Properties

Label 1.20.a.a.1.1
Level $1$
Weight $20$
Character 1.1
Self dual yes
Analytic conductor $2.288$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1 \)
Weight: \( k \) \(=\) \( 20 \)
Character orbit: \([\chi]\) \(=\) 1.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.28816696556\)
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 \(\chi\) \(=\) 1.1

$q$-expansion

\(f(q)\) \(=\) \(q+456.000 q^{2} +50652.0 q^{3} -316352. q^{4} -2.37741e6 q^{5} +2.30973e7 q^{6} -1.69175e7 q^{7} -3.83332e8 q^{8} +1.40336e9 q^{9} +O(q^{10})\) \(q+456.000 q^{2} +50652.0 q^{3} -316352. q^{4} -2.37741e6 q^{5} +2.30973e7 q^{6} -1.69175e7 q^{7} -3.83332e8 q^{8} +1.40336e9 q^{9} -1.08410e9 q^{10} -1.62121e7 q^{11} -1.60239e10 q^{12} +5.04216e10 q^{13} -7.71440e9 q^{14} -1.20421e11 q^{15} -8.93976e9 q^{16} +2.25070e11 q^{17} +6.39934e11 q^{18} -1.71028e12 q^{19} +7.52098e11 q^{20} -8.56907e11 q^{21} -7.39272e9 q^{22} +1.40365e13 q^{23} -1.94165e13 q^{24} -1.34214e13 q^{25} +2.29923e13 q^{26} +1.22123e13 q^{27} +5.35190e12 q^{28} +1.13784e12 q^{29} -5.49118e13 q^{30} -1.04627e14 q^{31} +1.96900e14 q^{32} -8.21176e11 q^{33} +1.02632e14 q^{34} +4.02199e13 q^{35} -4.43957e14 q^{36} -1.69392e14 q^{37} -7.79887e14 q^{38} +2.55396e15 q^{39} +9.11337e14 q^{40} -3.30998e15 q^{41} -3.90750e14 q^{42} +1.12791e15 q^{43} +5.12873e12 q^{44} -3.33637e15 q^{45} +6.40066e15 q^{46} +3.49869e15 q^{47} -4.52817e14 q^{48} -1.11127e16 q^{49} -6.12016e15 q^{50} +1.14003e16 q^{51} -1.59510e16 q^{52} +2.99563e16 q^{53} +5.56881e15 q^{54} +3.85428e13 q^{55} +6.48503e15 q^{56} -8.66290e16 q^{57} +5.18853e14 q^{58} +5.83914e16 q^{59} +3.80953e16 q^{60} +2.33737e16 q^{61} -4.77099e16 q^{62} -2.37415e16 q^{63} +9.44733e16 q^{64} -1.19873e17 q^{65} -3.74456e14 q^{66} -2.05103e17 q^{67} -7.12014e16 q^{68} +7.10979e17 q^{69} +1.83403e16 q^{70} -1.77902e17 q^{71} -5.37954e17 q^{72} +2.99854e17 q^{73} -7.72429e16 q^{74} -6.79821e17 q^{75} +5.41050e17 q^{76} +2.74269e14 q^{77} +1.16460e18 q^{78} -9.22271e16 q^{79} +2.12535e16 q^{80} -1.01250e18 q^{81} -1.50935e18 q^{82} +1.20854e18 q^{83} +2.71084e17 q^{84} -5.35084e17 q^{85} +5.14329e17 q^{86} +5.76336e16 q^{87} +6.21462e15 q^{88} +4.37120e18 q^{89} -1.52139e18 q^{90} -8.53010e17 q^{91} -4.44049e18 q^{92} -5.29956e18 q^{93} +1.59540e18 q^{94} +4.06603e18 q^{95} +9.97337e18 q^{96} -6.35013e17 q^{97} -5.06739e18 q^{98} -2.27515e16 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 456.000 0.629767 0.314883 0.949130i \(-0.398035\pi\)
0.314883 + 0.949130i \(0.398035\pi\)
\(3\) 50652.0 1.48575 0.742873 0.669432i \(-0.233463\pi\)
0.742873 + 0.669432i \(0.233463\pi\)
\(4\) −316352. −0.603394
\(5\) −2.37741e6 −0.544364 −0.272182 0.962246i \(-0.587745\pi\)
−0.272182 + 0.962246i \(0.587745\pi\)
\(6\) 2.30973e7 0.935674
\(7\) −1.69175e7 −0.158455 −0.0792275 0.996857i \(-0.525245\pi\)
−0.0792275 + 0.996857i \(0.525245\pi\)
\(8\) −3.83332e8 −1.00976
\(9\) 1.40336e9 1.20744
\(10\) −1.08410e9 −0.342822
\(11\) −1.62121e7 −0.00207305 −0.00103652 0.999999i \(-0.500330\pi\)
−0.00103652 + 0.999999i \(0.500330\pi\)
\(12\) −1.60239e10 −0.896490
\(13\) 5.04216e10 1.31873 0.659364 0.751824i \(-0.270826\pi\)
0.659364 + 0.751824i \(0.270826\pi\)
\(14\) −7.71440e9 −0.0997897
\(15\) −1.20421e11 −0.808786
\(16\) −8.93976e9 −0.0325227
\(17\) 2.25070e11 0.460313 0.230156 0.973154i \(-0.426076\pi\)
0.230156 + 0.973154i \(0.426076\pi\)
\(18\) 6.39934e11 0.760407
\(19\) −1.71028e12 −1.21593 −0.607964 0.793965i \(-0.708013\pi\)
−0.607964 + 0.793965i \(0.708013\pi\)
\(20\) 7.52098e11 0.328465
\(21\) −8.56907e11 −0.235424
\(22\) −7.39272e9 −0.00130554
\(23\) 1.40365e13 1.62497 0.812485 0.582982i \(-0.198114\pi\)
0.812485 + 0.582982i \(0.198114\pi\)
\(24\) −1.94165e13 −1.50025
\(25\) −1.34214e13 −0.703668
\(26\) 2.29923e13 0.830491
\(27\) 1.22123e13 0.308207
\(28\) 5.35190e12 0.0956107
\(29\) 1.13784e12 0.0145646 0.00728230 0.999973i \(-0.497682\pi\)
0.00728230 + 0.999973i \(0.497682\pi\)
\(30\) −5.49118e13 −0.509347
\(31\) −1.04627e14 −0.710734 −0.355367 0.934727i \(-0.615644\pi\)
−0.355367 + 0.934727i \(0.615644\pi\)
\(32\) 1.96900e14 0.989283
\(33\) −8.21176e11 −0.00308002
\(34\) 1.02632e14 0.289890
\(35\) 4.02199e13 0.0862571
\(36\) −4.43957e14 −0.728563
\(37\) −1.69392e14 −0.214278 −0.107139 0.994244i \(-0.534169\pi\)
−0.107139 + 0.994244i \(0.534169\pi\)
\(38\) −7.79887e14 −0.765751
\(39\) 2.55396e15 1.95929
\(40\) 9.11337e14 0.549679
\(41\) −3.30998e15 −1.57899 −0.789495 0.613757i \(-0.789657\pi\)
−0.789495 + 0.613757i \(0.789657\pi\)
\(42\) −3.90750e14 −0.148262
\(43\) 1.12791e15 0.342236 0.171118 0.985251i \(-0.445262\pi\)
0.171118 + 0.985251i \(0.445262\pi\)
\(44\) 5.12873e12 0.00125086
\(45\) −3.33637e15 −0.657288
\(46\) 6.40066e15 1.02335
\(47\) 3.49869e15 0.456012 0.228006 0.973660i \(-0.426779\pi\)
0.228006 + 0.973660i \(0.426779\pi\)
\(48\) −4.52817e14 −0.0483204
\(49\) −1.11127e16 −0.974892
\(50\) −6.12016e15 −0.443147
\(51\) 1.14003e16 0.683908
\(52\) −1.59510e16 −0.795711
\(53\) 2.99563e16 1.24700 0.623501 0.781822i \(-0.285710\pi\)
0.623501 + 0.781822i \(0.285710\pi\)
\(54\) 5.56881e15 0.194098
\(55\) 3.85428e13 0.00112849
\(56\) 6.48503e15 0.160002
\(57\) −8.66290e16 −1.80656
\(58\) 5.18853e14 0.00917230
\(59\) 5.83914e16 0.877515 0.438758 0.898605i \(-0.355419\pi\)
0.438758 + 0.898605i \(0.355419\pi\)
\(60\) 3.80953e16 0.488016
\(61\) 2.33737e16 0.255914 0.127957 0.991780i \(-0.459158\pi\)
0.127957 + 0.991780i \(0.459158\pi\)
\(62\) −4.77099e16 −0.447597
\(63\) −2.37415e16 −0.191325
\(64\) 9.44733e16 0.655540
\(65\) −1.19873e17 −0.717867
\(66\) −3.74456e14 −0.00193970
