LMFDB - Embedded newform 810.2.e.d.271.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [810,2,Mod(271,810)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(810, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([2, 0]))

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

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

chi := DirichletCharacter("810.271");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 810 = 2 \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	810.e (of order $3$, 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:	$6.46788256372$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 270)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			271.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	810.271
Dual form			810.2.e.d.541.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 - 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(0.500000 - 0.866025i) q^{5} +(-1.00000 - 1.73205i) q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+(-0.500000 - 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(0.500000 - 0.866025i) q^{5} +(-1.00000 - 1.73205i) q^{7} +1.00000 q^{8} -1.00000 q^{10} +(-1.50000 - 2.59808i) q^{11} +(-2.50000 + 4.33013i) q^{13} +(-1.00000 + 1.73205i) q^{14} +(-0.500000 - 0.866025i) q^{16} -3.00000 q^{17} -4.00000 q^{19} +(0.500000 + 0.866025i) q^{20} +(-1.50000 + 2.59808i) q^{22} +(4.50000 - 7.79423i) q^{23} +(-0.500000 - 0.866025i) q^{25} +5.00000 q^{26} +2.00000 q^{28} +(1.50000 + 2.59808i) q^{29} +(-2.50000 + 4.33013i) q^{31} +(-0.500000 + 0.866025i) q^{32} +(1.50000 + 2.59808i) q^{34} -2.00000 q^{35} -10.0000 q^{37} +(2.00000 + 3.46410i) q^{38} +(0.500000 - 0.866025i) q^{40} +(0.500000 + 0.866025i) q^{43} +3.00000 q^{44} -9.00000 q^{46} +(-4.50000 - 7.79423i) q^{47} +(1.50000 - 2.59808i) q^{49} +(-0.500000 + 0.866025i) q^{50} +(-2.50000 - 4.33013i) q^{52} -12.0000 q^{53} -3.00000 q^{55} +(-1.00000 - 1.73205i) q^{56} +(1.50000 - 2.59808i) q^{58} +(-6.00000 + 10.3923i) q^{59} +(-1.00000 - 1.73205i) q^{61} +5.00000 q^{62} +1.00000 q^{64} +(2.50000 + 4.33013i) q^{65} +(2.00000 - 3.46410i) q^{67} +(1.50000 - 2.59808i) q^{68} +(1.00000 + 1.73205i) q^{70} +12.0000 q^{71} -10.0000 q^{73} +(5.00000 + 8.66025i) q^{74} +(2.00000 - 3.46410i) q^{76} +(-3.00000 + 5.19615i) q^{77} +(6.50000 + 11.2583i) q^{79} -1.00000 q^{80} +(-3.00000 - 5.19615i) q^{83} +(-1.50000 + 2.59808i) q^{85} +(0.500000 - 0.866025i) q^{86} +(-1.50000 - 2.59808i) q^{88} -12.0000 q^{89} +10.0000 q^{91} +(4.50000 + 7.79423i) q^{92} +(-4.50000 + 7.79423i) q^{94} +(-2.00000 + 3.46410i) q^{95} +(-1.00000 - 1.73205i) q^{97} -3.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} - q^{4} + q^{5} - 2 q^{7} + 2 q^{8}+O(q^{10}) $ 2 * q - q^2 - q^4 + q^5 - 2 * q^7 + 2 * q^8	$ 2 q - q^{2} - q^{4} + q^{5} - 2 q^{7} + 2 q^{8} - 2 q^{10} - 3 q^{11} - 5 q^{13} - 2 q^{14} - q^{16} - 6 q^{17} - 8 q^{19} + q^{20} - 3 q^{22} + 9 q^{23} - q^{25} + 10 q^{26} + 4 q^{28} + 3 q^{29} - 5 q^{31} - q^{32} + 3 q^{34} - 4 q^{35} - 20 q^{37} + 4 q^{38} + q^{40} + q^{43} + 6 q^{44} - 18 q^{46} - 9 q^{47} + 3 q^{49} - q^{50} - 5 q^{52} - 24 q^{53} - 6 q^{55} - 2 q^{56} + 3 q^{58} - 12 q^{59} - 2 q^{61} + 10 q^{62} + 2 q^{64} + 5 q^{65} + 4 q^{67} + 3 q^{68} + 2 q^{70} + 24 q^{71} - 20 q^{73} + 10 q^{74} + 4 q^{76} - 6 q^{77} + 13 q^{79} - 2 q^{80} - 6 q^{83} - 3 q^{85} + q^{86} - 3 q^{88} - 24 q^{89} + 20 q^{91} + 9 q^{92} - 9 q^{94} - 4 q^{95} - 2 q^{97} - 6 q^{98}+O(q^{100}) $ 2 * q - q^2 - q^4 + q^5 - 2 * q^7 + 2 * q^8 - 2 * q^10 - 3 * q^11 - 5 * q^13 - 2 * q^14 - q^16 - 6 * q^17 - 8 * q^19 + q^20 - 3 * q^22 + 9 * q^23 - q^25 + 10 * q^26 + 4 * q^28 + 3 * q^29 - 5 * q^31 - q^32 + 3 * q^34 - 4 * q^35 - 20 * q^37 + 4 * q^38 + q^40 + q^43 + 6 * q^44 - 18 * q^46 - 9 * q^47 + 3 * q^49 - q^50 - 5 * q^52 - 24 * q^53 - 6 * q^55 - 2 * q^56 + 3 * q^58 - 12 * q^59 - 2 * q^61 + 10 * q^62 + 2 * q^64 + 5 * q^65 + 4 * q^67 + 3 * q^68 + 2 * q^70 + 24 * q^71 - 20 * q^73 + 10 * q^74 + 4 * q^76 - 6 * q^77 + 13 * q^79 - 2 * q^80 - 6 * q^83 - 3 * q^85 + q^86 - 3 * q^88 - 24 * q^89 + 20 * q^91 + 9 * q^92 - 9 * q^94 - 4 * q^95 - 2 * q^97 - 6 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/810\mathbb{Z}\right)^\times$.

$n$	$487$	$731$
$\chi(n)$	$1$	$e\left(\frac{1}{3}\right)$

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 $

$2$	−0.500000	−	0.866025i	−0.353553	−	0.612372i
$2$	−0.500000	−	0.866025i	−0.353553	−	0.612372i
$3$		0			0
$3$		0			0
$4$	−0.500000	+	0.866025i	−0.250000	+	0.433013i
$5$	0.500000	−	0.866025i	0.223607	−	0.387298i
$5$	0.500000	−	0.866025i	0.223607	−	0.387298i
$6$		0			0
$7$	−1.00000	−	1.73205i	−0.377964	−	0.654654i	0.612801	−	0.790237i	$-0.290043\pi$
$7$	−1.00000	−	1.73205i	−0.377964	−	0.654654i	−0.990766	+	0.135583i	$0.956709\pi$
$8$	1.00000			0.353553
$9$		0			0
$10$	−1.00000			−0.316228
$11$	−1.50000	−	2.59808i	−0.452267	−	0.783349i	0.546259	−	0.837616i	$-0.316051\pi$
$11$	−1.50000	−	2.59808i	−0.452267	−	0.783349i	−0.998526	+	0.0542666i	$0.982718\pi$
$12$		0			0
$13$	−2.50000	+	4.33013i	−0.693375	+	1.20096i	0.277350	+	0.960769i	$0.410544\pi$
$13$	−2.50000	+	4.33013i	−0.693375	+	1.20096i	−0.970725	+	0.240192i	$0.922790\pi$
$14$	−1.00000	+	1.73205i	−0.267261	+	0.462910i
$15$		0			0
$16$	−0.500000	−	0.866025i	−0.125000	−	0.216506i
$17$	−3.00000			−0.727607			−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000			−0.727607			−0.363803	+	0.931476i	$0.618522\pi$
$18$		0			0
$19$	−4.00000			−0.917663			−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000			−0.917663			−0.458831	+	0.888523i	$0.651732\pi$
$20$	0.500000	+	0.866025i	0.111803	+	0.193649i
$21$		0			0
$22$	−1.50000	+	2.59808i	−0.319801	+	0.553912i
$23$	4.50000	−	7.79423i	0.938315	−	1.62521i	0.169701	−	0.985496i	$-0.445720\pi$
$23$	4.50000	−	7.79423i	0.938315	−	1.62521i	0.768613	−	0.639713i	$-0.220947\pi$
$24$		0			0
$25$	−0.500000	−	0.866025i	−0.100000	−	0.173205i
$26$	5.00000			0.980581
$27$		0			0
$28$	2.00000			0.377964
$29$	1.50000	+	2.59808i	0.278543	+	0.482451i	0.971023	−	0.238987i	$-0.0768152\pi$
$29$	1.50000	+	2.59808i	0.278543	+	0.482451i	−0.692480	+	0.721437i	$0.743482\pi$
$30$		0			0
$31$	−2.50000	+	4.33013i	−0.449013	+	0.777714i	−0.998322	−	0.0579057i	$-0.981558\pi$
$31$	−2.50000	+	4.33013i	−0.449013	+	0.777714i	0.549309	+	0.835619i	$0.314891\pi$
$32$	−0.500000	+	0.866025i	−0.0883883	+	0.153093i
$33$		0			0
$34$	1.50000	+	2.59808i	0.257248	+	0.445566i
$35$	−2.00000			−0.338062
$36$		0			0
$37$	−10.0000			−1.64399			−0.821995	−	0.569495i	$-0.807139\pi$
$37$	−10.0000			−1.64399			−0.821995	+	0.569495i	$0.807139\pi$
$38$	2.00000	+	3.46410i	0.324443	+	0.561951i
$39$		0			0
$40$	0.500000	−	0.866025i	0.0790569	−	0.136931i
$41$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$41$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$42$		0			0
$43$	0.500000	+	0.866025i	0.0762493	+	0.132068i	0.901629	−	0.432511i	$-0.142372\pi$
$43$	0.500000	+	0.866025i	0.0762493	+	0.132068i	−0.825380	+	0.564578i	$0.809039\pi$
$44$	3.00000			0.452267
$45$		0			0
$46$	−9.00000			−1.32698
$47$	−4.50000	−	7.79423i	−0.656392	−	1.13691i	−0.981543	−	0.191243i	$-0.938748\pi$
$47$	−4.50000	−	7.79423i	−0.656392	−	1.13691i	0.325150	−	0.945662i	$-0.394585\pi$
$48$		0			0
$49$	1.50000	−	2.59808i	0.214286	−	0.371154i
