Properties

Label 50.26.a.l.1.3
Level $50$
Weight $26$
Character 50.1
Self dual yes
Analytic conductor $197.998$
Analytic rank $1$
Dimension $6$
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] = [50,26,Mod(1,50)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(50, 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("50.1");
 
S:= CuspForms(chi, 26);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 26 \)
Character orbit: \([\chi]\) \(=\) 50.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: \(197.998389976\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2 x^{5} - 27590779188 x^{4} + 26487255863952 x^{3} + \cdots - 30\!\cdots\!08 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{20}\cdot 3^{4}\cdot 5^{18} \)
Twist minimal: no (minimal twist has level 10)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(10012.7\) of defining polynomial
Character \(\chi\) \(=\) 50.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+4096.00 q^{2} -233693. q^{3} +1.67772e7 q^{4} -9.57206e8 q^{6} +6.39090e10 q^{7} +6.87195e10 q^{8} -7.92676e11 q^{9} +6.04007e12 q^{11} -3.92071e12 q^{12} -1.13809e14 q^{13} +2.61771e14 q^{14} +2.81475e14 q^{16} -9.52541e14 q^{17} -3.24680e15 q^{18} +8.44950e15 q^{19} -1.49351e16 q^{21} +2.47401e16 q^{22} +1.51928e16 q^{23} -1.60592e16 q^{24} -4.66162e17 q^{26} +3.83248e17 q^{27} +1.07222e18 q^{28} -1.63360e18 q^{29} -7.82464e18 q^{31} +1.15292e18 q^{32} -1.41152e18 q^{33} -3.90161e18 q^{34} -1.32989e19 q^{36} -3.53342e19 q^{37} +3.46091e19 q^{38} +2.65964e19 q^{39} -2.35937e20 q^{41} -6.11741e19 q^{42} -3.29832e19 q^{43} +1.01335e20 q^{44} +6.22296e19 q^{46} -1.82685e20 q^{47} -6.57787e19 q^{48} +2.74329e21 q^{49} +2.22602e20 q^{51} -1.90940e21 q^{52} +5.40172e21 q^{53} +1.56978e21 q^{54} +4.39179e21 q^{56} -1.97459e21 q^{57} -6.69124e21 q^{58} -1.13226e22 q^{59} -2.83642e22 q^{61} -3.20497e22 q^{62} -5.06592e22 q^{63} +4.72237e21 q^{64} -5.78158e21 q^{66} +1.16196e23 q^{67} -1.59810e22 q^{68} -3.55044e21 q^{69} +1.14601e23 q^{71} -5.44723e22 q^{72} +8.85118e21 q^{73} -1.44729e23 q^{74} +1.41759e23 q^{76} +3.86015e23 q^{77} +1.08939e23 q^{78} +2.17397e23 q^{79} +5.82063e23 q^{81} -9.66398e23 q^{82} -6.46451e22 q^{83} -2.50569e23 q^{84} -1.35099e23 q^{86} +3.81761e23 q^{87} +4.15070e23 q^{88} +3.57337e24 q^{89} -7.27343e24 q^{91} +2.54893e23 q^{92} +1.82856e24 q^{93} -7.48277e23 q^{94} -2.69429e23 q^{96} -2.49389e24 q^{97} +1.12365e25 q^{98} -4.78782e24 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 24576 q^{2} - 801416 q^{3} + 100663296 q^{4} - 3282599936 q^{6} - 34007705352 q^{7} + 412316860416 q^{8} + 541468782118 q^{9} + 9861544614312 q^{11} - 13445529337856 q^{12} - 30787386783696 q^{13}+ \cdots - 20\!\cdots\!64 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) −233693. −0.253881 −0.126940 0.991910i \(-0.540516\pi\)
−0.126940 + 0.991910i \(0.540516\pi\)
\(4\) 1.67772e7 0.500000
\(5\) 0 0
\(6\) −9.57206e8 −0.179521
\(7\) 6.39090e10 1.74517 0.872583 0.488466i \(-0.162443\pi\)
0.872583 + 0.488466i \(0.162443\pi\)
\(8\) 6.87195e10 0.353553
\(9\) −7.92676e11 −0.935545
\(10\) 0 0
\(11\) 6.04007e12 0.580274 0.290137 0.956985i \(-0.406299\pi\)
0.290137 + 0.956985i \(0.406299\pi\)
\(12\) −3.92071e12 −0.126940
\(13\) −1.13809e14 −1.35483 −0.677415 0.735601i \(-0.736900\pi\)
−0.677415 + 0.735601i \(0.736900\pi\)
\(14\) 2.61771e14 1.23402
\(15\) 0 0
\(16\) 2.81475e14 0.250000
\(17\) −9.52541e14 −0.396526 −0.198263 0.980149i \(-0.563530\pi\)
−0.198263 + 0.980149i \(0.563530\pi\)
\(18\) −3.24680e15 −0.661530
\(19\) 8.44950e15 0.875812 0.437906 0.899021i \(-0.355720\pi\)
0.437906 + 0.899021i \(0.355720\pi\)
\(20\) 0 0
\(21\) −1.49351e16 −0.443064
\(22\) 2.47401e16 0.410316
\(23\) 1.51928e16 0.144557 0.0722786 0.997384i \(-0.476973\pi\)
0.0722786 + 0.997384i \(0.476973\pi\)
\(24\) −1.60592e16 −0.0897604
\(25\) 0 0
\(26\) −4.66162e17 −0.958010
\(27\) 3.83248e17 0.491397
\(28\) 1.07222e18 0.872583
\(29\) −1.63360e18 −0.857376 −0.428688 0.903452i \(-0.641024\pi\)
−0.428688 + 0.903452i \(0.641024\pi\)
\(30\) 0 0
\(31\) −7.82464e18 −1.78420 −0.892099 0.451840i \(-0.850768\pi\)
−0.892099 + 0.451840i \(0.850768\pi\)
\(32\) 1.15292e18 0.176777
\(33\) −1.41152e18 −0.147320
\(34\) −3.90161e18 −0.280386
\(35\) 0 0
\(36\) −1.32989e19 −0.467772
\(37\) −3.53342e19 −0.882417 −0.441208 0.897405i \(-0.645450\pi\)
−0.441208 + 0.897405i \(0.645450\pi\)
\(38\) 3.46091e19 0.619293
\(39\) 2.65964e19 0.343965
\(40\) 0 0
\(41\) −2.35937e20 −1.63304 −0.816522 0.577315i \(-0.804101\pi\)
−0.816522 + 0.577315i \(0.804101\pi\)
\(42\) −6.11741e19 −0.313293
\(43\) −3.29832e19 −0.125874 −0.0629372 0.998017i \(-0.520047\pi\)
−0.0629372 + 0.998017i \(0.520047\pi\)
\(44\) 1.01335e20 0.290137
\(45\) 0 0
\(46\) 6.22296e19 0.102217
\(47\) −1.82685e20 −0.229340 −0.114670 0.993404i \(-0.536581\pi\)
−0.114670 + 0.993404i \(0.536581\pi\)
\(48\) −6.57787e19 −0.0634702
\(49\) 2.74329e21 2.04560
\(50\) 0 0
\(51\) 2.22602e20 0.100670
\(52\) −1.90940e21 −0.677415
\(53\) 5.40172e21 1.51037 0.755186 0.655511i \(-0.227547\pi\)
0.755186 + 0.655511i \(0.227547\pi\)
\(54\) 1.56978e21 0.347470
\(55\) 0 0
\(56\) 4.39179e21 0.617009