\(67\) −2.05103e17 −0.921002 −0.460501 0.887659i \(-0.652330\pi\)
−0.460501 + 0.887659i \(0.652330\pi\)
\(68\) −7.12014e16 −0.277750
\(69\) 7.10979e17 2.41429
\(70\) 1.83403e16 0.0543219
\(71\) −1.77902e17 −0.460498 −0.230249 0.973132i \(-0.573954\pi\)
−0.230249 + 0.973132i \(0.573954\pi\)
\(72\) −5.37954e17 −1.21923
\(73\) 2.99854e17 0.596132 0.298066 0.954545i \(-0.403658\pi\)
0.298066 + 0.954545i \(0.403658\pi\)
\(74\) −7.72429e16 −0.134945
\(75\) −6.79821e17 −1.04547
\(76\) 5.41050e17 0.733683
\(77\) 2.74269e14 0.000328485 0
\(78\) 1.16460e18 1.23390
\(79\) −9.22271e16 −0.0865767 −0.0432884 0.999063i \(-0.513783\pi\)
−0.0432884 + 0.999063i \(0.513783\pi\)
\(80\) 2.12535e16 0.0177042
\(81\) −1.01250e18 −0.749525
\(82\) −1.50935e18 −0.994396
\(83\) 1.20854e18 0.709611 0.354805 0.934940i \(-0.384547\pi\)
0.354805 + 0.934940i \(0.384547\pi\)
\(84\) 2.71084e17 0.142053
\(85\) −5.35084e17 −0.250578
\(86\) 5.14329e17 0.215529
\(87\) 5.76336e16 0.0216393
\(88\) 6.21462e15 0.00209329
\(89\) 4.37120e18 1.32250 0.661250 0.750166i \(-0.270026\pi\)
0.661250 + 0.750166i \(0.270026\pi\)
\(90\) −1.52139e18 −0.413938
\(91\) −8.53010e17 −0.208959
\(92\) −4.44049e18 −0.980497
\(93\) −5.29956e18 −1.05597
\(94\) 1.59540e18 0.287181
\(95\) 4.06603e18 0.661907
\(96\) 9.97337e18 1.46982
\(97\) −6.35013e17 −0.0848108 −0.0424054 0.999100i \(-0.513502\pi\)
−0.0424054 + 0.999100i \(0.513502\pi\)
\(98\) −5.06739e18 −0.613955
\(99\) −2.27515e16 −0.00250308
\(100\) 4.24589e18 0.424589
\(101\) −1.42252e19 −1.29421 −0.647105 0.762401i \(-0.724021\pi\)
−0.647105 + 0.762401i \(0.724021\pi\)
\(102\) 5.19851e18 0.430703
\(103\) 4.90729e18 0.370586 0.185293 0.982683i \(-0.440677\pi\)
0.185293 + 0.982683i \(0.440677\pi\)
\(104\) −1.93282e19 −1.33160
\(105\) 2.03722e18 0.128156
\(106\) 1.36601e19 0.785321
\(107\) 2.64625e19 1.39151 0.695753 0.718281i \(-0.255071\pi\)
0.695753 + 0.718281i \(0.255071\pi\)
\(108\) −3.86339e18 −0.185970
\(109\) −1.84178e19 −0.812242 −0.406121 0.913819i \(-0.633119\pi\)
−0.406121 + 0.913819i \(0.633119\pi\)
\(110\) 1.75755e16 0.000710686 0
\(111\) −8.58006e18 −0.318363
\(112\) 1.51239e17 0.00515338
\(113\) 2.57421e19 0.806118 0.403059 0.915174i \(-0.367947\pi\)
0.403059 + 0.915174i \(0.367947\pi\)
\(114\) −3.95028e19 −1.13771
\(115\) −3.33706e19 −0.884575
\(116\) −3.59956e17 −0.00878818
\(117\) 7.07599e19 1.59229
\(118\) 2.66265e19 0.552630
\(119\) −3.80763e18 −0.0729389
\(120\) 4.61610e19 0.816683
\(121\) −6.11588e19 −0.999996
\(122\) 1.06584e19 0.161166
\(123\) −1.67657e20 −2.34598
\(124\) 3.30989e19 0.428852
\(125\) 7.72537e19 0.927415
\(126\) −1.08261e19 −0.120490
\(127\) 8.80720e19 0.909290 0.454645 0.890673i \(-0.349766\pi\)
0.454645 + 0.890673i \(0.349766\pi\)
\(128\) −6.01524e19 −0.576445
\(129\) 5.71311e19 0.508476
\(130\) −5.46620e19 −0.452089
\(131\) 7.19289e19 0.553129 0.276564 0.960995i \(-0.410804\pi\)
0.276564 + 0.960995i \(0.410804\pi\)
\(132\) 2.59781e17 0.00185846
\(133\) 2.89337e19 0.192670
\(134\) −9.35268e19 −0.580016
\(135\) −2.90337e19 −0.167776
\(136\) −8.62765e19 −0.464807
\(137\) −2.95426e20 −1.48458 −0.742290 0.670079i \(-0.766260\pi\)
−0.742290 + 0.670079i \(0.766260\pi\)
\(138\) 3.24206e20 1.52044
\(139\) 1.38478e20 0.606375 0.303187 0.952931i \(-0.401949\pi\)
0.303187 + 0.952931i \(0.401949\pi\)
\(140\) −1.27237e19 −0.0520470
\(141\) 1.77216e20 0.677518
\(142\) −8.11235e19 −0.290006
\(143\) −8.17441e17 −0.00273378
\(144\) −1.25457e19 −0.0392692
\(145\) −2.70510e18 −0.00792843
\(146\) 1.36733e20 0.375424
\(147\) −5.62880e20 −1.44844
\(148\) 5.35876e19 0.129294
\(149\) −2.66021e20 −0.602070 −0.301035 0.953613i \(-0.597332\pi\)
−0.301035 + 0.953613i \(0.597332\pi\)
\(150\) −3.09998e20 −0.658404
\(151\) 5.75578e20 1.14769 0.573844 0.818965i \(-0.305452\pi\)
0.573844 + 0.818965i \(0.305452\pi\)
\(152\) 6.55604e20 1.22780
\(153\) 3.15855e20 0.555801
\(154\) 1.25067e17 0.000206869 0
\(155\) 2.48741e20 0.386898
\(156\) −8.07949e20 −1.18223
\(157\) −1.07238e21 −1.47673 −0.738363 0.674403i \(-0.764401\pi\)
−0.738363 + 0.674403i \(0.764401\pi\)
\(158\) −4.20556e19 −0.0545232
\(159\) 1.51735e21 1.85273
\(160\) −4.68111e20 −0.538529
\(161\) −2.37464e20 −0.257485
\(162\) −4.61699e20 −0.472026
\(163\) −5.80765e20 −0.560039 −0.280019 0.959994i \(-0.590341\pi\)
−0.280019 + 0.959994i \(0.590341\pi\)
\(164\) 1.04712e21 0.952752
\(165\) 1.95227e18 0.00167665
\(166\) 5.51096e20 0.446889
\(167\) 2.43392e20 0.186423 0.0932117 0.995646i \(-0.470287\pi\)
0.0932117 + 0.995646i \(0.470287\pi\)
\(168\) 3.28480e20 0.237723
\(169\) 1.08042e21 0.739041
\(170\) −2.43998e20 −0.157805
\(171\) −2.40014e21 −1.46816
\(172\) −3.56818e20 −0.206503
\(173\) −1.19350e21 −0.653711 −0.326855 0.945074i \(-0.605989\pi\)
−0.326855 + 0.945074i \(0.605989\pi\)
\(174\) 2.62809e19 0.0136277
\(175\) 2.27057e20 0.111500
\(176\) 1.44932e17 6.74210e−5 0
\(177\) 2.95764e21 1.30377
\(178\) 1.99327e21 0.832867
\(179\) −4.14664e21 −1.64283 −0.821415 0.570331i \(-0.806815\pi\)
−0.821415 + 0.570331i \(0.806815\pi\)
\(180\) 1.05547e21 0.396603
\(181\) 3.32364e21 1.18486 0.592430 0.805622i \(-0.298169\pi\)
0.592430 + 0.805622i \(0.298169\pi\)
\(182\) −3.88973e20 −0.131595
\(183\) 1.18392e21 0.380223
\(184\) −5.38065e21 −1.64084
\(185\) 4.02715e20 0.116645
\(186\) −2.41660e21 −0.665015
\(187\) −3.64886e18 −0.000954250 0
\(188\) −1.10682e21 −0.275155
\(189\) −2.06602e20 −0.0488369
\(190\) 1.85411e21 0.416847
\(191\) 6.19380e21 1.32477 0.662384 0.749164i \(-0.269545\pi\)
0.662384 + 0.749164i \(0.269545\pi\)
\(192\) 4.78526e21 0.973966
\(193\) −5.20697e21 −1.00877 −0.504383 0.863480i \(-0.668280\pi\)
