LMFDB - Embedded newform 112.9.s.b.33.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [112,9,Mod(17,112)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(112, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 9, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("112.17");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 112 = 2^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	112.s (of order $6$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$45.6264043268$
Analytic rank:	$0$
Dimension:	$10$
Relative dimension:	$5$ over $\Q(\zeta_{6})$
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} + 1442 x^{8} + 59551 x^{7} + 2229058 x^{6} + 41253567 x^{5} + 582209889 x^{4} + \cdots + 63214027776 $ x^10 - x^9 + 1442x^8 + 59551x^7 + 2229058x^6 + 41253567x^5 + 582209889x^4 + 4552713792x^3 + 25685059104x^2 + 45666643968x + 63214027776
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{12}\cdot 3^{6}\cdot 7^{6} $
Twist minimal:	no (minimal twist has level 28)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			33.2
Root			$-0.957903 - 1.65914i$ of defining polynomial
Character	$\chi$	$=$	112.33
Dual form			112.9.s.b.17.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-67.4293 + 38.9303i) q^{3} +(-616.084 - 355.696i) q^{5} +(2382.47 - 297.754i) q^{7} +(-249.357 + 431.899i) q^{9} +O(q^{10})$	\(q+(-67.4293 + 38.9303i) q^{3} +(-616.084 - 355.696i) q^{5} +(2382.47 - 297.754i) q^{7} +(-249.357 + 431.899i) q^{9} +(1141.87 + 1977.78i) q^{11} +8314.85i q^{13} +55389.5 q^{15} +(-63255.0 + 36520.3i) q^{17} +(218262. + 126014. i) q^{19} +(-149056. + 112828. i) q^{21} +(-142303. + 246477. i) q^{23} +(57726.8 + 99985.8i) q^{25} -549674. i q^{27} -105335. q^{29} +(-743496. + 429258. i) q^{31} +(-153991. - 88906.9i) q^{33} +(-1.57371e6 - 663992. i) q^{35} +(1.41911e6 - 2.45797e6i) q^{37} +(-323700. - 560665. i) q^{39} +1.75485e6i q^{41} +1.66109e6 q^{43} +(307250. - 177391. i) q^{45} +(-3.13716e6 - 1.81124e6i) q^{47} +(5.58749e6 - 1.41878e6i) q^{49} +(2.84350e6 - 4.92508e6i) q^{51} +(-5.68440e6 - 9.84568e6i) q^{53} -1.62464e6i q^{55} -1.96230e7 q^{57} +(1.15832e7 - 6.68757e6i) q^{59} +(-1.24328e7 - 7.17810e6i) q^{61} +(-465485. + 1.10323e6i) q^{63} +(2.95756e6 - 5.12265e6i) q^{65} +(-1.55171e7 - 2.68764e7i) q^{67} -2.21597e7i q^{69} -3.33725e7 q^{71} +(-7.19825e6 + 4.15591e6i) q^{73} +(-7.78496e6 - 4.49465e6i) q^{75} +(3.30936e6 + 4.37199e6i) q^{77} +(-3.40981e6 + 5.90597e6i) q^{79} +(1.97630e7 + 3.42305e7i) q^{81} +4.05040e7i q^{83} +5.19605e7 q^{85} +(7.10268e6 - 4.10074e6i) q^{87} +(-3.65699e7 - 2.11136e7i) q^{89} +(2.47578e6 + 1.98099e7i) q^{91} +(3.34223e7 - 5.78891e7i) q^{93} +(-8.96452e7 - 1.55270e8i) q^{95} -9.29688e7i q^{97} -1.13893e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 81 q^{3} - 837 q^{5} - 1526 q^{7} + 1902 q^{9}+O(q^{10}) $ 10 * q + 81 * q^3 - 837 * q^5 - 1526 * q^7 + 1902 * q^9	$ 10 q + 81 q^{3} - 837 q^{5} - 1526 q^{7} + 1902 q^{9} - 3705 q^{11} - 76134 q^{15} + 78003 q^{17} + 96741 q^{19} - 153153 q^{21} - 208533 q^{23} + 367978 q^{25} + 754764 q^{29} + 1053717 q^{31} - 1032993 q^{33} + 1306389 q^{35} - 998075 q^{37} - 1431900 q^{39} - 738292 q^{43} - 7432758 q^{45} - 710883 q^{47} + 13288114 q^{49} + 2571909 q^{51} + 10501461 q^{53} - 2744514 q^{57} + 37089081 q^{59} - 8180481 q^{61} - 47152518 q^{63} + 21459108 q^{65} - 48020189 q^{67} + 31918236 q^{71} - 133345593 q^{73} + 119504178 q^{75} + 188477625 q^{77} - 53590181 q^{79} + 173295063 q^{81} - 157179282 q^{85} + 413284806 q^{87} - 241368273 q^{89} - 420709128 q^{91} + 137961999 q^{93} - 347126775 q^{95} + 117796500 q^{99}+O(q^{100}) $ 10 * q + 81 * q^3 - 837 * q^5 - 1526 * q^7 + 1902 * q^9 - 3705 * q^11 - 76134 * q^15 + 78003 * q^17 + 96741 * q^19 - 153153 * q^21 - 208533 * q^23 + 367978 * q^25 + 754764 * q^29 + 1053717 * q^31 - 1032993 * q^33 + 1306389 * q^35 - 998075 * q^37 - 1431900 * q^39 - 738292 * q^43 - 7432758 * q^45 - 710883 * q^47 + 13288114 * q^49 + 2571909 * q^51 + 10501461 * q^53 - 2744514 * q^57 + 37089081 * q^59 - 8180481 * q^61 - 47152518 * q^63 + 21459108 * q^65 - 48020189 * q^67 + 31918236 * q^71 - 133345593 * q^73 + 119504178 * q^75 + 188477625 * q^77 - 53590181 * q^79 + 173295063 * q^81 - 157179282 * q^85 + 413284806 * q^87 - 241368273 * q^89 - 420709128 * q^91 + 137961999 * q^93 - 347126775 * q^95 + 117796500 * q^99

Display 100 coefficients 10 coefficients

Character values

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

$n$	$15$	$17$	$85$
$\chi(n)$	$1$	$e\left(\frac{5}{6}\right)$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−67.4293	+	38.9303i	−0.832461	+	0.480621i	−0.854694	−	0.519131i	$-0.826256\pi$
$3$	−67.4293	+	38.9303i	−0.832461	+	0.480621i	0.0222337	+	0.999753i	$0.492922\pi$
$4$		0			0
$5$	−616.084	−	355.696i	−0.985734	−	0.569114i	−0.0817374	−	0.996654i	$-0.526047\pi$
$5$	−616.084	−	355.696i	−0.985734	−	0.569114i	−0.903996	+	0.427540i	$0.859380\pi$
$6$		0			0
$7$	2382.47	−	297.754i	0.992281	−	0.124013i
$7$	2382.47	−	297.754i	0.992281	−	0.124013i
$8$		0			0
$9$	−249.357	+	431.899i	−0.0380060	+	0.0658283i
$10$		0			0
$11$	1141.87	+	1977.78i	0.0779913	+	0.135085i	0.902383	−	0.430935i	$-0.141816\pi$
$11$	1141.87	+	1977.78i	0.0779913	+	0.135085i	−0.824392	+	0.566019i	$0.808483\pi$
$12$		0			0
$13$			8314.85i			0.291126i	0.989349	+	0.145563i	$0.0464994\pi$
$13$			8314.85i			0.291126i	−0.989349	+	0.145563i	$0.953501\pi$
$14$		0			0
$15$	55389.5			1.09411
$16$		0			0
$17$	−63255.0	+	36520.3i	−0.757355	+	0.437259i	−0.828345	−	0.560218i	$-0.810717\pi$
$17$	−63255.0	+	36520.3i	−0.757355	+	0.437259i	0.0709904	+	0.997477i	$0.477384\pi$
$18$		0			0
$19$	218262.	+	126014.i	1.67481	+	0.966949i	0.964887	+	0.262665i	$0.0846015\pi$
$19$	218262.	+	126014.i	1.67481	+	0.966949i	0.709918	+	0.704284i	$0.248732\pi$
$20$		0			0
$21$	−149056.	+	112828.i	−0.766432	+	0.580147i
$22$		0			0
$23$	−142303.	+	246477.i	−0.508515	+	0.880773i	0.491437	+	0.870913i	$0.336472\pi$
$23$	−142303.	+	246477.i	−0.508515	+	0.880773i	−0.999951	+	0.00986012i	$0.996861\pi$
$24$		0			0
$25$	57726.8	+	99985.8i	0.147781	+	0.255964i
$26$		0			0
$27$		−	549674.i		−	1.03431i
$28$		0			0
$29$	−105335.			−0.148930			−0.0744649	−	0.997224i	$-0.523725\pi$
$29$	−105335.			−0.148930			−0.0744649	+	0.997224i	$0.523725\pi$
$30$		0			0
$31$	−743496.	+	429258.i	−0.805067	+	0.464805i	−0.845240	−	0.534387i	$-0.820542\pi$
$31$	−743496.	+	429258.i	−0.805067	+	0.464805i	0.0401732	+	0.999193i	$0.487209\pi$
$32$		0			0
$33$	−153991.	−	88906.9i	−0.129849	−	0.0749686i
$34$		0			0
$35$	−1.57371e6	−	663992.i	−1.04870	−	0.442477i
$36$		0			0
$37$	1.41911e6	−	2.45797e6i	0.757196	−	1.31150i	−0.187079	−	0.982345i	$-0.559902\pi$
$37$	1.41911e6	−	2.45797e6i	0.757196	−	1.31150i	0.944275	−	0.329157i	$-0.106765\pi$
$38$		0			0
$39$	−323700.	−	560665.i	−0.139921	−	0.242351i
$40$		0			0
$41$			1.75485e6i			0.621017i	0.950571	+	0.310509i	$0.100499\pi$
$41$			1.75485e6i			0.621017i	−0.950571	+	0.310509i	$0.899501\pi$
$42$		0			0
$43$	1.66109e6			0.485868			0.242934	−	0.970043i	$-0.421890\pi$
$43$	1.66109e6			0.485868			0.242934	+	0.970043i	$0.421890\pi$
$44$		0			0
$45$	307250.	−	177391.i	0.0749275	−	0.0432594i
$46$		0			0
$47$	−3.13716e6	−	1.81124e6i	−0.642904	−	0.371181i	0.142829	−	0.989747i	$-0.454380\pi$
