LMFDB - Embedded newform 112.10.i.b.65.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [112,10,Mod(65,112)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(112, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2])) N = Newforms(chi, 10, names="a")

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

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

Newform invariants

comment:select newform

sage:traces = [6,0,233] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$57.6840136504$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} + 1116x^{4} - 3085x^{3} + 1245325x^{2} - 2341500x + 4410000 $ x^6 - x^5 + 1116x^4 - 3085x^3 + 1245325x^2 - 2341500x + 4410000
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{6}\cdot 3\cdot 7^{2} $
Twist minimal:	no (minimal twist has level 14)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			65.1
Root			$-16.9063 + 29.2826i$ of defining polynomial
Character	$\chi$	$=$	112.65
Dual form			112.10.i.b.81.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-11.9224 - 20.6501i) q^{3} +(-541.184 + 937.358i) q^{5} +(5624.74 + 2952.27i) q^{7} +(9557.22 - 16553.6i) q^{9} +(37454.4 + 64872.9i) q^{11} -50060.6 q^{13} +25808.7 q^{15} +(-59482.6 - 103027. i) q^{17} +(402181. - 696599. i) q^{19} +(-6095.40 - 151350. i) q^{21} +(134394. - 232777. i) q^{23} +(390803. + 676891. i) q^{25} -925113. q^{27} -5.29735e6 q^{29} +(2.73293e6 + 4.73357e6i) q^{31} +(893089. - 1.54688e6i) q^{33} +(-5.81135e6 + 3.67467e6i) q^{35} +(-1.55446e6 + 2.69240e6i) q^{37} +(596840. + 1.03376e6i) q^{39} +2.07841e7 q^{41} -4.76437e6 q^{43} +(1.03444e7 + 1.79171e7i) q^{45} +(1.50766e7 - 2.61135e7i) q^{47} +(2.29218e7 + 3.32115e7i) q^{49} +(-1.41835e6 + 2.45665e6i) q^{51} +(-2.85871e7 - 4.95142e7i) q^{53} -8.10788e7 q^{55} -1.91798e7 q^{57} +(9.18379e7 + 1.59068e8i) q^{59} +(-2.65083e7 + 4.59138e7i) q^{61} +(1.02628e8 - 6.48941e7i) q^{63} +(2.70920e7 - 4.69247e7i) q^{65} +(8.67433e7 + 1.50244e8i) q^{67} -6.40918e6 q^{69} +1.68342e8 q^{71} +(2.32340e8 + 4.02424e8i) q^{73} +(9.31858e6 - 1.61403e7i) q^{75} +(1.91489e7 + 4.75469e8i) q^{77} +(-2.06864e8 + 3.58299e8i) q^{79} +(-1.77085e8 - 3.06720e8i) q^{81} -3.50584e8 q^{83} +1.28764e8 q^{85} +(6.31569e7 + 1.09391e8i) q^{87} +(-2.58736e8 + 4.48144e8i) q^{89} +(-2.81578e8 - 1.47792e8i) q^{91} +(6.51659e7 - 1.12871e8i) q^{93} +(4.35308e8 + 7.53976e8i) q^{95} +1.32529e9 q^{97} +1.43184e9 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 233 q^{3} - 733 q^{5} - 5012 q^{7} - 15058 q^{9} - 7339 q^{11} + 197036 q^{13} - 738238 q^{15} - 306665 q^{17} + 377991 q^{19} - 1585927 q^{21} + 2267255 q^{23} - 142612 q^{25} - 21348358 q^{27} - 13085956 q^{29}+ \cdots + 1256218868 q^{99}+O(q^{100}) $ 6 * q + 233 * q^3 - 733 * q^5 - 5012 * q^7 - 15058 * q^9 - 7339 * q^11 + 197036 * q^13 - 738238 * q^15 - 306665 * q^17 + 377991 * q^19 - 1585927 * q^21 + 2267255 * q^23 - 142612 * q^25 - 21348358 * q^27 - 13085956 * q^29 + 6654517 * q^31 + 3747977 * q^33 - 22864807 * q^35 - 22287969 * q^37 - 3151974 * q^39 + 68096196 * q^41 + 125648280 * q^43 - 86300638 * q^45 + 52703019 * q^47 + 57596070 * q^49 - 97736333 * q^51 - 12091125 * q^53 - 377435398 * q^55 - 158517670 * q^57 + 12949897 * q^59 - 160252153 * q^61 + 356021666 * q^63 + 334191270 * q^65 + 480890225 * q^67 + 1131233162 * q^69 + 74421440 * q^71 + 251382283 * q^73 - 16322324 * q^75 - 527045155 * q^77 - 286494785 * q^79 - 2165894587 * q^81 - 2295182344 * q^83 - 2389185074 * q^85 - 1412199358 * q^87 - 901243845 * q^89 - 1026798920 * q^91 + 710456889 * q^93 + 887366177 * q^95 + 629707876 * q^97 + 1256218868 * q^99

Character values

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

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

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−11.9224	−	20.6501i	−0.0849799	−	0.147190i	0.820403	−	0.571786i	$-0.193749\pi$
$3$	−11.9224	−	20.6501i	−0.0849799	−	0.147190i	−0.905383	+	0.424597i	$0.860416\pi$
$4$		0			0
$5$	−541.184	+	937.358i	−0.387239	+	0.670718i	−0.992077	−	0.125630i	$-0.959905\pi$
$5$	−541.184	+	937.358i	−0.387239	+	0.670718i	0.604838	+	0.796349i	$0.293238\pi$
$6$		0			0
$7$	5624.74	+	2952.27i	0.885444	+	0.464745i
$7$	5624.74	+	2952.27i	0.885444	+	0.464745i
$8$		0			0
$9$	9557.22	−	16553.6i	0.485557	−	0.841009i
$10$		0			0
$11$	37454.4	+	64872.9i	0.771322	+	1.33597i	0.936839	+	0.349762i	$0.113737\pi$
$11$	37454.4	+	64872.9i	0.771322	+	1.33597i	−0.165517	+	0.986207i	$0.552929\pi$
$12$		0			0
$13$	−50060.6			−0.486128			−0.243064	−	0.970010i	$-0.578153\pi$
$13$	−50060.6			−0.486128			−0.243064	+	0.970010i	$0.578153\pi$
$14$		0			0
$15$	25808.7			0.131630
$16$		0			0
$17$	−59482.6	−	103027.i	−0.172731	−	0.299179i	0.766643	−	0.642074i	$-0.221926\pi$
$17$	−59482.6	−	103027.i	−0.172731	−	0.299179i	−0.939374	+	0.342895i	$0.888592\pi$
$18$		0			0
$19$	402181.	−	696599.i	0.707996	−	1.22629i	−0.257603	−	0.966251i	$-0.582933\pi$
$19$	402181.	−	696599.i	0.707996	−	1.22629i	0.965599	−	0.260034i	$-0.0837339\pi$
$20$		0			0
$21$	−6095.40	−	151350.i	−0.00683936	−	0.169822i
$22$		0			0
$23$	134394.	−	232777.i	0.100139	−	0.173447i	−0.811603	−	0.584210i	$-0.801404\pi$
$23$	134394.	−	232777.i	0.100139	−	0.173447i	0.911742	+	0.410763i	$0.134738\pi$
$24$		0			0
$25$	390803.	+	676891.i	0.200091	+	0.346568i
$26$		0			0
$27$	−925113.			−0.335010
$28$		0			0
$29$	−5.29735e6			−1.39081			−0.695405	−	0.718618i	$-0.744775\pi$
$29$	−5.29735e6			−1.39081			−0.695405	+	0.718618i	$0.744775\pi$
$30$		0			0
$31$	2.73293e6	+	4.73357e6i	0.531497	+	0.920579i	0.999324	+	0.0367596i	$0.0117036\pi$
$31$	2.73293e6	+	4.73357e6i	0.531497	+	0.920579i	−0.467827	+	0.883820i	$0.654963\pi$
$32$		0			0
$33$	893089.	−	1.54688e6i	0.131094	−	0.227061i
$34$		0			0
$35$	−5.81135e6	+	3.67467e6i	−0.654592	+	0.413916i
$36$		0			0
$37$	−1.55446e6	+	2.69240e6i	−0.136355	+	0.236174i	−0.926114	−	0.377243i	$-0.876872\pi$
$37$	−1.55446e6	+	2.69240e6i	−0.136355	+	0.236174i	0.789759	+	0.613417i	$0.210206\pi$
$38$		0			0
$39$	596840.	+	1.03376e6i	0.0413111	+	0.0715530i
$40$		0			0
$41$	2.07841e7			1.14869			0.574345	−	0.818613i	$-0.305257\pi$
$41$	2.07841e7			1.14869			0.574345	+	0.818613i	$0.305257\pi$
$42$		0			0
$43$	−4.76437e6			−0.212519			−0.106259	−	0.994338i	$-0.533887\pi$
$43$	−4.76437e6			−0.212519			−0.106259	+	0.994338i	$0.533887\pi$
$44$		0			0
$45$	1.03444e7	+	1.79171e7i	0.376054	+	0.651344i
$46$		0			0
$47$	1.50766e7	−	2.61135e7i	0.450675	−	0.780592i	−0.547753	−	0.836640i	$-0.684517\pi$
$47$	1.50766e7	−	2.61135e7i	0.450675	−	0.780592i	0.998428	+	0.0560482i	$0.0178501\pi$
$48$		0			0
$49$	2.29218e7	+	3.32115e7i	0.568024	+	0.823012i
$50$		0			0
$51$	−1.41835e6	+	2.45665e6i	−0.0293573	+	0.0508483i
$52$		0			0
$53$	−2.85871e7	−	4.95142e7i	−0.497655	−	0.861963i	0.502342	−	0.864669i	$-0.332472\pi$
$53$	−2.85871e7	−	4.95142e7i	−0.497655	−	0.861963i	−0.999996	+	0.00270617i	$0.999139\pi$
$54$		0			0
$55$	−8.10788e7			−1.19475
$56$		0			0
$57$	−1.91798e7			−0.240662
$58$		0			0
$59$	9.18379e7	+	1.59068e8i	0.986706	+	1.70902i	0.634097	+	0.773254i	$0.281372\pi$
