LMFDB - Embedded newform 138.8.a.g.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [138,8,Mod(1,138)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(138, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

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

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

chi := DirichletCharacter("138.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 138 = 2 \cdot 3 \cdot 23 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	138.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:	$43.1091335168$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5167x^{2} - 24752x + 5245058 $ x^4 - 5167x^2 - 24752x + 5245058
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$65.8209$ of defining polynomial
Character	$\chi$	$=$	138.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} +7.78122 q^{5} -216.000 q^{6} +1390.34 q^{7} +512.000 q^{8} +729.000 q^{9} +O(q^{10})$	\(q+8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} +7.78122 q^{5} -216.000 q^{6} +1390.34 q^{7} +512.000 q^{8} +729.000 q^{9} +62.2498 q^{10} -8614.99 q^{11} -1728.00 q^{12} +9780.79 q^{13} +11122.7 q^{14} -210.093 q^{15} +4096.00 q^{16} +38428.9 q^{17} +5832.00 q^{18} -48016.2 q^{19} +497.998 q^{20} -37539.2 q^{21} -68919.9 q^{22} +12167.0 q^{23} -13824.0 q^{24} -78064.5 q^{25} +78246.3 q^{26} -19683.0 q^{27} +88981.8 q^{28} +170782. q^{29} -1680.74 q^{30} +37933.7 q^{31} +32768.0 q^{32} +232605. q^{33} +307431. q^{34} +10818.6 q^{35} +46656.0 q^{36} +238706. q^{37} -384130. q^{38} -264081. q^{39} +3983.98 q^{40} +779678. q^{41} -300314. q^{42} -236609. q^{43} -551359. q^{44} +5672.51 q^{45} +97336.0 q^{46} +1.06282e6 q^{47} -110592. q^{48} +1.10951e6 q^{49} -624516. q^{50} -1.03758e6 q^{51} +625970. q^{52} +687276. q^{53} -157464. q^{54} -67035.1 q^{55} +711855. q^{56} +1.29644e6 q^{57} +1.36626e6 q^{58} +1.82775e6 q^{59} -13445.9 q^{60} +239458. q^{61} +303470. q^{62} +1.01356e6 q^{63} +262144. q^{64} +76106.4 q^{65} +1.86084e6 q^{66} -1.97106e6 q^{67} +2.45945e6 q^{68} -328509. q^{69} +86548.4 q^{70} -43787.6 q^{71} +373248. q^{72} -856768. q^{73} +1.90965e6 q^{74} +2.10774e6 q^{75} -3.07304e6 q^{76} -1.19778e7 q^{77} -2.11265e6 q^{78} +3.82561e6 q^{79} +31871.9 q^{80} +531441. q^{81} +6.23742e6 q^{82} +4.70673e6 q^{83} -2.40251e6 q^{84} +299024. q^{85} -1.89287e6 q^{86} -4.61113e6 q^{87} -4.41087e6 q^{88} -5.03979e6 q^{89} +45380.1 q^{90} +1.35986e7 q^{91} +778688. q^{92} -1.02421e6 q^{93} +8.50258e6 q^{94} -373625. q^{95} -884736. q^{96} -4.37484e6 q^{97} +8.87604e6 q^{98} -6.28032e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 32 q^{2} - 108 q^{3} + 256 q^{4} + 342 q^{5} - 864 q^{6} - 30 q^{7} + 2048 q^{8} + 2916 q^{9}+O(q^{10}) $ 4 * q + 32 * q^2 - 108 * q^3 + 256 * q^4 + 342 * q^5 - 864 * q^6 - 30 * q^7 + 2048 * q^8 + 2916 * q^9	\( 4 q + 32 q^{2} - 108 q^{3} + 256 q^{4} + 342 q^{5} - 864 q^{6} - 30 q^{7} + 2048 q^{8} + 2916 q^{9} + 2736 q^{10} - 2280 q^{11} - 6912 q^{12} + 9204 q^{13} - 240 q^{14} - 9234 q^{15} + 16384 q^{16} + 21638 q^{17} + 23328 q^{18} - 58158 q^{19} + 21888 q^{20} + 810 q^{21} - 18240 q^{22} + 48668 q^{23} - 55296 q^{24} + 246104 q^{25} + 73632 q^{26} - 78732 q^{27} - 1920 q^{28} + 252372 q^{29} - 73872 q^{30} + 364604 q^{31} + 131072 q^{32} + 61560 q^{33} + 173104 q^{34} - 197064 q^{35} + 186624 q^{36} + 357596 q^{37} - 465264 q^{38} - 248508 q^{39} + 175104 q^{40} + 818768 q^{41} + 6480 q^{42} + 561066 q^{43} - 145920 q^{44} + 249318 q^{45} + 389344 q^{46} + 2359976 q^{47} - 442368 q^{48} + 3660776 q^{49} + 1968832 q^{50} - 584226 q^{51} + 589056 q^{52} + 4246586 q^{53} - 629856 q^{54} + 3314024 q^{55} - 15360 q^{56} + 1570266 q^{57} + 2018976 q^{58} + 5065776 q^{59} - 590976 q^{60} - 1403844 q^{61} + 2916832 q^{62} - 21870 q^{63} + 1048576 q^{64} + 5152364 q^{65} + 492480 q^{66} + 3643934 q^{67} + 1384832 q^{68} - 1314036 q^{69} - 1576512 q^{70} - 4894464 q^{71} + 1492992 q^{72} + 4426192 q^{73} + 2860768 q^{74} - 6644808 q^{75} - 3722112 q^{76} - 11044616 q^{77} - 1988064 q^{78} - 1095946 q^{79} + 1400832 q^{80} + 2125764 q^{81} + 6550144 q^{82} - 10412840 q^{83} + 51840 q^{84} + 7941764 q^{85} + 4488528 q^{86} - 6814044 q^{87} - 1167360 q^{88} + 4632090 q^{89} + 1994544 q^{90} + 2135500 q^{91} + 3114752 q^{92} - 9844308 q^{93} + 18879808 q^{94} - 2017088 q^{95} - 3538944 q^{96} - 15402348 q^{97} + 29286208 q^{98} - 1662120 q^{99}+O(q^{100}) \) 4 * q + 32 * q^2 - 108 * q^3 + 256 * q^4 + 342 * q^5 - 864 * q^6 - 30 * q^7 + 2048 * q^8 + 2916 * q^9 + 2736 * q^10 - 2280 * q^11 - 6912 * q^12 + 9204 * q^13 - 240 * q^14 - 9234 * q^15 + 16384 * q^16 + 21638 * q^17 + 23328 * q^18 - 58158 * q^19 + 21888 * q^20 + 810 * q^21 - 18240 * q^22 + 48668 * q^23 - 55296 * q^24 + 246104 * q^25 + 73632 * q^26 - 78732 * q^27 - 1920 * q^28 + 252372 * q^29 - 73872 * q^30 + 364604 * q^31 + 131072 * q^32 + 61560 * q^33 + 173104 * q^34 - 197064 * q^35 + 186624 * q^36 + 357596 * q^37 - 465264 * q^38 - 248508 * q^39 + 175104 * q^40 + 818768 * q^41 + 6480 * q^42 + 561066 * q^43 - 145920 * q^44 + 249318 * q^45 + 389344 * q^46 + 2359976 * q^47 - 442368 * q^48 + 3660776 * q^49 + 1968832 * q^50 - 584226 * q^51 + 589056 * q^52 + 4246586 * q^53 - 629856 * q^54 + 3314024 * q^55 - 15360 * q^56 + 1570266 * q^57 + 2018976 * q^58 + 5065776 * q^59 - 590976 * q^60 - 1403844 * q^61 + 2916832 * q^62 - 21870 * q^63 + 1048576 * q^64 + 5152364 * q^65 + 492480 * q^66 + 3643934 * q^67 + 1384832 * q^68 - 1314036 * q^69 - 1576512 * q^70 - 4894464 * q^71 + 1492992 * q^72 + 4426192 * q^73 + 2860768 * q^74 - 6644808 * q^75 - 3722112 * q^76 - 11044616 * q^77 - 1988064 * q^78 - 1095946 * q^79 + 1400832 * q^80 + 2125764 * q^81 + 6550144 * q^82 - 10412840 * q^83 + 51840 * q^84 + 7941764 * q^85 + 4488528 * q^86 - 6814044 * q^87 - 1167360 * q^88 + 4632090 * q^89 + 1994544 * q^90 + 2135500 * q^91 + 3114752 * q^92 - 9844308 * q^93 + 18879808 * q^94 - 2017088 * q^95 - 3538944 * q^96 - 15402348 * q^97 + 29286208 * q^98 - 1662120 * 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
$3$	−27.0000	−0.577350
$3$	−27.0000	−0.577350
$4$	64.0000	0.500000
$5$	7.78122	0.0278389	0.0139195	−	0.999903i	$-0.495569\pi$
$5$	7.78122	0.0278389	0.0139195	+	0.999903i	$0.495569\pi$
$6$	−216.000	−0.408248
$7$	1390.34	1.53207	0.766034	−	0.642800i	$-0.222227\pi$
$7$	1390.34	1.53207	0.766034	+	0.642800i	$0.222227\pi$
$8$	512.000	0.353553
$9$	729.000	0.333333
$10$	62.2498	0.0196851
$11$	−8614.99	−1.95155	−0.975776	−	0.218772i	$-0.929795\pi$
$11$	−8614.99	−1.95155	−0.975776	+	0.218772i	$0.929795\pi$
$12$	−1728.00	−0.288675
$13$	9780.79	1.23473	0.617365	−	0.786677i	$-0.288200\pi$
$13$	9780.79	1.23473	0.617365	+	0.786677i	$0.288200\pi$
$14$	11122.7	1.08334
$15$	−210.093	−0.0160728
$16$	4096.00	0.250000
