Properties

Label 147.2.a.d.1.1
Level $147$
Weight $2$
Character 147.1
Self dual yes
Analytic conductor $1.174$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 147 = 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 147.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.17380090971\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 147.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.41421 q^{2} -1.00000 q^{3} +3.82843 q^{4} -0.585786 q^{5} +2.41421 q^{6} -4.41421 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-2.41421 q^{2} -1.00000 q^{3} +3.82843 q^{4} -0.585786 q^{5} +2.41421 q^{6} -4.41421 q^{8} +1.00000 q^{9} +1.41421 q^{10} -2.00000 q^{11} -3.82843 q^{12} -5.41421 q^{13} +0.585786 q^{15} +3.00000 q^{16} -6.24264 q^{17} -2.41421 q^{18} -2.82843 q^{19} -2.24264 q^{20} +4.82843 q^{22} +3.65685 q^{23} +4.41421 q^{24} -4.65685 q^{25} +13.0711 q^{26} -1.00000 q^{27} -1.17157 q^{29} -1.41421 q^{30} +6.82843 q^{31} +1.58579 q^{32} +2.00000 q^{33} +15.0711 q^{34} +3.82843 q^{36} -4.00000 q^{37} +6.82843 q^{38} +5.41421 q^{39} +2.58579 q^{40} +2.24264 q^{41} -5.65685 q^{43} -7.65685 q^{44} -0.585786 q^{45} -8.82843 q^{46} -2.82843 q^{47} -3.00000 q^{48} +11.2426 q^{50} +6.24264 q^{51} -20.7279 q^{52} -2.00000 q^{53} +2.41421 q^{54} +1.17157 q^{55} +2.82843 q^{57} +2.82843 q^{58} +6.82843 q^{59} +2.24264 q^{60} -3.75736 q^{61} -16.4853 q^{62} -9.82843 q^{64} +3.17157 q^{65} -4.82843 q^{66} +5.65685 q^{67} -23.8995 q^{68} -3.65685 q^{69} -13.3137 q^{71} -4.41421 q^{72} +5.89949 q^{73} +9.65685 q^{74} +4.65685 q^{75} -10.8284 q^{76} -13.0711 q^{78} +2.34315 q^{79} -1.75736 q^{80} +1.00000 q^{81} -5.41421 q^{82} +15.3137 q^{83} +3.65685 q^{85} +13.6569 q^{86} +1.17157 q^{87} +8.82843 q^{88} +5.75736 q^{89} +1.41421 q^{90} +14.0000 q^{92} -6.82843 q^{93} +6.82843 q^{94} +1.65685 q^{95} -1.58579 q^{96} -5.41421 q^{97} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} - 2q^{3} + 2q^{4} - 4q^{5} + 2q^{6} - 6q^{8} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{2} - 2q^{3} + 2q^{4} - 4q^{5} + 2q^{6} - 6q^{8} + 2q^{9} - 4q^{11} - 2q^{12} - 8q^{13} + 4q^{15} + 6q^{16} - 4q^{17} - 2q^{18} + 4q^{20} + 4q^{22} - 4q^{23} + 6q^{24} + 2q^{25} + 12q^{26} - 2q^{27} - 8q^{29} + 8q^{31} + 6q^{32} + 4q^{33} + 16q^{34} + 2q^{36} - 8q^{37} + 8q^{38} + 8q^{39} + 8q^{40} - 4q^{41} - 4q^{44} - 4q^{45} - 12q^{46} - 6q^{48} + 14q^{50} + 4q^{51} - 16q^{52} - 4q^{53} + 2q^{54} + 8q^{55} + 8q^{59} - 4q^{60} - 16q^{61} - 16q^{62} - 14q^{64} + 12q^{65} - 4q^{66} - 28q^{68} + 4q^{69} - 4q^{71} - 6q^{72} - 8q^{73} + 8q^{74} - 2q^{75} - 16q^{76} - 12q^{78} + 16q^{79} - 12q^{80} + 2q^{81} - 8q^{82} + 8q^{83} - 4q^{85} + 16q^{86} + 8q^{87} + 12q^{88} + 20q^{89} + 28q^{92} - 8q^{93} + 8q^{94} - 8q^{95} - 6q^{96} - 8q^{97} - 4q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.41421 −1.70711 −0.853553 0.521005i \(-0.825557\pi\)
−0.853553 + 0.521005i \(0.825557\pi\)
\(3\) −1.00000 −0.577350
\(4\) 3.82843 1.91421
\(5\) −0.585786 −0.261972 −0.130986 0.991384i \(-0.541814\pi\)
−0.130986 + 0.991384i \(0.541814\pi\)
\(6\) 2.41421 0.985599
\(7\) 0 0
\(8\) −4.41421 −1.56066
\(9\) 1.00000 0.333333
\(10\) 1.41421 0.447214
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) −3.82843 −1.10517
\(13\) −5.41421 −1.50163 −0.750816 0.660511i \(-0.770340\pi\)
−0.750816 + 0.660511i \(0.770340\pi\)
\(14\) 0 0
\(15\) 0.585786 0.151249
\(16\) 3.00000 0.750000
\(17\) −6.24264 −1.51406 −0.757031 0.653379i \(-0.773351\pi\)
−0.757031 + 0.653379i \(0.773351\pi\)
\(18\) −2.41421 −0.569036
\(19\) −2.82843 −0.648886 −0.324443 0.945905i \(-0.605177\pi\)
−0.324443 + 0.945905i \(0.605177\pi\)
\(20\) −2.24264 −0.501470
\(21\) 0 0
\(22\) 4.82843 1.02942
\(23\) 3.65685 0.762507 0.381253 0.924471i \(-0.375493\pi\)
0.381253 + 0.924471i \(0.375493\pi\)
\(24\) 4.41421 0.901048
\(25\) −4.65685 −0.931371
\(26\) 13.0711 2.56345
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −1.17157 −0.217556 −0.108778 0.994066i \(-0.534694\pi\)
−0.108778 + 0.994066i \(0.534694\pi\)
\(30\) −1.41421 −0.258199
\(31\) 6.82843 1.22642 0.613211 0.789919i \(-0.289878\pi\)
0.613211 + 0.789919i \(0.289878\pi\)
\(32\) 1.58579 0.280330
\(33\) 2.00000 0.348155
\(34\) 15.0711 2.58467
\(35\) 0 0
\(36\) 3.82843 0.638071
\(37\) −4.00000 −0.657596 −0.328798 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) 6.82843 1.10772
\(39\) 5.41421 0.866968
\(40\) 2.58579 0.408849
\(41\) 2.24264 0.350242 0.175121 0.984547i \(-0.443968\pi\)
0.175121 + 0.984547i \(0.443968\pi\)
\(42\) 0 0
\(43\) −5.65685 −0.862662 −0.431331 0.902194i \(-0.641956\pi\)
−0.431331 + 0.902194i \(0.641956\pi\)
\(44\) −7.65685 −1.15431
\(45\) −0.585786 −0.0873239
\(46\) −8.82843 −1.30168
\(47\) −2.82843 −0.412568 −0.206284 0.978492i \(-0.566137\pi\)
−0.206284 + 0.978492i \(0.566137\pi\)
\(48\) −3.00000 −0.433013
\(49\) 0 0
\(50\) 11.2426 1.58995
\(51\) 6.24264 0.874145
\(52\) −20.7279 −2.87445
\(53\) −2.00000 −0.274721 −0.137361 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 2.41421 0.328533
\(55\) 1.17157 0.157975
\(56\) 0 0
\(57\) 2.82843 0.374634
