LMFDB - Embedded newform 38.9.b.a.37.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [38,9,Mod(37,38)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(38, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("38.37");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 38 = 2 \cdot 19 $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	38.b (of order $2$, degree $1$, 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:	$15.4803871823$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 46118 x^{10} + 738386961 x^{8} + 5214446299656 x^{6} + \cdots + 92\!\cdots\!64 $ x^12 + 46118x^10 + 738386961x^8 + 5214446299656x^6 + 17370647163698184x^4 + 24830681474333400768*x^2 + 9218779084612644462864
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{21} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			37.7
Root			$-111.533i$ of defining polynomial
Character	$\chi$	$=$	38.37
Dual form			38.9.b.a.37.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+11.3137i q^{2} -124.261i q^{3} -128.000 q^{4} -419.091 q^{5} +1405.85 q^{6} -2431.38 q^{7} -1448.15i q^{8} -8879.72 q^{9} +O(q^{10})$	\(q+11.3137i q^{2} -124.261i q^{3} -128.000 q^{4} -419.091 q^{5} +1405.85 q^{6} -2431.38 q^{7} -1448.15i q^{8} -8879.72 q^{9} -4741.48i q^{10} +18851.2 q^{11} +15905.4i q^{12} +52190.4i q^{13} -27507.9i q^{14} +52076.6i q^{15} +16384.0 q^{16} +163673. q^{17} -100463. i q^{18} +(-127361. + 27617.3i) q^{19} +53643.7 q^{20} +302125. i q^{21} +213277. i q^{22} -294819. q^{23} -179949. q^{24} -214988. q^{25} -590467. q^{26} +288125. i q^{27} +311217. q^{28} +1.01197e6i q^{29} -589179. q^{30} -188192. i q^{31} +185364. i q^{32} -2.34246e6i q^{33} +1.85175e6i q^{34} +1.01897e6 q^{35} +1.13660e6 q^{36} +1.31226e6i q^{37} +(-312455. - 1.44093e6i) q^{38} +6.48522e6 q^{39} +606909. i q^{40} +662199. i q^{41} -3.41815e6 q^{42} -1.24898e6 q^{43} -2.41295e6 q^{44} +3.72141e6 q^{45} -3.33550e6i q^{46} -3.35309e6 q^{47} -2.03589e6i q^{48} +146807. q^{49} -2.43231e6i q^{50} -2.03381e7i q^{51} -6.68038e6i q^{52} +6.52766e6i q^{53} -3.25977e6 q^{54} -7.90038e6 q^{55} +3.52101e6i q^{56} +(3.43175e6 + 1.58260e7i) q^{57} -1.14492e7 q^{58} +5.48176e6i q^{59} -6.66580e6i q^{60} +8.15096e6 q^{61} +2.12914e6 q^{62} +2.15900e7 q^{63} -2.09715e6 q^{64} -2.18726e7i q^{65} +2.65020e7 q^{66} -1.27314e7i q^{67} -2.09502e7 q^{68} +3.66344e7i q^{69} +1.15283e7i q^{70} -2.99568e7i q^{71} +1.28592e7i q^{72} +2.59027e7 q^{73} -1.48465e7 q^{74} +2.67145e7i q^{75} +(1.63022e7 - 3.53502e6i) q^{76} -4.58344e7 q^{77} +7.33719e7i q^{78} +5.24410e7i q^{79} -6.86639e6 q^{80} -2.24572e7 q^{81} -7.49193e6 q^{82} -3.87316e7 q^{83} -3.86720e7i q^{84} -6.85940e7 q^{85} -1.41306e7i q^{86} +1.25749e8 q^{87} -2.72995e7i q^{88} +4.25848e7i q^{89} +4.21030e7i q^{90} -1.26895e8i q^{91} +3.77368e7 q^{92} -2.33848e7 q^{93} -3.79359e7i q^{94} +(5.33759e7 - 1.15742e7i) q^{95} +2.30334e7 q^{96} +1.29599e8i q^{97} +1.66093e6i q^{98} -1.67393e8 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 1536 q^{4} + 558 q^{5} + 1792 q^{6} - 5422 q^{7} - 15592 q^{9}+O(q^{10}) $ 12 * q - 1536 * q^4 + 558 * q^5 + 1792 * q^6 - 5422 * q^7 - 15592 * q^9	$ 12 q - 1536 q^{4} + 558 q^{5} + 1792 q^{6} - 5422 q^{7} - 15592 q^{9} - 12546 q^{11} + 196608 q^{16} + 270810 q^{17} + 41512 q^{19} - 71424 q^{20} - 823956 q^{23} - 229376 q^{24} + 865538 q^{25} - 431616 q^{26} + 694016 q^{28} + 71168 q^{30} - 1194378 q^{35} + 1995776 q^{36} + 998784 q^{38} + 5786100 q^{39} - 8383744 q^{42} + 7586646 q^{43} + 1605888 q^{44} + 2226046 q^{45} - 20260530 q^{47} - 19498842 q^{49} + 16933888 q^{54} - 14858554 q^{55} + 14430564 q^{57} - 5506560 q^{58} - 41363266 q^{61} + 32266752 q^{62} + 84235798 q^{63} - 25165824 q^{64} + 14371328 q^{66} - 34663680 q^{68} + 87906498 q^{73} - 2149632 q^{74} - 5313536 q^{76} - 78817962 q^{77} + 9142272 q^{80} - 100904812 q^{81} - 49609728 q^{82} - 55944960 q^{83} + 25440254 q^{85} + 119189604 q^{87} + 105466368 q^{92} + 105500856 q^{93} + 81396774 q^{95} + 29360128 q^{96} - 85554938 q^{99}+O(q^{100}) $ 12 * q - 1536 * q^4 + 558 * q^5 + 1792 * q^6 - 5422 * q^7 - 15592 * q^9 - 12546 * q^11 + 196608 * q^16 + 270810 * q^17 + 41512 * q^19 - 71424 * q^20 - 823956 * q^23 - 229376 * q^24 + 865538 * q^25 - 431616 * q^26 + 694016 * q^28 + 71168 * q^30 - 1194378 * q^35 + 1995776 * q^36 + 998784 * q^38 + 5786100 * q^39 - 8383744 * q^42 + 7586646 * q^43 + 1605888 * q^44 + 2226046 * q^45 - 20260530 * q^47 - 19498842 * q^49 + 16933888 * q^54 - 14858554 * q^55 + 14430564 * q^57 - 5506560 * q^58 - 41363266 * q^61 + 32266752 * q^62 + 84235798 * q^63 - 25165824 * q^64 + 14371328 * q^66 - 34663680 * q^68 + 87906498 * q^73 - 2149632 * q^74 - 5313536 * q^76 - 78817962 * q^77 + 9142272 * q^80 - 100904812 * q^81 - 49609728 * q^82 - 55944960 * q^83 + 25440254 * q^85 + 119189604 * q^87 + 105466368 * q^92 + 105500856 * q^93 + 81396774 * q^95 + 29360128 * q^96 - 85554938 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/38\mathbb{Z}\right)^\times$.

$n$	$21$
$\chi(n)$	$-1$

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 $

$2$			11.3137i			0.707107i
$2$			11.3137i			0.707107i
$3$		−	124.261i		−	1.53408i	−0.641598	−	0.767041i	$-0.721728\pi$
$3$		−	124.261i		−	1.53408i	0.641598	−	0.767041i	$-0.278272\pi$
$4$	−128.000			−0.500000
$5$	−419.091			−0.670546			−0.335273	−	0.942121i	$-0.608829\pi$
$5$	−419.091			−0.670546			−0.335273	+	0.942121i	$0.608829\pi$
$6$	1405.85			1.08476
$7$	−2431.38			−1.01265			−0.506326	−	0.862342i	$-0.668997\pi$
$7$	−2431.38			−1.01265			−0.506326	+	0.862342i	$0.668997\pi$
$8$		−	1448.15i		−	0.353553i
$9$	−8879.72			−1.35341
$10$		−	4741.48i		−	0.474148i
$11$	18851.2			1.28756			0.643781	−	0.765209i	$-0.277365\pi$
$11$	18851.2			1.28756			0.643781	+	0.765209i	$0.277365\pi$
$12$			15905.4i			0.767041i
$13$			52190.4i			1.82733i	0.406466	+	0.913666i	$0.366761\pi$
$13$			52190.4i			1.82733i	−0.406466	+	0.913666i	$0.633239\pi$
$14$		−	27507.9i		−	0.716054i
$15$			52076.6i			1.02867i
$16$	16384.0			0.250000
$17$	163673.			1.95967			0.979833	−	0.199819i	$-0.0640356\pi$
$17$	163673.			1.95967			0.979833	+	0.199819i	$0.0640356\pi$
$18$		−	100463.i		−	0.957005i
$19$	−127361.	+	27617.3i	−0.977287	+	0.211918i
$19$	−127361.	+	27617.3i	−0.977287	+	0.211918i
$20$	53643.7			0.335273
$21$			302125.i			1.55349i
$22$			213277.i			0.910444i
$23$	−294819.			−1.05352			−0.526762	−	0.850013i	$-0.676594\pi$
$23$	−294819.			−1.05352			−0.526762	+	0.850013i	$0.676594\pi$
$24$	−179949.			−0.542380
$25$	−214988.			−0.550368
$26$	−590467.			−1.29212
$27$			288125.i			0.542159i
$28$	311217.			0.506326
$29$			1.01197e6i			1.43080i	0.698717	+	0.715398i	$0.253754\pi$
$29$			1.01197e6i			1.43080i	−0.698717	+	0.715398i	$0.746246\pi$
$30$	−589179.			−0.727382
$31$		−	188192.i		−	0.203776i	−0.994796	−	0.101888i	$-0.967512\pi$
$31$		−	188192.i		−	0.203776i	0.994796	−	0.101888i	$-0.0324884\pi$
$32$			185364.i			0.176777i
$33$		−	2.34246e6i		−	1.97523i
$34$			1.85175e6i			1.38569i
$35$	1.01897e6			0.679030
$36$	1.13660e6			0.676705
$37$			1.31226e6i			0.700184i	0.936715	+	0.350092i	$0.113850\pi$
$37$			1.31226e6i			0.700184i	−0.936715	+	0.350092i	$0.886150\pi$
$38$	−312455.	−	1.44093e6i	−0.149849	−	0.691047i
$39$	6.48522e6			2.80328
$40$			606909.i			0.237074i
$41$			662199.i			0.234344i	0.993112	+	0.117172i	$0.0373828\pi$
$41$			662199.i			0.234344i	−0.993112	+	0.117172i	$0.962617\pi$
