Properties

Label 9408.2.a.ei.1.3
Level $9408$
Weight $2$
Character 9408.1
Self dual yes
Analytic conductor $75.123$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 9408 = 2^{6} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9408.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(75.1232582216\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.621.1
Defining polynomial: \(x^{3} - 6 x - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 672)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-2.14510\) of defining polynomial
Character \(\chi\) \(=\) 9408.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} +2.74657 q^{5} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +2.74657 q^{5} +1.00000 q^{9} -1.54364 q^{11} +6.03677 q^{13} +2.74657 q^{15} -7.49314 q^{17} -6.03677 q^{19} -7.49314 q^{23} +2.54364 q^{25} +1.00000 q^{27} -1.25343 q^{29} -5.29021 q^{31} -1.54364 q^{33} -4.94950 q^{37} +6.03677 q^{39} +5.08727 q^{41} -3.45636 q^{43} +2.74657 q^{45} -9.49314 q^{47} -7.49314 q^{51} +3.83384 q^{53} -4.23970 q^{55} -6.03677 q^{57} +5.54364 q^{59} -14.5804 q^{61} +16.5804 q^{65} +4.03677 q^{67} -7.49314 q^{69} -5.49314 q^{71} -12.5436 q^{73} +2.54364 q^{75} -7.79707 q^{79} +1.00000 q^{81} -6.52991 q^{83} -20.5804 q^{85} -1.25343 q^{87} +9.49314 q^{89} -5.29021 q^{93} -16.5804 q^{95} -1.54364 q^{97} -1.54364 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{3} + 3q^{9} + O(q^{10}) \) \( 3q + 3q^{3} + 3q^{9} - 3q^{13} - 6q^{17} + 3q^{19} - 6q^{23} + 3q^{25} + 3q^{27} - 12q^{29} - 3q^{31} - 3q^{37} - 3q^{39} + 6q^{41} - 15q^{43} - 12q^{47} - 6q^{51} - 6q^{53} + 12q^{55} + 3q^{57} + 12q^{59} - 18q^{61} + 24q^{65} - 9q^{67} - 6q^{69} - 33q^{73} + 3q^{75} - 27q^{79} + 3q^{81} + 18q^{83} - 36q^{85} - 12q^{87} + 12q^{89} - 3q^{93} - 24q^{95} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) 2.74657 1.22830 0.614151 0.789188i \(-0.289498\pi\)
0.614151 + 0.789188i \(0.289498\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.54364 −0.465424 −0.232712 0.972546i \(-0.574760\pi\)
−0.232712 + 0.972546i \(0.574760\pi\)
\(12\) 0 0
\(13\) 6.03677 1.67430 0.837150 0.546974i \(-0.184220\pi\)
0.837150 + 0.546974i \(0.184220\pi\)
\(14\) 0 0
\(15\) 2.74657 0.709161
\(16\) 0 0
\(17\) −7.49314 −1.81735 −0.908676 0.417501i \(-0.862906\pi\)
−0.908676 + 0.417501i \(0.862906\pi\)
\(18\) 0 0
\(19\) −6.03677 −1.38493 −0.692465 0.721451i \(-0.743475\pi\)
−0.692465 + 0.721451i \(0.743475\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −7.49314 −1.56243 −0.781213 0.624264i \(-0.785399\pi\)
−0.781213 + 0.624264i \(0.785399\pi\)
\(24\) 0 0
\(25\) 2.54364 0.508727
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −1.25343 −0.232756 −0.116378 0.993205i \(-0.537128\pi\)
−0.116378 + 0.993205i \(0.537128\pi\)
\(30\) 0 0
\(31\) −5.29021 −0.950149 −0.475074 0.879946i \(-0.657579\pi\)
−0.475074 + 0.879946i \(0.657579\pi\)
\(32\) 0 0
\(33\) −1.54364 −0.268713
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −4.94950 −0.813693 −0.406846 0.913497i \(-0.633372\pi\)
−0.406846 + 0.913497i \(0.633372\pi\)
\(38\) 0 0
\(39\) 6.03677 0.966657
\(40\) 0 0
\(41\) 5.08727 0.794499 0.397249 0.917711i \(-0.369965\pi\)
0.397249 + 0.917711i \(0.369965\pi\)
\(42\) 0 0
\(43\) −3.45636 −0.527090 −0.263545 0.964647i \(-0.584892\pi\)
−0.263545 + 0.964647i \(0.584892\pi\)
\(44\) 0 0
\(45\) 2.74657 0.409434
\(46\) 0 0
\(47\) −9.49314 −1.38472 −0.692358 0.721554i \(-0.743428\pi\)
−0.692358 + 0.721554i \(0.743428\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −7.49314 −1.04925
\(52\) 0 0
\(53\) 3.83384 0.526619 0.263309 0.964711i \(-0.415186\pi\)
0.263309 + 0.964711i \(0.415186\pi\)
\(54\) 0 0
\(55\) −4.23970 −0.571682
\(56\) 0 0
\(57\) −6.03677 −0.799590
\(58\) 0 0
\(59\) 5.54364 0.721720 0.360860 0.932620i \(-0.382483\pi\)
0.360860 + 0.932620i \(0.382483\pi\)
\(60\) 0 0
\(61\) −14.5804 −1.86683 −0.933415 0.358798i \(-0.883187\pi\)
−0.933415 + 0.358798i \(0.883187\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 16.5804 2.05655
\(66\) 0 0
\(67\) 4.03677 0.493170 0.246585 0.969121i \(-0.420691\pi\)
0.246585 + 0.969121i \(0.420691\pi\)
\(68\) 0 0
\(69\) −7.49314 −0.902068
\(70\) 0 0
\(71\) −5.49314 −0.651915 −0.325958 0.945384i \(-0.605687\pi\)