\(58\) 2.82843 0.371391
\(59\) 6.82843 0.888985 0.444493 0.895782i \(-0.353384\pi\)
0.444493 + 0.895782i \(0.353384\pi\)
\(60\) 2.24264 0.289524
\(61\) −3.75736 −0.481081 −0.240540 0.970639i \(-0.577325\pi\)
−0.240540 + 0.970639i \(0.577325\pi\)
\(62\) −16.4853 −2.09363
\(63\) 0 0
\(64\) −9.82843 −1.22855
\(65\) 3.17157 0.393385
\(66\) −4.82843 −0.594338
\(67\) 5.65685 0.691095 0.345547 0.938401i \(-0.387693\pi\)
0.345547 + 0.938401i \(0.387693\pi\)
\(68\) −23.8995 −2.89824
\(69\) −3.65685 −0.440234
\(70\) 0 0
\(71\) −13.3137 −1.58005 −0.790023 0.613077i \(-0.789932\pi\)
−0.790023 + 0.613077i \(0.789932\pi\)
\(72\) −4.41421 −0.520220
\(73\) 5.89949 0.690484 0.345242 0.938514i \(-0.387797\pi\)
0.345242 + 0.938514i \(0.387797\pi\)
\(74\) 9.65685 1.12259
\(75\) 4.65685 0.537727
\(76\) −10.8284 −1.24211
\(77\) 0 0
\(78\) −13.0711 −1.48001
\(79\) 2.34315 0.263624 0.131812 0.991275i \(-0.457920\pi\)
0.131812 + 0.991275i \(0.457920\pi\)
\(80\) −1.75736 −0.196479
\(81\) 1.00000 0.111111
\(82\) −5.41421 −0.597900
\(83\) 15.3137 1.68090 0.840449 0.541891i \(-0.182291\pi\)
0.840449 + 0.541891i \(0.182291\pi\)
\(84\) 0 0
\(85\) 3.65685 0.396642
\(86\) 13.6569 1.47266
\(87\) 1.17157 0.125606
\(88\) 8.82843 0.941113
\(89\) 5.75736 0.610279 0.305139 0.952308i \(-0.401297\pi\)
0.305139 + 0.952308i \(0.401297\pi\)
\(90\) 1.41421 0.149071
\(91\) 0 0
\(92\) 14.0000 1.45960
\(93\) −6.82843 −0.708075
\(94\) 6.82843 0.704298
\(95\) 1.65685 0.169990
\(96\) −1.58579 −0.161849
\(97\) −5.41421 −0.549730 −0.274865 0.961483i \(-0.588633\pi\)
−0.274865 + 0.961483i \(0.588633\pi\)
\(98\) 0 0
\(99\) −2.00000 −0.201008
\(100\) −17.8284 −1.78284
\(101\) −17.0711 −1.69863 −0.849317 0.527883i \(-0.822986\pi\)
−0.849317 + 0.527883i \(0.822986\pi\)
\(102\) −15.0711 −1.49226
\(103\) 12.4853 1.23021 0.615106 0.788445i \(-0.289113\pi\)
0.615106 + 0.788445i \(0.289113\pi\)
\(104\) 23.8995 2.34354
\(105\) 0 0
\(106\) 4.82843 0.468978
\(107\) −11.6569 −1.12691 −0.563455 0.826147i \(-0.690528\pi\)
−0.563455 + 0.826147i \(0.690528\pi\)
\(108\) −3.82843 −0.368391
\(109\) 5.65685 0.541828 0.270914 0.962604i \(-0.412674\pi\)
0.270914 + 0.962604i \(0.412674\pi\)
\(110\) −2.82843 −0.269680
\(111\) 4.00000 0.379663
\(112\) 0 0
\(113\) 17.3137 1.62874 0.814368 0.580348i \(-0.197084\pi\)
0.814368 + 0.580348i \(0.197084\pi\)
\(114\) −6.82843 −0.639541
\(115\) −2.14214 −0.199755
\(116\) −4.48528 −0.416448
\(117\) −5.41421 −0.500544
\(118\) −16.4853 −1.51759
\(119\) 0 0
\(120\) −2.58579 −0.236049
\(121\) −7.00000 −0.636364
\(122\) 9.07107 0.821256
\(123\) −2.24264 −0.202212
\(124\) 26.1421 2.34763
\(125\) 5.65685 0.505964
\(126\) 0 0
\(127\) 9.65685 0.856907 0.428454 0.903564i \(-0.359059\pi\)
0.428454 + 0.903564i \(0.359059\pi\)
\(128\) 20.5563 1.81694
\(129\) 5.65685 0.498058
\(130\) −7.65685 −0.671551
\(131\) −7.31371 −0.639002 −0.319501 0.947586i \(-0.603515\pi\)
−0.319501 + 0.947586i \(0.603515\pi\)
\(132\) 7.65685 0.666444
\(133\) 0 0
\(134\) −13.6569 −1.17977
\(135\) 0.585786 0.0504165
\(136\) 27.5563 2.36294
\(137\) −14.1421 −1.20824 −0.604122 0.796892i \(-0.706476\pi\)
−0.604122 + 0.796892i \(0.706476\pi\)
\(138\) 8.82843 0.751526
\(139\) 6.34315 0.538019 0.269009 0.963138i \(-0.413304\pi\)
0.269009 + 0.963138i \(0.413304\pi\)
\(140\) 0 0
\(141\) 2.82843 0.238197
\(142\) 32.1421 2.69731
\(143\) 10.8284 0.905519
\(144\) 3.00000 0.250000
\(145\) 0.686292 0.0569934
\(146\) −14.2426 −1.17873
\(147\) 0 0
\(148\) −15.3137 −1.25878
\(149\) −5.31371 −0.435316 −0.217658 0.976025i \(-0.569842\pi\)
−0.217658 + 0.976025i \(0.569842\pi\)
\(150\) −11.2426 −0.917958
\(151\) 12.0000 0.976546 0.488273 0.872691i \(-0.337627\pi\)
0.488273 + 0.872691i \(0.337627\pi\)
\(152\) 12.4853 1.01269
\(153\) −6.24264 −0.504688
\(154\) 0 0
\(155\) −4.00000 −0.321288
\(156\) 20.7279 1.65956
\(157\) −20.2426 −1.61554 −0.807769 0.589499i \(-0.799325\pi\)
−0.807769 + 0.589499i \(0.799325\pi\)
\(158\) −5.65685 −0.450035
\(159\) 2.00000 0.158610
\(160\) −0.928932 −0.0734385
\(161\) 0 0
\(162\) −2.41421 −0.189679
\(163\) 11.3137 0.886158 0.443079 0.896483i \(-0.353886\pi\)
0.443079 + 0.896483i \(0.353886\pi\)
\(164\) 8.58579 0.670437
\(165\) −1.17157 −0.0912068
\(166\) −36.9706 −2.86947
\(167\) −19.7990 −1.53209 −0.766046 0.642786i \(-0.777779\pi\)
−0.766046 + 0.642786i \(0.777779\pi\)
\(168\) 0 0
\(169\) 16.3137 1.25490
\(170\) −8.82843 −0.677109
\(171\) −2.82843 −0.216295
\(172\) −21.6569 −1.65132
\(173\) −6.92893 −0.526797 −0.263398 0.964687i \(-0.584843\pi\)
−0.263398 + 0.964687i \(0.584843\pi\)
\(174\) −2.82843 −0.214423
\(175\) 0 0
\(176\) −6.00000 −0.452267
\(177\) −6.82843 −0.513256
\(178\) −13.8995 −1.04181
\(179\) −8.34315 −0.623596 −0.311798 0.950148i \(-0.600931\pi\)
−0.311798 + 0.950148i \(0.600931\pi\)
\(180\) −2.24264 −0.167157
\(181\) 5.41421 0.402435 0.201218 0.979547i \(-0.435510\pi\)
0.201218 + 0.979547i \(0.435510\pi\)
\(182\) 0 0
\(183\) 3.75736 0.277752
\(184\) −16.1421 −1.19001
\(185\) 2.34315 0.172272
\(186\) 16.4853 1.20876
\(187\) 12.4853 0.913014
\(188\) −10.8284 −0.789744
\(189\) 0 0
