LMFDB - Embedded newform 110.2.b.b.89.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [110,2,Mod(89,110)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

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")

//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);

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: f = N[0] # Warning: the index may be different

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.b.89.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -2.00000i q^{3} -1.00000 q^{4} +(1.00000 - 2.00000i) q^{5} +2.00000 q^{6} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} -2.00000i q^{3} -1.00000 q^{4} +(1.00000 - 2.00000i) q^{5} +2.00000 q^{6} -1.00000i q^{8} -1.00000 q^{9} +(2.00000 + 1.00000i) q^{10} +1.00000 q^{11} +2.00000i q^{12} +2.00000i q^{13} +(-4.00000 - 2.00000i) q^{15} +1.00000 q^{16} +6.00000i q^{17} -1.00000i q^{18} -4.00000 q^{19} +(-1.00000 + 2.00000i) q^{20} +1.00000i q^{22} +2.00000i q^{23} -2.00000 q^{24} +(-3.00000 - 4.00000i) q^{25} -2.00000 q^{26} -4.00000i q^{27} +10.0000 q^{29} +(2.00000 - 4.00000i) q^{30} -8.00000 q^{31} +1.00000i q^{32} -2.00000i q^{33} -6.00000 q^{34} +1.00000 q^{36} +8.00000i q^{37} -4.00000i q^{38} +4.00000 q^{39} +(-2.00000 - 1.00000i) q^{40} -2.00000 q^{41} -1.00000 q^{44} +(-1.00000 + 2.00000i) q^{45} -2.00000 q^{46} +2.00000i q^{47} -2.00000i q^{48} +7.00000 q^{49} +(4.00000 - 3.00000i) q^{50} +12.0000 q^{51} -2.00000i q^{52} +4.00000 q^{54} +(1.00000 - 2.00000i) q^{55} +8.00000i q^{57} +10.0000i q^{58} +12.0000 q^{59} +(4.00000 + 2.00000i) q^{60} -10.0000 q^{61} -8.00000i q^{62} -1.00000 q^{64} +(4.00000 + 2.00000i) q^{65} +2.00000 q^{66} -6.00000i q^{67} -6.00000i q^{68} +4.00000 q^{69} +1.00000i q^{72} -6.00000i q^{73} -8.00000 q^{74} +(-8.00000 + 6.00000i) q^{75} +4.00000 q^{76} +4.00000i q^{78} -12.0000 q^{79} +(1.00000 - 2.00000i) q^{80} -11.0000 q^{81} -2.00000i q^{82} -16.0000i q^{83} +(12.0000 + 6.00000i) q^{85} -20.0000i q^{87} -1.00000i q^{88} -18.0000 q^{89} +(-2.00000 - 1.00000i) q^{90} -2.00000i q^{92} +16.0000i q^{93} -2.00000 q^{94} +(-4.00000 + 8.00000i) q^{95} +2.00000 q^{96} -12.0000i q^{97} +7.00000i q^{98} -1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} + 2 q^{5} + 4 q^{6} - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^4 + 2 * q^5 + 4 * q^6 - 2 * q^9	$ 2 q - 2 q^{4} + 2 q^{5} + 4 q^{6} - 2 q^{9} + 4 q^{10} + 2 q^{11} - 8 q^{15} + 2 q^{16} - 8 q^{19} - 2 q^{20} - 4 q^{24} - 6 q^{25} - 4 q^{26} + 20 q^{29} + 4 q^{30} - 16 q^{31} - 12 q^{34} + 2 q^{36} + 8 q^{39} - 4 q^{40} - 4 q^{41} - 2 q^{44} - 2 q^{45} - 4 q^{46} + 14 q^{49} + 8 q^{50} + 24 q^{51} + 8 q^{54} + 2 q^{55} + 24 q^{59} + 8 q^{60} - 20 q^{61} - 2 q^{64} + 8 q^{65} + 4 q^{66} + 8 q^{69} - 16 q^{74} - 16 q^{75} + 8 q^{76} - 24 q^{79} + 2 q^{80} - 22 q^{81} + 24 q^{85} - 36 q^{89} - 4 q^{90} - 4 q^{94} - 8 q^{95} + 4 q^{96} - 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^4 + 2 * q^5 + 4 * q^6 - 2 * q^9 + 4 * q^10 + 2 * q^11 - 8 * q^15 + 2 * q^16 - 8 * q^19 - 2 * q^20 - 4 * q^24 - 6 * q^25 - 4 * q^26 + 20 * q^29 + 4 * q^30 - 16 * q^31 - 12 * q^34 + 2 * q^36 + 8 * q^39 - 4 * q^40 - 4 * q^41 - 2 * q^44 - 2 * q^45 - 4 * q^46 + 14 * q^49 + 8 * q^50 + 24 * q^51 + 8 * q^54 + 2 * q^55 + 24 * q^59 + 8 * q^60 - 20 * q^61 - 2 * q^64 + 8 * q^65 + 4 * q^66 + 8 * q^69 - 16 * q^74 - 16 * q^75 + 8 * q^76 - 24 * q^79 + 2 * q^80 - 22 * q^81 + 24 * q^85 - 36 * q^89 - 4 * q^90 - 4 * q^94 - 8 * q^95 + 4 * q^96 - 2 * q^99

Display 100 coefficients 10 coefficients

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
$3$		−	2.00000i		−	1.15470i	−0.816497	−	0.577350i	$-0.804087\pi$
$3$		−	2.00000i		−	1.15470i	0.816497	−	0.577350i	$-0.195913\pi$
$4$	−1.00000			−0.500000
$5$	1.00000	−	2.00000i	0.447214	−	0.894427i
$5$	1.00000	−	2.00000i	0.447214	−	0.894427i
$6$	2.00000			0.816497
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$		−	1.00000i		−	0.353553i
$9$	−1.00000			−0.333333
$10$	2.00000	+	1.00000i	0.632456	+	0.316228i
$11$	1.00000			0.301511
$11$	1.00000			0.301511
$12$			2.00000i			0.577350i
$13$			2.00000i			0.554700i	0.960769	+	0.277350i	$0.0894562\pi$
$13$			2.00000i			0.554700i	−0.960769	+	0.277350i	$0.910544\pi$
$14$		0			0
$15$	−4.00000	−	2.00000i	−1.03280	−	0.516398i
$16$	1.00000			0.250000
$17$			6.00000i			1.45521i	0.685994	+	0.727607i	$0.259367\pi$
$17$			6.00000i			1.45521i	−0.685994	+	0.727607i	$0.740633\pi$
$18$		−	1.00000i		−	0.235702i
$19$	−4.00000			−0.917663			−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000			−0.917663			−0.458831	+	0.888523i	$0.651732\pi$
$20$	−1.00000	+	2.00000i	−0.223607	+	0.447214i
$21$		0			0
$22$			1.00000i			0.213201i
$23$			2.00000i			0.417029i	0.978019	+	0.208514i	$0.0668628\pi$
$23$			2.00000i			0.417029i	−0.978019	+	0.208514i	$0.933137\pi$
$24$	−2.00000			−0.408248
$25$	−3.00000	−	4.00000i	−0.600000	−	0.800000i
$26$	−2.00000			−0.392232
$27$		−	4.00000i		−	0.769800i
$28$		0			0
$29$	10.0000			1.85695			0.928477	−	0.371391i	$-0.121119\pi$
$29$	10.0000			1.85695			0.928477	+	0.371391i	$0.121119\pi$
$30$	2.00000	−	4.00000i	0.365148	−	0.730297i
$31$	−8.00000			−1.43684			−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000			−1.43684			−0.718421	+	0.695608i	$0.755135\pi$
$32$			1.00000i			0.176777i
$33$		−	2.00000i		−	0.348155i
$34$	−6.00000			−1.02899
$35$		0			0
$36$	1.00000			0.166667
$37$			8.00000i			1.31519i	0.753371	+	0.657596i	$0.228427\pi$
$37$			8.00000i			1.31519i	−0.753371	+	0.657596i	$0.771573\pi$
$38$		−	4.00000i		−	0.648886i
$39$	4.00000			0.640513
$40$	−2.00000	−	1.00000i	−0.316228	−	0.158114i
$41$	−2.00000			−0.312348			−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000			−0.312348			−0.156174	+	0.987730i	$0.549916\pi$
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$	−1.00000			−0.150756
$45$	−1.00000	+	2.00000i	−0.149071	+	0.298142i
$46$	−2.00000			−0.294884
$47$			2.00000i			0.291730i	0.989305	+	0.145865i	$0.0465965\pi$
$47$			2.00000i			0.291730i	−0.989305	+	0.145865i	$0.953403\pi$
$48$		−	2.00000i		−	0.288675i
$49$	7.00000			1.00000
$50$	4.00000	−	3.00000i	0.565685	−	0.424264i
$51$	12.0000			1.68034
$52$		−	2.00000i		−	0.277350i
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$	4.00000			0.544331
$55$	1.00000	−	2.00000i	0.134840	−	0.269680i
$56$		0			0
$57$			8.00000i			1.05963i
$58$			10.0000i			1.31306i
$59$	12.0000			1.56227			0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000			1.56227			0.781133	+	0.624364i	$0.214642\pi$
