LMFDB - Embedded newform 69.8.a.b.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [69,8,Mod(1,69)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(69, 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("69.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 69 = 3 \cdot 23 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	69.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:	$21.5545667584$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 466x^{4} + 540x^{3} + 48973x^{2} - 77282x - 1061812 $ x^6 - 2x^5 - 466x^4 + 540x^3 + 48973x^2 - 77282*x - 1061812
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{5}\cdot 3^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-17.1241$ of defining polynomial
Character	$\chi$	$=$	69.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+16.1241 q^{2} +27.0000 q^{3} +131.987 q^{4} -519.327 q^{5} +435.351 q^{6} -439.281 q^{7} +64.2798 q^{8} +729.000 q^{9} +O(q^{10})$	\(q+16.1241 q^{2} +27.0000 q^{3} +131.987 q^{4} -519.327 q^{5} +435.351 q^{6} -439.281 q^{7} +64.2798 q^{8} +729.000 q^{9} -8373.68 q^{10} -1239.00 q^{11} +3563.64 q^{12} -886.064 q^{13} -7083.01 q^{14} -14021.8 q^{15} -15857.8 q^{16} -33265.6 q^{17} +11754.5 q^{18} +24223.0 q^{19} -68544.2 q^{20} -11860.6 q^{21} -19977.7 q^{22} -12167.0 q^{23} +1735.55 q^{24} +191576. q^{25} -14287.0 q^{26} +19683.0 q^{27} -57979.2 q^{28} -26376.0 q^{29} -226089. q^{30} +58381.7 q^{31} -263921. q^{32} -33452.9 q^{33} -536379. q^{34} +228131. q^{35} +96218.2 q^{36} +72694.7 q^{37} +390574. q^{38} -23923.7 q^{39} -33382.2 q^{40} +175132. q^{41} -191241. q^{42} -302259. q^{43} -163531. q^{44} -378590. q^{45} -196182. q^{46} +715309. q^{47} -428161. q^{48} -630575. q^{49} +3.08899e6 q^{50} -898172. q^{51} -116948. q^{52} -1.15458e6 q^{53} +317371. q^{54} +643444. q^{55} -28236.9 q^{56} +654022. q^{57} -425289. q^{58} -1.13651e6 q^{59} -1.85069e6 q^{60} +699341. q^{61} +941353. q^{62} -320236. q^{63} -2.22569e6 q^{64} +460157. q^{65} -539397. q^{66} -4.85518e6 q^{67} -4.39062e6 q^{68} -328509. q^{69} +3.67840e6 q^{70} +369628. q^{71} +46860.0 q^{72} +1.06243e6 q^{73} +1.17214e6 q^{74} +5.17254e6 q^{75} +3.19711e6 q^{76} +544267. q^{77} -385748. q^{78} +6.34807e6 q^{79} +8.23540e6 q^{80} +531441. q^{81} +2.82384e6 q^{82} +9.31819e6 q^{83} -1.56544e6 q^{84} +1.72758e7 q^{85} -4.87366e6 q^{86} -712151. q^{87} -79642.3 q^{88} -1.25442e7 q^{89} -6.10441e6 q^{90} +389231. q^{91} -1.60588e6 q^{92} +1.57631e6 q^{93} +1.15337e7 q^{94} -1.25797e7 q^{95} -7.12587e6 q^{96} -1.21853e7 q^{97} -1.01675e7 q^{98} -903228. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 8 q^{2} + 162 q^{3} + 178 q^{4} - 372 q^{5} - 216 q^{6} - 1104 q^{7} - 1956 q^{8} + 4374 q^{9}+O(q^{10}) $ 6 * q - 8 * q^2 + 162 * q^3 + 178 * q^4 - 372 * q^5 - 216 * q^6 - 1104 * q^7 - 1956 * q^8 + 4374 * q^9	\( 6 q - 8 q^{2} + 162 q^{3} + 178 q^{4} - 372 q^{5} - 216 q^{6} - 1104 q^{7} - 1956 q^{8} + 4374 q^{9} - 13042 q^{10} - 14824 q^{11} + 4806 q^{12} - 756 q^{13} - 3926 q^{14} - 10044 q^{15} - 13022 q^{16} - 69484 q^{17} - 5832 q^{18} - 43864 q^{19} + 78886 q^{20} - 29808 q^{21} + 98204 q^{22} - 73002 q^{23} - 52812 q^{24} + 228018 q^{25} - 311956 q^{26} + 118098 q^{27} - 545442 q^{28} - 311100 q^{29} - 352134 q^{30} - 245248 q^{31} - 390156 q^{32} - 400248 q^{33} + 235834 q^{34} - 1331256 q^{35} + 129762 q^{36} - 630044 q^{37} + 80910 q^{38} - 20412 q^{39} - 2153982 q^{40} - 969204 q^{41} - 106002 q^{42} - 1770208 q^{43} - 1749140 q^{44} - 271188 q^{45} + 97336 q^{46} - 1400024 q^{47} - 351594 q^{48} + 1985598 q^{49} - 956660 q^{50} - 1876068 q^{51} + 3217272 q^{52} - 1573516 q^{53} - 157464 q^{54} - 431296 q^{55} + 7740702 q^{56} - 1184328 q^{57} + 5987188 q^{58} - 1410320 q^{59} + 2129922 q^{60} - 942172 q^{61} + 3334412 q^{62} - 804816 q^{63} + 1996866 q^{64} - 420944 q^{65} + 2651508 q^{66} - 452072 q^{67} - 9258254 q^{68} - 1971054 q^{69} + 21981136 q^{70} + 122928 q^{71} - 1425924 q^{72} + 16490716 q^{73} - 600104 q^{74} + 6156486 q^{75} + 7428658 q^{76} + 7239696 q^{77} - 8422812 q^{78} + 2458408 q^{79} + 19440230 q^{80} + 3188646 q^{81} + 20510784 q^{82} - 7566456 q^{83} - 14726934 q^{84} + 5817744 q^{85} - 669666 q^{86} - 8399700 q^{87} + 14775668 q^{88} - 20368036 q^{89} - 9507618 q^{90} + 8815576 q^{91} - 2165726 q^{92} - 6621696 q^{93} + 16952576 q^{94} + 5143832 q^{95} - 10534212 q^{96} + 12586972 q^{97} - 39164812 q^{98} - 10806696 q^{99}+O(q^{100}) \) 6 * q - 8 * q^2 + 162 * q^3 + 178 * q^4 - 372 * q^5 - 216 * q^6 - 1104 * q^7 - 1956 * q^8 + 4374 * q^9 - 13042 * q^10 - 14824 * q^11 + 4806 * q^12 - 756 * q^13 - 3926 * q^14 - 10044 * q^15 - 13022 * q^16 - 69484 * q^17 - 5832 * q^18 - 43864 * q^19 + 78886 * q^20 - 29808 * q^21 + 98204 * q^22 - 73002 * q^23 - 52812 * q^24 + 228018 * q^25 - 311956 * q^26 + 118098 * q^27 - 545442 * q^28 - 311100 * q^29 - 352134 * q^30 - 245248 * q^31 - 390156 * q^32 - 400248 * q^33 + 235834 * q^34 - 1331256 * q^35 + 129762 * q^36 - 630044 * q^37 + 80910 * q^38 - 20412 * q^39 - 2153982 * q^40 - 969204 * q^41 - 106002 * q^42 - 1770208 * q^43 - 1749140 * q^44 - 271188 * q^45 + 97336 * q^46 - 1400024 * q^47 - 351594 * q^48 + 1985598 * q^49 - 956660 * q^50 - 1876068 * q^51 + 3217272 * q^52 - 1573516 * q^53 - 157464 * q^54 - 431296 * q^55 + 7740702 * q^56 - 1184328 * q^57 + 5987188 * q^58 - 1410320 * q^59 + 2129922 * q^60 - 942172 * q^61 + 3334412 * q^62 - 804816 * q^63 + 1996866 * q^64 - 420944 * q^65 + 2651508 * q^66 - 452072 * q^67 - 9258254 * q^68 - 1971054 * q^69 + 21981136 * q^70 + 122928 * q^71 - 1425924 * q^72 + 16490716 * q^73 - 600104 * q^74 + 6156486 * q^75 + 7428658 * q^76 + 7239696 * q^77 - 8422812 * q^78 + 2458408 * q^79 + 19440230 * q^80 + 3188646 * q^81 + 20510784 * q^82 - 7566456 * q^83 - 14726934 * q^84 + 5817744 * q^85 - 669666 * q^86 - 8399700 * q^87 + 14775668 * q^88 - 20368036 * q^89 - 9507618 * q^90 + 8815576 * q^91 - 2165726 * q^92 - 6621696 * q^93 + 16952576 * q^94 + 5143832 * q^95 - 10534212 * q^96 + 12586972 * q^97 - 39164812 * q^98 - 10806696 * 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 $

$2$	16.1241	1.42518	0.712591	−	0.701579i	$-0.247521\pi$
$2$	16.1241	1.42518	0.712591	+	0.701579i	$0.247521\pi$
$3$	27.0000	0.577350
$3$	27.0000	0.577350
$4$	131.987	1.03115
$5$	−519.327	−1.85800	−0.929001	−	0.370078i	$-0.879331\pi$
$5$	−519.327	−1.85800	−0.929001	+	0.370078i	$0.879331\pi$
$6$	435.351	0.822829
$7$	−439.281	−0.484060	−0.242030	−	0.970269i	$-0.577813\pi$
$7$	−439.281	−0.484060	−0.242030	+	0.970269i	$0.577813\pi$
$8$	64.2798	0.0443874
$9$	729.000	0.333333
$10$	−8373.68	−2.64799
$11$	−1239.00	−0.280670	−0.140335	−	0.990104i	$-0.544818\pi$
$11$	−1239.00	−0.280670	−0.140335	+	0.990104i	$0.544818\pi$
$12$	3563.64	0.595332
$13$	−886.064	−0.111857	−0.0559285	−	0.998435i	$-0.517812\pi$