\(57\) −1.97459e21 −0.222352
\(58\) −6.69124e21 −0.606257
\(59\) −1.13226e22 −0.828509 −0.414254 0.910161i \(-0.635958\pi\)
−0.414254 + 0.910161i \(0.635958\pi\)
\(60\) 0 0
\(61\) −2.83642e22 −1.36820 −0.684098 0.729390i \(-0.739804\pi\)
−0.684098 + 0.729390i \(0.739804\pi\)
\(62\) −3.20497e22 −1.26162
\(63\) −5.06592e22 −1.63268
\(64\) 4.72237e21 0.125000
\(65\) 0 0
\(66\) −5.78158e21 −0.104171
\(67\) 1.16196e23 1.73483 0.867415 0.497586i \(-0.165780\pi\)
0.867415 + 0.497586i \(0.165780\pi\)
\(68\) −1.59810e22 −0.198263
\(69\) −3.55044e21 −0.0367003
\(70\) 0 0
\(71\) 1.14601e23 0.828816 0.414408 0.910091i \(-0.363989\pi\)
0.414408 + 0.910091i \(0.363989\pi\)
\(72\) −5.44723e22 −0.330765
\(73\) 8.85118e21 0.0452340 0.0226170 0.999744i \(-0.492800\pi\)
0.0226170 + 0.999744i \(0.492800\pi\)
\(74\) −1.44729e23 −0.623963
\(75\) 0 0
\(76\) 1.41759e23 0.437906
\(77\) 3.86015e23 1.01267
\(78\) 1.08939e23 0.243220
\(79\) 2.17397e23 0.413918 0.206959 0.978350i \(-0.433643\pi\)
0.206959 + 0.978350i \(0.433643\pi\)
\(80\) 0 0
\(81\) 5.82063e23 0.810788
\(82\) −9.66398e23 −1.15474
\(83\) −6.46451e22 −0.0663834 −0.0331917 0.999449i \(-0.510567\pi\)
−0.0331917 + 0.999449i \(0.510567\pi\)
\(84\) −2.50569e23 −0.221532
\(85\) 0 0
\(86\) −1.35099e23 −0.0890067
\(87\) 3.81761e23 0.217671
\(88\) 4.15070e23 0.205158
\(89\) 3.57337e24 1.53357 0.766783 0.641906i \(-0.221856\pi\)
0.766783 + 0.641906i \(0.221856\pi\)
\(90\) 0 0
\(91\) −7.27343e24 −2.36440
\(92\) 2.54893e23 0.0722786
\(93\) 1.82856e24 0.452973
\(94\) −7.48277e23 −0.162168
\(95\) 0 0
\(96\) −2.69429e23 −0.0448802
\(97\) −2.49389e24 −0.364948 −0.182474 0.983211i \(-0.558411\pi\)
−0.182474 + 0.983211i \(0.558411\pi\)
\(98\) 1.12365e25 1.44646
\(99\) −4.78782e24 −0.542872
\(100\) 0 0
\(101\) −1.34804e25 −1.19038 −0.595192 0.803584i \(-0.702924\pi\)
−0.595192 + 0.803584i \(0.702924\pi\)
\(102\) 9.11777e23 0.0711847
\(103\) −1.98475e25 −1.37164 −0.685822 0.727769i \(-0.740557\pi\)
−0.685822 + 0.727769i \(0.740557\pi\)
\(104\) −7.82090e24 −0.479005
\(105\) 0 0
\(106\) 2.21255e25 1.06799
\(107\) −6.60004e24 −0.283302 −0.141651 0.989917i \(-0.545241\pi\)
−0.141651 + 0.989917i \(0.545241\pi\)
\(108\) 6.42983e24 0.245699
\(109\) −6.35443e24 −0.216394 −0.108197 0.994129i \(-0.534508\pi\)
−0.108197 + 0.994129i \(0.534508\pi\)
\(110\) 0 0
\(111\) 8.25735e24 0.224029
\(112\) 1.79888e25 0.436291
\(113\) −5.12661e25 −1.11263 −0.556313 0.830973i \(-0.687785\pi\)
−0.556313 + 0.830973i \(0.687785\pi\)
\(114\) −8.08791e24 −0.157226
\(115\) 0 0
\(116\) −2.74073e25 −0.428688
\(117\) 9.02138e25 1.26750
\(118\) −4.63775e25 −0.585844
\(119\) −6.08759e25 −0.692004
\(120\) 0 0
\(121\) −7.18647e25 −0.663282
\(122\) −1.16180e26 −0.967460
\(123\) 5.51368e25 0.414598
\(124\) −1.31276e26 −0.892099
\(125\) 0 0
\(126\) −2.07500e26 −1.15448
\(127\) −1.52068e26 −0.766462 −0.383231 0.923653i \(-0.625189\pi\)
−0.383231 + 0.923653i \(0.625189\pi\)
\(128\) 1.93428e25 0.0883883
\(129\) 7.70794e24 0.0319571
\(130\) 0 0
\(131\) −4.54456e26 −1.55454 −0.777268 0.629170i \(-0.783395\pi\)
−0.777268 + 0.629170i \(0.783395\pi\)
\(132\) −2.36814e25 −0.0736602
\(133\) 5.39999e26 1.52844
\(134\) 4.75940e26 1.22671
\(135\) 0 0
\(136\) −6.54581e25 −0.140193
\(137\) −5.90117e25 −0.115327 −0.0576635 0.998336i \(-0.518365\pi\)
−0.0576635 + 0.998336i \(0.518365\pi\)
\(138\) −1.45426e25 −0.0259510
\(139\) −1.48479e26 −0.242091 −0.121046 0.992647i \(-0.538625\pi\)
−0.121046 + 0.992647i \(0.538625\pi\)
\(140\) 0 0
\(141\) 4.26921e25 0.0582249
\(142\) 4.69405e26 0.586062
\(143\) −6.87414e26 −0.786173
\(144\) −2.23119e26 −0.233886
\(145\) 0 0
\(146\) 3.62544e25 0.0319853
\(147\) −6.41088e26 −0.519339
\(148\) −5.92809e26 −0.441208
\(149\) −5.34888e26 −0.365961 −0.182981 0.983117i \(-0.558575\pi\)
−0.182981 + 0.983117i \(0.558575\pi\)
\(150\) 0 0
\(151\) 2.12275e27 1.22938 0.614691 0.788768i \(-0.289281\pi\)
0.614691 + 0.788768i \(0.289281\pi\)
\(152\) 5.80645e26 0.309646
\(153\) 7.55056e26 0.370968
\(154\) 1.58112e27 0.716069
\(155\) 0 0
\(156\) 4.46213e26 0.171983
\(157\) −1.66162e26 −0.0591271 −0.0295636 0.999563i \(-0.509412\pi\)
−0.0295636 + 0.999563i \(0.509412\pi\)
\(158\) 8.90456e26 0.292684
\(159\) −1.26234e27 −0.383454
\(160\) 0 0
\(161\) 9.70956e26 0.252276
\(162\) 2.38413e27 0.573314
\(163\) −2.97944e27 −0.663421 −0.331710 0.943381i \(-0.607626\pi\)
−0.331710 + 0.943381i \(0.607626\pi\)
\(164\) −3.95837e27 −0.816522
\(165\) 0 0
\(166\) −2.64786e26 −0.0469401
\(167\) −8.95870e27 −1.47329 −0.736647 0.676278i \(-0.763592\pi\)
−0.736647 + 0.676278i \(0.763592\pi\)
\(168\) −1.02633e27 −0.156647
\(169\) 5.89610e27 0.835567
\(170\) 0 0
\(171\) −6.69772e27 −0.819361
\(172\) −5.53367e26 −0.0629372
\(173\) −1.40879e28 −1.49028 −0.745142 0.666905i \(-0.767619\pi\)
−0.745142 + 0.666905i \(0.767619\pi\)
\(174\) 1.56369e27 0.153917
\(175\) 0 0
\(176\) 1.70013e27 0.145068
\(177\) 2.64602e27 0.210342
\(178\) 1.46365e28 1.08440
\(179\) 9.19738e27 0.635333 0.317667 0.948202i \(-0.397101\pi\)
0.317667 + 0.948202i \(0.397101\pi\)
\(180\) 0 0
\(181\) 2.77373e28 1.66756 0.833782 0.552094i \(-0.186171\pi\)
0.833782 + 0.552094i \(0.186171\pi\)
