Properties

Label 6480.2.a.bs.1.3
Level $6480$
Weight $2$
Character 6480.1
Self dual yes
Analytic conductor $51.743$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6480 = 2^{4} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6480.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(51.7430605098\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.564.1
Defining polynomial: \( x^{3} - x^{2} - 5x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 45)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(2.51414\) of defining polynomial
Character \(\chi\) \(=\) 6480.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{5} +0.514137 q^{7} +O(q^{10})\) \(q-1.00000 q^{5} +0.514137 q^{7} -3.32088 q^{11} -1.32088 q^{13} +3.32088 q^{17} +1.32088 q^{19} +4.12763 q^{23} +1.00000 q^{25} +1.38650 q^{29} -8.73566 q^{31} -0.514137 q^{35} +0.292611 q^{37} +11.3492 q^{41} -10.3492 q^{43} -4.86330 q^{47} -6.73566 q^{49} +5.02827 q^{53} +3.32088 q^{55} -5.02827 q^{59} +7.34916 q^{61} +1.32088 q^{65} -9.44852 q^{67} +8.99093 q^{71} +6.05655 q^{73} -1.70739 q^{77} +8.05655 q^{79} +1.54241 q^{83} -3.32088 q^{85} +3.00000 q^{89} -0.679116 q^{91} -1.32088 q^{95} -12.2553 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{5} - 5 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{5} - 5 q^{7} - 2 q^{11} + 4 q^{13} + 2 q^{17} - 4 q^{19} + 3 q^{23} + 3 q^{25} + 7 q^{29} - 8 q^{31} + 5 q^{35} + 6 q^{37} + 13 q^{41} - 10 q^{43} + 13 q^{47} - 2 q^{49} + 2 q^{53} + 2 q^{55} - 2 q^{59} + q^{61} - 4 q^{65} - 11 q^{67} - 10 q^{71} - 8 q^{73} - 2 q^{79} - 15 q^{83} - 2 q^{85} + 9 q^{89} - 10 q^{91} + 4 q^{95} - 18 q^{97}+O(q^{100}) \) 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
\(3\) 0 0
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 0.514137 0.194325 0.0971627 0.995269i \(-0.469023\pi\)
0.0971627 + 0.995269i \(0.469023\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −3.32088 −1.00128 −0.500642 0.865654i \(-0.666903\pi\)
−0.500642 + 0.865654i \(0.666903\pi\)
\(12\) 0 0
\(13\) −1.32088 −0.366347 −0.183174 0.983081i \(-0.558637\pi\)
−0.183174 + 0.983081i \(0.558637\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.32088 0.805433 0.402716 0.915325i \(-0.368066\pi\)
0.402716 + 0.915325i \(0.368066\pi\)
\(18\) 0 0
\(19\) 1.32088 0.303032 0.151516 0.988455i \(-0.451585\pi\)
0.151516 + 0.988455i \(0.451585\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.12763 0.860671 0.430335 0.902669i \(-0.358395\pi\)
0.430335 + 0.902669i \(0.358395\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 1.38650 0.257467 0.128734 0.991679i \(-0.458909\pi\)
0.128734 + 0.991679i \(0.458909\pi\)
\(30\) 0 0
\(31\) −8.73566 −1.56897 −0.784486 0.620147i \(-0.787073\pi\)
−0.784486 + 0.620147i \(0.787073\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −0.514137 −0.0869050
\(36\) 0 0
\(37\) 0.292611 0.0481049 0.0240524 0.999711i \(-0.492343\pi\)
0.0240524 + 0.999711i \(0.492343\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 11.3492 1.77244 0.886220 0.463264i \(-0.153322\pi\)
0.886220 + 0.463264i \(0.153322\pi\)
\(42\) 0 0
\(43\) −10.3492 −1.57823 −0.789116 0.614244i \(-0.789461\pi\)
−0.789116 + 0.614244i \(0.789461\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −4.86330 −0.709385 −0.354692 0.934983i \(-0.615414\pi\)
−0.354692 + 0.934983i \(0.615414\pi\)
\(48\) 0 0
\(49\) −6.73566 −0.962238
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 5.02827 0.690687 0.345343 0.938476i \(-0.387762\pi\)
0.345343 + 0.938476i \(0.387762\pi\)
\(54\) 0 0
\(55\) 3.32088 0.447788
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −5.02827 −0.654625 −0.327313 0.944916i \(-0.606143\pi\)
−0.327313 + 0.944916i \(0.606143\pi\)
\(60\) 0 0
\(61\) 7.34916 0.940963 0.470482 0.882410i \(-0.344080\pi\)
0.470482 + 0.882410i \(0.344080\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 1.32088 0.163836
\(66\) 0 0
\(67\) −9.44852 −1.15432 −0.577160 0.816631i \(-0.695839\pi\)
−0.577160 + 0.816631i \(0.695839\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 8.99093 1.06703 0.533513 0.845792i \(-0.320871\pi\)
0.533513 + 0.845792i \(0.320871\pi\)
\(72\) 0 0
\(73\) 6.05655 0.708865 0.354433 0.935082i \(-0.384674\pi\)
