LMFDB - Embedded newform 112.10.i.d.65.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [112,10,Mod(65,112)] 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([0, 0, 2])) N = Newforms(chi, 10, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("112.65"); S:= CuspForms(chi, 10); N := Newforms(S);

Level:	$ N $	$=$	$ 112 = 2^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	112.i (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,0] 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:	$12$
Relative dimension:	$6$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - x^{11} + 7779 x^{10} + 365650 x^{9} + 45150527 x^{8} + 2129694927 x^{7} + 167292926543 x^{6} + \cdots + 68\!\cdots\!56 $ x^12 - x^11 + 7779x^10 + 365650x^9 + 45150527x^8 + 2129694927x^7 + 167292926543x^6 + 8284983835540x^5 + 421203033146961x^4 + 13701568631176769x^3 + 374859098811070993x^2 + 5870183155171733832x + 68823015636381281856
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{28}\cdot 3^{4}\cdot 7^{7} $
Twist minimal:	no (minimal twist has level 28)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			65.5
Root			$34.9935 + 60.6105i$ of defining polynomial
Character	$\chi$	$=$	112.65
Dual form			112.10.i.d.81.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(82.4523 + 142.812i) q^{3} +(1030.62 - 1785.08i) q^{5} +(-6318.25 - 658.292i) q^{7} +(-3755.26 + 6504.30i) q^{9} +(19582.0 + 33917.0i) q^{11} -14379.0 q^{13} +339908. q^{15} +(-117601. - 203691. i) q^{17} +(271910. - 470962. i) q^{19} +(-426942. - 956597. i) q^{21} +(-576096. + 997828. i) q^{23} +(-1.14779e6 - 1.98803e6i) q^{25} +2.00730e6 q^{27} -6.02178e6 q^{29} +(-4.27494e6 - 7.40442e6i) q^{31} +(-3.22916e6 + 5.59307e6i) q^{33} +(-7.68681e6 + 1.06002e7i) q^{35} +(2.47645e6 - 4.28933e6i) q^{37} +(-1.18558e6 - 2.05349e6i) q^{39} +1.48341e7 q^{41} -3.80424e7 q^{43} +(7.74048e6 + 1.34069e7i) q^{45} +(2.38065e7 - 4.12341e7i) q^{47} +(3.94869e7 + 8.31850e6i) q^{49} +(1.93930e7 - 3.35896e7i) q^{51} +(-4.94404e7 - 8.56333e7i) q^{53} +8.07263e7 q^{55} +8.96785e7 q^{57} +(2.07780e7 + 3.59886e7i) q^{59} +(1.71098e7 - 2.96350e7i) q^{61} +(2.80084e7 - 3.86237e7i) q^{63} +(-1.48193e7 + 2.56677e7i) q^{65} +(7.66238e7 + 1.32716e8i) q^{67} -1.90002e8 q^{69} -1.59112e8 q^{71} +(-2.03375e8 - 3.52256e8i) q^{73} +(1.89275e8 - 3.27835e8i) q^{75} +(-1.01397e8 - 2.27187e8i) q^{77} +(4.27911e7 - 7.41163e7i) q^{79} +(2.39421e8 + 4.14689e8i) q^{81} -5.27310e8 q^{83} -4.84808e8 q^{85} +(-4.96509e8 - 8.59980e8i) q^{87} +(2.67165e8 - 4.62744e8i) q^{89} +(9.08500e7 + 9.46557e6i) q^{91} +(7.04958e8 - 1.22102e9i) q^{93} +(-5.60472e8 - 9.70765e8i) q^{95} +1.26391e9 q^{97} -2.94142e8 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 966 q^{5} - 7696 q^{7} - 21620 q^{9} + 47640 q^{11} + 103432 q^{13} + 400280 q^{15} - 402234 q^{17} + 519960 q^{19} - 1025858 q^{21} + 683124 q^{23} - 1134656 q^{25} + 7641648 q^{27} + 4252680 q^{29}+ \cdots - 8151941296 q^{99}+O(q^{100}) $ 12 * q - 966 * q^5 - 7696 * q^7 - 21620 * q^9 + 47640 * q^11 + 103432 * q^13 + 400280 * q^15 - 402234 * q^17 + 519960 * q^19 - 1025858 * q^21 + 683124 * q^23 - 1134656 * q^25 + 7641648 * q^27 + 4252680 * q^29 + 1040564 * q^31 + 5260178 * q^33 + 552600 * q^35 - 4886646 * q^37 - 12290272 * q^39 + 34183128 * q^41 - 55245072 * q^43 - 17040100 * q^45 + 26858748 * q^47 + 12089772 * q^49 - 57449552 * q^51 - 21717558 * q^53 - 54952184 * q^55 - 72948300 * q^57 - 31298904 * q^59 - 155592542 * q^61 + 400205088 * q^63 + 7729020 * q^65 - 212196992 * q^67 - 249902212 * q^69 - 648519648 * q^71 - 549596530 * q^73 + 647801672 * q^75 - 519615786 * q^77 - 227611204 * q^79 + 363916186 * q^81 - 989338224 * q^83 + 2712621236 * q^85 + 860715296 * q^87 - 1746541986 * q^89 + 2592411872 * q^91 + 1354546774 * q^93 - 1586101596 * q^95 + 2306715320 * q^97 - 8151941296 * q^99

Character values

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

$n$	$15$	$17$	$85$
$\chi(n)$	$1$	$e\left(\frac{1}{3}\right)$	$1$

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
$3$	82.4523	+	142.812i	0.587702	+	1.01793i	0.994533	+	0.104426i	$0.0333005\pi$
$3$	82.4523	+	142.812i	0.587702	+	1.01793i	−0.406831	+	0.913504i	$0.633366\pi$
$4$		0			0
$5$	1030.62	−	1785.08i	0.737451	−	1.27730i	−0.216189	−	0.976352i	$-0.569363\pi$
$5$	1030.62	−	1785.08i	0.737451	−	1.27730i	0.953640	−	0.300951i	$-0.0973039\pi$
$6$		0			0
$7$	−6318.25	−	658.292i	−0.994616	−	0.103628i
$7$	−6318.25	−	658.292i	−0.994616	−	0.103628i
$8$		0			0
$9$	−3755.26	+	6504.30i	−0.190787	+	0.330453i
$10$		0			0
$11$	19582.0	+	33917.0i	0.403264	+	0.698474i	0.994118	−	0.108305i	$-0.0345422\pi$
$11$	19582.0	+	33917.0i	0.403264	+	0.698474i	−0.590854	+	0.806779i	$0.701209\pi$
$12$		0			0
$13$	−14379.0			−0.139631			−0.0698157	−	0.997560i	$-0.522241\pi$
$13$	−14379.0			−0.139631			−0.0698157	+	0.997560i	$0.522241\pi$
$14$		0			0
$15$	339908.			1.73360
$16$		0			0
$17$	−117601.	−	203691.i	−0.341501	−	0.591497i	0.643211	−	0.765689i	$-0.277602\pi$
$17$	−117601.	−	203691.i	−0.341501	−	0.591497i	−0.984712	+	0.174192i	$0.944269\pi$
$18$		0			0
$19$	271910.	−	470962.i	0.478668	−	0.829077i	−0.521033	−	0.853537i	$-0.674453\pi$
$19$	271910.	−	470962.i	0.478668	−	0.829077i	0.999701	+	0.0244594i	$0.00778645\pi$
$20$		0			0
$21$	−426942.	−	956597.i	−0.479052	−	1.07335i
$22$		0			0
$23$	−576096.	+	997828.i	−0.429260	+	0.743499i	−0.996808	−	0.0798409i	$-0.974559\pi$
$23$	−576096.	+	997828.i	−0.429260	+	0.743499i	0.567548	+	0.823340i	$0.307892\pi$
$24$		0			0
$25$	−1.14779e6	−	1.98803e6i	−0.587667	−	1.01787i
$26$		0			0
$27$	2.00730e6			0.726900
$28$		0			0
$29$	−6.02178e6			−1.58101			−0.790504	−	0.612458i	$-0.790181\pi$
$29$	−6.02178e6			−1.58101			−0.790504	+	0.612458i	$0.790181\pi$
$30$		0			0
$31$	−4.27494e6	−	7.40442e6i	−0.831386	−	1.44000i	−0.896939	−	0.442154i	$-0.854215\pi$
$31$	−4.27494e6	−	7.40442e6i	−0.831386	−	1.44000i	0.0655534	−	0.997849i	$-0.479119\pi$
$32$		0			0
$33$	−3.22916e6	+	5.59307e6i	−0.473998	+	0.820989i
$34$		0			0
$35$	−7.68681e6	+	1.06002e7i	−0.865845	+	1.19400i
$36$		0			0
$37$	2.47645e6	−	4.28933e6i	0.217231	−	0.376255i	−0.736730	−	0.676188i	$-0.763631\pi$
$37$	2.47645e6	−	4.28933e6i	0.217231	−	0.376255i	0.953960	+	0.299933i	$0.0969642\pi$
$38$		0			0
$39$	−1.18558e6	−	2.05349e6i	−0.0820617	−	0.142135i
$40$		0			0
$41$	1.48341e7			0.819851			0.409925	−	0.912119i	$-0.365555\pi$
$41$	1.48341e7			0.819851			0.409925	+	0.912119i	$0.365555\pi$
$42$		0			0
$43$	−3.80424e7			−1.69691			−0.848456	−	0.529265i	$-0.822468\pi$
$43$	−3.80424e7			−1.69691			−0.848456	+	0.529265i	$0.822468\pi$
$44$		0			0
$45$	7.74048e6	+	1.34069e7i	0.281392	+	0.487385i
$46$		0			0
$47$	2.38065e7	−	4.12341e7i	0.711632	−	1.23258i	−0.252612	−	0.967568i	$-0.581290\pi$
$47$	2.38065e7	−	4.12341e7i	0.711632	−	1.23258i	0.964244	−	0.265015i	$-0.0853769\pi$
$48$		0			0
$49$	3.94869e7	+	8.31850e6i	0.978522	+	0.206140i
$50$		0			0
$51$	1.93930e7	−	3.35896e7i	0.401401	−	0.695248i
$52$		0			0
$53$	−4.94404e7	−	8.56333e7i	−0.860678	−	1.49074i	−0.871275	−	0.490795i	$-0.836707\pi$
$53$	−4.94404e7	−	8.56333e7i	−0.860678	−	1.49074i	0.0105971	−	0.999944i	$-0.496627\pi$
