Properties

Label 3696.2.a.bp.1.3
Level $3696$
Weight $2$
Character 3696.1
Self dual yes
Analytic conductor $29.513$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3696 = 2^{4} \cdot 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3696.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(29.5127085871\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.837.1
Defining polynomial: \( x^{3} - 6x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 231)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-0.167449\) of defining polynomial
Character \(\chi\) \(=\) 3696.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} +3.80451 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +3.80451 q^{5} +1.00000 q^{7} +1.00000 q^{9} -1.00000 q^{11} +3.80451 q^{13} +3.80451 q^{15} +0.334898 q^{17} -8.13941 q^{19} +1.00000 q^{21} +1.66510 q^{23} +9.47431 q^{25} +1.00000 q^{27} +0.195488 q^{29} +9.94392 q^{31} -1.00000 q^{33} +3.80451 q^{35} -4.47431 q^{37} +3.80451 q^{39} -6.27882 q^{41} -2.33490 q^{43} +3.80451 q^{45} +12.1394 q^{47} +1.00000 q^{49} +0.334898 q^{51} +7.94392 q^{53} -3.80451 q^{55} -8.13941 q^{57} -3.74843 q^{59} +6.00000 q^{61} +1.00000 q^{63} +14.4743 q^{65} +0.139410 q^{67} +1.66510 q^{69} -4.66980 q^{71} +4.19549 q^{73} +9.47431 q^{75} -1.00000 q^{77} -3.33020 q^{79} +1.00000 q^{81} +13.9439 q^{83} +1.27412 q^{85} +0.195488 q^{87} -9.88784 q^{89} +3.80451 q^{91} +9.94392 q^{93} -30.9665 q^{95} +0.0560785 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{3} + 3 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{3} + 3 q^{7} + 3 q^{9} - 3 q^{11} - 12 q^{19} + 3 q^{21} + 6 q^{23} + 15 q^{25} + 3 q^{27} + 12 q^{29} + 6 q^{31} - 3 q^{33} + 6 q^{41} - 6 q^{43} + 24 q^{47} + 3 q^{49} - 12 q^{57} + 24 q^{59} + 18 q^{61} + 3 q^{63} + 30 q^{65} - 12 q^{67} + 6 q^{69} - 12 q^{71} + 24 q^{73} + 15 q^{75} - 3 q^{77} - 12 q^{79} + 3 q^{81} + 18 q^{83} - 18 q^{85} + 12 q^{87} + 18 q^{89} + 6 q^{93} - 12 q^{95} + 24 q^{97} - 3 q^{99}+O(q^{100}) \) 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
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) 3.80451 1.70143 0.850715 0.525628i \(-0.176170\pi\)
0.850715 + 0.525628i \(0.176170\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 3.80451 1.05518 0.527591 0.849499i \(-0.323095\pi\)
0.527591 + 0.849499i \(0.323095\pi\)
\(14\) 0 0
\(15\) 3.80451 0.982321
\(16\) 0 0
\(17\) 0.334898 0.0812248 0.0406124 0.999175i \(-0.487069\pi\)
0.0406124 + 0.999175i \(0.487069\pi\)
\(18\) 0 0
\(19\) −8.13941 −1.86731 −0.933654 0.358175i \(-0.883399\pi\)
−0.933654 + 0.358175i \(0.883399\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) 1.66510 0.347198 0.173599 0.984816i \(-0.444460\pi\)
0.173599 + 0.984816i \(0.444460\pi\)
\(24\) 0 0
\(25\) 9.47431 1.89486
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 0.195488 0.0363013 0.0181506 0.999835i \(-0.494222\pi\)
0.0181506 + 0.999835i \(0.494222\pi\)
\(30\) 0 0
\(31\) 9.94392 1.78598 0.892991 0.450075i \(-0.148603\pi\)
0.892991 + 0.450075i \(0.148603\pi\)
\(32\) 0 0
\(33\) −1.00000 −0.174078
\(34\) 0 0
\(35\) 3.80451 0.643080
\(36\) 0 0
\(37\) −4.47431 −0.735572 −0.367786 0.929911i \(-0.619884\pi\)
−0.367786 + 0.929911i \(0.619884\pi\)
\(38\) 0 0
\(39\) 3.80451 0.609209
\(40\) 0 0
\(41\) −6.27882 −0.980587 −0.490293 0.871557i \(-0.663110\pi\)
−0.490293 + 0.871557i \(0.663110\pi\)
\(42\) 0 0
\(43\) −2.33490 −0.356069 −0.178034 0.984024i \(-0.556974\pi\)
−0.178034 + 0.984024i \(0.556974\pi\)
\(44\) 0 0
\(45\) 3.80451 0.567143
\(46\) 0 0
\(47\) 12.1394 1.77071 0.885357 0.464911i \(-0.153914\pi\)
0.885357 + 0.464911i \(0.153914\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0.334898 0.0468952
\(52\) 0 0
\(53\) 7.94392 1.09118 0.545591 0.838052i \(-0.316305\pi\)
0.545591 + 0.838052i \(0.316305\pi\)
\(54\) 0 0
\(55\) −3.80451 −0.513000
\(56\) 0 0
\(57\) −8.13941 −1.07809
\(58\) 0 0
\(59\) −3.74843 −0.488004 −0.244002 0.969775i \(-0.578460\pi\)
−0.244002 + 0.969775i \(0.578460\pi\)
\(60\) 0 0
\(61\) 6.00000 0.768221 0.384111 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 0 0
\(65\) 14.4743 1.79532
\(66\) 0 0
\(67\) 0.139410 0.0170316 0.00851582 0.999964i \(-0.497289\pi\)
