LMFDB - Embedded newform 112.10.i.c.81.2

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)

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);

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")

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 = [10,0,-161] 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:	$10$
Relative dimension:	$5$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - x^{9} + 430 x^{8} + 61 x^{7} + 146753 x^{6} + 23608 x^{5} + 16136944 x^{4} + 30575648 x^{3} + \cdots + 761760000 $ x^10 - x^9 + 430x^8 + 61x^7 + 146753x^6 + 23608x^5 + 16136944x^4 + 30575648x^3 + 1399072384x^2 + 1034227200x + 761760000
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{18}\cdot 3^{3}\cdot 7^{4} $
Twist minimal:	no (minimal twist has level 7)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			81.2
Root			$5.89912 + 10.2176i$ of defining polynomial
Character	$\chi$	$=$	112.81
Dual form			112.10.i.c.65.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-104.977 + 181.826i) q^{3} +(983.791 + 1703.98i) q^{5} +(5768.52 + 2660.41i) q^{7} +(-12198.9 - 21129.2i) q^{9} +(15898.9 - 27537.7i) q^{11} -100188. q^{13} -413103. q^{15} +(-169129. + 292940. i) q^{17} +(130759. + 226480. i) q^{19} +(-1.08930e6 + 769584. i) q^{21} +(-272337. - 471702. i) q^{23} +(-959125. + 1.66125e6i) q^{25} +989914. q^{27} +2.32128e6 q^{29} +(-2.53535e6 + 4.39136e6i) q^{31} +(3.33804e6 + 5.78166e6i) q^{33} +(1.14174e6 + 1.24467e7i) q^{35} +(-2.93235e6 - 5.07897e6i) q^{37} +(1.05174e7 - 1.82167e7i) q^{39} -2.81312e7 q^{41} -3.88522e7 q^{43} +(2.40024e7 - 4.15734e7i) q^{45} +(1.26741e7 + 2.19521e7i) q^{47} +(2.61980e7 + 3.06933e7i) q^{49} +(-3.55094e7 - 6.15041e7i) q^{51} +(2.26489e7 - 3.92291e7i) q^{53} +6.25647e7 q^{55} -5.49067e7 q^{57} +(9.44557e6 - 1.63602e7i) q^{59} +(-5.08255e7 - 8.80323e7i) q^{61} +(-1.41575e7 - 1.54338e8i) q^{63} +(-9.85638e7 - 1.70718e8i) q^{65} +(6.55496e7 - 1.13535e8i) q^{67} +1.14357e8 q^{69} -9.51959e7 q^{71} +(-1.19056e8 + 2.06211e8i) q^{73} +(-2.01373e8 - 3.48788e8i) q^{75} +(1.64975e8 - 1.16554e8i) q^{77} +(9.42601e7 + 1.63263e8i) q^{79} +(1.36193e8 - 2.35894e8i) q^{81} +1.76014e8 q^{83} -6.65551e8 q^{85} +(-2.43682e8 + 4.22070e8i) q^{87} +(2.41517e8 + 4.18319e8i) q^{89} +(-5.77936e8 - 2.66541e8i) q^{91} +(-5.32309e8 - 9.21986e8i) q^{93} +(-2.57278e8 + 4.45619e8i) q^{95} -2.28821e7 q^{97} -7.75799e8 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 161 q^{3} + 1533 q^{5} + 1036 q^{7} - 35734 q^{9} - 42213 q^{11} - 319676 q^{13} - 151394 q^{15} + 324681 q^{17} + 16121 q^{19} - 1557857 q^{21} - 2638863 q^{23} - 1304092 q^{25} + 18331558 q^{27}+ \cdots + 1900777180 q^{99}+O(q^{100}) $ 10 * q - 161 * q^3 + 1533 * q^5 + 1036 * q^7 - 35734 * q^9 - 42213 * q^11 - 319676 * q^13 - 151394 * q^15 + 324681 * q^17 + 16121 * q^19 - 1557857 * q^21 - 2638863 * q^23 - 1304092 * q^25 + 18331558 * q^27 + 15292500 * q^29 - 19179237 * q^31 + 1689359 * q^33 + 43746759 * q^35 + 39566985 * q^37 + 44299486 * q^39 - 53436852 * q^41 - 101835992 * q^43 + 85098230 * q^45 - 32509659 * q^47 - 49024598 * q^49 - 44168403 * q^51 - 25714707 * q^53 + 144695222 * q^55 - 121710346 * q^57 - 46776513 * q^59 - 113075039 * q^61 - 318071530 * q^63 - 338113566 * q^65 + 126707879 * q^67 + 1323616182 * q^69 + 1188736032 * q^71 - 859257651 * q^73 + 169061732 * q^75 + 1911891891 * q^77 + 527065417 * q^79 + 551662715 * q^81 + 144863208 * q^83 - 1197360222 * q^85 + 340781350 * q^87 + 1661554797 * q^89 - 726641384 * q^91 - 423057489 * q^93 + 1197123495 * q^95 + 869770188 * q^97 + 1900777180 * q^99

Character values

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

$n$	$15$	$17$	$85$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$	$1$

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
$3$	−104.977	+	181.826i	−0.748255	+	1.29602i	0.200404	+	0.979713i	$0.435775\pi$
$3$	−104.977	+	181.826i	−0.748255	+	1.29602i	−0.948659	+	0.316302i	$0.897559\pi$
$4$		0			0
$5$	983.791	+	1703.98i	0.703943	+	1.21927i	0.967072	+	0.254505i	$0.0819124\pi$
$5$	983.791	+	1703.98i	0.703943	+	1.21927i	−0.263128	+	0.964761i	$0.584754\pi$
$6$		0			0
$7$	5768.52	+	2660.41i	0.908078	+	0.418801i
$7$	5768.52	+	2660.41i	0.908078	+	0.418801i
$8$		0			0
$9$	−12198.9	−	21129.2i	−0.619771	−	1.07347i
$10$		0			0
$11$	15898.9	−	27537.7i	0.327416	−	0.567101i	−0.654583	−	0.755991i	$-0.727156\pi$
$11$	15898.9	−	27537.7i	0.327416	−	0.567101i	0.981998	+	0.188890i	$0.0604889\pi$
$12$		0			0
$13$	−100188.			−0.972904			−0.486452	−	0.873707i	$-0.661709\pi$
$13$	−100188.			−0.972904			−0.486452	+	0.873707i	$0.661709\pi$
$14$		0			0
$15$	−413103.			−2.10692
$16$		0			0
$17$	−169129.	+	292940.i	−0.491132	+	0.850666i	−0.999948	−	0.0102097i	$-0.996750\pi$
$17$	−169129.	+	292940.i	−0.491132	+	0.850666i	0.508816	+	0.860875i	$0.330083\pi$
$18$		0			0
$19$	130759.	+	226480.i	0.230186	+	0.398694i	0.957863	−	0.287226i	$-0.0927332\pi$
$19$	130759.	+	226480.i	0.230186	+	0.398694i	−0.727677	+	0.685920i	$0.759400\pi$
$20$		0			0
$21$	−1.08930e6	+	769584.i	−1.22225	+	0.863514i
$22$		0			0
$23$	−272337.	−	471702.i	−0.202923	−	0.351474i	0.746546	−	0.665334i	$-0.231711\pi$
$23$	−272337.	−	471702.i	−0.202923	−	0.351474i	−0.949469	+	0.313860i	$0.898378\pi$
$24$		0			0
$25$	−959125.	+	1.66125e6i	−0.491072	+	0.850562i
$26$		0			0
$27$	989914.			0.358476
$28$		0			0
$29$	2.32128e6			0.609449			0.304724	−	0.952441i	$-0.401436\pi$
$29$	2.32128e6			0.609449			0.304724	+	0.952441i	$0.401436\pi$
$30$		0			0
$31$	−2.53535e6	+	4.39136e6i	−0.493073	+	0.854027i	−0.999968	−	0.00798051i	$-0.997460\pi$
$31$	−2.53535e6	+	4.39136e6i	−0.493073	+	0.854027i	0.506895	+	0.862008i	$0.330793\pi$
$32$		0			0
$33$	3.33804e6	+	5.78166e6i	0.489981	+	0.848672i
$34$		0			0
$35$	1.14174e6	+	1.24467e7i	0.128606	+	1.40200i
$36$		0			0
$37$	−2.93235e6	−	5.07897e6i	−0.257222	−	0.445521i	0.708275	−	0.705937i	$-0.249474\pi$
$37$	−2.93235e6	−	5.07897e6i	−0.257222	−	0.445521i	−0.965497	+	0.260416i	$0.916140\pi$
$38$		0			0
$39$	1.05174e7	−	1.82167e7i	0.727980	−	1.26090i
$40$		0			0
$41$	−2.81312e7			−1.55475			−0.777376	−	0.629036i	$-0.783450\pi$
$41$	−2.81312e7			−1.55475			−0.777376	+	0.629036i	$0.783450\pi$
$42$		0			0
$43$	−3.88522e7			−1.73304			−0.866518	−	0.499147i	$-0.833647\pi$
$43$	−3.88522e7			−1.73304			−0.866518	+	0.499147i	$0.833647\pi$
$44$		0			0
$45$	2.40024e7	−	4.15734e7i	0.872567	−	1.51133i
$46$		0			0
$47$	1.26741e7	+	2.19521e7i	0.378857	+	0.656199i	0.990896	−	0.134628i	$-0.0429839\pi$
$47$	1.26741e7	+	2.19521e7i	0.378857	+	0.656199i	−0.612039	+	0.790827i	$0.709651\pi$
$48$		0			0
$49$	2.61980e7	+	3.06933e7i	0.649212	+	0.760608i
$50$		0			0
$51$	−3.55094e7	−	6.15041e7i	−0.734984	−	1.27303i
$52$		0			0
$53$	2.26489e7	−	3.92291e7i	0.394281	−	0.682915i	−0.598728	−	0.800953i	$-0.704327\pi$
$53$	2.26489e7	−	3.92291e7i	0.394281	−	0.682915i	0.993009	+	0.118037i	$0.0376602\pi$
$54$		0			0
$55$	6.25647e7			0.921928
$56$		0			0
$57$	−5.49067e7			−0.688951
$58$		0			0
$59$	9.44557e6	−	1.63602e7i	0.101483	−	0.175774i	−0.810813	−	0.585306i	$-0.800975\pi$