−0.325958 + 0.945384i \(0.605687\pi\)
\(72\) 0 0
\(73\) −12.5436 −1.46812 −0.734061 0.679084i \(-0.762377\pi\)
−0.734061 + 0.679084i \(0.762377\pi\)
\(74\) 0 0
\(75\) 2.54364 0.293714
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −7.79707 −0.877239 −0.438619 0.898673i \(-0.644532\pi\)
−0.438619 + 0.898673i \(0.644532\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −6.52991 −0.716751 −0.358375 0.933578i \(-0.616669\pi\)
−0.358375 + 0.933578i \(0.616669\pi\)
\(84\) 0 0
\(85\) −20.5804 −2.23226
\(86\) 0 0
\(87\) −1.25343 −0.134382
\(88\) 0 0
\(89\) 9.49314 1.00627 0.503135 0.864208i \(-0.332180\pi\)
0.503135 + 0.864208i \(0.332180\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −5.29021 −0.548569
\(94\) 0 0
\(95\) −16.5804 −1.70111
\(96\) 0 0
\(97\) −1.54364 −0.156733 −0.0783663 0.996925i \(-0.524970\pi\)
−0.0783663 + 0.996925i \(0.524970\pi\)
\(98\) 0 0
\(99\) −1.54364 −0.155141
\(100\) 0 0
\(101\) −2.58041 −0.256760 −0.128380 0.991725i \(-0.540978\pi\)
−0.128380 + 0.991725i \(0.540978\pi\)
\(102\) 0 0
\(103\) −6.54364 −0.644764 −0.322382 0.946610i \(-0.604484\pi\)
−0.322382 + 0.946610i \(0.604484\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 4.52991 0.437923 0.218961 0.975734i \(-0.429733\pi\)
0.218961 + 0.975734i \(0.429733\pi\)
\(108\) 0 0
\(109\) 6.44264 0.617093 0.308546 0.951209i \(-0.400158\pi\)
0.308546 + 0.951209i \(0.400158\pi\)
\(110\) 0 0
\(111\) −4.94950 −0.469786
\(112\) 0 0
\(113\) 8.00000 0.752577 0.376288 0.926503i \(-0.377200\pi\)
0.376288 + 0.926503i \(0.377200\pi\)
\(114\) 0 0
\(115\) −20.5804 −1.91913
\(116\) 0 0
\(117\) 6.03677 0.558100
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −8.61718 −0.783380
\(122\) 0 0
\(123\) 5.08727 0.458704
\(124\) 0 0
\(125\) −6.74657 −0.603431
\(126\) 0 0
\(127\) 3.79707 0.336935 0.168468 0.985707i \(-0.446118\pi\)
0.168468 + 0.985707i \(0.446118\pi\)
\(128\) 0 0
\(129\) −3.45636 −0.304316
\(130\) 0 0
\(131\) 1.54364 0.134868 0.0674341 0.997724i \(-0.478519\pi\)
0.0674341 + 0.997724i \(0.478519\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 2.74657 0.236387
\(136\) 0 0
\(137\) 1.49314 0.127567 0.0637836 0.997964i \(-0.479683\pi\)
0.0637836 + 0.997964i \(0.479683\pi\)
\(138\) 0 0
\(139\) −2.94950 −0.250173 −0.125087 0.992146i \(-0.539921\pi\)
−0.125087 + 0.992146i \(0.539921\pi\)
\(140\) 0 0
\(141\) −9.49314 −0.799466
\(142\) 0 0
\(143\) −9.31859 −0.779259
\(144\) 0 0
\(145\) −3.44264 −0.285895
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 1.49314 0.122323 0.0611613 0.998128i \(-0.480520\pi\)
0.0611613 + 0.998128i \(0.480520\pi\)
\(150\) 0 0
\(151\) 9.73284 0.792047 0.396024 0.918240i \(-0.370390\pi\)
0.396024 + 0.918240i \(0.370390\pi\)
\(152\) 0 0
\(153\) −7.49314 −0.605784
\(154\) 0 0
\(155\) −14.5299 −1.16707
\(156\) 0 0
\(157\) 9.08727 0.725243 0.362622 0.931936i \(-0.381882\pi\)
0.362622 + 0.931936i \(0.381882\pi\)
\(158\) 0 0
\(159\) 3.83384 0.304043
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −19.5667 −1.53258 −0.766290 0.642494i \(-0.777900\pi\)
−0.766290 + 0.642494i \(0.777900\pi\)
\(164\) 0 0
\(165\) −4.23970 −0.330061
\(166\) 0 0
\(167\) 11.8990 0.920772 0.460386 0.887719i \(-0.347711\pi\)
0.460386 + 0.887719i \(0.347711\pi\)
\(168\) 0 0
\(169\) 23.4426 1.80328
\(170\) 0 0
\(171\) −6.03677 −0.461644
\(172\) 0 0
\(173\) 2.58041 0.196185 0.0980925 0.995177i \(-0.468726\pi\)
0.0980925 + 0.995177i \(0.468726\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 5.54364 0.416685
\(178\) 0 0
\(179\) 2.98627 0.223205 0.111602 0.993753i \(-0.464402\pi\)
0.111602 + 0.993753i \(0.464402\pi\)
\(180\) 0 0
\(181\) −10.5436 −0.783702 −0.391851 0.920029i \(-0.628165\pi\)
−0.391851 + 0.920029i \(0.628165\pi\)
\(182\) 0 0
\(183\) −14.5804 −1.07781
\(184\) 0 0
\(185\) −13.5941 −0.999461
\(186\) 0 0
\(187\) 11.5667 0.845840
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 17.1608 1.24171 0.620857 0.783924i \(-0.286785\pi\)
0.620857 + 0.783924i \(0.286785\pi\)
\(192\) 0 0
\(193\) 10.8990 0.784527 0.392264 0.919853i \(-0.371692\pi\)
0.392264 + 0.919853i \(0.371692\pi\)
\(194\) 0 0
\(195\) 16.5804 1.18735
\(196\) 0 0
\(197\) −5.41959 −0.386130 −0.193065 0.981186i \(-0.561843\pi\)
−0.193065 + 0.981186i \(0.561843\pi\)
\(198\) 0 0