$17$	38428.9	1.89708	0.948542	−	0.316651i	$-0.102558\pi$
$17$	38428.9	1.89708	0.948542	+	0.316651i	$0.102558\pi$
$18$	5832.00	0.235702
$19$	−48016.2	−1.60602	−0.803009	−	0.595967i	$-0.796769\pi$
$19$	−48016.2	−1.60602	−0.803009	+	0.595967i	$0.796769\pi$
$20$	497.998	0.0139195
$21$	−37539.2	−0.884540
$22$	−68919.9	−1.37996
$23$	12167.0	0.208514
$23$	12167.0	0.208514
$24$	−13824.0	−0.204124
$25$	−78064.5	−0.999225
$26$	78246.3	0.873086
$27$	−19683.0	−0.192450
$28$	88981.8	0.766034
$29$	170782.	1.30032	0.650160	−	0.759797i	$-0.274702\pi$
$29$	170782.	1.30032	0.650160	+	0.759797i	$0.274702\pi$
$30$	−1680.74	−0.0113652
$31$	37933.7	0.228696	0.114348	−	0.993441i	$-0.463522\pi$
$31$	37933.7	0.228696	0.114348	+	0.993441i	$0.463522\pi$
$32$	32768.0	0.176777
$33$	232605.	1.12673
$34$	307431.	1.34144
$35$	10818.6	0.0426512
$36$	46656.0	0.166667
$37$	238706.	0.774743	0.387372	−	0.921924i	$-0.373383\pi$
$37$	238706.	0.774743	0.387372	+	0.921924i	$0.373383\pi$
$38$	−384130.	−1.13563
$39$	−264081.	−0.712872
$40$	3983.98	0.00984255
$41$	779678.	1.76674	0.883368	−	0.468680i	$-0.155270\pi$
$41$	779678.	1.76674	0.883368	+	0.468680i	$0.155270\pi$
$42$	−300314.	−0.625464
$43$	−236609.	−0.453828	−0.226914	−	0.973915i	$-0.572864\pi$
$43$	−236609.	−0.453828	−0.226914	+	0.973915i	$0.572864\pi$
$44$	−551359.	−0.975776
$45$	5672.51	0.00927965
$46$	97336.0	0.147442
$47$	1.06282e6	1.49320	0.746601	−	0.665272i	$-0.231685\pi$
$47$	1.06282e6	1.49320	0.746601	+	0.665272i	$0.231685\pi$
$48$	−110592.	−0.144338
$49$	1.10951e6	1.34723
$50$	−624516.	−0.706559
$51$	−1.03758e6	−1.09528
$52$	625970.	0.617365
$53$	687276.	0.634112	0.317056	−	0.948407i	$-0.397306\pi$
$53$	687276.	0.634112	0.317056	+	0.948407i	$0.397306\pi$
$54$	−157464.	−0.136083
$55$	−67035.1	−0.0543291
$56$	711855.	0.541668
$57$	1.29644e6	0.927235
$58$	1.36626e6	0.919465
$59$	1.82775e6	1.15860	0.579301	−	0.815114i	$-0.303326\pi$
$59$	1.82775e6	1.15860	0.579301	+	0.815114i	$0.303326\pi$
$60$	−13445.9	−0.00803641
$61$	239458.	0.135075	0.0675375	−	0.997717i	$-0.478486\pi$
$61$	239458.	0.135075	0.0675375	+	0.997717i	$0.478486\pi$
$62$	303470.	0.161713
$63$	1.01356e6	0.510690
$64$	262144.	0.125000
$65$	76106.4	0.0343736
$66$	1.86084e6	0.796718
$67$	−1.97106e6	−0.800643	−0.400321	−	0.916375i	$-0.631101\pi$
$67$	−1.97106e6	−0.800643	−0.400321	+	0.916375i	$0.631101\pi$
$68$	2.45945e6	0.948542
$69$	−328509.	−0.120386
$70$	86548.4	0.0301589
$71$	−43787.6	−0.0145193	−0.00725966	−	0.999974i	$-0.502311\pi$
$71$	−43787.6	−0.0145193	−0.00725966	+	0.999974i	$0.502311\pi$
$72$	373248.	0.117851
$73$	−856768.	−0.257770	−0.128885	−	0.991660i	$-0.541140\pi$
$73$	−856768.	−0.257770	−0.128885	+	0.991660i	$0.541140\pi$
$74$	1.90965e6	0.547826
$75$	2.10774e6	0.576903
$76$	−3.07304e6	−0.803009
$77$	−1.19778e7	−2.98991
$78$	−2.11265e6	−0.504076
$79$	3.82561e6	0.872983	0.436491	−	0.899708i	$-0.356221\pi$
$79$	3.82561e6	0.872983	0.436491	+	0.899708i	$0.356221\pi$
$80$	31871.9	0.00695973
$81$	531441.	0.111111
$82$	6.23742e6	1.24927
$83$	4.70673e6	0.903536	0.451768	−	0.892135i	$-0.350793\pi$
$83$	4.70673e6	0.903536	0.451768	+	0.892135i	$0.350793\pi$
$84$	−2.40251e6	−0.442270
$85$	299024.	0.0528128
$86$	−1.89287e6	−0.320905
$87$	−4.61113e6	−0.750740
$88$	−4.41087e6	−0.689978
$89$	−5.03979e6	−0.757787	−0.378894	−	0.925440i	$-0.623695\pi$
$89$	−5.03979e6	−0.757787	−0.378894	+	0.925440i	$0.623695\pi$
$90$	45380.1	0.00656170
$91$	1.35986e7	1.89169
$92$	778688.	0.104257
$93$	−1.02421e6	−0.132038
$94$	8.50258e6	1.05585
$95$	−373625.	−0.0447099
$96$	−884736.	−0.102062
$97$	−4.37484e6	−0.486700	−0.243350	−	0.969939i	$-0.578246\pi$
$97$	−4.37484e6	−0.486700	−0.243350	+	0.969939i	$0.578246\pi$
$98$	8.87604e6	0.952639
$99$	−6.28032e6	−0.650517
$100$	−4.99612e6	−0.499612
$101$	9.51261e6	0.918702	0.459351	−	0.888255i	$-0.348082\pi$
$101$	9.51261e6	0.918702	0.459351	+	0.888255i	$0.348082\pi$
$102$	−8.30064e6	−0.774481
$103$	3.88640e6	0.350443	0.175222	−	0.984529i	$-0.443936\pi$
$103$	3.88640e6	0.350443	0.175222	+	0.984529i	$0.443936\pi$
$104$	5.00776e6	0.436543
$105$	−292101.	−0.0246247
$106$	5.49821e6	0.448385
$107$	−1.03803e7	−0.819155	−0.409577	−	0.912275i	$-0.634324\pi$
$107$	−1.03803e7	−0.819155	−0.409577	+	0.912275i	$0.634324\pi$
$108$	−1.25971e6	−0.0962250
$109$	−1.69653e7	−1.25478	−0.627391	−	0.778704i	$-0.715877\pi$
$109$	−1.69653e7	−1.25478	−0.627391	+	0.778704i	$0.715877\pi$
$110$	−536281.	−0.0384165
$111$	−6.44507e6	−0.447298
$112$	5.69484e6	0.383017
$113$	−1.10301e7	−0.719126	−0.359563	−	0.933121i	$-0.617074\pi$
$113$	−1.10301e7	−0.719126	−0.359563	+	0.933121i	$0.617074\pi$
$114$	1.03715e7	0.655654
$115$	94674.1	0.00580482
$116$	1.09301e7	0.650160
$117$	7.13019e6	0.411577
$118$	1.46220e7	0.819255
$119$	5.34293e7	2.90646
$120$	−107568.	−0.00568260
$121$	5.47308e7	2.80856
$122$	1.91566e6	0.0955125
$123$	−2.10513e7	−1.02003
$124$	2.42776e6	0.114348
$125$	−1.21534e6	−0.0556563
$126$	8.10847e6	0.361112
$127$	−4.16071e7	−1.80241	−0.901207	−	0.433388i	$-0.857318\pi$
$127$	−4.16071e7	−1.80241	−0.901207	+	0.433388i	$0.857318\pi$
$128$	2.09715e6	0.0883883
$129$	6.38844e6	0.262018
$130$	608852.	0.0243058
$131$	3.91494e6	0.152151	0.0760756	−	0.997102i	$-0.475761\pi$
$131$	3.91494e6	0.152151	0.0760756	+	0.997102i	$0.475761\pi$
$132$	1.48867e7	0.563365
$133$	−6.67590e7	−2.46053
$134$	−1.57685e7	−0.566140
$135$	−153158.	−0.00535761
$136$	1.96756e7	0.670720
$137$	1.15484e7	0.383707	0.191853	−	0.981424i	$-0.438550\pi$
$137$	1.15484e7	0.383707	0.191853	+	0.981424i	$0.438550\pi$
$138$	−2.62807e6	−0.0851257
$139$	−2.82414e7	−0.891938	−0.445969	−	0.895048i	$-0.647141\pi$
$139$	−2.82414e7	−0.891938	−0.445969	+	0.895048i	$0.647141\pi$
$140$	692387.	0.0213256
$141$	−2.86962e7	−0.862100
$142$	−350300.	−0.0102667
$143$	−8.42613e7	−2.40964
$144$	2.98598e6	0.0833333
$145$	1.32890e6	0.0361995
$146$	−6.85414e6	−0.182271
$147$	−2.99566e7	−0.777826
$148$	1.52772e7	0.387372
$149$	−4.73262e7	−1.17206	−0.586030	−	0.810289i	$-0.699310\pi$
$149$	−4.73262e7	−1.17206	−0.586030	+	0.810289i	$0.699310\pi$
$150$	1.68619e7	0.407932
$151$	2.38498e7	0.563723	0.281862	−	0.959455i	$-0.409048\pi$
$151$	2.38498e7	0.563723	0.281862	+	0.959455i	$0.409048\pi$
$152$	−2.45843e7	−0.567813
$153$	2.80147e7	0.632361
$154$	−9.58222e7	−2.11419
$155$	295170.	0.00636666
$156$	−1.69012e7	−0.356436
$157$	2.24770e6	0.0463543	0.0231772	−	0.999731i	$-0.492622\pi$
$157$	2.24770e6	0.0463543	0.0231772	+	0.999731i	$0.492622\pi$
$158$	3.06049e7	0.617292
$159$	−1.85565e7	−0.366105
$160$	254975.	0.00492128
$161$	1.69163e7	0.319458
$162$	4.25153e6	0.0785674
$163$	3.22635e7	0.583519	0.291760	−	0.956492i	$-0.405759\pi$
$163$	3.22635e7	0.583519	0.291760	+	0.956492i	$0.405759\pi$
$164$	4.98994e7	0.883368
$165$	1.80995e6	0.0313669
$166$	3.76538e7	0.638897
