LMFDB - Embedded newform 552.2.f.a.277.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [552,2,Mod(277,552)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(552, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("552.277");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 552 = 2^{3} \cdot 3 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	552.f (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:	$4.40774219157$
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			277.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	552.277
Dual form			552.2.f.a.277.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.00000 + 1.00000i) q^{2} +1.00000i q^{3} +2.00000i q^{4} +2.00000i q^{5} +(-1.00000 + 1.00000i) q^{6} +4.00000 q^{7} +(-2.00000 + 2.00000i) q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+(1.00000 + 1.00000i) q^{2} +1.00000i q^{3} +2.00000i q^{4} +2.00000i q^{5} +(-1.00000 + 1.00000i) q^{6} +4.00000 q^{7} +(-2.00000 + 2.00000i) q^{8} -1.00000 q^{9} +(-2.00000 + 2.00000i) q^{10} +2.00000i q^{11} -2.00000 q^{12} -4.00000i q^{13} +(4.00000 + 4.00000i) q^{14} -2.00000 q^{15} -4.00000 q^{16} +(-1.00000 - 1.00000i) q^{18} -4.00000i q^{19} -4.00000 q^{20} +4.00000i q^{21} +(-2.00000 + 2.00000i) q^{22} +1.00000 q^{23} +(-2.00000 - 2.00000i) q^{24} +1.00000 q^{25} +(4.00000 - 4.00000i) q^{26} -1.00000i q^{27} +8.00000i q^{28} -2.00000i q^{29} +(-2.00000 - 2.00000i) q^{30} -6.00000 q^{31} +(-4.00000 - 4.00000i) q^{32} -2.00000 q^{33} +8.00000i q^{35} -2.00000i q^{36} +6.00000i q^{37} +(4.00000 - 4.00000i) q^{38} +4.00000 q^{39} +(-4.00000 - 4.00000i) q^{40} +2.00000 q^{41} +(-4.00000 + 4.00000i) q^{42} -8.00000i q^{43} -4.00000 q^{44} -2.00000i q^{45} +(1.00000 + 1.00000i) q^{46} -8.00000 q^{47} -4.00000i q^{48} +9.00000 q^{49} +(1.00000 + 1.00000i) q^{50} +8.00000 q^{52} +6.00000i q^{53} +(1.00000 - 1.00000i) q^{54} -4.00000 q^{55} +(-8.00000 + 8.00000i) q^{56} +4.00000 q^{57} +(2.00000 - 2.00000i) q^{58} -4.00000i q^{59} -4.00000i q^{60} +6.00000i q^{61} +(-6.00000 - 6.00000i) q^{62} -4.00000 q^{63} -8.00000i q^{64} +8.00000 q^{65} +(-2.00000 - 2.00000i) q^{66} -4.00000i q^{67} +1.00000i q^{69} +(-8.00000 + 8.00000i) q^{70} +12.0000 q^{71} +(2.00000 - 2.00000i) q^{72} +14.0000 q^{73} +(-6.00000 + 6.00000i) q^{74} +1.00000i q^{75} +8.00000 q^{76} +8.00000i q^{77} +(4.00000 + 4.00000i) q^{78} +8.00000 q^{79} -8.00000i q^{80} +1.00000 q^{81} +(2.00000 + 2.00000i) q^{82} -14.0000i q^{83} -8.00000 q^{84} +(8.00000 - 8.00000i) q^{86} +2.00000 q^{87} +(-4.00000 - 4.00000i) q^{88} +12.0000 q^{89} +(2.00000 - 2.00000i) q^{90} -16.0000i q^{91} +2.00000i q^{92} -6.00000i q^{93} +(-8.00000 - 8.00000i) q^{94} +8.00000 q^{95} +(4.00000 - 4.00000i) q^{96} -14.0000 q^{97} +(9.00000 + 9.00000i) q^{98} -2.00000i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} - 2 q^{6} + 8 q^{7} - 4 q^{8} - 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^2 - 2 * q^6 + 8 * q^7 - 4 * q^8 - 2 * q^9	$ 2 q + 2 q^{2} - 2 q^{6} + 8 q^{7} - 4 q^{8} - 2 q^{9} - 4 q^{10} - 4 q^{12} + 8 q^{14} - 4 q^{15} - 8 q^{16} - 2 q^{18} - 8 q^{20} - 4 q^{22} + 2 q^{23} - 4 q^{24} + 2 q^{25} + 8 q^{26} - 4 q^{30} - 12 q^{31} - 8 q^{32} - 4 q^{33} + 8 q^{38} + 8 q^{39} - 8 q^{40} + 4 q^{41} - 8 q^{42} - 8 q^{44} + 2 q^{46} - 16 q^{47} + 18 q^{49} + 2 q^{50} + 16 q^{52} + 2 q^{54} - 8 q^{55} - 16 q^{56} + 8 q^{57} + 4 q^{58} - 12 q^{62} - 8 q^{63} + 16 q^{65} - 4 q^{66} - 16 q^{70} + 24 q^{71} + 4 q^{72} + 28 q^{73} - 12 q^{74} + 16 q^{76} + 8 q^{78} + 16 q^{79} + 2 q^{81} + 4 q^{82} - 16 q^{84} + 16 q^{86} + 4 q^{87} - 8 q^{88} + 24 q^{89} + 4 q^{90} - 16 q^{94} + 16 q^{95} + 8 q^{96} - 28 q^{97} + 18 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 - 2 * q^6 + 8 * q^7 - 4 * q^8 - 2 * q^9 - 4 * q^10 - 4 * q^12 + 8 * q^14 - 4 * q^15 - 8 * q^16 - 2 * q^18 - 8 * q^20 - 4 * q^22 + 2 * q^23 - 4 * q^24 + 2 * q^25 + 8 * q^26 - 4 * q^30 - 12 * q^31 - 8 * q^32 - 4 * q^33 + 8 * q^38 + 8 * q^39 - 8 * q^40 + 4 * q^41 - 8 * q^42 - 8 * q^44 + 2 * q^46 - 16 * q^47 + 18 * q^49 + 2 * q^50 + 16 * q^52 + 2 * q^54 - 8 * q^55 - 16 * q^56 + 8 * q^57 + 4 * q^58 - 12 * q^62 - 8 * q^63 + 16 * q^65 - 4 * q^66 - 16 * q^70 + 24 * q^71 + 4 * q^72 + 28 * q^73 - 12 * q^74 + 16 * q^76 + 8 * q^78 + 16 * q^79 + 2 * q^81 + 4 * q^82 - 16 * q^84 + 16 * q^86 + 4 * q^87 - 8 * q^88 + 24 * q^89 + 4 * q^90 - 16 * q^94 + 16 * q^95 + 8 * q^96 - 28 * q^97 + 18 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/552\mathbb{Z}\right)^\times$.

$n$	$97$	$185$	$277$	$415$
$\chi(n)$	$1$	$1$	$-1$	$1$

Coefficient data

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

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

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

$2$	1.00000	+	1.00000i	0.707107	+	0.707107i
$2$	1.00000	+	1.00000i	0.707107	+	0.707107i
$3$			1.00000i			0.577350i
$3$			1.00000i			0.577350i
$4$			2.00000i			1.00000i
$5$			2.00000i			0.894427i	0.894427	+	0.447214i	$0.147584\pi$
$5$			2.00000i			0.894427i	−0.894427	+	0.447214i	$0.852416\pi$
$6$	−1.00000	+	1.00000i	−0.408248	+	0.408248i
$7$	4.00000			1.51186			0.755929	−	0.654654i	$-0.227186\pi$
$7$	4.00000			1.51186			0.755929	+	0.654654i	$0.227186\pi$
$8$	−2.00000	+	2.00000i	−0.707107	+	0.707107i
$9$	−1.00000			−0.333333
$10$	−2.00000	+	2.00000i	−0.632456	+	0.632456i
$11$			2.00000i			0.603023i	0.953463	+	0.301511i	$0.0974911\pi$
$11$			2.00000i			0.603023i	−0.953463	+	0.301511i	$0.902509\pi$
$12$	−2.00000			−0.577350
$13$		−	4.00000i		−	1.10940i	−0.832050	−	0.554700i	$-0.812833\pi$
$13$		−	4.00000i		−	1.10940i	0.832050	−	0.554700i	$-0.187167\pi$
$14$	4.00000	+	4.00000i	1.06904	+	1.06904i
$15$	−2.00000			−0.516398
$16$	−4.00000			−1.00000
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$	−1.00000	−	1.00000i	−0.235702	−	0.235702i
$19$		−	4.00000i		−	0.917663i	−0.888523	−	0.458831i	$-0.848268\pi$
$19$		−	4.00000i		−	0.917663i	0.888523	−	0.458831i	$-0.151732\pi$
$20$	−4.00000			−0.894427
$21$			4.00000i			0.872872i
$22$	−2.00000	+	2.00000i	−0.426401	+	0.426401i
$23$	1.00000			0.208514
$23$	1.00000			0.208514
$24$	−2.00000	−	2.00000i	−0.408248	−	0.408248i
$25$	1.00000			0.200000
$26$	4.00000	−	4.00000i	0.784465	−	0.784465i
$27$		−	1.00000i		−	0.192450i
$28$			8.00000i			1.51186i
$29$		−	2.00000i		−	0.371391i	−0.982607	−	0.185695i	$-0.940546\pi$
$29$		−	2.00000i		−	0.371391i	0.982607	−	0.185695i	$-0.0594537\pi$
$30$	−2.00000	−	2.00000i	−0.365148	−	0.365148i
$31$	−6.00000			−1.07763			−0.538816	−	0.842424i	$-0.681128\pi$
$31$	−6.00000			−1.07763			−0.538816	+	0.842424i	$0.681128\pi$
$32$	−4.00000	−	4.00000i	−0.707107	−	0.707107i
$33$	−2.00000			−0.348155
$34$		0			0
$35$			8.00000i			1.35225i
$36$		−	2.00000i		−	0.333333i
$37$			6.00000i			0.986394i	0.869918	+	0.493197i	$0.164172\pi$
$37$			6.00000i			0.986394i	−0.869918	+	0.493197i	$0.835828\pi$
$38$	4.00000	−	4.00000i	0.648886	−	0.648886i
$39$	4.00000			0.640513
$40$	−4.00000	−	4.00000i	−0.632456	−	0.632456i
$41$	2.00000			0.312348			0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000			0.312348			0.156174	+	0.987730i	$0.450084\pi$
$42$	−4.00000	+	4.00000i	−0.617213	+	0.617213i
$43$		−	8.00000i		−	1.21999i	−0.792406	−	0.609994i	$-0.791172\pi$
$43$		−	8.00000i		−	1.21999i	0.792406	−	0.609994i	$-0.208828\pi$
$44$	−4.00000			−0.603023
$45$		−	2.00000i		−	0.298142i
