LMFDB - Embedded newform 84.9.p.b.65.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [84,9,Mod(53,84)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(84, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 4]))

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

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

chi := DirichletCharacter("84.53");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 84 = 2^{2} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	84.p (of order $6$, degree $2$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$34.2198032451$
Analytic rank:	$0$
Dimension:	$40$
Relative dimension:	$20$ over $\Q(\zeta_{6})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			65.12
Character	$\chi$	$=$	84.65
Dual form			84.9.p.b.53.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(23.7435 - 77.4419i) q^{3} +(705.317 + 407.215i) q^{5} +(-2163.85 + 1040.46i) q^{7} +(-5433.50 - 3677.48i) q^{9} +O(q^{10})$	\(q+(23.7435 - 77.4419i) q^{3} +(705.317 + 407.215i) q^{5} +(-2163.85 + 1040.46i) q^{7} +(-5433.50 - 3677.48i) q^{9} +(19089.3 - 11021.2i) q^{11} -53303.6 q^{13} +(48282.2 - 44952.4i) q^{15} +(78994.6 - 45607.6i) q^{17} +(82298.4 - 142545. i) q^{19} +(29197.8 + 192277. i) q^{21} +(167463. + 96684.9i) q^{23} +(136335. + 236140. i) q^{25} +(-413801. + 333464. i) q^{27} -914552. i q^{29} +(-424400. - 735082. i) q^{31} +(-400258. - 1.73999e6i) q^{33} +(-1.94989e6 - 147299. i) q^{35} +(588097. - 1.01861e6i) q^{37} +(-1.26561e6 + 4.12793e6i) q^{39} -1.19595e6i q^{41} -118171. q^{43} +(-2.33481e6 - 4.80639e6i) q^{45} +(-876262. - 505910. i) q^{47} +(3.59969e6 - 4.50279e6i) q^{49} +(-1.65633e6 - 7.20038e6i) q^{51} +(-3.38202e6 + 1.95261e6i) q^{53} +1.79520e7 q^{55} +(-9.08491e6 - 9.75785e6i) q^{57} +(3.56686e6 - 2.05933e6i) q^{59} +(1.89260e6 - 3.27809e6i) q^{61} +(1.55835e7 + 2.30418e6i) q^{63} +(-3.75959e7 - 2.17060e7i) q^{65} +(-1.76167e7 - 3.05130e7i) q^{67} +(1.14636e7 - 1.06730e7i) q^{69} -2.14501e7i q^{71} +(-7.42242e6 - 1.28560e7i) q^{73} +(2.15242e7 - 4.95130e6i) q^{75} +(-2.98393e7 + 4.37099e7i) q^{77} +(-3.51298e7 + 6.08465e7i) q^{79} +(1.59991e7 + 3.99631e7i) q^{81} +7.08020e7i q^{83} +7.42883e7 q^{85} +(-7.08246e7 - 2.17146e7i) q^{87} +(-2.46090e6 - 1.42080e6i) q^{89} +(1.15341e8 - 5.54602e7i) q^{91} +(-6.70029e7 + 1.54129e7i) q^{93} +(1.16093e8 - 6.70263e7i) q^{95} -6.00540e7 q^{97} +(-1.44252e8 - 1.03167e7i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q + 81 q^{3} - 34 q^{7} + 4771 q^{9}+O(q^{10}) $ 40 * q + 81 * q^3 - 34 * q^7 + 4771 * q^9	$ 40 q + 81 q^{3} - 34 q^{7} + 4771 q^{9} - 55464 q^{13} + 68482 q^{15} + 311690 q^{19} - 172343 q^{21} + 1766792 q^{25} - 3451932 q^{27} + 31596 q^{31} + 1874885 q^{33} - 1853482 q^{37} + 11217526 q^{39} - 13372600 q^{43} - 527785 q^{45} - 12653462 q^{49} - 1103461 q^{51} + 71577224 q^{55} - 17195214 q^{57} - 21761970 q^{61} + 21945045 q^{63} - 26337350 q^{67} - 5588722 q^{69} + 41115682 q^{73} - 17971730 q^{75} - 120916932 q^{79} - 24550133 q^{81} + 139250060 q^{85} - 16321046 q^{87} + 345074940 q^{91} + 25774675 q^{93} - 707216948 q^{97} - 94510994 q^{99}+O(q^{100}) $ 40 * q + 81 * q^3 - 34 * q^7 + 4771 * q^9 - 55464 * q^13 + 68482 * q^15 + 311690 * q^19 - 172343 * q^21 + 1766792 * q^25 - 3451932 * q^27 + 31596 * q^31 + 1874885 * q^33 - 1853482 * q^37 + 11217526 * q^39 - 13372600 * q^43 - 527785 * q^45 - 12653462 * q^49 - 1103461 * q^51 + 71577224 * q^55 - 17195214 * q^57 - 21761970 * q^61 + 21945045 * q^63 - 26337350 * q^67 - 5588722 * q^69 + 41115682 * q^73 - 17971730 * q^75 - 120916932 * q^79 - 24550133 * q^81 + 139250060 * q^85 - 16321046 * q^87 + 345074940 * q^91 + 25774675 * q^93 - 707216948 * q^97 - 94510994 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/84\mathbb{Z}\right)^\times$.

$n$	$29$	$43$	$73$
$\chi(n)$	$-1$	$1$	$e\left(\frac{1}{3}\right)$

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
$3$	23.7435	−	77.4419i	0.293129	−	0.956073i
$3$	23.7435	−	77.4419i	0.293129	−	0.956073i
$4$		0			0
$5$	705.317	+	407.215i	1.12851	+	0.651544i	0.943559	−	0.331204i	$-0.107455\pi$
$5$	705.317	+	407.215i	1.12851	+	0.651544i	0.184948	+	0.982748i	$0.440788\pi$
$6$		0			0
$7$	−2163.85	+	1040.46i	−0.901229	+	0.433344i
$7$	−2163.85	+	1040.46i	−0.901229	+	0.433344i
$8$		0			0
$9$	−5433.50	−	3677.48i	−0.828151	−	0.560505i
$10$		0			0
$11$	19089.3	−	11021.2i	1.30382	−	0.752763i	0.322767	−	0.946479i	$-0.395387\pi$
$11$	19089.3	−	11021.2i	1.30382	−	0.752763i	0.981058	+	0.193715i	$0.0620537\pi$
$12$		0			0
$13$	−53303.6			−1.86631			−0.933154	−	0.359478i	$-0.882955\pi$
$13$	−53303.6			−1.86631			−0.933154	+	0.359478i	$0.882955\pi$
$14$		0			0
$15$	48282.2	−	44952.4i	0.953722	−	0.887949i
$16$		0			0
$17$	78994.6	−	45607.6i	0.945806	−	0.546061i	0.0540302	−	0.998539i	$-0.482793\pi$
$17$	78994.6	−	45607.6i	0.945806	−	0.546061i	0.891776	+	0.452478i	$0.149460\pi$
$18$		0			0
$19$	82298.4	−	142545.i	0.631505	−	1.09380i	−0.355739	−	0.934585i	$-0.615771\pi$
$19$	82298.4	−	142545.i	0.631505	−	1.09380i	0.987244	−	0.159214i	$-0.0508959\pi$
$20$		0			0
$21$	29197.8	+	192277.i	0.150132	+	0.988666i
$22$		0			0
$23$	167463.	+	96684.9i	0.598422	+	0.345499i	0.768421	−	0.639945i	$-0.221043\pi$
$23$	167463.	+	96684.9i	0.598422	+	0.345499i	−0.169998	+	0.985444i	$0.554376\pi$
$24$		0			0
$25$	136335.	+	236140.i	0.349019	+	0.604518i
$26$		0			0
$27$	−413801.	+	333464.i	−0.778639	+	0.627472i
$28$		0			0
$29$		−	914552.i		−	1.29305i	−0.762891	−	0.646527i	$-0.776221\pi$
$29$		−	914552.i		−	1.29305i	0.762891	−	0.646527i	$-0.223779\pi$
$30$		0			0
$31$	−424400.	−	735082.i	−0.459545	−	0.795956i	0.539391	−	0.842055i	$-0.318654\pi$
$31$	−424400.	−	735082.i	−0.459545	−	0.795956i	−0.998937	+	0.0460990i	$0.985321\pi$
$32$		0			0
$33$	−400258.	−	1.73999e6i	−0.337508	−	1.46721i
$34$		0			0
$35$	−1.94989e6	−	147299.i	−1.29939	−	0.0981584i
$36$		0			0
$37$	588097.	−	1.01861e6i	0.313792	−	0.543504i	−0.665388	−	0.746498i	$-0.731734\pi$
$37$	588097.	−	1.01861e6i	0.313792	−	0.543504i	0.979180	+	0.202994i	$0.0650671\pi$
$38$		0			0
$39$	−1.26561e6	+	4.12793e6i	−0.547069	+	1.78433i
$40$		0			0
$41$		−	1.19595e6i		−	0.423231i	−0.977353	−	0.211616i	$-0.932128\pi$
$41$		−	1.19595e6i		−	0.423231i	0.977353	−	0.211616i	$-0.0678725\pi$
$42$		0			0
$43$	−118171.			−0.0345651			−0.0172826	−	0.999851i	$-0.505501\pi$
$43$	−118171.			−0.0345651			−0.0172826	+	0.999851i	$0.505501\pi$
$44$		0			0
$45$	−2.33481e6	−	4.80639e6i	−0.569380	−	1.17211i
$46$		0			0
$47$	−876262.	−	505910.i	−0.179574	−	0.103677i	0.407519	−	0.913197i	$-0.366394\pi$
$47$	−876262.	−	505910.i	−0.179574	−	0.103677i	−0.587092	+	0.809520i	$0.699727\pi$
$48$		0			0
$49$	3.59969e6	−	4.50279e6i	0.624426	−	0.781084i
$50$		0			0
$51$	−1.65633e6	−	7.20038e6i	−0.244831	−	1.06433i
$52$		0			0
$53$	−3.38202e6	+	1.95261e6i	−0.428620	+	0.247464i	−0.698758	−	0.715358i	$-0.746264\pi$
$53$	−3.38202e6	+	1.95261e6i	−0.428620	+	0.247464i	0.270139	+	0.962821i	$0.412930\pi$
$54$		0			0
$55$	1.79520e7			1.96183
$56$		0			0
$57$	−9.08491e6	−	9.75785e6i	−0.860639	−	0.924389i
$58$		0			0
$59$	3.56686e6	−	2.05933e6i	0.294359	−	0.169948i	−0.345547	−	0.938402i	$-0.612307\pi$
$59$	3.56686e6	−	2.05933e6i	0.294359	−	0.169948i	0.639906	+	0.768453i	$0.278973\pi$
