Properties

Label 49.12.a.a.1.1
Level $49$
Weight $12$
Character 49.1
Self dual yes
Analytic conductor $37.649$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 49.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(37.6488158474\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 49.1

$q$-expansion

\(f(q)\) \(=\) \(q-24.0000 q^{2} -252.000 q^{3} -1472.00 q^{4} -4830.00 q^{5} +6048.00 q^{6} +84480.0 q^{8} -113643. q^{9} +O(q^{10})\) \(q-24.0000 q^{2} -252.000 q^{3} -1472.00 q^{4} -4830.00 q^{5} +6048.00 q^{6} +84480.0 q^{8} -113643. q^{9} +115920. q^{10} +534612. q^{11} +370944. q^{12} +577738. q^{13} +1.21716e6 q^{15} +987136. q^{16} +6.90593e6 q^{17} +2.72743e6 q^{18} -1.06614e7 q^{19} +7.10976e6 q^{20} -1.28307e7 q^{22} +1.86433e7 q^{23} -2.12890e7 q^{24} -2.54992e7 q^{25} -1.38657e7 q^{26} +7.32791e7 q^{27} +1.28407e8 q^{29} -2.92118e7 q^{30} +5.28432e7 q^{31} -1.96706e8 q^{32} -1.34722e8 q^{33} -1.65742e8 q^{34} +1.67282e8 q^{36} -1.82213e8 q^{37} +2.55874e8 q^{38} -1.45590e8 q^{39} -4.08038e8 q^{40} -3.08120e8 q^{41} -1.71257e7 q^{43} -7.86949e8 q^{44} +5.48896e8 q^{45} -4.47439e8 q^{46} -2.68735e9 q^{47} -2.48758e8 q^{48} +6.11981e8 q^{50} -1.74030e9 q^{51} -8.50430e8 q^{52} -1.59606e9 q^{53} -1.75870e9 q^{54} -2.58218e9 q^{55} +2.68668e9 q^{57} -3.08176e9 q^{58} +5.18920e9 q^{59} -1.79166e9 q^{60} -6.95648e9 q^{61} -1.26824e9 q^{62} +2.69930e9 q^{64} -2.79047e9 q^{65} +3.23333e9 q^{66} -1.54818e10 q^{67} -1.01655e10 q^{68} -4.69810e9 q^{69} +9.79149e9 q^{71} -9.60056e9 q^{72} -1.46379e9 q^{73} +4.37312e9 q^{74} +6.42580e9 q^{75} +1.56936e10 q^{76} +3.49416e9 q^{78} +3.81168e10 q^{79} -4.76787e9 q^{80} +1.66519e9 q^{81} +7.39489e9 q^{82} +2.93351e10 q^{83} -3.33557e10 q^{85} +4.11017e8 q^{86} -3.23585e10 q^{87} +4.51640e10 q^{88} +2.49929e10 q^{89} -1.31735e10 q^{90} -2.74429e10 q^{92} -1.33165e10 q^{93} +6.44964e10 q^{94} +5.14947e10 q^{95} +4.95700e10 q^{96} -7.50136e10 q^{97} -6.07549e10 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −24.0000 −0.530330 −0.265165 0.964203i \(-0.585426\pi\)
−0.265165 + 0.964203i \(0.585426\pi\)
\(3\) −252.000 −0.598734 −0.299367 0.954138i \(-0.596775\pi\)
−0.299367 + 0.954138i \(0.596775\pi\)
\(4\) −1472.00 −0.718750
\(5\) −4830.00 −0.691213 −0.345607 0.938379i \(-0.612327\pi\)
−0.345607 + 0.938379i \(0.612327\pi\)
\(6\) 6048.00 0.317526
\(7\) 0 0
\(8\) 84480.0 0.911505
\(9\) −113643. −0.641518
\(10\) 115920. 0.366571
\(11\) 534612. 1.00087 0.500436 0.865773i \(-0.333173\pi\)
0.500436 + 0.865773i \(0.333173\pi\)
\(12\) 370944. 0.430340
\(13\) 577738. 0.431561 0.215781 0.976442i \(-0.430770\pi\)
0.215781 + 0.976442i \(0.430770\pi\)
\(14\) 0 0
\(15\) 1.21716e6 0.413853
\(16\) 987136. 0.235352
\(17\) 6.90593e6 1.17965 0.589825 0.807531i \(-0.299197\pi\)
0.589825 + 0.807531i \(0.299197\pi\)
\(18\) 2.72743e6 0.340216
\(19\) −1.06614e7 −0.987803 −0.493901 0.869518i \(-0.664430\pi\)
−0.493901 + 0.869518i \(0.664430\pi\)
\(20\) 7.10976e6 0.496810
\(21\) 0 0
\(22\) −1.28307e7 −0.530793
\(23\) 1.86433e7 0.603975 0.301988 0.953312i \(-0.402350\pi\)
0.301988 + 0.953312i \(0.402350\pi\)
\(24\) −2.12890e7 −0.545749
\(25\) −2.54992e7 −0.522224
\(26\) −1.38657e7 −0.228870
\(27\) 7.32791e7 0.982832
\(28\) 0 0
\(29\) 1.28407e8 1.16251 0.581257 0.813720i \(-0.302561\pi\)
0.581257 + 0.813720i \(0.302561\pi\)
\(30\) −2.92118e7 −0.219479
\(31\) 5.28432e7 0.331512 0.165756 0.986167i \(-0.446994\pi\)
0.165756 + 0.986167i \(0.446994\pi\)
\(32\) −1.96706e8 −1.03632
\(33\) −1.34722e8 −0.599256
\(34\) −1.65742e8 −0.625604
\(35\) 0 0
\(36\) 1.67282e8 0.461091
\(37\) −1.82213e8 −0.431987 −0.215993 0.976395i \(-0.569299\pi\)
−0.215993 + 0.976395i \(0.569299\pi\)
\(38\) 2.55874e8 0.523862
\(39\) −1.45590e8 −0.258390
\(40\) −4.08038e8 −0.630044
\(41\) −3.08120e8 −0.415345 −0.207673 0.978198i \(-0.566589\pi\)
−0.207673 + 0.978198i \(0.566589\pi\)
\(42\) 0 0
\(43\) −1.71257e7 −0.0177653 −0.00888264 0.999961i \(-0.502827\pi\)
−0.00888264 + 0.999961i \(0.502827\pi\)
\(44\) −7.86949e8 −0.719377
\(45\) 5.48896e8 0.443426
\(46\) −4.47439e8 −0.320306
\(47\) −2.68735e9 −1.70917 −0.854586 0.519310i \(-0.826189\pi\)
−0.854586 + 0.519310i \(0.826189\pi\)
\(48\) −2.48758e8 −0.140913
\(49\) 0 0
\(50\) 6.11981e8 0.276951
\(51\) −1.74030e9 −0.706296
\(52\) −8.50430e8 −0.310185
\(53\) −1.59606e9 −0.524241 −0.262120 0.965035i \(-0.584422\pi\)
−0.262120 + 0.965035i \(0.584422\pi\)
\(54\) −1.75870e9 −0.521225
\(55\) −2.58218e9 −0.691817
\(56\) 0 0
\(57\) 2.68668e9 0.591431
\(58\) −3.08176e9 −0.616517
\(59\) 5.18920e9 0.944963 0.472481 0.881341i \(-0.343358\pi\)
0.472481 + 0.881341i \(0.343358\pi\)
\(60\) −1.79166e9 −0.297457
\(61\) −6.95648e9 −1.05457 −0.527285 0.849689i \(-0.676790\pi\)
−0.527285 + 0.849689i \(0.676790\pi\)
\(62\) −1.26824e9 −0.175811
\(63\) 0 0
\(64\) 2.69930e9 0.314240
\(65\) −2.79047e9 −0.298301
\(66\) 3.23333e9 0.317804
\(67\) −1.54818e10 −1.40091 −0.700456 0.713696i \(-0.747020\pi\)
−0.700456 + 0.713696i \(0.747020\pi\)
\(68\) −1.01655e10 −0.847874
\(69\) −4.69810e9 −0.361620
\(70\) 0 0
\(71\) 9.79149e9 0.644062 0.322031 0.946729i \(-0.395634\pi\)
0.322031 + 0.946729i \(0.395634\pi\)
\(72\) −9.60056e9 −0.584747
\(73\) −1.46379e9 −0.0826425 −0.0413212 0.999146i \(-0.513157\pi\)
−0.0413212 + 0.999146i \(0.513157\pi\)
\(74\) 4.37312e9 0.229096
\(75\) 6.42580e9 0.312673