$167$	3.52969e7	0.586447	0.293224	−	0.956044i	$-0.405272\pi$
$167$	3.52969e7	0.586447	0.293224	+	0.956044i	$0.405272\pi$
$168$	−1.92201e7	−0.312732
$169$	3.29152e7	0.524558
$170$	2.39219e6	0.0373443
$171$	−3.50038e7	−0.535340
$172$	−1.51430e7	−0.226914
$173$	−4.07765e7	−0.598754	−0.299377	−	0.954135i	$-0.596779\pi$
$173$	−4.07765e7	−0.598754	−0.299377	+	0.954135i	$0.596779\pi$
$174$	−3.68890e7	−0.530853
$175$	−1.08536e8	−1.53088
$176$	−3.52870e7	−0.487888
$177$	−4.93492e7	−0.668919
$178$	−4.03183e7	−0.535836
$179$	−1.07685e8	−1.40336	−0.701681	−	0.712491i	$-0.747567\pi$
$179$	−1.07685e8	−1.40336	−0.701681	+	0.712491i	$0.747567\pi$
$180$	363041.	0.00463982
$181$	7.09701e7	0.889612	0.444806	−	0.895627i	$-0.353273\pi$
$181$	7.09701e7	0.889612	0.444806	+	0.895627i	$0.353273\pi$
$182$	1.08789e8	1.33763
$183$	−6.46537e6	−0.0779856
$184$	6.22950e6	0.0737210
$185$	1.85743e6	0.0215680
$186$	−8.19368e6	−0.0933649
$187$	−3.31064e8	−3.70226
$188$	6.80207e7	0.746601
$189$	−2.73661e7	−0.294847
$190$	−2.98900e6	−0.0316146
$191$	1.32543e8	1.37639	0.688195	−	0.725526i	$-0.258404\pi$
$191$	1.32543e8	1.37639	0.688195	+	0.725526i	$0.258404\pi$
$192$	−7.07789e6	−0.0721688
$193$	3.96153e7	0.396655	0.198327	−	0.980136i	$-0.436449\pi$
$193$	3.96153e7	0.396655	0.198327	+	0.980136i	$0.436449\pi$
$194$	−3.49987e7	−0.344149
$195$	−2.05487e6	−0.0198456
$196$	7.10083e7	0.673617
$197$	−1.29821e8	−1.20980	−0.604898	−	0.796303i	$-0.706786\pi$
$197$	−1.29821e8	−1.20980	−0.604898	+	0.796303i	$0.706786\pi$
$198$	−5.02426e7	−0.459985
$199$	−1.05686e8	−0.950672	−0.475336	−	0.879804i	$-0.657673\pi$
$199$	−1.05686e8	−0.950672	−0.475336	+	0.879804i	$0.657673\pi$
$200$	−3.99690e7	−0.353279
$201$	5.32187e7	0.462251
$202$	7.61009e7	0.649621
$203$	2.37446e8	1.99218
$204$	−6.64051e7	−0.547641
$205$	6.06685e6	0.0491841
$206$	3.10912e7	0.247801
$207$	8.86974e6	0.0695048
$208$	4.00621e7	0.308682
$209$	4.13659e8	3.13423
$210$	−2.33681e6	−0.0174123
$211$	−1.62387e8	−1.19005	−0.595023	−	0.803709i	$-0.702857\pi$
$211$	−1.62387e8	−1.19005	−0.595023	+	0.803709i	$0.702857\pi$
$212$	4.39857e7	0.317056
$213$	1.18226e6	0.00838274
$214$	−8.30423e7	−0.579230
$215$	−1.84111e6	−0.0126341
$216$	−1.00777e7	−0.0680414
$217$	5.27408e7	0.350379
$218$	−1.35722e8	−0.887266
$219$	2.31327e7	0.148824
$220$	−4.29025e6	−0.0271646
$221$	3.75865e8	2.34239
$222$	−5.15606e7	−0.316288
$223$	−1.19141e8	−0.719438	−0.359719	−	0.933061i	$-0.617127\pi$
$223$	−1.19141e8	−0.719438	−0.359719	+	0.933061i	$0.617127\pi$
$224$	4.55587e7	0.270834
$225$	−5.69090e7	−0.333075
$226$	−8.82409e7	−0.508499
$227$	1.92774e8	1.09385	0.546924	−	0.837182i	$-0.315799\pi$
$227$	1.92774e8	1.09385	0.546924	+	0.837182i	$0.315799\pi$
$228$	8.29721e7	0.463618
$229$	−2.32843e8	−1.28127	−0.640633	−	0.767847i	$-0.721328\pi$
$229$	−2.32843e8	−1.28127	−0.640633	+	0.767847i	$0.721328\pi$
$230$	757393.	0.00410463
$231$	3.23400e8	1.72623
$232$	8.74406e7	0.459733
$233$	1.32846e8	0.688022	0.344011	−	0.938966i	$-0.388214\pi$
$233$	1.32846e8	0.688022	0.344011	+	0.938966i	$0.388214\pi$
$234$	5.70415e7	0.291029
$235$	8.27006e6	0.0415692
$236$	1.16976e8	0.579301
$237$	−1.03291e8	−0.504017
$238$	4.27434e8	2.05518
$239$	−1.52455e8	−0.722354	−0.361177	−	0.932497i	$-0.617625\pi$
$239$	−1.52455e8	−0.722354	−0.361177	+	0.932497i	$0.617625\pi$
$240$	−860541.	−0.00401820
$241$	2.91715e8	1.34245	0.671226	−	0.741252i	$-0.265768\pi$
$241$	2.91715e8	1.34245	0.671226	+	0.741252i	$0.265768\pi$
$242$	4.37846e8	1.98595
$243$	−1.43489e7	−0.0641500
$244$	1.53253e7	0.0675375
$245$	8.63331e6	0.0375056
$246$	−1.68410e8	−0.721267
$247$	−4.69637e8	−1.98300
$248$	1.94221e7	0.0808564
$249$	−1.27082e8	−0.521657
$250$	−9.72276e6	−0.0393549
$251$	2.12269e8	0.847282	0.423641	−	0.905830i	$-0.360752\pi$
$251$	2.12269e8	0.847282	0.423641	+	0.905830i	$0.360752\pi$
$252$	6.48678e7	0.255345
$253$	−1.04819e8	−0.406927
$254$	−3.32857e8	−1.27450
$255$	−8.07364e6	−0.0304915
$256$	1.67772e7	0.0625000
$257$	−1.73470e8	−0.637468	−0.318734	−	0.947844i	$-0.603258\pi$
$257$	−1.73470e8	−0.637468	−0.318734	+	0.947844i	$0.603258\pi$
$258$	5.11076e7	0.185275
$259$	3.31883e8	1.18696
$260$	4.87081e6	0.0171868
$261$	1.24500e8	0.433440
$262$	3.13195e7	0.107587
$263$	−3.02018e8	−1.02373	−0.511867	−	0.859065i	$-0.671046\pi$
$263$	−3.02018e8	−1.02373	−0.511867	+	0.859065i	$0.671046\pi$
$264$	1.19094e8	0.398359
$265$	5.34785e6	0.0176530
$266$	−5.34072e8	−1.73986
$267$	1.36074e8	0.437509
$268$	−1.26148e8	−0.400321
$269$	1.33112e8	0.416951	0.208475	−	0.978028i	$-0.433150\pi$
$269$	1.33112e8	0.416951	0.208475	+	0.978028i	$0.433150\pi$
$270$	−1.22526e6	−0.00378840
$271$	3.21436e8	0.981073	0.490537	−	0.871421i	$-0.336801\pi$
$271$	3.21436e8	0.981073	0.490537	+	0.871421i	$0.336801\pi$
$272$	1.57405e8	0.474271
$273$	−3.67163e8	−1.09217
$274$	9.23872e7	0.271322
$275$	6.72524e8	1.95004
$276$	−2.10246e7	−0.0601929
$277$	−3.88008e8	−1.09689	−0.548443	−	0.836188i	$-0.684779\pi$
$277$	−3.88008e8	−1.09689	−0.548443	+	0.836188i	$0.684779\pi$
$278$	−2.25931e8	−0.630696
$279$	2.76537e7	0.0762321
$280$	5.53910e6	0.0150795
$281$	−4.11514e8	−1.10640	−0.553200	−	0.833049i	$-0.686593\pi$
$281$	−4.11514e8	−1.10640	−0.553200	+	0.833049i	$0.686593\pi$
$282$	−2.29570e8	−0.609597
$283$	6.00982e8	1.57619	0.788096	−	0.615552i	$-0.211067\pi$
$283$	6.00982e8	1.57619	0.788096	+	0.615552i	$0.211067\pi$
$284$	−2.80240e6	−0.00725966
$285$	1.00879e7	0.0258132
$286$	−6.74091e8	−1.70387
$287$	1.08402e9	2.70676
$288$	2.38879e7	0.0589256
$289$	1.06644e9	2.59893
$290$	1.06312e7	0.0255969
$291$	1.18121e8	0.280996
$292$	−5.48331e7	−0.128885
$293$	−3.00074e8	−0.696934	−0.348467	−	0.937321i	$-0.613298\pi$
$293$	−3.00074e8	−0.696934	−0.348467	+	0.937321i	$0.613298\pi$
$294$	−2.39653e8	−0.550006
$295$	1.42221e7	0.0322542
$296$	1.22218e8	0.273913
$297$	1.69569e8	0.375576
$298$	−3.78610e8	−0.828771
$299$	1.19003e8	0.257459
$300$	1.34895e8	0.288451
$301$	−3.28967e8	−0.695296
$302$	1.90799e8	0.398613
$303$	−2.56840e8	−0.530413
$304$	−1.96675e8	−0.401505
$305$	1.86328e6	0.00376035
$306$	2.24117e8	0.447147
$307$	1.19924e8	0.236549	0.118275	−	0.992981i	$-0.462264\pi$
$307$	1.19924e8	0.236549	0.118275	+	0.992981i	$0.462264\pi$
$308$	−7.66577e8	−1.49496
$309$	−1.04933e8	−0.202328
$310$	2.36136e6	0.00450191
$311$	−3.17140e8	−0.597847	−0.298923	−	0.954277i	$-0.596628\pi$
$311$	−3.17140e8	−0.597847	−0.298923	+	0.954277i	$0.596628\pi$
$312$	−1.35210e8	−0.252038
$313$	2.15125e8	0.396539	0.198270	−	0.980148i	$-0.436468\pi$
$313$	2.15125e8	0.396539	0.198270	+	0.980148i	$0.436468\pi$
$314$	1.79816e7	0.0327775
$315$	7.88672e6	0.0142171
$316$	2.44839e8	0.436491
$317$	1.12932e9	1.99118	0.995588	−	0.0938311i	$-0.0299114\pi$
$317$	1.12932e9	1.99118	0.995588	+	0.0938311i	$0.0299114\pi$
$318$	−1.48452e8	−0.258875
$319$	−1.47129e9	−2.53764