$47$	−3.13716e6	−	1.81124e6i	−0.642904	−	0.371181i	−0.785732	+	0.618567i	$0.787714\pi$
$48$		0			0
$49$	5.58749e6	−	1.41878e6i	0.969242	−	0.246111i
$50$		0			0
$51$	2.84350e6	−	4.92508e6i	0.420312	−	0.728002i
$52$		0			0
$53$	−5.68440e6	−	9.84568e6i	−0.720413	−	1.24779i	−0.960834	−	0.277123i	$-0.910619\pi$
$53$	−5.68440e6	−	9.84568e6i	−0.720413	−	1.24779i	0.240422	−	0.970669i	$-0.422714\pi$
$54$		0			0
$55$		−	1.62464e6i		−	0.177544i
$56$		0			0
$57$	−1.96230e7			−1.85895
$58$		0			0
$59$	1.15832e7	−	6.68757e6i	0.955919	−	0.551900i	0.0610044	−	0.998137i	$-0.480570\pi$
$59$	1.15832e7	−	6.68757e6i	0.955919	−	0.551900i	0.894915	+	0.446237i	$0.147236\pi$
$60$		0			0
$61$	−1.24328e7	−	7.17810e6i	−0.897947	−	0.518430i	−0.0214132	−	0.999771i	$-0.506817\pi$
$61$	−1.24328e7	−	7.17810e6i	−0.897947	−	0.518430i	−0.876533	+	0.481341i	$0.840150\pi$
$62$		0			0
$63$	−465485.	+	1.10323e6i	−0.0295490	+	0.0700333i
$64$		0			0
$65$	2.95756e6	−	5.12265e6i	0.165684	−	0.286973i
$66$		0			0
$67$	−1.55171e7	−	2.68764e7i	−0.770036	−	1.33374i	−0.937543	−	0.347870i	$-0.886905\pi$
$67$	−1.55171e7	−	2.68764e7i	−0.770036	−	1.33374i	0.167507	−	0.985871i	$-0.446428\pi$
$68$		0			0
$69$		−	2.21597e7i		−	0.977613i
$70$		0			0
$71$	−3.33725e7			−1.31327			−0.656637	−	0.754207i	$-0.728022\pi$
$71$	−3.33725e7			−1.31327			−0.656637	+	0.754207i	$0.728022\pi$
$72$		0			0
$73$	−7.19825e6	+	4.15591e6i	−0.253475	+	0.146344i	−0.621354	−	0.783530i	$-0.713417\pi$
$73$	−7.19825e6	+	4.15591e6i	−0.253475	+	0.146344i	0.367879	+	0.929874i	$0.380084\pi$
$74$		0			0
$75$	−7.78496e6	−	4.49465e6i	−0.246043	−	0.142053i
$76$		0			0
$77$	3.30936e6	+	4.37199e6i	0.0941415	+	0.124370i
$78$		0			0
$79$	−3.40981e6	+	5.90597e6i	−0.0875431	+	0.151629i	−0.906472	−	0.422266i	$-0.861235\pi$
$79$	−3.40981e6	+	5.90597e6i	−0.0875431	+	0.151629i	0.818929	+	0.573895i	$0.194568\pi$
$80$		0			0
$81$	1.97630e7	+	3.42305e7i	0.459105	+	0.795193i
$82$		0			0
$83$			4.05040e7i			0.853465i	0.904378	+	0.426732i	$0.140335\pi$
$83$			4.05040e7i			0.853465i	−0.904378	+	0.426732i	$0.859665\pi$
$84$		0			0
$85$	5.19605e7			0.995400
$86$		0			0
$87$	7.10268e6	−	4.10074e6i	0.123978	−	0.0715789i
$88$		0			0
$89$	−3.65699e7	−	2.11136e7i	−0.582860	−	0.336514i	0.179409	−	0.983774i	$-0.442581\pi$
$89$	−3.65699e7	−	2.11136e7i	−0.582860	−	0.336514i	−0.762269	+	0.647260i	$0.775915\pi$
$90$		0			0
$91$	2.47578e6	+	1.98099e7i	0.0361033	+	0.288879i
$92$		0			0
$93$	3.34223e7	−	5.78891e7i	0.446791	−	0.773865i
$94$		0			0
$95$	−8.96452e7	−	1.55270e8i	−1.10061	−	1.90631i
$96$		0			0
$97$		−	9.29688e7i		−	1.05015i	−0.851057	−	0.525074i	$-0.824038\pi$
$97$		−	9.29688e7i		−	1.05015i	0.851057	−	0.525074i	$-0.175962\pi$
$98$		0			0
$99$	−1.13893e6			−0.0118565
$100$		0			0
$101$	−1.75882e8	+	1.01545e8i	−1.69019	+	0.975832i	−0.735834	+	0.677162i	$0.763210\pi$
$101$	−1.75882e8	+	1.01545e8i	−1.69019	+	0.975832i	−0.954356	+	0.298670i	$0.903457\pi$
$102$		0			0
$103$	4.00729e7	+	2.31361e7i	0.356042	+	0.205561i	0.667343	−	0.744750i	$-0.267431\pi$
$103$	4.00729e7	+	2.31361e7i	0.356042	+	0.205561i	−0.311301	+	0.950311i	$0.600765\pi$
$104$		0			0
$105$	1.31964e8	−	1.64925e7i	1.08567	−	0.135684i
$106$		0			0
$107$	1.08056e8	−	1.87159e8i	0.824357	−	1.42783i	−0.0780533	−	0.996949i	$-0.524870\pi$
$107$	1.08056e8	−	1.87159e8i	0.824357	−	1.42783i	0.902410	−	0.430878i	$-0.141796\pi$
$108$		0			0
$109$	−5.17692e7	−	8.96669e7i	−0.366746	−	0.635223i	0.622309	−	0.782772i	$-0.286195\pi$
$109$	−5.17692e7	−	8.96669e7i	−0.366746	−	0.635223i	−0.989055	+	0.147549i	$0.952862\pi$
$110$		0			0
$111$			2.20985e8i			1.45570i
$112$		0			0
$113$	−2.87471e8			−1.76311			−0.881556	−	0.472079i	$-0.843504\pi$
$113$	−2.87471e8			−1.76311			−0.881556	+	0.472079i	$0.843504\pi$
$114$		0			0
$115$	1.75341e8	−	1.01233e8i	1.00252	−	0.578805i
$116$		0			0
$117$	−3.59118e6	−	2.07337e6i	−0.0191643	−	0.0110645i
$118$		0			0
$119$	−1.39829e8	+	1.05843e8i	−0.697283	+	0.527805i
$120$		0			0
$121$	1.04572e8	−	1.81123e8i	0.487835	−	0.844954i
$122$		0			0
$123$	−6.83168e7	−	1.18328e8i	−0.298474	−	0.516972i
$124$		0			0
$125$			1.95755e8i			0.801811i
$126$		0			0
$127$	1.09348e8			0.420336			0.210168	−	0.977665i	$-0.432599\pi$
$127$	1.09348e8			0.420336			0.210168	+	0.977665i	$0.432599\pi$
$128$		0			0
$129$	−1.12006e8	+	6.46667e7i	−0.404466	+	0.233519i
$130$		0			0
$131$	1.61254e8	+	9.31002e7i	0.547553	+	0.316130i	0.748134	−	0.663547i	$-0.230950\pi$
$131$	1.61254e8	+	9.31002e7i	0.547553	+	0.316130i	−0.200582	+	0.979677i	$0.564283\pi$
$132$		0			0
$133$	5.57524e8	+	2.35235e8i	1.78179	+	0.751788i
$134$		0			0
$135$	−1.95517e8	+	3.38645e8i	−0.588639	+	1.01955i
$136$		0			0
$137$	3.19189e8	+	5.52852e8i	0.906078	+	1.56937i	0.819463	+	0.573132i	$0.194272\pi$
$137$	3.19189e8	+	5.52852e8i	0.906078	+	1.56937i	0.0866147	+	0.996242i	$0.472395\pi$
$138$		0			0
$139$			3.54755e8i			0.950318i	0.879900	+	0.475159i	$0.157609\pi$
$139$			3.54755e8i			0.950318i	−0.879900	+	0.475159i	$0.842391\pi$
$140$		0			0
$141$	2.82049e8			0.713589
$142$		0			0
$143$	−1.64449e7	+	9.49449e6i	−0.0393268	+	0.0227053i
$144$		0			0
$145$	6.48953e7	+	3.74673e7i	0.146805	+	0.0847580i
$146$		0			0
$147$	−3.21527e8	+	3.13190e8i	−0.688570	+	0.670716i
$148$		0			0
$149$	1.47313e8	−	2.55154e8i	0.298880	−	0.517675i	−0.677000	−	0.735983i	$-0.736720\pi$
$149$	1.47313e8	−	2.55154e8i	0.298880	−	0.517675i	0.975880	+	0.218308i	$0.0700537\pi$
$150$		0			0
$151$	−1.80004e8	−	3.11777e8i	−0.346238	−	0.599702i	0.639340	−	0.768925i	$-0.279208\pi$
$151$	−1.80004e8	−	3.11777e8i	−0.346238	−	0.599702i	−0.985578	+	0.169222i	$0.945874\pi$
$152$		0			0
$153$		−	3.64264e7i		−	0.0664738i
$154$		0			0
$155$	6.10741e8			1.05811
$156$		0			0
$157$	−5.46689e8	+	3.15631e8i	−0.899790	+	0.519494i	−0.877132	−	0.480249i	$-0.840546\pi$
$157$	−5.46689e8	+	3.15631e8i	−0.899790	+	0.519494i	−0.0226582	+	0.999743i	$0.507213\pi$
$158$		0			0
$159$	7.66591e8	+	4.42592e8i	1.19943	+	0.692492i
$160$		0			0
$161$	−2.65643e8	+	6.29593e8i	−0.395362	+	0.937037i
$162$		0			0
$163$	−3.28000e8	+	5.68113e8i	−0.464648	+	0.804793i	−0.999186	−	0.0403513i	$-0.987152\pi$
$163$	−3.28000e8	+	5.68113e8i	−0.464648	+	0.804793i	0.534538	+	0.845144i	$0.320486\pi$
$164$		0			0
$165$	6.32476e7	+	1.09548e8i	0.0853313	+	0.147798i
$166$		0			0
$167$		−	2.33854e8i		−	0.300662i	−0.988636	−	0.150331i	$-0.951966\pi$
$167$		−	2.33854e8i		−	0.300662i	0.988636	−	0.150331i	$-0.0480340\pi$
$168$		0			0
$169$	7.46594e8			0.915246
$170$		0			0
$171$	−1.08851e8	+	6.28449e7i	−0.127305	+	0.0734997i
$172$		0			0
$173$	−2.00353e8	−	1.15674e8i	−0.223672	−	0.129137i	0.383977	−	0.923343i	$-0.374554\pi$
$173$	−2.00353e8	−	1.15674e8i	−0.223672	−	0.129137i	−0.607649	+	0.794205i	$0.707887\pi$
$174$		0			0
$175$	1.67303e8	+	2.21024e8i	0.178383	+	0.235661i
$176$		0			0
$177$	−5.20699e8	+	9.01877e8i	−0.530510	+	0.918871i
$178$		0			0
$179$	−3.67392e8	−	6.36342e8i	−0.357864	−	0.619838i	0.629740	−	0.776806i	$-0.283161\pi$
$179$	−3.67392e8	−	6.36342e8i	−0.357864	−	0.619838i	−0.987604	+	0.156968i	$0.949828\pi$
$180$		0			0
$181$		−	1.76795e9i		−	1.64723i	−0.567147	−	0.823616i	$-0.691953\pi$
$181$		−	1.76795e9i		−	1.64723i	0.567147	−	0.823616i	$-0.308047\pi$
$182$		0			0
$183$	1.11778e9			0.996674
$184$		0			0
$185$	−1.74858e9	+	1.00954e9i	−1.49279	+	0.861861i