$60$		0			0
$61$	1.89260e6	−	3.27809e6i	0.136691	−	0.236756i	−0.789551	−	0.613685i	$-0.789687\pi$
$61$	1.89260e6	−	3.27809e6i	0.136691	−	0.236756i	0.926242	+	0.376929i	$0.123020\pi$
$62$		0			0
$63$	1.55835e7	+	2.30418e6i	0.989245	+	0.146270i
$64$		0			0
$65$	−3.75959e7	−	2.17060e7i	−2.10614	−	1.21598i
$66$		0			0
$67$	−1.76167e7	−	3.05130e7i	−0.874228	−	1.51421i	−0.857582	−	0.514347i	$-0.828034\pi$
$67$	−1.76167e7	−	3.05130e7i	−0.874228	−	1.51421i	−0.0166462	−	0.999861i	$-0.505299\pi$
$68$		0			0
$69$	1.14636e7	−	1.06730e7i	0.505738	−	0.470860i
$70$		0			0
$71$		−	2.14501e7i		−	0.844106i	−0.906571	−	0.422053i	$-0.861310\pi$
$71$		−	2.14501e7i		−	0.844106i	0.906571	−	0.422053i	$-0.138690\pi$
$72$		0			0
$73$	−7.42242e6	−	1.28560e7i	−0.261369	−	0.452704i	0.705237	−	0.708972i	$-0.250841\pi$
$73$	−7.42242e6	−	1.28560e7i	−0.261369	−	0.452704i	−0.966606	+	0.256267i	$0.917507\pi$
$74$		0			0
$75$	2.15242e7	−	4.95130e6i	0.680271	−	0.156486i
$76$		0			0
$77$	−2.98393e7	+	4.37099e7i	−0.848839	+	1.24342i
$78$		0			0
$79$	−3.51298e7	+	6.08465e7i	−0.901917	+	1.56217i	−0.0769150	+	0.997038i	$0.524507\pi$
$79$	−3.51298e7	+	6.08465e7i	−0.901917	+	1.56217i	−0.825002	+	0.565129i	$0.808826\pi$
$80$		0			0
$81$	1.59991e7	+	3.99631e7i	0.371667	+	0.928366i
$82$		0			0
$83$			7.08020e7i			1.49188i	0.666015	+	0.745939i	$0.267999\pi$
$83$			7.08020e7i			1.49188i	−0.666015	+	0.745939i	$0.732001\pi$
$84$		0			0
$85$	7.42883e7			1.42313
$86$		0			0
$87$	−7.08246e7	−	2.17146e7i	−1.23625	−	0.379031i
$88$		0			0
$89$	−2.46090e6	−	1.42080e6i	−0.0392224	−	0.0226451i	0.480261	−	0.877126i	$-0.340542\pi$
$89$	−2.46090e6	−	1.42080e6i	−0.0392224	−	0.0226451i	−0.519483	+	0.854481i	$0.673875\pi$
$90$		0			0
$91$	1.15341e8	−	5.54602e7i	1.68197	−	0.808753i
$92$		0			0
$93$	−6.70029e7	+	1.54129e7i	−0.895698	+	0.206041i
$94$		0			0
$95$	1.16093e8	−	6.70263e7i	1.42532	−	0.822907i
$96$		0			0
$97$	−6.00540e7			−0.678352			−0.339176	−	0.940723i	$-0.610148\pi$
$97$	−6.00540e7			−0.678352			−0.339176	+	0.940723i	$0.610148\pi$
$98$		0			0
$99$	−1.44252e8	−	1.03167e7i	−1.50169	−	0.107399i
$100$		0			0
$101$	9.12780e7	−	5.26994e7i	0.877163	−	0.506431i	0.00744136	−	0.999972i	$-0.497631\pi$
$101$	9.12780e7	−	5.26994e7i	0.877163	−	0.506431i	0.869722	+	0.493542i	$0.164298\pi$
$102$		0			0
$103$	−8.07487e7	+	1.39861e8i	−0.717441	+	1.24265i	0.244569	+	0.969632i	$0.421354\pi$
$103$	−8.07487e7	+	1.39861e8i	−0.717441	+	1.24265i	−0.962010	+	0.273013i	$0.911980\pi$
$104$		0			0
$105$	−5.77042e7	+	1.47506e8i	−0.474734	+	1.21353i
$106$		0			0
$107$	2.03259e8	+	1.17352e8i	1.55065	+	0.895270i	0.998089	+	0.0617995i	$0.0196839\pi$
$107$	2.03259e8	+	1.17352e8i	1.55065	+	0.895270i	0.552564	+	0.833470i	$0.313649\pi$
$108$		0			0
$109$	6.35008e7	+	1.09987e8i	0.449856	+	0.779173i	0.998376	−	0.0569641i	$-0.0181421\pi$
$109$	6.35008e7	+	1.09987e8i	0.449856	+	0.779173i	−0.548520	+	0.836137i	$0.684809\pi$
$110$		0			0
$111$	−6.49200e7	−	6.97288e7i	−0.427648	−	0.459325i
$112$		0			0
$113$		−	2.24265e8i		−	1.37546i	−0.725967	−	0.687730i	$-0.758607\pi$
$113$		−	2.24265e8i		−	1.37546i	0.725967	−	0.687730i	$-0.241393\pi$
$114$		0			0
$115$	7.87430e7	+	1.36387e8i	0.450216	+	0.779797i
$116$		0			0
$117$	2.89625e8	+	1.96023e8i	1.54558	+	1.04608i
$118$		0			0
$119$	−1.23480e8	+	1.80879e8i	−0.615755	+	0.901985i
$120$		0			0
$121$	1.35755e8	−	2.35134e8i	0.633306	−	1.09692i
$122$		0			0
$123$	−9.26166e7	−	2.83960e7i	−0.404640	−	0.124061i
$124$		0			0
$125$		−	9.60653e7i		−	0.393484i
$126$		0			0
$127$	−5.23635e7			−0.201286			−0.100643	−	0.994923i	$-0.532090\pi$
$127$	−5.23635e7			−0.201286			−0.100643	+	0.994923i	$0.532090\pi$
$128$		0			0
$129$	−2.80579e6	+	9.15141e6i	−0.0101320	+	0.0330468i
$130$		0			0
$131$	4.10445e8	+	2.36970e8i	1.39370	+	0.804654i	0.993723	−	0.111871i	$-0.0356843\pi$
$131$	4.10445e8	+	2.36970e8i	1.39370	+	0.804654i	0.399978	+	0.916525i	$0.369018\pi$
$132$		0			0
$133$	−2.97692e7	+	3.94074e8i	−0.0951395	+	1.25942i
$134$		0			0
$135$	−4.27652e8	+	6.66922e7i	−1.28753	+	0.200789i
$136$		0			0
$137$	−3.44715e7	+	1.99021e7i	−0.0978539	+	0.0564960i	−0.548128	−	0.836394i	$-0.684659\pi$
$137$	−3.44715e7	+	1.99021e7i	−0.0978539	+	0.0564960i	0.450274	+	0.892890i	$0.351326\pi$
$138$		0			0
$139$	3.61392e8			0.968097			0.484049	−	0.875041i	$-0.339166\pi$
$139$	3.61392e8			0.968097			0.484049	+	0.875041i	$0.339166\pi$
$140$		0			0
$141$	−5.99841e7	+	5.58473e7i	−0.151761	+	0.141295i
$142$		0			0
$143$	−1.01753e9	+	5.87470e8i	−2.43334	+	1.40489i
$144$		0			0
$145$	3.72419e8	−	6.45049e8i	0.842481	−	1.45922i
$146$		0			0
$147$	−2.63236e8	−	3.85679e8i	−0.563736	−	0.825955i
$148$		0			0
$149$	1.48818e8	+	8.59200e7i	0.301932	+	0.174321i	0.643311	−	0.765605i	$-0.277560\pi$
$149$	1.48818e8	+	8.59200e7i	0.301932	+	0.174321i	−0.341378	+	0.939926i	$0.610894\pi$
$150$		0			0
$151$	−2.38009e7	−	4.12244e7i	−0.0457811	−	0.0792951i	0.842227	−	0.539123i	$-0.181244\pi$
$151$	−2.38009e7	−	4.12244e7i	−0.0457811	−	0.0792951i	−0.888008	+	0.459828i	$0.847911\pi$
$152$		0			0
$153$	−5.96938e8	−	4.26923e7i	−1.08934	−	0.0779083i
$154$		0			0
$155$		−	6.91288e8i		−	1.19766i
$156$		0			0
$157$	6.95522e7	+	1.20468e8i	0.114475	+	0.198277i	0.917570	−	0.397574i	$-0.130148\pi$
$157$	6.95522e7	+	1.20468e8i	0.114475	+	0.198277i	−0.803094	+	0.595852i	$0.796815\pi$
$158$		0			0
$159$	7.09129e7	+	3.08271e8i	0.110952	+	0.482330i
$160$		0			0
$161$	−4.62962e8	−	3.49731e7i	−0.689035	−	0.0520512i
$162$		0			0
$163$	1.49479e8	−	2.58906e8i	0.211754	−	0.366768i	−0.740510	−	0.672046i	$-0.765416\pi$
$163$	1.49479e8	−	2.58906e8i	0.211754	−	0.366768i	0.952263	+	0.305278i	$0.0987492\pi$
$164$		0			0
$165$	4.26243e8	−	1.39024e9i	0.575070	−	1.87566i
$166$		0			0
$167$		−	5.62931e8i		−	0.723751i	−0.932226	−	0.361876i	$-0.882136\pi$
$167$		−	5.62931e8i		−	0.723751i	0.932226	−	0.361876i	$-0.117864\pi$
$168$		0			0
$169$	2.02554e9			2.48310
$170$		0			0
$171$	−9.71374e8	+	4.71867e8i	−1.13606	+	0.551868i
$172$		0			0
$173$	−8.09208e8	−	4.67197e8i	−0.903391	−	0.521573i	−0.0250925	−	0.999685i	$-0.507988\pi$
$173$	−8.09208e8	−	4.67197e8i	−0.903391	−	0.521573i	−0.878299	+	0.478112i	$0.841321\pi$
$174$		0			0
$175$	−5.40703e8	−	3.69120e8i	−0.576510	−	0.393564i
$176$		0			0
$177$	−7.47886e7	−	3.25120e8i	−0.0761978	−	0.331246i
$178$		0			0
$179$	−1.74901e8	+	1.00979e8i	−0.170364	+	0.0983600i	−0.582758	−	0.812646i	$-0.698026\pi$
$179$	−1.74901e8	+	1.00979e8i	−0.170364	+	0.0983600i	0.412393	+	0.911006i	$0.364693\pi$
$180$		0			0
$181$	9.27966e8			0.864605			0.432302	−	0.901729i	$-0.357701\pi$
$181$	9.27966e8			0.864605			0.432302	+	0.901729i	$0.357701\pi$
$182$		0			0
$183$	−2.08924e8	−	2.24400e8i	−0.186288	−	0.200087i
$184$		0			0
$185$	8.29590e8	−	4.78964e8i	0.708234	−	0.408899i
$186$		0			0
$187$	1.00530e9	−	1.74123e9i	0.822110	−	1.42394i
$188$		0			0
$189$	5.48447e8	−	1.15211e9i	0.429821	−	0.902914i
$190$		0			0
$191$	4.74216e8	+	2.73789e8i	0.356322	+	0.205723i	0.667466	−	0.744640i	$-0.267379\pi$
$191$	4.74216e8	+	2.73789e8i	0.356322	+	0.205723i	−0.311144	+	0.950363i	$0.600712\pi$
$192$		0			0
$193$	6.65044e8	+	1.15189e9i	0.479315	+	0.830198i	0.999719	−	0.0237228i	$-0.00755190\pi$