\(182\) −2.97920e28 −1.67189
\(183\) 6.62852e27 0.347358
\(184\) 1.04404e27 0.0511087
\(185\) 0 0
\(186\) 7.48979e27 0.320301
\(187\) −5.75341e27 −0.230094
\(188\) −3.06494e27 −0.114670
\(189\) 2.44930e28 0.857570
\(190\) 0 0
\(191\) −1.68120e28 −0.516064 −0.258032 0.966136i \(-0.583074\pi\)
−0.258032 + 0.966136i \(0.583074\pi\)
\(192\) −1.10358e27 −0.0317351
\(193\) 2.97967e28 0.802975 0.401487 0.915865i \(-0.368493\pi\)
0.401487 + 0.915865i \(0.368493\pi\)
\(194\) −1.02150e28 −0.258057
\(195\) 0 0
\(196\) 4.60248e28 1.02280
\(197\) 8.93281e27 0.186277 0.0931387 0.995653i \(-0.470310\pi\)
0.0931387 + 0.995653i \(0.470310\pi\)
\(198\) −1.96109e28 −0.383869
\(199\) −9.58646e28 −1.76196 −0.880978 0.473157i \(-0.843114\pi\)
−0.880978 + 0.473157i \(0.843114\pi\)
\(200\) 0 0
\(201\) −2.71542e28 −0.440440
\(202\) −5.52159e28 −0.841728
\(203\) −1.04402e29 −1.49626
\(204\) 3.73464e27 0.0503352
\(205\) 0 0
\(206\) −8.12956e28 −0.969899
\(207\) −1.20430e28 −0.135240
\(208\) −3.20344e28 −0.338708
\(209\) 5.10355e28 0.508211
\(210\) 0 0
\(211\) 5.69613e27 0.0503558 0.0251779 0.999683i \(-0.491985\pi\)
0.0251779 + 0.999683i \(0.491985\pi\)
\(212\) 9.06259e28 0.755186
\(213\) −2.67814e28 −0.210420
\(214\) −2.70338e28 −0.200325
\(215\) 0 0
\(216\) 2.63366e28 0.173735
\(217\) −5.00065e29 −3.11372
\(218\) −2.60278e28 −0.153014
\(219\) −2.06846e27 −0.0114840
\(220\) 0 0
\(221\) 1.08408e29 0.537226
\(222\) 3.38221e28 0.158412
\(223\) −5.84548e28 −0.258827 −0.129413 0.991591i \(-0.541309\pi\)
−0.129413 + 0.991591i \(0.541309\pi\)
\(224\) 7.36821e28 0.308505
\(225\) 0 0
\(226\) −2.09986e29 −0.786746
\(227\) −5.52270e29 −1.95807 −0.979035 0.203691i \(-0.934706\pi\)
−0.979035 + 0.203691i \(0.934706\pi\)
\(228\) −3.31281e28 −0.111176
\(229\) −5.80273e28 −0.184369 −0.0921847 0.995742i \(-0.529385\pi\)
−0.0921847 + 0.995742i \(0.529385\pi\)
\(230\) 0 0
\(231\) −9.02088e28 −0.257098
\(232\) −1.12260e29 −0.303128
\(233\) 1.93762e29 0.495816 0.247908 0.968784i \(-0.420257\pi\)
0.247908 + 0.968784i \(0.420257\pi\)
\(234\) 3.69516e29 0.896261
\(235\) 0 0
\(236\) −1.89962e29 −0.414254
\(237\) −5.08040e28 −0.105086
\(238\) −2.49348e29 −0.489321
\(239\) 7.15964e28 0.133327 0.0666634 0.997776i \(-0.478765\pi\)
0.0666634 + 0.997776i \(0.478765\pi\)
\(240\) 0 0
\(241\) −8.42045e29 −1.41293 −0.706467 0.707746i \(-0.749712\pi\)
−0.706467 + 0.707746i \(0.749712\pi\)
\(242\) −2.94358e29 −0.469011
\(243\) −4.60746e29 −0.697241
\(244\) −4.75873e29 −0.684098
\(245\) 0 0
\(246\) 2.25840e29 0.293165
\(247\) −9.61630e29 −1.18658
\(248\) −5.37705e29 −0.630809
\(249\) 1.51071e28 0.0168535
\(250\) 0 0
\(251\) −3.23321e28 −0.0326372 −0.0163186 0.999867i \(-0.505195\pi\)
−0.0163186 + 0.999867i \(0.505195\pi\)
\(252\) −8.49920e29 −0.816340
\(253\) 9.17654e28 0.0838827
\(254\) −6.22869e29 −0.541970
\(255\) 0 0
\(256\) 7.92282e28 0.0625000
\(257\) 2.39756e29 0.180138 0.0900689 0.995936i \(-0.471291\pi\)
0.0900689 + 0.995936i \(0.471291\pi\)
\(258\) 3.15717e28 0.0225971
\(259\) −2.25817e30 −1.53996
\(260\) 0 0
\(261\) 1.29492e30 0.802114
\(262\) −1.86145e30 −1.09922
\(263\) −2.50236e30 −1.40898 −0.704488 0.709716i \(-0.748823\pi\)
−0.704488 + 0.709716i \(0.748823\pi\)
\(264\) −9.69989e28 −0.0520856
\(265\) 0 0
\(266\) 2.21184e30 1.08077
\(267\) −8.35070e29 −0.389343
\(268\) 1.94945e30 0.867415
\(269\) 3.20998e30 1.36332 0.681662 0.731667i \(-0.261257\pi\)
0.681662 + 0.731667i \(0.261257\pi\)
\(270\) 0 0
\(271\) 1.88885e30 0.731277 0.365638 0.930757i \(-0.380851\pi\)
0.365638 + 0.930757i \(0.380851\pi\)
\(272\) −2.68116e29 −0.0991316
\(273\) 1.69975e30 0.600277
\(274\) −2.41712e29 −0.0815485
\(275\) 0 0
\(276\) −5.95666e28 −0.0183501
\(277\) −2.14925e29 −0.0632835 −0.0316417 0.999499i \(-0.510074\pi\)
−0.0316417 + 0.999499i \(0.510074\pi\)
\(278\) −6.08169e29 −0.171184
\(279\) 6.20241e30 1.66920
\(280\) 0 0
\(281\) −1.34905e30 −0.332048 −0.166024 0.986122i \(-0.553093\pi\)
−0.166024 + 0.986122i \(0.553093\pi\)
\(282\) 1.74867e29 0.0411712
\(283\) −7.01801e30 −1.58082 −0.790412 0.612576i \(-0.790133\pi\)
−0.790412 + 0.612576i \(0.790133\pi\)
\(284\) 1.92268e30 0.414408
\(285\) 0 0
\(286\) −2.81565e30 −0.555908
\(287\) −1.50785e31 −2.84993
\(288\) −9.13894e29 −0.165382
\(289\) −4.86329e30 −0.842767
\(290\) 0 0
\(291\) 5.82805e29 0.0926533
\(292\) 1.48498e29 0.0226170
\(293\) 4.23132e30 0.617491 0.308746 0.951145i \(-0.400091\pi\)
0.308746 + 0.951145i \(0.400091\pi\)
\(294\) −2.62590e30 −0.367228
\(295\) 0 0
\(296\) −2.42815e30 −0.311981
\(297\) 2.31484e30 0.285145
\(298\) −2.19090e30 −0.258774
\(299\) −1.72908e30 −0.195850
\(300\) 0 0
\(301\) −2.10793e30 −0.219672
\(302\) 8.69478e30 0.869304
\(303\) 3.15028e30 0.302215
\(304\) 2.37832e30 0.218953
\(305\) 0 0
\(306\) 3.09271e30 0.262314
\(307\) 3.15394e30 0.256817 0.128409 0.991721i \(-0.459013\pi\)
0.128409 + 0.991721i \(0.459013\pi\)
\(308\) 6.47625e30 0.506337
\(309\) 4.63823e30 0.348234
\(310\) 0 0
\(311\) 1.98559e31 1.37526 0.687628 0.726063i \(-0.258652\pi\)
0.687628 + 0.726063i \(0.258652\pi\)
\(312\) 1.82769e30 0.121610
\(313\) 5.47200e29 0.0349818 0.0174909 0.999847i \(-0.494432\pi\)
0.0174909 + 0.999847i \(0.494432\pi\)