0.00851582 + 0.999964i \(0.497289\pi\)
\(68\) 0 0
\(69\) 1.66510 0.200455
\(70\) 0 0
\(71\) −4.66980 −0.554203 −0.277101 0.960841i \(-0.589374\pi\)
−0.277101 + 0.960841i \(0.589374\pi\)
\(72\) 0 0
\(73\) 4.19549 0.491045 0.245522 0.969391i \(-0.421041\pi\)
0.245522 + 0.969391i \(0.421041\pi\)
\(74\) 0 0
\(75\) 9.47431 1.09400
\(76\) 0 0
\(77\) −1.00000 −0.113961
\(78\) 0 0
\(79\) −3.33020 −0.374677 −0.187339 0.982295i \(-0.559986\pi\)
−0.187339 + 0.982295i \(0.559986\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 13.9439 1.53054 0.765272 0.643707i \(-0.222604\pi\)
0.765272 + 0.643707i \(0.222604\pi\)
\(84\) 0 0
\(85\) 1.27412 0.138198
\(86\) 0 0
\(87\) 0.195488 0.0209586
\(88\) 0 0
\(89\) −9.88784 −1.04811 −0.524055 0.851685i \(-0.675581\pi\)
−0.524055 + 0.851685i \(0.675581\pi\)
\(90\) 0 0
\(91\) 3.80451 0.398821
\(92\) 0 0
\(93\) 9.94392 1.03114
\(94\) 0 0
\(95\) −30.9665 −3.17709
\(96\) 0 0
\(97\) 0.0560785 0.00569391 0.00284695 0.999996i \(-0.499094\pi\)
0.00284695 + 0.999996i \(0.499094\pi\)
\(98\) 0 0
\(99\) −1.00000 −0.100504
\(100\) 0 0
\(101\) −18.8831 −1.87894 −0.939472 0.342626i \(-0.888683\pi\)
−0.939472 + 0.342626i \(0.888683\pi\)
\(102\) 0 0
\(103\) 8.27882 0.815736 0.407868 0.913041i \(-0.366272\pi\)
0.407868 + 0.913041i \(0.366272\pi\)
\(104\) 0 0
\(105\) 3.80451 0.371282
\(106\) 0 0
\(107\) 8.13941 0.786866 0.393433 0.919353i \(-0.371287\pi\)
0.393433 + 0.919353i \(0.371287\pi\)
\(108\) 0 0
\(109\) 11.5529 1.10657 0.553286 0.832992i \(-0.313374\pi\)
0.553286 + 0.832992i \(0.313374\pi\)
\(110\) 0 0
\(111\) −4.47431 −0.424683
\(112\) 0 0
\(113\) −1.33020 −0.125135 −0.0625675 0.998041i \(-0.519929\pi\)
−0.0625675 + 0.998041i \(0.519929\pi\)
\(114\) 0 0
\(115\) 6.33490 0.590732
\(116\) 0 0
\(117\) 3.80451 0.351727
\(118\) 0 0
\(119\) 0.334898 0.0307001
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) −6.27882 −0.566142
\(124\) 0 0
\(125\) 17.0226 1.52254
\(126\) 0 0
\(127\) −14.6137 −1.29676 −0.648379 0.761318i \(-0.724553\pi\)
−0.648379 + 0.761318i \(0.724553\pi\)
\(128\) 0 0
\(129\) −2.33490 −0.205576
\(130\) 0 0
\(131\) −20.5576 −1.79613 −0.898065 0.439863i \(-0.855027\pi\)
−0.898065 + 0.439863i \(0.855027\pi\)
\(132\) 0 0
\(133\) −8.13941 −0.705776
\(134\) 0 0
\(135\) 3.80451 0.327440
\(136\) 0 0
\(137\) −16.2227 −1.38600 −0.693001 0.720936i \(-0.743712\pi\)
−0.693001 + 0.720936i \(0.743712\pi\)
\(138\) 0 0
\(139\) −22.5482 −1.91252 −0.956259 0.292522i \(-0.905506\pi\)
−0.956259 + 0.292522i \(0.905506\pi\)
\(140\) 0 0
\(141\) 12.1394 1.02232
\(142\) 0 0
\(143\) −3.80451 −0.318149
\(144\) 0 0
\(145\) 0.743738 0.0617641
\(146\) 0 0
\(147\) 1.00000 0.0824786
\(148\) 0 0
\(149\) 8.19549 0.671401 0.335700 0.941969i \(-0.391027\pi\)
0.335700 + 0.941969i \(0.391027\pi\)
\(150\) 0 0
\(151\) 13.2741 1.08023 0.540116 0.841590i \(-0.318380\pi\)
0.540116 + 0.841590i \(0.318380\pi\)
\(152\) 0 0
\(153\) 0.334898 0.0270749
\(154\) 0 0
\(155\) 37.8318 3.03872
\(156\) 0 0
\(157\) −16.9392 −1.35190 −0.675949 0.736949i \(-0.736266\pi\)
−0.675949 + 0.736949i \(0.736266\pi\)
\(158\) 0 0
\(159\) 7.94392 0.629994
\(160\) 0 0
\(161\) 1.66510 0.131228
\(162\) 0 0
\(163\) 6.79982 0.532603 0.266301 0.963890i \(-0.414198\pi\)
0.266301 + 0.963890i \(0.414198\pi\)
\(164\) 0 0
\(165\) −3.80451 −0.296181
\(166\) 0 0
\(167\) 18.2227 1.41012 0.705059 0.709149i \(-0.250920\pi\)
0.705059 + 0.709149i \(0.250920\pi\)
\(168\) 0 0
\(169\) 1.47431 0.113408
\(170\) 0 0
\(171\) −8.13941 −0.622436
\(172\) 0 0
\(173\) −1.72118 −0.130859 −0.0654294 0.997857i \(-0.520842\pi\)
−0.0654294 + 0.997857i \(0.520842\pi\)
\(174\) 0 0
\(175\) 9.47431 0.716190
\(176\) 0 0
\(177\) −3.74843 −0.281749
\(178\) 0 0
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) 0.725875 0.0539539 0.0269769 0.999636i \(-0.491412\pi\)
0.0269769 + 0.999636i \(0.491412\pi\)
\(182\) 0 0
\(183\) 6.00000 0.443533
\(184\) 0 0
\(185\) −17.0226 −1.25152
\(186\) 0 0
\(187\) −0.334898 −0.0244902
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) −5.27412 −0.381622 −0.190811 0.981627i \(-0.561112\pi\)
−0.190811 + 0.981627i \(0.561112\pi\)
\(192\) 0 0
\(193\) −19.8318 −1.42752 −0.713761 0.700390i \(-0.753010\pi\)
−0.713761 + 0.700390i \(0.753010\pi\)
\(194\) 0 0
