Properties

Label 6.26.a.b.1.1
Level $6$
Weight $26$
Character 6.1
Self dual yes
Analytic conductor $23.760$
Analytic rank $0$
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] = [6,26,Mod(1,6)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 26, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6.1");
 
S:= CuspForms(chi, 26);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6 = 2 \cdot 3 \)
Weight: \( k \) \(=\) \( 26 \)
Character orbit: \([\chi]\) \(=\) 6.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: \(23.7598067971\)
Analytic rank: \(0\)
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 \(\chi\) \(=\) 6.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-4096.00 q^{2} +531441. q^{3} +1.67772e7 q^{4} +5.90426e8 q^{5} -2.17678e9 q^{6} +5.78574e10 q^{7} -6.87195e10 q^{8} +2.82430e11 q^{9} +O(q^{10})\) \(q-4096.00 q^{2} +531441. q^{3} +1.67772e7 q^{4} +5.90426e8 q^{5} -2.17678e9 q^{6} +5.78574e10 q^{7} -6.87195e10 q^{8} +2.82430e11 q^{9} -2.41838e12 q^{10} +9.49427e12 q^{11} +8.91610e12 q^{12} -1.34968e14 q^{13} -2.36984e14 q^{14} +3.13776e14 q^{15} +2.81475e14 q^{16} -2.52611e15 q^{17} -1.15683e15 q^{18} +1.14688e16 q^{19} +9.90570e15 q^{20} +3.07478e16 q^{21} -3.88885e16 q^{22} +1.13343e17 q^{23} -3.65203e16 q^{24} +5.05793e16 q^{25} +5.52829e17 q^{26} +1.50095e17 q^{27} +9.70686e17 q^{28} +1.08135e18 q^{29} -1.28523e18 q^{30} +4.64909e18 q^{31} -1.15292e18 q^{32} +5.04564e18 q^{33} +1.03470e19 q^{34} +3.41605e19 q^{35} +4.73838e18 q^{36} -4.60934e19 q^{37} -4.69760e19 q^{38} -7.17275e19 q^{39} -4.05737e19 q^{40} +5.14492e19 q^{41} -1.25943e20 q^{42} -3.69343e20 q^{43} +1.59287e20 q^{44} +1.66754e20 q^{45} -4.64251e20 q^{46} -4.91066e19 q^{47} +1.49587e20 q^{48} +2.00641e21 q^{49} -2.07173e20 q^{50} -1.34248e21 q^{51} -2.26439e21 q^{52} +4.44008e21 q^{53} -6.14788e20 q^{54} +5.60566e21 q^{55} -3.97593e21 q^{56} +6.09497e21 q^{57} -4.42921e21 q^{58} +2.25493e22 q^{59} +5.26430e21 q^{60} -1.23106e22 q^{61} -1.90427e22 q^{62} +1.63406e22 q^{63} +4.72237e21 q^{64} -7.96886e22 q^{65} -2.06670e22 q^{66} +6.38592e21 q^{67} -4.23812e22 q^{68} +6.02349e22 q^{69} -1.39921e23 q^{70} +6.02768e22 q^{71} -1.94084e22 q^{72} -2.68812e23 q^{73} +1.88798e23 q^{74} +2.68799e22 q^{75} +1.92414e23 q^{76} +5.49314e23 q^{77} +2.93796e23 q^{78} -2.62861e23 q^{79} +1.66190e23 q^{80} +7.97664e22 q^{81} -2.10736e23 q^{82} +6.88506e23 q^{83} +5.15863e23 q^{84} -1.49148e24 q^{85} +1.51283e24 q^{86} +5.74673e23 q^{87} -6.52441e23 q^{88} -2.55875e24 q^{89} -6.83023e23 q^{90} -7.80890e24 q^{91} +1.90157e24 q^{92} +2.47072e24 q^{93} +2.01141e23 q^{94} +6.77145e24 q^{95} -6.12710e23 q^{96} +2.59521e24 q^{97} -8.21826e24 q^{98} +2.68146e24 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4096.00 −0.707107
\(3\) 531441. 0.577350
\(4\) 1.67772e7 0.500000
\(5\) 5.90426e8 1.08153 0.540767 0.841172i \(-0.318134\pi\)
0.540767 + 0.841172i \(0.318134\pi\)
\(6\) −2.17678e9 −0.408248
\(7\) 5.78574e10 1.57991 0.789957 0.613162i \(-0.210103\pi\)
0.789957 + 0.613162i \(0.210103\pi\)
\(8\) −6.87195e10 −0.353553
\(9\) 2.82430e11 0.333333
\(10\) −2.41838e12 −0.764760
\(11\) 9.49427e12 0.912122 0.456061 0.889949i \(-0.349260\pi\)
0.456061 + 0.889949i \(0.349260\pi\)
\(12\) 8.91610e12 0.288675
\(13\) −1.34968e14 −1.60672 −0.803358 0.595497i \(-0.796955\pi\)
−0.803358 + 0.595497i \(0.796955\pi\)
\(14\) −2.36984e14 −1.11717
\(15\) 3.13776e14 0.624424
\(16\) 2.81475e14 0.250000
\(17\) −2.52611e15 −1.05158 −0.525789 0.850615i \(-0.676230\pi\)
−0.525789 + 0.850615i \(0.676230\pi\)
\(18\) −1.15683e15 −0.235702
\(19\) 1.14688e16 1.18877 0.594383 0.804182i \(-0.297396\pi\)
0.594383 + 0.804182i \(0.297396\pi\)
\(20\) 9.90570e15 0.540767
\(21\) 3.07478e16 0.912164
\(22\) −3.88885e16 −0.644968
\(23\) 1.13343e17 1.07844 0.539219 0.842165i \(-0.318719\pi\)
0.539219 + 0.842165i \(0.318719\pi\)
\(24\) −3.65203e16 −0.204124
\(25\) 5.05793e16 0.169716
\(26\) 5.52829e17 1.13612
\(27\) 1.50095e17 0.192450
\(28\) 9.70686e17 0.789957
\(29\) 1.08135e18 0.567533 0.283766 0.958893i \(-0.408416\pi\)
0.283766 + 0.958893i \(0.408416\pi\)
\(30\) −1.28523e18 −0.441534
\(31\) 4.64909e18 1.06010 0.530050 0.847966i \(-0.322173\pi\)
0.530050 + 0.847966i \(0.322173\pi\)
\(32\) −1.15292e18 −0.176777
\(33\) 5.04564e18 0.526614
\(34\) 1.03470e19 0.743578
\(35\) 3.41605e19 1.70873
\(36\) 4.73838e18 0.166667
\(37\) −4.60934e19 −1.15111 −0.575555 0.817763i \(-0.695214\pi\)
−0.575555 + 0.817763i \(0.695214\pi\)
\(38\) −4.69760e19 −0.840585
\(39\) −7.17275e19 −0.927638
\(40\) −4.05737e19 −0.382380
\(41\) 5.14492e19 0.356107 0.178054 0.984021i \(-0.443020\pi\)
0.178054 + 0.984021i \(0.443020\pi\)
\(42\) −1.25943e20 −0.644997
\(43\) −3.69343e20 −1.40953 −0.704764 0.709442i \(-0.748947\pi\)
−0.704764 + 0.709442i \(0.748947\pi\)
\(44\) 1.59287e20 0.456061
\(45\) 1.66754e20 0.360511
\(46\) −4.64251e20 −0.762571
\(47\) −4.91066e19 −0.0616477 −0.0308239 0.999525i \(-0.509813\pi\)
−0.0308239 + 0.999525i \(0.509813\pi\)
\(48\) 1.49587e20 0.144338
\(49\) 2.00641e21 1.49613
\(50\) −2.07173e20 −0.120007
\(51\) −1.34248e21 −0.607129
\(52\) −2.26439e21 −0.803358
\(53\) 4.44008e21 1.24149 0.620744 0.784014i \(-0.286831\pi\)
0.620744 + 0.784014i \(0.286831\pi\)
\(54\) −6.14788e20 −0.136083
\(55\) 5.60566e21 0.986491
\(56\) −3.97593e21 −0.558584
\(57\) 6.09497e21 0.686334
\(58\) −4.42921e21 −0.401306
\(59\) 2.25493e22 1.65000 0.825000 0.565133i \(-0.191175\pi\)
0.825000 + 0.565133i \(0.191175\pi\)
\(60\) 5.26430e21 0.312212
\(61\) −1.23106e22 −0.593824 −0.296912 0.954905i \(-0.595957\pi\)