\(199\) 22.9863 1.62945 0.814727 0.579845i \(-0.196887\pi\)
0.814727 + 0.579845i \(0.196887\pi\)
\(200\) 0 0
\(201\) 4.03677 0.284732
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 13.9725 0.975885
\(206\) 0 0
\(207\) −7.49314 −0.520809
\(208\) 0 0
\(209\) 9.31859 0.644580
\(210\) 0 0
\(211\) −3.08727 −0.212537 −0.106268 0.994337i \(-0.533890\pi\)
−0.106268 + 0.994337i \(0.533890\pi\)
\(212\) 0 0
\(213\) −5.49314 −0.376384
\(214\) 0 0
\(215\) −9.49314 −0.647427
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −12.5436 −0.847620
\(220\) 0 0
\(221\) −45.2344 −3.04279
\(222\) 0 0
\(223\) 22.7466 1.52322 0.761611 0.648034i \(-0.224409\pi\)
0.761611 + 0.648034i \(0.224409\pi\)
\(224\) 0 0
\(225\) 2.54364 0.169576
\(226\) 0 0
\(227\) 22.6035 1.50024 0.750122 0.661299i \(-0.229995\pi\)
0.750122 + 0.661299i \(0.229995\pi\)
\(228\) 0 0
\(229\) −11.3554 −0.750383 −0.375192 0.926947i \(-0.622423\pi\)
−0.375192 + 0.926947i \(0.622423\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 3.41959 0.224025 0.112012 0.993707i \(-0.464270\pi\)
0.112012 + 0.993707i \(0.464270\pi\)
\(234\) 0 0
\(235\) −26.0735 −1.70085
\(236\) 0 0
\(237\) −7.79707 −0.506474
\(238\) 0 0
\(239\) −7.49314 −0.484691 −0.242345 0.970190i \(-0.577917\pi\)
−0.242345 + 0.970190i \(0.577917\pi\)
\(240\) 0 0
\(241\) 18.6035 1.19835 0.599177 0.800617i \(-0.295495\pi\)
0.599177 + 0.800617i \(0.295495\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −36.4426 −2.31879
\(248\) 0 0
\(249\) −6.52991 −0.413816
\(250\) 0 0
\(251\) 24.5299 1.54831 0.774157 0.632994i \(-0.218174\pi\)
0.774157 + 0.632994i \(0.218174\pi\)
\(252\) 0 0
\(253\) 11.5667 0.727191
\(254\) 0 0
\(255\) −20.5804 −1.28880
\(256\) 0 0
\(257\) 17.4931 1.09119 0.545596 0.838048i \(-0.316303\pi\)
0.545596 + 0.838048i \(0.316303\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −1.25343 −0.0775855
\(262\) 0 0
\(263\) −13.4931 −0.832022 −0.416011 0.909359i \(-0.636572\pi\)
−0.416011 + 0.909359i \(0.636572\pi\)
\(264\) 0 0
\(265\) 10.5299 0.646847
\(266\) 0 0
\(267\) 9.49314 0.580971
\(268\) 0 0
\(269\) 6.34071 0.386600 0.193300 0.981140i \(-0.438081\pi\)
0.193300 + 0.981140i \(0.438081\pi\)
\(270\) 0 0
\(271\) 20.2397 1.22947 0.614737 0.788732i \(-0.289262\pi\)
0.614737 + 0.788732i \(0.289262\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −3.92645 −0.236774
\(276\) 0 0
\(277\) 2.54364 0.152832 0.0764162 0.997076i \(-0.475652\pi\)
0.0764162 + 0.997076i \(0.475652\pi\)
\(278\) 0 0
\(279\) −5.29021 −0.316716
\(280\) 0 0
\(281\) 26.5804 1.58565 0.792827 0.609447i \(-0.208608\pi\)
0.792827 + 0.609447i \(0.208608\pi\)
\(282\) 0 0
\(283\) −20.5436 −1.22119 −0.610596 0.791942i \(-0.709070\pi\)
−0.610596 + 0.791942i \(0.709070\pi\)
\(284\) 0 0
\(285\) −16.5804 −0.982139
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 39.1471 2.30277
\(290\) 0 0
\(291\) −1.54364 −0.0904896
\(292\) 0 0
\(293\) 5.25343 0.306909 0.153454 0.988156i \(-0.450960\pi\)
0.153454 + 0.988156i \(0.450960\pi\)
\(294\) 0 0
\(295\) 15.2260 0.886491
\(296\) 0 0
\(297\) −1.54364 −0.0895709
\(298\) 0 0
\(299\) −45.2344 −2.61597
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −2.58041 −0.148241
\(304\) 0 0
\(305\) −40.0461 −2.29303
\(306\) 0 0
\(307\) 14.8485 0.847449 0.423724 0.905791i \(-0.360723\pi\)
0.423724 + 0.905791i \(0.360723\pi\)
\(308\) 0 0
\(309\) −6.54364 −0.372255
\(310\) 0 0
\(311\) 1.89900 0.107682 0.0538412 0.998550i \(-0.482854\pi\)
0.0538412 + 0.998550i \(0.482854\pi\)
\(312\) 0 0
\(313\) 0.0872743 0.00493303 0.00246652 0.999997i \(-0.499215\pi\)
0.00246652 + 0.999997i \(0.499215\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −8.23970 −0.462788 −0.231394 0.972860i \(-0.574329\pi\)
−0.231394 + 0.972860i \(0.574329\pi\)
\(318\) 0 0
\(319\) 1.93484 0.108330
\(320\) 0 0
\(321\) 4.52991 0.252835
\(322\) 0 0
\(323\) 45.2344 2.51691
\(324\) 0 0
\(325\) 15.3554 0.851762
\(326\) 0 0
\(327\) 6.44264 0.356279
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 5.52991 0.303951 0.151976 0.988384i \(-0.451436\pi\)
0.151976 + 0.988384i \(0.451436\pi\)
\(332\) 0 0
\(333\) −4.94950 −0.271231
\(334\) 0 0
\(335\) 11.0873 0.605763
\(336\) 0 0
\(337\) 7.17455 0.390823 0.195411 0.980721i \(-0.437396\pi\)
