LMFDB - Embedded newform 855.2.k.f.676.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [855,2,Mod(406,855)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("855.406"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(855, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 855 = 3^{2} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	855.k (of order $3$, degree $2$, minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$6.82720937282$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{-3})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 2x^{2} + 4 $ x^4 + 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 285)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			676.2
Root			$0.707107 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	855.676
Dual form			855.2.k.f.406.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(1.20711 - 2.09077i) q^{2} +(-1.91421 - 3.31552i) q^{4} +(-0.500000 + 0.866025i) q^{5} -3.82843 q^{7} -4.41421 q^{8} +(1.20711 + 2.09077i) q^{10} -2.82843 q^{11} +(-1.91421 - 3.31552i) q^{13} +(-4.62132 + 8.00436i) q^{14} +(-1.50000 + 2.59808i) q^{16} +(-3.41421 + 5.91359i) q^{17} +(4.00000 - 1.73205i) q^{19} +3.82843 q^{20} +(-3.41421 + 5.91359i) q^{22} +(2.41421 + 4.18154i) q^{23} +(-0.500000 - 0.866025i) q^{25} -9.24264 q^{26} +(7.32843 + 12.6932i) q^{28} +(-0.828427 - 1.43488i) q^{29} -5.00000 q^{31} +(-0.792893 - 1.37333i) q^{32} +(8.24264 + 14.2767i) q^{34} +(1.91421 - 3.31552i) q^{35} -7.82843 q^{37} +(1.20711 - 10.4539i) q^{38} +(2.20711 - 3.82282i) q^{40} +(-1.41421 + 2.44949i) q^{41} +(-1.08579 + 1.88064i) q^{43} +(5.41421 + 9.37769i) q^{44} +11.6569 q^{46} +(-4.41421 - 7.64564i) q^{47} +7.65685 q^{49} -2.41421 q^{50} +(-7.32843 + 12.6932i) q^{52} +(-1.00000 - 1.73205i) q^{53} +(1.41421 - 2.44949i) q^{55} +16.8995 q^{56} -4.00000 q^{58} +(-7.15685 - 12.3960i) q^{61} +(-6.03553 + 10.4539i) q^{62} -9.82843 q^{64} +3.82843 q^{65} +(5.74264 + 9.94655i) q^{67} +26.1421 q^{68} +(-4.62132 - 8.00436i) q^{70} +(5.00000 - 8.66025i) q^{71} +(3.74264 - 6.48244i) q^{73} +(-9.44975 + 16.3674i) q^{74} +(-13.3995 - 9.94655i) q^{76} +10.8284 q^{77} +(7.32843 - 12.6932i) q^{79} +(-1.50000 - 2.59808i) q^{80} +(3.41421 + 5.91359i) q^{82} +8.00000 q^{83} +(-3.41421 - 5.91359i) q^{85} +(2.62132 + 4.54026i) q^{86} +12.4853 q^{88} +(-2.24264 - 3.88437i) q^{89} +(7.32843 + 12.6932i) q^{91} +(9.24264 - 16.0087i) q^{92} -21.3137 q^{94} +(-0.500000 + 4.33013i) q^{95} +(3.00000 - 5.19615i) q^{97} +(9.24264 - 16.0087i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} - 2 q^{4} - 2 q^{5} - 4 q^{7} - 12 q^{8} + 2 q^{10} - 2 q^{13} - 10 q^{14} - 6 q^{16} - 8 q^{17} + 16 q^{19} + 4 q^{20} - 8 q^{22} + 4 q^{23} - 2 q^{25} - 20 q^{26} + 18 q^{28} + 8 q^{29}+ \cdots + 20 q^{98}+O(q^{100}) $ 4 * q + 2 * q^2 - 2 * q^4 - 2 * q^5 - 4 * q^7 - 12 * q^8 + 2 * q^10 - 2 * q^13 - 10 * q^14 - 6 * q^16 - 8 * q^17 + 16 * q^19 + 4 * q^20 - 8 * q^22 + 4 * q^23 - 2 * q^25 - 20 * q^26 + 18 * q^28 + 8 * q^29 - 20 * q^31 - 6 * q^32 + 16 * q^34 + 2 * q^35 - 20 * q^37 + 2 * q^38 + 6 * q^40 - 10 * q^43 + 16 * q^44 + 24 * q^46 - 12 * q^47 + 8 * q^49 - 4 * q^50 - 18 * q^52 - 4 * q^53 + 28 * q^56 - 16 * q^58 - 6 * q^61 - 10 * q^62 - 28 * q^64 + 4 * q^65 + 6 * q^67 + 48 * q^68 - 10 * q^70 + 20 * q^71 - 2 * q^73 - 18 * q^74 - 14 * q^76 + 32 * q^77 + 18 * q^79 - 6 * q^80 + 8 * q^82 + 32 * q^83 - 8 * q^85 + 2 * q^86 + 16 * q^88 + 8 * q^89 + 18 * q^91 + 20 * q^92 - 40 * q^94 - 2 * q^95 + 12 * q^97 + 20 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/855\mathbb{Z}\right)^\times$.

$n$	$172$	$191$	$496$
$\chi(n)$	$1$	$1$	$e\left(\frac{2}{3}\right)$

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 $

$2$	1.20711	−	2.09077i	0.853553	−	1.47840i	−0.0244272	−	0.999702i	$-0.507776\pi$
$2$	1.20711	−	2.09077i	0.853553	−	1.47840i	0.877981	−	0.478696i	$-0.158890\pi$
$3$		0			0
$3$		0			0
$4$	−1.91421	−	3.31552i	−0.957107	−	1.65776i
$5$	−0.500000	+	0.866025i	−0.223607	+	0.387298i
$5$	−0.500000	+	0.866025i	−0.223607	+	0.387298i
$6$		0			0
$7$	−3.82843			−1.44701			−0.723505	−	0.690319i	$-0.757470\pi$
$7$	−3.82843			−1.44701			−0.723505	+	0.690319i	$0.757470\pi$
$8$	−4.41421			−1.56066
$9$		0			0
$10$	1.20711	+	2.09077i	0.381721	+	0.661160i
$11$	−2.82843			−0.852803			−0.426401	−	0.904534i	$-0.640219\pi$
$11$	−2.82843			−0.852803			−0.426401	+	0.904534i	$0.640219\pi$
$12$		0			0
$13$	−1.91421	−	3.31552i	−0.530907	−	0.919558i	−0.999349	−	0.0360643i	$-0.988518\pi$
$13$	−1.91421	−	3.31552i	−0.530907	−	0.919558i	0.468442	−	0.883494i	$-0.344815\pi$
$14$	−4.62132	+	8.00436i	−1.23510	+	2.13926i
$15$		0			0
$16$	−1.50000	+	2.59808i	−0.375000	+	0.649519i
$17$	−3.41421	+	5.91359i	−0.828068	+	1.43426i	0.0714831	+	0.997442i	$0.477227\pi$
$17$	−3.41421	+	5.91359i	−0.828068	+	1.43426i	−0.899551	+	0.436815i	$0.856107\pi$
$18$		0			0
$19$	4.00000	−	1.73205i	0.917663	−	0.397360i
$19$	4.00000	−	1.73205i	0.917663	−	0.397360i
$20$	3.82843			0.856062
$21$		0			0
$22$	−3.41421	+	5.91359i	−0.727913	+	1.26078i
$23$	2.41421	+	4.18154i	0.503398	+	0.871911i	0.999992	+	0.00392850i	$0.00125049\pi$
$23$	2.41421	+	4.18154i	0.503398	+	0.871911i	−0.496594	+	0.867983i	$0.665416\pi$
$24$		0			0
$25$	−0.500000	−	0.866025i	−0.100000	−	0.173205i
$26$	−9.24264			−1.81263
$27$		0			0
$28$	7.32843	+	12.6932i	1.38494	+	2.39879i
$29$	−0.828427	−	1.43488i	−0.153835	−	0.266450i	0.778799	−	0.627273i	$-0.215829\pi$
$29$	−0.828427	−	1.43488i	−0.153835	−	0.266450i	−0.932634	+	0.360823i	$0.882496\pi$
$30$		0			0
$31$	−5.00000			−0.898027			−0.449013	−	0.893525i	$-0.648224\pi$
$31$	−5.00000			−0.898027			−0.449013	+	0.893525i	$0.648224\pi$
$32$	−0.792893	−	1.37333i	−0.140165	−	0.242773i
$33$		0			0
$34$	8.24264	+	14.2767i	1.41360	+	2.44843i
$35$	1.91421	−	3.31552i	0.323561	−	0.560424i
$36$		0			0
$37$	−7.82843			−1.28699			−0.643493	−	0.765452i	$-0.722515\pi$
$37$	−7.82843			−1.28699			−0.643493	+	0.765452i	$0.722515\pi$
$38$	1.20711	−	10.4539i	0.195819	−	1.69584i
$39$		0			0
$40$	2.20711	−	3.82282i	0.348974	−	0.604441i
$41$	−1.41421	+	2.44949i	−0.220863	+	0.382546i	−0.955070	−	0.296379i	$-0.904221\pi$
$41$	−1.41421	+	2.44949i	−0.220863	+	0.382546i	0.734207	+	0.678925i	$0.237554\pi$
$42$		0			0
$43$	−1.08579	+	1.88064i	−0.165581	+	0.286794i	−0.936861	−	0.349701i	$-0.886283\pi$
$43$	−1.08579	+	1.88064i	−0.165581	+	0.286794i	0.771281	+	0.636495i	$0.219617\pi$
$44$	5.41421	+	9.37769i	0.816223	+	1.41374i
$45$		0			0
$46$	11.6569			1.71871
$47$	−4.41421	−	7.64564i	−0.643879	−	1.11523i	−0.984559	−	0.175052i	$-0.943991\pi$
$47$	−4.41421	−	7.64564i	−0.643879	−	1.11523i	0.340680	−	0.940179i	$-0.389343\pi$
$48$		0			0
$49$	7.65685			1.09384
$50$	−2.41421			−0.341421
$51$		0			0
$52$	−7.32843	+	12.6932i	−1.01627	+	1.76023i
$53$	−1.00000	−	1.73205i	−0.137361	−	0.237915i	0.789136	−	0.614218i	$-0.210529\pi$
$53$	−1.00000	−	1.73205i	−0.137361	−	0.237915i	−0.926497	+	0.376303i	$0.877195\pi$
$54$		0			0
$55$	1.41421	−	2.44949i	0.190693	−	0.330289i
$56$	16.8995			2.25829
$57$		0			0
$58$	−4.00000			−0.525226
$59$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$59$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$60$		0			0
