LMFDB - Embedded newform 468.2.bz.a.461.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [468,2,Mod(41,468)] 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("468.41"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(468, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([0, 10, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 468 = 2^{2} \cdot 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	468.bz (of order $12$, degree $4$, minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$3.73699881460$
Analytic rank:	$0$
Dimension:	$56$
Relative dimension:	$14$ over $\Q(\zeta_{12})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			461.7
Character	$\chi$	$=$	468.461
Dual form			468.2.bz.a.401.7

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(0.0566324 - 1.73112i) q^{3} +(-1.01961 + 3.80525i) q^{5} +(-1.33162 - 1.33162i) q^{7} +(-2.99359 - 0.196076i) q^{9} +(4.35045 + 1.16570i) q^{11} +(3.41964 + 1.14283i) q^{13} +(6.52962 + 1.98058i) q^{15} +(2.31976 + 4.01793i) q^{17} +(7.37291 + 1.97557i) q^{19} +(-2.38061 + 2.22979i) q^{21} -0.999222 q^{23} +(-9.11020 - 5.25978i) q^{25} +(-0.508965 + 5.17117i) q^{27} +(0.0144776 - 0.00835862i) q^{29} +(-3.02992 - 0.811864i) q^{31} +(2.26435 - 7.46516i) q^{33} +(6.42489 - 3.70941i) q^{35} +(-5.97798 + 1.60179i) q^{37} +(2.17204 - 5.85510i) q^{39} +(8.54895 + 8.54895i) q^{41} +2.30282i q^{43} +(3.79842 - 11.1914i) q^{45} +(-0.0107778 - 0.0402231i) q^{47} -3.45357i q^{49} +(7.08692 - 3.78824i) q^{51} -1.73858i q^{53} +(-8.87157 + 15.3660i) q^{55} +(3.83750 - 12.6515i) q^{57} +(-2.25516 - 8.41636i) q^{59} -5.22569 q^{61} +(3.72522 + 4.24742i) q^{63} +(-7.83547 + 11.8473i) q^{65} +(-0.0524121 + 0.0524121i) q^{67} +(-0.0565884 + 1.72978i) q^{69} +(1.28159 - 4.78296i) q^{71} +(-0.0292789 - 0.0292789i) q^{73} +(-9.62127 + 15.4730i) q^{75} +(-4.24088 - 7.34542i) q^{77} +(-1.17774 + 2.03990i) q^{79} +(8.92311 + 1.17394i) q^{81} +(-4.57822 + 1.22673i) q^{83} +(-17.6545 + 4.73051i) q^{85} +(-0.0136499 - 0.0255358i) q^{87} +(-3.20395 - 11.9573i) q^{89} +(-3.03185 - 6.07548i) q^{91} +(-1.57703 + 5.19918i) q^{93} +(-15.0350 + 26.0415i) q^{95} +(6.01956 - 6.01956i) q^{97} +(-12.7949 - 4.34264i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 56 q + 4 q^{7} + 12 q^{11} + 12 q^{15} + 4 q^{19} - 12 q^{21} - 4 q^{31} + 18 q^{33} + 66 q^{35} + 2 q^{37} + 24 q^{39} + 24 q^{41} + 24 q^{45} - 36 q^{47} - 12 q^{57} + 6 q^{63} - 36 q^{65} - 14 q^{67}+ \cdots - 48 q^{99}+O(q^{100}) $ 56 * q + 4 * q^7 + 12 * q^11 + 12 * q^15 + 4 * q^19 - 12 * q^21 - 4 * q^31 + 18 * q^33 + 66 * q^35 + 2 * q^37 + 24 * q^39 + 24 * q^41 + 24 * q^45 - 36 * q^47 - 12 * q^57 + 6 * q^63 - 36 * q^65 - 14 * q^67 - 42 * q^69 - 24 * q^71 - 14 * q^73 - 90 * q^75 + 24 * q^77 + 12 * q^79 - 12 * q^81 - 42 * q^83 - 36 * q^85 - 48 * q^87 + 4 * q^91 + 36 * q^93 - 46 * q^97 - 48 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/468\mathbb{Z}\right)^\times$.

$n$	$145$	$209$	$235$
$\chi(n)$	$e\left(\frac{5}{12}\right)$	$e\left(\frac{1}{6}\right)$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	0.0566324	−	1.73112i	0.0326967	−	0.999465i
$3$	0.0566324	−	1.73112i	0.0326967	−	0.999465i
$4$		0			0
$5$	−1.01961	+	3.80525i	−0.455985	+	1.70176i	0.229189	+	0.973382i	$0.426392\pi$
$5$	−1.01961	+	3.80525i	−0.455985	+	1.70176i	−0.685175	+	0.728379i	$0.740274\pi$
$6$		0			0
$7$	−1.33162	−	1.33162i	−0.503305	−	0.503305i	0.409158	−	0.912463i	$-0.365822\pi$
$7$	−1.33162	−	1.33162i	−0.503305	−	0.503305i	−0.912463	+	0.409158i	$0.865822\pi$
$8$		0			0
$9$	−2.99359	−	0.196076i	−0.997862	−	0.0653585i
$10$		0			0
$11$	4.35045	+	1.16570i	1.31171	+	0.351472i	0.845866	−	0.533395i	$-0.179084\pi$
$11$	4.35045	+	1.16570i	1.31171	+	0.351472i	0.465844	+	0.884867i	$0.345751\pi$
$12$		0			0
$13$	3.41964	+	1.14283i	0.948438	+	0.316964i
$13$	3.41964	+	1.14283i	0.948438	+	0.316964i
$14$		0			0
$15$	6.52962	+	1.98058i	1.68594	+	0.511384i
$16$		0			0
$17$	2.31976	+	4.01793i	0.562623	+	0.974492i	0.997266	+	0.0738894i	$0.0235412\pi$
$17$	2.31976	+	4.01793i	0.562623	+	0.974492i	−0.434643	+	0.900603i	$0.643125\pi$
$18$		0			0
$19$	7.37291	+	1.97557i	1.69146	+	0.453226i	0.970766	−	0.240026i	$-0.0771561\pi$
$19$	7.37291	+	1.97557i	1.69146	+	0.453226i	0.720695	+	0.693252i	$0.243823\pi$
$20$		0			0
$21$	−2.38061	+	2.22979i	−0.519493	+	0.486580i
$22$		0			0
$23$	−0.999222			−0.208352			−0.104176	−	0.994559i	$-0.533221\pi$
$23$	−0.999222			−0.208352			−0.104176	+	0.994559i	$0.533221\pi$
$24$		0			0
$25$	−9.11020	−	5.25978i	−1.82204	−	1.05196i
$26$		0			0
$27$	−0.508965	+	5.17117i	−0.0979504	+	0.995191i
$28$		0			0
$29$	0.0144776	−	0.00835862i	0.00268841	−	0.00155216i	−0.498655	−	0.866800i	$-0.666173\pi$
$29$	0.0144776	−	0.00835862i	0.00268841	−	0.00155216i	0.501344	+	0.865248i	$0.332839\pi$
$30$		0			0
$31$	−3.02992	−	0.811864i	−0.544189	−	0.145815i	−0.0237564	−	0.999718i	$-0.507563\pi$
$31$	−3.02992	−	0.811864i	−0.544189	−	0.145815i	−0.520433	+	0.853903i	$0.674229\pi$
$32$		0			0
$33$	2.26435	−	7.46516i	0.394173	−	1.29952i
$34$		0			0
$35$	6.42489	−	3.70941i	1.08601	−	0.627005i
$36$		0			0
$37$	−5.97798	+	1.60179i	−0.982774	+	0.263333i	−0.714212	−	0.699929i	$-0.753215\pi$
$37$	−5.97798	+	1.60179i	−0.982774	+	0.263333i	−0.268561	+	0.963263i	$0.586548\pi$
$38$		0			0
$39$	2.17204	−	5.85510i	0.347805	−	0.937567i
$40$		0			0
$41$	8.54895	+	8.54895i	1.33512	+	1.33512i	0.900719	+	0.434403i	$0.143040\pi$
$41$	8.54895	+	8.54895i	1.33512	+	1.33512i	0.434403	+	0.900719i	$0.356960\pi$
$42$		0			0
$43$			2.30282i			0.351177i	0.984464	+	0.175588i	$0.0561828\pi$
$43$			2.30282i			0.351177i	−0.984464	+	0.175588i	$0.943817\pi$
$44$		0			0
$45$	3.79842	−	11.1914i	0.566235	−	1.66832i
$46$		0			0
$47$	−0.0107778	−	0.0402231i	−0.00157210	−	0.00586715i	0.965135	−	0.261751i	$-0.0843001\pi$
$47$	−0.0107778	−	0.0402231i	−0.00157210	−	0.00586715i	−0.966707	+	0.255884i	$0.917633\pi$
$48$		0			0
$49$		−	3.45357i		−	0.493367i
$50$		0			0
$51$	7.08692	−	3.78824i	0.992367	−	0.530460i
$52$		0			0
$53$		−	1.73858i		−	0.238812i	−0.992845	−	0.119406i	$-0.961901\pi$
$53$		−	1.73858i		−	0.238812i	0.992845	−	0.119406i	$-0.0380991\pi$
$54$		0			0
$55$	−8.87157	+	15.3660i	−1.19624	+	2.07195i
$56$		0			0
$57$	3.83750	−	12.6515i	0.508289	−	1.67574i
$58$		0			0
$59$	−2.25516	−	8.41636i	−0.293596	−	1.09572i	−0.942326	−	0.334697i	$-0.891366\pi$
$59$	−2.25516	−	8.41636i	−0.293596	−	1.09572i	0.648729	−	0.761019i	$-0.275301\pi$
$60$		0			0
$61$	−5.22569			−0.669082			−0.334541	−	0.942381i	$-0.608581\pi$
$61$	−5.22569			−0.669082			−0.334541	+	0.942381i	$0.608581\pi$
$62$		0			0
$63$	3.72522	+	4.24742i	0.469334	+	0.535125i
$64$		0			0
$65$	−7.83547	+	11.8473i	−0.971870	+	1.46948i
$66$		0			0
$67$	−0.0524121	+	0.0524121i	−0.00640316	+	0.00640316i	−0.710301	−	0.703898i	$-0.751441\pi$
$67$	−0.0524121	+	0.0524121i	−0.00640316	+	0.00640316i	0.703898	+	0.710301i	$0.251441\pi$