$50$	−0.500000	+	0.866025i	−0.0707107	+	0.122474i
$51$		0			0
$52$	−2.50000	−	4.33013i	−0.346688	−	0.600481i
$53$	−12.0000			−1.64833			−0.824163	−	0.566352i	$-0.808354\pi$
$53$	−12.0000			−1.64833			−0.824163	+	0.566352i	$0.808354\pi$
$54$		0			0
$55$	−3.00000			−0.404520
$56$	−1.00000	−	1.73205i	−0.133631	−	0.231455i
$57$		0			0
$58$	1.50000	−	2.59808i	0.196960	−	0.341144i
$59$	−6.00000	+	10.3923i	−0.781133	+	1.35296i	0.150148	+	0.988663i	$0.452025\pi$
$59$	−6.00000	+	10.3923i	−0.781133	+	1.35296i	−0.931282	+	0.364299i	$0.881308\pi$
$60$		0			0
$61$	−1.00000	−	1.73205i	−0.128037	−	0.221766i	0.794879	−	0.606768i	$-0.207534\pi$
$61$	−1.00000	−	1.73205i	−0.128037	−	0.221766i	−0.922916	+	0.385002i	$0.874201\pi$
$62$	5.00000			0.635001
$63$		0			0
$64$	1.00000			0.125000
$65$	2.50000	+	4.33013i	0.310087	+	0.537086i
$66$		0			0
$67$	2.00000	−	3.46410i	0.244339	−	0.423207i	−0.717607	−	0.696449i	$-0.754762\pi$
$67$	2.00000	−	3.46410i	0.244339	−	0.423207i	0.961946	+	0.273241i	$0.0880957\pi$
$68$	1.50000	−	2.59808i	0.181902	−	0.315063i
$69$		0			0
$70$	1.00000	+	1.73205i	0.119523	+	0.207020i
$71$	12.0000			1.42414			0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000			1.42414			0.712069	+	0.702109i	$0.247758\pi$
$72$		0			0
$73$	−10.0000			−1.17041			−0.585206	−	0.810885i	$-0.698986\pi$
$73$	−10.0000			−1.17041			−0.585206	+	0.810885i	$0.698986\pi$
$74$	5.00000	+	8.66025i	0.581238	+	1.00673i
$75$		0			0
$76$	2.00000	−	3.46410i	0.229416	−	0.397360i
$77$	−3.00000	+	5.19615i	−0.341882	+	0.592157i
$78$		0			0
$79$	6.50000	+	11.2583i	0.731307	+	1.26666i	0.956325	+	0.292306i	$0.0944227\pi$
$79$	6.50000	+	11.2583i	0.731307	+	1.26666i	−0.225018	+	0.974355i	$0.572244\pi$
$80$	−1.00000			−0.111803
$81$		0			0
$82$		0			0
$83$	−3.00000	−	5.19615i	−0.329293	−	0.570352i	0.653079	−	0.757290i	$-0.273477\pi$
$83$	−3.00000	−	5.19615i	−0.329293	−	0.570352i	−0.982372	+	0.186938i	$0.940144\pi$
$84$		0			0
$85$	−1.50000	+	2.59808i	−0.162698	+	0.281801i
$86$	0.500000	−	0.866025i	0.0539164	−	0.0933859i
$87$		0			0
$88$	−1.50000	−	2.59808i	−0.159901	−	0.276956i
$89$	−12.0000			−1.27200			−0.635999	−	0.771690i	$-0.719412\pi$
$89$	−12.0000			−1.27200			−0.635999	+	0.771690i	$0.719412\pi$
$90$		0			0
$91$	10.0000			1.04828
$92$	4.50000	+	7.79423i	0.469157	+	0.812605i
$93$		0			0
$94$	−4.50000	+	7.79423i	−0.464140	+	0.803913i
$95$	−2.00000	+	3.46410i	−0.205196	+	0.355409i
$96$		0			0
$97$	−1.00000	−	1.73205i	−0.101535	−	0.175863i	0.810782	−	0.585348i	$-0.199042\pi$
$97$	−1.00000	−	1.73205i	−0.101535	−	0.175863i	−0.912317	+	0.409484i	$0.865709\pi$
$98$	−3.00000			−0.303046
$99$		0			0
$100$	1.00000			0.100000
$101$	−7.50000	−	12.9904i	−0.746278	−	1.29259i	−0.949595	−	0.313478i	$-0.898506\pi$
$101$	−7.50000	−	12.9904i	−0.746278	−	1.29259i	0.203317	−	0.979113i	$-0.434828\pi$
$102$		0			0
$103$	−1.00000	+	1.73205i	−0.0985329	+	0.170664i	−0.911078	−	0.412235i	$-0.864748\pi$
$103$	−1.00000	+	1.73205i	−0.0985329	+	0.170664i	0.812545	+	0.582899i	$0.198082\pi$
$104$	−2.50000	+	4.33013i	−0.245145	+	0.424604i
$105$		0			0
$106$	6.00000	+	10.3923i	0.582772	+	1.00939i
$107$	6.00000			0.580042			0.290021	−	0.957020i	$-0.406338\pi$
$107$	6.00000			0.580042			0.290021	+	0.957020i	$0.406338\pi$
$108$		0			0
$109$	20.0000			1.91565			0.957826	−	0.287348i	$-0.0927736\pi$
$109$	20.0000			1.91565			0.957826	+	0.287348i	$0.0927736\pi$
$110$	1.50000	+	2.59808i	0.143019	+	0.247717i
$111$		0			0
$112$	−1.00000	+	1.73205i	−0.0944911	+	0.163663i
$113$	7.50000	−	12.9904i	0.705541	−	1.22203i	−0.260955	−	0.965351i	$-0.584038\pi$
$113$	7.50000	−	12.9904i	0.705541	−	1.22203i	0.966496	−	0.256681i	$-0.0826291\pi$
$114$		0			0
$115$	−4.50000	−	7.79423i	−0.419627	−	0.726816i
$116$	−3.00000			−0.278543
$117$		0			0
$118$	12.0000			1.10469
$119$	3.00000	+	5.19615i	0.275010	+	0.476331i
$120$		0			0
$121$	1.00000	−	1.73205i	0.0909091	−	0.157459i
$122$	−1.00000	+	1.73205i	−0.0905357	+	0.156813i
$123$		0			0
$124$	−2.50000	−	4.33013i	−0.224507	−	0.388857i
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	2.00000			0.177471			0.0887357	−	0.996055i	$-0.471717\pi$
$127$	2.00000			0.177471			0.0887357	+	0.996055i	$0.471717\pi$
$128$	−0.500000	−	0.866025i	−0.0441942	−	0.0765466i
$129$		0			0
$130$	2.50000	−	4.33013i	0.219265	−	0.379777i
$131$	1.50000	−	2.59808i	0.131056	−	0.226995i	−0.793028	−	0.609185i	$-0.791497\pi$
$131$	1.50000	−	2.59808i	0.131056	−	0.226995i	0.924084	+	0.382190i	$0.124830\pi$
$132$		0			0
$133$	4.00000	+	6.92820i	0.346844	+	0.600751i
$134$	−4.00000			−0.345547
$135$		0			0
$136$	−3.00000			−0.257248
$137$	3.00000	+	5.19615i	0.256307	+	0.443937i	0.965250	−	0.261329i	$-0.0841608\pi$
$137$	3.00000	+	5.19615i	0.256307	+	0.443937i	−0.708942	+	0.705266i	$0.750827\pi$
$138$		0			0
$139$	−1.00000	+	1.73205i	−0.0848189	+	0.146911i	−0.905314	−	0.424743i	$-0.860365\pi$
$139$	−1.00000	+	1.73205i	−0.0848189	+	0.146911i	0.820495	+	0.571654i	$0.193698\pi$
$140$	1.00000	−	1.73205i	0.0845154	−	0.146385i
$141$		0			0
$142$	−6.00000	−	10.3923i	−0.503509	−	0.872103i
$143$	15.0000			1.25436
$144$		0			0
$145$	3.00000			0.249136
$146$	5.00000	+	8.66025i	0.413803	+	0.716728i
$147$		0			0
$148$	5.00000	−	8.66025i	0.410997	−	0.711868i
$149$	−10.5000	+	18.1865i	−0.860194	+	1.48990i	0.0115483	+	0.999933i	$0.496324\pi$
$149$	−10.5000	+	18.1865i	−0.860194	+	1.48990i	−0.871742	+	0.489966i	$0.837009\pi$
$150$		0			0
$151$	−5.50000	−	9.52628i	−0.447584	−	0.775238i	0.550645	−	0.834740i	$-0.314382\pi$
$151$	−5.50000	−	9.52628i	−0.447584	−	0.775238i	−0.998228	+	0.0595022i	$0.981049\pi$
$152$	−4.00000			−0.324443
$153$		0			0
$154$	6.00000			0.483494
$155$	2.50000	+	4.33013i	0.200805	+	0.347804i
$156$		0			0
$157$	6.50000	−	11.2583i	0.518756	−	0.898513i	−0.481006	−	0.876717i	$-0.659728\pi$
$157$	6.50000	−	11.2583i	0.518756	−	0.898513i	0.999762	−	0.0217953i	$-0.00693820\pi$
$158$	6.50000	−	11.2583i	0.517112	−	0.895665i
$159$		0			0
$160$	0.500000	+	0.866025i	0.0395285	+	0.0684653i
$161$	−18.0000			−1.41860
$162$		0			0
$163$	11.0000			0.861586			0.430793	−	0.902451i	$-0.358234\pi$
$163$	11.0000			0.861586			0.430793	+	0.902451i	$0.358234\pi$
$164$		0			0
$165$		0			0
$166$	−3.00000	+	5.19615i	−0.232845	+	0.403300i
$167$	6.00000	−	10.3923i	0.464294	−	0.804181i	−0.534875	−	0.844931i	$-0.679641\pi$
$167$	6.00000	−	10.3923i	0.464294	−	0.804181i	0.999169	+	0.0407502i	$0.0129748\pi$
$168$		0			0
$169$	−6.00000	−	10.3923i	−0.461538	−	0.799408i
$170$	3.00000			0.230089
$171$		0			0
$172$	−1.00000			−0.0762493
$173$	−9.00000	−	15.5885i	−0.684257	−	1.18517i	−0.973670	−	0.227964i	$-0.926793\pi$
$173$	−9.00000	−	15.5885i	−0.684257	−	1.18517i	0.289412	−	0.957205i	$-0.406540\pi$
$174$		0			0
$175$	−1.00000	+	1.73205i	−0.0755929	+	0.130931i
$176$	−1.50000	+	2.59808i	−0.113067	+	0.195837i
$177$		0			0
$178$	6.00000	+	10.3923i	0.449719	+	0.778936i
$179$	12.0000			0.896922			0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000			0.896922			0.448461	+	0.893802i	$0.351972\pi$
$180$		0			0
$181$	20.0000			1.48659			0.743294	−	0.668965i	$-0.233262\pi$
$181$	20.0000			1.48659			0.743294	+	0.668965i	$0.233262\pi$
$182$	−5.00000	−	8.66025i	−0.370625	−	0.641941i
$183$		0			0
$184$	4.50000	−	7.79423i	0.331744	−	0.574598i
$185$	−5.00000	+	8.66025i	−0.367607	+	0.636715i
$186$		0			0
$187$	4.50000	+	7.79423i	0.329073	+	0.569970i
