Properties

Label 2601.2.a.t.1.1
Level $2601$
Weight $2$
Character 2601.1
Self dual yes
Analytic conductor $20.769$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2601 = 3^{2} \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2601.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(20.7690895657\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 51)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.56155\) of defining polynomial
Character \(\chi\) \(=\) 2601.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.56155 q^{2} +0.438447 q^{4} -0.561553 q^{5} +2.43845 q^{8} +O(q^{10})\) \(q-1.56155 q^{2} +0.438447 q^{4} -0.561553 q^{5} +2.43845 q^{8} +0.876894 q^{10} -2.56155 q^{11} +4.56155 q^{13} -4.68466 q^{16} +7.68466 q^{19} -0.246211 q^{20} +4.00000 q^{22} -6.56155 q^{23} -4.68466 q^{25} -7.12311 q^{26} +8.24621 q^{29} +5.12311 q^{31} +2.43845 q^{32} -3.12311 q^{37} -12.0000 q^{38} -1.36932 q^{40} +0.561553 q^{41} -7.68466 q^{43} -1.12311 q^{44} +10.2462 q^{46} +2.87689 q^{47} -7.00000 q^{49} +7.31534 q^{50} +2.00000 q^{52} +4.24621 q^{53} +1.43845 q^{55} -12.8769 q^{58} +1.12311 q^{59} -0.876894 q^{61} -8.00000 q^{62} +5.56155 q^{64} -2.56155 q^{65} +4.00000 q^{67} +10.2462 q^{71} -4.24621 q^{73} +4.87689 q^{74} +3.36932 q^{76} -15.3693 q^{79} +2.63068 q^{80} -0.876894 q^{82} +9.12311 q^{83} +12.0000 q^{86} -6.24621 q^{88} -7.12311 q^{89} -2.87689 q^{92} -4.49242 q^{94} -4.31534 q^{95} +11.1231 q^{97} +10.9309 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} + 5q^{4} + 3q^{5} + 9q^{8} + O(q^{10}) \) \( 2q + q^{2} + 5q^{4} + 3q^{5} + 9q^{8} + 10q^{10} - q^{11} + 5q^{13} + 3q^{16} + 3q^{19} + 16q^{20} + 8q^{22} - 9q^{23} + 3q^{25} - 6q^{26} + 2q^{31} + 9q^{32} + 2q^{37} - 24q^{38} + 22q^{40} - 3q^{41} - 3q^{43} + 6q^{44} + 4q^{46} + 14q^{47} - 14q^{49} + 27q^{50} + 4q^{52} - 8q^{53} + 7q^{55} - 34q^{58} - 6q^{59} - 10q^{61} - 16q^{62} + 7q^{64} - q^{65} + 8q^{67} + 4q^{71} + 8q^{73} + 18q^{74} - 18q^{76} - 6q^{79} + 30q^{80} - 10q^{82} + 10q^{83} + 24q^{86} + 4q^{88} - 6q^{89} - 14q^{92} + 24q^{94} - 21q^{95} + 14q^{97} - 7q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.56155 −1.10418 −0.552092 0.833783i \(-0.686170\pi\)
−0.552092 + 0.833783i \(0.686170\pi\)
\(3\) 0 0
\(4\) 0.438447 0.219224
\(5\) −0.561553 −0.251134 −0.125567 0.992085i \(-0.540075\pi\)
−0.125567 + 0.992085i \(0.540075\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 2.43845 0.862121
\(9\) 0 0
\(10\) 0.876894 0.277298
\(11\) −2.56155 −0.772337 −0.386169 0.922428i \(-0.626202\pi\)
−0.386169 + 0.922428i \(0.626202\pi\)
\(12\) 0 0
\(13\) 4.56155 1.26515 0.632574 0.774500i \(-0.281999\pi\)
0.632574 + 0.774500i \(0.281999\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −4.68466 −1.17116
\(17\) 0 0
\(18\) 0 0
\(19\) 7.68466 1.76298 0.881491 0.472201i \(-0.156540\pi\)
0.881491 + 0.472201i \(0.156540\pi\)
\(20\) −0.246211 −0.0550545
\(21\) 0 0
\(22\) 4.00000 0.852803
\(23\) −6.56155 −1.36818 −0.684089 0.729398i \(-0.739800\pi\)
−0.684089 + 0.729398i \(0.739800\pi\)
\(24\) 0 0
\(25\) −4.68466 −0.936932
\(26\) −7.12311 −1.39696
\(27\) 0 0
\(28\) 0 0
\(29\) 8.24621 1.53128 0.765641 0.643268i \(-0.222422\pi\)
0.765641 + 0.643268i \(0.222422\pi\)
\(30\) 0 0
\(31\) 5.12311 0.920137 0.460068 0.887883i \(-0.347825\pi\)
0.460068 + 0.887883i \(0.347825\pi\)
\(32\) 2.43845 0.431061
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −3.12311 −0.513435 −0.256718 0.966486i \(-0.582641\pi\)
−0.256718 + 0.966486i \(0.582641\pi\)
\(38\) −12.0000 −1.94666
\(39\) 0 0
\(40\) −1.36932 −0.216508
\(41\) 0.561553 0.0876998 0.0438499 0.999038i \(-0.486038\pi\)
0.0438499 + 0.999038i \(0.486038\pi\)
\(42\) 0 0
\(43\) −7.68466 −1.17190 −0.585950 0.810347i \(-0.699278\pi\)
−0.585950 + 0.810347i \(0.699278\pi\)
\(44\) −1.12311 −0.169315
\(45\) 0 0
\(46\) 10.2462 1.51072
\(47\) 2.87689 0.419638 0.209819 0.977740i \(-0.432712\pi\)
0.209819 + 0.977740i \(0.432712\pi\)
\(48\) 0 0
\(49\) −7.00000 −1.00000
\(50\) 7.31534 1.03455
\(51\) 0 0
\(52\) 2.00000 0.277350
\(53\) 4.24621 0.583262 0.291631 0.956531i \(-0.405802\pi\)
0.291631 + 0.956531i \(0.405802\pi\)
\(54\) 0 0
\(55\) 1.43845 0.193960
\(56\) 0 0
\(57\) 0 0
\(58\) −12.8769 −1.69082
\(59\) 1.12311 0.146216 0.0731079 0.997324i \(-0.476708\pi\)
0.0731079 + 0.997324i \(0.476708\pi\)
\(60\) 0 0
\(61\) −0.876894 −0.112275 −0.0561374 0.998423i \(-0.517878\pi\)
−0.0561374 + 0.998423i \(0.517878\pi\)
\(62\) −8.00000 −1.01600
\(63\) 0 0
\(64\) 5.56155 0.695194
\(65\) −2.56155 −0.317722
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 10.2462 1.21600 0.608001 0.793936i \(-0.291972\pi\)