$68$		0			0
$69$	−0.0565884	+	1.72978i	−0.00681244	+	0.208241i
$70$		0			0
$71$	1.28159	−	4.78296i	0.152097	−	0.567633i	−0.847240	−	0.531211i	$-0.821737\pi$
$71$	1.28159	−	4.78296i	0.152097	−	0.567633i	0.999336	−	0.0364221i	$-0.0115961\pi$
$72$		0			0
$73$	−0.0292789	−	0.0292789i	−0.00342684	−	0.00342684i	0.705391	−	0.708818i	$-0.250771\pi$
$73$	−0.0292789	−	0.0292789i	−0.00342684	−	0.00342684i	−0.708818	+	0.705391i	$0.750771\pi$
$74$		0			0
$75$	−9.62127	+	15.4730i	−1.11097	+	1.78667i
$76$		0			0
$77$	−4.24088	−	7.34542i	−0.483293	−	0.837089i
$78$		0			0
$79$	−1.17774	+	2.03990i	−0.132506	+	0.229507i	−0.924642	−	0.380837i	$-0.875636\pi$
$79$	−1.17774	+	2.03990i	−0.132506	+	0.229507i	0.792136	+	0.610345i	$0.208969\pi$
$80$		0			0
$81$	8.92311	+	1.17394i	0.991457	+	0.130438i
$82$		0			0
$83$	−4.57822	+	1.22673i	−0.502525	+	0.134651i	−0.501172	−	0.865348i	$-0.667097\pi$
$83$	−4.57822	+	1.22673i	−0.502525	+	0.134651i	−0.00135341	+	0.999999i	$0.500431\pi$
$84$		0			0
$85$	−17.6545	+	4.73051i	−1.91490	+	0.513096i
$86$		0			0
$87$	−0.0136499	−	0.0255358i	−0.00146342	−	0.00273773i
$88$		0			0
$89$	−3.20395	−	11.9573i	−0.339618	−	1.26747i	−0.898776	−	0.438409i	$-0.855542\pi$
$89$	−3.20395	−	11.9573i	−0.339618	−	1.26747i	0.559158	−	0.829061i	$-0.311125\pi$
$90$		0			0
$91$	−3.03185	−	6.07548i	−0.317824	−	0.636883i
$92$		0			0
$93$	−1.57703	+	5.19918i	−0.163530	+	0.539130i
$94$		0			0
$95$	−15.0350	+	26.0415i	−1.54256	+	2.67180i
$96$		0			0
$97$	6.01956	−	6.01956i	0.611194	−	0.611194i	−0.332063	−	0.943257i	$-0.607745\pi$
$97$	6.01956	−	6.01956i	0.611194	−	0.611194i	0.943257	+	0.332063i	$0.107745\pi$
$98$		0			0
$99$	−12.7949	−	4.34264i	−1.28593	−	0.436452i
$100$		0			0
$101$	6.89315	+	11.9393i	0.685894	+	1.18800i	0.973155	+	0.230151i	$0.0739218\pi$
$101$	6.89315	+	11.9393i	0.685894	+	1.18800i	−0.287261	+	0.957852i	$0.592745\pi$
$102$		0			0
$103$	−14.2999	+	8.25605i	−1.40901	+	0.813493i	−0.995293	−	0.0969123i	$-0.969103\pi$
$103$	−14.2999	+	8.25605i	−1.40901	+	0.813493i	−0.413718	+	0.910405i	$0.635770\pi$
$104$		0			0
$105$	−6.05760	−	11.3324i	−0.591161	−	1.10593i
$106$		0			0
$107$	8.79512	+	5.07786i	0.850256	+	0.490896i	0.860737	−	0.509049i	$-0.170003\pi$
$107$	8.79512	+	5.07786i	0.850256	+	0.490896i	−0.0104810	+	0.999945i	$0.503336\pi$
$108$		0			0
$109$	11.2739	−	11.2739i	1.07985	−	1.07985i	0.0833248	−	0.996522i	$-0.473446\pi$
$109$	11.2739	−	11.2739i	1.07985	−	1.07985i	0.996522	−	0.0833248i	$-0.0265539\pi$
$110$		0			0
$111$	2.43436	+	10.4393i	0.231059	+	0.990858i
$112$		0			0
$113$	−3.07280	−	1.77408i	−0.289065	−	0.166892i	0.348455	−	0.937325i	$-0.386706\pi$
$113$	−3.07280	−	1.77408i	−0.289065	−	0.166892i	−0.637520	+	0.770434i	$0.720040\pi$
$114$		0			0
$115$	1.01882	−	3.80229i	0.0950056	−	0.354566i
$116$		0			0
$117$	−10.0129	−	4.09167i	−0.925693	−	0.378275i
$118$		0			0
$119$	2.26133	−	8.43940i	0.207296	−	0.773638i
$120$		0			0
$121$	8.04130	+	4.64265i	0.731027	+	0.422059i
$122$		0			0
$123$	15.2834	−	14.3151i	1.37806	−	1.29075i
$124$		0			0
$125$	15.3755	−	15.3755i	1.37523	−	1.37523i
$126$		0			0
$127$	−7.78898	−	4.49697i	−0.691160	−	0.399042i	0.112886	−	0.993608i	$-0.463990\pi$
$127$	−7.78898	−	4.49697i	−0.691160	−	0.399042i	−0.804047	+	0.594566i	$0.797324\pi$
$128$		0			0
$129$	3.98647	+	0.130414i	0.350989	+	0.0114823i
$130$		0			0
$131$	8.85588	−	5.11294i	0.773742	−	0.446720i	−0.0604660	−	0.998170i	$-0.519259\pi$
$131$	8.85588	−	5.11294i	0.773742	−	0.446720i	0.834208	+	0.551450i	$0.185925\pi$
$132$		0			0
$133$	−7.18722	−	12.4486i	−0.623211	−	1.07943i
$134$		0			0
$135$	−19.1586	−	7.20934i	−1.64891	−	0.620481i
$136$		0			0
$137$	−12.5721	+	12.5721i	−1.07411	+	1.07411i	−0.0770844	+	0.997025i	$0.524561\pi$
$137$	−12.5721	+	12.5721i	−1.07411	+	1.07411i	−0.997025	+	0.0770844i	$0.975439\pi$
$138$		0			0
$139$	5.49827	−	9.52329i	0.466358	−	0.807755i	−0.532904	−	0.846176i	$-0.678899\pi$
$139$	5.49827	−	9.52329i	0.466358	−	0.807755i	0.999262	+	0.0384207i	$0.0122327\pi$
$140$		0			0
$141$	−0.0702416	+	0.0163797i	−0.00591541	+	0.00137942i
$142$		0			0
$143$	13.5448	+	8.95810i	1.13267	+	0.749114i
$144$		0			0
$145$	0.0170451	+	0.0636133i	0.00141552	+	0.00528280i
$146$		0			0
$147$	−5.97856	−	0.195584i	−0.493104	−	0.0161315i
$148$		0			0
$149$	−21.3835	+	5.72969i	−1.75181	+	0.469395i	−0.985010	−	0.172497i	$-0.944816\pi$
$149$	−21.3835	+	5.72969i	−1.75181	+	0.469395i	−0.766795	+	0.641892i	$0.778150\pi$
$150$		0			0
$151$	22.9143	−	6.13987i	1.86474	−	0.499655i	0.864742	−	0.502216i	$-0.167482\pi$
$151$	22.9143	−	6.13987i	1.86474	−	0.499655i	0.999997	+	0.00256111i	$0.000815229\pi$
$152$		0			0
$153$	−6.15657	−	12.4829i	−0.497729	−	1.00918i
$154$		0			0
$155$	6.17869	−	10.7018i	0.496284	−	0.859590i
$156$		0			0
$157$	−11.3850	−	19.7195i	−0.908626	−	1.57379i	−0.815975	−	0.578087i	$-0.803799\pi$
$157$	−11.3850	−	19.7195i	−0.908626	−	1.57379i	−0.0926507	−	0.995699i	$-0.529534\pi$
$158$		0			0
$159$	−3.00970	−	0.0984600i	−0.238685	−	0.00780839i
$160$		0			0
$161$	1.33058	+	1.33058i	0.104865	+	0.104865i
$162$		0			0
$163$	3.35459	−	12.5195i	0.262752	−	0.980604i	−0.700860	−	0.713299i	$-0.747200\pi$
$163$	3.35459	−	12.5195i	0.262752	−	0.980604i	0.963612	−	0.267305i	$-0.0861330\pi$
$164$		0			0
$165$	26.0981	+	16.2280i	2.03173	+	1.26335i
$166$		0			0
$167$	4.28123	−	4.28123i	0.331292	−	0.331292i	−0.521785	−	0.853077i	$-0.674734\pi$
$167$	4.28123	−	4.28123i	0.331292	−	0.331292i	0.853077	+	0.521785i	$0.174734\pi$
$168$		0			0
$169$	10.3879	+	7.81613i	0.799068	+	0.601241i
$170$		0			0
$171$	−21.6841	−	7.35967i	−1.65822	−	0.562808i
$172$		0			0
$173$	0.119661			0.00909766			0.00454883	−	0.999990i	$-0.498552\pi$
$173$	0.119661			0.00909766			0.00454883	+	0.999990i	$0.498552\pi$
$174$		0			0
$175$	5.12731	+	19.1354i	0.387588	+	1.44650i
$176$		0			0
$177$	−14.6975	+	3.42732i	−1.10473	+	0.257613i
$178$		0			0
$179$	−7.04673	+	12.2053i	−0.526698	+	0.912267i	0.472819	+	0.881160i	$0.343237\pi$
$179$	−7.04673	+	12.2053i	−0.526698	+	0.912267i	−0.999516	+	0.0311070i	$0.990097\pi$
$180$		0			0
$181$			5.37997i			0.399890i	0.979807	+	0.199945i	$0.0640764\pi$
$181$			5.37997i			0.399890i	−0.979807	+	0.199945i	$0.935924\pi$
$182$		0			0
$183$	−0.295944	+	9.04633i	−0.0218768	+	0.668724i
$184$		0			0
$185$		−	24.3809i		−	1.79252i
$186$		0			0
$187$	5.40828	+	20.1840i	0.395492	+	1.47600i
$188$		0			0
$189$	7.56378	−	6.20828i	0.550184	−	0.451586i
$190$		0			0
$191$			10.3416i			0.748291i	0.927370	+	0.374146i	$0.122064\pi$
$191$			10.3416i			0.748291i	−0.927370	+	0.374146i	$0.877936\pi$
$192$		0			0
$193$	11.2662	+	11.2662i	0.810960	+	0.810960i	0.984778	−	0.173818i	$-0.0556105\pi$
$193$	11.2662	+	11.2662i	0.810960	+	0.810960i	−0.173818	+	0.984778i	$0.555610\pi$
$194$		0			0
$195$	20.0655	+	14.2351i	1.43692	+	1.01940i
$196$		0			0
$197$	−8.40291	+	2.25155i	−0.598683	+	0.160417i	−0.545418	−	0.838164i	$-0.683629\pi$