$54$		0			0
$55$	8.07263e7			1.18955
$56$		0			0
$57$	8.96785e7			1.12526
$58$		0			0
$59$	2.07780e7	+	3.59886e7i	0.223239	+	0.386662i	0.955790	−	0.294051i	$-0.0950035\pi$
$59$	2.07780e7	+	3.59886e7i	0.223239	+	0.386662i	−0.732550	+	0.680713i	$0.761670\pi$
$60$		0			0
$61$	1.71098e7	−	2.96350e7i	0.158220	−	0.274044i	−0.776007	−	0.630724i	$-0.782758\pi$
$61$	1.71098e7	−	2.96350e7i	0.158220	−	0.274044i	0.934227	+	0.356680i	$0.116091\pi$
$62$		0			0
$63$	2.80084e7	−	3.86237e7i	0.224004	−	0.308903i
$64$		0			0
$65$	−1.48193e7	+	2.56677e7i	−0.102971	+	0.178352i
$66$		0			0
$67$	7.66238e7	+	1.32716e8i	0.464544	+	0.804614i	0.999181	−	0.0404678i	$-0.0128848\pi$
$67$	7.66238e7	+	1.32716e8i	0.464544	+	0.804614i	−0.534637	+	0.845082i	$0.679551\pi$
$68$		0			0
$69$	−1.90002e8			−1.00911
$70$		0			0
$71$	−1.59112e8			−0.743090			−0.371545	−	0.928415i	$-0.621172\pi$
$71$	−1.59112e8			−0.743090			−0.371545	+	0.928415i	$0.621172\pi$
$72$		0			0
$73$	−2.03375e8	−	3.52256e8i	−0.838194	−	1.45179i	−0.891403	−	0.453211i	$-0.850279\pi$
$73$	−2.03375e8	−	3.52256e8i	−0.838194	−	1.45179i	0.0532099	−	0.998583i	$-0.483055\pi$
$74$		0			0
$75$	1.89275e8	−	3.27835e8i	0.690746	−	1.19641i
$76$		0			0
$77$	−1.01397e8	−	2.27187e8i	−0.328712	−	0.736503i
$78$		0			0
$79$	4.27911e7	−	7.41163e7i	0.123604	−	0.214088i	−0.797583	−	0.603210i	$-0.793888\pi$
$79$	4.27911e7	−	7.41163e7i	0.123604	−	0.214088i	0.921186	+	0.389122i	$0.127222\pi$
$80$		0			0
$81$	2.39421e8	+	4.14689e8i	0.617988	+	1.07039i
$82$		0			0
$83$	−5.27310e8			−1.21959			−0.609796	−	0.792558i	$-0.708749\pi$
$83$	−5.27310e8			−1.21959			−0.609796	+	0.792558i	$0.708749\pi$
$84$		0			0
$85$	−4.84808e8			−1.00736
$86$		0			0
$87$	−4.96509e8	−	8.59980e8i	−0.929161	−	1.60935i
$88$		0			0
$89$	2.67165e8	−	4.62744e8i	0.451362	−	0.781782i	−0.547109	−	0.837062i	$-0.684272\pi$
$89$	2.67165e8	−	4.62744e8i	0.451362	−	0.781782i	0.998471	+	0.0552793i	$0.0176049\pi$
$90$		0			0
$91$	9.08500e7	+	9.46557e6i	0.138880	+	0.0144697i
$92$		0			0
$93$	7.04958e8	−	1.22102e9i	0.977214	−	1.69258i
$94$		0			0
$95$	−5.60472e8	−	9.70765e8i	−0.705988	−	1.22281i
$96$		0			0
$97$	1.26391e9			1.44958			0.724791	−	0.688969i	$-0.241936\pi$
$97$	1.26391e9			1.44958			0.724791	+	0.688969i	$0.241936\pi$
$98$		0			0
$99$	−2.94142e8			−0.307750
$100$		0			0
$101$	−4.76474e8	−	8.25277e8i	−0.455610	−	0.789139i	0.543113	−	0.839660i	$-0.317245\pi$
$101$	−4.76474e8	−	8.25277e8i	−0.455610	−	0.789139i	−0.998723	+	0.0505201i	$0.983912\pi$
$102$		0			0
$103$	8.03077e8	−	1.39097e9i	0.703055	−	1.21773i	−0.264333	−	0.964431i	$-0.585152\pi$
$103$	8.03077e8	−	1.39097e9i	0.703055	−	1.21773i	0.967389	−	0.253296i	$-0.0815147\pi$
$104$		0			0
$105$	−2.14762e9	−	2.23758e8i	−1.72427	−	0.179650i
$106$		0			0
$107$	1.52449e8	−	2.64049e8i	0.112434	−	0.194741i	−0.804317	−	0.594200i	$-0.797469\pi$
$107$	1.52449e8	−	2.64049e8i	0.112434	−	0.194741i	0.916751	+	0.399459i	$0.130802\pi$
$108$		0			0
$109$	1.01127e9	+	1.75158e9i	0.686198	+	1.18853i	0.973059	+	0.230558i	$0.0740551\pi$
$109$	1.01127e9	+	1.75158e9i	0.686198	+	1.18853i	−0.286860	+	0.957972i	$0.592612\pi$
$110$		0			0
$111$	8.16755e8			0.510668
$112$		0			0
$113$	1.80211e8			0.103975			0.0519874	−	0.998648i	$-0.483444\pi$
$113$	1.80211e8			0.103975			0.0519874	+	0.998648i	$0.483444\pi$
$114$		0			0
$115$	1.18747e9	+	2.05676e9i	0.633116	+	1.09659i
$116$		0			0
$117$	5.39968e7	−	9.35253e7i	0.0266399	−	0.0461416i
$118$		0			0
$119$	6.08946e8	+	1.36439e9i	0.278367	+	0.623701i
$120$		0			0
$121$	4.12065e8	−	7.13718e8i	0.174756	−	0.302686i
$122$		0			0
$123$	1.22311e9	+	2.11849e9i	0.481828	+	0.834550i
$124$		0			0
$125$	−7.05871e8			−0.258601
$126$		0			0
$127$	−3.68048e9			−1.25541			−0.627707	−	0.778449i	$-0.716007\pi$
$127$	−3.68048e9			−1.25541			−0.627707	+	0.778449i	$0.716007\pi$
$128$		0			0
$129$	−3.13668e9	−	5.43289e9i	−0.997279	−	1.72734i
$130$		0			0
$131$	8.71850e8	−	1.51009e9i	0.258655	−	0.448004i	−0.707227	−	0.706987i	$-0.750054\pi$
$131$	8.71850e8	−	1.51009e9i	0.258655	−	0.448004i	0.965882	+	0.258983i	$0.0833874\pi$
$132$		0			0
$133$	−2.02803e9	+	2.79666e9i	−0.562006	+	0.775010i
$134$		0			0
$135$	2.06876e9	−	3.58319e9i	0.536053	−	0.928471i
$136$		0			0
$137$	1.42574e9	+	2.46945e9i	0.345777	+	0.598904i	0.985495	−	0.169706i	$-0.0542819\pi$
$137$	1.42574e9	+	2.46945e9i	0.345777	+	0.598904i	−0.639717	+	0.768610i	$0.720949\pi$
$138$		0			0
$139$	9.71824e8			0.220811			0.110406	−	0.993887i	$-0.464785\pi$
$139$	9.71824e8			0.220811			0.110406	+	0.993887i	$0.464785\pi$
$140$		0			0
$141$	7.85161e9			1.67291
$142$		0			0
$143$	−2.81569e8	−	4.87692e8i	−0.0563084	−	0.0975290i
$144$		0			0
$145$	−6.20616e9	+	1.07494e10i	−1.16591	+	2.01942i
$146$		0			0
$147$	2.06781e9	+	6.32507e9i	0.365243	+	1.11722i
$148$		0			0
$149$	4.42276e9	−	7.66044e9i	0.735114	−	1.27325i	−0.219559	−	0.975599i	$-0.570462\pi$
$149$	4.42276e9	−	7.66044e9i	0.735114	−	1.27325i	0.954673	−	0.297656i	$-0.0962048\pi$
$150$		0			0
$151$	2.42594e9	+	4.20185e9i	0.379738	+	0.657725i	0.991024	−	0.133684i	$-0.0426808\pi$
$151$	2.42594e9	+	4.20185e9i	0.379738	+	0.657725i	−0.611286	+	0.791410i	$0.709348\pi$
$152$		0			0
$153$	1.76649e9			0.260616
$154$		0			0
$155$	−1.76233e10			−2.45242
$156$		0			0
$157$	2.92788e9	+	5.07124e9i	0.384596	+	0.666140i	0.991713	−	0.128473i	$-0.0410074\pi$
$157$	2.92788e9	+	5.07124e9i	0.384596	+	0.666140i	−0.607117	+	0.794612i	$0.707674\pi$
$158$		0			0
$159$	8.15295e9	−	1.41213e10i	1.01164	−	1.75222i
$160$		0			0
$161$	4.29678e9	−	5.92529e9i	0.503996	−	0.695013i
$162$		0			0
$163$	−3.24998e9	+	5.62913e9i	−0.360609	+	0.624592i	−0.988061	−	0.154062i	$-0.950764\pi$
$163$	−3.24998e9	+	5.62913e9i	−0.360609	+	0.624592i	0.627453	+	0.778655i	$0.284098\pi$
$164$		0			0
$165$	6.65606e9	+	1.15286e10i	0.699101	+	1.21088i
$166$		0			0
$167$	−1.60674e10			−1.59853			−0.799266	−	0.600977i	$-0.794778\pi$
$167$	−1.60674e10			−1.59853			−0.799266	+	0.600977i	$0.794778\pi$
$168$		0			0
$169$	−1.03977e10			−0.980503
$170$		0			0
$171$	2.04219e9	+	3.53717e9i	0.182647	+	0.316354i
$172$		0			0
$173$	−2.88515e9	+	4.99722e9i	−0.244884	+	0.424152i	−0.962099	−	0.272700i	$-0.912083\pi$
$173$	−2.88515e9	+	4.99722e9i	−0.244884	+	0.424152i	0.717215	+	0.696852i	$0.245417\pi$
$174$		0			0
$175$	5.94331e9	+	1.33164e10i	0.479023	+	1.07329i
$176$		0			0
$177$	−3.42639e9	+	5.93469e9i	−0.262396	+	0.454484i
$178$		0			0
$179$	6.20303e9	+	1.07440e10i	0.451612	+	0.782215i	0.998486	−	0.0549998i	$-0.0175158\pi$
$179$	6.20303e9	+	1.07440e10i	0.451612	+	0.782215i	−0.546874	+	0.837215i	$0.684182\pi$
$180$		0			0
$181$	−2.28094e10			−1.57965			−0.789823	−	0.613335i	$-0.789827\pi$
$181$	−2.28094e10			−1.57965			−0.789823	+	0.613335i	$0.789827\pi$
$182$		0			0
$183$	5.64296e9			0.371944
$184$		0			0
$185$	−5.10455e9	−	8.84134e9i	−0.320394	−	0.554939i
$186$		0			0
$187$	4.60573e9	−	7.97736e9i	0.275430	−	0.477059i
$188$		0			0
$189$	−1.26826e10	−	1.32139e9i	−0.722987	−	0.0753272i
$190$		0			0
$191$	−5.95381e9	+	1.03123e10i	−0.323701	+	0.560667i	−0.981249	−	0.192746i	$-0.938261\pi$