0.195411 + 0.980721i \(0.437396\pi\)
\(338\) 0 0
\(339\) 8.00000 0.434500
\(340\) 0 0
\(341\) 8.16616 0.442222
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −20.5804 −1.10801
\(346\) 0 0
\(347\) −24.0735 −1.29234 −0.646168 0.763195i \(-0.723629\pi\)
−0.646168 + 0.763195i \(0.723629\pi\)
\(348\) 0 0
\(349\) −31.1608 −1.66800 −0.834000 0.551764i \(-0.813955\pi\)
−0.834000 + 0.551764i \(0.813955\pi\)
\(350\) 0 0
\(351\) 6.03677 0.322219
\(352\) 0 0
\(353\) 10.1745 0.541537 0.270768 0.962645i \(-0.412722\pi\)
0.270768 + 0.962645i \(0.412722\pi\)
\(354\) 0 0
\(355\) −15.0873 −0.800749
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −32.2481 −1.70199 −0.850995 0.525174i \(-0.824000\pi\)
−0.850995 + 0.525174i \(0.824000\pi\)
\(360\) 0 0
\(361\) 17.4426 0.918033
\(362\) 0 0
\(363\) −8.61718 −0.452285
\(364\) 0 0
\(365\) −34.4520 −1.80330
\(366\) 0 0
\(367\) −3.87062 −0.202045 −0.101022 0.994884i \(-0.532211\pi\)
−0.101022 + 0.994884i \(0.532211\pi\)
\(368\) 0 0
\(369\) 5.08727 0.264833
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −34.6907 −1.79622 −0.898109 0.439773i \(-0.855059\pi\)
−0.898109 + 0.439773i \(0.855059\pi\)
\(374\) 0 0
\(375\) −6.74657 −0.348391
\(376\) 0 0
\(377\) −7.56668 −0.389704
\(378\) 0 0
\(379\) −13.4564 −0.691207 −0.345603 0.938381i \(-0.612326\pi\)
−0.345603 + 0.938381i \(0.612326\pi\)
\(380\) 0 0
\(381\) 3.79707 0.194530
\(382\) 0 0
\(383\) −4.17455 −0.213309 −0.106655 0.994296i \(-0.534014\pi\)
−0.106655 + 0.994296i \(0.534014\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −3.45636 −0.175697
\(388\) 0 0
\(389\) 18.6540 0.945793 0.472897 0.881118i \(-0.343208\pi\)
0.472897 + 0.881118i \(0.343208\pi\)
\(390\) 0 0
\(391\) 56.1471 2.83948
\(392\) 0 0
\(393\) 1.54364 0.0778662
\(394\) 0 0
\(395\) −21.4152 −1.07751
\(396\) 0 0
\(397\) −1.45636 −0.0730928 −0.0365464 0.999332i \(-0.511636\pi\)
−0.0365464 + 0.999332i \(0.511636\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −29.2344 −1.45989 −0.729947 0.683503i \(-0.760455\pi\)
−0.729947 + 0.683503i \(0.760455\pi\)
\(402\) 0 0
\(403\) −31.9358 −1.59083
\(404\) 0 0
\(405\) 2.74657 0.136478
\(406\) 0 0
\(407\) 7.64023 0.378712
\(408\) 0 0
\(409\) −26.0598 −1.28858 −0.644288 0.764783i \(-0.722846\pi\)
−0.644288 + 0.764783i \(0.722846\pi\)
\(410\) 0 0
\(411\) 1.49314 0.0736510
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −17.9348 −0.880387
\(416\) 0 0
\(417\) −2.94950 −0.144438
\(418\) 0 0
\(419\) 32.8853 1.60655 0.803275 0.595608i \(-0.203089\pi\)
0.803275 + 0.595608i \(0.203089\pi\)
\(420\) 0 0
\(421\) 17.1976 0.838160 0.419080 0.907949i \(-0.362353\pi\)
0.419080 + 0.907949i \(0.362353\pi\)
\(422\) 0 0
\(423\) −9.49314 −0.461572
\(424\) 0 0
\(425\) −19.0598 −0.924537
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −9.31859 −0.449906
\(430\) 0 0
\(431\) 2.98627 0.143844 0.0719219 0.997410i \(-0.477087\pi\)
0.0719219 + 0.997410i \(0.477087\pi\)
\(432\) 0 0
\(433\) −18.4426 −0.886297 −0.443148 0.896448i \(-0.646138\pi\)
−0.443148 + 0.896448i \(0.646138\pi\)
\(434\) 0 0
\(435\) −3.44264 −0.165062
\(436\) 0 0
\(437\) 45.2344 2.16385
\(438\) 0 0
\(439\) −0.572020 −0.0273010 −0.0136505 0.999907i \(-0.504345\pi\)
−0.0136505 + 0.999907i \(0.504345\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −36.5299 −1.73559 −0.867794 0.496924i \(-0.834463\pi\)
−0.867794 + 0.496924i \(0.834463\pi\)
\(444\) 0 0
\(445\) 26.0735 1.23600
\(446\) 0 0
\(447\) 1.49314 0.0706229
\(448\) 0 0
\(449\) −22.5530 −1.06434 −0.532170 0.846638i \(-0.678623\pi\)
−0.532170 + 0.846638i \(0.678623\pi\)
\(450\) 0 0
\(451\) −7.85291 −0.369779
\(452\) 0 0
\(453\) 9.73284 0.457289
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 20.2344 0.946524 0.473262 0.880922i \(-0.343076\pi\)
0.473262 + 0.880922i \(0.343076\pi\)
\(458\) 0 0
\(459\) −7.49314 −0.349750
\(460\) 0 0
\(461\) 25.4931 1.18733 0.593667 0.804711i \(-0.297680\pi\)
0.593667 + 0.804711i \(0.297680\pi\)
\(462\) 0 0
\(463\) −7.70446 −0.358057 −0.179028 0.983844i \(-0.557295\pi\)
−0.179028 + 0.983844i \(0.557295\pi\)
\(464\) 0 0
\(465\) −14.5299 −0.673808
\(466\) 0 0
\(467\) 36.1471 1.67269 0.836344 0.548205i \(-0.184689\pi\)
0.836344 + 0.548205i \(0.184689\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 9.08727 0.418719