$186$		0			0
$187$	−1.44458e8	−	8.34030e7i	−0.118134	−	0.0682048i
$188$		0			0
$189$	−1.63668e8	−	1.30958e9i	−0.128267	−	1.02632i
$190$		0			0
$191$	1.15726e9	−	2.00444e9i	0.869558	−	1.50612i	0.00710853	−	0.999975i	$-0.497737\pi$
$191$	1.15726e9	−	2.00444e9i	0.869558	−	1.50612i	0.862449	−	0.506144i	$-0.168929\pi$
$192$		0			0
$193$	−7.41515e8	−	1.28434e9i	−0.534430	−	0.925660i	−0.999191	−	0.0402236i	$-0.987193\pi$
$193$	−7.41515e8	−	1.28434e9i	−0.534430	−	0.925660i	0.464761	−	0.885436i	$-0.346140\pi$
$194$		0			0
$195$			4.60555e8i			0.318525i
$196$		0			0
$197$	9.91728e8			0.658458			0.329229	−	0.944250i	$-0.393211\pi$
$197$	9.91728e8			0.658458			0.329229	+	0.944250i	$0.393211\pi$
$198$		0			0
$199$	−9.44567e8	+	5.45346e8i	−0.602310	+	0.347744i	−0.769950	−	0.638104i	$-0.779719\pi$
$199$	−9.44567e8	+	5.45346e8i	−0.602310	+	0.347744i	0.167640	+	0.985848i	$0.446386\pi$
$200$		0			0
$201$	2.09261e9	+	1.20817e9i	1.28205	+	0.740191i
$202$		0			0
$203$	−2.50958e8	+	3.13640e7i	−0.147780	+	0.0184692i
$204$		0			0
$205$	6.24192e8	−	1.08113e9i	0.353429	−	0.612158i
$206$		0			0
$207$	−7.09687e7	−	1.22921e8i	−0.0386532	−	0.0669493i
$208$		0			0
$209$			5.75566e8i			0.301655i
$210$		0			0
$211$	−2.72265e9			−1.37361			−0.686804	−	0.726843i	$-0.740987\pi$
$211$	−2.72265e9			−1.37361			−0.686804	+	0.726843i	$0.740987\pi$
$212$		0			0
$213$	2.25028e9	−	1.29920e9i	1.09325	−	0.631188i
$214$		0			0
$215$	−1.02337e9	−	5.90842e8i	−0.478937	−	0.276514i
$216$		0			0
$217$	−1.64354e9	+	1.24407e9i	−0.741210	+	0.561056i
$218$		0			0
$219$	3.23582e8	−	5.60461e8i	0.140672	−	0.243651i
$220$		0			0
$221$	−3.03661e8	−	5.25956e8i	−0.127298	−	0.220486i
$222$		0			0
$223$		−	2.79895e9i		−	1.13181i	−0.824469	−	0.565907i	$-0.808526\pi$
$223$		−	2.79895e9i		−	1.13181i	0.824469	−	0.565907i	$-0.191474\pi$
$224$		0			0
$225$	−5.75784e7			−0.0224662
$226$		0			0
$227$	−2.83318e9	+	1.63574e9i	−1.06702	+	0.616042i	−0.927365	−	0.374159i	$-0.877932\pi$
$227$	−2.83318e9	+	1.63574e9i	−1.06702	+	0.616042i	−0.139651	+	0.990201i	$0.544598\pi$
$228$		0			0
$229$	−4.31582e8	−	2.49174e8i	−0.156936	−	0.0906068i	0.419476	−	0.907767i	$-0.362214\pi$
$229$	−4.31582e8	−	2.49174e8i	−0.156936	−	0.0906068i	−0.576411	+	0.817160i	$0.695547\pi$
$230$		0			0
$231$	−3.93351e8	−	1.65966e8i	−0.138144	−	0.0582869i
$232$		0			0
$233$	2.05020e8	−	3.55106e8i	0.0695622	−	0.120485i	−0.829146	−	0.559031i	$-0.811173\pi$
$233$	2.05020e8	−	3.55106e8i	0.0695622	−	0.120485i	0.898709	+	0.438546i	$0.144506\pi$
$234$		0			0
$235$	1.28850e9	+	2.23175e9i	0.422488	+	0.731770i
$236$		0			0
$237$		−	5.30981e8i		−	0.168300i
$238$		0			0
$239$	2.65129e9			0.812578			0.406289	−	0.913745i	$-0.366823\pi$
$239$	2.65129e9			0.812578			0.406289	+	0.913745i	$0.366823\pi$
$240$		0			0
$241$	−1.98398e9	+	1.14545e9i	−0.588125	+	0.339554i	−0.764356	−	0.644795i	$-0.776943\pi$
$241$	−1.98398e9	+	1.14545e9i	−0.588125	+	0.339554i	0.176231	+	0.984349i	$0.443610\pi$
$242$		0			0
$243$	4.58037e8	+	2.64448e8i	0.131364	+	0.0758429i
$244$		0			0
$245$	−3.94701e9	−	1.11336e9i	−1.09548	−	0.309009i
$246$		0			0
$247$	−1.04779e9	+	1.81482e9i	−0.281504	+	0.487580i
$248$		0			0
$249$	−1.57683e9	−	2.73116e9i	−0.410193	−	0.710476i
$250$		0			0
$251$		−	6.86113e9i		−	1.72862i	−0.502956	−	0.864312i	$-0.667754\pi$
$251$		−	6.86113e9i		−	1.72862i	0.502956	−	0.864312i	$-0.332246\pi$
$252$		0			0
$253$	−6.49968e8			−0.158639
$254$		0			0
$255$	−3.50366e9	+	2.02284e9i	−0.828632	+	0.478411i
$256$		0			0
$257$	−4.96090e9	−	2.86418e9i	−1.13718	−	0.656550i	−0.191447	−	0.981503i	$-0.561318\pi$
$257$	−4.96090e9	−	2.86418e9i	−1.13718	−	0.656550i	−0.945730	+	0.324953i	$0.894651\pi$
$258$		0			0
$259$	2.64910e9	−	6.27857e9i	0.588708	−	1.39528i
$260$		0			0
$261$	2.62661e7	−	4.54942e7i	0.00566022	−	0.00980379i
$262$		0			0
$263$	8.97165e8	+	1.55394e9i	0.187521	+	0.324796i	0.944423	−	0.328733i	$-0.106621\pi$
$263$	8.97165e8	+	1.55394e9i	0.187521	+	0.324796i	−0.756902	+	0.653528i	$0.773288\pi$
$264$		0			0
$265$			8.08768e9i			1.63999i
$266$		0			0
$267$	3.28785e9			0.646944
$268$		0			0
$269$	7.31848e9	−	4.22532e9i	1.39769	−	0.806958i	0.403542	−	0.914961i	$-0.367779\pi$
$269$	7.31848e9	−	4.22532e9i	1.39769	−	0.806958i	0.994151	+	0.108003i	$0.0344457\pi$
$270$		0			0
$271$	−8.03480e8	−	4.63890e8i	−0.148970	−	0.0860077i	0.423662	−	0.905820i	$-0.360744\pi$
$271$	−8.03480e8	−	4.63890e8i	−0.148970	−	0.0860077i	−0.572632	+	0.819812i	$0.694078\pi$
$272$		0			0
$273$	−9.38145e8	−	1.23938e9i	−0.168896	−	0.223128i
$274$		0			0
$275$	−1.31833e8	+	2.28342e8i	−0.0230512	+	0.0399259i
$276$		0			0
$277$	−4.40624e9	−	7.63184e9i	−0.748427	−	1.29631i	−0.948576	−	0.316548i	$-0.897476\pi$
$277$	−4.40624e9	−	7.63184e9i	−0.748427	−	1.29631i	0.200150	−	0.979765i	$-0.435857\pi$
$278$		0			0
$279$		−	4.28154e8i		−	0.0706615i
$280$		0			0
$281$	−9.34848e8			−0.149939			−0.0749697	−	0.997186i	$-0.523886\pi$
$281$	−9.34848e8			−0.149939			−0.0749697	+	0.997186i	$0.523886\pi$
$282$		0			0
$283$	2.07905e9	−	1.20034e9i	0.324130	−	0.187137i	−0.329102	−	0.944294i	$-0.606746\pi$
$283$	2.07905e9	−	1.20034e9i	0.324130	−	0.187137i	0.653232	+	0.757158i	$0.273413\pi$
$284$		0			0
$285$	1.20894e10	+	6.97984e9i	1.83243	+	1.05795i
$286$		0			0
$287$	5.22513e8	+	4.18086e9i	0.0770140	+	0.616223i
$288$		0			0
$289$	−8.20413e8	+	1.42100e9i	−0.117609	+	0.203705i
$290$		0			0
$291$	3.61931e9	+	6.26882e9i	0.504724	+	0.874207i
$292$		0			0
$293$		−	1.88044e9i		−	0.255146i	−0.991829	−	0.127573i	$-0.959281\pi$
$293$		−	1.88044e9i		−	0.255146i	0.991829	−	0.127573i	$-0.0407187\pi$
$294$		0			0
$295$	−9.51497e9			−1.25638
$296$		0			0
$297$	1.08713e9	−	6.27657e8i	0.139720	−	0.0806671i
$298$		0			0
$299$	−2.04942e9	−	1.18323e9i	−0.256416	−	0.148042i
$300$		0			0
$301$	3.95748e9	−	4.94596e8i	0.482118	−	0.0602538i
$302$		0			0
$303$	7.90640e9	−	1.36943e10i	0.938012	−	1.62468i
$304$		0			0
$305$	5.10644e9	+	8.84461e9i	0.590091	+	1.02207i
$306$		0			0
$307$			1.12605e10i			1.26766i	0.773471	+	0.633832i	$0.218519\pi$
$307$			1.12605e10i			1.26766i	−0.773471	+	0.633832i	$0.781481\pi$
$308$		0			0
$309$	−3.60278e9			−0.395188
$310$		0			0
$311$	−1.00193e10	+	5.78466e9i	−1.07102	+	0.618353i	−0.928459	−	0.371434i	$-0.878866\pi$
$311$	−1.00193e10	+	5.78466e9i	−1.07102	+	0.618353i	−0.142559	+	0.989786i	$0.545533\pi$
$312$		0			0
$313$	5.74494e9	+	3.31684e9i	0.598561	+	0.345579i	0.768475	−	0.639880i	$-0.221016\pi$
$313$	5.74494e9	+	3.31684e9i	0.598561	+	0.345579i	−0.169914	+	0.985459i	$0.554349\pi$
$314$		0			0
$315$	6.79193e8	−	5.14112e8i	0.0689844	−	0.0522174i
$316$		0			0
$317$	6.66436e9	−	1.15430e10i	0.659966	−	1.14310i	−0.320658	−	0.947195i	$-0.603904\pi$
$317$	6.66436e9	−	1.15430e10i	0.659966	−	1.14310i	0.980624	−	0.195900i	$-0.0627628\pi$
$318$		0			0
$319$	−1.20279e8	−	2.08330e8i	−0.0116152	−	0.0201182i
$320$		0			0
$321$			1.68267e10i			1.58481i
$322$		0			0
$323$	−1.84083e10			−1.69123
$324$		0			0
$325$	−8.31367e8	+	4.79990e8i	−0.0745177	+	0.0430228i
$326$		0			0
$327$	6.98152e9	+	4.03078e9i	0.610603	+	0.352532i
$328$		0			0
$329$	−8.01349e9	−	3.38112e9i	−0.683972	−	0.288587i
$330$		0			0
$331$	3.10229e9	−	5.37332e9i	0.258446	−	0.447642i	−0.707380	−	0.706834i	$-0.750123\pi$
$331$	3.10229e9	−	5.37332e9i	0.258446	−	0.447642i	0.965826	+	0.259192i	$0.0834562\pi$
