9} +(-1.00000 + 1.00000i) q^{10} +2.00000 q^{11} -2.00000 q^{12} +(-3.00000 + 2.00000i) q^{13} +(3.00000 + 3.00000i) q^{14} -1.00000i q^{15} -4.00000 q^{16} +3.00000 q^{17} +(-2.00000 + 2.00000i) q^{18} -2.00000i q^{20} -3.00000 q^{21} +(-2.00000 + 2.00000i) q^{22} -6.00000 q^{23} +(2.00000 - 2.00000i) q^{24} -4.00000 q^{25} +(1.00000 - 5.00000i) q^{26} -5.00000i q^{27} -6.00000 q^{28} +6.00000i q^{29} +(1.00000 + 1.00000i) q^{30} +(4.00000 - 4.00000i) q^{32} -2.00000i q^{33} +(-3.00000 + 3.00000i) q^{34} -3.00000i q^{35} -4.00000i q^{36} +3.00000 q^{37} +(2.00000 + 3.00000i) q^{39} +(2.00000 + 2.00000i) q^{40} +10.0000i q^{41} +(3.00000 - 3.00000i) q^{42} +9.00000i q^{43} -4.00000i q^{44} +2.00000 q^{45} +(6.00000 - 6.00000i) q^{46} +7.00000i q^{47} +4.00000i q^{48} -2.00000 q^{49} +(4.00000 - 4.00000i) q^{50} -3.00000i q^{51} +(4.00000 + 6.00000i) q^{52} -6.00000i q^{53} +(5.00000 + 5.00000i) q^{54} +2.00000 q^{55} +(6.00000 - 6.00000i) q^{56} +(-6.00000 - 6.00000i) q^{58} +10.0000 q^{59} -2.00000 q^{60} -10.0000i q^{61} -6.00000i q^{63} +8.00000i q^{64} +(-3.00000 + 2.00000i) q^{65} +(2.00000 + 2.00000i) q^{66} -12.0000 q^{67} -6.00000i q^{68} +6.00000i q^{69} +(3.00000 + 3.00000i) q^{70} +5.00000i q^{71} +(4.00000 + 4.00000i) q^{72} -6.00000i q^{73} +(-3.00000 + 3.00000i) q^{74} +4.00000i q^{75} -6.00000i q^{77} +(-5.00000 - 1.00000i) q^{78} -4.00000 q^{80} +1.00000 q^{81} +(-10.0000 - 10.0000i) q^{82} -16.0000 q^{83} +6.00000i q^{84} +3.00000 q^{85} +(-9.00000 - 9.00000i) q^{86} +6.00000 q^{87} +(4.00000 + 4.00000i) q^{88} -4.00000i q^{89} +(-2.00000 + 2.00000i) q^{90} +(6.00000 + 9.00000i) q^{91} +12.0000i q^{92} +(-7.00000 - 7.00000i) q^{94} +(-4.00000 - 4.00000i) q^{96} -18.0000i q^{97} +(2.00000 - 2.00000i) q^{98} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{5} + 2 q^{6} + 4 q^{8} + 4 q^{9} - 2 q^{10} + 4 q^{11} - 4 q^{12} - 6 q^{13} + 6 q^{14} - 8 q^{16} + 6 q^{17} - 4 q^{18} - 6 q^{21} - 4 q^{22} - 12 q^{23} + 4 q^{24} - 8 q^{25} + 2 q^{26}+ \cdots + 8 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^5 + 2 * q^6 + 4 * q^8 + 4 * q^9 - 2 * q^10 + 4 * q^11 - 4 * q^12 - 6 * q^13 + 6 * q^14 - 8 * q^16 + 6 * q^17 - 4 * q^18 - 6 * q^21 - 4 * q^22 - 12 * q^23 + 4 * q^24 - 8 * q^25 + 2 * q^26 - 12 * q^28 + 2 * q^30 + 8 * q^32 - 6 * q^34 + 6 * q^37 + 4 * q^39 + 4 * q^40 + 6 * q^42 + 4 * q^45 + 12 * q^46 - 4 * q^49 + 8 * q^50 + 8 * q^52 + 10 * q^54 + 4 * q^55 + 12 * q^56 - 12 * q^58 + 20 * q^59 - 4 * q^60 - 6 * q^65 + 4 * q^66 - 24 * q^67 + 6 * q^70 + 8 * q^72 - 6 * q^74 - 10 * q^78 - 8 * q^80 + 2 * q^81 - 20 * q^82 - 32 * q^83 + 6 * q^85 - 18 * q^86 + 12 * q^87 + 8 * q^88 - 4 * q^90 + 12 * q^91 - 14 * q^94 - 8 * q^96 + 4 * q^98 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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