$42$	−3.41815e6			−1.09849
$43$	−1.24898e6			−0.365328			−0.182664	−	0.983175i	$-0.558472\pi$
$43$	−1.24898e6			−0.365328			−0.182664	+	0.983175i	$0.558472\pi$
$44$	−2.41295e6			−0.643781
$45$	3.72141e6			0.907523
$46$		−	3.33550e6i		−	0.744954i
$47$	−3.35309e6			−0.687154			−0.343577	−	0.939124i	$-0.611639\pi$
$47$	−3.35309e6			−0.687154			−0.343577	+	0.939124i	$0.611639\pi$
$48$		−	2.03589e6i		−	0.383521i
$49$	146807.			0.0254661
$50$		−	2.43231e6i		−	0.389169i
$51$		−	2.03381e7i		−	3.00629i
$52$		−	6.68038e6i		−	0.913666i
$53$			6.52766e6i			0.827283i	0.910440	+	0.413641i	$0.135743\pi$
$53$			6.52766e6i			0.827283i	−0.910440	+	0.413641i	$0.864257\pi$
$54$	−3.25977e6			−0.383364
$55$	−7.90038e6			−0.863370
$56$			3.52101e6i			0.358027i
$57$	3.43175e6	+	1.58260e7i	0.325099	+	1.49924i
$58$	−1.14492e7			−1.01173
$59$			5.48176e6i			0.452389i	0.974082	+	0.226194i	$0.0726284\pi$
$59$			5.48176e6i			0.452389i	−0.974082	+	0.226194i	$0.927372\pi$
$60$		−	6.66580e6i		−	0.514336i
$61$	8.15096e6			0.588694			0.294347	−	0.955699i	$-0.404898\pi$
$61$	8.15096e6			0.588694			0.294347	+	0.955699i	$0.404898\pi$
$62$	2.12914e6			0.144092
$63$	2.15900e7			1.37053
$64$	−2.09715e6			−0.125000
$65$		−	2.18726e7i		−	1.22531i
$66$	2.65020e7			1.39670
$67$		−	1.27314e7i		−	0.631796i	−0.948793	−	0.315898i	$-0.897694\pi$
$67$		−	1.27314e7i		−	0.631796i	0.948793	−	0.315898i	$-0.102306\pi$
$68$	−2.09502e7			−0.979833
$69$			3.66344e7i			1.61619i
$70$			1.15283e7i			0.480147i
$71$		−	2.99568e7i		−	1.17886i	−0.807819	−	0.589430i	$-0.799352\pi$
$71$		−	2.99568e7i		−	1.17886i	0.807819	−	0.589430i	$-0.200648\pi$
$72$			1.28592e7i			0.478502i
$73$	2.59027e7			0.912124			0.456062	−	0.889948i	$-0.349260\pi$
$73$	2.59027e7			0.912124			0.456062	+	0.889948i	$0.349260\pi$
$74$	−1.48465e7			−0.495105
$75$			2.67145e7i			0.844310i
$76$	1.63022e7	−	3.53502e6i	0.488644	−	0.105959i
$77$	−4.58344e7			−1.30385
$78$			7.33719e7i			1.98222i
$79$			5.24410e7i			1.34636i	0.739477	+	0.673182i	$0.235073\pi$
$79$			5.24410e7i			1.34636i	−0.739477	+	0.673182i	$0.764927\pi$
$80$	−6.86639e6			−0.167636
$81$	−2.24572e7			−0.521693
$82$	−7.49193e6			−0.165706
$83$	−3.87316e7			−0.816118			−0.408059	−	0.912955i	$-0.633794\pi$
$83$	−3.87316e7			−0.816118			−0.408059	+	0.912955i	$0.633794\pi$
$84$		−	3.86720e7i		−	0.776747i
$85$	−6.85940e7			−1.31405
$86$		−	1.41306e7i		−	0.258326i
$87$	1.25749e8			2.19496
$88$		−	2.72995e7i		−	0.455222i
$89$			4.25848e7i			0.678726i	0.940655	+	0.339363i	$0.110212\pi$
$89$			4.25848e7i			0.678726i	−0.940655	+	0.339363i	$0.889788\pi$
$90$			4.21030e7i			0.641716i
$91$		−	1.26895e8i		−	1.85045i
$92$	3.77368e7			0.526762
$93$	−2.33848e7			−0.312609
$94$		−	3.79359e7i		−	0.485892i
$95$	5.33759e7	−	1.15742e7i	0.655316	−	0.142101i
$96$	2.30334e7			0.271190
$97$			1.29599e8i			1.46391i	0.681352	+	0.731956i	$0.261392\pi$
$97$			1.29599e8i			1.46391i	−0.681352	+	0.731956i	$0.738608\pi$
$98$			1.66093e6i			0.0180072i
$99$	−1.67393e8			−1.74260
$100$	2.75184e7			0.275184
$101$	−1.59646e8			−1.53417			−0.767083	−	0.641548i	$-0.778292\pi$
$101$	−1.59646e8			−1.53417			−0.767083	+	0.641548i	$0.778292\pi$
$102$	2.30100e8			2.12577
$103$			300515.i			0.00267003i	0.999999	+	0.00133502i	$0.000424949\pi$
$103$			300515.i			0.00267003i	−0.999999	+	0.00133502i	$0.999575\pi$
$104$	7.55798e7			0.646060
$105$		−	1.26618e8i		−	1.04169i
$106$	−7.38520e7			−0.584977
$107$			1.64311e8i			1.25352i	0.779211	+	0.626761i	$0.215620\pi$
$107$			1.64311e8i			1.25352i	−0.779211	+	0.626761i	$0.784380\pi$
$108$		−	3.68801e7i		−	0.271079i
$109$		−	6.14943e7i		−	0.435641i	−0.975989	−	0.217821i	$-0.930105\pi$
$109$		−	6.14943e7i		−	0.435641i	0.975989	−	0.217821i	$-0.0698948\pi$
$110$		−	8.93826e7i		−	0.610495i
$111$	1.63062e8			1.07414
$112$	−3.98357e7			−0.253163
$113$		−	1.30584e8i		−	0.800894i	−0.916320	−	0.400447i	$-0.868855\pi$
$113$		−	1.30584e8i		−	0.800894i	0.916320	−	0.400447i	$-0.131145\pi$
$114$	−1.79050e8	+	3.88258e7i	−1.06012	+	0.229880i
$115$	1.23556e8			0.706436
$116$		−	1.29533e8i		−	0.715398i
$117$		−	4.63436e8i		−	2.47313i
$118$	−6.20190e7			−0.319887
$119$	−3.97952e8			−1.98446
$120$	7.54149e7			0.363691
$121$	1.41009e8			0.657818
$122$			9.22176e7i			0.416269i
$123$	8.22854e7			0.359503
$124$			2.40885e7i			0.101888i
$125$	2.53807e8			1.03959
$126$			2.44263e8i			0.969114i
$127$		−	769315.i		−	0.00295726i	−0.999999	−	0.00147863i	$-0.999529\pi$
$127$		−	769315.i		−	0.00295726i	0.999999	−	0.00147863i	$-0.000470663\pi$
$128$		−	2.37266e7i		−	0.0883883i
$129$			1.55200e8i			0.560443i
$130$	2.47460e8			0.866425
$131$	−1.33442e8			−0.453114			−0.226557	−	0.973998i	$-0.572747\pi$
$131$	−1.33442e8			−0.453114			−0.226557	+	0.973998i	$0.572747\pi$
$132$			2.99835e8i			0.987614i
$133$	3.09663e8	−	6.71482e7i	0.989653	−	0.214599i
$134$	1.44039e8			0.446747
$135$		−	1.20751e8i		−	0.363542i
$136$		−	2.37024e8i		−	0.692846i
$137$	−4.62601e8			−1.31318			−0.656590	−	0.754247i	$-0.728002\pi$
$137$	−4.62601e8			−1.31318			−0.656590	+	0.754247i	$0.728002\pi$
$138$	−4.14471e8			−1.14282
$139$	2.41952e8			0.648142			0.324071	−	0.946033i	$-0.394948\pi$
$139$	2.41952e8			0.648142			0.324071	+	0.946033i	$0.394948\pi$
$140$	−1.30428e8			−0.339515
$141$			4.16658e8i			1.05415i
$142$	3.38923e8			0.833580
$143$			9.83853e8i			2.35281i
$144$	−1.45485e8			−0.338352
$145$		−	4.24110e8i		−	0.959414i
$146$			2.93056e8i			0.644969i
$147$		−	1.82423e7i		−	0.0390670i
$148$		−	1.67969e8i		−	0.350092i
$149$	6.56868e8			1.33270			0.666351	−	0.745639i	$-0.267855\pi$
$149$	6.56868e8			1.33270			0.666351	+	0.745639i	$0.267855\pi$
$150$	−3.02240e8			−0.597017
$151$			1.30987e8i			0.251953i	0.992033	+	0.125977i	$0.0402064\pi$
$151$			1.30987e8i			0.251953i	−0.992033	+	0.125977i	$0.959794\pi$
$152$	3.99942e7	+	1.84439e8i	0.0749243	+	0.345523i
$153$	−1.45337e9			−2.65223
$154$		−	5.18558e8i		−	0.921964i
$155$			7.88694e7i			0.136641i
$156$	−8.30108e8			−1.40164
$157$	−4.82803e8			−0.794641			−0.397321	−	0.917680i	$-0.630060\pi$
$157$	−4.82803e8			−0.794641			−0.397321	+	0.917680i	$0.630060\pi$
$158$	−5.93302e8			−0.952023
$159$	8.11131e8			1.26912
$160$		−	7.76843e7i		−	0.118537i
$161$	7.16817e8			1.06685
$162$		−	2.54074e8i		−	0.368893i
$163$	8.10364e8			1.14797			0.573984	−	0.818867i	$-0.305397\pi$
$163$	8.10364e8			1.14797			0.573984	+	0.818867i	$0.305397\pi$
$164$		−	8.47615e7i		−	0.117172i
$165$			9.81706e8i			1.32448i
$166$		−	4.38198e8i		−	0.577083i
$167$			1.43055e9i			1.83924i	0.392812	+	0.919619i	$0.371502\pi$
$167$			1.43055e9i			1.83924i	−0.392812	+	0.919619i	$0.628498\pi$
$168$	4.37524e8			0.549243
$169$	−1.90811e9			−2.33914
$170$		−	7.76053e8i		−	0.929171i
$171$	1.13093e9	−	2.45234e8i	1.32267	−	0.286811i
$172$	1.59870e8			0.182664
$173$		−	6.71000e8i		−	0.749097i	−0.927207	−	0.374548i	$-0.877798\pi$
$173$		−	6.71000e8i		−	0.749097i	0.927207	−	0.374548i	$-0.122202\pi$
$174$			1.42268e9i			1.55207i
$175$	5.22716e8			0.557332
$176$	3.08858e8			0.321891
$177$	6.81167e8			0.694001
$178$	−4.81792e8			−0.479932
$179$		−	6.56448e8i		−	0.639423i	−0.947515	−	0.319711i	$-0.896414\pi$
$179$		−	6.56448e8i		−	0.639423i	0.947515	−	0.319711i	$-0.103586\pi$
$180$	−4.76341e8			−0.453762
$181$		−	5.45978e8i		−	0.508699i	−0.967112	−	0.254349i	$-0.918139\pi$
$181$		−	5.45978e8i		−	0.508699i	0.967112	−	0.254349i	$-0.0818613\pi$