\(195\) 14.4743 1.03653
\(196\) 0 0
\(197\) 2.66980 0.190215 0.0951076 0.995467i \(-0.469680\pi\)
0.0951076 + 0.995467i \(0.469680\pi\)
\(198\) 0 0
\(199\) −13.5529 −0.960743 −0.480371 0.877065i \(-0.659498\pi\)
−0.480371 + 0.877065i \(0.659498\pi\)
\(200\) 0 0
\(201\) 0.139410 0.00983322
\(202\) 0 0
\(203\) 0.195488 0.0137206
\(204\) 0 0
\(205\) −23.8878 −1.66840
\(206\) 0 0
\(207\) 1.66510 0.115733
\(208\) 0 0
\(209\) 8.13941 0.563015
\(210\) 0 0
\(211\) 4.27882 0.294566 0.147283 0.989094i \(-0.452947\pi\)
0.147283 + 0.989094i \(0.452947\pi\)
\(212\) 0 0
\(213\) −4.66980 −0.319969
\(214\) 0 0
\(215\) −8.88315 −0.605826
\(216\) 0 0
\(217\) 9.94392 0.675037
\(218\) 0 0
\(219\) 4.19549 0.283505
\(220\) 0 0
\(221\) 1.27412 0.0857069
\(222\) 0 0
\(223\) 10.2694 0.687692 0.343846 0.939026i \(-0.388270\pi\)
0.343846 + 0.939026i \(0.388270\pi\)
\(224\) 0 0
\(225\) 9.47431 0.631621
\(226\) 0 0
\(227\) 0.390977 0.0259500 0.0129750 0.999916i \(-0.495870\pi\)
0.0129750 + 0.999916i \(0.495870\pi\)
\(228\) 0 0
\(229\) 7.94392 0.524949 0.262475 0.964939i \(-0.415461\pi\)
0.262475 + 0.964939i \(0.415461\pi\)
\(230\) 0 0
\(231\) −1.00000 −0.0657952
\(232\) 0 0
\(233\) 26.5576 1.73985 0.869924 0.493185i \(-0.164167\pi\)
0.869924 + 0.493185i \(0.164167\pi\)
\(234\) 0 0
\(235\) 46.1845 3.01275
\(236\) 0 0
\(237\) −3.33020 −0.216320
\(238\) 0 0
\(239\) −10.7998 −0.698582 −0.349291 0.937014i \(-0.613578\pi\)
−0.349291 + 0.937014i \(0.613578\pi\)
\(240\) 0 0
\(241\) −12.0833 −0.778356 −0.389178 0.921163i \(-0.627241\pi\)
−0.389178 + 0.921163i \(0.627241\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 3.80451 0.243061
\(246\) 0 0
\(247\) −30.9665 −1.97035
\(248\) 0 0
\(249\) 13.9439 0.883660
\(250\) 0 0
\(251\) −4.80921 −0.303554 −0.151777 0.988415i \(-0.548500\pi\)
−0.151777 + 0.988415i \(0.548500\pi\)
\(252\) 0 0
\(253\) −1.66510 −0.104684
\(254\) 0 0
\(255\) 1.27412 0.0797888
\(256\) 0 0
\(257\) −16.7531 −1.04503 −0.522516 0.852630i \(-0.675006\pi\)
−0.522516 + 0.852630i \(0.675006\pi\)
\(258\) 0 0
\(259\) −4.47431 −0.278020
\(260\) 0 0
\(261\) 0.195488 0.0121004
\(262\) 0 0
\(263\) 12.1394 0.748548 0.374274 0.927318i \(-0.377892\pi\)
0.374274 + 0.927318i \(0.377892\pi\)
\(264\) 0 0
\(265\) 30.2227 1.85657
\(266\) 0 0
\(267\) −9.88784 −0.605126
\(268\) 0 0
\(269\) −25.2180 −1.53757 −0.768786 0.639506i \(-0.779139\pi\)
−0.768786 + 0.639506i \(0.779139\pi\)
\(270\) 0 0
\(271\) −4.13941 −0.251451 −0.125726 0.992065i \(-0.540126\pi\)
−0.125726 + 0.992065i \(0.540126\pi\)
\(272\) 0 0
\(273\) 3.80451 0.230260
\(274\) 0 0
\(275\) −9.47431 −0.571322
\(276\) 0 0
\(277\) −18.2788 −1.09827 −0.549134 0.835734i \(-0.685042\pi\)
−0.549134 + 0.835734i \(0.685042\pi\)
\(278\) 0 0
\(279\) 9.94392 0.595327
\(280\) 0 0
\(281\) 6.74374 0.402298 0.201149 0.979561i \(-0.435533\pi\)
0.201149 + 0.979561i \(0.435533\pi\)
\(282\) 0 0
\(283\) −23.3575 −1.38846 −0.694228 0.719755i \(-0.744254\pi\)
−0.694228 + 0.719755i \(0.744254\pi\)
\(284\) 0 0
\(285\) −30.9665 −1.83430
\(286\) 0 0
\(287\) −6.27882 −0.370627
\(288\) 0 0
\(289\) −16.8878 −0.993403
\(290\) 0 0
\(291\) 0.0560785 0.00328738
\(292\) 0 0
\(293\) −14.1667 −0.827625 −0.413813 0.910362i \(-0.635803\pi\)
−0.413813 + 0.910362i \(0.635803\pi\)
\(294\) 0 0
\(295\) −14.2610 −0.830305
\(296\) 0 0
\(297\) −1.00000 −0.0580259
\(298\) 0 0
\(299\) 6.33490 0.366357
\(300\) 0 0
\(301\) −2.33490 −0.134581
\(302\) 0 0
\(303\) −18.8831 −1.08481
\(304\) 0 0
\(305\) 22.8271 1.30707
\(306\) 0 0
\(307\) 12.5576 0.716702 0.358351 0.933587i \(-0.383339\pi\)
0.358351 + 0.933587i \(0.383339\pi\)
\(308\) 0 0
\(309\) 8.27882 0.470966
\(310\) 0 0
\(311\) −9.33959 −0.529600 −0.264800 0.964303i \(-0.585306\pi\)
−0.264800 + 0.964303i \(0.585306\pi\)
\(312\) 0 0
\(313\) −2.99530 −0.169305 −0.0846523 0.996411i \(-0.526978\pi\)
−0.0846523 + 0.996411i \(0.526978\pi\)
\(314\) 0 0
\(315\) 3.80451 0.214360
\(316\) 0 0
\(317\) 9.93453 0.557979 0.278989 0.960294i \(-0.410001\pi\)
0.278989 + 0.960294i \(0.410001\pi\)
\(318\) 0 0
\(319\) −0.195488 −0.0109453
\(320\) 0 0
\(321\) 8.13941 0.454298
\(322\) 0 0
\(323\) −2.72588 −0.151672
\(324\) 0 0
\(325\) 36.0451 1.99942
\(326\) 0 0
\(327\) 11.5529 0.638879
\(328\) 0 0
