LMFDB - Embedded newform 354.8.a.d.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [354,8,Mod(1,354)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(354, 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("354.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 354 = 2 \cdot 3 \cdot 59 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	354.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:	$110.584299021$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 103558 x^{6} + 5805883 x^{5} + 2559087821 x^{4} - 196601024266 x^{3} + \cdots - 22\!\cdots\!40 $ x^8 - x^7 - 103558x^6 + 5805883x^5 + 2559087821x^4 - 196601024266x^3 - 17055507548256x^2 + 1662873307981032x - 22179668074797840
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{3}\cdot 3^{7}\cdot 5 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-135.103$ of defining polynomial
Character	$\chi$	$=$	354.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} -11.9651 q^{5} -216.000 q^{6} +1695.87 q^{7} +512.000 q^{8} +729.000 q^{9} +O(q^{10})$	\(q+8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} -11.9651 q^{5} -216.000 q^{6} +1695.87 q^{7} +512.000 q^{8} +729.000 q^{9} -95.7205 q^{10} -3616.52 q^{11} -1728.00 q^{12} -8260.06 q^{13} +13566.9 q^{14} +323.057 q^{15} +4096.00 q^{16} -11349.6 q^{17} +5832.00 q^{18} -27404.3 q^{19} -765.764 q^{20} -45788.4 q^{21} -28932.1 q^{22} +83995.4 q^{23} -13824.0 q^{24} -77981.8 q^{25} -66080.5 q^{26} -19683.0 q^{27} +108535. q^{28} -72295.0 q^{29} +2584.45 q^{30} -103683. q^{31} +32768.0 q^{32} +97646.0 q^{33} -90796.4 q^{34} -20291.1 q^{35} +46656.0 q^{36} -243228. q^{37} -219235. q^{38} +223022. q^{39} -6126.11 q^{40} +280719. q^{41} -366307. q^{42} -728026. q^{43} -231457. q^{44} -8722.53 q^{45} +671963. q^{46} +239556. q^{47} -110592. q^{48} +2.05242e6 q^{49} -623855. q^{50} +306438. q^{51} -528644. q^{52} +149239. q^{53} -157464. q^{54} +43271.9 q^{55} +868283. q^{56} +739917. q^{57} -578360. q^{58} -205379. q^{59} +20675.6 q^{60} +2.21215e6 q^{61} -829463. q^{62} +1.23629e6 q^{63} +262144. q^{64} +98832.1 q^{65} +781168. q^{66} -1.88718e6 q^{67} -726371. q^{68} -2.26788e6 q^{69} -162329. q^{70} -2.90393e6 q^{71} +373248. q^{72} +6.19009e6 q^{73} -1.94582e6 q^{74} +2.10551e6 q^{75} -1.75388e6 q^{76} -6.13313e6 q^{77} +1.78417e6 q^{78} -5.99559e6 q^{79} -49008.9 q^{80} +531441. q^{81} +2.24575e6 q^{82} -7.41091e6 q^{83} -2.93046e6 q^{84} +135798. q^{85} -5.82421e6 q^{86} +1.95197e6 q^{87} -1.85166e6 q^{88} -1.13925e7 q^{89} -69780.2 q^{90} -1.40079e7 q^{91} +5.37570e6 q^{92} +2.79944e6 q^{93} +1.91645e6 q^{94} +327894. q^{95} -884736. q^{96} -4.68549e6 q^{97} +1.64193e7 q^{98} -2.63644e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 64 q^{2} - 216 q^{3} + 512 q^{4} - 592 q^{5} - 1728 q^{6} - 340 q^{7} + 4096 q^{8} + 5832 q^{9}+O(q^{10}) $ 8 * q + 64 * q^2 - 216 * q^3 + 512 * q^4 - 592 * q^5 - 1728 * q^6 - 340 * q^7 + 4096 * q^8 + 5832 * q^9	\( 8 q + 64 q^{2} - 216 q^{3} + 512 q^{4} - 592 q^{5} - 1728 q^{6} - 340 q^{7} + 4096 q^{8} + 5832 q^{9} - 4736 q^{10} - 2852 q^{11} - 13824 q^{12} + 1142 q^{13} - 2720 q^{14} + 15984 q^{15} + 32768 q^{16} - 22528 q^{17} + 46656 q^{18} - 33528 q^{19} - 37888 q^{20} + 9180 q^{21} - 22816 q^{22} + 41330 q^{23} - 110592 q^{24} + 209004 q^{25} + 9136 q^{26} - 157464 q^{27} - 21760 q^{28} - 48334 q^{29} + 127872 q^{30} + 217552 q^{31} + 262144 q^{32} + 77004 q^{33} - 180224 q^{34} - 171714 q^{35} + 373248 q^{36} - 77966 q^{37} - 268224 q^{38} - 30834 q^{39} - 303104 q^{40} - 446410 q^{41} + 73440 q^{42} - 470890 q^{43} - 182528 q^{44} - 431568 q^{45} + 330640 q^{46} + 1876568 q^{47} - 884736 q^{48} + 1667480 q^{49} + 1672032 q^{50} + 608256 q^{51} + 73088 q^{52} - 1155672 q^{53} - 1259712 q^{54} + 112064 q^{55} - 174080 q^{56} + 905256 q^{57} - 386672 q^{58} - 1643032 q^{59} + 1022976 q^{60} - 9094962 q^{61} + 1740416 q^{62} - 247860 q^{63} + 2097152 q^{64} - 6726234 q^{65} + 616032 q^{66} - 8552352 q^{67} - 1441792 q^{68} - 1115910 q^{69} - 1373712 q^{70} - 5829156 q^{71} + 2985984 q^{72} - 7639392 q^{73} - 623728 q^{74} - 5643108 q^{75} - 2145792 q^{76} - 17178270 q^{77} - 246672 q^{78} - 12614888 q^{79} - 2424832 q^{80} + 4251528 q^{81} - 3571280 q^{82} - 19145486 q^{83} + 587520 q^{84} - 20127842 q^{85} - 3767120 q^{86} + 1305018 q^{87} - 1460224 q^{88} - 16050066 q^{89} - 3452544 q^{90} - 20086856 q^{91} + 2645120 q^{92} - 5873904 q^{93} + 15012544 q^{94} - 8130136 q^{95} - 7077888 q^{96} - 1961876 q^{97} + 13339840 q^{98} - 2079108 q^{99}+O(q^{100}) \) 8 * q + 64 * q^2 - 216 * q^3 + 512 * q^4 - 592 * q^5 - 1728 * q^6 - 340 * q^7 + 4096 * q^8 + 5832 * q^9 - 4736 * q^10 - 2852 * q^11 - 13824 * q^12 + 1142 * q^13 - 2720 * q^14 + 15984 * q^15 + 32768 * q^16 - 22528 * q^17 + 46656 * q^18 - 33528 * q^19 - 37888 * q^20 + 9180 * q^21 - 22816 * q^22 + 41330 * q^23 - 110592 * q^24 + 209004 * q^25 + 9136 * q^26 - 157464 * q^27 - 21760 * q^28 - 48334 * q^29 + 127872 * q^30 + 217552 * q^31 + 262144 * q^32 + 77004 * q^33 - 180224 * q^34 - 171714 * q^35 + 373248 * q^36 - 77966 * q^37 - 268224 * q^38 - 30834 * q^39 - 303104 * q^40 - 446410 * q^41 + 73440 * q^42 - 470890 * q^43 - 182528 * q^44 - 431568 * q^45 + 330640 * q^46 + 1876568 * q^47 - 884736 * q^48 + 1667480 * q^49 + 1672032 * q^50 + 608256 * q^51 + 73088 * q^52 - 1155672 * q^53 - 1259712 * q^54 + 112064 * q^55 - 174080 * q^56 + 905256 * q^57 - 386672 * q^58 - 1643032 * q^59 + 1022976 * q^60 - 9094962 * q^61 + 1740416 * q^62 - 247860 * q^63 + 2097152 * q^64 - 6726234 * q^65 + 616032 * q^66 - 8552352 * q^67 - 1441792 * q^68 - 1115910 * q^69 - 1373712 * q^70 - 5829156 * q^71 + 2985984 * q^72 - 7639392 * q^73 - 623728 * q^74 - 5643108 * q^75 - 2145792 * q^76 - 17178270 * q^77 - 246672 * q^78 - 12614888 * q^79 - 2424832 * q^80 + 4251528 * q^81 - 3571280 * q^82 - 19145486 * q^83 + 587520 * q^84 - 20127842 * q^85 - 3767120 * q^86 + 1305018 * q^87 - 1460224 * q^88 - 16050066 * q^89 - 3452544 * q^90 - 20086856 * q^91 + 2645120 * q^92 - 5873904 * q^93 + 15012544 * q^94 - 8130136 * q^95 - 7077888 * q^96 - 1961876 * q^97 + 13339840 * q^98 - 2079108 * 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
$5$	−11.9651	−0.0428075	−0.0214038	−	0.999771i	$-0.506814\pi$
$5$	−11.9651	−0.0428075	−0.0214038	+	0.999771i	$0.506814\pi$
$6$	−216.000	−0.408248
$7$	1695.87	1.86874	0.934369	−	0.356308i	$-0.115965\pi$
$7$	1695.87	1.86874	0.934369	+	0.356308i	$0.115965\pi$
$8$	512.000	0.353553
$9$	729.000	0.333333
$10$	−95.7205	−0.0302695
$11$	−3616.52	−0.819250	−0.409625	−	0.912254i	$-0.634340\pi$
$11$	−3616.52	−0.819250	−0.409625	+	0.912254i	$0.634340\pi$
$12$	−1728.00	−0.288675
$13$	−8260.06	−1.04275	−0.521376	−	0.853327i	$-0.674581\pi$
$13$	−8260.06	−1.04275	−0.521376	+	0.853327i	$0.674581\pi$