$182$	1.43565e9			1.30847
$183$		−	1.01284e9i		−	0.903105i
$184$			4.26944e8i			0.372477i
$185$		−	5.49956e8i		−	0.469506i
$186$		−	2.64569e8i		−	0.221048i
$187$	3.08544e9			2.52319
$188$	4.29196e8			0.343577
$189$		−	7.00542e8i		−	0.549019i
$190$	1.30947e8	+	6.03880e8i	0.100480	+	0.463379i
$191$	−1.08420e7			−0.00814658			−0.00407329	−	0.999992i	$-0.501297\pi$
$191$	−1.08420e7			−0.00814658			−0.00407329	+	0.999992i	$0.501297\pi$
$192$			2.60594e8i			0.191760i
$193$		−	6.56281e8i		−	0.472999i	−0.971632	−	0.236500i	$-0.924000\pi$
$193$		−	6.56281e8i		−	0.472999i	0.971632	−	0.236500i	$-0.0760002\pi$
$194$	−1.46625e9			−1.03514
$195$	−2.71790e9			−1.87973
$196$	−1.87913e7			−0.0127330
$197$	4.53130e8			0.300855			0.150428	−	0.988621i	$-0.451935\pi$
$197$	4.53130e8			0.300855			0.150428	+	0.988621i	$0.451935\pi$
$198$		−	1.89384e9i		−	1.23220i
$199$	−1.59794e9			−1.01894			−0.509470	−	0.860489i	$-0.670158\pi$
$199$	−1.59794e9			−1.01894			−0.509470	+	0.860489i	$0.670158\pi$
$200$			3.11335e8i			0.194585i
$201$	−1.58201e9			−0.969227
$202$		−	1.80619e9i		−	1.08482i
$203$		−	2.46049e9i		−	1.44890i
$204$			2.60328e9i			1.50314i
$205$		−	2.77522e8i		−	0.157138i
$206$	−3.39994e6			−0.00188800
$207$	2.61791e9			1.42585
$208$			8.55088e8i			0.456833i
$209$	−2.40091e9	+	5.20620e8i	−1.25832	+	0.272858i
$210$	1.43252e9			0.736585
$211$		−	2.51512e9i		−	1.26891i	−0.772961	−	0.634453i	$-0.781225\pi$
$211$		−	2.51512e9i		−	1.26891i	0.772961	−	0.634453i	$-0.218775\pi$
$212$		−	8.35540e8i		−	0.413641i
$213$	−3.72245e9			−1.80847
$214$	−1.85897e9			−0.886374
$215$	5.23438e8			0.244969
$216$	4.17250e8			0.191682
$217$			4.57565e8i			0.206355i
$218$	6.95729e8			0.308045
$219$		−	3.21869e9i		−	1.39927i
$220$	1.01125e9			0.431685
$221$			8.54218e9i			3.58096i
$222$			1.84484e9i			0.759532i
$223$			1.22087e9i			0.493687i	0.969055	+	0.246843i	$0.0793933\pi$
$223$			1.22087e9i			0.493687i	−0.969055	+	0.246843i	$0.920607\pi$
$224$		−	4.50690e8i		−	0.179013i
$225$	1.90903e9			0.744873
$226$	1.47739e9			0.566318
$227$			8.23442e8i			0.310120i	0.987905	+	0.155060i	$0.0495571\pi$
$227$			8.23442e8i			0.310120i	−0.987905	+	0.155060i	$0.950443\pi$
$228$	−4.39264e8	−	2.02572e9i	−0.162550	−	0.749620i
$229$	−5.17641e9			−1.88229			−0.941145	−	0.338002i	$-0.890249\pi$
$229$	−5.17641e9			−1.88229			−0.941145	+	0.338002i	$0.890249\pi$
$230$			1.39788e9i			0.499526i
$231$			5.69542e9i			2.00022i
$232$	1.46550e9			0.505863
$233$	1.44033e9			0.488697			0.244348	−	0.969687i	$-0.421426\pi$
$233$	1.44033e9			0.488697			0.244348	+	0.969687i	$0.421426\pi$
$234$	5.24318e9			1.74877
$235$	1.40525e9			0.460769
$236$		−	7.01665e8i		−	0.226194i
$237$	6.51636e9			2.06543
$238$		−	4.50231e9i		−	1.40323i
$239$	−1.19815e9			−0.367215			−0.183608	−	0.983000i	$-0.558778\pi$
$239$	−1.19815e9			−0.367215			−0.183608	+	0.983000i	$0.558778\pi$
$240$			8.53222e8i			0.257168i
$241$		−	4.95635e9i		−	1.46924i	−0.678477	−	0.734621i	$-0.737360\pi$
$241$		−	4.95635e9i		−	1.46924i	0.678477	−	0.734621i	$-0.262640\pi$
$242$			1.59534e9i			0.465148i
$243$			4.68093e9i			1.34248i
$244$	−1.04332e9			−0.294347
$245$	−6.15254e7			−0.0170762
$246$			9.30953e8i			0.254207i
$247$	−1.44136e9	−	6.64703e9i	−0.387244	−	1.78583i
$248$	−2.72531e8			−0.0720458
$249$			4.81282e9i			1.25199i
$250$			2.87150e9i			0.735103i
$251$	−4.34101e8			−0.109369			−0.0546847	−	0.998504i	$-0.517415\pi$
$251$	−4.34101e8			−0.109369			−0.0546847	+	0.998504i	$0.517415\pi$
$252$	−2.76352e9			−0.685267
$253$	−5.55770e9			−1.35648
$254$	8.70381e6			0.00209110
$255$			8.52354e9i			2.01585i
$256$	2.68435e8			0.0625000
$257$			5.24516e9i			1.20234i	0.799122	+	0.601169i	$0.205298\pi$
$257$			5.24516e9i			1.20234i	−0.799122	+	0.601169i	$0.794702\pi$
$258$	−1.75588e9			−0.396293
$259$		−	3.19060e9i		−	0.709044i
$260$			2.79969e9i			0.612655i
$261$		−	8.98605e9i		−	1.93645i
$262$		−	1.50973e9i		−	0.320400i
$263$	−2.01646e9			−0.421470			−0.210735	−	0.977543i	$-0.567586\pi$
$263$	−2.01646e9			−0.421470			−0.210735	+	0.977543i	$0.567586\pi$
$264$	−3.39225e9			−0.698348
$265$		−	2.73568e9i		−	0.554731i
$266$	7.59696e8	+	3.50344e9i	0.151745	+	0.699790i
$267$	5.29162e9			1.04122
$268$			1.62962e9i			0.315898i
$269$		−	6.18818e9i		−	1.18183i	−0.806735	−	0.590913i	$-0.798768\pi$
$269$		−	6.18818e9i		−	1.18183i	0.806735	−	0.590913i	$-0.201232\pi$
$270$	1.36614e9			0.257063
$271$	2.96901e8			0.0550471			0.0275235	−	0.999621i	$-0.491238\pi$
$271$	2.96901e8			0.0550471			0.0275235	+	0.999621i	$0.491238\pi$
$272$	2.68162e9			0.489916
$273$	−1.57680e10			−2.83875
$274$		−	5.23373e9i		−	0.928559i
$275$	−4.05278e9			−0.708634
$276$		−	4.68921e9i		−	0.808096i
$277$	3.28776e9			0.558446			0.279223	−	0.960226i	$-0.409923\pi$
$277$	3.28776e9			0.558446			0.279223	+	0.960226i	$0.409923\pi$
$278$			2.73738e9i			0.458306i
$279$			1.67109e9i			0.275793i
$280$		−	1.47563e9i		−	0.240073i
$281$		−	3.61700e9i		−	0.580127i	−0.957007	−	0.290063i	$-0.906324\pi$
$281$		−	3.61700e9i		−	0.580127i	0.957007	−	0.290063i	$-0.0936764\pi$
$282$	−4.71395e9			−0.745398
$283$	1.06697e10			1.66344			0.831720	−	0.555195i	$-0.187356\pi$
$283$	1.06697e10			1.66344			0.831720	+	0.555195i	$0.187356\pi$
$284$			3.83447e9i			0.589430i
$285$	−1.43822e9	−	6.63253e9i	−0.217994	−	1.00531i
$286$	−1.11310e10			−1.66368
$287$		−	1.61006e9i		−	0.237309i
$288$		−	1.64598e9i		−	0.239251i
$289$	1.98132e10			2.84029
$290$	4.79825e9			0.678408
$291$	1.61041e10			2.24576
$292$	−3.31555e9			−0.456062
$293$		−	1.30694e9i		−	0.177331i	−0.996061	−	0.0886653i	$-0.971740\pi$
$293$		−	1.30694e9i		−	0.177331i	0.996061	−	0.0886653i	$-0.0282602\pi$
$294$	2.06388e8			0.0276246
$295$		−	2.29736e9i		−	0.303347i
$296$	1.90035e9			0.247553
$297$			5.43151e9i			0.698064i
$298$			7.43161e9i			0.942362i
$299$		−	1.53867e10i		−	1.92514i
$300$		−	3.41946e9i		−	0.422155i
$301$	3.03675e9			0.369951
$302$	−1.48195e9			−0.178158
$303$			1.98377e10i			2.35354i
$304$	−2.08668e9	+	4.52482e8i	−0.244322	+	0.0529795i
$305$	−3.41600e9			−0.394746
$306$		−	1.64430e10i		−	1.87541i
$307$		−	8.88173e9i		−	0.999871i	−0.866063	−	0.499935i	$-0.833357\pi$
$307$		−	8.88173e9i		−	0.999871i	0.866063	−	0.499935i	$-0.166643\pi$
$308$	5.86681e9			0.651927
$309$	3.73422e7			0.00409605
$310$	−8.92306e8			−0.0966200
$311$	1.33648e9			0.142863			0.0714317	−	0.997445i	$-0.477243\pi$
$311$	1.33648e9			0.142863			0.0714317	+	0.997445i	$0.477243\pi$
$312$		−	9.39160e9i		−	0.991109i
$313$	−8.88761e9			−0.925993			−0.462996	−	0.886360i	$-0.653226\pi$
$313$	−8.88761e9			−0.925993			−0.462996	+	0.886360i	$0.653226\pi$
$314$		−	5.46229e9i		−	0.561896i
$315$	−9.04817e9			−0.919006
$316$		−	6.71245e9i		−	0.673182i
$317$			1.20507e10i			1.19337i	0.802475	+	0.596685i	$0.203516\pi$
$317$			1.20507e10i			1.19337i	−0.802475	+	0.596685i	$0.796484\pi$
$318$			9.17690e9i			0.897403i
$319$			1.90769e10i			1.84224i
$320$	8.78898e8			0.0838182
$321$	2.04174e10			1.92301
$322$			8.10986e9i			0.754380i
$323$	−2.08456e10	+	4.52022e9i	−1.91516	+	0.415288i
$324$	2.87452e9			0.260846
$325$		−	1.12203e10i		−	1.00571i
$326$			9.16822e9i			0.811736i
$327$	−7.64133e9			−0.668310
$328$	9.58967e8			0.0828530
$329$	8.15265e9			0.695849
$330$	−1.11067e10			−0.936549
$331$			7.18195e9i			0.598316i	0.954204	+	0.299158i	$0.0967058\pi$
$331$			7.18195e9i			0.598316i	−0.954204	+	0.299158i	$0.903294\pi$
$332$	4.95765e9			0.408059