$191$	−5.95381e9	+	1.03123e10i	−0.323701	+	0.560667i	0.657547	+	0.753413i	$0.271594\pi$
$192$		0			0
$193$	9.90850e9	+	1.71620e10i	0.514044	+	0.890350i	0.999867	+	0.0162931i	$0.00518647\pi$
$193$	9.90850e9	+	1.71620e10i	0.514044	+	0.890350i	−0.485823	+	0.874057i	$0.661480\pi$
$194$		0			0
$195$	−4.88753e9			−0.242066
$196$		0			0
$197$	2.29235e10			1.08439			0.542193	−	0.840254i	$-0.317594\pi$
$197$	2.29235e10			1.08439			0.542193	+	0.840254i	$0.317594\pi$
$198$		0			0
$199$	1.37918e10	+	2.38882e10i	0.623424	+	1.07980i	0.988843	+	0.148958i	$0.0475921\pi$
$199$	1.37918e10	+	2.38882e10i	0.623424	+	1.07980i	−0.365420	+	0.930843i	$0.619075\pi$
$200$		0			0
$201$	−1.26356e10	+	2.18855e10i	−0.546027	+	0.945747i
$202$		0			0
$203$	3.80471e10	+	3.96409e9i	1.57250	+	0.163837i
$204$		0			0
$205$	1.52883e10	−	2.64802e10i	0.604600	−	1.04720i
$206$		0			0
$207$	−4.32678e9	−	7.49421e9i	−0.163794	−	0.283700i
$208$		0			0
$209$	2.12982e10			0.772119
$210$		0			0
$211$	1.00553e9			0.0349241			0.0174620	−	0.999848i	$-0.494441\pi$
$211$	1.00553e9			0.0349241			0.0174620	+	0.999848i	$0.494441\pi$
$212$		0			0
$213$	−1.31192e10	−	2.27231e10i	−0.436715	−	0.756413i
$214$		0			0
$215$	−3.92072e10	+	6.79088e10i	−1.25139	+	2.16747i
$216$		0			0
$217$	2.21359e10	+	4.95971e10i	0.677685	+	1.51840i
$218$		0			0
$219$	3.35374e10	−	5.80886e10i	0.985216	−	1.70644i
$220$		0			0
$221$	1.69099e9	+	2.92888e9i	0.0476843	+	0.0825916i
$222$		0			0
$223$	6.84669e9			0.185400			0.0926998	−	0.995694i	$-0.470450\pi$
$223$	6.84669e9			0.185400			0.0926998	+	0.995694i	$0.470450\pi$
$224$		0			0
$225$	1.72410e10			0.448477
$226$		0			0
$227$	−1.59007e10	−	2.75408e10i	−0.397465	−	0.688430i	0.595947	−	0.803024i	$-0.296777\pi$
$227$	−1.59007e10	−	2.75408e10i	−0.397465	−	0.688430i	−0.993412	+	0.114594i	$0.963443\pi$
$228$		0			0
$229$	−4.85226e9	+	8.40436e9i	−0.116596	+	0.201951i	−0.918417	−	0.395614i	$-0.870532\pi$
$229$	−4.85226e9	+	8.40436e9i	−0.116596	+	0.201951i	0.801821	+	0.597565i	$0.203865\pi$
$230$		0			0
$231$	2.40845e10	−	3.32127e10i	0.556524	−	0.767449i
$232$		0			0
$233$	7.92657e9	−	1.37292e10i	0.176191	−	0.305172i	−0.764382	−	0.644764i	$-0.776956\pi$
$233$	7.92657e9	−	1.37292e10i	0.176191	−	0.305172i	0.940573	+	0.339592i	$0.110289\pi$
$234$		0			0
$235$	−4.90709e10	−	8.49932e10i	−1.04959	−	1.81794i
$236$		0			0
$237$	1.41129e10			0.290568
$238$		0			0
$239$	2.10542e9			0.0417395			0.0208698	−	0.999782i	$-0.493356\pi$
$239$	2.10542e9			0.0417395			0.0208698	+	0.999782i	$0.493356\pi$
$240$		0			0
$241$	−1.18980e10	−	2.06079e10i	−0.227194	−	0.393512i	0.729781	−	0.683681i	$-0.239622\pi$
$241$	−1.18980e10	−	2.06079e10i	−0.227194	−	0.393512i	−0.956976	+	0.290169i	$0.906289\pi$
$242$		0			0
$243$	−1.97268e10	+	3.41678e10i	−0.362935	+	0.628622i
$244$		0			0
$245$	5.55452e10	−	6.19143e10i	0.984915	−	1.09785i
$246$		0			0
$247$	−3.90979e9	+	6.77196e9i	−0.0668371	+	0.115765i
$248$		0			0
$249$	−4.34779e10	−	7.53059e10i	−0.716756	−	1.24146i
$250$		0			0
$251$	−9.31587e10			−1.48147			−0.740733	−	0.671799i	$-0.765522\pi$
$251$	−9.31587e10			−1.48147			−0.740733	+	0.671799i	$0.765522\pi$
$252$		0			0
$253$	−4.51245e10			−0.692420
$254$		0			0
$255$	−3.99736e10	−	6.92362e10i	−0.592028	−	1.02542i
$256$		0			0
$257$	−1.85628e10	+	3.21517e10i	−0.265426	+	0.459732i	−0.967675	−	0.252200i	$-0.918846\pi$
$257$	−1.85628e10	+	3.21517e10i	−0.265426	+	0.459732i	0.702249	+	0.711932i	$0.252179\pi$
$258$		0			0
$259$	−1.84704e10	+	2.54709e10i	−0.255052	+	0.351718i
$260$		0			0
$261$	2.26133e10	−	3.91675e10i	0.301636	−	0.522448i
$262$		0			0
$263$	2.52620e10	+	4.37551e10i	0.325587	+	0.563934i	0.981631	−	0.190789i	$-0.0611047\pi$
$263$	2.52620e10	+	4.37551e10i	0.325587	+	0.563934i	−0.656044	+	0.754723i	$0.727771\pi$
$264$		0			0
$265$	−2.03817e11			−2.53883
$266$		0			0
$267$	8.81136e10			1.06107
$268$		0			0
$269$	3.81145e10	+	6.60162e10i	0.443818	+	0.768715i	0.997969	−	0.0637010i	$-0.0202904\pi$
$269$	3.81145e10	+	6.60162e10i	0.443818	+	0.768715i	−0.554151	+	0.832416i	$0.686957\pi$
$270$		0			0
$271$	1.99087e10	−	3.44828e10i	0.224223	−	0.388366i	−0.731863	−	0.681452i	$-0.761349\pi$
$271$	1.99087e10	−	3.44828e10i	0.224223	−	0.388366i	0.956086	+	0.293086i	$0.0946822\pi$
$272$		0			0
$273$	6.13900e9	+	1.37549e10i	0.0668907	+	0.149874i
$274$		0			0
$275$	4.49519e10	−	7.78590e10i	0.473970	−	0.820941i
$276$		0			0
$277$	−5.36274e10	−	9.28853e10i	−0.547303	−	0.947956i	−0.998458	−	0.0555105i	$-0.982321\pi$
$277$	−5.36274e10	−	9.28853e10i	−0.547303	−	0.947956i	0.451156	−	0.892445i	$-0.351012\pi$
$278$		0			0
$279$	6.42141e10			0.634470
$280$		0			0
$281$	−1.52428e11			−1.45844			−0.729218	−	0.684281i	$-0.760116\pi$
$281$	−1.52428e11			−1.45844			−0.729218	+	0.684281i	$0.760116\pi$
$282$		0			0
$283$	−3.68540e10	−	6.38330e10i	−0.341543	−	0.591571i	0.643176	−	0.765718i	$-0.277616\pi$
$283$	−3.68540e10	−	6.38330e10i	−0.341543	−	0.591571i	−0.984720	+	0.174148i	$0.944283\pi$
$284$		0			0
$285$	9.24243e10	−	1.60084e11i	0.829821	−	1.43729i
$286$		0			0
$287$	−9.37257e10	−	9.76519e9i	−0.815437	−	0.0849595i
$288$		0			0
$289$	3.16338e10	−	5.47914e10i	0.266754	−	0.462032i
$290$		0			0
$291$	1.04212e11	+	1.80501e11i	0.851922	+	1.47557i
$292$		0			0
$293$	9.52294e10			0.754861			0.377430	−	0.926038i	$-0.376808\pi$
$293$	9.52294e10			0.754861			0.377430	+	0.926038i	$0.376808\pi$
$294$		0			0
$295$	8.56570e10			0.658512
$296$		0			0
$297$	3.93069e10	+	6.80815e10i	0.293133	+	0.507721i
$298$		0			0
$299$	8.28369e9	−	1.43478e10i	0.0599381	−	0.103816i
$300$		0			0
$301$	2.40361e11	+	2.50430e10i	1.68778	+	0.175848i
$302$		0			0
$303$	7.85728e10	−	1.36092e11i	0.535526	−	0.927557i
$304$		0			0
$305$	−3.52673e10	−	6.10848e10i	−0.233358	−	0.404188i
$306$		0			0
$307$	3.29114e10			0.211458			0.105729	−	0.994395i	$-0.466282\pi$
$307$	3.29114e10			0.211458			0.105729	+	0.994395i	$0.466282\pi$
$308$		0			0
$309$	2.64862e11			1.65275
$310$		0			0
$311$	4.50909e9	+	7.80997e9i	0.0273317	+	0.0473399i	0.879368	−	0.476143i	$-0.157966\pi$
$311$	4.50909e9	+	7.80997e9i	0.0273317	+	0.0473399i	−0.852036	+	0.523483i	$0.824632\pi$
$312$		0			0
$313$	1.28532e9	−	2.22623e9i	0.00756939	−	0.0131106i	−0.862216	−	0.506541i	$-0.830924\pi$
$313$	1.28532e9	−	2.22623e9i	0.00756939	−	0.0131106i	0.869785	+	0.493430i	$0.164257\pi$
$314$		0			0
$315$	−4.00806e10	−	8.98037e10i	−0.229370	−	0.513921i
$316$		0			0
$317$	−1.03394e11	+	1.79084e11i	−0.575082	+	0.996071i	0.420951	+	0.907083i	$0.361696\pi$
$317$	−1.03394e11	+	1.79084e11i	−0.575082	+	0.996071i	−0.996033	+	0.0889876i	$0.971637\pi$
$318$		0			0
$319$	−1.17918e11	−	2.04241e11i	−0.637564	−	1.10429i
$320$		0			0
$321$	5.02790e10			0.264310
$322$		0			0
$323$	−1.27908e11			−0.653862
$324$		0			0
$325$	1.65040e10	+	2.85858e10i	0.0820568	+	0.142127i
$326$		0			0
$327$	−1.66764e11	+	2.88843e11i	−0.806560	+	1.39700i
$328$		0			0
$329$	−1.77560e11	+	2.44856e11i	−0.835531	+	1.15220i
$330$		0			0
$331$	−9.32775e10	+	1.61561e11i	−0.427121	+	0.739795i	−0.996616	−	0.0821995i	$-0.973806\pi$
$331$	−9.32775e10	+	1.61561e11i	−0.427121	+	0.739795i	0.569495	+	0.821995i	$0.307139\pi$
$332$		0			0