$14$	13566.9	1.32140
$15$	323.057	0.0247149
$16$	4096.00	0.250000
$17$	−11349.6	−0.560283	−0.280141	−	0.959959i	$-0.590381\pi$
$17$	−11349.6	−0.560283	−0.280141	+	0.959959i	$0.590381\pi$
$18$	5832.00	0.235702
$19$	−27404.3	−0.916604	−0.458302	−	0.888797i	$-0.651542\pi$
$19$	−27404.3	−0.916604	−0.458302	+	0.888797i	$0.651542\pi$
$20$	−765.764	−0.0214038
$21$	−45788.4	−1.07892
$22$	−28932.1	−0.579297
$23$	83995.4	1.43949	0.719744	−	0.694240i	$-0.244259\pi$
$23$	83995.4	1.43949	0.719744	+	0.694240i	$0.244259\pi$
$24$	−13824.0	−0.204124
$25$	−77981.8	−0.998168
$26$	−66080.5	−0.737338
$27$	−19683.0	−0.192450
$28$	108535.	0.934369
$29$	−72295.0	−0.550447	−0.275223	−	0.961380i	$-0.588752\pi$
$29$	−72295.0	−0.550447	−0.275223	+	0.961380i	$0.588752\pi$
$30$	2584.45	0.0174761
$31$	−103683.	−0.625088	−0.312544	−	0.949903i	$-0.601181\pi$
$31$	−103683.	−0.625088	−0.312544	+	0.949903i	$0.601181\pi$
$32$	32768.0	0.176777
$33$	97646.0	0.472994
$34$	−90796.4	−0.396180
$35$	−20291.1	−0.0799960
$36$	46656.0	0.166667
$37$	−243228.	−0.789419	−0.394710	−	0.918806i	$-0.629155\pi$
$37$	−243228.	−0.789419	−0.394710	+	0.918806i	$0.629155\pi$
$38$	−219235.	−0.648137
$39$	223022.	0.602034
$40$	−6126.11	−0.0151347
$41$	280719.	0.636104	0.318052	−	0.948073i	$-0.396971\pi$
$41$	280719.	0.636104	0.318052	+	0.948073i	$0.396971\pi$
$42$	−366307.	−0.762909
$43$	−728026.	−1.39639	−0.698196	−	0.715907i	$-0.746014\pi$
$43$	−728026.	−1.39639	−0.698196	+	0.715907i	$0.746014\pi$
$44$	−231457.	−0.409625
$45$	−8722.53	−0.0142692
$46$	671963.	1.01787
$47$	239556.	0.336562	0.168281	−	0.985739i	$-0.446178\pi$
$47$	239556.	0.336562	0.168281	+	0.985739i	$0.446178\pi$
$48$	−110592.	−0.144338
$49$	2.05242e6	2.49218
$50$	−623855.	−0.705811
$51$	306438.	0.323480
$52$	−528644.	−0.521376
$53$	149239.	0.137695	0.0688473	−	0.997627i	$-0.478068\pi$
$53$	149239.	0.137695	0.0688473	+	0.997627i	$0.478068\pi$
$54$	−157464.	−0.136083
$55$	43271.9	0.0350700
$56$	868283.	0.660698
$57$	739917.	0.529201
$58$	−578360.	−0.389225
$59$	−205379.	−0.130189
$59$	−205379.	−0.130189
$60$	20675.6	0.0123575
$61$	2.21215e6	1.24784	0.623922	−	0.781487i	$-0.285538\pi$
$61$	2.21215e6	1.24784	0.623922	+	0.781487i	$0.285538\pi$
$62$	−829463.	−0.442004
$63$	1.23629e6	0.622912
$64$	262144.	0.125000
$65$	98832.1	0.0446376
$66$	781168.	0.334457
$67$	−1.88718e6	−0.766569	−0.383284	−	0.923630i	$-0.625207\pi$
$67$	−1.88718e6	−0.766569	−0.383284	+	0.923630i	$0.625207\pi$
$68$	−726371.	−0.280141
$69$	−2.26788e6	−0.831089
$70$	−162329.	−0.0565657
$71$	−2.90393e6	−0.962903	−0.481451	−	0.876473i	$-0.659890\pi$
$71$	−2.90393e6	−0.962903	−0.481451	+	0.876473i	$0.659890\pi$
$72$	373248.	0.117851
$73$	6.19009e6	1.86237	0.931187	−	0.364542i	$-0.118774\pi$
$73$	6.19009e6	1.86237	0.931187	+	0.364542i	$0.118774\pi$
$74$	−1.94582e6	−0.558204
$75$	2.10551e6	0.576292
$76$	−1.75388e6	−0.458302
$77$	−6.13313e6	−1.53096
$78$	1.78417e6	0.425702
$79$	−5.99559e6	−1.36816	−0.684080	−	0.729407i	$-0.739796\pi$
$79$	−5.99559e6	−1.36816	−0.684080	+	0.729407i	$0.739796\pi$
$80$	−49008.9	−0.0107019
$81$	531441.	0.111111
$82$	2.24575e6	0.449793
$83$	−7.41091e6	−1.42265	−0.711325	−	0.702863i	$-0.751905\pi$
$83$	−7.41091e6	−1.42265	−0.711325	+	0.702863i	$0.751905\pi$
$84$	−2.93046e6	−0.539458
$85$	135798.	0.0239843
$86$	−5.82421e6	−0.987398
$87$	1.95197e6	0.317801
$88$	−1.85166e6	−0.289648
$89$	−1.13925e7	−1.71298	−0.856490	−	0.516163i	$-0.827360\pi$
$89$	−1.13925e7	−1.71298	−0.856490	+	0.516163i	$0.827360\pi$
$90$	−69780.2	−0.0100898
$91$	−1.40079e7	−1.94863
$92$	5.37570e6	0.719744
$93$	2.79944e6	0.360895
$94$	1.91645e6	0.237985
$95$	327894.	0.0392375
$96$	−884736.	−0.102062
$97$	−4.68549e6	−0.521259	−0.260629	−	0.965439i	$-0.583930\pi$
$97$	−4.68549e6	−0.521259	−0.260629	+	0.965439i	$0.583930\pi$
$98$	1.64193e7	1.76224
$99$	−2.63644e6	−0.273083
$100$	−4.99084e6	−0.499084
$101$	−3.11574e6	−0.300910	−0.150455	−	0.988617i	$-0.548074\pi$
$101$	−3.11574e6	−0.300910	−0.150455	+	0.988617i	$0.548074\pi$
$102$	2.45150e6	0.228735
$103$	1.14028e7	1.02820	0.514102	−	0.857729i	$-0.328125\pi$
$103$	1.14028e7	1.02820	0.514102	+	0.857729i	$0.328125\pi$
$104$	−4.22915e6	−0.368669
$105$	547861.	0.0461857
$106$	1.19391e6	0.0973647
$107$	−5.80498e6	−0.458097	−0.229048	−	0.973415i	$-0.573561\pi$
$107$	−5.80498e6	−0.458097	−0.229048	+	0.973415i	$0.573561\pi$
$108$	−1.25971e6	−0.0962250
$109$	4.44758e6	0.328951	0.164475	−	0.986381i	$-0.447407\pi$
$109$	4.44758e6	0.328951	0.164475	+	0.986381i	$0.447407\pi$
$110$	346175.	0.0247983
$111$	6.56716e6	0.455771
$112$	6.94626e6	0.467184
$113$	−1.16019e7	−0.756408	−0.378204	−	0.925722i	$-0.623458\pi$
$113$	−1.16019e7	−0.756408	−0.378204	+	0.925722i	$0.623458\pi$
$114$	5.91934e6	0.374202
$115$	−1.00501e6	−0.0616209
$116$	−4.62688e6	−0.275223
$117$	−6.02158e6	−0.347584
$118$	−1.64303e6	−0.0920575
$119$	−1.92473e7	−1.04702
$120$	165405.	0.00873805
$121$	−6.40797e6	−0.328830
$122$	1.76972e7	0.882359
$123$	−7.57941e6	−0.367255
$124$	−6.63570e6	−0.312544
$125$	1.86783e6	0.0855366
$126$	9.89029e6	0.440466
$127$	3.43864e7	1.48961	0.744807	−	0.667280i	$-0.232542\pi$
$127$	3.43864e7	1.48961	0.744807	+	0.667280i	$0.232542\pi$
$128$	2.09715e6	0.0883883
$129$	1.96567e7	0.806207
$130$	790657.	0.0315636
$131$	−4.26157e7	−1.65623	−0.828114	−	0.560560i	$-0.810586\pi$
$131$	−4.26157e7	−1.65623	−0.828114	+	0.560560i	$0.810586\pi$
$132$	6.24934e6	0.236497
$133$	−4.64740e7	−1.71289
$134$	−1.50974e7	−0.542046
$135$	235508.	0.00823831
$136$	−5.81097e6	−0.198090
$137$	−3.64363e7	−1.21063	−0.605316	−	0.795985i	$-0.706953\pi$
$137$	−3.64363e7	−1.21063	−0.605316	+	0.795985i	$0.706953\pi$
$138$	−1.81430e7	−0.587668
$139$	−3.05132e7	−0.963687	−0.481843	−	0.876257i	$-0.660033\pi$
$139$	−3.05132e7	−0.963687	−0.481843	+	0.876257i	$0.660033\pi$
$140$	−1.29863e6	−0.0399980
$141$	−6.46801e6	−0.194314
$142$	−2.32315e7	−0.680875
$143$	2.98726e7	0.854275
$144$	2.98598e6	0.0833333
$145$	865014.	0.0235633
$146$	4.95207e7	1.31690
$147$	−5.54152e7	−1.43886
$148$	−1.55666e7	−0.394710
$149$	−3.97313e7	−0.983969	−0.491984	−	0.870604i	$-0.663728\pi$
$149$	−3.97313e7	−0.983969	−0.491984	+	0.870604i	$0.663728\pi$
$150$	1.68441e7	0.407500
$151$	−7.80517e6	−0.184486	−0.0922430	−	0.995737i	$-0.529404\pi$
$151$	−7.80517e6	−0.184486	−0.0922430	+	0.995737i	$0.529404\pi$
$152$	−1.40310e7	−0.324068
$153$	−8.27382e6	−0.186761
$154$	−4.90650e7	−1.08255
$155$	1.24057e6	0.0267584
$156$	1.42734e7	0.301017
$157$	−8.49038e6	−0.175097	−0.0875484	−	0.996160i	$-0.527903\pi$
$157$	−8.49038e6	−0.175097	−0.0875484	+	0.996160i	$0.527903\pi$
$158$	−4.79647e7	−0.967435
$159$	−4.02945e6	−0.0794980
$160$	−392071.	−0.00756737
$161$	1.42445e8	2.69002
$162$	4.25153e6	0.0785674
$163$	2.11330e7	0.382213	0.191106	−	0.981569i	$-0.438792\pi$
