LMFDB - Embedded newform 38.8.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [38,8,Mod(1,38)]

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([0]))

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("38.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 38 = 2 \cdot 19 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	38.a (trivial)

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:	yes
Analytic conductor:	$11.8706309684$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{633}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 158 $ x^2 - x - 158
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$13.0797$ of defining polynomial
Character	$\chi$	$=$	38.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+8.00000 q^{2} -72.2392 q^{3} +64.0000 q^{4} +467.472 q^{5} -577.914 q^{6} -1471.23 q^{7} +512.000 q^{8} +3031.51 q^{9} +O(q^{10})$	\(q+8.00000 q^{2} -72.2392 q^{3} +64.0000 q^{4} +467.472 q^{5} -577.914 q^{6} -1471.23 q^{7} +512.000 q^{8} +3031.51 q^{9} +3739.78 q^{10} -3547.04 q^{11} -4623.31 q^{12} -12361.8 q^{13} -11769.9 q^{14} -33769.8 q^{15} +4096.00 q^{16} -13486.3 q^{17} +24252.1 q^{18} -6859.00 q^{19} +29918.2 q^{20} +106281. q^{21} -28376.3 q^{22} -94084.9 q^{23} -36986.5 q^{24} +140405. q^{25} -98894.4 q^{26} -61006.6 q^{27} -94158.9 q^{28} +33541.0 q^{29} -270159. q^{30} +212096. q^{31} +32768.0 q^{32} +256236. q^{33} -107890. q^{34} -687760. q^{35} +194016. q^{36} +33656.5 q^{37} -54872.0 q^{38} +893007. q^{39} +239346. q^{40} -480531. q^{41} +850246. q^{42} -561340. q^{43} -227011. q^{44} +1.41715e6 q^{45} -752679. q^{46} +478445. q^{47} -295892. q^{48} +1.34098e6 q^{49} +1.12324e6 q^{50} +974237. q^{51} -791156. q^{52} -93127.0 q^{53} -488053. q^{54} -1.65814e6 q^{55} -753271. q^{56} +495489. q^{57} +268328. q^{58} +955034. q^{59} -2.16127e6 q^{60} -1.71632e6 q^{61} +1.69677e6 q^{62} -4.46005e6 q^{63} +262144. q^{64} -5.77880e6 q^{65} +2.04988e6 q^{66} +476255. q^{67} -863120. q^{68} +6.79662e6 q^{69} -5.50208e6 q^{70} +1.95518e6 q^{71} +1.55213e6 q^{72} +1.21632e6 q^{73} +269252. q^{74} -1.01428e7 q^{75} -438976. q^{76} +5.21852e6 q^{77} +7.14406e6 q^{78} +3.94544e6 q^{79} +1.91477e6 q^{80} -2.22284e6 q^{81} -3.84425e6 q^{82} -3.54252e6 q^{83} +6.80197e6 q^{84} -6.30445e6 q^{85} -4.49072e6 q^{86} -2.42298e6 q^{87} -1.81609e6 q^{88} -7.50202e6 q^{89} +1.13372e7 q^{90} +1.81871e7 q^{91} -6.02143e6 q^{92} -1.53216e7 q^{93} +3.82756e6 q^{94} -3.20639e6 q^{95} -2.36714e6 q^{96} +1.27974e7 q^{97} +1.07279e7 q^{98} -1.07529e7 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 16 q^{2} - 69 q^{3} + 128 q^{4} + 155 q^{5} - 552 q^{6} - 2238 q^{7} + 1024 q^{8} + 855 q^{9}+O(q^{10}) $ 2 * q + 16 * q^2 - 69 * q^3 + 128 * q^4 + 155 * q^5 - 552 * q^6 - 2238 * q^7 + 1024 * q^8 + 855 * q^9	\( 2 q + 16 q^{2} - 69 q^{3} + 128 q^{4} + 155 q^{5} - 552 q^{6} - 2238 q^{7} + 1024 q^{8} + 855 q^{9} + 1240 q^{10} - 3295 q^{11} - 4416 q^{12} - 13427 q^{13} - 17904 q^{14} - 34782 q^{15} + 8192 q^{16} - 32256 q^{17} + 6840 q^{18} - 13718 q^{19} + 9920 q^{20} + 103797 q^{21} - 26360 q^{22} - 82525 q^{23} - 35328 q^{24} + 159919 q^{25} - 107416 q^{26} - 75141 q^{27} - 143232 q^{28} - 12749 q^{29} - 278256 q^{30} + 258944 q^{31} + 65536 q^{32} + 257052 q^{33} - 258048 q^{34} - 448167 q^{35} + 54720 q^{36} - 149260 q^{37} - 109744 q^{38} + 889557 q^{39} + 79360 q^{40} + 339130 q^{41} + 830376 q^{42} - 83869 q^{43} - 210880 q^{44} + 2097243 q^{45} - 660200 q^{46} + 1471025 q^{47} - 282624 q^{48} + 1105372 q^{49} + 1279352 q^{50} + 913437 q^{51} - 859328 q^{52} - 945643 q^{53} - 601128 q^{54} - 1736899 q^{55} - 1145856 q^{56} + 473271 q^{57} - 101992 q^{58} - 969009 q^{59} - 2226048 q^{60} - 1506755 q^{61} + 2071552 q^{62} - 2791179 q^{63} + 524288 q^{64} - 5445956 q^{65} + 2056416 q^{66} - 1848219 q^{67} - 2064384 q^{68} + 6834063 q^{69} - 3585336 q^{70} - 3417184 q^{71} + 437760 q^{72} - 2499822 q^{73} - 1194080 q^{74} - 10079553 q^{75} - 877952 q^{76} + 5025267 q^{77} + 7116456 q^{78} + 2636926 q^{79} + 634880 q^{80} + 2491398 q^{81} + 2713040 q^{82} - 10059354 q^{83} + 6643008 q^{84} - 439425 q^{85} - 670952 q^{86} - 2572923 q^{87} - 1687040 q^{88} - 3506160 q^{89} + 16777944 q^{90} + 19003851 q^{91} - 5281600 q^{92} - 15169884 q^{93} + 11768200 q^{94} - 1063145 q^{95} - 2260992 q^{96} + 5893526 q^{97} + 8842976 q^{98} - 11301453 q^{99}+O(q^{100}) \) 2 * q + 16 * q^2 - 69 * q^3 + 128 * q^4 + 155 * q^5 - 552 * q^6 - 2238 * q^7 + 1024 * q^8 + 855 * q^9 + 1240 * q^10 - 3295 * q^11 - 4416 * q^12 - 13427 * q^13 - 17904 * q^14 - 34782 * q^15 + 8192 * q^16 - 32256 * q^17 + 6840 * q^18 - 13718 * q^19 + 9920 * q^20 + 103797 * q^21 - 26360 * q^22 - 82525 * q^23 - 35328 * q^24 + 159919 * q^25 - 107416 * q^26 - 75141 * q^27 - 143232 * q^28 - 12749 * q^29 - 278256 * q^30 + 258944 * q^31 + 65536 * q^32 + 257052 * q^33 - 258048 * q^34 - 448167 * q^35 + 54720 * q^36 - 149260 * q^37 - 109744 * q^38 + 889557 * q^39 + 79360 * q^40 + 339130 * q^41 + 830376 * q^42 - 83869 * q^43 - 210880 * q^44 + 2097243 * q^45 - 660200 * q^46 + 1471025 * q^47 - 282624 * q^48 + 1105372 * q^49 + 1279352 * q^50 + 913437 * q^51 - 859328 * q^52 - 945643 * q^53 - 601128 * q^54 - 1736899 * q^55 - 1145856 * q^56 + 473271 * q^57 - 101992 * q^58 - 969009 * q^59 - 2226048 * q^60 - 1506755 * q^61 + 2071552 * q^62 - 2791179 * q^63 + 524288 * q^64 - 5445956 * q^65 + 2056416 * q^66 - 1848219 * q^67 - 2064384 * q^68 + 6834063 * q^69 - 3585336 * q^70 - 3417184 * q^71 + 437760 * q^72 - 2499822 * q^73 - 1194080 * q^74 - 10079553 * q^75 - 877952 * q^76 + 5025267 * q^77 + 7116456 * q^78 + 2636926 * q^79 + 634880 * q^80 + 2491398 * q^81 + 2713040 * q^82 - 10059354 * q^83 + 6643008 * q^84 - 439425 * q^85 - 670952 * q^86 - 2572923 * q^87 - 1687040 * q^88 - 3506160 * q^89 + 16777944 * q^90 + 19003851 * q^91 - 5281600 * q^92 - 15169884 * q^93 + 11768200 * q^94 - 1063145 * q^95 - 2260992 * q^96 + 5893526 * q^97 + 8842976 * q^98 - 11301453 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	8.00000	0.707107
$2$	8.00000	0.707107
$3$	−72.2392	−1.54472	−0.772358	−	0.635187i	$-0.780923\pi$
$3$	−72.2392	−1.54472	−0.772358	+	0.635187i	$0.780923\pi$
$4$	64.0000	0.500000
$5$	467.472	1.67248	0.836240	−	0.548364i	$-0.184749\pi$
$5$	467.472	1.67248	0.836240	+	0.548364i	$0.184749\pi$
$6$	−577.914	−1.09228
$7$	−1471.23	−1.62121	−0.810603	−	0.585596i	$-0.800861\pi$
$7$	−1471.23	−1.62121	−0.810603	+	0.585596i	$0.800861\pi$
$8$	512.000	0.353553
$9$	3031.51	1.38615
$10$	3739.78	1.18262
$11$	−3547.04	−0.803511	−0.401756	−	0.915747i	$-0.631600\pi$
$11$	−3547.04	−0.803511	−0.401756	+	0.915747i	$0.631600\pi$
$12$	−4623.31	−0.772358
$13$	−12361.8	−1.56056	−0.780279	−	0.625431i	$-0.784923\pi$
$13$	−12361.8	−1.56056	−0.780279	+	0.625431i	$0.784923\pi$
