LMFDB - Embedded newform 138.8.a.e.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:	$1$
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} - x^{3} - 47804x^{2} - 3068607x + 114119793 $ x^4 - x^3 - 47804x^2 - 3068607x + 114119793
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4}\cdot 3 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-145.604$ 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)

$f(q)$	$=$	$q-8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} -177.054 q^{5} +216.000 q^{6} -891.394 q^{7} -512.000 q^{8} +729.000 q^{9} +O(q^{10})$	\(q-8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} -177.054 q^{5} +216.000 q^{6} -891.394 q^{7} -512.000 q^{8} +729.000 q^{9} +1416.44 q^{10} +6726.05 q^{11} -1728.00 q^{12} -2226.09 q^{13} +7131.15 q^{14} +4780.47 q^{15} +4096.00 q^{16} -11483.4 q^{17} -5832.00 q^{18} +23693.0 q^{19} -11331.5 q^{20} +24067.6 q^{21} -53808.4 q^{22} +12167.0 q^{23} +13824.0 q^{24} -46776.7 q^{25} +17808.7 q^{26} -19683.0 q^{27} -57049.2 q^{28} +184543. q^{29} -38243.8 q^{30} +276562. q^{31} -32768.0 q^{32} -181603. q^{33} +91867.4 q^{34} +157825. q^{35} +46656.0 q^{36} +373651. q^{37} -189544. q^{38} +60104.3 q^{39} +90651.9 q^{40} -625477. q^{41} -192541. q^{42} +262339. q^{43} +430467. q^{44} -129073. q^{45} -97336.0 q^{46} -1.26653e6 q^{47} -110592. q^{48} -28960.4 q^{49} +374214. q^{50} +310052. q^{51} -142469. q^{52} -1.88192e6 q^{53} +157464. q^{54} -1.19088e6 q^{55} +456394. q^{56} -639711. q^{57} -1.47634e6 q^{58} +1.38332e6 q^{59} +305950. q^{60} +575420. q^{61} -2.21249e6 q^{62} -649826. q^{63} +262144. q^{64} +394138. q^{65} +1.45283e6 q^{66} -358353. q^{67} -734939. q^{68} -328509. q^{69} -1.26260e6 q^{70} -4.29242e6 q^{71} -373248. q^{72} -330973. q^{73} -2.98921e6 q^{74} +1.26297e6 q^{75} +1.51635e6 q^{76} -5.99556e6 q^{77} -480834. q^{78} -8.20357e6 q^{79} -725215. q^{80} +531441. q^{81} +5.00382e6 q^{82} -4.15440e6 q^{83} +1.54033e6 q^{84} +2.03319e6 q^{85} -2.09871e6 q^{86} -4.98265e6 q^{87} -3.44374e6 q^{88} +3.65983e6 q^{89} +1.03258e6 q^{90} +1.98432e6 q^{91} +778688. q^{92} -7.46717e6 q^{93} +1.01322e7 q^{94} -4.19495e6 q^{95} +884736. q^{96} +1.46218e6 q^{97} +231683. q^{98} +4.90329e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 32 q^{2} - 108 q^{3} + 256 q^{4} - 90 q^{5} + 864 q^{6} - 222 q^{7} - 2048 q^{8} + 2916 q^{9}+O(q^{10}) $ 4 * q - 32 * q^2 - 108 * q^3 + 256 * q^4 - 90 * q^5 + 864 * q^6 - 222 * q^7 - 2048 * q^8 + 2916 * q^9	\( 4 q - 32 q^{2} - 108 q^{3} + 256 q^{4} - 90 q^{5} + 864 q^{6} - 222 q^{7} - 2048 q^{8} + 2916 q^{9} + 720 q^{10} - 4120 q^{11} - 6912 q^{12} - 6796 q^{13} + 1776 q^{14} + 2430 q^{15} + 16384 q^{16} + 18918 q^{17} - 23328 q^{18} + 19682 q^{19} - 5760 q^{20} + 5994 q^{21} + 32960 q^{22} + 48668 q^{23} + 55296 q^{24} + 261304 q^{25} + 54368 q^{26} - 78732 q^{27} - 14208 q^{28} + 12756 q^{29} - 19440 q^{30} + 440892 q^{31} - 131072 q^{32} + 111240 q^{33} - 151344 q^{34} + 150200 q^{35} + 186624 q^{36} + 567212 q^{37} - 157456 q^{38} + 183492 q^{39} + 46080 q^{40} + 275056 q^{41} - 47952 q^{42} + 149946 q^{43} - 263680 q^{44} - 65610 q^{45} - 389344 q^{46} - 2000728 q^{47} - 442368 q^{48} - 1623352 q^{49} - 2090432 q^{50} - 510786 q^{51} - 434944 q^{52} - 2778518 q^{53} + 629856 q^{54} - 2706008 q^{55} + 113664 q^{56} - 531414 q^{57} - 102048 q^{58} - 3656368 q^{59} + 155520 q^{60} - 989076 q^{61} - 3527136 q^{62} - 161838 q^{63} + 1048576 q^{64} - 7481556 q^{65} - 889920 q^{66} - 5755282 q^{67} + 1210752 q^{68} - 1314036 q^{69} - 1201600 q^{70} - 8843424 q^{71} - 1492992 q^{72} - 1100272 q^{73} - 4537696 q^{74} - 7055208 q^{75} + 1259648 q^{76} - 10973480 q^{77} - 1467936 q^{78} + 959830 q^{79} - 368640 q^{80} + 2125764 q^{81} - 2200448 q^{82} - 24990584 q^{83} + 383616 q^{84} - 7601436 q^{85} - 1199568 q^{86} - 344412 q^{87} + 2109440 q^{88} - 3623430 q^{89} + 524880 q^{90} - 9436372 q^{91} + 3114752 q^{92} - 11904084 q^{93} + 16005824 q^{94} - 12385728 q^{95} + 3538944 q^{96} + 14161716 q^{97} + 12986816 q^{98} - 3003480 q^{99}+O(q^{100}) \) 4 * q - 32 * q^2 - 108 * q^3 + 256 * q^4 - 90 * q^5 + 864 * q^6 - 222 * q^7 - 2048 * q^8 + 2916 * q^9 + 720 * q^10 - 4120 * q^11 - 6912 * q^12 - 6796 * q^13 + 1776 * q^14 + 2430 * q^15 + 16384 * q^16 + 18918 * q^17 - 23328 * q^18 + 19682 * q^19 - 5760 * q^20 + 5994 * q^21 + 32960 * q^22 + 48668 * q^23 + 55296 * q^24 + 261304 * q^25 + 54368 * q^26 - 78732 * q^27 - 14208 * q^28 + 12756 * q^29 - 19440 * q^30 + 440892 * q^31 - 131072 * q^32 + 111240 * q^33 - 151344 * q^34 + 150200 * q^35 + 186624 * q^36 + 567212 * q^37 - 157456 * q^38 + 183492 * q^39 + 46080 * q^40 + 275056 * q^41 - 47952 * q^42 + 149946 * q^43 - 263680 * q^44 - 65610 * q^45 - 389344 * q^46 - 2000728 * q^47 - 442368 * q^48 - 1623352 * q^49 - 2090432 * q^50 - 510786 * q^51 - 434944 * q^52 - 2778518 * q^53 + 629856 * q^54 - 2706008 * q^55 + 113664 * q^56 - 531414 * q^57 - 102048 * q^58 - 3656368 * q^59 + 155520 * q^60 - 989076 * q^61 - 3527136 * q^62 - 161838 * q^63 + 1048576 * q^64 - 7481556 * q^65 - 889920 * q^66 - 5755282 * q^67 + 1210752 * q^68 - 1314036 * q^69 - 1201600 * q^70 - 8843424 * q^71 - 1492992 * q^72 - 1100272 * q^73 - 4537696 * q^74 - 7055208 * q^75 + 1259648 * q^76 - 10973480 * q^77 - 1467936 * q^78 + 959830 * q^79 - 368640 * q^80 + 2125764 * q^81 - 2200448 * q^82 - 24990584 * q^83 + 383616 * q^84 - 7601436 * q^85 - 1199568 * q^86 - 344412 * q^87 + 2109440 * q^88 - 3623430 * q^89 + 524880 * q^90 - 9436372 * q^91 + 3114752 * q^92 - 11904084 * q^93 + 16005824 * q^94 - 12385728 * q^95 + 3538944 * q^96 + 14161716 * q^97 + 12986816 * q^98 - 3003480 * 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
$5$	−177.054	−0.633449	−0.316725	−	0.948518i	$-0.602583\pi$
$5$	−177.054	−0.633449	−0.316725	+	0.948518i	$0.602583\pi$
$6$	216.000	0.408248
$7$	−891.394	−0.982260	−0.491130	−	0.871086i	$-0.663416\pi$
$7$	−891.394	−0.982260	−0.491130	+	0.871086i	$0.663416\pi$
$8$	−512.000	−0.353553
$9$	729.000	0.333333
$10$	1416.44	0.447916
$11$	6726.05	1.52365	0.761826	−	0.647781i	$-0.224303\pi$
$11$	6726.05	1.52365	0.761826	+	0.647781i	$0.224303\pi$
$12$	−1728.00	−0.288675
$13$	−2226.09	−0.281022	−0.140511	−	0.990079i	$-0.544875\pi$
$13$	−2226.09	−0.281022	−0.140511	+	0.990079i	$0.544875\pi$
$14$	7131.15	0.694563
$15$	4780.47	0.365722
$16$	4096.00	0.250000
$17$	−11483.4	−0.566892	−0.283446	−	0.958988i	$-0.591478\pi$
$17$	−11483.4	−0.566892	−0.283446	+	0.958988i	$0.591478\pi$
$18$	−5832.00	−0.235702
$19$	23693.0	0.792470	0.396235	−	0.918149i	$-0.370317\pi$
$19$	23693.0	0.792470	0.396235	+	0.918149i	$0.370317\pi$
$20$	−11331.5	−0.316725
$21$	24067.6	0.567108
$22$	−53808.4	−1.07739
$23$	12167.0	0.208514