\(76\) 1.56936e10 0.709983
\(77\) 0 0
\(78\) 3.49416e9 0.137032
\(79\) 3.81168e10 1.39370 0.696848 0.717219i \(-0.254585\pi\)
0.696848 + 0.717219i \(0.254585\pi\)
\(80\) −4.76787e9 −0.162678
\(81\) 1.66519e9 0.0530635
\(82\) 7.39489e9 0.220270
\(83\) 2.93351e10 0.817444 0.408722 0.912659i \(-0.365975\pi\)
0.408722 + 0.912659i \(0.365975\pi\)
\(84\) 0 0
\(85\) −3.33557e10 −0.815390
\(86\) 4.11017e8 0.00942146
\(87\) −3.23585e10 −0.696037
\(88\) 4.51640e10 0.912300
\(89\) 2.49929e10 0.474430 0.237215 0.971457i \(-0.423765\pi\)
0.237215 + 0.971457i \(0.423765\pi\)
\(90\) −1.31735e10 −0.235162
\(91\) 0 0
\(92\) −2.74429e10 −0.434107
\(93\) −1.33165e10 −0.198488
\(94\) 6.44964e10 0.906425
\(95\) 5.14947e10 0.682782
\(96\) 4.95700e10 0.620479
\(97\) −7.50136e10 −0.886942 −0.443471 0.896289i \(-0.646253\pi\)
−0.443471 + 0.896289i \(0.646253\pi\)
\(98\) 0 0
\(99\) −6.07549e10 −0.642078
\(100\) 3.75349e10 0.375349
\(101\) −8.17430e10 −0.773896 −0.386948 0.922101i \(-0.626471\pi\)
−0.386948 + 0.922101i \(0.626471\pi\)
\(102\) 4.17671e10 0.374570
\(103\) 2.25755e11 1.91881 0.959407 0.282025i \(-0.0910061\pi\)
0.959407 + 0.282025i \(0.0910061\pi\)
\(104\) 4.88073e10 0.393370
\(105\) 0 0
\(106\) 3.83053e10 0.278021
\(107\) 9.02413e10 0.622006 0.311003 0.950409i \(-0.399335\pi\)
0.311003 + 0.950409i \(0.399335\pi\)
\(108\) −1.07867e11 −0.706411
\(109\) 7.34827e10 0.457445 0.228723 0.973492i \(-0.426545\pi\)
0.228723 + 0.973492i \(0.426545\pi\)
\(110\) 6.19722e10 0.366891
\(111\) 4.59178e10 0.258645
\(112\) 0 0
\(113\) −8.51469e10 −0.434748 −0.217374 0.976088i \(-0.569749\pi\)
−0.217374 + 0.976088i \(0.569749\pi\)
\(114\) −6.44803e10 −0.313654
\(115\) −9.00470e10 −0.417476
\(116\) −1.89015e11 −0.835557
\(117\) −6.56559e10 −0.276854
\(118\) −1.24541e11 −0.501142
\(119\) 0 0
\(120\) 1.02826e11 0.377229
\(121\) 4.98320e8 0.00174658
\(122\) 1.66955e11 0.559270
\(123\) 7.76464e10 0.248681
\(124\) −7.77851e10 −0.238274
\(125\) 3.59001e11 1.05218
\(126\) 0 0
\(127\) −2.62717e11 −0.705615 −0.352808 0.935696i \(-0.614773\pi\)
−0.352808 + 0.935696i \(0.614773\pi\)
\(128\) 3.38071e11 0.869668
\(129\) 4.31568e9 0.0106367
\(130\) 6.69714e10 0.158198
\(131\) −6.31529e11 −1.43021 −0.715107 0.699015i \(-0.753622\pi\)
−0.715107 + 0.699015i \(0.753622\pi\)
\(132\) 1.98311e11 0.430715
\(133\) 0 0
\(134\) 3.71564e11 0.742946
\(135\) −3.53938e11 −0.679347
\(136\) 5.83413e11 1.07526
\(137\) −2.97199e11 −0.526119 −0.263059 0.964780i \(-0.584732\pi\)
−0.263059 + 0.964780i \(0.584732\pi\)
\(138\) 1.12755e11 0.191778
\(139\) −5.96794e11 −0.975535 −0.487767 0.872974i \(-0.662189\pi\)
−0.487767 + 0.872974i \(0.662189\pi\)
\(140\) 0 0
\(141\) 6.77212e11 1.02334
\(142\) −2.34996e11 −0.341565
\(143\) 3.08866e11 0.431938
\(144\) −1.12181e11 −0.150982
\(145\) −6.20204e11 −0.803546
\(146\) 3.51310e10 0.0438278
\(147\) 0 0
\(148\) 2.68218e11 0.310491
\(149\) −1.11543e12 −1.24428 −0.622142 0.782905i \(-0.713737\pi\)
−0.622142 + 0.782905i \(0.713737\pi\)
\(150\) −1.54219e11 −0.165820
\(151\) −8.24447e11 −0.854653 −0.427326 0.904097i \(-0.640544\pi\)
−0.427326 + 0.904097i \(0.640544\pi\)
\(152\) −9.00677e11 −0.900387
\(153\) −7.84811e11 −0.756767
\(154\) 0 0
\(155\) −2.55233e11 −0.229146
\(156\) 2.14308e11 0.185718
\(157\) −1.31512e12 −1.10031 −0.550156 0.835062i \(-0.685432\pi\)
−0.550156 + 0.835062i \(0.685432\pi\)
\(158\) −9.14804e11 −0.739119
\(159\) 4.02206e11 0.313881
\(160\) 9.50091e11 0.716317
\(161\) 0 0
\(162\) −3.99645e10 −0.0281412
\(163\) −3.57833e11 −0.243584 −0.121792 0.992556i \(-0.538864\pi\)
−0.121792 + 0.992556i \(0.538864\pi\)
\(164\) 4.53553e11 0.298529
\(165\) 6.50708e11 0.414214
\(166\) −7.04042e11 −0.433515
\(167\) −2.75483e12 −1.64117 −0.820587 0.571521i \(-0.806354\pi\)
−0.820587 + 0.571521i \(0.806354\pi\)
\(168\) 0 0
\(169\) −1.45838e12 −0.813755
\(170\) 8.00536e11 0.432426
\(171\) 1.21160e12 0.633693
\(172\) 2.52090e10 0.0127688
\(173\) 9.50387e11 0.466280 0.233140 0.972443i \(-0.425100\pi\)
0.233140 + 0.972443i \(0.425100\pi\)
\(174\) 7.76603e11 0.369129
\(175\) 0 0
\(176\) 5.27735e11 0.235557
\(177\) −1.30768e12 −0.565781
\(178\) −5.99830e11 −0.251604
\(179\) 1.68138e12 0.683873 0.341936 0.939723i \(-0.388917\pi\)
0.341936 + 0.939723i \(0.388917\pi\)
\(180\) −8.07974e11 −0.318712
\(181\) 9.96774e11 0.381386 0.190693 0.981650i \(-0.438927\pi\)
0.190693 + 0.981650i \(0.438927\pi\)
\(182\) 0 0
\(183\) 1.75303e12 0.631406
\(184\) 1.57498e12 0.550526
\(185\) 8.80090e11 0.298595
\(186\) 3.19595e11 0.105264
\(187\) 3.69200e12 1.18068
\(188\) 3.95578e12 1.22847
\(189\) 0 0
\(190\) −1.23587e12 −0.362100
\(191\) 2.76240e12 0.786328 0.393164 0.919468i \(-0.371381\pi\)
0.393164 + 0.919468i \(0.371381\pi\)
\(192\) −6.80223e11 −0.188146
\(193\) 5.44239e12 1.46293 0.731466 0.681878i \(-0.238836\pi\)
0.731466 + 0.681878i \(0.238836\pi\)
\(194\) 1.80033e12 0.470372
\(195\) 7.03200e11 0.178603
\(196\) 0 0
\(197\) −2.87609e12 −0.690619 −0.345309 0.938489i \(-0.612226\pi\)
−0.345309 + 0.938489i \(0.612226\pi\)
\(198\) 1.45812e12 0.340513
\(199\) −7.28391e11 −0.165452 −0.0827262 0.996572i \(-0.526363\pi\)
−0.0827262 + 0.996572i \(0.526363\pi\)
\(200\) −2.15417e12 −0.476010
\(201\) 3.90142e12 0.838773
\(202\) 1.96183e12 0.410421
\(203\) 0 0
\(204\) 2.56171e12 0.507651
\(205\) 1.48822e12 0.287092
\(206\) −5.41812e12 −1.01760
\(207\) −2.11868e12 −0.387461