$333$		−	1.16525e10i		−	0.947636i
$334$	−1.61848e10			−1.30054
$335$			5.33562e9i			0.423648i
$336$			4.95001e9i			0.388373i
$337$			7.28217e9i			0.564601i	0.959326	+	0.282300i	$0.0910975\pi$
$337$			7.28217e9i			0.564601i	−0.959326	+	0.282300i	$0.908903\pi$
$338$		−	2.15878e10i		−	1.65402i
$339$	−1.62264e10			−1.22864
$340$	8.78003e9			0.657023
$341$		−	3.54764e9i		−	0.262375i
$342$	2.77451e9	+	1.27950e10i	0.202806	+	0.935269i
$343$	1.36595e10			0.986865
$344$			1.80872e9i			0.129163i
$345$		−	1.53532e10i		−	1.08373i
$346$	7.59150e9			0.529691
$347$	1.25319e8			0.00864367			0.00432183	−	0.999991i	$-0.498624\pi$
$347$	1.25319e8			0.00864367			0.00432183	+	0.999991i	$0.498624\pi$
$348$	−1.60958e10			−1.09748
$349$	1.83163e9			0.123463			0.0617313	−	0.998093i	$-0.480338\pi$
$349$	1.83163e9			0.123463			0.0617313	+	0.998093i	$0.480338\pi$
$350$			5.91386e9i			0.394093i
$351$	−1.50374e10			−0.990704
$352$			3.49433e9i			0.227611i
$353$	6.31655e9			0.406800			0.203400	−	0.979096i	$-0.434801\pi$
$353$	6.31655e9			0.406800			0.203400	+	0.979096i	$0.434801\pi$
$354$			7.70652e9i			0.490733i
$355$			1.25546e10i			0.790480i
$356$		−	5.45085e9i		−	0.339363i
$357$			4.94498e10i			3.04433i
$358$	7.42686e9			0.452140
$359$	7.50786e9			0.452000			0.226000	−	0.974127i	$-0.427435\pi$
$359$	7.50786e9			0.452000			0.226000	+	0.974127i	$0.427435\pi$
$360$		−	5.38918e9i		−	0.320858i
$361$	1.54581e10	−	7.03475e9i	0.910182	−	0.414209i
$362$	6.17704e9			0.359704
$363$		−	1.75219e10i		−	1.00915i
$364$			1.62425e10i			0.925227i
$365$	−1.08556e10			−0.611621
$366$	1.14590e10			0.638592
$367$	−3.53863e9			−0.195061			−0.0975306	−	0.995233i	$-0.531094\pi$
$367$	−3.53863e9			−0.195061			−0.0975306	+	0.995233i	$0.531094\pi$
$368$	−4.83032e9			−0.263381
$369$		−	5.88014e9i		−	0.317163i
$370$	6.22204e9			0.331991
$371$		−	1.58712e10i		−	0.837750i
$372$	2.99326e9			0.156305
$373$			3.19012e10i			1.64805i	0.566551	+	0.824026i	$0.308277\pi$
$373$			3.19012e10i			1.64805i	−0.566551	+	0.824026i	$0.691723\pi$
$374$			3.49077e10i			1.78417i
$375$		−	3.15382e10i		−	1.59482i
$376$			4.85580e9i			0.242946i
$377$	−5.28154e10			−2.61454
$378$	7.92573e9			0.388215
$379$		−	2.71860e10i		−	1.31762i	−0.752311	−	0.658808i	$-0.771061\pi$
$379$		−	2.71860e10i		−	1.31762i	0.752311	−	0.658808i	$-0.228939\pi$
$380$	−6.83212e9	+	1.48150e9i	−0.327658	+	0.0710503i
$381$	−9.55956e7			−0.00453668
$382$		−	1.22663e8i		−	0.00576050i
$383$			3.88214e9i			0.180416i	0.995923	+	0.0902082i	$0.0287532\pi$
$383$			3.88214e9i			0.180416i	−0.995923	+	0.0902082i	$0.971247\pi$
$384$	−2.94828e9			−0.135595
$385$	1.92088e10			0.874294
$386$	7.42497e9			0.334461
$387$	1.10906e10			0.494438
$388$		−	1.65887e10i		−	0.731956i
$389$	4.06187e10			1.77390			0.886948	−	0.461870i	$-0.152821\pi$
$389$	4.06187e10			1.77390			0.886948	+	0.461870i	$0.152821\pi$
$390$		−	3.07495e10i		−	1.32917i
$391$	−4.82540e10			−2.06455
$392$		−	2.12599e8i		−	0.00900361i
$393$			1.65816e10i			0.695115i
$394$			5.12658e9i			0.212737i
$395$		−	2.19776e10i		−	0.902799i
$396$	2.14264e10			0.871300
$397$	−3.37520e10			−1.35874			−0.679372	−	0.733794i	$-0.737748\pi$
$397$	−3.37520e10			−1.35874			−0.679372	+	0.733794i	$0.737748\pi$
$398$		−	1.80786e10i		−	0.720499i
$399$	−8.34389e9	−	3.84790e10i	−0.329213	−	1.51821i
$400$	−3.52236e9			−0.137592
$401$		−	2.89932e10i		−	1.12129i	−0.828055	−	0.560646i	$-0.810553\pi$
$401$		−	2.89932e10i		−	1.12129i	0.828055	−	0.560646i	$-0.189447\pi$
$402$		−	1.78984e10i		−	0.685347i
$403$	9.82180e9			0.372367
$404$	2.04347e10			0.767083
$405$	9.41160e9			0.349819
$406$	2.78373e10			1.02453
$407$			2.47377e10i			0.901531i
$408$	−2.94528e10			−1.06288
$409$			2.62952e10i			0.939686i	0.882750	+	0.469843i	$0.155690\pi$
$409$			2.62952e10i			0.939686i	−0.882750	+	0.469843i	$0.844310\pi$
$410$	3.13980e9			0.111114
$411$			5.74831e10i			2.01453i
$412$		−	3.84659e7i		−	0.00133502i
$413$		−	1.33282e10i		−	0.458113i
$414$			2.96183e10i			1.00823i
$415$	1.62321e10			0.547245
$416$	−9.67422e9			−0.323030
$417$		−	3.00651e10i		−	0.994303i
$418$	−5.89015e9	−	2.71632e10i	−0.192939	−	0.889766i
$419$	−3.04930e10			−0.989335			−0.494667	−	0.869082i	$-0.664710\pi$
$419$	−3.04930e10			−0.989335			−0.494667	+	0.869082i	$0.664710\pi$
$420$			1.62071e10i			0.520844i
$421$		−	1.37997e9i		−	0.0439280i	−0.999759	−	0.0219640i	$-0.993008\pi$
$421$		−	1.37997e9i		−	0.0439280i	0.999759	−	0.0219640i	$-0.00699192\pi$
$422$	2.84554e10			0.897252
$423$	2.97745e10			0.930001
$424$	9.45306e9			0.292489
$425$	−3.51877e10			−1.07854
$426$		−	4.21148e10i		−	1.27878i
$427$	−1.98181e10			−0.596143
$428$		−	2.10318e10i		−	0.626761i
$429$	1.22254e11			3.60940
$430$			5.92203e9i			0.173219i
$431$		−	5.60782e10i		−	1.62512i	−0.582879	−	0.812559i	$-0.698074\pi$
$431$		−	5.60782e10i		−	1.62512i	0.582879	−	0.812559i	$-0.301926\pi$
$432$			4.72065e9i			0.135540i
$433$			4.24360e10i			1.20721i	0.797284	+	0.603605i	$0.206269\pi$
$433$			4.24360e10i			1.20721i	−0.797284	+	0.603605i	$0.793731\pi$
$434$	−5.17676e9			−0.145915
$435$	−5.27002e10			−1.47182
$436$			7.87127e9i			0.217821i
$437$	3.75485e10	−	8.14212e9i	1.02960	−	0.223260i
$438$	3.64153e10			0.989435
$439$		−	4.34205e9i		−	0.116906i	−0.998290	−	0.0584531i	$-0.981383\pi$
$439$		−	4.34205e9i		−	0.116906i	0.998290	−	0.0584531i	$-0.0186168\pi$
$440$			1.14410e10i			0.305247i
$441$	−1.30360e9			−0.0344660
$442$	−9.66437e10			−2.53212
$443$	−2.62509e10			−0.681599			−0.340799	−	0.940136i	$-0.610698\pi$
$443$	−2.62509e10			−0.681599			−0.340799	+	0.940136i	$0.610698\pi$
$444$	−2.08719e10			−0.537070
$445$		−	1.78469e10i		−	0.455117i
$446$	−1.38126e10			−0.349089
$447$		−	8.16228e10i		−	2.04447i
$448$	5.09897e9			0.126582
$449$			6.19394e10i			1.52399i	0.647584	+	0.761994i	$0.275780\pi$
$449$			6.19394e10i			1.52399i	−0.647584	+	0.761994i	$0.724220\pi$
$450$			2.15982e10i			0.526705i
$451$			1.24833e10i			0.301732i
$452$			1.67147e10i			0.400447i
$453$	1.62765e10			0.386517
$454$	−9.31619e9			−0.219288
$455$			5.31805e10i			1.24081i
$456$	2.29185e10	−	4.96970e9i	0.530061	−	0.114940i
$457$	−1.13543e10			−0.260312			−0.130156	−	0.991494i	$-0.541548\pi$
$457$	−1.13543e10			−0.260312			−0.130156	+	0.991494i	$0.541548\pi$
$458$		−	5.85644e10i		−	1.33098i
$459$			4.71584e10i			1.06245i
$460$	−1.58152e10			−0.353218
$461$	−2.30267e10			−0.509833			−0.254917	−	0.966963i	$-0.582048\pi$
$461$	−2.30267e10			−0.509833			−0.254917	+	0.966963i	$0.582048\pi$
$462$	−6.44363e10			−1.41437
$463$	4.80160e10			1.04487			0.522435	−	0.852679i	$-0.325024\pi$
$463$	4.80160e10			1.04487			0.522435	+	0.852679i	$0.325024\pi$
$464$			1.65802e10i			0.357699i
$465$	9.80037e9			0.209619
$466$			1.62955e10i			0.345561i
$467$	−4.22787e10			−0.888903			−0.444452	−	0.895803i	$-0.646601\pi$
$467$	−4.22787e10			−0.888903			−0.444452	+	0.895803i	$0.646601\pi$
$468$			5.93199e10i			1.23656i
$469$			3.09549e10i			0.639790i
$470$			1.58986e10i			0.325813i
$471$			5.99934e10i			1.21905i
$472$	7.93843e9			0.159943
$473$	−2.35449e10			−0.470383
$474$			7.37241e10i			1.46048i
$475$	2.73810e10	−	5.93738e9i	0.537868	−	0.116633i
$476$	5.09378e10			0.992231
$477$		−	5.79638e10i		−	1.11965i
$478$		−	1.35556e10i		−	0.259661i
$479$	−5.20093e10			−0.987959			−0.493980	−	0.869474i	$-0.664458\pi$
$479$	−5.20093e10			−0.987959			−0.493980	+	0.869474i	$0.664458\pi$
$480$	−9.65311e9			−0.181845