$193$	6.65044e8	+	1.15189e9i	0.479315	+	0.830198i	−0.520404	+	0.853920i	$0.674219\pi$
$194$		0			0
$195$	−2.57361e9	+	2.39612e9i	−1.77994	+	1.65719i
$196$		0			0
$197$		−	2.83667e8i		−	0.188341i	−0.995556	−	0.0941703i	$-0.969980\pi$
$197$		−	2.83667e8i		−	0.188341i	0.995556	−	0.0941703i	$-0.0300198\pi$
$198$		0			0
$199$	2.14950e8	+	3.72304e8i	0.137064	+	0.237402i	0.926384	−	0.376580i	$-0.122900\pi$
$199$	2.14950e8	+	3.72304e8i	0.137064	+	0.237402i	−0.789320	+	0.613982i	$0.789567\pi$
$200$		0			0
$201$	−2.78127e9	+	6.39786e8i	−1.70396	+	0.391968i
$202$		0			0
$203$	9.51553e8	+	1.97895e9i	0.560337	+	1.16534i
$204$		0			0
$205$	4.87009e8	−	8.43524e8i	0.275754	−	0.477619i
$206$		0			0
$207$	−5.54354e8	−	1.14118e9i	−0.301930	−	0.621545i
$208$		0			0
$209$		−	3.62811e9i		−	1.90150i
$210$		0			0
$211$	−1.26984e9			−0.640650			−0.320325	−	0.947308i	$-0.603792\pi$
$211$	−1.26984e9			−0.640650			−0.320325	+	0.947308i	$0.603792\pi$
$212$		0			0
$213$	−1.66114e9	−	5.09300e8i	−0.807026	−	0.247432i
$214$		0			0
$215$	−8.33482e7	−	4.81211e7i	−0.0390070	−	0.0225207i
$216$		0			0
$217$	1.68316e9	+	1.14904e9i	0.759078	+	0.518197i
$218$		0			0
$219$	−1.17183e9	+	2.69560e8i	−0.509433	+	0.117187i
$220$		0			0
$221$	−4.21070e9	+	2.43105e9i	−1.76516	+	1.01912i
$222$		0			0
$223$	6.04371e8			0.244390			0.122195	−	0.992506i	$-0.461007\pi$
$223$	6.04371e8			0.244390			0.122195	+	0.992506i	$0.461007\pi$
$224$		0			0
$225$	1.27621e8	−	1.78444e9i	0.0497956	−	0.696259i
$226$		0			0
$227$	1.33407e9	−	7.70227e8i	0.502430	−	0.290078i	−0.227286	−	0.973828i	$-0.572985\pi$
$227$	1.33407e9	−	7.70227e8i	0.502430	−	0.290078i	0.729717	+	0.683750i	$0.239652\pi$
$228$		0			0
$229$	−1.36713e9	+	2.36794e9i	−0.497127	+	0.861050i	−0.999995	−	0.00331386i	$-0.998945\pi$
$229$	−1.36713e9	+	2.36794e9i	−0.497127	+	0.861050i	0.502867	+	0.864364i	$0.332278\pi$
$230$		0			0
$231$	2.67649e9	+	3.34863e9i	0.939977	+	1.17603i
$232$		0			0
$233$	4.04710e9	+	2.33659e9i	1.37316	+	0.792793i	0.991324	−	0.131438i	$-0.0419596\pi$
$233$	4.04710e9	+	2.33659e9i	1.37316	+	0.792793i	0.381833	+	0.924231i	$0.375293\pi$
$234$		0			0
$235$	−4.12028e8	−	7.13654e8i	−0.135100	−	0.234000i
$236$		0			0
$237$	3.87797e9	+	4.16522e9i	1.22917	+	1.32022i
$238$		0			0
$239$			3.25494e9i			0.997589i	0.866720	+	0.498794i	$0.166224\pi$
$239$			3.25494e9i			0.997589i	−0.866720	+	0.498794i	$0.833776\pi$
$240$		0			0
$241$	−2.90575e9	−	5.03291e9i	−0.861371	−	1.49194i	−0.870606	−	0.491981i	$-0.836273\pi$
$241$	−2.90575e9	−	5.03291e9i	−0.861371	−	1.49194i	0.00923496	−	0.999957i	$-0.497060\pi$
$242$		0			0
$243$	3.47469e9	−	2.90135e8i	0.996532	−	0.0832100i
$244$		0			0
$245$	4.37253e9	−	1.71005e9i	1.21358	−	0.474618i
$246$		0			0
$247$	−4.38680e9	+	7.59816e9i	−1.17858	+	2.04137i
$248$		0			0
$249$	5.48304e9	+	1.68108e9i	1.42634	+	0.437312i
$250$		0			0
$251$			2.84260e9i			0.716179i	0.933687	+	0.358089i	$0.116572\pi$
$251$			2.84260e9i			0.716179i	−0.933687	+	0.358089i	$0.883428\pi$
$252$		0			0
$253$	4.26234e9			1.04032
$254$		0			0
$255$	1.76386e9	−	5.75303e9i	0.417161	−	1.36062i
$256$		0			0
$257$	−4.59637e9	−	2.65372e9i	−1.05362	−	0.608306i	−0.129957	−	0.991520i	$-0.541484\pi$
$257$	−4.59637e9	−	2.65372e9i	−1.05362	−	0.608306i	−0.923660	+	0.383214i	$0.874817\pi$
$258$		0			0
$259$	−2.12728e8	+	2.81602e9i	−0.0472744	+	0.625802i
$260$		0			0
$261$	−3.36324e9	+	4.96922e9i	−0.724763	+	1.07084i
$262$		0			0
$263$	−3.73960e9	+	2.15906e9i	−0.781631	+	0.451275i	−0.837008	−	0.547191i	$-0.815697\pi$
$263$	−3.73960e9	+	2.15906e9i	−0.781631	+	0.451275i	0.0553769	+	0.998466i	$0.482364\pi$
$264$		0			0
$265$	−3.18052e9			−0.644934
$266$		0			0
$267$	−1.68460e8	+	1.56842e8i	−0.0331476	+	0.0308616i
$268$		0			0
$269$	4.65286e9	−	2.68633e9i	0.888610	−	0.513039i	0.0151223	−	0.999886i	$-0.495186\pi$
$269$	4.65286e9	−	2.68633e9i	0.888610	−	0.513039i	0.873488	+	0.486846i	$0.161853\pi$
$270$		0			0
$271$	−6.00046e7	+	1.03931e8i	−0.0111252	+	0.0192694i	−0.871534	−	0.490334i	$-0.836875\pi$
$271$	−6.00046e7	+	1.03931e8i	−0.0111252	+	0.0192694i	0.860409	+	0.509604i	$0.170208\pi$
$272$		0			0
$273$	−1.55635e9	−	1.02490e10i	−0.280193	−	1.84515i
$274$		0			0
$275$	5.20510e9	+	3.00516e9i	0.910119	+	0.525457i
$276$		0			0
$277$	9.72096e7	+	1.68372e8i	0.0165116	+	0.0285990i	0.874163	−	0.485632i	$-0.161411\pi$
$277$	9.72096e7	+	1.68372e8i	0.0165116	+	0.0285990i	−0.857652	+	0.514231i	$0.828077\pi$
$278$		0			0
$279$	−3.97272e8	+	5.55479e9i	−0.0655648	+	0.916749i
$280$		0			0
$281$		−	1.03596e10i		−	1.66156i	−0.556599	−	0.830781i	$-0.687894\pi$
$281$		−	1.03596e10i		−	1.66156i	0.556599	−	0.830781i	$-0.312106\pi$
$282$		0			0
$283$	1.76351e8	+	3.05449e8i	0.0274937	+	0.0476204i	0.879445	−	0.476001i	$-0.157914\pi$
$283$	1.76351e8	+	3.05449e8i	0.0274937	+	0.0476204i	−0.851951	+	0.523621i	$0.824581\pi$
$284$		0			0
$285$	−2.43420e9	−	1.05819e10i	−0.368957	−	1.60392i
$286$		0			0
$287$	1.24434e9	+	2.58786e9i	0.183405	+	0.381428i
$288$		0			0
$289$	6.72223e8	−	1.16432e9i	0.0963656	−	0.166910i
$290$		0			0
$291$	−1.42589e9	+	4.65069e9i	−0.198845	+	0.648554i
$292$		0			0
$293$			1.22776e10i			1.66588i	0.553361	+	0.832942i	$0.313345\pi$
$293$			1.22776e10i			1.66588i	−0.553361	+	0.832942i	$0.686655\pi$
$294$		0			0
$295$	3.35435e9			0.442915
$296$		0			0
$297$	−4.22398e9	+	1.09262e10i	−0.542871	+	1.40424i
$298$		0			0
$299$	−8.92639e9	−	5.15365e9i	−1.11684	−	0.644808i
$300$		0			0
$301$	2.55705e8	−	1.22952e8i	0.0311511	−	0.0149786i
$302$		0			0
$303$	−1.91388e9	−	8.32001e9i	−0.227062	−	0.987082i
$304$		0			0
$305$	2.66977e9	−	1.54139e9i	0.308514	−	0.178121i
$306$		0			0
$307$	−1.30877e10			−1.47336			−0.736682	−	0.676239i	$-0.763609\pi$
$307$	−1.30877e10			−1.47336			−0.736682	+	0.676239i	$0.763609\pi$
$308$		0			0
$309$	8.91383e9	+	9.57411e9i	0.977756	+	1.05018i
$310$		0			0
$311$	−1.49702e10	+	8.64304e9i	−1.60024	+	0.923900i	−0.608805	+	0.793320i	$0.708351\pi$
$311$	−1.49702e10	+	8.64304e9i	−1.60024	+	0.923900i	−0.991438	+	0.130580i	$0.958316\pi$
$312$		0			0
$313$	−1.91037e9	+	3.30885e9i	−0.199039	+	0.344746i	−0.948217	−	0.317623i	$-0.897116\pi$
$313$	−1.91037e9	+	3.30885e9i	−0.199039	+	0.344746i	0.749178	+	0.662369i	$0.230449\pi$
$314$		0			0
$315$	1.00530e10	+	7.97102e9i	1.02107	+	0.809603i
$316$		0			0
$317$	9.62627e9	+	5.55773e9i	0.953281	+	0.550377i	0.894099	−	0.447870i	$-0.147817\pi$
$317$	9.62627e9	+	5.55773e9i	0.953281	+	0.550377i	0.0591823	+	0.998247i	$0.481151\pi$
$318$		0			0
$319$	−1.00795e10	−	1.74582e10i	−0.973363	−	1.68591i
$320$		0			0
$321$	1.39140e10	−	1.29544e10i	1.31048	−	1.22011i
$322$		0			0
$323$		−	1.50137e10i		−	1.37936i
$324$		0			0
$325$	−7.26717e9	−	1.25871e10i	−0.651376	−	1.12822i
$326$		0			0
$327$	1.00253e10	−	2.30616e9i	0.876812	−	0.201697i
$328$		0			0
$329$	2.42248e9	+	1.82999e8i	0.206765	+	0.0156194i
$330$		0			0
$331$	−1.04847e9	+	1.81600e9i	−0.0873460	+	0.151288i	−0.906388	−	0.422445i	$-0.861172\pi$
$331$	−1.04847e9	+	1.81600e9i	−0.0873460	+	0.151288i	0.819042	+	0.573733i	$0.194505\pi$
$332$		0			0
$333$	−6.94136e9	+	3.37193e9i	−0.564505	+	0.274221i
$334$		0			0
$335$		−	2.86951e10i		−	2.27839i
$336$		0			0
$337$	1.23657e10			0.958735			0.479367	−	0.877614i	$-0.340866\pi$