$188$	9.00000			0.656392
$189$		0			0
$190$	4.00000			0.290191
$191$	−3.00000	−	5.19615i	−0.217072	−	0.375980i	0.736839	−	0.676068i	$-0.236317\pi$
$191$	−3.00000	−	5.19615i	−0.217072	−	0.375980i	−0.953912	+	0.300088i	$0.902984\pi$
$192$		0			0
$193$	5.00000	−	8.66025i	0.359908	−	0.623379i	−0.628037	−	0.778183i	$-0.716141\pi$
$193$	5.00000	−	8.66025i	0.359908	−	0.623379i	0.987945	+	0.154805i	$0.0494748\pi$
$194$	−1.00000	+	1.73205i	−0.0717958	+	0.124354i
$195$		0			0
$196$	1.50000	+	2.59808i	0.107143	+	0.185577i
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$		0			0
$199$	−7.00000			−0.496217			−0.248108	−	0.968732i	$-0.579809\pi$
$199$	−7.00000			−0.496217			−0.248108	+	0.968732i	$0.579809\pi$
$200$	−0.500000	−	0.866025i	−0.0353553	−	0.0612372i
$201$		0			0
$202$	−7.50000	+	12.9904i	−0.527698	+	0.914000i
$203$	3.00000	−	5.19615i	0.210559	−	0.364698i
$204$		0			0
$205$		0			0
$206$	2.00000			0.139347
$207$		0			0
$208$	5.00000			0.346688
$209$	6.00000	+	10.3923i	0.415029	+	0.718851i
$210$		0			0
$211$	11.0000	−	19.0526i	0.757271	−	1.31163i	−0.186966	−	0.982366i	$-0.559865\pi$
$211$	11.0000	−	19.0526i	0.757271	−	1.31163i	0.944237	−	0.329266i	$-0.106801\pi$
$212$	6.00000	−	10.3923i	0.412082	−	0.713746i
$213$		0			0
$214$	−3.00000	−	5.19615i	−0.205076	−	0.355202i
$215$	1.00000			0.0681994
$216$		0			0
$217$	10.0000			0.678844
$218$	−10.0000	−	17.3205i	−0.677285	−	1.17309i
$219$		0			0
$220$	1.50000	−	2.59808i	0.101130	−	0.175162i
$221$	7.50000	−	12.9904i	0.504505	−	0.873828i
$222$		0			0
$223$	−4.00000	−	6.92820i	−0.267860	−	0.463947i	0.700449	−	0.713702i	$-0.252983\pi$
$223$	−4.00000	−	6.92820i	−0.267860	−	0.463947i	−0.968309	+	0.249756i	$0.919650\pi$
$224$	2.00000			0.133631
$225$		0			0
$226$	−15.0000			−0.997785
$227$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$227$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$228$		0			0
$229$	2.00000	−	3.46410i	0.132164	−	0.228914i	−0.792347	−	0.610071i	$-0.791141\pi$
$229$	2.00000	−	3.46410i	0.132164	−	0.228914i	0.924510	+	0.381157i	$0.124474\pi$
$230$	−4.50000	+	7.79423i	−0.296721	+	0.513936i
$231$		0			0
$232$	1.50000	+	2.59808i	0.0984798	+	0.170572i
$233$	6.00000			0.393073			0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000			0.393073			0.196537	+	0.980497i	$0.437031\pi$
$234$		0			0
$235$	−9.00000			−0.587095
$236$	−6.00000	−	10.3923i	−0.390567	−	0.676481i
$237$		0			0
$238$	3.00000	−	5.19615i	0.194461	−	0.336817i
$239$	3.00000	−	5.19615i	0.194054	−	0.336111i	−0.752536	−	0.658551i	$-0.771170\pi$
$239$	3.00000	−	5.19615i	0.194054	−	0.336111i	0.946590	+	0.322440i	$0.104503\pi$
$240$		0			0
$241$	9.50000	+	16.4545i	0.611949	+	1.05993i	0.990912	+	0.134515i	$0.0429475\pi$
$241$	9.50000	+	16.4545i	0.611949	+	1.05993i	−0.378963	+	0.925412i	$0.623719\pi$
$242$	−2.00000			−0.128565
$243$		0			0
$244$	2.00000			0.128037
$245$	−1.50000	−	2.59808i	−0.0958315	−	0.165985i
$246$		0			0
$247$	10.0000	−	17.3205i	0.636285	−	1.10208i
$248$	−2.50000	+	4.33013i	−0.158750	+	0.274963i
$249$		0			0
$250$	0.500000	+	0.866025i	0.0316228	+	0.0547723i
$251$	−9.00000			−0.568075			−0.284037	−	0.958813i	$-0.591674\pi$
$251$	−9.00000			−0.568075			−0.284037	+	0.958813i	$0.591674\pi$
$252$		0			0
$253$	−27.0000			−1.69748
$254$	−1.00000	−	1.73205i	−0.0627456	−	0.108679i
$255$		0			0
$256$	−0.500000	+	0.866025i	−0.0312500	+	0.0541266i
$257$	7.50000	−	12.9904i	0.467837	−	0.810318i	−0.531487	−	0.847066i	$-0.678367\pi$
$257$	7.50000	−	12.9904i	0.467837	−	0.810318i	0.999325	+	0.0367485i	$0.0117000\pi$
$258$		0			0
$259$	10.0000	+	17.3205i	0.621370	+	1.07624i
$260$	−5.00000			−0.310087
$261$		0			0
$262$	−3.00000			−0.185341
$263$	−6.00000	−	10.3923i	−0.369976	−	0.640817i	0.619586	−	0.784929i	$-0.287301\pi$
$263$	−6.00000	−	10.3923i	−0.369976	−	0.640817i	−0.989561	+	0.144112i	$0.953967\pi$
$264$		0			0
$265$	−6.00000	+	10.3923i	−0.368577	+	0.638394i
$266$	4.00000	−	6.92820i	0.245256	−	0.424795i
$267$		0			0
$268$	2.00000	+	3.46410i	0.122169	+	0.211604i
$269$	−9.00000			−0.548740			−0.274370	−	0.961624i	$-0.588469\pi$
$269$	−9.00000			−0.548740			−0.274370	+	0.961624i	$0.588469\pi$
$270$		0			0
$271$	8.00000			0.485965			0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000			0.485965			0.242983	+	0.970031i	$0.421874\pi$
$272$	1.50000	+	2.59808i	0.0909509	+	0.157532i
$273$		0			0
$274$	3.00000	−	5.19615i	0.181237	−	0.313911i
$275$	−1.50000	+	2.59808i	−0.0904534	+	0.156670i
$276$		0			0
$277$	5.00000	+	8.66025i	0.300421	+	0.520344i	0.976231	−	0.216731i	$-0.0695395\pi$
$277$	5.00000	+	8.66025i	0.300421	+	0.520344i	−0.675810	+	0.737075i	$0.736206\pi$
$278$	2.00000			0.119952
$279$		0			0
$280$	−2.00000			−0.119523
$281$	15.0000	+	25.9808i	0.894825	+	1.54988i	0.834021	+	0.551733i	$0.186033\pi$
$281$	15.0000	+	25.9808i	0.894825	+	1.54988i	0.0608039	+	0.998150i	$0.480634\pi$
$282$		0			0
$283$	−16.0000	+	27.7128i	−0.951101	+	1.64736i	−0.208053	+	0.978117i	$0.566713\pi$
$283$	−16.0000	+	27.7128i	−0.951101	+	1.64736i	−0.743048	+	0.669238i	$0.766621\pi$
$284$	−6.00000	+	10.3923i	−0.356034	+	0.616670i
$285$		0			0
$286$	−7.50000	−	12.9904i	−0.443484	−	0.768137i
$287$		0			0
$288$		0			0
$289$	−8.00000			−0.470588
$290$	−1.50000	−	2.59808i	−0.0880830	−	0.152564i
$291$		0			0
$292$	5.00000	−	8.66025i	0.292603	−	0.506803i
$293$	−9.00000	+	15.5885i	−0.525786	+	0.910687i	0.473763	+	0.880652i	$0.342895\pi$
$293$	−9.00000	+	15.5885i	−0.525786	+	0.910687i	−0.999549	+	0.0300351i	$0.990438\pi$
$294$		0			0
$295$	6.00000	+	10.3923i	0.349334	+	0.605063i
$296$	−10.0000			−0.581238
$297$		0			0
$298$	21.0000			1.21650
$299$	22.5000	+	38.9711i	1.30121	+	2.25376i
$300$		0			0
$301$	1.00000	−	1.73205i	0.0576390	−	0.0998337i
$302$	−5.50000	+	9.52628i	−0.316489	+	0.548176i
$303$		0			0
$304$	2.00000	+	3.46410i	0.114708	+	0.198680i
$305$	−2.00000			−0.114520
$306$		0			0
$307$	11.0000			0.627803			0.313902	−	0.949456i	$-0.398364\pi$
$307$	11.0000			0.627803			0.313902	+	0.949456i	$0.398364\pi$
$308$	−3.00000	−	5.19615i	−0.170941	−	0.296078i
$309$		0			0
$310$	2.50000	−	4.33013i	0.141990	−	0.245935i
$311$	12.0000	−	20.7846i	0.680458	−	1.17859i	−0.294384	−	0.955687i	$-0.595114\pi$
$311$	12.0000	−	20.7846i	0.680458	−	1.17859i	0.974841	−	0.222900i	$-0.0715523\pi$
$312$		0			0
$313$	−4.00000	−	6.92820i	−0.226093	−	0.391605i	0.730554	−	0.682855i	$-0.239262\pi$
$313$	−4.00000	−	6.92820i	−0.226093	−	0.391605i	−0.956647	+	0.291250i	$0.905929\pi$
$314$	−13.0000			−0.733632
$315$		0			0
$316$	−13.0000			−0.731307
$317$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$317$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$318$		0			0
$319$	4.50000	−	7.79423i	0.251952	−	0.436393i
$320$	0.500000	−	0.866025i	0.0279508	−	0.0484123i
$321$		0			0
$322$	9.00000	+	15.5885i	0.501550	+	0.868711i
$323$	12.0000			0.667698
$324$		0			0
$325$	5.00000			0.277350
$326$	−5.50000	−	9.52628i	−0.304617	−	0.527612i
$327$		0			0
$328$		0			0
$329$	−9.00000	+	15.5885i	−0.496186	+	0.859419i
$330$		0			0
$331$	−7.00000	−	12.1244i	−0.384755	−	0.666415i	0.606980	−	0.794717i	$-0.292381\pi$
$331$	−7.00000	−	12.1244i	−0.384755	−	0.666415i	−0.991735	+	0.128302i	$0.959047\pi$
$332$	6.00000			0.329293
$333$		0			0
$334$	−12.0000			−0.656611
$335$	−2.00000	−	3.46410i	−0.109272	−	0.189264i
$336$		0			0
