LMFDB - Embedded newform 112.10.p.a.31.9

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [112,10,Mod(31,112)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(112, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 0, 1])) N = Newforms(chi, 10, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("112.31"); S:= CuspForms(chi, 10); N := Newforms(S);

Level:	$ N $	$=$	$ 112 = 2^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	112.p (of order $6$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [24,0,-162] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$57.6840136504$
Analytic rank:	$0$
Dimension:	$24$
Relative dimension:	$12$ over $\Q(\zeta_{6})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			31.9
Character	$\chi$	$=$	112.31
Dual form			112.10.p.a.47.9

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(44.5044 - 77.0839i) q^{3} +(1678.45 - 969.052i) q^{5} +(-6146.89 - 1602.93i) q^{7} +(5880.22 + 10184.8i) q^{9} +(-3551.98 - 2050.74i) q^{11} +93186.2i q^{13} -172508. i q^{15} +(539781. + 311643. i) q^{17} +(338564. + 586411. i) q^{19} +(-397124. + 402488. i) q^{21} +(281953. - 162785. i) q^{23} +(901560. - 1.56155e6i) q^{25} +2.79874e6 q^{27} +2.51930e6 q^{29} +(-3.02394e6 + 5.23761e6i) q^{31} +(-316158. + 182534. i) q^{33} +(-1.18705e7 + 3.26622e6i) q^{35} +(-8.27646e6 - 1.43352e7i) q^{37} +(7.18315e6 + 4.14720e6i) q^{39} +1.45617e7i q^{41} -2.07780e7i q^{43} +(1.97393e7 + 1.13965e7i) q^{45} +(908961. + 1.57437e6i) q^{47} +(3.52148e7 + 1.97061e7i) q^{49} +(4.80453e7 - 2.77389e7i) q^{51} +(-8.31708e6 + 1.44056e7i) q^{53} -7.94908e6 q^{55} +6.02704e7 q^{57} +(8.34468e7 - 1.44534e8i) q^{59} +(2.92586e7 - 1.68924e7i) q^{61} +(-1.98194e7 - 7.20306e7i) q^{63} +(9.03022e7 + 1.56408e8i) q^{65} +(-1.66747e8 - 9.62715e7i) q^{67} -2.89787e7i q^{69} +3.04210e8i q^{71} +(3.22473e7 + 1.86180e7i) q^{73} +(-8.02468e7 - 1.38992e8i) q^{75} +(1.85464e7 + 1.82992e7i) q^{77} +(3.13759e8 - 1.81149e8i) q^{79} +(8.81609e6 - 1.52699e7i) q^{81} -6.01201e7 q^{83} +1.20799e9 q^{85} +(1.12120e8 - 1.94197e8i) q^{87} +(7.38024e8 - 4.26098e8i) q^{89} +(1.49371e8 - 5.72805e8i) q^{91} +(2.69157e8 + 4.66194e8i) q^{93} +(1.13652e9 + 6.56173e8i) q^{95} +1.55512e9i q^{97} -4.82351e7i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 162 q^{3} - 852 q^{5} - 6744 q^{7} - 77884 q^{9} - 57534 q^{11} + 789336 q^{17} + 469098 q^{19} - 2104376 q^{21} + 1553682 q^{23} + 3602544 q^{25} + 6389244 q^{27} - 2462040 q^{29} - 10306686 q^{31}+ \cdots + 1433917218 q^{95}+O(q^{100}) $ 24 * q - 162 * q^3 - 852 * q^5 - 6744 * q^7 - 77884 * q^9 - 57534 * q^11 + 789336 * q^17 + 469098 * q^19 - 2104376 * q^21 + 1553682 * q^23 + 3602544 * q^25 + 6389244 * q^27 - 2462040 * q^29 - 10306686 * q^31 + 8789748 * q^33 + 21149802 * q^35 + 10422720 * q^37 + 54458664 * q^39 + 2549604 * q^45 + 12751230 * q^47 - 16137504 * q^49 - 107322414 * q^51 + 62342520 * q^53 - 422625348 * q^55 - 317200696 * q^57 - 101234394 * q^59 + 48881304 * q^61 + 726717996 * q^63 + 272864028 * q^65 - 48335310 * q^67 + 672073812 * q^73 + 950672484 * q^75 + 10755684 * q^77 + 144370122 * q^79 - 305389432 * q^81 - 568641984 * q^83 - 734871504 * q^85 + 33665484 * q^87 + 1630891500 * q^89 + 1012044192 * q^91 - 457741376 * q^93 + 1433917218 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/112\mathbb{Z}\right)^\times$.

$n$	$15$	$17$	$85$
$\chi(n)$	$-1$	$e\left(\frac{1}{6}\right)$	$1$

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
$3$	44.5044	−	77.0839i	0.317218	−	0.549437i	−0.662689	−	0.748895i	$-0.730585\pi$
$3$	44.5044	−	77.0839i	0.317218	−	0.549437i	0.979906	+	0.199458i	$0.0639181\pi$
$4$		0			0
$5$	1678.45	−	969.052i	1.20100	−	0.693397i	0.240222	−	0.970718i	$-0.422780\pi$
$5$	1678.45	−	969.052i	1.20100	−	0.693397i	0.960777	+	0.277321i	$0.0894466\pi$
$6$		0			0
$7$	−6146.89	−	1602.93i	−0.967641	−	0.252333i
$7$	−6146.89	−	1602.93i	−0.967641	−	0.252333i
$8$		0			0
$9$	5880.22	+	10184.8i	0.298746	+	0.517443i
$10$		0			0
$11$	−3551.98	−	2050.74i	−0.0731482	−	0.0422321i	0.462980	−	0.886369i	$-0.346780\pi$
$11$	−3551.98	−	2050.74i	−0.0731482	−	0.0422321i	−0.536128	+	0.844137i	$0.680114\pi$
$12$		0			0
$13$			93186.2i			0.904912i	0.891787	+	0.452456i	$0.149452\pi$
$13$			93186.2i			0.904912i	−0.891787	+	0.452456i	$0.850548\pi$
$14$		0			0
$15$		−	172508.i		−	0.879831i
$16$		0			0
$17$	539781.	+	311643.i	1.56746	+	0.904975i	0.996464	+	0.0840250i	$0.0267776\pi$
$17$	539781.	+	311643.i	1.56746	+	0.904975i	0.571000	+	0.820950i	$0.306556\pi$
$18$		0			0
$19$	338564.	+	586411.i	0.596005	+	1.03231i	0.993404	+	0.114665i	$0.0365796\pi$
$19$	338564.	+	586411.i	0.596005	+	1.03231i	−0.397399	+	0.917646i	$0.630087\pi$
$20$		0			0
$21$	−397124.	+	402488.i	−0.445594	+	0.451613i
$22$		0			0
$23$	281953.	−	162785.i	0.210088	−	0.121294i	−0.391264	−	0.920278i	$-0.627962\pi$
$23$	281953.	−	162785.i	0.210088	−	0.121294i	0.601352	+	0.798984i	$0.294629\pi$
$24$		0			0
$25$	901560.	−	1.56155e6i	0.461599	−	0.799513i
$26$		0			0
$27$	2.79874e6			1.01351
$28$		0			0
$29$	2.51930e6			0.661436			0.330718	−	0.943730i	$-0.392709\pi$
$29$	2.51930e6			0.661436			0.330718	+	0.943730i	$0.392709\pi$
$30$		0			0
$31$	−3.02394e6	+	5.23761e6i	−0.588092	+	1.01860i	0.406390	+	0.913699i	$0.366787\pi$
$31$	−3.02394e6	+	5.23761e6i	−0.588092	+	1.01860i	−0.994482	+	0.104905i	$0.966546\pi$
$32$		0			0
$33$	−316158.	+	182534.i	−0.0464078	+	0.0267935i
$34$		0			0
$35$	−1.18705e7	+	3.26622e6i	−1.33710	+	0.367908i
$36$		0			0
$37$	−8.27646e6	−	1.43352e7i	−0.726000	−	1.25747i	−0.958561	−	0.284887i	$-0.908044\pi$
$37$	−8.27646e6	−	1.43352e7i	−0.726000	−	1.25747i	0.232561	−	0.972582i	$-0.425289\pi$
$38$		0			0
$39$	7.18315e6	+	4.14720e6i	0.497192	+	0.287054i
$40$		0			0
$41$			1.45617e7i			0.804794i	0.915465	+	0.402397i	$0.131823\pi$
$41$			1.45617e7i			0.804794i	−0.915465	+	0.402397i	$0.868177\pi$
$42$		0			0
$43$		−	2.07780e7i		−	0.926820i	−0.886144	−	0.463410i	$-0.846626\pi$
$43$		−	2.07780e7i		−	0.926820i	0.886144	−	0.463410i	$-0.153374\pi$
$44$		0			0
$45$	1.97393e7	+	1.13965e7i	0.717587	+	0.414299i
$46$		0			0
$47$	908961.	+	1.57437e6i	0.0271709	+	0.0470615i	0.879291	−	0.476285i	$-0.158017\pi$
$47$	908961.	+	1.57437e6i	0.0271709	+	0.0470615i	−0.852120	+	0.523346i	$0.824683\pi$
$48$		0			0
$49$	3.52148e7	+	1.97061e7i	0.872656	+	0.488335i
$50$		0			0
$51$	4.80453e7	−	2.77389e7i	0.994454	−	0.574148i
$52$		0			0
$53$	−8.31708e6	+	1.44056e7i	−0.144787	+	0.250778i	−0.929293	−	0.369342i	$-0.879583\pi$
$53$	−8.31708e6	+	1.44056e7i	−0.144787	+	0.250778i	0.784506	+	0.620121i	$0.212916\pi$
$54$		0			0
$55$	−7.94908e6			−0.117134
$56$		0			0
$57$	6.02704e7			0.756254
$58$		0			0
$59$	8.34468e7	−	1.44534e8i	0.896553	−	1.55287i	0.0646816	−	0.997906i	$-0.479397\pi$
$59$	8.34468e7	−	1.44534e8i	0.896553	−	1.55287i	0.831871	−	0.554969i	$-0.187270\pi$
$60$		0			0
$61$	2.92586e7	−	1.68924e7i	0.270563	−	0.156210i	−0.358580	−	0.933499i	$-0.616739\pi$
$61$	2.92586e7	−	1.68924e7i	0.270563	−	0.156210i	0.629144	+	0.777289i	$0.283406\pi$
$62$		0			0