\(190\) −4.00000 −0.290191
\(191\) −18.0000 −1.30243 −0.651217 0.758891i \(-0.725741\pi\)
−0.651217 + 0.758891i \(0.725741\pi\)
\(192\) 9.82843 0.709306
\(193\) −17.3137 −1.24627 −0.623134 0.782115i \(-0.714141\pi\)
−0.623134 + 0.782115i \(0.714141\pi\)
\(194\) 13.0711 0.938448
\(195\) −3.17157 −0.227121
\(196\) 0 0
\(197\) 2.00000 0.142494 0.0712470 0.997459i \(-0.477302\pi\)
0.0712470 + 0.997459i \(0.477302\pi\)
\(198\) 4.82843 0.343141
\(199\) −10.3431 −0.733206 −0.366603 0.930377i \(-0.619479\pi\)
−0.366603 + 0.930377i \(0.619479\pi\)
\(200\) 20.5563 1.45355
\(201\) −5.65685 −0.399004
\(202\) 41.2132 2.89975
\(203\) 0 0
\(204\) 23.8995 1.67330
\(205\) −1.31371 −0.0917534
\(206\) −30.1421 −2.10010
\(207\) 3.65685 0.254169
\(208\) −16.2426 −1.12622
\(209\) 5.65685 0.391293
\(210\) 0 0
\(211\) −20.9706 −1.44367 −0.721837 0.692064i \(-0.756702\pi\)
−0.721837 + 0.692064i \(0.756702\pi\)
\(212\) −7.65685 −0.525875
\(213\) 13.3137 0.912240
\(214\) 28.1421 1.92376
\(215\) 3.31371 0.225993
\(216\) 4.41421 0.300349
\(217\) 0 0
\(218\) −13.6569 −0.924959
\(219\) −5.89949 −0.398651
\(220\) 4.48528 0.302398
\(221\) 33.7990 2.27357
\(222\) −9.65685 −0.648126
\(223\) 8.97056 0.600713 0.300357 0.953827i \(-0.402894\pi\)
0.300357 + 0.953827i \(0.402894\pi\)
\(224\) 0 0
\(225\) −4.65685 −0.310457
\(226\) −41.7990 −2.78043
\(227\) 15.7990 1.04862 0.524308 0.851529i \(-0.324324\pi\)
0.524308 + 0.851529i \(0.324324\pi\)
\(228\) 10.8284 0.717130
\(229\) −8.24264 −0.544689 −0.272345 0.962200i \(-0.587799\pi\)
−0.272345 + 0.962200i \(0.587799\pi\)
\(230\) 5.17157 0.341003
\(231\) 0 0
\(232\) 5.17157 0.339530
\(233\) 22.1421 1.45058 0.725290 0.688444i \(-0.241706\pi\)
0.725290 + 0.688444i \(0.241706\pi\)
\(234\) 13.0711 0.854482
\(235\) 1.65685 0.108081
\(236\) 26.1421 1.70171
\(237\) −2.34315 −0.152204
\(238\) 0 0
\(239\) −4.34315 −0.280935 −0.140467 0.990085i \(-0.544861\pi\)
−0.140467 + 0.990085i \(0.544861\pi\)
\(240\) 1.75736 0.113437
\(241\) 7.75736 0.499695 0.249848 0.968285i \(-0.419619\pi\)
0.249848 + 0.968285i \(0.419619\pi\)
\(242\) 16.8995 1.08634
\(243\) −1.00000 −0.0641500
\(244\) −14.3848 −0.920891
\(245\) 0 0
\(246\) 5.41421 0.345198
\(247\) 15.3137 0.974388
\(248\) −30.1421 −1.91403
\(249\) −15.3137 −0.970467
\(250\) −13.6569 −0.863735
\(251\) −4.48528 −0.283108 −0.141554 0.989931i \(-0.545210\pi\)
−0.141554 + 0.989931i \(0.545210\pi\)
\(252\) 0 0
\(253\) −7.31371 −0.459809
\(254\) −23.3137 −1.46283
\(255\) −3.65685 −0.229001
\(256\) −29.9706 −1.87316
\(257\) 19.2132 1.19849 0.599243 0.800567i \(-0.295468\pi\)
0.599243 + 0.800567i \(0.295468\pi\)
\(258\) −13.6569 −0.850239
\(259\) 0 0
\(260\) 12.1421 0.753023
\(261\) −1.17157 −0.0725185
\(262\) 17.6569 1.09084
\(263\) −17.3137 −1.06761 −0.533805 0.845608i \(-0.679238\pi\)
−0.533805 + 0.845608i \(0.679238\pi\)
\(264\) −8.82843 −0.543352
\(265\) 1.17157 0.0719691
\(266\) 0 0
\(267\) −5.75736 −0.352345
\(268\) 21.6569 1.32290
\(269\) −10.7279 −0.654093 −0.327046 0.945008i \(-0.606053\pi\)
−0.327046 + 0.945008i \(0.606053\pi\)
\(270\) −1.41421 −0.0860663
\(271\) −18.1421 −1.10206 −0.551028 0.834487i \(-0.685764\pi\)
−0.551028 + 0.834487i \(0.685764\pi\)
\(272\) −18.7279 −1.13555
\(273\) 0 0
\(274\) 34.1421 2.06260
\(275\) 9.31371 0.561638
\(276\) −14.0000 −0.842701
\(277\) 13.3137 0.799943 0.399972 0.916528i \(-0.369020\pi\)
0.399972 + 0.916528i \(0.369020\pi\)
\(278\) −15.3137 −0.918455
\(279\) 6.82843 0.408807
\(280\) 0 0
\(281\) −16.4853 −0.983429 −0.491715 0.870756i \(-0.663630\pi\)
−0.491715 + 0.870756i \(0.663630\pi\)
\(282\) −6.82843 −0.406627
\(283\) −8.48528 −0.504398 −0.252199 0.967675i \(-0.581154\pi\)
−0.252199 + 0.967675i \(0.581154\pi\)
\(284\) −50.9706 −3.02455
\(285\) −1.65685 −0.0981436
\(286\) −26.1421 −1.54582
\(287\) 0 0
\(288\) 1.58579 0.0934434
\(289\) 21.9706 1.29239
\(290\) −1.65685 −0.0972938
\(291\) 5.41421 0.317387
\(292\) 22.5858 1.32173
\(293\) −19.4142 −1.13419 −0.567095 0.823652i \(-0.691933\pi\)
−0.567095 + 0.823652i \(0.691933\pi\)
\(294\) 0 0
\(295\) −4.00000 −0.232889
\(296\) 17.6569 1.02628
\(297\) 2.00000 0.116052
\(298\) 12.8284 0.743131
\(299\) −19.7990 −1.14501
\(300\) 17.8284 1.02932
\(301\) 0 0
\(302\) −28.9706 −1.66707
\(303\) 17.0711 0.980707
\(304\) −8.48528 −0.486664
\(305\) 2.20101 0.126029
\(306\) 15.0711 0.861556
\(307\) −1.85786 −0.106034 −0.0530170 0.998594i \(-0.516884\pi\)
−0.0530170 + 0.998594i \(0.516884\pi\)
\(308\) 0 0
\(309\) −12.4853 −0.710263
\(310\) 9.65685 0.548472
\(311\) 22.1421 1.25557 0.627783 0.778389i \(-0.283963\pi\)
0.627783 + 0.778389i \(0.283963\pi\)
\(312\) −23.8995 −1.35304
\(313\) 17.8995 1.01174 0.505870 0.862610i \(-0.331172\pi\)
0.505870 + 0.862610i \(0.331172\pi\)
\(314\) 48.8701 2.75790
\(315\) 0 0
\(316\) 8.97056 0.504634
\(317\) 10.0000 0.561656 0.280828 0.959758i \(-0.409391\pi\)
0.280828 + 0.959758i \(0.409391\pi\)
\(318\) −4.82843 −0.270765
\(319\) 2.34315 0.131191
\(320\) 5.75736 0.321846
\(321\) 11.6569 0.650622
\(322\) 0 0
\(323\) 17.6569 0.982454
\(324\) 3.82843 0.212690
\(325\) 25.2132 1.39858
\(326\) −27.3137 −1.51277
