LMFDB - Embedded newform 138.8.a.e.1.3

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			1.3
Root			$-122.254$ 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)

$f(q)$	$=$	$q-8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} +176.808 q^{5} +216.000 q^{6} +889.767 q^{7} -512.000 q^{8} +729.000 q^{9} +O(q^{10})$	\(q-8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} +176.808 q^{5} +216.000 q^{6} +889.767 q^{7} -512.000 q^{8} +729.000 q^{9} -1414.46 q^{10} -6376.13 q^{11} -1728.00 q^{12} -13666.1 q^{13} -7118.14 q^{14} -4773.82 q^{15} +4096.00 q^{16} +19123.9 q^{17} -5832.00 q^{18} +56593.2 q^{19} +11315.7 q^{20} -24023.7 q^{21} +51009.0 q^{22} +12167.0 q^{23} +13824.0 q^{24} -46863.9 q^{25} +109329. q^{26} -19683.0 q^{27} +56945.1 q^{28} -197788. q^{29} +38190.5 q^{30} +95075.4 q^{31} -32768.0 q^{32} +172156. q^{33} -152991. q^{34} +157318. q^{35} +46656.0 q^{36} +312833. q^{37} -452746. q^{38} +368986. q^{39} -90525.7 q^{40} +111303. q^{41} +192190. q^{42} -454381. q^{43} -408072. q^{44} +128893. q^{45} -97336.0 q^{46} -777505. q^{47} -110592. q^{48} -31857.1 q^{49} +374911. q^{50} -516345. q^{51} -874633. q^{52} +803702. q^{53} +157464. q^{54} -1.12735e6 q^{55} -455561. q^{56} -1.52802e6 q^{57} +1.58230e6 q^{58} +148944. q^{59} -305524. q^{60} -1.67000e6 q^{61} -760603. q^{62} +648640. q^{63} +262144. q^{64} -2.41628e6 q^{65} -1.37724e6 q^{66} -2.91492e6 q^{67} +1.22393e6 q^{68} -328509. q^{69} -1.25854e6 q^{70} -1.69856e6 q^{71} -373248. q^{72} +3.52620e6 q^{73} -2.50266e6 q^{74} +1.26533e6 q^{75} +3.62196e6 q^{76} -5.67327e6 q^{77} -2.95189e6 q^{78} -1.54260e6 q^{79} +724206. q^{80} +531441. q^{81} -890421. q^{82} -8.38500e6 q^{83} -1.53752e6 q^{84} +3.38126e6 q^{85} +3.63505e6 q^{86} +5.34027e6 q^{87} +3.26458e6 q^{88} -7.11261e6 q^{89} -1.03114e6 q^{90} -1.21597e7 q^{91} +778688. q^{92} -2.56704e6 q^{93} +6.22004e6 q^{94} +1.00061e7 q^{95} +884736. q^{96} +1.78554e6 q^{97} +254857. q^{98} -4.64820e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	176.808	0.632568	0.316284	−	0.948665i	$-0.397565\pi$
$5$	176.808	0.632568	0.316284	+	0.948665i	$0.397565\pi$
$6$	216.000	0.408248
$7$	889.767	0.980468	0.490234	−	0.871591i	$-0.336911\pi$
$7$	889.767	0.980468	0.490234	+	0.871591i	$0.336911\pi$
$8$	−512.000	−0.353553
$9$	729.000	0.333333
$10$	−1414.46	−0.447293
$11$	−6376.13	−1.44438	−0.722192	−	0.691692i	$-0.756865\pi$
$11$	−6376.13	−1.44438	−0.722192	+	0.691692i	$0.756865\pi$
$12$	−1728.00	−0.288675
$13$	−13666.1	−1.72522	−0.862609	−	0.505871i	$-0.831171\pi$
$13$	−13666.1	−1.72522	−0.862609	+	0.505871i	$0.831171\pi$
$14$	−7118.14	−0.693295
$15$	−4773.82	−0.365213
$16$	4096.00	0.250000
$17$	19123.9	0.944072	0.472036	−	0.881579i	$-0.343519\pi$
$17$	19123.9	0.944072	0.472036	+	0.881579i	$0.343519\pi$
$18$	−5832.00	−0.235702
$19$	56593.2	1.89290	0.946448	−	0.322857i	$-0.104643\pi$
$19$	56593.2	1.89290	0.946448	+	0.322857i	$0.104643\pi$
$20$	11315.7	0.316284
$21$	−24023.7	−0.566073
$22$	51009.0	1.02133
$23$	12167.0	0.208514
$23$	12167.0	0.208514
$24$	13824.0	0.204124
$25$	−46863.9	−0.599858
$26$	109329.	1.21991
$27$	−19683.0	−0.192450
$28$	56945.1	0.490234
$29$	−197788.	−1.50594	−0.752969	−	0.658057i	$-0.771379\pi$
$29$	−197788.	−1.50594	−0.752969	+	0.658057i	$0.771379\pi$
$30$	38190.5	0.258245
$31$	95075.4	0.573195	0.286597	−	0.958051i	$-0.407476\pi$
$31$	95075.4	0.573195	0.286597	+	0.958051i	$0.407476\pi$
$32$	−32768.0	−0.176777
$33$	172156.	0.833916
$34$	−152991.	−0.667560
$35$	157318.	0.620212
$36$	46656.0	0.166667
$37$	312833.	1.01533	0.507664	−	0.861555i	$-0.330509\pi$
$37$	312833.	1.01533	0.507664	+	0.861555i	$0.330509\pi$
$38$	−452746.	−1.33848
$39$	368986.	0.996055
$40$	−90525.7	−0.223647
$41$	111303.	0.252210	0.126105	−	0.992017i	$-0.459752\pi$
$41$	111303.	0.252210	0.126105	+	0.992017i	$0.459752\pi$
$42$	192190.	0.400274
$43$	−454381.	−0.871526	−0.435763	−	0.900061i	$-0.643521\pi$
$43$	−454381.	−0.871526	−0.435763	+	0.900061i	$0.643521\pi$
$44$	−408072.	−0.722192
$45$	128893.	0.210856
$46$	−97336.0	−0.147442
$47$	−777505.	−1.09235	−0.546173	−	0.837672i	$-0.683916\pi$
$47$	−777505.	−1.09235	−0.546173	+	0.837672i	$0.683916\pi$
$48$	−110592.	−0.144338
$49$	−31857.1	−0.0386830
$50$	374911.	0.424164
$51$	−516345.	−0.545060
$52$	−874633.	−0.862609
$53$	803702.	0.741531	0.370766	−	0.928726i	$-0.379095\pi$
$53$	803702.	0.741531	0.370766	+	0.928726i	$0.379095\pi$
$54$	157464.	0.136083
$55$	−1.12735e6	−0.913671
$56$	−455561.	−0.346648
$57$	−1.52802e6	−1.09286
$58$	1.58230e6	1.06486
$59$	148944.	0.0944151	0.0472075	−	0.998885i	$-0.484968\pi$
$59$	148944.	0.0944151	0.0472075	+	0.998885i	$0.484968\pi$
$60$	−305524.	−0.182607
$61$	−1.67000e6	−0.942023	−0.471012	−	0.882127i	$-0.656111\pi$
$61$	−1.67000e6	−0.942023	−0.471012	+	0.882127i	$0.656111\pi$
$62$	−760603.	−0.405310
$63$	648640.	0.326823
$64$	262144.	0.125000
$65$	−2.41628e6	−1.09132
$66$	−1.37724e6	−0.589667
$67$	−2.91492e6	−1.18404	−0.592019	−	0.805924i	$-0.701669\pi$
$67$	−2.91492e6	−1.18404	−0.592019	+	0.805924i	$0.701669\pi$
$68$	1.22393e6	0.472036
$69$	−328509.	−0.120386
$70$	−1.25854e6	−0.438556
$71$	−1.69856e6	−0.563219	−0.281610	−	0.959529i	$-0.590868\pi$
$71$	−1.69856e6	−0.563219	−0.281610	+	0.959529i	$0.590868\pi$
$72$	−373248.	−0.117851
$73$	3.52620e6	1.06090	0.530452	−	0.847715i	$-0.322022\pi$
$73$	3.52620e6	1.06090	0.530452	+	0.847715i	$0.322022\pi$
$74$	−2.50266e6	−0.717945
$75$	1.26533e6	0.346328
$76$	3.62196e6	0.946448
$77$	−5.67327e6	−1.41617
$78$	−2.95189e6	−0.704317
$79$	−1.54260e6	−0.352013	−0.176007	−	0.984389i	$-0.556318\pi$
$79$	−1.54260e6	−0.352013	−0.176007	+	0.984389i	$0.556318\pi$
$80$	724206.	0.158142
$81$	531441.	0.111111
$82$	−890421.	−0.178339
$83$	−8.38500e6	−1.60964	−0.804822	−	0.593516i	$-0.797739\pi$
$83$	−8.38500e6	−1.60964	−0.804822	+	0.593516i	$0.797739\pi$