0.354433 + 0.935082i \(0.384674\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −1.70739 −0.194575
\(78\) 0 0
\(79\) 8.05655 0.906432 0.453216 0.891401i \(-0.350277\pi\)
0.453216 + 0.891401i \(0.350277\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 1.54241 0.169302 0.0846508 0.996411i \(-0.473023\pi\)
0.0846508 + 0.996411i \(0.473023\pi\)
\(84\) 0 0
\(85\) −3.32088 −0.360200
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 3.00000 0.317999 0.159000 0.987279i \(-0.449173\pi\)
0.159000 + 0.987279i \(0.449173\pi\)
\(90\) 0 0
\(91\) −0.679116 −0.0711906
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −1.32088 −0.135520
\(96\) 0 0
\(97\) −12.2553 −1.24433 −0.622167 0.782885i \(-0.713747\pi\)
−0.622167 + 0.782885i \(0.713747\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −11.6700 −1.16121 −0.580606 0.814184i \(-0.697184\pi\)
−0.580606 + 0.814184i \(0.697184\pi\)
\(102\) 0 0
\(103\) −0.292611 −0.0288318 −0.0144159 0.999896i \(-0.504589\pi\)
−0.0144159 + 0.999896i \(0.504589\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1.87237 0.181009 0.0905043 0.995896i \(-0.471152\pi\)
0.0905043 + 0.995896i \(0.471152\pi\)
\(108\) 0 0
\(109\) 5.54787 0.531390 0.265695 0.964057i \(-0.414399\pi\)
0.265695 + 0.964057i \(0.414399\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −7.80128 −0.733883 −0.366942 0.930244i \(-0.619595\pi\)
−0.366942 + 0.930244i \(0.619595\pi\)
\(114\) 0 0
\(115\) −4.12763 −0.384904
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 1.70739 0.156516
\(120\) 0 0
\(121\) 0.0282739 0.00257035
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −17.8916 −1.58762 −0.793810 0.608166i \(-0.791906\pi\)
−0.793810 + 0.608166i \(0.791906\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −6.00000 −0.524222 −0.262111 0.965038i \(-0.584419\pi\)
−0.262111 + 0.965038i \(0.584419\pi\)
\(132\) 0 0
\(133\) 0.679116 0.0588868
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −5.67004 −0.484424 −0.242212 0.970223i \(-0.577873\pi\)
−0.242212 + 0.970223i \(0.577873\pi\)
\(138\) 0 0
\(139\) −8.00000 −0.678551 −0.339276 0.940687i \(-0.610182\pi\)
−0.339276 + 0.940687i \(0.610182\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 4.38650 0.366818
\(144\) 0 0
\(145\) −1.38650 −0.115143
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −17.6610 −1.44684 −0.723422 0.690407i \(-0.757432\pi\)
−0.723422 + 0.690407i \(0.757432\pi\)
\(150\) 0 0
\(151\) −1.26434 −0.102890 −0.0514451 0.998676i \(-0.516383\pi\)
−0.0514451 + 0.998676i \(0.516383\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 8.73566 0.701665
\(156\) 0 0
\(157\) −15.6700 −1.25061 −0.625303 0.780382i \(-0.715025\pi\)
−0.625303 + 0.780382i \(0.715025\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 2.12217 0.167250
\(162\) 0 0
\(163\) −15.7074 −1.23030 −0.615149 0.788411i \(-0.710904\pi\)
−0.615149 + 0.788411i \(0.710904\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 6.16498 0.477060 0.238530 0.971135i \(-0.423334\pi\)
0.238530 + 0.971135i \(0.423334\pi\)
\(168\) 0 0
\(169\) −11.2553 −0.865790
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −8.58522 −0.652722 −0.326361 0.945245i \(-0.605823\pi\)
−0.326361 + 0.945245i \(0.605823\pi\)
\(174\) 0 0
\(175\) 0.514137 0.0388651
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −1.06562 −0.0796482 −0.0398241 0.999207i \(-0.512680\pi\)
−0.0398241 + 0.999207i \(0.512680\pi\)
\(180\) 0 0
\(181\) −12.6700 −0.941757 −0.470878 0.882198i \(-0.656063\pi\)
−0.470878 + 0.882198i \(0.656063\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −0.292611 −0.0215132
\(186\) 0 0
\(187\) −11.0283 −0.806467
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −16.9344 −1.22533 −0.612664 0.790343i \(-0.709902\pi\)
−0.612664 + 0.790343i \(0.709902\pi\)
\(192\) 0 0
\(193\) 26.7175 1.92317 0.961585 0.274509i \(-0.0885153\pi\)
0.961585 + 0.274509i \(0.0885153\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 14.2553 1.01565 0.507823 0.861462i \(-0.330450\pi\)
0.507823 + 0.861462i \(0.330450\pi\)
\(198\) 0 0
\(199\) 24.6610 1.74817 0.874085 0.485773i \(-0.161462\pi\)
0.874085 + 0.485773i \(0.161462\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0.712853 0.0500325