$59$	9.44557e6	−	1.63602e7i	0.101483	−	0.175774i	0.912296	+	0.409532i	$0.134308\pi$
$60$		0			0
$61$	−5.08255e7	−	8.80323e7i	−0.469999	−	0.814062i	0.529412	−	0.848365i	$-0.322412\pi$
$61$	−5.08255e7	−	8.80323e7i	−0.469999	−	0.814062i	−0.999412	+	0.0343023i	$0.989079\pi$
$62$		0			0
$63$	−1.41575e7	−	1.54338e8i	−0.113228	−	1.23436i
$64$		0			0
$65$	−9.85638e7	−	1.70718e8i	−0.684869	−	1.18623i
$66$		0			0
$67$	6.55496e7	−	1.13535e8i	0.397405	−	0.688326i	−0.596000	−	0.802985i	$-0.703244\pi$
$67$	6.55496e7	−	1.13535e8i	0.397405	−	0.688326i	0.993405	+	0.114659i	$0.0365774\pi$
$68$		0			0
$69$	1.14357e8			0.607354
$70$		0			0
$71$	−9.51959e7			−0.444586			−0.222293	−	0.974980i	$-0.571354\pi$
$71$	−9.51959e7			−0.444586			−0.222293	+	0.974980i	$0.571354\pi$
$72$		0			0
$73$	−1.19056e8	+	2.06211e8i	−0.490679	+	0.849881i	−0.999942	−	0.0107300i	$-0.996584\pi$
$73$	−1.19056e8	+	2.06211e8i	−0.490679	+	0.849881i	0.509264	+	0.860611i	$0.329918\pi$
$74$		0			0
$75$	−2.01373e8	−	3.48788e8i	−0.734894	−	1.27287i
$76$		0			0
$77$	1.64975e8	−	1.16554e8i	0.534821	−	0.377850i
$78$		0			0
$79$	9.42601e7	+	1.63263e8i	0.272274	+	0.471592i	0.969444	−	0.245314i	$-0.0788910\pi$
$79$	9.42601e7	+	1.63263e8i	0.272274	+	0.471592i	−0.697170	+	0.716906i	$0.745558\pi$
$80$		0			0
$81$	1.36193e8	−	2.35894e8i	0.351539	−	0.608884i
$82$		0			0
$83$	1.76014e8			0.407095			0.203547	−	0.979065i	$-0.434753\pi$
$83$	1.76014e8			0.407095			0.203547	+	0.979065i	$0.434753\pi$
$84$		0			0
$85$	−6.65551e8			−1.38292
$86$		0			0
$87$	−2.43682e8	+	4.22070e8i	−0.456023	+	0.789855i
$88$		0			0
$89$	2.41517e8	+	4.18319e8i	0.408030	+	0.706729i	0.994669	−	0.103120i	$-0.0328824\pi$
$89$	2.41517e8	+	4.18319e8i	0.408030	+	0.706729i	−0.586639	+	0.809849i	$0.699549\pi$
$90$		0			0
$91$	−5.77936e8	−	2.66541e8i	−0.883473	−	0.407453i
$92$		0			0
$93$	−5.32309e8	−	9.21986e8i	−0.737888	−	1.27806i
$94$		0			0
$95$	−2.57278e8	+	4.45619e8i	−0.324076	+	0.561316i
$96$		0			0
$97$	−2.28821e7			−0.0262436			−0.0131218	−	0.999914i	$-0.504177\pi$
$97$	−2.28821e7			−0.0262436			−0.0131218	+	0.999914i	$0.504177\pi$
$98$		0			0
$99$	−7.75799e8			−0.811691
$100$		0			0
$101$	−1.03216e8	+	1.78776e8i	−0.0986966	+	0.170948i	−0.911145	−	0.412085i	$-0.864801\pi$
$101$	−1.03216e8	+	1.78776e8i	−0.0986966	+	0.170948i	0.812449	+	0.583033i	$0.198134\pi$
$102$		0			0
$103$	−1.47590e8	−	2.55634e8i	−0.129208	−	0.223795i	0.794162	−	0.607706i	$-0.207910\pi$
$103$	−1.47590e8	−	2.55634e8i	−0.129208	−	0.223795i	−0.923370	+	0.383911i	$0.874577\pi$
$104$		0			0
$105$	−2.38299e9	−	1.09902e9i	−1.91324	−	0.882378i
$106$		0			0
$107$	−1.22501e9	−	2.12179e9i	−0.903471	−	1.56486i	−0.822956	−	0.568105i	$-0.807677\pi$
$107$	−1.22501e9	−	2.12179e9i	−0.903471	−	1.56486i	−0.0805150	−	0.996753i	$-0.525656\pi$
$108$		0			0
$109$	1.02280e9	−	1.77153e9i	0.694016	−	1.20207i	−0.276495	−	0.961015i	$-0.589173\pi$
$109$	1.02280e9	−	1.77153e9i	0.694016	−	1.20207i	0.970511	−	0.241056i	$-0.0774937\pi$
$110$		0			0
$111$	1.23132e9			0.769869
$112$		0			0
$113$	1.59772e8			0.0921822			0.0460911	−	0.998937i	$-0.485324\pi$
$113$	1.59772e8			0.0921822			0.0460911	+	0.998937i	$0.485324\pi$
$114$		0			0
$115$	5.35846e8	−	9.28113e8i	0.285693	−	0.494835i
$116$		0			0
$117$	1.22219e9	+	2.11689e9i	0.602977	+	1.04439i
$118$		0			0
$119$	−1.75497e9	+	1.23988e9i	−0.802246	+	0.566784i
$120$		0			0
$121$	6.73425e8	+	1.16641e9i	0.285598	+	0.494670i
$122$		0			0
$123$	2.95314e9	−	5.11499e9i	1.16335	−	2.01498i
$124$		0			0
$125$	6.86192e7			0.0251392
$126$		0			0
$127$	5.15074e9			1.75692			0.878462	−	0.477813i	$-0.158570\pi$
$127$	5.15074e9			1.75692			0.878462	+	0.477813i	$0.158570\pi$
$128$		0			0
$129$	4.07860e9	−	7.06433e9i	1.29675	−	2.24604i
$130$		0			0
$131$	5.08431e8	+	8.80628e8i	0.150838	+	0.261259i	0.931536	−	0.363650i	$-0.118469\pi$
$131$	5.08431e8	+	8.80628e8i	0.150838	+	0.261259i	−0.780698	+	0.624909i	$0.785136\pi$
$132$		0			0
$133$	1.51752e8	+	1.65433e9i	0.0420536	+	0.458447i
$134$		0			0
$135$	9.73868e8	+	1.68679e9i	0.252347	+	0.437078i
$136$		0			0
$137$	−1.60440e9	+	2.77891e9i	−0.389108	+	0.673956i	−0.992330	−	0.123618i	$-0.960550\pi$
$137$	−1.60440e9	+	2.77891e9i	−0.389108	+	0.673956i	0.603221	+	0.797574i	$0.293884\pi$
$138$		0			0
$139$	−6.36396e8			−0.144598			−0.0722988	−	0.997383i	$-0.523034\pi$
$139$	−6.36396e8			−0.144598			−0.0722988	+	0.997383i	$0.523034\pi$
$140$		0			0
$141$	−5.32195e9			−1.13393
$142$		0			0
$143$	−1.59287e9	+	2.75894e9i	−0.318544	+	0.551734i
$144$		0			0
$145$	2.28366e9	+	3.95541e9i	0.429017	+	0.743080i
$146$		0			0
$147$	−8.33103e9	+	1.54139e9i	−1.47154	+	0.272260i
$148$		0			0
$149$	−2.50376e9	−	4.33664e9i	−0.416154	−	0.720800i	0.579395	−	0.815047i	$-0.303289\pi$
$149$	−2.50376e9	−	4.33664e9i	−0.416154	−	0.720800i	−0.995549	+	0.0942469i	$0.969956\pi$
$150$		0			0
$151$	−1.27205e9	+	2.20326e9i	−0.199118	+	0.344882i	−0.948243	−	0.317547i	$-0.897141\pi$
$151$	−1.27205e9	+	2.20326e9i	−0.199118	+	0.344882i	0.749125	+	0.662429i	$0.230474\pi$
$152$		0			0
$153$	8.25279e9			1.21756
$154$		0			0
$155$	−9.97703e9			−1.38838
$156$		0			0
$157$	−5.71100e9	+	9.89174e9i	−0.750177	+	1.29934i	0.197560	+	0.980291i	$0.436698\pi$
$157$	−5.71100e9	+	9.89174e9i	−0.750177	+	1.29934i	−0.947737	+	0.319053i	$0.896635\pi$
$158$		0			0
$159$	4.75524e9	+	8.23632e9i	0.590046	+	1.02199i
$160$		0			0
$161$	−3.16062e8	−	3.44555e9i	−0.0370729	−	0.404150i
$162$		0			0
$163$	5.67212e9	+	9.82441e9i	0.629364	+	1.09009i	0.987680	+	0.156489i	$0.0500177\pi$
$163$	5.67212e9	+	9.82441e9i	0.629364	+	1.09009i	−0.358316	+	0.933600i	$0.616649\pi$
$164$		0			0
$165$	−6.56787e9	+	1.13759e10i	−0.689837	+	1.19483i
$166$		0			0
$167$	1.04555e10			1.04021			0.520103	−	0.854104i	$-0.325894\pi$
$167$	1.04555e10			1.04021			0.520103	+	0.854104i	$0.325894\pi$
$168$		0			0
$169$	−5.66897e8			−0.0534582
$170$		0			0
$171$	3.19023e9	−	5.52565e9i	0.285325	−	0.494198i
$172$		0			0
$173$	5.63685e9	+	9.76331e9i	0.478442	+	0.828685i	0.999694	−	0.0247172i	$-0.00786852\pi$
$173$	5.63685e9	+	9.76331e9i	0.478442	+	0.828685i	−0.521253	+	0.853402i	$0.674535\pi$
$174$		0			0
$175$	−9.95235e9	+	7.03131e9i	−0.802148	+	0.566715i
$176$		0			0
$177$	1.98314e9	+	3.43490e9i	0.151871	+	0.263048i
$178$		0			0
$179$	−6.78310e9	+	1.17487e10i	−0.493844	+	0.855363i	−0.999975	−	0.00709392i	$-0.997742\pi$
$179$	−6.78310e9	+	1.17487e10i	−0.493844	+	0.855363i	0.506131	+	0.862457i	$0.331075\pi$
$180$		0			0
$181$	2.03119e10			1.40668			0.703342	−	0.710851i	$-0.251690\pi$
$181$	2.03119e10			1.40668			0.703342	+	0.710851i	$0.251690\pi$
$182$		0			0
$183$	2.13421e10			1.40672
$184$		0			0
$185$	5.76963e9	−	9.99329e9i	0.362139	−	0.627243i
$186$		0			0
$187$	5.37793e9	+	9.31485e9i	0.321609	+	0.557043i
$188$		0			0
$189$	5.71034e9	+	2.63358e9i	0.325525	+	0.150130i
$190$		0			0
$191$	−1.71927e10	−	2.97787e10i	−0.934748	−	1.61903i	−0.775083	−	0.631859i	$-0.782292\pi$
$191$	−1.71927e10	−	2.97787e10i	−0.934748	−	1.61903i	−0.159664	−	0.987171i	$-0.551041\pi$
$192$		0			0
$193$	−1.36102e10	+	2.35735e10i	−0.706084	+	1.22297i	0.260215	+	0.965551i	$0.416207\pi$