−0.296912 + 0.954905i \(0.595957\pi\)
\(62\) −1.90427e22 −0.749604
\(63\) 1.63406e22 0.526638
\(64\) 4.72237e21 0.125000
\(65\) −7.96886e22 −1.73772
\(66\) −2.06670e22 −0.372372
\(67\) 6.38592e21 0.0953429 0.0476715 0.998863i \(-0.484820\pi\)
0.0476715 + 0.998863i \(0.484820\pi\)
\(68\) −4.23812e22 −0.525789
\(69\) 6.02349e22 0.622637
\(70\) −1.39921e23 −1.20826
\(71\) 6.02768e22 0.435934 0.217967 0.975956i \(-0.430058\pi\)
0.217967 + 0.975956i \(0.430058\pi\)
\(72\) −1.94084e22 −0.117851
\(73\) −2.68812e23 −1.37377 −0.686883 0.726768i \(-0.741022\pi\)
−0.686883 + 0.726768i \(0.741022\pi\)
\(74\) 1.88798e23 0.813958
\(75\) 2.68799e22 0.0979856
\(76\) 1.92414e23 0.594383
\(77\) 5.49314e23 1.44107
\(78\) 2.93796e23 0.655939
\(79\) −2.62861e23 −0.500481 −0.250241 0.968184i \(-0.580510\pi\)
−0.250241 + 0.968184i \(0.580510\pi\)
\(80\) 1.66190e23 0.270384
\(81\) 7.97664e22 0.111111
\(82\) −2.10736e23 −0.251806
\(83\) 6.88506e23 0.707020 0.353510 0.935431i \(-0.384988\pi\)
0.353510 + 0.935431i \(0.384988\pi\)
\(84\) 5.15863e23 0.456082
\(85\) −1.49148e24 −1.13732
\(86\) 1.51283e24 0.996687
\(87\) 5.74673e23 0.327665
\(88\) −6.52441e23 −0.322484
\(89\) −2.55875e24 −1.09813 −0.549064 0.835780i \(-0.685016\pi\)
−0.549064 + 0.835780i \(0.685016\pi\)
\(90\) −6.83023e23 −0.254920
\(91\) −7.80890e24 −2.53847
\(92\) 1.90157e24 0.539219
\(93\) 2.47072e24 0.612049
\(94\) 2.01141e23 0.0435915
\(95\) 6.77145e24 1.28569
\(96\) −6.12710e23 −0.102062
\(97\) 2.59521e24 0.379774 0.189887 0.981806i \(-0.439188\pi\)
0.189887 + 0.981806i \(0.439188\pi\)
\(98\) −8.21826e24 −1.05792
\(99\) 2.68146e24 0.304041
\(100\) 8.48580e23 0.0848580
\(101\) −1.74742e25 −1.54305 −0.771524 0.636200i \(-0.780505\pi\)
−0.771524 + 0.636200i \(0.780505\pi\)
\(102\) 5.49880e24 0.429305
\(103\) −1.32798e25 −0.917756 −0.458878 0.888499i \(-0.651748\pi\)
−0.458878 + 0.888499i \(0.651748\pi\)
\(104\) 9.27493e24 0.568060
\(105\) 1.81543e25 0.986536
\(106\) −1.81866e25 −0.877864
\(107\) 4.17957e25 1.79405 0.897025 0.441980i \(-0.145724\pi\)
0.897025 + 0.441980i \(0.145724\pi\)
\(108\) 2.51817e24 0.0962250
\(109\) 4.64640e25 1.58229 0.791143 0.611631i \(-0.209486\pi\)
0.791143 + 0.611631i \(0.209486\pi\)
\(110\) −2.29608e25 −0.697554
\(111\) −2.44959e25 −0.664594
\(112\) 1.62854e25 0.394979
\(113\) 3.27395e25 0.710544 0.355272 0.934763i \(-0.384388\pi\)
0.355272 + 0.934763i \(0.384388\pi\)
\(114\) −2.49650e25 −0.485312
\(115\) 6.69204e25 1.16637
\(116\) 1.81420e25 0.283766
\(117\) −3.81190e25 −0.535572
\(118\) −9.23621e25 −1.16673
\(119\) −1.46154e26 −1.66140
\(120\) −2.15626e25 −0.220767
\(121\) −1.82060e25 −0.168034
\(122\) 5.04244e25 0.419897
\(123\) 2.73422e25 0.205599
\(124\) 7.79988e25 0.530050
\(125\) −1.46097e26 −0.897980
\(126\) −6.69313e25 −0.372389
\(127\) −1.69226e26 −0.852944 −0.426472 0.904501i \(-0.640244\pi\)
−0.426472 + 0.904501i \(0.640244\pi\)
\(128\) −1.93428e25 −0.0883883
\(129\) −1.96284e26 −0.813791
\(130\) 3.26404e26 1.22875
\(131\) 1.90262e25 0.0650821 0.0325410 0.999470i \(-0.489640\pi\)
0.0325410 + 0.999470i \(0.489640\pi\)
\(132\) 8.46518e25 0.263307
\(133\) 6.63553e26 1.87815
\(134\) −2.61567e25 −0.0674176
\(135\) 8.86197e25 0.208141
\(136\) 1.73593e26 0.371789
\(137\) −3.48433e25 −0.0680945 −0.0340473 0.999420i \(-0.510840\pi\)
−0.0340473 + 0.999420i \(0.510840\pi\)
\(138\) −2.46722e26 −0.440271
\(139\) 1.79180e26 0.292150 0.146075 0.989274i \(-0.453336\pi\)
0.146075 + 0.989274i \(0.453336\pi\)
\(140\) 5.73118e26 0.854366
\(141\) −2.60973e25 −0.0355923
\(142\) −2.46894e26 −0.308252
\(143\) −1.28142e27 −1.46552
\(144\) 7.94968e25 0.0833333
\(145\) 6.38456e26 0.613806
\(146\) 1.10105e27 0.971400
\(147\) 1.06629e27 0.863791
\(148\) −7.73318e26 −0.575555
\(149\) −2.46268e27 −1.68492 −0.842459 0.538760i \(-0.818893\pi\)
−0.842459 + 0.538760i \(0.818893\pi\)
\(150\) −1.10100e26 −0.0692863
\(151\) 4.33646e26 0.251144 0.125572 0.992084i \(-0.459923\pi\)
0.125572 + 0.992084i \(0.459923\pi\)
\(152\) −7.88127e26 −0.420292
\(153\) −7.13449e26 −0.350526
\(154\) −2.24999e27 −1.01899
\(155\) 2.74494e27 1.14653
\(156\) −1.20339e27 −0.463819
\(157\) −3.54008e27 −1.25970 −0.629850 0.776717i \(-0.716884\pi\)
−0.629850 + 0.776717i \(0.716884\pi\)
\(158\) 1.07668e27 0.353894
\(159\) 2.35964e27 0.716773
\(160\) −6.80715e26 −0.191190
\(161\) 6.55771e27 1.70384
\(162\) −3.26723e26 −0.0785674
\(163\) −6.45978e27 −1.43838 −0.719189 0.694815i \(-0.755486\pi\)
−0.719189 + 0.694815i \(0.755486\pi\)
\(164\) 8.63175e26 0.178054
\(165\) 2.97908e27 0.569551
\(166\) −2.82012e27 −0.499938
\(167\) −4.48432e27 −0.737463 −0.368732 0.929536i \(-0.620208\pi\)
−0.368732 + 0.929536i \(0.620208\pi\)
\(168\) −2.11297e27 −0.322499
\(169\) 1.11600e28 1.58153
\(170\) 6.10911e27 0.804205
\(171\) 3.23912e27 0.396255
\(172\) −6.19654e27 −0.704764
\(173\) 1.21973e27 0.129029 0.0645147 0.997917i \(-0.479450\pi\)
0.0645147 + 0.997917i \(0.479450\pi\)
\(174\) −2.35386e27 −0.231694
\(175\) 2.92639e27 0.268137
\(176\) 2.67240e27 0.228030
\(177\) 1.19836e28 0.952628
\(178\) 1.04806e28 0.776493
\(179\) −2.09258e28 −1.44550 −0.722751 0.691109i \(-0.757122\pi\)
−0.722751 + 0.691109i \(0.757122\pi\)
\(180\) 2.79766e27 0.180256
\(181\) −1.48896e27 −0.0895158 −0.0447579 0.998998i \(-0.514252\pi\)
−0.0447579 + 0.998998i \(0.514252\pi\)
\(182\) 3.19853e28 1.79497
\(183\) −6.54238e27 −0.342844
\(184\) −7.78885e27 −0.381286
\(185\) −2.72147e28 −1.24496
\(186\) −1.01201e28 −0.432784
\(187\) −2.39836e28 −0.959167
\(188\) −8.23873e26 −0.0308239
\(189\) 8.68409e27 0.304055