0.608001 + 0.793936i \(0.291972\pi\)
\(72\) 0 0
\(73\) −4.24621 −0.496981 −0.248491 0.968634i \(-0.579935\pi\)
−0.248491 + 0.968634i \(0.579935\pi\)
\(74\) 4.87689 0.566927
\(75\) 0 0
\(76\) 3.36932 0.386487
\(77\) 0 0
\(78\) 0 0
\(79\) −15.3693 −1.72918 −0.864592 0.502475i \(-0.832423\pi\)
−0.864592 + 0.502475i \(0.832423\pi\)
\(80\) 2.63068 0.294119
\(81\) 0 0
\(82\) −0.876894 −0.0968368
\(83\) 9.12311 1.00139 0.500695 0.865624i \(-0.333078\pi\)
0.500695 + 0.865624i \(0.333078\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 12.0000 1.29399
\(87\) 0 0
\(88\) −6.24621 −0.665848
\(89\) −7.12311 −0.755048 −0.377524 0.926000i \(-0.623224\pi\)
−0.377524 + 0.926000i \(0.623224\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −2.87689 −0.299937
\(93\) 0 0
\(94\) −4.49242 −0.463358
\(95\) −4.31534 −0.442745
\(96\) 0 0
\(97\) 11.1231 1.12938 0.564690 0.825303i \(-0.308996\pi\)
0.564690 + 0.825303i \(0.308996\pi\)
\(98\) 10.9309 1.10418
\(99\) 0 0
\(100\) −2.05398 −0.205398
\(101\) −19.1231 −1.90282 −0.951410 0.307927i \(-0.900365\pi\)
−0.951410 + 0.307927i \(0.900365\pi\)
\(102\) 0 0
\(103\) 4.31534 0.425203 0.212602 0.977139i \(-0.431806\pi\)
0.212602 + 0.977139i \(0.431806\pi\)
\(104\) 11.1231 1.09071
\(105\) 0 0
\(106\) −6.63068 −0.644029
\(107\) 7.68466 0.742904 0.371452 0.928452i \(-0.378860\pi\)
0.371452 + 0.928452i \(0.378860\pi\)
\(108\) 0 0
\(109\) 15.1231 1.44853 0.724265 0.689521i \(-0.242179\pi\)
0.724265 + 0.689521i \(0.242179\pi\)
\(110\) −2.24621 −0.214168
\(111\) 0 0
\(112\) 0 0
\(113\) −4.56155 −0.429115 −0.214557 0.976711i \(-0.568831\pi\)
−0.214557 + 0.976711i \(0.568831\pi\)
\(114\) 0 0
\(115\) 3.68466 0.343596
\(116\) 3.61553 0.335693
\(117\) 0 0
\(118\) −1.75379 −0.161449
\(119\) 0 0
\(120\) 0 0
\(121\) −4.43845 −0.403495
\(122\) 1.36932 0.123972
\(123\) 0 0
\(124\) 2.24621 0.201716
\(125\) 5.43845 0.486430
\(126\) 0 0
\(127\) −0.807764 −0.0716775 −0.0358387 0.999358i \(-0.511410\pi\)
−0.0358387 + 0.999358i \(0.511410\pi\)
\(128\) −13.5616 −1.19868
\(129\) 0 0
\(130\) 4.00000 0.350823
\(131\) 18.5616 1.62173 0.810865 0.585233i \(-0.198997\pi\)
0.810865 + 0.585233i \(0.198997\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −6.24621 −0.539590
\(135\) 0 0
\(136\) 0 0
\(137\) 16.2462 1.38801 0.694004 0.719971i \(-0.255845\pi\)
0.694004 + 0.719971i \(0.255845\pi\)
\(138\) 0 0
\(139\) 9.12311 0.773812 0.386906 0.922119i \(-0.373544\pi\)
0.386906 + 0.922119i \(0.373544\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −16.0000 −1.34269
\(143\) −11.6847 −0.977120
\(144\) 0 0
\(145\) −4.63068 −0.384557
\(146\) 6.63068 0.548759
\(147\) 0 0
\(148\) −1.36932 −0.112557
\(149\) 4.24621 0.347863 0.173932 0.984758i \(-0.444353\pi\)
0.173932 + 0.984758i \(0.444353\pi\)
\(150\) 0 0
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) 18.7386 1.51990
\(153\) 0 0
\(154\) 0 0
\(155\) −2.87689 −0.231078
\(156\) 0 0
\(157\) −5.68466 −0.453685 −0.226843 0.973931i \(-0.572840\pi\)
−0.226843 + 0.973931i \(0.572840\pi\)
\(158\) 24.0000 1.90934
\(159\) 0 0
\(160\) −1.36932 −0.108254
\(161\) 0 0
\(162\) 0 0
\(163\) 6.87689 0.538640 0.269320 0.963051i \(-0.413201\pi\)
0.269320 + 0.963051i \(0.413201\pi\)
\(164\) 0.246211 0.0192259
\(165\) 0 0
\(166\) −14.2462 −1.10572
\(167\) −0.807764 −0.0625067 −0.0312533 0.999511i \(-0.509950\pi\)
−0.0312533 + 0.999511i \(0.509950\pi\)
\(168\) 0 0
\(169\) 7.80776 0.600597
\(170\) 0 0
\(171\) 0 0
\(172\) −3.36932 −0.256908
\(173\) −18.8078 −1.42993 −0.714964 0.699161i \(-0.753557\pi\)
−0.714964 + 0.699161i \(0.753557\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 12.0000 0.904534
\(177\) 0 0
\(178\) 11.1231 0.833712
\(179\) 9.12311 0.681893 0.340946 0.940083i \(-0.389253\pi\)
0.340946 + 0.940083i \(0.389253\pi\)
\(180\) 0 0
\(181\) −6.00000 −0.445976 −0.222988 0.974821i \(-0.571581\pi\)
−0.222988 + 0.974821i \(0.571581\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −16.0000 −1.17954
\(185\) 1.75379 0.128941
\(186\) 0 0
\(187\) 0 0
\(188\) 1.26137 0.0919946
\(189\) 0 0
\(190\) 6.73863 0.488872
\(191\) 13.1231 0.949555 0.474777 0.880106i \(-0.342529\pi\)
0.474777 + 0.880106i \(0.342529\pi\)
\(192\) 0 0
\(193\) 24.2462 1.74528 0.872640 0.488364i \(-0.162406\pi\)
0.872640 + 0.488364i \(0.162406\pi\)
\(194\) −17.3693 −1.24704
\(195\) 0 0
\(196\) −3.06913 −0.219224
\(197\) 19.9309 1.42002 0.710008 0.704194i \(-0.248691\pi\)
0.710008 + 0.704194i \(0.248691\pi\)
\(198\) 0 0
\(199\) −16.0000 −1.13421 −0.567105 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) −11.4233 −0.807749