$84$	−1.53752e6	−0.283037
$85$	3.38126e6	0.597190
$86$	3.63505e6	0.616262
$87$	5.34027e6	0.869453
$88$	3.26458e6	0.510667
$89$	−7.11261e6	−1.06946	−0.534730	−	0.845023i	$-0.679587\pi$
$89$	−7.11261e6	−1.06946	−0.534730	+	0.845023i	$0.679587\pi$
$90$	−1.03114e6	−0.149098
$91$	−1.21597e7	−1.69152
$92$	778688.	0.104257
$93$	−2.56704e6	−0.330934
$94$	6.22004e6	0.772406
$95$	1.00061e7	1.19738
$96$	884736.	0.102062
$97$	1.78554e6	0.198641	0.0993206	−	0.995055i	$-0.468333\pi$
$97$	1.78554e6	0.198641	0.0993206	+	0.995055i	$0.468333\pi$
$98$	254857.	0.0273530
$99$	−4.64820e6	−0.481461
$100$	−2.99929e6	−0.299929
$101$	−1.53784e7	−1.48520	−0.742602	−	0.669733i	$-0.766409\pi$
$101$	−1.53784e7	−1.48520	−0.742602	+	0.669733i	$0.766409\pi$
$102$	4.13076e6	0.385416
$103$	2.34834e6	0.211754	0.105877	−	0.994379i	$-0.466235\pi$
$103$	2.34834e6	0.211754	0.105877	+	0.994379i	$0.466235\pi$
$104$	6.99706e6	0.609957
$105$	−4.24759e6	−0.358080
$106$	−6.42962e6	−0.524342
$107$	−6.01455e6	−0.474635	−0.237318	−	0.971432i	$-0.576268\pi$
$107$	−6.01455e6	−0.474635	−0.237318	+	0.971432i	$0.576268\pi$
$108$	−1.25971e6	−0.0962250
$109$	2.58854e7	1.91453	0.957264	−	0.289214i	$-0.0933941\pi$
$109$	2.58854e7	1.91453	0.957264	+	0.289214i	$0.0933941\pi$
$110$	9.01881e6	0.646063
$111$	−8.44649e6	−0.586200
$112$	3.64449e6	0.245117
$113$	−2.39735e7	−1.56300	−0.781498	−	0.623908i	$-0.785544\pi$
$113$	−2.39735e7	−1.56300	−0.781498	+	0.623908i	$0.785544\pi$
$114$	1.22241e7	0.772771
$115$	2.15122e6	0.131900
$116$	−1.26584e7	−0.752969
$117$	−9.96261e6	−0.575073
$118$	−1.19155e6	−0.0667615
$119$	1.70158e7	0.925633
$120$	2.44420e6	0.129122
$121$	2.11679e7	1.08625
$122$	1.33600e7	0.666111
$123$	−3.00517e6	−0.145613
$124$	6.08483e6	0.286597
$125$	−2.20990e7	−1.01202
$126$	−5.18912e6	−0.231098
$127$	−2.64653e7	−1.14647	−0.573237	−	0.819389i	$-0.694313\pi$
$127$	−2.64653e7	−1.14647	−0.573237	+	0.819389i	$0.694313\pi$
$128$	−2.09715e6	−0.0883883
$129$	1.22683e7	0.503176
$130$	1.93303e7	0.771678
$131$	−2.82540e7	−1.09807	−0.549036	−	0.835799i	$-0.685005\pi$
$131$	−2.82540e7	−1.09807	−0.549036	+	0.835799i	$0.685005\pi$
$132$	1.10180e7	0.416958
$133$	5.03548e7	1.85592
$134$	2.33194e7	0.837241
$135$	−3.48011e6	−0.121738
$136$	−9.79144e6	−0.333780
$137$	−1.65906e7	−0.551239	−0.275619	−	0.961267i	$-0.588883\pi$
$137$	−1.65906e7	−0.551239	−0.275619	+	0.961267i	$0.588883\pi$
$138$	2.62807e6	0.0851257
$139$	−1.45106e7	−0.458284	−0.229142	−	0.973393i	$-0.573592\pi$
$139$	−1.45106e7	−0.458284	−0.229142	+	0.973393i	$0.573592\pi$
$140$	1.00684e7	0.310106
$141$	2.09926e7	0.630667
$142$	1.35885e7	0.398256
$143$	8.71371e7	2.49188
$144$	2.98598e6	0.0833333
$145$	−3.49705e7	−0.952607
$146$	−2.82096e7	−0.750173
$147$	860142.	0.0223336
$148$	2.00213e7	0.507664
$149$	5.53980e7	1.37196	0.685981	−	0.727619i	$-0.259373\pi$
$149$	5.53980e7	1.37196	0.685981	+	0.727619i	$0.259373\pi$
$150$	−1.01226e7	−0.244891
$151$	1.55684e7	0.367979	0.183990	−	0.982928i	$-0.441099\pi$
$151$	1.55684e7	0.367979	0.183990	+	0.982928i	$0.441099\pi$
$152$	−2.89757e7	−0.669240
$153$	1.39413e7	0.314691
$154$	4.53862e7	1.00139
$155$	1.68101e7	0.362585
$156$	2.36151e7	0.498028
$157$	−3.46974e6	−0.0715563	−0.0357782	−	0.999360i	$-0.511391\pi$
$157$	−3.46974e6	−0.0715563	−0.0357782	+	0.999360i	$0.511391\pi$
$158$	1.23408e7	0.248911
$159$	−2.17000e7	−0.428123
$160$	−5.79365e6	−0.111823
$161$	1.08258e7	0.204442
$162$	−4.25153e6	−0.0785674
$163$	−1.02375e7	−0.185156	−0.0925779	−	0.995705i	$-0.529511\pi$
$163$	−1.02375e7	−0.185156	−0.0925779	+	0.995705i	$0.529511\pi$
$164$	7.12337e6	0.126105
$165$	3.04385e7	0.527508
$166$	6.70800e7	1.13819
$167$	−9.11745e7	−1.51484	−0.757418	−	0.652930i	$-0.773540\pi$
$167$	−9.11745e7	−1.51484	−0.757418	+	0.652930i	$0.773540\pi$
$168$	1.23001e7	0.200137
$169$	1.24015e8	1.97638
$170$	−2.70501e7	−0.422277
$171$	4.12564e7	0.630965
$172$	−2.90804e7	−0.435763
$173$	−1.04330e8	−1.53196	−0.765981	−	0.642863i	$-0.777746\pi$
$173$	−1.04330e8	−1.53196	−0.765981	+	0.642863i	$0.777746\pi$
$174$	−4.27222e7	−0.614796
$175$	−4.16980e7	−0.588141
$176$	−2.61166e7	−0.361096
$177$	−4.02149e6	−0.0545106
$178$	5.69009e7	0.756222
$179$	−5.90974e7	−0.770164	−0.385082	−	0.922882i	$-0.625827\pi$
$179$	−5.90974e7	−0.770164	−0.385082	+	0.922882i	$0.625827\pi$
$180$	8.24916e6	0.105428
$181$	−1.45585e8	−1.82491	−0.912453	−	0.409182i	$-0.865814\pi$
$181$	−1.45585e8	−1.82491	−0.912453	+	0.409182i	$0.865814\pi$
$182$	9.72774e7	1.19609
$183$	4.50900e7	0.543877
$184$	−6.22950e6	−0.0737210
$185$	5.53114e7	0.642264
$186$	2.05363e7	0.234006
$187$	−1.21937e8	−1.36360
$188$	−4.97603e7	−0.546173
$189$	−1.75133e7	−0.188691
$190$	−8.00491e7	−0.846679
$191$	−3.60901e7	−0.374776	−0.187388	−	0.982286i	$-0.560002\pi$
$191$	−3.60901e7	−0.374776	−0.187388	+	0.982286i	$0.560002\pi$
$192$	−7.07789e6	−0.0721688
$193$	1.75911e8	1.76133	0.880666	−	0.473737i	$-0.157095\pi$
$193$	1.75911e8	1.76133	0.880666	+	0.473737i	$0.157095\pi$
$194$	−1.42843e7	−0.140461
$195$	6.52396e7	0.630072
$196$	−2.03885e6	−0.0193415
$197$	1.60867e8	1.49911	0.749557	−	0.661940i	$-0.230267\pi$
$197$	1.60867e8	1.49911	0.749557	+	0.661940i	$0.230267\pi$
$198$	3.71856e7	0.340445
$199$	8.09589e7	0.728247	0.364124	−	0.931351i	$-0.381369\pi$
$199$	8.09589e7	0.728247	0.364124	+	0.931351i	$0.381369\pi$
$200$	2.39943e7	0.212082
$201$	7.87030e7	0.683604
$202$	1.23027e8	1.05020
$203$	−1.75985e8	−1.47652
$204$	−3.30461e7	−0.272530
$205$	1.96792e7	0.159540
$206$	−1.87867e7	−0.149733
$207$	8.86974e6	0.0695048
$208$	−5.59765e7	−0.431304
$209$	−3.60846e8	−2.73407
$210$	3.39807e7	0.253201
$211$	−1.19741e8	−0.877514	−0.438757	−	0.898606i	$-0.644581\pi$
$211$	−1.19741e8	−0.877514	−0.438757	+	0.898606i	$0.644581\pi$
$212$	5.14369e7	0.370766
$213$	4.58612e7	0.325175
$214$	4.81164e7	0.335618
$215$	−8.03382e7	−0.551299
$216$	1.00777e7	0.0680414