\(190\) −2.77359e28 −0.909121
\(191\) −3.19546e28 −0.980880 −0.490440 0.871475i \(-0.663164\pi\)
−0.490440 + 0.871475i \(0.663164\pi\)
\(192\) 2.50966e27 0.0721688
\(193\) 3.16427e28 0.852721 0.426361 0.904553i \(-0.359795\pi\)
0.426361 + 0.904553i \(0.359795\pi\)
\(194\) −1.06300e28 −0.268541
\(195\) −4.23498e28 −1.00327
\(196\) 3.36620e28 0.748065
\(197\) −4.82083e28 −1.00530 −0.502649 0.864491i \(-0.667641\pi\)
−0.502649 + 0.864491i \(0.667641\pi\)
\(198\) −1.09833e28 −0.214989
\(199\) 9.63887e28 1.77159 0.885793 0.464080i \(-0.153615\pi\)
0.885793 + 0.464080i \(0.153615\pi\)
\(200\) −3.47578e27 −0.0600037
\(201\) 3.39374e27 0.0550463
\(202\) 7.15742e28 1.09110
\(203\) 6.25641e28 0.896653
\(204\) −2.25231e28 −0.303564
\(205\) 3.03770e28 0.385142
\(206\) 5.43942e28 0.648951
\(207\) 3.20113e28 0.359479
\(208\) −3.79901e28 −0.401679
\(209\) 1.08887e29 1.08430
\(210\) −7.43600e28 −0.697587
\(211\) −7.22567e28 −0.638774 −0.319387 0.947624i \(-0.603477\pi\)
−0.319387 + 0.947624i \(0.603477\pi\)
\(212\) 7.44921e28 0.620744
\(213\) 3.20335e28 0.251687
\(214\) −1.71195e29 −1.26859
\(215\) −2.18069e29 −1.52445
\(216\) −1.03144e28 −0.0680414
\(217\) 2.68984e29 1.67487
\(218\) −1.90316e29 −1.11885
\(219\) −1.42858e29 −0.793145
\(220\) 9.40474e28 0.493245
\(221\) 3.40945e29 1.68959
\(222\) 1.00335e29 0.469939
\(223\) −7.19979e28 −0.318793 −0.159396 0.987215i \(-0.550955\pi\)
−0.159396 + 0.987215i \(0.550955\pi\)
\(224\) −6.67051e28 −0.279292
\(225\) 1.42851e28 0.0565720
\(226\) −1.34101e29 −0.502431
\(227\) −4.13856e29 −1.46732 −0.733662 0.679514i \(-0.762191\pi\)
−0.733662 + 0.679514i \(0.762191\pi\)
\(228\) 1.02257e29 0.343167
\(229\) −4.93767e29 −1.56884 −0.784420 0.620230i \(-0.787039\pi\)
−0.784420 + 0.620230i \(0.787039\pi\)
\(230\) −2.74106e29 −0.824747
\(231\) 2.91928e29 0.832005
\(232\) −7.43097e28 −0.200653
\(233\) 2.65749e29 0.680020 0.340010 0.940422i \(-0.389569\pi\)
0.340010 + 0.940422i \(0.389569\pi\)
\(234\) 1.56135e29 0.378706
\(235\) −2.89938e28 −0.0666741
\(236\) 3.78315e29 0.825000
\(237\) −1.39695e29 −0.288953
\(238\) 5.98649e29 1.17479
\(239\) −2.43722e29 −0.453859 −0.226929 0.973911i \(-0.572869\pi\)
−0.226929 + 0.973911i \(0.572869\pi\)
\(240\) 8.83202e28 0.156106
\(241\) −2.75488e29 −0.462263 −0.231132 0.972922i \(-0.574243\pi\)
−0.231132 + 0.972922i \(0.574243\pi\)
\(242\) 7.45716e28 0.118818
\(243\) 4.23912e28 0.0641500
\(244\) −2.06538e29 −0.296912
\(245\) 1.18464e30 1.61811
\(246\) −1.11994e29 −0.145380
\(247\) −1.54792e30 −1.91001
\(248\) −3.19483e29 −0.374802
\(249\) 3.65900e29 0.408198
\(250\) 5.98414e29 0.634968
\(251\) 1.20863e30 1.22004 0.610018 0.792387i \(-0.291162\pi\)
0.610018 + 0.792387i \(0.291162\pi\)
\(252\) 2.74151e29 0.263319
\(253\) 1.07611e30 0.983667
\(254\) 6.93150e29 0.603123
\(255\) −7.92635e29 −0.656630
\(256\) 7.92282e28 0.0625000
\(257\) −2.27637e29 −0.171032 −0.0855162 0.996337i \(-0.527254\pi\)
−0.0855162 + 0.996337i \(0.527254\pi\)
\(258\) 8.03979e29 0.575437
\(259\) −2.66684e30 −1.81866
\(260\) −1.33695e30 −0.868859
\(261\) 3.05405e29 0.189178
\(262\) −7.79314e28 −0.0460200
\(263\) 7.19227e28 0.0404966 0.0202483 0.999795i \(-0.493554\pi\)
0.0202483 + 0.999795i \(0.493554\pi\)
\(264\) −3.46734e29 −0.186186
\(265\) 2.62154e30 1.34271
\(266\) −2.71791e30 −1.32805
\(267\) −1.35982e30 −0.634004
\(268\) 1.07138e29 0.0476715
\(269\) 2.58878e29 0.109949 0.0549745 0.998488i \(-0.482492\pi\)
0.0549745 + 0.998488i \(0.482492\pi\)
\(270\) −3.62986e29 −0.147178
\(271\) −1.02532e30 −0.396955 −0.198478 0.980105i \(-0.563600\pi\)
−0.198478 + 0.980105i \(0.563600\pi\)
\(272\) −7.11038e29 −0.262894
\(273\) −4.14997e30 −1.46559
\(274\) 1.42718e29 0.0481501
\(275\) 4.80214e29 0.154802
\(276\) 1.01057e30 0.311318
\(277\) 4.94719e30 1.45667 0.728336 0.685220i \(-0.240294\pi\)
0.728336 + 0.685220i \(0.240294\pi\)
\(278\) −7.33923e29 −0.206581
\(279\) 1.31304e30 0.353367
\(280\) −2.34749e30 −0.604128
\(281\) −3.59090e30 −0.883842 −0.441921 0.897054i \(-0.645703\pi\)
−0.441921 + 0.897054i \(0.645703\pi\)
\(282\) 1.06894e29 0.0251676
\(283\) 4.52174e30 1.01853 0.509266 0.860609i \(-0.329917\pi\)
0.509266 + 0.860609i \(0.329917\pi\)
\(284\) 1.01128e30 0.217967
\(285\) 3.59863e30 0.742294
\(286\) 5.24871e30 1.03628
\(287\) 2.97672e30 0.562619
\(288\) −3.25619e29 −0.0589256
\(289\) 6.10625e29 0.105816
\(290\) −2.61512e30 −0.434026
\(291\) 1.37920e30 0.219263
\(292\) −4.50992e30 −0.686883
\(293\) 5.67005e30 0.827450 0.413725 0.910402i \(-0.364227\pi\)
0.413725 + 0.910402i \(0.364227\pi\)
\(294\) −4.36752e30 −0.610792
\(295\) 1.33137e31 1.78453
\(296\) 3.16751e30 0.406979
\(297\) 1.42504e30 0.175538
\(298\) 1.00871e31 1.19142
\(299\) −1.52976e31 −1.73274
\(300\) 4.50970e29 0.0489928
\(301\) −2.13692e31 −2.22693
\(302\) −1.77621e30 −0.177586
\(303\) −9.28650e30 −0.890880
\(304\) 3.22817e30 0.297192
\(305\) −7.26852e30 −0.642241
\(306\) 2.92229e30 0.247859
\(307\) −2.05241e31 −1.67122 −0.835611 0.549322i \(-0.814886\pi\)
−0.835611 + 0.549322i \(0.814886\pi\)
\(308\) 9.21596e30 0.720537
\(309\) −7.05745e30 −0.529867
\(310\) −1.12433e31 −0.810722
\(311\) 2.33797e31 1.61933 0.809663 0.586895i \(-0.199650\pi\)
0.809663 + 0.586895i \(0.199650\pi\)
\(312\) 4.92908e30 0.327969
\(313\) −3.96767e30 −0.253648 −0.126824 0.991925i \(-0.540478\pi\)
−0.126824 + 0.991925i \(0.540478\pi\)
\(314\) 1.45002e31 0.890742
\(315\) 9.64794e30 0.569577
\(316\) −4.41008e30 −0.250241
\(317\) −6.40482e30 −0.349354 −0.174677 0.984626i \(-0.555888\pi\)
−0.174677 + 0.984626i \(0.555888\pi\)
\(318\) −9.66508e30 −0.506835
