LMFDB - Embedded newform 138.8.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("138.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 138 = 2 \cdot 3 \cdot 23 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	138.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$43.1091335168$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 47804x^{2} - 3068607x + 114119793 $ x^4 - x^3 - 47804x^2 - 3068607x + 114119793
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4}\cdot 3 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$26.4468$ of defining polynomial
Character	$\chi$	$=$	138.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} -548.447 q^{5} +216.000 q^{6} +63.4064 q^{7} -512.000 q^{8} +729.000 q^{9} +O(q^{10})$	\(q-8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} -548.447 q^{5} +216.000 q^{6} +63.4064 q^{7} -512.000 q^{8} +729.000 q^{9} +4387.58 q^{10} -1650.76 q^{11} -1728.00 q^{12} +9563.50 q^{13} -507.251 q^{14} +14808.1 q^{15} +4096.00 q^{16} +18059.8 q^{17} -5832.00 q^{18} -9533.78 q^{19} -35100.6 q^{20} -1711.97 q^{21} +13206.1 q^{22} +12167.0 q^{23} +13824.0 q^{24} +222670. q^{25} -76508.0 q^{26} -19683.0 q^{27} +4058.01 q^{28} -191307. q^{29} -118465. q^{30} +72333.1 q^{31} -32768.0 q^{32} +44570.5 q^{33} -144479. q^{34} -34775.1 q^{35} +46656.0 q^{36} +131816. q^{37} +76270.2 q^{38} -258214. q^{39} +280805. q^{40} +664463. q^{41} +13695.8 q^{42} +462761. q^{43} -105649. q^{44} -399818. q^{45} -97336.0 q^{46} +103529. q^{47} -110592. q^{48} -819523. q^{49} -1.78136e6 q^{50} -487616. q^{51} +612064. q^{52} +230650. q^{53} +157464. q^{54} +905355. q^{55} -32464.1 q^{56} +257412. q^{57} +1.53045e6 q^{58} -2.99541e6 q^{59} +947717. q^{60} -1.14386e6 q^{61} -578665. q^{62} +46223.3 q^{63} +262144. q^{64} -5.24507e6 q^{65} -356564. q^{66} -2.45653e6 q^{67} +1.15583e6 q^{68} -328509. q^{69} +278201. q^{70} -115899. q^{71} -373248. q^{72} -5.47323e6 q^{73} -1.05453e6 q^{74} -6.01208e6 q^{75} -610162. q^{76} -104669. q^{77} +2.06571e6 q^{78} +5.58010e6 q^{79} -2.24644e6 q^{80} +531441. q^{81} -5.31571e6 q^{82} -7.36714e6 q^{83} -109566. q^{84} -9.90488e6 q^{85} -3.70209e6 q^{86} +5.16528e6 q^{87} +845189. q^{88} +7.28416e6 q^{89} +3.19855e6 q^{90} +606387. q^{91} +778688. q^{92} -1.95299e6 q^{93} -828234. q^{94} +5.22878e6 q^{95} +884736. q^{96} +2.69084e6 q^{97} +6.55618e6 q^{98} -1.20340e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 32 q^{2} - 108 q^{3} + 256 q^{4} - 90 q^{5} + 864 q^{6} - 222 q^{7} - 2048 q^{8} + 2916 q^{9}+O(q^{10}) $ 4 * q - 32 * q^2 - 108 * q^3 + 256 * q^4 - 90 * q^5 + 864 * q^6 - 222 * q^7 - 2048 * q^8 + 2916 * q^9	\( 4 q - 32 q^{2} - 108 q^{3} + 256 q^{4} - 90 q^{5} + 864 q^{6} - 222 q^{7} - 2048 q^{8} + 2916 q^{9} + 720 q^{10} - 4120 q^{11} - 6912 q^{12} - 6796 q^{13} + 1776 q^{14} + 2430 q^{15} + 16384 q^{16} + 18918 q^{17} - 23328 q^{18} + 19682 q^{19} - 5760 q^{20} + 5994 q^{21} + 32960 q^{22} + 48668 q^{23} + 55296 q^{24} + 261304 q^{25} + 54368 q^{26} - 78732 q^{27} - 14208 q^{28} + 12756 q^{29} - 19440 q^{30} + 440892 q^{31} - 131072 q^{32} + 111240 q^{33} - 151344 q^{34} + 150200 q^{35} + 186624 q^{36} + 567212 q^{37} - 157456 q^{38} + 183492 q^{39} + 46080 q^{40} + 275056 q^{41} - 47952 q^{42} + 149946 q^{43} - 263680 q^{44} - 65610 q^{45} - 389344 q^{46} - 2000728 q^{47} - 442368 q^{48} - 1623352 q^{49} - 2090432 q^{50} - 510786 q^{51} - 434944 q^{52} - 2778518 q^{53} + 629856 q^{54} - 2706008 q^{55} + 113664 q^{56} - 531414 q^{57} - 102048 q^{58} - 3656368 q^{59} + 155520 q^{60} - 989076 q^{61} - 3527136 q^{62} - 161838 q^{63} + 1048576 q^{64} - 7481556 q^{65} - 889920 q^{66} - 5755282 q^{67} + 1210752 q^{68} - 1314036 q^{69} - 1201600 q^{70} - 8843424 q^{71} - 1492992 q^{72} - 1100272 q^{73} - 4537696 q^{74} - 7055208 q^{75} + 1259648 q^{76} - 10973480 q^{77} - 1467936 q^{78} + 959830 q^{79} - 368640 q^{80} + 2125764 q^{81} - 2200448 q^{82} - 24990584 q^{83} + 383616 q^{84} - 7601436 q^{85} - 1199568 q^{86} - 344412 q^{87} + 2109440 q^{88} - 3623430 q^{89} + 524880 q^{90} - 9436372 q^{91} + 3114752 q^{92} - 11904084 q^{93} + 16005824 q^{94} - 12385728 q^{95} + 3538944 q^{96} + 14161716 q^{97} + 12986816 q^{98} - 3003480 q^{99}+O(q^{100}) \) 4 * q - 32 * q^2 - 108 * q^3 + 256 * q^4 - 90 * q^5 + 864 * q^6 - 222 * q^7 - 2048 * q^8 + 2916 * q^9 + 720 * q^10 - 4120 * q^11 - 6912 * q^12 - 6796 * q^13 + 1776 * q^14 + 2430 * q^15 + 16384 * q^16 + 18918 * q^17 - 23328 * q^18 + 19682 * q^19 - 5760 * q^20 + 5994 * q^21 + 32960 * q^22 + 48668 * q^23 + 55296 * q^24 + 261304 * q^25 + 54368 * q^26 - 78732 * q^27 - 14208 * q^28 + 12756 * q^29 - 19440 * q^30 + 440892 * q^31 - 131072 * q^32 + 111240 * q^33 - 151344 * q^34 + 150200 * q^35 + 186624 * q^36 + 567212 * q^37 - 157456 * q^38 + 183492 * q^39 + 46080 * q^40 + 275056 * q^41 - 47952 * q^42 + 149946 * q^43 - 263680 * q^44 - 65610 * q^45 - 389344 * q^46 - 2000728 * q^47 - 442368 * q^48 - 1623352 * q^49 - 2090432 * q^50 - 510786 * q^51 - 434944 * q^52 - 2778518 * q^53 + 629856 * q^54 - 2706008 * q^55 + 113664 * q^56 - 531414 * q^57 - 102048 * q^58 - 3656368 * q^59 + 155520 * q^60 - 989076 * q^61 - 3527136 * q^62 - 161838 * q^63 + 1048576 * q^64 - 7481556 * q^65 - 889920 * q^66 - 5755282 * q^67 + 1210752 * q^68 - 1314036 * q^69 - 1201600 * q^70 - 8843424 * q^71 - 1492992 * q^72 - 1100272 * q^73 - 4537696 * q^74 - 7055208 * q^75 + 1259648 * q^76 - 10973480 * q^77 - 1467936 * q^78 + 959830 * q^79 - 368640 * q^80 + 2125764 * q^81 - 2200448 * q^82 - 24990584 * q^83 + 383616 * q^84 - 7601436 * q^85 - 1199568 * q^86 - 344412 * q^87 + 2109440 * q^88 - 3623430 * q^89 + 524880 * q^90 - 9436372 * q^91 + 3114752 * q^92 - 11904084 * q^93 + 16005824 * q^94 - 12385728 * q^95 + 3538944 * q^96 + 14161716 * q^97 + 12986816 * q^98 - 3003480 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−8.00000	−0.707107
$2$	−8.00000	−0.707107
$3$	−27.0000	−0.577350
$3$	−27.0000	−0.577350
$4$	64.0000	0.500000
$5$	−548.447	−1.96219	−0.981093	−	0.193539i	$-0.938003\pi$
$5$	−548.447	−1.96219	−0.981093	+	0.193539i	$0.938003\pi$
$6$	216.000	0.408248
$7$	63.4064	0.0698699	0.0349349	−	0.999390i	$-0.488878\pi$
$7$	63.4064	0.0698699	0.0349349	+	0.999390i	$0.488878\pi$
$8$	−512.000	−0.353553
$9$	729.000	0.333333
$10$	4387.58	1.38747
$11$	−1650.76	−0.373946	−0.186973	−	0.982365i	$-0.559868\pi$
$11$	−1650.76	−0.373946	−0.186973	+	0.982365i	$0.559868\pi$
$12$	−1728.00	−0.288675
$13$	9563.50	1.20730	0.603650	−	0.797250i	$-0.293713\pi$
$13$	9563.50	1.20730	0.603650	+	0.797250i	$0.293713\pi$
$14$	−507.251	−0.0494055
$15$	14808.1	1.13287
$16$	4096.00	0.250000
$17$	18059.8	0.891544	0.445772	−	0.895147i	$-0.352929\pi$
$17$	18059.8	0.891544	0.445772	+	0.895147i	$0.352929\pi$
$18$	−5832.00	−0.235702
