LMFDB - Embedded newform 38.8.a.b.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:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{2737}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 684 $ x^2 - x - 684
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-25.6582$ 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)

$f(q)$	$=$	$q-8.00000 q^{2} -4.34183 q^{3} +64.0000 q^{4} +479.873 q^{5} +34.7346 q^{6} -1139.05 q^{7} -512.000 q^{8} -2168.15 q^{9} +O(q^{10})$	\(q-8.00000 q^{2} -4.34183 q^{3} +64.0000 q^{4} +479.873 q^{5} +34.7346 q^{6} -1139.05 q^{7} -512.000 q^{8} -2168.15 q^{9} -3838.98 q^{10} +7663.68 q^{11} -277.877 q^{12} +5047.74 q^{13} +9112.41 q^{14} -2083.52 q^{15} +4096.00 q^{16} +17162.2 q^{17} +17345.2 q^{18} -6859.00 q^{19} +30711.8 q^{20} +4945.56 q^{21} -61309.4 q^{22} +71169.3 q^{23} +2223.02 q^{24} +152153. q^{25} -40381.9 q^{26} +18909.3 q^{27} -72899.3 q^{28} +222985. q^{29} +16668.2 q^{30} -227455. q^{31} -32768.0 q^{32} -33274.4 q^{33} -137298. q^{34} -546599. q^{35} -138762. q^{36} +344955. q^{37} +54872.0 q^{38} -21916.4 q^{39} -245695. q^{40} +120.393 q^{41} -39564.5 q^{42} +284134. q^{43} +490476. q^{44} -1.04044e6 q^{45} -569354. q^{46} +211864. q^{47} -17784.1 q^{48} +473894. q^{49} -1.21722e6 q^{50} -74515.5 q^{51} +323055. q^{52} +71171.1 q^{53} -151274. q^{54} +3.67759e6 q^{55} +583194. q^{56} +29780.6 q^{57} -1.78388e6 q^{58} -1.87033e6 q^{59} -133346. q^{60} -2.59442e6 q^{61} +1.81964e6 q^{62} +2.46963e6 q^{63} +262144. q^{64} +2.42227e6 q^{65} +266195. q^{66} -1.24281e6 q^{67} +1.09838e6 q^{68} -309005. q^{69} +4.37279e6 q^{70} -5.27533e6 q^{71} +1.11009e6 q^{72} +4.58875e6 q^{73} -2.75964e6 q^{74} -660621. q^{75} -438976. q^{76} -8.72932e6 q^{77} +175331. q^{78} +5.88956e6 q^{79} +1.96556e6 q^{80} +4.65964e6 q^{81} -963.147 q^{82} -3.30218e6 q^{83} +316516. q^{84} +8.23569e6 q^{85} -2.27307e6 q^{86} -968163. q^{87} -3.92380e6 q^{88} -302955. q^{89} +8.32348e6 q^{90} -5.74963e6 q^{91} +4.55483e6 q^{92} +987568. q^{93} -1.69491e6 q^{94} -3.29145e6 q^{95} +142273. q^{96} -1.12581e7 q^{97} -3.79115e6 q^{98} -1.66160e7 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 16 q^{2} - 61 q^{3} + 128 q^{4} + 175 q^{5} + 488 q^{6} - 2592 q^{7} - 1024 q^{8} - 1145 q^{9}+O(q^{10}) $ 2 * q - 16 * q^2 - 61 * q^3 + 128 * q^4 + 175 * q^5 + 488 * q^6 - 2592 * q^7 - 1024 * q^8 - 1145 * q^9	\( 2 q - 16 q^{2} - 61 q^{3} + 128 q^{4} + 175 q^{5} + 488 q^{6} - 2592 q^{7} - 1024 q^{8} - 1145 q^{9} - 1400 q^{10} + 1045 q^{11} - 3904 q^{12} + 14647 q^{13} + 20736 q^{14} + 15190 q^{15} + 8192 q^{16} + 29616 q^{17} + 9160 q^{18} - 13718 q^{19} + 11200 q^{20} + 87267 q^{21} - 8360 q^{22} + 69985 q^{23} + 31232 q^{24} + 166975 q^{25} - 117176 q^{26} + 84851 q^{27} - 165888 q^{28} + 138821 q^{29} - 121520 q^{30} - 199396 q^{31} - 65536 q^{32} + 341728 q^{33} - 236928 q^{34} - 103635 q^{35} - 73280 q^{36} - 67840 q^{37} + 109744 q^{38} - 565793 q^{39} - 89600 q^{40} - 539350 q^{41} - 698136 q^{42} + 602639 q^{43} + 66880 q^{44} - 1352365 q^{45} - 559880 q^{46} - 1031975 q^{47} - 249856 q^{48} + 1761412 q^{49} - 1335800 q^{50} - 780123 q^{51} + 937408 q^{52} + 2138263 q^{53} - 678808 q^{54} + 5695445 q^{55} + 1327104 q^{56} + 418399 q^{57} - 1110568 q^{58} - 3936369 q^{59} + 972160 q^{60} - 1027655 q^{61} + 1595168 q^{62} + 983049 q^{63} + 524288 q^{64} - 504280 q^{65} - 2733824 q^{66} + 764949 q^{67} + 1895424 q^{68} - 241907 q^{69} + 829080 q^{70} - 3572084 q^{71} + 586240 q^{72} + 9069522 q^{73} + 542720 q^{74} - 1500425 q^{75} - 877952 q^{76} + 887283 q^{77} + 4526344 q^{78} - 2753414 q^{79} + 716800 q^{80} - 1314122 q^{81} + 4314800 q^{82} - 7643046 q^{83} + 5585088 q^{84} + 4438875 q^{85} - 4821112 q^{86} + 3800423 q^{87} - 535040 q^{88} + 1393620 q^{89} + 10818920 q^{90} - 19696869 q^{91} + 4479040 q^{92} - 602176 q^{93} + 8255800 q^{94} - 1200325 q^{95} + 1998848 q^{96} - 6921466 q^{97} - 14091296 q^{98} - 23387893 q^{99}+O(q^{100}) \) 2 * q - 16 * q^2 - 61 * q^3 + 128 * q^4 + 175 * q^5 + 488 * q^6 - 2592 * q^7 - 1024 * q^8 - 1145 * q^9 - 1400 * q^10 + 1045 * q^11 - 3904 * q^12 + 14647 * q^13 + 20736 * q^14 + 15190 * q^15 + 8192 * q^16 + 29616 * q^17 + 9160 * q^18 - 13718 * q^19 + 11200 * q^20 + 87267 * q^21 - 8360 * q^22 + 69985 * q^23 + 31232 * q^24 + 166975 * q^25 - 117176 * q^26 + 84851 * q^27 - 165888 * q^28 + 138821 * q^29 - 121520 * q^30 - 199396 * q^31 - 65536 * q^32 + 341728 * q^33 - 236928 * q^34 - 103635 * q^35 - 73280 * q^36 - 67840 * q^37 + 109744 * q^38 - 565793 * q^39 - 89600 * q^40 - 539350 * q^41 - 698136 * q^42 + 602639 * q^43 + 66880 * q^44 - 1352365 * q^45 - 559880 * q^46 - 1031975 * q^47 - 249856 * q^48 + 1761412 * q^49 - 1335800 * q^50 - 780123 * q^51 + 937408 * q^52 + 2138263 * q^53 - 678808 * q^54 + 5695445 * q^55 + 1327104 * q^56 + 418399 * q^57 - 1110568 * q^58 - 3936369 * q^59 + 972160 * q^60 - 1027655 * q^61 + 1595168 * q^62 + 983049 * q^63 + 524288 * q^64 - 504280 * q^65 - 2733824 * q^66 + 764949 * q^67 + 1895424 * q^68 - 241907 * q^69 + 829080 * q^70 - 3572084 * q^71 + 586240 * q^72 + 9069522 * q^73 + 542720 * q^74 - 1500425 * q^75 - 877952 * q^76 + 887283 * q^77 + 4526344 * q^78 - 2753414 * q^79 + 716800 * q^80 - 1314122 * q^81 + 4314800 * q^82 - 7643046 * q^83 + 5585088 * q^84 + 4438875 * q^85 - 4821112 * q^86 + 3800423 * q^87 - 535040 * q^88 + 1393620 * q^89 + 10818920 * q^90 - 19696869 * q^91 + 4479040 * q^92 - 602176 * q^93 + 8255800 * q^94 - 1200325 * q^95 + 1998848 * q^96 - 6921466 * q^97 - 14091296 * q^98 - 23387893 * 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
$3$	−4.34183	−0.0928428	−0.0464214	−	0.998922i	$-0.514782\pi$
$3$	−4.34183	−0.0928428	−0.0464214	+	0.998922i	$0.514782\pi$
$4$	64.0000	0.500000
$5$	479.873	1.71684	0.858422	−	0.512944i	$-0.171445\pi$
$5$	479.873	1.71684	0.858422	+	0.512944i	$0.171445\pi$
$6$	34.7346	0.0656498
$7$	−1139.05	−1.25516	−0.627581	−	0.778551i	$-0.715955\pi$
$7$	−1139.05	−1.25516	−0.627581	+	0.778551i	$0.715955\pi$
$8$	−512.000	−0.353553
$9$	−2168.15	−0.991380
$10$	−3838.98	−1.21399
$11$	7663.68	1.73605	0.868027	−	0.496518i	$-0.165388\pi$
$11$	7663.68	1.73605	0.868027	+	0.496518i	$0.165388\pi$
$12$	−277.877	−0.0464214
$13$	5047.74	0.637228	0.318614	−	0.947884i	$-0.396783\pi$
$13$	5047.74	0.637228	0.318614	+	0.947884i	$0.396783\pi$
$14$	9112.41	0.887534
$15$	−2083.52	−0.159397
$16$	4096.00	0.250000
$17$	17162.2	0.847233	0.423616	−	0.905842i	$-0.360761\pi$
$17$	17162.2	0.847233	0.423616	+	0.905842i	$0.360761\pi$
$18$	17345.2	0.701012
$19$	−6859.00	−0.229416
$19$	−6859.00	−0.229416
$20$	30711.8	0.858422
$21$	4945.56	0.116533
$22$	−61309.4	−1.22757
$23$	71169.3	1.21968	0.609839	−	0.792526i	$-0.291234\pi$
