LMFDB - Embedded newform 867.2.e.d.616.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [867,2,Mod(616,867)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(867, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 3]))

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

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

chi := DirichletCharacter("867.616");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 867 = 3 \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	867.e (of order $4$, degree $2$, not 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:	$6.92302985525$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 51)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			616.1
Root			$-0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	867.616
Dual form			867.2.e.d.829.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} +(-0.707107 + 0.707107i) q^{3} +1.00000 q^{4} +(0.707107 + 0.707107i) q^{6} +(2.82843 + 2.82843i) q^{7} -3.00000i q^{8} -1.00000i q^{9} +O(q^{10})$	\(q-1.00000i q^{2} +(-0.707107 + 0.707107i) q^{3} +1.00000 q^{4} +(0.707107 + 0.707107i) q^{6} +(2.82843 + 2.82843i) q^{7} -3.00000i q^{8} -1.00000i q^{9} +(2.82843 + 2.82843i) q^{11} +(-0.707107 + 0.707107i) q^{12} -2.00000 q^{13} +(2.82843 - 2.82843i) q^{14} -1.00000 q^{16} -1.00000 q^{18} +4.00000i q^{19} -4.00000 q^{21} +(2.82843 - 2.82843i) q^{22} +(-2.82843 - 2.82843i) q^{23} +(2.12132 + 2.12132i) q^{24} +5.00000i q^{25} +2.00000i q^{26} +(0.707107 + 0.707107i) q^{27} +(2.82843 + 2.82843i) q^{28} +(2.82843 - 2.82843i) q^{31} -5.00000i q^{32} -4.00000 q^{33} -1.00000i q^{36} +(-5.65685 + 5.65685i) q^{37} +4.00000 q^{38} +(1.41421 - 1.41421i) q^{39} +(5.65685 + 5.65685i) q^{41} +4.00000i q^{42} -4.00000i q^{43} +(2.82843 + 2.82843i) q^{44} +(-2.82843 + 2.82843i) q^{46} +8.00000 q^{47} +(0.707107 - 0.707107i) q^{48} +9.00000i q^{49} +5.00000 q^{50} -2.00000 q^{52} -6.00000i q^{53} +(0.707107 - 0.707107i) q^{54} +(8.48528 - 8.48528i) q^{56} +(-2.82843 - 2.82843i) q^{57} -12.0000i q^{59} +(5.65685 + 5.65685i) q^{61} +(-2.82843 - 2.82843i) q^{62} +(2.82843 - 2.82843i) q^{63} -7.00000 q^{64} +4.00000i q^{66} +12.0000 q^{67} +4.00000 q^{69} +(-8.48528 + 8.48528i) q^{71} -3.00000 q^{72} +(5.65685 + 5.65685i) q^{74} +(-3.53553 - 3.53553i) q^{75} +4.00000i q^{76} +16.0000i q^{77} +(-1.41421 - 1.41421i) q^{78} +(-2.82843 - 2.82843i) q^{79} -1.00000 q^{81} +(5.65685 - 5.65685i) q^{82} -12.0000i q^{83} -4.00000 q^{84} -4.00000 q^{86} +(8.48528 - 8.48528i) q^{88} +10.0000 q^{89} +(-5.65685 - 5.65685i) q^{91} +(-2.82843 - 2.82843i) q^{92} +4.00000i q^{93} -8.00000i q^{94} +(3.53553 + 3.53553i) q^{96} +(11.3137 - 11.3137i) q^{97} +9.00000 q^{98} +(2.82843 - 2.82843i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{4}+O(q^{10}) $ 4 * q + 4 * q^4	$ 4 q + 4 q^{4} - 8 q^{13} - 4 q^{16} - 4 q^{18} - 16 q^{21} - 16 q^{33} + 16 q^{38} + 32 q^{47} + 20 q^{50} - 8 q^{52} - 28 q^{64} + 48 q^{67} + 16 q^{69} - 12 q^{72} - 4 q^{81} - 16 q^{84} - 16 q^{86} + 40 q^{89} + 36 q^{98}+O(q^{100}) $ 4 * q + 4 * q^4 - 8 * q^13 - 4 * q^16 - 4 * q^18 - 16 * q^21 - 16 * q^33 + 16 * q^38 + 32 * q^47 + 20 * q^50 - 8 * q^52 - 28 * q^64 + 48 * q^67 + 16 * q^69 - 12 * q^72 - 4 * q^81 - 16 * q^84 - 16 * q^86 + 40 * q^89 + 36 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/867\mathbb{Z}\right)^\times$.

$n$	$290$	$292$
$\chi(n)$	$1$	$e\left(\frac{3}{4}\right)$

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	−0.935414	−	0.353553i	$-0.884973\pi$
$2$		−	1.00000i		−	0.707107i	0.935414	−	0.353553i	$-0.115027\pi$
$3$	−0.707107	+	0.707107i	−0.408248	+	0.408248i
$3$	−0.707107	+	0.707107i	−0.408248	+	0.408248i
$4$	1.00000			0.500000
$5$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$5$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$6$	0.707107	+	0.707107i	0.288675	+	0.288675i
$7$	2.82843	+	2.82843i	1.06904	+	1.06904i	0.997433	+	0.0716124i	$0.0228145\pi$
$7$	2.82843	+	2.82843i	1.06904	+	1.06904i	0.0716124	+	0.997433i	$0.477186\pi$
$8$		−	3.00000i		−	1.06066i
$9$		−	1.00000i		−	0.333333i
$10$		0			0
$11$	2.82843	+	2.82843i	0.852803	+	0.852803i	0.990478	−	0.137675i	$-0.0439628\pi$
$11$	2.82843	+	2.82843i	0.852803	+	0.852803i	−0.137675	+	0.990478i	$0.543963\pi$
$12$	−0.707107	+	0.707107i	−0.204124	+	0.204124i
$13$	−2.00000			−0.554700			−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000			−0.554700			−0.277350	+	0.960769i	$0.589456\pi$
$14$	2.82843	−	2.82843i	0.755929	−	0.755929i
$15$		0			0
$16$	−1.00000			−0.250000
$17$		0			0
$17$		0			0
$18$	−1.00000			−0.235702
$19$			4.00000i			0.917663i	0.888523	+	0.458831i	$0.151732\pi$
$19$			4.00000i			0.917663i	−0.888523	+	0.458831i	$0.848268\pi$
$20$		0			0
$21$	−4.00000			−0.872872
$22$	2.82843	−	2.82843i	0.603023	−	0.603023i
$23$	−2.82843	−	2.82843i	−0.589768	−	0.589768i	0.347801	−	0.937568i	$-0.386929\pi$
$23$	−2.82843	−	2.82843i	−0.589768	−	0.589768i	−0.937568	+	0.347801i	$0.886929\pi$
$24$	2.12132	+	2.12132i	0.433013	+	0.433013i
$25$			5.00000i			1.00000i
$26$			2.00000i			0.392232i
$27$	0.707107	+	0.707107i	0.136083	+	0.136083i
$28$	2.82843	+	2.82843i	0.534522	+	0.534522i
$29$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$29$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$30$		0			0
$31$	2.82843	−	2.82843i	0.508001	−	0.508001i	−0.405912	−	0.913912i	$-0.633046\pi$
$31$	2.82843	−	2.82843i	0.508001	−	0.508001i	0.913912	+	0.405912i	$0.133046\pi$
$32$		−	5.00000i		−	0.883883i
$33$	−4.00000			−0.696311
$34$		0			0
$35$		0			0
$36$		−	1.00000i		−	0.166667i
$37$	−5.65685	+	5.65685i	−0.929981	+	0.929981i	−0.997704	−	0.0677230i	$-0.978427\pi$
$37$	−5.65685	+	5.65685i	−0.929981	+	0.929981i	0.0677230	+	0.997704i	$0.478427\pi$
$38$	4.00000			0.648886
$39$	1.41421	−	1.41421i	0.226455	−	0.226455i
$40$		0			0
$41$	5.65685	+	5.65685i	0.883452	+	0.883452i	0.993884	−	0.110432i	$-0.0352233\pi$
$41$	5.65685	+	5.65685i	0.883452	+	0.883452i	−0.110432	+	0.993884i	$0.535223\pi$
$42$			4.00000i			0.617213i
$43$		−	4.00000i		−	0.609994i	−0.952353	−	0.304997i	$-0.901344\pi$
$43$		−	4.00000i		−	0.609994i	0.952353	−	0.304997i	$-0.0986555\pi$
$44$	2.82843	+	2.82843i	0.426401	+	0.426401i
$45$		0			0
$46$	−2.82843	+	2.82843i	−0.417029	+	0.417029i
$47$	8.00000			1.16692			0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000			1.16692			0.583460	+	0.812142i	$0.301699\pi$
$48$	0.707107	−	0.707107i	0.102062	−	0.102062i
$49$			9.00000i			1.28571i
$50$	5.00000			0.707107
$51$		0			0
$52$	−2.00000			−0.277350
$53$		−	6.00000i		−	0.824163i	−0.911147	−	0.412082i	$-0.864802\pi$
$53$		−	6.00000i		−	0.824163i	0.911147	−	0.412082i	$-0.135198\pi$
$54$	0.707107	−	0.707107i	0.0962250	−	0.0962250i
$55$		0			0
$56$	8.48528	−	8.48528i	1.13389	−	1.13389i
$57$	−2.82843	−	2.82843i	−0.374634	−	0.374634i
$58$		0			0
$59$		−	12.0000i		−	1.56227i	−0.624364	−	0.781133i	$-0.714642\pi$
$59$		−	12.0000i		−	1.56227i	0.624364	−	0.781133i	$-0.285358\pi$
$60$		0			0
$61$	5.65685	+	5.65685i	0.724286	+	0.724286i	0.969475	−	0.245189i	$-0.0788501\pi$
$61$	5.65685	+	5.65685i	0.724286	+	0.724286i	−0.245189	+	0.969475i	$0.578850\pi$
$62$	−2.82843	−	2.82843i	−0.359211	−	0.359211i
$63$	2.82843	−	2.82843i	0.356348	−	0.356348i