$193$	−1.36102e10	+	2.35735e10i	−0.706084	+	1.22297i	−0.966299	+	0.257422i	$0.917127\pi$
$194$		0			0
$195$	4.13878e10			2.04983
$196$		0			0
$197$	−1.52311e9			−0.0720499			−0.0360249	−	0.999351i	$-0.511470\pi$
$197$	−1.52311e9			−0.0720499			−0.0360249	+	0.999351i	$0.511470\pi$
$198$		0			0
$199$	−6.15674e9	+	1.06638e10i	−0.278299	+	0.482028i	−0.970962	−	0.239233i	$-0.923104\pi$
$199$	−6.15674e9	+	1.06638e10i	−0.278299	+	0.482028i	0.692663	+	0.721261i	$0.256437\pi$
$200$		0			0
$201$	1.37624e10	+	2.38372e10i	0.594721	+	1.03009i
$202$		0			0
$203$	1.33904e10	+	6.17557e9i	0.553427	+	0.255238i
$204$		0			0
$205$	−2.76752e10	−	4.79349e10i	−1.09446	−	1.89566i
$206$		0			0
$207$	−6.64446e9	+	1.15085e10i	−0.251532	+	0.435666i
$208$		0			0
$209$	8.31566e9			0.301466
$210$		0			0
$211$	2.60114e10			0.903426			0.451713	−	0.892163i	$-0.350813\pi$
$211$	2.60114e10			0.903426			0.451713	+	0.892163i	$0.350813\pi$
$212$		0			0
$213$	9.99341e9	−	1.73091e10i	0.332664	−	0.576190i
$214$		0			0
$215$	−3.82224e10	−	6.62031e10i	−1.21996	−	2.11303i
$216$		0			0
$217$	−2.63081e10	+	1.85866e10i	−0.805416	+	0.569024i
$218$		0			0
$219$	−2.49963e10	−	4.32948e10i	−0.734306	−	1.27185i
$220$		0			0
$221$	1.69447e10	−	2.93491e10i	0.477824	−	0.827616i
$222$		0			0
$223$	−1.83087e10			−0.495777			−0.247889	−	0.968789i	$-0.579737\pi$
$223$	−1.83087e10			−0.495777			−0.247889	+	0.968789i	$0.579737\pi$
$224$		0			0
$225$	4.68013e10			1.21741
$226$		0			0
$227$	−1.19844e10	+	2.07575e10i	−0.299570	+	0.518871i	−0.976038	−	0.217602i	$-0.930177\pi$
$227$	−1.19844e10	+	2.07575e10i	−0.299570	+	0.518871i	0.676467	+	0.736473i	$0.263510\pi$
$228$		0			0
$229$	−1.50424e10	−	2.60541e10i	−0.361457	−	0.626061i	0.626744	−	0.779225i	$-0.284387\pi$
$229$	−1.50424e10	−	2.60541e10i	−0.361457	−	0.626061i	−0.988201	+	0.153164i	$0.951054\pi$
$230$		0			0
$231$	3.87398e9	+	4.22322e10i	0.0895166	+	0.975865i
$232$		0			0
$233$	−6.75130e9	−	1.16936e10i	−0.150067	−	0.259924i	0.781185	−	0.624300i	$-0.214616\pi$
$233$	−6.75130e9	−	1.16936e10i	−0.150067	−	0.259924i	−0.931252	+	0.364376i	$0.881282\pi$
$234$		0			0
$235$	−2.49372e10	+	4.31926e10i	−0.533388	+	0.923854i
$236$		0			0
$237$	−3.95806e10			−0.814921
$238$		0			0
$239$	3.75005e9			0.0743440			0.0371720	−	0.999309i	$-0.488165\pi$
$239$	3.75005e9			0.0743440			0.0371720	+	0.999309i	$0.488165\pi$
$240$		0			0
$241$	1.34568e10	−	2.33078e10i	0.256960	−	0.445067i	−0.708466	−	0.705745i	$-0.750613\pi$
$241$	1.34568e10	−	2.33078e10i	0.256960	−	0.445067i	0.965426	+	0.260678i	$0.0839459\pi$
$242$		0			0
$243$	3.83367e10	+	6.64011e10i	0.705320	+	1.22165i
$244$		0			0
$245$	−2.65272e10	+	7.48365e10i	−0.470374	+	1.32699i
$246$		0			0
$247$	−1.31004e10	−	2.26906e10i	−0.223949	−	0.387891i
$248$		0			0
$249$	−1.84774e10	+	3.20039e10i	−0.304610	+	0.527601i
$250$		0			0
$251$	−1.56458e10			−0.248810			−0.124405	−	0.992232i	$-0.539702\pi$
$251$	−1.56458e10			−0.248810			−0.124405	+	0.992232i	$0.539702\pi$
$252$		0			0
$253$	−1.73194e10			−0.265761
$254$		0			0
$255$	6.98677e10	−	1.21014e11i	1.03477	−	1.79228i
$256$		0			0
$257$	−3.63435e10	−	6.29487e10i	−0.519670	−	0.900094i	−0.999739	−	0.0228635i	$-0.992722\pi$
$257$	−3.63435e10	−	6.29487e10i	−0.519670	−	0.900094i	0.480069	−	0.877231i	$-0.340612\pi$
$258$		0			0
$259$	−3.40315e9	−	3.70994e10i	−0.0469928	−	0.512292i
$260$		0			0
$261$	−2.83172e10	−	4.90469e10i	−0.377719	−	0.654228i
$262$		0			0
$263$	−4.11041e10	+	7.11943e10i	−0.529766	+	0.917581i	0.469631	+	0.882863i	$0.344387\pi$
$263$	−4.11041e10	+	7.11943e10i	−0.529766	+	0.917581i	−0.999397	+	0.0347186i	$0.988946\pi$
$264$		0			0
$265$	8.91272e10			1.11021
$266$		0			0
$267$	−1.01415e11			−1.22124
$268$		0			0
$269$	−1.64537e10	+	2.84986e10i	−0.191592	+	0.331847i	−0.945778	−	0.324814i	$-0.894698\pi$
$269$	−1.64537e10	+	2.84986e10i	−0.191592	+	0.331847i	0.754186	+	0.656661i	$0.228032\pi$
$270$		0			0
$271$	−6.71241e10	−	1.16262e11i	−0.755991	−	1.30941i	−0.944880	−	0.327416i	$-0.893822\pi$
$271$	−6.71241e10	−	1.16262e11i	−0.755991	−	1.30941i	0.188889	−	0.981998i	$-0.439511\pi$
$272$		0			0
$273$	1.09134e11	−	7.71030e10i	1.18913	−	0.840116i
$274$		0			0
$275$	3.04980e10	+	5.28241e10i	0.321569	+	0.556975i
$276$		0			0
$277$	−9.65495e10	+	1.67229e11i	−0.985351	+	1.70668i	−0.344986	+	0.938608i	$0.612116\pi$
$277$	−9.65495e10	+	1.67229e11i	−0.985351	+	1.70668i	−0.640365	+	0.768071i	$0.721217\pi$
$278$		0			0
$279$	1.23715e11			1.22237
$280$		0			0
$281$	1.81503e10			0.173663			0.0868313	−	0.996223i	$-0.472326\pi$
$281$	1.81503e10			0.173663			0.0868313	+	0.996223i	$0.472326\pi$
$282$		0			0
$283$	6.52307e10	−	1.12983e11i	0.604523	−	1.04707i	−0.387603	−	0.921826i	$-0.626697\pi$
$283$	6.52307e10	−	1.12983e11i	0.604523	−	1.04707i	0.992127	−	0.125239i	$-0.0399697\pi$
$284$		0			0
$285$	−5.40167e10	−	9.35596e10i	−0.484982	−	0.840014i
$286$		0			0
$287$	−1.62276e11	−	7.48406e10i	−1.41184	−	0.651132i
$288$		0			0
$289$	2.08461e9	+	3.61064e9i	0.0175786	+	0.0304470i
$290$		0			0
$291$	2.40210e9	−	4.16056e9i	0.0196369	−	0.0340121i
$292$		0			0
$293$	−1.61941e11			−1.28367			−0.641834	−	0.766843i	$-0.721826\pi$
$293$	−1.61941e11			−1.28367			−0.641834	+	0.766843i	$0.721826\pi$
$294$		0			0
$295$	3.71699e10			0.285754
$296$		0			0
$297$	1.57385e10	−	2.72599e10i	0.117371	−	0.203292i
$298$		0			0
$299$	2.72849e10	+	4.72588e10i	0.197425	+	0.341950i
$300$		0			0
$301$	−2.24120e11	−	1.03363e11i	−1.57373	−	0.725796i
$302$		0			0
$303$	−2.16707e10	−	3.75348e10i	−0.147700	−	0.255825i
$304$		0			0
$305$	1.00003e11	−	1.73211e11i	0.661705	−	1.14611i
$306$		0			0
$307$	−2.13521e11			−1.37188			−0.685942	−	0.727656i	$-0.740610\pi$
$307$	−2.13521e11			−1.37188			−0.685942	+	0.727656i	$0.740610\pi$
$308$		0			0
$309$	6.19745e10			0.386723
$310$		0			0
$311$	−6.44473e10	+	1.11626e11i	−0.390646	+	0.676618i	−0.992535	−	0.121961i	$-0.961082\pi$
$311$	−6.44473e10	+	1.11626e11i	−0.390646	+	0.676618i	0.601889	+	0.798580i	$0.294415\pi$
$312$		0			0
$313$	−8.90227e10	−	1.54192e11i	−0.524265	−	0.908054i	−0.999601	−	0.0282496i	$-0.991007\pi$
$313$	−8.90227e10	−	1.54192e11i	−0.524265	−	0.908054i	0.475336	−	0.879805i	$-0.342327\pi$
$314$		0			0
$315$	2.49061e11	−	1.75961e11i	1.42531	−	1.00697i
$316$		0			0
$317$	7.34098e10	+	1.27149e11i	0.408307	+	0.707209i	0.994700	−	0.102818i	$-0.0327858\pi$
$317$	7.34098e10	+	1.27149e11i	0.408307	+	0.707209i	−0.586393	+	0.810027i	$0.699452\pi$
$318$		0			0
$319$	3.69058e10	−	6.39227e10i	0.199543	−	0.345619i
$320$		0			0
$321$	5.14395e11			2.70411
$322$		0			0
$323$	−8.84603e10			−0.452207
$324$		0			0
$325$	9.60927e10	−	1.66437e11i	0.477766	−	0.827515i
$326$		0			0
$327$	2.14740e11	+	3.71941e11i	1.03860	+	1.79891i
$328$		0			0
$329$	1.47089e10	+	1.60349e11i	0.0692149	+	0.754546i
$330$		0			0
$331$	−8.55668e10	−	1.48206e11i	−0.391813	−	0.678641i	0.600875	−	0.799343i	$-0.294819\pi$
$331$	−8.55668e10	−	1.48206e11i	−0.391813	−	0.678641i	−0.992689	+	0.120702i	$0.961485\pi$
$332$		0			0
$333$	−7.15431e10	+	1.23916e11i	−0.318837	+	0.552242i
$334$		0			0
$335$	2.57948e11			1.11900
$336$		0			0