$333$	1.85994e10	+	3.22151e10i	0.0828896	+	0.143569i
$334$		0			0
$335$	3.15880e11			1.37031
$336$		0			0
$337$	−2.98733e11			−1.26168			−0.630838	−	0.775914i	$-0.717289\pi$
$337$	−2.98733e11			−1.26168			−0.630838	+	0.775914i	$0.717289\pi$
$338$		0			0
$339$	1.48588e10	+	2.57362e10i	0.0611062	+	0.105839i
$340$		0			0
$341$	1.67424e11	−	2.89986e11i	0.670536	−	1.16140i
$342$		0			0
$343$	−2.44012e11	−	7.85523e10i	−0.951892	−	0.306433i
$344$		0			0
$345$	−1.95820e11	+	3.39169e11i	−0.744166	+	1.28893i
$346$		0			0
$347$	4.54874e10	+	7.87865e10i	0.168426	+	0.291722i	0.937867	−	0.346996i	$-0.112798\pi$
$347$	4.54874e10	+	7.87865e10i	0.168426	+	0.291722i	−0.769441	+	0.638718i	$0.779465\pi$
$348$		0			0
$349$	4.05187e11			1.46198			0.730989	−	0.682389i	$-0.239059\pi$
$349$	4.05187e11			1.46198			0.730989	+	0.682389i	$0.239059\pi$
$350$		0			0
$351$	−2.88629e10			−0.101498
$352$		0			0
$353$	1.14775e11	+	1.98796e11i	0.393424	+	0.681430i	0.992899	−	0.118964i	$-0.0379572\pi$
$353$	1.14775e11	+	1.98796e11i	0.393424	+	0.681430i	−0.599475	+	0.800394i	$0.704624\pi$
$354$		0			0
$355$	−1.63984e11	+	2.84029e11i	−0.547992	+	0.949150i
$356$		0			0
$357$	−1.44642e11	+	1.99461e11i	−0.471288	+	0.649908i
$358$		0			0
$359$	−1.31163e11	+	2.27182e11i	−0.416762	+	0.721853i	−0.995612	−	0.0935815i	$-0.970168\pi$
$359$	−1.31163e11	+	2.27182e11i	−0.416762	+	0.721853i	0.578850	+	0.815434i	$0.303502\pi$
$360$		0			0
$361$	1.34735e10	+	2.33368e10i	0.0417540	+	0.0723200i
$362$		0			0
$363$	1.35903e11			0.410818
$364$		0			0
$365$	−8.38408e11			−2.47251
$366$		0			0
$367$	2.14762e10	+	3.71978e10i	0.0617959	+	0.107034i	0.895268	−	0.445528i	$-0.146984\pi$
$367$	2.14762e10	+	3.71978e10i	0.0617959	+	0.107034i	−0.833472	+	0.552561i	$0.813651\pi$
$368$		0			0
$369$	−5.57060e10	+	9.64857e10i	−0.156417	+	0.270922i
$370$		0			0
$371$	2.56005e11	+	5.73599e11i	0.701562	+	1.57190i
$372$		0			0
$373$	3.10665e11	−	5.38087e11i	0.831003	−	1.43934i	−0.0662412	−	0.997804i	$-0.521101\pi$
$373$	3.10665e11	−	5.38087e11i	0.831003	−	1.43934i	0.897244	−	0.441535i	$-0.145566\pi$
$374$		0			0
$375$	−5.82007e10	−	1.00807e11i	−0.151980	−	0.263238i
$376$		0			0
$377$	8.65871e10			0.220758
$378$		0			0
$379$	2.63758e11			0.656644			0.328322	−	0.944566i	$-0.393517\pi$
$379$	2.63758e11			0.656644			0.328322	+	0.944566i	$0.393517\pi$
$380$		0			0
$381$	−3.03464e11	−	5.25615e11i	−0.737810	−	1.27792i
$382$		0			0
$383$	3.01108e11	−	5.21534e11i	0.715035	−	1.23848i	−0.247911	−	0.968783i	$-0.579744\pi$
$383$	3.01108e11	−	5.21534e11i	0.715035	−	1.23848i	0.962946	−	0.269694i	$-0.0869227\pi$
$384$		0			0
$385$	−5.10049e11	−	5.31414e10i	−1.18315	−	0.123271i
$386$		0			0
$387$	1.42859e11	−	2.47439e11i	0.323749	−	0.560750i
$388$		0			0
$389$	1.88469e11	+	3.26438e11i	0.417317	+	0.722815i	0.995669	−	0.0929731i	$-0.0296371\pi$
$389$	1.88469e11	+	3.26438e11i	0.417317	+	0.722815i	−0.578351	+	0.815788i	$0.696304\pi$
$390$		0			0
$391$	2.70999e11			0.586370
$392$		0			0
$393$	2.87544e11			0.608048
$394$		0			0
$395$	−8.82026e10	−	1.52771e11i	−0.182303	−	0.315758i
$396$		0			0
$397$	−1.46890e11	+	2.54422e11i	−0.296781	+	0.514040i	−0.975398	−	0.220453i	$-0.929246\pi$
$397$	−1.46890e11	+	2.54422e11i	−0.296781	+	0.514040i	0.678617	+	0.734493i	$0.262580\pi$
$398$		0			0
$399$	−5.66611e11	−	5.90346e10i	−1.11920	−	0.116608i
$400$		0			0
$401$	−2.33172e11	+	4.03866e11i	−0.450326	+	0.779987i	−0.998406	−	0.0564386i	$-0.982025\pi$
$401$	−2.33172e11	+	4.03866e11i	−0.450326	+	0.779987i	0.548080	+	0.836426i	$0.315359\pi$
$402$		0			0
$403$	6.14694e10	+	1.06468e11i	0.116088	+	0.201070i
$404$		0			0
$405$	9.87007e11			1.82294
$406$		0			0
$407$	1.93975e11			0.350406
$408$		0			0
$409$	2.42851e11	+	4.20630e11i	0.429126	+	0.743268i	0.996796	−	0.0799885i	$-0.0254884\pi$
$409$	2.42851e11	+	4.20630e11i	0.429126	+	0.743268i	−0.567670	+	0.823256i	$0.692155\pi$
$410$		0			0
$411$	−2.35110e11	+	4.07223e11i	−0.406428	+	0.703954i
$412$		0			0
$413$	−1.07590e11	−	2.41063e11i	−0.181968	−	0.407714i
$414$		0			0
$415$	−5.43455e11	+	9.41292e11i	−0.899389	+	1.55779i
$416$		0			0
$417$	8.01291e10	+	1.38788e11i	0.129771	+	0.224770i
$418$		0			0
$419$	3.95518e11			0.626908			0.313454	−	0.949603i	$-0.398514\pi$
$419$	3.95518e11			0.626908			0.313454	+	0.949603i	$0.398514\pi$
$420$		0			0
$421$	9.26613e11			1.43757			0.718785	−	0.695233i	$-0.244699\pi$
$421$	9.26613e11			1.43757			0.718785	+	0.695233i	$0.244699\pi$
$422$		0			0
$423$	1.78799e11	+	3.09689e11i	0.271540	+	0.470321i
$424$		0			0
$425$	−2.69963e11	+	4.67589e11i	−0.401378	+	0.695207i
$426$		0			0
$427$	−1.27612e11	+	1.75978e11i	−0.185766	+	0.256173i
$428$		0			0
$429$	4.64321e10	−	8.04227e10i	0.0661851	−	0.114636i
$430$		0			0
$431$	−1.80836e11	−	3.13217e11i	−0.252428	−	0.437218i	0.711766	−	0.702417i	$-0.247896\pi$
$431$	−1.80836e11	−	3.13217e11i	−0.252428	−	0.437218i	−0.964194	+	0.265199i	$0.914562\pi$
$432$		0			0
$433$	1.76404e11			0.241165			0.120582	−	0.992703i	$-0.461524\pi$
$433$	1.76404e11			0.241165			0.120582	+	0.992703i	$0.461524\pi$
$434$		0			0
$435$	−2.04685e12			−2.74084
$436$		0			0
$437$	3.13293e11	+	5.42639e11i	0.410946	+	0.711779i
$438$		0			0
$439$	5.83361e11	−	1.01041e12i	0.749630	−	1.29840i	−0.198371	−	0.980127i	$-0.563565\pi$
$439$	5.83361e11	−	1.01041e12i	0.749630	−	1.29840i	0.948000	−	0.318270i	$-0.103102\pi$
$440$		0			0
$441$	−2.02390e11	+	2.25597e11i	−0.254809	+	0.284027i
$442$		0			0
$443$	3.94986e11	−	6.84136e11i	0.487265	−	0.843968i	−0.512628	−	0.858611i	$-0.671328\pi$
$443$	3.94986e11	−	6.84136e11i	0.487265	−	0.843968i	0.999893	+	0.0146433i	$0.00466127\pi$
$444$		0			0
$445$	−5.50691e11	−	9.53825e11i	−0.665715	−	1.15305i
$446$		0			0
$447$	1.45867e12			1.72811
$448$		0			0
$449$	8.89802e11			1.03320			0.516600	−	0.856227i	$-0.327197\pi$
$449$	8.89802e11			1.03320			0.516600	+	0.856227i	$0.327197\pi$
$450$		0			0
$451$	2.90482e11	+	5.03129e11i	0.330617	+	0.572645i
$452$		0			0
$453$	−4.00049e11	+	6.92905e11i	−0.446345	+	0.773093i
$454$		0			0
$455$	1.10529e11	−	1.52420e11i	0.120899	−	0.166721i
$456$		0			0
$457$	−6.13078e11	+	1.06188e12i	−0.657496	+	1.13882i	0.323766	+	0.946137i	$0.395051\pi$
$457$	−6.13078e11	+	1.06188e12i	−0.657496	+	1.13882i	−0.981262	+	0.192679i	$0.938282\pi$
$458$		0			0
$459$	−2.36061e11	−	4.08869e11i	−0.248237	−	0.429959i
$460$		0			0
$461$	4.74151e11			0.488948			0.244474	−	0.969656i	$-0.421385\pi$
$461$	4.74151e11			0.488948			0.244474	+	0.969656i	$0.421385\pi$
$462$		0			0
$463$	−7.13144e11			−0.721212			−0.360606	−	0.932718i	$-0.617430\pi$
$463$	−7.13144e11			−0.721212			−0.360606	+	0.932718i	$0.617430\pi$
$464$		0			0
$465$	−1.45309e12	−	2.51682e12i	−1.44129	−	2.49640i
$466$		0			0
$467$	7.35581e11	−	1.27406e12i	0.715657	−	1.23955i	−0.247049	−	0.969003i	$-0.579461\pi$
$467$	7.35581e11	−	1.27406e12i	0.715657	−	1.23955i	0.962706	−	0.270551i	$-0.0872058\pi$
$468$		0			0
$469$	−3.96762e11	−	8.88976e11i	−0.378663	−	0.848422i
$470$		0			0
$471$	−4.82821e11	+	8.36270e11i	−0.452056	+	0.782983i
$472$		0			0
$473$	−7.44945e11	−	1.29028e12i	−0.684304	−	1.18525i
$474$		0			0
$475$	−1.24838e12			−1.12519
$476$		0			0