$23$	71169.3	1.21968	0.609839	+	0.792526i	$0.291234\pi$
$24$	2223.02	0.0328249
$25$	152153.	1.94755
$26$	−40381.9	−0.450589
$27$	18909.3	0.184885
$28$	−72899.3	−0.627581
$29$	222985.	1.69779	0.848893	−	0.528565i	$-0.177270\pi$
$29$	222985.	1.69779	0.848893	+	0.528565i	$0.177270\pi$
$30$	16668.2	0.112710
$31$	−227455.	−1.37129	−0.685644	−	0.727937i	$-0.740479\pi$
$31$	−227455.	−1.37129	−0.685644	+	0.727937i	$0.740479\pi$
$32$	−32768.0	−0.176777
$33$	−33274.4	−0.161180
$34$	−137298.	−0.599084
$35$	−546599.	−2.15492
$36$	−138762.	−0.495690
$37$	344955.	1.11958	0.559792	−	0.828633i	$-0.310881\pi$
$37$	344955.	1.11958	0.559792	+	0.828633i	$0.310881\pi$
$38$	54872.0	0.162221
$39$	−21916.4	−0.0591621
$40$	−245695.	−0.606996
$41$	120.393	0.000272809 0	0.000136405	−	1.00000i	$-0.499957\pi$
$41$	120.393	0.000272809 0	0.000136405		1.00000i	$0.499957\pi$
$42$	−39564.5	−0.0824011
$43$	284134.	0.544983	0.272492	−	0.962158i	$-0.412152\pi$
$43$	284134.	0.544983	0.272492	+	0.962158i	$0.412152\pi$
$44$	490476.	0.868027
$45$	−1.04044e6	−1.70205
$46$	−569354.	−0.862442
$47$	211864.	0.297656	0.148828	−	0.988863i	$-0.452450\pi$
$47$	211864.	0.297656	0.148828	+	0.988863i	$0.452450\pi$
$48$	−17784.1	−0.0232107
$49$	473894.	0.575433
$50$	−1.21722e6	−1.37713
$51$	−74515.5	−0.0786594
$52$	323055.	0.318614
$53$	71171.1	0.0656656	0.0328328	−	0.999461i	$-0.489547\pi$
$53$	71171.1	0.0656656	0.0328328	+	0.999461i	$0.489547\pi$
$54$	−151274.	−0.130734
$55$	3.67759e6	2.98053
$56$	583194.	0.443767
$57$	29780.6	0.0212996
$58$	−1.78388e6	−1.20052
$59$	−1.87033e6	−1.18559	−0.592796	−	0.805352i	$-0.701976\pi$
$59$	−1.87033e6	−1.18559	−0.592796	+	0.805352i	$0.701976\pi$
$60$	−133346.	−0.0796983
$61$	−2.59442e6	−1.46348	−0.731739	−	0.681585i	$-0.761291\pi$
$61$	−2.59442e6	−1.46348	−0.731739	+	0.681585i	$0.761291\pi$
$62$	1.81964e6	0.969647
$63$	2.46963e6	1.24434
$64$	262144.	0.125000
$65$	2.42227e6	1.09402
$66$	266195.	0.113971
$67$	−1.24281e6	−0.504828	−0.252414	−	0.967619i	$-0.581224\pi$
$67$	−1.24281e6	−0.504828	−0.252414	+	0.967619i	$0.581224\pi$
$68$	1.09838e6	0.423616
$69$	−309005.	−0.113238
$70$	4.37279e6	1.52376
$71$	−5.27533e6	−1.74922	−0.874612	−	0.484823i	$-0.838884\pi$
$71$	−5.27533e6	−1.74922	−0.874612	+	0.484823i	$0.838884\pi$
$72$	1.11009e6	0.350506
$73$	4.58875e6	1.38059	0.690295	−	0.723528i	$-0.257481\pi$
$73$	4.58875e6	1.38059	0.690295	+	0.723528i	$0.257481\pi$
$74$	−2.75964e6	−0.791665
$75$	−660621.	−0.180816
$76$	−438976.	−0.114708
$77$	−8.72932e6	−2.17903
$78$	175331.	0.0418339
$79$	5.88956e6	1.34397	0.671983	−	0.740567i	$-0.265443\pi$
$79$	5.88956e6	1.34397	0.671983	+	0.740567i	$0.265443\pi$
$80$	1.96556e6	0.429211
$81$	4.65964e6	0.974215
$82$	−963.147	−0.000192905 0
$83$	−3.30218e6	−0.633909	−0.316955	−	0.948441i	$-0.602660\pi$
$83$	−3.30218e6	−0.633909	−0.316955	+	0.948441i	$0.602660\pi$
$84$	316516.	0.0582664
$85$	8.23569e6	1.45457
$86$	−2.27307e6	−0.385361
$87$	−968163.	−0.157627
$88$	−3.92380e6	−0.613787
$89$	−302955.	−0.0455526	−0.0227763	−	0.999741i	$-0.507251\pi$
$89$	−302955.	−0.0455526	−0.0227763	+	0.999741i	$0.507251\pi$
$90$	8.32348e6	1.20353
$91$	−5.74963e6	−0.799825
$92$	4.55483e6	0.609839
$93$	987568.	0.127314
$94$	−1.69491e6	−0.210474
$95$	−3.29145e6	−0.393871
$96$	142273.	0.0164124
$97$	−1.12581e7	−1.25246	−0.626231	−	0.779637i	$-0.715403\pi$
$97$	−1.12581e7	−1.25246	−0.626231	+	0.779637i	$0.715403\pi$
$98$	−3.79115e6	−0.406893
$99$	−1.66160e7	−1.72109
$100$	9.73777e6	0.973777
$101$	1.27407e6	0.123046	0.0615232	−	0.998106i	$-0.480404\pi$
$101$	1.27407e6	0.123046	0.0615232	+	0.998106i	$0.480404\pi$
$102$	596124.	0.0556206
$103$	−9.85432e6	−0.888580	−0.444290	−	0.895883i	$-0.646544\pi$
$103$	−9.85432e6	−0.888580	−0.444290	+	0.895883i	$0.646544\pi$
$104$	−2.58444e6	−0.225294
$105$	2.37324e6	0.200069
$106$	−569368.	−0.0464326
$107$	−2.42751e6	−0.191566	−0.0957828	−	0.995402i	$-0.530535\pi$
$107$	−2.42751e6	−0.191566	−0.0957828	+	0.995402i	$0.530535\pi$
$108$	1.21020e6	0.0924426
$109$	−1.99116e7	−1.47270	−0.736348	−	0.676603i	$-0.763451\pi$
$109$	−1.99116e7	−1.47270	−0.736348	+	0.676603i	$0.763451\pi$
$110$	−2.94207e7	−2.10756
$111$	−1.49773e6	−0.103945
$112$	−4.66555e6	−0.313791
$113$	1.67103e7	1.08946	0.544728	−	0.838613i	$-0.316633\pi$
$113$	1.67103e7	1.08946	0.544728	+	0.838613i	$0.316633\pi$
$114$	−238245.	−0.0150611
$115$	3.41522e7	2.09400
$116$	1.42710e7	0.848893
$117$	−1.09442e7	−0.631736
$118$	1.49626e7	0.838341
$119$	−1.95487e7	−1.06341
$120$	1.06676e6	0.0563552
$121$	3.92448e7	2.01388
$122$	2.07554e7	1.03484
$123$	−522.727	−2.53284e−5 0
$124$	−1.45571e7	−0.685644
$125$	3.55239e7	1.62680
$126$	−1.97571e7	−0.879884
$127$	−1.54290e7	−0.668380	−0.334190	−	0.942506i	$-0.608463\pi$
$127$	−1.54290e7	−0.668380	−0.334190	+	0.942506i	$0.608463\pi$
$128$	−2.09715e6	−0.0883883
$129$	−1.23366e6	−0.0505977
$130$	−1.93782e7	−0.773590
$131$	−5.86941e6	−0.228110	−0.114055	−	0.993474i	$-0.536384\pi$
$131$	−5.86941e6	−0.228110	−0.114055	+	0.993474i	$0.536384\pi$
$132$	−2.12956e6	−0.0805900
$133$	7.81275e6	0.287954
$134$	9.94249e6	0.356967
$135$	9.07406e6	0.317419
$136$	−8.78706e6	−0.299542
$137$	1.70341e7	0.565976	0.282988	−	0.959123i	$-0.408674\pi$
$137$	1.70341e7	0.565976	0.282988	+	0.959123i	$0.408674\pi$
$138$	2.47204e6	0.0800715
$139$	1.27406e7	0.402380	0.201190	−	0.979552i	$-0.435519\pi$
$139$	1.27406e7	0.402380	0.201190	+	0.979552i	$0.435519\pi$
$140$	−3.49824e7	−1.07746
$141$	−919875.	−0.0276352
$142$	4.22027e7	1.23689
$143$	3.86843e7	1.10626
$144$	−8.88074e6	−0.247845
$145$	1.07004e8	2.91483
$146$	−3.67100e7	−0.976224
$147$	−2.05757e6	−0.0534248
$148$	2.20771e7	0.559792
$149$	2.58248e7	0.639567	0.319783	−	0.947491i	$-0.396390\pi$
$149$	2.58248e7	0.639567	0.319783	+	0.947491i	$0.396390\pi$
$150$	5.28497e6	0.127856
$151$	−9.22188e6	−0.217972	−0.108986	−	0.994043i	$-0.534760\pi$
$151$	−9.22188e6	−0.217972	−0.108986	+	0.994043i	$0.534760\pi$
$152$	3.51181e6	0.0811107
$153$	−3.72103e7	−0.839930
$154$	6.98346e7	1.54081
$155$	−1.09149e8	−2.35429
$156$	−1.40265e6	−0.0295810
$157$	−3.62179e7	−0.746920	−0.373460	−	0.927646i	$-0.621829\pi$
$157$	−3.62179e7	−0.746920	−0.373460	+	0.927646i	$0.621829\pi$
$158$	−4.71165e7	−0.950327
$159$	−309012.	−0.00609658
$160$	−1.57245e7	−0.303498
$161$	−8.10654e7	−1.53089
$162$	−3.72771e7	−0.688874
$163$	1.79802e7	0.325190	0.162595	−	0.986693i	$-0.448014\pi$
$163$	1.79802e7	0.325190	0.162595	+	0.986693i	$0.448014\pi$
$164$	7705.18	0.000136405 0
$165$	−1.59675e7	−0.276721
$166$	2.64174e7	0.448242
$167$	2.10162e7	0.349178	0.174589	−	0.984641i	$-0.444140\pi$
$167$	2.10162e7	0.349178	0.174589	+	0.984641i	$0.444140\pi$
$168$	−2.53213e6	−0.0412006
$169$	−3.72688e7	−0.593940
$170$	−6.58855e7	−1.02853
