Properties

Label 6160.2.a.bd.1.1
Level $6160$
Weight $2$
Character 6160.1
Self dual yes
Analytic conductor $49.188$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6160 = 2^{4} \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6160.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(49.1878476451\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.404.1
Defining polynomial: \( x^{3} - x^{2} - 5x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3080)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.65544\) of defining polynomial
Character \(\chi\) \(=\) 6160.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.65544 q^{3} -1.00000 q^{5} +1.00000 q^{7} +4.05137 q^{9} +O(q^{10})\) \(q-2.65544 q^{3} -1.00000 q^{5} +1.00000 q^{7} +4.05137 q^{9} +1.00000 q^{11} -2.65544 q^{13} +2.65544 q^{15} -5.44731 q^{17} +4.10275 q^{19} -2.65544 q^{21} +0.259511 q^{23} +1.00000 q^{25} -2.79186 q^{27} -2.00000 q^{29} +4.91495 q^{31} -2.65544 q^{33} -1.00000 q^{35} -2.25951 q^{37} +7.05137 q^{39} -1.08505 q^{41} +4.53235 q^{43} -4.05137 q^{45} -4.13642 q^{47} +1.00000 q^{49} +14.4650 q^{51} -4.36226 q^{53} -1.00000 q^{55} -10.8946 q^{57} -5.97966 q^{59} +3.80957 q^{61} +4.05137 q^{63} +2.65544 q^{65} -7.15412 q^{67} -0.689115 q^{69} +16.2055 q^{71} -2.55269 q^{73} -2.65544 q^{75} +1.00000 q^{77} +3.63774 q^{79} -4.74049 q^{81} -6.79186 q^{83} +5.44731 q^{85} +5.31088 q^{87} +5.31088 q^{89} -2.65544 q^{91} -13.0514 q^{93} -4.10275 q^{95} +17.5164 q^{97} +4.05137 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{3} - 3 q^{5} + 3 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 2 q^{3} - 3 q^{5} + 3 q^{7} + 3 q^{9} + 3 q^{11} - 2 q^{13} + 2 q^{15} - 4 q^{17} - 6 q^{19} - 2 q^{21} - 2 q^{23} + 3 q^{25} - 2 q^{27} - 6 q^{29} + 6 q^{31} - 2 q^{33} - 3 q^{35} - 4 q^{37} + 12 q^{39} - 12 q^{41} + 10 q^{43} - 3 q^{45} - 12 q^{47} + 3 q^{49} + 4 q^{51} + 8 q^{53} - 3 q^{55} - 8 q^{57} - 2 q^{59} - 22 q^{61} + 3 q^{63} + 2 q^{65} + 6 q^{67} - 14 q^{69} + 12 q^{71} - 20 q^{73} - 2 q^{75} + 3 q^{77} + 32 q^{79} - 17 q^{81} - 14 q^{83} + 4 q^{85} + 4 q^{87} + 4 q^{89} - 2 q^{91} - 30 q^{93} + 6 q^{95} + 4 q^{97} + 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −2.65544 −1.53312 −0.766560 0.642172i \(-0.778033\pi\)
−0.766560 + 0.642172i \(0.778033\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 4.05137 1.35046
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) −2.65544 −0.736487 −0.368244 0.929729i \(-0.620041\pi\)
−0.368244 + 0.929729i \(0.620041\pi\)
\(14\) 0 0
\(15\) 2.65544 0.685632
\(16\) 0 0
\(17\) −5.44731 −1.32117 −0.660583 0.750753i \(-0.729691\pi\)
−0.660583 + 0.750753i \(0.729691\pi\)
\(18\) 0 0
\(19\) 4.10275 0.941235 0.470618 0.882337i \(-0.344031\pi\)
0.470618 + 0.882337i \(0.344031\pi\)
\(20\) 0 0
\(21\) −2.65544 −0.579465
\(22\) 0 0
\(23\) 0.259511 0.0541117 0.0270558 0.999634i \(-0.491387\pi\)
0.0270558 + 0.999634i \(0.491387\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −2.79186 −0.537294
\(28\) 0 0
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) 4.91495 0.882752 0.441376 0.897322i \(-0.354491\pi\)
0.441376 + 0.897322i \(0.354491\pi\)
\(32\) 0 0
\(33\) −2.65544 −0.462253
\(34\) 0 0
\(35\) −1.00000 −0.169031
\(36\) 0 0
\(37\) −2.25951 −0.371461 −0.185731 0.982601i \(-0.559465\pi\)
−0.185731 + 0.982601i \(0.559465\pi\)
\(38\) 0 0
\(39\) 7.05137 1.12912
\(40\) 0 0
\(41\) −1.08505 −0.169456 −0.0847279 0.996404i \(-0.527002\pi\)
−0.0847279 + 0.996404i \(0.527002\pi\)
\(42\) 0 0
\(43\) 4.53235 0.691177 0.345589 0.938386i \(-0.387679\pi\)
0.345589 + 0.938386i \(0.387679\pi\)
\(44\) 0 0
\(45\) −4.05137 −0.603943
\(46\) 0 0
\(47\) −4.13642 −0.603359 −0.301680 0.953409i \(-0.597547\pi\)
−0.301680 + 0.953409i \(0.597547\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 14.4650 2.02551
\(52\) 0 0
\(53\) −4.36226 −0.599202 −0.299601 0.954065i \(-0.596854\pi\)
−0.299601 + 0.954065i \(0.596854\pi\)
\(54\) 0 0
\(55\) −1.00000 −0.134840
\(56\) 0 0
\(57\) −10.8946 −1.44303
\(58\) 0 0
\(59\) −5.97966 −0.778485 −0.389243 0.921135i \(-0.627263\pi\)
−0.389243 + 0.921135i \(0.627263\pi\)
\(60\) 0 0
\(61\) 3.80957 0.487765 0.243882 0.969805i \(-0.421579\pi\)