$320$	2.03980e6	0.00347987
$321$	2.80268e8	0.472939
$322$	1.35330e8	0.225891
$323$	−1.84521e9	−3.04675
$324$	3.40122e7	0.0555556
$325$	−7.63532e8	−1.23377
$326$	2.58108e8	0.412611
$327$	4.58063e8	0.724449
$328$	3.99195e8	0.624636
$329$	1.47769e9	2.28769
$330$	1.44796e7	0.0221798
$331$	−4.89675e8	−0.742181	−0.371090	−	0.928597i	$-0.621016\pi$
$331$	−4.89675e8	−0.742181	−0.371090	+	0.928597i	$0.621016\pi$
$332$	3.01231e8	0.451768
$333$	1.74017e8	0.258248
$334$	2.82375e8	0.414681
$335$	−1.53373e7	−0.0222890
$336$	−1.53761e8	−0.221135
$337$	1.07432e8	0.152908	0.0764539	−	0.997073i	$-0.475640\pi$
$337$	1.07432e8	0.152908	0.0764539	+	0.997073i	$0.475640\pi$
$338$	2.63322e8	0.370919
$339$	2.97813e8	0.415188
$340$	1.91375e7	0.0264064
$341$	−3.26798e8	−0.446313
$342$	−2.80031e8	−0.378542
$343$	3.97585e8	0.531987
$344$	−1.21144e8	−0.160453
$345$	−2.55620e6	−0.00335141
$346$	−3.26212e8	−0.423383
$347$	−8.31135e8	−1.06787	−0.533935	−	0.845526i	$-0.679287\pi$
$347$	−8.31135e8	−1.06787	−0.533935	+	0.845526i	$0.679287\pi$
$348$	−2.95112e8	−0.375370
$349$	−1.13375e9	−1.42767	−0.713837	−	0.700311i	$-0.753044\pi$
$349$	−1.13375e9	−1.42767	−0.713837	+	0.700311i	$0.753044\pi$
$350$	−8.68290e8	−1.08250
$351$	−1.92515e8	−0.237624
$352$	−2.82296e8	−0.344989
$353$	−1.42200e9	−1.72064	−0.860318	−	0.509757i	$-0.829735\pi$
$353$	−1.42200e9	−1.72064	−0.860318	+	0.509757i	$0.829735\pi$
$354$	−3.94793e8	−0.472997
$355$	−340721.	−0.000404203 0
$356$	−3.22546e8	−0.378894
$357$	−1.44259e9	−1.67805
$358$	−8.61481e8	−0.992327
$359$	2.17695e8	0.248323	0.124161	−	0.992262i	$-0.460376\pi$
$359$	2.17695e8	0.248323	0.124161	+	0.992262i	$0.460376\pi$
$360$	2.90432e6	0.00328085
$361$	1.41169e9	1.57930
$362$	5.67761e8	0.629050
$363$	−1.47773e9	−1.62152
$364$	8.70312e8	0.945846
$365$	−6.66670e6	−0.00717606
$366$	−5.17229e7	−0.0551442
$367$	−6.00895e8	−0.634553	−0.317276	−	0.948333i	$-0.602768\pi$
$367$	−6.00895e8	−0.634553	−0.317276	+	0.948333i	$0.602768\pi$
$368$	4.98360e7	0.0521286
$369$	5.68385e8	0.588912
$370$	1.48594e7	0.0152509
$371$	9.55549e8	0.971503
$372$	−6.55494e7	−0.0660189
$373$	1.19265e9	1.18995	0.594977	−	0.803743i	$-0.297161\pi$
$373$	1.19265e9	1.18995	0.594977	+	0.803743i	$0.297161\pi$
$374$	−2.64851e9	−2.61789
$375$	3.28143e7	0.0321332
$376$	5.44165e8	0.527927
$377$	1.67039e9	1.60554
$378$	−2.18929e8	−0.208488
$379$	−5.81844e8	−0.548996	−0.274498	−	0.961588i	$-0.588512\pi$
$379$	−5.81844e8	−0.548996	−0.274498	+	0.961588i	$0.588512\pi$
$380$	−2.39120e7	−0.0223549
$381$	1.12339e9	1.04062
$382$	1.06035e9	0.973254
$383$	715905.	0.000651118 0	0.000325559	−	1.00000i	$-0.499896\pi$
$383$	715905.	0.000651118 0	0.000325559		1.00000i	$0.499896\pi$
$384$	−5.66231e7	−0.0510310
$385$	−9.32017e7	−0.0832360
$386$	3.16923e8	0.280477
$387$	−1.72488e8	−0.151276
$388$	−2.79990e8	−0.243350
$389$	−1.08782e9	−0.936989	−0.468495	−	0.883466i	$-0.655204\pi$
$389$	−1.08782e9	−0.936989	−0.468495	+	0.883466i	$0.655204\pi$
$390$	−1.64390e7	−0.0140330
$391$	4.67564e8	0.395569
$392$	5.68067e8	0.476319
$393$	−1.05703e8	−0.0878445
$394$	−1.03857e9	−0.855456
$395$	2.97679e7	0.0243029
$396$	−4.01941e8	−0.325259
$397$	7.45984e8	0.598360	0.299180	−	0.954197i	$-0.403287\pi$
$397$	7.45984e8	0.598360	0.299180	+	0.954197i	$0.403287\pi$
$398$	−8.45486e8	−0.672226
$399$	1.80249e9	1.42059
$400$	−3.19752e8	−0.249806
$401$	−4.74819e7	−0.0367724	−0.0183862	−	0.999831i	$-0.505853\pi$
$401$	−4.74819e7	−0.0367724	−0.0183862	+	0.999831i	$0.505853\pi$
$402$	4.25750e8	0.326861
$403$	3.71021e8	0.282378
$404$	6.08807e8	0.459351
$405$	4.13526e6	0.00309322
$406$	1.89957e9	1.40868
$407$	−2.05645e9	−1.51195
$408$	−5.31241e8	−0.387241
$409$	−1.12933e9	−0.816187	−0.408093	−	0.912940i	$-0.633806\pi$
$409$	−1.12933e9	−0.816187	−0.408093	+	0.912940i	$0.633806\pi$
$410$	4.85348e7	0.0347784
$411$	−3.11807e8	−0.221533
$412$	2.48730e8	0.175222
$413$	2.54119e9	1.77506
$414$	7.09579e7	0.0491473
$415$	3.66241e7	0.0251535
$416$	3.20497e8	0.218271
$417$	7.62519e8	0.514961
$418$	3.30927e9	2.21623
$419$	−2.63859e9	−1.75236	−0.876178	−	0.481989i	$-0.839915\pi$
$419$	−2.63859e9	−1.75236	−0.876178	+	0.481989i	$0.839915\pi$
$420$	−1.86945e7	−0.0123123
$421$	−9.35694e8	−0.611148	−0.305574	−	0.952168i	$-0.598848\pi$
$421$	−9.35694e8	−0.611148	−0.305574	+	0.952168i	$0.598848\pi$
$422$	−1.29910e9	−0.841489
$423$	7.74798e8	0.497734
$424$	3.51886e8	0.224192
$425$	−2.99993e9	−1.89561
$426$	9.45811e6	0.00592749
$427$	3.32928e8	0.206944
$428$	−6.64338e8	−0.409577
$429$	2.27506e9	1.39121
$430$	−1.47289e7	−0.00893366
$431$	3.58066e8	0.215423	0.107712	−	0.994182i	$-0.465648\pi$
$431$	3.58066e8	0.215423	0.107712	+	0.994182i	$0.465648\pi$
$432$	−8.06216e7	−0.0481125
$433$	−1.00469e9	−0.594735	−0.297368	−	0.954763i	$-0.596109\pi$
$433$	−1.00469e9	−0.594735	−0.297368	+	0.954763i	$0.596109\pi$
$434$	4.21926e8	0.247755
$435$	−3.58802e7	−0.0208998
$436$	−1.08578e9	−0.627391
$437$	−5.84214e8	−0.334878
$438$	1.85062e8	0.105234
$439$	6.25866e8	0.353066	0.176533	−	0.984295i	$-0.443512\pi$
$439$	6.25866e8	0.353066	0.176533	+	0.984295i	$0.443512\pi$
$440$	−3.43220e7	−0.0192083
$441$	8.08829e8	0.449078
$442$	3.00692e9	1.65632
$443$	1.08909e9	0.595184	0.297592	−	0.954693i	$-0.403817\pi$
$443$	1.08909e9	0.595184	0.297592	+	0.954693i	$0.403817\pi$
$444$	−4.12485e8	−0.223649
$445$	−3.92157e7	−0.0210960
$446$	−9.53126e8	−0.508719
$447$	1.27781e9	0.676689
$448$	3.64470e8	0.191509
$449$	2.89062e8	0.150705	0.0753526	−	0.997157i	$-0.475992\pi$
$449$	2.89062e8	0.150705	0.0753526	+	0.997157i	$0.475992\pi$
$450$	−4.55272e8	−0.235520
$451$	−6.71691e9	−3.44788
$452$	−7.05927e8	−0.359563
$453$	−6.43945e8	−0.325466
$454$	1.54219e9	0.773468
$455$	1.05814e8	0.0526627
$456$	6.63777e8	0.327827
$457$	1.91786e8	0.0939959	0.0469980	−	0.998895i	$-0.485035\pi$
$457$	1.91786e8	0.0939959	0.0469980	+	0.998895i	$0.485035\pi$
$458$	−1.86274e9	−0.905992
$459$	−7.56396e8	−0.365094
$460$	6.05914e6	0.00290241
$461$	−2.42679e9	−1.15366	−0.576831	−	0.816864i	$-0.695711\pi$
$461$	−2.42679e9	−1.15366	−0.576831	+	0.816864i	$0.695711\pi$
$462$	2.58720e9	1.22063
$463$	2.42784e9	1.13680	0.568402	−	0.822751i	$-0.307562\pi$
$463$	2.42784e9	1.13680	0.568402	+	0.822751i	$0.307562\pi$
$464$	6.99525e8	0.325080
$465$	−7.96960e6	−0.00367580
$466$	1.06277e9	0.486505
$467$	9.35757e7	0.0425162	0.0212581	−	0.999774i	$-0.493233\pi$
$467$	9.35757e7	0.0425162	0.0212581	+	0.999774i	$0.493233\pi$
$468$	4.56332e8	0.205788
$469$	−2.74045e9	−1.22664
$470$	6.61605e7	0.0293938
$471$	−6.06880e7	−0.0267627
$472$	9.35806e8	0.409627
$473$	2.03838e9	0.885670
$474$	−8.26332e8	−0.356394
$475$	3.74836e9	1.60477
$476$	3.41947e9	1.45323
$477$	5.01025e8	0.211371
$478$	−1.21964e9	−0.510781
$479$	−2.21042e9	−0.918966	−0.459483	−	0.888186i	$-0.651965\pi$
$479$	−2.21042e9	−0.918966	−0.459483	+	0.888186i	$0.651965\pi$