$14$	−11769.9	−1.14637
$15$	−33769.8	−2.58351
$16$	4096.00	0.250000
$17$	−13486.3	−0.665764	−0.332882	−	0.942969i	$-0.608021\pi$
$17$	−13486.3	−0.665764	−0.332882	+	0.942969i	$0.608021\pi$
$18$	24252.1	0.980155
$19$	−6859.00	−0.229416
$19$	−6859.00	−0.229416
$20$	29918.2	0.836240
$21$	106281.	2.50430
$22$	−28376.3	−0.568168
$23$	−94084.9	−1.61240	−0.806199	−	0.591644i	$-0.798479\pi$
$23$	−94084.9	−1.61240	−0.806199	+	0.591644i	$0.798479\pi$
$24$	−36986.5	−0.546140
$25$	140405.	1.79719
$26$	−98894.4	−1.10348
$27$	−61006.6	−0.596490
$28$	−94158.9	−0.810603
$29$	33541.0	0.255378	0.127689	−	0.991814i	$-0.459244\pi$
$29$	33541.0	0.255378	0.127689	+	0.991814i	$0.459244\pi$
$30$	−270159.	−1.82681
$31$	212096.	1.27869	0.639346	−	0.768919i	$-0.279205\pi$
$31$	212096.	1.27869	0.639346	+	0.768919i	$0.279205\pi$
$32$	32768.0	0.176777
$33$	256236.	1.24120
$34$	−107890.	−0.470766
$35$	−687760.	−2.71143
$36$	194016.	0.693074
$37$	33656.5	0.109235	0.0546176	−	0.998507i	$-0.482606\pi$
$37$	33656.5	0.109235	0.0546176	+	0.998507i	$0.482606\pi$
$38$	−54872.0	−0.162221
$39$	893007.	2.41062
$40$	239346.	0.591311
$41$	−480531.	−1.08887	−0.544437	−	0.838801i	$-0.683257\pi$
$41$	−480531.	−1.08887	−0.544437	+	0.838801i	$0.683257\pi$
$42$	850246.	1.77081
$43$	−561340.	−1.07668	−0.538339	−	0.842728i	$-0.680948\pi$
$43$	−561340.	−1.07668	−0.538339	+	0.842728i	$0.680948\pi$
$44$	−227011.	−0.401756
$45$	1.41715e6	2.31830
$46$	−752679.	−1.14014
$47$	478445.	0.672187	0.336093	−	0.941829i	$-0.390894\pi$
$47$	478445.	0.672187	0.336093	+	0.941829i	$0.390894\pi$
$48$	−295892.	−0.386179
$49$	1.34098e6	1.62831
$50$	1.12324e6	1.27080
$51$	974237.	1.02842
$52$	−791156.	−0.780279
$53$	−93127.0	−0.0859232	−0.0429616	−	0.999077i	$-0.513679\pi$
$53$	−93127.0	−0.0859232	−0.0429616	+	0.999077i	$0.513679\pi$
$54$	−488053.	−0.421782
$55$	−1.65814e6	−1.34386
$56$	−753271.	−0.573183
$57$	495489.	0.354382
$58$	268328.	0.180579
$59$	955034.	0.605392	0.302696	−	0.953087i	$-0.402113\pi$
$59$	955034.	0.605392	0.302696	+	0.953087i	$0.402113\pi$
$60$	−2.16127e6	−1.29175
$61$	−1.71632e6	−0.968152	−0.484076	−	0.875026i	$-0.660844\pi$
$61$	−1.71632e6	−0.968152	−0.484076	+	0.875026i	$0.660844\pi$
$62$	1.69677e6	0.904172
$63$	−4.46005e6	−2.24723
$64$	262144.	0.125000
$65$	−5.77880e6	−2.61000
$66$	2.04988e6	0.877659
$67$	476255.	0.193454	0.0967270	−	0.995311i	$-0.469163\pi$
$67$	476255.	0.193454	0.0967270	+	0.995311i	$0.469163\pi$
$68$	−863120.	−0.332882
$69$	6.79662e6	2.49070
$70$	−5.50208e6	−1.91727
$71$	1.95518e6	0.648311	0.324155	−	0.946004i	$-0.394920\pi$
$71$	1.95518e6	0.648311	0.324155	+	0.946004i	$0.394920\pi$
$72$	1.55213e6	0.490078
$73$	1.21632e6	0.365948	0.182974	−	0.983118i	$-0.441428\pi$
$73$	1.21632e6	0.365948	0.182974	+	0.983118i	$0.441428\pi$
$74$	269252.	0.0772409
$75$	−1.01428e7	−2.77614
$76$	−438976.	−0.114708
$77$	5.21852e6	1.30266
$78$	7.14406e6	1.70457
$79$	3.94544e6	0.900328	0.450164	−	0.892946i	$-0.351366\pi$
$79$	3.94544e6	0.900328	0.450164	+	0.892946i	$0.351366\pi$
$80$	1.91477e6	0.418120
$81$	−2.22284e6	−0.464740
$82$	−3.84425e6	−0.769951
$83$	−3.54252e6	−0.680048	−0.340024	−	0.940417i	$-0.610435\pi$
$83$	−3.54252e6	−0.680048	−0.340024	+	0.940417i	$0.610435\pi$
$84$	6.80197e6	1.25215
$85$	−6.30445e6	−1.11348
$86$	−4.49072e6	−0.761327
$87$	−2.42298e6	−0.394487
$88$	−1.81609e6	−0.284084
$89$	−7.50202e6	−1.12801	−0.564006	−	0.825771i	$-0.690740\pi$
$89$	−7.50202e6	−1.12801	−0.564006	+	0.825771i	$0.690740\pi$
$90$	1.13372e7	1.63929
$91$	1.81871e7	2.52999
$92$	−6.02143e6	−0.806199
$93$	−1.53216e7	−1.97522
$94$	3.82756e6	0.475308
$95$	−3.20639e6	−0.383693
$96$	−2.36714e6	−0.273070
$97$	1.27974e7	1.42371	0.711855	−	0.702326i	$-0.247855\pi$
$97$	1.27974e7	1.42371	0.711855	+	0.702326i	$0.247855\pi$
$98$	1.07279e7	1.15139
$99$	−1.07529e7	−1.11379
$100$	8.98593e6	0.898593
$101$	332855.	0.0321463	0.0160731	−	0.999871i	$-0.494884\pi$
$101$	332855.	0.0321463	0.0160731	+	0.999871i	$0.494884\pi$
$102$	7.79389e6	0.727200
$103$	4.52517e6	0.408042	0.204021	−	0.978967i	$-0.434599\pi$
$103$	4.52517e6	0.408042	0.204021	+	0.978967i	$0.434599\pi$
$104$	−6.32924e6	−0.551741
$105$	4.96833e7	4.18840
$106$	−745016.	−0.0607568
$107$	7.33885e6	0.579141	0.289571	−	0.957157i	$-0.406487\pi$
$107$	7.33885e6	0.579141	0.289571	+	0.957157i	$0.406487\pi$
$108$	−3.90442e6	−0.298245
$109$	−1.03920e7	−0.768612	−0.384306	−	0.923206i	$-0.625559\pi$
$109$	−1.03920e7	−0.768612	−0.384306	+	0.923206i	$0.625559\pi$
$110$	−1.32651e7	−0.950249
$111$	−2.43132e6	−0.168737
$112$	−6.02617e6	−0.405302
$113$	5.33971e6	0.348131	0.174066	−	0.984734i	$-0.444309\pi$
$113$	5.33971e6	0.348131	0.174066	+	0.984734i	$0.444309\pi$
$114$	3.96391e6	0.250586
$115$	−4.39820e7	−2.69670
$116$	2.14663e6	0.127689
$117$	−3.74749e7	−2.16317
$118$	7.64027e6	0.428077
$119$	1.98414e7	1.07934
$120$	−1.72902e7	−0.913407
$121$	−6.90567e6	−0.354370
$122$	−1.37306e7	−0.684587
$123$	3.47132e7	1.68200
$124$	1.35741e7	0.639346
$125$	2.91142e7	1.33328
$126$	−3.56804e7	−1.58903
$127$	−3.43331e7	−1.48730	−0.743652	−	0.668566i	$-0.766908\pi$
$127$	−3.43331e7	−1.48730	−0.743652	+	0.668566i	$0.766908\pi$
$128$	2.09715e6	0.0883883
$129$	4.05507e7	1.66316
$130$	−4.62304e7	−1.84555
$131$	250925.	0.00975202	0.00487601	−	0.999988i	$-0.498448\pi$
$131$	250925.	0.00975202	0.00487601	+	0.999988i	$0.498448\pi$
$132$	1.63991e7	0.620598
$133$	1.00912e7	0.371930
$134$	3.81004e6	0.136793
$135$	−2.85189e7	−0.997618
$136$	−6.90496e6	−0.235383
$137$	−2.54613e7	−0.845978	−0.422989	−	0.906135i	$-0.639019\pi$
$137$	−2.54613e7	−0.845978	−0.422989	+	0.906135i	$0.639019\pi$
$138$	5.43729e7	1.76119
$139$	4.33237e7	1.36828	0.684138	−	0.729352i	$-0.260178\pi$
$139$	4.33237e7	1.36828	0.684138	+	0.729352i	$0.260178\pi$
$140$	−4.40167e7	−1.35572
$141$	−3.45625e7	−1.03834
$142$	1.56415e7	0.458425
$143$	4.38478e7	1.25393
$144$	1.24171e7	0.346537
$145$	1.56795e7	0.427114
$146$	9.73058e6	0.258764
$147$	−9.68716e7	−2.51528
$148$	2.15401e6	0.0546176
$149$	−3.98620e7	−0.987204	−0.493602	−	0.869688i	$-0.664320\pi$
$149$	−3.98620e7	−0.987204	−0.493602	+	0.869688i	$0.664320\pi$
$150$	−8.11421e7	−1.96303
$151$	−2.13810e6	−0.0505369	−0.0252684	−	0.999681i	$-0.508044\pi$
$151$	−2.13810e6	−0.0505369	−0.0252684	+	0.999681i	$0.508044\pi$
$152$	−3.51181e6	−0.0811107
$153$	−4.08837e7	−0.922847
$154$	4.17482e7	0.921118
$155$	9.91489e7	2.13859
$156$	5.71525e7	1.20531
$157$	−3.04702e7	−0.628387	−0.314193	−	0.949359i	$-0.601734\pi$
$157$	−3.04702e7	−0.628387	−0.314193	+	0.949359i	$0.601734\pi$
$158$	3.15635e7	0.636628
$159$	6.72743e6	0.132727
$160$	1.53181e7	0.295655
$161$	1.38421e8	2.61403
$162$	−1.77827e7	−0.328621
$163$	−9.48952e7	−1.71628	−0.858139	−	0.513417i	$-0.828379\pi$
$163$	−9.48952e7	−1.71628	−0.858139	+	0.513417i	$0.828379\pi$
