LMFDB - Embedded newform 110.2.b.a.89.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [110,2,Mod(89,110)] 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("110.89"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(110, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 110 = 2 \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	110.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,-2,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$0.878354422234$
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			89.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	110.89
Dual form			110.2.b.a.89.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +(-2.00000 + 1.00000i) q^{5} -1.00000 q^{6} +3.00000i q^{7} -1.00000i q^{8} +2.00000 q^{9} +(-1.00000 - 2.00000i) q^{10} +1.00000 q^{11} -1.00000i q^{12} -4.00000i q^{13} -3.00000 q^{14} +(-1.00000 - 2.00000i) q^{15} +1.00000 q^{16} +3.00000i q^{17} +2.00000i q^{18} +5.00000 q^{19} +(2.00000 - 1.00000i) q^{20} -3.00000 q^{21} +1.00000i q^{22} -4.00000i q^{23} +1.00000 q^{24} +(3.00000 - 4.00000i) q^{25} +4.00000 q^{26} +5.00000i q^{27} -3.00000i q^{28} -5.00000 q^{29} +(2.00000 - 1.00000i) q^{30} +7.00000 q^{31} +1.00000i q^{32} +1.00000i q^{33} -3.00000 q^{34} +(-3.00000 - 6.00000i) q^{35} -2.00000 q^{36} -7.00000i q^{37} +5.00000i q^{38} +4.00000 q^{39} +(1.00000 + 2.00000i) q^{40} -8.00000 q^{41} -3.00000i q^{42} +6.00000i q^{43} -1.00000 q^{44} +(-4.00000 + 2.00000i) q^{45} +4.00000 q^{46} +8.00000i q^{47} +1.00000i q^{48} -2.00000 q^{49} +(4.00000 + 3.00000i) q^{50} -3.00000 q^{51} +4.00000i q^{52} -9.00000i q^{53} -5.00000 q^{54} +(-2.00000 + 1.00000i) q^{55} +3.00000 q^{56} +5.00000i q^{57} -5.00000i q^{58} +(1.00000 + 2.00000i) q^{60} -13.0000 q^{61} +7.00000i q^{62} +6.00000i q^{63} -1.00000 q^{64} +(4.00000 + 8.00000i) q^{65} -1.00000 q^{66} -12.0000i q^{67} -3.00000i q^{68} +4.00000 q^{69} +(6.00000 - 3.00000i) q^{70} -3.00000 q^{71} -2.00000i q^{72} +6.00000i q^{73} +7.00000 q^{74} +(4.00000 + 3.00000i) q^{75} -5.00000 q^{76} +3.00000i q^{77} +4.00000i q^{78} +(-2.00000 + 1.00000i) q^{80} +1.00000 q^{81} -8.00000i q^{82} -4.00000i q^{83} +3.00000 q^{84} +(-3.00000 - 6.00000i) q^{85} -6.00000 q^{86} -5.00000i q^{87} -1.00000i q^{88} +15.0000 q^{89} +(-2.00000 - 4.00000i) q^{90} +12.0000 q^{91} +4.00000i q^{92} +7.00000i q^{93} -8.00000 q^{94} +(-10.0000 + 5.00000i) q^{95} -1.00000 q^{96} -12.0000i q^{97} -2.00000i q^{98} +2.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} - 4 q^{5} - 2 q^{6} + 4 q^{9} - 2 q^{10} + 2 q^{11} - 6 q^{14} - 2 q^{15} + 2 q^{16} + 10 q^{19} + 4 q^{20} - 6 q^{21} + 2 q^{24} + 6 q^{25} + 8 q^{26} - 10 q^{29} + 4 q^{30} + 14 q^{31} - 6 q^{34}+ \cdots + 4 q^{99}+O(q^{100}) $ 2 * q - 2 * q^4 - 4 * q^5 - 2 * q^6 + 4 * q^9 - 2 * q^10 + 2 * q^11 - 6 * q^14 - 2 * q^15 + 2 * q^16 + 10 * q^19 + 4 * q^20 - 6 * q^21 + 2 * q^24 + 6 * q^25 + 8 * q^26 - 10 * q^29 + 4 * q^30 + 14 * q^31 - 6 * q^34 - 6 * q^35 - 4 * q^36 + 8 * q^39 + 2 * q^40 - 16 * q^41 - 2 * q^44 - 8 * q^45 + 8 * q^46 - 4 * q^49 + 8 * q^50 - 6 * q^51 - 10 * q^54 - 4 * q^55 + 6 * q^56 + 2 * q^60 - 26 * q^61 - 2 * q^64 + 8 * q^65 - 2 * q^66 + 8 * q^69 + 12 * q^70 - 6 * q^71 + 14 * q^74 + 8 * q^75 - 10 * q^76 - 4 * q^80 + 2 * q^81 + 6 * q^84 - 6 * q^85 - 12 * q^86 + 30 * q^89 - 4 * q^90 + 24 * q^91 - 16 * q^94 - 20 * q^95 - 2 * q^96 + 4 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/110\mathbb{Z}\right)^\times$.

$n$	$67$	$101$
$\chi(n)$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$			1.00000i			0.707107i
$2$			1.00000i			0.707107i
$3$			1.00000i			0.577350i	0.957427	+	0.288675i	$0.0932147\pi$
$3$			1.00000i			0.577350i	−0.957427	+	0.288675i	$0.906785\pi$
$4$	−1.00000			−0.500000
$5$	−2.00000	+	1.00000i	−0.894427	+	0.447214i
$5$	−2.00000	+	1.00000i	−0.894427	+	0.447214i
$6$	−1.00000			−0.408248
$7$			3.00000i			1.13389i	0.823754	+	0.566947i	$0.191875\pi$
$7$			3.00000i			1.13389i	−0.823754	+	0.566947i	$0.808125\pi$
$8$		−	1.00000i		−	0.353553i
$9$	2.00000			0.666667
$10$	−1.00000	−	2.00000i	−0.316228	−	0.632456i
$11$	1.00000			0.301511
$11$	1.00000			0.301511
$12$		−	1.00000i		−	0.288675i
$13$		−	4.00000i		−	1.10940i	−0.832050	−	0.554700i	$-0.812833\pi$
$13$		−	4.00000i		−	1.10940i	0.832050	−	0.554700i	$-0.187167\pi$
$14$	−3.00000			−0.801784
$15$	−1.00000	−	2.00000i	−0.258199	−	0.516398i
$16$	1.00000			0.250000
$17$			3.00000i			0.727607i	0.931476	+	0.363803i	$0.118522\pi$
$17$			3.00000i			0.727607i	−0.931476	+	0.363803i	$0.881478\pi$
$18$			2.00000i			0.471405i
$19$	5.00000			1.14708			0.573539	−	0.819178i	$-0.305570\pi$
$19$	5.00000			1.14708			0.573539	+	0.819178i	$0.305570\pi$
$20$	2.00000	−	1.00000i	0.447214	−	0.223607i
$21$	−3.00000			−0.654654
$22$			1.00000i			0.213201i
$23$		−	4.00000i		−	0.834058i	−0.908893	−	0.417029i	$-0.863071\pi$
$23$		−	4.00000i		−	0.834058i	0.908893	−	0.417029i	$-0.136929\pi$
$24$	1.00000			0.204124
$25$	3.00000	−	4.00000i	0.600000	−	0.800000i
$26$	4.00000			0.784465
$27$			5.00000i			0.962250i
$28$		−	3.00000i		−	0.566947i
$29$	−5.00000			−0.928477			−0.464238	−	0.885710i	$-0.653672\pi$
$29$	−5.00000			−0.928477			−0.464238	+	0.885710i	$0.653672\pi$
$30$	2.00000	−	1.00000i	0.365148	−	0.182574i
$31$	7.00000			1.25724			0.628619	−	0.777714i	$-0.283621\pi$
$31$	7.00000			1.25724			0.628619	+	0.777714i	$0.283621\pi$
$32$			1.00000i			0.176777i
$33$			1.00000i			0.174078i
$34$	−3.00000			−0.514496
$35$	−3.00000	−	6.00000i	−0.507093	−	1.01419i
$36$	−2.00000			−0.333333
$37$		−	7.00000i		−	1.15079i	−0.817875	−	0.575396i	$-0.804848\pi$
$37$		−	7.00000i		−	1.15079i	0.817875	−	0.575396i	$-0.195152\pi$
$38$			5.00000i			0.811107i
$39$	4.00000			0.640513
$40$	1.00000	+	2.00000i	0.158114	+	0.316228i
$41$	−8.00000			−1.24939			−0.624695	−	0.780869i	$-0.714777\pi$
$41$	−8.00000			−1.24939			−0.624695	+	0.780869i	$0.714777\pi$
$42$		−	3.00000i		−	0.462910i
$43$			6.00000i			0.914991i	0.889212	+	0.457496i	$0.151253\pi$
$43$			6.00000i			0.914991i	−0.889212	+	0.457496i	$0.848747\pi$
$44$	−1.00000			−0.150756
$45$	−4.00000	+	2.00000i	−0.596285	+	0.298142i
$46$	4.00000			0.589768
$47$			8.00000i			1.16692i	0.812142	+	0.583460i	$0.198301\pi$
$47$			8.00000i			1.16692i	−0.812142	+	0.583460i	$0.801699\pi$
$48$			1.00000i			0.144338i
$49$	−2.00000			−0.285714
$50$	4.00000	+	3.00000i	0.565685	+	0.424264i
$51$	−3.00000			−0.420084
$52$			4.00000i			0.554700i
$53$		−	9.00000i		−	1.23625i	−0.786082	−	0.618123i	$-0.787894\pi$
$53$		−	9.00000i		−	1.23625i	0.786082	−	0.618123i	$-0.212106\pi$
$54$	−5.00000			−0.680414
$55$	−2.00000	+	1.00000i	−0.269680	+	0.134840i
$56$	3.00000			0.400892
$57$			5.00000i			0.662266i
$58$		−	5.00000i		−	0.656532i
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$	1.00000	+	2.00000i	0.129099	+	0.258199i
$61$	−13.0000			−1.66448			−0.832240	−	0.554416i	$-0.812942\pi$
$61$	−13.0000			−1.66448			−0.832240	+	0.554416i	$0.812942\pi$