$13$	−886.064	−0.111857	−0.0559285	+	0.998435i	$0.517812\pi$
$14$	−7083.01	−0.689874
$15$	−14021.8	−1.07272
$16$	−15857.8	−0.967885
$17$	−33265.6	−1.64219	−0.821097	−	0.570788i	$-0.806638\pi$
$17$	−33265.6	−1.64219	−0.821097	+	0.570788i	$0.806638\pi$
$18$	11754.5	0.475061
$19$	24223.0	0.810197	0.405099	−	0.914273i	$-0.367237\pi$
$19$	24223.0	0.810197	0.405099	+	0.914273i	$0.367237\pi$
$20$	−68544.2	−1.91587
$21$	−11860.6	−0.279472
$22$	−19977.7	−0.400005
$23$	−12167.0	−0.208514
$23$	−12167.0	−0.208514
$24$	1735.55	0.0256271
$25$	191576.	2.45217
$26$	−14287.0	−0.159417
$27$	19683.0	0.192450
$28$	−57979.2	−0.499136
$29$	−26376.0	−0.200824	−0.100412	−	0.994946i	$-0.532016\pi$
$29$	−26376.0	−0.200824	−0.100412	+	0.994946i	$0.532016\pi$
$30$	−226089.	−1.52882
$31$	58381.7	0.351974	0.175987	−	0.984392i	$-0.443688\pi$
$31$	58381.7	0.351974	0.175987	+	0.984392i	$0.443688\pi$
$32$	−263921.	−1.42380
$33$	−33452.9	−0.162045
$34$	−536379.	−2.34043
$35$	228131.	0.899385
$36$	96218.2	0.343715
$37$	72694.7	0.235937	0.117969	−	0.993017i	$-0.462362\pi$
$37$	72694.7	0.235937	0.117969	+	0.993017i	$0.462362\pi$
$38$	390574.	1.15468
$39$	−23923.7	−0.0645807
$40$	−33382.2	−0.0824718
$41$	175132.	0.396846	0.198423	−	0.980117i	$-0.436418\pi$
$41$	175132.	0.396846	0.198423	+	0.980117i	$0.436418\pi$
$42$	−191241.	−0.398299
$43$	−302259.	−0.579749	−0.289875	−	0.957065i	$-0.593614\pi$
$43$	−302259.	−0.579749	−0.289875	+	0.957065i	$0.593614\pi$
$44$	−163531.	−0.289411
$45$	−378590.	−0.619334
$46$	−196182.	−0.297171
$47$	715309.	1.00497	0.502483	−	0.864587i	$-0.332420\pi$
$47$	715309.	1.00497	0.502483	+	0.864587i	$0.332420\pi$
$48$	−428161.	−0.558809
$49$	−630575.	−0.765686
$50$	3.08899e6	3.49479
$51$	−898172.	−0.948122
$52$	−116948.	−0.115341
$53$	−1.15458e6	−1.06526	−0.532632	−	0.846347i	$-0.678797\pi$
$53$	−1.15458e6	−1.06526	−0.532632	+	0.846347i	$0.678797\pi$
$54$	317371.	0.274276
$55$	643444.	0.521484
$56$	−28236.9	−0.0214862
$57$	654022.	0.467768
$58$	−425289.	−0.286211
$59$	−1.13651e6	−0.720429	−0.360215	−	0.932869i	$-0.617297\pi$
$59$	−1.13651e6	−0.720429	−0.360215	+	0.932869i	$0.617297\pi$
$60$	−1.85069e6	−1.10613
$61$	699341.	0.394489	0.197244	−	0.980354i	$-0.436801\pi$
$61$	699341.	0.394489	0.197244	+	0.980354i	$0.436801\pi$
$62$	941353.	0.501628
$63$	−320236.	−0.161353
$64$	−2.22569e6	−1.06129
$65$	460157.	0.207830
$66$	−539397.	−0.230943
$67$	−4.85518e6	−1.97217	−0.986083	−	0.166252i	$-0.946833\pi$
$67$	−4.85518e6	−1.97217	−0.986083	+	0.166252i	$0.946833\pi$
$68$	−4.39062e6	−1.69334
$69$	−328509.	−0.120386
$70$	3.67840e6	1.28179
$71$	369628.	0.122563	0.0612817	−	0.998121i	$-0.480481\pi$
$71$	369628.	0.122563	0.0612817	+	0.998121i	$0.480481\pi$
$72$	46860.0	0.0147958
$73$	1.06243e6	0.319646	0.159823	−	0.987146i	$-0.448908\pi$
$73$	1.06243e6	0.319646	0.159823	+	0.987146i	$0.448908\pi$
$74$	1.17214e6	0.336254
$75$	5.17254e6	1.41576
$76$	3.19711e6	0.835431
$77$	544267.	0.135861
$78$	−385748.	−0.0920392
$79$	6.34807e6	1.44860	0.724298	−	0.689487i	$-0.242164\pi$
$79$	6.34807e6	1.44860	0.724298	+	0.689487i	$0.242164\pi$
$80$	8.23540e6	1.79833
$81$	531441.	0.111111
$82$	2.82384e6	0.565577
$83$	9.31819e6	1.78879	0.894393	−	0.447282i	$-0.147608\pi$
$83$	9.31819e6	1.78879	0.894393	+	0.447282i	$0.147608\pi$
$84$	−1.56544e6	−0.288176
$85$	1.72758e7	3.05120
$86$	−4.87366e6	−0.826249
$87$	−712151.	−0.115946
$88$	−79642.3	−0.0124582
$89$	−1.25442e7	−1.88615	−0.943077	−	0.332573i	$-0.892083\pi$
$89$	−1.25442e7	−1.88615	−0.943077	+	0.332573i	$0.892083\pi$
$90$	−6.10441e6	−0.882664
$91$	389231.	0.0541455
$92$	−1.60588e6	−0.215009
$93$	1.57631e6	0.203213
$94$	1.15337e7	1.43226
$95$	−1.25797e7	−1.50535
$96$	−7.12587e6	−0.822031
$97$	−1.21853e7	−1.35562	−0.677808	−	0.735239i	$-0.737070\pi$
$97$	−1.21853e7	−1.35562	−0.677808	+	0.735239i	$0.737070\pi$
$98$	−1.01675e7	−1.09124
$99$	−903228.	−0.0935565
$100$	2.52854e7	2.52854
$101$	4.72794e6	0.456612	0.228306	−	0.973589i	$-0.426681\pi$
$101$	4.72794e6	0.456612	0.228306	+	0.973589i	$0.426681\pi$
$102$	−1.44822e7	−1.35125
$103$	1.88105e7	1.69617	0.848086	−	0.529859i	$-0.177755\pi$
$103$	1.88105e7	1.69617	0.848086	+	0.529859i	$0.177755\pi$
$104$	−56956.0	−0.00496504
$105$	6.15953e6	0.519260
$106$	−1.86165e7	−1.51820
$107$	6.37897e6	0.503393	0.251697	−	0.967806i	$-0.419011\pi$
$107$	6.37897e6	0.503393	0.251697	+	0.967806i	$0.419011\pi$
$108$	2.59789e6	0.198444
$109$	−2.00961e7	−1.48634	−0.743172	−	0.669101i	$-0.766679\pi$
$109$	−2.00961e7	−1.48634	−0.743172	+	0.669101i	$0.766679\pi$
$110$	1.03750e7	0.743210
$111$	1.96276e6	0.136219
$112$	6.96604e6	0.468515
$113$	−2.60610e7	−1.69909	−0.849545	−	0.527516i	$-0.823123\pi$
$113$	−2.60610e7	−1.69909	−0.849545	+	0.527516i	$0.823123\pi$
$114$	1.05455e7	0.666654
$115$	6.31865e6	0.387420
$116$	−3.48127e6	−0.207079
$117$	−645940.	−0.0372857
$118$	−1.83252e7	−1.02674
$119$	1.46130e7	0.794921
$120$	−901320.	−0.0476151
$121$	−1.79521e7	−0.921225
$122$	1.12763e7	0.562219
$123$	4.72856e6	0.229119
$124$	7.70561e6	0.362937
$125$	−5.89180e7	−2.69813
$126$	−5.16352e6	−0.229958
$127$	2.50989e7	1.08728	0.543640	−	0.839319i	$-0.317046\pi$
$127$	2.50989e7	1.08728	0.543640	+	0.839319i	$0.317046\pi$
$128$	−2.10530e6	−0.0887317
$129$	−8.16101e6	−0.334718
$130$	7.41961e6	0.296196
$131$	−3.64373e7	−1.41611	−0.708055	−	0.706157i	$-0.750427\pi$
$131$	−3.64373e7	−1.41611	−0.708055	+	0.706157i	$0.750427\pi$
$132$	−4.41533e6	−0.167092
$133$	−1.06407e7	−0.392184
$134$	−7.82854e7	−2.81070
$135$	−1.02219e7	−0.357573
$136$	−2.13831e6	−0.0728927
$137$	−1.69030e7	−0.561620	−0.280810	−	0.959763i	$-0.590603\pi$
$137$	−1.69030e7	−0.561620	−0.280810	+	0.959763i	$0.590603\pi$
$138$	−5.29691e6	−0.171572
$139$	−8.54424e6	−0.269849	−0.134925	−	0.990856i	$-0.543079\pi$
$139$	−8.54424e6	−0.269849	−0.134925	+	0.990856i	$0.543079\pi$
$140$	3.01102e7	0.927396
$141$	1.93134e7	0.580217
$142$	5.95992e6	0.174675
$143$	1.09783e6	0.0313948
$144$	−1.15604e7	−0.322628
$145$	1.36978e7	0.373131
$146$	1.71307e7	0.455554
$147$	−1.70255e7	−0.442069
$148$	9.59473e6	0.243286
$149$	−3.88029e7	−0.960975	−0.480487	−	0.877002i	$-0.659540\pi$
$149$	−3.88029e7	−0.960975	−0.480487	+	0.877002i	$0.659540\pi$
$150$	8.34026e7	2.01772
$151$	−6.01649e7	−1.42208	−0.711040	−	0.703152i	$-0.751775\pi$
$151$	−6.01649e7	−1.42208	−0.711040	+	0.703152i	$0.751775\pi$
$152$	1.55705e6	0.0359625
$153$	−2.42507e7	−0.547398
$154$	8.77582e6	0.193627
$155$	−3.03192e7	−0.653969
$156$	−3.15761e6	−0.0665920
$157$	5.28204e7	1.08931	0.544657	−	0.838659i	$-0.316660\pi$
$157$	5.28204e7	1.08931	0.544657	+	0.838659i	$0.316660\pi$
$158$	1.02357e8	2.06451
$159$	−3.11736e7	−0.615030
$160$	1.37061e8	2.64542
$161$	5.34473e6	0.100934
$162$	8.56901e6	0.158354
$163$	7.62576e7	1.37920	0.689599	−	0.724192i	$-0.257787\pi$
