LMFDB - Embedded newform 47.8.a.b.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [47,8,Mod(1,47)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(47, 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("47.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 47 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	47.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:	$14.6820961978$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - x^{15} - 1579 x^{14} + 411 x^{13} + 976985 x^{12} + 344128 x^{11} - 300557483 x^{10} + \cdots - 870476371764660 $ x^16 - x^15 - 1579x^14 + 411x^13 + 976985x^12 + 344128x^11 - 300557483x^10 - 241372726x^9 + 48208527759x^8 + 51058288103x^7 - 3836832303201x^6 - 4025612987345x^5 + 126622461371211x^4 + 84396440456938x^3 - 1031790019238985x^2 - 1863919661016228x - 870476371764660
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{9}\cdot 3 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$-1.15943$ of defining polynomial
Character	$\chi$	$=$	47.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.15943 q^{2} -69.5079 q^{3} -123.337 q^{4} -36.4103 q^{5} -150.097 q^{6} -1189.62 q^{7} -542.744 q^{8} +2644.35 q^{9} +O(q^{10})$	\(q+2.15943 q^{2} -69.5079 q^{3} -123.337 q^{4} -36.4103 q^{5} -150.097 q^{6} -1189.62 q^{7} -542.744 q^{8} +2644.35 q^{9} -78.6256 q^{10} -7630.65 q^{11} +8572.89 q^{12} +6111.31 q^{13} -2568.90 q^{14} +2530.81 q^{15} +14615.1 q^{16} +31529.4 q^{17} +5710.30 q^{18} +7558.44 q^{19} +4490.74 q^{20} +82688.0 q^{21} -16477.8 q^{22} -97061.2 q^{23} +37725.0 q^{24} -76799.3 q^{25} +13196.9 q^{26} -31789.8 q^{27} +146724. q^{28} +55495.8 q^{29} +5465.10 q^{30} -133849. q^{31} +101032. q^{32} +530391. q^{33} +68085.6 q^{34} +43314.5 q^{35} -326146. q^{36} -533182. q^{37} +16321.9 q^{38} -424785. q^{39} +19761.5 q^{40} +694729. q^{41} +178559. q^{42} +9029.52 q^{43} +941140. q^{44} -96281.9 q^{45} -209597. q^{46} -103823. q^{47} -1.01587e6 q^{48} +591652. q^{49} -165843. q^{50} -2.19155e6 q^{51} -753750. q^{52} -892837. q^{53} -68647.8 q^{54} +277835. q^{55} +645659. q^{56} -525372. q^{57} +119839. q^{58} +335079. q^{59} -312142. q^{60} +459478. q^{61} -289038. q^{62} -3.14578e6 q^{63} -1.65256e6 q^{64} -222515. q^{65} +1.14534e6 q^{66} +3.41231e6 q^{67} -3.88874e6 q^{68} +6.74653e6 q^{69} +93534.5 q^{70} -1.80416e6 q^{71} -1.43521e6 q^{72} -860792. q^{73} -1.15137e6 q^{74} +5.33816e6 q^{75} -932235. q^{76} +9.07757e6 q^{77} -917292. q^{78} +7.53835e6 q^{79} -532141. q^{80} -3.57356e6 q^{81} +1.50022e6 q^{82} +7.59654e6 q^{83} -1.01985e7 q^{84} -1.14800e6 q^{85} +19498.6 q^{86} -3.85740e6 q^{87} +4.14149e6 q^{88} -146145. q^{89} -207914. q^{90} -7.27013e6 q^{91} +1.19712e7 q^{92} +9.30360e6 q^{93} -224198. q^{94} -275206. q^{95} -7.02249e6 q^{96} +1.24917e6 q^{97} +1.27763e6 q^{98} -2.01781e7 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 15 q^{2} + 40 q^{3} + 1125 q^{4} + 444 q^{5} - 128 q^{6} + 1860 q^{7} + 2145 q^{8} + 16986 q^{9}+O(q^{10}) $ 16 * q + 15 * q^2 + 40 * q^3 + 1125 * q^4 + 444 * q^5 - 128 * q^6 + 1860 * q^7 + 2145 * q^8 + 16986 * q^9	\( 16 q + 15 q^{2} + 40 q^{3} + 1125 q^{4} + 444 q^{5} - 128 q^{6} + 1860 q^{7} + 2145 q^{8} + 16986 q^{9} + 7680 q^{10} + 4776 q^{11} - 20035 q^{12} + 16074 q^{13} - 7749 q^{14} + 22614 q^{15} + 131873 q^{16} + 67106 q^{17} + 243090 q^{18} + 69730 q^{19} + 300392 q^{20} + 247700 q^{21} + 357976 q^{22} + 101110 q^{23} + 167915 q^{24} + 542284 q^{25} + 424864 q^{26} + 253084 q^{27} + 383052 q^{28} + 199976 q^{29} + 252626 q^{30} + 373036 q^{31} + 463950 q^{32} + 64236 q^{33} + 17022 q^{34} - 140956 q^{35} + 694529 q^{36} + 596106 q^{37} - 488694 q^{38} - 352018 q^{39} - 188562 q^{40} - 66678 q^{41} - 776155 q^{42} + 1037964 q^{43} - 1598298 q^{44} + 905532 q^{45} - 2460646 q^{46} - 1661168 q^{47} - 9341196 q^{48} - 1580054 q^{49} - 7616723 q^{50} - 4897564 q^{51} + 1045432 q^{52} - 1253454 q^{53} - 11836385 q^{54} - 1489932 q^{55} - 9175912 q^{56} + 1630346 q^{57} + 1640408 q^{58} - 1977256 q^{59} - 12723470 q^{60} - 2381646 q^{61} - 5329480 q^{62} + 320028 q^{63} + 10719545 q^{64} + 9219376 q^{65} - 13727472 q^{66} + 11455752 q^{67} - 6720409 q^{68} + 9814322 q^{69} - 6175334 q^{70} + 5467788 q^{71} + 19454169 q^{72} + 12652462 q^{73} + 3377814 q^{74} + 14522488 q^{75} - 4047624 q^{76} + 12951326 q^{77} - 8107404 q^{78} + 34366848 q^{79} + 19400408 q^{80} + 24384080 q^{81} - 16612272 q^{82} + 27856008 q^{83} + 18267309 q^{84} + 8672002 q^{85} - 34325926 q^{86} + 41013236 q^{87} + 36623538 q^{88} + 14703638 q^{89} + 20243114 q^{90} + 11783202 q^{91} + 4020484 q^{92} + 35185978 q^{93} - 1557345 q^{94} + 14274548 q^{95} - 56487828 q^{96} + 51072842 q^{97} - 22140217 q^{98} + 55509018 q^{99}+O(q^{100}) \) 16 * q + 15 * q^2 + 40 * q^3 + 1125 * q^4 + 444 * q^5 - 128 * q^6 + 1860 * q^7 + 2145 * q^8 + 16986 * q^9 + 7680 * q^10 + 4776 * q^11 - 20035 * q^12 + 16074 * q^13 - 7749 * q^14 + 22614 * q^15 + 131873 * q^16 + 67106 * q^17 + 243090 * q^18 + 69730 * q^19 + 300392 * q^20 + 247700 * q^21 + 357976 * q^22 + 101110 * q^23 + 167915 * q^24 + 542284 * q^25 + 424864 * q^26 + 253084 * q^27 + 383052 * q^28 + 199976 * q^29 + 252626 * q^30 + 373036 * q^31 + 463950 * q^32 + 64236 * q^33 + 17022 * q^34 - 140956 * q^35 + 694529 * q^36 + 596106 * q^37 - 488694 * q^38 - 352018 * q^39 - 188562 * q^40 - 66678 * q^41 - 776155 * q^42 + 1037964 * q^43 - 1598298 * q^44 + 905532 * q^45 - 2460646 * q^46 - 1661168 * q^47 - 9341196 * q^48 - 1580054 * q^49 - 7616723 * q^50 - 4897564 * q^51 + 1045432 * q^52 - 1253454 * q^53 - 11836385 * q^54 - 1489932 * q^55 - 9175912 * q^56 + 1630346 * q^57 + 1640408 * q^58 - 1977256 * q^59 - 12723470 * q^60 - 2381646 * q^61 - 5329480 * q^62 + 320028 * q^63 + 10719545 * q^64 + 9219376 * q^65 - 13727472 * q^66 + 11455752 * q^67 - 6720409 * q^68 + 9814322 * q^69 - 6175334 * q^70 + 5467788 * q^71 + 19454169 * q^72 + 12652462 * q^73 + 3377814 * q^74 + 14522488 * q^75 - 4047624 * q^76 + 12951326 * q^77 - 8107404 * q^78 + 34366848 * q^79 + 19400408 * q^80 + 24384080 * q^81 - 16612272 * q^82 + 27856008 * q^83 + 18267309 * q^84 + 8672002 * q^85 - 34325926 * q^86 + 41013236 * q^87 + 36623538 * q^88 + 14703638 * q^89 + 20243114 * q^90 + 11783202 * q^91 + 4020484 * q^92 + 35185978 * q^93 - 1557345 * q^94 + 14274548 * q^95 - 56487828 * q^96 + 51072842 * q^97 - 22140217 * q^98 + 55509018 * 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 $

$2$	2.15943	0.190868	0.0954342	−	0.995436i	$-0.469576\pi$
$2$	2.15943	0.190868	0.0954342	+	0.995436i	$0.469576\pi$
$3$	−69.5079	−1.48631	−0.743156	−	0.669118i	$-0.766672\pi$
$3$	−69.5079	−1.48631	−0.743156	+	0.669118i	$0.766672\pi$
$4$	−123.337	−0.963569
$5$	−36.4103	−0.130266	−0.0651328	−	0.997877i	$-0.520747\pi$
$5$	−36.4103	−0.130266	−0.0651328	+	0.997877i	$0.520747\pi$
$6$	−150.097	−0.283690