$477$	7.42647e11			0.656825
$478$		0			0
$479$	6.00766e11	+	1.04056e12i	0.521429	+	0.903142i	0.999689	+	0.0249235i	$0.00793421\pi$
$479$	6.00766e11	+	1.04056e12i	0.521429	+	0.903142i	−0.478260	+	0.878218i	$0.658732\pi$
$480$		0			0
$481$	−3.56088e10	+	6.16763e10i	−0.0303323	+	0.0525370i
$482$		0			0
$483$	1.20048e12	+	1.25077e11i	1.00367	+	0.104572i
$484$		0			0
$485$	1.30261e12	−	2.25618e12i	1.06899	−	1.85155i
$486$		0			0
$487$	−1.21602e12	−	2.10621e12i	−0.979627	−	1.69676i	−0.663733	−	0.747970i	$-0.731029\pi$
$487$	−1.21602e12	−	2.10621e12i	−0.979627	−	1.69676i	−0.315895	−	0.948794i	$-0.602305\pi$
$488$		0			0
$489$	−1.07187e12			−0.847721
$490$		0			0
$491$	1.50023e12			1.16490			0.582452	−	0.812865i	$-0.302094\pi$
$491$	1.50023e12			1.16490			0.582452	+	0.812865i	$0.302094\pi$
$492$		0			0
$493$	7.08169e11	+	1.22658e12i	0.539915	+	0.935161i
$494$		0			0
$495$	−3.03148e11	+	5.25068e11i	−0.226951	+	0.393090i
$496$		0			0
$497$	1.00531e12	+	1.04742e11i	0.739089	+	0.0770049i
$498$		0			0
$499$	1.29362e12	−	2.24061e12i	0.934016	−	1.61776i	0.157636	−	0.987497i	$-0.449613\pi$
$499$	1.29362e12	−	2.24061e12i	0.934016	−	1.61776i	0.776380	−	0.630265i	$-0.217054\pi$
$500$		0			0
$501$	−1.32479e12	−	2.29461e12i	−0.939461	−	1.62719i
$502$		0			0
$503$	1.09942e12			0.765787			0.382893	−	0.923793i	$-0.374928\pi$
$503$	1.09942e12			0.765787			0.382893	+	0.923793i	$0.374928\pi$
$504$		0			0
$505$	−1.96425e12			−1.34396
$506$		0			0
$507$	−8.57318e11	−	1.48492e12i	−0.576243	−	0.998083i
$508$		0			0
$509$	−9.57644e11	+	1.65869e12i	−0.632374	+	1.09530i	0.354691	+	0.934984i	$0.384586\pi$
$509$	−9.57644e11	+	1.65869e12i	−0.632374	+	1.09530i	−0.987065	+	0.160320i	$0.948747\pi$
$510$		0			0
$511$	1.05309e12	+	2.35952e12i	0.683234	+	1.53084i
$512$		0			0
$513$	5.45805e11	−	9.45362e11i	0.347944	−	0.602656i
$514$		0			0
$515$	−1.65533e12	−	2.86712e12i	−1.03694	−	1.79603i
$516$		0			0
$517$	1.86472e12			1.14790
$518$		0			0
$519$	−9.51549e11			−0.575676
$520$		0			0
$521$	−6.37094e11	−	1.10348e12i	−0.378821	−	0.656137i	0.612070	−	0.790804i	$-0.290337\pi$
$521$	−6.37094e11	−	1.10348e12i	−0.378821	−	0.656137i	−0.990891	+	0.134667i	$0.957004\pi$
$522$		0			0
$523$	−8.00035e11	+	1.38570e12i	−0.467575	+	0.809864i	−0.999314	−	0.0370446i	$-0.988206\pi$
$523$	−8.00035e11	+	1.38570e12i	−0.467575	+	0.809864i	0.531738	+	0.846909i	$0.321539\pi$
$524$		0			0
$525$	−1.41170e12	+	1.94674e12i	−0.811009	+	1.11839i
$526$		0			0
$527$	−1.00548e12	+	1.74154e12i	−0.567838	+	0.983525i
$528$		0			0
$529$	2.36802e11	+	4.10153e11i	0.131473	+	0.227717i
$530$		0			0
$531$	−3.12108e11			−0.170365
$532$		0			0
$533$	−2.13300e11			−0.114477
$534$		0			0
$535$	−3.14233e11	−	5.44268e11i	−0.165829	−	0.287224i
$536$		0			0
$537$	−1.02291e12	+	1.77173e12i	−0.530826	+	0.919418i
$538$		0			0
$539$	4.91094e11	+	1.50217e12i	0.250619	+	0.766602i
$540$		0			0
$541$	7.53297e11	−	1.30475e12i	0.378075	−	0.654846i	−0.612707	−	0.790310i	$-0.709919\pi$
$541$	7.53297e11	−	1.30475e12i	0.378075	−	0.654846i	0.990782	+	0.135465i	$0.0432527\pi$
$542$		0			0
$543$	−1.88068e12	−	3.25744e12i	−0.928360	−	1.60797i
$544$		0			0
$545$	4.16895e12			2.02415
$546$		0			0
$547$	−2.78838e11			−0.133171			−0.0665853	−	0.997781i	$-0.521210\pi$
$547$	−2.78838e11			−0.133171			−0.0665853	+	0.997781i	$0.521210\pi$
$548$		0			0
$549$	1.28503e11	+	2.22574e11i	0.0603725	+	0.104568i
$550$		0			0
$551$	−1.63738e12	+	2.83603e12i	−0.756777	+	1.31078i
$552$		0			0
$553$	−3.19155e11	+	4.40116e11i	−0.145124	+	0.200126i
$554$		0			0
$555$	8.41763e11	−	1.45798e12i	0.376592	−	0.652277i
$556$		0			0
$557$	1.39313e12	+	2.41298e12i	0.613260	+	1.06220i	0.990687	+	0.136158i	$0.0434754\pi$
$557$	1.39313e12	+	2.41298e12i	0.613260	+	1.06220i	−0.377427	+	0.926039i	$0.623191\pi$
$558$		0			0
$559$	5.47011e11			0.236942
$560$		0			0
$561$	1.51901e12			0.647483
$562$		0			0
$563$	−3.34082e11	−	5.78647e11i	−0.140141	−	0.242732i	0.787409	−	0.616432i	$-0.211422\pi$
$563$	−3.34082e11	−	5.78647e11i	−0.140141	−	0.242732i	−0.927550	+	0.373700i	$0.878089\pi$
$564$		0			0
$565$	1.85729e11	−	3.21692e11i	0.0766763	−	0.132807i
$566$		0			0
$567$	−1.23974e12	−	2.77772e12i	−0.503738	−	1.12866i
$568$		0			0
$569$	5.40880e11	−	9.36831e11i	0.216319	−	0.374676i	−0.737360	−	0.675499i	$-0.763928\pi$
$569$	5.40880e11	−	9.36831e11i	0.216319	−	0.374676i	0.953680	+	0.300823i	$0.0972614\pi$
$570$		0			0
$571$	2.45658e12	+	4.25491e12i	0.967092	+	1.67505i	0.703887	+	0.710312i	$0.251446\pi$
$571$	2.45658e12	+	4.25491e12i	0.967092	+	1.67505i	0.263204	+	0.964740i	$0.415221\pi$
$572$		0			0
$573$	−1.96362e12			−0.760960
$574$		0			0
$575$	2.64495e12			1.00905
$576$		0			0
$577$	2.71516e11	+	4.70280e11i	0.101978	+	0.176630i	0.912499	−	0.409078i	$-0.134150\pi$
$577$	2.71516e11	+	4.70280e11i	0.101978	+	0.176630i	−0.810522	+	0.585709i	$0.800816\pi$
$578$		0			0
$579$	−1.63396e12	+	2.83010e12i	−0.604209	+	1.04652i
$580$		0			0
$581$	3.33167e12	+	3.47124e11i	1.21303	+	0.126384i
$582$		0			0
$583$	1.93628e12	−	3.35374e12i	0.694161	−	1.20232i
$584$		0			0
$585$	−1.11300e11	−	1.92778e11i	−0.0392912	−	0.0680543i
$586$		0			0
$587$	3.39997e11			0.118196			0.0590982	−	0.998252i	$-0.481178\pi$
$587$	3.39997e11			0.118196			0.0590982	+	0.998252i	$0.481178\pi$
$588$		0			0
$589$	−4.64960e12			−1.59183
$590$		0			0
$591$	1.89010e12	+	3.27375e12i	0.637295	+	1.10383i
$592$		0			0
$593$	7.79908e11	−	1.35084e12i	0.258999	−	0.448599i	−0.706975	−	0.707238i	$-0.749941\pi$
$593$	7.79908e11	−	1.35084e12i	0.258999	−	0.448599i	0.965974	+	0.258639i	$0.0832741\pi$
$594$		0			0
$595$	3.06314e12	+	3.19145e11i	1.00194	+	0.104391i
$596$		0			0
$597$	−2.27434e12	+	3.93927e12i	−0.732774	+	1.26920i
$598$		0			0
$599$	1.93115e12	+	3.34484e12i	0.612906	+	1.06159i	0.990748	+	0.135715i	$0.0433331\pi$
$599$	1.93115e12	+	3.34484e12i	0.612906	+	1.06159i	−0.377841	+	0.925870i	$0.623334\pi$
$600$		0			0
$601$	−2.20866e12			−0.690547			−0.345274	−	0.938502i	$-0.612214\pi$
$601$	−2.20866e12			−0.690547			−0.345274	+	0.938502i	$0.612214\pi$
$602$		0			0
$603$	−1.15097e12			−0.354516
$604$		0			0
$605$	−8.49365e11	−	1.47114e12i	−0.257748	−	0.446432i
$606$		0			0
$607$	1.57656e11	−	2.73069e11i	0.0471370	−	0.0816437i	−0.841494	−	0.540266i	$-0.818324\pi$
$607$	1.57656e11	−	2.73069e11i	0.0471370	−	0.0816437i	0.888631	+	0.458622i	$0.151657\pi$
$608$		0			0
$609$	2.57095e12	+	5.76041e12i	0.757384	+	1.69698i
$610$		0			0
$611$	−3.42314e11	+	5.92905e11i	−0.0993662	+	0.172107i
$612$		0			0
$613$	−1.74242e12	−	3.01796e12i	−0.498402	−	0.863258i	0.501596	−	0.865102i	$-0.332746\pi$
$613$	−1.74242e12	−	3.01796e12i	−0.498402	−	0.863258i	−0.999998	+	0.00184393i	$0.999413\pi$
$614$		0			0
$615$	5.04223e12			1.42130
$616$		0			0
$617$	6.18646e12			1.71854			0.859269	−	0.511524i	$-0.170919\pi$
$617$	6.18646e12			1.71854			0.859269	+	0.511524i	$0.170919\pi$
$618$		0			0
$619$	−1.97843e12	−	3.42674e12i	−0.541643	−	0.938153i	−0.998810	−	0.0487722i	$-0.984469\pi$
$619$	−1.97843e12	−	3.42674e12i	−0.541643	−	0.938153i	0.457167	−	0.889381i	$-0.348864\pi$
$620$		0			0
$621$	−1.15640e12	+	2.00294e12i	−0.312029	+	0.540450i