$60$	4.00000	+	2.00000i	0.516398	+	0.258199i
$61$	−10.0000			−1.28037			−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000			−1.28037			−0.640184	+	0.768221i	$0.721142\pi$
$62$		−	8.00000i		−	1.01600i
$63$		0			0
$64$	−1.00000			−0.125000
$65$	4.00000	+	2.00000i	0.496139	+	0.248069i
$66$	2.00000			0.246183
$67$		−	6.00000i		−	0.733017i	−0.930415	−	0.366508i	$-0.880553\pi$
$67$		−	6.00000i		−	0.733017i	0.930415	−	0.366508i	$-0.119447\pi$
$68$		−	6.00000i		−	0.727607i
$69$	4.00000			0.481543
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$			1.00000i			0.117851i
$73$		−	6.00000i		−	0.702247i	−0.936329	−	0.351123i	$-0.885800\pi$
$73$		−	6.00000i		−	0.702247i	0.936329	−	0.351123i	$-0.114200\pi$
$74$	−8.00000			−0.929981
$75$	−8.00000	+	6.00000i	−0.923760	+	0.692820i
$76$	4.00000			0.458831
$77$		0			0
$78$			4.00000i			0.452911i
$79$	−12.0000			−1.35011			−0.675053	−	0.737769i	$-0.735879\pi$
$79$	−12.0000			−1.35011			−0.675053	+	0.737769i	$0.735879\pi$
$80$	1.00000	−	2.00000i	0.111803	−	0.223607i
$81$	−11.0000			−1.22222
$82$		−	2.00000i		−	0.220863i
$83$		−	16.0000i		−	1.75623i	−0.478451	−	0.878114i	$-0.658802\pi$
$83$		−	16.0000i		−	1.75623i	0.478451	−	0.878114i	$-0.341198\pi$
$84$		0			0
$85$	12.0000	+	6.00000i	1.30158	+	0.650791i
$86$		0			0
$87$		−	20.0000i		−	2.14423i
$88$		−	1.00000i		−	0.106600i
$89$	−18.0000			−1.90800			−0.953998	−	0.299813i	$-0.903076\pi$
$89$	−18.0000			−1.90800			−0.953998	+	0.299813i	$0.903076\pi$
$90$	−2.00000	−	1.00000i	−0.210819	−	0.105409i
$91$		0			0
$92$		−	2.00000i		−	0.208514i
$93$			16.0000i			1.65912i
$94$	−2.00000			−0.206284
$95$	−4.00000	+	8.00000i	−0.410391	+	0.820783i
$96$	2.00000			0.204124
$97$		−	12.0000i		−	1.21842i	−0.793011	−	0.609208i	$-0.791488\pi$
$97$		−	12.0000i		−	1.21842i	0.793011	−	0.609208i	$-0.208512\pi$
$98$			7.00000i			0.707107i
$99$	−1.00000			−0.100504
$100$	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	$0.468274\pi$
$102$			12.0000i			1.18818i
$103$			14.0000i			1.37946i	0.724066	+	0.689730i	$0.242271\pi$
$103$			14.0000i			1.37946i	−0.724066	+	0.689730i	$0.757729\pi$
$104$	2.00000			0.196116
$105$		0			0
$106$		0			0
$107$			12.0000i			1.16008i	0.814587	+	0.580042i	$0.196964\pi$
$107$			12.0000i			1.16008i	−0.814587	+	0.580042i	$0.803036\pi$
$108$			4.00000i			0.384900i
$109$	−2.00000			−0.191565			−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000			−0.191565			−0.0957826	+	0.995402i	$0.530535\pi$
$110$	2.00000	+	1.00000i	0.190693	+	0.0953463i
$111$	16.0000			1.51865
$112$		0			0
$113$		−	12.0000i		−	1.12887i	−0.825479	−	0.564433i	$-0.809095\pi$
$113$		−	12.0000i		−	1.12887i	0.825479	−	0.564433i	$-0.190905\pi$
$114$	−8.00000			−0.749269
$115$	4.00000	+	2.00000i	0.373002	+	0.186501i
$116$	−10.0000			−0.928477
$117$		−	2.00000i		−	0.184900i
$118$			12.0000i			1.10469i
$119$		0			0
$120$	−2.00000	+	4.00000i	−0.182574	+	0.365148i
$121$	1.00000			0.0909091
$122$		−	10.0000i		−	0.905357i
$123$			4.00000i			0.360668i
$124$	8.00000			0.718421
$125$	−11.0000	+	2.00000i	−0.983870	+	0.178885i
$126$		0			0
$127$		−	4.00000i		−	0.354943i	−0.984126	−	0.177471i	$-0.943208\pi$
$127$		−	4.00000i		−	0.354943i	0.984126	−	0.177471i	$-0.0567917\pi$
$128$		−	1.00000i		−	0.0883883i
$129$		0			0
$130$	−2.00000	+	4.00000i	−0.175412	+	0.350823i
$131$	−8.00000			−0.698963			−0.349482	−	0.936943i	$-0.613642\pi$
$131$	−8.00000			−0.698963			−0.349482	+	0.936943i	$0.613642\pi$
$132$			2.00000i			0.174078i
$133$		0			0
$134$	6.00000			0.518321
$135$	−8.00000	−	4.00000i	−0.688530	−	0.344265i
$136$	6.00000			0.514496
$137$		−	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
$139$	−8.00000			−0.678551			−0.339276	−	0.940687i	$-0.610182\pi$
$139$	−8.00000			−0.678551			−0.339276	+	0.940687i	$0.610182\pi$
$140$		0			0
$141$	4.00000			0.336861
$142$		0			0
$143$			2.00000i			0.167248i
$144$	−1.00000			−0.0833333
$145$	10.0000	−	20.0000i	0.830455	−	1.66091i
$146$	6.00000			0.496564
$147$		−	14.0000i		−	1.15470i
$148$		−	8.00000i		−	0.657596i
$149$	14.0000			1.14692			0.573462	−	0.819232i	$-0.305600\pi$
$149$	14.0000			1.14692			0.573462	+	0.819232i	$0.305600\pi$
$150$	−6.00000	−	8.00000i	−0.489898	−	0.653197i
$151$	8.00000			0.651031			0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000			0.651031			0.325515	+	0.945537i	$0.394462\pi$
$152$			4.00000i			0.324443i
$153$		−	6.00000i		−	0.485071i
$154$		0			0
$155$	−8.00000	+	16.0000i	−0.642575	+	1.28515i
$156$	−4.00000			−0.320256
$157$			4.00000i			0.319235i	0.987179	+	0.159617i	$0.0510260\pi$
$157$			4.00000i			0.319235i	−0.987179	+	0.159617i	$0.948974\pi$
$158$		−	12.0000i		−	0.954669i
$159$		0			0
$160$	2.00000	+	1.00000i	0.158114	+	0.0790569i
$161$		0			0
$162$		−	11.0000i		−	0.864242i
$163$			6.00000i			0.469956i	0.972001	+	0.234978i	$0.0755019\pi$
$163$			6.00000i			0.469956i	−0.972001	+	0.234978i	$0.924498\pi$
$164$	2.00000			0.156174
$165$	−4.00000	−	2.00000i	−0.311400	−	0.155700i
$166$	16.0000			1.24184
$167$			20.0000i			1.54765i	0.633402	+	0.773823i	$0.281658\pi$
$167$			20.0000i			1.54765i	−0.633402	+	0.773823i	$0.718342\pi$
$168$		0			0
$169$	9.00000			0.692308
$170$	−6.00000	+	12.0000i	−0.460179	+	0.920358i
$171$	4.00000			0.305888
$172$		0			0
$173$		−	2.00000i		−	0.152057i	−0.997106	−	0.0760286i	$-0.975776\pi$
$173$		−	2.00000i		−	0.152057i	0.997106	−	0.0760286i	$-0.0242240\pi$
$174$	20.0000			1.51620
$175$		0			0
$176$	1.00000			0.0753778
$177$		−	24.0000i		−	1.80395i
$178$		−	18.0000i		−	1.34916i
$179$	−4.00000			−0.298974			−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000			−0.298974			−0.149487	+	0.988764i	$0.547762\pi$
$180$	1.00000	−	2.00000i	0.0745356	−	0.149071i
$181$	−6.00000			−0.445976			−0.222988	−	0.974821i	$-0.571581\pi$
$181$	−6.00000			−0.445976			−0.222988	+	0.974821i	$0.571581\pi$
$182$		0			0
$183$			20.0000i			1.47844i
$184$	2.00000			0.147442
$185$	16.0000	+	8.00000i	1.17634	+	0.588172i
$186$	−16.0000			−1.17318
$187$			6.00000i			0.438763i
$188$		−	2.00000i		−	0.145865i
$189$		0			0
$190$	−8.00000	−	4.00000i	−0.580381	−	0.290191i
$191$	16.0000			1.15772			0.578860	−	0.815427i	$-0.303498\pi$
$191$	16.0000			1.15772			0.578860	+	0.815427i	$0.303498\pi$
$192$			2.00000i			0.144338i
$193$			10.0000i			0.719816i	0.932988	+	0.359908i	$0.117192\pi$
$193$			10.0000i			0.719816i	−0.932988	+	0.359908i	$0.882808\pi$
$194$	12.0000			0.861550
$195$	4.00000	−	8.00000i	0.286446	−	0.572892i