−0.504383 + 0.863480i \(0.668280\pi\)
\(194\) −2.89566e20 −0.0534111
\(195\) −6.07180e21 −1.06657
\(196\) 3.51552e21 0.588244
\(197\) 2.42384e21 0.386433 0.193216 0.981156i \(-0.438108\pi\)
0.193216 + 0.981156i \(0.438108\pi\)
\(198\) −1.03747e19 −0.00157636
\(199\) −1.05907e21 −0.153399 −0.0766993 0.997054i \(-0.524438\pi\)
−0.0766993 + 0.997054i \(0.524438\pi\)
\(200\) 5.14485e21 0.710539
\(201\) −1.03889e22 −1.36837
\(202\) −6.48668e21 −0.815051
\(203\) −1.92494e19 −0.00230783
\(204\) −3.60649e21 −0.412666
\(205\) 7.86919e21 0.859545
\(206\) 2.23773e21 0.233383
\(207\) 1.96984e22 1.96206
\(208\) −4.50757e20 −0.0428885
\(209\) 2.77272e19 0.00252067
\(210\) 9.28972e20 0.0807085
\(211\) −1.32424e22 −1.09972 −0.549861 0.835256i \(-0.685319\pi\)
−0.549861 + 0.835256i \(0.685319\pi\)
\(212\) −9.47673e21 −0.752433
\(213\) −9.01111e21 −0.684183
\(214\) 1.20669e22 0.876325
\(215\) −2.68151e21 −0.186301
\(216\) −4.68137e21 −0.311216
\(217\) 1.77003e21 0.112619
\(218\) −8.39851e21 −0.511523
\(219\) 1.51882e22 0.885701
\(220\) −1.21931e19 −0.000680924 0
\(221\) 1.13484e22 0.607027
\(222\) −3.91251e21 −0.200494
\(223\) 2.00921e22 0.986575 0.493287 0.869866i \(-0.335795\pi\)
0.493287 + 0.869866i \(0.335795\pi\)
\(224\) −3.33106e21 −0.156757
\(225\) −1.88351e22 −0.849639
\(226\) 1.17384e22 0.507667
\(227\) −2.03494e22 −0.843929 −0.421965 0.906612i \(-0.638659\pi\)
−0.421965 + 0.906612i \(0.638659\pi\)
\(228\) 2.74053e22 1.09007
\(229\) −3.99900e22 −1.52586 −0.762930 0.646481i \(-0.776240\pi\)
−0.762930 + 0.646481i \(0.776240\pi\)
\(230\) −1.52170e22 −0.557076
\(231\) 1.38923e19 0.000488045 0
\(232\) −4.36168e20 −0.0147068
\(233\) 2.42170e22 0.783862 0.391931 0.919995i \(-0.371807\pi\)
0.391931 + 0.919995i \(0.371807\pi\)
\(234\) 3.22665e22 1.00277
\(235\) −8.31783e21 −0.248236
\(236\) −1.84722e22 −0.529487
\(237\) −4.67149e21 −0.128631
\(238\) −1.73628e21 −0.0459345
\(239\) 2.62411e22 0.667116 0.333558 0.942730i \(-0.391751\pi\)
0.333558 + 0.942730i \(0.391751\pi\)
\(240\) 1.07653e21 0.0263039
\(241\) 7.36445e22 1.72973 0.864865 0.502004i \(-0.167404\pi\)
0.864865 + 0.502004i \(0.167404\pi\)
\(242\) −2.78884e22 −0.629764
\(243\) −6.54789e22 −1.42181
\(244\) −7.39431e21 −0.154417
\(245\) 2.64194e22 0.530696
\(246\) −7.64518e22 −1.47742
\(247\) −8.62350e22 −1.60348
\(248\) 4.01068e22 0.717674
\(249\) 6.12151e22 1.05430
\(250\) 3.52277e22 0.584055
\(251\) 7.29309e22 1.16416 0.582078 0.813133i \(-0.302240\pi\)
0.582078 + 0.813133i \(0.302240\pi\)
\(252\) 7.51066e21 0.115444
\(253\) −2.27562e20 −0.00336864
\(254\) 4.01608e22 0.572641
\(255\) −2.71031e22 −0.372295
\(256\) −7.69607e22 −1.01857
\(257\) −6.38120e22 −0.813838 −0.406919 0.913464i \(-0.633397\pi\)
−0.406919 + 0.913464i \(0.633397\pi\)
\(258\) 2.60518e22 0.320221
\(259\) 2.86570e21 0.0339534
\(260\) 3.79220e22 0.433156
\(261\) 1.59680e21 0.0175859
\(262\) 3.27996e22 0.348342
\(263\) −1.35820e23 −1.39118 −0.695590 0.718439i \(-0.744857\pi\)
−0.695590 + 0.718439i \(0.744857\pi\)
\(264\) 3.14783e20 0.00311010
\(265\) −7.12184e22 −0.678823
\(266\) 1.31938e22 0.121337
\(267\) 2.21410e23 1.96490
\(268\) 6.48846e22 0.555726
\(269\) −1.33672e23 −1.10508 −0.552540 0.833486i \(-0.686341\pi\)
−0.552540 + 0.833486i \(0.686341\pi\)
\(270\) −1.32393e22 −0.105660
\(271\) 2.00548e23 1.54529 0.772643 0.634840i \(-0.218934\pi\)
0.772643 + 0.634840i \(0.218934\pi\)
\(272\) −2.01207e21 −0.0149706
\(273\) −4.32067e22 −0.310460
\(274\) −1.34714e23 −0.934939
\(275\) 2.17589e20 0.00145874
\(276\) −2.24919e23 −1.45677
\(277\) 2.00223e23 1.25301 0.626507 0.779416i \(-0.284484\pi\)
0.626507 + 0.779416i \(0.284484\pi\)
\(278\) 6.31461e22 0.381875
\(279\) −1.46830e23 −0.858170
\(280\) −1.54176e22 −0.0870994
\(281\) −1.19239e23 −0.651194 −0.325597 0.945509i \(-0.605565\pi\)
−0.325597 + 0.945509i \(0.605565\pi\)
\(282\) 8.08104e22 0.426679
\(283\) −3.46108e21 −0.0176702 −0.00883509 0.999961i \(-0.502812\pi\)
−0.00883509 + 0.999961i \(0.502812\pi\)
\(284\) 5.62798e22 0.277861
\(285\) 2.05953e23 0.983425
\(286\) −3.72753e20 −0.00172165
\(287\) 5.59968e22 0.250199
\(288\) 2.76322e23 1.19450
\(289\) −1.88416e23 −0.788112
\(290\) −1.23353e21 −0.00499307
\(291\) −3.21647e22 −0.126007
\(292\) −9.48593e22 −0.359702
\(293\) −2.13236e23 −0.782739 −0.391370 0.920234i \(-0.627999\pi\)
−0.391370 + 0.920234i \(0.627999\pi\)
\(294\) −2.56673e23 −0.912181
\(295\) −1.38820e23 −0.477687
\(296\) 6.49335e22 0.216370
\(297\) −1.97987e20 −0.000638927 0
\(298\) −1.21306e23 −0.379164
\(299\) 7.07745e23 2.14289
\(300\) 2.15063e23 0.630831
\(301\) −1.90815e22 −0.0542290
\(302\) 2.62464e23 0.722776
\(303\) −7.20534e23 −1.92287
\(304\) 1.52895e22 0.0395452
\(305\) −5.55688e22 −0.139310
\(306\) 1.44030e23 0.350025
\(307\) 1.91887e23 0.452097 0.226048 0.974116i \(-0.427419\pi\)
0.226048 + 0.974116i \(0.427419\pi\)
\(308\) −8.67656e19 −0.000198205 0
\(309\) 2.48564e23 0.550596
\(310\) 1.13426e23 0.243655
\(311\) −1.54522e23 −0.321933 −0.160967 0.986960i \(-0.551461\pi\)
−0.160967 + 0.986960i \(0.551461\pi\)
\(312\) −9.79013e23 −1.97843
\(313\) 2.87408e23 0.563413 0.281707 0.959501i \(-0.409100\pi\)
0.281707 + 0.959501i \(0.409100\pi\)
\(314\) −4.89003e23 −0.929994
\(315\) 5.64432e22 0.104150
\(316\) 2.91762e22 0.0522398
\(317\) 2.63533e22 0.0457901 0.0228950 0.999738i \(-0.492712\pi\)
0.0228950 + 0.999738i \(0.492712\pi\)
\(318\) 6.91910e23 1.16679
\(319\) −1.84467e19 −3.01931e−5 0
\(320\) −2.24602e23 −0.356852
\(321\) 1.34038e24 2.06743
\(322\) −1.08283e23 −0.162155
\(323\) −3.84933e23 −0.559707
\(324\) 3.20306e23 0.452259
\(325\) −6.76729e23 −0.927946
\(326\) −2.64829e23 −0.352694