$46$	1.00000	+	1.00000i	0.147442	+	0.147442i
$47$	−8.00000			−1.16692			−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000			−1.16692			−0.583460	+	0.812142i	$0.698301\pi$
$48$		−	4.00000i		−	0.577350i
$49$	9.00000			1.28571
$50$	1.00000	+	1.00000i	0.141421	+	0.141421i
$51$		0			0
$52$	8.00000			1.10940
$53$			6.00000i			0.824163i	0.911147	+	0.412082i	$0.135198\pi$
$53$			6.00000i			0.824163i	−0.911147	+	0.412082i	$0.864802\pi$
$54$	1.00000	−	1.00000i	0.136083	−	0.136083i
$55$	−4.00000			−0.539360
$56$	−8.00000	+	8.00000i	−1.06904	+	1.06904i
$57$	4.00000			0.529813
$58$	2.00000	−	2.00000i	0.262613	−	0.262613i
$59$		−	4.00000i		−	0.520756i	−0.965507	−	0.260378i	$-0.916153\pi$
$59$		−	4.00000i		−	0.520756i	0.965507	−	0.260378i	$-0.0838471\pi$
$60$		−	4.00000i		−	0.516398i
$61$			6.00000i			0.768221i	0.923287	+	0.384111i	$0.125492\pi$
$61$			6.00000i			0.768221i	−0.923287	+	0.384111i	$0.874508\pi$
$62$	−6.00000	−	6.00000i	−0.762001	−	0.762001i
$63$	−4.00000			−0.503953
$64$		−	8.00000i		−	1.00000i
$65$	8.00000			0.992278
$66$	−2.00000	−	2.00000i	−0.246183	−	0.246183i
$67$		−	4.00000i		−	0.488678i	−0.969690	−	0.244339i	$-0.921429\pi$
$67$		−	4.00000i		−	0.488678i	0.969690	−	0.244339i	$-0.0785709\pi$
$68$		0			0
$69$			1.00000i			0.120386i
$70$	−8.00000	+	8.00000i	−0.956183	+	0.956183i
$71$	12.0000			1.42414			0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000			1.42414			0.712069	+	0.702109i	$0.247758\pi$
$72$	2.00000	−	2.00000i	0.235702	−	0.235702i
$73$	14.0000			1.63858			0.819288	−	0.573382i	$-0.194369\pi$
$73$	14.0000			1.63858			0.819288	+	0.573382i	$0.194369\pi$
$74$	−6.00000	+	6.00000i	−0.697486	+	0.697486i
$75$			1.00000i			0.115470i
$76$	8.00000			0.917663
$77$			8.00000i			0.911685i
$78$	4.00000	+	4.00000i	0.452911	+	0.452911i
$79$	8.00000			0.900070			0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000			0.900070			0.450035	+	0.893011i	$0.351411\pi$
$80$		−	8.00000i		−	0.894427i
$81$	1.00000			0.111111
$82$	2.00000	+	2.00000i	0.220863	+	0.220863i
$83$		−	14.0000i		−	1.53670i	−0.640030	−	0.768350i	$-0.721078\pi$
$83$		−	14.0000i		−	1.53670i	0.640030	−	0.768350i	$-0.278922\pi$
$84$	−8.00000			−0.872872
$85$		0			0
$86$	8.00000	−	8.00000i	0.862662	−	0.862662i
$87$	2.00000			0.214423
$88$	−4.00000	−	4.00000i	−0.426401	−	0.426401i
$89$	12.0000			1.27200			0.635999	−	0.771690i	$-0.280588\pi$
$89$	12.0000			1.27200			0.635999	+	0.771690i	$0.280588\pi$
$90$	2.00000	−	2.00000i	0.210819	−	0.210819i
$91$		−	16.0000i		−	1.67726i
$92$			2.00000i			0.208514i
$93$		−	6.00000i		−	0.622171i
$94$	−8.00000	−	8.00000i	−0.825137	−	0.825137i
$95$	8.00000			0.820783
$96$	4.00000	−	4.00000i	0.408248	−	0.408248i
$97$	−14.0000			−1.42148			−0.710742	−	0.703452i	$-0.751641\pi$
$97$	−14.0000			−1.42148			−0.710742	+	0.703452i	$0.751641\pi$
$98$	9.00000	+	9.00000i	0.909137	+	0.909137i
$99$		−	2.00000i		−	0.201008i
$100$			2.00000i			0.200000i
$101$			18.0000i			1.79107i	0.444994	+	0.895533i	$0.353206\pi$
$101$			18.0000i			1.79107i	−0.444994	+	0.895533i	$0.646794\pi$
$102$		0			0
$103$		0			0			−	1.00000i	$-0.5\pi$
$103$		0			0				1.00000i	$0.5\pi$
$104$	8.00000	+	8.00000i	0.784465	+	0.784465i
$105$	−8.00000			−0.780720
$106$	−6.00000	+	6.00000i	−0.582772	+	0.582772i
$107$			2.00000i			0.193347i	0.995316	+	0.0966736i	$0.0308203\pi$
$107$			2.00000i			0.193347i	−0.995316	+	0.0966736i	$0.969180\pi$
$108$	2.00000			0.192450
$109$		−	2.00000i		−	0.191565i	−0.995402	−	0.0957826i	$-0.969465\pi$
$109$		−	2.00000i		−	0.191565i	0.995402	−	0.0957826i	$-0.0305354\pi$
$110$	−4.00000	−	4.00000i	−0.381385	−	0.381385i
$111$	−6.00000			−0.569495
$112$	−16.0000			−1.51186
$113$	−4.00000			−0.376288			−0.188144	−	0.982141i	$-0.560247\pi$
$113$	−4.00000			−0.376288			−0.188144	+	0.982141i	$0.560247\pi$
$114$	4.00000	+	4.00000i	0.374634	+	0.374634i
$115$			2.00000i			0.186501i
$116$	4.00000			0.371391
$117$			4.00000i			0.369800i
$118$	4.00000	−	4.00000i	0.368230	−	0.368230i
$119$		0			0
$120$	4.00000	−	4.00000i	0.365148	−	0.365148i
$121$	7.00000			0.636364
$122$	−6.00000	+	6.00000i	−0.543214	+	0.543214i
$123$			2.00000i			0.180334i
$124$		−	12.0000i		−	1.07763i
$125$			12.0000i			1.07331i
$126$	−4.00000	−	4.00000i	−0.356348	−	0.356348i
$127$	−14.0000			−1.24230			−0.621150	−	0.783692i	$-0.713334\pi$
$127$	−14.0000			−1.24230			−0.621150	+	0.783692i	$0.713334\pi$
$128$	8.00000	−	8.00000i	0.707107	−	0.707107i
$129$	8.00000			0.704361
$130$	8.00000	+	8.00000i	0.701646	+	0.701646i
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		−	4.00000i		−	0.348155i
$133$		−	16.0000i		−	1.38738i
$134$	4.00000	−	4.00000i	0.345547	−	0.345547i
$135$	2.00000			0.172133
$136$		0			0
$137$	−12.0000			−1.02523			−0.512615	−	0.858619i	$-0.671323\pi$
$137$	−12.0000			−1.02523			−0.512615	+	0.858619i	$0.671323\pi$
$138$	−1.00000	+	1.00000i	−0.0851257	+	0.0851257i
$139$		−	4.00000i		−	0.339276i	−0.985506	−	0.169638i	$-0.945740\pi$
$139$		−	4.00000i		−	0.339276i	0.985506	−	0.169638i	$-0.0542598\pi$
$140$	−16.0000			−1.35225
$141$		−	8.00000i		−	0.673722i
$142$	12.0000	+	12.0000i	1.00702	+	1.00702i
$143$	8.00000			0.668994
$144$	4.00000			0.333333
$145$	4.00000			0.332182
$146$	14.0000	+	14.0000i	1.15865	+	1.15865i
$147$			9.00000i			0.742307i
$148$	−12.0000			−0.986394
$149$		−	10.0000i		−	0.819232i	−0.912258	−	0.409616i	$-0.865663\pi$
$149$		−	10.0000i		−	0.819232i	0.912258	−	0.409616i	$-0.134337\pi$
$150$	−1.00000	+	1.00000i	−0.0816497	+	0.0816497i
$151$	−22.0000			−1.79033			−0.895167	−	0.445730i	$-0.852944\pi$
$151$	−22.0000			−1.79033			−0.895167	+	0.445730i	$0.852944\pi$
$152$	8.00000	+	8.00000i	0.648886	+	0.648886i
$153$		0			0
$154$	−8.00000	+	8.00000i	−0.644658	+	0.644658i
$155$		−	12.0000i		−	0.963863i
$156$			8.00000i			0.640513i
$157$		−	22.0000i		−	1.75579i	−0.478852	−	0.877896i	$-0.658947\pi$
$157$		−	22.0000i		−	1.75579i	0.478852	−	0.877896i	$-0.341053\pi$
$158$	8.00000	+	8.00000i	0.636446	+	0.636446i
$159$	−6.00000			−0.475831
$160$	8.00000	−	8.00000i	0.632456	−	0.632456i
$161$	4.00000			0.315244
$162$	1.00000	+	1.00000i	0.0785674	+	0.0785674i
$163$			20.0000i			1.56652i	0.621694	+	0.783260i	$0.286445\pi$
$163$			20.0000i			1.56652i	−0.621694	+	0.783260i	$0.713555\pi$
$164$			4.00000i			0.312348i
$165$		−	4.00000i		−	0.311400i
$166$	14.0000	−	14.0000i	1.08661	−	1.08661i
$167$	8.00000			0.619059			0.309529	−	0.950890i	$-0.399829\pi$
$167$	8.00000			0.619059			0.309529	+	0.950890i	$0.399829\pi$
$168$	−8.00000	−	8.00000i	−0.617213	−	0.617213i
$169$	−3.00000			−0.230769
$170$		0			0
$171$			4.00000i			0.305888i
$172$	16.0000			1.21999
$173$		−	6.00000i		−	0.456172i	−0.973641	−	0.228086i	$-0.926753\pi$
$173$		−	6.00000i		−	0.456172i	0.973641	−	0.228086i	$-0.0732467\pi$
$174$	2.00000	+	2.00000i	0.151620	+	0.151620i
$175$	4.00000			0.302372
$176$		−	8.00000i		−	0.603023i
$177$	4.00000			0.300658
$178$	12.0000	+	12.0000i	0.899438	+	0.899438i
$179$			16.0000i			1.19590i	0.801535	+	0.597948i	$0.204017\pi$
$179$			16.0000i			1.19590i	−0.801535	+	0.597948i	$0.795983\pi$
$180$	4.00000			0.298142
$181$			2.00000i			0.148659i	0.997234	+	0.0743294i	$0.0236816\pi$
$181$			2.00000i			0.148659i	−0.997234	+	0.0743294i	$0.976318\pi$
$182$	16.0000	−	16.0000i	1.18600	−	1.18600i
$183$	−6.00000			−0.443533
$184$	−2.00000	+	2.00000i	−0.147442	+	0.147442i
$185$	−12.0000			−0.882258