$337$	1.14022e11			0.481564			0.240782	−	0.970579i	$-0.422596\pi$
$337$	1.14022e11			0.481564			0.240782	+	0.970579i	$0.422596\pi$
$338$		0			0
$339$	−1.67724e10	+	2.90506e10i	−0.0689758	+	0.119470i
$340$		0			0
$341$	8.06186e10	+	1.39635e11i	0.322880	+	0.559244i
$342$		0			0
$343$	6.94673e10	+	2.46752e11i	0.270992	+	0.962582i
$344$		0			0
$345$	1.12503e11	+	1.94861e11i	0.427543	+	0.740525i
$346$		0			0
$347$	1.43423e10	−	2.48416e10i	0.0531051	−	0.0919808i	−0.838251	−	0.545285i	$-0.816421\pi$
$347$	1.43423e10	−	2.48416e10i	0.0531051	−	0.0919808i	0.891356	+	0.453304i	$0.149755\pi$
$348$		0			0
$349$	−3.10537e11			−1.12047			−0.560234	−	0.828334i	$-0.689289\pi$
$349$	−3.10537e11			−1.12047			−0.560234	+	0.828334i	$0.689289\pi$
$350$		0			0
$351$	−9.91774e10			−0.348763
$352$		0			0
$353$	−2.42567e11	+	4.20138e11i	−0.831467	+	1.44014i	0.0654074	+	0.997859i	$0.479165\pi$
$353$	−2.42567e11	+	4.20138e11i	−0.831467	+	1.44014i	−0.896875	+	0.442285i	$0.854168\pi$
$354$		0			0
$355$	−9.36528e10	−	1.62211e11i	−0.312963	−	0.542068i
$356$		0			0
$357$	−4.12106e10	−	4.49257e11i	−0.134277	−	1.46382i
$358$		0			0
$359$	1.53427e11	+	2.65743e11i	0.487501	+	0.844377i	0.999897	−	0.0143725i	$-0.00457505\pi$
$359$	1.53427e11	+	2.65743e11i	0.487501	+	0.844377i	−0.512395	+	0.858750i	$0.671242\pi$
$360$		0			0
$361$	1.27148e11	−	2.20227e11i	0.394029	−	0.682478i
$362$		0			0
$363$	−2.82777e11			−0.854800
$364$		0			0
$365$	−4.68503e11			−1.38164
$366$		0			0
$367$	−2.54740e11	+	4.41223e11i	−0.732993	+	1.26958i	0.222605	+	0.974909i	$0.428544\pi$
$367$	−2.54740e11	+	4.41223e11i	−0.732993	+	1.26958i	−0.955598	+	0.294673i	$0.904789\pi$
$368$		0			0
$369$	3.43171e11	+	5.94390e11i	0.963590	+	1.66899i
$370$		0			0
$371$	2.35016e11	−	1.66038e11i	0.644044	−	0.455015i
$372$		0			0
$373$	2.13628e11	+	3.70015e11i	0.571438	+	0.989759i	0.996419	+	0.0845567i	$0.0269474\pi$
$373$	2.13628e11	+	3.70015e11i	0.571438	+	0.989759i	−0.424981	+	0.905202i	$0.639719\pi$
$374$		0			0
$375$	−7.20346e9	+	1.24768e10i	−0.0188105	+	0.0325807i
$376$		0			0
$377$	−2.32564e11			−0.592935
$378$		0			0
$379$	−4.28378e7			−0.000106647			−5.33237e−5	−	1.00000i	$-0.500017\pi$
$379$	−4.28378e7			−0.000106647			−5.33237e−5		1.00000i	$0.500017\pi$
$380$		0			0
$381$	−5.40710e11	+	9.36538e11i	−1.31463	+	2.27700i
$382$		0			0
$383$	8.72269e9	+	1.51081e10i	0.0207136	+	0.0358770i	0.876196	−	0.481954i	$-0.160073\pi$
$383$	8.72269e9	+	1.51081e10i	0.0207136	+	0.0358770i	−0.855483	+	0.517831i	$0.826740\pi$
$384$		0			0
$385$	3.60906e11	+	1.66448e11i	0.837183	+	0.386104i
$386$		0			0
$387$	4.73956e11	+	8.20915e11i	1.07408	+	1.86037i
$388$		0			0
$389$	3.04707e11	−	5.27768e11i	0.674698	−	1.16861i	−0.301859	−	0.953353i	$-0.597607\pi$
$389$	3.04707e11	−	5.27768e11i	0.674698	−	1.16861i	0.976557	−	0.215259i	$-0.0690594\pi$
$390$		0			0
$391$	1.84241e11			0.398649
$392$		0			0
$393$	−2.13495e11			−0.451462
$394$		0			0
$395$	−1.85464e11	+	3.21234e11i	−0.383330	+	0.663948i
$396$		0			0
$397$	−2.73104e11	−	4.73030e11i	−0.551786	−	0.955721i	−0.998146	−	0.0608675i	$-0.980613\pi$
$397$	−2.73104e11	−	4.73030e11i	−0.551786	−	0.955721i	0.446360	−	0.894853i	$-0.352720\pi$
$398$		0			0
$399$	−3.16730e11	−	1.46074e11i	−0.625622	−	0.288533i
$400$		0			0
$401$	2.93880e11	+	5.09014e11i	0.567570	+	0.983061i	0.996805	+	0.0798679i	$0.0254498\pi$
$401$	2.93880e11	+	5.09014e11i	0.567570	+	0.983061i	−0.429235	+	0.903193i	$0.641217\pi$
$402$		0			0
$403$	2.54012e11	−	4.39961e11i	0.479712	−	0.830886i
$404$		0			0
$405$	5.35943e11			0.989854
$406$		0			0
$407$	−1.86484e11			−0.336874
$408$		0			0
$409$	2.06201e11	−	3.57151e11i	0.364365	−	0.631099i	−0.624309	−	0.781178i	$-0.714619\pi$
$409$	2.06201e11	−	3.57151e11i	0.364365	−	0.631099i	0.988674	+	0.150079i	$0.0479527\pi$
$410$		0			0
$411$	−3.36851e11	−	5.83444e11i	−0.582305	−	1.00858i
$412$		0			0
$413$	9.80118e10	−	6.92451e10i	0.165769	−	0.117115i
$414$		0			0
$415$	1.73161e11	+	2.99923e11i	0.286571	+	0.496356i
$416$		0			0
$417$	6.68071e10	−	1.15713e11i	0.108196	−	0.187401i
$418$		0			0
$419$	4.82213e11			0.764322			0.382161	−	0.924096i	$-0.375180\pi$
$419$	4.82213e11			0.764322			0.382161	+	0.924096i	$0.375180\pi$
$420$		0			0
$421$	−2.29026e11			−0.355317			−0.177658	−	0.984092i	$-0.556852\pi$
$421$	−2.29026e11			−0.355317			−0.177658	+	0.984092i	$0.556852\pi$
$422$		0			0
$423$	3.09220e11	−	5.35585e11i	0.469609	−	0.813387i
$424$		0			0
$425$	−3.24432e11	−	5.61933e11i	−0.482362	−	0.835476i
$426$		0			0
$427$	−5.89857e10	−	6.43032e11i	−0.0858660	−	0.936068i
$428$		0			0
$429$	−3.34431e11	−	5.79252e11i	−0.476704	−	0.825676i
$430$		0			0
$431$	−4.94384e10	+	8.56299e10i	−0.0690108	+	0.119530i	−0.898466	−	0.439043i	$-0.855318\pi$
$431$	−4.94384e10	+	8.56299e10i	−0.0690108	+	0.119530i	0.829455	+	0.558573i	$0.188651\pi$
$432$		0			0
$433$	1.10144e12			1.50580			0.752899	−	0.658136i	$-0.228655\pi$
$433$	1.10144e12			1.50580			0.752899	+	0.658136i	$0.228655\pi$
$434$		0			0
$435$	−9.58928e11			−1.28406
$436$		0			0
$437$	7.12209e10	−	1.23358e11i	0.0934202	−	0.161809i
$438$		0			0
$439$	−8.29491e10	−	1.43672e11i	−0.106591	−	0.184621i	0.807796	−	0.589462i	$-0.200660\pi$
$439$	−8.29491e10	−	1.43672e11i	−0.106591	−	0.184621i	−0.914387	+	0.404841i	$0.867327\pi$
$440$		0			0
$441$	3.28935e11	−	9.27969e11i	0.414130	−	1.16831i
$442$		0			0
$443$	7.64532e11	+	1.32421e12i	0.943146	+	1.63358i	0.759422	+	0.650599i	$0.225482\pi$
$443$	7.64532e11	+	1.32421e12i	0.943146	+	1.63358i	0.183724	+	0.982978i	$0.441185\pi$
$444$		0			0
$445$	−4.75204e11	+	8.23077e11i	−0.574460	+	0.994994i
$446$		0			0
$447$	1.05135e12			1.24556
$448$		0			0
$449$	1.61276e11			0.187268			0.0936338	−	0.995607i	$-0.470152\pi$
$449$	1.61276e11			0.187268			0.0936338	+	0.995607i	$0.470152\pi$
$450$		0			0
$451$	−4.47255e11	+	7.74668e11i	−0.509050	+	0.881701i
$452$		0			0
$453$	−2.67074e11	−	4.62585e11i	−0.297981	−	0.516119i
$454$		0			0
$455$	−1.14389e11	−	1.24701e12i	−0.125121	−	1.36401i
$456$		0			0
$457$	4.91038e11	+	8.50503e11i	0.526614	+	0.912122i	0.999519	+	0.0310087i	$0.00987197\pi$
$457$	4.91038e11	+	8.50503e11i	0.526614	+	0.912122i	−0.472905	+	0.881113i	$0.656795\pi$
$458$		0			0
$459$	−1.67423e11	+	2.89986e11i	−0.176059	+	0.304944i
$460$		0			0
$461$	−1.27083e12			−1.31049			−0.655245	−	0.755417i	$-0.727435\pi$
$461$	−1.27083e12			−1.31049			−0.655245	+	0.755417i	$0.727435\pi$
$462$		0			0
$463$	1.18923e12			1.20268			0.601342	−	0.798992i	$-0.294633\pi$
$463$	1.18923e12			1.20268			0.601342	+	0.798992i	$0.294633\pi$
$464$		0			0
$465$	1.04736e12	−	1.81408e12i	1.03886	−	1.79936i
$466$		0			0
$467$	−8.47027e11	−	1.46709e12i	−0.824083	−	1.42735i	−0.902618	−	0.430443i	$-0.858357\pi$
$467$	−8.47027e11	−	1.46709e12i	−0.824083	−	1.42735i	0.0785347	−	0.996911i	$-0.474976\pi$
$468$		0			0
$469$	6.80175e11	−	4.80542e11i	0.649146	−	0.458620i
$470$		0			0
$471$	−1.19905e12	−	2.07681e12i	−1.12265	−	1.94448i
$472$		0			0
$473$	−6.17706e11	+	1.06990e12i	−0.567423	+	0.982805i
$474$		0			0
$475$	−5.01655e11			−0.452152
$476$		0			0
$477$	−1.10517e12			−0.977456
$478$		0			0
