Properties

Label 285.10.a.a.1.1
Level $285$
Weight $10$
Character 285.1
Self dual yes
Analytic conductor $146.785$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [285,10,Mod(1,285)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(285, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("285.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 285 = 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 285.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: \(146.785213307\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 285.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-6.00000 q^{2} +81.0000 q^{3} -476.000 q^{4} -625.000 q^{5} -486.000 q^{6} -5866.00 q^{7} +5928.00 q^{8} +6561.00 q^{9} +O(q^{10})\) \(q-6.00000 q^{2} +81.0000 q^{3} -476.000 q^{4} -625.000 q^{5} -486.000 q^{6} -5866.00 q^{7} +5928.00 q^{8} +6561.00 q^{9} +3750.00 q^{10} -17982.0 q^{11} -38556.0 q^{12} -61720.0 q^{13} +35196.0 q^{14} -50625.0 q^{15} +208144. q^{16} +290010. q^{17} -39366.0 q^{18} +130321. q^{19} +297500. q^{20} -475146. q^{21} +107892. q^{22} -670584. q^{23} +480168. q^{24} +390625. q^{25} +370320. q^{26} +531441. q^{27} +2.79222e6 q^{28} +2.41854e6 q^{29} +303750. q^{30} +1.32334e6 q^{31} -4.28400e6 q^{32} -1.45654e6 q^{33} -1.74006e6 q^{34} +3.66625e6 q^{35} -3.12304e6 q^{36} +1.02391e7 q^{37} -781926. q^{38} -4.99932e6 q^{39} -3.70500e6 q^{40} +2.48450e7 q^{41} +2.85088e6 q^{42} -2.22809e7 q^{43} +8.55943e6 q^{44} -4.10062e6 q^{45} +4.02350e6 q^{46} -9.03929e6 q^{47} +1.68597e7 q^{48} -5.94365e6 q^{49} -2.34375e6 q^{50} +2.34908e7 q^{51} +2.93787e7 q^{52} +9.51654e7 q^{53} -3.18865e6 q^{54} +1.12388e7 q^{55} -3.47736e7 q^{56} +1.05560e7 q^{57} -1.45112e7 q^{58} +8.80645e7 q^{59} +2.40975e7 q^{60} +1.59151e8 q^{61} -7.94006e6 q^{62} -3.84868e7 q^{63} -8.08657e7 q^{64} +3.85750e7 q^{65} +8.73925e6 q^{66} -1.65220e8 q^{67} -1.38045e8 q^{68} -5.43173e7 q^{69} -2.19975e7 q^{70} -2.71185e8 q^{71} +3.88936e7 q^{72} -1.67589e8 q^{73} -6.14344e7 q^{74} +3.16406e7 q^{75} -6.20328e7 q^{76} +1.05482e8 q^{77} +2.99959e7 q^{78} -2.61760e8 q^{79} -1.30090e8 q^{80} +4.30467e7 q^{81} -1.49070e8 q^{82} -3.20535e8 q^{83} +2.26169e8 q^{84} -1.81256e8 q^{85} +1.33686e8 q^{86} +1.95902e8 q^{87} -1.06597e8 q^{88} +4.43005e8 q^{89} +2.46038e7 q^{90} +3.62050e8 q^{91} +3.19198e8 q^{92} +1.07191e8 q^{93} +5.42357e7 q^{94} -8.14506e7 q^{95} -3.47004e8 q^{96} -1.24603e9 q^{97} +3.56619e7 q^{98} -1.17980e8 q^{99} +O(q^{100})\)

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 \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −6.00000 −0.265165 −0.132583 0.991172i \(-0.542327\pi\)
−0.132583 + 0.991172i \(0.542327\pi\)
\(3\) 81.0000 0.577350
\(4\) −476.000 −0.929688
\(5\) −625.000 −0.447214
\(6\) −486.000 −0.153093
\(7\) −5866.00 −0.923423 −0.461712 0.887030i \(-0.652765\pi\)
−0.461712 + 0.887030i \(0.652765\pi\)
\(8\) 5928.00 0.511686
\(9\) 6561.00 0.333333
\(10\) 3750.00 0.118585
\(11\) −17982.0 −0.370315 −0.185157 0.982709i \(-0.559279\pi\)
−0.185157 + 0.982709i \(0.559279\pi\)
\(12\) −38556.0 −0.536755
\(13\) −61720.0 −0.599350 −0.299675 0.954041i \(-0.596878\pi\)
−0.299675 + 0.954041i \(0.596878\pi\)
\(14\) 35196.0 0.244860
\(15\) −50625.0 −0.258199
\(16\) 208144. 0.794006
\(17\) 290010. 0.842157 0.421078 0.907024i \(-0.361652\pi\)
0.421078 + 0.907024i \(0.361652\pi\)
\(18\) −39366.0 −0.0883883
\(19\) 130321. 0.229416
\(20\) 297500. 0.415769
\(21\) −475146. −0.533139
\(22\) 107892. 0.0981945
\(23\) −670584. −0.499664 −0.249832 0.968289i \(-0.580375\pi\)
−0.249832 + 0.968289i \(0.580375\pi\)
\(24\) 480168. 0.295422
\(25\) 390625. 0.200000
\(26\) 370320. 0.158927
\(27\) 531441. 0.192450
\(28\) 2.79222e6 0.858495
\(29\) 2.41854e6 0.634983 0.317492 0.948261i \(-0.397159\pi\)
0.317492 + 0.948261i \(0.397159\pi\)
\(30\) 303750. 0.0684653
\(31\) 1.32334e6 0.257362 0.128681 0.991686i \(-0.458926\pi\)
0.128681 + 0.991686i \(0.458926\pi\)
\(32\) −4.28400e6 −0.722228
\(33\) −1.45654e6 −0.213801
\(34\) −1.74006e6 −0.223310
\(35\) 3.66625e6 0.412968
\(36\) −3.12304e6 −0.309896
\(37\) 1.02391e7 0.898158 0.449079 0.893492i \(-0.351752\pi\)
0.449079 + 0.893492i \(0.351752\pi\)
\(38\) −781926. −0.0608330
\(39\) −4.99932e6 −0.346035
\(40\) −3.70500e6 −0.228833
\(41\) 2.48450e7 1.37313 0.686565 0.727068i \(-0.259118\pi\)
0.686565 + 0.727068i \(0.259118\pi\)
\(42\) 2.85088e6 0.141370
\(43\) −2.22809e7 −0.993860 −0.496930 0.867790i \(-0.665540\pi\)
−0.496930 + 0.867790i \(0.665540\pi\)
\(44\) 8.55943e6 0.344277
\(45\) −4.10062e6 −0.149071
\(46\) 4.02350e6 0.132493
\(47\) −9.03929e6 −0.270205 −0.135103 0.990832i \(-0.543136\pi\)
−0.135103 + 0.990832i \(0.543136\pi\)
\(48\) 1.68597e7 0.458420
\(49\) −5.94365e6 −0.147289
\(50\) −2.34375e6 −0.0530330
\(51\) 2.34908e7 0.486219
\(52\) 2.93787e7 0.557209
\(53\) 9.51654e7 1.65668 0.828338 0.560229i \(-0.189287\pi\)
0.828338 + 0.560229i \(0.189287\pi\)
\(54\) −3.18865e6 −0.0510310
\(55\) 1.12388e7 0.165610
\(56\) −3.47736e7 −0.472503
\(57\) 1.05560e7 0.132453
\(58\) −1.45112e7 −0.168375
\(59\) 8.80645e7 0.946165 0.473082 0.881018i \(-0.343141\pi\)
0.473082 + 0.881018i \(0.343141\pi\)
\(60\) 2.40975e7 0.240044
\(61\) 1.59151e8 1.47172 0.735858 0.677136i \(-0.236779\pi\)
0.735858 + 0.677136i \(0.236779\pi\)
\(62\) −7.94006e6 −0.0682435
\(63\) −3.84868e7 −0.307808
\(64\) −8.08657e7 −0.602497
\(65\) 3.85750e7 0.268038
\(66\) 8.73925e6 0.0566926
\(67\) −1.65220e8 −1.00167 −0.500835 0.865543i \(-0.666974\pi\)
−0.500835 + 0.865543i \(0.666974\pi\)
