LMFDB - Embedded newform 420.2.bn.a.269.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [420,2,Mod(89,420)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(420, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 3, 3, 5])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("420.89"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 420 = 2^{2} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	420.bn (of order $6$, degree $2$, 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:	$3.35371688489$
Analytic rank:	$0$
Dimension:	$32$
Relative dimension:	$16$ over $\Q(\zeta_{6})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			269.3
Character	$\chi$	$=$	420.269
Dual form			420.2.bn.a.89.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-1.48906 + 0.884700i) q^{3} +(-1.06703 + 1.96506i) q^{5} +(0.0973684 + 2.64396i) q^{7} +(1.43461 - 2.63475i) q^{9} +(-2.62913 - 1.51793i) q^{11} +2.31738 q^{13} +(-0.149616 - 3.87009i) q^{15} +(-4.49684 - 2.59625i) q^{17} +(-5.58515 + 3.22459i) q^{19} +(-2.48410 - 3.85088i) q^{21} +(-2.43442 - 4.21654i) q^{23} +(-2.72290 - 4.19354i) q^{25} +(0.194737 + 5.19250i) q^{27} +3.48622i q^{29} +(1.16615 + 0.673279i) q^{31} +(5.25784 - 0.0657018i) q^{33} +(-5.29943 - 2.62984i) q^{35} +(-2.43579 + 1.40630i) q^{37} +(-3.45072 + 2.05018i) q^{39} +1.06401 q^{41} +2.42063i q^{43} +(3.64666 + 5.63044i) q^{45} +(-3.30233 + 1.90660i) q^{47} +(-6.98104 + 0.514876i) q^{49} +(8.99298 - 0.112376i) q^{51} +(3.52984 - 6.11386i) q^{53} +(5.78817 - 3.54672i) q^{55} +(5.46384 - 9.74279i) q^{57} +(-6.18029 + 10.7046i) q^{59} +(11.4447 - 6.60758i) q^{61} +(7.10585 + 3.53651i) q^{63} +(-2.47270 + 4.55378i) q^{65} +(-3.72081 - 2.14821i) q^{67} +(7.35538 + 4.12496i) q^{69} +7.08288i q^{71} +(-0.0763237 + 0.132196i) q^{73} +(7.76460 + 3.83549i) q^{75} +(3.75734 - 7.09910i) q^{77} +(-3.04462 - 5.27344i) q^{79} +(-4.88378 - 7.55967i) q^{81} +10.2758i q^{83} +(9.90003 - 6.06628i) q^{85} +(-3.08426 - 5.19119i) q^{87} +(7.10913 + 12.3134i) q^{89} +(0.225639 + 6.12705i) q^{91} +(-2.33212 + 0.0291421i) q^{93} +(-0.376991 - 14.4159i) q^{95} +8.63266 q^{97} +(-7.77113 + 4.74945i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q + 2 q^{15} + 12 q^{19} - 8 q^{21} + 6 q^{25} - 12 q^{31} + 24 q^{39} + 33 q^{45} - 44 q^{49} - 10 q^{51} - 24 q^{61} + 21 q^{75} - 28 q^{79} - 20 q^{81} - 4 q^{85} + 16 q^{91} - 28 q^{99}+O(q^{100}) $ 32 * q + 2 * q^15 + 12 * q^19 - 8 * q^21 + 6 * q^25 - 12 * q^31 + 24 * q^39 + 33 * q^45 - 44 * q^49 - 10 * q^51 - 24 * q^61 + 21 * q^75 - 28 * q^79 - 20 * q^81 - 4 * q^85 + 16 * q^91 - 28 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/420\mathbb{Z}\right)^\times$.

$n$	$211$	$241$	$281$	$337$
$\chi(n)$	$1$	$e\left(\frac{1}{6}\right)$	$-1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−1.48906	+	0.884700i	−0.859710	+	0.510782i
$3$	−1.48906	+	0.884700i	−0.859710	+	0.510782i
$4$		0			0
$5$	−1.06703	+	1.96506i	−0.477189	+	0.878801i
$5$	−1.06703	+	1.96506i	−0.477189	+	0.878801i
$6$		0			0
$7$	0.0973684	+	2.64396i	0.0368018	+	0.999323i
$7$	0.0973684	+	2.64396i	0.0368018	+	0.999323i
$8$		0			0
$9$	1.43461	−	2.63475i	0.478204	−	0.878249i
$10$		0			0
$11$	−2.62913	−	1.51793i	−0.792712	−	0.457672i	0.0482045	−	0.998837i	$-0.484650\pi$
$11$	−2.62913	−	1.51793i	−0.792712	−	0.457672i	−0.840916	+	0.541165i	$0.817983\pi$
$12$		0			0
$13$	2.31738			0.642724			0.321362	−	0.946956i	$-0.395859\pi$
$13$	2.31738			0.642724			0.321362	+	0.946956i	$0.395859\pi$
$14$		0			0
$15$	−0.149616	−	3.87009i	−0.0386308	−	0.999254i
$16$		0			0
$17$	−4.49684	−	2.59625i	−1.09064	−	0.629683i	−0.156896	−	0.987615i	$-0.550149\pi$
$17$	−4.49684	−	2.59625i	−1.09064	−	0.629683i	−0.933748	+	0.357932i	$0.883482\pi$
$18$		0			0
$19$	−5.58515	+	3.22459i	−1.28132	+	0.739771i	−0.977090	−	0.212827i	$-0.931733\pi$
$19$	−5.58515	+	3.22459i	−1.28132	+	0.739771i	−0.304231	+	0.952598i	$0.598400\pi$
$20$		0			0
$21$	−2.48410	−	3.85088i	−0.542075	−	0.840330i
$22$		0			0
$23$	−2.43442	−	4.21654i	−0.507612	−	0.879209i	−0.999961	−	0.00881178i	$-0.997195\pi$
$23$	−2.43442	−	4.21654i	−0.507612	−	0.879209i	0.492349	−	0.870398i	$-0.336138\pi$
$24$		0			0
$25$	−2.72290	−	4.19354i	−0.544581	−	0.838708i
$26$		0			0
$27$	0.194737	+	5.19250i	0.0374771	+	0.999297i
$28$		0			0
$29$			3.48622i			0.647374i	0.946164	+	0.323687i	$0.104923\pi$
$29$			3.48622i			0.647374i	−0.946164	+	0.323687i	$0.895077\pi$
$30$		0			0
$31$	1.16615	+	0.673279i	0.209447	+	0.120924i	0.601054	−	0.799208i	$-0.294747\pi$
$31$	1.16615	+	0.673279i	0.209447	+	0.120924i	−0.391607	+	0.920132i	$0.628081\pi$
$32$		0			0
$33$	5.25784	−	0.0657018i	0.915273	−	0.0114372i
$34$		0			0
$35$	−5.29943	−	2.62984i	−0.895767	−	0.444525i
$36$		0			0
$37$	−2.43579	+	1.40630i	−0.400441	+	0.231195i	−0.686674	−	0.726965i	$-0.740930\pi$
$37$	−2.43579	+	1.40630i	−0.400441	+	0.231195i	0.286233	+	0.958160i	$0.407597\pi$
$38$		0			0
$39$	−3.45072	+	2.05018i	−0.552557	+	0.328292i
$40$		0			0
$41$	1.06401			0.166171			0.0830853	−	0.996542i	$-0.473523\pi$
$41$	1.06401			0.166171			0.0830853	+	0.996542i	$0.473523\pi$
$42$		0			0
$43$			2.42063i			0.369142i	0.982819	+	0.184571i	$0.0590896\pi$
$43$			2.42063i			0.369142i	−0.982819	+	0.184571i	$0.940910\pi$
$44$		0			0
$45$	3.64666	+	5.63044i	0.543612	+	0.839337i
$46$		0			0
$47$	−3.30233	+	1.90660i	−0.481695	+	0.278106i	−0.721122	−	0.692808i	$-0.756373\pi$
$47$	−3.30233	+	1.90660i	−0.481695	+	0.278106i	0.239428	+	0.970914i	$0.423040\pi$
$48$		0			0
$49$	−6.98104	+	0.514876i	−0.997291	+	0.0735537i
$50$		0			0
$51$	8.99298	−	0.112376i	1.25927	−	0.0157358i
$52$		0			0
$53$	3.52984	−	6.11386i	0.484861	−	0.839803i	−0.514988	−	0.857197i	$-0.672204\pi$
$53$	3.52984	−	6.11386i	0.484861	−	0.839803i	0.999849	+	0.0173942i	$0.00553703\pi$
$54$		0			0
$55$	5.78817	−	3.54672i	0.780476	−	0.478239i
$56$		0			0
$57$	5.46384	−	9.74279i	0.723703	−	1.29046i
$58$		0			0
$59$	−6.18029	+	10.7046i	−0.804605	+	1.39362i	0.111953	+	0.993714i	$0.464289\pi$
$59$	−6.18029	+	10.7046i	−0.804605	+	1.39362i	−0.916558	+	0.399903i	$0.869044\pi$
$60$		0			0
$61$	11.4447	−	6.60758i	1.46534	−	0.846014i	0.466088	−	0.884738i	$-0.345663\pi$
$61$	11.4447	−	6.60758i	1.46534	−	0.846014i	0.999250	+	0.0387246i	$0.0123295\pi$
$62$		0			0
$63$	7.10585	+	3.53651i	0.895253	+	0.445559i
$64$		0			0
$65$	−2.47270	+	4.55378i	−0.306701	+	0.564826i
$66$		0			0
$67$	−3.72081	−	2.14821i	−0.454569	−	0.262446i	0.255189	−	0.966891i	$-0.417862\pi$
$67$	−3.72081	−	2.14821i	−0.454569	−	0.262446i	−0.709758	+	0.704446i	$0.751196\pi$
$68$		0			0
$69$	7.35538	+	4.12496i	0.885483	+	0.496586i
$70$		0			0
$71$			7.08288i			0.840583i	0.907389	+	0.420292i	$0.138072\pi$
$71$			7.08288i			0.840583i	−0.907389	+	0.420292i	$0.861928\pi$
$72$		0			0
$73$	−0.0763237	+	0.132196i	−0.00893301	+	0.0154724i	−0.870457	−	0.492244i	$-0.836177\pi$
