LMFDB - Embedded newform 112.12.i.b.81.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [112,12,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, 12); 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, 12, names="a")

Level:	$ N $	$=$	$ 112 = 2^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	112.i (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,266] 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:	$86.0544362227$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} + 149344 x^{6} + 5578711 x^{5} + 20557200983 x^{4} + 408905884576 x^{3} + \cdots + 30\!\cdots\!24 $ x^8 - x^7 + 149344x^6 + 5578711x^5 + 20557200983x^4 + 408905884576x^3 + 267736482219682x^2 - 4732823471679688x + 3039499286293024324
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{12}\cdot 3\cdot 7^{3} $
Twist minimal:	no (minimal twist has level 14)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			81.4
Root			$-178.705 + 309.527i$ of defining polynomial
Character	$\chi$	$=$	112.81
Dual form			112.12.i.b.65.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(390.910 - 677.077i) q^{3} +(4992.28 + 8646.88i) q^{5} +(-24330.9 - 37220.1i) q^{7} +(-217048. - 375939. i) q^{9} +(-79178.4 + 137141. i) q^{11} +1.79763e6 q^{13} +7.80614e6 q^{15} +(1.39628e6 - 2.41842e6i) q^{17} +(-374482. - 648622. i) q^{19} +(-3.47121e7 + 1.92417e6i) q^{21} +(-1.66245e7 - 2.87945e7i) q^{23} +(-2.54317e7 + 4.40490e7i) q^{25} -2.00889e8 q^{27} +9.28902e7 q^{29} +(1.41494e7 - 2.45075e7i) q^{31} +(6.19033e7 + 1.07220e8i) q^{33} +(2.00371e8 - 3.96200e8i) q^{35} +(-3.28309e8 - 5.68647e8i) q^{37} +(7.02712e8 - 1.21713e9i) q^{39} -5.18887e8 q^{41} -4.47337e7 q^{43} +(2.16713e9 - 3.75359e9i) q^{45} +(3.99885e8 + 6.92620e8i) q^{47} +(-7.93341e8 + 1.81120e9i) q^{49} +(-1.09164e9 - 1.89077e9i) q^{51} +(5.54293e8 - 9.60064e8i) q^{53} -1.58112e9 q^{55} -5.85556e8 q^{57} +(2.10667e9 - 3.64887e9i) q^{59} +(-3.73122e9 - 6.46267e9i) q^{61} +(-8.71149e9 + 1.72255e10i) q^{63} +(8.97427e9 + 1.55439e10i) q^{65} +(5.49559e9 - 9.51864e9i) q^{67} -2.59948e10 q^{69} -1.63957e10 q^{71} +(3.74721e9 - 6.49036e9i) q^{73} +(1.98830e10 + 3.44384e10i) q^{75} +(7.03088e9 - 3.89739e8i) q^{77} +(7.98611e9 + 1.38323e10i) q^{79} +(-4.00801e10 + 6.94207e10i) q^{81} -6.04748e10 q^{83} +2.78825e10 q^{85} +(3.63118e10 - 6.28938e10i) q^{87} +(-1.08709e10 - 1.88289e10i) q^{89} +(-4.37379e10 - 6.69079e10i) q^{91} +(-1.10623e10 - 1.91604e10i) q^{93} +(3.73904e9 - 6.47621e9i) q^{95} +9.95342e10 q^{97} +6.87422e10 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 266 q^{3} + 3808 q^{5} - 110328 q^{7} - 503848 q^{9} - 920150 q^{11} + 997472 q^{13} + 500444 q^{15} + 1333724 q^{17} + 21551726 q^{19} - 90442480 q^{21} - 72510158 q^{23} - 77154744 q^{25} - 156683212 q^{27}+ \cdots + 545487662552 q^{99}+O(q^{100}) $ 8 * q + 266 * q^3 + 3808 * q^5 - 110328 * q^7 - 503848 * q^9 - 920150 * q^11 + 997472 * q^13 + 500444 * q^15 + 1333724 * q^17 + 21551726 * q^19 - 90442480 * q^21 - 72510158 * q^23 - 77154744 * q^25 - 156683212 * q^27 + 213575104 * q^29 + 194359774 * q^31 - 164547124 * q^33 + 981719074 * q^35 + 171517048 * q^37 + 653125236 * q^39 - 3312095136 * q^41 - 850279648 * q^43 + 6784014272 * q^45 - 2223880974 * q^47 - 4232044312 * q^49 - 6968362082 * q^51 - 7185483360 * q^53 - 624915620 * q^55 + 4959469112 * q^57 - 6997401502 * q^59 - 6476463280 * q^61 + 17252507684 * q^63 + 30625528248 * q^65 + 18660972186 * q^67 + 31319762096 * q^69 - 23224449248 * q^71 + 3731641452 * q^73 + 91204646176 * q^75 - 29345595440 * q^77 - 12221157926 * q^79 - 97333443028 * q^81 - 316739523968 * q^83 - 147794862544 * q^85 + 218122807364 * q^87 - 71204406084 * q^89 - 189816116528 * q^91 + 9838293624 * q^93 + 215091896614 * q^95 + 688251797184 * q^97 + 545487662552 * 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
$3$	390.910	−	677.077i	0.928775	−	1.60869i	0.143400	−	0.989665i	$-0.454197\pi$
$3$	390.910	−	677.077i	0.928775	−	1.60869i	0.785375	−	0.619020i	$-0.212470\pi$
$4$		0			0
$5$	4992.28	+	8646.88i	0.714437	+	1.23744i	0.963176	+	0.268871i	$0.0866505\pi$
$5$	4992.28	+	8646.88i	0.714437	+	1.23744i	−0.248739	+	0.968571i	$0.580016\pi$
$6$		0			0
$7$	−24330.9	−	37220.1i	−0.547166	−	0.837024i
$7$	−24330.9	−	37220.1i	−0.547166	−	0.837024i
$8$		0			0
$9$	−217048.	−	375939.i	−1.22525	−	2.12219i
$10$		0			0
$11$	−79178.4	+	137141.i	−0.148234	+	0.256748i	−0.930575	−	0.366102i	$-0.880692\pi$
$11$	−79178.4	+	137141.i	−0.148234	+	0.256748i	0.782341	+	0.622850i	$0.214025\pi$
$12$		0			0
$13$	1.79763e6			1.34280			0.671401	−	0.741095i	$-0.265693\pi$
$13$	1.79763e6			1.34280			0.671401	+	0.741095i	$0.265693\pi$
$14$		0			0
$15$	7.80614e6			2.65420
$16$		0			0
$17$	1.39628e6	−	2.41842e6i	0.238508	−	0.413108i	−0.721778	−	0.692124i	$-0.756675\pi$
$17$	1.39628e6	−	2.41842e6i	0.238508	−	0.413108i	0.960286	+	0.279016i	$0.0900084\pi$
$18$		0			0
$19$	−374482.	−	648622.i	−0.0346965	−	0.0600962i	0.848156	−	0.529747i	$-0.177713\pi$
$19$	−374482.	−	648622.i	−0.0346965	−	0.0600962i	−0.882852	+	0.469651i	$0.844380\pi$
$20$		0			0
$21$	−3.47121e7	+	1.92417e6i	−1.85470	+	0.102811i
$22$		0			0
$23$	−1.66245e7	−	2.87945e7i	−0.538575	−	0.932840i	−0.998981	−	0.0451313i	$-0.985629\pi$
$23$	−1.66245e7	−	2.87945e7i	−0.538575	−	0.932840i	0.460406	−	0.887709i	$-0.347704\pi$
$24$		0			0
$25$	−2.54317e7	+	4.40490e7i	−0.520841	+	0.902123i
$26$		0			0
$27$	−2.00889e8			−2.69436
$28$		0			0
$29$	9.28902e7			0.840971			0.420485	−	0.907299i	$-0.361860\pi$
$29$	9.28902e7			0.840971			0.420485	+	0.907299i	$0.361860\pi$
$30$		0			0
$31$	1.41494e7	−	2.45075e7i	0.0887663	−	0.153748i	−0.818224	−	0.574900i	$-0.805041\pi$
$31$	1.41494e7	−	2.45075e7i	0.0887663	−	0.153748i	0.906990	+	0.421152i	$0.138374\pi$
$32$		0			0
$33$	6.19033e7	+	1.07220e8i	0.275351	+	0.476923i
$34$		0			0
$35$	2.00371e8	−	3.96200e8i	0.644853	−	1.27509i
$36$		0			0
$37$	−3.28309e8	−	5.68647e8i	−0.778346	−	1.34814i	−0.932894	−	0.360150i	$-0.882725\pi$
$37$	−3.28309e8	−	5.68647e8i	−0.778346	−	1.34814i	0.154548	−	0.987985i	$-0.450608\pi$
$38$		0			0
$39$	7.02712e8	−	1.21713e9i	1.24716	−	2.16014i
$40$		0			0
$41$	−5.18887e8			−0.699458			−0.349729	−	0.936851i	$-0.613726\pi$
$41$	−5.18887e8			−0.699458			−0.349729	+	0.936851i	$0.613726\pi$
$42$		0			0
$43$	−4.47337e7			−0.0464043			−0.0232022	−	0.999731i	$-0.507386\pi$
$43$	−4.47337e7			−0.0464043			−0.0232022	+	0.999731i	$0.507386\pi$
$44$		0			0
$45$	2.16713e9	−	3.75359e9i	1.75072	−	3.03234i
$46$		0			0
$47$	3.99885e8	+	6.92620e8i	0.254329	+	0.440511i	0.964713	−	0.263303i	$-0.0848120\pi$
$47$	3.99885e8	+	6.92620e8i	0.254329	+	0.440511i	−0.710384	+	0.703815i	$0.751479\pi$
$48$		0			0
$49$	−7.93341e8	+	1.81120e9i	−0.401219	+	0.915982i
$50$		0			0
$51$	−1.09164e9	−	1.89077e9i	−0.443040	−	0.767368i
$52$		0			0
$53$	5.54293e8	−	9.60064e8i	0.182063	−	0.315343i	−0.760520	−	0.649315i	$-0.775056\pi$
$53$	5.54293e8	−	9.60064e8i	0.182063	−	0.315343i	0.942583	+	0.333972i	$0.108389\pi$
$54$		0			0
$55$	−1.58112e9			−0.423615
$56$		0			0
$57$	−5.85556e8			−0.128901
$58$		0			0
$59$	2.10667e9	−	3.64887e9i	0.383629	−	0.664465i	−0.607949	−	0.793976i	$-0.708007\pi$