$62$			7.00000i			0.889001i
$63$			6.00000i			0.755929i
$64$	−1.00000			−0.125000
$65$	4.00000	+	8.00000i	0.496139	+	0.992278i
$66$	−1.00000			−0.123091
$67$		−	12.0000i		−	1.46603i	−0.680211	−	0.733017i	$-0.738112\pi$
$67$		−	12.0000i		−	1.46603i	0.680211	−	0.733017i	$-0.261888\pi$
$68$		−	3.00000i		−	0.363803i
$69$	4.00000			0.481543
$70$	6.00000	−	3.00000i	0.717137	−	0.358569i
$71$	−3.00000			−0.356034			−0.178017	−	0.984027i	$-0.556968\pi$
$71$	−3.00000			−0.356034			−0.178017	+	0.984027i	$0.556968\pi$
$72$		−	2.00000i		−	0.235702i
$73$			6.00000i			0.702247i	0.936329	+	0.351123i	$0.114200\pi$
$73$			6.00000i			0.702247i	−0.936329	+	0.351123i	$0.885800\pi$
$74$	7.00000			0.813733
$75$	4.00000	+	3.00000i	0.461880	+	0.346410i
$76$	−5.00000			−0.573539
$77$			3.00000i			0.341882i
$78$			4.00000i			0.452911i
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$	−2.00000	+	1.00000i	−0.223607	+	0.111803i
$81$	1.00000			0.111111
$82$		−	8.00000i		−	0.883452i
$83$		−	4.00000i		−	0.439057i	−0.975606	−	0.219529i	$-0.929548\pi$
$83$		−	4.00000i		−	0.439057i	0.975606	−	0.219529i	$-0.0704519\pi$
$84$	3.00000			0.327327
$85$	−3.00000	−	6.00000i	−0.325396	−	0.650791i
$86$	−6.00000			−0.646997
$87$		−	5.00000i		−	0.536056i
$88$		−	1.00000i		−	0.106600i
$89$	15.0000			1.59000			0.794998	−	0.606612i	$-0.207472\pi$
$89$	15.0000			1.59000			0.794998	+	0.606612i	$0.207472\pi$
$90$	−2.00000	−	4.00000i	−0.210819	−	0.421637i
$91$	12.0000			1.25794
$92$			4.00000i			0.417029i
$93$			7.00000i			0.725866i
$94$	−8.00000			−0.825137
$95$	−10.0000	+	5.00000i	−1.02598	+	0.512989i
$96$	−1.00000			−0.102062
$97$		−	12.0000i		−	1.21842i	−0.793011	−	0.609208i	$-0.791488\pi$
$97$		−	12.0000i		−	1.21842i	0.793011	−	0.609208i	$-0.208512\pi$
$98$		−	2.00000i		−	0.202031i
$99$	2.00000			0.201008
$100$	−3.00000	+	4.00000i	−0.300000	+	0.400000i
$101$	2.00000			0.199007			0.0995037	−	0.995037i	$-0.468274\pi$
$101$	2.00000			0.199007			0.0995037	+	0.995037i	$0.468274\pi$
$102$		−	3.00000i		−	0.297044i
$103$		−	4.00000i		−	0.394132i	−0.980390	−	0.197066i	$-0.936859\pi$
$103$		−	4.00000i		−	0.394132i	0.980390	−	0.197066i	$-0.0631413\pi$
$104$	−4.00000			−0.392232
$105$	6.00000	−	3.00000i	0.585540	−	0.292770i
$106$	9.00000			0.874157
$107$		−	12.0000i		−	1.16008i	−0.814587	−	0.580042i	$-0.803036\pi$
$107$		−	12.0000i		−	1.16008i	0.814587	−	0.580042i	$-0.196964\pi$
$108$		−	5.00000i		−	0.481125i
$109$	10.0000			0.957826			0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000			0.957826			0.478913	+	0.877862i	$0.341031\pi$
$110$	−1.00000	−	2.00000i	−0.0953463	−	0.190693i
$111$	7.00000			0.664411
$112$			3.00000i			0.283473i
$113$			6.00000i			0.564433i	0.959351	+	0.282216i	$0.0910696\pi$
$113$			6.00000i			0.564433i	−0.959351	+	0.282216i	$0.908930\pi$
$114$	−5.00000			−0.468293
$115$	4.00000	+	8.00000i	0.373002	+	0.746004i
$116$	5.00000			0.464238
$117$		−	8.00000i		−	0.739600i
$118$		0			0
$119$	−9.00000			−0.825029
$120$	−2.00000	+	1.00000i	−0.182574	+	0.0912871i
$121$	1.00000			0.0909091
$122$		−	13.0000i		−	1.17696i
$123$		−	8.00000i		−	0.721336i
$124$	−7.00000			−0.628619
$125$	−2.00000	+	11.0000i	−0.178885	+	0.983870i
$126$	−6.00000			−0.534522
$127$			8.00000i			0.709885i	0.934888	+	0.354943i	$0.115500\pi$
$127$			8.00000i			0.709885i	−0.934888	+	0.354943i	$0.884500\pi$
$128$		−	1.00000i		−	0.0883883i
$129$	−6.00000			−0.528271
$130$	−8.00000	+	4.00000i	−0.701646	+	0.350823i
$131$	7.00000			0.611593			0.305796	−	0.952097i	$-0.401077\pi$
$131$	7.00000			0.611593			0.305796	+	0.952097i	$0.401077\pi$
$132$		−	1.00000i		−	0.0870388i
$133$			15.0000i			1.30066i
$134$	12.0000			1.03664
$135$	−5.00000	−	10.0000i	−0.430331	−	0.860663i
$136$	3.00000			0.257248
$137$		−	12.0000i		−	1.02523i	−0.858619	−	0.512615i	$-0.828677\pi$
$137$		−	12.0000i		−	1.02523i	0.858619	−	0.512615i	$-0.171323\pi$
$138$			4.00000i			0.340503i
$139$	−20.0000			−1.69638			−0.848189	−	0.529694i	$-0.822307\pi$
$139$	−20.0000			−1.69638			−0.848189	+	0.529694i	$0.822307\pi$
$140$	3.00000	+	6.00000i	0.253546	+	0.507093i
$141$	−8.00000			−0.673722
$142$		−	3.00000i		−	0.251754i
$143$		−	4.00000i		−	0.334497i
$144$	2.00000			0.166667
$145$	10.0000	−	5.00000i	0.830455	−	0.415227i
$146$	−6.00000			−0.496564
$147$		−	2.00000i		−	0.164957i
$148$			7.00000i			0.575396i
$149$	5.00000			0.409616			0.204808	−	0.978802i	$-0.434343\pi$
$149$	5.00000			0.409616			0.204808	+	0.978802i	$0.434343\pi$
$150$	−3.00000	+	4.00000i	−0.244949	+	0.326599i
$151$	2.00000			0.162758			0.0813788	−	0.996683i	$-0.474068\pi$
$151$	2.00000			0.162758			0.0813788	+	0.996683i	$0.474068\pi$
$152$		−	5.00000i		−	0.405554i
$153$			6.00000i			0.485071i
$154$	−3.00000			−0.241747
$155$	−14.0000	+	7.00000i	−1.12451	+	0.562254i
$156$	−4.00000			−0.320256
$157$			13.0000i			1.03751i	0.854922	+	0.518756i	$0.173605\pi$
$157$			13.0000i			1.03751i	−0.854922	+	0.518756i	$0.826395\pi$
$158$		0			0
$159$	9.00000			0.713746
$160$	−1.00000	−	2.00000i	−0.0790569	−	0.158114i
$161$	12.0000			0.945732
$162$			1.00000i			0.0785674i
$163$			21.0000i			1.64485i	0.568876	+	0.822423i	$0.307379\pi$
$163$			21.0000i			1.64485i	−0.568876	+	0.822423i	$0.692621\pi$
$164$	8.00000			0.624695
$165$	−1.00000	−	2.00000i	−0.0778499	−	0.155700i
$166$	4.00000			0.310460
$167$			23.0000i			1.77979i	0.456162	+	0.889897i	$0.349224\pi$
$167$			23.0000i			1.77979i	−0.456162	+	0.889897i	$0.650776\pi$
$168$			3.00000i			0.231455i
$169$	−3.00000			−0.230769
$170$	6.00000	−	3.00000i	0.460179	−	0.230089i
$171$	10.0000			0.764719
$172$		−	6.00000i		−	0.457496i
$173$		−	14.0000i		−	1.06440i	−0.846619	−	0.532200i	$-0.821365\pi$
$173$		−	14.0000i		−	1.06440i	0.846619	−	0.532200i	$-0.178635\pi$
$174$	5.00000			0.379049
$175$	12.0000	+	9.00000i	0.907115	+	0.680336i
$176$	1.00000			0.0753778
$177$		0			0
$178$			15.0000i			1.12430i
$179$	−10.0000			−0.747435			−0.373718	−	0.927543i	$-0.621917\pi$
$179$	−10.0000			−0.747435			−0.373718	+	0.927543i	$0.621917\pi$
$180$	4.00000	−	2.00000i	0.298142	−	0.149071i
$181$	−18.0000			−1.33793			−0.668965	−	0.743294i	$-0.733262\pi$
$181$	−18.0000			−1.33793			−0.668965	+	0.743294i	$0.733262\pi$
$182$			12.0000i			0.889499i
$183$		−	13.0000i		−	0.960988i
$184$	−4.00000			−0.294884
$185$	7.00000	+	14.0000i	0.514650	+	1.02930i
$186$	−7.00000			−0.513265
$187$			3.00000i			0.219382i
$188$		−	8.00000i		−	0.583460i
$189$	−15.0000			−1.09109
$190$	−5.00000	−	10.0000i	−0.362738	−	0.725476i
$191$	−8.00000			−0.578860			−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000			−0.578860			−0.289430	+	0.957199i	$0.593466\pi$
$192$		−	1.00000i		−	0.0721688i
$193$			1.00000i			0.0719816i	0.999352	+	0.0359908i	$0.0114587\pi$
$193$			1.00000i			0.0719816i	−0.999352	+	0.0359908i	$0.988541\pi$
$194$	12.0000			0.861550
$195$	−8.00000	+	4.00000i	−0.572892	+	0.286446i
$196$	2.00000			0.142857
$197$		−	22.0000i		−	1.56744i	−0.621117	−	0.783718i	$-0.713321\pi$