\(204\) 0 0
\(205\) −11.3492 −0.792660
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −4.38650 −0.303421
\(210\) 0 0
\(211\) 5.37743 0.370198 0.185099 0.982720i \(-0.440739\pi\)
0.185099 + 0.982720i \(0.440739\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 10.3492 0.705807
\(216\) 0 0
\(217\) −4.49133 −0.304891
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −4.38650 −0.295068
\(222\) 0 0
\(223\) −8.66458 −0.580223 −0.290112 0.956993i \(-0.593692\pi\)
−0.290112 + 0.956993i \(0.593692\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 3.32088 0.220415 0.110207 0.993909i \(-0.464848\pi\)
0.110207 + 0.993909i \(0.464848\pi\)
\(228\) 0 0
\(229\) −25.3118 −1.67265 −0.836326 0.548233i \(-0.815301\pi\)
−0.836326 + 0.548233i \(0.815301\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −27.6327 −1.81028 −0.905139 0.425116i \(-0.860233\pi\)
−0.905139 + 0.425116i \(0.860233\pi\)
\(234\) 0 0
\(235\) 4.86330 0.317246
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −4.19872 −0.271592 −0.135796 0.990737i \(-0.543359\pi\)
−0.135796 + 0.990737i \(0.543359\pi\)
\(240\) 0 0
\(241\) 3.60442 0.232181 0.116091 0.993239i \(-0.462964\pi\)
0.116091 + 0.993239i \(0.462964\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 6.73566 0.430326
\(246\) 0 0
\(247\) −1.74474 −0.111015
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −6.87783 −0.434125 −0.217062 0.976158i \(-0.569648\pi\)
−0.217062 + 0.976158i \(0.569648\pi\)
\(252\) 0 0
\(253\) −13.7074 −0.861776
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 18.0000 1.12281 0.561405 0.827541i \(-0.310261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(258\) 0 0
\(259\) 0.150442 0.00934801
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 6.23606 0.384532 0.192266 0.981343i \(-0.438416\pi\)
0.192266 + 0.981343i \(0.438416\pi\)
\(264\) 0 0
\(265\) −5.02827 −0.308884
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −9.92345 −0.605044 −0.302522 0.953142i \(-0.597828\pi\)
−0.302522 + 0.953142i \(0.597828\pi\)
\(270\) 0 0
\(271\) −6.60442 −0.401190 −0.200595 0.979674i \(-0.564288\pi\)
−0.200595 + 0.979674i \(0.564288\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −3.32088 −0.200257
\(276\) 0 0
\(277\) 22.6610 1.36157 0.680783 0.732485i \(-0.261640\pi\)
0.680783 + 0.732485i \(0.261640\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 15.5479 0.927508 0.463754 0.885964i \(-0.346502\pi\)
0.463754 + 0.885964i \(0.346502\pi\)
\(282\) 0 0
\(283\) −0.645378 −0.0383637 −0.0191819 0.999816i \(-0.506106\pi\)
−0.0191819 + 0.999816i \(0.506106\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 5.83502 0.344430
\(288\) 0 0
\(289\) −5.97173 −0.351278
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 1.37743 0.0804704 0.0402352 0.999190i \(-0.487189\pi\)
0.0402352 + 0.999190i \(0.487189\pi\)
\(294\) 0 0
\(295\) 5.02827 0.292757
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −5.45213 −0.315305
\(300\) 0 0
\(301\) −5.32088 −0.306691
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −7.34916 −0.420812
\(306\) 0 0
\(307\) 7.98546 0.455754 0.227877 0.973690i \(-0.426822\pi\)
0.227877 + 0.973690i \(0.426822\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 9.63270 0.546220 0.273110 0.961983i \(-0.411948\pi\)
0.273110 + 0.961983i \(0.411948\pi\)
\(312\) 0 0
\(313\) −24.5369 −1.38691 −0.693455 0.720500i \(-0.743912\pi\)
−0.693455 + 0.720500i \(0.743912\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 20.3492 1.14292 0.571461 0.820629i \(-0.306377\pi\)
0.571461 + 0.820629i \(0.306377\pi\)
\(318\) 0 0
\(319\) −4.60442 −0.257798
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 4.38650 0.244072
\(324\) 0 0
\(325\) −1.32088 −0.0732695
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −2.50040 −0.137851
\(330\) 0 0
\(331\) −16.4431 −0.903792 −0.451896 0.892071i \(-0.649252\pi\)
−0.451896 + 0.892071i \(0.649252\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 9.44852 0.516228
\(336\) 0 0
\(337\) 4.89703 0.266758 0.133379 0.991065i \(-0.457417\pi\)
0.133379 + 0.991065i \(0.457417\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 29.0101 1.57099
