LMFDB - Embedded newform 38.8.a.d.1.2

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			1.2
Root			$-12.0797$ 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)

$f(q)$	$=$	$q+8.00000 q^{2} +3.23924 q^{3} +64.0000 q^{4} -312.472 q^{5} +25.9139 q^{6} -766.767 q^{7} +512.000 q^{8} -2176.51 q^{9} +O(q^{10})$	\(q+8.00000 q^{2} +3.23924 q^{3} +64.0000 q^{4} -312.472 q^{5} +25.9139 q^{6} -766.767 q^{7} +512.000 q^{8} -2176.51 q^{9} -2499.78 q^{10} +252.042 q^{11} +207.311 q^{12} -1065.19 q^{13} -6134.14 q^{14} -1012.17 q^{15} +4096.00 q^{16} -18769.7 q^{17} -17412.1 q^{18} -6859.00 q^{19} -19998.2 q^{20} -2483.74 q^{21} +2016.33 q^{22} +11559.9 q^{23} +1658.49 q^{24} +19513.8 q^{25} -8521.55 q^{26} -14134.4 q^{27} -49073.1 q^{28} -46290.0 q^{29} -8097.37 q^{30} +46848.2 q^{31} +32768.0 q^{32} +816.422 q^{33} -150158. q^{34} +239593. q^{35} -139296. q^{36} -182916. q^{37} -54872.0 q^{38} -3450.42 q^{39} -159986. q^{40} +819661. q^{41} -19869.9 q^{42} +477471. q^{43} +16130.7 q^{44} +680098. q^{45} +92478.8 q^{46} +992580. q^{47} +13267.9 q^{48} -235611. q^{49} +156111. q^{50} -60799.7 q^{51} -68172.4 q^{52} -852516. q^{53} -113075. q^{54} -78756.0 q^{55} -392585. q^{56} -22217.9 q^{57} -370320. q^{58} -1.92404e6 q^{59} -64779.0 q^{60} +209564. q^{61} +374786. q^{62} +1.66887e6 q^{63} +262144. q^{64} +332843. q^{65} +6531.38 q^{66} -2.32447e6 q^{67} -1.20126e6 q^{68} +37445.1 q^{69} +1.91675e6 q^{70} -5.37237e6 q^{71} -1.11437e6 q^{72} -3.71614e6 q^{73} -1.46333e6 q^{74} +63209.9 q^{75} -438976. q^{76} -193257. q^{77} -27603.3 q^{78} -1.30851e6 q^{79} -1.27989e6 q^{80} +4.71424e6 q^{81} +6.55729e6 q^{82} -6.51683e6 q^{83} -158959. q^{84} +5.86502e6 q^{85} +3.81976e6 q^{86} -149944. q^{87} +129045. q^{88} +3.99586e6 q^{89} +5.44078e6 q^{90} +816756. q^{91} +739831. q^{92} +151753. q^{93} +7.94064e6 q^{94} +2.14325e6 q^{95} +106143. q^{96} -6.90391e6 q^{97} -1.88489e6 q^{98} -548570. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	3.23924	0.0692657	0.0346329	−	0.999400i	$-0.488974\pi$
$3$	3.23924	0.0692657	0.0346329	+	0.999400i	$0.488974\pi$
$4$	64.0000	0.500000
$5$	−312.472	−1.11793	−0.558967	−	0.829190i	$-0.688802\pi$
$5$	−312.472	−1.11793	−0.558967	+	0.829190i	$0.688802\pi$
$6$	25.9139	0.0489783
$7$	−766.767	−0.844929	−0.422465	−	0.906379i	$-0.638835\pi$
$7$	−766.767	−0.844929	−0.422465	+	0.906379i	$0.638835\pi$
$8$	512.000	0.353553
$9$	−2176.51	−0.995202
$10$	−2499.78	−0.790499
$11$	252.042	0.0570950	0.0285475	−	0.999592i	$-0.490912\pi$
$11$	252.042	0.0570950	0.0285475	+	0.999592i	$0.490912\pi$
$12$	207.311	0.0346329
$13$	−1065.19	−0.134471	−0.0672353	−	0.997737i	$-0.521418\pi$
$13$	−1065.19	−0.134471	−0.0672353	+	0.997737i	$0.521418\pi$
$14$	−6134.14	−0.597455
$15$	−1012.17	−0.0774345
$16$	4096.00	0.250000
$17$	−18769.7	−0.926589	−0.463295	−	0.886204i	$-0.653333\pi$
$17$	−18769.7	−0.926589	−0.463295	+	0.886204i	$0.653333\pi$
$18$	−17412.1	−0.703714
$19$	−6859.00	−0.229416
$19$	−6859.00	−0.229416
$20$	−19998.2	−0.558967
$21$	−2483.74	−0.0585246
$22$	2016.33	0.0403722
$23$	11559.9	0.198109	0.0990546	−	0.995082i	$-0.468418\pi$
$23$	11559.9	0.198109	0.0990546	+	0.995082i	$0.468418\pi$
$24$	1658.49	0.0244891
$25$	19513.8	0.249777
$26$	−8521.55	−0.0950850
$27$	−14134.4	−0.138199
$28$	−49073.1	−0.422465
$29$	−46290.0	−0.352448	−0.176224	−	0.984350i	$-0.556388\pi$
$29$	−46290.0	−0.352448	−0.176224	+	0.984350i	$0.556388\pi$
$30$	−8097.37	−0.0547545
$31$	46848.2	0.282441	0.141220	−	0.989978i	$-0.454897\pi$
$31$	46848.2	0.282441	0.141220	+	0.989978i	$0.454897\pi$
$32$	32768.0	0.176777
$33$	816.422	0.00395472
$34$	−150158.	−0.655197
$35$	239593.	0.944575
$36$	−139296.	−0.497601
$37$	−182916.	−0.593672	−0.296836	−	0.954928i	$-0.595932\pi$
$37$	−182916.	−0.593672	−0.296836	+	0.954928i	$0.595932\pi$
$38$	−54872.0	−0.162221
$39$	−3450.42	−0.00931420
$40$	−159986.	−0.395249
$41$	819661.	1.85734	0.928669	−	0.370910i	$-0.120954\pi$
$41$	819661.	1.85734	0.928669	+	0.370910i	$0.120954\pi$
$42$	−19869.9	−0.0413832
$43$	477471.	0.915814	0.457907	−	0.889000i	$-0.348599\pi$
$43$	477471.	0.915814	0.457907	+	0.889000i	$0.348599\pi$
$44$	16130.7	0.0285475
$45$	680098.	1.11257
$46$	92478.8	0.140084
$47$	992580.	1.39451	0.697257	−	0.716821i	$-0.254404\pi$
$47$	992580.	1.39451	0.697257	+	0.716821i	$0.254404\pi$
$48$	13267.9	0.0173164
$49$	−235611.	−0.286095
$50$	156111.	0.176619
$51$	−60799.7	−0.0641809
$52$	−68172.4	−0.0672353
$53$	−852516.	−0.786569	−0.393285	−	0.919417i	$-0.628661\pi$
$53$	−852516.	−0.786569	−0.393285	+	0.919417i	$0.628661\pi$
$54$	−113075.	−0.0977215
$55$	−78756.0	−0.0638284
$56$	−392585.	−0.298728
$57$	−22217.9	−0.0158906
$58$	−370320.	−0.249218
$59$	−1.92404e6	−1.21964	−0.609821	−	0.792539i	$-0.708759\pi$
$59$	−1.92404e6	−1.21964	−0.609821	+	0.792539i	$0.708759\pi$
$60$	−64779.0	−0.0387173
$61$	209564.	0.118212	0.0591062	−	0.998252i	$-0.481175\pi$
$61$	209564.	0.118212	0.0591062	+	0.998252i	$0.481175\pi$
$62$	374786.	0.199716
$63$	1.66887e6	0.840875
$64$	262144.	0.125000
$65$	332843.	0.150329
$66$	6531.38	0.00279641
$67$	−2.32447e6	−0.944198	−0.472099	−	0.881546i	$-0.656503\pi$
$67$	−2.32447e6	−0.944198	−0.472099	+	0.881546i	$0.656503\pi$
$68$	−1.20126e6	−0.463295
$69$	37445.1	0.0137222
$70$	1.91675e6	0.667916
$71$	−5.37237e6	−1.78140	−0.890700	−	0.454591i	$-0.849785\pi$
$71$	−5.37237e6	−1.78140	−0.890700	+	0.454591i	$0.849785\pi$
$72$	−1.11437e6	−0.351857
$73$	−3.71614e6	−1.11805	−0.559027	−	0.829150i	$-0.688825\pi$
$73$	−3.71614e6	−1.11805	−0.559027	+	0.829150i	$0.688825\pi$
$74$	−1.46333e6	−0.419790
$75$	63209.9	0.0173010
$76$	−438976.	−0.114708
$77$	−193257.	−0.0482412
$78$	−27603.3	−0.00658613
$79$	−1.30851e6	−0.298596	−0.149298	−	0.988792i	$-0.547701\pi$
$79$	−1.30851e6	−0.298596	−0.149298	+	0.988792i	$0.547701\pi$
$80$	−1.27989e6	−0.279484
$81$	4.71424e6	0.985630
$82$	6.55729e6	1.31334