$64$	−7.00000			−0.875000
$65$		0			0
$66$			4.00000i			0.492366i
$67$	12.0000			1.46603			0.733017	−	0.680211i	$-0.238112\pi$
$67$	12.0000			1.46603			0.733017	+	0.680211i	$0.238112\pi$
$68$		0			0
$69$	4.00000			0.481543
$70$		0			0
$71$	−8.48528	+	8.48528i	−1.00702	+	1.00702i	−0.00704243	+	0.999975i	$0.502242\pi$
$71$	−8.48528	+	8.48528i	−1.00702	+	1.00702i	−0.999975	+	0.00704243i	$0.997758\pi$
$72$	−3.00000			−0.353553
$73$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$73$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$74$	5.65685	+	5.65685i	0.657596	+	0.657596i
$75$	−3.53553	−	3.53553i	−0.408248	−	0.408248i
$76$			4.00000i			0.458831i
$77$			16.0000i			1.82337i
$78$	−1.41421	−	1.41421i	−0.160128	−	0.160128i
$79$	−2.82843	−	2.82843i	−0.318223	−	0.318223i	0.529861	−	0.848084i	$-0.322244\pi$
$79$	−2.82843	−	2.82843i	−0.318223	−	0.318223i	−0.848084	+	0.529861i	$0.822244\pi$
$80$		0			0
$81$	−1.00000			−0.111111
$82$	5.65685	−	5.65685i	0.624695	−	0.624695i
$83$		−	12.0000i		−	1.31717i	−0.752506	−	0.658586i	$-0.771155\pi$
$83$		−	12.0000i		−	1.31717i	0.752506	−	0.658586i	$-0.228845\pi$
$84$	−4.00000			−0.436436
$85$		0			0
$86$	−4.00000			−0.431331
$87$		0			0
$88$	8.48528	−	8.48528i	0.904534	−	0.904534i
$89$	10.0000			1.06000			0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000			1.06000			0.529999	+	0.847998i	$0.322192\pi$
$90$		0			0
$91$	−5.65685	−	5.65685i	−0.592999	−	0.592999i
$92$	−2.82843	−	2.82843i	−0.294884	−	0.294884i
$93$			4.00000i			0.414781i
$94$		−	8.00000i		−	0.825137i
$95$		0			0
$96$	3.53553	+	3.53553i	0.360844	+	0.360844i
$97$	11.3137	−	11.3137i	1.14873	−	1.14873i	0.161931	−	0.986802i	$-0.448228\pi$
$97$	11.3137	−	11.3137i	1.14873	−	1.14873i	0.986802	−	0.161931i	$-0.0517722\pi$
$98$	9.00000			0.909137
$99$	2.82843	−	2.82843i	0.284268	−	0.284268i
$100$			5.00000i			0.500000i
$101$	−6.00000			−0.597022			−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000			−0.597022			−0.298511	+	0.954406i	$0.596490\pi$
$102$		0			0
$103$		0			0			−	1.00000i	$-0.5\pi$
$103$		0			0				1.00000i	$0.5\pi$
$104$			6.00000i			0.588348i
$105$		0			0
$106$	−6.00000			−0.582772
$107$	−8.48528	+	8.48528i	−0.820303	+	0.820303i	−0.986151	−	0.165848i	$-0.946964\pi$
$107$	−8.48528	+	8.48528i	−0.820303	+	0.820303i	0.165848	+	0.986151i	$0.446964\pi$
$108$	0.707107	+	0.707107i	0.0680414	+	0.0680414i
$109$	−5.65685	−	5.65685i	−0.541828	−	0.541828i	0.382236	−	0.924065i	$-0.375154\pi$
$109$	−5.65685	−	5.65685i	−0.541828	−	0.541828i	−0.924065	+	0.382236i	$0.875154\pi$
$110$		0			0
$111$		−	8.00000i		−	0.759326i
$112$	−2.82843	−	2.82843i	−0.267261	−	0.267261i
$113$	−5.65685	−	5.65685i	−0.532152	−	0.532152i	0.389060	−	0.921212i	$-0.372800\pi$
$113$	−5.65685	−	5.65685i	−0.532152	−	0.532152i	−0.921212	+	0.389060i	$0.872800\pi$
$114$	−2.82843	+	2.82843i	−0.264906	+	0.264906i
$115$		0			0
$116$		0			0
$117$			2.00000i			0.184900i
$118$	−12.0000			−1.10469
$119$		0			0
$120$		0			0
$121$			5.00000i			0.454545i
$122$	5.65685	−	5.65685i	0.512148	−	0.512148i
$123$	−8.00000			−0.721336
$124$	2.82843	−	2.82843i	0.254000	−	0.254000i
$125$		0			0
$126$	−2.82843	−	2.82843i	−0.251976	−	0.251976i
$127$			8.00000i			0.709885i	0.934888	+	0.354943i	$0.115500\pi$
$127$			8.00000i			0.709885i	−0.934888	+	0.354943i	$0.884500\pi$
$128$		−	3.00000i		−	0.265165i
$129$	2.82843	+	2.82843i	0.249029	+	0.249029i
$130$		0			0
$131$	2.82843	−	2.82843i	0.247121	−	0.247121i	−0.572667	−	0.819788i	$-0.694091\pi$
$131$	2.82843	−	2.82843i	0.247121	−	0.247121i	0.819788	+	0.572667i	$0.194091\pi$
$132$	−4.00000			−0.348155
$133$	−11.3137	+	11.3137i	−0.981023	+	0.981023i
$134$		−	12.0000i		−	1.03664i
$135$		0			0
$136$		0			0
$137$	−10.0000			−0.854358			−0.427179	−	0.904167i	$-0.640493\pi$
$137$	−10.0000			−0.854358			−0.427179	+	0.904167i	$0.640493\pi$
$138$		−	4.00000i		−	0.340503i
$139$	−2.82843	+	2.82843i	−0.239904	+	0.239904i	−0.816810	−	0.576906i	$-0.804260\pi$
$139$	−2.82843	+	2.82843i	−0.239904	+	0.239904i	0.576906	+	0.816810i	$0.304260\pi$
$140$		0			0
$141$	−5.65685	+	5.65685i	−0.476393	+	0.476393i
$142$	8.48528	+	8.48528i	0.712069	+	0.712069i
$143$	−5.65685	−	5.65685i	−0.473050	−	0.473050i
$144$			1.00000i			0.0833333i
$145$		0			0
$146$		0			0
$147$	−6.36396	−	6.36396i	−0.524891	−	0.524891i
$148$	−5.65685	+	5.65685i	−0.464991	+	0.464991i
$149$	−6.00000			−0.491539			−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000			−0.491539			−0.245770	+	0.969328i	$0.579041\pi$
$150$	−3.53553	+	3.53553i	−0.288675	+	0.288675i
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$	12.0000			0.973329
$153$		0			0
$154$	16.0000			1.28932
$155$		0			0
$156$	1.41421	−	1.41421i	0.113228	−	0.113228i
$157$	2.00000			0.159617			0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000			0.159617			0.0798087	+	0.996810i	$0.474569\pi$
$158$	−2.82843	+	2.82843i	−0.225018	+	0.225018i
$159$	4.24264	+	4.24264i	0.336463	+	0.336463i
$160$		0			0
$161$		−	16.0000i		−	1.26098i
$162$			1.00000i			0.0785674i
$163$	−14.1421	−	14.1421i	−1.10770	−	1.10770i	−0.993453	−	0.114245i	$-0.963555\pi$
$163$	−14.1421	−	14.1421i	−1.10770	−	1.10770i	−0.114245	−	0.993453i	$-0.536445\pi$
$164$	5.65685	+	5.65685i	0.441726	+	0.441726i
$165$		0			0
$166$	−12.0000			−0.931381
$167$	−8.48528	+	8.48528i	−0.656611	+	0.656611i	−0.954577	−	0.297966i	$-0.903692\pi$
$167$	−8.48528	+	8.48528i	−0.656611	+	0.656611i	0.297966	+	0.954577i	$0.403692\pi$
$168$			12.0000i			0.925820i
$169$	−9.00000			−0.692308
$170$		0			0
$171$	4.00000			0.305888
$172$		−	4.00000i		−	0.304997i
$173$	11.3137	−	11.3137i	0.860165	−	0.860165i	−0.131192	−	0.991357i	$-0.541880\pi$
$173$	11.3137	−	11.3137i	0.860165	−	0.860165i	0.991357	+	0.131192i	$0.0418803\pi$
$174$		0			0
$175$	−14.1421	+	14.1421i	−1.06904	+	1.06904i
$176$	−2.82843	−	2.82843i	−0.213201	−	0.213201i
$177$	8.48528	+	8.48528i	0.637793	+	0.637793i
$178$		−	10.0000i		−	0.749532i
$179$			4.00000i			0.298974i	0.988764	+	0.149487i	$0.0477622\pi$
$179$			4.00000i			0.298974i	−0.988764	+	0.149487i	$0.952238\pi$
$180$		0			0
$181$	−5.65685	−	5.65685i	−0.420471	−	0.420471i	0.464895	−	0.885366i	$-0.346092\pi$
$181$	−5.65685	−	5.65685i	−0.420471	−	0.420471i	−0.885366	+	0.464895i	$0.846092\pi$
$182$	−5.65685	+	5.65685i	−0.419314	+	0.419314i
$183$	−8.00000			−0.591377
$184$	−8.48528	+	8.48528i	−0.625543	+	0.625543i
$185$		0			0
$186$	4.00000			0.293294
$187$		0			0
$188$	8.00000			0.583460
$189$			4.00000i			0.290957i
$190$		0			0
$191$	8.00000			0.578860			0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000			0.578860			0.289430	+	0.957199i	$0.406534\pi$
$192$	4.94975	−	4.94975i	0.357217	−	0.357217i
$193$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$193$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$194$	−11.3137	−	11.3137i	−0.812277	−	0.812277i
$195$		0			0
$196$			9.00000i			0.642857i
$197$	−11.3137	−	11.3137i	−0.806068	−	0.806068i	0.177968	−	0.984036i	$-0.443048\pi$
$197$	−11.3137	−	11.3137i	−0.806068	−	0.806068i	−0.984036	+	0.177968i	$0.943048\pi$
$198$	−2.82843	−	2.82843i	−0.201008	−	0.201008i
$199$	−14.1421	+	14.1421i	−1.00251	+	1.00251i	−0.00251257	+	0.999997i	$0.500800\pi$
$199$	−14.1421	+	14.1421i	−1.00251	+	1.00251i	−0.999997	+	0.00251257i	$0.999200\pi$