$163$	7.62576e7	1.37920	0.689599	+	0.724192i	$0.257787\pi$
$164$	2.31150e7	0.409205
$165$	1.73730e7	0.301079
$166$	1.50247e8	2.54935
$167$	−3.06894e7	−0.509895	−0.254947	−	0.966955i	$-0.582058\pi$
$167$	−3.06894e7	−0.509895	−0.254947	+	0.966955i	$0.582058\pi$
$168$	−762396.	−0.0124050
$169$	−6.19634e7	−0.987488
$170$	2.78556e8	4.34852
$171$	1.76586e7	0.270066
$172$	−3.98942e7	−0.597806
$173$	9.31796e6	0.136823	0.0684116	−	0.997657i	$-0.478207\pi$
$173$	9.31796e6	0.136823	0.0684116	+	0.997657i	$0.478207\pi$
$174$	−1.14828e7	−0.165244
$175$	−8.41556e7	−1.18700
$176$	1.96478e7	0.271656
$177$	−3.06858e7	−0.415940
$178$	−2.02264e8	−2.68811
$179$	5.74511e7	0.748708	0.374354	−	0.927286i	$-0.377865\pi$
$179$	5.74511e7	0.748708	0.374354	+	0.927286i	$0.377865\pi$
$180$	−4.99687e7	−0.638623
$181$	−8.31396e7	−1.04216	−0.521078	−	0.853509i	$-0.674470\pi$
$181$	−8.31396e7	−1.04216	−0.521078	+	0.853509i	$0.674470\pi$
$182$	6.27600e6	0.0771672
$183$	1.88822e7	0.227758
$184$	−782092.	−0.00925541
$185$	−3.77524e7	−0.438372
$186$	2.54165e7	0.289615
$187$	4.12160e7	0.460914
$188$	9.44112e7	1.03627
$189$	−8.64637e6	−0.0931574
$190$	−2.02836e8	−2.14539
$191$	1.24826e7	0.129625	0.0648124	−	0.997897i	$-0.479355\pi$
$191$	1.24826e7	0.129625	0.0648124	+	0.997897i	$0.479355\pi$
$192$	−6.00935e7	−0.612736
$193$	1.10312e8	1.10452	0.552259	−	0.833672i	$-0.313766\pi$
$193$	1.10312e8	1.10452	0.552259	+	0.833672i	$0.313766\pi$
$194$	−1.96478e8	−1.93200
$195$	1.24242e7	0.119991
$196$	−8.32274e7	−0.789533
$197$	7.99268e7	0.744836	0.372418	−	0.928065i	$-0.378529\pi$
$197$	7.99268e7	0.744836	0.372418	+	0.928065i	$0.378529\pi$
$198$	−1.45637e7	−0.133335
$199$	8.75763e7	0.787773	0.393886	−	0.919159i	$-0.371130\pi$
$199$	8.75763e7	0.787773	0.393886	+	0.919159i	$0.371130\pi$
$200$	1.23144e7	0.108845
$201$	−1.31090e8	−1.13863
$202$	7.62338e7	0.650756
$203$	1.15865e7	0.0972109
$204$	−1.18547e8	−0.977651
$205$	−9.09507e7	−0.737340
$206$	3.03302e8	2.41735
$207$	−8.86974e6	−0.0695048
$208$	1.40510e7	0.108265
$209$	−3.00122e7	−0.227398
$210$	9.93168e7	0.740040
$211$	−1.01709e8	−0.745366	−0.372683	−	0.927959i	$-0.621562\pi$
$211$	−1.01709e8	−0.745366	−0.372683	+	0.927959i	$0.621562\pi$
$212$	−1.52389e8	−1.09844
$213$	9.97996e6	0.0707620
$214$	1.02855e8	0.717427
$215$	1.56972e8	1.07718
$216$	1.26522e6	0.00854235
$217$	−2.56460e7	−0.170377
$218$	−3.24032e8	−2.11831
$219$	2.86856e7	0.184548
$220$	8.49260e7	0.537726
$221$	2.94755e7	0.183691
$222$	3.16477e7	0.194136
$223$	−1.43993e8	−0.869508	−0.434754	−	0.900549i	$-0.643165\pi$
$223$	−1.43993e8	−0.869508	−0.434754	+	0.900549i	$0.643165\pi$
$224$	1.15936e8	0.689205
$225$	1.39659e8	0.817390
$226$	−4.20210e8	−2.42151
$227$	1.92424e8	1.09187	0.545933	−	0.837829i	$-0.316175\pi$
$227$	1.92424e8	1.09187	0.545933	+	0.837829i	$0.316175\pi$
$228$	8.63221e7	0.482336
$229$	6.02604e7	0.331595	0.165797	−	0.986160i	$-0.446980\pi$
$229$	6.02604e7	0.331595	0.165797	+	0.986160i	$0.446980\pi$
$230$	1.01883e8	0.552144
$231$	1.46952e7	0.0784394
$232$	−1.69544e6	−0.00891405
$233$	−2.88983e8	−1.49667	−0.748336	−	0.663320i	$-0.769147\pi$
$233$	−2.88983e8	−1.49667	−0.748336	+	0.663320i	$0.769147\pi$
$234$	−1.04152e7	−0.0531389
$235$	−3.71480e8	−1.86723
$236$	−1.50004e8	−0.742867
$237$	1.71398e8	0.836347
$238$	2.35621e8	1.13291
$239$	137677.	0.000652333 0	0.000326167	−	1.00000i	$-0.499896\pi$
$239$	137677.	0.000652333 0	0.000326167		1.00000i	$0.499896\pi$
$240$	2.22356e8	1.03827
$241$	3.89474e8	1.79233	0.896167	−	0.443717i	$-0.146340\pi$
$241$	3.89474e8	1.79233	0.896167	+	0.443717i	$0.146340\pi$
$242$	−2.89461e8	−1.31291
$243$	1.43489e7	0.0641500
$244$	9.23037e7	0.406775
$245$	3.27475e8	1.42265
$246$	7.62438e7	0.326536
$247$	−2.14631e7	−0.0906262
$248$	3.75277e6	0.0156232
$249$	2.51591e8	1.03276
$250$	−9.50000e8	−3.84533
$251$	−3.07150e8	−1.22600	−0.613002	−	0.790081i	$-0.710038\pi$
$251$	−3.07150e8	−1.22600	−0.613002	+	0.790081i	$0.710038\pi$
$252$	−4.22668e7	−0.166379
$253$	1.50749e7	0.0585236
$254$	4.04697e8	1.54957
$255$	4.66445e8	1.76161
$256$	2.50942e8	0.934831
$257$	2.58814e7	0.0951092	0.0475546	−	0.998869i	$-0.484857\pi$
$257$	2.58814e7	0.0951092	0.0475546	+	0.998869i	$0.484857\pi$
$258$	−1.31589e8	−0.477035
$259$	−3.19334e7	−0.114208
$260$	6.07345e7	0.214303
$261$	−1.92281e7	−0.0669413
$262$	−5.87519e8	−2.01821
$263$	−5.03935e8	−1.70816	−0.854082	−	0.520139i	$-0.825880\pi$
$263$	−5.03935e8	−1.70816	−0.854082	+	0.520139i	$0.825880\pi$
$264$	−2.15034e6	−0.00719273
$265$	5.99603e8	1.97926
$266$	−1.71572e8	−0.558934
$267$	−3.38693e8	−1.08897
$268$	−6.40819e8	−2.03359
$269$	−3.95543e8	−1.23897	−0.619486	−	0.785008i	$-0.712659\pi$
$269$	−3.95543e8	−1.23897	−0.619486	+	0.785008i	$0.712659\pi$
$270$	−1.64819e8	−0.509606
$271$	2.90728e8	0.887348	0.443674	−	0.896188i	$-0.353675\pi$
$271$	2.90728e8	0.887348	0.443674	+	0.896188i	$0.353675\pi$
$272$	5.27521e8	1.58946
$273$	1.05092e7	0.0312609
$274$	−2.72546e8	−0.800411
$275$	−2.37361e8	−0.688249
$276$	−4.33588e7	−0.124135
$277$	6.25916e8	1.76944	0.884722	−	0.466120i	$-0.154348\pi$
$277$	6.25916e8	1.76944	0.884722	+	0.466120i	$0.154348\pi$
$278$	−1.37768e8	−0.384585
$279$	4.25603e7	0.117325
$280$	1.46642e7	0.0399213
$281$	−4.60295e8	−1.23755	−0.618777	−	0.785566i	$-0.712372\pi$
$281$	−4.60295e8	−1.23755	−0.618777	+	0.785566i	$0.712372\pi$
$282$	3.11410e8	0.826916
$283$	8.37117e7	0.219550	0.109775	−	0.993956i	$-0.464987\pi$
$283$	8.37117e7	0.219550	0.109775	+	0.993956i	$0.464987\pi$
$284$	4.87860e7	0.126381
$285$	−3.39651e8	−0.869113
$286$	1.77015e7	0.0447434
$287$	−7.69321e7	−0.192097
$288$	−1.92398e8	−0.474600
$289$	6.96264e8	1.69680
$290$	2.20864e8	0.531780
$291$	−3.29004e8	−0.782665
$292$	1.40226e8	0.329602
$293$	5.37739e8	1.24892	0.624461	−	0.781056i	$-0.285319\pi$
$293$	5.37739e8	1.24892	0.624461	+	0.781056i	$0.285319\pi$
$294$	−2.74521e8	−0.630029
$295$	5.90221e8	1.33856
$296$	4.67280e6	0.0104726
$297$	−2.43871e7	−0.0540149
$298$	−6.25662e8	−1.36956
$299$	1.07807e7	0.0233238
$300$	6.82706e8	1.45985
$301$	1.32777e8	0.280634
$302$	−9.70105e8	−2.02672
$303$	1.27654e8	0.263625
$304$	−3.84125e8	−0.784178
$305$	−3.63187e8	−0.732961
$306$	−3.91020e8	−0.780142
$307$	1.42374e7	0.0280832	0.0140416	−	0.999901i	$-0.495530\pi$
$307$	1.42374e7	0.0280832	0.0140416	+	0.999901i	$0.495530\pi$
$308$	7.18360e7	0.140092
$309$	5.07883e8	0.979285
$310$	−4.88870e8	−0.932025
$311$	2.30383e8	0.434300	0.217150	−	0.976138i	$-0.430324\pi$
$311$	2.30383e8	0.434300	0.217150	+	0.976138i	$0.430324\pi$
$312$	−1.53781e6	−0.00286657
$313$	2.77390e8	0.511312	0.255656	−	0.966768i	$-0.417709\pi$
$313$	2.77390e8	0.511312	0.255656	+	0.966768i	$0.417709\pi$
$314$	8.51681e8	1.55247
$315$	1.66307e8	0.299795
$316$	8.37860e8	1.49371
$317$	−6.75974e8	−1.19185	−0.595926	−	0.803039i	$-0.703215\pi$
$317$	−6.75974e8	−1.19185	−0.595926	+	0.803039i	$0.703215\pi$