$59$	9.18379e7	+	1.59068e8i	0.986706	+	1.70902i	0.352609	+	0.935771i	$0.385295\pi$
$60$		0			0
$61$	−2.65083e7	+	4.59138e7i	−0.245131	+	0.424579i	−0.962168	−	0.272455i	$-0.912164\pi$
$61$	−2.65083e7	+	4.59138e7i	−0.245131	+	0.424579i	0.717038	+	0.697035i	$0.245498\pi$
$62$		0			0
$63$	1.02628e8	−	6.48941e7i	0.820789	−	0.519007i
$64$		0			0
$65$	2.70920e7	−	4.69247e7i	0.188248	−	0.326055i
$66$		0			0
$67$	8.67433e7	+	1.50244e8i	0.525896	+	0.910878i	0.999545	+	0.0301643i	$0.00960304\pi$
$67$	8.67433e7	+	1.50244e8i	0.525896	+	0.910878i	−0.473649	+	0.880713i	$0.657064\pi$
$68$		0			0
$69$	−6.40918e6			−0.0340394
$70$		0			0
$71$	1.68342e8			0.786195			0.393097	−	0.919497i	$-0.371404\pi$
$71$	1.68342e8			0.786195			0.393097	+	0.919497i	$0.371404\pi$
$72$		0			0
$73$	2.32340e8	+	4.02424e8i	0.957570	+	1.65856i	0.728375	+	0.685179i	$0.240276\pi$
$73$	2.32340e8	+	4.02424e8i	0.957570	+	1.65856i	0.229195	+	0.973381i	$0.426391\pi$
$74$		0			0
$75$	9.31858e6	−	1.61403e7i	0.0340075	−	0.0589026i
$76$		0			0
$77$	1.91489e7	+	4.75469e8i	0.0620776	+	1.54139i
$78$		0			0
$79$	−2.06864e8	+	3.58299e8i	−0.597535	+	1.03496i	0.395649	+	0.918402i	$0.370520\pi$
$79$	−2.06864e8	+	3.58299e8i	−0.597535	+	1.03496i	−0.993184	+	0.116559i	$0.962814\pi$
$80$		0			0
$81$	−1.77085e8	−	3.06720e8i	−0.457088	−	0.791699i
$82$		0			0
$83$	−3.50584e8			−0.810850			−0.405425	−	0.914128i	$-0.632876\pi$
$83$	−3.50584e8			−0.810850			−0.405425	+	0.914128i	$0.632876\pi$
$84$		0			0
$85$	1.28764e8			0.267553
$86$		0			0
$87$	6.31569e7	+	1.09391e8i	0.118191	+	0.204713i
$88$		0			0
$89$	−2.58736e8	+	4.48144e8i	−0.437122	+	0.757117i	−0.997466	−	0.0711428i	$-0.977335\pi$
$89$	−2.58736e8	+	4.48144e8i	−0.437122	+	0.757117i	0.560345	+	0.828260i	$0.310669\pi$
$90$		0			0
$91$	−2.81578e8	−	1.47792e8i	−0.430439	−	0.225926i
$92$		0			0
$93$	6.51659e7	−	1.12871e8i	0.0903331	−	0.156462i
$94$		0			0
$95$	4.35308e8	+	7.53976e8i	0.548328	+	0.949732i
$96$		0			0
$97$	1.32529e9			1.51998			0.759990	−	0.649935i	$-0.225204\pi$
$97$	1.32529e9			1.51998			0.759990	+	0.649935i	$0.225204\pi$
$98$		0			0
$99$	1.43184e9			1.49808
$100$		0			0
$101$	−3.16581e7	−	5.48335e7i	−0.0302718	−	0.0524324i	0.850493	−	0.525987i	$-0.176304\pi$
$101$	−3.16581e7	−	5.48335e7i	−0.0302718	−	0.0524324i	−0.880764	+	0.473555i	$0.842971\pi$
$102$		0			0
$103$	−5.64570e8	+	9.77863e8i	−0.494254	+	0.856073i	−0.999978	−	0.00662253i	$-0.997892\pi$
$103$	−5.64570e8	+	9.77863e8i	−0.494254	+	0.856073i	0.505724	+	0.862695i	$0.331225\pi$
$104$		0			0
$105$	1.45167e8	+	7.61943e7i	0.116551	+	0.0611746i
$106$		0			0
$107$	9.23383e7	−	1.59935e8i	0.0681012	−	0.117955i	−0.829964	−	0.557817i	$-0.811639\pi$
$107$	9.23383e7	−	1.59935e8i	0.0681012	−	0.117955i	0.898065	+	0.439862i	$0.144973\pi$
$108$		0			0
$109$	6.03407e8	+	1.04513e9i	0.409441	+	0.709172i	0.994827	−	0.101582i	$-0.0323905\pi$
$109$	6.03407e8	+	1.04513e9i	0.409441	+	0.709172i	−0.585386	+	0.810754i	$0.699057\pi$
$110$		0			0
$111$	7.41313e7			0.0463498
$112$		0			0
$113$	3.17054e8			0.182928			0.0914639	−	0.995808i	$-0.470845\pi$
$113$	3.17054e8			0.182928			0.0914639	+	0.995808i	$0.470845\pi$
$114$		0			0
$115$	1.45464e8	+	2.51951e8i	0.0775559	+	0.134331i
$116$		0			0
$117$	−4.78440e8	+	8.28682e8i	−0.236043	+	0.408838i
$118$		0			0
$119$	−3.04110e7	−	7.55108e8i	−0.0139017	−	0.345182i
$120$		0			0
$121$	−1.62669e9	+	2.81751e9i	−0.689875	+	1.19490i
$122$		0			0
$123$	−2.47795e8	−	4.29193e8i	−0.0976156	−	0.169075i
$124$		0			0
$125$	−2.95998e9			−1.08441
$126$		0			0
$127$	1.91107e9			0.651868			0.325934	−	0.945392i	$-0.394321\pi$
$127$	1.91107e9			0.651868			0.325934	+	0.945392i	$0.394321\pi$
$128$		0			0
$129$	5.68025e7	+	9.83848e7i	0.0180598	+	0.0312806i
$130$		0			0
$131$	−3.02645e9	+	5.24197e9i	−0.897869	+	1.55516i	−0.0676566	+	0.997709i	$0.521552\pi$
$131$	−3.02645e9	+	5.24197e9i	−0.897869	+	1.55516i	−0.830213	+	0.557447i	$0.811781\pi$
$132$		0			0
$133$	4.31871e9	−	2.73084e9i	1.19680	−	0.756770i
$134$		0			0
$135$	5.00656e8	−	8.67162e8i	0.129729	−	0.224698i
$136$		0			0
$137$	−2.49781e9	−	4.32634e9i	−0.605783	−	1.04925i	−0.991927	−	0.126809i	$-0.959527\pi$
$137$	−2.49781e9	−	4.32634e9i	−0.605783	−	1.04925i	0.386144	−	0.922438i	$-0.373807\pi$
$138$		0			0
$139$	5.39864e9			1.22664			0.613321	−	0.789834i	$-0.289833\pi$
$139$	5.39864e9			1.22664			0.613321	+	0.789834i	$0.289833\pi$
$140$		0			0
$141$	−7.18995e8			−0.153193
$142$		0			0
$143$	−1.87499e9	−	3.24758e9i	−0.374961	−	0.649452i
$144$		0			0
$145$	2.86684e9	−	4.96551e9i	0.538577	−	0.932842i
$146$		0			0
$147$	4.12540e8	−	8.69297e8i	0.0728682	−	0.153547i
$148$		0			0
$149$	−3.12472e8	+	5.41218e8i	−0.0519366	+	0.0899568i	−0.890825	−	0.454347i	$-0.849873\pi$
$149$	−3.12472e8	+	5.41218e8i	−0.0519366	+	0.0899568i	0.838888	+	0.544304i	$0.183206\pi$
$150$		0			0
$151$	−5.50838e9	−	9.54079e9i	−0.862238	−	1.49344i	−0.869763	−	0.493469i	$-0.835729\pi$
$151$	−5.50838e9	−	9.54079e9i	−0.862238	−	1.49344i	0.00752500	−	0.999972i	$-0.497605\pi$
$152$		0			0
$153$	−2.27395e9			−0.335483
$154$		0			0
$155$	−5.91606e9			−0.823266
$156$		0			0
$157$	3.40159e9	+	5.89172e9i	0.446821	+	0.773916i	0.998177	−	0.0603538i	$-0.0192229\pi$
$157$	3.40159e9	+	5.89172e9i	0.446821	+	0.773916i	−0.551356	+	0.834270i	$0.685890\pi$
$158$		0			0
$159$	−6.81650e8	+	1.18065e9i	−0.0845813	+	0.146499i
$160$		0			0
$161$	1.44315e9	−	9.12545e8i	0.169276	−	0.107038i
$162$		0			0
$163$	1.03329e9	−	1.78972e9i	0.114651	−	0.198582i	−0.802989	−	0.595994i	$-0.796758\pi$
$163$	1.03329e9	−	1.78972e9i	0.114651	−	0.198582i	0.917640	+	0.397412i	$0.130092\pi$
$164$		0			0
$165$	9.66650e8	+	1.67429e9i	0.101529	+	0.175854i
$166$		0			0
$167$	6.75774e8			0.0672322			0.0336161	−	0.999435i	$-0.489298\pi$
$167$	6.75774e8			0.0672322			0.0336161	+	0.999435i	$0.489298\pi$
$168$		0			0
$169$	−8.09844e9			−0.763679
$170$		0			0
$171$	−7.68747e9	−	1.33151e10i	−0.687545	−	1.19086i
$172$		0			0
$173$	9.72079e9	−	1.68369e10i	0.825076	−	1.42907i	−0.0767850	−	0.997048i	$-0.524466\pi$
$173$	9.72079e9	−	1.68369e10i	0.825076	−	1.42907i	0.901861	−	0.432026i	$-0.142201\pi$
$174$		0			0
$175$	1.99801e8	+	4.96109e9i	0.0161037	+	0.399858i
$176$		0			0
$177$	2.18985e9	−	3.79293e9i	0.167700	−	0.290466i
$178$		0			0
$179$	2.04825e9	+	3.54767e9i	0.149123	+	0.258288i	0.930903	−	0.365265i	$-0.119022\pi$
$179$	2.04825e9	+	3.54767e9i	0.149123	+	0.258288i	−0.781781	+	0.623553i	$0.785688\pi$
$180$		0			0
$181$	−2.41926e10			−1.67544			−0.837720	−	0.546101i	$-0.816112\pi$
$181$	−2.41926e10			−1.67544			−0.837720	+	0.546101i	$0.816112\pi$
$182$		0			0
$183$	1.26417e9			0.0833248
$184$		0			0
$185$	−1.68250e9	−	2.91417e9i	−0.105604	−	0.182912i
$186$		0			0
$187$	4.45577e9	−	7.71762e9i	0.266462	−	0.461526i
$188$		0			0
$189$	−5.20352e9	−	2.73118e9i	−0.296633	−	0.155694i
$190$		0			0
$191$	2.90999e9	−	5.04025e9i	0.158213	−	0.274033i	−0.776011	−	0.630719i	$-0.782760\pi$
$191$	2.90999e9	−	5.04025e9i	0.158213	−	0.274033i	0.934224	+	0.356686i	$0.116093\pi$
$192$		0			0
$193$	−1.19292e10	−	2.06619e10i	−0.618874	−	1.07192i	−0.989692	−	0.143215i	$-0.954256\pi$