$481$	−6.84873e10			−1.27947
$482$	5.60747e10			1.03891
$483$		−	8.90722e10i		−	1.63664i
$484$	−1.80492e10			−0.328909
$485$		−	5.43138e10i		−	0.981620i
$486$	−5.29587e10			−0.949276
$487$			3.08222e10i			0.547959i	0.961735	+	0.273980i	$0.0883401\pi$
$487$			3.08222e10i			0.547959i	−0.961735	+	0.273980i	$0.911660\pi$
$488$		−	1.18039e10i		−	0.208135i
$489$		−	1.00696e11i		−	1.76108i
$490$		−	6.96081e8i		−	0.0120747i
$491$	8.16859e10			1.40547			0.702735	−	0.711452i	$-0.251962\pi$
$491$	8.16859e10			1.40547			0.702735	+	0.711452i	$0.251962\pi$
$492$	−1.05325e10			−0.179751
$493$			1.65633e11i			2.80388i
$494$	7.52026e10	−	1.63071e10i	1.26277	−	0.273823i
$495$	7.01531e10			1.16849
$496$		−	3.08333e9i		−	0.0509440i
$497$			7.28364e10i			1.19378i
$498$	−5.44508e10			−0.885293
$499$	7.92783e10			1.27865			0.639326	−	0.768936i	$-0.279214\pi$
$499$	7.92783e10			1.27865			0.639326	+	0.768936i	$0.279214\pi$
$500$	−3.24873e10			−0.519797
$501$	1.77761e11			2.82154
$502$		−	4.91129e9i		−	0.0773359i
$503$	2.38054e10			0.371881			0.185940	−	0.982561i	$-0.440467\pi$
$503$	2.38054e10			0.371881			0.185940	+	0.982561i	$0.440467\pi$
$504$		−	3.12656e10i		−	0.484557i
$505$	6.69062e10			1.02873
$506$		−	6.28782e10i		−	0.959175i
$507$			2.37103e11i			3.58844i
$508$			9.84724e7i			0.00147863i
$509$		−	1.02167e11i		−	1.52208i	−0.648702	−	0.761042i	$-0.724688\pi$
$509$		−	1.02167e11i		−	1.52208i	0.648702	−	0.761042i	$-0.275312\pi$
$510$	−9.64328e10			−1.42542
$511$	−6.29793e10			−0.923665
$512$			3.03700e9i			0.0441942i
$513$	−7.95726e9	−	3.66960e10i	−0.114893	−	0.529845i
$514$	−5.93422e10			−0.850181
$515$		−	1.25943e8i		−	0.00179038i
$516$		−	1.98656e10i		−	0.280222i
$517$	−6.32099e10			−0.884755
$518$	3.60975e10			0.501370
$519$	−8.33789e10			−1.14918
$520$	−3.16748e10			−0.433213
$521$		−	1.32024e10i		−	0.179186i	−0.995978	−	0.0895929i	$-0.971443\pi$
$521$		−	1.32024e10i		−	0.179186i	0.995978	−	0.0895929i	$-0.0285566\pi$
$522$	1.01666e11			1.36928
$523$			8.84421e10i			1.18210i	0.806637	+	0.591048i	$0.201285\pi$
$523$			8.84421e10i			1.18210i	−0.806637	+	0.591048i	$0.798715\pi$
$524$	1.70806e10			0.226557
$525$		−	6.49531e10i		−	0.854993i
$526$		−	2.28136e10i		−	0.298024i
$527$		−	3.08019e10i		−	0.399333i
$528$		−	3.83789e10i		−	0.493807i
$529$	8.60731e9			0.109912
$530$	3.09507e10			0.392254
$531$		−	4.86764e10i		−	0.612267i
$532$	−3.96369e10	+	8.59498e9i	−0.494827	+	0.107300i
$533$	−3.45605e10			−0.428224
$534$			5.98678e10i			0.736255i
$535$		−	6.88614e10i		−	0.840544i
$536$	−1.84370e10			−0.223374
$537$	−8.15706e10			−0.980927
$538$	7.00112e10			0.835677
$539$	2.76749e9			0.0327892
$540$			1.54561e10i			0.181771i
$541$	−7.15785e10			−0.835591			−0.417796	−	0.908541i	$-0.637197\pi$
$541$	−7.15785e10			−0.835591			−0.417796	+	0.908541i	$0.637197\pi$
$542$			3.35905e9i			0.0389242i
$543$	−6.78436e10			−0.780386
$544$			3.03391e10i			0.346423i
$545$			2.57717e10i			0.292117i
$546$		−	1.78395e11i		−	2.00730i
$547$			4.33427e10i			0.484135i	0.970259	+	0.242068i	$0.0778256\pi$
$547$			4.33427e10i			0.484135i	−0.970259	+	0.242068i	$0.922174\pi$
$548$	5.92129e10			0.656590
$549$	−7.23782e10			−0.796744
$550$		−	4.58519e10i		−	0.501080i
$551$	−2.79480e10	−	1.28886e11i	−0.303211	−	1.39830i
$552$	5.30523e10			0.571410
$553$		−	1.27504e11i		−	1.36340i
$554$			3.71968e10i			0.394881i
$555$	−6.83379e10			−0.720261
$556$	−3.09699e10			−0.324071
$557$	−2.63069e9			−0.0273306			−0.0136653	−	0.999907i	$-0.504350\pi$
$557$	−2.63069e9			−0.0273306			−0.0136653	+	0.999907i	$0.504350\pi$
$558$	−1.89062e10			−0.195015
$559$		−	6.51850e10i		−	0.667576i
$560$	1.66948e10			0.169758
$561$		−	3.83399e11i		−	3.87079i
$562$	4.09217e10			0.410212
$563$			4.17573e10i			0.415622i	0.978169	+	0.207811i	$0.0666340\pi$
$563$			4.17573e10i			0.415622i	−0.978169	+	0.207811i	$0.933366\pi$
$564$		−	5.33322e10i		−	0.527076i
$565$			5.47265e10i			0.537036i
$566$			1.20714e11i			1.17623i
$567$	5.46019e10			0.528294
$568$	−4.33821e10			−0.416790
$569$			1.72406e11i			1.64476i	0.568938	+	0.822381i	$0.307355\pi$
$569$			1.72406e11i			1.64476i	−0.568938	+	0.822381i	$0.692645\pi$
$570$	7.50385e10	−	1.62716e10i	0.710861	−	0.154145i
$571$	7.55025e10			0.710259			0.355129	−	0.934817i	$-0.384437\pi$
$571$	7.55025e10			0.710259			0.355129	+	0.934817i	$0.384437\pi$
$572$		−	1.25933e11i		−	1.17640i
$573$			1.34723e9i			0.0124975i
$574$	1.82157e10			0.167803
$575$	6.33824e10			0.579826
$576$	1.86221e10			0.169176
$577$	6.69917e10			0.604391			0.302195	−	0.953246i	$-0.402280\pi$
$577$	6.69917e10			0.604391			0.302195	+	0.953246i	$0.402280\pi$
$578$			2.24160e11i			2.00839i
$579$	−8.15499e10			−0.725620
$580$			5.42860e10i			0.479707i
$581$	9.41713e10			0.826445
$582$			1.82197e11i			1.58799i
$583$			1.23054e11i			1.06518i
$584$		−	3.75111e10i		−	0.322484i
$585$			1.94222e11i			1.65835i
$586$	1.47863e10			0.125392
$587$	3.78872e10			0.319110			0.159555	−	0.987189i	$-0.448994\pi$
$587$	3.78872e10			0.319110			0.159555	+	0.987189i	$0.448994\pi$
$588$			2.33502e9i			0.0195335i
$589$	5.19735e9	+	2.39683e10i	0.0431838	+	0.199148i
$590$	2.59916e10			0.214499
$591$		−	5.63062e10i		−	0.461537i
$592$			2.15000e10i			0.175046i
$593$	4.38042e10			0.354239			0.177120	−	0.984189i	$-0.443322\pi$
$593$	4.38042e10			0.354239			0.177120	+	0.984189i	$0.443322\pi$
$594$	−6.14505e10			−0.493605
$595$	1.66778e11			1.33067
$596$	−8.40790e10			−0.666351
$597$			1.98561e11i			1.56314i
$598$	1.74081e11			1.36128
$599$		−	1.32046e11i		−	1.02569i	−0.858480	−	0.512847i	$-0.828591\pi$
$599$		−	1.32046e11i		−	1.02569i	0.858480	−	0.512847i	$-0.171409\pi$
$600$	3.86867e10			0.298509
$601$		−	2.59370e10i		−	0.198802i	−0.995047	−	0.0994012i	$-0.968307\pi$
$601$		−	2.59370e10i		−	0.198802i	0.995047	−	0.0994012i	$-0.0316927\pi$
$602$			3.43570e10i			0.261595i
$603$			1.13051e11i			0.855079i
$604$		−	1.67663e10i		−	0.125977i
$605$	−5.90957e10			−0.441097
$606$	−2.24438e11			−1.66420
$607$		−	1.84790e11i		−	1.36120i	−0.732654	−	0.680601i	$-0.761719\pi$
$607$		−	1.84790e11i		−	1.36120i	0.732654	−	0.680601i	$-0.238281\pi$
$608$	−5.11926e9	−	2.36081e10i	−0.0374621	−	0.172762i
$609$	−3.05743e11			−2.22273
$610$		−	3.86476e10i		−	0.279128i
$611$		−	1.74999e11i		−	1.25566i
$612$	1.86032e11			1.32611
$613$	1.13917e11			0.806767			0.403383	−	0.915031i	$-0.367834\pi$
$613$	1.13917e11			0.806767			0.403383	+	0.915031i	$0.367834\pi$
$614$	1.00485e11			0.707015
$615$	−3.44851e10			−0.241063
$616$			6.63754e10i			0.460982i
$617$	−2.19273e11			−1.51302			−0.756511	−	0.653981i	$-0.773098\pi$
$617$	−2.19273e11			−1.51302			−0.756511	+	0.653981i	$0.773098\pi$
$618$			4.22478e8i			0.00289635i
$619$	1.11363e11			0.758540			0.379270	−	0.925286i	$-0.376175\pi$
$619$	1.11363e11			0.758540			0.379270	+	0.925286i	$0.376175\pi$
$620$		−	1.00953e10i		−	0.0683206i
$621$		−	8.49449e10i		−	0.571177i
$622$			1.51206e10i			0.101020i
$623$		−	1.03540e11i		−	0.687314i
$624$	1.06254e11			0.700820
$625$	−2.23887e10			−0.146727
$626$		−	1.00552e11i		−	0.654776i
$627$	6.46926e10	+	2.98339e11i	0.418586	+	1.93037i
$628$	6.17987e10			0.397321
$629$			2.14782e11i			1.37213i
$630$		−	1.02368e11i		−	0.649835i
$631$	−1.41858e11			−0.894819			−0.447409	−	0.894329i	$-0.647653\pi$
$631$	−1.41858e11			−0.894819			−0.447409	+	0.894329i	$0.647653\pi$
$632$	7.59427e10			0.476012
$633$	−3.12531e11			−1.94661
$634$	−1.36338e11			−0.843841