$63$	−1.98194e7	−	7.20306e7i	−0.158511	−	0.576082i
$64$		0			0
$65$	9.03022e7	+	1.56408e8i	0.627463	+	1.08680i
$66$		0			0
$67$	−1.66747e8	−	9.62715e7i	−1.01093	−	0.583662i	−0.0994675	−	0.995041i	$-0.531714\pi$
$67$	−1.66747e8	−	9.62715e7i	−1.01093	−	0.583662i	−0.911464	+	0.411379i	$0.865047\pi$
$68$		0			0
$69$		−	2.89787e7i		−	0.153907i
$70$		0			0
$71$			3.04210e8i			1.42073i	0.703834	+	0.710364i	$0.251470\pi$
$71$			3.04210e8i			1.42073i	−0.703834	+	0.710364i	$0.748530\pi$
$72$		0			0
$73$	3.22473e7	+	1.86180e7i	0.132905	+	0.0767325i	0.564978	−	0.825106i	$-0.308885\pi$
$73$	3.22473e7	+	1.86180e7i	0.132905	+	0.0767325i	−0.432074	+	0.901838i	$0.642218\pi$
$74$		0			0
$75$	−8.02468e7	−	1.38992e8i	−0.292855	−	0.507239i
$76$		0			0
$77$	1.85464e7	+	1.82992e7i	0.0601246	+	0.0593232i
$78$		0			0
$79$	3.13759e8	−	1.81149e8i	0.906303	−	0.523255i	0.0270635	−	0.999634i	$-0.491384\pi$
$79$	3.13759e8	−	1.81149e8i	0.906303	−	0.523255i	0.879240	+	0.476379i	$0.158051\pi$
$80$		0			0
$81$	8.81609e6	−	1.52699e7i	0.0227559	−	0.0394143i
$82$		0			0
$83$	−6.01201e7			−0.139049			−0.0695245	−	0.997580i	$-0.522148\pi$
$83$	−6.01201e7			−0.139049			−0.0695245	+	0.997580i	$0.522148\pi$
$84$		0			0
$85$	1.20799e9			2.51003
$86$		0			0
$87$	1.12120e8	−	1.94197e8i	0.209819	−	0.363418i
$88$		0			0
$89$	7.38024e8	−	4.26098e8i	1.24685	−	0.719871i	0.276373	−	0.961051i	$-0.410868\pi$
$89$	7.38024e8	−	4.26098e8i	1.24685	−	0.719871i	0.970481	+	0.241179i	$0.0775342\pi$
$90$		0			0
$91$	1.49371e8	−	5.72805e8i	0.228339	−	0.875630i
$92$		0			0
$93$	2.69157e8	+	4.66194e8i	0.373106	+	0.646239i
$94$		0			0
$95$	1.13652e9	+	6.56173e8i	1.43160	+	0.826537i
$96$		0			0
$97$			1.55512e9i			1.78358i	0.452452	+	0.891789i	$0.350549\pi$
$97$			1.55512e9i			1.78358i	−0.452452	+	0.891789i	$0.649451\pi$
$98$		0			0
$99$		−	4.82351e7i		−	0.0504667i
$100$		0			0
$101$	1.07368e9	+	6.19891e8i	1.02667	+	0.592747i	0.916028	−	0.401114i	$-0.131377\pi$
$101$	1.07368e9	+	6.19891e8i	1.02667	+	0.592747i	0.110639	+	0.993861i	$0.464710\pi$
$102$		0			0
$103$	−5.46543e8	−	9.46640e8i	−0.478472	−	0.828738i	0.521223	−	0.853421i	$-0.325476\pi$
$103$	−5.46543e8	−	9.46640e8i	−0.478472	−	0.828738i	−0.999695	+	0.0246821i	$0.992143\pi$
$104$		0			0
$105$	−2.76519e8	+	1.06039e9i	−0.222010	+	0.851360i
$106$		0			0
$107$	1.02523e9	−	5.91916e8i	0.756125	−	0.436549i	−0.0717779	−	0.997421i	$-0.522867\pi$
$107$	1.02523e9	−	5.91916e8i	0.756125	−	0.436549i	0.827903	+	0.560872i	$0.189534\pi$
$108$		0			0
$109$	8.31971e8	−	1.44102e9i	0.564533	−	0.977799i	−0.432560	−	0.901605i	$-0.642390\pi$
$109$	8.31971e8	−	1.44102e9i	0.564533	−	0.977799i	0.997093	−	0.0761941i	$-0.0242769\pi$
$110$		0			0
$111$	−1.47336e9			−0.921200
$112$		0			0
$113$	1.49759e9			0.864054			0.432027	−	0.901861i	$-0.357799\pi$
$113$	1.49759e9			0.864054			0.432027	+	0.901861i	$0.357799\pi$
$114$		0			0
$115$	3.15495e8	−	5.46453e8i	0.168210	−	0.291348i
$116$		0			0
$117$	−9.49085e8	+	5.47955e8i	−0.468240	+	0.270339i
$118$		0			0
$119$	−2.81843e9	−	2.78086e9i	−1.28839	−	1.27121i
$120$		0			0
$121$	−1.17056e9	−	2.02747e9i	−0.496433	−	0.859847i
$122$		0			0
$123$	1.12247e9	+	6.48060e8i	0.442184	+	0.255295i
$124$		0			0
$125$			2.90724e8i			0.106509i
$126$		0			0
$127$			4.17913e8i			0.142551i	0.997457	+	0.0712753i	$0.0227069\pi$
$127$			4.17913e8i			0.142551i	−0.997457	+	0.0712753i	$0.977293\pi$
$128$		0			0
$129$	−1.60165e9	−	9.24712e8i	−0.509229	−	0.294004i
$130$		0			0
$131$	2.64239e9	+	4.57675e9i	0.783927	+	1.35780i	0.929638	+	0.368474i	$0.120120\pi$
$131$	2.64239e9	+	4.57675e9i	0.783927	+	1.35780i	−0.145711	+	0.989327i	$0.546547\pi$
$132$		0			0
$133$	−1.14114e9	−	4.14730e9i	−0.316233	−	1.14930i
$134$		0			0
$135$	4.69754e9	−	2.71213e9i	1.21722	−	0.702762i
$136$		0			0
$137$	−2.88346e9	+	4.99431e9i	−0.699314	+	1.21125i	0.269391	+	0.963031i	$0.413178\pi$
$137$	−2.88346e9	+	4.99431e9i	−0.699314	+	1.21125i	−0.968705	+	0.248216i	$0.920156\pi$
$138$		0			0
$139$	−8.10344e9			−1.84121			−0.920604	−	0.390497i	$-0.872303\pi$
$139$	−8.10344e9			−1.84121			−0.920604	+	0.390497i	$0.872303\pi$
$140$		0			0
$141$	1.61811e8			0.0344764
$142$		0			0
$143$	1.91100e8	−	3.30995e8i	0.0382163	−	0.0661927i
$144$		0			0
$145$	4.22850e9	−	2.44133e9i	0.794384	−	0.458638i
$146$		0			0
$147$	3.08624e9	−	1.83749e9i	0.545131	−	0.324561i
$148$		0			0
$149$	5.43731e8	+	9.41770e8i	0.0903745	+	0.156533i	0.907669	−	0.419687i	$-0.137860\pi$
$149$	5.43731e8	+	9.41770e8i	0.0903745	+	0.156533i	−0.817294	+	0.576221i	$0.804527\pi$
$150$		0			0
$151$	−1.14310e9	−	6.59970e8i	−0.178932	−	0.103307i	0.407859	−	0.913045i	$-0.366276\pi$
$151$	−1.14310e9	−	6.59970e8i	−0.178932	−	0.103307i	−0.586791	+	0.809738i	$0.699609\pi$
$152$		0			0
$153$			7.33010e9i			1.08143i
$154$		0			0
$155$			1.17214e10i			1.63112i
$156$		0			0
$157$	−1.02002e10	−	5.88909e9i	−1.33986	−	0.773571i	−0.353077	−	0.935594i	$-0.614865\pi$
$157$	−1.02002e10	−	5.88909e9i	−1.33986	−	0.773571i	−0.986787	+	0.162023i	$0.948198\pi$
$158$		0			0
$159$	7.40293e8	+	1.28223e9i	0.0918580	+	0.159103i
$160$		0			0
$161$	−1.99406e9	+	5.48673e8i	−0.233896	+	0.0643572i
$162$		0			0
$163$	−3.69689e9	+	2.13440e9i	−0.410196	+	0.236827i	−0.690874	−	0.722975i	$-0.742774\pi$
$163$	−3.69689e9	+	2.13440e9i	−0.410196	+	0.236827i	0.280678	+	0.959802i	$0.409441\pi$
$164$		0			0
$165$	−3.53769e8	+	6.12746e8i	−0.0371571	+	0.0643580i
$166$		0			0
$167$	−1.03702e9			−0.103172			−0.0515861	−	0.998669i	$-0.516428\pi$
$167$	−1.03702e9			−0.103172			−0.0515861	+	0.998669i	$0.516428\pi$
$168$		0			0
$169$	1.92084e9			0.181134
$170$		0			0
$171$	−3.98166e9	+	6.89644e9i	−0.356108	+	0.616798i
$172$		0			0
$173$	−3.30550e9	+	1.90843e9i	−0.280562	+	0.161983i	−0.633678	−	0.773597i	$-0.718456\pi$
$173$	−3.30550e9	+	1.90843e9i	−0.280562	+	0.161983i	0.353116	+	0.935580i	$0.385122\pi$
$174$		0			0
$175$	−8.04484e9	+	8.15352e9i	−0.648405	+	0.657164i
$176$		0			0
$177$	−7.42750e9	−	1.28648e10i	−0.568805	−	0.985199i
$178$		0			0
$179$	1.19270e10	+	6.88608e9i	0.868349	+	0.501341i	0.866799	−	0.498657i	$-0.166173\pi$
$179$	1.19270e10	+	6.88608e9i	0.868349	+	0.501341i	0.00154950	+	0.999999i	$0.499507\pi$
$180$		0			0
$181$		−	1.47399e10i		−	1.02080i	−0.859937	−	0.510399i	$-0.829498\pi$
$181$		−	1.47399e10i		−	1.02080i	0.859937	−	0.510399i	$-0.170502\pi$
$182$		0			0
$183$		−	3.00715e9i		−	0.198210i
$184$		0			0
$185$	−2.77832e10	−	1.60406e10i	−1.74385	−	1.00681i
$186$		0			0
$187$	−1.27819e9	−	2.21390e9i	−0.0764380	−	0.132395i
$188$		0			0
$189$	−1.72036e10	−	4.48619e9i	−0.980709	−	0.255741i
$190$		0			0
$191$	−9.59224e9	+	5.53808e9i	−0.521518	+	0.301099i	−0.737556	−	0.675286i	$-0.764020\pi$
$191$	−9.59224e9	+	5.53808e9i	−0.521518	+	0.301099i	0.216037	+	0.976385i	$0.430687\pi$
$192$		0			0
$193$	1.04466e9	−	1.80940e9i	0.0541958	−	0.0938699i	−0.837655	−	0.546200i	$-0.816074\pi$
$193$	1.04466e9	−	1.80940e9i	0.0541958	−	0.0938699i	0.891851	+	0.452330i	$0.149407\pi$
$194$		0			0
$195$	1.60754e10			0.796170
$196$		0			0
$197$	−1.07436e10			−0.508222			−0.254111	−	0.967175i	$-0.581783\pi$