\(319\) 1.02666e31 0.517659
\(320\) 2.78821e30 0.135192
\(321\) 2.22120e31 1.03580
\(322\) −2.68604e31 −1.20480
\(323\) −2.89714e31 −1.25008
\(324\) 1.33826e30 0.0555556
\(325\) −6.82659e30 −0.272685
\(326\) 2.64593e31 1.01709
\(327\) 2.46929e31 0.913533
\(328\) −3.53556e30 −0.125903
\(329\) −2.84118e30 −0.0973981
\(330\) −1.22023e31 −0.402733
\(331\) 3.45945e31 1.09940 0.549702 0.835361i \(-0.314741\pi\)
0.549702 + 0.835361i \(0.314741\pi\)
\(332\) 1.15512e31 0.353510
\(333\) −1.30181e31 −0.383703
\(334\) 1.83678e31 0.521465
\(335\) 3.77041e30 0.103117
\(336\) 8.65474e30 0.228041
\(337\) −2.03336e31 −0.516228 −0.258114 0.966114i \(-0.583101\pi\)
−0.258114 + 0.966114i \(0.583101\pi\)
\(338\) −4.57112e31 −1.11831
\(339\) 1.73991e31 0.410233
\(340\) −2.50229e31 −0.568659
\(341\) 4.41397e31 0.966940
\(342\) −1.32674e31 −0.280195
\(343\) 3.84951e31 0.783842
\(344\) 2.53810e31 0.498343
\(345\) 3.55642e31 0.673403
\(346\) −4.99603e30 −0.0912376
\(347\) 1.40061e31 0.246718 0.123359 0.992362i \(-0.460633\pi\)
0.123359 + 0.992362i \(0.460633\pi\)
\(348\) 9.64142e30 0.163833
\(349\) −1.88041e31 −0.308273 −0.154136 0.988050i \(-0.549260\pi\)
−0.154136 + 0.988050i \(0.549260\pi\)
\(350\) −1.19865e31 −0.189601
\(351\) −2.02580e31 −0.309213
\(352\) −1.09461e31 −0.161242
\(353\) −1.05140e32 −1.49481 −0.747404 0.664370i \(-0.768700\pi\)
−0.747404 + 0.664370i \(0.768700\pi\)
\(354\) −4.90850e31 −0.673609
\(355\) 3.55890e31 0.471477
\(356\) −4.29287e31 −0.549064
\(357\) −7.76725e31 −0.959211
\(358\) 8.57119e31 1.02212
\(359\) −2.39508e31 −0.275828 −0.137914 0.990444i \(-0.544040\pi\)
−0.137914 + 0.990444i \(0.544040\pi\)
\(360\) −1.14592e31 −0.127460
\(361\) 3.84559e31 0.413165
\(362\) 6.09876e30 0.0632972
\(363\) −9.67540e30 −0.0970144
\(364\) −1.31012e32 −1.26924
\(365\) −1.58714e32 −1.48578
\(366\) 2.67976e31 0.242428
\(367\) 1.16817e31 0.102136 0.0510680 0.998695i \(-0.483737\pi\)
0.0510680 + 0.998695i \(0.483737\pi\)
\(368\) 3.19031e31 0.269610
\(369\) 1.45308e31 0.118702
\(370\) 1.11471e32 0.880323
\(371\) 2.56891e32 1.96144
\(372\) 4.14518e31 0.306025
\(373\) −6.29652e31 −0.449511 −0.224755 0.974415i \(-0.572158\pi\)
−0.224755 + 0.974415i \(0.572158\pi\)
\(374\) 9.82368e31 0.678234
\(375\) −7.76421e31 −0.518449
\(376\) 3.37458e30 0.0217958
\(377\) −1.45948e32 −0.911864
\(378\) −3.55700e31 −0.214999
\(379\) 7.98394e31 0.466903 0.233452 0.972368i \(-0.424998\pi\)
0.233452 + 0.972368i \(0.424998\pi\)
\(380\) 1.13606e32 0.642846
\(381\) −8.99337e31 −0.492448
\(382\) 1.30886e32 0.693587
\(383\) 1.16203e32 0.595980 0.297990 0.954569i \(-0.403684\pi\)
0.297990 + 0.954569i \(0.403684\pi\)
\(384\) −1.02796e31 −0.0510310
\(385\) 3.24329e32 1.55857
\(386\) −1.29608e32 −0.602965
\(387\) −1.04313e32 −0.469843
\(388\) 4.35404e31 0.189887
\(389\) 1.94162e32 0.819964 0.409982 0.912094i \(-0.365535\pi\)
0.409982 + 0.912094i \(0.365535\pi\)
\(390\) 1.73465e32 0.709420
\(391\) −2.86316e32 −1.13406
\(392\) −1.37880e32 −0.528962
\(393\) 1.01113e31 0.0375751
\(394\) 1.97461e32 0.710852
\(395\) −1.55200e32 −0.541288
\(396\) 4.49875e31 0.152020
\(397\) −1.56136e32 −0.511237 −0.255618 0.966778i \(-0.582279\pi\)
−0.255618 + 0.966778i \(0.582279\pi\)
\(398\) −3.94808e32 −1.25270
\(399\) 3.52639e32 1.08435
\(400\) 1.42368e31 0.0424290
\(401\) 2.04402e32 0.590447 0.295223 0.955428i \(-0.404606\pi\)
0.295223 + 0.955428i \(0.404606\pi\)
\(402\) −1.39008e31 −0.0389236
\(403\) −6.27479e32 −1.70328
\(404\) −2.93168e32 −0.771524
\(405\) 4.70962e31 0.120170
\(406\) −2.56262e32 −0.634029
\(407\) −4.37623e32 −1.04995
\(408\) 9.22546e31 0.214652
\(409\) −5.80074e32 −1.30901 −0.654503 0.756059i \(-0.727122\pi\)
−0.654503 + 0.756059i \(0.727122\pi\)
\(410\) −1.24424e32 −0.272337
\(411\) −1.85172e31 −0.0393144
\(412\) −2.22799e32 −0.458878
\(413\) 1.30465e33 2.60686
\(414\) −1.31118e32 −0.254190
\(415\) 4.06512e32 0.764666
\(416\) 1.55608e32 0.284030
\(417\) 9.52238e31 0.168673
\(418\) −4.46003e32 −0.766716
\(419\) 9.59622e31 0.160112 0.0800562 0.996790i \(-0.474490\pi\)
0.0800562 + 0.996790i \(0.474490\pi\)
\(420\) 3.04579e32 0.493268
\(421\) 8.99240e32 1.41367 0.706837 0.707377i \(-0.250121\pi\)
0.706837 + 0.707377i \(0.250121\pi\)
\(422\) 2.95963e32 0.451682
\(423\) −1.38692e31 −0.0205492
\(424\) −3.05120e32 −0.438932
\(425\) −1.27769e32 −0.178470
\(426\) −1.31209e32 −0.177969
\(427\) −7.12262e32 −0.938191
\(428\) 7.01216e32 0.897025
\(429\) −6.81000e32 −0.846118
\(430\) 8.93212e32 1.07795
\(431\) 4.87745e31 0.0571778 0.0285889 0.999591i \(-0.490899\pi\)
0.0285889 + 0.999591i \(0.490899\pi\)
\(432\) 4.22479e31 0.0481125
\(433\) −3.43563e32 −0.380109 −0.190054 0.981774i \(-0.560866\pi\)
−0.190054 + 0.981774i \(0.560866\pi\)
\(434\) −1.10176e33 −1.18431
\(435\) 3.39302e32 0.354381
\(436\) 7.79536e32 0.791143
\(437\) 1.29990e33 1.28201
\(438\) 5.85145e32 0.560838
\(439\) 9.52703e32 0.887465 0.443732 0.896159i \(-0.353654\pi\)
0.443732 + 0.896159i \(0.353654\pi\)
\(440\) −3.85218e32 −0.348777
\(441\) 5.66670e32 0.498710
\(442\) −1.39651e33 −1.19472
\(443\) 3.31350e32 0.275576 0.137788 0.990462i \(-0.456001\pi\)
0.137788 + 0.990462i \(0.456001\pi\)
\(444\) −4.10973e32 −0.332297
\(445\) −1.51075e33 −1.18766
\(446\) 2.94903e32 0.225421
\(447\) −1.30877e33 −0.972788
\(448\) 2.73224e32 0.197489
\(449\) −1.03610e33 −0.728318 −0.364159 0.931337i \(-0.618644\pi\)
−0.364159 + 0.931337i \(0.618644\pi\)
\(450\) −5.85117e31 −0.0400025
\(451\) 4.88473e32 0.324813
\(452\) 5.49277e32 0.355272
\(453\) 2.30457e32 0.144998
\(454\) 1.69515e33 1.03756
\(455\) −4.61058e33 −2.74544