$196$	−7.00000			−0.500000
$197$			2.00000i			0.142494i	0.997459	+	0.0712470i	$0.0226979\pi$
$197$			2.00000i			0.142494i	−0.997459	+	0.0712470i	$0.977302\pi$
$198$		−	1.00000i		−	0.0710669i
$199$	8.00000			0.567105			0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000			0.567105			0.283552	+	0.958957i	$0.408487\pi$
$200$	−4.00000	+	3.00000i	−0.282843	+	0.212132i
$201$	−12.0000			−0.846415
$202$			2.00000i			0.140720i
$203$		0			0
$204$	−12.0000			−0.840168
$205$	−2.00000	+	4.00000i	−0.139686	+	0.279372i
$206$	−14.0000			−0.975426
$207$		−	2.00000i		−	0.139010i
$208$			2.00000i			0.138675i
$209$	−4.00000			−0.276686
$210$		0			0
$211$	8.00000			0.550743			0.275371	−	0.961338i	$-0.411199\pi$
$211$	8.00000			0.550743			0.275371	+	0.961338i	$0.411199\pi$
$212$		0			0
$213$		0			0
$214$	−12.0000			−0.820303
$215$		0			0
$216$	−4.00000			−0.272166
$217$		0			0
$218$		−	2.00000i		−	0.135457i
$219$	−12.0000			−0.810885
$220$	−1.00000	+	2.00000i	−0.0674200	+	0.134840i
$221$	−12.0000			−0.807207
$222$			16.0000i			1.07385i
$223$		−	14.0000i		−	0.937509i	−0.883328	−	0.468755i	$-0.844703\pi$
$223$		−	14.0000i		−	0.937509i	0.883328	−	0.468755i	$-0.155297\pi$
$224$		0			0
$225$	3.00000	+	4.00000i	0.200000	+	0.266667i
$226$	12.0000			0.798228
$227$			24.0000i			1.59294i	0.604681	+	0.796468i	$0.293301\pi$
$227$			24.0000i			1.59294i	−0.604681	+	0.796468i	$0.706699\pi$
$228$		−	8.00000i		−	0.529813i
$229$	6.00000			0.396491			0.198246	−	0.980152i	$-0.436476\pi$
$229$	6.00000			0.396491			0.198246	+	0.980152i	$0.436476\pi$
$230$	−2.00000	+	4.00000i	−0.131876	+	0.263752i
$231$		0			0
$232$		−	10.0000i		−	0.656532i
$233$		−	10.0000i		−	0.655122i	−0.944830	−	0.327561i	$-0.893773\pi$
$233$		−	10.0000i		−	0.655122i	0.944830	−	0.327561i	$-0.106227\pi$
$234$	2.00000			0.130744
$235$	4.00000	+	2.00000i	0.260931	+	0.130466i
$236$	−12.0000			−0.781133
$237$			24.0000i			1.55897i
$238$		0			0
$239$	−12.0000			−0.776215			−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000			−0.776215			−0.388108	+	0.921614i	$0.626871\pi$
$240$	−4.00000	−	2.00000i	−0.258199	−	0.129099i
$241$	−2.00000			−0.128831			−0.0644157	−	0.997923i	$-0.520518\pi$
$241$	−2.00000			−0.128831			−0.0644157	+	0.997923i	$0.520518\pi$
$242$			1.00000i			0.0642824i
$243$			10.0000i			0.641500i
$244$	10.0000			0.640184
$245$	7.00000	−	14.0000i	0.447214	−	0.894427i
$246$	−4.00000			−0.255031
$247$		−	8.00000i		−	0.509028i
$248$			8.00000i			0.508001i
$249$	−32.0000			−2.02792
$250$	−2.00000	−	11.0000i	−0.126491	−	0.695701i
$251$	20.0000			1.26239			0.631194	−	0.775625i	$-0.282565\pi$
$251$	20.0000			1.26239			0.631194	+	0.775625i	$0.282565\pi$
$252$		0			0
$253$			2.00000i			0.125739i
$254$	4.00000			0.250982
$255$	12.0000	−	24.0000i	0.751469	−	1.50294i
$256$	1.00000			0.0625000
$257$		−	8.00000i		−	0.499026i	−0.968371	−	0.249513i	$-0.919729\pi$
$257$		−	8.00000i		−	0.499026i	0.968371	−	0.249513i	$-0.0802706\pi$
$258$		0			0
$259$		0			0
$260$	−4.00000	−	2.00000i	−0.248069	−	0.124035i
$261$	−10.0000			−0.618984
$262$		−	8.00000i		−	0.494242i
$263$			12.0000i			0.739952i	0.929041	+	0.369976i	$0.120634\pi$
$263$			12.0000i			0.739952i	−0.929041	+	0.369976i	$0.879366\pi$
$264$	−2.00000			−0.123091
$265$		0			0
$266$		0			0
$267$			36.0000i			2.20316i
$268$			6.00000i			0.366508i
$269$	−18.0000			−1.09748			−0.548740	−	0.835993i	$-0.684892\pi$
$269$	−18.0000			−1.09748			−0.548740	+	0.835993i	$0.684892\pi$
$270$	4.00000	−	8.00000i	0.243432	−	0.486864i
$271$	−8.00000			−0.485965			−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000			−0.485965			−0.242983	+	0.970031i	$0.578126\pi$
$272$			6.00000i			0.363803i
$273$		0			0
$274$	12.0000			0.724947
$275$	−3.00000	−	4.00000i	−0.180907	−	0.241209i
$276$	−4.00000			−0.240772
$277$			22.0000i			1.32185i	0.750451	+	0.660926i	$0.229836\pi$
$277$			22.0000i			1.32185i	−0.750451	+	0.660926i	$0.770164\pi$
$278$		−	8.00000i		−	0.479808i
$279$	8.00000			0.478947
$280$		0			0
$281$	−18.0000			−1.07379			−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000			−1.07379			−0.536895	+	0.843649i	$0.680403\pi$
$282$			4.00000i			0.238197i
$283$		−	32.0000i		−	1.90220i	−0.308879	−	0.951101i	$-0.599954\pi$
$283$		−	32.0000i		−	1.90220i	0.308879	−	0.951101i	$-0.400046\pi$
$284$		0			0
$285$	16.0000	+	8.00000i	0.947758	+	0.473879i
$286$	−2.00000			−0.118262
$287$		0			0
$288$		−	1.00000i		−	0.0589256i
$289$	−19.0000			−1.11765
$290$	20.0000	+	10.0000i	1.17444	+	0.587220i
$291$	−24.0000			−1.40690
$292$			6.00000i			0.351123i
$293$		−	14.0000i		−	0.817889i	−0.912559	−	0.408944i	$-0.865897\pi$
$293$		−	14.0000i		−	0.817889i	0.912559	−	0.408944i	$-0.134103\pi$
$294$	14.0000			0.816497
$295$	12.0000	−	24.0000i	0.698667	−	1.39733i
$296$	8.00000			0.464991
$297$		−	4.00000i		−	0.232104i
$298$			14.0000i			0.810998i
$299$	−4.00000			−0.231326
$300$	8.00000	−	6.00000i	0.461880	−	0.346410i
$301$		0			0
$302$			8.00000i			0.460348i
$303$		−	4.00000i		−	0.229794i
$304$	−4.00000			−0.229416
$305$	−10.0000	+	20.0000i	−0.572598	+	1.14520i
$306$	6.00000			0.342997
$307$		−	4.00000i		−	0.228292i	−0.993464	−	0.114146i	$-0.963587\pi$
$307$		−	4.00000i		−	0.228292i	0.993464	−	0.114146i	$-0.0364132\pi$
$308$		0			0
$309$	28.0000			1.59286
$310$	−16.0000	−	8.00000i	−0.908739	−	0.454369i
$311$	32.0000			1.81455			0.907277	−	0.420534i	$-0.138157\pi$
$311$	32.0000			1.81455			0.907277	+	0.420534i	$0.138157\pi$
$312$		−	4.00000i		−	0.226455i
$313$			16.0000i			0.904373i	0.891923	+	0.452187i	$0.149356\pi$
$313$			16.0000i			0.904373i	−0.891923	+	0.452187i	$0.850644\pi$
$314$	−4.00000			−0.225733
$315$		0			0
$316$	12.0000			0.675053
$317$		−	28.0000i		−	1.57264i	−0.617822	−	0.786318i	$-0.711985\pi$
$317$		−	28.0000i		−	1.57264i	0.617822	−	0.786318i	$-0.288015\pi$
$318$		0			0
$319$	10.0000			0.559893
$320$	−1.00000	+	2.00000i	−0.0559017	+	0.111803i
$321$	24.0000			1.33955
$322$		0			0
$323$		−	24.0000i		−	1.33540i
$324$	11.0000			0.611111
$325$	8.00000	−	6.00000i	0.443760	−	0.332820i
$326$	−6.00000			−0.332309
$327$			4.00000i			0.221201i
$328$			2.00000i			0.110432i
$329$		0			0
$330$	2.00000	−	4.00000i	0.110096	−	0.220193i
$331$	12.0000			0.659580			0.329790	−	0.944054i	$-0.393022\pi$
$331$	12.0000			0.659580			0.329790	+	0.944054i	$0.393022\pi$
$332$			16.0000i			0.878114i
$333$		−	8.00000i		−	0.438397i
$334$	−20.0000			−1.09435
$335$	−12.0000	−	6.00000i	−0.655630	−	0.327815i
$336$		0			0