$197$	−1.07436e10			−0.508222			−0.254111	+	0.967175i	$0.581783\pi$
$198$		0			0
$199$	−1.30054e10	+	2.25260e10i	−0.587874	+	1.01823i	0.406636	+	0.913590i	$0.366702\pi$
$199$	−1.30054e10	+	2.25260e10i	−0.587874	+	1.01823i	−0.994510	+	0.104637i	$0.966632\pi$
$200$		0			0
$201$	−1.48420e10	+	8.56901e9i	−0.641371	+	0.370296i
$202$		0			0
$203$	−1.54858e10	−	4.03826e9i	−0.640033	−	0.166902i
$204$		0			0
$205$	1.41110e10	+	2.44410e10i	0.558042	+	0.966557i
$206$		0			0
$207$	3.31588e9	+	1.91443e9i	0.125526	+	0.0724723i
$208$		0			0
$209$		−	2.77723e9i		−	0.100682i
$210$		0			0
$211$		−	4.24346e10i		−	1.47383i	−0.675983	−	0.736917i	$-0.736281\pi$
$211$		−	4.24346e10i		−	1.47383i	0.675983	−	0.736917i	$-0.263719\pi$
$212$		0			0
$213$	2.34497e10	+	1.35387e10i	0.780601	+	0.450680i
$214$		0			0
$215$	−2.01349e10	−	3.48747e10i	−0.642654	−	1.11311i
$216$		0			0
$217$	2.69833e10	−	2.73478e10i	0.826089	−	0.837248i
$218$		0			0
$219$	2.87029e9	−	1.65716e9i	0.0843193	−	0.0486818i
$220$		0			0
$221$	−2.90408e10	+	5.03001e10i	−0.818923	+	1.41842i
$222$		0			0
$223$	4.63089e10			1.25399			0.626993	−	0.779025i	$-0.284285\pi$
$223$	4.63089e10			1.25399			0.626993	+	0.779025i	$0.284285\pi$
$224$		0			0
$225$	2.12055e10			0.551603
$226$		0			0
$227$	−1.95744e9	+	3.39039e9i	−0.0489297	+	0.0847487i	−0.889453	−	0.457027i	$-0.848914\pi$
$227$	−1.95744e9	+	3.39039e9i	−0.0489297	+	0.0847487i	0.840523	+	0.541775i	$0.182248\pi$
$228$		0			0
$229$	4.42299e9	−	2.55361e9i	0.106281	−	0.0613614i	−0.445917	−	0.895074i	$-0.647122\pi$
$229$	4.42299e9	−	2.55361e9i	0.106281	−	0.0613614i	0.552198	+	0.833713i	$0.313789\pi$
$230$		0			0
$231$	2.23597e9	−	6.15235e8i	0.0516669	−	0.0142163i
$232$		0			0
$233$	1.58263e10	+	2.74120e10i	0.351786	+	0.609311i	0.986562	−	0.163385i	$-0.0522412\pi$
$233$	1.58263e10	+	2.74120e10i	0.351786	+	0.609311i	−0.634777	+	0.772696i	$0.718908\pi$
$234$		0			0
$235$	3.05128e9	+	1.76166e9i	0.0652645	+	0.0376805i
$236$		0			0
$237$		−	3.22476e10i		−	0.663942i
$238$		0			0
$239$		−	1.41427e10i		−	0.280378i	−0.990125	−	0.140189i	$-0.955229\pi$
$239$		−	1.41427e10i		−	0.280378i	0.990125	−	0.140189i	$-0.0447710\pi$
$240$		0			0
$241$	−4.02266e9	−	2.32248e9i	−0.0768133	−	0.0443482i	0.461101	−	0.887347i	$-0.347454\pi$
$241$	−4.02266e9	−	2.32248e9i	−0.0768133	−	0.0443482i	−0.537915	+	0.842999i	$0.680788\pi$
$242$		0			0
$243$	2.67591e10	+	4.63481e10i	0.492315	+	0.852715i
$244$		0			0
$245$	7.82024e10	−	1.04940e9i	1.38667	−	0.0186078i
$246$		0			0
$247$	−5.46454e10	+	3.15495e10i	−0.934151	+	0.539332i
$248$		0			0
$249$	−2.67561e9	+	4.63429e9i	−0.0441088	+	0.0763987i
$250$		0			0
$251$	5.82163e10			0.925791			0.462896	−	0.886413i	$-0.346811\pi$
$251$	5.82163e10			0.925791			0.462896	+	0.886413i	$0.346811\pi$
$252$		0			0
$253$	−1.33532e9			−0.0204900
$254$		0			0
$255$	5.37609e10	−	9.31167e10i	0.796225	−	1.37910i
$256$		0			0
$257$	7.77875e10	−	4.49106e10i	1.11227	−	0.642170i	0.172854	−	0.984947i	$-0.444701\pi$
$257$	7.77875e10	−	4.49106e10i	1.11227	−	0.642170i	0.939417	+	0.342777i	$0.111368\pi$
$258$		0			0
$259$	2.78960e10	+	1.01384e11i	0.385206	+	1.39997i
$260$		0			0
$261$	1.48140e10	+	2.56586e10i	0.197601	+	0.342256i
$262$		0			0
$263$	−1.01612e11	−	5.86656e10i	−1.30961	−	0.756106i	−0.327583	−	0.944822i	$-0.606234\pi$
$263$	−1.01612e11	−	5.86656e10i	−1.30961	−	0.756106i	−0.982032	+	0.188716i	$0.939567\pi$
$264$		0			0
$265$			3.22387e10i			0.401579i
$266$		0			0
$267$		−	7.58530e10i		−	0.913423i
$268$		0			0
$269$	9.43971e9	+	5.45002e9i	0.109919	+	0.0634618i	0.553952	−	0.832549i	$-0.313119\pi$
$269$	9.43971e9	+	5.45002e9i	0.109919	+	0.0634618i	−0.444033	+	0.896011i	$0.646453\pi$
$270$		0			0
$271$	2.16086e9	+	3.74272e9i	0.0243369	+	0.0421527i	0.877937	−	0.478776i	$-0.158919\pi$
$271$	2.16086e9	+	3.74272e9i	0.0243369	+	0.0421527i	−0.853600	+	0.520928i	$0.825586\pi$
$272$		0			0
$273$	−3.75064e10	−	3.70064e10i	−0.408670	−	0.403223i
$274$		0			0
$275$	−6.40465e9	+	3.69773e9i	−0.0675302	+	0.0389886i
$276$		0			0
$277$	−1.36050e10	+	2.35645e10i	−0.138848	+	0.240492i	−0.927061	−	0.374911i	$-0.877673\pi$
$277$	−1.36050e10	+	2.35645e10i	−0.138848	+	0.240492i	0.788213	+	0.615403i	$0.211007\pi$
$278$		0			0
$279$	−7.11256e10			−0.702760
$280$		0			0
$281$	1.30723e11			1.25076			0.625379	−	0.780321i	$-0.284944\pi$
$281$	1.30723e11			1.25076			0.625379	+	0.780321i	$0.284944\pi$
$282$		0			0
$283$	−4.79663e10	+	8.30801e10i	−0.444526	+	0.769942i	−0.998019	−	0.0629119i	$-0.979961\pi$
$283$	−4.79663e10	+	8.30801e10i	−0.444526	+	0.769942i	0.553493	+	0.832854i	$0.313295\pi$
$284$		0			0
$285$	1.01161e11	−	5.84052e10i	0.908260	−	0.524384i
$286$		0			0
$287$	2.33414e10	−	8.95091e10i	0.203076	−	0.778751i
$288$		0			0
$289$	1.34948e11	+	2.33737e11i	1.13796	+	1.97101i
$290$		0			0
$291$	1.19875e11	+	6.92098e10i	0.979964	+	0.565782i
$292$		0			0
$293$			7.13085e10i			0.565245i	0.959231	+	0.282623i	$0.0912044\pi$
$293$			7.13085e10i			0.565245i	−0.959231	+	0.282623i	$0.908796\pi$
$294$		0			0
$295$		−	3.23457e11i		−	2.48667i
$296$		0			0
$297$	−9.94108e9	−	5.73948e9i	−0.0741361	−	0.0428025i
$298$		0			0
$299$	1.51693e10	+	2.62741e10i	0.109761	+	0.190111i
$300$		0			0
$301$	−3.33057e10	+	1.27720e11i	−0.233867	+	0.896828i
$302$		0			0
$303$	9.55672e10	−	5.51758e10i	0.651354	−	0.376059i
$304$		0			0
$305$	3.27393e10	−	5.67061e10i	0.216631	−	0.375215i
$306$		0			0
$307$	2.35875e11			1.51551			0.757756	−	0.652538i	$-0.226296\pi$
$307$	2.35875e11			1.51551			0.757756	+	0.652538i	$0.226296\pi$
$308$		0			0
$309$	−9.72943e10			−0.607119
$310$		0			0
$311$	−1.06230e9	+	1.83996e9i	−0.00643909	+	0.0111528i	−0.869227	−	0.494413i	$-0.835383\pi$
$311$	−1.06230e9	+	1.83996e9i	−0.00643909	+	0.0111528i	0.862788	+	0.505566i	$0.168716\pi$
$312$		0			0
$313$	−1.22659e11	+	7.08172e10i	−0.722353	+	0.417051i	−0.815618	−	0.578590i	$-0.803603\pi$
$313$	−1.22659e11	+	7.08172e10i	−0.722353	+	0.417051i	0.0932648	+	0.995641i	$0.470270\pi$
$314$		0			0
$315$	−1.03067e11	−	1.01693e11i	−0.589825	−	0.581963i
$316$		0			0
$317$	−1.53831e11	−	2.66443e11i	−0.855612	−	1.48196i	−0.876076	−	0.482173i	$-0.839848\pi$
$317$	−1.53831e11	−	2.66443e11i	−0.855612	−	1.48196i	0.0204637	−	0.999791i	$-0.493486\pi$
$318$		0			0
$319$	−8.94849e9	−	5.16641e9i	−0.0483829	−	0.0279339i
$320$		0			0
$321$		−	1.05371e11i		−	0.553924i
$322$		0			0
$323$			4.22044e11i			2.15748i
$324$		0			0
$325$	1.45515e11	+	8.40130e10i	0.723489	+	0.417706i
$326$		0			0
$327$	−7.40528e10	−	1.28263e11i	−0.358159	−	0.620350i
$328$		0			0
$329$	−3.06368e9	−	1.11345e10i	−0.0144166	−	0.0523947i
$330$		0			0
$331$	−1.34210e11	+	7.74864e10i	−0.614555	+	0.354813i	−0.774746	−	0.632273i	$-0.782122\pi$
$331$	−1.34210e11	+	7.74864e10i	−0.614555	+	0.354813i	0.160191	+	0.987086i	$0.448789\pi$
$332$		0			0
$333$	9.73347e10	−	1.68589e11i	0.433779	−	0.751327i
$334$		0			0
$335$	−3.73168e11			−1.61884
$336$		0			0
$337$	−3.23927e11			−1.36808			−0.684041	−	0.729444i	$-0.739779\pi$
$337$	−3.23927e11			−1.36808			−0.684041	+	0.729444i	$0.739779\pi$