\(327\) −5.65685 −0.312825
\(328\) −9.89949 −0.546608
\(329\) 0 0
\(330\) 2.82843 0.155700
\(331\) −4.00000 −0.219860 −0.109930 0.993939i \(-0.535063\pi\)
−0.109930 + 0.993939i \(0.535063\pi\)
\(332\) 58.6274 3.21760
\(333\) −4.00000 −0.219199
\(334\) 47.7990 2.61544
\(335\) −3.31371 −0.181047
\(336\) 0 0
\(337\) −18.3431 −0.999215 −0.499607 0.866252i \(-0.666522\pi\)
−0.499607 + 0.866252i \(0.666522\pi\)
\(338\) −39.3848 −2.14225
\(339\) −17.3137 −0.940352
\(340\) 14.0000 0.759257
\(341\) −13.6569 −0.739560
\(342\) 6.82843 0.369239
\(343\) 0 0
\(344\) 24.9706 1.34632
\(345\) 2.14214 0.115329
\(346\) 16.7279 0.899299
\(347\) 10.6863 0.573670 0.286835 0.957980i \(-0.407397\pi\)
0.286835 + 0.957980i \(0.407397\pi\)
\(348\) 4.48528 0.240436
\(349\) −9.89949 −0.529908 −0.264954 0.964261i \(-0.585357\pi\)
−0.264954 + 0.964261i \(0.585357\pi\)
\(350\) 0 0
\(351\) 5.41421 0.288989
\(352\) −3.17157 −0.169045
\(353\) −10.7279 −0.570990 −0.285495 0.958380i \(-0.592158\pi\)
−0.285495 + 0.958380i \(0.592158\pi\)
\(354\) 16.4853 0.876183
\(355\) 7.79899 0.413927
\(356\) 22.0416 1.16820
\(357\) 0 0
\(358\) 20.1421 1.06454
\(359\) −11.6569 −0.615225 −0.307613 0.951512i \(-0.599530\pi\)
−0.307613 + 0.951512i \(0.599530\pi\)
\(360\) 2.58579 0.136283
\(361\) −11.0000 −0.578947
\(362\) −13.0711 −0.687000
\(363\) 7.00000 0.367405
\(364\) 0 0
\(365\) −3.45584 −0.180887
\(366\) −9.07107 −0.474152
\(367\) −19.3137 −1.00817 −0.504084 0.863655i \(-0.668170\pi\)
−0.504084 + 0.863655i \(0.668170\pi\)
\(368\) 10.9706 0.571880
\(369\) 2.24264 0.116747
\(370\) −5.65685 −0.294086
\(371\) 0 0
\(372\) −26.1421 −1.35541
\(373\) −33.3137 −1.72492 −0.862459 0.506127i \(-0.831077\pi\)
−0.862459 + 0.506127i \(0.831077\pi\)
\(374\) −30.1421 −1.55861
\(375\) −5.65685 −0.292119
\(376\) 12.4853 0.643879
\(377\) 6.34315 0.326689
\(378\) 0 0
\(379\) 31.3137 1.60848 0.804239 0.594307i \(-0.202573\pi\)
0.804239 + 0.594307i \(0.202573\pi\)
\(380\) 6.34315 0.325397
\(381\) −9.65685 −0.494736
\(382\) 43.4558 2.22339
\(383\) 29.6569 1.51539 0.757697 0.652606i \(-0.226324\pi\)
0.757697 + 0.652606i \(0.226324\pi\)
\(384\) −20.5563 −1.04901
\(385\) 0 0
\(386\) 41.7990 2.12751
\(387\) −5.65685 −0.287554
\(388\) −20.7279 −1.05230
\(389\) 10.1421 0.514227 0.257113 0.966381i \(-0.417229\pi\)
0.257113 + 0.966381i \(0.417229\pi\)
\(390\) 7.65685 0.387720
\(391\) −22.8284 −1.15448
\(392\) 0 0
\(393\) 7.31371 0.368928
\(394\) −4.82843 −0.243253
\(395\) −1.37258 −0.0690621
\(396\) −7.65685 −0.384771
\(397\) −34.3848 −1.72572 −0.862861 0.505441i \(-0.831330\pi\)
−0.862861 + 0.505441i \(0.831330\pi\)
\(398\) 24.9706 1.25166
\(399\) 0 0
\(400\) −13.9706 −0.698528
\(401\) 22.1421 1.10573 0.552863 0.833272i \(-0.313535\pi\)
0.552863 + 0.833272i \(0.313535\pi\)
\(402\) 13.6569 0.681142
\(403\) −36.9706 −1.84163
\(404\) −65.3553 −3.25155
\(405\) −0.585786 −0.0291080
\(406\) 0 0
\(407\) 8.00000 0.396545
\(408\) −27.5563 −1.36424
\(409\) 18.5858 0.919008 0.459504 0.888176i \(-0.348027\pi\)
0.459504 + 0.888176i \(0.348027\pi\)
\(410\) 3.17157 0.156633
\(411\) 14.1421 0.697580
\(412\) 47.7990 2.35489
\(413\) 0 0
\(414\) −8.82843 −0.433894
\(415\) −8.97056 −0.440348
\(416\) −8.58579 −0.420953
\(417\) −6.34315 −0.310625
\(418\) −13.6569 −0.667979
\(419\) −38.8284 −1.89689 −0.948446 0.316938i \(-0.897345\pi\)
−0.948446 + 0.316938i \(0.897345\pi\)
\(420\) 0 0
\(421\) −28.6274 −1.39521 −0.697607 0.716480i \(-0.745752\pi\)
−0.697607 + 0.716480i \(0.745752\pi\)
\(422\) 50.6274 2.46450
\(423\) −2.82843 −0.137523
\(424\) 8.82843 0.428746
\(425\) 29.0711 1.41015
\(426\) −32.1421 −1.55729
\(427\) 0 0
\(428\) −44.6274 −2.15715
\(429\) −10.8284 −0.522801
\(430\) −8.00000 −0.385794
\(431\) 6.97056 0.335760 0.167880 0.985807i \(-0.446308\pi\)
0.167880 + 0.985807i \(0.446308\pi\)
\(432\) −3.00000 −0.144338
\(433\) 11.7574 0.565023 0.282511 0.959264i \(-0.408833\pi\)
0.282511 + 0.959264i \(0.408833\pi\)
\(434\) 0 0
\(435\) −0.686292 −0.0329052
\(436\) 21.6569 1.03718
\(437\) −10.3431 −0.494780
\(438\) 14.2426 0.680540
\(439\) −35.3137 −1.68543 −0.842716 0.538359i \(-0.819044\pi\)
−0.842716 + 0.538359i \(0.819044\pi\)
\(440\) −5.17157 −0.246545
\(441\) 0 0
\(442\) −81.5980 −3.88122
\(443\) 1.02944 0.0489100 0.0244550 0.999701i \(-0.492215\pi\)
0.0244550 + 0.999701i \(0.492215\pi\)
\(444\) 15.3137 0.726756
\(445\) −3.37258 −0.159876
\(446\) −21.6569 −1.02548
\(447\) 5.31371 0.251330
\(448\) 0 0
\(449\) 17.3137 0.817084 0.408542 0.912739i \(-0.366037\pi\)
0.408542 + 0.912739i \(0.366037\pi\)
\(450\) 11.2426 0.529983
\(451\) −4.48528 −0.211204
\(452\) 66.2843 3.11775
\(453\) −12.0000 −0.563809
\(454\) −38.1421 −1.79010
\(455\) 0 0
\(456\) −12.4853 −0.584677
\(457\) −18.0000 −0.842004 −0.421002 0.907060i \(-0.638322\pi\)
−0.421002 + 0.907060i \(0.638322\pi\)
\(458\) 19.8995 0.929842
\(459\) 6.24264 0.291382
\(460\) −8.20101 −0.382374
\(461\) 19.4142 0.904210 0.452105 0.891965i \(-0.350673\pi\)
0.452105 + 0.891965i \(0.350673\pi\)
\(462\) 0 0
\(463\) 18.6274 0.865689 0.432845 0.901468i \(-0.357510\pi\)
0.432845 + 0.901468i \(0.357510\pi\)