$217$	8.45950e7	0.561999
$218$	−2.07083e8	−1.35378
$219$	−9.52073e7	−0.612514
$220$	−7.21505e7	−0.456836
$221$	−2.61350e8	−1.62873
$222$	6.75719e7	0.414506
$223$	2.06333e8	1.24595	0.622976	−	0.782241i	$-0.285924\pi$
$223$	2.06333e8	1.24595	0.622976	+	0.782241i	$0.285924\pi$
$224$	−2.91559e7	−0.173324
$225$	−3.41638e7	−0.199953
$226$	1.91788e8	1.10520
$227$	−1.47889e8	−0.839162	−0.419581	−	0.907718i	$-0.637823\pi$
$227$	−1.47889e8	−0.839162	−0.419581	+	0.907718i	$0.637823\pi$
$228$	−9.77930e7	−0.546432
$229$	−2.54998e8	−1.40318	−0.701589	−	0.712582i	$-0.747526\pi$
$229$	−2.54998e8	−1.40318	−0.701589	+	0.712582i	$0.747526\pi$
$230$	−1.72098e7	−0.0932670
$231$	1.53178e8	0.817627
$232$	1.01267e8	0.532429
$233$	2.37159e8	1.22827	0.614136	−	0.789200i	$-0.289505\pi$
$233$	2.37159e8	1.22827	0.614136	+	0.789200i	$0.289505\pi$
$234$	7.97009e7	0.406638
$235$	−1.37469e8	−0.690983
$236$	9.53242e6	0.0472075
$237$	4.16502e7	0.203235
$238$	−1.36127e8	−0.654521
$239$	−1.14391e7	−0.0541999	−0.0271000	−	0.999633i	$-0.508627\pi$
$239$	−1.14391e7	−0.0541999	−0.0271000	+	0.999633i	$0.508627\pi$
$240$	−1.95536e7	−0.0913033
$241$	1.93228e8	0.889221	0.444610	−	0.895724i	$-0.353342\pi$
$241$	1.93228e8	0.889221	0.444610	+	0.895724i	$0.353342\pi$
$242$	−1.69343e8	−0.768092
$243$	−1.43489e7	−0.0641500
$244$	−1.06880e8	−0.471012
$245$	−5.63259e6	−0.0244696
$246$	2.40414e7	0.102964
$247$	−7.73410e8	−3.26566
$248$	−4.86786e7	−0.202655
$249$	2.26395e8	0.929329
$250$	1.76792e8	0.715605
$251$	2.96716e8	1.18436	0.592179	−	0.805806i	$-0.298268\pi$
$251$	2.96716e8	1.18436	0.592179	+	0.805806i	$0.298268\pi$
$252$	4.15130e7	0.163411
$253$	−7.75784e7	−0.301175
$254$	2.11723e8	0.810680
$255$	−9.12940e7	−0.344788
$256$	1.67772e7	0.0625000
$257$	1.91976e8	0.705474	0.352737	−	0.935722i	$-0.385251\pi$
$257$	1.91976e8	0.705474	0.352737	+	0.935722i	$0.385251\pi$
$258$	−9.81462e7	−0.355799
$259$	2.78348e8	0.995496
$260$	−1.54642e8	−0.545659
$261$	−1.44187e8	−0.501979
$262$	2.26032e8	0.776454
$263$	1.21866e8	0.413084	0.206542	−	0.978438i	$-0.433779\pi$
$263$	1.21866e8	0.413084	0.206542	+	0.978438i	$0.433779\pi$
$264$	−8.81436e7	−0.294834
$265$	1.42101e8	0.469069
$266$	−4.02838e8	−1.31234
$267$	1.92041e8	0.617453
$268$	−1.86555e8	−0.592019
$269$	9.60669e7	0.300913	0.150456	−	0.988617i	$-0.451926\pi$
$269$	9.60669e7	0.300913	0.150456	+	0.988617i	$0.451926\pi$
$270$	2.78409e7	0.0860816
$271$	4.12417e8	1.25876	0.629381	−	0.777097i	$-0.283308\pi$
$271$	4.12417e8	1.25876	0.629381	+	0.777097i	$0.283308\pi$
$272$	7.83315e7	0.236018
$273$	3.28311e8	0.976600
$274$	1.32725e8	0.389785
$275$	2.98810e8	0.866425
$276$	−2.10246e7	−0.0601929
$277$	−5.90482e8	−1.66927	−0.834637	−	0.550800i	$-0.814323\pi$
$277$	−5.90482e8	−1.66927	−0.834637	+	0.550800i	$0.814323\pi$
$278$	1.16085e8	0.324056
$279$	6.93100e7	0.191065
$280$	−8.05468e7	−0.219278
$281$	−9.00038e7	−0.241985	−0.120993	−	0.992653i	$-0.538608\pi$
$281$	−9.00038e7	−0.241985	−0.120993	+	0.992653i	$0.538608\pi$
$282$	−1.67941e8	−0.445949
$283$	−3.29536e8	−0.864272	−0.432136	−	0.901809i	$-0.642240\pi$
$283$	−3.29536e8	−0.864272	−0.432136	+	0.901809i	$0.642240\pi$
$284$	−1.08708e8	−0.281610
$285$	−2.70166e8	−0.691310
$286$	−6.97096e8	−1.76202
$287$	9.90335e7	0.247284
$288$	−2.38879e7	−0.0589256
$289$	−4.46150e7	−0.108727
$290$	2.79764e8	0.673595
$291$	−4.82097e7	−0.114686
$292$	2.25676e8	0.530452
$293$	−2.81602e8	−0.654031	−0.327016	−	0.945019i	$-0.606043\pi$
$293$	−2.81602e8	−0.654031	−0.327016	+	0.945019i	$0.606043\pi$
$294$	−6.88114e6	−0.0157923
$295$	2.63345e7	0.0597239
$296$	−1.60170e8	−0.358973
$297$	1.25501e8	0.277972
$298$	−4.43184e8	−0.970124
$299$	−1.66276e8	−0.359733
$300$	8.09808e7	0.173164
$301$	−4.04293e8	−0.854503
$302$	−1.24547e8	−0.260201
$303$	4.15217e8	0.857483
$304$	2.31806e8	0.473224
$305$	−2.95269e8	−0.595894
$306$	−1.11531e8	−0.222520
$307$	3.58992e7	0.0708110	0.0354055	−	0.999373i	$-0.488728\pi$
$307$	3.58992e7	0.0708110	0.0354055	+	0.999373i	$0.488728\pi$
$308$	−3.63089e8	−0.708086
$309$	−6.34053e7	−0.122256
$310$	−1.34481e8	−0.256386
$311$	−8.16219e8	−1.53867	−0.769335	−	0.638846i	$-0.779412\pi$
$311$	−8.16219e8	−1.53867	−0.769335	+	0.638846i	$0.779412\pi$
$312$	−1.88921e8	−0.352159
$313$	4.35005e8	0.801843	0.400921	−	0.916112i	$-0.368690\pi$
$313$	4.35005e8	0.801843	0.400921	+	0.916112i	$0.368690\pi$
$314$	2.77579e7	0.0505980
$315$	1.14685e8	0.206737
$316$	−9.87265e7	−0.176007
$317$	1.54740e8	0.272833	0.136416	−	0.990652i	$-0.456442\pi$
$317$	1.54740e8	0.272833	0.136416	+	0.990652i	$0.456442\pi$
$318$	1.73600e8	0.302729
$319$	1.26112e9	2.17515
$320$	4.63492e7	0.0790710
$321$	1.62393e8	0.274031
$322$	−8.66064e7	−0.144562
$323$	1.08228e9	1.78703
$324$	3.40122e7	0.0555556
$325$	6.40448e8	1.03489
$326$	8.19000e7	0.130925
$327$	−6.98905e8	−1.10535
$328$	−5.69870e7	−0.0891697
$329$	−6.91798e8	−1.07101
$330$	−2.43508e8	−0.373005
$331$	2.46117e8	0.373029	0.186515	−	0.982452i	$-0.440281\pi$
$331$	2.46117e8	0.373029	0.186515	+	0.982452i	$0.440281\pi$
$332$	−5.36640e8	−0.804822
$333$	2.28055e8	0.338443
$334$	7.29396e8	1.07115
$335$	−5.15382e8	−0.748984
$336$	−9.84011e7	−0.141518
$337$	−7.28943e8	−1.03750	−0.518751	−	0.854926i	$-0.673603\pi$
$337$	−7.28943e8	−1.03750	−0.518751	+	0.854926i	$0.673603\pi$
$338$	−9.92118e8	−1.39751
$339$	6.47286e8	0.902396
$340$	2.16401e8	0.298595
$341$	−6.06213e8	−0.827914
$342$	−3.30051e8	−0.446160
$343$	−7.61107e8	−1.01840
$344$	2.32643e8	0.308131
$345$	−5.80831e7	−0.0761522
$346$	8.34640e8	1.08326
$347$	5.43931e8	0.698861	0.349430	−	0.936962i	$-0.386375\pi$
$347$	5.43931e8	0.698861	0.349430	+	0.936962i	$0.386375\pi$
$348$	3.41778e8	0.434727
$349$	8.79794e8	1.10788	0.553939	−	0.832557i	$-0.313124\pi$
$349$	8.79794e8	1.10788	0.553939	+	0.832557i	$0.313124\pi$
$350$	3.33584e8	0.415879
$351$	2.68991e8	0.332018
$352$	2.08933e8	0.255334