\(208\) 5.70306e11 0.101569
\(209\) −5.69972e12 −0.988665
\(210\) 0 0
\(211\) −6.79317e12 −1.11820 −0.559099 0.829101i \(-0.688853\pi\)
−0.559099 + 0.829101i \(0.688853\pi\)
\(212\) 2.34939e12 0.376798
\(213\) −2.46745e12 −0.385622
\(214\) −2.16579e12 −0.329868
\(215\) 8.27172e10 0.0122796
\(216\) 6.19062e12 0.895856
\(217\) 0 0
\(218\) −1.76358e12 −0.242597
\(219\) 3.68875e11 0.0494808
\(220\) 3.80096e12 0.497243
\(221\) 3.98982e12 0.509092
\(222\) −1.10203e12 −0.137167
\(223\) −7.33486e12 −0.890667 −0.445333 0.895365i \(-0.646915\pi\)
−0.445333 + 0.895365i \(0.646915\pi\)
\(224\) 0 0
\(225\) 2.89781e12 0.335016
\(226\) 2.04352e12 0.230560
\(227\) 1.35984e12 0.149743 0.0748713 0.997193i \(-0.476145\pi\)
0.0748713 + 0.997193i \(0.476145\pi\)
\(228\) −3.95479e12 −0.425091
\(229\) 1.18244e13 1.24075 0.620375 0.784305i \(-0.286980\pi\)
0.620375 + 0.784305i \(0.286980\pi\)
\(230\) 2.16113e12 0.221400
\(231\) 0 0
\(232\) 1.08478e13 1.05964
\(233\) −1.75634e13 −1.67552 −0.837761 0.546038i \(-0.816135\pi\)
−0.837761 + 0.546038i \(0.816135\pi\)
\(234\) 1.57574e12 0.146824
\(235\) 1.29799e13 1.18140
\(236\) −7.63851e12 −0.679192
\(237\) −9.60545e12 −0.834452
\(238\) 0 0
\(239\) −7.13958e12 −0.592221 −0.296111 0.955154i \(-0.595690\pi\)
−0.296111 + 0.955154i \(0.595690\pi\)
\(240\) 1.20150e12 0.0974009
\(241\) 2.31307e11 0.0183271 0.00916357 0.999958i \(-0.497083\pi\)
0.00916357 + 0.999958i \(0.497083\pi\)
\(242\) −1.19597e10 −0.000926264 0
\(243\) −1.34008e13 −1.01460
\(244\) 1.02399e13 0.757972
\(245\) 0 0
\(246\) −1.86351e12 −0.131883
\(247\) −6.15951e12 −0.426297
\(248\) 4.46419e12 0.302175
\(249\) −7.39245e12 −0.489431
\(250\) −8.61603e12 −0.558004
\(251\) −1.29831e13 −0.822567 −0.411284 0.911507i \(-0.634919\pi\)
−0.411284 + 0.911507i \(0.634919\pi\)
\(252\) 0 0
\(253\) 9.96692e12 0.604502
\(254\) 6.30521e12 0.374209
\(255\) 8.40563e12 0.488201
\(256\) −1.36419e13 −0.775451
\(257\) −2.39612e13 −1.33314 −0.666571 0.745442i \(-0.732239\pi\)
−0.666571 + 0.745442i \(0.732239\pi\)
\(258\) −1.03576e11 −0.00564095
\(259\) 0 0
\(260\) 4.10758e12 0.214404
\(261\) −1.45925e13 −0.745774
\(262\) 1.51567e13 0.758485
\(263\) −2.42737e13 −1.18954 −0.594771 0.803895i \(-0.702757\pi\)
−0.594771 + 0.803895i \(0.702757\pi\)
\(264\) −1.13813e13 −0.546225
\(265\) 7.70895e12 0.362362
\(266\) 0 0
\(267\) −6.29822e12 −0.284057
\(268\) 2.27892e13 1.00691
\(269\) −2.58377e13 −1.11845 −0.559225 0.829016i \(-0.688901\pi\)
−0.559225 + 0.829016i \(0.688901\pi\)
\(270\) 8.49451e12 0.360278
\(271\) 3.76793e12 0.156593 0.0782964 0.996930i \(-0.475052\pi\)
0.0782964 + 0.996930i \(0.475052\pi\)
\(272\) 6.81710e12 0.277633
\(273\) 0 0
\(274\) 7.13277e12 0.279017
\(275\) −1.36322e13 −0.522680
\(276\) 6.91561e12 0.259915
\(277\) −1.64189e13 −0.604931 −0.302466 0.953160i \(-0.597810\pi\)
−0.302466 + 0.953160i \(0.597810\pi\)
\(278\) 1.43230e13 0.517355
\(279\) −6.00526e12 −0.212671
\(280\) 0 0
\(281\) 2.10357e13 0.716263 0.358132 0.933671i \(-0.383414\pi\)
0.358132 + 0.933671i \(0.383414\pi\)
\(282\) −1.62531e13 −0.542707
\(283\) −1.67132e13 −0.547310 −0.273655 0.961828i \(-0.588233\pi\)
−0.273655 + 0.961828i \(0.588233\pi\)
\(284\) −1.44131e13 −0.462920
\(285\) −1.29767e13 −0.408805
\(286\) −7.41278e12 −0.229070
\(287\) 0 0
\(288\) 2.23543e13 0.664817
\(289\) 1.34200e13 0.391575
\(290\) 1.48849e13 0.426144
\(291\) 1.89034e13 0.531042
\(292\) 2.15470e12 0.0593993
\(293\) 2.39269e13 0.647312 0.323656 0.946175i \(-0.395088\pi\)
0.323656 + 0.946175i \(0.395088\pi\)
\(294\) 0 0
\(295\) −2.50639e13 −0.653171
\(296\) −1.53934e13 −0.393758
\(297\) 3.91759e13 0.983690
\(298\) 2.67704e13 0.659881
\(299\) 1.07709e13 0.260652
\(300\) −9.45878e12 −0.224734
\(301\) 0 0
\(302\) 1.97867e13 0.453248
\(303\) 2.05992e13 0.463358
\(304\) −1.05243e13 −0.232481
\(305\) 3.35998e13 0.728933
\(306\) 1.88355e13 0.401336
\(307\) −1.53111e13 −0.320439 −0.160219 0.987081i \(-0.551220\pi\)
−0.160219 + 0.987081i \(0.551220\pi\)
\(308\) 0 0
\(309\) −5.68903e13 −1.14886
\(310\) 6.12558e12 0.121523
\(311\) −4.98752e13 −0.972080 −0.486040 0.873936i \(-0.661559\pi\)
−0.486040 + 0.873936i \(0.661559\pi\)
\(312\) −1.22994e13 −0.235524
\(313\) 9.94808e13 1.87174 0.935870 0.352345i \(-0.114616\pi\)
0.935870 + 0.352345i \(0.114616\pi\)
\(314\) 3.15628e13 0.583529
\(315\) 0 0
\(316\) −5.61080e13 −1.00172
\(317\) 8.33692e13 1.46278 0.731392 0.681958i \(-0.238871\pi\)
0.731392 + 0.681958i \(0.238871\pi\)
\(318\) −9.65294e12 −0.166460
\(319\) 6.86477e13 1.16353
\(320\) −1.30376e13 −0.217207
\(321\) −2.27408e13 −0.372416
\(322\) 0 0
\(323\) −7.36271e13 −1.16526
\(324\) −2.45116e12 −0.0381394
\(325\) −1.47319e13 −0.225372
\(326\) 8.58799e12 0.129180
\(327\) −1.85176e13 −0.273888
\(328\) −2.60300e13 −0.378589
\(329\) 0 0
\(330\) −1.56170e13 −0.219670
\(331\) −6.35840e13 −0.879618 −0.439809 0.898091i \(-0.644954\pi\)
−0.439809 + 0.898091i \(0.644954\pi\)
\(332\) −4.31813e13 −0.587538
\(333\) 2.07073e13 0.277127
\(334\) 6.61160e13 0.870364
\(335\) 7.47772e13 0.968329
\(336\) 0 0
\(337\) 1.21001e14 1.51644 0.758221 0.651997i \(-0.226069\pi\)
0.758221 + 0.651997i \(0.226069\pi\)
\(338\) 3.50011e13 0.431559
\(339\) 2.14570e13 0.260298
\(340\) 4.90995e13 0.586062
\(341\) 2.82506e13 0.331802
\(342\) −2.90783e13 −0.336067
\(343\) 0 0
\(344\) −1.44678e12 −0.0161931
\(345\) 2.26918e13 0.249957
\(346\) −2.28093e13 −0.247283