$23$	12167.0	0.208514
$24$	13824.0	0.204124
$25$	−46776.7	−0.598742
$26$	17808.7	0.198712
$27$	−19683.0	−0.192450
$28$	−57049.2	−0.491130
$29$	184543.	1.40509	0.702544	−	0.711640i	$-0.252047\pi$
$29$	184543.	1.40509	0.702544	+	0.711640i	$0.252047\pi$
$30$	−38243.8	−0.258605
$31$	276562.	1.66735	0.833674	−	0.552257i	$-0.186233\pi$
$31$	276562.	1.66735	0.833674	+	0.552257i	$0.186233\pi$
$32$	−32768.0	−0.176777
$33$	−181603.	−0.879681
$34$	91867.4	0.400853
$35$	157825.	0.622212
$36$	46656.0	0.166667
$37$	373651.	1.21272	0.606359	−	0.795191i	$-0.292629\pi$
$37$	373651.	1.21272	0.606359	+	0.795191i	$0.292629\pi$
$38$	−189544.	−0.560361
$39$	60104.3	0.162248
$40$	90651.9	0.223958
$41$	−625477.	−1.41732	−0.708660	−	0.705550i	$-0.750700\pi$
$41$	−625477.	−1.41732	−0.708660	+	0.705550i	$0.750700\pi$
$42$	−192541.	−0.401006
$43$	262339.	0.503180	0.251590	−	0.967834i	$-0.419046\pi$
$43$	262339.	0.503180	0.251590	+	0.967834i	$0.419046\pi$
$44$	430467.	0.761826
$45$	−129073.	−0.211150
$46$	−97336.0	−0.147442
$47$	−1.26653e6	−1.77940	−0.889698	−	0.456549i	$-0.849085\pi$
$47$	−1.26653e6	−1.77940	−0.889698	+	0.456549i	$0.849085\pi$
$48$	−110592.	−0.144338
$49$	−28960.4	−0.0351656
$50$	374214.	0.423375
$51$	310052.	0.327295
$52$	−142469.	−0.140511
$53$	−1.88192e6	−1.73634	−0.868172	−	0.496264i	$-0.834705\pi$
$53$	−1.88192e6	−1.73634	−0.868172	+	0.496264i	$0.834705\pi$
$54$	157464.	0.136083
$55$	−1.19088e6	−0.965157
$56$	456394.	0.347281
$57$	−639711.	−0.457533
$58$	−1.47634e6	−0.993548
$59$	1.38332e6	0.876883	0.438441	−	0.898760i	$-0.355531\pi$
$59$	1.38332e6	0.876883	0.438441	+	0.898760i	$0.355531\pi$
$60$	305950.	0.182861
$61$	575420.	0.324586	0.162293	−	0.986743i	$-0.448111\pi$
$61$	575420.	0.324586	0.162293	+	0.986743i	$0.448111\pi$
$62$	−2.21249e6	−1.17899
$63$	−649826.	−0.327420
$64$	262144.	0.125000
$65$	394138.	0.178013
$66$	1.45283e6	0.622029
$67$	−358353.	−0.145562	−0.0727812	−	0.997348i	$-0.523187\pi$
$67$	−358353.	−0.145562	−0.0727812	+	0.997348i	$0.523187\pi$
$68$	−734939.	−0.283446
$69$	−328509.	−0.120386
$70$	−1.26260e6	−0.439970
$71$	−4.29242e6	−1.42330	−0.711652	−	0.702532i	$-0.752053\pi$
$71$	−4.29242e6	−1.42330	−0.711652	+	0.702532i	$0.752053\pi$
$72$	−373248.	−0.117851
$73$	−330973.	−0.0995778	−0.0497889	−	0.998760i	$-0.515855\pi$
$73$	−330973.	−0.0995778	−0.0497889	+	0.998760i	$0.515855\pi$
$74$	−2.98921e6	−0.857522
$75$	1.26297e6	0.345684
$76$	1.51635e6	0.396235
$77$	−5.99556e6	−1.49662
$78$	−480834.	−0.114727
$79$	−8.20357e6	−1.87201	−0.936005	−	0.351987i	$-0.885506\pi$
$79$	−8.20357e6	−1.87201	−0.936005	+	0.351987i	$0.885506\pi$
$80$	−725215.	−0.158362
$81$	531441.	0.111111
$82$	5.00382e6	1.00220
$83$	−4.15440e6	−0.797507	−0.398754	−	0.917058i	$-0.630557\pi$
$83$	−4.15440e6	−0.797507	−0.398754	+	0.917058i	$0.630557\pi$
$84$	1.54033e6	0.283554
$85$	2.03319e6	0.359097
$86$	−2.09871e6	−0.355802
$87$	−4.98265e6	−0.811228
$88$	−3.44374e6	−0.538693
$89$	3.65983e6	0.550296	0.275148	−	0.961402i	$-0.411273\pi$
$89$	3.65983e6	0.550296	0.275148	+	0.961402i	$0.411273\pi$
$90$	1.03258e6	0.149305
$91$	1.98432e6	0.276036
$92$	778688.	0.104257
$93$	−7.46717e6	−0.962643
$94$	1.01322e7	1.25822
$95$	−4.19495e6	−0.501989
$96$	884736.	0.102062
$97$	1.46218e6	0.162668	0.0813338	−	0.996687i	$-0.474082\pi$
$97$	1.46218e6	0.162668	0.0813338	+	0.996687i	$0.474082\pi$
$98$	231683.	0.0248659
$99$	4.90329e6	0.507884
$100$	−2.99371e6	−0.299371
$101$	1.96844e7	1.90106	0.950532	−	0.310628i	$-0.100539\pi$
$101$	1.96844e7	1.90106	0.950532	+	0.310628i	$0.100539\pi$
$102$	−2.48042e6	−0.231433
$103$	−786558.	−0.0709252	−0.0354626	−	0.999371i	$-0.511290\pi$
$103$	−786558.	−0.0709252	−0.0354626	+	0.999371i	$0.511290\pi$
$104$	1.13976e6	0.0993562
$105$	−4.26128e6	−0.359234
$106$	1.50554e7	1.22778
$107$	−1.05476e7	−0.832358	−0.416179	−	0.909283i	$-0.636631\pi$
$107$	−1.05476e7	−0.832358	−0.416179	+	0.909283i	$0.636631\pi$
$108$	−1.25971e6	−0.0962250
$109$	−1.97355e7	−1.45967	−0.729835	−	0.683623i	$-0.760403\pi$
$109$	−1.97355e7	−1.45967	−0.729835	+	0.683623i	$0.760403\pi$
$110$	9.52702e6	0.682469
$111$	−1.00886e7	−0.700164
$112$	−3.65115e6	−0.245565
$113$	−2.44414e7	−1.59350	−0.796750	−	0.604309i	$-0.793449\pi$
$113$	−2.44414e7	−1.59350	−0.796750	+	0.604309i	$0.793449\pi$
$114$	5.11769e6	0.323524
$115$	−2.15422e6	−0.132083
$116$	1.18107e7	0.702544
$117$	−1.62282e6	−0.0936739
$118$	−1.10666e7	−0.620050
$119$	1.02362e7	0.556835
$120$	−2.44760e6	−0.129302
$121$	2.57526e7	1.32152
$122$	−4.60336e6	−0.229517
$123$	1.68879e7	0.818290
$124$	1.76999e7	0.833674
$125$	2.21144e7	1.01272
$126$	5.19861e6	0.231521
$127$	2.08098e7	0.901478	0.450739	−	0.892656i	$-0.351160\pi$
$127$	2.08098e7	0.901478	0.450739	+	0.892656i	$0.351160\pi$
$128$	−2.09715e6	−0.0883883
$129$	−7.08316e6	−0.290511
$130$	−3.15311e6	−0.125874
$131$	−5.00111e7	−1.94364	−0.971822	−	0.235715i	$-0.924257\pi$
$131$	−5.00111e7	−1.94364	−0.971822	+	0.235715i	$0.924257\pi$
$132$	−1.16226e7	−0.439841
$133$	−2.11198e7	−0.778411
$134$	2.86682e6	0.102928
$135$	3.48496e6	0.121907
$136$	5.87951e6	0.200426
$137$	−6.54042e6	−0.217312	−0.108656	−	0.994079i	$-0.534655\pi$
$137$	−6.54042e6	−0.217312	−0.108656	+	0.994079i	$0.534655\pi$
$138$	2.62807e6	0.0851257
$139$	1.87466e7	0.592066	0.296033	−	0.955178i	$-0.404336\pi$
$139$	1.87466e7	0.592066	0.296033	+	0.955178i	$0.404336\pi$
$140$	1.01008e7	0.311106
$141$	3.41963e7	1.02734
$142$	3.43393e7	1.00643
$143$	−1.49728e7	−0.428180
$144$	2.98598e6	0.0833333
$145$	−3.26741e7	−0.890052
$146$	2.64778e6	0.0704122
$147$	781931.	0.0203029
$148$	2.39137e7	0.606359
$149$	3.50139e6	0.0867140	0.0433570	−	0.999060i	$-0.486195\pi$
$149$	3.50139e6	0.0867140	0.0433570	+	0.999060i	$0.486195\pi$
$150$	−1.01038e7	−0.244435
$151$	2.59201e7	0.612658	0.306329	−	0.951926i	$-0.400899\pi$
$151$	2.59201e7	0.612658	0.306329	+	0.951926i	$0.400899\pi$
$152$	−1.21308e7	−0.280180
$153$	−8.37141e6	−0.188964
$154$	4.79645e7	1.05827
$155$	−4.89665e7	−1.05618
$156$	3.84668e6	0.0811240
$157$	7.38016e7	1.52201	0.761004	−	0.648747i	$-0.224707\pi$
$157$	7.38016e7	1.52201	0.761004	+	0.648747i	$0.224707\pi$
$158$	6.56286e7	1.32371
$159$	5.08119e7	1.00248
$160$	5.80172e6	0.111979
$161$	−1.08456e7	−0.204815
$162$	−4.25153e6	−0.0785674
$163$	1.01244e8	1.83110	0.915550	−	0.402205i	$-0.131756\pi$
$163$	1.01244e8	1.83110	0.915550	+	0.402205i	$0.131756\pi$
$164$	−4.00305e7	−0.708660
$165$	3.21537e7	0.557233
$166$	3.32352e7	0.563923
$167$	1.25895e7	0.209170	0.104585	−	0.994516i	$-0.466649\pi$
$167$	1.25895e7	0.209170	0.104585	+	0.994516i	$0.466649\pi$
$168$	−1.23226e7	−0.200503
$169$	−5.77931e7	−0.921027
$170$	−1.62655e7	−0.253920
$171$	1.72722e7	0.264157
$172$	1.67897e7	0.251590