$83$	−6.51683e6	−1.25102	−0.625508	−	0.780218i	$-0.715108\pi$
$83$	−6.51683e6	−1.25102	−0.625508	+	0.780218i	$0.715108\pi$
$84$	−158959.	−0.0292623
$85$	5.86502e6	1.03587
$86$	3.81976e6	0.647578
$87$	−149944.	−0.0244125
$88$	129045.	0.0201861
$89$	3.99586e6	0.600822	0.300411	−	0.953810i	$-0.402876\pi$
$89$	3.99586e6	0.600822	0.300411	+	0.953810i	$0.402876\pi$
$90$	5.44078e6	0.786706
$91$	816756.	0.113618
$92$	739831.	0.0990546
$93$	151753.	0.0195635
$94$	7.94064e6	0.986070
$95$	2.14325e6	0.256472
$96$	106143.	0.0122446
$97$	−6.90391e6	−0.768058	−0.384029	−	0.923321i	$-0.625464\pi$
$97$	−6.90391e6	−0.768058	−0.384029	+	0.923321i	$0.625464\pi$
$98$	−1.88489e6	−0.202299
$99$	−548570.	−0.0568210
$100$	1.24888e6	0.124888
$101$	2.81610e6	0.271971	0.135986	−	0.990711i	$-0.456580\pi$
$101$	2.81610e6	0.271971	0.135986	+	0.990711i	$0.456580\pi$
$102$	−486397.	−0.0453827
$103$	7.86273e6	0.708995	0.354497	−	0.935057i	$-0.384652\pi$
$103$	7.86273e6	0.708995	0.354497	+	0.935057i	$0.384652\pi$
$104$	−545379.	−0.0475425
$105$	776100.	0.0654267
$106$	−6.82013e6	−0.556188
$107$	611626.	0.0482662	0.0241331	−	0.999709i	$-0.492317\pi$
$107$	611626.	0.0482662	0.0241331	+	0.999709i	$0.492317\pi$
$108$	−904604.	−0.0690996
$109$	5.23210e6	0.386976	0.193488	−	0.981103i	$-0.438020\pi$
$109$	5.23210e6	0.386976	0.193488	+	0.981103i	$0.438020\pi$
$110$	−630048.	−0.0451335
$111$	−592510.	−0.0411211
$112$	−3.14068e6	−0.211232
$113$	−3.58527e6	−0.233747	−0.116874	−	0.993147i	$-0.537287\pi$
$113$	−3.58527e6	−0.233747	−0.116874	+	0.993147i	$0.537287\pi$
$114$	−177743.	−0.0112364
$115$	−3.61213e6	−0.221473
$116$	−2.96256e6	−0.176224
$117$	2.31840e6	0.133825
$118$	−1.53923e7	−0.862418
$119$	1.43920e7	0.782902
$120$	−518232.	−0.0273772
$121$	−1.94236e7	−0.996740
$122$	1.67651e6	0.0835888
$123$	2.65508e6	0.128650
$124$	2.99829e6	0.141220
$125$	1.83144e7	0.838700
$126$	1.33510e7	0.594589
$127$	−2.99465e7	−1.29728	−0.648639	−	0.761096i	$-0.724661\pi$
$127$	−2.99465e7	−1.29728	−0.648639	+	0.761096i	$0.724661\pi$
$128$	2.09715e6	0.0883883
$129$	1.54664e6	0.0634345
$130$	2.66275e6	0.106299
$131$	−8.11292e6	−0.315303	−0.157651	−	0.987495i	$-0.550392\pi$
$131$	−8.11292e6	−0.315303	−0.157651	+	0.987495i	$0.550392\pi$
$132$	52251.0	0.00197736
$133$	5.25926e6	0.193840
$134$	−1.85958e7	−0.667649
$135$	4.41662e6	0.154498
$136$	−9.61011e6	−0.327599
$137$	1.44157e7	0.478977	0.239488	−	0.970899i	$-0.423020\pi$
$137$	1.44157e7	0.478977	0.239488	+	0.970899i	$0.423020\pi$
$138$	299561.	0.00970305
$139$	−3.77446e7	−1.19207	−0.596036	−	0.802957i	$-0.703259\pi$
$139$	−3.77446e7	−1.19207	−0.596036	+	0.802957i	$0.703259\pi$
$140$	1.53340e7	0.472288
$141$	3.21520e6	0.0965920
$142$	−4.29789e7	−1.25964
$143$	−268473.	−0.00767759
$144$	−8.91497e6	−0.248801
$145$	1.44643e7	0.394013
$146$	−2.97292e7	−0.790583
$147$	−763200.	−0.0198165
$148$	−1.17067e7	−0.296836
$149$	7.30679e7	1.80957	0.904784	−	0.425872i	$-0.140033\pi$
$149$	7.30679e7	1.80957	0.904784	+	0.425872i	$0.140033\pi$
$150$	505679.	0.0122336
$151$	−4.14056e7	−0.978678	−0.489339	−	0.872094i	$-0.662762\pi$
$151$	−4.14056e7	−0.978678	−0.489339	+	0.872094i	$0.662762\pi$
$152$	−3.51181e6	−0.0811107
$153$	4.08525e7	0.922144
$154$	−1.54606e6	−0.0341117
$155$	−1.46388e7	−0.315750
$156$	−220827.	−0.00465710
$157$	2.93945e7	0.606203	0.303101	−	0.952958i	$-0.401978\pi$
$157$	2.93945e7	0.606203	0.303101	+	0.952958i	$0.401978\pi$
$158$	−1.04681e7	−0.211139
$159$	−2.76150e6	−0.0544823
$160$	−1.02391e7	−0.197625
$161$	−8.86371e6	−0.167388
$162$	3.77139e7	0.696946
$163$	9.03924e6	0.163484	0.0817420	−	0.996654i	$-0.473952\pi$
$163$	9.03924e6	0.163484	0.0817420	+	0.996654i	$0.473952\pi$
$164$	5.24583e7	0.928669
$165$	−255109.	−0.00442112
$166$	−5.21346e7	−0.884602
$167$	−8.16728e7	−1.35697	−0.678484	−	0.734615i	$-0.737363\pi$
$167$	−8.16728e7	−1.35697	−0.678484	+	0.734615i	$0.737363\pi$
$168$	−1.27168e6	−0.0206916
$169$	−6.16139e7	−0.981918
$170$	4.69202e7	0.732468
$171$	1.49287e7	0.228315
$172$	3.05581e7	0.457907
$173$	1.33191e8	1.95575	0.977874	−	0.209197i	$-0.0670849\pi$
$173$	1.33191e8	1.95575	0.977874	+	0.209197i	$0.0670849\pi$
$174$	−1.19956e6	−0.0172623
$175$	−1.49626e7	−0.211044
$176$	1.03236e6	0.0142737
$177$	−6.23243e6	−0.0844794
$178$	3.19669e7	0.424845
$179$	−4.90910e7	−0.639759	−0.319879	−	0.947458i	$-0.603642\pi$
$179$	−4.90910e7	−0.639759	−0.319879	+	0.947458i	$0.603642\pi$
$180$	4.35263e7	0.556285
$181$	8.11979e7	1.01782	0.508908	−	0.860821i	$-0.330049\pi$
$181$	8.11979e7	1.01782	0.508908	+	0.860821i	$0.330049\pi$
$182$	6.53405e6	0.0803401
$183$	678828.	0.00818806
$184$	5.91864e6	0.0700422
$185$	5.71563e7	0.663686
$186$	1.21402e6	0.0138335
$187$	−4.73076e6	−0.0529036
$188$	6.35251e7	0.697257
$189$	1.08378e7	0.116768
$190$	1.71460e7	0.181353
$191$	7.40713e7	0.769190	0.384595	−	0.923086i	$-0.374341\pi$
$191$	7.40713e7	0.769190	0.384595	+	0.923086i	$0.374341\pi$
$192$	849147.	0.00865821
$193$	−9.70125e7	−0.971353	−0.485676	−	0.874139i	$-0.661427\pi$
$193$	−9.70125e7	−0.971353	−0.485676	+	0.874139i	$0.661427\pi$
$194$	−5.52313e7	−0.543099
$195$	1.07816e6	0.0104127
$196$	−1.50791e7	−0.143047
$197$	−4.34770e7	−0.405161	−0.202581	−	0.979266i	$-0.564933\pi$
$197$	−4.34770e7	−0.405161	−0.202581	+	0.979266i	$0.564933\pi$
$198$	−4.38856e6	−0.0401785
$199$	3.89651e7	0.350502	0.175251	−	0.984524i	$-0.443926\pi$
$199$	3.89651e7	0.350502	0.175251	+	0.984524i	$0.443926\pi$
$200$	9.99108e6	0.0883095
$201$	−7.52952e6	−0.0654005
$202$	2.25288e7	0.192313
$203$	3.54937e7	0.297793
$204$	−3.89118e6	−0.0320904
$205$	−2.56121e8	−2.07638
$206$	6.29018e7	0.501335
$207$	−2.51601e7	−0.197159
$208$	−4.36304e6	−0.0336176
$209$	−1.72875e6	−0.0130985
$210$	6.20880e6	0.0462637
$211$	−3.85264e7	−0.282339	−0.141169	−	0.989985i	$-0.545086\pi$
$211$	−3.85264e7	−0.282339	−0.141169	+	0.989985i	$0.545086\pi$
$212$	−5.45610e7	−0.393285
$213$	−1.74024e7	−0.123390
