Properties

Label 5.12.a.b.1.1
Level 5
Weight 12
Character 5.1
Self dual yes
Analytic conductor 3.842
Analytic rank 0
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 5 \)
Weight: \( k \) = \( 12 \)
Character orbit: \([\chi]\) = 5.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.84171590280\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{151}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-12.2882\)
Character \(\chi\) = 5.1

$q$-expansion

\(f(q)\) \(=\) \(q-83.7292 q^{2} -503.223 q^{3} +4962.58 q^{4} -3125.00 q^{5} +42134.4 q^{6} +15973.7 q^{7} -244036. q^{8} +76086.0 q^{9} +O(q^{10})\) \(q-83.7292 q^{2} -503.223 q^{3} +4962.58 q^{4} -3125.00 q^{5} +42134.4 q^{6} +15973.7 q^{7} -244036. q^{8} +76086.0 q^{9} +261654. q^{10} +339729. q^{11} -2.49728e6 q^{12} +2.02328e6 q^{13} -1.33746e6 q^{14} +1.57257e6 q^{15} +1.02696e7 q^{16} -2.45063e6 q^{17} -6.37062e6 q^{18} -4.08504e6 q^{19} -1.55081e7 q^{20} -8.03830e6 q^{21} -2.84453e7 q^{22} +2.86497e7 q^{23} +1.22804e8 q^{24} +9.76562e6 q^{25} -1.69408e8 q^{26} +5.08562e7 q^{27} +7.92706e7 q^{28} -9.41230e6 q^{29} -1.31670e8 q^{30} +2.99399e8 q^{31} -3.60078e8 q^{32} -1.70959e8 q^{33} +2.05190e8 q^{34} -4.99177e7 q^{35} +3.77583e8 q^{36} -4.57279e8 q^{37} +3.42037e8 q^{38} -1.01816e9 q^{39} +7.62612e8 q^{40} +1.83814e8 q^{41} +6.73041e8 q^{42} +6.56811e8 q^{43} +1.68594e9 q^{44} -2.37769e8 q^{45} -2.39882e9 q^{46} -1.97090e8 q^{47} -5.16788e9 q^{48} -1.72217e9 q^{49} -8.17668e8 q^{50} +1.23321e9 q^{51} +1.00407e10 q^{52} +5.15890e9 q^{53} -4.25815e9 q^{54} -1.06165e9 q^{55} -3.89815e9 q^{56} +2.05568e9 q^{57} +7.88085e8 q^{58} -6.62200e8 q^{59} +7.80401e9 q^{60} +5.58296e8 q^{61} -2.50684e10 q^{62} +1.21537e9 q^{63} +9.11694e9 q^{64} -6.32275e9 q^{65} +1.43143e10 q^{66} +1.01206e10 q^{67} -1.21615e10 q^{68} -1.44172e10 q^{69} +4.17957e9 q^{70} +1.78161e10 q^{71} -1.85677e10 q^{72} -2.33380e10 q^{73} +3.82876e10 q^{74} -4.91428e9 q^{75} -2.02724e10 q^{76} +5.42672e9 q^{77} +8.52498e10 q^{78} +1.24957e10 q^{79} -3.20924e10 q^{80} -3.90704e10 q^{81} -1.53906e10 q^{82} +3.37037e10 q^{83} -3.98908e10 q^{84} +7.65823e9 q^{85} -5.49943e10 q^{86} +4.73648e9 q^{87} -8.29062e10 q^{88} +2.94282e10 q^{89} +1.99082e10 q^{90} +3.23192e10 q^{91} +1.42177e11 q^{92} -1.50664e11 q^{93} +1.65022e10 q^{94} +1.27657e10 q^{95} +1.81199e11 q^{96} -1.13262e11 q^{97} +1.44196e11 q^{98} +2.58486e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 20q^{2} - 220q^{3} + 6976q^{4} - 6250q^{5} + 60184q^{6} + 57900q^{7} - 246240q^{8} - 20846q^{9} + O(q^{10}) \) \( 2q - 20q^{2} - 220q^{3} + 6976q^{4} - 6250q^{5} + 60184q^{6} + 57900q^{7} - 246240q^{8} - 20846q^{9} + 62500q^{10} - 618176q^{11} - 1927040q^{12} + 3414260q^{13} + 1334472q^{14} + 687500q^{15} + 6005632q^{16} + 1317940q^{17} - 12548020q^{18} + 5325320q^{19} - 21800000q^{20} + 3836184q^{21} - 89491840q^{22} + 58943940q^{23} + 122180160q^{24} + 19531250q^{25} - 80761736q^{26} - 26769160q^{27} + 163685760q^{28} + 94140380q^{29} - 188075000q^{30} + 244543464q^{31} - 627301120q^{32} - 442259840q^{33} + 445358072q^{34} - 180937500q^{35} + 182418752q^{36} + 21003220q^{37} + 941752240q^{38} - 624203992q^{39} + 769500000q^{40} - 745743316q^{41} + 1429793040q^{42} + 629950100q^{43} - 242725888q^{44} + 65143750q^{45} - 468194856q^{46} - 1402061540q^{47} - 6375522560q^{48} - 1941677414q^{49} - 195312500q^{50} + 2300559784q^{51} + 12841321600q^{52} + 1138320580q^{53} - 9205154480q^{54} + 1931800000q^{55} - 3990553920q^{56} + 4720910480q^{57} + 7387417960q^{58} + 7317515560q^{59} + 6022000000q^{60} - 1516425676q^{61} - 28564327440q^{62} - 2848632180q^{63} + 819531776q^{64} - 10669562500q^{65} - 2975464192q^{66} + 15734290140q^{67} - 4573774720q^{68} - 5837195832q^{69} - 4170225000q^{70} + 32938471544q^{71} - 18354067680q^{72} - 29982848860q^{73} + 68768198072q^{74} - 2148437500q^{75} - 1325392640q^{76} - 34734748800q^{77} + 110356370800q^{78} - 3302823120q^{79} - 18767600000q^{80} - 43884431798q^{81} - 74630515640q^{82} + 13299102420q^{83} - 15982487808q^{84} - 4118562500q^{85} - 56706093896q^{86} + 34064940920q^{87} - 80794874880q^{88} - 12674770860q^{89} + 39212562500q^{90} + 90637859064q^{91} + 203171571840q^{92} - 166200542640q^{93} - 60289765528q^{94} - 16641625000q^{95} + 105515416064q^{96} - 3080703740q^{97} + 130206802940q^{98} + 118700272448q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −83.7292 −1.85017 −0.925086 0.379757i \(-0.876007\pi\)
−0.925086 + 0.379757i \(0.876007\pi\)
\(3\) −503.223 −1.19562 −0.597810 0.801638i \(-0.703962\pi\)
−0.597810 + 0.801638i \(0.703962\pi\)
\(4\) 4962.58 2.42314
\(5\) −3125.00 −0.447214
\(6\) 42134.4 2.21210
\(7\) 15973.7 0.359224 0.179612 0.983738i \(-0.442516\pi\)
0.179612 + 0.983738i \(0.442516\pi\)
\(8\) −244036. −2.63305
\(9\) 76086.0 0.429508
\(10\) 261654. 0.827422
\(11\) 339729. 0.636024 0.318012 0.948087i \(-0.396985\pi\)
0.318012 + 0.948087i \(0.396985\pi\)
\(12\) −2.49728e6 −2.89715
\(13\) 2.02328e6 1.51136 0.755680 0.654941i \(-0.227307\pi\)
0.755680 + 0.654941i \(0.227307\pi\)
\(14\) −1.33746e6 −0.664626
\(15\) 1.57257e6 0.534698
\(16\) 1.02696e7 2.44846
\(17\) −2.45063e6 −0.418610 −0.209305 0.977850i \(-0.567120\pi\)
−0.209305 + 0.977850i \(0.567120\pi\)
\(18\) −6.37062e6 −0.794663
\(19\) −4.08504e6 −0.378487 −0.189244 0.981930i \(-0.560604\pi\)
−0.189244 + 0.981930i \(0.560604\pi\)
\(20\) −1.55081e7 −1.08366
\(21\) −8.03830e6 −0.429495
\(22\) −2.84453e7 −1.17675
\(23\) 2.86497e7 0.928149 0.464074 0.885796i \(-0.346387\pi\)
0.464074 + 0.885796i \(0.346387\pi\)
\(24\) 1.22804e8 3.14813
\(25\) 9.76562e6 0.200000
\(26\) −1.69408e8 −2.79628
\(27\) 5.08562e7 0.682092
\(28\) 7.92706e7 0.870448
\(29\) −9.41230e6 −0.0852132 −0.0426066 0.999092i \(-0.513566\pi\)
−0.0426066 + 0.999092i \(0.513566\pi\)