\(327\) −9.32897e23 −1.20679
\(328\) 1.26882e24 1.59441
\(329\) −5.91893e22 −0.0722574
\(330\) 8.90236e20 0.00105590
\(331\) −1.05338e24 −1.21400 −0.606998 0.794703i \(-0.707626\pi\)
−0.606998 + 0.794703i \(0.707626\pi\)
\(332\) −3.82325e23 −0.428175
\(333\) −2.37719e23 −0.258728
\(334\) 1.10987e23 0.117403
\(335\) 4.87613e23 0.501360
\(336\) 7.66055e21 0.00765661
\(337\) 6.76160e23 0.657000 0.328500 0.944504i \(-0.393457\pi\)
0.328500 + 0.944504i \(0.393457\pi\)
\(338\) 4.92671e23 0.465424
\(339\) 1.30389e24 1.19769
\(340\) 1.69275e23 0.151197
\(341\) 1.69622e21 0.00147338
\(342\) −1.09447e24 −0.924600
\(343\) 3.80841e23 0.312932
\(344\) −4.32365e23 −0.345578
\(345\) −1.69029e24 −1.31425
\(346\) −5.44237e23 −0.411685
\(347\) 1.06325e24 0.782535 0.391268 0.920277i \(-0.372037\pi\)
0.391268 + 0.920277i \(0.372037\pi\)
\(348\) −1.82325e22 −0.0130570
\(349\) −7.14667e23 −0.498037 −0.249019 0.968499i \(-0.580108\pi\)
−0.249019 + 0.968499i \(0.580108\pi\)
\(350\) 1.03538e23 0.0702189
\(351\) 6.15764e23 0.406440
\(352\) −3.19216e21 −0.00205083
\(353\) −5.07321e23 −0.317266 −0.158633 0.987338i \(-0.550709\pi\)
−0.158633 + 0.987338i \(0.550709\pi\)
\(354\) 1.34868e24 0.821068
\(355\) 4.22947e23 0.250678
\(356\) −1.38284e24 −0.797988
\(357\) −1.92864e23 −0.108369
\(358\) −1.89087e24 −1.03460
\(359\) 3.32006e24 1.76909 0.884544 0.466458i \(-0.154470\pi\)
0.884544 + 0.466458i \(0.154470\pi\)
\(360\) 1.27894e24 0.663706
\(361\) 9.46633e23 0.478479
\(362\) 1.51558e24 0.746186
\(363\) −3.09782e24 −1.48574
\(364\) 2.69851e23 0.126084
\(365\) −7.12875e23 −0.324512
\(366\) 5.39869e23 0.239452
\(367\) −2.24430e24 −0.969958 −0.484979 0.874526i \(-0.661173\pi\)
−0.484979 + 0.874526i \(0.661173\pi\)
\(368\) −1.25483e23 −0.0528484
\(369\) −4.64511e24 −1.90654
\(370\) 1.83638e23 0.0734592
\(371\) −5.06787e23 −0.197594
\(372\) 1.67653e24 0.637166
\(373\) 5.10606e24 1.89170 0.945850 0.324603i \(-0.105231\pi\)
0.945850 + 0.324603i \(0.105231\pi\)
\(374\) −1.66388e21 −0.000600955 0
\(375\) 3.91305e24 1.37790
\(376\) −1.34116e24 −0.460465
\(377\) 5.73715e22 0.0192067
\(378\) −9.42106e22 −0.0307559
\(379\) −4.28975e24 −1.36571 −0.682857 0.730552i \(-0.739263\pi\)
−0.682857 + 0.730552i \(0.739263\pi\)
\(380\) −1.28630e24 −0.399390
\(381\) 4.46102e24 1.35098
\(382\) 2.82437e24 0.834295
\(383\) −1.86803e24 −0.538264 −0.269132 0.963103i \(-0.586737\pi\)
−0.269132 + 0.963103i \(0.586737\pi\)
\(384\) −3.04684e24 −0.856451
\(385\) −6.52050e20 −0.000178815 0
\(386\) −2.37438e24 −0.635288
\(387\) 1.58287e24 0.413230
\(388\) 2.00888e23 0.0511743
\(389\) −6.47448e24 −1.60947 −0.804737 0.593632i \(-0.797694\pi\)
−0.804737 + 0.593632i \(0.797694\pi\)
\(390\) −2.76874e24 −0.671689
\(391\) 3.15920e24 0.747995
\(392\) 4.25985e24 0.984411
\(393\) 3.64334e24 0.821809
\(394\) 1.10527e24 0.243363
\(395\) 2.19262e23 0.0471292
\(396\) 7.19748e21 0.00151034
\(397\) 1.22760e24 0.251505 0.125752 0.992062i \(-0.459866\pi\)
0.125752 + 0.992062i \(0.459866\pi\)
\(398\) −4.82937e23 −0.0966054
\(399\) 1.46555e24 0.286258
\(400\) 1.19984e23 0.0228852
\(401\) −5.17895e23 −0.0964651 −0.0482326 0.998836i \(-0.515359\pi\)
−0.0482326 + 0.998836i \(0.515359\pi\)
\(402\) −4.73732e24 −0.861757
\(403\) −5.27546e24 −0.937264
\(404\) 4.50017e24 0.780918
\(405\) 2.40712e24 0.408014
\(406\) −8.77772e21 −0.00145340
\(407\) 2.74621e21 0.000444208 0
\(408\) −4.37008e24 −0.690586
\(409\) 2.81877e24 0.435199 0.217599 0.976038i \(-0.430177\pi\)
0.217599 + 0.976038i \(0.430177\pi\)
\(410\) 3.58835e24 0.541313
\(411\) −1.49639e25 −2.20571
\(412\) −1.55243e24 −0.223609
\(413\) −9.87839e23 −0.139047
\(414\) 8.98245e24 1.23564
\(415\) −2.87320e24 −0.386286
\(416\) 9.92800e24 1.30459
\(417\) 7.01420e24 0.900919
\(418\) 1.26436e22 0.00158744
\(419\) 8.65571e24 1.06235 0.531177 0.847261i \(-0.321750\pi\)
0.531177 + 0.847261i \(0.321750\pi\)
\(420\) −6.44479e23 −0.0773286
\(421\) −5.21652e24 −0.611929 −0.305964 0.952043i \(-0.598979\pi\)
−0.305964 + 0.952043i \(0.598979\pi\)
\(422\) −6.03853e24 −0.692569
\(423\) 4.90994e24 0.550608
\(424\) −1.14832e25 −1.25918
\(425\) −3.02076e24 −0.323908
\(426\) −4.10907e24 −0.430876
\(427\) −3.95425e23 −0.0405508
\(428\) −8.37147e24 −0.839626
\(429\) −4.14050e22 −0.00406171
\(430\) −1.22277e24 −0.117326
\(431\) 1.04364e25 0.979526 0.489763 0.871856i \(-0.337083\pi\)
0.489763 + 0.871856i \(0.337083\pi\)
\(432\) −1.09175e23 −0.0100237
\(433\) 1.45110e25 1.30335 0.651675 0.758499i \(-0.274067\pi\)
0.651675 + 0.758499i \(0.274067\pi\)
\(434\) 8.07134e23 0.0709240
\(435\) −1.37019e23 −0.0117796
\(436\) 5.82650e24 0.490102
\(437\) −2.40064e25 −1.97585
\(438\) 6.92582e24 0.557785
\(439\) 1.79857e25 1.41747 0.708735 0.705474i \(-0.249266\pi\)
0.708735 + 0.705474i \(0.249266\pi\)
\(440\) −1.47747e22 −0.00113951
\(441\) −1.55951e25 −1.17713
\(442\) 5.17487e24 0.382286
\(443\) 8.73685e24 0.631712 0.315856 0.948807i \(-0.397708\pi\)
0.315856 + 0.948807i \(0.397708\pi\)
\(444\) 2.71432e24 0.192098
\(445\) −1.03921e25 −0.719921
\(446\) 9.16201e24 0.621312
\(447\) −1.34745e25 −0.894523
\(448\) −1.59826e24 −0.103874
\(449\) −1.60576e25 −1.02174 −0.510870 0.859658i \(-0.670677\pi\)
−0.510870 + 0.859658i \(0.670677\pi\)
\(450\) −8.58881e24 −0.535075
\(451\) 5.36618e22 0.00327332
\(452\) −8.14357e24 −0.486407
\(453\) 2.91542e25 1.70517
\(454\) −9.27933e24 −0.531479
\(455\) 2.02795e24 0.113750
\(456\) 3.32077e25 1.82420
\(457\) −2.67587e24 −0.143966 −0.0719831 0.997406i \(-0.522933\pi\)
−0.0719831 + 0.997406i \(0.522933\pi\)
\(458\) −1.82355e25 −0.960936
\(459\) 2.74863e24 0.141871
\(460\) 1.05569e25 0.533747