$164$	−3.07540e7	−0.544437
$165$	1.19783e8	2.07588
$166$	−2.83402e7	−0.480867
$167$	−1.10332e8	−1.83313	−0.916563	−	0.399890i	$-0.869048\pi$
$167$	−1.10332e8	−1.83313	−0.916563	+	0.399890i	$0.869048\pi$
$168$	5.44157e7	0.885405
$169$	9.00657e7	1.43534
$170$	−5.04356e7	−0.787346
$171$	−2.07931e7	−0.318004
$172$	−3.59257e7	−0.538339
$173$	1.06112e8	1.55813	0.779066	−	0.626942i	$-0.215694\pi$
$173$	1.06112e8	1.55813	0.779066	+	0.626942i	$0.215694\pi$
$174$	−1.93838e7	−0.278944
$175$	−2.06569e8	−2.91361
$176$	−1.45287e7	−0.200878
$177$	−6.89909e7	−0.935159
$178$	−6.00162e7	−0.797624
$179$	4.88351e7	0.636424	0.318212	−	0.948020i	$-0.396918\pi$
$179$	4.88351e7	0.636424	0.318212	+	0.948020i	$0.396918\pi$
$180$	9.06973e7	1.15915
$181$	5.52074e6	0.0692026	0.0346013	−	0.999401i	$-0.488984\pi$
$181$	5.52074e6	0.0692026	0.0346013	+	0.999401i	$0.488984\pi$
$182$	1.45497e8	1.78897
$183$	1.23986e8	1.49552
$184$	−4.81714e7	−0.570069
$185$	1.57335e7	0.182693
$186$	−1.22573e8	−1.39669
$187$	4.78363e7	0.534948
$188$	3.06205e7	0.336093
$189$	8.97549e7	0.967034
$190$	−2.56511e7	−0.271312
$191$	−1.15562e8	−1.20005	−0.600023	−	0.799983i	$-0.704842\pi$
$191$	−1.15562e8	−1.20005	−0.600023	+	0.799983i	$0.704842\pi$
$192$	−1.89371e7	−0.193090
$193$	4.01954e6	0.0402463	0.0201232	−	0.999798i	$-0.493594\pi$
$193$	4.01954e6	0.0402463	0.0201232	+	0.999798i	$0.493594\pi$
$194$	1.02379e8	1.00672
$195$	4.17456e8	4.03171
$196$	8.58229e7	0.814155
$197$	8.92289e7	0.831522	0.415761	−	0.909474i	$-0.363515\pi$
$197$	8.92289e7	0.831522	0.415761	+	0.909474i	$0.363515\pi$
$198$	−8.60231e7	−0.787566
$199$	−1.73811e8	−1.56348	−0.781740	−	0.623604i	$-0.785668\pi$
$199$	−1.73811e8	−1.56348	−0.781740	+	0.623604i	$0.785668\pi$
$200$	7.18875e7	0.635401
$201$	−3.44043e7	−0.298832
$202$	2.66284e6	0.0227308
$203$	−4.93467e7	−0.414020
$204$	6.23511e7	0.514208
$205$	−2.24635e8	−1.82112
$206$	3.62014e7	0.288529
$207$	−2.85219e8	−2.23502
$208$	−5.06340e7	−0.390140
$209$	2.43292e7	0.184338
$210$	3.97466e8	2.96164
$211$	−1.48188e8	−1.08598	−0.542992	−	0.839738i	$-0.682709\pi$
$211$	−1.48188e8	−1.08598	−0.542992	+	0.839738i	$0.682709\pi$
$212$	−5.96013e6	−0.0429616
$213$	−1.41241e8	−1.00146
$214$	5.87108e7	0.409515
$215$	−2.62411e8	−1.80072
$216$	−3.12354e7	−0.210891
$217$	−3.12042e8	−2.07302
$218$	−8.31361e7	−0.543491
$219$	−8.78662e7	−0.565285
$220$	−1.06121e8	−0.671928
$221$	1.66714e8	1.03896
$222$	−1.94505e7	−0.119315
$223$	1.38594e6	0.00836907	0.00418454	−	0.999991i	$-0.498668\pi$
$223$	1.38594e6	0.00836907	0.00418454	+	0.999991i	$0.498668\pi$
$224$	−4.82094e7	−0.286591
$225$	4.25639e8	2.49117
$226$	4.27177e7	0.246166
$227$	1.35147e7	0.0766858	0.0383429	−	0.999265i	$-0.487792\pi$
$227$	1.35147e7	0.0766858	0.0383429	+	0.999265i	$0.487792\pi$
$228$	3.17113e7	0.177191
$229$	3.09882e7	0.170519	0.0852594	−	0.996359i	$-0.472828\pi$
$229$	3.09882e7	0.170519	0.0852594	+	0.996359i	$0.472828\pi$
$230$	−3.51856e8	−1.90686
$231$	−3.76982e8	−2.01224
$232$	1.71730e7	0.0902897
$233$	−2.04142e8	−1.05727	−0.528635	−	0.848849i	$-0.677296\pi$
$233$	−2.04142e8	−1.05727	−0.528635	+	0.848849i	$0.677296\pi$
$234$	−2.99799e8	−1.52959
$235$	2.23660e8	1.12422
$236$	6.11221e7	0.302696
$237$	−2.85016e8	−1.39075
$238$	1.58731e8	0.763209
$239$	−2.25595e8	−1.06890	−0.534449	−	0.845201i	$-0.679481\pi$
$239$	−2.25595e8	−1.06890	−0.534449	+	0.845201i	$0.679481\pi$
$240$	−1.38321e8	−0.645876
$241$	4.16156e7	0.191512	0.0957562	−	0.995405i	$-0.469473\pi$
$241$	4.16156e7	0.191512	0.0957562	+	0.995405i	$0.469473\pi$
$242$	−5.52453e7	−0.250577
$243$	2.93998e8	1.31438
$244$	−1.09844e8	−0.484076
$245$	6.26872e8	2.72331
$246$	2.77706e8	1.18936
$247$	8.47896e7	0.358017
$248$	1.08593e8	0.452086
$249$	2.55909e8	1.05048
$250$	2.32914e8	0.942769
$251$	4.55579e8	1.81847	0.909233	−	0.416287i	$-0.136669\pi$
$251$	4.55579e8	1.81847	0.909233	+	0.416287i	$0.136669\pi$
$252$	−2.85443e8	−1.12362
$253$	3.33723e8	1.29558
$254$	−2.74665e8	−1.05168
$255$	4.55428e8	1.72000
$256$	1.67772e7	0.0625000
$257$	−2.46264e8	−0.904972	−0.452486	−	0.891772i	$-0.649463\pi$
$257$	−2.46264e8	−0.904972	−0.452486	+	0.891772i	$0.649463\pi$
$258$	3.24406e8	1.17603
$259$	−4.95165e7	−0.177093
$260$	−3.69843e8	−1.30500
$261$	1.01680e8	0.353992
$262$	2.00740e6	0.00689572
$263$	4.04810e8	1.37217	0.686083	−	0.727523i	$-0.259329\pi$
$263$	4.04810e8	1.37217	0.686083	+	0.727523i	$0.259329\pi$
$264$	1.31193e8	0.438829
$265$	−4.35343e7	−0.143705
$266$	8.07295e7	0.262994
$267$	5.41940e8	1.74246
$268$	3.04803e7	0.0967270
$269$	−2.44584e8	−0.766118	−0.383059	−	0.923724i	$-0.625129\pi$
$269$	−2.44584e8	−0.766118	−0.383059	+	0.923724i	$0.625129\pi$
$270$	−2.28151e8	−0.705422
$271$	7.88689e7	0.240721	0.120360	−	0.992730i	$-0.461595\pi$
$271$	7.88689e7	0.240721	0.120360	+	0.992730i	$0.461595\pi$
$272$	−5.52397e7	−0.166441
$273$	−1.31382e9	−3.90811
$274$	−2.03691e8	−0.598196
$275$	−4.98023e8	−1.44406
$276$	4.34984e8	1.24535
$277$	−2.30329e8	−0.651133	−0.325567	−	0.945519i	$-0.605555\pi$
$277$	−2.30329e8	−0.651133	−0.325567	+	0.945519i	$0.605555\pi$
$278$	3.46590e8	0.967518
$279$	6.42970e8	1.77246
$280$	−3.52133e8	−0.958637
$281$	−5.44589e8	−1.46419	−0.732094	−	0.681203i	$-0.761457\pi$
$281$	−5.44589e8	−1.46419	−0.732094	+	0.681203i	$0.761457\pi$
$282$	−2.76500e8	−0.734216
$283$	−3.69887e7	−0.0970101	−0.0485051	−	0.998823i	$-0.515446\pi$
$283$	−3.69887e7	−0.0970101	−0.0485051	+	0.998823i	$0.515446\pi$
$284$	1.25132e8	0.324155
$285$	2.31627e8	0.592697
$286$	3.50783e8	0.886660
$287$	7.06973e8	1.76529
$288$	9.93364e7	0.245039
$289$	−2.28460e8	−0.556759
$290$	1.25436e8	0.302015
$291$	−9.24477e8	−2.19923
$292$	7.78447e7	0.182974
$293$	1.98050e8	0.459979	0.229990	−	0.973193i	$-0.426131\pi$
$293$	1.98050e8	0.459979	0.229990	+	0.973193i	$0.426131\pi$
$294$	−7.74973e8	−1.77857
$295$	4.46452e8	1.01251
$296$	1.72321e7	0.0386204
$297$	2.16393e8	0.479287
$298$	−3.18896e8	−0.698058
$299$	1.16306e9	2.51624
$300$	−6.49137e8	−1.38807
$301$	8.25861e8	1.74552
$302$	−1.71048e7	−0.0357350
$303$	−2.40452e7	−0.0496569
$304$	−2.80945e7	−0.0573539
$305$	−8.02331e8	−1.61921
$306$	−3.27069e8	−0.652552
$307$	3.28496e8	0.647957	0.323979	−	0.946064i	$-0.394979\pi$
$307$	3.28496e8	0.647957	0.323979	+	0.946064i	$0.394979\pi$
$308$	3.33986e8	0.651329
$309$	−3.26895e8	−0.630309
$310$	7.93191e8	1.51221
$311$	−5.87246e8	−1.10703	−0.553514	−	0.832840i	$-0.686713\pi$
$311$	−5.87246e8	−1.10703	−0.553514	+	0.832840i	$0.686713\pi$
$312$	4.57220e8	0.852283
$313$	6.29976e8	1.16123	0.580616	−	0.814178i	$-0.302812\pi$
$313$	6.29976e8	1.16123	0.580616	+	0.814178i	$0.302812\pi$
$314$	−2.43762e8	−0.444337
$315$	−2.08495e9	−3.75845
$316$	2.52508e8	0.450164
$317$	8.00885e8	1.41209	0.706046	−	0.708166i	$-0.250477\pi$