$353$	8.23969e8	0.997010	0.498505	−	0.866887i	$-0.333883\pi$
$353$	8.23969e8	0.997010	0.498505	+	0.866887i	$0.333883\pi$
$354$	3.21719e7	0.0385448
$355$	−3.00320e8	−0.356274
$356$	−4.55207e8	−0.534730
$357$	−4.59427e8	−0.534414
$358$	4.72779e8	0.544588
$359$	−4.60720e8	−0.525541	−0.262770	−	0.964858i	$-0.584636\pi$
$359$	−4.60720e8	−0.525541	−0.262770	+	0.964858i	$0.584636\pi$
$360$	−6.59933e7	−0.0745488
$361$	2.30892e9	2.58305
$362$	1.16468e9	1.29040
$363$	−5.71532e8	−0.627145
$364$	−7.78220e8	−0.845760
$365$	6.23460e8	0.671094
$366$	−3.60720e8	−0.384579
$367$	−6.60262e8	−0.697244	−0.348622	−	0.937263i	$-0.613350\pi$
$367$	−6.60262e8	−0.697244	−0.348622	+	0.937263i	$0.613350\pi$
$368$	4.98360e7	0.0521286
$369$	8.11397e7	0.0840700
$370$	−4.42491e8	−0.454149
$371$	7.15108e8	0.727047
$372$	−1.64290e8	−0.165467
$373$	6.94343e8	0.692776	0.346388	−	0.938091i	$-0.387408\pi$
$373$	6.94343e8	0.692776	0.346388	+	0.938091i	$0.387408\pi$
$374$	9.75492e8	0.964213
$375$	5.96674e8	0.584289
$376$	3.98082e8	0.386203
$377$	2.70300e9	2.59807
$378$	1.40106e8	0.133425
$379$	−3.83102e8	−0.361474	−0.180737	−	0.983531i	$-0.557848\pi$
$379$	−3.83102e8	−0.361474	−0.180737	+	0.983531i	$0.557848\pi$
$380$	6.40393e8	0.598692
$381$	7.14564e8	0.661917
$382$	2.88721e8	0.265007
$383$	−1.67196e8	−0.152065	−0.0760325	−	0.997105i	$-0.524225\pi$
$383$	−1.67196e8	−0.152065	−0.0760325	+	0.997105i	$0.524225\pi$
$384$	5.66231e7	0.0510310
$385$	−1.00308e9	−0.895825
$386$	−1.40728e9	−1.24545
$387$	−3.31244e8	−0.290509
$388$	1.14275e8	0.0993206
$389$	1.13867e9	0.980789	0.490395	−	0.871500i	$-0.336853\pi$
$389$	1.13867e9	0.980789	0.490395	+	0.871500i	$0.336853\pi$
$390$	−5.21917e8	−0.445528
$391$	2.32681e8	0.196853
$392$	1.63108e7	0.0136765
$393$	7.62858e8	0.633972
$394$	−1.28693e9	−1.06003
$395$	−2.72744e8	−0.222672
$396$	−2.97485e8	−0.240731
$397$	8.73518e8	0.700657	0.350328	−	0.936627i	$-0.386070\pi$
$397$	8.73518e8	0.700657	0.350328	+	0.936627i	$0.386070\pi$
$398$	−6.47671e8	−0.514949
$399$	−1.35958e9	−1.07152
$400$	−1.91955e8	−0.149964
$401$	2.30067e9	1.78176	0.890879	−	0.454241i	$-0.150090\pi$
$401$	2.30067e9	1.78176	0.890879	+	0.454241i	$0.150090\pi$
$402$	−6.29624e8	−0.483381
$403$	−1.29931e9	−0.988886
$404$	−9.84218e8	−0.742602
$405$	9.39631e7	0.0702853
$406$	1.40788e9	1.04406
$407$	−1.99466e9	−1.46652
$408$	2.64369e8	0.192708
$409$	−3.08439e8	−0.222914	−0.111457	−	0.993769i	$-0.535552\pi$
$409$	−3.08439e8	−0.222914	−0.111457	+	0.993769i	$0.535552\pi$
$410$	−1.57434e8	−0.112812
$411$	4.47946e8	0.318258
$412$	1.50294e8	0.105877
$413$	1.32526e8	0.0925709
$414$	−7.09579e7	−0.0491473
$415$	−1.48254e9	−1.01821
$416$	4.47812e8	0.304978
$417$	3.91787e8	0.264590
$418$	2.88676e9	1.93328
$419$	2.54726e9	1.69171	0.845853	−	0.533416i	$-0.179092\pi$
$419$	2.54726e9	1.69171	0.845853	+	0.533416i	$0.179092\pi$
$420$	−2.71846e8	−0.179040
$421$	2.92536e9	1.91070	0.955349	−	0.295480i	$-0.0954796\pi$
$421$	2.92536e9	1.91070	0.955349	+	0.295480i	$0.0954796\pi$
$422$	9.57928e8	0.620496
$423$	−5.66801e8	−0.364115
$424$	−4.11495e8	−0.262171
$425$	−8.96221e8	−0.566309
$426$	−3.66890e8	−0.229933
$427$	−1.48591e9	−0.923623
$428$	−3.84931e8	−0.237318
$429$	−2.35270e9	−1.43869
$430$	6.42705e8	0.389827
$431$	4.47518e8	0.269240	0.134620	−	0.990897i	$-0.457019\pi$
$431$	4.47518e8	0.269240	0.134620	+	0.990897i	$0.457019\pi$
$432$	−8.06216e7	−0.0481125
$433$	−2.94458e9	−1.74307	−0.871537	−	0.490329i	$-0.836877\pi$
$433$	−2.94458e9	−1.74307	−0.871537	+	0.490329i	$0.836877\pi$
$434$	−6.76760e8	−0.397393
$435$	9.44204e8	0.549988
$436$	1.65666e9	0.957264
$437$	6.88569e8	0.394696
$438$	7.61658e8	0.433112
$439$	1.89103e9	1.06677	0.533387	−	0.845871i	$-0.320919\pi$
$439$	1.89103e9	1.06677	0.533387	+	0.845871i	$0.320919\pi$
$440$	5.77204e8	0.323032
$441$	−2.32238e7	−0.0128943
$442$	2.09080e9	1.15169
$443$	5.89082e8	0.321931	0.160965	−	0.986960i	$-0.448539\pi$
$443$	5.89082e8	0.321931	0.160965	+	0.986960i	$0.448539\pi$
$444$	−5.40575e8	−0.293100
$445$	−1.25757e9	−0.676506
$446$	−1.65066e9	−0.881021
$447$	−1.49575e9	−0.792103
$448$	2.33247e8	0.122558
$449$	−2.45083e9	−1.27777	−0.638884	−	0.769303i	$-0.720603\pi$
$449$	−2.45083e9	−1.27777	−0.638884	+	0.769303i	$0.720603\pi$
$450$	2.73310e8	0.141388
$451$	−7.09680e8	−0.364288
$452$	−1.53431e9	−0.781498
$453$	−4.20346e8	−0.212453
$454$	1.18311e9	0.593377
$455$	−2.14993e9	−1.07000
$456$	7.82344e8	0.386386
$457$	−2.80311e9	−1.37383	−0.686915	−	0.726738i	$-0.741036\pi$
$457$	−2.80311e9	−1.37383	−0.686915	+	0.726738i	$0.741036\pi$
$458$	2.03998e9	0.992196
$459$	−3.76416e8	−0.181687
$460$	1.37678e8	0.0659498
$461$	−2.16343e8	−0.102846	−0.0514232	−	0.998677i	$-0.516376\pi$
$461$	−2.16343e8	−0.102846	−0.0514232	+	0.998677i	$0.516376\pi$
$462$	−1.22543e9	−0.578150
$463$	−3.51451e9	−1.64562	−0.822812	−	0.568313i	$-0.807596\pi$
$463$	−3.51451e9	−1.64562	−0.822812	+	0.568313i	$0.807596\pi$
$464$	−8.10139e8	−0.376484
$465$	−4.53873e8	−0.209338
$466$	−1.89728e9	−0.868520
$467$	−1.65663e8	−0.0752689	−0.0376345	−	0.999292i	$-0.511982\pi$
$467$	−1.65663e8	−0.0752689	−0.0376345	+	0.999292i	$0.511982\pi$
$468$	−6.37607e8	−0.287536
$469$	−2.59360e9	−1.16091
$470$	1.09975e9	0.488599
$471$	9.36830e7	0.0413131
$472$	−7.62594e7	−0.0333808
$473$	2.89719e9	1.25882
$474$	−3.33202e8	−0.143709
$475$	−2.65218e9	−1.13547
$476$	1.08901e9	0.462816
$477$	5.85899e8	0.247177
$478$	9.15127e7	0.0383251
$479$	2.71633e9	1.12930	0.564649	−	0.825331i	$-0.309011\pi$
$479$	2.71633e9	1.12930	0.564649	+	0.825331i	$0.309011\pi$
$480$	1.56428e8	0.0645612
$481$	−4.27522e9	−1.75166
$482$	−1.54582e9	−0.628774
$483$	−2.92297e8	−0.118034
$484$	1.35474e9	0.543123
$485$	3.15699e8	0.125654
$486$	1.14791e8	0.0453609
$487$	−3.60293e9	−1.41353	−0.706765	−	0.707448i	$-0.749846\pi$
$487$	−3.60293e9	−1.41353	−0.706765	+	0.707448i	$0.749846\pi$
$488$	8.55039e8	0.333056
$489$	2.76413e8	0.106900