$73$	−0.0763237	+	0.132196i	−0.00893301	+	0.0154724i	0.861524	+	0.507716i	$0.169510\pi$
$74$		0			0
$75$	7.76460	+	3.83549i	0.896579	+	0.442884i
$76$		0			0
$77$	3.75734	−	7.09910i	0.428189	−	0.809018i
$78$		0			0
$79$	−3.04462	−	5.27344i	−0.342547	−	0.593309i	0.642358	−	0.766405i	$-0.277956\pi$
$79$	−3.04462	−	5.27344i	−0.342547	−	0.593309i	−0.984905	+	0.173096i	$0.944623\pi$
$80$		0			0
$81$	−4.88378	−	7.55967i	−0.542643	−	0.839964i
$82$		0			0
$83$			10.2758i			1.12791i	0.825804	+	0.563957i	$0.190722\pi$
$83$			10.2758i			1.12791i	−0.825804	+	0.563957i	$0.809278\pi$
$84$		0			0
$85$	9.90003	−	6.06628i	1.07381	−	0.657980i
$86$		0			0
$87$	−3.08426	−	5.19119i	−0.330667	−	0.556555i
$88$		0			0
$89$	7.10913	+	12.3134i	0.753567	+	1.30522i	0.946084	+	0.323922i	$0.105002\pi$
$89$	7.10913	+	12.3134i	0.753567	+	1.30522i	−0.192517	+	0.981294i	$0.561665\pi$
$90$		0			0
$91$	0.225639	+	6.12705i	0.0236534	+	0.642289i
$92$		0			0
$93$	−2.33212	+	0.0291421i	−0.241830	+	0.00302189i
$94$		0			0
$95$	−0.376991	−	14.4159i	−0.0386784	−	1.47904i
$96$		0			0
$97$	8.63266			0.876514			0.438257	−	0.898850i	$-0.355596\pi$
$97$	8.63266			0.876514			0.438257	+	0.898850i	$0.355596\pi$
$98$		0			0
$99$	−7.77113	+	4.74945i	−0.781028	+	0.477338i
$100$		0			0
$101$	−8.23108	+	14.2566i	−0.819023	+	1.41859i	0.0873801	+	0.996175i	$0.472151\pi$
$101$	−8.23108	+	14.2566i	−0.819023	+	1.41859i	−0.906403	+	0.422414i	$0.861183\pi$
$102$		0			0
$103$	−3.30237	−	5.71987i	−0.325392	−	0.563596i	0.656199	−	0.754588i	$-0.272163\pi$
$103$	−3.30237	−	5.71987i	−0.325392	−	0.563596i	−0.981592	+	0.190992i	$0.938830\pi$
$104$		0			0
$105$	10.2178	−	0.772405i	0.997155	−	0.0753790i
$106$		0			0
$107$	9.26659	+	16.0502i	0.895835	+	1.55163i	0.832768	+	0.553623i	$0.186755\pi$
$107$	9.26659	+	16.0502i	0.895835	+	1.55163i	0.0630674	+	0.998009i	$0.479912\pi$
$108$		0			0
$109$	−9.25130	+	16.0237i	−0.886114	+	1.53479i	−0.0416824	+	0.999131i	$0.513272\pi$
$109$	−9.25130	+	16.0237i	−0.886114	+	1.53479i	−0.844432	+	0.535664i	$0.820062\pi$
$110$		0			0
$111$	2.38288	−	4.24901i	0.226173	−	0.403299i
$112$		0			0
$113$	5.86968			0.552173			0.276087	−	0.961133i	$-0.410962\pi$
$113$	5.86968			0.552173			0.276087	+	0.961133i	$0.410962\pi$
$114$		0			0
$115$	10.8833	−	0.284611i	1.01488	−	0.0265401i
$116$		0			0
$117$	3.32453	−	6.10570i	0.307353	−	0.564472i
$118$		0			0
$119$	6.42653	−	12.1423i	0.589119	−	1.11308i
$120$		0			0
$121$	−0.891792	−	1.54463i	−0.0810720	−	0.140421i
$122$		0			0
$123$	−1.58438	+	0.941331i	−0.142859	+	0.0848769i
$124$		0			0
$125$	11.1460	−	0.876035i	0.996926	−	0.0783550i
$126$		0			0
$127$			14.5896i			1.29462i	0.762226	+	0.647311i	$0.224106\pi$
$127$			14.5896i			1.29462i	−0.762226	+	0.647311i	$0.775894\pi$
$128$		0			0
$129$	−2.14153	−	3.60446i	−0.188551	−	0.317355i
$130$		0			0
$131$	−10.0356	−	17.3822i	−0.876817	−	1.51869i	−0.854815	−	0.518933i	$-0.826329\pi$
$131$	−10.0356	−	17.3822i	−0.876817	−	1.51869i	−0.0220016	−	0.999758i	$-0.507004\pi$
$132$		0			0
$133$	−9.06949	−	14.4529i	−0.786425	−	1.25323i
$134$		0			0
$135$	−10.4114	−	5.15788i	−0.896067	−	0.443919i
$136$		0			0
$137$	2.33533	−	4.04491i	0.199521	−	0.345580i	−0.748852	−	0.662737i	$-0.769395\pi$
$137$	2.33533	−	4.04491i	0.199521	−	0.345580i	0.948373	+	0.317157i	$0.102728\pi$
$138$		0			0
$139$		−	5.51296i		−	0.467603i	−0.972284	−	0.233801i	$-0.924883\pi$
$139$		−	5.51296i		−	0.467603i	0.972284	−	0.233801i	$-0.0751166\pi$
$140$		0			0
$141$	3.23060	−	5.76062i	0.272066	−	0.485132i
$142$		0			0
$143$	−6.09268	−	3.51761i	−0.509495	−	0.294157i
$144$		0			0
$145$	−6.85062	−	3.71989i	−0.568913	−	0.308920i
$146$		0			0
$147$	9.93969	−	6.94281i	0.819812	−	0.572633i
$148$		0			0
$149$	−12.6579	+	7.30806i	−1.03698	+	0.598700i	−0.918976	−	0.394313i	$-0.870982\pi$
$149$	−12.6579	+	7.30806i	−1.03698	+	0.598700i	−0.118003	+	0.993013i	$0.537649\pi$
$150$		0			0
$151$	−4.48104	+	7.76139i	−0.364662	+	0.631613i	−0.988722	−	0.149764i	$-0.952149\pi$
$151$	−4.48104	+	7.76139i	−0.364662	+	0.631613i	0.624060	+	0.781376i	$0.285482\pi$
$152$		0			0
$153$	−13.2917	+	8.12342i	−1.07457	+	0.656740i
$154$		0			0
$155$	−2.56735	+	1.57315i	−0.206214	+	0.126358i
$156$		0			0
$157$	−10.2703	+	17.7886i	−0.819657	+	1.41969i	0.0862784	+	0.996271i	$0.472503\pi$
$157$	−10.2703	+	17.7886i	−0.819657	+	1.41969i	−0.905935	+	0.423416i	$0.860831\pi$
$158$		0			0
$159$	0.152785	+	12.2268i	0.0121166	+	0.969645i
$160$		0			0
$161$	10.9113	−	6.84707i	0.859933	−	0.539624i
$162$		0			0
$163$	17.3645	−	10.0254i	1.36009	−	0.785250i	0.370457	−	0.928850i	$-0.379201\pi$
$163$	17.3645	−	10.0254i	1.36009	−	0.785250i	0.989636	+	0.143600i	$0.0458678\pi$
$164$		0			0
$165$	−5.48116	+	10.4021i	−0.426708	+	0.809800i
$166$		0			0
$167$		−	12.2463i		−	0.947646i	−0.880620	−	0.473823i	$-0.842874\pi$
$167$		−	12.2463i		−	0.947646i	0.880620	−	0.473823i	$-0.157126\pi$
$168$		0			0
$169$	−7.62977			−0.586906
$170$		0			0
$171$	0.483456	+	19.3415i	0.0369708	+	1.47908i
$172$		0			0
$173$	−18.9007	+	10.9123i	−1.43699	+	0.829647i	−0.997640	−	0.0686676i	$-0.978125\pi$
$173$	−18.9007	+	10.9123i	−1.43699	+	0.829647i	−0.439352	+	0.898315i	$0.644792\pi$
$174$		0			0
$175$	10.8224	−	7.60756i	0.818099	−	0.575078i
$176$		0			0
$177$	−0.267507	−	21.4075i	−0.0201070	−	1.60908i
$178$		0			0
$179$	−1.11034	−	0.641056i	−0.0829909	−	0.0479148i	0.457930	−	0.888988i	$-0.348591\pi$
$179$	−1.11034	−	0.641056i	−0.0829909	−	0.0479148i	−0.540921	+	0.841073i	$0.681924\pi$
$180$		0			0
$181$		−	14.5617i		−	1.08236i	−0.840906	−	0.541182i	$-0.817977\pi$
$181$		−	14.5617i		−	1.08236i	0.840906	−	0.541182i	$-0.182023\pi$
$182$		0			0
$183$	−11.1961	+	19.9642i	−0.827638	+	1.47579i
$184$		0			0
$185$	−0.164413	−	6.28703i	−0.0120879	−	0.462232i
$186$		0			0
$187$	7.88184	+	13.6518i	0.576377	+	0.998315i
$188$		0			0
$189$	−13.7098	+	1.02046i	−0.997241	+	0.0742277i
$190$		0			0
$191$	20.7027	−	11.9527i	1.49800	−	0.864869i	0.498000	−	0.867177i	$-0.334068\pi$
$191$	20.7027	−	11.9527i	1.49800	−	0.864869i	0.999997	+	0.00230818i	$0.000734719\pi$
$192$		0			0
$193$	7.30736	+	4.21891i	0.525996	+	0.303684i	0.739384	−	0.673284i	$-0.235117\pi$
$193$	7.30736	+	4.21891i	0.525996	+	0.303684i	−0.213389	+	0.976967i	$0.568450\pi$
$194$		0			0
$195$	−0.346718	−	8.96846i	−0.0248290	−	0.642244i
$196$		0			0
$197$	−8.25869			−0.588408			−0.294204	−	0.955743i	$-0.595054\pi$
$197$	−8.25869			−0.588408			−0.294204	+	0.955743i	$0.595054\pi$
$198$		0			0
$199$	8.94051	+	5.16181i	0.633776	+	0.365911i	0.782213	−	0.623011i	$-0.214091\pi$
$199$	8.94051	+	5.16181i	0.633776	+	0.365911i	−0.148437	+	0.988922i	$0.547424\pi$
$200$		0			0
$201$	7.44104	−	0.0929828i	0.524850	−	0.00655850i
$202$		0			0
$203$	−9.21742	+	0.339448i	−0.646936	+	0.0238245i
$204$		0			0
$205$	−1.13533	+	2.09084i	−0.0792948	+	0.146031i
$206$		0			0
$207$	−14.6020	+	0.364988i	−1.01491	+	0.0253684i