$332$		0			0
$333$	7.07729e8	+	1.22582e9i	0.0575559	+	0.0996898i
$334$		0			0
$335$			2.20775e10i			1.75295i
$336$		0			0
$337$	−1.25909e10			−0.976196			−0.488098	−	0.872789i	$-0.662309\pi$
$337$	−1.25909e10			−0.976196			−0.488098	+	0.872789i	$0.662309\pi$
$338$		0			0
$339$	1.93840e10	−	1.11913e10i	1.46772	−	0.847390i
$340$		0			0
$341$	−1.69795e9	−	9.80314e8i	−0.125576	−	0.0725016i
$342$		0			0
$343$	1.28895e10	−	5.04389e9i	0.931239	−	0.364409i
$344$		0			0
$345$	−7.88210e9	+	1.36522e10i	−0.556373	+	0.963666i
$346$		0			0
$347$	8.11191e9	+	1.40502e10i	0.559507	+	0.969094i	0.997538	+	0.0701340i	$0.0223427\pi$
$347$	8.11191e9	+	1.40502e10i	0.559507	+	0.969094i	−0.438031	+	0.898960i	$0.644324\pi$
$348$		0			0
$349$			1.03313e9i			0.0696391i	0.999394	+	0.0348196i	$0.0110857\pi$
$349$			1.03313e9i			0.0696391i	−0.999394	+	0.0348196i	$0.988914\pi$
$350$		0			0
$351$	4.57046e9			0.301114
$352$		0			0
$353$	−1.02373e10	+	5.91051e9i	−0.659306	+	0.380651i	−0.792013	−	0.610505i	$-0.790967\pi$
$353$	−1.02373e10	+	5.91051e9i	−0.659306	+	0.380651i	0.132706	+	0.991155i	$0.457633\pi$
$354$		0			0
$355$	2.05602e10	+	1.18705e10i	1.29454	+	0.747402i
$356$		0			0
$357$	5.30807e9	−	1.25805e10i	0.326786	−	0.774506i
$358$		0			0
$359$	−7.42552e9	+	1.28614e10i	−0.447042	+	0.774300i	−0.998192	−	0.0601062i	$-0.980856\pi$
$359$	−7.42552e9	+	1.28614e10i	−0.447042	+	0.774300i	0.551149	+	0.834407i	$0.314189\pi$
$360$		0			0
$361$	2.32672e10	+	4.02999e10i	1.36998	+	2.37288i
$362$		0			0
$363$			1.62840e10i			0.937855i
$364$		0			0
$365$	5.91296e9			0.333145
$366$		0			0
$367$	−1.92235e8	+	1.10987e8i	−0.0105966	+	0.00611796i	−0.505289	−	0.862950i	$-0.668614\pi$
$367$	−1.92235e8	+	1.10987e8i	−0.0105966	+	0.00611796i	0.494692	+	0.869068i	$0.335281\pi$
$368$		0			0
$369$	−7.57917e8	−	4.37583e8i	−0.0408805	−	0.0236024i
$370$		0			0
$371$	−1.64745e10	−	2.17644e10i	−0.869594	−	1.14882i
$372$		0			0
$373$	1.16778e10	−	2.02265e10i	0.603288	−	1.04493i	−0.389031	−	0.921225i	$-0.627190\pi$
$373$	1.16778e10	−	2.02265e10i	0.603288	−	1.04493i	0.992319	−	0.123701i	$-0.0394765\pi$
$374$		0			0
$375$	−7.62080e9	−	1.31996e10i	−0.385368	−	0.667477i
$376$		0			0
$377$		−	8.75847e8i		−	0.0433574i
$378$		0			0
$379$	−4.13272e8			−0.0200299			−0.0100150	−	0.999950i	$-0.503188\pi$
$379$	−4.13272e8			−0.0200299			−0.0100150	+	0.999950i	$0.503188\pi$
$380$		0			0
$381$	−7.37328e9	+	4.25696e9i	−0.349914	+	0.202023i
$382$		0			0
$383$	1.36941e10	+	7.90631e9i	0.636414	+	0.367434i	0.783232	−	0.621730i	$-0.213570\pi$
$383$	1.36941e10	+	7.90631e9i	0.636414	+	0.367434i	−0.146818	+	0.989164i	$0.546903\pi$
$384$		0			0
$385$	−4.83743e8	−	3.87064e9i	−0.0220177	−	0.176173i
$386$		0			0
$387$	−4.14204e8	+	7.17422e8i	−0.0184659	+	0.0319839i
$388$		0			0
$389$	1.91657e8	+	3.31960e8i	0.00837001	+	0.0144973i	0.870180	−	0.492734i	$-0.164002\pi$
$389$	1.91657e8	+	3.31960e8i	0.00837001	+	0.0144973i	−0.861810	+	0.507231i	$0.830669\pi$
$390$		0			0
$391$		−	2.07878e10i		−	0.889411i
$392$		0			0
$393$	−1.44977e10			−0.607755
$394$		0			0
$395$	4.20146e9	−	2.42571e9i	0.172588	−	0.0996440i
$396$		0			0
$397$	3.61525e9	+	2.08727e9i	0.145538	+	0.0840265i	0.571001	−	0.820949i	$-0.306555\pi$
$397$	3.61525e9	+	2.08727e9i	0.145538	+	0.0840265i	−0.425463	+	0.904976i	$0.639889\pi$
$398$		0			0
$399$	−4.67512e10	+	5.84285e9i	−1.84460	+	0.230533i
$400$		0			0
$401$	1.51066e10	−	2.61654e10i	0.584238	−	1.01193i	−0.410732	−	0.911756i	$-0.634727\pi$
$401$	1.51066e10	−	2.61654e10i	0.584238	−	1.01193i	0.994970	−	0.100174i	$-0.0319400\pi$
$402$		0			0
$403$	−3.56921e9	−	6.18206e9i	−0.135317	−	0.234376i
$404$		0			0
$405$		−	2.81184e10i		−	1.04513i
$406$		0			0
$407$	6.48175e9			0.236219
$408$		0			0
$409$	2.37437e9	−	1.37084e9i	0.0848507	−	0.0489886i	−0.456974	−	0.889480i	$-0.651067\pi$
$409$	2.37437e9	−	1.37084e9i	0.0848507	−	0.0489886i	0.541825	+	0.840491i	$0.317734\pi$
$410$		0			0
$411$	−4.30454e10	−	2.48523e10i	−1.50855	−	0.870961i
$412$		0			0
$413$	2.56054e10	−	1.93819e10i	0.880097	−	0.666186i
$414$		0			0
$415$	1.44071e10	−	2.49538e10i	0.485718	−	0.841289i
$416$		0			0
$417$	−1.38107e10	−	2.39209e10i	−0.456743	−	0.791102i
$418$		0			0
$419$		−	3.48634e9i		−	0.113113i	−0.998399	−	0.0565567i	$-0.981988\pi$
$419$		−	3.48634e9i		−	0.113113i	0.998399	−	0.0565567i	$-0.0180122\pi$
$420$		0			0
$421$	−1.91451e10			−0.609438			−0.304719	−	0.952442i	$-0.598563\pi$
$421$	−1.91451e10			−0.609438			−0.304719	+	0.952442i	$0.598563\pi$
$422$		0			0
$423$	1.56455e9	−	9.03292e8i	0.0488683	−	0.0282141i
$424$		0			0
$425$	−7.30302e9	−	4.21640e9i	−0.223845	−	0.129237i
$426$		0			0
$427$	−3.17581e10	−	1.33996e10i	−0.955307	−	0.403071i
$428$		0			0
$429$	7.39248e8	−	1.28041e9i	0.0218253	−	0.0378026i
$430$		0			0
$431$	−2.38075e10	−	4.12358e10i	−0.689930	−	1.19499i	−0.971860	−	0.235558i	$-0.924308\pi$
$431$	−2.38075e10	−	4.12358e10i	−0.689930	−	1.19499i	0.281931	−	0.959435i	$-0.409025\pi$
$432$		0			0
$433$		−	1.83719e10i		−	0.522640i	−0.965252	−	0.261320i	$-0.915842\pi$
$433$		−	1.83719e10i		−	0.522640i	0.965252	−	0.261320i	$-0.0841579\pi$
$434$		0			0
$435$	−5.83446e9			−0.162946
$436$		0			0
$437$	−6.21189e10	+	3.58644e10i	−1.70333	+	0.983416i
$438$		0			0
$439$	5.33233e10	+	3.07862e10i	1.43568	+	0.828892i	0.997546	−	0.0700162i	$-0.0223051\pi$
$439$	5.33233e10	+	3.07862e10i	1.43568	+	0.828892i	0.438137	+	0.898908i	$0.355638\pi$
$440$		0			0
$441$	−7.80510e8	+	2.76701e9i	−0.0206359	+	0.0731572i
$442$		0			0
$443$	8.71680e8	−	1.50979e9i	0.0226330	−	0.0392015i	−0.854487	−	0.519473i	$-0.826128\pi$
$443$	8.71680e8	−	1.50979e9i	0.0226330	−	0.0392015i	0.877120	+	0.480271i	$0.159462\pi$
$444$		0			0
$445$	1.50201e10	+	2.60155e10i	0.383030	+	0.663427i
$446$		0			0
$447$			2.29398e10i			0.574592i
$448$		0			0
$449$	−2.37418e10			−0.584156			−0.292078	−	0.956395i	$-0.594347\pi$
$449$	−2.37418e10			−0.584156			−0.292078	+	0.956395i	$0.594347\pi$
$450$		0			0
$451$	−3.47070e9	+	2.00381e9i	−0.0838901	+	0.0484340i
$452$		0			0
$453$	2.42751e10	+	1.40153e10i	0.576460	+	0.332819i
$454$		0			0
$455$	5.52100e9	−	1.30852e10i	0.128817	−	0.305305i
$456$		0			0
$457$	3.05435e10	−	5.29030e10i	0.700253	−	1.21287i	−0.268125	−	0.963384i	$-0.586404\pi$
$457$	3.05435e10	−	5.29030e10i	0.700253	−	1.21287i	0.968378	−	0.249489i	$-0.0802626\pi$
$458$		0			0
$459$	2.00743e10	+	3.47697e10i	0.452261	+	0.783339i
$460$		0			0
$461$			3.26327e10i			0.722518i	0.932465	+	0.361259i	$0.117653\pi$
$461$			3.26327e10i			0.722518i	−0.932465	+	0.361259i	$0.882347\pi$
$462$		0			0
$463$	4.98874e10			1.08559			0.542797	−	0.839864i	$-0.317365\pi$
$463$	4.98874e10			1.08559			0.542797	+	0.839864i	$0.317365\pi$
$464$		0			0
$465$	−4.11818e10	+	2.37763e10i	−0.880834	+	0.508550i
$466$		0			0
$467$	6.22519e10	+	3.59412e10i	1.30884	+	0.755657i	0.981902	−	0.189391i	$-0.0606513\pi$
$467$	6.22519e10	+	3.59412e10i	1.30884	+	0.755657i	0.326934	+	0.945047i	$0.393985\pi$
$468$		0			0
$469$	−4.49715e10	−	5.94118e10i	−0.929492	−	1.22795i
$470$		0			0
$471$	2.45752e10	−	4.25655e10i	0.499360	−	0.864917i
$472$		0			0
$473$	1.89675e9	+	3.28526e9i	0.0378935	+	0.0656335i
$474$		0			0
$475$			2.90975e10i			0.571586i
$476$		0			0
$477$	5.66979e9			0.109520
$478$		0			0
$479$	−4.14306e10	+	2.39199e10i	−0.787007	+	0.454379i	−0.838908	−	0.544273i	$-0.816805\pi$