\(342\) 0 0
\(343\) −7.06201 −0.381313
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 22.2745 1.19576 0.597878 0.801587i \(-0.296011\pi\)
0.597878 + 0.801587i \(0.296011\pi\)
\(348\) 0 0
\(349\) −2.94345 −0.157559 −0.0787797 0.996892i \(-0.525102\pi\)
−0.0787797 + 0.996892i \(0.525102\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −18.8296 −1.00220 −0.501098 0.865390i \(-0.667070\pi\)
−0.501098 + 0.865390i \(0.667070\pi\)
\(354\) 0 0
\(355\) −8.99093 −0.477189
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −31.8770 −1.68241 −0.841203 0.540720i \(-0.818152\pi\)
−0.841203 + 0.540720i \(0.818152\pi\)
\(360\) 0 0
\(361\) −17.2553 −0.908172
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −6.05655 −0.317014
\(366\) 0 0
\(367\) 18.3492 0.957818 0.478909 0.877864i \(-0.341032\pi\)
0.478909 + 0.877864i \(0.341032\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 2.58522 0.134218
\(372\) 0 0
\(373\) −2.19872 −0.113845 −0.0569226 0.998379i \(-0.518129\pi\)
−0.0569226 + 0.998379i \(0.518129\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −1.83141 −0.0943226
\(378\) 0 0
\(379\) −15.4713 −0.794709 −0.397354 0.917665i \(-0.630072\pi\)
−0.397354 + 0.917665i \(0.630072\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 7.70739 0.393829 0.196915 0.980421i \(-0.436908\pi\)
0.196915 + 0.980421i \(0.436908\pi\)
\(384\) 0 0
\(385\) 1.70739 0.0870166
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 24.6327 1.24893 0.624464 0.781054i \(-0.285318\pi\)
0.624464 + 0.781054i \(0.285318\pi\)
\(390\) 0 0
\(391\) 13.7074 0.693212
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −8.05655 −0.405369
\(396\) 0 0
\(397\) −6.77301 −0.339928 −0.169964 0.985450i \(-0.554365\pi\)
−0.169964 + 0.985450i \(0.554365\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 18.4996 0.923826 0.461913 0.886925i \(-0.347163\pi\)
0.461913 + 0.886925i \(0.347163\pi\)
\(402\) 0 0
\(403\) 11.5388 0.574789
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −0.971726 −0.0481667
\(408\) 0 0
\(409\) −13.4148 −0.663318 −0.331659 0.943399i \(-0.607608\pi\)
−0.331659 + 0.943399i \(0.607608\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −2.58522 −0.127210
\(414\) 0 0
\(415\) −1.54241 −0.0757140
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −33.1150 −1.61777 −0.808886 0.587966i \(-0.799929\pi\)
−0.808886 + 0.587966i \(0.799929\pi\)
\(420\) 0 0
\(421\) −14.6983 −0.716352 −0.358176 0.933654i \(-0.616601\pi\)
−0.358176 + 0.933654i \(0.616601\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 3.32088 0.161087
\(426\) 0 0
\(427\) 3.77847 0.182853
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −32.7549 −1.57775 −0.788873 0.614556i \(-0.789335\pi\)
−0.788873 + 0.614556i \(0.789335\pi\)
\(432\) 0 0
\(433\) −11.8314 −0.568581 −0.284291 0.958738i \(-0.591758\pi\)
−0.284291 + 0.958738i \(0.591758\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 5.45213 0.260811
\(438\) 0 0
\(439\) −8.31181 −0.396701 −0.198351 0.980131i \(-0.563558\pi\)
−0.198351 + 0.980131i \(0.563558\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −29.1751 −1.38615 −0.693076 0.720865i \(-0.743745\pi\)
−0.693076 + 0.720865i \(0.743745\pi\)
\(444\) 0 0
\(445\) −3.00000 −0.142214
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −18.9717 −0.895331 −0.447666 0.894201i \(-0.647744\pi\)
−0.447666 + 0.894201i \(0.647744\pi\)
\(450\) 0 0
\(451\) −37.6892 −1.77472
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0.679116 0.0318374
\(456\) 0 0
\(457\) −23.2353 −1.08690 −0.543450 0.839442i \(-0.682882\pi\)
−0.543450 + 0.839442i \(0.682882\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −4.42571 −0.206126 −0.103063 0.994675i \(-0.532864\pi\)
−0.103063 + 0.994675i \(0.532864\pi\)
\(462\) 0 0
\(463\) 19.5087 0.906645 0.453322 0.891347i \(-0.350239\pi\)
0.453322 + 0.891347i \(0.350239\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −24.5935 −1.13805 −0.569026 0.822320i \(-0.692679\pi\)
−0.569026 + 0.822320i \(0.692679\pi\)
\(468\) 0 0
\(469\) −4.85783 −0.224314
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 34.3684 1.58026
\(474\) 0 0
\(475\) 1.32088 0.0606063