$208$		0			0
$209$	19.5788			1.35429
$210$		0			0
$211$	−5.75694			−0.396324			−0.198162	−	0.980169i	$-0.563497\pi$
$211$	−5.75694			−0.396324			−0.198162	+	0.980169i	$0.563497\pi$
$212$		0			0
$213$	−6.26623	−	10.5468i	−0.429355	−	0.722658i
$214$		0			0
$215$	−4.75667	−	2.58287i	−0.324402	−	0.176151i
$216$		0			0
$217$	−1.66657	+	3.14882i	−0.113134	+	0.213756i
$218$		0			0
$219$	−0.00330358	−	0.264372i	−0.000223236	−	0.0178646i
$220$		0			0
$221$	−10.4209	−	6.01649i	−0.700983	−	0.404713i
$222$		0			0
$223$	−27.0927			−1.81426			−0.907129	−	0.420852i	$-0.861731\pi$
$223$	−27.0927			−1.81426			−0.907129	+	0.420852i	$0.861731\pi$
$224$		0			0
$225$	−14.9552	+	1.15806i	−0.997015	+	0.0772040i
$226$		0			0
$227$	6.20333	+	3.58150i	0.411730	+	0.237712i	0.691533	−	0.722345i	$-0.256936\pi$
$227$	6.20333	+	3.58150i	0.411730	+	0.237712i	−0.279803	+	0.960057i	$0.590269\pi$
$228$		0			0
$229$	4.16615	−	2.40533i	0.275307	−	0.158949i	−0.355990	−	0.934490i	$-0.615856\pi$
$229$	4.16615	−	2.40533i	0.275307	−	0.158949i	0.631297	+	0.775541i	$0.282523\pi$
$230$		0			0
$231$	0.685661	+	13.8951i	0.0451132	+	0.914232i
$232$		0			0
$233$	8.90616	+	15.4259i	0.583462	+	1.01059i	0.995065	+	0.0992223i	$0.0316355\pi$
$233$	8.90616	+	15.4259i	0.583462	+	1.01059i	−0.411604	+	0.911363i	$0.635031\pi$
$234$		0			0
$235$	−0.222903	−	8.52367i	−0.0145406	−	0.556023i
$236$		0			0
$237$	9.19905	+	5.15890i	0.597542	+	0.335107i
$238$		0			0
$239$		−	13.7867i		−	0.891789i	−0.895086	−	0.445894i	$-0.852886\pi$
$239$		−	13.7867i		−	0.891789i	0.895086	−	0.445894i	$-0.147114\pi$
$240$		0			0
$241$	−11.7785	−	6.80032i	−0.758720	−	0.438047i	0.0701158	−	0.997539i	$-0.477663\pi$
$241$	−11.7785	−	6.80032i	−0.758720	−	0.438047i	−0.828836	+	0.559492i	$0.810996\pi$
$242$		0			0
$243$	13.9603	+	6.93614i	0.895554	+	0.444953i
$244$		0			0
$245$	6.43720	−	14.2675i	0.411258	−	0.911519i
$246$		0			0
$247$	−12.9429	+	7.47258i	−0.823536	+	0.475469i
$248$		0			0
$249$	−9.09100	−	15.3013i	−0.576119	−	0.969680i
$250$		0			0
$251$	7.89600			0.498391			0.249196	−	0.968453i	$-0.419834\pi$
$251$	7.89600			0.498391			0.249196	+	0.968453i	$0.419834\pi$
$252$		0			0
$253$			14.7811i			0.929280i
$254$		0			0
$255$	−9.37493	+	17.7916i	−0.587081	+	1.11415i
$256$		0			0
$257$	−15.9997	+	9.23741i	−0.998032	+	0.576214i	−0.907665	−	0.419695i	$-0.862137\pi$
$257$	−15.9997	+	9.23741i	−0.998032	+	0.576214i	−0.0903666	+	0.995909i	$0.528804\pi$
$258$		0			0
$259$	−3.95538	−	6.30320i	−0.245775	−	0.391661i
$260$		0			0
$261$	9.18530	+	5.00137i	0.568556	+	0.309577i
$262$		0			0
$263$	10.7794	−	18.6705i	0.664687	−	1.15127i	−0.314683	−	0.949197i	$-0.601898\pi$
$263$	10.7794	−	18.6705i	0.664687	−	1.15127i	0.979370	−	0.202075i	$-0.0647686\pi$
$264$		0			0
$265$	8.24765	+	13.4600i	0.506649	+	0.826841i
$266$		0			0
$267$	−21.4796	−	12.0459i	−1.31453	−	0.737199i
$268$		0			0
$269$	7.98483	−	13.8301i	0.486844	−	0.843238i	−0.513042	−	0.858364i	$-0.671481\pi$
$269$	7.98483	−	13.8301i	0.486844	−	0.843238i	0.999886	+	0.0151253i	$0.00481473\pi$
$270$		0			0
$271$	−4.72410	+	2.72746i	−0.286968	+	0.165681i	−0.636574	−	0.771216i	$-0.719649\pi$
$271$	−4.72410	+	2.72746i	−0.286968	+	0.165681i	0.349605	+	0.936897i	$0.386316\pi$
$272$		0			0
$273$	−5.75659	−	8.92393i	−0.348405	−	0.540101i
$274$		0			0
$275$	0.793367	+	15.1585i	0.0478418	+	0.914094i
$276$		0			0
$277$	6.72315	+	3.88162i	0.403955	+	0.233224i	0.688189	−	0.725531i	$-0.258406\pi$
$277$	6.72315	+	3.88162i	0.403955	+	0.233224i	−0.284234	+	0.958755i	$0.591739\pi$
$278$		0			0
$279$	3.44689	−	2.10662i	0.206360	−	0.126120i
$280$		0			0
$281$			0.742571i			0.0442981i	0.999755	+	0.0221490i	$0.00705084\pi$
$281$			0.742571i			0.0442981i	−0.999755	+	0.0221490i	$0.992949\pi$
$282$		0			0
$283$	4.79525	−	8.30562i	0.285048	−	0.493718i	−0.687573	−	0.726116i	$-0.741324\pi$
$283$	4.79525	−	8.30562i	0.285048	−	0.493718i	0.972621	+	0.232398i	$0.0746571\pi$
$284$		0			0
$285$	13.3151	+	21.1326i	0.788717	+	1.25179i
$286$		0			0
$287$	0.103601	+	2.81320i	0.00611538	+	0.166058i
$288$		0			0
$289$	4.98104	+	8.62741i	0.293002	+	0.507495i
$290$		0			0
$291$	−12.8546	+	7.63732i	−0.753548	+	0.447707i
$292$		0			0
$293$			0.569685i			0.0332814i	0.999862	+	0.0166407i	$0.00529714\pi$
$293$			0.569685i			0.0332814i	−0.999862	+	0.0166407i	$0.994703\pi$
$294$		0			0
$295$	−14.4406	−	23.5667i	−0.840762	−	1.37211i
$296$		0			0
$297$	7.36985	−	13.9473i	0.427642	−	0.809307i
$298$		0			0
$299$	−5.64147	−	9.77131i	−0.326254	−	0.565089i
$300$		0			0
$301$	−6.40003	+	0.235692i	−0.368892	+	0.0135851i
$302$		0			0
$303$	−0.356273	−	28.5111i	−0.0204673	−	1.63792i
$304$		0			0
$305$	0.772500	+	29.5399i	0.0442332	+	1.69145i
$306$		0			0
$307$	−9.46424			−0.540153			−0.270076	−	0.962839i	$-0.587049\pi$
$307$	−9.46424			−0.540153			−0.270076	+	0.962839i	$0.587049\pi$
$308$		0			0
$309$	9.97781	+	5.59564i	0.567618	+	0.318325i
$310$		0			0
$311$	−3.25858	+	5.64403i	−0.184777	+	0.320044i	−0.943501	−	0.331368i	$-0.892490\pi$
$311$	−3.25858	+	5.64403i	−0.184777	+	0.320044i	0.758724	+	0.651412i	$0.225823\pi$
$312$		0			0
$313$	11.0684	+	19.1711i	0.625626	+	1.08362i	0.988420	+	0.151746i	$0.0484896\pi$
$313$	11.0684	+	19.1711i	0.625626	+	1.08362i	−0.362794	+	0.931869i	$0.618177\pi$
$314$		0			0
$315$	−14.5316	+	10.1898i	−0.818762	+	0.574133i
$316$		0			0
$317$	2.10782	+	3.65086i	0.118387	+	0.205053i	0.919129	−	0.393957i	$-0.128894\pi$
$317$	2.10782	+	3.65086i	0.118387	+	0.205053i	−0.800741	+	0.599010i	$0.795561\pi$
$318$		0			0
$319$	5.29183	−	9.16571i	0.296285	−	0.513181i
$320$		0			0
$321$	−27.9981	−	15.7016i	−1.56270	−	0.876378i
$322$		0			0
$323$	33.4874			1.86329
$324$		0			0
$325$	−6.30999	−	9.71801i	−0.350015	−	0.539058i
$326$		0			0
$327$	−0.400432	−	32.0449i	−0.0221439	−	1.77209i
$328$		0			0
$329$	−5.36252	−	8.54559i	−0.295645	−	0.471133i
$330$		0			0
$331$	12.0041	+	20.7918i	0.659807	+	1.14282i	0.980665	+	0.195693i	$0.0626955\pi$
$331$	12.0041	+	20.7918i	0.659807	+	1.14282i	−0.320858	+	0.947127i	$0.603971\pi$
$332$		0			0
$333$	0.210844	+	8.43518i	0.0115542	+	0.462245i
$334$		0			0
$335$	8.19156	−	5.01940i	0.447553	−	0.274239i
$336$		0			0
$337$		−	9.42028i		−	0.513155i	−0.966524	−	0.256578i	$-0.917405\pi$
$337$		−	9.42028i		−	0.513155i	0.966524	−	0.256578i	$-0.0825949\pi$
$338$		0			0
$339$	−8.74032	+	5.19291i	−0.474709	+	0.282040i
$340$		0			0
$341$	−2.04398	−	3.54027i	−0.110688	−	0.191716i
$342$		0			0
$343$	−2.04104	−	18.4074i	−0.110206	−	0.993909i
$344$		0			0
$345$	−15.9542	+	10.0523i	−0.858944	+	0.541197i
$346$		0			0
$347$	11.1961	−	19.3922i	0.601037	−	1.04103i	−0.391627	−	0.920124i	$-0.628088\pi$
$347$	11.1961	−	19.3922i	0.601037	−	1.04103i	0.992664	−	0.120903i	$-0.0385789\pi$
$348$		0			0
$349$		−	12.0663i		−	0.645895i	−0.946417	−	0.322948i	$-0.895326\pi$
$349$		−	12.0663i		−	0.645895i	0.946417	−	0.322948i	$-0.104674\pi$