$197$	−8.40291	+	2.25155i	−0.598683	+	0.160417i	−0.0532643	+	0.998580i	$0.516963\pi$
$198$		0			0
$199$	12.1832	−	7.03398i	0.863645	−	0.498626i	−0.00158596	−	0.999999i	$-0.500505\pi$
$199$	12.1832	−	7.03398i	0.863645	−	0.498626i	0.865231	+	0.501373i	$0.167171\pi$
$200$		0			0
$201$	0.0877637	+	0.0937002i	0.00619038	+	0.00660910i
$202$		0			0
$203$	−0.0304091	−	0.00814810i	−0.00213430	−	0.000571884i
$204$		0			0
$205$	−41.2475	+	23.8143i	−2.88085	+	1.66326i
$206$		0			0
$207$	2.99126	+	0.195923i	0.207907	+	0.0136176i
$208$		0			0
$209$	29.7726	+	17.1892i	2.05941	+	1.18900i
$210$		0			0
$211$	−1.89164			−0.130226			−0.0651131	−	0.997878i	$-0.520741\pi$
$211$	−1.89164			−0.130226			−0.0651131	+	0.997878i	$0.520741\pi$
$212$		0			0
$213$	−8.20732	−	2.48946i	−0.562356	−	0.170575i
$214$		0			0
$215$	−8.76281	−	2.34799i	−0.597619	−	0.160131i
$216$		0			0
$217$	2.95360	+	5.11579i	0.200504	+	0.347283i
$218$		0			0
$219$	−0.0523436	+	0.0490273i	−0.00353705	+	0.00331296i
$220$		0			0
$221$	3.34091	+	16.3910i	0.224734	+	1.10258i
$222$		0			0
$223$	18.4580	+	4.94580i	1.23604	+	0.331195i	0.816928	−	0.576740i	$-0.195675\pi$
$223$	18.4580	+	4.94580i	1.23604	+	0.331195i	0.419110	+	0.907935i	$0.362342\pi$
$224$		0			0
$225$	26.2409	+	17.5319i	1.74939	+	1.16879i
$226$		0			0
$227$	−9.53495	−	9.53495i	−0.632857	−	0.632857i	0.315927	−	0.948784i	$-0.397685\pi$
$227$	−9.53495	−	9.53495i	−0.632857	−	0.632857i	−0.948784	+	0.315927i	$0.897685\pi$
$228$		0			0
$229$	2.94216	−	10.9803i	0.194424	−	0.725599i	−0.797992	−	0.602669i	$-0.794104\pi$
$229$	2.94216	−	10.9803i	0.194424	−	0.725599i	0.992415	−	0.122930i	$-0.0392292\pi$
$230$		0			0
$231$	−12.9560	+	6.92551i	−0.852443	+	0.455665i
$232$		0			0
$233$	−8.36969			−0.548316			−0.274158	−	0.961685i	$-0.588399\pi$
$233$	−8.36969			−0.548316			−0.274158	+	0.961685i	$0.588399\pi$
$234$		0			0
$235$	0.164048			0.0107013
$236$		0			0
$237$	3.46463	+	2.15434i	0.225052	+	0.139939i
$238$		0			0
$239$	4.29278	−	16.0209i	0.277677	−	1.03630i	−0.676349	−	0.736581i	$-0.736439\pi$
$239$	4.29278	−	16.0209i	0.277677	−	1.03630i	0.954026	−	0.299724i	$-0.0968945\pi$
$240$		0			0
$241$	0.480854	+	0.480854i	0.0309745	+	0.0309745i	0.722424	−	0.691450i	$-0.243028\pi$
$241$	0.480854	+	0.480854i	0.0309745	+	0.0309745i	−0.691450	+	0.722424i	$0.743028\pi$
$242$		0			0
$243$	2.53757	−	15.3805i	0.162785	−	0.986662i
$244$		0			0
$245$	13.1417	+	3.52131i	0.839593	+	0.224968i
$246$		0			0
$247$	22.9550	+	15.1817i	1.46059	+	0.965989i
$248$		0			0
$249$	1.86435	+	7.99494i	0.118148	+	0.506659i
$250$		0			0
$251$	−14.0317	−	24.3035i	−0.885670	−	1.53403i	−0.844944	−	0.534855i	$-0.820366\pi$
$251$	−14.0317	−	24.3035i	−0.885670	−	1.53403i	−0.0407262	−	0.999170i	$-0.512967\pi$
$252$		0			0
$253$	−4.34707	−	1.16479i	−0.273298	−	0.0732299i
$254$		0			0
$255$	7.18929	+	30.8301i	0.450211	+	1.93065i
$256$		0			0
$257$	−7.89235			−0.492311			−0.246155	−	0.969230i	$-0.579167\pi$
$257$	−7.89235			−0.492311			−0.246155	+	0.969230i	$0.579167\pi$
$258$		0			0
$259$	10.0934	+	5.82742i	0.627172	+	0.362098i
$260$		0			0
$261$	−0.0449787	+	0.0221835i	−0.00278411	+	0.00137313i
$262$		0			0
$263$	5.14048	−	2.96786i	0.316976	−	0.183006i	−0.333068	−	0.942903i	$-0.608084\pi$
$263$	5.14048	−	2.96786i	0.316976	−	0.183006i	0.650044	+	0.759897i	$0.274751\pi$
$264$		0			0
$265$	6.61574	+	1.77268i	0.406401	+	0.108895i
$266$		0			0
$267$	−20.8810	+	4.86926i	−1.27790	+	0.297994i
$268$		0			0
$269$	7.40210	−	4.27360i	0.451314	−	0.260566i	−0.257071	−	0.966392i	$-0.582757\pi$
$269$	7.40210	−	4.27360i	0.451314	−	0.260566i	0.708385	+	0.705826i	$0.249424\pi$
$270$		0			0
$271$	−5.18870	+	1.39031i	−0.315191	+	0.0844552i	−0.412946	−	0.910755i	$-0.635500\pi$
$271$	−5.18870	+	1.39031i	−0.315191	+	0.0844552i	0.0977555	+	0.995210i	$0.468834\pi$
$272$		0			0
$273$	−10.6891	+	4.90444i	−0.646935	+	0.296830i
$274$		0			0
$275$	−33.5022	−	33.5022i	−2.02026	−	2.02026i
$276$		0			0
$277$		−	23.1882i		−	1.39324i	−0.717439	−	0.696621i	$-0.754686\pi$
$277$		−	23.1882i		−	1.39324i	0.717439	−	0.696621i	$-0.245314\pi$
$278$		0			0
$279$	8.91113	+	3.02448i	0.533495	+	0.181071i
$280$		0			0
$281$	−0.981285	−	3.66221i	−0.0585386	−	0.218469i	0.930460	−	0.366393i	$-0.119407\pi$
$281$	−0.981285	−	3.66221i	−0.0585386	−	0.218469i	−0.988999	+	0.147924i	$0.952741\pi$
$282$		0			0
$283$		−	28.0666i		−	1.66838i	−0.551474	−	0.834192i	$-0.685934\pi$
$283$		−	28.0666i		−	1.66838i	0.551474	−	0.834192i	$-0.314066\pi$
$284$		0			0
$285$	44.2296	+	27.5023i	2.61993	+	1.62910i
$286$		0			0
$287$		−	22.7679i		−	1.34395i
$288$		0			0
$289$	−2.26253	+	3.91882i	−0.133090	+	0.230519i
$290$		0			0
$291$	−10.0797	−	10.7615i	−0.590883	−	0.630851i
$292$		0			0
$293$	−3.45012	−	12.8760i	−0.201558	−	0.752225i	−0.990471	−	0.137720i	$-0.956022\pi$
$293$	−3.45012	−	12.8760i	−0.201558	−	0.752225i	0.788913	−	0.614505i	$-0.210644\pi$
$294$		0			0
$295$	34.3258			1.99852
$296$		0			0
$297$	−8.24226	+	21.9036i	−0.478264	+	1.27098i
$298$		0			0
$299$	−3.41698	−	1.14194i	−0.197609	−	0.0660401i
$300$		0			0
$301$	3.06648	−	3.06648i	0.176749	−	0.176749i
$302$		0			0
$303$	21.0588	−	11.2567i	1.20979	−	0.646683i
$304$		0			0
$305$	5.32819	−	19.8851i	0.305091	−	1.13862i
$306$		0			0
$307$	9.09884	+	9.09884i	0.519298	+	0.519298i	0.917359	−	0.398061i	$-0.130317\pi$
$307$	9.09884	+	9.09884i	0.519298	+	0.519298i	−0.398061	+	0.917359i	$0.630317\pi$
$308$		0			0
$309$	13.4824	+	25.2225i	0.766988	+	1.43486i
$310$		0			0
$311$	1.39052	+	2.40845i	0.0788492	+	0.136571i	0.902754	−	0.430158i	$-0.141542\pi$
$311$	1.39052	+	2.40845i	0.0788492	+	0.136571i	−0.823904	+	0.566729i	$0.808209\pi$
$312$		0			0
$313$	−8.75421	+	15.1627i	−0.494817	+	0.857049i	−0.999982	−	0.00597416i	$-0.998098\pi$
$313$	−8.75421	+	15.1627i	−0.494817	+	0.857049i	0.505165	+	0.863023i	$0.331432\pi$
$314$		0			0
$315$	−19.9608	+	9.84468i	−1.12466	+	0.554685i
$316$		0			0
$317$	20.3076	−	5.44141i	1.14059	−	0.305620i	0.361402	−	0.932410i	$-0.382298\pi$
$317$	20.3076	−	5.44141i	1.14059	−	0.305620i	0.779188	+	0.626790i	$0.215632\pi$
$318$		0			0
$319$	0.0727275	−	0.0194873i	0.00407196	−	0.00109108i
$320$		0			0
$321$	9.28851	−	14.9379i	0.518434	−	0.833751i
$322$		0			0
$323$	9.16566	+	34.2067i	0.509991	+	1.90331i
$324$		0			0
$325$	−25.1426	−	28.3980i	−1.39466	−	1.57524i
$326$		0			0
$327$	−18.8781	−	20.1551i	−1.04396	−	1.11458i
$328$		0			0
$329$	−0.0392101	+	0.0679138i	−0.00216172	+	0.00374421i
$330$		0			0
$331$	0.792923	−	0.792923i	0.0435830	−	0.0435830i	−0.684979	−	0.728562i	$-0.740189\pi$
$331$	0.792923	−	0.792923i	0.0435830	−	0.0435830i	0.728562	+	0.684979i	$0.240189\pi$
$332$		0			0
$333$	18.2097	−	3.62297i	0.997883	−	0.198538i
$334$		0			0
$335$	−0.146001	−	0.252882i	−0.00797690	−	0.0138164i
$336$		0			0
$337$	−1.39429	+	0.804994i	−0.0759519	+	0.0438508i	−0.537495	−	0.843267i	$-0.680629\pi$
$337$	−1.39429	+	0.804994i	−0.0759519	+	0.0438508i	0.461543	+	0.887118i	$0.347296\pi$