$59$	2.10667e9	−	3.64887e9i	0.383629	−	0.664465i	0.991578	+	0.129511i	$0.0413408\pi$
$60$		0			0
$61$	−3.73122e9	−	6.46267e9i	−0.565636	−	0.979711i	−0.996990	−	0.0775280i	$-0.975297\pi$
$61$	−3.73122e9	−	6.46267e9i	−0.565636	−	0.979711i	0.431354	−	0.902183i	$-0.358036\pi$
$62$		0			0
$63$	−8.71149e9	+	1.72255e10i	−1.10591	+	2.18675i
$64$		0			0
$65$	8.97427e9	+	1.55439e10i	0.959347	+	1.66164i
$66$		0			0
$67$	5.49559e9	−	9.51864e9i	0.497282	−	0.861318i	−0.502713	−	0.864453i	$-0.667665\pi$
$67$	5.49559e9	−	9.51864e9i	0.497282	−	0.861318i	0.999995	+	0.00313534i	$0.000998011\pi$
$68$		0			0
$69$	−2.59948e10			−2.00086
$70$		0			0
$71$	−1.63957e10			−1.07847			−0.539237	−	0.842154i	$-0.681287\pi$
$71$	−1.63957e10			−1.07847			−0.539237	+	0.842154i	$0.681287\pi$
$72$		0			0
$73$	3.74721e9	−	6.49036e9i	0.211560	−	0.366432i	−0.740643	−	0.671898i	$-0.765479\pi$
$73$	3.74721e9	−	6.49036e9i	0.211560	−	0.366432i	0.952203	+	0.305467i	$0.0988124\pi$
$74$		0			0
$75$	1.98830e10	+	3.44384e10i	0.967488	+	1.67574i
$76$		0			0
$77$	7.03088e9	−	3.89739e8i	0.296013	−	0.0164087i
$78$		0			0
$79$	7.98611e9	+	1.38323e10i	0.292002	+	0.505763i	0.974283	−	0.225327i	$-0.0723452\pi$
$79$	7.98611e9	+	1.38323e10i	0.292002	+	0.505763i	−0.682281	+	0.731090i	$0.739012\pi$
$80$		0			0
$81$	−4.00801e10	+	6.94207e10i	−1.27721	+	2.21219i
$82$		0			0
$83$	−6.04748e10			−1.68517			−0.842587	−	0.538560i	$-0.818968\pi$
$83$	−6.04748e10			−1.68517			−0.842587	+	0.538560i	$0.818968\pi$
$84$		0			0
$85$	2.78825e10			0.681596
$86$		0			0
$87$	3.63118e10	−	6.28938e10i	0.781073	−	1.35286i
$88$		0			0
$89$	−1.08709e10	−	1.88289e10i	−0.206357	−	0.357420i	0.744207	−	0.667949i	$-0.232827\pi$
$89$	−1.08709e10	−	1.88289e10i	−0.206357	−	0.357420i	−0.950564	+	0.310528i	$0.899494\pi$
$90$		0			0
$91$	−4.37379e10	−	6.69079e10i	−0.734735	−	1.12396i
$92$		0			0
$93$	−1.10623e10	−	1.91604e10i	−0.164888	−	0.285594i
$94$		0			0
$95$	3.73904e9	−	6.47621e9i	0.0495770	−	0.0858699i
$96$		0			0
$97$	9.95342e10			1.17687			0.588434	−	0.808545i	$-0.299745\pi$
$97$	9.95342e10			1.17687			0.588434	+	0.808545i	$0.299745\pi$
$98$		0			0
$99$	6.87422e10			0.726491
$100$		0			0
$101$	6.53712e10	−	1.13226e11i	0.618897	−	1.07196i	−0.370790	−	0.928717i	$-0.620913\pi$
$101$	6.53712e10	−	1.13226e11i	0.618897	−	1.07196i	0.989687	−	0.143245i	$-0.0457537\pi$
$102$		0			0
$103$	−6.41218e10	−	1.11062e11i	−0.545006	−	0.943978i	−0.998607	−	0.0527728i	$-0.983194\pi$
$103$	−6.41218e10	−	1.11062e11i	−0.545006	−	0.943978i	0.453601	−	0.891205i	$-0.350139\pi$
$104$		0			0
$105$	−1.89930e11	−	2.90545e11i	−1.45229	−	2.22163i
$106$		0			0
$107$	5.93975e10	+	1.02879e11i	0.409409	+	0.709117i	0.994824	−	0.101617i	$-0.0324017\pi$
$107$	5.93975e10	+	1.02879e11i	0.409409	+	0.709117i	−0.585415	+	0.810734i	$0.699068\pi$
$108$		0			0
$109$	−1.22410e11	+	2.12020e11i	−0.762028	+	1.31987i	0.179775	+	0.983708i	$0.442463\pi$
$109$	−1.22410e11	+	2.12020e11i	−0.762028	+	1.31987i	−0.941803	+	0.336164i	$0.890870\pi$
$110$		0			0
$111$	−5.13357e11			−2.89163
$112$		0			0
$113$	−2.65326e11			−1.35472			−0.677358	−	0.735653i	$-0.736875\pi$
$113$	−2.65326e11			−1.35472			−0.677358	+	0.735653i	$0.736875\pi$
$114$		0			0
$115$	1.65989e11	−	2.87501e11i	0.769556	−	1.33291i
$116$		0			0
$117$	−3.90173e11	−	6.75799e11i	−1.64526	−	2.84967i
$118$		0			0
$119$	−1.23987e11	+	6.87288e9i	−0.476285	+	0.0264016i
$120$		0			0
$121$	1.30117e11	+	2.25370e11i	0.456054	+	0.789908i
$122$		0			0
$123$	−2.02838e11	+	3.51326e11i	−0.649639	+	1.12521i
$124$		0			0
$125$	−2.03210e10			−0.0595579
$126$		0			0
$127$	2.24538e11			0.603073			0.301536	−	0.953455i	$-0.402501\pi$
$127$	2.24538e11			0.603073			0.301536	+	0.953455i	$0.402501\pi$
$128$		0			0
$129$	−1.74869e10	+	3.02882e10i	−0.0430992	+	0.0746500i
$130$		0			0
$131$	9.59825e9	+	1.66247e10i	0.0217370	+	0.0376496i	0.876689	−	0.481057i	$-0.159747\pi$
$131$	9.59825e9	+	1.66247e10i	0.0217370	+	0.0376496i	−0.854952	+	0.518707i	$0.826414\pi$
$132$		0			0
$133$	−1.50303e10	+	2.97198e10i	−0.0313172	+	0.0619244i
$134$		0			0
$135$	−1.00289e12	−	1.73706e12i	−1.92495	−	3.33411i
$136$		0			0
$137$	6.08474e10	−	1.05391e11i	0.107716	−	0.186569i	−0.807129	−	0.590375i	$-0.798980\pi$
$137$	6.08474e10	−	1.05391e11i	0.107716	−	0.186569i	0.914844	+	0.403806i	$0.132313\pi$
$138$		0			0
$139$	1.46631e11			0.239687			0.119843	−	0.992793i	$-0.461761\pi$
$139$	1.46631e11			0.239687			0.119843	+	0.992793i	$0.461761\pi$
$140$		0			0
$141$	6.25276e11			0.944859
$142$		0			0
$143$	−1.42333e11	+	2.46529e11i	−0.199048	+	0.344762i
$144$		0			0
$145$	4.63734e11	+	8.03211e11i	0.600821	+	1.04065i
$146$		0			0
$147$	9.16194e11	+	1.24517e12i	1.10088	+	1.49618i
$148$		0			0
$149$	−6.05258e11	−	1.04834e12i	−0.675174	−	1.16944i	−0.976418	−	0.215889i	$-0.930735\pi$
$149$	−6.05258e11	−	1.04834e12i	−0.675174	−	1.16944i	0.301243	−	0.953547i	$-0.402598\pi$
$150$		0			0
$151$	−3.28533e11	+	5.69037e11i	−0.340570	+	0.589885i	−0.984539	−	0.175167i	$-0.943953\pi$
$151$	−3.28533e11	+	5.69037e11i	−0.340570	+	0.589885i	0.643969	+	0.765052i	$0.277287\pi$
$152$		0			0
$153$	−1.21224e12			−1.16892
$154$		0			0
$155$	2.82551e11			0.253672
$156$		0			0
$157$	−9.53281e11	+	1.65113e12i	−0.797578	+	1.38145i	0.123612	+	0.992331i	$0.460552\pi$
$157$	−9.53281e11	+	1.65113e12i	−0.797578	+	1.38145i	−0.921189	+	0.389114i	$0.872781\pi$
$158$		0			0
$159$	−4.33358e11	−	7.50598e11i	−0.338192	−	0.585765i
$160$		0			0
$161$	−6.67245e11	+	1.31936e12i	−0.486119	+	0.961219i
$162$		0			0
$163$	5.12933e11	+	8.88426e11i	0.349164	+	0.604769i	0.986101	−	0.166146i	$-0.0531324\pi$
$163$	5.12933e11	+	8.88426e11i	0.349164	+	0.604769i	−0.636938	+	0.770915i	$0.719799\pi$
$164$		0			0
$165$	−6.18078e11	+	1.07054e12i	−0.393443	+	0.681463i
$166$		0			0
$167$	−2.39071e12			−1.42425			−0.712126	−	0.702051i	$-0.752268\pi$
$167$	−2.39071e12			−1.42425			−0.712126	+	0.702051i	$0.752268\pi$
$168$		0			0
$169$	1.43931e12			0.803115
$170$		0			0
$171$	−1.62562e11	+	2.81565e11i	−0.0850235	+	0.147265i
$172$		0			0
$173$	−1.43391e12	−	2.48361e12i	−0.703507	−	1.21851i	−0.967228	−	0.253911i	$-0.918283\pi$
$173$	−1.43391e12	−	2.48361e12i	−0.703507	−	1.21851i	0.263721	−	0.964599i	$-0.415050\pi$
$174$		0			0
$175$	2.25828e12	−	1.25182e11i	1.04008	−	0.0576545i
$176$		0			0
$177$	−1.64704e12	−	2.85276e12i	−0.712610	−	1.23428i
$178$		0			0
$179$	−1.36744e11	+	2.36847e11i	−0.0556180	+	0.0963332i	−0.892494	−	0.451059i	$-0.851046\pi$
$179$	−1.36744e11	+	2.36847e11i	−0.0556180	+	0.0963332i	0.836876	+	0.547393i	$0.184380\pi$
$180$		0			0
$181$	2.88576e12			1.10415			0.552074	−	0.833795i	$-0.313836\pi$
$181$	2.88576e12			1.10415			0.552074	+	0.833795i	$0.313836\pi$
$182$		0			0
$183$	−5.83430e12			−2.10139
$184$		0			0
$185$	3.27802e12	−	5.67769e12i	1.11216	−	1.92632i
$186$		0			0
$187$	2.21110e11	+	3.82974e11i	0.0707098	+	0.122473i
$188$		0			0
$189$	4.88781e12	+	7.47710e12i	1.47426	+	2.25524i
$190$		0			0
$191$	−1.62685e12	−	2.81779e12i	−0.463089	−	0.802094i	0.536024	−	0.844203i	$-0.319926\pi$
$191$	−1.62685e12	−	2.81779e12i	−0.463089	−	0.802094i	−0.999113	+	0.0421086i	$0.986592\pi$