\(68\) −1.38045e8 −0.782942
\(69\) −5.43173e7 −0.288481
\(70\) −2.19975e7 −0.109505
\(71\) −2.71185e8 −1.26650 −0.633248 0.773949i \(-0.718279\pi\)
−0.633248 + 0.773949i \(0.718279\pi\)
\(72\) 3.88936e7 0.170562
\(73\) −1.67589e8 −0.690704 −0.345352 0.938473i \(-0.612240\pi\)
−0.345352 + 0.938473i \(0.612240\pi\)
\(74\) −6.14344e7 −0.238160
\(75\) 3.16406e7 0.115470
\(76\) −6.20328e7 −0.213285
\(77\) 1.05482e8 0.341957
\(78\) 2.99959e7 0.0917564
\(79\) −2.61760e8 −0.756102 −0.378051 0.925785i \(-0.623406\pi\)
−0.378051 + 0.925785i \(0.623406\pi\)
\(80\) −1.30090e8 −0.355090
\(81\) 4.30467e7 0.111111
\(82\) −1.49070e8 −0.364106
\(83\) −3.20535e8 −0.741351 −0.370675 0.928762i \(-0.620874\pi\)
−0.370675 + 0.928762i \(0.620874\pi\)
\(84\) 2.26169e8 0.495652
\(85\) −1.81256e8 −0.376624
\(86\) 1.33686e8 0.263537
\(87\) 1.95902e8 0.366608
\(88\) −1.06597e8 −0.189485
\(89\) 4.43005e8 0.748434 0.374217 0.927341i \(-0.377911\pi\)
0.374217 + 0.927341i \(0.377911\pi\)
\(90\) 2.46038e7 0.0395285
\(91\) 3.62050e8 0.553454
\(92\) 3.19198e8 0.464531
\(93\) 1.07191e8 0.148588
\(94\) 5.42357e7 0.0716490
\(95\) −8.14506e7 −0.102598
\(96\) −3.47004e8 −0.416979
\(97\) −1.24603e9 −1.42908 −0.714538 0.699596i \(-0.753363\pi\)
−0.714538 + 0.699596i \(0.753363\pi\)
\(98\) 3.56619e7 0.0390560
\(99\) −1.17980e8 −0.123438
\(100\) −1.85938e8 −0.185938
\(101\) 3.76457e8 0.359973 0.179986 0.983669i \(-0.442395\pi\)
0.179986 + 0.983669i \(0.442395\pi\)
\(102\) −1.40945e8 −0.128928
\(103\) 2.47036e8 0.216268 0.108134 0.994136i \(-0.465512\pi\)
0.108134 + 0.994136i \(0.465512\pi\)
\(104\) −3.65876e8 −0.306679
\(105\) 2.96966e8 0.238427
\(106\) −5.70992e8 −0.439293
\(107\) −4.47583e8 −0.330101 −0.165050 0.986285i \(-0.552779\pi\)
−0.165050 + 0.986285i \(0.552779\pi\)
\(108\) −2.52966e8 −0.178918
\(109\) 2.01615e9 1.36805 0.684027 0.729456i \(-0.260227\pi\)
0.684027 + 0.729456i \(0.260227\pi\)
\(110\) −6.74325e7 −0.0439139
\(111\) 8.29365e8 0.518552
\(112\) −1.22097e9 −0.733204
\(113\) 2.69208e9 1.55323 0.776615 0.629976i \(-0.216935\pi\)
0.776615 + 0.629976i \(0.216935\pi\)
\(114\) −6.33360e7 −0.0351220
\(115\) 4.19115e8 0.223456
\(116\) −1.15123e9 −0.590336
\(117\) −4.04945e8 −0.199783
\(118\) −5.28387e8 −0.250890
\(119\) −1.70120e9 −0.777667
\(120\) −3.00105e8 −0.132117
\(121\) −2.03460e9 −0.862867
\(122\) −9.54903e8 −0.390247
\(123\) 2.01245e9 0.792777
\(124\) −6.29912e8 −0.239267
\(125\) −2.44141e8 −0.0894427
\(126\) 2.30921e8 0.0816199
\(127\) −2.53375e9 −0.864265 −0.432133 0.901810i \(-0.642239\pi\)
−0.432133 + 0.901810i \(0.642239\pi\)
\(128\) 2.67860e9 0.881989
\(129\) −1.80476e9 −0.573806
\(130\) −2.31450e8 −0.0710742
\(131\) 1.04001e9 0.308545 0.154272 0.988028i \(-0.450697\pi\)
0.154272 + 0.988028i \(0.450697\pi\)
\(132\) 6.93314e8 0.198768
\(133\) −7.64463e8 −0.211848
\(134\) 9.91318e8 0.265608
\(135\) −3.32151e8 −0.0860663
\(136\) 1.71918e9 0.430919
\(137\) −5.13752e9 −1.24598 −0.622989 0.782230i \(-0.714082\pi\)
−0.622989 + 0.782230i \(0.714082\pi\)
\(138\) 3.25904e8 0.0764951
\(139\) 5.72487e8 0.130077 0.0650383 0.997883i \(-0.479283\pi\)
0.0650383 + 0.997883i \(0.479283\pi\)
\(140\) −1.74513e9 −0.383931
\(141\) −7.32182e8 −0.156003
\(142\) 1.62711e9 0.335830
\(143\) 1.10985e9 0.221948
\(144\) 1.36563e9 0.264669
\(145\) −1.51159e9 −0.283973
\(146\) 1.00553e9 0.183150
\(147\) −4.81436e8 −0.0850375
\(148\) −4.87380e9 −0.835006
\(149\) 4.32784e9 0.719338 0.359669 0.933080i \(-0.382890\pi\)
0.359669 + 0.933080i \(0.382890\pi\)
\(150\) −1.89844e8 −0.0306186
\(151\) −7.64525e9 −1.19673 −0.598364 0.801225i \(-0.704182\pi\)
−0.598364 + 0.801225i \(0.704182\pi\)
\(152\) 7.72543e8 0.117389
\(153\) 1.90276e9 0.280719
\(154\) −6.32894e8 −0.0906751
\(155\) −8.27090e8 −0.115096
\(156\) 2.37968e9 0.321705
\(157\) 6.29944e9 0.827473 0.413737 0.910397i \(-0.364223\pi\)
0.413737 + 0.910397i \(0.364223\pi\)
\(158\) 1.57056e9 0.200492
\(159\) 7.70840e9 0.956482
\(160\) 2.67750e9 0.322990
\(161\) 3.93365e9 0.461401
\(162\) −2.58280e8 −0.0294628
\(163\) 6.71844e9 0.745460 0.372730 0.927940i \(-0.378422\pi\)
0.372730 + 0.927940i \(0.378422\pi\)
\(164\) −1.18262e10 −1.27658
\(165\) 9.10339e8 0.0956148
\(166\) 1.92321e9 0.196580
\(167\) −1.42492e10 −1.41764 −0.708820 0.705390i \(-0.750772\pi\)
−0.708820 + 0.705390i \(0.750772\pi\)
\(168\) −2.81667e9 −0.272799
\(169\) −6.79514e9 −0.640779
\(170\) 1.08754e9 0.0998675
\(171\) 8.55036e8 0.0764719
\(172\) 1.06057e10 0.923980
\(173\) −3.10361e9 −0.263427 −0.131713 0.991288i \(-0.542048\pi\)
−0.131713 + 0.991288i \(0.542048\pi\)
\(174\) −1.17541e9 −0.0972116
\(175\) −2.29141e9 −0.184685
\(176\) −3.74285e9 −0.294032
\(177\) 7.13323e9 0.546269
\(178\) −2.65803e9 −0.198459
\(179\) 1.03107e10 0.750670 0.375335 0.926889i \(-0.377528\pi\)
0.375335 + 0.926889i \(0.377528\pi\)
\(180\) 1.95190e9 0.138590
\(181\) −2.30100e10 −1.59354 −0.796771 0.604282i \(-0.793460\pi\)
−0.796771 + 0.604282i \(0.793460\pi\)
\(182\) −2.17230e9 −0.146757
\(183\) 1.28912e10 0.849695
\(184\) −3.97522e9 −0.255671
\(185\) −6.39942e9 −0.401668
\(186\) −6.43145e8 −0.0394004
\(187\) −5.21496e9 −0.311863
\(188\) 4.30270e9 0.251206
\(189\) −3.11743e9 −0.177713
\(190\) 4.88704e8 0.0272054
\(191\) −2.28816e10 −1.24404 −0.622021 0.783000i \(-0.713688\pi\)
−0.622021 + 0.783000i \(0.713688\pi\)
\(192\) −6.55012e9 −0.347852
\(193\) 2.29137e10 1.18874 0.594372 0.804190i \(-0.297401\pi\)
0.594372 + 0.804190i \(0.297401\pi\)
\(194\) 7.47618e9 0.378941
\(195\) 3.12458e9 0.154752
\(196\) 2.82918e9 0.136933