$193$	−1.19292e10	−	2.06619e10i	−0.618874	−	1.07192i	0.370818	−	0.928706i	$-0.379077\pi$
$194$		0			0
$195$	−1.29200e9			−0.0639892
$196$		0			0
$197$	−1.13267e10			−0.535802			−0.267901	−	0.963446i	$-0.586330\pi$
$197$	−1.13267e10			−0.535802			−0.267901	+	0.963446i	$0.586330\pi$
$198$		0			0
$199$	8.76433e9	+	1.51803e10i	0.396168	+	0.686184i	0.993250	−	0.115997i	$-0.0370063\pi$
$199$	8.76433e9	+	1.51803e10i	0.396168	+	0.686184i	−0.597081	+	0.802181i	$0.703673\pi$
$200$		0			0
$201$	2.06837e9	−	3.58252e9i	0.0893811	−	0.154813i
$202$		0			0
$203$	−2.97962e10	−	1.56392e10i	−1.23149	−	0.646372i
$204$		0			0
$205$	−1.12480e10	+	1.94821e10i	−0.444818	+	0.770448i
$206$		0			0
$207$	−2.56887e9	−	4.44941e9i	−0.0972468	−	0.168436i
$208$		0			0
$209$	6.02538e10			2.18437
$210$		0			0
$211$	3.12620e10			1.08579			0.542895	−	0.839801i	$-0.317328\pi$
$211$	3.12620e10			1.08579			0.542895	+	0.839801i	$0.317328\pi$
$212$		0			0
$213$	−2.00703e9	−	3.47628e9i	−0.0668108	−	0.115720i
$214$		0			0
$215$	2.57840e9	−	4.46592e9i	0.0822957	−	0.142540i
$216$		0			0
$217$	1.39723e9	+	3.46935e10i	0.0427759	+	1.06213i
$218$		0			0
$219$	5.54007e9	−	9.59568e9i	0.162748	−	0.281889i
$220$		0			0
$221$	2.97773e9	+	5.15759e9i	0.0839693	+	0.145439i
$222$		0			0
$223$	−2.74321e10			−0.742825			−0.371413	−	0.928468i	$-0.621127\pi$
$223$	−2.74321e10			−0.742825			−0.371413	+	0.928468i	$0.621127\pi$
$224$		0			0
$225$	1.49400e10			0.388622
$226$		0			0
$227$	5.78305e9	+	1.00165e10i	0.144557	+	0.250381i	0.929208	−	0.369558i	$-0.120491\pi$
$227$	5.78305e9	+	1.00165e10i	0.144557	+	0.250381i	−0.784650	+	0.619939i	$0.787157\pi$
$228$		0			0
$229$	−2.11891e10	+	3.67007e10i	−0.509159	+	0.881890i	0.490785	+	0.871281i	$0.336710\pi$
$229$	−2.11891e10	+	3.67007e10i	−0.509159	+	0.881890i	−0.999944	+	0.0106087i	$0.996623\pi$
$230$		0			0
$231$	9.59019e9	−	6.06413e9i	0.221602	−	0.140125i
$232$		0			0
$233$	4.26184e10	−	7.38173e10i	0.947318	−	1.64080i	0.196276	−	0.980549i	$-0.437115\pi$
$233$	4.26184e10	−	7.38173e10i	0.947318	−	1.64080i	0.751042	−	0.660254i	$-0.229552\pi$
$234$		0			0
$235$	1.63184e10	+	2.82643e10i	0.349038	+	0.604552i
$236$		0			0
$237$	9.86522e9			0.203114
$238$		0			0
$239$	−1.81331e10			−0.359486			−0.179743	−	0.983714i	$-0.557527\pi$
$239$	−1.81331e10			−0.359486			−0.179743	+	0.983714i	$0.557527\pi$
$240$		0			0
$241$	−1.24598e10	−	2.15810e10i	−0.237921	−	0.412092i	0.722196	−	0.691688i	$-0.243133\pi$
$241$	−1.24598e10	−	2.15810e10i	−0.237921	−	0.412092i	−0.960118	+	0.279596i	$0.909799\pi$
$242$		0			0
$243$	−1.33270e10	+	2.30831e10i	−0.245192	+	0.424684i
$244$		0			0
$245$	−4.35360e10	+	3.51240e9i	−0.771971	+	0.0622812i
$246$		0			0
$247$	−2.01334e10	+	3.48721e10i	−0.344177	+	0.596132i
$248$		0			0
$249$	4.17979e9	+	7.23960e9i	0.0689060	+	0.119349i
$250$		0			0
$251$	3.47730e9			0.0552981			0.0276491	−	0.999618i	$-0.491198\pi$
$251$	3.47730e9			0.0552981			0.0276491	+	0.999618i	$0.491198\pi$
$252$		0			0
$253$	2.01346e10			0.308959
$254$		0			0
$255$	−1.53517e9	−	2.65899e9i	−0.0227366	−	0.0393810i
$256$		0			0
$257$	−3.77069e10	+	6.53102e10i	−0.539164	+	0.933860i	0.459785	+	0.888030i	$0.347927\pi$
$257$	−3.77069e10	+	6.53102e10i	−0.539164	+	0.933860i	−0.998949	+	0.0458298i	$0.985407\pi$
$258$		0			0
$259$	−1.66921e10	+	1.05549e10i	−0.230496	+	0.145749i
$260$		0			0
$261$	−5.06279e10	+	8.76901e10i	−0.675317	+	1.16968i
$262$		0			0
$263$	5.88025e10	+	1.01849e11i	0.757870	+	1.31267i	0.943935	+	0.330132i	$0.107093\pi$
$263$	5.88025e10	+	1.01849e11i	0.757870	+	1.31267i	−0.186064	+	0.982538i	$0.559573\pi$
$264$		0			0
$265$	6.18834e10			0.770846
$266$		0			0
$267$	1.23390e10			0.148586
$268$		0			0
$269$	3.71283e10	+	6.43082e10i	0.432335	+	0.748826i	0.997074	−	0.0764436i	$-0.0243565\pi$
$269$	3.71283e10	+	6.43082e10i	0.432335	+	0.748826i	−0.564739	+	0.825270i	$0.691023\pi$
$270$		0			0
$271$	4.01996e10	−	6.96277e10i	0.452751	−	0.784188i	−0.545805	−	0.837913i	$-0.683776\pi$
$271$	4.01996e10	−	6.96277e10i	0.452751	−	0.784188i	0.998556	+	0.0537244i	$0.0171092\pi$
$272$		0			0
$273$	3.05139e8	+	7.57665e9i	0.00332480	+	0.0825553i
$274$		0			0
$275$	−2.92746e10	+	5.07051e10i	−0.308669	+	0.534631i
$276$		0			0
$277$	6.52889e10	+	1.13084e11i	0.666316	+	1.15409i	0.978927	+	0.204212i	$0.0654631\pi$
$277$	6.52889e10	+	1.13084e11i	0.666316	+	1.15409i	−0.312611	+	0.949881i	$0.601204\pi$
$278$		0			0
$279$	1.04477e11			1.03229
$280$		0			0
$281$	−6.23328e10			−0.596401			−0.298200	−	0.954503i	$-0.596386\pi$
$281$	−6.23328e10			−0.596401			−0.298200	+	0.954503i	$0.596386\pi$
$282$		0			0
$283$	3.39157e8	+	5.87437e8i	0.00314312	+	0.00544405i	0.867593	−	0.497276i	$-0.165666\pi$
$283$	3.39157e8	+	5.87437e8i	0.00314312	+	0.00544405i	−0.864450	+	0.502720i	$0.832333\pi$
$284$		0			0
$285$	1.03798e10	−	1.79783e10i	0.0931938	−	0.161416i
$286$		0			0
$287$	1.16905e11	+	6.13601e10i	1.01710	+	0.533848i
$288$		0			0
$289$	5.22176e10	−	9.04435e10i	0.440328	−	0.762671i
$290$		0			0
$291$	−1.58006e10	−	2.73674e10i	−0.129168	−	0.223725i
$292$		0			0
$293$	5.67165e9			0.0449578			0.0224789	−	0.999747i	$-0.492844\pi$
$293$	5.67165e9			0.0449578			0.0224789	+	0.999747i	$0.492844\pi$
$294$		0			0
$295$	−1.98805e11			−1.52837
$296$		0			0
$297$	−3.46496e10	−	6.00148e10i	−0.258401	−	0.447563i
$298$		0			0
$299$	−6.72785e9	+	1.16530e10i	−0.0486806	+	0.0843173i
$300$		0			0
$301$	−2.67983e10	−	1.40657e10i	−0.188174	−	0.0987671i
$302$		0			0
$303$	−7.54878e8	+	1.30749e9i	−0.00514500	+	0.00891139i
$304$		0			0
$305$	−2.86917e10	−	4.96956e10i	−0.189849	−	0.328828i
$306$		0			0
$307$	1.81171e11			1.16403			0.582017	−	0.813176i	$-0.302264\pi$
$307$	1.81171e11			1.16403			0.582017	+	0.813176i	$0.302264\pi$
$308$		0			0
$309$	2.69240e10			0.168007
$310$		0			0
$311$	5.19032e10	+	8.98990e10i	0.314610	+	0.544920i	0.979354	−	0.202150i	$-0.0647930\pi$
$311$	5.19032e10	+	8.98990e10i	0.314610	+	0.544920i	−0.664745	+	0.747071i	$0.731460\pi$
$312$		0			0
$313$	7.44320e10	−	1.28920e11i	0.438339	−	0.759226i	−0.559222	−	0.829018i	$-0.688900\pi$
$313$	7.44320e10	−	1.28920e11i	0.438339	−	0.759226i	0.997562	+	0.0697919i	$0.0222335\pi$
$314$		0			0
$315$	5.28867e9	+	1.31318e11i	0.0302656	+	0.751498i
$316$		0			0
$317$	−9.88493e10	+	1.71212e11i	−0.549803	+	0.952286i	0.448485	+	0.893790i	$0.351964\pi$
$317$	−9.88493e10	+	1.71212e11i	−0.549803	+	0.952286i	−0.998288	+	0.0584960i	$0.981370\pi$
$318$		0			0
$319$	−1.98409e11	−	3.43655e11i	−1.07276	−	1.85808i
$320$		0			0
$321$	−4.40356e9			−0.0231489
$322$		0			0
$323$	−9.56912e10			−0.489171
$324$		0			0
$325$	−1.95638e10	−	3.38855e10i	−0.0972699	−	0.168476i
$326$		0			0
$327$	1.43881e10	−	2.49208e10i	0.0695885	−	0.120531i
$328$		0			0
$329$	1.61896e11	−	1.02371e11i	0.761824	−	0.481722i
$330$		0			0
$331$	2.78940e10	−	4.83139e10i	0.127728	−	0.221231i	−0.795068	−	0.606520i	$-0.792565\pi$
$331$	2.78940e10	−	4.83139e10i	0.127728	−	0.221231i	0.922796	+	0.385289i	$0.125898\pi$
$332$		0			0
$333$	2.97126e10	+	5.14638e10i	0.132416	+	0.229352i
$334$		0			0
$335$	−1.87776e11			−0.814590
$336$		0			0
$337$	4.01748e11			1.69676			0.848378	−	0.529391i	$-0.177580\pi$