$214$	4.89301e6	0.0341293
$215$	−1.49196e8	−1.02382
$216$	−7.23683e6	−0.0488608
$217$	−3.59217e7	−0.238642
$218$	4.18568e7	0.273633
$219$	−1.20375e7	−0.0774428
$220$	−5.04038e6	−0.0319142
$221$	1.99934e7	0.124599
$222$	−4.74008e6	−0.0290770
$223$	−2.03401e8	−1.22825	−0.614124	−	0.789209i	$-0.710491\pi$
$223$	−2.03401e8	−1.22825	−0.614124	+	0.789209i	$0.710491\pi$
$224$	−2.51254e7	−0.149364
$225$	−4.24720e7	−0.248579
$226$	−2.86821e7	−0.165284
$227$	2.24944e8	1.27639	0.638195	−	0.769875i	$-0.279681\pi$
$227$	2.24944e8	1.27639	0.638195	+	0.769875i	$0.279681\pi$
$228$	−1.42195e6	−0.00794532
$229$	−2.38882e8	−1.31450	−0.657248	−	0.753674i	$-0.728280\pi$
$229$	−2.38882e8	−1.31450	−0.657248	+	0.753674i	$0.728280\pi$
$230$	−2.88971e7	−0.156605
$231$	−626006.	−0.00334146
$232$	−2.37005e7	−0.124609
$233$	3.23019e8	1.67295	0.836475	−	0.548005i	$-0.184613\pi$
$233$	3.23019e8	1.67295	0.836475	+	0.548005i	$0.184613\pi$
$234$	1.85472e7	0.0946288
$235$	−3.10153e8	−1.55897
$236$	−1.23139e8	−0.609821
$237$	−4.23859e6	−0.0206825
$238$	1.15136e8	0.553595
$239$	−5.18160e7	−0.245511	−0.122756	−	0.992437i	$-0.539173\pi$
$239$	−5.18160e7	−0.245511	−0.122756	+	0.992437i	$0.539173\pi$
$240$	−4.14585e6	−0.0193586
$241$	1.15926e8	0.533484	0.266742	−	0.963768i	$-0.414053\pi$
$241$	1.15926e8	0.533484	0.266742	+	0.963768i	$0.414053\pi$
$242$	−1.55389e8	−0.704802
$243$	4.61825e7	0.206469
$244$	1.34121e7	0.0591062
$245$	7.36219e7	0.319835
$246$	2.12406e7	0.0909692
$247$	7.30617e6	0.0308497
$248$	2.39863e7	0.0998579
$249$	−2.11096e7	−0.0866525
$250$	1.46515e8	0.593050
$251$	−4.03761e7	−0.161163	−0.0805817	−	0.996748i	$-0.525678\pi$
$251$	−4.03761e7	−0.161163	−0.0805817	+	0.996748i	$0.525678\pi$
$252$	1.06808e8	0.420438
$253$	2.91356e6	0.0113110
$254$	−2.39572e8	−0.917314
$255$	1.89982e7	0.0717500
$256$	1.67772e7	0.0625000
$257$	2.67661e8	0.983602	0.491801	−	0.870708i	$-0.336339\pi$
$257$	2.67661e8	0.983602	0.491801	+	0.870708i	$0.336339\pi$
$258$	1.23731e7	0.0448549
$259$	1.40254e8	0.501611
$260$	2.13020e7	0.0751646
$261$	1.00751e8	0.350757
$262$	−6.49034e7	−0.222953
$263$	5.12778e8	1.73814	0.869068	−	0.494692i	$-0.164719\pi$
$263$	5.12778e8	1.73814	0.869068	+	0.494692i	$0.164719\pi$
$264$	418008.	0.00139821
$265$	2.66387e8	0.879333
$266$	4.20740e7	0.137066
$267$	1.29435e7	0.0416163
$268$	−1.48766e8	−0.472099
$269$	5.75792e8	1.80357	0.901784	−	0.432187i	$-0.142258\pi$
$269$	5.75792e8	1.80357	0.901784	+	0.432187i	$0.142258\pi$
$270$	3.53329e7	0.109246
$271$	5.24839e8	1.60189	0.800946	−	0.598737i	$-0.204330\pi$
$271$	5.24839e8	1.60189	0.800946	+	0.598737i	$0.204330\pi$
$272$	−7.68809e7	−0.231647
$273$	2.64567e6	0.00786984
$274$	1.15326e8	0.338688
$275$	4.91829e6	0.0142610
$276$	2.39649e6	0.00686109
$277$	−4.64620e8	−1.31347	−0.656733	−	0.754123i	$-0.728062\pi$
$277$	−4.64620e8	−1.31347	−0.656733	+	0.754123i	$0.728062\pi$
$278$	−3.01957e8	−0.842923
$279$	−1.01966e8	−0.281086
$280$	1.22672e8	0.333958
$281$	−1.75429e8	−0.471660	−0.235830	−	0.971794i	$-0.575781\pi$
$281$	−1.75429e8	−0.471660	−0.235830	+	0.971794i	$0.575781\pi$
$282$	2.57216e7	0.0683009
$283$	1.44948e8	0.380154	0.190077	−	0.981769i	$-0.439126\pi$
$283$	1.44948e8	0.380154	0.190077	+	0.981769i	$0.439126\pi$
$284$	−3.43832e8	−0.890700
$285$	6.94248e6	0.0177647
$286$	−2.14779e6	−0.00542888
$287$	−6.28489e8	−1.56932
$288$	−7.13198e7	−0.175929
$289$	−5.80353e7	−0.141433
$290$	1.15715e8	0.278609
$291$	−2.23634e7	−0.0532001
$292$	−2.37833e8	−0.559027
$293$	−4.24522e8	−0.985970	−0.492985	−	0.870038i	$-0.664094\pi$
$293$	−4.24522e8	−0.985970	−0.492985	+	0.870038i	$0.664094\pi$
$294$	−6.10560e6	−0.0140124
$295$	6.01210e8	1.36348
$296$	−9.36532e7	−0.209895
$297$	−3.56247e6	−0.00789047
$298$	5.84543e8	1.27956
$299$	−1.23135e7	−0.0266399
$300$	4.04543e6	0.00865049
$301$	−3.66109e8	−0.773798
$302$	−3.31245e8	−0.692030
$303$	9.12200e6	0.0188383
$304$	−2.80945e7	−0.0573539
$305$	−6.54830e7	−0.132154
$306$	3.26820e8	0.652054
$307$	4.27528e8	0.843297	0.421649	−	0.906759i	$-0.361452\pi$
$307$	4.27528e8	0.843297	0.421649	+	0.906759i	$0.361452\pi$
$308$	−1.23685e7	−0.0241206
$309$	2.54692e7	0.0491090
$310$	−1.17110e8	−0.223269
$311$	−1.70275e8	−0.320989	−0.160495	−	0.987037i	$-0.551309\pi$
$311$	−1.70275e8	−0.320989	−0.160495	+	0.987037i	$0.551309\pi$
$312$	−1.76661e6	−0.00329307
$313$	−4.02143e8	−0.741269	−0.370634	−	0.928779i	$-0.620860\pi$
$313$	−4.02143e8	−0.741269	−0.370634	+	0.928779i	$0.620860\pi$
$314$	2.35156e8	0.428650
$315$	−5.21477e8	−0.940043
$316$	−8.37450e7	−0.149298
$317$	−1.90630e8	−0.336112	−0.168056	−	0.985777i	$-0.553749\pi$
$317$	−1.90630e8	−0.336112	−0.168056	+	0.985777i	$0.553749\pi$
$318$	−2.20920e7	−0.0385248
$319$	−1.16670e7	−0.0201230
$320$	−8.19127e7	−0.139742
$321$	1.98120e6	0.00334319
$322$	−7.09097e7	−0.118361
$323$	1.28742e8	0.212574
$324$	3.01711e8	0.492815
$325$	−2.07860e7	−0.0335876
$326$	7.23139e7	0.115601
$327$	1.69480e7	0.0268041
$328$	4.19666e8	0.656668
$329$	−7.61077e8	−1.17827
$330$	−2.04087e6	−0.00312620
$331$	−4.12008e8	−0.624464	−0.312232	−	0.950006i	$-0.601077\pi$
$331$	−4.12008e8	−0.624464	−0.312232	+	0.950006i	$0.601077\pi$
$332$	−4.17077e8	−0.625508
$333$	3.98119e8	0.590824
$334$	−6.53382e8	−0.959522
$335$	7.26333e8	1.05555
$336$	−1.01734e7	−0.0146312
$337$	−6.99272e8	−0.995271	−0.497636	−	0.867386i	$-0.665798\pi$
$337$	−6.99272e8	−0.995271	−0.497636	+	0.867386i	$0.665798\pi$
$338$	−4.92911e8	−0.694321
$339$	−1.16135e7	−0.0161907
$340$	3.75361e8	0.517933
$341$	1.18077e7	0.0161259
$342$	1.19429e8	0.161443
$343$	8.12125e8	1.08666
$344$	2.44465e8	0.323789
$345$	−1.17005e7	−0.0153405
$346$	1.06553e9	1.38292
$347$	−9.29885e8	−1.19475	−0.597374	−	0.801963i	$-0.703789\pi$
$347$	−9.29885e8	−1.19475	−0.597374	+	0.801963i	$0.703789\pi$
$348$	−9.59644e6	−0.0122063
$349$	−4.43776e8	−0.558823	−0.279412	−	0.960171i	$-0.590139\pi$