$61$	−7.15685	−	12.3960i	−0.916341	−	1.58715i	−0.804926	−	0.593376i	$-0.797795\pi$
$61$	−7.15685	−	12.3960i	−0.916341	−	1.58715i	−0.111416	−	0.993774i	$-0.535539\pi$
$62$	−6.03553	+	10.4539i	−0.766514	+	1.32764i
$63$		0			0
$64$	−9.82843			−1.22855
$65$	3.82843			0.474858
$66$		0			0
$67$	5.74264	+	9.94655i	0.701575	+	1.21516i	0.967913	+	0.251284i	$0.0808529\pi$
$67$	5.74264	+	9.94655i	0.701575	+	1.21516i	−0.266338	+	0.963880i	$0.585814\pi$
$68$	26.1421			3.17020
$69$		0			0
$70$	−4.62132	−	8.00436i	−0.552353	−	0.956704i
$71$	5.00000	−	8.66025i	0.593391	−	1.02778i	−0.400381	−	0.916349i	$-0.631122\pi$
$71$	5.00000	−	8.66025i	0.593391	−	1.02778i	0.993772	−	0.111434i	$-0.0355445\pi$
$72$		0			0
$73$	3.74264	−	6.48244i	0.438043	−	0.758713i	−0.559496	−	0.828833i	$-0.689005\pi$
$73$	3.74264	−	6.48244i	0.438043	−	0.758713i	0.997539	+	0.0701207i	$0.0223384\pi$
$74$	−9.44975	+	16.3674i	−1.09851	+	1.90268i
$75$		0			0
$76$	−13.3995	−	9.94655i	−1.53703	−	1.14095i
$77$	10.8284			1.23401
$78$		0			0
$79$	7.32843	−	12.6932i	0.824512	−	1.42810i	−0.0777789	−	0.996971i	$-0.524783\pi$
$79$	7.32843	−	12.6932i	0.824512	−	1.42810i	0.902291	−	0.431127i	$-0.141884\pi$
$80$	−1.50000	−	2.59808i	−0.167705	−	0.290474i
$81$		0			0
$82$	3.41421	+	5.91359i	0.377037	+	0.653047i
$83$	8.00000			0.878114			0.439057	−	0.898459i	$-0.355313\pi$
$83$	8.00000			0.878114			0.439057	+	0.898459i	$0.355313\pi$
$84$		0			0
$85$	−3.41421	−	5.91359i	−0.370323	−	0.641419i
$86$	2.62132	+	4.54026i	0.282664	+	0.489589i
$87$		0			0
$88$	12.4853			1.33094
$89$	−2.24264	−	3.88437i	−0.237719	−	0.411742i	0.722340	−	0.691538i	$-0.243067\pi$
$89$	−2.24264	−	3.88437i	−0.237719	−	0.411742i	−0.960060	+	0.279796i	$0.909733\pi$
$90$		0			0
$91$	7.32843	+	12.6932i	0.768228	+	1.33061i
$92$	9.24264	−	16.0087i	0.963612	−	1.66902i
$93$		0			0
$94$	−21.3137			−2.19834
$95$	−0.500000	+	4.33013i	−0.0512989	+	0.444262i
$96$		0			0
$97$	3.00000	−	5.19615i	0.304604	−	0.527589i	−0.672569	−	0.740034i	$-0.734809\pi$
$97$	3.00000	−	5.19615i	0.304604	−	0.527589i	0.977173	+	0.212445i	$0.0681426\pi$
$98$	9.24264	−	16.0087i	0.933648	−	1.61713i
$99$		0			0
$100$	−1.91421	+	3.31552i	−0.191421	+	0.331552i
$101$	−8.07107	−	13.9795i	−0.803101	−	1.39101i	−0.917565	−	0.397585i	$-0.869848\pi$
$101$	−8.07107	−	13.9795i	−0.803101	−	1.39101i	0.114464	−	0.993427i	$-0.463485\pi$
$102$		0			0
$103$	4.17157			0.411037			0.205519	−	0.978653i	$-0.434112\pi$
$103$	4.17157			0.411037			0.205519	+	0.978653i	$0.434112\pi$
$104$	8.44975	+	14.6354i	0.828566	+	1.43512i
$105$		0			0
$106$	−4.82843			−0.468978
$107$	−14.0000			−1.35343			−0.676716	−	0.736245i	$-0.736597\pi$
$107$	−14.0000			−1.35343			−0.676716	+	0.736245i	$0.736597\pi$
$108$		0			0
$109$	−2.65685	+	4.60181i	−0.254480	+	0.440773i	−0.964754	−	0.263152i	$-0.915238\pi$
$109$	−2.65685	+	4.60181i	−0.254480	+	0.440773i	0.710274	+	0.703926i	$0.248571\pi$
$110$	−3.41421	−	5.91359i	−0.325532	−	0.563839i
$111$		0			0
$112$	5.74264	−	9.94655i	0.542629	−	0.939860i
$113$	−4.00000			−0.376288			−0.188144	−	0.982141i	$-0.560247\pi$
$113$	−4.00000			−0.376288			−0.188144	+	0.982141i	$0.560247\pi$
$114$		0			0
$115$	−4.82843			−0.450253
$116$	−3.17157	+	5.49333i	−0.294473	+	0.510042i
$117$		0			0
$118$		0			0
$119$	13.0711	−	22.6398i	1.19822	−	2.07538i
$120$		0			0
$121$	−3.00000			−0.272727
$122$	−34.5563			−3.12858
$123$		0			0
$124$	9.57107	+	16.5776i	0.859507	+	1.48871i
$125$	1.00000			0.0894427
$126$		0			0
$127$	10.4853	+	18.1610i	0.930418	+	1.61153i	0.782607	+	0.622516i	$0.213889\pi$
$127$	10.4853	+	18.1610i	0.930418	+	1.61153i	0.147811	+	0.989016i	$0.452777\pi$
$128$	−10.2782	+	17.8023i	−0.908471	+	1.57352i
$129$		0			0
$130$	4.62132	−	8.00436i	0.405317	−	0.702029i
$131$	−5.24264	+	9.08052i	−0.458052	+	0.793369i	−0.998858	−	0.0477784i	$-0.984786\pi$
$131$	−5.24264	+	9.08052i	−0.458052	+	0.793369i	0.540806	+	0.841147i	$0.318119\pi$
$132$		0			0
$133$	−15.3137	+	6.63103i	−1.32787	+	0.574983i
$134$	27.7279			2.39533
$135$		0			0
$136$	15.0711	−	26.1039i	1.29233	−	2.23839i
$137$	3.24264	+	5.61642i	0.277037	+	0.479843i	0.970647	−	0.240508i	$-0.0773140\pi$
$137$	3.24264	+	5.61642i	0.277037	+	0.479843i	−0.693610	+	0.720351i	$0.743981\pi$
$138$		0			0
$139$	−3.15685	−	5.46783i	−0.267761	−	0.463775i	0.700522	−	0.713630i	$-0.252950\pi$
$139$	−3.15685	−	5.46783i	−0.267761	−	0.463775i	−0.968283	+	0.249855i	$0.919617\pi$
$140$	−14.6569			−1.23873
$141$		0			0
$142$	−12.0711	−	20.9077i	−1.01298	−	1.75454i
$143$	5.41421	+	9.37769i	0.452759	+	0.784202i
$144$		0			0
$145$	1.65685			0.137594
$146$	−9.03553	−	15.6500i	−0.747786	−	1.29520i
$147$		0			0
$148$	14.9853	+	25.9553i	1.23178	+	2.13351i
$149$	−3.41421	+	5.91359i	−0.279703	+	0.484460i	−0.971311	−	0.237813i	$-0.923569\pi$
$149$	−3.41421	+	5.91359i	−0.279703	+	0.484460i	0.691608	+	0.722273i	$0.256903\pi$
$150$		0			0
$151$	21.6569			1.76241			0.881205	−	0.472735i	$-0.156733\pi$
$151$	21.6569			1.76241			0.881205	+	0.472735i	$0.156733\pi$
$152$	−17.6569	+	7.64564i	−1.43216	+	0.620143i
$153$		0			0
$154$	13.0711	−	22.6398i	1.05330	−	1.82436i
$155$	2.50000	−	4.33013i	0.200805	−	0.347804i
$156$		0			0
$157$	7.08579	−	12.2729i	0.565507	−	0.979487i	−0.431495	−	0.902115i	$-0.642014\pi$
$157$	7.08579	−	12.2729i	0.565507	−	0.979487i	0.997002	−	0.0773721i	$-0.0246529\pi$
$158$	−17.6924	−	30.6441i	−1.40753	−	2.43791i
$159$		0			0
$160$	1.58579			0.125367
$161$	−9.24264	−	16.0087i	−0.728422	−	1.26166i
$162$		0			0
$163$	−13.1421			−1.02937			−0.514686	−	0.857379i	$-0.672091\pi$
$163$	−13.1421			−1.02937			−0.514686	+	0.857379i	$0.672091\pi$
$164$	10.8284			0.845558
$165$		0			0
$166$	9.65685	−	16.7262i	0.749517	−	1.29820i
$167$	4.48528	+	7.76874i	0.347081	+	0.601163i	0.985730	−	0.168336i	$-0.0538394\pi$
$167$	4.48528	+	7.76874i	0.347081	+	0.601163i	−0.638648	+	0.769499i	$0.720506\pi$
$168$		0			0
$169$	−0.828427	+	1.43488i	−0.0637252	+	0.110375i
$170$	−16.4853			−1.26436
$171$		0			0
$172$	8.31371			0.633914
$173$	−7.65685	+	13.2621i	−0.582140	+	1.00830i	0.413086	+	0.910692i	$0.364451\pi$
$173$	−7.65685	+	13.2621i	−0.582140	+	1.00830i	−0.995225	+	0.0976036i	$0.968882\pi$
$174$		0			0
$175$	1.91421	+	3.31552i	0.144701	+	0.250629i
$176$	4.24264	−	7.34847i	0.319801	−	0.553912i
$177$		0			0
$178$	−10.8284			−0.811625
$179$	12.1421			0.907546			0.453773	−	0.891117i	$-0.350078\pi$
$179$	12.1421			0.907546			0.453773	+	0.891117i	$0.350078\pi$
$180$		0			0
$181$	−1.34315	−	2.32640i	−0.0998352	−	0.172920i	0.811781	−	0.583962i	$-0.198498\pi$
$181$	−1.34315	−	2.32640i	−0.0998352	−	0.172920i	−0.911616	+	0.411042i	$0.865165\pi$
$182$	35.3848			2.62289
$183$		0			0
$184$	−10.6569	−	18.4582i	−0.785634	−	1.36076i
$185$	3.91421	−	6.77962i	0.287779	−	0.498447i
$186$		0			0
$187$	9.65685	−	16.7262i	0.706179	−	1.22314i
$188$	−16.8995	+	29.2708i	−1.23252	+	2.13479i
$189$		0			0
$190$	8.44975	+	6.27231i	0.613009	+	0.455041i
$191$	11.1716			0.808347			0.404173	−	0.914682i	$-0.367559\pi$
$191$	11.1716			0.808347			0.404173	+	0.914682i	$0.367559\pi$
$192$		0			0
$193$	−7.39949	+	12.8163i	−0.532627	+	0.922538i	0.466647	+	0.884444i	$0.345462\pi$