$338$		0			0
$339$	6.66495e10	−	1.15440e11i	0.274093	−	0.474743i
$340$		0			0
$341$	2.14819e10	−	1.24026e10i	0.0860357	−	0.0496727i
$342$		0			0
$343$	−1.84874e11	−	1.77578e11i	−0.721195	−	0.692732i
$344$		0			0
$345$	−2.80818e10	−	4.86392e10i	−0.106718	−	0.184842i
$346$		0			0
$347$	2.15404e11	+	1.24364e11i	0.797575	+	0.460480i	0.842622	−	0.538505i	$-0.181011\pi$
$347$	2.15404e11	+	1.24364e11i	0.797575	+	0.460480i	−0.0450477	+	0.998985i	$0.514344\pi$
$348$		0			0
$349$		−	1.33550e11i		−	0.481871i	−0.970541	−	0.240935i	$-0.922546\pi$
$349$		−	1.33550e11i		−	0.481871i	0.970541	−	0.240935i	$-0.0774542\pi$
$350$		0			0
$351$			2.60804e11i			0.917133i
$352$		0			0
$353$	−4.44948e11	−	2.56891e11i	−1.52519	−	0.880566i	−0.999554	−	0.0298541i	$-0.990496\pi$
$353$	−4.44948e11	−	2.56891e11i	−1.52519	−	0.880566i	−0.525632	−	0.850712i	$-0.676171\pi$
$354$		0			0
$355$	2.94795e11	+	5.10601e11i	0.985129	+	1.70629i
$356$		0			0
$357$	−3.39792e11	+	9.34949e10i	−1.10715	+	0.304636i
$358$		0			0
$359$	−4.04926e11	+	2.33784e11i	−1.28662	+	0.742832i	−0.978050	−	0.208369i	$-0.933184\pi$
$359$	−4.04926e11	+	2.33784e11i	−1.28662	+	0.742832i	−0.308572	+	0.951201i	$0.599851\pi$
$360$		0			0
$361$	−6.79078e10	+	1.17620e11i	−0.210444	+	0.364501i
$362$		0			0
$363$	−2.08381e11			−0.629909
$364$		0			0
$365$	7.21671e10			0.212824
$366$		0			0
$367$	1.44777e11	−	2.50761e11i	0.416583	−	0.721542i	−0.579010	−	0.815320i	$-0.696561\pi$
$367$	1.44777e11	−	2.50761e11i	0.416583	−	0.721542i	0.995593	+	0.0937777i	$0.0298943\pi$
$368$		0			0
$369$	−1.48308e11	+	8.56259e10i	−0.416435	+	0.240429i
$370$		0			0
$371$	7.42153e10	−	7.52179e10i	0.203381	−	0.206129i
$372$		0			0
$373$	−4.54927e10	−	7.87957e10i	−0.121689	−	0.210772i	0.798745	−	0.601670i	$-0.205498\pi$
$373$	−4.54927e10	−	7.87957e10i	−0.121689	−	0.210772i	−0.920434	+	0.390898i	$0.872164\pi$
$374$		0			0
$375$	2.24101e10	+	1.29385e10i	0.0585199	+	0.0337865i
$376$		0			0
$377$			2.34763e11i			0.598542i
$378$		0			0
$379$			6.69685e11i			1.66722i	0.552351	+	0.833612i	$0.313731\pi$
$379$			6.69685e11i			1.66722i	−0.552351	+	0.833612i	$0.686269\pi$
$380$		0			0
$381$	3.22144e10	+	1.85990e10i	0.0783226	+	0.0452196i
$382$		0			0
$383$	1.85802e11	+	3.21818e11i	0.441220	+	0.764216i	0.997780	−	0.0665913i	$-0.0212124\pi$
$383$	1.85802e11	+	3.21818e11i	0.441220	+	0.764216i	−0.556560	+	0.830807i	$0.687879\pi$
$384$		0			0
$385$	4.88621e10	+	1.27418e10i	0.113344	+	0.0295569i
$386$		0			0
$387$	2.11620e11	−	1.22179e11i	0.479577	−	0.276884i
$388$		0			0
$389$	2.95600e11	−	5.11995e11i	0.654533	−	1.13369i	−0.327477	−	0.944859i	$-0.606198\pi$
$389$	2.95600e11	−	5.11995e11i	0.654533	−	1.13369i	0.982011	−	0.188826i	$-0.0604682\pi$
$390$		0			0
$391$	2.02923e11			0.439073
$392$		0			0
$393$	4.70391e11			0.994702
$394$		0			0
$395$	3.51085e11	−	6.08097e11i	0.725646	−	1.25686i
$396$		0			0
$397$	−1.25499e11	+	7.24571e10i	−0.253562	+	0.146394i	−0.621394	−	0.783498i	$-0.713433\pi$
$397$	−1.25499e11	+	7.24571e10i	−0.253562	+	0.146394i	0.367832	+	0.929892i	$0.380100\pi$
$398$		0			0
$399$	−3.70476e11	−	9.66093e10i	−0.731782	−	0.190828i
$400$		0			0
$401$	−1.95373e11	−	3.38396e11i	−0.377324	−	0.653544i	0.613348	−	0.789813i	$-0.289822\pi$
$401$	−1.95373e11	−	3.38396e11i	−0.377324	−	0.653544i	−0.990672	+	0.136269i	$0.956489\pi$
$402$		0			0
$403$	−4.88073e11	−	2.81789e11i	−0.921748	−	0.532171i
$404$		0			0
$405$		−	3.41730e10i		−	0.0631154i
$406$		0			0
$407$			6.78913e10i			0.122642i
$408$		0			0
$409$	−4.10530e11	−	2.37020e11i	−0.725421	−	0.418822i	0.0913238	−	0.995821i	$-0.470890\pi$
$409$	−4.10530e11	−	2.37020e11i	−0.725421	−	0.418822i	−0.816745	+	0.576999i	$0.804223\pi$
$410$		0			0
$411$	2.56654e11	+	4.44537e11i	0.443669	+	0.768458i
$412$		0			0
$413$	−7.44617e11	+	7.54676e11i	−1.25938	+	1.27640i
$414$		0			0
$415$	−1.00908e11	+	5.82595e10i	−0.166998	+	0.0964162i
$416$		0			0
$417$	−3.60639e11	+	6.24645e11i	−0.584064	+	1.01163i
$418$		0			0
$419$	−1.49336e11			−0.236702			−0.118351	−	0.992972i	$-0.537761\pi$
$419$	−1.49336e11			−0.236702			−0.118351	+	0.992972i	$0.537761\pi$
$420$		0			0
$421$	−8.57426e11			−1.33023			−0.665116	−	0.746740i	$-0.731618\pi$
$421$	−8.57426e11			−1.33023			−0.665116	+	0.746740i	$0.731618\pi$
$422$		0			0
$423$	−1.06898e10	+	1.85152e10i	−0.0162344	+	0.0281188i
$424$		0			0
$425$	9.73290e11	−	5.61929e11i	1.44708	−	0.835471i
$426$		0			0
$427$	−2.06926e11	+	5.69364e10i	−0.301225	+	0.0828829i
$428$		0			0
$429$	−1.70096e10	−	2.94615e10i	−0.0242458	−	0.0419950i
$430$		0			0
$431$	−4.15497e11	−	2.39887e11i	−0.579989	−	0.334857i	0.181140	−	0.983457i	$-0.442021\pi$
$431$	−4.15497e11	−	2.39887e11i	−0.579989	−	0.334857i	−0.761129	+	0.648600i	$0.775355\pi$
$432$		0			0
$433$		−	3.26515e11i		−	0.446383i	−0.974775	−	0.223192i	$-0.928352\pi$
$433$		−	3.26515e11i		−	0.446383i	0.974775	−	0.223192i	$-0.0716476\pi$
$434$		0			0
$435$		−	4.34599e11i		−	0.581952i
$436$		0			0
$437$	1.90918e11	+	1.10227e11i	0.250427	+	0.144584i
$438$		0			0
$439$	−2.06834e11	−	3.58247e11i	−0.265786	−	0.460355i	0.701983	−	0.712193i	$-0.252298\pi$
$439$	−2.06834e11	−	3.58247e11i	−0.265786	−	0.460355i	−0.967769	+	0.251839i	$0.918965\pi$
$440$		0			0
$441$	6.36779e9	+	4.74533e11i	0.00801706	+	0.597438i
$442$		0			0
$443$	−5.14995e11	+	2.97333e11i	−0.635311	+	0.366797i	−0.782806	−	0.622266i	$-0.786212\pi$
$443$	−5.14995e11	+	2.97333e11i	−0.635311	+	0.366797i	0.147495	+	0.989063i	$0.452879\pi$
$444$		0			0
$445$	8.25823e11	−	1.43037e12i	0.998313	−	1.72913i
$446$		0			0
$447$	9.67937e10			0.114674
$448$		0			0
$449$	−8.08300e11			−0.938564			−0.469282	−	0.883048i	$-0.655487\pi$
$449$	−8.08300e11			−0.938564			−0.469282	+	0.883048i	$0.655487\pi$
$450$		0			0
$451$	2.98622e10	−	5.17229e10i	0.0339882	−	0.0588692i
$452$		0			0
$453$	−1.01746e11	+	5.87432e10i	−0.113521	+	0.0655414i
$454$		0			0
$455$	−3.04366e11	−	1.10617e12i	−0.332924	−	1.20996i
$456$		0			0
$457$	4.86355e11	+	8.42392e11i	0.521592	+	0.903423i	0.999685	+	0.0251138i	$0.00799482\pi$
$457$	4.86355e11	+	8.42392e11i	0.521592	+	0.903423i	−0.478093	+	0.878309i	$0.658672\pi$
$458$		0			0
$459$	1.51071e12	+	8.72208e11i	1.58863	+	0.917197i
$460$		0			0
$461$		−	5.63074e11i		−	0.580646i	−0.956929	−	0.290323i	$-0.906237\pi$
$461$		−	5.63074e11i		−	0.580646i	0.956929	−	0.290323i	$-0.0937628\pi$
$462$		0			0
$463$		−	1.60048e12i		−	1.61859i	−0.587406	−	0.809293i	$-0.699851\pi$
$463$		−	1.60048e12i		−	1.61859i	0.587406	−	0.809293i	$-0.300149\pi$
$464$		0			0
$465$	9.03532e11	+	5.21654e11i	0.896200	+	0.517421i
$466$		0			0
$467$	7.87544e11	+	1.36407e12i	0.766212	+	1.32712i	0.939603	+	0.342266i	$0.111194\pi$
$467$	7.87544e11	+	1.36407e12i	0.766212	+	1.32712i	−0.173391	+	0.984853i	$0.555472\pi$
$468$		0			0
$469$	8.70659e11	+	8.59054e11i	0.830942	+	0.819866i
$470$		0			0
$471$	−9.07909e11	+	5.24181e11i	−0.850057	+	0.490781i
$472$		0			0
$473$	−4.26102e10	+	7.38030e10i	−0.0391416	+	0.0677952i
$474$		0			0
$475$	1.22094e12			1.10046
$476$		0			0
$477$	−1.95625e11			−0.173018
$478$		0			0