$479$	−4.14306e10	+	2.39199e10i	−0.787007	+	0.454379i	0.0519006	+	0.998652i	$0.483472\pi$
$480$		0			0
$481$	2.04376e10	+	1.17997e10i	0.381813	+	0.220440i
$482$		0			0
$483$	−6.59814e9	−	5.27946e10i	−0.121236	−	0.970066i
$484$		0			0
$485$	−3.30686e10	+	5.72766e10i	−0.597653	+	1.03517i
$486$		0			0
$487$	−1.96693e10	−	3.40683e10i	−0.349683	−	0.605668i	0.636510	−	0.771268i	$-0.280377\pi$
$487$	−1.96693e10	−	3.40683e10i	−0.349683	−	0.605668i	−0.986193	+	0.165600i	$0.947044\pi$
$488$		0			0
$489$		−	5.10766e10i		−	0.893278i
$490$		0			0
$491$	−1.00258e11			−1.72502			−0.862509	−	0.506041i	$-0.831108\pi$
$491$	−1.00258e11			−1.72502			−0.862509	+	0.506041i	$0.831108\pi$
$492$		0			0
$493$	6.66298e9	−	3.84688e9i	0.112793	−	0.0651209i
$494$		0			0
$495$	7.01679e8	+	4.05115e8i	0.0116874	+	0.00674772i
$496$		0			0
$497$	−7.95088e10	+	9.93681e9i	−1.30314	+	0.162863i
$498$		0			0
$499$	−7.63552e9	+	1.32251e10i	−0.123151	+	0.213303i	−0.921009	−	0.389542i	$-0.872633\pi$
$499$	−7.63552e9	+	1.32251e10i	−0.123151	+	0.213303i	0.797858	+	0.602846i	$0.205966\pi$
$500$		0			0
$501$	9.10402e9	+	1.57686e10i	0.144505	+	0.250290i
$502$		0			0
$503$		−	4.70851e10i		−	0.735549i	−0.929915	−	0.367774i	$-0.880120\pi$
$503$		−	4.70851e10i		−	0.735549i	0.929915	−	0.367774i	$-0.119880\pi$
$504$		0			0
$505$	1.44477e11			2.22144
$506$		0			0
$507$	−5.03423e10	+	2.90652e10i	−0.761906	+	0.439887i
$508$		0			0
$509$	4.97031e10	+	2.86961e10i	0.740478	+	0.427515i	0.822243	−	0.569136i	$-0.192722\pi$
$509$	4.97031e10	+	2.86961e10i	0.740478	+	0.427515i	−0.0817648	+	0.996652i	$0.526056\pi$
$510$		0			0
$511$	−1.59121e10	+	1.20446e10i	−0.233370	+	0.176648i
$512$		0			0
$513$	6.92665e10	−	1.19973e11i	1.00012	−	1.73227i
$514$		0			0
$515$	−1.64588e10	−	2.85075e10i	−0.233975	−	0.405257i
$516$		0			0
$517$		−	8.27282e9i		−	0.115795i
$518$		0			0
$519$	1.80129e10			0.248264
$520$		0			0
$521$	−4.54486e10	+	2.62397e10i	−0.616836	+	0.356130i	−0.775636	−	0.631181i	$-0.782571\pi$
$521$	−4.54486e10	+	2.62397e10i	−0.616836	+	0.356130i	0.158800	+	0.987311i	$0.449237\pi$
$522$		0			0
$523$	1.95547e10	+	1.12899e10i	0.261363	+	0.150898i	0.624956	−	0.780660i	$-0.285117\pi$
$523$	1.95547e10	+	1.12899e10i	0.261363	+	0.150898i	−0.363593	+	0.931558i	$0.618450\pi$
$524$		0			0
$525$	−1.98857e10	−	8.39034e9i	−0.261760	−	0.110444i
$526$		0			0
$527$	3.13532e10	−	5.43054e10i	0.406481	−	0.704045i
$528$		0			0
$529$	−1.34496e9	−	2.32954e9i	−0.0171746	−	0.0297473i
$530$		0			0
$531$			6.67038e9i			0.0839020i
$532$		0			0
$533$	−1.45913e10			−0.180794
$534$		0			0
$535$	−1.33143e11	+	7.68704e10i	−1.62519	+	0.938305i
$536$		0			0
$537$	4.95460e10	+	2.86054e10i	0.595815	+	0.343994i
$538$		0			0
$539$	9.18622e9	+	9.43075e9i	0.108838	+	0.111735i
$540$		0			0
$541$	−3.77813e10	+	6.54391e10i	−0.441050	+	0.763920i	−0.997768	−	0.0667812i	$-0.978727\pi$
$541$	−3.77813e10	+	6.54391e10i	−0.441050	+	0.763920i	0.556718	+	0.830702i	$0.312060\pi$
$542$		0			0
$543$	6.88268e10	+	1.19211e11i	0.791696	+	1.37126i
$544$		0			0
$545$			7.36564e10i			0.834881i
$546$		0			0
$547$	1.03691e11			1.15823			0.579113	−	0.815247i	$-0.303399\pi$
$547$	1.03691e11			1.15823			0.579113	+	0.815247i	$0.303399\pi$
$548$		0			0
$549$	6.20043e9	−	3.57982e9i	0.0682546	−	0.0394068i
$550$		0			0
$551$	−2.29907e10	−	1.32737e10i	−0.249428	−	0.144008i
$552$		0			0
$553$	−6.36523e9	+	1.50861e10i	−0.0680634	+	0.161315i
$554$		0			0
$555$	7.86036e10	−	1.36145e11i	0.828458	−	1.43493i
$556$		0			0
$557$	−8.78292e10	−	1.52125e11i	−0.912469	−	1.58044i	−0.810565	−	0.585649i	$-0.800840\pi$
$557$	−8.78292e10	−	1.52125e11i	−0.912469	−	1.58044i	−0.101905	−	0.994794i	$-0.532494\pi$
$558$		0			0
$559$			1.38117e10i			0.141449i
$560$		0			0
$561$	1.29876e10			0.131123
$562$		0			0
$563$	9.89713e8	−	5.71411e8i	0.00985089	−	0.00568741i	−0.495066	−	0.868855i	$-0.664856\pi$
$563$	9.89713e8	−	5.71411e8i	0.00985089	−	0.00568741i	0.504917	+	0.863168i	$0.331523\pi$
$564$		0			0
$565$	1.77106e11	+	1.02252e11i	1.73796	+	1.00341i
$566$		0			0
$567$	5.72769e10	+	7.56684e10i	0.554175	+	0.732120i
$568$		0			0
$569$	2.53958e10	−	4.39868e10i	0.242277	−	0.419636i	−0.719085	−	0.694922i	$-0.755439\pi$
$569$	2.53958e10	−	4.39868e10i	0.242277	−	0.419636i	0.961363	+	0.275285i	$0.0887724\pi$
$570$		0			0
$571$	5.58711e10	+	9.67715e10i	0.525584	+	0.910339i	0.999556	+	0.0297987i	$0.00948663\pi$
$571$	5.58711e10	+	9.67715e10i	0.525584	+	0.910339i	−0.473972	+	0.880540i	$0.657180\pi$
$572$		0			0
$573$			1.80211e11i			1.67171i
$574$		0			0
$575$	−3.28589e10			−0.300595
$576$		0			0
$577$	6.57318e10	−	3.79503e10i	0.593024	−	0.342382i	−0.173269	−	0.984875i	$-0.555433\pi$
$577$	6.57318e10	−	3.79503e10i	0.593024	−	0.342382i	0.766292	+	0.642492i	$0.222100\pi$
$578$		0			0
$579$	9.99997e10	+	5.77349e10i	0.889784	+	0.513717i
$580$		0			0
$581$	1.20602e10	+	9.64994e10i	0.105840	+	0.846876i
$582$		0			0
$583$	1.29817e10	−	2.24850e10i	0.112372	−	0.194634i
$584$		0			0
$585$	1.47498e9	+	2.55474e9i	0.0125939	+	0.0218134i
$586$		0			0
$587$		−	2.81096e10i		−	0.236756i	−0.992969	−	0.118378i	$-0.962230\pi$
$587$		−	2.81096e10i		−	0.236756i	0.992969	−	0.118378i	$-0.0377695\pi$
$588$		0			0
$589$	−2.16369e11			−1.79777
$590$		0			0
$591$	−6.68716e10	+	3.86083e10i	−0.548140	+	0.316469i
$592$		0			0
$593$	−7.48692e10	−	4.32257e10i	−0.605458	−	0.349561i	0.165728	−	0.986172i	$-0.447003\pi$
$593$	−7.48692e10	−	4.32257e10i	−0.605458	−	0.349561i	−0.771186	+	0.636610i	$0.780336\pi$
$594$		0			0
$595$	1.23794e11	−	1.54715e10i	0.987716	−	0.123442i
$596$		0			0
$597$	4.24610e10	−	7.35446e10i	0.334266	−	0.578967i
$598$		0			0
$599$	2.30484e10	+	3.99210e10i	0.179033	+	0.310094i	0.941550	−	0.336874i	$-0.109370\pi$
$599$	2.30484e10	+	3.99210e10i	0.179033	+	0.310094i	−0.762517	+	0.646969i	$0.776036\pi$
$600$		0			0
$601$			1.65816e11i			1.27095i	0.772120	+	0.635476i	$0.219196\pi$
$601$			1.65816e11i			1.27095i	−0.772120	+	0.635476i	$0.780804\pi$
$602$		0			0
$603$	1.54772e10			0.117064
$604$		0			0
$605$	−1.28850e11	+	7.43915e10i	−0.961750	+	0.555267i
$606$		0			0
$607$	1.04541e11	+	6.03566e10i	0.770071	+	0.444601i	0.832900	−	0.553424i	$-0.186679\pi$
$607$	1.04541e11	+	6.03566e10i	0.770071	+	0.444601i	−0.0628290	+	0.998024i	$0.520012\pi$
$608$		0			0
$609$	1.57009e10	−	1.18847e10i	0.114145	−	0.0864012i
$610$		0			0
$611$	1.50602e10	−	2.60851e10i	0.108060	−	0.187166i
$612$		0			0
$613$	−7.32931e10	−	1.26947e11i	−0.519064	−	0.899045i	−0.999755	−	0.0221550i	$-0.992947\pi$
$613$	−7.32931e10	−	1.26947e11i	−0.519064	−	0.899045i	0.480691	−	0.876890i	$-0.340386\pi$
$614$		0			0
$615$			9.72000e10i			0.679463i
$616$		0			0
$617$	−4.34126e9			−0.0299554			−0.0149777	−	0.999888i	$-0.504768\pi$
$617$	−4.34126e9			−0.0299554			−0.0149777	+	0.999888i	$0.504768\pi$
$618$		0			0
$619$	−1.46823e11	+	8.47685e10i	−1.00007	+	0.577394i	−0.908270	−	0.418384i	$-0.862597\pi$
$619$	−1.46823e11	+	8.47685e10i	−1.00007	+	0.577394i	−0.0918045	+	0.995777i	$0.529263\pi$
$620$		0			0
$621$	1.35482e11	+	7.82204e10i	0.910992	+	0.525961i
$622$		0			0
$623$	−9.34133e10	−	3.94137e10i	−0.620092	−	0.261634i
$624$		0			0
$625$	9.21787e10	−	1.59658e11i	0.604102	−	1.04634i
$626$		0			0
$627$	−2.24070e10	−	3.88100e10i	−0.144982	−	0.251116i
$628$		0			0