$200$	15.0000			1.06066
$201$	−8.48528	+	8.48528i	−0.598506	+	0.598506i
$202$			6.00000i			0.422159i
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−2.82843	+	2.82843i	−0.196589	+	0.196589i
$208$	2.00000			0.138675
$209$	−11.3137	+	11.3137i	−0.782586	+	0.782586i
$210$		0			0
$211$	−2.82843	−	2.82843i	−0.194717	−	0.194717i	0.603014	−	0.797731i	$-0.293966\pi$
$211$	−2.82843	−	2.82843i	−0.194717	−	0.194717i	−0.797731	+	0.603014i	$0.793966\pi$
$212$		−	6.00000i		−	0.412082i
$213$		−	12.0000i		−	0.822226i
$214$	8.48528	+	8.48528i	0.580042	+	0.580042i
$215$		0			0
$216$	2.12132	−	2.12132i	0.144338	−	0.144338i
$217$	16.0000			1.08615
$218$	−5.65685	+	5.65685i	−0.383131	+	0.383131i
$219$		0			0
$220$		0			0
$221$		0			0
$222$	−8.00000			−0.536925
$223$		−	16.0000i		−	1.07144i	−0.844396	−	0.535720i	$-0.820040\pi$
$223$		−	16.0000i		−	1.07144i	0.844396	−	0.535720i	$-0.179960\pi$
$224$	14.1421	−	14.1421i	0.944911	−	0.944911i
$225$	5.00000			0.333333
$226$	−5.65685	+	5.65685i	−0.376288	+	0.376288i
$227$	−8.48528	−	8.48528i	−0.563188	−	0.563188i	0.367024	−	0.930212i	$-0.380377\pi$
$227$	−8.48528	−	8.48528i	−0.563188	−	0.563188i	−0.930212	+	0.367024i	$0.880377\pi$
$228$	−2.82843	−	2.82843i	−0.187317	−	0.187317i
$229$			10.0000i			0.660819i	0.943838	+	0.330409i	$0.107187\pi$
$229$			10.0000i			0.660819i	−0.943838	+	0.330409i	$0.892813\pi$
$230$		0			0
$231$	−11.3137	−	11.3137i	−0.744387	−	0.744387i
$232$		0			0
$233$	5.65685	−	5.65685i	0.370593	−	0.370593i	−0.497100	−	0.867693i	$-0.665602\pi$
$233$	5.65685	−	5.65685i	0.370593	−	0.370593i	0.867693	+	0.497100i	$0.165602\pi$
$234$	2.00000			0.130744
$235$		0			0
$236$		−	12.0000i		−	0.781133i
$237$	4.00000			0.259828
$238$		0			0
$239$	−8.00000			−0.517477			−0.258738	−	0.965947i	$-0.583307\pi$
$239$	−8.00000			−0.517477			−0.258738	+	0.965947i	$0.583307\pi$
$240$		0			0
$241$	11.3137	−	11.3137i	0.728780	−	0.728780i	−0.241597	−	0.970377i	$-0.577671\pi$
$241$	11.3137	−	11.3137i	0.728780	−	0.728780i	0.970377	+	0.241597i	$0.0776711\pi$
$242$	5.00000			0.321412
$243$	0.707107	−	0.707107i	0.0453609	−	0.0453609i
$244$	5.65685	+	5.65685i	0.362143	+	0.362143i
$245$		0			0
$246$			8.00000i			0.510061i
$247$		−	8.00000i		−	0.509028i
$248$	−8.48528	−	8.48528i	−0.538816	−	0.538816i
$249$	8.48528	+	8.48528i	0.537733	+	0.537733i
$250$		0			0
$251$	−12.0000			−0.757433			−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000			−0.757433			−0.378717	+	0.925513i	$0.623635\pi$
$252$	2.82843	−	2.82843i	0.178174	−	0.178174i
$253$		−	16.0000i		−	1.00591i
$254$	8.00000			0.501965
$255$		0			0
$256$	−17.0000			−1.06250
$257$			2.00000i			0.124757i	0.998053	+	0.0623783i	$0.0198685\pi$
$257$			2.00000i			0.124757i	−0.998053	+	0.0623783i	$0.980131\pi$
$258$	2.82843	−	2.82843i	0.176090	−	0.176090i
$259$	−32.0000			−1.98838
$260$		0			0
$261$		0			0
$262$	−2.82843	−	2.82843i	−0.174741	−	0.174741i
$263$			24.0000i			1.47990i	0.672660	+	0.739952i	$0.265152\pi$
$263$			24.0000i			1.47990i	−0.672660	+	0.739952i	$0.734848\pi$
$264$			12.0000i			0.738549i
$265$		0			0
$266$	11.3137	+	11.3137i	0.693688	+	0.693688i
$267$	−7.07107	+	7.07107i	−0.432742	+	0.432742i
$268$	12.0000			0.733017
$269$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$269$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$270$		0			0
$271$		0			0			−	1.00000i	$-0.5\pi$
$271$		0			0				1.00000i	$0.5\pi$
$272$		0			0
$273$	8.00000			0.484182
$274$			10.0000i			0.604122i
$275$	−14.1421	+	14.1421i	−0.852803	+	0.852803i
$276$	4.00000			0.240772
$277$	−5.65685	+	5.65685i	−0.339887	+	0.339887i	−0.856325	−	0.516437i	$-0.827258\pi$
$277$	−5.65685	+	5.65685i	−0.339887	+	0.339887i	0.516437	+	0.856325i	$0.327258\pi$
$278$	2.82843	+	2.82843i	0.169638	+	0.169638i
$279$	−2.82843	−	2.82843i	−0.169334	−	0.169334i
$280$		0			0
$281$			10.0000i			0.596550i	0.954480	+	0.298275i	$0.0964112\pi$
$281$			10.0000i			0.596550i	−0.954480	+	0.298275i	$0.903589\pi$
$282$	5.65685	+	5.65685i	0.336861	+	0.336861i
$283$	2.82843	+	2.82843i	0.168133	+	0.168133i	0.786158	−	0.618026i	$-0.212067\pi$
$283$	2.82843	+	2.82843i	0.168133	+	0.168133i	−0.618026	+	0.786158i	$0.712067\pi$
$284$	−8.48528	+	8.48528i	−0.503509	+	0.503509i
$285$		0			0
$286$	−5.65685	+	5.65685i	−0.334497	+	0.334497i
$287$			32.0000i			1.88890i
$288$	−5.00000			−0.294628
$289$		0			0
$290$		0			0
$291$			16.0000i			0.937937i
$292$		0			0
$293$	10.0000			0.584206			0.292103	−	0.956387i	$-0.405645\pi$
$293$	10.0000			0.584206			0.292103	+	0.956387i	$0.405645\pi$
$294$	−6.36396	+	6.36396i	−0.371154	+	0.371154i
$295$		0			0
$296$	16.9706	+	16.9706i	0.986394	+	0.986394i
$297$			4.00000i			0.232104i
$298$			6.00000i			0.347571i
$299$	5.65685	+	5.65685i	0.327144	+	0.327144i
$300$	−3.53553	−	3.53553i	−0.204124	−	0.204124i
$301$	11.3137	−	11.3137i	0.652111	−	0.652111i
$302$		0			0
$303$	4.24264	−	4.24264i	0.243733	−	0.243733i
$304$		−	4.00000i		−	0.229416i
$305$		0			0
$306$		0			0
$307$	−12.0000			−0.684876			−0.342438	−	0.939540i	$-0.611253\pi$
$307$	−12.0000			−0.684876			−0.342438	+	0.939540i	$0.611253\pi$
$308$			16.0000i			0.911685i
$309$		0			0
$310$		0			0
$311$	8.48528	−	8.48528i	0.481156	−	0.481156i	−0.424345	−	0.905501i	$-0.639495\pi$
$311$	8.48528	−	8.48528i	0.481156	−	0.481156i	0.905501	+	0.424345i	$0.139495\pi$
$312$	−4.24264	−	4.24264i	−0.240192	−	0.240192i
$313$	11.3137	+	11.3137i	0.639489	+	0.639489i	0.950429	−	0.310941i	$-0.100644\pi$
$313$	11.3137	+	11.3137i	0.639489	+	0.639489i	−0.310941	+	0.950429i	$0.600644\pi$
$314$		−	2.00000i		−	0.112867i
$315$		0			0
$316$	−2.82843	−	2.82843i	−0.159111	−	0.159111i
$317$	−22.6274	−	22.6274i	−1.27088	−	1.27088i	−0.945626	−	0.325257i	$-0.894549\pi$
$317$	−22.6274	−	22.6274i	−1.27088	−	1.27088i	−0.325257	−	0.945626i	$-0.605451\pi$
$318$	4.24264	−	4.24264i	0.237915	−	0.237915i
$319$		0			0
$320$		0			0
$321$		−	12.0000i		−	0.669775i
$322$	−16.0000			−0.891645
$323$		0			0
$324$	−1.00000			−0.0555556
$325$		−	10.0000i		−	0.554700i
$326$	−14.1421	+	14.1421i	−0.783260	+	0.783260i
$327$	8.00000			0.442401
$328$	16.9706	−	16.9706i	0.937043	−	0.937043i
$329$	22.6274	+	22.6274i	1.24749	+	1.24749i
$330$		0			0
$331$		−	20.0000i		−	1.09930i	−0.835395	−	0.549650i	$-0.814761\pi$
$331$		−	20.0000i		−	1.09930i	0.835395	−	0.549650i	$-0.185239\pi$
$332$		−	12.0000i		−	0.658586i
$333$	5.65685	+	5.65685i	0.309994	+	0.309994i
$334$	8.48528	+	8.48528i	0.464294	+	0.464294i
$335$		0			0
$336$	4.00000			0.218218
$337$	11.3137	−	11.3137i	0.616297	−	0.616297i	−0.328283	−	0.944580i	$-0.606470\pi$
$337$	11.3137	−	11.3137i	0.616297	−	0.616297i	0.944580	+	0.328283i	$0.106470\pi$
$338$			9.00000i			0.489535i
$339$	8.00000			0.434500
$340$		0			0
$341$	16.0000			0.866449
$342$		−	4.00000i		−	0.216295i
$343$	−5.65685	+	5.65685i	−0.305441	+	0.305441i
$344$	−12.0000			−0.646997
$345$		0			0
$346$	−11.3137	−	11.3137i	−0.608229	−	0.608229i
$347$	−2.82843	−	2.82843i	−0.151838	−	0.151838i	0.627100	−	0.778938i	$-0.284242\pi$
$347$	−2.82843	−	2.82843i	−0.151838	−	0.151838i	−0.778938	+	0.627100i	$0.784242\pi$
$348$		0			0
$349$		−	2.00000i		−	0.107058i	−0.998566	−	0.0535288i	$-0.982953\pi$
$349$		−	2.00000i		−	0.107058i	0.998566	−	0.0535288i	$-0.0170469\pi$