$337$	−16.0000	+	27.7128i	−0.871576	+	1.50961i	−0.0112091	+	0.999937i	$0.503568\pi$
$337$	−16.0000	+	27.7128i	−0.871576	+	1.50961i	−0.860366	+	0.509676i	$0.829765\pi$
$338$	−6.00000	+	10.3923i	−0.326357	+	0.565267i
$339$		0			0
$340$	−1.50000	−	2.59808i	−0.0813489	−	0.140900i
$341$	15.0000			0.812296
$342$		0			0
$343$	−20.0000			−1.07990
$344$	0.500000	+	0.866025i	0.0269582	+	0.0466930i
$345$		0			0
$346$	−9.00000	+	15.5885i	−0.483843	+	0.838041i
$347$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$347$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$348$		0			0
$349$	−7.00000	−	12.1244i	−0.374701	−	0.649002i	0.615581	−	0.788074i	$-0.288921\pi$
$349$	−7.00000	−	12.1244i	−0.374701	−	0.649002i	−0.990282	+	0.139072i	$0.955588\pi$
$350$	2.00000			0.106904
$351$		0			0
$352$	3.00000			0.159901
$353$	−1.50000	−	2.59808i	−0.0798369	−	0.138282i	0.823343	−	0.567545i	$-0.192107\pi$
$353$	−1.50000	−	2.59808i	−0.0798369	−	0.138282i	−0.903179	+	0.429263i	$0.858773\pi$
$354$		0			0
$355$	6.00000	−	10.3923i	0.318447	−	0.551566i
$356$	6.00000	−	10.3923i	0.317999	−	0.550791i
$357$		0			0
$358$	−6.00000	−	10.3923i	−0.317110	−	0.549250i
$359$	12.0000			0.633336			0.316668	−	0.948536i	$-0.397436\pi$
$359$	12.0000			0.633336			0.316668	+	0.948536i	$0.397436\pi$
$360$		0			0
$361$	−3.00000			−0.157895
$362$	−10.0000	−	17.3205i	−0.525588	−	0.910346i
$363$		0			0
$364$	−5.00000	+	8.66025i	−0.262071	+	0.453921i
$365$	−5.00000	+	8.66025i	−0.261712	+	0.453298i
$366$		0			0
$367$	−13.0000	−	22.5167i	−0.678594	−	1.17536i	−0.975404	−	0.220423i	$-0.929256\pi$
$367$	−13.0000	−	22.5167i	−0.678594	−	1.17536i	0.296810	−	0.954937i	$-0.404077\pi$
$368$	−9.00000			−0.469157
$369$		0			0
$370$	10.0000			0.519875
$371$	12.0000	+	20.7846i	0.623009	+	1.07908i
$372$		0			0
$373$	0.500000	−	0.866025i	0.0258890	−	0.0448411i	−0.852791	−	0.522253i	$-0.825092\pi$
$373$	0.500000	−	0.866025i	0.0258890	−	0.0448411i	0.878680	+	0.477412i	$0.158425\pi$
$374$	4.50000	−	7.79423i	0.232689	−	0.403030i
$375$		0			0
$376$	−4.50000	−	7.79423i	−0.232070	−	0.401957i
$377$	−15.0000			−0.772539
$378$		0			0
$379$	−16.0000			−0.821865			−0.410932	−	0.911666i	$-0.634797\pi$
$379$	−16.0000			−0.821865			−0.410932	+	0.911666i	$0.634797\pi$
$380$	−2.00000	−	3.46410i	−0.102598	−	0.177705i
$381$		0			0
$382$	−3.00000	+	5.19615i	−0.153493	+	0.265858i
$383$	4.50000	−	7.79423i	0.229939	−	0.398266i	−0.727851	−	0.685736i	$-0.759481\pi$
$383$	4.50000	−	7.79423i	0.229939	−	0.398266i	0.957790	+	0.287469i	$0.0928139\pi$
$384$		0			0
$385$	3.00000	+	5.19615i	0.152894	+	0.264820i
$386$	−10.0000			−0.508987
$387$		0			0
$388$	2.00000			0.101535
$389$	−4.50000	−	7.79423i	−0.228159	−	0.395183i	0.729103	−	0.684403i	$-0.239937\pi$
$389$	−4.50000	−	7.79423i	−0.228159	−	0.395183i	−0.957263	+	0.289220i	$0.906604\pi$
$390$		0			0
$391$	−13.5000	+	23.3827i	−0.682724	+	1.18251i
$392$	1.50000	−	2.59808i	0.0757614	−	0.131223i
$393$		0			0
$394$		0			0
$395$	13.0000			0.654101
$396$		0			0
$397$	−7.00000			−0.351320			−0.175660	−	0.984451i	$-0.556206\pi$
$397$	−7.00000			−0.351320			−0.175660	+	0.984451i	$0.556206\pi$
$398$	3.50000	+	6.06218i	0.175439	+	0.303870i
$399$		0			0
$400$	−0.500000	+	0.866025i	−0.0250000	+	0.0433013i
$401$	−18.0000	+	31.1769i	−0.898877	+	1.55690i	−0.0699455	+	0.997551i	$0.522283\pi$
$401$	−18.0000	+	31.1769i	−0.898877	+	1.55690i	−0.828932	+	0.559350i	$0.811051\pi$
$402$		0			0
$403$	−12.5000	−	21.6506i	−0.622669	−	1.07849i
$404$	15.0000			0.746278
$405$		0			0
$406$	−6.00000			−0.297775
$407$	15.0000	+	25.9808i	0.743522	+	1.28782i
$408$		0			0
$409$	−11.5000	+	19.9186i	−0.568638	+	0.984911i	0.428063	+	0.903749i	$0.359196\pi$
$409$	−11.5000	+	19.9186i	−0.568638	+	0.984911i	−0.996701	+	0.0811615i	$0.974137\pi$
$410$		0			0
$411$		0			0
$412$	−1.00000	−	1.73205i	−0.0492665	−	0.0853320i
$413$	24.0000			1.18096
$414$		0			0
$415$	−6.00000			−0.294528
$416$	−2.50000	−	4.33013i	−0.122573	−	0.212302i
$417$		0			0
$418$	6.00000	−	10.3923i	0.293470	−	0.508304i
$419$	1.50000	−	2.59808i	0.0732798	−	0.126924i	−0.827057	−	0.562118i	$-0.809987\pi$
$419$	1.50000	−	2.59808i	0.0732798	−	0.126924i	0.900337	+	0.435194i	$0.143320\pi$
$420$		0			0
$421$	2.00000	+	3.46410i	0.0974740	+	0.168830i	0.910638	−	0.413204i	$-0.135590\pi$
$421$	2.00000	+	3.46410i	0.0974740	+	0.168830i	−0.813164	+	0.582034i	$0.802257\pi$
$422$	−22.0000			−1.07094
$423$		0			0
$424$	−12.0000			−0.582772
$425$	1.50000	+	2.59808i	0.0727607	+	0.126025i
$426$		0			0
$427$	−2.00000	+	3.46410i	−0.0967868	+	0.167640i
$428$	−3.00000	+	5.19615i	−0.145010	+	0.251166i
$429$		0			0
$430$	−0.500000	−	0.866025i	−0.0241121	−	0.0417635i
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$		0			0
$433$	2.00000			0.0961139			0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000			0.0961139			0.0480569	+	0.998845i	$0.484697\pi$
$434$	−5.00000	−	8.66025i	−0.240008	−	0.415705i
$435$		0			0
$436$	−10.0000	+	17.3205i	−0.478913	+	0.829502i
$437$	−18.0000	+	31.1769i	−0.861057	+	1.49139i
$438$		0			0
$439$	−4.00000	−	6.92820i	−0.190910	−	0.330665i	0.754642	−	0.656136i	$-0.227810\pi$
$439$	−4.00000	−	6.92820i	−0.190910	−	0.330665i	−0.945552	+	0.325471i	$0.894477\pi$
$440$	−3.00000			−0.143019
$441$		0			0
$442$	−15.0000			−0.713477
$443$	6.00000	+	10.3923i	0.285069	+	0.493753i	0.972626	−	0.232377i	$-0.0746503\pi$
$443$	6.00000	+	10.3923i	0.285069	+	0.493753i	−0.687557	+	0.726130i	$0.741317\pi$
$444$		0			0
$445$	−6.00000	+	10.3923i	−0.284427	+	0.492642i
$446$	−4.00000	+	6.92820i	−0.189405	+	0.328060i
$447$		0			0
$448$	−1.00000	−	1.73205i	−0.0472456	−	0.0818317i
$449$	−6.00000			−0.283158			−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000			−0.283158			−0.141579	+	0.989927i	$0.545218\pi$
$450$		0			0
$451$		0			0
$452$	7.50000	+	12.9904i	0.352770	+	0.611016i
$453$		0			0
$454$		0			0
$455$	5.00000	−	8.66025i	0.234404	−	0.405999i
$456$		0			0
$457$	14.0000	+	24.2487i	0.654892	+	1.13431i	0.981921	+	0.189292i	$0.0606194\pi$
$457$	14.0000	+	24.2487i	0.654892	+	1.13431i	−0.327028	+	0.945015i	$0.606047\pi$
$458$	−4.00000			−0.186908
$459$		0			0
$460$	9.00000			0.419627
$461$	−9.00000	−	15.5885i	−0.419172	−	0.726027i	0.576685	−	0.816967i	$-0.304346\pi$
$461$	−9.00000	−	15.5885i	−0.419172	−	0.726027i	−0.995856	+	0.0909401i	$0.971013\pi$
$462$		0			0
$463$	−19.0000	+	32.9090i	−0.883005	+	1.52941i	−0.0350215	+	0.999387i	$0.511150\pi$
$463$	−19.0000	+	32.9090i	−0.883005	+	1.52941i	−0.847983	+	0.530023i	$0.822183\pi$
$464$	1.50000	−	2.59808i	0.0696358	−	0.120613i
$465$		0			0
$466$	−3.00000	−	5.19615i	−0.138972	−	0.240707i
$467$	12.0000			0.555294			0.277647	−	0.960683i	$-0.410445\pi$
$467$	12.0000			0.555294			0.277647	+	0.960683i	$0.410445\pi$
$468$		0			0
$469$	−8.00000			−0.369406
$470$	4.50000	+	7.79423i	0.207570	+	0.359521i
$471$		0			0
$472$	−6.00000	+	10.3923i	−0.276172	+	0.478345i
$473$	1.50000	−	2.59808i	0.0689701	−	0.119460i
$474$		0			0
$475$	2.00000	+	3.46410i	0.0917663	+	0.158944i
$476$	−6.00000			−0.275010
$477$		0			0
$478$	−6.00000			−0.274434
$479$	−18.0000	−	31.1769i	−0.822441	−	1.42451i	−0.903859	−	0.427830i	$-0.859278\pi$
$479$	−18.0000	−	31.1769i	−0.822441	−	1.42451i	0.0814184	−	0.996680i	$-0.474055\pi$
$480$		0			0
$481$	25.0000	−	43.3013i	1.13990	−	1.97437i
$482$	9.50000	−	16.4545i	0.432713	−	0.749481i