\(30\) −1.31670e8 −0.989283
\(31\) 2.99399e8 1.87828 0.939141 0.343532i \(-0.111623\pi\)
0.939141 + 0.343532i \(0.111623\pi\)
\(32\) −3.60078e8 −1.89702
\(33\) −1.70959e8 −0.760443
\(34\) 2.05190e8 0.774500
\(35\) −4.99177e7 −0.160650
\(36\) 3.77583e8 1.04076
\(37\) −4.57279e8 −1.08411 −0.542053 0.840344i \(-0.682353\pi\)
−0.542053 + 0.840344i \(0.682353\pi\)
\(38\) 3.42037e8 0.700267
\(39\) −1.01816e9 −1.80701
\(40\) 7.62612e8 1.17754
\(41\) 1.83814e8 0.247780 0.123890 0.992296i \(-0.460463\pi\)
0.123890 + 0.992296i \(0.460463\pi\)
\(42\) 6.73041e8 0.794640
\(43\) 6.56811e8 0.681340 0.340670 0.940183i \(-0.389346\pi\)
0.340670 + 0.940183i \(0.389346\pi\)
\(44\) 1.68594e9 1.54117
\(45\) −2.37769e8 −0.192082
\(46\) −2.39882e9 −1.71724
\(47\) −1.97090e8 −0.125350 −0.0626752 0.998034i \(-0.519963\pi\)
−0.0626752 + 0.998034i \(0.519963\pi\)
\(48\) −5.16788e9 −2.92742
\(49\) −1.72217e9 −0.870958
\(50\) −8.17668e8 −0.370034
\(51\) 1.23321e9 0.500498
\(52\) 1.00407e10 3.66223
\(53\) 5.15890e9 1.69449 0.847247 0.531199i \(-0.178258\pi\)
0.847247 + 0.531199i \(0.178258\pi\)
\(54\) −4.25815e9 −1.26199
\(55\) −1.06165e9 −0.284438
\(56\) −3.89815e9 −0.945854
\(57\) 2.05568e9 0.452527
\(58\) 7.88085e8 0.157659
\(59\) −6.62200e8 −0.120588 −0.0602939 0.998181i \(-0.519204\pi\)
−0.0602939 + 0.998181i \(0.519204\pi\)
\(60\) 7.80401e9 1.29565
\(61\) 5.58296e8 0.0846351 0.0423176 0.999104i \(-0.486526\pi\)
0.0423176 + 0.999104i \(0.486526\pi\)
\(62\) −2.50684e10 −3.47515
\(63\) 1.21537e9 0.154289
\(64\) 9.11694e9 1.06135
\(65\) −6.32275e9 −0.675900
\(66\) 1.43143e10 1.40695
\(67\) 1.01206e10 0.915787 0.457894 0.889007i \(-0.348604\pi\)
0.457894 + 0.889007i \(0.348604\pi\)
\(68\) −1.21615e10 −1.01435
\(69\) −1.44172e10 −1.10971
\(70\) 4.17957e9 0.297230
\(71\) 1.78161e10 1.17190 0.585952 0.810346i \(-0.300721\pi\)
0.585952 + 0.810346i \(0.300721\pi\)
\(72\) −1.85677e10 −1.13091
\(73\) −2.33380e10 −1.31761 −0.658807 0.752312i \(-0.728939\pi\)
−0.658807 + 0.752312i \(0.728939\pi\)
\(74\) 3.82876e10 2.00578
\(75\) −4.91428e9 −0.239124
\(76\) −2.02724e10 −0.917127
\(77\) 5.42672e9 0.228475
\(78\) 8.52498e10 3.34328
\(79\) 1.24957e10 0.456888 0.228444 0.973557i \(-0.426636\pi\)
0.228444 + 0.973557i \(0.426636\pi\)
\(80\) −3.20924e10 −1.09498
\(81\) −3.90704e10 −1.24503
\(82\) −1.53906e10 −0.458436
\(83\) 3.37037e10 0.939179 0.469589 0.882885i \(-0.344402\pi\)
0.469589 + 0.882885i \(0.344402\pi\)
\(84\) −3.98908e10 −1.04073
\(85\) 7.65823e9 0.187208
\(86\) −5.49943e10 −1.26060
\(87\) 4.73648e9 0.101883
\(88\) −8.29062e10 −1.67468
\(89\) 2.94282e10 0.558623 0.279311 0.960201i \(-0.409894\pi\)
0.279311 + 0.960201i \(0.409894\pi\)
\(90\) 1.99082e10 0.355384
\(91\) 3.23192e10 0.542916
\(92\) 1.42177e11 2.24903
\(93\) −1.50664e11 −2.24571
\(94\) 1.65022e10 0.231920
\(95\) 1.27657e10 0.169265
\(96\) 1.81199e11 2.26811
\(97\) −1.13262e11 −1.33918 −0.669588 0.742732i \(-0.733529\pi\)
−0.669588 + 0.742732i \(0.733529\pi\)
\(98\) 1.44196e11 1.61142
\(99\) 2.58486e10 0.273177
\(100\) 4.84627e10 0.484627
\(101\) 1.19564e11 1.13196 0.565982 0.824417i \(-0.308497\pi\)
0.565982 + 0.824417i \(0.308497\pi\)
\(102\) −1.03256e11 −0.926008
\(103\) −9.46874e10 −0.804799 −0.402399 0.915464i \(-0.631824\pi\)
−0.402399 + 0.915464i \(0.631824\pi\)
\(104\) −4.93753e11 −3.97948
\(105\) 2.51197e10 0.192076
\(106\) −4.31951e11 −3.13511
\(107\) 1.52769e11 1.05299 0.526497 0.850177i \(-0.323505\pi\)
0.526497 + 0.850177i \(0.323505\pi\)
\(108\) 2.52378e11 1.65280
\(109\) 2.85078e11 1.77467 0.887337 0.461122i \(-0.152553\pi\)
0.887337 + 0.461122i \(0.152553\pi\)
\(110\) 8.88915e10 0.526260
\(111\) 2.30113e11 1.29618
\(112\) 1.64043e11 0.879544
\(113\) −2.35563e11 −1.20275 −0.601374 0.798968i \(-0.705380\pi\)
−0.601374 + 0.798968i \(0.705380\pi\)
\(114\) −1.72121e11 −0.837253
\(115\) −8.95304e10 −0.415081
\(116\) −4.67093e10 −0.206483
\(117\) 1.53943e11 0.649140
\(118\) 5.54455e10 0.223108
\(119\) −3.91456e10 −0.150375
\(120\) −3.83764e11 −1.40788
\(121\) −1.69896e11 −0.595474
\(122\) −4.67457e10 −0.156590
\(123\) −9.24991e10 −0.296251
\(124\) 1.48579e12 4.55133
\(125\) −3.05176e10 −0.0894427
\(126\) −1.01762e11 −0.285462
\(127\) −1.25786e11 −0.337840 −0.168920 0.985630i \(-0.554028\pi\)
−0.168920 + 0.985630i \(0.554028\pi\)
\(128\) −2.59156e10 −0.0666664
\(129\) −3.30522e11 −0.814624
\(130\) 5.29399e11 1.25053
\(131\) −2.97347e11 −0.673397 −0.336698 0.941613i \(-0.609310\pi\)
−0.336698 + 0.941613i \(0.609310\pi\)
\(132\) −8.48401e11 −1.84266
\(133\) −6.52530e10 −0.135962
\(134\) −8.47389e11 −1.69436
\(135\) −1.58926e11 −0.305041
\(136\) 5.98043e11 1.10222
\(137\) −6.29203e10 −0.111385 −0.0556926 0.998448i \(-0.517737\pi\)
−0.0556926 + 0.998448i \(0.517737\pi\)
\(138\) 1.20714e12 2.05316
\(139\) −5.11416e11 −0.835974 −0.417987 0.908453i \(-0.637264\pi\)
−0.417987 + 0.908453i \(0.637264\pi\)
\(140\) −2.47721e11 −0.389276
\(141\) 9.91800e10 0.149871
\(142\) −1.49173e12 −2.16823
\(143\) 6.87368e11 0.961260
\(144\) 7.81370e11 1.05163
\(145\) 2.94134e10 0.0381085
\(146\) 1.95407e12 2.43781
\(147\) 8.66634e11 1.04134
\(148\) −2.26929e12 −2.62694
\(149\) 7.94154e11 0.885891 0.442945 0.896549i \(-0.353934\pi\)
0.442945 + 0.896549i \(0.353934\pi\)
\(150\) 4.11469e11 0.442421
\(151\) 1.24629e12 1.29195 0.645974 0.763360i \(-0.276452\pi\)
0.645974 + 0.763360i \(0.276452\pi\)
\(152\) 9.96896e11 0.996576
\(153\) −1.86459e11 −0.179796
\(154\) −4.54375e11 −0.422718
\(155\) −9.35621e11 −0.839993
\(156\) −5.05271e12 −4.37864
\(157\) 9.73519e11 0.814510 0.407255 0.913315i \(-0.366486\pi\)
0.407255 + 0.913315i \(0.366486\pi\)
\(158\) −1.04625e12 −0.845322
\(159\) −2.59608e12 −2.02597
\(160\) 1.12524e12 0.848372
\(161\) 4.57641e11 0.333413
\(162\) 3.27133e12 2.30352