$19$	−9533.78	−0.318880	−0.159440	−	0.987208i	$-0.550969\pi$
$19$	−9533.78	−0.318880	−0.159440	+	0.987208i	$0.550969\pi$
$20$	−35100.6	−0.981093
$21$	−1711.97	−0.0403394
$22$	13206.1	0.264420
$23$	12167.0	0.208514
$23$	12167.0	0.208514
$24$	13824.0	0.204124
$25$	222670.	2.85017
$26$	−76508.0	−0.853689
$27$	−19683.0	−0.192450
$28$	4058.01	0.0349349
$29$	−191307.	−1.45659	−0.728294	−	0.685264i	$-0.759687\pi$
$29$	−191307.	−1.45659	−0.728294	+	0.685264i	$0.759687\pi$
$30$	−118465.	−0.801059
$31$	72333.1	0.436085	0.218043	−	0.975939i	$-0.430033\pi$
$31$	72333.1	0.436085	0.218043	+	0.975939i	$0.430033\pi$
$32$	−32768.0	−0.176777
$33$	44570.5	0.215898
$34$	−144479.	−0.630417
$35$	−34775.1	−0.137098
$36$	46656.0	0.166667
$37$	131816.	0.427820	0.213910	−	0.976853i	$-0.431380\pi$
$37$	131816.	0.427820	0.213910	+	0.976853i	$0.431380\pi$
$38$	76270.2	0.225482
$39$	−258214.	−0.697035
$40$	280805.	0.693737
$41$	664463.	1.50566	0.752831	−	0.658214i	$-0.228688\pi$
$41$	664463.	1.50566	0.752831	+	0.658214i	$0.228688\pi$
$42$	13695.8	0.0285243
$43$	462761.	0.887600	0.443800	−	0.896126i	$-0.353630\pi$
$43$	462761.	0.887600	0.443800	+	0.896126i	$0.353630\pi$
$44$	−105649.	−0.186973
$45$	−399818.	−0.654062
$46$	−97336.0	−0.147442
$47$	103529.	0.145452	0.0727262	−	0.997352i	$-0.476830\pi$
$47$	103529.	0.145452	0.0727262	+	0.997352i	$0.476830\pi$
$48$	−110592.	−0.144338
$49$	−819523.	−0.995118
$50$	−1.78136e6	−2.01538
$51$	−487616.	−0.514733
$52$	612064.	0.603650
$53$	230650.	0.212808	0.106404	−	0.994323i	$-0.466066\pi$
$53$	230650.	0.212808	0.106404	+	0.994323i	$0.466066\pi$
$54$	157464.	0.136083
$55$	905355.	0.733752
$56$	−32464.1	−0.0247027
$57$	257412.	0.184106
$58$	1.53045e6	1.02996
$59$	−2.99541e6	−1.89878	−0.949389	−	0.314103i	$-0.898296\pi$
$59$	−2.99541e6	−1.89878	−0.949389	+	0.314103i	$0.898296\pi$
$60$	947717.	0.566434
$61$	−1.14386e6	−0.645238	−0.322619	−	0.946529i	$-0.604563\pi$
$61$	−1.14386e6	−0.645238	−0.322619	+	0.946529i	$0.604563\pi$
$62$	−578665.	−0.308359
$63$	46223.3	0.0232900
$64$	262144.	0.125000
$65$	−5.24507e6	−2.36894
$66$	−356564.	−0.152663
$67$	−2.45653e6	−0.997838	−0.498919	−	0.866649i	$-0.666270\pi$
$67$	−2.45653e6	−0.997838	−0.498919	+	0.866649i	$0.666270\pi$
$68$	1.15583e6	0.445772
$69$	−328509.	−0.120386
$70$	278201.	0.0969427
$71$	−115899.	−0.0384305	−0.0192153	−	0.999815i	$-0.506117\pi$
$71$	−115899.	−0.0384305	−0.0192153	+	0.999815i	$0.506117\pi$
$72$	−373248.	−0.117851
$73$	−5.47323e6	−1.64670	−0.823348	−	0.567537i	$-0.807896\pi$
$73$	−5.47323e6	−1.64670	−0.823348	+	0.567537i	$0.807896\pi$
$74$	−1.05453e6	−0.302515
$75$	−6.01208e6	−1.64555
$76$	−610162.	−0.159440
$77$	−104669.	−0.0261276
$78$	2.06571e6	0.492878
$79$	5.58010e6	1.27335	0.636674	−	0.771133i	$-0.280310\pi$
$79$	5.58010e6	1.27335	0.636674	+	0.771133i	$0.280310\pi$
$80$	−2.24644e6	−0.490546
$81$	531441.	0.111111
$82$	−5.31571e6	−1.06466
$83$	−7.36714e6	−1.41425	−0.707124	−	0.707090i	$-0.750008\pi$
$83$	−7.36714e6	−1.41425	−0.707124	+	0.707090i	$0.750008\pi$
$84$	−109566.	−0.0201697
$85$	−9.90488e6	−1.74937
$86$	−3.70209e6	−0.627628
$87$	5.16528e6	0.840962
$88$	845189.	0.132210
$89$	7.28416e6	1.09525	0.547627	−	0.836723i	$-0.315531\pi$
$89$	7.28416e6	1.09525	0.547627	+	0.836723i	$0.315531\pi$
$90$	3.19855e6	0.462492
$91$	606387.	0.0843539
$92$	778688.	0.104257
$93$	−1.95299e6	−0.251774
$94$	−828234.	−0.102850
$95$	5.22878e6	0.625702
$96$	884736.	0.102062
$97$	2.69084e6	0.299355	0.149677	−	0.988735i	$-0.452176\pi$
$97$	2.69084e6	0.299355	0.149677	+	0.988735i	$0.452176\pi$
$98$	6.55618e6	0.703655
$99$	−1.20340e6	−0.124649
$100$	1.42509e7	1.42509
$101$	−1.01685e6	−0.0982048	−0.0491024	−	0.998794i	$-0.515636\pi$
$101$	−1.01685e6	−0.0982048	−0.0491024	+	0.998794i	$0.515636\pi$
$102$	3.90093e6	0.363971
$103$	6.97583e6	0.629022	0.314511	−	0.949254i	$-0.398160\pi$
$103$	6.97583e6	0.629022	0.314511	+	0.949254i	$0.398160\pi$
$104$	−4.89651e6	−0.426845
$105$	938927.	0.0791534
$106$	−1.84520e6	−0.150478
$107$	1.46700e7	1.15767	0.578836	−	0.815444i	$-0.303507\pi$
$107$	1.46700e7	1.15767	0.578836	+	0.815444i	$0.303507\pi$
$108$	−1.25971e6	−0.0962250
$109$	−1.54863e7	−1.14539	−0.572695	−	0.819768i	$-0.694102\pi$
$109$	−1.54863e7	−1.14539	−0.572695	+	0.819768i	$0.694102\pi$
$110$	−7.24284e6	−0.518841
$111$	−3.55902e6	−0.247002
$112$	259713.	0.0174675
$113$	6.32504e6	0.412371	0.206186	−	0.978513i	$-0.433895\pi$
$113$	6.32504e6	0.412371	0.206186	+	0.978513i	$0.433895\pi$
$114$	−2.05930e6	−0.130182
$115$	−6.67296e6	−0.409144
$116$	−1.22436e7	−0.728294
$117$	6.97179e6	0.402433
$118$	2.39633e7	1.34264
$119$	1.14511e6	0.0622921
$120$	−7.58174e6	−0.400529
$121$	−1.67622e7	−0.860164
$122$	9.15091e6	0.456252
$123$	−1.79405e7	−0.869294
$124$	4.62932e6	0.218043
$125$	−7.92751e7	−3.63038
$126$	−369786.	−0.0164685
$127$	−3.80752e6	−0.164941	−0.0824706	−	0.996593i	$-0.526281\pi$
$127$	−3.80752e6	−0.164941	−0.0824706	+	0.996593i	$0.526281\pi$
$128$	−2.09715e6	−0.0883883
$129$	−1.24945e7	−0.512456
$130$	4.19606e7	1.67510
$131$	3.41062e7	1.32551	0.662756	−	0.748835i	$-0.269387\pi$
$131$	3.41062e7	1.32551	0.662756	+	0.748835i	$0.269387\pi$
$132$	2.85251e6	0.107949
$133$	−604503.	−0.0222801
$134$	1.96522e7	0.705578
$135$	1.07951e7	0.377623
$136$	−9.24664e6	−0.315208
$137$	−3.89393e7	−1.29380	−0.646899	−	0.762576i	$-0.723935\pi$
$137$	−3.89393e7	−1.29380	−0.646899	+	0.762576i	$0.723935\pi$
$138$	2.62807e6	0.0851257
$139$	3.39199e7	1.07128	0.535640	−	0.844447i	$-0.320070\pi$
$139$	3.39199e7	1.07128	0.535640	+	0.844447i	$0.320070\pi$
$140$	−2.22561e6	−0.0685488
$141$	−2.79529e6	−0.0839770
$142$	927194.	0.0271745
$143$	−1.57870e7	−0.451465
$144$	2.98598e6	0.0833333
$145$	1.04922e8	2.85810
$146$	4.37858e7	1.16439
$147$	2.21271e7	0.574532
$148$	8.43621e6	0.213910
$149$	3.62501e7	0.897754	0.448877	−	0.893594i	$-0.351824\pi$
$149$	3.62501e7	0.897754	0.448877	+	0.893594i	$0.351824\pi$
$150$	4.80966e7	1.16358
$151$	2.25080e7	0.532007	0.266004	−	0.963972i	$-0.414297\pi$
$151$	2.25080e7	0.532007	0.266004	+	0.963972i	$0.414297\pi$
$152$	4.88129e6	0.112741
$153$	1.31656e7	0.297181
$154$	837350.	0.0184750
$155$	−3.96709e7	−0.855680
$156$	−1.65257e7	−0.348517
$157$	3.22829e7	0.665768	0.332884	−	0.942968i	$-0.391978\pi$
$157$	3.22829e7	0.665768	0.332884	+	0.942968i	$0.391978\pi$
$158$	−4.46408e7	−0.900394
$159$	−6.22755e6	−0.122865
$160$	1.79715e7	0.346869
$161$	771466.	0.0145689
$162$	−4.25153e6	−0.0785674
$163$	2.44202e7	0.441664	0.220832	−	0.975312i	$-0.429123\pi$
$163$	2.44202e7	0.441664	0.220832	+	0.975312i	$0.429123\pi$
$164$	4.25256e7	0.752831
$165$	−2.44446e7	−0.423632
$166$	5.89371e7	1.00002
$167$	−7.79177e7	−1.29458	−0.647290	−	0.762244i	$-0.724098\pi$
$167$	−7.79177e7	−1.29458	−0.647290	+	0.762244i	$0.724098\pi$