$349$	−4.43776e8	−0.558823	−0.279412	+	0.960171i	$0.590139\pi$
$350$	−1.19700e8	−0.149231
$351$	1.50559e7	0.0185837
$352$	8.25890e6	0.0100931
$353$	−1.49388e9	−1.80761	−0.903804	−	0.427946i	$-0.859237\pi$
$353$	−1.49388e9	−1.80761	−0.903804	+	0.427946i	$0.859237\pi$
$354$	−4.98594e7	−0.0597360
$355$	1.67872e9	1.99149
$356$	2.55735e8	0.300411
$357$	4.66192e7	0.0542283
$358$	−3.92728e8	−0.452378
$359$	−1.77937e8	−0.202972	−0.101486	−	0.994837i	$-0.532360\pi$
$359$	−1.77937e8	−0.202972	−0.101486	+	0.994837i	$0.532360\pi$
$360$	3.48210e8	0.393353
$361$	4.70459e7	0.0526316
$362$	6.49583e8	0.719705
$363$	−6.29178e7	−0.0690399
$364$	5.22724e7	0.0568090
$365$	1.16119e9	1.24991
$366$	5.43063e6	0.00578983
$367$	−1.37515e9	−1.45218	−0.726088	−	0.687601i	$-0.758664\pi$
$367$	−1.37515e9	−1.45218	−0.726088	+	0.687601i	$0.758664\pi$
$368$	4.73492e7	0.0495273
$369$	−1.78400e9	−1.84843
$370$	4.57250e8	0.469297
$371$	6.53681e8	0.664595
$372$	9.71216e6	0.00978173
$373$	1.55845e9	1.55493	0.777466	−	0.628925i	$-0.216505\pi$
$373$	1.55845e9	1.55493	0.777466	+	0.628925i	$0.216505\pi$
$374$	−3.78461e7	−0.0374085
$375$	5.93245e7	0.0580932
$376$	5.08201e8	0.493035
$377$	4.93079e7	0.0473938
$378$	8.67026e7	0.0825678
$379$	8.84549e8	0.834613	0.417306	−	0.908766i	$-0.362974\pi$
$379$	8.84549e8	0.834613	0.417306	+	0.908766i	$0.362974\pi$
$380$	1.37168e8	0.128236
$381$	−9.70038e7	−0.0898569
$382$	5.92571e8	0.543899
$383$	1.44573e9	1.31490	0.657449	−	0.753499i	$-0.271635\pi$
$383$	1.44573e9	1.31490	0.657449	+	0.753499i	$0.271635\pi$
$384$	6.79317e6	0.00612228
$385$	6.03875e7	0.0539305
$386$	−7.76100e8	−0.686850
$387$	−1.03922e9	−0.911420
$388$	−4.41850e8	−0.384029
$389$	−1.02164e9	−0.879986	−0.439993	−	0.898001i	$-0.645019\pi$
$389$	−1.02164e9	−0.879986	−0.439993	+	0.898001i	$0.645019\pi$
$390$	8.62527e6	0.00736286
$391$	−2.16975e8	−0.183566
$392$	−1.20633e8	−0.101150
$393$	−2.62797e7	−0.0218397
$394$	−3.47816e8	−0.286492
$395$	4.08874e8	0.333811
$396$	−3.51085e7	−0.0284105
$397$	1.80050e8	0.144419	0.0722097	−	0.997389i	$-0.476995\pi$
$397$	1.80050e8	0.144419	0.0722097	+	0.997389i	$0.476995\pi$
$398$	3.11721e8	0.247842
$399$	1.70360e7	0.0134265
$400$	7.99286e7	0.0624442
$401$	5.06670e8	0.392392	0.196196	−	0.980565i	$-0.437141\pi$
$401$	5.06670e8	0.392392	0.196196	+	0.980565i	$0.437141\pi$
$402$	−6.02362e7	−0.0462452
$403$	−4.99025e7	−0.0379799
$404$	1.80230e8	0.135986
$405$	−1.47307e9	−1.10187
$406$	2.83949e8	0.210572
$407$	−4.61026e7	−0.0338957
$408$	−3.11294e7	−0.0226914
$409$	1.14898e9	0.830388	0.415194	−	0.909733i	$-0.363714\pi$
$409$	1.14898e9	0.830388	0.415194	+	0.909733i	$0.363714\pi$
$410$	−2.04897e9	−1.46822
$411$	4.66959e7	0.0331767
$412$	5.03215e8	0.354497
$413$	1.47529e9	1.03051
$414$	−2.01281e8	−0.139412
$415$	2.03633e9	1.39855
$416$	−3.49043e7	−0.0237713
$417$	−1.22264e8	−0.0825698
$418$	−1.38300e7	−0.00926203
$419$	2.02289e9	1.34346	0.671729	−	0.740797i	$-0.265552\pi$
$419$	2.02289e9	1.34346	0.671729	+	0.740797i	$0.265552\pi$
$420$	4.96704e7	0.0327133
$421$	2.55503e9	1.66882	0.834408	−	0.551147i	$-0.185810\pi$
$421$	2.55503e9	1.66882	0.834408	+	0.551147i	$0.185810\pi$
$422$	−3.08211e8	−0.199644
$423$	−2.16036e9	−1.38782
$424$	−4.36488e8	−0.278094
$425$	−3.66269e8	−0.231441
$426$	−1.39219e8	−0.0872499
$427$	−1.60687e8	−0.0998811
$428$	3.91441e7	0.0241331
$429$	−869648.	−0.000531794 0
$430$	−1.19357e9	−0.723950
$431$	1.47316e9	0.886295	0.443147	−	0.896449i	$-0.353862\pi$
$431$	1.47316e9	0.886295	0.443147	+	0.896449i	$0.353862\pi$
$432$	−5.78946e7	−0.0345498
$433$	−9.33243e8	−0.552443	−0.276221	−	0.961094i	$-0.589082\pi$
$433$	−9.33243e8	−0.552443	−0.276221	+	0.961094i	$0.589082\pi$
$434$	−2.87373e8	−0.168746
$435$	4.68534e7	0.0272916
$436$	3.34855e8	0.193488
$437$	−7.92890e7	−0.0454494
$438$	−9.62998e7	−0.0547603
$439$	−1.94411e9	−1.09672	−0.548358	−	0.836244i	$-0.684747\pi$
$439$	−1.94411e9	−1.09672	−0.548358	+	0.836244i	$0.684747\pi$
$440$	−4.03231e7	−0.0225668
$441$	5.12809e8	0.284722
$442$	1.59947e8	0.0881047
$443$	−2.35592e9	−1.28750	−0.643751	−	0.765235i	$-0.722623\pi$
$443$	−2.35592e9	−1.28750	−0.643751	+	0.765235i	$0.722623\pi$
$444$	−3.79206e7	−0.0205606
$445$	−1.24860e9	−0.671679
$446$	−1.62721e9	−0.868503
$447$	2.36684e8	0.125341
$448$	−2.01003e8	−0.105616
$449$	−1.83374e9	−0.956042	−0.478021	−	0.878349i	$-0.658646\pi$
$449$	−1.83374e9	−0.956042	−0.478021	+	0.878349i	$0.658646\pi$
$450$	−3.39776e8	−0.175772
$451$	2.06589e8	0.106045
$452$	−2.29457e8	−0.116874
$453$	−1.34123e8	−0.0677889
$454$	1.79955e9	0.902544
$455$	−2.55213e8	−0.127018
$456$	−1.13756e7	−0.00561819
$457$	−3.76683e9	−1.84616	−0.923079	−	0.384611i	$-0.874336\pi$
$457$	−3.76683e9	−1.84616	−0.923079	+	0.384611i	$0.874336\pi$
$458$	−1.91106e9	−0.929489
$459$	2.65300e8	0.128054
$460$	−2.31176e8	−0.110737
$461$	3.30560e9	1.57144	0.785720	−	0.618583i	$-0.212293\pi$
$461$	3.30560e9	1.57144	0.785720	+	0.618583i	$0.212293\pi$
$462$	−5.00805e6	−0.00236277
$463$	−3.66326e9	−1.71528	−0.857638	−	0.514253i	$-0.828069\pi$
$463$	−3.66326e9	−1.71528	−0.857638	+	0.514253i	$0.828069\pi$
$464$	−1.89604e8	−0.0881119
$465$	−4.74184e7	−0.0218707
$466$	2.58416e9	1.18295
$467$	−2.22674e9	−1.01172	−0.505860	−	0.862616i	$-0.668825\pi$
$467$	−2.22674e9	−1.01172	−0.505860	+	0.862616i	$0.668825\pi$
$468$	1.48378e8	0.0669127
$469$	1.78233e9	0.797780
$470$	−2.48123e9	−1.10236
$471$	9.52159e7	0.0419891
$472$	−9.85110e8	−0.431209
$473$	1.20342e8	0.0522883
$474$	−3.39087e7	−0.0146247
$475$	−1.33845e8	−0.0573028
$476$	9.21090e8	0.391451
$477$	1.85551e9	0.782796
$478$	−4.14528e8	−0.173603
$479$	1.43508e9	0.596625	0.298313	−	0.954468i	$-0.403576\pi$
$479$	1.43508e9	0.596625	0.298313	+	0.954468i	$0.403576\pi$
$480$	−3.31668e7	−0.0136886
$481$	1.94842e8	0.0798314
$482$	9.27409e8	0.377230
$483$	−2.87117e7	−0.0115943
$484$	−1.24311e9	−0.498370
$485$	2.15728e9	0.858638
$486$	3.69460e8	0.145996