\(329\) 12.1394 0.669267
\(330\) 0 0
\(331\) −22.5482 −1.23936 −0.619682 0.784853i \(-0.712738\pi\)
−0.619682 + 0.784853i \(0.712738\pi\)
\(332\) 0 0
\(333\) −4.47431 −0.245191
\(334\) 0 0
\(335\) 0.530387 0.0289781
\(336\) 0 0
\(337\) 13.2835 0.723599 0.361800 0.932256i \(-0.382162\pi\)
0.361800 + 0.932256i \(0.382162\pi\)
\(338\) 0 0
\(339\) −1.33020 −0.0722467
\(340\) 0 0
\(341\) −9.94392 −0.538494
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 6.33490 0.341059
\(346\) 0 0
\(347\) 27.7757 1.49108 0.745538 0.666463i \(-0.232192\pi\)
0.745538 + 0.666463i \(0.232192\pi\)
\(348\) 0 0
\(349\) −2.85589 −0.152873 −0.0764363 0.997074i \(-0.524354\pi\)
−0.0764363 + 0.997074i \(0.524354\pi\)
\(350\) 0 0
\(351\) 3.80451 0.203070
\(352\) 0 0
\(353\) 24.6410 1.31151 0.655753 0.754975i \(-0.272351\pi\)
0.655753 + 0.754975i \(0.272351\pi\)
\(354\) 0 0
\(355\) −17.7663 −0.942937
\(356\) 0 0
\(357\) 0.334898 0.0177247
\(358\) 0 0
\(359\) 11.8878 0.627416 0.313708 0.949519i \(-0.398429\pi\)
0.313708 + 0.949519i \(0.398429\pi\)
\(360\) 0 0
\(361\) 47.2500 2.48684
\(362\) 0 0
\(363\) 1.00000 0.0524864
\(364\) 0 0
\(365\) 15.9618 0.835478
\(366\) 0 0
\(367\) −3.60902 −0.188389 −0.0941947 0.995554i \(-0.530028\pi\)
−0.0941947 + 0.995554i \(0.530028\pi\)
\(368\) 0 0
\(369\) −6.27882 −0.326862
\(370\) 0 0
\(371\) 7.94392 0.412428
\(372\) 0 0
\(373\) 12.3349 0.638677 0.319338 0.947641i \(-0.396539\pi\)
0.319338 + 0.947641i \(0.396539\pi\)
\(374\) 0 0
\(375\) 17.0226 0.879041
\(376\) 0 0
\(377\) 0.743738 0.0383045
\(378\) 0 0
\(379\) −23.3575 −1.19979 −0.599896 0.800078i \(-0.704791\pi\)
−0.599896 + 0.800078i \(0.704791\pi\)
\(380\) 0 0
\(381\) −14.6137 −0.748683
\(382\) 0 0
\(383\) 15.2180 0.777606 0.388803 0.921321i \(-0.372889\pi\)
0.388803 + 0.921321i \(0.372889\pi\)
\(384\) 0 0
\(385\) −3.80451 −0.193896
\(386\) 0 0
\(387\) −2.33490 −0.118690
\(388\) 0 0
\(389\) 3.73057 0.189147 0.0945737 0.995518i \(-0.469851\pi\)
0.0945737 + 0.995518i \(0.469851\pi\)
\(390\) 0 0
\(391\) 0.557640 0.0282011
\(392\) 0 0
\(393\) −20.5576 −1.03700
\(394\) 0 0
\(395\) −12.6698 −0.637487
\(396\) 0 0
\(397\) 20.8925 1.04857 0.524283 0.851544i \(-0.324333\pi\)
0.524283 + 0.851544i \(0.324333\pi\)
\(398\) 0 0
\(399\) −8.13941 −0.407480
\(400\) 0 0
\(401\) 23.2741 1.16225 0.581127 0.813813i \(-0.302612\pi\)
0.581127 + 0.813813i \(0.302612\pi\)
\(402\) 0 0
\(403\) 37.8318 1.88453
\(404\) 0 0
\(405\) 3.80451 0.189048
\(406\) 0 0
\(407\) 4.47431 0.221783
\(408\) 0 0
\(409\) −23.8972 −1.18164 −0.590821 0.806803i \(-0.701196\pi\)
−0.590821 + 0.806803i \(0.701196\pi\)
\(410\) 0 0
\(411\) −16.2227 −0.800209
\(412\) 0 0
\(413\) −3.74843 −0.184448
\(414\) 0 0
\(415\) 53.0498 2.60411
\(416\) 0 0
\(417\) −22.5482 −1.10419
\(418\) 0 0
\(419\) 30.2967 1.48009 0.740045 0.672557i \(-0.234804\pi\)
0.740045 + 0.672557i \(0.234804\pi\)
\(420\) 0 0
\(421\) −15.5257 −0.756676 −0.378338 0.925668i \(-0.623504\pi\)
−0.378338 + 0.925668i \(0.623504\pi\)
\(422\) 0 0
\(423\) 12.1394 0.590238
\(424\) 0 0
\(425\) 3.17293 0.153910
\(426\) 0 0
\(427\) 6.00000 0.290360
\(428\) 0 0
\(429\) −3.80451 −0.183684
\(430\) 0 0
\(431\) −3.07864 −0.148293 −0.0741463 0.997247i \(-0.523623\pi\)
−0.0741463 + 0.997247i \(0.523623\pi\)
\(432\) 0 0
\(433\) 16.9392 0.814047 0.407024 0.913418i \(-0.366567\pi\)
0.407024 + 0.913418i \(0.366567\pi\)
\(434\) 0 0
\(435\) 0.743738 0.0356595
\(436\) 0 0
\(437\) −13.5529 −0.648325
\(438\) 0 0
\(439\) 11.0786 0.528754 0.264377 0.964419i \(-0.414834\pi\)
0.264377 + 0.964419i \(0.414834\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 14.5482 0.691208 0.345604 0.938380i \(-0.387674\pi\)
0.345604 + 0.938380i \(0.387674\pi\)
\(444\) 0 0
\(445\) −37.6184 −1.78328
\(446\) 0 0
\(447\) 8.19549 0.387633
\(448\) 0 0
\(449\) 27.4875 1.29721 0.648607 0.761123i \(-0.275352\pi\)
0.648607 + 0.761123i \(0.275352\pi\)
\(450\) 0 0
\(451\) 6.27882 0.295658
\(452\) 0 0
\(453\) 13.2741 0.623673
\(454\) 0 0
\(455\) 14.4743 0.678566
\(456\) 0 0
\(457\) −7.94392 −0.371601 −0.185800 0.982587i \(-0.559488\pi\)
−0.185800 + 0.982587i \(0.559488\pi\)
\(458\) 0 0
\(459\) 0.334898 0.0156317
\(460\) 0 0
\(461\) 8.26943 0.385146 0.192573 0.981283i \(-0.438317\pi\)
0.192573 + 0.981283i \(0.438317\pi\)