$7$	−1189.62	−1.31089	−0.655443	−	0.755245i	$-0.727518\pi$
$7$	−1189.62	−1.31089	−0.655443	+	0.755245i	$0.727518\pi$
$8$	−542.744	−0.374783
$9$	2644.35	1.20912
$10$	−78.6256	−0.0248636
$11$	−7630.65	−1.72857	−0.864285	−	0.503002i	$-0.832229\pi$
$11$	−7630.65	−1.72857	−0.864285	+	0.503002i	$0.832229\pi$
$12$	8572.89	1.43216
$13$	6111.31	0.771494	0.385747	−	0.922605i	$-0.373944\pi$
$13$	6111.31	0.771494	0.385747	+	0.922605i	$0.373944\pi$
$14$	−2568.90	−0.250207
$15$	2530.81	0.193615
$16$	14615.1	0.892035
$17$	31529.4	1.55648	0.778242	−	0.627964i	$-0.216112\pi$
$17$	31529.4	1.55648	0.778242	+	0.627964i	$0.216112\pi$
$18$	5710.30	0.230784
$19$	7558.44	0.252810	0.126405	−	0.991979i	$-0.459656\pi$
$19$	7558.44	0.252810	0.126405	+	0.991979i	$0.459656\pi$
$20$	4490.74	0.125520
$21$	82688.0	1.94839
$22$	−16477.8	−0.329929
$23$	−97061.2	−1.66341	−0.831703	−	0.555220i	$-0.812634\pi$
$23$	−97061.2	−1.66341	−0.831703	+	0.555220i	$0.812634\pi$
$24$	37725.0	0.557045
$25$	−76799.3	−0.983031
$26$	13196.9	0.147254
$27$	−31789.8	−0.310824
$28$	146724.	1.26313
$29$	55495.8	0.422539	0.211270	−	0.977428i	$-0.432240\pi$
$29$	55495.8	0.422539	0.211270	+	0.977428i	$0.432240\pi$
$30$	5465.10	0.0369550
$31$	−133849.	−0.806957	−0.403479	−	0.914989i	$-0.632199\pi$
$31$	−133849.	−0.806957	−0.403479	+	0.914989i	$0.632199\pi$
$32$	101032.	0.545045
$33$	530391.	2.56920
$34$	68085.6	0.297084
$35$	43314.5	0.170763
$36$	−326146.	−1.16507
$37$	−533182.	−1.73049	−0.865246	−	0.501348i	$-0.832838\pi$
$37$	−533182.	−1.73049	−0.865246	+	0.501348i	$0.832838\pi$
$38$	16321.9	0.0482535
$39$	−424785.	−1.14668
$40$	19761.5	0.0488214
$41$	694729.	1.57424	0.787122	−	0.616798i	$-0.211570\pi$
$41$	694729.	1.57424	0.787122	+	0.616798i	$0.211570\pi$
$42$	178559.	0.371885
$43$	9029.52	0.0173191	0.00865955	−	0.999963i	$-0.497244\pi$
$43$	9029.52	0.0173191	0.00865955	+	0.999963i	$0.497244\pi$
$44$	941140.	1.66560
$45$	−96281.9	−0.157507
$46$	−209597.	−0.317492
$47$	−103823.	−0.145865
$47$	−103823.	−0.145865
$48$	−1.01587e6	−1.32584
$49$	591652.	0.718422
$50$	−165843.	−0.187629
$51$	−2.19155e6	−2.31342
$52$	−753750.	−0.743388
$53$	−892837.	−0.823772	−0.411886	−	0.911235i	$-0.635130\pi$
$53$	−892837.	−0.823772	−0.411886	+	0.911235i	$0.635130\pi$
$54$	−68647.8	−0.0593265
$55$	277835.	0.225173
$56$	645659.	0.491298
$57$	−525372.	−0.375755
$58$	119839.	0.0806494
$59$	335079.	0.212405	0.106203	−	0.994345i	$-0.466131\pi$
$59$	335079.	0.212405	0.106203	+	0.994345i	$0.466131\pi$
$60$	−312142.	−0.186562
$61$	459478.	0.259185	0.129593	−	0.991567i	$-0.458633\pi$
$61$	459478.	0.259185	0.129593	+	0.991567i	$0.458633\pi$
$62$	−289038.	−0.154023
$63$	−3.14578e6	−1.58502
$64$	−1.65256e6	−0.788003
$65$	−222515.	−0.100499
$66$	1.14534e6	0.490378
$67$	3.41231e6	1.38608	0.693038	−	0.720901i	$-0.256272\pi$
$67$	3.41231e6	1.38608	0.693038	+	0.720901i	$0.256272\pi$
$68$	−3.88874e6	−1.49978
$69$	6.74653e6	2.47234
$70$	93534.5	0.0325933
$71$	−1.80416e6	−0.598234	−0.299117	−	0.954217i	$-0.596692\pi$
$71$	−1.80416e6	−0.598234	−0.299117	+	0.954217i	$0.596692\pi$
$72$	−1.43521e6	−0.453160
$73$	−860792.	−0.258981	−0.129491	−	0.991581i	$-0.541334\pi$
$73$	−860792.	−0.258981	−0.129491	+	0.991581i	$0.541334\pi$
$74$	−1.15137e6	−0.330296
$75$	5.33816e6	1.46109
$76$	−932235.	−0.243600
$77$	9.07757e6	2.26596
$78$	−917292.	−0.218865
$79$	7.53835e6	1.72021	0.860105	−	0.510118i	$-0.170398\pi$
$79$	7.53835e6	1.72021	0.860105	+	0.510118i	$0.170398\pi$
$80$	−532141.	−0.116201
$81$	−3.57356e6	−0.747143
$82$	1.50022e6	0.300473
$83$	7.59654e6	1.45828	0.729142	−	0.684362i	$-0.239919\pi$
$83$	7.59654e6	1.45828	0.729142	+	0.684362i	$0.239919\pi$
$84$	−1.01985e7	−1.87740
$85$	−1.14800e6	−0.202756
$86$	19498.6	0.00330567
$87$	−3.85740e6	−0.628025
$88$	4.14149e6	0.647839
$89$	−146145.	−0.0219745	−0.0109873	−	0.999940i	$-0.503497\pi$
$89$	−146145.	−0.0219745	−0.0109873	+	0.999940i	$0.503497\pi$
$90$	−207914.	−0.0300632
$91$	−7.27013e6	−1.01134
$92$	1.19712e7	1.60281
$93$	9.30360e6	1.19939
$94$	−224198.	−0.0278410
$95$	−275206.	−0.0329325
$96$	−7.02249e6	−0.810106
$97$	1.24917e6	0.138970	0.0694852	−	0.997583i	$-0.477864\pi$
$97$	1.24917e6	0.138970	0.0694852	+	0.997583i	$0.477864\pi$
$98$	1.27763e6	0.137124
$99$	−2.01781e7	−2.09006
$100$	9.47218e6	0.947218
$101$	−1.04764e7	−1.01178	−0.505889	−	0.862598i	$-0.668836\pi$
$101$	−1.04764e7	−1.01178	−0.505889	+	0.862598i	$0.668836\pi$
$102$	−4.73249e6	−0.441559
$103$	1.74818e7	1.57636	0.788181	−	0.615443i	$-0.211023\pi$
$103$	1.74818e7	1.57636	0.788181	+	0.615443i	$0.211023\pi$
$104$	−3.31688e6	−0.289143
$105$	−3.01070e6	−0.253808
$106$	−1.92802e6	−0.157232
$107$	1.15857e6	0.0914281	0.0457140	−	0.998955i	$-0.485444\pi$
$107$	1.15857e6	0.0914281	0.0457140	+	0.998955i	$0.485444\pi$
$108$	3.92085e6	0.299500
$109$	−2.30271e7	−1.70312	−0.851561	−	0.524256i	$-0.824343\pi$
$109$	−2.30271e7	−1.70312	−0.851561	+	0.524256i	$0.824343\pi$
$110$	599964.	0.0429784
$111$	3.70604e7	2.57205
$112$	−1.73864e7	−1.16936
$113$	1.54381e7	1.00651	0.503257	−	0.864137i	$-0.332135\pi$
$113$	1.54381e7	1.00651	0.503257	+	0.864137i	$0.332135\pi$
$114$	−1.13450e6	−0.0717198
$115$	3.53403e6	0.216685
$116$	−6.84468e6	−0.407146
$117$	1.61605e7	0.932832
$118$	723580.	0.0405414
$119$	−3.75080e7	−2.04037
$120$	−1.37358e6	−0.0725638
$121$	3.87396e7	1.98796
$122$	992210.	0.0494702
$123$	−4.82892e7	−2.33982
$124$	1.65086e7	0.777559
$125$	5.64085e6	0.258321
$126$	−6.79308e6	−0.302531
$127$	−1.57537e7	−0.682447	−0.341223	−	0.939982i	$-0.610841\pi$
$127$	−1.57537e7	−0.682447	−0.341223	+	0.939982i	$0.610841\pi$
$128$	−1.65006e7	−0.695449
$129$	−627623.	−0.0257416
$130$	−480505.	−0.0191821
$131$	−1.23671e7	−0.480639	−0.240320	−	0.970694i	$-0.577252\pi$
$131$	−1.23671e7	−0.480639	−0.240320	+	0.970694i	$0.577252\pi$
$132$	−6.54167e7	−2.47560
$133$	−8.99167e6	−0.331406
$134$	7.36865e6	0.264558
$135$	1.15748e6	0.0404897
$136$	−1.71124e7	−0.583344
$137$	1.65308e7	0.549254	0.274627	−	0.961551i	$-0.411446\pi$
$137$	1.65308e7	0.549254	0.274627	+	0.961551i	$0.411446\pi$
$138$	1.45686e7	0.471892
$139$	1.56754e7	0.495070	0.247535	−	0.968879i	$-0.420380\pi$
$139$	1.56754e7	0.495070	0.247535	+	0.968879i	$0.420380\pi$
$140$	−5.34227e6	−0.164542
$141$	7.21652e6	0.216801
$142$	−3.89595e6	−0.114184
$143$	−4.66333e7	−1.33358
$144$	3.86475e7	1.07858
$145$	−2.02062e6	−0.0550423
$146$	−1.85882e6	−0.0494313
$147$	−4.11245e7	−1.06780
$148$	6.57610e7	1.66745
$149$	−2.42495e7	−0.600552	−0.300276	−	0.953852i	$-0.597079\pi$
$149$	−2.42495e7	−0.600552	−0.300276	+	0.953852i	$0.597079\pi$
$150$	1.15274e7	0.278876
$151$	−7.61645e6	−0.180025	−0.0900126	−	0.995941i	$-0.528691\pi$
$151$	−7.61645e6	−0.180025	−0.0900126	+	0.995941i	$0.528691\pi$
$152$	−4.10230e6	−0.0947491
$153$	8.33750e7	1.88198
$154$	1.96024e7	0.432500
$155$	4.87350e6	0.105119
$156$	5.23916e7	1.10491
$157$	−2.09205e7	−0.431443	−0.215722	−	0.976455i	$-0.569210\pi$