\(456\) −4.18843e32 −0.242656
\(457\) 2.36104e33 1.33092 0.665458 0.746436i \(-0.268236\pi\)
0.665458 + 0.746436i \(0.268236\pi\)
\(458\) 2.02247e33 1.10934
\(459\) −3.79156e32 −0.202376
\(460\) 1.12274e33 0.583184
\(461\) −3.86492e32 −0.195380 −0.0976898 0.995217i \(-0.531145\pi\)
−0.0976898 + 0.995217i \(0.531145\pi\)
\(462\) −1.19574e33 −0.588316
\(463\) −1.13691e33 −0.544456 −0.272228 0.962233i \(-0.587761\pi\)
−0.272228 + 0.962233i \(0.587761\pi\)
\(464\) 3.04373e32 0.141883
\(465\) 1.45878e33 0.661952
\(466\) −1.08851e33 −0.480847
\(467\) −1.38364e33 −0.595061 −0.297530 0.954712i \(-0.596163\pi\)
−0.297530 + 0.954712i \(0.596163\pi\)
\(468\) −6.39530e32 −0.267786
\(469\) 3.69473e32 0.150634
\(470\) 1.18759e32 0.0471457
\(471\) −1.88134e33 −0.727288
\(472\) −1.54958e33 −0.583363
\(473\) −3.50664e33 −1.28566
\(474\) 5.72192e32 0.204321
\(475\) 5.80082e32 0.201753
\(476\) −2.45206e33 −0.830701
\(477\) 1.25401e33 0.413829
\(478\) 9.98284e32 0.320926
\(479\) 2.24544e33 0.703248 0.351624 0.936141i \(-0.385630\pi\)
0.351624 + 0.936141i \(0.385630\pi\)
\(480\) −3.61760e32 −0.110384
\(481\) 6.22113e33 1.84951
\(482\) 1.12840e33 0.326870
\(483\) 3.48504e33 0.983713
\(484\) −3.05445e32 −0.0840169
\(485\) 1.53228e33 0.410739
\(486\) −1.73634e32 −0.0453609
\(487\) 5.22121e33 1.32941 0.664706 0.747105i \(-0.268557\pi\)
0.664706 + 0.747105i \(0.268557\pi\)
\(488\) 8.45981e32 0.209948
\(489\) −3.43299e33 −0.830448
\(490\) −4.85227e33 −1.14418
\(491\) 2.55379e33 0.587039 0.293519 0.955953i \(-0.405173\pi\)
0.293519 + 0.955953i \(0.405173\pi\)
\(492\) 4.58727e32 0.102799
\(493\) −2.73161e33 −0.596805
\(494\) 6.34026e33 1.35058
\(495\) 1.58320e33 0.328830
\(496\) 1.30860e33 0.265025
\(497\) 3.48746e33 0.688738
\(498\) −1.49873e33 −0.288640
\(499\) 7.23080e32 0.135809 0.0679047 0.997692i \(-0.478369\pi\)
0.0679047 + 0.997692i \(0.478369\pi\)
\(500\) −2.45111e33 −0.448990
\(501\) −2.38315e33 −0.425775
\(502\) −4.95056e33 −0.862696
\(503\) 3.39140e33 0.576474 0.288237 0.957559i \(-0.406931\pi\)
0.288237 + 0.957559i \(0.406931\pi\)
\(504\) −1.12292e33 −0.186195
\(505\) −1.03172e34 −1.66886
\(506\) −4.40773e33 −0.695558
\(507\) 5.93086e33 0.913099
\(508\) −2.83914e33 −0.426472
\(509\) 5.96782e31 0.00874668 0.00437334 0.999990i \(-0.498608\pi\)
0.00437334 + 0.999990i \(0.498608\pi\)
\(510\) 3.24663e33 0.464308
\(511\) −1.55528e34 −2.17043
\(512\) −3.24519e32 −0.0441942
\(513\) 1.72140e33 0.228778
\(514\) 9.32400e32 0.120938
\(515\) −7.84075e33 −0.992584
\(516\) −3.29310e33 −0.406896
\(517\) −4.66231e32 −0.0562302
\(518\) 1.09234e34 1.28598
\(519\) 6.48217e32 0.0744952
\(520\) 5.47616e33 0.614376
\(521\) 6.90204e32 0.0755972 0.0377986 0.999285i \(-0.487965\pi\)
0.0377986 + 0.999285i \(0.487965\pi\)
\(522\) −1.25094e33 −0.133769
\(523\) −8.36286e32 −0.0873139 −0.0436570 0.999047i \(-0.513901\pi\)
−0.0436570 + 0.999047i \(0.513901\pi\)
\(524\) 3.19207e32 0.0325410
\(525\) 1.55520e33 0.154809
\(526\) −2.94595e32 −0.0286355
\(527\) −1.17441e34 −1.11478
\(528\) 1.42022e33 0.131653
\(529\) 1.80078e33 0.163029
\(530\) −1.07378e34 −0.949440
\(531\) 6.36860e33 0.550000
\(532\) 1.11326e34 0.939074
\(533\) −6.94400e33 −0.572163
\(534\) 5.56984e33 0.448309
\(535\) 2.46773e34 1.94033
\(536\) −4.38837e32 −0.0337088
\(537\) −1.11208e34 −0.834561
\(538\) −1.06036e33 −0.0777456
\(539\) 1.90494e34 1.36465
\(540\) 1.48679e33 0.104071
\(541\) −1.79946e34 −1.23077 −0.615384 0.788228i \(-0.710999\pi\)
−0.615384 + 0.788228i \(0.710999\pi\)
\(542\) 4.19969e33 0.280690
\(543\) −7.91292e32 −0.0516820
\(544\) 2.91241e33 0.185894
\(545\) 2.74335e34 1.71130
\(546\) 1.69983e34 1.03633
\(547\) −8.89037e33 −0.529759 −0.264879 0.964282i \(-0.585332\pi\)
−0.264879 + 0.964282i \(0.585332\pi\)
\(548\) −5.84573e32 −0.0340473
\(549\) −3.47689e33 −0.197941
\(550\) −1.96695e33 −0.109461
\(551\) 1.24017e34 0.674664
\(552\) −4.13931e33 −0.220135
\(553\) −1.52085e34 −0.790718
\(554\) −2.02637e34 −1.03002
\(555\) −1.44630e34 −0.718781
\(556\) 3.00615e33 0.146075
\(557\) −3.01551e34 −1.43275 −0.716376 0.697714i \(-0.754201\pi\)
−0.716376 + 0.697714i \(0.754201\pi\)
\(558\) −5.37821e33 −0.249868
\(559\) 4.98494e34 2.26471
\(560\) 9.61533e33 0.427183
\(561\) −1.27459e34 −0.553775
\(562\) 1.47083e34 0.624970
\(563\) −1.95891e34 −0.814067 −0.407033 0.913413i \(-0.633437\pi\)
−0.407033 + 0.913413i \(0.633437\pi\)
\(564\) −4.37840e32 −0.0177962
\(565\) 1.93302e34 0.768478
\(566\) −1.85210e34 −0.720211
\(567\) 4.61508e33 0.175546
\(568\) −4.14219e33 −0.154126
\(569\) −1.37387e33 −0.0500085 −0.0250042 0.999687i \(-0.507960\pi\)
−0.0250042 + 0.999687i \(0.507960\pi\)
\(570\) −1.47400e34 −0.524881
\(571\) −6.79583e33 −0.236751 −0.118375 0.992969i \(-0.537769\pi\)
−0.118375 + 0.992969i \(0.537769\pi\)
\(572\) −2.14987e34 −0.732760
\(573\) −1.69820e34 −0.566311
\(574\) −1.21926e34 −0.397832
\(575\) 5.73279e33 0.183028
\(576\) 1.33374e33 0.0416667
\(577\) −4.58517e34 −1.40171 −0.700855 0.713304i \(-0.747198\pi\)
−0.700855 + 0.713304i \(0.747198\pi\)
\(578\) −2.50112e33 −0.0748232
\(579\) 1.68162e34 0.492319
\(580\) 1.07115e34 0.306903
\(581\) 3.98352e34 1.11703
\(582\) −5.64920e33 −0.155042
\(583\) 4.21553e34 1.13239
\(584\) 1.84726e34 0.485700
\(585\) −2.25064e34 −0.579239
\(586\) −2.32245e34 −0.585096
\(587\) 1.77739e34 0.438334 0.219167 0.975687i \(-0.429666\pi\)
0.219167 + 0.975687i \(0.429666\pi\)
\(588\) 1.78894e34 0.431895
\(589\) 5.33193e34 1.26021
\(590\) −5.45330e34 −1.26185
\(591\) −2.56199e34 −0.580409
\(592\) −1.29741e34 −0.287778