$186$	6.00000	−	6.00000i	0.439941	−	0.439941i
$187$		0			0
$188$		−	16.0000i		−	1.16692i
$189$		−	4.00000i		−	0.290957i
$190$	8.00000	+	8.00000i	0.580381	+	0.580381i
$191$	−16.0000			−1.15772			−0.578860	−	0.815427i	$-0.696502\pi$
$191$	−16.0000			−1.15772			−0.578860	+	0.815427i	$0.696502\pi$
$192$	8.00000			0.577350
$193$	14.0000			1.00774			0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000			1.00774			0.503871	+	0.863779i	$0.331909\pi$
$194$	−14.0000	−	14.0000i	−1.00514	−	1.00514i
$195$			8.00000i			0.572892i
$196$			18.0000i			1.28571i
$197$			18.0000i			1.28245i	0.767354	+	0.641223i	$0.221573\pi$
$197$			18.0000i			1.28245i	−0.767354	+	0.641223i	$0.778427\pi$
$198$	2.00000	−	2.00000i	0.142134	−	0.142134i
$199$	−20.0000			−1.41776			−0.708881	−	0.705328i	$-0.750800\pi$
$199$	−20.0000			−1.41776			−0.708881	+	0.705328i	$0.750800\pi$
$200$	−2.00000	+	2.00000i	−0.141421	+	0.141421i
$201$	4.00000			0.282138
$202$	−18.0000	+	18.0000i	−1.26648	+	1.26648i
$203$		−	8.00000i		−	0.561490i
$204$		0			0
$205$			4.00000i			0.279372i
$206$		0			0
$207$	−1.00000			−0.0695048
$208$			16.0000i			1.10940i
$209$	8.00000			0.553372
$210$	−8.00000	−	8.00000i	−0.552052	−	0.552052i
$211$		−	12.0000i		−	0.826114i	−0.910705	−	0.413057i	$-0.864461\pi$
$211$		−	12.0000i		−	0.826114i	0.910705	−	0.413057i	$-0.135539\pi$
$212$	−12.0000			−0.824163
$213$			12.0000i			0.822226i
$214$	−2.00000	+	2.00000i	−0.136717	+	0.136717i
$215$	16.0000			1.09119
$216$	2.00000	+	2.00000i	0.136083	+	0.136083i
$217$	−24.0000			−1.62923
$218$	2.00000	−	2.00000i	0.135457	−	0.135457i
$219$			14.0000i			0.946032i
$220$		−	8.00000i		−	0.539360i
$221$		0			0
$222$	−6.00000	−	6.00000i	−0.402694	−	0.402694i
$223$	10.0000			0.669650			0.334825	−	0.942280i	$-0.391323\pi$
$223$	10.0000			0.669650			0.334825	+	0.942280i	$0.391323\pi$
$224$	−16.0000	−	16.0000i	−1.06904	−	1.06904i
$225$	−1.00000			−0.0666667
$226$	−4.00000	−	4.00000i	−0.266076	−	0.266076i
$227$		−	6.00000i		−	0.398234i	−0.979976	−	0.199117i	$-0.936193\pi$
$227$		−	6.00000i		−	0.398234i	0.979976	−	0.199117i	$-0.0638074\pi$
$228$			8.00000i			0.529813i
$229$		−	22.0000i		−	1.45380i	−0.686743	−	0.726900i	$-0.740960\pi$
$229$		−	22.0000i		−	1.45380i	0.686743	−	0.726900i	$-0.259040\pi$
$230$	−2.00000	+	2.00000i	−0.131876	+	0.131876i
$231$	−8.00000			−0.526361
$232$	4.00000	+	4.00000i	0.262613	+	0.262613i
$233$	−18.0000			−1.17922			−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000			−1.17922			−0.589610	+	0.807688i	$0.700718\pi$
$234$	−4.00000	+	4.00000i	−0.261488	+	0.261488i
$235$		−	16.0000i		−	1.04372i
$236$	8.00000			0.520756
$237$			8.00000i			0.519656i
$238$		0			0
$239$	−28.0000			−1.81117			−0.905585	−	0.424165i	$-0.860568\pi$
$239$	−28.0000			−1.81117			−0.905585	+	0.424165i	$0.860568\pi$
$240$	8.00000			0.516398
$241$	10.0000			0.644157			0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000			0.644157			0.322078	+	0.946713i	$0.395619\pi$
$242$	7.00000	+	7.00000i	0.449977	+	0.449977i
$243$			1.00000i			0.0641500i
$244$	−12.0000			−0.768221
$245$			18.0000i			1.14998i
$246$	−2.00000	+	2.00000i	−0.127515	+	0.127515i
$247$	−16.0000			−1.01806
$248$	12.0000	−	12.0000i	0.762001	−	0.762001i
$249$	14.0000			0.887214
$250$	−12.0000	+	12.0000i	−0.758947	+	0.758947i
$251$		−	30.0000i		−	1.89358i	−0.321847	−	0.946792i	$-0.604304\pi$
$251$		−	30.0000i		−	1.89358i	0.321847	−	0.946792i	$-0.395696\pi$
$252$		−	8.00000i		−	0.503953i
$253$			2.00000i			0.125739i
$254$	−14.0000	−	14.0000i	−0.878438	−	0.878438i
$255$		0			0
$256$	16.0000			1.00000
$257$	−6.00000			−0.374270			−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000			−0.374270			−0.187135	+	0.982334i	$0.559920\pi$
$258$	8.00000	+	8.00000i	0.498058	+	0.498058i
$259$			24.0000i			1.49129i
$260$			16.0000i			0.992278i
$261$			2.00000i			0.123797i
$262$		0			0
$263$	24.0000			1.47990			0.739952	−	0.672660i	$-0.234848\pi$
$263$	24.0000			1.47990			0.739952	+	0.672660i	$0.234848\pi$
$264$	4.00000	−	4.00000i	0.246183	−	0.246183i
$265$	−12.0000			−0.737154
$266$	16.0000	−	16.0000i	0.981023	−	0.981023i
$267$			12.0000i			0.734388i
$268$	8.00000			0.488678
$269$			6.00000i			0.365826i	0.983129	+	0.182913i	$0.0585527\pi$
$269$			6.00000i			0.365826i	−0.983129	+	0.182913i	$0.941447\pi$
$270$	2.00000	+	2.00000i	0.121716	+	0.121716i
$271$	−6.00000			−0.364474			−0.182237	−	0.983255i	$-0.558334\pi$
$271$	−6.00000			−0.364474			−0.182237	+	0.983255i	$0.558334\pi$
$272$		0			0
$273$	16.0000			0.968364
$274$	−12.0000	−	12.0000i	−0.724947	−	0.724947i
$275$			2.00000i			0.120605i
$276$	−2.00000			−0.120386
$277$			8.00000i			0.480673i	0.970690	+	0.240337i	$0.0772579\pi$
$277$			8.00000i			0.480673i	−0.970690	+	0.240337i	$0.922742\pi$
$278$	4.00000	−	4.00000i	0.239904	−	0.239904i
$279$	6.00000			0.359211
$280$	−16.0000	−	16.0000i	−0.956183	−	0.956183i
$281$	−12.0000			−0.715860			−0.357930	−	0.933748i	$-0.616517\pi$
$281$	−12.0000			−0.715860			−0.357930	+	0.933748i	$0.616517\pi$
$282$	8.00000	−	8.00000i	0.476393	−	0.476393i
$283$			32.0000i			1.90220i	0.308879	+	0.951101i	$0.400046\pi$
$283$			32.0000i			1.90220i	−0.308879	+	0.951101i	$0.599954\pi$
$284$			24.0000i			1.42414i
$285$			8.00000i			0.473879i
$286$	8.00000	+	8.00000i	0.473050	+	0.473050i
$287$	8.00000			0.472225
$288$	4.00000	+	4.00000i	0.235702	+	0.235702i
$289$	−17.0000			−1.00000
$290$	4.00000	+	4.00000i	0.234888	+	0.234888i
$291$		−	14.0000i		−	0.820695i
$292$			28.0000i			1.63858i
$293$		−	22.0000i		−	1.28525i	−0.766179	−	0.642627i	$-0.777845\pi$
$293$		−	22.0000i		−	1.28525i	0.766179	−	0.642627i	$-0.222155\pi$
$294$	−9.00000	+	9.00000i	−0.524891	+	0.524891i
$295$	8.00000			0.465778
$296$	−12.0000	−	12.0000i	−0.697486	−	0.697486i
$297$	2.00000			0.116052
$298$	10.0000	−	10.0000i	0.579284	−	0.579284i
$299$		−	4.00000i		−	0.231326i
$300$	−2.00000			−0.115470
$301$		−	32.0000i		−	1.84445i
$302$	−22.0000	−	22.0000i	−1.26596	−	1.26596i
$303$	−18.0000			−1.03407
$304$			16.0000i			0.917663i
$305$	−12.0000			−0.687118
$306$		0			0
$307$		−	28.0000i		−	1.59804i	−0.601302	−	0.799022i	$-0.705351\pi$
$307$		−	28.0000i		−	1.59804i	0.601302	−	0.799022i	$-0.294649\pi$
$308$	−16.0000			−0.911685
$309$		0			0
$310$	12.0000	−	12.0000i	0.681554	−	0.681554i
$311$	−24.0000			−1.36092			−0.680458	−	0.732787i	$-0.738219\pi$
$311$	−24.0000			−1.36092			−0.680458	+	0.732787i	$0.738219\pi$
$312$	−8.00000	+	8.00000i	−0.452911	+	0.452911i
$313$	−6.00000			−0.339140			−0.169570	−	0.985518i	$-0.554238\pi$
$313$	−6.00000			−0.339140			−0.169570	+	0.985518i	$0.554238\pi$
$314$	22.0000	−	22.0000i	1.24153	−	1.24153i
$315$		−	8.00000i		−	0.450749i
$316$			16.0000i			0.900070i
$317$			22.0000i			1.23564i	0.786318	+	0.617822i	$0.211985\pi$
$317$			22.0000i			1.23564i	−0.786318	+	0.617822i	$0.788015\pi$
$318$	−6.00000	−	6.00000i	−0.336463	−	0.336463i
$319$	4.00000			0.223957
$320$	16.0000			0.894427
$321$	−2.00000			−0.111629
$322$	4.00000	+	4.00000i	0.222911	+	0.222911i
$323$		0			0
$324$			2.00000i			0.111111i
$325$		−	4.00000i		−	0.221880i
$326$	−20.0000	+	20.0000i	−1.10770	+	1.10770i
$327$	2.00000			0.110600
$328$	−4.00000	+	4.00000i	−0.220863	+	0.220863i
$329$	−32.0000			−1.76422
$330$	4.00000	−	4.00000i	0.220193	−	0.220193i
$331$			4.00000i			0.219860i	0.993939	+	0.109930i	$0.0350627\pi$