$629$			2.07305e11i			1.32436i
$630$		0			0
$631$	−1.67082e11			−1.05393			−0.526967	−	0.849886i	$-0.676671\pi$
$631$	−1.67082e11			−1.05393			−0.526967	+	0.849886i	$0.676671\pi$
$632$		0			0
$633$	1.83587e11	−	1.05994e11i	1.14347	−	0.660185i
$634$		0			0
$635$	−6.73677e10	−	3.88947e10i	−0.414340	−	0.239219i
$636$		0			0
$637$	1.17969e10	+	4.64591e10i	0.0716493	+	0.282172i
$638$		0			0
$639$	8.32167e9	−	1.44136e10i	0.0499122	−	0.0864505i
$640$		0			0
$641$	1.44089e10	+	2.49570e10i	0.0853491	+	0.147829i	0.905540	−	0.424261i	$-0.139466\pi$
$641$	1.44089e10	+	2.49570e10i	0.0853491	+	0.147829i	−0.820191	+	0.572090i	$0.806133\pi$
$642$		0			0
$643$			9.88987e9i			0.0578558i	0.999582	+	0.0289279i	$0.00920931\pi$
$643$			9.88987e9i			0.0578558i	−0.999582	+	0.0289279i	$0.990791\pi$
$644$		0			0
$645$	9.20067e10			0.531595
$646$		0			0
$647$	−1.71054e11	+	9.87580e10i	−0.976148	+	0.563579i	−0.901105	−	0.433601i	$-0.857243\pi$
$647$	−1.71054e11	+	9.87580e10i	−0.976148	+	0.563579i	−0.0750432	+	0.997180i	$0.523909\pi$
$648$		0			0
$649$	2.64531e10	+	1.52727e10i	0.149107	+	0.0860869i
$650$		0			0
$651$	6.23907e10	−	1.47870e11i	0.347373	−	0.823299i
$652$		0			0
$653$	−5.38657e8	+	9.32982e8i	−0.00296251	+	0.00513122i	−0.867503	−	0.497432i	$-0.834276\pi$
$653$	−5.38657e8	+	9.32982e8i	−0.00296251	+	0.00513122i	0.864540	+	0.502563i	$0.167610\pi$
$654$		0			0
$655$	−6.62307e10	−	1.14715e11i	−0.359828	−	0.623240i
$656$		0			0
$657$		−	4.14522e9i		−	0.0222478i
$658$		0			0
$659$	−1.24072e11			−0.657859			−0.328929	−	0.944355i	$-0.606688\pi$
$659$	−1.24072e11			−0.657859			−0.328929	+	0.944355i	$0.606688\pi$
$660$		0			0
$661$	−1.51218e11	+	8.73060e10i	−0.792134	+	0.457339i	−0.840713	−	0.541481i	$-0.817864\pi$
$661$	−1.51218e11	+	8.73060e10i	−0.792134	+	0.457339i	0.0485793	+	0.998819i	$0.484531\pi$
$662$		0			0
$663$	4.09513e10	+	2.36433e10i	0.211940	+	0.122364i
$664$		0			0
$665$	−2.59809e11	−	3.43233e11i	−1.32852	−	1.75510i
$666$		0			0
$667$	1.49896e10	−	2.59627e10i	0.0757330	−	0.131173i
$668$		0			0
$669$	1.08964e11	+	1.88731e11i	0.543974	+	0.942191i
$670$		0			0
$671$		−	3.27858e10i		−	0.161732i
$672$		0			0
$673$	1.01995e11			0.497184			0.248592	−	0.968608i	$-0.420032\pi$
$673$	1.01995e11			0.497184			0.248592	+	0.968608i	$0.420032\pi$
$674$		0			0
$675$	5.49596e10	−	3.17309e10i	0.264745	−	0.152851i
$676$		0			0
$677$	−7.52406e10	−	4.34402e10i	−0.358177	−	0.206794i	0.310104	−	0.950703i	$-0.399636\pi$
$677$	−7.52406e10	−	4.34402e10i	−0.358177	−	0.206794i	−0.668281	+	0.743909i	$0.732969\pi$
$678$		0			0
$679$	−2.76819e10	−	2.21495e11i	−0.130232	−	1.04204i
$680$		0			0
$681$	1.27360e11	−	2.20593e11i	0.592166	−	1.02566i
$682$		0			0
$683$	8.12762e9	+	1.40774e10i	0.0373491	+	0.0646906i	0.884096	−	0.467306i	$-0.154775\pi$
$683$	8.12762e9	+	1.40774e10i	0.0373491	+	0.0646906i	−0.846747	+	0.531996i	$0.821442\pi$
$684$		0			0
$685$		−	4.54137e11i		−	2.06265i
$686$		0			0
$687$	3.88017e10			0.174190
$688$		0			0
$689$	8.18654e10	−	4.72650e10i	0.363265	−	0.209731i
$690$		0			0
$691$	−2.95833e10	−	1.70800e10i	−0.129758	−	0.0749160i	0.433716	−	0.901050i	$-0.357202\pi$
$691$	−2.95833e10	−	1.70800e10i	−0.129758	−	0.0749160i	−0.563474	+	0.826134i	$0.690536\pi$
$692$		0			0
$693$	−2.71347e9	+	3.39123e8i	−0.0117650	+	0.00147036i
$694$		0			0
$695$	1.26185e11	−	2.18558e11i	0.540839	−	0.936760i
$696$		0			0
$697$	−6.40875e10	−	1.11003e11i	−0.271545	−	0.470330i
$698$		0			0
$699$			3.19261e10i			0.133732i
$700$		0			0
$701$	−2.18076e11			−0.903100			−0.451550	−	0.892246i	$-0.649129\pi$
$701$	−2.18076e11			−0.903100			−0.451550	+	0.892246i	$0.649129\pi$
$702$		0			0
$703$	6.19475e11	−	3.57654e11i	2.53631	−	1.46434i
$704$		0			0
$705$	−1.73766e11	−	1.00324e11i	−0.703409	−	0.406113i
$706$		0			0
$707$	−3.88797e11	+	2.94298e11i	−1.55613	+	1.17790i
$708$		0			0
$709$	−2.33656e11	+	4.04704e11i	−0.924681	+	1.60159i	−0.132606	+	0.991169i	$0.542335\pi$
$709$	−2.33656e11	+	4.04704e11i	−0.924681	+	1.60159i	−0.792074	+	0.610425i	$0.790999\pi$
$710$		0			0
$711$	−1.70052e9	−	2.94539e9i	−0.00665432	−	0.0115256i
$712$		0			0
$713$		−	2.44339e11i		−	0.945442i
$714$		0			0
$715$	1.35086e10			0.0516876
$716$		0			0
$717$	−1.78774e11	+	1.03215e11i	−0.676439	+	0.390542i
$718$		0			0
$719$	−4.77167e10	−	2.75492e10i	−0.178548	−	0.103085i	0.408062	−	0.912954i	$-0.366205\pi$
$719$	−4.77167e10	−	2.75492e10i	−0.178548	−	0.103085i	−0.586610	+	0.809869i	$0.699538\pi$
$720$		0			0
$721$	1.02361e11	+	4.31890e10i	0.378786	+	0.159821i
$722$		0			0
$723$	8.91857e10	−	1.54474e11i	0.326394	−	0.565331i
$724$		0			0
$725$	−6.08067e9	−	1.05320e10i	−0.0220089	−	0.0381206i
$726$		0			0
$727$			1.20385e10i			0.0430957i	0.999768	+	0.0215478i	$0.00685942\pi$
$727$			1.20385e10i			0.0430957i	−0.999768	+	0.0215478i	$0.993141\pi$
$728$		0			0
$729$	−3.00510e11			−1.06402
$730$		0			0
$731$	−1.05072e11	+	6.06634e10i	−0.367975	+	0.212450i
$732$		0			0
$733$	2.82913e11	+	1.63340e11i	0.980023	+	0.565817i	0.902277	−	0.431157i	$-0.141894\pi$
$733$	2.82913e11	+	1.63340e11i	0.980023	+	0.565817i	0.0777461	+	0.996973i	$0.475228\pi$
$734$		0			0
$735$	3.09488e11	−	7.85854e10i	1.06046	−	0.269273i
$736$		0			0
$737$	3.54370e10	−	6.13787e10i	0.120112	−	0.208040i
$738$		0			0
$739$	−1.01067e11	−	1.75053e11i	−0.338869	−	0.586938i	0.645351	−	0.763886i	$-0.276711\pi$
$739$	−1.01067e11	−	1.75053e11i	−0.338869	−	0.586938i	−0.984220	+	0.176948i	$0.943378\pi$
$740$		0			0
$741$		−	1.63163e11i		−	0.541188i
$742$		0			0
$743$	−4.17987e11			−1.37154			−0.685768	−	0.727820i	$-0.740534\pi$
$743$	−4.17987e11			−1.37154			−0.685768	+	0.727820i	$0.740534\pi$
$744$		0			0
$745$	−1.81514e11	+	1.04797e11i	−0.589232	+	0.340193i
$746$		0			0
$747$	−1.74936e10	−	1.01000e10i	−0.0561821	−	0.0324367i
$748$		0			0
$749$	2.01713e11	−	4.78074e11i	0.640924	−	1.51904i
$750$		0			0
$751$	−1.43927e11	+	2.49289e11i	−0.452462	+	0.783688i	−0.998538	−	0.0540477i	$-0.982788\pi$
$751$	−1.43927e11	+	2.49289e11i	−0.452462	+	0.783688i	0.546076	+	0.837736i	$0.316121\pi$
$752$		0			0
$753$	2.67106e11	+	4.62641e11i	0.830814	+	1.43901i
$754$		0			0
$755$			2.56107e11i			0.788196i
$756$		0			0
$757$	−3.76242e9			−0.0114574			−0.00572868	−	0.999984i	$-0.501824\pi$
$757$	−3.76242e9			−0.0114574			−0.00572868	+	0.999984i	$0.501824\pi$
$758$		0			0
$759$	4.38269e10	−	2.53035e10i	0.132061	−	0.0762453i
$760$		0			0
$761$	−5.20846e11	−	3.00710e11i	−1.55300	−	0.896623i	−0.997896	−	0.0648394i	$-0.979346\pi$
$761$	−5.20846e11	−	3.00710e11i	−1.55300	−	0.896623i	−0.555100	−	0.831783i	$-0.687320\pi$
$762$		0			0
$763$	−1.50037e11	−	1.98214e11i	−0.442691	−	0.584838i
$764$		0			0
$765$	−1.29567e10	+	2.24417e10i	−0.0378311	+	0.0655255i
$766$		0			0
$767$	5.56062e10	+	9.63128e10i	0.160673	+	0.278293i
$768$		0			0
$769$			2.32644e10i			0.0665252i	0.999447	+	0.0332626i	$0.0105898\pi$
$769$			2.32644e10i			0.0665252i	−0.999447	+	0.0332626i	$0.989410\pi$
$770$		0			0
$771$	4.46014e11			1.26221
$772$		0			0
$773$	2.95209e10	−	1.70439e10i	0.0826820	−	0.0477365i	−0.458089	−	0.888906i	$-0.651466\pi$
$773$	2.95209e10	−	1.70439e10i	0.0826820	−	0.0477365i	0.540771	+	0.841170i	$0.318133\pi$
$774$		0			0
$775$	−8.58393e10	−	4.95593e10i	−0.237946	−	0.137378i
$776$		0			0
$777$	6.57994e10	+	5.26490e11i	0.180525	+	1.44446i
$778$		0			0