$168$	876530.	0.0142621
$169$	2.87119e7	0.457571
$170$	7.92390e7	1.23699
$171$	−6.95012e6	−0.106293
$172$	2.96167e7	0.443800
$173$	2.89222e7	0.424688	0.212344	−	0.977195i	$-0.431890\pi$
$173$	2.89222e7	0.424688	0.212344	+	0.977195i	$0.431890\pi$
$174$	−4.13222e7	−0.594650
$175$	1.41187e7	0.199141
$176$	−6.76151e6	−0.0934866
$177$	8.08761e7	1.09626
$178$	−5.82733e7	−0.774461
$179$	−1.27465e8	−1.66113	−0.830567	−	0.556919i	$-0.811983\pi$
$179$	−1.27465e8	−1.66113	−0.830567	+	0.556919i	$0.811983\pi$
$180$	−2.55884e7	−0.327031
$181$	9.31642e7	1.16782	0.583908	−	0.811820i	$-0.301523\pi$
$181$	9.31642e7	1.16782	0.583908	+	0.811820i	$0.301523\pi$
$182$	−4.85109e6	−0.0596472
$183$	3.08843e7	0.372528
$184$	−6.22950e6	−0.0737210
$185$	−7.22940e7	−0.839462
$186$	1.56240e7	0.178031
$187$	−2.98125e7	−0.333390
$188$	6.62588e6	0.0727262
$189$	−1.24803e6	−0.0134465
$190$	−4.18302e7	−0.442438
$191$	−1.02331e8	−1.06265	−0.531325	−	0.847168i	$-0.678306\pi$
$191$	−1.02331e8	−1.06265	−0.531325	+	0.847168i	$0.678306\pi$
$192$	−7.07789e6	−0.0721688
$193$	−1.20185e8	−1.20337	−0.601684	−	0.798735i	$-0.705503\pi$
$193$	−1.20185e8	−1.20337	−0.601684	+	0.798735i	$0.705503\pi$
$194$	−2.15267e7	−0.211676
$195$	1.41617e8	1.36771
$196$	−5.24494e7	−0.497559
$197$	3.41701e7	0.318430	0.159215	−	0.987244i	$-0.449104\pi$
$197$	3.41701e7	0.318430	0.159215	+	0.987244i	$0.449104\pi$
$198$	9.62723e6	0.0881400
$199$	2.23148e7	0.200727	0.100364	−	0.994951i	$-0.467999\pi$
$199$	2.23148e7	0.200727	0.100364	+	0.994951i	$0.467999\pi$
$200$	−1.14007e8	−1.00769
$201$	6.63263e7	0.576102
$202$	8.13481e6	0.0694413
$203$	−1.21301e7	−0.101772
$204$	−3.12074e7	−0.257367
$205$	−3.64423e8	−2.95439
$206$	−5.58066e7	−0.444786
$207$	8.86974e6	0.0695048
$208$	3.91721e7	0.301825
$209$	1.57380e7	0.119244
$210$	−7.51142e6	−0.0559699
$211$	−3.93838e7	−0.288621	−0.144311	−	0.989532i	$-0.546096\pi$
$211$	−3.93838e7	−0.288621	−0.144311	+	0.989532i	$0.546096\pi$
$212$	1.47616e7	0.106404
$213$	3.12928e6	0.0221879
$214$	−1.17360e8	−0.818598
$215$	−2.53800e8	−1.74164
$216$	1.00777e7	0.0680414
$217$	4.58638e6	0.0304692
$218$	1.23890e8	0.809914
$219$	1.47777e8	0.950720
$220$	5.79427e7	0.366876
$221$	1.72715e8	1.07636
$222$	2.84722e7	0.174657
$223$	−2.58686e8	−1.56209	−0.781044	−	0.624476i	$-0.785313\pi$
$223$	−2.58686e8	−1.56209	−0.781044	+	0.624476i	$0.785313\pi$
$224$	−2.07770e6	−0.0123514
$225$	1.62326e8	0.950057
$226$	−5.06003e7	−0.291591
$227$	−8.45563e7	−0.479795	−0.239897	−	0.970798i	$-0.577114\pi$
$227$	−8.45563e7	−0.479795	−0.239897	+	0.970798i	$0.577114\pi$
$228$	1.64744e7	0.0920528
$229$	2.97083e8	1.63476	0.817380	−	0.576099i	$-0.195426\pi$
$229$	2.97083e8	1.63476	0.817380	+	0.576099i	$0.195426\pi$
$230$	5.33837e7	0.289308
$231$	2.82605e6	0.0150848
$232$	9.79490e7	0.514982
$233$	2.14814e7	0.111254	0.0556272	−	0.998452i	$-0.482284\pi$
$233$	2.14814e7	0.111254	0.0556272	+	0.998452i	$0.482284\pi$
$234$	−5.57743e7	−0.284563
$235$	−5.67804e7	−0.285405
$236$	−1.91706e8	−0.949389
$237$	−1.50663e8	−0.735168
$238$	−9.16088e6	−0.0440471
$239$	6.37762e7	0.302180	0.151090	−	0.988520i	$-0.451722\pi$
$239$	6.37762e7	0.302180	0.151090	+	0.988520i	$0.451722\pi$
$240$	6.06539e7	0.283217
$241$	9.16866e7	0.421935	0.210968	−	0.977493i	$-0.432338\pi$
$241$	9.16866e7	0.421935	0.210968	+	0.977493i	$0.432338\pi$
$242$	1.34097e8	0.608228
$243$	−1.43489e7	−0.0641500
$244$	−7.32073e7	−0.322619
$245$	4.49465e8	1.95261
$246$	1.43524e8	0.614684
$247$	−9.11762e7	−0.384984
$248$	−3.70346e7	−0.154179
$249$	1.98913e8	0.816516
$250$	6.34201e8	2.56707
$251$	−4.82480e8	−1.92584	−0.962922	−	0.269778i	$-0.913050\pi$
$251$	−4.82480e8	−1.92584	−0.962922	+	0.269778i	$0.913050\pi$
$252$	2.95829e6	0.0116450
$253$	−2.00848e7	−0.0779732
$254$	3.04602e7	0.116631
$255$	2.67432e8	1.01000
$256$	1.67772e7	0.0625000
$257$	−2.43009e8	−0.893011	−0.446505	−	0.894781i	$-0.647332\pi$
$257$	−2.43009e8	−0.893011	−0.446505	+	0.894781i	$0.647332\pi$
$258$	9.99564e7	0.362361
$259$	8.35796e6	0.0298917
$260$	−3.35685e8	−1.18447
$261$	−1.39463e8	−0.485530
$262$	−2.72849e8	−0.937278
$263$	−2.13243e8	−0.722818	−0.361409	−	0.932407i	$-0.617704\pi$
$263$	−2.13243e8	−0.722818	−0.361409	+	0.932407i	$0.617704\pi$
$264$	−2.28201e7	−0.0763315
$265$	−1.26499e8	−0.417569
$266$	4.83602e6	0.0157544
$267$	−1.96672e8	−0.632345
$268$	−1.57218e8	−0.498919
$269$	−2.17798e8	−0.682213	−0.341107	−	0.940025i	$-0.610802\pi$
$269$	−2.17798e8	−0.682213	−0.341107	+	0.940025i	$0.610802\pi$
$270$	−8.63607e7	−0.267020
$271$	−4.78480e8	−1.46040	−0.730200	−	0.683234i	$-0.760573\pi$
$271$	−4.78480e8	−1.46040	−0.730200	+	0.683234i	$0.760573\pi$
$272$	7.39731e7	0.222886
$273$	−1.63724e7	−0.0487017
$274$	3.11515e8	0.914854
$275$	−3.67574e8	−1.06581
$276$	−2.10246e7	−0.0601929
$277$	−2.56098e8	−0.723980	−0.361990	−	0.932182i	$-0.617903\pi$
$277$	−2.56098e8	−0.723980	−0.361990	+	0.932182i	$0.617903\pi$
$278$	−2.71359e8	−0.757509
$279$	5.27309e7	0.145362
$280$	1.78048e7	0.0484713
$281$	6.88981e8	1.85240	0.926200	−	0.377031i	$-0.123055\pi$
$281$	6.88981e8	1.85240	0.926200	+	0.377031i	$0.123055\pi$
$282$	2.23623e7	0.0593807
$283$	−1.23713e8	−0.324460	−0.162230	−	0.986753i	$-0.551869\pi$
$283$	−1.23713e8	−0.324460	−0.162230	+	0.986753i	$0.551869\pi$
$284$	−7.41755e6	−0.0192153
$285$	−1.41177e8	−0.361249
$286$	1.26296e8	0.319234
$287$	4.21312e7	0.105200
$288$	−2.38879e7	−0.0589256
$289$	−8.41807e7	−0.205149
$290$	−8.39373e8	−2.02098
$291$	−7.26526e7	−0.172833
$292$	−3.50286e8	−0.823348
$293$	4.41949e8	1.02644	0.513222	−	0.858256i	$-0.328452\pi$
$293$	4.41949e8	1.02644	0.513222	+	0.858256i	$0.328452\pi$
$294$	−1.77017e8	−0.406255
$295$	1.64282e9	3.72575
$296$	−6.74897e7	−0.151257
$297$	3.24919e7	0.0719660
$298$	−2.90001e8	−0.634808
$299$	1.16359e8	0.251739
$300$	−3.84773e8	−0.822774
$301$	2.93420e7	0.0620165
$302$	−1.80064e8	−0.376186
$303$	2.74550e7	0.0566986
$304$	−3.90504e7	−0.0797200
$305$	6.27349e8	1.26608
$306$	−1.05325e8	−0.210139
$307$	−8.81424e8	−1.73860	−0.869301	−	0.494282i	$-0.835431\pi$
$307$	−8.81424e8	−1.73860	−0.869301	+	0.494282i	$0.835431\pi$
$308$	−6.69880e6	−0.0130638
$309$	−1.88347e8	−0.363166
$310$	3.17367e8	0.605057
$311$	2.89587e8	0.545905	0.272953	−	0.962027i	$-0.412000\pi$
$311$	2.89587e8	0.545905	0.272953	+	0.962027i	$0.412000\pi$
$312$	1.32206e8	0.246439
$313$	−2.52352e8	−0.465160	−0.232580	−	0.972577i	$-0.574717\pi$
$313$	−2.52352e8	−0.465160	−0.232580	+	0.972577i	$0.574717\pi$
$314$	−2.58263e8	−0.470769
$315$	−2.53510e7	−0.0456992
$316$	3.57127e8	0.636674
$317$	−8.24512e8	−1.45375	−0.726874	−	0.686770i	$-0.759028\pi$
$317$	−8.24512e8	−1.45375	−0.726874	+	0.686770i	$0.759028\pi$
$318$	4.98204e7	0.0868785
$319$	3.15801e8	0.544686
$320$	−1.43772e8	−0.245273
$321$	−3.96089e8	−0.668383