$331$			4.00000i			0.219860i	−0.993939	+	0.109930i	$0.964937\pi$
$332$	28.0000			1.53670
$333$		−	6.00000i		−	0.328798i
$334$	8.00000	+	8.00000i	0.437741	+	0.437741i
$335$	8.00000			0.437087
$336$		−	16.0000i		−	0.872872i
$337$	22.0000			1.19842			0.599208	−	0.800593i	$-0.295482\pi$
$337$	22.0000			1.19842			0.599208	+	0.800593i	$0.295482\pi$
$338$	−3.00000	−	3.00000i	−0.163178	−	0.163178i
$339$		−	4.00000i		−	0.217250i
$340$		0			0
$341$		−	12.0000i		−	0.649836i
$342$	−4.00000	+	4.00000i	−0.216295	+	0.216295i
$343$	8.00000			0.431959
$344$	16.0000	+	16.0000i	0.862662	+	0.862662i
$345$	−2.00000			−0.107676
$346$	6.00000	−	6.00000i	0.322562	−	0.322562i
$347$		−	12.0000i		−	0.644194i	−0.946707	−	0.322097i	$-0.895612\pi$
$347$		−	12.0000i		−	0.644194i	0.946707	−	0.322097i	$-0.104388\pi$
$348$			4.00000i			0.214423i
$349$			20.0000i			1.07058i	0.844670	+	0.535288i	$0.179797\pi$
$349$			20.0000i			1.07058i	−0.844670	+	0.535288i	$0.820203\pi$
$350$	4.00000	+	4.00000i	0.213809	+	0.213809i
$351$	−4.00000			−0.213504
$352$	8.00000	−	8.00000i	0.426401	−	0.426401i
$353$	−10.0000			−0.532246			−0.266123	−	0.963939i	$-0.585743\pi$
$353$	−10.0000			−0.532246			−0.266123	+	0.963939i	$0.585743\pi$
$354$	4.00000	+	4.00000i	0.212598	+	0.212598i
$355$			24.0000i			1.27379i
$356$			24.0000i			1.27200i
$357$		0			0
$358$	−16.0000	+	16.0000i	−0.845626	+	0.845626i
$359$	−24.0000			−1.26667			−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000			−1.26667			−0.633336	+	0.773877i	$0.718315\pi$
$360$	4.00000	+	4.00000i	0.210819	+	0.210819i
$361$	3.00000			0.157895
$362$	−2.00000	+	2.00000i	−0.105118	+	0.105118i
$363$			7.00000i			0.367405i
$364$	32.0000			1.67726
$365$			28.0000i			1.46559i
$366$	−6.00000	−	6.00000i	−0.313625	−	0.313625i
$367$	8.00000			0.417597			0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000			0.417597			0.208798	+	0.977959i	$0.433045\pi$
$368$	−4.00000			−0.208514
$369$	−2.00000			−0.104116
$370$	−12.0000	−	12.0000i	−0.623850	−	0.623850i
$371$			24.0000i			1.24602i
$372$	12.0000			0.622171
$373$			6.00000i			0.310668i	0.987862	+	0.155334i	$0.0496454\pi$
$373$			6.00000i			0.310668i	−0.987862	+	0.155334i	$0.950355\pi$
$374$		0			0
$375$	−12.0000			−0.619677
$376$	16.0000	−	16.0000i	0.825137	−	0.825137i
$377$	−8.00000			−0.412021
$378$	4.00000	−	4.00000i	0.205738	−	0.205738i
$379$		−	12.0000i		−	0.616399i	−0.951322	−	0.308199i	$-0.900274\pi$
$379$		−	12.0000i		−	0.616399i	0.951322	−	0.308199i	$-0.0997264\pi$
$380$			16.0000i			0.820783i
$381$		−	14.0000i		−	0.717242i
$382$	−16.0000	−	16.0000i	−0.818631	−	0.818631i
$383$	24.0000			1.22634			0.613171	−	0.789950i	$-0.289894\pi$
$383$	24.0000			1.22634			0.613171	+	0.789950i	$0.289894\pi$
$384$	8.00000	+	8.00000i	0.408248	+	0.408248i
$385$	−16.0000			−0.815436
$386$	14.0000	+	14.0000i	0.712581	+	0.712581i
$387$			8.00000i			0.406663i
$388$		−	28.0000i		−	1.42148i
$389$		−	10.0000i		−	0.507020i	−0.967333	−	0.253510i	$-0.918415\pi$
$389$		−	10.0000i		−	0.507020i	0.967333	−	0.253510i	$-0.0815851\pi$
$390$	−8.00000	+	8.00000i	−0.405096	+	0.405096i
$391$		0			0
$392$	−18.0000	+	18.0000i	−0.909137	+	0.909137i
$393$		0			0
$394$	−18.0000	+	18.0000i	−0.906827	+	0.906827i
$395$			16.0000i			0.805047i
$396$	4.00000			0.201008
$397$		−	8.00000i		−	0.401508i	−0.979642	−	0.200754i	$-0.935661\pi$
$397$		−	8.00000i		−	0.401508i	0.979642	−	0.200754i	$-0.0643393\pi$
$398$	−20.0000	−	20.0000i	−1.00251	−	1.00251i
$399$	16.0000			0.801002
$400$	−4.00000			−0.200000
$401$	−24.0000			−1.19850			−0.599251	−	0.800561i	$-0.704535\pi$
$401$	−24.0000			−1.19850			−0.599251	+	0.800561i	$0.704535\pi$
$402$	4.00000	+	4.00000i	0.199502	+	0.199502i
$403$			24.0000i			1.19553i
$404$	−36.0000			−1.79107
$405$			2.00000i			0.0993808i
$406$	8.00000	−	8.00000i	0.397033	−	0.397033i
$407$	−12.0000			−0.594818
$408$		0			0
$409$	22.0000			1.08783			0.543915	−	0.839140i	$-0.316941\pi$
$409$	22.0000			1.08783			0.543915	+	0.839140i	$0.316941\pi$
$410$	−4.00000	+	4.00000i	−0.197546	+	0.197546i
$411$		−	12.0000i		−	0.591916i
$412$		0			0
$413$		−	16.0000i		−	0.787309i
$414$	−1.00000	−	1.00000i	−0.0491473	−	0.0491473i
$415$	28.0000			1.37447
$416$	−16.0000	+	16.0000i	−0.784465	+	0.784465i
$417$	4.00000			0.195881
$418$	8.00000	+	8.00000i	0.391293	+	0.391293i
$419$			18.0000i			0.879358i	0.898155	+	0.439679i	$0.144908\pi$
$419$			18.0000i			0.879358i	−0.898155	+	0.439679i	$0.855092\pi$
$420$		−	16.0000i		−	0.780720i
$421$			22.0000i			1.07221i	0.844150	+	0.536107i	$0.180106\pi$
$421$			22.0000i			1.07221i	−0.844150	+	0.536107i	$0.819894\pi$
$422$	12.0000	−	12.0000i	0.584151	−	0.584151i
$423$	8.00000			0.388973
$424$	−12.0000	−	12.0000i	−0.582772	−	0.582772i
$425$		0			0
$426$	−12.0000	+	12.0000i	−0.581402	+	0.581402i
$427$			24.0000i			1.16144i
$428$	−4.00000			−0.193347
$429$			8.00000i			0.386244i
$430$	16.0000	+	16.0000i	0.771589	+	0.771589i
$431$	16.0000			0.770693			0.385346	−	0.922772i	$-0.374082\pi$
$431$	16.0000			0.770693			0.385346	+	0.922772i	$0.374082\pi$
$432$			4.00000i			0.192450i
$433$	−34.0000			−1.63394			−0.816968	−	0.576683i	$-0.804347\pi$
$433$	−34.0000			−1.63394			−0.816968	+	0.576683i	$0.804347\pi$
$434$	−24.0000	−	24.0000i	−1.15204	−	1.15204i
$435$			4.00000i			0.191785i
$436$	4.00000			0.191565
$437$		−	4.00000i		−	0.191346i
$438$	−14.0000	+	14.0000i	−0.668946	+	0.668946i
$439$	34.0000			1.62273			0.811366	−	0.584539i	$-0.198725\pi$
$439$	34.0000			1.62273			0.811366	+	0.584539i	$0.198725\pi$
$440$	8.00000	−	8.00000i	0.381385	−	0.381385i
$441$	−9.00000			−0.428571
$442$		0			0
$443$		−	24.0000i		−	1.14027i	−0.821549	−	0.570137i	$-0.806890\pi$
$443$		−	24.0000i		−	1.14027i	0.821549	−	0.570137i	$-0.193110\pi$
$444$		−	12.0000i		−	0.569495i
$445$			24.0000i			1.13771i
$446$	10.0000	+	10.0000i	0.473514	+	0.473514i
$447$	10.0000			0.472984
$448$		−	32.0000i		−	1.51186i
$449$	−26.0000			−1.22702			−0.613508	−	0.789689i	$-0.710242\pi$
$449$	−26.0000			−1.22702			−0.613508	+	0.789689i	$0.710242\pi$
$450$	−1.00000	−	1.00000i	−0.0471405	−	0.0471405i
$451$			4.00000i			0.188353i
$452$		−	8.00000i		−	0.376288i
$453$		−	22.0000i		−	1.03365i
$454$	6.00000	−	6.00000i	0.281594	−	0.281594i
$455$	32.0000			1.50018
$456$	−8.00000	+	8.00000i	−0.374634	+	0.374634i
$457$	38.0000			1.77757			0.888783	−	0.458329i	$-0.151552\pi$
$457$	38.0000			1.77757			0.888783	+	0.458329i	$0.151552\pi$
$458$	22.0000	−	22.0000i	1.02799	−	1.02799i
$459$		0			0
$460$	−4.00000			−0.186501
$461$			10.0000i			0.465746i	0.972507	+	0.232873i	$0.0748127\pi$
$461$			10.0000i			0.465746i	−0.972507	+	0.232873i	$0.925187\pi$
$462$	−8.00000	−	8.00000i	−0.372194	−	0.372194i
$463$	6.00000			0.278844			0.139422	−	0.990233i	$-0.455476\pi$
$463$	6.00000			0.278844			0.139422	+	0.990233i	$0.455476\pi$
$464$			8.00000i			0.371391i
$465$	12.0000			0.556487
$466$	−18.0000	−	18.0000i	−0.833834	−	0.833834i
$467$		−	18.0000i		−	0.832941i	−0.909149	−	0.416470i	$-0.863267\pi$
$467$		−	18.0000i		−	0.832941i	0.909149	−	0.416470i	$-0.136733\pi$
$468$	−8.00000			−0.369800
$469$		−	16.0000i		−	0.738811i
$470$	16.0000	−	16.0000i	0.738025	−	0.738025i
$471$	22.0000			1.01371
$472$	8.00000	+	8.00000i	0.368230	+	0.368230i
$473$	16.0000			0.735681
$474$	−8.00000	+	8.00000i	−0.367452	+	0.367452i
$475$		−	4.00000i		−	0.183533i
$476$		0			0
$477$		−	6.00000i		−	0.274721i
$478$	−28.0000	−	28.0000i	−1.28069	−	1.28069i