0.243882 + 0.969805i \(0.421579\pi\)
\(62\) 0 0
\(63\) 4.05137 0.510425
\(64\) 0 0
\(65\) 2.65544 0.329367
\(66\) 0 0
\(67\) −7.15412 −0.874015 −0.437008 0.899458i \(-0.643962\pi\)
−0.437008 + 0.899458i \(0.643962\pi\)
\(68\) 0 0
\(69\) −0.689115 −0.0829597
\(70\) 0 0
\(71\) 16.2055 1.92324 0.961619 0.274387i \(-0.0884750\pi\)
0.961619 + 0.274387i \(0.0884750\pi\)
\(72\) 0 0
\(73\) −2.55269 −0.298770 −0.149385 0.988779i \(-0.547729\pi\)
−0.149385 + 0.988779i \(0.547729\pi\)
\(74\) 0 0
\(75\) −2.65544 −0.306624
\(76\) 0 0
\(77\) 1.00000 0.113961
\(78\) 0 0
\(79\) 3.63774 0.409278 0.204639 0.978838i \(-0.434398\pi\)
0.204639 + 0.978838i \(0.434398\pi\)
\(80\) 0 0
\(81\) −4.74049 −0.526721
\(82\) 0 0
\(83\) −6.79186 −0.745504 −0.372752 0.927931i \(-0.621586\pi\)
−0.372752 + 0.927931i \(0.621586\pi\)
\(84\) 0 0
\(85\) 5.44731 0.590843
\(86\) 0 0
\(87\) 5.31088 0.569387
\(88\) 0 0
\(89\) 5.31088 0.562953 0.281476 0.959568i \(-0.409176\pi\)
0.281476 + 0.959568i \(0.409176\pi\)
\(90\) 0 0
\(91\) −2.65544 −0.278366
\(92\) 0 0
\(93\) −13.0514 −1.35336
\(94\) 0 0
\(95\) −4.10275 −0.420933
\(96\) 0 0
\(97\) 17.5164 1.77852 0.889260 0.457403i \(-0.151220\pi\)
0.889260 + 0.457403i \(0.151220\pi\)
\(98\) 0 0
\(99\) 4.05137 0.407178
\(100\) 0 0
\(101\) −7.43397 −0.739708 −0.369854 0.929090i \(-0.620592\pi\)
−0.369854 + 0.929090i \(0.620592\pi\)
\(102\) 0 0
\(103\) −4.23917 −0.417698 −0.208849 0.977948i \(-0.566972\pi\)
−0.208849 + 0.977948i \(0.566972\pi\)
\(104\) 0 0
\(105\) 2.65544 0.259145
\(106\) 0 0
\(107\) 0.519021 0.0501757 0.0250878 0.999685i \(-0.492013\pi\)
0.0250878 + 0.999685i \(0.492013\pi\)
\(108\) 0 0
\(109\) 3.31088 0.317125 0.158563 0.987349i \(-0.449314\pi\)
0.158563 + 0.987349i \(0.449314\pi\)
\(110\) 0 0
\(111\) 6.00000 0.569495
\(112\) 0 0
\(113\) 11.5164 1.08337 0.541685 0.840582i \(-0.317787\pi\)
0.541685 + 0.840582i \(0.317787\pi\)
\(114\) 0 0
\(115\) −0.259511 −0.0241995
\(116\) 0 0
\(117\) −10.7582 −0.994595
\(118\) 0 0
\(119\) −5.44731 −0.499354
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 2.88128 0.259796
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 20.9974 1.86321 0.931607 0.363467i \(-0.118407\pi\)
0.931607 + 0.363467i \(0.118407\pi\)
\(128\) 0 0
\(129\) −12.0354 −1.05966
\(130\) 0 0
\(131\) 1.58373 0.138371 0.0691855 0.997604i \(-0.477960\pi\)
0.0691855 + 0.997604i \(0.477960\pi\)
\(132\) 0 0
\(133\) 4.10275 0.355753
\(134\) 0 0
\(135\) 2.79186 0.240285
\(136\) 0 0
\(137\) 8.15676 0.696879 0.348440 0.937331i \(-0.386712\pi\)
0.348440 + 0.937331i \(0.386712\pi\)
\(138\) 0 0
\(139\) −11.9327 −1.01211 −0.506057 0.862500i \(-0.668898\pi\)
−0.506057 + 0.862500i \(0.668898\pi\)
\(140\) 0 0
\(141\) 10.9840 0.925022
\(142\) 0 0
\(143\) −2.65544 −0.222059
\(144\) 0 0
\(145\) 2.00000 0.166091
\(146\) 0 0
\(147\) −2.65544 −0.219017
\(148\) 0 0
\(149\) −4.17009 −0.341627 −0.170814 0.985303i \(-0.554640\pi\)
−0.170814 + 0.985303i \(0.554640\pi\)
\(150\) 0 0
\(151\) 21.8786 1.78046 0.890229 0.455513i \(-0.150544\pi\)
0.890229 + 0.455513i \(0.150544\pi\)
\(152\) 0 0
\(153\) −22.0691 −1.78418
\(154\) 0 0
\(155\) −4.91495 −0.394779
\(156\) 0 0
\(157\) −12.2055 −0.974105 −0.487052 0.873373i \(-0.661928\pi\)
−0.487052 + 0.873373i \(0.661928\pi\)
\(158\) 0 0
\(159\) 11.5837 0.918649
\(160\) 0 0
\(161\) 0.259511 0.0204523
\(162\) 0 0
\(163\) 19.1541 1.50027 0.750133 0.661287i \(-0.229989\pi\)
0.750133 + 0.661287i \(0.229989\pi\)
\(164\) 0 0
\(165\) 2.65544 0.206726
\(166\) 0 0
\(167\) −3.03804 −0.235091 −0.117545 0.993068i \(-0.537503\pi\)
−0.117545 + 0.993068i \(0.537503\pi\)
\(168\) 0 0
\(169\) −5.94863 −0.457587
\(170\) 0 0
\(171\) 16.6218 1.27110
\(172\) 0 0
\(173\) −3.07171 −0.233538 −0.116769 0.993159i \(-0.537254\pi\)
−0.116769 + 0.993159i \(0.537254\pi\)
\(174\) 0 0
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) 15.8786 1.19351
\(178\) 0 0
\(179\) −21.7626 −1.62661 −0.813305 0.581838i \(-0.802334\pi\)
−0.813305 + 0.581838i \(0.802334\pi\)
\(180\) 0 0
\(181\) −14.3082 −1.06352 −0.531762 0.846894i \(-0.678470\pi\)
−0.531762 + 0.846894i \(0.678470\pi\)
\(182\) 0 0
\(183\) −10.1161 −0.747802
\(184\) 0 0
\(185\) 2.25951 0.166123
\(186\) 0 0
\(187\) −5.44731 −0.398346