$483$		0			0
$484$	1.00000	+	1.73205i	0.0454545	+	0.0787296i
$485$	−2.00000			−0.0908153
$486$		0			0
$487$	20.0000			0.906287			0.453143	−	0.891438i	$-0.350303\pi$
$487$	20.0000			0.906287			0.453143	+	0.891438i	$0.350303\pi$
$488$	−1.00000	−	1.73205i	−0.0452679	−	0.0784063i
$489$		0			0
$490$	−1.50000	+	2.59808i	−0.0677631	+	0.117369i
$491$	6.00000	−	10.3923i	0.270776	−	0.468998i	−0.698285	−	0.715820i	$-0.746053\pi$
$491$	6.00000	−	10.3923i	0.270776	−	0.468998i	0.969061	+	0.246822i	$0.0793863\pi$
$492$		0			0
$493$	−4.50000	−	7.79423i	−0.202670	−	0.351034i
$494$	−20.0000			−0.899843
$495$		0			0
$496$	5.00000			0.224507
$497$	−12.0000	−	20.7846i	−0.538274	−	0.932317i
$498$		0			0
$499$	2.00000	−	3.46410i	0.0895323	−	0.155074i	−0.817781	−	0.575529i	$-0.804796\pi$
$499$	2.00000	−	3.46410i	0.0895323	−	0.155074i	0.907314	+	0.420455i	$0.138129\pi$
$500$	0.500000	−	0.866025i	0.0223607	−	0.0387298i
$501$		0			0
$502$	4.50000	+	7.79423i	0.200845	+	0.347873i
$503$	−9.00000			−0.401290			−0.200645	−	0.979664i	$-0.564304\pi$
$503$	−9.00000			−0.401290			−0.200645	+	0.979664i	$0.564304\pi$
$504$		0			0
$505$	−15.0000			−0.667491
$506$	13.5000	+	23.3827i	0.600148	+	1.03949i
$507$		0			0
$508$	−1.00000	+	1.73205i	−0.0443678	+	0.0768473i
$509$	−4.50000	+	7.79423i	−0.199459	+	0.345473i	−0.948353	−	0.317217i	$-0.897252\pi$
$509$	−4.50000	+	7.79423i	−0.199459	+	0.345473i	0.748894	+	0.662690i	$0.230585\pi$
$510$		0			0
$511$	10.0000	+	17.3205i	0.442374	+	0.766214i
$512$	1.00000			0.0441942
$513$		0			0
$514$	−15.0000			−0.661622
$515$	1.00000	+	1.73205i	0.0440653	+	0.0763233i
$516$		0			0
$517$	−13.5000	+	23.3827i	−0.593729	+	1.02837i
$518$	10.0000	−	17.3205i	0.439375	−	0.761019i
$519$		0			0
$520$	2.50000	+	4.33013i	0.109632	+	0.189889i
$521$	−12.0000			−0.525730			−0.262865	−	0.964833i	$-0.584667\pi$
$521$	−12.0000			−0.525730			−0.262865	+	0.964833i	$0.584667\pi$
$522$		0			0
$523$	−31.0000			−1.35554			−0.677768	−	0.735276i	$-0.737052\pi$
$523$	−31.0000			−1.35554			−0.677768	+	0.735276i	$0.737052\pi$
$524$	1.50000	+	2.59808i	0.0655278	+	0.113497i
$525$		0			0
$526$	−6.00000	+	10.3923i	−0.261612	+	0.453126i
$527$	7.50000	−	12.9904i	0.326705	−	0.565870i
$528$		0			0
$529$	−29.0000	−	50.2295i	−1.26087	−	2.18389i
$530$	12.0000			0.521247
$531$		0			0
$532$	−8.00000			−0.346844
$533$		0			0
$534$		0			0
$535$	3.00000	−	5.19615i	0.129701	−	0.224649i
$536$	2.00000	−	3.46410i	0.0863868	−	0.149626i
$537$		0			0
$538$	4.50000	+	7.79423i	0.194009	+	0.336033i
$539$	−9.00000			−0.387657
$540$		0			0
$541$	−22.0000			−0.945854			−0.472927	−	0.881102i	$-0.656803\pi$
$541$	−22.0000			−0.945854			−0.472927	+	0.881102i	$0.656803\pi$
$542$	−4.00000	−	6.92820i	−0.171815	−	0.297592i
$543$		0			0
$544$	1.50000	−	2.59808i	0.0643120	−	0.111392i
$545$	10.0000	−	17.3205i	0.428353	−	0.741929i
$546$		0			0
$547$	6.50000	+	11.2583i	0.277920	+	0.481371i	0.970868	−	0.239616i	$-0.0770217\pi$
$547$	6.50000	+	11.2583i	0.277920	+	0.481371i	−0.692948	+	0.720988i	$0.743688\pi$
$548$	−6.00000			−0.256307
$549$		0			0
$550$	3.00000			0.127920
$551$	−6.00000	−	10.3923i	−0.255609	−	0.442727i
$552$		0			0
$553$	13.0000	−	22.5167i	0.552816	−	0.957506i
$554$	5.00000	−	8.66025i	0.212430	−	0.367939i
$555$		0			0
$556$	−1.00000	−	1.73205i	−0.0424094	−	0.0734553i
$557$	12.0000			0.508456			0.254228	−	0.967144i	$-0.418179\pi$
$557$	12.0000			0.508456			0.254228	+	0.967144i	$0.418179\pi$
$558$		0			0
$559$	−5.00000			−0.211477
$560$	1.00000	+	1.73205i	0.0422577	+	0.0731925i
$561$		0			0
$562$	15.0000	−	25.9808i	0.632737	−	1.09593i
$563$	−3.00000	+	5.19615i	−0.126435	+	0.218992i	−0.922293	−	0.386492i	$-0.873687\pi$
$563$	−3.00000	+	5.19615i	−0.126435	+	0.218992i	0.795858	+	0.605483i	$0.207020\pi$
$564$		0			0
$565$	−7.50000	−	12.9904i	−0.315527	−	0.546509i
$566$	32.0000			1.34506
$567$		0			0
$568$	12.0000			0.503509
$569$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$569$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$570$		0			0
$571$	5.00000	−	8.66025i	0.209243	−	0.362420i	−0.742233	−	0.670142i	$-0.766233\pi$
$571$	5.00000	−	8.66025i	0.209243	−	0.362420i	0.951476	+	0.307722i	$0.0995665\pi$
$572$	−7.50000	+	12.9904i	−0.313591	+	0.543155i
$573$		0			0
$574$		0			0
$575$	−9.00000			−0.375326
$576$		0			0
$577$	32.0000			1.33218			0.666089	−	0.745873i	$-0.267967\pi$
$577$	32.0000			1.33218			0.666089	+	0.745873i	$0.267967\pi$
$578$	4.00000	+	6.92820i	0.166378	+	0.288175i
$579$		0			0
$580$	−1.50000	+	2.59808i	−0.0622841	+	0.107879i
$581$	−6.00000	+	10.3923i	−0.248922	+	0.431145i
$582$		0			0
$583$	18.0000	+	31.1769i	0.745484	+	1.29122i
$584$	−10.0000			−0.413803
$585$		0			0
$586$	18.0000			0.743573
$587$	9.00000	+	15.5885i	0.371470	+	0.643404i	0.989792	−	0.142520i	$-0.0455206\pi$
$587$	9.00000	+	15.5885i	0.371470	+	0.643404i	−0.618322	+	0.785925i	$0.712187\pi$
$588$		0			0
$589$	10.0000	−	17.3205i	0.412043	−	0.713679i
$590$	6.00000	−	10.3923i	0.247016	−	0.427844i
$591$		0			0
$592$	5.00000	+	8.66025i	0.205499	+	0.355934i
$593$	−27.0000			−1.10876			−0.554379	−	0.832265i	$-0.687044\pi$
$593$	−27.0000			−1.10876			−0.554379	+	0.832265i	$0.687044\pi$
$594$		0			0
$595$	6.00000			0.245976
$596$	−10.5000	−	18.1865i	−0.430097	−	0.744949i
$597$		0			0
$598$	22.5000	−	38.9711i	0.920093	−	1.59365i
$599$	15.0000	−	25.9808i	0.612883	−	1.06155i	−0.377869	−	0.925859i	$-0.623343\pi$
$599$	15.0000	−	25.9808i	0.612883	−	1.06155i	0.990752	−	0.135686i	$-0.0433238\pi$
$600$		0			0
$601$	9.50000	+	16.4545i	0.387513	+	0.671192i	0.992114	−	0.125336i	$-0.0400009\pi$
$601$	9.50000	+	16.4545i	0.387513	+	0.671192i	−0.604601	+	0.796528i	$0.706668\pi$
$602$	−2.00000			−0.0815139
$603$		0			0
$604$	11.0000			0.447584
$605$	−1.00000	−	1.73205i	−0.0406558	−	0.0704179i
$606$		0			0
$607$	17.0000	−	29.4449i	0.690009	−	1.19513i	−0.281826	−	0.959466i	$-0.590940\pi$
$607$	17.0000	−	29.4449i	0.690009	−	1.19513i	0.971834	−	0.235665i	$-0.0757267\pi$
$608$	2.00000	−	3.46410i	0.0811107	−	0.140488i
$609$		0			0
$610$	1.00000	+	1.73205i	0.0404888	+	0.0701287i
$611$	45.0000			1.82051
$612$		0			0
$613$	−19.0000			−0.767403			−0.383701	−	0.923457i	$-0.625351\pi$
$613$	−19.0000			−0.767403			−0.383701	+	0.923457i	$0.625351\pi$
$614$	−5.50000	−	9.52628i	−0.221962	−	0.384449i
$615$		0			0
$616$	−3.00000	+	5.19615i	−0.120873	+	0.209359i
$617$	−1.50000	+	2.59808i	−0.0603877	+	0.104595i	−0.894639	−	0.446790i	$-0.852567\pi$
$617$	−1.50000	+	2.59808i	−0.0603877	+	0.104595i	0.834251	+	0.551385i	$0.185900\pi$
$618$		0			0
$619$	17.0000	+	29.4449i	0.683288	+	1.18349i	0.973972	+	0.226670i	$0.0727838\pi$
$619$	17.0000	+	29.4449i	0.683288	+	1.18349i	−0.290684	+	0.956819i	$0.593883\pi$
$620$	−5.00000			−0.200805
$621$		0			0
$622$	−24.0000			−0.962312
$623$	12.0000	+	20.7846i	0.480770	+	0.832718i
$624$		0			0
$625$	−0.500000	+	0.866025i	−0.0200000	+	0.0346410i
$626$	−4.00000	+	6.92820i	−0.159872	+	0.276907i
$627$		0			0
$628$	6.50000	+	11.2583i	0.259378	+	0.449256i
$629$	30.0000			1.19618
$630$		0			0
$631$	−16.0000			−0.636950			−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000			−0.636950			−0.318475	+	0.947931i	$0.603171\pi$
$632$	6.50000	+	11.2583i	0.258556	+	0.447832i
$633$		0			0
$634$		0			0
$635$	1.00000	−	1.73205i	0.0396838	−	0.0687343i
$636$		0			0
$637$	7.50000	+	12.9904i	0.297161	+	0.514698i