\(314\) −6.80601e29 −0.0418092
\(315\) 0 0
\(316\) 3.64731e30 0.206959
\(317\) 2.18550e31 1.19209 0.596046 0.802950i \(-0.296737\pi\)
0.596046 + 0.802950i \(0.296737\pi\)
\(318\) −5.17056e30 −0.271143
\(319\) −9.86707e30 −0.497513
\(320\) 0 0
\(321\) 1.54238e30 0.0719249
\(322\) 3.97704e30 0.178386
\(323\) −8.04849e30 −0.347282
\(324\) 9.76540e30 0.405394
\(325\) 0 0
\(326\) −1.22038e31 −0.469109
\(327\) 1.48499e30 0.0549383
\(328\) −1.62135e31 −0.577368
\(329\) −1.16752e31 −0.400236
\(330\) 0 0
\(331\) −2.93057e31 −0.931325 −0.465663 0.884962i \(-0.654184\pi\)
−0.465663 + 0.884962i \(0.654184\pi\)
\(332\) −1.08456e30 −0.0331917
\(333\) 2.80086e31 0.825540
\(334\) −3.66949e31 −1.04178
\(335\) 0 0
\(336\) −4.20385e30 −0.110766
\(337\) 4.60143e31 1.16821 0.584103 0.811680i \(-0.301447\pi\)
0.584103 + 0.811680i \(0.301447\pi\)
\(338\) 2.41504e31 0.590835
\(339\) 1.19805e31 0.282474
\(340\) 0 0
\(341\) −4.72613e31 −1.03532
\(342\) −2.74338e31 −0.579376
\(343\) 8.96148e31 1.82475
\(344\) −2.26659e30 −0.0445033
\(345\) 0 0
\(346\) −5.77039e31 −1.05379
\(347\) 7.38077e31 1.30012 0.650060 0.759883i \(-0.274744\pi\)
0.650060 + 0.759883i \(0.274744\pi\)
\(348\) 6.40489e30 0.108836
\(349\) 3.56553e31 0.584530 0.292265 0.956337i \(-0.405591\pi\)
0.292265 + 0.956337i \(0.405591\pi\)
\(350\) 0 0
\(351\) −4.36171e31 −0.665760
\(352\) 6.96372e30 0.102579
\(353\) −8.69355e31 −1.23599 −0.617993 0.786184i \(-0.712054\pi\)
−0.617993 + 0.786184i \(0.712054\pi\)
\(354\) 1.08381e31 0.148735
\(355\) 0 0
\(356\) 5.99512e31 0.766783
\(357\) 1.42263e31 0.175686
\(358\) 3.76725e31 0.449248
\(359\) 2.26294e30 0.0260611 0.0130305 0.999915i \(-0.495852\pi\)
0.0130305 + 0.999915i \(0.495852\pi\)
\(360\) 0 0
\(361\) −2.16825e31 −0.232954
\(362\) 1.13612e32 1.17915
\(363\) 1.67943e31 0.168395
\(364\) −1.22028e32 −1.18220
\(365\) 0 0
\(366\) 2.71504e31 0.245619
\(367\) 3.34192e31 0.292193 0.146096 0.989270i \(-0.453329\pi\)
0.146096 + 0.989270i \(0.453329\pi\)
\(368\) 4.27639e30 0.0361393
\(369\) 1.87022e32 1.52779
\(370\) 0 0
\(371\) 3.45219e32 2.63585
\(372\) 3.06782e31 0.226487
\(373\) −1.79027e32 −1.27808 −0.639041 0.769172i \(-0.720669\pi\)
−0.639041 + 0.769172i \(0.720669\pi\)
\(374\) −2.35660e31 −0.162701
\(375\) 0 0
\(376\) −1.25540e31 −0.0810839
\(377\) 1.85919e32 1.16160
\(378\) 1.00323e32 0.606393
\(379\) −9.45760e31 −0.553084 −0.276542 0.961002i \(-0.589188\pi\)
−0.276542 + 0.961002i \(0.589188\pi\)
\(380\) 0 0
\(381\) 3.55371e31 0.194590
\(382\) −6.88621e31 −0.364912
\(383\) 2.01782e32 1.03490 0.517449 0.855714i \(-0.326882\pi\)
0.517449 + 0.855714i \(0.326882\pi\)
\(384\) −4.52028e30 −0.0224401
\(385\) 0 0
\(386\) 1.22047e32 0.567789
\(387\) 2.61450e31 0.117761
\(388\) −4.18406e31 −0.182474
\(389\) 1.16247e32 0.490921 0.245460 0.969407i \(-0.421061\pi\)
0.245460 + 0.969407i \(0.421061\pi\)
\(390\) 0 0
\(391\) −1.44717e31 −0.0573207
\(392\) 1.88518e32 0.723230
\(393\) 1.06203e32 0.394667
\(394\) 3.65888e31 0.131718
\(395\) 0 0
\(396\) −8.03262e31 −0.271436
\(397\) 2.03760e32 0.667172 0.333586 0.942720i \(-0.391741\pi\)
0.333586 + 0.942720i \(0.391741\pi\)
\(398\) −3.92662e32 −1.24589
\(399\) −1.26194e32 −0.388041
\(400\) 0 0
\(401\) −1.67418e32 −0.483613 −0.241806 0.970324i \(-0.577740\pi\)
−0.241806 + 0.970324i \(0.577740\pi\)
\(402\) −1.11224e32 −0.311438
\(403\) 8.90515e32 2.41729
\(404\) −2.26164e32 −0.595192
\(405\) 0 0
\(406\) −4.27630e32 −1.05802
\(407\) −2.13421e32 −0.512043
\(408\) 1.52971e31 0.0355924
\(409\) 7.52295e32 1.69765 0.848823 0.528678i \(-0.177312\pi\)
0.848823 + 0.528678i \(0.177312\pi\)
\(410\) 0 0
\(411\) 1.37906e31 0.0292793
\(412\) −3.32987e32 −0.685822
\(413\) −7.23618e32 −1.44589
\(414\) −4.93280e31 −0.0956288
\(415\) 0 0
\(416\) −1.31213e32 −0.239503
\(417\) 3.46984e31 0.0614623
\(418\) 2.09041e32 0.359359
\(419\) −6.25496e31 −0.104364 −0.0521818 0.998638i \(-0.516618\pi\)
−0.0521818 + 0.998638i \(0.516618\pi\)
\(420\) 0 0
\(421\) 8.02159e32 1.26106 0.630528 0.776167i \(-0.282838\pi\)
0.630528 + 0.776167i \(0.282838\pi\)
\(422\) 2.33314e31 0.0356069
\(423\) 1.44810e32 0.214558
\(424\) 3.71204e32 0.533997
\(425\) 0 0
\(426\) −1.09697e32 −0.148790
\(427\) −1.81273e33 −2.38773
\(428\) −1.10730e32 −0.141651
\(429\) 1.60644e32 0.199594
\(430\) 0 0
\(431\) 1.32299e33 1.55092 0.775460 0.631397i \(-0.217518\pi\)
0.775460 + 0.631397i \(0.217518\pi\)
\(432\) 1.07875e32 0.122849
\(433\) −1.44206e33 −1.59546 −0.797730 0.603015i \(-0.793966\pi\)
−0.797730 + 0.603015i \(0.793966\pi\)
\(434\) −2.04827e33 −2.20173
\(435\) 0 0
\(436\) −1.06610e32 −0.108197
\(437\) 1.28371e32 0.126605
\(438\) −8.47240e30 −0.00812045
\(439\) 9.88019e32 0.920362 0.460181 0.887825i \(-0.347785\pi\)
0.460181 + 0.887825i \(0.347785\pi\)
\(440\) 0 0
\(441\) −2.17454e33 −1.91375
\(442\) 4.44038e32 0.379876
\(443\) −6.78172e32 −0.564018 −0.282009 0.959412i \(-0.591001\pi\)
−0.282009 + 0.959412i \(0.591001\pi\)
\(444\) 1.38535e32 0.112014
\(445\) 0 0
\(446\) −2.39431e32 −0.183018
\(447\) 1.25000e32 0.0929104
\(448\) 3.01802e32 0.218146
\(449\) −1.40094e33 −0.984784 −0.492392 0.870374i \(-0.663877\pi\)
−0.492392 + 0.870374i \(0.663877\pi\)