$337$		−	10.0000i		−	0.544735i	−0.962193	−	0.272367i	$-0.912193\pi$
$337$		−	10.0000i		−	0.544735i	0.962193	−	0.272367i	$-0.0878066\pi$
$338$			9.00000i			0.489535i
$339$	−24.0000			−1.30350
$340$	−12.0000	−	6.00000i	−0.650791	−	0.325396i
$341$	−8.00000			−0.433224
$342$			4.00000i			0.216295i
$343$		0			0
$344$		0			0
$345$	4.00000	−	8.00000i	0.215353	−	0.430706i
$346$	2.00000			0.107521
$347$		−	12.0000i		−	0.644194i	−0.946707	−	0.322097i	$-0.895612\pi$
$347$		−	12.0000i		−	0.644194i	0.946707	−	0.322097i	$-0.104388\pi$
$348$			20.0000i			1.07211i
$349$	34.0000			1.81998			0.909989	−	0.414632i	$-0.136090\pi$
$349$	34.0000			1.81998			0.909989	+	0.414632i	$0.136090\pi$
$350$		0			0
$351$	8.00000			0.427008
$352$			1.00000i			0.0533002i
$353$		−	8.00000i		−	0.425797i	−0.977074	−	0.212899i	$-0.931710\pi$
$353$		−	8.00000i		−	0.425797i	0.977074	−	0.212899i	$-0.0682904\pi$
$354$	24.0000			1.27559
$355$		0			0
$356$	18.0000			0.953998
$357$		0			0
$358$		−	4.00000i		−	0.211407i
$359$	−12.0000			−0.633336			−0.316668	−	0.948536i	$-0.602564\pi$
$359$	−12.0000			−0.633336			−0.316668	+	0.948536i	$0.602564\pi$
$360$	2.00000	+	1.00000i	0.105409	+	0.0527046i
$361$	−3.00000			−0.157895
$362$		−	6.00000i		−	0.315353i
$363$		−	2.00000i		−	0.104973i
$364$		0			0
$365$	−12.0000	−	6.00000i	−0.628109	−	0.314054i
$366$	−20.0000			−1.04542
$367$			26.0000i			1.35719i	0.734513	+	0.678594i	$0.237411\pi$
$367$			26.0000i			1.35719i	−0.734513	+	0.678594i	$0.762589\pi$
$368$			2.00000i			0.104257i
$369$	2.00000			0.104116
$370$	−8.00000	+	16.0000i	−0.415900	+	0.831800i
$371$		0			0
$372$		−	16.0000i		−	0.829561i
$373$			14.0000i			0.724893i	0.932005	+	0.362446i	$0.118058\pi$
$373$			14.0000i			0.724893i	−0.932005	+	0.362446i	$0.881942\pi$
$374$	−6.00000			−0.310253
$375$	4.00000	+	22.0000i	0.206559	+	1.13608i
$376$	2.00000			0.103142
$377$			20.0000i			1.03005i
$378$		0			0
$379$	4.00000			0.205466			0.102733	−	0.994709i	$-0.467241\pi$
$379$	4.00000			0.205466			0.102733	+	0.994709i	$0.467241\pi$
$380$	4.00000	−	8.00000i	0.205196	−	0.410391i
$381$	−8.00000			−0.409852
$382$			16.0000i			0.818631i
$383$		−	18.0000i		−	0.919757i	−0.887982	−	0.459879i	$-0.847893\pi$
$383$		−	18.0000i		−	0.919757i	0.887982	−	0.459879i	$-0.152107\pi$
$384$	−2.00000			−0.102062
$385$		0			0
$386$	−10.0000			−0.508987
$387$		0			0
$388$			12.0000i			0.609208i
$389$	6.00000			0.304212			0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000			0.304212			0.152106	+	0.988364i	$0.451394\pi$
$390$	8.00000	+	4.00000i	0.405096	+	0.202548i
$391$	−12.0000			−0.606866
$392$		−	7.00000i		−	0.353553i
$393$			16.0000i			0.807093i
$394$	−2.00000			−0.100759
$395$	−12.0000	+	24.0000i	−0.603786	+	1.20757i
$396$	1.00000			0.0502519
$397$		−	28.0000i		−	1.40528i	−0.711546	−	0.702640i	$-0.752005\pi$
$397$		−	28.0000i		−	1.40528i	0.711546	−	0.702640i	$-0.247995\pi$
$398$			8.00000i			0.401004i
$399$		0			0
$400$	−3.00000	−	4.00000i	−0.150000	−	0.200000i
$401$	30.0000			1.49813			0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000			1.49813			0.749064	+	0.662497i	$0.230503\pi$
$402$		−	12.0000i		−	0.598506i
$403$		−	16.0000i		−	0.797017i
$404$	−2.00000			−0.0995037
$405$	−11.0000	+	22.0000i	−0.546594	+	1.09319i
$406$		0			0
$407$			8.00000i			0.396545i
$408$		−	12.0000i		−	0.594089i
$409$	−2.00000			−0.0988936			−0.0494468	−	0.998777i	$-0.515746\pi$
$409$	−2.00000			−0.0988936			−0.0494468	+	0.998777i	$0.515746\pi$
$410$	−4.00000	−	2.00000i	−0.197546	−	0.0987730i
$411$	−24.0000			−1.18383
$412$		−	14.0000i		−	0.689730i
$413$		0			0
$414$	2.00000			0.0982946
$415$	−32.0000	−	16.0000i	−1.57082	−	0.785409i
$416$	−2.00000			−0.0980581
$417$			16.0000i			0.783523i
$418$		−	4.00000i		−	0.195646i
$419$	20.0000			0.977064			0.488532	−	0.872546i	$-0.337533\pi$
$419$	20.0000			0.977064			0.488532	+	0.872546i	$0.337533\pi$
$420$		0			0
$421$	2.00000			0.0974740			0.0487370	−	0.998812i	$-0.484480\pi$
$421$	2.00000			0.0974740			0.0487370	+	0.998812i	$0.484480\pi$
$422$			8.00000i			0.389434i
$423$		−	2.00000i		−	0.0972433i
$424$		0			0
$425$	24.0000	−	18.0000i	1.16417	−	0.873128i
$426$		0			0
$427$		0			0
$428$		−	12.0000i		−	0.580042i
$429$	4.00000			0.193122
$430$		0			0
$431$	−28.0000			−1.34871			−0.674356	−	0.738406i	$-0.735579\pi$
$431$	−28.0000			−1.34871			−0.674356	+	0.738406i	$0.735579\pi$
$432$		−	4.00000i		−	0.192450i
$433$			4.00000i			0.192228i	0.995370	+	0.0961139i	$0.0306413\pi$
$433$			4.00000i			0.192228i	−0.995370	+	0.0961139i	$0.969359\pi$
$434$		0			0
$435$	−40.0000	−	20.0000i	−1.91785	−	0.958927i
$436$	2.00000			0.0957826
$437$		−	8.00000i		−	0.382692i
$438$		−	12.0000i		−	0.573382i
$439$	−16.0000			−0.763638			−0.381819	−	0.924237i	$-0.624702\pi$
$439$	−16.0000			−0.763638			−0.381819	+	0.924237i	$0.624702\pi$
$440$	−2.00000	−	1.00000i	−0.0953463	−	0.0476731i
$441$	−7.00000			−0.333333
$442$		−	12.0000i		−	0.570782i
$443$			14.0000i			0.665160i	0.943075	+	0.332580i	$0.107919\pi$
$443$			14.0000i			0.665160i	−0.943075	+	0.332580i	$0.892081\pi$
$444$	−16.0000			−0.759326
$445$	−18.0000	+	36.0000i	−0.853282	+	1.70656i
$446$	14.0000			0.662919
$447$		−	28.0000i		−	1.32435i
$448$		0			0
$449$	−18.0000			−0.849473			−0.424736	−	0.905317i	$-0.639633\pi$
$449$	−18.0000			−0.849473			−0.424736	+	0.905317i	$0.639633\pi$
$450$	−4.00000	+	3.00000i	−0.188562	+	0.141421i
$451$	−2.00000			−0.0941763
$452$			12.0000i			0.564433i
$453$		−	16.0000i		−	0.751746i
$454$	−24.0000			−1.12638
$455$		0			0
$456$	8.00000			0.374634
$457$			18.0000i			0.842004i	0.907060	+	0.421002i	$0.138322\pi$
$457$			18.0000i			0.842004i	−0.907060	+	0.421002i	$0.861678\pi$
$458$			6.00000i			0.280362i
$459$	24.0000			1.12022
$460$	−4.00000	−	2.00000i	−0.186501	−	0.0932505i
$461$	2.00000			0.0931493			0.0465746	−	0.998915i	$-0.485169\pi$
$461$	2.00000			0.0931493			0.0465746	+	0.998915i	$0.485169\pi$
$462$		0			0
$463$			34.0000i			1.58011i	0.613033	+	0.790057i	$0.289949\pi$
$463$			34.0000i			1.58011i	−0.613033	+	0.790057i	$0.710051\pi$
$464$	10.0000			0.464238
$465$	32.0000	+	16.0000i	1.48396	+	0.741982i
$466$	10.0000			0.463241
$467$		−	6.00000i		−	0.277647i	−0.990317	−	0.138823i	$-0.955668\pi$
$467$		−	6.00000i		−	0.277647i	0.990317	−	0.138823i	$-0.0443321\pi$
$468$			2.00000i			0.0924500i
$469$		0			0
$470$	−2.00000	+	4.00000i	−0.0922531	+	0.184506i
$471$	8.00000			0.368621
$472$		−	12.0000i		−	0.552345i
$473$		0			0
$474$	−24.0000			−1.10236