$480$	−6.88433e6	−0.00284130
$481$	2.33474e9	0.956599
$482$	2.33372e9	0.949258
$483$	−4.56740e8	−0.184439
$484$	3.50277e9	1.40428
$485$	−3.40416e7	−0.0135492
$486$	−1.14791e8	−0.0453609
$487$	−5.51385e8	−0.216323	−0.108162	−	0.994133i	$-0.534496\pi$
$487$	−5.51385e8	−0.216323	−0.108162	+	0.994133i	$0.534496\pi$
$488$	1.22603e8	0.0477562
$489$	−8.71115e8	−0.336895
$490$	6.90664e7	0.0265204
$491$	1.43771e9	0.548132	0.274066	−	0.961711i	$-0.411631\pi$
$491$	1.43771e9	0.548132	0.274066	+	0.961711i	$0.411631\pi$
$492$	−1.34728e9	−0.510013
$493$	6.56298e9	2.46682
$494$	−3.75709e9	−1.40219
$495$	−4.88686e7	−0.0181097
$496$	1.55376e8	0.0571741
$497$	−6.08796e7	−0.0222446
$498$	−1.01665e9	−0.368867
$499$	−1.02653e9	−0.369847	−0.184923	−	0.982753i	$-0.559204\pi$
$499$	−1.02653e9	−0.369847	−0.184923	+	0.982753i	$0.559204\pi$
$500$	−7.77820e7	−0.0278282
$501$	−9.53016e8	−0.338585
$502$	1.69815e9	0.599119
$503$	−1.73757e8	−0.0608773	−0.0304386	−	0.999537i	$-0.509690\pi$
$503$	−1.73757e8	−0.0608773	−0.0304386	+	0.999537i	$0.509690\pi$
$504$	5.18942e8	0.180556
$505$	7.40197e7	0.0255757
$506$	−8.38548e8	−0.287741
$507$	−8.88712e8	−0.302854
$508$	−2.66286e9	−0.901207
$509$	4.90001e9	1.64697	0.823483	−	0.567341i	$-0.192028\pi$
$509$	4.90001e9	1.64697	0.823483	+	0.567341i	$0.192028\pi$
$510$	−6.45891e7	−0.0215607
$511$	−1.19120e9	−0.394922
$512$	1.34218e8	0.0441942
$513$	9.45104e8	0.309078
$514$	−1.38776e9	−0.450758
$515$	3.02410e7	0.00975597
$516$	4.08860e8	0.131009
$517$	−9.15621e9	−2.91406
$518$	2.65507e9	0.839307
$519$	1.10097e9	0.345691
$520$	3.89665e7	0.0121529
$521$	−2.45922e9	−0.761843	−0.380922	−	0.924607i	$-0.624393\pi$
$521$	−2.45922e9	−0.761843	−0.380922	+	0.924607i	$0.624393\pi$
$522$	9.96003e8	0.306488
$523$	−8.80733e8	−0.269208	−0.134604	−	0.990899i	$-0.542976\pi$
$523$	−8.80733e8	−0.269208	−0.134604	+	0.990899i	$0.542976\pi$
$524$	2.50556e8	0.0760756
$525$	2.93048e9	0.883855
$526$	−2.41614e9	−0.723890
$527$	1.45775e9	0.433856
$528$	9.52748e8	0.281682
$529$	1.48036e8	0.0434783
$530$	4.27828e7	0.0124826
$531$	1.33243e9	0.386200
$532$	−4.27257e9	−1.23027
$533$	7.62586e9	2.18144
$534$	1.08859e9	0.309365
$535$	−8.07713e7	−0.0228044
$536$	−1.00918e9	−0.283070
$537$	2.90750e9	0.810232
$538$	1.06490e9	0.294829
$539$	−9.55837e9	−2.62920
$540$	−9.80210e6	−0.00267880
$541$	−2.26240e9	−0.614298	−0.307149	−	0.951661i	$-0.599375\pi$
$541$	−2.26240e9	−0.614298	−0.307149	+	0.951661i	$0.599375\pi$
$542$	2.57148e9	0.693723
$543$	−1.91619e9	−0.513617
$544$	1.25924e9	0.335360
$545$	−1.32011e8	−0.0349318
$546$	−2.93730e9	−0.772280
$547$	2.40632e9	0.628633	0.314317	−	0.949318i	$-0.398225\pi$
$547$	2.40632e9	0.628633	0.314317	+	0.949318i	$0.398225\pi$
$548$	7.39097e8	0.191853
$549$	1.74565e8	0.0450250
$550$	5.38019e9	1.37889
$551$	−8.20033e9	−2.08834
$552$	−1.68197e8	−0.0425628
$553$	5.31890e9	1.33747
$554$	−3.10406e9	−0.775615
$555$	−5.01505e7	−0.0124523
$556$	−1.80745e9	−0.445969
$557$	−3.95823e9	−0.970528	−0.485264	−	0.874368i	$-0.661277\pi$
$557$	−3.95823e9	−0.970528	−0.485264	+	0.874368i	$0.661277\pi$
$558$	2.21229e8	0.0539042
$559$	−2.31422e9	−0.560356
$560$	4.43128e7	0.0106628
$561$	8.93874e9	2.13750
$562$	−3.29211e9	−0.782343
$563$	1.30128e9	0.307320	0.153660	−	0.988124i	$-0.450894\pi$
$563$	1.30128e9	0.307320	0.153660	+	0.988124i	$0.450894\pi$
$564$	−1.83656e9	−0.431050
$565$	−8.58277e7	−0.0200197
$566$	4.80786e9	1.11454
$567$	7.38884e8	0.170230
$568$	−2.24192e7	−0.00513336
$569$	−4.20804e9	−0.957606	−0.478803	−	0.877922i	$-0.658929\pi$
$569$	−4.20804e9	−0.957606	−0.478803	+	0.877922i	$0.658929\pi$
$570$	8.07030e7	0.0182527
$571$	8.85997e9	1.99162	0.995809	−	0.0914620i	$-0.0291540\pi$
$571$	8.85997e9	1.99162	0.995809	+	0.0914620i	$0.0291540\pi$
$572$	−5.39272e9	−1.20482
$573$	−3.57867e9	−0.794659
$574$	8.67215e9	1.91397
$575$	−9.49810e8	−0.208353
$576$	1.91103e8	0.0416667
$577$	−5.36089e9	−1.16177	−0.580887	−	0.813984i	$-0.697294\pi$
$577$	−5.36089e9	−1.16177	−0.580887	+	0.813984i	$0.697294\pi$
$578$	8.53152e9	1.83772
$579$	−1.06961e9	−0.229009
$580$	8.50493e7	0.0180998
$581$	6.54396e9	1.38428
$582$	9.44966e8	0.198694
$583$	−5.92088e9	−1.23750
$584$	−4.38665e8	−0.0911356
$585$	5.54816e7	0.0114579
$586$	−2.40059e9	−0.492807
$587$	9.29319e9	1.89641	0.948204	−	0.317663i	$-0.102898\pi$
$587$	9.29319e9	1.89641	0.948204	+	0.317663i	$0.102898\pi$
$588$	−1.91723e9	−0.388913
$589$	−1.82143e9	−0.367291
$590$	1.13777e8	0.0228072
$591$	3.50516e9	0.698476
$592$	9.77741e8	0.193686
$593$	−3.50365e9	−0.689968	−0.344984	−	0.938609i	$-0.612116\pi$
$593$	−3.50365e9	−0.689968	−0.344984	+	0.938609i	$0.612116\pi$
$594$	1.35655e9	0.265573
$595$	4.15745e8	0.0809128
$596$	−3.02888e9	−0.586030
$597$	2.85351e9	0.548871
$598$	9.52023e8	0.182051
$599$	−7.21918e9	−1.37244	−0.686221	−	0.727393i	$-0.740732\pi$
$599$	−7.21918e9	−1.37244	−0.686221	+	0.727393i	$0.740732\pi$
$600$	1.07916e9	0.203966
$601$	−1.57303e9	−0.295582	−0.147791	−	0.989019i	$-0.547216\pi$
$601$	−1.57303e9	−0.295582	−0.147791	+	0.989019i	$0.547216\pi$
$602$	−2.63174e9	−0.491649
$603$	−1.43691e9	−0.266881
$604$	1.52639e9	0.281862
$605$	4.25872e8	0.0781872
$606$	−2.05472e9	−0.375059
$607$	−6.90549e9	−1.25324	−0.626620	−	0.779325i	$-0.715562\pi$
$607$	−6.90549e9	−1.25324	−0.626620	+	0.779325i	$0.715562\pi$
$608$	−1.57340e9	−0.283907
$609$	−6.41104e9	−1.15019
$610$	1.49062e7	0.00265897
$611$	1.03952e10	1.84370
$612$	1.79294e9	0.316181
$613$	−3.96481e9	−0.695201	−0.347600	−	0.937643i	$-0.613003\pi$
$613$	−3.96481e9	−0.695201	−0.347600	+	0.937643i	$0.613003\pi$
$614$	9.59391e8	0.167265
$615$	−1.63805e8	−0.0283964
$616$	−6.13262e9	−1.05709
$617$	3.02591e8	0.0518630	0.0259315	−	0.999664i	$-0.491745\pi$
$617$	3.02591e8	0.0518630	0.0259315	+	0.999664i	$0.491745\pi$
$618$	−8.39463e8	−0.143068
$619$	−2.90927e9	−0.493023	−0.246511	−	0.969140i	$-0.579284\pi$
$619$	−2.90927e9	−0.493023	−0.246511	+	0.969140i	$0.579284\pi$
$620$	1.88909e7	0.00318333
$621$	−2.39483e8	−0.0401286
$622$	−2.53712e9	−0.422742
$623$	−7.00703e9	−1.16098
$624$	−1.08168e9	−0.178218
$625$	6.08933e9	0.997676
$626$	1.72100e9	0.280395
$627$	−1.11688e10	−1.80955
$628$	1.43853e8	0.0231772
$629$	9.17322e9	1.46975
$630$	6.30938e7	0.0100530
$631$	8.50191e9	1.34714	0.673572	−	0.739122i	$-0.264759\pi$
$631$	8.50191e9	1.34714	0.673572	+	0.739122i	$0.264759\pi$
$632$	1.95871e9	0.308646
$633$	4.38446e9	0.687073
$634$	9.03456e9	1.40797
$635$	−3.23754e8	−0.0501773
$636$	−1.18761e9	−0.183052
$637$	1.08518e10	1.66347
$638$	−1.17703e10	−1.79438
$639$	−3.19211e7	−0.00483978
$640$	1.63184e7	0.00246064
$641$	−3.13720e9	−0.470479	−0.235239	−	0.971937i	$-0.575587\pi$
$641$	−3.13720e9	−0.470479	−0.235239	+	0.971937i	$0.575587\pi$
$642$	2.24214e9	0.334419
$643$	−1.32367e10	−1.96355	−0.981776	−	0.190039i	$-0.939138\pi$