\(461\) 6.96807e24 0.345107 0.172553 0.985000i \(-0.444798\pi\)
0.172553 + 0.985000i \(0.444798\pi\)
\(462\) 6.33488e21 0.000307354 0
\(463\) −2.49843e25 −1.18754 −0.593770 0.804635i \(-0.702361\pi\)
−0.593770 + 0.804635i \(0.702361\pi\)
\(464\) −1.01720e22 −0.000473679 0
\(465\) 1.25992e25 0.574832
\(466\) 1.10430e25 0.493650
\(467\) 1.94531e25 0.852075 0.426037 0.904706i \(-0.359909\pi\)
0.426037 + 0.904706i \(0.359909\pi\)
\(468\) −2.23850e25 −0.960776
\(469\) 3.46983e24 0.145937
\(470\) −3.79293e24 −0.156331
\(471\) −5.43180e25 −2.19404
\(472\) −2.23833e25 −0.886084
\(473\) −1.82859e22 −0.000709471 0
\(474\) −2.13020e24 −0.0810076
\(475\) 2.29543e25 0.855610
\(476\) 1.20455e24 0.0440108
\(477\) 4.20396e25 1.50568
\(478\) 1.19659e25 0.420128
\(479\) 3.34153e25 1.15016 0.575080 0.818097i \(-0.304971\pi\)
0.575080 + 0.818097i \(0.304971\pi\)
\(480\) −2.37108e25 −0.800118
\(481\) −8.54103e24 −0.282574
\(482\) 3.35819e25 1.08933
\(483\) −1.20280e25 −0.382557
\(484\) 1.93477e25 0.603391
\(485\) 1.50969e24 0.0461679
\(486\) −2.98584e25 −0.895410
\(487\) −4.86181e25 −1.42979 −0.714896 0.699231i \(-0.753526\pi\)
−0.714896 + 0.699231i \(0.753526\pi\)
\(488\) −8.95988e24 −0.258413
\(489\) −2.94169e25 −0.832075
\(490\) 1.20473e25 0.334215
\(491\) 9.86730e24 0.268488 0.134244 0.990948i \(-0.457139\pi\)
0.134244 + 0.990948i \(0.457139\pi\)
\(492\) 5.30387e25 1.41555
\(493\) 2.56093e23 0.00670427
\(494\) −3.93232e25 −1.00982
\(495\) 5.40896e22 0.00136259
\(496\) 9.35339e23 0.0231150
\(497\) 3.00967e24 0.0729681
\(498\) 2.79141e25 0.663964
\(499\) −3.54150e25 −0.826481 −0.413240 0.910622i \(-0.635603\pi\)
−0.413240 + 0.910622i \(0.635603\pi\)
\(500\) −2.44394e25 −0.559596
\(501\) 1.23283e25 0.276978
\(502\) 3.32565e25 0.733147
\(503\) 1.47204e25 0.318436 0.159218 0.987243i \(-0.449103\pi\)
0.159218 + 0.987243i \(0.449103\pi\)
\(504\) 9.10086e24 0.193193
\(505\) 3.38191e25 0.704521
\(506\) −1.03768e23 −0.00212146
\(507\) 5.47254e25 1.09803
\(508\) −2.78618e25 −0.548660
\(509\) −4.88290e25 −0.943754 −0.471877 0.881664i \(-0.656423\pi\)
−0.471877 + 0.881664i \(0.656423\pi\)
\(510\) −1.23590e25 −0.234459
\(511\) −5.07279e24 −0.0944601
\(512\) −3.55692e24 −0.0650143
\(513\) −2.08864e25 −0.374757
\(514\) −2.90983e25 −0.512528
\(515\) −1.16667e25 −0.201733
\(516\) −1.80735e25 −0.306811
\(517\) −5.67212e22 −0.000945334 0
\(518\) 1.30676e24 0.0213827
\(519\) −6.04533e25 −0.971248
\(520\) 4.59511e25 0.724876
\(521\) 7.14445e25 1.10665 0.553325 0.832965i \(-0.313359\pi\)
0.553325 + 0.832965i \(0.313359\pi\)
\(522\) 7.28139e23 0.0110750
\(523\) 8.99895e25 1.34408 0.672041 0.740514i \(-0.265418\pi\)
0.672041 + 0.740514i \(0.265418\pi\)
\(524\) −2.27548e25 −0.333754
\(525\) 1.15009e25 0.165660
\(526\) −6.19338e25 −0.876120
\(527\) −2.35484e25 −0.327160
\(528\) 7.34111e21 0.000100170 0
\(529\) 1.22409e26 1.64053
\(530\) −3.24756e25 −0.427500
\(531\) 8.19444e25 1.05955
\(532\) −9.15324e24 −0.116256
\(533\) −1.66895e26 −2.08226
\(534\) 1.00963e26 1.23743
\(535\) −6.29123e25 −0.757486
\(536\) 7.86223e25 0.929995
\(537\) −2.10036e26 −2.44083
\(538\) −6.09544e25 −0.695943
\(539\) 1.80160e23 0.00202100
\(540\) 9.18486e24 0.101235
\(541\) −9.33602e25 −1.01109 −0.505543 0.862802i \(-0.668708\pi\)
−0.505543 + 0.862802i \(0.668708\pi\)
\(542\) 9.14497e25 0.973171
\(543\) 1.68349e26 1.76040
\(544\) 4.43162e25 0.455379
\(545\) 4.37866e25 0.442155
\(546\) −1.97022e25 −0.195517
\(547\) −2.74670e25 −0.267875 −0.133938 0.990990i \(-0.542762\pi\)
−0.133938 + 0.990990i \(0.542762\pi\)
\(548\) 9.34587e25 0.895786
\(549\) 3.28018e25 0.309001
\(550\) 9.92207e22 0.000918664 0
\(551\) −1.94602e24 −0.0177095
\(552\) −2.72541e26 −2.43787
\(553\) 1.56026e24 0.0137185
\(554\) 9.13016e25 0.789106
\(555\) 2.03983e25 0.173305
\(556\) −4.38079e25 −0.365882
\(557\) 1.42589e26 1.17074 0.585370 0.810766i \(-0.300949\pi\)
0.585370 + 0.810766i \(0.300949\pi\)
\(558\) −6.69543e25 −0.540447
\(559\) 5.68712e25 0.451316
\(560\) −3.59557e23 −0.00280531
\(561\) −1.84822e23 −0.00141777
\(562\) −5.43732e25 −0.410100
\(563\) −1.72252e26 −1.27742 −0.638712 0.769446i \(-0.720533\pi\)
−0.638712 + 0.769446i \(0.720533\pi\)
\(564\) −5.60626e25 −0.408810
\(565\) −6.11995e25 −0.438821
\(566\) −1.57825e24 −0.0111281
\(567\) 1.71290e25 0.118766
\(568\) 6.81956e25 0.464994
\(569\) 5.24893e25 0.351969 0.175984 0.984393i \(-0.443689\pi\)
0.175984 + 0.984393i \(0.443689\pi\)
\(570\) 9.39144e25 0.619329
\(571\) −3.21674e24 −0.0208628 −0.0104314 0.999946i \(-0.503320\pi\)
−0.0104314 + 0.999946i \(0.503320\pi\)
\(572\) 2.58599e23 0.00164955
\(573\) 3.13728e26 1.96827
\(574\) 2.55345e25 0.157567
\(575\) −1.88390e26 −1.14344
\(576\) 1.32580e26 0.791527
\(577\) 1.17453e26 0.689752 0.344876 0.938648i \(-0.387921\pi\)
0.344876 + 0.938648i \(0.387921\pi\)
\(578\) −8.59176e25 −0.496327
\(579\) −2.63744e26 −1.49877
\(580\) 8.55764e23 0.00478397
\(581\) −2.04456e25 −0.112441
\(582\) −1.46671e25 −0.0793553
\(583\) −4.85655e23 −0.00258509
\(584\) −1.14943e26 −0.601953
\(585\) −1.68225e26 −0.866783
\(586\) −9.72354e25 −0.492943
\(587\) 1.86886e24 0.00932213 0.00466107 0.999989i \(-0.498516\pi\)
0.00466107 + 0.999989i \(0.498516\pi\)
\(588\) 1.78068e26 0.873981
\(589\) 1.78941e26 0.864201
\(590\) −6.33021e25 −0.300832
\(591\) 1.22772e26 0.574141
\(592\) 1.51433e24 0.00696889
\(593\) −1.50165e26 −0.680063 −0.340032 0.940414i \(-0.610438\pi\)
−0.340032 + 0.940414i \(0.610438\pi\)
\(594\) −9.02822e22 −0.000402375 0
\(595\) 9.05231e24 0.0397053
\(596\) 8.41563e25 0.363285
\(597\) −5.36441e25 −0.227911
\(598\) 3.22732e26 1.34952
\(599\) 1.65804e26 0.682403 0.341201 0.939990i \(-0.389166\pi\)