\(450\) 0 0
\(451\) −1.42507e33 −0.947613
\(452\) −8.60102e32 −0.556313
\(453\) −4.96071e32 −0.312116
\(454\) −2.26210e33 −1.38456
\(455\) 0 0
\(456\) −1.35693e32 −0.0786132
\(457\) −2.87797e33 −1.62231 −0.811156 0.584830i \(-0.801161\pi\)
−0.811156 + 0.584830i \(0.801161\pi\)
\(458\) −2.37680e32 −0.130369
\(459\) −3.65059e32 −0.194852
\(460\) 0 0
\(461\) −4.62750e32 −0.233930 −0.116965 0.993136i \(-0.537316\pi\)
−0.116965 + 0.993136i \(0.537316\pi\)
\(462\) −3.69495e32 −0.181796
\(463\) −2.53220e32 −0.121265 −0.0606325 0.998160i \(-0.519312\pi\)
−0.0606325 + 0.998160i \(0.519312\pi\)
\(464\) −4.59818e32 −0.214344
\(465\) 0 0
\(466\) 7.93650e32 0.350595
\(467\) 1.56052e33 0.671134 0.335567 0.942016i \(-0.391072\pi\)
0.335567 + 0.942016i \(0.391072\pi\)
\(468\) 1.51354e33 0.633752
\(469\) 7.42598e33 3.02757
\(470\) 0 0
\(471\) 3.88309e31 0.0150112
\(472\) −7.78085e32 −0.292922
\(473\) −1.99221e32 −0.0730416
\(474\) −2.08093e32 −0.0743068
\(475\) 0 0
\(476\) −1.02133e33 −0.346002
\(477\) −4.28182e33 −1.41302
\(478\) 2.93259e32 0.0942763
\(479\) 7.23567e32 0.226613 0.113306 0.993560i \(-0.463856\pi\)
0.113306 + 0.993560i \(0.463856\pi\)
\(480\) 0 0
\(481\) 4.02135e33 1.19553
\(482\) −3.44901e33 −0.999096
\(483\) −2.26905e32 −0.0640480
\(484\) −1.20569e33 −0.331641
\(485\) 0 0
\(486\) −1.88721e33 −0.493024
\(487\) −3.11091e33 −0.792091 −0.396046 0.918231i \(-0.629618\pi\)
−0.396046 + 0.918231i \(0.629618\pi\)
\(488\) −1.94918e33 −0.483730
\(489\) 6.96273e32 0.168430
\(490\) 0 0
\(491\) 3.51132e33 0.807146 0.403573 0.914948i \(-0.367768\pi\)
0.403573 + 0.914948i \(0.367768\pi\)
\(492\) 9.25042e32 0.207299
\(493\) 1.55607e33 0.339972
\(494\) −3.93883e33 −0.839037
\(495\) 0 0
\(496\) −2.20244e33 −0.446050
\(497\) 7.32402e33 1.44642
\(498\) 6.18787e31 0.0119172
\(499\) 8.78119e33 1.64929 0.824644 0.565652i \(-0.191376\pi\)
0.824644 + 0.565652i \(0.191376\pi\)
\(500\) 0 0
\(501\) 2.09358e33 0.374041
\(502\) −1.32432e32 −0.0230780
\(503\) 1.68186e33 0.285883 0.142942 0.989731i \(-0.454344\pi\)
0.142942 + 0.989731i \(0.454344\pi\)
\(504\) −3.48127e33 −0.577240
\(505\) 0 0
\(506\) 3.75871e32 0.0593140
\(507\) −1.37788e33 −0.212134
\(508\) −2.55127e33 −0.383231
\(509\) −5.67785e33 −0.832169 −0.416084 0.909326i \(-0.636598\pi\)
−0.416084 + 0.909326i \(0.636598\pi\)
\(510\) 0 0
\(511\) 5.65670e32 0.0789409
\(512\) 3.24519e32 0.0441942
\(513\) 3.23825e33 0.430372
\(514\) 9.82039e32 0.127377
\(515\) 0 0
\(516\) 1.29318e32 0.0159785
\(517\) −1.10343e33 −0.133080
\(518\) −9.24948e33 −1.08892
\(519\) 3.29224e33 0.378355
\(520\) 0 0
\(521\) 8.57848e33 0.939591 0.469796 0.882775i \(-0.344328\pi\)
0.469796 + 0.882775i \(0.344328\pi\)
\(522\) 5.30399e33 0.567180
\(523\) −1.65708e34 −1.73010 −0.865052 0.501683i \(-0.832714\pi\)
−0.865052 + 0.501683i \(0.832714\pi\)
\(524\) −7.62451e33 −0.777268
\(525\) 0 0
\(526\) −1.02497e34 −0.996296
\(527\) 7.45329e33 0.707482
\(528\) −3.97307e32 −0.0368301
\(529\) −1.08149e34 −0.979103
\(530\) 0 0
\(531\) 8.97518e33 0.775107
\(532\) 9.05968e33 0.764218
\(533\) 2.68518e34 2.21250
\(534\) −3.42045e33 −0.275307
\(535\) 0 0
\(536\) 7.98494e33 0.613355
\(537\) −2.14936e33 −0.161299
\(538\) 1.31481e34 0.964016
\(539\) 1.65697e34 1.18701
\(540\) 0 0
\(541\) −4.23198e33 −0.289453 −0.144727 0.989472i \(-0.546230\pi\)
−0.144727 + 0.989472i \(0.546230\pi\)
\(542\) 7.73674e33 0.517091
\(543\) −6.48202e33 −0.423362
\(544\) −1.09820e33 −0.0700966
\(545\) 0 0
\(546\) 6.96217e33 0.424460
\(547\) 5.57179e33 0.332011 0.166006 0.986125i \(-0.446913\pi\)
0.166006 + 0.986125i \(0.446913\pi\)
\(548\) −9.90052e32 −0.0576635
\(549\) 2.24837e34 1.28001
\(550\) 0 0
\(551\) −1.38031e34 −0.750900
\(552\) −2.43985e32 −0.0129755
\(553\) 1.38936e34 0.722355
\(554\) −8.80334e32 −0.0447482
\(555\) 0 0
\(556\) −2.49106e33 −0.121046
\(557\) 1.51207e34 0.718427 0.359213 0.933255i \(-0.383045\pi\)
0.359213 + 0.933255i \(0.383045\pi\)
\(558\) 2.54051e34 1.18030
\(559\) 3.75379e33 0.170539
\(560\) 0 0
\(561\) 1.34453e33 0.0584164
\(562\) −5.52573e33 −0.234793
\(563\) −1.21471e34 −0.504799 −0.252400 0.967623i \(-0.581220\pi\)
−0.252400 + 0.967623i \(0.581220\pi\)
\(564\) 7.16255e32 0.0291125
\(565\) 0 0
\(566\) −2.87458e34 −1.11781
\(567\) 3.71991e34 1.41496
\(568\) 7.87531e33 0.293031
\(569\) 2.58692e34 0.941628 0.470814 0.882232i \(-0.343960\pi\)
0.470814 + 0.882232i \(0.343960\pi\)
\(570\) 0 0
\(571\) −3.34607e34 −1.16569 −0.582846 0.812583i \(-0.698061\pi\)
−0.582846 + 0.812583i \(0.698061\pi\)
\(572\) −1.15329e34 −0.393086
\(573\) 3.92885e33 0.131019
\(574\) −6.17615e34 −2.01521
\(575\) 0 0
\(576\) −3.74331e33 −0.116943
\(577\) 1.34968e34 0.412603 0.206302 0.978488i \(-0.433857\pi\)
0.206302 + 0.978488i \(0.433857\pi\)
\(578\) −1.99201e34 −0.595926
\(579\) −6.96328e33 −0.203860
\(580\) 0 0
\(581\) −4.13140e33 −0.115850
\(582\) 2.38717e33 0.0655158
\(583\) 3.26268e34 0.876429
\(584\) 6.08248e32 0.0159926
\(585\) 0 0
\(586\) 1.73315e34 0.436632
\(587\) 4.17737e34 1.03021 0.515107 0.857126i \(-0.327752\pi\)
0.515107 + 0.857126i \(0.327752\pi\)
\(588\) −1.07557e34 −0.259670
\(589\) −6.61143e34 −1.56262
\(590\) 0 0
\(591\) −2.08753e33 −0.0472923