$197$		−	22.0000i		−	1.56744i	0.621117	−	0.783718i	$-0.286679\pi$
$198$			2.00000i			0.142134i
$199$	−25.0000			−1.77220			−0.886102	−	0.463491i	$-0.846597\pi$
$199$	−25.0000			−1.77220			−0.886102	+	0.463491i	$0.846597\pi$
$200$	−4.00000	−	3.00000i	−0.282843	−	0.212132i
$201$	12.0000			0.846415
$202$			2.00000i			0.140720i
$203$		−	15.0000i		−	1.05279i
$204$	3.00000			0.210042
$205$	16.0000	−	8.00000i	1.11749	−	0.558744i
$206$	4.00000			0.278693
$207$		−	8.00000i		−	0.556038i
$208$		−	4.00000i		−	0.277350i
$209$	5.00000			0.345857
$210$	3.00000	+	6.00000i	0.207020	+	0.414039i
$211$	−13.0000			−0.894957			−0.447478	−	0.894295i	$-0.647678\pi$
$211$	−13.0000			−0.894957			−0.447478	+	0.894295i	$0.647678\pi$
$212$			9.00000i			0.618123i
$213$		−	3.00000i		−	0.205557i
$214$	12.0000			0.820303
$215$	−6.00000	−	12.0000i	−0.409197	−	0.818393i
$216$	5.00000			0.340207
$217$			21.0000i			1.42557i
$218$			10.0000i			0.677285i
$219$	−6.00000			−0.405442
$220$	2.00000	−	1.00000i	0.134840	−	0.0674200i
$221$	12.0000			0.807207
$222$			7.00000i			0.469809i
$223$		−	14.0000i		−	0.937509i	−0.883328	−	0.468755i	$-0.844703\pi$
$223$		−	14.0000i		−	0.937509i	0.883328	−	0.468755i	$-0.155297\pi$
$224$	−3.00000			−0.200446
$225$	6.00000	−	8.00000i	0.400000	−	0.533333i
$226$	−6.00000			−0.399114
$227$		−	12.0000i		−	0.796468i	−0.917284	−	0.398234i	$-0.869623\pi$
$227$		−	12.0000i		−	0.796468i	0.917284	−	0.398234i	$-0.130377\pi$
$228$		−	5.00000i		−	0.331133i
$229$		0			0			−	1.00000i	$-0.5\pi$
$229$		0			0				1.00000i	$0.5\pi$
$230$	−8.00000	+	4.00000i	−0.527504	+	0.263752i
$231$	−3.00000			−0.197386
$232$			5.00000i			0.328266i
$233$		−	19.0000i		−	1.24473i	−0.782727	−	0.622366i	$-0.786172\pi$
$233$		−	19.0000i		−	1.24473i	0.782727	−	0.622366i	$-0.213828\pi$
$234$	8.00000			0.522976
$235$	−8.00000	−	16.0000i	−0.521862	−	1.04372i
$236$		0			0
$237$		0			0
$238$		−	9.00000i		−	0.583383i
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$	−1.00000	−	2.00000i	−0.0645497	−	0.129099i
$241$	−8.00000			−0.515325			−0.257663	−	0.966235i	$-0.582952\pi$
$241$	−8.00000			−0.515325			−0.257663	+	0.966235i	$0.582952\pi$
$242$			1.00000i			0.0642824i
$243$			16.0000i			1.02640i
$244$	13.0000			0.832240
$245$	4.00000	−	2.00000i	0.255551	−	0.127775i
$246$	8.00000			0.510061
$247$		−	20.0000i		−	1.27257i
$248$		−	7.00000i		−	0.444500i
$249$	4.00000			0.253490
$250$	−11.0000	−	2.00000i	−0.695701	−	0.126491i
$251$	2.00000			0.126239			0.0631194	−	0.998006i	$-0.479895\pi$
$251$	2.00000			0.126239			0.0631194	+	0.998006i	$0.479895\pi$
$252$		−	6.00000i		−	0.377964i
$253$		−	4.00000i		−	0.251478i
$254$	−8.00000			−0.501965
$255$	6.00000	−	3.00000i	0.375735	−	0.187867i
$256$	1.00000			0.0625000
$257$			28.0000i			1.74659i	0.487190	+	0.873296i	$0.338022\pi$
$257$			28.0000i			1.74659i	−0.487190	+	0.873296i	$0.661978\pi$
$258$		−	6.00000i		−	0.373544i
$259$	21.0000			1.30488
$260$	−4.00000	−	8.00000i	−0.248069	−	0.496139i
$261$	−10.0000			−0.618984
$262$			7.00000i			0.432461i
$263$		−	9.00000i		−	0.554964i	−0.960731	−	0.277482i	$-0.910500\pi$
$263$		−	9.00000i		−	0.554964i	0.960731	−	0.277482i	$-0.0894999\pi$
$264$	1.00000			0.0615457
$265$	9.00000	+	18.0000i	0.552866	+	1.10573i
$266$	−15.0000			−0.919709
$267$			15.0000i			0.917985i
$268$			12.0000i			0.733017i
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$	10.0000	−	5.00000i	0.608581	−	0.304290i
$271$	22.0000			1.33640			0.668202	−	0.743980i	$-0.267064\pi$
$271$	22.0000			1.33640			0.668202	+	0.743980i	$0.267064\pi$
$272$			3.00000i			0.181902i
$273$			12.0000i			0.726273i
$274$	12.0000			0.724947
$275$	3.00000	−	4.00000i	0.180907	−	0.241209i
$276$	−4.00000			−0.240772
$277$			28.0000i			1.68236i	0.540758	+	0.841178i	$0.318138\pi$
$277$			28.0000i			1.68236i	−0.540758	+	0.841178i	$0.681862\pi$
$278$		−	20.0000i		−	1.19952i
$279$	14.0000			0.838158
$280$	−6.00000	+	3.00000i	−0.358569	+	0.179284i
$281$	−18.0000			−1.07379			−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000			−1.07379			−0.536895	+	0.843649i	$0.680403\pi$
$282$		−	8.00000i		−	0.476393i
$283$			16.0000i			0.951101i	0.879688	+	0.475551i	$0.157751\pi$
$283$			16.0000i			0.951101i	−0.879688	+	0.475551i	$0.842249\pi$
$284$	3.00000			0.178017
$285$	−5.00000	−	10.0000i	−0.296174	−	0.592349i
$286$	4.00000			0.236525
$287$		−	24.0000i		−	1.41668i
$288$			2.00000i			0.117851i
$289$	8.00000			0.470588
$290$	5.00000	+	10.0000i	0.293610	+	0.587220i
$291$	12.0000			0.703452
$292$		−	6.00000i		−	0.351123i
$293$			16.0000i			0.934730i	0.884064	+	0.467365i	$0.154797\pi$
$293$			16.0000i			0.934730i	−0.884064	+	0.467365i	$0.845203\pi$
$294$	2.00000			0.116642
$295$		0			0
$296$	−7.00000			−0.406867
$297$			5.00000i			0.290129i
$298$			5.00000i			0.289642i
$299$	−16.0000			−0.925304
$300$	−4.00000	−	3.00000i	−0.230940	−	0.173205i
$301$	−18.0000			−1.03750
$302$			2.00000i			0.115087i
$303$			2.00000i			0.114897i
$304$	5.00000			0.286770
$305$	26.0000	−	13.0000i	1.48876	−	0.744378i
$306$	−6.00000			−0.342997
$307$			8.00000i			0.456584i	0.973593	+	0.228292i	$0.0733141\pi$
$307$			8.00000i			0.456584i	−0.973593	+	0.228292i	$0.926686\pi$
$308$		−	3.00000i		−	0.170941i
$309$	4.00000			0.227552
$310$	−7.00000	−	14.0000i	−0.397573	−	0.795147i
$311$	−13.0000			−0.737162			−0.368581	−	0.929596i	$-0.620156\pi$
$311$	−13.0000			−0.737162			−0.368581	+	0.929596i	$0.620156\pi$
$312$		−	4.00000i		−	0.226455i
$313$		−	14.0000i		−	0.791327i	−0.918396	−	0.395663i	$-0.870515\pi$
$313$		−	14.0000i		−	0.791327i	0.918396	−	0.395663i	$-0.129485\pi$
$314$	−13.0000			−0.733632
$315$	−6.00000	−	12.0000i	−0.338062	−	0.676123i
$316$		0			0
$317$			23.0000i			1.29181i	0.763418	+	0.645904i	$0.223520\pi$
$317$			23.0000i			1.29181i	−0.763418	+	0.645904i	$0.776480\pi$
$318$			9.00000i			0.504695i
$319$	−5.00000			−0.279946
$320$	2.00000	−	1.00000i	0.111803	−	0.0559017i
$321$	12.0000			0.669775
$322$			12.0000i			0.668734i
$323$			15.0000i			0.834622i
$324$	−1.00000			−0.0555556
$325$	−16.0000	−	12.0000i	−0.887520	−	0.665640i
$326$	−21.0000			−1.16308
$327$			10.0000i			0.553001i
$328$			8.00000i			0.441726i
$329$	−24.0000			−1.32316
$330$	2.00000	−	1.00000i	0.110096	−	0.0550482i
$331$	−18.0000			−0.989369			−0.494685	−	0.869072i	$-0.664716\pi$
$331$	−18.0000			−0.989369			−0.494685	+	0.869072i	$0.664716\pi$
$332$			4.00000i			0.219529i
$333$		−	14.0000i		−	0.767195i
$334$	−23.0000			−1.25850
$335$	12.0000	+	24.0000i	0.655630	+	1.31126i
$336$	−3.00000			−0.163663
$337$		−	7.00000i		−	0.381314i	−0.981657	−	0.190657i	$-0.938938\pi$
$337$		−	7.00000i		−	0.381314i	0.981657	−	0.190657i	$-0.0610619\pi$
$338$		−	3.00000i		−	0.163178i
$339$	−6.00000			−0.325875
$340$	3.00000	+	6.00000i	0.162698	+	0.325396i
$341$	7.00000			0.379071
$342$			10.0000i			0.540738i
$343$			15.0000i			0.809924i
$344$	6.00000			0.323498
$345$	−8.00000	+	4.00000i	−0.430706	+	0.215353i
$346$	14.0000			0.752645
$347$			18.0000i			0.966291i	0.875540	+	0.483145i	$0.160506\pi$