$479$	9.11211e11	−	1.57826e12i	0.790877	−	1.36984i	−0.134547	−	0.990907i	$-0.542958\pi$
$479$	9.11211e11	−	1.57826e12i	0.790877	−	1.36984i	0.925424	−	0.378933i	$-0.123709\pi$
$480$		0			0
$481$	1.33585e12	−	7.71251e11i	1.13790	−	0.656966i
$482$		0			0
$483$	−4.64508e10	+	1.78129e11i	−0.0388357	+	0.148926i
$484$		0			0
$485$	1.50700e12	+	2.61019e12i	1.23673	+	2.14207i
$486$		0			0
$487$	4.31571e11	+	2.49167e11i	0.347673	+	0.200729i	0.663660	−	0.748034i	$-0.269002\pi$
$487$	4.31571e11	+	2.49167e11i	0.347673	+	0.200729i	−0.315987	+	0.948764i	$0.602335\pi$
$488$		0			0
$489$			3.79961e11i			0.300503i
$490$		0			0
$491$		−	1.46306e12i		−	1.13605i	−0.823013	−	0.568023i	$-0.807709\pi$
$491$		−	1.46306e12i		−	1.13605i	0.823013	−	0.568023i	$-0.192291\pi$
$492$		0			0
$493$	1.35987e12	+	7.85120e11i	1.03678	+	0.598584i
$494$		0			0
$495$	−4.67423e10	−	8.09601e10i	−0.0349934	−	0.0606104i
$496$		0			0
$497$	4.87628e11	−	1.86995e12i	0.358496	−	1.37475i
$498$		0			0
$499$	−1.25906e12	+	7.26918e11i	−0.909063	+	0.524848i	−0.880129	−	0.474734i	$-0.842544\pi$
$499$	−1.25906e12	+	7.26918e11i	−0.909063	+	0.524848i	−0.0289331	+	0.999581i	$0.509211\pi$
$500$		0			0
$501$	−4.61519e10	+	7.99375e10i	−0.0327281	+	0.0566867i
$502$		0			0
$503$	1.47160e12			1.02502			0.512511	−	0.858681i	$-0.328716\pi$
$503$	1.47160e12			1.02502			0.512511	+	0.858681i	$0.328716\pi$
$504$		0			0
$505$	2.40283e12			1.64404
$506$		0			0
$507$	8.54857e10	−	1.48066e11i	0.0574590	−	0.0995218i
$508$		0			0
$509$	−6.23026e11	+	3.59704e11i	−0.411411	+	0.237529i	−0.691396	−	0.722476i	$-0.743004\pi$
$509$	−6.23026e11	+	3.59704e11i	−0.411411	+	0.237529i	0.279985	+	0.960005i	$0.409671\pi$
$510$		0			0
$511$	−1.68377e11	−	1.66133e11i	−0.109242	−	0.107786i
$512$		0			0
$513$	9.47555e11	+	1.64121e12i	0.604054	+	1.04625i
$514$		0			0
$515$	−1.83469e12	−	1.05926e12i	−1.14929	−	0.663543i
$516$		0			0
$517$		−	7.45616e9i		−	0.00458995i
$518$		0			0
$519$			3.39734e11i			0.205535i
$520$		0			0
$521$	−2.28654e12	−	1.32013e12i	−1.35959	−	0.784961i	−0.370024	−	0.929022i	$-0.620650\pi$
$521$	−2.28654e12	−	1.32013e12i	−1.35959	−	0.784961i	−0.989569	+	0.144061i	$0.953984\pi$
$522$		0			0
$523$	9.31384e11	+	1.61320e12i	0.544341	+	0.942827i	0.998648	+	0.0519813i	$0.0165536\pi$
$523$	9.31384e11	+	1.61320e12i	0.544341	+	0.942827i	−0.454307	+	0.890845i	$0.650113\pi$
$524$		0			0
$525$	2.70474e11	+	9.82996e11i	0.155385	+	0.564722i
$526$		0			0
$527$	−3.26453e12	+	1.88478e12i	−1.84362	+	1.06442i
$528$		0			0
$529$	−8.47578e11	+	1.46805e12i	−0.470575	+	0.815061i
$530$		0			0
$531$	1.96274e12			1.07137
$532$		0			0
$533$	−1.35695e12			−0.728268
$534$		0			0
$535$	1.14719e12	−	1.98700e12i	0.605403	−	1.04859i
$536$		0			0
$537$	1.06161e12	−	6.12922e11i	0.550911	−	0.318069i
$538$		0			0
$539$	−8.46704e10	−	1.42212e11i	−0.0432098	−	0.0725749i
$540$		0			0
$541$	−9.07717e11	−	1.57221e12i	−0.455578	−	0.789084i	0.543143	−	0.839640i	$-0.317234\pi$
$541$	−9.07717e11	−	1.57221e12i	−0.455578	−	0.789084i	−0.998721	+	0.0505558i	$0.983901\pi$
$542$		0			0
$543$	−1.13621e12	−	6.55989e11i	−0.560865	−	0.323815i
$544$		0			0
$545$		−	3.22489e12i		−	1.56578i
$546$		0			0
$547$		−	2.91430e12i		−	1.39185i	−0.718116	−	0.695923i	$-0.754995\pi$
$547$		−	2.91430e12i		−	1.39185i	0.718116	−	0.695923i	$-0.245005\pi$
$548$		0			0
$549$	3.44093e11	+	1.98662e11i	0.161659	+	0.0933340i
$550$		0			0
$551$	8.52944e11	+	1.47734e12i	0.394220	+	0.682808i
$552$		0			0
$553$	−2.21901e12	+	6.10567e11i	−1.00901	+	0.277632i
$554$		0			0
$555$	−2.47295e12	+	1.42776e12i	−1.10636	+	0.638757i
$556$		0			0
$557$	6.37219e11	−	1.10370e12i	0.280505	−	0.485849i	−0.691004	−	0.722851i	$-0.742831\pi$
$557$	6.37219e11	−	1.10370e12i	0.280505	−	0.485849i	0.971509	+	0.237002i	$0.0761648\pi$
$558$		0			0
$559$	1.93622e12			0.838690
$560$		0			0
$561$	−2.27541e11			−0.0969900
$562$		0			0
$563$	2.99481e11	−	5.18716e11i	0.125626	−	0.217591i	−0.796351	−	0.604835i	$-0.793239\pi$
$563$	2.99481e11	−	5.18716e11i	0.125626	−	0.217591i	0.921978	+	0.387243i	$0.126573\pi$
$564$		0			0
$565$	2.51363e12	−	1.45124e12i	1.03773	−	0.599132i
$566$		0			0
$567$	−7.86681e10	+	7.97309e10i	−0.0319650	+	0.0323968i
$568$		0			0
$569$	6.38022e11	+	1.10509e12i	0.255170	+	0.441968i	0.964942	−	0.262464i	$-0.0845351\pi$
$569$	6.38022e11	+	1.10509e12i	0.255170	+	0.441968i	−0.709771	+	0.704432i	$0.751202\pi$
$570$		0			0
$571$	1.51715e12	+	8.75926e11i	0.597263	+	0.344830i	0.767964	−	0.640493i	$-0.221270\pi$
$571$	1.51715e12	+	8.75926e11i	0.597263	+	0.344830i	−0.170701	+	0.985323i	$0.554603\pi$
$572$		0			0
$573$			9.85876e11i			0.382055i
$574$		0			0
$575$		−	5.87043e11i		−	0.223957i
$576$		0			0
$577$	−4.18956e12	−	2.41884e12i	−1.57354	−	0.908482i	−0.995731	−	0.0923074i	$-0.970576\pi$
$577$	−4.18956e12	−	2.41884e12i	−1.57354	−	0.908482i	−0.577806	−	0.816174i	$-0.696091\pi$
$578$		0			0
$579$	−9.29836e10	−	1.61052e11i	−0.0343837	−	0.0595544i
$580$		0			0
$581$	3.69551e11	+	9.63683e10i	0.134550	+	0.0350866i
$582$		0			0
$583$	5.90842e10	−	3.41123e10i	0.0211818	−	0.0122293i
$584$		0			0
$585$	−1.06199e12	+	1.83943e12i	−0.374904	+	0.649353i
$586$		0			0
$587$	2.28419e11			0.0794073			0.0397036	−	0.999211i	$-0.487359\pi$
$587$	2.28419e11			0.0794073			0.0397036	+	0.999211i	$0.487359\pi$
$588$		0			0
$589$	−4.09519e12			−1.40202
$590$		0			0
$591$	−4.78139e11	+	8.28162e11i	−0.161217	+	0.279236i
$592$		0			0
$593$	1.98325e11	−	1.14503e11i	0.0658616	−	0.0380252i	−0.466708	−	0.884412i	$-0.654560\pi$
$593$	1.98325e11	−	1.14503e11i	0.0658616	−	0.0380252i	0.532569	+	0.846386i	$0.321227\pi$
$594$		0			0
$595$	−7.42539e12	−	1.93633e12i	−2.42881	−	0.633362i
$596$		0			0
$597$	1.15759e12	+	2.00501e12i	0.372968	+	0.646000i
$598$		0			0
$599$	2.88988e12	+	1.66847e12i	0.917190	+	0.529540i	0.882738	−	0.469866i	$-0.155698\pi$
$599$	2.88988e12	+	1.66847e12i	0.917190	+	0.529540i	0.0344526	+	0.999406i	$0.489031\pi$
$600$		0			0
$601$			2.59183e12i			0.810348i	0.914240	+	0.405174i	$0.132789\pi$
$601$			2.59183e12i			0.810348i	−0.914240	+	0.405174i	$0.867211\pi$
$602$		0			0
$603$		−	2.26439e12i		−	0.697466i
$604$		0			0
$605$	−3.92946e12	−	2.26867e12i	−1.19243	−	0.688450i
$606$		0			0
$607$	3.20276e12	+	5.54734e12i	0.957580	+	1.65858i	0.728351	+	0.685204i	$0.240287\pi$
$607$	3.20276e12	+	5.54734e12i	0.957580	+	1.65858i	0.229228	+	0.973373i	$0.426380\pi$
$608$		0			0
$609$	−1.00047e12	+	1.01399e12i	−0.294732	+	0.298713i
$610$		0			0
$611$	−1.46709e11	+	8.47026e10i	−0.0425865	+	0.0245873i
$612$		0			0
$613$	−1.39443e12	+	2.41522e12i	−0.398863	+	0.690851i	−0.993586	−	0.113079i	$-0.963929\pi$
$613$	−1.39443e12	+	2.41522e12i	−0.398863	+	0.690851i	0.594723	+	0.803931i	$0.297262\pi$
$614$		0			0
$615$	2.51201e12			0.708083
$616$		0			0
$617$	2.93747e12			0.816001			0.408000	−	0.912982i	$-0.366226\pi$
$617$	2.93747e12			0.816001			0.408000	+	0.912982i	$0.366226\pi$
$618$		0			0
$619$	1.85916e12	−	3.22016e12i	0.508990	−	0.881596i	−0.490956	−	0.871184i	$-0.663352\pi$
$619$	1.85916e12	−	3.22016e12i	0.508990	−	0.881596i	0.999946	−	0.0104116i	$-0.00331419\pi$