\(347\) −1.55662e14 −1.66100 −0.830499 0.557020i \(-0.811945\pi\)
−0.830499 + 0.557020i \(0.811945\pi\)
\(348\) 4.76317e13 0.500276
\(349\) 2.56430e13 0.265112 0.132556 0.991176i \(-0.457682\pi\)
0.132556 + 0.991176i \(0.457682\pi\)
\(350\) 0 0
\(351\) 4.23361e13 0.424152
\(352\) −1.05162e14 −1.03722
\(353\) −2.49098e13 −0.241885 −0.120943 0.992659i \(-0.538592\pi\)
−0.120943 + 0.992659i \(0.538592\pi\)
\(354\) 3.13843e13 0.300051
\(355\) −4.72929e13 −0.445184
\(356\) −3.67896e13 −0.340996
\(357\) 0 0
\(358\) −4.03532e13 −0.362678
\(359\) 1.57584e14 1.39474 0.697370 0.716712i \(-0.254354\pi\)
0.697370 + 0.716712i \(0.254354\pi\)
\(360\) 4.63707e13 0.404185
\(361\) −2.82438e12 −0.0242457
\(362\) −2.39226e13 −0.202260
\(363\) −1.25577e11 −0.00104574
\(364\) 0 0
\(365\) 7.07011e12 0.0571236
\(366\) −4.20728e13 −0.334854
\(367\) 1.77901e14 1.39481 0.697406 0.716676i \(-0.254338\pi\)
0.697406 + 0.716676i \(0.254338\pi\)
\(368\) 1.84034e13 0.142146
\(369\) 3.50157e13 0.266452
\(370\) −2.11222e13 −0.158354
\(371\) 0 0
\(372\) 1.96019e13 0.142663
\(373\) −5.51617e13 −0.395585 −0.197792 0.980244i \(-0.563377\pi\)
−0.197792 + 0.980244i \(0.563377\pi\)
\(374\) −8.86079e13 −0.626150
\(375\) −9.04683e13 −0.629976
\(376\) −2.27027e14 −1.55792
\(377\) 7.41854e13 0.501696
\(378\) 0 0
\(379\) 1.46463e14 0.962083 0.481042 0.876698i \(-0.340259\pi\)
0.481042 + 0.876698i \(0.340259\pi\)
\(380\) −7.58001e13 −0.490750
\(381\) 6.62047e13 0.422476
\(382\) −6.62977e13 −0.417013
\(383\) −2.31450e14 −1.43504 −0.717519 0.696539i \(-0.754722\pi\)
−0.717519 + 0.696539i \(0.754722\pi\)
\(384\) −8.51940e13 −0.520700
\(385\) 0 0
\(386\) −1.30617e14 −0.775837
\(387\) 1.94622e12 0.0113967
\(388\) 1.10420e14 0.637490
\(389\) −1.49872e14 −0.853093 −0.426547 0.904466i \(-0.640270\pi\)
−0.426547 + 0.904466i \(0.640270\pi\)
\(390\) −1.68768e13 −0.0947184
\(391\) 1.28749e14 0.712480
\(392\) 0 0
\(393\) 1.59145e14 0.856317
\(394\) 6.90262e13 0.366256
\(395\) −1.84104e14 −0.963341
\(396\) 8.94312e13 0.461494
\(397\) −2.08111e14 −1.05912 −0.529562 0.848271i \(-0.677644\pi\)
−0.529562 + 0.848271i \(0.677644\pi\)
\(398\) 1.74814e13 0.0877443
\(399\) 0 0
\(400\) −2.51712e13 −0.122906
\(401\) −1.33408e14 −0.642521 −0.321261 0.946991i \(-0.604107\pi\)
−0.321261 + 0.946991i \(0.604107\pi\)
\(402\) −9.36341e13 −0.444827
\(403\) 3.05295e13 0.143068
\(404\) 1.20326e14 0.556238
\(405\) −8.04286e12 −0.0366782
\(406\) 0 0
\(407\) −9.74134e13 −0.432364
\(408\) −1.47020e14 −0.643793
\(409\) 2.06168e14 0.890722 0.445361 0.895351i \(-0.353075\pi\)
0.445361 + 0.895351i \(0.353075\pi\)
\(410\) −3.57173e13 −0.152254
\(411\) 7.48941e13 0.315005
\(412\) −3.32312e14 −1.37915
\(413\) 0 0
\(414\) 5.08483e13 0.205482
\(415\) −1.41689e14 −0.565028
\(416\) −1.13645e14 −0.447235
\(417\) 1.50392e14 0.584085
\(418\) 1.36793e14 0.524319
\(419\) −7.34035e13 −0.277677 −0.138838 0.990315i \(-0.544337\pi\)
−0.138838 + 0.990315i \(0.544337\pi\)
\(420\) 0 0
\(421\) 1.71112e14 0.630563 0.315282 0.948998i \(-0.397901\pi\)
0.315282 + 0.948998i \(0.397901\pi\)
\(422\) 1.63036e14 0.593014
\(423\) 3.05398e14 1.09646
\(424\) −1.34835e14 −0.477848
\(425\) −1.76096e14 −0.616042
\(426\) 5.92189e13 0.204507
\(427\) 0 0
\(428\) −1.32835e14 −0.447067
\(429\) −7.78341e13 −0.258616
\(430\) −1.98521e12 −0.00651224
\(431\) −7.17758e13 −0.232463 −0.116231 0.993222i \(-0.537081\pi\)
−0.116231 + 0.993222i \(0.537081\pi\)
\(432\) 7.23364e13 0.231311
\(433\) −9.98812e13 −0.315356 −0.157678 0.987491i \(-0.550401\pi\)
−0.157678 + 0.987491i \(0.550401\pi\)
\(434\) 0 0
\(435\) 1.56291e14 0.481110
\(436\) −1.08166e14 −0.328789
\(437\) −1.98764e14 −0.596608
\(438\) −8.85301e12 −0.0262412
\(439\) 2.90312e13 0.0849788 0.0424894 0.999097i \(-0.486471\pi\)
0.0424894 + 0.999097i \(0.486471\pi\)
\(440\) −2.18142e14 −0.630594
\(441\) 0 0
\(442\) −9.57557e13 −0.269987
\(443\) 3.28370e14 0.914414 0.457207 0.889360i \(-0.348850\pi\)
0.457207 + 0.889360i \(0.348850\pi\)
\(444\) −6.75909e13 −0.185901
\(445\) −1.20716e14 −0.327932
\(446\) 1.76037e14 0.472347
\(447\) 2.81089e14 0.744994
\(448\) 0 0
\(449\) −6.12368e14 −1.58364 −0.791822 0.610752i \(-0.790867\pi\)
−0.791822 + 0.610752i \(0.790867\pi\)
\(450\) −6.95474e13 −0.177669
\(451\) −1.64725e14 −0.415708
\(452\) 1.25336e14 0.312475
\(453\) 2.07761e14 0.511709
\(454\) −3.26361e13 −0.0794130
\(455\) 0 0
\(456\) 2.26971e14 0.539092
\(457\) 3.03483e14 0.712189 0.356095 0.934450i \(-0.384108\pi\)
0.356095 + 0.934450i \(0.384108\pi\)
\(458\) −2.83786e14 −0.658007
\(459\) 5.06060e14 1.15940
\(460\) 1.32549e14 0.300061
\(461\) 7.29308e14 1.63138 0.815691 0.578487i \(-0.196357\pi\)
0.815691 + 0.578487i \(0.196357\pi\)
\(462\) 0 0
\(463\) 1.22188e14 0.266891 0.133445 0.991056i \(-0.457396\pi\)
0.133445 + 0.991056i \(0.457396\pi\)
\(464\) 1.26755e14 0.273600
\(465\) 6.43186e13 0.137197
\(466\) 4.21520e14 0.888579
\(467\) 6.17381e14 1.28621 0.643103 0.765780i \(-0.277647\pi\)
0.643103 + 0.765780i \(0.277647\pi\)
\(468\) 9.66455e13 0.198989
\(469\) 0 0
\(470\) −3.11517e14 −0.626533
\(471\) 3.31409e14 0.658794
\(472\) 4.38384e14 0.861338
\(473\) −9.15561e12 −0.0177808
\(474\) 2.30531e14 0.442535
\(475\) 2.71858e14 0.515854
\(476\) 0 0
\(477\) 1.81381e14 0.336310
\(478\) 1.71350e14 0.314073
\(479\) −1.05084e15 −1.90410 −0.952052 0.305938i \(-0.901030\pi\)
−0.952052 + 0.305938i \(0.901030\pi\)