0.341201 + 0.939990i \(0.389166\pi\)
\(600\) 2.60597e26 1.05568
\(601\) −2.54795e26 −1.01598 −0.507989 0.861364i \(-0.669611\pi\)
−0.507989 + 0.861364i \(0.669611\pi\)
\(602\) −8.70118e24 −0.0341516
\(603\) −2.87833e26 −1.11206
\(604\) −1.82085e26 −0.692507
\(605\) 1.45400e26 0.544361
\(606\) −3.28564e26 −1.21096
\(607\) 3.14191e26 1.13999 0.569996 0.821648i \(-0.306945\pi\)
0.569996 + 0.821648i \(0.306945\pi\)
\(608\) −3.36753e26 −1.20290
\(609\) −9.75020e23 −0.00342885
\(610\) −2.53394e25 −0.0877329
\(611\) 1.76410e26 0.601355
\(612\) −9.99214e25 −0.335367
\(613\) −3.91478e26 −1.29370 −0.646848 0.762619i \(-0.723913\pi\)
−0.646848 + 0.762619i \(0.723913\pi\)
\(614\) 8.75005e25 0.284715
\(615\) 3.98590e26 1.27707
\(616\) −1.05136e23 −0.000331692 0
\(617\) 4.86066e26 1.51003 0.755016 0.655706i \(-0.227629\pi\)
0.755016 + 0.655706i \(0.227629\pi\)
\(618\) 1.13345e26 0.346747
\(619\) −4.51098e26 −1.35897 −0.679485 0.733689i \(-0.737797\pi\)
−0.679485 + 0.733689i \(0.737797\pi\)
\(620\) −7.86897e25 −0.233452
\(621\) 1.71418e26 0.500827
\(622\) −7.04619e25 −0.202743
\(623\) −7.39500e25 −0.209557
\(624\) −2.28318e25 −0.0637215
\(625\) 7.23294e25 0.198817
\(626\) 1.31058e26 0.354819
\(627\) 1.40444e24 0.00374508
\(628\) 3.39248e26 0.891047
\(629\) −3.81251e25 −0.0986349
\(630\) 2.57381e25 0.0655905
\(631\) 2.03805e26 0.511607 0.255803 0.966729i \(-0.417660\pi\)
0.255803 + 0.966729i \(0.417660\pi\)
\(632\) 3.53536e25 0.0874221
\(633\) −6.70753e26 −1.63391
\(634\) 1.20171e25 0.0288371
\(635\) −2.09383e26 −0.494985
\(636\) −4.80016e26 −1.11793
\(637\) −5.60320e26 −1.28562
\(638\) −8.41170e21 −1.90146e−5 0
\(639\) −2.49662e26 −0.556024
\(640\) 1.43007e26 0.313796
\(641\) 5.77927e26 1.24946 0.624729 0.780842i \(-0.285209\pi\)
0.624729 + 0.780842i \(0.285209\pi\)
\(642\) 6.11213e26 1.30200
\(643\) 7.72049e26 1.62047 0.810235 0.586106i \(-0.199340\pi\)
0.810235 + 0.586106i \(0.199340\pi\)
\(644\) 7.51221e25 0.155365
\(645\) −1.35824e26 −0.276796
\(646\) −1.75529e26 −0.352485
\(647\) −1.39226e24 −0.00275504 −0.00137752 0.999999i \(-0.500438\pi\)
−0.00137752 + 0.999999i \(0.500438\pi\)
\(648\) 3.88123e26 0.756844
\(649\) −9.46648e23 −0.00181913
\(650\) −3.08588e26 −0.584390
\(651\) 8.96556e25 0.167324
\(652\) 1.83726e26 0.337924
\(653\) −6.19470e26 −1.12291 −0.561455 0.827507i \(-0.689758\pi\)
−0.561455 + 0.827507i \(0.689758\pi\)
\(654\) −4.25401e26 −0.759994
\(655\) −1.71004e26 −0.301103
\(656\) 2.95905e25 0.0513530
\(657\) 4.20804e26 0.719795
\(658\) −2.69903e25 −0.0455053
\(659\) −7.96627e25 −0.132386 −0.0661932 0.997807i \(-0.521085\pi\)
−0.0661932 + 0.997807i \(0.521085\pi\)
\(660\) −6.17605e23 −0.00101168
\(661\) −1.85436e26 −0.299420 −0.149710 0.988730i \(-0.547834\pi\)
−0.149710 + 0.988730i \(0.547834\pi\)
\(662\) −4.80339e26 −0.764535
\(663\) 5.74819e26 0.901888
\(664\) −4.63273e26 −0.716540
\(665\) −6.87873e25 −0.104882
\(666\) −1.08400e26 −0.162938
\(667\) 1.59713e25 0.0236670
\(668\) −7.69977e25 −0.112487
\(669\) 1.01771e27 1.46580
\(670\) 2.22351e26 0.315740
\(671\) −3.78937e23 −0.000530521 0
\(672\) −1.68725e26 −0.232901
\(673\) 5.91532e26 0.805073 0.402536 0.915404i \(-0.368129\pi\)
0.402536 + 0.915404i \(0.368129\pi\)
\(674\) 3.08329e26 0.413757
\(675\) −1.63906e26 −0.216875
\(676\) −3.41793e26 −0.445933
\(677\) −2.97418e26 −0.382626 −0.191313 0.981529i \(-0.561275\pi\)
−0.191313 + 0.981529i \(0.561275\pi\)
\(678\) 5.94573e26 0.754264
\(679\) 1.07429e25 0.0134387
\(680\) 2.05115e26 0.253024
\(681\) −1.03074e27 −1.25387
\(682\) 7.73477e23 0.000927889 0
\(683\) −1.25943e27 −1.48997 −0.744983 0.667084i \(-0.767542\pi\)
−0.744983 + 0.667084i \(0.767542\pi\)
\(684\) 7.59290e26 0.885880
\(685\) 7.02349e26 0.808151
\(686\) 1.73663e26 0.197074
\(687\) −2.02558e27 −2.26704
\(688\) −1.00833e25 −0.0111304
\(689\) 1.51044e27 1.64446
\(690\) −7.70771e26 −0.827674
\(691\) 3.19965e25 0.0338891 0.0169446 0.999856i \(-0.494606\pi\)
0.0169446 + 0.999856i \(0.494606\pi\)
\(692\) 3.77567e26 0.394445
\(693\) 3.84899e23 0.000396626 0
\(694\) 4.84840e26 0.492815
\(695\) −3.29220e26 −0.330088
\(696\) −2.20928e25 −0.0218506
\(697\) −7.44979e26 −0.726829
\(698\) −3.25888e26 −0.313648
\(699\) 1.22664e27 1.16462
\(700\) −7.18300e25 −0.0672782
\(701\) 2.06463e27 1.90775 0.953874 0.300208i \(-0.0970562\pi\)
0.953874 + 0.300208i \(0.0970562\pi\)
\(702\) 2.80788e26 0.255963
\(703\) 2.89708e26 0.260546
\(704\) −1.53161e24 −0.00135897
\(705\) −4.21315e26 −0.368816
\(706\) −2.31338e26 −0.199803
\(707\) 2.40655e26 0.205074
\(708\) −9.35656e26 −0.786684
\(709\) −1.44096e27 −1.19539 −0.597697 0.801722i \(-0.703918\pi\)
−0.597697 + 0.801722i \(0.703918\pi\)
\(710\) 1.92864e26 0.157869
\(711\) −1.29428e26 −0.104536
\(712\) −1.67562e27 −1.33541
\(713\) −1.46860e27 −1.15492
\(714\) −8.79461e25 −0.0682470
\(715\) 1.94339e24 0.00148817
\(716\) 1.31180e27 0.991273
\(717\) 1.32916e27 0.991165
\(718\) 1.51395e27 1.11411
\(719\) −1.26393e27 −0.917907 −0.458953 0.888460i \(-0.651776\pi\)
−0.458953 + 0.888460i \(0.651776\pi\)
\(720\) 2.98264e25 0.0213767
\(721\) −8.30194e25 −0.0587211
\(722\) 4.31665e26 0.301331
\(723\) 3.73024e27 2.56994
\(724\) −1.05144e27 −0.714937
\(725\) −1.52714e25 −0.0102486
\(726\) −1.41260e27 −0.935670
\(727\) 5.54468e26 0.362493 0.181246 0.983438i \(-0.441987\pi\)
0.181246 + 0.983438i \(0.441987\pi\)
\(728\) 3.26986e26 0.210999
\(729\) −2.13985e27 −1.36293
\(730\) −3.25071e26 −0.204367
\(731\) 2.53860e26 0.157536
\(732\) −3.74537e26 −0.229424
\(733\) 8.06563e26 0.487698 0.243849 0.969813i \(-0.421590\pi\)
0.243849 + 0.969813i \(0.421590\pi\)
\(734\) −1.02340e27 −0.610848
\(735\) 1.33820e27 0.788479