$193$	−7.39949	+	12.8163i	−0.532627	+	0.922538i	−0.999274	+	0.0380938i	$0.987871\pi$
$194$	−7.24264	−	12.5446i	−0.519991	−	0.900651i
$195$		0			0
$196$	−14.6569	−	25.3864i	−1.04692	−	1.81332i
$197$	−6.34315			−0.451930			−0.225965	−	0.974135i	$-0.572554\pi$
$197$	−6.34315			−0.451930			−0.225965	+	0.974135i	$0.572554\pi$
$198$		0			0
$199$	8.50000	+	14.7224i	0.602549	+	1.04365i	0.992434	+	0.122782i	$0.0391815\pi$
$199$	8.50000	+	14.7224i	0.602549	+	1.04365i	−0.389885	+	0.920864i	$0.627485\pi$
$200$	2.20711	+	3.82282i	0.156066	+	0.270314i
$201$		0			0
$202$	−38.9706			−2.74196
$203$	3.17157	+	5.49333i	0.222601	+	0.385556i
$204$		0			0
$205$	−1.41421	−	2.44949i	−0.0987730	−	0.171080i
$206$	5.03553	−	8.72180i	0.350842	−	0.607677i
$207$		0			0
$208$	11.4853			0.796361
$209$	−11.3137	+	4.89898i	−0.782586	+	0.338869i
$210$		0			0
$211$	−11.1569	+	19.3242i	−0.768070	+	1.33034i	0.170539	+	0.985351i	$0.445449\pi$
$211$	−11.1569	+	19.3242i	−0.768070	+	1.33034i	−0.938608	+	0.344984i	$0.887884\pi$
$212$	−3.82843	+	6.63103i	−0.262937	+	0.455421i
$213$		0			0
$214$	−16.8995	+	29.2708i	−1.15523	+	2.00091i
$215$	−1.08579	−	1.88064i	−0.0740500	−	0.128258i
$216$		0			0
$217$	19.1421			1.29945
$218$	6.41421	+	11.1097i	0.434425	+	0.752447i
$219$		0			0
$220$	−10.8284			−0.730052
$221$	26.1421			1.75851
$222$		0			0
$223$	−0.0857864	+	0.148586i	−0.00574468	+	0.00995009i	−0.868883	−	0.495017i	$-0.835162\pi$
$223$	−0.0857864	+	0.148586i	−0.00574468	+	0.00995009i	0.863139	+	0.504967i	$0.168495\pi$
$224$	3.03553	+	5.25770i	0.202820	+	0.351295i
$225$		0			0
$226$	−4.82843	+	8.36308i	−0.321182	+	0.556304i
$227$	−14.9706			−0.993631			−0.496816	−	0.867856i	$-0.665497\pi$
$227$	−14.9706			−0.993631			−0.496816	+	0.867856i	$0.665497\pi$
$228$		0			0
$229$	6.65685			0.439897			0.219949	−	0.975511i	$-0.429411\pi$
$229$	6.65685			0.439897			0.219949	+	0.975511i	$0.429411\pi$
$230$	−5.82843	+	10.0951i	−0.384315	+	0.665653i
$231$		0			0
$232$	3.65685	+	6.33386i	0.240084	+	0.415838i
$233$	−1.34315	+	2.32640i	−0.0879924	+	0.152407i	−0.906662	−	0.421857i	$-0.861378\pi$
$233$	−1.34315	+	2.32640i	−0.0879924	+	0.152407i	0.818670	+	0.574264i	$0.194712\pi$
$234$		0			0
$235$	8.82843			0.575903
$236$		0			0
$237$		0			0
$238$	−31.5563	−	54.6572i	−2.04549	−	3.54290i
$239$	−20.6274			−1.33428			−0.667138	−	0.744934i	$-0.732481\pi$
$239$	−20.6274			−1.33428			−0.667138	+	0.744934i	$0.732481\pi$
$240$		0			0
$241$	−2.50000	−	4.33013i	−0.161039	−	0.278928i	0.774202	−	0.632938i	$-0.218151\pi$
$241$	−2.50000	−	4.33013i	−0.161039	−	0.278928i	−0.935242	+	0.354010i	$0.884818\pi$
$242$	−3.62132	+	6.27231i	−0.232787	+	0.403199i
$243$		0			0
$244$	−27.3995	+	47.4573i	−1.75407	+	3.03814i
$245$	−3.82843	+	6.63103i	−0.244589	+	0.423641i
$246$		0			0
$247$	−13.3995	−	9.94655i	−0.852589	−	0.632884i
$248$	22.0711			1.40151
$249$		0			0
$250$	1.20711	−	2.09077i	0.0763441	−	0.132232i
$251$	−13.8284	−	23.9515i	−0.872843	−	1.51181i	−0.859043	−	0.511903i	$-0.828941\pi$
$251$	−13.8284	−	23.9515i	−0.872843	−	1.51181i	−0.0137993	−	0.999905i	$-0.504393\pi$
$252$		0			0
$253$	−6.82843	−	11.8272i	−0.429300	−	0.743569i
$254$	50.6274			3.17665
$255$		0			0
$256$	14.9853	+	25.9553i	0.936580	+	1.62220i
$257$	−5.58579	−	9.67487i	−0.348432	−	0.603502i	0.637539	−	0.770418i	$-0.279952\pi$
$257$	−5.58579	−	9.67487i	−0.348432	−	0.603502i	−0.985971	+	0.166916i	$0.946619\pi$
$258$		0			0
$259$	29.9706			1.86228
$260$	−7.32843	−	12.6932i	−0.454490	−	0.787199i
$261$		0			0
$262$	12.6569	+	21.9223i	0.781943	+	1.35437i
$263$	−6.89949	+	11.9503i	−0.425441	+	0.736886i	−0.996462	−	0.0840503i	$-0.973214\pi$
$263$	−6.89949	+	11.9503i	−0.425441	+	0.736886i	0.571020	+	0.820936i	$0.306548\pi$
$264$		0			0
$265$	2.00000			0.122859
$266$	−4.62132	+	40.0218i	−0.283351	+	2.45389i
$267$		0			0
$268$	21.9853	−	38.0796i	1.34296	−	2.32608i
$269$	−4.82843	+	8.36308i	−0.294394	+	0.509906i	−0.974844	−	0.222889i	$-0.928451\pi$
$269$	−4.82843	+	8.36308i	−0.294394	+	0.509906i	0.680449	+	0.732795i	$0.261785\pi$
$270$		0			0
$271$	−1.17157	+	2.02922i	−0.0711680	+	0.123267i	−0.899413	−	0.437099i	$-0.856006\pi$
$271$	−1.17157	+	2.02922i	−0.0711680	+	0.123267i	0.828245	+	0.560365i	$0.189339\pi$
$272$	−10.2426	−	17.7408i	−0.621051	−	1.07569i
$273$		0			0
$274$	15.6569			0.945865
$275$	1.41421	+	2.44949i	0.0852803	+	0.147710i
$276$		0			0
$277$	6.00000			0.360505			0.180253	−	0.983620i	$-0.442309\pi$
$277$	6.00000			0.360505			0.180253	+	0.983620i	$0.442309\pi$
$278$	−15.2426			−0.914193
$279$		0			0
$280$	−8.44975	+	14.6354i	−0.504969	+	0.874632i
$281$	13.4853	+	23.3572i	0.804464	+	1.39337i	0.916652	+	0.399686i	$0.130881\pi$
$281$	13.4853	+	23.3572i	0.804464	+	1.39337i	−0.112188	+	0.993687i	$0.535786\pi$
$282$		0			0
$283$	−6.48528	+	11.2328i	−0.385510	+	0.667723i	−0.991840	−	0.127490i	$-0.959308\pi$
$283$	−6.48528	+	11.2328i	−0.385510	+	0.667723i	0.606330	+	0.795213i	$0.292641\pi$
$284$	−38.2843			−2.27175
$285$		0			0
$286$	26.1421			1.54582
$287$	5.41421	−	9.37769i	0.319591	−	0.553548i
$288$		0			0
$289$	−14.8137	−	25.6581i	−0.871395	−	1.50930i
$290$	2.00000	−	3.46410i	0.117444	−	0.203419i
$291$		0			0
$292$	−28.6569			−1.67702
$293$	−17.3137			−1.01148			−0.505739	−	0.862687i	$-0.668780\pi$
$293$	−17.3137			−1.01148			−0.505739	+	0.862687i	$0.668780\pi$
$294$		0			0
$295$		0			0
$296$	34.5563			2.00855
$297$		0			0
$298$	8.24264	+	14.2767i	0.477483	+	0.827025i
$299$	9.24264	−	16.0087i	0.534516	−	0.925808i
$300$		0			0
$301$	4.15685	−	7.19988i	0.239597	−	0.414994i
$302$	26.1421	−	45.2795i	1.50431	−	2.60554i
$303$		0			0
$304$	−1.50000	+	12.9904i	−0.0860309	+	0.745049i
$305$	14.3137			0.819601
$306$		0			0
$307$	3.17157	−	5.49333i	0.181011	−	0.313521i	−0.761214	−	0.648501i	$-0.775396\pi$
$307$	3.17157	−	5.49333i	0.181011	−	0.313521i	0.942225	+	0.334980i	$0.108730\pi$
$308$	−20.7279	−	35.9018i	−1.18108	−	2.04570i
$309$		0			0
$310$	−6.03553	−	10.4539i	−0.342795	−	0.593739i
$311$	4.00000			0.226819			0.113410	−	0.993548i	$-0.463823\pi$
$311$	4.00000			0.226819			0.113410	+	0.993548i	$0.463823\pi$
$312$		0			0
$313$	3.00000	+	5.19615i	0.169570	+	0.293704i	0.938269	−	0.345907i	$-0.112429\pi$
$313$	3.00000	+	5.19615i	0.169570	+	0.293704i	−0.768699	+	0.639611i	$0.779095\pi$
$314$	−17.1066	−	29.6295i	−0.965381	−	1.67209i
$315$		0			0
$316$	−56.1127			−3.15659
$317$	−8.65685	−	14.9941i	−0.486217	−	0.842153i	0.513657	−	0.857996i	$-0.328290\pi$
$317$	−8.65685	−	14.9941i	−0.486217	−	0.842153i	−0.999875	+	0.0158423i	$0.994957\pi$
$318$		0			0
$319$	2.34315	+	4.05845i	0.131191	+	0.227229i
$320$	4.91421	−	8.51167i	0.274713	−	0.475817i
$321$		0			0
$322$	−44.6274			−2.48699
$323$	−3.41421	+	29.5680i	−0.189972	+	1.64521i
$324$		0			0
$325$	−1.91421	+	3.31552i	−0.106181	+	0.183912i
$326$	−15.8640	+	27.4772i	−0.878624	+	1.52182i
$327$		0			0
$328$	6.24264	−	10.8126i	0.344692	−	0.597024i
$329$	16.8995	+	29.2708i	0.931699	+	1.61375i
$330$		0			0
$331$	−16.6569			−0.915544			−0.457772	−	0.889070i	$-0.651352\pi$
$331$	−16.6569			−0.915544			−0.457772	+	0.889070i	$0.651352\pi$
$332$	−15.3137	−	26.5241i	−0.840449	−	1.45570i
$333$		0			0