\(188\) 0 0
\(189\) −2.79186 −0.203078
\(190\) 0 0
\(191\) 17.3109 1.25257 0.626286 0.779594i \(-0.284574\pi\)
0.626286 + 0.779594i \(0.284574\pi\)
\(192\) 0 0
\(193\) −4.42960 −0.318850 −0.159425 0.987210i \(-0.550964\pi\)
−0.159425 + 0.987210i \(0.550964\pi\)
\(194\) 0 0
\(195\) −7.05137 −0.504959
\(196\) 0 0
\(197\) 3.57040 0.254380 0.127190 0.991878i \(-0.459404\pi\)
0.127190 + 0.991878i \(0.459404\pi\)
\(198\) 0 0
\(199\) 10.0824 0.714723 0.357361 0.933966i \(-0.383676\pi\)
0.357361 + 0.933966i \(0.383676\pi\)
\(200\) 0 0
\(201\) 18.9974 1.33997
\(202\) 0 0
\(203\) −2.00000 −0.140372
\(204\) 0 0
\(205\) 1.08505 0.0757830
\(206\) 0 0
\(207\) 1.05137 0.0730756
\(208\) 0 0
\(209\) 4.10275 0.283793
\(210\) 0 0
\(211\) −15.9593 −1.09868 −0.549342 0.835597i \(-0.685122\pi\)
−0.549342 + 0.835597i \(0.685122\pi\)
\(212\) 0 0
\(213\) −43.0328 −2.94856
\(214\) 0 0
\(215\) −4.53235 −0.309104
\(216\) 0 0
\(217\) 4.91495 0.333649
\(218\) 0 0
\(219\) 6.77853 0.458051
\(220\) 0 0
\(221\) 14.4650 0.973022
\(222\) 0 0
\(223\) −21.2772 −1.42483 −0.712414 0.701760i \(-0.752398\pi\)
−0.712414 + 0.701760i \(0.752398\pi\)
\(224\) 0 0
\(225\) 4.05137 0.270092
\(226\) 0 0
\(227\) 2.82727 0.187652 0.0938261 0.995589i \(-0.470090\pi\)
0.0938261 + 0.995589i \(0.470090\pi\)
\(228\) 0 0
\(229\) 2.24618 0.148432 0.0742158 0.997242i \(-0.476355\pi\)
0.0742158 + 0.997242i \(0.476355\pi\)
\(230\) 0 0
\(231\) −2.65544 −0.174715
\(232\) 0 0
\(233\) −26.9840 −1.76778 −0.883891 0.467692i \(-0.845085\pi\)
−0.883891 + 0.467692i \(0.845085\pi\)
\(234\) 0 0
\(235\) 4.13642 0.269830
\(236\) 0 0
\(237\) −9.65981 −0.627472
\(238\) 0 0
\(239\) −15.6191 −1.01032 −0.505159 0.863026i \(-0.668566\pi\)
−0.505159 + 0.863026i \(0.668566\pi\)
\(240\) 0 0
\(241\) 2.60143 0.167573 0.0837864 0.996484i \(-0.473299\pi\)
0.0837864 + 0.996484i \(0.473299\pi\)
\(242\) 0 0
\(243\) 20.9637 1.34482
\(244\) 0 0
\(245\) −1.00000 −0.0638877
\(246\) 0 0
\(247\) −10.8946 −0.693208
\(248\) 0 0
\(249\) 18.0354 1.14295
\(250\) 0 0
\(251\) −21.1878 −1.33736 −0.668681 0.743549i \(-0.733141\pi\)
−0.668681 + 0.743549i \(0.733141\pi\)
\(252\) 0 0
\(253\) 0.259511 0.0163153
\(254\) 0 0
\(255\) −14.4650 −0.905834
\(256\) 0 0
\(257\) −1.06471 −0.0664146 −0.0332073 0.999448i \(-0.510572\pi\)
−0.0332073 + 0.999448i \(0.510572\pi\)
\(258\) 0 0
\(259\) −2.25951 −0.140399
\(260\) 0 0
\(261\) −8.10275 −0.501548
\(262\) 0 0
\(263\) −24.6572 −1.52043 −0.760213 0.649674i \(-0.774906\pi\)
−0.760213 + 0.649674i \(0.774906\pi\)
\(264\) 0 0
\(265\) 4.36226 0.267971
\(266\) 0 0
\(267\) −14.1027 −0.863074
\(268\) 0 0
\(269\) 16.4110 1.00060 0.500298 0.865853i \(-0.333224\pi\)
0.500298 + 0.865853i \(0.333224\pi\)
\(270\) 0 0
\(271\) −16.4783 −1.00099 −0.500494 0.865740i \(-0.666848\pi\)
−0.500494 + 0.865740i \(0.666848\pi\)
\(272\) 0 0
\(273\) 7.05137 0.426769
\(274\) 0 0
\(275\) 1.00000 0.0603023
\(276\) 0 0
\(277\) 5.74049 0.344913 0.172456 0.985017i \(-0.444830\pi\)
0.172456 + 0.985017i \(0.444830\pi\)
\(278\) 0 0
\(279\) 19.9123 1.19212
\(280\) 0 0
\(281\) −13.6865 −0.816467 −0.408233 0.912878i \(-0.633855\pi\)
−0.408233 + 0.912878i \(0.633855\pi\)
\(282\) 0 0
\(283\) −11.1408 −0.662251 −0.331126 0.943587i \(-0.607428\pi\)
−0.331126 + 0.943587i \(0.607428\pi\)
\(284\) 0 0
\(285\) 10.8946 0.645341
\(286\) 0 0
\(287\) −1.08505 −0.0640483
\(288\) 0 0
\(289\) 12.6731 0.745479
\(290\) 0 0
\(291\) −46.5137 −2.72668
\(292\) 0 0
\(293\) −25.9663 −1.51697 −0.758485 0.651691i \(-0.774060\pi\)
−0.758485 + 0.651691i \(0.774060\pi\)
\(294\) 0 0
\(295\) 5.97966 0.348149
\(296\) 0 0
\(297\) −2.79186 −0.162000
\(298\) 0 0
\(299\) −0.689115 −0.0398526
\(300\) 0 0
\(301\) 4.53235 0.261240
\(302\) 0 0
\(303\) 19.7405 1.13406
\(304\) 0 0
\(305\) −3.80957 −0.218135
\(306\) 0 0
\(307\) −15.5571 −0.887888 −0.443944 0.896054i \(-0.646421\pi\)
−0.443944 + 0.896054i \(0.646421\pi\)
\(308\) 0 0
\(309\) 11.2569 0.640381
\(310\) 0 0
\(311\) −4.53936 −0.257404 −0.128702 0.991683i \(-0.541081\pi\)
−0.128702 + 0.991683i \(0.541081\pi\)
\(312\) 0 0
\(313\) −23.7626 −1.34314 −0.671570 0.740941i \(-0.734380\pi\)
−0.671570 + 0.740941i \(0.734380\pi\)
\(314\) 0 0
\(315\) −4.05137 −0.228269
\(316\) 0 0