$322$	−6.17173e6	−0.0103018
$323$	−1.72179e8	−0.284296
$324$	3.40122e7	0.0555556
$325$	2.12950e9	3.44101
$326$	−1.95361e8	−0.312304
$327$	4.18129e8	0.661292
$328$	−3.40205e8	−0.532332
$329$	6.56442e6	0.0101627
$330$	1.95557e8	0.299553
$331$	4.75019e7	0.0719968	0.0359984	−	0.999352i	$-0.488539\pi$
$331$	4.75019e7	0.0719968	0.0359984	+	0.999352i	$0.488539\pi$
$332$	−4.71497e8	−0.707124
$333$	9.60937e7	0.142607
$334$	6.23342e8	0.915406
$335$	1.34728e9	1.95794
$336$	−7.01224e6	−0.0100848
$337$	6.77271e8	0.963958	0.481979	−	0.876183i	$-0.339918\pi$
$337$	6.77271e8	0.963958	0.481979	+	0.876183i	$0.339918\pi$
$338$	−2.29695e8	−0.323552
$339$	−1.70776e8	−0.238083
$340$	−6.33912e8	−0.874687
$341$	−1.19405e8	−0.163072
$342$	5.56010e7	0.0751608
$343$	−1.04181e8	−0.139399
$344$	−2.36934e8	−0.313814
$345$	1.80170e8	0.236219
$346$	−2.31378e8	−0.300300
$347$	−3.74073e8	−0.480621	−0.240311	−	0.970696i	$-0.577249\pi$
$347$	−3.74073e8	−0.480621	−0.240311	+	0.970696i	$0.577249\pi$
$348$	3.30578e8	0.420481
$349$	−2.29630e8	−0.289160	−0.144580	−	0.989493i	$-0.546183\pi$
$349$	−2.29630e8	−0.289160	−0.144580	+	0.989493i	$0.546183\pi$
$350$	−1.12949e8	−0.140814
$351$	−1.88238e8	−0.232345
$352$	5.40921e7	0.0661050
$353$	6.87036e8	0.831320	0.415660	−	0.909520i	$-0.363551\pi$
$353$	6.87036e8	0.831320	0.415660	+	0.909520i	$0.363551\pi$
$354$	−6.47008e8	−0.775173
$355$	6.35646e7	0.0754078
$356$	4.66186e8	0.547627
$357$	−3.09180e7	−0.0359643
$358$	1.01972e9	1.17460
$359$	1.46020e8	0.166564	0.0832819	−	0.996526i	$-0.473460\pi$
$359$	1.46020e8	0.166564	0.0832819	+	0.996526i	$0.473460\pi$
$360$	2.04707e8	0.231246
$361$	−8.02979e8	−0.898315
$362$	−7.45314e8	−0.825770
$363$	4.52578e8	0.496616
$364$	3.88088e7	0.0421769
$365$	3.00178e9	3.23112
$366$	−2.47075e8	−0.263417
$367$	−1.61937e9	−1.71008	−0.855038	−	0.518565i	$-0.826466\pi$
$367$	−1.61937e9	−1.71008	−0.855038	+	0.518565i	$0.826466\pi$
$368$	4.98360e7	0.0521286
$369$	4.84394e8	0.501887
$370$	5.78352e8	0.593590
$371$	1.46247e7	0.0148689
$372$	−1.24992e8	−0.125887
$373$	−1.59164e9	−1.58805	−0.794027	−	0.607883i	$-0.792019\pi$
$373$	−1.59164e9	−1.58805	−0.794027	+	0.607883i	$0.792019\pi$
$374$	2.38500e8	0.235742
$375$	2.14043e9	2.09600
$376$	−5.30070e7	−0.0514252
$377$	−1.82956e9	−1.75854
$378$	9.98423e6	0.00950809
$379$	−5.95198e8	−0.561596	−0.280798	−	0.959767i	$-0.590599\pi$
$379$	−5.95198e8	−0.561596	−0.280798	+	0.959767i	$0.590599\pi$
$380$	3.34642e8	0.312851
$381$	1.02803e8	0.0952289
$382$	8.18648e8	0.751407
$383$	4.04524e8	0.367916	0.183958	−	0.982934i	$-0.441109\pi$
$383$	4.04524e8	0.367916	0.183958	+	0.982934i	$0.441109\pi$
$384$	5.66231e7	0.0510310
$385$	5.74053e7	0.0512672
$386$	9.61477e8	0.850909
$387$	3.37353e8	0.295867
$388$	1.72214e8	0.149677
$389$	4.48079e8	0.385950	0.192975	−	0.981204i	$-0.438186\pi$
$389$	4.48079e8	0.385950	0.192975	+	0.981204i	$0.438186\pi$
$390$	−1.13294e9	−0.967118
$391$	2.19734e8	0.185900
$392$	4.19596e8	0.351827
$393$	−9.20867e8	−0.765285
$394$	−2.73361e8	−0.225164
$395$	−3.06039e9	−2.49855
$396$	−7.70178e7	−0.0623244
$397$	1.35233e9	1.08472	0.542358	−	0.840148i	$-0.317532\pi$
$397$	1.35233e9	1.08472	0.542358	+	0.840148i	$0.317532\pi$
$398$	−1.78518e8	−0.141936
$399$	1.63216e7	0.0128634
$400$	9.12055e8	0.712543
$401$	1.34846e9	1.04432	0.522160	−	0.852847i	$-0.325126\pi$
$401$	1.34846e9	1.04432	0.522160	+	0.852847i	$0.325126\pi$
$402$	−5.30610e8	−0.407366
$403$	6.91758e8	0.526485
$404$	−6.50785e7	−0.0491024
$405$	−2.91467e8	−0.218021
$406$	9.70405e7	0.0719635
$407$	−2.17596e8	−0.159982
$408$	2.49659e8	0.181986
$409$	−2.13765e9	−1.54492	−0.772459	−	0.635065i	$-0.780973\pi$
$409$	−2.13765e9	−1.54492	−0.772459	+	0.635065i	$0.780973\pi$
$410$	2.91539e9	2.08907
$411$	1.05136e9	0.746975
$412$	4.46453e8	0.314511
$413$	−1.89928e8	−0.132667
$414$	−7.09579e7	−0.0491473
$415$	4.04049e9	2.77502
$416$	−3.13377e8	−0.213422
$417$	−9.15838e8	−0.618504
$418$	−1.25904e8	−0.0843183
$419$	2.00943e9	1.33452	0.667259	−	0.744826i	$-0.267467\pi$
$419$	2.00943e9	1.33452	0.667259	+	0.744826i	$0.267467\pi$
$420$	6.00913e7	0.0395767
$421$	−2.92407e9	−1.90986	−0.954928	−	0.296838i	$-0.904068\pi$
$421$	−2.92407e9	−1.90986	−0.954928	+	0.296838i	$0.904068\pi$
$422$	3.15070e8	0.204086
$423$	7.54729e7	0.0484841
$424$	−1.18093e8	−0.0752390
$425$	4.02138e9	2.54105
$426$	−2.50342e7	−0.0156892
$427$	−7.25283e7	−0.0450827
$428$	9.38878e8	0.578836
$429$	4.26250e8	0.260653
$430$	2.03040e9	1.23152
$431$	1.56942e9	0.944209	0.472104	−	0.881543i	$-0.343495\pi$
$431$	1.56942e9	0.944209	0.472104	+	0.881543i	$0.343495\pi$
$432$	−8.06216e7	−0.0481125
$433$	−8.87073e8	−0.525111	−0.262556	−	0.964917i	$-0.584565\pi$
$433$	−8.87073e8	−0.525111	−0.262556	+	0.964917i	$0.584565\pi$
$434$	−3.66911e7	−0.0215450
$435$	−2.83288e9	−1.65012
$436$	−9.91121e8	−0.572695
$437$	−1.15997e8	−0.0664911
$438$	−1.18222e9	−0.672261
$439$	1.57654e9	0.889362	0.444681	−	0.895689i	$-0.353317\pi$
$439$	1.57654e9	0.889362	0.444681	+	0.895689i	$0.353317\pi$
$440$	−4.63542e8	−0.259420
$441$	−5.97432e8	−0.331706
$442$	−1.38172e9	−0.761102
$443$	−1.68300e9	−0.919754	−0.459877	−	0.887983i	$-0.652106\pi$
$443$	−1.68300e9	−0.919754	−0.459877	+	0.887983i	$0.652106\pi$
$444$	−2.27778e8	−0.123501
$445$	−3.99498e9	−2.14909
$446$	2.06949e9	1.10456
$447$	−9.78753e8	−0.518318
$448$	1.66216e7	0.00873374
$449$	−2.28484e9	−1.19123	−0.595613	−	0.803272i	$-0.703091\pi$
$449$	−2.28484e9	−1.19123	−0.595613	+	0.803272i	$0.703091\pi$
$450$	−1.29861e9	−0.671792
$451$	−1.09687e9	−0.563037
$452$	4.04802e8	0.206186
$453$	−6.07716e8	−0.307154
$454$	6.76450e8	0.339266
$455$	−3.32571e8	−0.165518
$456$	−1.31795e8	−0.0650911
$457$	2.17527e9	1.06612	0.533061	−	0.846077i	$-0.321041\pi$
$457$	2.17527e9	1.06612	0.533061	+	0.846077i	$0.321041\pi$
$458$	−2.37666e9	−1.15595
$459$	−3.55472e8	−0.171578
$460$	−4.27069e8	−0.204572
$461$	−8.71164e8	−0.414139	−0.207070	−	0.978326i	$-0.566393\pi$
$461$	−8.71164e8	−0.414139	−0.207070	+	0.978326i	$0.566393\pi$
$462$	−2.26084e7	−0.0106665
$463$	−5.07281e8	−0.237528	−0.118764	−	0.992923i	$-0.537893\pi$
$463$	−5.07281e8	−0.237528	−0.118764	+	0.992923i	$0.537893\pi$
$464$	−7.83592e8	−0.364147
$465$	1.07112e9	0.494027
$466$	−1.71851e8	−0.0786687
$467$	1.86207e9	0.846033	0.423017	−	0.906122i	$-0.360971\pi$
$467$	1.86207e9	0.846033	0.423017	+	0.906122i	$0.360971\pi$
$468$	4.46194e8	0.201217
$469$	−1.55760e8	−0.0697188
$470$	4.54243e8	0.201811
$471$	−8.71637e8	−0.384382
$472$	1.53365e9	0.671319
$473$	−7.63907e8	−0.331915
$474$	1.20530e9	0.519843
$475$	−2.12288e9	−0.908863
$476$	7.32870e7	0.0311460
$477$	1.68144e8	0.0709360
$478$	−5.10209e8	−0.213674
$479$	−4.64799e9	−1.93237	−0.966187	−	0.257841i	$-0.916989\pi$
$479$	−4.64799e9	−1.93237	−0.966187	+	0.257841i	$0.916989\pi$
$480$	−4.85231e8	−0.200265