$334$	21.6569			1.18501
$335$	−11.4853			−0.627508
$336$		0			0
$337$	−0.257359	+	0.445759i	−0.0140193	+	0.0242821i	−0.872950	−	0.487810i	$-0.837796\pi$
$337$	−0.257359	+	0.445759i	−0.0140193	+	0.0242821i	0.858931	+	0.512092i	$0.171129\pi$
$338$	2.00000	+	3.46410i	0.108786	+	0.188422i
$339$		0			0
$340$	−13.0711	+	22.6398i	−0.708878	+	1.22781i
$341$	14.1421			0.765840
$342$		0			0
$343$	−2.51472			−0.135782
$344$	4.79289	−	8.30153i	0.258415	−	0.447589i
$345$		0			0
$346$	18.4853	+	32.0174i	0.993775	+	1.72127i
$347$	7.72792	−	13.3852i	0.414857	−	0.718553i	−0.580557	−	0.814220i	$-0.697165\pi$
$347$	7.72792	−	13.3852i	0.414857	−	0.718553i	0.995413	+	0.0956671i	$0.0304984\pi$
$348$		0			0
$349$	13.6274			0.729459			0.364729	−	0.931114i	$-0.381161\pi$
$349$	13.6274			0.729459			0.364729	+	0.931114i	$0.381161\pi$
$350$	9.24264			0.494040
$351$		0			0
$352$	2.24264	+	3.88437i	0.119533	+	0.207037i
$353$	−21.3137			−1.13441			−0.567207	−	0.823575i	$-0.691976\pi$
$353$	−21.3137			−1.13441			−0.567207	+	0.823575i	$0.691976\pi$
$354$		0			0
$355$	5.00000	+	8.66025i	0.265372	+	0.459639i
$356$	−8.58579	+	14.8710i	−0.455046	+	0.788162i
$357$		0			0
$358$	14.6569	−	25.3864i	0.774639	−	1.34171i
$359$	7.41421	−	12.8418i	0.391307	−	0.677764i	−0.601315	−	0.799012i	$-0.705356\pi$
$359$	7.41421	−	12.8418i	0.391307	−	0.677764i	0.992622	+	0.121248i	$0.0386896\pi$
$360$		0			0
$361$	13.0000	−	13.8564i	0.684211	−	0.729285i
$362$	−6.48528			−0.340859
$363$		0			0
$364$	28.0563	−	48.5950i	1.47055	−	2.54707i
$365$	3.74264	+	6.48244i	0.195899	+	0.339307i
$366$		0			0
$367$	−14.0563	−	24.3463i	−0.733735	−	1.27087i	−0.955276	−	0.295716i	$-0.904442\pi$
$367$	−14.0563	−	24.3463i	−0.733735	−	1.27087i	0.221540	−	0.975151i	$-0.428892\pi$
$368$	−14.4853			−0.755097
$369$		0			0
$370$	−9.44975	−	16.3674i	−0.491269	−	0.850903i
$371$	3.82843	+	6.63103i	0.198762	+	0.344266i
$372$		0			0
$373$	−0.343146			−0.0177674			−0.00888371	−	0.999961i	$-0.502828\pi$
$373$	−0.343146			−0.0177674			−0.00888371	+	0.999961i	$0.502828\pi$
$374$	−23.3137	−	40.3805i	−1.20552	−	2.08803i
$375$		0			0
$376$	19.4853	+	33.7495i	1.00488	+	1.74050i
$377$	−3.17157	+	5.49333i	−0.163344	+	0.282921i
$378$		0			0
$379$	24.3137			1.24891			0.624456	−	0.781060i	$-0.285321\pi$
$379$	24.3137			1.24891			0.624456	+	0.781060i	$0.285321\pi$
$380$	15.3137	−	6.63103i	0.785577	−	0.340165i
$381$		0			0
$382$	13.4853	−	23.3572i	0.689967	−	1.19506i
$383$	16.9706	−	29.3939i	0.867155	−	1.50196i	0.00226413	−	0.999997i	$-0.499279\pi$
$383$	16.9706	−	29.3939i	0.867155	−	1.50196i	0.864891	−	0.501960i	$-0.167387\pi$
$384$		0			0
$385$	−5.41421	+	9.37769i	−0.275934	+	0.477931i
$386$	17.8640	+	30.9413i	0.909252	+	1.57487i
$387$		0			0
$388$	−22.9706			−1.16615
$389$	17.7279	+	30.7057i	0.898841	+	1.55684i	0.828977	+	0.559282i	$0.188923\pi$
$389$	17.7279	+	30.7057i	0.898841	+	1.55684i	0.0698641	+	0.997557i	$0.477743\pi$
$390$		0			0
$391$	−32.9706			−1.66739
$392$	−33.7990			−1.70711
$393$		0			0
$394$	−7.65685	+	13.2621i	−0.385747	+	0.668133i
$395$	7.32843	+	12.6932i	0.368733	+	0.638665i
$396$		0			0
$397$	9.08579	−	15.7370i	0.456003	−	0.789820i	−0.542743	−	0.839899i	$-0.682614\pi$
$397$	9.08579	−	15.7370i	0.456003	−	0.789820i	0.998745	+	0.0500794i	$0.0159475\pi$
$398$	41.0416			2.05723
$399$		0			0
$400$	3.00000			0.150000
$401$	11.3137	−	19.5959i	0.564980	−	0.978573i	−0.432072	−	0.901839i	$-0.642217\pi$
$401$	11.3137	−	19.5959i	0.564980	−	0.978573i	0.997052	−	0.0767343i	$-0.0244493\pi$
$402$		0			0
$403$	9.57107	+	16.5776i	0.476769	+	0.825788i
$404$	−30.8995	+	53.5195i	−1.53731	+	2.66269i
$405$		0			0
$406$	15.3137			0.760007
$407$	22.1421			1.09754
$408$		0			0
$409$	−3.00000	−	5.19615i	−0.148340	−	0.256933i	0.782274	−	0.622935i	$-0.214060\pi$
$409$	−3.00000	−	5.19615i	−0.148340	−	0.256933i	−0.930614	+	0.366002i	$0.880726\pi$
$410$	−6.82843			−0.337232
$411$		0			0
$412$	−7.98528	−	13.8309i	−0.393407	−	0.681400i
$413$		0			0
$414$		0			0
$415$	−4.00000	+	6.92820i	−0.196352	+	0.340092i
$416$	−3.03553	+	5.25770i	−0.148829	+	0.257780i
$417$		0			0
$418$	−3.41421	+	29.5680i	−0.166995	+	1.44622i
$419$	19.3137			0.943536			0.471768	−	0.881723i	$-0.343616\pi$
$419$	19.3137			0.943536			0.471768	+	0.881723i	$0.343616\pi$
$420$		0			0
$421$	−1.82843	+	3.16693i	−0.0891121	+	0.154347i	−0.907136	−	0.420837i	$-0.861736\pi$
$421$	−1.82843	+	3.16693i	−0.0891121	+	0.154347i	0.818024	+	0.575184i	$0.195070\pi$
$422$	26.9350	+	46.6528i	1.31118	+	2.27102i
$423$		0			0
$424$	4.41421	+	7.64564i	0.214373	+	0.371305i
$425$	6.82843			0.331227
$426$		0			0
$427$	27.3995	+	47.4573i	1.32595	+	2.29662i
$428$	26.7990	+	46.4172i	1.29538	+	2.24366i
$429$		0			0
$430$	−5.24264			−0.252823
$431$	7.24264	+	12.5446i	0.348866	+	0.604253i	0.986048	−	0.166460i	$-0.0532336\pi$
$431$	7.24264	+	12.5446i	0.348866	+	0.604253i	−0.637183	+	0.770713i	$0.719900\pi$
$432$		0			0
$433$	17.0858	+	29.5935i	0.821090	+	1.42217i	0.904871	+	0.425686i	$0.139967\pi$
$433$	17.0858	+	29.5935i	0.821090	+	1.42217i	−0.0837804	+	0.996484i	$0.526699\pi$
$434$	23.1066	−	40.0218i	1.10915	−	1.92111i
$435$		0			0
$436$	20.3431			0.974260
$437$	16.8995	+	12.5446i	0.808412	+	0.600091i
$438$		0			0
$439$	−8.50000	+	14.7224i	−0.405683	+	0.702663i	−0.994401	−	0.105675i	$-0.966300\pi$
$439$	−8.50000	+	14.7224i	−0.405683	+	0.702663i	0.588718	+	0.808339i	$0.299633\pi$
$440$	−6.24264	+	10.8126i	−0.297606	+	0.515469i
$441$		0			0
$442$	31.5563	−	54.6572i	1.50098	−	2.59978i
$443$	−5.07107	−	8.78335i	−0.240934	−	0.417309i	0.720047	−	0.693925i	$-0.244120\pi$
$443$	−5.07107	−	8.78335i	−0.240934	−	0.417309i	−0.960981	+	0.276616i	$0.910787\pi$
$444$		0			0
$445$	4.48528			0.212623
$446$	0.207107	+	0.358719i	0.00980679	+	0.0169859i
$447$		0			0
$448$	37.6274			1.77773
$449$	−18.8284			−0.888568			−0.444284	−	0.895886i	$-0.646542\pi$
$449$	−18.8284			−0.888568			−0.444284	+	0.895886i	$0.646542\pi$
$450$		0			0
$451$	4.00000	−	6.92820i	0.188353	−	0.326236i
$452$	7.65685	+	13.2621i	0.360148	+	0.623795i
$453$		0			0
$454$	−18.0711	+	31.3000i	−0.848117	+	1.46898i
$455$	−14.6569			−0.687124
$456$		0			0
$457$	−9.82843			−0.459754			−0.229877	−	0.973220i	$-0.573832\pi$
$457$	−9.82843			−0.459754			−0.229877	+	0.973220i	$0.573832\pi$
$458$	8.03553	−	13.9180i	0.375476	−	0.650343i
$459$		0			0
$460$	9.24264	+	16.0087i	0.430940	+	0.746411i
$461$	0.928932	−	1.60896i	0.0432647	−	0.0749366i	−0.843582	−	0.537000i	$-0.819557\pi$
$461$	0.928932	−	1.60896i	0.0432647	−	0.0749366i	0.886847	+	0.462064i	$0.152891\pi$
$462$		0			0
$463$	16.7990			0.780715			0.390358	−	0.920663i	$-0.372351\pi$
$463$	16.7990			0.780715			0.390358	+	0.920663i	$0.372351\pi$
$464$	4.97056			0.230753
$465$		0			0
$466$	3.24264	+	5.61642i	0.150212	+	0.260176i
$467$	15.7990			0.731090			0.365545	−	0.930794i	$-0.380883\pi$
$467$	15.7990			0.731090			0.365545	+	0.930794i	$0.380883\pi$
$468$		0			0
$469$	−21.9853	−	38.0796i	−1.01519	−	1.75835i
$470$	10.6569	−	18.4582i	0.491564	−	0.851414i
$471$		0			0
$472$		0			0
$473$	3.07107	−	5.31925i	0.141208	−	0.244579i
$474$		0			0
$475$	−3.50000	−	2.59808i	−0.160591	−	0.119208i
$476$	−100.083			−4.58731