$490$	4.50608e7	0.0173026
$491$	9.60286e8	0.366113	0.183057	−	0.983102i	$-0.441401\pi$
$491$	9.60286e8	0.366113	0.183057	+	0.983102i	$0.441401\pi$
$492$	−1.92331e8	−0.0728067
$493$	−3.78248e9	−1.42171
$494$	6.18728e9	2.30917
$495$	−8.21839e8	−0.304557
$496$	3.89429e8	0.143299
$497$	−1.51133e9	−0.552218
$498$	−1.81116e9	−0.657135
$499$	−1.09580e9	−0.394801	−0.197401	−	0.980323i	$-0.563250\pi$
$499$	−1.09580e9	−0.394801	−0.197401	+	0.980323i	$0.563250\pi$
$500$	−1.41434e9	−0.506009
$501$	2.46171e9	0.874591
$502$	−2.37373e9	−0.837468
$503$	2.21120e9	0.774713	0.387357	−	0.921930i	$-0.373388\pi$
$503$	2.21120e9	0.774713	0.387357	+	0.921930i	$0.373388\pi$
$504$	−3.32104e8	−0.115549
$505$	−2.71903e9	−0.939493
$506$	6.20627e8	0.212963
$507$	−3.34840e9	−1.14106
$508$	−1.69378e9	−0.573237
$509$	1.74713e9	0.587235	0.293618	−	0.955923i	$-0.405141\pi$
$509$	1.74713e9	0.587235	0.293618	+	0.955923i	$0.405141\pi$
$510$	7.30352e8	0.243802
$511$	3.13749e9	1.04018
$512$	−1.34218e8	−0.0441942
$513$	−1.11392e9	−0.364288
$514$	−1.53581e9	−0.498846
$515$	4.15206e8	0.133949
$516$	7.85170e8	0.251588
$517$	4.95747e9	1.57777
$518$	−2.22679e9	−0.703922
$519$	2.81691e9	0.884479
$520$	1.23714e9	0.385839
$521$	−2.85519e9	−0.884510	−0.442255	−	0.896889i	$-0.645821\pi$
$521$	−2.85519e9	−0.884510	−0.442255	+	0.896889i	$0.645821\pi$
$522$	1.15350e9	0.354953
$523$	−2.07580e9	−0.634496	−0.317248	−	0.948343i	$-0.602759\pi$
$523$	−2.07580e9	−0.634496	−0.317248	+	0.948343i	$0.602759\pi$
$524$	−1.80826e9	−0.549036
$525$	1.12585e9	0.339564
$526$	−9.74929e8	−0.292094
$527$	1.81821e9	0.541137
$528$	7.05149e8	0.208479
$529$	1.48036e8	0.0434783
$530$	−1.13681e9	−0.331682
$531$	1.08580e8	0.0314717
$532$	3.22271e9	0.927961
$533$	−1.52108e9	−0.435117
$534$	−1.53632e9	−0.436605
$535$	−1.06342e9	−0.300239
$536$	1.49244e9	0.418620
$537$	1.59563e9	0.444654
$538$	−7.68535e8	−0.212777
$539$	2.03125e8	0.0558731
$540$	−2.22727e8	−0.0608689
$541$	1.48257e9	0.402553	0.201277	−	0.979534i	$-0.435491\pi$
$541$	1.48257e9	0.402553	0.201277	+	0.979534i	$0.435491\pi$
$542$	−3.29933e9	−0.890079
$543$	3.93078e9	1.05361
$544$	−6.26652e8	−0.166890
$545$	4.57675e9	1.21107
$546$	−2.62649e9	−0.690560
$547$	−1.08691e9	−0.283947	−0.141974	−	0.989870i	$-0.545345\pi$
$547$	−1.08691e9	−0.283947	−0.141974	+	0.989870i	$0.545345\pi$
$548$	−1.06180e9	−0.275619
$549$	−1.21743e9	−0.314008
$550$	−2.39048e9	−0.612655
$551$	−1.11935e10	−2.85058
$552$	1.68197e8	0.0425628
$553$	−1.37256e9	−0.345138
$554$	4.72386e9	1.18036
$555$	−1.49341e9	−0.370811
$556$	−9.28680e8	−0.229142
$557$	−9.77168e8	−0.239594	−0.119797	−	0.992798i	$-0.538224\pi$
$557$	−9.77168e8	−0.239594	−0.119797	+	0.992798i	$0.538224\pi$
$558$	−5.54480e8	−0.135103
$559$	6.20963e9	1.50357
$560$	6.44375e8	0.155053
$561$	3.29229e9	0.787277
$562$	7.20031e8	0.171109
$563$	−5.95321e9	−1.40596	−0.702978	−	0.711211i	$-0.748147\pi$
$563$	−5.95321e9	−1.40596	−0.702978	+	0.711211i	$0.748147\pi$
$564$	1.34353e9	0.315333
$565$	−4.23872e9	−0.988701
$566$	2.63629e9	0.611132
$567$	4.72859e8	0.108941
$568$	8.69664e8	0.199128
$569$	5.77624e9	1.31448	0.657238	−	0.753683i	$-0.271725\pi$
$569$	5.77624e9	1.31448	0.657238	+	0.753683i	$0.271725\pi$
$570$	2.16133e9	0.488830
$571$	−6.03334e8	−0.135622	−0.0678111	−	0.997698i	$-0.521602\pi$
$571$	−6.03334e8	−0.135622	−0.0678111	+	0.997698i	$0.521602\pi$
$572$	5.57677e9	1.24594
$573$	9.74434e8	0.216377
$574$	−7.92268e8	−0.174856
$575$	−5.70193e8	−0.125079
$576$	1.91103e8	0.0416667
$577$	7.20695e9	1.56184	0.780919	−	0.624632i	$-0.214751\pi$
$577$	7.20695e9	1.56184	0.780919	+	0.624632i	$0.214751\pi$
$578$	3.56920e8	0.0768818
$579$	−4.74959e9	−1.01691
$580$	−2.23811e9	−0.476304
$581$	−7.46070e9	−1.57820
$582$	3.85677e8	0.0810949
$583$	−5.12451e9	−1.07106
$584$	−1.80541e9	−0.375086
$585$	−1.76147e9	−0.363772
$586$	2.25281e9	0.462470
$587$	2.60285e9	0.531148	0.265574	−	0.964090i	$-0.414439\pi$
$587$	2.60285e9	0.531148	0.265574	+	0.964090i	$0.414439\pi$
$588$	5.50491e7	0.0111668
$589$	5.38062e9	1.08500
$590$	−2.10676e8	−0.0422312
$591$	−4.34340e9	−0.865514
$592$	1.28136e9	0.253832
$593$	7.06897e9	1.39208	0.696041	−	0.718002i	$-0.254943\pi$
$593$	7.06897e9	1.39208	0.696041	+	0.718002i	$0.254943\pi$
$594$	−1.00401e9	−0.196556
$595$	3.00854e9	0.585525
$596$	3.54547e9	0.685981
$597$	−2.18589e9	−0.420454
$598$	1.33021e9	0.254370
$599$	6.57856e9	1.25065	0.625327	−	0.780363i	$-0.284966\pi$
$599$	6.57856e9	1.25065	0.625327	+	0.780363i	$0.284966\pi$
$600$	−6.47847e8	−0.122445
$601$	9.48877e9	1.78299	0.891496	−	0.453029i	$-0.149656\pi$
$601$	9.48877e9	1.78299	0.891496	+	0.453029i	$0.149656\pi$
$602$	3.23434e9	0.604225
$603$	−2.12498e9	−0.394679
$604$	9.96375e8	0.183990
$605$	3.74265e9	0.687125
$606$	−3.32173e9	−0.606332
$607$	−9.67981e8	−0.175674	−0.0878368	−	0.996135i	$-0.527995\pi$
$607$	−9.67981e8	−0.175674	−0.0878368	+	0.996135i	$0.527995\pi$
$608$	−1.85445e9	−0.334620
$609$	4.75160e9	0.852471
$610$	2.36215e9	0.421360
$611$	1.06255e10	1.88454
$612$	8.92245e8	0.157345
$613$	7.87191e8	0.138028	0.0690142	−	0.997616i	$-0.478015\pi$
$613$	7.87191e8	0.138028	0.0690142	+	0.997616i	$0.478015\pi$
$614$	−2.87194e8	−0.0500710
$615$	−5.31339e8	−0.0921104
$616$	2.90472e9	0.500693
$617$	2.62861e8	0.0450534	0.0225267	−	0.999746i	$-0.492829\pi$
$617$	2.62861e8	0.0450534	0.0225267	+	0.999746i	$0.492829\pi$
$618$	5.07242e8	0.0864481
$619$	1.78204e9	0.301996	0.150998	−	0.988534i	$-0.451751\pi$
$619$	1.78204e9	0.301996	0.150998	+	0.988534i	$0.451751\pi$
$620$	1.07585e9	0.181292
$621$	−2.39483e8	−0.0401286
$622$	6.52975e9	1.08800
$623$	−6.32857e9	−1.04857
$624$	1.51137e9	0.249014
$625$	−2.46049e8	−0.0403126
$626$	−3.48004e9	−0.566988
$627$	9.74283e9	1.57852
$628$	−2.22063e8	−0.0357782
$629$	5.98258e9	0.958543
$630$	−9.17479e8	−0.146185
$631$	−9.28943e9	−1.47193	−0.735963	−	0.677021i	$-0.763270\pi$