$337$	4.01748e11			1.69676			0.848378	+	0.529391i	$0.177580\pi$
$338$		0			0
$339$	−3.78003e9	−	6.54720e9i	−0.0155452	−	0.0269251i
$340$		0			0
$341$	−2.04720e11	+	3.54586e11i	−0.819910	+	1.42013i
$342$		0			0
$343$	3.08798e10	+	2.54477e11i	0.120462	+	0.992718i
$344$		0			0
$345$	3.46854e9	−	6.00769e9i	0.0131814	−	0.0228308i
$346$		0			0
$347$	−1.68936e11	−	2.92606e11i	−0.625518	−	1.08343i	−0.988440	−	0.151610i	$-0.951554\pi$
$347$	−1.68936e11	−	2.92606e11i	−0.625518	−	1.08343i	0.362923	−	0.931819i	$-0.381779\pi$
$348$		0			0
$349$	−3.59959e11			−1.29879			−0.649395	−	0.760451i	$-0.724978\pi$
$349$	−3.59959e11			−1.29879			−0.649395	+	0.760451i	$0.724978\pi$
$350$		0			0
$351$	4.63117e10			0.162858
$352$		0			0
$353$	−9.51732e10	−	1.64845e11i	−0.326233	−	0.565053i	0.655528	−	0.755171i	$-0.272446\pi$
$353$	−9.51732e10	−	1.64845e11i	−0.326233	−	0.565053i	−0.981761	+	0.190118i	$0.939113\pi$
$354$		0			0
$355$	−9.11040e10	+	1.57797e11i	−0.304446	+	0.527315i
$356$		0			0
$357$	−1.52305e10	+	9.63066e9i	−0.0496258	+	0.0313797i
$358$		0			0
$359$	1.96349e11	−	3.40086e11i	0.623884	−	1.08060i	−0.364872	−	0.931058i	$-0.618887\pi$
$359$	1.96349e11	−	3.40086e11i	0.623884	−	1.08060i	0.988756	−	0.149540i	$-0.0477794\pi$
$360$		0			0
$361$	−1.62156e11	−	2.80862e11i	−0.502517	−	0.870385i
$362$		0			0
$363$	7.75758e10			0.234502
$364$		0			0
$365$	−5.02954e11			−1.48324
$366$		0			0
$367$	−1.62237e10	−	2.81002e10i	−0.0466823	−	0.0808561i	0.841740	−	0.539883i	$-0.181532\pi$
$367$	−1.62237e10	−	2.81002e10i	−0.0466823	−	0.0808561i	−0.888422	+	0.459027i	$0.848198\pi$
$368$		0			0
$369$	1.98638e11	−	3.44051e11i	0.557754	−	0.966059i
$370$		0			0
$371$	−1.46154e10	−	3.62901e11i	−0.0400523	−	0.994503i
$372$		0			0
$373$	2.19500e11	−	3.80185e11i	0.587143	−	1.01696i	−0.407461	−	0.913223i	$-0.633586\pi$
$373$	2.19500e11	−	3.80185e11i	0.587143	−	1.01696i	0.994604	−	0.103740i	$-0.0330809\pi$
$374$		0			0
$375$	3.52900e10	+	6.11240e10i	0.0921532	+	0.159614i
$376$		0			0
$377$	2.65189e11			0.676112
$378$		0			0
$379$	3.56933e11			0.888607			0.444304	−	0.895876i	$-0.353451\pi$
$379$	3.56933e11			0.888607			0.444304	+	0.895876i	$0.353451\pi$
$380$		0			0
$381$	−2.27845e10	−	3.94638e10i	−0.0553957	−	0.0959482i
$382$		0			0
$383$	−7.62364e10	+	1.32045e11i	−0.181037	+	0.313566i	−0.942234	−	0.334955i	$-0.891279\pi$
$383$	−7.62364e10	+	1.32045e11i	−0.181037	+	0.313566i	0.761197	+	0.648521i	$0.224612\pi$
$384$		0			0
$385$	−4.56047e11	−	2.39367e11i	−1.05788	−	0.555252i
$386$		0			0
$387$	−4.55341e10	+	7.88674e10i	−0.103190	+	0.178730i
$388$		0			0
$389$	−2.78559e11	−	4.82478e11i	−0.616800	−	1.06833i	−0.990066	−	0.140604i	$-0.955096\pi$
$389$	−2.78559e11	−	4.82478e11i	−0.616800	−	1.06833i	0.373266	−	0.927724i	$-0.378238\pi$
$390$		0			0
$391$	−3.19765e10			−0.0691887
$392$		0			0
$393$	1.44330e11			0.305203
$394$		0			0
$395$	−2.23903e11	−	3.87811e11i	−0.462778	−	0.801555i
$396$		0			0
$397$	−4.32956e11	+	7.49901e11i	−0.874755	+	1.51512i	−0.0177309	+	0.999843i	$0.505644\pi$
$397$	−4.32956e11	+	7.49901e11i	−0.874755	+	1.51512i	−0.857024	+	0.515277i	$0.827689\pi$
$398$		0			0
$399$	−1.07881e11	−	5.66239e10i	−0.213093	−	0.111846i
$400$		0			0
$401$	3.38246e11	−	5.85858e11i	0.653254	−	1.13147i	−0.329074	−	0.944304i	$-0.606737\pi$
$401$	3.38246e11	−	5.85858e11i	0.653254	−	1.13147i	0.982328	−	0.187166i	$-0.0599301\pi$
$402$		0			0
$403$	−1.36812e11	−	2.36965e11i	−0.258376	−	0.447520i
$404$		0			0
$405$	3.83342e11			0.708010
$406$		0			0
$407$	−2.32885e11			−0.420695
$408$		0			0
$409$	1.39521e11	+	2.41658e11i	0.246539	+	0.427018i	0.962563	−	0.271058i	$-0.0873734\pi$
$409$	1.39521e11	+	2.41658e11i	0.246539	+	0.427018i	−0.716024	+	0.698075i	$0.754040\pi$
$410$		0			0
$411$	−5.95596e10	+	1.03160e11i	−0.102959	+	0.178330i
$412$		0			0
$413$	4.69528e10	+	1.16585e12i	0.0794121	+	1.97181i
$414$		0			0
$415$	1.89730e11	−	3.28623e11i	0.313993	−	0.543852i
$416$		0			0
$417$	−6.43645e10	−	1.11483e11i	−0.104240	−	0.180549i
$418$		0			0
$419$	9.62344e10			0.152534			0.0762671	−	0.997087i	$-0.475700\pi$
$419$	9.62344e10			0.152534			0.0762671	+	0.997087i	$0.475700\pi$
$420$		0			0
$421$	4.35437e10			0.0675547			0.0337774	−	0.999429i	$-0.489246\pi$
$421$	4.35437e10			0.0675547			0.0337774	+	0.999429i	$0.489246\pi$
$422$		0			0
$423$	−2.88181e11	−	4.99144e11i	−0.437657	−	0.758043i
$424$		0			0
$425$	4.64920e10	−	8.05265e10i	0.0691238	−	0.119726i
$426$		0			0
$427$	−2.84652e11	+	1.79993e11i	−0.414371	+	0.262018i
$428$		0			0
$429$	−4.47085e10	+	7.74375e10i	−0.0637284	+	0.110381i
$430$		0			0
$431$	3.88074e11	+	6.72163e11i	0.541709	+	0.938268i	0.998806	+	0.0488511i	$0.0155560\pi$
$431$	3.88074e11	+	6.72163e11i	0.541709	+	0.938268i	−0.457097	+	0.889417i	$0.651111\pi$
$432$		0			0
$433$	5.97634e11			0.817033			0.408517	−	0.912751i	$-0.366046\pi$
$433$	5.97634e11			0.817033			0.408517	+	0.912751i	$0.366046\pi$
$434$		0			0
$435$	−1.36718e11			−0.183073
$436$		0			0
$437$	−1.08102e11	−	1.87238e11i	−0.141797	−	0.245599i
$438$		0			0
$439$	−1.59917e11	+	2.76984e11i	−0.205496	+	0.355930i	−0.950291	−	0.311364i	$-0.899214\pi$
$439$	−1.59917e11	+	2.76984e11i	−0.205496	+	0.355930i	0.744795	+	0.667294i	$0.232547\pi$
$440$		0			0
$441$	7.68838e11	−	6.20284e10i	0.967969	−	0.0780939i
$442$		0			0
$443$	−2.21049e11	+	3.82869e11i	−0.272692	+	0.472317i	−0.969550	−	0.244892i	$-0.921247\pi$
$443$	−2.21049e11	+	3.82869e11i	−0.272692	+	0.472317i	0.696858	+	0.717209i	$0.254581\pi$
$444$		0			0
$445$	−2.80048e11	−	4.85057e11i	−0.338541	−	0.586371i
$446$		0			0
$447$	1.49016e10			0.0176543
$448$		0			0
$449$	−7.93959e11			−0.921913			−0.460956	−	0.887423i	$-0.652494\pi$
$449$	−7.93959e11			−0.921913			−0.460956	+	0.887423i	$0.652494\pi$
$450$		0			0
$451$	7.78454e11	+	1.34832e12i	0.886010	+	1.53461i
$452$		0			0
$453$	−1.31346e11	+	2.27497e11i	−0.146546	+	0.253825i
$454$		0			0
$455$	2.90920e11	−	1.83956e11i	0.318216	−	0.201216i
$456$		0			0
$457$	−4.85797e10	+	8.41426e10i	−0.0520993	+	0.0902387i	−0.890899	−	0.454202i	$-0.849925\pi$
$457$	−4.85797e10	+	8.41426e10i	−0.0520993	+	0.0902387i	0.838800	+	0.544440i	$0.183258\pi$
$458$		0			0
$459$	5.50282e10	+	9.53116e10i	0.0578666	+	0.100228i
$460$		0			0
$461$	8.96191e11			0.924158			0.462079	−	0.886839i	$-0.347104\pi$
$461$	8.96191e11			0.924158			0.462079	+	0.886839i	$0.347104\pi$
$462$		0			0
$463$	6.89366e11			0.697164			0.348582	−	0.937278i	$-0.386663\pi$
$463$	6.89366e11			0.697164			0.348582	+	0.937278i	$0.386663\pi$
$464$		0			0
$465$	7.05334e10	+	1.22167e11i	0.0699611	+	0.121176i
$466$		0			0
$467$	−1.87790e11	+	3.25262e11i	−0.182703	+	0.316451i	−0.942800	−	0.333358i	$-0.891818\pi$
$467$	−1.87790e11	+	3.25262e11i	−0.182703	+	0.316451i	0.760097	+	0.649810i	$0.225151\pi$
$468$		0			0
$469$	4.43482e10	+	1.10117e12i	0.0423251	+	1.05094i
$470$		0			0
$471$	8.11098e10	−	1.40486e11i	0.0759416	−	0.131535i
$472$		0			0
$473$	−1.78447e11	−	3.09079e11i	−0.163920	−	0.283919i
$474$		0			0
$475$	6.28695e11			0.566655
$476$		0			0
$477$	−1.09285e12			−0.966558
$478$		0			0
$479$	−5.90992e11	−	1.02363e12i	−0.512946	−	0.888448i	−0.999887	−	0.0150137i	$-0.995221\pi$