$487$	−2.11642e9	−0.830330	−0.415165	−	0.909746i	$-0.636276\pi$
$487$	−2.11642e9	−0.830330	−0.415165	+	0.909746i	$0.636276\pi$
$488$	1.07297e8	0.0417944
$489$	2.92802e7	0.0113238
$490$	5.88975e8	0.226157
$491$	4.07105e9	1.55211	0.776053	−	0.630668i	$-0.217219\pi$
$491$	4.07105e9	1.55211	0.776053	+	0.630668i	$0.217219\pi$
$492$	1.69925e8	0.0643249
$493$	8.68852e8	0.326574
$494$	5.84493e7	0.0218140
$495$	1.71413e8	0.0635222
$496$	1.91890e8	0.0706102
$497$	4.11935e9	1.50516
$498$	−1.68876e8	−0.0612726
$499$	3.93039e9	1.41607	0.708034	−	0.706178i	$-0.249582\pi$
$499$	3.93039e9	1.41607	0.708034	+	0.706178i	$0.249582\pi$
$500$	1.17212e9	0.419350
$501$	−2.64558e8	−0.0939914
$502$	−3.23009e8	−0.113960
$503$	−1.70030e9	−0.595714	−0.297857	−	0.954610i	$-0.596272\pi$
$503$	−1.70030e9	−0.595714	−0.297857	+	0.954610i	$0.596272\pi$
$504$	8.54464e8	0.297294
$505$	−8.79952e8	−0.304046
$506$	2.33085e7	0.00799812
$507$	−1.99582e8	−0.0680132
$508$	−1.91658e9	−0.648639
$509$	−2.62955e8	−0.0883830	−0.0441915	−	0.999023i	$-0.514071\pi$
$509$	−2.62955e8	−0.0883830	−0.0441915	+	0.999023i	$0.514071\pi$
$510$	1.51986e8	0.0507349
$511$	2.84942e9	0.944676
$512$	1.34218e8	0.0441942
$513$	9.69481e7	0.0317051
$514$	2.14129e9	0.695512
$515$	−2.45688e9	−0.792609
$516$	9.89850e7	0.0317172
$517$	2.50171e8	0.0796197
$518$	1.12203e9	0.354693
$519$	4.31436e8	0.135466
$520$	1.70416e8	0.0531494
$521$	−1.86990e9	−0.579277	−0.289639	−	0.957136i	$-0.593535\pi$
$521$	−1.86990e9	−0.579277	−0.289639	+	0.957136i	$0.593535\pi$
$522$	8.06005e8	0.248022
$523$	−5.49460e9	−1.67950	−0.839750	−	0.542972i	$-0.817299\pi$
$523$	−5.49460e9	−1.67950	−0.839750	+	0.542972i	$0.817299\pi$
$524$	−5.19227e8	−0.157651
$525$	−4.84673e7	−0.0146181
$526$	4.10222e9	1.22905
$527$	−8.79329e8	−0.261706
$528$	3.34407e6	0.000988681 0
$529$	−3.27120e9	−0.960753
$530$	2.13110e9	0.621782
$531$	4.18769e9	1.21379
$532$	3.36592e8	0.0969200
$533$	−8.73098e8	−0.249757
$534$	1.03548e8	0.0294272
$535$	−1.91116e8	−0.0539584
$536$	−1.19013e9	−0.333824
$537$	−1.59017e8	−0.0443133
$538$	4.60634e9	1.27532
$539$	−5.93838e7	−0.0163346
$540$	2.82663e8	0.0772488
$541$	6.97892e9	1.89495	0.947475	−	0.319830i	$-0.103626\pi$
$541$	6.97892e9	1.89495	0.947475	+	0.319830i	$0.103626\pi$
$542$	4.19871e9	1.13271
$543$	2.63019e8	0.0704998
$544$	−6.15047e8	−0.163799
$545$	−1.63489e9	−0.432613
$546$	2.11653e7	0.00556481
$547$	−5.87859e9	−1.53574	−0.767870	−	0.640606i	$-0.778683\pi$
$547$	−5.87859e9	−1.53574	−0.767870	+	0.640606i	$0.778683\pi$
$548$	9.22606e8	0.239488
$549$	−4.56118e8	−0.117645
$550$	3.93464e7	0.0100841
$551$	3.17503e8	0.0808570
$552$	1.91719e7	0.00485152
$553$	1.00333e9	0.252292
$554$	−3.71696e9	−0.928761
$555$	1.85143e8	0.0459707
$556$	−2.41565e9	−0.596036
$557$	−2.03155e9	−0.498119	−0.249060	−	0.968488i	$-0.580122\pi$
$557$	−2.03155e9	−0.498119	−0.249060	+	0.968488i	$0.580122\pi$
$558$	−8.15724e8	−0.198758
$559$	−5.08599e8	−0.123150
$560$	9.81374e8	0.236144
$561$	−1.53240e7	−0.00366440
$562$	−1.40343e9	−0.333514
$563$	1.68161e9	0.397142	0.198571	−	0.980087i	$-0.436370\pi$
$563$	1.68161e9	0.397142	0.198571	+	0.980087i	$0.436370\pi$
$564$	2.05773e8	0.0482960
$565$	1.12030e9	0.261314
$566$	1.15958e9	0.268810
$567$	−3.61472e9	−0.832787
$568$	−2.75065e9	−0.629820
$569$	−4.57746e9	−1.04167	−0.520837	−	0.853656i	$-0.674380\pi$
$569$	−4.57746e9	−1.04167	−0.520837	+	0.853656i	$0.674380\pi$
$570$	5.55399e7	0.0125615
$571$	−5.79221e9	−1.30202	−0.651010	−	0.759069i	$-0.725655\pi$
$571$	−5.79221e9	−1.30202	−0.651010	+	0.759069i	$0.725655\pi$
$572$	−1.71823e7	−0.00383879
$573$	2.39935e8	0.0532785
$574$	−5.02791e9	−1.10968
$575$	2.25577e8	0.0494831
$576$	−5.70558e8	−0.124400
$577$	−3.74692e9	−0.812005	−0.406003	−	0.913872i	$-0.633078\pi$
$577$	−3.74692e9	−0.812005	−0.406003	+	0.913872i	$0.633078\pi$
$578$	−4.64282e8	−0.100008
$579$	−3.14246e8	−0.0672814
$580$	9.25718e8	0.197007
$581$	4.99689e9	1.05702
$582$	−1.78907e8	−0.0376181
$583$	−2.14869e8	−0.0449091
$584$	−1.90267e9	−0.395292
$585$	−7.24436e8	−0.149608
$586$	−3.39618e9	−0.697186
$587$	−5.69193e9	−1.16152	−0.580760	−	0.814075i	$-0.697244\pi$
$587$	−5.69193e9	−1.16152	−0.580760	+	0.814075i	$0.697244\pi$
$588$	−4.88448e7	−0.00990827
$589$	−3.21332e8	−0.0647963
$590$	4.80968e9	0.964126
$591$	−1.40832e8	−0.0280638
$592$	−7.49226e8	−0.148418
$593$	1.45596e9	0.286720	0.143360	−	0.989671i	$-0.454209\pi$
$593$	1.45596e9	0.286720	0.143360	+	0.989671i	$0.454209\pi$
$594$	−2.84997e7	−0.00557941
$595$	−4.49711e9	−0.875233
$596$	4.67635e9	0.904784
$597$	1.26217e8	0.0242778
$598$	−9.85079e7	−0.0188372
$599$	−6.21735e9	−1.18198	−0.590992	−	0.806677i	$-0.701264\pi$
$599$	−6.21735e9	−1.18198	−0.590992	+	0.806677i	$0.701264\pi$
$600$	3.23635e7	0.00611682
$601$	9.98776e9	1.87675	0.938377	−	0.345613i	$-0.112329\pi$
$601$	9.98776e9	1.87675	0.938377	+	0.345613i	$0.112329\pi$
$602$	−2.92887e9	−0.547158
$603$	5.05924e9	0.939668
$604$	−2.64996e9	−0.489339
$605$	6.06935e9	1.11429
$606$	7.29760e7	0.0133207
$607$	3.17827e9	0.576807	0.288404	−	0.957509i	$-0.406876\pi$
$607$	3.17827e9	0.576807	0.288404	+	0.957509i	$0.406876\pi$
$608$	−2.24756e8	−0.0405554
$609$	1.14972e8	0.0206269
$610$	−5.23864e8	−0.0934467
$611$	−1.05729e9	−0.187521
$612$	2.61456e9	0.461072
$613$	−6.34969e8	−0.111337	−0.0556687	−	0.998449i	$-0.517729\pi$
$613$	−6.34969e8	−0.111337	−0.0556687	+	0.998449i	$0.517729\pi$
$614$	3.42023e9	0.596301
$615$	−8.29637e8	−0.143822
$616$	−9.89477e7	−0.0170558
$617$	−4.48835e7	−0.00769287	−0.00384643	−	0.999993i	$-0.501224\pi$
$617$	−4.48835e7	−0.00769287	−0.00384643	+	0.999993i	$0.501224\pi$
$618$	2.03754e8	0.0347253
$619$	−9.79132e8	−0.165930	−0.0829648	−	0.996552i	$-0.526439\pi$
$619$	−9.79132e8	−0.165930	−0.0829648	+	0.996552i	$0.526439\pi$
$620$	−9.36881e8	−0.157875
$621$	−1.63392e8	−0.0273785
$622$	−1.36220e9	−0.226974
$623$	−3.06390e9	−0.507652
$624$	−1.41329e7	−0.00232855