$347$			18.0000i			0.966291i	−0.875540	+	0.483145i	$0.839494\pi$
$348$			5.00000i			0.268028i
$349$	10.0000			0.535288			0.267644	−	0.963518i	$-0.413755\pi$
$349$	10.0000			0.535288			0.267644	+	0.963518i	$0.413755\pi$
$350$	−9.00000	+	12.0000i	−0.481070	+	0.641427i
$351$	20.0000			1.06752
$352$			1.00000i			0.0533002i
$353$		−	14.0000i		−	0.745145i	−0.928003	−	0.372572i	$-0.878476\pi$
$353$		−	14.0000i		−	0.745145i	0.928003	−	0.372572i	$-0.121524\pi$
$354$		0			0
$355$	6.00000	−	3.00000i	0.318447	−	0.159223i
$356$	−15.0000			−0.794998
$357$		−	9.00000i		−	0.476331i
$358$		−	10.0000i		−	0.528516i
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$	2.00000	+	4.00000i	0.105409	+	0.210819i
$361$	6.00000			0.315789
$362$		−	18.0000i		−	0.946059i
$363$			1.00000i			0.0524864i
$364$	−12.0000			−0.628971
$365$	−6.00000	−	12.0000i	−0.314054	−	0.628109i
$366$	13.0000			0.679521
$367$			8.00000i			0.417597i	0.977959	+	0.208798i	$0.0669552\pi$
$367$			8.00000i			0.417597i	−0.977959	+	0.208798i	$0.933045\pi$
$368$		−	4.00000i		−	0.208514i
$369$	−16.0000			−0.832927
$370$	−14.0000	+	7.00000i	−0.727825	+	0.363913i
$371$	27.0000			1.40177
$372$		−	7.00000i		−	0.362933i
$373$		−	4.00000i		−	0.207112i	−0.994624	−	0.103556i	$-0.966978\pi$
$373$		−	4.00000i		−	0.207112i	0.994624	−	0.103556i	$-0.0330221\pi$
$374$	−3.00000			−0.155126
$375$	−11.0000	−	2.00000i	−0.568038	−	0.103280i
$376$	8.00000			0.412568
$377$			20.0000i			1.03005i
$378$		−	15.0000i		−	0.771517i
$379$	10.0000			0.513665			0.256833	−	0.966456i	$-0.417321\pi$
$379$	10.0000			0.513665			0.256833	+	0.966456i	$0.417321\pi$
$380$	10.0000	−	5.00000i	0.512989	−	0.256495i
$381$	−8.00000			−0.409852
$382$		−	8.00000i		−	0.409316i
$383$			6.00000i			0.306586i	0.988181	+	0.153293i	$0.0489878\pi$
$383$			6.00000i			0.306586i	−0.988181	+	0.153293i	$0.951012\pi$
$384$	1.00000			0.0510310
$385$	−3.00000	−	6.00000i	−0.152894	−	0.305788i
$386$	−1.00000			−0.0508987
$387$			12.0000i			0.609994i
$388$			12.0000i			0.609208i
$389$	30.0000			1.52106			0.760530	−	0.649303i	$-0.224939\pi$
$389$	30.0000			1.52106			0.760530	+	0.649303i	$0.224939\pi$
$390$	−4.00000	−	8.00000i	−0.202548	−	0.405096i
$391$	12.0000			0.606866
$392$			2.00000i			0.101015i
$393$			7.00000i			0.353103i
$394$	22.0000			1.10834
$395$		0			0
$396$	−2.00000			−0.100504
$397$		−	22.0000i		−	1.10415i	−0.833795	−	0.552074i	$-0.813837\pi$
$397$		−	22.0000i		−	1.10415i	0.833795	−	0.552074i	$-0.186163\pi$
$398$		−	25.0000i		−	1.25314i
$399$	−15.0000			−0.750939
$400$	3.00000	−	4.00000i	0.150000	−	0.200000i
$401$	27.0000			1.34832			0.674158	−	0.738587i	$-0.264507\pi$
$401$	27.0000			1.34832			0.674158	+	0.738587i	$0.264507\pi$
$402$			12.0000i			0.598506i
$403$		−	28.0000i		−	1.39478i
$404$	−2.00000			−0.0995037
$405$	−2.00000	+	1.00000i	−0.0993808	+	0.0496904i
$406$	15.0000			0.744438
$407$		−	7.00000i		−	0.346977i
$408$			3.00000i			0.148522i
$409$	−20.0000			−0.988936			−0.494468	−	0.869196i	$-0.664637\pi$
$409$	−20.0000			−0.988936			−0.494468	+	0.869196i	$0.664637\pi$
$410$	8.00000	+	16.0000i	0.395092	+	0.790184i
$411$	12.0000			0.591916
$412$			4.00000i			0.197066i
$413$		0			0
$414$	8.00000			0.393179
$415$	4.00000	+	8.00000i	0.196352	+	0.392705i
$416$	4.00000			0.196116
$417$		−	20.0000i		−	0.979404i
$418$			5.00000i			0.244558i
$419$	20.0000			0.977064			0.488532	−	0.872546i	$-0.337533\pi$
$419$	20.0000			0.977064			0.488532	+	0.872546i	$0.337533\pi$
$420$	−6.00000	+	3.00000i	−0.292770	+	0.146385i
$421$	−28.0000			−1.36464			−0.682318	−	0.731055i	$-0.739028\pi$
$421$	−28.0000			−1.36464			−0.682318	+	0.731055i	$0.739028\pi$
$422$		−	13.0000i		−	0.632830i
$423$			16.0000i			0.777947i
$424$	−9.00000			−0.437079
$425$	12.0000	+	9.00000i	0.582086	+	0.436564i
$426$	3.00000			0.145350
$427$		−	39.0000i		−	1.88734i
$428$			12.0000i			0.580042i
$429$	4.00000			0.193122
$430$	12.0000	−	6.00000i	0.578691	−	0.289346i
$431$	32.0000			1.54139			0.770693	−	0.637207i	$-0.219910\pi$
$431$	32.0000			1.54139			0.770693	+	0.637207i	$0.219910\pi$
$432$			5.00000i			0.240563i
$433$		−	14.0000i		−	0.672797i	−0.941720	−	0.336399i	$-0.890791\pi$
$433$		−	14.0000i		−	0.672797i	0.941720	−	0.336399i	$-0.109209\pi$
$434$	−21.0000			−1.00803
$435$	5.00000	+	10.0000i	0.239732	+	0.479463i
$436$	−10.0000			−0.478913
$437$		−	20.0000i		−	0.956730i
$438$		−	6.00000i		−	0.286691i
$439$	−10.0000			−0.477274			−0.238637	−	0.971109i	$-0.576701\pi$
$439$	−10.0000			−0.477274			−0.238637	+	0.971109i	$0.576701\pi$
$440$	1.00000	+	2.00000i	0.0476731	+	0.0953463i
$441$	−4.00000			−0.190476
$442$			12.0000i			0.570782i
$443$		−	4.00000i		−	0.190046i	−0.995475	−	0.0950229i	$-0.969708\pi$
$443$		−	4.00000i		−	0.190046i	0.995475	−	0.0950229i	$-0.0302924\pi$
$444$	−7.00000			−0.332205
$445$	−30.0000	+	15.0000i	−1.42214	+	0.711068i
$446$	14.0000			0.662919
$447$			5.00000i			0.236492i
$448$		−	3.00000i		−	0.141737i
$449$	−30.0000			−1.41579			−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000			−1.41579			−0.707894	+	0.706319i	$0.750354\pi$
$450$	8.00000	+	6.00000i	0.377124	+	0.282843i
$451$	−8.00000			−0.376705
$452$		−	6.00000i		−	0.282216i
$453$			2.00000i			0.0939682i
$454$	12.0000			0.563188
$455$	−24.0000	+	12.0000i	−1.12514	+	0.562569i
$456$	5.00000			0.234146
$457$		−	27.0000i		−	1.26301i	−0.775373	−	0.631503i	$-0.782438\pi$
$457$		−	27.0000i		−	1.26301i	0.775373	−	0.631503i	$-0.217562\pi$
$458$		0			0
$459$	−15.0000			−0.700140
$460$	−4.00000	−	8.00000i	−0.186501	−	0.373002i
$461$	−13.0000			−0.605470			−0.302735	−	0.953075i	$-0.597900\pi$
$461$	−13.0000			−0.605470			−0.302735	+	0.953075i	$0.597900\pi$
$462$		−	3.00000i		−	0.139573i
$463$			16.0000i			0.743583i	0.928316	+	0.371792i	$0.121256\pi$
$463$			16.0000i			0.743583i	−0.928316	+	0.371792i	$0.878744\pi$
$464$	−5.00000			−0.232119
$465$	−7.00000	−	14.0000i	−0.324617	−	0.649234i
$466$	19.0000			0.880158
$467$			3.00000i			0.138823i	0.997588	+	0.0694117i	$0.0221122\pi$
$467$			3.00000i			0.138823i	−0.997588	+	0.0694117i	$0.977888\pi$
$468$			8.00000i			0.369800i
$469$	36.0000			1.66233
$470$	16.0000	−	8.00000i	0.738025	−	0.369012i
$471$	−13.0000			−0.599008
$472$		0			0
$473$			6.00000i			0.275880i
$474$		0			0
$475$	15.0000	−	20.0000i	0.688247	−	0.917663i
$476$	9.00000			0.412514
$477$		−	18.0000i		−	0.824163i
$478$		0			0
$479$	10.0000			0.456912			0.228456	−	0.973554i	$-0.426632\pi$
$479$	10.0000			0.456912			0.228456	+	0.973554i	$0.426632\pi$
$480$	2.00000	−	1.00000i	0.0912871	−	0.0456435i
$481$	−28.0000			−1.27669
$482$		−	8.00000i		−	0.364390i
$483$			12.0000i			0.546019i
$484$	−1.00000			−0.0454545
$485$	12.0000	+	24.0000i	0.544892	+	1.08978i
$486$	−16.0000			−0.725775
$487$		−	22.0000i		−	0.996915i	−0.866914	−	0.498458i	$-0.833900\pi$
$487$		−	22.0000i		−	0.996915i	0.866914	−	0.498458i	$-0.166100\pi$
$488$			13.0000i			0.588482i
$489$	−21.0000			−0.949653
$490$	2.00000	+	4.00000i	0.0903508	+	0.180702i