$157$	−2.09205e7	−0.431443	−0.215722	+	0.976455i	$0.569210\pi$
$158$	1.62785e7	0.328334
$159$	6.20593e7	1.22438
$160$	−3.67859e6	−0.0710006
$161$	1.15466e8	2.18054
$162$	−7.71685e6	−0.142606
$163$	8.68280e7	1.57037	0.785187	−	0.619259i	$-0.212567\pi$
$163$	8.68280e7	1.57037	0.785187	+	0.619259i	$0.212567\pi$
$164$	−8.56857e7	−1.51689
$165$	−1.93117e7	−0.334678
$166$	1.64042e7	0.278340
$167$	−4.26264e7	−0.708224	−0.354112	−	0.935203i	$-0.615217\pi$
$167$	−4.26264e7	−0.708224	−0.354112	+	0.935203i	$0.615217\pi$
$168$	−4.48784e7	−0.730222
$169$	−2.54004e7	−0.404797
$170$	−2.47902e6	−0.0386998
$171$	1.99872e7	0.305679
$172$	−1.11367e6	−0.0166881
$173$	4.24371e7	0.623138	0.311569	−	0.950224i	$-0.399145\pi$
$173$	4.24371e7	0.623138	0.311569	+	0.950224i	$0.399145\pi$
$174$	−8.32978e6	−0.119870
$175$	9.13619e7	1.28864
$176$	−1.11523e8	−1.54195
$177$	−2.32907e7	−0.315701
$178$	−315590.	−0.00419424
$179$	−8.19247e6	−0.106765	−0.0533825	−	0.998574i	$-0.517000\pi$
$179$	−8.19247e6	−0.106765	−0.0533825	+	0.998574i	$0.517000\pi$
$180$	1.18751e7	0.151769
$181$	−2.32497e7	−0.291435	−0.145717	−	0.989326i	$-0.546549\pi$
$181$	−2.32497e7	−0.291435	−0.145717	+	0.989326i	$0.546549\pi$
$182$	−1.56993e7	−0.193033
$183$	−3.19374e7	−0.385230
$184$	5.26794e7	0.623417
$185$	1.94133e7	0.225423
$186$	2.00905e7	0.228926
$187$	−2.40590e8	−2.69049
$188$	1.28052e7	0.140551
$189$	3.78178e7	0.407455
$190$	−594287.	−0.00628577
$191$	−3.42698e7	−0.355873	−0.177937	−	0.984042i	$-0.556942\pi$
$191$	−3.42698e7	−0.355873	−0.177937	+	0.984042i	$0.556942\pi$
$192$	1.14866e8	1.17122
$193$	1.15197e8	1.15343	0.576714	−	0.816946i	$-0.304335\pi$
$193$	1.15197e8	1.15343	0.576714	+	0.816946i	$0.304335\pi$
$194$	2.69750e6	0.0265250
$195$	1.54666e7	0.149373
$196$	−7.29724e7	−0.692249
$197$	−1.27276e7	−0.118608	−0.0593042	−	0.998240i	$-0.518888\pi$
$197$	−1.27276e7	−0.118608	−0.0593042	+	0.998240i	$0.518888\pi$
$198$	−4.35733e7	−0.398926
$199$	−791355.	−0.00711846	−0.00355923	−	0.999994i	$-0.501133\pi$
$199$	−791355.	−0.00711846	−0.00355923	+	0.999994i	$0.501133\pi$
$200$	4.16824e7	0.368424
$201$	−2.37183e8	−2.06014
$202$	−2.26230e7	−0.193117
$203$	−6.60189e7	−0.553901
$204$	2.70298e8	2.22914
$205$	−2.52953e7	−0.205070
$206$	3.77507e7	0.300878
$207$	−2.56664e8	−2.01127
$208$	8.93174e7	0.688200
$209$	−5.76758e7	−0.437000
$210$	−6.50139e6	−0.0484439
$211$	−2.80618e7	−0.205649	−0.102825	−	0.994699i	$-0.532788\pi$
$211$	−2.80618e7	−0.205649	−0.102825	+	0.994699i	$0.532788\pi$
$212$	1.10120e8	0.793761
$213$	1.25403e8	0.889162
$214$	2.50185e6	0.0174507
$215$	−328768.	−0.00225608
$216$	1.72537e7	0.116492
$217$	1.59230e8	1.05783
$218$	−4.97253e7	−0.325072
$219$	5.98319e7	0.384927
$220$	−3.42672e7	−0.216970
$221$	1.92686e8	1.20082
$222$	8.00293e7	0.490923
$223$	2.64158e8	1.59514	0.797568	−	0.603230i	$-0.206120\pi$
$223$	2.64158e8	1.59514	0.797568	+	0.603230i	$0.206120\pi$
$224$	−1.20189e8	−0.714491
$225$	−2.03085e8	−1.18861
$226$	3.33375e7	0.192112
$227$	−2.35421e8	−1.33584	−0.667921	−	0.744232i	$-0.732816\pi$
$227$	−2.35421e8	−1.33584	−0.667921	+	0.744232i	$0.732816\pi$
$228$	6.47977e7	0.362066
$229$	−1.49149e8	−0.820724	−0.410362	−	0.911923i	$-0.634598\pi$
$229$	−1.49149e8	−0.820724	−0.410362	+	0.911923i	$0.634598\pi$
$230$	7.63149e6	0.0413582
$231$	−6.30963e8	−3.36792
$232$	−3.01200e7	−0.158361
$233$	−3.34112e8	−1.73040	−0.865201	−	0.501425i	$-0.832809\pi$
$233$	−3.34112e8	−1.73040	−0.865201	+	0.501425i	$0.832809\pi$
$234$	3.48974e7	0.178048
$235$	3.78023e6	0.0190012
$236$	−4.13276e7	−0.204667
$237$	−5.23975e8	−2.55677
$238$	−8.09959e7	−0.389443
$239$	1.06998e8	0.506971	0.253485	−	0.967339i	$-0.418423\pi$
$239$	1.06998e8	0.506971	0.253485	+	0.967339i	$0.418423\pi$
$240$	3.69880e7	0.172712
$241$	1.28460e8	0.591165	0.295583	−	0.955317i	$-0.404486\pi$
$241$	1.28460e8	0.591165	0.295583	+	0.955317i	$0.404486\pi$
$242$	8.36555e7	0.379438
$243$	3.17915e8	1.42131
$244$	−5.66706e7	−0.249743
$245$	−2.15422e7	−0.0935857
$246$	−1.04277e8	−0.446597
$247$	4.61920e7	0.195042
$248$	7.26460e7	0.302434
$249$	−5.28020e8	−2.16747
$250$	1.21810e7	0.0493053
$251$	−5.60087e7	−0.223562	−0.111781	−	0.993733i	$-0.535655\pi$
$251$	−5.60087e7	−0.223562	−0.111781	+	0.993733i	$0.535655\pi$
$252$	3.87990e8	1.52728
$253$	7.40640e8	2.87532
$254$	−3.40189e7	−0.130257
$255$	7.97949e7	0.301359
$256$	1.75896e8	0.655264
$257$	3.62340e8	1.33153	0.665763	−	0.746163i	$-0.268106\pi$
$257$	3.62340e8	1.33153	0.665763	+	0.746163i	$0.268106\pi$
$258$	−1.35531e6	−0.00491325
$259$	6.34284e8	2.26848
$260$	2.74443e7	0.0968379
$261$	1.46751e8	0.510902
$262$	−2.67059e7	−0.0917388
$263$	1.96680e7	0.0666675	0.0333338	−	0.999444i	$-0.489388\pi$
$263$	1.96680e7	0.0666675	0.0333338	+	0.999444i	$0.489388\pi$
$264$	−2.87866e8	−0.962891
$265$	3.25085e7	0.107309
$266$	−1.94169e7	−0.0632548
$267$	1.01583e7	0.0326610
$268$	−4.20864e8	−1.33558
$269$	−5.14123e8	−1.61040	−0.805200	−	0.593004i	$-0.797942\pi$
$269$	−5.14123e8	−1.61040	−0.805200	+	0.593004i	$0.797942\pi$
$270$	2.49949e6	0.00772820
$271$	9.08702e7	0.277351	0.138675	−	0.990338i	$-0.455716\pi$
$271$	9.08702e7	0.277351	0.138675	+	0.990338i	$0.455716\pi$
$272$	4.60806e8	1.38844
$273$	5.05332e8	1.50317
$274$	3.56972e7	0.104835
$275$	5.86028e8	1.69924
$276$	−8.32096e8	−2.38227
$277$	4.49211e8	1.26991	0.634953	−	0.772551i	$-0.281019\pi$
$277$	4.49211e8	1.26991	0.634953	+	0.772551i	$0.281019\pi$
$278$	3.38499e7	0.0944932
$279$	−3.53945e8	−0.975711
$280$	−2.35087e7	−0.0639992
$281$	6.13513e7	0.164950	0.0824749	−	0.996593i	$-0.473718\pi$
$281$	6.13513e7	0.164950	0.0824749	+	0.996593i	$0.473718\pi$
$282$	1.55836e7	0.0413804
$283$	5.32194e8	1.39578	0.697891	−	0.716204i	$-0.254122\pi$
$283$	5.32194e8	1.39578	0.697891	+	0.716204i	$0.254122\pi$
$284$	2.22519e8	0.576439
$285$	1.91290e7	0.0489480
$286$	−1.00701e8	−0.254539
$287$	−8.26463e8	−2.06365
$288$	2.67163e8	0.659027
$289$	5.83766e8	1.42264
$290$	−4.36339e6	−0.0105058
$291$	−8.68276e7	−0.206553
$292$	1.06167e8	0.249546
$293$	−2.22346e8	−0.516407	−0.258203	−	0.966091i	$-0.583130\pi$
$293$	−2.22346e8	−0.516407	−0.258203	+	0.966091i	$0.583130\pi$
$294$	−8.88054e7	−0.203809
$295$	−1.22003e7	−0.0276691
$296$	2.89381e8	0.648559
$297$	2.42577e8	0.537281
$298$	−5.23651e7	−0.114626
$299$	−5.93171e8	−1.28331
$300$	−6.58392e8	−1.40786
$301$	−1.07417e7	−0.0227034
$302$	−1.64472e7	−0.0343611
$303$	7.28190e8	1.50382
$304$	1.10467e8	0.225516
$305$	−1.67297e7	−0.0337629
$306$	1.80042e8	0.359211
$307$	1.05630e8	0.208355	0.104177	−	0.994559i	$-0.466779\pi$
$307$	1.05630e8	0.208355	0.104177	+	0.994559i	$0.466779\pi$
$308$	−1.11960e9	−2.18341
$309$	−1.21512e9	−2.34297
$310$	1.05240e7	0.0200638
$311$	9.42851e7	0.177739	0.0888693	−	0.996043i	$-0.471675\pi$
$311$	9.42851e7	0.177739	0.0888693	+	0.996043i	$0.471675\pi$
$312$	2.30549e8	0.429757