$171$	1.48713e7	0.227438
$172$	1.81845e7	0.272492
$173$	1.11056e7	0.163072	0.0815361	−	0.996670i	$-0.474017\pi$
$173$	1.11056e7	0.163072	0.0815361	+	0.996670i	$0.474017\pi$
$174$	7.74530e6	0.111459
$175$	−1.73310e8	−2.44450
$176$	3.13904e7	0.434013
$177$	8.12064e6	0.110074
$178$	2.42364e6	0.0322106
$179$	1.63871e7	0.213558	0.106779	−	0.994283i	$-0.465946\pi$
$179$	1.63871e7	0.213558	0.106779	+	0.994283i	$0.465946\pi$
$180$	−6.65878e7	−0.851023
$181$	2.34586e7	0.294054	0.147027	−	0.989132i	$-0.453030\pi$
$181$	2.34586e7	0.294054	0.147027	+	0.989132i	$0.453030\pi$
$182$	4.59971e7	0.565562
$183$	1.12645e7	0.135873
$184$	−3.64387e7	−0.431221
$185$	1.65534e8	1.92215
$186$	−7.90055e6	−0.0900247
$187$	1.31526e8	1.47084
$188$	1.35593e7	0.148828
$189$	−2.15387e7	−0.232061
$190$	2.63316e7	0.278509
$191$	−1.32760e8	−1.37864	−0.689320	−	0.724457i	$-0.742091\pi$
$191$	−1.32760e8	−1.37864	−0.689320	+	0.724457i	$0.742091\pi$
$192$	−1.13818e6	−0.0116053
$193$	−3.52001e7	−0.352446	−0.176223	−	0.984350i	$-0.556388\pi$
$193$	−3.52001e7	−0.352446	−0.176223	+	0.984350i	$0.556388\pi$
$194$	9.00649e7	0.885625
$195$	−1.05171e7	−0.101572
$196$	3.03292e7	0.287717
$197$	6.57848e6	0.0613047	0.0306524	−	0.999530i	$-0.490242\pi$
$197$	6.57848e6	0.0613047	0.0306524	+	0.999530i	$0.490242\pi$
$198$	1.32928e8	1.21699
$199$	3.78753e7	0.340698	0.170349	−	0.985384i	$-0.445510\pi$
$199$	3.78753e7	0.340698	0.170349	+	0.985384i	$0.445510\pi$
$200$	−7.79022e7	−0.688565
$201$	5.39607e6	0.0468696
$202$	−1.01926e7	−0.0870069
$203$	−2.53991e8	−2.13100
$204$	−4.76899e6	−0.0393297
$205$	57773.5	0.000468371 0
$206$	7.88346e7	0.628321
$207$	−1.54306e8	−1.20916
$208$	2.06755e7	0.159307
$209$	−5.25652e7	−0.398278
$210$	−1.89859e7	−0.141470
$211$	8.71990e6	0.0639032	0.0319516	−	0.999489i	$-0.489828\pi$
$211$	8.71990e6	0.0639032	0.0319516	+	0.999489i	$0.489828\pi$
$212$	4.55495e6	0.0328328
$213$	2.29046e7	0.162403
$214$	1.94201e7	0.135457
$215$	1.36348e8	0.935651
$216$	−9.68156e6	−0.0653668
$217$	2.59082e8	1.72119
$218$	1.59293e8	1.04135
$219$	−1.99236e7	−0.128178
$220$	2.35366e8	1.49027
$221$	8.66305e7	0.539881
$222$	1.19819e7	0.0735004
$223$	7.65311e6	0.0462137	0.0231069	−	0.999733i	$-0.492644\pi$
$223$	7.65311e6	0.0462137	0.0231069	+	0.999733i	$0.492644\pi$
$224$	3.73244e7	0.221884
$225$	−3.29890e8	−1.93077
$226$	−1.33682e8	−0.770362
$227$	2.23572e7	0.126861	0.0634303	−	0.997986i	$-0.479796\pi$
$227$	2.23572e7	0.126861	0.0634303	+	0.997986i	$0.479796\pi$
$228$	1.90596e6	0.0106498
$229$	1.26153e8	0.694184	0.347092	−	0.937831i	$-0.387169\pi$
$229$	1.26153e8	0.694184	0.347092	+	0.937831i	$0.387169\pi$
$230$	−2.73217e8	−1.48068
$231$	3.79012e7	0.202307
$232$	−1.14168e8	−0.600258
$233$	3.21102e8	1.66302	0.831509	−	0.555512i	$-0.187478\pi$
$233$	3.21102e8	1.66302	0.831509	+	0.555512i	$0.187478\pi$
$234$	8.75540e7	0.446705
$235$	1.01668e8	0.511028
$236$	−1.19701e8	−0.592796
$237$	−2.55715e7	−0.124778
$238$	1.56389e8	0.751948
$239$	8.75889e7	0.415008	0.207504	−	0.978234i	$-0.433466\pi$
$239$	8.75889e7	0.415008	0.207504	+	0.978234i	$0.433466\pi$
$240$	−8.53411e6	−0.0398492
$241$	−1.40775e8	−0.647836	−0.323918	−	0.946085i	$-0.605000\pi$
$241$	−1.40775e8	−0.647836	−0.323918	+	0.946085i	$0.605000\pi$
$242$	−3.13959e8	−1.42403
$243$	−6.15860e7	−0.275334
$244$	−1.66043e8	−0.731739
$245$	2.27409e8	0.987929
$246$	4181.82	1.79099e−5 0
$247$	−3.46224e7	−0.146190
$248$	1.16457e8	0.484823
$249$	1.43375e7	0.0588539
$250$	−2.84191e8	−1.15032
$251$	−5.26041e7	−0.209972	−0.104986	−	0.994474i	$-0.533480\pi$
$251$	−5.26041e7	−0.209972	−0.104986	+	0.994474i	$0.533480\pi$
$252$	1.58056e8	0.622172
$253$	5.45418e8	2.11742
$254$	1.23432e8	0.472616
$255$	−3.57579e7	−0.135046
$256$	1.67772e7	0.0625000
$257$	−5.06891e8	−1.86273	−0.931363	−	0.364092i	$-0.881379\pi$
$257$	−5.06891e8	−1.86273	−0.931363	+	0.364092i	$0.881379\pi$
$258$	9.86927e6	0.0357780
$259$	−3.92921e8	−1.40526
$260$	1.55025e8	0.547011
$261$	−4.83465e8	−1.68315
$262$	4.69553e7	0.161298
$263$	1.19806e8	0.406102	0.203051	−	0.979168i	$-0.434914\pi$
$263$	1.19806e8	0.406102	0.203051	+	0.979168i	$0.434914\pi$
$264$	1.70365e7	0.0569857
$265$	3.41530e7	0.112738
$266$	−6.25020e7	−0.203614
$267$	1.31538e6	0.00422923
$268$	−7.95399e7	−0.252414
$269$	−2.98358e8	−0.934555	−0.467277	−	0.884111i	$-0.654765\pi$
$269$	−2.98358e8	−0.934555	−0.467277	+	0.884111i	$0.654765\pi$
$270$	−7.25924e7	−0.224449
$271$	−2.22282e8	−0.678439	−0.339220	−	0.940707i	$-0.610163\pi$
$271$	−2.22282e8	−0.678439	−0.339220	+	0.940707i	$0.610163\pi$
$272$	7.02965e7	0.211808
$273$	2.49639e7	0.0742580
$274$	−1.36273e8	−0.400206
$275$	1.16605e9	3.38106
$276$	−1.97763e7	−0.0566191
$277$	1.84740e8	0.522254	0.261127	−	0.965304i	$-0.415906\pi$
$277$	1.84740e8	0.522254	0.261127	+	0.965304i	$0.415906\pi$
$278$	−1.01925e8	−0.284526
$279$	4.93155e8	1.35947
$280$	2.79859e8	0.761879
$281$	−6.85728e8	−1.84366	−0.921828	−	0.387599i	$-0.873305\pi$
$281$	−6.85728e8	−1.84366	−0.921828	+	0.387599i	$0.873305\pi$
$282$	7.35900e6	0.0195410
$283$	−2.31722e8	−0.607735	−0.303867	−	0.952714i	$-0.598278\pi$
$283$	−2.31722e8	−0.607735	−0.303867	+	0.952714i	$0.598278\pi$
$284$	−3.37621e8	−0.874612
$285$	1.42909e7	0.0365681
$286$	−3.09474e8	−0.782246
$287$	−137134.	−0.000342420 0
$288$	7.10459e7	0.175253
$289$	−1.15796e8	−0.282197
$290$	−8.56036e8	−2.06110
$291$	4.88808e7	0.116282
$292$	2.93680e8	0.690295
$293$	8.38834e8	1.94823	0.974113	−	0.226062i	$-0.0725851\pi$
$293$	8.38834e8	1.94823	0.974113	+	0.226062i	$0.0725851\pi$
$294$	1.64605e7	0.0377771
$295$	−8.97519e8	−2.03548
$296$	−1.76617e8	−0.395832
$297$	1.44915e8	0.320971
$298$	−2.06599e8	−0.452242
$299$	3.59244e8	0.777213
$300$	−4.22797e7	−0.0904082
$301$	−3.23643e8	−0.684042
$302$	7.37750e7	0.154129
$303$	−5.53180e6	−0.0114240
$304$	−2.80945e7	−0.0573539
$305$	−1.24499e9	−2.51256
$306$	2.97682e8	0.593920
$307$	7.62332e7	0.150370	0.0751848	−	0.997170i	$-0.476045\pi$
$307$	7.62332e7	0.150370	0.0751848	+	0.997170i	$0.476045\pi$
$308$	−5.58677e8	−1.08951
$309$	4.27858e7	0.0824982
$310$	8.73194e8	1.66473
$311$	5.29430e8	0.998038	0.499019	−	0.866591i	$-0.333694\pi$
$311$	5.29430e8	0.998038	0.499019	+	0.866591i	$0.333694\pi$
$312$	1.12212e7	0.0209169
$313$	−6.48301e8	−1.19501	−0.597505	−	0.801865i	$-0.703841\pi$
$313$	−6.48301e8	−1.19501	−0.597505	+	0.801865i	$0.703841\pi$
$314$	2.89743e8	0.528152
$315$	1.18511e9	2.13634
$316$	3.76932e8	0.671983
$317$	2.48405e8	0.437978	0.218989	−	0.975727i	$-0.429724\pi$
$317$	2.48405e8	0.437978	0.218989	+	0.975727i	$0.429724\pi$
$318$	2.47210e6	0.00431093
$319$	1.70889e9	2.94745
$320$	1.25796e8	0.214606
$321$	1.05398e7	0.0177855
$322$	6.48523e8	1.08251
$323$	−1.17716e8	−0.194368
$324$	2.98217e8	0.487107
$325$	7.68027e8	1.24104