$638$	−9.00000			−0.356313
$639$		0			0
$640$	−1.00000			−0.0395285
$641$	−15.0000	−	25.9808i	−0.592464	−	1.02618i	−0.993899	−	0.110291i	$-0.964822\pi$
$641$	−15.0000	−	25.9808i	−0.592464	−	1.02618i	0.401435	−	0.915888i	$-0.368512\pi$
$642$		0			0
$643$	−11.5000	+	19.9186i	−0.453516	+	0.785512i	−0.998602	−	0.0528680i	$-0.983164\pi$
$643$	−11.5000	+	19.9186i	−0.453516	+	0.785512i	0.545086	+	0.838380i	$0.316497\pi$
$644$	9.00000	−	15.5885i	0.354650	−	0.614271i
$645$		0			0
$646$	−6.00000	−	10.3923i	−0.236067	−	0.408880i
$647$	12.0000			0.471769			0.235884	−	0.971781i	$-0.424201\pi$
$647$	12.0000			0.471769			0.235884	+	0.971781i	$0.424201\pi$
$648$		0			0
$649$	36.0000			1.41312
$650$	−2.50000	−	4.33013i	−0.0980581	−	0.169842i
$651$		0			0
$652$	−5.50000	+	9.52628i	−0.215397	+	0.373078i
$653$	−18.0000	+	31.1769i	−0.704394	+	1.22005i	0.262515	+	0.964928i	$0.415448\pi$
$653$	−18.0000	+	31.1769i	−0.704394	+	1.22005i	−0.966910	+	0.255119i	$0.917885\pi$
$654$		0			0
$655$	−1.50000	−	2.59808i	−0.0586098	−	0.101515i
$656$		0			0
$657$		0			0
$658$	18.0000			0.701713
$659$	−24.0000	−	41.5692i	−0.934907	−	1.61931i	−0.774799	−	0.632207i	$-0.782149\pi$
$659$	−24.0000	−	41.5692i	−0.934907	−	1.61931i	−0.160108	−	0.987099i	$-0.551184\pi$
$660$		0			0
$661$	−1.00000	+	1.73205i	−0.0388955	+	0.0673690i	−0.884818	−	0.465937i	$-0.845717\pi$
$661$	−1.00000	+	1.73205i	−0.0388955	+	0.0673690i	0.845922	+	0.533306i	$0.179051\pi$
$662$	−7.00000	+	12.1244i	−0.272063	+	0.471226i
$663$		0			0
$664$	−3.00000	−	5.19615i	−0.116423	−	0.201650i
$665$	8.00000			0.310227
$666$		0			0
$667$	27.0000			1.04544
$668$	6.00000	+	10.3923i	0.232147	+	0.402090i
$669$		0			0
$670$	−2.00000	+	3.46410i	−0.0772667	+	0.133830i
$671$	−3.00000	+	5.19615i	−0.115814	+	0.200595i
$672$		0			0
$673$	11.0000	+	19.0526i	0.424019	+	0.734422i	0.996328	−	0.0856156i	$-0.0272857\pi$
$673$	11.0000	+	19.0526i	0.424019	+	0.734422i	−0.572309	+	0.820038i	$0.693952\pi$
$674$	32.0000			1.23259
$675$		0			0
$676$	12.0000			0.461538
$677$	−6.00000	−	10.3923i	−0.230599	−	0.399409i	0.727386	−	0.686229i	$-0.240735\pi$
$677$	−6.00000	−	10.3923i	−0.230599	−	0.399409i	−0.957984	+	0.286820i	$0.907402\pi$
$678$		0			0
$679$	−2.00000	+	3.46410i	−0.0767530	+	0.132940i
$680$	−1.50000	+	2.59808i	−0.0575224	+	0.0996317i
$681$		0			0
$682$	−7.50000	−	12.9904i	−0.287190	−	0.497427i
$683$	−48.0000			−1.83667			−0.918334	−	0.395805i	$-0.870466\pi$
$683$	−48.0000			−1.83667			−0.918334	+	0.395805i	$0.870466\pi$
$684$		0			0
$685$	6.00000			0.229248
$686$	10.0000	+	17.3205i	0.381802	+	0.661300i
$687$		0			0
$688$	0.500000	−	0.866025i	0.0190623	−	0.0330169i
$689$	30.0000	−	51.9615i	1.14291	−	1.97958i
$690$		0			0
$691$	5.00000	+	8.66025i	0.190209	+	0.329452i	0.945319	−	0.326146i	$-0.105750\pi$
$691$	5.00000	+	8.66025i	0.190209	+	0.329452i	−0.755110	+	0.655598i	$0.772417\pi$
$692$	18.0000			0.684257
$693$		0			0
$694$		0			0
$695$	1.00000	+	1.73205i	0.0379322	+	0.0657004i
$696$		0			0
$697$		0			0
$698$	−7.00000	+	12.1244i	−0.264954	+	0.458914i
$699$		0			0
$700$	−1.00000	−	1.73205i	−0.0377964	−	0.0654654i
$701$	21.0000			0.793159			0.396580	−	0.918000i	$-0.370197\pi$
$701$	21.0000			0.793159			0.396580	+	0.918000i	$0.370197\pi$
$702$		0			0
$703$	40.0000			1.50863
$704$	−1.50000	−	2.59808i	−0.0565334	−	0.0979187i
$705$		0			0
$706$	−1.50000	+	2.59808i	−0.0564532	+	0.0977799i
$707$	−15.0000	+	25.9808i	−0.564133	+	0.977107i
$708$		0			0
$709$	−10.0000	−	17.3205i	−0.375558	−	0.650485i	0.614852	−	0.788642i	$-0.289216\pi$
$709$	−10.0000	−	17.3205i	−0.375558	−	0.650485i	−0.990410	+	0.138157i	$0.955882\pi$
$710$	−12.0000			−0.450352
$711$		0			0
$712$	−12.0000			−0.449719
$713$	22.5000	+	38.9711i	0.842632	+	1.45948i
$714$		0			0
$715$	7.50000	−	12.9904i	0.280484	−	0.485813i
$716$	−6.00000	+	10.3923i	−0.224231	+	0.388379i
$717$		0			0
$718$	−6.00000	−	10.3923i	−0.223918	−	0.387837i
$719$	−30.0000			−1.11881			−0.559406	−	0.828894i	$-0.688971\pi$
$719$	−30.0000			−1.11881			−0.559406	+	0.828894i	$0.688971\pi$
$720$		0			0
$721$	4.00000			0.148968
$722$	1.50000	+	2.59808i	0.0558242	+	0.0966904i
$723$		0			0
$724$	−10.0000	+	17.3205i	−0.371647	+	0.643712i
$725$	1.50000	−	2.59808i	0.0557086	−	0.0964901i
$726$		0			0
$727$	−22.0000	−	38.1051i	−0.815935	−	1.41324i	−0.908655	−	0.417548i	$-0.862889\pi$
$727$	−22.0000	−	38.1051i	−0.815935	−	1.41324i	0.0927199	−	0.995692i	$-0.470444\pi$
$728$	10.0000			0.370625
$729$		0			0
$730$	10.0000			0.370117
$731$	−1.50000	−	2.59808i	−0.0554795	−	0.0960933i
$732$		0			0
$733$	11.0000	−	19.0526i	0.406294	−	0.703722i	−0.588177	−	0.808732i	$-0.700154\pi$
$733$	11.0000	−	19.0526i	0.406294	−	0.703722i	0.994471	+	0.105010i	$0.0334875\pi$
$734$	−13.0000	+	22.5167i	−0.479839	+	0.831105i
$735$		0			0
$736$	4.50000	+	7.79423i	0.165872	+	0.287299i
$737$	−12.0000			−0.442026
$738$		0			0
$739$	20.0000			0.735712			0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000			0.735712			0.367856	+	0.929883i	$0.380092\pi$
$740$	−5.00000	−	8.66025i	−0.183804	−	0.318357i
$741$		0			0
$742$	12.0000	−	20.7846i	0.440534	−	0.763027i
$743$	13.5000	−	23.3827i	0.495267	−	0.857828i	−0.504718	−	0.863284i	$-0.668404\pi$
$743$	13.5000	−	23.3827i	0.495267	−	0.857828i	0.999985	+	0.00545664i	$0.00173691\pi$
$744$		0			0
$745$	10.5000	+	18.1865i	0.384690	+	0.666303i
$746$	−1.00000			−0.0366126
$747$		0			0
$748$	−9.00000			−0.329073
$749$	−6.00000	−	10.3923i	−0.219235	−	0.379727i
$750$		0			0
$751$	12.5000	−	21.6506i	0.456131	−	0.790043i	−0.542621	−	0.839978i	$-0.682568\pi$
$751$	12.5000	−	21.6506i	0.456131	−	0.790043i	0.998752	+	0.0499348i	$0.0159013\pi$
$752$	−4.50000	+	7.79423i	−0.164098	+	0.284226i
$753$		0			0
$754$	7.50000	+	12.9904i	0.273134	+	0.473082i
$755$	−11.0000			−0.400331
$756$		0			0
$757$	−13.0000			−0.472493			−0.236247	−	0.971693i	$-0.575917\pi$
$757$	−13.0000			−0.472493			−0.236247	+	0.971693i	$0.575917\pi$
$758$	8.00000	+	13.8564i	0.290573	+	0.503287i
$759$		0			0
$760$	−2.00000	+	3.46410i	−0.0725476	+	0.125656i
$761$	9.00000	−	15.5885i	0.326250	−	0.565081i	−0.655515	−	0.755182i	$-0.727548\pi$
$761$	9.00000	−	15.5885i	0.326250	−	0.565081i	0.981764	+	0.190101i	$0.0608816\pi$
$762$		0			0
$763$	−20.0000	−	34.6410i	−0.724049	−	1.25409i
$764$	6.00000			0.217072
$765$		0			0
$766$	−9.00000			−0.325183
$767$	−30.0000	−	51.9615i	−1.08324	−	1.87622i
$768$		0			0
$769$	−11.5000	+	19.9186i	−0.414701	+	0.718283i	−0.995397	−	0.0958377i	$-0.969447\pi$
$769$	−11.5000	+	19.9186i	−0.414701	+	0.718283i	0.580696	+	0.814120i	$0.302780\pi$
$770$	3.00000	−	5.19615i	0.108112	−	0.187256i
$771$		0			0
$772$	5.00000	+	8.66025i	0.179954	+	0.311689i
$773$	−42.0000			−1.51064			−0.755318	−	0.655359i	$-0.772517\pi$
$773$	−42.0000			−1.51064			−0.755318	+	0.655359i	$0.772517\pi$
$774$		0			0
$775$	5.00000			0.179605
$776$	−1.00000	−	1.73205i	−0.0358979	−	0.0621770i
$777$		0			0
$778$	−4.50000	+	7.79423i	−0.161333	+	0.279437i
$779$		0			0
$780$		0			0
$781$	−18.0000	−	31.1769i	−0.644091	−	1.11560i
$782$	27.0000			0.965518
$783$		0			0
$784$	−3.00000			−0.107143
$785$	−6.50000	−	11.2583i	−0.231995	−	0.401827i
$786$		0			0
$787$	0.500000	−	0.866025i	0.0178231	−	0.0308705i	−0.856976	−	0.515356i	$-0.827660\pi$
$787$	0.500000	−	0.866025i	0.0178231	−	0.0308705i	0.874799	+	0.484485i	$0.160993\pi$