$163$	2.11330e7	0.382213	0.191106	+	0.981569i	$0.438792\pi$
$164$	1.79660e7	0.318052
$165$	−1.16834e6	−0.0202477
$166$	−5.92873e7	−1.00597
$167$	1.52238e7	0.252939	0.126470	−	0.991970i	$-0.459635\pi$
$167$	1.52238e7	0.252939	0.126470	+	0.991970i	$0.459635\pi$
$168$	−2.34436e7	−0.381454
$169$	5.48004e6	0.0873334
$170$	1.08638e6	0.0169595
$171$	−1.99778e7	−0.305535
$172$	−4.65937e7	−0.698196
$173$	−8.53928e7	−1.25389	−0.626945	−	0.779063i	$-0.715695\pi$
$173$	−8.53928e7	−1.25389	−0.626945	+	0.779063i	$0.715695\pi$
$174$	1.56157e7	0.224719
$175$	−1.32247e8	−1.86531
$176$	−1.48133e7	−0.204812
$177$	5.54523e6	0.0751646
$178$	−9.11397e7	−1.21126
$179$	9.26658e7	1.20763	0.603815	−	0.797124i	$-0.293646\pi$
$179$	9.26658e7	1.20763	0.603815	+	0.797124i	$0.293646\pi$
$180$	−558242.	−0.00713458
$181$	5.88861e7	0.738138	0.369069	−	0.929402i	$-0.379677\pi$
$181$	5.88861e7	0.738138	0.369069	+	0.929402i	$0.379677\pi$
$182$	−1.12064e8	−1.37789
$183$	−5.97280e7	−0.720443
$184$	4.30056e7	0.508936
$185$	2.91024e6	0.0337931
$186$	2.23955e7	0.255191
$187$	4.10459e7	0.459012
$188$	1.53316e7	0.168281
$189$	−3.33797e7	−0.359639
$190$	2.62316e6	0.0277451
$191$	6.31905e7	0.656198	0.328099	−	0.944643i	$-0.393592\pi$
$191$	6.31905e7	0.656198	0.328099	+	0.944643i	$0.393592\pi$
$192$	−7.07789e6	−0.0721688
$193$	9.95959e7	0.997220	0.498610	−	0.866827i	$-0.333844\pi$
$193$	9.95959e7	0.997220	0.498610	+	0.866827i	$0.333844\pi$
$194$	−3.74839e7	−0.368586
$195$	−2.66847e6	−0.0257716
$196$	1.31355e8	1.24609
$197$	−1.22406e8	−1.14070	−0.570350	−	0.821402i	$-0.693192\pi$
$197$	−1.22406e8	−1.14070	−0.570350	+	0.821402i	$0.693192\pi$
$198$	−2.10915e7	−0.193099
$199$	−1.17487e8	−1.05683	−0.528413	−	0.848987i	$-0.677213\pi$
$199$	−1.17487e8	−1.05683	−0.528413	+	0.848987i	$0.677213\pi$
$200$	−3.99267e7	−0.352906
$201$	5.09538e7	0.442579
$202$	−2.49259e7	−0.212775
$203$	−1.22603e8	−1.02864
$204$	1.96120e7	0.161740
$205$	−3.35882e6	−0.0272300
$206$	9.12221e7	0.727051
$207$	6.12326e7	0.479829
$208$	−3.38332e7	−0.260688
$209$	9.91082e7	0.750927
$210$	4.38288e6	0.0326582
$211$	2.47048e8	1.81048	0.905238	−	0.424905i	$-0.139692\pi$
$211$	2.47048e8	1.81048	0.905238	+	0.424905i	$0.139692\pi$
$212$	9.55129e6	0.0688473
$213$	7.84062e7	0.555932
$214$	−4.64398e7	−0.323923
$215$	8.71088e6	0.0597761
$216$	−1.00777e7	−0.0680414
$217$	−1.75832e8	−1.16812
$218$	3.55806e7	0.232603
$219$	−1.67132e8	−1.07524
$220$	2.76940e6	0.0175350
$221$	9.37479e7	0.584237
$222$	5.25373e7	0.322279
$223$	4.33051e7	0.261500	0.130750	−	0.991415i	$-0.458261\pi$
$223$	4.33051e7	0.261500	0.130750	+	0.991415i	$0.458261\pi$
$224$	5.55701e7	0.330349
$225$	−5.68488e7	−0.332723
$226$	−9.28155e7	−0.534861
$227$	−2.30558e8	−1.30824	−0.654122	−	0.756389i	$-0.726962\pi$
$227$	−2.30558e8	−1.30824	−0.654122	+	0.756389i	$0.726962\pi$
$228$	4.73547e7	0.264601
$229$	2.29077e8	1.26054	0.630271	−	0.776376i	$-0.282944\pi$
$229$	2.29077e8	1.26054	0.630271	+	0.776376i	$0.282944\pi$
$230$	−8.04008e6	−0.0435725
$231$	1.65594e8	0.883901
$232$	−3.70150e7	−0.194612
$233$	1.73389e8	0.898001	0.449000	−	0.893532i	$-0.351780\pi$
$233$	1.73389e8	0.898001	0.449000	+	0.893532i	$0.351780\pi$
$234$	−4.81727e7	−0.245779
$235$	−2.86630e6	−0.0144074
$236$	−1.31443e7	−0.0650945
$237$	1.61881e8	0.789907
$238$	−1.53978e8	−0.740356
$239$	−2.13585e8	−1.01199	−0.505997	−	0.862535i	$-0.668875\pi$
$239$	−2.13585e8	−1.01199	−0.505997	+	0.862535i	$0.668875\pi$
$240$	1.32324e6	0.00617873
$241$	1.92729e8	0.886924	0.443462	−	0.896293i	$-0.353750\pi$
$241$	1.92729e8	0.886924	0.443462	+	0.896293i	$0.353750\pi$
$242$	−5.12637e7	−0.232518
$243$	−1.43489e7	−0.0641500
$244$	1.41578e8	0.623922
$245$	−2.45573e7	−0.106684
$246$	−6.06353e7	−0.259688
$247$	2.26361e8	0.955791
$248$	−5.30856e7	−0.221002
$249$	2.00095e8	0.821368
$250$	1.49426e7	0.0604835
$251$	1.33276e8	0.531978	0.265989	−	0.963976i	$-0.414301\pi$
$251$	1.33276e8	0.531978	0.265989	+	0.963976i	$0.414301\pi$
$252$	7.91223e7	0.311456
$253$	−3.03771e8	−1.17930
$254$	2.75091e8	1.05332
$255$	−3.66655e6	−0.0138474
$256$	1.67772e7	0.0625000
$257$	−2.28924e8	−0.841252	−0.420626	−	0.907234i	$-0.638189\pi$
$257$	−2.28924e8	−0.841252	−0.420626	+	0.907234i	$0.638189\pi$
$258$	1.57254e8	0.570075
$259$	−4.12482e8	−1.47522
$260$	6.32525e6	0.0223188
$261$	−5.27031e7	−0.183482
$262$	−3.40926e8	−1.17113
$263$	−6.51060e7	−0.220686	−0.110343	−	0.993894i	$-0.535195\pi$
$263$	−6.51060e7	−0.220686	−0.110343	+	0.993894i	$0.535195\pi$
$264$	4.99947e7	0.167229
$265$	−1.78565e6	−0.00589436
$266$	−3.71792e8	−1.21120
$267$	3.07596e8	0.988990
$268$	−1.20779e8	−0.383284
$269$	1.25159e8	0.392038	0.196019	−	0.980600i	$-0.437199\pi$
$269$	1.25159e8	0.392038	0.196019	+	0.980600i	$0.437199\pi$
$270$	1.88407e6	0.00582536
$271$	−4.99612e8	−1.52490	−0.762448	−	0.647050i	$-0.776003\pi$
$271$	−4.99612e8	−1.52490	−0.762448	+	0.647050i	$0.776003\pi$
$272$	−4.64878e7	−0.140071
$273$	3.78215e8	1.12504
$274$	−2.91490e8	−0.856046
$275$	2.82023e8	0.817748
$276$	−1.45144e8	−0.415544
$277$	1.20544e8	0.340773	0.170387	−	0.985377i	$-0.445498\pi$
$277$	1.20544e8	0.340773	0.170387	+	0.985377i	$0.445498\pi$
$278$	−2.44106e8	−0.681429
$279$	−7.55848e7	−0.208363
$280$	−1.03891e7	−0.0282828
$281$	3.23072e8	0.868614	0.434307	−	0.900765i	$-0.356993\pi$
$281$	3.23072e8	0.868614	0.434307	+	0.900765i	$0.356993\pi$
$282$	−5.17441e7	−0.137401
$283$	−2.55769e8	−0.670803	−0.335401	−	0.942075i	$-0.608872\pi$
$283$	−2.55769e8	−0.670803	−0.335401	+	0.942075i	$0.608872\pi$
$284$	−1.85852e8	−0.481451
$285$	−8.85315e6	−0.0226538
$286$	2.38981e8	0.604064
$287$	4.76061e8	1.18871
$288$	2.38879e7	0.0589256
$289$	−2.81526e8	−0.686083
$290$	6.92011e6	0.0166617
$291$	1.26508e8	0.300949
$292$	3.96166e8	0.931187
$293$	−3.81427e8	−0.885879	−0.442940	−	0.896551i	$-0.646064\pi$
$293$	−3.81427e8	−0.885879	−0.442940	+	0.896551i	$0.646064\pi$
$294$	−4.43322e8	−1.01743
$295$	2.45737e6	0.00557306
$296$	−1.24533e8	−0.279102
$297$	7.11839e7	0.157665
$298$	−3.17851e8	−0.695771
$299$	−6.93807e8	−1.50103
$300$	1.34753e8	0.288146
$301$	−1.23463e9	−2.60949
$302$	−6.24414e7	−0.130451
$303$	8.41250e7	0.173730
$304$	−1.12248e8	−0.229151
$305$	−2.64685e7	−0.0534171
$306$	−6.61906e7	−0.132060
$307$	−4.47132e8	−0.881964	−0.440982	−	0.897516i	$-0.645370\pi$
$307$	−4.47132e8	−0.881964	−0.440982	+	0.897516i	$0.645370\pi$
$308$	−3.92520e8	−0.765481
$309$	−3.07874e8	−0.593634
$310$	9.92457e6	0.0189211
$311$	−6.53277e8	−1.23150	−0.615752	−	0.787940i	$-0.711148\pi$
$311$	−6.53277e8	−1.23150	−0.615752	+	0.787940i	$0.711148\pi$
$312$	1.14187e8	0.212851
$313$	−4.10084e8	−0.755907	−0.377953	−	0.925825i	$-0.623372\pi$
$313$	−4.10084e8	−0.755907	−0.377953	+	0.925825i	$0.623372\pi$
$314$	−6.79230e7	−0.123812
$315$	−1.47922e7	−0.0266653
$316$	−3.83717e8	−0.684080