$479$	−3.77462e11	+	6.53783e11i	−0.327614	+	0.567445i	−0.982038	−	0.188683i	$-0.939578\pi$
$479$	−3.77462e11	+	6.53783e11i	−0.327614	+	0.567445i	0.654423	+	0.756128i	$0.272911\pi$
$480$		0			0
$481$	2.93785e11	+	5.08851e11i	0.250252	+	0.433449i
$482$		0			0
$483$	6.59670e11	+	3.04236e11i	0.551525	+	0.254360i
$484$		0			0
$485$	−2.25112e10	−	3.89906e10i	−0.0184740	−	0.0319979i
$486$		0			0
$487$	−5.45763e11	+	9.45290e11i	−0.439667	+	0.761526i	−0.997664	−	0.0683175i	$-0.978237\pi$
$487$	−5.45763e11	+	9.45290e11i	−0.439667	+	0.761526i	0.557997	+	0.829843i	$0.311570\pi$
$488$		0			0
$489$	−2.38178e12			−1.88370
$490$		0			0
$491$	6.37184e11			0.494764			0.247382	−	0.968918i	$-0.420430\pi$
$491$	6.37184e11			0.494764			0.247382	+	0.968918i	$0.420430\pi$
$492$		0			0
$493$	−3.92597e11	+	6.79997e11i	−0.299320	+	0.518437i
$494$		0			0
$495$	−7.63223e11	−	1.32194e12i	−0.571384	−	0.989666i
$496$		0			0
$497$	−5.49140e11	−	2.53260e11i	−0.403719	−	0.186193i
$498$		0			0
$499$	7.17606e11	+	1.24293e12i	0.518124	+	0.897417i	0.999778	+	0.0210554i	$0.00670263\pi$
$499$	7.17606e11	+	1.24293e12i	0.518124	+	0.897417i	−0.481655	+	0.876361i	$0.659964\pi$
$500$		0			0
$501$	−1.09759e12	+	1.90107e12i	−0.778339	+	1.34812i
$502$		0			0
$503$	−1.29165e12			−0.899683			−0.449842	−	0.893108i	$-0.648520\pi$
$503$	−1.29165e12			−0.899683			−0.449842	+	0.893108i	$0.648520\pi$
$504$		0			0
$505$	−4.06173e11			−0.277907
$506$		0			0
$507$	5.95113e10	−	1.03077e11i	0.0400003	−	0.0692826i
$508$		0			0
$509$	−1.06901e11	−	1.85158e11i	−0.0705914	−	0.122268i	0.828569	−	0.559887i	$-0.189155\pi$
$509$	−1.06901e11	−	1.85158e11i	−0.0705914	−	0.122268i	−0.899161	+	0.437619i	$0.855822\pi$
$510$		0			0
$511$	−1.23538e12	+	8.72792e11i	−0.801505	+	0.566261i
$512$		0			0
$513$	1.29440e11	+	2.24196e11i	0.0825163	+	0.142922i
$514$		0			0
$515$	2.90396e11	−	5.02981e11i	0.181911	−	0.315078i
$516$		0			0
$517$	8.06013e11			0.496175
$518$		0			0
$519$	−2.36696e12			−1.43198
$520$		0			0
$521$	2.67755e11	−	4.63764e11i	0.159209	−	0.275758i	−0.775375	−	0.631501i	$-0.782439\pi$
$521$	2.67755e11	−	4.63764e11i	0.159209	−	0.275758i	0.934584	+	0.355744i	$0.115772\pi$
$522$		0			0
$523$	1.84565e11	+	3.19677e11i	0.107868	+	0.186833i	0.914906	−	0.403666i	$-0.132264\pi$
$523$	1.84565e11	+	3.19677e11i	0.107868	+	0.186833i	−0.807038	+	0.590499i	$0.798931\pi$
$524$		0			0
$525$	−2.33704e11	−	2.54772e12i	−0.134261	−	1.46364i
$526$		0			0
$527$	−8.57605e11	−	1.48541e12i	−0.484328	−	0.838880i
$528$		0			0
$529$	7.52241e11	−	1.30292e12i	0.417644	−	0.723381i
$530$		0			0
$531$	−4.60904e11			−0.251585
$532$		0			0
$533$	2.81841e12			1.51262
$534$		0			0
$535$	2.41032e12	−	4.17479e12i	1.27198	−	2.20314i
$536$		0			0
$537$	−1.42414e12	−	2.46669e12i	−0.739042	−	1.28006i
$538$		0			0
$539$	1.26174e12	−	2.33445e11i	0.643903	−	0.119134i
$540$		0			0
$541$	−1.01264e12	−	1.75394e12i	−0.508237	−	0.880293i	−0.999955	−	0.00953769i	$-0.996964\pi$
$541$	−1.01264e12	−	1.75394e12i	−0.508237	−	0.880293i	0.491717	−	0.870755i	$-0.336369\pi$
$542$		0			0
$543$	−2.13229e12	+	3.69323e12i	−1.05256	+	1.82308i
$544$		0			0
$545$	4.02487e12			1.95419
$546$		0			0
$547$	3.52443e12			1.68324			0.841621	−	0.540069i	$-0.181602\pi$
$547$	3.52443e12			1.68324			0.841621	+	0.540069i	$0.181602\pi$
$548$		0			0
$549$	−1.24003e12	+	2.14780e12i	−0.582583	+	1.00906i
$550$		0			0
$551$	3.03528e11	+	5.25725e11i	0.140287	+	0.242983i
$552$		0			0
$553$	1.09394e11	+	1.19256e12i	0.0497428	+	0.542271i
$554$		0			0
$555$	1.21136e12	+	2.09814e12i	0.541944	+	0.938675i
$556$		0			0
$557$	−3.55817e11	+	6.16292e11i	−0.156631	+	0.271293i	−0.933652	−	0.358182i	$-0.883397\pi$
$557$	−3.55817e11	+	6.16292e11i	−0.156631	+	0.271293i	0.777021	+	0.629475i	$0.216730\pi$
$558$		0			0
$559$	3.89252e12			1.68608
$560$		0			0
$561$	−2.25824e12			−0.962581
$562$		0			0
$563$	−1.82812e12	+	3.16640e12i	−0.766863	+	1.32825i	0.172393	+	0.985028i	$0.444850\pi$
$563$	−1.82812e12	+	3.16640e12i	−0.766863	+	1.32825i	−0.939256	+	0.343218i	$0.888483\pi$
$564$		0			0
$565$	1.57182e11	+	2.72247e11i	0.0648910	+	0.112395i
$566$		0			0
$567$	1.41321e12	−	9.98429e11i	0.574226	−	0.405689i
$568$		0			0
$569$	5.35022e11	+	9.26686e11i	0.213977	+	0.370619i	0.952956	−	0.303110i	$-0.0980250\pi$
$569$	5.35022e11	+	9.26686e11i	0.213977	+	0.370619i	−0.738979	+	0.673729i	$0.764692\pi$
$570$		0			0
$571$	4.09702e11	−	7.09624e11i	0.161289	−	0.279361i	−0.774042	−	0.633134i	$-0.781768\pi$
$571$	4.09702e11	−	7.09624e11i	0.161289	−	0.279361i	0.935331	+	0.353773i	$0.115101\pi$
$572$		0			0
$573$	7.21938e12			2.79772
$574$		0			0
$575$	1.04482e12			0.398600
$576$		0			0
$577$	1.30100e12	−	2.25340e12i	0.488637	−	0.846343i	−0.511278	−	0.859415i	$-0.670828\pi$
$577$	1.30100e12	−	2.25340e12i	0.488637	−	0.846343i	0.999915	+	0.0130720i	$0.00416107\pi$
$578$		0			0
$579$	−2.85752e12	−	4.94937e12i	−1.05666	−	1.83019i
$580$		0			0
$581$	1.01534e12	+	4.68269e11i	0.369674	+	0.170491i
$582$		0			0
$583$	−7.20185e11	−	1.24740e12i	−0.258188	−	0.447194i
$584$		0			0
$585$	−2.40475e12	+	4.16515e12i	−0.848924	+	1.47038i
$586$		0			0
$587$	5.66219e11			0.196840			0.0984199	−	0.995145i	$-0.468621\pi$
$587$	5.66219e11			0.196840			0.0984199	+	0.995145i	$0.468621\pi$
$588$		0			0
$589$	−1.32608e12			−0.453994
$590$		0			0
$591$	1.59892e11	−	2.76941e11i	0.0539117	−	0.0933777i
$592$		0			0
$593$	−1.44623e12	−	2.50495e12i	−0.480278	−	0.831865i	0.519466	−	0.854491i	$-0.326131\pi$
$593$	−1.44623e12	−	2.50495e12i	−0.480278	−	0.831865i	−0.999744	+	0.0226255i	$0.992797\pi$
$594$		0			0
$595$	−3.83924e12	−	1.77064e12i	−1.25580	−	0.579166i
$596$		0			0
$597$	−1.29263e12	−	2.23891e12i	−0.416477	−	0.721360i
$598$		0			0
$599$	−2.96101e12	+	5.12862e12i	−0.939764	+	1.62772i	−0.173855	+	0.984771i	$0.555622\pi$
$599$	−2.96101e12	+	5.12862e12i	−0.939764	+	1.62772i	−0.765910	+	0.642948i	$0.777711\pi$
$600$		0			0
$601$	2.24993e12			0.703450			0.351725	−	0.936103i	$-0.385595\pi$
$601$	2.24993e12			0.703450			0.351725	+	0.936103i	$0.385595\pi$
$602$		0			0
$603$	−3.19855e12			−0.985201
$604$		0			0
$605$	−1.32502e12	+	2.29500e12i	−0.402089	+	0.696439i
$606$		0			0
$607$	4.72003e11	+	8.17534e11i	0.141122	+	0.244431i	0.927920	−	0.372780i	$-0.121596\pi$
$607$	4.72003e11	+	8.17534e11i	0.141122	+	0.244431i	−0.786797	+	0.617212i	$0.788262\pi$
$608$		0			0
$609$	−2.52856e12	+	1.78642e12i	−0.744896	+	0.526267i
$610$		0			0
$611$	−1.26979e12	−	2.19933e12i	−0.368591	−	0.638419i
$612$		0			0
$613$	−9.56847e11	+	1.65731e12i	−0.273697	+	0.474058i	−0.969806	−	0.243879i	$-0.921580\pi$
$613$	−9.56847e11	+	1.65731e12i	−0.273697	+	0.474058i	0.696108	+	0.717937i	$0.254913\pi$
$614$		0			0
$615$	1.16211e13			3.27573
$616$		0			0
$617$	−5.85573e12			−1.62666			−0.813332	−	0.581800i	$-0.802349\pi$
$617$	−5.85573e12			−1.62666			−0.813332	+	0.581800i	$0.802349\pi$
$618$		0			0
$619$	3.15408e11	−	5.46303e11i	0.0863505	−	0.149563i	−0.819615	−	0.572914i	$-0.805813\pi$
$619$	3.15408e11	−	5.46303e11i	0.0863505	−	0.149563i	0.905966	+	0.423351i	$0.139146\pi$
$620$		0			0
$621$	−2.69591e11	−	4.66945e11i	−0.0727433	−	0.125995i