$622$		0			0
$623$	−1.99264e12	+	2.74786e12i	−0.529947	+	0.730799i
$624$		0			0
$625$	1.51429e12	−	2.62282e12i	0.396962	−	0.687558i
$626$		0			0
$627$	1.75608e12	+	3.04162e12i	0.453776	+	0.785962i
$628$		0			0
$629$	−1.16493e12			−0.296738
$630$		0			0
$631$	−2.94437e12			−0.739368			−0.369684	−	0.929158i	$-0.620534\pi$
$631$	−2.94437e12			−0.739368			−0.369684	+	0.929158i	$0.620534\pi$
$632$		0			0
$633$	8.29085e10	+	1.43602e11i	0.0205250	+	0.0355503i
$634$		0			0
$635$	−3.79317e12	+	6.56996e12i	−0.925807	+	1.60354i
$636$		0			0
$637$	−5.67782e11	−	1.19612e11i	−0.136633	−	0.0287837i
$638$		0			0
$639$	5.97508e11	−	1.03491e12i	0.141772	−	0.245556i
$640$		0			0
$641$	−1.84567e12	−	3.19679e12i	−0.431810	−	0.747917i	0.565219	−	0.824941i	$-0.308792\pi$
$641$	−1.84567e12	−	3.19679e12i	−0.431810	−	0.747917i	−0.997029	+	0.0770236i	$0.975458\pi$
$642$		0			0
$643$	3.35353e11			0.0773664			0.0386832	−	0.999252i	$-0.487684\pi$
$643$	3.35353e11			0.0773664			0.0386832	+	0.999252i	$0.487684\pi$
$644$		0			0
$645$	−1.29309e13			−2.94178
$646$		0			0
$647$	−3.45282e12	−	5.98046e12i	−0.774649	−	1.34173i	−0.934992	−	0.354670i	$-0.884593\pi$
$647$	−3.45282e12	−	5.98046e12i	−0.774649	−	1.34173i	0.160343	−	0.987061i	$-0.448740\pi$
$648$		0			0
$649$	−8.13751e11	+	1.40946e12i	−0.180049	+	0.311854i
$650$		0			0
$651$	−5.25789e12	+	7.25065e12i	−1.14735	+	1.58221i
$652$		0			0
$653$	−2.56965e11	+	4.45076e11i	−0.0553050	+	0.0957911i	−0.892352	−	0.451339i	$-0.850946\pi$
$653$	−2.56965e11	+	4.45076e11i	−0.0553050	+	0.0957911i	0.837047	+	0.547130i	$0.184280\pi$
$654$		0			0
$655$	−1.79709e12	−	3.11265e12i	−0.381491	−	0.660761i
$656$		0			0
$657$	3.05490e12			0.639666
$658$		0			0
$659$	1.01010e12			0.208631			0.104316	−	0.994544i	$-0.466735\pi$
$659$	1.01010e12			0.208631			0.104316	+	0.994544i	$0.466735\pi$
$660$		0			0
$661$	3.27523e12	+	5.67287e12i	0.667322	+	1.15584i	0.978650	+	0.205534i	$0.0658930\pi$
$661$	3.27523e12	+	5.67287e12i	0.667322	+	1.15584i	−0.311328	+	0.950303i	$0.600774\pi$
$662$		0			0
$663$	−2.78852e11	+	4.82985e11i	−0.0560483	+	0.0970785i
$664$		0			0
$665$	2.90215e12	+	6.50249e12i	0.575470	+	1.28938i
$666$		0			0
$667$	3.46913e12	−	6.00870e12i	0.678662	−	1.17548i
$668$		0			0
$669$	5.64525e11	+	9.77786e11i	0.108960	+	0.188724i
$670$		0			0
$671$	1.34017e12			0.255217
$672$		0			0
$673$	−1.72770e12			−0.324639			−0.162319	−	0.986738i	$-0.551897\pi$
$673$	−1.72770e12			−0.324639			−0.162319	+	0.986738i	$0.551897\pi$
$674$		0			0
$675$	−2.30395e12	−	3.99056e12i	−0.427175	−	0.739890i
$676$		0			0
$677$	−2.64558e12	+	4.58228e12i	−0.484029	+	0.838364i	−0.999832	−	0.0183440i	$-0.994161\pi$
$677$	−2.64558e12	+	4.58228e12i	−0.484029	+	0.838364i	0.515802	+	0.856708i	$0.327494\pi$
$678$		0			0
$679$	−7.98568e12	−	8.32020e11i	−1.44178	−	0.150217i
$680$		0			0
$681$	2.62209e12	−	4.54160e12i	0.467182	−	0.809183i
$682$		0			0
$683$	1.43432e12	+	2.48431e12i	0.252204	+	0.436830i	0.964132	−	0.265422i	$-0.0855113\pi$
$683$	1.43432e12	+	2.48431e12i	0.252204	+	0.436830i	−0.711928	+	0.702252i	$0.752178\pi$
$684$		0			0
$685$	5.87756e12			1.01997
$686$		0			0
$687$	−1.60032e12			−0.274095
$688$		0			0
$689$	7.10903e11	+	1.23132e12i	0.120178	+	0.208154i
$690$		0			0
$691$	−1.13357e12	+	1.96340e12i	−0.189146	+	0.327610i	−0.944966	−	0.327169i	$-0.893905\pi$
$691$	−1.13357e12	+	1.96340e12i	−0.189146	+	0.327610i	0.755820	+	0.654780i	$0.227239\pi$
$692$		0			0
$693$	1.85846e12	+	1.93631e11i	0.306093	+	0.0318916i
$694$		0			0
$695$	1.00158e12	−	1.73479e12i	0.162837	−	0.282043i
$696$		0			0
$697$	−1.74451e12	−	3.02159e12i	−0.279980	−	0.484939i
$698$		0			0
$699$	2.61425e12			0.414191
$700$		0			0
$701$	−8.37724e12			−1.31030			−0.655149	−	0.755500i	$-0.727394\pi$
$701$	−8.37724e12			−1.31030			−0.655149	+	0.755500i	$0.727394\pi$
$702$		0			0
$703$	−1.34674e12	−	2.33263e12i	−0.207963	−	0.360202i
$704$		0			0
$705$	8.09201e12	−	1.40158e13i	1.23369	−	2.13681i
$706$		0			0
$707$	2.46721e12	+	5.52797e12i	0.371380	+	0.832105i
$708$		0			0
$709$	4.96129e12	−	8.59321e12i	0.737372	−	1.27717i	−0.216302	−	0.976326i	$-0.569400\pi$
$709$	4.96129e12	−	8.59321e12i	0.737372	−	1.27717i	0.953675	−	0.300840i	$-0.0972670\pi$
$710$		0			0
$711$	3.21383e11	+	5.56652e11i	0.0471639	+	0.0816903i
$712$		0			0
$713$	9.85112e12			1.42752
$714$		0			0
$715$	−1.16076e12			−0.166099
$716$		0			0
$717$	1.73597e11	+	3.00678e11i	0.0245304	+	0.0424879i
$718$		0			0
$719$	1.93655e12	−	3.35420e12i	0.270239	−	0.468068i	−0.698684	−	0.715430i	$-0.746231\pi$
$719$	1.93655e12	−	3.35420e12i	0.270239	−	0.468068i	0.968923	+	0.247363i	$0.0795639\pi$
$720$		0			0
$721$	−5.98970e12	+	8.25983e12i	−0.825461	+	1.13832i
$722$		0			0
$723$	1.96204e12	−	3.39835e12i	0.267045	−	0.462536i
$724$		0			0
$725$	6.91172e12	+	1.19715e13i	0.929106	+	1.60926i
$726$		0			0
$727$	8.04443e12			1.06805			0.534023	−	0.845470i	$-0.320679\pi$
$727$	8.04443e12			1.06805			0.534023	+	0.845470i	$0.320679\pi$
$728$		0			0
$729$	2.91897e12			0.382785
$730$		0			0
$731$	4.47383e12	+	7.74890e12i	0.579497	+	1.00372i
$732$		0			0
$733$	−1.30235e12	+	2.25573e12i	−0.166632	+	0.288615i	−0.937234	−	0.348702i	$-0.886623\pi$
$733$	−1.30235e12	+	2.25573e12i	−0.166632	+	0.288615i	0.770602	+	0.637317i	$0.219956\pi$
$734$		0			0
$735$	1.34219e13	+	2.82752e12i	1.69637	+	0.357366i
$736$		0			0
$737$	−3.00089e12	+	5.19770e12i	−0.374668	+	0.648944i
$738$		0			0
$739$	−3.37471e12	−	5.84516e12i	−0.416233	−	0.720936i	0.579324	−	0.815097i	$-0.303316\pi$
$739$	−3.37471e12	−	5.84516e12i	−0.416233	−	0.720936i	−0.995557	+	0.0941611i	$0.969983\pi$
$740$		0			0
$741$	−1.28949e12			−0.157121
$742$		0			0
$743$	4.97147e12			0.598459			0.299230	−	0.954181i	$-0.403270\pi$
$743$	4.97147e12			0.598459			0.299230	+	0.954181i	$0.403270\pi$
$744$		0			0
$745$	−9.11635e12	−	1.57900e13i	−1.08422	−	1.87793i
$746$		0			0
$747$	1.98019e12	−	3.42978e12i	0.232682	−	0.403017i
$748$		0			0
$749$	−1.13703e12	+	1.56797e12i	−0.132009	+	0.182041i
$750$		0			0
$751$	1.16982e11	−	2.02618e11i	0.0134196	−	0.0232434i	−0.859238	−	0.511577i	$-0.829062\pi$
$751$	1.16982e11	−	2.02618e11i	0.0134196	−	0.0232434i	0.872657	+	0.488333i	$0.162395\pi$
$752$		0			0
$753$	−7.68115e12	−	1.33041e13i	−0.870661	−	1.50803i
$754$		0			0
$755$	1.00009e13			1.12015
$756$		0			0
$757$	−6.59200e12			−0.729602			−0.364801	−	0.931086i	$-0.618863\pi$
$757$	−6.59200e12			−0.729602			−0.364801	+	0.931086i	$0.618863\pi$
$758$		0			0
$759$	−3.72061e12	−	6.44429e12i	−0.406937	−	0.704835i
$760$		0			0
$761$	5.48492e12	−	9.50015e12i	0.592842	−	1.02683i	−0.401006	−	0.916076i	$-0.631339\pi$
$761$	5.48492e12	−	9.50015e12i	0.592842	−	1.02683i	0.993848	−	0.110757i	$-0.0353275\pi$
$762$		0			0
$763$	−5.23643e12	−	1.17326e13i	−0.559339	−	1.25324i
$764$		0			0
$765$	1.82058e12	−	3.15334e12i	0.192191	−	0.332885i
$766$		0			0
$767$	−2.98767e11	−	5.17480e11i	−0.0311712	−	0.0539901i
$768$		0			0
$769$	−6.05536e12			−0.624412			−0.312206	−	0.950014i	$-0.601068\pi$
$769$	−6.05536e12			−0.624412			−0.312206	+	0.950014i	$0.601068\pi$
$770$		0			0
$771$	−6.12218e12			−0.623966
$772$		0			0