$475$	12.0000	+	16.0000i	0.550598	+	0.734130i
$476$		0			0
$477$		0			0
$478$		−	12.0000i		−	0.548867i
$479$	4.00000			0.182765			0.0913823	−	0.995816i	$-0.470871\pi$
$479$	4.00000			0.182765			0.0913823	+	0.995816i	$0.470871\pi$
$480$	2.00000	−	4.00000i	0.0912871	−	0.182574i
$481$	−16.0000			−0.729537
$482$		−	2.00000i		−	0.0910975i
$483$		0			0
$484$	−1.00000			−0.0454545
$485$	−24.0000	−	12.0000i	−1.08978	−	0.544892i
$486$	−10.0000			−0.453609
$487$			2.00000i			0.0906287i	0.998973	+	0.0453143i	$0.0144289\pi$
$487$			2.00000i			0.0906287i	−0.998973	+	0.0453143i	$0.985571\pi$
$488$			10.0000i			0.452679i
$489$	12.0000			0.542659
$490$	14.0000	+	7.00000i	0.632456	+	0.316228i
$491$	36.0000			1.62466			0.812329	−	0.583200i	$-0.198200\pi$
$491$	36.0000			1.62466			0.812329	+	0.583200i	$0.198200\pi$
$492$		−	4.00000i		−	0.180334i
$493$			60.0000i			2.70226i
$494$	8.00000			0.359937
$495$	−1.00000	+	2.00000i	−0.0449467	+	0.0898933i
$496$	−8.00000			−0.359211
$497$		0			0
$498$		−	32.0000i		−	1.43395i
$499$	−20.0000			−0.895323			−0.447661	−	0.894203i	$-0.647743\pi$
$499$	−20.0000			−0.895323			−0.447661	+	0.894203i	$0.647743\pi$
$500$	11.0000	−	2.00000i	0.491935	−	0.0894427i
$501$	40.0000			1.78707
$502$			20.0000i			0.892644i
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$		0			0
$505$	2.00000	−	4.00000i	0.0889988	−	0.177998i
$506$	−2.00000			−0.0889108
$507$		−	18.0000i		−	0.799408i
$508$			4.00000i			0.177471i
$509$	14.0000			0.620539			0.310270	−	0.950649i	$-0.399581\pi$
$509$	14.0000			0.620539			0.310270	+	0.950649i	$0.399581\pi$
$510$	24.0000	+	12.0000i	1.06274	+	0.531369i
$511$		0			0
$512$			1.00000i			0.0441942i
$513$			16.0000i			0.706417i
$514$	8.00000			0.352865
$515$	28.0000	+	14.0000i	1.23383	+	0.616914i
$516$		0			0
$517$			2.00000i			0.0879599i
$518$		0			0
$519$	−4.00000			−0.175581
$520$	2.00000	−	4.00000i	0.0877058	−	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
$523$			4.00000i			0.174908i	0.996169	+	0.0874539i	$0.0278730\pi$
$523$			4.00000i			0.174908i	−0.996169	+	0.0874539i	$0.972127\pi$
$524$	8.00000			0.349482
$525$		0			0
$526$	−12.0000			−0.523225
$527$		−	48.0000i		−	2.09091i
$528$		−	2.00000i		−	0.0870388i
$529$	19.0000			0.826087
$530$		0			0
$531$	−12.0000			−0.520756
$532$		0			0
$533$		−	4.00000i		−	0.173259i
$534$	−36.0000			−1.55787
$535$	24.0000	+	12.0000i	1.03761	+	0.518805i
$536$	−6.00000			−0.259161
$537$			8.00000i			0.345225i
$538$		−	18.0000i		−	0.776035i
$539$	7.00000			0.301511
$540$	8.00000	+	4.00000i	0.344265	+	0.172133i
$541$	−22.0000			−0.945854			−0.472927	−	0.881102i	$-0.656803\pi$
$541$	−22.0000			−0.945854			−0.472927	+	0.881102i	$0.656803\pi$
$542$		−	8.00000i		−	0.343629i
$543$			12.0000i			0.514969i
$544$	−6.00000			−0.257248
$545$	−2.00000	+	4.00000i	−0.0856706	+	0.171341i
$546$		0			0
$547$		−	16.0000i		−	0.684111i	−0.939680	−	0.342055i	$-0.888877\pi$
$547$		−	16.0000i		−	0.684111i	0.939680	−	0.342055i	$-0.111123\pi$
$548$			12.0000i			0.512615i
$549$	10.0000			0.426790
$550$	4.00000	−	3.00000i	0.170561	−	0.127920i
$551$	−40.0000			−1.70406
$552$		−	4.00000i		−	0.170251i
$553$		0			0
$554$	−22.0000			−0.934690
$555$	16.0000	−	32.0000i	0.679162	−	1.35832i
$556$	8.00000			0.339276
$557$		−	38.0000i		−	1.61011i	−0.593199	−	0.805056i	$-0.702135\pi$
$557$		−	38.0000i		−	1.61011i	0.593199	−	0.805056i	$-0.297865\pi$
$558$			8.00000i			0.338667i
$559$		0			0
$560$		0			0
$561$	12.0000			0.506640
$562$		−	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	$-0.0268622\pi$
$564$	−4.00000			−0.168430
$565$	−24.0000	−	12.0000i	−1.00969	−	0.504844i
$566$	32.0000			1.34506
$567$		0			0
$568$		0			0
$569$	−26.0000			−1.08998			−0.544988	−	0.838444i	$-0.683466\pi$
$569$	−26.0000			−1.08998			−0.544988	+	0.838444i	$0.683466\pi$
$570$	−8.00000	+	16.0000i	−0.335083	+	0.670166i
$571$	12.0000			0.502184			0.251092	−	0.967963i	$-0.419210\pi$
$571$	12.0000			0.502184			0.251092	+	0.967963i	$0.419210\pi$
$572$		−	2.00000i		−	0.0836242i
$573$		−	32.0000i		−	1.33682i
$574$		0			0
$575$	8.00000	−	6.00000i	0.333623	−	0.250217i
$576$	1.00000			0.0416667
$577$		−	4.00000i		−	0.166522i	−0.996528	−	0.0832611i	$-0.973466\pi$
$577$		−	4.00000i		−	0.166522i	0.996528	−	0.0832611i	$-0.0265335\pi$
$578$		−	19.0000i		−	0.790296i
$579$	20.0000			0.831172
$580$	−10.0000	+	20.0000i	−0.415227	+	0.830455i
$581$		0			0
$582$		−	24.0000i		−	0.994832i
$583$		0			0
$584$	−6.00000			−0.248282
$585$	−4.00000	−	2.00000i	−0.165380	−	0.0826898i
$586$	14.0000			0.578335
$587$		−	22.0000i		−	0.908037i	−0.890992	−	0.454019i	$-0.849990\pi$
$587$		−	22.0000i		−	0.908037i	0.890992	−	0.454019i	$-0.150010\pi$
$588$			14.0000i			0.577350i
$589$	32.0000			1.31854
$590$	24.0000	+	12.0000i	0.988064	+	0.494032i
$591$	4.00000			0.164538
$592$			8.00000i			0.328798i
$593$			18.0000i			0.739171i	0.929197	+	0.369586i	$0.120500\pi$
$593$			18.0000i			0.739171i	−0.929197	+	0.369586i	$0.879500\pi$
$594$	4.00000			0.164122
$595$		0			0
$596$	−14.0000			−0.573462
$597$		−	16.0000i		−	0.654836i
$598$		−	4.00000i		−	0.163572i
$599$	8.00000			0.326871			0.163436	−	0.986554i	$-0.447742\pi$
$599$	8.00000			0.326871			0.163436	+	0.986554i	$0.447742\pi$
$600$	6.00000	+	8.00000i	0.244949	+	0.326599i
$601$	−34.0000			−1.38689			−0.693444	−	0.720510i	$-0.743908\pi$
$601$	−34.0000			−1.38689			−0.693444	+	0.720510i	$0.743908\pi$
$602$		0			0
$603$			6.00000i			0.244339i
$604$	−8.00000			−0.325515
$605$	1.00000	−	2.00000i	0.0406558	−	0.0813116i
$606$	4.00000			0.162489
$607$		−	28.0000i		−	1.13648i	−0.822861	−	0.568242i	$-0.807624\pi$
$607$		−	28.0000i		−	1.13648i	0.822861	−	0.568242i	$-0.192376\pi$
$608$		−	4.00000i		−	0.162221i
$609$		0			0
$610$	−20.0000	−	10.0000i	−0.809776	−	0.404888i
$611$	−4.00000			−0.161823
$612$			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	$0.961336\pi$
$614$	4.00000			0.161427
$615$	8.00000	+	4.00000i	0.322591	+	0.161296i
$616$		0			0
$617$			36.0000i			1.44931i	0.689114	+	0.724653i	$0.258000\pi$
$617$			36.0000i			1.44931i	−0.689114	+	0.724653i	$0.742000\pi$
$618$			28.0000i			1.12633i
$619$	4.00000			0.160774			0.0803868	−	0.996764i	$-0.474384\pi$
$619$	4.00000			0.160774			0.0803868	+	0.996764i	$0.474384\pi$
$620$	8.00000	−	16.0000i	0.321288	−	0.642575i
$621$	8.00000			0.321029
$622$			32.0000i			1.28308i
$623$		0			0
$624$	4.00000			0.160128
$625$	−7.00000	+	24.0000i	−0.280000	+	0.960000i
$626$	−16.0000			−0.639489
$627$			8.00000i			0.319489i