\(163\) 1.26084e12 0.858281 0.429140 0.903238i \(-0.358817\pi\)
0.429140 + 0.903238i \(0.358817\pi\)
\(164\) 9.12190e11 0.600405
\(165\) 5.34248e11 0.340080
\(166\) −2.82199e12 −1.73764
\(167\) −3.01398e12 −1.79556 −0.897780 0.440443i \(-0.854821\pi\)
−0.897780 + 0.440443i \(0.854821\pi\)
\(168\) 1.96164e12 1.13088
\(169\) 2.30151e12 1.28421
\(170\) −6.41218e11 −0.346367
\(171\) −3.10814e11 −0.162563
\(172\) 3.25948e12 1.65098
\(173\) −3.07482e11 −0.150857 −0.0754285 0.997151i \(-0.524032\pi\)
−0.0754285 + 0.997151i \(0.524032\pi\)
\(174\) −3.96582e11 −0.188500
\(175\) 1.55993e11 0.0718448
\(176\) 3.48887e12 1.55728
\(177\) 3.33234e11 0.144177
\(178\) −2.46400e12 −1.03355
\(179\) −1.91470e12 −0.778769 −0.389384 0.921075i \(-0.627312\pi\)
−0.389384 + 0.921075i \(0.627312\pi\)
\(180\) −1.17995e12 −0.465440
\(181\) 1.10300e11 0.0422032 0.0211016 0.999777i \(-0.493283\pi\)
0.0211016 + 0.999777i \(0.493283\pi\)
\(182\) −2.70606e12 −1.00449
\(183\) −2.80947e11 −0.101191
\(184\) −6.99157e12 −2.44386
\(185\) 1.42900e12 0.484827
\(186\) 1.26150e13 4.15495
\(187\) −8.32552e11 −0.266246
\(188\) −9.78074e11 −0.303741
\(189\) 8.12359e11 0.245024
\(190\) −1.06887e12 −0.313169
\(191\) −4.43991e12 −1.26384 −0.631918 0.775035i \(-0.717732\pi\)
−0.631918 + 0.775035i \(0.717732\pi\)
\(192\) −4.58785e12 −1.26897
\(193\) 3.15713e12 0.848647 0.424324 0.905511i \(-0.360512\pi\)
0.424324 + 0.905511i \(0.360512\pi\)
\(194\) 9.48330e12 2.47771
\(195\) 3.18175e12 0.808120
\(196\) −8.54641e12 −2.11045
\(197\) −3.58626e12 −0.861147 −0.430574 0.902555i \(-0.641689\pi\)
−0.430574 + 0.902555i \(0.641689\pi\)
\(198\) −2.16429e12 −0.505424
\(199\) 4.14823e12 0.942260 0.471130 0.882064i \(-0.343846\pi\)
0.471130 + 0.882064i \(0.343846\pi\)
\(200\) −2.38316e12 −0.526610
\(201\) −5.09291e12 −1.09493
\(202\) −1.00110e13 −2.09433
\(203\) −1.50349e11 −0.0306106
\(204\) 6.11993e12 1.21278
\(205\) −5.74417e11 −0.110811
\(206\) 7.92810e12 1.48902
\(207\) 2.17984e12 0.398647
\(208\) 2.07782e13 3.70050
\(209\) −1.38781e12 −0.240727
\(210\) −2.10325e12 −0.355374
\(211\) 1.14275e12 0.188104 0.0940519 0.995567i \(-0.470018\pi\)
0.0940519 + 0.995567i \(0.470018\pi\)
\(212\) 2.56015e13 4.10599
\(213\) −8.96547e12 −1.40115
\(214\) −1.27913e13 −1.94822
\(215\) −2.05253e12 −0.304704
\(216\) −1.24107e13 −1.79598
\(217\) 4.78249e12 0.674724
\(218\) −2.38694e13 −3.28345
\(219\) 1.17442e13 1.57537
\(220\) −5.26855e12 −0.689233
\(221\) −4.95832e12 −0.632670
\(222\) −1.92672e13 −2.39816
\(223\) 3.54889e12 0.430939 0.215469 0.976511i \(-0.430872\pi\)
0.215469 + 0.976511i \(0.430872\pi\)
\(224\) −5.75175e12 −0.681454
\(225\) 7.43027e11 0.0859015
\(226\) 1.97235e13 2.22529
\(227\) 3.90543e12 0.430058 0.215029 0.976608i \(-0.431015\pi\)
0.215029 + 0.976608i \(0.431015\pi\)
\(228\) 1.02015e13 1.09654
\(229\) −4.85126e12 −0.509048 −0.254524 0.967066i \(-0.581919\pi\)
−0.254524 + 0.967066i \(0.581919\pi\)
\(230\) 7.49632e12 0.767971
\(231\) −2.73085e12 −0.273169
\(232\) 2.29694e12 0.224370
\(233\) 1.03473e13 0.987115 0.493557 0.869713i \(-0.335696\pi\)
0.493557 + 0.869713i \(0.335696\pi\)
\(234\) −1.28896e13 −1.20102
\(235\) 6.15905e11 0.0560584
\(236\) −3.28622e12 −0.292201
\(237\) −6.28810e12 −0.546265
\(238\) 3.27763e12 0.278219
\(239\) 1.85140e12 0.153572 0.0767859 0.997048i \(-0.475534\pi\)
0.0767859 + 0.997048i \(0.475534\pi\)
\(240\) 1.61496e13 1.30918
\(241\) 2.73018e12 0.216321 0.108160 0.994133i \(-0.465504\pi\)
0.108160 + 0.994133i \(0.465504\pi\)
\(242\) 1.42252e13 1.10173
\(243\) 1.06521e13 0.806492
\(244\) 2.77059e12 0.205083
\(245\) 5.38178e12 0.389504
\(246\) 7.74488e12 0.548115
\(247\) −8.26518e12 −0.572031
\(248\) −7.30641e13 −4.94561
\(249\) −1.69605e13 −1.12290
\(250\) 2.55521e12 0.165484
\(251\) 1.18976e13 0.753796 0.376898 0.926255i \(-0.376991\pi\)
0.376898 + 0.926255i \(0.376991\pi\)
\(252\) 6.03138e12 0.373864
\(253\) 9.73316e12 0.590324
\(254\) 1.05319e13 0.625062
\(255\) −3.85380e12 −0.223830
\(256\) −1.65016e13 −0.938007
\(257\) 3.53878e10 0.00196889 0.000984443 1.00000i \(-0.499687\pi\)
0.000984443 1.00000i \(0.499687\pi\)
\(258\) 2.76744e13 1.50719
\(259\) −7.30442e12 −0.389437
\(260\) −3.13772e13 −1.63780
\(261\) −7.16144e11 −0.0365997
\(262\) 2.48966e13 1.24590
\(263\) −1.19904e13 −0.587592 −0.293796 0.955868i \(-0.594919\pi\)
−0.293796 + 0.955868i \(0.594919\pi\)
\(264\) 4.17202e13 2.00228
\(265\) −1.61216e13 −0.757801
\(266\) 5.46358e12 0.251553
\(267\) −1.48089e13 −0.667901
\(268\) 5.02243e13 2.21908
\(269\) −3.81149e13 −1.64990 −0.824948 0.565208i \(-0.808796\pi\)
−0.824948 + 0.565208i \(0.808796\pi\)
\(270\) 1.33067e13 0.564378
\(271\) −1.13510e13 −0.471740 −0.235870 0.971785i \(-0.575794\pi\)
−0.235870 + 0.971785i \(0.575794\pi\)
\(272\) −2.51670e13 −1.02495
\(273\) −1.62637e13 −0.649122
\(274\) 5.26827e12 0.206082
\(275\) 3.31767e12 0.127205
\(276\) −7.15466e13 −2.68899
\(277\) 3.02533e13 1.11464 0.557320 0.830298i \(-0.311830\pi\)
0.557320 + 0.830298i \(0.311830\pi\)
\(278\) 4.28205e13 1.54670
\(279\) 2.27801e13 0.806736
\(280\) 1.21817e13 0.422999
\(281\) 4.72047e13 1.60731 0.803657 0.595093i \(-0.202885\pi\)
0.803657 + 0.595093i \(0.202885\pi\)
\(282\) −8.30426e12 −0.277288
\(283\) −8.68805e12 −0.284510 −0.142255 0.989830i \(-0.545435\pi\)
−0.142255 + 0.989830i \(0.545435\pi\)
\(284\) 8.84140e13 2.83969
\(285\) −6.42401e12 −0.202376
\(286\) −5.75528e13 −1.77850
\(287\) 2.93617e12 0.0890085
\(288\) −2.73969e13 −0.814783
\(289\) −2.82663e13 −0.824766
\(290\) −2.46277e12 −0.0705073
\(291\) 5.69958e13 1.60115
\(292\) −1.15817e14 −3.19276
\(293\) 2.17869e13 0.589417 0.294709 0.955587i \(-0.404777\pi\)
0.294709 + 0.955587i \(0.404777\pi\)
\(294\) −7.25626e13 −1.92665
\(295\) 2.06938e12 0.0539285
\(296\) 1.11593e14 2.85450
\(297\) 1.72773e13 0.433827
\(298\) −6.64939e13 −1.63905