\(201\) 0 0
\(202\) 29.8617 2.10106
\(203\) 0 0
\(204\) 0 0
\(205\) −0.315342 −0.0220244
\(206\) −6.73863 −0.469503
\(207\) 0 0
\(208\) −21.3693 −1.48170
\(209\) −19.6847 −1.36162
\(210\) 0 0
\(211\) 11.3693 0.782696 0.391348 0.920243i \(-0.372009\pi\)
0.391348 + 0.920243i \(0.372009\pi\)
\(212\) 1.86174 0.127865
\(213\) 0 0
\(214\) −12.0000 −0.820303
\(215\) 4.31534 0.294304
\(216\) 0 0
\(217\) 0 0
\(218\) −23.6155 −1.59945
\(219\) 0 0
\(220\) 0.630683 0.0425206
\(221\) 0 0
\(222\) 0 0
\(223\) 13.9309 0.932880 0.466440 0.884553i \(-0.345536\pi\)
0.466440 + 0.884553i \(0.345536\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 7.12311 0.473822
\(227\) 23.0540 1.53015 0.765073 0.643944i \(-0.222703\pi\)
0.765073 + 0.643944i \(0.222703\pi\)
\(228\) 0 0
\(229\) 6.00000 0.396491 0.198246 0.980152i \(-0.436476\pi\)
0.198246 + 0.980152i \(0.436476\pi\)
\(230\) −5.75379 −0.379394
\(231\) 0 0
\(232\) 20.1080 1.32015
\(233\) 0.561553 0.0367885 0.0183943 0.999831i \(-0.494145\pi\)
0.0183943 + 0.999831i \(0.494145\pi\)
\(234\) 0 0
\(235\) −1.61553 −0.105385
\(236\) 0.492423 0.0320540
\(237\) 0 0
\(238\) 0 0
\(239\) −10.2462 −0.662772 −0.331386 0.943495i \(-0.607516\pi\)
−0.331386 + 0.943495i \(0.607516\pi\)
\(240\) 0 0
\(241\) 21.3693 1.37652 0.688259 0.725465i \(-0.258375\pi\)
0.688259 + 0.725465i \(0.258375\pi\)
\(242\) 6.93087 0.445533
\(243\) 0 0
\(244\) −0.384472 −0.0246133
\(245\) 3.93087 0.251134
\(246\) 0 0
\(247\) 35.0540 2.23043
\(248\) 12.4924 0.793270
\(249\) 0 0
\(250\) −8.49242 −0.537108
\(251\) 24.4924 1.54595 0.772974 0.634438i \(-0.218768\pi\)
0.772974 + 0.634438i \(0.218768\pi\)
\(252\) 0 0
\(253\) 16.8078 1.05670
\(254\) 1.26137 0.0791452
\(255\) 0 0
\(256\) 10.0540 0.628373
\(257\) −9.36932 −0.584442 −0.292221 0.956351i \(-0.594394\pi\)
−0.292221 + 0.956351i \(0.594394\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −1.12311 −0.0696521
\(261\) 0 0
\(262\) −28.9848 −1.79069
\(263\) −12.4924 −0.770316 −0.385158 0.922851i \(-0.625853\pi\)
−0.385158 + 0.922851i \(0.625853\pi\)
\(264\) 0 0
\(265\) −2.38447 −0.146477
\(266\) 0 0
\(267\) 0 0
\(268\) 1.75379 0.107130
\(269\) 20.5616 1.25366 0.626830 0.779156i \(-0.284352\pi\)
0.626830 + 0.779156i \(0.284352\pi\)
\(270\) 0 0
\(271\) −0.807764 −0.0490682 −0.0245341 0.999699i \(-0.507810\pi\)
−0.0245341 + 0.999699i \(0.507810\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) −25.3693 −1.53262
\(275\) 12.0000 0.723627
\(276\) 0 0
\(277\) −6.00000 −0.360505 −0.180253 0.983620i \(-0.557691\pi\)
−0.180253 + 0.983620i \(0.557691\pi\)
\(278\) −14.2462 −0.854431
\(279\) 0 0
\(280\) 0 0
\(281\) 19.1231 1.14079 0.570394 0.821371i \(-0.306790\pi\)
0.570394 + 0.821371i \(0.306790\pi\)
\(282\) 0 0
\(283\) 3.36932 0.200285 0.100143 0.994973i \(-0.468070\pi\)
0.100143 + 0.994973i \(0.468070\pi\)
\(284\) 4.49242 0.266576
\(285\) 0 0
\(286\) 18.2462 1.07892
\(287\) 0 0
\(288\) 0 0
\(289\) 0 0
\(290\) 7.23106 0.424622
\(291\) 0 0
\(292\) −1.86174 −0.108950
\(293\) 7.12311 0.416136 0.208068 0.978114i \(-0.433282\pi\)
0.208068 + 0.978114i \(0.433282\pi\)
\(294\) 0 0
\(295\) −0.630683 −0.0367198
\(296\) −7.61553 −0.442644
\(297\) 0 0
\(298\) −6.63068 −0.384105
\(299\) −29.9309 −1.73095
\(300\) 0 0
\(301\) 0 0
\(302\) −12.4924 −0.718858
\(303\) 0 0
\(304\) −36.0000 −2.06474
\(305\) 0.492423 0.0281960
\(306\) 0 0
\(307\) −0.492423 −0.0281040 −0.0140520 0.999901i \(-0.504473\pi\)
−0.0140520 + 0.999901i \(0.504473\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 4.49242 0.255152
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 7.61553 0.430455 0.215228 0.976564i \(-0.430951\pi\)
0.215228 + 0.976564i \(0.430951\pi\)
\(314\) 8.87689 0.500952
\(315\) 0 0
\(316\) −6.73863 −0.379078
\(317\) −18.0000 −1.01098 −0.505490 0.862832i \(-0.668688\pi\)
−0.505490 + 0.862832i \(0.668688\pi\)
\(318\) 0 0
\(319\) −21.1231 −1.18267
\(320\) −3.12311 −0.174587
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −21.3693 −1.18536
\(326\) −10.7386 −0.594758
\(327\) 0 0
\(328\) 1.36932 0.0756079
\(329\) 0 0
\(330\) 0 0
\(331\) −6.06913 −0.333590 −0.166795 0.985992i \(-0.553342\pi\)
−0.166795 + 0.985992i \(0.553342\pi\)
\(332\) 4.00000 0.219529
\(333\) 0 0
\(334\) 1.26137 0.0690189
\(335\) −2.24621 −0.122724
\(336\) 0 0
\(337\) −32.7386 −1.78339 −0.891694 0.452640i \(-0.850482\pi\)
−0.891694 + 0.452640i \(0.850482\pi\)
\(338\) −12.1922 −0.663170
\(339\) 0 0
\(340\) 0 0
\(341\) −13.1231 −0.710656
\(342\) 0 0
\(343\) 0 0
\(344\) −18.7386 −1.01032
\(345\) 0 0
\(346\) 29.3693 1.57890