\(317\) 11.5837 0.650607 0.325303 0.945610i \(-0.394534\pi\)
0.325303 + 0.945610i \(0.394534\pi\)
\(318\) 0 0
\(319\) −2.00000 −0.111979
\(320\) 0 0
\(321\) −1.37823 −0.0769253
\(322\) 0 0
\(323\) −22.3489 −1.24353
\(324\) 0 0
\(325\) −2.65544 −0.147297
\(326\) 0 0
\(327\) −8.79186 −0.486191
\(328\) 0 0
\(329\) −4.13642 −0.228048
\(330\) 0 0
\(331\) 26.0354 1.43104 0.715518 0.698595i \(-0.246191\pi\)
0.715518 + 0.698595i \(0.246191\pi\)
\(332\) 0 0
\(333\) −9.15412 −0.501643
\(334\) 0 0
\(335\) 7.15412 0.390871
\(336\) 0 0
\(337\) −3.36490 −0.183298 −0.0916488 0.995791i \(-0.529214\pi\)
−0.0916488 + 0.995791i \(0.529214\pi\)
\(338\) 0 0
\(339\) −30.5811 −1.66094
\(340\) 0 0
\(341\) 4.91495 0.266160
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 0.689115 0.0371007
\(346\) 0 0
\(347\) −17.0514 −0.915366 −0.457683 0.889116i \(-0.651320\pi\)
−0.457683 + 0.889116i \(0.651320\pi\)
\(348\) 0 0
\(349\) −14.4987 −0.776097 −0.388048 0.921639i \(-0.626851\pi\)
−0.388048 + 0.921639i \(0.626851\pi\)
\(350\) 0 0
\(351\) 7.41363 0.395710
\(352\) 0 0
\(353\) −1.14079 −0.0607182 −0.0303591 0.999539i \(-0.509665\pi\)
−0.0303591 + 0.999539i \(0.509665\pi\)
\(354\) 0 0
\(355\) −16.2055 −0.860098
\(356\) 0 0
\(357\) 14.4650 0.765569
\(358\) 0 0
\(359\) 11.5297 0.608515 0.304258 0.952590i \(-0.401592\pi\)
0.304258 + 0.952590i \(0.401592\pi\)
\(360\) 0 0
\(361\) −2.16745 −0.114077
\(362\) 0 0
\(363\) −2.65544 −0.139375
\(364\) 0 0
\(365\) 2.55269 0.133614
\(366\) 0 0
\(367\) −26.2392 −1.36967 −0.684837 0.728697i \(-0.740127\pi\)
−0.684837 + 0.728697i \(0.740127\pi\)
\(368\) 0 0
\(369\) −4.39593 −0.228843
\(370\) 0 0
\(371\) −4.36226 −0.226477
\(372\) 0 0
\(373\) −30.1922 −1.56329 −0.781646 0.623723i \(-0.785619\pi\)
−0.781646 + 0.623723i \(0.785619\pi\)
\(374\) 0 0
\(375\) 2.65544 0.137126
\(376\) 0 0
\(377\) 5.31088 0.273524
\(378\) 0 0
\(379\) 6.62177 0.340137 0.170069 0.985432i \(-0.445601\pi\)
0.170069 + 0.985432i \(0.445601\pi\)
\(380\) 0 0
\(381\) −55.7573 −2.85653
\(382\) 0 0
\(383\) 31.7910 1.62444 0.812221 0.583350i \(-0.198258\pi\)
0.812221 + 0.583350i \(0.198258\pi\)
\(384\) 0 0
\(385\) −1.00000 −0.0509647
\(386\) 0 0
\(387\) 18.3623 0.933406
\(388\) 0 0
\(389\) −22.9707 −1.16466 −0.582330 0.812952i \(-0.697859\pi\)
−0.582330 + 0.812952i \(0.697859\pi\)
\(390\) 0 0
\(391\) −1.41363 −0.0714905
\(392\) 0 0
\(393\) −4.20550 −0.212139
\(394\) 0 0
\(395\) −3.63774 −0.183035
\(396\) 0 0
\(397\) −20.7245 −1.04013 −0.520067 0.854126i \(-0.674093\pi\)
−0.520067 + 0.854126i \(0.674093\pi\)
\(398\) 0 0
\(399\) −10.8946 −0.545413
\(400\) 0 0
\(401\) 3.24354 0.161975 0.0809873 0.996715i \(-0.474193\pi\)
0.0809873 + 0.996715i \(0.474193\pi\)
\(402\) 0 0
\(403\) −13.0514 −0.650135
\(404\) 0 0
\(405\) 4.74049 0.235557
\(406\) 0 0
\(407\) −2.25951 −0.112000
\(408\) 0 0
\(409\) −2.77152 −0.137043 −0.0685215 0.997650i \(-0.521828\pi\)
−0.0685215 + 0.997650i \(0.521828\pi\)
\(410\) 0 0
\(411\) −21.6598 −1.06840
\(412\) 0 0
\(413\) −5.97966 −0.294240
\(414\) 0 0
\(415\) 6.79186 0.333399
\(416\) 0 0
\(417\) 31.6865 1.55169
\(418\) 0 0
\(419\) −27.2586 −1.33167 −0.665835 0.746099i \(-0.731924\pi\)
−0.665835 + 0.746099i \(0.731924\pi\)
\(420\) 0 0
\(421\) −5.63774 −0.274767 −0.137383 0.990518i \(-0.543869\pi\)
−0.137383 + 0.990518i \(0.543869\pi\)
\(422\) 0 0
\(423\) −16.7582 −0.814811
\(424\) 0 0
\(425\) −5.44731 −0.264233
\(426\) 0 0
\(427\) 3.80957 0.184358
\(428\) 0 0
\(429\) 7.05137 0.340444
\(430\) 0 0
\(431\) −14.1515 −0.681653 −0.340826 0.940126i \(-0.610707\pi\)
−0.340826 + 0.940126i \(0.610707\pi\)
\(432\) 0 0
\(433\) −13.7272 −0.659685 −0.329843 0.944036i \(-0.606996\pi\)
−0.329843 + 0.944036i \(0.606996\pi\)
\(434\) 0 0
\(435\) −5.31088 −0.254637
\(436\) 0 0
\(437\) 1.06471 0.0509318
\(438\) 0 0
\(439\) −19.8246 −0.946178 −0.473089 0.881015i \(-0.656861\pi\)
−0.473089 + 0.881015i \(0.656861\pi\)
\(440\) 0 0
\(441\) 4.05137 0.192923
\(442\) 0 0
\(443\) −7.74049 −0.367762 −0.183881 0.982949i \(-0.558866\pi\)
−0.183881 + 0.982949i \(0.558866\pi\)
\(444\) 0 0
\(445\) −5.31088 −0.251760
\(446\) 0 0
\(447\) 11.0734 0.523756
\(448\) 0 0
\(449\) 12.7785 0.603056 0.301528 0.953457i \(-0.402503\pi\)
0.301528 + 0.953457i \(0.402503\pi\)