$779$	−2.21135e11	+	3.83017e11i	−0.600492	+	1.04008i
$780$		0			0
$781$	−3.81071e10	−	6.60034e10i	−0.102424	−	0.177404i
$782$		0			0
$783$			5.79001e10i			0.154039i
$784$		0			0
$785$	4.49074e11			1.18260
$786$		0			0
$787$	6.83036e10	−	3.94351e10i	0.178051	−	0.102798i	−0.408326	−	0.912836i	$-0.633887\pi$
$787$	6.83036e10	−	3.94351e10i	0.178051	−	0.102798i	0.586377	+	0.810039i	$0.300554\pi$
$788$		0			0
$789$	−1.20991e11	−	6.98539e10i	−0.312208	−	0.180253i
$790$		0			0
$791$	−6.84890e11	+	8.55957e10i	−1.74950	+	0.218648i
$792$		0			0
$793$	5.96848e10	−	1.03377e11i	0.150928	−	0.261416i
$794$		0			0
$795$	−3.14856e11	−	5.45347e11i	−0.788213	−	1.36522i
$796$		0			0
$797$			2.31195e11i			0.572988i	0.958082	+	0.286494i	$0.0924898\pi$
$797$			2.31195e11i			0.572988i	−0.958082	+	0.286494i	$0.907510\pi$
$798$		0			0
$799$	2.64589e11			0.649208
$800$		0			0
$801$	1.82379e10	−	1.05297e10i	0.0443043	−	0.0255791i
$802$		0			0
$803$	−1.64389e10	−	9.49103e9i	−0.0395377	−	0.0228271i
$804$		0			0
$805$	3.87602e11	−	2.93394e11i	0.923002	−	0.698663i
$806$		0			0
$807$	−3.28987e11	+	5.69822e11i	−0.775683	+	1.34352i
$808$		0			0
$809$	3.77818e11	+	6.54400e11i	0.882040	+	1.52774i	0.849069	+	0.528282i	$0.177164\pi$
$809$	3.77818e11	+	6.54400e11i	0.882040	+	1.52774i	0.0329710	+	0.999456i	$0.489503\pi$
$810$		0			0
$811$			4.12335e10i			0.0953162i	0.998864	+	0.0476581i	$0.0151758\pi$
$811$			4.12335e10i			0.0953162i	−0.998864	+	0.0476581i	$0.984824\pi$
$812$		0			0
$813$	7.22375e10			0.165349
$814$		0			0
$815$	4.04151e11	−	2.33337e11i	0.916037	−	0.528874i
$816$		0			0
$817$	3.62553e11	+	2.09320e11i	0.813735	+	0.469810i
$818$		0			0
$819$	−9.17322e9	−	3.87044e9i	−0.0203885	−	0.00860250i
$820$		0			0
$821$	7.04220e10	−	1.21974e11i	0.155001	−	0.268470i	−0.778058	−	0.628192i	$-0.783795\pi$
$821$	7.04220e10	−	1.21974e11i	0.155001	−	0.268470i	0.933060	+	0.359722i	$0.117128\pi$
$822$		0			0
$823$	2.94058e11	+	5.09323e11i	0.640964	+	1.11018i	0.985218	+	0.171305i	$0.0547984\pi$
$823$	2.94058e11	+	5.09323e11i	0.640964	+	1.11018i	−0.344254	+	0.938876i	$0.611868\pi$
$824$		0			0
$825$		−	2.05292e10i		−	0.0443156i
$826$		0			0
$827$	2.48519e10			0.0531297			0.0265648	−	0.999647i	$-0.491543\pi$
$827$	2.48519e10			0.0531297			0.0265648	+	0.999647i	$0.491543\pi$
$828$		0			0
$829$	4.39196e11	−	2.53570e11i	0.929909	−	0.536883i	0.0431265	−	0.999070i	$-0.486268\pi$
$829$	4.39196e11	−	2.53570e11i	0.929909	−	0.536883i	0.886783	+	0.462186i	$0.152935\pi$
$830$		0			0
$831$	5.94220e11	+	3.43073e11i	1.24607	+	0.719420i
$832$		0			0
$833$	−3.01622e11	+	2.93802e11i	−0.626446	+	0.610203i
$834$		0			0
$835$	−8.31810e10	+	1.44074e11i	−0.171111	+	0.296373i
$836$		0			0
$837$	2.35952e11	+	4.08680e11i	0.480752	+	0.832687i
$838$		0			0
$839$		−	5.64346e11i		−	1.13893i	−0.822015	−	0.569465i	$-0.807150\pi$
$839$		−	5.64346e11i		−	1.13893i	0.822015	−	0.569465i	$-0.192850\pi$
$840$		0			0
$841$	−4.89151e11			−0.977820
$842$		0			0
$843$	6.30362e10	−	3.63939e10i	0.124819	−	0.0720641i
$844$		0			0
$845$	−4.59964e11	−	2.65560e11i	−0.902188	−	0.520879i
$846$		0			0
$847$	1.95208e11	−	4.62657e11i	0.379284	−	0.898930i
$848$		0			0
$849$	−9.34594e10	+	1.61876e11i	−0.179884	+	0.311568i
$850$		0			0
$851$	4.03887e11	+	6.99553e11i	0.770091	+	1.33384i
$852$		0			0
$853$		−	4.33682e11i		−	0.819173i	−0.912271	−	0.409586i	$-0.865673\pi$
$853$		−	4.33682e11i		−	0.819173i	0.912271	−	0.409586i	$-0.134327\pi$
$854$		0			0
$855$	8.94147e10			0.167319
$856$		0			0
$857$	−5.92795e11	+	3.42250e11i	−1.09896	+	0.634484i	−0.935947	−	0.352141i	$-0.885454\pi$
$857$	−5.92795e11	+	3.42250e11i	−1.09896	+	0.634484i	−0.163011	+	0.986624i	$0.552121\pi$
$858$		0			0
$859$	4.14525e11	+	2.39326e11i	0.761339	+	0.439559i	0.829776	−	0.558096i	$-0.188468\pi$
$859$	4.14525e11	+	2.39326e11i	0.761339	+	0.439559i	−0.0684372	+	0.997655i	$0.521801\pi$
$860$		0			0
$861$	−1.97995e11	−	2.61571e11i	−0.360281	−	0.475967i
$862$		0			0
$863$	4.96342e10	−	8.59690e10i	0.0894825	−	0.154988i	−0.817810	−	0.575488i	$-0.804812\pi$
$863$	4.96342e10	−	8.59690e10i	0.0894825	−	0.154988i	0.907292	+	0.420500i	$0.138145\pi$
$864$		0			0
$865$	8.22896e10	+	1.42530e11i	0.146987	+	0.254590i
$866$		0			0
$867$		−	1.27756e11i		−	0.226102i
$868$		0			0
$869$	−1.55743e10			−0.0273104
$870$		0			0
$871$	2.23473e11	−	1.29022e11i	0.388287	−	0.224177i
$872$		0			0
$873$	4.01532e10	+	2.31824e10i	0.0691294	+	0.0399119i
$874$		0			0
$875$	5.82868e10	+	4.66379e11i	0.0994348	+	0.795622i
$876$		0			0
$877$	−3.51149e11	+	6.08207e11i	−0.593598	+	1.02814i	0.400145	+	0.916452i	$0.368960\pi$
$877$	−3.51149e11	+	6.08207e11i	−0.593598	+	1.02814i	−0.993743	+	0.111691i	$0.964373\pi$
$878$		0			0
$879$	7.32060e10	+	1.26797e11i	0.122629	+	0.212399i
$880$		0			0
$881$			5.35262e10i			0.0888512i	0.999013	+	0.0444256i	$0.0141458\pi$
$881$			5.35262e10i			0.0888512i	−0.999013	+	0.0444256i	$0.985854\pi$
$882$		0			0
$883$	7.23059e11			1.18941			0.594704	−	0.803945i	$-0.297269\pi$
$883$	7.23059e11			1.18941			0.594704	+	0.803945i	$0.297269\pi$
$884$		0			0
$885$	6.41588e11	−	3.70421e11i	1.04588	−	0.603841i
$886$		0			0
$887$	−7.11223e11	−	4.10625e11i	−1.14898	−	0.663362i	−0.200339	−	0.979727i	$-0.564204\pi$
$887$	−7.11223e11	−	4.10625e11i	−1.14898	−	0.663362i	−0.948638	+	0.316365i	$0.897538\pi$
$888$		0			0
$889$	2.60518e11	−	3.25589e10i	0.417092	−	0.0521270i
$890$		0			0
$891$	−4.51335e10	+	7.81736e10i	−0.0716124	+	0.124036i
$892$		0			0
$893$	−4.56483e11	−	7.90652e11i	−0.717825	−	1.24331i
$894$		0			0
$895$			5.22720e11i			0.814660i
$896$		0			0
$897$	1.84254e11			0.284609
$898$		0			0
$899$	7.83163e10	−	4.52159e10i	0.119898	−	0.0692234i
$900$		0			0
$901$	7.19134e11	+	4.15192e11i	1.09122	+	0.630014i
$902$		0			0
$903$	−2.47596e11	+	1.87416e11i	−0.372385	+	0.281875i
$904$		0			0
$905$	−6.28852e11	+	1.08920e12i	−0.937463	+	1.62373i
$906$		0			0
$907$	−4.58264e11	−	7.93736e11i	−0.677153	−	1.17286i	−0.975835	−	0.218510i	$-0.929880\pi$
$907$	−4.58264e11	−	7.93736e11i	−0.677153	−	1.17286i	0.298682	−	0.954353i	$-0.403453\pi$
$908$		0			0
$909$		−	1.01284e11i		−	0.148350i
$910$		0			0
$911$	−6.64275e11			−0.964438			−0.482219	−	0.876051i	$-0.660169\pi$
$911$	−6.64275e11			−0.964438			−0.482219	+	0.876051i	$0.660169\pi$
$912$		0			0
$913$	−8.01079e10	+	4.62503e10i	−0.115290	+	0.0665628i
$914$		0			0
$915$	−6.88648e11	−	3.97591e11i	−0.982455	−	0.567221i
$916$		0			0
$917$	4.11904e11	+	1.73794e11i	0.582530	+	0.245786i
$918$		0			0
$919$	−5.26364e10	+	9.11688e10i	−0.0737945	+	0.127816i	−0.900561	−	0.434729i	$-0.856844\pi$
$919$	−5.26364e10	+	9.11688e10i	−0.0737945	+	0.127816i	0.826767	+	0.562545i	$0.190178\pi$
$920$		0			0
$921$	−4.38375e11	−	7.59288e11i	−0.609267	−	1.05528i
$922$		0			0
$923$		−	2.77487e11i		−	0.382328i
$924$		0			0
$925$	3.27682e11			0.447596
$926$		0			0
$927$	−1.99849e10	+	1.15383e10i	−0.0270635	+	0.0156251i
$928$		0			0
$929$	−4.33087e11	−	2.50043e11i	−0.581449	−	0.335700i	0.180260	−	0.983619i	$-0.442306\pi$
$929$	−4.33087e11	−	2.50043e11i	−0.581449	−	0.335700i	−0.761709	+	0.647919i	$0.775639\pi$
$930$		0			0
$931$	1.39832e12	+	3.94434e11i	1.86127	+	0.525020i
$932$		0			0
$933$	4.50397e11	−	7.80111e11i	0.594387	−	1.02951i
$934$		0			0
$935$	5.93322e10	+	1.02766e11i	0.0776326	+	0.134464i
$936$		0			0