$622$		0			0
$623$	2.80293e11	+	3.05562e12i	0.0745447	+	0.812649i
$624$		0			0
$625$	1.94080e12	+	3.36156e12i	0.508769	+	0.881213i
$626$		0			0
$627$	−8.72955e11	+	1.51200e12i	−0.225573	+	0.390705i
$628$		0			0
$629$	1.98378e12			0.505319
$630$		0			0
$631$	−5.39720e12			−1.35530			−0.677652	−	0.735383i	$-0.737002\pi$
$631$	−5.39720e12			−1.35530			−0.677652	+	0.735383i	$0.737002\pi$
$632$		0			0
$633$	−2.73060e12	+	4.72955e12i	−0.675993	+	1.17085i
$634$		0			0
$635$	5.06725e12	+	8.77673e12i	1.23677	+	2.14216i
$636$		0			0
$637$	−2.62472e12	−	3.07509e12i	−0.631621	−	0.739998i
$638$		0			0
$639$	1.16129e12	+	2.01141e12i	0.275541	+	0.477252i
$640$		0			0
$641$	−4.05089e12	+	7.01635e12i	−0.947741	+	1.64154i	−0.197572	+	0.980288i	$0.563306\pi$
$641$	−4.05089e12	+	7.01635e12i	−0.947741	+	1.64154i	−0.750169	+	0.661247i	$0.770028\pi$
$642$		0			0
$643$	7.06646e12			1.63024			0.815122	−	0.579290i	$-0.196670\pi$
$643$	7.06646e12			1.63024			0.815122	+	0.579290i	$0.196670\pi$
$644$		0			0
$645$	1.60499e13			3.65136
$646$		0			0
$647$	−1.87113e12	+	3.24089e12i	−0.419792	+	0.727100i	−0.995918	−	0.0902600i	$-0.971230\pi$
$647$	−1.87113e12	+	3.24089e12i	−0.419792	+	0.727100i	0.576127	+	0.817360i	$0.304564\pi$
$648$		0			0
$649$	−3.00348e11	−	5.20218e11i	−0.0664544	−	0.115102i
$650$		0			0
$651$	−6.17774e11	−	6.73466e12i	−0.134808	−	1.46961i
$652$		0			0
$653$	2.90962e12	+	5.03961e12i	0.626220	+	1.08465i	0.988304	+	0.152500i	$0.0487323\pi$
$653$	2.90962e12	+	5.03961e12i	0.626220	+	1.08465i	−0.362083	+	0.932146i	$0.617934\pi$
$654$		0			0
$655$	−1.00038e12	+	1.73271e12i	−0.212363	+	0.367824i
$656$		0			0
$657$	5.80942e12			1.21643
$658$		0			0
$659$	−5.67402e12			−1.17194			−0.585971	−	0.810332i	$-0.699287\pi$
$659$	−5.67402e12			−1.17194			−0.585971	+	0.810332i	$0.699287\pi$
$660$		0			0
$661$	3.81176e12	−	6.60217e12i	0.776640	−	1.34518i	−0.157229	−	0.987562i	$-0.550256\pi$
$661$	3.81176e12	−	6.60217e12i	0.776640	−	1.34518i	0.933868	−	0.357617i	$-0.116411\pi$
$662$		0			0
$663$	3.55761e12	+	6.16197e12i	0.715069	+	1.23854i
$664$		0			0
$665$	−2.66964e12	+	1.88609e12i	−0.529365	+	0.373995i
$666$		0			0
$667$	−6.32173e11	−	1.09495e12i	−0.123671	−	0.214205i
$668$		0			0
$669$	1.92200e12	−	3.32900e12i	0.370968	−	0.642535i
$670$		0			0
$671$	−3.23227e12			−0.615540
$672$		0			0
$673$	−6.37901e12			−1.19863			−0.599315	−	0.800513i	$-0.704560\pi$
$673$	−6.37901e12			−1.19863			−0.599315	+	0.800513i	$0.704560\pi$
$674$		0			0
$675$	−9.49451e11	+	1.64450e12i	−0.176038	+	0.304906i
$676$		0			0
$677$	4.97630e12	+	8.61921e12i	0.910454	+	1.57695i	0.813425	+	0.581670i	$0.197601\pi$
$677$	4.97630e12	+	8.61921e12i	0.910454	+	1.57695i	0.0970291	+	0.995282i	$0.469066\pi$
$678$		0			0
$679$	−1.31996e11	−	6.08758e10i	−0.0238312	−	0.0109908i
$680$		0			0
$681$	−2.51617e12	−	4.35814e12i	−0.448310	−	0.776495i
$682$		0			0
$683$	−1.37743e12	+	2.38579e12i	−0.242202	+	0.419506i	−0.961341	−	0.275360i	$-0.911203\pi$
$683$	−1.37743e12	+	2.38579e12i	−0.242202	+	0.419506i	0.719139	+	0.694866i	$0.244536\pi$
$684$		0			0
$685$	−6.31358e12			−1.09564
$686$		0			0
$687$	6.31642e12			1.08185
$688$		0			0
$689$	−2.26915e12	+	3.93028e12i	−0.383598	+	0.664411i
$690$		0			0
$691$	−4.92067e12	−	8.52285e12i	−0.821057	−	1.42211i	−0.904896	−	0.425633i	$-0.860051\pi$
$691$	−4.92067e12	−	8.52285e12i	−0.821057	−	1.42211i	0.0838390	−	0.996479i	$-0.473282\pi$
$692$		0			0
$693$	−4.47521e12	−	2.06394e12i	−0.737079	−	0.339937i
$694$		0			0
$695$	−6.26081e11	−	1.08440e12i	−0.101788	−	0.176303i
$696$		0			0
$697$	4.75781e12	−	8.24077e12i	0.763589	−	1.32257i
$698$		0			0
$699$	2.83493e12			0.449154
$700$		0			0
$701$	4.83994e12			0.757022			0.378511	−	0.925597i	$-0.376436\pi$
$701$	4.83994e12			0.757022			0.378511	+	0.925597i	$0.376436\pi$
$702$		0			0
$703$	7.66859e11	−	1.32824e12i	0.118418	−	0.205105i
$704$		0			0
$705$	−5.23568e12	−	9.06847e12i	−0.798220	−	1.38256i
$706$		0			0
$707$	−1.07102e12	+	7.56675e11i	−0.161217	+	0.113900i
$708$		0			0
$709$	−2.36329e12	−	4.09334e12i	−0.351244	−	0.608372i	0.635224	−	0.772328i	$-0.280908\pi$
$709$	−2.36329e12	−	4.09334e12i	−0.351244	−	0.608372i	−0.986468	+	0.163956i	$0.947575\pi$
$710$		0			0
$711$	2.29975e12	−	3.98328e12i	0.337495	−	0.584558i
$712$		0			0
$713$	2.76189e12			0.400224
$714$		0			0
$715$	−6.26822e12			−0.896947
$716$		0			0
$717$	−3.93669e11	+	6.81855e11i	−0.0556283	+	0.0963510i
$718$		0			0
$719$	−3.31967e11	−	5.74984e11i	−0.0463250	−	0.0802372i	0.841933	−	0.539582i	$-0.181418\pi$
$719$	−3.31967e11	−	5.74984e11i	−0.0463250	−	0.0802372i	−0.888258	+	0.459345i	$0.848084\pi$
$720$		0			0
$721$	−1.71287e11	−	1.86728e12i	−0.0236056	−	0.257336i
$722$		0			0
$723$	2.82531e12	+	4.89359e12i	0.384543	+	0.666047i
$724$		0			0
$725$	−2.22640e12	+	3.85624e12i	−0.299283	+	0.518374i
$726$		0			0
$727$	−5.45701e11			−0.0724520			−0.0362260	−	0.999344i	$-0.511534\pi$
$727$	−5.45701e11			−0.0724520			−0.0362260	+	0.999344i	$0.511534\pi$
$728$		0			0
$729$	−1.07365e13			−1.40796
$730$		0			0
$731$	6.57104e12	−	1.13814e13i	0.851149	−	1.47423i
$732$		0			0
$733$	3.78254e12	+	6.55155e12i	0.483967	+	0.838255i	0.999830	−	0.0184156i	$-0.00586221\pi$
$733$	3.78254e12	+	6.55155e12i	0.483967	+	0.838255i	−0.515864	+	0.856671i	$0.672529\pi$
$734$		0			0
$735$	−1.08225e13	−	1.26795e13i	−1.36783	−	1.60254i
$736$		0			0
$737$	−2.08433e12	−	3.61017e12i	−0.260233	−	0.450737i
$738$		0			0
$739$	1.29852e11	−	2.24911e11i	0.0160158	−	0.0277402i	−0.857906	−	0.513806i	$-0.828235\pi$
$739$	1.29852e11	−	2.24911e11i	0.0160158	−	0.0277402i	0.873922	+	0.486066i	$0.161568\pi$
$740$		0			0
$741$	5.50098e12			0.670283
$742$		0			0
$743$	−2.96088e12			−0.356427			−0.178214	−	0.983992i	$-0.557032\pi$
$743$	−2.96088e12			−0.356427			−0.178214	+	0.983992i	$0.557032\pi$
$744$		0			0
$745$	4.92635e12	−	8.53269e12i	0.585898	−	1.01480i
$746$		0			0
$747$	−2.14718e12	−	3.71903e12i	−0.252305	−	0.437006i
$748$		0			0
$749$	−1.42170e12	−	1.54986e13i	−0.165059	−	1.79939i
$750$		0			0
$751$	3.27838e12	+	5.67833e12i	0.376080	+	0.651390i	0.990488	−	0.137599i	$-0.0439385\pi$
$751$	3.27838e12	+	5.67833e12i	0.376080	+	0.651390i	−0.614408	+	0.788988i	$0.710605\pi$
$752$		0			0
$753$	1.64246e12	−	2.84482e12i	0.186173	−	0.322461i
$754$		0			0
$755$	−5.00574e12			−0.560670
$756$		0			0
$757$	−7.49463e12			−0.829504			−0.414752	−	0.909934i	$-0.636132\pi$
$757$	−7.49463e12			−0.829504			−0.414752	+	0.909934i	$0.636132\pi$
$758$		0			0
$759$	1.81815e12	−	3.14912e12i	0.198857	−	0.344431i
$760$		0			0
$761$	−3.14710e12	−	5.45093e12i	−0.340157	−	0.589169i	0.644305	−	0.764769i	$-0.277147\pi$
$761$	−3.14710e12	−	5.45093e12i	−0.340157	−	0.589169i	−0.984462	+	0.175600i	$0.943814\pi$
$762$		0			0
$763$	1.06130e13	−	7.49807e12i	1.13365	−	0.800920i
$764$		0			0
$765$	8.11902e12	+	1.40626e13i	0.857091	+	1.48453i
$766$		0			0
$767$	−9.46331e11	+	1.63909e12i	−0.0987334	+	0.171011i
$768$		0			0
$769$	1.16579e13			1.20214			0.601068	−	0.799198i	$-0.294742\pi$
$769$	1.16579e13			1.20214			0.601068	+	0.799198i	$0.294742\pi$
$770$		0			0