$337$	1.23657e10			0.958735			0.479367	+	0.877614i	$0.340866\pi$
$338$		0			0
$339$	−1.73675e10	−	5.32483e9i	−1.31504	−	0.403187i
$340$		0			0
$341$	−1.62030e10	−	9.35480e9i	−1.19833	−	0.691858i
$342$		0			0
$343$	−3.10423e9	+	1.34887e10i	−0.224273	+	0.974526i
$344$		0			0
$345$	1.24317e10	−	2.85971e9i	0.877514	−	0.201858i
$346$		0			0
$347$	2.26748e9	−	1.30913e9i	0.156396	−	0.0902951i	−0.419760	−	0.907635i	$-0.637886\pi$
$347$	2.26748e9	−	1.30913e9i	0.156396	−	0.0902951i	0.576155	+	0.817340i	$0.304552\pi$
$348$		0			0
$349$	−1.89034e10			−1.27420			−0.637100	−	0.770781i	$-0.719866\pi$
$349$	−1.89034e10			−1.27420			−0.637100	+	0.770781i	$0.719866\pi$
$350$		0			0
$351$	2.20571e10	−	1.77749e10i	1.45318	−	1.17106i
$352$		0			0
$353$	−1.19714e10	+	6.91169e9i	−0.770985	+	0.445128i	−0.833226	−	0.552933i	$-0.813509\pi$
$353$	−1.19714e10	+	6.91169e9i	−0.770985	+	0.445128i	0.0622408	+	0.998061i	$0.480175\pi$
$354$		0			0
$355$	8.73482e9	−	1.51291e10i	0.549972	−	0.952579i
$356$		0			0
$357$	1.10757e10	+	1.38572e10i	0.681868	+	0.853105i
$358$		0			0
$359$	1.42902e10	+	8.25044e9i	0.860320	+	0.496706i	0.864119	−	0.503287i	$-0.167876\pi$
$359$	1.42902e10	+	8.25044e9i	0.860320	+	0.496706i	−0.00379931	+	0.999993i	$0.501209\pi$
$360$		0			0
$361$	−5.05427e9	−	8.75425e9i	−0.297598	−	0.515454i
$362$		0			0
$363$	−1.49859e10	−	1.60960e10i	−0.863093	−	0.927025i
$364$		0			0
$365$		−	1.20901e10i		−	0.681174i
$366$		0			0
$367$	1.39115e9	+	2.40955e9i	0.0766850	+	0.132822i	0.901818	−	0.432117i	$-0.142233\pi$
$367$	1.39115e9	+	2.40955e9i	0.0766850	+	0.132822i	−0.825133	+	0.564939i	$0.808900\pi$
$368$		0			0
$369$	−4.39808e9	+	6.49819e9i	−0.237223	+	0.350499i
$370$		0			0
$371$	5.28657e9	−	7.74400e9i	0.279047	−	0.408761i
$372$		0			0
$373$	−2.57377e9	+	4.45790e9i	−0.132964	+	0.230301i	−0.924818	−	0.380410i	$-0.875783\pi$
$373$	−2.57377e9	+	4.45790e9i	−0.132964	+	0.230301i	0.791854	+	0.610711i	$0.209116\pi$
$374$		0			0
$375$	−7.43948e9	−	2.28092e9i	−0.376199	−	0.115341i
$376$		0			0
$377$			4.87489e10i			2.41323i
$378$		0			0
$379$	8.70105e9			0.421711			0.210855	−	0.977517i	$-0.432375\pi$
$379$	8.70105e9			0.421711			0.210855	+	0.977517i	$0.432375\pi$
$380$		0			0
$381$	−1.24329e9	+	4.05513e9i	−0.0590028	+	0.192444i
$382$		0			0
$383$	2.69528e10	+	1.55612e10i	1.25259	+	0.723183i	0.971623	−	0.236534i	$-0.0760115\pi$
$383$	2.69528e10	+	1.55612e10i	1.25259	+	0.723183i	0.280967	+	0.959717i	$0.409345\pi$
$384$		0			0
$385$	−3.88454e10	+	1.86783e10i	−1.76806	+	0.850149i
$386$		0			0
$387$	6.42083e8	+	4.34572e8i	0.0286251	+	0.0193739i
$388$		0			0
$389$	5.62573e9	−	3.24802e9i	0.245686	−	0.141847i	−0.372101	−	0.928192i	$-0.621363\pi$
$389$	5.62573e9	−	3.24802e9i	0.245686	−	0.141847i	0.617787	+	0.786345i	$0.288029\pi$
$390$		0			0
$391$	1.76383e10			0.754655
$392$		0			0
$393$	2.80968e10	−	2.61591e10i	1.17784	−	1.09661i
$394$		0			0
$395$	−4.95552e10	+	2.86107e10i	−2.03564	+	1.17528i
$396$		0			0
$397$	1.02791e10	−	1.78040e10i	0.413803	−	0.716728i	−0.581499	−	0.813547i	$-0.697534\pi$
$397$	1.02791e10	−	1.78040e10i	0.413803	−	0.716728i	0.995302	+	0.0968192i	$0.0308669\pi$
$398$		0			0
$399$	2.98110e10	+	1.16621e10i	1.17621	+	0.460133i
$400$		0			0
$401$	−1.97066e10	−	1.13776e10i	−0.762140	−	0.440022i	0.0679234	−	0.997691i	$-0.478363\pi$
$401$	−1.97066e10	−	1.13776e10i	−0.762140	−	0.440022i	−0.830064	+	0.557669i	$0.811696\pi$
$402$		0			0
$403$	2.26220e10	+	3.91825e10i	0.857653	+	1.48550i
$404$		0			0
$405$	−4.98917e9	+	3.47017e10i	−0.185442	+	1.28983i
$406$		0			0
$407$		−	2.59262e10i		−	0.944846i
$408$		0			0
$409$	−1.13163e10	−	1.96004e10i	−0.404399	−	0.700440i	0.589852	−	0.807511i	$-0.299186\pi$
$409$	−1.13163e10	−	1.96004e10i	−0.404399	−	0.700440i	−0.994251	+	0.107071i	$0.965853\pi$
$410$		0			0
$411$	7.22787e8	+	3.14208e9i	0.0253304	+	0.110116i
$412$		0			0
$413$	−5.57550e9	+	8.16724e9i	−0.191639	+	0.280721i
$414$		0			0
$415$	−2.88316e10	+	4.99378e10i	−0.972023	+	1.68359i
$416$		0			0
$417$	8.58069e9	−	2.79869e10i	0.283777	−	0.925572i
$418$		0			0
$419$			9.16215e9i			0.297263i	0.988893	+	0.148632i	$0.0474869\pi$
$419$			9.16215e9i			0.297263i	−0.988893	+	0.148632i	$0.952513\pi$
$420$		0			0
$421$	−1.95687e10			−0.622921			−0.311461	−	0.950259i	$-0.600818\pi$
$421$	−1.95687e10			−0.622921			−0.311461	+	0.950259i	$0.600818\pi$
$422$		0			0
$423$	2.90069e9	+	5.97129e9i	0.0906026	+	0.186512i
$424$		0			0
$425$	2.15395e10	+	1.24359e10i	0.660208	+	0.381171i
$426$		0			0
$427$	−6.84598e8	+	9.06247e9i	−0.0205932	+	0.272606i
$428$		0			0
$429$	2.13352e10	+	9.27479e10i	0.629894	+	2.73826i
$430$		0			0
$431$	2.34384e10	−	1.35322e10i	0.679232	−	0.392155i	−0.120333	−	0.992734i	$-0.538396\pi$
$431$	2.34384e10	−	1.35322e10i	0.679232	−	0.392155i	0.799566	+	0.600579i	$0.205063\pi$
$432$		0			0
$433$	−3.31493e10			−0.943025			−0.471512	−	0.881859i	$-0.656292\pi$
$433$	−3.31493e10			−0.943025			−0.471512	+	0.881859i	$0.656292\pi$
$434$		0			0
$435$	−4.11113e10	−	4.41565e10i	−1.14816	−	1.23321i
$436$		0			0
$437$	2.75639e10	−	1.59140e10i	0.755814	−	0.436369i
$438$		0			0
$439$	1.04594e10	−	1.81163e10i	0.281611	−	0.487765i	−0.690171	−	0.723647i	$-0.742465\pi$
$439$	1.04594e10	−	1.81163e10i	0.281611	−	0.487765i	0.971782	+	0.235882i	$0.0757979\pi$
$440$		0			0
$441$	−3.61178e10	+	1.12281e10i	−0.954921	+	0.296861i
$442$		0			0
$443$	−3.25483e9	−	1.87918e9i	−0.0845110	−	0.0487924i	0.457149	−	0.889390i	$-0.348871\pi$
$443$	−3.25483e9	−	1.87918e9i	−0.0845110	−	0.0487924i	−0.541660	+	0.840598i	$0.682204\pi$
$444$		0			0
$445$	−1.15714e9	−	2.00423e9i	−0.0295085	−	0.0511102i
$446$		0			0
$447$	1.01873e10	−	9.48469e9i	0.255168	−	0.237571i
$448$		0			0
$449$		−	3.68428e10i		−	0.906500i	−0.891383	−	0.453250i	$-0.850264\pi$
$449$		−	3.68428e10i		−	0.906500i	0.891383	−	0.453250i	$-0.149736\pi$
$450$		0			0
$451$	−1.31808e10	−	2.28298e10i	−0.318593	−	0.551819i
$452$		0			0
$453$	−3.75761e9	+	8.64379e8i	−0.0892317	+	0.0205263i
$454$		0			0
$455$	1.03936e11	+	7.85156e9i	2.42505	+	0.183194i
$456$		0			0
$457$	3.29317e9	−	5.70394e9i	0.0755004	−	0.130771i	−0.825803	−	0.563958i	$-0.809278\pi$
$457$	3.29317e9	−	5.70394e9i	0.0755004	−	0.130771i	0.901304	+	0.433188i	$0.142611\pi$
$458$		0			0
$459$	−1.74795e10	+	4.52143e10i	−0.393803	+	1.01865i
$460$		0			0
$461$			2.81459e10i			0.623176i	0.950217	+	0.311588i	$0.100861\pi$
$461$			2.81459e10i			0.623176i	−0.950217	+	0.311588i	$0.899139\pi$
$462$		0			0
$463$	6.32873e10			1.37719			0.688593	−	0.725148i	$-0.258229\pi$
$463$	6.32873e10			1.37719			0.688593	+	0.725148i	$0.258229\pi$
$464$		0			0
$465$	−5.35346e10	−	1.64136e10i	−1.14505	−	0.351068i
$466$		0			0
$467$	4.75325e10	+	2.74429e10i	0.999362	+	0.576982i	0.908059	−	0.418841i	$-0.137564\pi$
$467$	4.75325e10	+	2.74429e10i	0.999362	+	0.576982i	0.0913026	+	0.995823i	$0.470897\pi$
$468$		0			0
$469$	6.98674e10	+	4.76961e10i	1.44405	+	0.985806i
$470$		0			0
$471$	1.09807e10	−	2.52593e9i	0.223124	−	0.0513261i
$472$		0			0
$473$	−2.25581e9	+	1.30239e9i	−0.0450669	+	0.0260194i
$474$		0			0
$475$	4.48808e10			0.881629
$476$		0			0
$477$	2.55568e10	+	1.82779e9i	0.493666	+	0.0353064i
$478$		0			0
$479$	−1.93737e10	+	1.11854e10i	−0.368018	+	0.212476i	−0.672592	−	0.740013i	$-0.734819\pi$