\(197\) 5.50918e9 0.260609 0.130304 0.991474i \(-0.458405\pi\)
0.130304 + 0.991474i \(0.458405\pi\)
\(198\) 7.07879e8 0.0327315
\(199\) −9.55562e9 −0.431937 −0.215968 0.976400i \(-0.569291\pi\)
−0.215968 + 0.976400i \(0.569291\pi\)
\(200\) 2.31562e9 0.102337
\(201\) −1.33828e10 −0.578315
\(202\) −2.25874e9 −0.0954521
\(203\) −1.41872e10 −0.586358
\(204\) −1.11816e10 −0.452032
\(205\) −1.55281e10 −0.614083
\(206\) −1.48221e9 −0.0573467
\(207\) −4.39970e9 −0.166555
\(208\) −1.28466e10 −0.475888
\(209\) −2.34343e9 −0.0849560
\(210\) −1.78180e9 −0.0632225
\(211\) −3.34512e10 −1.16182 −0.580912 0.813967i \(-0.697304\pi\)
−0.580912 + 0.813967i \(0.697304\pi\)
\(212\) −4.52987e10 −1.54019
\(213\) −2.19660e10 −0.731212
\(214\) 2.68550e9 0.0875312
\(215\) 1.39256e10 0.444468
\(216\) 3.15038e9 0.0984740
\(217\) −7.76274e9 −0.237654
\(218\) −1.20969e10 −0.362760
\(219\) −1.35747e10 −0.398778
\(220\) −5.34964e9 −0.153965
\(221\) −1.78994e10 −0.504747
\(222\) −4.97619e9 −0.137502
\(223\) −4.73013e10 −1.28086 −0.640429 0.768018i \(-0.721243\pi\)
−0.640429 + 0.768018i \(0.721243\pi\)
\(224\) 2.51299e10 0.666923
\(225\) 2.56289e9 0.0666667
\(226\) −1.61525e10 −0.411862
\(227\) −2.11631e10 −0.529009 −0.264504 0.964384i \(-0.585208\pi\)
−0.264504 + 0.964384i \(0.585208\pi\)
\(228\) −5.02466e9 −0.123140
\(229\) 3.74951e10 0.900980 0.450490 0.892782i \(-0.351249\pi\)
0.450490 + 0.892782i \(0.351249\pi\)
\(230\) −2.51469e9 −0.0592528
\(231\) 8.54408e9 0.197429
\(232\) 1.43371e10 0.324912
\(233\) −8.75711e10 −1.94652 −0.973260 0.229704i \(-0.926224\pi\)
−0.973260 + 0.229704i \(0.926224\pi\)
\(234\) 2.42967e9 0.0529756
\(235\) 5.64956e9 0.120839
\(236\) −4.19187e10 −0.879638
\(237\) −2.12025e10 −0.436536
\(238\) 1.02072e10 0.206210
\(239\) 5.58298e9 0.110682 0.0553408 0.998468i \(-0.482375\pi\)
0.0553408 + 0.998468i \(0.482375\pi\)
\(240\) −1.05373e10 −0.205012
\(241\) −5.94161e10 −1.13456 −0.567280 0.823525i \(-0.692004\pi\)
−0.567280 + 0.823525i \(0.692004\pi\)
\(242\) 1.22076e10 0.228802
\(243\) 3.48678e9 0.0641500
\(244\) −7.57557e10 −1.36824
\(245\) 3.71478e9 0.0658697
\(246\) −1.20747e10 −0.210217
\(247\) −8.04341e9 −0.137500
\(248\) 7.84478e9 0.131689
\(249\) −2.59633e10 −0.428019
\(250\) 1.46484e9 0.0237171
\(251\) −5.31574e10 −0.845340 −0.422670 0.906284i \(-0.638907\pi\)
−0.422670 + 0.906284i \(0.638907\pi\)
\(252\) 1.83197e10 0.286165
\(253\) 1.20584e10 0.185033
\(254\) 1.52025e10 0.229173
\(255\) −1.46818e10 −0.217444
\(256\) 2.53316e10 0.368624
\(257\) 4.74074e10 0.677870 0.338935 0.940810i \(-0.389933\pi\)
0.338935 + 0.940810i \(0.389933\pi\)
\(258\) 1.08285e10 0.152153
\(259\) −6.00624e10 −0.829380
\(260\) −1.83617e10 −0.249191
\(261\) 1.58680e10 0.211661
\(262\) −6.24008e9 −0.0818153
\(263\) −3.21309e10 −0.414116 −0.207058 0.978329i \(-0.566389\pi\)
−0.207058 + 0.978329i \(0.566389\pi\)
\(264\) −8.63438e9 −0.109399
\(265\) −5.94784e10 −0.740888
\(266\) 4.58678e9 0.0561746
\(267\) 3.58834e10 0.432109
\(268\) 7.86445e10 0.931241
\(269\) −6.91081e10 −0.804718 −0.402359 0.915482i \(-0.631810\pi\)
−0.402359 + 0.915482i \(0.631810\pi\)
\(270\) 1.99290e9 0.0228218
\(271\) −1.35561e11 −1.52677 −0.763384 0.645945i \(-0.776464\pi\)
−0.763384 + 0.645945i \(0.776464\pi\)
\(272\) 6.03638e10 0.668678
\(273\) 2.93260e10 0.319537
\(274\) 3.08251e10 0.330390
\(275\) −7.02422e9 −0.0740629
\(276\) 2.58550e10 0.268197
\(277\) 2.35359e10 0.240199 0.120100 0.992762i \(-0.461679\pi\)
0.120100 + 0.992762i \(0.461679\pi\)
\(278\) −3.43492e9 −0.0344918
\(279\) 8.68246e9 0.0857875
\(280\) 2.17335e10 0.211310
\(281\) 8.02523e10 0.767855 0.383927 0.923363i \(-0.374571\pi\)
0.383927 + 0.923363i \(0.374571\pi\)
\(282\) 4.39309e9 0.0413666
\(283\) −5.07944e10 −0.470735 −0.235367 0.971906i \(-0.575629\pi\)
−0.235367 + 0.971906i \(0.575629\pi\)
\(284\) 1.29084e11 1.17745
\(285\) −6.59750e9 −0.0592349
\(286\) −6.65909e9 −0.0588529
\(287\) −1.45741e11 −1.26798
\(288\) −2.81073e10 −0.240743
\(289\) −3.44821e10 −0.290772
\(290\) 9.06952e9 0.0752998
\(291\) −1.00928e11 −0.825078
\(292\) 7.97722e10 0.642139
\(293\) 2.18892e11 1.73510 0.867552 0.497346i \(-0.165692\pi\)
0.867552 + 0.497346i \(0.165692\pi\)
\(294\) 2.88861e9 0.0225490
\(295\) −5.50403e10 −0.423138
\(296\) 6.06972e10 0.459574
\(297\) −9.55637e9 −0.0712671
\(298\) −2.59670e10 −0.190743
\(299\) 4.13884e10 0.299474
\(300\) −1.50609e10 −0.107351
\(301\) 1.30700e11 0.917754
\(302\) 4.58715e10 0.317330
\(303\) 3.04930e10 0.207830
\(304\) 2.71255e10 0.182158
\(305\) −9.94691e10 −0.658171
\(306\) −1.14165e10 −0.0744368
\(307\) 2.23471e11 1.43581 0.717906 0.696140i \(-0.245101\pi\)
0.717906 + 0.696140i \(0.245101\pi\)
\(308\) −5.02096e10 −0.317913
\(309\) 2.00099e10 0.124862
\(310\) 4.96254e9 0.0305194
\(311\) −2.46225e11 −1.49249 −0.746244 0.665672i \(-0.768145\pi\)
−0.746244 + 0.665672i \(0.768145\pi\)
\(312\) −2.96360e10 −0.177061
\(313\) −7.19669e10 −0.423822 −0.211911 0.977289i \(-0.567969\pi\)
−0.211911 + 0.977289i \(0.567969\pi\)
\(314\) −3.77967e10 −0.219417
\(315\) 2.40543e10 0.137656
\(316\) 1.24598e11 0.702939
\(317\) −2.43373e11 −1.35365 −0.676823 0.736145i \(-0.736644\pi\)
−0.676823 + 0.736145i \(0.736644\pi\)
\(318\) −4.62504e10 −0.253626
\(319\) −4.34902e10 −0.235144
\(320\) 5.05411e10 0.269445
\(321\) −3.62542e10 −0.190584
\(322\) −2.36019e10 −0.122347
\(323\) 3.77944e10 0.193204
\(324\) −2.04902e10 −0.103299
\(325\) −2.41094e10 −0.119870
\(326\) −4.03107e10 −0.197670
\(327\) 1.63308e11 0.789847
\(328\) 1.47281e11 0.702611