$317$	8.00885e8	1.41209	0.706046	+	0.708166i	$0.250477\pi$
$318$	5.38194e7	0.0938521
$319$	−1.18971e8	−0.205199
$320$	1.22545e8	0.209060
$321$	−5.30153e8	−0.894609
$322$	1.10737e9	1.84840
$323$	9.25022e7	0.152737
$324$	−1.42262e8	−0.232370
$325$	−1.73566e9	−2.80462
$326$	−7.59162e8	−1.21359
$327$	7.50711e8	1.18729
$328$	−2.46032e8	−0.384975
$329$	−7.03905e8	−1.08975
$330$	9.58264e8	1.46787
$331$	−4.54262e8	−0.688507	−0.344253	−	0.938877i	$-0.611868\pi$
$331$	−4.54262e8	−0.688507	−0.344253	+	0.938877i	$0.611868\pi$
$332$	−2.26722e8	−0.340024
$333$	1.02030e8	0.151416
$334$	−8.82653e8	−1.29622
$335$	2.22636e8	0.323548
$336$	4.35326e8	0.626076
$337$	−9.81183e8	−1.39651	−0.698257	−	0.715847i	$-0.746041\pi$
$337$	−9.81183e8	−1.39651	−0.698257	+	0.715847i	$0.746041\pi$
$338$	7.20526e8	1.01494
$339$	−3.85737e8	−0.537764
$340$	−4.03485e8	−0.556738
$341$	−7.52313e8	−1.02744
$342$	−1.66345e8	−0.224863
$343$	−7.61275e8	−1.01862
$344$	−2.87406e8	−0.380663
$345$	3.17723e9	4.16564
$346$	8.48898e8	1.10177
$347$	2.69918e8	0.346800	0.173400	−	0.984852i	$-0.444525\pi$
$347$	2.69918e8	0.346800	0.173400	+	0.984852i	$0.444525\pi$
$348$	−1.55071e8	−0.197243
$349$	1.09409e9	1.37773	0.688863	−	0.724892i	$-0.258110\pi$
$349$	1.09409e9	1.37773	0.688863	+	0.724892i	$0.258110\pi$
$350$	−1.65255e9	−2.06023
$351$	7.54151e8	0.930858
$352$	−1.16229e8	−0.142042
$353$	5.02386e8	0.607891	0.303946	−	0.952689i	$-0.401696\pi$
$353$	5.02386e8	0.607891	0.303946	+	0.952689i	$0.401696\pi$
$354$	−5.51927e8	−0.661257
$355$	9.13994e8	1.08429
$356$	−4.80130e8	−0.564006
$357$	−1.43333e9	−1.66727
$358$	3.90681e8	0.450020
$359$	−4.89464e8	−0.558329	−0.279165	−	0.960243i	$-0.590058\pi$
$359$	−4.89464e8	−0.558329	−0.279165	+	0.960243i	$0.590058\pi$
$360$	7.25578e8	0.819645
$361$	4.70459e7	0.0526316
$362$	4.41659e7	0.0489336
$363$	4.98860e8	0.547401
$364$	1.16397e9	1.26499
$365$	5.68597e8	0.612040
$366$	9.91885e8	1.05749
$367$	−5.54974e8	−0.586059	−0.293029	−	0.956103i	$-0.594663\pi$
$367$	−5.54974e8	−0.586059	−0.293029	+	0.956103i	$0.594663\pi$
$368$	−3.85372e8	−0.403100
$369$	−1.45673e9	−1.50934
$370$	1.25868e8	0.129184
$371$	1.37012e8	0.139299
$372$	−9.80585e8	−0.987609
$373$	−1.29835e9	−1.29542	−0.647708	−	0.761889i	$-0.724272\pi$
$373$	−1.29835e9	−1.29542	−0.647708	+	0.761889i	$0.724272\pi$
$374$	3.82690e8	0.378266
$375$	−2.10319e9	−2.05954
$376$	2.44964e8	0.237654
$377$	−4.14628e8	−0.398532
$378$	7.18039e8	0.683796
$379$	7.49227e8	0.706929	0.353465	−	0.935448i	$-0.385003\pi$
$379$	7.49227e8	0.706929	0.353465	+	0.935448i	$0.385003\pi$
$380$	−2.05209e8	−0.191847
$381$	2.48020e9	2.29746
$382$	−9.24495e8	−0.848561
$383$	2.12814e8	0.193555	0.0967774	−	0.995306i	$-0.469147\pi$
$383$	2.12814e8	0.193555	0.0967774	+	0.995306i	$0.469147\pi$
$384$	−1.51497e8	−0.136535
$385$	2.43951e9	2.17867
$386$	3.21563e7	0.0284584
$387$	−1.70171e9	−1.49244
$388$	8.19036e8	0.711855
$389$	3.13367e8	0.269917	0.134958	−	0.990851i	$-0.456910\pi$
$389$	3.13367e8	0.269917	0.134958	+	0.990851i	$0.456910\pi$
$390$	3.33965e9	2.85085
$391$	1.26885e9	1.07348
$392$	6.86583e8	0.575694
$393$	−1.81266e7	−0.0150641
$394$	7.13831e8	0.587975
$395$	1.84438e9	1.50578
$396$	−6.88184e8	−0.556893
$397$	1.91013e9	1.53213	0.766066	−	0.642762i	$-0.222211\pi$
$397$	1.91013e9	1.53213	0.766066	+	0.642762i	$0.222211\pi$
$398$	−1.39049e9	−1.10555
$399$	−7.28980e8	−0.574527
$400$	5.75100e8	0.449297
$401$	−1.27797e9	−0.989726	−0.494863	−	0.868971i	$-0.664782\pi$
$401$	−1.27797e9	−0.989726	−0.494863	+	0.868971i	$0.664782\pi$
$402$	−2.75234e8	−0.211306
$403$	−2.62189e9	−1.99547
$404$	2.13027e7	0.0160731
$405$	−1.03912e9	−0.777269
$406$	−3.94773e8	−0.292757
$407$	−1.19381e8	−0.0877716
$408$	4.98809e8	0.363600
$409$	2.92650e8	0.211503	0.105752	−	0.994393i	$-0.466275\pi$
$409$	2.92650e8	0.211503	0.105752	+	0.994393i	$0.466275\pi$
$410$	−1.79708e9	−1.28773
$411$	1.83931e9	1.30680
$412$	2.89611e8	0.204021
$413$	−1.40508e9	−0.981465
$414$	−2.28175e9	−1.58040
$415$	−1.65603e9	−1.13737
$416$	−4.05072e8	−0.275870
$417$	−3.12967e9	−2.11360
$418$	1.94633e8	0.130347
$419$	3.81573e8	0.253413	0.126706	−	0.991940i	$-0.459559\pi$
$419$	3.81573e8	0.253413	0.126706	+	0.991940i	$0.459559\pi$
$420$	3.17973e9	2.09420
$421$	2.64639e9	1.72849	0.864244	−	0.503073i	$-0.167797\pi$
$421$	2.64639e9	1.72849	0.864244	+	0.503073i	$0.167797\pi$
$422$	−1.18550e9	−0.767906
$423$	1.45041e9	0.931751
$424$	−4.76810e7	−0.0303784
$425$	−1.89354e9	−1.19650
$426$	−1.12993e9	−0.708137
$427$	2.52511e9	1.56957
$428$	4.69686e8	0.289571
$429$	−3.16753e9	−1.93696
$430$	−2.09928e9	−1.27330
$431$	1.69877e9	1.02203	0.511015	−	0.859572i	$-0.329270\pi$
$431$	1.69877e9	1.02203	0.511015	+	0.859572i	$0.329270\pi$
$432$	−2.49883e8	−0.149123
$433$	2.37930e9	1.40845	0.704226	−	0.709975i	$-0.251294\pi$
$433$	2.37930e9	1.40845	0.704226	+	0.709975i	$0.251294\pi$
$434$	−2.49634e9	−1.46585
$435$	−1.13267e9	−0.659770
$436$	−6.65089e8	−0.384306
$437$	6.45328e8	0.369909
$438$	−7.02930e8	−0.399717
$439$	2.18333e9	1.23167	0.615835	−	0.787875i	$-0.288819\pi$
$439$	2.18333e9	1.23167	0.615835	+	0.787875i	$0.288819\pi$
$440$	−8.48969e8	−0.475125
$441$	4.06520e9	2.25708
$442$	1.33372e9	0.734658
$443$	−2.41144e9	−1.31784	−0.658922	−	0.752212i	$-0.728987\pi$
$443$	−2.41144e9	−1.31784	−0.658922	+	0.752212i	$0.728987\pi$
$444$	−1.55604e8	−0.0843686
$445$	−3.50699e9	−1.88658
$446$	1.10875e7	0.00591783
$447$	2.87960e9	1.52495
$448$	−3.85675e8	−0.202651
$449$	−6.02088e8	−0.313905	−0.156952	−	0.987606i	$-0.550167\pi$
$449$	−6.02088e8	−0.313905	−0.156952	+	0.987606i	$0.550167\pi$
$450$	3.40511e9	1.76152
$451$	1.70446e9	0.874923
$452$	3.41741e8	0.174066
$453$	1.54455e8	0.0780652
$454$	1.08117e8	0.0542251
$455$	8.50196e9	4.23135
$456$	2.53690e8	0.125293
$457$	−3.28085e9	−1.60798	−0.803989	−	0.594644i	$-0.797293\pi$
$457$	−3.28085e9	−1.60798	−0.803989	+	0.594644i	$0.797293\pi$
$458$	2.47906e8	0.120575
$459$	8.22750e8	0.397122
$460$	−2.81485e9	−1.34835
$461$	−3.50631e9	−1.66685	−0.833426	−	0.552632i	$-0.813624\pi$
$461$	−3.50631e9	−1.66685	−0.833426	+	0.552632i	$0.813624\pi$
$462$	−3.01586e9	−1.42287
$463$	−3.85170e9	−1.80351	−0.901757	−	0.432244i	$-0.857722\pi$
$463$	−3.85170e9	−1.80351	−0.901757	+	0.432244i	$0.857722\pi$
$464$	1.37384e8	0.0638445
$465$	−7.16244e9	−3.30351
$466$	−1.63313e9	−0.747603
$467$	4.73318e8	0.215052	0.107526	−	0.994202i	$-0.465707\pi$
$467$	4.73318e8	0.215052	0.107526	+	0.994202i	$0.465707\pi$
$468$	−2.39839e9	−1.08158
$469$	−7.00682e8	−0.313629
$470$	1.78928e9	0.794942
$471$	2.20115e9	0.970680
$472$	4.88977e8	0.214038
$473$	1.99109e9	0.865123
$474$	−2.28013e9	−0.983410
$475$	−9.63039e8	−0.412303
$476$	1.26985e9	0.539670
$477$	−2.82315e8	−0.119102
$478$	−1.80476e9	−0.755826
$479$	2.43257e9	1.01132	0.505662	−	0.862732i	$-0.331248\pi$