\(593\) 8.33392e34 1.80995 0.904973 0.425469i \(-0.139891\pi\)
0.904973 + 0.425469i \(0.139891\pi\)
\(594\) −5.83696e33 −0.124124
\(595\) −8.62933e34 −1.79686
\(596\) −4.13168e34 −0.842459
\(597\) 5.12249e34 1.02283
\(598\) 6.26591e34 1.22523
\(599\) −7.75840e34 −1.48572 −0.742860 0.669447i \(-0.766531\pi\)
−0.742860 + 0.669447i \(0.766531\pi\)
\(600\) −1.84717e33 −0.0346431
\(601\) −6.07045e34 −1.11504 −0.557519 0.830164i \(-0.688247\pi\)
−0.557519 + 0.830164i \(0.688247\pi\)
\(602\) 8.75283e34 1.57468
\(603\) 1.80357e33 0.0317810
\(604\) 7.27537e33 0.125572
\(605\) −1.07493e34 −0.181734
\(606\) 3.80375e34 0.629947
\(607\) 1.40609e34 0.228116 0.114058 0.993474i \(-0.463615\pi\)
0.114058 + 0.993474i \(0.463615\pi\)
\(608\) −1.32226e34 −0.210146
\(609\) 3.32491e34 0.517683
\(610\) 2.97719e34 0.454133
\(611\) 6.62783e33 0.0990503
\(612\) −1.19697e34 −0.175263
\(613\) 1.12382e35 1.61229 0.806143 0.591721i \(-0.201551\pi\)
0.806143 + 0.591721i \(0.201551\pi\)
\(614\) 8.40666e34 1.18173
\(615\) 1.61436e34 0.222362
\(616\) −3.77486e34 −0.509497
\(617\) −5.19277e34 −0.686807 −0.343403 0.939188i \(-0.611580\pi\)
−0.343403 + 0.939188i \(0.611580\pi\)
\(618\) 2.89073e34 0.374672
\(619\) −4.73877e33 −0.0601911 −0.0300956 0.999547i \(-0.509581\pi\)
−0.0300956 + 0.999547i \(0.509581\pi\)
\(620\) 4.60525e34 0.573267
\(621\) 1.70121e34 0.207546
\(622\) −9.57633e34 −1.14504
\(623\) −1.48043e35 −1.73495
\(624\) −2.01895e34 −0.231909
\(625\) −1.01333e35 −1.14091
\(626\) 1.62516e34 0.179356
\(627\) 5.78673e34 0.626021
\(628\) −5.93926e34 −0.629850
\(629\) 1.16437e35 1.21048
\(630\) −3.95179e34 −0.402752
\(631\) −3.76797e33 −0.0376479 −0.0188240 0.999823i \(-0.505992\pi\)
−0.0188240 + 0.999823i \(0.505992\pi\)
\(632\) 1.80637e34 0.176947
\(633\) −3.84002e34 −0.368796
\(634\) 2.62342e34 0.247031
\(635\) −9.99154e34 −0.922488
\(636\) 3.95882e34 0.358386
\(637\) −2.70801e35 −2.40385
\(638\) −4.20521e34 −0.366040
\(639\) 1.70239e34 0.145311
\(640\) −1.14205e34 −0.0955950
\(641\) 6.21895e34 0.510495 0.255248 0.966876i \(-0.417843\pi\)
0.255248 + 0.966876i \(0.417843\pi\)
\(642\) −9.09802e34 −0.732418
\(643\) −2.73698e34 −0.216090 −0.108045 0.994146i \(-0.534459\pi\)
−0.108045 + 0.994146i \(0.534459\pi\)
\(644\) 1.10020e35 0.851920
\(645\) −1.15891e35 −0.880143
\(646\) 1.18667e35 0.883940
\(647\) −1.42584e35 −1.04176 −0.520879 0.853631i \(-0.674396\pi\)
−0.520879 + 0.853631i \(0.674396\pi\)
\(648\) −5.48151e33 −0.0392837
\(649\) 2.14089e35 1.50500
\(650\) 2.79617e34 0.192818
\(651\) 1.42949e35 0.966985
\(652\) −1.08377e35 −0.719189
\(653\) 2.51493e35 1.63724 0.818619 0.574337i \(-0.194740\pi\)
0.818619 + 0.574337i \(0.194740\pi\)
\(654\) −1.01142e35 −0.645966
\(655\) 1.12336e34 0.0703885
\(656\) 1.44817e34 0.0890268
\(657\) −7.59205e34 −0.457922
\(658\) 1.16375e34 0.0688709
\(659\) −1.23988e35 −0.719968 −0.359984 0.932959i \(-0.617218\pi\)
−0.359984 + 0.932959i \(0.617218\pi\)
\(660\) 4.99806e34 0.284775
\(661\) 2.28549e35 1.27779 0.638896 0.769293i \(-0.279391\pi\)
0.638896 + 0.769293i \(0.279391\pi\)
\(662\) −1.41699e35 −0.777395
\(663\) 1.81192e35 0.975483
\(664\) −4.73138e34 −0.249969
\(665\) 3.91779e35 2.03128
\(666\) 5.33223e34 0.271319
\(667\) 1.22563e35 0.612049
\(668\) −7.52344e34 −0.368732
\(669\) −3.82626e34 −0.184055
\(670\) −1.54436e34 −0.0729145
\(671\) −1.16881e35 −0.541640
\(672\) −3.54498e34 −0.161249
\(673\) 2.09133e35 0.933756 0.466878 0.884322i \(-0.345379\pi\)
0.466878 + 0.884322i \(0.345379\pi\)
\(674\) 8.32866e34 0.365028
\(675\) 7.59169e33 0.0326619
\(676\) 1.87233e35 0.790767
\(677\) −3.93166e35 −1.63011 −0.815057 0.579380i \(-0.803295\pi\)
−0.815057 + 0.579380i \(0.803295\pi\)
\(678\) −7.12667e34 −0.290078
\(679\) 1.50152e35 0.600011
\(680\) 1.02494e35 0.402102
\(681\) −2.19940e35 −0.847160
\(682\) −1.80796e35 −0.683730
\(683\) −1.01629e35 −0.377364 −0.188682 0.982038i \(-0.560422\pi\)
−0.188682 + 0.982038i \(0.560422\pi\)
\(684\) 5.43434e34 0.198128
\(685\) −2.05724e34 −0.0736465
\(686\) −1.57676e35 −0.554260
\(687\) −2.62408e35 −0.905770
\(688\) −1.03961e35 −0.352382
\(689\) −5.99268e35 −1.99472
\(690\) −1.45671e35 −0.476168
\(691\) 4.93084e35 1.58287 0.791435 0.611253i \(-0.209334\pi\)
0.791435 + 0.611253i \(0.209334\pi\)
\(692\) 2.04637e34 0.0645147
\(693\) 1.55142e35 0.480358
\(694\) −5.73692e34 −0.174456
\(695\) 1.05793e35 0.315970
\(696\) −3.94912e34 −0.115847
\(697\) −1.29967e35 −0.374474
\(698\) 7.70217e34 0.217982
\(699\) 1.41230e35 0.392610
\(700\) 4.90967e34 0.134068
\(701\) 6.82364e35 1.83038 0.915189 0.403025i \(-0.132041\pi\)
0.915189 + 0.403025i \(0.132041\pi\)
\(702\) 8.29767e34 0.218646
\(703\) −5.28634e35 −1.36840
\(704\) 4.48354e34 0.114015
\(705\) −1.54085e34 −0.0384943
\(706\) 4.30655e35 1.05699
\(707\) −1.01101e36 −2.43788
\(708\) 2.01052e35 0.476314
\(709\) −2.78374e35 −0.647964 −0.323982 0.946063i \(-0.605022\pi\)
−0.323982 + 0.946063i \(0.605022\pi\)
\(710\) −1.45772e35 −0.333385
\(711\) −7.42398e34 −0.166827
\(712\) 1.75836e35 0.388247
\(713\) 5.26940e35 1.14325
\(714\) 3.18146e35 0.678265
\(715\) −7.56585e35 −1.58501
\(716\) −3.51076e35 −0.722751
\(717\) −1.29524e35 −0.262035
\(718\) 9.81024e34 0.195040
\(719\) 8.78510e35 1.71647 0.858234 0.513259i \(-0.171562\pi\)
0.858234 + 0.513259i \(0.171562\pi\)
\(720\) 4.69370e34 0.0901278
\(721\) −7.68337e35 −1.44998
\(722\) −1.57516e35 −0.292152
\(723\) −1.46406e35 −0.266888
\(724\) −2.49805e34 −0.0447579
\(725\) 5.46939e34 0.0963194
\(726\) 3.96304e34 0.0685995
\(727\) 8.08759e35 1.37607 0.688033 0.725680i \(-0.258475\pi\)
0.688033 + 0.725680i \(0.258475\pi\)