$481$	1.26062e9	0.516507
$482$	−7.33493e8	−0.298353
$483$	−2.08296e7	−0.00841135
$484$	−1.07278e9	−0.430082
$485$	−1.47578e9	−0.587390
$486$	1.14791e8	0.0453609
$487$	−2.57082e9	−1.00860	−0.504301	−	0.863528i	$-0.668250\pi$
$487$	−2.57082e9	−1.00860	−0.504301	+	0.863528i	$0.668250\pi$
$488$	5.85658e8	0.228126
$489$	−6.59345e8	−0.254995
$490$	−3.59572e9	−1.38070
$491$	−2.96037e9	−1.12865	−0.564326	−	0.825552i	$-0.690864\pi$
$491$	−2.96037e9	−1.12865	−0.564326	+	0.825552i	$0.690864\pi$
$492$	−1.14819e9	−0.434647
$493$	−3.45497e9	−1.29861
$494$	7.29410e8	0.272225
$495$	6.60003e8	0.244584
$496$	2.96277e8	0.109021
$497$	−7.34875e6	−0.00268514
$498$	−1.59130e9	−0.577364
$499$	−1.68201e9	−0.606004	−0.303002	−	0.952990i	$-0.597989\pi$
$499$	−1.68201e9	−0.606004	−0.303002	+	0.952990i	$0.597989\pi$
$500$	−5.07361e9	−1.81519
$501$	2.10378e9	0.747426
$502$	3.85984e9	1.36178
$503$	−1.53429e9	−0.537552	−0.268776	−	0.963203i	$-0.586619\pi$
$503$	−1.53429e9	−0.537552	−0.268776	+	0.963203i	$0.586619\pi$
$504$	−2.36663e7	−0.00823424
$505$	5.57690e8	0.192696
$506$	1.60678e8	0.0551354
$507$	−7.75222e8	−0.264179
$508$	−2.43681e8	−0.0824706
$509$	5.02515e9	1.68903	0.844514	−	0.535533i	$-0.179889\pi$
$509$	5.02515e9	1.68903	0.844514	+	0.535533i	$0.179889\pi$
$510$	−2.13945e9	−0.714179
$511$	−3.47038e8	−0.115054
$512$	−1.34218e8	−0.0441942
$513$	1.87653e8	0.0613685
$514$	1.94407e9	0.631454
$515$	−3.82588e9	−1.23426
$516$	−7.99651e8	−0.256228
$517$	−1.70902e8	−0.0543914
$518$	−6.68637e7	−0.0211367
$519$	−7.80899e8	−0.245194
$520$	2.68548e9	0.837548
$521$	3.12710e9	0.968744	0.484372	−	0.874862i	$-0.339048\pi$
$521$	3.12710e9	0.968744	0.484372	+	0.874862i	$0.339048\pi$
$522$	1.11570e9	0.343321
$523$	−2.97480e9	−0.909289	−0.454644	−	0.890673i	$-0.650234\pi$
$523$	−2.97480e9	−0.909289	−0.454644	+	0.890673i	$0.650234\pi$
$524$	2.18280e9	0.662756
$525$	−3.81204e8	−0.114974
$526$	1.70594e9	0.511109
$527$	1.30633e9	0.388789
$528$	1.82561e8	0.0539745
$529$	1.48036e8	0.0434783
$530$	1.01200e9	0.295266
$531$	−2.18365e9	−0.632926
$532$	−3.86882e7	−0.0111401
$533$	6.35459e9	1.81778
$534$	1.57338e9	0.447135
$535$	−8.04571e9	−2.27157
$536$	1.25774e9	0.352789
$537$	3.44155e9	0.959056
$538$	1.74238e9	0.482398
$539$	1.35283e9	0.372121
$540$	6.90886e8	0.188811
$541$	−1.96307e9	−0.533021	−0.266511	−	0.963832i	$-0.585871\pi$
$541$	−1.96307e9	−0.533021	−0.266511	+	0.963832i	$0.585871\pi$
$542$	3.82784e9	1.03266
$543$	−2.51543e9	−0.674239
$544$	−5.91785e8	−0.157604
$545$	8.49340e9	2.24747
$546$	1.30980e8	0.0344373
$547$	−6.68929e9	−1.74753	−0.873764	−	0.486350i	$-0.838328\pi$
$547$	−6.68929e9	−1.74753	−0.873764	+	0.486350i	$0.838328\pi$
$548$	−2.49212e9	−0.646899
$549$	−8.33877e8	−0.215079
$550$	2.94059e9	0.753642
$551$	1.82387e9	0.464477
$552$	1.68197e8	0.0425628
$553$	3.53814e8	0.0889687
$554$	2.04878e9	0.511931
$555$	1.95194e9	0.484664
$556$	2.17087e9	0.535640
$557$	3.66844e9	0.899473	0.449737	−	0.893161i	$-0.351518\pi$
$557$	3.66844e9	0.899473	0.449737	+	0.893161i	$0.351518\pi$
$558$	−4.21847e8	−0.102786
$559$	4.42561e9	1.07160
$560$	−1.42439e8	−0.0342744
$561$	8.04936e8	0.192483
$562$	−5.51185e9	−1.30985
$563$	9.57221e8	0.226065	0.113032	−	0.993591i	$-0.463944\pi$
$563$	9.57221e8	0.226065	0.113032	+	0.993591i	$0.463944\pi$
$564$	−1.78899e8	−0.0419885
$565$	−3.46895e9	−0.809149
$566$	9.89700e8	0.229428
$567$	3.36968e7	0.00776332
$568$	5.93404e7	0.0135872
$569$	−4.24668e9	−0.966399	−0.483199	−	0.875510i	$-0.660525\pi$
$569$	−4.24668e9	−0.966399	−0.483199	+	0.875510i	$0.660525\pi$
$570$	1.12942e9	0.255442
$571$	−7.03031e9	−1.58033	−0.790165	−	0.612894i	$-0.790005\pi$
$571$	−7.03031e9	−1.58033	−0.790165	+	0.612894i	$0.790005\pi$
$572$	−1.01037e9	−0.225733
$573$	2.76294e9	0.613521
$574$	−3.37050e8	−0.0743879
$575$	2.70922e9	0.594302
$576$	1.91103e8	0.0416667
$577$	3.83605e9	0.831320	0.415660	−	0.909520i	$-0.363551\pi$
$577$	3.83605e9	0.831320	0.415660	+	0.909520i	$0.363551\pi$
$578$	6.73445e8	0.145062
$579$	3.24498e9	0.694764
$580$	6.71498e9	1.42905
$581$	−4.67124e8	−0.0988133
$582$	5.81221e8	0.122211
$583$	−3.80748e8	−0.0795788
$584$	2.80229e9	0.582195
$585$	−3.82366e9	−0.789648
$586$	−3.53559e9	−0.725806
$587$	−2.55520e9	−0.521425	−0.260713	−	0.965416i	$-0.583958\pi$
$587$	−2.55520e9	−0.521425	−0.260713	+	0.965416i	$0.583958\pi$
$588$	1.41614e9	0.287266
$589$	−6.89608e8	−0.139059
$590$	−1.31426e10	−2.63451
$591$	−9.22593e8	−0.183846
$592$	5.39917e8	0.106955
$593$	−6.35795e9	−1.25206	−0.626031	−	0.779798i	$-0.715322\pi$
$593$	−6.35795e9	−1.25206	−0.626031	+	0.779798i	$0.715322\pi$
$594$	−2.59935e8	−0.0508876
$595$	−6.28033e8	−0.122229
$596$	2.32001e9	0.448877
$597$	−6.02498e8	−0.115890
$598$	−9.30872e8	−0.178007
$599$	3.84866e9	0.731672	0.365836	−	0.930679i	$-0.380783\pi$
$599$	3.84866e9	0.731672	0.365836	+	0.930679i	$0.380783\pi$
$600$	3.07818e9	0.581789
$601$	3.19100e9	0.599607	0.299803	−	0.954001i	$-0.403079\pi$
$601$	3.19100e9	0.599607	0.299803	+	0.954001i	$0.403079\pi$
$602$	−2.34736e8	−0.0438523
$603$	−1.79081e9	−0.332613
$604$	1.44051e9	0.266004
$605$	9.19317e9	1.68780
$606$	−2.19640e8	−0.0400919
$607$	3.17653e9	0.576490	0.288245	−	0.957557i	$-0.406928\pi$
$607$	3.17653e9	0.576490	0.288245	+	0.957557i	$0.406928\pi$
$608$	3.12403e8	0.0563706
$609$	3.27512e8	0.0587579
$610$	−5.01879e9	−0.895251
$611$	9.90102e8	0.175605
$612$	8.42600e8	0.148591
$613$	−5.74452e9	−1.00726	−0.503631	−	0.863919i	$-0.668003\pi$
$613$	−5.74452e9	−1.00726	−0.503631	+	0.863919i	$0.668003\pi$
$614$	7.05139e9	1.22938
$615$	9.83942e9	1.70572
$616$	5.35904e7	0.00923750
$617$	−2.88104e9	−0.493800	−0.246900	−	0.969041i	$-0.579412\pi$
$617$	−2.88104e9	−0.493800	−0.246900	+	0.969041i	$0.579412\pi$
$618$	1.50678e9	0.256797
$619$	−5.67714e9	−0.962083	−0.481042	−	0.876698i	$-0.659741\pi$
$619$	−5.67714e9	−0.962083	−0.481042	+	0.876698i	$0.659741\pi$
$620$	−2.53894e9	−0.427840
$621$	−2.39483e8	−0.0401286
$622$	−2.31669e9	−0.386013
$623$	4.61863e8	0.0765252
$624$	−1.05765e9	−0.174259
$625$	2.60822e10	4.27330
$626$	2.01882e9	0.328918
$627$	−4.24925e8	−0.0688456
$628$	2.06610e9	0.332884
$629$	2.38057e9	0.381420
$630$	2.02808e8	0.0323142
$631$	7.98061e9	1.26454	0.632271	−	0.774747i	$-0.282123\pi$
$631$	7.98061e9	1.26454	0.632271	+	0.774747i	$0.282123\pi$
$632$	−2.85701e9	−0.450197
$633$	1.06336e9	0.166636
$634$	6.59609e9	1.02796
$635$	2.08823e9	0.323645
$636$	−3.98563e8	−0.0614324
$637$	−7.83750e9	−1.20141
$638$	−2.52641e9	−0.385151
$639$	−8.44905e7	−0.0128102
$640$	1.15018e9	0.173434
$641$	−8.53014e9	−1.27924	−0.639622	−	0.768690i	$-0.720909\pi$
$641$	−8.53014e9	−1.27924	−0.639622	+	0.768690i	$0.720909\pi$
$642$	3.16871e9	0.472618
$643$	−3.54651e9	−0.526094	−0.263047	−	0.964783i	$-0.584727\pi$