\(299\) 5.79665e13 1.40277
\(300\) −2.43875e13 −0.579430
\(301\) 1.04917e13 0.244754
\(302\) −1.04351e14 −2.39033
\(303\) −6.01673e13 −1.35340
\(304\) −4.19516e13 −0.926710
\(305\) −1.74468e12 −0.0378500
\(306\) 1.56121e13 0.332654
\(307\) 2.05380e13 0.429829 0.214915 0.976633i \(-0.431053\pi\)
0.214915 + 0.976633i \(0.431053\pi\)
\(308\) 2.69305e13 0.553626
\(309\) 4.76488e13 0.962234
\(310\) 7.83389e13 1.55413
\(311\) −7.37156e13 −1.43674 −0.718369 0.695663i \(-0.755111\pi\)
−0.718369 + 0.695663i \(0.755111\pi\)
\(312\) 2.48468e14 4.75795
\(313\) −2.82026e13 −0.530634 −0.265317 0.964161i \(-0.585477\pi\)
−0.265317 + 0.964161i \(0.585477\pi\)
\(314\) −8.15120e13 −1.50698
\(315\) −3.79803e12 −0.0690003
\(316\) 6.20108e13 1.10710
\(317\) 2.56938e13 0.450819 0.225410 0.974264i \(-0.427628\pi\)
0.225410 + 0.974264i \(0.427628\pi\)
\(318\) 2.17367e14 3.74840
\(319\) −3.19763e12 −0.0541976
\(320\) −2.84904e13 −0.474651
\(321\) −7.68771e13 −1.25898
\(322\) −3.83179e13 −0.616872
\(323\) 1.00109e13 0.158439
\(324\) −1.93890e14 −3.01688
\(325\) 1.97586e13 0.302272
\(326\) −1.05569e14 −1.58797
\(327\) −1.43458e14 −2.12184
\(328\) −4.48571e13 −0.652417
\(329\) −3.14824e12 −0.0450288
\(330\) −4.47322e13 −0.629207
\(331\) −6.14309e13 −0.849831 −0.424916 0.905233i \(-0.639696\pi\)
−0.424916 + 0.905233i \(0.639696\pi\)
\(332\) 1.67258e14 2.27576
\(333\) −3.47925e13 −0.465632
\(334\) 2.52358e14 3.32210
\(335\) −3.16269e13 −0.409552
\(336\) −8.25499e13 −1.05160
\(337\) 1.35566e14 1.69898 0.849488 0.527608i \(-0.176911\pi\)
0.849488 + 0.527608i \(0.176911\pi\)
\(338\) −1.92703e14 −2.37600
\(339\) 1.18540e14 1.43803
\(340\) 3.80046e13 0.453631
\(341\) 1.01715e14 1.19463
\(342\) 2.60242e13 0.300770
\(343\) −5.90945e13 −0.672093
\(344\) −1.60285e14 −1.79400
\(345\) 4.50537e13 0.496279
\(346\) 2.57452e13 0.279111
\(347\) −2.88020e13 −0.307334 −0.153667 0.988123i \(-0.549108\pi\)
−0.153667 + 0.988123i \(0.549108\pi\)
\(348\) 2.35052e13 0.246876
\(349\) 3.70742e13 0.383294 0.191647 0.981464i \(-0.438617\pi\)
0.191647 + 0.981464i \(0.438617\pi\)
\(350\) −1.30612e13 −0.132925
\(351\) 1.02896e14 1.03089
\(352\) −1.22329e14 −1.20655
\(353\) −5.64627e13 −0.548278 −0.274139 0.961690i \(-0.588393\pi\)
−0.274139 + 0.961690i \(0.588393\pi\)
\(354\) −2.79014e13 −0.266753
\(355\) −5.56754e13 −0.524092
\(356\) 1.46040e14 1.35362
\(357\) 1.96989e13 0.179791
\(358\) 1.60316e14 1.44086
\(359\) −1.96868e14 −1.74243 −0.871217 0.490899i \(-0.836668\pi\)
−0.871217 + 0.490899i \(0.836668\pi\)
\(360\) 5.80241e13 0.505760
\(361\) −9.98027e13 −0.856747
\(362\) −9.23537e12 −0.0780831
\(363\) 8.54954e13 0.711961
\(364\) 1.60387e14 1.31556
\(365\) 7.29313e13 0.589255
\(366\) 2.35235e13 0.187222
\(367\) −1.00444e14 −0.787514 −0.393757 0.919214i \(-0.628825\pi\)
−0.393757 + 0.919214i \(0.628825\pi\)
\(368\) 2.94221e14 2.27253
\(369\) 1.39856e13 0.106423
\(370\) −1.19649e14 −0.897014
\(371\) 8.24065e13 0.608703
\(372\) −7.47684e14 −5.44167
\(373\) −7.09849e13 −0.509058 −0.254529 0.967065i \(-0.581921\pi\)
−0.254529 + 0.967065i \(0.581921\pi\)
\(374\) 6.97090e13 0.492600
\(375\) 1.53571e13 0.106940
\(376\) 4.80970e13 0.330054
\(377\) −1.90437e13 −0.128788
\(378\) −6.80182e13 −0.453336
\(379\) 6.79976e13 0.446661 0.223330 0.974743i \(-0.428307\pi\)
0.223330 + 0.974743i \(0.428307\pi\)
\(380\) 6.33511e13 0.410152
\(381\) 6.32982e13 0.403928
\(382\) 3.71750e14 2.33831
\(383\) 1.89007e14 1.17188 0.585942 0.810353i \(-0.300725\pi\)
0.585942 + 0.810353i \(0.300725\pi\)
\(384\) 1.30413e13 0.0797076
\(385\) −1.69585e13 −0.102177
\(386\) −2.64344e14 −1.57014
\(387\) 4.99741e13 0.292641
\(388\) −5.62070e14 −3.24501
\(389\) −2.90163e14 −1.65165 −0.825826 0.563924i \(-0.809291\pi\)
−0.825826 + 0.563924i \(0.809291\pi\)
\(390\) −2.66406e14 −1.49516
\(391\) −7.02100e13 −0.388532
\(392\) 4.20271e14 2.29328
\(393\) 1.49632e14 0.805127
\(394\) 3.00275e14 1.59327
\(395\) −3.90489e13 −0.204327
\(396\) 1.28276e14 0.661945
\(397\) −5.15115e13 −0.262154 −0.131077 0.991372i \(-0.541843\pi\)
−0.131077 + 0.991372i \(0.541843\pi\)
\(398\) −3.47328e14 −1.74334
\(399\) 3.28368e13 0.162559
\(400\) 1.00289e14 0.489691
\(401\) −1.11012e14 −0.534657 −0.267329 0.963605i \(-0.586141\pi\)
−0.267329 + 0.963605i \(0.586141\pi\)
\(402\) 4.26426e14 2.02582
\(403\) 6.05768e14 2.83876
\(404\) 5.93346e14 2.74291
\(405\) 1.22095e14 0.556795
\(406\) 1.25886e13 0.0566349
\(407\) −1.55351e14 −0.689517
\(408\) −3.00949e14 −1.31784
\(409\) −1.92951e14 −0.833623 −0.416811 0.908993i \(-0.636852\pi\)
−0.416811 + 0.908993i \(0.636852\pi\)
\(410\) 4.80955e13 0.205019
\(411\) 3.16629e13 0.133174
\(412\) −4.69894e14 −1.95014
\(413\) −1.05778e13 −0.0433180
\(414\) −1.82517e14 −0.737565
\(415\) −1.05324e14 −0.420013
\(416\) −7.28538e14 −2.86707
\(417\) 2.57356e14 0.999508
\(418\) 1.16200e14 0.445386
\(419\) 2.28750e14 0.865336 0.432668 0.901553i \(-0.357572\pi\)
0.432668 + 0.901553i \(0.357572\pi\)
\(420\) 1.24659e14 0.465427
\(421\) −4.00332e14 −1.47526 −0.737630 0.675205i \(-0.764055\pi\)
−0.737630 + 0.675205i \(0.764055\pi\)
\(422\) −9.56816e13 −0.348024
\(423\) −1.49958e13 −0.0538389
\(424\) −1.25896e15 −4.46169
\(425\) −2.39320e13 −0.0837220
\(426\) 7.50672e14 2.59237
\(427\) 8.91803e12 0.0304030
\(428\) 7.58132e14 2.55155
\(429\) −3.45899e14 −1.14930
\(430\) 1.71857e14 0.563756
\(431\) 1.29757e14 0.420250 0.210125 0.977675i \(-0.432613\pi\)
0.210125 + 0.977675i \(0.432613\pi\)
\(432\) 5.22271e14 1.67007
\(433\) −3.09354e14 −0.976725 −0.488363 0.872641i \(-0.662406\pi\)
−0.488363 + 0.872641i \(0.662406\pi\)
\(434\) −4.00435e14 −1.24835
\(435\) −1.48015e13 −0.0455633
\(436\) 1.41473e15 4.30028
\(437\) −1.17035e14 −0.351293
\(438\) −9.83334e14 −2.91470
\(439\) −6.15652e13 −0.180211 −0.0901054 0.995932i \(-0.528720\pi\)