$173$	5.61749e7	0.824862	0.412431	−	0.910989i	$-0.364680\pi$
$173$	5.61749e7	0.824862	0.412431	+	0.910989i	$0.364680\pi$
$174$	3.98612e7	0.573625
$175$	4.16965e7	0.588120
$176$	2.75499e7	0.380913
$177$	−3.73497e7	−0.506268
$178$	−2.92787e7	−0.389118
$179$	−7.36073e7	−0.959257	−0.479629	−	0.877472i	$-0.659229\pi$
$179$	−7.36073e7	−0.959257	−0.479629	+	0.877472i	$0.659229\pi$
$180$	−8.26065e6	−0.105575
$181$	7.38871e6	0.0926176	0.0463088	−	0.998927i	$-0.485254\pi$
$181$	7.38871e6	0.0926176	0.0463088	+	0.998927i	$0.485254\pi$
$182$	−1.58745e7	−0.195187
$183$	−1.55363e7	−0.187400
$184$	−6.22950e6	−0.0737210
$185$	−6.61566e7	−0.768196
$186$	5.97373e7	0.680692
$187$	−7.72381e7	−0.863746
$188$	−8.10579e7	−0.889698
$189$	1.75453e7	0.189036
$190$	3.35596e7	0.354960
$191$	−1.56311e6	−0.0162321	−0.00811604	−	0.999967i	$-0.502583\pi$
$191$	−1.56311e6	−0.0162321	−0.00811604	+	0.999967i	$0.502583\pi$
$192$	−7.07789e6	−0.0721688
$193$	−6.97825e7	−0.698708	−0.349354	−	0.936991i	$-0.613599\pi$
$193$	−6.97825e7	−0.698708	−0.349354	+	0.936991i	$0.613599\pi$
$194$	−1.16975e7	−0.115023
$195$	−1.06417e7	−0.102776
$196$	−1.85347e6	−0.0175828
$197$	−2.03065e8	−1.89236	−0.946180	−	0.323640i	$-0.895093\pi$
$197$	−2.03065e8	−1.89236	−0.946180	+	0.323640i	$0.895093\pi$
$198$	−3.92263e7	−0.359128
$199$	−4.61295e6	−0.0414947	−0.0207474	−	0.999785i	$-0.506605\pi$
$199$	−4.61295e6	−0.0414947	−0.0207474	+	0.999785i	$0.506605\pi$
$200$	2.39497e7	0.211687
$201$	9.67553e6	0.0840405
$202$	−1.57475e8	−1.34425
$203$	−1.64500e8	−1.38016
$204$	1.98434e7	0.163648
$205$	1.10743e8	0.897800
$206$	6.29246e6	0.0501517
$207$	8.86974e6	0.0695048
$208$	−9.11804e6	−0.0702555
$209$	1.59360e8	1.20745
$210$	3.40902e7	0.254017
$211$	−4.53083e7	−0.332039	−0.166020	−	0.986122i	$-0.553092\pi$
$211$	−4.53083e7	−0.332039	−0.166020	+	0.986122i	$0.553092\pi$
$212$	−1.20443e8	−0.868172
$213$	1.15895e8	0.821745
$214$	8.43808e7	0.588566
$215$	−4.64483e7	−0.318739
$216$	1.00777e7	0.0680414
$217$	−2.46525e8	−1.63777
$218$	1.57884e8	1.03214
$219$	8.93627e6	0.0574913
$220$	−7.62162e7	−0.482578
$221$	2.55631e7	0.159309
$222$	8.07086e7	0.495090
$223$	−2.59176e8	−1.56505	−0.782523	−	0.622621i	$-0.786068\pi$
$223$	−2.59176e8	−1.56505	−0.782523	+	0.622621i	$0.786068\pi$
$224$	2.92092e7	0.173641
$225$	−3.41002e7	−0.199581
$226$	1.95531e8	1.12677
$227$	1.06375e8	0.603599	0.301799	−	0.953371i	$-0.402413\pi$
$227$	1.06375e8	0.603599	0.301799	+	0.953371i	$0.402413\pi$
$228$	−4.09415e7	−0.228766
$229$	−1.76192e7	−0.0969532	−0.0484766	−	0.998824i	$-0.515437\pi$
$229$	−1.76192e7	−0.0969532	−0.0484766	+	0.998824i	$0.515437\pi$
$230$	1.72338e7	0.0933970
$231$	1.61880e8	0.864076
$232$	−9.44858e7	−0.496774
$233$	4.29365e7	0.222372	0.111186	−	0.993800i	$-0.464535\pi$
$233$	4.29365e7	0.222372	0.111186	+	0.993800i	$0.464535\pi$
$234$	1.29825e7	0.0662375
$235$	2.24245e8	1.12716
$236$	8.85327e7	0.438441
$237$	2.21496e8	1.08081
$238$	−8.18900e7	−0.393742
$239$	−1.83028e8	−0.867213	−0.433606	−	0.901102i	$-0.642759\pi$
$239$	−1.83028e8	−0.867213	−0.433606	+	0.901102i	$0.642759\pi$
$240$	1.95808e7	0.0914305
$241$	7.65600e7	0.352324	0.176162	−	0.984361i	$-0.443632\pi$
$241$	7.65600e7	0.352324	0.176162	+	0.984361i	$0.443632\pi$
$242$	−2.06021e8	−0.934454
$243$	−1.43489e7	−0.0641500
$244$	3.68269e7	0.162293
$245$	5.12757e6	0.0222756
$246$	−1.35103e8	−0.578618
$247$	−5.27427e7	−0.222701
$248$	−1.41600e8	−0.589496
$249$	1.12169e8	0.460441
$250$	−1.76915e8	−0.716103
$251$	−2.37957e8	−0.949819	−0.474909	−	0.880035i	$-0.657519\pi$
$251$	−2.37957e8	−0.949819	−0.474909	+	0.880035i	$0.657519\pi$
$252$	−4.15889e7	−0.163710
$253$	8.18359e7	0.317704
$254$	−1.66479e8	−0.637442
$255$	−5.48961e7	−0.207325
$256$	1.67772e7	0.0625000
$257$	3.45055e8	1.26801	0.634005	−	0.773329i	$-0.281410\pi$
$257$	3.45055e8	1.26801	0.634005	+	0.773329i	$0.281410\pi$
$258$	5.66653e7	0.205423
$259$	−3.33070e8	−1.19121
$260$	2.52248e7	0.0890065
$261$	1.34532e8	0.468363
$262$	4.00089e8	1.37436
$263$	−4.83348e8	−1.63838	−0.819191	−	0.573521i	$-0.805577\pi$
$263$	−4.83348e8	−1.63838	−0.819191	+	0.573521i	$0.805577\pi$
$264$	9.29810e7	0.311014
$265$	3.33202e8	1.09989
$266$	1.68958e8	0.550420
$267$	−9.88155e7	−0.317714
$268$	−2.29346e7	−0.0727812
$269$	−2.52332e8	−0.790386	−0.395193	−	0.918598i	$-0.629322\pi$
$269$	−2.52332e8	−0.790386	−0.395193	+	0.918598i	$0.629322\pi$
$270$	−2.78797e7	−0.0862015
$271$	1.58841e8	0.484809	0.242405	−	0.970175i	$-0.422064\pi$
$271$	1.58841e8	0.484809	0.242405	+	0.970175i	$0.422064\pi$
$272$	−4.70361e7	−0.141723
$273$	−5.35766e7	−0.159370
$274$	5.23234e7	0.153663
$275$	−3.14623e8	−0.912275
$276$	−2.10246e7	−0.0601929
$277$	6.37039e7	0.180089	0.0900444	−	0.995938i	$-0.471299\pi$
$277$	6.37039e7	0.180089	0.0900444	+	0.995938i	$0.471299\pi$
$278$	−1.49973e8	−0.418654
$279$	2.01613e8	0.555782
$280$	−8.08065e7	−0.219985
$281$	−3.38308e8	−0.909579	−0.454789	−	0.890599i	$-0.650285\pi$
$281$	−3.38308e8	−0.909579	−0.454789	+	0.890599i	$0.650285\pi$
$282$	−2.73570e8	−0.726436
$283$	−3.99842e8	−1.04866	−0.524331	−	0.851515i	$-0.675685\pi$
$283$	−3.99842e8	−1.04866	−0.524331	+	0.851515i	$0.675685\pi$
$284$	−2.74715e8	−0.711652
$285$	1.13264e8	0.289824
$286$	1.19782e8	0.302769
$287$	5.57546e8	1.39218
$288$	−2.38879e7	−0.0589256
$289$	−2.78470e8	−0.678634
$290$	2.61393e8	0.629362
$291$	−3.94790e7	−0.0939161
$292$	−2.11823e7	−0.0497889
$293$	5.78855e8	1.34441	0.672207	−	0.740363i	$-0.265347\pi$
$293$	5.78855e8	1.34441	0.672207	+	0.740363i	$0.265347\pi$
$294$	−6.25545e6	−0.0143563
$295$	−2.44923e8	−0.555461
$296$	−1.91309e8	−0.428761
$297$	−1.32389e8	−0.293227
$298$	−2.80112e7	−0.0613160
$299$	−2.70848e7	−0.0585971
$300$	8.08302e7	0.172842
$301$	−2.33848e8	−0.494254
$302$	−2.07361e8	−0.433215
$303$	−5.31478e8	−1.09758
$304$	9.70466e7	0.198117
$305$	−1.01881e8	−0.205609
$306$	6.69713e7	0.133618
$307$	3.84602e8	0.758625	0.379313	−	0.925269i	$-0.376160\pi$
$307$	3.84602e8	0.758625	0.379313	+	0.925269i	$0.376160\pi$
$308$	−3.83716e8	−0.748311
$309$	2.12371e7	0.0409487
$310$	3.91732e8	0.746832
$311$	−5.48997e7	−0.103492	−0.0517462	−	0.998660i	$-0.516479\pi$
$311$	−5.48997e7	−0.103492	−0.0517462	+	0.998660i	$0.516479\pi$
$312$	−3.07734e7	−0.0573633
$313$	1.23637e8	0.227900	0.113950	−	0.993487i	$-0.463650\pi$
$313$	1.23637e8	0.227900	0.113950	+	0.993487i	$0.463650\pi$
$314$	−5.90413e8	−1.07622
$315$	1.15055e8	0.207404
$316$	−5.25029e8	−0.936005
$317$	4.15729e8	0.732998	0.366499	−	0.930419i	$-0.380556\pi$
$317$	4.15729e8	0.732998	0.366499	+	0.930419i	$0.380556\pi$
$318$	−4.06495e8	−0.708859
$319$	1.24124e9	2.14087
$320$	−4.64138e7	−0.0791812
$321$	2.84785e8	0.480562
$322$	8.67647e7	0.144826
$323$	−2.72077e8	−0.449244
$324$	3.40122e7	0.0555556