\(736\) 2.76379e27 1.60755
\(737\) 3.32514e24 0.00190928
\(738\) −2.11817e27 −1.20068
\(739\) 6.63850e26 0.371490 0.185745 0.982598i \(-0.440530\pi\)
0.185745 + 0.982598i \(0.440530\pi\)
\(740\) −1.27400e26 −0.0703829
\(741\) −4.36798e27 −2.38236
\(742\) −2.31095e26 −0.124438
\(743\) −3.64095e27 −1.93562 −0.967811 0.251677i \(-0.919018\pi\)
−0.967811 + 0.251677i \(0.919018\pi\)
\(744\) 2.03149e27 1.06628
\(745\) 6.32441e26 0.327745
\(746\) 2.32837e27 1.19133
\(747\) 1.69603e27 0.856814
\(748\) 1.15432e24 0.000575788 0
\(749\) −4.47681e26 −0.220491
\(750\) 1.78435e27 0.867758
\(751\) −1.55458e27 −0.746505 −0.373253 0.927730i \(-0.621758\pi\)
−0.373253 + 0.927730i \(0.621758\pi\)
\(752\) −3.12775e25 −0.0148307
\(753\) 3.69410e27 1.72964
\(754\) 2.61614e25 0.0120958
\(755\) −1.36839e27 −0.624759
\(756\) 6.53590e25 0.0294679
\(757\) 1.33406e27 0.593970 0.296985 0.954882i \(-0.404019\pi\)
0.296985 + 0.954882i \(0.404019\pi\)
\(758\) −1.95613e27 −0.860081
\(759\) −1.15265e25 −0.00500494
\(760\) −1.55864e27 −0.668370
\(761\) −6.89829e25 −0.0292137 −0.0146069 0.999893i \(-0.504650\pi\)
−0.0146069 + 0.999893i \(0.504650\pi\)
\(762\) 2.03423e27 0.850799
\(763\) 3.11584e26 0.128704
\(764\) −1.95942e27 −0.799357
\(765\) −7.50917e26 −0.302558
\(766\) −8.51822e26 −0.338981
\(767\) 2.94419e27 1.15720
\(768\) −3.89821e27 −1.51333
\(769\) −8.09673e26 −0.310463 −0.155231 0.987878i \(-0.549612\pi\)
−0.155231 + 0.987878i \(0.549612\pi\)
\(770\) −2.97335e23 −0.000112612 0
\(771\) −3.23221e27 −1.20916
\(772\) 1.64724e27 0.608683
\(773\) 1.00924e27 0.368374 0.184187 0.982891i \(-0.441035\pi\)
0.184187 + 0.982891i \(0.441035\pi\)
\(774\) 7.21790e26 0.260239
\(775\) 1.40424e27 0.500121
\(776\) 2.43421e26 0.0856389
\(777\) 1.45154e26 0.0504461
\(778\) −2.95236e27 −1.01359
\(779\) 5.66100e27 1.91994
\(780\) 1.92083e27 0.643560
\(781\) 2.88417e24 0.000954633 0
\(782\) 1.44060e27 0.471062
\(783\) 1.38956e25 0.00448890
\(784\) 9.93448e25 0.0317061
\(785\) 2.54948e27 0.803876
\(786\) 1.66136e27 0.517548
\(787\) −4.88307e27 −1.50291 −0.751456 0.659784i \(-0.770648\pi\)
−0.751456 + 0.659784i \(0.770648\pi\)
\(788\) −7.66785e26 −0.233171
\(789\) −6.87955e27 −2.06694
\(790\) 9.99833e25 0.0296804
\(791\) −4.35493e26 −0.127733
\(792\) 8.72137e24 0.00252752
\(793\) 1.17854e27 0.337480
\(794\) 5.59784e26 0.158389
\(795\) −3.60735e27 −1.00856
\(796\) 3.35040e26 0.0925598
\(797\) 1.18346e27 0.323072 0.161536 0.986867i \(-0.448355\pi\)
0.161536 + 0.986867i \(0.448355\pi\)
\(798\) 6.68291e26 0.180276
\(799\) 7.87451e26 0.209908
\(800\) −2.64267e27 −0.696127
\(801\) 6.13438e27 1.59684
\(802\) −2.36160e26 −0.0607506
\(803\) −4.86126e24 −0.00123581
\(804\) 3.28653e27 0.825669
\(805\) 5.64549e26 0.140165
\(806\) −2.40561e27 −0.590258
\(807\) −6.77075e27 −1.64187
\(808\) 5.45297e27 1.30685
\(809\) −5.08495e27 −1.20441 −0.602207 0.798340i \(-0.705712\pi\)
−0.602207 + 0.798340i \(0.705712\pi\)
\(810\) 1.09765e27 0.256954
\(811\) −2.72133e25 −0.00629627 −0.00314813 0.999995i \(-0.501002\pi\)
−0.00314813 + 0.999995i \(0.501002\pi\)
\(812\) 6.08958e24 0.00139253
\(813\) 1.01581e28 2.29590
\(814\) 1.25227e24 0.000279748 0
\(815\) 1.38072e27 0.304865
\(816\) −1.01916e26 −0.0222425
\(817\) −1.92905e27 −0.416134
\(818\) 1.28536e27 0.274074
\(819\) −1.19708e27 −0.252306
\(820\) −2.48943e27 −0.518644
\(821\) −5.16381e26 −0.106343 −0.0531716 0.998585i \(-0.516933\pi\)
−0.0531716 + 0.998585i \(0.516933\pi\)
\(822\) −6.82355e27 −1.38908
\(823\) −1.97624e27 −0.397687 −0.198843 0.980031i \(-0.563718\pi\)
−0.198843 + 0.980031i \(0.563718\pi\)
\(824\) −1.88112e27 −0.374204
\(825\) 1.10213e25 0.00216731
\(826\) −4.50455e26 −0.0875670
\(827\) −1.16795e27 −0.224450 −0.112225 0.993683i \(-0.535798\pi\)
−0.112225 + 0.993683i \(0.535798\pi\)
\(828\) −6.23162e27 −1.18389
\(829\) 4.20809e27 0.790345 0.395173 0.918607i \(-0.370685\pi\)
0.395173 + 0.918607i \(0.370685\pi\)
\(830\) −1.31018e27 −0.243270
\(831\) 1.01417e28 1.86166
\(832\) 4.76350e27 0.864479
\(833\) −2.50113e27 −0.448755
\(834\) 3.19848e27 0.567369
\(835\) −5.78644e26 −0.101482
\(836\) −8.77156e24 −0.00152096
\(837\) −1.27774e27 −0.219053
\(838\) 3.94700e27 0.669036
\(839\) 8.49254e27 1.42331 0.711654 0.702530i \(-0.247947\pi\)
0.711654 + 0.702530i \(0.247947\pi\)
\(840\) −7.80931e26 −0.129408
\(841\) −6.10197e27 −0.999788
\(842\) −2.37873e27 −0.385372
\(843\) −6.03972e27 −0.967508
\(844\) 4.18926e27 0.663565
\(845\) −2.56860e27 −0.402307
\(846\) 2.23893e27 0.346755
\(847\) 1.03466e27 0.158454
\(848\) −2.67802e26 −0.0405558
\(849\) −1.75311e26 −0.0262534
\(850\) −1.37747e27 −0.203986
\(851\) −2.37768e27 −0.348195
\(852\) 2.85068e27 0.412831
\(853\) −9.33345e27 −1.33668 −0.668338 0.743857i \(-0.732994\pi\)
−0.668338 + 0.743857i \(0.732994\pi\)
\(854\) −1.80314e26 −0.0255376
\(855\) 5.70612e27 0.799214
\(856\) −1.01439e28 −1.40509
\(857\) −1.11037e28 −1.52108 −0.760539 0.649292i \(-0.775065\pi\)
−0.760539 + 0.649292i \(0.775065\pi\)
\(858\) −1.88807e25 −0.00255793
\(859\) 1.11478e28 1.49367 0.746834 0.665011i \(-0.231573\pi\)
0.746834 + 0.665011i \(0.231573\pi\)
\(860\) 8.48302e26 0.112413
\(861\) 2.83635e27 0.371732
\(862\) 4.75899e27 0.616873
\(863\) 1.15445e28 1.48004 0.740021 0.672583i \(-0.234815\pi\)
0.740021 + 0.672583i \(0.234815\pi\)
\(864\) 2.40460e27 0.304903
\(865\) 2.83745e27 0.355856
\(866\) 6.61699e27 0.820806
\(867\) −9.54364e27 −1.17093
\(868\) −5.59952e26 −0.0679538
\(869\) 1.49520e24 0.000179478 0
\(870\) −6.24806e25 −0.00741843
\(871\) −1.03416e28 −1.21455
\(872\) 7.06012e27 0.820173
\(873\) −8.91154e26 −0.102404
\(874\) −1.09469e28 −1.24432