$318$	−5.02646e8	−0.876531
$319$	3.26797e7	0.0563652
$320$	1.15586e9	1.97188
$321$	1.72232e8	0.290634
$322$	8.61790e7	0.143849
$323$	−8.05794e8	−1.33050
$324$	7.01431e7	0.114572
$325$	−1.69748e8	−0.274292
$326$	1.22958e9	1.96561
$327$	−5.42595e8	−0.858141
$328$	1.12574e7	0.0176149
$329$	−3.14222e8	−0.486464
$330$	2.80124e8	0.429093
$331$	−1.06087e9	−1.60792	−0.803958	−	0.594687i	$-0.797276\pi$
$331$	−1.06087e9	−1.60792	−0.803958	+	0.594687i	$0.797276\pi$
$332$	1.22988e9	1.84450
$333$	5.29945e7	0.0786458
$334$	−4.94839e8	−0.726693
$335$	2.52143e9	3.66429
$336$	1.88083e8	0.270497
$337$	6.27245e8	0.892755	0.446378	−	0.894845i	$-0.352714\pi$
$337$	6.27245e8	0.892755	0.446378	+	0.894845i	$0.352714\pi$
$338$	−9.99104e8	−1.40735
$339$	−7.03647e8	−0.980970
$340$	2.28017e9	3.14623
$341$	−7.23347e7	−0.0987885
$342$	2.84729e8	0.384893
$343$	6.38767e8	0.854698
$344$	−1.94292e7	−0.0257335
$345$	1.70604e8	0.223677
$346$	1.50244e8	0.194998
$347$	2.38728e8	0.306726	0.153363	−	0.988170i	$-0.450990\pi$
$347$	2.38728e8	0.306726	0.153363	+	0.988170i	$0.450990\pi$
$348$	−9.39944e7	−0.119557
$349$	−4.28787e8	−0.539949	−0.269974	−	0.962867i	$-0.587015\pi$
$349$	−4.28787e8	−0.539949	−0.269974	+	0.962867i	$0.587015\pi$
$350$	−1.35693e9	−1.69169
$351$	−1.74404e7	−0.0215269
$352$	3.26997e8	0.399617
$353$	−5.91768e8	−0.716045	−0.358022	−	0.933713i	$-0.616549\pi$
$353$	−5.91768e8	−0.716045	−0.358022	+	0.933713i	$0.616549\pi$
$354$	−4.94781e8	−0.592791
$355$	−1.91958e8	−0.227723
$356$	−1.65566e9	−1.94490
$357$	3.94550e8	0.458948
$358$	9.26346e8	1.06705
$359$	3.75613e8	0.428460	0.214230	−	0.976783i	$-0.431276\pi$
$359$	3.75613e8	0.428460	0.214230	+	0.976783i	$0.431276\pi$
$360$	−2.43356e7	−0.0274906
$361$	−3.07117e8	−0.343580
$362$	−1.34055e9	−1.48526
$363$	−4.84706e8	−0.531869
$364$	5.13733e7	0.0558319
$365$	−5.51748e8	−0.593903
$366$	3.04459e8	0.324597
$367$	6.77196e8	0.715127	0.357563	−	0.933889i	$-0.383608\pi$
$367$	6.77196e8	0.715127	0.357563	+	0.933889i	$0.383608\pi$
$368$	1.92942e8	0.201818
$369$	1.27671e8	0.132282
$370$	−6.08723e8	−0.624760
$371$	5.07184e8	0.515652
$372$	2.08051e8	0.209542
$373$	−5.35696e8	−0.534488	−0.267244	−	0.963629i	$-0.586113\pi$
$373$	−5.35696e8	−0.534488	−0.267244	+	0.963629i	$0.586113\pi$
$374$	6.64571e8	0.656887
$375$	−1.59079e9	−1.55777
$376$	4.59799e7	0.0446078
$377$	2.33708e7	0.0224636
$378$	−1.39415e8	−0.132766
$379$	−2.59380e8	−0.244737	−0.122368	−	0.992485i	$-0.539049\pi$
$379$	−2.59380e8	−0.244737	−0.122368	+	0.992485i	$0.539049\pi$
$380$	−1.66035e9	−1.55223
$381$	6.77669e8	0.627741
$382$	2.01271e8	0.184739
$383$	1.26184e9	1.14765	0.573824	−	0.818979i	$-0.305459\pi$
$383$	1.26184e9	1.14765	0.573824	+	0.818979i	$0.305459\pi$
$384$	−5.68430e7	−0.0512293
$385$	−2.82653e8	−0.252430
$386$	1.77869e9	1.57414
$387$	−2.20347e8	−0.193250
$388$	−1.60830e9	−1.39784
$389$	2.21921e9	1.91150	0.955749	−	0.294182i	$-0.0950471\pi$
$389$	2.21921e9	1.91150	0.955749	+	0.294182i	$0.0950471\pi$
$390$	2.00330e8	0.171009
$391$	4.04743e8	0.342421
$392$	−4.05332e7	−0.0339868
$393$	−9.83808e8	−0.817591
$394$	1.28875e9	1.06153
$395$	−3.29673e9	−2.69149
$396$	−1.19214e8	−0.0964703
$397$	−1.73593e9	−1.39241	−0.696203	−	0.717845i	$-0.745129\pi$
$397$	−1.73593e9	−1.39241	−0.696203	+	0.717845i	$0.745129\pi$
$398$	1.41209e9	1.12272
$399$	−2.87299e8	−0.226428
$400$	−3.03797e9	−2.37342
$401$	4.16615e8	0.322648	0.161324	−	0.986901i	$-0.448424\pi$
$401$	4.16615e8	0.322648	0.161324	+	0.986901i	$0.448424\pi$
$402$	−2.11371e9	−1.62276
$403$	−5.17299e7	−0.0393708
$404$	6.24025e8	0.470833
$405$	−2.75992e8	−0.206445
$406$	1.86821e8	0.138543
$407$	−9.00684e7	−0.0662205
$408$	−5.77343e7	−0.0420846
$409$	1.78390e9	1.28925	0.644626	−	0.764498i	$-0.277013\pi$
$409$	1.78390e9	1.28925	0.644626	+	0.764498i	$0.277013\pi$
$410$	−1.46650e9	−1.05084
$411$	−4.56382e8	−0.324251
$412$	2.48273e9	1.74900
$413$	4.99248e8	0.348731
$414$	−1.43017e8	−0.0990570
$415$	−4.83919e9	−3.32357
$416$	2.33851e8	0.159262
$417$	−2.30694e8	−0.155798
$418$	−4.83920e8	−0.324083
$419$	2.49057e9	1.65405	0.827026	−	0.562163i	$-0.190031\pi$
$419$	2.49057e9	1.65405	0.827026	+	0.562163i	$0.190031\pi$
$420$	8.12975e8	0.535432
$421$	6.45056e8	0.421318	0.210659	−	0.977560i	$-0.432439\pi$
$421$	6.45056e8	0.421318	0.210659	+	0.977560i	$0.432439\pi$
$422$	−1.63996e9	−1.06228
$423$	5.21460e8	0.334989
$424$	−7.42159e7	−0.0472843
$425$	−6.37289e9	−4.02694
$426$	1.60918e8	0.100849
$427$	−3.07208e8	−0.190956
$428$	8.41938e8	0.519071
$429$	2.96414e7	0.0181258
$430$	2.53102e9	1.53517
$431$	1.98666e8	0.119523	0.0597617	−	0.998213i	$-0.480966\pi$
$431$	1.98666e8	0.119523	0.0597617	+	0.998213i	$0.480966\pi$
$432$	−3.12130e8	−0.186270
$433$	−2.95758e8	−0.175077	−0.0875385	−	0.996161i	$-0.527900\pi$
$433$	−2.95758e8	−0.175077	−0.0875385	+	0.996161i	$0.527900\pi$
$434$	−4.13519e8	−0.242818
$435$	3.69840e8	0.215427
$436$	−2.65242e9	−1.53264
$437$	−2.94722e8	−0.168938
$438$	4.62529e8	0.263014
$439$	−6.11806e6	−0.00345134	−0.00172567	−	0.999999i	$-0.500549\pi$
$439$	−6.11806e6	−0.00345134	−0.00172567	+	0.999999i	$0.500549\pi$
$440$	4.13604e7	0.0231473
$441$	−4.59689e8	−0.255229
$442$	4.75265e8	0.261793
$443$	1.76550e9	0.964841	0.482421	−	0.875940i	$-0.339758\pi$
$443$	1.76550e9	0.964841	0.482421	+	0.875940i	$0.339758\pi$
$444$	2.59058e8	0.140461
$445$	6.51453e9	3.50448
$446$	−2.32175e9	−1.23921
$447$	−1.04768e9	−0.554819
$448$	9.77702e8	0.513728
$449$	3.77930e9	1.97037	0.985187	−	0.171484i	$-0.0548562\pi$
$449$	3.77930e9	1.97037	0.985187	+	0.171484i	$0.0548562\pi$
$450$	2.25187e9	1.16493
$451$	−2.16988e8	−0.111382
$452$	−3.43970e9	−1.75201
$453$	−1.62445e9	−0.821038
$454$	3.10267e9	1.55611
$455$	−2.02138e8	−0.100602
$456$	4.20404e7	0.0207630
$457$	1.20572e9	0.590933	0.295467	−	0.955353i	$-0.404525\pi$
$457$	1.20572e9	0.590933	0.295467	+	0.955353i	$0.404525\pi$
$458$	9.71645e8	0.472583
$459$	−6.54768e8	−0.316041
$460$	8.33977e8	0.399486
$461$	−7.73586e8	−0.367752	−0.183876	−	0.982949i	$-0.558865\pi$
$461$	−7.73586e8	−0.367752	−0.183876	+	0.982949i	$0.558865\pi$
$462$	2.36947e8	0.111790
$463$	1.53598e8	0.0719204	0.0359602	−	0.999353i	$-0.488551\pi$
$463$	1.53598e8	0.0719204	0.0359602	+	0.999353i	$0.488551\pi$
$464$	4.18266e8	0.194375
$465$	−8.18619e8	−0.377569
$466$	−4.65959e9	−2.13303
$467$	−3.04134e9	−1.38184	−0.690918	−	0.722933i	$-0.742794\pi$
$467$	−3.04134e9	−1.38184	−0.690918	+	0.722933i	$0.742794\pi$
$468$	−8.52554e7	−0.0384469
$469$	2.13279e9	0.954647
$470$	−5.98977e9	−2.66114
$471$	1.42615e9	0.628915
$472$	−7.30546e7	−0.0319780
$473$	3.74498e8	0.162718
$474$	2.76364e9	1.19195
$475$	4.64054e9	1.98674
$476$	1.92872e9	0.819679
$477$	−8.41686e8	−0.355088
$478$	2.21992e6	0.000929694 0
$479$	2.70620e9	1.12509	0.562543	−	0.826768i	$-0.309823\pi$
$479$	2.70620e9	1.12509	0.562543	+	0.826768i	$0.309823\pi$