$325$	1.04129e8	0.168260
$326$	−8.09951e8	−1.29478
$327$	5.32858e8	0.842741
$328$	3.20244e8	0.501098
$329$	1.12898e9	1.74783
$330$	−2.57230e8	−0.394024
$331$	−8.80573e8	−1.33465	−0.667325	−	0.744767i	$-0.732561\pi$
$331$	−8.80573e8	−1.33465	−0.667325	+	0.744767i	$0.732561\pi$
$332$	−2.65881e8	−0.398754
$333$	2.72392e8	0.404240
$334$	−1.00716e8	−0.147906
$335$	6.34480e7	0.0922063
$336$	9.85810e7	0.141777
$337$	−6.20707e8	−0.883450	−0.441725	−	0.897150i	$-0.645633\pi$
$337$	−6.20707e8	−0.883450	−0.441725	+	0.897150i	$0.645633\pi$
$338$	4.62344e8	0.651264
$339$	6.59918e8	0.920007
$340$	1.30124e8	0.179549
$341$	1.86017e9	2.54046
$342$	−1.38178e8	−0.186787
$343$	7.59916e8	1.01680
$344$	−1.34318e8	−0.177901
$345$	5.81640e7	0.0762583
$346$	−4.49399e8	−0.583265
$347$	−2.58523e8	−0.332159	−0.166080	−	0.986112i	$-0.553111\pi$
$347$	−2.58523e8	−0.332159	−0.166080	+	0.986112i	$0.553111\pi$
$348$	−3.18890e8	−0.405614
$349$	−6.41058e8	−0.807251	−0.403626	−	0.914924i	$-0.632250\pi$
$349$	−6.41058e8	−0.807251	−0.403626	+	0.914924i	$0.632250\pi$
$350$	−3.33572e8	−0.415864
$351$	4.38160e7	0.0540827
$352$	−2.20399e8	−0.269346
$353$	−7.37610e8	−0.892514	−0.446257	−	0.894905i	$-0.647243\pi$
$353$	−7.37610e8	−0.892514	−0.446257	+	0.894905i	$0.647243\pi$
$354$	2.98798e8	0.357986
$355$	7.59992e8	0.901591
$356$	2.34229e8	0.275148
$357$	−2.76379e8	−0.321489
$358$	5.88858e8	0.678297
$359$	4.34082e8	0.495155	0.247578	−	0.968868i	$-0.420365\pi$
$359$	4.34082e8	0.495155	0.247578	+	0.968868i	$0.420365\pi$
$360$	6.60852e7	0.0746527
$361$	−3.32513e8	−0.371992
$362$	−5.91097e7	−0.0654906
$363$	−6.95321e8	−0.762978
$364$	1.26996e8	0.138018
$365$	5.86003e7	0.0630775
$366$	1.24291e8	0.132512
$367$	−1.16469e9	−1.22992	−0.614962	−	0.788557i	$-0.710829\pi$
$367$	−1.16469e9	−1.22992	−0.614962	+	0.788557i	$0.710829\pi$
$368$	4.98360e7	0.0521286
$369$	−4.55973e8	−0.472440
$370$	5.29253e8	0.543197
$371$	1.67753e9	1.70554
$372$	−4.77899e8	−0.481322
$373$	1.48969e9	1.48633	0.743167	−	0.669106i	$-0.233323\pi$
$373$	1.48969e9	1.48633	0.743167	+	0.669106i	$0.233323\pi$
$374$	6.17905e8	0.610761
$375$	−5.97089e8	−0.584695
$376$	6.48463e8	0.629112
$377$	−4.10808e8	−0.394861
$378$	−1.40362e8	−0.133669
$379$	5.49598e8	0.518570	0.259285	−	0.965801i	$-0.416513\pi$
$379$	5.49598e8	0.518570	0.259285	+	0.965801i	$0.416513\pi$
$380$	−2.68477e8	−0.250995
$381$	−5.61865e8	−0.520469
$382$	1.25049e7	0.0114778
$383$	4.27406e8	0.388727	0.194364	−	0.980930i	$-0.437736\pi$
$383$	4.27406e8	0.388727	0.194364	+	0.980930i	$0.437736\pi$
$384$	5.66231e7	0.0510310
$385$	1.06154e9	0.948034
$386$	5.58260e8	0.494061
$387$	1.91245e8	0.167727
$388$	9.35798e7	0.0813338
$389$	−1.96677e9	−1.69407	−0.847033	−	0.531540i	$-0.821614\pi$
$389$	−1.96677e9	−1.69407	−0.847033	+	0.531540i	$0.821614\pi$
$390$	8.51339e7	0.0726735
$391$	−1.39719e8	−0.118205
$392$	1.48277e7	0.0124329
$393$	1.35030e9	1.12216
$394$	1.62452e9	1.33810
$395$	1.45248e9	1.18582
$396$	3.13811e8	0.253942
$397$	9.23087e8	0.740416	0.370208	−	0.928949i	$-0.379286\pi$
$397$	9.23087e8	0.740416	0.370208	+	0.928949i	$0.379286\pi$
$398$	3.69036e7	0.0293412
$399$	5.70235e8	0.449416
$400$	−1.91597e8	−0.149686
$401$	−4.90877e8	−0.380160	−0.190080	−	0.981769i	$-0.560875\pi$
$401$	−4.90877e8	−0.380160	−0.190080	+	0.981769i	$0.560875\pi$
$402$	−7.74042e7	−0.0594256
$403$	−6.15650e8	−0.468561
$404$	1.25980e9	0.950532
$405$	−9.40940e7	−0.0703832
$406$	1.31600e9	0.975922
$407$	2.51320e9	1.84776
$408$	−1.58747e8	−0.115716
$409$	−1.76026e9	−1.27217	−0.636084	−	0.771620i	$-0.719447\pi$
$409$	−1.76026e9	−1.27217	−0.636084	+	0.771620i	$0.719447\pi$
$410$	−8.85948e8	−0.634840
$411$	1.76591e8	0.125465
$412$	−5.03397e7	−0.0354626
$413$	−1.23309e9	−0.861327
$414$	−7.09579e7	−0.0491473
$415$	7.35554e8	0.505180
$416$	7.29444e7	0.0496781
$417$	−5.06158e8	−0.341830
$418$	−1.27488e9	−0.853795
$419$	2.63900e8	0.175263	0.0876316	−	0.996153i	$-0.472070\pi$
$419$	2.63900e8	0.175263	0.0876316	+	0.996153i	$0.472070\pi$
$420$	−2.72722e8	−0.179617
$421$	−9.01342e7	−0.0588711	−0.0294356	−	0.999567i	$-0.509371\pi$
$421$	−9.01342e7	−0.0588711	−0.0294356	+	0.999567i	$0.509371\pi$
$422$	3.62467e8	0.234787
$423$	−9.23300e8	−0.593132
$424$	9.63543e8	0.613890
$425$	5.37157e8	0.339422
$426$	−9.27162e8	−0.581062
$427$	−5.12925e8	−0.318828
$428$	−6.75046e8	−0.416179
$429$	4.04265e8	0.247210
$430$	3.71587e8	0.225383
$431$	−1.74027e9	−1.04700	−0.523499	−	0.852026i	$-0.675374\pi$
$431$	−1.74027e9	−1.04700	−0.523499	+	0.852026i	$0.675374\pi$
$432$	−8.06216e7	−0.0481125
$433$	7.25207e8	0.429294	0.214647	−	0.976692i	$-0.431140\pi$
$433$	7.25207e8	0.429294	0.214647	+	0.976692i	$0.431140\pi$
$434$	1.97220e9	1.15808
$435$	8.82200e8	0.513872
$436$	−1.26307e9	−0.729835
$437$	2.88273e8	0.165241
$438$	−7.14902e7	−0.0406525
$439$	−1.56064e9	−0.880394	−0.440197	−	0.897901i	$-0.645091\pi$
$439$	−1.56064e9	−0.880394	−0.440197	+	0.897901i	$0.645091\pi$
$440$	6.09729e8	0.341234
$441$	−2.11121e7	−0.0117219
$442$	−2.04505e8	−0.112648
$443$	−2.62583e9	−1.43501	−0.717504	−	0.696555i	$-0.754715\pi$
$443$	−2.62583e9	−1.43501	−0.717504	+	0.696555i	$0.754715\pi$
$444$	−6.45669e8	−0.350082
$445$	−6.47990e8	−0.348585
$446$	2.07341e9	1.10665
$447$	−9.45376e7	−0.0500643
$448$	−2.33673e8	−0.122782
$449$	2.51957e9	1.31360	0.656801	−	0.754064i	$-0.271909\pi$
$449$	2.51957e9	1.31360	0.656801	+	0.754064i	$0.271909\pi$
$450$	2.72802e8	0.141125
$451$	−4.20699e9	−2.15950
$452$	−1.56425e9	−0.796750
$453$	−6.99844e8	−0.353718
$454$	−8.50999e8	−0.426809
$455$	−3.51332e8	−0.174855
$456$	3.27532e8	0.161762
$457$	1.34459e8	0.0658998	0.0329499	−	0.999457i	$-0.489510\pi$
$457$	1.34459e8	0.0658998	0.0329499	+	0.999457i	$0.489510\pi$
$458$	1.40954e8	0.0685563
$459$	2.26028e8	0.109098
$460$	−1.37870e8	−0.0660416
$461$	1.79566e9	0.853631	0.426816	−	0.904339i	$-0.359635\pi$
$461$	1.79566e9	0.853631	0.426816	+	0.904339i	$0.359635\pi$
$462$	−1.29504e9	−0.610994
$463$	−3.36418e9	−1.57524	−0.787618	−	0.616164i	$-0.788686\pi$
$463$	−3.36418e9	−1.57524	−0.787618	+	0.616164i	$0.788686\pi$
$464$	7.55887e8	0.351272
$465$	1.32209e9	0.609786
$466$	−3.43492e8	−0.157241
$467$	−2.39764e9	−1.08937	−0.544685	−	0.838640i	$-0.683351\pi$
$467$	−2.39764e9	−1.08937	−0.544685	+	0.838640i	$0.683351\pi$
$468$	−1.03860e8	−0.0468370
$469$	3.19433e8	0.142980
$470$	−1.79396e9	−0.797021
$471$	−1.99264e9	−0.878732
$472$	−7.08261e8	−0.310025
$473$	1.76451e9	0.766672
$474$	−1.77197e9	−0.764245
$475$	−1.10828e9	−0.474485
$476$	6.55120e8	0.278417
$477$	−1.37192e9	−0.578781
$478$	1.46423e9	0.613212
$479$	−1.98943e9	−0.827092	−0.413546	−	0.910483i	$-0.635710\pi$
$479$	−1.98943e9	−0.827092	−0.413546	+	0.910483i	$0.635710\pi$
$480$	−1.56646e8	−0.0646511
$481$	−8.31779e8	−0.340800