\(480\) −2.39423e14 −0.428883
\(481\) −1.05272e14 −0.186429
\(482\) −5.55137e12 −0.00971944
\(483\) 0 0
\(484\) −7.33527e11 −0.00125536
\(485\) 3.62316e14 0.613066
\(486\) 3.21619e14 0.538074
\(487\) −2.19910e14 −0.363777 −0.181889 0.983319i \(-0.558221\pi\)
−0.181889 + 0.983319i \(0.558221\pi\)
\(488\) −5.87683e14 −0.961246
\(489\) 9.01739e13 0.145842
\(490\) 0 0
\(491\) −4.83863e14 −0.765199 −0.382599 0.923914i \(-0.624971\pi\)
−0.382599 + 0.923914i \(0.624971\pi\)
\(492\) −1.14295e14 −0.178740
\(493\) 8.86768e14 1.37136
\(494\) 1.47828e14 0.226078
\(495\) 2.93446e14 0.443813
\(496\) 5.21634e13 0.0780219
\(497\) 0 0
\(498\) 1.77419e14 0.259560
\(499\) −1.08878e14 −0.157538 −0.0787691 0.996893i \(-0.525099\pi\)
−0.0787691 + 0.996893i \(0.525099\pi\)
\(500\) −5.28450e14 −0.756256
\(501\) 6.94218e14 0.982626
\(502\) 3.11593e14 0.436232
\(503\) −5.06588e14 −0.701506 −0.350753 0.936468i \(-0.614074\pi\)
−0.350753 + 0.936468i \(0.614074\pi\)
\(504\) 0 0
\(505\) 3.94818e14 0.534927
\(506\) −2.39206e14 −0.320586
\(507\) 3.67512e14 0.487222
\(508\) 3.86720e14 0.507161
\(509\) −8.57534e13 −0.111251 −0.0556254 0.998452i \(-0.517715\pi\)
−0.0556254 + 0.998452i \(0.517715\pi\)
\(510\) −2.01735e14 −0.258908
\(511\) 0 0
\(512\) −3.64965e14 −0.458423
\(513\) −7.81259e14 −0.970844
\(514\) 5.75069e14 0.707005
\(515\) −1.09040e15 −1.32631
\(516\) −6.35268e12 −0.00764511
\(517\) −1.43669e15 −1.71066
\(518\) 0 0
\(519\) −2.39498e14 −0.279178
\(520\) −2.35739e14 −0.271903
\(521\) −9.27575e14 −1.05862 −0.529312 0.848428i \(-0.677550\pi\)
−0.529312 + 0.848428i \(0.677550\pi\)
\(522\) 3.50220e14 0.395506
\(523\) 2.18187e13 0.0243820 0.0121910 0.999926i \(-0.496119\pi\)
0.0121910 + 0.999926i \(0.496119\pi\)
\(524\) 9.29610e14 1.02797
\(525\) 0 0
\(526\) 5.82569e14 0.630850
\(527\) 3.64931e14 0.391069
\(528\) −1.32989e14 −0.141036
\(529\) −6.05238e14 −0.635214
\(530\) −1.85015e14 −0.192172
\(531\) −5.89717e14 −0.606211
\(532\) 0 0
\(533\) −1.78013e14 −0.179247
\(534\) 1.51157e14 0.150644
\(535\) −4.35865e14 −0.429939
\(536\) −1.30790e15 −1.27694
\(537\) −4.23709e14 −0.409458
\(538\) 6.20105e14 0.593147
\(539\) 0 0
\(540\) 5.20997e14 0.488280
\(541\) −1.69527e15 −1.57273 −0.786363 0.617765i \(-0.788038\pi\)
−0.786363 + 0.617765i \(0.788038\pi\)
\(542\) −9.04304e13 −0.0830459
\(543\) −2.51187e14 −0.228349
\(544\) −1.35844e15 −1.22249
\(545\) −3.54921e14 −0.316192
\(546\) 0 0
\(547\) 7.52145e14 0.656706 0.328353 0.944555i \(-0.393506\pi\)
0.328353 + 0.944555i \(0.393506\pi\)
\(548\) 4.37477e14 0.378148
\(549\) 7.90555e14 0.676526
\(550\) 3.27173e14 0.277193
\(551\) −1.36900e15 −1.14834
\(552\) −3.96896e14 −0.329619
\(553\) 0 0
\(554\) 3.94054e14 0.320813
\(555\) −2.21783e14 −0.178779
\(556\) 8.78480e14 0.701166
\(557\) 1.87489e14 0.148174 0.0740870 0.997252i \(-0.476396\pi\)
0.0740870 + 0.997252i \(0.476396\pi\)
\(558\) 1.44126e14 0.112786
\(559\) −9.89417e12 −0.00766681
\(560\) 0 0
\(561\) −9.30383e14 −0.706913
\(562\) −5.04857e14 −0.379856
\(563\) −2.44971e14 −0.182524 −0.0912618 0.995827i \(-0.529090\pi\)
−0.0912618 + 0.995827i \(0.529090\pi\)
\(564\) −9.96856e14 −0.735525
\(565\) 4.11259e14 0.300503
\(566\) 4.01116e14 0.290255
\(567\) 0 0
\(568\) 8.27185e14 0.587066
\(569\) 1.35243e15 0.950596 0.475298 0.879825i \(-0.342340\pi\)
0.475298 + 0.879825i \(0.342340\pi\)
\(570\) 3.11440e14 0.216801
\(571\) 1.43223e15 0.987447 0.493723 0.869619i \(-0.335636\pi\)
0.493723 + 0.869619i \(0.335636\pi\)
\(572\) −4.54650e14 −0.310455
\(573\) −6.96126e14 −0.470801
\(574\) 0 0
\(575\) −4.75389e14 −0.315410
\(576\) −3.06756e14 −0.201590
\(577\) 8.77659e14 0.571293 0.285647 0.958335i \(-0.407792\pi\)
0.285647 + 0.958335i \(0.407792\pi\)
\(578\) −3.22081e14 −0.207664
\(579\) −1.37148e15 −0.875907
\(580\) 9.12940e14 0.577548
\(581\) 0 0
\(582\) −4.53682e14 −0.281628
\(583\) −8.53271e14 −0.524698
\(584\) −1.23661e14 −0.0753290
\(585\) 3.17118e14 0.191365
\(586\) −5.74245e14 −0.343289
\(587\) 2.43425e15 1.44164 0.720818 0.693124i \(-0.243766\pi\)
0.720818 + 0.693124i \(0.243766\pi\)
\(588\) 0 0
\(589\) −5.63383e14 −0.327469
\(590\) 6.01532e14 0.346396
\(591\) 7.24775e14 0.413497
\(592\) −1.79869e14 −0.101669
\(593\) 3.03318e14 0.169863 0.0849313 0.996387i \(-0.472933\pi\)
0.0849313 + 0.996387i \(0.472933\pi\)
\(594\) −9.40221e14 −0.521680
\(595\) 0 0
\(596\) 1.64192e15 0.894329
\(597\) 1.83555e14 0.0990619
\(598\) −2.58502e14 −0.138232
\(599\) −1.70198e15 −0.901795 −0.450898 0.892576i \(-0.648896\pi\)
−0.450898 + 0.892576i \(0.648896\pi\)
\(600\) 5.42852e14 0.285003
\(601\) −2.33922e15 −1.21692 −0.608458 0.793586i \(-0.708212\pi\)
−0.608458 + 0.793586i \(0.708212\pi\)
\(602\) 0 0
\(603\) 1.75940e15 0.898710
\(604\) 1.21359e15 0.614282
\(605\) −2.40689e12 −0.00120726
\(606\) −4.94381e14 −0.245733
\(607\) 2.49607e15 1.22947 0.614737 0.788732i \(-0.289262\pi\)
0.614737 + 0.788732i \(0.289262\pi\)
\(608\) 2.09717e15 1.02368
\(609\) 0 0
\(610\) −8.06395e14 −0.386575
\(611\) −1.55258e15 −0.737612
\(612\) 1.15524e15 0.543926
\(613\) 2.47301e15 1.15397 0.576983 0.816756i \(-0.304230\pi\)
0.576983 + 0.816756i \(0.304230\pi\)
\(614\) 3.67466e14 0.169938
\(615\) −3.75032e14 −0.171892
\(616\) 0 0
\(617\) 2.43368e13 0.0109571 0.00547854 0.999985i \(-0.498256\pi\)
0.00547854 + 0.999985i \(0.498256\pi\)
\(618\) 1.36537e15 0.609274