\(329\) 5.30245e10 0.249514
\(330\) −5.46203e9 −0.0253537
\(331\) 1.08623e10 0.0497387 0.0248694 0.999691i \(-0.492083\pi\)
0.0248694 + 0.999691i \(0.492083\pi\)
\(332\) 1.52575e11 0.689225
\(333\) 6.71785e10 0.299386
\(334\) 8.54951e10 0.375908
\(335\) 1.03262e11 0.447961
\(336\) −9.88988e10 −0.423316
\(337\) 1.47875e10 0.0624539 0.0312269 0.999512i \(-0.490059\pi\)
0.0312269 + 0.999512i \(0.490059\pi\)
\(338\) 4.07708e10 0.169912
\(339\) 2.18059e11 0.896757
\(340\) 8.62780e10 0.350142
\(341\) −2.37964e10 −0.0953051
\(342\) −5.13022e9 −0.0202777
\(343\) 2.71580e11 1.05943
\(344\) −1.32081e11 −0.508544
\(345\) 3.39483e10 0.129013
\(346\) 1.86217e10 0.0698516
\(347\) −6.12334e10 −0.226728 −0.113364 0.993553i \(-0.536163\pi\)
−0.113364 + 0.993553i \(0.536163\pi\)
\(348\) −9.32492e10 −0.340831
\(349\) −2.47994e11 −0.894800 −0.447400 0.894334i \(-0.647650\pi\)
−0.447400 + 0.894334i \(0.647650\pi\)
\(350\) 1.37484e10 0.0489719
\(351\) −3.28005e10 −0.115345
\(352\) 7.70349e10 0.267452
\(353\) 1.04820e11 0.359300 0.179650 0.983731i \(-0.442503\pi\)
0.179650 + 0.983731i \(0.442503\pi\)
\(354\) −4.27994e10 −0.144851
\(355\) 1.69491e11 0.566394
\(356\) −2.10870e11 −0.695810
\(357\) −1.37797e11 −0.448986
\(358\) −6.18642e10 −0.199052
\(359\) −3.37195e11 −1.07141 −0.535705 0.844405i \(-0.679954\pi\)
−0.535705 + 0.844405i \(0.679954\pi\)
\(360\) −2.43085e10 −0.0762776
\(361\) 1.69836e10 0.0526316
\(362\) 1.38060e11 0.422551
\(363\) −1.64802e11 −0.498177
\(364\) −1.72336e11 −0.514539
\(365\) 1.04743e11 0.308892
\(366\) −7.73472e10 −0.225309
\(367\) 5.35281e11 1.54023 0.770114 0.637907i \(-0.220200\pi\)
0.770114 + 0.637907i \(0.220200\pi\)
\(368\) −1.39578e11 −0.396736
\(369\) 1.63008e11 0.457710
\(370\) 3.83965e10 0.106508
\(371\) −5.58240e11 −1.52981
\(372\) −5.10229e10 −0.138141
\(373\) 3.83382e11 1.02552 0.512758 0.858533i \(-0.328624\pi\)
0.512758 + 0.858533i \(0.328624\pi\)
\(374\) 3.12898e10 0.0826951
\(375\) −1.97754e10 −0.0516398
\(376\) −5.35849e10 −0.138260
\(377\) −1.49272e11 −0.380578
\(378\) 1.87046e10 0.0471233
\(379\) 6.81052e11 1.69552 0.847762 0.530377i \(-0.177950\pi\)
0.847762 + 0.530377i \(0.177950\pi\)
\(380\) 3.87705e10 0.0953839
\(381\) −2.05234e11 −0.498984
\(382\) 1.37289e11 0.329877
\(383\) −2.13428e11 −0.506824 −0.253412 0.967358i \(-0.581553\pi\)
−0.253412 + 0.967358i \(0.581553\pi\)
\(384\) 2.16967e11 0.509217
\(385\) −6.59265e10 −0.152928
\(386\) −1.37482e11 −0.315213
\(387\) −1.46185e11 −0.331287
\(388\) 5.93110e11 1.32859
\(389\) 5.04049e10 0.111609 0.0558046 0.998442i \(-0.482228\pi\)
0.0558046 + 0.998442i \(0.482228\pi\)
\(390\) −1.87474e10 −0.0410347
\(391\) −1.94476e11 −0.420795
\(392\) −3.52340e10 −0.0753658
\(393\) 8.42411e10 0.178138
\(394\) −3.30551e10 −0.0691043
\(395\) 1.63600e11 0.338139
\(396\) 5.61584e10 0.114759
\(397\) 2.31641e11 0.468013 0.234007 0.972235i \(-0.424816\pi\)
0.234007 + 0.972235i \(0.424816\pi\)
\(398\) 5.73337e10 0.114535
\(399\) −6.19215e10 −0.122310
\(400\) 8.13062e10 0.158801
\(401\) 8.74014e10 0.168799 0.0843993 0.996432i \(-0.473103\pi\)
0.0843993 + 0.996432i \(0.473103\pi\)
\(402\) 8.02967e10 0.153349
\(403\) −8.16768e10 −0.154250
\(404\) −1.79194e11 −0.334662
\(405\) −2.69042e10 −0.0496904
\(406\) 8.51229e10 0.155482
\(407\) −1.84119e11 −0.332601
\(408\) 1.39254e11 0.248791
\(409\) 4.88717e11 0.863580 0.431790 0.901974i \(-0.357882\pi\)
0.431790 + 0.901974i \(0.357882\pi\)
\(410\) 9.31688e10 0.162833
\(411\) −4.16139e11 −0.719366
\(412\) −1.17589e11 −0.201062
\(413\) −5.16586e11 −0.873711
\(414\) 2.63982e10 0.0441645
\(415\) 2.00334e11 0.331542
\(416\) 2.64408e11 0.432868
\(417\) 4.63715e10 0.0750997
\(418\) 1.40606e10 0.0225274
\(419\) −1.93528e11 −0.306747 −0.153373 0.988168i \(-0.549014\pi\)
−0.153373 + 0.988168i \(0.549014\pi\)
\(420\) −1.41356e11 −0.221663
\(421\) 1.22667e12 1.90308 0.951541 0.307521i \(-0.0994993\pi\)
0.951541 + 0.307521i \(0.0994993\pi\)
\(422\) 2.00707e11 0.308075
\(423\) −5.93068e10 −0.0900684
\(424\) 5.64140e11 0.847697
\(425\) 1.13285e11 0.168431
\(426\) 1.31796e11 0.193892
\(427\) −9.33577e11 −1.35902
\(428\) 2.13049e11 0.306891
\(429\) 8.98978e10 0.128142
\(430\) −8.35535e10 −0.117857
\(431\) 1.27648e12 1.78183 0.890914 0.454171i \(-0.150065\pi\)
0.890914 + 0.454171i \(0.150065\pi\)
\(432\) 1.10616e11 0.152807
\(433\) −3.04014e11 −0.415622 −0.207811 0.978169i \(-0.566634\pi\)
−0.207811 + 0.978169i \(0.566634\pi\)
\(434\) 4.65764e10 0.0630177
\(435\) −1.22439e11 −0.163952
\(436\) −9.59687e11 −1.27186
\(437\) −8.73912e10 −0.114631
\(438\) 8.14481e10 0.105742
\(439\) 1.97425e10 0.0253695 0.0126847 0.999920i \(-0.495962\pi\)
0.0126847 + 0.999920i \(0.495962\pi\)
\(440\) 6.66233e10 0.0847401
\(441\) −3.89963e10 −0.0490964
\(442\) 1.07397e11 0.133841
\(443\) 9.84601e11 1.21463 0.607314 0.794462i \(-0.292247\pi\)
0.607314 + 0.794462i \(0.292247\pi\)
\(444\) −3.94778e11 −0.482091
\(445\) −2.76878e11 −0.334710
\(446\) 2.83808e11 0.339639
\(447\) 3.50555e11 0.415310
\(448\) 4.74358e11 0.556359
\(449\) 6.13059e11 0.711859 0.355929 0.934513i \(-0.384164\pi\)
0.355929 + 0.934513i \(0.384164\pi\)
\(450\) −1.53773e10 −0.0176777
\(451\) −4.46763e11 −0.508490
\(452\) −1.28143e12 −1.44402
\(453\) −6.19265e11 −0.690931
\(454\) 1.26979e11 0.140275
\(455\) −2.26281e11 −0.247512
\(456\) 6.25760e10 0.0677744
\(457\) −9.50605e11 −1.01948 −0.509738 0.860329i \(-0.670258\pi\)
−0.509738 + 0.860329i \(0.670258\pi\)
\(458\) −2.24971e11 −0.238908
\(459\) 1.54123e11 0.162073
\(460\) −1.99499e11 −0.207745