$317$	1.70403e8	0.300448	0.150224	−	0.988652i	$-0.452001\pi$
$317$	1.70403e8	0.300448	0.150224	+	0.988652i	$0.452001\pi$
$318$	−3.22356e7	−0.0562136
$319$	2.61456e8	0.450953
$320$	−3.13657e6	−0.00535094
$321$	1.56734e8	0.264482
$322$	1.13956e9	1.90213
$323$	3.11027e8	0.513557
$324$	3.40122e7	0.0555556
$325$	6.44134e8	1.04084
$326$	1.69064e8	0.270265
$327$	−1.20085e8	−0.189920
$328$	1.43728e8	0.224897
$329$	4.06255e8	0.628945
$330$	−9.34672e6	−0.0143173
$331$	8.59030e8	1.30200	0.650999	−	0.759079i	$-0.274350\pi$
$331$	8.59030e8	1.30200	0.650999	+	0.759079i	$0.274350\pi$
$332$	−4.74298e8	−0.711325
$333$	−1.77313e8	−0.263140
$334$	1.21791e8	0.178855
$335$	2.25802e7	0.0328149
$336$	−1.87549e8	−0.269729
$337$	8.70059e8	1.23835	0.619176	−	0.785253i	$-0.287467\pi$
$337$	8.70059e8	1.23835	0.619176	+	0.785253i	$0.287467\pi$
$338$	4.38403e7	0.0617540
$339$	3.13252e8	0.436712
$340$	8.69108e6	0.0119922
$341$	3.74971e8	0.512103
$342$	−1.59822e8	−0.216046
$343$	2.08400e9	2.78849
$344$	−3.72749e8	−0.493699
$345$	2.71353e7	0.0355768
$346$	−6.83142e8	−0.886635
$347$	9.20496e8	1.18268	0.591342	−	0.806421i	$-0.298598\pi$
$347$	9.20496e8	1.18268	0.591342	+	0.806421i	$0.298598\pi$
$348$	1.24926e8	0.158900
$349$	9.86675e8	1.24247	0.621234	−	0.783625i	$-0.286632\pi$
$349$	9.86675e8	1.24247	0.621234	+	0.783625i	$0.286632\pi$
$350$	−1.05797e9	−1.31898
$351$	1.62583e8	0.200678
$352$	−1.18506e8	−0.144824
$353$	−8.24246e8	−0.997345	−0.498672	−	0.866791i	$-0.666179\pi$
$353$	−8.24246e8	−0.997345	−0.498672	+	0.866791i	$0.666179\pi$
$354$	4.43619e7	0.0531494
$355$	3.47457e7	0.0412195
$356$	−7.29117e8	−0.856490
$357$	5.19677e8	0.604498
$358$	7.41327e8	0.853924
$359$	4.70332e8	0.536505	0.268253	−	0.963349i	$-0.413554\pi$
$359$	4.70332e8	0.536505	0.268253	+	0.963349i	$0.413554\pi$
$360$	−4.46594e6	−0.00504491
$361$	−1.42874e8	−0.159838
$362$	4.71089e8	0.521942
$363$	1.73015e8	0.189850
$364$	−8.96509e8	−0.974315
$365$	−7.40648e7	−0.0797236
$366$	−4.77824e8	−0.509430
$367$	1.75455e8	0.185282	0.0926412	−	0.995700i	$-0.470469\pi$
$367$	1.75455e8	0.185282	0.0926412	+	0.995700i	$0.470469\pi$
$368$	3.44045e8	0.359872
$369$	2.04644e8	0.212035
$370$	2.32819e7	0.0238953
$371$	2.53089e8	0.257315
$372$	1.79164e8	0.180447
$373$	−1.65672e9	−1.65298	−0.826492	−	0.562948i	$-0.809667\pi$
$373$	−1.65672e9	−1.65298	−0.826492	+	0.562948i	$0.809667\pi$
$374$	3.28367e8	0.324570
$375$	−5.04314e7	−0.0493846
$376$	1.22653e8	0.118992
$377$	5.97161e8	0.573980
$378$	−2.67038e8	−0.254303
$379$	4.74567e8	0.447775	0.223888	−	0.974615i	$-0.428125\pi$
$379$	4.74567e8	0.447775	0.223888	+	0.974615i	$0.428125\pi$
$380$	2.09852e7	0.0196188
$381$	−9.28433e8	−0.860029
$382$	5.05524e8	0.464002
$383$	−3.20160e8	−0.291187	−0.145593	−	0.989345i	$-0.546509\pi$
$383$	−3.20160e8	−0.291187	−0.145593	+	0.989345i	$0.546509\pi$
$384$	−5.66231e7	−0.0510310
$385$	7.33832e7	0.0655367
$386$	7.96767e8	0.705141
$387$	−5.30731e8	−0.465464
$388$	−2.99871e8	−0.260629
$389$	1.97910e9	1.70468	0.852342	−	0.522984i	$-0.175181\pi$
$389$	1.97910e9	1.70468	0.852342	+	0.522984i	$0.175181\pi$
$390$	−2.13477e7	−0.0182232
$391$	−9.53310e8	−0.806521
$392$	1.05084e9	0.881118
$393$	1.15062e9	0.956224
$394$	−9.79249e8	−0.806597
$395$	7.17375e7	0.0585675
$396$	−1.68732e8	−0.136542
$397$	1.17913e9	0.945792	0.472896	−	0.881118i	$-0.343209\pi$
$397$	1.17913e9	0.945792	0.472896	+	0.881118i	$0.343209\pi$
$398$	−9.39896e8	−0.747289
$399$	1.25480e9	0.988938
$400$	−3.19414e8	−0.249542
$401$	1.56202e9	1.20971	0.604854	−	0.796336i	$-0.293231\pi$
$401$	1.56202e9	1.20971	0.604854	+	0.796336i	$0.293231\pi$
$402$	4.07631e8	0.312950
$403$	8.56426e8	0.651812
$404$	−1.99407e8	−0.150455
$405$	−6.35872e6	−0.00475639
$406$	−9.80821e8	−0.727358
$407$	8.79639e8	0.646731
$408$	1.56896e8	0.114367
$409$	9.96457e8	0.720157	0.360078	−	0.932922i	$-0.382750\pi$
$409$	9.96457e8	0.720157	0.360078	+	0.932922i	$0.382750\pi$
$410$	−2.68705e7	−0.0192545
$411$	9.83780e8	0.698959
$412$	7.29777e8	0.514102
$413$	−3.48295e8	−0.243289
$414$	4.89861e8	0.339291
$415$	8.86720e7	0.0609001
$416$	−2.70666e8	−0.184334
$417$	8.23856e8	0.556385
$418$	7.92866e8	0.530986
$419$	−7.27246e8	−0.482984	−0.241492	−	0.970403i	$-0.577637\pi$
$419$	−7.27246e8	−0.482984	−0.241492	+	0.970403i	$0.577637\pi$
$420$	3.50631e7	0.0230928
$421$	−2.08826e9	−1.36395	−0.681974	−	0.731376i	$-0.738878\pi$
$421$	−2.08826e9	−1.36395	−0.681974	+	0.731376i	$0.738878\pi$
$422$	1.97638e9	1.28020
$423$	1.74636e8	0.112187
$424$	7.64103e7	0.0486824
$425$	8.85059e8	0.559256
$426$	6.27250e8	0.393103
$427$	3.75151e9	2.33189
$428$	−3.71518e8	−0.229048
$429$	−8.06562e8	−0.493216
$430$	6.96870e7	0.0422681
$431$	−2.36825e9	−1.42481	−0.712405	−	0.701768i	$-0.752394\pi$
$431$	−2.36825e9	−1.42481	−0.712405	+	0.701768i	$0.752394\pi$
$432$	−8.06216e7	−0.0481125
$433$	−1.83604e8	−0.108686	−0.0543430	−	0.998522i	$-0.517306\pi$
$433$	−1.83604e8	−0.108686	−0.0543430	+	0.998522i	$0.517306\pi$
$434$	−1.40666e9	−0.825989
$435$	−2.33554e7	−0.0136043
$436$	2.84645e8	0.164475
$437$	−2.30184e9	−1.31944
$438$	−1.33706e9	−0.760311
$439$	2.49819e9	1.40929	0.704643	−	0.709562i	$-0.251107\pi$
$439$	2.49819e9	1.40929	0.704643	+	0.709562i	$0.251107\pi$
$440$	2.21552e7	0.0123991
$441$	1.49621e9	0.830726
$442$	7.49984e8	0.413118
$443$	−3.60407e9	−1.96961	−0.984806	−	0.173659i	$-0.944441\pi$
$443$	−3.60407e9	−1.96961	−0.984806	+	0.173659i	$0.944441\pi$
$444$	4.20298e8	0.227886
$445$	1.36311e8	0.0733284
$446$	3.46441e8	0.184909
$447$	1.07275e9	0.568095
$448$	4.44561e8	0.233592
$449$	3.80429e9	1.98341	0.991704	−	0.128545i	$-0.0410306\pi$
$449$	3.80429e9	1.98341	0.991704	+	0.128545i	$0.0410306\pi$
$450$	−4.54790e8	−0.235270
$451$	−1.01522e9	−0.521128
$452$	−7.42524e8	−0.378204
$453$	2.10740e8	0.106513
$454$	−1.84446e9	−0.925068
$455$	1.67606e8	0.0834160
$456$	3.78837e8	0.187101
$457$	−3.60461e9	−1.76666	−0.883328	−	0.468755i	$-0.844703\pi$
$457$	−3.60461e9	−1.76666	−0.883328	+	0.468755i	$0.844703\pi$
$458$	1.83262e9	0.891337
$459$	2.23393e8	0.107827
$460$	−6.43206e7	−0.0308104
$461$	−1.09859e9	−0.522255	−0.261128	−	0.965304i	$-0.584094\pi$
$461$	−1.09859e9	−0.522255	−0.261128	+	0.965304i	$0.584094\pi$
$462$	1.32476e9	0.625013
$463$	3.00053e9	1.40496	0.702481	−	0.711702i	$-0.252076\pi$
$463$	3.00053e9	1.40496	0.702481	+	0.711702i	$0.252076\pi$
$464$	−2.96120e8	−0.137612
$465$	−3.34954e7	−0.0154490
$466$	1.38711e9	0.634982
$467$	3.30473e7	0.0150151	0.00750753	−	0.999972i	$-0.497610\pi$
$467$	3.30473e7	0.0150151	0.00750753	+	0.999972i	$0.497610\pi$
$468$	−3.85381e8	−0.173792
$469$	−3.20040e9	−1.43252
$470$	−2.29304e7	−0.0101875
$471$	2.29240e8	0.101092
$472$	−1.05154e8	−0.0460287
$473$	2.63292e9	1.14399
$474$	1.29505e9	0.558549
$475$	2.13704e9	0.914924
$476$	−1.23183e9	−0.523511
$477$	1.08795e8	0.0458982
$478$	−1.70868e9	−0.715588