$479$	−24.0000			−1.09659			−0.548294	−	0.836286i	$-0.684723\pi$
$479$	−24.0000			−1.09659			−0.548294	+	0.836286i	$0.684723\pi$
$480$	8.00000	+	8.00000i	0.365148	+	0.365148i
$481$	24.0000			1.09431
$482$	10.0000	+	10.0000i	0.455488	+	0.455488i
$483$			4.00000i			0.182006i
$484$			14.0000i			0.636364i
$485$		−	28.0000i		−	1.27141i
$486$	−1.00000	+	1.00000i	−0.0453609	+	0.0453609i
$487$	38.0000			1.72194			0.860972	−	0.508652i	$-0.169856\pi$
$487$	38.0000			1.72194			0.860972	+	0.508652i	$0.169856\pi$
$488$	−12.0000	−	12.0000i	−0.543214	−	0.543214i
$489$	−20.0000			−0.904431
$490$	−18.0000	+	18.0000i	−0.813157	+	0.813157i
$491$		−	40.0000i		−	1.80517i	−0.430507	−	0.902587i	$-0.641665\pi$
$491$		−	40.0000i		−	1.80517i	0.430507	−	0.902587i	$-0.358335\pi$
$492$	−4.00000			−0.180334
$493$		0			0
$494$	−16.0000	−	16.0000i	−0.719874	−	0.719874i
$495$	4.00000			0.179787
$496$	24.0000			1.07763
$497$	48.0000			2.15309
$498$	14.0000	+	14.0000i	0.627355	+	0.627355i
$499$			20.0000i			0.895323i	0.894203	+	0.447661i	$0.147743\pi$
$499$			20.0000i			0.895323i	−0.894203	+	0.447661i	$0.852257\pi$
$500$	−24.0000			−1.07331
$501$			8.00000i			0.357414i
$502$	30.0000	−	30.0000i	1.33897	−	1.33897i
$503$	8.00000			0.356702			0.178351	−	0.983967i	$-0.442924\pi$
$503$	8.00000			0.356702			0.178351	+	0.983967i	$0.442924\pi$
$504$	8.00000	−	8.00000i	0.356348	−	0.356348i
$505$	−36.0000			−1.60198
$506$	−2.00000	+	2.00000i	−0.0889108	+	0.0889108i
$507$		−	3.00000i		−	0.133235i
$508$		−	28.0000i		−	1.24230i
$509$		−	42.0000i		−	1.86162i	−0.365507	−	0.930809i	$-0.619104\pi$
$509$		−	42.0000i		−	1.86162i	0.365507	−	0.930809i	$-0.380896\pi$
$510$		0			0
$511$	56.0000			2.47729
$512$	16.0000	+	16.0000i	0.707107	+	0.707107i
$513$	−4.00000			−0.176604
$514$	−6.00000	−	6.00000i	−0.264649	−	0.264649i
$515$		0			0
$516$			16.0000i			0.704361i
$517$		−	16.0000i		−	0.703679i
$518$	−24.0000	+	24.0000i	−1.05450	+	1.05450i
$519$	6.00000			0.263371
$520$	−16.0000	+	16.0000i	−0.701646	+	0.701646i
$521$	12.0000			0.525730			0.262865	−	0.964833i	$-0.415333\pi$
$521$	12.0000			0.525730			0.262865	+	0.964833i	$0.415333\pi$
$522$	−2.00000	+	2.00000i	−0.0875376	+	0.0875376i
$523$		−	8.00000i		−	0.349816i	−0.984585	−	0.174908i	$-0.944037\pi$
$523$		−	8.00000i		−	0.349816i	0.984585	−	0.174908i	$-0.0559627\pi$
$524$		0			0
$525$			4.00000i			0.174574i
$526$	24.0000	+	24.0000i	1.04645	+	1.04645i
$527$		0			0
$528$	8.00000			0.348155
$529$	1.00000			0.0434783
$530$	−12.0000	−	12.0000i	−0.521247	−	0.521247i
$531$			4.00000i			0.173585i
$532$	32.0000			1.38738
$533$		−	8.00000i		−	0.346518i
$534$	−12.0000	+	12.0000i	−0.519291	+	0.519291i
$535$	−4.00000			−0.172935
$536$	8.00000	+	8.00000i	0.345547	+	0.345547i
$537$	−16.0000			−0.690451
$538$	−6.00000	+	6.00000i	−0.258678	+	0.258678i
$539$			18.0000i			0.775315i
$540$			4.00000i			0.172133i
$541$			8.00000i			0.343947i	0.985102	+	0.171973i	$0.0550143\pi$
$541$			8.00000i			0.343947i	−0.985102	+	0.171973i	$0.944986\pi$
$542$	−6.00000	−	6.00000i	−0.257722	−	0.257722i
$543$	−2.00000			−0.0858282
$544$		0			0
$545$	4.00000			0.171341
$546$	16.0000	+	16.0000i	0.684737	+	0.684737i
$547$			20.0000i			0.855138i	0.903983	+	0.427569i	$0.140630\pi$
$547$			20.0000i			0.855138i	−0.903983	+	0.427569i	$0.859370\pi$
$548$		−	24.0000i		−	1.02523i
$549$		−	6.00000i		−	0.256074i
$550$	−2.00000	+	2.00000i	−0.0852803	+	0.0852803i
$551$	−8.00000			−0.340811
$552$	−2.00000	−	2.00000i	−0.0851257	−	0.0851257i
$553$	32.0000			1.36078
$554$	−8.00000	+	8.00000i	−0.339887	+	0.339887i
$555$		−	12.0000i		−	0.509372i
$556$	8.00000			0.339276
$557$			14.0000i			0.593199i	0.955002	+	0.296600i	$0.0958526\pi$
$557$			14.0000i			0.593199i	−0.955002	+	0.296600i	$0.904147\pi$
$558$	6.00000	+	6.00000i	0.254000	+	0.254000i
$559$	−32.0000			−1.35346
$560$		−	32.0000i		−	1.35225i
$561$		0			0
$562$	−12.0000	−	12.0000i	−0.506189	−	0.506189i
$563$		−	22.0000i		−	0.927189i	−0.886047	−	0.463595i	$-0.846559\pi$
$563$		−	22.0000i		−	0.927189i	0.886047	−	0.463595i	$-0.153441\pi$
$564$	16.0000			0.673722
$565$		−	8.00000i		−	0.336563i
$566$	−32.0000	+	32.0000i	−1.34506	+	1.34506i
$567$	4.00000			0.167984
$568$	−24.0000	+	24.0000i	−1.00702	+	1.00702i
$569$	−8.00000			−0.335377			−0.167689	−	0.985840i	$-0.553630\pi$
$569$	−8.00000			−0.335377			−0.167689	+	0.985840i	$0.553630\pi$
$570$	−8.00000	+	8.00000i	−0.335083	+	0.335083i
$571$		−	12.0000i		−	0.502184i	−0.967963	−	0.251092i	$-0.919210\pi$
$571$		−	12.0000i		−	0.502184i	0.967963	−	0.251092i	$-0.0807897\pi$
$572$			16.0000i			0.668994i
$573$		−	16.0000i		−	0.668410i
$574$	8.00000	+	8.00000i	0.333914	+	0.333914i
$575$	1.00000			0.0417029
$576$			8.00000i			0.333333i
$577$	−34.0000			−1.41544			−0.707719	−	0.706494i	$-0.750276\pi$
$577$	−34.0000			−1.41544			−0.707719	+	0.706494i	$0.750276\pi$
$578$	−17.0000	−	17.0000i	−0.707107	−	0.707107i
$579$			14.0000i			0.581820i
$580$			8.00000i			0.332182i
$581$		−	56.0000i		−	2.32327i
$582$	14.0000	−	14.0000i	0.580319	−	0.580319i
$583$	−12.0000			−0.496989
$584$	−28.0000	+	28.0000i	−1.15865	+	1.15865i
$585$	−8.00000			−0.330759
$586$	22.0000	−	22.0000i	0.908812	−	0.908812i
$587$			24.0000i			0.990586i	0.868726	+	0.495293i	$0.164939\pi$
$587$			24.0000i			0.990586i	−0.868726	+	0.495293i	$0.835061\pi$
$588$	−18.0000			−0.742307
$589$			24.0000i			0.988903i
$590$	8.00000	+	8.00000i	0.329355	+	0.329355i
$591$	−18.0000			−0.740421
$592$		−	24.0000i		−	0.986394i
$593$	34.0000			1.39621			0.698106	−	0.715994i	$-0.254026\pi$
$593$	34.0000			1.39621			0.698106	+	0.715994i	$0.254026\pi$
$594$	2.00000	+	2.00000i	0.0820610	+	0.0820610i
$595$		0			0
$596$	20.0000			0.819232
$597$		−	20.0000i		−	0.818546i
$598$	4.00000	−	4.00000i	0.163572	−	0.163572i
$599$	32.0000			1.30748			0.653742	−	0.756717i	$-0.273198\pi$
$599$	32.0000			1.30748			0.653742	+	0.756717i	$0.273198\pi$
$600$	−2.00000	−	2.00000i	−0.0816497	−	0.0816497i
$601$	26.0000			1.06056			0.530281	−	0.847822i	$-0.322086\pi$
$601$	26.0000			1.06056			0.530281	+	0.847822i	$0.322086\pi$
$602$	32.0000	−	32.0000i	1.30422	−	1.30422i
$603$			4.00000i			0.162893i
$604$		−	44.0000i		−	1.79033i
$605$			14.0000i			0.569181i
$606$	−18.0000	−	18.0000i	−0.731200	−	0.731200i
$607$	−22.0000			−0.892952			−0.446476	−	0.894795i	$-0.647321\pi$
$607$	−22.0000			−0.892952			−0.446476	+	0.894795i	$0.647321\pi$
$608$	−16.0000	+	16.0000i	−0.648886	+	0.648886i
$609$	8.00000			0.324176
$610$	−12.0000	−	12.0000i	−0.485866	−	0.485866i
$611$			32.0000i			1.29458i
$612$		0			0
$613$		−	6.00000i		−	0.242338i	−0.992632	−	0.121169i	$-0.961336\pi$
$613$		−	6.00000i		−	0.242338i	0.992632	−	0.121169i	$-0.0386643\pi$
$614$	28.0000	−	28.0000i	1.12999	−	1.12999i
$615$	−4.00000			−0.161296
$616$	−16.0000	−	16.0000i	−0.644658	−	0.644658i
$617$	−12.0000			−0.483102			−0.241551	−	0.970388i	$-0.577656\pi$
$617$	−12.0000			−0.483102			−0.241551	+	0.970388i	$0.577656\pi$
$618$		0			0
$619$		−	8.00000i		−	0.321547i	−0.986991	−	0.160774i	$-0.948601\pi$
$619$		−	8.00000i		−	0.321547i	0.986991	−	0.160774i	$-0.0513989\pi$
$620$	24.0000			0.963863
$621$		−	1.00000i		−	0.0401286i
$622$	−24.0000	−	24.0000i	−0.962312	−	0.962312i
$623$	48.0000			1.92308
$624$	−16.0000			−0.640513
$625$	−19.0000			−0.760000
$626$	−6.00000	−	6.00000i	−0.239808	−	0.239808i
$627$			8.00000i			0.319489i