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 32.7549 1.49661 0.748304 0.663356i \(-0.230868\pi\)
0.748304 + 0.663356i \(0.230868\pi\)
\(480\) 0 0
\(481\) −0.386505 −0.0176231
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 12.2553 0.556483
\(486\) 0 0
\(487\) 6.03735 0.273578 0.136789 0.990600i \(-0.456322\pi\)
0.136789 + 0.990600i \(0.456322\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 14.4431 0.651806 0.325903 0.945403i \(-0.394332\pi\)
0.325903 + 0.945403i \(0.394332\pi\)
\(492\) 0 0
\(493\) 4.60442 0.207373
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 4.62257 0.207351
\(498\) 0 0
\(499\) −20.9717 −0.938823 −0.469412 0.882979i \(-0.655534\pi\)
−0.469412 + 0.882979i \(0.655534\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −5.31728 −0.237086 −0.118543 0.992949i \(-0.537822\pi\)
−0.118543 + 0.992949i \(0.537822\pi\)
\(504\) 0 0
\(505\) 11.6700 0.519310
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −18.2270 −0.807897 −0.403949 0.914782i \(-0.632363\pi\)
−0.403949 + 0.914782i \(0.632363\pi\)
\(510\) 0 0
\(511\) 3.11389 0.137751
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0.292611 0.0128940
\(516\) 0 0
\(517\) 16.1504 0.710296
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −40.1232 −1.75783 −0.878915 0.476978i \(-0.841732\pi\)
−0.878915 + 0.476978i \(0.841732\pi\)
\(522\) 0 0
\(523\) −18.9873 −0.830257 −0.415129 0.909763i \(-0.636263\pi\)
−0.415129 + 0.909763i \(0.636263\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −29.0101 −1.26370
\(528\) 0 0
\(529\) −5.96265 −0.259246
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −14.9909 −0.649329
\(534\) 0 0
\(535\) −1.87237 −0.0809495
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 22.3684 0.963473
\(540\) 0 0
\(541\) 16.5279 0.710589 0.355294 0.934754i \(-0.384381\pi\)
0.355294 + 0.934754i \(0.384381\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −5.54787 −0.237645
\(546\) 0 0
\(547\) −17.6737 −0.755671 −0.377835 0.925873i \(-0.623331\pi\)
−0.377835 + 0.925873i \(0.623331\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 1.83141 0.0780208
\(552\) 0 0
\(553\) 4.14217 0.176143
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −17.3401 −0.734723 −0.367362 0.930078i \(-0.619739\pi\)
−0.367362 + 0.930078i \(0.619739\pi\)
\(558\) 0 0
\(559\) 13.6700 0.578181
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 12.9945 0.547654 0.273827 0.961779i \(-0.411710\pi\)
0.273827 + 0.961779i \(0.411710\pi\)
\(564\) 0 0
\(565\) 7.80128 0.328202
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 16.6802 0.699269 0.349635 0.936886i \(-0.386306\pi\)
0.349635 + 0.936886i \(0.386306\pi\)
\(570\) 0 0
\(571\) −20.0000 −0.836974 −0.418487 0.908223i \(-0.637439\pi\)
−0.418487 + 0.908223i \(0.637439\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 4.12763 0.172134
\(576\) 0 0
\(577\) −23.5953 −0.982287 −0.491144 0.871079i \(-0.663421\pi\)
−0.491144 + 0.871079i \(0.663421\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0.793010 0.0328996
\(582\) 0 0
\(583\) −16.6983 −0.691574
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 28.1276 1.16095 0.580476 0.814277i \(-0.302867\pi\)
0.580476 + 0.814277i \(0.302867\pi\)
\(588\) 0 0
\(589\) −11.5388 −0.475448
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 9.17872 0.376925 0.188462 0.982080i \(-0.439650\pi\)
0.188462 + 0.982080i \(0.439650\pi\)
\(594\) 0 0
\(595\) −1.70739 −0.0699961
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −31.4713 −1.28588 −0.642942 0.765915i \(-0.722286\pi\)
−0.642942 + 0.765915i \(0.722286\pi\)
\(600\) 0 0
\(601\) −29.2654 −1.19376 −0.596880 0.802330i \(-0.703593\pi\)
−0.596880 + 0.802330i \(0.703593\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −0.0282739 −0.00114950
\(606\) 0 0
\(607\) 44.2034 1.79416 0.897080 0.441868i \(-0.145684\pi\)
0.897080 + 0.441868i \(0.145684\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 6.42385 0.259881
\(612\) 0 0
\(613\) −35.1715 −1.42056 −0.710282 0.703918i \(-0.751432\pi\)
−0.710282 + 0.703918i \(0.751432\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 7.42571 0.298948 0.149474 0.988766i \(-0.452242\pi\)
0.149474 + 0.988766i \(0.452242\pi\)
\(618\) 0 0