$479$	1.21379e9	0.504625	0.252313	−	0.967646i	$-0.418809\pi$
$479$	1.21379e9	0.504625	0.252313	+	0.967646i	$0.418809\pi$
$480$	1.05859e7	0.00436902
$481$	2.00908e9	0.823169
$482$	1.54183e9	0.627150
$483$	−3.84601e9	−1.55309
$484$	−4.10110e8	−0.164415
$485$	5.60621e7	0.0223138
$486$	−1.14791e8	−0.0453609
$487$	−5.09145e7	−0.0199751	−0.00998757	−	0.999950i	$-0.503179\pi$
$487$	−5.09145e7	−0.0199751	−0.00998757	+	0.999950i	$0.503179\pi$
$488$	1.13262e9	0.441179
$489$	−5.70591e8	−0.220670
$490$	−1.96458e8	−0.0754369
$491$	2.54896e9	0.971802	0.485901	−	0.874014i	$-0.338492\pi$
$491$	2.54896e9	0.971802	0.485901	+	0.874014i	$0.338492\pi$
$492$	−4.85082e8	−0.183627
$493$	8.20516e8	0.308406
$494$	1.81089e9	0.675846
$495$	3.15452e7	0.0116900
$496$	−4.24685e8	−0.156272
$497$	−4.92468e9	−1.79941
$498$	1.60076e9	0.580795
$499$	5.33033e9	1.92045	0.960223	−	0.279234i	$-0.0900805\pi$
$499$	5.33033e9	1.92045	0.960223	+	0.279234i	$0.0900805\pi$
$500$	1.19541e8	0.0427683
$501$	−4.11043e8	−0.146034
$502$	1.06621e9	0.376166
$503$	6.00314e8	0.210325	0.105162	−	0.994455i	$-0.466464\pi$
$503$	6.00314e8	0.210325	0.105162	+	0.994455i	$0.466464\pi$
$504$	6.32978e8	0.220233
$505$	3.72800e7	0.0128812
$506$	−2.43017e9	−0.833891
$507$	−1.47961e8	−0.0504220
$508$	2.20073e9	0.744807
$509$	−4.58092e9	−1.53972	−0.769858	−	0.638215i	$-0.779673\pi$
$509$	−4.58092e9	−1.53972	−0.769858	+	0.638215i	$0.779673\pi$
$510$	−2.93324e7	−0.00979156
$511$	1.04976e10	3.48029
$512$	1.34218e8	0.0441942
$513$	5.39399e8	0.176400
$514$	−1.83139e9	−0.594855
$515$	−1.36435e8	−0.0440149
$516$	1.25803e9	0.403104
$517$	−8.66359e8	−0.275728
$518$	−3.29986e9	−1.04314
$519$	2.30560e9	0.723934
$520$	5.06020e7	0.0157818
$521$	−5.20691e9	−1.61305	−0.806525	−	0.591200i	$-0.798654\pi$
$521$	−5.20691e9	−1.61305	−0.806525	+	0.591200i	$0.798654\pi$
$522$	−4.21624e8	−0.129742
$523$	−2.35948e9	−0.721208	−0.360604	−	0.932719i	$-0.617429\pi$
$523$	−2.35948e9	−0.721208	−0.360604	+	0.932719i	$0.617429\pi$
$524$	−2.72740e9	−0.828114
$525$	3.57066e9	1.07694
$526$	−5.20848e8	−0.156049
$527$	1.17675e9	0.350226
$528$	3.99958e8	0.118248
$529$	3.65040e9	1.07213
$530$	−1.42852e7	−0.00416794
$531$	−1.49721e8	−0.0433963
$532$	−2.97434e9	−0.856446
$533$	−2.31875e9	−0.663299
$534$	2.46077e9	0.699321
$535$	6.94569e7	0.0196100
$536$	−9.66236e8	−0.271023
$537$	−2.50198e9	−0.697226
$538$	1.00127e9	0.277213
$539$	−7.42260e9	−2.04172
$540$	1.50725e7	0.00411915
$541$	1.05074e8	0.0285302	0.0142651	−	0.999898i	$-0.495459\pi$
$541$	1.05074e8	0.0285302	0.0142651	+	0.999898i	$0.495459\pi$
$542$	−3.99689e9	−1.07826
$543$	−1.58992e9	−0.426164
$544$	−3.71902e8	−0.0990450
$545$	−5.32155e7	−0.0140816
$546$	3.02572e9	0.795525
$547$	6.35452e9	1.66007	0.830037	−	0.557709i	$-0.188319\pi$
$547$	6.35452e9	1.66007	0.830037	+	0.557709i	$0.188319\pi$
$548$	−2.33192e9	−0.605316
$549$	1.61266e9	0.415948
$550$	2.25618e9	0.578235
$551$	1.98120e9	0.504541
$552$	−1.16115e9	−0.293834
$553$	−1.01677e10	−2.55673
$554$	9.64351e8	0.240963
$555$	−7.85765e7	−0.0195104
$556$	−1.95284e9	−0.481843
$557$	−1.12995e9	−0.277055	−0.138528	−	0.990359i	$-0.544237\pi$
$557$	−1.12995e9	−0.277055	−0.138528	+	0.990359i	$0.544237\pi$
$558$	−6.04678e8	−0.147335
$559$	6.01354e9	1.45609
$560$	−8.31125e7	−0.0199990
$561$	−1.10824e9	−0.265010
$562$	2.58457e9	0.614203
$563$	−6.50041e9	−1.53519	−0.767593	−	0.640937i	$-0.778546\pi$
$563$	−6.50041e9	−1.53519	−0.767593	+	0.640937i	$0.778546\pi$
$564$	−4.13953e8	−0.0971570
$565$	1.38818e8	0.0323799
$566$	−2.04615e9	−0.474329
$567$	9.01252e8	0.207637
$568$	−1.48681e9	−0.340438
$569$	−2.72529e9	−0.620182	−0.310091	−	0.950707i	$-0.600360\pi$
$569$	−2.72529e9	−0.620182	−0.310091	+	0.950707i	$0.600360\pi$
$570$	−7.08252e7	−0.0160186
$571$	3.41306e9	0.767216	0.383608	−	0.923496i	$-0.374681\pi$
$571$	3.41306e9	0.767216	0.383608	+	0.923496i	$0.374681\pi$
$572$	1.91185e9	0.427137
$573$	−1.70614e9	−0.378856
$574$	3.80849e9	0.840546
$575$	−6.55011e9	−1.43685
$576$	1.91103e8	0.0416667
$577$	−2.15916e9	−0.467919	−0.233959	−	0.972246i	$-0.575168\pi$
$577$	−2.15916e9	−0.467919	−0.233959	+	0.972246i	$0.575168\pi$
$578$	−2.25221e9	−0.485134
$579$	−2.68909e9	−0.575745
$580$	5.53609e7	0.0117816
$581$	−1.25679e10	−2.65856
$582$	1.01206e9	0.212803
$583$	−5.39725e8	−0.112806
$584$	3.16933e9	0.658449
$585$	7.20486e7	0.0148792
$586$	−3.05141e9	−0.626411
$587$	4.46492e9	0.911130	0.455565	−	0.890202i	$-0.349437\pi$
$587$	4.46492e9	0.911130	0.455565	+	0.890202i	$0.349437\pi$
$588$	−3.54657e9	−0.719430
$589$	2.84136e9	0.572958
$590$	1.96590e7	0.00394075
$591$	3.30497e9	0.658584
$592$	−9.96262e8	−0.197355
$593$	−4.42853e9	−0.872104	−0.436052	−	0.899922i	$-0.643624\pi$
$593$	−4.42853e9	−0.872104	−0.436052	+	0.899922i	$0.643624\pi$
$594$	5.69471e8	0.111486
$595$	2.30295e8	0.0448204
$596$	−2.54281e9	−0.491984
$597$	3.17215e9	0.610159
$598$	−5.55045e9	−1.06139
$599$	1.77104e9	0.336694	0.168347	−	0.985728i	$-0.446157\pi$
$599$	1.77104e9	0.336694	0.168347	+	0.985728i	$0.446157\pi$
$600$	1.07802e9	0.203750
$601$	7.09086e9	1.33241	0.666205	−	0.745768i	$-0.267917\pi$
$601$	7.09086e9	1.33241	0.666205	+	0.745768i	$0.267917\pi$
$602$	−9.87707e9	−1.84519
$603$	−1.37575e9	−0.255523
$604$	−4.99531e8	−0.0922430
$605$	7.66717e7	0.0140764
$606$	6.73000e8	0.122846
$607$	9.04567e8	0.164165	0.0820825	−	0.996626i	$-0.473843\pi$
$607$	9.04567e8	0.164165	0.0820825	+	0.996626i	$0.473843\pi$
$608$	−8.97985e8	−0.162034
$609$	3.31027e9	0.593886
$610$	−2.11748e8	−0.0377716
$611$	−1.97875e9	−0.350951
$612$	−5.29525e8	−0.0933805
$613$	−5.50419e8	−0.0965121	−0.0482560	−	0.998835i	$-0.515366\pi$
$613$	−5.50419e8	−0.0965121	−0.0482560	+	0.998835i	$0.515366\pi$
$614$	−3.57705e9	−0.623643
$615$	9.06881e7	0.0157213
$616$	−3.14016e9	−0.541277
$617$	1.91894e9	0.328900	0.164450	−	0.986385i	$-0.447415\pi$
$617$	1.91894e9	0.328900	0.164450	+	0.986385i	$0.447415\pi$
$618$	−2.46300e9	−0.419763
$619$	−2.98749e9	−0.506278	−0.253139	−	0.967430i	$-0.581463\pi$
$619$	−2.98749e9	−0.506278	−0.253139	+	0.967430i	$0.581463\pi$
$620$	7.93966e7	0.0133792
$621$	−1.65328e9	−0.277030
$622$	−5.22622e9	−0.870805
$623$	−1.93201e10	−3.20111
$624$	9.13496e8	0.150508
$625$	6.06998e9	0.994506
$626$	−3.28068e9	−0.534507
$627$	−2.67592e9	−0.433548
$628$	−5.43384e8	−0.0875484
$629$	2.76053e9	0.442298
$630$	−1.18338e8	−0.0188552
$631$	−1.42011e9	−0.225019	−0.112509	−	0.993651i	$-0.535889\pi$
$631$	−1.42011e9	−0.225019	−0.112509	+	0.993651i	$0.535889\pi$
$632$	−3.06974e9	−0.483717
$633$	−6.67030e9	−1.04528
$634$	1.36322e9	0.212448
$635$	−4.11435e8	−0.0637666
$636$	−2.57885e8	−0.0397490
$637$	−1.69531e10	−2.59873
$638$	2.09165e9	0.318872
$639$	−2.11697e9	−0.320968
$640$	−2.50926e7	−0.00378368
$641$	−2.33545e9	−0.350242	−0.175121	−	0.984547i	$-0.556032\pi$