$350$		0			0
$351$	0.451278	+	12.0330i	0.0240875	+	0.642273i
$352$		0			0
$353$	7.19515	+	4.15412i	0.382959	+	0.221102i	0.679105	−	0.734041i	$-0.262368\pi$
$353$	7.19515	+	4.15412i	0.382959	+	0.221102i	−0.296146	+	0.955143i	$0.595701\pi$
$354$		0			0
$355$	−13.9183	−	7.55763i	−0.738705	−	0.401117i
$356$		0			0
$357$	1.17275	+	23.7661i	0.0620684	+	1.25784i
$358$		0			0
$359$	6.27295	−	3.62169i	0.331074	−	0.191145i	−0.325244	−	0.945630i	$-0.605446\pi$
$359$	6.27295	−	3.62169i	0.331074	−	0.191145i	0.656318	+	0.754485i	$0.272113\pi$
$360$		0			0
$361$	11.2959	−	19.5651i	0.594522	−	1.02974i
$362$		0			0
$363$	2.69447	+	1.51108i	0.141423	+	0.0793111i
$364$		0			0
$365$	−0.178334	−	0.291038i	−0.00933444	−	0.0152336i
$366$		0			0
$367$	0.624486	−	1.08164i	0.0325979	−	0.0564612i	−0.849266	−	0.527965i	$-0.822955\pi$
$367$	0.624486	−	1.08164i	0.0325979	−	0.0564612i	0.881864	+	0.471504i	$0.156289\pi$
$368$		0			0
$369$	1.52644	−	2.80340i	0.0794634	−	0.145939i
$370$		0			0
$371$	16.5085	+	8.73745i	0.857078	+	0.453626i
$372$		0			0
$373$	26.7505	−	15.4444i	1.38509	−	0.799682i	0.392334	−	0.919823i	$-0.371668\pi$
$373$	26.7505	−	15.4444i	1.38509	−	0.799682i	0.992757	+	0.120141i	$0.0383346\pi$
$374$		0			0
$375$	−15.8220	+	11.1653i	−0.817045	+	0.576574i
$376$		0			0
$377$			8.07888i			0.416083i
$378$		0			0
$379$	29.6463			1.52283			0.761413	−	0.648267i	$-0.224506\pi$
$379$	29.6463			1.52283			0.761413	+	0.648267i	$0.224506\pi$
$380$		0			0
$381$	−12.9075	−	21.7249i	−0.661269	−	1.11300i
$382$		0			0
$383$	18.1740	−	10.4927i	0.928646	−	0.536154i	0.0422631	−	0.999107i	$-0.486543\pi$
$383$	18.1740	−	10.4927i	0.928646	−	0.536154i	0.886383	+	0.462952i	$0.153210\pi$
$384$		0			0
$385$	9.94096	+	14.9583i	0.506638	+	0.762348i
$386$		0			0
$387$	6.37774	+	3.47266i	0.324199	+	0.176525i
$388$		0			0
$389$	14.2540	+	8.22952i	0.722704	+	0.417253i	0.815747	−	0.578409i	$-0.196326\pi$
$389$	14.2540	+	8.22952i	0.722704	+	0.417253i	−0.0930430	+	0.995662i	$0.529659\pi$
$390$		0			0
$391$			25.2815i			1.27854i
$392$		0			0
$393$	30.3217	+	17.0047i	1.52953	+	0.857772i
$394$		0			0
$395$	13.6113	−	0.355951i	0.684860	−	0.0179098i
$396$		0			0
$397$	2.04631	+	3.54432i	0.102702	+	0.177884i	0.912797	−	0.408414i	$-0.133918\pi$
$397$	2.04631	+	3.54432i	0.102702	+	0.177884i	−0.810095	+	0.586298i	$0.800585\pi$
$398$		0			0
$399$	26.2916	+	13.4975i	1.31622	+	0.675722i
$400$		0			0
$401$	−22.5314	+	13.0085i	−1.12516	+	0.649613i	−0.942714	−	0.333602i	$-0.891736\pi$
$401$	−22.5314	+	13.0085i	−1.12516	+	0.649613i	−0.182449	+	0.983215i	$0.558402\pi$
$402$		0			0
$403$	2.70241	+	1.56024i	0.134617	+	0.0777210i
$404$		0			0
$405$	20.0663	−	1.53053i	0.997104	−	0.0760528i
$406$		0			0
$407$	8.53866			0.423246
$408$		0			0
$409$	30.5083	+	17.6140i	1.50854	+	0.870955i	0.999951	+	0.00994431i	$0.00316542\pi$
$409$	30.5083	+	17.6140i	1.50854	+	0.870955i	0.508587	+	0.861010i	$0.330168\pi$
$410$		0			0
$411$	0.101082	+	8.08919i	0.00498601	+	0.399010i
$412$		0			0
$413$	−28.9042	−	15.2981i	−1.42228	−	0.752772i
$414$		0			0
$415$	−20.1925	−	10.9646i	−0.991212	−	0.538229i
$416$		0			0
$417$	4.87731	+	8.20913i	0.238843	+	0.402003i
$418$		0			0
$419$	−40.3278			−1.97014			−0.985072	−	0.172146i	$-0.944930\pi$
$419$	−40.3278			−1.97014			−0.985072	+	0.172146i	$0.944930\pi$
$420$		0			0
$421$	0.216416			0.0105475			0.00527375	−	0.999986i	$-0.498321\pi$
$421$	0.216416			0.0105475			0.00527375	+	0.999986i	$0.498321\pi$
$422$		0			0
$423$	0.285853	+	11.4360i	0.0138986	+	0.556039i
$424$		0			0
$425$	1.35697	+	25.9270i	0.0658226	+	1.25765i
$426$		0			0
$427$	18.5845	+	29.6158i	0.899367	+	1.43321i
$428$		0			0
$429$	12.1844	−	0.152256i	0.588268	−	0.00735097i
$430$		0			0
$431$	25.3179	+	14.6173i	1.21952	+	0.704090i	0.964815	−	0.262928i	$-0.0846883\pi$
$431$	25.3179	+	14.6173i	1.21952	+	0.704090i	0.254705	+	0.967019i	$0.418022\pi$
$432$		0			0
$433$	−19.8669			−0.954743			−0.477371	−	0.878702i	$-0.658410\pi$
$433$	−19.8669			−0.954743			−0.477371	+	0.878702i	$0.658410\pi$
$434$		0			0
$435$	13.4920	−	0.521596i	0.646891	−	0.0250086i
$436$		0			0
$437$	27.1932	+	15.7000i	1.30083	+	0.751033i
$438$		0			0
$439$	19.0134	−	10.9774i	0.907459	−	0.523921i	0.0278460	−	0.999612i	$-0.491135\pi$
$439$	19.0134	−	10.9774i	0.907459	−	0.523921i	0.879613	+	0.475691i	$0.157802\pi$
$440$		0			0
$441$	−8.65851	+	19.1319i	−0.412310	+	0.911044i
$442$		0			0
$443$	4.36391	+	7.55852i	0.207336	+	0.359116i	0.950874	−	0.309577i	$-0.100187\pi$
$443$	4.36391	+	7.55852i	0.207336	+	0.359116i	−0.743539	+	0.668693i	$0.766854\pi$
$444$		0			0
$445$	−31.7821	+	0.831138i	−1.50662	+	0.0393997i
$446$		0			0
$447$	12.3830	−	22.0806i	0.585696	−	1.04438i
$448$		0			0
$449$			1.03244i			0.0487238i	0.999703	+	0.0243619i	$0.00775540\pi$
$449$			1.03244i			0.0487238i	−0.999703	+	0.0243619i	$0.992245\pi$
$450$		0			0
$451$	−2.79742	−	1.61509i	−0.131725	−	0.0760517i
$452$		0			0
$453$	−0.193957	−	15.5216i	−0.00911288	−	0.729267i
$454$		0			0
$455$	−12.2808	−	6.09433i	−0.575731	−	0.285707i
$456$		0			0
$457$	−27.2085	+	15.7088i	−1.27276	+	0.734828i	−0.975506	−	0.219971i	$-0.929404\pi$
$457$	−27.2085	+	15.7088i	−1.27276	+	0.734828i	−0.297253	+	0.954799i	$0.596070\pi$
$458$		0			0
$459$	12.6053	−	23.8554i	0.588367	−	1.11348i
$460$		0			0
$461$	−3.22290			−0.150105			−0.0750526	−	0.997180i	$-0.523912\pi$
$461$	−3.22290			−0.150105			−0.0750526	+	0.997180i	$0.523912\pi$
$462$		0			0
$463$			5.06917i			0.235584i	0.993038	+	0.117792i	$0.0375817\pi$
$463$			5.06917i			0.235584i	−0.993038	+	0.117792i	$0.962418\pi$
$464$		0			0
$465$	2.43117	−	4.61385i	0.112743	−	0.213962i
$466$		0			0
$467$	−7.81525	+	4.51214i	−0.361647	+	0.208797i	−0.669803	−	0.742539i	$-0.733621\pi$
$467$	−7.81525	+	4.51214i	−0.361647	+	0.208797i	0.308156	+	0.951336i	$0.400288\pi$
$468$		0			0
$469$	5.31749	−	10.0468i	0.245539	−	0.463920i
$470$		0			0
$471$	−0.444537	−	35.5745i	−0.0204832	−	1.63919i
$472$		0			0
$473$	3.67433	−	6.36413i	0.168946	−	0.292623i
$474$		0			0
$475$	28.7303	+	14.6413i	1.31823	+	0.671790i
$476$		0			0
$477$	−11.0445	−	18.0712i	−0.505694	−	0.827425i
$478$		0			0
$479$	1.26094	−	2.18401i	0.0576137	−	0.0997899i	−0.835780	−	0.549064i	$-0.814984\pi$
$479$	1.26094	−	2.18401i	0.0576137	−	0.0997899i	0.893394	+	0.449275i	$0.148317\pi$
$480$		0			0
$481$	−5.64464	+	3.25893i	−0.257373	+	0.148594i
$482$		0			0
$483$	−10.1900	+	19.8490i	−0.463663	+	0.903159i
$484$		0			0
$485$	−9.21129	+	16.9637i	−0.418263	+	0.770281i
$486$		0			0
$487$	2.12791	+	1.22855i	0.0964247	+	0.0556709i	0.547437	−	0.836847i	$-0.315604\pi$
$487$	2.12791	+	1.22855i	0.0964247	+	0.0556709i	−0.451012	+	0.892518i	$0.648937\pi$
$488$		0			0
$489$	−16.9873	+	30.2908i	−0.768194	+	1.36980i
$490$		0			0
$491$			14.3239i			0.646429i	0.946326	+	0.323215i	$0.104764\pi$