\(619\) −8.54787 −0.343568 −0.171784 0.985135i \(-0.554953\pi\)
−0.171784 + 0.985135i \(0.554953\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 1.54241 0.0617954
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0.971726 0.0387453
\(630\) 0 0
\(631\) 2.36836 0.0942829 0.0471415 0.998888i \(-0.484989\pi\)
0.0471415 + 0.998888i \(0.484989\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 17.8916 0.710005
\(636\) 0 0
\(637\) 8.89703 0.352513
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0.133096 0.00525698 0.00262849 0.999997i \(-0.499163\pi\)
0.00262849 + 0.999997i \(0.499163\pi\)
\(642\) 0 0
\(643\) 22.6464 0.893088 0.446544 0.894762i \(-0.352655\pi\)
0.446544 + 0.894762i \(0.352655\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 46.3912 1.82383 0.911913 0.410385i \(-0.134606\pi\)
0.911913 + 0.410385i \(0.134606\pi\)
\(648\) 0 0
\(649\) 16.6983 0.655466
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −36.4057 −1.42467 −0.712333 0.701842i \(-0.752361\pi\)
−0.712333 + 0.701842i \(0.752361\pi\)
\(654\) 0 0
\(655\) 6.00000 0.234439
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 19.1414 0.745642 0.372821 0.927903i \(-0.378391\pi\)
0.372821 + 0.927903i \(0.378391\pi\)
\(660\) 0 0
\(661\) 39.9072 1.55221 0.776104 0.630605i \(-0.217193\pi\)
0.776104 + 0.630605i \(0.217193\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −0.679116 −0.0263350
\(666\) 0 0
\(667\) 5.72298 0.221595
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −24.4057 −0.942172
\(672\) 0 0
\(673\) 23.6508 0.911673 0.455836 0.890064i \(-0.349340\pi\)
0.455836 + 0.890064i \(0.349340\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 14.8031 0.568931 0.284465 0.958686i \(-0.408184\pi\)
0.284465 + 0.958686i \(0.408184\pi\)
\(678\) 0 0
\(679\) −6.30088 −0.241806
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −4.95252 −0.189503 −0.0947515 0.995501i \(-0.530206\pi\)
−0.0947515 + 0.995501i \(0.530206\pi\)
\(684\) 0 0
\(685\) 5.67004 0.216641
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −6.64177 −0.253031
\(690\) 0 0
\(691\) 19.2088 0.730739 0.365369 0.930863i \(-0.380943\pi\)
0.365369 + 0.930863i \(0.380943\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 8.00000 0.303457
\(696\) 0 0
\(697\) 37.6892 1.42758
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 29.3492 1.10850 0.554251 0.832349i \(-0.313005\pi\)
0.554251 + 0.832349i \(0.313005\pi\)
\(702\) 0 0
\(703\) 0.386505 0.0145773
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −6.00000 −0.225653
\(708\) 0 0
\(709\) 38.7266 1.45441 0.727204 0.686422i \(-0.240820\pi\)
0.727204 + 0.686422i \(0.240820\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −36.0576 −1.35037
\(714\) 0 0
\(715\) −4.38650 −0.164046
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −15.0848 −0.562569 −0.281284 0.959624i \(-0.590760\pi\)
−0.281284 + 0.959624i \(0.590760\pi\)
\(720\) 0 0
\(721\) −0.150442 −0.00560275
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 1.38650 0.0514935
\(726\) 0 0
\(727\) −12.3455 −0.457871 −0.228936 0.973442i \(-0.573525\pi\)
−0.228936 + 0.973442i \(0.573525\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −34.3684 −1.27116
\(732\) 0 0
\(733\) −22.0000 −0.812589 −0.406294 0.913742i \(-0.633179\pi\)
−0.406294 + 0.913742i \(0.633179\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 31.3774 1.15580
\(738\) 0 0
\(739\) −29.7266 −1.09351 −0.546755 0.837293i \(-0.684137\pi\)
−0.546755 + 0.837293i \(0.684137\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 48.3648 1.77433 0.887165 0.461452i \(-0.152671\pi\)
0.887165 + 0.461452i \(0.152671\pi\)
\(744\) 0 0
\(745\) 17.6610 0.647048
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0.962653 0.0351746
\(750\) 0 0
\(751\) 31.8205 1.16115 0.580573 0.814208i \(-0.302829\pi\)
0.580573 + 0.814208i \(0.302829\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 1.26434 0.0460139
\(756\) 0 0
\(757\) 4.94531 0.179740 0.0898701 0.995953i \(-0.471355\pi\)
0.0898701 + 0.995953i \(0.471355\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −35.4249 −1.28415 −0.642076 0.766641i \(-0.721927\pi\)
−0.642076 + 0.766641i \(0.721927\pi\)
\(762\) 0 0