$628$		−	4.00000i		−	0.159617i
$629$	−48.0000			−1.91389
$630$		0			0
$631$	−40.0000			−1.59237			−0.796187	−	0.605050i	$-0.793153\pi$
$631$	−40.0000			−1.59237			−0.796187	+	0.605050i	$0.793153\pi$
$632$			12.0000i			0.477334i
$633$		−	16.0000i		−	0.635943i
$634$	28.0000			1.11202
$635$	−8.00000	−	4.00000i	−0.317470	−	0.158735i
$636$		0			0
$637$			14.0000i			0.554700i
$638$			10.0000i			0.395904i
$639$		0			0
$640$	−2.00000	−	1.00000i	−0.0790569	−	0.0395285i
$641$	−18.0000			−0.710957			−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000			−0.710957			−0.355479	+	0.934684i	$0.615682\pi$
$642$			24.0000i			0.947204i
$643$		−	26.0000i		−	1.02534i	−0.858586	−	0.512670i	$-0.828656\pi$
$643$		−	26.0000i		−	1.02534i	0.858586	−	0.512670i	$-0.171344\pi$
$644$		0			0
$645$		0			0
$646$	24.0000			0.944267
$647$			30.0000i			1.17942i	0.807614	+	0.589711i	$0.200758\pi$
$647$			30.0000i			1.17942i	−0.807614	+	0.589711i	$0.799242\pi$
$648$			11.0000i			0.432121i
$649$	12.0000			0.471041
$650$	6.00000	+	8.00000i	0.235339	+	0.313786i
$651$		0			0
$652$		−	6.00000i		−	0.234978i
$653$		−	24.0000i		−	0.939193i	−0.882881	−	0.469596i	$-0.844399\pi$
$653$		−	24.0000i		−	0.939193i	0.882881	−	0.469596i	$-0.155601\pi$
$654$	−4.00000			−0.156412
$655$	−8.00000	+	16.0000i	−0.312586	+	0.625172i
$656$	−2.00000			−0.0780869
$657$			6.00000i			0.234082i
$658$		0			0
$659$	32.0000			1.24654			0.623272	−	0.782006i	$-0.285803\pi$
$659$	32.0000			1.24654			0.623272	+	0.782006i	$0.285803\pi$
$660$	4.00000	+	2.00000i	0.155700	+	0.0778499i
$661$	−42.0000			−1.63361			−0.816805	−	0.576913i	$-0.804257\pi$
$661$	−42.0000			−1.63361			−0.816805	+	0.576913i	$0.804257\pi$
$662$			12.0000i			0.466393i
$663$			24.0000i			0.932083i
$664$	−16.0000			−0.620920
$665$		0			0
$666$	8.00000			0.309994
$667$			20.0000i			0.774403i
$668$		−	20.0000i		−	0.773823i
$669$	−28.0000			−1.08254
$670$	6.00000	−	12.0000i	0.231800	−	0.463600i
$671$	−10.0000			−0.386046
$672$		0			0
$673$			10.0000i			0.385472i	0.981251	+	0.192736i	$0.0617360\pi$
$673$			10.0000i			0.385472i	−0.981251	+	0.192736i	$0.938264\pi$
$674$	10.0000			0.385186
$675$	−16.0000	+	12.0000i	−0.615840	+	0.461880i
$676$	−9.00000			−0.346154
$677$			14.0000i			0.538064i	0.963131	+	0.269032i	$0.0867037\pi$
$677$			14.0000i			0.538064i	−0.963131	+	0.269032i	$0.913296\pi$
$678$		−	24.0000i		−	0.921714i
$679$		0			0
$680$	6.00000	−	12.0000i	0.230089	−	0.460179i
$681$	48.0000			1.83936
$682$		−	8.00000i		−	0.306336i
$683$			26.0000i			0.994862i	0.867503	+	0.497431i	$0.165723\pi$
$683$			26.0000i			0.994862i	−0.867503	+	0.497431i	$0.834277\pi$
$684$	−4.00000			−0.152944
$685$	−24.0000	−	12.0000i	−0.916993	−	0.458496i
$686$		0			0
$687$		−	12.0000i		−	0.457829i
$688$		0			0
$689$		0			0
$690$	8.00000	+	4.00000i	0.304555	+	0.152277i
$691$	−20.0000			−0.760836			−0.380418	−	0.924815i	$-0.624220\pi$
$691$	−20.0000			−0.760836			−0.380418	+	0.924815i	$0.624220\pi$
$692$			2.00000i			0.0760286i
$693$		0			0
$694$	12.0000			0.455514
$695$	−8.00000	+	16.0000i	−0.303457	+	0.606915i
$696$	−20.0000			−0.758098
$697$		−	12.0000i		−	0.454532i
$698$			34.0000i			1.28692i
$699$	−20.0000			−0.756469
$700$		0			0
$701$	6.00000			0.226617			0.113308	−	0.993560i	$-0.463855\pi$
$701$	6.00000			0.226617			0.113308	+	0.993560i	$0.463855\pi$
$702$			8.00000i			0.301941i
$703$		−	32.0000i		−	1.20690i
$704$	−1.00000			−0.0376889
$705$	4.00000	−	8.00000i	0.150649	−	0.301297i
$706$	8.00000			0.301084
$707$		0			0
$708$			24.0000i			0.901975i
$709$	26.0000			0.976450			0.488225	−	0.872718i	$-0.337644\pi$
$709$	26.0000			0.976450			0.488225	+	0.872718i	$0.337644\pi$
$710$		0			0
$711$	12.0000			0.450035
$712$			18.0000i			0.674579i
$713$		−	16.0000i		−	0.599205i
$714$		0			0
$715$	4.00000	+	2.00000i	0.149592	+	0.0747958i
$716$	4.00000			0.149487
$717$			24.0000i			0.896296i
$718$		−	12.0000i		−	0.447836i
$719$	24.0000			0.895049			0.447524	−	0.894272i	$-0.352306\pi$
$719$	24.0000			0.895049			0.447524	+	0.894272i	$0.352306\pi$
$720$	−1.00000	+	2.00000i	−0.0372678	+	0.0745356i
$721$		0			0
$722$		−	3.00000i		−	0.111648i
$723$			4.00000i			0.148762i
$724$	6.00000			0.222988
$725$	−30.0000	−	40.0000i	−1.11417	−	1.48556i
$726$	2.00000			0.0742270
$727$			30.0000i			1.11264i	0.830969	+	0.556319i	$0.187787\pi$
$727$			30.0000i			1.11264i	−0.830969	+	0.556319i	$0.812213\pi$
$728$		0			0
$729$	−13.0000			−0.481481
$730$	6.00000	−	12.0000i	0.222070	−	0.444140i
$731$		0			0
$732$		−	20.0000i		−	0.739221i
$733$		−	50.0000i		−	1.84679i	−0.383849	−	0.923396i	$-0.625402\pi$
$733$		−	50.0000i		−	1.84679i	0.383849	−	0.923396i	$-0.374598\pi$
$734$	−26.0000			−0.959678
$735$	−28.0000	−	14.0000i	−1.03280	−	0.516398i
$736$	−2.00000			−0.0737210
$737$		−	6.00000i		−	0.221013i
$738$			2.00000i			0.0736210i
$739$	32.0000			1.17714			0.588570	−	0.808447i	$-0.299691\pi$
$739$	32.0000			1.17714			0.588570	+	0.808447i	$0.299691\pi$
$740$	−16.0000	−	8.00000i	−0.588172	−	0.294086i
$741$	−16.0000			−0.587775
$742$		0			0
$743$			28.0000i			1.02722i	0.858024	+	0.513610i	$0.171692\pi$
$743$			28.0000i			1.02722i	−0.858024	+	0.513610i	$0.828308\pi$
$744$	16.0000			0.586588
$745$	14.0000	−	28.0000i	0.512920	−	1.02584i
$746$	−14.0000			−0.512576
$747$			16.0000i			0.585409i
$748$		−	6.00000i		−	0.219382i
$749$		0			0
$750$	−22.0000	+	4.00000i	−0.803326	+	0.146059i
$751$		0			0			−	1.00000i	$-0.5\pi$
$751$		0			0				1.00000i	$0.5\pi$
$752$			2.00000i			0.0729325i
$753$		−	40.0000i		−	1.45768i
$754$	−20.0000			−0.728357
$755$	8.00000	−	16.0000i	0.291150	−	0.582300i
$756$		0			0
$757$		−	12.0000i		−	0.436147i	−0.975932	−	0.218074i	$-0.930023\pi$
$757$		−	12.0000i		−	0.436147i	0.975932	−	0.218074i	$-0.0699773\pi$
$758$			4.00000i			0.145287i
$759$	4.00000			0.145191
$760$	8.00000	+	4.00000i	0.290191	+	0.145095i
$761$	42.0000			1.52250			0.761249	−	0.648459i	$-0.224586\pi$
$761$	42.0000			1.52250			0.761249	+	0.648459i	$0.224586\pi$
$762$		−	8.00000i		−	0.289809i
$763$		0			0
$764$	−16.0000			−0.578860
$765$	−12.0000	−	6.00000i	−0.433861	−	0.216930i
$766$	18.0000			0.650366
$767$			24.0000i			0.866590i
$768$		−	2.00000i		−	0.0721688i
$769$	−30.0000			−1.08183			−0.540914	−	0.841078i	$-0.681921\pi$
$769$	−30.0000			−1.08183			−0.540914	+	0.841078i	$0.681921\pi$
$770$		0			0
$771$	−16.0000			−0.576226
$772$		−	10.0000i		−	0.359908i
$773$			32.0000i			1.15096i	0.817816	+	0.575480i	$0.195185\pi$
$773$			32.0000i			1.15096i	−0.817816	+	0.575480i	$0.804815\pi$