−0.0901054 + 0.995932i \(0.528720\pi\)
\(440\) 2.59082e14 0.748940
\(441\) −1.31033e14 −0.374083
\(442\) 4.15156e14 1.17055
\(443\) −2.42894e13 −0.0676389 −0.0338194 0.999428i \(-0.510767\pi\)
−0.0338194 + 0.999428i \(0.510767\pi\)
\(444\) 1.14196e15 3.14082
\(445\) −9.19631e13 −0.249824
\(446\) −2.97146e14 −0.797311
\(447\) −3.99636e14 −1.05919
\(448\) 1.45631e14 0.381263
\(449\) 6.65728e14 1.72164 0.860820 0.508910i \(-0.169951\pi\)
0.860820 + 0.508910i \(0.169951\pi\)
\(450\) −6.22131e13 −0.158933
\(451\) 6.24468e13 0.157594
\(452\) −1.16900e15 −2.91442
\(453\) −6.27160e14 −1.54468
\(454\) −3.26999e14 −0.795681
\(455\) −1.00997e14 −0.242800
\(456\) −5.01661e14 −1.19153
\(457\) 6.99716e14 1.64204 0.821018 0.570902i \(-0.193406\pi\)
0.821018 + 0.570902i \(0.193406\pi\)
\(458\) 4.06192e14 0.941827
\(459\) −1.24630e14 −0.285531
\(460\) −4.44302e14 −1.00580
\(461\) 3.43320e14 0.767971 0.383985 0.923339i \(-0.374551\pi\)
0.383985 + 0.923339i \(0.374551\pi\)
\(462\) 2.28652e14 0.505410
\(463\) 3.77708e14 0.825012 0.412506 0.910955i \(-0.364654\pi\)
0.412506 + 0.910955i \(0.364654\pi\)
\(464\) −9.66603e13 −0.208641
\(465\) 4.70826e14 1.00431
\(466\) −8.66368e14 −1.82633
\(467\) −5.01733e14 −1.04527 −0.522637 0.852555i \(-0.675052\pi\)
−0.522637 + 0.852555i \(0.675052\pi\)
\(468\) 7.63957e14 1.57296
\(469\) 1.61663e14 0.328972
\(470\) −5.15693e13 −0.103718
\(471\) −4.89897e14 −0.973844
\(472\) 1.61601e14 0.317513
\(473\) 2.23138e14 0.433348
\(474\) 5.26498e14 1.01068
\(475\) −3.98930e13 −0.0756975
\(476\) −1.94263e14 −0.364378
\(477\) 3.92520e14 0.727798
\(478\) −1.55016e14 −0.284134
\(479\) 9.56649e14 1.73343 0.866717 0.498800i \(-0.166226\pi\)
0.866717 + 0.498800i \(0.166226\pi\)
\(480\) −5.66247e14 −1.01433
\(481\) −9.25204e14 −1.63847
\(482\) −2.28596e14 −0.400231
\(483\) −2.30295e14 −0.398635
\(484\) −8.43122e14 −1.44292
\(485\) 3.53942e14 0.598898
\(486\) −8.91890e14 −1.49215
\(487\) 1.81856e14 0.300828 0.150414 0.988623i \(-0.451939\pi\)
0.150414 + 0.988623i \(0.451939\pi\)
\(488\) −1.36244e14 −0.222848
\(489\) −6.34485e14 −1.02618
\(490\) −4.50612e14 −0.720650
\(491\) 9.28536e14 1.46842 0.734210 0.678922i \(-0.237553\pi\)
0.734210 + 0.678922i \(0.237553\pi\)
\(492\) −4.59035e14 −0.717856
\(493\) 2.30661e13 0.0356711
\(494\) 6.92037e14 1.05835
\(495\) −8.07770e13 −0.122168
\(496\) 3.07470e15 4.59889
\(497\) 2.84589e14 0.420976
\(498\) 1.42009e15 2.07756
\(499\) 1.10604e15 1.60036 0.800178 0.599763i \(-0.204739\pi\)
0.800178 + 0.599763i \(0.204739\pi\)
\(500\) −1.51446e14 −0.216732
\(501\) 1.51670e15 2.14681
\(502\) −9.96177e14 −1.39465
\(503\) −1.07467e15 −1.48816 −0.744081 0.668089i \(-0.767112\pi\)
−0.744081 + 0.668089i \(0.767112\pi\)
\(504\) −2.96594e14 −0.406251
\(505\) −3.73637e14 −0.506230
\(506\) −8.14950e14 −1.09220
\(507\) −1.15817e15 −1.53542
\(508\) −6.24222e14 −0.818632
\(509\) −1.50116e15 −1.94750 −0.973752 0.227610i \(-0.926909\pi\)
−0.973752 + 0.227610i \(0.926909\pi\)
\(510\) 3.22675e14 0.414123
\(511\) −3.72793e14 −0.473318
\(512\) 1.43474e15 1.80214
\(513\) −2.07750e14 −0.258163
\(514\) −2.96299e12 −0.00364278
\(515\) 2.95898e14 0.359917
\(516\) −1.64024e15 −1.97395
\(517\) −6.69571e13 −0.0797258
\(518\) 6.11593e14 0.720525
\(519\) 1.54732e14 0.180368
\(520\) 1.54298e15 1.77968
\(521\) −8.31060e13 −0.0948473 −0.0474237 0.998875i \(-0.515101\pi\)
−0.0474237 + 0.998875i \(0.515101\pi\)
\(522\) 5.99622e13 0.0677158
\(523\) −9.42223e14 −1.05292 −0.526459 0.850201i \(-0.676481\pi\)
−0.526459 + 0.850201i \(0.676481\pi\)
\(524\) −1.47561e15 −1.63173
\(525\) −7.84991e13 −0.0858990
\(526\) 1.00394e15 1.08715
\(527\) −7.33717e14 −0.786267
\(528\) −1.75568e15 −1.86191
\(529\) −1.32002e14 −0.138540
\(530\) 1.34985e15 1.40206
\(531\) −5.03841e13 −0.0517933
\(532\) −3.23824e14 −0.329454
\(533\) 3.71906e14 0.374485
\(534\) 1.23994e15 1.23573
\(535\) −4.77405e14 −0.470913
\(536\) −2.46979e15 −2.41131
\(537\) 9.63519e14 0.931112
\(538\) 3.19133e15 3.05259
\(539\) −5.85071e14 −0.553950
\(540\) −7.88682e14 −0.739156
\(541\) 1.18229e15 1.09683 0.548416 0.836206i \(-0.315231\pi\)
0.548416 + 0.836206i \(0.315231\pi\)
\(542\) 9.50410e14 0.872800
\(543\) −5.55056e13 −0.0504589
\(544\) 8.82418e14 0.794110
\(545\) −8.90870e14 −0.793658
\(546\) 1.36175e15 1.20099
\(547\) 1.57701e15 1.37691 0.688453 0.725281i \(-0.258290\pi\)
0.688453 + 0.725281i \(0.258290\pi\)
\(548\) −3.12247e14 −0.269902
\(549\) 4.24785e13 0.0363514
\(550\) −2.77786e14 −0.235351
\(551\) 3.84496e13 0.0322521
\(552\) 3.51831e15 2.92193
\(553\) 1.99601e14 0.164125
\(554\) −2.53309e15 −2.06228
\(555\) −7.19104e14 −0.579669
\(556\) −2.53795e15 −2.02568
\(557\) 4.53070e14 0.358065 0.179033 0.983843i \(-0.442703\pi\)
0.179033 + 0.983843i \(0.442703\pi\)
\(558\) −1.90736e15 −1.49260
\(559\) 1.32891e15 1.02975
\(560\) −5.12633e14 −0.393344
\(561\) 4.18959e14 0.318329
\(562\) −3.95241e15 −2.97381
\(563\) 4.23450e13 0.0315505 0.0157752 0.999876i \(-0.494978\pi\)
0.0157752 + 0.999876i \(0.494978\pi\)
\(564\) 4.92189e14 0.363159
\(565\) 7.36133e14 0.537885
\(566\) 7.27444e14 0.526392
\(567\) −6.24097e14 −0.447245
\(568\) −4.34777e15 −3.08568
\(569\) −9.72594e14 −0.683619 −0.341810 0.939769i \(-0.611040\pi\)
−0.341810 + 0.939769i \(0.611040\pi\)
\(570\) 5.37878e14 0.374431
\(571\) −2.07663e15 −1.43173 −0.715863 0.698241i \(-0.753966\pi\)
−0.715863 + 0.698241i \(0.753966\pi\)
\(572\) 3.41112e15 2.32927
\(573\) 2.23426e15 1.51107
\(574\) −2.45844e14 −0.164681
\(575\) 2.79783e14 0.185630
\(576\) 6.93671e14 0.455858
\(577\) 2.62999e15 1.71194 0.855968 0.517028i \(-0.172962\pi\)
0.855968 + 0.517028i \(0.172962\pi\)
\(578\) 2.36671e15 1.52596
\(579\) −1.58874e15 −1.01466
\(580\) 1.45967e14 0.0923421
\(581\) 5.38371e14 0.337375
\(582\) −4.77221e15 −2.96240
\(583\) 1.75263e15 1.07774
\(584\) 5.69531e15 3.46934