\(592\) −9.94569e33 −0.220604
\(593\) 2.55607e33 0.0555123 0.0277562 0.999615i \(-0.491164\pi\)
0.0277562 + 0.999615i \(0.491164\pi\)
\(594\) 9.48160e33 0.201628
\(595\) 0 0
\(596\) −8.97394e33 −0.182981
\(597\) 2.24029e34 0.447327
\(598\) −7.08230e33 −0.138487
\(599\) 4.70324e34 0.900662 0.450331 0.892862i \(-0.351306\pi\)
0.450331 + 0.892862i \(0.351306\pi\)
\(600\) 0 0
\(601\) 5.96787e34 1.09620 0.548099 0.836414i \(-0.315352\pi\)
0.548099 + 0.836414i \(0.315352\pi\)
\(602\) −8.63406e33 −0.155331
\(603\) −9.21060e34 −1.62301
\(604\) 3.56138e34 0.614691
\(605\) 0 0
\(606\) 1.29036e34 0.213699
\(607\) 2.92667e34 0.474804 0.237402 0.971411i \(-0.423704\pi\)
0.237402 + 0.971411i \(0.423704\pi\)
\(608\) 9.74161e33 0.154823
\(609\) 2.43980e34 0.379872
\(610\) 0 0
\(611\) 2.07912e34 0.310717
\(612\) 1.26677e34 0.185484
\(613\) −8.48844e34 −1.21779 −0.608895 0.793251i \(-0.708387\pi\)
−0.608895 + 0.793251i \(0.708387\pi\)
\(614\) 1.29186e34 0.181597
\(615\) 0 0
\(616\) 2.65267e34 0.358034
\(617\) −1.30414e35 −1.72488 −0.862442 0.506156i \(-0.831066\pi\)
−0.862442 + 0.506156i \(0.831066\pi\)
\(618\) 1.89982e34 0.246239
\(619\) −8.02670e34 −1.01954 −0.509770 0.860311i \(-0.670269\pi\)
−0.509770 + 0.860311i \(0.670269\pi\)
\(620\) 0 0
\(621\) 5.82260e33 0.0710350
\(622\) 8.13296e34 0.972453
\(623\) 2.28370e35 2.67633
\(624\) 7.48621e33 0.0859913
\(625\) 0 0
\(626\) 2.24133e33 0.0247359
\(627\) −1.19266e34 −0.129025
\(628\) −2.78774e33 −0.0295636
\(629\) 3.36573e34 0.349901
\(630\) 0 0
\(631\) −3.51125e34 −0.350828 −0.175414 0.984495i \(-0.556126\pi\)
−0.175414 + 0.984495i \(0.556126\pi\)
\(632\) 1.49394e34 0.146342
\(633\) −1.33115e33 −0.0127844
\(634\) 8.95181e34 0.842937
\(635\) 0 0
\(636\) −2.11786e34 −0.191727
\(637\) −3.12212e35 −2.77145
\(638\) −4.04155e34 −0.351795
\(639\) −9.08413e34 −0.775394
\(640\) 0 0
\(641\) −1.10476e35 −0.906862 −0.453431 0.891291i \(-0.649800\pi\)
−0.453431 + 0.891291i \(0.649800\pi\)
\(642\) 6.31760e33 0.0508586
\(643\) −5.33371e34 −0.421107 −0.210553 0.977582i \(-0.567527\pi\)
−0.210553 + 0.977582i \(0.567527\pi\)
\(644\) 1.62899e34 0.126138
\(645\) 0 0
\(646\) −3.29666e34 −0.245566
\(647\) −2.39579e34 −0.175043 −0.0875216 0.996163i \(-0.527895\pi\)
−0.0875216 + 0.996163i \(0.527895\pi\)
\(648\) 3.99991e34 0.286657
\(649\) −6.83894e34 −0.480762
\(650\) 0 0
\(651\) 1.16862e35 0.790514
\(652\) −4.99866e34 −0.331710
\(653\) −6.46858e34 −0.421109 −0.210554 0.977582i \(-0.567527\pi\)
−0.210554 + 0.977582i \(0.567527\pi\)
\(654\) 6.08250e33 0.0388472
\(655\) 0 0
\(656\) −6.64104e34 −0.408261
\(657\) −7.01612e33 −0.0423185
\(658\) −4.78217e34 −0.283010
\(659\) 1.74748e35 1.01471 0.507357 0.861736i \(-0.330623\pi\)
0.507357 + 0.861736i \(0.330623\pi\)
\(660\) 0 0
\(661\) 2.05707e35 1.15009 0.575044 0.818123i \(-0.304985\pi\)
0.575044 + 0.818123i \(0.304985\pi\)
\(662\) −1.20036e35 −0.658547
\(663\) −2.53341e34 −0.136391
\(664\) −4.44238e33 −0.0234701
\(665\) 0 0
\(666\) 1.14723e35 0.583745
\(667\) −2.48190e34 −0.123940
\(668\) −1.50302e35 −0.736647
\(669\) 1.36605e34 0.0657112
\(670\) 0 0
\(671\) −1.71322e35 −0.793928
\(672\) −1.72190e34 −0.0783234
\(673\) −1.94000e35 −0.866190 −0.433095 0.901348i \(-0.642579\pi\)
−0.433095 + 0.901348i \(0.642579\pi\)
\(674\) 1.88475e35 0.826046
\(675\) 0 0
\(676\) 9.89201e34 0.417783
\(677\) 1.50061e35 0.622172 0.311086 0.950382i \(-0.399307\pi\)
0.311086 + 0.950382i \(0.399307\pi\)
\(678\) 4.90722e34 0.199740
\(679\) −1.59382e35 −0.636895
\(680\) 0 0
\(681\) 1.29062e35 0.497116
\(682\) −1.93582e35 −0.732084
\(683\) 2.23175e35 0.828679 0.414339 0.910122i \(-0.364013\pi\)
0.414339 + 0.910122i \(0.364013\pi\)
\(684\) −1.12369e35 −0.409681
\(685\) 0 0
\(686\) 3.67062e35 1.29029
\(687\) 1.35606e34 0.0468078
\(688\) −9.28395e33 −0.0314686
\(689\) −6.14765e35 −2.04630
\(690\) 0 0
\(691\) 4.24403e34 0.136240 0.0681198 0.997677i \(-0.478300\pi\)
0.0681198 + 0.997677i \(0.478300\pi\)
\(692\) −2.36355e35 −0.745142
\(693\) −3.05985e35 −0.947402
\(694\) 3.02316e35 0.919324
\(695\) 0 0
\(696\) 2.62344e34 0.0769584
\(697\) 2.24740e35 0.647545
\(698\) 1.46044e35 0.413325
\(699\) −4.52808e34 −0.125878
\(700\) 0 0
\(701\) −6.67347e35 −1.79010 −0.895048 0.445969i \(-0.852859\pi\)
−0.895048 + 0.445969i \(0.852859\pi\)
\(702\) −1.78656e35 −0.470764
\(703\) −2.98556e35 −0.772831
\(704\) 2.85234e34 0.0725342
\(705\) 0 0
\(706\) −3.56088e35 −0.873974
\(707\) −8.61522e35 −2.07742
\(708\) 4.43928e34 0.105171
\(709\) 5.82085e35 1.35490 0.677452 0.735567i \(-0.263084\pi\)
0.677452 + 0.735567i \(0.263084\pi\)
\(710\) 0 0
\(711\) −1.72325e35 −0.387238
\(712\) 2.45560e35 0.542198
\(713\) −1.18878e35 −0.257919
\(714\) 5.82708e34 0.124229
\(715\) 0 0
\(716\) 1.54307e35 0.317667
\(717\) −1.67316e34 −0.0338491
\(718\) 9.26900e33 0.0184280
\(719\) −4.08106e35 −0.797374 −0.398687 0.917087i \(-0.630534\pi\)
−0.398687 + 0.917087i \(0.630534\pi\)
\(720\) 0 0
\(721\) −1.26844e36 −2.39375
\(722\) −8.88115e34 −0.164723
\(723\) 1.96780e35 0.358717
\(724\) 4.65355e35 0.833782
\(725\) 0 0
\(726\) 6.87893e34 0.119073
\(727\) −7.44759e34 −0.126717 −0.0633586 0.997991i \(-0.520181\pi\)