$774$		0			0
$775$	24.0000	+	32.0000i	0.862105	+	1.14947i
$776$	−12.0000			−0.430775
$777$		0			0
$778$			6.00000i			0.215110i
$779$	8.00000			0.286630
$780$	−4.00000	+	8.00000i	−0.143223	+	0.286446i
$781$		0			0
$782$		−	12.0000i		−	0.429119i
$783$		−	40.0000i		−	1.42948i
$784$	7.00000			0.250000
$785$	8.00000	+	4.00000i	0.285532	+	0.142766i
$786$	−16.0000			−0.570701
$787$		−	52.0000i		−	1.85360i	−0.375555	−	0.926800i	$-0.622548\pi$
$787$		−	52.0000i		−	1.85360i	0.375555	−	0.926800i	$-0.377452\pi$
$788$		−	2.00000i		−	0.0712470i
$789$	24.0000			0.854423
$790$	−24.0000	−	12.0000i	−0.853882	−	0.426941i
$791$		0			0
$792$			1.00000i			0.0355335i
$793$		−	20.0000i		−	0.710221i
$794$	28.0000			0.993683
$795$		0			0
$796$	−8.00000			−0.283552
$797$			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
$799$	−12.0000			−0.424529
$800$	4.00000	−	3.00000i	0.141421	−	0.106066i
$801$	18.0000			0.635999
$802$			30.0000i			1.05934i
$803$		−	6.00000i		−	0.211735i
$804$	12.0000			0.423207
$805$		0			0
$806$	16.0000			0.563576
$807$			36.0000i			1.26726i
$808$		−	2.00000i		−	0.0703598i
$809$	26.0000			0.914111			0.457056	−	0.889438i	$-0.348904\pi$
$809$	26.0000			0.914111			0.457056	+	0.889438i	$0.348904\pi$
$810$	−22.0000	−	11.0000i	−0.773001	−	0.386501i
$811$	28.0000			0.983213			0.491606	−	0.870817i	$-0.336410\pi$
$811$	28.0000			0.983213			0.491606	+	0.870817i	$0.336410\pi$
$812$		0			0
$813$			16.0000i			0.561144i
$814$	−8.00000			−0.280400
$815$	12.0000	+	6.00000i	0.420342	+	0.210171i
$816$	12.0000			0.420084
$817$		0			0
$818$		−	2.00000i		−	0.0699284i
$819$		0			0
$820$	2.00000	−	4.00000i	0.0698430	−	0.139686i
$821$	−2.00000			−0.0698005			−0.0349002	−	0.999391i	$-0.511111\pi$
$821$	−2.00000			−0.0698005			−0.0349002	+	0.999391i	$0.511111\pi$
$822$		−	24.0000i		−	0.837096i
$823$		−	34.0000i		−	1.18517i	−0.805510	−	0.592583i	$-0.798108\pi$
$823$		−	34.0000i		−	1.18517i	0.805510	−	0.592583i	$-0.201892\pi$
$824$	14.0000			0.487713
$825$	−8.00000	+	6.00000i	−0.278524	+	0.208893i
$826$		0			0
$827$		−	4.00000i		−	0.139094i	−0.997579	−	0.0695468i	$-0.977845\pi$
$827$		−	4.00000i		−	0.139094i	0.997579	−	0.0695468i	$-0.0221553\pi$
$828$			2.00000i			0.0695048i
$829$	26.0000			0.903017			0.451509	−	0.892267i	$-0.350886\pi$
$829$	26.0000			0.903017			0.451509	+	0.892267i	$0.350886\pi$
$830$	16.0000	−	32.0000i	0.555368	−	1.11074i
$831$	44.0000			1.52634
$832$		−	2.00000i		−	0.0693375i
$833$			42.0000i			1.45521i
$834$	−16.0000			−0.554035
$835$	40.0000	+	20.0000i	1.38426	+	0.692129i
$836$	4.00000			0.138343
$837$			32.0000i			1.10608i
$838$			20.0000i			0.690889i
$839$	24.0000			0.828572			0.414286	−	0.910147i	$-0.364031\pi$
$839$	24.0000			0.828572			0.414286	+	0.910147i	$0.364031\pi$
$840$		0			0
$841$	71.0000			2.44828
$842$			2.00000i			0.0689246i
$843$			36.0000i			1.23991i
$844$	−8.00000			−0.275371
$845$	9.00000	−	18.0000i	0.309609	−	0.619219i
$846$	2.00000			0.0687614
$847$		0			0
$848$		0			0
$849$	−64.0000			−2.19647
$850$	18.0000	+	24.0000i	0.617395	+	0.823193i
$851$	−16.0000			−0.548473
$852$		0			0
$853$		−	26.0000i		−	0.890223i	−0.895475	−	0.445112i	$-0.853164\pi$
$853$		−	26.0000i		−	0.890223i	0.895475	−	0.445112i	$-0.146836\pi$
$854$		0			0
$855$	4.00000	−	8.00000i	0.136797	−	0.273594i
$856$	12.0000			0.410152
$857$			46.0000i			1.57133i	0.618652	+	0.785665i	$0.287679\pi$
$857$			46.0000i			1.57133i	−0.618652	+	0.785665i	$0.712321\pi$
$858$			4.00000i			0.136558i
$859$	−20.0000			−0.682391			−0.341196	−	0.939992i	$-0.610832\pi$
$859$	−20.0000			−0.682391			−0.341196	+	0.939992i	$0.610832\pi$
$860$		0			0
$861$		0			0
$862$		−	28.0000i		−	0.953684i
$863$			6.00000i			0.204242i	0.994772	+	0.102121i	$0.0325630\pi$
$863$			6.00000i			0.204242i	−0.994772	+	0.102121i	$0.967437\pi$
$864$	4.00000			0.136083
$865$	−4.00000	−	2.00000i	−0.136004	−	0.0680020i
$866$	−4.00000			−0.135926
$867$			38.0000i			1.29055i
$868$		0			0
$869$	−12.0000			−0.407072
$870$	20.0000	−	40.0000i	0.678064	−	1.35613i
$871$	12.0000			0.406604
$872$			2.00000i			0.0677285i
$873$			12.0000i			0.406138i
$874$	8.00000			0.270604
$875$		0			0
$876$	12.0000			0.405442
$877$		−	38.0000i		−	1.28317i	−0.767052	−	0.641584i	$-0.778277\pi$
$877$		−	38.0000i		−	1.28317i	0.767052	−	0.641584i	$-0.221723\pi$
$878$		−	16.0000i		−	0.539974i
$879$	−28.0000			−0.944417
$880$	1.00000	−	2.00000i	0.0337100	−	0.0674200i
$881$	18.0000			0.606435			0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000			0.606435			0.303218	+	0.952921i	$0.401939\pi$
$882$		−	7.00000i		−	0.235702i
$883$			30.0000i			1.00958i	0.863242	+	0.504790i	$0.168430\pi$
$883$			30.0000i			1.00958i	−0.863242	+	0.504790i	$0.831570\pi$
$884$	12.0000			0.403604
$885$	−48.0000	−	24.0000i	−1.61350	−	0.806751i
$886$	−14.0000			−0.470339
$887$			4.00000i			0.134307i	0.997743	+	0.0671534i	$0.0213917\pi$
$887$			4.00000i			0.134307i	−0.997743	+	0.0671534i	$0.978608\pi$
$888$		−	16.0000i		−	0.536925i
$889$		0			0
$890$	−36.0000	−	18.0000i	−1.20672	−	0.603361i
$891$	−11.0000			−0.368514
$892$			14.0000i			0.468755i
$893$		−	8.00000i		−	0.267710i
$894$	28.0000			0.936460
$895$	−4.00000	+	8.00000i	−0.133705	+	0.267411i
$896$		0			0
$897$			8.00000i			0.267112i
$898$		−	18.0000i		−	0.600668i
$899$	−80.0000			−2.66815
$900$	−3.00000	−	4.00000i	−0.100000	−	0.133333i
$901$		0			0
$902$		−	2.00000i		−	0.0665927i
$903$		0			0
$904$	−12.0000			−0.399114
$905$	−6.00000	+	12.0000i	−0.199447	+	0.398893i
$906$	16.0000			0.531564
$907$			50.0000i			1.66022i	0.557598	+	0.830111i	$0.311723\pi$
$907$			50.0000i			1.66022i	−0.557598	+	0.830111i	$0.688277\pi$
$908$		−	24.0000i		−	0.796468i
$909$	−2.00000			−0.0663358
$910$		0			0
$911$	−32.0000			−1.06021			−0.530104	−	0.847933i	$-0.677847\pi$
$911$	−32.0000			−1.06021			−0.530104	+	0.847933i	$0.677847\pi$
$912$			8.00000i			0.264906i
$913$		−	16.0000i		−	0.529523i
$914$	−18.0000			−0.595387
$915$	40.0000	+	20.0000i	1.32236	+	0.661180i
$916$	−6.00000			−0.198246
$917$		0			0
$918$			24.0000i			0.792118i
$919$	−8.00000			−0.263896			−0.131948	−	0.991257i	$-0.542123\pi$
$919$	−8.00000			−0.263896			−0.131948	+	0.991257i	$0.542123\pi$
$920$	2.00000	−	4.00000i	0.0659380	−	0.131876i
$921$	−8.00000			−0.263609
$922$			2.00000i			0.0658665i
$923$		0			0
$924$		0			0
$925$	32.0000	−	24.0000i	1.05215	−	0.789115i
$926$	−34.0000			−1.11731
$927$		−	14.0000i		−	0.459820i
$928$			10.0000i			0.328266i