$313$	2.74750e8	0.506446	0.253223	−	0.967408i	$-0.418509\pi$
$313$	2.74750e8	0.506446	0.253223	+	0.967408i	$0.418509\pi$
$314$	−4.51764e7	−0.0823489
$315$	1.14539e8	0.206474
$316$	−9.29756e8	−1.65754
$317$	−1.67438e8	−0.295220	−0.147610	−	0.989046i	$-0.547158\pi$
$317$	−1.67438e8	−0.295220	−0.147610	+	0.989046i	$0.547158\pi$
$318$	1.34013e8	0.233696
$319$	−4.23469e8	−0.730389
$320$	6.01704e7	0.102650
$321$	−8.05299e7	−0.135891
$322$	2.49341e8	0.416195
$323$	2.38313e8	0.393495
$324$	4.40752e8	0.719924
$325$	−4.69344e8	−0.758403
$326$	1.87499e8	0.299735
$327$	1.60056e9	2.53137
$328$	−3.77060e8	−0.590000
$329$	1.23510e8	0.191212
$330$	−4.17023e7	−0.0638794
$331$	−2.52949e8	−0.383385	−0.191692	−	0.981455i	$-0.561398\pi$
$331$	−2.52949e8	−0.383385	−0.191692	+	0.981455i	$0.561398\pi$
$332$	−9.36933e8	−1.40516
$333$	−1.40992e9	−2.09238
$334$	−9.20487e7	−0.135178
$335$	−1.24244e8	−0.180558
$336$	1.20849e9	1.73803
$337$	−1.04815e9	−1.49184	−0.745918	−	0.666038i	$-0.767989\pi$
$337$	−1.04815e9	−1.49184	−0.745918	+	0.666038i	$0.767989\pi$
$338$	−5.48504e7	−0.0772629
$339$	−1.07307e9	−1.49599
$340$	1.41590e8	0.195370
$341$	1.02136e9	1.39488
$342$	4.31610e7	0.0583445
$343$	2.75863e8	0.369116
$344$	−4.90072e6	−0.00649091
$345$	−2.45643e8	−0.322061
$346$	9.16399e7	0.118937
$347$	−9.49757e8	−1.22028	−0.610140	−	0.792294i	$-0.708887\pi$
$347$	−9.49757e8	−1.22028	−0.610140	+	0.792294i	$0.708887\pi$
$348$	4.75759e8	0.605146
$349$	6.65286e8	0.837760	0.418880	−	0.908042i	$-0.362423\pi$
$349$	6.65286e8	0.837760	0.418880	+	0.908042i	$0.362423\pi$
$350$	1.97290e8	0.245961
$351$	−1.94277e8	−0.239799
$352$	−7.70936e8	−0.942148
$353$	1.54255e9	1.86650	0.933249	−	0.359231i	$-0.116961\pi$
$353$	1.54255e9	1.86650	0.933249	+	0.359231i	$0.116961\pi$
$354$	−5.02945e7	−0.0602573
$355$	6.56901e7	0.0779292
$356$	1.80251e7	0.0211740
$357$	2.60711e9	3.03263
$358$	−1.76911e7	−0.0203781
$359$	3.93118e8	0.448427	0.224214	−	0.974540i	$-0.428019\pi$
$359$	3.93118e8	0.448427	0.224214	+	0.974540i	$0.428019\pi$
$360$	5.22564e7	0.0590311
$361$	−8.36742e8	−0.936087
$362$	−5.02060e7	−0.0556257
$363$	−2.69271e9	−2.95472
$364$	8.96676e8	0.974497
$365$	3.13417e7	0.0337363
$366$	−6.89665e7	−0.0735282
$367$	−7.90351e8	−0.834620	−0.417310	−	0.908764i	$-0.637027\pi$
$367$	−7.90351e8	−0.834620	−0.417310	+	0.908764i	$0.637027\pi$
$368$	−1.41856e9	−1.48382
$369$	1.83711e9	1.90346
$370$	4.19217e7	0.0430262
$371$	1.06214e9	1.07987
$372$	−1.14748e9	−1.15570
$373$	−7.56336e8	−0.754629	−0.377315	−	0.926085i	$-0.623152\pi$
$373$	−7.56336e8	−0.754629	−0.377315	+	0.926085i	$0.623152\pi$
$374$	−5.19537e8	−0.513530
$375$	−3.92084e8	−0.383945
$376$	5.63493e7	0.0546678
$377$	3.39152e8	0.325987
$378$	8.16647e7	0.0777702
$379$	−5.24042e8	−0.494458	−0.247229	−	0.968957i	$-0.579520\pi$
$379$	−5.24042e8	−0.494458	−0.247229	+	0.968957i	$0.579520\pi$
$380$	3.39430e7	0.0317327
$381$	1.09501e9	1.01433
$382$	−7.40033e7	−0.0679249
$383$	8.44774e8	0.768325	0.384162	−	0.923266i	$-0.374490\pi$
$383$	8.44774e8	0.768325	0.384162	+	0.923266i	$0.374490\pi$
$384$	1.14692e9	1.03366
$385$	−3.30517e8	−0.295176
$386$	2.48760e8	0.220153
$387$	2.38773e7	0.0209409
$388$	−1.54069e8	−0.133908
$389$	4.32014e8	0.372112	0.186056	−	0.982539i	$-0.440429\pi$
$389$	4.32014e8	0.372112	0.186056	+	0.982539i	$0.440429\pi$
$390$	3.33989e7	0.0285106
$391$	−3.06029e9	−2.58907
$392$	−3.21115e8	−0.269253
$393$	8.59613e8	0.714380
$394$	−2.74844e7	−0.0226386
$395$	−2.74474e8	−0.224084
$396$	2.48871e9	2.01391
$397$	2.02282e9	1.62252	0.811259	−	0.584687i	$-0.198783\pi$
$397$	2.02282e9	1.62252	0.811259	+	0.584687i	$0.198783\pi$
$398$	−1.70888e6	−0.00135869
$399$	6.24993e8	0.492572
$400$	−1.12243e9	−0.876898
$401$	1.78796e8	0.138469	0.0692346	−	0.997600i	$-0.477944\pi$
$401$	1.78796e8	0.138469	0.0692346	+	0.997600i	$0.477944\pi$
$402$	−5.12180e8	−0.393216
$403$	−8.17995e8	−0.622563
$404$	1.29212e9	0.974919
$405$	1.30115e8	0.0973270
$406$	−1.42563e8	−0.105722
$407$	4.06852e9	2.99128
$408$	1.18945e9	0.867032
$409$	−1.74004e8	−0.125755	−0.0628777	−	0.998021i	$-0.520028\pi$
$409$	−1.74004e8	−0.125755	−0.0628777	+	0.998021i	$0.520028\pi$
$410$	−5.46235e7	−0.0391413
$411$	−1.14903e9	−0.816363
$412$	−2.15615e9	−1.51893
$413$	−3.98617e8	−0.278439
$414$	−5.54248e8	−0.383887
$415$	−2.76592e8	−0.189964
$416$	6.17435e8	0.420499
$417$	−1.08956e9	−0.735828
$418$	−1.24547e8	−0.0834096
$419$	−5.89618e8	−0.391581	−0.195790	−	0.980646i	$-0.562727\pi$
$419$	−5.89618e8	−0.391581	−0.195790	+	0.980646i	$0.562727\pi$
$420$	3.71330e8	0.244561
$421$	2.77275e9	1.81102	0.905510	−	0.424324i	$-0.139488\pi$
$421$	2.77275e9	1.81102	0.905510	+	0.424324i	$0.139488\pi$
$422$	−6.05976e7	−0.0392520
$423$	−2.74545e8	−0.176369
$424$	4.84582e8	0.308736
$425$	−2.42144e9	−1.53007
$426$	2.70800e8	0.169713
$427$	−5.46604e8	−0.339762
$428$	−1.42895e8	−0.0880973
$429$	3.24138e9	1.98212
$430$	−709951.	−0.000430615 0
$431$	−2.59603e9	−1.56185	−0.780924	−	0.624627i	$-0.785251\pi$
$431$	−2.59603e9	−1.56185	−0.780924	+	0.624627i	$0.785251\pi$
$432$	−4.64611e8	−0.277266
$433$	2.37138e9	1.40376	0.701881	−	0.712294i	$-0.252344\pi$
$433$	2.37138e9	1.40376	0.701881	+	0.712294i	$0.252344\pi$
$434$	3.43846e8	0.201906
$435$	1.40449e8	0.0818101
$436$	2.84008e9	1.64108
$437$	−7.33632e8	−0.420526
$438$	1.29203e8	0.0734704
$439$	−1.09900e9	−0.619971	−0.309986	−	0.950741i	$-0.600324\pi$
$439$	−1.09900e9	−0.619971	−0.309986	+	0.950741i	$0.600324\pi$
$440$	−1.50793e8	−0.0843912
$441$	1.56454e9	0.868662
$442$	4.16092e8	0.229198
$443$	6.57647e8	0.359402	0.179701	−	0.983721i	$-0.442487\pi$
$443$	6.57647e8	0.359402	0.179701	+	0.983721i	$0.442487\pi$
$444$	−4.57091e9	−2.47835
$445$	5.32120e6	0.00286253
$446$	5.70431e8	0.304461
$447$	1.68553e9	0.892608
$448$	1.96592e9	1.03298
$449$	−2.43931e9	−1.27176	−0.635880	−	0.771788i	$-0.719363\pi$
$449$	−2.43931e9	−1.27176	−0.635880	+	0.771788i	$0.719363\pi$
$450$	−4.38547e8	−0.226867
$451$	−5.30123e9	−2.72119
$452$	−1.90409e9	−0.969846
$453$	5.29404e8	0.267574
$454$	−5.08375e8	−0.254970
$455$	2.64708e8	0.131743
$456$	2.85143e8	0.140827
$457$	3.09201e9	1.51543	0.757713	−	0.652588i	$-0.226317\pi$
$457$	3.09201e9	1.51543	0.757713	+	0.652588i	$0.226317\pi$
$458$	−3.22077e8	−0.156650
$459$	−1.00231e9	−0.483793
$460$	−4.35877e8	−0.208791
$461$	2.32618e9	1.10583	0.552917	−	0.833236i	$-0.313515\pi$
$461$	2.32618e9	1.10583	0.552917	+	0.833236i	$0.313515\pi$
$462$	−1.36252e9	−0.642830
$463$	3.82398e9	1.79053	0.895267	−	0.445530i	$-0.146985\pi$
$463$	3.82398e9	1.79053	0.895267	+	0.445530i	$0.146985\pi$
$464$	8.11077e8	0.376920
$465$	−3.38747e8	−0.156239
$466$	−7.21492e8	−0.330279
$467$	−3.30404e9	−1.50119	−0.750597	−	0.660761i	$-0.770234\pi$
$467$	−3.30404e9	−1.50119	−0.750597	+	0.660761i	$0.770234\pi$
$468$	−1.99318e9	−0.898849
$469$	−4.05935e9	−1.81699
$470$	8.16314e6	0.00362673
$471$	1.45414e9	0.641260
$472$	−1.81862e8	−0.0796059