$643$	−3.54651e9	−0.526094	−0.263047	+	0.964783i	$0.584727\pi$
$644$	4.93738e7	0.00728444
$645$	6.85260e9	1.00553
$646$	1.37743e9	0.201027
$647$	6.01216e9	0.872702	0.436351	−	0.899777i	$-0.356271\pi$
$647$	6.01216e9	0.872702	0.436351	+	0.899777i	$0.356271\pi$
$648$	−2.72098e8	−0.0392837
$649$	4.94470e9	0.710041
$650$	−1.70360e10	−2.43316
$651$	−1.23832e8	−0.0175914
$652$	1.56289e9	0.220832
$653$	6.26225e9	0.880104	0.440052	−	0.897972i	$-0.354960\pi$
$653$	6.26225e9	0.880104	0.440052	+	0.897972i	$0.354960\pi$
$654$	−3.34503e9	−0.467604
$655$	−1.87055e10	−2.60090
$656$	2.72164e9	0.376415
$657$	−3.98998e9	−0.548899
$658$	−5.25154e7	−0.00718614
$659$	−1.16455e10	−1.58511	−0.792556	−	0.609799i	$-0.791250\pi$
$659$	−1.16455e10	−1.58511	−0.792556	+	0.609799i	$0.791250\pi$
$660$	−1.56445e9	−0.211816
$661$	6.24384e9	0.840905	0.420453	−	0.907315i	$-0.361871\pi$
$661$	6.24384e9	0.840905	0.420453	+	0.907315i	$0.361871\pi$
$662$	−3.80016e8	−0.0509094
$663$	−4.66331e9	−0.621437
$664$	3.77197e9	0.500012
$665$	3.31538e8	0.0437177
$666$	−7.68749e8	−0.100838
$667$	−2.32763e9	−0.303720
$668$	−4.98674e9	−0.647290
$669$	6.98452e9	0.901872
$670$	−1.07782e10	−1.38448
$671$	1.88824e9	0.241284
$672$	5.60979e7	0.00713106
$673$	1.43122e10	1.80990	0.904951	−	0.425516i	$-0.139907\pi$
$673$	1.43122e10	1.80990	0.904951	+	0.425516i	$0.139907\pi$
$674$	−5.41817e9	−0.681621
$675$	−4.38281e9	−0.548516
$676$	1.83756e9	0.228786
$677$	−6.54450e9	−0.810618	−0.405309	−	0.914180i	$-0.632836\pi$
$677$	−6.54450e9	−0.810618	−0.405309	+	0.914180i	$0.632836\pi$
$678$	1.36621e9	0.168350
$679$	1.70616e8	0.0209159
$680$	5.07130e9	0.618497
$681$	2.28302e9	0.277010
$682$	9.55237e8	0.115310
$683$	8.54497e9	1.02621	0.513107	−	0.858325i	$-0.328494\pi$
$683$	8.54497e9	1.02621	0.513107	+	0.858325i	$0.328494\pi$
$684$	−4.44808e8	−0.0531467
$685$	2.13562e10	2.53867
$686$	8.33447e8	0.0985697
$687$	−8.02124e9	−0.943829
$688$	1.89547e9	0.221900
$689$	2.20582e9	0.256923
$690$	−1.44136e9	−0.167032
$691$	1.59908e10	1.84373	0.921865	−	0.387511i	$-0.126665\pi$
$691$	1.59908e10	1.84373	0.921865	+	0.387511i	$0.126665\pi$
$692$	1.85102e9	0.212344
$693$	−7.63035e7	−0.00870919
$694$	2.99258e9	0.339851
$695$	−1.86033e10	−2.10205
$696$	−2.64462e9	−0.297325
$697$	1.20001e10	1.34236
$698$	1.83704e9	0.204467
$699$	−5.79998e8	−0.0642328
$700$	9.03596e8	0.0995706
$701$	7.66410e9	0.840327	0.420163	−	0.907449i	$-0.361973\pi$
$701$	7.66410e9	0.840327	0.420163	+	0.907449i	$0.361973\pi$
$702$	1.50591e9	0.164293
$703$	−1.25670e9	−0.136423
$704$	−4.32737e8	−0.0467433
$705$	1.53307e9	0.164778
$706$	−5.49629e9	−0.587832
$707$	−6.44749e7	−0.00686156
$708$	5.17607e9	0.548130
$709$	−1.34684e10	−1.41924	−0.709620	−	0.704585i	$-0.751133\pi$
$709$	−1.34684e10	−1.41924	−0.709620	+	0.704585i	$0.751133\pi$
$710$	−5.08517e8	−0.0533214
$711$	4.06790e9	0.424450
$712$	−3.72949e9	−0.387231
$713$	8.80077e8	0.0909301
$714$	2.47344e8	0.0254306
$715$	8.65835e9	0.885858
$716$	−8.15774e9	−0.830567
$717$	−1.72196e9	−0.174464
$718$	−1.16816e9	−0.117778
$719$	1.61467e10	1.62007	0.810033	−	0.586384i	$-0.199449\pi$
$719$	1.61467e10	1.62007	0.810033	+	0.586384i	$0.199449\pi$
$720$	−1.63766e9	−0.163515
$721$	4.42312e8	0.0439497
$722$	6.42383e9	0.635205
$723$	−2.47554e9	−0.243605
$724$	5.96251e9	0.583908
$725$	−4.25982e10	−4.15153
$726$	−3.62063e9	−0.351161
$727$	−7.36170e9	−0.710572	−0.355286	−	0.934758i	$-0.615616\pi$
$727$	−7.36170e9	−0.710572	−0.355286	+	0.934758i	$0.615616\pi$
$728$	−3.10470e8	−0.0298236
$729$	3.87420e8	0.0370370
$730$	−2.40142e10	−2.28475
$731$	8.35739e9	0.791334
$732$	1.97660e9	0.186264
$733$	1.02774e10	0.963873	0.481936	−	0.876206i	$-0.339934\pi$
$733$	1.02774e10	0.963873	0.481936	+	0.876206i	$0.339934\pi$
$734$	1.29550e10	1.20921
$735$	−1.21356e10	−1.12734
$736$	−3.98688e8	−0.0368605
$737$	4.05514e9	0.373138
$738$	−3.87515e9	−0.354888
$739$	1.14006e10	1.03913	0.519567	−	0.854430i	$-0.326093\pi$
$739$	1.14006e10	1.03913	0.519567	+	0.854430i	$0.326093\pi$
$740$	−4.62682e9	−0.419731
$741$	2.46176e9	0.222270
$742$	−1.16998e8	−0.0105139
$743$	−1.70791e10	−1.52758	−0.763792	−	0.645463i	$-0.776665\pi$
$743$	−1.70791e10	−1.52758	−0.763792	+	0.645463i	$0.776665\pi$
$744$	9.99933e8	0.0890155
$745$	−1.98813e10	−1.76156
$746$	1.27332e10	1.12292
$747$	−5.37064e9	−0.471416
$748$	−1.90800e9	−0.166695
$749$	9.30170e8	0.0808865
$750$	−1.71234e10	−1.48210
$751$	4.53499e9	0.390694	0.195347	−	0.980734i	$-0.437417\pi$
$751$	4.53499e9	0.390694	0.195347	+	0.980734i	$0.437417\pi$
$752$	4.24056e8	0.0363631
$753$	1.30270e10	1.11189
$754$	1.46365e10	1.24347
$755$	−1.23445e10	−1.04390
$756$	−7.98738e7	−0.00672323
$757$	1.01610e10	0.851338	0.425669	−	0.904879i	$-0.360039\pi$
$757$	1.01610e10	0.851338	0.425669	+	0.904879i	$0.360039\pi$
$758$	4.76158e9	0.397109
$759$	5.42289e8	0.0450178
$760$	−2.67713e9	−0.221219
$761$	−1.15049e10	−0.946316	−0.473158	−	0.880978i	$-0.656886\pi$
$761$	−1.15049e10	−0.946316	−0.473158	+	0.880978i	$0.656886\pi$
$762$	−8.22425e8	−0.0673370
$763$	−9.81928e8	−0.0800283
$764$	−6.54918e9	−0.531325
$765$	−7.22065e9	−0.583125
$766$	−3.23619e9	−0.260156
$767$	−2.86466e10	−2.29239
$768$	−4.52985e8	−0.0360844
$769$	−6.51988e9	−0.517008	−0.258504	−	0.966010i	$-0.583230\pi$
$769$	−6.51988e9	−0.517008	−0.258504	+	0.966010i	$0.583230\pi$
$770$	−4.59242e8	−0.0362514
$771$	6.56125e9	0.515580
$772$	−7.69181e9	−0.601684
$773$	1.59454e9	0.124168	0.0620838	−	0.998071i	$-0.480225\pi$
$773$	1.59454e9	0.124168	0.0620838	+	0.998071i	$0.480225\pi$
$774$	−2.69882e9	−0.209209
$775$	1.61064e10	1.24292
$776$	−1.37771e9	−0.105838
$777$	−2.25665e8	−0.0172580
$778$	−3.58463e9	−0.272908
$779$	−6.33484e9	−0.480126
$780$	9.06349e9	0.683855
$781$	1.91322e8	0.0143710
$782$	−1.75787e9	−0.131451
$783$	3.76549e9	0.280321
$784$	−3.35676e9	−0.248780
$785$	−1.77054e10	−1.30636
$786$	7.36694e9	0.541138
$787$	8.83747e9	0.646274	0.323137	−	0.946352i	$-0.395263\pi$
$787$	8.83747e9	0.646274	0.323137	+	0.946352i	$0.395263\pi$
$788$	2.18689e9	0.159215
$789$	5.75755e9	0.417319
$790$	2.44831e10	1.76674
$791$	4.01048e8	0.0288123
$792$	6.16142e8	0.0440700
$793$	−1.09393e10	−0.778995
$794$	−1.08186e10	−0.767009
$795$	3.41548e9	0.241083
$796$	1.42814e9	0.100364
$797$	−1.03548e9	−0.0724499	−0.0362250	−	0.999344i	$-0.511533\pi$
$797$	−1.03548e9	−0.0724499	−0.0362250	+	0.999344i	$0.511533\pi$
$798$	−1.30573e8	−0.00909582
$799$	1.86972e9	0.129677
$800$	−7.29644e9	−0.503844
$801$	5.31015e9	0.365084
$802$	−1.07877e10	−0.738446
$803$	9.03498e9	0.615776
$804$	4.24488e9	0.288051
$805$	−4.23108e8	−0.0285868
$806$	−5.53406e9	−0.372281
$807$	5.88054e9	0.393876
$808$	5.20628e8	0.0347206
$809$	1.94015e10	1.28830	0.644148	−	0.764901i	$-0.277212\pi$
$809$	1.94015e10	1.28830	0.644148	+	0.764901i	$0.277212\pi$
$810$	2.33174e9	0.154164