$929$	−30.0000			−0.984268			−0.492134	−	0.870519i	$-0.663783\pi$
$929$	−30.0000			−0.984268			−0.492134	+	0.870519i	$0.663783\pi$
$930$	−16.0000	+	32.0000i	−0.524661	+	1.04932i
$931$	−28.0000			−0.917663
$932$			10.0000i			0.327561i
$933$		−	64.0000i		−	2.09527i
$934$	6.00000			0.196326
$935$	12.0000	+	6.00000i	0.392442	+	0.196221i
$936$	−2.00000			−0.0653720
$937$			18.0000i			0.588034i	0.955800	+	0.294017i	$0.0949923\pi$
$937$			18.0000i			0.588034i	−0.955800	+	0.294017i	$0.905008\pi$
$938$		0			0
$939$	32.0000			1.04428
$940$	−4.00000	−	2.00000i	−0.130466	−	0.0652328i
$941$	−22.0000			−0.717180			−0.358590	−	0.933495i	$-0.616742\pi$
$941$	−22.0000			−0.717180			−0.358590	+	0.933495i	$0.616742\pi$
$942$			8.00000i			0.260654i
$943$		−	4.00000i		−	0.130258i
$944$	12.0000			0.390567
$945$		0			0
$946$		0			0
$947$		−	22.0000i		−	0.714904i	−0.933932	−	0.357452i	$-0.883646\pi$
$947$		−	22.0000i		−	0.714904i	0.933932	−	0.357452i	$-0.116354\pi$
$948$		−	24.0000i		−	0.779484i
$949$	12.0000			0.389536
$950$	−16.0000	+	12.0000i	−0.519109	+	0.389331i
$951$	−56.0000			−1.81592
$952$		0			0
$953$			46.0000i			1.49009i	0.667016	+	0.745043i	$0.267571\pi$
$953$			46.0000i			1.49009i	−0.667016	+	0.745043i	$0.732429\pi$
$954$		0			0
$955$	16.0000	−	32.0000i	0.517748	−	1.03550i
$956$	12.0000			0.388108
$957$		−	20.0000i		−	0.646508i
$958$			4.00000i			0.129234i
$959$		0			0
$960$	4.00000	+	2.00000i	0.129099	+	0.0645497i
$961$	33.0000			1.06452
$962$		−	16.0000i		−	0.515861i
$963$		−	12.0000i		−	0.386695i
$964$	2.00000			0.0644157
$965$	20.0000	+	10.0000i	0.643823	+	0.321911i
$966$		0			0
$967$			4.00000i			0.128631i	0.997930	+	0.0643157i	$0.0204865\pi$
$967$			4.00000i			0.128631i	−0.997930	+	0.0643157i	$0.979514\pi$
$968$		−	1.00000i		−	0.0321412i
$969$	−48.0000			−1.54198
$970$	12.0000	−	24.0000i	0.385297	−	0.770594i
$971$	20.0000			0.641831			0.320915	−	0.947108i	$-0.396010\pi$
$971$	20.0000			0.641831			0.320915	+	0.947108i	$0.396010\pi$
$972$		−	10.0000i		−	0.320750i
$973$		0			0
$974$	−2.00000			−0.0640841
$975$	−12.0000	−	16.0000i	−0.384308	−	0.512410i
$976$	−10.0000			−0.320092
$977$		−	20.0000i		−	0.639857i	−0.947442	−	0.319928i	$-0.896341\pi$
$977$		−	20.0000i		−	0.639857i	0.947442	−	0.319928i	$-0.103659\pi$
$978$			12.0000i			0.383718i
$979$	−18.0000			−0.575282
$980$	−7.00000	+	14.0000i	−0.223607	+	0.447214i
$981$	2.00000			0.0638551
$982$			36.0000i			1.14881i
$983$			22.0000i			0.701691i	0.936433	+	0.350846i	$0.114106\pi$
$983$			22.0000i			0.701691i	−0.936433	+	0.350846i	$0.885894\pi$
$984$	4.00000			0.127515
$985$	4.00000	+	2.00000i	0.127451	+	0.0637253i
$986$	−60.0000			−1.91079
$987$		0			0
$988$			8.00000i			0.254514i
$989$		0			0
$990$	−2.00000	−	1.00000i	−0.0635642	−	0.0317821i
$991$	40.0000			1.27064			0.635321	−	0.772248i	$-0.280868\pi$
$991$	40.0000			1.27064			0.635321	+	0.772248i	$0.280868\pi$
$992$		−	8.00000i		−	0.254000i
$993$		−	24.0000i		−	0.761617i
$994$		0			0
$995$	8.00000	−	16.0000i	0.253617	−	0.507234i
$996$	32.0000			1.01396
$997$			58.0000i			1.83688i	0.395562	+	0.918439i	$0.370550\pi$
$997$			58.0000i			1.83688i	−0.395562	+	0.918439i	$0.629450\pi$
$998$		−	20.0000i		−	0.633089i
$999$	32.0000			1.01244

Twists

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

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

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

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -2.00000i q^{3} -1.00000 q^{4} +(1.00000 - 2.00000i) q^{5} +2.00000 q^{6} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} -2.00000i q^{3} -1.00000 q^{4} +(1.00000 - 2.00000i) q^{5} +2.00000 q^{6} -1.00000i q^{8} -1.00000 q^{9} +(2.00000 + 1.00000i) q^{10} +1.00000 q^{11} +2.00000i q^{12} +2.00000i q^{13} +(-4.00000 - 2.00000i) q^{15} +1.00000 q^{16} +6.00000i q^{17} -1.00000i q^{18} -4.00000 q^{19} +(-1.00000 + 2.00000i) q^{20} +1.00000i q^{22} +2.00000i q^{23} -2.00000 q^{24} +(-3.00000 - 4.00000i) q^{25} -2.00000 q^{26} -4.00000i q^{27} +10.0000 q^{29} +(2.00000 - 4.00000i) q^{30} -8.00000 q^{31} +1.00000i q^{32} -2.00000i q^{33} -6.00000 q^{34} +1.00000 q^{36} +8.00000i q^{37} -4.00000i q^{38} +4.00000 q^{39} +(-2.00000 - 1.00000i) q^{40} -2.00000 q^{41} -1.00000 q^{44} +(-1.00000 + 2.00000i) q^{45} -2.00000 q^{46} +2.00000i q^{47} -2.00000i q^{48} +7.00000 q^{49} +(4.00000 - 3.00000i) q^{50} +12.0000 q^{51} -2.00000i q^{52} +4.00000 q^{54} +(1.00000 - 2.00000i) q^{55} +8.00000i q^{57} +10.0000i q^{58} +12.0000 q^{59} +(4.00000 + 2.00000i) q^{60} -10.0000 q^{61} -8.00000i q^{62} -1.00000 q^{64} +(4.00000 + 2.00000i) q^{65} +2.00000 q^{66} -6.00000i q^{67} -6.00000i q^{68} +4.00000 q^{69} +1.00000i q^{72} -6.00000i q^{73} -8.00000 q^{74} +(-8.00000 + 6.00000i) q^{75} +4.00000 q^{76} +4.00000i q^{78} -12.0000 q^{79} +(1.00000 - 2.00000i) q^{80} -11.0000 q^{81} -2.00000i q^{82} -16.0000i q^{83} +(12.0000 + 6.00000i) q^{85} -20.0000i q^{87} -1.00000i q^{88} -18.0000 q^{89} +(-2.00000 - 1.00000i) q^{90} -2.00000i q^{92} +16.0000i q^{93} -2.00000 q^{94} +(-4.00000 + 8.00000i) q^{95} +2.00000 q^{96} -12.0000i q^{97} +7.00000i q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + 2 q^{5} + 4 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 + 2 * q^5 + 4 * q^6 - 2 * q^9	\( 2 q - 2 q^{4} + 2 q^{5} + 4 q^{6} - 2 q^{9} + 4 q^{10} + 2 q^{11} - 8 q^{15} + 2 q^{16} - 8 q^{19} - 2 q^{20} - 4 q^{24} - 6 q^{25} - 4 q^{26} + 20 q^{29} + 4 q^{30} - 16 q^{31} - 12 q^{34} + 2 q^{36} + 8 q^{39} - 4 q^{40} - 4 q^{41} - 2 q^{44} - 2 q^{45} - 4 q^{46} + 14 q^{49} + 8 q^{50} + 24 q^{51} + 8 q^{54} + 2 q^{55} + 24 q^{59} + 8 q^{60} - 20 q^{61} - 2 q^{64} + 8 q^{65} + 4 q^{66} + 8 q^{69} - 16 q^{74} - 16 q^{75} + 8 q^{76} - 24 q^{79} + 2 q^{80} - 22 q^{81} + 24 q^{85} - 36 q^{89} - 4 q^{90} - 4 q^{94} - 8 q^{95} + 4 q^{96} - 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 + 2 * q^5 + 4 * q^6 - 2 * q^9 + 4 * q^10 + 2 * q^11 - 8 * q^15 + 2 * q^16 - 8 * q^19 - 2 * q^20 - 4 * q^24 - 6 * q^25 - 4 * q^26 + 20 * q^29 + 4 * q^30 - 16 * q^31 - 12 * q^34 + 2 * q^36 + 8 * q^39 - 4 * q^40 - 4 * q^41 - 2 * q^44 - 2 * q^45 - 4 * q^46 + 14 * q^49 + 8 * q^50 + 24 * q^51 + 8 * q^54 + 2 * q^55 + 24 * q^59 + 8 * q^60 - 20 * q^61 - 2 * q^64 + 8 * q^65 + 4 * q^66 + 8 * q^69 - 16 * q^74 - 16 * q^75 + 8 * q^76 - 24 * q^79 + 2 * q^80 - 22 * q^81 + 24 * q^85 - 36 * q^89 - 4 * q^90 - 4 * q^94 - 8 * q^95 + 4 * q^96 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more