$477$		0			0
$478$	−24.8995	+	43.1272i	−1.13888	+	1.97259i
$479$	−7.24264	−	12.5446i	−0.330925	−	0.573178i	0.651769	−	0.758418i	$-0.274027\pi$
$479$	−7.24264	−	12.5446i	−0.330925	−	0.573178i	−0.982693	+	0.185239i	$0.940694\pi$
$480$		0			0
$481$	14.9853	+	25.9553i	0.683270	+	1.18346i
$482$	−12.0711			−0.549822
$483$		0			0
$484$	5.74264	+	9.94655i	0.261029	+	0.452116i
$485$	3.00000	+	5.19615i	0.136223	+	0.235945i
$486$		0			0
$487$	−17.6569			−0.800108			−0.400054	−	0.916491i	$-0.631009\pi$
$487$	−17.6569			−0.800108			−0.400054	+	0.916491i	$0.631009\pi$
$488$	31.5919	+	54.7187i	1.43010	+	2.47700i
$489$		0			0
$490$	9.24264	+	16.0087i	0.417540	+	0.723200i
$491$	6.17157	−	10.6895i	0.278519	−	0.482409i	−0.692498	−	0.721420i	$-0.743490\pi$
$491$	6.17157	−	10.6895i	0.278519	−	0.482409i	0.971017	+	0.239011i	$0.0768231\pi$
$492$		0			0
$493$	11.3137			0.509544
$494$	−36.9706	+	16.0087i	−1.66338	+	0.720267i
$495$		0			0
$496$	7.50000	−	12.9904i	0.336760	−	0.583285i
$497$	−19.1421	+	33.1552i	−0.858642	+	1.48721i
$498$		0			0
$499$	9.32843	−	16.1573i	0.417598	−	0.723301i	−0.578100	−	0.815966i	$-0.696205\pi$
$499$	9.32843	−	16.1573i	0.417598	−	0.723301i	0.995697	+	0.0926657i	$0.0295388\pi$
$500$	−1.91421	−	3.31552i	−0.0856062	−	0.148274i
$501$		0			0
$502$	−66.7696			−2.98007
$503$	21.0711	+	36.4962i	0.939512	+	1.62728i	0.766383	+	0.642384i	$0.222055\pi$
$503$	21.0711	+	36.4962i	0.939512	+	1.62728i	0.173130	+	0.984899i	$0.444612\pi$
$504$		0			0
$505$	16.1421			0.718316
$506$	−32.9706			−1.46572
$507$		0			0
$508$	40.1421	−	69.5282i	1.78102	−	3.08482i
$509$	−14.5563	−	25.2123i	−0.645199	−	1.11752i	−0.984256	−	0.176751i	$-0.943441\pi$
$509$	−14.5563	−	25.2123i	−0.645199	−	1.11752i	0.339057	−	0.940766i	$-0.389892\pi$
$510$		0			0
$511$	−14.3284	+	24.8176i	−0.633852	+	1.09786i
$512$	31.2426			1.38074
$513$		0			0
$514$	−26.9706			−1.18962
$515$	−2.08579	+	3.61269i	−0.0919107	+	0.159194i
$516$		0			0
$517$	12.4853	+	21.6251i	0.549102	+	0.951073i
$518$	36.1777	−	62.6616i	1.58956	−	2.75319i
$519$		0			0
$520$	−16.8995			−0.741092
$521$	−17.6569			−0.773561			−0.386780	−	0.922172i	$-0.626413\pi$
$521$	−17.6569			−0.773561			−0.386780	+	0.922172i	$0.626413\pi$
$522$		0			0
$523$	−21.8848	−	37.9055i	−0.956954	−	1.65749i	−0.729831	−	0.683628i	$-0.760401\pi$
$523$	−21.8848	−	37.9055i	−0.956954	−	1.65749i	−0.227124	−	0.973866i	$-0.572932\pi$
$524$	40.1421			1.75362
$525$		0			0
$526$	16.6569	+	28.8505i	0.726273	+	1.25794i
$527$	17.0711	−	29.5680i	0.743627	−	1.28800i
$528$		0			0
$529$	−0.156854	+	0.271680i	−0.00681975	+	0.0118122i
$530$	2.41421	−	4.18154i	0.104867	−	0.181635i
$531$		0			0
$532$	51.2990	+	38.0796i	2.22409	+	1.65096i
$533$	10.8284			0.469031
$534$		0			0
$535$	7.00000	−	12.1244i	0.302636	−	0.524182i
$536$	−25.3492	−	43.9062i	−1.09492	−	1.89646i
$537$		0			0
$538$	11.6569	+	20.1903i	0.502563	+	0.870464i
$539$	−21.6569			−0.932827
$540$		0			0
$541$	3.15685	+	5.46783i	0.135724	+	0.235080i	0.925874	−	0.377833i	$-0.123331\pi$
$541$	3.15685	+	5.46783i	0.135724	+	0.235080i	−0.790150	+	0.612914i	$0.789997\pi$
$542$	2.82843	+	4.89898i	0.121491	+	0.210429i
$543$		0			0
$544$	10.8284			0.464265
$545$	−2.65685	−	4.60181i	−0.113807	−	0.197120i
$546$		0			0
$547$	−7.57107	−	13.1135i	−0.323715	−	0.560692i	0.657536	−	0.753423i	$-0.271599\pi$
$547$	−7.57107	−	13.1135i	−0.323715	−	0.560692i	−0.981252	+	0.192731i	$0.938265\pi$
$548$	12.4142	−	21.5020i	0.530309	−	0.918522i
$549$		0			0
$550$	6.82843			0.291165
$551$	−5.79899	−	4.30463i	−0.247045	−	0.183384i
$552$		0			0
$553$	−28.0563	+	48.5950i	−1.19308	+	2.06647i
$554$	7.24264	−	12.5446i	0.307710	−	0.532970i
$555$		0			0
$556$	−12.0858	+	20.9332i	−0.512552	+	0.887765i
$557$	−21.4853	−	37.2136i	−0.910361	−	1.57679i	−0.813555	−	0.581487i	$-0.802471\pi$
$557$	−21.4853	−	37.2136i	−0.910361	−	1.57679i	−0.0968052	−	0.995303i	$-0.530862\pi$
$558$		0			0
$559$	8.31371			0.351632
$560$	5.74264	+	9.94655i	0.242671	+	0.420318i
$561$		0			0
$562$	65.1127			2.74661
$563$	−16.1421			−0.680310			−0.340155	−	0.940369i	$-0.610480\pi$
$563$	−16.1421			−0.680310			−0.340155	+	0.940369i	$0.610480\pi$
$564$		0			0
$565$	2.00000	−	3.46410i	0.0841406	−	0.145736i
$566$	15.6569	+	27.1185i	0.658107	+	1.13987i
$567$		0			0
$568$	−22.0711	+	38.2282i	−0.926081	+	1.60402i
$569$	28.9706			1.21451			0.607255	−	0.794507i	$-0.292271\pi$
$569$	28.9706			1.21451			0.607255	+	0.794507i	$0.292271\pi$
$570$		0			0
$571$	−8.31371			−0.347918			−0.173959	−	0.984753i	$-0.555656\pi$
$571$	−8.31371			−0.347918			−0.173959	+	0.984753i	$0.555656\pi$
$572$	20.7279	−	35.9018i	0.866678	−	1.50113i
$573$		0			0
$574$	−13.0711	−	22.6398i	−0.545576	−	0.944965i
$575$	2.41421	−	4.18154i	0.100680	−	0.174382i
$576$		0			0
$577$	−47.6569			−1.98398			−0.991990	−	0.126313i	$-0.959686\pi$
$577$	−47.6569			−1.98398			−0.991990	+	0.126313i	$0.959686\pi$
$578$	−71.5269			−2.97513
$579$		0			0
$580$	−3.17157	−	5.49333i	−0.131692	−	0.228098i
$581$	−30.6274			−1.27064
$582$		0			0
$583$	2.82843	+	4.89898i	0.117141	+	0.202895i
$584$	−16.5208	+	28.6149i	−0.683636	+	1.18409i
$585$		0			0
$586$	−20.8995	+	36.1990i	−0.863350	+	1.49537i
$587$	−17.3137	+	29.9882i	−0.714613	+	1.23775i	0.248495	+	0.968633i	$0.420064\pi$
$587$	−17.3137	+	29.9882i	−0.714613	+	1.23775i	−0.963109	+	0.269113i	$0.913269\pi$
$588$		0			0
$589$	−20.0000	+	8.66025i	−0.824086	+	0.356840i
$590$		0			0
$591$		0			0
$592$	11.7426	−	20.3389i	0.482620	−	0.835922i
$593$	−19.0711	−	33.0321i	−0.783155	−	1.35646i	−0.930095	−	0.367319i	$-0.880276\pi$
$593$	−19.0711	−	33.0321i	−0.783155	−	1.35646i	0.146940	−	0.989145i	$-0.453057\pi$
$594$		0			0
$595$	13.0711	+	22.6398i	0.535862	+	0.928139i
$596$	26.1421			1.07082
$597$		0			0
$598$	−22.3137	−	38.6485i	−0.912475	−	1.58045i
$599$	8.48528	+	14.6969i	0.346699	+	0.600501i	0.985661	−	0.168738i	$-0.0539691\pi$
$599$	8.48528	+	14.6969i	0.346699	+	0.600501i	−0.638962	+	0.769238i	$0.720636\pi$
$600$		0			0
$601$	−46.6569			−1.90317			−0.951586	−	0.307381i	$-0.900547\pi$
$601$	−46.6569			−1.90317			−0.951586	+	0.307381i	$0.900547\pi$
$602$	−10.0355	−	17.3821i	−0.409018	−	0.708440i
$603$		0			0
$604$	−41.4558	−	71.8036i	−1.68681	−	2.92165i
$605$	1.50000	−	2.59808i	0.0609837	−	0.105627i
$606$		0			0
$607$	−11.8284			−0.480101			−0.240051	−	0.970760i	$-0.577164\pi$
$607$	−11.8284			−0.480101			−0.240051	+	0.970760i	$0.577164\pi$
$608$	−5.55025	−	4.11999i	−0.225092	−	0.167088i
$609$		0			0
$610$	17.2782	−	29.9267i	0.699573	−	1.21170i
$611$	−16.8995	+	29.2708i	−0.683680	+	1.18417i
$612$		0			0
$613$	22.7990	−	39.4890i	0.920843	−	1.59495i	0.122728	−	0.992440i	$-0.460836\pi$
$613$	22.7990	−	39.4890i	0.920843	−	1.59495i	0.798115	−	0.602506i	$-0.205831\pi$
$614$	−7.65685	−	13.2621i	−0.309005	−	0.535213i
$615$		0			0
$616$	−47.7990			−1.92588
$617$	7.17157	+	12.4215i	0.288717	+	0.500072i	0.973504	−	0.228671i	$-0.0734381\pi$
$617$	7.17157	+	12.4215i	0.288717	+	0.500072i	−0.684787	+	0.728743i	$0.740105\pi$
$618$		0			0
$619$	38.3137			1.53996			0.769979	−	0.638069i	$-0.220267\pi$
$619$	38.3137			1.53996			0.769979	+	0.638069i	$0.220267\pi$
$620$	−19.1421			−0.768767