$788$		0			0
$789$		0			0
$790$	−6.50000	−	11.2583i	−0.231260	−	0.400553i
$791$	−30.0000			−1.06668
$792$		0			0
$793$	10.0000			0.355110
$794$	3.50000	+	6.06218i	0.124210	+	0.215139i
$795$		0			0
$796$	3.50000	−	6.06218i	0.124054	−	0.214868i
$797$	18.0000	−	31.1769i	0.637593	−	1.10434i	−0.348367	−	0.937358i	$-0.613264\pi$
$797$	18.0000	−	31.1769i	0.637593	−	1.10434i	0.985959	−	0.166985i	$-0.0534030\pi$
$798$		0			0
$799$	13.5000	+	23.3827i	0.477596	+	0.827220i
$800$	1.00000			0.0353553
$801$		0			0
$802$	36.0000			1.27120
$803$	15.0000	+	25.9808i	0.529339	+	0.916841i
$804$		0			0
$805$	−9.00000	+	15.5885i	−0.317208	+	0.549421i
$806$	−12.5000	+	21.6506i	−0.440294	+	0.762611i
$807$		0			0
$808$	−7.50000	−	12.9904i	−0.263849	−	0.457000i
$809$	24.0000			0.843795			0.421898	−	0.906644i	$-0.361364\pi$
$809$	24.0000			0.843795			0.421898	+	0.906644i	$0.361364\pi$
$810$		0			0
$811$	38.0000			1.33436			0.667180	−	0.744896i	$-0.267501\pi$
$811$	38.0000			1.33436			0.667180	+	0.744896i	$0.267501\pi$
$812$	3.00000	+	5.19615i	0.105279	+	0.182349i
$813$		0			0
$814$	15.0000	−	25.9808i	0.525750	−	0.910625i
$815$	5.50000	−	9.52628i	0.192657	−	0.333691i
$816$		0			0
$817$	−2.00000	−	3.46410i	−0.0699711	−	0.121194i
$818$	23.0000			0.804176
$819$		0			0
$820$		0			0
$821$	21.0000	+	36.3731i	0.732905	+	1.26943i	0.955636	+	0.294549i	$0.0951694\pi$
$821$	21.0000	+	36.3731i	0.732905	+	1.26943i	−0.222731	+	0.974880i	$0.571497\pi$
$822$		0			0
$823$	2.00000	−	3.46410i	0.0697156	−	0.120751i	−0.829060	−	0.559159i	$-0.811124\pi$
$823$	2.00000	−	3.46410i	0.0697156	−	0.120751i	0.898776	+	0.438408i	$0.144457\pi$
$824$	−1.00000	+	1.73205i	−0.0348367	+	0.0603388i
$825$		0			0
$826$	−12.0000	−	20.7846i	−0.417533	−	0.723189i
$827$	−42.0000			−1.46048			−0.730242	−	0.683189i	$-0.760592\pi$
$827$	−42.0000			−1.46048			−0.730242	+	0.683189i	$0.760592\pi$
$828$		0			0
$829$	44.0000			1.52818			0.764092	−	0.645108i	$-0.223188\pi$
$829$	44.0000			1.52818			0.764092	+	0.645108i	$0.223188\pi$
$830$	3.00000	+	5.19615i	0.104132	+	0.180361i
$831$		0			0
$832$	−2.50000	+	4.33013i	−0.0866719	+	0.150120i
$833$	−4.50000	+	7.79423i	−0.155916	+	0.270054i
$834$		0			0
$835$	−6.00000	−	10.3923i	−0.207639	−	0.359641i
$836$	−12.0000			−0.415029
$837$		0			0
$838$	−3.00000			−0.103633
$839$	−3.00000	−	5.19615i	−0.103572	−	0.179391i	0.809582	−	0.587007i	$-0.199694\pi$
$839$	−3.00000	−	5.19615i	−0.103572	−	0.179391i	−0.913154	+	0.407615i	$0.866360\pi$
$840$		0			0
$841$	10.0000	−	17.3205i	0.344828	−	0.597259i
$842$	2.00000	−	3.46410i	0.0689246	−	0.119381i
$843$		0			0
$844$	11.0000	+	19.0526i	0.378636	+	0.655816i
$845$	−12.0000			−0.412813
$846$		0			0
$847$	−4.00000			−0.137442
$848$	6.00000	+	10.3923i	0.206041	+	0.356873i
$849$		0			0
$850$	1.50000	−	2.59808i	0.0514496	−	0.0891133i
$851$	−45.0000	+	77.9423i	−1.54258	+	2.67183i
$852$		0			0
$853$	9.50000	+	16.4545i	0.325274	+	0.563391i	0.981568	−	0.191115i	$-0.0612102\pi$
$853$	9.50000	+	16.4545i	0.325274	+	0.563391i	−0.656294	+	0.754505i	$0.727877\pi$
$854$	4.00000			0.136877
$855$		0			0
$856$	6.00000			0.205076
$857$	3.00000	+	5.19615i	0.102478	+	0.177497i	0.912705	−	0.408619i	$-0.133990\pi$
$857$	3.00000	+	5.19615i	0.102478	+	0.177497i	−0.810227	+	0.586116i	$0.800656\pi$
$858$		0			0
$859$	−1.00000	+	1.73205i	−0.0341196	+	0.0590968i	−0.882581	−	0.470160i	$-0.844196\pi$
$859$	−1.00000	+	1.73205i	−0.0341196	+	0.0590968i	0.848461	+	0.529257i	$0.177529\pi$
$860$	−0.500000	+	0.866025i	−0.0170499	+	0.0295312i
$861$		0			0
$862$		0			0
$863$	21.0000			0.714848			0.357424	−	0.933942i	$-0.383655\pi$
$863$	21.0000			0.714848			0.357424	+	0.933942i	$0.383655\pi$
$864$		0			0
$865$	−18.0000			−0.612018
$866$	−1.00000	−	1.73205i	−0.0339814	−	0.0588575i
$867$		0			0
$868$	−5.00000	+	8.66025i	−0.169711	+	0.293948i
$869$	19.5000	−	33.7750i	0.661492	−	1.14574i
$870$		0			0
$871$	10.0000	+	17.3205i	0.338837	+	0.586883i
$872$	20.0000			0.677285
$873$		0			0
$874$	36.0000			1.21772
$875$	1.00000	+	1.73205i	0.0338062	+	0.0585540i
$876$		0			0
$877$	0.500000	−	0.866025i	0.0168838	−	0.0292436i	−0.857460	−	0.514551i	$-0.827959\pi$
$877$	0.500000	−	0.866025i	0.0168838	−	0.0292436i	0.874344	+	0.485307i	$0.161292\pi$
$878$	−4.00000	+	6.92820i	−0.134993	+	0.233816i
$879$		0			0
$880$	1.50000	+	2.59808i	0.0505650	+	0.0875811i
$881$	−18.0000			−0.606435			−0.303218	−	0.952921i	$-0.598061\pi$
$881$	−18.0000			−0.606435			−0.303218	+	0.952921i	$0.598061\pi$
$882$		0			0
$883$	−4.00000			−0.134611			−0.0673054	−	0.997732i	$-0.521440\pi$
$883$	−4.00000			−0.134611			−0.0673054	+	0.997732i	$0.521440\pi$
$884$	7.50000	+	12.9904i	0.252252	+	0.436914i
$885$		0			0
$886$	6.00000	−	10.3923i	0.201574	−	0.349136i
$887$	1.50000	−	2.59808i	0.0503651	−	0.0872349i	−0.839744	−	0.542983i	$-0.817295\pi$
$887$	1.50000	−	2.59808i	0.0503651	−	0.0872349i	0.890109	+	0.455748i	$0.150628\pi$
$888$		0			0
$889$	−2.00000	−	3.46410i	−0.0670778	−	0.116182i
$890$	12.0000			0.402241
$891$		0			0
$892$	8.00000			0.267860
$893$	18.0000	+	31.1769i	0.602347	+	1.04330i
$894$		0			0
$895$	6.00000	−	10.3923i	0.200558	−	0.347376i
$896$	−1.00000	+	1.73205i	−0.0334077	+	0.0578638i
$897$		0			0
$898$	3.00000	+	5.19615i	0.100111	+	0.173398i
$899$	−15.0000			−0.500278
$900$		0			0
$901$	36.0000			1.19933
$902$		0			0
$903$		0			0
$904$	7.50000	−	12.9904i	0.249446	−	0.432054i
$905$	10.0000	−	17.3205i	0.332411	−	0.575753i
$906$		0			0
$907$	−29.5000	−	51.0955i	−0.979531	−	1.69660i	−0.664089	−	0.747653i	$-0.731180\pi$
$907$	−29.5000	−	51.0955i	−0.979531	−	1.69660i	−0.315442	−	0.948945i	$-0.602153\pi$
$908$		0			0
$909$		0			0
$910$	−10.0000			−0.331497
$911$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$911$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$912$		0			0
$913$	−9.00000	+	15.5885i	−0.297857	+	0.515903i
$914$	14.0000	−	24.2487i	0.463079	−	0.802076i
$915$		0			0
$916$	2.00000	+	3.46410i	0.0660819	+	0.114457i
$917$	−6.00000			−0.198137
$918$		0			0
$919$	−25.0000			−0.824674			−0.412337	−	0.911031i	$-0.635287\pi$
$919$	−25.0000			−0.824674			−0.412337	+	0.911031i	$0.635287\pi$
$920$	−4.50000	−	7.79423i	−0.148361	−	0.256968i
$921$		0			0
$922$	−9.00000	+	15.5885i	−0.296399	+	0.513378i
$923$	−30.0000	+	51.9615i	−0.987462	+	1.71033i
$924$		0			0
$925$	5.00000	+	8.66025i	0.164399	+	0.284747i
$926$	38.0000			1.24876
$927$		0			0
$928$	−3.00000			−0.0984798
$929$	−18.0000	−	31.1769i	−0.590561	−	1.02288i	−0.994157	−	0.107944i	$-0.965573\pi$
$929$	−18.0000	−	31.1769i	−0.590561	−	1.02288i	0.403596	−	0.914937i	$-0.367760\pi$
$930$		0			0
$931$	−6.00000	+	10.3923i	−0.196642	+	0.340594i
$932$	−3.00000	+	5.19615i	−0.0982683	+	0.170206i
$933$		0			0
$934$	−6.00000	−	10.3923i	−0.196326	−	0.340047i
$935$	9.00000			0.294331
$936$		0			0
$937$	−22.0000			−0.718709			−0.359354	−	0.933201i	$-0.617003\pi$
$937$	−22.0000			−0.718709			−0.359354	+	0.933201i	$0.617003\pi$
$938$	4.00000	+	6.92820i	0.130605	+	0.226214i
$939$		0			0
$940$	4.50000	−	7.79423i	0.146774	−	0.254220i
$941$	−16.5000	+	28.5788i	−0.537885	+	0.931644i	0.461133	+	0.887331i	$0.347443\pi$
$941$	−16.5000	+	28.5788i	−0.537885	+	0.931644i	−0.999018	+	0.0443125i	$0.985890\pi$
$942$		0			0
$943$		0			0
$944$	12.0000			0.390567
$945$		0			0
$946$	−3.00000			−0.0975384