$350$	14.1421	+	14.1421i	0.755929	+	0.755929i
$351$	−1.41421	−	1.41421i	−0.0754851	−	0.0754851i
$352$	14.1421	−	14.1421i	0.753778	−	0.753778i
$353$	−18.0000			−0.958043			−0.479022	−	0.877803i	$-0.659008\pi$
$353$	−18.0000			−0.958043			−0.479022	+	0.877803i	$0.659008\pi$
$354$	8.48528	−	8.48528i	0.450988	−	0.450988i
$355$		0			0
$356$	10.0000			0.529999
$357$		0			0
$358$	4.00000			0.211407
$359$		−	32.0000i		−	1.68890i	−0.535638	−	0.844448i	$-0.679929\pi$
$359$		−	32.0000i		−	1.68890i	0.535638	−	0.844448i	$-0.320071\pi$
$360$		0			0
$361$	3.00000			0.157895
$362$	−5.65685	+	5.65685i	−0.297318	+	0.297318i
$363$	−3.53553	−	3.53553i	−0.185567	−	0.185567i
$364$	−5.65685	−	5.65685i	−0.296500	−	0.296500i
$365$		0			0
$366$			8.00000i			0.418167i
$367$	−8.48528	−	8.48528i	−0.442928	−	0.442928i	0.450067	−	0.892995i	$-0.351400\pi$
$367$	−8.48528	−	8.48528i	−0.442928	−	0.442928i	−0.892995	+	0.450067i	$0.851400\pi$
$368$	2.82843	+	2.82843i	0.147442	+	0.147442i
$369$	5.65685	−	5.65685i	0.294484	−	0.294484i
$370$		0			0
$371$	16.9706	−	16.9706i	0.881068	−	0.881068i
$372$			4.00000i			0.207390i
$373$	26.0000			1.34623			0.673114	−	0.739538i	$-0.264956\pi$
$373$	26.0000			1.34623			0.673114	+	0.739538i	$0.264956\pi$
$374$		0			0
$375$		0			0
$376$		−	24.0000i		−	1.23771i
$377$		0			0
$378$	4.00000			0.205738
$379$	14.1421	−	14.1421i	0.726433	−	0.726433i	−0.243475	−	0.969907i	$-0.578287\pi$
$379$	14.1421	−	14.1421i	0.726433	−	0.726433i	0.969907	+	0.243475i	$0.0782872\pi$
$380$		0			0
$381$	−5.65685	−	5.65685i	−0.289809	−	0.289809i
$382$		−	8.00000i		−	0.409316i
$383$		−	16.0000i		−	0.817562i	−0.912633	−	0.408781i	$-0.865954\pi$
$383$		−	16.0000i		−	0.817562i	0.912633	−	0.408781i	$-0.134046\pi$
$384$	2.12132	+	2.12132i	0.108253	+	0.108253i
$385$		0			0
$386$		0			0
$387$	−4.00000			−0.203331
$388$	11.3137	−	11.3137i	0.574367	−	0.574367i
$389$			6.00000i			0.304212i	0.988364	+	0.152106i	$0.0486055\pi$
$389$			6.00000i			0.304212i	−0.988364	+	0.152106i	$0.951394\pi$
$390$		0			0
$391$		0			0
$392$	27.0000			1.36371
$393$			4.00000i			0.201773i
$394$	−11.3137	+	11.3137i	−0.569976	+	0.569976i
$395$		0			0
$396$	2.82843	−	2.82843i	0.142134	−	0.142134i
$397$	−5.65685	−	5.65685i	−0.283909	−	0.283909i	0.550757	−	0.834666i	$-0.314339\pi$
$397$	−5.65685	−	5.65685i	−0.283909	−	0.283909i	−0.834666	+	0.550757i	$0.814339\pi$
$398$	14.1421	+	14.1421i	0.708881	+	0.708881i
$399$		−	16.0000i		−	0.801002i
$400$		−	5.00000i		−	0.250000i
$401$	16.9706	+	16.9706i	0.847469	+	0.847469i	0.989817	−	0.142347i	$-0.0454650\pi$
$401$	16.9706	+	16.9706i	0.847469	+	0.847469i	−0.142347	+	0.989817i	$0.545465\pi$
$402$	8.48528	+	8.48528i	0.423207	+	0.423207i
$403$	−5.65685	+	5.65685i	−0.281788	+	0.281788i
$404$	−6.00000			−0.298511
$405$		0			0
$406$		0			0
$407$	−32.0000			−1.58618
$408$		0			0
$409$	10.0000			0.494468			0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000			0.494468			0.247234	+	0.968956i	$0.420478\pi$
$410$		0			0
$411$	7.07107	−	7.07107i	0.348790	−	0.348790i
$412$		0			0
$413$	33.9411	−	33.9411i	1.67013	−	1.67013i
$414$	2.82843	+	2.82843i	0.139010	+	0.139010i
$415$		0			0
$416$			10.0000i			0.490290i
$417$		−	4.00000i		−	0.195881i
$418$	11.3137	+	11.3137i	0.553372	+	0.553372i
$419$	2.82843	+	2.82843i	0.138178	+	0.138178i	0.772812	−	0.634635i	$-0.218849\pi$
$419$	2.82843	+	2.82843i	0.138178	+	0.138178i	−0.634635	+	0.772812i	$0.718849\pi$
$420$		0			0
$421$	22.0000			1.07221			0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000			1.07221			0.536107	+	0.844150i	$0.319894\pi$
$422$	−2.82843	+	2.82843i	−0.137686	+	0.137686i
$423$		−	8.00000i		−	0.388973i
$424$	−18.0000			−0.874157
$425$		0			0
$426$	−12.0000			−0.581402
$427$			32.0000i			1.54859i
$428$	−8.48528	+	8.48528i	−0.410152	+	0.410152i
$429$	8.00000			0.386244
$430$		0			0
$431$	−14.1421	−	14.1421i	−0.681203	−	0.681203i	0.279068	−	0.960271i	$-0.409974\pi$
$431$	−14.1421	−	14.1421i	−0.681203	−	0.681203i	−0.960271	+	0.279068i	$0.909974\pi$
$432$	−0.707107	−	0.707107i	−0.0340207	−	0.0340207i
$433$		−	2.00000i		−	0.0961139i	−0.998845	−	0.0480569i	$-0.984697\pi$
$433$		−	2.00000i		−	0.0961139i	0.998845	−	0.0480569i	$-0.0153029\pi$
$434$		−	16.0000i		−	0.768025i
$435$		0			0
$436$	−5.65685	−	5.65685i	−0.270914	−	0.270914i
$437$	11.3137	−	11.3137i	0.541208	−	0.541208i
$438$		0			0
$439$	25.4558	−	25.4558i	1.21494	−	1.21494i	0.245560	−	0.969381i	$-0.421028\pi$
$439$	25.4558	−	25.4558i	1.21494	−	1.21494i	0.969381	−	0.245560i	$-0.0789717\pi$
$440$		0			0
$441$	9.00000			0.428571
$442$		0			0
$443$	−20.0000			−0.950229			−0.475114	−	0.879924i	$-0.657593\pi$
$443$	−20.0000			−0.950229			−0.475114	+	0.879924i	$0.657593\pi$
$444$		−	8.00000i		−	0.379663i
$445$		0			0
$446$	−16.0000			−0.757622
$447$	4.24264	−	4.24264i	0.200670	−	0.200670i
$448$	−19.7990	−	19.7990i	−0.935414	−	0.935414i
$449$	−5.65685	−	5.65685i	−0.266963	−	0.266963i	0.560912	−	0.827875i	$-0.310451\pi$
$449$	−5.65685	−	5.65685i	−0.266963	−	0.266963i	−0.827875	+	0.560912i	$0.810451\pi$
$450$		−	5.00000i		−	0.235702i
$451$			32.0000i			1.50682i
$452$	−5.65685	−	5.65685i	−0.266076	−	0.266076i
$453$		0			0
$454$	−8.48528	+	8.48528i	−0.398234	+	0.398234i
$455$		0			0
$456$	−8.48528	+	8.48528i	−0.397360	+	0.397360i
$457$		−	38.0000i		−	1.77757i	−0.458329	−	0.888783i	$-0.651552\pi$
$457$		−	38.0000i		−	1.77757i	0.458329	−	0.888783i	$-0.348448\pi$
$458$	10.0000			0.467269
$459$		0			0
$460$		0			0
$461$			34.0000i			1.58354i	0.610821	+	0.791769i	$0.290840\pi$
$461$			34.0000i			1.58354i	−0.610821	+	0.791769i	$0.709160\pi$
$462$	−11.3137	+	11.3137i	−0.526361	+	0.526361i
$463$	40.0000			1.85896			0.929479	−	0.368875i	$-0.120257\pi$
$463$	40.0000			1.85896			0.929479	+	0.368875i	$0.120257\pi$
$464$		0			0
$465$		0			0
$466$	−5.65685	−	5.65685i	−0.262049	−	0.262049i
$467$			4.00000i			0.185098i	0.995708	+	0.0925490i	$0.0295015\pi$
$467$			4.00000i			0.185098i	−0.995708	+	0.0925490i	$0.970499\pi$
$468$			2.00000i			0.0924500i
$469$	33.9411	+	33.9411i	1.56726	+	1.56726i
$470$		0			0
$471$	−1.41421	+	1.41421i	−0.0651635	+	0.0651635i
$472$	−36.0000			−1.65703
$473$	11.3137	−	11.3137i	0.520205	−	0.520205i
$474$		−	4.00000i		−	0.183726i
$475$	−20.0000			−0.917663
$476$		0			0
$477$	−6.00000			−0.274721
$478$			8.00000i			0.365911i
$479$	−8.48528	+	8.48528i	−0.387702	+	0.387702i	−0.873867	−	0.486165i	$-0.838395\pi$
$479$	−8.48528	+	8.48528i	−0.387702	+	0.387702i	0.486165	+	0.873867i	$0.338395\pi$
$480$		0			0
$481$	11.3137	−	11.3137i	0.515861	−	0.515861i
$482$	−11.3137	−	11.3137i	−0.515325	−	0.515325i
$483$	11.3137	+	11.3137i	0.514792	+	0.514792i
$484$			5.00000i			0.227273i
$485$		0			0
$486$	−0.707107	−	0.707107i	−0.0320750	−	0.0320750i
$487$	−2.82843	−	2.82843i	−0.128168	−	0.128168i	0.640113	−	0.768281i	$-0.278888\pi$
$487$	−2.82843	−	2.82843i	−0.128168	−	0.128168i	−0.768281	+	0.640113i	$0.778888\pi$
$488$	16.9706	−	16.9706i	0.768221	−	0.768221i
$489$	20.0000			0.904431
$490$		0			0
$491$			20.0000i			0.902587i	0.892375	+	0.451294i	$0.149037\pi$
$491$			20.0000i			0.902587i	−0.892375	+	0.451294i	$0.850963\pi$
$492$	−8.00000			−0.360668
$493$		0			0
$494$	−8.00000			−0.359937
$495$		0			0
$496$	−2.82843	+	2.82843i	−0.127000	+	0.127000i