\(461\) −1.11955e12 −1.15448 −0.577242 0.816573i \(-0.695871\pi\)
−0.577242 + 0.816573i \(0.695871\pi\)
\(462\) −5.12645e10 −0.0523513
\(463\) 3.49143e11 0.353093 0.176546 0.984292i \(-0.443507\pi\)
0.176546 + 0.984292i \(0.443507\pi\)
\(464\) 5.03405e11 0.504181
\(465\) −6.69943e10 −0.0664507
\(466\) 5.25426e11 0.516149
\(467\) −1.58589e12 −1.54294 −0.771468 0.636269i \(-0.780477\pi\)
−0.771468 + 0.636269i \(0.780477\pi\)
\(468\) 1.92754e11 0.185736
\(469\) 9.69178e11 0.924966
\(470\) −3.38973e10 −0.0320424
\(471\) 5.10255e11 0.477742
\(472\) 5.22046e11 0.484139
\(473\) 4.00656e11 0.368041
\(474\) 1.27215e11 0.115754
\(475\) 5.09066e10 0.0458831
\(476\) 8.09771e11 0.722987
\(477\) 6.24380e11 0.552225
\(478\) −3.34979e10 −0.0293489
\(479\) −9.64004e11 −0.836699 −0.418349 0.908286i \(-0.637391\pi\)
−0.418349 + 0.908286i \(0.637391\pi\)
\(480\) 2.16878e11 0.186479
\(481\) −6.31955e11 −0.538311
\(482\) 3.56497e11 0.300846
\(483\) 3.18625e11 0.266390
\(484\) 9.68467e11 0.802197
\(485\) 7.78768e11 0.639102
\(486\) −2.09207e10 −0.0170103
\(487\) −7.74475e11 −0.623917 −0.311959 0.950096i \(-0.600985\pi\)
−0.311959 + 0.950096i \(0.600985\pi\)
\(488\) 9.43444e11 0.753056
\(489\) 5.44194e11 0.430392
\(490\) −2.22887e10 −0.0174664
\(491\) −1.69635e11 −0.131719 −0.0658597 0.997829i \(-0.520979\pi\)
−0.0658597 + 0.997829i \(0.520979\pi\)
\(492\) −9.57924e11 −0.737035
\(493\) 7.01401e11 0.534755
\(494\) 4.82605e10 0.0364603
\(495\) 7.37374e10 0.0552033
\(496\) 2.75446e11 0.204347
\(497\) 1.59077e12 1.16951
\(498\) 1.55780e11 0.113496
\(499\) 2.81692e11 0.203387 0.101693 0.994816i \(-0.467574\pi\)
0.101693 + 0.994816i \(0.467574\pi\)
\(500\) 1.16211e11 0.0831538
\(501\) −1.15418e12 −0.818475
\(502\) 3.18944e11 0.224155
\(503\) 2.22525e11 0.154997 0.0774986 0.996992i \(-0.475307\pi\)
0.0774986 + 0.996992i \(0.475307\pi\)
\(504\) −2.28150e11 −0.157501
\(505\) −2.35286e11 −0.160985
\(506\) −7.23506e10 −0.0490642
\(507\) −5.50406e11 −0.369954
\(508\) 1.20607e12 0.803497
\(509\) 1.85160e12 1.22269 0.611345 0.791364i \(-0.290629\pi\)
0.611345 + 0.791364i \(0.290629\pi\)
\(510\) 8.80905e10 0.0576585
\(511\) 9.83075e11 0.637812
\(512\) −1.52343e12 −0.979736
\(513\) 6.92579e10 0.0441511
\(514\) −2.84444e11 −0.179748
\(515\) −1.54397e11 −0.0967180
\(516\) 8.59064e11 0.533460
\(517\) 1.62544e11 0.100061
\(518\) 3.60374e11 0.219923
\(519\) −2.51393e11 −0.152090
\(520\) 2.28673e11 0.137151
\(521\) 2.54579e12 1.51374 0.756872 0.653563i \(-0.226726\pi\)
0.756872 + 0.653563i \(0.226726\pi\)
\(522\) −9.52082e10 −0.0561251
\(523\) 4.65313e11 0.271949 0.135975 0.990712i \(-0.456583\pi\)
0.135975 + 0.990712i \(0.456583\pi\)
\(524\) −4.95046e11 −0.286850
\(525\) −1.85604e11 −0.106628
\(526\) 1.92786e11 0.109809
\(527\) 3.83783e11 0.216739
\(528\) −3.03170e11 −0.169760
\(529\) −1.35147e12 −0.750336
\(530\) 3.56870e11 0.196458
\(531\) 5.77791e11 0.315388
\(532\) 3.63884e11 0.196952
\(533\) −1.53343e12 −0.822987
\(534\) −2.15300e11 −0.114580
\(535\) 2.79739e11 0.147626
\(536\) −9.79422e11 −0.512541
\(537\) 8.35166e11 0.433400
\(538\) 4.14649e11 0.213383
\(539\) 1.06879e11 0.0545434
\(540\) 1.58104e11 0.0800148
\(541\) 3.16133e12 1.58665 0.793326 0.608797i \(-0.208348\pi\)
0.793326 + 0.608797i \(0.208348\pi\)
\(542\) 8.13367e11 0.404846
\(543\) −1.86381e12 −0.920031
\(544\) −1.24240e12 −0.608229
\(545\) −1.26009e12 −0.611813
\(546\) −1.75956e11 −0.0847300
\(547\) −3.55796e11 −0.169925 −0.0849627 0.996384i \(-0.527077\pi\)
−0.0849627 + 0.996384i \(0.527077\pi\)
\(548\) 2.44546e12 1.15837
\(549\) 1.04419e12 0.490572
\(550\) 4.21453e10 0.0196389
\(551\) 3.15187e11 0.145675
\(552\) −3.21993e11 −0.147612
\(553\) 1.53548e12 0.698203
\(554\) −1.41215e11 −0.0636925
\(555\) −5.18353e11 −0.231903
\(556\) −2.72504e11 −0.120931
\(557\) −2.40107e12 −1.05696 −0.528478 0.848947i \(-0.677237\pi\)
−0.528478 + 0.848947i \(0.677237\pi\)
\(558\) −5.20948e10 −0.0227478
\(559\) 1.37518e12 0.595671
\(560\) 7.63108e11 0.327899
\(561\) −4.22412e11 −0.180054
\(562\) −4.81514e11 −0.203608
\(563\) −7.02837e10 −0.0294827 −0.0147413 0.999891i \(-0.504692\pi\)
−0.0147413 + 0.999891i \(0.504692\pi\)
\(564\) 3.48519e11 0.145034
\(565\) −1.68255e12 −0.694625
\(566\) 3.04766e11 0.124822
\(567\) −2.52512e11 −0.102603
\(568\) −1.60759e12 −0.648048
\(569\) 1.18396e12 0.473514 0.236757 0.971569i \(-0.423915\pi\)
0.236757 + 0.971569i \(0.423915\pi\)
\(570\) 3.95850e10 0.0157070
\(571\) −1.52494e11 −0.0600328 −0.0300164 0.999549i \(-0.509556\pi\)
−0.0300164 + 0.999549i \(0.509556\pi\)
\(572\) −5.28288e11 −0.206343
\(573\) −1.85341e12 −0.718248
\(574\) 8.74445e11 0.336224
\(575\) −2.61947e11 −0.0999328
\(576\) −5.30560e11 −0.200832
\(577\) −8.73507e11 −0.328076 −0.164038 0.986454i \(-0.552452\pi\)
−0.164038 + 0.986454i \(0.552452\pi\)
\(578\) 2.06892e11 0.0771027
\(579\) 1.85601e12 0.686321
\(580\) 7.19516e11 0.264006
\(581\) 1.88026e12 0.684581
\(582\) 6.05570e11 0.218782
\(583\) −1.71126e12 −0.613491
\(584\) −9.93466e11 −0.353423
\(585\) 2.53091e11 0.0893459
\(586\) −1.31335e12 −0.460089
\(587\) −2.95130e12 −1.02599 −0.512994 0.858392i \(-0.671464\pi\)
−0.512994 + 0.858392i \(0.671464\pi\)
\(588\) 2.29163e11 0.0790583
\(589\) 1.72460e11 0.0590430
\(590\) 3.30242e11 0.112201
\(591\) 4.46244e11 0.150462
\(592\) 2.13120e12 0.713143
\(593\) −4.55212e12 −1.51171 −0.755853 0.654742i \(-0.772777\pi\)
−0.755853 + 0.654742i \(0.772777\pi\)
\(594\) 5.73382e10 0.0188975
\(595\) 1.06325e12 0.347783
\(596\) −2.06005e12 −0.668760
\(597\) −7.74006e11 −0.249379