$620$		0			0
$621$	7.89113e11	−	4.55594e11i	0.212925	−	0.122932i
$622$		0			0
$623$	−5.21956e12	+	1.43618e12i	−1.38815	+	0.381955i
$624$		0			0
$625$	2.04259e12	+	3.53786e12i	0.535452	+	0.927430i
$626$		0			0
$627$	−2.14079e11	−	1.23599e11i	−0.0553186	−	0.0319382i
$628$		0			0
$629$		−	1.03172e13i		−	2.62805i
$630$		0			0
$631$			5.54087e12i			1.39138i	0.718342	+	0.695690i	$0.244901\pi$
$631$			5.54087e12i			1.39138i	−0.718342	+	0.695690i	$0.755099\pi$
$632$		0			0
$633$	−3.27102e12	−	1.88853e12i	−0.809780	−	0.467526i
$634$		0			0
$635$	4.04979e11	+	7.01445e11i	0.0988442	+	0.171203i
$636$		0			0
$637$	−1.83633e12	+	3.28154e12i	−0.441900	+	0.789677i
$638$		0			0
$639$	−3.09833e12	+	1.78882e12i	−0.735146	+	0.424437i
$640$		0			0
$641$	1.65826e12	−	2.87219e12i	0.387964	−	0.671973i	−0.604212	−	0.796824i	$-0.706512\pi$
$641$	1.65826e12	−	2.87219e12i	0.387964	−	0.671973i	0.992176	+	0.124851i	$0.0398452\pi$
$642$		0			0
$643$	−6.25708e12			−1.44352			−0.721759	−	0.692144i	$-0.756666\pi$
$643$	−6.25708e12			−1.44352			−0.721759	+	0.692144i	$0.756666\pi$
$644$		0			0
$645$	−3.58437e12			−0.815445
$646$		0			0
$647$	−2.54826e12	+	4.41371e12i	−0.571708	+	0.990227i	0.424683	+	0.905342i	$0.360386\pi$
$647$	−2.54826e12	+	4.41371e12i	−0.571708	+	0.990227i	−0.996391	+	0.0848847i	$0.972948\pi$
$648$		0			0
$649$	−5.92803e11	+	3.42255e11i	−0.131162	+	0.0757266i
$650$		0			0
$651$	−9.07202e11	−	3.29708e12i	−0.197965	−	0.719474i
$652$		0			0
$653$	−3.01691e12	−	5.22544e12i	−0.649311	−	1.12464i	−0.983288	−	0.182058i	$-0.941724\pi$
$653$	−3.01691e12	−	5.22544e12i	−0.649311	−	1.12464i	0.333977	−	0.942581i	$-0.391609\pi$
$654$		0			0
$655$	8.87021e12	+	5.12122e12i	1.88299	+	1.08715i
$656$		0			0
$657$			4.37910e11i			0.0916940i
$658$		0			0
$659$			3.07274e12i			0.634660i	0.948315	+	0.317330i	$0.102786\pi$
$659$			3.07274e12i			0.634660i	−0.948315	+	0.317330i	$0.897214\pi$
$660$		0			0
$661$	−2.97147e12	−	1.71558e12i	−0.605431	−	0.349546i	0.165744	−	0.986169i	$-0.446997\pi$
$661$	−2.97147e12	−	1.71558e12i	−0.605431	−	0.349546i	−0.771175	+	0.636623i	$0.780331\pi$
$662$		0			0
$663$	2.58489e12	+	4.47715e12i	0.519554	+	0.899893i
$664$		0			0
$665$	−5.93429e12	−	5.85519e12i	−1.17672	−	1.16103i
$666$		0			0
$667$	7.10322e11	−	4.10105e11i	0.138960	−	0.0802284i
$668$		0			0
$669$	2.06095e12	−	3.56967e12i	0.397786	−	0.688986i
$670$		0			0
$671$	−1.38568e11			−0.0263883
$672$		0			0
$673$	1.60656e12			0.301876			0.150938	−	0.988543i	$-0.451771\pi$
$673$	1.60656e12			0.301876			0.150938	+	0.988543i	$0.451771\pi$
$674$		0			0
$675$	2.52324e12	−	4.37037e12i	0.467833	−	0.810310i
$676$		0			0
$677$	4.30495e12	−	2.48546e12i	0.787624	−	0.454735i	−0.0515011	−	0.998673i	$-0.516401\pi$
$677$	4.30495e12	−	2.48546e12i	0.787624	−	0.454735i	0.839126	+	0.543938i	$0.183067\pi$
$678$		0			0
$679$	2.49276e12	−	9.55917e12i	0.450055	−	1.72586i
$680$		0			0
$681$	1.74230e11	+	3.01774e11i	0.0310427	+	0.0537676i
$682$		0			0
$683$	−5.58468e12	−	3.22432e12i	−0.981985	−	0.566949i	−0.0791163	−	0.996865i	$-0.525210\pi$
$683$	−5.58468e12	−	3.22432e12i	−0.981985	−	0.566949i	−0.902869	+	0.429916i	$0.858543\pi$
$684$		0			0
$685$			1.11769e13i			1.93961i
$686$		0			0
$687$		−	4.54588e11i		−	0.0778597i
$688$		0			0
$689$	−1.34240e12	−	7.75037e11i	−0.226932	−	0.131019i
$690$		0			0
$691$	5.39277e12	+	9.34055e12i	0.899830	+	1.55855i	0.827711	+	0.561155i	$0.189643\pi$
$691$	5.39277e12	+	9.34055e12i	0.899830	+	1.55855i	0.0721192	+	0.997396i	$0.477024\pi$
$692$		0			0
$693$	−7.73175e10	+	2.96496e11i	−0.0127344	+	0.0488336i
$694$		0			0
$695$	−1.36012e13	+	7.85265e12i	−2.21129	+	1.27669i
$696$		0			0
$697$	−4.53805e12	+	7.86013e12i	−0.728319	+	1.26149i
$698$		0			0
$699$	2.81736e12			0.446371
$700$		0			0
$701$	7.58708e12			1.18671			0.593354	−	0.804942i	$-0.297804\pi$
$701$	7.58708e12			1.18671			0.593354	+	0.804942i	$0.297804\pi$
$702$		0			0
$703$	5.60423e12	−	9.70681e12i	0.865400	−	1.49892i
$704$		0			0
$705$	2.71591e11	−	1.56803e11i	0.0414061	−	0.0239058i
$706$		0			0
$707$	−5.60616e12	−	5.53144e12i	−0.843876	−	0.832628i
$708$		0			0
$709$	3.76590e12	+	6.52272e12i	0.559706	+	0.969440i	0.997521	+	0.0703743i	$0.0224194\pi$
$709$	3.76590e12	+	6.52272e12i	0.559706	+	0.969440i	−0.437814	+	0.899065i	$0.644247\pi$
$710$		0			0
$711$	3.68994e12	+	2.13039e12i	0.541509	+	0.312640i
$712$		0			0
$713$			1.96901e12i			0.285329i
$714$		0			0
$715$		−	7.40744e11i		−	0.105996i
$716$		0			0
$717$	−1.09018e12	−	6.29415e11i	−0.154050	−	0.0889407i
$718$		0			0
$719$	−1.33565e12	−	2.31341e12i	−0.186385	−	0.322829i	0.757657	−	0.652653i	$-0.226344\pi$
$719$	−1.33565e12	−	2.31341e12i	−0.186385	−	0.322829i	−0.944043	+	0.329824i	$0.893011\pi$
$720$		0			0
$721$	1.84214e12	+	6.69496e12i	0.253871	+	0.922655i
$722$		0			0
$723$	−3.58052e11	+	2.06721e11i	−0.0487331	+	0.0281361i
$724$		0			0
$725$	2.27130e12	−	3.93400e12i	0.305318	−	0.528827i
$726$		0			0
$727$	−4.80151e11			−0.0637490			−0.0318745	−	0.999492i	$-0.510148\pi$
$727$	−4.80151e11			−0.0637490			−0.0318745	+	0.999492i	$0.510148\pi$
$728$		0			0
$729$	5.11065e12			0.670196
$730$		0			0
$731$	6.47531e12	−	1.12156e13i	0.838749	−	1.45276i
$732$		0			0
$733$	7.07802e12	−	4.08650e12i	0.905616	−	0.522858i	0.0265980	−	0.999646i	$-0.491533\pi$
$733$	7.07802e12	−	4.08650e12i	0.905616	−	0.522858i	0.879018	+	0.476789i	$0.158199\pi$
$734$		0			0
$735$	3.39946e12	−	6.07485e12i	0.429652	−	0.767790i
$736$		0			0
$737$	3.94855e11	+	6.83909e11i	0.0492985	+	0.0853876i
$738$		0			0
$739$	−6.66043e12	−	3.84540e12i	−0.821490	−	0.474288i	0.0294397	−	0.999567i	$-0.490628\pi$
$739$	−6.66043e12	−	3.84540e12i	−0.821490	−	0.474288i	−0.850930	+	0.525279i	$0.823961\pi$
$740$		0			0
$741$			5.61637e12i			0.684343i
$742$		0			0
$743$		−	6.90697e12i		−	0.831453i	−0.909490	−	0.415727i	$-0.863527\pi$
$743$		−	6.90697e12i		−	0.831453i	0.909490	−	0.415727i	$-0.136473\pi$
$744$		0			0
$745$	1.82525e12	+	1.05381e12i	0.217079	+	0.125331i
$746$		0			0
$747$	−3.53519e11	−	6.12313e11i	−0.0415403	−	0.0719500i
$748$		0			0
$749$	−7.25076e12	+	1.99507e12i	−0.841812	+	0.231627i
$750$		0			0
$751$	1.07624e13	−	6.21366e12i	1.23461	−	0.712801i	0.266620	−	0.963802i	$-0.414093\pi$
$751$	1.07624e13	−	6.21366e12i	1.23461	−	0.712801i	0.967987	+	0.251001i	$0.0807598\pi$
$752$		0			0
$753$	2.59088e12	−	4.48754e12i	0.293677	−	0.508664i
$754$		0			0
$755$	−2.55818e12			−0.286530
$756$		0			0
$757$	7.06812e12			0.782299			0.391149	−	0.920327i	$-0.372078\pi$
$757$	7.06812e12			0.782299			0.391149	+	0.920327i	$0.372078\pi$
$758$		0			0
$759$	−5.94276e10	+	1.02932e11i	−0.00649981	+	0.0112580i
$760$		0			0
$761$	1.01979e13	−	5.88773e12i	1.10224	−	0.636381i	0.165435	−	0.986221i	$-0.447097\pi$
$761$	1.01979e13	−	5.88773e12i	1.10224	−	0.636381i	0.936810	+	0.349840i	$0.113764\pi$
$762$		0			0
$763$	−7.42388e12	+	7.52417e12i	−0.792995	+	0.803708i
$764$		0			0
$765$	7.10325e12	+	1.23032e13i	0.749861	+	1.29880i
$766$		0			0
$767$	1.34686e13	+	7.77609e12i	1.40522	+	0.811301i
$768$		0			0