$479$	−5.90992e11	−	1.02363e12i	−0.512946	−	0.888448i	0.486941	−	0.873435i	$-0.338113\pi$
$480$		0			0
$481$	7.78172e10	−	1.34783e11i	0.0662861	−	0.114811i
$482$		0			0
$483$	−3.60500e10	−	1.89216e10i	−0.0301400	−	0.0158196i
$484$		0			0
$485$	−7.17225e11	+	1.24227e12i	−0.588596	+	1.01948i
$486$		0			0
$487$	−1.39234e11	−	2.41160e11i	−0.112167	−	0.194278i	0.804477	−	0.593984i	$-0.202446\pi$
$487$	−1.39234e11	−	2.41160e11i	−0.112167	−	0.194278i	−0.916644	+	0.399706i	$0.869112\pi$
$488$		0			0
$489$	−4.92771e10			−0.0389722
$490$		0			0
$491$	−7.96998e10			−0.0618857			−0.0309429	−	0.999521i	$-0.509851\pi$
$491$	−7.96998e10			−0.0618857			−0.0309429	+	0.999521i	$0.509851\pi$
$492$		0			0
$493$	3.15100e11	+	5.45770e11i	0.240236	+	0.416101i
$494$		0			0
$495$	−7.74888e11	+	1.34214e12i	−0.580117	+	1.00479i
$496$		0			0
$497$	9.46880e11	+	4.96991e11i	0.696132	+	0.365380i
$498$		0			0
$499$	1.59936e11	−	2.77017e11i	0.115477	−	0.200011i	−0.802494	−	0.596661i	$-0.796494\pi$
$499$	1.59936e11	−	2.77017e11i	0.115477	−	0.200011i	0.917970	+	0.396649i	$0.129827\pi$
$500$		0			0
$501$	−8.05682e9	−	1.39548e10i	−0.00571339	−	0.00989588i
$502$		0			0
$503$	1.67487e12			1.16661			0.583306	−	0.812253i	$-0.301759\pi$
$503$	1.67487e12			1.16661			0.583306	+	0.812253i	$0.301759\pi$
$504$		0			0
$505$	6.85314e10			0.0468898
$506$		0			0
$507$	9.65524e10	+	1.67234e11i	0.0648974	+	0.112406i
$508$		0			0
$509$	3.12305e11	−	5.40927e11i	0.206228	−	0.357198i	−0.744295	−	0.667851i	$-0.767214\pi$
$509$	3.12305e11	−	5.40927e11i	0.206228	−	0.357198i	0.950523	+	0.310653i	$0.100548\pi$
$510$		0			0
$511$	1.18786e11	+	2.94946e12i	0.0770672	+	1.91359i
$512$		0			0
$513$	−3.72063e11	+	6.44433e11i	−0.237186	+	0.410818i
$514$		0			0
$515$	−6.11072e11	−	1.05841e12i	−0.382789	−	0.663010i
$516$		0			0
$517$	2.25874e12			1.39046
$518$		0			0
$519$	−4.63579e11			−0.280460
$520$		0			0
$521$	−3.21473e11	−	5.56808e11i	−0.191150	−	0.331082i	0.754481	−	0.656321i	$-0.227888\pi$
$521$	−3.21473e11	−	5.56808e11i	−0.191150	−	0.331082i	−0.945632	+	0.325239i	$0.894555\pi$
$522$		0			0
$523$	8.08666e11	−	1.40065e12i	0.472619	−	0.818601i	−0.526890	−	0.849934i	$-0.676642\pi$
$523$	8.08666e11	−	1.40065e12i	0.472619	−	0.818601i	0.999509	+	0.0313330i	$0.00997522\pi$
$524$		0			0
$525$	1.00065e11	−	6.32738e10i	0.0574864	−	0.0363502i
$526$		0			0
$527$	3.25123e11	−	5.63130e11i	0.183612	−	0.318025i
$528$		0			0
$529$	8.64453e11	+	1.49728e12i	0.479944	+	0.831288i
$530$		0			0
$531$	3.51086e12			1.91641
$532$		0			0
$533$	−1.04046e12			−0.558411
$534$		0			0
$535$	9.99439e10	+	1.73108e11i	0.0527430	+	0.0913535i
$536$		0			0
$537$	4.88398e10	−	8.45930e10i	0.0253448	−	0.0438986i
$538$		0			0
$539$	−1.29600e12	+	2.73092e12i	−0.661389	+	1.39367i
$540$		0			0
$541$	1.21728e12	−	2.10840e12i	0.610948	−	1.05819i	−0.380132	−	0.924932i	$-0.624122\pi$
$541$	1.21728e12	−	2.10840e12i	0.610948	−	1.05819i	0.991081	−	0.133262i	$-0.0425451\pi$
$542$		0			0
$543$	2.88432e11	+	4.99580e11i	0.142379	+	0.246607i
$544$		0			0
$545$	−1.30622e12			−0.634206
$546$		0			0
$547$	−3.63462e12			−1.73587			−0.867934	−	0.496680i	$-0.834552\pi$
$547$	−3.63462e12			−1.73587			−0.867934	+	0.496680i	$0.834552\pi$
$548$		0			0
$549$	5.06692e11	+	8.77616e11i	0.238050	+	0.412315i
$550$		0			0
$551$	−2.13050e12	+	3.69013e12i	−0.984688	+	1.70553i
$552$		0			0
$553$	−2.22135e12	+	1.40462e12i	−1.01008	+	0.638699i
$554$		0			0
$555$	−4.01186e10	+	6.94875e10i	−0.0179485	+	0.0310877i
$556$		0			0
$557$	2.22234e10	+	3.84921e10i	0.00978278	+	0.0169443i	0.870875	−	0.491504i	$-0.163553\pi$
$557$	2.22234e10	+	3.84921e10i	0.00978278	+	0.0169443i	−0.861093	+	0.508448i	$0.830219\pi$
$558$		0			0
$559$	2.38507e11			0.103311
$560$		0			0
$561$	−2.12493e11			−0.0905757
$562$		0			0
$563$	−1.93179e12	−	3.34596e12i	−0.810349	−	1.40357i	−0.912620	−	0.408810i	$-0.865944\pi$
$563$	−1.93179e12	−	3.34596e12i	−0.810349	−	1.40357i	0.102270	−	0.994757i	$-0.467389\pi$
$564$		0			0
$565$	−1.71584e11	+	2.97193e11i	−0.0708369	+	0.122693i
$566$		0			0
$567$	−9.05362e10	−	2.24803e12i	−0.0367874	−	0.913435i
$568$		0			0
$569$	1.03247e12	−	1.78828e12i	0.412925	−	0.715206i	−0.582284	−	0.812986i	$-0.697841\pi$
$569$	1.03247e12	−	1.78828e12i	0.412925	−	0.715206i	0.995208	+	0.0977795i	$0.0311740\pi$
$570$		0			0
$571$	−1.55084e12	−	2.68613e12i	−0.610527	−	1.05746i	−0.991152	−	0.132734i	$-0.957624\pi$
$571$	−1.55084e12	−	2.68613e12i	−0.610527	−	1.05746i	0.380625	−	0.924729i	$-0.375709\pi$
$572$		0			0
$573$	−1.38776e11			−0.0537796
$574$		0			0
$575$	2.10087e11			0.0801480
$576$		0			0
$577$	−1.04111e11	−	1.80326e11i	−0.0391027	−	0.0677279i	0.845812	−	0.533482i	$-0.179117\pi$
$577$	−1.04111e11	−	1.80326e11i	−0.0391027	−	0.0677279i	−0.884914	+	0.465754i	$0.845783\pi$
$578$		0			0
$579$	−2.84447e11	+	4.92677e11i	−0.105184	+	0.182184i
$580$		0			0
$581$	−1.97194e12	−	1.03502e12i	−0.717963	−	0.376839i
$582$		0			0
$583$	2.14142e12	−	3.70905e12i	0.767704	−	1.32970i
$584$		0			0
$585$	−5.17848e11	−	8.96938e11i	−0.182810	−	0.316637i
$586$		0			0
$587$	2.56958e12			0.893286			0.446643	−	0.894712i	$-0.352619\pi$
$587$	2.56958e12			0.893286			0.446643	+	0.894712i	$0.352619\pi$
$588$		0			0
$589$	4.39653e12			1.50519
$590$		0			0
$591$	1.35041e11	+	2.33897e11i	0.0455324	+	0.0788644i
$592$		0			0
$593$	−1.01774e12	+	1.76278e12i	−0.337980	+	0.585399i	−0.984053	−	0.177877i	$-0.943077\pi$
$593$	−1.01774e12	+	1.76278e12i	−0.337980	+	0.585399i	0.646073	+	0.763276i	$0.276410\pi$
$594$		0			0
$595$	7.24265e11	+	3.80146e11i	0.236903	+	0.124344i
$596$		0			0
$597$	2.08983e11	−	3.61969e11i	0.0673327	−	0.116624i
$598$		0			0
$599$	−1.40874e12	−	2.44002e12i	−0.447107	−	0.774412i	0.551090	−	0.834446i	$-0.314212\pi$
$599$	−1.40874e12	−	2.44002e12i	−0.447107	−	0.774412i	−0.998196	+	0.0600345i	$0.980879\pi$
$600$		0			0
$601$	2.84630e12			0.889908			0.444954	−	0.895553i	$-0.353220\pi$
$601$	2.84630e12			0.889908			0.444954	+	0.895553i	$0.353220\pi$
$602$		0			0
$603$	3.31610e12			1.02141
$604$		0			0
$605$	−1.76067e12	−	3.04958e12i	−0.534293	−	0.925423i
$606$		0			0
$607$	1.51972e12	−	2.63223e12i	0.454375	−	0.787000i	−0.544277	−	0.838905i	$-0.683196\pi$
$607$	1.51972e12	−	2.63223e12i	0.454375	−	0.787000i	0.998652	+	0.0519053i	$0.0165294\pi$
$608$		0			0
$609$	3.22895e10	+	8.01752e11i	0.00951225	+	0.236190i
$610$		0			0
$611$	−7.54744e11	+	1.30725e12i	−0.219086	+	0.379468i
$612$		0			0
$613$	−2.85482e12	−	4.94469e12i	−0.816594	−	1.41438i	−0.908178	−	0.418584i	$-0.862526\pi$
$613$	−2.85482e12	−	4.94469e12i	−0.816594	−	1.41438i	0.0915841	−	0.995797i	$-0.470807\pi$
$614$		0			0
$615$	5.36410e11			0.151202
$616$		0			0
$617$	−2.92315e12			−0.812023			−0.406011	−	0.913868i	$-0.633081\pi$
$617$	−2.92315e12			−0.812023			−0.406011	+	0.913868i	$0.633081\pi$
$618$		0			0
$619$	2.88780e12	+	5.00181e12i	0.790604	+	1.36937i	0.925593	+	0.378519i	$0.123567\pi$
$619$	2.88780e12	+	5.00181e12i	0.790604	+	1.36937i	−0.134989	+	0.990847i	$0.543100\pi$
$620$		0			0
$621$	−1.24330e11	+	2.15346e11i	−0.0335477	+	0.0581064i
$622$		0			0
$623$	−2.77837e12	+	1.75684e12i	−0.738913	+	0.467235i