$641$	−2.33545e9	−0.350242	−0.175121	+	0.984547i	$0.556032\pi$
$642$	1.25387e9	0.187017
$643$	−2.49002e9	−0.369373	−0.184687	−	0.982797i	$-0.559127\pi$
$643$	−2.49002e9	−0.369373	−0.184687	+	0.982797i	$0.559127\pi$
$644$	9.11647e9	1.34501
$645$	−2.35194e8	−0.0345117
$646$	2.48821e9	0.363140
$647$	8.29895e9	1.20464	0.602322	−	0.798254i	$-0.294243\pi$
$647$	8.29895e9	1.20464	0.602322	+	0.798254i	$0.294243\pi$
$648$	2.72098e8	0.0392837
$649$	7.42757e8	0.106657
$650$	5.15308e9	0.735986
$651$	4.74747e9	0.674417
$652$	1.35251e9	0.191106
$653$	8.07189e9	1.13443	0.567217	−	0.823568i	$-0.308020\pi$
$653$	8.07189e9	1.13443	0.567217	+	0.823568i	$0.308020\pi$
$654$	−9.60677e8	−0.134294
$655$	5.09899e8	0.0708990
$656$	1.14982e9	0.159026
$657$	4.51258e9	0.620791
$658$	3.25004e9	0.444731
$659$	3.89132e9	0.529661	0.264831	−	0.964295i	$-0.414684\pi$
$659$	3.89132e9	0.529661	0.264831	+	0.964295i	$0.414684\pi$
$660$	−7.47738e7	−0.0101238
$661$	8.03071e9	1.08156	0.540778	−	0.841165i	$-0.318130\pi$
$661$	8.03071e9	1.08156	0.540778	+	0.841165i	$0.318130\pi$
$662$	6.87224e9	0.920651
$663$	−2.53119e9	−0.337309
$664$	−3.79439e9	−0.502983
$665$	5.56065e8	0.0733246
$666$	−1.41851e9	−0.186068
$667$	−6.07245e9	−0.792361
$668$	9.74325e8	0.126470
$669$	−1.16924e9	−0.150977
$670$	1.80642e8	0.0232036
$671$	−8.00028e9	−1.02230
$672$	−1.50039e9	−0.190727
$673$	1.09345e10	1.38276	0.691378	−	0.722493i	$-0.257004\pi$
$673$	1.09345e10	1.38276	0.691378	+	0.722493i	$0.257004\pi$
$674$	6.96047e9	0.875647
$675$	1.53492e9	0.192097
$676$	3.50723e8	0.0436667
$677$	5.95836e9	0.738017	0.369009	−	0.929426i	$-0.379697\pi$
$677$	5.95836e9	0.738017	0.369009	+	0.929426i	$0.379697\pi$
$678$	2.50602e9	0.308802
$679$	−7.94595e9	−0.974096
$680$	6.95286e7	0.00847974
$681$	6.22505e9	0.755315
$682$	2.99977e9	0.362111
$683$	7.07369e9	0.849520	0.424760	−	0.905306i	$-0.360358\pi$
$683$	7.07369e9	0.849520	0.424760	+	0.905306i	$0.360358\pi$
$684$	−1.27858e9	−0.152767
$685$	4.35962e8	0.0518241
$686$	1.66720e10	1.97176
$687$	−6.18508e9	−0.727774
$688$	−2.98199e9	−0.349098
$689$	−1.23272e9	−0.143581
$690$	2.17082e8	0.0251566
$691$	−4.36208e9	−0.502945	−0.251472	−	0.967864i	$-0.580915\pi$
$691$	−4.36208e9	−0.502945	−0.251472	+	0.967864i	$0.580915\pi$
$692$	−5.46514e9	−0.626945
$693$	−4.47105e9	−0.510321
$694$	7.36397e9	0.836284
$695$	3.65092e8	0.0412530
$696$	9.99406e8	0.112359
$697$	−3.18603e9	−0.356398
$698$	7.89340e9	0.878557
$699$	−4.68151e9	−0.518461
$700$	−8.46379e9	−0.932656
$701$	−1.30719e10	−1.43327	−0.716633	−	0.697450i	$-0.754318\pi$
$701$	−1.30719e10	−1.43327	−0.716633	+	0.697450i	$0.754318\pi$
$702$	1.30066e9	0.141901
$703$	6.66550e9	0.723584
$704$	−9.48048e8	−0.102406
$705$	7.73902e7	0.00831809
$706$	−6.59397e9	−0.705229
$707$	−5.28387e9	−0.562321
$708$	3.54895e8	0.0375823
$709$	−8.88303e9	−0.936051	−0.468025	−	0.883715i	$-0.655034\pi$
$709$	−8.88303e9	−0.936051	−0.468025	+	0.883715i	$0.655034\pi$
$710$	2.77966e8	0.0291466
$711$	−4.37078e9	−0.456053
$712$	−5.83294e9	−0.605630
$713$	−8.70888e9	−0.899806
$714$	4.15742e9	0.427445
$715$	−3.57428e8	−0.0365694
$716$	5.93061e9	0.603815
$717$	5.76680e9	0.584276
$718$	3.76266e9	0.379366
$719$	−2.41314e9	−0.242121	−0.121060	−	0.992645i	$-0.538629\pi$
$719$	−2.41314e9	−0.242121	−0.121060	+	0.992645i	$0.538629\pi$
$720$	−3.57275e7	−0.00356729
$721$	1.93375e10	1.92144
$722$	−1.14300e9	−0.113022
$723$	−5.20367e9	−0.512066
$724$	3.76871e9	0.369069
$725$	5.63770e9	0.549438
$726$	1.38412e9	0.134244
$727$	−1.55317e10	−1.49916	−0.749582	−	0.661911i	$-0.769746\pi$
$727$	−1.55317e10	−1.49916	−0.749582	+	0.661911i	$0.769746\pi$
$728$	−7.17207e9	−0.688945
$729$	3.87420e8	0.0370370
$730$	−5.92518e8	−0.0563731
$731$	8.26277e9	0.782375
$732$	−3.82259e9	−0.360221
$733$	−3.21864e9	−0.301862	−0.150931	−	0.988544i	$-0.548227\pi$
$733$	−3.21864e9	−0.301862	−0.150931	+	0.988544i	$0.548227\pi$
$734$	1.40364e9	0.131014
$735$	6.63047e8	0.0615940
$736$	2.75236e9	0.254468
$737$	6.82502e9	0.628011
$738$	1.63715e9	0.149931
$739$	−7.22523e9	−0.658561	−0.329280	−	0.944232i	$-0.606806\pi$
$739$	−7.22523e9	−0.658561	−0.329280	+	0.944232i	$0.606806\pi$
$740$	1.86255e8	0.0168965
$741$	−6.11176e9	−0.551826
$742$	2.02471e9	0.181949
$743$	−2.02064e10	−1.80729	−0.903646	−	0.428281i	$-0.859119\pi$
$743$	−2.02064e10	−1.80729	−0.903646	+	0.428281i	$0.859119\pi$
$744$	1.43331e9	0.127595
$745$	4.75388e8	0.0421212
$746$	−1.32538e10	−1.16884
$747$	−5.40255e9	−0.474217
$748$	2.62693e9	0.229506
$749$	−9.84446e9	−0.856062
$750$	−4.03451e8	−0.0349202
$751$	2.27679e9	0.196147	0.0980737	−	0.995179i	$-0.468732\pi$
$751$	2.27679e9	0.196147	0.0980737	+	0.995179i	$0.468732\pi$
$752$	9.81221e8	0.0841404
$753$	−3.59845e9	−0.307138
$754$	4.77729e9	0.405865
$755$	9.33894e7	0.00789738
$756$	−2.13630e9	−0.179819
$757$	−1.44699e10	−1.21236	−0.606178	−	0.795329i	$-0.707298\pi$
$757$	−1.44699e10	−1.21236	−0.606178	+	0.795329i	$0.707298\pi$
$758$	3.79654e9	0.316625
$759$	8.20181e9	0.680869
$760$	1.67882e8	0.0138726
$761$	9.74821e9	0.801823	0.400911	−	0.916117i	$-0.368694\pi$
$761$	9.74821e9	0.801823	0.400911	+	0.916117i	$0.368694\pi$
$762$	−7.42746e9	−0.608132
$763$	7.54249e9	0.614722
$764$	4.04419e9	0.328099
$765$	9.89968e7	0.00799477
$766$	−2.56128e9	−0.205900
$767$	1.69644e9	0.135755
$768$	−4.52985e8	−0.0360844
$769$	−1.26232e10	−1.00099	−0.500494	−	0.865740i	$-0.666848\pi$
$769$	−1.26232e10	−1.00099	−0.500494	+	0.865740i	$0.666848\pi$
$770$	5.87066e8	0.0463414
$771$	6.18096e9	0.485697
$772$	6.37414e9	0.498610
$773$	−1.75808e10	−1.36902	−0.684510	−	0.729003i	$-0.739984\pi$
$773$	−1.75808e10	−1.36902	−0.684510	+	0.729003i	$0.739984\pi$
$774$	−4.24585e9	−0.329133
$775$	8.08538e9	0.623942
$776$	−2.39897e9	−0.184293
$777$	1.11370e10	0.851717
$778$	1.58328e10	1.20539
$779$	−7.69291e9	−0.583055
$780$	−1.70782e8	−0.0128858
$781$	1.05021e10	0.788858
$782$	−7.62648e9	−0.570296
$783$	1.42298e9	0.105934
$784$	8.40670e9	0.623045
$785$	1.01588e8	0.00749546
$786$	9.20499e9	0.676152
$787$	−9.05631e9	−0.662277	−0.331138	−	0.943582i	$-0.607433\pi$
$787$	−9.05631e9	−0.662277	−0.331138	+	0.943582i	$0.607433\pi$
$788$	−7.83399e9	−0.570350
$789$	1.75786e9	0.127413
$790$	5.73900e8	0.0414135
$791$	−1.96753e10	−1.41353
$792$	−1.34986e9	−0.0965495
$793$	−1.82725e10	−1.30119
$794$	9.43305e9	0.668776
$795$	4.82126e7	0.00340311
$796$	−7.51916e9	−0.528413
$797$	8.52956e9	0.596791	0.298396	−	0.954442i	$-0.403549\pi$
$797$	8.52956e9	0.596791	0.298396	+	0.954442i	$0.403549\pi$
$798$	1.00384e10	0.699285
$799$	−2.71885e9	−0.188570
$800$	−2.55531e9	−0.176453
$801$	−8.30510e9	−0.570994
$802$	1.24961e10	0.855393
$803$	−2.23866e10	−1.52575
$804$	3.26105e9	0.221289
$805$	−1.70436e9	−0.115153
$806$	6.85141e9	0.460901
$807$	−3.37929e9	−0.226343
$808$	−1.59526e9	−0.106388
$809$	1.45511e10	0.966218	0.483109	−	0.875560i	$-0.339507\pi$