$769$		−	6.85806e12i		−	0.707185i	−0.935400	−	0.353592i	$-0.884960\pi$
$769$		−	6.85806e12i		−	0.707185i	0.935400	−	0.353592i	$-0.115040\pi$
$770$		0			0
$771$		−	7.99488e12i		−	0.814831i
$772$		0			0
$773$	1.74638e12	+	1.00827e12i	0.175926	+	0.101571i	0.585377	−	0.810761i	$-0.300946\pi$
$773$	1.74638e12	+	1.00827e12i	0.175926	+	0.101571i	−0.409451	+	0.912332i	$0.634280\pi$
$774$		0			0
$775$	5.45252e12	+	9.44405e12i	0.542925	+	0.940374i
$776$		0			0
$777$	9.05655e12	+	2.36169e12i	0.891391	+	0.232449i
$778$		0			0
$779$	−8.53914e12	+	4.93007e12i	−0.830798	+	0.479661i
$780$		0			0
$781$	6.23855e11	−	1.08055e12i	0.0600004	−	0.103924i
$782$		0			0
$783$	7.05086e12			0.670369
$784$		0			0
$785$	−2.28274e13			−2.14557
$786$		0			0
$787$	−5.94681e11	+	1.03002e12i	−0.0552584	+	0.0957103i	−0.892331	−	0.451381i	$-0.850932\pi$
$787$	−5.94681e11	+	1.03002e12i	−0.0552584	+	0.0957103i	0.837073	+	0.547091i	$0.184265\pi$
$788$		0			0
$789$	−9.04435e12	+	5.22176e12i	−0.830866	+	0.479701i
$790$		0			0
$791$	−9.20553e12	−	2.40054e12i	−0.836093	−	0.218029i
$792$		0			0
$793$	1.57414e12	+	2.72649e12i	0.141356	+	0.244836i
$794$		0			0
$795$	2.48509e12	+	1.43477e12i	0.220643	+	0.127388i
$796$		0			0
$797$		−	1.56560e13i		−	1.37442i	−0.726460	−	0.687209i	$-0.758836\pi$
$797$		−	1.56560e13i		−	1.37442i	0.726460	−	0.687209i	$-0.241164\pi$
$798$		0			0
$799$			1.13308e12i			0.0983561i
$800$		0			0
$801$	8.67948e12	+	5.01110e12i	0.744985	+	0.430117i
$802$		0			0
$803$	−7.63611e10	−	1.32261e11i	−0.00648115	−	0.0112257i
$804$		0			0
$805$	−2.81524e12	+	2.85327e12i	−0.236284	+	0.239476i
$806$		0			0
$807$	8.40217e11	−	4.85100e11i	0.0697366	−	0.0402624i
$808$		0			0
$809$	9.65511e12	−	1.67231e13i	0.792481	−	1.37262i	−0.131945	−	0.991257i	$-0.542122\pi$
$809$	9.65511e12	−	1.67231e13i	0.792481	−	1.37262i	0.924426	−	0.381361i	$-0.124544\pi$
$810$		0			0
$811$	1.51386e12			0.122883			0.0614414	−	0.998111i	$-0.480430\pi$
$811$	1.51386e12			0.122883			0.0614414	+	0.998111i	$0.480430\pi$
$812$		0			0
$813$	3.84671e11			0.0308804
$814$		0			0
$815$	−4.13669e12	+	7.16495e12i	−0.328430	+	0.568858i
$816$		0			0
$817$	1.21844e13	−	7.03468e12i	0.956767	−	0.552389i
$818$		0			0
$819$	6.71225e12	−	1.84690e12i	0.521304	−	0.143438i
$820$		0			0
$821$	−9.93346e11	−	1.72053e12i	−0.0763056	−	0.132165i	0.825348	−	0.564625i	$-0.190979\pi$
$821$	−9.93346e11	−	1.72053e12i	−0.0763056	−	0.132165i	−0.901653	+	0.432460i	$0.857646\pi$
$822$		0			0
$823$	−2.11949e13	−	1.22369e13i	−1.61039	−	0.929760i	−0.989279	−	0.146040i	$-0.953347\pi$
$823$	−2.11949e13	−	1.22369e13i	−1.61039	−	0.929760i	−0.621113	−	0.783721i	$-0.713319\pi$
$824$		0			0
$825$			6.58260e11i			0.0494715i
$826$		0			0
$827$		−	1.27673e13i		−	0.949125i	−0.880222	−	0.474563i	$-0.842606\pi$
$827$		−	1.27673e13i		−	0.949125i	0.880222	−	0.474563i	$-0.157394\pi$
$828$		0			0
$829$	1.85602e13	+	1.07157e13i	1.36486	+	0.788000i	0.990266	−	0.139190i	$-0.0444499\pi$
$829$	1.85602e13	+	1.07157e13i	1.36486	+	0.788000i	0.374591	+	0.927190i	$0.377783\pi$
$830$		0			0
$831$	1.21096e12	+	2.09745e12i	0.0880901	+	0.152576i
$832$		0			0
$833$	1.28670e13	+	2.16114e13i	0.925926	+	1.55518i
$834$		0			0
$835$	−1.74058e12	+	1.00493e12i	−0.123910	+	0.0715393i
$836$		0			0
$837$	−8.46322e12	+	1.46587e13i	−0.596034	+	1.03236i
$838$		0			0
$839$	9.18960e12			0.640277			0.320138	−	0.947371i	$-0.396271\pi$
$839$	9.18960e12			0.640277			0.320138	+	0.947371i	$0.396271\pi$
$840$		0			0
$841$	−8.16030e12			−0.562502
$842$		0			0
$843$	5.81774e12	−	1.00766e13i	0.396762	−	0.687213i
$844$		0			0
$845$	3.22402e12	−	1.86139e12i	0.217542	−	0.125598i
$846$		0			0
$847$	3.94542e12	+	1.43390e13i	0.263401	+	0.957289i
$848$		0			0
$849$	4.26942e12	+	7.39486e12i	0.282023	+	0.488478i
$850$		0			0
$851$	−4.66714e12	−	2.69457e12i	−0.305047	−	0.176119i
$852$		0			0
$853$		−	3.86566e12i		−	0.250007i	−0.992156	−	0.125004i	$-0.960106\pi$
$853$		−	3.86566e12i		−	0.250007i	0.992156	−	0.125004i	$-0.0398943\pi$
$854$		0			0
$855$			1.54338e13i			0.987698i
$856$		0			0
$857$	−7.45970e12	−	4.30686e12i	−0.472398	−	0.272739i	0.244845	−	0.969562i	$-0.421263\pi$
$857$	−7.45970e12	−	4.30686e12i	−0.472398	−	0.272739i	−0.717243	+	0.696823i	$0.754596\pi$
$858$		0			0
$859$	6.40133e12	+	1.10874e13i	0.401145	+	0.694803i	0.993864	−	0.110606i	$-0.0352792\pi$
$859$	6.40133e12	+	1.10874e13i	0.401145	+	0.694803i	−0.592720	+	0.805409i	$0.701946\pi$
$860$		0			0
$861$	−5.86092e12	−	5.78280e12i	−0.363456	−	0.358611i
$862$		0			0
$863$	−2.77648e13	+	1.60300e13i	−1.70391	+	0.983752i	−0.762187	+	0.647356i	$0.775875\pi$
$863$	−2.77648e13	+	1.60300e13i	−1.70391	+	0.983752i	−0.941721	+	0.336396i	$0.890792\pi$
$864$		0			0
$865$	−3.69874e12	+	6.40640e12i	−0.224637	+	0.389082i
$866$		0			0
$867$	2.40232e13			1.44392
$868$		0			0
$869$	−1.48595e12			−0.0883926
$870$		0			0
$871$	8.97118e12	−	1.55385e13i	0.528163	−	0.914804i
$872$		0			0
$873$	−1.58387e13	+	9.14446e12i	−0.922900	+	0.532837i
$874$		0			0
$875$	4.66010e11	−	1.78705e12i	0.0268757	−	0.103062i
$876$		0			0
$877$	−1.74935e12	−	3.02997e12i	−0.0998572	−	0.172958i	0.811768	−	0.583980i	$-0.198505\pi$
$877$	−1.74935e12	−	3.02997e12i	−0.0998572	−	0.172958i	−0.911625	+	0.411022i	$0.865172\pi$
$878$		0			0
$879$	5.49674e12	+	3.17354e12i	0.310567	+	0.179306i
$880$		0			0
$881$		−	1.63962e13i		−	0.916962i	−0.888704	−	0.458481i	$-0.848394\pi$
$881$		−	1.63962e13i		−	0.916962i	0.888704	−	0.458481i	$-0.151606\pi$
$882$		0			0
$883$		−	1.88925e13i		−	1.04584i	−0.852381	−	0.522921i	$-0.824842\pi$
$883$		−	1.88925e13i		−	1.04584i	0.852381	−	0.522921i	$-0.175158\pi$
$884$		0			0
$885$	−2.49333e13	−	1.43953e13i	−1.36627	−	0.788815i
$886$		0			0
$887$	−1.06658e13	−	1.84737e13i	−0.578545	−	1.00207i	−0.995647	−	0.0932096i	$-0.970287\pi$
$887$	−1.06658e13	−	1.84737e13i	−0.578545	−	1.00207i	0.417101	−	0.908860i	$-0.363046\pi$
$888$		0			0
$889$	6.69886e11	−	2.56886e12i	0.0359702	−	0.137938i
$890$		0			0
$891$	−6.26291e10	+	3.61590e10i	−0.00332910	+	0.00192206i
$892$		0			0
$893$	−6.15483e11	+	1.06605e12i	−0.0323881	+	0.0560977i
$894$		0			0
$895$	2.66919e13			1.39051
$896$		0			0
$897$	2.70041e12			0.139272
$898$		0			0
$899$	−7.61819e12	+	1.31951e13i	−0.388985	+	0.673742i
$900$		0			0
$901$	−8.97880e12	+	5.18391e12i	−0.453896	+	0.262057i
$902$		0			0
$903$	8.36290e12	+	8.25143e12i	0.418564	+	0.412985i
$904$		0			0
$905$	−1.42837e13	−	2.47401e13i	−0.707819	−	1.22598i
$906$		0			0
$907$	−1.76767e13	−	1.02057e13i	−0.867300	−	0.500736i	−0.000849825	−	1.00000i	$-0.500271\pi$
$907$	−1.76767e13	−	1.02057e13i	−0.867300	−	0.500736i	−0.866450	+	0.499264i	$0.833604\pi$
$908$		0			0
$909$			1.45804e13i			0.708323i
$910$		0			0
$911$			1.91454e13i			0.920939i	0.887675	+	0.460470i	$0.152319\pi$
$911$			1.91454e13i			0.920939i	−0.887675	+	0.460470i	$0.847681\pi$
$912$		0			0
$913$	2.13545e11	+	1.23290e11i	0.0101712	+	0.00587234i
$914$		0			0
$915$	−2.91409e12	−	5.04734e12i	−0.137438	−	0.238050i
$916$		0			0
$917$	−8.90624e12	−	3.23683e13i	−0.415942	−	1.51167i
$918$		0			0