\(462\) 0 0
\(463\) −9.73904 −0.452612 −0.226306 0.974056i \(-0.572665\pi\)
−0.226306 + 0.974056i \(0.572665\pi\)
\(464\) 0 0
\(465\) 37.8318 1.75441
\(466\) 0 0
\(467\) 40.6970 1.88323 0.941617 0.336685i \(-0.109306\pi\)
0.941617 + 0.336685i \(0.109306\pi\)
\(468\) 0 0
\(469\) 0.139410 0.00643735
\(470\) 0 0
\(471\) −16.9392 −0.780518
\(472\) 0 0
\(473\) 2.33490 0.107359
\(474\) 0 0
\(475\) −77.1153 −3.53829
\(476\) 0 0
\(477\) 7.94392 0.363727
\(478\) 0 0
\(479\) −37.8318 −1.72858 −0.864289 0.502996i \(-0.832231\pi\)
−0.864289 + 0.502996i \(0.832231\pi\)
\(480\) 0 0
\(481\) −17.0226 −0.776162
\(482\) 0 0
\(483\) 1.66510 0.0757647
\(484\) 0 0
\(485\) 0.213351 0.00968778
\(486\) 0 0
\(487\) −40.5576 −1.83784 −0.918921 0.394442i \(-0.870938\pi\)
−0.918921 + 0.394442i \(0.870938\pi\)
\(488\) 0 0
\(489\) 6.79982 0.307498
\(490\) 0 0
\(491\) −31.0786 −1.40256 −0.701280 0.712886i \(-0.747388\pi\)
−0.701280 + 0.712886i \(0.747388\pi\)
\(492\) 0 0
\(493\) 0.0654688 0.00294856
\(494\) 0 0
\(495\) −3.80451 −0.171000
\(496\) 0 0
\(497\) −4.66980 −0.209469
\(498\) 0 0
\(499\) −15.3575 −0.687494 −0.343747 0.939062i \(-0.611696\pi\)
−0.343747 + 0.939062i \(0.611696\pi\)
\(500\) 0 0
\(501\) 18.2227 0.814132
\(502\) 0 0
\(503\) −14.8925 −0.664025 −0.332013 0.943275i \(-0.607728\pi\)
−0.332013 + 0.943275i \(0.607728\pi\)
\(504\) 0 0
\(505\) −71.8412 −3.19689
\(506\) 0 0
\(507\) 1.47431 0.0654763
\(508\) 0 0
\(509\) −18.0000 −0.797836 −0.398918 0.916987i \(-0.630614\pi\)
−0.398918 + 0.916987i \(0.630614\pi\)
\(510\) 0 0
\(511\) 4.19549 0.185597
\(512\) 0 0
\(513\) −8.13941 −0.359364
\(514\) 0 0
\(515\) 31.4969 1.38792
\(516\) 0 0
\(517\) −12.1394 −0.533891
\(518\) 0 0
\(519\) −1.72118 −0.0755514
\(520\) 0 0
\(521\) 27.6924 1.21322 0.606612 0.794998i \(-0.292528\pi\)
0.606612 + 0.794998i \(0.292528\pi\)
\(522\) 0 0
\(523\) −18.4088 −0.804962 −0.402481 0.915428i \(-0.631852\pi\)
−0.402481 + 0.915428i \(0.631852\pi\)
\(524\) 0 0
\(525\) 9.47431 0.413493
\(526\) 0 0
\(527\) 3.33020 0.145066
\(528\) 0 0
\(529\) −20.2274 −0.879454
\(530\) 0 0
\(531\) −3.74843 −0.162668
\(532\) 0 0
\(533\) −23.8878 −1.03470
\(534\) 0 0
\(535\) 30.9665 1.33880
\(536\) 0 0
\(537\) 12.0000 0.517838
\(538\) 0 0
\(539\) −1.00000 −0.0430730
\(540\) 0 0
\(541\) −4.26943 −0.183557 −0.0917786 0.995779i \(-0.529255\pi\)
−0.0917786 + 0.995779i \(0.529255\pi\)
\(542\) 0 0
\(543\) 0.725875 0.0311503
\(544\) 0 0
\(545\) 43.9533 1.88275
\(546\) 0 0
\(547\) −20.0000 −0.855138 −0.427569 0.903983i \(-0.640630\pi\)
−0.427569 + 0.903983i \(0.640630\pi\)
\(548\) 0 0
\(549\) 6.00000 0.256074
\(550\) 0 0
\(551\) −1.59116 −0.0677857
\(552\) 0 0
\(553\) −3.33020 −0.141615
\(554\) 0 0
\(555\) −17.0226 −0.722567
\(556\) 0 0
\(557\) −12.4743 −0.528553 −0.264277 0.964447i \(-0.585133\pi\)
−0.264277 + 0.964447i \(0.585133\pi\)
\(558\) 0 0
\(559\) −8.88315 −0.375717
\(560\) 0 0
\(561\) −0.334898 −0.0141394
\(562\) 0 0
\(563\) 32.7710 1.38113 0.690566 0.723269i \(-0.257361\pi\)
0.690566 + 0.723269i \(0.257361\pi\)
\(564\) 0 0
\(565\) −5.06077 −0.212908
\(566\) 0 0
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) −29.2180 −1.22488 −0.612442 0.790515i \(-0.709813\pi\)
−0.612442 + 0.790515i \(0.709813\pi\)
\(570\) 0 0
\(571\) −32.4455 −1.35780 −0.678901 0.734230i \(-0.737543\pi\)
−0.678901 + 0.734230i \(0.737543\pi\)
\(572\) 0 0
\(573\) −5.27412 −0.220330
\(574\) 0 0
\(575\) 15.7757 0.657892
\(576\) 0 0
\(577\) −14.8831 −0.619594 −0.309797 0.950803i \(-0.600261\pi\)
−0.309797 + 0.950803i \(0.600261\pi\)
\(578\) 0 0
\(579\) −19.8318 −0.824180
\(580\) 0 0
\(581\) 13.9439 0.578491
\(582\) 0 0
\(583\) −7.94392 −0.329004
\(584\) 0 0
\(585\) 14.4743 0.598439
\(586\) 0 0
\(587\) 4.53039 0.186989 0.0934945 0.995620i \(-0.470196\pi\)
0.0934945 + 0.995620i \(0.470196\pi\)
\(588\) 0 0
\(589\) −80.9377 −3.33498
\(590\) 0 0
\(591\) 2.66980 0.109821
\(592\) 0 0
\(593\) 8.28821 0.340356 0.170178 0.985413i \(-0.445566\pi\)
0.170178 + 0.985413i \(0.445566\pi\)
\(594\) 0 0
\(595\) 1.27412 0.0522340
\(596\) 0 0
\(597\) −13.5529 −0.554685
\(598\) 0 0
\(599\) −1.99061 −0.0813341 −0.0406671 0.999173i \(-0.512948\pi\)
−0.0406671 + 0.999173i \(0.512948\pi\)
\(600\) 0 0