\(598\) −2.48331e11 −0.0794100
\(599\) −1.79480e12 −0.569634 −0.284817 0.958582i \(-0.591933\pi\)
−0.284817 + 0.958582i \(0.591933\pi\)
\(600\) 1.87566e11 0.0590844
\(601\) −4.11804e12 −1.28753 −0.643763 0.765225i \(-0.722628\pi\)
−0.643763 + 0.765225i \(0.722628\pi\)
\(602\) −7.84200e11 −0.243356
\(603\) −1.08401e12 −0.333890
\(604\) 3.63914e12 1.11258
\(605\) 1.27162e12 0.385886
\(606\) −1.82958e11 −0.0551093
\(607\) 4.59570e12 1.37405 0.687026 0.726633i \(-0.258916\pi\)
0.687026 + 0.726633i \(0.258916\pi\)
\(608\) −5.58295e11 −0.165691
\(609\) −1.14916e12 −0.338534
\(610\) 5.96814e11 0.174524
\(611\) 5.57905e11 0.161948
\(612\) −9.05712e11 −0.260981
\(613\) −1.80405e12 −0.516032 −0.258016 0.966141i \(-0.583069\pi\)
−0.258016 + 0.966141i \(0.583069\pi\)
\(614\) −1.34082e12 −0.380727
\(615\) −1.25778e12 −0.354541
\(616\) 6.25300e11 0.174975
\(617\) −2.72290e12 −0.756396 −0.378198 0.925725i \(-0.623456\pi\)
−0.378198 + 0.925725i \(0.623456\pi\)
\(618\) −1.20059e11 −0.0331091
\(619\) −2.39188e12 −0.654835 −0.327417 0.944880i \(-0.606178\pi\)
−0.327417 + 0.944880i \(0.606178\pi\)
\(620\) 3.93695e11 0.107003
\(621\) −3.56376e11 −0.0961604
\(622\) 1.47735e12 0.395756
\(623\) −2.59867e12 −0.691122
\(624\) −1.04058e12 −0.274754
\(625\) 1.52588e11 0.0400000
\(626\) 4.31802e11 0.112383
\(627\) −1.89818e11 −0.0490494
\(628\) −2.99854e12 −0.769291
\(629\) 2.96943e12 0.756390
\(630\) −1.44326e11 −0.0365015
\(631\) 6.89708e12 1.73194 0.865971 0.500094i \(-0.166701\pi\)
0.865971 + 0.500094i \(0.166701\pi\)
\(632\) −1.55171e12 −0.386887
\(633\) −2.70954e12 −0.670779
\(634\) 1.46024e12 0.358940
\(635\) 1.58359e12 0.386511
\(636\) −3.66920e12 −0.889230
\(637\) 3.66842e11 0.0882779
\(638\) 2.60941e11 0.0623519
\(639\) −1.77925e12 −0.422165
\(640\) −1.67413e12 −0.394438
\(641\) −5.01889e12 −1.17421 −0.587106 0.809510i \(-0.699733\pi\)
−0.587106 + 0.809510i \(0.699733\pi\)
\(642\) 2.17525e11 0.0505361
\(643\) −4.22828e12 −0.975471 −0.487736 0.872991i \(-0.662177\pi\)
−0.487736 + 0.872991i \(0.662177\pi\)
\(644\) −1.87242e12 −0.428959
\(645\) 1.12797e12 0.256614
\(646\) −2.26766e11 −0.0512309
\(647\) −4.69992e12 −1.05444 −0.527219 0.849729i \(-0.676765\pi\)
−0.527219 + 0.849729i \(0.676765\pi\)
\(648\) 2.55181e11 0.0568540
\(649\) −1.58358e12 −0.350379
\(650\) 1.44656e11 0.0317854
\(651\) −6.28782e11 −0.137210
\(652\) −3.19798e12 −0.693045
\(653\) −7.92054e11 −0.170469 −0.0852345 0.996361i \(-0.527164\pi\)
−0.0852345 + 0.996361i \(0.527164\pi\)
\(654\) −9.79849e11 −0.209440
\(655\) −6.50008e11 −0.137985
\(656\) 5.17134e12 1.09027
\(657\) −1.09955e12 −0.230235
\(658\) −3.18147e11 −0.0661624
\(659\) −3.67245e12 −0.758528 −0.379264 0.925288i \(-0.623823\pi\)
−0.379264 + 0.925288i \(0.623823\pi\)
\(660\) −4.33321e11 −0.0888919
\(661\) −2.78193e12 −0.566813 −0.283406 0.959000i \(-0.591465\pi\)
−0.283406 + 0.959000i \(0.591465\pi\)
\(662\) −6.51736e10 −0.0131890
\(663\) −1.44985e12 −0.291416
\(664\) −1.90013e12 −0.379339
\(665\) 4.77789e11 0.0947412
\(666\) −4.03071e11 −0.0793867
\(667\) −1.62183e12 −0.317278
\(668\) 6.78261e12 1.31796
\(669\) −3.83140e12 −0.739503
\(670\) −6.19574e11 −0.118784
\(671\) −2.86184e12 −0.544998
\(672\) 2.03553e12 0.385048
\(673\) 4.17416e12 0.784335 0.392167 0.919894i \(-0.371725\pi\)
0.392167 + 0.919894i \(0.371725\pi\)
\(674\) −8.87248e10 −0.0165606
\(675\) 2.07594e11 0.0384900
\(676\) 3.23449e12 0.595724
\(677\) −3.11414e12 −0.569756 −0.284878 0.958564i \(-0.591953\pi\)
−0.284878 + 0.958564i \(0.591953\pi\)
\(678\) −1.30835e12 −0.237789
\(679\) 7.30921e12 1.31964
\(680\) −1.07449e12 −0.192713
\(681\) −1.71421e12 −0.305423
\(682\) 1.42778e11 0.0252716
\(683\) −2.50273e12 −0.440069 −0.220034 0.975492i \(-0.570617\pi\)
−0.220034 + 0.975492i \(0.570617\pi\)
\(684\) −4.06997e11 −0.0710950
\(685\) 3.21095e12 0.557219
\(686\) −1.62948e12 −0.280925
\(687\) 3.03710e12 0.520181
\(688\) −4.63764e12 −0.789132
\(689\) −5.87361e12 −0.992930
\(690\) −2.03690e11 −0.0342096
\(691\) 9.18260e12 1.53220 0.766099 0.642723i \(-0.222195\pi\)
0.766099 + 0.642723i \(0.222195\pi\)
\(692\) 1.47732e12 0.244905
\(693\) 6.92070e11 0.113986
\(694\) 3.67401e11 0.0601205
\(695\) −3.57804e11 −0.0581720
\(696\) 1.16131e12 0.187588
\(697\) 7.20530e12 1.15639
\(698\) 1.48796e12 0.237270
\(699\) −7.09326e12 −1.12382
\(700\) 1.09071e12 0.171699
\(701\) 2.43652e11 0.0381100 0.0190550 0.999818i \(-0.493934\pi\)
0.0190550 + 0.999818i \(0.493934\pi\)
\(702\) 1.96803e11 0.0305855
\(703\) 1.33437e12 0.206052
\(704\) 1.45413e12 0.223113
\(705\) 4.57614e11 0.0697667
\(706\) −6.28920e11 −0.0952739
\(707\) −2.20830e12 −0.332407
\(708\) −3.39542e12 −0.507859
\(709\) 6.80072e12 1.01076 0.505379 0.862898i \(-0.331353\pi\)
0.505379 + 0.862898i \(0.331353\pi\)
\(710\) −1.01695e12 −0.150188
\(711\) −1.71740e12 −0.252034
\(712\) 2.62613e12 0.382963
\(713\) −8.87413e11 −0.128595
\(714\) 8.26783e11 0.119055
\(715\) −6.93656e11 −0.0992583
\(716\) −4.90789e12 −0.697889
\(717\) 4.52222e11 0.0639021
\(718\) 2.02317e12 0.284101
\(719\) 2.77354e12 0.387039 0.193520 0.981096i \(-0.438010\pi\)
0.193520 + 0.981096i \(0.438010\pi\)
\(720\) −8.53520e11 −0.118363
\(721\) −1.44911e12 −0.199707
\(722\) −1.01901e11 −0.0139561
\(723\) −4.81270e12 −0.655039
\(724\) 1.09528e13 1.48150
\(725\) 9.44742e11 0.126997
\(726\) 9.88813e11 0.132099
\(727\) 6.29829e12 0.836215 0.418107 0.908398i \(-0.362694\pi\)
0.418107 + 0.908398i \(0.362694\pi\)
\(728\) 2.14623e12 0.283195
\(729\) 2.82430e11 0.0370370
\(730\) −6.28458e11 −0.0819074