$628$	44.0000			1.75579
$629$		0			0
$630$	8.00000	−	8.00000i	0.318728	−	0.318728i
$631$	−44.0000			−1.75161			−0.875806	−	0.482663i	$-0.839670\pi$
$631$	−44.0000			−1.75161			−0.875806	+	0.482663i	$0.839670\pi$
$632$	−16.0000	+	16.0000i	−0.636446	+	0.636446i
$633$	12.0000			0.476957
$634$	−22.0000	+	22.0000i	−0.873732	+	0.873732i
$635$		−	28.0000i		−	1.11115i
$636$		−	12.0000i		−	0.475831i
$637$		−	36.0000i		−	1.42637i
$638$	4.00000	+	4.00000i	0.158362	+	0.158362i
$639$	−12.0000			−0.474713
$640$	16.0000	+	16.0000i	0.632456	+	0.632456i
$641$	−16.0000			−0.631962			−0.315981	−	0.948766i	$-0.602334\pi$
$641$	−16.0000			−0.631962			−0.315981	+	0.948766i	$0.602334\pi$
$642$	−2.00000	−	2.00000i	−0.0789337	−	0.0789337i
$643$		−	8.00000i		−	0.315489i	−0.987480	−	0.157745i	$-0.949578\pi$
$643$		−	8.00000i		−	0.315489i	0.987480	−	0.157745i	$-0.0504223\pi$
$644$			8.00000i			0.315244i
$645$			16.0000i			0.629999i
$646$		0			0
$647$	12.0000			0.471769			0.235884	−	0.971781i	$-0.424201\pi$
$647$	12.0000			0.471769			0.235884	+	0.971781i	$0.424201\pi$
$648$	−2.00000	+	2.00000i	−0.0785674	+	0.0785674i
$649$	8.00000			0.314027
$650$	4.00000	−	4.00000i	0.156893	−	0.156893i
$651$		−	24.0000i		−	0.940634i
$652$	−40.0000			−1.56652
$653$			10.0000i			0.391330i	0.980671	+	0.195665i	$0.0626866\pi$
$653$			10.0000i			0.391330i	−0.980671	+	0.195665i	$0.937313\pi$
$654$	2.00000	+	2.00000i	0.0782062	+	0.0782062i
$655$		0			0
$656$	−8.00000			−0.312348
$657$	−14.0000			−0.546192
$658$	−32.0000	−	32.0000i	−1.24749	−	1.24749i
$659$		−	6.00000i		−	0.233727i	−0.993148	−	0.116863i	$-0.962716\pi$
$659$		−	6.00000i		−	0.233727i	0.993148	−	0.116863i	$-0.0372840\pi$
$660$	8.00000			0.311400
$661$			10.0000i			0.388955i	0.980907	+	0.194477i	$0.0623011\pi$
$661$			10.0000i			0.388955i	−0.980907	+	0.194477i	$0.937699\pi$
$662$	−4.00000	+	4.00000i	−0.155464	+	0.155464i
$663$		0			0
$664$	28.0000	+	28.0000i	1.08661	+	1.08661i
$665$	32.0000			1.24091
$666$	6.00000	−	6.00000i	0.232495	−	0.232495i
$667$		−	2.00000i		−	0.0774403i
$668$			16.0000i			0.619059i
$669$			10.0000i			0.386622i
$670$	8.00000	+	8.00000i	0.309067	+	0.309067i
$671$	−12.0000			−0.463255
$672$	16.0000	−	16.0000i	0.617213	−	0.617213i
$673$	30.0000			1.15642			0.578208	−	0.815890i	$-0.303752\pi$
$673$	30.0000			1.15642			0.578208	+	0.815890i	$0.303752\pi$
$674$	22.0000	+	22.0000i	0.847408	+	0.847408i
$675$		−	1.00000i		−	0.0384900i
$676$		−	6.00000i		−	0.230769i
$677$			6.00000i			0.230599i	0.993331	+	0.115299i	$0.0367827\pi$
$677$			6.00000i			0.230599i	−0.993331	+	0.115299i	$0.963217\pi$
$678$	4.00000	−	4.00000i	0.153619	−	0.153619i
$679$	−56.0000			−2.14908
$680$		0			0
$681$	6.00000			0.229920
$682$	12.0000	−	12.0000i	0.459504	−	0.459504i
$683$			12.0000i			0.459167i	0.973289	+	0.229584i	$0.0737364\pi$
$683$			12.0000i			0.459167i	−0.973289	+	0.229584i	$0.926264\pi$
$684$	−8.00000			−0.305888
$685$		−	24.0000i		−	0.916993i
$686$	8.00000	+	8.00000i	0.305441	+	0.305441i
$687$	22.0000			0.839352
$688$			32.0000i			1.21999i
$689$	24.0000			0.914327
$690$	−2.00000	−	2.00000i	−0.0761387	−	0.0761387i
$691$		−	44.0000i		−	1.67384i	−0.547326	−	0.836919i	$-0.684354\pi$
$691$		−	44.0000i		−	1.67384i	0.547326	−	0.836919i	$-0.315646\pi$
$692$	12.0000			0.456172
$693$		−	8.00000i		−	0.303895i
$694$	12.0000	−	12.0000i	0.455514	−	0.455514i
$695$	8.00000			0.303457
$696$	−4.00000	+	4.00000i	−0.151620	+	0.151620i
$697$		0			0
$698$	−20.0000	+	20.0000i	−0.757011	+	0.757011i
$699$		−	18.0000i		−	0.680823i
$700$			8.00000i			0.302372i
$701$		−	2.00000i		−	0.0755390i	−0.999286	−	0.0377695i	$-0.987975\pi$
$701$		−	2.00000i		−	0.0755390i	0.999286	−	0.0377695i	$-0.0120253\pi$
$702$	−4.00000	−	4.00000i	−0.150970	−	0.150970i
$703$	24.0000			0.905177
$704$	16.0000			0.603023
$705$	16.0000			0.602595
$706$	−10.0000	−	10.0000i	−0.376355	−	0.376355i
$707$			72.0000i			2.70784i
$708$			8.00000i			0.300658i
$709$			38.0000i			1.42712i	0.700594	+	0.713560i	$0.252918\pi$
$709$			38.0000i			1.42712i	−0.700594	+	0.713560i	$0.747082\pi$
$710$	−24.0000	+	24.0000i	−0.900704	+	0.900704i
$711$	−8.00000			−0.300023
$712$	−24.0000	+	24.0000i	−0.899438	+	0.899438i
$713$	−6.00000			−0.224702
$714$		0			0
$715$			16.0000i			0.598366i
$716$	−32.0000			−1.19590
$717$		−	28.0000i		−	1.04568i
$718$	−24.0000	−	24.0000i	−0.895672	−	0.895672i
$719$	36.0000			1.34257			0.671287	−	0.741198i	$-0.265742\pi$
$719$	36.0000			1.34257			0.671287	+	0.741198i	$0.265742\pi$
$720$			8.00000i			0.298142i
$721$		0			0
$722$	3.00000	+	3.00000i	0.111648	+	0.111648i
$723$			10.0000i			0.371904i
$724$	−4.00000			−0.148659
$725$		−	2.00000i		−	0.0742781i
$726$	−7.00000	+	7.00000i	−0.259794	+	0.259794i
$727$	−12.0000			−0.445055			−0.222528	−	0.974926i	$-0.571431\pi$
$727$	−12.0000			−0.445055			−0.222528	+	0.974926i	$0.571431\pi$
$728$	32.0000	+	32.0000i	1.18600	+	1.18600i
$729$	−1.00000			−0.0370370
$730$	−28.0000	+	28.0000i	−1.03633	+	1.03633i
$731$		0			0
$732$		−	12.0000i		−	0.443533i
$733$		−	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	+	8.00000i	0.295285	+	0.295285i
$735$	−18.0000			−0.663940
$736$	−4.00000	−	4.00000i	−0.147442	−	0.147442i
$737$	8.00000			0.294684
$738$	−2.00000	−	2.00000i	−0.0736210	−	0.0736210i
$739$			20.0000i			0.735712i	0.929883	+	0.367856i	$0.119908\pi$
$739$			20.0000i			0.735712i	−0.929883	+	0.367856i	$0.880092\pi$
$740$		−	24.0000i		−	0.882258i
$741$		−	16.0000i		−	0.587775i
$742$	−24.0000	+	24.0000i	−0.881068	+	0.881068i
$743$	16.0000			0.586983			0.293492	−	0.955962i	$-0.405183\pi$
$743$	16.0000			0.586983			0.293492	+	0.955962i	$0.405183\pi$
$744$	12.0000	+	12.0000i	0.439941	+	0.439941i
$745$	20.0000			0.732743
$746$	−6.00000	+	6.00000i	−0.219676	+	0.219676i
$747$			14.0000i			0.512233i
$748$		0			0
$749$			8.00000i			0.292314i
$750$	−12.0000	−	12.0000i	−0.438178	−	0.438178i
$751$	−32.0000			−1.16770			−0.583848	−	0.811863i	$-0.698454\pi$
$751$	−32.0000			−1.16770			−0.583848	+	0.811863i	$0.698454\pi$
$752$	32.0000			1.16692
$753$	30.0000			1.09326
$754$	−8.00000	−	8.00000i	−0.291343	−	0.291343i
$755$		−	44.0000i		−	1.60132i
$756$	8.00000			0.290957
$757$			34.0000i			1.23575i	0.786276	+	0.617876i	$0.212006\pi$
$757$			34.0000i			1.23575i	−0.786276	+	0.617876i	$0.787994\pi$
$758$	12.0000	−	12.0000i	0.435860	−	0.435860i
$759$	−2.00000			−0.0725954
$760$	−16.0000	+	16.0000i	−0.580381	+	0.580381i
$761$	−2.00000			−0.0724999			−0.0362500	−	0.999343i	$-0.511541\pi$
$761$	−2.00000			−0.0724999			−0.0362500	+	0.999343i	$0.511541\pi$
$762$	14.0000	−	14.0000i	0.507166	−	0.507166i
$763$		−	8.00000i		−	0.289619i
$764$		−	32.0000i		−	1.15772i
$765$		0			0
$766$	24.0000	+	24.0000i	0.867155	+	0.867155i
$767$	−16.0000			−0.577727
$768$			16.0000i			0.577350i
$769$	−50.0000			−1.80305			−0.901523	−	0.432731i	$-0.857550\pi$
$769$	−50.0000			−1.80305			−0.901523	+	0.432731i	$0.857550\pi$
$770$	−16.0000	−	16.0000i	−0.576600	−	0.576600i
$771$		−	6.00000i		−	0.216085i
$772$			28.0000i			1.00774i
$773$		−	38.0000i		−	1.36677i	−0.730061	−	0.683383i	$-0.760508\pi$
$773$		−	38.0000i		−	1.36677i	0.730061	−	0.683383i	$-0.239492\pi$
$774$	−8.00000	+	8.00000i	−0.287554	+	0.287554i
$775$	−6.00000			−0.215526
$776$	28.0000	−	28.0000i	1.00514	−	1.00514i
$777$	−24.0000			−0.860995
$778$	10.0000	−	10.0000i	0.358517	−	0.358517i
$779$		−	8.00000i		−	0.286630i