$624$		0			0
$625$	8.38607e11	−	1.45251e12i	0.219836	−	0.380767i
$626$		0			0
$627$	−7.18368e11	−	1.24425e12i	−0.185628	−	0.321517i
$628$		0			0
$629$	3.69853e11			0.0942110
$630$		0			0
$631$	−2.44150e12			−0.613090			−0.306545	−	0.951856i	$-0.599173\pi$
$631$	−2.44150e12			−0.613090			−0.306545	+	0.951856i	$0.599173\pi$
$632$		0			0
$633$	−3.72716e11	−	6.45564e11i	−0.0922703	−	0.159817i
$634$		0			0
$635$	−1.03424e12	+	1.79136e12i	−0.252429	+	0.437220i
$636$		0			0
$637$	−1.14748e12	−	1.66259e12i	−0.276132	−	0.400089i
$638$		0			0
$639$	1.60888e12	−	2.78666e12i	0.381742	−	0.661197i
$640$		0			0
$641$	−3.95349e12	−	6.84764e12i	−0.924952	−	1.60206i	−0.791639	−	0.610989i	$-0.790772\pi$
$641$	−3.95349e12	−	6.84764e12i	−0.924952	−	1.60206i	−0.133313	−	0.991074i	$-0.542562\pi$
$642$		0			0
$643$	−1.32729e12			−0.306208			−0.153104	−	0.988210i	$-0.548927\pi$
$643$	−1.32729e12			−0.306208			−0.153104	+	0.988210i	$0.548927\pi$
$644$		0			0
$645$	−1.22962e11			−0.0279739
$646$		0			0
$647$	−1.61610e12	−	2.79917e12i	−0.362576	−	0.628000i	0.625808	−	0.779977i	$-0.284769\pi$
$647$	−1.61610e12	−	2.79917e12i	−0.362576	−	0.628000i	−0.988384	+	0.151977i	$0.951436\pi$
$648$		0			0
$649$	−6.87946e12	+	1.19156e13i	−1.52214	+	2.63642i
$650$		0			0
$651$	6.99766e11	−	4.42481e11i	0.152700	−	0.0965561i
$652$		0			0
$653$	9.19644e11	−	1.59287e12i	0.197929	−	0.342824i	−0.749927	−	0.661520i	$-0.769912\pi$
$653$	9.19644e11	−	1.59287e12i	0.197929	−	0.342824i	0.947857	+	0.318696i	$0.103245\pi$
$654$		0			0
$655$	−3.27573e12	−	5.67374e12i	−0.695381	−	1.20444i
$656$		0			0
$657$	8.88208e12			1.85982
$658$		0			0
$659$	−5.84606e12			−1.20748			−0.603738	−	0.797182i	$-0.706323\pi$
$659$	−5.84606e12			−1.20748			−0.603738	+	0.797182i	$0.706323\pi$
$660$		0			0
$661$	−2.82594e12	−	4.89468e12i	−0.575780	−	0.997280i	−0.995956	−	0.0898379i	$-0.971365\pi$
$661$	−2.82594e12	−	4.89468e12i	−0.575780	−	0.997280i	0.420176	−	0.907443i	$-0.361968\pi$
$662$		0			0
$663$	7.10032e10	−	1.22981e11i	0.0142714	−	0.0247188i
$664$		0			0
$665$	2.22555e11	+	5.52606e12i	0.0441306	+	1.09577i
$666$		0			0
$667$	−7.11933e11	+	1.23310e12i	−0.139275	+	0.241231i
$668$		0			0
$669$	3.27055e11	+	5.66476e11i	0.0631252	+	0.109336i
$670$		0			0
$671$	−3.97141e12			−0.756299
$672$		0			0
$673$	−1.86409e12			−0.350267			−0.175133	−	0.984545i	$-0.556036\pi$
$673$	−1.86409e12			−0.350267			−0.175133	+	0.984545i	$0.556036\pi$
$674$		0			0
$675$	−3.61537e11	−	6.26201e11i	−0.0670326	−	0.116104i
$676$		0			0
$677$	1.14713e12	−	1.98689e12i	0.209877	−	0.363517i	−0.741799	−	0.670622i	$-0.766027\pi$
$677$	1.14713e12	−	1.98689e12i	0.209877	−	0.363517i	0.951675	+	0.307106i	$0.0993604\pi$
$678$		0			0
$679$	7.45441e12	+	3.91261e12i	1.34586	+	0.706403i
$680$		0			0
$681$	1.37895e11	−	2.38841e11i	0.0245690	−	0.0425547i
$682$		0			0
$683$	−2.46771e12	−	4.27420e12i	−0.433911	−	0.751556i	0.563295	−	0.826256i	$-0.309533\pi$
$683$	−2.46771e12	−	4.27420e12i	−0.433911	−	0.751556i	−0.997206	+	0.0746998i	$0.976200\pi$
$684$		0			0
$685$	5.40710e12			0.938332
$686$		0			0
$687$	1.01050e12			0.173073
$688$		0			0
$689$	1.43108e12	+	2.47871e12i	0.241924	+	0.419024i
$690$		0			0
$691$	3.95992e12	−	6.85879e12i	0.660747	−	1.14445i	−0.319672	−	0.947528i	$-0.603573\pi$
$691$	3.95992e12	−	6.85879e12i	0.660747	−	1.14445i	0.980420	−	0.196920i	$-0.0630939\pi$
$692$		0			0
$693$	8.05372e12	+	4.22717e12i	1.32647	+	0.696227i
$694$		0			0
$695$	−2.92166e12	+	5.06046e12i	−0.475004	+	0.822731i
$696$		0			0
$697$	−1.23629e12	−	2.14132e12i	−0.198414	−	0.343664i
$698$		0			0
$699$	−2.03245e12			−0.322012
$700$		0			0
$701$	−1.24405e12			−0.194584			−0.0972919	−	0.995256i	$-0.531018\pi$
$701$	−1.24405e12			−0.194584			−0.0972919	+	0.995256i	$0.531018\pi$
$702$		0			0
$703$	1.25035e12	+	2.16567e12i	0.193078	+	0.334421i
$704$		0			0
$705$	3.89108e11	−	6.73955e11i	0.0593225	−	0.102750i
$706$		0			0
$707$	−1.61855e10	−	4.01887e11i	−0.00243634	−	0.0604946i
$708$		0			0
$709$	3.64164e12	−	6.30751e12i	0.541239	−	0.937454i	−0.457594	−	0.889161i	$-0.651289\pi$
$709$	3.64164e12	−	6.30751e12i	0.541239	−	0.937454i	0.998833	−	0.0482926i	$-0.0153780\pi$
$710$		0			0
$711$	3.95409e12	+	6.84868e12i	0.580274	+	1.00506i
$712$		0			0
$713$	1.46916e12			0.212895
$714$		0			0
$715$	4.05885e12			0.580799
$716$		0			0
$717$	2.16190e11	+	3.74451e11i	0.0305491	+	0.0529126i
$718$		0			0
$719$	−1.97345e12	+	3.41811e12i	−0.275388	+	0.476986i	−0.970233	−	0.242174i	$-0.922140\pi$
$719$	−1.97345e12	+	3.41811e12i	−0.275388	+	0.476986i	0.694845	+	0.719160i	$0.255473\pi$
$720$		0			0
$721$	−6.06247e12	+	3.83346e12i	−0.835490	+	0.528303i
$722$		0			0
$723$	−2.97100e11	+	5.14592e11i	−0.0404371	+	0.0700391i
$724$		0			0
$725$	−2.07022e12	−	3.58573e12i	−0.278289	−	0.482010i
$726$		0			0
$727$	1.29298e13			1.71667			0.858335	−	0.513089i	$-0.171499\pi$
$727$	1.29298e13			1.71667			0.858335	+	0.513089i	$0.171499\pi$
$728$		0			0
$729$	−6.33557e12			−0.830830
$730$		0			0
$731$	2.83397e11	+	4.90858e11i	0.0367086	+	0.0635811i
$732$		0			0
$733$	3.52694e11	−	6.10884e11i	0.0451263	−	0.0781611i	−0.842580	−	0.538571i	$-0.818964\pi$
$733$	3.52694e11	−	6.10884e11i	0.0451263	−	0.0781611i	0.887706	+	0.460410i	$0.152298\pi$
$734$		0			0
$735$	5.91583e11	+	8.57147e11i	0.0747691	+	0.108333i
$736$		0			0
$737$	−6.49784e12	+	1.12546e13i	−0.811269	+	1.40516i
$738$		0			0
$739$	1.24236e12	+	2.15182e12i	0.153231	+	0.265403i	0.932413	−	0.361394i	$-0.117699\pi$
$739$	1.24236e12	+	2.15182e12i	0.153231	+	0.265403i	−0.779183	+	0.626797i	$0.784366\pi$
$740$		0			0
$741$	9.60152e11			0.116992
$742$		0			0
$743$	−7.27764e12			−0.876074			−0.438037	−	0.898957i	$-0.644326\pi$
$743$	−7.27764e12			−0.876074			−0.438037	+	0.898957i	$0.644326\pi$
$744$		0			0
$745$	−3.38210e11	−	5.85797e11i	−0.0402238	−	0.0696697i
$746$		0			0
$747$	−3.35061e12	+	5.80342e12i	−0.393714	+	0.681932i
$748$		0			0
$749$	9.91549e11	−	6.26983e11i	0.115119	−	0.0727927i
$750$		0			0
$751$	−2.75196e12	+	4.76654e12i	−0.315691	+	0.546794i	−0.979584	−	0.201034i	$-0.935570\pi$
$751$	−2.75196e12	+	4.76654e12i	−0.315691	+	0.546794i	0.663893	+	0.747828i	$0.268903\pi$
$752$		0			0
$753$	−4.14576e10	−	7.18067e10i	−0.00469923	−	0.00813931i
$754$		0			0
$755$	1.19242e13			1.33557
$756$		0			0
$757$	1.41842e13			1.56991			0.784955	−	0.619553i	$-0.212686\pi$
$757$	1.41842e13			1.56991			0.784955	+	0.619553i	$0.212686\pi$
$758$		0			0
$759$	−2.40052e11	−	4.15782e11i	−0.0262553	−	0.0454755i
$760$		0			0
$761$	−5.00695e12	+	8.67229e12i	−0.541180	+	0.937352i	0.457656	+	0.889129i	$0.348689\pi$
$761$	−5.00695e12	+	8.67229e12i	−0.541180	+	0.937352i	−0.998837	+	0.0482227i	$0.984644\pi$
$762$		0			0
$763$	3.08497e11	+	7.66001e12i	0.0329526	+	0.818218i
$764$		0			0
$765$	1.23063e12	−	2.13151e12i	0.129912	−	0.225014i
$766$		0			0
$767$	−4.59746e12	−	7.96303e12i	−0.479665	−	0.830805i
$768$		0			0
$769$	1.39641e12			0.143994			0.0719968	−	0.997405i	$-0.477063\pi$
$769$	1.39641e12			0.143994			0.0719968	+	0.997405i	$0.477063\pi$
$770$		0			0
$771$	1.79822e12			0.183273
$772$		0			0