$497$	−48.0000			−2.15309
$498$	8.48528	−	8.48528i	0.380235	−	0.380235i
$499$	25.4558	+	25.4558i	1.13956	+	1.13956i	0.988529	+	0.151031i	$0.0482594\pi$
$499$	25.4558	+	25.4558i	1.13956	+	1.13956i	0.151031	+	0.988529i	$0.451741\pi$
$500$		0			0
$501$		−	12.0000i		−	0.536120i
$502$			12.0000i			0.535586i
$503$	−8.48528	−	8.48528i	−0.378340	−	0.378340i	0.492163	−	0.870503i	$-0.336206\pi$
$503$	−8.48528	−	8.48528i	−0.378340	−	0.378340i	−0.870503	+	0.492163i	$0.836206\pi$
$504$	−8.48528	−	8.48528i	−0.377964	−	0.377964i
$505$		0			0
$506$	−16.0000			−0.711287
$507$	6.36396	−	6.36396i	0.282633	−	0.282633i
$508$			8.00000i			0.354943i
$509$	−18.0000			−0.797836			−0.398918	−	0.916987i	$-0.630614\pi$
$509$	−18.0000			−0.797836			−0.398918	+	0.916987i	$0.630614\pi$
$510$		0			0
$511$		0			0
$512$			11.0000i			0.486136i
$513$	−2.82843	+	2.82843i	−0.124878	+	0.124878i
$514$	2.00000			0.0882162
$515$		0			0
$516$	2.82843	+	2.82843i	0.124515	+	0.124515i
$517$	22.6274	+	22.6274i	0.995153	+	0.995153i
$518$			32.0000i			1.40600i
$519$			16.0000i			0.702322i
$520$		0			0
$521$	16.9706	+	16.9706i	0.743494	+	0.743494i	0.973249	−	0.229755i	$-0.0737924\pi$
$521$	16.9706	+	16.9706i	0.743494	+	0.743494i	−0.229755	+	0.973249i	$0.573792\pi$
$522$		0			0
$523$	12.0000			0.524723			0.262362	−	0.964970i	$-0.415499\pi$
$523$	12.0000			0.524723			0.262362	+	0.964970i	$0.415499\pi$
$524$	2.82843	−	2.82843i	0.123560	−	0.123560i
$525$		−	20.0000i		−	0.872872i
$526$	24.0000			1.04645
$527$		0			0
$528$	4.00000			0.174078
$529$		−	7.00000i		−	0.304348i
$530$		0			0
$531$	−12.0000			−0.520756
$532$	−11.3137	+	11.3137i	−0.490511	+	0.490511i
$533$	−11.3137	−	11.3137i	−0.490051	−	0.490051i
$534$	7.07107	+	7.07107i	0.305995	+	0.305995i
$535$		0			0
$536$		−	36.0000i		−	1.55496i
$537$	−2.82843	−	2.82843i	−0.122056	−	0.122056i
$538$		0			0
$539$	−25.4558	+	25.4558i	−1.09646	+	1.09646i
$540$		0			0
$541$	−28.2843	+	28.2843i	−1.21604	+	1.21604i	−0.247027	+	0.969009i	$0.579454\pi$
$541$	−28.2843	+	28.2843i	−1.21604	+	1.21604i	−0.969009	+	0.247027i	$0.920546\pi$
$542$		0			0
$543$	8.00000			0.343313
$544$		0			0
$545$		0			0
$546$		−	8.00000i		−	0.342368i
$547$	−25.4558	+	25.4558i	−1.08841	+	1.08841i	−0.0927212	+	0.995692i	$0.529557\pi$
$547$	−25.4558	+	25.4558i	−1.08841	+	1.08841i	−0.995692	+	0.0927212i	$0.970443\pi$
$548$	−10.0000			−0.427179
$549$	5.65685	−	5.65685i	0.241429	−	0.241429i
$550$	14.1421	+	14.1421i	0.603023	+	0.603023i
$551$		0			0
$552$		−	12.0000i		−	0.510754i
$553$		−	16.0000i		−	0.680389i
$554$	5.65685	+	5.65685i	0.240337	+	0.240337i
$555$		0			0
$556$	−2.82843	+	2.82843i	−0.119952	+	0.119952i
$557$	−18.0000			−0.762684			−0.381342	−	0.924434i	$-0.624538\pi$
$557$	−18.0000			−0.762684			−0.381342	+	0.924434i	$0.624538\pi$
$558$	−2.82843	+	2.82843i	−0.119737	+	0.119737i
$559$			8.00000i			0.338364i
$560$		0			0
$561$		0			0
$562$	10.0000			0.421825
$563$		−	44.0000i		−	1.85438i	−0.374593	−	0.927189i	$-0.622217\pi$
$563$		−	44.0000i		−	1.85438i	0.374593	−	0.927189i	$-0.377783\pi$
$564$	−5.65685	+	5.65685i	−0.238197	+	0.238197i
$565$		0			0
$566$	2.82843	−	2.82843i	0.118888	−	0.118888i
$567$	−2.82843	−	2.82843i	−0.118783	−	0.118783i
$568$	25.4558	+	25.4558i	1.06810	+	1.06810i
$569$			26.0000i			1.08998i	0.838444	+	0.544988i	$0.183466\pi$
$569$			26.0000i			1.08998i	−0.838444	+	0.544988i	$0.816534\pi$
$570$		0			0
$571$	−14.1421	−	14.1421i	−0.591830	−	0.591830i	0.346296	−	0.938125i	$-0.387439\pi$
$571$	−14.1421	−	14.1421i	−0.591830	−	0.591830i	−0.938125	+	0.346296i	$0.887439\pi$
$572$	−5.65685	−	5.65685i	−0.236525	−	0.236525i
$573$	−5.65685	+	5.65685i	−0.236318	+	0.236318i
$574$	32.0000			1.33565
$575$	14.1421	−	14.1421i	0.589768	−	0.589768i
$576$			7.00000i			0.291667i
$577$	30.0000			1.24892			0.624458	−	0.781058i	$-0.285320\pi$
$577$	30.0000			1.24892			0.624458	+	0.781058i	$0.285320\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	33.9411	−	33.9411i	1.40812	−	1.40812i
$582$	16.0000			0.663221
$583$	16.9706	−	16.9706i	0.702849	−	0.702849i
$584$		0			0
$585$		0			0
$586$		−	10.0000i		−	0.413096i
$587$			12.0000i			0.495293i	0.968850	+	0.247647i	$0.0796572\pi$
$587$			12.0000i			0.495293i	−0.968850	+	0.247647i	$0.920343\pi$
$588$	−6.36396	−	6.36396i	−0.262445	−	0.262445i
$589$	11.3137	+	11.3137i	0.466173	+	0.466173i
$590$		0			0
$591$	16.0000			0.658152
$592$	5.65685	−	5.65685i	0.232495	−	0.232495i
$593$			46.0000i			1.88899i	0.328521	+	0.944497i	$0.393450\pi$
$593$			46.0000i			1.88899i	−0.328521	+	0.944497i	$0.606550\pi$
$594$	4.00000			0.164122
$595$		0			0
$596$	−6.00000			−0.245770
$597$		−	20.0000i		−	0.818546i
$598$	5.65685	−	5.65685i	0.231326	−	0.231326i
$599$	8.00000			0.326871			0.163436	−	0.986554i	$-0.447742\pi$
$599$	8.00000			0.326871			0.163436	+	0.986554i	$0.447742\pi$
$600$	−10.6066	+	10.6066i	−0.433013	+	0.433013i
$601$	11.3137	+	11.3137i	0.461496	+	0.461496i	0.899146	−	0.437650i	$-0.144189\pi$
$601$	11.3137	+	11.3137i	0.461496	+	0.461496i	−0.437650	+	0.899146i	$0.644189\pi$
$602$	−11.3137	−	11.3137i	−0.461112	−	0.461112i
$603$		−	12.0000i		−	0.488678i
$604$		0			0
$605$		0			0
$606$	−4.24264	−	4.24264i	−0.172345	−	0.172345i
$607$	−2.82843	+	2.82843i	−0.114802	+	0.114802i	−0.762174	−	0.647372i	$-0.775868\pi$
$607$	−2.82843	+	2.82843i	−0.114802	+	0.114802i	0.647372	+	0.762174i	$0.275868\pi$
$608$	20.0000			0.811107
$609$		0			0
$610$		0			0
$611$	−16.0000			−0.647291
$612$		0			0
$613$	6.00000			0.242338			0.121169	−	0.992632i	$-0.461336\pi$
$613$	6.00000			0.242338			0.121169	+	0.992632i	$0.461336\pi$
$614$			12.0000i			0.484281i
$615$		0			0
$616$	48.0000			1.93398
$617$	−16.9706	+	16.9706i	−0.683209	+	0.683209i	−0.960722	−	0.277513i	$-0.910490\pi$
$617$	−16.9706	+	16.9706i	−0.683209	+	0.683209i	0.277513	+	0.960722i	$0.410490\pi$
$618$		0			0
$619$	−25.4558	−	25.4558i	−1.02316	−	1.02316i	−0.999725	−	0.0234313i	$-0.992541\pi$
$619$	−25.4558	−	25.4558i	−1.02316	−	1.02316i	−0.0234313	−	0.999725i	$-0.507459\pi$
$620$		0			0
$621$		−	4.00000i		−	0.160514i
$622$	−8.48528	−	8.48528i	−0.340229	−	0.340229i
$623$	28.2843	+	28.2843i	1.13319	+	1.13319i
$624$	−1.41421	+	1.41421i	−0.0566139	+	0.0566139i
$625$	−25.0000			−1.00000
$626$	11.3137	−	11.3137i	0.452187	−	0.452187i
$627$		−	16.0000i		−	0.638978i
$628$	2.00000			0.0798087
$629$		0			0
$630$		0			0
$631$			8.00000i			0.318475i	0.987240	+	0.159237i	$0.0509036\pi$
$631$			8.00000i			0.318475i	−0.987240	+	0.159237i	$0.949096\pi$
$632$	−8.48528	+	8.48528i	−0.337526	+	0.337526i
$633$	4.00000			0.158986
$634$	−22.6274	+	22.6274i	−0.898650	+	0.898650i
$635$		0			0
$636$	4.24264	+	4.24264i	0.168232	+	0.168232i
$637$		−	18.0000i		−	0.713186i
$638$		0			0
$639$	8.48528	+	8.48528i	0.335673	+	0.335673i
$640$		0			0
$641$	5.65685	−	5.65685i	0.223432	−	0.223432i	−0.586510	−	0.809942i	$-0.699498\pi$
$641$	5.65685	−	5.65685i	0.223432	−	0.223432i	0.809942	+	0.586510i	$0.199498\pi$
$642$	−12.0000			−0.473602
$643$	−2.82843	+	2.82843i	−0.111542	+	0.111542i	−0.760675	−	0.649133i	$-0.775132\pi$
$643$	−2.82843	+	2.82843i	−0.111542	+	0.111542i	0.649133	+	0.760675i	$0.275132\pi$
$644$		−	16.0000i		−	0.630488i
$645$		0			0
$646$		0			0
$647$	32.0000			1.25805			0.629025	−	0.777385i	$-0.283454\pi$