\(728\) 5.36624e35 0.897486
\(729\) 2.25284e34 0.0370370
\(730\) 6.50091e35 1.05060
\(731\) 9.33002e35 1.48223
\(732\) −1.09763e35 −0.171422
\(733\) −2.44873e35 −0.375960 −0.187980 0.982173i \(-0.560194\pi\)
−0.187980 + 0.982173i \(0.560194\pi\)
\(734\) −4.78481e34 −0.0722211
\(735\) 6.29565e35 0.934219
\(736\) −1.30675e35 −0.190643
\(737\) 6.06296e34 0.0869644
\(738\) −5.95181e34 −0.0839353
\(739\) 9.48378e35 1.31500 0.657500 0.753454i \(-0.271614\pi\)
0.657500 + 0.753454i \(0.271614\pi\)
\(740\) −4.56587e35 −0.622482
\(741\) −8.22626e35 −1.10274
\(742\) −1.05223e36 −1.38695
\(743\) −5.64636e35 −0.731827 −0.365914 0.930649i \(-0.619243\pi\)
−0.365914 + 0.930649i \(0.619243\pi\)
\(744\) −1.69786e35 −0.216392
\(745\) −1.45403e36 −1.82230
\(746\) 2.57905e35 0.317852
\(747\) 1.94454e35 0.235673
\(748\) −4.02378e35 −0.479584
\(749\) 2.41819e36 2.83445
\(750\) 3.18022e35 0.366599
\(751\) −2.89134e35 −0.327793 −0.163897 0.986478i \(-0.552406\pi\)
−0.163897 + 0.986478i \(0.552406\pi\)
\(752\) −1.38223e34 −0.0154119
\(753\) 6.42317e35 0.704388
\(754\) 5.97801e35 0.644785
\(755\) 2.56036e35 0.271621
\(756\) 1.45695e35 0.152027
\(757\) −1.58236e36 −1.62408 −0.812040 0.583602i \(-0.801643\pi\)
−0.812040 + 0.583602i \(0.801643\pi\)
\(758\) −3.27022e35 −0.330151
\(759\) 5.71886e35 0.567921
\(760\) −4.65331e35 −0.454560
\(761\) −1.92995e36 −1.85454 −0.927272 0.374389i \(-0.877853\pi\)
−0.927272 + 0.374389i \(0.877853\pi\)
\(762\) 3.68368e35 0.348213
\(763\) 2.68828e36 2.49988
\(764\) −5.36108e35 −0.490440
\(765\) −4.21239e35 −0.379106
\(766\) −4.75966e35 −0.421422
\(767\) −3.04344e36 −2.65108
\(768\) 4.21051e34 0.0360844
\(769\) 1.00839e35 0.0850252 0.0425126 0.999096i \(-0.486464\pi\)
0.0425126 + 0.999096i \(0.486464\pi\)
\(770\) −1.32845e36 −1.10208
\(771\) −1.20975e35 −0.0987456
\(772\) 5.30876e35 0.426361
\(773\) 2.06169e36 1.62922 0.814610 0.580009i \(-0.196951\pi\)
0.814610 + 0.580009i \(0.196951\pi\)
\(774\) 4.27267e35 0.332229
\(775\) 2.35148e35 0.179916
\(776\) −1.78341e35 −0.134270
\(777\) −1.41727e36 −1.05000
\(778\) −7.95288e35 −0.579802
\(779\) 5.90059e35 0.423328
\(780\) −7.10511e35 −0.501636
\(781\) 5.72284e35 0.397625
\(782\) 1.17275e36 0.801903
\(783\) 1.62305e35 0.109222
\(784\) 5.64755e35 0.374032
\(785\) −2.09015e36 −1.36241
\(786\) −4.14159e34 −0.0265696
\(787\) −1.65623e36 −1.04577 −0.522885 0.852404i \(-0.675144\pi\)
−0.522885 + 0.852404i \(0.675144\pi\)
\(788\) −8.08802e35 −0.502649
\(789\) 3.82227e34 0.0233807
\(790\) 6.35699e35 0.382748
\(791\) 1.89422e36 1.12260
\(792\) −1.84269e35 −0.107495
\(793\) 1.66154e36 0.954106
\(794\) 6.39534e35 0.361499
\(795\) 1.39319e36 0.775214
\(796\) 1.61713e36 0.885793
\(797\) 1.70830e36 0.921163 0.460581 0.887618i \(-0.347641\pi\)
0.460581 + 0.887618i \(0.347641\pi\)
\(798\) −1.44441e36 −0.766751
\(799\) 1.24049e35 0.0648274
\(800\) −5.83140e34 −0.0300018
\(801\) −7.22667e35 −0.366042
\(802\) −8.37231e35 −0.417509
\(803\) −2.55217e36 −1.25304
\(804\) 5.69375e34 0.0275231
\(805\) 3.87184e36 1.84276
\(806\) 2.57015e36 1.20440
\(807\) 1.37578e35 0.0634790
\(808\) 1.20082e36 0.545550
\(809\) 1.58337e36 0.708314 0.354157 0.935186i \(-0.384768\pi\)
0.354157 + 0.935186i \(0.384768\pi\)
\(810\) −1.92906e35 −0.0849733
\(811\) 2.16537e36 0.939228 0.469614 0.882872i \(-0.344393\pi\)
0.469614 + 0.882872i \(0.344393\pi\)
\(812\) 1.04965e36 0.448327
\(813\) −5.44895e35 −0.229182
\(814\) 1.79250e36 0.742429
\(815\) −3.81402e36 −1.55565
\(816\) −3.77875e35 −0.151782
\(817\) −4.23590e36 −1.67560
\(818\) 2.37598e36 0.925608
\(819\) −2.20546e36 −0.846158
\(820\) 5.09641e35 0.192571
\(821\) −3.49750e36 −1.30157 −0.650785 0.759262i \(-0.725560\pi\)
−0.650785 + 0.759262i \(0.725560\pi\)
\(822\) 7.58463e34 0.0277995
\(823\) −1.05634e36 −0.381333 −0.190666 0.981655i \(-0.561065\pi\)
−0.190666 + 0.981655i \(0.561065\pi\)
\(824\) 9.12583e35 0.324476
\(825\) 2.55205e35 0.0893748
\(826\) −5.34383e36 −1.84333
\(827\) −2.02229e35 −0.0687106 −0.0343553 0.999410i \(-0.510938\pi\)
−0.0343553 + 0.999410i \(0.510938\pi\)
\(828\) 5.37061e35 0.179740
\(829\) 2.37604e36 0.783288 0.391644 0.920117i \(-0.371906\pi\)
0.391644 + 0.920117i \(0.371906\pi\)
\(830\) −1.66507e36 −0.540700
\(831\) 2.62914e36 0.841010
\(832\) −6.37368e35 −0.200839
\(833\) −5.06843e36 −1.57330
\(834\) −3.90037e35 −0.119270
\(835\) −2.64766e36 −0.797592
\(836\) 1.82683e36 0.542150
\(837\) 6.97804e35 0.204016
\(838\) −3.93061e35 −0.113217
\(839\) 2.93630e36 0.833252 0.416626 0.909078i \(-0.363212\pi\)
0.416626 + 0.909078i \(0.363212\pi\)
\(840\) −1.24755e36 −0.348793
\(841\) −2.46105e36 −0.677907
\(842\) −3.68329e36 −0.999618
\(843\) −1.90835e36 −0.510286
\(844\) −1.21227e36 −0.319387
\(845\) 6.58913e36 1.71048
\(846\) 5.68081e34 0.0145305
\(847\) −1.05335e36 −0.265479
\(848\) 1.24977e36 0.310372
\(849\) 2.40304e36 0.588050
\(850\) 5.23342e35 0.126197
\(851\) −5.22434e36 −1.24140
\(852\) 5.37434e35 0.125843
\(853\) 4.86448e36 1.12247 0.561234 0.827657i \(-0.310327\pi\)
0.561234 + 0.827657i \(0.310327\pi\)
\(854\) 2.91742e36 0.663401
\(855\) 1.91246e36 0.428564
\(856\) −2.87218e36 −0.634293
\(857\) 6.23674e36 1.35737 0.678684 0.734430i \(-0.262551\pi\)
0.678684 + 0.734430i \(0.262551\pi\)
\(858\) 2.78938e36 0.598296
\(859\) −2.04246e36 −0.431758 −0.215879 0.976420i \(-0.569262\pi\)
−0.215879 + 0.976420i \(0.569262\pi\)
\(860\) −3.65860e36 −0.762226
\(861\) 1.58195e36 0.324828
\(862\) −1.99780e35 −0.0404308
\(863\) 3.83159e36 0.764266 0.382133 0.924107i \(-0.375190\pi\)
0.382133 + 0.924107i \(0.375190\pi\)
\(864\) −1.73047e35 −0.0340207