$480$	3.70066e9	1.52734
$481$	−6.44122e7	−0.0263913
$482$	6.27992e9	2.55440
$483$	1.44308e8	0.0582740
$484$	−2.36943e9	−0.949916
$485$	6.32818e9	2.51874
$486$	2.31363e8	0.0914255
$487$	3.18750e8	0.125054	0.0625272	−	0.998043i	$-0.480084\pi$
$487$	3.18750e8	0.125054	0.0625272	+	0.998043i	$0.480084\pi$
$488$	4.49535e7	0.0175103
$489$	2.05895e9	0.796280
$490$	5.28024e9	2.02753
$491$	1.69191e9	0.645047	0.322524	−	0.946561i	$-0.395469\pi$
$491$	1.69191e9	0.645047	0.322524	+	0.946561i	$0.395469\pi$
$492$	6.24106e8	0.236255
$493$	8.77414e8	0.329792
$494$	−3.46074e8	−0.129159
$495$	4.69071e8	0.173828
$496$	−9.25808e8	−0.340671
$497$	−1.62371e8	−0.0593281
$498$	4.05668e9	1.47187
$499$	−7.59141e8	−0.273508	−0.136754	−	0.990605i	$-0.543667\pi$
$499$	−7.59141e8	−0.273508	−0.136754	+	0.990605i	$0.543667\pi$
$500$	−7.77639e9	−2.78217
$501$	−8.28613e8	−0.294388
$502$	−4.95251e9	−1.74728
$503$	−2.36181e9	−0.827477	−0.413739	−	0.910396i	$-0.635777\pi$
$503$	−2.36181e9	−0.827477	−0.413739	+	0.910396i	$0.635777\pi$
$504$	−2.05847e7	−0.00716205
$505$	−2.45535e9	−0.848386
$506$	2.43068e8	0.0834069
$507$	−1.67301e9	−0.570126
$508$	3.31271e9	1.12114
$509$	8.81763e8	0.296374	0.148187	−	0.988959i	$-0.452656\pi$
$509$	8.81763e8	0.296374	0.148187	+	0.988959i	$0.452656\pi$
$510$	7.52101e9	2.51062
$511$	−4.66705e8	−0.154728
$512$	4.31569e9	1.42104
$513$	4.76782e8	0.155923
$514$	4.17315e8	0.135548
$515$	−9.76880e9	−3.15149
$516$	−1.07714e9	−0.345143
$517$	−8.86265e8	−0.282063
$518$	−5.14898e8	−0.162767
$519$	2.51585e8	0.0789949
$520$	2.95788e7	0.00922505
$521$	−1.05646e9	−0.327281	−0.163640	−	0.986520i	$-0.552324\pi$
$521$	−1.05646e9	−0.327281	−0.163640	+	0.986520i	$0.552324\pi$
$522$	−3.10036e8	−0.0954036
$523$	−3.04696e9	−0.931346	−0.465673	−	0.884957i	$-0.654188\pi$
$523$	−3.04696e9	−0.931346	−0.465673	+	0.884957i	$0.654188\pi$
$524$	−4.80924e9	−1.46021
$525$	−2.27220e9	−0.685313
$526$	−8.12550e9	−2.43444
$527$	−1.94211e9	−0.578011
$528$	5.30490e8	0.156841
$529$	1.48036e8	0.0434783
$530$	9.66806e9	2.82081
$531$	−8.28516e8	−0.240143
$532$	−1.40443e9	−0.404399
$533$	−1.55178e8	−0.0443900
$534$	−5.46112e9	−1.55198
$535$	−3.31277e9	−0.935305
$536$	−3.12090e8	−0.0875393
$537$	1.55118e9	0.432267
$538$	−6.37778e9	−1.76576
$539$	7.81280e8	0.214905
$540$	−1.34916e9	−0.368709
$541$	−4.51721e9	−1.22653	−0.613267	−	0.789875i	$-0.710145\pi$
$541$	−4.51721e9	−1.22653	−0.613267	+	0.789875i	$0.710145\pi$
$542$	4.68772e9	1.26463
$543$	−2.24477e9	−0.601689
$544$	8.77950e9	2.33816
$545$	1.04365e10	2.76163
$546$	1.69452e8	0.0445525
$547$	−3.07485e9	−0.803283	−0.401641	−	0.915797i	$-0.631560\pi$
$547$	−3.07485e9	−0.803283	−0.401641	+	0.915797i	$0.631560\pi$
$548$	−2.23097e9	−0.579112
$549$	5.09820e8	0.131496
$550$	−3.82724e9	−0.980881
$551$	−6.38906e8	−0.162707
$552$	−2.11165e7	−0.00534361
$553$	−2.78859e9	−0.701207
$554$	1.00923e10	2.52178
$555$	−1.01931e9	−0.253094
$556$	−1.12772e9	−0.278254
$557$	−4.09467e9	−1.00398	−0.501991	−	0.864873i	$-0.667399\pi$
$557$	−4.09467e9	−1.00398	−0.501991	+	0.864873i	$0.667399\pi$
$558$	6.86246e8	0.167209
$559$	2.67821e8	0.0648490
$560$	−3.61766e9	−0.870501
$561$	1.11283e9	0.266109
$562$	−7.42185e9	−1.76374
$563$	−8.37874e8	−0.197879	−0.0989394	−	0.995093i	$-0.531545\pi$
$563$	−8.37874e8	−0.197879	−0.0989394	+	0.995093i	$0.531545\pi$
$564$	2.54910e9	0.598288
$565$	1.35342e10	3.15691
$566$	1.34978e9	0.312899
$567$	−2.33452e8	−0.0537845
$568$	2.37596e7	0.00544027
$569$	−5.29170e9	−1.20421	−0.602106	−	0.798417i	$-0.705671\pi$
$569$	−5.29170e9	−1.20421	−0.602106	+	0.798417i	$0.705671\pi$
$570$	−5.47657e9	−1.23864
$571$	−2.99442e9	−0.673110	−0.336555	−	0.941664i	$-0.609262\pi$
$571$	−2.99442e9	−0.673110	−0.336555	+	0.941664i	$0.609262\pi$
$572$	1.44899e8	0.0323726
$573$	3.37030e8	0.0748389
$574$	−1.24046e9	−0.273774
$575$	−2.33090e9	−0.511313
$576$	−1.62253e9	−0.353763
$577$	−3.24689e9	−0.703642	−0.351821	−	0.936067i	$-0.614437\pi$
$577$	−3.24689e9	−0.703642	−0.351821	+	0.936067i	$0.614437\pi$
$578$	1.12266e10	2.41826
$579$	2.97843e9	0.637694
$580$	1.80792e9	0.384752
$581$	−4.09331e9	−0.865880
$582$	−5.30490e9	−1.11544
$583$	1.43052e9	0.298987
$584$	6.82927e7	0.0141883
$585$	3.35454e8	0.0692768
$586$	8.67056e9	1.77994
$587$	−3.25912e9	−0.665069	−0.332534	−	0.943091i	$-0.607904\pi$
$587$	−3.25912e9	−0.665069	−0.332534	+	0.943091i	$0.607904\pi$
$588$	−2.24714e9	−0.455837
$589$	1.41418e9	0.285169
$590$	9.51678e9	1.90769
$591$	2.15802e9	0.430031
$592$	−1.15278e9	−0.228360
$593$	3.83139e9	0.754510	0.377255	−	0.926109i	$-0.376868\pi$
$593$	3.83139e9	0.754510	0.377255	+	0.926109i	$0.376868\pi$
$594$	−3.93221e8	−0.0769811
$595$	−7.58891e9	−1.47696
$596$	−5.12146e9	−0.990905
$597$	2.36456e9	0.454821
$598$	1.73830e8	0.0332407
$599$	−2.13369e8	−0.0405637	−0.0202818	−	0.999794i	$-0.506456\pi$
$599$	−2.13369e8	−0.0405637	−0.0202818	+	0.999794i	$0.506456\pi$
$600$	3.32490e8	0.0628419
$601$	−8.28292e9	−1.55641	−0.778203	−	0.628013i	$-0.783869\pi$
$601$	−8.28292e9	−1.55641	−0.778203	+	0.628013i	$0.783869\pi$
$602$	2.14091e9	0.399954
$603$	−3.53943e9	−0.657389
$604$	−7.94096e9	−1.46637
$605$	9.32299e9	1.71164
$606$	2.05831e9	0.375714
$607$	7.09347e9	1.28735	0.643677	−	0.765297i	$-0.277408\pi$
$607$	7.09347e9	1.28735	0.643677	+	0.765297i	$0.277408\pi$
$608$	−6.39296e9	−1.15356
$609$	3.12835e8	0.0561247
$610$	−5.85606e9	−1.04460
$611$	−6.33809e8	−0.112412
$612$	−3.20076e9	−0.564447
$613$	−7.47850e9	−1.31130	−0.655651	−	0.755064i	$-0.727606\pi$
$613$	−7.47850e9	−1.31130	−0.655651	+	0.755064i	$0.727606\pi$
$614$	2.29565e8	0.0400237
$615$	−2.45567e9	−0.425703
$616$	3.49854e7	0.00603051
$617$	5.98175e9	1.02525	0.512626	−	0.858612i	$-0.328673\pi$
$617$	5.98175e9	1.02525	0.512626	+	0.858612i	$0.328673\pi$
$618$	8.18916e9	1.39566
$619$	4.19998e9	0.711755	0.355877	−	0.934533i	$-0.384182\pi$
$619$	4.19998e9	0.711755	0.355877	+	0.934533i	$0.384182\pi$
$620$	−4.00173e9	−0.674337
$621$	−2.39483e8	−0.0401286
$622$	3.71473e9	0.618957
$623$	5.51042e9	0.913013
$624$	3.79378e8	0.0625066
$625$	1.56309e10	2.56096
$626$	4.47267e9	0.728713
$627$	−8.10330e8	−0.131288
$628$	6.97158e9	1.12324
$629$	−2.41824e9	−0.387455
$630$	2.68155e9	0.427262
$631$	−1.03015e10	−1.63229	−0.816146	−	0.577845i	$-0.803894\pi$
$631$	−1.03015e10	−1.63229	−0.816146	+	0.577845i	$0.803894\pi$
$632$	4.08053e8	0.0642993
$633$	−2.74614e9	−0.430337
$634$	−1.08995e10	−1.69861
$635$	−1.30345e10	−2.02017
$636$	−4.11449e9	−0.634186
$637$	5.58730e8	0.0856473
$638$	5.26931e8	0.0803307
$639$	2.69459e8	0.0408545
$640$	1.09334e9	0.164864
$641$	−1.07358e9	−0.161003	−0.0805013	−	0.996755i	$-0.525652\pi$
$641$	−1.07358e9	−0.161003	−0.0805013	+	0.996755i	$0.525652\pi$
$642$	2.77709e9	0.414207
$643$	−3.34469e8	−0.0496155	−0.0248078	−	0.999692i	$-0.507897\pi$