\(875\) −1.30694e27 −0.146954
\(876\) −4.80482e27 −0.534426
\(877\) 1.55912e28 1.71547 0.857734 0.514094i \(-0.171872\pi\)
0.857734 + 0.514094i \(0.171872\pi\)
\(878\) 8.20147e27 0.892676
\(879\) −1.08008e28 −1.16295
\(880\) −3.44564e23 −3.67015e−5 0
\(881\) 3.35920e26 0.0353969 0.0176985 0.999843i \(-0.494366\pi\)
0.0176985 + 0.999843i \(0.494366\pi\)
\(882\) −7.11139e27 −0.741315
\(883\) −1.34374e26 −0.0138576 −0.00692882 0.999976i \(-0.502206\pi\)
−0.00692882 + 0.999976i \(0.502206\pi\)
\(884\) −3.59009e27 −0.366276
\(885\) −7.03153e27 −0.709722
\(886\) 3.98400e27 0.397832
\(887\) 9.23887e27 0.912735 0.456367 0.889791i \(-0.349150\pi\)
0.456367 + 0.889791i \(0.349150\pi\)
\(888\) 3.28901e27 0.321471
\(889\) −1.48996e27 −0.144082
\(890\) −4.73881e27 −0.453382
\(891\) 1.64147e25 0.00155380
\(892\) −6.35619e27 −0.595293
\(893\) −5.98374e27 −0.554478
\(894\) −6.14437e27 −0.563341
\(895\) 9.85826e27 0.894296
\(896\) 1.01763e27 0.0913406
\(897\) 3.58487e28 3.18379
\(898\) −7.32226e27 −0.643458
\(899\) −1.19048e26 −0.0103516
\(900\) 5.95853e27 0.512667
\(901\) 6.74227e27 0.574011
\(902\) 2.44698e25 0.00206143
\(903\) −9.66517e26 −0.0805705
\(904\) −9.86777e27 −0.813990
\(905\) −7.90165e27 −0.644995
\(906\) 1.32943e28 1.07386
\(907\) −1.80585e28 −1.44349 −0.721743 0.692161i \(-0.756659\pi\)
−0.721743 + 0.692161i \(0.756659\pi\)
\(908\) 6.43758e27 0.509222
\(909\) −1.99631e28 −1.56268
\(910\) 9.24747e26 0.0716357
\(911\) −1.99304e28 −1.52789 −0.763943 0.645283i \(-0.776739\pi\)
−0.763943 + 0.645283i \(0.776739\pi\)
\(912\) 7.74443e26 0.0587541
\(913\) −1.95930e25 −0.00147106
\(914\) −1.22020e27 −0.0906651
\(915\) −2.81467e27 −0.206980
\(916\) 1.26509e28 0.920694
\(917\) −1.21686e27 −0.0876460
\(918\) 1.25337e27 0.0893460
\(919\) 1.04320e28 0.735984 0.367992 0.929829i \(-0.380045\pi\)
0.367992 + 0.929829i \(0.380045\pi\)
\(920\) 1.27920e28 0.893212
\(921\) 9.71946e27 0.671701
\(922\) 3.17744e27 0.217337
\(923\) −8.97012e27 −0.607271
\(924\) −4.39485e24 −0.000294483 0
\(925\) 2.27348e27 0.150781
\(926\) −1.13929e28 −0.747874
\(927\) 6.88672e27 0.447461
\(928\) 2.24039e26 0.0144085
\(929\) −5.17736e27 −0.329578 −0.164789 0.986329i \(-0.552694\pi\)
−0.164789 + 0.986329i \(0.552694\pi\)
\(930\) 5.74525e27 0.362010
\(931\) 1.90058e28 1.18540
\(932\) −7.66111e27 −0.472977
\(933\) −7.82683e27 −0.478311
\(934\) 8.87060e27 0.536609
\(935\) 8.67484e24 0.000519459 0
\(936\) −2.71245e28 −1.60783
\(937\) 9.33202e27 0.547582 0.273791 0.961789i \(-0.411722\pi\)
0.273791 + 0.961789i \(0.411722\pi\)
\(938\) 1.58224e27 0.0919065
\(939\) 1.45578e28 0.837089
\(940\) 2.63136e27 0.149784
\(941\) 4.50115e27 0.253642 0.126821 0.991926i \(-0.459523\pi\)
0.126821 + 0.991926i \(0.459523\pi\)
\(942\) −2.47690e28 −1.38173
\(943\) −4.64607e28 −2.56581
\(944\) −5.22005e26 −0.0285391
\(945\) 4.91178e26 0.0265850
\(946\) −8.33835e24 −0.000446801 0
\(947\) 1.76126e28 0.934324 0.467162 0.884172i \(-0.345276\pi\)
0.467162 + 0.884172i \(0.345276\pi\)
\(948\) 1.47783e27 0.0776151
\(949\) 1.51191e28 0.786135
\(950\) 1.04672e28 0.538835
\(951\) 1.33485e27 0.0680325
\(952\) 1.45959e27 0.0736511
\(953\) 5.93730e27 0.296624 0.148312 0.988941i \(-0.452616\pi\)
0.148312 + 0.988941i \(0.452616\pi\)
\(954\) 1.91700e28 0.948230
\(955\) −1.47252e28 −0.721155
\(956\) −8.30141e27 −0.402534
\(957\) −9.34363e23 −4.48593e−5 0
\(958\) 1.52374e28 0.724333
\(959\) 4.99789e27 0.235239
\(960\) −1.13765e28 −0.530192
\(961\) −1.07239e28 −0.494857
\(962\) −3.89471e27 −0.177956
\(963\) 3.71365e28 1.68016
\(964\) −2.32976e28 −1.04371
\(965\) 1.23791e28 0.549136
\(966\) −5.48477e27 −0.240922
\(967\) −2.58489e28 −1.12432 −0.562160 0.827028i \(-0.690030\pi\)
−0.562160 + 0.827028i \(0.690030\pi\)
\(968\) 2.34441e28 1.00976
\(969\) −1.94976e28 −0.831583
\(970\) 6.88417e26 0.0290750
\(971\) 2.05732e28 0.860438 0.430219 0.902725i \(-0.358436\pi\)
0.430219 + 0.902725i \(0.358436\pi\)
\(972\) 2.07144e28 0.857912
\(973\) −2.34271e27 −0.0960831
\(974\) −2.21698e28 −0.900435
\(975\) −3.42777e28 −1.37869
\(976\) −2.08955e26 −0.00832300
\(977\) −2.34524e28 −0.925099 −0.462550 0.886593i \(-0.653065\pi\)
−0.462550 + 0.886593i \(0.653065\pi\)
\(978\) −1.34141e28 −0.524014
\(979\) −7.08664e25 −0.00274160
\(980\) −8.35784e27 −0.320218
\(981\) −2.58468e28 −0.980736
\(982\) 4.49949e27 0.169085
\(983\) 4.25526e28 1.58368 0.791840 0.610729i \(-0.209123\pi\)
0.791840 + 0.610729i \(0.209123\pi\)
\(984\) 6.42684e28 2.36889
\(985\) −5.76245e27 −0.210360
\(986\) 1.16778e26 0.00422213
\(987\) −2.99806e27 −0.107356
\(988\) 2.72806e28 0.967527
\(989\) 1.58320e28 0.556123
\(990\) 2.46649e25 0.000858113 0
\(991\) 5.21472e28 1.79693 0.898466 0.439044i \(-0.144683\pi\)
0.898466 + 0.439044i \(0.144683\pi\)
\(992\) −2.06010e28 −0.703117
\(993\) −5.33556e28 −1.80369
\(994\) 1.37241e27 0.0459529
\(995\) 2.51785e27 0.0835046
\(996\) −1.93655e28 −0.636159
\(997\) −1.61541e27 −0.0525629 −0.0262815 0.999655i \(-0.508367\pi\)
−0.0262815 + 0.999655i \(0.508367\pi\)
\(998\) −1.61493e28 −0.520490
\(999\) −2.06867e27 −0.0660419
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 1.20.a.a.1.1 1
3.2 odd 2 9.20.a.a.1.1 1
4.3 odd 2 16.20.a.a.1.1 1
5.2 odd 4 25.20.b.a.24.2 2
5.3 odd 4 25.20.b.a.24.1 2
5.4 even 2 25.20.a.a.1.1 1
7.6 odd 2 49.20.a.b.1.1 1
8.3 odd 2 64.20.a.h.1.1 1
8.5 even 2 64.20.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1.20.a.a.1.1 1 1.1 even 1 trivial
9.20.a.a.1.1 1 3.2 odd 2
16.20.a.a.1.1 1 4.3 odd 2
25.20.a.a.1.1 1 5.4 even 2
25.20.b.a.24.1 2 5.3 odd 4
25.20.b.a.24.2 2 5.2 odd 4
49.20.a.b.1.1 1 7.6 odd 2
64.20.a.b.1.1 1 8.5 even 2
64.20.a.h.1.1 1 8.3 odd 2