$192$		0			0
$193$	−4.75159e11	+	8.23000e11i	−0.127724	+	0.221225i	−0.922795	−	0.385292i	$-0.874101\pi$
$193$	−4.75159e11	+	8.23000e11i	−0.127724	+	0.221225i	0.795070	+	0.606517i	$0.207434\pi$
$194$		0			0
$195$	1.40325e13			3.56407
$196$		0			0
$197$	1.00428e10			0.00241152			0.00120576	−	0.999999i	$-0.499616\pi$
$197$	1.00428e10			0.00241152			0.00120576	+	0.999999i	$0.499616\pi$
$198$		0			0
$199$	3.15890e12	−	5.47138e12i	0.717537	−	1.24281i	−0.244436	−	0.969665i	$-0.578603\pi$
$199$	3.15890e12	−	5.47138e12i	0.717537	−	1.24281i	0.961973	−	0.273145i	$-0.0880638\pi$
$200$		0			0
$201$	−4.29657e12	−	7.44187e12i	−0.923726	−	1.59994i
$202$		0			0
$203$	−2.26010e12	−	3.45738e12i	−0.460151	−	0.703913i
$204$		0			0
$205$	−2.59043e12	−	4.48676e12i	−0.499719	−	0.865538i
$206$		0			0
$207$	−7.21666e12	+	1.24996e13i	−1.31977	+	2.28592i
$208$		0			0
$209$	1.18604e11			0.0205728
$210$		0			0
$211$	5.42132e12			0.892383			0.446191	−	0.894938i	$-0.352780\pi$
$211$	5.42132e12			0.892383			0.446191	+	0.894938i	$0.352780\pi$
$212$		0			0
$213$	−6.40927e12	+	1.11012e13i	−1.00166	+	1.73493i
$214$		0			0
$215$	−2.23323e11	−	3.86807e11i	−0.0331530	−	0.0574227i
$216$		0			0
$217$	−1.25644e12	+	6.96474e10i	−0.177261	+	0.00982598i
$218$		0			0
$219$	−2.92965e12	−	5.07430e12i	−0.392982	−	0.680665i
$220$		0			0
$221$	2.50999e12	−	4.34743e12i	0.320269	−	0.554722i
$222$		0			0
$223$	9.98983e12			1.21306			0.606528	−	0.795062i	$-0.292562\pi$
$223$	9.98983e12			1.21306			0.606528	+	0.795062i	$0.292562\pi$
$224$		0			0
$225$	2.20796e13			2.55263
$226$		0			0
$227$	−1.20640e12	+	2.08955e12i	−0.132846	+	0.230096i	−0.924773	−	0.380520i	$-0.875745\pi$
$227$	−1.20640e12	+	2.08955e12i	−0.132846	+	0.230096i	0.791926	+	0.610616i	$0.209078\pi$
$228$		0			0
$229$	7.44079e12	+	1.28878e13i	0.780771	+	1.35234i	0.931493	+	0.363759i	$0.118507\pi$
$229$	7.44079e12	+	1.28878e13i	0.780771	+	1.35234i	−0.150722	+	0.988576i	$0.548160\pi$
$230$		0			0
$231$	2.48456e12	−	4.91280e12i	0.248533	−	0.491432i
$232$		0			0
$233$	7.96979e12	+	1.38041e13i	0.760308	+	1.31689i	0.942692	+	0.333664i	$0.108285\pi$
$233$	7.96979e12	+	1.38041e13i	0.760308	+	1.31689i	−0.182384	+	0.983227i	$0.558381\pi$
$234$		0			0
$235$	−3.99267e12	+	6.91551e12i	−0.363405	+	0.629435i
$236$		0			0
$237$	1.24874e13			1.08482
$238$		0			0
$239$	1.48885e13			1.23499			0.617494	−	0.786576i	$-0.288148\pi$
$239$	1.48885e13			1.23499			0.617494	+	0.786576i	$0.288148\pi$
$240$		0			0
$241$	8.18935e12	−	1.41844e13i	0.648867	−	1.12387i	−0.334526	−	0.942386i	$-0.608576\pi$
$241$	8.18935e12	−	1.41844e13i	0.648867	−	1.12387i	0.983394	−	0.181485i	$-0.0580903\pi$
$242$		0			0
$243$	1.35420e13	+	2.34555e13i	1.02529	+	1.77586i
$244$		0			0
$245$	−1.96218e13	+	2.18207e12i	−1.42012	+	0.157927i
$246$		0			0
$247$	−6.73180e11	−	1.16598e12i	−0.0465906	−	0.0806972i
$248$		0			0
$249$	−2.36402e13	+	4.09461e13i	−1.56515	+	2.71091i
$250$		0			0
$251$	−1.36880e13			−0.867232			−0.433616	−	0.901098i	$-0.642762\pi$
$251$	−1.36880e13			−0.867232			−0.433616	+	0.901098i	$0.642762\pi$
$252$		0			0
$253$	5.26522e12			0.319340
$254$		0			0
$255$	1.08995e13	−	1.88786e13i	0.633049	−	1.09647i
$256$		0			0
$257$	1.27360e13	+	2.20593e13i	0.708597	+	1.22733i	0.965378	+	0.260857i	$0.0840050\pi$
$257$	1.27360e13	+	2.20593e13i	0.708597	+	1.22733i	−0.256780	+	0.966470i	$0.582662\pi$
$258$		0			0
$259$	−1.31770e13	+	2.60554e13i	−0.702537	+	1.38915i
$260$		0			0
$261$	−2.01617e13	−	3.49211e13i	−1.03040	−	1.78470i
$262$		0			0
$263$	−1.52870e13	+	2.64778e13i	−0.749144	+	1.29756i	0.199089	+	0.979981i	$0.436202\pi$
$263$	−1.52870e13	+	2.64778e13i	−0.749144	+	1.29756i	−0.948233	+	0.317574i	$0.897132\pi$
$264$		0			0
$265$	1.10688e13			0.520291
$266$		0			0
$267$	−1.69981e13			−0.766635
$268$		0			0
$269$	9.40859e12	−	1.62961e13i	0.407274	−	0.705419i	−0.587309	−	0.809363i	$-0.699813\pi$
$269$	9.40859e12	−	1.62961e13i	0.407274	−	0.705419i	0.994583	+	0.103943i	$0.0331460\pi$
$270$		0			0
$271$	1.96569e13	+	3.40468e13i	0.816930	+	1.41496i	0.907934	+	0.419114i	$0.137659\pi$
$271$	1.96569e13	+	3.40468e13i	0.816930	+	1.41496i	−0.0910034	+	0.995851i	$0.529007\pi$
$272$		0			0
$273$	−6.23994e13	+	3.45895e12i	−2.49050	+	0.138054i
$274$		0			0
$275$	−4.02728e12	−	6.97545e12i	−0.154412	−	0.267450i
$276$		0			0
$277$	1.91126e13	−	3.31040e13i	0.704176	−	1.21967i	−0.262813	−	0.964847i	$-0.584650\pi$
$277$	1.91126e13	−	3.31040e13i	0.704176	−	1.21967i	0.966988	−	0.254821i	$-0.0820165\pi$
$278$		0			0
$279$	−1.22844e13			−0.435042
$280$		0			0
$281$	2.43790e13			0.830103			0.415051	−	0.909798i	$-0.363764\pi$
$281$	2.43790e13			0.830103			0.415051	+	0.909798i	$0.363764\pi$
$282$		0			0
$283$	2.21500e13	−	3.83649e13i	0.725350	−	1.25634i	−0.233480	−	0.972362i	$-0.575011\pi$
$283$	2.21500e13	−	3.83649e13i	0.725350	−	1.25634i	0.958830	−	0.283981i	$-0.0916554\pi$
$284$		0			0
$285$	−2.92326e12	−	5.06323e12i	−0.0920917	−	0.159508i
$286$		0			0
$287$	1.26250e13	+	1.93130e13i	0.382719	+	0.585463i
$288$		0			0
$289$	1.32368e13	+	2.29267e13i	0.386228	+	0.668966i
$290$		0			0
$291$	3.89090e13	−	6.73923e13i	1.09305	−	1.89321i
$292$		0			0
$293$	3.62215e13			0.979930			0.489965	−	0.871742i	$-0.337010\pi$
$293$	3.62215e13			0.979930			0.489965	+	0.871742i	$0.337010\pi$
$294$		0			0
$295$	4.20684e13			1.09632
$296$		0			0
$297$	1.59061e13	−	2.75501e13i	0.399395	−	0.691772i
$298$		0			0
$299$	−2.98848e13	−	5.17619e13i	−0.723200	−	1.25262i
$300$		0			0
$301$	1.08841e12	+	1.66499e12i	0.0253909	+	0.0388416i
$302$		0			0
$303$	−5.11085e13	−	8.85226e13i	−1.14963	−	1.99122i
$304$		0			0
$305$	3.72546e13	−	6.45269e13i	0.808223	−	1.39988i
$306$		0			0
$307$	1.51099e13			0.316229			0.158115	−	0.987421i	$-0.449458\pi$
$307$	1.51099e13			0.316229			0.158115	+	0.987421i	$0.449458\pi$
$308$		0			0
$309$	−1.00264e14			−2.02475
$310$		0			0
$311$	2.68576e13	−	4.65186e13i	0.523461	−	0.906661i	−0.476166	−	0.879355i	$-0.657974\pi$
$311$	2.68576e13	−	4.65186e13i	0.523461	−	0.906661i	0.999627	−	0.0273057i	$-0.00869275\pi$
$312$		0			0
$313$	−2.70143e13	−	4.67901e13i	−0.508275	−	0.880359i	−0.999954	−	0.00958212i	$-0.996950\pi$
$313$	−2.70143e13	−	4.67901e13i	−0.508275	−	0.880359i	0.491679	−	0.870777i	$-0.336383\pi$
$314$		0			0
$315$	−1.92437e14	+	1.06673e13i	−3.49608	+	0.193796i
$316$		0			0
$317$	−1.72833e13	−	2.99356e13i	−0.303251	−	0.525245i	0.673620	−	0.739078i	$-0.264739\pi$
$317$	−1.72833e13	−	2.99356e13i	−0.303251	−	0.525245i	−0.976870	+	0.213833i	$0.931405\pi$
$318$		0			0
$319$	−7.35490e12	+	1.27391e13i	−0.124660	+	0.215918i
$320$		0			0
$321$	9.28763e13			1.52099
$322$		0			0
$323$	−2.09152e12			−0.0331016
$324$		0			0
$325$	−4.57167e13	+	7.91837e13i	−0.699386	+	1.21137i
$326$		0			0
$327$	9.57027e13	+	1.65762e14i	1.41551	+	2.45173i
$328$		0			0
$329$	1.60498e13	−	3.17358e13i	0.229558	−	0.453913i
$330$		0			0
$331$	−6.42548e13	−	1.11293e14i	−0.888897	−	1.53961i	−0.841181	−	0.540754i	$-0.818139\pi$
$331$	−6.42548e13	−	1.11293e14i	−0.888897	−	1.53961i	−0.0477161	−	0.998861i	$-0.515194\pi$
$332$		0			0
$333$	−1.42518e14	+	2.46848e14i	−1.90733	+	3.30359i
$334$		0			0