$647$	32.0000			1.25805			0.629025	+	0.777385i	$0.283454\pi$
$648$			3.00000i			0.117851i
$649$	33.9411	−	33.9411i	1.33231	−	1.33231i
$650$	−10.0000			−0.392232
$651$	−11.3137	+	11.3137i	−0.443419	+	0.443419i
$652$	−14.1421	−	14.1421i	−0.553849	−	0.553849i
$653$	22.6274	+	22.6274i	0.885479	+	0.885479i	0.994085	−	0.108606i	$-0.0346386\pi$
$653$	22.6274	+	22.6274i	0.885479	+	0.885479i	−0.108606	+	0.994085i	$0.534639\pi$
$654$		−	8.00000i		−	0.312825i
$655$		0			0
$656$	−5.65685	−	5.65685i	−0.220863	−	0.220863i
$657$		0			0
$658$	22.6274	−	22.6274i	0.882109	−	0.882109i
$659$	4.00000			0.155818			0.0779089	−	0.996960i	$-0.475176\pi$
$659$	4.00000			0.155818			0.0779089	+	0.996960i	$0.475176\pi$
$660$		0			0
$661$			42.0000i			1.63361i	0.576913	+	0.816805i	$0.304257\pi$
$661$			42.0000i			1.63361i	−0.576913	+	0.816805i	$0.695743\pi$
$662$	−20.0000			−0.777322
$663$		0			0
$664$	−36.0000			−1.39707
$665$		0			0
$666$	5.65685	−	5.65685i	0.219199	−	0.219199i
$667$		0			0
$668$	−8.48528	+	8.48528i	−0.328305	+	0.328305i
$669$	11.3137	+	11.3137i	0.437413	+	0.437413i
$670$		0			0
$671$			32.0000i			1.23535i
$672$			20.0000i			0.771517i
$673$	−22.6274	−	22.6274i	−0.872223	−	0.872223i	0.120492	−	0.992714i	$-0.461553\pi$
$673$	−22.6274	−	22.6274i	−0.872223	−	0.872223i	−0.992714	+	0.120492i	$0.961553\pi$
$674$	−11.3137	−	11.3137i	−0.435788	−	0.435788i
$675$	−3.53553	+	3.53553i	−0.136083	+	0.136083i
$676$	−9.00000			−0.346154
$677$	−33.9411	+	33.9411i	−1.30446	+	1.30446i	−0.379113	+	0.925350i	$0.623771\pi$
$677$	−33.9411	+	33.9411i	−1.30446	+	1.30446i	−0.925350	+	0.379113i	$0.876229\pi$
$678$		−	8.00000i		−	0.307238i
$679$	64.0000			2.45609
$680$		0			0
$681$	12.0000			0.459841
$682$		−	16.0000i		−	0.612672i
$683$	8.48528	−	8.48528i	0.324680	−	0.324680i	−0.525879	−	0.850559i	$-0.676264\pi$
$683$	8.48528	−	8.48528i	0.324680	−	0.324680i	0.850559	+	0.525879i	$0.176264\pi$
$684$	4.00000			0.152944
$685$		0			0
$686$	5.65685	+	5.65685i	0.215980	+	0.215980i
$687$	−7.07107	−	7.07107i	−0.269778	−	0.269778i
$688$			4.00000i			0.152499i
$689$			12.0000i			0.457164i
$690$		0			0
$691$	14.1421	+	14.1421i	0.537992	+	0.537992i	0.922939	−	0.384947i	$-0.125780\pi$
$691$	14.1421	+	14.1421i	0.537992	+	0.537992i	−0.384947	+	0.922939i	$0.625780\pi$
$692$	11.3137	−	11.3137i	0.430083	−	0.430083i
$693$	16.0000			0.607790
$694$	−2.82843	+	2.82843i	−0.107366	+	0.107366i
$695$		0			0
$696$		0			0
$697$		0			0
$698$	−2.00000			−0.0757011
$699$			8.00000i			0.302588i
$700$	−14.1421	+	14.1421i	−0.534522	+	0.534522i
$701$	−34.0000			−1.28416			−0.642081	−	0.766637i	$-0.721929\pi$
$701$	−34.0000			−1.28416			−0.642081	+	0.766637i	$0.721929\pi$
$702$	−1.41421	+	1.41421i	−0.0533761	+	0.0533761i
$703$	−22.6274	−	22.6274i	−0.853409	−	0.853409i
$704$	−19.7990	−	19.7990i	−0.746203	−	0.746203i
$705$		0			0
$706$			18.0000i			0.677439i
$707$	−16.9706	−	16.9706i	−0.638244	−	0.638244i
$708$	8.48528	+	8.48528i	0.318896	+	0.318896i
$709$	−28.2843	+	28.2843i	−1.06224	+	1.06224i	−0.0643080	+	0.997930i	$0.520484\pi$
$709$	−28.2843	+	28.2843i	−1.06224	+	1.06224i	−0.997930	+	0.0643080i	$0.979516\pi$
$710$		0			0
$711$	−2.82843	+	2.82843i	−0.106074	+	0.106074i
$712$		−	30.0000i		−	1.12430i
$713$	−16.0000			−0.599205
$714$		0			0
$715$		0			0
$716$			4.00000i			0.149487i
$717$	5.65685	−	5.65685i	0.211259	−	0.211259i
$718$	−32.0000			−1.19423
$719$	−25.4558	+	25.4558i	−0.949343	+	0.949343i	−0.998777	−	0.0494346i	$-0.984258\pi$
$719$	−25.4558	+	25.4558i	−0.949343	+	0.949343i	0.0494346	+	0.998777i	$0.484258\pi$
$720$		0			0
$721$		0			0
$722$		−	3.00000i		−	0.111648i
$723$			16.0000i			0.595046i
$724$	−5.65685	−	5.65685i	−0.210235	−	0.210235i
$725$		0			0
$726$	−3.53553	+	3.53553i	−0.131216	+	0.131216i
$727$	−8.00000			−0.296704			−0.148352	−	0.988935i	$-0.547397\pi$
$727$	−8.00000			−0.296704			−0.148352	+	0.988935i	$0.547397\pi$
$728$	−16.9706	+	16.9706i	−0.628971	+	0.628971i
$729$			1.00000i			0.0370370i
$730$		0			0
$731$		0			0
$732$	−8.00000			−0.295689
$733$			34.0000i			1.25582i	0.778287	+	0.627909i	$0.216089\pi$
$733$			34.0000i			1.25582i	−0.778287	+	0.627909i	$0.783911\pi$
$734$	−8.48528	+	8.48528i	−0.313197	+	0.313197i
$735$		0			0
$736$	−14.1421	+	14.1421i	−0.521286	+	0.521286i
$737$	33.9411	+	33.9411i	1.25024	+	1.25024i
$738$	−5.65685	−	5.65685i	−0.208232	−	0.208232i
$739$		−	4.00000i		−	0.147142i	−0.997290	−	0.0735712i	$-0.976560\pi$
$739$		−	4.00000i		−	0.147142i	0.997290	−	0.0735712i	$-0.0234396\pi$
$740$		0			0
$741$	5.65685	+	5.65685i	0.207810	+	0.207810i
$742$	−16.9706	−	16.9706i	−0.623009	−	0.623009i
$743$	−25.4558	+	25.4558i	−0.933884	+	0.933884i	−0.997946	−	0.0640616i	$-0.979595\pi$
$743$	−25.4558	+	25.4558i	−0.933884	+	0.933884i	0.0640616	+	0.997946i	$0.479595\pi$
$744$	12.0000			0.439941
$745$		0			0
$746$		−	26.0000i		−	0.951928i
$747$	−12.0000			−0.439057
$748$		0			0
$749$	−48.0000			−1.75388
$750$		0			0
$751$	14.1421	−	14.1421i	0.516054	−	0.516054i	−0.400321	−	0.916375i	$-0.631101\pi$
$751$	14.1421	−	14.1421i	0.516054	−	0.516054i	0.916375	+	0.400321i	$0.131101\pi$
$752$	−8.00000			−0.291730
$753$	8.48528	−	8.48528i	0.309221	−	0.309221i
$754$		0			0
$755$		0			0
$756$			4.00000i			0.145479i
$757$		−	38.0000i		−	1.38113i	−0.723269	−	0.690567i	$-0.757361\pi$
$757$		−	38.0000i		−	1.38113i	0.723269	−	0.690567i	$-0.242639\pi$
$758$	−14.1421	−	14.1421i	−0.513665	−	0.513665i
$759$	11.3137	+	11.3137i	0.410662	+	0.410662i
$760$		0			0
$761$	−6.00000			−0.217500			−0.108750	−	0.994069i	$-0.534685\pi$
$761$	−6.00000			−0.217500			−0.108750	+	0.994069i	$0.534685\pi$
$762$	−5.65685	+	5.65685i	−0.204926	+	0.204926i
$763$		−	32.0000i		−	1.15848i
$764$	8.00000			0.289430
$765$		0			0
$766$	−16.0000			−0.578103
$767$			24.0000i			0.866590i
$768$	12.0208	−	12.0208i	0.433764	−	0.433764i
$769$	34.0000			1.22607			0.613036	−	0.790055i	$-0.289948\pi$
$769$	34.0000			1.22607			0.613036	+	0.790055i	$0.289948\pi$
$770$		0			0
$771$	−1.41421	−	1.41421i	−0.0509317	−	0.0509317i
$772$		0			0
$773$		−	6.00000i		−	0.215805i	−0.994161	−	0.107903i	$-0.965587\pi$
$773$		−	6.00000i		−	0.215805i	0.994161	−	0.107903i	$-0.0344134\pi$
$774$			4.00000i			0.143777i
$775$	14.1421	+	14.1421i	0.508001	+	0.508001i
$776$	−33.9411	−	33.9411i	−1.21842	−	1.21842i
$777$	22.6274	−	22.6274i	0.811754	−	0.811754i
$778$	6.00000			0.215110
$779$	−22.6274	+	22.6274i	−0.810711	+	0.810711i
$780$		0			0
$781$	−48.0000			−1.71758
$782$		0			0
$783$		0			0
$784$		−	9.00000i		−	0.321429i
$785$		0			0
$786$	4.00000			0.142675
$787$	−8.48528	+	8.48528i	−0.302468	+	0.302468i	−0.841979	−	0.539511i	$-0.818609\pi$
$787$	−8.48528	+	8.48528i	−0.302468	+	0.302468i	0.539511	+	0.841979i	$0.318609\pi$
$788$	−11.3137	−	11.3137i	−0.403034	−	0.403034i
$789$	−16.9706	−	16.9706i	−0.604168	−	0.604168i
$790$		0			0
$791$		−	32.0000i		−	1.13779i
$792$	−8.48528	−	8.48528i	−0.301511	−	0.301511i
$793$	−11.3137	−	11.3137i	−0.401762	−	0.401762i
$794$	−5.65685	+	5.65685i	−0.200754	+	0.200754i
$795$		0			0
$796$	−14.1421	+	14.1421i	−0.501255	+	0.501255i
$797$			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			−0.566394
$799$		0			0
$800$	25.0000			0.883883
$801$		−	10.0000i		−	0.353333i
$802$	16.9706	−	16.9706i	0.599251	−	0.599251i