$491$			14.3239i			0.646429i	−0.946326	+	0.323215i	$0.895236\pi$
$492$		0			0
$493$	9.05110	−	15.6770i	0.407641	−	0.706055i
$494$		0			0
$495$	−1.04093	−	20.3385i	−0.0467864	−	0.914148i
$496$		0			0
$497$	−18.7268	+	0.689649i	−0.840014	+	0.0309350i
$498$		0			0
$499$	−19.8770	−	34.4279i	−0.889816	−	1.54121i	−0.840093	−	0.542443i	$-0.817500\pi$
$499$	−19.8770	−	34.4279i	−0.889816	−	1.54121i	−0.0497229	−	0.998763i	$-0.515834\pi$
$500$		0			0
$501$	10.8343	+	18.2355i	0.484040	+	0.814701i
$502$		0			0
$503$			10.6486i			0.474796i	0.971412	+	0.237398i	$0.0762946\pi$
$503$			10.6486i			0.474796i	−0.971412	+	0.237398i	$0.923705\pi$
$504$		0			0
$505$	−19.2323	−	31.3868i	−0.855828	−	1.39669i
$506$		0			0
$507$	11.3612	−	6.75006i	0.504569	−	0.299781i
$508$		0			0
$509$	−18.5593	−	32.1456i	−0.822626	−	1.42483i	−0.903721	−	0.428123i	$-0.859175\pi$
$509$	−18.5593	−	32.1456i	−0.822626	−	1.42483i	0.0810950	−	0.996706i	$-0.474158\pi$
$510$		0			0
$511$	−0.356954	−	0.188925i	−0.0157907	−	0.00835755i
$512$		0			0
$513$	−17.8313	−	28.3730i	−0.787272	−	1.25270i
$514$		0			0
$515$	14.7636	−	0.386084i	0.650562	−	0.0170129i
$516$		0			0
$517$	11.5763			0.509127
$518$		0			0
$519$	18.4901	−	32.9705i	0.811627	−	1.44725i
$520$		0			0
$521$	8.12000	−	14.0642i	0.355744	−	0.616166i	−0.631501	−	0.775375i	$-0.717561\pi$
$521$	8.12000	−	14.0642i	0.355744	−	0.616166i	0.987245	+	0.159209i	$0.0508943\pi$
$522$		0			0
$523$	−8.87476	−	15.3715i	−0.388066	−	0.672150i	0.604123	−	0.796891i	$-0.293523\pi$
$523$	−8.87476	−	15.3715i	−0.388066	−	0.672150i	−0.992189	+	0.124741i	$0.960190\pi$
$524$		0			0
$525$	−9.38485	+	20.9027i	−0.409589	+	0.912270i
$526$		0			0
$527$	−3.49600	−	6.05525i	−0.152288	−	0.263771i
$528$		0			0
$529$	−0.352809	+	0.611082i	−0.0153395	+	0.0265688i
$530$		0			0
$531$	19.3375	+	31.6404i	0.839177	+	1.37308i
$532$		0			0
$533$	2.46571			0.106802
$534$		0			0
$535$	−41.4273	+	1.08337i	−1.79106	+	0.0468381i
$536$		0			0
$537$	2.22051	−	0.0277474i	0.0958221	−	0.00119739i
$538$		0			0
$539$	19.1356	+	9.24304i	0.824228	+	0.398126i
$540$		0			0
$541$	−0.167695	−	0.290456i	−0.00720976	−	0.0124877i	0.862398	−	0.506231i	$-0.168962\pi$
$541$	−0.167695	−	0.290456i	−0.00720976	−	0.0124877i	−0.869608	+	0.493743i	$0.835628\pi$
$542$		0			0
$543$	12.8827	+	21.6833i	0.552852	+	0.930519i
$544$		0			0
$545$	−21.6161	−	35.2771i	−0.925934	−	1.51110i
$546$		0			0
$547$		−	11.7041i		−	0.500433i	−0.968190	−	0.250216i	$-0.919498\pi$
$547$		−	11.7041i		−	0.500433i	0.968190	−	0.250216i	$-0.0805018\pi$
$548$		0			0
$549$	−0.990661	−	39.6331i	−0.0422804	−	1.69150i
$550$		0			0
$551$	−11.2416	−	19.4710i	−0.478909	−	0.829495i
$552$		0			0
$553$	13.6463	−	8.56333i	0.580300	−	0.364150i
$554$		0			0
$555$	5.80696	+	9.21632i	0.246492	+	0.391211i
$556$		0			0
$557$	−2.55833	+	4.43115i	−0.108400	+	0.187754i	−0.915122	−	0.403177i	$-0.867906\pi$
$557$	−2.55833	+	4.43115i	−0.108400	+	0.187754i	0.806722	+	0.590931i	$0.201239\pi$
$558$		0			0
$559$			5.60950i			0.237256i
$560$		0			0
$561$	−23.8143	−	13.3552i	−1.00544	−	0.563858i
$562$		0			0
$563$	−16.8089	−	9.70464i	−0.708412	−	0.409002i	0.102061	−	0.994778i	$-0.467456\pi$
$563$	−16.8089	−	9.70464i	−0.708412	−	0.409002i	−0.810473	+	0.585776i	$0.800790\pi$
$564$		0			0
$565$	−6.26311	+	11.5343i	−0.263491	+	0.485250i
$566$		0			0
$567$	19.5119	−	13.6486i	0.819424	−	0.573187i
$568$		0			0
$569$	−20.1553	+	11.6367i	−0.844954	+	0.487834i	−0.858945	−	0.512068i	$-0.828880\pi$
$569$	−20.1553	+	11.6367i	−0.844954	+	0.487834i	0.0139912	+	0.999902i	$0.495546\pi$
$570$		0			0
$571$	7.04462	−	12.2016i	0.294808	−	0.510623i	−0.680132	−	0.733090i	$-0.738078\pi$
$571$	7.04462	−	12.2016i	0.294808	−	0.510623i	0.974940	+	0.222467i	$0.0714109\pi$
$572$		0			0
$573$	−20.2531	+	36.1141i	−0.846084	+	1.50869i
$574$		0			0
$575$	−11.0535	+	21.6901i	−0.460965	+	0.904539i
$576$		0			0
$577$	−11.0816	+	19.1938i	−0.461332	+	0.799050i	−0.999028	−	0.0440886i	$-0.985962\pi$
$577$	−11.0816	+	19.1938i	−0.461332	+	0.799050i	0.537696	+	0.843139i	$0.319295\pi$
$578$		0			0
$579$	−14.6136	+	0.182611i	−0.607320	+	0.00758904i
$580$		0			0
$581$	−27.1688	+	1.00054i	−1.12715	+	0.0415093i
$582$		0			0
$583$	−18.5608	+	10.7161i	−0.768709	+	0.443815i
$584$		0			0
$585$	8.45068	+	13.0478i	0.349393	+	0.539462i
$586$		0			0
$587$		−	11.5459i		−	0.476550i	−0.971198	−	0.238275i	$-0.923418\pi$
$587$		−	11.5459i		−	0.476550i	0.971198	−	0.238275i	$-0.0765819\pi$
$588$		0			0
$589$	−8.68418			−0.357825
$590$		0			0
$591$	12.2977	−	7.30647i	0.505860	−	0.300548i
$592$		0			0
$593$	−18.5778	+	10.7259i	−0.762901	+	0.440461i	−0.830336	−	0.557263i	$-0.811852\pi$
$593$	−18.5778	+	10.7259i	−0.762901	+	0.440461i	0.0674357	+	0.997724i	$0.478518\pi$
$594$		0			0
$595$	17.0029	+	25.5846i	0.697052	+	1.04887i
$596$		0			0
$597$	−17.8796	+	0.223423i	−0.731765	+	0.00914409i
$598$		0			0
$599$	3.19001	+	1.84175i	0.130340	+	0.0752519i	0.563752	−	0.825944i	$-0.309357\pi$
$599$	3.19001	+	1.84175i	0.130340	+	0.0752519i	−0.433412	+	0.901196i	$0.642691\pi$
$600$		0			0
$601$			8.20607i			0.334733i	0.985895	+	0.167366i	$0.0535263\pi$
$601$			8.20607i			0.334733i	−0.985895	+	0.167366i	$0.946474\pi$
$602$		0			0
$603$	−10.9979	+	6.72154i	−0.447869	+	0.273722i
$604$		0			0
$605$	3.98685	−	0.104260i	0.162089	−	0.00423879i
$606$		0			0
$607$	−3.52607	−	6.10733i	−0.143119	−	0.247889i	0.785551	−	0.618797i	$-0.212380\pi$
$607$	−3.52607	−	6.10733i	−0.143119	−	0.247889i	−0.928670	+	0.370908i	$0.879046\pi$
$608$		0			0
$609$	13.4250	−	8.66011i	0.544008	−	0.350925i
$610$		0			0
$611$	−7.65274	+	4.41831i	−0.309597	+	0.178746i
$612$		0			0
$613$	17.0882	+	9.86586i	0.690185	+	0.398478i	0.803681	−	0.595060i	$-0.202872\pi$
$613$	17.0882	+	9.86586i	0.690185	+	0.398478i	−0.113496	+	0.993538i	$0.536205\pi$
$614$		0			0
$615$	−0.159194	−	4.11782i	−0.00641930	−	0.166047i
$616$		0			0
$617$	7.15802			0.288171			0.144086	−	0.989565i	$-0.453976\pi$
$617$	7.15802			0.288171			0.144086	+	0.989565i	$0.453976\pi$
$618$		0			0
$619$	−8.50675	−	4.91137i	−0.341915	−	0.197405i	0.319204	−	0.947686i	$-0.396585\pi$
$619$	−8.50675	−	4.91137i	−0.341915	−	0.197405i	−0.661119	+	0.750281i	$0.729918\pi$
$620$		0			0
$621$	21.4203	−	13.4619i	0.859568	−	0.540205i
$622$		0			0
$623$	−31.8639	+	19.9952i	−1.27660	+	0.801090i
$624$		0			0
$625$	−10.1716	+	22.8372i	−0.406864	+	0.913489i
$626$		0			0
$627$	−29.1540	+	17.3213i	−1.16430	+	0.691747i
$628$		0			0
$629$	14.6045			0.582318
$630$		0			0
$631$	−13.2595			−0.527854			−0.263927	−	0.964543i	$-0.585018\pi$
$631$	−13.2595			−0.527854			−0.263927	+	0.964543i	$0.585018\pi$
$632$		0			0
$633$	8.57244	−	5.09317i	0.340724	−	0.202435i
$634$		0			0
$635$	−28.6695	−	15.5676i	−1.13771	−	0.617780i
$636$		0			0
$637$	−16.1777	+	1.19316i	−0.640983	+	0.0472748i
$638$		0			0
$639$	18.6616	+	10.1612i	0.738242	+	0.401970i
$640$		0			0