$947$	21.0000	+	36.3731i	0.682408	+	1.18197i	0.974244	+	0.225497i	$0.0724007\pi$
$947$	21.0000	+	36.3731i	0.682408	+	1.18197i	−0.291835	+	0.956469i	$0.594266\pi$
$948$		0			0
$949$	25.0000	−	43.3013i	0.811534	−	1.40562i
$950$	2.00000	−	3.46410i	0.0648886	−	0.112390i
$951$		0			0
$952$	3.00000	+	5.19615i	0.0972306	+	0.168408i
$953$	15.0000			0.485898			0.242949	−	0.970039i	$-0.421885\pi$
$953$	15.0000			0.485898			0.242949	+	0.970039i	$0.421885\pi$
$954$		0			0
$955$	−6.00000			−0.194155
$956$	3.00000	+	5.19615i	0.0970269	+	0.168056i
$957$		0			0
$958$	−18.0000	+	31.1769i	−0.581554	+	1.00728i
$959$	6.00000	−	10.3923i	0.193750	−	0.335585i
$960$		0			0
$961$	3.00000	+	5.19615i	0.0967742	+	0.167618i
$962$	−50.0000			−1.61206
$963$		0			0
$964$	−19.0000			−0.611949
$965$	−5.00000	−	8.66025i	−0.160956	−	0.278783i
$966$		0			0
$967$	−1.00000	+	1.73205i	−0.0321578	+	0.0556990i	−0.881656	−	0.471892i	$-0.843571\pi$
$967$	−1.00000	+	1.73205i	−0.0321578	+	0.0556990i	0.849499	+	0.527591i	$0.176905\pi$
$968$	1.00000	−	1.73205i	0.0321412	−	0.0556702i
$969$		0			0
$970$	1.00000	+	1.73205i	0.0321081	+	0.0556128i
$971$	−57.0000			−1.82922			−0.914609	−	0.404341i	$-0.867501\pi$
$971$	−57.0000			−1.82922			−0.914609	+	0.404341i	$0.867501\pi$
$972$		0			0
$973$	4.00000			0.128234
$974$	−10.0000	−	17.3205i	−0.320421	−	0.554985i
$975$		0			0
$976$	−1.00000	+	1.73205i	−0.0320092	+	0.0554416i
$977$	−16.5000	+	28.5788i	−0.527882	+	0.914318i	0.471590	+	0.881818i	$0.343680\pi$
$977$	−16.5000	+	28.5788i	−0.527882	+	0.914318i	−0.999472	+	0.0325001i	$0.989653\pi$
$978$		0			0
$979$	18.0000	+	31.1769i	0.575282	+	0.996419i
$980$	3.00000			0.0958315
$981$		0			0
$982$	−12.0000			−0.382935
$983$	−19.5000	−	33.7750i	−0.621953	−	1.07725i	−0.989122	−	0.147100i	$-0.953006\pi$
$983$	−19.5000	−	33.7750i	−0.621953	−	1.07725i	0.367168	−	0.930155i	$-0.380327\pi$
$984$		0			0
$985$		0			0
$986$	−4.50000	+	7.79423i	−0.143309	+	0.248219i
$987$		0			0
$988$	10.0000	+	17.3205i	0.318142	+	0.551039i
$989$	9.00000			0.286183
$990$		0			0
$991$	5.00000			0.158830			0.0794151	−	0.996842i	$-0.474695\pi$
$991$	5.00000			0.158830			0.0794151	+	0.996842i	$0.474695\pi$
$992$	−2.50000	−	4.33013i	−0.0793751	−	0.137482i
$993$		0			0
$994$	−12.0000	+	20.7846i	−0.380617	+	0.659248i
$995$	−3.50000	+	6.06218i	−0.110957	+	0.192184i
$996$		0			0
$997$	18.5000	+	32.0429i	0.585901	+	1.01481i	0.994762	+	0.102214i	$0.0325925\pi$
$997$	18.5000	+	32.0429i	0.585901	+	1.01481i	−0.408862	+	0.912596i	$0.634074\pi$
$998$	−4.00000			−0.126618
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	810.2.e.d.271.1		2
3.2	odd	2		810.2.e.i.271.1		2
9.2	odd	6		810.2.e.i.541.1		2
9.4	even	3		270.2.a.c.1.1	yes	1
9.5	odd	6		270.2.a.b.1.1	✓	1
9.7	even	3	inner	810.2.e.d.541.1		2
36.23	even	6		2160.2.a.q.1.1		1
36.31	odd	6		2160.2.a.b.1.1		1
45.4	even	6		1350.2.a.d.1.1		1
45.13	odd	12		1350.2.c.j.649.1		2
45.14	odd	6		1350.2.a.o.1.1		1
45.22	odd	12		1350.2.c.j.649.2		2
45.23	even	12		1350.2.c.c.649.2		2
45.32	even	12		1350.2.c.c.649.1		2
72.5	odd	6		8640.2.a.y.1.1		1
72.13	even	6		8640.2.a.bx.1.1		1
72.59	even	6		8640.2.a.e.1.1		1
72.67	odd	6		8640.2.a.bn.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
270.2.a.b.1.1	✓	1	9.5	odd	6
270.2.a.c.1.1	yes	1	9.4	even	3
810.2.e.d.271.1		2	1.1	even	1	trivial
810.2.e.d.541.1		2	9.7	even	3	inner
810.2.e.i.271.1		2	3.2	odd	2
810.2.e.i.541.1		2	9.2	odd	6
1350.2.a.d.1.1		1	45.4	even	6
1350.2.a.o.1.1		1	45.14	odd	6
1350.2.c.c.649.1		2	45.32	even	12
1350.2.c.c.649.2		2	45.23	even	12
1350.2.c.j.649.1		2	45.13	odd	12
1350.2.c.j.649.2		2	45.22	odd	12
2160.2.a.b.1.1		1	36.31	odd	6
2160.2.a.q.1.1		1	36.23	even	6
8640.2.a.e.1.1		1	72.59	even	6
8640.2.a.y.1.1		1	72.5	odd	6
8640.2.a.bn.1.1		1	72.67	odd	6
8640.2.a.bx.1.1		1	72.13	even	6

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:	\( N \)	\(=\)	\( 810 = 2 \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	810.e (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 - 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(0.500000 - 0.866025i) q^{5} +(-1.00000 - 1.73205i) q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+(-0.500000 - 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(0.500000 - 0.866025i) q^{5} +(-1.00000 - 1.73205i) q^{7} +1.00000 q^{8} -1.00000 q^{10} +(-1.50000 - 2.59808i) q^{11} +(-2.50000 + 4.33013i) q^{13} +(-1.00000 + 1.73205i) q^{14} +(-0.500000 - 0.866025i) q^{16} -3.00000 q^{17} -4.00000 q^{19} +(0.500000 + 0.866025i) q^{20} +(-1.50000 + 2.59808i) q^{22} +(4.50000 - 7.79423i) q^{23} +(-0.500000 - 0.866025i) q^{25} +5.00000 q^{26} +2.00000 q^{28} +(1.50000 + 2.59808i) q^{29} +(-2.50000 + 4.33013i) q^{31} +(-0.500000 + 0.866025i) q^{32} +(1.50000 + 2.59808i) q^{34} -2.00000 q^{35} -10.0000 q^{37} +(2.00000 + 3.46410i) q^{38} +(0.500000 - 0.866025i) q^{40} +(0.500000 + 0.866025i) q^{43} +3.00000 q^{44} -9.00000 q^{46} +(-4.50000 - 7.79423i) q^{47} +(1.50000 - 2.59808i) q^{49} +(-0.500000 + 0.866025i) q^{50} +(-2.50000 - 4.33013i) q^{52} -12.0000 q^{53} -3.00000 q^{55} +(-1.00000 - 1.73205i) q^{56} +(1.50000 - 2.59808i) q^{58} +(-6.00000 + 10.3923i) q^{59} +(-1.00000 - 1.73205i) q^{61} +5.00000 q^{62} +1.00000 q^{64} +(2.50000 + 4.33013i) q^{65} +(2.00000 - 3.46410i) q^{67} +(1.50000 - 2.59808i) q^{68} +(1.00000 + 1.73205i) q^{70} +12.0000 q^{71} -10.0000 q^{73} +(5.00000 + 8.66025i) q^{74} +(2.00000 - 3.46410i) q^{76} +(-3.00000 + 5.19615i) q^{77} +(6.50000 + 11.2583i) q^{79} -1.00000 q^{80} +(-3.00000 - 5.19615i) q^{83} +(-1.50000 + 2.59808i) q^{85} +(0.500000 - 0.866025i) q^{86} +(-1.50000 - 2.59808i) q^{88} -12.0000 q^{89} +10.0000 q^{91} +(4.50000 + 7.79423i) q^{92} +(-4.50000 + 7.79423i) q^{94} +(-2.00000 + 3.46410i) q^{95} +(-1.00000 - 1.73205i) q^{97} -3.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} - q^{4} + q^{5} - 2 q^{7} + 2 q^{8}+O(q^{10}) \) 2 * q - q^2 - q^4 + q^5 - 2 * q^7 + 2 * q^8	\( 2 q - q^{2} - q^{4} + q^{5} - 2 q^{7} + 2 q^{8} - 2 q^{10} - 3 q^{11} - 5 q^{13} - 2 q^{14} - q^{16} - 6 q^{17} - 8 q^{19} + q^{20} - 3 q^{22} + 9 q^{23} - q^{25} + 10 q^{26} + 4 q^{28} + 3 q^{29} - 5 q^{31} - q^{32} + 3 q^{34} - 4 q^{35} - 20 q^{37} + 4 q^{38} + q^{40} + q^{43} + 6 q^{44} - 18 q^{46} - 9 q^{47} + 3 q^{49} - q^{50} - 5 q^{52} - 24 q^{53} - 6 q^{55} - 2 q^{56} + 3 q^{58} - 12 q^{59} - 2 q^{61} + 10 q^{62} + 2 q^{64} + 5 q^{65} + 4 q^{67} + 3 q^{68} + 2 q^{70} + 24 q^{71} - 20 q^{73} + 10 q^{74} + 4 q^{76} - 6 q^{77} + 13 q^{79} - 2 q^{80} - 6 q^{83} - 3 q^{85} + q^{86} - 3 q^{88} - 24 q^{89} + 20 q^{91} + 9 q^{92} - 9 q^{94} - 4 q^{95} - 2 q^{97} - 6 q^{98}+O(q^{100}) \) 2 * q - q^2 - q^4 + q^5 - 2 * q^7 + 2 * q^8 - 2 * q^10 - 3 * q^11 - 5 * q^13 - 2 * q^14 - q^16 - 6 * q^17 - 8 * q^19 + q^20 - 3 * q^22 + 9 * q^23 - q^25 + 10 * q^26 + 4 * q^28 + 3 * q^29 - 5 * q^31 - q^32 + 3 * q^34 - 4 * q^35 - 20 * q^37 + 4 * q^38 + q^40 + q^43 + 6 * q^44 - 18 * q^46 - 9 * q^47 + 3 * q^49 - q^50 - 5 * q^52 - 24 * q^53 - 6 * q^55 - 2 * q^56 + 3 * q^58 - 12 * q^59 - 2 * q^61 + 10 * q^62 + 2 * q^64 + 5 * q^65 + 4 * q^67 + 3 * q^68 + 2 * q^70 + 24 * q^71 - 20 * q^73 + 10 * q^74 + 4 * q^76 - 6 * q^77 + 13 * q^79 - 2 * q^80 - 6 * q^83 - 3 * q^85 + q^86 - 3 * q^88 - 24 * q^89 + 20 * q^91 + 9 * q^92 - 9 * q^94 - 4 * q^95 - 2 * q^97 - 6 * q^98

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