\(763\) 2.85237 0.103263
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 6.64177 0.239820
\(768\) 0 0
\(769\) 49.4249 1.78231 0.891154 0.453701i \(-0.149897\pi\)
0.891154 + 0.453701i \(0.149897\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −12.6599 −0.455345 −0.227673 0.973738i \(-0.573112\pi\)
−0.227673 + 0.973738i \(0.573112\pi\)
\(774\) 0 0
\(775\) −8.73566 −0.313794
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 14.9909 0.537106
\(780\) 0 0
\(781\) −29.8578 −1.06840
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 15.6700 0.559288
\(786\) 0 0
\(787\) −30.9344 −1.10269 −0.551346 0.834277i \(-0.685885\pi\)
−0.551346 + 0.834277i \(0.685885\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −4.01093 −0.142612
\(792\) 0 0
\(793\) −9.70739 −0.344720
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 30.5935 1.08368 0.541839 0.840483i \(-0.317728\pi\)
0.541839 + 0.840483i \(0.317728\pi\)
\(798\) 0 0
\(799\) −16.1504 −0.571362
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −20.1131 −0.709776
\(804\) 0 0
\(805\) −2.12217 −0.0747966
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 2.89703 0.101854 0.0509271 0.998702i \(-0.483782\pi\)
0.0509271 + 0.998702i \(0.483782\pi\)
\(810\) 0 0
\(811\) 14.8861 0.522722 0.261361 0.965241i \(-0.415829\pi\)
0.261361 + 0.965241i \(0.415829\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 15.7074 0.550206
\(816\) 0 0
\(817\) −13.6700 −0.478254
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 8.95173 0.312417 0.156209 0.987724i \(-0.450073\pi\)
0.156209 + 0.987724i \(0.450073\pi\)
\(822\) 0 0
\(823\) −2.99454 −0.104383 −0.0521915 0.998637i \(-0.516621\pi\)
−0.0521915 + 0.998637i \(0.516621\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 31.9663 1.11158 0.555788 0.831324i \(-0.312417\pi\)
0.555788 + 0.831324i \(0.312417\pi\)
\(828\) 0 0
\(829\) 22.7458 0.789994 0.394997 0.918682i \(-0.370746\pi\)
0.394997 + 0.918682i \(0.370746\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −22.3684 −0.775018
\(834\) 0 0
\(835\) −6.16498 −0.213348
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −23.2643 −0.803174 −0.401587 0.915821i \(-0.631541\pi\)
−0.401587 + 0.915821i \(0.631541\pi\)
\(840\) 0 0
\(841\) −27.0776 −0.933710
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 11.2553 0.387193
\(846\) 0 0
\(847\) 0.0145366 0.000499485 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 1.20779 0.0414025
\(852\) 0 0
\(853\) 10.9909 0.376322 0.188161 0.982138i \(-0.439747\pi\)
0.188161 + 0.982138i \(0.439747\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 16.1504 0.551689 0.275844 0.961202i \(-0.411043\pi\)
0.275844 + 0.961202i \(0.411043\pi\)
\(858\) 0 0
\(859\) −28.5188 −0.973049 −0.486524 0.873667i \(-0.661736\pi\)
−0.486524 + 0.873667i \(0.661736\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 12.2890 0.418322 0.209161 0.977881i \(-0.432927\pi\)
0.209161 + 0.977881i \(0.432927\pi\)
\(864\) 0 0
\(865\) 8.58522 0.291906
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −26.7549 −0.907597
\(870\) 0 0
\(871\) 12.4804 0.422882
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −0.514137 −0.0173810
\(876\) 0 0
\(877\) −39.7002 −1.34058 −0.670290 0.742099i \(-0.733830\pi\)
−0.670290 + 0.742099i \(0.733830\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −32.1040 −1.08161 −0.540806 0.841147i \(-0.681881\pi\)
−0.540806 + 0.841147i \(0.681881\pi\)
\(882\) 0 0
\(883\) −13.5051 −0.454482 −0.227241 0.973839i \(-0.572970\pi\)
−0.227241 + 0.973839i \(0.572970\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 35.1222 1.17929 0.589643 0.807664i \(-0.299268\pi\)
0.589643 + 0.807664i \(0.299268\pi\)
\(888\) 0 0
\(889\) −9.19872 −0.308515
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −6.42385 −0.214966
\(894\) 0 0
\(895\) 1.06562 0.0356198
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −12.1120 −0.403959
\(900\) 0 0
\(901\) 16.6983 0.556302
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 12.6700 0.421166
\(906\) 0 0
\(907\) −15.1186 −0.502004 −0.251002 0.967987i \(-0.580760\pi\)
−0.251002 + 0.967987i \(0.580760\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 52.5561 1.74126 0.870631 0.491936i \(-0.163711\pi\)