$338$		0			0
$339$	−3.24518	+	5.21893i	−0.176254	+	0.283453i
$340$		0			0
$341$	−12.2351	−	7.06395i	−0.662569	−	0.382534i
$342$		0			0
$343$	−13.9202	+	13.9202i	−0.751620	+	0.751620i
$344$		0			0
$345$	−6.52454	−	1.97904i	−0.351270	−	0.106548i
$346$		0			0
$347$	−2.60090	−	1.50163i	−0.139624	−	0.0806119i	0.428561	−	0.903513i	$-0.359021\pi$
$347$	−2.60090	−	1.50163i	−0.139624	−	0.0806119i	−0.568185	+	0.822901i	$0.692354\pi$
$348$		0			0
$349$	−6.17845	+	23.0583i	−0.330725	+	1.23428i	0.577706	+	0.816245i	$0.303948\pi$
$349$	−6.17845	+	23.0583i	−0.330725	+	1.23428i	−0.908430	+	0.418036i	$0.862718\pi$
$350$		0			0
$351$	−7.65024	+	17.1019i	−0.408340	+	0.912830i
$352$		0			0
$353$	−4.88095	+	18.2159i	−0.259787	+	0.969537i	0.705578	+	0.708632i	$0.250687\pi$
$353$	−4.88095	+	18.2159i	−0.259787	+	0.969537i	−0.965365	+	0.260904i	$0.915979\pi$
$354$		0			0
$355$	16.8936	+	9.75355i	0.896621	+	0.517664i
$356$		0			0
$357$	−14.4816	−	4.39259i	−0.766447	−	0.232480i
$358$		0			0
$359$	−1.19442	+	1.19442i	−0.0630389	+	0.0630389i	−0.737923	−	0.674885i	$-0.764193\pi$
$359$	−1.19442	+	1.19442i	−0.0630389	+	0.0630389i	0.674885	+	0.737923i	$0.264193\pi$
$360$		0			0
$361$	34.0025	+	19.6313i	1.78960	+	1.03323i
$362$		0			0
$363$	8.49240	−	13.6576i	0.445735	−	0.716836i
$364$		0			0
$365$	0.141267	−	0.0815605i	0.00739425	−	0.00426907i
$366$		0			0
$367$	−10.1517	−	17.5832i	−0.529913	−	0.917836i	−0.999391	−	0.0348923i	$-0.988891\pi$
$367$	−10.1517	−	17.5832i	−0.529913	−	0.917836i	0.469478	−	0.882944i	$-0.344442\pi$
$368$		0			0
$369$	−23.9158	−	27.2682i	−1.24501	−	1.41953i
$370$		0			0
$371$	−2.31513	+	2.31513i	−0.120196	+	0.120196i
$372$		0			0
$373$	−1.06229	+	1.83995i	−0.0550034	+	0.0952688i	−0.892216	−	0.451609i	$-0.850850\pi$
$373$	−1.06229	+	1.83995i	−0.0550034	+	0.0952688i	0.837213	+	0.546877i	$0.184184\pi$
$374$		0			0
$375$	−25.7461	−	27.4876i	−1.32952	−	1.41946i
$376$		0			0
$377$	0.0590605	−	0.0120381i	0.00304177	−	0.000619993i
$378$		0			0
$379$	−4.82922	−	18.0229i	−0.248060	−	0.925774i	−0.971820	−	0.235722i	$-0.924254\pi$
$379$	−4.82922	−	18.0229i	−0.248060	−	0.925774i	0.723760	−	0.690052i	$-0.242412\pi$
$380$		0			0
$381$	−8.22593	+	13.2290i	−0.421427	+	0.677743i
$382$		0			0
$383$	−0.642392	+	0.172129i	−0.0328247	+	0.00879536i	−0.275194	−	0.961389i	$-0.588742\pi$
$383$	−0.642392	+	0.172129i	−0.0328247	+	0.00879536i	0.242369	+	0.970184i	$0.422075\pi$
$384$		0			0
$385$	32.2753	−	8.64813i	1.64490	−	0.440749i
$386$		0			0
$387$	0.451527	−	6.89369i	0.0229524	−	0.350426i
$388$		0			0
$389$	−11.6152	+	20.1182i	−0.588916	+	1.02003i	0.405458	+	0.914113i	$0.367112\pi$
$389$	−11.6152	+	20.1182i	−0.588916	+	1.02003i	−0.994375	+	0.105919i	$0.966221\pi$
$390$		0			0
$391$	−2.31795	−	4.01481i	−0.117224	−	0.203038i
$392$		0			0
$393$	−8.34961	−	15.6202i	−0.421182	−	0.787934i
$394$		0			0
$395$	−6.56151	−	6.56151i	−0.330145	−	0.330145i
$396$		0			0
$397$	4.19965	−	15.6733i	0.210774	−	0.786621i	−0.776837	−	0.629702i	$-0.783177\pi$
$397$	4.19965	−	15.6733i	0.210774	−	0.786621i	0.987611	−	0.156919i	$-0.0501562\pi$
$398$		0			0
$399$	−21.9571	+	11.7370i	−1.09923	+	0.587584i
$400$		0			0
$401$	−20.0928	+	20.0928i	−1.00339	+	1.00339i	−0.00339153	+	0.999994i	$0.501080\pi$
$401$	−20.0928	+	20.0928i	−1.00339	+	1.00339i	−0.999994	+	0.00339153i	$0.998920\pi$
$402$		0			0
$403$	−9.43340	−	6.23896i	−0.469911	−	0.310785i
$404$		0			0
$405$	−13.5653	+	32.7577i	−0.674063	+	1.62774i
$406$		0			0
$407$	−27.8741			−1.38167
$408$		0			0
$409$	−3.16669	−	11.8182i	−0.156583	−	0.584375i	−0.998965	−	0.0454938i	$-0.985514\pi$
$409$	−3.16669	−	11.8182i	−0.156583	−	0.584375i	0.842382	−	0.538881i	$-0.181153\pi$
$410$		0			0
$411$	21.0519	+	22.4759i	1.03841	+	1.10865i
$412$		0			0
$413$	−8.20438	+	14.2104i	−0.403711	+	0.699249i
$414$		0			0
$415$		−	18.6721i		−	0.916576i
$416$		0			0
$417$	−16.1746	−	10.0575i	−0.792075	−	0.492519i
$418$		0			0
$419$			0.438006i			0.0213980i	0.999943	+	0.0106990i	$0.00340566\pi$
$419$			0.438006i			0.0213980i	−0.999943	+	0.0106990i	$0.996594\pi$
$420$		0			0
$421$	1.63663	+	6.10799i	0.0797645	+	0.297685i	0.994271	−	0.106885i	$-0.0340877\pi$
$421$	1.63663	+	6.10799i	0.0797645	+	0.297685i	−0.914507	+	0.404570i	$0.867421\pi$
$422$		0			0
$423$	0.0243774	+	0.122525i	0.00118527	+	0.00595735i
$424$		0			0
$425$		−	48.8056i		−	2.36742i
$426$		0			0
$427$	6.95864	+	6.95864i	0.336752	+	0.336752i
$428$		0			0
$429$	16.2747	−	22.9404i	0.785748	−	1.10757i
$430$		0			0
$431$	22.5907	−	6.05315i	1.08815	−	0.291570i	0.330222	−	0.943903i	$-0.392876\pi$
$431$	22.5907	−	6.05315i	1.08815	−	0.291570i	0.757932	+	0.652333i	$0.226210\pi$
$432$		0			0
$433$	−7.49672	+	4.32823i	−0.360269	+	0.208002i	−0.669199	−	0.743083i	$-0.733363\pi$
$433$	−7.49672	+	4.32823i	−0.360269	+	0.208002i	0.308930	+	0.951085i	$0.400029\pi$
$434$		0			0
$435$	0.111088	−	0.0259047i	0.00532626	−	0.00124203i
$436$		0			0
$437$	−7.36717	−	1.97403i	−0.352420	−	0.0944306i
$438$		0			0
$439$	−1.49405	+	0.862589i	−0.0713070	+	0.0411691i	−0.535230	−	0.844707i	$-0.679775\pi$
$439$	−1.49405	+	0.862589i	−0.0713070	+	0.0411691i	0.463923	+	0.885876i	$0.346442\pi$
$440$		0			0
$441$	−0.677161	+	10.3386i	−0.0322458	+	0.492313i
$442$		0			0
$443$	2.81328	+	1.62425i	0.133663	+	0.0771704i	0.565340	−	0.824858i	$-0.308745\pi$
$443$	2.81328	+	1.62425i	0.133663	+	0.0771704i	−0.431677	+	0.902028i	$0.642078\pi$
$444$		0			0
$445$	48.7673			2.31179
$446$		0			0
$447$	8.70781	+	37.3420i	0.411865	+	1.76622i
$448$		0			0
$449$	−4.05170	−	1.08565i	−0.191212	−	0.0512350i	0.161942	−	0.986800i	$-0.448224\pi$
$449$	−4.05170	−	1.08565i	−0.191212	−	0.0512350i	−0.353154	+	0.935565i	$0.614891\pi$
$450$		0			0
$451$	27.2263	+	47.1573i	1.28204	+	2.22055i
$452$		0			0
$453$	−9.33118	−	40.0152i	−0.438417	−	1.88008i
$454$		0			0
$455$	26.2101	−	5.34230i	1.22875	−	0.250451i
$456$		0			0
$457$	1.25001	+	0.334939i	0.0584730	+	0.0156678i	0.287937	−	0.957649i	$-0.407031\pi$
$457$	1.25001	+	0.334939i	0.0584730	+	0.0156678i	−0.229464	+	0.973317i	$0.573697\pi$
$458$		0			0
$459$	−21.9581	+	9.95085i	−1.02492	+	0.464466i
$460$		0			0
$461$	9.85374	+	9.85374i	0.458934	+	0.458934i	0.898306	−	0.439371i	$-0.144799\pi$
$461$	9.85374	+	9.85374i	0.458934	+	0.458934i	−0.439371	+	0.898306i	$0.644799\pi$
$462$		0			0
$463$	9.59936	−	35.8253i	0.446120	−	1.66494i	−0.266842	−	0.963740i	$-0.585980\pi$
$463$	9.59936	−	35.8253i	0.446120	−	1.66494i	0.712962	−	0.701203i	$-0.247353\pi$
$464$		0			0
$465$	−18.1762	−	11.3022i	−0.842903	−	0.524125i
$466$		0			0
$467$	−20.2780			−0.938353			−0.469176	−	0.883104i	$-0.655449\pi$
$467$	−20.2780			−0.938353			−0.469176	+	0.883104i	$0.655449\pi$
$468$		0			0
$469$	0.139586			0.00644549
$470$		0			0
$471$	−34.7816	+	18.5922i	−1.60265	+	0.856682i
$472$		0			0
$473$	−2.68440	+	10.0183i	−0.123429	+	0.460642i
$474$		0			0
$475$	−56.7777	−	56.7777i	−2.60514	−	2.60514i
$476$		0			0