$625$	−7.24724e9	−1.18739
$626$	−3.21715e9	−0.524156
$627$	−5.59984e6	−0.000907276 0
$628$	1.88125e9	0.303101
$629$	3.43330e9	0.550090
$630$	−4.17181e9	−0.664711
$631$	−2.42563e9	−0.384345	−0.192172	−	0.981361i	$-0.561553\pi$
$631$	−2.42563e9	−0.384345	−0.192172	+	0.981361i	$0.561553\pi$
$632$	−6.69960e8	−0.105570
$633$	−1.24796e8	−0.0195564
$634$	−1.52504e9	−0.237667
$635$	9.35744e9	1.45027
$636$	−1.76736e8	−0.0272411
$637$	2.50972e8	0.0384713
$638$	−9.33361e7	−0.0142291
$639$	1.16930e10	1.77285
$640$	−6.55302e8	−0.0988124
$641$	9.23132e8	0.138440	0.0692199	−	0.997601i	$-0.477949\pi$
$641$	9.23132e8	0.138440	0.0692199	+	0.997601i	$0.477949\pi$
$642$	1.58496e7	0.00236399
$643$	6.86334e9	1.01812	0.509058	−	0.860732i	$-0.329994\pi$
$643$	6.86334e9	1.01812	0.509058	+	0.860732i	$0.329994\pi$
$644$	−5.67278e8	−0.0836942
$645$	−4.83282e8	−0.0709156
$646$	1.02993e9	0.150313
$647$	5.02399e8	0.0729262	0.0364631	−	0.999335i	$-0.488391\pi$
$647$	5.02399e8	0.0729262	0.0364631	+	0.999335i	$0.488391\pi$
$648$	2.41369e9	0.348473
$649$	−4.84939e8	−0.0696355
$650$	−1.66288e8	−0.0237500
$651$	−1.16359e8	−0.0165297
$652$	5.78512e8	0.0817420
$653$	4.23523e9	0.595225	0.297612	−	0.954687i	$-0.403810\pi$
$653$	4.23523e9	0.595225	0.297612	+	0.954687i	$0.403810\pi$
$654$	1.35584e8	0.0189534
$655$	2.53506e9	0.352488
$656$	3.35733e9	0.464334
$657$	8.08822e9	1.11269
$658$	−6.08862e9	−0.833160
$659$	−1.34554e10	−1.83146	−0.915731	−	0.401791i	$-0.868388\pi$
$659$	−1.34554e10	−1.83146	−0.915731	+	0.401791i	$0.868388\pi$
$660$	−1.63270e7	−0.00221056
$661$	1.23518e10	1.66351	0.831753	−	0.555146i	$-0.187338\pi$
$661$	1.23518e10	1.66351	0.831753	+	0.555146i	$0.187338\pi$
$662$	−3.29606e9	−0.441563
$663$	6.47634e7	0.00863043
$664$	−3.33662e9	−0.442301
$665$	−1.64337e9	−0.216700
$666$	3.18495e9	0.417776
$667$	−5.35106e8	−0.0698231
$668$	−5.22706e9	−0.678484
$669$	−6.58864e8	−0.0850755
$670$	5.81067e9	0.746387
$671$	5.28189e7	0.00674933
$672$	−8.13872e7	−0.0103458
$673$	−1.52638e9	−0.193024	−0.0965119	−	0.995332i	$-0.530769\pi$
$673$	−1.52638e9	−0.193024	−0.0965119	+	0.995332i	$0.530769\pi$
$674$	−5.59417e9	−0.703763
$675$	−2.75817e8	−0.0345189
$676$	−3.94329e9	−0.490959
$677$	−1.55415e9	−0.192500	−0.0962502	−	0.995357i	$-0.530685\pi$
$677$	−1.55415e9	−0.192500	−0.0962502	+	0.995357i	$0.530685\pi$
$678$	−9.29082e7	−0.0114485
$679$	5.29369e9	0.648954
$680$	3.00289e9	0.366234
$681$	7.28646e8	0.0884100
$682$	9.44616e7	0.0114028
$683$	−3.21824e9	−0.386497	−0.193249	−	0.981150i	$-0.561902\pi$
$683$	−3.21824e9	−0.386497	−0.193249	+	0.981150i	$0.561902\pi$
$684$	9.55434e8	0.114158
$685$	−4.50451e9	−0.535465
$686$	6.49700e9	0.768384
$687$	−7.73796e8	−0.0910495
$688$	1.95572e9	0.228953
$689$	9.08095e8	0.105770
$690$	−9.36044e7	−0.0108474
$691$	9.15999e9	1.05614	0.528071	−	0.849200i	$-0.322916\pi$
$691$	9.15999e9	1.05614	0.528071	+	0.849200i	$0.322916\pi$
$692$	8.52421e9	0.977874
$693$	4.20626e8	0.0480098
$694$	−7.43908e9	−0.844814
$695$	1.17941e10	1.33266
$696$	−7.67715e7	−0.00863113
$697$	−1.53848e10	−1.72099
$698$	−3.55021e9	−0.395148
$699$	1.04634e9	0.115878
$700$	−9.57604e8	−0.105522
$701$	−7.97717e9	−0.874653	−0.437327	−	0.899303i	$-0.644075\pi$
$701$	−7.97717e9	−0.874653	−0.437327	+	0.899303i	$0.644075\pi$
$702$	1.20447e8	0.0131407
$703$	1.25462e9	0.136198
$704$	6.60712e7	0.00713687
$705$	−1.00466e9	−0.107984
$706$	−1.19510e10	−1.27817
$707$	−2.15929e9	−0.229796
$708$	−3.98875e8	−0.0422397
$709$	7.97010e9	0.839851	0.419925	−	0.907559i	$-0.362056\pi$
$709$	7.97010e9	0.839851	0.419925	+	0.907559i	$0.362056\pi$
$710$	1.34297e10	1.40819
$711$	2.84799e9	0.297163
$712$	2.04588e9	0.212423
$713$	5.41559e8	0.0559541
$714$	3.72953e8	0.0383452
$715$	8.38904e7	0.00858304
$716$	−3.14182e9	−0.319879
$717$	−1.67844e8	−0.0170055
$718$	−1.42350e9	−0.143523
$719$	1.00263e10	1.00598	0.502991	−	0.864292i	$-0.332233\pi$
$719$	1.00263e10	1.00598	0.502991	+	0.864292i	$0.332233\pi$
$720$	2.78568e9	0.278143
$721$	−6.02888e9	−0.599050
$722$	3.76367e8	0.0372161
$723$	3.75512e8	0.0369522
$724$	5.19666e9	0.508908
$725$	−9.03295e8	−0.0880333
$726$	−5.03342e8	−0.0488186
$727$	−1.65401e10	−1.59650	−0.798248	−	0.602329i	$-0.794240\pi$
$727$	−1.65401e10	−1.59650	−0.798248	+	0.602329i	$0.794240\pi$
$728$	4.18179e8	0.0401701
$729$	−1.01604e10	−0.971329
$730$	9.28953e9	0.883820
$731$	−8.96200e9	−0.848583
$732$	4.34450e7	0.00409403
$733$	9.81994e9	0.920969	0.460484	−	0.887668i	$-0.347676\pi$
$733$	9.81994e9	0.920969	0.460484	+	0.887668i	$0.347676\pi$
$734$	−1.10012e10	−1.02684
$735$	2.38479e8	0.0221536
$736$	3.78793e8	0.0350211
$737$	−5.85864e8	−0.0539089
$738$	−1.42720e10	−1.30703
$739$	5.41304e9	0.493385	0.246692	−	0.969094i	$-0.420656\pi$
$739$	5.41304e9	0.493385	0.246692	+	0.969094i	$0.420656\pi$
$740$	3.65800e9	0.331843
$741$	2.36664e7	0.00213682
$742$	5.22945e9	0.469940
$743$	−8.26105e9	−0.738881	−0.369440	−	0.929254i	$-0.620451\pi$
$743$	−8.26105e9	−0.738881	−0.369440	+	0.929254i	$0.620451\pi$
$744$	7.76973e7	0.00691673
$745$	−2.28317e10	−2.02298
$746$	1.24676e10	1.09950
$747$	1.41839e10	1.24501
$748$	−3.02768e8	−0.0264518
$749$	−4.68975e8	−0.0407815
$750$	4.74596e8	0.0410781
$751$	−1.44332e10	−1.24343	−0.621715	−	0.783243i	$-0.713564\pi$
$751$	−1.44332e10	−1.24343	−0.621715	+	0.783243i	$0.713564\pi$
$752$	4.06561e9	0.348628
$753$	−1.30788e8	−0.0111631
$754$	3.94463e8	0.0335125
$755$	1.29381e10	1.09410
$756$	6.93620e8	0.0583842
$757$	−8.31294e9	−0.696496	−0.348248	−	0.937402i	$-0.613223\pi$
$757$	−8.31294e9	−0.696496	−0.348248	+	0.937402i	$0.613223\pi$
$758$	7.07640e9	0.590160
$759$	9.43772e6	0.000783467 0
$760$	1.09734e9	0.0906764
$761$	−1.96636e10	−1.61739	−0.808697	−	0.588226i	$-0.799827\pi$
$761$	−1.96636e10	−1.61739	−0.808697	+	0.588226i	$0.799827\pi$
$762$	−7.76030e8	−0.0635384
$763$	−4.01180e9	−0.326967
$764$	4.74057e9	0.384595
$765$	−1.27653e10	−1.03090
$766$	1.15659e10	0.929774