\(472\) 0 0
\(473\) 5.33537 0.245321
\(474\) 0 0
\(475\) −15.3554 −0.704552
\(476\) 0 0
\(477\) 3.83384 0.175540
\(478\) 0 0
\(479\) −26.5804 −1.21449 −0.607245 0.794515i \(-0.707725\pi\)
−0.607245 + 0.794515i \(0.707725\pi\)
\(480\) 0 0
\(481\) −29.8790 −1.36237
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −4.23970 −0.192515
\(486\) 0 0
\(487\) −40.4236 −1.83177 −0.915883 0.401444i \(-0.868508\pi\)
−0.915883 + 0.401444i \(0.868508\pi\)
\(488\) 0 0
\(489\) −19.5667 −0.884836
\(490\) 0 0
\(491\) −14.6309 −0.660284 −0.330142 0.943931i \(-0.607097\pi\)
−0.330142 + 0.943931i \(0.607097\pi\)
\(492\) 0 0
\(493\) 9.39214 0.423000
\(494\) 0 0
\(495\) −4.23970 −0.190561
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 36.1103 1.61652 0.808260 0.588826i \(-0.200410\pi\)
0.808260 + 0.588826i \(0.200410\pi\)
\(500\) 0 0
\(501\) 11.8990 0.531608
\(502\) 0 0
\(503\) −19.4931 −0.869156 −0.434578 0.900634i \(-0.643102\pi\)
−0.434578 + 0.900634i \(0.643102\pi\)
\(504\) 0 0
\(505\) −7.08727 −0.315380
\(506\) 0 0
\(507\) 23.4426 1.04112
\(508\) 0 0
\(509\) −23.9074 −1.05968 −0.529838 0.848099i \(-0.677747\pi\)
−0.529838 + 0.848099i \(0.677747\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −6.03677 −0.266530
\(514\) 0 0
\(515\) −17.9725 −0.791965
\(516\) 0 0
\(517\) 14.6540 0.644480
\(518\) 0 0
\(519\) 2.58041 0.113267
\(520\) 0 0
\(521\) 7.08727 0.310499 0.155250 0.987875i \(-0.450382\pi\)
0.155250 + 0.987875i \(0.450382\pi\)
\(522\) 0 0
\(523\) −28.7475 −1.25704 −0.628520 0.777793i \(-0.716339\pi\)
−0.628520 + 0.777793i \(0.716339\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 39.6402 1.72676
\(528\) 0 0
\(529\) 33.1471 1.44118
\(530\) 0 0
\(531\) 5.54364 0.240573
\(532\) 0 0
\(533\) 30.7107 1.33023
\(534\) 0 0
\(535\) 12.4417 0.537902
\(536\) 0 0
\(537\) 2.98627 0.128867
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 7.02305 0.301944 0.150972 0.988538i \(-0.451760\pi\)
0.150972 + 0.988538i \(0.451760\pi\)
\(542\) 0 0
\(543\) −10.5436 −0.452471
\(544\) 0 0
\(545\) 17.6951 0.757976
\(546\) 0 0
\(547\) −5.59414 −0.239188 −0.119594 0.992823i \(-0.538159\pi\)
−0.119594 + 0.992823i \(0.538159\pi\)
\(548\) 0 0
\(549\) −14.5804 −0.622277
\(550\) 0 0
\(551\) 7.56668 0.322352
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −13.5941 −0.577039
\(556\) 0 0
\(557\) −28.6456 −1.21375 −0.606876 0.794797i \(-0.707577\pi\)
−0.606876 + 0.794797i \(0.707577\pi\)
\(558\) 0 0
\(559\) −20.8653 −0.882507
\(560\) 0 0
\(561\) 11.5667 0.488346
\(562\) 0 0
\(563\) −19.8927 −0.838379 −0.419189 0.907899i \(-0.637686\pi\)
−0.419189 + 0.907899i \(0.637686\pi\)
\(564\) 0 0
\(565\) 21.9725 0.924392
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 35.6402 1.49412 0.747058 0.664759i \(-0.231466\pi\)
0.747058 + 0.664759i \(0.231466\pi\)
\(570\) 0 0
\(571\) 27.6770 1.15825 0.579123 0.815240i \(-0.303395\pi\)
0.579123 + 0.815240i \(0.303395\pi\)
\(572\) 0 0
\(573\) 17.1608 0.716904
\(574\) 0 0
\(575\) −19.0598 −0.794849
\(576\) 0 0
\(577\) 6.01373 0.250355 0.125177 0.992134i \(-0.460050\pi\)
0.125177 + 0.992134i \(0.460050\pi\)
\(578\) 0 0
\(579\) 10.8990 0.452947
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −5.91806 −0.245101
\(584\) 0 0
\(585\) 16.5804 0.685516
\(586\) 0 0
\(587\) 15.5436 0.641555 0.320777 0.947155i \(-0.396056\pi\)
0.320777 + 0.947155i \(0.396056\pi\)
\(588\) 0 0
\(589\) 31.9358 1.31589
\(590\) 0 0
\(591\) −5.41959 −0.222932
\(592\) 0 0
\(593\) 48.6540 1.99798 0.998989 0.0449488i \(-0.0143125\pi\)
0.998989 + 0.0449488i \(0.0143125\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 22.9863 0.940766
\(598\) 0 0
\(599\) −20.9127 −0.854471 −0.427235 0.904140i \(-0.640512\pi\)
−0.427235 + 0.904140i \(0.640512\pi\)
\(600\) 0 0
\(601\) 7.98627 0.325767 0.162883 0.986645i \(-0.447921\pi\)
0.162883 + 0.986645i \(0.447921\pi\)
\(602\) 0 0
\(603\) 4.03677 0.164390
\(604\) 0 0
\(605\) −23.6677 −0.962228
\(606\) 0 0
\(607\) −2.70979 −0.109987 −0.0549936 0.998487i \(-0.517514\pi\)
−0.0549936 + 0.998487i \(0.517514\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −57.3079 −2.31843
\(612\) 0 0
\(613\) 16.5069 0.666706 0.333353 0.942802i \(-0.391820\pi\)
0.333353 + 0.942802i \(0.391820\pi\)
\(614\) 0 0