\(347\) 24.4924 1.31482 0.657411 0.753532i \(-0.271652\pi\)
0.657411 + 0.753532i \(0.271652\pi\)
\(348\) 0 0
\(349\) 7.43845 0.398171 0.199085 0.979982i \(-0.436203\pi\)
0.199085 + 0.979982i \(0.436203\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −6.24621 −0.332924
\(353\) −22.4924 −1.19715 −0.598575 0.801066i \(-0.704266\pi\)
−0.598575 + 0.801066i \(0.704266\pi\)
\(354\) 0 0
\(355\) −5.75379 −0.305379
\(356\) −3.12311 −0.165524
\(357\) 0 0
\(358\) −14.2462 −0.752936
\(359\) 2.24621 0.118550 0.0592752 0.998242i \(-0.481121\pi\)
0.0592752 + 0.998242i \(0.481121\pi\)
\(360\) 0 0
\(361\) 40.0540 2.10810
\(362\) 9.36932 0.492440
\(363\) 0 0
\(364\) 0 0
\(365\) 2.38447 0.124809
\(366\) 0 0
\(367\) 18.2462 0.952444 0.476222 0.879325i \(-0.342006\pi\)
0.476222 + 0.879325i \(0.342006\pi\)
\(368\) 30.7386 1.60236
\(369\) 0 0
\(370\) −2.73863 −0.142375
\(371\) 0 0
\(372\) 0 0
\(373\) 16.2462 0.841197 0.420598 0.907247i \(-0.361820\pi\)
0.420598 + 0.907247i \(0.361820\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 7.01515 0.361779
\(377\) 37.6155 1.93730
\(378\) 0 0
\(379\) 12.0000 0.616399 0.308199 0.951322i \(-0.400274\pi\)
0.308199 + 0.951322i \(0.400274\pi\)
\(380\) −1.89205 −0.0970601
\(381\) 0 0
\(382\) −20.4924 −1.04848
\(383\) 10.2462 0.523557 0.261778 0.965128i \(-0.415691\pi\)
0.261778 + 0.965128i \(0.415691\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −37.8617 −1.92711
\(387\) 0 0
\(388\) 4.87689 0.247587
\(389\) 21.8617 1.10843 0.554217 0.832372i \(-0.313018\pi\)
0.554217 + 0.832372i \(0.313018\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −17.0691 −0.862121
\(393\) 0 0
\(394\) −31.1231 −1.56796
\(395\) 8.63068 0.434257
\(396\) 0 0
\(397\) −5.36932 −0.269478 −0.134739 0.990881i \(-0.543020\pi\)
−0.134739 + 0.990881i \(0.543020\pi\)
\(398\) 24.9848 1.25238
\(399\) 0 0
\(400\) 21.9460 1.09730
\(401\) −6.17708 −0.308469 −0.154234 0.988034i \(-0.549291\pi\)
−0.154234 + 0.988034i \(0.549291\pi\)
\(402\) 0 0
\(403\) 23.3693 1.16411
\(404\) −8.38447 −0.417143
\(405\) 0 0
\(406\) 0 0
\(407\) 8.00000 0.396545
\(408\) 0 0
\(409\) −2.31534 −0.114486 −0.0572431 0.998360i \(-0.518231\pi\)
−0.0572431 + 0.998360i \(0.518231\pi\)
\(410\) 0.492423 0.0243190
\(411\) 0 0
\(412\) 1.89205 0.0932146
\(413\) 0 0
\(414\) 0 0
\(415\) −5.12311 −0.251483
\(416\) 11.1231 0.545355
\(417\) 0 0
\(418\) 30.7386 1.50348
\(419\) 32.4924 1.58736 0.793679 0.608336i \(-0.208163\pi\)
0.793679 + 0.608336i \(0.208163\pi\)
\(420\) 0 0
\(421\) 28.5616 1.39200 0.696002 0.718039i \(-0.254960\pi\)
0.696002 + 0.718039i \(0.254960\pi\)
\(422\) −17.7538 −0.864241
\(423\) 0 0
\(424\) 10.3542 0.502843
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 3.36932 0.162862
\(429\) 0 0
\(430\) −6.73863 −0.324966
\(431\) −24.0000 −1.15604 −0.578020 0.816023i \(-0.696174\pi\)
−0.578020 + 0.816023i \(0.696174\pi\)
\(432\) 0 0
\(433\) 14.3153 0.687951 0.343976 0.938979i \(-0.388226\pi\)
0.343976 + 0.938979i \(0.388226\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 6.63068 0.317552
\(437\) −50.4233 −2.41207
\(438\) 0 0
\(439\) 5.75379 0.274613 0.137307 0.990529i \(-0.456155\pi\)
0.137307 + 0.990529i \(0.456155\pi\)
\(440\) 3.50758 0.167217
\(441\) 0 0
\(442\) 0 0
\(443\) 22.8769 1.08691 0.543457 0.839437i \(-0.317115\pi\)
0.543457 + 0.839437i \(0.317115\pi\)
\(444\) 0 0
\(445\) 4.00000 0.189618
\(446\) −21.7538 −1.03007
\(447\) 0 0
\(448\) 0 0
\(449\) −12.7386 −0.601173 −0.300587 0.953755i \(-0.597182\pi\)
−0.300587 + 0.953755i \(0.597182\pi\)
\(450\) 0 0
\(451\) −1.43845 −0.0677338
\(452\) −2.00000 −0.0940721
\(453\) 0 0
\(454\) −36.0000 −1.68956
\(455\) 0 0
\(456\) 0 0
\(457\) −6.80776 −0.318454 −0.159227 0.987242i \(-0.550900\pi\)
−0.159227 + 0.987242i \(0.550900\pi\)
\(458\) −9.36932 −0.437799
\(459\) 0 0
\(460\) 1.61553 0.0753244
\(461\) −8.24621 −0.384064 −0.192032 0.981389i \(-0.561508\pi\)
−0.192032 + 0.981389i \(0.561508\pi\)
\(462\) 0 0
\(463\) 24.9848 1.16114 0.580572 0.814209i \(-0.302829\pi\)
0.580572 + 0.814209i \(0.302829\pi\)
\(464\) −38.6307 −1.79338
\(465\) 0 0
\(466\) −0.876894 −0.0406213
\(467\) 3.36932 0.155913 0.0779567 0.996957i \(-0.475160\pi\)
0.0779567 + 0.996957i \(0.475160\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 2.52273 0.116365
\(471\) 0 0
\(472\) 2.73863 0.126056
\(473\) 19.6847 0.905102
\(474\) 0 0
\(475\) −36.0000 −1.65179
\(476\) 0 0
\(477\) 0 0
\(478\) 16.0000 0.731823
\(479\) −29.3002 −1.33876 −0.669380 0.742920i \(-0.733440\pi\)
−0.669380 + 0.742920i \(0.733440\pi\)
\(480\) 0 0
\(481\) −14.2462 −0.649571
\(482\) −33.3693 −1.51993
\(483\) 0 0