$643$	−1.32367e10	−1.96355	−0.981776	+	0.190039i	$0.939138\pi$
$644$	1.08264e9	0.159729
$645$	4.97099e7	0.00729430
$646$	−1.47617e10	−2.15438
$647$	−8.09375e8	−0.117486	−0.0587428	−	0.998273i	$-0.518709\pi$
$647$	−8.09375e8	−0.117486	−0.0587428	+	0.998273i	$0.518709\pi$
$648$	2.72098e8	0.0392837
$649$	−1.57460e10	−2.26107
$650$	−6.10825e9	−0.872409
$651$	−1.42400e9	−0.202291
$652$	2.06487e9	0.291760
$653$	8.50711e9	1.19560	0.597800	−	0.801645i	$-0.296042\pi$
$653$	8.50711e9	1.19560	0.597800	+	0.801645i	$0.296042\pi$
$654$	3.66450e9	0.512263
$655$	3.04630e7	0.00423573
$656$	3.19356e9	0.441684
$657$	−6.24584e8	−0.0859235
$658$	1.18215e10	1.61764
$659$	3.95637e9	0.538515	0.269257	−	0.963068i	$-0.413222\pi$
$659$	3.95637e9	0.538515	0.269257	+	0.963068i	$0.413222\pi$
$660$	1.15837e8	0.0156835
$661$	−6.83238e9	−0.920168	−0.460084	−	0.887875i	$-0.652181\pi$
$661$	−6.83238e9	−0.920168	−0.460084	+	0.887875i	$0.652181\pi$
$662$	−3.91740e9	−0.524801
$663$	−1.01483e10	−1.35238
$664$	2.40984e9	0.319448
$665$	−5.19466e8	−0.0684986
$666$	1.39214e9	0.182609
$667$	2.07791e9	0.271135
$668$	2.25900e9	0.293224
$669$	3.21680e9	0.415368
$670$	−1.22698e8	−0.0157607
$671$	−2.06293e9	−0.263606
$672$	−1.23008e9	−0.156366
$673$	1.48903e9	0.188300	0.0941499	−	0.995558i	$-0.469987\pi$
$673$	1.48903e9	0.188300	0.0941499	+	0.995558i	$0.469987\pi$
$674$	8.59457e8	0.108122
$675$	1.53654e9	0.192301
$676$	2.10658e9	0.262279
$677$	−1.33089e10	−1.64847	−0.824236	−	0.566246i	$-0.808395\pi$
$677$	−1.33089e10	−1.64847	−0.824236	+	0.566246i	$0.808395\pi$
$678$	2.38250e9	0.293582
$679$	−6.08252e9	−0.745657
$680$	1.53100e8	0.0186721
$681$	−5.20489e9	−0.631534
$682$	−2.61439e9	−0.315591
$683$	5.45708e8	0.0655372	0.0327686	−	0.999463i	$-0.489568\pi$
$683$	5.45708e8	0.0655372	0.0327686	+	0.999463i	$0.489568\pi$
$684$	−2.24025e9	−0.267670
$685$	8.98606e7	0.0106820
$686$	3.18068e9	0.376172
$687$	6.28676e9	0.739739
$688$	−9.69151e8	−0.113457
$689$	6.72210e9	0.782957
$690$	−2.04496e7	−0.00236981
$691$	−5.71714e9	−0.659182	−0.329591	−	0.944124i	$-0.606911\pi$
$691$	−5.71714e9	−0.659182	−0.329591	+	0.944124i	$0.606911\pi$
$692$	−2.60970e9	−0.299377
$693$	−8.73179e9	−0.996637
$694$	−6.64908e9	−0.755098
$695$	−2.19753e8	−0.0248306
$696$	−2.36090e9	−0.265427
$697$	2.99622e10	3.35165
$698$	−9.07002e9	−1.00952
$699$	−3.58684e9	−0.397230
$700$	−6.94632e9	−0.765441
$701$	1.46364e10	1.60480	0.802402	−	0.596784i	$-0.203555\pi$
$701$	1.46364e10	1.60480	0.802402	+	0.596784i	$0.203555\pi$
$702$	−1.54012e9	−0.168025
$703$	−1.14618e10	−1.24425
$704$	−2.25837e9	−0.243944
$705$	−2.23292e8	−0.0240000
$706$	−1.13760e10	−1.21667
$707$	1.32258e10	1.40752
$708$	−3.15835e9	−0.334459
$709$	9.83890e9	1.03678	0.518388	−	0.855146i	$-0.326532\pi$
$709$	9.83890e9	1.03678	0.518388	+	0.855146i	$0.326532\pi$
$710$	−2.72576e6	−0.000285814 0
$711$	2.78887e9	0.290994
$712$	−2.58037e9	−0.267918
$713$	4.61539e8	0.0476865
$714$	−1.15407e10	−1.18656
$715$	−6.55656e8	−0.0670818
$716$	−6.89185e9	−0.701681
$717$	4.11629e9	0.417051
$718$	1.74156e9	0.175591
$719$	1.29432e8	0.0129864	0.00649321	−	0.999979i	$-0.497933\pi$
$719$	1.29432e8	0.0129864	0.00649321	+	0.999979i	$0.497933\pi$
$720$	2.32346e7	0.00231991
$721$	5.40343e9	0.536903
$722$	1.12935e10	1.11673
$723$	−7.87630e9	−0.775066
$724$	4.54209e9	0.444806
$725$	−1.33320e10	−1.29931
$726$	−1.18219e10	−1.14659
$727$	−3.49368e9	−0.337219	−0.168610	−	0.985683i	$-0.553928\pi$
$727$	−3.49368e9	−0.337219	−0.168610	+	0.985683i	$0.553928\pi$
$728$	6.96250e9	0.668814
$729$	3.87420e8	0.0370370
$730$	−5.33336e7	−0.00507424
$731$	−9.09262e9	−0.860951
$732$	−4.13784e8	−0.0389928
$733$	−4.76309e9	−0.446709	−0.223355	−	0.974737i	$-0.571701\pi$
$733$	−4.76309e9	−0.446709	−0.223355	+	0.974737i	$0.571701\pi$
$734$	−4.80716e9	−0.448696
$735$	−2.33099e8	−0.0216539
$736$	3.98688e8	0.0368605
$737$	1.69807e10	1.56250
$738$	4.54708e9	0.416424
$739$	1.43304e10	1.30618	0.653090	−	0.757281i	$-0.273472\pi$
$739$	1.43304e10	1.30618	0.653090	+	0.757281i	$0.273472\pi$
$740$	1.18875e8	0.0107840
$741$	1.26802e10	1.14489
$742$	7.64439e9	0.686956
$743$	7.90577e9	0.707104	0.353552	−	0.935415i	$-0.384974\pi$
$743$	7.90577e9	0.707104	0.353552	+	0.935415i	$0.384974\pi$
$744$	−5.24395e8	−0.0466824
$745$	−3.68256e8	−0.0326289
$746$	9.54116e9	0.841425
$747$	3.43120e9	0.301179
$748$	−2.11881e10	−1.85113
$749$	−1.44321e10	−1.25500
$750$	2.62514e8	0.0227216
$751$	−1.56808e10	−1.35092	−0.675460	−	0.737397i	$-0.736055\pi$
$751$	−1.56808e10	−1.35092	−0.675460	+	0.737397i	$0.736055\pi$
$752$	4.35332e9	0.373300
$753$	−5.73126e9	−0.489179
$754$	1.33631e10	1.13529
$755$	1.85581e8	0.0156935
$756$	−1.75143e9	−0.147423
$757$	1.37498e10	1.15202	0.576010	−	0.817443i	$-0.304609\pi$
$757$	1.37498e10	1.15202	0.576010	+	0.817443i	$0.304609\pi$
$758$	−4.65475e9	−0.388199
$759$	2.83010e9	0.234939
$760$	−1.91296e8	−0.0158073
$761$	2.27732e10	1.87317	0.936586	−	0.350437i	$-0.113967\pi$
$761$	2.27732e10	1.87317	0.936586	+	0.350437i	$0.113967\pi$
$762$	8.98714e9	0.735833
$763$	−2.35875e10	−1.92241
$764$	8.48278e9	0.688195
$765$	2.17988e8	0.0176043
$766$	5.72724e6	0.000460410 0
$767$	1.78768e10	1.43056
$768$	−4.52985e8	−0.0360844
$769$	1.42336e10	1.12869	0.564343	−	0.825541i	$-0.309130\pi$
$769$	1.42336e10	1.12869	0.564343	+	0.825541i	$0.309130\pi$
$770$	−7.45613e8	−0.0588567
$771$	4.68369e9	0.368042
$772$	2.53538e9	0.198327
$773$	6.91927e9	0.538805	0.269403	−	0.963028i	$-0.413174\pi$
$773$	6.91927e9	0.538805	0.269403	+	0.963028i	$0.413174\pi$
$774$	−1.37990e9	−0.106968
$775$	−2.96127e9	−0.228519
$776$	−2.23992e9	−0.172074
$777$	−8.96085e9	−0.685292
$778$	−8.70259e9	−0.662552
$779$	−3.74372e10	−2.83741
$780$	−1.31512e8	−0.00992280
$781$	3.77229e8	0.0283352
$782$	3.74051e9	0.279710
$783$	−3.36151e9	−0.250247
$784$	4.54453e9	0.336809
$785$	1.74899e7	0.00129046
$786$	−8.45626e8	−0.0621154
$787$	9.35393e9	0.684042	0.342021	−	0.939692i	$-0.388889\pi$
$787$	9.35393e9	0.684042	0.342021	+	0.939692i	$0.388889\pi$
$788$	−8.30853e9	−0.604898
$789$	8.15448e9	0.591053
$790$	2.38143e8	0.0171848
$791$	−1.53356e10	−1.10175
$792$	−3.21553e9	−0.229993
$793$	2.34209e9	0.166781
$794$	5.96787e9	0.423105
$795$	−1.44392e8	−0.0101920
$796$	−6.76389e9	−0.475336
$797$	2.37193e9	0.165958	0.0829789	−	0.996551i	$-0.473557\pi$
$797$	2.37193e9	0.165958	0.0829789	+	0.996551i	$0.473557\pi$
$798$	1.44199e10	1.00451
$799$	4.08431e10	2.83273
$800$	−2.55802e9	−0.176640
$801$	−3.67401e9	−0.252596
$802$	−3.79855e8	−0.0260020
$803$	7.38104e9	0.503053
$804$	3.40600e9	0.231126
$805$	1.31629e8	0.00889338
$806$	2.96817e9	0.199672
$807$	−3.59403e9	−0.240727
$808$	4.87046e9	0.324810
$809$	2.83538e10	1.88274	0.941371	−	0.337373i	$-0.109538\pi$
$809$	2.83538e10	1.88274	0.941371	+	0.337373i	$0.109538\pi$
$810$	3.30821e7	0.00218723