\(865\) 7.20162e35 0.139550
\(866\) 1.40724e36 0.268778
\(867\) 3.24511e35 0.0610929
\(868\) 4.51281e36 0.837434
\(869\) −2.49567e36 −0.456500
\(870\) −1.38978e36 −0.250585
\(871\) −8.61895e35 −0.153189
\(872\) −3.19298e36 −0.559423
\(873\) 7.32963e35 0.126591
\(874\) −5.32439e36 −0.906519
\(875\) −8.45281e36 −1.41873
\(876\) −2.39676e36 −0.396572
\(877\) 8.65600e36 1.41196 0.705980 0.708232i \(-0.250507\pi\)
0.705980 + 0.708232i \(0.250507\pi\)
\(878\) −3.90227e36 −0.627532
\(879\) 3.01330e36 0.477729
\(880\) 1.57785e36 0.246623
\(881\) 6.20228e36 0.955767 0.477884 0.878423i \(-0.341404\pi\)
0.477884 + 0.878423i \(0.341404\pi\)
\(882\) −2.32108e36 −0.352641
\(883\) −1.95836e35 −0.0293349 −0.0146675 0.999892i \(-0.504669\pi\)
−0.0146675 + 0.999892i \(0.504669\pi\)
\(884\) 5.72010e36 0.844793
\(885\) 7.07545e36 1.03030
\(886\) −1.35721e36 −0.194861
\(887\) 6.06391e36 0.858434 0.429217 0.903201i \(-0.358789\pi\)
0.429217 + 0.903201i \(0.358789\pi\)
\(888\) 1.68335e36 0.234969
\(889\) −9.79098e36 −1.34758
\(890\) 6.18804e36 0.839804
\(891\) 7.57324e35 0.101347
\(892\) −1.20792e36 −0.159396
\(893\) −5.63192e35 −0.0732847
\(894\) 5.36071e36 0.687865
\(895\) −1.23551e37 −1.56336
\(896\) −1.11913e36 −0.139646
\(897\) −8.12979e36 −1.00040
\(898\) 4.24385e36 0.514999
\(899\) 5.02729e36 0.601641
\(900\) 2.39664e35 0.0282860
\(901\) −1.12161e37 −1.30552
\(902\) −2.00078e36 −0.229678
\(903\) −1.13565e37 −1.28572
\(904\) −2.24984e36 −0.251215
\(905\) −8.79118e35 −0.0968144
\(906\) −9.43953e35 −0.102529
\(907\) 7.32294e36 0.784503 0.392251 0.919858i \(-0.371696\pi\)
0.392251 + 0.919858i \(0.371696\pi\)
\(908\) −6.94335e36 −0.733662
\(909\) −4.93522e36 −0.514350
\(910\) 1.88849e37 1.94132
\(911\) 7.08337e36 0.718225 0.359112 0.933294i \(-0.383079\pi\)
0.359112 + 0.933294i \(0.383079\pi\)
\(912\) 1.71558e36 0.171584
\(913\) 6.53686e36 0.644888
\(914\) −9.67081e36 −0.941099
\(915\) −3.86279e36 −0.370798
\(916\) −8.28404e36 −0.784420
\(917\) 1.10081e36 0.102824
\(918\) 1.55302e36 0.143102
\(919\) 6.34698e36 0.576930 0.288465 0.957490i \(-0.406855\pi\)
0.288465 + 0.957490i \(0.406855\pi\)
\(920\) −4.59874e36 −0.412373
\(921\) −1.09073e37 −0.964880
\(922\) 1.58307e36 0.138154
\(923\) −8.13544e36 −0.700422
\(924\) 4.89774e36 0.416002
\(925\) −2.33137e36 −0.195362
\(926\) 4.65678e36 0.384989
\(927\) −3.75062e36 −0.305919
\(928\) −1.24671e36 −0.100327
\(929\) −2.21142e37 −1.75580 −0.877900 0.478844i \(-0.841056\pi\)
−0.877900 + 0.478844i \(0.841056\pi\)
\(930\) −5.97514e36 −0.468071
\(931\) 2.30111e37 1.77855
\(932\) 4.45852e36 0.340010
\(933\) 1.24249e37 0.934919
\(934\) 5.66738e36 0.420772
\(935\) −1.41605e37 −1.03737
\(936\) 2.61951e36 0.189353
\(937\) −2.06870e37 −1.47554 −0.737772 0.675050i \(-0.764122\pi\)
−0.737772 + 0.675050i \(0.764122\pi\)
\(938\) −1.51336e36 −0.106514
\(939\) −2.10858e36 −0.146444
\(940\) −4.86436e35 −0.0333371
\(941\) 1.47946e37 1.00054 0.500268 0.865870i \(-0.333235\pi\)
0.500268 + 0.865870i \(0.333235\pi\)
\(942\) 7.70597e36 0.514270
\(943\) 5.83139e36 0.384040
\(944\) 6.34708e36 0.412500
\(945\) 5.12731e36 0.328845
\(946\) 1.43632e37 0.909100
\(947\) 1.88544e37 1.17771 0.588853 0.808240i \(-0.299580\pi\)
0.588853 + 0.808240i \(0.299580\pi\)
\(948\) −2.34370e36 −0.144477
\(949\) 3.62810e37 2.20725
\(950\) −2.37602e36 −0.142661
\(951\) −3.40379e36 −0.201700
\(952\) 1.00437e37 0.587395
\(953\) 8.92812e35 0.0515346 0.0257673 0.999668i \(-0.491797\pi\)
0.0257673 + 0.999668i \(0.491797\pi\)
\(954\) −5.13642e36 −0.292621
\(955\) −1.88668e37 −1.06085
\(956\) −4.08897e36 −0.226929
\(957\) 5.45610e36 0.298871
\(958\) −9.19734e36 −0.497271
\(959\) −2.01594e36 −0.107583
\(960\) 1.48177e36 0.0780530
\(961\) 2.38125e36 0.123812
\(962\) −2.54818e37 −1.30780
\(963\) 1.18043e37 0.598017
\(964\) −4.62192e36 −0.231132
\(965\) 1.86827e37 0.922247
\(966\) −1.42747e37 −0.695590
\(967\) −5.32831e36 −0.256306 −0.128153 0.991754i \(-0.540905\pi\)
−0.128153 + 0.991754i \(0.540905\pi\)
\(968\) 1.25110e36 0.0594089
\(969\) −1.53966e37 −0.721734
\(970\) −6.27621e36 −0.290436
\(971\) 3.40804e37 1.55691 0.778456 0.627699i \(-0.216003\pi\)
0.778456 + 0.627699i \(0.216003\pi\)
\(972\) 7.11206e35 0.0320750
\(973\) 1.03669e37 0.461571
\(974\) −2.13861e37 −0.940036
\(975\) −3.62793e36 −0.157435
\(976\) −3.46514e36 −0.148456
\(977\) −4.16179e37 −1.76035 −0.880173 0.474653i \(-0.842573\pi\)
−0.880173 + 0.474653i \(0.842573\pi\)
\(978\) 1.40615e37 0.587215
\(979\) −2.42935e37 −1.00163
\(980\) 1.98749e37 0.809057
\(981\) 1.31228e37 0.527429
\(982\) −1.04603e37 −0.415099
\(983\) 1.59521e37 0.625026 0.312513 0.949913i \(-0.398829\pi\)
0.312513 + 0.949913i \(0.398829\pi\)
\(984\) −1.87894e36 −0.0726901
\(985\) −2.84634e37 −1.08726
\(986\) 1.11887e37 0.422005
\(987\) −1.50992e36 −0.0562328
\(988\) −2.59697e37 −0.955004
\(989\) −4.18623e37 −1.52009
\(990\) −6.48480e36 −0.232518
\(991\) 3.15449e36 0.111689 0.0558443 0.998439i \(-0.482215\pi\)
0.0558443 + 0.998439i \(0.482215\pi\)
\(992\) −5.36004e36 −0.187401
\(993\) 1.83849e37 0.634741
\(994\) −1.42846e37 −0.487011
\(995\) 5.69103e37 1.91603
\(996\) 6.13879e36 0.204099
\(997\) 1.19794e37 0.393321 0.196660 0.980472i \(-0.436990\pi\)
0.196660 + 0.980472i \(0.436990\pi\)
\(998\) −2.96174e36 −0.0960317
\(999\) −6.91837e36 −0.221531
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 6.26.a.b.1.1 1
3.2 odd 2 18.26.a.b.1.1 1
4.3 odd 2 48.26.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6.26.a.b.1.1 1 1.1 even 1 trivial
18.26.a.b.1.1 1 3.2 odd 2
48.26.a.a.1.1 1 4.3 odd 2