\(484\) −1.94602 −0.0884557
\(485\) −6.24621 −0.283626
\(486\) 0 0
\(487\) 7.36932 0.333936 0.166968 0.985962i \(-0.446602\pi\)
0.166968 + 0.985962i \(0.446602\pi\)
\(488\) −2.13826 −0.0967945
\(489\) 0 0
\(490\) −6.13826 −0.277298
\(491\) −3.36932 −0.152055 −0.0760276 0.997106i \(-0.524224\pi\)
−0.0760276 + 0.997106i \(0.524224\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) −54.7386 −2.46281
\(495\) 0 0
\(496\) −24.0000 −1.07763
\(497\) 0 0
\(498\) 0 0
\(499\) 11.3693 0.508961 0.254480 0.967078i \(-0.418096\pi\)
0.254480 + 0.967078i \(0.418096\pi\)
\(500\) 2.38447 0.106637
\(501\) 0 0
\(502\) −38.2462 −1.70701
\(503\) −25.4384 −1.13424 −0.567122 0.823634i \(-0.691943\pi\)
−0.567122 + 0.823634i \(0.691943\pi\)
\(504\) 0 0
\(505\) 10.7386 0.477863
\(506\) −26.2462 −1.16679
\(507\) 0 0
\(508\) −0.354162 −0.0157134
\(509\) −16.8769 −0.748055 −0.374028 0.927418i \(-0.622023\pi\)
−0.374028 + 0.927418i \(0.622023\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 11.4233 0.504843
\(513\) 0 0
\(514\) 14.6307 0.645332
\(515\) −2.42329 −0.106783
\(516\) 0 0
\(517\) −7.36932 −0.324102
\(518\) 0 0
\(519\) 0 0
\(520\) −6.24621 −0.273914
\(521\) −31.4384 −1.37734 −0.688672 0.725073i \(-0.741806\pi\)
−0.688672 + 0.725073i \(0.741806\pi\)
\(522\) 0 0
\(523\) −20.0000 −0.874539 −0.437269 0.899331i \(-0.644054\pi\)
−0.437269 + 0.899331i \(0.644054\pi\)
\(524\) 8.13826 0.355522
\(525\) 0 0
\(526\) 19.5076 0.850571
\(527\) 0 0
\(528\) 0 0
\(529\) 20.0540 0.871912
\(530\) 3.72348 0.161738
\(531\) 0 0
\(532\) 0 0
\(533\) 2.56155 0.110953
\(534\) 0 0
\(535\) −4.31534 −0.186568
\(536\) 9.75379 0.421300
\(537\) 0 0
\(538\) −32.1080 −1.38427
\(539\) 17.9309 0.772337
\(540\) 0 0
\(541\) 40.1080 1.72438 0.862188 0.506589i \(-0.169094\pi\)
0.862188 + 0.506589i \(0.169094\pi\)
\(542\) 1.26137 0.0541803
\(543\) 0 0
\(544\) 0 0
\(545\) −8.49242 −0.363775
\(546\) 0 0
\(547\) −28.0000 −1.19719 −0.598597 0.801050i \(-0.704275\pi\)
−0.598597 + 0.801050i \(0.704275\pi\)
\(548\) 7.12311 0.304284
\(549\) 0 0
\(550\) −18.7386 −0.799018
\(551\) 63.3693 2.69962
\(552\) 0 0
\(553\) 0 0
\(554\) 9.36932 0.398064
\(555\) 0 0
\(556\) 4.00000 0.169638
\(557\) 6.49242 0.275093 0.137546 0.990495i \(-0.456078\pi\)
0.137546 + 0.990495i \(0.456078\pi\)
\(558\) 0 0
\(559\) −35.0540 −1.48263
\(560\) 0 0
\(561\) 0 0
\(562\) −29.8617 −1.25964
\(563\) −22.8769 −0.964146 −0.482073 0.876131i \(-0.660116\pi\)
−0.482073 + 0.876131i \(0.660116\pi\)
\(564\) 0 0
\(565\) 2.56155 0.107765
\(566\) −5.26137 −0.221152
\(567\) 0 0
\(568\) 24.9848 1.04834
\(569\) −12.8769 −0.539827 −0.269914 0.962885i \(-0.586995\pi\)
−0.269914 + 0.962885i \(0.586995\pi\)
\(570\) 0 0
\(571\) −18.7386 −0.784187 −0.392094 0.919925i \(-0.628249\pi\)
−0.392094 + 0.919925i \(0.628249\pi\)
\(572\) −5.12311 −0.214208
\(573\) 0 0
\(574\) 0 0
\(575\) 30.7386 1.28189
\(576\) 0 0
\(577\) −41.0540 −1.70910 −0.854550 0.519370i \(-0.826167\pi\)
−0.854550 + 0.519370i \(0.826167\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) −2.03031 −0.0843040
\(581\) 0 0
\(582\) 0 0
\(583\) −10.8769 −0.450475
\(584\) −10.3542 −0.428458
\(585\) 0 0
\(586\) −11.1231 −0.459491
\(587\) −36.9848 −1.52653 −0.763264 0.646087i \(-0.776404\pi\)
−0.763264 + 0.646087i \(0.776404\pi\)
\(588\) 0 0
\(589\) 39.3693 1.62218
\(590\) 0.984845 0.0405454
\(591\) 0 0
\(592\) 14.6307 0.601317
\(593\) −44.2462 −1.81697 −0.908487 0.417913i \(-0.862762\pi\)
−0.908487 + 0.417913i \(0.862762\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 1.86174 0.0762598
\(597\) 0 0
\(598\) 46.7386 1.91128
\(599\) 41.6155 1.70036 0.850182 0.526489i \(-0.176492\pi\)
0.850182 + 0.526489i \(0.176492\pi\)
\(600\) 0 0
\(601\) −34.9848 −1.42706 −0.713531 0.700624i \(-0.752905\pi\)
−0.713531 + 0.700624i \(0.752905\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 3.50758 0.142721
\(605\) 2.49242 0.101331
\(606\) 0 0
\(607\) −15.3693 −0.623821 −0.311911 0.950111i \(-0.600969\pi\)
−0.311911 + 0.950111i \(0.600969\pi\)
\(608\) 18.7386 0.759952
\(609\) 0 0
\(610\) −0.768944 −0.0311336
\(611\) 13.1231 0.530904
\(612\) 0 0
\(613\) 2.31534 0.0935158 0.0467579 0.998906i \(-0.485111\pi\)
0.0467579 + 0.998906i \(0.485111\pi\)
\(614\) 0.768944 0.0310320
\(615\) 0 0
\(616\) 0 0
\(617\) −27.7538 −1.11733 −0.558663 0.829395i \(-0.688685\pi\)
−0.558663 + 0.829395i \(0.688685\pi\)
\(618\) 0 0
\(619\) 19.3693 0.778519 0.389259 0.921128i \(-0.372731\pi\)
0.389259 + 0.921128i \(0.372731\pi\)
\(620\) −1.26137 −0.0506577
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 20.3693 0.814773