$479$	2.43257e9	1.01132	0.505662	+	0.862732i	$0.331248\pi$
$480$	−1.10657e9	−0.456704
$481$	−4.16055e8	−0.170468
$482$	3.32925e8	0.135420
$483$	−9.99941e9	−4.03793
$484$	−4.41963e8	−0.177185
$485$	5.98244e9	2.38113
$486$	2.35198e9	0.929409
$487$	−3.37943e9	−1.32584	−0.662921	−	0.748689i	$-0.730683\pi$
$487$	−3.37943e9	−1.32584	−0.662921	+	0.748689i	$0.730683\pi$
$488$	−8.78755e8	−0.342294
$489$	6.85516e9	2.65116
$490$	5.01498e9	1.92567
$491$	−1.62343e9	−0.618938	−0.309469	−	0.950910i	$-0.600151\pi$
$491$	−1.62343e9	−0.618938	−0.309469	+	0.950910i	$0.600151\pi$
$492$	2.22164e9	0.841001
$493$	−4.52343e8	−0.170021
$494$	6.78317e8	0.253156
$495$	−5.02667e9	−1.86278
$496$	8.68744e8	0.319673
$497$	−2.87653e9	−1.05105
$498$	2.04727e9	0.742802
$499$	3.88153e9	1.39846	0.699231	−	0.714896i	$-0.253526\pi$
$499$	3.88153e9	1.39846	0.699231	+	0.714896i	$0.253526\pi$
$500$	1.86331e9	0.666639
$501$	7.97027e9	2.83166
$502$	3.64463e9	1.28585
$503$	2.38583e9	0.835893	0.417947	−	0.908472i	$-0.362750\pi$
$503$	2.38583e9	0.835893	0.417947	+	0.908472i	$0.362750\pi$
$504$	−2.28355e9	−0.794517
$505$	1.55600e8	0.0537639
$506$	2.66978e9	0.916113
$507$	−6.50628e9	−2.21720
$508$	−2.19732e9	−0.743652
$509$	−3.37295e9	−1.13370	−0.566849	−	0.823822i	$-0.691838\pi$
$509$	−3.37295e9	−1.13370	−0.566849	+	0.823822i	$0.691838\pi$
$510$	3.64343e9	1.21623
$511$	−1.78949e9	−0.593277
$512$	1.34218e8	0.0441942
$513$	4.18444e8	0.136844
$514$	−1.97011e9	−0.639912
$515$	2.11539e9	0.682441
$516$	2.59525e9	0.831582
$517$	−1.69707e9	−0.540109
$518$	−3.96132e8	−0.125223
$519$	−7.66547e9	−2.40687
$520$	−2.95875e9	−0.922775
$521$	5.32384e9	1.64927	0.824637	−	0.565662i	$-0.191379\pi$
$521$	5.32384e9	1.64927	0.824637	+	0.565662i	$0.191379\pi$
$522$	8.13439e8	0.250310
$523$	4.07644e9	1.24602	0.623010	−	0.782214i	$-0.285910\pi$
$523$	4.07644e9	1.24602	0.623010	+	0.782214i	$0.285910\pi$
$524$	1.60592e7	0.00487601
$525$	1.49224e10	4.50070
$526$	3.23848e9	0.970268
$527$	−2.86038e9	−0.851307
$528$	1.04954e9	0.310299
$529$	5.44713e9	1.59983
$530$	−3.48274e8	−0.101615
$531$	2.89519e9	0.839163
$532$	6.45836e8	0.185965
$533$	5.94023e9	1.69925
$534$	4.33552e9	1.23210
$535$	3.43071e9	0.968602
$536$	2.43843e8	0.0683964
$537$	−3.52781e9	−0.983095
$538$	−1.95668e9	−0.541728
$539$	−4.75652e9	−1.30837
$540$	−1.82521e9	−0.498809
$541$	−3.63310e9	−0.986477	−0.493239	−	0.869894i	$-0.664187\pi$
$541$	−3.63310e9	−0.986477	−0.493239	+	0.869894i	$0.664187\pi$
$542$	6.30951e8	0.170215
$543$	−3.98814e8	−0.106898
$544$	−4.41918e8	−0.117692
$545$	−4.85798e9	−1.28549
$546$	−1.05106e10	−2.76345
$547$	−5.74113e9	−1.49983	−0.749914	−	0.661535i	$-0.769905\pi$
$547$	−5.74113e9	−1.49983	−0.749914	+	0.661535i	$0.769905\pi$
$548$	−1.62952e9	−0.422989
$549$	−5.20303e9	−1.34200
$550$	−3.98418e9	−1.02110
$551$	−2.30058e8	−0.0585877
$552$	3.47987e9	0.880595
$553$	−5.80466e9	−1.45962
$554$	−1.84263e9	−0.460421
$555$	−1.13657e9	−0.282210
$556$	2.77272e9	0.684138
$557$	9.95936e8	0.244196	0.122098	−	0.992518i	$-0.461038\pi$
$557$	9.95936e8	0.244196	0.122098	+	0.992518i	$0.461038\pi$
$558$	5.14376e9	1.25332
$559$	6.93917e9	1.68022
$560$	−2.81707e9	−0.677858
$561$	−3.45566e9	−0.826344
$562$	−4.35671e9	−1.03534
$563$	−4.05283e9	−0.957147	−0.478573	−	0.878048i	$-0.658846\pi$
$563$	−4.05283e9	−0.957147	−0.478573	+	0.878048i	$0.658846\pi$
$564$	−2.21200e9	−0.519169
$565$	2.49617e9	0.582242
$566$	−2.95910e8	−0.0685965
$567$	3.27031e9	0.753440
$568$	1.00105e9	0.229212
$569$	3.35141e9	0.762666	0.381333	−	0.924438i	$-0.375465\pi$
$569$	3.35141e9	0.762666	0.381333	+	0.924438i	$0.375465\pi$
$570$	1.85302e9	0.419100
$571$	4.22454e9	0.949627	0.474813	−	0.880087i	$-0.342516\pi$
$571$	4.22454e9	0.949627	0.474813	+	0.880087i	$0.342516\pi$
$572$	2.80626e9	0.626963
$573$	8.34810e9	1.85373
$574$	5.65579e9	1.24825
$575$	−1.32100e10	−2.89778
$576$	7.94691e8	0.173269
$577$	−6.43062e9	−1.39360	−0.696799	−	0.717266i	$-0.745393\pi$
$577$	−6.43062e9	−1.39360	−0.696799	+	0.717266i	$0.745393\pi$
$578$	−1.82768e9	−0.393688
$579$	−2.90369e8	−0.0621691
$580$	1.00349e9	0.213557
$581$	5.21188e9	1.10250
$582$	−7.39581e9	−1.55509
$583$	3.30325e8	0.0690402
$584$	6.22757e8	0.129382
$585$	−1.75185e10	−3.61785
$586$	1.58440e9	0.325254
$587$	−2.90754e9	−0.593326	−0.296663	−	0.954982i	$-0.595874\pi$
$587$	−2.90754e9	−0.593326	−0.296663	+	0.954982i	$0.595874\pi$
$588$	−6.19978e9	−1.25764
$589$	−1.45476e9	−0.293352
$590$	3.57161e9	0.715949
$591$	−6.44582e9	−1.28447
$592$	1.37857e8	0.0273088
$593$	−5.13346e8	−0.101093	−0.0505463	−	0.998722i	$-0.516096\pi$
$593$	−5.13346e8	−0.101093	−0.0505463	+	0.998722i	$0.516096\pi$
$594$	1.73114e9	0.338907
$595$	9.27531e9	1.80517
$596$	−2.55117e9	−0.493602
$597$	1.25560e10	2.41513
$598$	9.30447e9	1.77925
$599$	−3.44859e9	−0.655613	−0.327807	−	0.944745i	$-0.606309\pi$
$599$	−3.44859e9	−0.655613	−0.327807	+	0.944745i	$0.606309\pi$
$600$	−5.19309e9	−0.981515
$601$	−5.96985e9	−1.12177	−0.560883	−	0.827895i	$-0.689538\pi$
$601$	−5.96985e9	−1.12177	−0.560883	+	0.827895i	$0.689538\pi$
$602$	6.60689e9	1.23427
$603$	1.44377e9	0.268156
$604$	−1.36838e8	−0.0252684
$605$	−3.22821e9	−0.592676
$606$	−1.92362e8	−0.0351127
$607$	−1.00863e10	−1.83050	−0.915249	−	0.402888i	$-0.868007\pi$
$607$	−1.00863e10	−1.83050	−0.915249	+	0.402888i	$0.868007\pi$
$608$	−2.24756e8	−0.0405554
$609$	3.56477e9	0.639544
$610$	−6.41865e9	−1.14496
$611$	−5.91445e9	−1.04899
$612$	−2.61656e9	−0.461424
$613$	1.24543e9	0.218378	0.109189	−	0.994021i	$-0.465175\pi$
$613$	1.24543e9	0.218378	0.109189	+	0.994021i	$0.465175\pi$
$614$	2.62797e9	0.458175
$615$	1.62275e10	2.81311
$616$	2.67188e9	0.460559
$617$	−5.96038e8	−0.102159	−0.0510794	−	0.998695i	$-0.516266\pi$
$617$	−5.96038e8	−0.102159	−0.0510794	+	0.998695i	$0.516266\pi$
$618$	−2.61516e9	−0.445696
$619$	−3.90038e9	−0.660981	−0.330491	−	0.943809i	$-0.607214\pi$
$619$	−3.90038e9	−0.660981	−0.330491	+	0.943809i	$0.607214\pi$
$620$	6.34553e9	1.06929
$621$	5.73979e9	0.961780
$622$	−4.69797e9	−0.782787
$623$	1.10372e10	1.82874
$624$	3.65776e9	0.602655
$625$	2.64094e9	0.432692
$626$	5.03981e9	0.821115
$627$	−1.75752e9	−0.284750
$628$	−1.95010e9	−0.314193
$629$	−4.53899e8	−0.0727248
$630$	−1.66796e10	−2.65763
$631$	4.64916e9	0.736668	0.368334	−	0.929694i	$-0.379928\pi$
$631$	4.64916e9	0.736668	0.368334	+	0.929694i	$0.379928\pi$
$632$	2.02007e9	0.318314
$633$	1.07050e10	1.67754
$634$	6.40708e9	0.998499
$635$	−1.60498e10	−2.48749
$636$	4.30555e8	0.0663635
$637$	−1.65770e10	−2.54107
$638$	−9.51772e8	−0.145098
$639$	5.92715e9	0.898655
$640$	9.80360e8	0.147828
$641$	9.04607e9	1.35662	0.678308	−	0.734778i	$-0.262714\pi$
$641$	9.04607e9	1.35662	0.678308	+	0.734778i	$0.262714\pi$
$642$	−4.24122e9	−0.632584
$643$	1.11196e10	1.64949	0.824746	−	0.565503i	$-0.191318\pi$