$326$	−1.43841e8	−0.229944
$327$	8.64526e7	0.136729
$328$	−61641.4	−9.64527e−5 0
$329$	−2.41324e8	−0.373606
$330$	1.27740e8	0.195671
$331$	−7.25869e8	−1.10017	−0.550085	−	0.835108i	$-0.685405\pi$
$331$	−7.25869e8	−1.10017	−0.550085	+	0.835108i	$0.685405\pi$
$332$	−2.11339e8	−0.316955
$333$	−7.47914e8	−1.10993
$334$	−1.68130e8	−0.246906
$335$	−5.96391e8	−0.866711
$336$	2.02570e7	0.0291332
$337$	1.09423e9	1.55741	0.778705	−	0.627390i	$-0.215877\pi$
$337$	1.09423e9	1.55741	0.778705	+	0.627390i	$0.215877\pi$
$338$	2.98151e8	0.419979
$339$	−7.25532e7	−0.101148
$340$	5.27084e8	0.727283
$341$	−1.74314e9	−2.38063
$342$	−1.18971e8	−0.160823
$343$	3.98268e8	0.532900
$344$	−1.45476e8	−0.192681
$345$	−1.48283e8	−0.194412
$346$	−8.88446e7	−0.115309
$347$	−4.45339e8	−0.572186	−0.286093	−	0.958202i	$-0.592357\pi$
$347$	−4.45339e8	−0.572186	−0.286093	+	0.958202i	$0.592357\pi$
$348$	−6.19624e7	−0.0788136
$349$	4.56049e8	0.574278	0.287139	−	0.957889i	$-0.407296\pi$
$349$	4.56049e8	0.574278	0.287139	+	0.957889i	$0.407296\pi$
$350$	1.38648e9	1.72852
$351$	9.54492e7	0.117814
$352$	−2.51124e8	−0.306894
$353$	1.29420e9	1.56600	0.782998	−	0.622024i	$-0.213689\pi$
$353$	1.29420e9	1.56600	0.782998	+	0.622024i	$0.213689\pi$
$354$	−6.49651e7	−0.0778339
$355$	−2.53149e9	−3.00315
$356$	−1.93891e7	−0.0227763
$357$	8.48769e7	0.0987304
$358$	−1.31096e8	−0.151008
$359$	3.37354e8	0.384818	0.192409	−	0.981315i	$-0.438370\pi$
$359$	3.37354e8	0.384818	0.192409	+	0.981315i	$0.438370\pi$
$360$	5.32703e8	0.601764
$361$	4.70459e7	0.0526316
$362$	−1.87669e8	−0.207927
$363$	−1.70394e8	−0.186974
$364$	−3.67976e8	−0.399913
$365$	2.20202e9	2.37026
$366$	−9.01163e7	−0.0960770
$367$	−1.44933e9	−1.53051	−0.765256	−	0.643726i	$-0.777387\pi$
$367$	−1.44933e9	−1.53051	−0.765256	+	0.643726i	$0.777387\pi$
$368$	2.91509e8	0.304919
$369$	−261031.	−0.000270458 0
$370$	−1.32428e9	−1.35917
$371$	−8.10675e7	−0.0824210
$372$	6.32044e7	0.0636571
$373$	−1.27740e9	−1.27452	−0.637258	−	0.770650i	$-0.719932\pi$
$373$	−1.27740e9	−1.27452	−0.637258	+	0.770650i	$0.719932\pi$
$374$	−1.05221e9	−1.04004
$375$	−1.54238e8	−0.151037
$376$	−1.08474e8	−0.105237
$377$	1.12557e9	1.08188
$378$	1.72309e8	0.164092
$379$	6.43803e7	0.0607458	0.0303729	−	0.999539i	$-0.490331\pi$
$379$	6.43803e7	0.0607458	0.0303729	+	0.999539i	$0.490331\pi$
$380$	−2.10653e8	−0.196936
$381$	6.69899e7	0.0620543
$382$	1.06208e9	0.974846
$383$	−3.08830e8	−0.280881	−0.140441	−	0.990089i	$-0.544852\pi$
$383$	−3.08830e8	−0.280881	−0.140441	+	0.990089i	$0.544852\pi$
$384$	9.10547e6	0.00820622
$385$	−4.18896e9	−3.74105
$386$	2.81601e8	0.249217
$387$	−6.16044e8	−0.540285
$388$	−7.20520e8	−0.626231
$389$	−1.77108e9	−1.52550	−0.762752	−	0.646691i	$-0.776152\pi$
$389$	−1.77108e9	−1.52550	−0.762752	+	0.646691i	$0.776152\pi$
$390$	8.41367e7	0.0718223
$391$	1.22142e9	1.03335
$392$	−2.42634e8	−0.203446
$393$	2.54840e7	0.0211784
$394$	−5.26278e7	−0.0433490
$395$	2.82624e9	2.30738
$396$	−1.06342e9	−0.860544
$397$	−6.10056e8	−0.489331	−0.244666	−	0.969607i	$-0.578678\pi$
$397$	−6.10056e8	−0.489331	−0.244666	+	0.969607i	$0.578678\pi$
$398$	−3.03002e8	−0.240910
$399$	−3.39216e7	−0.0267345
$400$	6.23217e8	0.486889
$401$	−1.88893e9	−1.46289	−0.731445	−	0.681901i	$-0.761154\pi$
$401$	−1.88893e9	−1.46289	−0.731445	+	0.681901i	$0.761154\pi$
$402$	−4.31686e7	−0.0331418
$403$	−1.14813e9	−0.873824
$404$	8.15406e7	0.0615232
$405$	2.23603e9	1.67258
$406$	2.03193e9	1.50684
$407$	2.64362e9	1.94366
$408$	3.81519e7	0.0278103
$409$	−1.78957e9	−1.29335	−0.646677	−	0.762764i	$-0.723842\pi$
$409$	−1.78957e9	−1.29335	−0.646677	+	0.762764i	$0.723842\pi$
$410$	−462188.	−0.000331188 0
$411$	−7.39593e7	−0.0525468
$412$	−6.30677e8	−0.444290
$413$	2.13040e9	1.48811
$414$	1.23444e9	0.855008
$415$	−1.58463e9	−1.08832
$416$	−1.65404e8	−0.112647
$417$	−5.53173e7	−0.0373581
$418$	4.20522e8	0.281625
$419$	1.80427e9	1.19826	0.599131	−	0.800651i	$-0.295513\pi$
$419$	1.80427e9	1.19826	0.599131	+	0.800651i	$0.295513\pi$
$420$	1.51887e8	0.100034
$421$	−9.51064e8	−0.621187	−0.310593	−	0.950543i	$-0.600528\pi$
$421$	−9.51064e8	−0.621187	−0.310593	+	0.950543i	$0.600528\pi$
$422$	−6.97592e7	−0.0451864
$423$	−4.59352e8	−0.295090
$424$	−3.64396e7	−0.0232163
$425$	2.61128e9	1.65003
$426$	−1.83237e8	−0.114836
$427$	2.95518e9	1.83690
$428$	−1.55360e8	−0.0957828
$429$	−1.67960e8	−0.102708
$430$	−1.09078e9	−0.661605
$431$	1.03659e9	0.623643	0.311821	−	0.950141i	$-0.399061\pi$
$431$	1.03659e9	0.623643	0.311821	+	0.950141i	$0.399061\pi$
$432$	7.74525e7	0.0462213
$433$	−767017.	−0.000454043 0	−0.000227022	−	1.00000i	$-0.500072\pi$
$433$	−767017.	−0.000454043 0	−0.000227022		1.00000i	$0.500072\pi$
$434$	−2.07266e9	−1.21706
$435$	−4.64595e8	−0.270621
$436$	−1.27434e9	−0.736348
$437$	−4.88150e8	−0.279813
$438$	1.59389e8	0.0906354
$439$	3.11236e9	1.75576	0.877878	−	0.478885i	$-0.158959\pi$
$439$	3.11236e9	1.75576	0.877878	+	0.478885i	$0.158959\pi$
$440$	−1.88293e9	−1.05378
$441$	−1.02747e9	−0.570473
$442$	−6.93044e8	−0.381753
$443$	2.06182e9	1.12678	0.563389	−	0.826192i	$-0.309497\pi$
$443$	2.06182e9	1.12678	0.563389	+	0.826192i	$0.309497\pi$
$444$	−9.58550e7	−0.0519726
$445$	−1.45380e8	−0.0782068
$446$	−6.12249e7	−0.0326780
$447$	−1.12127e8	−0.0593791
$448$	−2.98595e8	−0.156895
$449$	1.78694e9	0.931640	0.465820	−	0.884879i	$-0.345759\pi$
$449$	1.78694e9	0.931640	0.465820	+	0.884879i	$0.345759\pi$
$450$	2.63912e9	1.36526
$451$	922657.	0.000473612 0
$452$	1.06946e9	0.544728
$453$	4.00398e7	0.0202371
$454$	−1.78858e8	−0.0897040
$455$	−2.75909e9	−1.37318
$456$	−1.52477e7	−0.00753054
$457$	−1.12258e9	−0.550187	−0.275093	−	0.961418i	$-0.588709\pi$
$457$	−1.12258e9	−0.550187	−0.275093	+	0.961418i	$0.588709\pi$
$458$	−1.00923e9	−0.490862
$459$	3.24526e8	0.156641
$460$	2.18574e9	1.04700
$461$	8.66502e8	0.411923	0.205962	−	0.978560i	$-0.433968\pi$
$461$	8.66502e8	0.411923	0.205962	+	0.978560i	$0.433968\pi$
$462$	−3.03210e8	−0.143053
$463$	−7.06994e8	−0.331042	−0.165521	−	0.986206i	$-0.552930\pi$
$463$	−7.06994e8	−0.331042	−0.165521	+	0.986206i	$0.552930\pi$
$464$	9.13347e8	0.424446
$465$	4.73907e8	0.218579
$466$	−2.56881e9	−1.17593
$467$	−6.53807e8	−0.297058	−0.148529	−	0.988908i	$-0.547454\pi$
$467$	−6.53807e8	−0.297058	−0.148529	+	0.988908i	$0.547454\pi$
$468$	−7.00432e8	−0.315868
$469$	1.41563e9	0.633641
$470$	−8.13340e8	−0.361352
$471$	1.57252e8	0.0693462
$472$	9.57607e8	0.419170
$473$	2.17751e9	0.946120
$474$	2.04572e8	0.0882310
$475$	−1.04362e9	−0.446800
$476$	−1.25111e9	−0.531707
$477$	−1.54309e8	−0.0650996
$478$	−7.00711e8	−0.293455
$479$	−2.08278e9	−0.865902	−0.432951	−	0.901418i	$-0.642528\pi$
$479$	−2.08278e9	−0.865902	−0.432951	+	0.901418i	$0.642528\pi$
$480$	6.82729e7	0.0281776