$631$	−9.28943e9	−1.47193	−0.735963	+	0.677021i	$0.763270\pi$
$632$	7.89812e8	0.124455
$633$	3.23301e9	0.506633
$634$	−1.23792e9	−0.192922
$635$	−4.67929e9	−0.725223
$636$	−1.38880e9	−0.214062
$637$	4.35364e8	0.0667366
$638$	−1.00890e10	−1.53806
$639$	−1.23825e9	−0.187740
$640$	−3.70793e8	−0.0559116
$641$	1.30833e9	0.196208	0.0981038	−	0.995176i	$-0.468722\pi$
$641$	1.30833e9	0.196208	0.0981038	+	0.995176i	$0.468722\pi$
$642$	−1.29914e9	−0.193769
$643$	−3.36391e8	−0.0499006	−0.0249503	−	0.999689i	$-0.507943\pi$
$643$	−3.36391e8	−0.0499006	−0.0249503	+	0.999689i	$0.507943\pi$
$644$	6.92851e8	0.102221
$645$	2.16913e9	0.318293
$646$	−8.65826e9	−1.26362
$647$	−7.51516e9	−1.09087	−0.545435	−	0.838153i	$-0.683636\pi$
$647$	−7.51516e9	−1.09087	−0.545435	+	0.838153i	$0.683636\pi$
$648$	−2.72098e8	−0.0392837
$649$	−9.49687e8	−0.136372
$650$	−5.12359e9	−0.731775
$651$	−2.28406e9	−0.324470
$652$	−6.55200e8	−0.0925779
$653$	3.78570e9	0.532047	0.266024	−	0.963966i	$-0.414290\pi$
$653$	3.78570e9	0.532047	0.266024	+	0.963966i	$0.414290\pi$
$654$	5.59124e9	0.781603
$655$	−4.99554e9	−0.694605
$656$	4.55896e8	0.0630525
$657$	2.57060e9	0.353635
$658$	5.53438e9	0.757319
$659$	−5.51041e9	−0.750041	−0.375021	−	0.927016i	$-0.622364\pi$
$659$	−5.51041e9	−0.750041	−0.375021	+	0.927016i	$0.622364\pi$
$660$	1.94806e9	0.263754
$661$	3.93515e9	0.529976	0.264988	−	0.964252i	$-0.414632\pi$
$661$	3.93515e9	0.529976	0.264988	+	0.964252i	$0.414632\pi$
$662$	−1.96893e9	−0.263772
$663$	7.05645e9	0.940348
$664$	4.29312e9	0.569095
$665$	8.90313e9	1.17400
$666$	−1.82444e9	−0.239315
$667$	−2.40649e9	−0.314010
$668$	−5.83517e9	−0.757418
$669$	−5.57098e9	−0.719350
$670$	4.12306e9	0.529612
$671$	1.06481e10	1.36064
$672$	7.87209e8	0.100069
$673$	−1.47823e10	−1.86935	−0.934674	−	0.355505i	$-0.884309\pi$
$673$	−1.47823e10	−1.86935	−0.934674	+	0.355505i	$0.884309\pi$
$674$	5.83154e9	0.733624
$675$	9.22422e8	0.115443
$676$	7.93694e9	0.988189
$677$	−3.19850e9	−0.396174	−0.198087	−	0.980184i	$-0.563473\pi$
$677$	−3.19850e9	−0.396174	−0.198087	+	0.980184i	$0.563473\pi$
$678$	−5.17828e9	−0.638090
$679$	1.58872e9	0.194761
$680$	−1.73121e9	−0.211139
$681$	3.99300e9	0.484490
$682$	4.84971e9	0.585423
$683$	−6.85312e9	−0.823030	−0.411515	−	0.911403i	$-0.635000\pi$
$683$	−6.85312e9	−0.823030	−0.411515	+	0.911403i	$0.635000\pi$
$684$	2.64041e9	0.315483
$685$	−2.93335e9	−0.348696
$686$	6.08886e9	0.720114
$687$	6.88495e9	0.810125
$688$	−1.86114e9	−0.217881
$689$	−1.09835e10	−1.27930
$690$	4.64664e8	0.0538478
$691$	6.19614e9	0.714411	0.357206	−	0.934026i	$-0.383730\pi$
$691$	6.19614e9	0.714411	0.357206	+	0.934026i	$0.383730\pi$
$692$	−6.67712e9	−0.765981
$693$	−4.13582e9	−0.472057
$694$	−4.35145e9	−0.494169
$695$	−2.56560e9	−0.289896
$696$	−2.73422e9	−0.307398
$697$	2.12854e9	0.238104
$698$	−7.03835e9	−0.783388
$699$	−6.40331e9	−0.709143
$700$	−2.66867e9	−0.294071
$701$	−5.07699e9	−0.556664	−0.278332	−	0.960485i	$-0.589782\pi$
$701$	−5.07699e9	−0.556664	−0.278332	+	0.960485i	$0.589782\pi$
$702$	−2.15192e9	−0.234772
$703$	1.77042e10	1.92191
$704$	−1.67146e9	−0.180548
$705$	3.71167e9	0.398939
$706$	−6.59175e9	−0.704992
$707$	−1.36832e10	−1.45620
$708$	−2.57375e8	−0.0272553
$709$	1.23277e9	0.129903	0.0649515	−	0.997888i	$-0.479311\pi$
$709$	1.23277e9	0.129903	0.0649515	+	0.997888i	$0.479311\pi$
$710$	2.40256e9	0.251924
$711$	−1.12456e9	−0.117338
$712$	3.64166e9	0.378111
$713$	1.15678e9	0.119519
$714$	3.67542e9	0.377888
$715$	1.54065e10	1.57628
$716$	−3.78224e9	−0.385082
$717$	3.08855e8	0.0312923
$718$	3.68576e9	0.371613
$719$	−7.56289e8	−0.0758816	−0.0379408	−	0.999280i	$-0.512080\pi$
$719$	−7.56289e8	−0.0758816	−0.0379408	+	0.999280i	$0.512080\pi$
$720$	5.27946e8	0.0527140
$721$	2.08948e9	0.207618
$722$	−1.84713e10	−1.82649
$723$	−5.21715e9	−0.513392
$724$	−9.31741e9	−0.912453
$725$	9.26911e9	0.903348
$726$	4.57226e9	0.443458
$727$	−1.58504e10	−1.52992	−0.764962	−	0.644075i	$-0.777242\pi$
$727$	−1.58504e10	−1.52992	−0.764962	+	0.644075i	$0.777242\pi$
$728$	6.22576e9	0.598043
$729$	3.87420e8	0.0370370
$730$	−4.98768e9	−0.474535
$731$	−8.68953e9	−0.822783
$732$	2.88576e9	0.271939
$733$	7.96077e8	0.0746606	0.0373303	−	0.999303i	$-0.488115\pi$
$733$	7.96077e8	0.0746606	0.0373303	+	0.999303i	$0.488115\pi$
$734$	5.28210e9	0.493026
$735$	1.52080e8	0.0141275
$736$	−3.98688e8	−0.0368605
$737$	1.85859e10	1.71021
$738$	−6.49117e8	−0.0594464
$739$	1.55180e10	1.41442	0.707212	−	0.707002i	$-0.249953\pi$
$739$	1.55180e10	1.41442	0.707212	+	0.707002i	$0.249953\pi$
$740$	3.53993e9	0.321132
$741$	2.08821e10	1.88543
$742$	−5.72086e9	−0.514100
$743$	−1.40109e10	−1.25316	−0.626579	−	0.779358i	$-0.715546\pi$
$743$	−1.40109e10	−1.25316	−0.626579	+	0.779358i	$0.715546\pi$
$744$	1.31432e9	0.117003
$745$	9.79482e9	0.867860
$746$	−5.55474e9	−0.489867
$747$	−6.11267e9	−0.536548
$748$	−7.80394e9	−0.681802
$749$	−5.35155e9	−0.465365
$750$	−4.77339e9	−0.413155
$751$	7.98903e9	0.688262	0.344131	−	0.938922i	$-0.388173\pi$
$751$	7.98903e9	0.688262	0.344131	+	0.938922i	$0.388173\pi$
$752$	−3.18466e9	−0.273087
$753$	−8.01133e9	−0.683789
$754$	−2.16240e10	−1.83711
$755$	2.75261e9	0.232772
$756$	−1.12085e9	−0.0943456
$757$	1.43727e10	1.20421	0.602107	−	0.798415i	$-0.294328\pi$
$757$	1.43727e10	1.20421	0.602107	+	0.798415i	$0.294328\pi$
$758$	3.06481e9	0.255601
$759$	2.09462e9	0.173883
$760$	−5.12314e9	−0.423339
$761$	1.05070e10	0.864239	0.432120	−	0.901816i	$-0.357766\pi$
$761$	1.05070e10	0.864239	0.432120	+	0.901816i	$0.357766\pi$
$762$	−5.71651e9	−0.468046
$763$	2.30320e10	1.87713
$764$	−2.30977e9	−0.187388
$765$	2.46494e9	0.199063
$766$	1.33756e9	0.107526
$767$	−2.03549e9	−0.162887
$768$	−4.52985e8	−0.0360844
$769$	4.56545e9	0.362028	0.181014	−	0.983481i	$-0.442062\pi$
$769$	4.56545e9	0.362028	0.181014	+	0.983481i	$0.442062\pi$
$770$	8.02464e9	0.633444
$771$	−5.18336e9	−0.407306