$643$	1.11196e10	1.64949	0.824746	+	0.565503i	$0.191318\pi$
$644$	8.85893e9	1.30701
$645$	1.89563e10	2.78161
$646$	7.40018e8	0.108001
$647$	−3.91564e9	−0.568379	−0.284189	−	0.958768i	$-0.591724\pi$
$647$	−3.91564e9	−0.568379	−0.284189	+	0.958768i	$0.591724\pi$
$648$	−1.13809e9	−0.164311
$649$	−3.38754e9	−0.486439
$650$	−1.38853e10	−1.98316
$651$	2.25417e10	3.20223
$652$	−6.07329e9	−0.858139
$653$	−1.07233e10	−1.50707	−0.753536	−	0.657407i	$-0.771653\pi$
$653$	−1.07233e10	−1.50707	−0.753536	+	0.657407i	$0.771653\pi$
$654$	6.00569e9	0.839539
$655$	1.17300e8	0.0163100
$656$	−1.96826e9	−0.272219
$657$	3.68729e9	0.507258
$658$	−5.63124e9	−0.770572
$659$	9.55747e8	0.130090	0.0650450	−	0.997882i	$-0.479281\pi$
$659$	9.55747e8	0.130090	0.0650450	+	0.997882i	$0.479281\pi$
$660$	7.66611e9	1.03794
$661$	−1.17758e10	−1.58594	−0.792971	−	0.609260i	$-0.791467\pi$
$661$	−1.17758e10	−1.58594	−0.792971	+	0.609260i	$0.791467\pi$
$662$	−3.63409e9	−0.486848
$663$	−1.20433e10	−1.60490
$664$	−1.81377e9	−0.240433
$665$	4.71735e9	0.622046
$666$	8.16238e8	0.107067
$667$	−3.15570e9	−0.411771
$668$	−7.06122e9	−0.916563
$669$	−1.00119e8	−0.0129278
$670$	1.78109e9	0.228783
$671$	6.08786e9	0.777921
$672$	3.48261e9	0.442703
$673$	5.31058e9	0.671567	0.335783	−	0.941939i	$-0.390999\pi$
$673$	5.31058e9	0.671567	0.335783	+	0.941939i	$0.390999\pi$
$674$	−7.84946e9	−0.987485
$675$	−8.56564e9	−1.07200
$676$	5.76421e9	0.717672
$677$	1.06518e10	1.31936	0.659680	−	0.751547i	$-0.270692\pi$
$677$	1.06518e10	1.31936	0.659680	+	0.751547i	$0.270692\pi$
$678$	−3.08589e9	−0.380257
$679$	−1.88280e10	−2.30813
$680$	−3.22788e9	−0.393673
$681$	−9.76290e8	−0.118458
$682$	−6.01850e9	−0.726512
$683$	−3.14122e8	−0.0377247	−0.0188624	−	0.999822i	$-0.506004\pi$
$683$	−3.14122e8	−0.0377247	−0.0188624	+	0.999822i	$0.506004\pi$
$684$	−1.33076e9	−0.159002
$685$	−1.19025e10	−1.41488
$686$	−6.09020e9	−0.720273
$687$	−2.23856e9	−0.263403
$688$	−2.29925e9	−0.269170
$689$	1.15122e9	0.134088
$690$	2.54178e10	2.94555
$691$	−1.44774e10	−1.66923	−0.834615	−	0.550833i	$-0.814310\pi$
$691$	−1.44774e10	−1.66923	−0.834615	+	0.550833i	$0.814310\pi$
$692$	6.79119e9	0.779066
$693$	1.58200e10	1.80568
$694$	2.15934e9	0.245224
$695$	2.02526e10	2.28841
$696$	−1.24057e9	−0.139472
$697$	6.48056e9	0.724933
$698$	8.75269e9	0.974199
$699$	1.47470e10	1.63318
$700$	−1.32204e10	−1.45680
$701$	1.22852e10	1.34701	0.673503	−	0.739185i	$-0.264789\pi$
$701$	1.22852e10	1.34701	0.673503	+	0.739185i	$0.264789\pi$
$702$	6.03321e9	0.658216
$703$	−2.30850e8	−0.0250603
$704$	−9.29836e8	−0.100439
$705$	−1.61570e10	−1.73660
$706$	4.01909e9	0.429844
$707$	−4.89707e8	−0.0521157
$708$	−4.41542e9	−0.467579
$709$	−3.48632e9	−0.367371	−0.183686	−	0.982985i	$-0.558803\pi$
$709$	−3.48632e9	−0.367371	−0.183686	+	0.982985i	$0.558803\pi$
$710$	7.31195e9	0.766706
$711$	1.19606e10	1.24799
$712$	−3.84104e9	−0.398812
$713$	−1.99550e10	−2.06176
$714$	−1.14666e10	−1.17894
$715$	2.04976e10	2.09717
$716$	3.12545e9	0.318212
$717$	1.62968e10	1.65115
$718$	−3.91571e9	−0.394798
$719$	6.30005e9	0.632110	0.316055	−	0.948741i	$-0.397642\pi$
$719$	6.30005e9	0.632110	0.316055	+	0.948741i	$0.397642\pi$
$720$	5.80463e9	0.579576
$721$	−6.65758e9	−0.661520
$722$	3.76367e8	0.0372161
$723$	−3.00628e9	−0.295832
$724$	3.53328e8	0.0346013
$725$	4.70933e9	0.458962
$726$	3.99088e9	0.387071
$727$	−4.41318e9	−0.425973	−0.212986	−	0.977055i	$-0.568319\pi$
$727$	−4.41318e9	−0.425973	−0.212986	+	0.977055i	$0.568319\pi$
$728$	9.31179e9	0.894486
$729$	−1.63768e10	−1.56561
$730$	4.54878e9	0.432777
$731$	7.57037e9	0.716814
$732$	7.93508e9	0.747760
$733$	1.15847e10	1.08648	0.543240	−	0.839577i	$-0.317197\pi$
$733$	1.15847e10	1.08648	0.543240	+	0.839577i	$0.317197\pi$
$734$	−4.43979e9	−0.414406
$735$	−4.52848e10	−4.20675
$736$	−3.08297e9	−0.285034
$737$	−1.68930e9	−0.155443
$738$	−1.16539e10	−1.06727
$739$	1.60174e10	1.45995	0.729974	−	0.683475i	$-0.239532\pi$
$739$	1.60174e10	1.45995	0.729974	+	0.683475i	$0.239532\pi$
$740$	1.00694e9	0.0913467
$741$	−6.12514e9	−0.553034
$742$	1.09609e9	0.0984994
$743$	6.54024e9	0.584969	0.292484	−	0.956270i	$-0.405518\pi$
$743$	6.54024e9	0.584969	0.292484	+	0.956270i	$0.405518\pi$
$744$	−7.84468e9	−0.698345
$745$	−1.86344e10	−1.65108
$746$	−1.03868e10	−0.915997
$747$	−1.07392e10	−0.942648
$748$	3.06152e9	0.267474
$749$	−1.07972e10	−0.938908
$750$	−1.68255e10	−1.45631
$751$	1.04818e10	0.903015	0.451508	−	0.892267i	$-0.350886\pi$
$751$	1.04818e10	0.903015	0.451508	+	0.892267i	$0.350886\pi$
$752$	1.95971e9	0.168047
$753$	−3.29107e10	−2.80901
$754$	−3.31702e9	−0.281805
$755$	−9.99502e8	−0.0845219
$756$	5.74431e9	0.483517
$757$	4.11773e9	0.345003	0.172501	−	0.985009i	$-0.444815\pi$
$757$	4.11773e9	0.345003	0.172501	+	0.985009i	$0.444815\pi$
$758$	5.99381e9	0.499874
$759$	−2.41079e10	−2.00130
$760$	−1.64167e9	−0.135656
$761$	−1.90960e10	−1.57071	−0.785357	−	0.619043i	$-0.787521\pi$
$761$	−1.90960e10	−1.57071	−0.785357	+	0.619043i	$0.787521\pi$
$762$	1.98416e10	1.62455
$763$	1.52891e10	1.24608
$764$	−7.39596e9	−0.600023
$765$	−1.91120e10	−1.54344
$766$	1.70251e9	0.136864
$767$	−1.18059e10	−0.944750
$768$	−1.21197e9	−0.0965448
$769$	−1.20473e10	−0.955315	−0.477658	−	0.878546i	$-0.658514\pi$
$769$	−1.20473e10	−0.955315	−0.477658	+	0.878546i	$0.658514\pi$
$770$	1.95161e10	1.54055
$771$	1.77899e10	1.39792
$772$	2.57251e8	0.0201232
$773$	4.62608e9	0.360234	0.180117	−	0.983645i	$-0.442352\pi$
$773$	4.62608e9	0.360234	0.180117	+	0.983645i	$0.442352\pi$
$774$	−1.36136e10	−1.05531
$775$	2.97793e10	2.29805
$776$	6.55229e9	0.503358
$777$	3.57703e9	0.273558
$778$	2.50694e9	0.190860
$779$	3.29596e9	0.249805
$780$	2.67172e10	2.01586
$781$	−6.93512e9	−0.520925
$782$	1.01508e10	0.759062
$783$	−2.04622e9	−0.152330
$784$	5.49267e9	0.407077
$785$	−1.42440e10	−1.05096
$786$	−1.45013e8	−0.0106519
$787$	5.71149e9	0.417675	0.208837	−	0.977950i	$-0.433032\pi$
$787$	5.71149e9	0.417675	0.208837	+	0.977950i	$0.433032\pi$
$788$	5.71065e9	0.415761
$789$	−2.92432e10	−2.11961
$790$	1.47551e10	1.06475
$791$	−7.85596e9	−0.564393
$792$	−5.50548e9	−0.393783
$793$	2.12168e10	1.51086
$794$	1.52810e10	1.08338
$795$	3.14488e9	0.221983
$796$	−1.11239e10	−0.781740
$797$	−1.15032e10	−0.804848	−0.402424	−	0.915453i	$-0.631832\pi$
$797$	−1.15032e10	−0.804848	−0.402424	+	0.915453i	$0.631832\pi$
$798$	−5.83184e9	−0.406252
$799$	−6.45244e9	−0.447517
$800$	4.60080e9	0.317701
$801$	−2.27424e10	−1.56359
$802$	−1.02237e10	−0.699842
$803$	−4.31435e9	−0.294043
$804$	−2.20188e9	−0.149416
$805$	6.47078e10	4.37191
$806$	−2.09751e10	−1.41101
$807$	1.76686e10	1.18344
$808$	1.70422e8	0.0113654
$809$	1.91918e10	1.27437	0.637184	−	0.770712i	$-0.280099\pi$
$809$	1.91918e10	1.27437	0.637184	+	0.770712i	$0.280099\pi$
$810$	−8.31292e9	−0.549612