\(601\) −8.47431 −0.345674 −0.172837 0.984950i \(-0.555293\pi\)
−0.172837 + 0.984950i \(0.555293\pi\)
\(602\) 0 0
\(603\) 0.139410 0.00567721
\(604\) 0 0
\(605\) 3.80451 0.154675
\(606\) 0 0
\(607\) 22.9665 0.932181 0.466090 0.884737i \(-0.345662\pi\)
0.466090 + 0.884737i \(0.345662\pi\)
\(608\) 0 0
\(609\) 0.195488 0.00792159
\(610\) 0 0
\(611\) 46.1845 1.86843
\(612\) 0 0
\(613\) 3.55294 0.143502 0.0717510 0.997423i \(-0.477141\pi\)
0.0717510 + 0.997423i \(0.477141\pi\)
\(614\) 0 0
\(615\) −23.8878 −0.963251
\(616\) 0 0
\(617\) 22.6698 0.912652 0.456326 0.889813i \(-0.349165\pi\)
0.456326 + 0.889813i \(0.349165\pi\)
\(618\) 0 0
\(619\) −8.71648 −0.350345 −0.175173 0.984538i \(-0.556048\pi\)
−0.175173 + 0.984538i \(0.556048\pi\)
\(620\) 0 0
\(621\) 1.66510 0.0668182
\(622\) 0 0
\(623\) −9.88784 −0.396148
\(624\) 0 0
\(625\) 17.3910 0.695639
\(626\) 0 0
\(627\) 8.13941 0.325057
\(628\) 0 0
\(629\) −1.49844 −0.0597467
\(630\) 0 0
\(631\) 11.8878 0.473248 0.236624 0.971601i \(-0.423959\pi\)
0.236624 + 0.971601i \(0.423959\pi\)
\(632\) 0 0
\(633\) 4.27882 0.170068
\(634\) 0 0
\(635\) −55.5981 −2.20634
\(636\) 0 0
\(637\) 3.80451 0.150740
\(638\) 0 0
\(639\) −4.66980 −0.184734
\(640\) 0 0
\(641\) −4.89254 −0.193244 −0.0966218 0.995321i \(-0.530804\pi\)
−0.0966218 + 0.995321i \(0.530804\pi\)
\(642\) 0 0
\(643\) −22.7804 −0.898371 −0.449185 0.893439i \(-0.648286\pi\)
−0.449185 + 0.893439i \(0.648286\pi\)
\(644\) 0 0
\(645\) −8.88315 −0.349774
\(646\) 0 0
\(647\) −31.2453 −1.22838 −0.614190 0.789158i \(-0.710517\pi\)
−0.614190 + 0.789158i \(0.710517\pi\)
\(648\) 0 0
\(649\) 3.74843 0.147139
\(650\) 0 0
\(651\) 9.94392 0.389733
\(652\) 0 0
\(653\) 28.2882 1.10700 0.553502 0.832848i \(-0.313291\pi\)
0.553502 + 0.832848i \(0.313291\pi\)
\(654\) 0 0
\(655\) −78.2118 −3.05599
\(656\) 0 0
\(657\) 4.19549 0.163682
\(658\) 0 0
\(659\) −30.2967 −1.18019 −0.590096 0.807333i \(-0.700910\pi\)
−0.590096 + 0.807333i \(0.700910\pi\)
\(660\) 0 0
\(661\) −35.7196 −1.38933 −0.694666 0.719333i \(-0.744448\pi\)
−0.694666 + 0.719333i \(0.744448\pi\)
\(662\) 0 0
\(663\) 1.27412 0.0494829
\(664\) 0 0
\(665\) −30.9665 −1.20083
\(666\) 0 0
\(667\) 0.325508 0.0126037
\(668\) 0 0
\(669\) 10.2694 0.397039
\(670\) 0 0
\(671\) −6.00000 −0.231627
\(672\) 0 0
\(673\) 34.9377 1.34675 0.673374 0.739302i \(-0.264844\pi\)
0.673374 + 0.739302i \(0.264844\pi\)
\(674\) 0 0
\(675\) 9.47431 0.364666
\(676\) 0 0
\(677\) 20.0094 0.769023 0.384512 0.923120i \(-0.374370\pi\)
0.384512 + 0.923120i \(0.374370\pi\)
\(678\) 0 0
\(679\) 0.0560785 0.00215209
\(680\) 0 0
\(681\) 0.390977 0.0149823
\(682\) 0 0
\(683\) −25.0498 −0.958504 −0.479252 0.877677i \(-0.659092\pi\)
−0.479252 + 0.877677i \(0.659092\pi\)
\(684\) 0 0
\(685\) −61.7196 −2.35818
\(686\) 0 0
\(687\) 7.94392 0.303080
\(688\) 0 0
\(689\) 30.2227 1.15139
\(690\) 0 0
\(691\) 38.8271 1.47705 0.738526 0.674225i \(-0.235522\pi\)
0.738526 + 0.674225i \(0.235522\pi\)
\(692\) 0 0
\(693\) −1.00000 −0.0379869
\(694\) 0 0
\(695\) −85.7851 −3.25401
\(696\) 0 0
\(697\) −2.10277 −0.0796480
\(698\) 0 0
\(699\) 26.5576 1.00450
\(700\) 0 0
\(701\) −41.7757 −1.57785 −0.788923 0.614492i \(-0.789361\pi\)
−0.788923 + 0.614492i \(0.789361\pi\)
\(702\) 0 0
\(703\) 36.4182 1.37354
\(704\) 0 0
\(705\) 46.1845 1.73941
\(706\) 0 0
\(707\) −18.8831 −0.710174
\(708\) 0 0
\(709\) −13.4041 −0.503403 −0.251702 0.967805i \(-0.580990\pi\)
−0.251702 + 0.967805i \(0.580990\pi\)
\(710\) 0 0
\(711\) −3.33020 −0.124892
\(712\) 0 0
\(713\) 16.5576 0.620088
\(714\) 0 0
\(715\) −14.4743 −0.541308
\(716\) 0 0
\(717\) −10.7998 −0.403327
\(718\) 0 0
\(719\) −5.47900 −0.204332 −0.102166 0.994767i \(-0.532577\pi\)
−0.102166 + 0.994767i \(0.532577\pi\)
\(720\) 0 0
\(721\) 8.27882 0.308319
\(722\) 0 0
\(723\) −12.0833 −0.449384
\(724\) 0 0
\(725\) 1.85212 0.0687859
\(726\) 0 0
\(727\) 32.8831 1.21957 0.609784 0.792567i \(-0.291256\pi\)
0.609784 + 0.792567i \(0.291256\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −0.781954 −0.0289216
\(732\) 0 0
\(733\) −35.8972 −1.32589 −0.662947 0.748666i \(-0.730695\pi\)
−0.662947 + 0.748666i \(0.730695\pi\)
\(734\) 0 0
\(735\) 3.80451 0.140332
\(736\) 0 0
\(737\) −0.139410 −0.00513523
\(738\) 0 0