$772$	1.12583e10	0.880666
$773$	−2.30678e10	−1.79630	−0.898148	−	0.439693i	$-0.855087\pi$
$773$	−2.30678e10	−1.79630	−0.898148	+	0.439693i	$0.855087\pi$
$774$	2.64995e9	0.205421
$775$	−4.45560e9	−0.343835
$776$	−9.14198e8	−0.0702303
$777$	−7.51541e9	−0.574750
$778$	−9.10939e9	−0.693523
$779$	6.29897e9	0.477407
$780$	4.17534e9	0.315036
$781$	1.08303e10	0.813505
$782$	−1.86144e9	−0.139196
$783$	3.89306e9	0.289818
$784$	−1.30487e8	−0.00967075
$785$	−6.13478e8	−0.0452642
$786$	−6.10287e9	−0.448286
$787$	−8.42377e9	−0.616020	−0.308010	−	0.951383i	$-0.599663\pi$
$787$	−8.42377e9	−0.616020	−0.308010	+	0.951383i	$0.599663\pi$
$788$	1.02955e10	0.749557
$789$	−3.29039e9	−0.238494
$790$	2.18196e9	0.157453
$791$	−2.13309e10	−1.53247
$792$	2.37988e9	0.170222
$793$	2.28224e10	1.62520
$794$	−6.98815e9	−0.495439
$795$	−3.83673e9	−0.270817
$796$	5.18137e9	0.364124
$797$	−4.55898e9	−0.318980	−0.159490	−	0.987200i	$-0.550985\pi$
$797$	−4.55898e9	−0.318980	−0.159490	+	0.987200i	$0.550985\pi$
$798$	1.08766e10	0.757677
$799$	−1.48689e10	−1.03125
$800$	1.53564e9	0.106041
$801$	−5.18510e9	−0.356486
$802$	−1.84053e10	−1.25989
$803$	−2.24835e10	−1.53235
$804$	5.03699e9	0.341802
$805$	1.91409e9	0.129323
$806$	1.03945e10	0.699248
$807$	−2.59381e9	−0.173732
$808$	7.87374e9	0.525099
$809$	−2.23983e9	−0.148729	−0.0743646	−	0.997231i	$-0.523693\pi$
$809$	−2.23983e9	−0.148729	−0.0743646	+	0.997231i	$0.523693\pi$
$810$	−7.51705e8	−0.0496992
$811$	−2.00573e10	−1.32038	−0.660190	−	0.751099i	$-0.729524\pi$
$811$	−2.00573e10	−1.32038	−0.660190	+	0.751099i	$0.729524\pi$
$812$	−1.12631e10	−0.738261
$813$	−1.11353e10	−0.726747
$814$	1.59573e10	1.03699
$815$	−1.81007e9	−0.117124
$816$	−2.11495e9	−0.136265
$817$	−2.57149e10	−1.64971
$818$	2.46751e9	0.157624
$819$	−8.86441e9	−0.563840
$820$	1.25947e9	0.0797699
$821$	2.05767e10	1.29770	0.648851	−	0.760915i	$-0.275250\pi$
$821$	2.05767e10	1.29770	0.648851	+	0.760915i	$0.275250\pi$
$822$	−3.58357e9	−0.225042
$823$	−1.04251e10	−0.651899	−0.325950	−	0.945387i	$-0.605684\pi$
$823$	−1.04251e10	−0.651899	−0.325950	+	0.945387i	$0.605684\pi$
$824$	−1.20235e9	−0.0748663
$825$	−8.06788e9	−0.500231
$826$	−1.06020e9	−0.0654575
$827$	−1.15446e10	−0.709758	−0.354879	−	0.934912i	$-0.615478\pi$
$827$	−1.15446e10	−0.709758	−0.354879	+	0.934912i	$0.615478\pi$
$828$	5.67664e8	0.0347524
$829$	−2.77852e10	−1.69384	−0.846922	−	0.531717i	$-0.821547\pi$
$829$	−2.77852e10	−1.69384	−0.846922	+	0.531717i	$0.821547\pi$
$830$	1.18603e10	0.719983
$831$	1.59430e10	0.963756
$832$	−3.58250e9	−0.215652
$833$	−6.09232e8	−0.0365195
$834$	−3.13430e9	−0.187094
$835$	−1.61204e10	−0.958237
$836$	−2.30941e10	−1.36703
$837$	−1.87137e9	−0.110311
$838$	−2.03781e10	−1.19622
$839$	−2.04260e10	−1.19404	−0.597018	−	0.802228i	$-0.703648\pi$
$839$	−2.04260e10	−1.19404	−0.597018	+	0.802228i	$0.703648\pi$
$840$	2.17476e9	0.126600
$841$	2.18702e10	1.26785
$842$	−2.34029e10	−1.35107
$843$	2.43010e9	0.139710
$844$	−7.66342e9	−0.438757
$845$	2.19268e10	1.25019
$846$	4.53441e9	0.257469
$847$	1.88345e10	1.06503
$848$	3.29196e9	0.185383
$849$	8.89747e9	0.498987
$850$	7.16977e9	0.400441
$851$	3.80624e9	0.211710
$852$	2.93512e9	0.162587
$853$	3.26267e10	1.79991	0.899955	−	0.435982i	$-0.143599\pi$
$853$	3.26267e10	1.79991	0.899955	+	0.435982i	$0.143599\pi$
$854$	1.18873e10	0.653100
$855$	7.29447e9	0.399128
$856$	3.07945e9	0.167809
$857$	−4.12935e9	−0.224104	−0.112052	−	0.993702i	$-0.535742\pi$
$857$	−4.12935e9	−0.224104	−0.112052	+	0.993702i	$0.535742\pi$
$858$	1.88216e10	1.01730
$859$	−5.65664e8	−0.0304496	−0.0152248	−	0.999884i	$-0.504846\pi$
$859$	−5.65664e8	−0.0304496	−0.0152248	+	0.999884i	$0.504846\pi$
$860$	−5.14164e9	−0.275650
$861$	−2.67390e9	−0.142769
$862$	−3.58014e9	−0.190381
$863$	1.27842e10	0.677076	0.338538	−	0.940953i	$-0.390068\pi$
$863$	1.27842e10	0.677076	0.338538	+	0.940953i	$0.390068\pi$
$864$	6.44973e8	0.0340207
$865$	−1.84464e10	−0.969070
$866$	2.35567e10	1.23254
$867$	1.20461e9	0.0627737
$868$	5.41408e9	0.281000
$869$	9.83583e9	0.508442
$870$	−7.55363e9	−0.388900
$871$	3.98357e10	2.04272
$872$	−1.32533e10	−0.676888
$873$	1.30166e9	0.0662137
$874$	−5.50855e9	−0.279092
$875$	−1.96630e10	−0.992252
$876$	−6.09327e9	−0.306257
$877$	−5.24615e9	−0.262629	−0.131314	−	0.991341i	$-0.541920\pi$
$877$	−5.24615e9	−0.262629	−0.131314	+	0.991341i	$0.541920\pi$
$878$	−1.51282e10	−0.754323
$879$	7.60324e9	0.377605
$880$	−4.61763e9	−0.228418
$881$	1.42763e10	0.703399	0.351699	−	0.936113i	$-0.385604\pi$
$881$	1.42763e10	0.703399	0.351699	+	0.936113i	$0.385604\pi$
$882$	1.85791e8	0.00911767
$883$	−2.08590e10	−1.01960	−0.509802	−	0.860292i	$-0.670281\pi$
$883$	−2.08590e10	−1.01960	−0.509802	+	0.860292i	$0.670281\pi$
$884$	−1.67264e10	−0.814365
$885$	−7.11032e8	−0.0344816
$886$	−4.71265e9	−0.227639
$887$	2.86909e10	1.38042	0.690211	−	0.723608i	$-0.257518\pi$
$887$	2.86909e10	1.38042	0.690211	+	0.723608i	$0.257518\pi$
$888$	4.32460e9	0.207253
$889$	−2.35480e10	−1.12408
$890$	1.00605e10	0.478362
$891$	−3.38854e9	−0.160487
$892$	1.32053e10	0.622976
$893$	−4.40015e10	−2.06770
$894$	1.19660e10	0.560102
$895$	−1.04489e10	−0.487181
$896$	−1.86598e9	−0.0866619
$897$	4.48945e9	0.207692
$898$	1.96067e10	0.903518
$899$	−1.88048e10	−0.863195
$900$	−2.18648e9	−0.0999763
$901$	1.53699e10	0.700059
$902$	5.67744e9	0.257591
$903$	1.09159e10	0.493348
$904$	1.22745e10	0.552602
$905$	−2.57405e10	−1.15438
$906$	3.36277e9	0.150227
$907$	4.21469e10	1.87560	0.937799	−	0.347179i	$-0.112860\pi$
$907$	4.21469e10	1.87560	0.937799	+	0.347179i	$0.112860\pi$
$908$	−9.46490e9	−0.419581
$909$	−1.12109e10	−0.495068
$910$	1.71994e10	0.756605
$911$	2.16175e10	0.947308	0.473654	−	0.880711i	$-0.342935\pi$
$911$	2.16175e10	0.947308	0.473654	+	0.880711i	$0.342935\pi$
$912$	−6.25875e9	−0.273216
$913$	5.34639e10	2.32495
$914$	2.24249e10	0.971445
$915$	7.97227e9	0.344039