\(585\) −4.81073e14 −0.290304
\(586\) −1.82420e15 −1.09052
\(587\) −2.89275e15 −1.71317 −0.856587 0.516003i \(-0.827419\pi\)
−0.856587 + 0.516003i \(0.827419\pi\)
\(588\) 4.30075e15 2.52330
\(589\) −1.22306e15 −0.710906
\(590\) −1.73267e14 −0.0997770
\(591\) 1.80469e15 1.02960
\(592\) −4.69606e15 −2.65439
\(593\) −1.15734e15 −0.648125 −0.324063 0.946036i \(-0.605049\pi\)
−0.324063 + 0.946036i \(0.605049\pi\)
\(594\) −1.44662e15 −0.802654
\(595\) 1.22330e14 0.0672496
\(596\) 3.94106e15 2.14663
\(597\) −2.08748e15 −1.12659
\(598\) −4.85349e15 −2.59536
\(599\) 2.42056e14 0.128253 0.0641265 0.997942i \(-0.479574\pi\)
0.0641265 + 0.997942i \(0.479574\pi\)
\(600\) 1.19926e15 0.629625
\(601\) −3.08465e15 −1.60471 −0.802354 0.596848i \(-0.796419\pi\)
−0.802354 + 0.596848i \(0.796419\pi\)
\(602\) −8.78460e14 −0.452836
\(603\) 7.70035e14 0.393337
\(604\) 6.18480e15 3.13057
\(605\) 5.30924e14 0.266304
\(606\) 5.03776e15 2.50402
\(607\) 2.71223e15 1.33595 0.667973 0.744186i \(-0.267162\pi\)
0.667973 + 0.744186i \(0.267162\pi\)
\(608\) 1.47093e15 0.717997
\(609\) 7.56589e13 0.0365987
\(610\) 1.46080e14 0.0700290
\(611\) −3.98768e14 −0.189449
\(612\) −9.25318e14 −0.435671
\(613\) −1.80466e15 −0.842097 −0.421049 0.907038i \(-0.638338\pi\)
−0.421049 + 0.907038i \(0.638338\pi\)
\(614\) −1.71963e15 −0.795258
\(615\) 2.89060e14 0.132487
\(616\) −1.32431e15 −0.601585
\(617\) 4.28183e14 0.192780 0.0963898 0.995344i \(-0.469270\pi\)
0.0963898 + 0.995344i \(0.469270\pi\)
\(618\) −3.98960e15 −1.78030
\(619\) 2.39110e15 1.05754 0.528772 0.848764i \(-0.322653\pi\)
0.528772 + 0.848764i \(0.322653\pi\)
\(620\) −4.64310e15 −2.03542
\(621\) 1.45702e15 0.633083
\(622\) 6.17215e15 2.65821
\(623\) 4.70076e14 0.200671
\(624\) −1.04561e16 −4.42439
\(625\) 9.53674e13 0.0400000
\(626\) 2.36138e15 0.981763
\(627\) 6.98376e14 0.287818
\(628\) 4.83117e15 1.97367
\(629\) 1.12062e15 0.453818
\(630\) 3.18007e14 0.127662
\(631\) 3.26028e15 1.29746 0.648730 0.761019i \(-0.275301\pi\)
0.648730 + 0.761019i \(0.275301\pi\)
\(632\) −3.04939e15 −1.20301
\(633\) −5.75057e14 −0.224901
\(634\) −2.15132e15 −0.834093
\(635\) 3.93080e14 0.151087
\(636\) −1.28832e16 −4.90921
\(637\) −3.48443e15 −1.31633
\(638\) 2.67735e14 0.100275
\(639\) 1.35556e15 0.503342
\(640\) 8.09863e13 0.0298141
\(641\) −2.32104e15 −0.847157 −0.423579 0.905859i \(-0.639226\pi\)
−0.423579 + 0.905859i \(0.639226\pi\)
\(642\) 6.43686e15 2.32933
\(643\) −2.55397e15 −0.916336 −0.458168 0.888866i \(-0.651494\pi\)
−0.458168 + 0.888866i \(0.651494\pi\)
\(644\) 2.27108e15 0.807906
\(645\) 1.03288e15 0.364311
\(646\) −8.38208e14 −0.293139
\(647\) −1.70408e15 −0.590902 −0.295451 0.955358i \(-0.595470\pi\)
−0.295451 + 0.955358i \(0.595470\pi\)
\(648\) 9.53458e15 3.27823
\(649\) −2.24969e14 −0.0766966
\(650\) −1.65437e15 −0.559255
\(651\) −2.40666e15 −0.806713
\(652\) 6.25704e15 2.07973
\(653\) −2.20439e15 −0.726552 −0.363276 0.931682i \(-0.618342\pi\)
−0.363276 + 0.931682i \(0.618342\pi\)
\(654\) 1.20116e16 3.92576
\(655\) 9.29208e14 0.301152
\(656\) 1.88769e15 0.606679
\(657\) −1.77570e15 −0.565925
\(658\) 2.63600e14 0.0833111
\(659\) 1.97759e14 0.0619821 0.0309910 0.999520i \(-0.490134\pi\)
0.0309910 + 0.999520i \(0.490134\pi\)
\(660\) 2.65125e15 0.824061
\(661\) −3.43414e15 −1.05855 −0.529274 0.848451i \(-0.677536\pi\)
−0.529274 + 0.848451i \(0.677536\pi\)
\(662\) 5.14356e15 1.57233
\(663\) 2.49514e15 0.756433
\(664\) −8.22492e15 −2.47290
\(665\) 2.03916e14 0.0608039
\(666\) 2.91315e15 0.861499
\(667\) −2.69660e14 −0.0790905
\(668\) −1.49571e16 −4.35089
\(669\) −1.78588e15 −0.515239
\(670\) 2.64809e15 0.757743
\(671\) 1.89670e14 0.0538299
\(672\) 2.89441e15 0.814760
\(673\) 2.01250e15 0.561892 0.280946 0.959724i \(-0.409352\pi\)
0.280946 + 0.959724i \(0.409352\pi\)
\(674\) −1.13509e16 −3.14340
\(675\) 4.96643e14 0.136418
\(676\) 1.14214e16 3.11181
\(677\) −1.01903e15 −0.275392 −0.137696 0.990475i \(-0.543970\pi\)
−0.137696 + 0.990475i \(0.543970\pi\)
\(678\) −9.92530e15 −2.66060
\(679\) −1.80920e15 −0.481064
\(680\) −1.86888e15 −0.492928
\(681\) −1.96530e15 −0.514186
\(682\) −8.51648e15 −2.21027
\(683\) 2.86885e15 0.738575 0.369287 0.929315i \(-0.379602\pi\)
0.369287 + 0.929315i \(0.379602\pi\)
\(684\) −1.54244e15 −0.393913
\(685\) 1.96626e14 0.0498130
\(686\) 4.94793e15 1.24349
\(687\) 2.44126e15 0.608629
\(688\) 6.74517e15 1.66823
\(689\) 1.04379e16 2.56099
\(690\) −3.77232e15 −0.918201
\(691\) −1.79931e15 −0.434487 −0.217243 0.976117i \(-0.569707\pi\)
−0.217243 + 0.976117i \(0.569707\pi\)
\(692\) −1.52590e15 −0.365547
\(693\) 4.12897e14 0.0981316
\(694\) 2.41157e15 0.568621
\(695\) 1.59817e15 0.373859
\(696\) −1.15587e15 −0.268262
\(697\) −4.50460e14 −0.103723
\(698\) −3.10420e15 −0.709160
\(699\) −5.20697e15 −1.18021
\(700\) 7.74127e14 0.174090
\(701\) 2.99337e15 0.667899 0.333949 0.942591i \(-0.391619\pi\)
0.333949 + 0.942591i \(0.391619\pi\)
\(702\) −8.61543e15 −1.90732
\(703\) 1.86800e15 0.410321
\(704\) 3.09729e15 0.675045
\(705\) −3.09937e14 −0.0670245
\(706\) 4.72758e15 1.01441
\(707\) 1.90987e15 0.406629
\(708\) 1.65370e15 0.349361
\(709\) 2.74187e15 0.574769 0.287384 0.957815i \(-0.407214\pi\)
0.287384 + 0.957815i \(0.407214\pi\)
\(710\) 4.66166e15 0.969660
\(711\) 9.50744e14 0.196237
\(712\) −7.18154e15 −1.47088
\(713\) 8.57770e15 1.74333
\(714\) −1.64938e15 −0.332644
\(715\) −2.14802e15 −0.429889
\(716\) −9.50185e15 −1.88706
\(717\) −9.31666e14 −0.183614
\(718\) 1.64836e16 3.22380
\(719\) −4.38021e15 −0.850131 −0.425065 0.905163i \(-0.639749\pi\)
−0.425065 + 0.905163i \(0.639749\pi\)
\(720\) −2.44178e15 −0.470303
\(721\) −1.51250e15 −0.289103
\(722\) 8.35640e15 1.58513
\(723\) −1.37389e15 −0.258637
\(724\) 5.47375e14 0.102264
\(725\) −9.19170e13 −0.0170426
\(726\) −7.15846e15 −1.31725