\(739\) −29.6651 −1.09125 −0.545624 0.838030i \(-0.683707\pi\)
−0.545624 + 0.838030i \(0.683707\pi\)
\(740\) 0 0
\(741\) −30.9665 −1.13758
\(742\) 0 0
\(743\) 18.7998 0.689698 0.344849 0.938658i \(-0.387930\pi\)
0.344849 + 0.938658i \(0.387930\pi\)
\(744\) 0 0
\(745\) 31.1798 1.14234
\(746\) 0 0
\(747\) 13.9439 0.510181
\(748\) 0 0
\(749\) 8.13941 0.297408
\(750\) 0 0
\(751\) 9.59116 0.349986 0.174993 0.984570i \(-0.444010\pi\)
0.174993 + 0.984570i \(0.444010\pi\)
\(752\) 0 0
\(753\) −4.80921 −0.175257
\(754\) 0 0
\(755\) 50.5016 1.83794
\(756\) 0 0
\(757\) 2.74374 0.0997229 0.0498614 0.998756i \(-0.484122\pi\)
0.0498614 + 0.998756i \(0.484122\pi\)
\(758\) 0 0
\(759\) −1.66510 −0.0604394
\(760\) 0 0
\(761\) 6.39098 0.231673 0.115836 0.993268i \(-0.463045\pi\)
0.115836 + 0.993268i \(0.463045\pi\)
\(762\) 0 0
\(763\) 11.5529 0.418245
\(764\) 0 0
\(765\) 1.27412 0.0460661
\(766\) 0 0
\(767\) −14.2610 −0.514933
\(768\) 0 0
\(769\) 17.7951 0.641708 0.320854 0.947129i \(-0.396030\pi\)
0.320854 + 0.947129i \(0.396030\pi\)
\(770\) 0 0
\(771\) −16.7531 −0.603349
\(772\) 0 0
\(773\) −31.5257 −1.13390 −0.566950 0.823752i \(-0.691877\pi\)
−0.566950 + 0.823752i \(0.691877\pi\)
\(774\) 0 0
\(775\) 94.2118 3.38419
\(776\) 0 0
\(777\) −4.47431 −0.160515
\(778\) 0 0
\(779\) 51.1059 1.83106
\(780\) 0 0
\(781\) 4.66980 0.167098
\(782\) 0 0
\(783\) 0.195488 0.00698619
\(784\) 0 0
\(785\) −64.4455 −2.30016
\(786\) 0 0
\(787\) −31.8606 −1.13571 −0.567854 0.823130i \(-0.692226\pi\)
−0.567854 + 0.823130i \(0.692226\pi\)
\(788\) 0 0
\(789\) 12.1394 0.432174
\(790\) 0 0
\(791\) −1.33020 −0.0472966
\(792\) 0 0
\(793\) 22.8271 0.810613
\(794\) 0 0
\(795\) 30.2227 1.07189
\(796\) 0 0
\(797\) 43.8590 1.55357 0.776783 0.629768i \(-0.216850\pi\)
0.776783 + 0.629768i \(0.216850\pi\)
\(798\) 0 0
\(799\) 4.06547 0.143826
\(800\) 0 0
\(801\) −9.88784 −0.349370
\(802\) 0 0
\(803\) −4.19549 −0.148056
\(804\) 0 0
\(805\) 6.33490 0.223276
\(806\) 0 0
\(807\) −25.2180 −0.887717
\(808\) 0 0
\(809\) 6.74374 0.237097 0.118549 0.992948i \(-0.462176\pi\)
0.118549 + 0.992948i \(0.462176\pi\)
\(810\) 0 0
\(811\) −3.97275 −0.139502 −0.0697510 0.997564i \(-0.522220\pi\)
−0.0697510 + 0.997564i \(0.522220\pi\)
\(812\) 0 0
\(813\) −4.13941 −0.145175
\(814\) 0 0
\(815\) 25.8700 0.906186
\(816\) 0 0
\(817\) 19.0047 0.664890
\(818\) 0 0
\(819\) 3.80451 0.132940
\(820\) 0 0
\(821\) 49.5896 1.73069 0.865344 0.501178i \(-0.167100\pi\)
0.865344 + 0.501178i \(0.167100\pi\)
\(822\) 0 0
\(823\) −23.1908 −0.808380 −0.404190 0.914675i \(-0.632447\pi\)
−0.404190 + 0.914675i \(0.632447\pi\)
\(824\) 0 0
\(825\) −9.47431 −0.329853
\(826\) 0 0
\(827\) −27.7484 −0.964908 −0.482454 0.875921i \(-0.660254\pi\)
−0.482454 + 0.875921i \(0.660254\pi\)
\(828\) 0 0
\(829\) 0.269430 0.00935768 0.00467884 0.999989i \(-0.498511\pi\)
0.00467884 + 0.999989i \(0.498511\pi\)
\(830\) 0 0
\(831\) −18.2788 −0.634085
\(832\) 0 0
\(833\) 0.334898 0.0116035
\(834\) 0 0
\(835\) 69.3286 2.39922
\(836\) 0 0
\(837\) 9.94392 0.343712
\(838\) 0 0
\(839\) 27.3575 0.944484 0.472242 0.881469i \(-0.343445\pi\)
0.472242 + 0.881469i \(0.343445\pi\)
\(840\) 0 0
\(841\) −28.9618 −0.998682
\(842\) 0 0
\(843\) 6.74374 0.232267
\(844\) 0 0
\(845\) 5.60902 0.192956
\(846\) 0 0
\(847\) 1.00000 0.0343604
\(848\) 0 0
\(849\) −23.3575 −0.801626
\(850\) 0 0
\(851\) −7.45018 −0.255389
\(852\) 0 0
\(853\) −28.5482 −0.977473 −0.488737 0.872431i \(-0.662542\pi\)
−0.488737 + 0.872431i \(0.662542\pi\)
\(854\) 0 0
\(855\) −30.9665 −1.05903
\(856\) 0 0
\(857\) −39.8972 −1.36286 −0.681432 0.731882i \(-0.738642\pi\)
−0.681432 + 0.731882i \(0.738642\pi\)
\(858\) 0 0
\(859\) −20.5576 −0.701418 −0.350709 0.936485i \(-0.614059\pi\)
−0.350709 + 0.936485i \(0.614059\pi\)
\(860\) 0 0
\(861\) −6.27882 −0.213982
\(862\) 0 0
\(863\) 9.66510 0.329004 0.164502 0.986377i \(-0.447398\pi\)
0.164502 + 0.986377i \(0.447398\pi\)
\(864\) 0 0
\(865\) −6.54825 −0.222647
\(866\) 0 0
\(867\) −16.8878 −0.573541
\(868\) 0 0
\(869\) 3.33020 0.112969
\(870\) 0 0
\(871\) 0.530387 0.0179715
\(872\) 0 0
\(873\) 0.0560785 0.00189797
\(874\) 0 0
\(875\) 17.0226 0.575467
\(876\) 0 0
\(877\) −25.7212 −0.868543 −0.434271 0.900782i \(-0.642994\pi\)
−0.434271 + 0.900782i \(0.642994\pi\)