\(619\) −4.22545e15 −1.86885 −0.934425 0.356160i \(-0.884086\pi\)
−0.934425 + 0.356160i \(0.884086\pi\)
\(620\) 3.75702e14 0.164698
\(621\) 1.36616e15 0.593606
\(622\) 1.19700e15 0.515523
\(623\) 0 0
\(624\) −1.43717e14 −0.0608126
\(625\) −4.88896e14 −0.205058
\(626\) −2.38754e15 −0.992640
\(627\) 1.43633e15 0.591947
\(628\) 1.93585e15 0.790850
\(629\) −1.25835e15 −0.509594
\(630\) 0 0
\(631\) −4.26326e15 −1.69660 −0.848302 0.529513i \(-0.822375\pi\)
−0.848302 + 0.529513i \(0.822375\pi\)
\(632\) 3.22011e15 1.27036
\(633\) 1.71188e15 0.669503
\(634\) −2.00086e15 −0.775758
\(635\) 1.26892e15 0.487731
\(636\) −5.92047e14 −0.225602
\(637\) 0 0
\(638\) −1.64755e15 −0.617055
\(639\) −1.11273e15 −0.413177
\(640\) −1.63288e15 −0.601126
\(641\) 1.00830e15 0.368018 0.184009 0.982925i \(-0.441092\pi\)
0.184009 + 0.982925i \(0.441092\pi\)
\(642\) 5.45779e14 0.197503
\(643\) −3.03982e14 −0.109066 −0.0545328 0.998512i \(-0.517367\pi\)
−0.0545328 + 0.998512i \(0.517367\pi\)
\(644\) 0 0
\(645\) −2.08447e13 −0.00735221
\(646\) 1.76705e15 0.617974
\(647\) −3.43583e15 −1.19140 −0.595700 0.803207i \(-0.703125\pi\)
−0.595700 + 0.803207i \(0.703125\pi\)
\(648\) 1.40675e14 0.0483676
\(649\) 2.77421e15 0.945788
\(650\) 3.53565e14 0.119521
\(651\) 0 0
\(652\) 5.26730e14 0.175076
\(653\) −1.18539e15 −0.390695 −0.195347 0.980734i \(-0.562583\pi\)
−0.195347 + 0.980734i \(0.562583\pi\)
\(654\) 4.44423e14 0.145251
\(655\) 3.05028e15 0.988583
\(656\) −3.04157e14 −0.0977522
\(657\) 1.66350e14 0.0530167
\(658\) 0 0
\(659\) −2.26510e15 −0.709934 −0.354967 0.934879i \(-0.615508\pi\)
−0.354967 + 0.934879i \(0.615508\pi\)
\(660\) −9.57843e14 −0.297716
\(661\) 5.33012e15 1.64297 0.821484 0.570232i \(-0.193147\pi\)
0.821484 + 0.570232i \(0.193147\pi\)
\(662\) 1.52602e15 0.466488
\(663\) −1.00543e15 −0.304810
\(664\) 2.47823e15 0.745104
\(665\) 0 0
\(666\) −4.96974e14 −0.146969
\(667\) 2.39392e15 0.702130
\(668\) 4.05512e15 1.17959
\(669\) 1.84839e15 0.533272
\(670\) −1.79465e15 −0.513534
\(671\) −3.71902e15 −1.05549
\(672\) 0 0
\(673\) 4.74120e15 1.32375 0.661874 0.749615i \(-0.269761\pi\)
0.661874 + 0.749615i \(0.269761\pi\)
\(674\) −2.90403e15 −0.804215
\(675\) −1.86856e15 −0.513259
\(676\) 2.14673e15 0.584886
\(677\) 1.41307e15 0.381880 0.190940 0.981602i \(-0.438846\pi\)
0.190940 + 0.981602i \(0.438846\pi\)
\(678\) −5.14968e14 −0.138044
\(679\) 0 0
\(680\) −2.81789e15 −0.743232
\(681\) −3.42680e14 −0.0896559
\(682\) −6.78014e14 −0.175964
\(683\) −3.03116e15 −0.780359 −0.390180 0.920739i \(-0.627587\pi\)
−0.390180 + 0.920739i \(0.627587\pi\)
\(684\) −1.78347e15 −0.455467
\(685\) 1.43547e15 0.363660
\(686\) 0 0
\(687\) −2.97975e15 −0.742879
\(688\) −1.69054e13 −0.00418109
\(689\) −9.22102e14 −0.226242
\(690\) −5.44604e14 −0.132560
\(691\) 2.74731e15 0.663405 0.331703 0.943384i \(-0.392377\pi\)
0.331703 + 0.943384i \(0.392377\pi\)
\(692\) −1.39897e15 −0.335139
\(693\) 0 0
\(694\) 3.73588e15 0.880878
\(695\) 2.88251e15 0.674303
\(696\) −2.73364e15 −0.634441
\(697\) −2.12786e15 −0.489962
\(698\) −6.15433e14 −0.140597
\(699\) 4.42597e15 1.00319
\(700\) 0 0
\(701\) 5.72747e15 1.27795 0.638974 0.769228i \(-0.279359\pi\)
0.638974 + 0.769228i \(0.279359\pi\)
\(702\) −1.01607e15 −0.224941
\(703\) 1.94265e15 0.426718
\(704\) 1.44308e15 0.314514
\(705\) −3.27093e15 −0.707345
\(706\) 5.97836e14 0.128279
\(707\) 0 0
\(708\) 1.92490e15 0.406655
\(709\) 6.98326e14 0.146388 0.0731938 0.997318i \(-0.476681\pi\)
0.0731938 + 0.997318i \(0.476681\pi\)
\(710\) 1.13503e15 0.236095
\(711\) −4.33171e15 −0.894081
\(712\) 2.11140e15 0.432445
\(713\) 9.85170e14 0.200225
\(714\) 0 0
\(715\) −1.49182e15 −0.298561
\(716\) −2.47500e15 −0.491534
\(717\) 1.79917e15 0.354583
\(718\) −3.78202e15 −0.739672
\(719\) −9.70979e15 −1.88452 −0.942260 0.334882i \(-0.891304\pi\)
−0.942260 + 0.334882i \(0.891304\pi\)
\(720\) 5.41835e14 0.104361
\(721\) 0 0
\(722\) 6.77852e13 0.0128582
\(723\) −5.82893e13 −0.0109731
\(724\) −1.46725e15 −0.274121
\(725\) −3.27427e15 −0.607093
\(726\) 3.01384e12 0.000554586 0
\(727\) −2.46469e15 −0.450114 −0.225057 0.974346i \(-0.572257\pi\)
−0.225057 + 0.974346i \(0.572257\pi\)
\(728\) 0 0
\(729\) 3.08202e15 0.554413
\(730\) −1.69683e14 −0.0302944
\(731\) −1.18269e14 −0.0209568
\(732\) −2.58046e15 −0.453823
\(733\) −7.91285e15 −1.38121 −0.690607 0.723230i \(-0.742657\pi\)
−0.690607 + 0.723230i \(0.742657\pi\)
\(734\) −4.26963e15 −0.739711
\(735\) 0 0
\(736\) −3.66725e15 −0.625911
\(737\) −8.27677e15 −1.40213
\(738\) −8.40378e14 −0.141307
\(739\) −8.40694e15 −1.40312 −0.701558 0.712613i \(-0.747512\pi\)
−0.701558 + 0.712613i \(0.747512\pi\)
\(740\) −1.29549e15 −0.214615
\(741\) 1.55220e15 0.255239
\(742\) 0 0
\(743\) 1.36287e15 0.220809 0.110404 0.993887i \(-0.464785\pi\)
0.110404 + 0.993887i \(0.464785\pi\)
\(744\) −1.12498e15 −0.180922
\(745\) 5.38754e15 0.860065
\(746\) 1.32388e15 0.209790
\(747\) −3.33373e15 −0.524405
\(748\) −5.43462e15 −0.848614
\(749\) 0 0
\(750\) 2.17124e15 0.334095
\(751\) 6.81722e15 1.04133 0.520664 0.853762i \(-0.325684\pi\)
0.520664 + 0.853762i \(0.325684\pi\)
\(752\) −2.65278e15 −0.402256
\(753\) 3.27173e15 0.492499
\(754\) −1.78045e15 −0.266065
\(755\) 3.98208e15 0.590747
\(756\) 0 0
\(757\) −6.67049e14 −0.0975282 −0.0487641 0.998810i \(-0.515528\pi\)
−0.0487641 + 0.998810i \(0.515528\pi\)
\(758\) −3.51511e15 −0.510222
\(759\) −2.51166e15 −0.361936