$621$		0			0
$622$	4.82843	−	8.36308i	0.193602	−	0.335329i
$623$	8.58579	+	14.8710i	0.343982	+	0.595795i
$624$		0			0
$625$	−0.500000	+	0.866025i	−0.0200000	+	0.0346410i
$626$	14.4853			0.578948
$627$		0			0
$628$	−54.2548			−2.16500
$629$	26.7279	−	46.2941i	1.06571	−	1.84587i
$630$		0			0
$631$	−21.6421	−	37.4853i	−0.861560	−	1.49227i	−0.870423	−	0.492305i	$-0.836155\pi$
$631$	−21.6421	−	37.4853i	−0.861560	−	1.49227i	0.00886314	−	0.999961i	$-0.497179\pi$
$632$	−32.3492	+	56.0305i	−1.28678	+	2.22877i
$633$		0			0
$634$	−41.7990			−1.66005
$635$	−20.9706			−0.832191
$636$		0			0
$637$	−14.6569	−	25.3864i	−0.580726	−	1.00585i
$638$	11.3137			0.447914
$639$		0			0
$640$	−10.2782	−	17.8023i	−0.406281	−	0.703699i
$641$	−0.0710678	+	0.123093i	−0.00280701	+	0.00486188i	−0.867425	−	0.497567i	$-0.834227\pi$
$641$	−0.0710678	+	0.123093i	−0.00280701	+	0.00486188i	0.864618	+	0.502429i	$0.167560\pi$
$642$		0			0
$643$	−0.0857864	+	0.148586i	−0.00338309	+	0.00585968i	−0.867712	−	0.497067i	$-0.834410\pi$
$643$	−0.0857864	+	0.148586i	−0.00338309	+	0.00585968i	0.864329	+	0.502927i	$0.167744\pi$
$644$	−35.3848	+	61.2882i	−1.39436	+	2.41509i
$645$		0			0
$646$	57.6985	+	42.8300i	2.27012	+	1.68512i
$647$	−24.2843			−0.954713			−0.477357	−	0.878710i	$-0.658405\pi$
$647$	−24.2843			−0.954713			−0.477357	+	0.878710i	$0.658405\pi$
$648$		0			0
$649$		0			0
$650$	4.62132	+	8.00436i	0.181263	+	0.313957i
$651$		0			0
$652$	25.1569	+	43.5729i	0.985218	+	1.70645i
$653$	27.9411			1.09342			0.546710	−	0.837322i	$-0.315880\pi$
$653$	27.9411			1.09342			0.546710	+	0.837322i	$0.315880\pi$
$654$		0			0
$655$	−5.24264	−	9.08052i	−0.204847	−	0.354805i
$656$	−4.24264	−	7.34847i	−0.165647	−	0.286910i
$657$		0			0
$658$	81.5980			3.18102
$659$	12.2426	+	21.2049i	0.476906	+	0.826025i	0.999650	−	0.0264649i	$-0.00842503\pi$
$659$	12.2426	+	21.2049i	0.476906	+	0.826025i	−0.522744	+	0.852490i	$0.675092\pi$
$660$		0			0
$661$	−13.4853	−	23.3572i	−0.524517	−	0.908489i	−0.999593	−	0.0285447i	$-0.990913\pi$
$661$	−13.4853	−	23.3572i	−0.524517	−	0.908489i	0.475076	−	0.879945i	$-0.342421\pi$
$662$	−20.1066	+	34.8257i	−0.781465	+	1.35354i
$663$		0			0
$664$	−35.3137			−1.37044
$665$	1.91421	−	16.5776i	0.0742300	−	0.642851i
$666$		0			0
$667$	4.00000	−	6.92820i	0.154881	−	0.268261i
$668$	17.1716	−	29.7420i	0.664388	−	1.15075i
$669$		0			0
$670$	−13.8640	+	24.0131i	−0.535612	+	0.927706i
$671$	20.2426	+	35.0613i	0.781458	+	1.35353i
$672$		0			0
$673$	−18.1716			−0.700463			−0.350231	−	0.936663i	$-0.613897\pi$
$673$	−18.1716			−0.700463			−0.350231	+	0.936663i	$0.613897\pi$
$674$	0.621320	+	1.07616i	0.0239324	+	0.0414521i
$675$		0			0
$676$	6.34315			0.243967
$677$	−27.4558			−1.05521			−0.527607	−	0.849489i	$-0.676911\pi$
$677$	−27.4558			−1.05521			−0.527607	+	0.849489i	$0.676911\pi$
$678$		0			0
$679$	−11.4853	+	19.8931i	−0.440765	+	0.763427i
$680$	15.0711	+	26.1039i	0.577949	+	1.00104i
$681$		0			0
$682$	17.0711	−	29.5680i	0.653685	−	1.13222i
$683$	−15.8579			−0.606784			−0.303392	−	0.952866i	$-0.598119\pi$
$683$	−15.8579			−0.606784			−0.303392	+	0.952866i	$0.598119\pi$
$684$		0			0
$685$	−6.48528			−0.247790
$686$	−3.03553	+	5.25770i	−0.115897	+	0.200740i
$687$		0			0
$688$	−3.25736	−	5.64191i	−0.124186	−	0.215096i
$689$	−3.82843	+	6.63103i	−0.145851	+	0.252622i
$690$		0			0
$691$	8.68629			0.330442			0.165221	−	0.986257i	$-0.447166\pi$
$691$	8.68629			0.330442			0.165221	+	0.986257i	$0.447166\pi$
$692$	58.6274			2.22868
$693$		0			0
$694$	−18.6569	−	32.3146i	−0.708205	−	1.22665i
$695$	6.31371			0.239493
$696$		0			0
$697$	−9.65685	−	16.7262i	−0.365779	−	0.633549i
$698$	16.4497	−	28.4918i	0.622632	−	1.07843i
$699$		0			0
$700$	7.32843	−	12.6932i	0.276989	−	0.479758i
$701$	1.92893	−	3.34101i	0.0728548	−	0.126188i	−0.827297	−	0.561765i	$-0.810122\pi$
$701$	1.92893	−	3.34101i	0.0728548	−	0.126188i	0.900151	+	0.435577i	$0.143456\pi$
$702$		0			0
$703$	−31.3137	+	13.5592i	−1.18102	+	0.511396i
$704$	27.7990			1.04771
$705$		0			0
$706$	−25.7279	+	44.5621i	−0.968283	+	1.67712i
$707$	30.8995	+	53.5195i	1.16210	+	2.01281i
$708$		0			0
$709$	6.81371	+	11.8017i	0.255894	+	0.443222i	0.965138	−	0.261742i	$-0.0842968\pi$
$709$	6.81371	+	11.8017i	0.255894	+	0.443222i	−0.709244	+	0.704963i	$0.750963\pi$
$710$	24.1421			0.906038
$711$		0			0
$712$	9.89949	+	17.1464i	0.370999	+	0.642590i
$713$	−12.0711	−	20.9077i	−0.452065	−	0.783000i
$714$		0			0
$715$	−10.8284			−0.404960
$716$	−23.2426	−	40.2574i	−0.868618	−	1.50449i
$717$		0			0
$718$	−17.8995	−	31.0028i	−0.668003	−	1.15702i
$719$	−14.4142	+	24.9662i	−0.537559	+	0.931080i	0.461475	+	0.887153i	$0.347320\pi$
$719$	−14.4142	+	24.9662i	−0.537559	+	0.931080i	−0.999035	+	0.0439272i	$0.986013\pi$
$720$		0			0
$721$	−15.9706			−0.594775
$722$	−13.2782	−	43.9062i	−0.494162	−	1.63402i
$723$		0			0
$724$	−5.14214	+	8.90644i	−0.191106	+	0.331005i
$725$	−0.828427	+	1.43488i	−0.0307670	+	0.0532900i
$726$		0			0
$727$	−20.9142	+	36.2245i	−0.775665	+	1.34349i	0.158755	+	0.987318i	$0.449252\pi$
$727$	−20.9142	+	36.2245i	−0.775665	+	1.34349i	−0.934420	+	0.356174i	$0.884081\pi$
$728$	−32.3492	−	56.0305i	−1.19894	−	2.07663i
$729$		0			0
$730$	18.0711			0.668840
$731$	−7.41421	−	12.8418i	−0.274225	−	0.474971i
$732$		0			0
$733$	−7.65685			−0.282812			−0.141406	−	0.989952i	$-0.545162\pi$
$733$	−7.65685			−0.282812			−0.141406	+	0.989952i	$0.545162\pi$
$734$	−67.8701			−2.50513
$735$		0			0
$736$	3.82843	−	6.63103i	0.141118	−	0.244423i
$737$	−16.2426	−	28.1331i	−0.598305	−	1.03630i
$738$		0			0
$739$	−1.81371	+	3.14144i	−0.0667183	+	0.115560i	−0.897455	−	0.441106i	$-0.854586\pi$
$739$	−1.81371	+	3.14144i	−0.0667183	+	0.115560i	0.830737	+	0.556666i	$0.187920\pi$
$740$	−29.9706			−1.10174
$741$		0			0
$742$	18.4853			0.678616
$743$	2.92893	−	5.07306i	0.107452	−	0.186112i	−0.807285	−	0.590161i	$-0.799064\pi$
$743$	2.92893	−	5.07306i	0.107452	−	0.186112i	0.914737	+	0.404049i	$0.132397\pi$
$744$		0			0
$745$	−3.41421	−	5.91359i	−0.125087	−	0.216657i
$746$	−0.414214	+	0.717439i	−0.0151654	+	0.0262673i
$747$		0			0
$748$	−73.9411			−2.70356
$749$	53.5980			1.95843
$750$		0			0
$751$	−1.84315	−	3.19242i	−0.0672573	−	0.116493i	0.830436	−	0.557114i	$-0.188092\pi$
$751$	−1.84315	−	3.19242i	−0.0672573	−	0.116493i	−0.897693	+	0.440621i	$0.854758\pi$
$752$	26.4853			0.965819
$753$		0			0
$754$	7.65685	+	13.2621i	0.278846	+	0.482976i
$755$	−10.8284	+	18.7554i	−0.394087	+	0.682578i
$756$		0			0
$757$	2.42893	−	4.20703i	0.0882810	−	0.152907i	−0.818504	−	0.574501i	$-0.805196\pi$
$757$	2.42893	−	4.20703i	0.0882810	−	0.152907i	0.906785	+	0.421594i	$0.138529\pi$
$758$	29.3492	−	50.8344i	1.06601	−	1.84639i
$759$		0			0
$760$	2.20711	−	19.1141i	0.0800602	−	0.693341i
$761$	27.5980			1.00043			0.500213	−	0.865902i	$-0.333255\pi$
$761$	27.5980			1.00043			0.500213	+	0.865902i	$0.333255\pi$
$762$		0			0
$763$	10.1716	−	17.6177i	0.368236	−	0.637803i
$764$	−21.3848	−	37.0395i	−0.773674	−	1.34004i
$765$		0			0
$766$	−40.9706	−	70.9631i	−1.48033	−	2.56400i
$767$		0			0
$768$		0			0
$769$	4.15685	+	7.19988i	0.149900	+	0.259634i	0.931190	−	0.364533i	$-0.118771\pi$