$477$	−0.340893	+	5.20459i	−0.0156084	+	0.238302i
$478$		0			0
$479$	19.6844	+	5.27443i	0.899405	+	0.240995i	0.678761	−	0.734360i	$-0.262517\pi$
$479$	19.6844	+	5.27443i	0.899405	+	0.240995i	0.220644	+	0.975354i	$0.429184\pi$
$480$		0			0
$481$	−22.2731	−	1.35425i	−1.01557	−	0.0617485i
$482$		0			0
$483$	2.37876	−	2.22805i	0.108237	−	0.101380i
$484$		0			0
$485$	16.7683	+	29.0436i	0.761411	+	1.31880i
$486$		0			0
$487$	−5.24083	−	1.40428i	−0.237485	−	0.0636338i	0.138114	−	0.990416i	$-0.455896\pi$
$487$	−5.24083	−	1.40428i	−0.237485	−	0.0636338i	−0.375598	+	0.926783i	$0.622563\pi$
$488$		0			0
$489$	−21.4829	−	6.51623i	−0.971488	−	0.294674i
$490$		0			0
$491$	23.2041			1.04719			0.523594	−	0.851968i	$-0.324591\pi$
$491$	23.2041			1.04719			0.523594	+	0.851968i	$0.324591\pi$
$492$		0			0
$493$	0.0671688	+	0.0387799i	0.00302513	+	0.00174656i
$494$		0			0
$495$	29.5707	−	44.2599i	1.32910	−	1.98934i
$496$		0			0
$497$	−8.07568	+	4.66250i	−0.362244	+	0.209142i
$498$		0			0
$499$	−30.6811	−	8.22097i	−1.37347	−	0.368021i	−0.504728	−	0.863279i	$-0.668407\pi$
$499$	−30.6811	−	8.22097i	−1.37347	−	0.368021i	−0.868746	+	0.495257i	$0.835074\pi$
$500$		0			0
$501$	−7.16889	−	7.65380i	−0.320282	−	0.341947i
$502$		0			0
$503$	−23.3710	+	13.4932i	−1.04206	+	0.601634i	−0.920416	−	0.390940i	$-0.872150\pi$
$503$	−23.3710	+	13.4932i	−1.04206	+	0.601634i	−0.121644	+	0.992574i	$0.538817\pi$
$504$		0			0
$505$	−52.4603	+	14.0567i	−2.33445	+	0.625515i
$506$		0			0
$507$	14.1190	−	17.5401i	0.627046	−	0.778982i
$508$		0			0
$509$	−7.32135	−	7.32135i	−0.324513	−	0.324513i	0.525982	−	0.850495i	$-0.323698\pi$
$509$	−7.32135	−	7.32135i	−0.324513	−	0.324513i	−0.850495	+	0.525982i	$0.823698\pi$
$510$		0			0
$511$			0.0779769i			0.00344949i
$512$		0			0
$513$	−13.9685	+	37.1210i	−0.616726	+	1.63893i
$514$		0			0
$515$	−16.8360	−	62.8327i	−0.741882	−	2.76874i
$516$		0			0
$517$		−	0.187552i		−	0.00824855i
$518$		0			0
$519$	0.00677670	−	0.207148i	0.000297464	−	0.00909280i
$520$		0			0
$521$		−	13.8080i		−	0.604939i	−0.953159	−	0.302469i	$-0.902189\pi$
$521$		−	13.8080i		−	0.604939i	0.953159	−	0.302469i	$-0.0978110\pi$
$522$		0			0
$523$	−4.75258	+	8.23170i	−0.207816	+	0.359947i	−0.951026	−	0.309110i	$-0.899969\pi$
$523$	−4.75258	+	8.23170i	−0.207816	+	0.359947i	0.743211	+	0.669058i	$0.233302\pi$
$524$		0			0
$525$	33.4161	−	7.79232i	1.45840	−	0.340085i
$526$		0			0
$527$	−3.76665	−	14.0573i	−0.164078	−	0.612347i
$528$		0			0
$529$	−22.0016			−0.956589
$530$		0			0
$531$	5.10076	+	25.6373i	0.221354	+	1.11256i
$532$		0			0
$533$	19.4643	+	39.0043i	0.843094	+	1.68946i
$534$		0			0
$535$	−28.2902	+	28.2902i	−1.22309	+	1.22309i
$536$		0			0
$537$	20.7298	+	12.8900i	0.894558	+	0.556244i
$538$		0			0
$539$	4.02583	−	15.0246i	0.173405	−	0.647155i
$540$		0			0
$541$	19.2507	+	19.2507i	0.827651	+	0.827651i	0.987191	−	0.159540i	$-0.0510012\pi$
$541$	19.2507	+	19.2507i	0.827651	+	0.827651i	−0.159540	+	0.987191i	$0.551001\pi$
$542$		0			0
$543$	9.31340	+	0.304681i	0.399676	+	0.0130751i
$544$		0			0
$545$	31.4051	+	54.3952i	1.34525	+	2.33004i
$546$		0			0
$547$	−20.0559	+	34.7379i	−0.857530	+	1.48529i	0.0167471	+	0.999860i	$0.494669\pi$
$547$	−20.0559	+	34.7379i	−0.857530	+	1.48529i	−0.874277	+	0.485426i	$0.838664\pi$
$548$		0			0
$549$	15.6436	+	1.02463i	0.667651	+	0.0437302i
$550$		0			0
$551$	0.123255	−	0.0330260i	0.00525083	−	0.00140695i
$552$		0			0
$553$	4.28468	−	1.14808i	0.182203	−	0.0488212i
$554$		0			0
$555$	−42.2064	−	1.38075i	−1.79156	−	0.0586096i
$556$		0			0
$557$	1.96905	+	7.34860i	0.0834313	+	0.311370i	0.995013	−	0.0997501i	$-0.0318043\pi$
$557$	1.96905	+	7.34860i	0.0834313	+	0.311370i	−0.911581	+	0.411120i	$0.865138\pi$
$558$		0			0
$559$	−2.63173	+	7.87481i	−0.111310	+	0.333069i
$560$		0			0
$561$	35.2473	−	8.21934i	1.48814	−	0.347021i
$562$		0			0
$563$	4.29228	−	7.43445i	0.180898	−	0.313325i	−0.761289	−	0.648413i	$-0.775433\pi$
$563$	4.29228	−	7.43445i	0.180898	−	0.313325i	0.942187	+	0.335089i	$0.108766\pi$
$564$		0			0
$565$	9.88390	−	9.88390i	0.415819	−	0.415819i
$566$		0			0
$567$	−10.3190	−	13.4454i	−0.433355	−	0.564655i
$568$		0			0
$569$	−17.9702	−	31.1253i	−0.753351	−	1.30484i	−0.946190	−	0.323611i	$-0.895103\pi$
$569$	−17.9702	−	31.1253i	−0.753351	−	1.30484i	0.192839	−	0.981230i	$-0.438230\pi$
$570$		0			0
$571$	9.68042	−	5.58899i	0.405113	−	0.233892i	−0.283575	−	0.958950i	$-0.591520\pi$
$571$	9.68042	−	5.58899i	0.405113	−	0.233892i	0.688688	+	0.725058i	$0.258187\pi$
$572$		0			0
$573$	17.9026	+	0.585669i	0.747891	+	0.0244667i
$574$		0			0
$575$	9.10312	+	5.25569i	0.379626	+	0.219177i
$576$		0			0
$577$	12.4732	−	12.4732i	0.519266	−	0.519266i	−0.398083	−	0.917349i	$-0.630325\pi$
$577$	12.4732	−	12.4732i	0.519266	−	0.519266i	0.917349	+	0.398083i	$0.130325\pi$
$578$		0			0
$579$	20.1413	−	18.8652i	0.837042	−	0.784010i
$580$		0			0
$581$	7.72999	+	4.46291i	0.320694	+	0.185153i
$582$		0			0
$583$	2.02666	−	7.56361i	0.0839358	−	0.313253i
$584$		0			0
$585$	25.7791	−	33.9297i	1.06584	−	1.40282i
$586$		0			0
$587$	3.48074	−	12.9903i	0.143666	−	0.536167i	−0.856146	−	0.516735i	$-0.827147\pi$
$587$	3.48074	−	12.9903i	0.143666	−	0.536167i	0.999811	−	0.0194325i	$-0.00618594\pi$
$588$		0			0
$589$	−20.7354	−	11.9716i	−0.854388	−	0.493281i
$590$		0			0
$591$	3.42184	+	14.6740i	0.140756	+	0.603608i
$592$		0			0
$593$	−4.12229	+	4.12229i	−0.169282	+	0.169282i	−0.786664	−	0.617382i	$-0.788193\pi$
$593$	−4.12229	+	4.12229i	−0.169282	+	0.169282i	0.617382	+	0.786664i	$0.288193\pi$
$594$		0			0
$595$	29.8084	+	17.2099i	1.22202	+	0.705536i
$596$		0			0
$597$	−11.4867	−	21.4890i	−0.470121	−	0.879487i
$598$		0			0
$599$	−26.3160	+	15.1935i	−1.07524	+	0.620791i	−0.929609	−	0.368547i	$-0.879855\pi$
$599$	−26.3160	+	15.1935i	−1.07524	+	0.620791i	−0.145633	+	0.989339i	$0.546522\pi$
$600$		0			0
$601$	8.54362	+	14.7980i	0.348502	+	0.603622i	0.985983	−	0.166843i	$-0.0533573\pi$
$601$	8.54362	+	14.7980i	0.348502	+	0.603622i	−0.637482	+	0.770465i	$0.720024\pi$
$602$		0			0
$603$	0.167177	−	0.146624i	0.00680797	−	0.00597097i
$604$		0			0
$605$	−25.8655	+	25.8655i	−1.05158	+	1.05158i
$606$		0			0
$607$	14.2310	−	24.6488i	0.577619	−	1.00047i	−0.418133	−	0.908386i	$-0.637315\pi$
$607$	14.2310	−	24.6488i	0.577619	−	1.00047i	0.995752	−	0.0920795i	$-0.0293514\pi$
$608$		0			0
$609$	−0.0158275	+	0.0521805i	−0.000641363	+	0.00211446i
$610$		0			0
$611$	0.00911214	−	0.149866i	0.000368638	−	0.00606292i
$612$		0			0
$613$	8.00452	+	29.8733i	0.323299	+	1.20657i	0.916010	+	0.401155i	$0.131391\pi$
$613$	8.00452	+	29.8733i	0.323299	+	1.20657i	−0.592711	+	0.805415i	$0.701942\pi$
$614$		0			0
$615$	38.8895	+	72.7533i	1.56818	+	2.93370i
$616$		0			0
$617$	26.9588	−	7.22358i	1.08532	−	0.290810i	0.328546	−	0.944488i	$-0.393441\pi$
$617$	26.9588	−	7.22358i	1.08532	−	0.290810i	0.756773	+	0.653677i	$0.226775\pi$
$618$		0			0