$811$	−3.37608e9	−0.222249	−0.111125	−	0.993806i	$-0.535445\pi$
$811$	−3.37608e9	−0.222249	−0.111125	+	0.993806i	$0.535445\pi$
$812$	−7.76324e8	−0.0508858
$813$	1.29190e10	0.843162
$814$	1.74077e9	0.113124
$815$	−1.33932e10	−0.866627
$816$	−1.99727e9	−0.128683
$817$	−4.41186e9	−0.283038
$818$	1.71012e10	1.09242
$819$	4.42056e8	0.0281180
$820$	−2.33231e10	−1.47719
$821$	−2.21974e9	−0.139991	−0.0699956	−	0.997547i	$-0.522299\pi$
$821$	−2.21974e9	−0.139991	−0.0699956	+	0.997547i	$0.522299\pi$
$822$	−8.41090e9	−0.528191
$823$	5.13698e9	0.321224	0.160612	−	0.987018i	$-0.448653\pi$
$823$	5.13698e9	0.321224	0.160612	+	0.987018i	$0.448653\pi$
$824$	−3.57163e9	−0.222393
$825$	9.92449e9	0.615346
$826$	1.51943e9	0.0938100
$827$	2.06839e10	1.27164	0.635818	−	0.771839i	$-0.280663\pi$
$827$	2.06839e10	1.27164	0.635818	+	0.771839i	$0.280663\pi$
$828$	5.67664e8	0.0347524
$829$	−9.72254e8	−0.0592706	−0.0296353	−	0.999561i	$-0.509435\pi$
$829$	−9.72254e8	−0.0592706	−0.0296353	+	0.999561i	$0.509435\pi$
$830$	−3.23239e10	−1.96223
$831$	6.91464e9	0.417990
$832$	2.50701e9	0.150912
$833$	−1.48005e10	−0.887192
$834$	7.32670e9	0.437348
$835$	4.27338e10	2.54020
$836$	1.00723e9	0.0596220
$837$	−1.42373e9	−0.0839247
$838$	−1.60755e10	−0.943646
$839$	3.17070e10	1.85348	0.926740	−	0.375703i	$-0.122599\pi$
$839$	3.17070e10	1.85348	0.926740	+	0.375703i	$0.122599\pi$
$840$	−4.80731e8	−0.0279849
$841$	1.93483e10	1.12165
$842$	2.33926e10	1.35047
$843$	−1.86025e10	−1.06948
$844$	−2.52056e9	−0.144311
$845$	−1.57470e10	−0.897840
$846$	−6.03783e8	−0.0342835
$847$	−1.06283e9	−0.0600996
$848$	9.44743e8	0.0532020
$849$	3.34024e9	0.187327
$850$	−3.21710e10	−1.79680
$851$	1.60380e9	0.0892067
$852$	2.00274e8	0.0110939
$853$	−1.50663e10	−0.831158	−0.415579	−	0.909557i	$-0.636421\pi$
$853$	−1.50663e10	−0.831158	−0.415579	+	0.909557i	$0.636421\pi$
$854$	5.80226e8	0.0318783
$855$	3.81178e9	0.208567
$856$	−7.51102e9	−0.409299
$857$	6.26684e8	0.0340107	0.0170054	−	0.999855i	$-0.494587\pi$
$857$	6.26684e8	0.0340107	0.0170054	+	0.999855i	$0.494587\pi$
$858$	−3.41000e9	−0.184310
$859$	−3.61840e10	−1.94778	−0.973891	−	0.227015i	$-0.927103\pi$
$859$	−3.61840e10	−1.94778	−0.973891	+	0.227015i	$0.927103\pi$
$860$	−1.62432e10	−0.870818
$861$	−1.13754e9	−0.0607375
$862$	−1.25553e10	−0.667656
$863$	−1.58528e10	−0.839593	−0.419797	−	0.907618i	$-0.637899\pi$
$863$	−1.58528e10	−0.839593	−0.419797	+	0.907618i	$0.637899\pi$
$864$	6.44973e8	0.0340207
$865$	−1.58623e10	−0.833316
$866$	7.09658e9	0.371310
$867$	2.27288e9	0.118443
$868$	2.93529e8	0.0152346
$869$	−9.21141e9	−0.476164
$870$	2.26631e10	1.16681
$871$	−2.34930e10	−1.20469
$872$	7.92896e9	0.404957
$873$	1.96162e9	0.0997849
$874$	9.27980e8	0.0470163
$875$	−5.02655e9	−0.253654
$876$	9.45773e9	0.475360
$877$	−7.20558e9	−0.360720	−0.180360	−	0.983601i	$-0.557726\pi$
$877$	−7.20558e9	−0.360720	−0.180360	+	0.983601i	$0.557726\pi$
$878$	−1.26123e10	−0.628874
$879$	−1.19326e10	−0.592618
$880$	3.70833e9	0.183438
$881$	2.57658e10	1.26948	0.634742	−	0.772724i	$-0.281106\pi$
$881$	2.57658e10	1.26948	0.634742	+	0.772724i	$0.281106\pi$
$882$	4.77946e9	0.234552
$883$	−2.43904e10	−1.19222	−0.596110	−	0.802902i	$-0.703288\pi$
$883$	−2.43904e10	−1.19222	−0.596110	+	0.802902i	$0.703288\pi$
$884$	1.10538e10	0.538180
$885$	−4.43563e10	−2.15106
$886$	1.34640e10	0.650364
$887$	3.24346e10	1.56054	0.780271	−	0.625441i	$-0.215081\pi$
$887$	3.24346e10	1.56054	0.780271	+	0.625441i	$0.215081\pi$
$888$	1.82222e9	0.0873284
$889$	−2.41421e8	−0.0115244
$890$	3.19598e10	1.51964
$891$	−8.77281e8	−0.0415496
$892$	−1.65559e10	−0.781044
$893$	−9.87025e8	−0.0463819
$894$	7.83002e9	0.366506
$895$	6.99077e10	3.25945
$896$	−1.32973e8	−0.00617568
$897$	−3.14169e9	−0.145342
$898$	1.82787e10	0.842324
$899$	−1.38378e10	−0.635197
$900$	1.03889e10	0.475029
$901$	4.16550e9	0.189728
$902$	8.77495e9	0.398127
$903$	−7.92234e8	−0.0358052
$904$	−3.23842e9	−0.145795
$905$	−5.10957e10	−2.29147
$906$	4.86173e9	0.217191
$907$	−1.90648e10	−0.848413	−0.424207	−	0.905565i	$-0.639447\pi$
$907$	−1.90648e10	−0.848413	−0.424207	+	0.905565i	$0.639447\pi$
$908$	−5.41160e9	−0.239897
$909$	−7.41285e8	−0.0327349
$910$	2.66057e9	0.117039
$911$	−8.45592e9	−0.370550	−0.185275	−	0.982687i	$-0.559318\pi$
$911$	−8.45592e9	−0.370550	−0.185275	+	0.982687i	$0.559318\pi$
$912$	1.05436e9	0.0460264
$913$	1.21614e10	0.528853
$914$	−1.74022e10	−0.753863
$915$	−1.69384e10	−0.730970
$916$	1.90133e10	0.817380
$917$	2.16255e9	0.0926134
$918$	2.84378e9	0.121324
$919$	−2.46250e10	−1.04658	−0.523288	−	0.852156i	$-0.675295\pi$
$919$	−2.46250e10	−1.04658	−0.523288	+	0.852156i	$0.675295\pi$
$920$	3.41656e9	0.144654
$921$	2.37984e10	1.00378
$922$	6.96931e9	0.292841
$923$	−1.10840e9	−0.0463971
$924$	1.80867e8	0.00754238
$925$	2.93514e10	1.21936
$926$	4.05825e9	0.167958
$927$	5.08538e9	0.209674
$928$	6.26874e9	0.257491
$929$	−3.43924e9	−0.140737	−0.0703684	−	0.997521i	$-0.522417\pi$
$929$	−3.43924e9	−0.140737	−0.0703684	+	0.997521i	$0.522417\pi$
$930$	−8.56892e9	−0.349330
$931$	7.81315e9	0.317323
$932$	1.37481e9	0.0556272
$933$	−7.81884e9	−0.315178
$934$	−1.48966e10	−0.598236
$935$	1.63506e10	0.654172
$936$	−3.56956e9	−0.142282
$937$	1.18204e10	0.469400	0.234700	−	0.972068i	$-0.424589\pi$
$937$	1.18204e10	0.469400	0.234700	+	0.972068i	$0.424589\pi$
$938$	1.24608e9	0.0492987
$939$	6.81352e9	0.268560
$940$	−3.63394e9	−0.142702
$941$	2.51251e10	0.982979	0.491490	−	0.870883i	$-0.336453\pi$
$941$	2.51251e10	0.982979	0.491490	+	0.870883i	$0.336453\pi$
$942$	6.97310e9	0.271799
$943$	8.08452e9	0.313952
$944$	−1.22692e10	−0.474694
$945$	6.84478e8	0.0263845
$946$	6.11126e9	0.234699
$947$	−8.76211e9	−0.335262	−0.167631	−	0.985850i	$-0.553612\pi$
$947$	−8.76211e9	−0.335262	−0.167631	+	0.985850i	$0.553612\pi$
$948$	−9.64242e9	−0.367584
$949$	−5.23432e10	−1.98805
$950$	1.69831e10	0.642663
$951$	2.22618e10	0.839322
$952$	−5.86296e8	−0.0220236
$953$	−2.46574e10	−0.922829	−0.461415	−	0.887185i	$-0.652658\pi$
$953$	−2.46574e10	−0.922829	−0.461415	+	0.887185i	$0.652658\pi$
$954$	−1.34515e9	−0.0501593
$955$	5.61232e10	2.08512
$956$	4.08167e9	0.151090
$957$	−8.52663e9	−0.314475
$958$	3.71840e10	1.36640
$959$	−2.46900e9	−0.0903975
$960$	3.88185e9	0.141609
$961$	−2.22805e10	−0.809830
$962$	−1.00850e10	−0.365226
$963$	1.06944e10	0.385891
$964$	5.86794e9	0.210968
$965$	6.59149e10	2.36123
$966$	1.66637e8	0.00594772
$967$	−3.75887e10	−1.33680	−0.668398	−	0.743804i	$-0.733020\pi$
$967$	−3.75887e10	−1.33680	−0.668398	+	0.743804i	$0.733020\pi$
$968$	8.58223e9	0.304114
$969$	4.64882e9	0.164138
$970$	1.18063e10	0.415347
$971$	3.59460e10	1.26004	0.630019	−	0.776580i	$-0.283047\pi$
$971$	3.59460e10	1.26004	0.630019	+	0.776580i	$0.283047\pi$
$972$	−9.18330e8	−0.0320750
$973$	2.15074e9	0.0748502