$481$	1.74124e9	0.713430
$482$	1.12620e9	0.458089
$483$	3.51972e8	0.142132
$484$	2.51167e9	1.00694
$485$	−5.40246e9	−2.15028
$486$	4.92688e8	0.194691
$487$	4.64872e9	1.82382	0.911910	−	0.410390i	$-0.134607\pi$
$487$	4.64872e9	1.82382	0.911910	+	0.410390i	$0.134607\pi$
$488$	1.32834e9	0.517418
$489$	−7.80667e7	−0.0301915
$490$	−1.81927e9	−0.698572
$491$	−1.97975e9	−0.754787	−0.377394	−	0.926053i	$-0.623180\pi$
$491$	−1.97975e9	−0.754787	−0.377394	+	0.926053i	$0.623180\pi$
$492$	−33454.6	−1.26642e−5 0
$493$	3.82692e9	1.43842
$494$	2.76980e8	0.103372
$495$	−7.97356e9	−2.95484
$496$	−9.31654e8	−0.342822
$497$	6.00887e9	2.19556
$498$	−1.14700e8	−0.0416160
$499$	−3.36644e9	−1.21288	−0.606441	−	0.795128i	$-0.707403\pi$
$499$	−3.36644e9	−1.21288	−0.606441	+	0.795128i	$0.707403\pi$
$500$	2.27353e9	0.813402
$501$	−9.12489e7	−0.0324187
$502$	4.20833e8	0.148473
$503$	−2.64080e8	−0.0925225	−0.0462612	−	0.998929i	$-0.514731\pi$
$503$	−2.64080e8	−0.0925225	−0.0462612	+	0.998929i	$0.514731\pi$
$504$	−1.26445e9	−0.439942
$505$	6.11392e8	0.211251
$506$	−4.36335e9	−1.49725
$507$	1.61815e8	0.0551430
$508$	−9.87453e8	−0.334190
$509$	−3.60140e9	−1.21048	−0.605242	−	0.796041i	$-0.706924\pi$
$509$	−3.60140e9	−1.21048	−0.605242	+	0.796041i	$0.706924\pi$
$510$	2.86063e8	0.0954919
$511$	−5.22682e9	−1.73286
$512$	−1.34218e8	−0.0441942
$513$	−1.29699e8	−0.0424156
$514$	4.05513e9	1.31715
$515$	−4.72882e9	−1.52555
$516$	−7.89542e7	−0.0252989
$517$	1.62366e9	0.516746
$518$	3.14337e9	0.993668
$519$	−4.82185e7	−0.0151401
$520$	−1.24020e9	−0.386795
$521$	−6.11173e8	−0.189336	−0.0946678	−	0.995509i	$-0.530179\pi$
$521$	−6.11173e8	−0.189336	−0.0946678	+	0.995509i	$0.530179\pi$
$522$	3.86772e9	1.19017
$523$	3.04248e8	0.0929976	0.0464988	−	0.998918i	$-0.485194\pi$
$523$	3.04248e8	0.0929976	0.0464988	+	0.998918i	$0.485194\pi$
$524$	−3.75642e8	−0.114055
$525$	7.52481e8	0.226954
$526$	−9.58451e8	−0.287157
$527$	−3.90363e9	−1.16180
$528$	−1.36292e8	−0.0402950
$529$	1.66024e9	0.487613
$530$	−2.73224e8	−0.0797175
$531$	4.05515e9	1.17537
$532$	5.00016e8	0.143977
$533$	607715.	0.000173842 0
$534$	−1.05230e7	−0.00299052
$535$	−1.16489e9	−0.328888
$536$	6.36319e8	0.178484
$537$	−7.11498e7	−0.0198273
$538$	2.38686e9	0.660830
$539$	3.63177e9	0.998983
$540$	5.80740e8	0.158710
$541$	1.14881e8	0.0311931	0.0155966	−	0.999878i	$-0.495035\pi$
$541$	1.14881e8	0.0311931	0.0155966	+	0.999878i	$0.495035\pi$
$542$	1.77825e9	0.479729
$543$	−1.01853e8	−0.0273008
$544$	−5.62372e8	−0.149771
$545$	−9.55502e9	−2.52839
$546$	−1.99711e8	−0.0525083
$547$	4.68520e9	1.22398	0.611988	−	0.790867i	$-0.290370\pi$
$547$	4.68520e9	1.22398	0.611988	+	0.790867i	$0.290370\pi$
$548$	1.09019e9	0.282988
$549$	5.62509e9	1.45086
$550$	−9.32840e9	−2.39077
$551$	−1.52946e9	−0.389499
$552$	1.58210e8	0.0400358
$553$	−6.70851e9	−1.68690
$554$	−1.47792e9	−0.369289
$555$	−7.18722e8	−0.178458
$556$	8.15396e8	0.201190
$557$	5.76492e9	1.41351	0.706757	−	0.707456i	$-0.250157\pi$
$557$	5.76492e9	1.41351	0.706757	+	0.707456i	$0.250157\pi$
$558$	−3.94524e9	−0.961289
$559$	1.43423e9	0.347279
$560$	−2.23887e9	−0.538730
$561$	−5.71063e8	−0.136557
$562$	5.48583e9	1.30366
$563$	−6.34304e9	−1.49802	−0.749011	−	0.662558i	$-0.769471\pi$
$563$	−6.34304e9	−1.49802	−0.749011	+	0.662558i	$0.769471\pi$
$564$	−5.88720e7	−0.0138176
$565$	8.01881e9	1.87043
$566$	1.85377e9	0.429733
$567$	−5.30757e9	−1.22280
$568$	2.70097e9	0.618444
$569$	−6.02589e8	−0.137129	−0.0685644	−	0.997647i	$-0.521842\pi$
$569$	−6.02589e8	−0.137129	−0.0685644	+	0.997647i	$0.521842\pi$
$570$	−1.14327e8	−0.0258575
$571$	−2.03817e9	−0.458157	−0.229079	−	0.973408i	$-0.573571\pi$
$571$	−2.03817e9	−0.458157	−0.229079	+	0.973408i	$0.573571\pi$
$572$	2.47579e9	0.553131
$573$	5.76422e8	0.127997
$574$	1.09707e6	0.000242128 0
$575$	1.08286e10	2.37539
$576$	−5.68367e8	−0.123923
$577$	−2.06212e9	−0.446887	−0.223443	−	0.974717i	$-0.571730\pi$
$577$	−2.06212e9	−0.446887	−0.223443	+	0.974717i	$0.571730\pi$
$578$	9.26371e8	0.199543
$579$	1.52833e8	0.0327221
$580$	6.84828e9	1.45742
$581$	3.76135e9	0.795660
$582$	−3.91046e8	−0.0822239
$583$	5.45432e8	0.113999
$584$	−2.34944e9	−0.488112
$585$	−5.25184e9	−1.08459
$586$	−6.71067e9	−1.37760
$587$	−7.21880e9	−1.47310	−0.736549	−	0.676385i	$-0.763546\pi$
$587$	−7.21880e9	−1.47310	−0.736549	+	0.676385i	$0.763546\pi$
$588$	−1.31684e8	−0.0267124
$589$	1.56011e9	0.314595
$590$	7.18015e9	1.43930
$591$	−2.85626e7	−0.00569170
$592$	1.41294e9	0.279896
$593$	−6.06919e9	−1.19520	−0.597598	−	0.801796i	$-0.703878\pi$
$593$	−6.06919e9	−1.19520	−0.597598	+	0.801796i	$0.703878\pi$
$594$	−1.15932e9	−0.226961
$595$	−9.38087e9	−1.82572
$596$	1.65279e9	0.319783
$597$	−1.64448e8	−0.0316314
$598$	−2.87395e9	−0.549573
$599$	6.66887e9	1.26782	0.633911	−	0.773406i	$-0.281449\pi$
$599$	6.66887e9	1.26782	0.633911	+	0.773406i	$0.281449\pi$
$600$	3.38238e8	0.0639282
$601$	3.17679e9	0.596936	0.298468	−	0.954420i	$-0.403524\pi$
$601$	3.17679e9	0.596936	0.298468	+	0.954420i	$0.403524\pi$
$602$	2.58914e9	0.483691
$603$	2.69460e9	0.500476
$604$	−5.90200e8	−0.108986
$605$	1.88325e10	3.45752
$606$	4.42544e7	0.00807797
$607$	6.54344e9	1.18753	0.593766	−	0.804638i	$-0.297640\pi$
$607$	6.54344e9	1.18753	0.593766	+	0.804638i	$0.297640\pi$
$608$	2.24756e8	0.0405554
$609$	1.10279e9	0.197848
$610$	9.95994e9	1.77665
$611$	1.06943e9	0.189675
$612$	−2.38146e9	−0.419965
$613$	−4.18496e9	−0.733803	−0.366902	−	0.930260i	$-0.619581\pi$
$613$	−4.18496e9	−0.733803	−0.366902	+	0.930260i	$0.619581\pi$
$614$	−6.09866e8	−0.106327
$615$	−250843.	−4.34849e−5 0
$616$	4.46941e9	0.770403
$617$	−6.17748e9	−1.05880	−0.529400	−	0.848373i	$-0.677583\pi$
$617$	−6.17748e9	−1.05880	−0.529400	+	0.848373i	$0.677583\pi$
$618$	−3.42286e8	−0.0583351
$619$	7.53499e9	1.27693	0.638463	−	0.769653i	$-0.279571\pi$
$619$	7.53499e9	1.27693	0.638463	+	0.769653i	$0.279571\pi$
$620$	−6.98555e9	−1.17714
$621$	1.34576e9	0.225500
$622$	−4.23544e9	−0.705719
$623$	3.45082e8	0.0571760
$624$	−8.97696e7	−0.0147905
$625$	5.16000e9	0.845414
$626$	5.18641e9	0.845000
$627$	2.28229e8	0.0369772
$628$	−2.31794e9	−0.373460
$629$	5.92020e9	0.948547
$630$	−9.48087e9	−1.51062
$631$	5.78958e9	0.917370	0.458685	−	0.888599i	$-0.348321\pi$
$631$	5.78958e9	0.917370	0.458685	+	0.888599i	$0.348321\pi$
$632$	−3.01546e9	−0.475164
$633$	−3.78603e7	−0.00593296
$634$	−1.98724e9	−0.309697
$635$	−7.40393e9	−1.14750
$636$	−1.97768e7	−0.00304829
$637$	2.39209e9	0.366682
$638$	−1.36711e10	−2.08416
$639$	1.14377e10	1.73415
$640$	−1.00637e9	−0.151749
$641$	−6.47867e9	−0.971589	−0.485794	−	0.874073i	$-0.661470\pi$
$641$	−6.47867e9	−0.971589	−0.485794	+	0.874073i	$0.661470\pi$
$642$	−8.43185e7	−0.0125762
$643$	9.45772e8	0.140297	0.0701484	−	0.997537i	$-0.477653\pi$