$479$	−1.93737e10	+	1.11854e10i	−0.368018	+	0.212476i	0.304574	+	0.952489i	$0.401486\pi$
$480$		0			0
$481$	−3.13477e10	+	5.42958e10i	−0.585633	+	1.01435i
$482$		0			0
$483$	−1.37007e10	+	3.50223e10i	−0.251741	+	0.643510i
$484$		0			0
$485$	−4.23571e10	−	2.44549e10i	−0.765525	−	0.441976i
$486$		0			0
$487$	−1.53548e10	−	2.65953e10i	−0.272979	−	0.472813i	0.696645	−	0.717416i	$-0.254675\pi$
$487$	−1.53548e10	−	2.65953e10i	−0.272979	−	0.472813i	−0.969623	+	0.244604i	$0.921342\pi$
$488$		0			0
$489$	−1.65010e10	−	1.77233e10i	−0.288586	−	0.309962i
$490$		0			0
$491$		−	5.04115e10i		−	0.867369i	−0.901065	−	0.433685i	$-0.857213\pi$
$491$		−	5.04115e10i		−	0.867369i	0.901065	−	0.433685i	$-0.142787\pi$
$492$		0			0
$493$	−4.17105e10	−	7.22447e10i	−0.706086	−	1.22298i
$494$		0			0
$495$	−9.75422e10	−	6.60181e10i	−1.62469	−	1.09962i
$496$		0			0
$497$	2.23180e10	+	4.64149e10i	0.365788	+	0.760732i
$498$		0			0
$499$	3.31587e10	−	5.74326e10i	0.534806	−	0.926310i	−0.464367	−	0.885643i	$-0.653718\pi$
$499$	3.31587e10	−	5.74326e10i	0.534806	−	0.926310i	0.999173	−	0.0406676i	$-0.0129485\pi$
$500$		0			0
$501$	−4.35945e10	−	1.33659e10i	−0.691959	−	0.212153i
$502$		0			0
$503$		−	6.40152e10i		−	1.00003i	−0.866018	−	0.500013i	$-0.833329\pi$
$503$		−	6.40152e10i		−	1.00003i	0.866018	−	0.500013i	$-0.166671\pi$
$504$		0			0
$505$	8.58399e10			1.31985
$506$		0			0
$507$	4.80934e10	−	1.56862e11i	0.727870	−	2.37403i
$508$		0			0
$509$	−9.43891e10	−	5.44956e10i	−1.40621	−	0.811877i	−0.411192	−	0.911549i	$-0.634887\pi$
$509$	−9.43891e10	−	5.44956e10i	−1.40621	−	0.811877i	−0.995020	+	0.0996719i	$0.968221\pi$
$510$		0			0
$511$	2.94372e10	+	2.00958e10i	0.431730	+	0.294728i
$512$		0			0
$513$	1.34785e10	+	8.64288e10i	0.194614	+	1.24793i
$514$		0			0
$515$	−1.13907e11	+	6.57641e10i	−1.61928	+	0.934889i
$516$		0			0
$517$	−2.23030e10			−0.312177
$518$		0			0
$519$	−5.53940e10	+	5.15738e10i	−0.763472	+	0.710820i
$520$		0			0
$521$	−8.45887e10	+	4.88373e10i	−1.14805	+	0.662828i	−0.948412	−	0.317041i	$-0.897311\pi$
$521$	−8.45887e10	+	4.88373e10i	−1.14805	+	0.662828i	−0.199640	+	0.979869i	$0.563977\pi$
$522$		0			0
$523$	2.58970e10	−	4.48550e10i	0.346133	−	0.599520i	−0.639426	−	0.768853i	$-0.720828\pi$
$523$	2.58970e10	−	4.48550e10i	0.346133	−	0.599520i	0.985559	+	0.169333i	$0.0541612\pi$
$524$		0			0
$525$	−4.14235e10	+	3.31089e10i	−0.545268	+	0.435821i
$526$		0			0
$527$	−6.70506e10	−	3.87117e10i	−0.869282	−	0.501880i
$528$		0			0
$529$	−2.04596e10	−	3.54370e10i	−0.261260	−	0.452516i
$530$		0			0
$531$	−2.69536e10	−	1.92769e9i	−0.339031	−	0.0242471i
$532$		0			0
$533$			6.37484e10i			0.789879i
$534$		0			0
$535$	9.55747e10	+	1.65540e11i	1.16662	+	2.02064i
$536$		0			0
$537$	3.66725e9	+	1.59422e10i	0.0441005	+	0.191713i
$538$		0			0
$539$	1.90894e10	−	1.25628e11i	0.226171	−	1.48844i
$540$		0			0
$541$	2.27103e10	−	3.93354e10i	0.265115	−	0.459193i	−0.702479	−	0.711705i	$-0.747923\pi$
$541$	2.27103e10	−	3.93354e10i	0.265115	−	0.459193i	0.967594	+	0.252512i	$0.0812567\pi$
$542$		0			0
$543$	2.20331e10	−	7.18634e10i	0.253441	−	0.826625i
$544$		0			0
$545$			1.03434e11i			1.17240i
$546$		0			0
$547$	1.69431e11			1.89253			0.946266	−	0.323390i	$-0.104822\pi$
$547$	1.69431e11			1.89253			0.946266	+	0.323390i	$0.104822\pi$
$548$		0			0
$549$	−2.23386e10	+	1.08515e10i	−0.245904	+	0.119454i
$550$		0			0
$551$	−1.30365e11	−	7.52662e10i	−1.41434	−	0.816570i
$552$		0			0
$553$	1.27072e10	−	1.68214e11i	0.135878	−	1.79871i
$554$		0			0
$555$	−1.73946e10	−	7.56173e10i	−0.183333	−	0.796983i
$556$		0			0
$557$	1.06588e11	−	6.15388e10i	1.10736	−	0.639334i	0.169215	−	0.985579i	$-0.445877\pi$
$557$	1.06588e11	−	6.15388e10i	1.10736	−	0.639334i	0.938144	+	0.346245i	$0.112543\pi$
$558$		0			0
$559$	6.29895e9			0.0645091
$560$		0			0
$561$	−1.10975e11	−	1.19195e11i	−1.12040	−	1.20339i
$562$		0			0
$563$	−5.49202e10	+	3.17082e10i	−0.546636	+	0.315600i	−0.747764	−	0.663965i	$-0.768873\pi$
$563$	−5.49202e10	+	3.17082e10i	−0.546636	+	0.315600i	0.201128	+	0.979565i	$0.435539\pi$
$564$		0			0
$565$	9.13241e10	−	1.58178e11i	0.896172	−	1.55222i
$566$		0			0
$567$	−7.61995e10	−	6.98278e10i	−0.737259	−	0.675610i
$568$		0			0
$569$	1.75800e11	+	1.01498e11i	1.67715	+	0.968301i	0.963467	+	0.267829i	$0.0863062\pi$
$569$	1.75800e11	+	1.01498e11i	1.67715	+	0.968301i	0.713680	+	0.700472i	$0.247027\pi$
$570$		0			0
$571$	7.75044e9	+	1.34242e10i	0.0729091	+	0.126282i	0.900175	−	0.435528i	$-0.143438\pi$
$571$	7.75044e9	+	1.34242e10i	0.0729091	+	0.126282i	−0.827266	+	0.561810i	$0.810105\pi$
$572$		0			0
$573$	3.24623e10	−	3.02235e10i	0.301135	−	0.280367i
$574$		0			0
$575$			5.27263e10i			0.482343i
$576$		0			0
$577$	6.03987e10	+	1.04614e11i	0.544910	+	0.943811i	0.998613	+	0.0526582i	$0.0167694\pi$
$577$	6.03987e10	+	1.04614e11i	0.544910	+	0.943811i	−0.453703	+	0.891153i	$0.649897\pi$
$578$		0			0
$579$	1.04995e11	−	2.41524e10i	0.934230	−	0.214905i
$580$		0			0
$581$	−7.36665e10	−	1.53205e11i	−0.646496	−	1.34452i
$582$		0			0
$583$	−4.30402e10	+	7.45478e10i	−0.372563	+	0.645298i
$584$		0			0
$585$	1.24454e11	+	2.56198e11i	1.06264	+	2.18752i
$586$		0			0
$587$			7.44864e10i			0.627371i	0.949527	+	0.313686i	$0.101564\pi$
$587$			7.44864e10i			0.627371i	−0.949527	+	0.313686i	$0.898436\pi$
$588$		0			0
$589$	−1.39710e11			−1.16082
$590$		0			0
$591$	−2.19677e10	−	6.73523e9i	−0.180067	−	0.0552081i
$592$		0			0
$593$	9.28707e10	+	5.36189e10i	0.751034	+	0.433610i	0.826068	−	0.563571i	$-0.190573\pi$
$593$	9.28707e10	+	5.36189e10i	0.751034	+	0.433610i	−0.0750331	+	0.997181i	$0.523906\pi$
$594$		0			0
$595$	−1.60749e11	+	7.72939e10i	−1.28257	+	0.616705i
$596$		0			0
$597$	3.39356e10	−	7.80634e9i	0.267152	−	0.0614540i
$598$		0			0
$599$	3.47649e10	−	2.00715e10i	0.270043	−	0.155910i	−0.358864	−	0.933390i	$-0.616836\pi$
$599$	3.47649e10	−	2.00715e10i	0.270043	−	0.155910i	0.628907	+	0.777480i	$0.283503\pi$
$600$		0			0
$601$	2.65618e10			0.203592			0.101796	−	0.994805i	$-0.467541\pi$
$601$	2.65618e10			0.203592			0.101796	+	0.994805i	$0.467541\pi$
$602$		0			0
$603$	−1.64906e10	+	2.30577e11i	−0.124729	+	1.74400i
$604$		0			0
$605$	1.91500e11	−	1.10563e11i	1.42938	−	0.825253i
$606$		0			0
$607$	−1.21213e11	+	2.09946e11i	−0.892879	+	1.54651i	−0.0564714	+	0.998404i	$0.517985\pi$
$607$	−1.21213e11	+	2.09946e11i	−0.892879	+	1.54651i	−0.836408	+	0.548108i	$0.815348\pi$
$608$		0			0
$609$	1.75847e11	−	2.67029e10i	1.27840	−	0.194129i
$610$		0			0
$611$	4.67079e10	+	2.69668e10i	0.335140	+	0.193493i
$612$		0			0
$613$	−9.13060e10	−	1.58147e11i	−0.646632	−	1.12000i	−0.983922	−	0.178599i	$-0.942843\pi$
$613$	−9.13060e10	−	1.58147e11i	−0.646632	−	1.12000i	0.337290	−	0.941401i	$-0.390490\pi$
$614$		0			0
$615$	−5.37608e10	−	5.77430e10i	−0.375807	−	0.403645i
$616$		0			0
$617$			7.57874e10i			0.522945i	0.965211	+	0.261473i	$0.0842081\pi$
$617$			7.57874e10i			0.522945i	−0.965211	+	0.261473i	$0.915792\pi$
$618$		0			0
$619$	−3.05655e10	−	5.29410e10i	−0.208194	−	0.360603i	0.742952	−	0.669345i	$-0.233425\pi$
$619$	−3.05655e10	−	5.29410e10i	−0.208194	−	0.360603i	−0.951146	+	0.308742i	$0.900092\pi$
$620$		0			0
$621$	−1.01537e11	+	1.58347e10i	−0.682746	+	0.106474i
$622$		0			0
$623$	6.80331e9	+	5.13937e8i	0.0451615	+	0.00341160i