$643$	−3.34469e8	−0.0496155	−0.0248078	+	0.999692i	$0.507897\pi$
$644$	7.05433e8	0.104077
$645$	4.23823e9	0.621907
$646$	−1.29927e10	−1.89621
$647$	−6.21703e9	−0.902440	−0.451220	−	0.892413i	$-0.649011\pi$
$647$	−6.21703e9	−0.902440	−0.451220	+	0.892413i	$0.649011\pi$
$648$	3.41609e7	0.00493193
$649$	1.40813e9	0.202203
$650$	−2.73704e9	−0.390916
$651$	−6.92442e8	−0.0983671
$652$	1.00650e10	1.42215
$653$	1.19060e10	1.67329	0.836645	−	0.547745i	$-0.184514\pi$
$653$	1.19060e10	1.67329	0.836645	+	0.547745i	$0.184514\pi$
$654$	−8.74885e9	−1.22301
$655$	1.89229e10	2.63113
$656$	−2.77721e9	−0.384101
$657$	7.74511e8	0.106549
$658$	−5.06654e9	−0.693300
$659$	−9.67726e9	−1.31721	−0.658603	−	0.752491i	$-0.728852\pi$
$659$	−9.67726e9	−1.31721	−0.658603	+	0.752491i	$0.728852\pi$
$660$	2.29300e9	0.310456
$661$	6.44298e9	0.867724	0.433862	−	0.900979i	$-0.357151\pi$
$661$	6.44298e9	0.867724	0.433862	+	0.900979i	$0.357151\pi$
$662$	−1.71055e10	−2.29157
$663$	7.95838e8	0.106054
$664$	5.98971e8	0.0793995
$665$	5.52601e9	0.728679
$666$	8.54488e8	0.112085
$667$	3.20917e8	0.0418747
$668$	−4.05059e9	−0.525775
$669$	−3.88781e9	−0.502011
$670$	4.06557e10	5.22228
$671$	−8.66481e8	−0.110721
$672$	3.13026e9	0.397913
$673$	−4.78618e9	−0.605252	−0.302626	−	0.953109i	$-0.597863\pi$
$673$	−4.78618e9	−0.605252	−0.302626	+	0.953109i	$0.597863\pi$
$674$	1.01138e10	1.27234
$675$	3.77078e9	0.471920
$676$	−8.17834e9	−1.01824
$677$	−2.69329e9	−0.333598	−0.166799	−	0.985991i	$-0.553343\pi$
$677$	−2.69329e9	−0.333598	−0.166799	+	0.985991i	$0.553343\pi$
$678$	−1.13457e10	−1.39806
$679$	5.35279e9	0.656200
$680$	1.11048e9	0.135435
$681$	5.19546e9	0.630389
$682$	−1.16633e9	−0.140792
$683$	−5.44458e9	−0.653871	−0.326935	−	0.945047i	$-0.606016\pi$
$683$	−5.44458e9	−0.653871	−0.326935	+	0.945047i	$0.606016\pi$
$684$	2.33070e9	0.278477
$685$	8.77820e9	1.04349
$686$	1.02995e10	1.21810
$687$	1.62703e9	0.191446
$688$	4.79318e9	0.561131
$689$	1.02303e9	0.119157
$690$	2.75083e9	0.318781
$691$	−1.35606e10	−1.56353	−0.781763	−	0.623576i	$-0.785679\pi$
$691$	−1.35606e10	−1.56353	−0.781763	+	0.623576i	$0.785679\pi$
$692$	1.22985e9	0.141084
$693$	3.96771e8	0.0452870
$694$	3.84927e9	0.437140
$695$	4.43725e9	0.501381
$696$	−4.57769e7	−0.00514653
$697$	−5.82587e9	−0.651698
$698$	−6.91380e9	−0.769526
$699$	−7.80254e9	−0.864104
$700$	−1.11074e10	−1.22397
$701$	−1.21383e10	−1.33090	−0.665450	−	0.746443i	$-0.731760\pi$
$701$	−1.21383e10	−1.33090	−0.665450	+	0.746443i	$0.731760\pi$
$702$	−2.81211e8	−0.0306797
$703$	1.76089e9	0.191156
$704$	2.75761e9	0.297872
$705$	−1.00299e10	−1.07804
$706$	−9.54173e9	−1.02049
$707$	−2.07690e9	−0.221028
$708$	−4.05011e9	−0.428895
$709$	−3.91398e9	−0.412436	−0.206218	−	0.978506i	$-0.566116\pi$
$709$	−3.91398e9	−0.412436	−0.206218	+	0.978506i	$0.566116\pi$
$710$	−3.09515e9	−0.324547
$711$	4.62775e9	0.482865
$712$	−8.06337e8	−0.0837214
$713$	−7.10331e8	−0.0733918
$714$	6.36177e9	0.654085
$715$	−5.70132e8	−0.0583317
$716$	7.58277e9	0.772026
$717$	3.71729e6	0.000376625 0
$718$	6.05642e9	0.610634
$719$	8.59067e8	0.0861938	0.0430969	−	0.999071i	$-0.486278\pi$
$719$	8.59067e8	0.0861938	0.0430969	+	0.999071i	$0.486278\pi$
$720$	6.00361e9	0.599444
$721$	−8.26309e9	−0.821049
$722$	−4.95198e9	−0.489665
$723$	1.05158e10	1.03480
$724$	−1.09733e10	−1.07461
$725$	−5.05300e9	−0.492454
$726$	−7.81544e9	−0.758011
$727$	3.24989e9	0.313688	0.156844	−	0.987623i	$-0.449868\pi$
$727$	3.24989e9	0.313688	0.156844	+	0.987623i	$0.449868\pi$
$728$	2.50197e7	0.00240338
$729$	3.87420e8	0.0370370
$730$	−8.89644e9	−0.846421
$731$	1.00549e10	0.952062
$732$	2.49220e9	0.234852
$733$	−1.31932e9	−0.123733	−0.0618666	−	0.998084i	$-0.519705\pi$
$733$	−1.31932e9	−0.123733	−0.0618666	+	0.998084i	$0.519705\pi$
$734$	1.09192e10	1.01919
$735$	8.84182e9	0.821364
$736$	3.21113e9	0.296883
$737$	6.01555e9	0.553527
$738$	2.05858e9	0.188526
$739$	6.75718e9	0.615900	0.307950	−	0.951403i	$-0.400357\pi$
$739$	6.75718e9	0.615900	0.307950	+	0.951403i	$0.400357\pi$
$740$	−4.98280e9	−0.452025
$741$	−5.79505e8	−0.0523231
$742$	8.17788e9	0.734898
$743$	8.31567e8	0.0743766	0.0371883	−	0.999308i	$-0.488160\pi$
$743$	8.31567e8	0.0743766	0.0371883	+	0.999308i	$0.488160\pi$
$744$	1.01325e8	0.00902007
$745$	2.01514e10	1.78549
$746$	−8.63762e9	−0.761743
$747$	6.79296e9	0.596262
$748$	5.43996e9	0.475269
$749$	−2.80216e9	−0.243673
$750$	−2.56500e10	−2.22010
$751$	−1.24613e10	−1.07355	−0.536777	−	0.843724i	$-0.680358\pi$
$751$	−1.24613e10	−1.07355	−0.536777	+	0.843724i	$0.680358\pi$
$752$	−1.13433e10	−0.972691
$753$	−8.29304e9	−0.707834
$754$	3.76833e8	0.0320147
$755$	3.12453e10	2.64223
$756$	−1.14120e9	−0.0960588
$757$	1.69129e10	1.41704	0.708520	−	0.705691i	$-0.249363\pi$
$757$	1.69129e10	1.41704	0.708520	+	0.705691i	$0.249363\pi$
$758$	−4.18227e9	−0.348795
$759$	4.07021e8	0.0337886
$760$	−8.08619e8	−0.0668184
$761$	−4.92489e9	−0.405089	−0.202544	−	0.979273i	$-0.564921\pi$
$761$	−4.92489e9	−0.405089	−0.202544	+	0.979273i	$0.564921\pi$
$762$	1.09268e10	0.894645
$763$	8.82784e9	0.719480
$764$	1.64754e9	0.133662
$765$	1.25940e10	1.01707
$766$	2.03460e10	1.63561
$767$	1.00702e9	0.0805851
$768$	6.77543e9	0.539725
$769$	1.38388e10	1.09737	0.548687	−	0.836028i	$-0.315128\pi$
$769$	1.38388e10	1.09737	0.548687	+	0.836028i	$0.315128\pi$
$770$	−4.55752e9	−0.359759
$771$	6.98799e8	0.0549113
$772$	1.45597e10	1.13892
$773$	−7.78161e8	−0.0605956	−0.0302978	−	0.999541i	$-0.509646\pi$
$773$	−7.78161e8	−0.0605956	−0.0302978	+	0.999541i	$0.509646\pi$
$774$	−3.55290e9	−0.275416
$775$	1.11845e10	0.863101
$776$	−7.83271e8	−0.0601722
$777$	−8.62203e8	−0.0659380
$778$	3.57827e10	2.72423
$779$	4.24222e9	0.321523
$780$	1.63983e9	0.123728
$781$	−4.57968e8	−0.0343998
$782$	6.52612e9	0.488013
$783$	−5.19158e8	−0.0386486
$784$	9.99955e9	0.741096
$785$	−2.74311e10	−2.02395
$786$	−1.58630e10	−1.16522
$787$	1.34125e10	0.980839	0.490420	−	0.871486i	$-0.336844\pi$
$787$	1.34125e10	0.980839	0.490420	+	0.871486i	$0.336844\pi$
$788$	1.05493e10	0.768034
$789$	−1.36062e10	−0.986209
$790$	−5.31567e10	−3.83587
$791$	1.14481e10	0.822462
$792$	−5.80593e7	−0.00415273
$793$	−6.19661e8	−0.0441263
$794$	−2.79903e10	−1.98443
$795$	1.61893e10	1.14273
$796$	1.15589e10	0.812308
$797$	−2.23182e10	−1.56155	−0.780775	−	0.624813i	$-0.785175\pi$
$797$	−2.23182e10	−1.56155	−0.780775	+	0.624813i	$0.785175\pi$
$798$	−4.63244e9	−0.322701
$799$	−2.37952e10	−1.65035
$800$	−5.05608e10	−3.49140
$801$	−9.14471e9	−0.628718
$802$	6.71754e9	0.459833
$803$	−1.31634e9	−0.0897150
$804$	−1.73021e10	−1.17409
$805$	−2.77567e9	−0.187535
$806$	−8.34099e8	−0.0561106
$807$	−1.06797e10	−0.715320
$808$	3.03911e8	0.0202678
$809$	2.62752e10	1.74472	0.872360	−	0.488864i	$-0.162589\pi$
$809$	2.62752e10	1.74472	0.872360	+	0.488864i	$0.162589\pi$
$810$	−4.45012e9	−0.294221
$811$	−7.12447e9	−0.469008	−0.234504	−	0.972115i	$-0.575347\pi$