\(626\) −11.8920 −0.475302
\(627\) 0 0
\(628\) −2.49242 −0.0994585
\(629\) 0 0
\(630\) 0 0
\(631\) 11.6847 0.465159 0.232579 0.972577i \(-0.425283\pi\)
0.232579 + 0.972577i \(0.425283\pi\)
\(632\) −37.4773 −1.49077
\(633\) 0 0
\(634\) 28.1080 1.11631
\(635\) 0.453602 0.0180007
\(636\) 0 0
\(637\) −31.9309 −1.26515
\(638\) 32.9848 1.30588
\(639\) 0 0
\(640\) 7.61553 0.301030
\(641\) −0.0691303 −0.00273048 −0.00136524 0.999999i \(-0.500435\pi\)
−0.00136524 + 0.999999i \(0.500435\pi\)
\(642\) 0 0
\(643\) 30.2462 1.19279 0.596397 0.802690i \(-0.296598\pi\)
0.596397 + 0.802690i \(0.296598\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −15.3693 −0.604230 −0.302115 0.953271i \(-0.597693\pi\)
−0.302115 + 0.953271i \(0.597693\pi\)
\(648\) 0 0
\(649\) −2.87689 −0.112928
\(650\) 33.3693 1.30885
\(651\) 0 0
\(652\) 3.01515 0.118083
\(653\) −4.06913 −0.159237 −0.0796187 0.996825i \(-0.525370\pi\)
−0.0796187 + 0.996825i \(0.525370\pi\)
\(654\) 0 0
\(655\) −10.4233 −0.407272
\(656\) −2.63068 −0.102711
\(657\) 0 0
\(658\) 0 0
\(659\) −47.8617 −1.86443 −0.932214 0.361907i \(-0.882126\pi\)
−0.932214 + 0.361907i \(0.882126\pi\)
\(660\) 0 0
\(661\) 25.6847 0.999017 0.499509 0.866309i \(-0.333514\pi\)
0.499509 + 0.866309i \(0.333514\pi\)
\(662\) 9.47727 0.368344
\(663\) 0 0
\(664\) 22.2462 0.863320
\(665\) 0 0
\(666\) 0 0
\(667\) −54.1080 −2.09507
\(668\) −0.354162 −0.0137029
\(669\) 0 0
\(670\) 3.50758 0.135510
\(671\) 2.24621 0.0867140
\(672\) 0 0
\(673\) −48.7386 −1.87874 −0.939368 0.342910i \(-0.888587\pi\)
−0.939368 + 0.342910i \(0.888587\pi\)
\(674\) 51.1231 1.96919
\(675\) 0 0
\(676\) 3.42329 0.131665
\(677\) −13.6847 −0.525944 −0.262972 0.964803i \(-0.584703\pi\)
−0.262972 + 0.964803i \(0.584703\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 20.4924 0.784695
\(683\) −5.43845 −0.208096 −0.104048 0.994572i \(-0.533180\pi\)
−0.104048 + 0.994572i \(0.533180\pi\)
\(684\) 0 0
\(685\) −9.12311 −0.348576
\(686\) 0 0
\(687\) 0 0
\(688\) 36.0000 1.37249
\(689\) 19.3693 0.737912
\(690\) 0 0
\(691\) −36.9848 −1.40697 −0.703485 0.710710i \(-0.748374\pi\)
−0.703485 + 0.710710i \(0.748374\pi\)
\(692\) −8.24621 −0.313474
\(693\) 0 0
\(694\) −38.2462 −1.45181
\(695\) −5.12311 −0.194330
\(696\) 0 0
\(697\) 0 0
\(698\) −11.6155 −0.439654
\(699\) 0 0
\(700\) 0 0
\(701\) 9.36932 0.353874 0.176937 0.984222i \(-0.443381\pi\)
0.176937 + 0.984222i \(0.443381\pi\)
\(702\) 0 0
\(703\) −24.0000 −0.905177
\(704\) −14.2462 −0.536924
\(705\) 0 0
\(706\) 35.1231 1.32188
\(707\) 0 0
\(708\) 0 0
\(709\) −4.73863 −0.177963 −0.0889816 0.996033i \(-0.528361\pi\)
−0.0889816 + 0.996033i \(0.528361\pi\)
\(710\) 8.98485 0.337195
\(711\) 0 0
\(712\) −17.3693 −0.650943
\(713\) −33.6155 −1.25891
\(714\) 0 0
\(715\) 6.56155 0.245388
\(716\) 4.00000 0.149487
\(717\) 0 0
\(718\) −3.50758 −0.130902
\(719\) 8.80776 0.328474 0.164237 0.986421i \(-0.447484\pi\)
0.164237 + 0.986421i \(0.447484\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −62.5464 −2.32774
\(723\) 0 0
\(724\) −2.63068 −0.0977686
\(725\) −38.6307 −1.43471
\(726\) 0 0
\(727\) −8.00000 −0.296704 −0.148352 0.988935i \(-0.547397\pi\)
−0.148352 + 0.988935i \(0.547397\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −3.72348 −0.137812
\(731\) 0 0
\(732\) 0 0
\(733\) −28.2462 −1.04330 −0.521649 0.853160i \(-0.674683\pi\)
−0.521649 + 0.853160i \(0.674683\pi\)
\(734\) −28.4924 −1.05167
\(735\) 0 0
\(736\) −16.0000 −0.589768
\(737\) −10.2462 −0.377424
\(738\) 0 0
\(739\) −8.31534 −0.305885 −0.152942 0.988235i \(-0.548875\pi\)
−0.152942 + 0.988235i \(0.548875\pi\)
\(740\) 0.768944 0.0282669
\(741\) 0 0
\(742\) 0 0
\(743\) −4.49242 −0.164811 −0.0824055 0.996599i \(-0.526260\pi\)
−0.0824055 + 0.996599i \(0.526260\pi\)
\(744\) 0 0
\(745\) −2.38447 −0.0873603
\(746\) −25.3693 −0.928837
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0.630683 0.0230140 0.0115070 0.999934i \(-0.496337\pi\)
0.0115070 + 0.999934i \(0.496337\pi\)
\(752\) −13.4773 −0.491465
\(753\) 0 0
\(754\) −58.7386 −2.13913
\(755\) −4.49242 −0.163496
\(756\) 0 0
\(757\) −21.0540 −0.765220 −0.382610 0.923910i \(-0.624975\pi\)
−0.382610 + 0.923910i \(0.624975\pi\)
\(758\) −18.7386 −0.680618
\(759\) 0 0
\(760\) −10.5227 −0.381700
\(761\) 32.2462 1.16892 0.584462 0.811421i \(-0.301306\pi\)
0.584462 + 0.811421i \(0.301306\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 5.75379 0.208165
\(765\) 0 0
\(766\) −16.0000 −0.578103
\(767\) 5.12311 0.184985
\(768\) 0 0
\(769\) −29.5464 −1.06547 −0.532735 0.846282i \(-0.678836\pi\)
−0.532735 + 0.846282i \(0.678836\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 10.6307 0.382607