$773$	−3.18511e12	−	5.51677e12i	−0.320861	−	0.555747i	0.659805	−	0.751437i	$-0.270639\pi$
$773$	−3.18511e12	−	5.51677e12i	−0.320861	−	0.555747i	−0.980666	+	0.195689i	$0.937306\pi$
$774$		0			0
$775$	−9.81345e12	+	1.69974e13i	−0.977157	+	1.69248i
$776$		0			0
$777$	−5.16046e12	−	5.37663e11i	−0.507919	−	0.0529195i
$778$		0			0
$779$	4.03355e12	−	6.98632e12i	0.392436	−	0.679720i
$780$		0			0
$781$	−3.11574e12	−	5.39661e12i	−0.299661	−	0.519029i
$782$		0			0
$783$	−1.20875e13			−1.14923
$784$		0			0
$785$	1.20701e13			1.13448
$786$		0			0
$787$	5.09973e12	+	8.83298e12i	0.473871	+	0.820769i	0.999553	−	0.0299123i	$-0.00952280\pi$
$787$	5.09973e12	+	8.83298e12i	0.473871	+	0.820769i	−0.525681	+	0.850682i	$0.676189\pi$
$788$		0			0
$789$	−4.16582e12	+	7.21542e12i	−0.382696	+	0.662850i
$790$		0			0
$791$	−1.13862e12	−	1.18631e11i	−0.103415	−	0.0107747i
$792$		0			0
$793$	−2.46021e11	+	4.26122e11i	−0.0220924	+	0.0382652i
$794$		0			0
$795$	−1.68052e13	−	2.91074e13i	−1.49208	−	2.58435i
$796$		0			0
$797$	8.51564e12			0.747575			0.373788	−	0.927514i	$-0.378059\pi$
$797$	8.51564e12			0.747575			0.373788	+	0.927514i	$0.378059\pi$
$798$		0			0
$799$	−1.11987e13			−0.972092
$800$		0			0
$801$	2.00655e12	+	3.47545e12i	0.172228	+	0.298308i
$802$		0			0
$803$	7.96497e12	−	1.37957e13i	0.676027	−	1.17091i
$804$		0			0
$805$	−6.14879e12	−	1.37768e13i	−0.516070	−	1.15629i
$806$		0			0
$807$	−6.28525e12	+	1.08864e13i	−0.521665	+	0.903551i
$808$		0			0
$809$	9.25476e10	+	1.60297e11i	0.00759621	+	0.0131570i	0.869799	−	0.493407i	$-0.164249\pi$
$809$	9.25476e10	+	1.60297e11i	0.00759621	+	0.0131570i	−0.862202	+	0.506564i	$0.830915\pi$
$810$		0			0
$811$	−2.36011e12			−0.191575			−0.0957874	−	0.995402i	$-0.530537\pi$
$811$	−2.36011e12			−0.191575			−0.0957874	+	0.995402i	$0.530537\pi$
$812$		0			0
$813$	6.56606e12			0.527106
$814$		0			0
$815$	6.69898e12	+	1.16030e13i	0.531862	+	0.921212i
$816$		0			0
$817$	−1.03441e13	+	1.79165e13i	−0.812258	+	1.40687i
$818$		0			0
$819$	−4.02732e11	+	5.55370e11i	−0.0312780	+	0.0431325i
$820$		0			0
$821$	−7.18240e12	+	1.24403e13i	−0.551728	+	0.955621i	0.446422	+	0.894823i	$0.352698\pi$
$821$	−7.18240e12	+	1.24403e13i	−0.551728	+	0.955621i	−0.998150	+	0.0607986i	$0.980635\pi$
$822$		0			0
$823$	−4.79729e12	−	8.30914e12i	−0.364499	−	0.631331i	0.624197	−	0.781267i	$-0.285426\pi$
$823$	−4.79729e12	−	8.30914e12i	−0.364499	−	0.631331i	−0.988696	+	0.149936i	$0.952093\pi$
$824$		0			0
$825$	1.48256e13			1.11421
$826$		0			0
$827$	6.35481e12			0.472419			0.236210	−	0.971702i	$-0.424095\pi$
$827$	6.35481e12			0.472419			0.236210	+	0.971702i	$0.424095\pi$
$828$		0			0
$829$	4.91698e12	+	8.51646e12i	0.361579	+	0.626273i	0.988221	−	0.153034i	$-0.0489045\pi$
$829$	4.91698e12	+	8.51646e12i	0.361579	+	0.626273i	−0.626642	+	0.779307i	$0.715571\pi$
$830$		0			0
$831$	8.84340e12	−	1.53172e13i	0.643301	−	1.11423i
$832$		0			0
$833$	−2.94930e12	−	9.02141e12i	−0.212235	−	0.649190i
$834$		0			0
$835$	−1.65594e13	+	2.86817e13i	−1.17884	+	2.04181i
$836$		0			0
$837$	−8.58108e12	−	1.48629e13i	−0.604335	−	1.04674i
$838$		0			0
$839$	2.48146e12			0.172893			0.0864467	−	0.996256i	$-0.472449\pi$
$839$	2.48146e12			0.172893			0.0864467	+	0.996256i	$0.472449\pi$
$840$		0			0
$841$	2.17547e13			1.49958
$842$		0			0
$843$	−1.25681e13	−	2.17685e13i	−0.857126	−	1.48459i
$844$		0			0
$845$	−1.07161e13	+	1.85608e13i	−0.723073	+	1.25240i
$846$		0			0
$847$	−3.07337e12	+	4.23819e12i	−0.205182	+	0.282947i
$848$		0			0
$849$	6.07740e12	−	1.05264e13i	0.401451	−	0.695334i
$850$		0			0
$851$	2.85335e12	+	4.94214e12i	0.186497	+	0.323022i
$852$		0			0
$853$	2.28133e11			0.0147542			0.00737712	−	0.999973i	$-0.497652\pi$
$853$	2.28133e11			0.0147542			0.00737712	+	0.999973i	$0.497652\pi$
$854$		0			0
$855$	8.41887e12			0.538773
$856$		0			0
$857$	−2.67503e12	−	4.63328e12i	−0.169400	−	0.293410i	0.768809	−	0.639479i	$-0.220850\pi$
$857$	−2.67503e12	−	4.63328e12i	−0.169400	−	0.293410i	−0.938209	+	0.346069i	$0.887516\pi$
$858$		0			0
$859$	1.81361e12	−	3.14127e12i	0.113651	−	0.196850i	−0.803588	−	0.595185i	$-0.797079\pi$
$859$	1.81361e12	−	3.14127e12i	0.113651	−	0.196850i	0.917240	+	0.398335i	$0.130412\pi$
$860$		0			0
$861$	−6.33332e12	−	1.41903e13i	−0.392751	−	0.879988i
$862$		0			0
$863$	1.23004e13	−	2.13049e13i	0.754865	−	1.30746i	−0.190576	−	0.981672i	$-0.561036\pi$
$863$	1.23004e13	−	2.13049e13i	0.754865	−	1.30746i	0.945441	−	0.325792i	$-0.105631\pi$
$864$		0			0
$865$	5.94698e12	+	1.03005e13i	0.361180	+	0.625582i
$866$		0			0
$867$	1.04331e13			0.627088
$868$		0			0
$869$	3.35174e12			0.199380
$870$		0			0
$871$	−1.10177e12	−	1.90833e12i	−0.0648650	−	0.112349i
$872$		0			0
$873$	−4.74630e12	+	8.22084e12i	−0.276561	+	0.479018i
$874$		0			0
$875$	4.45987e12	+	4.64669e11i	0.257209	+	0.0267983i
$876$		0			0
$877$	9.50281e12	−	1.64593e13i	0.542443	−	0.939538i	−0.456321	−	0.889815i	$-0.650833\pi$
$877$	9.50281e12	−	1.64593e13i	0.542443	−	0.939538i	0.998763	−	0.0497226i	$-0.0158337\pi$
$878$		0			0
$879$	7.85188e12	+	1.35999e13i	0.443633	+	0.768395i
$880$		0			0
$881$	2.07924e12			0.116282			0.0581412	−	0.998308i	$-0.481483\pi$
$881$	2.07924e12			0.116282			0.0581412	+	0.998308i	$0.481483\pi$
$882$		0			0
$883$	−1.86533e13			−1.03260			−0.516301	−	0.856407i	$-0.672691\pi$
$883$	−1.86533e13			−1.03260			−0.516301	+	0.856407i	$0.672691\pi$
$884$		0			0
$885$	7.06261e12	+	1.22328e13i	0.387009	+	0.670319i
$886$		0			0
$887$	−4.02512e12	+	6.97171e12i	−0.218335	+	0.378167i	−0.954299	−	0.298854i	$-0.903396\pi$
$887$	−4.02512e12	+	6.97171e12i	−0.218335	+	0.378167i	0.735964	+	0.677020i	$0.236729\pi$
$888$		0			0
$889$	2.32542e13	+	2.42283e12i	1.24866	+	0.130096i
$890$		0			0
$891$	−9.37668e12	+	1.62409e13i	−0.498425	+	0.863297i
$892$		0			0
$893$	−1.29465e13	−	2.24239e13i	−0.681271	−	1.18000i
$894$		0			0
$895$	2.55718e13			1.33217
$896$		0			0
$897$	2.73204e12			0.140903
$898$		0			0
$899$	2.57428e13	+	4.45878e13i	1.31443	+	2.27665i
$900$		0			0
$901$	−1.16285e13	+	2.01412e13i	−0.587845	+	1.01818i
$902$		0			0
$903$	1.62419e13	+	3.63912e13i	0.812909	+	1.82138i
$904$		0			0
$905$	−2.35078e13	+	4.07166e13i	−1.16491	+	2.01768i
$906$		0			0
$907$	−5.01433e12	−	8.68508e12i	−0.246026	−	0.426129i	0.716394	−	0.697696i	$-0.245791\pi$
$907$	−5.01433e12	−	8.68508e12i	−0.246026	−	0.426129i	−0.962419	+	0.271567i	$0.912458\pi$
$908$		0			0
$909$	7.15714e12			0.347698
$910$		0			0
$911$	3.43460e12			0.165213			0.0826064	−	0.996582i	$-0.473676\pi$
$911$	3.43460e12			0.165213			0.0826064	+	0.996582i	$0.473676\pi$
$912$		0			0
$913$	−1.03258e13	−	1.78848e13i	−0.491818	−	0.851853i
$914$		0			0
$915$	5.81574e12	−	1.00732e13i	0.274290	−	0.475085i
$916$		0			0
$917$	−6.50264e12	+	8.96718e12i	−0.303688	+	0.418788i
$918$		0			0
$919$	−6.84732e12	+	1.18599e13i	−0.316666	+	0.548481i	−0.979790	−	0.200028i	$-0.935897\pi$
$919$	−6.84732e12	+	1.18599e13i	−0.316666	+	0.548481i	0.663125	+	0.748509i	$0.269230\pi$
$920$		0			0
$921$	2.71362e12	+	4.70012e12i	0.124274	+	0.215249i
$922$		0			0
$923$	2.28787e12			0.103759
$924$		0			0
$925$	−1.13697e13			−0.510638
$926$		0			0
$927$	6.03152e12	+	1.04469e13i	0.268268	+	0.464653i