$473$	−6.89011e7	−0.0299373
$474$	−1.13149e9	−0.488006
$475$	−5.80483e8	−0.248520
$476$	4.62612e9	1.96604
$477$	−2.36098e9	−0.996042
$478$	2.31054e8	0.0967646
$479$	6.60692e7	0.0274678	0.0137339	−	0.999906i	$-0.495628\pi$
$479$	6.60692e7	0.0274678	0.0137339	+	0.999906i	$0.495628\pi$
$480$	2.55691e8	0.105529
$481$	−3.25844e9	−1.33506
$482$	2.77401e8	0.112835
$483$	−8.02580e9	−3.24096
$484$	−4.77802e9	−1.91553
$485$	−4.54829e7	−0.0181031
$486$	6.86515e8	0.271283
$487$	2.90864e9	1.14114	0.570570	−	0.821249i	$-0.306722\pi$
$487$	2.90864e9	1.14114	0.570570	+	0.821249i	$0.306722\pi$
$488$	−2.49379e8	−0.0971383
$489$	−6.03523e9	−2.33407
$490$	−4.65189e7	−0.0178625
$491$	3.05841e9	1.16603	0.583017	−	0.812460i	$-0.301872\pi$
$491$	3.05841e9	1.16603	0.583017	+	0.812460i	$0.301872\pi$
$492$	5.95584e9	2.25458
$493$	1.74975e9	0.657676
$494$	9.97484e7	0.0372273
$495$	7.34693e8	0.272262
$496$	−1.95622e9	−0.719834
$497$	2.14626e9	0.784216
$498$	−1.14022e9	−0.413701
$499$	−2.69474e9	−0.970880	−0.485440	−	0.874270i	$-0.661341\pi$
$499$	−2.69474e9	−0.970880	−0.485440	+	0.874270i	$0.661341\pi$
$500$	−6.95724e8	−0.248910
$501$	2.96287e9	1.05264
$502$	−1.20947e8	−0.0426709
$503$	2.83909e8	0.0994699	0.0497350	−	0.998762i	$-0.484162\pi$
$503$	2.83909e8	0.0994699	0.0497350	+	0.998762i	$0.484162\pi$
$504$	1.70735e9	0.594040
$505$	3.81448e8	0.131800
$506$	1.59936e9	0.548807
$507$	1.76553e9	0.601654
$508$	1.94301e9	0.657585
$509$	5.69582e8	0.191445	0.0957225	−	0.995408i	$-0.469484\pi$
$509$	5.69582e8	0.191445	0.0957225	+	0.995408i	$0.469484\pi$
$510$	1.72312e8	0.0575200
$511$	1.02402e9	0.339495
$512$	2.49192e9	0.820519
$513$	−2.40281e8	−0.0785795
$514$	7.82447e8	0.254146
$515$	−6.36519e8	−0.205346
$516$	7.74091e7	0.0248038
$517$	7.92237e8	0.252138
$518$	1.36969e9	0.432980
$519$	−2.94972e9	−0.926178
$520$	1.20769e8	0.0376654
$521$	3.83385e9	1.18769	0.593845	−	0.804580i	$-0.297609\pi$
$521$	3.83385e9	1.18769	0.593845	+	0.804580i	$0.297609\pi$
$522$	3.16897e8	0.0975151
$523$	−3.07751e9	−0.940684	−0.470342	−	0.882484i	$-0.655869\pi$
$523$	−3.07751e9	−0.940684	−0.470342	+	0.882484i	$0.655869\pi$
$524$	1.52532e9	0.463129
$525$	−6.35038e9	−1.91532
$526$	4.24716e7	0.0127247
$527$	−4.22019e9	−1.25602
$528$	7.75171e9	2.29181
$529$	6.01606e9	1.76692
$530$	7.01998e7	0.0204819
$531$	8.86068e8	0.256824
$532$	1.10900e9	0.319332
$533$	4.24571e9	1.21452
$534$	2.19360e7	0.00623396
$535$	−4.21840e7	−0.0119099
$536$	−1.85201e9	−0.519478
$537$	5.69442e8	0.158686
$538$	−1.11021e9	−0.307374
$539$	−4.51468e9	−1.24184
$540$	−1.42760e8	−0.0390146
$541$	−5.23400e7	−0.0142116	−0.00710580	−	0.999975i	$-0.502262\pi$
$541$	−5.23400e7	−0.0142116	−0.00710580	+	0.999975i	$0.502262\pi$
$542$	1.96228e8	0.0529375
$543$	1.61604e9	0.433163
$544$	3.18547e9	0.848353
$545$	8.38423e8	0.221858
$546$	1.09123e9	0.286907
$547$	−6.19021e9	−1.61715	−0.808574	−	0.588394i	$-0.799760\pi$
$547$	−6.19021e9	−1.61715	−0.808574	+	0.588394i	$0.799760\pi$
$548$	−2.03886e9	−0.529244
$549$	1.21502e9	0.313387
$550$	1.26549e9	0.324331
$551$	4.19462e8	0.106822
$552$	−3.66164e9	−0.926592
$553$	−8.96776e9	−2.25500
$554$	9.70040e8	0.242385
$555$	−1.34938e9	−0.335050
$556$	−1.93335e9	−0.477034
$557$	−6.60405e9	−1.61926	−0.809632	−	0.586938i	$-0.800333\pi$
$557$	−6.60405e9	−1.61926	−0.809632	+	0.586938i	$0.800333\pi$
$558$	−7.64320e8	−0.186232
$559$	5.51822e7	0.0133616
$560$	6.33045e8	0.152327
$561$	1.67229e10	3.99891
$562$	1.32484e8	0.0314837
$563$	−1.72734e9	−0.407941	−0.203971	−	0.978977i	$-0.565385\pi$
$563$	−1.72734e9	−0.407941	−0.203971	+	0.978977i	$0.565385\pi$
$564$	−8.90063e8	−0.208903
$565$	−5.62107e8	−0.131114
$566$	1.14924e9	0.266411
$567$	4.25118e9	0.979419
$568$	9.79197e8	0.224208
$569$	−4.86630e8	−0.110740	−0.0553702	−	0.998466i	$-0.517634\pi$
$569$	−4.86630e8	−0.110740	−0.0553702	+	0.998466i	$0.517634\pi$
$570$	4.13077e7	0.00934262
$571$	6.23981e9	1.40264	0.701318	−	0.712848i	$-0.252595\pi$
$571$	6.23981e9	1.40264	0.701318	+	0.712848i	$0.252595\pi$
$572$	5.75160e9	1.28500
$573$	2.38203e9	0.528939
$574$	−1.78469e9	−0.393886
$575$	7.45423e9	1.63518
$576$	−4.36996e9	−0.952794
$577$	−2.77623e9	−0.601644	−0.300822	−	0.953680i	$-0.597261\pi$
$577$	−2.77623e9	−0.601644	−0.300822	+	0.953680i	$0.597261\pi$
$578$	1.26060e9	0.271538
$579$	−8.00711e9	−1.71436
$580$	2.49217e8	0.0530371
$581$	−9.03699e9	−1.91164
$582$	−1.87498e8	−0.0394245
$583$	6.81293e9	1.42395
$584$	4.67190e8	0.0970618
$585$	−5.88408e8	−0.121516
$586$	−4.80140e8	−0.0985657
$587$	−4.82309e8	−0.0984221	−0.0492110	−	0.998788i	$-0.515671\pi$
$587$	−4.82309e8	−0.0984221	−0.0492110	+	0.998788i	$0.515671\pi$
$588$	5.07216e9	1.02890
$589$	−1.01169e9	−0.204007
$590$	−2.63458e7	−0.00528116
$591$	8.84671e8	0.176289
$592$	−7.79251e9	−1.54366
$593$	2.49409e9	0.491158	0.245579	−	0.969377i	$-0.421022\pi$
$593$	2.49409e9	0.491158	0.245579	+	0.969377i	$0.421022\pi$
$594$	5.23827e8	0.102550
$595$	1.36568e9	0.265791
$596$	2.99086e9	0.578674
$597$	5.50055e7	0.0105802
$598$	−1.28091e9	−0.244943
$599$	5.56502e8	0.105797	0.0528985	−	0.998600i	$-0.483154\pi$
$599$	5.56502e8	0.105797	0.0528985	+	0.998600i	$0.483154\pi$
$600$	−2.89726e9	−0.547592
$601$	−8.43542e9	−1.58506	−0.792531	−	0.609832i	$-0.791237\pi$
$601$	−8.43542e9	−1.58506	−0.792531	+	0.609832i	$0.791237\pi$
$602$	−2.31959e7	−0.00433335
$603$	9.02337e9	1.67594
$604$	9.39389e8	0.173467
$605$	−1.41052e9	−0.258962
$606$	1.57248e9	0.287032
$607$	6.83608e9	1.24064	0.620321	−	0.784348i	$-0.287002\pi$
$607$	6.83608e9	1.24064	0.620321	+	0.784348i	$0.287002\pi$
$608$	7.63641e8	0.137793
$609$	4.58884e9	0.823270
$610$	−3.61267e7	−0.00644427
$611$	−6.34495e8	−0.112534
$612$	−1.02832e10	−1.81342
$613$	3.67761e9	0.644842	0.322421	−	0.946596i	$-0.395503\pi$
$613$	3.67761e9	0.644842	0.322421	+	0.946596i	$0.395503\pi$
$614$	2.28101e8	0.0397683
$615$	1.75823e9	0.304798
$616$	−4.92680e9	−0.849243
$617$	−1.53685e8	−0.0263411	−0.0131705	−	0.999913i	$-0.504192\pi$
$617$	−1.53685e8	−0.0263411	−0.0131705	+	0.999913i	$0.504192\pi$
$618$	−2.62397e9	−0.447198
$619$	4.38503e9	0.743114	0.371557	−	0.928410i	$-0.378824\pi$
$619$	4.38503e9	0.743114	0.371557	+	0.928410i	$0.378824\pi$
$620$	−6.01082e8	−0.101289
$621$	3.08556e9	0.517027
$622$	2.03602e8	0.0339247
$623$	1.73857e8	0.0288061
$624$	−6.20827e9	−1.02288
$625$	5.79456e9	0.949381
$626$	5.93304e8	0.0966645
$627$	4.00893e9	0.649519
$628$	2.58027e9	0.415726
$629$	−1.68109e10	−2.69348
$630$	2.47338e8	0.0394094
$631$	−5.10841e8	−0.0809436	−0.0404718	−	0.999181i	$-0.512886\pi$
$631$	−5.10841e8	−0.0809436	−0.0404718	+	0.999181i	$0.512886\pi$
$632$	−4.09139e9	−0.644706
$633$	1.95052e9	0.305659
$634$	−3.61570e8	−0.0563482
$635$	5.73597e8	0.0888993
$636$	−7.65420e9	−1.17978
$637$	3.61577e9	0.554258
$638$	−9.14451e8	−0.139408
$639$	−4.77084e9	−0.723339
$640$	6.00793e8	0.0905931
$641$	−3.65945e9	−0.548798	−0.274399	−	0.961616i	$-0.588479\pi$