$771$	1.52610e13			1.55538
$772$		0			0
$773$	−8.64199e12	+	1.49684e13i	−0.870574	+	1.50788i	−0.00917093	+	0.999958i	$0.502919\pi$
$773$	−8.64199e12	+	1.49684e13i	−0.870574	+	1.50788i	−0.861404	+	0.507921i	$0.830414\pi$
$774$		0			0
$775$	−4.86344e12	−	8.42373e12i	−0.484268	−	0.838778i
$776$		0			0
$777$	7.10289e12	+	3.27581e12i	0.699102	+	0.322422i
$778$		0			0
$779$	−3.67840e12	−	6.37117e12i	−0.357882	−	0.619870i
$780$		0			0
$781$	−1.51351e12	+	2.62147e12i	−0.145564	+	0.252125i
$782$		0			0
$783$	2.29787e12			0.218473
$784$		0			0
$785$	−2.24737e13			−2.11233
$786$		0			0
$787$	−1.92920e12	+	3.34147e12i	−0.179263	+	0.310492i	−0.941628	−	0.336654i	$-0.890705\pi$
$787$	−1.92920e12	+	3.34147e12i	−0.179263	+	0.310492i	0.762365	+	0.647147i	$0.224038\pi$
$788$		0			0
$789$	−8.62998e12	−	1.49476e13i	−0.792800	−	1.37317i
$790$		0			0
$791$	9.21646e11	+	4.25058e11i	0.0837086	+	0.0386060i
$792$		0			0
$793$	5.09209e12	+	8.81976e12i	0.457264	+	0.792004i
$794$		0			0
$795$	−9.35633e12	+	1.62056e13i	−0.830717	+	1.43884i
$796$		0			0
$797$	1.07296e12			0.0941935			0.0470967	−	0.998890i	$-0.485003\pi$
$797$	1.07296e12			0.0941935			0.0470967	+	0.998890i	$0.485003\pi$
$798$		0			0
$799$	−8.57421e12			−0.744275
$800$		0			0
$801$	5.89250e12	−	1.02061e13i	0.505771	−	0.876020i
$802$		0			0
$803$	3.78570e12	+	6.55703e12i	0.321312	+	0.556528i
$804$		0			0
$805$	5.56020e12	−	3.92827e12i	0.466669	−	0.329700i
$806$		0			0
$807$	−3.45452e12	−	5.98341e12i	−0.286719	−	0.496613i
$808$		0			0
$809$	−4.81093e12	+	8.33277e12i	−0.394876	+	0.683945i	−0.993085	−	0.117395i	$-0.962546\pi$
$809$	−4.81093e12	+	8.33277e12i	−0.394876	+	0.683945i	0.598210	+	0.801340i	$0.295879\pi$
$810$		0			0
$811$	1.42997e12			0.116074			0.0580369	−	0.998314i	$-0.481516\pi$
$811$	1.42997e12			0.116074			0.0580369	+	0.998314i	$0.481516\pi$
$812$		0			0
$813$	2.81860e13			2.26270
$814$		0			0
$815$	−1.11604e13	+	1.93303e13i	−0.886072	+	1.53472i
$816$		0			0
$817$	−5.08025e12	−	8.79926e12i	−0.398920	−	0.690950i
$818$		0			0
$819$	1.41841e12	+	1.54628e13i	0.110160	+	1.20091i
$820$		0			0
$821$	−8.91175e12	−	1.54356e13i	−0.684571	−	1.18571i	−0.973571	−	0.228383i	$-0.926656\pi$
$821$	−8.91175e12	−	1.54356e13i	−0.684571	−	1.18571i	0.289000	−	0.957329i	$-0.406677\pi$
$822$		0			0
$823$	6.67366e12	−	1.15591e13i	0.507066	−	0.878264i	−0.492900	−	0.870086i	$-0.664063\pi$
$823$	6.67366e12	−	1.15591e13i	0.507066	−	0.878264i	0.999967	−	0.00817863i	$-0.00260337\pi$
$824$		0			0
$825$	−1.28064e13			−0.962463
$826$		0			0
$827$	−1.34978e13			−1.00344			−0.501718	−	0.865031i	$-0.667298\pi$
$827$	−1.34978e13			−1.00344			−0.501718	+	0.865031i	$0.667298\pi$
$828$		0			0
$829$	−4.99503e12	+	8.65164e12i	−0.367318	+	0.636214i	−0.989145	−	0.146941i	$-0.953057\pi$
$829$	−4.99503e12	+	8.65164e12i	−0.367318	+	0.636214i	0.621827	+	0.783155i	$0.286391\pi$
$830$		0			0
$831$	−2.02710e13	−	3.51104e13i	−1.47459	−	2.55406i
$832$		0			0
$833$	−1.34221e13	+	2.48334e12i	−0.965872	+	0.178703i
$834$		0			0
$835$	1.02860e13	+	1.78158e13i	0.732245	+	1.26829i
$836$		0			0
$837$	−2.50978e12	+	4.34707e12i	−0.176755	+	0.306149i
$838$		0			0
$839$	4.86136e12			0.338711			0.169355	−	0.985555i	$-0.445831\pi$
$839$	4.86136e12			0.338711			0.169355	+	0.985555i	$0.445831\pi$
$840$		0			0
$841$	−9.11879e12			−0.628572
$842$		0			0
$843$	−1.90537e12	+	3.30020e12i	−0.129944	+	0.225069i
$844$		0			0
$845$	−5.57708e11	−	9.65979e11i	−0.0376315	−	0.0651797i
$846$		0			0
$847$	7.81546e11	+	8.52002e12i	0.0521770	+	0.568808i
$848$		0			0
$849$	1.36955e13	+	2.37213e13i	0.904675	+	1.56694i
$850$		0			0
$851$	−1.59718e12	+	2.76639e12i	−0.104393	+	0.180813i
$852$		0			0
$853$	−2.85346e12			−0.184544			−0.0922722	−	0.995734i	$-0.529413\pi$
$853$	−2.85346e12			−0.184544			−0.0922722	+	0.995734i	$0.529413\pi$
$854$		0			0
$855$	1.25541e13			0.803411
$856$		0			0
$857$	6.98986e12	−	1.21068e13i	0.442644	−	0.766682i	−0.555241	−	0.831690i	$-0.687374\pi$
$857$	6.98986e12	−	1.21068e13i	0.442644	−	0.766682i	0.997885	+	0.0650077i	$0.0207072\pi$
$858$		0			0
$859$	−5.05054e12	−	8.74780e12i	−0.316496	−	0.548188i	0.663258	−	0.748391i	$-0.269173\pi$
$859$	−5.05054e12	−	8.74780e12i	−0.316496	−	0.548188i	−0.979754	+	0.200203i	$0.935840\pi$
$860$		0			0
$861$	3.06432e13	−	2.16493e13i	1.90029	−	1.34255i
$862$		0			0
$863$	1.00407e13	+	1.73910e13i	0.616190	+	1.06727i	0.990174	+	0.139838i	$0.0446580\pi$
$863$	1.00407e13	+	1.73910e13i	0.616190	+	1.06727i	−0.373984	+	0.927435i	$0.622009\pi$
$864$		0			0
$865$	−1.10910e13	+	1.92101e13i	−0.673591	+	1.16669i
$866$		0			0
$867$	−8.75345e11			−0.0526130
$868$		0			0
$869$	5.99452e12			0.356587
$870$		0			0
$871$	−6.56727e12	+	1.13749e13i	−0.386637	+	0.669675i
$872$		0			0
$873$	2.79138e11	+	4.83481e11i	0.0162650	+	0.0281718i
$874$		0			0
$875$	3.95831e11	+	1.82555e11i	0.0228283	+	0.0105283i
$876$		0			0
$877$	−1.07426e13	−	1.86068e13i	−0.613215	−	1.06212i	−0.990695	−	0.136102i	$-0.956542\pi$
$877$	−1.07426e13	−	1.86068e13i	−0.613215	−	1.06212i	0.377480	−	0.926018i	$-0.376791\pi$
$878$		0			0
$879$	1.70001e13	−	2.94451e13i	0.960511	−	1.66365i
$880$		0			0
$881$	−2.78108e13			−1.55533			−0.777664	−	0.628680i	$-0.783596\pi$
$881$	−2.78108e13			−1.55533			−0.777664	+	0.628680i	$0.783596\pi$
$882$		0			0
$883$	−3.49281e12			−0.193353			−0.0966767	−	0.995316i	$-0.530821\pi$
$883$	−3.49281e12			−0.193353			−0.0966767	+	0.995316i	$0.530821\pi$
$884$		0			0
$885$	−3.90199e12	+	6.75844e12i	−0.213817	+	0.370341i
$886$		0			0
$887$	−1.49021e12	−	2.58112e12i	−0.0808336	−	0.140008i	0.822775	−	0.568368i	$-0.192425\pi$
$887$	−1.49021e12	−	2.58112e12i	−0.0808336	−	0.140008i	−0.903608	+	0.428360i	$0.859092\pi$
$888$		0			0
$889$	2.97121e13	+	1.37031e13i	1.59542	+	0.735801i
$890$		0			0
$891$	−4.33065e12	−	7.50090e12i	−0.230199	−	0.398716i
$892$		0			0
$893$	−3.31448e12	+	5.74085e12i	−0.174415	+	0.302096i
$894$		0			0
$895$	−2.66926e13			−1.39055
$896$		0			0
$897$	−1.14572e13			−0.590897
$898$		0			0
$899$	−5.88528e12	+	1.01936e13i	−0.300503	+	0.520486i
$900$		0			0
$901$	7.66119e12	+	1.32696e13i	0.387288	+	0.670803i
$902$		0			0
$903$	4.23215e13	−	2.99000e13i	2.11820	−	1.49650i
$904$		0			0
$905$	1.99826e13	+	3.46110e13i	0.990226	+	1.71512i
$906$		0			0
$907$	1.17327e13	−	2.03217e13i	0.575661	−	0.997075i	−0.420308	−	0.907381i	$-0.638078\pi$
$907$	1.17327e13	−	2.03217e13i	0.575661	−	0.997075i	0.995969	−	0.0896931i	$-0.0285886\pi$
$908$		0			0
$909$	5.03652e12			0.244677
$910$		0			0
$911$	1.39221e13			0.669688			0.334844	−	0.942274i	$-0.391316\pi$
$911$	1.39221e13			0.669688			0.334844	+	0.942274i	$0.391316\pi$
$912$		0			0
$913$	2.79842e12	−	4.84701e12i	0.133289	−	0.230864i
$914$		0			0
$915$	2.09961e13	+	3.63664e13i	0.990248	+	1.71516i
$916$		0			0
$917$	5.90062e11	+	6.43256e12i	0.0275572	+	0.300415i
$918$		0			0
$919$	8.05476e12	+	1.39513e13i	0.372506	+	0.645199i	0.989950	−	0.141416i	$-0.0451654\pi$
$919$	8.05476e12	+	1.39513e13i	0.372506	+	0.645199i	−0.617445	+	0.786614i	$0.711832\pi$
$920$		0			0
$921$	2.24148e13	−	3.88236e13i	1.02652	−	1.77798i
$922$		0			0
$923$	9.53747e12			0.432539