$635$			3.22413e8i			0.00198298i
$636$	−1.03825e11			−0.634560
$637$			7.66191e9i			0.0465350i
$638$	−2.15831e11			−1.30266
$639$			2.66008e11i			1.59548i
$640$			9.94360e9i			0.0592684i
$641$			1.83546e11i			1.08721i	0.839341	+	0.543606i	$0.182941\pi$
$641$			1.83546e11i			1.08721i	−0.839341	+	0.543606i	$0.817059\pi$
$642$			2.30997e11i			1.35977i
$643$	−2.98123e11			−1.74402			−0.872011	−	0.489487i	$-0.837184\pi$
$643$	−2.98123e11			−1.74402			−0.872011	+	0.489487i	$0.837184\pi$
$644$	−9.17526e10			−0.533427
$645$		−	6.50428e10i		−	0.375803i
$646$	−5.11404e10	−	2.35841e11i	−0.293653	−	1.35422i
$647$	−4.95090e9			−0.0282531			−0.0141266	−	0.999900i	$-0.504497\pi$
$647$	−4.95090e9			−0.0282531			−0.0141266	+	0.999900i	$0.504497\pi$
$648$			3.25214e10i			0.184446i
$649$			1.03338e11i			0.582479i
$650$	1.26943e11			0.711141
$651$	5.68574e10			0.316565
$652$	−1.03727e11			−0.573984
$653$	−6.17884e10			−0.339824			−0.169912	−	0.985459i	$-0.554348\pi$
$653$	−6.17884e10			−0.339824			−0.169912	+	0.985459i	$0.554348\pi$
$654$		−	8.64517e10i		−	0.472566i
$655$	5.59244e10			0.303834
$656$			1.08495e10i			0.0585859i
$657$	−2.30009e11			−1.23448
$658$			9.22367e10i			0.492040i
$659$			2.67718e11i			1.41950i	0.704453	+	0.709751i	$0.251193\pi$
$659$			2.67718e11i			1.41950i	−0.704453	+	0.709751i	$0.748807\pi$
$660$		−	1.25658e11i		−	0.662240i
$661$		−	8.84369e10i		−	0.463263i	−0.972804	−	0.231631i	$-0.925594\pi$
$661$		−	8.84369e10i		−	0.463263i	0.972804	−	0.231631i	$-0.0744063\pi$
$662$	−8.12545e10			−0.423073
$663$	1.06146e12			5.49349
$664$			5.60894e10i			0.288541i
$665$	−1.29777e11	+	2.81412e10i	−0.663608	+	0.143899i
$666$	1.31833e11			0.670080
$667$		−	2.98349e11i		−	1.50738i
$668$		−	1.83111e11i		−	0.919619i
$669$	1.51707e11			0.757356
$670$	−6.03656e10			−0.299565
$671$	1.53655e11			0.757980
$672$	−5.60030e10			−0.274621
$673$		−	3.47693e11i		−	1.69487i	−0.530902	−	0.847433i	$-0.678147\pi$
$673$		−	3.47693e11i		−	1.69487i	0.530902	−	0.847433i	$-0.321853\pi$
$674$	−8.23884e10			−0.399233
$675$		−	6.19434e10i		−	0.298387i
$676$	2.44238e11			1.16957
$677$			3.61885e10i			0.172273i	0.996283	+	0.0861364i	$0.0274521\pi$
$677$			3.61885e10i			0.172273i	−0.996283	+	0.0861364i	$0.972548\pi$
$678$		−	1.83581e11i		−	0.868778i
$679$		−	3.15104e11i		−	1.48243i
$680$			9.93347e10i			0.464585i
$681$	1.02322e11			0.475750
$682$	4.01370e10			0.185527
$683$		−	1.32775e11i		−	0.610144i	−0.952329	−	0.305072i	$-0.901320\pi$
$683$		−	1.32775e11i		−	0.610144i	0.952329	−	0.305072i	$-0.0986805\pi$
$684$	−1.44759e11	+	3.13900e10i	−0.661335	+	0.143406i
$685$	1.93872e11			0.880548
$686$			1.54539e11i			0.697819i
$687$			6.43224e11i			2.88759i
$688$	−2.04634e10			−0.0913320
$689$	−3.40681e11			−1.51172
$690$	1.73701e11			0.766314
$691$	−1.82168e11			−0.799025			−0.399512	−	0.916728i	$-0.630821\pi$
$691$	−1.82168e11			−0.799025			−0.399512	+	0.916728i	$0.630821\pi$
$692$			8.58880e10i			0.374548i
$693$	4.06997e11			1.76465
$694$			1.41782e9i			0.00611200i
$695$	−1.01400e11			−0.434609
$696$		−	1.82103e11i		−	0.776035i
$697$			1.08384e11i			0.459235i
$698$			2.07225e10i			0.0873012i
$699$		−	1.78977e11i		−	0.749701i
$700$	−6.69077e10			−0.278666
$701$	1.01373e11			0.419808			0.209904	−	0.977722i	$-0.432685\pi$
$701$	1.01373e11			0.419808			0.209904	+	0.977722i	$0.432685\pi$
$702$		−	1.70129e11i		−	0.700534i
$703$	−3.62411e10	−	1.67131e11i	−0.148382	−	0.684281i
$704$	−3.95338e10			−0.160945
$705$		−	1.74618e11i		−	0.706857i
$706$			7.14636e10i			0.287651i
$707$	3.88160e11			1.55358
$708$	−8.71893e10			−0.347001
$709$	4.52517e11			1.79081			0.895407	−	0.445249i	$-0.146885\pi$
$709$	4.52517e11			1.79081			0.895407	+	0.445249i	$0.146885\pi$
$710$	−1.42040e11			−0.558954
$711$		−	4.65661e11i		−	1.82218i
$712$	6.16694e10			0.239966
$713$			5.54825e10i			0.214683i
$714$	−5.59460e11			−2.15266
$715$		−	4.12324e11i		−	1.57766i
$716$			8.40253e10i			0.319711i
$717$			1.48883e11i			0.563339i
$718$			8.49417e10i			0.319612i
$719$	−1.80755e11			−0.676353			−0.338177	−	0.941083i	$-0.609810\pi$
$719$	−1.80755e11			−0.676353			−0.338177	+	0.941083i	$0.609810\pi$
$720$	6.09716e10			0.226881
$721$		−	7.30665e8i		−	0.00270382i
$722$	7.95891e10	+	1.74889e11i	0.292890	+	0.643596i
$723$	−6.15879e11			−2.25394
$724$			6.98852e10i			0.254349i
$725$		−	2.17562e11i		−	0.787464i
$726$	1.98238e11			0.713575
$727$	3.76607e11			1.34819			0.674094	−	0.738645i	$-0.264534\pi$
$727$	3.76607e11			1.34819			0.674094	+	0.738645i	$0.264534\pi$
$728$	−1.83763e11			−0.654234
$729$	4.34315e11			1.53778
$730$		−	1.22817e11i		−	0.432481i
$731$	−2.04425e11			−0.715921
$732$			1.29644e11i			0.451552i
$733$	−1.71833e11			−0.595238			−0.297619	−	0.954685i	$-0.596192\pi$
$733$	−1.71833e11			−0.595238			−0.297619	+	0.954685i	$0.596192\pi$
$734$		−	4.00350e10i		−	0.137929i
$735$			7.64519e9i			0.0261962i
$736$		−	5.46488e10i		−	0.186238i
$737$		−	2.40002e11i		−	0.813477i
$738$	6.65262e10			0.224268
$739$	−1.46848e11			−0.492368			−0.246184	−	0.969223i	$-0.579177\pi$
$739$	−1.46848e11			−0.492368			−0.246184	+	0.969223i	$0.579177\pi$
$740$			7.03944e10i			0.234753i
$741$	−8.25965e11	+	1.79105e11i	−2.73961	+	0.594065i
$742$	1.79562e11			0.592379
$743$			4.41397e11i			1.44835i	0.689615	+	0.724177i	$0.257780\pi$
$743$			4.41397e11i			1.44835i	−0.689615	+	0.724177i	$0.742220\pi$
$744$			3.38648e10i			0.110524i
$745$	−2.75287e11			−0.893637
$746$	−3.60920e11			−1.16535
$747$	3.43926e11			1.10454
$748$	−3.94936e11			−1.26160
$749$		−	3.99503e11i		−	1.26938i
$750$	3.56814e11			1.12771
$751$			4.93724e11i			1.55212i	0.630661	+	0.776059i	$0.282784\pi$
$751$			4.93724e11i			1.55212i	−0.630661	+	0.776059i	$0.717216\pi$
$752$	−5.49371e10			−0.171789
$753$			5.39417e10i			0.167782i
$754$		−	5.97538e11i		−	1.84876i
$755$		−	5.48954e10i		−	0.168946i
$756$			8.96694e10i			0.274509i
$757$	1.16205e11			0.353868			0.176934	−	0.984223i	$-0.443382\pi$
$757$	1.16205e11			0.353868			0.176934	+	0.984223i	$0.443382\pi$
$758$	3.07575e11			0.931695
$759$			6.90603e11i			2.08095i
$760$	−1.67612e10	−	7.72966e10i	−0.0502402	−	0.231689i
$761$	−2.30030e10			−0.0685875			−0.0342938	−	0.999412i	$-0.510918\pi$
$761$	−2.30030e10			−0.0685875			−0.0342938	+	0.999412i	$0.510918\pi$
$762$		−	1.08154e9i		−	0.00320792i
$763$			1.49516e11i			0.441153i
$764$	1.38777e9			0.00407329
$765$	6.09095e11			1.77844
$766$	−4.39214e10			−0.127574
$767$	−2.86095e11			−0.826664
$768$		−	3.33560e10i		−	0.0958802i
$769$	−2.82954e11			−0.809115			−0.404557	−	0.914513i	$-0.632574\pi$
$769$	−2.82954e11			−0.809115			−0.404557	+	0.914513i	$0.632574\pi$
$770$			2.17323e11i			0.618219i
$771$	6.51767e11			1.84449
$772$			8.40040e10i			0.236500i
$773$			9.47008e10i			0.265238i	0.991167	+	0.132619i	$0.0423387\pi$
$773$			9.47008e10i			0.265238i	−0.991167	+	0.132619i	$0.957661\pi$
$774$			1.25476e11i			0.349621i
$775$			4.04588e10i			0.112152i
$776$	1.87679e11			0.517571
$777$	−3.96466e11			−1.08773
$778$			4.59549e11i			1.25433i
$779$	−1.82882e10	−	8.43384e10i	−0.0496616	−	0.229021i
$780$	3.47891e11			0.939864
$781$		−	5.64722e11i		−	1.51786i
$782$		−	5.45932e11i		−	1.45986i
$783$	−2.91576e11			−0.775718
$784$	2.40528e9			0.00636651
$785$	2.02338e11			0.532843
$786$	−1.87600e11			−0.491520
$787$			5.15172e11i			1.34293i	0.741037	+	0.671464i	$0.234334\pi$
$787$			5.15172e11i			1.34293i	−0.741037	+	0.671464i	$0.765666\pi$