$937$			3.30356e11i			0.428572i	0.976771	+	0.214286i	$0.0687424\pi$
$937$			3.30356e11i			0.428572i	−0.976771	+	0.214286i	$0.931258\pi$
$938$		0			0
$939$	−5.16503e11			−0.664371
$940$		0			0
$941$	1.89310e11	−	1.09298e11i	0.241444	−	0.139398i	−0.374396	−	0.927269i	$-0.622150\pi$
$941$	1.89310e11	−	1.09298e11i	0.241444	−	0.139398i	0.615840	+	0.787871i	$0.288817\pi$
$942$		0			0
$943$	−4.32528e11	−	2.49720e11i	−0.546975	−	0.315796i
$944$		0			0
$945$	−3.64979e11	+	8.65027e11i	−0.457658	+	1.08468i
$946$		0			0
$947$	4.29630e11	−	7.44142e11i	0.534189	−	0.925243i	−0.465013	−	0.885304i	$-0.653950\pi$
$947$	4.29630e11	−	7.44142e11i	0.534189	−	0.925243i	0.999202	−	0.0399390i	$-0.0127164\pi$
$948$		0			0
$949$	−3.45558e10	−	5.98524e10i	−0.0426046	−	0.0737933i
$950$		0			0
$951$			1.03778e12i			1.26878i
$952$		0			0
$953$	1.26575e12			1.53453			0.767265	−	0.641330i	$-0.221617\pi$
$953$	1.26575e12			1.53453			0.767265	+	0.641330i	$0.221617\pi$
$954$		0			0
$955$	−1.42594e12	+	8.23267e11i	−1.71430	+	0.989754i
$956$		0			0
$957$	1.62207e10	+	9.36503e9i	0.0193385	+	0.0111651i
$958$		0			0
$959$	9.25071e11	+	1.22211e12i	1.09371	+	1.44489i
$960$		0			0
$961$	−5.79215e10	+	1.00323e11i	−0.0679119	+	0.117627i
$962$		0			0
$963$	5.38892e10	+	9.33389e10i	0.0626609	+	0.108532i
$964$		0			0
$965$			1.05502e12i			1.21661i
$966$		0			0
$967$	−7.22731e11			−0.826553			−0.413276	−	0.910606i	$-0.635616\pi$
$967$	−7.22731e11			−0.826553			−0.413276	+	0.910606i	$0.635616\pi$
$968$		0			0
$969$	1.24126e12	−	7.16640e11i	1.40788	−	0.812841i
$970$		0			0
$971$	−2.64175e11	−	1.52522e11i	−0.297177	−	0.171575i	0.343997	−	0.938971i	$-0.388219\pi$
$971$	−2.64175e11	−	1.52522e11i	−0.297177	−	0.171575i	−0.641174	+	0.767395i	$0.721552\pi$
$972$		0			0
$973$	1.05630e11	+	8.45191e11i	0.117851	+	0.942982i
$974$		0			0
$975$	3.73724e10	−	6.47308e10i	0.0413554	−	0.0716296i
$976$		0			0
$977$	−3.67849e10	−	6.37133e10i	−0.0403730	−	0.0699282i	0.845133	−	0.534556i	$-0.179521\pi$
$977$	−3.67849e10	−	6.37133e10i	−0.0403730	−	0.0699282i	−0.885506	+	0.464628i	$0.846188\pi$
$978$		0			0
$979$		−	9.64363e10i		−	0.104981i
$980$		0			0
$981$	5.16361e10			0.0557541
$982$		0			0
$983$	5.99150e11	−	3.45919e11i	0.641685	−	0.370477i	−0.143579	−	0.989639i	$-0.545861\pi$
$983$	5.99150e11	−	3.45919e11i	0.641685	−	0.370477i	0.785263	+	0.619162i	$0.212528\pi$
$984$		0			0
$985$	−6.10987e11	−	3.52754e11i	−0.649064	−	0.374737i
$986$		0			0
$987$	6.71972e11	−	8.39814e10i	0.708081	−	0.0884941i
$988$		0			0
$989$	−2.36378e11	+	4.09419e11i	−0.247071	+	0.427940i
$990$		0			0
$991$	5.34640e9	+	9.26023e9i	0.00554328	+	0.00960124i	0.868784	−	0.495192i	$-0.164902\pi$
$991$	5.34640e9	+	9.26023e9i	0.00554328	+	0.00960124i	−0.863240	+	0.504793i	$0.831569\pi$
$992$		0			0
$993$			4.83093e11i			0.496859i
$994$		0			0
$995$	7.75909e11			0.791623
$996$		0			0
$997$	5.71798e11	−	3.30128e11i	0.578711	−	0.334119i	−0.181910	−	0.983315i	$-0.558228\pi$
$997$	5.71798e11	−	3.30128e11i	0.578711	−	0.334119i	0.760621	+	0.649196i	$0.224895\pi$
$998$		0			0
$999$	−1.35108e12	−	7.80047e11i	−1.35650	−	0.783175i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	112.9.s.b.33.2		10
4.3	odd	2		28.9.h.a.5.4	✓	10
7.3	odd	6	inner	112.9.s.b.17.2		10
12.11	even	2		252.9.z.c.145.4		10
28.3	even	6		28.9.h.a.17.4	yes	10
28.11	odd	6		196.9.h.a.129.2		10
28.19	even	6		196.9.b.a.97.4		10
28.23	odd	6		196.9.b.a.97.7		10
28.27	even	2		196.9.h.a.117.2		10
84.59	odd	6		252.9.z.c.73.4		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
28.9.h.a.5.4	✓	10	4.3	odd	2
28.9.h.a.17.4	yes	10	28.3	even	6
112.9.s.b.17.2		10	7.3	odd	6	inner
112.9.s.b.33.2		10	1.1	even	1	trivial
196.9.b.a.97.4		10	28.19	even	6
196.9.b.a.97.7		10	28.23	odd	6
196.9.h.a.117.2		10	28.27	even	2
196.9.h.a.129.2		10	28.11	odd	6
252.9.z.c.73.4		10	84.59	odd	6
252.9.z.c.145.4		10	12.11	even	2

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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 112 = 2^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	112.s (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-67.4293 + 38.9303i) q^{3} +(-616.084 - 355.696i) q^{5} +(2382.47 - 297.754i) q^{7} +(-249.357 + 431.899i) q^{9} +O(q^{10})\)	\(q+(-67.4293 + 38.9303i) q^{3} +(-616.084 - 355.696i) q^{5} +(2382.47 - 297.754i) q^{7} +(-249.357 + 431.899i) q^{9} +(1141.87 + 1977.78i) q^{11} +8314.85i q^{13} +55389.5 q^{15} +(-63255.0 + 36520.3i) q^{17} +(218262. + 126014. i) q^{19} +(-149056. + 112828. i) q^{21} +(-142303. + 246477. i) q^{23} +(57726.8 + 99985.8i) q^{25} -549674. i q^{27} -105335. q^{29} +(-743496. + 429258. i) q^{31} +(-153991. - 88906.9i) q^{33} +(-1.57371e6 - 663992. i) q^{35} +(1.41911e6 - 2.45797e6i) q^{37} +(-323700. - 560665. i) q^{39} +1.75485e6i q^{41} +1.66109e6 q^{43} +(307250. - 177391. i) q^{45} +(-3.13716e6 - 1.81124e6i) q^{47} +(5.58749e6 - 1.41878e6i) q^{49} +(2.84350e6 - 4.92508e6i) q^{51} +(-5.68440e6 - 9.84568e6i) q^{53} -1.62464e6i q^{55} -1.96230e7 q^{57} +(1.15832e7 - 6.68757e6i) q^{59} +(-1.24328e7 - 7.17810e6i) q^{61} +(-465485. + 1.10323e6i) q^{63} +(2.95756e6 - 5.12265e6i) q^{65} +(-1.55171e7 - 2.68764e7i) q^{67} -2.21597e7i q^{69} -3.33725e7 q^{71} +(-7.19825e6 + 4.15591e6i) q^{73} +(-7.78496e6 - 4.49465e6i) q^{75} +(3.30936e6 + 4.37199e6i) q^{77} +(-3.40981e6 + 5.90597e6i) q^{79} +(1.97630e7 + 3.42305e7i) q^{81} +4.05040e7i q^{83} +5.19605e7 q^{85} +(7.10268e6 - 4.10074e6i) q^{87} +(-3.65699e7 - 2.11136e7i) q^{89} +(2.47578e6 + 1.98099e7i) q^{91} +(3.34223e7 - 5.78891e7i) q^{93} +(-8.96452e7 - 1.55270e8i) q^{95} -9.29688e7i q^{97} -1.13893e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 81 q^{3} - 837 q^{5} - 1526 q^{7} + 1902 q^{9}+O(q^{10}) \) 10 * q + 81 * q^3 - 837 * q^5 - 1526 * q^7 + 1902 * q^9	\( 10 q + 81 q^{3} - 837 q^{5} - 1526 q^{7} + 1902 q^{9} - 3705 q^{11} - 76134 q^{15} + 78003 q^{17} + 96741 q^{19} - 153153 q^{21} - 208533 q^{23} + 367978 q^{25} + 754764 q^{29} + 1053717 q^{31} - 1032993 q^{33} + 1306389 q^{35} - 998075 q^{37} - 1431900 q^{39} - 738292 q^{43} - 7432758 q^{45} - 710883 q^{47} + 13288114 q^{49} + 2571909 q^{51} + 10501461 q^{53} - 2744514 q^{57} + 37089081 q^{59} - 8180481 q^{61} - 47152518 q^{63} + 21459108 q^{65} - 48020189 q^{67} + 31918236 q^{71} - 133345593 q^{73} + 119504178 q^{75} + 188477625 q^{77} - 53590181 q^{79} + 173295063 q^{81} - 157179282 q^{85} + 413284806 q^{87} - 241368273 q^{89} - 420709128 q^{91} + 137961999 q^{93} - 347126775 q^{95} + 117796500 q^{99}+O(q^{100}) \) 10 * q + 81 * q^3 - 837 * q^5 - 1526 * q^7 + 1902 * q^9 - 3705 * q^11 - 76134 * q^15 + 78003 * q^17 + 96741 * q^19 - 153153 * q^21 - 208533 * q^23 + 367978 * q^25 + 754764 * q^29 + 1053717 * q^31 - 1032993 * q^33 + 1306389 * q^35 - 998075 * q^37 - 1431900 * q^39 - 738292 * q^43 - 7432758 * q^45 - 710883 * q^47 + 13288114 * q^49 + 2571909 * q^51 + 10501461 * q^53 - 2744514 * q^57 + 37089081 * q^59 - 8180481 * q^61 - 47152518 * q^63 + 21459108 * q^65 - 48020189 * q^67 + 31918236 * q^71 - 133345593 * q^73 + 119504178 * q^75 + 188477625 * q^77 - 53590181 * q^79 + 173295063 * q^81 - 157179282 * q^85 + 413284806 * q^87 - 241368273 * q^89 - 420709128 * q^91 + 137961999 * q^93 - 347126775 * q^95 + 117796500 * q^99

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