$619$	−13.2777	+	3.55774i	−0.533674	+	0.142998i	−0.515584	−	0.856839i	$-0.672425\pi$
$619$	−13.2777	+	3.55774i	−0.533674	+	0.142998i	−0.0180898	+	0.999836i	$0.505758\pi$
$620$		0			0
$621$	0.508569	−	5.16714i	0.0204082	−	0.207350i
$622$		0			0
$623$	−11.6561	+	20.1890i	−0.466993	+	0.808856i
$624$		0			0
$625$	16.5317	+	28.6337i	0.661266	+	1.14535i
$626$		0			0
$627$	31.4427	−	50.5666i	1.25570	−	2.01943i
$628$		0			0
$629$	−20.3034	−	20.3034i	−0.809548	−	0.809548i
$630$		0			0
$631$	−2.32391	+	8.67295i	−0.0925133	+	0.345265i	−0.996631	−	0.0820197i	$-0.973863\pi$
$631$	−2.32391	+	8.67295i	−0.0925133	+	0.345265i	0.904117	+	0.427284i	$0.140530\pi$
$632$		0			0
$633$	−0.107128	+	3.27467i	−0.00425797	+	0.130157i
$634$		0			0
$635$	25.0539	−	25.0539i	0.994232	−	0.994232i
$636$		0			0
$637$	3.94684	−	11.8100i	0.156380	−	0.467928i
$638$		0			0
$639$	−4.77437	+	14.0669i	−0.188871	+	0.556478i
$640$		0			0
$641$	−41.4190			−1.63595			−0.817976	−	0.575253i	$-0.804904\pi$
$641$	−41.4190			−1.63595			−0.817976	+	0.575253i	$0.804904\pi$
$642$		0			0
$643$	−12.7764	−	47.6823i	−0.503854	−	1.88041i	−0.473352	−	0.880874i	$-0.656956\pi$
$643$	−12.7764	−	47.6823i	−0.503854	−	1.88041i	−0.0305022	−	0.999535i	$-0.509711\pi$
$644$		0			0
$645$	−4.56092	+	15.0365i	−0.179586	+	0.592063i
$646$		0			0
$647$	−1.03827	+	1.79833i	−0.0408184	+	0.0706996i	−0.885713	−	0.464233i	$-0.846330\pi$
$647$	−1.03827	+	1.79833i	−0.0408184	+	0.0706996i	0.844894	+	0.534933i	$0.179663\pi$
$648$		0			0
$649$		−	39.2438i		−	1.54045i
$650$		0			0
$651$	9.02335	−	4.82334i	0.353653	−	0.189042i
$652$		0			0
$653$		−	5.31179i		−	0.207867i	−0.994584	−	0.103933i	$-0.966857\pi$
$653$		−	5.31179i		−	0.207867i	0.994584	−	0.103933i	$-0.0331428\pi$
$654$		0			0
$655$	10.4265	+	38.9121i	0.407396	+	1.52042i
$656$		0			0
$657$	0.0819081	+	0.0933899i	0.00319554	+	0.00364349i
$658$		0			0
$659$			7.00574i			0.272905i	0.990647	+	0.136452i	$0.0435701\pi$
$659$			7.00574i			0.272905i	−0.990647	+	0.136452i	$0.956430\pi$
$660$		0			0
$661$	23.2482	+	23.2482i	0.904249	+	0.904249i	0.995800	−	0.0915510i	$-0.0291824\pi$
$661$	23.2482	+	23.2482i	0.904249	+	0.904249i	−0.0915510	+	0.995800i	$0.529182\pi$
$662$		0			0
$663$	28.5640	−	4.85528i	1.10933	−	0.188563i
$664$		0			0
$665$	54.6983	−	14.6564i	2.12111	−	0.568350i
$666$		0			0
$667$	−0.0144663	+	0.00835212i	−0.000560137	+	0.000323395i
$668$		0			0
$669$	9.60712	−	31.6730i	0.371433	−	1.22455i
$670$		0			0
$671$	−22.7341	−	6.09159i	−0.877641	−	0.235163i
$672$		0			0
$673$	−1.68972	+	0.975559i	−0.0651338	+	0.0376050i	−0.532213	−	0.846610i	$-0.678640\pi$
$673$	−1.68972	+	0.975559i	−0.0651338	+	0.0376050i	0.467079	+	0.884215i	$0.345306\pi$
$674$		0			0
$675$	31.8360	−	44.4333i	1.22537	−	1.71024i
$676$		0			0
$677$	13.1079	+	7.56783i	0.503776	+	0.290855i	0.730272	−	0.683157i	$-0.239393\pi$
$677$	13.1079	+	7.56783i	0.503776	+	0.290855i	−0.226495	+	0.974012i	$0.572727\pi$
$678$		0			0
$679$	−16.0316			−0.615235
$680$		0			0
$681$	−17.0462	+	15.9662i	−0.653211	+	0.611826i
$682$		0			0
$683$	26.1296	+	7.00141i	0.999822	+	0.267901i	0.721371	−	0.692549i	$-0.243512\pi$
$683$	26.1296	+	7.00141i	0.999822	+	0.267901i	0.278451	+	0.960450i	$0.410179\pi$
$684$		0			0
$685$	−35.0214	−	60.6588i	−1.33810	−	2.31765i
$686$		0			0
$687$	−18.8417	−	5.71509i	−0.718854	−	0.218044i
$688$		0			0
$689$	1.98690	−	5.94532i	0.0756949	−	0.226499i
$690$		0			0
$691$	35.8188	+	9.59761i	1.36261	+	0.365110i	0.864774	−	0.502161i	$-0.167462\pi$
$691$	35.8188	+	9.59761i	1.36261	+	0.365110i	0.497836	+	0.867271i	$0.334128\pi$
$692$		0			0
$693$	11.2552	+	22.8207i	0.427549	+	0.866886i
$694$		0			0
$695$	30.6324	+	30.6324i	1.16195	+	1.16195i
$696$		0			0
$697$	−14.5176	+	54.1806i	−0.549895	+	2.05224i
$698$		0			0
$699$	−0.473996	+	14.4890i	−0.0179282	+	0.548023i
$700$		0			0
$701$	7.64065			0.288583			0.144292	−	0.989535i	$-0.453910\pi$
$701$	7.64065			0.288583			0.144292	+	0.989535i	$0.453910\pi$
$702$		0			0
$703$	−47.2395			−1.78167
$704$		0			0
$705$	0.00929045	−	0.283988i	0.000349899	−	0.0106956i
$706$		0			0
$707$	6.71954	−	25.0777i	0.252714	−	0.943142i
$708$		0			0
$709$	24.1779	+	24.1779i	0.908021	+	0.908021i	0.996112	−	0.0880916i	$-0.0280768\pi$
$709$	24.1779	+	24.1779i	0.908021	+	0.908021i	−0.0880916	+	0.996112i	$0.528077\pi$
$710$		0			0
$711$	3.92564	−	5.87570i	0.147223	−	0.220356i
$712$		0			0
$713$	3.02756	+	0.811232i	0.113383	+	0.0303809i
$714$		0			0
$715$	−47.8983	+	42.4075i	−1.79129	+	1.58595i
$716$		0			0
$717$	−27.4910	−	8.33864i	−1.02667	−	0.311412i
$718$		0			0
$719$	2.01084	+	3.48288i	0.0749917	+	0.129889i	0.901083	−	0.433647i	$-0.142774\pi$
$719$	2.01084	+	3.48288i	0.0749917	+	0.129889i	−0.826091	+	0.563537i	$0.809440\pi$
$720$		0			0
$721$	30.0360	+	8.04811i	1.11860	+	0.299727i
$722$		0			0
$723$	0.859650	−	0.805187i	0.0319707	−	0.0299452i
$724$		0			0
$725$	−0.175858			−0.00653120
$726$		0			0
$727$	−31.9389	−	18.4399i	−1.18455	−	0.683900i	−0.227486	−	0.973781i	$-0.573051\pi$
$727$	−31.9389	−	18.4399i	−1.18455	−	0.683900i	−0.957062	+	0.289882i	$0.906384\pi$
$728$		0			0
$729$	−26.4819	−	5.26389i	−0.980811	−	0.194959i
$730$		0			0
$731$	−9.25258	+	5.34198i	−0.342219	+	0.197580i
$732$		0			0
$733$	−38.9036	−	10.4242i	−1.43694	−	0.385026i	−0.545478	−	0.838125i	$-0.683652\pi$
$733$	−38.9036	−	10.4242i	−1.43694	−	0.385026i	−0.891460	+	0.453099i	$0.850318\pi$
$734$		0			0
$735$	6.84008	−	22.5505i	0.252300	−	0.831789i
$736$		0			0
$737$	−0.289113	+	0.166920i	−0.0106496	+	0.00614857i
$738$		0			0
$739$	10.9711	−	2.93969i	0.403578	−	0.108138i	−0.0513194	−	0.998682i	$-0.516343\pi$
$739$	10.9711	−	2.93969i	0.403578	−	0.108138i	0.454897	+	0.890544i	$0.349676\pi$
$740$		0			0
$741$	27.5814	−	38.8781i	1.01323	−	1.42822i
$742$		0			0
$743$	4.17617	+	4.17617i	0.153209	+	0.153209i	0.779549	−	0.626341i	$-0.215448\pi$
$743$	4.17617	+	4.17617i	0.153209	+	0.153209i	−0.626341	+	0.779549i	$0.715448\pi$
$744$		0			0
$745$		−	87.2117i		−	3.19519i
$746$		0			0
$747$	13.9458	−	2.77465i	0.510251	−	0.101519i
$748$		0			0
$749$	−4.94997	−	18.4736i	−0.180868	−	0.675009i
$750$		0			0
$751$			44.7900i			1.63441i	0.576348	+	0.817205i	$0.304477\pi$
$751$			44.7900i			1.63441i	−0.576348	+	0.817205i	$0.695523\pi$
$752$		0			0
$753$	−42.8671	+	22.9142i	−1.56216	+	0.835039i
$754$		0			0
$755$			93.4550i			3.40117i
$756$		0			0
$757$	13.4400	−	23.2787i	0.488484	−	0.846079i	−0.511428	−	0.859326i	$-0.670883\pi$
$757$	13.4400	−	23.2787i	0.488484	−	0.846079i	0.999912	+	0.0132468i	$0.00421670\pi$
$758$		0			0
$759$	−2.26259	+	7.45935i	−0.0821267	+	0.270757i
$760$		0			0
$761$	9.52394	+	35.5438i	0.345242	+	1.28846i	0.892329	+	0.451386i	$0.149070\pi$
$761$	9.52394	+	35.5438i	0.345242	+	1.28846i	−0.547086	+	0.837076i	$0.684263\pi$
$762$		0			0
$763$	−30.0252			−1.08699
$764$		0			0
$765$	53.7778	−	10.6996i	1.94434	−	0.386844i
$766$		0			0
$767$	1.90664	−	31.3582i	0.0688448	−	1.13228i
$768$		0			0