\(773\) 33.3693 1.20021 0.600105 0.799921i \(-0.295125\pi\)
0.600105 + 0.799921i \(0.295125\pi\)
\(774\) 0 0
\(775\) −24.0000 −0.862105
\(776\) 27.1231 0.973663
\(777\) 0 0
\(778\) −34.1383 −1.22392
\(779\) 4.31534 0.154613
\(780\) 0 0
\(781\) −26.2462 −0.939163
\(782\) 0 0
\(783\) 0 0
\(784\) 32.7926 1.17116
\(785\) 3.19224 0.113936
\(786\) 0 0
\(787\) −6.24621 −0.222653 −0.111327 0.993784i \(-0.535510\pi\)
−0.111327 + 0.993784i \(0.535510\pi\)
\(788\) 8.73863 0.311301
\(789\) 0 0
\(790\) −13.4773 −0.479500
\(791\) 0 0
\(792\) 0 0
\(793\) −4.00000 −0.142044
\(794\) 8.38447 0.297554
\(795\) 0 0
\(796\) −7.01515 −0.248646
\(797\) −31.6155 −1.11988 −0.559940 0.828533i \(-0.689176\pi\)
−0.559940 + 0.828533i \(0.689176\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −11.4233 −0.403874
\(801\) 0 0
\(802\) 9.64584 0.340606
\(803\) 10.8769 0.383837
\(804\) 0 0
\(805\) 0 0
\(806\) −36.4924 −1.28539
\(807\) 0 0
\(808\) −46.6307 −1.64046
\(809\) 53.0540 1.86528 0.932639 0.360810i \(-0.117500\pi\)
0.932639 + 0.360810i \(0.117500\pi\)
\(810\) 0 0
\(811\) 20.6307 0.724441 0.362221 0.932092i \(-0.382019\pi\)
0.362221 + 0.932092i \(0.382019\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −12.4924 −0.437859
\(815\) −3.86174 −0.135271
\(816\) 0 0
\(817\) −59.0540 −2.06604
\(818\) 3.61553 0.126414
\(819\) 0 0
\(820\) −0.138261 −0.00482827
\(821\) −16.5616 −0.578002 −0.289001 0.957329i \(-0.593323\pi\)
−0.289001 + 0.957329i \(0.593323\pi\)
\(822\) 0 0
\(823\) −36.4924 −1.27205 −0.636023 0.771670i \(-0.719422\pi\)
−0.636023 + 0.771670i \(0.719422\pi\)
\(824\) 10.5227 0.366577
\(825\) 0 0
\(826\) 0 0
\(827\) −14.4233 −0.501547 −0.250774 0.968046i \(-0.580685\pi\)
−0.250774 + 0.968046i \(0.580685\pi\)
\(828\) 0 0
\(829\) 50.4924 1.75367 0.876837 0.480787i \(-0.159649\pi\)
0.876837 + 0.480787i \(0.159649\pi\)
\(830\) 8.00000 0.277684
\(831\) 0 0
\(832\) 25.3693 0.879523
\(833\) 0 0
\(834\) 0 0
\(835\) 0.453602 0.0156976
\(836\) −8.63068 −0.298498
\(837\) 0 0
\(838\) −50.7386 −1.75274
\(839\) 11.0540 0.381626 0.190813 0.981626i \(-0.438888\pi\)
0.190813 + 0.981626i \(0.438888\pi\)
\(840\) 0 0
\(841\) 39.0000 1.34483
\(842\) −44.6004 −1.53703
\(843\) 0 0
\(844\) 4.98485 0.171585
\(845\) −4.38447 −0.150830
\(846\) 0 0
\(847\) 0 0
\(848\) −19.8920 −0.683096
\(849\) 0 0
\(850\) 0 0
\(851\) 20.4924 0.702471
\(852\) 0 0
\(853\) −20.7386 −0.710077 −0.355039 0.934852i \(-0.615532\pi\)
−0.355039 + 0.934852i \(0.615532\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 18.7386 0.640473
\(857\) −6.00000 −0.204956 −0.102478 0.994735i \(-0.532677\pi\)
−0.102478 + 0.994735i \(0.532677\pi\)
\(858\) 0 0
\(859\) 12.0000 0.409435 0.204717 0.978821i \(-0.434372\pi\)
0.204717 + 0.978821i \(0.434372\pi\)
\(860\) 1.89205 0.0645183
\(861\) 0 0
\(862\) 37.4773 1.27648
\(863\) 26.2462 0.893431 0.446716 0.894676i \(-0.352594\pi\)
0.446716 + 0.894676i \(0.352594\pi\)
\(864\) 0 0
\(865\) 10.5616 0.359104
\(866\) −22.3542 −0.759625
\(867\) 0 0
\(868\) 0 0
\(869\) 39.3693 1.33551
\(870\) 0 0
\(871\) 18.2462 0.618249
\(872\) 36.8769 1.24881
\(873\) 0 0
\(874\) 78.7386 2.66337
\(875\) 0 0
\(876\) 0 0
\(877\) 34.0000 1.14810 0.574049 0.818821i \(-0.305372\pi\)
0.574049 + 0.818821i \(0.305372\pi\)
\(878\) −8.98485 −0.303224
\(879\) 0 0
\(880\) −6.73863 −0.227159
\(881\) 23.7538 0.800285 0.400143 0.916453i \(-0.368961\pi\)
0.400143 + 0.916453i \(0.368961\pi\)
\(882\) 0 0
\(883\) 38.4233 1.29305 0.646523 0.762894i \(-0.276222\pi\)
0.646523 + 0.762894i \(0.276222\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −35.7235 −1.20015
\(887\) 22.5616 0.757543 0.378771 0.925490i \(-0.376347\pi\)
0.378771 + 0.925490i \(0.376347\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −6.24621 −0.209373
\(891\) 0 0
\(892\) 6.10795 0.204509
\(893\) 22.1080 0.739814
\(894\) 0 0
\(895\) −5.12311 −0.171247
\(896\) 0 0
\(897\) 0 0
\(898\) 19.8920 0.663806
\(899\) 42.2462 1.40899
\(900\) 0 0
\(901\) 0 0
\(902\) 2.24621 0.0747907
\(903\) 0 0
\(904\) −11.1231 −0.369949
\(905\) 3.36932 0.112000
\(906\) 0 0
\(907\) −47.8617 −1.58922 −0.794611 0.607118i \(-0.792325\pi\)
−0.794611 + 0.607118i \(0.792325\pi\)
\(908\) 10.1080 0.335444
\(909\) 0 0
\(910\) 0 0
\(911\) −29.3002 −0.970758 −0.485379 0.874304i \(-0.661318\pi\)
−0.485379 + 0.874304i \(0.661318\pi\)
\(912\) 0 0
\(913\) −23.3693 −0.773412
\(914\) 10.6307 0.351632
\(915\) 0 0
\(916\) 2.63068 0.0869202
\(917\) 0 0
\(918\) 0 0
\(919\) −4.31534 −0.142350 −0.0711750 0.997464i \(-0.522675\pi\)
−0.0711750 + 0.997464i \(0.522675\pi\)