$803$		0			0
$804$	−8.48528	+	8.48528i	−0.299253	+	0.299253i
$805$		0			0
$806$	5.65685	+	5.65685i	0.199254	+	0.199254i
$807$		0			0
$808$			18.0000i			0.633238i
$809$	28.2843	+	28.2843i	0.994422	+	0.994422i	0.999985	−	0.00556251i	$-0.00177061\pi$
$809$	28.2843	+	28.2843i	0.994422	+	0.994422i	−0.00556251	+	0.999985i	$0.501771\pi$
$810$		0			0
$811$	2.82843	−	2.82843i	0.0993195	−	0.0993195i	−0.655701	−	0.755021i	$-0.727627\pi$
$811$	2.82843	−	2.82843i	0.0993195	−	0.0993195i	0.755021	+	0.655701i	$0.227627\pi$
$812$		0			0
$813$		0			0
$814$			32.0000i			1.12160i
$815$		0			0
$816$		0			0
$817$	16.0000			0.559769
$818$		−	10.0000i		−	0.349642i
$819$	−5.65685	+	5.65685i	−0.197666	+	0.197666i
$820$		0			0
$821$	22.6274	−	22.6274i	0.789702	−	0.789702i	−0.191743	−	0.981445i	$-0.561414\pi$
$821$	22.6274	−	22.6274i	0.789702	−	0.789702i	0.981445	+	0.191743i	$0.0614140\pi$
$822$	−7.07107	−	7.07107i	−0.246632	−	0.246632i
$823$	−31.1127	−	31.1127i	−1.08452	−	1.08452i	−0.996082	−	0.0884389i	$-0.971812\pi$
$823$	−31.1127	−	31.1127i	−1.08452	−	1.08452i	−0.0884389	−	0.996082i	$-0.528188\pi$
$824$		0			0
$825$		−	20.0000i		−	0.696311i
$826$	−33.9411	−	33.9411i	−1.18096	−	1.18096i
$827$	−8.48528	−	8.48528i	−0.295062	−	0.295062i	0.544014	−	0.839076i	$-0.316904\pi$
$827$	−8.48528	−	8.48528i	−0.295062	−	0.295062i	−0.839076	+	0.544014i	$0.816904\pi$
$828$	−2.82843	+	2.82843i	−0.0982946	+	0.0982946i
$829$	2.00000			0.0694629			0.0347314	−	0.999397i	$-0.488942\pi$
$829$	2.00000			0.0694629			0.0347314	+	0.999397i	$0.488942\pi$
$830$		0			0
$831$		−	8.00000i		−	0.277517i
$832$	14.0000			0.485363
$833$		0			0
$834$	−4.00000			−0.138509
$835$		0			0
$836$	−11.3137	+	11.3137i	−0.391293	+	0.391293i
$837$	4.00000			0.138260
$838$	2.82843	−	2.82843i	0.0977064	−	0.0977064i
$839$	31.1127	+	31.1127i	1.07413	+	1.07413i	0.997023	+	0.0771068i	$0.0245682\pi$
$839$	31.1127	+	31.1127i	1.07413	+	1.07413i	0.0771068	+	0.997023i	$0.475432\pi$
$840$		0			0
$841$			29.0000i			1.00000i
$842$		−	22.0000i		−	0.758170i
$843$	−7.07107	−	7.07107i	−0.243541	−	0.243541i
$844$	−2.82843	−	2.82843i	−0.0973585	−	0.0973585i
$845$		0			0
$846$	−8.00000			−0.275046
$847$	−14.1421	+	14.1421i	−0.485930	+	0.485930i
$848$			6.00000i			0.206041i
$849$	−4.00000			−0.137280
$850$		0			0
$851$	32.0000			1.09695
$852$		−	12.0000i		−	0.411113i
$853$	16.9706	−	16.9706i	0.581061	−	0.581061i	−0.354134	−	0.935195i	$-0.615224\pi$
$853$	16.9706	−	16.9706i	0.581061	−	0.581061i	0.935195	+	0.354134i	$0.115224\pi$
$854$	32.0000			1.09502
$855$		0			0
$856$	25.4558	+	25.4558i	0.870063	+	0.870063i
$857$	−5.65685	−	5.65685i	−0.193234	−	0.193234i	0.603858	−	0.797092i	$-0.293630\pi$
$857$	−5.65685	−	5.65685i	−0.193234	−	0.193234i	−0.797092	+	0.603858i	$0.793630\pi$
$858$		−	8.00000i		−	0.273115i
$859$		−	4.00000i		−	0.136478i	−0.997669	−	0.0682391i	$-0.978262\pi$
$859$		−	4.00000i		−	0.136478i	0.997669	−	0.0682391i	$-0.0217381\pi$
$860$		0			0
$861$	−22.6274	−	22.6274i	−0.771140	−	0.771140i
$862$	−14.1421	+	14.1421i	−0.481683	+	0.481683i
$863$		0			0			−	1.00000i	$-0.5\pi$
$863$		0			0				1.00000i	$0.5\pi$
$864$	3.53553	−	3.53553i	0.120281	−	0.120281i
$865$		0			0
$866$	−2.00000			−0.0679628
$867$		0			0
$868$	16.0000			0.543075
$869$		−	16.0000i		−	0.542763i
$870$		0			0
$871$	−24.0000			−0.813209
$872$	−16.9706	+	16.9706i	−0.574696	+	0.574696i
$873$	−11.3137	−	11.3137i	−0.382911	−	0.382911i
$874$	−11.3137	−	11.3137i	−0.382692	−	0.382692i
$875$		0			0
$876$		0			0
$877$	28.2843	+	28.2843i	0.955092	+	0.955092i	0.999034	−	0.0439421i	$-0.0139917\pi$
$877$	28.2843	+	28.2843i	0.955092	+	0.955092i	−0.0439421	+	0.999034i	$0.513992\pi$
$878$	−25.4558	−	25.4558i	−0.859093	−	0.859093i
$879$	−7.07107	+	7.07107i	−0.238501	+	0.238501i
$880$		0			0
$881$	−16.9706	+	16.9706i	−0.571753	+	0.571753i	−0.932618	−	0.360865i	$-0.882481\pi$
$881$	−16.9706	+	16.9706i	−0.571753	+	0.571753i	0.360865	+	0.932618i	$0.382481\pi$
$882$		−	9.00000i		−	0.303046i
$883$	−20.0000			−0.673054			−0.336527	−	0.941674i	$-0.609252\pi$
$883$	−20.0000			−0.673054			−0.336527	+	0.941674i	$0.609252\pi$
$884$		0			0
$885$		0			0
$886$			20.0000i			0.671913i
$887$	2.82843	−	2.82843i	0.0949693	−	0.0949693i	−0.658026	−	0.752995i	$-0.728608\pi$
$887$	2.82843	−	2.82843i	0.0949693	−	0.0949693i	0.752995	+	0.658026i	$0.228608\pi$
$888$	−24.0000			−0.805387
$889$	−22.6274	+	22.6274i	−0.758899	+	0.758899i
$890$		0			0
$891$	−2.82843	−	2.82843i	−0.0947559	−	0.0947559i
$892$		−	16.0000i		−	0.535720i
$893$			32.0000i			1.07084i
$894$	−4.24264	−	4.24264i	−0.141895	−	0.141895i
$895$		0			0
$896$	8.48528	−	8.48528i	0.283473	−	0.283473i
$897$	−8.00000			−0.267112
$898$	−5.65685	+	5.65685i	−0.188772	+	0.188772i
$899$		0			0
$900$	5.00000			0.166667
$901$		0			0
$902$	32.0000			1.06548
$903$			16.0000i			0.532447i
$904$	−16.9706	+	16.9706i	−0.564433	+	0.564433i
$905$		0			0
$906$		0			0
$907$	−19.7990	−	19.7990i	−0.657415	−	0.657415i	0.297353	−	0.954768i	$-0.403896\pi$
$907$	−19.7990	−	19.7990i	−0.657415	−	0.657415i	−0.954768	+	0.297353i	$0.903896\pi$
$908$	−8.48528	−	8.48528i	−0.281594	−	0.281594i
$909$			6.00000i			0.199007i
$910$		0			0
$911$	−19.7990	−	19.7990i	−0.655970	−	0.655970i	0.298454	−	0.954424i	$-0.403529\pi$
$911$	−19.7990	−	19.7990i	−0.655970	−	0.655970i	−0.954424	+	0.298454i	$0.903529\pi$
$912$	2.82843	+	2.82843i	0.0936586	+	0.0936586i
$913$	33.9411	−	33.9411i	1.12329	−	1.12329i
$914$	−38.0000			−1.25693
$915$		0			0
$916$			10.0000i			0.330409i
$917$	16.0000			0.528367
$918$		0			0
$919$	24.0000			0.791687			0.395843	−	0.918318i	$-0.370452\pi$
$919$	24.0000			0.791687			0.395843	+	0.918318i	$0.370452\pi$
$920$		0			0
$921$	8.48528	−	8.48528i	0.279600	−	0.279600i
$922$	34.0000			1.11973
$923$	16.9706	−	16.9706i	0.558593	−	0.558593i
$924$	−11.3137	−	11.3137i	−0.372194	−	0.372194i
$925$	−28.2843	−	28.2843i	−0.929981	−	0.929981i
$926$		−	40.0000i		−	1.31448i
$927$		0			0
$928$		0			0
$929$	16.9706	+	16.9706i	0.556786	+	0.556786i	0.928391	−	0.371605i	$-0.121192\pi$
$929$	16.9706	+	16.9706i	0.556786	+	0.556786i	−0.371605	+	0.928391i	$0.621192\pi$
$930$		0			0
$931$	−36.0000			−1.17985
$932$	5.65685	−	5.65685i	0.185296	−	0.185296i
$933$			12.0000i			0.392862i
$934$	4.00000			0.130884
$935$		0			0
$936$	6.00000			0.196116
$937$		−	26.0000i		−	0.849383i	−0.905338	−	0.424691i	$-0.860383\pi$
$937$		−	26.0000i		−	0.849383i	0.905338	−	0.424691i	$-0.139617\pi$
$938$	33.9411	−	33.9411i	1.10822	−	1.10822i
$939$	−16.0000			−0.522140
$940$		0			0
$941$	−33.9411	−	33.9411i	−1.10645	−	1.10645i	−0.993614	−	0.112835i	$-0.964007\pi$
$941$	−33.9411	−	33.9411i	−1.10645	−	1.10645i	−0.112835	−	0.993614i	$-0.535993\pi$
$942$	1.41421	+	1.41421i	0.0460776	+	0.0460776i
$943$		−	32.0000i		−	1.04206i
$944$			12.0000i			0.390567i
$945$		0			0
$946$	−11.3137	−	11.3137i	−0.367840	−	0.367840i
$947$	−8.48528	+	8.48528i	−0.275735	+	0.275735i	−0.831404	−	0.555669i	$-0.812462\pi$
$947$	−8.48528	+	8.48528i	−0.275735	+	0.275735i	0.555669	+	0.831404i	$0.312462\pi$
$948$	4.00000			0.129914
$949$		0			0
$950$			20.0000i			0.648886i
$951$	32.0000			1.03767
$952$		0			0
$953$	−42.0000			−1.36051			−0.680257	−	0.732974i	$-0.738132\pi$
$953$	−42.0000			−1.36051			−0.680257	+	0.732974i	$0.738132\pi$