$788$	−5.80006e10			−0.150428
$789$			2.50567e11i			0.646570i
$790$	2.48648e11			0.638375
$791$			3.17499e11i			0.811028i
$792$			2.42412e11i			0.616102i
$793$			4.25402e11i			1.07574i
$794$		−	3.81860e11i		−	0.960777i
$795$	−3.39938e11			−0.851003
$796$	2.04536e11			0.509470
$797$			1.73952e11i			0.431119i	0.976491	+	0.215559i	$0.0691575\pi$
$797$			1.73952e11i			0.431119i	−0.976491	+	0.215559i	$0.930843\pi$
$798$	4.35340e11	−	9.44003e10i	1.07354	−	0.232789i
$799$	−5.48812e11			−1.34659
$800$		−	3.98509e10i		−	0.0972923i
$801$		−	3.78141e11i		−	0.918594i
$802$	3.28021e11			0.792874
$803$	4.88297e11			1.17442
$804$	2.02498e11			0.484614
$805$	−3.00412e11			−0.715375
$806$			1.11121e11i			0.263303i
$807$	−7.68947e11			−1.81302
$808$			2.31192e11i			0.542409i
$809$	5.80910e11			1.35617			0.678086	−	0.734983i	$-0.262810\pi$
$809$	5.80910e11			1.35617			0.678086	+	0.734983i	$0.262810\pi$
$810$			1.06480e11i			0.247359i
$811$		−	6.95097e11i		−	1.60680i	−0.595439	−	0.803401i	$-0.703022\pi$
$811$		−	6.95097e11i		−	1.60680i	0.595439	−	0.803401i	$-0.296978\pi$
$812$			3.14943e11i			0.724450i
$813$		−	3.68931e10i		−	0.0844468i
$814$	−2.79875e11			−0.637479
$815$	−3.39616e11			−0.769765
$816$		−	3.33220e11i		−	0.751572i
$817$	1.59072e11	−	3.44936e10i	0.357031	−	0.0774195i
$818$	−2.97496e11			−0.664459
$819$			1.12679e12i			2.50442i
$820$			3.55228e10i			0.0785691i
$821$	1.93809e11			0.426581			0.213290	−	0.976989i	$-0.431582\pi$
$821$	1.93809e11			0.426581			0.213290	+	0.976989i	$0.431582\pi$
$822$	−6.50347e11			−1.42449
$823$	3.76429e10			0.0820511			0.0410255	−	0.999158i	$-0.486938\pi$
$823$	3.76429e10			0.0820511			0.0410255	+	0.999158i	$0.486938\pi$
$824$	4.35192e8			0.000944000
$825$			5.03601e11i			1.08710i
$826$	1.50792e11			0.323935
$827$			1.92643e11i			0.411843i	0.978569	+	0.205921i	$0.0660191\pi$
$827$			1.92643e11i			0.411843i	−0.978569	+	0.205921i	$0.933981\pi$
$828$	−3.35093e11			−0.712924
$829$			1.61600e11i			0.342156i	0.985258	+	0.171078i	$0.0547250\pi$
$829$			1.61600e11i			0.342156i	−0.985258	+	0.171078i	$0.945275\pi$
$830$			1.83645e11i			0.386961i
$831$		−	4.08540e11i		−	0.856702i
$832$		−	1.09451e11i		−	0.228417i
$833$	2.40283e10			0.0499050
$834$	3.40148e11			0.703079
$835$		−	5.99532e11i		−	1.23329i
$836$	3.07317e11	−	6.66394e10i	0.629160	−	0.136429i
$837$	5.42228e10			0.110479
$838$		−	3.44988e11i		−	0.699565i
$839$		−	4.81679e11i		−	0.972097i	−0.873932	−	0.486049i	$-0.838438\pi$
$839$		−	4.81679e11i		−	0.972097i	0.873932	−	0.486049i	$-0.161562\pi$
$840$	−1.83362e11			−0.368293
$841$	−5.23846e11			−1.04718
$842$	1.56126e10			0.0310618
$843$	−4.49451e11			−0.889962
$844$			3.21936e11i			0.634453i
$845$	7.99673e11			1.56850
$846$			3.36860e11i			0.657610i
$847$	−3.42847e11			−0.666142
$848$			1.06949e11i			0.206821i
$849$		−	1.32583e12i		−	2.55185i
$850$		−	3.98103e11i		−	0.762641i
$851$		−	3.86879e11i		−	0.737661i
$852$	4.76474e11			0.904234
$853$	6.06370e11			1.14536			0.572679	−	0.819780i	$-0.305904\pi$
$853$	6.06370e11			1.14536			0.572679	+	0.819780i	$0.305904\pi$
$854$		−	2.24216e11i		−	0.421536i
$855$	−4.73963e11	+	1.02775e11i	−0.886911	+	0.192320i
$856$	2.37948e11			0.443187
$857$			5.32627e11i			0.987416i	0.869628	+	0.493708i	$0.164359\pi$
$857$			5.32627e11i			0.987416i	−0.869628	+	0.493708i	$0.835641\pi$
$858$			1.38315e12i			2.55223i
$859$	1.13776e11			0.208966			0.104483	−	0.994527i	$-0.466681\pi$
$859$	1.13776e11			0.208966			0.104483	+	0.994527i	$0.466681\pi$
$860$	−6.70001e10			−0.122485
$861$	−2.00067e11			−0.364051
$862$	6.34452e11			1.14913
$863$			5.64549e10i			0.101779i	0.998704	+	0.0508896i	$0.0162057\pi$
$863$			5.64549e10i			0.101779i	−0.998704	+	0.0508896i	$0.983794\pi$
$864$	−5.34080e10			−0.0958410
$865$			2.81210e11i			0.502304i
$866$	−4.80108e11			−0.853626
$867$		−	2.46200e12i		−	4.35724i
$868$		−	5.85683e10i		−	0.103177i
$869$			9.88576e11i			1.73353i
$870$		−	5.96234e11i		−	1.04073i
$871$	6.64457e11			1.15450
$872$	−8.90533e10			−0.154022
$873$		−	1.15080e12i		−	1.98127i
$874$	9.21176e10	+	4.24813e11i	0.157869	+	0.728034i
$875$	−6.17101e11			−1.05275
$876$			4.11992e11i			0.699636i
$877$		−	5.39767e11i		−	0.912447i	−0.889865	−	0.456223i	$-0.849202\pi$
$877$		−	5.39767e11i		−	0.912447i	0.889865	−	0.456223i	$-0.150798\pi$
$878$	4.91247e10			0.0826651
$879$	−1.62401e11			−0.272040
$880$	−1.29440e11			−0.215843
$881$	1.13071e12			1.87692			0.938461	−	0.345385i	$-0.112252\pi$
$881$	1.13071e12			1.87692			0.938461	+	0.345385i	$0.112252\pi$
$882$		−	1.47486e10i		−	0.0243711i
$883$	7.01399e11			1.15378			0.576889	−	0.816822i	$-0.304266\pi$
$883$	7.01399e11			1.15378			0.576889	+	0.816822i	$0.304266\pi$
$884$		−	1.09340e12i		−	1.79048i
$885$	−2.85471e11			−0.465360
$886$		−	2.96995e11i		−	0.481963i
$887$			9.54811e11i			1.54249i	0.636537	+	0.771246i	$0.280366\pi$
$887$			9.54811e11i			1.54249i	−0.636537	+	0.771246i	$0.719634\pi$
$888$		−	2.36139e11i		−	0.379766i
$889$			1.87050e9i			0.00299468i
$890$	2.01915e11			0.321816
$891$	−4.23345e11			−0.671712
$892$		−	1.56272e11i		−	0.246843i
$893$	4.27054e11	−	9.26036e10i	0.671547	−	0.145620i
$894$	9.23457e11			1.44566
$895$			2.75111e11i			0.428762i
$896$			5.76883e10i			0.0895067i
$897$	−1.91197e12			−2.95332
$898$	−7.00764e11			−1.07762
$899$	1.90445e11			0.291562
$900$	−2.44356e11			−0.372437
$901$			1.06840e12i			1.62120i
$902$	−1.41232e11			−0.213357
$903$		−	3.77349e11i		−	0.567535i
$904$	−1.89105e11			−0.283159
$905$			2.28815e11i			0.341106i
$906$			1.84148e11i			0.273309i
$907$			6.62886e10i			0.0979511i	0.998800	+	0.0489756i	$0.0155956\pi$
$907$			6.62886e10i			0.0979511i	−0.998800	+	0.0489756i	$0.984404\pi$
$908$		−	1.05401e11i		−	0.155060i
$909$	1.41761e12			2.07635
$910$	−6.01669e11			−0.877388
$911$		−	4.30099e11i		−	0.624446i	−0.950009	−	0.312223i	$-0.898926\pi$
$911$		−	4.30099e11i		−	0.624446i	0.950009	−	0.312223i	$-0.101074\pi$
$912$	5.62258e10	+	2.59293e11i	0.0812748	+	0.374810i
$913$	−7.30138e11			−1.05080
$914$		−	1.28459e11i		−	0.184069i
$915$			4.24474e11i			0.605573i
$916$	6.62580e11			0.941145
$917$	3.24449e11			0.458848
$918$	−5.33537e11			−0.751266
$919$	6.09038e11			0.853852			0.426926	−	0.904287i	$-0.359597\pi$
$919$	6.09038e11			0.853852			0.426926	+	0.904287i	$0.359597\pi$
$920$		−	1.78928e11i		−	0.249763i
$921$	−1.10365e12			−1.53388
$922$		−	2.60517e11i		−	0.360506i
$923$	1.56346e12			2.15417
$924$		−	7.29014e11i		−	1.00011i
$925$		−	2.82119e11i		−	0.385359i
$926$			5.43240e11i			0.738835i
$927$		−	2.66849e9i		−	0.00361365i
$928$	−1.87583e11			−0.252931
$929$	3.42134e11			0.459338			0.229669	−	0.973269i	$-0.426236\pi$
$929$	3.42134e11			0.459338			0.229669	+	0.973269i	$0.426236\pi$
$930$			1.10879e11i			0.148223i
$931$	−1.86975e10	+	4.05441e9i	−0.0248877	+	0.00539671i
$932$	−1.84363e11			−0.244348
$933$		−	1.66072e11i		−	0.219164i
$934$		−	4.78329e11i		−	0.628550i
$935$	−1.29308e12			−1.69192
$936$	−6.71127e11			−0.874383
$937$	−6.89868e11			−0.894968			−0.447484	−	0.894292i	$-0.647680\pi$
$937$	−6.89868e11			−0.894968			−0.447484	+	0.894292i	$0.647680\pi$
$938$	−3.50214e11			−0.452400
$939$			1.10438e12i			1.42055i
$940$	−1.79872e11			−0.230384
$941$			4.74893e11i			0.605672i	0.953043	+	0.302836i	$0.0979334\pi$
$941$			4.74893e11i			0.605672i	−0.953043	+	0.302836i	$0.902067\pi$
$942$	−6.78748e11			−0.861995
$943$		−	1.95229e11i		−	0.246887i