$491$	−33.0000			−1.48927			−0.744635	−	0.667472i	$-0.767376\pi$
$491$	−33.0000			−1.48927			−0.744635	+	0.667472i	$0.767376\pi$
$492$			8.00000i			0.360668i
$493$		−	15.0000i		−	0.675566i
$494$	20.0000			0.899843
$495$	−4.00000	+	2.00000i	−0.179787	+	0.0898933i
$496$	7.00000			0.314309
$497$		−	9.00000i		−	0.403705i
$498$			4.00000i			0.179244i
$499$	40.0000			1.79065			0.895323	−	0.445418i	$-0.146945\pi$
$499$	40.0000			1.79065			0.895323	+	0.445418i	$0.146945\pi$
$500$	2.00000	−	11.0000i	0.0894427	−	0.491935i
$501$	−23.0000			−1.02756
$502$			2.00000i			0.0892644i
$503$		−	24.0000i		−	1.07011i	−0.844818	−	0.535054i	$-0.820291\pi$
$503$		−	24.0000i		−	1.07011i	0.844818	−	0.535054i	$-0.179709\pi$
$504$	6.00000			0.267261
$505$	−4.00000	+	2.00000i	−0.177998	+	0.0889988i
$506$	4.00000			0.177822
$507$		−	3.00000i		−	0.133235i
$508$		−	8.00000i		−	0.354943i
$509$	20.0000			0.886484			0.443242	−	0.896402i	$-0.353828\pi$
$509$	20.0000			0.886484			0.443242	+	0.896402i	$0.353828\pi$
$510$	3.00000	+	6.00000i	0.132842	+	0.265684i
$511$	−18.0000			−0.796273
$512$			1.00000i			0.0441942i
$513$			25.0000i			1.10378i
$514$	−28.0000			−1.23503
$515$	4.00000	+	8.00000i	0.176261	+	0.352522i
$516$	6.00000			0.264135
$517$			8.00000i			0.351840i
$518$			21.0000i			0.922687i
$519$	14.0000			0.614532
$520$	8.00000	−	4.00000i	0.350823	−	0.175412i
$521$	22.0000			0.963837			0.481919	−	0.876216i	$-0.339940\pi$
$521$	22.0000			0.963837			0.481919	+	0.876216i	$0.339940\pi$
$522$		−	10.0000i		−	0.437688i
$523$			16.0000i			0.699631i	0.936819	+	0.349816i	$0.113756\pi$
$523$			16.0000i			0.699631i	−0.936819	+	0.349816i	$0.886244\pi$
$524$	−7.00000			−0.305796
$525$	−9.00000	+	12.0000i	−0.392792	+	0.523723i
$526$	9.00000			0.392419
$527$			21.0000i			0.914774i
$528$			1.00000i			0.0435194i
$529$	7.00000			0.304348
$530$	−18.0000	+	9.00000i	−0.781870	+	0.390935i
$531$		0			0
$532$		−	15.0000i		−	0.650332i
$533$			32.0000i			1.38607i
$534$	−15.0000			−0.649113
$535$	12.0000	+	24.0000i	0.518805	+	1.03761i
$536$	−12.0000			−0.518321
$537$		−	10.0000i		−	0.431532i
$538$		0			0
$539$	−2.00000			−0.0861461
$540$	5.00000	+	10.0000i	0.215166	+	0.430331i
$541$	17.0000			0.730887			0.365444	−	0.930834i	$-0.380917\pi$
$541$	17.0000			0.730887			0.365444	+	0.930834i	$0.380917\pi$
$542$			22.0000i			0.944981i
$543$		−	18.0000i		−	0.772454i
$544$	−3.00000			−0.128624
$545$	−20.0000	+	10.0000i	−0.856706	+	0.428353i
$546$	−12.0000			−0.513553
$547$			8.00000i			0.342055i	0.985266	+	0.171028i	$0.0547087\pi$
$547$			8.00000i			0.342055i	−0.985266	+	0.171028i	$0.945291\pi$
$548$			12.0000i			0.512615i
$549$	−26.0000			−1.10965
$550$	4.00000	+	3.00000i	0.170561	+	0.127920i
$551$	−25.0000			−1.06504
$552$		−	4.00000i		−	0.170251i
$553$		0			0
$554$	−28.0000			−1.18961
$555$	−14.0000	+	7.00000i	−0.594267	+	0.297133i
$556$	20.0000			0.848189
$557$			28.0000i			1.18640i	0.805056	+	0.593199i	$0.202135\pi$
$557$			28.0000i			1.18640i	−0.805056	+	0.593199i	$0.797865\pi$
$558$			14.0000i			0.592667i
$559$	24.0000			1.01509
$560$	−3.00000	−	6.00000i	−0.126773	−	0.253546i
$561$	−3.00000			−0.126660
$562$		−	18.0000i		−	0.759284i
$563$		−	4.00000i		−	0.168580i	−0.996441	−	0.0842900i	$-0.973138\pi$
$563$		−	4.00000i		−	0.168580i	0.996441	−	0.0842900i	$-0.0268622\pi$
$564$	8.00000			0.336861
$565$	−6.00000	−	12.0000i	−0.252422	−	0.504844i
$566$	−16.0000			−0.672530
$567$			3.00000i			0.125988i
$568$			3.00000i			0.125877i
$569$	−20.0000			−0.838444			−0.419222	−	0.907884i	$-0.637697\pi$
$569$	−20.0000			−0.838444			−0.419222	+	0.907884i	$0.637697\pi$
$570$	10.0000	−	5.00000i	0.418854	−	0.209427i
$571$	−3.00000			−0.125546			−0.0627730	−	0.998028i	$-0.519994\pi$
$571$	−3.00000			−0.125546			−0.0627730	+	0.998028i	$0.519994\pi$
$572$			4.00000i			0.167248i
$573$		−	8.00000i		−	0.334205i
$574$	24.0000			1.00174
$575$	−16.0000	−	12.0000i	−0.667246	−	0.500435i
$576$	−2.00000			−0.0833333
$577$			8.00000i			0.333044i	0.986038	+	0.166522i	$0.0532537\pi$
$577$			8.00000i			0.333044i	−0.986038	+	0.166522i	$0.946746\pi$
$578$			8.00000i			0.332756i
$579$	−1.00000			−0.0415586
$580$	−10.0000	+	5.00000i	−0.415227	+	0.207614i
$581$	12.0000			0.497844
$582$			12.0000i			0.497416i
$583$		−	9.00000i		−	0.372742i
$584$	6.00000			0.248282
$585$	8.00000	+	16.0000i	0.330759	+	0.661519i
$586$	−16.0000			−0.660954
$587$		−	7.00000i		−	0.288921i	−0.989511	−	0.144460i	$-0.953855\pi$
$587$		−	7.00000i		−	0.288921i	0.989511	−	0.144460i	$-0.0461446\pi$
$588$			2.00000i			0.0824786i
$589$	35.0000			1.44215
$590$		0			0
$591$	22.0000			0.904959
$592$		−	7.00000i		−	0.287698i
$593$			6.00000i			0.246390i	0.992382	+	0.123195i	$0.0393141\pi$
$593$			6.00000i			0.246390i	−0.992382	+	0.123195i	$0.960686\pi$
$594$	−5.00000			−0.205152
$595$	18.0000	−	9.00000i	0.737928	−	0.368964i
$596$	−5.00000			−0.204808
$597$		−	25.0000i		−	1.02318i
$598$		−	16.0000i		−	0.654289i
$599$	5.00000			0.204294			0.102147	−	0.994769i	$-0.467429\pi$
$599$	5.00000			0.204294			0.102147	+	0.994769i	$0.467429\pi$
$600$	3.00000	−	4.00000i	0.122474	−	0.163299i
$601$	2.00000			0.0815817			0.0407909	−	0.999168i	$-0.487012\pi$
$601$	2.00000			0.0815817			0.0407909	+	0.999168i	$0.487012\pi$
$602$		−	18.0000i		−	0.733625i
$603$		−	24.0000i		−	0.977356i
$604$	−2.00000			−0.0813788
$605$	−2.00000	+	1.00000i	−0.0813116	+	0.0406558i
$606$	−2.00000			−0.0812444
$607$		−	7.00000i		−	0.284121i	−0.989858	−	0.142061i	$-0.954627\pi$
$607$		−	7.00000i		−	0.284121i	0.989858	−	0.142061i	$-0.0453728\pi$
$608$			5.00000i			0.202777i
$609$	15.0000			0.607831
$610$	13.0000	+	26.0000i	0.526355	+	1.05271i
$611$	32.0000			1.29458
$612$		−	6.00000i		−	0.242536i
$613$			6.00000i			0.242338i	0.992632	+	0.121169i	$0.0386643\pi$
$613$			6.00000i			0.242338i	−0.992632	+	0.121169i	$0.961336\pi$
$614$	−8.00000			−0.322854
$615$	8.00000	+	16.0000i	0.322591	+	0.645182i
$616$	3.00000			0.120873
$617$		−	12.0000i		−	0.483102i	−0.970388	−	0.241551i	$-0.922344\pi$
$617$		−	12.0000i		−	0.483102i	0.970388	−	0.241551i	$-0.0776561\pi$
$618$			4.00000i			0.160904i
$619$	−20.0000			−0.803868			−0.401934	−	0.915669i	$-0.631662\pi$
$619$	−20.0000			−0.803868			−0.401934	+	0.915669i	$0.631662\pi$
$620$	14.0000	−	7.00000i	0.562254	−	0.281127i
$621$	20.0000			0.802572
$622$		−	13.0000i		−	0.521253i
$623$			45.0000i			1.80289i
$624$	4.00000			0.160128
$625$	−7.00000	−	24.0000i	−0.280000	−	0.960000i
$626$	14.0000			0.559553
$627$			5.00000i			0.199681i
$628$		−	13.0000i		−	0.518756i
$629$	21.0000			0.837325
$630$	12.0000	−	6.00000i	0.478091	−	0.239046i
$631$	−43.0000			−1.71180			−0.855901	−	0.517139i	$-0.826997\pi$
$631$	−43.0000			−1.71180			−0.855901	+	0.517139i	$0.826997\pi$
$632$		0			0
$633$		−	13.0000i		−	0.516704i
$634$	−23.0000			−0.913447
$635$	−8.00000	−	16.0000i	−0.317470	−	0.634941i
$636$	−9.00000			−0.356873
$637$			8.00000i			0.316972i
$638$		−	5.00000i		−	0.197952i