$335$	1.09742e14			1.42111
$336$		0			0
$337$	−1.03299e14			−1.29459			−0.647293	−	0.762241i	$-0.724099\pi$
$337$	−1.03299e14			−1.29459			−0.647293	+	0.762241i	$0.724099\pi$
$338$		0			0
$339$	−1.03719e14	+	1.79646e14i	−1.25823	+	2.17931i
$340$		0			0
$341$	2.24065e12	+	3.88092e12i	0.0263163	+	0.0455812i
$342$		0			0
$343$	8.67156e13	−	1.45398e13i	0.986233	−	0.165364i
$344$		0			0
$345$	−1.29773e14	−	2.24774e14i	−1.42949	−	2.47595i
$346$		0			0
$347$	−6.83703e12	+	1.18421e13i	−0.0729551	+	0.126362i	−0.900195	−	0.435487i	$-0.856576\pi$
$347$	−6.83703e12	+	1.18421e13i	−0.0729551	+	0.126362i	0.827240	+	0.561849i	$0.189910\pi$
$348$		0			0
$349$	3.25361e13			0.336376			0.168188	−	0.985755i	$-0.446208\pi$
$349$	3.25361e13			0.336376			0.168188	+	0.985755i	$0.446208\pi$
$350$		0			0
$351$	−3.61124e14			−3.61799
$352$		0			0
$353$	−9.46425e12	+	1.63926e13i	−0.0919021	+	0.159179i	−0.908311	−	0.418294i	$-0.862628\pi$
$353$	−9.46425e12	+	1.63926e13i	−0.0919021	+	0.159179i	0.816409	+	0.577474i	$0.195961\pi$
$354$		0			0
$355$	−8.18521e13	−	1.41772e14i	−0.770503	−	1.33455i
$356$		0			0
$357$	−4.38142e13	+	8.66352e13i	−0.399889	+	0.790713i
$358$		0			0
$359$	7.36621e13	+	1.27586e14i	0.651965	+	1.12924i	0.982645	+	0.185494i	$0.0593886\pi$
$359$	7.36621e13	+	1.27586e14i	0.651965	+	1.12924i	−0.330680	+	0.943743i	$0.607278\pi$
$360$		0			0
$361$	5.79647e13	−	1.00398e14i	0.497592	−	0.861855i
$362$		0			0
$363$	2.03457e14			1.69428
$364$		0			0
$365$	7.48286e13			0.604584
$366$		0			0
$367$	2.11924e12	−	3.67063e12i	0.0166156	−	0.0287791i	−0.857598	−	0.514321i	$-0.828044\pi$
$367$	2.11924e12	−	3.67063e12i	0.0166156	−	0.0287791i	0.874214	+	0.485541i	$0.161378\pi$
$368$		0			0
$369$	1.12624e14	+	1.95070e14i	0.857007	+	1.48438i
$370$		0			0
$371$	−4.92201e13	+	2.72839e12i	−0.363568	+	0.0201535i
$372$		0			0
$373$	−3.52994e13	−	6.11403e13i	−0.253144	−	0.438459i	0.711246	−	0.702944i	$-0.248131\pi$
$373$	−3.52994e13	−	6.11403e13i	−0.253144	−	0.438459i	−0.964390	+	0.264485i	$0.914798\pi$
$374$		0			0
$375$	−7.94368e12	+	1.37589e13i	−0.0553159	+	0.0958100i
$376$		0			0
$377$	1.66982e14			1.12926
$378$		0			0
$379$	2.89337e14			1.90059			0.950294	−	0.311354i	$-0.100782\pi$
$379$	2.89337e14			1.90059			0.950294	+	0.311354i	$0.100782\pi$
$380$		0			0
$381$	8.77743e13	−	1.52030e14i	0.560119	−	0.970154i
$382$		0			0
$383$	−5.67291e13	−	9.82577e13i	−0.351733	−	0.609219i	0.634821	−	0.772660i	$-0.281074\pi$
$383$	−5.67291e13	−	9.82577e13i	−0.351733	−	0.609219i	−0.986553	+	0.163441i	$0.947741\pi$
$384$		0			0
$385$	3.84702e13	+	5.88495e13i	0.231788	+	0.354576i
$386$		0			0
$387$	9.70939e12	+	1.68172e13i	0.0568567	+	0.0984787i
$388$		0			0
$389$	6.63469e13	−	1.14916e14i	0.377657	−	0.654122i	−0.613063	−	0.790034i	$-0.710063\pi$
$389$	6.63469e13	−	1.14916e14i	0.377657	−	0.654122i	0.990721	+	0.135912i	$0.0433964\pi$
$390$		0			0
$391$	−9.28499e13			−0.513818
$392$		0			0
$393$	1.50082e13			0.0807552
$394$		0			0
$395$	−7.97378e13	+	1.38110e14i	−0.417235	+	0.722671i
$396$		0			0
$397$	7.36118e13	+	1.27499e14i	0.374627	+	0.648874i	0.990271	−	0.139151i	$-0.0444373\pi$
$397$	7.36118e13	+	1.27499e14i	0.374627	+	0.648874i	−0.615644	+	0.788025i	$0.711104\pi$
$398$		0			0
$399$	1.42471e13	+	2.17944e13i	0.0705303	+	0.107893i
$400$		0			0
$401$	−5.51572e13	−	9.55351e13i	−0.265649	−	0.460118i	0.702084	−	0.712094i	$-0.252253\pi$
$401$	−5.51572e13	−	9.55351e13i	−0.265649	−	0.460118i	−0.967733	+	0.251976i	$0.918920\pi$
$402$		0			0
$403$	2.54353e13	−	4.40553e13i	0.119196	−	0.206453i
$404$		0			0
$405$	−8.00364e14			−3.64993
$406$		0			0
$407$	1.03980e14			0.461509
$408$		0			0
$409$	5.62078e13	−	9.73547e13i	0.242839	−	0.420609i	−0.718683	−	0.695338i	$-0.755255\pi$
$409$	5.62078e13	−	9.73547e13i	0.242839	−	0.420609i	0.961522	+	0.274729i	$0.0885880\pi$
$410$		0			0
$411$	−4.75718e13	−	8.23967e13i	−0.200087	−	0.346561i
$412$		0			0
$413$	−1.87068e14	+	1.03697e13i	−0.766082	+	0.0424658i
$414$		0			0
$415$	−3.01907e14	−	5.22918e14i	−1.20395	−	2.08530i
$416$		0			0
$417$	5.73196e13	−	9.92804e13i	0.222615	−	0.385581i
$418$		0			0
$419$	1.89825e14			0.718085			0.359043	−	0.933321i	$-0.383103\pi$
$419$	1.89825e14			0.718085			0.359043	+	0.933321i	$0.383103\pi$
$420$		0			0
$421$	1.99203e14			0.734082			0.367041	−	0.930205i	$-0.380371\pi$
$421$	1.99203e14			0.734082			0.367041	+	0.930205i	$0.380371\pi$
$422$		0			0
$423$	1.73589e14	−	3.00664e14i	0.623232	−	1.07947i
$424$		0			0
$425$	7.10194e13	+	1.23009e14i	0.248449	+	0.430327i
$426$		0			0
$427$	−1.49757e14	+	2.96119e14i	−0.510545	+	1.00952i
$428$		0			0
$429$	1.11279e14	+	1.92741e14i	0.369742	+	0.640412i
$430$		0			0
$431$	−3.43729e13	+	5.95356e13i	−0.111325	+	0.192820i	−0.916305	−	0.400482i	$-0.868843\pi$
$431$	−3.43729e13	+	5.95356e13i	−0.111325	+	0.192820i	0.804980	+	0.593302i	$0.202176\pi$
$432$		0			0
$433$	−2.77431e13			−0.0875935			−0.0437968	−	0.999040i	$-0.513945\pi$
$433$	−2.77431e13			−0.0875935			−0.0437968	+	0.999040i	$0.513945\pi$
$434$		0			0
$435$	7.25114e14			2.23211
$436$		0			0
$437$	−1.24512e13	+	2.15661e13i	−0.0373734	+	0.0647326i
$438$		0			0
$439$	4.16339e13	+	7.21121e13i	0.121869	+	0.211083i	0.920505	−	0.390732i	$-0.127778\pi$
$439$	4.16339e13	+	7.21121e13i	0.121869	+	0.211083i	−0.798636	+	0.601815i	$0.794445\pi$
$440$		0			0
$441$	8.53093e14	−	9.48695e13i	2.43548	−	0.270841i
$442$		0			0
$443$	2.69384e14	+	4.66587e14i	0.750155	+	1.29931i	0.947747	+	0.319023i	$0.103355\pi$
$443$	2.69384e14	+	4.66587e14i	0.750155	+	1.29931i	−0.197592	+	0.980284i	$0.563312\pi$
$444$		0			0
$445$	1.08541e14	−	1.87998e14i	0.294858	−	0.510709i
$446$		0			0
$447$	−9.46407e14			−2.50834
$448$		0			0
$449$	1.14100e14			0.295074			0.147537	−	0.989057i	$-0.452865\pi$
$449$	1.14100e14			0.295074			0.147537	+	0.989057i	$0.452865\pi$
$450$		0			0
$451$	4.10846e13	−	7.11607e13i	0.103683	−	0.179585i
$452$		0			0
$453$	2.56854e14	+	4.44885e14i	0.632626	+	1.09574i
$454$		0			0
$455$	3.60193e14	−	7.12220e14i	0.865909	−	1.71219i
$456$		0			0
$457$	2.27653e14	+	3.94307e14i	0.534237	+	0.925326i	0.999200	+	0.0399959i	$0.0127345\pi$
$457$	2.27653e14	+	3.94307e14i	0.534237	+	0.925326i	−0.464962	+	0.885330i	$0.653932\pi$
$458$		0			0
$459$	−2.80497e14	+	4.85835e14i	−0.642626	+	1.11306i
$460$		0			0
$461$	−1.77822e14			−0.397768			−0.198884	−	0.980023i	$-0.563732\pi$
$461$	−1.77822e14			−0.397768			−0.198884	+	0.980023i	$0.563732\pi$
$462$		0			0
$463$	4.30595e13			0.0940531			0.0470266	−	0.998894i	$-0.485025\pi$
$463$	4.30595e13			0.0940531			0.0470266	+	0.998894i	$0.485025\pi$
$464$		0			0
$465$	1.10452e14	−	1.91309e14i	0.235604	−	0.408078i
$466$		0			0
$467$	−2.86216e14	−	4.95741e14i	−0.596281	−	1.03279i	−0.993365	−	0.115007i	$-0.963311\pi$
$467$	−2.86216e14	−	4.95741e14i	−0.596281	−	1.03279i	0.397083	−	0.917783i	$-0.370022\pi$
$468$		0			0
$469$	−4.87997e14	+	2.70509e13i	−0.993040	+	0.0550466i
$470$		0			0
$471$	7.45295e14	+	1.29089e15i	1.48154	+	2.56610i
$472$		0			0
$473$	3.54195e12	−	6.13483e12i	0.00687869	−	0.0119142i
$474$		0			0
$475$	3.80948e13			0.0722855
$476$		0			0
$477$	−4.81234e14			−0.892288
$478$		0			0