$773$	3.71378e12	+	6.43246e12i	0.374118	+	0.647992i	0.990195	−	0.139695i	$-0.0446121\pi$
$773$	3.71378e12	+	6.43246e12i	0.374118	+	0.647992i	−0.616077	+	0.787686i	$0.711279\pi$
$774$		0			0
$775$	−2.13607e12	+	3.69979e12i	−0.212696	+	0.368400i
$776$		0			0
$777$	4.16969e11	+	2.18856e11i	0.0410402	+	0.0215409i
$778$		0			0
$779$	8.35896e12	−	1.44781e13i	0.813268	−	1.40862i
$780$		0			0
$781$	6.30515e12	+	1.09208e13i	0.606409	+	1.05033i
$782$		0			0
$783$	4.90065e12			0.465935
$784$		0			0
$785$	−7.36353e12			−0.692106
$786$		0			0
$787$	−2.15657e12	−	3.73529e12i	−0.200391	−	0.347087i	0.748264	−	0.663402i	$-0.230888\pi$
$787$	−2.15657e12	−	3.73529e12i	−0.200391	−	0.347087i	−0.948654	+	0.316314i	$0.897555\pi$
$788$		0			0
$789$	1.40213e12	−	2.42856e12i	0.128808	−	0.223101i
$790$		0			0
$791$	1.78335e12	+	9.36029e11i	0.161972	+	0.0850149i
$792$		0			0
$793$	1.32702e12	−	2.29847e12i	0.119165	−	0.206400i
$794$		0			0
$795$	−7.37796e11	−	1.27790e12i	−0.0655064	−	0.113460i
$796$		0			0
$797$	−6.97760e12			−0.612553			−0.306276	−	0.951943i	$-0.599083\pi$
$797$	−6.97760e12			−0.612553			−0.306276	+	0.951943i	$0.599083\pi$
$798$		0			0
$799$	−3.58718e12			−0.311382
$800$		0			0
$801$	4.94560e12	+	8.56602e12i	0.424495	+	0.735246i
$802$		0			0
$803$	−1.74043e13	+	3.01451e13i	−1.47719	+	2.55857i
$804$		0			0
$805$	7.43696e10	+	1.84661e12i	0.00624186	+	0.154986i
$806$		0			0
$807$	8.85314e11	−	1.53341e12i	0.0734796	−	0.127270i
$808$		0			0
$809$	−1.14195e11	−	1.97791e11i	−0.00937296	−	0.0162344i	0.861301	−	0.508095i	$-0.169650\pi$
$809$	−1.14195e11	−	1.97791e11i	−0.00937296	−	0.0162344i	−0.870674	+	0.491861i	$0.836317\pi$
$810$		0			0
$811$	1.24712e12			0.101231			0.0506156	−	0.998718i	$-0.483882\pi$
$811$	1.24712e12			0.101231			0.0506156	+	0.998718i	$0.483882\pi$
$812$		0			0
$813$	−1.91709e12			−0.153899
$814$		0			0
$815$	1.11840e12	+	1.93713e12i	0.0887950	+	0.153798i
$816$		0			0
$817$	−1.91614e12	+	3.31885e12i	−0.150463	+	0.260609i
$818$		0			0
$819$	−5.13759e12	+	3.24864e12i	−0.399008	+	0.252304i
$820$		0			0
$821$	2.13203e11	−	3.69278e11i	0.0163775	−	0.0283667i	−0.857720	−	0.514116i	$-0.828120\pi$
$821$	2.13203e11	−	3.69278e11i	0.0163775	−	0.0283667i	0.874098	+	0.485750i	$0.161453\pi$
$822$		0			0
$823$	−5.10619e11	−	8.84418e11i	−0.0387970	−	0.0671983i	0.845975	−	0.533223i	$-0.179019\pi$
$823$	−5.10619e11	−	8.84418e11i	−0.0387970	−	0.0671983i	−0.884772	+	0.466024i	$0.845686\pi$
$824$		0			0
$825$	1.39609e12			0.104923
$826$		0			0
$827$	5.09034e12			0.378418			0.189209	−	0.981937i	$-0.439408\pi$
$827$	5.09034e12			0.378418			0.189209	+	0.981937i	$0.439408\pi$
$828$		0			0
$829$	−6.45038e12	−	1.11724e13i	−0.474340	−	0.821582i	0.525228	−	0.850962i	$-0.323980\pi$
$829$	−6.45038e12	−	1.11724e13i	−0.474340	−	0.821582i	−0.999568	+	0.0293800i	$0.990647\pi$
$830$		0			0
$831$	1.55679e12	−	2.69645e12i	0.113247	−	0.196149i
$832$		0			0
$833$	2.05823e12	−	4.33707e12i	0.148112	−	0.312100i
$834$		0			0
$835$	−3.65718e11	+	6.33442e11i	−0.0260350	+	0.0450939i
$836$		0			0
$837$	−2.52827e12	−	4.37909e12i	−0.178057	−	0.308403i
$838$		0			0
$839$	6.08459e12			0.423938			0.211969	−	0.977276i	$-0.432012\pi$
$839$	6.08459e12			0.423938			0.211969	+	0.977276i	$0.432012\pi$
$840$		0			0
$841$	1.35548e13			0.934353
$842$		0			0
$843$	7.43153e11	+	1.28718e12i	0.0506821	+	0.0877839i
$844$		0			0
$845$	4.38274e12	−	7.59113e12i	0.295727	−	0.512214i
$846$		0			0
$847$	−1.74677e13	+	1.10453e13i	−1.16617	+	0.737400i
$848$		0			0
$849$	8.08709e9	−	1.40073e10i	0.000534205	−	0.000925270i
$850$		0			0
$851$	4.17821e11	+	7.23686e11i	0.0273091	+	0.0473007i
$852$		0			0
$853$	1.69683e13			1.09741			0.548703	−	0.836017i	$-0.315122\pi$
$853$	1.69683e13			1.09741			0.548703	+	0.836017i	$0.315122\pi$
$854$		0			0
$855$	1.66413e13			1.06498
$856$		0			0
$857$	2.35529e12	+	4.07948e12i	0.149152	+	0.258339i	0.930914	−	0.365237i	$-0.119012\pi$
$857$	2.35529e12	+	4.07948e12i	0.149152	+	0.258339i	−0.781762	+	0.623577i	$0.785679\pi$
$858$		0			0
$859$	4.02258e12	−	6.96732e12i	0.252079	−	0.436613i	−0.712019	−	0.702160i	$-0.752219\pi$
$859$	4.02258e12	−	6.96732e12i	0.252079	−	0.436613i	0.964098	+	0.265547i	$0.0855525\pi$
$860$		0			0
$861$	−1.26687e11	−	3.14566e12i	−0.00785630	−	0.195073i
$862$		0			0
$863$	8.78137e12	−	1.52098e13i	0.538907	−	0.933414i	−0.460056	−	0.887890i	$-0.652171\pi$
$863$	8.78137e12	−	1.52098e13i	0.538907	−	0.933414i	0.998963	−	0.0455244i	$-0.0144959\pi$
$864$		0			0
$865$	1.05215e13	+	1.82237e13i	0.639004	+	1.10679i
$866$		0			0
$867$	−2.49023e12			−0.149676
$868$		0			0
$869$	−3.09919e13			−1.84357
$870$		0			0
$871$	−4.34242e12	−	7.52129e12i	−0.255653	−	0.442803i
$872$		0			0
$873$	1.26661e13	−	2.19383e13i	0.738037	−	1.27832i
$874$		0			0
$875$	−1.66491e13	−	8.73867e12i	−0.960186	−	0.503975i
$876$		0			0
$877$	−4.21010e12	+	7.29210e12i	−0.240322	+	0.416250i	−0.960806	−	0.277221i	$-0.910586\pi$
$877$	−4.21010e12	+	7.29210e12i	−0.240322	+	0.416250i	0.720484	+	0.693472i	$0.243920\pi$
$878$		0			0
$879$	−6.76194e10	−	1.17120e11i	−0.00382051	−	0.00661732i
$880$		0			0
$881$	−3.03091e13			−1.69504			−0.847522	−	0.530760i	$-0.821907\pi$
$881$	−3.03091e13			−1.69504			−0.847522	+	0.530760i	$0.821907\pi$
$882$		0			0
$883$	2.44649e12			0.135432			0.0677159	−	0.997705i	$-0.478429\pi$
$883$	2.44649e12			0.135432			0.0677159	+	0.997705i	$0.478429\pi$
$884$		0			0
$885$	2.37022e12	+	4.10534e12i	0.129880	+	0.224959i
$886$		0			0
$887$	1.02686e13	−	1.77858e13i	0.557001	−	0.964753i	−0.440744	−	0.897633i	$-0.645285\pi$
$887$	1.02686e13	−	1.77858e13i	0.557001	−	0.964753i	0.997745	−	0.0671207i	$-0.0213813\pi$
$888$		0			0
$889$	1.07493e13	+	5.64200e12i	0.577193	+	0.302953i
$890$		0			0
$891$	1.32652e13	−	2.29761e13i	0.705123	−	1.22131i
$892$		0			0
$893$	−1.21271e13	−	2.10047e13i	−0.638152	−	1.10531i
$894$		0			0
$895$	−4.43391e12			−0.230985
$896$		0			0
$897$	3.20847e11			0.0165475
$898$		0			0
$899$	−1.44773e13	−	2.50754e13i	−0.739211	−	1.28035i
$900$		0			0
$901$	−3.40087e12	+	5.89047e12i	−0.171921	+	0.297775i
$902$		0			0
$903$	2.90407e10	+	7.21085e11i	0.00145349	+	0.0360904i
$904$		0			0
$905$	1.30926e13	−	2.26771e13i	0.648796	−	1.12375i
$906$		0			0
$907$	6.75232e12	+	1.16954e13i	0.331299	+	0.573827i	0.982767	−	0.184849i	$-0.0591797\pi$
$907$	6.75232e12	+	1.16954e13i	0.331299	+	0.573827i	−0.651468	+	0.758676i	$0.725846\pi$
$908$		0			0
$909$	−1.21025e12			−0.0587948
$910$		0			0
$911$	−2.83771e13			−1.36501			−0.682504	−	0.730882i	$-0.739109\pi$
$911$	−2.83771e13			−1.36501			−0.682504	+	0.730882i	$0.739109\pi$
$912$		0			0
$913$	−1.31309e13	−	2.27434e13i	−0.625426	−	1.08327i
$914$		0			0
$915$	−6.84146e11	+	1.18498e12i	−0.0322667	+	0.0558875i
$916$		0			0
$917$	−3.24987e13	+	2.05498e13i	−1.51776	+	0.959723i
$918$		0			0
$919$	−1.30106e13	+	2.25350e13i	−0.601696	+	1.04217i	0.390868	+	0.920447i	$0.372175\pi$
$919$	−1.30106e13	+	2.25350e13i	−0.601696	+	1.04217i	−0.992564	+	0.121722i	$0.961158\pi$
$920$		0			0
$921$	−2.15998e12	−	3.74120e12i	−0.0989196	−	0.171334i
$922$		0			0
$923$	−8.42730e12			−0.382191
$924$		0			0
$925$	−2.42995e12			−0.109134