$780$	−16.0000			−0.572892
$781$			24.0000i			0.858788i
$782$		0			0
$783$	−2.00000			−0.0714742
$784$	−36.0000			−1.28571
$785$	44.0000			1.57043
$786$		0			0
$787$			40.0000i			1.42585i	0.701242	+	0.712923i	$0.252629\pi$
$787$			40.0000i			1.42585i	−0.701242	+	0.712923i	$0.747371\pi$
$788$	−36.0000			−1.28245
$789$			24.0000i			0.854423i
$790$	−16.0000	+	16.0000i	−0.569254	+	0.569254i
$791$	−16.0000			−0.568895
$792$	4.00000	+	4.00000i	0.142134	+	0.142134i
$793$	24.0000			0.852265
$794$	8.00000	−	8.00000i	0.283909	−	0.283909i
$795$		−	12.0000i		−	0.425596i
$796$		−	40.0000i		−	1.41776i
$797$			18.0000i			0.637593i	0.947823	+	0.318796i	$0.103279\pi$
$797$			18.0000i			0.637593i	−0.947823	+	0.318796i	$0.896721\pi$
$798$	16.0000	+	16.0000i	0.566394	+	0.566394i
$799$		0			0
$800$	−4.00000	−	4.00000i	−0.141421	−	0.141421i
$801$	−12.0000			−0.423999
$802$	−24.0000	−	24.0000i	−0.847469	−	0.847469i
$803$			28.0000i			0.988099i
$804$			8.00000i			0.282138i
$805$			8.00000i			0.281963i
$806$	−24.0000	+	24.0000i	−0.845364	+	0.845364i
$807$	−6.00000			−0.211210
$808$	−36.0000	−	36.0000i	−1.26648	−	1.26648i
$809$	−26.0000			−0.914111			−0.457056	−	0.889438i	$-0.651096\pi$
$809$	−26.0000			−0.914111			−0.457056	+	0.889438i	$0.651096\pi$
$810$	−2.00000	+	2.00000i	−0.0702728	+	0.0702728i
$811$			4.00000i			0.140459i	0.997531	+	0.0702295i	$0.0223732\pi$
$811$			4.00000i			0.140459i	−0.997531	+	0.0702295i	$0.977627\pi$
$812$	16.0000			0.561490
$813$		−	6.00000i		−	0.210429i
$814$	−12.0000	−	12.0000i	−0.420600	−	0.420600i
$815$	−40.0000			−1.40114
$816$		0			0
$817$	−32.0000			−1.11954
$818$	22.0000	+	22.0000i	0.769212	+	0.769212i
$819$			16.0000i			0.559085i
$820$	−8.00000			−0.279372
$821$			34.0000i			1.18661i	0.804978	+	0.593304i	$0.202177\pi$
$821$			34.0000i			1.18661i	−0.804978	+	0.593304i	$0.797823\pi$
$822$	12.0000	−	12.0000i	0.418548	−	0.418548i
$823$	14.0000			0.488009			0.244005	−	0.969774i	$-0.421539\pi$
$823$	14.0000			0.488009			0.244005	+	0.969774i	$0.421539\pi$
$824$		0			0
$825$	−2.00000			−0.0696311
$826$	16.0000	−	16.0000i	0.556711	−	0.556711i
$827$			14.0000i			0.486828i	0.969923	+	0.243414i	$0.0782673\pi$
$827$			14.0000i			0.486828i	−0.969923	+	0.243414i	$0.921733\pi$
$828$		−	2.00000i		−	0.0695048i
$829$			40.0000i			1.38926i	0.719368	+	0.694629i	$0.244431\pi$
$829$			40.0000i			1.38926i	−0.719368	+	0.694629i	$0.755569\pi$
$830$	28.0000	+	28.0000i	0.971894	+	0.971894i
$831$	−8.00000			−0.277517
$832$	−32.0000			−1.10940
$833$		0			0
$834$	4.00000	+	4.00000i	0.138509	+	0.138509i
$835$			16.0000i			0.553703i
$836$			16.0000i			0.553372i
$837$			6.00000i			0.207390i
$838$	−18.0000	+	18.0000i	−0.621800	+	0.621800i
$839$	−24.0000			−0.828572			−0.414286	−	0.910147i	$-0.635969\pi$
$839$	−24.0000			−0.828572			−0.414286	+	0.910147i	$0.635969\pi$
$840$	16.0000	−	16.0000i	0.552052	−	0.552052i
$841$	25.0000			0.862069
$842$	−22.0000	+	22.0000i	−0.758170	+	0.758170i
$843$		−	12.0000i		−	0.413302i
$844$	24.0000			0.826114
$845$		−	6.00000i		−	0.206406i
$846$	8.00000	+	8.00000i	0.275046	+	0.275046i
$847$	28.0000			0.962091
$848$		−	24.0000i		−	0.824163i
$849$	−32.0000			−1.09824
$850$		0			0
$851$			6.00000i			0.205677i
$852$	−24.0000			−0.822226
$853$			16.0000i			0.547830i	0.961754	+	0.273915i	$0.0883186\pi$
$853$			16.0000i			0.547830i	−0.961754	+	0.273915i	$0.911681\pi$
$854$	−24.0000	+	24.0000i	−0.821263	+	0.821263i
$855$	−8.00000			−0.273594
$856$	−4.00000	−	4.00000i	−0.136717	−	0.136717i
$857$	−22.0000			−0.751506			−0.375753	−	0.926720i	$-0.622616\pi$
$857$	−22.0000			−0.751506			−0.375753	+	0.926720i	$0.622616\pi$
$858$	−8.00000	+	8.00000i	−0.273115	+	0.273115i
$859$			28.0000i			0.955348i	0.878537	+	0.477674i	$0.158520\pi$
$859$			28.0000i			0.955348i	−0.878537	+	0.477674i	$0.841480\pi$
$860$			32.0000i			1.09119i
$861$			8.00000i			0.272639i
$862$	16.0000	+	16.0000i	0.544962	+	0.544962i
$863$	16.0000			0.544646			0.272323	−	0.962206i	$-0.412208\pi$
$863$	16.0000			0.544646			0.272323	+	0.962206i	$0.412208\pi$
$864$	−4.00000	+	4.00000i	−0.136083	+	0.136083i
$865$	12.0000			0.408012
$866$	−34.0000	−	34.0000i	−1.15537	−	1.15537i
$867$		−	17.0000i		−	0.577350i
$868$		−	48.0000i		−	1.62923i
$869$			16.0000i			0.542763i
$870$	−4.00000	+	4.00000i	−0.135613	+	0.135613i
$871$	−16.0000			−0.542139
$872$	4.00000	+	4.00000i	0.135457	+	0.135457i
$873$	14.0000			0.473828
$874$	4.00000	−	4.00000i	0.135302	−	0.135302i
$875$			48.0000i			1.62270i
$876$	−28.0000			−0.946032
$877$			28.0000i			0.945493i	0.881199	+	0.472746i	$0.156737\pi$
$877$			28.0000i			0.945493i	−0.881199	+	0.472746i	$0.843263\pi$
$878$	34.0000	+	34.0000i	1.14744	+	1.14744i
$879$	22.0000			0.742042
$880$	16.0000			0.539360
$881$	−52.0000			−1.75192			−0.875962	−	0.482380i	$-0.839773\pi$
$881$	−52.0000			−1.75192			−0.875962	+	0.482380i	$0.839773\pi$
$882$	−9.00000	−	9.00000i	−0.303046	−	0.303046i
$883$			52.0000i			1.74994i	0.484178	+	0.874970i	$0.339119\pi$
$883$			52.0000i			1.74994i	−0.484178	+	0.874970i	$0.660881\pi$
$884$		0			0
$885$			8.00000i			0.268917i
$886$	24.0000	−	24.0000i	0.806296	−	0.806296i
$887$	28.0000			0.940148			0.470074	−	0.882627i	$-0.344227\pi$
$887$	28.0000			0.940148			0.470074	+	0.882627i	$0.344227\pi$
$888$	12.0000	−	12.0000i	0.402694	−	0.402694i
$889$	−56.0000			−1.87818
$890$	−24.0000	+	24.0000i	−0.804482	+	0.804482i
$891$			2.00000i			0.0670025i
$892$			20.0000i			0.669650i
$893$			32.0000i			1.07084i
$894$	10.0000	+	10.0000i	0.334450	+	0.334450i
$895$	−32.0000			−1.06964
$896$	32.0000	−	32.0000i	1.06904	−	1.06904i
$897$	4.00000			0.133556
$898$	−26.0000	−	26.0000i	−0.867631	−	0.867631i
$899$			12.0000i			0.400222i
$900$		−	2.00000i		−	0.0666667i
$901$		0			0
$902$	−4.00000	+	4.00000i	−0.133185	+	0.133185i
$903$	32.0000			1.06489
$904$	8.00000	−	8.00000i	0.266076	−	0.266076i
$905$	−4.00000			−0.132964
$906$	22.0000	−	22.0000i	0.730901	−	0.730901i
$907$		−	36.0000i		−	1.19536i	−0.801735	−	0.597680i	$-0.796089\pi$
$907$		−	36.0000i		−	1.19536i	0.801735	−	0.597680i	$-0.203911\pi$
$908$	12.0000			0.398234
$909$		−	18.0000i		−	0.597022i
$910$	32.0000	+	32.0000i	1.06079	+	1.06079i
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$	−16.0000			−0.529813
$913$	28.0000			0.926665
$914$	38.0000	+	38.0000i	1.25693	+	1.25693i
$915$		−	12.0000i		−	0.396708i
$916$	44.0000			1.45380
$917$		0			0
$918$		0			0
$919$	−8.00000			−0.263896			−0.131948	−	0.991257i	$-0.542123\pi$
$919$	−8.00000			−0.263896			−0.131948	+	0.991257i	$0.542123\pi$
$920$	−4.00000	−	4.00000i	−0.131876	−	0.131876i
$921$	28.0000			0.922631
$922$	−10.0000	+	10.0000i	−0.329332	+	0.329332i
$923$		−	48.0000i		−	1.57994i
$924$		−	16.0000i		−	0.526361i
$925$			6.00000i			0.197279i
$926$	6.00000	+	6.00000i	0.197172	+	0.197172i
$927$		0			0
$928$	−8.00000	+	8.00000i	−0.262613	+	0.262613i
$929$	−50.0000			−1.64045			−0.820223	−	0.572043i	$-0.806151\pi$
$929$	−50.0000			−1.64045			−0.820223	+	0.572043i	$0.806151\pi$
$930$	12.0000	+	12.0000i	0.393496	+	0.393496i
$931$		−	36.0000i		−	1.17985i
$932$		−	36.0000i		−	1.17922i
$933$		−	24.0000i		−	0.785725i
$934$	18.0000	−	18.0000i	0.588978	−	0.588978i
$935$		0			0
$936$	−8.00000	−	8.00000i	−0.261488	−	0.261488i
$937$	−18.0000			−0.588034			−0.294017	−	0.955800i	$-0.594992\pi$
$937$	−18.0000			−0.588034			−0.294017	+	0.955800i	$0.594992\pi$