$479$	−7.88521e13	+	1.36576e14i	−0.142879	+	0.247474i	−0.928580	−	0.371133i	$-0.878969\pi$
$479$	−7.88521e13	+	1.36576e14i	−0.142879	+	0.247474i	0.785701	+	0.618607i	$0.212303\pi$
$480$		0			0
$481$	−5.90177e14	−	1.02222e15i	−1.04516	−	1.81028i
$482$		0			0
$483$	6.32478e14	+	9.67529e14i	1.09480	+	1.67477i
$484$		0			0
$485$	4.96903e14	+	8.60661e14i	0.840799	+	1.45631i
$486$		0			0
$487$	1.43196e14	−	2.48022e14i	0.236876	−	0.410281i	−0.722940	−	0.690910i	$-0.757210\pi$
$487$	1.43196e14	−	2.48022e14i	0.236876	−	0.410281i	0.959816	+	0.280629i	$0.0905432\pi$
$488$		0			0
$489$	8.02044e14			1.29718
$490$		0			0
$491$	4.94798e14			0.782491			0.391246	−	0.920286i	$-0.372044\pi$
$491$	4.94798e14			0.782491			0.391246	+	0.920286i	$0.372044\pi$
$492$		0			0
$493$	1.29701e14	−	2.24648e14i	0.200578	−	0.347412i
$494$		0			0
$495$	3.43181e14	+	5.94406e14i	0.519032	+	0.898990i
$496$		0			0
$497$	3.98923e14	+	6.10251e14i	0.590105	+	0.902710i
$498$		0			0
$499$	5.17660e14	+	8.96614e14i	0.749017	+	1.29734i	0.948294	+	0.317393i	$0.102807\pi$
$499$	5.17660e14	+	8.96614e14i	0.749017	+	1.29734i	−0.199277	+	0.979943i	$0.563859\pi$
$500$		0			0
$501$	−9.34555e14	+	1.61870e15i	−1.32281	+	2.29117i
$502$		0			0
$503$	−4.63983e14			−0.642508			−0.321254	−	0.946993i	$-0.604104\pi$
$503$	−4.63983e14			−0.642508			−0.321254	+	0.946993i	$0.604104\pi$
$504$		0			0
$505$	1.30541e15			1.76865
$506$		0			0
$507$	5.62642e14	−	9.74524e14i	0.745913	−	1.29196i
$508$		0			0
$509$	−3.91489e14	−	6.78079e14i	−0.507892	−	0.879695i	−0.999958	−	0.00913743i	$-0.997091\pi$
$509$	−3.91489e14	−	6.78079e14i	−0.507892	−	0.879695i	0.492066	−	0.870558i	$-0.336242\pi$
$510$		0			0
$511$	−3.32745e14	+	1.84449e13i	−0.422470	+	0.0234186i
$512$		0			0
$513$	7.52293e13	+	1.30301e14i	0.0934849	+	0.161921i
$514$		0			0
$515$	6.40228e14	−	1.10891e15i	0.778745	−	1.34883i
$516$		0			0
$517$	−1.26649e14			−0.150801
$518$		0			0
$519$	−2.24212e15			−2.61360
$520$		0			0
$521$	7.30719e13	−	1.26564e14i	0.0833956	−	0.144445i	−0.821311	−	0.570481i	$-0.806757\pi$
$521$	7.30719e13	−	1.26564e14i	0.0833956	−	0.144445i	0.904706	+	0.426036i	$0.140090\pi$
$522$		0			0
$523$	6.08353e13	+	1.05370e14i	0.0679824	+	0.117749i	0.898013	−	0.439969i	$-0.145010\pi$
$523$	6.08353e13	+	1.05370e14i	0.0679824	+	0.117749i	−0.830031	+	0.557718i	$0.811677\pi$
$524$		0			0
$525$	7.98028e14	−	1.57797e15i	0.873257	−	1.72672i
$526$		0			0
$527$	−3.95130e13	−	6.84384e13i	−0.0423430	−	0.0733401i
$528$		0			0
$529$	−7.63456e13	+	1.32235e14i	−0.0801268	+	0.138784i
$530$		0			0
$531$	−1.82900e15			−1.88016
$532$		0			0
$533$	−9.32766e14			−0.939233
$534$		0			0
$535$	−5.93058e14	+	1.02721e15i	−0.584994	+	1.01324i
$536$		0			0
$537$	1.06909e14	+	1.85172e14i	0.103313	+	0.178944i
$538$		0			0
$539$	−1.85574e14	−	2.52207e14i	−0.175703	−	0.238792i
$540$		0			0
$541$	2.27463e14	+	3.93977e14i	0.211021	+	0.365499i	0.952034	−	0.305991i	$-0.0989878\pi$
$541$	2.27463e14	+	3.93977e14i	0.211021	+	0.365499i	−0.741013	+	0.671490i	$0.765655\pi$
$542$		0			0
$543$	1.12807e15	−	1.95388e15i	1.02551	−	1.77623i
$544$		0			0
$545$	−2.44442e15			−2.17769
$546$		0			0
$547$	6.58812e14			0.575217			0.287608	−	0.957748i	$-0.407140\pi$
$547$	6.58812e14			0.575217			0.287608	+	0.957748i	$0.407140\pi$
$548$		0			0
$549$	−1.61971e15	+	2.80543e15i	−1.38609	+	2.40077i
$550$		0			0
$551$	−3.47857e13	−	6.02506e13i	−0.0291788	−	0.0505391i
$552$		0			0
$553$	3.20532e14	−	6.33797e14i	0.263562	−	0.521149i
$554$		0			0
$555$	−2.56282e15	−	4.43894e15i	−2.06589	−	3.57823i
$556$		0			0
$557$	1.12454e15	−	1.94777e15i	0.888735	−	1.53933i	0.0473637	−	0.998878i	$-0.484918\pi$
$557$	1.12454e15	−	1.94777e15i	0.888735	−	1.53933i	0.841372	−	0.540457i	$-0.181749\pi$
$558$		0			0
$559$	−8.04147e13			−0.0623118
$560$		0			0
$561$	3.45737e14			0.262694
$562$		0			0
$563$	−1.20750e15	+	2.09145e15i	−0.899684	+	1.55830i	−0.0717849	+	0.997420i	$0.522870\pi$
$563$	−1.20750e15	+	2.09145e15i	−0.899684	+	1.55830i	−0.827899	+	0.560878i	$0.810464\pi$
$564$		0			0
$565$	−1.32458e15	−	2.29424e15i	−0.967860	−	1.67638i
$566$		0			0
$567$	3.55903e15	−	1.97286e14i	2.55050	−	0.141380i
$568$		0			0
$569$	2.92094e14	+	5.05922e14i	0.205308	+	0.355603i	0.950231	−	0.311547i	$-0.100847\pi$
$569$	2.92094e14	+	5.05922e14i	0.205308	+	0.355603i	−0.744923	+	0.667150i	$0.767514\pi$
$570$		0			0
$571$	−4.51312e14	+	7.81696e14i	−0.311156	+	0.538938i	−0.978613	−	0.205710i	$-0.934050\pi$
$571$	−4.51312e14	+	7.81696e14i	−0.311156	+	0.538938i	0.667457	+	0.744649i	$0.267383\pi$
$572$		0			0
$573$	−2.54382e15			−1.72042
$574$		0			0
$575$	1.69116e15			1.12205
$576$		0			0
$577$	2.46431e14	−	4.26832e14i	0.160409	−	0.277837i	−0.774606	−	0.632444i	$-0.782052\pi$
$577$	2.46431e14	−	4.26832e14i	0.160409	−	0.277837i	0.935015	+	0.354607i	$0.115385\pi$
$578$		0			0
$579$	3.71489e14	+	6.43439e14i	0.237254	+	0.410937i
$580$		0			0
$581$	1.47141e15	+	2.25088e15i	0.922070	+	1.41053i
$582$		0			0
$583$	8.77761e13	+	1.52033e14i	0.0539758	+	0.0934889i
$584$		0			0
$585$	3.89570e15	−	6.74756e15i	2.35087	−	4.07183i
$586$		0			0
$587$	−4.72586e14			−0.279880			−0.139940	−	0.990160i	$-0.544691\pi$
$587$	−4.72586e14			−0.279880			−0.139940	+	0.990160i	$0.544691\pi$
$588$		0			0
$589$	−2.11948e13			−0.0123195
$590$		0			0
$591$	3.92585e12	−	6.79976e12i	0.00223976	−	0.00387938i
$592$		0			0
$593$	1.03063e14	+	1.78510e14i	0.0577167	+	0.0999682i	0.893440	−	0.449182i	$-0.148285\pi$
$593$	1.03063e14	+	1.78510e14i	0.0577167	+	0.0999682i	−0.835723	+	0.549151i	$0.814951\pi$
$594$		0			0
$595$	−6.78405e14	−	1.03779e15i	−0.372946	−	0.570512i
$596$		0			0
$597$	−2.46970e15	−	4.27764e15i	−1.33286	−	2.30858i
$598$		0			0
$599$	1.15771e15	−	2.00522e15i	0.613414	−	1.06246i	−0.377246	−	0.926113i	$-0.623129\pi$
$599$	1.15771e15	−	2.00522e15i	0.613414	−	1.06246i	0.990660	−	0.136352i	$-0.0435377\pi$
$600$		0			0
$601$	1.95172e15			1.01533			0.507665	−	0.861554i	$-0.330509\pi$
$601$	1.95172e15			1.01533			0.507665	+	0.861554i	$0.330509\pi$
$602$		0			0
$603$	−4.77124e15			−2.43717
$604$		0			0
$605$	−1.29917e15	+	2.25022e15i	−0.651643	+	1.12868i
$606$		0			0
$607$	3.33270e14	+	5.77241e14i	0.164157	+	0.284328i	0.936356	−	0.351053i	$-0.114176\pi$
$607$	3.33270e14	+	5.77241e14i	0.164157	+	0.284328i	−0.772199	+	0.635381i	$0.780843\pi$
$608$		0			0
$609$	−3.22441e15	+	1.78737e14i	−1.55975	+	0.0864608i
$610$		0			0
$611$	7.18844e14	+	1.24507e15i	0.341514	+	0.591519i
$612$		0			0
$613$	−1.04774e15	+	1.81474e15i	−0.488902	+	0.846803i	−0.999918	−	0.0127677i	$-0.995936\pi$
$613$	−1.04774e15	+	1.81474e15i	−0.488902	+	0.846803i	0.511016	+	0.859571i	$0.329269\pi$
$614$		0			0
$615$	−4.05050e15			−1.85650
$616$		0			0
$617$	−2.06813e15			−0.931128			−0.465564	−	0.885014i	$-0.654148\pi$
$617$	−2.06813e15			−0.931128			−0.465564	+	0.885014i	$0.654148\pi$
$618$		0			0
$619$	1.23313e15	−	2.13584e15i	0.545394	−	0.944650i	−0.453188	−	0.891415i	$-0.649713\pi$
$619$	1.23313e15	−	2.13584e15i	0.545394	−	0.944650i	0.998582	−	0.0532349i	$-0.0169532\pi$
$620$		0			0
$621$	3.33968e15	+	5.78450e15i	1.45111	+	2.51340i
$622$		0			0
$623$	−4.36314e14	+	8.62737e14i	−0.186258	+	0.368294i