\(920\) 8.98485 0.296222
\(921\) 0 0
\(922\) 12.8769 0.424078
\(923\) 46.7386 1.53842
\(924\) 0 0
\(925\) 14.6307 0.481054
\(926\) −39.0152 −1.28212
\(927\) 0 0
\(928\) 20.1080 0.660076
\(929\) 31.9309 1.04762 0.523809 0.851836i \(-0.324511\pi\)
0.523809 + 0.851836i \(0.324511\pi\)
\(930\) 0 0
\(931\) −53.7926 −1.76298
\(932\) 0.246211 0.00806492
\(933\) 0 0
\(934\) −5.26137 −0.172157
\(935\) 0 0
\(936\) 0 0
\(937\) −22.0000 −0.718709 −0.359354 0.933201i \(-0.617003\pi\)
−0.359354 + 0.933201i \(0.617003\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −0.708324 −0.0231030
\(941\) 30.0000 0.977972 0.488986 0.872292i \(-0.337367\pi\)
0.488986 + 0.872292i \(0.337367\pi\)
\(942\) 0 0
\(943\) −3.68466 −0.119989
\(944\) −5.26137 −0.171243
\(945\) 0 0
\(946\) −30.7386 −0.999399
\(947\) 12.0000 0.389948 0.194974 0.980808i \(-0.437538\pi\)
0.194974 + 0.980808i \(0.437538\pi\)
\(948\) 0 0
\(949\) −19.3693 −0.628755
\(950\) 56.2159 1.82388
\(951\) 0 0
\(952\) 0 0
\(953\) 54.3542 1.76070 0.880352 0.474321i \(-0.157306\pi\)
0.880352 + 0.474321i \(0.157306\pi\)
\(954\) 0 0
\(955\) −7.36932 −0.238465
\(956\) −4.49242 −0.145295
\(957\) 0 0
\(958\) 45.7538 1.47824
\(959\) 0 0
\(960\) 0 0
\(961\) −4.75379 −0.153348
\(962\) 22.2462 0.717247
\(963\) 0 0
\(964\) 9.36932 0.301765
\(965\) −13.6155 −0.438299
\(966\) 0 0
\(967\) 46.5616 1.49732 0.748659 0.662955i \(-0.230698\pi\)
0.748659 + 0.662955i \(0.230698\pi\)
\(968\) −10.8229 −0.347862
\(969\) 0 0
\(970\) 9.75379 0.313175
\(971\) 2.38447 0.0765213 0.0382607 0.999268i \(-0.487818\pi\)
0.0382607 + 0.999268i \(0.487818\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −11.5076 −0.368727
\(975\) 0 0
\(976\) 4.10795 0.131492
\(977\) 8.24621 0.263820 0.131910 0.991262i \(-0.457889\pi\)
0.131910 + 0.991262i \(0.457889\pi\)
\(978\) 0 0
\(979\) 18.2462 0.583151
\(980\) 1.72348 0.0550545
\(981\) 0 0
\(982\) 5.26137 0.167897
\(983\) 2.06913 0.0659950 0.0329975 0.999455i \(-0.489495\pi\)
0.0329975 + 0.999455i \(0.489495\pi\)
\(984\) 0 0
\(985\) −11.1922 −0.356614
\(986\) 0 0
\(987\) 0 0
\(988\) 15.3693 0.488963
\(989\) 50.4233 1.60337
\(990\) 0 0
\(991\) 6.73863 0.214060 0.107030 0.994256i \(-0.465866\pi\)
0.107030 + 0.994256i \(0.465866\pi\)
\(992\) 12.4924 0.396635
\(993\) 0 0
\(994\) 0 0
\(995\) 8.98485 0.284839
\(996\) 0 0
\(997\) 10.0000 0.316703 0.158352 0.987383i \(-0.449382\pi\)
0.158352 + 0.987383i \(0.449382\pi\)
\(998\) −17.7538 −0.561986
\(999\) 0 0
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 2601.2.a.t.1.1 2
3.2 odd 2 867.2.a.f.1.2 2
17.16 even 2 153.2.a.e.1.1 2
51.2 odd 8 867.2.e.f.616.4 8
51.5 even 16 867.2.h.j.688.1 16
51.8 odd 8 867.2.e.f.829.1 8
51.11 even 16 867.2.h.j.733.3 16
51.14 even 16 867.2.h.j.757.3 16
51.20 even 16 867.2.h.j.757.4 16
51.23 even 16 867.2.h.j.733.4 16
51.26 odd 8 867.2.e.f.829.2 8
51.29 even 16 867.2.h.j.688.2 16
51.32 odd 8 867.2.e.f.616.3 8
51.38 odd 4 867.2.d.c.577.1 4
51.41 even 16 867.2.h.j.712.1 16
51.44 even 16 867.2.h.j.712.2 16
51.47 odd 4 867.2.d.c.577.2 4
51.50 odd 2 51.2.a.b.1.2 2
68.67 odd 2 2448.2.a.v.1.2 2
85.84 even 2 3825.2.a.s.1.2 2
119.118 odd 2 7497.2.a.v.1.1 2
136.67 odd 2 9792.2.a.cz.1.1 2
136.101 even 2 9792.2.a.cy.1.1 2
204.203 even 2 816.2.a.m.1.1 2
255.152 even 4 1275.2.b.d.1174.3 4
255.203 even 4 1275.2.b.d.1174.2 4
255.254 odd 2 1275.2.a.n.1.1 2
357.356 even 2 2499.2.a.o.1.2 2
408.101 odd 2 3264.2.a.bl.1.2 2
408.203 even 2 3264.2.a.bg.1.2 2
561.560 even 2 6171.2.a.p.1.1 2
663.662 odd 2 8619.2.a.q.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
51.2.a.b.1.2 2 51.50 odd 2
153.2.a.e.1.1 2 17.16 even 2
816.2.a.m.1.1 2 204.203 even 2
867.2.a.f.1.2 2 3.2 odd 2
867.2.d.c.577.1 4 51.38 odd 4
867.2.d.c.577.2 4 51.47 odd 4
867.2.e.f.616.3 8 51.32 odd 8
867.2.e.f.616.4 8 51.2 odd 8
867.2.e.f.829.1 8 51.8 odd 8
867.2.e.f.829.2 8 51.26 odd 8
867.2.h.j.688.1 16 51.5 even 16
867.2.h.j.688.2 16 51.29 even 16
867.2.h.j.712.1 16 51.41 even 16
867.2.h.j.712.2 16 51.44 even 16
867.2.h.j.733.3 16 51.11 even 16
867.2.h.j.733.4 16 51.23 even 16
867.2.h.j.757.3 16 51.14 even 16
867.2.h.j.757.4 16 51.20 even 16
1275.2.a.n.1.1 2 255.254 odd 2
1275.2.b.d.1174.2 4 255.203 even 4
1275.2.b.d.1174.3 4 255.152 even 4
2448.2.a.v.1.2 2 68.67 odd 2
2499.2.a.o.1.2 2 357.356 even 2
2601.2.a.t.1.1 2 1.1 even 1 trivial
3264.2.a.bg.1.2 2 408.203 even 2
3264.2.a.bl.1.2 2 408.101 odd 2
3825.2.a.s.1.2 2 85.84 even 2
6171.2.a.p.1.1 2 561.560 even 2
7497.2.a.v.1.1 2 119.118 odd 2
8619.2.a.q.1.1 2 663.662 odd 2
9792.2.a.cy.1.1 2 136.101 even 2
9792.2.a.cz.1.1 2 136.67 odd 2