$809$	1.45511e10	0.966218	0.483109	+	0.875560i	$0.339507\pi$
$810$	−5.08698e7	−0.00336328
$811$	2.66639e10	1.75530	0.877649	−	0.479304i	$-0.159111\pi$
$811$	2.66639e10	1.75530	0.877649	+	0.479304i	$0.159111\pi$
$812$	−7.84657e9	−0.514320
$813$	1.34895e10	0.880399
$814$	7.03711e9	0.457308
$815$	−2.52858e8	−0.0163616
$816$	1.25517e9	0.0808699
$817$	1.99511e10	1.27994
$818$	7.97166e9	0.509228
$819$	−1.02118e10	−0.649544
$820$	−2.14964e8	−0.0136150
$821$	−9.80318e9	−0.618253	−0.309126	−	0.951021i	$-0.600037\pi$
$821$	−9.80318e9	−0.618253	−0.309126	+	0.951021i	$0.600037\pi$
$822$	7.87024e9	0.494238
$823$	2.20963e10	1.38172	0.690860	−	0.722989i	$-0.257232\pi$
$823$	2.20963e10	1.38172	0.690860	+	0.722989i	$0.257232\pi$
$824$	5.83821e9	0.363525
$825$	−7.61461e9	−0.472127
$826$	−2.78636e9	−0.172031
$827$	1.59307e10	0.979415	0.489707	−	0.871887i	$-0.337104\pi$
$827$	1.59307e10	0.979415	0.489707	+	0.871887i	$0.337104\pi$
$828$	3.91889e9	0.239915
$829$	−1.14501e9	−0.0698022	−0.0349011	−	0.999391i	$-0.511112\pi$
$829$	−1.14501e9	−0.0698022	−0.0349011	+	0.999391i	$0.511112\pi$
$830$	7.09376e8	0.0430629
$831$	−3.25468e9	−0.196746
$832$	−2.16532e9	−0.130344
$833$	−2.32940e10	−1.39633
$834$	6.59085e9	0.393423
$835$	−1.82154e8	−0.0108277
$836$	6.34293e9	0.375464
$837$	2.04079e9	0.120298
$838$	−5.81797e9	−0.341521
$839$	2.01454e10	1.17763	0.588814	−	0.808269i	$-0.299595\pi$
$839$	2.01454e10	1.17763	0.588814	+	0.808269i	$0.299595\pi$
$840$	2.80505e8	0.0163291
$841$	−1.20233e10	−0.697008
$842$	−1.67061e10	−0.964457
$843$	−8.72294e9	−0.501495
$844$	1.58111e10	0.905238
$845$	−6.55690e7	−0.00373852
$846$	1.39709e9	0.0793283
$847$	−1.08671e10	−0.614497
$848$	6.11283e8	0.0344236
$849$	6.90575e9	0.387288
$850$	7.08047e9	0.395454
$851$	−2.04300e10	−1.13636
$852$	5.01800e9	0.277966
$853$	9.79876e9	0.540567	0.270284	−	0.962781i	$-0.412883\pi$
$853$	9.79876e9	0.540567	0.270284	+	0.962781i	$0.412883\pi$
$854$	3.00121e10	1.64890
$855$	2.39035e8	0.0130792
$856$	−2.97215e9	−0.161962
$857$	1.92850e10	1.04661	0.523306	−	0.852145i	$-0.324698\pi$
$857$	1.92850e10	1.04661	0.523306	+	0.852145i	$0.324698\pi$
$858$	−6.45249e9	−0.348756
$859$	3.35706e8	0.0180710	0.00903552	−	0.999959i	$-0.497124\pi$
$859$	3.35706e8	0.0180710	0.00903552	+	0.999959i	$0.497124\pi$
$860$	5.57496e8	0.0298880
$861$	−1.28537e10	−0.686303
$862$	−1.89460e10	−1.00749
$863$	1.12418e10	0.595383	0.297692	−	0.954662i	$-0.403783\pi$
$863$	1.12418e10	0.595383	0.297692	+	0.954662i	$0.403783\pi$
$864$	−6.44973e8	−0.0340207
$865$	1.02173e9	0.0536759
$866$	−1.46883e9	−0.0768526
$867$	7.60121e9	0.396110
$868$	−1.12533e10	−0.584062
$869$	2.16831e10	1.12086
$870$	−1.86843e8	−0.00961966
$871$	1.55882e10	0.799342
$872$	2.27716e9	0.116302
$873$	−3.41572e9	−0.173753
$874$	−1.84147e10	−0.932985
$875$	3.16758e9	0.159845
$876$	−1.06965e10	−0.537621
$877$	−2.46550e10	−1.23426	−0.617129	−	0.786862i	$-0.711704\pi$
$877$	−2.46550e10	−1.23426	−0.617129	+	0.786862i	$0.711704\pi$
$878$	1.99855e10	0.996515
$879$	1.02985e10	0.511463
$880$	1.77242e8	0.00876751
$881$	−2.00251e10	−0.986641	−0.493320	−	0.869848i	$-0.664217\pi$
$881$	−2.00251e10	−0.986641	−0.493320	+	0.869848i	$0.664217\pi$
$882$	1.19697e10	0.587412
$883$	−4.41483e9	−0.215800	−0.107900	−	0.994162i	$-0.534413\pi$
$883$	−4.41483e9	−0.215800	−0.107900	+	0.994162i	$0.534413\pi$
$884$	5.99987e9	0.292118
$885$	−6.63491e7	−0.00321761
$886$	−2.88326e10	−1.39273
$887$	2.31004e10	1.11144	0.555722	−	0.831368i	$-0.312442\pi$
$887$	2.31004e10	1.11144	0.555722	+	0.831368i	$0.312442\pi$
$888$	3.36239e9	0.161140
$889$	5.83147e10	2.78370
$890$	1.09049e9	0.0518510
$891$	−1.92197e9	−0.0910277
$892$	2.77153e9	0.130750
$893$	−6.56487e9	−0.308494
$894$	8.58197e9	0.401704
$895$	−1.10875e9	−0.0516957
$896$	3.55649e9	0.165175
$897$	1.87328e10	0.866620
$898$	3.04344e10	1.40248
$899$	7.49575e9	0.344077
$900$	−3.63832e9	−0.166361
$901$	−1.69380e9	−0.0771479
$902$	−8.12180e9	−0.368493
$903$	3.33351e10	1.50659
$904$	−5.94019e9	−0.267431
$905$	−7.04575e8	−0.0315978
$906$	1.68592e9	0.0753161
$907$	7.20559e9	0.320660	0.160330	−	0.987063i	$-0.448744\pi$
$907$	7.20559e9	0.320660	0.160330	+	0.987063i	$0.448744\pi$
$908$	−1.47557e10	−0.654122
$909$	−2.27137e9	−0.100303
$910$	1.34085e9	0.0589840
$911$	3.20724e10	1.40546	0.702728	−	0.711459i	$-0.251965\pi$
$911$	3.20724e10	1.40546	0.702728	+	0.711459i	$0.251965\pi$
$912$	3.03070e9	0.132300
$913$	2.68017e10	1.16551
$914$	−2.88369e10	−1.24921
$915$	7.14650e8	0.0308404
$916$	1.46609e10	0.630271
$917$	−7.22705e10	−3.09505
$918$	1.78715e9	0.0762449
$919$	2.54076e6	0.000107984 0	5.39921e−5	−	1.00000i	$-0.499983\pi$
$919$	2.54076e6	0.000107984 0	5.39921e−5		1.00000i	$0.499983\pi$
$920$	−5.14565e8	−0.0217863
$921$	1.20726e10	0.509202
$922$	−8.78873e9	−0.369290
$923$	2.39867e10	1.00407
$924$	1.05980e10	0.441951
$925$	1.89674e10	0.787973
$926$	2.40042e10	0.993458
$927$	8.31261e9	0.342735
$928$	−2.36896e9	−0.0973062
$929$	2.96537e10	1.21346	0.606728	−	0.794909i	$-0.292482\pi$
$929$	2.96537e10	1.21346	0.606728	+	0.794909i	$0.292482\pi$
$930$	−2.67963e8	−0.0109241
$931$	−5.62451e10	−2.28434
$932$	1.10969e10	0.449000
$933$	1.76385e10	0.711010
$934$	2.64379e8	0.0106173
$935$	−4.91116e8	−0.0196491
$936$	−3.08305e9	−0.122890
$937$	−1.66140e10	−0.659759	−0.329880	−	0.944023i	$-0.607008\pi$
$937$	−1.66140e10	−0.659759	−0.329880	+	0.944023i	$0.607008\pi$
$938$	−2.56032e10	−1.01294
$939$	1.10723e10	0.436423
$940$	−1.83443e8	−0.00720368
$941$	2.40170e10	0.939624	0.469812	−	0.882766i	$-0.344322\pi$
$941$	2.40170e10	0.939624	0.469812	+	0.882766i	$0.344322\pi$
$942$	1.83392e9	0.0714830
$943$	2.35791e10	0.915664
$944$	−8.41232e8	−0.0325472
$945$	3.99390e8	0.0153952
$946$	2.10634e10	0.808926
$947$	5.08531e10	1.94577	0.972887	−	0.231282i	$-0.0742921\pi$
$947$	5.08531e10	1.94577	0.972887	+	0.231282i	$0.0742921\pi$
$948$	1.03604e10	0.394954
$949$	−5.11305e10	−1.94200
$950$	1.70963e10	0.646949
$951$	−4.60087e9	−0.173463
$952$	−9.85462e9	−0.370178
$953$	4.24841e9	0.159001	0.0795007	−	0.996835i	$-0.474667\pi$
$953$	4.24841e9	0.159001	0.0795007	+	0.996835i	$0.474667\pi$
$954$	8.70362e8	0.0324549
$955$	−7.56078e8	−0.0280902
$956$	−1.36694e10	−0.505997
$957$	−7.05932e9	−0.260358
$958$	9.71031e9	0.356824
$959$	−6.17910e10	−2.26235
$960$	8.46874e7	0.00308937
$961$	−1.67625e10	−0.609265
$962$	1.60726e10	0.582068
$963$	−4.23183e9	−0.152699
$964$	1.23346e10	0.443462
$965$	−1.19167e9	−0.0426885
$966$	−3.07681e10	−1.09820
$967$	3.98497e10	1.41720	0.708602	−	0.705608i	$-0.249326\pi$
$967$	3.98497e10	1.41720	0.708602	+	0.705608i	$0.249326\pi$
$968$	−3.28088e9	−0.116259
$969$	−8.39772e9	−0.296503
$970$	4.48497e8	0.0157782
$971$	3.45127e10	1.20979	0.604896	−	0.796304i	$-0.293215\pi$
$971$	3.45127e10	1.20979	0.604896	+	0.796304i	$0.293215\pi$