$624$		0			0
$625$	1.14033e15	+	1.97512e15i	0.478290	+	0.828423i
$626$		0			0
$627$	4.63634e13	−	8.03037e13i	0.0191075	−	0.0330951i
$628$		0			0
$629$	−1.83364e15			−0.742567
$630$		0			0
$631$	−1.15579e14			−0.0459958			−0.0229979	−	0.999736i	$-0.507321\pi$
$631$	−1.15579e14			−0.0459958			−0.0229979	+	0.999736i	$0.507321\pi$
$632$		0			0
$633$	2.11925e15	−	3.67065e15i	0.828823	−	1.43556i
$634$		0			0
$635$	1.12096e15	+	1.94156e15i	0.430858	+	0.746267i
$636$		0			0
$637$	−1.42613e15	+	3.25586e15i	−0.538757	+	1.22998i
$638$		0			0
$639$	3.55867e15	+	6.16380e15i	1.32140	+	2.28873i
$640$		0			0
$641$	−2.24626e14	+	3.89063e14i	−0.0819861	+	0.142004i	−0.904103	−	0.427315i	$-0.859460\pi$
$641$	−2.24626e14	+	3.89063e14i	−0.0819861	+	0.142004i	0.822117	+	0.569319i	$0.192793\pi$
$642$		0			0
$643$	−4.10162e14			−0.147162			−0.0735809	−	0.997289i	$-0.523443\pi$
$643$	−4.10162e14			−0.147162			−0.0735809	+	0.997289i	$0.523443\pi$
$644$		0			0
$645$	−3.49198e14			−0.123167
$646$		0			0
$647$	−2.10384e15	+	3.64396e15i	−0.729524	+	1.26357i	0.227560	+	0.973764i	$0.426925\pi$
$647$	−2.10384e15	+	3.64396e15i	−0.729524	+	1.26357i	−0.957084	+	0.289809i	$0.906408\pi$
$648$		0			0
$649$	3.33606e14	+	5.77823e14i	0.113733	+	0.196992i
$650$		0			0
$651$	−4.43998e14	+	8.77930e14i	−0.148828	+	0.294282i
$652$		0			0
$653$	5.69350e14	+	9.86142e14i	0.187653	+	0.325025i	0.944467	−	0.328605i	$-0.106578\pi$
$653$	5.69350e14	+	9.86142e14i	0.187653	+	0.325025i	−0.756814	+	0.653630i	$0.773245\pi$
$654$		0			0
$655$	−9.58343e13	+	1.65990e14i	−0.0310595	+	0.0537966i
$656$		0			0
$657$	−3.25331e15			−1.03685
$658$		0			0
$659$	−4.18339e15			−1.31117			−0.655584	−	0.755122i	$-0.727577\pi$
$659$	−4.18339e15			−1.31117			−0.655584	+	0.755122i	$0.727577\pi$
$660$		0			0
$661$	−2.81203e15	+	4.87057e15i	−0.866785	+	1.50132i	−0.00152075	+	0.999999i	$0.500484\pi$
$661$	−2.81203e15	+	4.87057e15i	−0.866785	+	1.50132i	−0.865264	+	0.501316i	$0.832849\pi$
$662$		0			0
$663$	−1.96236e15	−	3.39891e15i	−0.594915	−	1.03042i
$664$		0			0
$665$	−3.32019e14	+	1.84046e13i	−0.0990020	+	0.00548792i
$666$		0			0
$667$	−1.54426e15	−	2.67473e15i	−0.452926	−	0.784491i
$668$		0			0
$669$	3.90513e15	−	6.76388e15i	1.12666	−	1.95143i
$670$		0			0
$671$	1.18173e15			0.335386
$672$		0			0
$673$	5.05272e14			0.141072			0.0705362	−	0.997509i	$-0.477529\pi$
$673$	5.05272e14			0.141072			0.0705362	+	0.997509i	$0.477529\pi$
$674$		0			0
$675$	5.10894e15	−	8.84895e15i	1.40333	−	2.43064i
$676$		0			0
$677$	−7.63868e14	−	1.32306e15i	−0.206434	−	0.357553i	0.744155	−	0.668007i	$-0.232852\pi$
$677$	−7.63868e14	−	1.32306e15i	−0.206434	−	0.357553i	−0.950589	+	0.310454i	$0.899519\pi$
$678$		0			0
$679$	−2.42176e15	−	3.70467e15i	−0.643942	−	0.985067i
$680$		0			0
$681$	9.43188e14	+	1.63365e15i	0.246768	+	0.427415i
$682$		0			0
$683$	−3.40746e15	+	5.90190e15i	−0.877238	+	1.51942i	−0.0228783	+	0.999738i	$0.507283\pi$
$683$	−3.40746e15	+	5.90190e15i	−0.877238	+	1.51942i	−0.854360	+	0.519682i	$0.826050\pi$
$684$		0			0
$685$	1.21507e15			0.307824
$686$		0			0
$687$	1.16347e16			2.90064
$688$		0			0
$689$	9.96414e14	−	1.72584e15i	0.244475	−	0.423443i
$690$		0			0
$691$	2.80338e15	+	4.85560e15i	0.676944	+	1.17250i	0.975897	+	0.218234i	$0.0700294\pi$
$691$	2.80338e15	+	4.85560e15i	0.676944	+	1.17250i	−0.298953	+	0.954268i	$0.596637\pi$
$692$		0			0
$693$	−1.67256e15	−	2.55859e15i	−0.397511	−	0.608090i
$694$		0			0
$695$	7.32023e14	+	1.26790e15i	0.171241	+	0.296599i
$696$		0			0
$697$	−7.24511e14	+	1.25489e15i	−0.166826	+	0.288952i
$698$		0			0
$699$	1.24619e16			2.82462
$700$		0			0
$701$	7.51596e15			1.67701			0.838504	−	0.544895i	$-0.183430\pi$
$701$	7.51596e15			1.67701			0.838504	+	0.544895i	$0.183430\pi$
$702$		0			0
$703$	−2.45891e14	+	4.25896e14i	−0.0540119	+	0.0935513i
$704$		0			0
$705$	3.12156e15	+	5.40669e15i	0.675042	+	1.16921i
$706$		0			0
$707$	−5.80483e15	+	3.21776e14i	−1.23590	+	0.0685088i
$708$		0			0
$709$	−1.35794e15	−	2.35201e15i	−0.284659	−	0.493044i	0.687868	−	0.725836i	$-0.258547\pi$
$709$	−1.35794e15	−	2.35201e15i	−0.284659	−	0.493044i	−0.972526	+	0.232793i	$0.925214\pi$
$710$		0			0
$711$	3.46675e15	−	6.00458e15i	0.715549	−	1.23937i
$712$		0			0
$713$	−9.40908e14			−0.191229
$714$		0			0
$715$	−2.84227e15			−0.568830
$716$		0			0
$717$	5.82007e15	−	1.00807e16i	1.14703	−	1.98671i
$718$		0			0
$719$	−3.07684e15	−	5.32925e15i	−0.597168	−	1.03433i	−0.993237	−	0.116104i	$-0.962959\pi$
$719$	−3.07684e15	−	5.32925e15i	−0.597168	−	1.03433i	0.396069	−	0.918221i	$-0.370374\pi$
$720$		0			0
$721$	−2.57360e15	+	5.08886e15i	−0.491924	+	0.972696i
$722$		0			0
$723$	−6.40261e15	−	1.10896e16i	−1.20530	−	2.08765i
$724$		0			0
$725$	−2.36235e15	+	4.09172e15i	−0.438012	+	0.758659i
$726$		0			0
$727$	−3.95637e15			−0.722533			−0.361266	−	0.932463i	$-0.617655\pi$
$727$	−3.95637e15			−0.722533			−0.361266	+	0.932463i	$0.617655\pi$
$728$		0			0
$729$	6.97473e15			1.25466
$730$		0			0
$731$	−6.24607e13	+	1.08185e14i	−0.0110678	+	0.0191700i
$732$		0			0
$733$	−1.81738e15	−	3.14779e15i	−0.317229	−	0.549457i	0.662680	−	0.748903i	$-0.269419\pi$
$733$	−1.81738e15	−	3.14779e15i	−0.317229	−	0.549457i	−0.979909	+	0.199446i	$0.936086\pi$
$734$		0			0
$735$	−6.19293e15	+	1.41385e16i	−1.06492	+	2.43120i
$736$		0			0
$737$	8.70264e14	+	1.50734e15i	0.147428	+	0.255353i
$738$		0			0
$739$	2.53693e15	−	4.39409e15i	0.423413	−	0.733373i	−0.572858	−	0.819655i	$-0.694165\pi$
$739$	2.53693e15	−	4.39409e15i	0.423413	−	0.733373i	0.996271	+	0.0862819i	$0.0274986\pi$
$740$		0			0
$741$	−1.05261e15			−0.173089
$742$		0			0
$743$	−1.73028e15			−0.280335			−0.140168	−	0.990128i	$-0.544764\pi$
$743$	−1.73028e15			−0.280335			−0.140168	+	0.990128i	$0.544764\pi$
$744$		0			0
$745$	6.04324e15	−	1.04672e16i	0.964739	−	1.67098i
$746$		0			0
$747$	1.31260e16	+	2.27348e16i	2.06475	+	3.57625i
$748$		0			0
$749$	2.38399e15	−	4.71393e15i	0.369533	−	0.730689i
$750$		0			0
$751$	4.99394e14	+	8.64975e14i	0.0762822	+	0.132125i	0.901643	−	0.432481i	$-0.142362\pi$
$751$	4.99394e14	+	8.64975e14i	0.0762822	+	0.132125i	−0.825361	+	0.564605i	$0.809028\pi$
$752$		0			0
$753$	−5.35079e15	+	9.26784e15i	−0.805463	+	1.39510i
$754$		0			0
$755$	−6.56053e15			−0.973264
$756$		0			0
$757$	−1.72753e15			−0.252579			−0.126290	−	0.991993i	$-0.540307\pi$
$757$	−1.72753e15			−0.252579			−0.126290	+	0.991993i	$0.540307\pi$
$758$		0			0
$759$	2.05823e15	−	3.56496e15i	0.296595	−	0.513718i
$760$		0			0
$761$	7.75708e14	+	1.34357e15i	0.110175	+	0.190829i	0.915841	−	0.401542i	$-0.131526\pi$
$761$	7.75708e14	+	1.34357e15i	0.110175	+	0.190829i	−0.805666	+	0.592370i	$0.798192\pi$
$762$		0			0
$763$	1.08698e16	−	6.02537e14i	1.52172	−	0.0843527i
$764$		0			0
$765$	−6.05184e15	−	1.04821e16i	−0.835122	−	1.44647i
$766$		0			0
$767$	3.78702e15	−	6.55931e15i	0.515137	−	0.892244i
$768$		0			0
$769$	2.36048e15			0.316524			0.158262	−	0.987397i	$-0.449411\pi$
$769$	2.36048e15			0.316524			0.158262	+	0.987397i	$0.449411\pi$
$770$		0			0
$771$	1.99145e16			2.63251
$772$		0			0
$773$	−6.38424e13	+	1.10578e14i	−0.00831997	+	0.0144106i	−0.870155	−	0.492777i	$-0.835982\pi$