$811$	2.80518e10	1.84666	0.923330	−	0.384007i	$-0.125456\pi$
$811$	2.80518e10	1.84666	0.923330	+	0.384007i	$0.125456\pi$
$812$	1.51965e10	0.996090
$813$	−8.67876e9	−0.566423
$814$	−1.64516e10	−1.06911
$815$	2.51050e8	0.0162446
$816$	−4.24993e9	−0.273820
$817$	1.13611e10	0.728857
$818$	−9.03464e9	−0.577131
$819$	9.91340e9	0.630564
$820$	3.88278e8	0.0245920
$821$	4.41421e9	0.278389	0.139194	−	0.990265i	$-0.455549\pi$
$821$	4.41421e9	0.278389	0.139194	+	0.990265i	$0.455549\pi$
$822$	−2.49445e9	−0.156648
$823$	−4.77976e9	−0.298887	−0.149443	−	0.988770i	$-0.547748\pi$
$823$	−4.77976e9	−0.298887	−0.149443	+	0.988770i	$0.547748\pi$
$824$	1.98984e9	0.123900
$825$	−1.81582e10	−1.12586
$826$	2.03295e10	1.25515
$827$	−1.53856e10	−0.945897	−0.472949	−	0.881090i	$-0.656810\pi$
$827$	−1.53856e10	−0.945897	−0.472949	+	0.881090i	$0.656810\pi$
$828$	5.67664e8	0.0347524
$829$	−1.99325e10	−1.21513	−0.607563	−	0.794271i	$-0.707853\pi$
$829$	−1.99325e10	−1.21513	−0.607563	+	0.794271i	$0.707853\pi$
$830$	2.92993e8	0.0177862
$831$	1.04762e10	0.633287
$832$	2.56397e9	0.154341
$833$	4.26371e10	2.55582
$834$	6.10015e9	0.364132
$835$	2.74653e8	0.0163261
$836$	2.64742e10	1.56711
$837$	−7.46649e8	−0.0440126
$838$	−2.11087e10	−1.23910
$839$	9.82440e9	0.574301	0.287150	−	0.957886i	$-0.407292\pi$
$839$	9.82440e9	0.574301	0.287150	+	0.957886i	$0.407292\pi$
$840$	−1.49556e8	−0.00870613
$841$	1.19168e10	0.690832
$842$	−7.48555e9	−0.432147
$843$	1.11109e10	0.638780
$844$	−1.03928e10	−0.595023
$845$	2.56121e8	0.0146031
$846$	6.19838e9	0.351951
$847$	7.60945e10	4.30290
$848$	2.81508e9	0.158528
$849$	−1.62265e10	−0.910015
$850$	−2.39994e10	−1.34040
$851$	2.90434e9	0.161545
$852$	7.56649e7	0.00419137
$853$	1.19974e10	0.661861	0.330931	−	0.943655i	$-0.392637\pi$
$853$	1.19974e10	0.661861	0.330931	+	0.943655i	$0.392637\pi$
$854$	2.66343e9	0.146332
$855$	−2.72373e8	−0.0149033
$856$	−5.31471e9	−0.289615
$857$	5.67953e9	0.308233	0.154117	−	0.988053i	$-0.450747\pi$
$857$	5.67953e9	0.308233	0.154117	+	0.988053i	$0.450747\pi$
$858$	1.82004e10	0.983731
$859$	−1.03130e10	−0.555150	−0.277575	−	0.960704i	$-0.589531\pi$
$859$	−1.03130e10	−0.555150	−0.277575	+	0.960704i	$0.589531\pi$
$860$	−1.17831e8	−0.00631705
$861$	−2.92685e10	−1.56275
$862$	2.86453e9	0.152327
$863$	1.71356e10	0.907531	0.453766	−	0.891121i	$-0.350080\pi$
$863$	1.71356e10	0.907531	0.453766	+	0.891121i	$0.350080\pi$
$864$	−6.44973e8	−0.0340207
$865$	−3.17291e8	−0.0166687
$866$	−8.03750e9	−0.420541
$867$	−2.87939e10	−1.50049
$868$	3.37541e9	0.175189
$869$	−3.29576e10	−1.70367
$870$	−2.87042e8	−0.0147784
$871$	−1.92785e10	−0.988578
$872$	−8.68623e9	−0.443633
$873$	−3.18926e9	−0.162233
$874$	−4.67371e9	−0.236795
$875$	−1.68974e9	−0.0852693
$876$	1.48049e9	0.0744119
$877$	1.81350e10	0.907860	0.453930	−	0.891037i	$-0.350022\pi$
$877$	1.81350e10	0.907860	0.453930	+	0.891037i	$0.350022\pi$
$878$	5.00693e9	0.249655
$879$	8.10200e9	0.402375
$880$	−2.74576e8	−0.0135823
$881$	3.38467e9	0.166763	0.0833817	−	0.996518i	$-0.473428\pi$
$881$	3.38467e9	0.166763	0.0833817	+	0.996518i	$0.473428\pi$
$882$	6.47064e9	0.317546
$883$	−6.38903e9	−0.312301	−0.156150	−	0.987733i	$-0.549908\pi$
$883$	−6.38903e9	−0.312301	−0.156150	+	0.987733i	$0.549908\pi$
$884$	2.40553e10	1.17119
$885$	−3.83997e8	−0.0186220
$886$	8.71272e9	0.420858
$887$	−9.61718e9	−0.462716	−0.231358	−	0.972869i	$-0.574317\pi$
$887$	−9.61718e9	−0.462716	−0.231358	+	0.972869i	$0.574317\pi$
$888$	−3.29988e9	−0.158144
$889$	−5.78481e10	−2.76142
$890$	−3.13726e8	−0.0149171
$891$	−4.57836e9	−0.216839
$892$	−7.62501e9	−0.359719
$893$	−5.10328e10	−2.39811
$894$	1.02225e10	0.478491
$895$	−8.37921e8	−0.0390681
$896$	2.91576e9	0.135417
$897$	−3.21308e9	−0.148644
$898$	2.31249e9	0.106565
$899$	6.47841e9	0.297378
$900$	−3.64218e9	−0.166537
$901$	2.64113e10	1.20296
$902$	−5.37353e10	−2.43802
$903$	8.88212e9	0.401430
$904$	−5.64742e9	−0.254250
$905$	5.52234e8	0.0247658
$906$	−5.15156e9	−0.230139
$907$	−3.32697e10	−1.48055	−0.740276	−	0.672303i	$-0.765305\pi$
$907$	−3.32697e10	−1.48055	−0.740276	+	0.672303i	$0.765305\pi$
$908$	1.23375e10	0.546924
$909$	6.93469e9	0.306234
$910$	8.46511e8	0.0372381
$911$	−8.25922e9	−0.361930	−0.180965	−	0.983490i	$-0.557922\pi$
$911$	−8.25922e9	−0.361930	−0.180965	+	0.983490i	$0.557922\pi$
$912$	5.31021e9	0.231809
$913$	−4.05484e10	−1.76330
$914$	1.53428e9	0.0664652
$915$	−5.03085e7	−0.00217104
$916$	−1.49020e10	−0.640633
$917$	5.44310e9	0.233106
$918$	−6.05117e9	−0.258160
$919$	−1.61730e10	−0.687363	−0.343681	−	0.939086i	$-0.611674\pi$
$919$	−1.61730e10	−0.687363	−0.343681	+	0.939086i	$0.611674\pi$
$920$	4.84731e7	0.00205231
$921$	−3.23794e9	−0.136572
$922$	−1.94143e10	−0.815762
$923$	−4.28277e8	−0.0179274
$924$	2.06976e10	0.863113
$925$	−1.86345e10	−0.774143
$926$	1.94227e10	0.803842
$927$	2.83319e9	0.116814
$928$	5.59620e9	0.229866
$929$	8.36893e9	0.342464	0.171232	−	0.985231i	$-0.445225\pi$
$929$	8.36893e9	0.342464	0.171232	+	0.985231i	$0.445225\pi$
$930$	−6.37568e7	−0.00259918
$931$	−5.32743e10	−2.16368
$932$	8.50214e9	0.344011
$933$	8.56279e9	0.345167
$934$	7.48606e8	0.0300635
$935$	−2.57608e9	−0.103067
$936$	3.65066e9	0.145514
$937$	−4.57239e10	−1.81574	−0.907872	−	0.419248i	$-0.862294\pi$
$937$	−4.57239e10	−1.81574	−0.907872	+	0.419248i	$0.862294\pi$
$938$	−2.19236e10	−0.867365
$939$	−5.80838e9	−0.228942
$940$	5.29284e8	0.0207846
$941$	−2.84483e9	−0.111299	−0.0556497	−	0.998450i	$-0.517723\pi$
$941$	−2.84483e9	−0.111299	−0.0556497	+	0.998450i	$0.517723\pi$
$942$	−4.85504e8	−0.0189241
$943$	9.48634e9	0.368390
$944$	7.48645e9	0.289650
$945$	−2.12942e8	−0.00820822
$946$	1.63071e10	0.626263
$947$	9.79801e9	0.374898	0.187449	−	0.982274i	$-0.439978\pi$
$947$	9.79801e9	0.374898	0.187449	+	0.982274i	$0.439978\pi$
$948$	−6.61065e9	−0.252008
$949$	−8.37986e9	−0.318277
$950$	2.99869e10	1.13475
$951$	−3.04916e10	−1.14961
$952$	2.73558e10	1.02759
$953$	9.62304e9	0.360153	0.180077	−	0.983653i	$-0.442365\pi$
$953$	9.62304e9	0.360153	0.180077	+	0.983653i	$0.442365\pi$
$954$	4.00820e9	0.149462
$955$	1.03135e9	0.0383172
$956$	−9.75714e9	−0.361177
$957$	3.97248e10	1.46511
$958$	−1.76833e10	−0.649807
$959$	1.60562e10	0.587865
$960$	−5.50746e7	−0.00200910
$961$	−2.60736e10	−0.947698
$962$	1.86779e10	0.676418
$963$	−7.56723e9	−0.273052
$964$	1.86698e10	0.671226
$965$	3.08256e8	0.0110424
$966$	−3.65392e9	−0.130418
$967$	−1.21458e10	−0.431949	−0.215974	−	0.976399i	$-0.569293\pi$
$967$	−1.21458e10	−0.431949	−0.215974	+	0.976399i	$0.569293\pi$
$968$	2.80222e10	0.992974
$969$	4.98207e10	1.75904
$970$	−2.72333e8	−0.00958074
$971$	2.23958e10	0.785053	0.392526	−	0.919741i	$-0.371601\pi$
$971$	2.23958e10	0.785053	0.392526	+	0.919741i	$0.371601\pi$
$972$	−9.18330e8	−0.0320750
$973$	−3.92652e10	−1.36651
$974$	−4.41108e9	−0.152964