\(464\) −3.51472 −0.163167
\(465\) 4.00000 0.185496
\(466\) −53.4558 −2.47629
\(467\) −39.7990 −1.84168 −0.920839 0.389943i \(-0.872495\pi\)
−0.920839 + 0.389943i \(0.872495\pi\)
\(468\) −20.7279 −0.958149
\(469\) 0 0
\(470\) −4.00000 −0.184506
\(471\) 20.2426 0.932732
\(472\) −30.1421 −1.38740
\(473\) 11.3137 0.520205
\(474\) 5.65685 0.259828
\(475\) 13.1716 0.604353
\(476\) 0 0
\(477\) −2.00000 −0.0915737
\(478\) 10.4853 0.479586
\(479\) −30.1421 −1.37723 −0.688615 0.725127i \(-0.741781\pi\)
−0.688615 + 0.725127i \(0.741781\pi\)
\(480\) 0.928932 0.0423998
\(481\) 21.6569 0.987468
\(482\) −18.7279 −0.853033
\(483\) 0 0
\(484\) −26.7990 −1.21814
\(485\) 3.17157 0.144014
\(486\) 2.41421 0.109511
\(487\) −18.6274 −0.844089 −0.422044 0.906575i \(-0.638687\pi\)
−0.422044 + 0.906575i \(0.638687\pi\)
\(488\) 16.5858 0.750803
\(489\) −11.3137 −0.511624
\(490\) 0 0
\(491\) 38.9706 1.75872 0.879358 0.476160i \(-0.157972\pi\)
0.879358 + 0.476160i \(0.157972\pi\)
\(492\) −8.58579 −0.387077
\(493\) 7.31371 0.329393
\(494\) −36.9706 −1.66338
\(495\) 1.17157 0.0526583
\(496\) 20.4853 0.919816
\(497\) 0 0
\(498\) 36.9706 1.65669
\(499\) −19.3137 −0.864600 −0.432300 0.901730i \(-0.642298\pi\)
−0.432300 + 0.901730i \(0.642298\pi\)
\(500\) 21.6569 0.968524
\(501\) 19.7990 0.884554
\(502\) 10.8284 0.483296
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 10.0000 0.444994
\(506\) 17.6569 0.784943
\(507\) −16.3137 −0.724517
\(508\) 36.9706 1.64030
\(509\) 25.5563 1.13277 0.566383 0.824142i \(-0.308342\pi\)
0.566383 + 0.824142i \(0.308342\pi\)
\(510\) 8.82843 0.390929
\(511\) 0 0
\(512\) 31.2426 1.38074
\(513\) 2.82843 0.124878
\(514\) −46.3848 −2.04594
\(515\) −7.31371 −0.322281
\(516\) 21.6569 0.953390
\(517\) 5.65685 0.248788
\(518\) 0 0
\(519\) 6.92893 0.304146
\(520\) −14.0000 −0.613941
\(521\) 32.5858 1.42761 0.713805 0.700345i \(-0.246970\pi\)
0.713805 + 0.700345i \(0.246970\pi\)
\(522\) 2.82843 0.123797
\(523\) 14.3431 0.627182 0.313591 0.949558i \(-0.398468\pi\)
0.313591 + 0.949558i \(0.398468\pi\)
\(524\) −28.0000 −1.22319
\(525\) 0 0
\(526\) 41.7990 1.82252
\(527\) −42.6274 −1.85688
\(528\) 6.00000 0.261116
\(529\) −9.62742 −0.418583
\(530\) −2.82843 −0.122859
\(531\) 6.82843 0.296328
\(532\) 0 0
\(533\) −12.1421 −0.525934
\(534\) 13.8995 0.601490
\(535\) 6.82843 0.295219
\(536\) −24.9706 −1.07856
\(537\) 8.34315 0.360033
\(538\) 25.8995 1.11661
\(539\) 0 0
\(540\) 2.24264 0.0965079
\(541\) −5.31371 −0.228454 −0.114227 0.993455i \(-0.536439\pi\)
−0.114227 + 0.993455i \(0.536439\pi\)
\(542\) 43.7990 1.88133
\(543\) −5.41421 −0.232346
\(544\) −9.89949 −0.424437
\(545\) −3.31371 −0.141944
\(546\) 0 0
\(547\) −3.02944 −0.129529 −0.0647647 0.997901i \(-0.520630\pi\)
−0.0647647 + 0.997901i \(0.520630\pi\)
\(548\) −54.1421 −2.31284
\(549\) −3.75736 −0.160360
\(550\) −22.4853 −0.958776
\(551\) 3.31371 0.141169
\(552\) 16.1421 0.687055
\(553\) 0 0
\(554\) −32.1421 −1.36559
\(555\) −2.34315 −0.0994610
\(556\) 24.2843 1.02988
\(557\) 26.0000 1.10166 0.550828 0.834619i \(-0.314312\pi\)
0.550828 + 0.834619i \(0.314312\pi\)
\(558\) −16.4853 −0.697878
\(559\) 30.6274 1.29540
\(560\) 0 0
\(561\) −12.4853 −0.527129
\(562\) 39.7990 1.67882
\(563\) 6.82843 0.287784 0.143892 0.989593i \(-0.454038\pi\)
0.143892 + 0.989593i \(0.454038\pi\)
\(564\) 10.8284 0.455959
\(565\) −10.1421 −0.426683
\(566\) 20.4853 0.861061
\(567\) 0 0
\(568\) 58.7696 2.46592
\(569\) 0.485281 0.0203441 0.0101720 0.999948i \(-0.496762\pi\)
0.0101720 + 0.999948i \(0.496762\pi\)
\(570\) 4.00000 0.167542
\(571\) 33.6569 1.40850 0.704248 0.709954i \(-0.251284\pi\)
0.704248 + 0.709954i \(0.251284\pi\)
\(572\) 41.4558 1.73336
\(573\) 18.0000 0.751961
\(574\) 0 0
\(575\) −17.0294 −0.710177
\(576\) −9.82843 −0.409518
\(577\) 14.1005 0.587012 0.293506 0.955957i \(-0.405178\pi\)
0.293506 + 0.955957i \(0.405178\pi\)
\(578\) −53.0416 −2.20624
\(579\) 17.3137 0.719533
\(580\) 2.62742 0.109098
\(581\) 0 0
\(582\) −13.0711 −0.541813
\(583\) 4.00000 0.165663
\(584\) −26.0416 −1.07761
\(585\) 3.17157 0.131128
\(586\) 46.8701 1.93618
\(587\) 17.1716 0.708747 0.354373 0.935104i \(-0.384694\pi\)
0.354373 + 0.935104i \(0.384694\pi\)
\(588\) 0 0
\(589\) −19.3137 −0.795807
\(590\) 9.65685 0.397566
\(591\) −2.00000 −0.0822690
\(592\) −12.0000 −0.493197
\(593\) 21.0711 0.865285 0.432643 0.901566i \(-0.357581\pi\)
0.432643 + 0.901566i \(0.357581\pi\)
\(594\) −4.82843 −0.198113
\(595\) 0 0
\(596\) −20.3431 −0.833288
\(597\) 10.3431 0.423317
\(598\) 47.7990 1.95465
\(599\) −2.00000 −0.0817178 −0.0408589 0.999165i \(-0.513009\pi\)
−0.0408589 + 0.999165i \(0.513009\pi\)
\(600\) −20.5563 −0.839209
\(601\) 0.928932 0.0378919 0.0189460 0.999821i \(-0.493969\pi\)
0.0189460 + 0.999821i \(0.493969\pi\)
\(602\) 0 0
\(603\) 5.65685 0.230365
\(604\) 45.9411 1.86932
\(605\) 4.10051 0.166709
\(606\) −41.2132 −1.67417
\(607\) 29.6569 1.20373 0.601867 0.798596i \(-0.294424\pi\)
0.601867 + 0.798596i \(0.294424\pi\)
\(608\) −4.48528 −0.181902
\(609\) 0 0
\(610\) −5.31371 −0.215146
\(611\) 15.3137 0.619526
\(612\) −23.8995 −0.966080