$767$	2.04948e9	0.164006
$768$	5.43454e7	0.00432911
$769$	1.27102e10	1.00788	0.503942	−	0.863738i	$-0.331883\pi$
$769$	1.27102e10	1.00788	0.503942	+	0.863738i	$0.331883\pi$
$770$	4.83100e8	0.0381346
$771$	8.67018e8	0.0681299
$772$	−6.20880e9	−0.485676
$773$	−2.87614e9	−0.223966	−0.111983	−	0.993710i	$-0.535720\pi$
$773$	−2.87614e9	−0.223966	−0.111983	+	0.993710i	$0.535720\pi$
$774$	−8.31375e9	−0.644471
$775$	9.14188e8	0.0705472
$776$	−3.53480e9	−0.271549
$777$	4.54317e8	0.0347444
$778$	−8.17315e9	−0.622244
$779$	−5.62206e9	−0.426102
$780$	6.90022e7	0.00520633
$781$	−1.35406e9	−0.101709
$782$	−1.73580e9	−0.129801
$783$	6.54283e8	0.0487079
$784$	−9.65063e8	−0.0715236
$785$	−9.18498e9	−0.677695
$786$	−2.10237e8	−0.0154430
$787$	5.61813e9	0.410847	0.205424	−	0.978673i	$-0.434143\pi$
$787$	5.61813e9	0.410847	0.205424	+	0.978673i	$0.434143\pi$
$788$	−2.78253e9	−0.202581
$789$	1.66101e9	0.120393
$790$	3.27100e9	0.236040
$791$	2.74906e9	0.197500
$792$	−2.80868e8	−0.0200893
$793$	−2.23227e8	−0.0158961
$794$	1.44040e9	0.102120
$795$	8.62892e8	0.0609076
$796$	2.49377e9	0.175251
$797$	−4.98926e9	−0.349085	−0.174543	−	0.984650i	$-0.555845\pi$
$797$	−4.98926e9	−0.349085	−0.174543	+	0.984650i	$0.555845\pi$
$798$	1.36288e8	0.00949395
$799$	−1.86305e10	−1.29214
$800$	6.39429e8	0.0441547
$801$	−8.69703e9	−0.597939
$802$	4.05336e9	0.277463
$803$	−9.36623e8	−0.0638352
$804$	−4.81889e8	−0.0327003
$805$	2.76966e9	0.187129
$806$	−3.99220e8	−0.0268559
$807$	1.86513e9	0.124925
$808$	1.44184e9	0.0961563
$809$	2.60176e10	1.72762	0.863809	−	0.503820i	$-0.168073\pi$
$809$	2.60176e10	1.72762	0.863809	+	0.503820i	$0.168073\pi$
$810$	−1.17845e10	−0.779139
$811$	1.36033e10	0.895513	0.447756	−	0.894156i	$-0.352223\pi$
$811$	1.36033e10	0.895513	0.447756	+	0.894156i	$0.352223\pi$
$812$	2.27160e9	0.148897
$813$	1.70008e9	0.110956
$814$	−3.68820e8	−0.0239679
$815$	−2.82451e9	−0.182764
$816$	−2.49035e8	−0.0160452
$817$	−3.27497e9	−0.210102
$818$	9.19184e9	0.587173
$819$	−1.77768e9	−0.113073
$820$	−1.63918e10	−1.03819
$821$	5.59803e9	0.353048	0.176524	−	0.984296i	$-0.443515\pi$
$821$	5.59803e9	0.353048	0.176524	+	0.984296i	$0.443515\pi$
$822$	3.73568e8	0.0234595
$823$	2.83609e10	1.77346	0.886728	−	0.462292i	$-0.152973\pi$
$823$	2.83609e10	1.77346	0.886728	+	0.462292i	$0.152973\pi$
$824$	4.02572e9	0.250667
$825$	1.59315e7	0.000987799 0
$826$	1.18023e10	0.728682
$827$	−1.94087e10	−1.19324	−0.596619	−	0.802524i	$-0.703490\pi$
$827$	−1.94087e10	−1.19324	−0.596619	+	0.802524i	$0.703490\pi$
$828$	−1.61025e9	−0.0985794
$829$	1.25701e10	0.766301	0.383150	−	0.923686i	$-0.374839\pi$
$829$	1.25701e10	0.766301	0.383150	+	0.923686i	$0.374839\pi$
$830$	1.62906e10	0.988927
$831$	−1.50501e9	−0.0909781
$832$	−2.79234e8	−0.0168088
$833$	4.42236e9	0.265092
$834$	−9.78109e8	−0.0583856
$835$	2.55205e10	1.51700
$836$	−1.10640e8	−0.00654924
$837$	−6.62173e8	−0.0390331
$838$	1.61831e10	0.949968
$839$	1.95535e10	1.14303	0.571515	−	0.820592i	$-0.306356\pi$
$839$	1.95535e10	1.14303	0.571515	+	0.820592i	$0.306356\pi$
$840$	3.97363e8	0.0231318
$841$	−1.51071e10	−0.875781
$842$	2.04402e10	1.18003
$843$	−5.68256e8	−0.0326699
$844$	−2.46569e9	−0.141169
$845$	1.92526e10	1.09772
$846$	−1.72829e10	−0.981339
$847$	1.48934e10	0.842175
$848$	−3.49191e9	−0.196642
$849$	4.69521e8	0.0263316
$850$	−2.93016e9	−0.163653
$851$	−2.11449e9	−0.117612
$852$	−1.11375e9	−0.0616950
$853$	1.14196e10	0.629986	0.314993	−	0.949094i	$-0.397998\pi$
$853$	1.14196e10	0.629986	0.314993	+	0.949094i	$0.397998\pi$
$854$	−1.28550e9	−0.0706266
$855$	−4.66479e9	−0.255241
$856$	3.13153e8	0.0170647
$857$	−1.94401e10	−1.05503	−0.527516	−	0.849545i	$-0.676877\pi$
$857$	−1.94401e10	−1.05503	−0.527516	+	0.849545i	$0.676877\pi$
$858$	−6.95719e6	−0.000376035 0
$859$	1.07841e10	0.580509	0.290254	−	0.956950i	$-0.406260\pi$
$859$	1.07841e10	0.580509	0.290254	+	0.956950i	$0.406260\pi$
$860$	−9.54856e9	−0.511910
$861$	−2.03583e9	−0.108700
$862$	1.17852e10	0.626705
$863$	−8.45236e8	−0.0447651	−0.0223826	−	0.999749i	$-0.507125\pi$
$863$	−8.45236e8	−0.0447651	−0.0223826	+	0.999749i	$0.507125\pi$
$864$	−4.63157e8	−0.0244304
$865$	−4.16184e10	−2.18640
$866$	−7.46595e9	−0.390636
$867$	−1.87990e8	−0.00979643
$868$	−2.29899e9	−0.119321
$869$	−3.29800e8	−0.0170483
$870$	3.74828e8	0.0192981
$871$	2.47602e9	0.126967
$872$	2.67884e9	0.136817
$873$	1.50264e10	0.764373
$874$	−6.34312e8	−0.0321376
$875$	−1.40428e10	−0.708642
$876$	−7.70398e8	−0.0387214
$877$	2.17563e10	1.08915	0.544574	−	0.838713i	$-0.316691\pi$
$877$	2.17563e10	1.08915	0.544574	+	0.838713i	$0.316691\pi$
$878$	−1.55529e10	−0.775495
$879$	−1.37513e9	−0.0682939
$880$	−3.22584e8	−0.0159571
$881$	2.18056e10	1.07436	0.537182	−	0.843466i	$-0.319489\pi$
$881$	2.18056e10	1.07436	0.537182	+	0.843466i	$0.319489\pi$
$882$	4.10248e9	0.201329
$883$	2.25662e9	0.110305	0.0551526	−	0.998478i	$-0.482435\pi$
$883$	2.25662e9	0.110305	0.0551526	+	0.998478i	$0.482435\pi$
$884$	1.27958e9	0.0622995
$885$	1.94746e9	0.0944424
$886$	−1.88474e10	−0.910401
$887$	−1.72271e10	−0.828857	−0.414428	−	0.910082i	$-0.636018\pi$
$887$	−1.72271e10	−0.828857	−0.414428	+	0.910082i	$0.636018\pi$
$888$	−3.03365e8	−0.0145385
$889$	2.29620e10	1.09611
$890$	−9.98877e9	−0.474949
$891$	1.18818e9	0.0562745
$892$	−1.30177e10	−0.614124
$893$	−6.80810e9	−0.319923
$894$	1.89347e9	0.0886294
$895$	1.53396e10	0.715208
$896$	−1.60803e9	−0.0746819
$897$	−3.98863e7	−0.00184523
$898$	−1.46700e10	−0.676023
$899$	−2.16861e9	−0.0995455
$900$	−2.71821e9	−0.124289
$901$	1.60015e10	0.728827
$902$	1.65271e9	0.0749849
$903$	−1.18591e9	−0.0535976
$904$	−1.83566e9	−0.0826422
$905$	−2.53721e10	−1.13785
$906$	−1.07298e9	−0.0479340
$907$	2.32557e10	1.03491	0.517456	−	0.855710i	$-0.326879\pi$
$907$	2.32557e10	1.03491	0.517456	+	0.855710i	$0.326879\pi$
$908$	1.43964e10	0.638195
$909$	−6.12926e9	−0.270666
$910$	−2.04171e9	−0.0898150