$944$			8.98131e10i			0.113097i
$945$			2.93591e11i			0.368142i
$946$		−	2.66380e11i		−	0.332611i
$947$	−1.27186e12			−1.58140			−0.790699	−	0.612205i	$-0.790283\pi$
$947$	−1.27186e12			−1.58140			−0.790699	+	0.612205i	$0.790283\pi$
$948$	−8.34093e11			−1.03272
$949$			1.35187e12i			1.66675i
$950$	6.71738e10	+	3.09781e11i	0.0824718	+	0.380330i
$951$	1.49743e12			1.83073
$952$			5.76296e11i			0.701613i
$953$		−	4.65803e10i		−	0.0564716i	−0.999601	−	0.0282358i	$-0.991011\pi$
$953$		−	4.65803e10i		−	0.0564716i	0.999601	−	0.0282358i	$-0.00898893\pi$
$954$	6.55785e11			0.791714
$955$	4.54378e9			0.00546266
$956$	1.53364e11			0.183608
$957$	2.37051e12			2.82615
$958$		−	5.88418e11i		−	0.698593i
$959$	1.12476e12			1.32980
$960$		−	1.09212e11i		−	0.128584i
$961$	8.17475e11			0.958475
$962$		−	7.74846e11i		−	0.904722i
$963$		−	1.45904e12i		−	1.69653i
$964$			6.34413e11i			0.734621i
$965$			2.75042e11i			0.317168i
$966$	1.00774e12			1.15728
$967$	−1.15462e12			−1.32049			−0.660245	−	0.751050i	$-0.729547\pi$
$967$	−1.15462e12			−1.32049			−0.660245	+	0.751050i	$0.729547\pi$
$968$		−	2.04203e11i		−	0.232574i
$969$	5.61685e11	+	2.59029e12i	0.637086	+	2.93801i
$970$	6.14491e11			0.694110
$971$		−	6.13870e10i		−	0.0690557i	−0.999404	−	0.0345279i	$-0.989007\pi$
$971$		−	6.13870e10i		−	0.0690557i	0.999404	−	0.0345279i	$-0.0109927\pi$
$972$		−	5.99159e11i		−	0.671239i
$973$	−5.88277e11			−0.656343
$974$	−3.48714e11			−0.387466
$975$	−1.39424e12			−1.54284
$976$	1.33545e11			0.147173
$977$		−	6.15166e11i		−	0.675171i	−0.941295	−	0.337586i	$-0.890390\pi$
$977$		−	6.15166e11i		−	0.675171i	0.941295	−	0.337586i	$-0.109610\pi$
$978$	1.13925e12			1.24527
$979$			8.02775e11i			0.873903i
$980$	7.87525e9			0.00853808
$981$			5.46052e11i			0.589601i
$982$			9.24171e11i			0.993817i
$983$		−	3.06094e11i		−	0.327824i	−0.986475	−	0.163912i	$-0.947589\pi$
$983$		−	3.06094e11i		−	0.327824i	0.986475	−	0.163912i	$-0.0524114\pi$
$984$		−	1.19162e11i		−	0.127103i
$985$	−1.89903e11			−0.201737
$986$	−1.87392e12			−1.98264
$987$		−	1.01305e12i		−	1.06749i
$988$	1.84494e11	+	8.50820e11i	0.193622	+	0.892914i
$989$	3.68224e11			0.384882
$990$			7.93692e11i			0.826249i
$991$			1.16697e12i			1.20994i	0.796247	+	0.604972i	$0.206816\pi$
$991$			1.16697e12i			1.20994i	−0.796247	+	0.604972i	$0.793184\pi$
$992$	3.48839e10			0.0360229
$993$	8.92435e11			0.917866
$994$	−8.24050e11			−0.844127
$995$	6.69683e11			0.683246
$996$		−	6.16040e11i		−	0.625996i
$997$	−6.19793e11			−0.627287			−0.313643	−	0.949541i	$-0.601550\pi$
$997$	−6.19793e11			−0.627287			−0.313643	+	0.949541i	$0.601550\pi$
$998$			8.96931e11i			0.904143i
$999$	−3.78095e11			−0.379611

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.9.b.a.37.7	yes	12
3.2	odd	2		342.9.d.a.37.5		12
4.3	odd	2		304.9.e.e.113.11		12
19.18	odd	2	inner	38.9.b.a.37.6	✓	12
57.56	even	2		342.9.d.a.37.11		12
76.75	even	2		304.9.e.e.113.2		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.9.b.a.37.6	✓	12	19.18	odd	2	inner
38.9.b.a.37.7	yes	12	1.1	even	1	trivial
304.9.e.e.113.2		12	76.75	even	2
304.9.e.e.113.11		12	4.3	odd	2
342.9.d.a.37.5		12	3.2	odd	2
342.9.d.a.37.11		12	57.56	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 38 = 2 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	38.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+11.3137i q^{2} -124.261i q^{3} -128.000 q^{4} -419.091 q^{5} +1405.85 q^{6} -2431.38 q^{7} -1448.15i q^{8} -8879.72 q^{9} +O(q^{10})\)	\(q+11.3137i q^{2} -124.261i q^{3} -128.000 q^{4} -419.091 q^{5} +1405.85 q^{6} -2431.38 q^{7} -1448.15i q^{8} -8879.72 q^{9} -4741.48i q^{10} +18851.2 q^{11} +15905.4i q^{12} +52190.4i q^{13} -27507.9i q^{14} +52076.6i q^{15} +16384.0 q^{16} +163673. q^{17} -100463. i q^{18} +(-127361. + 27617.3i) q^{19} +53643.7 q^{20} +302125. i q^{21} +213277. i q^{22} -294819. q^{23} -179949. q^{24} -214988. q^{25} -590467. q^{26} +288125. i q^{27} +311217. q^{28} +1.01197e6i q^{29} -589179. q^{30} -188192. i q^{31} +185364. i q^{32} -2.34246e6i q^{33} +1.85175e6i q^{34} +1.01897e6 q^{35} +1.13660e6 q^{36} +1.31226e6i q^{37} +(-312455. - 1.44093e6i) q^{38} +6.48522e6 q^{39} +606909. i q^{40} +662199. i q^{41} -3.41815e6 q^{42} -1.24898e6 q^{43} -2.41295e6 q^{44} +3.72141e6 q^{45} -3.33550e6i q^{46} -3.35309e6 q^{47} -2.03589e6i q^{48} +146807. q^{49} -2.43231e6i q^{50} -2.03381e7i q^{51} -6.68038e6i q^{52} +6.52766e6i q^{53} -3.25977e6 q^{54} -7.90038e6 q^{55} +3.52101e6i q^{56} +(3.43175e6 + 1.58260e7i) q^{57} -1.14492e7 q^{58} +5.48176e6i q^{59} -6.66580e6i q^{60} +8.15096e6 q^{61} +2.12914e6 q^{62} +2.15900e7 q^{63} -2.09715e6 q^{64} -2.18726e7i q^{65} +2.65020e7 q^{66} -1.27314e7i q^{67} -2.09502e7 q^{68} +3.66344e7i q^{69} +1.15283e7i q^{70} -2.99568e7i q^{71} +1.28592e7i q^{72} +2.59027e7 q^{73} -1.48465e7 q^{74} +2.67145e7i q^{75} +(1.63022e7 - 3.53502e6i) q^{76} -4.58344e7 q^{77} +7.33719e7i q^{78} +5.24410e7i q^{79} -6.86639e6 q^{80} -2.24572e7 q^{81} -7.49193e6 q^{82} -3.87316e7 q^{83} -3.86720e7i q^{84} -6.85940e7 q^{85} -1.41306e7i q^{86} +1.25749e8 q^{87} -2.72995e7i q^{88} +4.25848e7i q^{89} +4.21030e7i q^{90} -1.26895e8i q^{91} +3.77368e7 q^{92} -2.33848e7 q^{93} -3.79359e7i q^{94} +(5.33759e7 - 1.15742e7i) q^{95} +2.30334e7 q^{96} +1.29599e8i q^{97} +1.66093e6i q^{98} -1.67393e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 1536 q^{4} + 558 q^{5} + 1792 q^{6} - 5422 q^{7} - 15592 q^{9}+O(q^{10}) \) 12 * q - 1536 * q^4 + 558 * q^5 + 1792 * q^6 - 5422 * q^7 - 15592 * q^9	\( 12 q - 1536 q^{4} + 558 q^{5} + 1792 q^{6} - 5422 q^{7} - 15592 q^{9} - 12546 q^{11} + 196608 q^{16} + 270810 q^{17} + 41512 q^{19} - 71424 q^{20} - 823956 q^{23} - 229376 q^{24} + 865538 q^{25} - 431616 q^{26} + 694016 q^{28} + 71168 q^{30} - 1194378 q^{35} + 1995776 q^{36} + 998784 q^{38} + 5786100 q^{39} - 8383744 q^{42} + 7586646 q^{43} + 1605888 q^{44} + 2226046 q^{45} - 20260530 q^{47} - 19498842 q^{49} + 16933888 q^{54} - 14858554 q^{55} + 14430564 q^{57} - 5506560 q^{58} - 41363266 q^{61} + 32266752 q^{62} + 84235798 q^{63} - 25165824 q^{64} + 14371328 q^{66} - 34663680 q^{68} + 87906498 q^{73} - 2149632 q^{74} - 5313536 q^{76} - 78817962 q^{77} + 9142272 q^{80} - 100904812 q^{81} - 49609728 q^{82} - 55944960 q^{83} + 25440254 q^{85} + 119189604 q^{87} + 105466368 q^{92} + 105500856 q^{93} + 81396774 q^{95} + 29360128 q^{96} - 85554938 q^{99}+O(q^{100}) \) 12 * q - 1536 * q^4 + 558 * q^5 + 1792 * q^6 - 5422 * q^7 - 15592 * q^9 - 12546 * q^11 + 196608 * q^16 + 270810 * q^17 + 41512 * q^19 - 71424 * q^20 - 823956 * q^23 - 229376 * q^24 + 865538 * q^25 - 431616 * q^26 + 694016 * q^28 + 71168 * q^30 - 1194378 * q^35 + 1995776 * q^36 + 998784 * q^38 + 5786100 * q^39 - 8383744 * q^42 + 7586646 * q^43 + 1605888 * q^44 + 2226046 * q^45 - 20260530 * q^47 - 19498842 * q^49 + 16933888 * q^54 - 14858554 * q^55 + 14430564 * q^57 - 5506560 * q^58 - 41363266 * q^61 + 32266752 * q^62 + 84235798 * q^63 - 25165824 * q^64 + 14371328 * q^66 - 34663680 * q^68 + 87906498 * q^73 - 2149632 * q^74 - 5313536 * q^76 - 78817962 * q^77 + 9142272 * q^80 - 100904812 * q^81 - 49609728 * q^82 - 55944960 * q^83 + 25440254 * q^85 + 119189604 * q^87 + 105466368 * q^92 + 105500856 * q^93 + 81396774 * q^95 + 29360128 * q^96 - 85554938 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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