$938$	16.0000	−	16.0000i	0.522419	−	0.522419i
$939$		−	6.00000i		−	0.195803i
$940$	32.0000			1.04372
$941$			18.0000i			0.586783i	0.955992	+	0.293392i	$0.0947840\pi$
$941$			18.0000i			0.586783i	−0.955992	+	0.293392i	$0.905216\pi$
$942$	22.0000	+	22.0000i	0.716799	+	0.716799i
$943$	2.00000			0.0651290
$944$			16.0000i			0.520756i
$945$	8.00000			0.260240
$946$	16.0000	+	16.0000i	0.520205	+	0.520205i
$947$			24.0000i			0.779895i	0.920837	+	0.389948i	$0.127507\pi$
$947$			24.0000i			0.779895i	−0.920837	+	0.389948i	$0.872493\pi$
$948$	−16.0000			−0.519656
$949$		−	56.0000i		−	1.81784i
$950$	4.00000	−	4.00000i	0.129777	−	0.129777i
$951$	−22.0000			−0.713399
$952$		0			0
$953$	−48.0000			−1.55487			−0.777436	−	0.628962i	$-0.783480\pi$
$953$	−48.0000			−1.55487			−0.777436	+	0.628962i	$0.783480\pi$
$954$	6.00000	−	6.00000i	0.194257	−	0.194257i
$955$		−	32.0000i		−	1.03550i
$956$		−	56.0000i		−	1.81117i
$957$			4.00000i			0.129302i
$958$	−24.0000	−	24.0000i	−0.775405	−	0.775405i
$959$	−48.0000			−1.55000
$960$			16.0000i			0.516398i
$961$	5.00000			0.161290
$962$	24.0000	+	24.0000i	0.773791	+	0.773791i
$963$		−	2.00000i		−	0.0644491i
$964$			20.0000i			0.644157i
$965$			28.0000i			0.901352i
$966$	−4.00000	+	4.00000i	−0.128698	+	0.128698i
$967$	−6.00000			−0.192947			−0.0964735	−	0.995336i	$-0.530756\pi$
$967$	−6.00000			−0.192947			−0.0964735	+	0.995336i	$0.530756\pi$
$968$	−14.0000	+	14.0000i	−0.449977	+	0.449977i
$969$		0			0
$970$	28.0000	−	28.0000i	0.899026	−	0.899026i
$971$		−	50.0000i		−	1.60458i	−0.596937	−	0.802288i	$-0.703616\pi$
$971$		−	50.0000i		−	1.60458i	0.596937	−	0.802288i	$-0.296384\pi$
$972$	−2.00000			−0.0641500
$973$		−	16.0000i		−	0.512936i
$974$	38.0000	+	38.0000i	1.21760	+	1.21760i
$975$	4.00000			0.128103
$976$		−	24.0000i		−	0.768221i
$977$	28.0000			0.895799			0.447900	−	0.894084i	$-0.352172\pi$
$977$	28.0000			0.895799			0.447900	+	0.894084i	$0.352172\pi$
$978$	−20.0000	−	20.0000i	−0.639529	−	0.639529i
$979$			24.0000i			0.767043i
$980$	−36.0000			−1.14998
$981$			2.00000i			0.0638551i
$982$	40.0000	−	40.0000i	1.27645	−	1.27645i
$983$	−32.0000			−1.02064			−0.510321	−	0.859984i	$-0.670473\pi$
$983$	−32.0000			−1.02064			−0.510321	+	0.859984i	$0.670473\pi$
$984$	−4.00000	−	4.00000i	−0.127515	−	0.127515i
$985$	−36.0000			−1.14706
$986$		0			0
$987$		−	32.0000i		−	1.01857i
$988$		−	32.0000i		−	1.01806i
$989$		−	8.00000i		−	0.254385i
$990$	4.00000	+	4.00000i	0.127128	+	0.127128i
$991$	10.0000			0.317660			0.158830	−	0.987306i	$-0.449228\pi$
$991$	10.0000			0.317660			0.158830	+	0.987306i	$0.449228\pi$
$992$	24.0000	+	24.0000i	0.762001	+	0.762001i
$993$	−4.00000			−0.126936
$994$	48.0000	+	48.0000i	1.52247	+	1.52247i
$995$		−	40.0000i		−	1.26809i
$996$			28.0000i			0.887214i
$997$		−	4.00000i		−	0.126681i	−0.997992	−	0.0633406i	$-0.979825\pi$
$997$		−	4.00000i		−	0.126681i	0.997992	−	0.0633406i	$-0.0201755\pi$
$998$	−20.0000	+	20.0000i	−0.633089	+	0.633089i
$999$	6.00000			0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	552.2.f.a.277.2	yes	2
4.3	odd	2		2208.2.f.a.1105.1		2
8.3	odd	2		2208.2.f.a.1105.2		2
8.5	even	2	inner	552.2.f.a.277.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
552.2.f.a.277.1	✓	2	8.5	even	2	inner
552.2.f.a.277.2	yes	2	1.1	even	1	trivial
2208.2.f.a.1105.1		2	4.3	odd	2
2208.2.f.a.1105.2		2	8.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 552 = 2^{3} \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	552.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 + 1.00000i) q^{2} +1.00000i q^{3} +2.00000i q^{4} +2.00000i q^{5} +(-1.00000 + 1.00000i) q^{6} +4.00000 q^{7} +(-2.00000 + 2.00000i) q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+(1.00000 + 1.00000i) q^{2} +1.00000i q^{3} +2.00000i q^{4} +2.00000i q^{5} +(-1.00000 + 1.00000i) q^{6} +4.00000 q^{7} +(-2.00000 + 2.00000i) q^{8} -1.00000 q^{9} +(-2.00000 + 2.00000i) q^{10} +2.00000i q^{11} -2.00000 q^{12} -4.00000i q^{13} +(4.00000 + 4.00000i) q^{14} -2.00000 q^{15} -4.00000 q^{16} +(-1.00000 - 1.00000i) q^{18} -4.00000i q^{19} -4.00000 q^{20} +4.00000i q^{21} +(-2.00000 + 2.00000i) q^{22} +1.00000 q^{23} +(-2.00000 - 2.00000i) q^{24} +1.00000 q^{25} +(4.00000 - 4.00000i) q^{26} -1.00000i q^{27} +8.00000i q^{28} -2.00000i q^{29} +(-2.00000 - 2.00000i) q^{30} -6.00000 q^{31} +(-4.00000 - 4.00000i) q^{32} -2.00000 q^{33} +8.00000i q^{35} -2.00000i q^{36} +6.00000i q^{37} +(4.00000 - 4.00000i) q^{38} +4.00000 q^{39} +(-4.00000 - 4.00000i) q^{40} +2.00000 q^{41} +(-4.00000 + 4.00000i) q^{42} -8.00000i q^{43} -4.00000 q^{44} -2.00000i q^{45} +(1.00000 + 1.00000i) q^{46} -8.00000 q^{47} -4.00000i q^{48} +9.00000 q^{49} +(1.00000 + 1.00000i) q^{50} +8.00000 q^{52} +6.00000i q^{53} +(1.00000 - 1.00000i) q^{54} -4.00000 q^{55} +(-8.00000 + 8.00000i) q^{56} +4.00000 q^{57} +(2.00000 - 2.00000i) q^{58} -4.00000i q^{59} -4.00000i q^{60} +6.00000i q^{61} +(-6.00000 - 6.00000i) q^{62} -4.00000 q^{63} -8.00000i q^{64} +8.00000 q^{65} +(-2.00000 - 2.00000i) q^{66} -4.00000i q^{67} +1.00000i q^{69} +(-8.00000 + 8.00000i) q^{70} +12.0000 q^{71} +(2.00000 - 2.00000i) q^{72} +14.0000 q^{73} +(-6.00000 + 6.00000i) q^{74} +1.00000i q^{75} +8.00000 q^{76} +8.00000i q^{77} +(4.00000 + 4.00000i) q^{78} +8.00000 q^{79} -8.00000i q^{80} +1.00000 q^{81} +(2.00000 + 2.00000i) q^{82} -14.0000i q^{83} -8.00000 q^{84} +(8.00000 - 8.00000i) q^{86} +2.00000 q^{87} +(-4.00000 - 4.00000i) q^{88} +12.0000 q^{89} +(2.00000 - 2.00000i) q^{90} -16.0000i q^{91} +2.00000i q^{92} -6.00000i q^{93} +(-8.00000 - 8.00000i) q^{94} +8.00000 q^{95} +(4.00000 - 4.00000i) q^{96} -14.0000 q^{97} +(9.00000 + 9.00000i) q^{98} -2.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - 2 q^{6} + 8 q^{7} - 4 q^{8} - 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^2 - 2 * q^6 + 8 * q^7 - 4 * q^8 - 2 * q^9	\( 2 q + 2 q^{2} - 2 q^{6} + 8 q^{7} - 4 q^{8} - 2 q^{9} - 4 q^{10} - 4 q^{12} + 8 q^{14} - 4 q^{15} - 8 q^{16} - 2 q^{18} - 8 q^{20} - 4 q^{22} + 2 q^{23} - 4 q^{24} + 2 q^{25} + 8 q^{26} - 4 q^{30} - 12 q^{31} - 8 q^{32} - 4 q^{33} + 8 q^{38} + 8 q^{39} - 8 q^{40} + 4 q^{41} - 8 q^{42} - 8 q^{44} + 2 q^{46} - 16 q^{47} + 18 q^{49} + 2 q^{50} + 16 q^{52} + 2 q^{54} - 8 q^{55} - 16 q^{56} + 8 q^{57} + 4 q^{58} - 12 q^{62} - 8 q^{63} + 16 q^{65} - 4 q^{66} - 16 q^{70} + 24 q^{71} + 4 q^{72} + 28 q^{73} - 12 q^{74} + 16 q^{76} + 8 q^{78} + 16 q^{79} + 2 q^{81} + 4 q^{82} - 16 q^{84} + 16 q^{86} + 4 q^{87} - 8 q^{88} + 24 q^{89} + 4 q^{90} - 16 q^{94} + 16 q^{95} + 8 q^{96} - 28 q^{97} + 18 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 - 2 * q^6 + 8 * q^7 - 4 * q^8 - 2 * q^9 - 4 * q^10 - 4 * q^12 + 8 * q^14 - 4 * q^15 - 8 * q^16 - 2 * q^18 - 8 * q^20 - 4 * q^22 + 2 * q^23 - 4 * q^24 + 2 * q^25 + 8 * q^26 - 4 * q^30 - 12 * q^31 - 8 * q^32 - 4 * q^33 + 8 * q^38 + 8 * q^39 - 8 * q^40 + 4 * q^41 - 8 * q^42 - 8 * q^44 + 2 * q^46 - 16 * q^47 + 18 * q^49 + 2 * q^50 + 16 * q^52 + 2 * q^54 - 8 * q^55 - 16 * q^56 + 8 * q^57 + 4 * q^58 - 12 * q^62 - 8 * q^63 + 16 * q^65 - 4 * q^66 - 16 * q^70 + 24 * q^71 + 4 * q^72 + 28 * q^73 - 12 * q^74 + 16 * q^76 + 8 * q^78 + 16 * q^79 + 2 * q^81 + 4 * q^82 - 16 * q^84 + 16 * q^86 + 4 * q^87 - 8 * q^88 + 24 * q^89 + 4 * q^90 - 16 * q^94 + 16 * q^95 + 8 * q^96 - 28 * q^97 + 18 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more