$641$	−3.65945e9	−0.548798	−0.274399	+	0.961616i	$0.588479\pi$
$642$	−1.73899e8	−0.0259372
$643$	−2.47340e9	−0.366906	−0.183453	−	0.983028i	$-0.558728\pi$
$643$	−2.47340e9	−0.366906	−0.183453	+	0.983028i	$0.558728\pi$
$644$	−1.42412e10	−2.10110
$645$	2.28520e7	0.00335324
$646$	5.14621e8	0.0751058
$647$	1.00782e10	1.46292	0.731458	−	0.681886i	$-0.238840\pi$
$647$	1.00782e10	1.46292	0.731458	+	0.681886i	$0.238840\pi$
$648$	1.93953e9	0.280017
$649$	−2.55687e9	−0.367157
$650$	−1.01352e9	−0.144755
$651$	−1.10677e10	−1.57226
$652$	−1.07091e10	−1.51316
$653$	−3.62017e9	−0.508783	−0.254391	−	0.967101i	$-0.581875\pi$
$653$	−3.62017e9	−0.508783	−0.254391	+	0.967101i	$0.581875\pi$
$654$	3.45630e9	0.483158
$655$	4.50291e8	0.0626107
$656$	1.01535e10	1.40428
$657$	−2.27624e9	−0.313141
$658$	2.66711e8	0.0364964
$659$	−2.47848e9	−0.337355	−0.168677	−	0.985671i	$-0.553950\pi$
$659$	−2.47848e9	−0.337355	−0.168677	+	0.985671i	$0.553950\pi$
$660$	2.38185e9	0.322485
$661$	−3.52073e9	−0.474162	−0.237081	−	0.971490i	$-0.576191\pi$
$661$	−3.52073e9	−0.474162	−0.237081	+	0.971490i	$0.576191\pi$
$662$	−5.46225e8	−0.0731760
$663$	−1.33932e10	−1.78479
$664$	−4.12297e9	−0.546541
$665$	3.27390e8	0.0431707
$666$	−3.04463e9	−0.399369
$667$	−5.38649e9	−0.702855
$668$	5.25741e9	0.682423
$669$	−1.83611e10	−2.37087
$670$	−2.68295e8	−0.0344628
$671$	−3.50611e9	−0.448020
$672$	8.35410e9	1.06196
$673$	9.71331e9	1.22833	0.614164	−	0.789178i	$-0.289493\pi$
$673$	9.71331e9	1.22833	0.614164	+	0.789178i	$0.289493\pi$
$674$	−2.26342e9	−0.284744
$675$	2.44143e9	0.305550
$676$	3.13281e9	0.390050
$677$	1.10185e10	1.36478	0.682389	−	0.730989i	$-0.260941\pi$
$677$	1.10185e10	1.36478	0.682389	+	0.730989i	$0.260941\pi$
$678$	−2.31722e9	−0.285538
$679$	−1.48604e9	−0.182174
$680$	6.23069e8	0.0759897
$681$	1.63636e10	1.98548
$682$	2.20555e9	0.266239
$683$	−3.08194e9	−0.370128	−0.185064	−	0.982726i	$-0.559249\pi$
$683$	−3.08194e9	−0.370128	−0.185064	+	0.982726i	$0.559249\pi$
$684$	−2.46516e9	−0.294543
$685$	−6.01894e8	−0.0715489
$686$	5.95706e8	0.0704527
$687$	1.03671e10	1.21985
$688$	1.31967e8	0.0154492
$689$	−5.45641e9	−0.635535
$690$	−5.30449e8	−0.0614713
$691$	3.95018e9	0.455453	0.227727	−	0.973725i	$-0.426871\pi$
$691$	3.95018e9	0.455453	0.227727	+	0.973725i	$0.426871\pi$
$692$	−5.23406e9	−0.600437
$693$	2.40043e10	2.73983
$694$	−2.05093e9	−0.232913
$695$	−5.70746e8	−0.0644906
$696$	2.09358e9	0.235373
$697$	2.19044e10	2.45029
$698$	1.43664e9	0.159902
$699$	2.32235e10	2.57192
$700$	−1.12683e10	−1.24170
$701$	−1.60069e10	−1.75507	−0.877534	−	0.479515i	$-0.840813\pi$
$701$	−1.60069e10	−1.75507	−0.877534	+	0.479515i	$0.840813\pi$
$702$	−4.19528e8	−0.0457700
$703$	−4.03003e9	−0.437486
$704$	1.26101e10	1.36212
$705$	−2.62756e8	−0.0282417
$706$	3.33102e9	0.356255
$707$	1.24629e10	1.32633
$708$	2.87260e9	0.304199
$709$	−8.29019e9	−0.873580	−0.436790	−	0.899564i	$-0.643885\pi$
$709$	−8.29019e9	−0.873580	−0.436790	+	0.899564i	$0.643885\pi$
$710$	1.41853e8	0.0148742
$711$	1.99341e10	2.07995
$712$	7.93195e7	0.00823569
$713$	1.29916e10	1.34230
$714$	5.62986e9	0.578834
$715$	1.69793e9	0.173720
$716$	1.01043e9	0.102876
$717$	−7.43721e9	−0.753517
$718$	8.48910e8	0.0855906
$719$	8.60894e9	0.863771	0.431885	−	0.901928i	$-0.357848\pi$
$719$	8.60894e9	0.863771	0.431885	+	0.901928i	$0.357848\pi$
$720$	−1.40717e9	−0.140502
$721$	−2.07967e10	−2.06643
$722$	−1.80688e9	−0.178669
$723$	−8.92900e9	−0.878656
$724$	2.86754e9	0.280818
$725$	−4.26204e9	−0.415369
$726$	−5.81472e9	−0.563963
$727$	−6.19733e9	−0.598183	−0.299092	−	0.954224i	$-0.596684\pi$
$727$	−6.19733e9	−0.598183	−0.299092	+	0.954224i	$0.596684\pi$
$728$	3.94582e9	0.379034
$729$	−1.42823e10	−1.36537
$730$	6.76803e7	0.00643920
$731$	2.84696e8	0.0269569
$732$	3.93905e9	0.371196
$733$	−5.39255e8	−0.0505743	−0.0252872	−	0.999680i	$-0.508050\pi$
$733$	−5.39255e8	−0.0505743	−0.0252872	+	0.999680i	$0.508050\pi$
$734$	−1.70671e9	−0.159303
$735$	1.49736e9	0.139098
$736$	−9.80624e9	−0.906631
$737$	−2.60382e10	−2.39593
$738$	3.96711e9	0.363310
$739$	1.36652e10	1.24554	0.622772	−	0.782403i	$-0.286006\pi$
$739$	1.36652e10	1.24554	0.622772	+	0.782403i	$0.286006\pi$
$740$	−2.39438e9	−0.217211
$741$	−3.21071e9	−0.289893
$742$	2.29361e9	0.206113
$743$	3.53677e9	0.316334	0.158167	−	0.987412i	$-0.449442\pi$
$743$	3.53677e9	0.316334	0.158167	+	0.987412i	$0.449442\pi$
$744$	−5.04947e9	−0.449511
$745$	8.82933e8	0.0782313
$746$	−1.63325e9	−0.144035
$747$	2.00879e10	1.76325
$748$	2.96736e10	2.59248
$749$	−1.37826e9	−0.119852
$750$	−8.46677e8	−0.0732830
$751$	9.97111e9	0.859021	0.429511	−	0.903062i	$-0.358686\pi$
$751$	9.97111e9	0.859021	0.429511	+	0.903062i	$0.358686\pi$
$752$	−1.51738e9	−0.130117
$753$	3.89305e9	0.332283
$754$	7.32375e8	0.0622205
$755$	2.77318e8	0.0234511
$756$	−4.66432e9	−0.392611
$757$	8.01079e9	0.671181	0.335591	−	0.942008i	$-0.391064\pi$
$757$	8.01079e9	0.671181	0.335591	+	0.942008i	$0.391064\pi$
$758$	−1.13163e9	−0.0943764
$759$	−5.14804e10	−4.27362
$760$	1.49366e8	0.0123425
$761$	−1.04835e10	−0.862300	−0.431150	−	0.902280i	$-0.641892\pi$
$761$	−1.04835e10	−0.862300	−0.431150	+	0.902280i	$0.641892\pi$
$762$	2.36459e9	0.193603
$763$	2.73934e10	2.23260
$764$	4.22673e9	0.342908
$765$	−3.03571e9	−0.245158
$766$	1.82423e9	0.146649
$767$	2.04777e9	0.163869
$768$	−1.22262e10	−0.973927
$769$	2.25020e10	1.78434	0.892172	−	0.451695i	$-0.149181\pi$
$769$	2.25020e10	1.78434	0.892172	+	0.451695i	$0.149181\pi$
$770$	−7.13729e8	−0.0563398
$771$	−2.51855e10	−1.97906
$772$	−1.42080e10	−1.11141
$773$	−8.15049e9	−0.634681	−0.317340	−	0.948312i	$-0.602790\pi$
$773$	−8.15049e9	−0.634681	−0.317340	+	0.948312i	$0.602790\pi$
$774$	5.15612e7	0.00399696
$775$	1.02795e10	0.793264
$776$	−6.77982e8	−0.0520838
$777$	−4.40878e10	−3.37166
$778$	9.32903e8	0.0710245
$779$	5.25107e9	0.397985
$780$	−1.90760e9	−0.143931
$781$	1.37669e10	1.03409
$782$	−6.60847e9	−0.494171
$783$	−1.76420e9	−0.131335
$784$	8.64705e9	0.640858
$785$	7.61724e8	0.0562022
$786$	1.85627e9	0.136353
$787$	4.72441e9	0.345490	0.172745	−	0.984967i	$-0.444736\pi$
$787$	4.72441e9	0.345490	0.172745	+	0.984967i	$0.444736\pi$
$788$	1.56979e9	0.114287
$789$	−1.36708e9	−0.0990888
$790$	−5.92707e8	−0.0427706
$791$	−1.83655e10	−1.31942
$792$	1.09516e10	0.783318
$793$	2.80801e9	0.199960
$794$	4.36813e9	0.309687
$795$	−2.25960e9	−0.159495
$796$	9.76033e7	0.00685913
$797$	7.67953e9	0.537317	0.268658	−	0.963236i	$-0.413420\pi$
$797$	7.67953e9	0.537317	0.268658	+	0.963236i	$0.413420\pi$
$798$	1.34963e9	0.0940164
$799$	−3.27348e9	−0.227037
$800$	−7.75915e9	−0.535796
$801$	−3.86460e8	−0.0265700
$802$	3.86098e8	0.0264294
$803$	6.56840e9	0.447667
$804$	2.92534e10	1.98509
$805$	−4.20415e9	−0.284049
$806$	−1.76640e9	−0.118828
$807$	3.57356e10	2.39356
$808$	5.68598e9	0.379198
$809$	1.43739e10	0.954453	0.477227	−	0.878780i	$-0.341642\pi$