\(760\) 4.35027e15 0.622360
\(761\) 7.74408e15 1.09990 0.549951 0.835197i \(-0.314646\pi\)
0.549951 + 0.835197i \(0.314646\pi\)
\(762\) −1.58891e15 −0.224052
\(763\) 0 0
\(764\) −4.06626e15 −0.565173
\(765\) 3.79064e15 0.523088
\(766\) 5.55479e15 0.761043
\(767\) 2.99800e15 0.407809
\(768\) 3.43775e15 0.464288
\(769\) −2.52411e15 −0.338465 −0.169232 0.985576i \(-0.554129\pi\)
−0.169232 + 0.985576i \(0.554129\pi\)
\(770\) 0 0
\(771\) 6.03822e15 0.798197
\(772\) −8.01119e15 −1.05148
\(773\) 1.11453e16 1.45246 0.726229 0.687453i \(-0.241271\pi\)
0.726229 + 0.687453i \(0.241271\pi\)
\(774\) −4.67092e13 −0.00604404
\(775\) −1.34746e15 −0.173124
\(776\) −6.33715e15 −0.808452
\(777\) 0 0
\(778\) 3.59692e15 0.452421
\(779\) 3.28500e15 0.410279
\(780\) −1.03511e15 −0.128371
\(781\) 5.23465e15 0.644624
\(782\) −3.08998e15 −0.377849
\(783\) 9.40952e15 1.14256
\(784\) 0 0
\(785\) 6.35201e15 0.760551
\(786\) −3.81949e15 −0.454131
\(787\) −1.32271e16 −1.56172 −0.780861 0.624705i \(-0.785219\pi\)
−0.780861 + 0.624705i \(0.785219\pi\)
\(788\) 4.23361e15 0.496382
\(789\) 6.11698e15 0.712219
\(790\) 4.41850e15 0.510889
\(791\) 0 0
\(792\) −5.13257e15 −0.585257
\(793\) −4.01902e15 −0.455112
\(794\) 4.99466e15 0.561685
\(795\) −1.94266e15 −0.216958
\(796\) 1.07219e15 0.118919
\(797\) −2.30248e15 −0.253615 −0.126807 0.991927i \(-0.540473\pi\)
−0.126807 + 0.991927i \(0.540473\pi\)
\(798\) 0 0
\(799\) −1.85587e16 −2.01623
\(800\) 5.01586e15 0.541191
\(801\) −2.84027e15 −0.304355
\(802\) 3.20179e15 0.340748
\(803\) −7.82560e14 −0.0827146
\(804\) −5.74289e15 −0.602868
\(805\) 0 0
\(806\) −7.32708e14 −0.0758732
\(807\) 6.51110e15 0.669653
\(808\) −6.90565e15 −0.705410
\(809\) 5.60472e15 0.568639 0.284320 0.958730i \(-0.408232\pi\)
0.284320 + 0.958730i \(0.408232\pi\)
\(810\) 1.93029e14 0.0194515
\(811\) 5.08516e15 0.508968 0.254484 0.967077i \(-0.418094\pi\)
0.254484 + 0.967077i \(0.418094\pi\)
\(812\) 0 0
\(813\) −9.49519e14 −0.0937574
\(814\) 2.33792e15 0.229296
\(815\) 1.72833e15 0.168368
\(816\) −1.71791e15 −0.166228
\(817\) 1.82584e14 0.0175486
\(818\) −4.94802e15 −0.472377
\(819\) 0 0
\(820\) −2.19066e15 −0.206348
\(821\) 2.79111e14 0.0261150 0.0130575 0.999915i \(-0.495844\pi\)
0.0130575 + 0.999915i \(0.495844\pi\)
\(822\) −1.79746e15 −0.167057
\(823\) −1.35265e16 −1.24878 −0.624391 0.781112i \(-0.714653\pi\)
−0.624391 + 0.781112i \(0.714653\pi\)
\(824\) 1.90718e16 1.74901
\(825\) 3.43531e15 0.312946
\(826\) 0 0
\(827\) 2.72544e14 0.0244994 0.0122497 0.999925i \(-0.496101\pi\)
0.0122497 + 0.999925i \(0.496101\pi\)
\(828\) 3.11869e15 0.278488
\(829\) −1.80459e16 −1.60077 −0.800385 0.599486i \(-0.795372\pi\)
−0.800385 + 0.599486i \(0.795372\pi\)
\(830\) 3.40052e15 0.299651
\(831\) 4.13757e15 0.362193
\(832\) 1.55949e15 0.135614
\(833\) 0 0
\(834\) −3.60941e15 −0.309758
\(835\) 1.33058e16 1.13440
\(836\) 8.38999e15 0.710603
\(837\) 3.87230e15 0.325821
\(838\) 1.76168e15 0.147260
\(839\) 7.96183e15 0.661184 0.330592 0.943774i \(-0.392752\pi\)
0.330592 + 0.943774i \(0.392752\pi\)
\(840\) 0 0
\(841\) 4.28775e15 0.351440
\(842\) −4.10669e15 −0.334407
\(843\) −5.30100e15 −0.428851
\(844\) 9.99954e15 0.803705
\(845\) 7.04397e15 0.562478
\(846\) −7.32956e15 −0.581488
\(847\) 0 0
\(848\) −1.57552e15 −0.123381
\(849\) 4.21172e15 0.327693
\(850\) 4.22630e15 0.326706
\(851\) −3.39705e15 −0.260909
\(852\) 3.63209e15 0.277165
\(853\) 1.49826e16 1.13598 0.567988 0.823037i \(-0.307722\pi\)
0.567988 + 0.823037i \(0.307722\pi\)
\(854\) 0 0
\(855\) −5.85201e15 −0.438017
\(856\) 7.62358e15 0.566961
\(857\) 2.22561e16 1.64458 0.822290 0.569068i \(-0.192696\pi\)
0.822290 + 0.569068i \(0.192696\pi\)
\(858\) 1.86802e15 0.137152
\(859\) −5.44237e15 −0.397032 −0.198516 0.980098i \(-0.563612\pi\)
−0.198516 + 0.980098i \(0.563612\pi\)
\(860\) −1.21760e14 −0.00882596
\(861\) 0 0
\(862\) 1.72262e15 0.123282
\(863\) 1.08110e16 0.768787 0.384393 0.923169i \(-0.374411\pi\)
0.384393 + 0.923169i \(0.374411\pi\)
\(864\) −1.44145e16 −1.01853
\(865\) −4.59037e15 −0.322299
\(866\) 2.39715e15 0.167243
\(867\) −3.38185e15 −0.234449
\(868\) 0 0
\(869\) 2.03777e16 1.39491
\(870\) −3.75099e15 −0.255147
\(871\) −8.94444e15 −0.604579
\(872\) 6.20782e15 0.416964
\(873\) 8.52477e15 0.568989
\(874\) 4.77033e15 0.316399
\(875\) 0 0
\(876\) −5.42985e14 −0.0355644
\(877\) −2.81024e16 −1.82914 −0.914568 0.404431i \(-0.867470\pi\)
−0.914568 + 0.404431i \(0.867470\pi\)
\(878\) −6.96749e14 −0.0450668
\(879\) −6.02957e15 −0.387568
\(880\) −2.54896e15 −0.162820
\(881\) −4.22209e15 −0.268016 −0.134008 0.990980i \(-0.542785\pi\)
−0.134008 + 0.990980i \(0.542785\pi\)
\(882\) 0 0
\(883\) 5.16092e14 0.0323551 0.0161776 0.999869i \(-0.494850\pi\)
0.0161776 + 0.999869i \(0.494850\pi\)
\(884\) −5.87302e15 −0.365910
\(885\) 6.31609e15 0.391075
\(886\) −7.88088e15 −0.484941
\(887\) −5.71906e15 −0.349740 −0.174870 0.984592i \(-0.555950\pi\)
−0.174870 + 0.984592i \(0.555950\pi\)
\(888\) 3.87913e15 0.235756
\(889\) 0 0
\(890\) 2.89718e15 0.173912
\(891\) 8.90230e14 0.0531098
\(892\) 1.07969e16 0.640167
\(893\) 2.86510e16 1.68832
\(894\) −6.74614e15 −0.395093
\(895\) −8.12109e15 −0.472702
\(896\) 0 0
\(897\) −2.71427e15 −0.156061
\(898\) 1.46968e16 0.839854
\(899\) 6.78541e15 0.385388
\(900\) −4.26557e15 −0.240793
\(901\) −1.10223e16 −0.618421
\(902\) 3.95340e15 0.220462
\(903\) 0 0