\(727\) 6.60205e15 1.20570 0.602851 0.797854i \(-0.294031\pi\)
0.602851 + 0.797854i \(0.294031\pi\)
\(728\) −7.88704e15 −1.42952
\(729\) 1.56084e15 0.280773
\(730\) −6.10648e15 −1.09022
\(731\) −1.60960e15 −0.285216
\(732\) −1.39422e15 −0.245201
\(733\) −8.22795e15 −1.43622 −0.718108 0.695932i \(-0.754992\pi\)
−0.718108 + 0.695932i \(0.754992\pi\)
\(734\) 8.41006e15 1.45704
\(735\) −2.70823e15 −0.465699
\(736\) −1.03161e16 −1.76071
\(737\) 3.43826e15 0.582462
\(738\) −1.17101e15 −0.196902
\(739\) −1.10770e16 −1.84875 −0.924376 0.381483i \(-0.875414\pi\)
−0.924376 + 0.381483i \(0.875414\pi\)
\(740\) 7.09152e15 1.17480
\(741\) 4.15923e15 0.683931
\(742\) −6.89984e15 −1.12620
\(743\) −3.86160e15 −0.625646 −0.312823 0.949811i \(-0.601275\pi\)
−0.312823 + 0.949811i \(0.601275\pi\)
\(744\) 3.67675e16 5.91307
\(745\) −2.48173e15 −0.396182
\(746\) 5.94352e15 0.941846
\(747\) 2.56438e15 0.403384
\(748\) −4.13161e15 −0.645150
\(749\) 2.44029e15 0.378260
\(750\) −1.28584e15 −0.197857
\(751\) 1.05947e15 0.161834 0.0809168 0.996721i \(-0.474215\pi\)
0.0809168 + 0.996721i \(0.474215\pi\)
\(752\) −2.02403e15 −0.306915
\(753\) −5.98714e15 −0.901254
\(754\) 1.59452e15 0.238280
\(755\) −3.89465e15 −0.577776
\(756\) 4.03140e15 0.593726
\(757\) −7.04375e15 −1.02986 −0.514928 0.857233i \(-0.672182\pi\)
−0.514928 + 0.857233i \(0.672182\pi\)
\(758\) −5.69339e15 −0.826400
\(759\) −4.89794e15 −0.705804
\(760\) −3.11530e15 −0.445682
\(761\) 3.63598e15 0.516424 0.258212 0.966088i \(-0.416867\pi\)
0.258212 + 0.966088i \(0.416867\pi\)
\(762\) −5.29991e15 −0.747337
\(763\) 4.55374e15 0.637505
\(764\) −2.20334e16 −3.06245
\(765\) 5.82684e14 0.0804072
\(766\) −1.58254e16 −2.16819
\(767\) −1.33982e15 −0.182251
\(768\) 8.30398e15 1.12150
\(769\) −5.34829e15 −0.717167 −0.358583 0.933498i \(-0.616740\pi\)
−0.358583 + 0.933498i \(0.616740\pi\)
\(770\) 1.41992e15 0.189045
\(771\) −1.78079e13 −0.00235404
\(772\) 1.56675e16 2.05639
\(773\) 8.06669e15 1.05126 0.525628 0.850715i \(-0.323831\pi\)
0.525628 + 0.850715i \(0.323831\pi\)
\(774\) −4.18429e15 −0.541436
\(775\) 2.92382e15 0.375656
\(776\) 2.76399e16 3.52612
\(777\) 3.67575e15 0.465618
\(778\) 2.42951e16 3.05584
\(779\) −7.50886e14 −0.0937816
\(780\) 1.57897e16 1.95819
\(781\) 6.05266e15 0.745359
\(782\) 5.87863e15 0.718851
\(783\) −4.78674e14 −0.0581233
\(784\) −1.76859e16 −2.13250
\(785\) −3.04225e15 −0.364260
\(786\) −1.25285e16 −1.48962
\(787\) −8.56068e15 −1.01076 −0.505379 0.862897i \(-0.668647\pi\)
−0.505379 + 0.862897i \(0.668647\pi\)
\(788\) −1.77971e16 −2.08668
\(789\) 6.03383e15 0.702537
\(790\) 3.26954e15 0.378040
\(791\) −3.76279e15 −0.432056
\(792\) −6.30799e15 −0.719288
\(793\) 1.12959e15 0.127914
\(794\) 4.31302e15 0.485030
\(795\) 8.11274e15 0.906042
\(796\) 2.05859e16 2.28323
\(797\) −1.47001e16 −1.61919 −0.809596 0.586987i \(-0.800314\pi\)
−0.809596 + 0.586987i \(0.800314\pi\)
\(798\) −2.74940e15 −0.300761
\(799\) 4.82995e14 0.0524729
\(800\) −3.51638e15 −0.379403
\(801\) 2.23907e15 0.239933
\(802\) 9.29494e15 0.989208
\(803\) −7.92861e15 −0.838033
\(804\) −2.52740e16 −2.65317
\(805\) −1.43013e15 −0.149107
\(806\) −5.07205e16 −5.25219
\(807\) 1.91803e16 1.97265
\(808\) −2.91779e16 −2.98052
\(809\) 5.20792e15 0.528382 0.264191 0.964470i \(-0.414895\pi\)
0.264191 + 0.964470i \(0.414895\pi\)
\(810\) −1.02229e16 −1.03017
\(811\) 1.95571e16 1.95745 0.978724 0.205183i \(-0.0657789\pi\)
0.978724 + 0.205183i \(0.0657789\pi\)
\(812\) −7.46119e14 −0.0741737
\(813\) 5.71207e15 0.564022
\(814\) 1.30074e16 1.27573
\(815\) −3.94014e15 −0.383835
\(816\) 1.26646e16 1.22545
\(817\) −2.68310e15 −0.257879
\(818\) 1.61557e16 1.54235
\(819\) 2.45904e15 0.233187
\(820\) −2.85059e15 −0.268509
\(821\) 1.02440e16 0.958476 0.479238 0.877685i \(-0.340913\pi\)
0.479238 + 0.877685i \(0.340913\pi\)
\(822\) −2.65111e15 −0.246396
\(823\) −1.40372e16 −1.29593 −0.647964 0.761671i \(-0.724379\pi\)
−0.647964 + 0.761671i \(0.724379\pi\)
\(824\) 2.31071e16 2.11907
\(825\) −1.66953e15 −0.152089
\(826\) 8.85667e14 0.0801457
\(827\) 8.80160e14 0.0791191 0.0395596 0.999217i \(-0.487405\pi\)
0.0395596 + 0.999217i \(0.487405\pi\)
\(828\) 1.08177e16 0.965976
\(829\) 4.54642e14 0.0403292 0.0201646 0.999797i \(-0.493581\pi\)
0.0201646 + 0.999797i \(0.493581\pi\)
\(830\) 8.81871e15 0.777097
\(831\) −1.52242e16 −1.33269
\(832\) 1.84461e16 1.60408
\(833\) 4.22041e15 0.364592
\(834\) −2.15482e16 −1.84926
\(835\) 9.41870e15 0.802999
\(836\) −6.88711e15 −0.583314
\(837\) 1.52263e16 1.28116
\(838\) −1.91531e16 −1.60102
\(839\) 1.32346e16 1.09906 0.549529 0.835474i \(-0.314807\pi\)
0.549529 + 0.835474i \(0.314807\pi\)
\(840\) −6.13011e15 −0.505746
\(841\) −1.21119e16 −0.992739
\(842\) 3.35195e16 2.72949
\(843\) −2.37545e16 −1.92174
\(844\) 5.67099e15 0.455801
\(845\) −7.19220e15 −0.574315
\(846\) 1.25558e15 0.0996113
\(847\) −2.71386e15 −0.213908
\(848\) 5.29797e16 4.14889
\(849\) 4.37202e15 0.340166
\(850\) 2.00381e15 0.154900
\(851\) −1.31009e16 −1.00621
\(852\) −4.44919e16 −3.39519
\(853\) −2.60074e15 −0.197186 −0.0985932 0.995128i \(-0.531434\pi\)
−0.0985932 + 0.995128i \(0.531434\pi\)
\(854\) −7.46700e14 −0.0562507
\(855\) 9.71294e14 0.0727005
\(856\) −3.72812e16 −2.77258
\(857\) −1.68895e16 −1.24802 −0.624011 0.781415i \(-0.714498\pi\)
−0.624011 + 0.781415i \(0.714498\pi\)
\(858\) 2.89619e16 2.12641
\(859\) 8.98241e15 0.655285 0.327643 0.944802i \(-0.393746\pi\)
0.327643 + 0.944802i \(0.393746\pi\)
\(860\) −1.01859e16 −0.738341
\(861\) −1.47755e15 −0.106420
\(862\) −1.08645e16 −0.777534
\(863\) −1.16178e16 −0.826164 −0.413082 0.910694i \(-0.635548\pi\)
−0.413082 + 0.910694i \(0.635548\pi\)
\(864\) −1.83122e16 −1.29394
\(865\) 9.60880e14 0.0674653
\(866\) 2.59020e16 1.80711
\(867\) 1.42242e16 0.986107
\(868\) 2.37335e16 1.63495
\(869\) 4.24514e15 0.290592