$809$	1.43739e10	0.954453	0.477227	+	0.878780i	$0.341642\pi$
$810$	2.80973e8	0.0185766
$811$	1.51163e10	0.995114	0.497557	−	0.867431i	$-0.334231\pi$
$811$	1.51163e10	0.995114	0.497557	+	0.867431i	$0.334231\pi$
$812$	8.14256e9	0.533722
$813$	−6.31620e9	−0.412230
$814$	8.78569e9	0.570940
$815$	−3.16144e9	−0.204566
$816$	−3.20297e10	−2.06365
$817$	6.82491e7	0.00437845
$818$	−3.75749e8	−0.0240027
$819$	−1.92248e10	−1.22284
$820$	3.11985e9	0.197599
$821$	−2.14679e10	−1.35391	−0.676953	−	0.736026i	$-0.736700\pi$
$821$	−2.14679e10	−1.35391	−0.676953	+	0.736026i	$0.736700\pi$
$822$	−2.48124e9	−0.155818
$823$	−1.16145e10	−0.726277	−0.363138	−	0.931735i	$-0.618295\pi$
$823$	−1.16145e10	−0.726277	−0.363138	+	0.931735i	$0.618295\pi$
$824$	−9.48815e9	−0.590794
$825$	−4.07336e10	−2.52560
$826$	−8.60784e8	−0.0531452
$827$	−8.81157e9	−0.541731	−0.270866	−	0.962617i	$-0.587310\pi$
$827$	−8.81157e9	−0.541731	−0.270866	+	0.962617i	$0.587310\pi$
$828$	3.16562e10	1.93799
$829$	2.09518e10	1.27726	0.638632	−	0.769513i	$-0.279501\pi$
$829$	2.09518e10	1.27726	0.638632	+	0.769513i	$0.279501\pi$
$830$	−5.97282e8	−0.0362582
$831$	−3.12238e10	−1.88748
$832$	−1.00993e10	−0.607940
$833$	1.86544e10	1.11821
$834$	−2.35284e9	−0.140446
$835$	1.55204e9	0.0922573
$836$	7.11356e9	0.421080
$837$	4.25504e9	0.250822
$838$	−1.27324e9	−0.0747404
$839$	5.66067e9	0.330903	0.165451	−	0.986218i	$-0.447092\pi$
$839$	5.66067e9	0.330903	0.165451	+	0.986218i	$0.447092\pi$
$840$	1.63404e9	0.0951229
$841$	−1.41701e10	−0.821461
$842$	5.98756e9	0.345667
$843$	−4.26440e9	−0.245167
$844$	3.46106e9	0.198157
$845$	9.24837e8	0.0527311
$846$	−5.92860e8	−0.0336632
$847$	−4.60854e10	−2.60598
$848$	−1.30489e10	−0.734833
$849$	−3.69917e10	−2.07457
$850$	−5.22892e9	−0.292042
$851$	5.17513e10	2.87851
$852$	−1.54669e10	−0.856769
$853$	2.58288e10	1.42489	0.712446	−	0.701727i	$-0.247587\pi$
$853$	2.58288e10	1.42489	0.712446	+	0.701727i	$0.247587\pi$
$854$	−1.18035e9	−0.0648499
$855$	−7.27741e8	−0.0398195
$856$	−6.28808e8	−0.0342657
$857$	−1.22207e10	−0.663227	−0.331613	−	0.943415i	$-0.607593\pi$
$857$	−1.22207e10	−0.663227	−0.331613	+	0.943415i	$0.607593\pi$
$858$	6.99954e9	0.378324
$859$	−2.04943e10	−1.10321	−0.551603	−	0.834106i	$-0.685984\pi$
$859$	−2.04943e10	−1.10321	−0.551603	+	0.834106i	$0.685984\pi$
$860$	4.05492e7	0.00217389
$861$	5.74458e10	3.06723
$862$	−5.60593e9	−0.298107
$863$	5.69617e9	0.301679	0.150839	−	0.988558i	$-0.451802\pi$
$863$	5.69617e9	0.301679	0.150839	+	0.988558i	$0.451802\pi$
$864$	−3.21177e9	−0.169413
$865$	−1.54515e9	−0.0811735
$866$	5.12083e9	0.267934
$867$	−4.05764e10	−2.11449
$868$	−1.96389e10	−1.01929
$869$	−5.75225e10	−2.97350
$870$	3.03290e8	0.0156150
$871$	2.08537e10	1.06935
$872$	1.24978e10	0.638301
$873$	3.30326e9	0.168032
$874$	−1.58423e9	−0.0802652
$875$	−6.71046e9	−0.338629
$876$	−7.37948e9	−0.370904
$877$	1.66087e10	0.831453	0.415726	−	0.909490i	$-0.363527\pi$
$877$	1.66087e10	0.831453	0.415726	+	0.909490i	$0.363527\pi$
$878$	−2.37321e9	−0.118333
$879$	1.54548e10	0.767542
$880$	4.06058e9	0.200862
$881$	3.51295e10	1.73084	0.865419	−	0.501049i	$-0.167052\pi$
$881$	3.51295e10	1.73084	0.865419	+	0.501049i	$0.167052\pi$
$882$	3.37851e9	0.165800
$883$	1.49258e10	0.729583	0.364791	−	0.931089i	$-0.381140\pi$
$883$	1.49258e10	0.729583	0.364791	+	0.931089i	$0.381140\pi$
$884$	−2.37653e10	−1.15707
$885$	8.48021e8	0.0411249
$886$	1.42014e9	0.0685984
$887$	1.68772e10	0.812020	0.406010	−	0.913869i	$-0.366920\pi$
$887$	1.68772e10	0.812020	0.406010	+	0.913869i	$0.366920\pi$
$888$	−2.01143e10	−0.963961
$889$	1.87409e10	0.894610
$890$	1.14908e7	0.000546366 0
$891$	2.72686e10	1.29149
$892$	−3.25805e10	−1.53702
$893$	−7.84740e8	−0.0368762
$894$	3.63979e9	0.170371
$895$	2.98291e8	0.0139078
$896$	1.96295e10	0.911655
$897$	4.12301e10	1.90740
$898$	−5.26752e9	−0.242739
$899$	−7.42808e9	−0.340971
$900$	2.50478e10	1.14530
$901$	−2.81507e10	−1.28219
$902$	−1.14476e10	−0.519389
$903$	7.46633e8	0.0337443
$904$	−8.37895e9	−0.377225
$905$	8.46528e8	0.0379639
$906$	1.14321e9	0.0510714
$907$	8.58988e9	0.382263	0.191131	−	0.981564i	$-0.438784\pi$
$907$	8.58988e9	0.382263	0.191131	+	0.981564i	$0.438784\pi$
$908$	2.90361e10	1.28718
$909$	−2.77032e10	−1.22337
$910$	5.71618e8	0.0251456
$911$	−3.88012e10	−1.70032	−0.850161	−	0.526522i	$-0.823496\pi$
$911$	−3.88012e10	−1.70032	−0.850161	+	0.526522i	$0.823496\pi$
$912$	−7.67836e9	−0.335187
$913$	−5.79665e10	−2.52075
$914$	6.67699e9	0.289247
$915$	1.16285e9	0.0501822
$916$	1.83956e10	0.790824
$917$	1.47122e10	0.630063
$918$	−2.16443e9	−0.0923407
$919$	2.34747e10	0.997690	0.498845	−	0.866691i	$-0.333758\pi$
$919$	2.34747e10	0.997690	0.498845	+	0.866691i	$0.333758\pi$
$920$	−1.91808e9	−0.0812098
$921$	−7.34213e9	−0.309680
$922$	5.02322e9	0.211069
$923$	−1.10258e10	−0.461534
$924$	7.78210e10	3.24523
$925$	4.09480e10	1.70113
$926$	8.25762e9	0.341756
$927$	4.62281e10	1.90602
$928$	5.60682e9	0.230303
$929$	4.43526e10	1.81495	0.907473	−	0.420111i	$-0.138009\pi$
$929$	4.43526e10	1.81495	0.907473	+	0.420111i	$0.138009\pi$
$930$	−7.31500e8	−0.0298211
$931$	4.47197e9	0.181625
$932$	4.12084e10	1.66736
$933$	−6.55357e9	−0.264175
$934$	−7.13485e9	−0.286530
$935$	8.75996e9	0.350479
$936$	−8.77100e9	−0.349610
$937$	9.83923e9	0.390726	0.195363	−	0.980731i	$-0.437411\pi$
$937$	9.83923e9	0.390726	0.195363	+	0.980731i	$0.437411\pi$
$938$	−8.76589e9	−0.346806
$939$	−1.90973e10	−0.752737
$940$	−4.66242e8	−0.0183090
$941$	−4.41431e9	−0.172703	−0.0863513	−	0.996265i	$-0.527521\pi$
$941$	−4.41431e9	−0.172703	−0.0863513	+	0.996265i	$0.527521\pi$
$942$	3.14012e9	0.122396
$943$	−6.74313e10	−2.61861
$944$	4.89722e9	0.189473
$945$	−1.37696e9	−0.0530773
$946$	−1.48787e8	−0.00571408
$947$	−4.02463e10	−1.53993	−0.769965	−	0.638086i	$-0.779726\pi$
$947$	−4.02463e10	−1.53993	−0.769965	+	0.638086i	$0.779726\pi$
$948$	6.46254e10	2.46362
$949$	−5.26057e9	−0.199803
$950$	−1.25351e9	−0.0474347
$951$	1.16383e10	0.438790
$952$	2.03573e10	0.764698
$953$	−2.65876e10	−0.995071	−0.497535	−	0.867444i	$-0.665762\pi$
$953$	−2.65876e10	−0.995071	−0.497535	+	0.867444i	$0.665762\pi$
$954$	−5.09837e9	−0.190113
$955$	1.24778e9	0.0463580
$956$	−1.31968e10	−0.488501
$957$	2.94345e10	1.08559
$958$	1.42672e8	0.00524274
$959$	−1.96654e10	−0.720009
$960$	−4.18232e9	−0.152570
$961$	−9.59696e9	−0.348820
$962$	−7.03637e9	−0.254821
$963$	3.06367e9	0.110548
$964$	−1.58439e10	−0.569629
$965$	−4.19436e9	−0.150252
$966$	−1.73311e10	−0.618596
$967$	2.50722e10	0.891661	0.445831	−	0.895117i	$-0.352908\pi$
$967$	2.50722e10	0.891661	0.445831	+	0.895117i	$0.352908\pi$
$968$	−2.10257e10	−0.745052
$969$	−1.65647e10	−0.584857
$970$	−9.82171e7	−0.00345530
$971$	−4.33015e10	−1.51787	−0.758937	−	0.651164i	$-0.774281\pi$
$971$	−4.33015e10	−1.51787	−0.758937	+	0.651164i	$0.774281\pi$
$972$	−3.92107e10	−1.36953
$973$	−1.86477e10	−0.648980
$974$	6.28101e9	0.217808