\(450\) 0 0
\(451\) −1.08505 −0.0510929
\(452\) 0 0
\(453\) −58.0975 −2.72966
\(454\) 0 0
\(455\) 2.65544 0.124489
\(456\) 0 0
\(457\) 2.19216 0.102545 0.0512726 0.998685i \(-0.483672\pi\)
0.0512726 + 0.998685i \(0.483672\pi\)
\(458\) 0 0
\(459\) 15.2081 0.709855
\(460\) 0 0
\(461\) 14.0824 0.655883 0.327942 0.944698i \(-0.393645\pi\)
0.327942 + 0.944698i \(0.393645\pi\)
\(462\) 0 0
\(463\) −28.6032 −1.32930 −0.664651 0.747154i \(-0.731420\pi\)
−0.664651 + 0.747154i \(0.731420\pi\)
\(464\) 0 0
\(465\) 13.0514 0.605243
\(466\) 0 0
\(467\) −5.13906 −0.237807 −0.118904 0.992906i \(-0.537938\pi\)
−0.118904 + 0.992906i \(0.537938\pi\)
\(468\) 0 0
\(469\) −7.15412 −0.330347
\(470\) 0 0
\(471\) 32.4110 1.49342
\(472\) 0 0
\(473\) 4.53235 0.208398
\(474\) 0 0
\(475\) 4.10275 0.188247
\(476\) 0 0
\(477\) −17.6731 −0.809198
\(478\) 0 0
\(479\) 20.6218 0.942233 0.471116 0.882071i \(-0.343851\pi\)
0.471116 + 0.882071i \(0.343851\pi\)
\(480\) 0 0
\(481\) 6.00000 0.273576
\(482\) 0 0
\(483\) −0.689115 −0.0313558
\(484\) 0 0
\(485\) −17.5164 −0.795378
\(486\) 0 0
\(487\) −10.0221 −0.454143 −0.227072 0.973878i \(-0.572915\pi\)
−0.227072 + 0.973878i \(0.572915\pi\)
\(488\) 0 0
\(489\) −50.8627 −2.30009
\(490\) 0 0
\(491\) 24.2276 1.09337 0.546687 0.837337i \(-0.315889\pi\)
0.546687 + 0.837337i \(0.315889\pi\)
\(492\) 0 0
\(493\) 10.8946 0.490669
\(494\) 0 0
\(495\) −4.05137 −0.182096
\(496\) 0 0
\(497\) 16.2055 0.726916
\(498\) 0 0
\(499\) 31.3463 1.40325 0.701626 0.712545i \(-0.252458\pi\)
0.701626 + 0.712545i \(0.252458\pi\)
\(500\) 0 0
\(501\) 8.06735 0.360422
\(502\) 0 0
\(503\) −37.4136 −1.66819 −0.834096 0.551620i \(-0.814010\pi\)
−0.834096 + 0.551620i \(0.814010\pi\)
\(504\) 0 0
\(505\) 7.43397 0.330807
\(506\) 0 0
\(507\) 15.7962 0.701535
\(508\) 0 0
\(509\) −23.7219 −1.05145 −0.525727 0.850653i \(-0.676207\pi\)
−0.525727 + 0.850653i \(0.676207\pi\)
\(510\) 0 0
\(511\) −2.55269 −0.112925
\(512\) 0 0
\(513\) −11.4543 −0.505720
\(514\) 0 0
\(515\) 4.23917 0.186800
\(516\) 0 0
\(517\) −4.13642 −0.181920
\(518\) 0 0
\(519\) 8.15676 0.358042
\(520\) 0 0
\(521\) −3.97334 −0.174075 −0.0870375 0.996205i \(-0.527740\pi\)
−0.0870375 + 0.996205i \(0.527740\pi\)
\(522\) 0 0
\(523\) 27.7626 1.21397 0.606986 0.794713i \(-0.292378\pi\)
0.606986 + 0.794713i \(0.292378\pi\)
\(524\) 0 0
\(525\) −2.65544 −0.115893
\(526\) 0 0
\(527\) −26.7733 −1.16626
\(528\) 0 0
\(529\) −22.9327 −0.997072
\(530\) 0 0
\(531\) −24.2258 −1.05131
\(532\) 0 0
\(533\) 2.88128 0.124802
\(534\) 0 0
\(535\) −0.519021 −0.0224392
\(536\) 0 0
\(537\) 57.7892 2.49379
\(538\) 0 0
\(539\) 1.00000 0.0430730
\(540\) 0 0
\(541\) 2.79186 0.120032 0.0600158 0.998197i \(-0.480885\pi\)
0.0600158 + 0.998197i \(0.480885\pi\)
\(542\) 0 0
\(543\) 37.9947 1.63051
\(544\) 0 0
\(545\) −3.31088 −0.141823
\(546\) 0 0
\(547\) −2.48362 −0.106192 −0.0530959 0.998589i \(-0.516909\pi\)
−0.0530959 + 0.998589i \(0.516909\pi\)
\(548\) 0 0
\(549\) 15.4340 0.658706
\(550\) 0 0
\(551\) −8.20550 −0.349566
\(552\) 0 0
\(553\) 3.63774 0.154692
\(554\) 0 0
\(555\) −6.00000 −0.254686
\(556\) 0 0
\(557\) 14.4650 0.612902 0.306451 0.951886i \(-0.400858\pi\)
0.306451 + 0.951886i \(0.400858\pi\)
\(558\) 0 0
\(559\) −12.0354 −0.509043
\(560\) 0 0
\(561\) 14.4650 0.610713
\(562\) 0 0
\(563\) 16.6572 0.702016 0.351008 0.936372i \(-0.385839\pi\)
0.351008 + 0.936372i \(0.385839\pi\)
\(564\) 0 0
\(565\) −11.5164 −0.484498
\(566\) 0 0
\(567\) −4.74049 −0.199082
\(568\) 0 0
\(569\) −5.72716 −0.240095 −0.120047 0.992768i \(-0.538305\pi\)
−0.120047 + 0.992768i \(0.538305\pi\)
\(570\) 0 0
\(571\) 28.1648 1.17866 0.589330 0.807892i \(-0.299392\pi\)
0.589330 + 0.807892i \(0.299392\pi\)
\(572\) 0 0
\(573\) −45.9681 −1.92034
\(574\) 0 0
\(575\) 0.259511 0.0108223
\(576\) 0 0
\(577\) 14.1701 0.589909 0.294954 0.955511i \(-0.404696\pi\)
0.294954 + 0.955511i \(0.404696\pi\)
\(578\) 0 0
\(579\) 11.7626 0.488835
\(580\) 0 0
\(581\) −6.79186 −0.281774
\(582\) 0 0
\(583\) −4.36226 −0.180666
\(584\) 0 0
\(585\) 10.7582 0.444796
\(586\) 0 0
\(587\) −15.1071 −0.623537 −0.311769 0.950158i \(-0.600921\pi\)
−0.311769 + 0.950158i \(0.600921\pi\)
\(588\) 0 0
\(589\) 20.1648 0.830877
\(590\) 0 0