$769$	4.15685	+	7.19988i	0.149900	+	0.259634i	−0.781290	+	0.624168i	$0.785438\pi$
$770$	13.0711	+	22.6398i	0.471049	+	0.815880i
$771$		0			0
$772$	56.6569			2.03912
$773$	−11.0000	−	19.0526i	−0.395643	−	0.685273i	0.597540	−	0.801839i	$-0.296145\pi$
$773$	−11.0000	−	19.0526i	−0.395643	−	0.685273i	−0.993183	+	0.116566i	$0.962811\pi$
$774$		0			0
$775$	2.50000	+	4.33013i	0.0898027	+	0.155543i
$776$	−13.2426	+	22.9369i	−0.475383	+	0.823388i
$777$		0			0
$778$	85.5980			3.06884
$779$	−1.41421	+	12.2474i	−0.0506695	+	0.438810i
$780$		0			0
$781$	−14.1421	+	24.4949i	−0.506045	+	0.876496i
$782$	−39.7990	+	68.9339i	−1.42321	+	2.46507i
$783$		0			0
$784$	−11.4853	+	19.8931i	−0.410189	+	0.710468i
$785$	7.08579	+	12.2729i	0.252903	+	0.438040i
$786$		0			0
$787$	11.4853			0.409406			0.204703	−	0.978824i	$-0.434377\pi$
$787$	11.4853			0.409406			0.204703	+	0.978824i	$0.434377\pi$
$788$	12.1421	+	21.0308i	0.432546	+	0.749191i
$789$		0			0
$790$	35.3848			1.25893
$791$	15.3137			0.544493
$792$		0			0
$793$	−27.3995	+	47.4573i	−0.972985	+	1.68526i
$794$	−21.9350	−	37.9926i	−0.778445	−	1.34831i
$795$		0			0
$796$	32.5416	−	56.3638i	1.15341	−	1.99776i
$797$	−9.11270			−0.322788			−0.161394	−	0.986890i	$-0.551599\pi$
$797$	−9.11270			−0.322788			−0.161394	+	0.986890i	$0.551599\pi$
$798$		0			0
$799$	60.2843			2.13270
$800$	−0.792893	+	1.37333i	−0.0280330	+	0.0485546i
$801$		0			0
$802$	−27.3137	−	47.3087i	−0.964481	−	1.67053i
$803$	−10.5858	+	18.3351i	−0.373564	+	0.647032i
$804$		0			0
$805$	18.4853			0.651521
$806$	46.2132			1.62779
$807$		0			0
$808$	35.6274	+	61.7085i	1.25337	+	2.17090i
$809$	−48.6274			−1.70965			−0.854824	−	0.518917i	$-0.826335\pi$
$809$	−48.6274			−1.70965			−0.854824	+	0.518917i	$0.826335\pi$
$810$		0			0
$811$	7.31371	+	12.6677i	0.256819	+	0.444824i	0.965388	−	0.260818i	$-0.0839922\pi$
$811$	7.31371	+	12.6677i	0.256819	+	0.444824i	−0.708569	+	0.705642i	$0.750659\pi$
$812$	12.1421	−	21.0308i	0.426105	−	0.738036i
$813$		0			0
$814$	26.7279	−	46.2941i	0.936813	−	1.62261i
$815$	6.57107	−	11.3814i	0.230174	−	0.398674i
$816$		0			0
$817$	−1.08579	+	9.40319i	−0.0379869	+	0.328976i
$818$	−14.4853			−0.506466
$819$		0			0
$820$	−5.41421	+	9.37769i	−0.189073	+	0.327483i
$821$	−26.0000	−	45.0333i	−0.907406	−	1.57167i	−0.817654	−	0.575710i	$-0.804726\pi$
$821$	−26.0000	−	45.0333i	−0.907406	−	1.57167i	−0.0897520	−	0.995964i	$-0.528607\pi$
$822$		0			0
$823$	20.9706	+	36.3221i	0.730988	+	1.26611i	0.956461	+	0.291859i	$0.0942737\pi$
$823$	20.9706	+	36.3221i	0.730988	+	1.26611i	−0.225474	+	0.974249i	$0.572393\pi$
$824$	−18.4142			−0.641490
$825$		0			0
$826$		0			0
$827$	9.65685	+	16.7262i	0.335802	+	0.581626i	0.983639	−	0.180154i	$-0.0576595\pi$
$827$	9.65685	+	16.7262i	0.335802	+	0.581626i	−0.647837	+	0.761779i	$0.724326\pi$
$828$		0			0
$829$	−55.9706			−1.94394			−0.971969	−	0.235109i	$-0.924455\pi$
$829$	−55.9706			−1.94394			−0.971969	+	0.235109i	$0.924455\pi$
$830$	9.65685	+	16.7262i	0.335194	+	0.580574i
$831$		0			0
$832$	18.8137	+	32.5863i	0.652248	+	1.12973i
$833$	−26.1421	+	45.2795i	−0.905771	+	1.56884i
$834$		0			0
$835$	−8.97056			−0.310439
$836$	37.8995	+	28.1331i	1.31078	+	0.973003i
$837$		0			0
$838$	23.3137	−	40.3805i	0.805359	−	1.39492i
$839$	16.8995	−	29.2708i	0.583435	−	1.01054i	−0.411633	−	0.911350i	$-0.635041\pi$
$839$	16.8995	−	29.2708i	0.583435	−	1.01054i	0.995069	−	0.0991901i	$-0.0316252\pi$
$840$		0			0
$841$	13.1274	−	22.7374i	0.452670	−	0.784047i
$842$	4.41421	+	7.64564i	0.152124	+	0.263486i
$843$		0			0
$844$	85.4264			2.94050
$845$	−0.828427	−	1.43488i	−0.0284988	−	0.0493613i
$846$		0			0
$847$	11.4853			0.394639
$848$	6.00000			0.206041
$849$		0			0
$850$	8.24264	−	14.2767i	0.282720	−	0.489686i
$851$	−18.8995	−	32.7349i	−0.647866	−	1.12214i
$852$		0			0
$853$	23.2279	−	40.2319i	0.795309	−	1.37752i	−0.127334	−	0.991860i	$-0.540642\pi$
$853$	23.2279	−	40.2319i	0.795309	−	1.37752i	0.922643	−	0.385656i	$-0.126025\pi$
$854$	132.296			4.52709
$855$		0			0
$856$	61.7990			2.11225
$857$	12.7574	−	22.0964i	0.435783	−	0.754798i	−0.561576	−	0.827425i	$-0.689805\pi$
$857$	12.7574	−	22.0964i	0.435783	−	0.754798i	0.997359	+	0.0726266i	$0.0231381\pi$
$858$		0			0
$859$	16.4706	+	28.5279i	0.561968	+	0.973358i	0.997325	+	0.0730995i	$0.0232891\pi$
$859$	16.4706	+	28.5279i	0.561968	+	0.973358i	−0.435356	+	0.900258i	$0.643378\pi$
$860$	−4.15685	+	7.19988i	−0.141748	+	0.245514i
$861$		0			0
$862$	34.9706			1.19110
$863$	2.68629			0.0914424			0.0457212	−	0.998954i	$-0.485441\pi$
$863$	2.68629			0.0914424			0.0457212	+	0.998954i	$0.485441\pi$
$864$		0			0
$865$	−7.65685	−	13.2621i	−0.260341	−	0.450924i
$866$	82.4975			2.80338
$867$		0			0
$868$	−36.6421	−	63.4660i	−1.24372	−	2.15418i
$869$	−20.7279	+	35.9018i	−0.703147	+	1.21789i
$870$		0			0
$871$	21.9853	−	38.0796i	0.744943	−	1.29028i
$872$	11.7279	−	20.3134i	0.397158	−	0.687897i
$873$		0			0
$874$	46.6274	−	20.1903i	1.57720	−	0.682946i
$875$	−3.82843			−0.129424
$876$		0			0
$877$	2.60051	−	4.50421i	0.0878128	−	0.152096i	−0.818774	−	0.574117i	$-0.805346\pi$
$877$	2.60051	−	4.50421i	0.0878128	−	0.152096i	0.906586	+	0.422020i	$0.138679\pi$
$878$	20.5208	+	35.5431i	0.692544	+	1.19952i
$879$		0			0
$880$	4.24264	+	7.34847i	0.143019	+	0.247717i
$881$	−9.02944			−0.304209			−0.152105	−	0.988364i	$-0.548605\pi$
$881$	−9.02944			−0.304209			−0.152105	+	0.988364i	$0.548605\pi$
$882$		0			0
$883$	2.42893	+	4.20703i	0.0817401	+	0.141578i	0.903997	−	0.427538i	$-0.140619\pi$
$883$	2.42893	+	4.20703i	0.0817401	+	0.141578i	−0.822257	+	0.569116i	$0.807286\pi$
$884$	−50.0416	−	86.6746i	−1.68308	−	2.91518i
$885$		0			0
$886$	−24.4853			−0.822599
$887$	−10.7279	−	18.5813i	−0.360208	−	0.623899i	0.627787	−	0.778386i	$-0.283961\pi$
$887$	−10.7279	−	18.5813i	−0.360208	−	0.623899i	−0.987995	+	0.154486i	$0.950628\pi$
$888$		0			0
$889$	−40.1421	−	69.5282i	−1.34632	−	2.33190i
$890$	5.41421	−	9.37769i	0.181485	−	0.314341i
$891$		0			0
$892$	0.656854			0.0219931
$893$	−30.8995	−	22.9369i	−1.03401	−	0.767555i
$894$		0			0
$895$	−6.07107	+	10.5154i	−0.202933	+	0.351491i
$896$	39.3492	−	68.1549i	1.31457	−	2.27689i
$897$		0			0
$898$	−22.7279	+	39.3659i	−0.758441	+	1.31366i
$899$	4.14214	+	7.17439i	0.138148	+	0.239279i
$900$		0			0
$901$	13.6569			0.454976
$902$	−9.65685	−	16.7262i	−0.321538	−	0.556920i
$903$		0			0
$904$	17.6569			0.587258
$905$	2.68629			0.0892954
$906$		0			0
$907$	20.8284	−	36.0759i	0.691597	−	1.19788i	−0.279718	−	0.960082i	$-0.590241\pi$
$907$	20.8284	−	36.0759i	0.691597	−	1.19788i	0.971315	−	0.237798i	$-0.0764257\pi$
$908$	28.6569	+	49.6351i	0.951011	+	1.64720i
$909$		0			0
$910$	−17.6924	+	30.6441i	−0.586497	+	1.01584i
$911$	−9.85786			−0.326606			−0.163303	−	0.986576i	$-0.552215\pi$
$911$	−9.85786			−0.326606			−0.163303	+	0.986576i	$0.552215\pi$
$912$		0			0
$913$	−22.6274			−0.748858
$914$	−11.8640	+	20.5490i	−0.392425	+	0.679700i
$915$		0			0
$916$	−12.7426	−	22.0709i	−0.421029	−	0.729243i
$917$	20.0711	−	34.7641i	0.662805	−	1.14801i
$918$		0			0
$919$	−3.00000			−0.0989609			−0.0494804	−	0.998775i	$-0.515757\pi$
$919$	−3.00000			−0.0989609			−0.0494804	+	0.998775i	$0.515757\pi$