−0.0633586 + 0.997991i \(0.520181\pi\)
\(728\) −4.99826e35 −0.835943
\(729\) −3.85503e35 −0.633772
\(730\) 0 0
\(731\) 3.14179e34 0.0499125
\(732\) 1.11208e35 0.173679
\(733\) 1.00546e36 1.54371 0.771853 0.635802i \(-0.219330\pi\)
0.771853 + 0.635802i \(0.219330\pi\)
\(734\) 1.36885e35 0.206612
\(735\) 0 0
\(736\) 1.75161e34 0.0255543
\(737\) 7.01833e35 1.00668
\(738\) 7.66041e35 1.08031
\(739\) −7.42029e35 −1.02888 −0.514441 0.857526i \(-0.672001\pi\)
−0.514441 + 0.857526i \(0.672001\pi\)
\(740\) 0 0
\(741\) 2.24726e35 0.301249
\(742\) 1.41402e36 1.86383
\(743\) 4.93877e35 0.640115 0.320058 0.947398i \(-0.396298\pi\)
0.320058 + 0.947398i \(0.396298\pi\)
\(744\) 1.25658e35 0.160150
\(745\) 0 0
\(746\) −7.33296e35 −0.903741
\(747\) 5.12426e34 0.0621046
\(748\) −9.65262e34 −0.115047
\(749\) −4.21802e35 −0.494409
\(750\) 0 0
\(751\) 1.60017e36 1.81413 0.907063 0.420995i \(-0.138319\pi\)
0.907063 + 0.420995i \(0.138319\pi\)
\(752\) −5.14212e34 −0.0573349
\(753\) 7.55579e33 0.00828595
\(754\) 7.61524e35 0.821375
\(755\) 0 0
\(756\) 4.10924e35 0.428785
\(757\) −9.77978e34 −0.100376 −0.0501881 0.998740i \(-0.515982\pi\)
−0.0501881 + 0.998740i \(0.515982\pi\)
\(758\) −3.87383e35 −0.391089
\(759\) −2.14449e34 −0.0212962
\(760\) 0 0
\(761\) 1.42035e36 1.36485 0.682427 0.730954i \(-0.260924\pi\)
0.682427 + 0.730954i \(0.260924\pi\)
\(762\) 1.45560e35 0.137596
\(763\) −4.06106e35 −0.377644
\(764\) −2.82059e35 −0.258032
\(765\) 0 0
\(766\) 8.26499e35 0.731784
\(767\) 1.28862e36 1.12249
\(768\) −1.85151e34 −0.0158675
\(769\) 4.43106e35 0.373619 0.186809 0.982396i \(-0.440185\pi\)
0.186809 + 0.982396i \(0.440185\pi\)
\(770\) 0 0
\(771\) −5.60292e34 −0.0457335
\(772\) 4.99906e35 0.401487
\(773\) −1.04683e36 −0.827241 −0.413621 0.910449i \(-0.635736\pi\)
−0.413621 + 0.910449i \(0.635736\pi\)
\(774\) 1.07090e35 0.0832697
\(775\) 0 0
\(776\) −1.71379e35 −0.129029
\(777\) 5.27719e35 0.390967
\(778\) 4.76147e35 0.347133
\(779\) −1.99355e36 −1.43024
\(780\) 0 0
\(781\) 6.92196e35 0.480940
\(782\) −5.92763e34 −0.0405319
\(783\) −6.26075e35 −0.421313
\(784\) 7.72169e35 0.511401
\(785\) 0 0
\(786\) 4.35008e35 0.279071
\(787\) 1.02940e36 0.649977 0.324989 0.945718i \(-0.394640\pi\)
0.324989 + 0.945718i \(0.394640\pi\)
\(788\) 1.49868e35 0.0931387
\(789\) 5.84784e35 0.357712
\(790\) 0 0
\(791\) −3.27636e36 −1.94172
\(792\) −3.29016e35 −0.191934
\(793\) 3.22811e36 1.85367
\(794\) 8.34601e35 0.471762
\(795\) 0 0
\(796\) −1.60834e36 −0.880978
\(797\) 2.74499e36 1.48017 0.740086 0.672512i \(-0.234785\pi\)
0.740086 + 0.672512i \(0.234785\pi\)
\(798\) −5.16890e35 −0.274386
\(799\) 1.74015e35 0.0909393
\(800\) 0 0
\(801\) −2.83252e36 −1.43472
\(802\) −6.85745e35 −0.341966
\(803\) 5.34617e34 0.0262481
\(804\) −4.55572e35 −0.220220
\(805\) 0 0
\(806\) 3.64755e36 1.70928
\(807\) −7.50150e35 −0.346122
\(808\) −9.26369e35 −0.420864
\(809\) 2.35511e36 1.05355 0.526776 0.850004i \(-0.323401\pi\)
0.526776 + 0.850004i \(0.323401\pi\)
\(810\) 0 0
\(811\) 3.33343e36 1.44587 0.722936 0.690915i \(-0.242792\pi\)
0.722936 + 0.690915i \(0.242792\pi\)
\(812\) −1.75157e36 −0.748132
\(813\) −4.41411e35 −0.185657
\(814\) −8.74172e35 −0.362069
\(815\) 0 0
\(816\) 6.26569e34 0.0251676
\(817\) −2.78692e35 −0.110242
\(818\) 3.08140e36 1.20042
\(819\) 5.76547e36 2.21201
\(820\) 0 0
\(821\) 1.82523e36 0.679249 0.339625 0.940561i \(-0.389700\pi\)
0.339625 + 0.940561i \(0.389700\pi\)
\(822\) 5.64864e34 0.0207036
\(823\) 1.63571e36 0.590483 0.295241 0.955423i \(-0.404600\pi\)
0.295241 + 0.955423i \(0.404600\pi\)
\(824\) −1.36391e36 −0.484950
\(825\) 0 0
\(826\) −2.96394e36 −1.02240
\(827\) 3.73907e36 1.27041 0.635206 0.772342i \(-0.280915\pi\)
0.635206 + 0.772342i \(0.280915\pi\)
\(828\) −2.02047e35 −0.0676198
\(829\) −3.08664e36 −1.01755 −0.508773 0.860901i \(-0.669901\pi\)
−0.508773 + 0.860901i \(0.669901\pi\)
\(830\) 0 0
\(831\) 5.02265e34 0.0160665
\(832\) −5.37448e35 −0.169354
\(833\) −2.61310e36 −0.811135
\(834\) 1.42125e35 0.0434604
\(835\) 0 0
\(836\) 8.56234e35 0.254105
\(837\) −2.99878e36 −0.876750
\(838\) −2.56203e35 −0.0737963
\(839\) −3.69450e36 −1.04841 −0.524205 0.851592i \(-0.675637\pi\)
−0.524205 + 0.851592i \(0.675637\pi\)
\(840\) 0 0
\(841\) −9.61704e35 −0.264906
\(842\) 3.28564e36 0.891701
\(843\) 3.15264e35 0.0843005
\(844\) 9.55652e34 0.0251779
\(845\) 0 0
\(846\) 5.93142e35 0.151715
\(847\) −4.59280e36 −1.15754
\(848\) 1.52045e36 0.377593
\(849\) 1.64006e36 0.401341
\(850\) 0 0
\(851\) −5.36825e35 −0.127560
\(852\) −4.49317e35 −0.105210
\(853\) −7.36060e35 −0.169844 −0.0849220 0.996388i \(-0.527064\pi\)
−0.0849220 + 0.996388i \(0.527064\pi\)
\(854\) −7.42494e36 −1.68838
\(855\) 0 0
\(856\) −4.53551e35 −0.100162
\(857\) 5.06811e35 0.110303 0.0551513 0.998478i \(-0.482436\pi\)
0.0551513 + 0.998478i \(0.482436\pi\)
\(858\) 6.57997e35 0.141134
\(859\) 7.80778e33 0.00165049 0.000825245 1.00000i \(-0.499737\pi\)
0.000825245 1.00000i \(0.499737\pi\)
\(860\) 0 0
\(861\) 3.52374e36 0.723543
\(862\) 5.41895e36 1.09667
\(863\) −8.60176e36 −1.71574 −0.857871 0.513864i \(-0.828213\pi\)
−0.857871 + 0.513864i \(0.828213\pi\)
\(864\) 4.41855e35 0.0868676
\(865\) 0 0