$641$	24.8302	+	14.3357i	0.980734	+	0.566227i	0.902492	−	0.430707i	$-0.141736\pi$
$641$	24.8302	+	14.3357i	0.980734	+	0.566227i	0.0782423	+	0.996934i	$0.475069\pi$
$642$		0			0
$643$	−38.1006			−1.50254			−0.751271	−	0.659994i	$-0.770559\pi$
$643$	−38.1006			−1.50254			−0.751271	+	0.659994i	$0.770559\pi$
$644$		0			0
$645$	9.36804	−	0.362166i	0.368866	−	0.0142603i
$646$		0			0
$647$	−33.4126	−	19.2908i	−1.31359	−	0.758399i	−0.330897	−	0.943667i	$-0.607351\pi$
$647$	−33.4126	−	19.2908i	−1.31359	−	0.758399i	−0.982688	+	0.185268i	$0.940685\pi$
$648$		0			0
$649$	32.4975	−	18.7625i	1.27564	−	0.736491i
$650$		0			0
$651$	−0.304126	−	6.16320i	−0.0119196	−	0.241555i
$652$		0			0
$653$	−21.2942	−	36.8826i	−0.833306	−	1.44333i	−0.895402	−	0.445259i	$-0.853112\pi$
$653$	−21.2942	−	36.8826i	−0.833306	−	1.44333i	0.0620956	−	0.998070i	$-0.480222\pi$
$654$		0			0
$655$	44.8653	−	1.17328i	1.75303	−	0.0458437i
$656$		0			0
$657$	0.238810	+	0.390744i	0.00931685	+	0.0152444i
$658$		0			0
$659$			13.0441i			0.508128i	0.967187	+	0.254064i	$0.0817674\pi$
$659$			13.0441i			0.508128i	−0.967187	+	0.254064i	$0.918233\pi$
$660$		0			0
$661$	−12.9159	−	7.45701i	−0.502371	−	0.290044i	0.227321	−	0.973820i	$-0.427003\pi$
$661$	−12.9159	−	7.45701i	−0.502371	−	0.290044i	−0.729692	+	0.683776i	$0.760337\pi$
$662$		0			0
$663$	20.8401	−	0.260417i	0.809362	−	0.0101138i
$664$		0			0
$665$	38.0782	−	2.40040i	1.47661	−	0.0930834i
$666$		0			0
$667$	14.6998	−	8.48692i	0.569178	−	0.328615i
$668$		0			0
$669$	40.3426	−	23.9689i	1.55974	−	0.926690i
$670$		0			0
$671$	−40.1193			−1.54879
$672$		0			0
$673$			51.0490i			1.96780i	0.178732	+	0.983898i	$0.442801\pi$
$673$			51.0490i			1.96780i	−0.178732	+	0.983898i	$0.557199\pi$
$674$		0			0
$675$	21.2447	−	14.9553i	0.817710	−	0.575630i
$676$		0			0
$677$	0.281190	−	0.162345i	0.0108070	−	0.00623942i	−0.494587	−	0.869128i	$-0.664681\pi$
$677$	0.281190	−	0.162345i	0.0108070	−	0.00623942i	0.505394	+	0.862889i	$0.331347\pi$
$678$		0			0
$679$	0.840548	+	22.8244i	0.0322573	+	0.875920i
$680$		0			0
$681$	−12.4057	+	0.155021i	−0.475387	+	0.00594042i
$682$		0			0
$683$	−14.8704	+	25.7563i	−0.569001	+	0.985538i	0.427665	+	0.903938i	$0.359336\pi$
$683$	−14.8704	+	25.7563i	−0.569001	+	0.985538i	−0.996665	+	0.0816004i	$0.973997\pi$
$684$		0			0
$685$	5.45662	+	8.90509i	0.208487	+	0.340246i
$686$		0			0
$687$	−4.07566	+	7.26748i	−0.155496	+	0.277272i
$688$		0			0
$689$	8.17996	−	14.1681i	0.311632	−	0.539762i
$690$		0			0
$691$	−25.2719	+	14.5907i	−0.961388	+	0.555058i	−0.896600	−	0.442842i	$-0.853970\pi$
$691$	−25.2719	+	14.5907i	−0.961388	+	0.555058i	−0.0647879	+	0.997899i	$0.520637\pi$
$692$		0			0
$693$	−13.3140	−	20.0841i	−0.505758	−	0.762932i
$694$		0			0
$695$	10.8333	+	5.88248i	0.410930	+	0.223135i
$696$		0			0
$697$	−4.78468	−	2.76244i	−0.181233	−	0.104635i
$698$		0			0
$699$	−26.9091	−	15.0909i	−1.01780	−	0.570789i
$700$		0			0
$701$		−	4.22879i		−	0.159719i	−0.996806	−	0.0798596i	$-0.974553\pi$
$701$		−	4.22879i		−	0.159719i	0.996806	−	0.0798596i	$-0.0254472\pi$
$702$		0			0
$703$	9.06949	−	15.7088i	0.342062	−	0.592470i
$704$		0			0
$705$	7.87281	+	12.4951i	0.296507	+	0.470592i
$706$		0			0
$707$	−38.4954	−	20.3745i	−1.44777	−	0.766261i
$708$		0			0
$709$	13.9852	+	24.2230i	0.525225	+	0.909716i	0.999568	+	0.0293760i	$0.00935203\pi$
$709$	13.9852	+	24.2230i	0.525225	+	0.909716i	−0.474344	+	0.880340i	$0.657315\pi$
$710$		0			0
$711$	−18.2620	+	0.456474i	−0.684880	+	0.0171191i
$712$		0			0
$713$		−	6.55617i		−	0.245531i
$714$		0			0
$715$	13.4134	−	8.21907i	0.501631	−	0.307376i
$716$		0			0
$717$	12.1971	+	20.5293i	0.455510	+	0.766680i
$718$		0			0
$719$	−2.81744	−	4.87995i	−0.105073	−	0.181991i	0.808695	−	0.588228i	$-0.200174\pi$
$719$	−2.81744	−	4.87995i	−0.105073	−	0.181991i	−0.913768	+	0.406237i	$0.866841\pi$
$720$		0			0
$721$	14.8016	−	9.28827i	0.551239	−	0.345913i
$722$		0			0
$723$	23.5552	−	0.294344i	0.876026	−	0.0109468i
$724$		0			0
$725$	14.6196	−	9.49263i	0.542958	−	0.352548i
$726$		0			0
$727$	36.1095			1.33923			0.669613	−	0.742710i	$-0.266460\pi$
$727$	36.1095			1.33923			0.669613	+	0.742710i	$0.266460\pi$
$728$		0			0
$729$	−26.9242	+	2.02234i	−0.997191	+	0.0749016i
$730$		0			0
$731$	6.28455	−	10.8852i	0.232443	−	0.402602i
$732$		0			0
$733$	2.99103	+	5.18062i	0.110476	+	0.191351i	0.915962	−	0.401264i	$-0.131429\pi$
$733$	2.99103	+	5.18062i	0.110476	+	0.191351i	−0.805486	+	0.592615i	$0.798096\pi$
$734$		0			0
$735$	3.03710	+	26.9402i	0.112025	+	0.993705i
$736$		0			0
$737$	6.52165	+	11.2958i	0.240228	+	0.416087i
$738$		0			0
$739$	−19.1257	+	33.1266i	−0.703549	+	1.21858i	0.263663	+	0.964615i	$0.415069\pi$
$739$	−19.1257	+	33.1266i	−0.703549	+	1.21858i	−0.967213	+	0.253968i	$0.918264\pi$
$740$		0			0
$741$	12.6618	−	22.5777i	0.465142	−	0.829413i
$742$		0			0
$743$	−3.60215			−0.132150			−0.0660750	−	0.997815i	$-0.521048\pi$
$743$	−3.60215			−0.132150			−0.0660750	+	0.997815i	$0.521048\pi$
$744$		0			0
$745$	−0.854395	−	32.6715i	−0.0313026	−	1.19699i
$746$		0			0
$747$	27.0741	+	14.7418i	0.990590	+	0.539373i
$748$		0			0
$749$	−41.5338	+	26.0633i	−1.51761	+	0.952331i
$750$		0			0
$751$	2.07691	+	3.59731i	0.0757874	+	0.131268i	0.901428	−	0.432928i	$-0.142520\pi$
$751$	2.07691	+	3.59731i	0.0757874	+	0.131268i	−0.825641	+	0.564196i	$0.809186\pi$
$752$		0			0
$753$	−11.7576	+	6.98559i	−0.428472	+	0.254569i
$754$		0			0
$755$	−10.4702	−	17.0871i	−0.381049	−	0.621864i
$756$		0			0
$757$		−	15.4535i		−	0.561668i	−0.959756	−	0.280834i	$-0.909389\pi$
$757$		−	15.4535i		−	0.561668i	0.959756	−	0.280834i	$-0.0906110\pi$
$758$		0			0
$759$	−13.0768	−	22.0100i	−0.474659	−	0.798911i
$760$		0			0
$761$	14.2676	+	24.7122i	0.517199	+	0.895816i	0.999800	+	0.0199755i	$0.00635881\pi$
$761$	14.2676	+	24.7122i	0.517199	+	0.895816i	−0.482601	+	0.875840i	$0.660308\pi$
$762$		0			0
$763$	−43.2669	−	22.8999i	−1.56637	−	0.829031i
$764$		0			0
$765$	−1.78040	−	34.7868i	−0.0643706	−	1.25772i
$766$		0			0
$767$	−14.3220	+	24.8065i	−0.517139	+	0.895711i
$768$		0			0
$769$		−	19.5067i		−	0.703431i	−0.936107	−	0.351716i	$-0.885598\pi$
$769$		−	19.5067i		−	0.703431i	0.936107	−	0.351716i	$-0.114402\pi$
$770$		0			0
$771$	15.6522	−	27.9100i	0.563699	−	1.00515i
$772$		0			0
$773$	18.2685	+	10.5473i	0.657074	+	0.379362i	0.791161	−	0.611608i	$-0.209477\pi$
$773$	18.2685	+	10.5473i	0.657074	+	0.379362i	−0.134087	+	0.990970i	$0.542810\pi$
$774$		0			0
$775$	−0.351899	−	6.72358i	−0.0126406	−	0.241518i
$776$		0			0
$777$	11.4662	+	5.88653i	0.411349	+	0.211178i
$778$		0			0
$779$	−5.94266	+	3.43100i	−0.212918	+	0.122928i
$780$		0			0
$781$	10.7513	−	18.6218i	0.384712	−	0.666340i
$782$		0			0
$783$	−18.1022	+	0.678895i	−0.646920	+	0.0242617i
$784$		0			0
$785$	−23.9970	−	39.1626i	−0.856490	−	1.39777i
$786$		0			0
$787$	2.60421	−	4.51062i	0.0928301	−	0.160786i	−0.815871	−	0.578234i	$-0.803742\pi$