$975$	3.26232e10	1.12722
$976$	6.71531e9	0.231202
$977$	4.07827e10	1.39909	0.699545	−	0.714589i	$-0.253386\pi$
$977$	4.07827e10	1.39909	0.699545	+	0.714589i	$0.253386\pi$
$978$	−1.30327e10	−0.445499
$979$	1.11518e9	0.0379845
$980$	2.65695e9	0.0901763
$981$	−6.08917e10	−2.05929
$982$	6.60443e9	0.222559
$983$	−5.48315e9	−0.184116	−0.0920582	−	0.995754i	$-0.529345\pi$
$983$	−5.48315e9	−0.184116	−0.0920582	+	0.995754i	$0.529345\pi$
$984$	2.62087e10	0.876924
$985$	4.63417e8	0.0154506
$986$	3.77846e9	0.125530
$987$	−8.58492e9	−0.284201
$988$	−5.69718e9	−0.187936
$989$	−8.76417e8	−0.0288087
$990$	1.58652e9	0.0519663
$991$	3.90652e10	1.27507	0.637533	−	0.770423i	$-0.279955\pi$
$991$	3.90652e10	1.27507	0.637533	+	0.770423i	$0.279955\pi$
$992$	−1.35230e10	−0.439828
$993$	1.75820e10	0.569829
$994$	4.63470e9	0.149682
$995$	2.88135e7	0.000927290 0
$996$	6.51243e10	2.08850
$997$	−3.82562e10	−1.22256	−0.611278	−	0.791416i	$-0.709344\pi$
$997$	−3.82562e10	−1.22256	−0.611278	+	0.791416i	$0.709344\pi$
$998$	−5.81911e9	−0.185310
$999$	1.69497e10	0.537878

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	47.8.a.b.1.9	✓	16
3.2	odd	2		423.8.a.f.1.8		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
47.8.a.b.1.9	✓	16	1.1	even	1	trivial
423.8.a.f.1.8		16	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:	\( N \)	\(=\)	\( 47 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	47.a (trivial)

\(f(q)\)	\(=\)	\(q+2.15943 q^{2} -69.5079 q^{3} -123.337 q^{4} -36.4103 q^{5} -150.097 q^{6} -1189.62 q^{7} -542.744 q^{8} +2644.35 q^{9} +O(q^{10})\)	\(q+2.15943 q^{2} -69.5079 q^{3} -123.337 q^{4} -36.4103 q^{5} -150.097 q^{6} -1189.62 q^{7} -542.744 q^{8} +2644.35 q^{9} -78.6256 q^{10} -7630.65 q^{11} +8572.89 q^{12} +6111.31 q^{13} -2568.90 q^{14} +2530.81 q^{15} +14615.1 q^{16} +31529.4 q^{17} +5710.30 q^{18} +7558.44 q^{19} +4490.74 q^{20} +82688.0 q^{21} -16477.8 q^{22} -97061.2 q^{23} +37725.0 q^{24} -76799.3 q^{25} +13196.9 q^{26} -31789.8 q^{27} +146724. q^{28} +55495.8 q^{29} +5465.10 q^{30} -133849. q^{31} +101032. q^{32} +530391. q^{33} +68085.6 q^{34} +43314.5 q^{35} -326146. q^{36} -533182. q^{37} +16321.9 q^{38} -424785. q^{39} +19761.5 q^{40} +694729. q^{41} +178559. q^{42} +9029.52 q^{43} +941140. q^{44} -96281.9 q^{45} -209597. q^{46} -103823. q^{47} -1.01587e6 q^{48} +591652. q^{49} -165843. q^{50} -2.19155e6 q^{51} -753750. q^{52} -892837. q^{53} -68647.8 q^{54} +277835. q^{55} +645659. q^{56} -525372. q^{57} +119839. q^{58} +335079. q^{59} -312142. q^{60} +459478. q^{61} -289038. q^{62} -3.14578e6 q^{63} -1.65256e6 q^{64} -222515. q^{65} +1.14534e6 q^{66} +3.41231e6 q^{67} -3.88874e6 q^{68} +6.74653e6 q^{69} +93534.5 q^{70} -1.80416e6 q^{71} -1.43521e6 q^{72} -860792. q^{73} -1.15137e6 q^{74} +5.33816e6 q^{75} -932235. q^{76} +9.07757e6 q^{77} -917292. q^{78} +7.53835e6 q^{79} -532141. q^{80} -3.57356e6 q^{81} +1.50022e6 q^{82} +7.59654e6 q^{83} -1.01985e7 q^{84} -1.14800e6 q^{85} +19498.6 q^{86} -3.85740e6 q^{87} +4.14149e6 q^{88} -146145. q^{89} -207914. q^{90} -7.27013e6 q^{91} +1.19712e7 q^{92} +9.30360e6 q^{93} -224198. q^{94} -275206. q^{95} -7.02249e6 q^{96} +1.24917e6 q^{97} +1.27763e6 q^{98} -2.01781e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 15 q^{2} + 40 q^{3} + 1125 q^{4} + 444 q^{5} - 128 q^{6} + 1860 q^{7} + 2145 q^{8} + 16986 q^{9}+O(q^{10}) \) 16 * q + 15 * q^2 + 40 * q^3 + 1125 * q^4 + 444 * q^5 - 128 * q^6 + 1860 * q^7 + 2145 * q^8 + 16986 * q^9	\( 16 q + 15 q^{2} + 40 q^{3} + 1125 q^{4} + 444 q^{5} - 128 q^{6} + 1860 q^{7} + 2145 q^{8} + 16986 q^{9} + 7680 q^{10} + 4776 q^{11} - 20035 q^{12} + 16074 q^{13} - 7749 q^{14} + 22614 q^{15} + 131873 q^{16} + 67106 q^{17} + 243090 q^{18} + 69730 q^{19} + 300392 q^{20} + 247700 q^{21} + 357976 q^{22} + 101110 q^{23} + 167915 q^{24} + 542284 q^{25} + 424864 q^{26} + 253084 q^{27} + 383052 q^{28} + 199976 q^{29} + 252626 q^{30} + 373036 q^{31} + 463950 q^{32} + 64236 q^{33} + 17022 q^{34} - 140956 q^{35} + 694529 q^{36} + 596106 q^{37} - 488694 q^{38} - 352018 q^{39} - 188562 q^{40} - 66678 q^{41} - 776155 q^{42} + 1037964 q^{43} - 1598298 q^{44} + 905532 q^{45} - 2460646 q^{46} - 1661168 q^{47} - 9341196 q^{48} - 1580054 q^{49} - 7616723 q^{50} - 4897564 q^{51} + 1045432 q^{52} - 1253454 q^{53} - 11836385 q^{54} - 1489932 q^{55} - 9175912 q^{56} + 1630346 q^{57} + 1640408 q^{58} - 1977256 q^{59} - 12723470 q^{60} - 2381646 q^{61} - 5329480 q^{62} + 320028 q^{63} + 10719545 q^{64} + 9219376 q^{65} - 13727472 q^{66} + 11455752 q^{67} - 6720409 q^{68} + 9814322 q^{69} - 6175334 q^{70} + 5467788 q^{71} + 19454169 q^{72} + 12652462 q^{73} + 3377814 q^{74} + 14522488 q^{75} - 4047624 q^{76} + 12951326 q^{77} - 8107404 q^{78} + 34366848 q^{79} + 19400408 q^{80} + 24384080 q^{81} - 16612272 q^{82} + 27856008 q^{83} + 18267309 q^{84} + 8672002 q^{85} - 34325926 q^{86} + 41013236 q^{87} + 36623538 q^{88} + 14703638 q^{89} + 20243114 q^{90} + 11783202 q^{91} + 4020484 q^{92} + 35185978 q^{93} - 1557345 q^{94} + 14274548 q^{95} - 56487828 q^{96} + 51072842 q^{97} - 22140217 q^{98} + 55509018 q^{99}+O(q^{100}) \) 16 * q + 15 * q^2 + 40 * q^3 + 1125 * q^4 + 444 * q^5 - 128 * q^6 + 1860 * q^7 + 2145 * q^8 + 16986 * q^9 + 7680 * q^10 + 4776 * q^11 - 20035 * q^12 + 16074 * q^13 - 7749 * q^14 + 22614 * q^15 + 131873 * q^16 + 67106 * q^17 + 243090 * q^18 + 69730 * q^19 + 300392 * q^20 + 247700 * q^21 + 357976 * q^22 + 101110 * q^23 + 167915 * q^24 + 542284 * q^25 + 424864 * q^26 + 253084 * q^27 + 383052 * q^28 + 199976 * q^29 + 252626 * q^30 + 373036 * q^31 + 463950 * q^32 + 64236 * q^33 + 17022 * q^34 - 140956 * q^35 + 694529 * q^36 + 596106 * q^37 - 488694 * q^38 - 352018 * q^39 - 188562 * q^40 - 66678 * q^41 - 776155 * q^42 + 1037964 * q^43 - 1598298 * q^44 + 905532 * q^45 - 2460646 * q^46 - 1661168 * q^47 - 9341196 * q^48 - 1580054 * q^49 - 7616723 * q^50 - 4897564 * q^51 + 1045432 * q^52 - 1253454 * q^53 - 11836385 * q^54 - 1489932 * q^55 - 9175912 * q^56 + 1630346 * q^57 + 1640408 * q^58 - 1977256 * q^59 - 12723470 * q^60 - 2381646 * q^61 - 5329480 * q^62 + 320028 * q^63 + 10719545 * q^64 + 9219376 * q^65 - 13727472 * q^66 + 11455752 * q^67 - 6720409 * q^68 + 9814322 * q^69 - 6175334 * q^70 + 5467788 * q^71 + 19454169 * q^72 + 12652462 * q^73 + 3377814 * q^74 + 14522488 * q^75 - 4047624 * q^76 + 12951326 * q^77 - 8107404 * q^78 + 34366848 * q^79 + 19400408 * q^80 + 24384080 * q^81 - 16612272 * q^82 + 27856008 * q^83 + 18267309 * q^84 + 8672002 * q^85 - 34325926 * q^86 + 41013236 * q^87 + 36623538 * q^88 + 14703638 * q^89 + 20243114 * q^90 + 11783202 * q^91 + 4020484 * q^92 + 35185978 * q^93 - 1557345 * q^94 + 14274548 * q^95 - 56487828 * q^96 + 51072842 * q^97 - 22140217 * q^98 + 55509018 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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