\(866\) −5.90669e36 −1.12816
\(867\) 1.13652e36 0.213962
\(868\) −8.38970e36 −1.55686
\(869\) 1.31309e36 0.240186
\(870\) 0 0
\(871\) −1.32242e37 −2.35040
\(872\) −4.36673e35 −0.0765069
\(873\) 1.97685e36 0.341425
\(874\) 5.25809e35 0.0895231
\(875\) 0 0
\(876\) −3.47029e34 −0.00574202
\(877\) −7.33903e36 −1.19714 −0.598568 0.801072i \(-0.704263\pi\)
−0.598568 + 0.801072i \(0.704263\pi\)
\(878\) 4.04692e36 0.650794
\(879\) −9.88829e35 −0.156769
\(880\) 0 0
\(881\) 7.73442e36 1.19187 0.595935 0.803032i \(-0.296781\pi\)
0.595935 + 0.803032i \(0.296781\pi\)
\(882\) −8.90693e36 −1.35323
\(883\) 1.11931e37 1.67664 0.838320 0.545178i \(-0.183538\pi\)
0.838320 + 0.545178i \(0.183538\pi\)
\(884\) 1.81878e36 0.268613
\(885\) 0 0
\(886\) −2.77779e36 −0.398821
\(887\) −9.94167e36 −1.40739 −0.703695 0.710502i \(-0.748468\pi\)
−0.703695 + 0.710502i \(0.748468\pi\)
\(888\) 5.67441e35 0.0792060
\(889\) −9.71850e36 −1.33760
\(890\) 0 0
\(891\) 3.51570e36 0.470479
\(892\) −9.80709e35 −0.129413
\(893\) −1.54360e36 −0.200858
\(894\) 5.11998e35 0.0656976
\(895\) 0 0
\(896\) 1.23618e36 0.154252
\(897\) 4.04073e35 0.0497226
\(898\) −5.73826e36 −0.696347
\(899\) 1.27824e37 1.52973
\(900\) 0 0
\(901\) −5.14536e36 −0.598902
\(902\) −5.83711e36 −0.670063
\(903\) 4.92607e35 0.0557704
\(904\) −3.52298e36 −0.393373
\(905\) 0 0
\(906\) −2.03191e36 −0.220700
\(907\) 2.49643e36 0.267441 0.133721 0.991019i \(-0.457308\pi\)
0.133721 + 0.991019i \(0.457308\pi\)
\(908\) −9.26555e36 −0.979035
\(909\) 1.06856e37 1.11366
\(910\) 0 0
\(911\) 9.34594e36 0.947640 0.473820 0.880622i \(-0.342875\pi\)
0.473820 + 0.880622i \(0.342875\pi\)
\(912\) −5.55797e35 −0.0555879
\(913\) −3.90461e35 −0.0385205
\(914\) −1.17882e37 −1.14715
\(915\) 0 0
\(916\) −9.73537e35 −0.0921847
\(917\) −2.90438e37 −2.71292
\(918\) −1.49528e36 −0.137781
\(919\) 2.08298e37 1.89340 0.946698 0.322122i \(-0.104396\pi\)
0.946698 + 0.322122i \(0.104396\pi\)
\(920\) 0 0
\(921\) −7.37054e35 −0.0652010
\(922\) −1.89543e36 −0.165413
\(923\) −1.30426e37 −1.12291
\(924\) −1.51345e36 −0.128549
\(925\) 0 0
\(926\) −1.03719e36 −0.0857474
\(927\) 1.57327e37 1.28323
\(928\) −1.88342e36 −0.151564
\(929\) −2.93833e36 −0.233294 −0.116647 0.993173i \(-0.537215\pi\)
−0.116647 + 0.993173i \(0.537215\pi\)
\(930\) 0 0
\(931\) 2.31795e37 1.79156
\(932\) 3.25079e36 0.247908
\(933\) −4.64017e36 −0.349151
\(934\) 6.39190e36 0.474563
\(935\) 0 0
\(936\) 6.19944e36 0.448131
\(937\) 1.33465e37 0.951968 0.475984 0.879454i \(-0.342092\pi\)
0.475984 + 0.879454i \(0.342092\pi\)
\(938\) 3.04168e37 2.14081
\(939\) −1.27877e35 −0.00888120
\(940\) 0 0
\(941\) −1.37070e37 −0.926983 −0.463492 0.886101i \(-0.653404\pi\)
−0.463492 + 0.886101i \(0.653404\pi\)
\(942\) 1.59051e35 0.0106145
\(943\) −3.58454e36 −0.236068
\(944\) −3.18704e36 −0.207127
\(945\) 0 0
\(946\) −8.16009e35 −0.0516482
\(947\) −3.73876e36 −0.233535 −0.116768 0.993159i \(-0.537253\pi\)
−0.116768 + 0.993159i \(0.537253\pi\)
\(948\) −8.52350e35 −0.0525429
\(949\) −1.00734e36 −0.0612845
\(950\) 0 0
\(951\) −5.10736e36 −0.302649
\(952\) −4.18336e36 −0.244660
\(953\) 4.80715e36 0.277477 0.138738 0.990329i \(-0.455695\pi\)
0.138738 + 0.990329i \(0.455695\pi\)
\(954\) −1.75383e37 −0.999156
\(955\) 0 0
\(956\) 1.20119e36 0.0666634
\(957\) 2.30586e36 0.126309
\(958\) 2.96373e36 0.160240
\(959\) −3.77138e36 −0.201265
\(960\) 0 0
\(961\) 4.19922e37 2.18336
\(962\) 1.64715e37 0.845364
\(963\) 5.23170e36 0.265042
\(964\) −1.41272e37 −0.706467
\(965\) 0 0
\(966\) −9.29405e35 −0.0452888
\(967\) −1.46042e37 −0.702498 −0.351249 0.936282i \(-0.614243\pi\)
−0.351249 + 0.936282i \(0.614243\pi\)
\(968\) −4.93850e36 −0.234506
\(969\) 1.88087e36 0.0881683
\(970\) 0 0
\(971\) −3.16381e37 −1.44534 −0.722670 0.691193i \(-0.757085\pi\)
−0.722670 + 0.691193i \(0.757085\pi\)
\(972\) −7.73003e36 −0.348620
\(973\) −9.48913e36 −0.422490
\(974\) −1.27423e37 −0.560093
\(975\) 0 0
\(976\) −7.98382e36 −0.342049
\(977\) 3.15815e37 1.33583 0.667914 0.744239i \(-0.267188\pi\)
0.667914 + 0.744239i \(0.267188\pi\)
\(978\) 2.85193e36 0.119098
\(979\) 2.15834e37 0.889889
\(980\) 0 0
\(981\) 5.03701e36 0.202446
\(982\) 1.43824e37 0.570738
\(983\) −1.30026e37 −0.509461 −0.254731 0.967012i \(-0.581987\pi\)
−0.254731 + 0.967012i \(0.581987\pi\)
\(984\) 3.78897e36 0.146583
\(985\) 0 0
\(986\) 6.37368e36 0.240397
\(987\) 2.72841e36 0.101612
\(988\) −1.61335e37 −0.593288
\(989\) −5.01107e35 −0.0181960
\(990\) 0 0
\(991\) 1.69845e37 0.601355 0.300677 0.953726i \(-0.402787\pi\)
0.300677 + 0.953726i \(0.402787\pi\)
\(992\) −9.02119e36 −0.315405
\(993\) 6.84852e36 0.236446
\(994\) 2.99992e37 1.02277
\(995\) 0 0
\(996\) 2.53455e35 0.00842673
\(997\) −1.43944e37 −0.472612 −0.236306 0.971679i \(-0.575937\pi\)
−0.236306 + 0.971679i \(0.575937\pi\)
\(998\) 3.59677e37 1.16622
\(999\) −1.35418e37 −0.433617
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 50.26.a.l.1.3 6
5.2 odd 4 10.26.b.a.9.10 yes 12
5.3 odd 4 10.26.b.a.9.3 12
5.4 even 2 50.26.a.k.1.4 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.26.b.a.9.3 12 5.3 odd 4
10.26.b.a.9.10 yes 12 5.2 odd 4
50.26.a.k.1.4 6 5.4 even 2
50.26.a.l.1.3 6 1.1 even 1 trivial