$769$	−20.8065	+	20.8065i	−0.750300	+	0.750300i	−0.974535	−	0.224235i	$-0.928012\pi$
$769$	−20.8065	+	20.8065i	−0.750300	+	0.750300i	0.224235	+	0.974535i	$0.428012\pi$
$770$		0			0
$771$	−0.446963	+	13.6626i	−0.0160970	+	0.492048i
$772$		0			0
$773$	−3.62390	+	13.5246i	−0.130343	+	0.486445i	−0.999974	−	0.00725702i	$-0.997690\pi$
$773$	−3.62390	+	13.5246i	−0.130343	+	0.486445i	0.869631	+	0.493702i	$0.164357\pi$
$774$		0			0
$775$	23.3329	+	23.3329i	0.838144	+	0.838144i
$776$		0			0
$777$	10.6596	−	17.1429i	0.382411	−	0.614998i
$778$		0			0
$779$	46.1416	+	79.9196i	1.65319	+	2.86342i
$780$		0			0
$781$	11.1510	−	19.3141i	0.399014	−	0.691112i
$782$		0			0
$783$	0.0358552	+	0.0791201i	0.00128136	+	0.00282752i
$784$		0			0
$785$	86.6460	−	23.2167i	3.09253	−	0.828640i
$786$		0			0
$787$	9.08397	−	2.43404i	0.323809	−	0.0867643i	−0.0932530	−	0.995642i	$-0.529727\pi$
$787$	9.08397	−	2.43404i	0.323809	−	0.0867643i	0.417062	+	0.908878i	$0.363060\pi$
$788$		0			0
$789$	−4.84661	−	9.06689i	−0.172544	−	0.322790i
$790$		0			0
$791$	1.72940	+	6.45421i	0.0614904	+	0.229485i
$792$		0			0
$793$	−17.8700	−	5.97208i	−0.634582	−	0.212075i
$794$		0			0
$795$	3.44340	−	11.3523i	0.122125	−	0.402624i
$796$		0			0
$797$	−19.9237	+	34.5089i	−0.705735	+	1.22237i	0.260690	+	0.965423i	$0.416050\pi$
$797$	−19.9237	+	34.5089i	−0.705735	+	1.22237i	−0.966426	+	0.256947i	$0.917283\pi$
$798$		0			0
$799$	0.136612	−	0.136612i	0.00483299	−	0.00483299i
$800$		0			0
$801$	7.24675	+	36.4234i	0.256051	+	1.28696i
$802$		0			0
$803$	−0.0932461	−	0.161507i	−0.00329058	−	0.00569946i
$804$		0			0
$805$	−6.41989	+	3.70653i	−0.226272	+	0.130638i
$806$		0			0
$807$	−6.97894	−	13.0560i	−0.245670	−	0.459592i
$808$		0			0
$809$	3.19032	+	1.84193i	0.112166	+	0.0647588i	0.555033	−	0.831828i	$-0.312706\pi$
$809$	3.19032	+	1.84193i	0.112166	+	0.0647588i	−0.442868	+	0.896587i	$0.646039\pi$
$810$		0			0
$811$	23.1531	−	23.1531i	0.813016	−	0.813016i	−0.172069	−	0.985085i	$-0.555045\pi$
$811$	23.1531	−	23.1531i	0.813016	−	0.813016i	0.985085	+	0.172069i	$0.0550453\pi$
$812$		0			0
$813$	2.11295	+	9.06102i	0.0741043	+	0.317784i
$814$		0			0
$815$	44.2195	+	25.5301i	1.54894	+	0.894282i
$816$		0			0
$817$	−4.54937	+	16.9785i	−0.159162	+	0.594002i
$818$		0			0
$819$	7.88484	+	18.7819i	0.275519	+	0.656294i
$820$		0			0
$821$	3.35025	−	12.5033i	0.116924	−	0.436368i	−0.882499	−	0.470314i	$-0.844141\pi$
$821$	3.35025	−	12.5033i	0.116924	−	0.436368i	0.999424	+	0.0339458i	$0.0108074\pi$
$822$		0			0
$823$	18.9105	+	10.9180i	0.659178	+	0.380577i	0.791964	−	0.610568i	$-0.209059\pi$
$823$	18.9105	+	10.9180i	0.659178	+	0.380577i	−0.132785	+	0.991145i	$0.542392\pi$
$824$		0			0
$825$	−59.8938	+	56.0991i	−2.08523	+	1.95312i
$826$		0			0
$827$	−16.4531	+	16.4531i	−0.572131	+	0.572131i	−0.932724	−	0.360592i	$-0.882575\pi$
$827$	−16.4531	+	16.4531i	−0.572131	+	0.572131i	0.360592	+	0.932724i	$0.382575\pi$
$828$		0			0
$829$	−21.9234	−	12.6575i	−0.761432	−	0.439613i	0.0683775	−	0.997660i	$-0.478218\pi$
$829$	−21.9234	−	12.6575i	−0.761432	−	0.439613i	−0.829810	+	0.558046i	$0.811551\pi$
$830$		0			0
$831$	−40.1416	−	1.31320i	−1.39250	−	0.0455545i
$832$		0			0
$833$	13.8762	−	8.01144i	0.480783	−	0.277580i
$834$		0			0
$835$	11.9260	+	20.6564i	0.412715	+	0.714843i
$836$		0			0
$837$	5.74040	−	15.2550i	0.198417	−	0.527289i
$838$		0			0
$839$	−15.3174	+	15.3174i	−0.528814	+	0.528814i	−0.920219	−	0.391404i	$-0.871989\pi$
$839$	−15.3174	+	15.3174i	−0.528814	+	0.528814i	0.391404	+	0.920219i	$0.371989\pi$
$840$		0			0
$841$	−14.4999	+	25.1145i	−0.499995	+	0.866017i
$842$		0			0
$843$	−6.39531	+	1.49133i	−0.220266	+	0.0513640i
$844$		0			0
$845$	−40.3340	+	31.5591i	−1.38753	+	1.08566i
$846$		0			0
$847$	−4.52572	−	16.8902i	−0.155505	−	0.580354i
$848$		0			0
$849$	−48.5867	−	1.58948i	−1.66749	−	0.0545507i
$850$		0			0
$851$	5.97333	−	1.60055i	0.204763	−	0.0548661i
$852$		0			0
$853$	12.3890	−	3.31961i	0.424190	−	0.113661i	−0.0404076	−	0.999183i	$-0.512866\pi$
$853$	12.3890	−	3.31961i	0.424190	−	0.113661i	0.464598	+	0.885522i	$0.346199\pi$
$854$		0			0
$855$	50.1148	−	75.0093i	1.71389	−	2.56527i
$856$		0			0
$857$	24.4649	−	42.3745i	0.835705	−	1.44748i	−0.0577497	−	0.998331i	$-0.518393\pi$
$857$	24.4649	−	42.3745i	0.835705	−	1.44748i	0.893455	−	0.449153i	$-0.148274\pi$
$858$		0			0
$859$	18.1131	+	31.3727i	0.618010	+	1.07042i	0.989848	+	0.142127i	$0.0453941\pi$
$859$	18.1131	+	31.3727i	0.618010	+	1.07042i	−0.371839	+	0.928297i	$0.621273\pi$
$860$		0			0
$861$	−39.4141	−	1.28940i	−1.34323	−	0.0439427i
$862$		0			0
$863$	25.6667	+	25.6667i	0.873704	+	0.873704i	0.992874	−	0.119170i	$-0.0380232\pi$
$863$	25.6667	+	25.6667i	0.873704	+	0.873704i	−0.119170	+	0.992874i	$0.538023\pi$
$864$		0			0
$865$	−0.122008	+	0.455341i	−0.00414840	+	0.0154820i
$866$		0			0
$867$	6.65583	+	4.13865i	0.226044	+	0.140556i
$868$		0			0
$869$	−7.50162	+	7.50162i	−0.254475	+	0.254475i
$870$		0			0
$871$	−0.239129	+	0.119333i	−0.00810257	+	0.00404343i
$872$		0			0
$873$	−19.2004	+	16.8398i	−0.649834	+	0.569941i
$874$		0			0
$875$	−40.9486			−1.38432
$876$		0			0
$877$	−7.39964	−	27.6158i	−0.249868	−	0.932521i	−0.970874	−	0.239592i	$-0.922986\pi$
$877$	−7.39964	−	27.6158i	−0.249868	−	0.932521i	0.721006	−	0.692929i	$-0.243680\pi$
$878$		0			0
$879$	−22.4854	+	5.24339i	−0.758413	+	0.176855i
$880$		0			0
$881$	10.2203	−	17.7021i	0.344331	−	0.596399i	−0.640901	−	0.767624i	$-0.721439\pi$
$881$	10.2203	−	17.7021i	0.344331	−	0.596399i	0.985232	+	0.171225i	$0.0547724\pi$
$882$		0			0
$883$		−	59.1925i		−	1.99199i	−0.0894367	−	0.995993i	$-0.528507\pi$
$883$		−	59.1925i		−	1.99199i	0.0894367	−	0.995993i	$-0.471493\pi$
$884$		0			0
$885$	1.94395	−	59.4222i	0.0653452	−	1.99745i
$886$		0			0
$887$			44.9612i			1.50965i	0.655927	+	0.754825i	$0.272278\pi$
$887$			44.9612i			1.50965i	−0.655927	+	0.754825i	$0.727722\pi$
$888$		0			0
$889$	4.38371	+	16.3602i	0.147025	+	0.548705i
$890$		0			0
$891$	37.4511	+	15.5088i	1.25466	+	0.519565i
$892$		0			0
$893$		−	0.317854i		−	0.0106366i
$894$		0			0
$895$	−39.2593	−	39.2593i	−1.31229	−	1.31229i
$896$		0			0
$897$	−2.17035	+	5.85055i	−0.0724660	+	0.195344i
$898$		0			0
$899$	−0.0506518	+	0.0135721i	−0.00168933	+	0.000452655i
$900$		0			0
$901$	6.98550	−	4.03308i	0.232721	−	0.134361i
$902$		0			0
$903$	−5.13480	−	5.48212i	−0.170875	−	0.182434i
$904$		0			0
$905$	−20.4721	−	5.48550i	−0.680517	−	0.182344i
$906$		0			0
$907$	−16.8952	+	9.75445i	−0.560996	+	0.323891i	−0.753545	−	0.657396i	$-0.771658\pi$
$907$	−16.8952	+	9.75445i	−0.560996	+	0.323891i	0.192549	+	0.981287i	$0.438325\pi$
$908$		0			0
$909$	−18.2942	−	37.0928i	−0.606781	−	1.23029i
$910$		0			0
$911$	−51.3564	−	29.6506i	−1.70151	−	0.982370i	−0.944231	−	0.329283i	$-0.893193\pi$
$911$	−51.3564	−	29.6506i	−1.70151	−	0.982370i	−0.757283	−	0.653087i	$-0.773474\pi$
$912$		0			0
$913$	−21.3473			−0.706494
$914$		0			0
$915$	−34.1218	−	10.3499i	−1.12803	−	0.342157i