$643$	9.45772e8	0.140297	0.0701484	+	0.997537i	$0.477653\pi$
$644$	−5.18819e9	−0.765447
$645$	−5.91999e8	−0.0868684
$646$	9.41726e8	0.137439
$647$	−1.01933e10	−1.47962	−0.739812	−	0.672814i	$-0.765085\pi$
$647$	−1.01933e10	−1.47962	−0.739812	+	0.672814i	$0.765085\pi$
$648$	−2.38574e9	−0.344437
$649$	−1.43336e10	−2.05825
$650$	−6.14422e9	−0.877546
$651$	−1.12489e9	−0.159800
$652$	1.15073e9	0.162595
$653$	−1.43721e9	−0.201987	−0.100994	−	0.994887i	$-0.532202\pi$
$653$	−1.43721e9	−0.201987	−0.100994	+	0.994887i	$0.532202\pi$
$654$	−6.91621e8	−0.0966821
$655$	−2.81657e9	−0.391630
$656$	493131.	6.82023e−5 0
$657$	−9.94909e9	−1.36869
$658$	1.93059e9	0.264179
$659$	−4.15841e9	−0.566015	−0.283008	−	0.959118i	$-0.591332\pi$
$659$	−4.15841e9	−0.566015	−0.283008	+	0.959118i	$0.591332\pi$
$660$	−1.02192e9	−0.138360
$661$	−1.66535e9	−0.224284	−0.112142	−	0.993692i	$-0.535771\pi$
$661$	−1.66535e9	−0.224284	−0.112142	+	0.993692i	$0.535771\pi$
$662$	5.80695e9	0.777938
$663$	−3.76135e8	−0.0501240
$664$	1.69072e9	0.224121
$665$	3.74912e9	0.494372
$666$	5.98331e9	0.784841
$667$	1.58697e10	2.07075
$668$	1.34504e9	0.174589
$669$	−3.32285e7	−0.00429061
$670$	4.77113e9	0.612857
$671$	−1.98828e10	−2.54068
$672$	−1.62056e8	−0.0206003
$673$	6.20239e8	0.0784343	0.0392172	−	0.999231i	$-0.487514\pi$
$673$	6.20239e8	0.0784343	0.0392172	+	0.999231i	$0.487514\pi$
$674$	−8.75382e9	−1.10126
$675$	2.87710e9	0.360074
$676$	−2.38521e9	−0.296970
$677$	3.41875e9	0.423455	0.211727	−	0.977329i	$-0.432091\pi$
$677$	3.41875e9	0.423455	0.211727	+	0.977329i	$0.432091\pi$
$678$	5.80426e8	0.0715225
$679$	1.28236e10	1.57204
$680$	−4.21667e9	−0.514267
$681$	−9.70711e7	−0.0117781
$682$	1.39451e10	1.68336
$683$	−1.24659e10	−1.49711	−0.748553	−	0.663075i	$-0.769251\pi$
$683$	−1.24659e10	−1.49711	−0.748553	+	0.663075i	$0.769251\pi$
$684$	9.51765e8	0.113719
$685$	8.17422e9	0.971693
$686$	−3.18614e9	−0.376817
$687$	−5.47736e8	−0.0644499
$688$	1.16381e9	0.136246
$689$	3.59253e8	0.0418440
$690$	1.18626e9	0.137470
$691$	−1.73188e10	−1.99685	−0.998423	−	0.0561331i	$-0.982123\pi$
$691$	−1.73188e10	−1.99685	−0.998423	+	0.0561331i	$0.982123\pi$
$692$	7.10757e8	0.0815361
$693$	1.89265e10	2.16025
$694$	3.56271e9	0.404597
$695$	6.11385e9	0.690825
$696$	4.95699e8	0.0557296
$697$	2.06622e6	0.000231133 0
$698$	−3.64839e9	−0.406076
$699$	−1.39417e9	−0.154399
$700$	−1.10918e10	−1.22225
$701$	−1.23706e10	−1.35637	−0.678185	−	0.734891i	$-0.737233\pi$
$701$	−1.23706e10	−1.35637	−0.678185	+	0.734891i	$0.737233\pi$
$702$	−7.63594e8	−0.0833072
$703$	−2.36605e9	−0.256850
$704$	2.00899e9	0.217007
$705$	−4.41423e8	−0.0474453
$706$	−1.03536e10	−1.10733
$707$	−1.45123e9	−0.154443
$708$	5.19721e8	0.0550369
$709$	−1.83088e10	−1.92930	−0.964648	−	0.263541i	$-0.915110\pi$
$709$	−1.83088e10	−1.92930	−0.964648	+	0.263541i	$0.915110\pi$
$710$	2.02519e10	2.12355
$711$	−1.27694e10	−1.33238
$712$	1.55113e8	0.0161053
$713$	−1.61878e10	−1.67253
$714$	−6.79015e8	−0.0698129
$715$	1.85635e10	1.89928
$716$	1.04877e9	0.106779
$717$	−3.80296e8	−0.0385305
$718$	−2.69884e9	−0.272108
$719$	8.55893e9	0.858753	0.429377	−	0.903126i	$-0.358733\pi$
$719$	8.55893e9	0.858753	0.429377	+	0.903126i	$0.358733\pi$
$720$	−4.26162e9	−0.425511
$721$	1.12246e10	1.11531
$722$	−3.76367e8	−0.0372161
$723$	6.11220e8	0.0601469
$724$	1.50135e9	0.147027
$725$	3.39278e10	3.30653
$726$	1.36315e9	0.132211
$727$	5.98649e9	0.577832	0.288916	−	0.957354i	$-0.406705\pi$
$727$	5.98649e9	0.577832	0.288916	+	0.957354i	$0.406705\pi$
$728$	2.94381e9	0.282781
$729$	−9.92324e9	−0.948652
$730$	−1.76161e10	−1.67602
$731$	4.87637e9	0.461727
$732$	7.20930e8	0.0679367
$733$	5.03464e9	0.472177	0.236088	−	0.971732i	$-0.424134\pi$
$733$	5.03464e9	0.472177	0.236088	+	0.971732i	$0.424134\pi$
$734$	1.15947e10	1.08224
$735$	−9.87370e8	−0.0917221
$736$	−2.33207e9	−0.215611
$737$	−9.52451e9	−0.876408
$738$	2.08825e6	0.000191243 0
$739$	−7.60690e9	−0.693350	−0.346675	−	0.937985i	$-0.612689\pi$
$739$	−7.60690e9	−0.693350	−0.346675	+	0.937985i	$0.612689\pi$
$740$	1.05942e10	0.961075
$741$	1.50325e8	0.0135727
$742$	6.48540e8	0.0582804
$743$	2.17110e9	0.194186	0.0970931	−	0.995275i	$-0.469046\pi$
$743$	2.17110e9	0.194186	0.0970931	+	0.995275i	$0.469046\pi$
$744$	−5.05635e8	−0.0450124
$745$	1.23926e10	1.09804
$746$	1.02192e10	0.901219
$747$	7.15961e9	0.628445
$748$	8.41766e9	0.735420
$749$	2.76505e9	0.240446
$750$	1.23391e9	0.106799
$751$	1.73532e10	1.49499	0.747497	−	0.664265i	$-0.231255\pi$
$751$	1.73532e10	1.49499	0.747497	+	0.664265i	$0.231255\pi$
$752$	8.67794e8	0.0744139
$753$	2.28398e8	0.0194944
$754$	−9.00457e9	−0.765003
$755$	−4.42533e9	−0.374223
$756$	−1.37847e9	−0.116031
$757$	9.38319e9	0.786167	0.393084	−	0.919503i	$-0.371408\pi$
$757$	9.38319e9	0.786167	0.393084	+	0.919503i	$0.371408\pi$
$758$	−5.15043e8	−0.0429537
$759$	−2.36811e9	−0.196588
$760$	1.68522e9	0.139254
$761$	5.53583e9	0.455341	0.227670	−	0.973738i	$-0.426889\pi$
$761$	5.53583e9	0.455341	0.227670	+	0.973738i	$0.426889\pi$
$762$	−5.35919e8	−0.0438790
$763$	2.26803e10	1.84847
$764$	−8.49665e9	−0.689320
$765$	−1.78562e10	−1.44203
$766$	2.47064e9	0.198613
$767$	−9.44092e9	−0.755493
$768$	−7.28438e7	−0.00580267
$769$	−5.35575e9	−0.424696	−0.212348	−	0.977194i	$-0.568111\pi$
$769$	−5.35575e9	−0.424696	−0.212348	+	0.977194i	$0.568111\pi$
$770$	3.35117e10	2.64532
$771$	2.20083e9	0.172941
$772$	−2.25281e9	−0.176223
$773$	1.88126e10	1.46494	0.732470	−	0.680799i	$-0.238367\pi$
$773$	1.88126e10	1.46494	0.732470	+	0.680799i	$0.238367\pi$
$774$	4.92835e9	0.382039
$775$	−3.46078e10	−2.67066
$776$	5.76416e9	0.442812
$777$	1.70600e9	0.130468
$778$	1.41686e10	1.07869
$779$	−825778.	−6.25868e−5 0
$780$	−6.73093e8	−0.0507860
$781$	−4.04285e10	−3.03675
$782$	−9.77139e9	−0.730689
$783$	4.21649e9	0.313896
$784$	1.94107e9	0.143858
$785$	−1.73800e10	−1.28235
$786$	−2.03872e8	−0.0149754
$787$	3.31420e9	0.242363	0.121182	−	0.992630i	$-0.461332\pi$
$787$	3.31420e9	0.242363	0.121182	+	0.992630i	$0.461332\pi$
$788$	4.21023e8	0.0306524
$789$	−5.20179e8	−0.0377036
$790$	−2.26099e10	−1.63156
$791$	−1.90339e10	−1.36744
$792$	8.50739e9	0.608497
$793$	−1.30960e10	−0.932570
$794$	4.88045e9	0.346010
$795$	−1.48287e8	−0.0104669
$796$	2.42402e9	0.170349
$797$	−2.11139e10	−1.47728	−0.738642	−	0.674098i	$-0.764532\pi$
$797$	−2.11139e10	−1.47728	−0.738642	+	0.674098i	$0.764532\pi$
$798$	2.71373e8	0.0189041
$799$	3.63605e9	0.252183
$800$	−4.98574e9	−0.344282
$801$	6.56852e8	0.0451600
$802$	1.51115e10	1.03442
$803$	3.51667e10	2.39678
$804$	3.45349e8	0.0234348
$805$	−3.89011e10	−2.62831
$806$	9.18505e9	0.617887
$807$	1.29542e9	0.0867667
$808$	−6.52324e8	−0.0435035
$809$	−1.25988e10	−0.836584	−0.418292	−	0.908313i	$-0.637371\pi$