$482$	−6.12480e8	−0.249131
$483$	2.92831e8	0.118250
$484$	1.64817e9	0.660759
$485$	−2.58886e8	−0.103042
$486$	1.14791e8	0.0453609
$487$	2.51147e9	0.985318	0.492659	−	0.870223i	$-0.336025\pi$
$487$	2.51147e9	0.985318	0.492659	+	0.870223i	$0.336025\pi$
$488$	−2.94615e8	−0.114759
$489$	−2.73358e9	−1.05719
$490$	−4.10206e7	−0.0157513
$491$	−2.11295e9	−0.805572	−0.402786	−	0.915294i	$-0.631958\pi$
$491$	−2.11295e9	−0.805572	−0.402786	+	0.915294i	$0.631958\pi$
$492$	1.08082e9	0.409145
$493$	−2.11918e9	−0.796533
$494$	4.21941e8	0.157474
$495$	−8.68150e8	−0.321719
$496$	1.13280e9	0.416837
$497$	3.82623e9	1.39805
$498$	−8.97350e8	−0.325581
$499$	−2.01930e9	−0.727525	−0.363763	−	0.931492i	$-0.618508\pi$
$499$	−2.01930e9	−0.727525	−0.363763	+	0.931492i	$0.618508\pi$
$500$	1.41532e9	0.506361
$501$	−3.39916e8	−0.120764
$502$	1.90366e9	0.671623
$503$	2.64421e9	0.926422	0.463211	−	0.886248i	$-0.346697\pi$
$503$	2.64421e9	0.926422	0.463211	+	0.886248i	$0.346697\pi$
$504$	3.32711e8	0.115760
$505$	−3.48520e9	−1.20423
$506$	−6.54687e8	−0.224650
$507$	1.56041e9	0.531755
$508$	1.33183e9	0.450739
$509$	3.89059e9	1.30769	0.653843	−	0.756631i	$-0.273156\pi$
$509$	3.89059e9	1.30769	0.653843	+	0.756631i	$0.273156\pi$
$510$	4.39169e8	0.146601
$511$	2.95027e8	0.0978113
$512$	−1.34218e8	−0.0441942
$513$	−4.66349e8	−0.152511
$514$	−2.76044e9	−0.896619
$515$	1.39264e8	0.0449275
$516$	−4.53322e8	−0.145256
$517$	−8.51875e9	−2.71118
$518$	2.66456e9	0.842309
$519$	−1.51672e9	−0.476234
$520$	−2.01799e8	−0.0629371
$521$	−3.00219e9	−0.930049	−0.465025	−	0.885298i	$-0.653955\pi$
$521$	−3.00219e9	−0.930049	−0.465025	+	0.885298i	$0.653955\pi$
$522$	−1.07625e9	−0.331183
$523$	−2.66144e9	−0.813506	−0.406753	−	0.913538i	$-0.633339\pi$
$523$	−2.66144e9	−0.813506	−0.406753	+	0.913538i	$0.633339\pi$
$524$	−3.20071e9	−0.971822
$525$	−1.12580e9	−0.339551
$526$	3.86679e9	1.15851
$527$	−3.17587e9	−0.945205
$528$	−7.43848e8	−0.219920
$529$	1.48036e8	0.0434783
$530$	−2.66562e9	−0.777737
$531$	1.00844e9	0.292294
$532$	−1.35167e9	−0.389206
$533$	1.39236e9	0.398298
$534$	7.90524e8	0.224657
$535$	1.86750e9	0.527257
$536$	1.83477e8	0.0514641
$537$	1.98740e9	0.553827
$538$	2.01865e9	0.558887
$539$	−1.94789e8	−0.0535802
$540$	2.23038e8	0.0609537
$541$	−6.10584e9	−1.65789	−0.828944	−	0.559331i	$-0.811058\pi$
$541$	−6.10584e9	−1.65789	−0.828944	+	0.559331i	$0.811058\pi$
$542$	−1.27073e9	−0.342812
$543$	−1.99495e8	−0.0534728
$544$	3.76289e8	0.100213
$545$	3.49425e9	0.924627
$546$	4.28613e8	0.112691
$547$	−4.14216e9	−1.08211	−0.541055	−	0.840987i	$-0.681975\pi$
$547$	−4.14216e9	−1.08211	−0.541055	+	0.840987i	$0.681975\pi$
$548$	−4.18587e8	−0.108656
$549$	4.19481e8	0.108195
$550$	2.51698e9	0.645076
$551$	4.37237e9	1.11349
$552$	1.68197e8	0.0425628
$553$	7.31261e9	1.83880
$554$	−5.09631e8	−0.127342
$555$	1.78623e9	0.443518
$556$	1.19978e9	0.296033
$557$	1.35314e9	0.331779	0.165890	−	0.986144i	$-0.446950\pi$
$557$	1.35314e9	0.331779	0.165890	+	0.986144i	$0.446950\pi$
$558$	−1.61291e9	−0.392998
$559$	−5.83990e8	−0.141405
$560$	6.46452e8	0.155553
$561$	2.08543e9	0.498684
$562$	2.70647e9	0.643169
$563$	−4.11885e8	−0.0972740	−0.0486370	−	0.998817i	$-0.515488\pi$
$563$	−4.11885e8	−0.0972740	−0.0486370	+	0.998817i	$0.515488\pi$
$564$	2.18856e9	0.513668
$565$	4.32746e9	1.00940
$566$	3.19873e9	0.741516
$567$	−4.73723e8	−0.109140
$568$	2.19772e9	0.503214
$569$	4.50430e9	1.02502	0.512512	−	0.858680i	$-0.328715\pi$
$569$	4.50430e9	1.02502	0.512512	+	0.858680i	$0.328715\pi$
$570$	−9.06110e8	−0.204936
$571$	−4.67876e9	−1.05173	−0.525865	−	0.850568i	$-0.676258\pi$
$571$	−4.67876e9	−1.05173	−0.525865	+	0.850568i	$0.676258\pi$
$572$	−9.58257e8	−0.214090
$573$	4.22041e7	0.00937159
$574$	−4.46037e9	−0.984417
$575$	−5.69132e8	−0.124846
$576$	1.91103e8	0.0416667
$577$	−1.12946e9	−0.244768	−0.122384	−	0.992483i	$-0.539054\pi$
$577$	−1.12946e9	−0.244768	−0.122384	+	0.992483i	$0.539054\pi$
$578$	2.22776e9	0.479867
$579$	1.88413e9	0.403399
$580$	−2.09114e9	−0.445026
$581$	3.70320e9	0.783359
$582$	3.15832e8	0.0664087
$583$	−1.26579e10	−2.64558
$584$	1.69458e8	0.0352061
$585$	2.87327e8	0.0593377
$586$	−4.63084e9	−0.950644
$587$	−2.87656e9	−0.587002	−0.293501	−	0.955959i	$-0.594820\pi$
$587$	−2.87656e9	−0.587002	−0.293501	+	0.955959i	$0.594820\pi$
$588$	5.00436e7	0.0101514
$589$	6.55258e9	1.32132
$590$	1.95939e9	0.392770
$591$	5.48276e9	1.09256
$592$	1.53047e9	0.303180
$593$	6.05269e9	1.19195	0.595974	−	0.803004i	$-0.296766\pi$
$593$	6.05269e9	1.19195	0.595974	+	0.803004i	$0.296766\pi$
$594$	1.05911e9	0.207343
$595$	−1.81237e9	−0.352727
$596$	2.24089e8	0.0433570
$597$	1.24550e8	0.0239570
$598$	2.16678e8	0.0414344
$599$	2.68933e9	0.511270	0.255635	−	0.966773i	$-0.417715\pi$
$599$	2.68933e9	0.511270	0.255635	+	0.966773i	$0.417715\pi$
$600$	−6.46641e8	−0.122218
$601$	8.41460e9	1.58115	0.790574	−	0.612366i	$-0.209782\pi$
$601$	8.41460e9	1.58115	0.790574	+	0.612366i	$0.209782\pi$
$602$	1.87078e9	0.349490
$603$	−2.61239e8	−0.0485208
$604$	1.65889e9	0.306329
$605$	−4.55962e9	−0.837114
$606$	4.25182e9	0.776106
$607$	−2.05559e9	−0.373057	−0.186529	−	0.982450i	$-0.559724\pi$
$607$	−2.05559e9	−0.373057	−0.186529	+	0.982450i	$0.559724\pi$
$608$	−7.76373e8	−0.140090
$609$	4.44150e9	0.796837
$610$	8.15045e8	0.145387
$611$	2.81940e9	0.500049
$612$	−5.35771e8	−0.0944819
$613$	8.59032e9	1.50625	0.753126	−	0.657876i	$-0.228545\pi$
$613$	8.59032e9	1.50625	0.753126	+	0.657876i	$0.228545\pi$
$614$	−3.07682e9	−0.536429
$615$	−2.99007e9	−0.518345
$616$	3.06973e9	0.529136
$617$	−5.89223e9	−1.00991	−0.504954	−	0.863146i	$-0.668491\pi$
$617$	−5.89223e9	−1.00991	−0.504954	+	0.863146i	$0.668491\pi$
$618$	−1.69896e8	−0.0289551
$619$	4.97442e9	0.842995	0.421497	−	0.906830i	$-0.361505\pi$
$619$	4.97442e9	0.842995	0.421497	+	0.906830i	$0.361505\pi$
$620$	−3.13385e9	−0.528090
$621$	−2.39483e8	−0.0401286
$622$	4.39197e8	0.0731802
$623$	−3.26235e9	−0.540534
$624$	2.46187e8	0.0405620
$625$	−2.61022e8	−0.0427658
$626$	−9.89097e8	−0.161149
$627$	−4.30273e9	−0.697121
$628$	4.72330e9	0.761004
$629$	−4.29079e9	−0.687480
$630$	−9.20437e8	−0.146657
$631$	−2.26042e9	−0.358167	−0.179084	−	0.983834i	$-0.557313\pi$
$631$	−2.26042e9	−0.358167	−0.179084	+	0.983834i	$0.557313\pi$
$632$	4.20023e9	0.661855
$633$	1.22333e9	0.191703
$634$	−3.32583e9	−0.518308
$635$	−3.68447e9	−0.571041
$636$	3.25196e9	0.501239
$637$	6.44684e7	0.00988231
$638$	−9.92995e9	−1.51382
$639$	−3.12917e9	−0.474435
$640$	3.71310e8	0.0559895
$641$	3.13969e9	0.470852	0.235426	−	0.971892i	$-0.424352\pi$
$641$	3.13969e9	0.470852	0.235426	+	0.971892i	$0.424352\pi$
$642$	−2.27828e9	−0.339809
$643$	3.14001e9	0.465792	0.232896	−	0.972502i	$-0.425180\pi$