$811$	−5.04377e9	−0.332034	−0.166017	−	0.986123i	$-0.553091\pi$
$811$	−5.04377e9	−0.332034	−0.166017	+	0.986123i	$0.553091\pi$
$812$	−3.15819e9	−0.207010
$813$	−5.69743e9	−0.371845
$814$	−9.55047e8	−0.0620639
$815$	−4.43609e10	−2.87044
$816$	3.99047e9	0.257104
$817$	3.85023e9	0.247007
$818$	2.34120e9	0.149556
$819$	5.51343e10	3.50694
$820$	−1.43766e10	−0.910560
$821$	2.85707e10	1.80185	0.900927	−	0.433972i	$-0.142888\pi$
$821$	2.85707e10	1.80185	0.900927	+	0.433972i	$0.142888\pi$
$822$	1.47144e10	0.924044
$823$	1.82564e10	1.14161	0.570803	−	0.821087i	$-0.306632\pi$
$823$	1.82564e10	1.14161	0.570803	+	0.821087i	$0.306632\pi$
$824$	2.31689e9	0.144265
$825$	3.59768e10	2.23066
$826$	−1.12406e10	−0.694001
$827$	−1.92073e9	−0.118086	−0.0590428	−	0.998255i	$-0.518805\pi$
$827$	−1.92073e9	−0.118086	−0.0590428	+	0.998255i	$0.518805\pi$
$828$	−1.82540e10	−1.11751
$829$	4.75211e9	0.289698	0.144849	−	0.989454i	$-0.453730\pi$
$829$	4.75211e9	0.289698	0.144849	+	0.989454i	$0.453730\pi$
$830$	−1.32483e10	−0.804239
$831$	1.66388e10	1.00582
$832$	−3.24057e9	−0.195070
$833$	−1.80848e10	−1.08407
$834$	−2.50374e10	−1.49454
$835$	−5.15770e10	−3.06587
$836$	1.55707e9	0.0921690
$837$	−1.29392e10	−0.762728
$838$	3.05258e9	0.179190
$839$	−1.17511e10	−0.686927	−0.343463	−	0.939166i	$-0.611600\pi$
$839$	−1.17511e10	−0.686927	−0.343463	+	0.939166i	$0.611600\pi$
$840$	2.54378e10	1.48082
$841$	−1.61249e10	−0.934782
$842$	2.11711e10	1.22223
$843$	3.93407e10	2.26176
$844$	−9.48400e9	−0.542992
$845$	4.21032e10	2.40058
$846$	1.16033e10	0.658847
$847$	1.01598e10	0.574507
$848$	−3.81448e8	−0.0214808
$849$	2.67204e9	0.149853
$850$	−1.51483e10	−0.846054
$851$	−3.16656e9	−0.176130
$852$	−9.03942e9	−0.500728
$853$	−1.98045e10	−1.09255	−0.546276	−	0.837605i	$-0.683955\pi$
$853$	−1.98045e10	−1.09255	−0.546276	+	0.837605i	$0.683955\pi$
$854$	2.02008e10	1.10986
$855$	−9.72020e9	−0.531856
$856$	3.75749e9	0.204757
$857$	1.83237e10	0.994445	0.497223	−	0.867623i	$-0.334353\pi$
$857$	1.83237e10	0.994445	0.497223	+	0.867623i	$0.334353\pi$
$858$	−2.53403e10	−1.36964
$859$	2.65234e10	1.42775	0.713876	−	0.700272i	$-0.246938\pi$
$859$	2.65234e10	1.42775	0.713876	+	0.700272i	$0.246938\pi$
$860$	−1.67943e10	−0.900361
$861$	−5.10712e10	−2.72687
$862$	1.35901e10	0.722684
$863$	−1.77522e10	−0.940188	−0.470094	−	0.882616i	$-0.655780\pi$
$863$	−1.77522e10	−0.940188	−0.470094	+	0.882616i	$0.655780\pi$
$864$	−1.99906e9	−0.105446
$865$	4.96045e10	2.60594
$866$	1.90344e10	0.995927
$867$	1.65038e10	0.860034
$868$	−1.99707e10	−1.03651
$869$	−1.39946e10	−0.723423
$870$	−9.06140e9	−0.466528
$871$	−5.88737e9	−0.301897
$872$	−5.32071e9	−0.271745
$873$	3.87955e10	1.97348
$874$	5.16262e9	0.261566
$875$	−4.28338e10	−2.16152
$876$	−5.62344e9	−0.282643
$877$	1.22608e9	0.0613792	0.0306896	−	0.999529i	$-0.490230\pi$
$877$	1.22608e9	0.0613792	0.0306896	+	0.999529i	$0.490230\pi$
$878$	1.74667e10	0.870922
$879$	−1.43070e10	−0.710537
$880$	−6.79175e9	−0.335964
$881$	6.34486e9	0.312612	0.156306	−	0.987709i	$-0.450041\pi$
$881$	6.34486e9	0.312612	0.156306	+	0.987709i	$0.450041\pi$
$882$	3.25216e10	1.59600
$883$	−2.24216e10	−1.09598	−0.547992	−	0.836484i	$-0.684608\pi$
$883$	−2.24216e10	−1.09598	−0.547992	+	0.836484i	$0.684608\pi$
$884$	1.06697e10	0.519482
$885$	−3.22513e10	−1.56403
$886$	−1.92915e10	−0.931856
$887$	3.38047e10	1.62647	0.813233	−	0.581938i	$-0.197705\pi$
$887$	3.38047e10	1.62647	0.813233	+	0.581938i	$0.197705\pi$
$888$	−1.24483e9	−0.0596576
$889$	5.05120e10	2.41123
$890$	−2.80559e10	−1.33401
$891$	7.88450e9	0.373424
$892$	8.87002e7	0.00418454
$893$	−3.28166e9	−0.154210
$894$	2.30368e10	1.07830
$895$	2.28291e10	1.06441
$896$	−3.08540e9	−0.143296
$897$	−8.40185e10	−3.88688
$898$	−4.81671e9	−0.221964
$899$	7.11391e9	0.326550
$900$	2.72409e10	1.24558
$901$	1.25593e9	0.0572045
$902$	1.36357e10	0.618664
$903$	−5.96596e10	−2.69633
$904$	2.73393e9	0.123083
$905$	2.58079e9	0.115740
$906$	1.23564e9	0.0552004
$907$	−1.56762e10	−0.697616	−0.348808	−	0.937194i	$-0.613413\pi$
$907$	−1.56762e10	−0.697616	−0.348808	+	0.937194i	$0.613413\pi$
$908$	8.64939e8	0.0383429
$909$	1.00905e9	0.0445595
$910$	6.80157e10	2.99202
$911$	2.35920e10	1.03383	0.516916	−	0.856036i	$-0.327080\pi$
$911$	2.35920e10	1.03383	0.516916	+	0.856036i	$0.327080\pi$
$912$	2.02952e9	0.0885956
$913$	1.25655e10	0.546426
$914$	−2.62468e10	−1.13701
$915$	5.79598e10	2.50123
$916$	1.98325e9	0.0852594
$917$	−3.69169e8	−0.0158100
$918$	6.58200e9	0.280807
$919$	−3.39353e10	−1.44227	−0.721137	−	0.692793i	$-0.756380\pi$
$919$	−3.39353e10	−1.44227	−0.721137	+	0.692793i	$0.756380\pi$
$920$	−2.25188e10	−0.953428
$921$	−2.37303e10	−1.00091
$922$	−2.80505e10	−1.17864
$923$	−2.41696e10	−1.01173
$924$	−2.41269e10	−1.00612
$925$	4.72554e9	0.196316
$926$	−3.08136e10	−1.27528
$927$	1.37181e10	0.565607
$928$	1.09907e9	0.0451449
$929$	7.75319e9	0.317267	0.158634	−	0.987338i	$-0.449291\pi$
$929$	7.75319e9	0.317267	0.158634	+	0.987338i	$0.449291\pi$
$930$	−5.72995e10	−2.33593
$931$	−9.19780e9	−0.373560
$932$	−1.30651e10	−0.528635
$933$	4.24222e10	1.71004
$934$	3.78654e9	0.152065
$935$	2.23621e10	0.894690
$936$	−1.91872e10	−0.764795
$937$	−3.05613e10	−1.21362	−0.606811	−	0.794846i	$-0.707552\pi$
$937$	−3.05613e10	−1.21362	−0.606811	+	0.794846i	$0.707552\pi$
$938$	−5.60546e9	−0.221769
$939$	−4.55090e10	−1.79377
$940$	1.43142e10	0.562109
$941$	−4.59802e10	−1.79890	−0.899451	−	0.437022i	$-0.856033\pi$
$941$	−4.59802e10	−1.79890	−0.899451	+	0.437022i	$0.856033\pi$
$942$	1.76092e10	0.686374
$943$	4.52107e10	1.75570
$944$	3.91182e9	0.151348
$945$	4.19579e10	1.61734
$946$	1.59288e10	0.611735
$947$	−3.07383e10	−1.17613	−0.588065	−	0.808813i	$-0.700110\pi$
$947$	−3.07383e10	−1.17613	−0.588065	+	0.808813i	$0.700110\pi$
$948$	−1.82410e10	−0.695376
$949$	−1.50359e10	−0.571083
$950$	−7.70431e9	−0.291542
$951$	−5.78553e10	−2.18128
$952$	1.01588e10	0.381604
$953$	1.04326e10	0.390451	0.195226	−	0.980758i	$-0.437456\pi$
$953$	1.04326e10	0.390451	0.195226	+	0.980758i	$0.437456\pi$
$954$	−2.25852e9	−0.0842180
$955$	−5.40220e10	−2.00705
$956$	−1.44381e10	−0.534449
$957$	8.59441e9	0.316974
$958$	1.94605e10	0.715114
$959$	3.74595e10	1.37150
$960$	−8.85256e9	−0.322938
$961$	1.74720e10	0.635054
$962$	−3.32844e9	−0.120539
$963$	2.22478e10	0.802776
$964$	2.66340e9	0.0957562
$965$	1.87902e9	0.0673111
$966$	−7.99953e10	−2.85525
$967$	1.29860e10	0.461829	0.230915	−	0.972974i	$-0.425828\pi$
$967$	1.29860e10	0.461829	0.230915	+	0.972974i	$0.425828\pi$
$968$	−3.53570e9	−0.125289
$969$	−6.68229e9	−0.235935
$970$	4.78595e10	1.68371
$971$	−5.17386e10	−1.81362	−0.906812	−	0.421536i	$-0.861491\pi$
$971$	−5.17386e10	−1.81362	−0.906812	+	0.421536i	$0.861491\pi$
$972$	1.88158e10	0.657191
$973$	−6.37393e10	−2.21826
$974$	−2.70354e10	−0.937512
$975$	1.25383e11	4.33234