$639$	−6.00000			−0.237356
$640$	1.00000	+	2.00000i	0.0395285	+	0.0790569i
$641$	−33.0000			−1.30342			−0.651711	−	0.758468i	$-0.725948\pi$
$641$	−33.0000			−1.30342			−0.651711	+	0.758468i	$0.725948\pi$
$642$			12.0000i			0.473602i
$643$			1.00000i			0.0394362i	0.999806	+	0.0197181i	$0.00627687\pi$
$643$			1.00000i			0.0394362i	−0.999806	+	0.0197181i	$0.993723\pi$
$644$	−12.0000			−0.472866
$645$	12.0000	−	6.00000i	0.472500	−	0.236250i
$646$	−15.0000			−0.590167
$647$		−	12.0000i		−	0.471769i	−0.971781	−	0.235884i	$-0.924201\pi$
$647$		−	12.0000i		−	0.471769i	0.971781	−	0.235884i	$-0.0757987\pi$
$648$		−	1.00000i		−	0.0392837i
$649$		0			0
$650$	12.0000	−	16.0000i	0.470679	−	0.627572i
$651$	−21.0000			−0.823055
$652$		−	21.0000i		−	0.822423i
$653$			21.0000i			0.821794i	0.911682	+	0.410897i	$0.134784\pi$
$653$			21.0000i			0.821794i	−0.911682	+	0.410897i	$0.865216\pi$
$654$	−10.0000			−0.391031
$655$	−14.0000	+	7.00000i	−0.547025	+	0.273513i
$656$	−8.00000			−0.312348
$657$			12.0000i			0.468165i
$658$		−	24.0000i		−	0.935617i
$659$	35.0000			1.36341			0.681703	−	0.731629i	$-0.261240\pi$
$659$	35.0000			1.36341			0.681703	+	0.731629i	$0.261240\pi$
$660$	1.00000	+	2.00000i	0.0389249	+	0.0778499i
$661$	12.0000			0.466746			0.233373	−	0.972387i	$-0.425024\pi$
$661$	12.0000			0.466746			0.233373	+	0.972387i	$0.425024\pi$
$662$		−	18.0000i		−	0.699590i
$663$			12.0000i			0.466041i
$664$	−4.00000			−0.155230
$665$	−15.0000	−	30.0000i	−0.581675	−	1.16335i
$666$	14.0000			0.542489
$667$			20.0000i			0.774403i
$668$		−	23.0000i		−	0.889897i
$669$	14.0000			0.541271
$670$	−24.0000	+	12.0000i	−0.927201	+	0.463600i
$671$	−13.0000			−0.501859
$672$		−	3.00000i		−	0.115728i
$673$			1.00000i			0.0385472i	0.999814	+	0.0192736i	$0.00613535\pi$
$673$			1.00000i			0.0385472i	−0.999814	+	0.0192736i	$0.993865\pi$
$674$	7.00000			0.269630
$675$	20.0000	+	15.0000i	0.769800	+	0.577350i
$676$	3.00000			0.115385
$677$			8.00000i			0.307465i	0.988113	+	0.153732i	$0.0491294\pi$
$677$			8.00000i			0.307465i	−0.988113	+	0.153732i	$0.950871\pi$
$678$		−	6.00000i		−	0.230429i
$679$	36.0000			1.38155
$680$	−6.00000	+	3.00000i	−0.230089	+	0.115045i
$681$	12.0000			0.459841
$682$			7.00000i			0.268044i
$683$		−	49.0000i		−	1.87493i	−0.348076	−	0.937466i	$-0.613165\pi$
$683$		−	49.0000i		−	1.87493i	0.348076	−	0.937466i	$-0.386835\pi$
$684$	−10.0000			−0.382360
$685$	12.0000	+	24.0000i	0.458496	+	0.916993i
$686$	−15.0000			−0.572703
$687$		0			0
$688$			6.00000i			0.228748i
$689$	−36.0000			−1.37149
$690$	−4.00000	−	8.00000i	−0.152277	−	0.304555i
$691$	−8.00000			−0.304334			−0.152167	−	0.988355i	$-0.548625\pi$
$691$	−8.00000			−0.304334			−0.152167	+	0.988355i	$0.548625\pi$
$692$			14.0000i			0.532200i
$693$			6.00000i			0.227921i
$694$	−18.0000			−0.683271
$695$	40.0000	−	20.0000i	1.51729	−	0.758643i
$696$	−5.00000			−0.189525
$697$		−	24.0000i		−	0.909065i
$698$			10.0000i			0.378506i
$699$	19.0000			0.718646
$700$	−12.0000	−	9.00000i	−0.453557	−	0.340168i
$701$	27.0000			1.01978			0.509888	−	0.860241i	$-0.329687\pi$
$701$	27.0000			1.01978			0.509888	+	0.860241i	$0.329687\pi$
$702$			20.0000i			0.754851i
$703$		−	35.0000i		−	1.32005i
$704$	−1.00000			−0.0376889
$705$	16.0000	−	8.00000i	0.602595	−	0.301297i
$706$	14.0000			0.526897
$707$			6.00000i			0.225653i
$708$		0			0
$709$	−10.0000			−0.375558			−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000			−0.375558			−0.187779	+	0.982211i	$0.560129\pi$
$710$	3.00000	+	6.00000i	0.112588	+	0.225176i
$711$		0			0
$712$		−	15.0000i		−	0.562149i
$713$		−	28.0000i		−	1.04861i
$714$	9.00000			0.336817
$715$	4.00000	+	8.00000i	0.149592	+	0.299183i
$716$	10.0000			0.373718
$717$		0			0
$718$		0			0
$719$	−15.0000			−0.559406			−0.279703	−	0.960087i	$-0.590236\pi$
$719$	−15.0000			−0.559406			−0.279703	+	0.960087i	$0.590236\pi$
$720$	−4.00000	+	2.00000i	−0.149071	+	0.0745356i
$721$	12.0000			0.446903
$722$			6.00000i			0.223297i
$723$		−	8.00000i		−	0.297523i
$724$	18.0000			0.668965
$725$	−15.0000	+	20.0000i	−0.557086	+	0.742781i
$726$	−1.00000			−0.0371135
$727$			18.0000i			0.667583i	0.942647	+	0.333792i	$0.108328\pi$
$727$			18.0000i			0.667583i	−0.942647	+	0.333792i	$0.891672\pi$
$728$		−	12.0000i		−	0.444750i
$729$	−13.0000			−0.481481
$730$	12.0000	−	6.00000i	0.444140	−	0.222070i
$731$	−18.0000			−0.665754
$732$			13.0000i			0.480494i
$733$		−	14.0000i		−	0.517102i	−0.965998	−	0.258551i	$-0.916755\pi$
$733$		−	14.0000i		−	0.517102i	0.965998	−	0.258551i	$-0.0832450\pi$
$734$	−8.00000			−0.295285
$735$	2.00000	+	4.00000i	0.0737711	+	0.147542i
$736$	4.00000			0.147442
$737$		−	12.0000i		−	0.442026i
$738$		−	16.0000i		−	0.588968i
$739$	−40.0000			−1.47142			−0.735712	−	0.677295i	$-0.763152\pi$
$739$	−40.0000			−1.47142			−0.735712	+	0.677295i	$0.763152\pi$
$740$	−7.00000	−	14.0000i	−0.257325	−	0.514650i
$741$	20.0000			0.734718
$742$			27.0000i			0.991201i
$743$			1.00000i			0.0366864i	0.999832	+	0.0183432i	$0.00583916\pi$
$743$			1.00000i			0.0366864i	−0.999832	+	0.0183432i	$0.994161\pi$
$744$	7.00000			0.256632
$745$	−10.0000	+	5.00000i	−0.366372	+	0.183186i
$746$	4.00000			0.146450
$747$		−	8.00000i		−	0.292705i
$748$		−	3.00000i		−	0.109691i
$749$	36.0000			1.31541
$750$	2.00000	−	11.0000i	0.0730297	−	0.401663i
$751$	27.0000			0.985244			0.492622	−	0.870243i	$-0.336039\pi$
$751$	27.0000			0.985244			0.492622	+	0.870243i	$0.336039\pi$
$752$			8.00000i			0.291730i
$753$			2.00000i			0.0728841i
$754$	−20.0000			−0.728357
$755$	−4.00000	+	2.00000i	−0.145575	+	0.0727875i
$756$	15.0000			0.545545
$757$			18.0000i			0.654221i	0.944986	+	0.327111i	$0.106075\pi$
$757$			18.0000i			0.654221i	−0.944986	+	0.327111i	$0.893925\pi$
$758$			10.0000i			0.363216i
$759$	4.00000			0.145191
$760$	5.00000	+	10.0000i	0.181369	+	0.362738i
$761$	12.0000			0.435000			0.217500	−	0.976060i	$-0.430210\pi$
$761$	12.0000			0.435000			0.217500	+	0.976060i	$0.430210\pi$
$762$		−	8.00000i		−	0.289809i
$763$			30.0000i			1.08607i
$764$	8.00000			0.289430
$765$	−6.00000	−	12.0000i	−0.216930	−	0.433861i
$766$	−6.00000			−0.216789
$767$		0			0
$768$			1.00000i			0.0360844i
$769$	−30.0000			−1.08183			−0.540914	−	0.841078i	$-0.681921\pi$
$769$	−30.0000			−1.08183			−0.540914	+	0.841078i	$0.681921\pi$
$770$	6.00000	−	3.00000i	0.216225	−	0.108112i
$771$	−28.0000			−1.00840
$772$		−	1.00000i		−	0.0359908i
$773$			11.0000i			0.395643i	0.980238	+	0.197821i	$0.0633866\pi$
$773$			11.0000i			0.395643i	−0.980238	+	0.197821i	$0.936613\pi$
$774$	−12.0000			−0.431331
$775$	21.0000	−	28.0000i	0.754342	−	1.00579i
$776$	−12.0000			−0.430775
$777$			21.0000i			0.753371i
$778$			30.0000i			1.07555i
$779$	−40.0000			−1.43315
$780$	8.00000	−	4.00000i	0.286446	−	0.143223i
$781$	−3.00000			−0.107348
$782$			12.0000i			0.429119i
$783$		−	25.0000i		−	0.893427i
$784$	−2.00000			−0.0714286
$785$	−13.0000	−	26.0000i	−0.463990	−	0.927980i
$786$	−7.00000			−0.249682