$919$	−3.35564e13	+	1.93738e13i	−1.55187	+	0.895973i	−0.553881	+	0.832596i	$0.686854\pi$
$919$	−3.35564e13	+	1.93738e13i	−1.55187	+	0.895973i	−0.997990	+	0.0633775i	$0.979813\pi$
$920$		0			0
$921$	1.04975e13	−	1.81822e13i	0.480747	−	0.832679i
$922$		0			0
$923$	−2.83482e13			−1.28563
$924$		0			0
$925$	−2.98469e13			−1.34048
$926$		0			0
$927$	6.42758e12	−	1.11329e13i	0.285883	−	0.495164i
$928$		0			0
$929$	−2.77886e13	+	1.60437e13i	−1.22404	+	0.706700i	−0.965777	−	0.259374i	$-0.916484\pi$
$929$	−2.77886e13	+	1.60437e13i	−1.22404	+	0.706700i	−0.258264	+	0.966074i	$0.583150\pi$
$930$		0			0
$931$	3.66638e11	+	2.73221e13i	0.0159942	+	1.19190i
$932$		0			0
$933$	9.45539e10	+	1.63772e11i	0.00408519	+	0.00707576i
$934$		0			0
$935$	−4.29076e12	−	2.47727e12i	−0.183604	−	0.106004i
$936$		0			0
$937$		−	1.55947e13i		−	0.660921i	−0.943820	−	0.330460i	$-0.892796\pi$
$937$		−	1.55947e13i		−	0.660921i	0.943820	−	0.330460i	$-0.107204\pi$
$938$		0			0
$939$			1.26067e13i			0.529184i
$940$		0			0
$941$	−6.37940e12	−	3.68315e12i	−0.265233	−	0.153132i	0.361487	−	0.932377i	$-0.382269\pi$
$941$	−6.37940e12	−	3.68315e12i	−0.265233	−	0.153132i	−0.626719	+	0.779245i	$0.715603\pi$
$942$		0			0
$943$	2.37043e12	+	4.10571e12i	0.0976169	+	0.169077i
$944$		0			0
$945$	−3.32226e13	+	9.14130e12i	−1.35516	+	0.372876i
$946$		0			0
$947$	−5.37263e12	+	3.10189e12i	−0.217076	+	0.125329i	−0.604596	−	0.796532i	$-0.706665\pi$
$947$	−5.37263e12	+	3.10189e12i	−0.217076	+	0.125329i	0.387519	+	0.921862i	$0.373332\pi$
$948$		0			0
$949$	−1.73494e12	+	3.00500e12i	−0.0694361	+	0.120267i
$950$		0			0
$951$	−2.73846e13			−1.08566
$952$		0			0
$953$	−3.03409e13			−1.19154			−0.595772	−	0.803153i	$-0.703154\pi$
$953$	−3.03409e13			−1.19154			−0.595772	+	0.803153i	$0.703154\pi$
$954$		0			0
$955$	−1.07334e13	+	1.85907e13i	−0.417562	+	0.723239i
$956$		0			0
$957$	−7.96494e11	+	4.59856e11i	−0.0306958	+	0.0177222i
$958$		0			0
$959$	2.57299e13	−	2.60774e13i	0.982322	−	0.995592i
$960$		0			0
$961$	−5.06858e12	−	8.77903e12i	−0.191704	−	0.332041i
$962$		0			0
$963$	1.20571e13	+	6.96118e12i	0.451778	+	0.260834i
$964$		0			0
$965$		−	4.04930e12i		−	0.150317i
$966$		0			0
$967$		−	3.32759e13i		−	1.22380i	−0.790935	−	0.611901i	$-0.790405\pi$
$967$		−	3.32759e13i		−	1.22380i	0.790935	−	0.611901i	$-0.209595\pi$
$968$		0			0
$969$	3.25328e13	+	1.87828e13i	1.18540	+	0.684391i
$970$		0			0
$971$	1.47854e13	+	2.56090e13i	0.533759	+	0.924498i	0.999222	+	0.0394309i	$0.0125545\pi$
$971$	1.47854e13	+	2.56090e13i	0.533759	+	0.924498i	−0.465463	+	0.885067i	$0.654112\pi$
$972$		0			0
$973$	4.98109e13	+	1.29893e13i	1.78163	+	0.464597i
$974$		0			0
$975$	1.29521e13	−	7.47789e12i	0.459007	−	0.265008i
$976$		0			0
$977$	1.45727e13	−	2.52406e13i	0.511698	−	0.886288i	−0.488210	−	0.872726i	$-0.662350\pi$
$977$	1.45727e13	−	2.52406e13i	0.511698	−	0.886288i	0.999908	−	0.0135612i	$-0.00431681\pi$
$978$		0			0
$979$	−3.49526e12			−0.121607
$980$		0			0
$981$	1.95687e13			0.674607
$982$		0			0
$983$	−1.94903e13	+	3.37582e13i	−0.665776	+	1.15316i	0.313299	+	0.949655i	$0.398566\pi$
$983$	−1.94903e13	+	3.37582e13i	−0.665776	+	1.15316i	−0.979074	+	0.203503i	$0.934767\pi$
$984$		0			0
$985$	−1.80326e13	+	1.04111e13i	−0.610374	+	0.352400i
$986$		0			0
$987$	−9.94634e11	−	2.59372e11i	−0.0333608	−	0.00869953i
$988$		0			0
$989$	−3.38235e12	−	5.85841e12i	−0.112418	−	0.194714i
$990$		0			0
$991$	−9.11133e11	−	5.26043e11i	−0.0300089	−	0.0173256i	0.484920	−	0.874558i	$-0.338849\pi$
$991$	−9.11133e11	−	5.26043e11i	−0.0300089	−	0.0173256i	−0.514929	+	0.857233i	$0.672182\pi$
$992$		0			0
$993$			1.37940e13i			0.450212i
$994$		0			0
$995$			5.04115e13i			1.63052i
$996$		0			0
$997$	−1.72411e13	−	9.95415e12i	−0.552633	−	0.319063i	0.197551	−	0.980293i	$-0.436701\pi$
$997$	−1.72411e13	−	9.95415e12i	−0.552633	−	0.319063i	−0.750183	+	0.661230i	$0.770035\pi$
$998$		0			0
$999$	−2.31637e13	−	4.01207e13i	−0.735805	−	1.27445i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	112.10.p.a.31.9	✓	24
4.3	odd	2		112.10.p.c.31.4	yes	24
7.5	odd	6		112.10.p.c.47.4	yes	24
28.19	even	6	inner	112.10.p.a.47.9	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.10.p.a.31.9	✓	24	1.1	even	1	trivial
112.10.p.a.47.9	yes	24	28.19	even	6	inner
112.10.p.c.31.4	yes	24	4.3	odd	2
112.10.p.c.47.4	yes	24	7.5	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:	\( N \)	\(=\)	\( 112 = 2^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	112.p (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(44.5044 - 77.0839i) q^{3} +(1678.45 - 969.052i) q^{5} +(-6146.89 - 1602.93i) q^{7} +(5880.22 + 10184.8i) q^{9} +(-3551.98 - 2050.74i) q^{11} +93186.2i q^{13} -172508. i q^{15} +(539781. + 311643. i) q^{17} +(338564. + 586411. i) q^{19} +(-397124. + 402488. i) q^{21} +(281953. - 162785. i) q^{23} +(901560. - 1.56155e6i) q^{25} +2.79874e6 q^{27} +2.51930e6 q^{29} +(-3.02394e6 + 5.23761e6i) q^{31} +(-316158. + 182534. i) q^{33} +(-1.18705e7 + 3.26622e6i) q^{35} +(-8.27646e6 - 1.43352e7i) q^{37} +(7.18315e6 + 4.14720e6i) q^{39} +1.45617e7i q^{41} -2.07780e7i q^{43} +(1.97393e7 + 1.13965e7i) q^{45} +(908961. + 1.57437e6i) q^{47} +(3.52148e7 + 1.97061e7i) q^{49} +(4.80453e7 - 2.77389e7i) q^{51} +(-8.31708e6 + 1.44056e7i) q^{53} -7.94908e6 q^{55} +6.02704e7 q^{57} +(8.34468e7 - 1.44534e8i) q^{59} +(2.92586e7 - 1.68924e7i) q^{61} +(-1.98194e7 - 7.20306e7i) q^{63} +(9.03022e7 + 1.56408e8i) q^{65} +(-1.66747e8 - 9.62715e7i) q^{67} -2.89787e7i q^{69} +3.04210e8i q^{71} +(3.22473e7 + 1.86180e7i) q^{73} +(-8.02468e7 - 1.38992e8i) q^{75} +(1.85464e7 + 1.82992e7i) q^{77} +(3.13759e8 - 1.81149e8i) q^{79} +(8.81609e6 - 1.52699e7i) q^{81} -6.01201e7 q^{83} +1.20799e9 q^{85} +(1.12120e8 - 1.94197e8i) q^{87} +(7.38024e8 - 4.26098e8i) q^{89} +(1.49371e8 - 5.72805e8i) q^{91} +(2.69157e8 + 4.66194e8i) q^{93} +(1.13652e9 + 6.56173e8i) q^{95} +1.55512e9i q^{97} -4.82351e7i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 162 q^{3} - 852 q^{5} - 6744 q^{7} - 77884 q^{9} - 57534 q^{11} + 789336 q^{17} + 469098 q^{19} - 2104376 q^{21} + 1553682 q^{23} + 3602544 q^{25} + 6389244 q^{27} - 2462040 q^{29} - 10306686 q^{31}+ \cdots + 1433917218 q^{95}+O(q^{100}) \) 24 * q - 162 * q^3 - 852 * q^5 - 6744 * q^7 - 77884 * q^9 - 57534 * q^11 + 789336 * q^17 + 469098 * q^19 - 2104376 * q^21 + 1553682 * q^23 + 3602544 * q^25 + 6389244 * q^27 - 2462040 * q^29 - 10306686 * q^31 + 8789748 * q^33 + 21149802 * q^35 + 10422720 * q^37 + 54458664 * q^39 + 2549604 * q^45 + 12751230 * q^47 - 16137504 * q^49 - 107322414 * q^51 + 62342520 * q^53 - 422625348 * q^55 - 317200696 * q^57 - 101234394 * q^59 + 48881304 * q^61 + 726717996 * q^63 + 272864028 * q^65 - 48335310 * q^67 + 672073812 * q^73 + 950672484 * q^75 + 10755684 * q^77 + 144370122 * q^79 - 305389432 * q^81 - 568641984 * q^83 - 734871504 * q^85 + 33665484 * q^87 + 1630891500 * q^89 + 1012044192 * q^91 - 457741376 * q^93 + 1433917218 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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