\(615\) 13.9725 0.563427
\(616\) 0 0
\(617\) −23.1608 −0.932420 −0.466210 0.884674i \(-0.654381\pi\)
−0.466210 + 0.884674i \(0.654381\pi\)
\(618\) 0 0
\(619\) −29.4289 −1.18285 −0.591424 0.806361i \(-0.701434\pi\)
−0.591424 + 0.806361i \(0.701434\pi\)
\(620\) 0 0
\(621\) −7.49314 −0.300689
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −31.2481 −1.24992
\(626\) 0 0
\(627\) 9.31859 0.372149
\(628\) 0 0
\(629\) 37.0873 1.47877
\(630\) 0 0
\(631\) −35.7054 −1.42141 −0.710705 0.703491i \(-0.751624\pi\)
−0.710705 + 0.703491i \(0.751624\pi\)
\(632\) 0 0
\(633\) −3.08727 −0.122708
\(634\) 0 0
\(635\) 10.4289 0.413859
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −5.49314 −0.217305
\(640\) 0 0
\(641\) −3.36282 −0.132824 −0.0664118 0.997792i \(-0.521155\pi\)
−0.0664118 + 0.997792i \(0.521155\pi\)
\(642\) 0 0
\(643\) 10.2113 0.402695 0.201348 0.979520i \(-0.435468\pi\)
0.201348 + 0.979520i \(0.435468\pi\)
\(644\) 0 0
\(645\) −9.49314 −0.373792
\(646\) 0 0
\(647\) 20.5530 0.808020 0.404010 0.914755i \(-0.367616\pi\)
0.404010 + 0.914755i \(0.367616\pi\)
\(648\) 0 0
\(649\) −8.55736 −0.335906
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 11.3544 0.444333 0.222167 0.975009i \(-0.428687\pi\)
0.222167 + 0.975009i \(0.428687\pi\)
\(654\) 0 0
\(655\) 4.23970 0.165659
\(656\) 0 0
\(657\) −12.5436 −0.489374
\(658\) 0 0
\(659\) −21.2344 −0.827174 −0.413587 0.910465i \(-0.635724\pi\)
−0.413587 + 0.910465i \(0.635724\pi\)
\(660\) 0 0
\(661\) −28.9220 −1.12494 −0.562469 0.826819i \(-0.690148\pi\)
−0.562469 + 0.826819i \(0.690148\pi\)
\(662\) 0 0
\(663\) −45.2344 −1.75676
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 9.39214 0.363665
\(668\) 0 0
\(669\) 22.7466 0.879433
\(670\) 0 0
\(671\) 22.5069 0.868868
\(672\) 0 0
\(673\) −34.2618 −1.32070 −0.660348 0.750960i \(-0.729591\pi\)
−0.660348 + 0.750960i \(0.729591\pi\)
\(674\) 0 0
\(675\) 2.54364 0.0979046
\(676\) 0 0
\(677\) 34.7466 1.33542 0.667710 0.744422i \(-0.267275\pi\)
0.667710 + 0.744422i \(0.267275\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 22.6035 0.866166
\(682\) 0 0
\(683\) −0.281814 −0.0107833 −0.00539166 0.999985i \(-0.501716\pi\)
−0.00539166 + 0.999985i \(0.501716\pi\)
\(684\) 0 0
\(685\) 4.10100 0.156691
\(686\) 0 0
\(687\) −11.3554 −0.433234
\(688\) 0 0
\(689\) 23.1440 0.881718
\(690\) 0 0
\(691\) 34.4152 1.30922 0.654608 0.755969i \(-0.272834\pi\)
0.654608 + 0.755969i \(0.272834\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −8.10100 −0.307288
\(696\) 0 0
\(697\) −38.1196 −1.44388
\(698\) 0 0
\(699\) 3.41959 0.129341
\(700\) 0 0
\(701\) 0.498472 0.0188270 0.00941352 0.999956i \(-0.497004\pi\)
0.00941352 + 0.999956i \(0.497004\pi\)
\(702\) 0 0
\(703\) 29.8790 1.12691
\(704\) 0 0
\(705\) −26.0735 −0.981987
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 4.17455 0.156778 0.0783892 0.996923i \(-0.475022\pi\)
0.0783892 + 0.996923i \(0.475022\pi\)
\(710\) 0 0
\(711\) −7.79707 −0.292413
\(712\) 0 0
\(713\) 39.6402 1.48454
\(714\) 0 0
\(715\) −25.5941 −0.957166
\(716\) 0 0
\(717\) −7.49314 −0.279836
\(718\) 0 0
\(719\) 0.986273 0.0367818 0.0183909 0.999831i \(-0.494146\pi\)
0.0183909 + 0.999831i \(0.494146\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 18.6035 0.691870
\(724\) 0 0
\(725\) −3.18828 −0.118410
\(726\) 0 0
\(727\) 34.9304 1.29550 0.647749 0.761854i \(-0.275711\pi\)
0.647749 + 0.761854i \(0.275711\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 25.8990 0.957909
\(732\) 0 0
\(733\) 5.12405 0.189261 0.0946305 0.995512i \(-0.469833\pi\)
0.0946305 + 0.995512i \(0.469833\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −6.23131 −0.229533
\(738\) 0 0
\(739\) 16.9220 0.622488 0.311244 0.950330i \(-0.399254\pi\)
0.311244 + 0.950330i \(0.399254\pi\)
\(740\) 0 0
\(741\) −36.4426 −1.33875
\(742\) 0 0
\(743\) 6.81172 0.249898 0.124949 0.992163i \(-0.460123\pi\)
0.124949 + 0.992163i \(0.460123\pi\)
\(744\) 0 0
\(745\) 4.10100 0.150249
\(746\) 0 0
\(747\) −6.52991 −0.238917
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −26.1755 −0.955157 −0.477578 0.878589i \(-0.658485\pi\)
−0.477578 + 0.878589i \(0.658485\pi\)
\(752\) 0 0
\(753\) 24.5299 0.893920
\(754\) 0 0
\(755\) 26.7319 0.972874
\(756\) 0 0