$787$	2.60421	−	4.51062i	0.0928301	−	0.160786i	0.908701	+	0.417448i	$0.137075\pi$
$788$		0			0
$789$	0.466575	+	37.3381i	0.0166105	+	1.32927i
$790$		0			0
$791$	0.571521	+	15.5192i	0.0203210	+	0.551799i
$792$		0			0
$793$	26.5216	−	15.3122i	0.941808	−	0.543753i
$794$		0			0
$795$	−24.1893	−	12.7461i	−0.857907	−	0.452056i
$796$		0			0
$797$		−	41.1247i		−	1.45671i	−0.685199	−	0.728356i	$-0.740285\pi$
$797$		−	41.1247i		−	1.45671i	0.685199	−	0.728356i	$-0.259715\pi$
$798$		0			0
$799$	19.8001			0.700476
$800$		0			0
$801$	42.6415	−	1.06586i	1.50666	−	0.0376602i
$802$		0			0
$803$	0.401329	−	0.231708i	0.0141626	−	0.00817679i
$804$		0			0
$805$	1.81219	+	28.7474i	0.0638714	+	1.01321i
$806$		0			0
$807$	0.345614	+	27.6581i	0.0121662	+	0.973612i
$808$		0			0
$809$	8.18379	+	4.72491i	0.287727	+	0.166119i	0.636916	−	0.770933i	$-0.280210\pi$
$809$	8.18379	+	4.72491i	0.287727	+	0.166119i	−0.349190	+	0.937052i	$0.613543\pi$
$810$		0			0
$811$			13.1110i			0.460388i	0.973145	+	0.230194i	$0.0739361\pi$
$811$			13.1110i			0.460388i	−0.973145	+	0.230194i	$0.926064\pi$
$812$		0			0
$813$	4.62149	−	8.24076i	0.162083	−	0.289016i
$814$		0			0
$815$	1.17208	+	44.8196i	0.0410562	+	1.56996i
$816$		0			0
$817$	−7.80552	−	13.5196i	−0.273081	−	0.472989i
$818$		0			0
$819$	16.4669	+	8.19542i	0.575401	+	0.286371i
$820$		0			0
$821$	−24.7530	+	14.2911i	−0.863885	+	0.498764i	−0.865311	−	0.501235i	$-0.832879\pi$
$821$	−24.7530	+	14.2911i	−0.863885	+	0.498764i	0.00142663	+	0.999999i	$0.499546\pi$
$822$		0			0
$823$	21.4406	+	12.3787i	0.747371	+	0.431495i	0.824743	−	0.565508i	$-0.191320\pi$
$823$	21.4406	+	12.3787i	0.747371	+	0.431495i	−0.0773725	+	0.997002i	$0.524653\pi$
$824$		0			0
$825$	−14.5921	−	21.8701i	−0.508033	−	0.761419i
$826$		0			0
$827$	35.9582			1.25039			0.625194	−	0.780470i	$-0.285020\pi$
$827$	35.9582			1.25039			0.625194	+	0.780470i	$0.285020\pi$
$828$		0			0
$829$	−12.8068	−	7.39399i	−0.444798	−	0.256804i	0.260833	−	0.965384i	$-0.416003\pi$
$829$	−12.8068	−	7.39399i	−0.444798	−	0.256804i	−0.705631	+	0.708580i	$0.749336\pi$
$830$		0			0
$831$	−13.4453	+	0.168011i	−0.466411	+	0.00582825i
$832$		0			0
$833$	32.7294	+	15.8092i	1.13400	+	0.547757i
$834$		0			0
$835$	24.0647	+	13.0671i	0.832792	+	0.452207i
$836$		0			0
$837$	−3.26891	+	6.18636i	−0.112990	+	0.213832i
$838$		0			0
$839$	0.979006			0.0337991			0.0168995	−	0.999857i	$-0.494620\pi$
$839$	0.979006			0.0337991			0.0168995	+	0.999857i	$0.494620\pi$
$840$		0			0
$841$	16.8463			0.580906
$842$		0			0
$843$	−0.656953	−	1.10573i	−0.0226267	−	0.0380835i
$844$		0			0
$845$	8.14118	−	14.9929i	0.280065	−	0.515773i
$846$		0			0
$847$	3.99710	−	2.50826i	0.137342	−	0.0861848i
$848$		0			0
$849$	0.207557	+	16.6099i	0.00712334	+	0.570052i
$850$		0			0
$851$	11.8595	+	6.84707i	0.406537	+	0.234714i
$852$		0			0
$853$	−31.6889			−1.08501			−0.542504	−	0.840053i	$-0.682524\pi$
$853$	−31.6889			−1.08501			−0.542504	+	0.840053i	$0.682524\pi$
$854$		0			0
$855$	−38.5230	−	19.6879i	−1.31746	−	0.673311i
$856$		0			0
$857$	28.0280	+	16.1820i	0.957419	+	0.552766i	0.895378	−	0.445307i	$-0.146906\pi$
$857$	28.0280	+	16.1820i	0.957419	+	0.552766i	0.0620413	+	0.998074i	$0.480239\pi$
$858$		0			0
$859$	−41.2350	+	23.8070i	−1.40692	+	0.812285i	−0.995090	−	0.0989753i	$-0.968444\pi$
$859$	−41.2350	+	23.8070i	−1.40692	+	0.812285i	−0.411830	+	0.911261i	$0.635110\pi$
$860$		0			0
$861$	−2.64311	−	4.09737i	−0.0900769	−	0.139638i
$862$		0			0
$863$	−2.98476	−	5.16975i	−0.101602	−	0.175980i	0.810743	−	0.585403i	$-0.199064\pi$
$863$	−2.98476	−	5.16975i	−0.101602	−	0.175980i	−0.912345	+	0.409422i	$0.865730\pi$
$864$		0			0
$865$	−1.27577	−	48.7846i	−0.0433775	−	1.65873i
$866$		0			0
$867$	−15.0497	−	8.44002i	−0.511116	−	0.286638i
$868$		0			0
$869$			18.4861i			0.627097i
$870$		0			0
$871$	−8.62251	−	4.97821i	−0.292163	−	0.168680i
$872$		0			0
$873$	12.3845	−	22.7449i	0.419152	−	0.769797i
$874$		0			0
$875$	3.40147	+	29.3842i	0.114991	+	0.993367i
$876$		0			0
$877$	14.7945	−	8.54159i	0.499574	−	0.288429i	−0.228964	−	0.973435i	$-0.573534\pi$
$877$	14.7945	−	8.54159i	0.499574	−	0.288429i	0.728537	+	0.685006i	$0.240200\pi$
$878$		0			0
$879$	−0.504001	−	0.848296i	−0.0169995	−	0.0286123i
$880$		0			0
$881$	13.7490			0.463215			0.231607	−	0.972809i	$-0.425601\pi$
$881$	13.7490			0.463215			0.231607	+	0.972809i	$0.425601\pi$
$882$		0			0
$883$			29.4325i			0.990482i	0.868756	+	0.495241i	$0.164920\pi$
$883$			29.4325i			0.990482i	−0.868756	+	0.495241i	$0.835080\pi$
$884$		0			0
$885$	42.3523	+	22.3167i	1.42366	+	0.750168i
$886$		0			0
$887$	14.5904	−	8.42380i	0.489899	−	0.282843i	−0.234633	−	0.972084i	$-0.575389\pi$
$887$	14.5904	−	8.42380i	0.489899	−	0.282843i	0.724533	+	0.689240i	$0.242056\pi$
$888$		0			0
$889$	−38.5744	+	1.42057i	−1.29374	+	0.0476444i
$890$		0			0
$891$	1.36505	+	27.2886i	0.0457310	+	0.914202i
$892$		0			0
$893$	12.2960	−	21.2973i	0.411470	−	0.712687i
$894$		0			0
$895$	2.44448	−	1.49786i	0.0817099	−	0.0500680i
$896$		0			0
$897$	17.0452	+	9.55907i	0.569122	+	0.319168i
$898$		0			0
$899$	−2.34720	+	4.06546i	−0.0782834	+	0.135591i
$900$		0			0
$901$	−31.7462	+	18.3287i	−1.05762	+	0.610617i
$902$		0			0
$903$	9.32153	−	6.01307i	0.310201	−	0.200103i
$904$		0			0
$905$	28.6146	+	15.5378i	0.951182	+	0.516492i
$906$		0			0
$907$	−14.0050	−	8.08581i	−0.465030	−	0.268485i	0.249127	−	0.968471i	$-0.419856\pi$
$907$	−14.0050	−	8.08581i	−0.465030	−	0.268485i	−0.714157	+	0.699986i	$0.753190\pi$
$908$		0			0
$909$	25.7543	+	42.1395i	0.854215	+	1.39768i
$910$		0			0
$911$		−	13.3127i		−	0.441071i	−0.975379	−	0.220535i	$-0.929220\pi$
$911$		−	13.3127i		−	0.441071i	0.975379	−	0.220535i	$-0.0707805\pi$
$912$		0			0
$913$	15.5979	−	27.0164i	0.516216	−	0.894111i
$914$		0			0
$915$	−27.2842	−	43.3033i	−0.901989	−	1.43156i
$916$		0			0
$917$	44.9807	−	28.2263i	1.48539	−	0.932113i
$918$		0			0
$919$	6.00415	+	10.3995i	0.198058	+	0.343047i	0.947899	−	0.318571i	$-0.103203\pi$
$919$	6.00415	+	10.3995i	0.198058	+	0.343047i	−0.749840	+	0.661619i	$0.769870\pi$
$920$		0			0
$921$	14.0928	−	8.37301i	0.464375	−	0.275900i
$922$		0			0
$923$			16.4137i			0.540263i
$924$		0			0
$925$	12.5298	+	6.38535i	0.411978	+	0.209949i
$926$		0			0
$927$	−19.8080	+	0.495118i	−0.650581	+	0.0162618i
$928$		0			0
$929$	3.50004	+	6.06225i	0.114833	+	0.198896i	0.917713	−	0.397244i	$-0.130034\pi$
$929$	3.50004	+	6.06225i	0.114833	+	0.198896i	−0.802880	+	0.596140i	$0.796700\pi$
$930$		0			0
$931$	37.3299	−	25.3866i	1.22344	−	0.832013i
$932$		0			0
$933$	−0.141044	−	11.2872i	−0.00461758	−	0.369526i
$934$		0			0
$935$	−35.2366	+	0.921476i	−1.15236	+	0.0301355i
$936$		0			0
$937$	−20.7984			−0.679456			−0.339728	−	0.940524i	$-0.610335\pi$
$937$	−20.7984			−0.679456			−0.339728	+	0.940524i	$0.610335\pi$
$938$		0			0
$939$	−33.4423	−	18.7547i	−1.09135	−	0.612037i