\(878\) 0 0
\(879\) −14.1667 −0.477830
\(880\) 0 0
\(881\) −14.6316 −0.492950 −0.246475 0.969149i \(-0.579272\pi\)
−0.246475 + 0.969149i \(0.579272\pi\)
\(882\) 0 0
\(883\) −18.6877 −0.628890 −0.314445 0.949276i \(-0.601818\pi\)
−0.314445 + 0.949276i \(0.601818\pi\)
\(884\) 0 0
\(885\) −14.2610 −0.479377
\(886\) 0 0
\(887\) −38.6137 −1.29652 −0.648261 0.761418i \(-0.724503\pi\)
−0.648261 + 0.761418i \(0.724503\pi\)
\(888\) 0 0
\(889\) −14.6137 −0.490128
\(890\) 0 0
\(891\) −1.00000 −0.0335013
\(892\) 0 0
\(893\) −98.8076 −3.30647
\(894\) 0 0
\(895\) 45.6541 1.52605
\(896\) 0 0
\(897\) 6.33490 0.211516
\(898\) 0 0
\(899\) 1.94392 0.0648334
\(900\) 0 0
\(901\) 2.66041 0.0886310
\(902\) 0 0
\(903\) −2.33490 −0.0777006
\(904\) 0 0
\(905\) 2.76160 0.0917987
\(906\) 0 0
\(907\) −49.0965 −1.63022 −0.815111 0.579304i \(-0.803324\pi\)
−0.815111 + 0.579304i \(0.803324\pi\)
\(908\) 0 0
\(909\) −18.8831 −0.626314
\(910\) 0 0
\(911\) −29.3753 −0.973248 −0.486624 0.873612i \(-0.661772\pi\)
−0.486624 + 0.873612i \(0.661772\pi\)
\(912\) 0 0
\(913\) −13.9439 −0.461476
\(914\) 0 0
\(915\) 22.8271 0.754640
\(916\) 0 0
\(917\) −20.5576 −0.678873
\(918\) 0 0
\(919\) 27.0047 0.890803 0.445401 0.895331i \(-0.353061\pi\)
0.445401 + 0.895331i \(0.353061\pi\)
\(920\) 0 0
\(921\) 12.5576 0.413788
\(922\) 0 0
\(923\) −17.7663 −0.584785
\(924\) 0 0
\(925\) −42.3910 −1.39381
\(926\) 0 0
\(927\) 8.27882 0.271912
\(928\) 0 0
\(929\) 57.8496 1.89798 0.948992 0.315299i \(-0.102105\pi\)
0.948992 + 0.315299i \(0.102105\pi\)
\(930\) 0 0
\(931\) −8.13941 −0.266758
\(932\) 0 0
\(933\) −9.33959 −0.305765
\(934\) 0 0
\(935\) −1.27412 −0.0416683
\(936\) 0 0
\(937\) 25.2180 0.823838 0.411919 0.911221i \(-0.364859\pi\)
0.411919 + 0.911221i \(0.364859\pi\)
\(938\) 0 0
\(939\) −2.99530 −0.0977481
\(940\) 0 0
\(941\) −34.2788 −1.11746 −0.558729 0.829350i \(-0.688711\pi\)
−0.558729 + 0.829350i \(0.688711\pi\)
\(942\) 0 0
\(943\) −10.4549 −0.340458
\(944\) 0 0
\(945\) 3.80451 0.123761
\(946\) 0 0
\(947\) 18.1573 0.590032 0.295016 0.955492i \(-0.404675\pi\)
0.295016 + 0.955492i \(0.404675\pi\)
\(948\) 0 0
\(949\) 15.9618 0.518141
\(950\) 0 0
\(951\) 9.93453 0.322149
\(952\) 0 0
\(953\) −19.8045 −0.641531 −0.320766 0.947159i \(-0.603940\pi\)
−0.320766 + 0.947159i \(0.603940\pi\)
\(954\) 0 0
\(955\) −20.0655 −0.649303
\(956\) 0 0
\(957\) −0.195488 −0.00631924
\(958\) 0 0
\(959\) −16.2227 −0.523860
\(960\) 0 0
\(961\) 67.8816 2.18973
\(962\) 0 0
\(963\) 8.13941 0.262289
\(964\) 0 0
\(965\) −75.4502 −2.42883
\(966\) 0 0
\(967\) 0.278820 0.00896624 0.00448312 0.999990i \(-0.498573\pi\)
0.00448312 + 0.999990i \(0.498573\pi\)
\(968\) 0 0
\(969\) −2.72588 −0.0875677
\(970\) 0 0
\(971\) −25.3481 −0.813458 −0.406729 0.913549i \(-0.633331\pi\)
−0.406729 + 0.913549i \(0.633331\pi\)
\(972\) 0 0
\(973\) −22.5482 −0.722864
\(974\) 0 0
\(975\) 36.0451 1.15437
\(976\) 0 0
\(977\) 6.71648 0.214879 0.107440 0.994212i \(-0.465735\pi\)
0.107440 + 0.994212i \(0.465735\pi\)
\(978\) 0 0
\(979\) 9.88784 0.316017
\(980\) 0 0
\(981\) 11.5529 0.368857
\(982\) 0 0
\(983\) −9.99061 −0.318651 −0.159325 0.987226i \(-0.550932\pi\)
−0.159325 + 0.987226i \(0.550932\pi\)
\(984\) 0 0
\(985\) 10.1573 0.323638
\(986\) 0 0
\(987\) 12.1394 0.386402
\(988\) 0 0
\(989\) −3.88784 −0.123626
\(990\) 0 0
\(991\) 26.7064 0.848358 0.424179 0.905578i \(-0.360563\pi\)
0.424179 + 0.905578i \(0.360563\pi\)
\(992\) 0 0
\(993\) −22.5482 −0.715547
\(994\) 0 0
\(995\) −51.5623 −1.63464
\(996\) 0 0
\(997\) 54.3333 1.72075 0.860377 0.509658i \(-0.170228\pi\)
0.860377 + 0.509658i \(0.170228\pi\)
\(998\) 0 0
\(999\) −4.47431 −0.141561
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 3696.2.a.bp.1.3 3
4.3 odd 2 231.2.a.d.1.2 3
12.11 even 2 693.2.a.m.1.2 3
20.19 odd 2 5775.2.a.bw.1.2 3
28.27 even 2 1617.2.a.s.1.2 3
44.43 even 2 2541.2.a.bi.1.2 3
84.83 odd 2 4851.2.a.bp.1.2 3
132.131 odd 2 7623.2.a.cb.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
231.2.a.d.1.2 3 4.3 odd 2
693.2.a.m.1.2 3 12.11 even 2
1617.2.a.s.1.2 3 28.27 even 2
2541.2.a.bi.1.2 3 44.43 even 2
3696.2.a.bp.1.3 3 1.1 even 1 trivial
4851.2.a.bp.1.2 3 84.83 odd 2
5775.2.a.bw.1.2 3 20.19 odd 2
7623.2.a.cb.1.2 3 132.131 odd 2