$976$	−7.03004e9	−0.242038
$977$	3.29063e10	1.12888	0.564440	−	0.825474i	$-0.309092\pi$
$977$	3.29063e10	1.12888	0.564440	+	0.825474i	$0.309092\pi$
$978$	5.48413e10	1.87466
$979$	2.66100e10	0.906370
$980$	4.01198e10	1.36166
$981$	−3.15035e10	−1.06541
$982$	−1.29874e10	−0.437655
$983$	−2.58802e9	−0.0869021	−0.0434510	−	0.999056i	$-0.513835\pi$
$983$	−2.58802e9	−0.0869021	−0.0434510	+	0.999056i	$0.513835\pi$
$984$	1.77732e10	0.594678
$985$	4.17120e10	1.39070
$986$	−3.61874e9	−0.120223
$987$	5.08495e10	1.68336
$988$	5.42654e9	0.179008
$989$	5.28136e10	1.73603
$990$	−4.02134e10	−1.31719
$991$	−4.62321e10	−1.50899	−0.754494	−	0.656306i	$-0.772118\pi$
$991$	−4.62321e10	−1.50899	−0.754494	+	0.656306i	$0.772118\pi$
$992$	6.94995e9	0.226043
$993$	3.28155e10	1.06355
$994$	−2.30122e10	−0.743201
$995$	−8.12520e10	−2.61489
$996$	1.63782e10	0.525241
$997$	2.54695e10	0.813930	0.406965	−	0.913444i	$-0.366587\pi$
$997$	2.54695e10	0.813930	0.406965	+	0.913444i	$0.366587\pi$
$998$	3.10522e10	0.988862
$999$	−2.05326e9	−0.0651577

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.8.a.d.1.1	✓	2
3.2	odd	2		342.8.a.g.1.1		2
4.3	odd	2		304.8.a.d.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.8.a.d.1.1	✓	2	1.1	even	1	trivial
304.8.a.d.1.2		2	4.3	odd	2
342.8.a.g.1.1		2	3.2	odd	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:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	38.a (trivial)

\(f(q)\)	\(=\)	\(q+8.00000 q^{2} -72.2392 q^{3} +64.0000 q^{4} +467.472 q^{5} -577.914 q^{6} -1471.23 q^{7} +512.000 q^{8} +3031.51 q^{9} +O(q^{10})\)	\(q+8.00000 q^{2} -72.2392 q^{3} +64.0000 q^{4} +467.472 q^{5} -577.914 q^{6} -1471.23 q^{7} +512.000 q^{8} +3031.51 q^{9} +3739.78 q^{10} -3547.04 q^{11} -4623.31 q^{12} -12361.8 q^{13} -11769.9 q^{14} -33769.8 q^{15} +4096.00 q^{16} -13486.3 q^{17} +24252.1 q^{18} -6859.00 q^{19} +29918.2 q^{20} +106281. q^{21} -28376.3 q^{22} -94084.9 q^{23} -36986.5 q^{24} +140405. q^{25} -98894.4 q^{26} -61006.6 q^{27} -94158.9 q^{28} +33541.0 q^{29} -270159. q^{30} +212096. q^{31} +32768.0 q^{32} +256236. q^{33} -107890. q^{34} -687760. q^{35} +194016. q^{36} +33656.5 q^{37} -54872.0 q^{38} +893007. q^{39} +239346. q^{40} -480531. q^{41} +850246. q^{42} -561340. q^{43} -227011. q^{44} +1.41715e6 q^{45} -752679. q^{46} +478445. q^{47} -295892. q^{48} +1.34098e6 q^{49} +1.12324e6 q^{50} +974237. q^{51} -791156. q^{52} -93127.0 q^{53} -488053. q^{54} -1.65814e6 q^{55} -753271. q^{56} +495489. q^{57} +268328. q^{58} +955034. q^{59} -2.16127e6 q^{60} -1.71632e6 q^{61} +1.69677e6 q^{62} -4.46005e6 q^{63} +262144. q^{64} -5.77880e6 q^{65} +2.04988e6 q^{66} +476255. q^{67} -863120. q^{68} +6.79662e6 q^{69} -5.50208e6 q^{70} +1.95518e6 q^{71} +1.55213e6 q^{72} +1.21632e6 q^{73} +269252. q^{74} -1.01428e7 q^{75} -438976. q^{76} +5.21852e6 q^{77} +7.14406e6 q^{78} +3.94544e6 q^{79} +1.91477e6 q^{80} -2.22284e6 q^{81} -3.84425e6 q^{82} -3.54252e6 q^{83} +6.80197e6 q^{84} -6.30445e6 q^{85} -4.49072e6 q^{86} -2.42298e6 q^{87} -1.81609e6 q^{88} -7.50202e6 q^{89} +1.13372e7 q^{90} +1.81871e7 q^{91} -6.02143e6 q^{92} -1.53216e7 q^{93} +3.82756e6 q^{94} -3.20639e6 q^{95} -2.36714e6 q^{96} +1.27974e7 q^{97} +1.07279e7 q^{98} -1.07529e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 16 q^{2} - 69 q^{3} + 128 q^{4} + 155 q^{5} - 552 q^{6} - 2238 q^{7} + 1024 q^{8} + 855 q^{9}+O(q^{10}) \) 2 * q + 16 * q^2 - 69 * q^3 + 128 * q^4 + 155 * q^5 - 552 * q^6 - 2238 * q^7 + 1024 * q^8 + 855 * q^9	\( 2 q + 16 q^{2} - 69 q^{3} + 128 q^{4} + 155 q^{5} - 552 q^{6} - 2238 q^{7} + 1024 q^{8} + 855 q^{9} + 1240 q^{10} - 3295 q^{11} - 4416 q^{12} - 13427 q^{13} - 17904 q^{14} - 34782 q^{15} + 8192 q^{16} - 32256 q^{17} + 6840 q^{18} - 13718 q^{19} + 9920 q^{20} + 103797 q^{21} - 26360 q^{22} - 82525 q^{23} - 35328 q^{24} + 159919 q^{25} - 107416 q^{26} - 75141 q^{27} - 143232 q^{28} - 12749 q^{29} - 278256 q^{30} + 258944 q^{31} + 65536 q^{32} + 257052 q^{33} - 258048 q^{34} - 448167 q^{35} + 54720 q^{36} - 149260 q^{37} - 109744 q^{38} + 889557 q^{39} + 79360 q^{40} + 339130 q^{41} + 830376 q^{42} - 83869 q^{43} - 210880 q^{44} + 2097243 q^{45} - 660200 q^{46} + 1471025 q^{47} - 282624 q^{48} + 1105372 q^{49} + 1279352 q^{50} + 913437 q^{51} - 859328 q^{52} - 945643 q^{53} - 601128 q^{54} - 1736899 q^{55} - 1145856 q^{56} + 473271 q^{57} - 101992 q^{58} - 969009 q^{59} - 2226048 q^{60} - 1506755 q^{61} + 2071552 q^{62} - 2791179 q^{63} + 524288 q^{64} - 5445956 q^{65} + 2056416 q^{66} - 1848219 q^{67} - 2064384 q^{68} + 6834063 q^{69} - 3585336 q^{70} - 3417184 q^{71} + 437760 q^{72} - 2499822 q^{73} - 1194080 q^{74} - 10079553 q^{75} - 877952 q^{76} + 5025267 q^{77} + 7116456 q^{78} + 2636926 q^{79} + 634880 q^{80} + 2491398 q^{81} + 2713040 q^{82} - 10059354 q^{83} + 6643008 q^{84} - 439425 q^{85} - 670952 q^{86} - 2572923 q^{87} - 1687040 q^{88} - 3506160 q^{89} + 16777944 q^{90} + 19003851 q^{91} - 5281600 q^{92} - 15169884 q^{93} + 11768200 q^{94} - 1063145 q^{95} - 2260992 q^{96} + 5893526 q^{97} + 8842976 q^{98} - 11301453 q^{99}+O(q^{100}) \) 2 * q + 16 * q^2 - 69 * q^3 + 128 * q^4 + 155 * q^5 - 552 * q^6 - 2238 * q^7 + 1024 * q^8 + 855 * q^9 + 1240 * q^10 - 3295 * q^11 - 4416 * q^12 - 13427 * q^13 - 17904 * q^14 - 34782 * q^15 + 8192 * q^16 - 32256 * q^17 + 6840 * q^18 - 13718 * q^19 + 9920 * q^20 + 103797 * q^21 - 26360 * q^22 - 82525 * q^23 - 35328 * q^24 + 159919 * q^25 - 107416 * q^26 - 75141 * q^27 - 143232 * q^28 - 12749 * q^29 - 278256 * q^30 + 258944 * q^31 + 65536 * q^32 + 257052 * q^33 - 258048 * q^34 - 448167 * q^35 + 54720 * q^36 - 149260 * q^37 - 109744 * q^38 + 889557 * q^39 + 79360 * q^40 + 339130 * q^41 + 830376 * q^42 - 83869 * q^43 - 210880 * q^44 + 2097243 * q^45 - 660200 * q^46 + 1471025 * q^47 - 282624 * q^48 + 1105372 * q^49 + 1279352 * q^50 + 913437 * q^51 - 859328 * q^52 - 945643 * q^53 - 601128 * q^54 - 1736899 * q^55 - 1145856 * q^56 + 473271 * q^57 - 101992 * q^58 - 969009 * q^59 - 2226048 * q^60 - 1506755 * q^61 + 2071552 * q^62 - 2791179 * q^63 + 524288 * q^64 - 5445956 * q^65 + 2056416 * q^66 - 1848219 * q^67 - 2064384 * q^68 + 6834063 * q^69 - 3585336 * q^70 - 3417184 * q^71 + 437760 * q^72 - 2499822 * q^73 - 1194080 * q^74 - 10079553 * q^75 - 877952 * q^76 + 5025267 * q^77 + 7116456 * q^78 + 2636926 * q^79 + 634880 * q^80 + 2491398 * q^81 + 2713040 * q^82 - 10059354 * q^83 + 6643008 * q^84 - 439425 * q^85 - 670952 * q^86 - 2572923 * q^87 - 1687040 * q^88 - 3506160 * q^89 + 16777944 * q^90 + 19003851 * q^91 - 5281600 * q^92 - 15169884 * q^93 + 11768200 * q^94 - 1063145 * q^95 - 2260992 * q^96 + 5893526 * q^97 + 8842976 * q^98 - 11301453 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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