$924$		0			0
$925$	1.12499e13			0.505257
$926$		0			0
$927$	−3.60089e12	+	6.23693e12i	−0.160159	+	0.277404i
$928$		0			0
$929$	3.96121e12	+	6.86101e12i	0.174485	+	0.302216i	0.939983	−	0.341222i	$-0.110841\pi$
$929$	3.96121e12	+	6.86101e12i	0.174485	+	0.302216i	−0.765498	+	0.643438i	$0.777507\pi$
$930$		0			0
$931$	−3.52581e12	+	9.94675e12i	−0.153810	+	0.433918i
$932$		0			0
$933$	−1.35310e13	−	2.34364e13i	−0.584605	−	1.01257i
$934$		0			0
$935$	−1.05815e13	+	1.83277e13i	−0.452789	+	0.784253i
$936$		0			0
$937$	8.84621e12			0.374912			0.187456	−	0.982273i	$-0.439976\pi$
$937$	8.84621e12			0.374912			0.187456	+	0.982273i	$0.439976\pi$
$938$		0			0
$939$	3.73814e13			1.56914
$940$		0			0
$941$	−1.75672e13	+	3.04272e13i	−0.730379	+	1.26505i	0.226342	+	0.974048i	$0.427323\pi$
$941$	−1.75672e13	+	3.04272e13i	−0.730379	+	1.26505i	−0.956721	+	0.291006i	$0.906010\pi$
$942$		0			0
$943$	7.66119e12	+	1.32696e13i	0.315496	+	0.546455i
$944$		0			0
$945$	1.13023e12	+	1.23212e13i	0.0461023	+	0.502584i
$946$		0			0
$947$	−1.08238e13	−	1.87474e13i	−0.437325	−	0.757470i	0.560157	−	0.828387i	$-0.310741\pi$
$947$	−1.08238e13	−	1.87474e13i	−0.437325	−	0.757470i	−0.997482	+	0.0709167i	$0.977408\pi$
$948$		0			0
$949$	1.19279e13	−	2.06598e13i	0.477383	−	0.826852i
$950$		0			0
$951$	−3.08254e13			−1.22207
$952$		0			0
$953$	4.63945e13			1.82200			0.911001	−	0.412404i	$-0.135311\pi$
$953$	4.63945e13			1.82200			0.911001	+	0.412404i	$0.135311\pi$
$954$		0			0
$955$	3.38281e13	−	5.85919e13i	1.31602	−	2.27941i
$956$		0			0
$957$	7.74854e12	+	1.34209e13i	0.298618	+	0.517222i
$958$		0			0
$959$	−1.66481e13	+	1.17618e13i	−0.635594	+	0.449045i
$960$		0			0
$961$	3.63770e11	+	6.30068e11i	0.0137585	+	0.0238305i
$962$		0			0
$963$	−2.98878e13	+	5.17672e13i	−1.11989	+	1.93971i
$964$		0			0
$965$	−5.35583e13			−1.98817
$966$		0			0
$967$	−2.17411e12			−0.0799581			−0.0399791	−	0.999201i	$-0.512729\pi$
$967$	−2.17411e12			−0.0799581			−0.0399791	+	0.999201i	$0.512729\pi$
$968$		0			0
$969$	9.28632e12	−	1.60844e13i	0.338366	−	0.586067i
$970$		0			0
$971$	−8.88797e12	−	1.53944e13i	−0.320860	−	0.555746i	0.659805	−	0.751436i	$-0.270639\pi$
$971$	−8.88797e12	−	1.53944e13i	−0.320860	−	0.555746i	−0.980666	+	0.195690i	$0.937305\pi$
$972$		0			0
$973$	−3.67106e12	−	1.69307e12i	−0.131306	−	0.0605576i
$974$		0			0
$975$	2.01751e13	+	3.49443e13i	0.714981	+	1.23838i
$976$		0			0
$977$	−6.14244e12	+	1.06390e13i	−0.215683	+	0.373574i	−0.953484	−	0.301445i	$-0.902531\pi$
$977$	−6.14244e12	+	1.06390e13i	−0.215683	+	0.373574i	0.737801	+	0.675019i	$0.235864\pi$
$978$		0			0
$979$	1.53594e13			0.534382
$980$		0			0
$981$	−4.99081e13			−1.72052
$982$		0			0
$983$	−1.38063e13	+	2.39131e13i	−0.471612	+	0.816856i	−0.999473	−	0.0324748i	$-0.989661\pi$
$983$	−1.38063e13	+	2.39131e13i	−0.471612	+	0.816856i	0.527860	+	0.849331i	$0.322994\pi$
$984$		0			0
$985$	−1.49842e12	−	2.59534e12i	−0.0507190	−	0.0878479i
$986$		0			0
$987$	−3.06998e13	−	1.41586e13i	−1.02969	−	0.474889i
$988$		0			0
$989$	1.05809e13	+	1.83267e13i	0.351673	+	0.609116i
$990$		0			0
$991$	−1.44913e13	+	2.50997e13i	−0.477284	+	0.826680i	−0.999661	−	0.0260347i	$-0.991712\pi$
$991$	−1.44913e13	+	2.50997e13i	−0.477284	+	0.826680i	0.522377	+	0.852714i	$0.325045\pi$
$992$		0			0
$993$	3.59303e13			1.17271
$994$		0			0
$995$	−2.42278e13			−0.783627
$996$		0			0
$997$	−7.66674e12	+	1.32792e13i	−0.245744	+	0.425641i	−0.962340	−	0.271847i	$-0.912366\pi$
$997$	−7.66674e12	+	1.32792e13i	−0.245744	+	0.425641i	0.716597	+	0.697488i	$0.245699\pi$
$998$		0			0
$999$	−2.90277e12	−	5.02775e12i	−0.0922079	−	0.159709i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	112.10.i.c.81.2		10
4.3	odd	2		7.10.c.a.4.4	yes	10
7.2	even	3	inner	112.10.i.c.65.2		10
12.11	even	2		63.10.e.b.46.2		10
28.3	even	6		49.10.a.f.1.2		5
28.11	odd	6		49.10.a.e.1.2		5
28.19	even	6		49.10.c.g.30.4		10
28.23	odd	6		7.10.c.a.2.4	✓	10
28.27	even	2		49.10.c.g.18.4		10
84.23	even	6		63.10.e.b.37.2		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7.10.c.a.2.4	✓	10	28.23	odd	6
7.10.c.a.4.4	yes	10	4.3	odd	2
49.10.a.e.1.2		5	28.11	odd	6
49.10.a.f.1.2		5	28.3	even	6
49.10.c.g.18.4		10	28.27	even	2
49.10.c.g.30.4		10	28.19	even	6
63.10.e.b.37.2		10	84.23	even	6
63.10.e.b.46.2		10	12.11	even	2
112.10.i.c.65.2		10	7.2	even	3	inner
112.10.i.c.81.2		10	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+(-104.977 + 181.826i) q^{3} +(983.791 + 1703.98i) q^{5} +(5768.52 + 2660.41i) q^{7} +(-12198.9 - 21129.2i) q^{9} +(15898.9 - 27537.7i) q^{11} -100188. q^{13} -413103. q^{15} +(-169129. + 292940. i) q^{17} +(130759. + 226480. i) q^{19} +(-1.08930e6 + 769584. i) q^{21} +(-272337. - 471702. i) q^{23} +(-959125. + 1.66125e6i) q^{25} +989914. q^{27} +2.32128e6 q^{29} +(-2.53535e6 + 4.39136e6i) q^{31} +(3.33804e6 + 5.78166e6i) q^{33} +(1.14174e6 + 1.24467e7i) q^{35} +(-2.93235e6 - 5.07897e6i) q^{37} +(1.05174e7 - 1.82167e7i) q^{39} -2.81312e7 q^{41} -3.88522e7 q^{43} +(2.40024e7 - 4.15734e7i) q^{45} +(1.26741e7 + 2.19521e7i) q^{47} +(2.61980e7 + 3.06933e7i) q^{49} +(-3.55094e7 - 6.15041e7i) q^{51} +(2.26489e7 - 3.92291e7i) q^{53} +6.25647e7 q^{55} -5.49067e7 q^{57} +(9.44557e6 - 1.63602e7i) q^{59} +(-5.08255e7 - 8.80323e7i) q^{61} +(-1.41575e7 - 1.54338e8i) q^{63} +(-9.85638e7 - 1.70718e8i) q^{65} +(6.55496e7 - 1.13535e8i) q^{67} +1.14357e8 q^{69} -9.51959e7 q^{71} +(-1.19056e8 + 2.06211e8i) q^{73} +(-2.01373e8 - 3.48788e8i) q^{75} +(1.64975e8 - 1.16554e8i) q^{77} +(9.42601e7 + 1.63263e8i) q^{79} +(1.36193e8 - 2.35894e8i) q^{81} +1.76014e8 q^{83} -6.65551e8 q^{85} +(-2.43682e8 + 4.22070e8i) q^{87} +(2.41517e8 + 4.18319e8i) q^{89} +(-5.77936e8 - 2.66541e8i) q^{91} +(-5.32309e8 - 9.21986e8i) q^{93} +(-2.57278e8 + 4.45619e8i) q^{95} -2.28821e7 q^{97} -7.75799e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 161 q^{3} + 1533 q^{5} + 1036 q^{7} - 35734 q^{9} - 42213 q^{11} - 319676 q^{13} - 151394 q^{15} + 324681 q^{17} + 16121 q^{19} - 1557857 q^{21} - 2638863 q^{23} - 1304092 q^{25} + 18331558 q^{27}+ \cdots + 1900777180 q^{99}+O(q^{100}) \) 10 * q - 161 * q^3 + 1533 * q^5 + 1036 * q^7 - 35734 * q^9 - 42213 * q^11 - 319676 * q^13 - 151394 * q^15 + 324681 * q^17 + 16121 * q^19 - 1557857 * q^21 - 2638863 * q^23 - 1304092 * q^25 + 18331558 * q^27 + 15292500 * q^29 - 19179237 * q^31 + 1689359 * q^33 + 43746759 * q^35 + 39566985 * q^37 + 44299486 * q^39 - 53436852 * q^41 - 101835992 * q^43 + 85098230 * q^45 - 32509659 * q^47 - 49024598 * q^49 - 44168403 * q^51 - 25714707 * q^53 + 144695222 * q^55 - 121710346 * q^57 - 46776513 * q^59 - 113075039 * q^61 - 318071530 * q^63 - 338113566 * q^65 + 126707879 * q^67 + 1323616182 * q^69 + 1188736032 * q^71 - 859257651 * q^73 + 169061732 * q^75 + 1911891891 * q^77 + 527065417 * q^79 + 551662715 * q^81 + 144863208 * q^83 - 1197360222 * q^85 + 340781350 * q^87 + 1661554797 * q^89 - 726641384 * q^91 - 423057489 * q^93 + 1197123495 * q^95 + 869770188 * q^97 + 1900777180 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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