$926$		0			0
$927$	1.07914e13	+	1.86913e13i	0.479977	+	0.831344i
$928$		0			0
$929$	8.52205e12	−	1.47606e13i	0.375382	−	0.650181i	−0.615002	−	0.788525i	$-0.710845\pi$
$929$	8.52205e12	−	1.47606e13i	0.375382	−	0.650181i	0.990384	+	0.138345i	$0.0441782\pi$
$930$		0			0
$931$	3.23538e13	−	2.61025e12i	1.41141	−	0.113870i
$932$		0			0
$933$	1.23762e12	−	2.14361e12i	0.0534710	−	0.0926145i
$934$		0			0
$935$	4.82278e12	+	8.35330e12i	0.206369	+	0.357442i
$936$		0			0
$937$	1.17078e13			0.496188			0.248094	−	0.968736i	$-0.420196\pi$
$937$	1.17078e13			0.496188			0.248094	+	0.968736i	$0.420196\pi$
$938$		0			0
$939$	−3.54962e12			−0.149000
$940$		0			0
$941$	7.99721e12	+	1.38516e13i	0.332495	+	0.575898i	0.983000	−	0.183603i	$-0.0587762\pi$
$941$	7.99721e12	+	1.38516e13i	0.332495	+	0.575898i	−0.650505	+	0.759502i	$0.725443\pi$
$942$		0			0
$943$	2.79325e12	−	4.83806e12i	0.115029	−	0.199236i
$944$		0			0
$945$	5.37616e12	−	3.39949e12i	0.219295	−	0.138666i
$946$		0			0
$947$	−1.35192e13	+	2.34159e13i	−0.546231	+	0.946099i	0.452298	+	0.891867i	$0.350604\pi$
$947$	−1.35192e13	+	2.34159e13i	−0.546231	+	0.946099i	−0.998528	+	0.0542323i	$0.982729\pi$
$948$		0			0
$949$	−1.16311e13	−	2.01456e13i	−0.465502	−	0.806273i
$950$		0			0
$951$	4.71407e12			0.186889
$952$		0			0
$953$	−3.10710e13			−1.22022			−0.610108	−	0.792318i	$-0.708874\pi$
$953$	−3.10710e13			−1.22022			−0.610108	+	0.792318i	$0.708874\pi$
$954$		0			0
$955$	3.14968e12	+	5.45541e12i	0.122532	+	0.212233i
$956$		0			0
$957$	−4.73101e12	+	8.19434e12i	−0.182326	+	0.315799i
$958$		0			0
$959$	−1.27703e12	−	3.17087e13i	−0.0487547	−	1.21058i
$960$		0			0
$961$	−1.71799e12	+	2.97564e12i	−0.0649777	+	0.112545i
$962$		0			0
$963$	−1.76499e12	−	3.05706e12i	−0.0661340	−	0.114547i
$964$		0			0
$965$	2.58235e13			0.958610
$966$		0			0
$967$	1.42603e13			0.524457			0.262228	−	0.965006i	$-0.415543\pi$
$967$	1.42603e13			0.524457			0.262228	+	0.965006i	$0.415543\pi$
$968$		0			0
$969$	1.14086e12	+	1.97604e12i	0.0415697	+	0.0720009i
$970$		0			0
$971$	1.29280e13	−	2.23920e13i	0.466709	−	0.808364i	−0.532568	−	0.846387i	$-0.678773\pi$
$971$	1.29280e13	−	2.23920e13i	0.466709	−	0.808364i	0.999277	+	0.0380235i	$0.0121062\pi$
$972$		0			0
$973$	3.03660e13	+	1.59382e13i	1.08612	+	0.570076i
$974$		0			0
$975$	−4.66494e11	+	8.07991e11i	−0.0165320	+	0.0286342i
$976$		0			0
$977$	−5.41303e12	−	9.37565e12i	−0.190071	−	0.329212i	0.755203	−	0.655491i	$-0.227538\pi$
$977$	−5.41303e12	−	9.37565e12i	−0.190071	−	0.329212i	−0.945273	+	0.326279i	$0.894205\pi$
$978$		0			0
$979$	−3.87632e13			−1.34865
$980$		0			0
$981$	2.30676e13			0.795227
$982$		0			0
$983$	−1.38301e13	−	2.39545e13i	−0.472428	−	0.818269i	0.527074	−	0.849819i	$-0.323289\pi$
$983$	−1.38301e13	−	2.39545e13i	−0.472428	−	0.818269i	−0.999502	+	0.0315503i	$0.989956\pi$
$984$		0			0
$985$	6.12981e12	−	1.06171e13i	0.207484	−	0.359372i
$986$		0			0
$987$	−4.04416e12	−	2.12267e12i	−0.135644	−	0.0711958i
$988$		0			0
$989$	−6.40303e11	+	1.10904e12i	−0.0212815	+	0.0368607i
$990$		0			0
$991$	−1.35483e13	−	2.34663e13i	−0.446223	−	0.772881i	0.551914	−	0.833901i	$-0.313898\pi$
$991$	−1.35483e13	−	2.34663e13i	−0.446223	−	0.772881i	−0.998137	+	0.0610206i	$0.980564\pi$
$992$		0			0
$993$	−1.33025e12			−0.0434172
$994$		0			0
$995$	−1.89724e13			−0.613648
$996$		0			0
$997$	−1.56566e13	−	2.71180e13i	−0.501845	−	0.869221i	−0.999998	−	0.00213142i	$-0.999322\pi$
$997$	−1.56566e13	−	2.71180e13i	−0.501845	−	0.869221i	0.498153	−	0.867089i	$-0.334012\pi$
$998$		0			0
$999$	1.43805e12	−	2.49078e12i	0.0456804	−	0.0791207i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	112.10.i.b.65.1		6
4.3	odd	2		14.10.c.a.9.3	✓	6
7.4	even	3	inner	112.10.i.b.81.1		6
12.11	even	2		126.10.g.f.37.2		6
28.3	even	6		98.10.c.k.67.1		6
28.11	odd	6		14.10.c.a.11.3	yes	6
28.19	even	6		98.10.a.i.1.3		3
28.23	odd	6		98.10.a.j.1.1		3
28.27	even	2		98.10.c.k.79.1		6
84.11	even	6		126.10.g.f.109.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.10.c.a.9.3	✓	6	4.3	odd	2
14.10.c.a.11.3	yes	6	28.11	odd	6
98.10.a.i.1.3		3	28.19	even	6
98.10.a.j.1.1		3	28.23	odd	6
98.10.c.k.67.1		6	28.3	even	6
98.10.c.k.79.1		6	28.27	even	2
112.10.i.b.65.1		6	1.1	even	1	trivial
112.10.i.b.81.1		6	7.4	even	3	inner
126.10.g.f.37.2		6	12.11	even	2
126.10.g.f.109.2		6	84.11	even	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(-11.9224 - 20.6501i) q^{3} +(-541.184 + 937.358i) q^{5} +(5624.74 + 2952.27i) q^{7} +(9557.22 - 16553.6i) q^{9} +(37454.4 + 64872.9i) q^{11} -50060.6 q^{13} +25808.7 q^{15} +(-59482.6 - 103027. i) q^{17} +(402181. - 696599. i) q^{19} +(-6095.40 - 151350. i) q^{21} +(134394. - 232777. i) q^{23} +(390803. + 676891. i) q^{25} -925113. q^{27} -5.29735e6 q^{29} +(2.73293e6 + 4.73357e6i) q^{31} +(893089. - 1.54688e6i) q^{33} +(-5.81135e6 + 3.67467e6i) q^{35} +(-1.55446e6 + 2.69240e6i) q^{37} +(596840. + 1.03376e6i) q^{39} +2.07841e7 q^{41} -4.76437e6 q^{43} +(1.03444e7 + 1.79171e7i) q^{45} +(1.50766e7 - 2.61135e7i) q^{47} +(2.29218e7 + 3.32115e7i) q^{49} +(-1.41835e6 + 2.45665e6i) q^{51} +(-2.85871e7 - 4.95142e7i) q^{53} -8.10788e7 q^{55} -1.91798e7 q^{57} +(9.18379e7 + 1.59068e8i) q^{59} +(-2.65083e7 + 4.59138e7i) q^{61} +(1.02628e8 - 6.48941e7i) q^{63} +(2.70920e7 - 4.69247e7i) q^{65} +(8.67433e7 + 1.50244e8i) q^{67} -6.40918e6 q^{69} +1.68342e8 q^{71} +(2.32340e8 + 4.02424e8i) q^{73} +(9.31858e6 - 1.61403e7i) q^{75} +(1.91489e7 + 4.75469e8i) q^{77} +(-2.06864e8 + 3.58299e8i) q^{79} +(-1.77085e8 - 3.06720e8i) q^{81} -3.50584e8 q^{83} +1.28764e8 q^{85} +(6.31569e7 + 1.09391e8i) q^{87} +(-2.58736e8 + 4.48144e8i) q^{89} +(-2.81578e8 - 1.47792e8i) q^{91} +(6.51659e7 - 1.12871e8i) q^{93} +(4.35308e8 + 7.53976e8i) q^{95} +1.32529e9 q^{97} +1.43184e9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 233 q^{3} - 733 q^{5} - 5012 q^{7} - 15058 q^{9} - 7339 q^{11} + 197036 q^{13} - 738238 q^{15} - 306665 q^{17} + 377991 q^{19} - 1585927 q^{21} + 2267255 q^{23} - 142612 q^{25} - 21348358 q^{27} - 13085956 q^{29}+ \cdots + 1256218868 q^{99}+O(q^{100}) \) 6 * q + 233 * q^3 - 733 * q^5 - 5012 * q^7 - 15058 * q^9 - 7339 * q^11 + 197036 * q^13 - 738238 * q^15 - 306665 * q^17 + 377991 * q^19 - 1585927 * q^21 + 2267255 * q^23 - 142612 * q^25 - 21348358 * q^27 - 13085956 * q^29 + 6654517 * q^31 + 3747977 * q^33 - 22864807 * q^35 - 22287969 * q^37 - 3151974 * q^39 + 68096196 * q^41 + 125648280 * q^43 - 86300638 * q^45 + 52703019 * q^47 + 57596070 * q^49 - 97736333 * q^51 - 12091125 * q^53 - 377435398 * q^55 - 158517670 * q^57 + 12949897 * q^59 - 160252153 * q^61 + 356021666 * q^63 + 334191270 * q^65 + 480890225 * q^67 + 1131233162 * q^69 + 74421440 * q^71 + 251382283 * q^73 - 16322324 * q^75 - 527045155 * q^77 - 286494785 * q^79 - 2165894587 * q^81 - 2295182344 * q^83 - 2389185074 * q^85 - 1412199358 * q^87 - 901243845 * q^89 - 1026798920 * q^91 + 710456889 * q^93 + 887366177 * q^95 + 629707876 * q^97 + 1256218868 * q^99

Introduction

L-functions

Modular forms

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Database

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