$787$			38.0000i			1.35455i	0.735728	+	0.677277i	$0.236840\pi$
$787$			38.0000i			1.35455i	−0.735728	+	0.677277i	$0.763160\pi$
$788$			22.0000i			0.783718i
$789$	9.00000			0.320408
$790$		0			0
$791$	−18.0000			−0.640006
$792$		−	2.00000i		−	0.0710669i
$793$			52.0000i			1.84657i
$794$	22.0000			0.780751
$795$	−18.0000	+	9.00000i	−0.638394	+	0.319197i
$796$	25.0000			0.886102
$797$			18.0000i			0.637593i	0.947823	+	0.318796i	$0.103279\pi$
$797$			18.0000i			0.637593i	−0.947823	+	0.318796i	$0.896721\pi$
$798$		−	15.0000i		−	0.530994i
$799$	−24.0000			−0.849059
$800$	4.00000	+	3.00000i	0.141421	+	0.106066i
$801$	30.0000			1.06000
$802$			27.0000i			0.953403i
$803$			6.00000i			0.211735i
$804$	−12.0000			−0.423207
$805$	−24.0000	+	12.0000i	−0.845889	+	0.422944i
$806$	28.0000			0.986258
$807$		0			0
$808$		−	2.00000i		−	0.0703598i
$809$	−10.0000			−0.351581			−0.175791	−	0.984428i	$-0.556248\pi$
$809$	−10.0000			−0.351581			−0.175791	+	0.984428i	$0.556248\pi$
$810$	−1.00000	−	2.00000i	−0.0351364	−	0.0702728i
$811$	37.0000			1.29925			0.649623	−	0.760257i	$-0.274927\pi$
$811$	37.0000			1.29925			0.649623	+	0.760257i	$0.274927\pi$
$812$			15.0000i			0.526397i
$813$			22.0000i			0.771574i
$814$	7.00000			0.245350
$815$	−21.0000	−	42.0000i	−0.735598	−	1.47120i
$816$	−3.00000			−0.105021
$817$			30.0000i			1.04957i
$818$		−	20.0000i		−	0.699284i
$819$	24.0000			0.838628
$820$	−16.0000	+	8.00000i	−0.558744	+	0.279372i
$821$	22.0000			0.767805			0.383903	−	0.923374i	$-0.374580\pi$
$821$	22.0000			0.767805			0.383903	+	0.923374i	$0.374580\pi$
$822$			12.0000i			0.418548i
$823$			26.0000i			0.906303i	0.891434	+	0.453152i	$0.149700\pi$
$823$			26.0000i			0.906303i	−0.891434	+	0.453152i	$0.850300\pi$
$824$	−4.00000			−0.139347
$825$	4.00000	+	3.00000i	0.139262	+	0.104447i
$826$		0			0
$827$		−	22.0000i		−	0.765015i	−0.923952	−	0.382507i	$-0.875061\pi$
$827$		−	22.0000i		−	0.765015i	0.923952	−	0.382507i	$-0.124939\pi$
$828$			8.00000i			0.278019i
$829$	20.0000			0.694629			0.347314	−	0.937749i	$-0.387094\pi$
$829$	20.0000			0.694629			0.347314	+	0.937749i	$0.387094\pi$
$830$	−8.00000	+	4.00000i	−0.277684	+	0.138842i
$831$	−28.0000			−0.971309
$832$			4.00000i			0.138675i
$833$		−	6.00000i		−	0.207888i
$834$	20.0000			0.692543
$835$	−23.0000	−	46.0000i	−0.795948	−	1.59190i
$836$	−5.00000			−0.172929
$837$			35.0000i			1.20978i
$838$			20.0000i			0.690889i
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$	−3.00000	−	6.00000i	−0.103510	−	0.207020i
$841$	−4.00000			−0.137931
$842$		−	28.0000i		−	0.964944i
$843$		−	18.0000i		−	0.619953i
$844$	13.0000			0.447478
$845$	6.00000	−	3.00000i	0.206406	−	0.103203i
$846$	−16.0000			−0.550091
$847$			3.00000i			0.103081i
$848$		−	9.00000i		−	0.309061i
$849$	−16.0000			−0.549119
$850$	−9.00000	+	12.0000i	−0.308697	+	0.411597i
$851$	−28.0000			−0.959828
$852$			3.00000i			0.102778i
$853$		−	44.0000i		−	1.50653i	−0.657716	−	0.753266i	$-0.728477\pi$
$853$		−	44.0000i		−	1.50653i	0.657716	−	0.753266i	$-0.271523\pi$
$854$	39.0000			1.33455
$855$	−20.0000	+	10.0000i	−0.683986	+	0.341993i
$856$	−12.0000			−0.410152
$857$		−	17.0000i		−	0.580709i	−0.956919	−	0.290354i	$-0.906227\pi$
$857$		−	17.0000i		−	0.580709i	0.956919	−	0.290354i	$-0.0937732\pi$
$858$			4.00000i			0.136558i
$859$	10.0000			0.341196			0.170598	−	0.985341i	$-0.445430\pi$
$859$	10.0000			0.341196			0.170598	+	0.985341i	$0.445430\pi$
$860$	6.00000	+	12.0000i	0.204598	+	0.409197i
$861$	24.0000			0.817918
$862$			32.0000i			1.08992i
$863$			36.0000i			1.22545i	0.790295	+	0.612727i	$0.209928\pi$
$863$			36.0000i			1.22545i	−0.790295	+	0.612727i	$0.790072\pi$
$864$	−5.00000			−0.170103
$865$	14.0000	+	28.0000i	0.476014	+	0.952029i
$866$	14.0000			0.475739
$867$			8.00000i			0.271694i
$868$		−	21.0000i		−	0.712786i
$869$		0			0
$870$	−10.0000	+	5.00000i	−0.339032	+	0.169516i
$871$	−48.0000			−1.62642
$872$		−	10.0000i		−	0.338643i
$873$		−	24.0000i		−	0.812277i
$874$	20.0000			0.676510
$875$	−33.0000	−	6.00000i	−1.11560	−	0.202837i
$876$	6.00000			0.202721
$877$		−	2.00000i		−	0.0675352i	−0.999430	−	0.0337676i	$-0.989249\pi$
$877$		−	2.00000i		−	0.0675352i	0.999430	−	0.0337676i	$-0.0107506\pi$
$878$		−	10.0000i		−	0.337484i
$879$	−16.0000			−0.539667
$880$	−2.00000	+	1.00000i	−0.0674200	+	0.0337100i
$881$	−18.0000			−0.606435			−0.303218	−	0.952921i	$-0.598061\pi$
$881$	−18.0000			−0.606435			−0.303218	+	0.952921i	$0.598061\pi$
$882$		−	4.00000i		−	0.134687i
$883$		−	9.00000i		−	0.302874i	−0.988467	−	0.151437i	$-0.951610\pi$
$883$		−	9.00000i		−	0.302874i	0.988467	−	0.151437i	$-0.0483901\pi$
$884$	−12.0000			−0.403604
$885$		0			0
$886$	4.00000			0.134383
$887$		−	32.0000i		−	1.07445i	−0.843437	−	0.537227i	$-0.819472\pi$
$887$		−	32.0000i		−	1.07445i	0.843437	−	0.537227i	$-0.180528\pi$
$888$		−	7.00000i		−	0.234905i
$889$	−24.0000			−0.804934
$890$	−15.0000	−	30.0000i	−0.502801	−	1.00560i
$891$	1.00000			0.0335013
$892$			14.0000i			0.468755i
$893$			40.0000i			1.33855i
$894$	−5.00000			−0.167225
$895$	20.0000	−	10.0000i	0.668526	−	0.334263i
$896$	3.00000			0.100223
$897$		−	16.0000i		−	0.534224i
$898$		−	30.0000i		−	1.00111i
$899$	−35.0000			−1.16732
$900$	−6.00000	+	8.00000i	−0.200000	+	0.266667i
$901$	27.0000			0.899500
$902$		−	8.00000i		−	0.266371i
$903$		−	18.0000i		−	0.599002i
$904$	6.00000			0.199557
$905$	36.0000	−	18.0000i	1.19668	−	0.598340i
$906$	−2.00000			−0.0664455
$907$			23.0000i			0.763702i	0.924224	+	0.381851i	$0.124713\pi$
$907$			23.0000i			0.763702i	−0.924224	+	0.381851i	$0.875287\pi$
$908$			12.0000i			0.398234i
$909$	4.00000			0.132672
$910$	−12.0000	−	24.0000i	−0.397796	−	0.795592i
$911$	37.0000			1.22586			0.612932	−	0.790135i	$-0.289990\pi$
$911$	37.0000			1.22586			0.612932	+	0.790135i	$0.289990\pi$
$912$			5.00000i			0.165567i
$913$		−	4.00000i		−	0.132381i
$914$	27.0000			0.893081
$915$	13.0000	+	26.0000i	0.429767	+	0.859533i
$916$		0			0
$917$			21.0000i			0.693481i
$918$		−	15.0000i		−	0.495074i
$919$	−50.0000			−1.64935			−0.824674	−	0.565608i	$-0.808641\pi$
$919$	−50.0000			−1.64935			−0.824674	+	0.565608i	$0.808641\pi$
$920$	8.00000	−	4.00000i	0.263752	−	0.131876i
$921$	−8.00000			−0.263609
$922$		−	13.0000i		−	0.428132i
$923$			12.0000i			0.394985i
$924$	3.00000			0.0986928
$925$	−28.0000	−	21.0000i	−0.920634	−	0.690476i
$926$	−16.0000			−0.525793
$927$		−	8.00000i		−	0.262754i
$928$		−	5.00000i		−	0.164133i
$929$	−15.0000			−0.492134			−0.246067	−	0.969253i	$-0.579138\pi$
$929$	−15.0000			−0.492134			−0.246067	+	0.969253i	$0.579138\pi$
$930$	14.0000	−	7.00000i	0.459078	−	0.229539i
$931$	−10.0000			−0.327737
$932$			19.0000i			0.622366i
$933$		−	13.0000i		−	0.425601i
$934$	−3.00000			−0.0981630
$935$	−3.00000	−	6.00000i	−0.0981105	−	0.196221i
$936$	−8.00000			−0.261488
$937$		−	42.0000i		−	1.37208i	−0.727564	−	0.686040i	$-0.759347\pi$
$937$		−	42.0000i		−	1.37208i	0.727564	−	0.686040i	$-0.240653\pi$
$938$			36.0000i			1.17544i
$939$	14.0000			0.456873