$911$	−4.22775e10	−1.85266	−0.926329	−	0.376715i	$-0.877054\pi$
$911$	−4.22775e10	−1.85266	−0.926329	+	0.376715i	$0.877054\pi$
$912$	−9.10046e7	−0.00397266
$913$	−1.64251e9	−0.0714267
$914$	−3.01346e10	−1.30543
$915$	−2.12115e8	−0.00915372
$916$	−1.52885e10	−0.657248
$917$	6.22072e9	0.266408
$918$	2.12240e9	0.0905477
$919$	2.74897e10	1.16833	0.584164	−	0.811635i	$-0.301422\pi$
$919$	2.74897e10	1.16833	0.584164	+	0.811635i	$0.301422\pi$
$920$	−1.84941e9	−0.0783026
$921$	1.38487e9	0.0584116
$922$	2.64448e10	1.11118
$923$	5.72261e9	0.239546
$924$	−4.00644e7	−0.00167073
$925$	−3.56940e9	−0.148286
$926$	−2.93061e10	−1.21288
$927$	−1.71133e10	−0.705593
$928$	−1.51683e9	−0.0623045
$929$	−2.42814e10	−0.993614	−0.496807	−	0.867861i	$-0.665494\pi$
$929$	−2.42814e10	−0.993614	−0.496807	+	0.867861i	$0.665494\pi$
$930$	−3.79347e8	−0.0154649
$931$	1.61606e9	0.0656346
$932$	2.06732e10	0.836475
$933$	−5.51563e8	−0.0222336
$934$	−1.78139e10	−0.715393
$935$	1.47823e9	0.0591427
$936$	1.18702e9	0.0473144
$937$	−3.53708e10	−1.40461	−0.702305	−	0.711876i	$-0.747846\pi$
$937$	−3.53708e10	−1.40461	−0.702305	+	0.711876i	$0.747846\pi$
$938$	1.42586e10	0.564116
$939$	−1.30264e9	−0.0513445
$940$	−1.98498e10	−0.779487
$941$	−3.60255e10	−1.40944	−0.704719	−	0.709486i	$-0.748927\pi$
$941$	−3.60255e10	−1.40944	−0.704719	+	0.709486i	$0.748927\pi$
$942$	7.61727e8	0.0296908
$943$	9.47516e9	0.367956
$944$	−7.88088e9	−0.304911
$945$	−3.38652e9	−0.130539
$946$	9.62740e8	0.0369734
$947$	−2.20930e10	−0.845336	−0.422668	−	0.906285i	$-0.638906\pi$
$947$	−2.20930e10	−0.845336	−0.422668	+	0.906285i	$0.638906\pi$
$948$	−2.71270e8	−0.0103412
$949$	3.95842e9	0.150345
$950$	−1.07076e9	−0.0405192
$951$	−6.17496e8	−0.0232810
$952$	7.36872e9	0.276798
$953$	−2.36409e9	−0.0884787	−0.0442393	−	0.999021i	$-0.514086\pi$
$953$	−2.36409e9	−0.0884787	−0.0442393	+	0.999021i	$0.514086\pi$
$954$	1.48441e10	0.553520
$955$	−2.31452e10	−0.859903
$956$	−3.31623e9	−0.122756
$957$	−3.77922e7	−0.00139383
$958$	1.14806e10	0.421878
$959$	−1.10535e10	−0.404702
$960$	−2.65335e8	−0.00967931
$961$	−2.53179e10	−0.920227
$962$	1.55873e9	0.0564493
$963$	−1.33121e9	−0.0480346
$964$	7.41927e9	0.266742
$965$	3.03137e10	1.08591
$966$	−2.29693e8	−0.00819839
$967$	2.69098e10	0.957014	0.478507	−	0.878084i	$-0.341178\pi$
$967$	2.69098e10	0.957014	0.478507	+	0.878084i	$0.341178\pi$
$968$	−9.94491e9	−0.352401
$969$	4.17025e8	0.0147241
$970$	1.72582e10	0.607149
$971$	−4.71594e9	−0.165311	−0.0826553	−	0.996578i	$-0.526340\pi$
$971$	−4.71594e9	−0.165311	−0.0826553	+	0.996578i	$0.526340\pi$
$972$	2.95568e9	0.103235
$973$	2.89413e10	1.00722
$974$	−1.69314e10	−0.587132
$975$	−6.73308e7	−0.00232647
$976$	8.58375e8	0.0295531
$977$	1.79514e10	0.615839	0.307919	−	0.951412i	$-0.400367\pi$
$977$	1.79514e10	0.615839	0.307919	+	0.951412i	$0.400367\pi$
$978$	2.34242e8	0.00800716
$979$	1.00712e9	0.0343039
$980$	4.71180e9	0.159917
$981$	−1.13877e10	−0.385119
$982$	3.25684e10	1.09750
$983$	2.39450e10	0.804039	0.402019	−	0.915631i	$-0.368308\pi$
$983$	2.39450e10	0.804039	0.402019	+	0.915631i	$0.368308\pi$
$984$	1.35940e9	0.0454846
$985$	1.35854e10	0.452944
$986$	6.95082e9	0.230923
$987$	−2.46531e9	−0.0816134
$988$	4.67595e8	0.0154248
$989$	5.51949e9	0.181431
$990$	1.37130e9	0.0449170
$991$	1.07956e10	0.352362	0.176181	−	0.984358i	$-0.443626\pi$
$991$	1.07956e10	0.352362	0.176181	+	0.984358i	$0.443626\pi$
$992$	1.53512e9	0.0499289
$993$	−1.33459e9	−0.0432540
$994$	3.29548e10	1.06431
$995$	−1.21755e10	−0.391838
$996$	−1.35101e9	−0.0433263
$997$	−3.54933e10	−1.13426	−0.567131	−	0.823627i	$-0.691947\pi$
$997$	−3.54933e10	−1.13426	−0.567131	+	0.823627i	$0.691947\pi$
$998$	3.14431e10	1.00131
$999$	2.58542e9	0.0820450

Twists

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

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

\(f(q)\)	\(=\)	\(q+8.00000 q^{2} +3.23924 q^{3} +64.0000 q^{4} -312.472 q^{5} +25.9139 q^{6} -766.767 q^{7} +512.000 q^{8} -2176.51 q^{9} +O(q^{10})\)	\(q+8.00000 q^{2} +3.23924 q^{3} +64.0000 q^{4} -312.472 q^{5} +25.9139 q^{6} -766.767 q^{7} +512.000 q^{8} -2176.51 q^{9} -2499.78 q^{10} +252.042 q^{11} +207.311 q^{12} -1065.19 q^{13} -6134.14 q^{14} -1012.17 q^{15} +4096.00 q^{16} -18769.7 q^{17} -17412.1 q^{18} -6859.00 q^{19} -19998.2 q^{20} -2483.74 q^{21} +2016.33 q^{22} +11559.9 q^{23} +1658.49 q^{24} +19513.8 q^{25} -8521.55 q^{26} -14134.4 q^{27} -49073.1 q^{28} -46290.0 q^{29} -8097.37 q^{30} +46848.2 q^{31} +32768.0 q^{32} +816.422 q^{33} -150158. q^{34} +239593. q^{35} -139296. q^{36} -182916. q^{37} -54872.0 q^{38} -3450.42 q^{39} -159986. q^{40} +819661. q^{41} -19869.9 q^{42} +477471. q^{43} +16130.7 q^{44} +680098. q^{45} +92478.8 q^{46} +992580. q^{47} +13267.9 q^{48} -235611. q^{49} +156111. q^{50} -60799.7 q^{51} -68172.4 q^{52} -852516. q^{53} -113075. q^{54} -78756.0 q^{55} -392585. q^{56} -22217.9 q^{57} -370320. q^{58} -1.92404e6 q^{59} -64779.0 q^{60} +209564. q^{61} +374786. q^{62} +1.66887e6 q^{63} +262144. q^{64} +332843. q^{65} +6531.38 q^{66} -2.32447e6 q^{67} -1.20126e6 q^{68} +37445.1 q^{69} +1.91675e6 q^{70} -5.37237e6 q^{71} -1.11437e6 q^{72} -3.71614e6 q^{73} -1.46333e6 q^{74} +63209.9 q^{75} -438976. q^{76} -193257. q^{77} -27603.3 q^{78} -1.30851e6 q^{79} -1.27989e6 q^{80} +4.71424e6 q^{81} +6.55729e6 q^{82} -6.51683e6 q^{83} -158959. q^{84} +5.86502e6 q^{85} +3.81976e6 q^{86} -149944. q^{87} +129045. q^{88} +3.99586e6 q^{89} +5.44078e6 q^{90} +816756. q^{91} +739831. q^{92} +151753. q^{93} +7.94064e6 q^{94} +2.14325e6 q^{95} +106143. q^{96} -6.90391e6 q^{97} -1.88489e6 q^{98} -548570. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 16 q^{2} - 69 q^{3} + 128 q^{4} + 155 q^{5} - 552 q^{6} - 2238 q^{7} + 1024 q^{8} + 855 q^{9}+O(q^{10}) \) 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

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