\(591\) −9.48098 −0.389995
\(592\) 0 0
\(593\) 3.96633 0.162878 0.0814388 0.996678i \(-0.474048\pi\)
0.0814388 + 0.996678i \(0.474048\pi\)
\(594\) 0 0
\(595\) 5.44731 0.223318
\(596\) 0 0
\(597\) −26.7733 −1.09576
\(598\) 0 0
\(599\) −14.1027 −0.576223 −0.288111 0.957597i \(-0.593027\pi\)
−0.288111 + 0.957597i \(0.593027\pi\)
\(600\) 0 0
\(601\) −4.22584 −0.172376 −0.0861878 0.996279i \(-0.527468\pi\)
−0.0861878 + 0.996279i \(0.527468\pi\)
\(602\) 0 0
\(603\) −28.9840 −1.18032
\(604\) 0 0
\(605\) −1.00000 −0.0406558
\(606\) 0 0
\(607\) 38.4110 1.55905 0.779527 0.626369i \(-0.215459\pi\)
0.779527 + 0.626369i \(0.215459\pi\)
\(608\) 0 0
\(609\) 5.31088 0.215208
\(610\) 0 0
\(611\) 10.9840 0.444366
\(612\) 0 0
\(613\) 18.8052 0.759535 0.379767 0.925082i \(-0.376004\pi\)
0.379767 + 0.925082i \(0.376004\pi\)
\(614\) 0 0
\(615\) −2.88128 −0.116184
\(616\) 0 0
\(617\) 6.77853 0.272893 0.136447 0.990647i \(-0.456432\pi\)
0.136447 + 0.990647i \(0.456432\pi\)
\(618\) 0 0
\(619\) 18.5341 0.744948 0.372474 0.928043i \(-0.378510\pi\)
0.372474 + 0.928043i \(0.378510\pi\)
\(620\) 0 0
\(621\) −0.724518 −0.0290739
\(622\) 0 0
\(623\) 5.31088 0.212776
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −10.8946 −0.435089
\(628\) 0 0
\(629\) 12.3082 0.490762
\(630\) 0 0
\(631\) −13.9947 −0.557121 −0.278561 0.960419i \(-0.589857\pi\)
−0.278561 + 0.960419i \(0.589857\pi\)
\(632\) 0 0
\(633\) 42.3791 1.68442
\(634\) 0 0
\(635\) −20.9974 −0.833255
\(636\) 0 0
\(637\) −2.65544 −0.105212
\(638\) 0 0
\(639\) 65.6545 2.59725
\(640\) 0 0
\(641\) −28.9840 −1.14480 −0.572400 0.819974i \(-0.693988\pi\)
−0.572400 + 0.819974i \(0.693988\pi\)
\(642\) 0 0
\(643\) 17.2011 0.678346 0.339173 0.940724i \(-0.389853\pi\)
0.339173 + 0.940724i \(0.389853\pi\)
\(644\) 0 0
\(645\) 12.0354 0.473894
\(646\) 0 0
\(647\) −8.38260 −0.329554 −0.164777 0.986331i \(-0.552690\pi\)
−0.164777 + 0.986331i \(0.552690\pi\)
\(648\) 0 0
\(649\) −5.97966 −0.234722
\(650\) 0 0
\(651\) −13.0514 −0.511524
\(652\) 0 0
\(653\) 0.348927 0.0136546 0.00682728 0.999977i \(-0.497827\pi\)
0.00682728 + 0.999977i \(0.497827\pi\)
\(654\) 0 0
\(655\) −1.58373 −0.0618814
\(656\) 0 0
\(657\) −10.3419 −0.403477
\(658\) 0 0
\(659\) −19.7492 −0.769321 −0.384660 0.923058i \(-0.625681\pi\)
−0.384660 + 0.923058i \(0.625681\pi\)
\(660\) 0 0
\(661\) −1.33755 −0.0520246 −0.0260123 0.999662i \(-0.508281\pi\)
−0.0260123 + 0.999662i \(0.508281\pi\)
\(662\) 0 0
\(663\) −38.4110 −1.49176
\(664\) 0 0
\(665\) −4.10275 −0.159098
\(666\) 0 0
\(667\) −0.519021 −0.0200966
\(668\) 0 0
\(669\) 56.5004 2.18443
\(670\) 0 0
\(671\) 3.80957 0.147067
\(672\) 0 0
\(673\) −51.1488 −1.97164 −0.985822 0.167797i \(-0.946335\pi\)
−0.985822 + 0.167797i \(0.946335\pi\)
\(674\) 0 0
\(675\) −2.79186 −0.107459
\(676\) 0 0
\(677\) 22.9016 0.880181 0.440090 0.897953i \(-0.354946\pi\)
0.440090 + 0.897953i \(0.354946\pi\)
\(678\) 0 0
\(679\) 17.5164 0.672217
\(680\) 0 0
\(681\) −7.50764 −0.287694
\(682\) 0 0
\(683\) −43.8520 −1.67795 −0.838975 0.544171i \(-0.816844\pi\)
−0.838975 + 0.544171i \(0.816844\pi\)
\(684\) 0 0
\(685\) −8.15676 −0.311654
\(686\) 0 0
\(687\) −5.96460 −0.227564
\(688\) 0 0
\(689\) 11.5837 0.441305
\(690\) 0 0
\(691\) −16.9504 −0.644822 −0.322411 0.946600i \(-0.604493\pi\)
−0.322411 + 0.946600i \(0.604493\pi\)
\(692\) 0 0
\(693\) 4.05137 0.153899
\(694\) 0 0
\(695\) 11.9327 0.452631
\(696\) 0 0
\(697\) 5.91058 0.223879
\(698\) 0 0
\(699\) 71.6545 2.71022
\(700\) 0 0
\(701\) 7.96460 0.300819 0.150409 0.988624i \(-0.451941\pi\)
0.150409 + 0.988624i \(0.451941\pi\)
\(702\) 0 0
\(703\) −9.27020 −0.349632
\(704\) 0 0
\(705\) −10.9840 −0.413682
\(706\) 0 0
\(707\) −7.43397 −0.279583
\(708\) 0 0
\(709\) 32.3349 1.21436 0.607182 0.794563i \(-0.292300\pi\)
0.607182 + 0.794563i \(0.292300\pi\)
\(710\) 0 0
\(711\) 14.7379 0.552713
\(712\) 0 0
\(713\) 1.27548 0.0477672
\(714\) 0 0
\(715\) 2.65544 0.0993079
\(716\) 0 0
\(717\) 41.4757 1.54894
\(718\) 0 0
\(719\) 34.2206 1.27621 0.638106 0.769949i \(-0.279718\pi\)
0.638106 + 0.769949i \(0.279718\pi\)
\(720\) 0 0
\(721\) −4.23917 −0.157875
\(722\) 0 0
\(723\) −6.90794 −0.256909
\(724\) 0 0
\(725\) −2.00000 −0.0742781
\(726\) 0 0