\(904\) −7.19321e15 −0.396275
\(905\) −4.81442e15 −0.263619
\(906\) −4.98626e15 −0.271375
\(907\) −8.43778e13 −0.00456445 −0.00228222 0.999997i \(-0.500726\pi\)
−0.00228222 + 0.999997i \(0.500726\pi\)
\(908\) −2.00168e15 −0.107628
\(909\) 9.28952e15 0.496468
\(910\) 0 0
\(911\) −1.10091e16 −0.581298 −0.290649 0.956830i \(-0.593871\pi\)
−0.290649 + 0.956830i \(0.593871\pi\)
\(912\) 2.65212e15 0.139194
\(913\) 1.56829e16 0.818158
\(914\) −7.28359e15 −0.377695
\(915\) −8.46715e15 −0.436437
\(916\) −1.74055e16 −0.891789
\(917\) 0 0
\(918\) −1.21455e16 −0.614864
\(919\) −4.86351e15 −0.244746 −0.122373 0.992484i \(-0.539050\pi\)
−0.122373 + 0.992484i \(0.539050\pi\)
\(920\) −7.60717e15 −0.380531
\(921\) 3.85840e15 0.191857
\(922\) −1.75034e16 −0.865171
\(923\) 5.65691e15 0.277952
\(924\) 0 0
\(925\) 4.64630e15 0.225594
\(926\) −2.93252e15 −0.141540
\(927\) −2.56555e16 −1.23095
\(928\) −2.52584e16 −1.20474
\(929\) −3.57534e15 −0.169524 −0.0847620 0.996401i \(-0.527013\pi\)
−0.0847620 + 0.996401i \(0.527013\pi\)
\(930\) −1.54365e15 −0.0727598
\(931\) 0 0
\(932\) 2.58533e16 1.20428
\(933\) 1.25685e16 0.582017
\(934\) −1.48171e16 −0.682113
\(935\) −1.78323e16 −0.816102
\(936\) −5.54661e15 −0.252354
\(937\) −3.86373e16 −1.74759 −0.873795 0.486295i \(-0.838348\pi\)
−0.873795 + 0.486295i \(0.838348\pi\)
\(938\) 0 0
\(939\) −2.50692e16 −1.12067
\(940\) −1.91064e16 −0.849133
\(941\) 3.48997e16 1.54198 0.770991 0.636846i \(-0.219761\pi\)
0.770991 + 0.636846i \(0.219761\pi\)
\(942\) −7.95383e15 −0.349378
\(943\) −5.74437e15 −0.250858
\(944\) 5.12245e15 0.222398
\(945\) 0 0
\(946\) 2.19735e14 0.00942969
\(947\) −2.85123e16 −1.21649 −0.608243 0.793751i \(-0.708125\pi\)
−0.608243 + 0.793751i \(0.708125\pi\)
\(948\) 1.41392e16 0.599763
\(949\) −8.45688e14 −0.0356653
\(950\) −6.52459e15 −0.273573
\(951\) −2.10091e16 −0.875817
\(952\) 0 0
\(953\) 4.00334e16 1.64973 0.824863 0.565332i \(-0.191252\pi\)
0.824863 + 0.565332i \(0.191252\pi\)
\(954\) −4.35313e15 −0.178355
\(955\) −1.33424e16 −0.543520
\(956\) 1.05095e16 0.425659
\(957\) −1.72992e16 −0.696644
\(958\) 2.52201e16 1.00980
\(959\) 0 0
\(960\) 3.28548e15 0.130049
\(961\) −2.26161e16 −0.890100
\(962\) 2.52652e15 0.0988688
\(963\) −1.02553e16 −0.399028
\(964\) −3.40484e14 −0.0131726
\(965\) −2.62867e16 −1.01120
\(966\) 0 0
\(967\) 1.84953e16 0.703422 0.351711 0.936109i \(-0.385600\pi\)
0.351711 + 0.936109i \(0.385600\pi\)
\(968\) 4.20981e13 0.00159202
\(969\) 1.85540e16 0.697682
\(970\) −8.69557e15 −0.325127
\(971\) 2.14877e16 0.798884 0.399442 0.916759i \(-0.369204\pi\)
0.399442 + 0.916759i \(0.369204\pi\)
\(972\) 1.97260e16 0.729246
\(973\) 0 0
\(974\) 5.27784e15 0.192922
\(975\) 3.71243e15 0.134938
\(976\) −6.86699e15 −0.248195
\(977\) −8.73880e15 −0.314074 −0.157037 0.987593i \(-0.550194\pi\)
−0.157037 + 0.987593i \(0.550194\pi\)
\(978\) −2.16417e15 −0.0773443
\(979\) 1.33615e16 0.474844
\(980\) 0 0
\(981\) −8.35079e15 −0.293459
\(982\) 1.16127e16 0.405808
\(983\) −1.18924e16 −0.413263 −0.206631 0.978419i \(-0.566250\pi\)
−0.206631 + 0.978419i \(0.566250\pi\)
\(984\) 6.55956e15 0.226674
\(985\) 1.38915e16 0.477365
\(986\) −2.12824e16 −0.727274
\(987\) 0 0
\(988\) 9.06679e15 0.306401
\(989\) −3.19279e14 −0.0107298
\(990\) −7.04271e15 −0.235367
\(991\) 2.34409e16 0.779056 0.389528 0.921015i \(-0.372638\pi\)
0.389528 + 0.921015i \(0.372638\pi\)
\(992\) −1.03946e16 −0.343552
\(993\) 1.60232e16 0.526657
\(994\) 0 0
\(995\) 3.51813e15 0.114363
\(996\) 1.08817e16 0.351779
\(997\) 2.14004e16 0.688016 0.344008 0.938967i \(-0.388215\pi\)
0.344008 + 0.938967i \(0.388215\pi\)
\(998\) 2.61307e15 0.0835473
\(999\) −1.33524e16 −0.424571
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 49.12.a.a.1.1 1
7.2 even 3 49.12.c.c.18.1 2
7.3 odd 6 49.12.c.b.30.1 2
7.4 even 3 49.12.c.c.30.1 2
7.5 odd 6 49.12.c.b.18.1 2
7.6 odd 2 1.12.a.a.1.1 1
21.20 even 2 9.12.a.b.1.1 1
28.27 even 2 16.12.a.a.1.1 1
35.13 even 4 25.12.b.b.24.2 2
35.27 even 4 25.12.b.b.24.1 2
35.34 odd 2 25.12.a.b.1.1 1
56.13 odd 2 64.12.a.b.1.1 1
56.27 even 2 64.12.a.f.1.1 1
63.13 odd 6 81.12.c.d.55.1 2
63.20 even 6 81.12.c.b.28.1 2
63.34 odd 6 81.12.c.d.28.1 2
63.41 even 6 81.12.c.b.55.1 2
77.76 even 2 121.12.a.b.1.1 1
84.83 odd 2 144.12.a.d.1.1 1
91.90 odd 2 169.12.a.a.1.1 1
105.62 odd 4 225.12.b.d.199.2 2
105.83 odd 4 225.12.b.d.199.1 2
105.104 even 2 225.12.a.b.1.1 1
112.13 odd 4 256.12.b.e.129.2 2
112.27 even 4 256.12.b.c.129.2 2
112.69 odd 4 256.12.b.e.129.1 2
112.83 even 4 256.12.b.c.129.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1.12.a.a.1.1 1 7.6 odd 2
9.12.a.b.1.1 1 21.20 even 2
16.12.a.a.1.1 1 28.27 even 2
25.12.a.b.1.1 1 35.34 odd 2
25.12.b.b.24.1 2 35.27 even 4
25.12.b.b.24.2 2 35.13 even 4
49.12.a.a.1.1 1 1.1 even 1 trivial
49.12.c.b.18.1 2 7.5 odd 6
49.12.c.b.30.1 2 7.3 odd 6
49.12.c.c.18.1 2 7.2 even 3
49.12.c.c.30.1 2 7.4 even 3
64.12.a.b.1.1 1 56.13 odd 2
64.12.a.f.1.1 1 56.27 even 2
81.12.c.b.28.1 2 63.20 even 6
81.12.c.b.55.1 2 63.41 even 6
81.12.c.d.28.1 2 63.34 odd 6
81.12.c.d.55.1 2 63.13 odd 6
121.12.a.b.1.1 1 77.76 even 2
144.12.a.d.1.1 1 84.83 odd 2
169.12.a.a.1.1 1 91.90 odd 2
225.12.a.b.1.1 1 105.104 even 2
225.12.b.d.199.1 2 105.83 odd 4
225.12.b.d.199.2 2 105.62 odd 4
256.12.b.c.129.1 2 112.83 even 4
256.12.b.c.129.2 2 112.27 even 4
256.12.b.e.129.1 2 112.69 odd 4
256.12.b.e.129.2 2 112.13 odd 4