\(731\) −6.46169e12 −0.836986
\(732\) −6.13621e12 −0.789951
\(733\) 5.64464e10 0.00722218 0.00361109 0.999993i \(-0.498851\pi\)
0.00361109 + 0.999993i \(0.498851\pi\)
\(734\) −3.21169e12 −0.408414
\(735\) 3.00897e11 0.0380299
\(736\) 2.87278e12 0.360871
\(737\) 2.97098e12 0.370933
\(738\) −9.78049e11 −0.121369
\(739\) 6.09347e12 0.751562 0.375781 0.926708i \(-0.377374\pi\)
0.375781 + 0.926708i \(0.377374\pi\)
\(740\) 3.04612e12 0.373426
\(741\) −6.51516e11 −0.0793859
\(742\) 3.34944e12 0.405653
\(743\) −1.38725e13 −1.66995 −0.834977 0.550285i \(-0.814519\pi\)
−0.834977 + 0.550285i \(0.814519\pi\)
\(744\) 6.35427e11 0.0760305
\(745\) −2.70490e12 −0.321698
\(746\) −2.30029e12 −0.271931
\(747\) −2.10303e12 −0.247117
\(748\) 2.48232e12 0.289935
\(749\) 2.62552e12 0.304823
\(750\) 1.18652e11 0.0136931
\(751\) 2.41345e12 0.276859 0.138430 0.990372i \(-0.455794\pi\)
0.138430 + 0.990372i \(0.455794\pi\)
\(752\) −1.88147e12 −0.214545
\(753\) −4.30575e12 −0.488058
\(754\) 8.95634e11 0.100916
\(755\) 4.77828e12 0.535193
\(756\) 1.48390e12 0.165217
\(757\) −6.28091e12 −0.695170 −0.347585 0.937649i \(-0.612998\pi\)
−0.347585 + 0.937649i \(0.612998\pi\)
\(758\) −4.08631e12 −0.449594
\(759\) 9.76734e11 0.106829
\(760\) −4.82839e11 −0.0524978
\(761\) −3.30751e12 −0.357495 −0.178748 0.983895i \(-0.557205\pi\)
−0.178748 + 0.983895i \(0.557205\pi\)
\(762\) 1.23140e12 0.132313
\(763\) −1.18267e13 −1.26329
\(764\) 1.08916e13 1.15657
\(765\) −1.18922e12 −0.125541
\(766\) 1.28057e12 0.134392
\(767\) −5.43534e12 −0.567084
\(768\) 2.05186e12 0.212825
\(769\) 4.17246e12 0.430253 0.215127 0.976586i \(-0.430984\pi\)
0.215127 + 0.976586i \(0.430984\pi\)
\(770\) 3.95559e11 0.0405511
\(771\) 3.84000e12 0.391369
\(772\) −1.09069e13 −1.10516
\(773\) −5.76746e12 −0.581001 −0.290500 0.956875i \(-0.593822\pi\)
−0.290500 + 0.956875i \(0.593822\pi\)
\(774\) 8.77111e11 0.0878457
\(775\) 5.16931e11 0.0514725
\(776\) −7.38646e12 −0.731238
\(777\) −4.86505e12 −0.478843
\(778\) −3.02429e11 −0.0295948
\(779\) 3.23783e12 0.315018
\(780\) −1.48730e12 −0.143871
\(781\) 4.87646e12 0.469002
\(782\) 1.16686e12 0.111580
\(783\) 1.28531e12 0.122203
\(784\) −1.23714e12 −0.116949
\(785\) −3.93715e12 −0.370057
\(786\) −5.05446e11 −0.0472361
\(787\) 1.10224e13 1.02421 0.512105 0.858923i \(-0.328866\pi\)
0.512105 + 0.858923i \(0.328866\pi\)
\(788\) −2.62237e12 −0.242285
\(789\) −2.60261e12 −0.239090
\(790\) −9.81598e11 −0.0896627
\(791\) −1.57918e13 −1.43429
\(792\) −6.99385e11 −0.0631616
\(793\) −9.82277e12 −0.882073
\(794\) −1.38985e12 −0.124101
\(795\) −4.81775e12 −0.427752
\(796\) 4.54848e12 0.401566
\(797\) 2.60120e12 0.228356 0.114178 0.993460i \(-0.463577\pi\)
0.114178 + 0.993460i \(0.463577\pi\)
\(798\) 3.71529e11 0.0324324
\(799\) −2.62148e12 −0.227555
\(800\) −1.67344e12 −0.144446
\(801\) 2.90656e12 0.249478
\(802\) −5.24409e11 −0.0447595
\(803\) 3.01358e12 0.255778
\(804\) 6.37021e12 0.537652
\(805\) −2.45853e12 −0.206345
\(806\) 4.90061e11 0.0409018
\(807\) −5.59776e12 −0.464604
\(808\) 2.23164e12 0.184193
\(809\) 8.25982e12 0.677957 0.338979 0.940794i \(-0.389919\pi\)
0.338979 + 0.940794i \(0.389919\pi\)
\(810\) 1.61425e11 0.0131762
\(811\) −1.11201e13 −0.902638 −0.451319 0.892363i \(-0.649046\pi\)
−0.451319 + 0.892363i \(0.649046\pi\)
\(812\) 6.75309e12 0.545130
\(813\) −1.09804e13 −0.881480
\(814\) 1.10471e12 0.0881942
\(815\) −4.19903e12 −0.333380
\(816\) 4.88947e12 0.386061
\(817\) −2.90367e12 −0.228007
\(818\) −2.93230e12 −0.228991
\(819\) 2.37541e12 0.184485
\(820\) 7.39139e12 0.570905
\(821\) −1.36720e13 −1.05024 −0.525119 0.851029i \(-0.675979\pi\)
−0.525119 + 0.851029i \(0.675979\pi\)
\(822\) 2.49683e12 0.190751
\(823\) 5.80997e12 0.441443 0.220721 0.975337i \(-0.429159\pi\)
0.220721 + 0.975337i \(0.429159\pi\)
\(824\) 1.46443e12 0.110661
\(825\) −5.68962e11 −0.0427603
\(826\) 3.09952e12 0.231678
\(827\) 1.72434e13 1.28188 0.640942 0.767589i \(-0.278544\pi\)
0.640942 + 0.767589i \(0.278544\pi\)
\(828\) 2.09426e12 0.154844
\(829\) 2.94113e12 0.216281 0.108141 0.994136i \(-0.465510\pi\)
0.108141 + 0.994136i \(0.465510\pi\)
\(830\) −1.20201e12 −0.0879134
\(831\) 1.90641e12 0.138679
\(832\) 4.99103e12 0.361107
\(833\) −1.72372e12 −0.124041
\(834\) −2.78229e11 −0.0199138
\(835\) 8.90574e12 0.633988
\(836\) 1.11547e12 0.0789825
\(837\) 7.03279e11 0.0495294
\(838\) 1.16117e12 0.0813385
\(839\) 5.11516e12 0.356394 0.178197 0.983995i \(-0.442974\pi\)
0.178197 + 0.983995i \(0.442974\pi\)
\(840\) 1.76042e12 0.122000
\(841\) −8.65781e12 −0.596796
\(842\) −7.36001e12 −0.504631
\(843\) 6.50044e12 0.443321
\(844\) 1.59228e13 1.08013
\(845\) 4.24696e12 0.286565
\(846\) 3.55841e11 0.0238830
\(847\) 1.19349e13 0.796792
\(848\) 1.98081e13 1.31541
\(849\) −4.11434e12 −0.271779
\(850\) −6.79711e11 −0.0446621
\(851\) −6.86616e12 −0.448777
\(852\) 1.04558e13 0.679798
\(853\) −9.00696e12 −0.582516 −0.291258 0.956645i \(-0.594074\pi\)
−0.291258 + 0.956645i \(0.594074\pi\)
\(854\) 5.60146e12 0.360364
\(855\) −5.34398e11 −0.0341993
\(856\) −2.65327e12 −0.168908
\(857\) −1.72770e13 −1.09409 −0.547047 0.837102i \(-0.684248\pi\)
−0.547047 + 0.837102i \(0.684248\pi\)
\(858\) −5.39387e11 −0.0339787
\(859\) −1.73475e13 −1.08710 −0.543549 0.839378i \(-0.682920\pi\)
−0.543549 + 0.839378i \(0.682920\pi\)
\(860\) −6.62858e12 −0.413216
\(861\) −1.18050e13 −0.732069
\(862\) −7.65887e12 −0.472479
\(863\) 1.87251e12 0.114915 0.0574574 0.998348i \(-0.481701\pi\)
0.0574574 + 0.998348i \(0.481701\pi\)
\(864\) −2.27669e12 −0.138993