\(727\) 27.2772 1.01166 0.505828 0.862634i \(-0.331187\pi\)
0.505828 + 0.862634i \(0.331187\pi\)
\(728\) 0 0
\(729\) −41.4464 −1.53505
\(730\) 0 0
\(731\) −24.6891 −0.913160
\(732\) 0 0
\(733\) 25.9256 0.957586 0.478793 0.877928i \(-0.341075\pi\)
0.478793 + 0.877928i \(0.341075\pi\)
\(734\) 0 0
\(735\) 2.65544 0.0979475
\(736\) 0 0
\(737\) −7.15412 −0.263525
\(738\) 0 0
\(739\) 20.5457 0.755785 0.377893 0.925849i \(-0.376649\pi\)
0.377893 + 0.925849i \(0.376649\pi\)
\(740\) 0 0
\(741\) 28.9300 1.06277
\(742\) 0 0
\(743\) −42.3082 −1.55214 −0.776069 0.630647i \(-0.782789\pi\)
−0.776069 + 0.630647i \(0.782789\pi\)
\(744\) 0 0
\(745\) 4.17009 0.152780
\(746\) 0 0
\(747\) −27.5164 −1.00677
\(748\) 0 0
\(749\) 0.519021 0.0189646
\(750\) 0 0
\(751\) −37.9274 −1.38399 −0.691995 0.721902i \(-0.743268\pi\)
−0.691995 + 0.721902i \(0.743268\pi\)
\(752\) 0 0
\(753\) 56.2630 2.05034
\(754\) 0 0
\(755\) −21.8786 −0.796245
\(756\) 0 0
\(757\) 36.3756 1.32209 0.661047 0.750345i \(-0.270113\pi\)
0.661047 + 0.750345i \(0.270113\pi\)
\(758\) 0 0
\(759\) −0.689115 −0.0250133
\(760\) 0 0
\(761\) 29.6661 1.07540 0.537698 0.843137i \(-0.319294\pi\)
0.537698 + 0.843137i \(0.319294\pi\)
\(762\) 0 0
\(763\) 3.31088 0.119862
\(764\) 0 0
\(765\) 22.0691 0.797909
\(766\) 0 0
\(767\) 15.8786 0.573344
\(768\) 0 0
\(769\) −35.9797 −1.29746 −0.648730 0.761019i \(-0.724699\pi\)
−0.648730 + 0.761019i \(0.724699\pi\)
\(770\) 0 0
\(771\) 2.82727 0.101822
\(772\) 0 0
\(773\) 36.3436 1.30719 0.653595 0.756844i \(-0.273260\pi\)
0.653595 + 0.756844i \(0.273260\pi\)
\(774\) 0 0
\(775\) 4.91495 0.176550
\(776\) 0 0
\(777\) 6.00000 0.215249
\(778\) 0 0
\(779\) −4.45168 −0.159498
\(780\) 0 0
\(781\) 16.2055 0.579878
\(782\) 0 0
\(783\) 5.58373 0.199546
\(784\) 0 0
\(785\) 12.2055 0.435633
\(786\) 0 0
\(787\) −24.1648 −0.861383 −0.430691 0.902499i \(-0.641730\pi\)
−0.430691 + 0.902499i \(0.641730\pi\)
\(788\) 0 0
\(789\) 65.4757 2.33100
\(790\) 0 0
\(791\) 11.5164 0.409475
\(792\) 0 0
\(793\) −10.1161 −0.359233
\(794\) 0 0
\(795\) −11.5837 −0.410832
\(796\) 0 0
\(797\) −3.79450 −0.134408 −0.0672041 0.997739i \(-0.521408\pi\)
−0.0672041 + 0.997739i \(0.521408\pi\)
\(798\) 0 0
\(799\) 22.5324 0.797137
\(800\) 0 0
\(801\) 21.5164 0.760244
\(802\) 0 0
\(803\) −2.55269 −0.0900826
\(804\) 0 0
\(805\) −0.259511 −0.00914654
\(806\) 0 0
\(807\) −43.5784 −1.53403
\(808\) 0 0
\(809\) −10.9300 −0.384279 −0.192139 0.981368i \(-0.561543\pi\)
−0.192139 + 0.981368i \(0.561543\pi\)
\(810\) 0 0
\(811\) −3.31088 −0.116261 −0.0581304 0.998309i \(-0.518514\pi\)
−0.0581304 + 0.998309i \(0.518514\pi\)
\(812\) 0 0
\(813\) 43.7573 1.53463
\(814\) 0 0
\(815\) −19.1541 −0.670940
\(816\) 0 0
\(817\) 18.5951 0.650560
\(818\) 0 0
\(819\) −10.7582 −0.375922
\(820\) 0 0
\(821\) −39.8566 −1.39100 −0.695502 0.718524i \(-0.744818\pi\)
−0.695502 + 0.718524i \(0.744818\pi\)
\(822\) 0 0
\(823\) −3.18079 −0.110875 −0.0554376 0.998462i \(-0.517655\pi\)
−0.0554376 + 0.998462i \(0.517655\pi\)
\(824\) 0 0
\(825\) −2.65544 −0.0924506
\(826\) 0 0
\(827\) 46.7059 1.62412 0.812062 0.583571i \(-0.198345\pi\)
0.812062 + 0.583571i \(0.198345\pi\)
\(828\) 0 0
\(829\) −33.0328 −1.14728 −0.573638 0.819109i \(-0.694468\pi\)
−0.573638 + 0.819109i \(0.694468\pi\)
\(830\) 0 0
\(831\) −15.2435 −0.528793
\(832\) 0 0
\(833\) −5.44731 −0.188738
\(834\) 0 0
\(835\) 3.03804 0.105136
\(836\) 0 0
\(837\) −13.7219 −0.474298
\(838\) 0 0
\(839\) −40.0505 −1.38270 −0.691348 0.722522i \(-0.742983\pi\)
−0.691348 + 0.722522i \(0.742983\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 0 0
\(843\) 36.3436 1.25174
\(844\) 0 0
\(845\) 5.94863 0.204639
\(846\) 0 0
\(847\) 1.00000 0.0343604
\(848\) 0 0
\(849\) 29.5837 1.01531
\(850\) 0 0
\(851\) −0.586367 −0.0201004
\(852\) 0 0
\(853\) 21.2099 0.726212 0.363106 0.931748i \(-0.381716\pi\)
0.363106 + 0.931748i \(0.381716\pi\)
\(854\) 0 0
\(855\) −16.6218 −0.568453
\(856\) 0 0
\(857\) −20.3686 −0.695778 −0.347889 0.937536i \(-0.613101\pi\)
−0.347889 + 0.937536i \(0.613101\pi\)
\(858\) 0 0
\(859\) 14.9910 0.511488 0.255744 0.966745i \(-0.417680\pi\)
0.255744 + 0.966745i \(0.417680\pi\)
\(860\) 0 0
\(861\) 2.88128 0.0981938
\(862\) 0 0
\(863\) −12.9840 −0.441981 −0.220991 0.975276i \(-0.570929\pi\)