$928$		0			0
$929$	−9.37653e12	+	1.62406e13i	−0.413020	+	0.715372i	−0.995218	−	0.0976747i	$-0.968860\pi$
$929$	−9.37653e12	+	1.62406e13i	−0.413020	+	0.715372i	0.582198	+	0.813047i	$0.302193\pi$
$930$		0			0
$931$	1.46546e13	−	1.63350e13i	0.639293	−	0.712598i
$932$		0			0
$933$	−7.43569e11	+	1.28790e12i	−0.0321258	+	0.0556435i
$934$		0			0
$935$	−9.49351e12	−	1.64432e13i	−0.406232	−	0.703615i
$936$		0			0
$937$	−2.39729e13			−1.01600			−0.507998	−	0.861358i	$-0.669614\pi$
$937$	−2.39729e13			−1.01600			−0.507998	+	0.861358i	$0.669614\pi$
$938$		0			0
$939$	4.23909e11			0.0177942
$940$		0			0
$941$	1.29891e13	+	2.24979e13i	0.540042	+	0.935380i	0.998901	+	0.0468708i	$0.0149249\pi$
$941$	1.29891e13	+	2.24979e13i	0.540042	+	0.935380i	−0.458859	+	0.888509i	$0.651742\pi$
$942$		0			0
$943$	−8.54589e12	+	1.48019e13i	−0.351929	+	0.609559i
$944$		0			0
$945$	−1.54297e13	+	2.12777e13i	−0.629383	+	0.867922i
$946$		0			0
$947$	−7.86515e12	+	1.36228e13i	−0.317784	+	0.550418i	−0.980025	−	0.198873i	$-0.936272\pi$
$947$	−7.86515e12	+	1.36228e13i	−0.317784	+	0.550418i	0.662241	+	0.749291i	$0.269605\pi$
$948$		0			0
$949$	2.92432e12	+	5.06508e12i	0.117038	+	0.202716i
$950$		0			0
$951$	−3.41004e13			−1.35191
$952$		0			0
$953$	4.40371e13			1.72942			0.864710	−	0.502272i	$-0.167502\pi$
$953$	4.40371e13			1.72942			0.864710	+	0.502272i	$0.167502\pi$
$954$		0			0
$955$	1.22722e13	+	2.12561e13i	0.477428	+	0.826929i
$956$		0			0
$957$	1.94453e13	−	3.36802e13i	0.749395	−	1.29799i
$958$		0			0
$959$	−7.38253e12	−	1.65411e13i	−0.281852	−	0.631512i
$960$		0			0
$961$	−2.33305e13	+	4.04095e13i	−0.882405	+	1.52837i
$962$		0			0
$963$	1.14497e12	+	1.98315e12i	0.0429018	+	0.0743081i
$964$		0			0
$965$	4.08476e13			1.51633
$966$		0			0
$967$	−4.82779e13			−1.77553			−0.887767	−	0.460293i	$-0.847744\pi$
$967$	−4.82779e13			−1.77553			−0.887767	+	0.460293i	$0.847744\pi$
$968$		0			0
$969$	−1.05463e13	−	1.82667e13i	−0.384276	−	0.665586i
$970$		0			0
$971$	−5.53815e12	+	9.59235e12i	−0.199930	+	0.346289i	−0.948506	−	0.316761i	$-0.897405\pi$
$971$	−5.53815e12	+	9.59235e12i	−0.199930	+	0.346289i	0.748576	+	0.663050i	$0.230738\pi$
$972$		0			0
$973$	−6.14023e12	−	6.39744e11i	−0.219622	−	0.0228822i
$974$		0			0
$975$	−2.72159e12	+	4.71393e12i	−0.0964499	+	0.167056i
$976$		0			0
$977$	5.17338e12	+	8.96055e12i	0.181656	+	0.314637i	0.942444	−	0.334363i	$-0.108521\pi$
$977$	5.17338e12	+	8.96055e12i	0.181656	+	0.314637i	−0.760789	+	0.648999i	$0.775188\pi$
$978$		0			0
$979$	2.09265e13			0.728073
$980$		0			0
$981$	−1.51904e13			−0.523671
$982$		0			0
$983$	−7.99880e12	−	1.38543e13i	−0.273234	−	0.473254i	0.696454	−	0.717601i	$-0.254760\pi$
$983$	−7.99880e12	−	1.38543e13i	−0.273234	−	0.473254i	−0.969688	+	0.244347i	$0.921427\pi$
$984$		0			0
$985$	2.36254e13	−	4.09204e13i	0.799681	−	1.38509i
$986$		0			0
$987$	−4.96084e13	−	5.16865e12i	−1.66390	−	0.173360i
$988$		0			0
$989$	2.19161e13	−	3.79598e13i	0.728416	−	1.26165i
$990$		0			0
$991$	6.80092e12	+	1.17795e13i	0.223994	+	0.387969i	0.956017	−	0.293311i	$-0.0947571\pi$
$991$	6.80092e12	+	1.17795e13i	0.223994	+	0.387969i	−0.732023	+	0.681280i	$0.761424\pi$
$992$		0			0
$993$	−3.07638e13			−1.00408
$994$		0			0
$995$	5.68565e13			1.83898
$996$		0			0
$997$	2.86340e13	+	4.95956e13i	0.917812	+	1.58970i	0.802731	+	0.596342i	$0.203380\pi$
$997$	2.86340e13	+	4.95956e13i	0.917812	+	1.58970i	0.115082	+	0.993356i	$0.463287\pi$
$998$		0			0
$999$	4.97097e12	−	8.60997e12i	0.157905	−	0.273500i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	112.10.i.d.65.5		12
4.3	odd	2		28.10.e.a.9.2	✓	12
7.4	even	3	inner	112.10.i.d.81.5		12
12.11	even	2		252.10.k.e.37.1		12
28.3	even	6		196.10.e.h.165.5		12
28.11	odd	6		28.10.e.a.25.2	yes	12
28.19	even	6		196.10.a.e.1.2		6
28.23	odd	6		196.10.a.f.1.5		6
28.27	even	2		196.10.e.h.177.5		12
84.11	even	6		252.10.k.e.109.1		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
28.10.e.a.9.2	✓	12	4.3	odd	2
28.10.e.a.25.2	yes	12	28.11	odd	6
112.10.i.d.65.5		12	1.1	even	1	trivial
112.10.i.d.81.5		12	7.4	even	3	inner
196.10.a.e.1.2		6	28.19	even	6
196.10.a.f.1.5		6	28.23	odd	6
196.10.e.h.165.5		12	28.3	even	6
196.10.e.h.177.5		12	28.27	even	2
252.10.k.e.37.1		12	12.11	even	2
252.10.k.e.109.1		12	84.11	even	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 112 = 2^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	112.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(82.4523 + 142.812i) q^{3} +(1030.62 - 1785.08i) q^{5} +(-6318.25 - 658.292i) q^{7} +(-3755.26 + 6504.30i) q^{9} +(19582.0 + 33917.0i) q^{11} -14379.0 q^{13} +339908. q^{15} +(-117601. - 203691. i) q^{17} +(271910. - 470962. i) q^{19} +(-426942. - 956597. i) q^{21} +(-576096. + 997828. i) q^{23} +(-1.14779e6 - 1.98803e6i) q^{25} +2.00730e6 q^{27} -6.02178e6 q^{29} +(-4.27494e6 - 7.40442e6i) q^{31} +(-3.22916e6 + 5.59307e6i) q^{33} +(-7.68681e6 + 1.06002e7i) q^{35} +(2.47645e6 - 4.28933e6i) q^{37} +(-1.18558e6 - 2.05349e6i) q^{39} +1.48341e7 q^{41} -3.80424e7 q^{43} +(7.74048e6 + 1.34069e7i) q^{45} +(2.38065e7 - 4.12341e7i) q^{47} +(3.94869e7 + 8.31850e6i) q^{49} +(1.93930e7 - 3.35896e7i) q^{51} +(-4.94404e7 - 8.56333e7i) q^{53} +8.07263e7 q^{55} +8.96785e7 q^{57} +(2.07780e7 + 3.59886e7i) q^{59} +(1.71098e7 - 2.96350e7i) q^{61} +(2.80084e7 - 3.86237e7i) q^{63} +(-1.48193e7 + 2.56677e7i) q^{65} +(7.66238e7 + 1.32716e8i) q^{67} -1.90002e8 q^{69} -1.59112e8 q^{71} +(-2.03375e8 - 3.52256e8i) q^{73} +(1.89275e8 - 3.27835e8i) q^{75} +(-1.01397e8 - 2.27187e8i) q^{77} +(4.27911e7 - 7.41163e7i) q^{79} +(2.39421e8 + 4.14689e8i) q^{81} -5.27310e8 q^{83} -4.84808e8 q^{85} +(-4.96509e8 - 8.59980e8i) q^{87} +(2.67165e8 - 4.62744e8i) q^{89} +(9.08500e7 + 9.46557e6i) q^{91} +(7.04958e8 - 1.22102e9i) q^{93} +(-5.60472e8 - 9.70765e8i) q^{95} +1.26391e9 q^{97} -2.94142e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 966 q^{5} - 7696 q^{7} - 21620 q^{9} + 47640 q^{11} + 103432 q^{13} + 400280 q^{15} - 402234 q^{17} + 519960 q^{19} - 1025858 q^{21} + 683124 q^{23} - 1134656 q^{25} + 7641648 q^{27} + 4252680 q^{29}+ \cdots - 8151941296 q^{99}+O(q^{100}) \) 12 * q - 966 * q^5 - 7696 * q^7 - 21620 * q^9 + 47640 * q^11 + 103432 * q^13 + 400280 * q^15 - 402234 * q^17 + 519960 * q^19 - 1025858 * q^21 + 683124 * q^23 - 1134656 * q^25 + 7641648 * q^27 + 4252680 * q^29 + 1040564 * q^31 + 5260178 * q^33 + 552600 * q^35 - 4886646 * q^37 - 12290272 * q^39 + 34183128 * q^41 - 55245072 * q^43 - 17040100 * q^45 + 26858748 * q^47 + 12089772 * q^49 - 57449552 * q^51 - 21717558 * q^53 - 54952184 * q^55 - 72948300 * q^57 - 31298904 * q^59 - 155592542 * q^61 + 400205088 * q^63 + 7729020 * q^65 - 212196992 * q^67 - 249902212 * q^69 - 648519648 * q^71 - 549596530 * q^73 + 647801672 * q^75 - 519615786 * q^77 - 227611204 * q^79 + 363916186 * q^81 - 989338224 * q^83 + 2712621236 * q^85 + 860715296 * q^87 - 1746541986 * q^89 + 2592411872 * q^91 + 1354546774 * q^93 - 1586101596 * q^95 + 2306715320 * q^97 - 8151941296 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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