$974$	2.05665e10	0.713189
$975$	−5.74965e10	−1.98667
$976$	−4.68527e9	−0.161310
$977$	−6.88565e9	−0.236219	−0.118109	−	0.993001i	$-0.537683\pi$
$977$	−6.88565e9	−0.236219	−0.118109	+	0.993001i	$0.537683\pi$
$978$	5.27476e9	0.180309
$979$	−1.20244e10	−0.409566
$980$	2.87658e10	0.976303
$981$	−1.12895e10	−0.381797
$982$	2.36829e10	0.798078
$983$	−3.85623e10	−1.29487	−0.647435	−	0.762121i	$-0.724158\pi$
$983$	−3.85623e10	−1.29487	−0.647435	+	0.762121i	$0.724158\pi$
$984$	9.18554e9	0.307342
$985$	−1.87405e10	−0.624819
$986$	2.76397e10	0.918258
$987$	−1.77239e8	−0.00586746
$988$	−5.83528e9	−0.192492
$989$	5.63041e9	0.185077
$990$	−5.28003e9	−0.172947
$991$	3.07752e10	1.00449	0.502243	−	0.864727i	$-0.332508\pi$
$991$	3.07752e10	1.00449	0.502243	+	0.864727i	$0.332508\pi$
$992$	−2.37021e9	−0.0770897
$993$	−1.28255e9	−0.0415674
$994$	5.87900e7	0.00189868
$995$	−1.22385e10	−0.393864
$996$	1.27304e10	0.408258
$997$	2.17354e10	0.694599	0.347299	−	0.937754i	$-0.387099\pi$
$997$	2.17354e10	0.694599	0.347299	+	0.937754i	$0.387099\pi$
$998$	1.34561e10	0.428510
$999$	−2.59453e9	−0.0823340

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	138.8.a.e.1.1	✓	4
3.2	odd	2		414.8.a.k.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.8.a.e.1.1	✓	4	1.1	even	1	trivial
414.8.a.k.1.4		4	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 138 = 2 \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	138.a (trivial)

\(f(q)\)	\(=\)	\(q-8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} -548.447 q^{5} +216.000 q^{6} +63.4064 q^{7} -512.000 q^{8} +729.000 q^{9} +O(q^{10})\)	\(q-8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} -548.447 q^{5} +216.000 q^{6} +63.4064 q^{7} -512.000 q^{8} +729.000 q^{9} +4387.58 q^{10} -1650.76 q^{11} -1728.00 q^{12} +9563.50 q^{13} -507.251 q^{14} +14808.1 q^{15} +4096.00 q^{16} +18059.8 q^{17} -5832.00 q^{18} -9533.78 q^{19} -35100.6 q^{20} -1711.97 q^{21} +13206.1 q^{22} +12167.0 q^{23} +13824.0 q^{24} +222670. q^{25} -76508.0 q^{26} -19683.0 q^{27} +4058.01 q^{28} -191307. q^{29} -118465. q^{30} +72333.1 q^{31} -32768.0 q^{32} +44570.5 q^{33} -144479. q^{34} -34775.1 q^{35} +46656.0 q^{36} +131816. q^{37} +76270.2 q^{38} -258214. q^{39} +280805. q^{40} +664463. q^{41} +13695.8 q^{42} +462761. q^{43} -105649. q^{44} -399818. q^{45} -97336.0 q^{46} +103529. q^{47} -110592. q^{48} -819523. q^{49} -1.78136e6 q^{50} -487616. q^{51} +612064. q^{52} +230650. q^{53} +157464. q^{54} +905355. q^{55} -32464.1 q^{56} +257412. q^{57} +1.53045e6 q^{58} -2.99541e6 q^{59} +947717. q^{60} -1.14386e6 q^{61} -578665. q^{62} +46223.3 q^{63} +262144. q^{64} -5.24507e6 q^{65} -356564. q^{66} -2.45653e6 q^{67} +1.15583e6 q^{68} -328509. q^{69} +278201. q^{70} -115899. q^{71} -373248. q^{72} -5.47323e6 q^{73} -1.05453e6 q^{74} -6.01208e6 q^{75} -610162. q^{76} -104669. q^{77} +2.06571e6 q^{78} +5.58010e6 q^{79} -2.24644e6 q^{80} +531441. q^{81} -5.31571e6 q^{82} -7.36714e6 q^{83} -109566. q^{84} -9.90488e6 q^{85} -3.70209e6 q^{86} +5.16528e6 q^{87} +845189. q^{88} +7.28416e6 q^{89} +3.19855e6 q^{90} +606387. q^{91} +778688. q^{92} -1.95299e6 q^{93} -828234. q^{94} +5.22878e6 q^{95} +884736. q^{96} +2.69084e6 q^{97} +6.55618e6 q^{98} -1.20340e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 32 q^{2} - 108 q^{3} + 256 q^{4} - 90 q^{5} + 864 q^{6} - 222 q^{7} - 2048 q^{8} + 2916 q^{9}+O(q^{10}) \) 4 * q - 32 * q^2 - 108 * q^3 + 256 * q^4 - 90 * q^5 + 864 * q^6 - 222 * q^7 - 2048 * q^8 + 2916 * q^9	\( 4 q - 32 q^{2} - 108 q^{3} + 256 q^{4} - 90 q^{5} + 864 q^{6} - 222 q^{7} - 2048 q^{8} + 2916 q^{9} + 720 q^{10} - 4120 q^{11} - 6912 q^{12} - 6796 q^{13} + 1776 q^{14} + 2430 q^{15} + 16384 q^{16} + 18918 q^{17} - 23328 q^{18} + 19682 q^{19} - 5760 q^{20} + 5994 q^{21} + 32960 q^{22} + 48668 q^{23} + 55296 q^{24} + 261304 q^{25} + 54368 q^{26} - 78732 q^{27} - 14208 q^{28} + 12756 q^{29} - 19440 q^{30} + 440892 q^{31} - 131072 q^{32} + 111240 q^{33} - 151344 q^{34} + 150200 q^{35} + 186624 q^{36} + 567212 q^{37} - 157456 q^{38} + 183492 q^{39} + 46080 q^{40} + 275056 q^{41} - 47952 q^{42} + 149946 q^{43} - 263680 q^{44} - 65610 q^{45} - 389344 q^{46} - 2000728 q^{47} - 442368 q^{48} - 1623352 q^{49} - 2090432 q^{50} - 510786 q^{51} - 434944 q^{52} - 2778518 q^{53} + 629856 q^{54} - 2706008 q^{55} + 113664 q^{56} - 531414 q^{57} - 102048 q^{58} - 3656368 q^{59} + 155520 q^{60} - 989076 q^{61} - 3527136 q^{62} - 161838 q^{63} + 1048576 q^{64} - 7481556 q^{65} - 889920 q^{66} - 5755282 q^{67} + 1210752 q^{68} - 1314036 q^{69} - 1201600 q^{70} - 8843424 q^{71} - 1492992 q^{72} - 1100272 q^{73} - 4537696 q^{74} - 7055208 q^{75} + 1259648 q^{76} - 10973480 q^{77} - 1467936 q^{78} + 959830 q^{79} - 368640 q^{80} + 2125764 q^{81} - 2200448 q^{82} - 24990584 q^{83} + 383616 q^{84} - 7601436 q^{85} - 1199568 q^{86} - 344412 q^{87} + 2109440 q^{88} - 3623430 q^{89} + 524880 q^{90} - 9436372 q^{91} + 3114752 q^{92} - 11904084 q^{93} + 16005824 q^{94} - 12385728 q^{95} + 3538944 q^{96} + 14161716 q^{97} + 12986816 q^{98} - 3003480 q^{99}+O(q^{100}) \) 4 * q - 32 * q^2 - 108 * q^3 + 256 * q^4 - 90 * q^5 + 864 * q^6 - 222 * q^7 - 2048 * q^8 + 2916 * q^9 + 720 * q^10 - 4120 * q^11 - 6912 * q^12 - 6796 * q^13 + 1776 * q^14 + 2430 * q^15 + 16384 * q^16 + 18918 * q^17 - 23328 * q^18 + 19682 * q^19 - 5760 * q^20 + 5994 * q^21 + 32960 * q^22 + 48668 * q^23 + 55296 * q^24 + 261304 * q^25 + 54368 * q^26 - 78732 * q^27 - 14208 * q^28 + 12756 * q^29 - 19440 * q^30 + 440892 * q^31 - 131072 * q^32 + 111240 * q^33 - 151344 * q^34 + 150200 * q^35 + 186624 * q^36 + 567212 * q^37 - 157456 * q^38 + 183492 * q^39 + 46080 * q^40 + 275056 * q^41 - 47952 * q^42 + 149946 * q^43 - 263680 * q^44 - 65610 * q^45 - 389344 * q^46 - 2000728 * q^47 - 442368 * q^48 - 1623352 * q^49 - 2090432 * q^50 - 510786 * q^51 - 434944 * q^52 - 2778518 * q^53 + 629856 * q^54 - 2706008 * q^55 + 113664 * q^56 - 531414 * q^57 - 102048 * q^58 - 3656368 * q^59 + 155520 * q^60 - 989076 * q^61 - 3527136 * q^62 - 161838 * q^63 + 1048576 * q^64 - 7481556 * q^65 - 889920 * q^66 - 5755282 * q^67 + 1210752 * q^68 - 1314036 * q^69 - 1201600 * q^70 - 8843424 * q^71 - 1492992 * q^72 - 1100272 * q^73 - 4537696 * q^74 - 7055208 * q^75 + 1259648 * q^76 - 10973480 * q^77 - 1467936 * q^78 + 959830 * q^79 - 368640 * q^80 + 2125764 * q^81 - 2200448 * q^82 - 24990584 * q^83 + 383616 * q^84 - 7601436 * q^85 - 1199568 * q^86 - 344412 * q^87 + 2109440 * q^88 - 3623430 * q^89 + 524880 * q^90 - 9436372 * q^91 + 3114752 * q^92 - 11904084 * q^93 + 16005824 * q^94 - 12385728 * q^95 + 3538944 * q^96 + 14161716 * q^97 + 12986816 * q^98 - 3003480 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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