−0.220991 + 0.975276i \(0.570929\pi\)
\(864\) 0 0
\(865\) 3.07171 0.104441
\(866\) 0 0
\(867\) −33.6528 −1.14291
\(868\) 0 0
\(869\) 3.63774 0.123402
\(870\) 0 0
\(871\) 18.9974 0.643701
\(872\) 0 0
\(873\) 70.9654 2.40182
\(874\) 0 0
\(875\) −1.00000 −0.0338062
\(876\) 0 0
\(877\) −52.1462 −1.76085 −0.880426 0.474183i \(-0.842743\pi\)
−0.880426 + 0.474183i \(0.842743\pi\)
\(878\) 0 0
\(879\) 68.9521 2.32570
\(880\) 0 0
\(881\) −23.6865 −0.798018 −0.399009 0.916947i \(-0.630646\pi\)
−0.399009 + 0.916947i \(0.630646\pi\)
\(882\) 0 0
\(883\) −22.5004 −0.757199 −0.378600 0.925561i \(-0.623594\pi\)
−0.378600 + 0.925561i \(0.623594\pi\)
\(884\) 0 0
\(885\) −15.8786 −0.533755
\(886\) 0 0
\(887\) 12.1382 0.407559 0.203780 0.979017i \(-0.434677\pi\)
0.203780 + 0.979017i \(0.434677\pi\)
\(888\) 0 0
\(889\) 20.9974 0.704229
\(890\) 0 0
\(891\) −4.74049 −0.158812
\(892\) 0 0
\(893\) −16.9707 −0.567903
\(894\) 0 0
\(895\) 21.7626 0.727442
\(896\) 0 0
\(897\) 1.82991 0.0610988
\(898\) 0 0
\(899\) −9.82991 −0.327846
\(900\) 0 0
\(901\) 23.7626 0.791646
\(902\) 0 0
\(903\) −12.0354 −0.400513
\(904\) 0 0
\(905\) 14.3082 0.475622
\(906\) 0 0
\(907\) −19.6058 −0.651000 −0.325500 0.945542i \(-0.605533\pi\)
−0.325500 + 0.945542i \(0.605533\pi\)
\(908\) 0 0
\(909\) −30.1178 −0.998945
\(910\) 0 0
\(911\) 47.5251 1.57458 0.787289 0.616584i \(-0.211484\pi\)
0.787289 + 0.616584i \(0.211484\pi\)
\(912\) 0 0
\(913\) −6.79186 −0.224778
\(914\) 0 0
\(915\) 10.1161 0.334427
\(916\) 0 0
\(917\) 1.58373 0.0522993
\(918\) 0 0
\(919\) 8.38087 0.276459 0.138230 0.990400i \(-0.455859\pi\)
0.138230 + 0.990400i \(0.455859\pi\)
\(920\) 0 0
\(921\) 41.3109 1.36124
\(922\) 0 0
\(923\) −43.0328 −1.41644
\(924\) 0 0
\(925\) −2.25951 −0.0742922
\(926\) 0 0
\(927\) −17.1745 −0.564083
\(928\) 0 0
\(929\) 5.61039 0.184071 0.0920355 0.995756i \(-0.470663\pi\)
0.0920355 + 0.995756i \(0.470663\pi\)
\(930\) 0 0
\(931\) 4.10275 0.134462
\(932\) 0 0
\(933\) 12.0540 0.394631
\(934\) 0 0
\(935\) 5.44731 0.178146
\(936\) 0 0
\(937\) 43.5448 1.42255 0.711273 0.702916i \(-0.248119\pi\)
0.711273 + 0.702916i \(0.248119\pi\)
\(938\) 0 0
\(939\) 63.1001 2.05919
\(940\) 0 0
\(941\) −12.3959 −0.404096 −0.202048 0.979376i \(-0.564760\pi\)
−0.202048 + 0.979376i \(0.564760\pi\)
\(942\) 0 0
\(943\) −0.281581 −0.00916954
\(944\) 0 0
\(945\) 2.79186 0.0908193
\(946\) 0 0
\(947\) 16.4917 0.535907 0.267954 0.963432i \(-0.413653\pi\)
0.267954 + 0.963432i \(0.413653\pi\)
\(948\) 0 0
\(949\) 6.77853 0.220040
\(950\) 0 0
\(951\) −30.7599 −0.997459
\(952\) 0 0
\(953\) −12.5944 −0.407973 −0.203987 0.978974i \(-0.565390\pi\)
−0.203987 + 0.978974i \(0.565390\pi\)
\(954\) 0 0
\(955\) −17.3109 −0.560167
\(956\) 0 0
\(957\) 5.31088 0.171677
\(958\) 0 0
\(959\) 8.15676 0.263396
\(960\) 0 0
\(961\) −6.84324 −0.220750
\(962\) 0 0
\(963\) 2.10275 0.0677601
\(964\) 0 0
\(965\) 4.42960 0.142594
\(966\) 0 0
\(967\) −6.42960 −0.206762 −0.103381 0.994642i \(-0.532966\pi\)
−0.103381 + 0.994642i \(0.532966\pi\)
\(968\) 0 0
\(969\) 59.3463 1.90648
\(970\) 0 0
\(971\) −30.3959 −0.975452 −0.487726 0.872997i \(-0.662173\pi\)
−0.487726 + 0.872997i \(0.662173\pi\)
\(972\) 0 0
\(973\) −11.9327 −0.382543
\(974\) 0 0
\(975\) 7.05137 0.225825
\(976\) 0 0
\(977\) 10.5597 0.337835 0.168917 0.985630i \(-0.445973\pi\)
0.168917 + 0.985630i \(0.445973\pi\)
\(978\) 0 0
\(979\) 5.31088 0.169737
\(980\) 0 0
\(981\) 13.4136 0.428264
\(982\) 0 0
\(983\) −55.8670 −1.78188 −0.890941 0.454119i \(-0.849954\pi\)
−0.890941 + 0.454119i \(0.849954\pi\)
\(984\) 0 0
\(985\) −3.57040 −0.113762
\(986\) 0 0
\(987\) 10.9840 0.349625
\(988\) 0 0
\(989\) 1.17619 0.0374008
\(990\) 0 0
\(991\) 50.0354 1.58943 0.794713 0.606985i \(-0.207621\pi\)
0.794713 + 0.606985i \(0.207621\pi\)
\(992\) 0 0
\(993\) −69.1355 −2.19395
\(994\) 0 0
\(995\) −10.0824 −0.319634
\(996\) 0 0
\(997\) −36.5474 −1.15747 −0.578734 0.815516i \(-0.696453\pi\)
−0.578734 + 0.815516i \(0.696453\pi\)
\(998\) 0 0
\(999\) 6.30825 0.199584
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 6160.2.a.bd.1.1 3
4.3 odd 2 3080.2.a.m.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3080.2.a.m.1.3 3 4.3 odd 2
6160.2.a.bd.1.1 3 1.1 even 1 trivial