$624$		0			0
$625$	9.23753e10	−	1.59999e11i	0.605391	−	1.04857i
$626$		0			0
$627$	−2.80968e11	−	8.61439e10i	−1.81797	−	0.557384i
$628$		0			0
$629$		−	1.07287e11i		−	0.685399i
$630$		0			0
$631$	1.49762e11			0.944676			0.472338	−	0.881417i	$-0.343410\pi$
$631$	1.49762e11			0.944676			0.472338	+	0.881417i	$0.343410\pi$
$632$		0			0
$633$	−3.01505e10	+	9.83392e10i	−0.187793	+	0.612508i
$634$		0			0
$635$	−3.69329e10	−	2.13232e10i	−0.227153	−	0.131147i
$636$		0			0
$637$	−1.91877e11	+	2.40015e11i	−1.16537	+	1.45774i
$638$		0			0
$639$	−7.88824e10	+	1.16549e11i	−0.473126	+	0.699047i
$640$		0			0
$641$	−1.46292e10	+	8.44619e9i	−0.0866542	+	0.0500298i	−0.542701	−	0.839926i	$-0.682598\pi$
$641$	−1.46292e10	+	8.44619e9i	−0.0866542	+	0.0500298i	0.456047	+	0.889956i	$0.349265\pi$
$642$		0			0
$643$	−9.99179e10			−0.584520			−0.292260	−	0.956339i	$-0.594407\pi$
$643$	−9.99179e10			−0.584520			−0.292260	+	0.956339i	$0.594407\pi$
$644$		0			0
$645$	−5.70556e9	+	5.31208e9i	−0.0329655	+	0.0306921i
$646$		0			0
$647$	1.85800e11	−	1.07272e11i	1.06030	−	0.612165i	0.134786	−	0.990875i	$-0.456965\pi$
$647$	1.85800e11	−	1.07272e11i	1.06030	−	0.612165i	0.925516	+	0.378710i	$0.123632\pi$
$648$		0			0
$649$	4.53925e10	−	7.86222e10i	0.255862	−	0.443166i
$650$		0			0
$651$	1.28948e11	−	1.03065e11i	0.717942	−	0.573835i
$652$		0			0
$653$	−2.93043e10	−	1.69189e10i	−0.161168	−	0.0930504i	0.417247	−	0.908793i	$-0.362995\pi$
$653$	−2.93043e10	−	1.69189e10i	−0.161168	−	0.0930504i	−0.578415	+	0.815743i	$0.696328\pi$
$654$		0			0
$655$	1.92996e11	+	3.34279e11i	1.04853	+	1.81611i
$656$		0			0
$657$	−6.94797e9	+	9.71489e10i	−0.0372904	+	0.521406i
$658$		0			0
$659$			3.40336e11i			1.80454i	0.431174	+	0.902269i	$0.358099\pi$
$659$			3.40336e11i			1.80454i	−0.431174	+	0.902269i	$0.641901\pi$
$660$		0			0
$661$	1.64319e11	+	2.84610e11i	0.860762	+	1.49088i	0.871194	+	0.490938i	$0.163346\pi$
$661$	1.64319e11	+	2.84610e11i	0.860762	+	1.49088i	−0.0104324	+	0.999946i	$0.503321\pi$
$662$		0			0
$663$	8.82885e10	+	3.83806e11i	0.456930	+	1.98636i
$664$		0			0
$665$	−1.81470e11	+	2.65825e11i	−0.927934	+	1.35928i
$666$		0			0
$667$	8.84233e10	−	1.53154e11i	0.446749	−	0.773792i
$668$		0			0
$669$	1.43499e10	−	4.68036e10i	0.0716379	−	0.233655i
$670$		0			0
$671$		−	8.34352e10i		−	0.411585i
$672$		0			0
$673$	−2.84489e11			−1.38677			−0.693387	−	0.720565i	$-0.743882\pi$
$673$	−2.84489e11			−1.38677			−0.693387	+	0.720565i	$0.743882\pi$
$674$		0			0
$675$	−1.35160e11	−	5.22519e10i	−0.651078	−	0.251702i
$676$		0			0
$677$	−1.08407e9	−	6.25890e8i	−0.00516065	−	0.00297950i	0.497417	−	0.867511i	$-0.334282\pi$
$677$	−1.08407e9	−	6.25890e8i	−0.00516065	−	0.00297950i	−0.502578	+	0.864532i	$0.667615\pi$
$678$		0			0
$679$	1.29948e11	−	6.24837e10i	0.611350	−	0.293960i
$680$		0			0
$681$	−2.79724e10	−	1.21601e11i	−0.130059	−	0.565390i
$682$		0			0
$683$	−8.87260e10	+	5.12260e10i	−0.407726	+	0.235401i	−0.689812	−	0.723988i	$-0.742307\pi$
$683$	−8.87260e10	+	5.12260e10i	−0.407726	+	0.235401i	0.282086	+	0.959389i	$0.408974\pi$
$684$		0			0
$685$	−3.24178e10			−0.147238
$686$		0			0
$687$	1.50917e11	+	1.62096e11i	0.677504	+	0.727689i
$688$		0			0
$689$	1.80274e11	−	1.04081e11i	0.799936	−	0.461843i
$690$		0			0
$691$	−8.13000e10	+	1.40816e11i	−0.356597	+	0.617645i	−0.987390	−	0.158307i	$-0.949397\pi$
$691$	−8.13000e10	+	1.40816e11i	−0.356597	+	0.617645i	0.630793	+	0.775951i	$0.282730\pi$
$692$		0			0
$693$	3.22874e11	−	1.27764e11i	1.39991	−	0.553958i
$694$		0			0
$695$	2.54896e11	+	1.47164e11i	1.09250	+	0.630758i
$696$		0			0
$697$	−5.45444e10	−	9.44736e10i	−0.231110	−	0.400294i
$698$		0			0
$699$	2.77042e11	−	2.57936e11i	1.16048	−	1.08045i
$700$		0			0
$701$		−	3.23306e11i		−	1.33888i	−0.742867	−	0.669439i	$-0.766534\pi$
$701$		−	3.23306e11i		−	1.33888i	0.742867	−	0.669439i	$-0.233466\pi$
$702$		0			0
$703$	−9.67989e10	−	1.67661e11i	−0.396323	−	0.686452i
$704$		0			0
$705$	−6.50497e10	+	1.49636e10i	−0.263323	+	0.0605733i
$706$		0			0
$707$	−1.42680e11	+	2.09004e11i	−0.571066	+	0.836523i
$708$		0			0
$709$	−2.66527e10	+	4.61638e10i	−0.105477	+	0.182691i	−0.913933	−	0.405866i	$-0.866970\pi$
$709$	−2.66527e10	+	4.61638e10i	−0.105477	+	0.182691i	0.808456	+	0.588556i	$0.200303\pi$
$710$		0			0
$711$	4.14639e11	−	2.01421e11i	1.62253	−	0.788180i
$712$		0			0
$713$		−	1.64132e11i		−	0.635091i
$714$		0			0
$715$	−9.56907e11			−3.66138
$716$		0			0
$717$	2.52069e11	+	7.72835e10i	0.953768	+	0.292422i
$718$		0			0
$719$	2.14532e11	+	1.23860e11i	0.802744	+	0.463464i	0.844430	−	0.535666i	$-0.179940\pi$
$719$	2.14532e11	+	1.23860e11i	0.802744	+	0.463464i	−0.0416858	+	0.999131i	$0.513273\pi$
$720$		0			0
$721$	2.92086e10	−	3.86653e11i	0.108086	−	1.43081i
$722$		0			0
$723$	−4.58751e11	+	1.05528e11i	−1.67889	+	0.386203i
$724$		0			0
$725$	2.15962e11	−	1.24686e11i	0.781674	−	0.451300i
$726$		0			0
$727$	−2.33638e11			−0.836383			−0.418192	−	0.908359i	$-0.637336\pi$
$727$	−2.33638e11			−0.836383			−0.418192	+	0.908359i	$0.637336\pi$
$728$		0			0
$729$	6.00326e10	−	2.75976e11i	0.212558	−	0.977149i
$730$		0			0
$731$	−9.33490e9	+	5.38951e9i	−0.0326919	+	0.0188747i
$732$		0			0
$733$	−4.59222e10	+	7.95396e10i	−0.159077	+	0.275529i	−0.934536	−	0.355869i	$-0.884185\pi$
$733$	−4.59222e10	+	7.95396e10i	−0.159077	+	0.275529i	0.775459	+	0.631398i	$0.217518\pi$
$734$		0			0
$735$	−2.86104e10	−	3.79219e11i	−0.0980335	−	1.29939i
$736$		0			0
$737$	−6.72580e11	−	3.88314e11i	−2.27968	−	1.31617i
$738$		0			0
$739$	3.88598e10	+	6.73072e10i	0.130294	+	0.225675i	0.923790	−	0.382900i	$-0.125075\pi$
$739$	3.88598e10	+	6.73072e10i	0.130294	+	0.225675i	−0.793496	+	0.608575i	$0.791741\pi$
$740$		0			0
$741$	4.84258e11	+	5.20129e11i	1.60622	+	1.72519i
$742$		0			0
$743$			7.78337e10i			0.255395i	0.991813	+	0.127698i	$0.0407587\pi$
$743$			7.78337e10i			0.255395i	−0.991813	+	0.127698i	$0.959241\pi$
$744$		0			0
$745$	6.99758e10	+	1.21202e11i	0.227155	+	0.393444i
$746$		0			0
$747$	2.60373e11	−	3.84702e11i	0.836205	−	1.23550i
$748$		0			0
$749$	−5.61921e11	−	4.24488e10i	−1.78545	−	0.134877i
$750$		0			0
$751$	2.11799e11	−	3.66846e11i	0.665831	−	1.15325i	−0.313228	−	0.949678i	$-0.601411\pi$
$751$	2.11799e11	−	3.66846e11i	0.665831	−	1.15325i	0.979059	−	0.203575i	$-0.0652561\pi$
$752$		0			0
$753$	2.20137e11	+	6.74932e10i	0.684719	+	0.209933i
$754$		0			0
$755$		−	3.87683e10i		−	0.119313i
$756$		0			0
$757$	5.14290e11			1.56612			0.783058	−	0.621948i	$-0.213659\pi$
$757$	5.14290e11			1.56612			0.783058	+	0.621948i	$0.213659\pi$
$758$		0			0
$759$	1.01203e11	−	3.30084e11i	0.304947	−	0.994619i
$760$		0			0
$761$	1.03436e11	+	5.97185e10i	0.308412	+	0.178062i	0.646216	−	0.763155i	$-0.276351\pi$
$761$	1.03436e11	+	5.97185e10i	0.308412	+	0.178062i	−0.337804	+	0.941217i	$0.609684\pi$
$762$		0			0
$763$	−2.51843e11	−	1.71925e11i	−0.743073	−	0.507271i
$764$		0			0
$765$	−4.03645e11	−	2.73194e11i	−1.17857	−	0.797673i
$766$		0			0
$767$	−1.90126e11	+	1.09770e11i	−0.549365	+	0.317176i
$768$		0			0
$769$	1.97516e9			0.00564804			0.00282402	−	0.999996i	$-0.499101\pi$
$769$	1.97516e9			0.00564804			0.00282402	+	0.999996i	$0.499101\pi$
$770$		0			0
$771$	−3.14643e11	+	2.92943e11i	−0.890430	+	0.829022i