$643$	3.14001e9	0.465792	0.232896	+	0.972502i	$0.425180\pi$
$644$	−6.94118e8	−0.102408
$645$	1.25410e9	0.184024
$646$	2.17661e9	0.317664
$647$	4.90048e9	0.711334	0.355667	−	0.934613i	$-0.384254\pi$
$647$	4.90048e9	0.711334	0.355667	+	0.934613i	$0.384254\pi$
$648$	−2.72098e8	−0.0392837
$649$	9.30430e9	1.33606
$650$	−8.33032e8	−0.118977
$651$	6.65618e9	0.945566
$652$	6.47960e9	0.915550
$653$	−5.55177e9	−0.780252	−0.390126	−	0.920761i	$-0.627569\pi$
$653$	−5.55177e9	−0.780252	−0.390126	+	0.920761i	$0.627569\pi$
$654$	−4.26286e9	−0.595908
$655$	8.85469e9	1.23120
$656$	−2.56195e9	−0.354330
$657$	−2.41279e8	−0.0331926
$658$	−9.03181e9	−1.23590
$659$	−8.33071e9	−1.13392	−0.566961	−	0.823745i	$-0.691881\pi$
$659$	−8.33071e9	−1.13392	−0.566961	+	0.823745i	$0.691881\pi$
$660$	2.05784e9	0.278617
$661$	3.71918e9	0.500890	0.250445	−	0.968131i	$-0.419423\pi$
$661$	3.71918e9	0.500890	0.250445	+	0.968131i	$0.419423\pi$
$662$	7.04458e9	0.943740
$663$	−6.90203e8	−0.0919771
$664$	2.12705e9	0.281961
$665$	3.73935e9	0.493084
$666$	−2.17913e9	−0.285841
$667$	2.24533e9	0.292981
$668$	8.05726e8	0.104585
$669$	6.99774e9	0.903580
$670$	−5.07584e8	−0.0651997
$671$	3.87030e9	0.494557
$672$	−7.88648e8	−0.100251
$673$	−8.18257e9	−1.03475	−0.517377	−	0.855758i	$-0.673091\pi$
$673$	−8.18257e9	−1.03475	−0.517377	+	0.855758i	$0.673091\pi$
$674$	4.96566e9	0.624694
$675$	9.20706e8	0.115228
$676$	−3.69876e9	−0.460513
$677$	−2.64604e9	−0.327745	−0.163873	−	0.986482i	$-0.552399\pi$
$677$	−2.64604e9	−0.327745	−0.163873	+	0.986482i	$0.552399\pi$
$678$	−5.27935e9	−0.650543
$679$	−1.30338e9	−0.159782
$680$	−1.04099e9	−0.126960
$681$	−2.87212e9	−0.348488
$682$	−1.48814e10	−1.79638
$683$	1.08885e9	0.130767	0.0653833	−	0.997860i	$-0.479173\pi$
$683$	1.08885e9	0.130767	0.0653833	+	0.997860i	$0.479173\pi$
$684$	1.10542e9	0.132078
$685$	1.15801e9	0.137656
$686$	−6.07933e9	−0.718987
$687$	4.75719e8	0.0559760
$688$	1.07454e9	0.125795
$689$	4.18932e9	0.487950
$690$	−4.65312e8	−0.0539228
$691$	−2.30229e9	−0.265453	−0.132727	−	0.991153i	$-0.542373\pi$
$691$	−2.30229e9	−0.265453	−0.132727	+	0.991153i	$0.542373\pi$
$692$	3.59520e9	0.412431
$693$	−4.37076e9	−0.498874
$694$	2.06819e9	0.234872
$695$	−3.31917e9	−0.375044
$696$	2.55112e9	0.286813
$697$	7.18261e9	0.803467
$698$	5.12847e9	0.570813
$699$	−1.15928e9	−0.128387
$700$	2.66857e9	0.294060
$701$	1.11511e10	1.22265	0.611326	−	0.791379i	$-0.290636\pi$
$701$	1.11511e10	1.22265	0.611326	+	0.791379i	$0.290636\pi$
$702$	−3.50528e8	−0.0382422
$703$	8.85292e9	0.961043
$704$	1.76319e9	0.190457
$705$	−6.05461e9	−0.650765
$706$	5.90088e9	0.631103
$707$	−1.75465e10	−1.86734
$708$	−2.39038e9	−0.253134
$709$	1.01449e10	1.06902	0.534509	−	0.845163i	$-0.320497\pi$
$709$	1.01449e10	1.06902	0.534509	+	0.845163i	$0.320497\pi$
$710$	−6.07993e9	−0.637521
$711$	−5.98040e9	−0.624003
$712$	−1.87384e9	−0.194559
$713$	3.36493e9	0.347666
$714$	2.21103e9	0.227327
$715$	2.65100e9	0.271230
$716$	−4.71087e9	−0.479629
$717$	4.94177e9	0.500686
$718$	−3.47266e9	−0.350128
$719$	−1.39291e10	−1.39757	−0.698784	−	0.715332i	$-0.746275\pi$
$719$	−1.39291e10	−1.39757	−0.698784	+	0.715332i	$0.746275\pi$
$720$	−5.28682e8	−0.0527874
$721$	7.01133e8	0.0696669
$722$	2.66010e9	0.263038
$723$	−2.06712e9	−0.203414
$724$	4.72878e8	0.0463088
$725$	−8.63230e9	−0.841286
$726$	5.56257e9	0.539507
$727$	9.71242e9	0.937469	0.468735	−	0.883339i	$-0.344710\pi$
$727$	9.71242e9	0.937469	0.468735	+	0.883339i	$0.344710\pi$
$728$	−1.01597e9	−0.0975936
$729$	3.87420e8	0.0370370
$730$	−4.68802e8	−0.0446025
$731$	−3.01255e9	−0.285249
$732$	−9.94325e8	−0.0937000
$733$	−1.23697e10	−1.16010	−0.580049	−	0.814581i	$-0.696967\pi$
$733$	−1.23697e10	−1.16010	−0.580049	+	0.814581i	$0.696967\pi$
$734$	9.31750e9	0.869688
$735$	−1.38444e8	−0.0128609
$736$	−3.98688e8	−0.0368605
$737$	−2.41030e9	−0.221786
$738$	3.64778e9	0.334065
$739$	−5.31249e8	−0.0484220	−0.0242110	−	0.999707i	$-0.507707\pi$
$739$	−5.31249e8	−0.0484220	−0.0242110	+	0.999707i	$0.507707\pi$
$740$	−4.23402e9	−0.384098
$741$	1.42405e9	0.128577
$742$	−1.34203e10	−1.20600
$743$	2.10796e10	1.88539	0.942694	−	0.333658i	$-0.108283\pi$
$743$	2.10796e10	1.88539	0.942694	+	0.333658i	$0.108283\pi$
$744$	3.82319e9	0.340346
$745$	−6.19937e8	−0.0549289
$746$	−1.19176e10	−1.05100
$747$	−3.02856e9	−0.265836
$748$	−4.94324e9	−0.431873
$749$	9.40206e9	0.817592
$750$	4.77671e9	0.413442
$751$	−1.58195e10	−1.36287	−0.681434	−	0.731880i	$-0.738643\pi$
$751$	−1.58195e10	−1.36287	−0.681434	+	0.731880i	$0.738643\pi$
$752$	−5.18770e9	−0.444849
$753$	6.42484e9	0.548378
$754$	3.28646e9	0.279209
$755$	−4.58928e9	−0.388088
$756$	1.12290e9	0.0945180
$757$	2.32217e10	1.94562	0.972809	−	0.231607i	$-0.0743985\pi$
$757$	2.32217e10	1.94562	0.972809	+	0.231607i	$0.0743985\pi$
$758$	−4.39678e9	−0.366685
$759$	−2.20957e9	−0.183426
$760$	2.14782e9	0.177480
$761$	−7.64170e9	−0.628556	−0.314278	−	0.949331i	$-0.601762\pi$
$761$	−7.64170e9	−0.628556	−0.314278	+	0.949331i	$0.601762\pi$
$762$	4.49492e9	0.368027
$763$	1.75921e10	1.43378
$764$	−1.00039e8	−0.00811604
$765$	1.48220e9	0.119699
$766$	−3.41925e9	−0.274872
$767$	−3.07939e9	−0.246423
$768$	−4.52985e8	−0.0360844
$769$	7.52939e9	0.597060	0.298530	−	0.954400i	$-0.403504\pi$
$769$	7.52939e9	0.597060	0.298530	+	0.954400i	$0.403504\pi$
$770$	−8.49233e9	−0.670362
$771$	−9.31650e9	−0.732086
$772$	−4.46608e9	−0.349354
$773$	−1.49235e10	−1.16210	−0.581049	−	0.813869i	$-0.697357\pi$
$773$	−1.49235e10	−1.16210	−0.581049	+	0.813869i	$0.697357\pi$
$774$	−1.52996e9	−0.118601
$775$	−1.29367e10	−0.998311
$776$	−7.48638e8	−0.0575117
$777$	8.99289e9	0.687743
$778$	1.57342e10	1.19789
$779$	−1.48194e10	−1.12318
$780$	−6.81071e8	−0.0513879
$781$	−2.88710e10	−2.16862
$782$	1.11775e9	0.0835836
$783$	−3.63235e9	−0.270409
$784$	−1.18622e8	−0.00879141
$785$	−1.30669e10	−0.964115
$786$	−1.08024e10	−0.793489
$787$	7.64151e8	0.0558814	0.0279407	−	0.999610i	$-0.491105\pi$
$787$	7.64151e8	0.0558814	0.0279407	+	0.999610i	$0.491105\pi$
$788$	−1.29962e10	−0.946180
$789$	1.30504e10	0.945920
$790$	−1.16198e10	−0.838504
$791$	2.17869e10	1.56523
$792$	−2.51049e9	−0.179564
$793$	−1.28093e9	−0.0912158
$794$	−7.38470e9	−0.523553
$795$	−8.99646e9	−0.635019
$796$	−2.95229e8	−0.0207474
$797$	−5.46728e9	−0.382532	−0.191266	−	0.981538i	$-0.561259\pi$
$797$	−5.46728e9	−0.382532	−0.191266	+	0.981538i	$0.561259\pi$
$798$	−4.56188e9	−0.317785
$799$	1.45441e10	1.00873
$800$	1.53278e9	0.105844
$801$	2.66802e9	0.183432
$802$	3.92701e9	0.268814
$803$	−2.22614e9	−0.151722
$804$	6.19234e8	0.0420202
$805$	1.92026e9	0.129740
$806$	4.92520e9	0.331323
$807$	6.81296e9	0.456329
$808$	−1.00784e10	−0.672127
$809$	−1.69364e10	−1.12461	−0.562305	−	0.826930i	$-0.690085\pi$