\(613\) 27.3137 1.10319 0.551595 0.834112i \(-0.314019\pi\)
0.551595 + 0.834112i \(0.314019\pi\)
\(614\) 4.48528 0.181011
\(615\) 1.31371 0.0529738
\(616\) 0 0
\(617\) −7.51472 −0.302531 −0.151266 0.988493i \(-0.548335\pi\)
−0.151266 + 0.988493i \(0.548335\pi\)
\(618\) 30.1421 1.21249
\(619\) 4.97056 0.199784 0.0998919 0.994998i \(-0.468150\pi\)
0.0998919 + 0.994998i \(0.468150\pi\)
\(620\) −15.3137 −0.615013
\(621\) −3.65685 −0.146745
\(622\) −53.4558 −2.14338
\(623\) 0 0
\(624\) 16.2426 0.650226
\(625\) 19.9706 0.798823
\(626\) −43.2132 −1.72715
\(627\) −5.65685 −0.225913
\(628\) −77.4975 −3.09249
\(629\) 24.9706 0.995642
\(630\) 0 0
\(631\) 0.686292 0.0273208 0.0136604 0.999907i \(-0.495652\pi\)
0.0136604 + 0.999907i \(0.495652\pi\)
\(632\) −10.3431 −0.411428
\(633\) 20.9706 0.833505
\(634\) −24.1421 −0.958807
\(635\) −5.65685 −0.224485
\(636\) 7.65685 0.303614
\(637\) 0 0
\(638\) −5.65685 −0.223957
\(639\) −13.3137 −0.526682
\(640\) −12.0416 −0.475987
\(641\) −5.17157 −0.204265 −0.102132 0.994771i \(-0.532567\pi\)
−0.102132 + 0.994771i \(0.532567\pi\)
\(642\) −28.1421 −1.11068
\(643\) −50.4264 −1.98862 −0.994312 0.106510i \(-0.966033\pi\)
−0.994312 + 0.106510i \(0.966033\pi\)
\(644\) 0 0
\(645\) −3.31371 −0.130477
\(646\) −42.6274 −1.67715
\(647\) −21.1716 −0.832340 −0.416170 0.909287i \(-0.636628\pi\)
−0.416170 + 0.909287i \(0.636628\pi\)
\(648\) −4.41421 −0.173407
\(649\) −13.6569 −0.536078
\(650\) −60.8701 −2.38752
\(651\) 0 0
\(652\) 43.3137 1.69630
\(653\) 19.5147 0.763670 0.381835 0.924231i \(-0.375292\pi\)
0.381835 + 0.924231i \(0.375292\pi\)
\(654\) 13.6569 0.534025
\(655\) 4.28427 0.167400
\(656\) 6.72792 0.262681
\(657\) 5.89949 0.230161
\(658\) 0 0
\(659\) −13.3137 −0.518628 −0.259314 0.965793i \(-0.583497\pi\)
−0.259314 + 0.965793i \(0.583497\pi\)
\(660\) −4.48528 −0.174589
\(661\) −7.55635 −0.293908 −0.146954 0.989143i \(-0.546947\pi\)
−0.146954 + 0.989143i \(0.546947\pi\)
\(662\) 9.65685 0.375324
\(663\) −33.7990 −1.31264
\(664\) −67.5980 −2.62331
\(665\) 0 0
\(666\) 9.65685 0.374196
\(667\) −4.28427 −0.165888
\(668\) −75.7990 −2.93275
\(669\) −8.97056 −0.346822
\(670\) 8.00000 0.309067
\(671\) 7.51472 0.290102
\(672\) 0 0
\(673\) 0.686292 0.0264546 0.0132273 0.999913i \(-0.495789\pi\)
0.0132273 + 0.999913i \(0.495789\pi\)
\(674\) 44.2843 1.70577
\(675\) 4.65685 0.179242
\(676\) 62.4558 2.40215
\(677\) −28.5858 −1.09864 −0.549321 0.835612i \(-0.685113\pi\)
−0.549321 + 0.835612i \(0.685113\pi\)
\(678\) 41.7990 1.60528
\(679\) 0 0
\(680\) −16.1421 −0.619023
\(681\) −15.7990 −0.605419
\(682\) 32.9706 1.26251
\(683\) −8.34315 −0.319242 −0.159621 0.987178i \(-0.551027\pi\)
−0.159621 + 0.987178i \(0.551027\pi\)
\(684\) −10.8284 −0.414035
\(685\) 8.28427 0.316526
\(686\) 0 0
\(687\) 8.24264 0.314476
\(688\) −16.9706 −0.646997
\(689\) 10.8284 0.412530
\(690\) −5.17157 −0.196878
\(691\) 23.3137 0.886895 0.443448 0.896300i \(-0.353755\pi\)
0.443448 + 0.896300i \(0.353755\pi\)
\(692\) −26.5269 −1.00840
\(693\) 0 0
\(694\) −25.7990 −0.979316
\(695\) −3.71573 −0.140946
\(696\) −5.17157 −0.196028
\(697\) −14.0000 −0.530288
\(698\) 23.8995 0.904609
\(699\) −22.1421 −0.837492
\(700\) 0 0
\(701\) −22.8284 −0.862218 −0.431109 0.902300i \(-0.641878\pi\)
−0.431109 + 0.902300i \(0.641878\pi\)
\(702\) −13.0711 −0.493336
\(703\) 11.3137 0.426705
\(704\) 19.6569 0.740846
\(705\) −1.65685 −0.0624007
\(706\) 25.8995 0.974740
\(707\) 0 0
\(708\) −26.1421 −0.982482
\(709\) 20.2843 0.761792 0.380896 0.924618i \(-0.375616\pi\)
0.380896 + 0.924618i \(0.375616\pi\)
\(710\) −18.8284 −0.706618
\(711\) 2.34315 0.0878748
\(712\) −25.4142 −0.952438
\(713\) 24.9706 0.935155
\(714\) 0 0
\(715\) −6.34315 −0.237220
\(716\) −31.9411 −1.19370
\(717\) 4.34315 0.162198
\(718\) 28.1421 1.05026
\(719\) 25.9411 0.967441 0.483720 0.875223i \(-0.339285\pi\)
0.483720 + 0.875223i \(0.339285\pi\)
\(720\) −1.75736 −0.0654929
\(721\) 0 0
\(722\) 26.5563 0.988325
\(723\) −7.75736 −0.288499
\(724\) 20.7279 0.770347
\(725\) 5.45584 0.202625
\(726\) −16.8995 −0.627199
\(727\) 4.48528 0.166350 0.0831749 0.996535i \(-0.473494\pi\)
0.0831749 + 0.996535i \(0.473494\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 8.34315 0.308794
\(731\) 35.3137 1.30612
\(732\) 14.3848 0.531677
\(733\) 9.69848 0.358222 0.179111 0.983829i \(-0.442678\pi\)
0.179111 + 0.983829i \(0.442678\pi\)
\(734\) 46.6274 1.72105
\(735\) 0 0
\(736\) 5.79899 0.213754
\(737\) −11.3137 −0.416746
\(738\) −5.41421 −0.199300
\(739\) 27.3137 1.00475 0.502376 0.864650i \(-0.332460\pi\)
0.502376 + 0.864650i \(0.332460\pi\)
\(740\) 8.97056 0.329764
\(741\) −15.3137 −0.562563
\(742\) 0 0
\(743\) −17.0294 −0.624749 −0.312375 0.949959i \(-0.601124\pi\)
−0.312375 + 0.949959i \(0.601124\pi\)
\(744\) 30.1421 1.10506
\(745\) 3.11270 0.114040
\(746\) 80.4264 2.94462
\(747\) 15.3137 0.560299
\(748\) 47.7990 1.74770
\(749\) 0 0
\(750\) 13.6569 0.498678
\(751\) 2.34315 0.0855026 0.0427513 0.999086i \(-0.486388\pi\)
0.0427513 + 0.999086i \(0.486388\pi\)
\(752\) −8.48528 −0.309426
\(753\) 4.48528 0.163453
\(754\) −15.3137 −0.557692