$954$			6.00000i			0.194257i
$955$		0			0
$956$	−8.00000			−0.258738
$957$		0			0
$958$	8.48528	+	8.48528i	0.274147	+	0.274147i
$959$	−28.2843	−	28.2843i	−0.913347	−	0.913347i
$960$		0			0
$961$			15.0000i			0.483871i
$962$	−11.3137	−	11.3137i	−0.364769	−	0.364769i
$963$	8.48528	+	8.48528i	0.273434	+	0.273434i
$964$	11.3137	−	11.3137i	0.364390	−	0.364390i
$965$		0			0
$966$	11.3137	−	11.3137i	0.364013	−	0.364013i
$967$		−	24.0000i		−	0.771788i	−0.922543	−	0.385894i	$-0.873893\pi$
$967$		−	24.0000i		−	0.771788i	0.922543	−	0.385894i	$-0.126107\pi$
$968$	15.0000			0.482118
$969$		0			0
$970$		0			0
$971$		−	36.0000i		−	1.15529i	−0.816286	−	0.577647i	$-0.803971\pi$
$971$		−	36.0000i		−	1.15529i	0.816286	−	0.577647i	$-0.196029\pi$
$972$	0.707107	−	0.707107i	0.0226805	−	0.0226805i
$973$	−16.0000			−0.512936
$974$	−2.82843	+	2.82843i	−0.0906287	+	0.0906287i
$975$	7.07107	+	7.07107i	0.226455	+	0.226455i
$976$	−5.65685	−	5.65685i	−0.181071	−	0.181071i
$977$		−	46.0000i		−	1.47167i	−0.677161	−	0.735835i	$-0.736790\pi$
$977$		−	46.0000i		−	1.47167i	0.677161	−	0.735835i	$-0.263210\pi$
$978$		−	20.0000i		−	0.639529i
$979$	28.2843	+	28.2843i	0.903969	+	0.903969i
$980$		0			0
$981$	−5.65685	+	5.65685i	−0.180609	+	0.180609i
$982$	20.0000			0.638226
$983$	14.1421	−	14.1421i	0.451064	−	0.451064i	−0.444644	−	0.895708i	$-0.646670\pi$
$983$	14.1421	−	14.1421i	0.451064	−	0.451064i	0.895708	+	0.444644i	$0.146670\pi$
$984$			24.0000i			0.765092i
$985$		0			0
$986$		0			0
$987$	−32.0000			−1.01857
$988$		−	8.00000i		−	0.254514i
$989$	−11.3137	+	11.3137i	−0.359755	+	0.359755i
$990$		0			0
$991$	8.48528	−	8.48528i	0.269544	−	0.269544i	−0.559373	−	0.828916i	$-0.688958\pi$
$991$	8.48528	−	8.48528i	0.269544	−	0.269544i	0.828916	+	0.559373i	$0.188958\pi$
$992$	−14.1421	−	14.1421i	−0.449013	−	0.449013i
$993$	14.1421	+	14.1421i	0.448787	+	0.448787i
$994$			48.0000i			1.52247i
$995$		0			0
$996$	8.48528	+	8.48528i	0.268866	+	0.268866i
$997$	5.65685	+	5.65685i	0.179154	+	0.179154i	0.790987	−	0.611833i	$-0.209567\pi$
$997$	5.65685	+	5.65685i	0.179154	+	0.179154i	−0.611833	+	0.790987i	$0.709567\pi$
$998$	25.4558	−	25.4558i	0.805791	−	0.805791i
$999$	−8.00000			−0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	867.2.e.d.616.1		4
17.2	even	8		51.2.d.b.16.2	yes	2
17.3	odd	16		867.2.h.d.733.1		8
17.4	even	4	inner	867.2.e.d.829.2		4
17.5	odd	16		867.2.h.d.712.1		8
17.6	odd	16		867.2.h.d.688.1		8
17.7	odd	16		867.2.h.d.757.2		8
17.8	even	8		867.2.a.b.1.1		1
17.9	even	8		867.2.a.a.1.1		1
17.10	odd	16		867.2.h.d.757.1		8
17.11	odd	16		867.2.h.d.688.2		8
17.12	odd	16		867.2.h.d.712.2		8
17.13	even	4	inner	867.2.e.d.829.1		4
17.14	odd	16		867.2.h.d.733.2		8
17.15	even	8		51.2.d.b.16.1	✓	2
17.16	even	2	inner	867.2.e.d.616.2		4
51.2	odd	8		153.2.d.a.118.1		2
51.8	odd	8		2601.2.a.j.1.1		1
51.26	odd	8		2601.2.a.i.1.1		1
51.32	odd	8		153.2.d.a.118.2		2
68.15	odd	8		816.2.c.c.577.2		2
68.19	odd	8		816.2.c.c.577.1		2
85.2	odd	8		1275.2.d.d.424.2		2
85.19	even	8		1275.2.g.a.526.1		2
85.32	odd	8		1275.2.d.b.424.2		2
85.49	even	8		1275.2.g.a.526.2		2
85.53	odd	8		1275.2.d.b.424.1		2
85.83	odd	8		1275.2.d.d.424.1		2
136.19	odd	8		3264.2.c.d.577.2		2
136.53	even	8		3264.2.c.e.577.1		2
136.83	odd	8		3264.2.c.d.577.1		2
136.117	even	8		3264.2.c.e.577.2		2
204.83	even	8		2448.2.c.j.577.1		2
204.155	even	8		2448.2.c.j.577.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
51.2.d.b.16.1	✓	2	17.15	even	8
51.2.d.b.16.2	yes	2	17.2	even	8
153.2.d.a.118.1		2	51.2	odd	8
153.2.d.a.118.2		2	51.32	odd	8
816.2.c.c.577.1		2	68.19	odd	8
816.2.c.c.577.2		2	68.15	odd	8
867.2.a.a.1.1		1	17.9	even	8
867.2.a.b.1.1		1	17.8	even	8
867.2.e.d.616.1		4	1.1	even	1	trivial
867.2.e.d.616.2		4	17.16	even	2	inner
867.2.e.d.829.1		4	17.13	even	4	inner
867.2.e.d.829.2		4	17.4	even	4	inner
867.2.h.d.688.1		8	17.6	odd	16
867.2.h.d.688.2		8	17.11	odd	16
867.2.h.d.712.1		8	17.5	odd	16
867.2.h.d.712.2		8	17.12	odd	16
867.2.h.d.733.1		8	17.3	odd	16
867.2.h.d.733.2		8	17.14	odd	16
867.2.h.d.757.1		8	17.10	odd	16
867.2.h.d.757.2		8	17.7	odd	16
1275.2.d.b.424.1		2	85.53	odd	8
1275.2.d.b.424.2		2	85.32	odd	8
1275.2.d.d.424.1		2	85.83	odd	8
1275.2.d.d.424.2		2	85.2	odd	8
1275.2.g.a.526.1		2	85.19	even	8
1275.2.g.a.526.2		2	85.49	even	8
2448.2.c.j.577.1		2	204.83	even	8
2448.2.c.j.577.2		2	204.155	even	8
2601.2.a.i.1.1		1	51.26	odd	8
2601.2.a.j.1.1		1	51.8	odd	8
3264.2.c.d.577.1		2	136.83	odd	8
3264.2.c.d.577.2		2	136.19	odd	8
3264.2.c.e.577.1		2	136.53	even	8
3264.2.c.e.577.2		2	136.117	even	8

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:	\( N \)	\(=\)	\( 867 = 3 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	867.e (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +(-0.707107 + 0.707107i) q^{3} +1.00000 q^{4} +(0.707107 + 0.707107i) q^{6} +(2.82843 + 2.82843i) q^{7} -3.00000i q^{8} -1.00000i q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} +(-0.707107 + 0.707107i) q^{3} +1.00000 q^{4} +(0.707107 + 0.707107i) q^{6} +(2.82843 + 2.82843i) q^{7} -3.00000i q^{8} -1.00000i q^{9} +(2.82843 + 2.82843i) q^{11} +(-0.707107 + 0.707107i) q^{12} -2.00000 q^{13} +(2.82843 - 2.82843i) q^{14} -1.00000 q^{16} -1.00000 q^{18} +4.00000i q^{19} -4.00000 q^{21} +(2.82843 - 2.82843i) q^{22} +(-2.82843 - 2.82843i) q^{23} +(2.12132 + 2.12132i) q^{24} +5.00000i q^{25} +2.00000i q^{26} +(0.707107 + 0.707107i) q^{27} +(2.82843 + 2.82843i) q^{28} +(2.82843 - 2.82843i) q^{31} -5.00000i q^{32} -4.00000 q^{33} -1.00000i q^{36} +(-5.65685 + 5.65685i) q^{37} +4.00000 q^{38} +(1.41421 - 1.41421i) q^{39} +(5.65685 + 5.65685i) q^{41} +4.00000i q^{42} -4.00000i q^{43} +(2.82843 + 2.82843i) q^{44} +(-2.82843 + 2.82843i) q^{46} +8.00000 q^{47} +(0.707107 - 0.707107i) q^{48} +9.00000i q^{49} +5.00000 q^{50} -2.00000 q^{52} -6.00000i q^{53} +(0.707107 - 0.707107i) q^{54} +(8.48528 - 8.48528i) q^{56} +(-2.82843 - 2.82843i) q^{57} -12.0000i q^{59} +(5.65685 + 5.65685i) q^{61} +(-2.82843 - 2.82843i) q^{62} +(2.82843 - 2.82843i) q^{63} -7.00000 q^{64} +4.00000i q^{66} +12.0000 q^{67} +4.00000 q^{69} +(-8.48528 + 8.48528i) q^{71} -3.00000 q^{72} +(5.65685 + 5.65685i) q^{74} +(-3.53553 - 3.53553i) q^{75} +4.00000i q^{76} +16.0000i q^{77} +(-1.41421 - 1.41421i) q^{78} +(-2.82843 - 2.82843i) q^{79} -1.00000 q^{81} +(5.65685 - 5.65685i) q^{82} -12.0000i q^{83} -4.00000 q^{84} -4.00000 q^{86} +(8.48528 - 8.48528i) q^{88} +10.0000 q^{89} +(-5.65685 - 5.65685i) q^{91} +(-2.82843 - 2.82843i) q^{92} +4.00000i q^{93} -8.00000i q^{94} +(3.53553 + 3.53553i) q^{96} +(11.3137 - 11.3137i) q^{97} +9.00000 q^{98} +(2.82843 - 2.82843i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{4}+O(q^{10}) \) 4 * q + 4 * q^4	\( 4 q + 4 q^{4} - 8 q^{13} - 4 q^{16} - 4 q^{18} - 16 q^{21} - 16 q^{33} + 16 q^{38} + 32 q^{47} + 20 q^{50} - 8 q^{52} - 28 q^{64} + 48 q^{67} + 16 q^{69} - 12 q^{72} - 4 q^{81} - 16 q^{84} - 16 q^{86} + 40 q^{89} + 36 q^{98}+O(q^{100}) \) 4 * q + 4 * q^4 - 8 * q^13 - 4 * q^16 - 4 * q^18 - 16 * q^21 - 16 * q^33 + 16 * q^38 + 32 * q^47 + 20 * q^50 - 8 * q^52 - 28 * q^64 + 48 * q^67 + 16 * q^69 - 12 * q^72 - 4 * q^81 - 16 * q^84 - 16 * q^86 + 40 * q^89 + 36 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more