$811$	−7.12447e9	−0.469008	−0.234504	+	0.972115i	$0.575347\pi$
$812$	1.52926e9	0.100239
$813$	7.84965e9	0.512311
$814$	−1.45227e9	−0.0943762
$815$	−3.96026e10	−2.56255
$816$	1.42431e10	0.917673
$817$	−7.32164e9	−0.469711
$818$	2.87637e10	1.83742
$819$	2.83749e8	0.0180485
$820$	−1.20043e10	−0.760304
$821$	−1.18524e10	−0.747492	−0.373746	−	0.927531i	$-0.621927\pi$
$821$	−1.18524e10	−0.747492	−0.373746	+	0.927531i	$0.621927\pi$
$822$	−7.35874e9	−0.462117
$823$	−1.39730e10	−0.873755	−0.436877	−	0.899521i	$-0.643916\pi$
$823$	−1.39730e10	−0.873755	−0.436877	+	0.899521i	$0.643916\pi$
$824$	1.20913e9	0.0752886
$825$	−6.40876e9	−0.397361
$826$	8.04992e9	0.497006
$827$	−2.17594e10	−1.33776	−0.668879	−	0.743371i	$-0.733226\pi$
$827$	−2.17594e10	−1.33776	−0.668879	+	0.743371i	$0.733226\pi$
$828$	−1.17069e9	−0.0716695
$829$	−1.90784e10	−1.16306	−0.581528	−	0.813527i	$-0.697545\pi$
$829$	−1.90784e10	−1.16306	−0.581528	+	0.813527i	$0.697545\pi$
$830$	−7.80276e10	−4.73669
$831$	1.68997e10	1.02159
$832$	1.97210e9	0.118713
$833$	2.09765e10	1.25741
$834$	−3.71974e9	−0.222040
$835$	1.59378e10	0.947385
$836$	−3.96121e9	−0.234480
$837$	1.14913e9	0.0677375
$838$	4.01582e10	2.35733
$839$	−7.66588e9	−0.448121	−0.224060	−	0.974575i	$-0.571931\pi$
$839$	−7.66588e9	−0.448121	−0.224060	+	0.974575i	$0.571931\pi$
$840$	3.95933e8	0.0230486
$841$	−1.65542e10	−0.959670
$842$	1.04010e10	0.600456
$843$	−1.24280e10	−0.714503
$844$	−1.34242e10	−0.768581
$845$	3.21793e10	1.83475
$846$	8.40808e9	0.477420
$847$	7.88600e9	0.445928
$848$	1.83091e10	1.03105
$849$	2.26022e9	0.126757
$850$	−1.02757e11	−5.73912
$851$	−8.84477e8	−0.0491964
$852$	1.31722e9	0.0729659
$853$	−4.17913e9	−0.230549	−0.115275	−	0.993334i	$-0.536775\pi$
$853$	−4.17913e9	−0.230549	−0.115275	+	0.993334i	$0.536775\pi$
$854$	−4.95344e9	−0.272148
$855$	−9.17058e9	−0.501783
$856$	4.10039e8	0.0223443
$857$	2.49437e10	1.35372	0.676859	−	0.736112i	$-0.263341\pi$
$857$	2.49437e10	1.35372	0.676859	+	0.736112i	$0.263341\pi$
$858$	4.77940e8	0.0258326
$859$	−2.11511e10	−1.13856	−0.569280	−	0.822144i	$-0.692778\pi$
$859$	−2.11511e10	−1.13856	−0.569280	+	0.822144i	$0.692778\pi$
$860$	2.07181e10	1.11072
$861$	−2.07717e9	−0.110907
$862$	3.20331e9	0.170343
$863$	2.03614e10	1.07838	0.539189	−	0.842185i	$-0.318731\pi$
$863$	2.03614e10	1.07838	0.539189	+	0.842185i	$0.318731\pi$
$864$	−5.19476e9	−0.274010
$865$	−4.83907e9	−0.254218
$866$	−4.76883e9	−0.249517
$867$	1.87991e10	0.979650
$868$	−3.38493e9	−0.175683
$869$	−7.86523e9	−0.406577
$870$	5.96333e9	0.307023
$871$	4.30200e9	0.220601
$872$	−1.29177e9	−0.0659749
$873$	−8.88312e9	−0.451872
$874$	−4.75212e9	−0.240767
$875$	2.58816e10	1.30606
$876$	3.78611e9	0.190296
$877$	8.64182e9	0.432620	0.216310	−	0.976325i	$-0.430598\pi$
$877$	8.64182e9	0.432620	0.216310	+	0.976325i	$0.430598\pi$
$878$	−9.86482e7	−0.00491879
$879$	1.45190e10	0.721065
$880$	−1.02036e10	−0.504737
$881$	2.32377e10	1.14493	0.572464	−	0.819930i	$-0.305988\pi$
$881$	2.32377e10	1.14493	0.572464	+	0.819930i	$0.305988\pi$
$882$	−7.41207e9	−0.363747
$883$	−1.80558e10	−0.882582	−0.441291	−	0.897364i	$-0.645479\pi$
$883$	−1.80558e10	−0.882582	−0.441291	+	0.897364i	$0.645479\pi$
$884$	3.89037e9	0.189412
$885$	1.59360e10	0.772817
$886$	2.84672e10	1.37507
$887$	−1.68574e9	−0.0811067	−0.0405534	−	0.999177i	$-0.512912\pi$
$887$	−1.68574e9	−0.0811067	−0.0405534	+	0.999177i	$0.512912\pi$
$888$	1.26166e8	0.00604638
$889$	−1.10255e10	−0.526309
$890$	1.05041e11	4.99452
$891$	−6.58453e8	−0.0311855
$892$	−1.90051e10	−0.896589
$893$	1.73270e10	0.814221
$894$	−1.68929e10	−0.790719
$895$	−2.98359e10	−1.39110
$896$	9.24818e8	0.0429515
$897$	2.91080e8	0.0134660
$898$	6.09377e10	2.80814
$899$	−1.53988e9	−0.0706849
$900$	1.84331e10	0.842847
$901$	3.84077e10	1.74937
$902$	−3.49873e9	−0.158740
$903$	3.58498e9	0.162024
$904$	−1.67519e9	−0.0754181
$905$	4.31766e10	1.93633
$906$	−2.61928e10	−1.17013
$907$	3.69615e10	1.64484	0.822422	−	0.568878i	$-0.192622\pi$
$907$	3.69615e10	1.64484	0.822422	+	0.568878i	$0.192622\pi$
$908$	2.53974e10	1.12587
$909$	3.44667e9	0.152204
$910$	−3.25930e9	−0.143377
$911$	−2.56946e10	−1.12597	−0.562986	−	0.826466i	$-0.690348\pi$
$911$	−2.56946e10	−1.12597	−0.562986	+	0.826466i	$0.690348\pi$
$912$	−1.03714e10	−0.452745
$913$	−1.15452e10	−0.502058
$914$	1.94411e10	0.842187
$915$	−9.80605e9	−0.423175
$916$	7.95356e9	0.341922
$917$	1.60062e10	0.685482
$918$	−1.05575e10	−0.450415
$919$	−3.71651e10	−1.57954	−0.789772	−	0.613401i	$-0.789801\pi$
$919$	−3.71651e10	−1.57954	−0.789772	+	0.613401i	$0.789801\pi$
$920$	4.06162e8	0.0171966
$921$	3.84410e8	0.0162138
$922$	−1.24734e10	−0.524114
$923$	−3.27514e8	−0.0137096
$924$	1.93957e9	0.0808824
$925$	1.39265e10	0.578559
$926$	2.47663e9	0.102500
$927$	1.37128e10	0.565390
$928$	6.96117e9	0.285933
$929$	−1.03012e10	−0.421536	−0.210768	−	0.977536i	$-0.567596\pi$
$929$	−1.03012e10	−0.421536	−0.210768	+	0.977536i	$0.567596\pi$
$930$	−1.31995e10	−0.538105
$931$	−1.52744e10	−0.620356
$932$	−3.81419e10	−1.54329
$933$	6.22035e9	0.250743
$934$	−4.90389e10	−1.96937
$935$	−2.14046e10	−0.856379
$936$	−4.15209e7	−0.00165501
$937$	−1.80935e10	−0.718511	−0.359255	−	0.933239i	$-0.616969\pi$
$937$	−1.80935e10	−0.718511	−0.359255	+	0.933239i	$0.616969\pi$
$938$	3.43893e10	1.36055
$939$	7.48953e9	0.295206
$940$	−4.90303e10	−1.92538
$941$	−9.79964e9	−0.383395	−0.191698	−	0.981454i	$-0.561399\pi$
$941$	−9.79964e9	−0.383395	−0.191698	+	0.981454i	$0.561399\pi$
$942$	2.29954e10	0.896319
$943$	−2.13083e9	−0.0827480
$944$	1.80226e10	0.697293
$945$	4.49030e9	0.173087
$946$	6.03844e9	0.231903
$947$	−2.14911e10	−0.822307	−0.411153	−	0.911566i	$-0.634874\pi$
$947$	−2.14911e10	−0.822307	−0.411153	+	0.911566i	$0.634874\pi$
$948$	2.26222e10	0.862395
$949$	−9.41379e8	−0.0357547
$950$	7.48246e10	2.83147
$951$	−1.82513e10	−0.688116
$952$	9.39318e8	0.0352845
$953$	−4.88212e10	−1.82719	−0.913593	−	0.406629i	$-0.866704\pi$
$953$	−4.88212e10	−1.82719	−0.913593	+	0.406629i	$0.866704\pi$
$954$	−1.35714e10	−0.506065
$955$	−6.48255e9	−0.240843
$956$	1.81715e7	0.000672650 0
$957$	8.82352e8	0.0325424
$958$	4.36350e10	1.60345
$959$	7.42518e9	0.271858
$960$	3.12082e10	1.13846
$961$	−2.41042e10	−0.876114
$962$	−1.03859e9	−0.0376124
$963$	4.65027e9	0.167798
$964$	5.14054e10	1.84816
$965$	−5.72881e10	−2.05220
$966$	2.32683e9	0.0830511
$967$	−1.44801e10	−0.514967	−0.257483	−	0.966283i	$-0.582893\pi$
$967$	−1.44801e10	−0.514967	−0.257483	+	0.966283i	$0.582893\pi$
$968$	−1.15395e9	−0.0408907
$969$	−2.17565e10	−0.768166
$970$	1.02036e11	3.58966
$971$	1.52131e10	0.533274	0.266637	−	0.963797i	$-0.414087\pi$
$971$	1.52131e10	0.533274	0.266637	+	0.963797i	$0.414087\pi$
$972$	1.89386e9	0.0661480
$973$	3.75332e9	0.130623
$974$	5.13956e9	0.178225
$975$	−4.58320e9	−0.158363