\(755\) −7.02944 −0.255827
\(756\) 0 0
\(757\) 37.6569 1.36866 0.684331 0.729172i \(-0.260094\pi\)
0.684331 + 0.729172i \(0.260094\pi\)
\(758\) −75.5980 −2.74584
\(759\) 7.31371 0.265471
\(760\) −7.31371 −0.265296
\(761\) −46.5269 −1.68660 −0.843300 0.537444i \(-0.819390\pi\)
−0.843300 + 0.537444i \(0.819390\pi\)
\(762\) 23.3137 0.844567
\(763\) 0 0
\(764\) −68.9117 −2.49314
\(765\) 3.65685 0.132214
\(766\) −71.5980 −2.58694
\(767\) −36.9706 −1.33493
\(768\) 29.9706 1.08147
\(769\) 29.6985 1.07095 0.535477 0.844550i \(-0.320132\pi\)
0.535477 + 0.844550i \(0.320132\pi\)
\(770\) 0 0
\(771\) −19.2132 −0.691947
\(772\) −66.2843 −2.38562
\(773\) −21.5563 −0.775328 −0.387664 0.921801i \(-0.626718\pi\)
−0.387664 + 0.921801i \(0.626718\pi\)
\(774\) 13.6569 0.490885
\(775\) −31.7990 −1.14225
\(776\) 23.8995 0.857942
\(777\) 0 0
\(778\) −24.4853 −0.877840
\(779\) −6.34315 −0.227267
\(780\) −12.1421 −0.434758
\(781\) 26.6274 0.952804
\(782\) 55.1127 1.97083
\(783\) 1.17157 0.0418686
\(784\) 0 0
\(785\) 11.8579 0.423225
\(786\) −17.6569 −0.629799
\(787\) −47.3137 −1.68655 −0.843276 0.537481i \(-0.819376\pi\)
−0.843276 + 0.537481i \(0.819376\pi\)
\(788\) 7.65685 0.272764
\(789\) 17.3137 0.616384
\(790\) 3.31371 0.117896
\(791\) 0 0
\(792\) 8.82843 0.313704
\(793\) 20.3431 0.722406
\(794\) 83.0122 2.94599
\(795\) −1.17157 −0.0415514
\(796\) −39.5980 −1.40351
\(797\) −28.3848 −1.00544 −0.502720 0.864449i \(-0.667667\pi\)
−0.502720 + 0.864449i \(0.667667\pi\)
\(798\) 0 0
\(799\) 17.6569 0.624655
\(800\) −7.38478 −0.261091
\(801\) 5.75736 0.203426
\(802\) −53.4558 −1.88759
\(803\) −11.7990 −0.416377
\(804\) −21.6569 −0.763778
\(805\) 0 0
\(806\) 89.2548 3.14387
\(807\) 10.7279 0.377641
\(808\) 75.3553 2.65099
\(809\) −47.9411 −1.68552 −0.842760 0.538289i \(-0.819071\pi\)
−0.842760 + 0.538289i \(0.819071\pi\)
\(810\) 1.41421 0.0496904
\(811\) 6.34315 0.222738 0.111369 0.993779i \(-0.464476\pi\)
0.111369 + 0.993779i \(0.464476\pi\)
\(812\) 0 0
\(813\) 18.1421 0.636272
\(814\) −19.3137 −0.676945
\(815\) −6.62742 −0.232148
\(816\) 18.7279 0.655608
\(817\) 16.0000 0.559769
\(818\) −44.8701 −1.56884
\(819\) 0 0
\(820\) −5.02944 −0.175636
\(821\) −33.3137 −1.16266 −0.581328 0.813669i \(-0.697467\pi\)
−0.581328 + 0.813669i \(0.697467\pi\)
\(822\) −34.1421 −1.19084
\(823\) −24.9706 −0.870419 −0.435210 0.900329i \(-0.643326\pi\)
−0.435210 + 0.900329i \(0.643326\pi\)
\(824\) −55.1127 −1.91994
\(825\) −9.31371 −0.324262
\(826\) 0 0
\(827\) 36.3431 1.26378 0.631888 0.775060i \(-0.282280\pi\)
0.631888 + 0.775060i \(0.282280\pi\)
\(828\) 14.0000 0.486534
\(829\) −24.7279 −0.858836 −0.429418 0.903106i \(-0.641281\pi\)
−0.429418 + 0.903106i \(0.641281\pi\)
\(830\) 21.6569 0.751720
\(831\) −13.3137 −0.461847
\(832\) 53.2132 1.84484
\(833\) 0 0
\(834\) 15.3137 0.530270
\(835\) 11.5980 0.401365
\(836\) 21.6569 0.749018
\(837\) −6.82843 −0.236025
\(838\) 93.7401 3.23820
\(839\) −45.1716 −1.55950 −0.779748 0.626094i \(-0.784653\pi\)
−0.779748 + 0.626094i \(0.784653\pi\)
\(840\) 0 0
\(841\) −27.6274 −0.952670
\(842\) 69.1127 2.38178
\(843\) 16.4853 0.567783
\(844\) −80.2843 −2.76350
\(845\) −9.55635 −0.328748
\(846\) 6.82843 0.234766
\(847\) 0 0
\(848\) −6.00000 −0.206041
\(849\) 8.48528 0.291214
\(850\) −70.1838 −2.40728
\(851\) −14.6274 −0.501421
\(852\) 50.9706 1.74622
\(853\) −49.4975 −1.69476 −0.847381 0.530986i \(-0.821822\pi\)
−0.847381 + 0.530986i \(0.821822\pi\)
\(854\) 0 0
\(855\) 1.65685 0.0566632
\(856\) 51.4558 1.75872
\(857\) −12.5858 −0.429922 −0.214961 0.976623i \(-0.568962\pi\)
−0.214961 + 0.976623i \(0.568962\pi\)
\(858\) 26.1421 0.892478
\(859\) −6.54416 −0.223284 −0.111642 0.993749i \(-0.535611\pi\)
−0.111642 + 0.993749i \(0.535611\pi\)
\(860\) 12.6863 0.432599
\(861\) 0 0
\(862\) −16.8284 −0.573179
\(863\) −5.31371 −0.180881 −0.0904404 0.995902i \(-0.528827\pi\)
−0.0904404 + 0.995902i \(0.528827\pi\)
\(864\) −1.58579 −0.0539496
\(865\) 4.05887 0.138006
\(866\) −28.3848 −0.964554
\(867\) −21.9706 −0.746159
\(868\) 0 0
\(869\) −4.68629 −0.158972
\(870\) 1.65685 0.0561726
\(871\) −30.6274 −1.03777
\(872\) −24.9706 −0.845610
\(873\) −5.41421 −0.183243
\(874\) 24.9706 0.844642
\(875\) 0 0
\(876\) −22.5858 −0.763103
\(877\) 11.3137 0.382037 0.191018 0.981586i \(-0.438821\pi\)
0.191018 + 0.981586i \(0.438821\pi\)
\(878\) 85.2548 2.87721
\(879\) 19.4142 0.654825
\(880\) 3.51472 0.118481
\(881\) −30.2426 −1.01890 −0.509450 0.860500i \(-0.670151\pi\)
−0.509450 + 0.860500i \(0.670151\pi\)
\(882\) 0 0
\(883\) −27.3137 −0.919179 −0.459590 0.888131i \(-0.652004\pi\)
−0.459590 + 0.888131i \(0.652004\pi\)
\(884\) 129.397 4.35209
\(885\) 4.00000 0.134459
\(886\) −2.48528 −0.0834947
\(887\) −2.82843 −0.0949693 −0.0474846 0.998872i \(-0.515121\pi\)
−0.0474846 + 0.998872i \(0.515121\pi\)
\(888\) −17.6569 −0.592525
\(889\) 0 0
\(890\) 8.14214 0.272925
\(891\) −2.00000 −0.0670025
\(892\) 34.3431 1.14989
\(893\) 8.00000 0.267710
\(894\) −12.8284 −0.429047
\(895\) 4.88730 0.163364
\(896\) 0 0
\(897\) 19.7990 0.661069
\(898\) −41.7990 −1.39485
\(899\) −8.00000 −0.266815
\(900\) −17.8284 −0.594281