$809$	−1.25988e10	−0.836584	−0.418292	+	0.908313i	$0.637371\pi$
$810$	−1.78883e10	−1.18269
$811$	6.19737e9	0.407976	0.203988	−	0.978973i	$-0.434610\pi$
$811$	6.19737e9	0.407976	0.203988	+	0.978973i	$0.434610\pi$
$812$	−1.62555e10	−1.06550
$813$	9.65108e8	0.0629882
$814$	−2.11490e10	−1.37437
$815$	8.62818e9	0.558300
$816$	−3.05215e8	−0.0196649
$817$	−1.94887e9	−0.125028
$818$	1.43166e10	0.914540
$819$	1.24661e10	0.792931
$820$	3.69750e6	0.000234186 0
$821$	8.18458e9	0.516173	0.258086	−	0.966122i	$-0.416908\pi$
$821$	8.18458e9	0.516173	0.258086	+	0.966122i	$0.416908\pi$
$822$	5.91675e8	0.0371562
$823$	2.64926e10	1.65663	0.828315	−	0.560262i	$-0.189300\pi$
$823$	2.64926e10	1.65663	0.828315	+	0.560262i	$0.189300\pi$
$824$	5.04541e9	0.314160
$825$	−5.06279e9	−0.313907
$826$	−1.70432e10	−1.05225
$827$	−6.84453e9	−0.420799	−0.210399	−	0.977616i	$-0.567476\pi$
$827$	−6.84453e9	−0.420799	−0.210399	+	0.977616i	$0.567476\pi$
$828$	−9.87555e9	−0.604582
$829$	8.57259e8	0.0522602	0.0261301	−	0.999659i	$-0.491682\pi$
$829$	8.57259e8	0.0522602	0.0261301	+	0.999659i	$0.491682\pi$
$830$	1.26770e10	0.769561
$831$	−8.02109e8	−0.0484875
$832$	1.32323e9	0.0796536
$833$	8.13308e9	0.487526
$834$	4.42539e8	0.0264162
$835$	1.00851e10	0.599485
$836$	−3.36417e9	−0.199139
$837$	−4.30101e9	−0.253531
$838$	−1.44341e10	−0.847299
$839$	3.09097e10	1.80687	0.903436	−	0.428722i	$-0.141036\pi$
$839$	3.09097e10	1.80687	0.903436	+	0.428722i	$0.141036\pi$
$840$	−1.21510e9	−0.0707350
$841$	3.24725e10	1.88248
$842$	7.60851e9	0.439245
$843$	2.97731e9	0.171170
$844$	5.58074e8	0.0319516
$845$	−1.78843e10	−1.01970
$846$	3.67482e9	0.208660
$847$	−4.47019e10	−2.52775
$848$	2.91517e8	0.0164164
$849$	1.00610e9	0.0564238
$850$	−2.08902e10	−1.16675
$851$	2.45502e10	1.36553
$852$	1.46589e9	0.0812015
$853$	−1.99363e10	−1.09983	−0.549913	−	0.835222i	$-0.685339\pi$
$853$	−1.99363e10	−1.09983	−0.549913	+	0.835222i	$0.685339\pi$
$854$	−2.36414e10	−1.29889
$855$	7.13634e9	0.390476
$856$	1.24288e9	0.0677286
$857$	−1.52939e10	−0.830011	−0.415006	−	0.909819i	$-0.636220\pi$
$857$	−1.52939e10	−0.830011	−0.415006	+	0.909819i	$0.636220\pi$
$858$	1.34368e9	0.0726259
$859$	−2.10493e10	−1.13308	−0.566541	−	0.824034i	$-0.691719\pi$
$859$	−2.10493e10	−1.13308	−0.566541	+	0.824034i	$0.691719\pi$
$860$	8.72627e9	0.467826
$861$	595413.	3.17912e−5 0
$862$	−8.29271e9	−0.440982
$863$	−2.26703e10	−1.20066	−0.600330	−	0.799752i	$-0.704964\pi$
$863$	−2.26703e10	−1.20066	−0.600330	+	0.799752i	$0.704964\pi$
$864$	−6.19620e8	−0.0326834
$865$	5.32926e9	0.279969
$866$	6.13613e6	0.000321057 0
$867$	5.02768e8	0.0262000
$868$	1.65813e10	0.860595
$869$	4.51357e10	2.33320
$870$	3.71676e9	0.191358
$871$	−6.27339e9	−0.321691
$872$	1.01947e10	0.520677
$873$	2.44093e10	1.24167
$874$	3.90520e9	0.197858
$875$	−4.04635e10	−2.04190
$876$	−1.27511e9	−0.0640889
$877$	−3.86231e9	−0.193352	−0.0966759	−	0.995316i	$-0.530821\pi$
$877$	−3.86231e9	−0.193352	−0.0966759	+	0.995316i	$0.530821\pi$
$878$	−2.48989e10	−1.24151
$879$	−3.64207e9	−0.180879
$880$	1.50634e10	0.745133
$881$	1.19539e7	0.000588973 0	0.000294486	−	1.00000i	$-0.499906\pi$
$881$	1.19539e7	0.000588973 0	0.000294486		1.00000i	$0.499906\pi$
$882$	8.21978e9	0.403385
$883$	3.11683e10	1.52353	0.761763	−	0.647855i	$-0.224334\pi$
$883$	3.11683e10	1.52353	0.761763	+	0.647855i	$0.224334\pi$
$884$	5.54435e9	0.269940
$885$	3.89687e9	0.188979
$886$	−1.64946e10	−0.796753
$887$	−8.84392e9	−0.425512	−0.212756	−	0.977105i	$-0.568244\pi$
$887$	−8.84392e9	−0.425512	−0.212756	+	0.977105i	$0.568244\pi$
$888$	7.66840e8	0.0367502
$889$	1.75744e10	0.838926
$890$	1.16304e9	0.0553006
$891$	3.57100e10	1.69129
$892$	4.89799e8	0.0231069
$893$	−1.45317e9	−0.0682869
$894$	8.97016e8	0.0419874
$895$	7.86370e9	0.366646
$896$	2.38876e9	0.110942
$897$	−1.55977e9	−0.0721586
$898$	−1.42955e10	−0.658769
$899$	−5.07190e10	−2.32815
$900$	−2.11129e10	−0.965384
$901$	1.22145e9	0.0556340
$902$	−7.38125e6	−0.000334894 0
$903$	1.40520e9	0.0635084
$904$	−8.55567e9	−0.385181
$905$	1.12571e10	0.504845
$906$	−3.20318e8	−0.0143098
$907$	−1.60496e10	−0.714230	−0.357115	−	0.934060i	$-0.616240\pi$
$907$	−1.60496e10	−0.714230	−0.357115	+	0.934060i	$0.616240\pi$
$908$	1.43086e9	0.0634303
$909$	−2.76238e9	−0.121986
$910$	2.20727e10	0.970982
$911$	2.60981e10	1.14366	0.571828	−	0.820374i	$-0.306234\pi$
$911$	2.60981e10	1.14366	0.571828	+	0.820374i	$0.306234\pi$
$912$	1.21981e8	0.00532490
$913$	−2.53068e10	−1.10050
$914$	8.98063e9	0.389041
$915$	5.40554e9	0.233273
$916$	8.07381e9	0.347092
$917$	6.68556e9	0.286316
$918$	−2.59621e9	−0.110762
$919$	1.28704e10	0.547001	0.273501	−	0.961872i	$-0.411818\pi$
$919$	1.28704e10	0.547001	0.273501	+	0.961872i	$0.411818\pi$
$920$	−1.74859e10	−0.740339
$921$	−3.30991e8	−0.0139607
$922$	−6.93201e9	−0.291274
$923$	−2.66285e10	−1.11466
$924$	2.42568e9	0.101154
$925$	5.24858e10	2.18045
$926$	5.65596e9	0.234082
$927$	2.13656e10	0.880921
$928$	−7.30678e9	−0.300129
$929$	2.49717e9	0.102186	0.0510932	−	0.998694i	$-0.483729\pi$
$929$	2.49717e9	0.102186	0.0510932	+	0.998694i	$0.483729\pi$
$930$	−3.79126e9	−0.154558
$931$	−3.25044e9	−0.132013
$932$	2.05505e10	0.831509
$933$	−2.29869e9	−0.0926606
$934$	5.23046e9	0.210051
$935$	6.31157e10	2.52520
$936$	5.60345e9	0.223352
$937$	−2.58901e10	−1.02812	−0.514062	−	0.857753i	$-0.671860\pi$
$937$	−2.58901e10	−1.02812	−0.514062	+	0.857753i	$0.671860\pi$
$938$	−1.13250e10	−0.448052
$939$	2.81481e9	0.110948
$940$	6.50672e9	0.255514
$941$	2.12968e10	0.833203	0.416601	−	0.909089i	$-0.363221\pi$
$941$	2.12968e10	0.833203	0.416601	+	0.909089i	$0.363221\pi$
$942$	−1.25801e9	−0.0490351
$943$	8.56831e6	0.000332739 0
$944$	−7.66086e9	−0.296398
$945$	−1.03358e10	−0.398413
$946$	−1.74201e10	−0.669008
$947$	1.53310e10	0.586605	0.293303	−	0.956020i	$-0.405246\pi$
$947$	1.53310e10	0.586605	0.293303	+	0.956020i	$0.405246\pi$
$948$	−1.63657e9	−0.0623888
$949$	2.31628e10	0.879751
$950$	8.34892e9	0.315935
$951$	−1.07853e9	−0.0406631
$952$	1.00089e10	0.375974
$953$	4.58699e9	0.171673	0.0858367	−	0.996309i	$-0.472644\pi$
$953$	4.58699e9	0.171673	0.0858367	+	0.996309i	$0.472644\pi$
$954$	1.23448e9	0.0460323
$955$	−6.37080e10	−2.36691
$956$	5.60569e9	0.207504
$957$	−7.41969e9	−0.273649
$958$	1.66622e10	0.612285
$959$	−1.94028e10	−0.710392
$960$	−5.46183e8	−0.0199246
$961$	2.42229e10	0.880430
$962$	−1.39299e10	−0.504471
$963$	5.26320e9	0.189914
$964$	−9.00959e9	−0.323918
$965$	−1.68916e10	−0.605096
$966$	−2.81578e9	−0.100503
$967$	−6.92287e9	−0.246203	−0.123102	−	0.992394i	$-0.539284\pi$
$967$	−6.92287e9	−0.246203	−0.123102	+	0.992394i	$0.539284\pi$
$968$	−2.00934e10	−0.712014
$969$	5.11102e8	0.0180457
$970$	4.32197e10	1.52048
$971$	−4.15887e10	−1.45783	−0.728917	−	0.684603i	$-0.759976\pi$