$809$	−1.69364e10	−1.12461	−0.562305	+	0.826930i	$0.690085\pi$
$810$	7.52752e8	0.0497685
$811$	2.88458e10	1.89894	0.949468	−	0.313865i	$-0.101624\pi$
$811$	2.88458e10	1.89894	0.949468	+	0.313865i	$0.101624\pi$
$812$	−1.05280e10	−0.690081
$813$	−4.28871e9	−0.279905
$814$	−2.01056e10	−1.30657
$815$	−1.79257e10	−1.15991
$816$	1.26997e9	0.0818238
$817$	6.21561e9	0.398755
$818$	1.40821e10	0.899559
$819$	1.44657e9	0.0920121
$820$	7.08758e9	0.448900
$821$	1.78036e10	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$821$	1.78036e10	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$822$	−1.41273e9	−0.0887173
$823$	2.08377e10	1.30302	0.651509	−	0.758641i	$-0.274136\pi$
$823$	2.08377e10	1.30302	0.651509	+	0.758641i	$0.274136\pi$
$824$	4.02718e8	0.0250758
$825$	8.49482e9	0.526702
$826$	9.86468e9	0.609050
$827$	5.38425e9	0.331021	0.165511	−	0.986208i	$-0.447073\pi$
$827$	5.38425e9	0.331021	0.165511	+	0.986208i	$0.447073\pi$
$828$	5.67664e8	0.0347524
$829$	−2.11026e8	−0.0128646	−0.00643229	−	0.999979i	$-0.502047\pi$
$829$	−2.11026e8	−0.0128646	−0.00643229	+	0.999979i	$0.502047\pi$
$830$	−5.88444e9	−0.357217
$831$	−1.72000e9	−0.103974
$832$	−5.83555e8	−0.0351277
$833$	3.32565e8	0.0199351
$834$	4.04926e9	0.241710
$835$	−2.22902e9	−0.132499
$836$	1.01991e10	0.603724
$837$	−5.44356e9	−0.320881
$838$	−2.11120e9	−0.123930
$839$	−1.91066e10	−1.11690	−0.558452	−	0.829537i	$-0.688605\pi$
$839$	−1.91066e10	−1.11690	−0.558452	+	0.829537i	$0.688605\pi$
$840$	2.18178e9	0.127008
$841$	1.68061e10	0.974274
$842$	7.21074e8	0.0416282
$843$	9.13432e9	0.525146
$844$	−2.89973e9	−0.166020
$845$	1.02325e10	0.583424
$846$	7.38640e9	0.419408
$847$	−2.29557e10	−1.29807
$848$	−7.70835e9	−0.434086
$849$	1.07957e10	0.605445
$850$	−4.29725e9	−0.240008
$851$	4.54621e9	0.252869
$852$	7.41730e9	0.410873
$853$	−1.42188e10	−0.784406	−0.392203	−	0.919879i	$-0.628287\pi$
$853$	−1.42188e10	−0.784406	−0.392203	+	0.919879i	$0.628287\pi$
$854$	4.10340e9	0.225446
$855$	−3.05812e9	−0.167330
$856$	5.40037e9	0.294283
$857$	−1.26011e10	−0.683872	−0.341936	−	0.939723i	$-0.611083\pi$
$857$	−1.26011e10	−0.683872	−0.341936	+	0.939723i	$0.611083\pi$
$858$	−3.23412e9	−0.174804
$859$	2.64100e10	1.42165	0.710825	−	0.703369i	$-0.248322\pi$
$859$	2.64100e10	1.42165	0.710825	+	0.703369i	$0.248322\pi$
$860$	−2.97269e9	−0.159370
$861$	−1.50537e10	−0.803773
$862$	1.39221e10	0.740339
$863$	−1.51664e10	−0.803240	−0.401620	−	0.915806i	$-0.631553\pi$
$863$	−1.51664e10	−0.803240	−0.401620	+	0.915806i	$0.631553\pi$
$864$	6.44973e8	0.0340207
$865$	−9.94602e9	−0.522508
$866$	−5.80166e9	−0.303557
$867$	7.51868e9	0.391809
$868$	−1.57776e10	−0.818884
$869$	−5.51777e10	−2.85229
$870$	−7.05760e9	−0.363362
$871$	7.97724e8	0.0409062
$872$	1.01046e10	0.516071
$873$	1.06593e9	0.0542225
$874$	−2.30618e9	−0.116843
$875$	−1.97126e10	−0.994756
$876$	5.71922e8	0.0287456
$877$	−8.15789e9	−0.408394	−0.204197	−	0.978930i	$-0.565458\pi$
$877$	−8.15789e9	−0.408394	−0.204197	+	0.978930i	$0.565458\pi$
$878$	1.24851e10	0.622533
$879$	−1.56291e10	−0.776198
$880$	−4.87784e9	−0.241289
$881$	2.12806e10	1.04850	0.524250	−	0.851564i	$-0.324346\pi$
$881$	2.12806e10	1.04850	0.524250	+	0.851564i	$0.324346\pi$
$882$	1.68897e8	0.00828862
$883$	−3.76680e10	−1.84124	−0.920619	−	0.390461i	$-0.872315\pi$
$883$	−3.76680e10	−1.84124	−0.920619	+	0.390461i	$0.872315\pi$
$884$	1.63604e9	0.0796545
$885$	6.61293e9	0.320695
$886$	2.10067e10	1.01470
$887$	1.83865e10	0.884638	0.442319	−	0.896858i	$-0.354156\pi$
$887$	1.83865e10	0.884638	0.442319	+	0.896858i	$0.354156\pi$
$888$	5.16535e9	0.247545
$889$	−1.85497e10	−0.885486
$890$	5.18392e9	0.246487
$891$	3.57450e9	0.169295
$892$	−1.65872e10	−0.782523
$893$	−3.00079e10	−1.41012
$894$	7.56301e8	0.0354008
$895$	1.30325e10	0.607641
$896$	1.86939e9	0.0868203
$897$	7.31289e8	0.0338311
$898$	−2.01565e10	−0.928857
$899$	5.10374e10	2.34277
$900$	−2.18241e9	−0.0997903
$901$	2.16109e10	0.984319
$902$	3.36559e10	1.52700
$903$	6.31388e9	0.285358
$904$	1.25140e10	0.563387
$905$	−1.30820e9	−0.0586686
$906$	5.59875e9	0.250117
$907$	−3.61205e10	−1.60741	−0.803707	−	0.595025i	$-0.797142\pi$
$907$	−3.61205e10	−1.60741	−0.803707	+	0.595025i	$0.797142\pi$
$908$	6.80799e9	0.301799
$909$	1.43499e10	0.633688
$910$	2.81066e9	0.123641
$911$	−9.58003e9	−0.419810	−0.209905	−	0.977722i	$-0.567315\pi$
$911$	−9.58003e9	−0.419810	−0.209905	+	0.977722i	$0.567315\pi$
$912$	−2.62026e9	−0.114383
$913$	−2.79427e10	−1.21512
$914$	−1.07567e9	−0.0465982
$915$	2.75078e9	0.118708
$916$	−1.12763e9	−0.0484766
$917$	4.45796e10	1.90916
$918$	−1.80823e9	−0.0771442
$919$	7.21849e9	0.306790	0.153395	−	0.988165i	$-0.450979\pi$
$919$	7.21849e9	0.306790	0.153395	+	0.988165i	$0.450979\pi$
$920$	1.10296e9	0.0466985
$921$	−1.03843e10	−0.437993
$922$	−1.43653e10	−0.603608
$923$	9.55529e9	0.399980
$924$	1.03603e10	0.432038
$925$	−1.74782e10	−0.726106
$926$	2.69134e10	1.11386
$927$	−5.73401e8	−0.0236417
$928$	−6.04709e9	−0.248387
$929$	−4.53114e10	−1.85418	−0.927091	−	0.374836i	$-0.877699\pi$
$929$	−4.53114e10	−1.85418	−0.927091	+	0.374836i	$0.877699\pi$
$930$	−1.05768e10	−0.431184
$931$	−6.86159e8	−0.0278677
$932$	2.74793e9	0.111186
$933$	1.48229e9	0.0597513
$934$	1.91812e10	0.770301
$935$	1.36754e10	0.547139
$936$	8.30882e8	0.0331187
$937$	1.76024e10	0.699010	0.349505	−	0.936935i	$-0.386350\pi$
$937$	1.76024e10	0.699010	0.349505	+	0.936935i	$0.386350\pi$
$938$	−2.55547e9	−0.101102
$939$	−3.33820e9	−0.131578
$940$	1.43517e10	0.563579
$941$	4.43297e10	1.73433	0.867164	−	0.498022i	$-0.165940\pi$
$941$	4.43297e10	1.73433	0.867164	+	0.498022i	$0.165940\pi$
$942$	1.59411e10	0.621357
$943$	−7.61018e9	−0.295532
$944$	5.66609e9	0.219221
$945$	−3.10647e9	−0.119745
$946$	−1.41161e10	−0.542119
$947$	7.36652e9	0.281863	0.140931	−	0.990019i	$-0.454990\pi$
$947$	7.36652e9	0.281863	0.140931	+	0.990019i	$0.454990\pi$
$948$	1.41758e10	0.540403
$949$	7.36774e8	0.0279835
$950$	8.86625e9	0.335511
$951$	−1.12247e10	−0.423196
$952$	−5.24096e9	−0.196871
$953$	2.18031e10	0.816006	0.408003	−	0.912981i	$-0.366225\pi$
$953$	2.18031e10	0.816006	0.408003	+	0.912981i	$0.366225\pi$
$954$	1.09754e10	0.409260
$955$	2.76756e8	0.0102822
$956$	−1.17138e10	−0.433606
$957$	−3.35136e10	−1.23603
$958$	1.59154e10	0.584842
$959$	5.83009e9	0.213457
$960$	1.25317e9	0.0457153
$961$	4.89738e10	1.78005
$962$	6.65423e9	0.240982
$963$	−7.68920e9	−0.277453
$964$	4.89984e9	0.176162
$965$	1.23553e10	0.442596
$966$	−2.34265e9	−0.0836155
$967$	1.35229e10	0.480925	0.240462	−	0.970658i	$-0.422701\pi$
$967$	1.35229e10	0.480925	0.240462	+	0.970658i	$0.422701\pi$
$968$	−1.31853e10	−0.467227
$969$	7.34607e9	0.259371
$970$	2.07109e9	0.0728614
$971$	−1.20514e10	−0.422446	−0.211223	−	0.977438i	$-0.567745\pi$