$773$	−6.38424e13	+	1.10578e14i	−0.00831997	+	0.0144106i	0.861835	+	0.507188i	$0.169315\pi$
$774$		0			0
$775$	7.19685e14	+	1.24653e15i	0.0924663	+	0.160156i
$776$		0			0
$777$	1.24904e16	+	1.91072e16i	1.58220	+	2.42037i
$778$		0			0
$779$	1.94314e14	+	3.36561e14i	0.0242688	+	0.0420347i
$780$		0			0
$781$	1.29819e15	−	2.24853e15i	0.159866	−	0.276897i
$782$		0			0
$783$	−1.86606e16			−2.26588
$784$		0			0
$785$	−1.90362e16			−2.27928
$786$		0			0
$787$	−3.90777e15	+	6.76846e15i	−0.461390	+	0.799151i	−0.999031	−	0.0440233i	$-0.985982\pi$
$787$	−3.90777e15	+	6.76846e15i	−0.461390	+	0.799151i	0.537641	+	0.843174i	$0.319316\pi$
$788$		0			0
$789$	1.19517e16	+	2.07009e16i	1.39157	+	2.41027i
$790$		0			0
$791$	6.45562e15	+	9.87545e15i	0.741255	+	1.13393i
$792$		0			0
$793$	−6.70736e15	−	1.16175e16i	−0.759537	−	1.31556i
$794$		0			0
$795$	4.32689e15	−	7.49439e15i	0.483233	−	0.836984i
$796$		0			0
$797$	6.95757e15			0.766368			0.383184	−	0.923672i	$-0.374828\pi$
$797$	6.95757e15			0.766368			0.383184	+	0.923672i	$0.374828\pi$
$798$		0			0
$799$	2.23340e15			0.242638
$800$		0			0
$801$	−4.71900e15	+	8.17355e15i	−0.505675	+	0.875855i
$802$		0			0
$803$	5.93397e14	+	1.02779e15i	0.0627205	+	0.108635i
$804$		0			0
$805$	−1.47395e16	+	8.17044e14i	−1.53675	+	0.0851860i
$806$		0			0
$807$	−7.35583e15	−	1.27407e16i	−0.756532	−	1.31035i
$808$		0			0
$809$	−3.61624e15	+	6.26351e15i	−0.366894	+	0.635478i	−0.989078	−	0.147392i	$-0.952912\pi$
$809$	−3.61624e15	+	6.26351e15i	−0.366894	+	0.635478i	0.622185	+	0.782871i	$0.286245\pi$
$810$		0			0
$811$	−4.95682e15			−0.496122			−0.248061	−	0.968744i	$-0.579793\pi$
$811$	−4.95682e15			−0.496122			−0.248061	+	0.968744i	$0.579793\pi$
$812$		0			0
$813$	3.07364e16			3.03498
$814$		0			0
$815$	−5.12141e15	+	8.87055e15i	−0.498911	+	0.864139i
$816$		0			0
$817$	1.67520e13	+	2.90153e13i	0.00161007	+	0.00278872i
$818$		0			0
$819$	−1.56600e16	+	3.09651e16i	−1.48502	+	2.93637i
$820$		0			0
$821$	−3.69548e14	−	6.40076e14i	−0.0345767	−	0.0598886i	0.848219	−	0.529645i	$-0.177675\pi$
$821$	−3.69548e14	−	6.40076e14i	−0.0345767	−	0.0598886i	−0.882796	+	0.469757i	$0.844342\pi$
$822$		0			0
$823$	−8.05747e15	+	1.39560e16i	−0.743875	+	1.28843i	0.206844	+	0.978374i	$0.433681\pi$
$823$	−8.05747e15	+	1.39560e16i	−0.743875	+	1.28843i	−0.950719	+	0.310055i	$0.899653\pi$
$824$		0			0
$825$	−6.29722e15			−0.573657
$826$		0			0
$827$	1.78503e16			1.60459			0.802296	−	0.596926i	$-0.203611\pi$
$827$	1.78503e16			1.60459			0.802296	+	0.596926i	$0.203611\pi$
$828$		0			0
$829$	−6.49860e15	+	1.12559e16i	−0.576461	+	0.998460i	0.419420	+	0.907792i	$0.362233\pi$
$829$	−6.49860e15	+	1.12559e16i	−0.576461	+	0.998460i	−0.995881	+	0.0906676i	$0.971100\pi$
$830$		0			0
$831$	−1.49426e16	−	2.58814e16i	−1.30804	−	2.26559i
$832$		0			0
$833$	3.27252e15	+	4.44757e15i	0.282706	+	0.384216i
$834$		0			0
$835$	−1.19351e16	−	2.06722e16i	−1.01754	−	1.76243i
$836$		0			0
$837$	−2.84245e15	+	4.92328e15i	−0.239168	+	0.414252i
$838$		0			0
$839$	3.83501e15			0.318475			0.159238	−	0.987240i	$-0.449096\pi$
$839$	3.83501e15			0.318475			0.159238	+	0.987240i	$0.449096\pi$
$840$		0			0
$841$	−3.57192e15			−0.292768
$842$		0			0
$843$	9.53002e15	−	1.65065e16i	0.770979	−	1.33537i
$844$		0			0
$845$	7.18544e15	+	1.24456e16i	0.573775	+	0.993808i
$846$		0			0
$847$	5.22241e15	−	1.03264e16i	0.411635	−	0.813938i
$848$		0			0
$849$	−1.73173e16	−	2.99945e16i	−1.34737	−	2.33372i
$850$		0			0
$851$	−1.09160e16	+	1.89070e16i	−0.838396	+	1.45215i
$852$		0			0
$853$	−8.88285e15			−0.673492			−0.336746	−	0.941596i	$-0.609326\pi$
$853$	−8.88285e15			−0.673492			−0.336746	+	0.941596i	$0.609326\pi$
$854$		0			0
$855$	−3.24621e15			−0.242976
$856$		0			0
$857$	−4.30831e15	+	7.46222e15i	−0.318356	+	0.551409i	−0.980145	−	0.198281i	$-0.936464\pi$
$857$	−4.30831e15	+	7.46222e15i	−0.318356	+	0.551409i	0.661789	+	0.749690i	$0.269797\pi$
$858$		0			0
$859$	4.98251e15	+	8.62996e15i	0.363484	+	0.629573i	0.988532	−	0.151014i	$-0.0482537\pi$
$859$	4.98251e15	+	8.62996e15i	0.363484	+	0.629573i	−0.625047	+	0.780587i	$0.714920\pi$
$860$		0			0
$861$	1.80116e16	−	9.98429e14i	1.29729	−	0.0719117i
$862$		0			0
$863$	−1.52755e15	−	2.64580e15i	−0.108627	−	0.188147i	0.806587	−	0.591115i	$-0.201312\pi$
$863$	−1.52755e15	−	2.64580e15i	−0.108627	−	0.188147i	−0.915214	+	0.402968i	$0.867979\pi$
$864$		0			0
$865$	1.43170e16	−	2.47977e16i	1.00522	−	1.74110i
$866$		0			0
$867$	2.06976e16			1.43487
$868$		0			0
$869$	−2.52931e15			−0.173138
$870$		0			0
$871$	9.87903e15	−	1.71110e16i	0.667751	−	1.15658i
$872$		0			0
$873$	−2.16038e16	−	3.74188e16i	−1.44195	−	2.49753i
$874$		0			0
$875$	4.94428e14	+	7.56348e14i	0.0325881	+	0.0498514i
$876$		0			0
$877$	−2.06631e15	−	3.57895e15i	−0.134492	−	0.232948i	0.790911	−	0.611931i	$-0.209607\pi$
$877$	−2.06631e15	−	3.57895e15i	−0.134492	−	0.232948i	−0.925403	+	0.378983i	$0.876274\pi$
$878$		0			0
$879$	1.41594e16	−	2.45248e16i	0.910134	−	1.57640i
$880$		0			0
$881$	2.04897e16			1.30067			0.650336	−	0.759647i	$-0.274628\pi$
$881$	2.04897e16			1.30067			0.650336	+	0.759647i	$0.274628\pi$
$882$		0			0
$883$	−1.67624e16			−1.05088			−0.525438	−	0.850832i	$-0.676098\pi$
$883$	−1.67624e16			−1.05088			−0.525438	+	0.850832i	$0.676098\pi$
$884$		0			0
$885$	1.64450e16	−	2.84836e16i	1.01823	−	1.76363i
$886$		0			0
$887$	9.84570e15	+	1.70533e16i	0.602097	+	1.04286i	0.992503	+	0.122220i	$0.0390014\pi$
$887$	9.84570e15	+	1.70533e16i	0.602097	+	1.04286i	−0.390406	+	0.920643i	$0.627665\pi$
$888$		0			0
$889$	−5.46322e15	−	8.35733e15i	−0.329981	−	0.504787i
$890$		0			0
$891$	−6.34695e15	−	1.09932e16i	−0.378650	−	0.655841i
$892$		0			0
$893$	2.99499e14	−	5.18748e14i	0.0176487	−	0.0305684i
$894$		0			0
$895$	−2.73065e15			−0.158942
$896$		0			0
$897$	−4.67291e16			−2.68676
$898$		0			0
$899$	1.31434e15	−	2.27650e15i	0.0746499	−	0.129297i
$900$		0			0
$901$	−1.54790e15	−	2.68103e15i	−0.0868471	−	0.150424i
$902$		0			0
$903$	1.55280e15	−	8.60755e13i	0.0860662	−	0.00477086i
$904$		0			0
$905$	1.44065e16	+	2.49528e16i	0.788845	+	1.36632i
$906$		0			0
$907$	−7.98542e15	+	1.38311e16i	−0.431974	+	0.748201i	−0.997043	−	0.0768426i	$-0.975516\pi$
$907$	−7.98542e15	+	1.38311e16i	−0.431974	+	0.748201i	0.565069	+	0.825043i	$0.308849\pi$
$908$		0			0
$909$	−5.67549e16			−3.03320
$910$		0			0
$911$	7.80377e15			0.412053			0.206027	−	0.978546i	$-0.433947\pi$
$911$	7.80377e15			0.412053			0.206027	+	0.978546i	$0.433947\pi$
$912$		0			0
$913$	4.78830e15	−	8.29357e15i	0.249800	−	0.432666i
$914$		0			0
$915$	−2.91265e16	−	5.04485e16i	−1.50131	−	2.60035i
$916$		0			0
$917$	3.85237e14	−	7.61740e14i	0.0196199	−	0.0387950i
$918$		0			0
$919$	−4.34918e15	−	7.53299e15i	−0.218863	−	0.379081i	0.735598	−	0.677418i	$-0.236901\pi$
$919$	−4.34918e15	−	7.53299e15i	−0.218863	−	0.379081i	−0.954461	+	0.298337i	$0.903568\pi$
$920$		0			0
$921$	5.90664e15	−	1.02306e16i	0.293706	−	0.508713i
$922$		0			0
$923$	−2.94735e16			−1.44818
$924$		0			0
$925$	3.33978e16			1.62158
$926$		0			0
$927$	−2.78351e16	+	4.82118e16i	−1.33553	+	2.31321i
$928$		0			0
$929$	2.05086e16	+	3.55220e16i	0.972412	+	1.68427i	0.688222	+	0.725500i	$0.258391\pi$