$972$	−9.18330e8	−0.0320750
$973$	−5.17463e10	−1.80088
$974$	−4.07316e8	−0.0141246
$975$	−1.73916e10	−0.600930
$976$	9.06097e9	0.311961
$977$	1.27453e10	0.437238	0.218619	−	0.975810i	$-0.429845\pi$
$977$	1.27453e10	0.437238	0.218619	+	0.975810i	$0.429845\pi$
$978$	−4.56473e9	−0.156038
$979$	4.12010e10	1.40336
$980$	−1.57167e9	−0.0533420
$981$	3.24228e9	0.109650
$982$	2.03917e10	0.687168
$983$	−2.33918e9	−0.0785464	−0.0392732	−	0.999229i	$-0.512504\pi$
$983$	−2.33918e9	−0.0785464	−0.0392732	+	0.999229i	$0.512504\pi$
$984$	−3.88066e9	−0.129844
$985$	1.46460e9	0.0488305
$986$	6.56413e9	0.218076
$987$	−1.09689e10	−0.363122
$988$	1.44871e10	0.477895
$989$	−6.11508e10	−2.01009
$990$	2.52362e8	0.00826609
$991$	−5.47515e10	−1.78706	−0.893528	−	0.449008i	$-0.851777\pi$
$991$	−5.47515e10	−1.78706	−0.893528	+	0.449008i	$0.851777\pi$
$992$	−3.39748e9	−0.110501
$993$	−2.31938e10	−0.751709
$994$	−3.93974e10	−1.27238
$995$	1.40574e9	0.0452401
$996$	1.28061e10	0.410684
$997$	8.50092e9	0.271665	0.135832	−	0.990732i	$-0.456629\pi$
$997$	8.50092e9	0.271665	0.135832	+	0.990732i	$0.456629\pi$
$998$	4.26426e10	1.35796
$999$	4.78746e9	0.151924

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	354.8.a.d.1.4	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
354.8.a.d.1.4	✓	8	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 354 = 2 \cdot 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	354.a (trivial)

\(f(q)\)	\(=\)	\(q+8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} -11.9651 q^{5} -216.000 q^{6} +1695.87 q^{7} +512.000 q^{8} +729.000 q^{9} +O(q^{10})\)	\(q+8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} -11.9651 q^{5} -216.000 q^{6} +1695.87 q^{7} +512.000 q^{8} +729.000 q^{9} -95.7205 q^{10} -3616.52 q^{11} -1728.00 q^{12} -8260.06 q^{13} +13566.9 q^{14} +323.057 q^{15} +4096.00 q^{16} -11349.6 q^{17} +5832.00 q^{18} -27404.3 q^{19} -765.764 q^{20} -45788.4 q^{21} -28932.1 q^{22} +83995.4 q^{23} -13824.0 q^{24} -77981.8 q^{25} -66080.5 q^{26} -19683.0 q^{27} +108535. q^{28} -72295.0 q^{29} +2584.45 q^{30} -103683. q^{31} +32768.0 q^{32} +97646.0 q^{33} -90796.4 q^{34} -20291.1 q^{35} +46656.0 q^{36} -243228. q^{37} -219235. q^{38} +223022. q^{39} -6126.11 q^{40} +280719. q^{41} -366307. q^{42} -728026. q^{43} -231457. q^{44} -8722.53 q^{45} +671963. q^{46} +239556. q^{47} -110592. q^{48} +2.05242e6 q^{49} -623855. q^{50} +306438. q^{51} -528644. q^{52} +149239. q^{53} -157464. q^{54} +43271.9 q^{55} +868283. q^{56} +739917. q^{57} -578360. q^{58} -205379. q^{59} +20675.6 q^{60} +2.21215e6 q^{61} -829463. q^{62} +1.23629e6 q^{63} +262144. q^{64} +98832.1 q^{65} +781168. q^{66} -1.88718e6 q^{67} -726371. q^{68} -2.26788e6 q^{69} -162329. q^{70} -2.90393e6 q^{71} +373248. q^{72} +6.19009e6 q^{73} -1.94582e6 q^{74} +2.10551e6 q^{75} -1.75388e6 q^{76} -6.13313e6 q^{77} +1.78417e6 q^{78} -5.99559e6 q^{79} -49008.9 q^{80} +531441. q^{81} +2.24575e6 q^{82} -7.41091e6 q^{83} -2.93046e6 q^{84} +135798. q^{85} -5.82421e6 q^{86} +1.95197e6 q^{87} -1.85166e6 q^{88} -1.13925e7 q^{89} -69780.2 q^{90} -1.40079e7 q^{91} +5.37570e6 q^{92} +2.79944e6 q^{93} +1.91645e6 q^{94} +327894. q^{95} -884736. q^{96} -4.68549e6 q^{97} +1.64193e7 q^{98} -2.63644e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 64 q^{2} - 216 q^{3} + 512 q^{4} - 592 q^{5} - 1728 q^{6} - 340 q^{7} + 4096 q^{8} + 5832 q^{9}+O(q^{10}) \) 8 * q + 64 * q^2 - 216 * q^3 + 512 * q^4 - 592 * q^5 - 1728 * q^6 - 340 * q^7 + 4096 * q^8 + 5832 * q^9	\( 8 q + 64 q^{2} - 216 q^{3} + 512 q^{4} - 592 q^{5} - 1728 q^{6} - 340 q^{7} + 4096 q^{8} + 5832 q^{9} - 4736 q^{10} - 2852 q^{11} - 13824 q^{12} + 1142 q^{13} - 2720 q^{14} + 15984 q^{15} + 32768 q^{16} - 22528 q^{17} + 46656 q^{18} - 33528 q^{19} - 37888 q^{20} + 9180 q^{21} - 22816 q^{22} + 41330 q^{23} - 110592 q^{24} + 209004 q^{25} + 9136 q^{26} - 157464 q^{27} - 21760 q^{28} - 48334 q^{29} + 127872 q^{30} + 217552 q^{31} + 262144 q^{32} + 77004 q^{33} - 180224 q^{34} - 171714 q^{35} + 373248 q^{36} - 77966 q^{37} - 268224 q^{38} - 30834 q^{39} - 303104 q^{40} - 446410 q^{41} + 73440 q^{42} - 470890 q^{43} - 182528 q^{44} - 431568 q^{45} + 330640 q^{46} + 1876568 q^{47} - 884736 q^{48} + 1667480 q^{49} + 1672032 q^{50} + 608256 q^{51} + 73088 q^{52} - 1155672 q^{53} - 1259712 q^{54} + 112064 q^{55} - 174080 q^{56} + 905256 q^{57} - 386672 q^{58} - 1643032 q^{59} + 1022976 q^{60} - 9094962 q^{61} + 1740416 q^{62} - 247860 q^{63} + 2097152 q^{64} - 6726234 q^{65} + 616032 q^{66} - 8552352 q^{67} - 1441792 q^{68} - 1115910 q^{69} - 1373712 q^{70} - 5829156 q^{71} + 2985984 q^{72} - 7639392 q^{73} - 623728 q^{74} - 5643108 q^{75} - 2145792 q^{76} - 17178270 q^{77} - 246672 q^{78} - 12614888 q^{79} - 2424832 q^{80} + 4251528 q^{81} - 3571280 q^{82} - 19145486 q^{83} + 587520 q^{84} - 20127842 q^{85} - 3767120 q^{86} + 1305018 q^{87} - 1460224 q^{88} - 16050066 q^{89} - 3452544 q^{90} - 20086856 q^{91} + 2645120 q^{92} - 5873904 q^{93} + 15012544 q^{94} - 8130136 q^{95} - 7077888 q^{96} - 1961876 q^{97} + 13339840 q^{98} - 2079108 q^{99}+O(q^{100}) \) 8 * q + 64 * q^2 - 216 * q^3 + 512 * q^4 - 592 * q^5 - 1728 * q^6 - 340 * q^7 + 4096 * q^8 + 5832 * q^9 - 4736 * q^10 - 2852 * q^11 - 13824 * q^12 + 1142 * q^13 - 2720 * q^14 + 15984 * q^15 + 32768 * q^16 - 22528 * q^17 + 46656 * q^18 - 33528 * q^19 - 37888 * q^20 + 9180 * q^21 - 22816 * q^22 + 41330 * q^23 - 110592 * q^24 + 209004 * q^25 + 9136 * q^26 - 157464 * q^27 - 21760 * q^28 - 48334 * q^29 + 127872 * q^30 + 217552 * q^31 + 262144 * q^32 + 77004 * q^33 - 180224 * q^34 - 171714 * q^35 + 373248 * q^36 - 77966 * q^37 - 268224 * q^38 - 30834 * q^39 - 303104 * q^40 - 446410 * q^41 + 73440 * q^42 - 470890 * q^43 - 182528 * q^44 - 431568 * q^45 + 330640 * q^46 + 1876568 * q^47 - 884736 * q^48 + 1667480 * q^49 + 1672032 * q^50 + 608256 * q^51 + 73088 * q^52 - 1155672 * q^53 - 1259712 * q^54 + 112064 * q^55 - 174080 * q^56 + 905256 * q^57 - 386672 * q^58 - 1643032 * q^59 + 1022976 * q^60 - 9094962 * q^61 + 1740416 * q^62 - 247860 * q^63 + 2097152 * q^64 - 6726234 * q^65 + 616032 * q^66 - 8552352 * q^67 - 1441792 * q^68 - 1115910 * q^69 - 1373712 * q^70 - 5829156 * q^71 + 2985984 * q^72 - 7639392 * q^73 - 623728 * q^74 - 5643108 * q^75 - 2145792 * q^76 - 17178270 * q^77 - 246672 * q^78 - 12614888 * q^79 - 2424832 * q^80 + 4251528 * q^81 - 3571280 * q^82 - 19145486 * q^83 + 587520 * q^84 - 20127842 * q^85 - 3767120 * q^86 + 1305018 * q^87 - 1460224 * q^88 - 16050066 * q^89 - 3452544 * q^90 - 20086856 * q^91 + 2645120 * q^92 - 5873904 * q^93 + 15012544 * q^94 - 8130136 * q^95 - 7077888 * q^96 - 1961876 * q^97 + 13339840 * q^98 - 2079108 * q^99

Introduction

L-functions

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