\(870\) 1.23932e15 0.0842999
\(871\) 2.04768e16 1.38408
\(872\) −6.95694e16 −4.67280
\(873\) −8.61761e15 −0.575186
\(874\) 9.79928e15 0.649952
\(875\) −4.87477e14 −0.0321300
\(876\) 5.82817e16 3.81733
\(877\) 4.73202e15 0.307999 0.153999 0.988071i \(-0.450785\pi\)
0.153999 + 0.988071i \(0.450785\pi\)
\(878\) 5.15481e15 0.333421
\(879\) −1.09636e16 −0.704719
\(880\) −1.09027e16 −0.696435
\(881\) 2.81982e15 0.179000 0.0895002 0.995987i \(-0.471473\pi\)
0.0895002 + 0.995987i \(0.471473\pi\)
\(882\) 1.09713e16 0.692118
\(883\) 1.91741e16 1.20208 0.601038 0.799221i \(-0.294754\pi\)
0.601038 + 0.799221i \(0.294754\pi\)
\(884\) −2.46061e16 −1.53305
\(885\) −1.04136e15 −0.0644780
\(886\) 2.03373e15 0.125144
\(887\) 2.08389e15 0.127437 0.0637183 0.997968i \(-0.479704\pi\)
0.0637183 + 0.997968i \(0.479704\pi\)
\(888\) −5.61559e16 −3.41290
\(889\) −2.00926e15 −0.121360
\(890\) 7.70000e15 0.462217
\(891\) −1.32734e16 −0.791869
\(892\) 1.76117e16 1.04422
\(893\) 8.05119e14 0.0474435
\(894\) 3.34612e16 1.95968
\(895\) 5.98343e15 0.348276
\(896\) −4.13967e14 −0.0239481
\(897\) −2.91700e16 −1.67718
\(898\) −5.57409e16 −3.18533
\(899\) −2.81803e15 −0.160054
\(900\) 3.68733e15 0.208151
\(901\) −1.26426e16 −0.709332
\(902\) −5.22863e15 −0.291576
\(903\) −5.27965e15 −0.292632
\(904\) 5.74857e16 3.16689
\(905\) −3.44689e14 −0.0188738
\(906\) 5.25116e16 2.85792
\(907\) 3.20443e16 1.73345 0.866723 0.498791i \(-0.166222\pi\)
0.866723 + 0.498791i \(0.166222\pi\)
\(908\) 1.93810e16 1.04209
\(909\) 9.09714e15 0.486187
\(910\) 8.45644e15 0.449221
\(911\) −8.44282e15 −0.445797 −0.222898 0.974842i \(-0.571552\pi\)
−0.222898 + 0.974842i \(0.571552\pi\)
\(912\) 2.11110e16 1.10799
\(913\) 1.14501e16 0.597340
\(914\) −5.85867e16 −3.03805
\(915\) 8.77960e14 0.0452542
\(916\) −2.40748e16 −1.23349
\(917\) −4.74971e15 −0.241900
\(918\) 1.04352e16 0.528281
\(919\) −1.08277e16 −0.544880 −0.272440 0.962173i \(-0.587831\pi\)
−0.272440 + 0.962173i \(0.587831\pi\)
\(920\) 2.18486e16 1.09293
\(921\) −1.03352e16 −0.513913
\(922\) −2.87460e16 −1.42088
\(923\) 3.60470e16 1.77117
\(924\) −1.35521e16 −0.661926
\(925\) −4.46562e15 −0.216821
\(926\) −3.16252e16 −1.52641
\(927\) −7.20438e15 −0.345667
\(928\) 3.38916e15 0.161651
\(929\) −1.30338e16 −0.617994 −0.308997 0.951063i \(-0.599993\pi\)
−0.308997 + 0.951063i \(0.599993\pi\)
\(930\) −3.94219e16 −1.85815
\(931\) 7.03513e15 0.329647
\(932\) 5.13491e16 2.39191
\(933\) 3.70954e16 1.71779
\(934\) 4.20098e16 1.93394
\(935\) 2.60173e15 0.119069
\(936\) −3.75677e16 −1.70922
\(937\) −2.06679e16 −0.934819 −0.467410 0.884041i \(-0.654813\pi\)
−0.467410 + 0.884041i \(0.654813\pi\)
\(938\) −1.35359e16 −0.608656
\(939\) 1.41922e16 0.634436
\(940\) 3.05648e15 0.135837
\(941\) −3.48692e16 −1.54063 −0.770317 0.637661i \(-0.779902\pi\)
−0.770317 + 0.637661i \(0.779902\pi\)
\(942\) 4.10187e16 1.80178
\(943\) 5.26621e15 0.229977
\(944\) −6.80051e15 −0.295254
\(945\) −2.53862e15 −0.109578
\(946\) −1.86832e16 −0.801769
\(947\) −2.53080e15 −0.107977 −0.0539887 0.998542i \(-0.517193\pi\)
−0.0539887 + 0.998542i \(0.517193\pi\)
\(948\) −3.12052e16 −1.32368
\(949\) −4.72194e16 −1.99139
\(950\) 3.34021e15 0.140053
\(951\) −1.29297e16 −0.539008
\(952\) 9.55293e15 0.395944
\(953\) −1.68411e16 −0.693999 −0.347000 0.937865i \(-0.612799\pi\)
−0.347000 + 0.937865i \(0.612799\pi\)
\(954\) −3.28654e16 −1.34655
\(955\) 1.38747e16 0.565205
\(956\) 9.18773e15 0.372126
\(957\) 1.60912e15 0.0647997
\(958\) −8.00995e16 −3.20715
\(959\) −1.00507e15 −0.0400122
\(960\) 1.43370e16 0.567502
\(961\) 6.42312e16 2.52794
\(962\) 7.74666e16 3.03146
\(963\) 1.16236e16 0.452269
\(964\) 1.35488e16 0.524175
\(965\) −9.86603e15 −0.379527
\(966\) 1.92825e16 0.737544
\(967\) −2.82761e16 −1.07541 −0.537705 0.843133i \(-0.680708\pi\)
−0.537705 + 0.843133i \(0.680708\pi\)
\(968\) 4.14607e16 1.56791
\(969\) −5.03773e15 −0.189432
\(970\) −2.96353e16 −1.10806
\(971\) 2.55089e16 0.948386 0.474193 0.880421i \(-0.342740\pi\)
0.474193 + 0.880421i \(0.342740\pi\)
\(972\) 5.28618e16 1.95424
\(973\) −8.16918e15 −0.300302
\(974\) −1.52266e16 −0.556583
\(975\) −9.94298e15 −0.361402
\(976\) 5.73346e15 0.207225
\(977\) 5.16659e16 1.85688 0.928441 0.371480i \(-0.121150\pi\)
0.928441 + 0.371480i \(0.121150\pi\)
\(978\) 5.31249e16 1.89861
\(979\) 9.99762e15 0.355297
\(980\) 2.67075e16 0.943823
\(981\) 2.16905e16 0.762236
\(982\) −7.77456e16 −2.71683
\(983\) 1.06975e15 0.0371740 0.0185870 0.999827i \(-0.494083\pi\)
0.0185870 + 0.999827i \(0.494083\pi\)
\(984\) 2.25731e16 0.780043
\(985\) 1.12071e16 0.385117
\(986\) −1.93131e15 −0.0659976
\(987\) 1.58427e15 0.0538374
\(988\) −4.10167e16 −1.38611
\(989\) 1.88175e16 0.632385
\(990\) 6.76339e15 0.226033
\(991\) 2.05333e16 0.682423 0.341211 0.939987i \(-0.389163\pi\)
0.341211 + 0.939987i \(0.389163\pi\)
\(992\) −1.07807e17 −3.56313
\(993\) 3.09134e16 1.01608
\(994\) −2.38284e16 −0.778878
\(995\) −1.29632e16 −0.421391
\(996\) −8.41678e16 −2.72094
\(997\) −1.61931e16 −0.520603 −0.260301 0.965527i \(-0.583822\pi\)
−0.260301 + 0.965527i \(0.583822\pi\)
\(998\) −9.26076e16 −2.96093
\(999\) −2.32555e16 −0.739461
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 5.12.a.b.1.1 2
3.2 odd 2 45.12.a.d.1.2 2
4.3 odd 2 80.12.a.j.1.2 2
5.2 odd 4 25.12.b.c.24.1 4
5.3 odd 4 25.12.b.c.24.4 4
5.4 even 2 25.12.a.c.1.2 2
7.6 odd 2 245.12.a.b.1.1 2
15.2 even 4 225.12.b.f.199.4 4
15.8 even 4 225.12.b.f.199.1 4
15.14 odd 2 225.12.a.h.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5.12.a.b.1.1 2 1.1 even 1 trivial
25.12.a.c.1.2 2 5.4 even 2
25.12.b.c.24.1 4 5.2 odd 4
25.12.b.c.24.4 4 5.3 odd 4
45.12.a.d.1.2 2 3.2 odd 2
80.12.a.j.1.2 2 4.3 odd 2
225.12.a.h.1.1 2 15.14 odd 2
225.12.b.f.199.1 4 15.8 even 4
225.12.b.f.199.4 4 15.2 even 4
245.12.a.b.1.1 2 7.6 odd 2