$929$	2.05086e16	+	3.55220e16i	0.972412	+	1.68427i	0.284190	+	0.958768i	$0.408275\pi$
$930$		0			0
$931$	1.47187e15	−	1.63682e14i	0.0689679	−	0.00766969i
$932$		0			0
$933$	−2.09978e16	−	3.63693e16i	−0.972355	−	1.68417i
$934$		0			0
$935$	−2.20769e15	+	3.82383e15i	−0.101035	+	0.174999i
$936$		0			0
$937$	−7.62475e15			−0.344872			−0.172436	−	0.985021i	$-0.555164\pi$
$937$	−7.62475e15			−0.344872			−0.172436	+	0.985021i	$0.555164\pi$
$938$		0			0
$939$	−4.22406e16			−1.88829
$940$		0			0
$941$	1.36567e16	−	2.36540e16i	0.603395	−	1.04511i	−0.388908	−	0.921277i	$-0.627148\pi$
$941$	1.36567e16	−	2.36540e16i	0.603395	−	1.04511i	0.992303	−	0.123834i	$-0.0395190\pi$
$942$		0			0
$943$	8.62626e15	+	1.49411e16i	0.376711	+	0.652482i
$944$		0			0
$945$	−4.02523e16	+	7.95921e16i	−1.73746	+	3.43554i
$946$		0			0
$947$	1.29620e16	+	2.24508e16i	0.553027	+	0.957871i	0.998054	+	0.0623538i	$0.0198607\pi$
$947$	1.29620e16	+	2.24508e16i	0.553027	+	0.957871i	−0.445027	+	0.895517i	$0.646806\pi$
$948$		0			0
$949$	6.73610e15	−	1.16673e16i	0.284082	−	0.492045i
$950$		0			0
$951$	−2.70249e16			−1.12661
$952$		0			0
$953$	−3.31043e16			−1.36419			−0.682093	−	0.731265i	$-0.738930\pi$
$953$	−3.31043e16			−1.36419			−0.682093	+	0.731265i	$0.738930\pi$
$954$		0			0
$955$	1.62434e16	−	2.81344e16i	0.661697	−	1.14609i
$956$		0			0
$957$	5.75022e15	+	9.95966e15i	0.231563	+	0.401078i
$958$		0			0
$959$	−5.40313e15	+	2.99508e14i	−0.215101	+	0.0119236i
$960$		0			0
$961$	1.23038e16	+	2.13109e16i	0.484241	+	0.838730i
$962$		0			0
$963$	2.57843e16	−	4.46596e16i	1.00325	−	1.73768i
$964$		0			0
$965$	−9.48851e15			−0.365004
$966$		0			0
$967$	−1.03306e16			−0.392897			−0.196449	−	0.980514i	$-0.562941\pi$
$967$	−1.03306e16			−0.392897			−0.196449	+	0.980514i	$0.562941\pi$
$968$		0			0
$969$	−8.17599e14	+	1.41612e15i	−0.0307439	+	0.0532501i
$970$		0			0
$971$	4.84130e14	+	8.38537e14i	0.0179993	+	0.0311757i	0.874885	−	0.484331i	$-0.160937\pi$
$971$	4.84130e14	+	8.38537e14i	0.0179993	+	0.0311757i	−0.856885	+	0.515507i	$0.827604\pi$
$972$		0			0
$973$	−3.56766e15	−	5.45762e15i	−0.131149	−	0.200624i
$974$		0			0
$975$	3.57423e16	+	6.19075e16i	1.29914	+	2.25018i
$976$		0			0
$977$	−7.09083e15	+	1.22817e16i	−0.254846	+	0.441406i	−0.964854	−	0.262788i	$-0.915358\pi$
$977$	−7.09083e15	+	1.22817e16i	−0.254846	+	0.441406i	0.710008	+	0.704194i	$0.248691\pi$
$978$		0			0
$979$	3.44295e15			0.122356
$980$		0			0
$981$	1.06276e17			3.73469
$982$		0			0
$983$	2.43090e16	−	4.21044e16i	0.844738	−	1.46313i	−0.0411098	−	0.999155i	$-0.513089\pi$
$983$	2.43090e16	−	4.21044e16i	0.844738	−	1.46313i	0.885848	−	0.463975i	$-0.153577\pi$
$984$		0			0
$985$	5.01366e13	+	8.68392e13i	0.00172288	+	0.00298412i
$986$		0			0
$987$	−1.52135e16	−	2.32728e16i	−0.516994	−	0.790870i
$988$		0			0
$989$	7.43678e14	+	1.28809e15i	0.0249922	+	0.0432878i
$990$		0			0
$991$	1.72625e16	−	2.98995e16i	0.573718	−	0.993709i	−0.422461	−	0.906381i	$-0.638834\pi$
$991$	1.72625e16	−	2.98995e16i	0.573718	−	0.993709i	0.996180	−	0.0873282i	$-0.0278329\pi$
$992$		0			0
$993$	−1.00471e17			−3.30234
$994$		0			0
$995$	6.30805e16			2.05054
$996$		0			0
$997$	−1.72913e16	+	2.99493e16i	−0.555909	+	0.962862i	0.441924	+	0.897053i	$0.354296\pi$
$997$	−1.72913e16	+	2.99493e16i	−0.555909	+	0.962862i	−0.997832	+	0.0658092i	$0.979037\pi$
$998$		0			0
$999$	6.59536e16	+	1.14235e17i	2.09714	+	3.63236i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	112.12.i.b.81.4		8
4.3	odd	2		14.12.c.b.11.1	yes	8
7.2	even	3	inner	112.12.i.b.65.4		8
12.11	even	2		126.12.g.c.109.1		8
28.3	even	6		98.12.a.h.1.1		4
28.11	odd	6		98.12.a.i.1.4		4
28.19	even	6		98.12.c.n.79.4		8
28.23	odd	6		14.12.c.b.9.1	✓	8
28.27	even	2		98.12.c.n.67.4		8
84.23	even	6		126.12.g.c.37.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.12.c.b.9.1	✓	8	28.23	odd	6
14.12.c.b.11.1	yes	8	4.3	odd	2
98.12.a.h.1.1		4	28.3	even	6
98.12.a.i.1.4		4	28.11	odd	6
98.12.c.n.67.4		8	28.27	even	2
98.12.c.n.79.4		8	28.19	even	6
112.12.i.b.65.4		8	7.2	even	3	inner
112.12.i.b.81.4		8	1.1	even	1	trivial
126.12.g.c.37.1		8	84.23	even	6
126.12.g.c.109.1		8	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 112 = 2^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	112.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(390.910 - 677.077i) q^{3} +(4992.28 + 8646.88i) q^{5} +(-24330.9 - 37220.1i) q^{7} +(-217048. - 375939. i) q^{9} +(-79178.4 + 137141. i) q^{11} +1.79763e6 q^{13} +7.80614e6 q^{15} +(1.39628e6 - 2.41842e6i) q^{17} +(-374482. - 648622. i) q^{19} +(-3.47121e7 + 1.92417e6i) q^{21} +(-1.66245e7 - 2.87945e7i) q^{23} +(-2.54317e7 + 4.40490e7i) q^{25} -2.00889e8 q^{27} +9.28902e7 q^{29} +(1.41494e7 - 2.45075e7i) q^{31} +(6.19033e7 + 1.07220e8i) q^{33} +(2.00371e8 - 3.96200e8i) q^{35} +(-3.28309e8 - 5.68647e8i) q^{37} +(7.02712e8 - 1.21713e9i) q^{39} -5.18887e8 q^{41} -4.47337e7 q^{43} +(2.16713e9 - 3.75359e9i) q^{45} +(3.99885e8 + 6.92620e8i) q^{47} +(-7.93341e8 + 1.81120e9i) q^{49} +(-1.09164e9 - 1.89077e9i) q^{51} +(5.54293e8 - 9.60064e8i) q^{53} -1.58112e9 q^{55} -5.85556e8 q^{57} +(2.10667e9 - 3.64887e9i) q^{59} +(-3.73122e9 - 6.46267e9i) q^{61} +(-8.71149e9 + 1.72255e10i) q^{63} +(8.97427e9 + 1.55439e10i) q^{65} +(5.49559e9 - 9.51864e9i) q^{67} -2.59948e10 q^{69} -1.63957e10 q^{71} +(3.74721e9 - 6.49036e9i) q^{73} +(1.98830e10 + 3.44384e10i) q^{75} +(7.03088e9 - 3.89739e8i) q^{77} +(7.98611e9 + 1.38323e10i) q^{79} +(-4.00801e10 + 6.94207e10i) q^{81} -6.04748e10 q^{83} +2.78825e10 q^{85} +(3.63118e10 - 6.28938e10i) q^{87} +(-1.08709e10 - 1.88289e10i) q^{89} +(-4.37379e10 - 6.69079e10i) q^{91} +(-1.10623e10 - 1.91604e10i) q^{93} +(3.73904e9 - 6.47621e9i) q^{95} +9.95342e10 q^{97} +6.87422e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 266 q^{3} + 3808 q^{5} - 110328 q^{7} - 503848 q^{9} - 920150 q^{11} + 997472 q^{13} + 500444 q^{15} + 1333724 q^{17} + 21551726 q^{19} - 90442480 q^{21} - 72510158 q^{23} - 77154744 q^{25} - 156683212 q^{27}+ \cdots + 545487662552 q^{99}+O(q^{100}) \) 8 * q + 266 * q^3 + 3808 * q^5 - 110328 * q^7 - 503848 * q^9 - 920150 * q^11 + 997472 * q^13 + 500444 * q^15 + 1333724 * q^17 + 21551726 * q^19 - 90442480 * q^21 - 72510158 * q^23 - 77154744 * q^25 - 156683212 * q^27 + 213575104 * q^29 + 194359774 * q^31 - 164547124 * q^33 + 981719074 * q^35 + 171517048 * q^37 + 653125236 * q^39 - 3312095136 * q^41 - 850279648 * q^43 + 6784014272 * q^45 - 2223880974 * q^47 - 4232044312 * q^49 - 6968362082 * q^51 - 7185483360 * q^53 - 624915620 * q^55 + 4959469112 * q^57 - 6997401502 * q^59 - 6476463280 * q^61 + 17252507684 * q^63 + 30625528248 * q^65 + 18660972186 * q^67 + 31319762096 * q^69 - 23224449248 * q^71 + 3731641452 * q^73 + 91204646176 * q^75 - 29345595440 * q^77 - 12221157926 * q^79 - 97333443028 * q^81 - 316739523968 * q^83 - 147794862544 * q^85 + 218122807364 * q^87 - 71204406084 * q^89 - 189816116528 * q^91 + 9838293624 * q^93 + 215091896614 * q^95 + 688251797184 * q^97 + 545487662552 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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