\(865\) 1.93976e12 0.117808
\(866\) 1.82408e12 0.110208
\(867\) −2.79305e12 −0.167877
\(868\) 3.69506e12 0.220944
\(869\) 4.70696e12 0.279996
\(870\) 7.34632e11 0.0434743
\(871\) 1.01974e13 0.600352
\(872\) 1.19517e13 0.700014
\(873\) −8.17520e12 −0.476359
\(874\) 5.24347e11 0.0303961
\(875\) 1.43213e12 0.0825935
\(876\) 6.46155e12 0.370739
\(877\) −2.62343e13 −1.49752 −0.748758 0.662844i \(-0.769349\pi\)
−0.748758 + 0.662844i \(0.769349\pi\)
\(878\) −1.18455e11 −0.00672709
\(879\) 1.77303e13 1.00176
\(880\) 2.33928e12 0.131495
\(881\) −2.36010e13 −1.31989 −0.659947 0.751312i \(-0.729421\pi\)
−0.659947 + 0.751312i \(0.729421\pi\)
\(882\) 2.33978e11 0.0130187
\(883\) −2.76189e13 −1.52891 −0.764457 0.644675i \(-0.776993\pi\)
−0.764457 + 0.644675i \(0.776993\pi\)
\(884\) 8.52012e12 0.469257
\(885\) −4.45827e12 −0.244299
\(886\) −5.90761e12 −0.322077
\(887\) 1.88352e13 1.02168 0.510838 0.859677i \(-0.329335\pi\)
0.510838 + 0.859677i \(0.329335\pi\)
\(888\) 4.91647e12 0.265335
\(889\) 1.48630e13 0.798083
\(890\) 1.66127e12 0.0887534
\(891\) −7.74066e11 −0.0411461
\(892\) 2.25154e13 1.19080
\(893\) −1.17801e12 −0.0619893
\(894\) −2.10333e12 −0.110126
\(895\) −6.44418e12 −0.335710
\(896\) −1.57127e13 −0.814450
\(897\) 3.35246e12 0.172901
\(898\) −3.67836e12 −0.188760
\(899\) 3.20056e12 0.163421
\(900\) −1.21994e12 −0.0619792
\(901\) 2.75989e13 1.39518
\(902\) 2.68058e12 0.134834
\(903\) 1.05867e13 0.529866
\(904\) 1.59587e13 0.794765
\(905\) 1.43813e13 0.712653
\(906\) 3.71559e12 0.183211
\(907\) −4.94068e12 −0.242412 −0.121206 0.992627i \(-0.538676\pi\)
−0.121206 + 0.992627i \(0.538676\pi\)
\(908\) 1.00736e13 0.491813
\(909\) 2.46994e12 0.119991
\(910\) 1.35769e12 0.0656316
\(911\) 9.90652e12 0.476528 0.238264 0.971200i \(-0.423422\pi\)
0.238264 + 0.971200i \(0.423422\pi\)
\(912\) 2.19717e12 0.105169
\(913\) 5.76386e12 0.274533
\(914\) 5.70363e12 0.270330
\(915\) −8.05700e12 −0.379995
\(916\) −1.78477e13 −0.837629
\(917\) −6.10072e12 −0.284917
\(918\) −9.24739e11 −0.0429761
\(919\) 3.20599e13 1.48266 0.741331 0.671139i \(-0.234195\pi\)
0.741331 + 0.671139i \(0.234195\pi\)
\(920\) 2.48451e12 0.114339
\(921\) 1.81011e13 0.828967
\(922\) 6.71728e12 0.306129
\(923\) 1.67376e13 0.759075
\(924\) −4.06698e12 −0.183547
\(925\) 3.99964e12 0.179632
\(926\) −2.09486e12 −0.0936278
\(927\) 1.62080e12 0.0720893
\(928\) −1.03610e13 −0.458603
\(929\) 5.29359e12 0.233174 0.116587 0.993181i \(-0.462805\pi\)
0.116587 + 0.993181i \(0.462805\pi\)
\(930\) 4.01966e11 0.0176204
\(931\) −7.74583e11 −0.0337905
\(932\) 4.16838e13 1.80966
\(933\) −1.99443e13 −0.861689
\(934\) 9.51535e12 0.409133
\(935\) 3.25935e12 0.139469
\(936\) −2.40051e12 −0.102226
\(937\) −1.51048e13 −0.640158 −0.320079 0.947391i \(-0.603709\pi\)
−0.320079 + 0.947391i \(0.603709\pi\)
\(938\) −5.81507e12 −0.245269
\(939\) −5.82932e12 −0.244694
\(940\) −2.68919e12 −0.112343
\(941\) −2.90124e13 −1.20623 −0.603115 0.797655i \(-0.706074\pi\)
−0.603115 + 0.797655i \(0.706074\pi\)
\(942\) −3.06153e12 −0.126680
\(943\) −1.66607e13 −0.686104
\(944\) 1.83301e13 0.751261
\(945\) 1.94840e12 0.0794756
\(946\) −2.40393e12 −0.0975916
\(947\) −2.65673e13 −1.07343 −0.536713 0.843765i \(-0.680334\pi\)
−0.536713 + 0.843765i \(0.680334\pi\)
\(948\) 1.00924e13 0.405842
\(949\) 1.03436e13 0.413974
\(950\) −3.05440e11 −0.0121666
\(951\) −1.97132e13 −0.781528
\(952\) −1.00847e13 −0.397921
\(953\) 1.59521e13 0.626469 0.313235 0.949676i \(-0.398587\pi\)
0.313235 + 0.949676i \(0.398587\pi\)
\(954\) −3.74628e12 −0.146431
\(955\) 1.43010e13 0.556353
\(956\) −2.65750e12 −0.102899
\(957\) −3.52271e12 −0.135760
\(958\) 5.78402e12 0.221863
\(959\) 3.01367e13 1.15057
\(960\) 4.09383e12 0.155564
\(961\) −2.46884e13 −0.933765
\(962\) 3.79173e12 0.142741
\(963\) −2.93659e12 −0.110034
\(964\) 2.82821e13 1.05479
\(965\) −1.43211e13 −0.531622
\(966\) −1.91175e12 −0.0706374
\(967\) 3.00890e13 1.10660 0.553298 0.832984i \(-0.313369\pi\)
0.553298 + 0.832984i \(0.313369\pi\)
\(968\) −1.20611e13 −0.441517
\(969\) 3.06135e12 0.111546
\(970\) −4.67261e12 −0.169468
\(971\) 1.55700e13 0.562084 0.281042 0.959696i \(-0.409320\pi\)
0.281042 + 0.959696i \(0.409320\pi\)
\(972\) −1.65971e12 −0.0596395
\(973\) −3.35821e12 −0.120116
\(974\) 4.64685e12 0.165441
\(975\) −1.95286e12 −0.0692070
\(976\) 3.31262e13 1.16855
\(977\) −4.75797e13 −1.67069 −0.835345 0.549726i \(-0.814732\pi\)
−0.835345 + 0.549726i \(0.814732\pi\)
\(978\) −3.26516e12 −0.114125
\(979\) −7.96612e12 −0.277156
\(980\) −1.76824e12 −0.0612383
\(981\) 1.32280e13 0.456018
\(982\) 1.01781e12 0.0349274
\(983\) 4.55928e13 1.55742 0.778709 0.627386i \(-0.215875\pi\)
0.778709 + 0.627386i \(0.215875\pi\)
\(984\) 1.19298e13 0.405653
\(985\) −3.44324e12 −0.116548
\(986\) −4.20840e12 −0.141798
\(987\) 4.29498e12 0.144057
\(988\) 3.82866e12 0.127832
\(989\) 1.49412e13 0.496596
\(990\) −4.42425e11 −0.0146380
\(991\) 3.10435e13 1.02244 0.511221 0.859449i \(-0.329193\pi\)
0.511221 + 0.859449i \(0.329193\pi\)
\(992\) −5.66921e12 −0.185874
\(993\) 8.79844e11 0.0287167
\(994\) −9.54464e12 −0.310114
\(995\) 5.97226e12 0.193168
\(996\) 1.23585e13 0.397924
\(997\) 2.46892e13 0.791369 0.395684 0.918387i \(-0.370507\pi\)
0.395684 + 0.918387i \(0.370507\pi\)
\(998\) −1.69015e12 −0.0539310
\(999\) 5.44146e12 0.172851
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 \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 285.10.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
285.10.a.a.1.1 1 1.1 even 1 trivial