$772$		0			0
$773$	−2.08647e11	+	1.20462e11i	−0.584379	+	0.337391i	−0.762872	−	0.646550i	$-0.776211\pi$
$773$	−2.08647e11	+	1.20462e11i	−0.584379	+	0.337391i	0.178493	+	0.983941i	$0.442878\pi$
$774$		0			0
$775$	1.15722e11	−	2.00436e11i	0.320780	−	0.555607i
$776$		0			0
$777$	2.13027e11	+	8.33361e10i	0.584455	+	0.228638i
$778$		0			0
$779$	−1.70477e11	−	9.84247e10i	−0.462930	−	0.267273i
$780$		0			0
$781$	−2.36406e11	−	4.09468e11i	−0.635412	−	1.10057i
$782$		0			0
$783$	3.04971e11	+	3.78442e11i	0.811355	+	1.00682i
$784$		0			0
$785$			1.13291e11i			0.298343i
$786$		0			0
$787$	3.06555e11	+	5.30968e11i	0.799114	+	1.38411i	0.920194	+	0.391464i	$0.128031\pi$
$787$	3.06555e11	+	5.30968e11i	0.799114	+	1.38411i	−0.121079	+	0.992643i	$0.538636\pi$
$788$		0			0
$789$	7.84106e10	+	3.40865e11i	0.202333	+	0.879578i
$790$		0			0
$791$	2.33339e11	+	4.85276e11i	0.596047	+	1.23960i
$792$		0			0
$793$	−1.00883e11	+	1.74734e11i	−0.255108	+	0.441860i
$794$		0			0
$795$	−7.55166e10	+	2.46306e11i	−0.189049	+	0.616604i
$796$		0			0
$797$		−	4.84244e11i		−	1.20014i	−0.799948	−	0.600069i	$-0.795140\pi$
$797$		−	4.84244e11i		−	1.20014i	0.799948	−	0.600069i	$-0.204860\pi$
$798$		0			0
$799$	−9.22933e10			−0.226456
$800$		0			0
$801$	8.14634e9	+	1.67698e10i	0.0197894	+	0.0407379i
$802$		0			0
$803$	−2.83378e11	−	1.63608e11i	−0.681559	−	0.393498i
$804$		0			0
$805$	−3.12293e11	−	2.13192e11i	−0.743668	−	0.507677i
$806$		0			0
$807$	−9.75596e10	−	4.24109e11i	−0.230025	−	0.999962i
$808$		0			0
$809$	−4.39425e11	+	2.53702e11i	−1.02587	+	0.592284i	−0.915798	−	0.401640i	$-0.868440\pi$
$809$	−4.39425e11	+	2.53702e11i	−1.02587	+	0.592284i	−0.110069	+	0.993924i	$0.535107\pi$
$810$		0			0
$811$	1.15970e11			0.268079			0.134040	−	0.990976i	$-0.457205\pi$
$811$	1.15970e11			0.268079			0.134040	+	0.990976i	$0.457205\pi$
$812$		0			0
$813$	6.62390e9	+	7.11455e9i	0.0151618	+	0.0162849i
$814$		0			0
$815$	2.10861e11	−	1.21740e11i	0.477931	−	0.275933i
$816$		0			0
$817$	−9.72530e9	+	1.68447e10i	−0.0218281	+	0.0378073i
$818$		0			0
$819$	−8.30659e11	−	1.22821e11i	−1.84623	−	0.272984i
$820$		0			0
$821$	8.63980e10	+	4.98819e10i	0.190165	+	0.109792i	0.592060	−	0.805894i	$-0.298315\pi$
$821$	8.63980e10	+	4.98819e10i	0.190165	+	0.109792i	−0.401895	+	0.915686i	$0.631648\pi$
$822$		0			0
$823$	9.14646e10	+	1.58421e11i	0.199367	+	0.345314i	0.948323	−	0.317305i	$-0.102778\pi$
$823$	9.14646e10	+	1.58421e11i	0.199367	+	0.345314i	−0.748956	+	0.662619i	$0.769445\pi$
$824$		0			0
$825$	3.56313e11	−	3.31740e11i	0.769158	−	0.716113i
$826$		0			0
$827$		−	4.96107e11i		−	1.06060i	−0.847809	−	0.530302i	$-0.822079\pi$
$827$		−	4.96107e11i		−	1.06060i	0.847809	−	0.530302i	$-0.177921\pi$
$828$		0			0
$829$	−2.32536e11	−	4.02764e11i	−0.492347	−	0.852770i	0.507614	−	0.861585i	$-0.330528\pi$
$829$	−2.32536e11	−	4.02764e11i	−0.492347	−	0.852770i	−0.999961	+	0.00881421i	$0.997194\pi$
$830$		0			0
$831$	1.53471e10	−	3.53036e9i	0.0321828	−	0.00740313i
$832$		0			0
$833$	7.89950e10	−	5.19870e11i	0.164066	−	1.07973i
$834$		0			0
$835$	2.29234e11	−	3.97045e11i	0.471556	−	0.816759i
$836$		0			0
$837$	4.20741e11	+	1.62655e11i	0.857260	+	0.331411i
$838$		0			0
$839$		−	5.59458e11i		−	1.12907i	−0.825410	−	0.564533i	$-0.809056\pi$
$839$		−	5.59458e11i		−	1.12907i	0.825410	−	0.564533i	$-0.190944\pi$
$840$		0			0
$841$	−3.36159e11			−0.671987
$842$		0			0
$843$	−8.02265e11	−	2.45972e11i	−1.58857	−	0.487052i
$844$		0			0
$845$	1.42865e12	+	8.24832e11i	2.80220	+	1.61785i
$846$		0			0
$847$	−4.91056e10	+	6.50042e11i	−0.0954107	+	1.26301i
$848$		0			0
$849$	2.78417e10	−	6.40455e9i	0.0535878	−	0.0123270i
$850$		0			0
$851$	1.96969e11	−	1.13720e11i	0.375561	−	0.216830i
$852$		0			0
$853$	−3.63372e11			−0.686365			−0.343183	−	0.939269i	$-0.611505\pi$
$853$	−3.63372e11			−0.686365			−0.343183	+	0.939269i	$0.611505\pi$
$854$		0			0
$855$	−8.77278e11	−	6.27418e10i	−1.64162	−	0.117407i
$856$		0			0
$857$	8.56740e11	−	4.94639e11i	1.58827	−	0.916991i	0.594683	−	0.803960i	$-0.297278\pi$
$857$	8.56740e11	−	4.94639e11i	1.58827	−	0.916991i	0.993592	−	0.113030i	$-0.0360557\pi$
$858$		0			0
$859$	−3.35996e11	+	5.81963e11i	−0.617109	+	1.06886i	0.372902	+	0.927871i	$0.378363\pi$
$859$	−3.35996e11	+	5.81963e11i	−0.617109	+	1.06886i	−0.990011	+	0.140993i	$0.954971\pi$
$860$		0			0
$861$	2.29953e11	−	3.49191e10i	0.418434	−	0.0635405i
$862$		0			0
$863$	1.00483e11	+	5.80140e10i	0.181155	+	0.104590i	0.587835	−	0.808981i	$-0.299980\pi$
$863$	1.00483e11	+	5.80140e10i	0.181155	+	0.104590i	−0.406680	+	0.913571i	$0.633314\pi$
$864$		0			0
$865$	−3.80499e11	−	6.59043e11i	−0.679656	−	1.17720i
$866$		0			0
$867$	−7.42066e10	−	7.97033e10i	−0.131331	−	0.141059i
$868$		0			0
$869$			1.54869e12i			2.71572i
$870$		0			0
$871$	9.39033e11	+	1.62645e12i	1.63158	+	2.82598i
$872$		0			0
$873$	3.26303e11	+	2.20847e11i	0.561777	+	0.380220i
$874$		0			0
$875$	9.99520e10	+	2.07871e11i	0.170514	+	0.354619i
$876$		0			0
$877$	3.52252e11	−	6.10118e11i	0.595463	−	1.03137i	−0.398018	−	0.917378i	$-0.630302\pi$
$877$	3.52252e11	−	6.10118e11i	0.595463	−	1.03137i	0.993481	−	0.113995i	$-0.0363649\pi$
$878$		0			0
$879$	9.50804e11	+	2.91514e11i	1.59271	+	0.488319i
$880$		0			0
$881$		−	5.53017e11i		−	0.917983i	−0.888441	−	0.458991i	$-0.848211\pi$
$881$		−	5.53017e11i		−	0.917983i	0.888441	−	0.458991i	$-0.151789\pi$
$882$		0			0
$883$	2.35957e11			0.388141			0.194070	−	0.980988i	$-0.437831\pi$
$883$	2.35957e11			0.388141			0.194070	+	0.980988i	$0.437831\pi$
$884$		0			0
$885$	7.96439e10	−	2.59768e11i	0.129831	−	0.423459i
$886$		0			0
$887$	−2.54930e11	−	1.47184e11i	−0.411838	−	0.237775i	0.279741	−	0.960075i	$-0.409751\pi$
$887$	−2.54930e11	−	1.47184e11i	−0.411838	−	0.237775i	−0.691579	+	0.722301i	$0.743085\pi$
$888$		0			0
$889$	1.13307e11	−	5.44821e10i	0.181405	−	0.0872261i
$890$		0			0
$891$	7.45853e11	+	5.86539e11i	1.18343	+	0.930649i
$892$		0			0
$893$	−1.44230e11	+	8.32712e10i	−0.226803	+	0.130945i
$894$		0			0
$895$	−1.64480e11			−0.256343
$896$		0			0
$897$	−6.11052e11	+	5.68911e11i	−0.943862	+	0.878769i
$898$		0			0
$899$	−6.72271e11	+	3.88136e11i	−1.02921	+	0.594217i
$900$		0			0
$901$	−1.78107e11	+	3.08491e11i	−0.270261	+	0.468105i
$902$		0			0
$903$	−3.45034e9	−	2.27216e10i	−0.00518933	−	0.0341734i
$904$		0			0
$905$	6.54510e11	+	3.77882e11i	0.975713	+	0.563328i
$906$		0			0
$907$	−2.42858e11	−	4.20643e11i	−0.358859	−	0.621562i	0.628911	−	0.777477i	$-0.283501\pi$
$907$	−2.42858e11	−	4.20643e11i	−0.358859	−	0.621562i	−0.987771	+	0.155915i	$0.950167\pi$
$908$		0			0
$909$	−6.89759e11	−	4.93308e10i	−1.01028	−	0.0722540i
$910$		0			0
$911$		−	3.03963e11i		−	0.441314i	−0.975351	−	0.220657i	$-0.929180\pi$
$911$		−	3.03963e11i		−	0.441314i	0.975351	−	0.220657i	$-0.0708201\pi$
$912$		0			0
$913$	7.80323e11	+	1.35156e12i	1.12303	+	1.94515i
$914$		0			0
$915$	−5.59789e10	−	2.43350e11i	−0.0798619	−	0.347174i
$916$		0			0
$917$	−1.13470e12	−	8.57176e10i	−1.60474	−	0.121225i
$918$		0			0
$919$	3.69640e11	−	6.40235e11i	0.518223	−	0.897588i	−0.481553	−	0.876417i	$-0.659927\pi$
$919$	3.69640e11	−	6.40235e11i	0.518223	−	0.897588i	0.999776	−	0.0211713i	$-0.00673953\pi$
$920$		0			0
$921$	−3.10747e11	+	1.01354e12i	−0.431886	+	1.40864i