$971$	−1.20514e10	−0.422446	−0.211223	+	0.977438i	$0.567745\pi$
$972$	−9.18330e8	−0.0320750
$973$	−1.67106e10	−0.581563
$974$	−2.00917e10	−0.696725
$975$	−2.81148e9	−0.0971447
$976$	2.35692e9	0.0811466
$977$	2.07643e10	0.712340	0.356170	−	0.934421i	$-0.384082\pi$
$977$	2.07643e10	0.712340	0.356170	+	0.934421i	$0.384082\pi$
$978$	2.18687e10	0.747543
$979$	2.46162e10	0.838460
$980$	3.28165e8	0.0111378
$981$	−1.43872e10	−0.486557
$982$	1.69036e10	0.569625
$983$	−3.92557e10	−1.31815	−0.659076	−	0.752076i	$-0.729053\pi$
$983$	−3.92557e10	−1.31815	−0.659076	+	0.752076i	$0.729053\pi$
$984$	−8.64659e9	−0.289309
$985$	3.59536e10	1.19871
$986$	1.69534e10	0.563234
$987$	−3.04824e10	−1.00911
$988$	−3.37553e9	−0.111351
$989$	3.19188e9	0.104920
$990$	6.94520e9	0.227490
$991$	3.90476e10	1.27449	0.637245	−	0.770661i	$-0.280074\pi$
$991$	3.90476e10	1.27449	0.637245	+	0.770661i	$0.280074\pi$
$992$	−9.06237e9	−0.294748
$993$	2.37755e10	0.770560
$994$	−3.06099e10	−0.988574
$995$	8.16744e8	0.0262848
$996$	7.17880e9	0.230221
$997$	−5.45514e10	−1.74330	−0.871651	−	0.490127i	$-0.836950\pi$
$997$	−5.45514e10	−1.74330	−0.871651	+	0.490127i	$0.836950\pi$
$998$	1.61544e10	0.514438
$999$	−7.35457e9	−0.233388

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.8.a.e.1.2	✓	4	1.1	even	1	trivial
414.8.a.k.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)

\(f(q)\)	\(=\)	\(q-8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} -177.054 q^{5} +216.000 q^{6} -891.394 q^{7} -512.000 q^{8} +729.000 q^{9} +O(q^{10})\)	\(q-8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} -177.054 q^{5} +216.000 q^{6} -891.394 q^{7} -512.000 q^{8} +729.000 q^{9} +1416.44 q^{10} +6726.05 q^{11} -1728.00 q^{12} -2226.09 q^{13} +7131.15 q^{14} +4780.47 q^{15} +4096.00 q^{16} -11483.4 q^{17} -5832.00 q^{18} +23693.0 q^{19} -11331.5 q^{20} +24067.6 q^{21} -53808.4 q^{22} +12167.0 q^{23} +13824.0 q^{24} -46776.7 q^{25} +17808.7 q^{26} -19683.0 q^{27} -57049.2 q^{28} +184543. q^{29} -38243.8 q^{30} +276562. q^{31} -32768.0 q^{32} -181603. q^{33} +91867.4 q^{34} +157825. q^{35} +46656.0 q^{36} +373651. q^{37} -189544. q^{38} +60104.3 q^{39} +90651.9 q^{40} -625477. q^{41} -192541. q^{42} +262339. q^{43} +430467. q^{44} -129073. q^{45} -97336.0 q^{46} -1.26653e6 q^{47} -110592. q^{48} -28960.4 q^{49} +374214. q^{50} +310052. q^{51} -142469. q^{52} -1.88192e6 q^{53} +157464. q^{54} -1.19088e6 q^{55} +456394. q^{56} -639711. q^{57} -1.47634e6 q^{58} +1.38332e6 q^{59} +305950. q^{60} +575420. q^{61} -2.21249e6 q^{62} -649826. q^{63} +262144. q^{64} +394138. q^{65} +1.45283e6 q^{66} -358353. q^{67} -734939. q^{68} -328509. q^{69} -1.26260e6 q^{70} -4.29242e6 q^{71} -373248. q^{72} -330973. q^{73} -2.98921e6 q^{74} +1.26297e6 q^{75} +1.51635e6 q^{76} -5.99556e6 q^{77} -480834. q^{78} -8.20357e6 q^{79} -725215. q^{80} +531441. q^{81} +5.00382e6 q^{82} -4.15440e6 q^{83} +1.54033e6 q^{84} +2.03319e6 q^{85} -2.09871e6 q^{86} -4.98265e6 q^{87} -3.44374e6 q^{88} +3.65983e6 q^{89} +1.03258e6 q^{90} +1.98432e6 q^{91} +778688. q^{92} -7.46717e6 q^{93} +1.01322e7 q^{94} -4.19495e6 q^{95} +884736. q^{96} +1.46218e6 q^{97} +231683. q^{98} +4.90329e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 32 q^{2} - 108 q^{3} + 256 q^{4} - 90 q^{5} + 864 q^{6} - 222 q^{7} - 2048 q^{8} + 2916 q^{9}+O(q^{10}) \) 4 * q - 32 * q^2 - 108 * q^3 + 256 * q^4 - 90 * q^5 + 864 * q^6 - 222 * q^7 - 2048 * q^8 + 2916 * q^9	\( 4 q - 32 q^{2} - 108 q^{3} + 256 q^{4} - 90 q^{5} + 864 q^{6} - 222 q^{7} - 2048 q^{8} + 2916 q^{9} + 720 q^{10} - 4120 q^{11} - 6912 q^{12} - 6796 q^{13} + 1776 q^{14} + 2430 q^{15} + 16384 q^{16} + 18918 q^{17} - 23328 q^{18} + 19682 q^{19} - 5760 q^{20} + 5994 q^{21} + 32960 q^{22} + 48668 q^{23} + 55296 q^{24} + 261304 q^{25} + 54368 q^{26} - 78732 q^{27} - 14208 q^{28} + 12756 q^{29} - 19440 q^{30} + 440892 q^{31} - 131072 q^{32} + 111240 q^{33} - 151344 q^{34} + 150200 q^{35} + 186624 q^{36} + 567212 q^{37} - 157456 q^{38} + 183492 q^{39} + 46080 q^{40} + 275056 q^{41} - 47952 q^{42} + 149946 q^{43} - 263680 q^{44} - 65610 q^{45} - 389344 q^{46} - 2000728 q^{47} - 442368 q^{48} - 1623352 q^{49} - 2090432 q^{50} - 510786 q^{51} - 434944 q^{52} - 2778518 q^{53} + 629856 q^{54} - 2706008 q^{55} + 113664 q^{56} - 531414 q^{57} - 102048 q^{58} - 3656368 q^{59} + 155520 q^{60} - 989076 q^{61} - 3527136 q^{62} - 161838 q^{63} + 1048576 q^{64} - 7481556 q^{65} - 889920 q^{66} - 5755282 q^{67} + 1210752 q^{68} - 1314036 q^{69} - 1201600 q^{70} - 8843424 q^{71} - 1492992 q^{72} - 1100272 q^{73} - 4537696 q^{74} - 7055208 q^{75} + 1259648 q^{76} - 10973480 q^{77} - 1467936 q^{78} + 959830 q^{79} - 368640 q^{80} + 2125764 q^{81} - 2200448 q^{82} - 24990584 q^{83} + 383616 q^{84} - 7601436 q^{85} - 1199568 q^{86} - 344412 q^{87} + 2109440 q^{88} - 3623430 q^{89} + 524880 q^{90} - 9436372 q^{91} + 3114752 q^{92} - 11904084 q^{93} + 16005824 q^{94} - 12385728 q^{95} + 3538944 q^{96} + 14161716 q^{97} + 12986816 q^{98} - 3003480 q^{99}+O(q^{100}) \) 4 * q - 32 * q^2 - 108 * q^3 + 256 * q^4 - 90 * q^5 + 864 * q^6 - 222 * q^7 - 2048 * q^8 + 2916 * q^9 + 720 * q^10 - 4120 * q^11 - 6912 * q^12 - 6796 * q^13 + 1776 * q^14 + 2430 * q^15 + 16384 * q^16 + 18918 * q^17 - 23328 * q^18 + 19682 * q^19 - 5760 * q^20 + 5994 * q^21 + 32960 * q^22 + 48668 * q^23 + 55296 * q^24 + 261304 * q^25 + 54368 * q^26 - 78732 * q^27 - 14208 * q^28 + 12756 * q^29 - 19440 * q^30 + 440892 * q^31 - 131072 * q^32 + 111240 * q^33 - 151344 * q^34 + 150200 * q^35 + 186624 * q^36 + 567212 * q^37 - 157456 * q^38 + 183492 * q^39 + 46080 * q^40 + 275056 * q^41 - 47952 * q^42 + 149946 * q^43 - 263680 * q^44 - 65610 * q^45 - 389344 * q^46 - 2000728 * q^47 - 442368 * q^48 - 1623352 * q^49 - 2090432 * q^50 - 510786 * q^51 - 434944 * q^52 - 2778518 * q^53 + 629856 * q^54 - 2706008 * q^55 + 113664 * q^56 - 531414 * q^57 - 102048 * q^58 - 3656368 * q^59 + 155520 * q^60 - 989076 * q^61 - 3527136 * q^62 - 161838 * q^63 + 1048576 * q^64 - 7481556 * q^65 - 889920 * q^66 - 5755282 * q^67 + 1210752 * q^68 - 1314036 * q^69 - 1201600 * q^70 - 8843424 * q^71 - 1492992 * q^72 - 1100272 * q^73 - 4537696 * q^74 - 7055208 * q^75 + 1259648 * q^76 - 10973480 * q^77 - 1467936 * q^78 + 959830 * q^79 - 368640 * q^80 + 2125764 * q^81 - 2200448 * q^82 - 24990584 * q^83 + 383616 * q^84 - 7601436 * q^85 - 1199568 * q^86 - 344412 * q^87 + 2109440 * q^88 - 3623430 * q^89 + 524880 * q^90 - 9436372 * q^91 + 3114752 * q^92 - 11904084 * q^93 + 16005824 * q^94 - 12385728 * q^95 + 3538944 * q^96 + 14161716 * q^97 + 12986816 * q^98 - 3003480 * q^99

Introduction

L-functions

Modular forms

Varieties

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