$976$	−1.10900e10	−0.381820
$977$	−7.61072e9	−0.261093	−0.130546	−	0.991442i	$-0.541673\pi$
$977$	−7.61072e9	−0.261093	−0.130546	+	0.991442i	$0.541673\pi$
$978$	3.31988e10	1.13484
$979$	1.55422e10	0.529386
$980$	4.32223e10	1.46695
$981$	−1.46501e10	−0.495448
$982$	2.72805e10	0.919310
$983$	5.78044e10	1.94099	0.970495	−	0.241121i	$-0.0775149\pi$
$983$	5.78044e10	1.94099	0.970495	+	0.241121i	$0.0775149\pi$
$984$	3.03951e8	0.0101700
$985$	−4.15081e10	−1.38391
$986$	1.41475e10	0.470014
$987$	−8.48399e9	−0.280860
$988$	−2.83285e9	−0.0934488
$989$	3.67759e9	0.120886
$990$	7.56334e9	0.247737
$991$	−9.60474e9	−0.313493	−0.156747	−	0.987639i	$-0.550101\pi$
$991$	−9.60474e9	−0.313493	−0.156747	+	0.987639i	$0.550101\pi$
$992$	−1.54082e10	−0.501141
$993$	−2.86434e10	−0.928330
$994$	−2.61808e9	−0.0845534
$995$	−4.54808e10	−1.46368
$996$	3.32067e10	1.06492
$997$	−5.00503e10	−1.59946	−0.799730	−	0.600360i	$-0.795024\pi$
$997$	−5.00503e10	−1.59946	−0.799730	+	0.600360i	$0.795024\pi$
$998$	−1.22405e10	−0.389799
$999$	1.43085e9	0.0454062

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.8.a.b.1.6	✓	6
3.2	odd	2		207.8.a.c.1.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.8.a.b.1.6	✓	6	1.1	even	1	trivial
207.8.a.c.1.1		6	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:	\( N \)	\(=\)	\( 69 = 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	69.a (trivial)

\(f(q)\)	\(=\)	\(q+16.1241 q^{2} +27.0000 q^{3} +131.987 q^{4} -519.327 q^{5} +435.351 q^{6} -439.281 q^{7} +64.2798 q^{8} +729.000 q^{9} +O(q^{10})\)	\(q+16.1241 q^{2} +27.0000 q^{3} +131.987 q^{4} -519.327 q^{5} +435.351 q^{6} -439.281 q^{7} +64.2798 q^{8} +729.000 q^{9} -8373.68 q^{10} -1239.00 q^{11} +3563.64 q^{12} -886.064 q^{13} -7083.01 q^{14} -14021.8 q^{15} -15857.8 q^{16} -33265.6 q^{17} +11754.5 q^{18} +24223.0 q^{19} -68544.2 q^{20} -11860.6 q^{21} -19977.7 q^{22} -12167.0 q^{23} +1735.55 q^{24} +191576. q^{25} -14287.0 q^{26} +19683.0 q^{27} -57979.2 q^{28} -26376.0 q^{29} -226089. q^{30} +58381.7 q^{31} -263921. q^{32} -33452.9 q^{33} -536379. q^{34} +228131. q^{35} +96218.2 q^{36} +72694.7 q^{37} +390574. q^{38} -23923.7 q^{39} -33382.2 q^{40} +175132. q^{41} -191241. q^{42} -302259. q^{43} -163531. q^{44} -378590. q^{45} -196182. q^{46} +715309. q^{47} -428161. q^{48} -630575. q^{49} +3.08899e6 q^{50} -898172. q^{51} -116948. q^{52} -1.15458e6 q^{53} +317371. q^{54} +643444. q^{55} -28236.9 q^{56} +654022. q^{57} -425289. q^{58} -1.13651e6 q^{59} -1.85069e6 q^{60} +699341. q^{61} +941353. q^{62} -320236. q^{63} -2.22569e6 q^{64} +460157. q^{65} -539397. q^{66} -4.85518e6 q^{67} -4.39062e6 q^{68} -328509. q^{69} +3.67840e6 q^{70} +369628. q^{71} +46860.0 q^{72} +1.06243e6 q^{73} +1.17214e6 q^{74} +5.17254e6 q^{75} +3.19711e6 q^{76} +544267. q^{77} -385748. q^{78} +6.34807e6 q^{79} +8.23540e6 q^{80} +531441. q^{81} +2.82384e6 q^{82} +9.31819e6 q^{83} -1.56544e6 q^{84} +1.72758e7 q^{85} -4.87366e6 q^{86} -712151. q^{87} -79642.3 q^{88} -1.25442e7 q^{89} -6.10441e6 q^{90} +389231. q^{91} -1.60588e6 q^{92} +1.57631e6 q^{93} +1.15337e7 q^{94} -1.25797e7 q^{95} -7.12587e6 q^{96} -1.21853e7 q^{97} -1.01675e7 q^{98} -903228. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 8 q^{2} + 162 q^{3} + 178 q^{4} - 372 q^{5} - 216 q^{6} - 1104 q^{7} - 1956 q^{8} + 4374 q^{9}+O(q^{10}) \) 6 * q - 8 * q^2 + 162 * q^3 + 178 * q^4 - 372 * q^5 - 216 * q^6 - 1104 * q^7 - 1956 * q^8 + 4374 * q^9	\( 6 q - 8 q^{2} + 162 q^{3} + 178 q^{4} - 372 q^{5} - 216 q^{6} - 1104 q^{7} - 1956 q^{8} + 4374 q^{9} - 13042 q^{10} - 14824 q^{11} + 4806 q^{12} - 756 q^{13} - 3926 q^{14} - 10044 q^{15} - 13022 q^{16} - 69484 q^{17} - 5832 q^{18} - 43864 q^{19} + 78886 q^{20} - 29808 q^{21} + 98204 q^{22} - 73002 q^{23} - 52812 q^{24} + 228018 q^{25} - 311956 q^{26} + 118098 q^{27} - 545442 q^{28} - 311100 q^{29} - 352134 q^{30} - 245248 q^{31} - 390156 q^{32} - 400248 q^{33} + 235834 q^{34} - 1331256 q^{35} + 129762 q^{36} - 630044 q^{37} + 80910 q^{38} - 20412 q^{39} - 2153982 q^{40} - 969204 q^{41} - 106002 q^{42} - 1770208 q^{43} - 1749140 q^{44} - 271188 q^{45} + 97336 q^{46} - 1400024 q^{47} - 351594 q^{48} + 1985598 q^{49} - 956660 q^{50} - 1876068 q^{51} + 3217272 q^{52} - 1573516 q^{53} - 157464 q^{54} - 431296 q^{55} + 7740702 q^{56} - 1184328 q^{57} + 5987188 q^{58} - 1410320 q^{59} + 2129922 q^{60} - 942172 q^{61} + 3334412 q^{62} - 804816 q^{63} + 1996866 q^{64} - 420944 q^{65} + 2651508 q^{66} - 452072 q^{67} - 9258254 q^{68} - 1971054 q^{69} + 21981136 q^{70} + 122928 q^{71} - 1425924 q^{72} + 16490716 q^{73} - 600104 q^{74} + 6156486 q^{75} + 7428658 q^{76} + 7239696 q^{77} - 8422812 q^{78} + 2458408 q^{79} + 19440230 q^{80} + 3188646 q^{81} + 20510784 q^{82} - 7566456 q^{83} - 14726934 q^{84} + 5817744 q^{85} - 669666 q^{86} - 8399700 q^{87} + 14775668 q^{88} - 20368036 q^{89} - 9507618 q^{90} + 8815576 q^{91} - 2165726 q^{92} - 6621696 q^{93} + 16952576 q^{94} + 5143832 q^{95} - 10534212 q^{96} + 12586972 q^{97} - 39164812 q^{98} - 10806696 q^{99}+O(q^{100}) \) 6 * q - 8 * q^2 + 162 * q^3 + 178 * q^4 - 372 * q^5 - 216 * q^6 - 1104 * q^7 - 1956 * q^8 + 4374 * q^9 - 13042 * q^10 - 14824 * q^11 + 4806 * q^12 - 756 * q^13 - 3926 * q^14 - 10044 * q^15 - 13022 * q^16 - 69484 * q^17 - 5832 * q^18 - 43864 * q^19 + 78886 * q^20 - 29808 * q^21 + 98204 * q^22 - 73002 * q^23 - 52812 * q^24 + 228018 * q^25 - 311956 * q^26 + 118098 * q^27 - 545442 * q^28 - 311100 * q^29 - 352134 * q^30 - 245248 * q^31 - 390156 * q^32 - 400248 * q^33 + 235834 * q^34 - 1331256 * q^35 + 129762 * q^36 - 630044 * q^37 + 80910 * q^38 - 20412 * q^39 - 2153982 * q^40 - 969204 * q^41 - 106002 * q^42 - 1770208 * q^43 - 1749140 * q^44 - 271188 * q^45 + 97336 * q^46 - 1400024 * q^47 - 351594 * q^48 + 1985598 * q^49 - 956660 * q^50 - 1876068 * q^51 + 3217272 * q^52 - 1573516 * q^53 - 157464 * q^54 - 431296 * q^55 + 7740702 * q^56 - 1184328 * q^57 + 5987188 * q^58 - 1410320 * q^59 + 2129922 * q^60 - 942172 * q^61 + 3334412 * q^62 - 804816 * q^63 + 1996866 * q^64 - 420944 * q^65 + 2651508 * q^66 - 452072 * q^67 - 9258254 * q^68 - 1971054 * q^69 + 21981136 * q^70 + 122928 * q^71 - 1425924 * q^72 + 16490716 * q^73 - 600104 * q^74 + 6156486 * q^75 + 7428658 * q^76 + 7239696 * q^77 - 8422812 * q^78 + 2458408 * q^79 + 19440230 * q^80 + 3188646 * q^81 + 20510784 * q^82 - 7566456 * q^83 - 14726934 * q^84 + 5817744 * q^85 - 669666 * q^86 - 8399700 * q^87 + 14775668 * q^88 - 20368036 * q^89 - 9507618 * q^90 + 8815576 * q^91 - 2165726 * q^92 - 6621696 * q^93 + 16952576 * q^94 + 5143832 * q^95 - 10534212 * q^96 + 12586972 * q^97 - 39164812 * q^98 - 10806696 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

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Database

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