$975$	2.06154e10	0.712319
$976$	9.80820e8	0.0337688
$977$	−4.89988e10	−1.68095	−0.840475	−	0.541851i	$-0.817724\pi$
$977$	−4.89988e10	−1.68095	−0.840475	+	0.541851i	$0.817724\pi$
$978$	−6.96892e9	−0.238221
$979$	4.34177e10	1.47886
$980$	5.52532e8	0.0187528
$981$	−1.23677e10	−0.418261
$982$	1.15016e10	0.387588
$983$	−4.01113e10	−1.34688	−0.673441	−	0.739241i	$-0.735184\pi$
$983$	−4.01113e10	−1.34688	−0.673441	+	0.739241i	$0.735184\pi$
$984$	−1.07783e10	−0.360634
$985$	−1.01016e9	−0.0336795
$986$	5.25038e10	1.74430
$987$	−3.98975e10	−1.32080
$988$	−3.00567e10	−0.991500
$989$	−2.87882e9	−0.0946298
$990$	−3.90949e8	−0.0128055
$991$	−3.20824e9	−0.104715	−0.0523576	−	0.998628i	$-0.516674\pi$
$991$	−3.20824e9	−0.104715	−0.0523576	+	0.998628i	$0.516674\pi$
$992$	1.24301e9	0.0404282
$993$	1.32212e10	0.428498
$994$	−4.87037e8	−0.0157293
$995$	−8.22364e8	−0.0264657
$996$	−8.13322e9	−0.260829
$997$	−5.79883e9	−0.185314	−0.0926568	−	0.995698i	$-0.529536\pi$
$997$	−5.79883e9	−0.185314	−0.0926568	+	0.995698i	$0.529536\pi$
$998$	−8.21228e9	−0.261521
$999$	−4.69846e9	−0.149099

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	138.8.a.g.1.2	✓	4
3.2	odd	2		414.8.a.h.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.8.a.g.1.2	✓	4	1.1	even	1	trivial
414.8.a.h.1.3		4	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:	\( N \)	\(=\)	\( 138 = 2 \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	138.a (trivial)

\(f(q)\)	\(=\)	\(q+8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} +7.78122 q^{5} -216.000 q^{6} +1390.34 q^{7} +512.000 q^{8} +729.000 q^{9} +O(q^{10})\)	\(q+8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} +7.78122 q^{5} -216.000 q^{6} +1390.34 q^{7} +512.000 q^{8} +729.000 q^{9} +62.2498 q^{10} -8614.99 q^{11} -1728.00 q^{12} +9780.79 q^{13} +11122.7 q^{14} -210.093 q^{15} +4096.00 q^{16} +38428.9 q^{17} +5832.00 q^{18} -48016.2 q^{19} +497.998 q^{20} -37539.2 q^{21} -68919.9 q^{22} +12167.0 q^{23} -13824.0 q^{24} -78064.5 q^{25} +78246.3 q^{26} -19683.0 q^{27} +88981.8 q^{28} +170782. q^{29} -1680.74 q^{30} +37933.7 q^{31} +32768.0 q^{32} +232605. q^{33} +307431. q^{34} +10818.6 q^{35} +46656.0 q^{36} +238706. q^{37} -384130. q^{38} -264081. q^{39} +3983.98 q^{40} +779678. q^{41} -300314. q^{42} -236609. q^{43} -551359. q^{44} +5672.51 q^{45} +97336.0 q^{46} +1.06282e6 q^{47} -110592. q^{48} +1.10951e6 q^{49} -624516. q^{50} -1.03758e6 q^{51} +625970. q^{52} +687276. q^{53} -157464. q^{54} -67035.1 q^{55} +711855. q^{56} +1.29644e6 q^{57} +1.36626e6 q^{58} +1.82775e6 q^{59} -13445.9 q^{60} +239458. q^{61} +303470. q^{62} +1.01356e6 q^{63} +262144. q^{64} +76106.4 q^{65} +1.86084e6 q^{66} -1.97106e6 q^{67} +2.45945e6 q^{68} -328509. q^{69} +86548.4 q^{70} -43787.6 q^{71} +373248. q^{72} -856768. q^{73} +1.90965e6 q^{74} +2.10774e6 q^{75} -3.07304e6 q^{76} -1.19778e7 q^{77} -2.11265e6 q^{78} +3.82561e6 q^{79} +31871.9 q^{80} +531441. q^{81} +6.23742e6 q^{82} +4.70673e6 q^{83} -2.40251e6 q^{84} +299024. q^{85} -1.89287e6 q^{86} -4.61113e6 q^{87} -4.41087e6 q^{88} -5.03979e6 q^{89} +45380.1 q^{90} +1.35986e7 q^{91} +778688. q^{92} -1.02421e6 q^{93} +8.50258e6 q^{94} -373625. q^{95} -884736. q^{96} -4.37484e6 q^{97} +8.87604e6 q^{98} -6.28032e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 32 q^{2} - 108 q^{3} + 256 q^{4} + 342 q^{5} - 864 q^{6} - 30 q^{7} + 2048 q^{8} + 2916 q^{9}+O(q^{10}) \) 4 * q + 32 * q^2 - 108 * q^3 + 256 * q^4 + 342 * q^5 - 864 * q^6 - 30 * q^7 + 2048 * q^8 + 2916 * q^9	\( 4 q + 32 q^{2} - 108 q^{3} + 256 q^{4} + 342 q^{5} - 864 q^{6} - 30 q^{7} + 2048 q^{8} + 2916 q^{9} + 2736 q^{10} - 2280 q^{11} - 6912 q^{12} + 9204 q^{13} - 240 q^{14} - 9234 q^{15} + 16384 q^{16} + 21638 q^{17} + 23328 q^{18} - 58158 q^{19} + 21888 q^{20} + 810 q^{21} - 18240 q^{22} + 48668 q^{23} - 55296 q^{24} + 246104 q^{25} + 73632 q^{26} - 78732 q^{27} - 1920 q^{28} + 252372 q^{29} - 73872 q^{30} + 364604 q^{31} + 131072 q^{32} + 61560 q^{33} + 173104 q^{34} - 197064 q^{35} + 186624 q^{36} + 357596 q^{37} - 465264 q^{38} - 248508 q^{39} + 175104 q^{40} + 818768 q^{41} + 6480 q^{42} + 561066 q^{43} - 145920 q^{44} + 249318 q^{45} + 389344 q^{46} + 2359976 q^{47} - 442368 q^{48} + 3660776 q^{49} + 1968832 q^{50} - 584226 q^{51} + 589056 q^{52} + 4246586 q^{53} - 629856 q^{54} + 3314024 q^{55} - 15360 q^{56} + 1570266 q^{57} + 2018976 q^{58} + 5065776 q^{59} - 590976 q^{60} - 1403844 q^{61} + 2916832 q^{62} - 21870 q^{63} + 1048576 q^{64} + 5152364 q^{65} + 492480 q^{66} + 3643934 q^{67} + 1384832 q^{68} - 1314036 q^{69} - 1576512 q^{70} - 4894464 q^{71} + 1492992 q^{72} + 4426192 q^{73} + 2860768 q^{74} - 6644808 q^{75} - 3722112 q^{76} - 11044616 q^{77} - 1988064 q^{78} - 1095946 q^{79} + 1400832 q^{80} + 2125764 q^{81} + 6550144 q^{82} - 10412840 q^{83} + 51840 q^{84} + 7941764 q^{85} + 4488528 q^{86} - 6814044 q^{87} - 1167360 q^{88} + 4632090 q^{89} + 1994544 q^{90} + 2135500 q^{91} + 3114752 q^{92} - 9844308 q^{93} + 18879808 q^{94} - 2017088 q^{95} - 3538944 q^{96} - 15402348 q^{97} + 29286208 q^{98} - 1662120 q^{99}+O(q^{100}) \) 4 * q + 32 * q^2 - 108 * q^3 + 256 * q^4 + 342 * q^5 - 864 * q^6 - 30 * q^7 + 2048 * q^8 + 2916 * q^9 + 2736 * q^10 - 2280 * q^11 - 6912 * q^12 + 9204 * q^13 - 240 * q^14 - 9234 * q^15 + 16384 * q^16 + 21638 * q^17 + 23328 * q^18 - 58158 * q^19 + 21888 * q^20 + 810 * q^21 - 18240 * q^22 + 48668 * q^23 - 55296 * q^24 + 246104 * q^25 + 73632 * q^26 - 78732 * q^27 - 1920 * q^28 + 252372 * q^29 - 73872 * q^30 + 364604 * q^31 + 131072 * q^32 + 61560 * q^33 + 173104 * q^34 - 197064 * q^35 + 186624 * q^36 + 357596 * q^37 - 465264 * q^38 - 248508 * q^39 + 175104 * q^40 + 818768 * q^41 + 6480 * q^42 + 561066 * q^43 - 145920 * q^44 + 249318 * q^45 + 389344 * q^46 + 2359976 * q^47 - 442368 * q^48 + 3660776 * q^49 + 1968832 * q^50 - 584226 * q^51 + 589056 * q^52 + 4246586 * q^53 - 629856 * q^54 + 3314024 * q^55 - 15360 * q^56 + 1570266 * q^57 + 2018976 * q^58 + 5065776 * q^59 - 590976 * q^60 - 1403844 * q^61 + 2916832 * q^62 - 21870 * q^63 + 1048576 * q^64 + 5152364 * q^65 + 492480 * q^66 + 3643934 * q^67 + 1384832 * q^68 - 1314036 * q^69 - 1576512 * q^70 - 4894464 * q^71 + 1492992 * q^72 + 4426192 * q^73 + 2860768 * q^74 - 6644808 * q^75 - 3722112 * q^76 - 11044616 * q^77 - 1988064 * q^78 - 1095946 * q^79 + 1400832 * q^80 + 2125764 * q^81 + 6550144 * q^82 - 10412840 * q^83 + 51840 * q^84 + 7941764 * q^85 + 4488528 * q^86 - 6814044 * q^87 - 1167360 * q^88 + 4632090 * q^89 + 1994544 * q^90 + 2135500 * q^91 + 3114752 * q^92 - 9844308 * q^93 + 18879808 * q^94 - 2017088 * q^95 - 3538944 * q^96 - 15402348 * q^97 + 29286208 * q^98 - 1662120 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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