\(757\) −49.7138 −1.80688 −0.903439 0.428717i \(-0.858966\pi\)
−0.903439 + 0.428717i \(0.858966\pi\)
\(758\) 0 0
\(759\) 11.5667 0.419844
\(760\) 0 0
\(761\) 12.6540 0.458706 0.229353 0.973343i \(-0.426339\pi\)
0.229353 + 0.973343i \(0.426339\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −20.5804 −0.744086
\(766\) 0 0
\(767\) 33.4657 1.20838
\(768\) 0 0
\(769\) 20.2344 0.729670 0.364835 0.931072i \(-0.381125\pi\)
0.364835 + 0.931072i \(0.381125\pi\)
\(770\) 0 0
\(771\) 17.4931 0.630000
\(772\) 0 0
\(773\) −19.7412 −0.710043 −0.355021 0.934858i \(-0.615526\pi\)
−0.355021 + 0.934858i \(0.615526\pi\)
\(774\) 0 0
\(775\) −13.4564 −0.483367
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −30.7107 −1.10033
\(780\) 0 0
\(781\) 8.47941 0.303417
\(782\) 0 0
\(783\) −1.25343 −0.0447940
\(784\) 0 0
\(785\) 24.9588 0.890818
\(786\) 0 0
\(787\) −16.9127 −0.602874 −0.301437 0.953486i \(-0.597466\pi\)
−0.301437 + 0.953486i \(0.597466\pi\)
\(788\) 0 0
\(789\) −13.4931 −0.480368
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −88.0186 −3.12563
\(794\) 0 0
\(795\) 10.5299 0.373457
\(796\) 0 0
\(797\) 3.48475 0.123436 0.0617180 0.998094i \(-0.480342\pi\)
0.0617180 + 0.998094i \(0.480342\pi\)
\(798\) 0 0
\(799\) 71.1334 2.51652
\(800\) 0 0
\(801\) 9.49314 0.335423
\(802\) 0 0
\(803\) 19.3628 0.683299
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 6.34071 0.223203
\(808\) 0 0
\(809\) −19.1883 −0.674624 −0.337312 0.941393i \(-0.609518\pi\)
−0.337312 + 0.941393i \(0.609518\pi\)
\(810\) 0 0
\(811\) −48.8285 −1.71460 −0.857300 0.514817i \(-0.827860\pi\)
−0.857300 + 0.514817i \(0.827860\pi\)
\(812\) 0 0
\(813\) 20.2397 0.709837
\(814\) 0 0
\(815\) −53.7412 −1.88247
\(816\) 0 0
\(817\) 20.8653 0.729984
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 26.5446 0.926412 0.463206 0.886251i \(-0.346699\pi\)
0.463206 + 0.886251i \(0.346699\pi\)
\(822\) 0 0
\(823\) 16.9588 0.591147 0.295574 0.955320i \(-0.404489\pi\)
0.295574 + 0.955320i \(0.404489\pi\)
\(824\) 0 0
\(825\) −3.92645 −0.136702
\(826\) 0 0
\(827\) −42.4289 −1.47540 −0.737699 0.675130i \(-0.764088\pi\)
−0.737699 + 0.675130i \(0.764088\pi\)
\(828\) 0 0
\(829\) −31.9358 −1.10918 −0.554588 0.832125i \(-0.687124\pi\)
−0.554588 + 0.832125i \(0.687124\pi\)
\(830\) 0 0
\(831\) 2.54364 0.0882378
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 32.6814 1.13099
\(836\) 0 0
\(837\) −5.29021 −0.182856
\(838\) 0 0
\(839\) −49.5392 −1.71028 −0.855142 0.518394i \(-0.826530\pi\)
−0.855142 + 0.518394i \(0.826530\pi\)
\(840\) 0 0
\(841\) −27.4289 −0.945824
\(842\) 0 0
\(843\) 26.5804 0.915478
\(844\) 0 0
\(845\) 64.3868 2.21497
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −20.5436 −0.705056
\(850\) 0 0
\(851\) 37.0873 1.27134
\(852\) 0 0
\(853\) 11.4564 0.392258 0.196129 0.980578i \(-0.437163\pi\)
0.196129 + 0.980578i \(0.437163\pi\)
\(854\) 0 0
\(855\) −16.5804 −0.567038
\(856\) 0 0
\(857\) 32.2481 1.10157 0.550787 0.834646i \(-0.314328\pi\)
0.550787 + 0.834646i \(0.314328\pi\)
\(858\) 0 0
\(859\) −3.08727 −0.105336 −0.0526682 0.998612i \(-0.516773\pi\)
−0.0526682 + 0.998612i \(0.516773\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −18.1471 −0.617734 −0.308867 0.951105i \(-0.599950\pi\)
−0.308867 + 0.951105i \(0.599950\pi\)
\(864\) 0 0
\(865\) 7.08727 0.240975
\(866\) 0 0
\(867\) 39.1471 1.32951
\(868\) 0 0
\(869\) 12.0358 0.408288
\(870\) 0 0
\(871\) 24.3691 0.825715
\(872\) 0 0
\(873\) −1.54364 −0.0522442
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −8.88527 −0.300034 −0.150017 0.988683i \(-0.547933\pi\)
−0.150017 + 0.988683i \(0.547933\pi\)
\(878\) 0 0
\(879\) 5.25343 0.177194
\(880\) 0 0
\(881\) 41.5667 1.40042 0.700209 0.713938i \(-0.253090\pi\)
0.700209 + 0.713938i \(0.253090\pi\)
\(882\) 0 0
\(883\) 28.7182 0.966444 0.483222 0.875498i \(-0.339466\pi\)
0.483222 + 0.875498i \(0.339466\pi\)
\(884\) 0 0
\(885\) 15.2260 0.511816
\(886\) 0 0
\(887\) 41.8148 1.40400 0.702001 0.712176i \(-0.252290\pi\)
0.702001 + 0.712176i \(0.252290\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −1.54364 −0.0517138
\(892\) 0 0
\(893\) 57.3079 1.91774
\(894\) 0 0
\(895\) 8.20200 0.274163
\(896\) 0 0
\(897\) −45.2344 −1.51033
\(898\) 0 0
\(899\) 6.63091 0.221153
\(900\) 0 0
\(901\) −28.7275 −0.957052