$916$		0			0
$917$	−18.6012	−	4.98417i	−0.614265	−	0.164592i
$918$		0			0
$919$	21.8304	+	37.8114i	0.720119	+	1.24728i	0.960952	+	0.276716i	$0.0892463\pi$
$919$	21.8304	+	37.8114i	0.720119	+	1.24728i	−0.240832	+	0.970567i	$0.577420\pi$
$920$		0			0
$921$	16.2665	−	15.2359i	0.536000	−	0.502041i
$922$		0			0
$923$	9.84868	−	14.8914i	0.324173	−	0.490155i
$924$		0			0
$925$	62.8857	+	16.8502i	2.06767	+	0.554030i
$926$		0			0
$927$	44.4268	−	21.9113i	1.45917	−	0.719663i
$928$		0			0
$929$	−12.0748	−	12.0748i	−0.396160	−	0.396160i	0.480716	−	0.876876i	$-0.340377\pi$
$929$	−12.0748	−	12.0748i	−0.396160	−	0.396160i	−0.876876	+	0.480716i	$0.840377\pi$
$930$		0			0
$931$	6.82276	−	25.4629i	0.223607	−	0.834512i
$932$		0			0
$933$	4.24808	−	2.27077i	0.139076	−	0.0743416i
$934$		0			0
$935$	−82.3195			−2.69213
$936$		0			0
$937$	41.1938			1.34574			0.672871	−	0.739759i	$-0.265061\pi$
$937$	41.1938			1.34574			0.672871	+	0.739759i	$0.265061\pi$
$938$		0			0
$939$	25.7528	+	16.0133i	0.840412	+	0.522575i
$940$		0			0
$941$	10.3736	−	38.7147i	0.338169	−	1.26206i	−0.562224	−	0.826985i	$-0.690054\pi$
$941$	10.3736	−	38.7147i	0.338169	−	1.26206i	0.900393	−	0.435078i	$-0.143279\pi$
$942$		0			0
$943$	−8.54230	−	8.54230i	−0.278175	−	0.278175i
$944$		0			0
$945$	15.9119	+	35.1121i	0.517616	+	1.14220i
$946$		0			0
$947$	24.5842	+	6.58731i	0.798879	+	0.214059i	0.635092	−	0.772437i	$-0.280962\pi$
$947$	24.5842	+	6.58731i	0.798879	+	0.214059i	0.163787	+	0.986496i	$0.447629\pi$
$948$		0			0
$949$	−0.0666626	−	0.133584i	−0.00216396	−	0.00433633i
$950$		0			0
$951$	−8.26969	−	35.4632i	−0.268163	−	1.14997i
$952$		0			0
$953$	−11.7501	−	20.3518i	−0.380624	−	0.659260i	0.610527	−	0.791995i	$-0.290957\pi$
$953$	−11.7501	−	20.3518i	−0.380624	−	0.659260i	−0.991152	+	0.132735i	$0.957624\pi$
$954$		0			0
$955$	−39.3524	−	10.5444i	−1.27341	−	0.341210i
$956$		0			0
$957$	−0.0296162	−	0.127004i	−0.000957355	−	0.00410546i
$958$		0			0
$959$	33.4826			1.08121
$960$		0			0
$961$	−18.3255	−	10.5802i	−0.591146	−	0.341298i
$962$		0			0
$963$	−25.3333	−	16.9255i	−0.816354	−	0.545418i
$964$		0			0
$965$	−54.3580	+	31.3836i	−1.74984	+	1.01027i
$966$		0			0
$967$	−0.799247	−	0.214158i	−0.0257021	−	0.00688685i	0.245945	−	0.969284i	$-0.420902\pi$
$967$	−0.799247	−	0.214158i	−0.0257021	−	0.00688685i	−0.271647	+	0.962397i	$0.587568\pi$
$968$		0			0
$969$	59.7351	−	13.9297i	1.91897	−	0.447486i
$970$		0			0
$971$	41.6673	−	24.0566i	1.33717	−	0.772013i	0.350780	−	0.936458i	$-0.385916\pi$
$971$	41.6673	−	24.0566i	1.33717	−	0.772013i	0.986386	+	0.164444i	$0.0525831\pi$
$972$		0			0
$973$	−20.0030	+	5.35980i	−0.641268	+	0.171827i
$974$		0			0
$975$	−50.5843	+	41.9167i	−1.61999	+	1.34241i
$976$		0			0
$977$	−8.77017	−	8.77017i	−0.280583	−	0.280583i	0.552759	−	0.833341i	$-0.313575\pi$
$977$	−8.77017	−	8.77017i	−0.280583	−	0.280583i	−0.833341	+	0.552759i	$0.813575\pi$
$978$		0			0
$979$		−	55.7545i		−	1.78192i
$980$		0			0
$981$	−35.9600	+	31.5389i	−1.14812	+	1.00696i
$982$		0			0
$983$	−11.2667	−	42.0477i	−0.359350	−	1.34111i	−0.874920	−	0.484267i	$-0.839086\pi$
$983$	−11.2667	−	42.0477i	−0.359350	−	1.34111i	0.515570	−	0.856847i	$-0.327580\pi$
$984$		0			0
$985$		−	34.2709i		−	1.09196i
$986$		0			0
$987$	0.115347	+	0.0717237i	0.00367153	+	0.00228299i
$988$		0			0
$989$		−	2.30103i		−	0.0731684i
$990$		0			0
$991$	2.07400	−	3.59227i	0.0658826	−	0.114112i	−0.831203	−	0.555970i	$-0.812347\pi$
$991$	2.07400	−	3.59227i	0.0658826	−	0.114112i	0.897085	+	0.441858i	$0.145680\pi$
$992$		0			0
$993$	−1.32774	−	1.41755i	−0.0421347	−	0.0449847i
$994$		0			0
$995$	14.3439	+	53.5322i	0.454732	+	1.69708i
$996$		0			0
$997$	25.4139			0.804868			0.402434	−	0.915449i	$-0.368164\pi$
$997$	25.4139			0.804868			0.402434	+	0.915449i	$0.368164\pi$
$998$		0			0
$999$	−5.24056	−	31.7284i	−0.165804	−	1.00384i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	468.2.bz.a.461.7	yes	56
3.2	odd	2		1404.2.cc.a.305.14		56
9.4	even	3		1404.2.bz.a.773.14		56
9.5	odd	6		468.2.bw.a.149.12	✓	56
13.11	odd	12		468.2.bw.a.245.12	yes	56
39.11	even	12		1404.2.bz.a.89.14		56
117.50	even	12	inner	468.2.bz.a.401.7	yes	56
117.76	odd	12		1404.2.cc.a.557.14		56

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
468.2.bw.a.149.12	✓	56	9.5	odd	6
468.2.bw.a.245.12	yes	56	13.11	odd	12
468.2.bz.a.401.7	yes	56	117.50	even	12	inner
468.2.bz.a.461.7	yes	56	1.1	even	1	trivial
1404.2.bz.a.89.14		56	39.11	even	12
1404.2.bz.a.773.14		56	9.4	even	3
1404.2.cc.a.305.14		56	3.2	odd	2
1404.2.cc.a.557.14		56	117.76	odd	12

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:	\( N \)	\(=\)	\( 468 = 2^{2} \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	468.bz (of order \(12\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\(q+(0.0566324 - 1.73112i) q^{3} +(-1.01961 + 3.80525i) q^{5} +(-1.33162 - 1.33162i) q^{7} +(-2.99359 - 0.196076i) q^{9} +(4.35045 + 1.16570i) q^{11} +(3.41964 + 1.14283i) q^{13} +(6.52962 + 1.98058i) q^{15} +(2.31976 + 4.01793i) q^{17} +(7.37291 + 1.97557i) q^{19} +(-2.38061 + 2.22979i) q^{21} -0.999222 q^{23} +(-9.11020 - 5.25978i) q^{25} +(-0.508965 + 5.17117i) q^{27} +(0.0144776 - 0.00835862i) q^{29} +(-3.02992 - 0.811864i) q^{31} +(2.26435 - 7.46516i) q^{33} +(6.42489 - 3.70941i) q^{35} +(-5.97798 + 1.60179i) q^{37} +(2.17204 - 5.85510i) q^{39} +(8.54895 + 8.54895i) q^{41} +2.30282i q^{43} +(3.79842 - 11.1914i) q^{45} +(-0.0107778 - 0.0402231i) q^{47} -3.45357i q^{49} +(7.08692 - 3.78824i) q^{51} -1.73858i q^{53} +(-8.87157 + 15.3660i) q^{55} +(3.83750 - 12.6515i) q^{57} +(-2.25516 - 8.41636i) q^{59} -5.22569 q^{61} +(3.72522 + 4.24742i) q^{63} +(-7.83547 + 11.8473i) q^{65} +(-0.0524121 + 0.0524121i) q^{67} +(-0.0565884 + 1.72978i) q^{69} +(1.28159 - 4.78296i) q^{71} +(-0.0292789 - 0.0292789i) q^{73} +(-9.62127 + 15.4730i) q^{75} +(-4.24088 - 7.34542i) q^{77} +(-1.17774 + 2.03990i) q^{79} +(8.92311 + 1.17394i) q^{81} +(-4.57822 + 1.22673i) q^{83} +(-17.6545 + 4.73051i) q^{85} +(-0.0136499 - 0.0255358i) q^{87} +(-3.20395 - 11.9573i) q^{89} +(-3.03185 - 6.07548i) q^{91} +(-1.57703 + 5.19918i) q^{93} +(-15.0350 + 26.0415i) q^{95} +(6.01956 - 6.01956i) q^{97} +(-12.7949 - 4.34264i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q + 4 q^{7} + 12 q^{11} + 12 q^{15} + 4 q^{19} - 12 q^{21} - 4 q^{31} + 18 q^{33} + 66 q^{35} + 2 q^{37} + 24 q^{39} + 24 q^{41} + 24 q^{45} - 36 q^{47} - 12 q^{57} + 6 q^{63} - 36 q^{65} - 14 q^{67}+ \cdots - 48 q^{99}+O(q^{100}) \) 56 * q + 4 * q^7 + 12 * q^11 + 12 * q^15 + 4 * q^19 - 12 * q^21 - 4 * q^31 + 18 * q^33 + 66 * q^35 + 2 * q^37 + 24 * q^39 + 24 * q^41 + 24 * q^45 - 36 * q^47 - 12 * q^57 + 6 * q^63 - 36 * q^65 - 14 * q^67 - 42 * q^69 - 24 * q^71 - 14 * q^73 - 90 * q^75 + 24 * q^77 + 12 * q^79 - 12 * q^81 - 42 * q^83 - 36 * q^85 - 48 * q^87 + 4 * q^91 + 36 * q^93 - 46 * q^97 - 48 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more