$971$	−4.15887e10	−1.45783	−0.728917	+	0.684603i	$0.759976\pi$
$972$	−3.94150e9	−0.137667
$973$	−1.45122e10	−0.505053
$974$	−3.71898e10	−1.28964
$975$	−3.33464e9	−0.115221
$976$	−1.06268e10	−0.365870
$977$	2.28939e9	0.0785398	0.0392699	−	0.999229i	$-0.487497\pi$
$977$	2.28939e9	0.0785398	0.0392699	+	0.999229i	$0.487497\pi$
$978$	6.24534e8	0.0213486
$979$	−2.32175e9	−0.0790818
$980$	1.45542e10	0.493965
$981$	4.31713e10	1.46000
$982$	1.58380e10	0.533715
$983$	4.47036e10	1.50108	0.750542	−	0.660823i	$-0.229793\pi$
$983$	4.47036e10	1.50108	0.750542	+	0.660823i	$0.229793\pi$
$984$	267636.	8.95493e−6 0
$985$	3.15683e9	0.105251
$986$	−3.06154e10	−1.01712
$987$	1.04778e9	0.0346866
$988$	−2.21584e9	−0.0730951
$989$	2.02216e10	0.664703
$990$	6.37885e10	2.08939
$991$	4.73079e10	1.54410	0.772051	−	0.635561i	$-0.219231\pi$
$991$	4.73079e10	1.54410	0.772051	+	0.635561i	$0.219231\pi$
$992$	7.45323e9	0.242412
$993$	3.15160e9	0.102143
$994$	−4.80710e10	−1.55250
$995$	1.81753e10	0.584926
$996$	9.17599e8	0.0294270
$997$	−4.76756e10	−1.52357	−0.761786	−	0.647829i	$-0.775677\pi$
$997$	−4.76756e10	−1.52357	−0.761786	+	0.647829i	$0.775677\pi$
$998$	2.69315e10	0.857637
$999$	6.52286e9	0.206994

Twists

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

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

\(f(q)\)	\(=\)	\(q-8.00000 q^{2} -4.34183 q^{3} +64.0000 q^{4} +479.873 q^{5} +34.7346 q^{6} -1139.05 q^{7} -512.000 q^{8} -2168.15 q^{9} +O(q^{10})\)	\(q-8.00000 q^{2} -4.34183 q^{3} +64.0000 q^{4} +479.873 q^{5} +34.7346 q^{6} -1139.05 q^{7} -512.000 q^{8} -2168.15 q^{9} -3838.98 q^{10} +7663.68 q^{11} -277.877 q^{12} +5047.74 q^{13} +9112.41 q^{14} -2083.52 q^{15} +4096.00 q^{16} +17162.2 q^{17} +17345.2 q^{18} -6859.00 q^{19} +30711.8 q^{20} +4945.56 q^{21} -61309.4 q^{22} +71169.3 q^{23} +2223.02 q^{24} +152153. q^{25} -40381.9 q^{26} +18909.3 q^{27} -72899.3 q^{28} +222985. q^{29} +16668.2 q^{30} -227455. q^{31} -32768.0 q^{32} -33274.4 q^{33} -137298. q^{34} -546599. q^{35} -138762. q^{36} +344955. q^{37} +54872.0 q^{38} -21916.4 q^{39} -245695. q^{40} +120.393 q^{41} -39564.5 q^{42} +284134. q^{43} +490476. q^{44} -1.04044e6 q^{45} -569354. q^{46} +211864. q^{47} -17784.1 q^{48} +473894. q^{49} -1.21722e6 q^{50} -74515.5 q^{51} +323055. q^{52} +71171.1 q^{53} -151274. q^{54} +3.67759e6 q^{55} +583194. q^{56} +29780.6 q^{57} -1.78388e6 q^{58} -1.87033e6 q^{59} -133346. q^{60} -2.59442e6 q^{61} +1.81964e6 q^{62} +2.46963e6 q^{63} +262144. q^{64} +2.42227e6 q^{65} +266195. q^{66} -1.24281e6 q^{67} +1.09838e6 q^{68} -309005. q^{69} +4.37279e6 q^{70} -5.27533e6 q^{71} +1.11009e6 q^{72} +4.58875e6 q^{73} -2.75964e6 q^{74} -660621. q^{75} -438976. q^{76} -8.72932e6 q^{77} +175331. q^{78} +5.88956e6 q^{79} +1.96556e6 q^{80} +4.65964e6 q^{81} -963.147 q^{82} -3.30218e6 q^{83} +316516. q^{84} +8.23569e6 q^{85} -2.27307e6 q^{86} -968163. q^{87} -3.92380e6 q^{88} -302955. q^{89} +8.32348e6 q^{90} -5.74963e6 q^{91} +4.55483e6 q^{92} +987568. q^{93} -1.69491e6 q^{94} -3.29145e6 q^{95} +142273. q^{96} -1.12581e7 q^{97} -3.79115e6 q^{98} -1.66160e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 16 q^{2} - 61 q^{3} + 128 q^{4} + 175 q^{5} + 488 q^{6} - 2592 q^{7} - 1024 q^{8} - 1145 q^{9}+O(q^{10}) \) 2 * q - 16 * q^2 - 61 * q^3 + 128 * q^4 + 175 * q^5 + 488 * q^6 - 2592 * q^7 - 1024 * q^8 - 1145 * q^9	\( 2 q - 16 q^{2} - 61 q^{3} + 128 q^{4} + 175 q^{5} + 488 q^{6} - 2592 q^{7} - 1024 q^{8} - 1145 q^{9} - 1400 q^{10} + 1045 q^{11} - 3904 q^{12} + 14647 q^{13} + 20736 q^{14} + 15190 q^{15} + 8192 q^{16} + 29616 q^{17} + 9160 q^{18} - 13718 q^{19} + 11200 q^{20} + 87267 q^{21} - 8360 q^{22} + 69985 q^{23} + 31232 q^{24} + 166975 q^{25} - 117176 q^{26} + 84851 q^{27} - 165888 q^{28} + 138821 q^{29} - 121520 q^{30} - 199396 q^{31} - 65536 q^{32} + 341728 q^{33} - 236928 q^{34} - 103635 q^{35} - 73280 q^{36} - 67840 q^{37} + 109744 q^{38} - 565793 q^{39} - 89600 q^{40} - 539350 q^{41} - 698136 q^{42} + 602639 q^{43} + 66880 q^{44} - 1352365 q^{45} - 559880 q^{46} - 1031975 q^{47} - 249856 q^{48} + 1761412 q^{49} - 1335800 q^{50} - 780123 q^{51} + 937408 q^{52} + 2138263 q^{53} - 678808 q^{54} + 5695445 q^{55} + 1327104 q^{56} + 418399 q^{57} - 1110568 q^{58} - 3936369 q^{59} + 972160 q^{60} - 1027655 q^{61} + 1595168 q^{62} + 983049 q^{63} + 524288 q^{64} - 504280 q^{65} - 2733824 q^{66} + 764949 q^{67} + 1895424 q^{68} - 241907 q^{69} + 829080 q^{70} - 3572084 q^{71} + 586240 q^{72} + 9069522 q^{73} + 542720 q^{74} - 1500425 q^{75} - 877952 q^{76} + 887283 q^{77} + 4526344 q^{78} - 2753414 q^{79} + 716800 q^{80} - 1314122 q^{81} + 4314800 q^{82} - 7643046 q^{83} + 5585088 q^{84} + 4438875 q^{85} - 4821112 q^{86} + 3800423 q^{87} - 535040 q^{88} + 1393620 q^{89} + 10818920 q^{90} - 19696869 q^{91} + 4479040 q^{92} - 602176 q^{93} + 8255800 q^{94} - 1200325 q^{95} + 1998848 q^{96} - 6921466 q^{97} - 14091296 q^{98} - 23387893 q^{99}+O(q^{100}) \) 2 * q - 16 * q^2 - 61 * q^3 + 128 * q^4 + 175 * q^5 + 488 * q^6 - 2592 * q^7 - 1024 * q^8 - 1145 * q^9 - 1400 * q^10 + 1045 * q^11 - 3904 * q^12 + 14647 * q^13 + 20736 * q^14 + 15190 * q^15 + 8192 * q^16 + 29616 * q^17 + 9160 * q^18 - 13718 * q^19 + 11200 * q^20 + 87267 * q^21 - 8360 * q^22 + 69985 * q^23 + 31232 * q^24 + 166975 * q^25 - 117176 * q^26 + 84851 * q^27 - 165888 * q^28 + 138821 * q^29 - 121520 * q^30 - 199396 * q^31 - 65536 * q^32 + 341728 * q^33 - 236928 * q^34 - 103635 * q^35 - 73280 * q^36 - 67840 * q^37 + 109744 * q^38 - 565793 * q^39 - 89600 * q^40 - 539350 * q^41 - 698136 * q^42 + 602639 * q^43 + 66880 * q^44 - 1352365 * q^45 - 559880 * q^46 - 1031975 * q^47 - 249856 * q^48 + 1761412 * q^49 - 1335800 * q^50 - 780123 * q^51 + 937408 * q^52 + 2138263 * q^53 - 678808 * q^54 + 5695445 * q^55 + 1327104 * q^56 + 418399 * q^57 - 1110568 * q^58 - 3936369 * q^59 + 972160 * q^60 - 1027655 * q^61 + 1595168 * q^62 + 983049 * q^63 + 524288 * q^64 - 504280 * q^65 - 2733824 * q^66 + 764949 * q^67 + 1895424 * q^68 - 241907 * q^69 + 829080 * q^70 - 3572084 * q^71 + 586240 * q^72 + 9069522 * q^73 + 542720 * q^74 - 1500425 * q^75 - 877952 * q^76 + 887283 * q^77 + 4526344 * q^78 - 2753414 * q^79 + 716800 * q^80 - 1314122 * q^81 + 4314800 * q^82 - 7643046 * q^83 + 5585088 * q^84 + 4438875 * q^85 - 4821112 * q^86 + 3800423 * q^87 - 535040 * q^88 + 1393620 * q^89 + 10818920 * q^90 - 19696869 * q^91 + 4479040 * q^92 - 602176 * q^93 + 8255800 * q^94 - 1200325 * q^95 + 1998848 * q^96 - 6921466 * q^97 - 14091296 * q^98 - 23387893 * q^99

Introduction

L-functions

Modular forms

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Database

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