$916$	−1.63199e10	−0.701589
$917$	−2.51395e10	−1.07662
$918$	3.01133e9	0.128472
$919$	1.63161e10	0.693443	0.346722	−	0.937968i	$-0.387295\pi$
$919$	1.63161e10	0.693443	0.346722	+	0.937968i	$0.387295\pi$
$920$	−1.10143e9	−0.0466335
$921$	−9.69279e8	−0.0408828
$922$	1.73074e9	0.0727234
$923$	2.32128e10	0.971676
$924$	9.80342e9	0.408814
$925$	−1.46606e10	−0.609052
$926$	2.81160e10	1.16363
$927$	1.71194e9	0.0705846
$928$	6.48112e9	0.266215
$929$	−9.58818e9	−0.392357	−0.196178	−	0.980568i	$-0.562853\pi$
$929$	−9.58818e9	−0.392357	−0.196178	+	0.980568i	$0.562853\pi$
$930$	3.63098e9	0.148025
$931$	−1.80290e9	−0.0732229
$932$	1.51782e10	0.614136
$933$	2.20379e10	0.888351
$934$	1.32530e9	0.0532232
$935$	−2.15594e10	−0.862572
$936$	5.10086e9	0.203319
$937$	−3.04396e10	−1.20879	−0.604394	−	0.796685i	$-0.706585\pi$
$937$	−3.04396e10	−1.20879	−0.604394	+	0.796685i	$0.706585\pi$
$938$	2.07488e10	0.820888
$939$	−1.17451e10	−0.462944
$940$	−8.79802e9	−0.345492
$941$	−8.58720e9	−0.335960	−0.167980	−	0.985790i	$-0.553724\pi$
$941$	−8.58720e9	−0.335960	−0.167980	+	0.985790i	$0.553724\pi$
$942$	−7.49464e8	−0.0292128
$943$	1.35422e9	0.0525894
$944$	6.10075e8	0.0236038
$945$	−3.09649e9	−0.119360
$946$	−2.31775e10	−0.890119
$947$	3.05722e10	1.16977	0.584887	−	0.811115i	$-0.301139\pi$
$947$	3.05722e10	1.16977	0.584887	+	0.811115i	$0.301139\pi$
$948$	2.66562e9	0.101617
$949$	−4.81895e10	−1.83029
$950$	2.12174e10	0.802897
$951$	−4.17799e9	−0.157520
$952$	−8.71210e9	−0.327261
$953$	−3.43363e10	−1.28508	−0.642538	−	0.766254i	$-0.722118\pi$
$953$	−3.43363e10	−1.28508	−0.642538	+	0.766254i	$0.722118\pi$
$954$	−4.68719e9	−0.174781
$955$	−6.38103e9	−0.237071
$956$	−7.32101e8	−0.0271000
$957$	−3.40503e10	−1.25582
$958$	−2.17307e10	−0.798535
$959$	−1.47618e10	−0.540472
$960$	−1.25143e9	−0.0456517
$961$	−1.84733e10	−0.671448
$962$	3.42017e10	1.23861
$963$	−4.38461e9	−0.158212
$964$	1.23666e10	0.444610
$965$	3.11024e10	1.11416
$966$	2.33837e9	0.0834630
$967$	−2.41866e10	−0.860166	−0.430083	−	0.902789i	$-0.641516\pi$
$967$	−2.41866e10	−0.860166	−0.430083	+	0.902789i	$0.641516\pi$
$968$	−1.08379e10	−0.384046
$969$	−2.92216e10	−1.03174
$970$	−2.52559e9	−0.0888508
$971$	−6.67980e9	−0.234151	−0.117076	−	0.993123i	$-0.537352\pi$
$971$	−6.67980e9	−0.234151	−0.117076	+	0.993123i	$0.537352\pi$
$972$	−9.18330e8	−0.0320750
$973$	−1.29111e10	−0.449332
$974$	2.88235e10	0.999517
$975$	−1.72921e10	−0.597491
$976$	−6.84031e9	−0.235506
$977$	−2.35203e9	−0.0806884	−0.0403442	−	0.999186i	$-0.512845\pi$
$977$	−2.35203e9	−0.0806884	−0.0403442	+	0.999186i	$0.512845\pi$
$978$	−2.21130e9	−0.0755896
$979$	4.53510e10	1.54471
$980$	−3.60486e8	−0.0122348
$981$	1.88704e10	0.638176
$982$	−7.68229e9	−0.258881
$983$	3.23883e10	1.08756	0.543778	−	0.839229i	$-0.316993\pi$
$983$	3.23883e10	1.08756	0.543778	+	0.839229i	$0.316993\pi$
$984$	1.53865e9	0.0514821
$985$	2.84426e10	0.948291
$986$	3.02598e10	1.00530
$987$	1.86785e10	0.618348
$988$	−4.94983e10	−1.63283
$989$	−5.52845e9	−0.181726
$990$	6.57471e9	0.215354
$991$	−4.53195e10	−1.47920	−0.739600	−	0.673047i	$-0.764985\pi$
$991$	−4.53195e10	−1.47920	−0.739600	+	0.673047i	$0.764985\pi$
$992$	−3.11543e9	−0.101327
$993$	−6.64515e9	−0.215369
$994$	1.20906e10	0.390477
$995$	1.43142e10	0.460666
$996$	1.44893e10	0.464664
$997$	3.13597e9	0.100216	0.0501082	−	0.998744i	$-0.484043\pi$
$997$	3.13597e9	0.100216	0.0501082	+	0.998744i	$0.484043\pi$
$998$	8.76638e9	0.279166
$999$	−6.15749e9	−0.195400

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.8.a.e.1.3	✓	4	1.1	even	1	trivial
414.8.a.k.1.2		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)

\(f(q)\)	\(=\)	\(q-8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} +176.808 q^{5} +216.000 q^{6} +889.767 q^{7} -512.000 q^{8} +729.000 q^{9} +O(q^{10})\)	\(q-8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} +176.808 q^{5} +216.000 q^{6} +889.767 q^{7} -512.000 q^{8} +729.000 q^{9} -1414.46 q^{10} -6376.13 q^{11} -1728.00 q^{12} -13666.1 q^{13} -7118.14 q^{14} -4773.82 q^{15} +4096.00 q^{16} +19123.9 q^{17} -5832.00 q^{18} +56593.2 q^{19} +11315.7 q^{20} -24023.7 q^{21} +51009.0 q^{22} +12167.0 q^{23} +13824.0 q^{24} -46863.9 q^{25} +109329. q^{26} -19683.0 q^{27} +56945.1 q^{28} -197788. q^{29} +38190.5 q^{30} +95075.4 q^{31} -32768.0 q^{32} +172156. q^{33} -152991. q^{34} +157318. q^{35} +46656.0 q^{36} +312833. q^{37} -452746. q^{38} +368986. q^{39} -90525.7 q^{40} +111303. q^{41} +192190. q^{42} -454381. q^{43} -408072. q^{44} +128893. q^{45} -97336.0 q^{46} -777505. q^{47} -110592. q^{48} -31857.1 q^{49} +374911. q^{50} -516345. q^{51} -874633. q^{52} +803702. q^{53} +157464. q^{54} -1.12735e6 q^{55} -455561. q^{56} -1.52802e6 q^{57} +1.58230e6 q^{58} +148944. q^{59} -305524. q^{60} -1.67000e6 q^{61} -760603. q^{62} +648640. q^{63} +262144. q^{64} -2.41628e6 q^{65} -1.37724e6 q^{66} -2.91492e6 q^{67} +1.22393e6 q^{68} -328509. q^{69} -1.25854e6 q^{70} -1.69856e6 q^{71} -373248. q^{72} +3.52620e6 q^{73} -2.50266e6 q^{74} +1.26533e6 q^{75} +3.62196e6 q^{76} -5.67327e6 q^{77} -2.95189e6 q^{78} -1.54260e6 q^{79} +724206. q^{80} +531441. q^{81} -890421. q^{82} -8.38500e6 q^{83} -1.53752e6 q^{84} +3.38126e6 q^{85} +3.63505e6 q^{86} +5.34027e6 q^{87} +3.26458e6 q^{88} -7.11261e6 q^{89} -1.03114e6 q^{90} -1.21597e7 q^{91} +778688. q^{92} -2.56704e6 q^{93} +6.22004e6 q^{94} +1.00061e7 q^{95} +884736. q^{96} +1.78554e6 q^{97} +254857. q^{98} -4.64820e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 32 q^{2} - 108 q^{3} + 256 q^{4} - 90 q^{5} + 864 q^{6} - 222 q^{7} - 2048 q^{8} + 2916 q^{9}+O(q^{10}) \) 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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Representations

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Database

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