$940$		0			0
$941$	14.5688	−	25.2339i	0.474929	−	0.822600i	−0.524659	−	0.851312i	$-0.675807\pi$
$941$	14.5688	−	25.2339i	0.474929	−	0.822600i	0.999588	+	0.0287120i	$0.00914058\pi$
$942$		0			0
$943$	−2.59025	−	4.48644i	−0.0843501	−	0.146099i
$944$		0			0
$945$	12.6235	−	28.0294i	0.410642	−	0.911797i
$946$		0			0
$947$	7.98578	+	13.8318i	0.259503	+	0.449472i	0.966109	−	0.258135i	$-0.0831079\pi$
$947$	7.98578	+	13.8318i	0.259503	+	0.449472i	−0.706606	+	0.707607i	$0.749775\pi$
$948$		0			0
$949$	−0.176871	+	0.306349i	−0.00574146	+	0.00994451i
$950$		0			0
$951$	−6.36860	−	3.57156i	−0.206516	−	0.115816i
$952$		0			0
$953$	24.9487			0.808167			0.404084	−	0.914722i	$-0.367590\pi$
$953$	24.9487			0.808167			0.404084	+	0.914722i	$0.367590\pi$
$954$		0			0
$955$	1.39741	+	53.4360i	0.0452191	+	1.72915i
$956$		0			0
$957$	0.229051	+	18.3300i	0.00740416	+	0.592525i
$958$		0			0
$959$	10.9220	+	5.78067i	0.352689	+	0.186668i
$960$		0			0
$961$	−14.5934	−	25.2765i	−0.470755	−	0.815371i
$962$		0			0
$963$	55.5822	−	1.38932i	1.79111	−	0.0447703i
$964$		0			0
$965$	−16.0876	+	9.85770i	−0.517877	+	0.317331i
$966$		0			0
$967$			7.93647i			0.255220i	0.991824	+	0.127610i	$0.0407305\pi$
$967$			7.93647i			0.255220i	−0.991824	+	0.127610i	$0.959269\pi$
$968$		0			0
$969$	−49.8647	+	29.6263i	−1.60189	+	0.951733i
$970$		0			0
$971$	17.5378	+	30.3763i	0.562814	+	0.974822i	0.997249	+	0.0741192i	$0.0236145\pi$
$971$	17.5378	+	30.3763i	0.562814	+	0.974822i	−0.434436	+	0.900703i	$0.643052\pi$
$972$		0			0
$973$	14.5760	−	0.536788i	0.467286	−	0.0172086i
$974$		0			0
$975$	17.9935	+	8.88827i	0.576253	+	0.284653i
$976$		0			0
$977$	−1.56200	+	2.70546i	−0.0499728	+	0.0865554i	−0.889930	−	0.456098i	$-0.849247\pi$
$977$	−1.56200	+	2.70546i	−0.0499728	+	0.0865554i	0.839957	+	0.542653i	$0.182580\pi$
$978$		0			0
$979$		−	43.1646i		−	1.37955i
$980$		0			0
$981$	28.9464	+	47.3626i	0.924189	+	1.51217i
$982$		0			0
$983$	19.8301	+	11.4489i	0.632481	+	0.365163i	0.781712	−	0.623639i	$-0.214347\pi$
$983$	19.8301	+	11.4489i	0.632481	+	0.365163i	−0.149231	+	0.988802i	$0.547680\pi$
$984$		0			0
$985$	8.81226	−	16.2288i	0.280782	−	0.517093i
$986$		0			0
$987$	15.5454	+	7.98068i	0.494816	+	0.254028i
$988$		0			0
$989$	10.2067	−	5.89282i	0.324553	−	0.187381i
$990$		0			0
$991$	−5.34209	+	9.25277i	−0.169697	+	0.293924i	−0.938313	−	0.345786i	$-0.887612\pi$
$991$	−5.34209	+	9.25277i	−0.169697	+	0.293924i	0.768616	+	0.639710i	$0.220946\pi$
$992$		0			0
$993$	−36.2694	−	20.3402i	−1.15098	−	0.645476i
$994$		0			0
$995$	−19.6830	+	12.0608i	−0.623994	+	0.382354i
$996$		0			0
$997$	25.1937	−	43.6367i	0.797892	−	1.38199i	−0.123094	−	0.992395i	$-0.539282\pi$
$997$	25.1937	−	43.6367i	0.797892	−	1.38199i	0.920986	−	0.389595i	$-0.127385\pi$
$998$		0			0
$999$	−7.77657	−	12.3740i	−0.246040	−	0.391495i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	420.2.bn.a.269.3	yes	32
3.2	odd	2	inner	420.2.bn.a.269.8	yes	32
5.2	odd	4		2100.2.bi.n.101.11		32
5.3	odd	4		2100.2.bi.n.101.6		32
5.4	even	2	inner	420.2.bn.a.269.14	yes	32
7.3	odd	6		2940.2.f.a.1469.6		32
7.4	even	3		2940.2.f.a.1469.27		32
7.5	odd	6	inner	420.2.bn.a.89.9	yes	32
15.2	even	4		2100.2.bi.n.101.16		32
15.8	even	4		2100.2.bi.n.101.1		32
15.14	odd	2	inner	420.2.bn.a.269.9	yes	32
21.5	even	6	inner	420.2.bn.a.89.14	yes	32
21.11	odd	6		2940.2.f.a.1469.26		32
21.17	even	6		2940.2.f.a.1469.7		32
35.4	even	6		2940.2.f.a.1469.5		32
35.12	even	12		2100.2.bi.n.1601.16		32
35.19	odd	6	inner	420.2.bn.a.89.8	yes	32
35.24	odd	6		2940.2.f.a.1469.28		32
35.33	even	12		2100.2.bi.n.1601.1		32
105.47	odd	12		2100.2.bi.n.1601.11		32
105.59	even	6		2940.2.f.a.1469.25		32
105.68	odd	12		2100.2.bi.n.1601.6		32
105.74	odd	6		2940.2.f.a.1469.8		32
105.89	even	6	inner	420.2.bn.a.89.3	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
420.2.bn.a.89.3	✓	32	105.89	even	6	inner
420.2.bn.a.89.8	yes	32	35.19	odd	6	inner
420.2.bn.a.89.9	yes	32	7.5	odd	6	inner
420.2.bn.a.89.14	yes	32	21.5	even	6	inner
420.2.bn.a.269.3	yes	32	1.1	even	1	trivial
420.2.bn.a.269.8	yes	32	3.2	odd	2	inner
420.2.bn.a.269.9	yes	32	15.14	odd	2	inner
420.2.bn.a.269.14	yes	32	5.4	even	2	inner
2100.2.bi.n.101.1		32	15.8	even	4
2100.2.bi.n.101.6		32	5.3	odd	4
2100.2.bi.n.101.11		32	5.2	odd	4
2100.2.bi.n.101.16		32	15.2	even	4
2100.2.bi.n.1601.1		32	35.33	even	12
2100.2.bi.n.1601.6		32	105.68	odd	12
2100.2.bi.n.1601.11		32	105.47	odd	12
2100.2.bi.n.1601.16		32	35.12	even	12
2940.2.f.a.1469.5		32	35.4	even	6
2940.2.f.a.1469.6		32	7.3	odd	6
2940.2.f.a.1469.7		32	21.17	even	6
2940.2.f.a.1469.8		32	105.74	odd	6
2940.2.f.a.1469.25		32	105.59	even	6
2940.2.f.a.1469.26		32	21.11	odd	6
2940.2.f.a.1469.27		32	7.4	even	3
2940.2.f.a.1469.28		32	35.24	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:	\( N \)	\(=\)	\( 420 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	420.bn (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.48906 + 0.884700i) q^{3} +(-1.06703 + 1.96506i) q^{5} +(0.0973684 + 2.64396i) q^{7} +(1.43461 - 2.63475i) q^{9} +(-2.62913 - 1.51793i) q^{11} +2.31738 q^{13} +(-0.149616 - 3.87009i) q^{15} +(-4.49684 - 2.59625i) q^{17} +(-5.58515 + 3.22459i) q^{19} +(-2.48410 - 3.85088i) q^{21} +(-2.43442 - 4.21654i) q^{23} +(-2.72290 - 4.19354i) q^{25} +(0.194737 + 5.19250i) q^{27} +3.48622i q^{29} +(1.16615 + 0.673279i) q^{31} +(5.25784 - 0.0657018i) q^{33} +(-5.29943 - 2.62984i) q^{35} +(-2.43579 + 1.40630i) q^{37} +(-3.45072 + 2.05018i) q^{39} +1.06401 q^{41} +2.42063i q^{43} +(3.64666 + 5.63044i) q^{45} +(-3.30233 + 1.90660i) q^{47} +(-6.98104 + 0.514876i) q^{49} +(8.99298 - 0.112376i) q^{51} +(3.52984 - 6.11386i) q^{53} +(5.78817 - 3.54672i) q^{55} +(5.46384 - 9.74279i) q^{57} +(-6.18029 + 10.7046i) q^{59} +(11.4447 - 6.60758i) q^{61} +(7.10585 + 3.53651i) q^{63} +(-2.47270 + 4.55378i) q^{65} +(-3.72081 - 2.14821i) q^{67} +(7.35538 + 4.12496i) q^{69} +7.08288i q^{71} +(-0.0763237 + 0.132196i) q^{73} +(7.76460 + 3.83549i) q^{75} +(3.75734 - 7.09910i) q^{77} +(-3.04462 - 5.27344i) q^{79} +(-4.88378 - 7.55967i) q^{81} +10.2758i q^{83} +(9.90003 - 6.06628i) q^{85} +(-3.08426 - 5.19119i) q^{87} +(7.10913 + 12.3134i) q^{89} +(0.225639 + 6.12705i) q^{91} +(-2.33212 + 0.0291421i) q^{93} +(-0.376991 - 14.4159i) q^{95} +8.63266 q^{97} +(-7.77113 + 4.74945i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 2 q^{15} + 12 q^{19} - 8 q^{21} + 6 q^{25} - 12 q^{31} + 24 q^{39} + 33 q^{45} - 44 q^{49} - 10 q^{51} - 24 q^{61} + 21 q^{75} - 28 q^{79} - 20 q^{81} - 4 q^{85} + 16 q^{91} - 28 q^{99}+O(q^{100}) \) 32 * q + 2 * q^15 + 12 * q^19 - 8 * q^21 + 6 * q^25 - 12 * q^31 + 24 * q^39 + 33 * q^45 - 44 * q^49 - 10 * q^51 - 24 * q^61 + 21 * q^75 - 28 * q^79 - 20 * q^81 - 4 * q^85 + 16 * q^91 - 28 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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