$920$	21.3137			0.702692
$921$		0			0
$922$	−2.24264	−	3.88437i	−0.0738574	−	0.127925i
$923$	−38.2843			−1.26014
$924$		0			0
$925$	3.91421	+	6.77962i	0.128699	+	0.222912i
$926$	20.2782	−	35.1228i	0.666382	−	1.15421i
$927$		0			0
$928$	−1.31371	+	2.27541i	−0.0431246	+	0.0746940i
$929$	−2.75736	+	4.77589i	−0.0904660	+	0.156692i	−0.907707	−	0.419604i	$-0.862169\pi$
$929$	−2.75736	+	4.77589i	−0.0904660	+	0.156692i	0.817241	+	0.576296i	$0.195502\pi$
$930$		0			0
$931$	30.6274	−	13.2621i	1.00377	−	0.434646i
$932$	10.2843			0.336873
$933$		0			0
$934$	19.0711	−	33.0321i	0.624024	−	1.08084i
$935$	9.65685	+	16.7262i	0.315813	+	0.547004i
$936$		0			0
$937$	15.0858	+	26.1293i	0.492831	+	0.853609i	0.999966	−	0.00825817i	$-0.00262869\pi$
$937$	15.0858	+	26.1293i	0.492831	+	0.853609i	−0.507135	+	0.861867i	$0.669295\pi$
$938$	−106.154			−3.46606
$939$		0			0
$940$	−16.8995	−	29.2708i	−0.551201	−	0.954708i
$941$	−2.58579	−	4.47871i	−0.0842942	−	0.146002i	0.820796	−	0.571221i	$-0.193530\pi$
$941$	−2.58579	−	4.47871i	−0.0842942	−	0.146002i	−0.905090	+	0.425220i	$0.860197\pi$
$942$		0			0
$943$	−13.6569			−0.444728
$944$		0			0
$945$		0			0
$946$	−7.41421	−	12.8418i	−0.241057	−	0.417523i
$947$	2.55635	−	4.42773i	0.0830702	−	0.143882i	−0.821497	−	0.570213i	$-0.806861\pi$
$947$	2.55635	−	4.42773i	0.0830702	−	0.143882i	0.904567	+	0.426331i	$0.140194\pi$
$948$		0			0
$949$	−28.6569			−0.930241
$950$	−9.65685	+	4.18154i	−0.313310	+	0.135667i
$951$		0			0
$952$	−57.6985	+	99.9367i	−1.87002	+	3.23897i
$953$	−23.7990	+	41.2211i	−0.770925	+	1.33528i	0.166132	+	0.986104i	$0.446872\pi$
$953$	−23.7990	+	41.2211i	−0.770925	+	1.33528i	−0.937057	+	0.349178i	$0.886461\pi$
$954$		0			0
$955$	−5.58579	+	9.67487i	−0.180752	+	0.313071i
$956$	39.4853	+	68.3905i	1.27705	+	2.21191i
$957$		0			0
$958$	−34.9706			−1.12985
$959$	−12.4142	−	21.5020i	−0.400876	−	0.694337i
$960$		0			0
$961$	−6.00000			−0.193548
$962$	72.3553			2.33283
$963$		0			0
$964$	−9.57107	+	16.5776i	−0.308263	+	0.533928i
$965$	−7.39949	−	12.8163i	−0.238198	−	0.412571i
$966$		0			0
$967$	11.3995	−	19.7445i	0.366583	−	0.634940i	−0.622446	−	0.782663i	$-0.713861\pi$
$967$	11.3995	−	19.7445i	0.366583	−	0.634940i	0.989029	+	0.147722i	$0.0471942\pi$
$968$	13.2426			0.425635
$969$		0			0
$970$	14.4853			0.465094
$971$	−7.58579	+	13.1390i	−0.243439	+	0.421650i	−0.961692	−	0.274133i	$-0.911609\pi$
$971$	−7.58579	+	13.1390i	−0.243439	+	0.421650i	0.718252	+	0.695783i	$0.244942\pi$
$972$		0			0
$973$	12.0858	+	20.9332i	0.387453	+	0.671087i
$974$	−21.3137	+	36.9164i	−0.682935	+	1.18288i
$975$		0			0
$976$	42.9411			1.37451
$977$	−22.2843			−0.712937			−0.356468	−	0.934307i	$-0.616019\pi$
$977$	−22.2843			−0.712937			−0.356468	+	0.934307i	$0.616019\pi$
$978$		0			0
$979$	6.34315	+	10.9867i	0.202728	+	0.351135i
$980$	29.3137			0.936392
$981$		0			0
$982$	−14.8995	−	25.8067i	−0.475462	−	0.823524i
$983$	−1.51472	+	2.62357i	−0.0483120	+	0.0836789i	−0.889170	−	0.457577i	$-0.848718\pi$
$983$	−1.51472	+	2.62357i	−0.0483120	+	0.0836789i	0.840858	+	0.541256i	$0.182051\pi$
$984$		0			0
$985$	3.17157	−	5.49333i	0.101055	−	0.175032i
$986$	13.6569	−	23.6544i	0.434923	−	0.753308i
$987$		0			0
$988$	−7.32843	+	63.4660i	−0.233148	+	2.01912i
$989$	−10.4853			−0.333413
$990$		0			0
$991$	14.6716	−	25.4119i	0.466058	−	0.807236i	−0.533191	−	0.845995i	$-0.679007\pi$
$991$	14.6716	−	25.4119i	0.466058	−	0.807236i	0.999249	+	0.0387592i	$0.0123405\pi$
$992$	3.96447	+	6.86666i	0.125872	+	0.218017i
$993$		0			0
$994$	46.2132	+	80.0436i	1.46579	+	2.53883i
$995$	−17.0000			−0.538936
$996$		0			0
$997$	6.25736	+	10.8381i	0.198173	+	0.343245i	0.947936	−	0.318461i	$-0.103166\pi$
$997$	6.25736	+	10.8381i	0.198173	+	0.343245i	−0.749763	+	0.661706i	$0.769833\pi$
$998$	−22.5208	−	39.0072i	−0.712884	−	1.23475i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	855.2.k.f.676.2		4
3.2	odd	2		285.2.i.d.106.1	✓	4
19.7	even	3	inner	855.2.k.f.406.2		4
57.8	even	6		5415.2.a.o.1.1		2
57.11	odd	6		5415.2.a.u.1.2		2
57.26	odd	6		285.2.i.d.121.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
285.2.i.d.106.1	✓	4	3.2	odd	2
285.2.i.d.121.1	yes	4	57.26	odd	6
855.2.k.f.406.2		4	19.7	even	3	inner
855.2.k.f.676.2		4	1.1	even	1	trivial
5415.2.a.o.1.1		2	57.8	even	6
5415.2.a.u.1.2		2	57.11	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 855 = 3^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	855.k (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.20711 - 2.09077i) q^{2} +(-1.91421 - 3.31552i) q^{4} +(-0.500000 + 0.866025i) q^{5} -3.82843 q^{7} -4.41421 q^{8} +(1.20711 + 2.09077i) q^{10} -2.82843 q^{11} +(-1.91421 - 3.31552i) q^{13} +(-4.62132 + 8.00436i) q^{14} +(-1.50000 + 2.59808i) q^{16} +(-3.41421 + 5.91359i) q^{17} +(4.00000 - 1.73205i) q^{19} +3.82843 q^{20} +(-3.41421 + 5.91359i) q^{22} +(2.41421 + 4.18154i) q^{23} +(-0.500000 - 0.866025i) q^{25} -9.24264 q^{26} +(7.32843 + 12.6932i) q^{28} +(-0.828427 - 1.43488i) q^{29} -5.00000 q^{31} +(-0.792893 - 1.37333i) q^{32} +(8.24264 + 14.2767i) q^{34} +(1.91421 - 3.31552i) q^{35} -7.82843 q^{37} +(1.20711 - 10.4539i) q^{38} +(2.20711 - 3.82282i) q^{40} +(-1.41421 + 2.44949i) q^{41} +(-1.08579 + 1.88064i) q^{43} +(5.41421 + 9.37769i) q^{44} +11.6569 q^{46} +(-4.41421 - 7.64564i) q^{47} +7.65685 q^{49} -2.41421 q^{50} +(-7.32843 + 12.6932i) q^{52} +(-1.00000 - 1.73205i) q^{53} +(1.41421 - 2.44949i) q^{55} +16.8995 q^{56} -4.00000 q^{58} +(-7.15685 - 12.3960i) q^{61} +(-6.03553 + 10.4539i) q^{62} -9.82843 q^{64} +3.82843 q^{65} +(5.74264 + 9.94655i) q^{67} +26.1421 q^{68} +(-4.62132 - 8.00436i) q^{70} +(5.00000 - 8.66025i) q^{71} +(3.74264 - 6.48244i) q^{73} +(-9.44975 + 16.3674i) q^{74} +(-13.3995 - 9.94655i) q^{76} +10.8284 q^{77} +(7.32843 - 12.6932i) q^{79} +(-1.50000 - 2.59808i) q^{80} +(3.41421 + 5.91359i) q^{82} +8.00000 q^{83} +(-3.41421 - 5.91359i) q^{85} +(2.62132 + 4.54026i) q^{86} +12.4853 q^{88} +(-2.24264 - 3.88437i) q^{89} +(7.32843 + 12.6932i) q^{91} +(9.24264 - 16.0087i) q^{92} -21.3137 q^{94} +(-0.500000 + 4.33013i) q^{95} +(3.00000 - 5.19615i) q^{97} +(9.24264 - 16.0087i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} - 2 q^{4} - 2 q^{5} - 4 q^{7} - 12 q^{8} + 2 q^{10} - 2 q^{13} - 10 q^{14} - 6 q^{16} - 8 q^{17} + 16 q^{19} + 4 q^{20} - 8 q^{22} + 4 q^{23} - 2 q^{25} - 20 q^{26} + 18 q^{28} + 8 q^{29}+ \cdots + 20 q^{98}+O(q^{100}) \) 4 * q + 2 * q^2 - 2 * q^4 - 2 * q^5 - 4 * q^7 - 12 * q^8 + 2 * q^10 - 2 * q^13 - 10 * q^14 - 6 * q^16 - 8 * q^17 + 16 * q^19 + 4 * q^20 - 8 * q^22 + 4 * q^23 - 2 * q^25 - 20 * q^26 + 18 * q^28 + 8 * q^29 - 20 * q^31 - 6 * q^32 + 16 * q^34 + 2 * q^35 - 20 * q^37 + 2 * q^38 + 6 * q^40 - 10 * q^43 + 16 * q^44 + 24 * q^46 - 12 * q^47 + 8 * q^49 - 4 * q^50 - 18 * q^52 - 4 * q^53 + 28 * q^56 - 16 * q^58 - 6 * q^61 - 10 * q^62 - 28 * q^64 + 4 * q^65 + 6 * q^67 + 48 * q^68 - 10 * q^70 + 20 * q^71 - 2 * q^73 - 18 * q^74 - 14 * q^76 + 32 * q^77 + 18 * q^79 - 6 * q^80 + 8 * q^82 + 32 * q^83 - 8 * q^85 + 2 * q^86 + 16 * q^88 + 8 * q^89 + 18 * q^91 + 20 * q^92 - 40 * q^94 - 2 * q^95 + 12 * q^97 + 20 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more