$922$		0			0
$923$			1.14337e12i			1.57536i
$924$		0			0
$925$	3.20714e11			0.438078
$926$		0			0
$927$	9.53082e11	−	4.62982e11i	1.29066	−	0.626968i
$928$		0			0
$929$	−6.96111e11	−	4.01900e11i	−0.934578	−	0.539579i	−0.0463214	−	0.998927i	$-0.514750\pi$
$929$	−6.96111e11	−	4.01900e11i	−0.934578	−	0.539579i	−0.888257	+	0.459348i	$0.848083\pi$
$930$		0			0
$931$	−3.45602e11	−	8.83691e11i	−0.460020	−	1.17626i
$932$		0			0
$933$	3.13890e11	+	1.36454e12i	0.414239	+	1.80077i
$934$		0			0
$935$	1.41811e12	−	8.18747e11i	1.85551	−	1.07128i
$936$		0			0
$937$	−1.12614e12			−1.46094			−0.730470	−	0.682945i	$-0.760699\pi$
$937$	−1.12614e12			−1.46094			−0.730470	+	0.682945i	$0.760699\pi$
$938$		0			0
$939$	2.10885e11	+	2.26506e11i	0.271258	+	0.291351i
$940$		0			0
$941$	−1.86163e11	+	1.07481e11i	−0.237430	+	0.137080i	−0.613995	−	0.789310i	$-0.710438\pi$
$941$	−1.86163e11	+	1.07481e11i	−0.237430	+	0.137080i	0.376565	+	0.926390i	$0.377105\pi$
$942$		0			0
$943$	1.15630e11	−	2.00277e11i	0.146226	−	0.253271i
$944$		0			0
$945$	8.55985e11	−	5.89267e11i	1.07334	−	0.738898i
$946$		0			0
$947$	−9.31322e11	−	5.37699e11i	−1.15798	−	0.668558i	−0.207158	−	0.978307i	$-0.566421\pi$
$947$	−9.31322e11	−	5.37699e11i	−1.15798	−	0.668558i	−0.950818	+	0.309749i	$0.899755\pi$
$948$		0			0
$949$	3.95642e11	+	6.85272e11i	0.487795	+	0.844886i
$950$		0			0
$951$	6.58962e11	−	6.13517e11i	0.805635	−	0.750074i
$952$		0			0
$953$			1.39130e12i			1.68674i	0.537330	+	0.843372i	$0.319433\pi$
$953$			1.39130e12i			1.68674i	−0.537330	+	0.843372i	$0.680567\pi$
$954$		0			0
$955$	2.22982e11	+	3.86216e11i	0.268075	+	0.464319i
$956$		0			0
$957$	−1.59131e12	+	3.66056e11i	−1.89718	+	0.436416i
$958$		0			0
$959$	5.38838e10	−	7.89314e10i	0.0637065	−	0.0933202i
$960$		0			0
$961$	6.62149e10	−	1.14688e11i	0.0776359	−	0.134469i
$962$		0			0
$963$	−6.72849e11	−	1.38511e12i	−0.782371	−	1.61057i
$964$		0			0
$965$			1.08326e12i			1.24918i
$966$		0			0
$967$	8.29360e11			0.948500			0.474250	−	0.880390i	$-0.342719\pi$
$967$	8.29360e11			0.948500			0.474250	+	0.880390i	$0.342719\pi$
$968$		0			0
$969$	−1.16269e12	−	3.56478e11i	−1.31877	−	0.404331i
$970$		0			0
$971$	−3.10342e11	−	1.79176e11i	−0.349112	−	0.201560i	0.315182	−	0.949031i	$-0.397934\pi$
$971$	−3.10342e11	−	1.79176e11i	−0.349112	−	0.201560i	−0.664294	+	0.747471i	$0.731268\pi$
$972$		0			0
$973$	−7.81998e11	+	3.76013e11i	−0.872477	+	0.419519i
$974$		0			0
$975$	−1.14732e12	+	2.63922e11i	−1.26959	+	0.292050i
$976$		0			0
$977$	−5.08613e11	+	2.93648e11i	−0.558225	+	0.322291i	−0.752433	−	0.658669i	$-0.771120\pi$
$977$	−5.08613e11	+	2.93648e11i	−0.558225	+	0.322291i	0.194208	+	0.980960i	$0.437786\pi$
$978$		0			0
$979$	−6.26359e10			−0.0681855
$980$		0			0
$981$	5.94418e10	−	8.31135e11i	0.0641823	−	0.897419i
$982$		0			0
$983$	7.15591e11	−	4.13147e11i	0.766392	−	0.442477i	−0.0651939	−	0.997873i	$-0.520767\pi$
$983$	7.15591e11	−	4.13147e11i	0.766392	−	0.442477i	0.831586	+	0.555396i	$0.187433\pi$
$984$		0			0
$985$	1.15513e11	−	2.00075e11i	0.122712	−	0.212544i
$986$		0			0
$987$	7.16898e10	−	1.83256e11i	0.0755420	−	0.193104i
$988$		0			0
$989$	−1.97893e10	−	1.14254e10i	−0.0206845	−	0.0119422i
$990$		0			0
$991$	2.39476e11	+	4.14784e11i	0.248295	+	0.430059i	0.963053	−	0.269313i	$-0.0867966\pi$
$991$	2.39476e11	+	4.14784e11i	0.248295	+	0.430059i	−0.714758	+	0.699372i	$0.753463\pi$
$992$		0			0
$993$	1.15740e11	+	1.24313e11i	0.119038	+	0.127856i
$994$		0			0
$995$			3.50123e11i			0.357214i
$996$		0			0
$997$	−2.46523e11	−	4.26990e11i	−0.249503	−	0.432152i	0.713885	−	0.700263i	$-0.246934\pi$
$997$	−2.46523e11	−	4.26990e11i	−0.249503	−	0.432152i	−0.963388	+	0.268111i	$0.913601\pi$
$998$		0			0
$999$	9.63165e10	+	6.17613e11i	0.0967028	+	0.620090i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	84.9.p.b.65.12	yes	40
3.2	odd	2	inner	84.9.p.b.65.1	yes	40
7.4	even	3	inner	84.9.p.b.53.1	✓	40
21.11	odd	6	inner	84.9.p.b.53.12	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.9.p.b.53.1	✓	40	7.4	even	3	inner
84.9.p.b.53.12	yes	40	21.11	odd	6	inner
84.9.p.b.65.1	yes	40	3.2	odd	2	inner
84.9.p.b.65.12	yes	40	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	84.p (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(23.7435 - 77.4419i) q^{3} +(705.317 + 407.215i) q^{5} +(-2163.85 + 1040.46i) q^{7} +(-5433.50 - 3677.48i) q^{9} +O(q^{10})\)	\(q+(23.7435 - 77.4419i) q^{3} +(705.317 + 407.215i) q^{5} +(-2163.85 + 1040.46i) q^{7} +(-5433.50 - 3677.48i) q^{9} +(19089.3 - 11021.2i) q^{11} -53303.6 q^{13} +(48282.2 - 44952.4i) q^{15} +(78994.6 - 45607.6i) q^{17} +(82298.4 - 142545. i) q^{19} +(29197.8 + 192277. i) q^{21} +(167463. + 96684.9i) q^{23} +(136335. + 236140. i) q^{25} +(-413801. + 333464. i) q^{27} -914552. i q^{29} +(-424400. - 735082. i) q^{31} +(-400258. - 1.73999e6i) q^{33} +(-1.94989e6 - 147299. i) q^{35} +(588097. - 1.01861e6i) q^{37} +(-1.26561e6 + 4.12793e6i) q^{39} -1.19595e6i q^{41} -118171. q^{43} +(-2.33481e6 - 4.80639e6i) q^{45} +(-876262. - 505910. i) q^{47} +(3.59969e6 - 4.50279e6i) q^{49} +(-1.65633e6 - 7.20038e6i) q^{51} +(-3.38202e6 + 1.95261e6i) q^{53} +1.79520e7 q^{55} +(-9.08491e6 - 9.75785e6i) q^{57} +(3.56686e6 - 2.05933e6i) q^{59} +(1.89260e6 - 3.27809e6i) q^{61} +(1.55835e7 + 2.30418e6i) q^{63} +(-3.75959e7 - 2.17060e7i) q^{65} +(-1.76167e7 - 3.05130e7i) q^{67} +(1.14636e7 - 1.06730e7i) q^{69} -2.14501e7i q^{71} +(-7.42242e6 - 1.28560e7i) q^{73} +(2.15242e7 - 4.95130e6i) q^{75} +(-2.98393e7 + 4.37099e7i) q^{77} +(-3.51298e7 + 6.08465e7i) q^{79} +(1.59991e7 + 3.99631e7i) q^{81} +7.08020e7i q^{83} +7.42883e7 q^{85} +(-7.08246e7 - 2.17146e7i) q^{87} +(-2.46090e6 - 1.42080e6i) q^{89} +(1.15341e8 - 5.54602e7i) q^{91} +(-6.70029e7 + 1.54129e7i) q^{93} +(1.16093e8 - 6.70263e7i) q^{95} -6.00540e7 q^{97} +(-1.44252e8 - 1.03167e7i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 81 q^{3} - 34 q^{7} + 4771 q^{9}+O(q^{10}) \) 40 * q + 81 * q^3 - 34 * q^7 + 4771 * q^9	\( 40 q + 81 q^{3} - 34 q^{7} + 4771 q^{9} - 55464 q^{13} + 68482 q^{15} + 311690 q^{19} - 172343 q^{21} + 1766792 q^{25} - 3451932 q^{27} + 31596 q^{31} + 1874885 q^{33} - 1853482 q^{37} + 11217526 q^{39} - 13372600 q^{43} - 527785 q^{45} - 12653462 q^{49} - 1103461 q^{51} + 71577224 q^{55} - 17195214 q^{57} - 21761970 q^{61} + 21945045 q^{63} - 26337350 q^{67} - 5588722 q^{69} + 41115682 q^{73} - 17971730 q^{75} - 120916932 q^{79} - 24550133 q^{81} + 139250060 q^{85} - 16321046 q^{87} + 345074940 q^{91} + 25774675 q^{93} - 707216948 q^{97} - 94510994 q^{99}+O(q^{100}) \) 40 * q + 81 * q^3 - 34 * q^7 + 4771 * q^9 - 55464 * q^13 + 68482 * q^15 + 311690 * q^19 - 172343 * q^21 + 1766792 * q^25 - 3451932 * q^27 + 31596 * q^31 + 1874885 * q^33 - 1853482 * q^37 + 11217526 * q^39 - 13372600 * q^43 - 527785 * q^45 - 12653462 * q^49 - 1103461 * q^51 + 71577224 * q^55 - 17195214 * q^57 - 21761970 * q^61 + 21945045 * q^63 - 26337350 * q^67 - 5588722 * q^69 + 41115682 * q^73 - 17971730 * q^75 - 120916932 * q^79 - 24550133 * q^81 + 139250060 * q^85 - 16321046 * q^87 + 345074940 * q^91 + 25774675 * q^93 - 707216948 * q^97 - 94510994 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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