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −28.9588 −0.962624
\(906\) 0 0
\(907\) −57.1701 −1.89830 −0.949152 0.314819i \(-0.898056\pi\)
−0.949152 + 0.314819i \(0.898056\pi\)
\(908\) 0 0
\(909\) −2.58041 −0.0855868
\(910\) 0 0
\(911\) 24.4059 0.808602 0.404301 0.914626i \(-0.367515\pi\)
0.404301 + 0.914626i \(0.367515\pi\)
\(912\) 0 0
\(913\) 10.0798 0.333593
\(914\) 0 0
\(915\) −40.0461 −1.32388
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −47.7045 −1.57362 −0.786812 0.617192i \(-0.788270\pi\)
−0.786812 + 0.617192i \(0.788270\pi\)
\(920\) 0 0
\(921\) 14.8485 0.489275
\(922\) 0 0
\(923\) −33.1608 −1.09150
\(924\) 0 0
\(925\) −12.5897 −0.413948
\(926\) 0 0
\(927\) −6.54364 −0.214921
\(928\) 0 0
\(929\) −39.2344 −1.28724 −0.643619 0.765346i \(-0.722568\pi\)
−0.643619 + 0.765346i \(0.722568\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 1.89900 0.0621704
\(934\) 0 0
\(935\) 31.7687 1.03895
\(936\) 0 0
\(937\) −15.3491 −0.501433 −0.250717 0.968061i \(-0.580666\pi\)
−0.250717 + 0.968061i \(0.580666\pi\)
\(938\) 0 0
\(939\) 0.0872743 0.00284809
\(940\) 0 0
\(941\) −12.1662 −0.396605 −0.198303 0.980141i \(-0.563543\pi\)
−0.198303 + 0.980141i \(0.563543\pi\)
\(942\) 0 0
\(943\) −38.1196 −1.24135
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −19.0598 −0.619361 −0.309680 0.950841i \(-0.600222\pi\)
−0.309680 + 0.950841i \(0.600222\pi\)
\(948\) 0 0
\(949\) −75.7231 −2.45808
\(950\) 0 0
\(951\) −8.23970 −0.267191
\(952\) 0 0
\(953\) 17.1334 0.555004 0.277502 0.960725i \(-0.410493\pi\)
0.277502 + 0.960725i \(0.410493\pi\)
\(954\) 0 0
\(955\) 47.1334 1.52520
\(956\) 0 0
\(957\) 1.93484 0.0625446
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −3.01373 −0.0972170
\(962\) 0 0
\(963\) 4.52991 0.145974
\(964\) 0 0
\(965\) 29.9348 0.963637
\(966\) 0 0
\(967\) −26.7833 −0.861294 −0.430647 0.902520i \(-0.641715\pi\)
−0.430647 + 0.902520i \(0.641715\pi\)
\(968\) 0 0
\(969\) 45.2344 1.45314
\(970\) 0 0
\(971\) −11.5162 −0.369572 −0.184786 0.982779i \(-0.559159\pi\)
−0.184786 + 0.982779i \(0.559159\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 15.3554 0.491765
\(976\) 0 0
\(977\) 21.3354 0.682579 0.341289 0.939958i \(-0.389136\pi\)
0.341289 + 0.939958i \(0.389136\pi\)
\(978\) 0 0
\(979\) −14.6540 −0.468343
\(980\) 0 0
\(981\) 6.44264 0.205698
\(982\) 0 0
\(983\) −50.3952 −1.60736 −0.803678 0.595064i \(-0.797127\pi\)
−0.803678 + 0.595064i \(0.797127\pi\)
\(984\) 0 0
\(985\) −14.8853 −0.474284
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 25.8990 0.823540
\(990\) 0 0
\(991\) −12.6823 −0.402868 −0.201434 0.979502i \(-0.564560\pi\)
−0.201434 + 0.979502i \(0.564560\pi\)
\(992\) 0 0
\(993\) 5.52991 0.175486
\(994\) 0 0
\(995\) 63.1334 2.00146
\(996\) 0 0
\(997\) −40.3416 −1.27763 −0.638816 0.769359i \(-0.720576\pi\)
−0.638816 + 0.769359i \(0.720576\pi\)
\(998\) 0 0
\(999\) −4.94950 −0.156595
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 9408.2.a.ei.1.3 3
4.3 odd 2 9408.2.a.eg.1.3 3
7.3 odd 6 1344.2.q.z.961.3 6
7.5 odd 6 1344.2.q.z.193.3 6
7.6 odd 2 9408.2.a.eh.1.1 3
8.3 odd 2 4704.2.a.bv.1.1 3
8.5 even 2 4704.2.a.bt.1.1 3
28.3 even 6 1344.2.q.y.961.3 6
28.19 even 6 1344.2.q.y.193.3 6
28.27 even 2 9408.2.a.ej.1.1 3
56.3 even 6 672.2.q.l.289.1 yes 6
56.5 odd 6 672.2.q.k.193.1 6
56.13 odd 2 4704.2.a.bu.1.3 3
56.19 even 6 672.2.q.l.193.1 yes 6
56.27 even 2 4704.2.a.bs.1.3 3
56.45 odd 6 672.2.q.k.289.1 yes 6
168.5 even 6 2016.2.s.u.865.3 6
168.59 odd 6 2016.2.s.v.289.3 6
168.101 even 6 2016.2.s.u.289.3 6
168.131 odd 6 2016.2.s.v.865.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
672.2.q.k.193.1 6 56.5 odd 6
672.2.q.k.289.1 yes 6 56.45 odd 6
672.2.q.l.193.1 yes 6 56.19 even 6
672.2.q.l.289.1 yes 6 56.3 even 6
1344.2.q.y.193.3 6 28.19 even 6
1344.2.q.y.961.3 6 28.3 even 6
1344.2.q.z.193.3 6 7.5 odd 6
1344.2.q.z.961.3 6 7.3 odd 6
2016.2.s.u.289.3 6 168.101 even 6
2016.2.s.u.865.3 6 168.5 even 6
2016.2.s.v.289.3 6 168.59 odd 6
2016.2.s.v.865.3 6 168.131 odd 6
4704.2.a.bs.1.3 3 56.27 even 2
4704.2.a.bt.1.1 3 8.5 even 2
4704.2.a.bu.1.3 3 56.13 odd 2
4704.2.a.bv.1.1 3 8.3 odd 2
9408.2.a.eg.1.3 3 4.3 odd 2
9408.2.a.eh.1.1 3 7.6 odd 2
9408.2.a.ei.1.3 3 1.1 even 1 trivial
9408.2.a.ej.1.1 3 28.27 even 2