$940$	8.00000	+	16.0000i	0.260931	+	0.521862i
$941$	17.0000			0.554184			0.277092	−	0.960843i	$-0.410629\pi$
$941$	17.0000			0.554184			0.277092	+	0.960843i	$0.410629\pi$
$942$		−	13.0000i		−	0.423563i
$943$			32.0000i			1.04206i
$944$		0			0
$945$	30.0000	−	15.0000i	0.975900	−	0.487950i
$946$	−6.00000			−0.195077
$947$			53.0000i			1.72227i	0.508378	+	0.861134i	$0.330245\pi$
$947$			53.0000i			1.72227i	−0.508378	+	0.861134i	$0.669755\pi$
$948$		0			0
$949$	24.0000			0.779073
$950$	20.0000	+	15.0000i	0.648886	+	0.486664i
$951$	−23.0000			−0.745826
$952$			9.00000i			0.291692i
$953$		−	29.0000i		−	0.939402i	−0.882826	−	0.469701i	$-0.844362\pi$
$953$		−	29.0000i		−	0.939402i	0.882826	−	0.469701i	$-0.155638\pi$
$954$	18.0000			0.582772
$955$	16.0000	−	8.00000i	0.517748	−	0.258874i
$956$		0			0
$957$		−	5.00000i		−	0.161627i
$958$			10.0000i			0.323085i
$959$	36.0000			1.16250
$960$	1.00000	+	2.00000i	0.0322749	+	0.0645497i
$961$	18.0000			0.580645
$962$		−	28.0000i		−	0.902756i
$963$		−	24.0000i		−	0.773389i
$964$	8.00000			0.257663
$965$	−1.00000	−	2.00000i	−0.0321911	−	0.0643823i
$966$	−12.0000			−0.386094
$967$			13.0000i			0.418052i	0.977910	+	0.209026i	$0.0670293\pi$
$967$			13.0000i			0.418052i	−0.977910	+	0.209026i	$0.932971\pi$
$968$		−	1.00000i		−	0.0321412i
$969$	−15.0000			−0.481869
$970$	−24.0000	+	12.0000i	−0.770594	+	0.385297i
$971$	−28.0000			−0.898563			−0.449281	−	0.893390i	$-0.648320\pi$
$971$	−28.0000			−0.898563			−0.449281	+	0.893390i	$0.648320\pi$
$972$		−	16.0000i		−	0.513200i
$973$		−	60.0000i		−	1.92351i
$974$	22.0000			0.704925
$975$	12.0000	−	16.0000i	0.384308	−	0.512410i
$976$	−13.0000			−0.416120
$977$			28.0000i			0.895799i	0.894084	+	0.447900i	$0.147828\pi$
$977$			28.0000i			0.895799i	−0.894084	+	0.447900i	$0.852172\pi$
$978$		−	21.0000i		−	0.671506i
$979$	15.0000			0.479402
$980$	−4.00000	+	2.00000i	−0.127775	+	0.0638877i
$981$	20.0000			0.638551
$982$		−	33.0000i		−	1.05307i
$983$		−	14.0000i		−	0.446531i	−0.974758	−	0.223265i	$-0.928328\pi$
$983$		−	14.0000i		−	0.446531i	0.974758	−	0.223265i	$-0.0716716\pi$
$984$	−8.00000			−0.255031
$985$	22.0000	+	44.0000i	0.700978	+	1.40196i
$986$	15.0000			0.477697
$987$		−	24.0000i		−	0.763928i
$988$			20.0000i			0.636285i
$989$	24.0000			0.763156
$990$	−2.00000	−	4.00000i	−0.0635642	−	0.127128i
$991$	−8.00000			−0.254128			−0.127064	−	0.991894i	$-0.540555\pi$
$991$	−8.00000			−0.254128			−0.127064	+	0.991894i	$0.540555\pi$
$992$			7.00000i			0.222250i
$993$		−	18.0000i		−	0.571213i
$994$	9.00000			0.285463
$995$	50.0000	−	25.0000i	1.58511	−	0.792553i
$996$	−4.00000			−0.126745
$997$		−	2.00000i		−	0.0633406i	−0.999498	−	0.0316703i	$-0.989917\pi$
$997$		−	2.00000i		−	0.0633406i	0.999498	−	0.0316703i	$-0.0100827\pi$
$998$			40.0000i			1.26618i
$999$	35.0000			1.10735

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	110.2.b.a.89.2	yes	2
3.2	odd	2		990.2.c.d.199.1		2
4.3	odd	2		880.2.b.a.529.1		2
5.2	odd	4		550.2.a.e.1.1		1
5.3	odd	4		550.2.a.j.1.1		1
5.4	even	2	inner	110.2.b.a.89.1	✓	2
11.10	odd	2		1210.2.b.a.969.1		2
15.2	even	4		4950.2.a.ba.1.1		1
15.8	even	4		4950.2.a.q.1.1		1
15.14	odd	2		990.2.c.d.199.2		2
20.3	even	4		4400.2.a.s.1.1		1
20.7	even	4		4400.2.a.k.1.1		1
20.19	odd	2		880.2.b.a.529.2		2
55.32	even	4		6050.2.a.bk.1.1		1
55.43	even	4		6050.2.a.f.1.1		1
55.54	odd	2		1210.2.b.a.969.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
110.2.b.a.89.1	✓	2	5.4	even	2	inner
110.2.b.a.89.2	yes	2	1.1	even	1	trivial
550.2.a.e.1.1		1	5.2	odd	4
550.2.a.j.1.1		1	5.3	odd	4
880.2.b.a.529.1		2	4.3	odd	2
880.2.b.a.529.2		2	20.19	odd	2
990.2.c.d.199.1		2	3.2	odd	2
990.2.c.d.199.2		2	15.14	odd	2
1210.2.b.a.969.1		2	11.10	odd	2
1210.2.b.a.969.2		2	55.54	odd	2
4400.2.a.k.1.1		1	20.7	even	4
4400.2.a.s.1.1		1	20.3	even	4
4950.2.a.q.1.1		1	15.8	even	4
4950.2.a.ba.1.1		1	15.2	even	4
6050.2.a.f.1.1		1	55.43	even	4
6050.2.a.bk.1.1		1	55.32	even	4

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:	\( N \)	\(=\)	\( 110 = 2 \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	110.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +(-2.00000 + 1.00000i) q^{5} -1.00000 q^{6} +3.00000i q^{7} -1.00000i q^{8} +2.00000 q^{9} +(-1.00000 - 2.00000i) q^{10} +1.00000 q^{11} -1.00000i q^{12} -4.00000i q^{13} -3.00000 q^{14} +(-1.00000 - 2.00000i) q^{15} +1.00000 q^{16} +3.00000i q^{17} +2.00000i q^{18} +5.00000 q^{19} +(2.00000 - 1.00000i) q^{20} -3.00000 q^{21} +1.00000i q^{22} -4.00000i q^{23} +1.00000 q^{24} +(3.00000 - 4.00000i) q^{25} +4.00000 q^{26} +5.00000i q^{27} -3.00000i q^{28} -5.00000 q^{29} +(2.00000 - 1.00000i) q^{30} +7.00000 q^{31} +1.00000i q^{32} +1.00000i q^{33} -3.00000 q^{34} +(-3.00000 - 6.00000i) q^{35} -2.00000 q^{36} -7.00000i q^{37} +5.00000i q^{38} +4.00000 q^{39} +(1.00000 + 2.00000i) q^{40} -8.00000 q^{41} -3.00000i q^{42} +6.00000i q^{43} -1.00000 q^{44} +(-4.00000 + 2.00000i) q^{45} +4.00000 q^{46} +8.00000i q^{47} +1.00000i q^{48} -2.00000 q^{49} +(4.00000 + 3.00000i) q^{50} -3.00000 q^{51} +4.00000i q^{52} -9.00000i q^{53} -5.00000 q^{54} +(-2.00000 + 1.00000i) q^{55} +3.00000 q^{56} +5.00000i q^{57} -5.00000i q^{58} +(1.00000 + 2.00000i) q^{60} -13.0000 q^{61} +7.00000i q^{62} +6.00000i q^{63} -1.00000 q^{64} +(4.00000 + 8.00000i) q^{65} -1.00000 q^{66} -12.0000i q^{67} -3.00000i q^{68} +4.00000 q^{69} +(6.00000 - 3.00000i) q^{70} -3.00000 q^{71} -2.00000i q^{72} +6.00000i q^{73} +7.00000 q^{74} +(4.00000 + 3.00000i) q^{75} -5.00000 q^{76} +3.00000i q^{77} +4.00000i q^{78} +(-2.00000 + 1.00000i) q^{80} +1.00000 q^{81} -8.00000i q^{82} -4.00000i q^{83} +3.00000 q^{84} +(-3.00000 - 6.00000i) q^{85} -6.00000 q^{86} -5.00000i q^{87} -1.00000i q^{88} +15.0000 q^{89} +(-2.00000 - 4.00000i) q^{90} +12.0000 q^{91} +4.00000i q^{92} +7.00000i q^{93} -8.00000 q^{94} +(-10.0000 + 5.00000i) q^{95} -1.00000 q^{96} -12.0000i q^{97} -2.00000i q^{98} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} - 4 q^{5} - 2 q^{6} + 4 q^{9} - 2 q^{10} + 2 q^{11} - 6 q^{14} - 2 q^{15} + 2 q^{16} + 10 q^{19} + 4 q^{20} - 6 q^{21} + 2 q^{24} + 6 q^{25} + 8 q^{26} - 10 q^{29} + 4 q^{30} + 14 q^{31} - 6 q^{34}+ \cdots + 4 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 - 4 * q^5 - 2 * q^6 + 4 * q^9 - 2 * q^10 + 2 * q^11 - 6 * q^14 - 2 * q^15 + 2 * q^16 + 10 * q^19 + 4 * q^20 - 6 * q^21 + 2 * q^24 + 6 * q^25 + 8 * q^26 - 10 * q^29 + 4 * q^30 + 14 * q^31 - 6 * q^34 - 6 * q^35 - 4 * q^36 + 8 * q^39 + 2 * q^40 - 16 * q^41 - 2 * q^44 - 8 * q^45 + 8 * q^46 - 4 * q^49 + 8 * q^50 - 6 * q^51 - 10 * q^54 - 4 * q^55 + 6 * q^56 + 2 * q^60 - 26 * q^61 - 2 * q^64 + 8 * q^65 - 2 * q^66 + 8 * q^69 + 12 * q^70 - 6 * q^71 + 14 * q^74 + 8 * q^75 - 10 * q^76 - 4 * q^80 + 2 * q^81 + 6 * q^84 - 6 * q^85 - 12 * q^86 + 30 * q^89 - 4 * q^90 + 24 * q^91 - 16 * q^94 - 20 * q^95 - 2 * q^96 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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