\(901\) 12.4853 0.415945
\(902\) 10.8284 0.360547
\(903\) 0 0
\(904\) −76.4264 −2.54190
\(905\) −3.17157 −0.105427
\(906\) 28.9706 0.962482
\(907\) −16.0000 −0.531271 −0.265636 0.964073i \(-0.585582\pi\)
−0.265636 + 0.964073i \(0.585582\pi\)
\(908\) 60.4853 2.00727
\(909\) −17.0711 −0.566212
\(910\) 0 0
\(911\) −34.9706 −1.15863 −0.579313 0.815105i \(-0.696679\pi\)
−0.579313 + 0.815105i \(0.696679\pi\)
\(912\) 8.48528 0.280976
\(913\) −30.6274 −1.01362
\(914\) 43.4558 1.43739
\(915\) −2.20101 −0.0727632
\(916\) −31.5563 −1.04265
\(917\) 0 0
\(918\) −15.0711 −0.497419
\(919\) 48.2843 1.59275 0.796376 0.604802i \(-0.206748\pi\)
0.796376 + 0.604802i \(0.206748\pi\)
\(920\) 9.45584 0.311750
\(921\) 1.85786 0.0612187
\(922\) −46.8701 −1.54358
\(923\) 72.0833 2.37265
\(924\) 0 0
\(925\) 18.6274 0.612466
\(926\) −44.9706 −1.47782
\(927\) 12.4853 0.410070
\(928\) −1.85786 −0.0609874
\(929\) −3.21320 −0.105422 −0.0527109 0.998610i \(-0.516786\pi\)
−0.0527109 + 0.998610i \(0.516786\pi\)
\(930\) −9.65685 −0.316661
\(931\) 0 0
\(932\) 84.7696 2.77672
\(933\) −22.1421 −0.724901
\(934\) 96.0833 3.14394
\(935\) −7.31371 −0.239184
\(936\) 23.8995 0.781179
\(937\) −33.4142 −1.09159 −0.545797 0.837917i \(-0.683773\pi\)
−0.545797 + 0.837917i \(0.683773\pi\)
\(938\) 0 0
\(939\) −17.8995 −0.584128
\(940\) 6.34315 0.206891
\(941\) 7.21320 0.235144 0.117572 0.993064i \(-0.462489\pi\)
0.117572 + 0.993064i \(0.462489\pi\)
\(942\) −48.8701 −1.59227
\(943\) 8.20101 0.267062
\(944\) 20.4853 0.666739
\(945\) 0 0
\(946\) −27.3137 −0.888045
\(947\) 53.3137 1.73246 0.866231 0.499643i \(-0.166535\pi\)
0.866231 + 0.499643i \(0.166535\pi\)
\(948\) −8.97056 −0.291350
\(949\) −31.9411 −1.03685
\(950\) −31.7990 −1.03170
\(951\) −10.0000 −0.324272
\(952\) 0 0
\(953\) 2.00000 0.0647864 0.0323932 0.999475i \(-0.489687\pi\)
0.0323932 + 0.999475i \(0.489687\pi\)
\(954\) 4.82843 0.156326
\(955\) 10.5442 0.341201
\(956\) −16.6274 −0.537769
\(957\) −2.34315 −0.0757431
\(958\) 72.7696 2.35108
\(959\) 0 0
\(960\) −5.75736 −0.185818
\(961\) 15.6274 0.504110
\(962\) −52.2843 −1.68571
\(963\) −11.6569 −0.375637
\(964\) 29.6985 0.956524
\(965\) 10.1421 0.326487
\(966\) 0 0
\(967\) 22.3431 0.718507 0.359254 0.933240i \(-0.383031\pi\)
0.359254 + 0.933240i \(0.383031\pi\)
\(968\) 30.8995 0.993147
\(969\) −17.6569 −0.567220
\(970\) −7.65685 −0.245847
\(971\) 5.37258 0.172414 0.0862072 0.996277i \(-0.472525\pi\)
0.0862072 + 0.996277i \(0.472525\pi\)
\(972\) −3.82843 −0.122797
\(973\) 0 0
\(974\) 44.9706 1.44095
\(975\) −25.2132 −0.807469
\(976\) −11.2721 −0.360810
\(977\) −26.8284 −0.858317 −0.429159 0.903229i \(-0.641190\pi\)
−0.429159 + 0.903229i \(0.641190\pi\)
\(978\) 27.3137 0.873396
\(979\) −11.5147 −0.368012
\(980\) 0 0
\(981\) 5.65685 0.180609
\(982\) −94.0833 −3.00232
\(983\) −37.2548 −1.18824 −0.594122 0.804375i \(-0.702501\pi\)
−0.594122 + 0.804375i \(0.702501\pi\)
\(984\) 9.89949 0.315584
\(985\) −1.17157 −0.0373294
\(986\) −17.6569 −0.562309
\(987\) 0 0
\(988\) 58.6274 1.86519
\(989\) −20.6863 −0.657786
\(990\) −2.82843 −0.0898933
\(991\) 20.9706 0.666152 0.333076 0.942900i \(-0.391913\pi\)
0.333076 + 0.942900i \(0.391913\pi\)
\(992\) 10.8284 0.343803
\(993\) 4.00000 0.126936
\(994\) 0 0
\(995\) 6.05887 0.192079
\(996\) −58.6274 −1.85768
\(997\) 10.3848 0.328889 0.164445 0.986386i \(-0.447417\pi\)
0.164445 + 0.986386i \(0.447417\pi\)
\(998\) 46.6274 1.47597
\(999\) 4.00000 0.126554
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 147.2.a.d.1.1 2
3.2 odd 2 441.2.a.j.1.2 2
4.3 odd 2 2352.2.a.be.1.2 2
5.4 even 2 3675.2.a.bf.1.2 2
7.2 even 3 147.2.e.e.67.2 4
7.3 odd 6 147.2.e.d.79.2 4
7.4 even 3 147.2.e.e.79.2 4
7.5 odd 6 147.2.e.d.67.2 4
7.6 odd 2 147.2.a.e.1.1 yes 2
8.3 odd 2 9408.2.a.dq.1.1 2
8.5 even 2 9408.2.a.ef.1.1 2
12.11 even 2 7056.2.a.cv.1.1 2
21.2 odd 6 441.2.e.f.361.1 4
21.5 even 6 441.2.e.g.361.1 4
21.11 odd 6 441.2.e.f.226.1 4
21.17 even 6 441.2.e.g.226.1 4
21.20 even 2 441.2.a.i.1.2 2
28.3 even 6 2352.2.q.bd.961.2 4
28.11 odd 6 2352.2.q.bb.961.1 4
28.19 even 6 2352.2.q.bd.1537.2 4
28.23 odd 6 2352.2.q.bb.1537.1 4
28.27 even 2 2352.2.a.bc.1.1 2
35.34 odd 2 3675.2.a.bd.1.2 2
56.13 odd 2 9408.2.a.di.1.2 2
56.27 even 2 9408.2.a.dt.1.2 2
84.83 odd 2 7056.2.a.cf.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
147.2.a.d.1.1 2 1.1 even 1 trivial
147.2.a.e.1.1 yes 2 7.6 odd 2
147.2.e.d.67.2 4 7.5 odd 6
147.2.e.d.79.2 4 7.3 odd 6
147.2.e.e.67.2 4 7.2 even 3
147.2.e.e.79.2 4 7.4 even 3
441.2.a.i.1.2 2 21.20 even 2
441.2.a.j.1.2 2 3.2 odd 2
441.2.e.f.226.1 4 21.11 odd 6
441.2.e.f.361.1 4 21.2 odd 6
441.2.e.g.226.1 4 21.17 even 6
441.2.e.g.361.1 4 21.5 even 6
2352.2.a.bc.1.1 2 28.27 even 2
2352.2.a.be.1.2 2 4.3 odd 2
2352.2.q.bb.961.1 4 28.11 odd 6
2352.2.q.bb.1537.1 4 28.23 odd 6
2352.2.q.bd.961.2 4 28.3 even 6
2352.2.q.bd.1537.2 4 28.19 even 6
3675.2.a.bd.1.2 2 35.34 odd 2
3675.2.a.bf.1.2 2 5.4 even 2
7056.2.a.cf.1.2 2 84.83 odd 2
7056.2.a.cv.1.1 2 12.11 even 2
9408.2.a.di.1.2 2 56.13 odd 2
9408.2.a.dq.1.1 2 8.3 odd 2
9408.2.a.dt.1.2 2 56.27 even 2
9408.2.a.ef.1.1 2 8.5 even 2