0.870631 + 0.491936i \(0.163711\pi\)
\(912\) 0 0
\(913\) −5.12217 −0.169519
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −3.08482 −0.101870
\(918\) 0 0
\(919\) 54.5489 1.79940 0.899702 0.436505i \(-0.143784\pi\)
0.899702 + 0.436505i \(0.143784\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −11.8760 −0.390903
\(924\) 0 0
\(925\) 0.292611 0.00962098
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −20.3793 −0.668623 −0.334311 0.942463i \(-0.608504\pi\)
−0.334311 + 0.942463i \(0.608504\pi\)
\(930\) 0 0
\(931\) −8.89703 −0.291588
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 11.0283 0.360663
\(936\) 0 0
\(937\) 49.1979 1.60723 0.803613 0.595152i \(-0.202908\pi\)
0.803613 + 0.595152i \(0.202908\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 23.2371 0.757508 0.378754 0.925497i \(-0.376353\pi\)
0.378754 + 0.925497i \(0.376353\pi\)
\(942\) 0 0
\(943\) 46.8452 1.52549
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 37.1642 1.20767 0.603837 0.797108i \(-0.293638\pi\)
0.603837 + 0.797108i \(0.293638\pi\)
\(948\) 0 0
\(949\) −8.00000 −0.259691
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 23.5761 0.763706 0.381853 0.924223i \(-0.375286\pi\)
0.381853 + 0.924223i \(0.375286\pi\)
\(954\) 0 0
\(955\) 16.9344 0.547984
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −2.91518 −0.0941360
\(960\) 0 0
\(961\) 45.3118 1.46167
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −26.7175 −0.860067
\(966\) 0 0
\(967\) −8.38290 −0.269576 −0.134788 0.990874i \(-0.543035\pi\)
−0.134788 + 0.990874i \(0.543035\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 13.2078 0.423858 0.211929 0.977285i \(-0.432025\pi\)
0.211929 + 0.977285i \(0.432025\pi\)
\(972\) 0 0
\(973\) −4.11310 −0.131860
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 14.3310 0.458490 0.229245 0.973369i \(-0.426374\pi\)
0.229245 + 0.973369i \(0.426374\pi\)
\(978\) 0 0
\(979\) −9.96265 −0.318408
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −32.3082 −1.03047 −0.515236 0.857048i \(-0.672296\pi\)
−0.515236 + 0.857048i \(0.672296\pi\)
\(984\) 0 0
\(985\) −14.2553 −0.454210
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −42.7175 −1.35834
\(990\) 0 0
\(991\) 39.6700 1.26016 0.630080 0.776530i \(-0.283022\pi\)
0.630080 + 0.776530i \(0.283022\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −24.6610 −0.781805
\(996\) 0 0
\(997\) 38.6874 1.22524 0.612621 0.790377i \(-0.290115\pi\)
0.612621 + 0.790377i \(0.290115\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 6480.2.a.bs.1.3 3
3.2 odd 2 6480.2.a.bv.1.3 3
4.3 odd 2 405.2.a.i.1.1 3
9.2 odd 6 720.2.q.i.481.1 6
9.4 even 3 2160.2.q.k.721.1 6
9.5 odd 6 720.2.q.i.241.1 6
9.7 even 3 2160.2.q.k.1441.1 6
12.11 even 2 405.2.a.j.1.3 3
20.3 even 4 2025.2.b.m.649.6 6
20.7 even 4 2025.2.b.m.649.1 6
20.19 odd 2 2025.2.a.o.1.3 3
36.7 odd 6 135.2.e.b.91.3 6
36.11 even 6 45.2.e.b.31.1 yes 6
36.23 even 6 45.2.e.b.16.1 6
36.31 odd 6 135.2.e.b.46.3 6
60.23 odd 4 2025.2.b.l.649.1 6
60.47 odd 4 2025.2.b.l.649.6 6
60.59 even 2 2025.2.a.n.1.1 3
180.7 even 12 675.2.k.b.199.1 12
180.23 odd 12 225.2.k.b.124.6 12
180.43 even 12 675.2.k.b.199.6 12
180.47 odd 12 225.2.k.b.49.6 12
180.59 even 6 225.2.e.b.151.3 6
180.67 even 12 675.2.k.b.424.6 12
180.79 odd 6 675.2.e.b.226.1 6
180.83 odd 12 225.2.k.b.49.1 12
180.103 even 12 675.2.k.b.424.1 12
180.119 even 6 225.2.e.b.76.3 6
180.139 odd 6 675.2.e.b.451.1 6
180.167 odd 12 225.2.k.b.124.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
45.2.e.b.16.1 6 36.23 even 6
45.2.e.b.31.1 yes 6 36.11 even 6
135.2.e.b.46.3 6 36.31 odd 6
135.2.e.b.91.3 6 36.7 odd 6
225.2.e.b.76.3 6 180.119 even 6
225.2.e.b.151.3 6 180.59 even 6
225.2.k.b.49.1 12 180.83 odd 12
225.2.k.b.49.6 12 180.47 odd 12
225.2.k.b.124.1 12 180.167 odd 12
225.2.k.b.124.6 12 180.23 odd 12
405.2.a.i.1.1 3 4.3 odd 2
405.2.a.j.1.3 3 12.11 even 2
675.2.e.b.226.1 6 180.79 odd 6
675.2.e.b.451.1 6 180.139 odd 6
675.2.k.b.199.1 12 180.7 even 12
675.2.k.b.199.6 12 180.43 even 12
675.2.k.b.424.1 12 180.103 even 12
675.2.k.b.424.6 12 180.67 even 12
720.2.q.i.241.1 6 9.5 odd 6
720.2.q.i.481.1 6 9.2 odd 6
2025.2.a.n.1.1 3 60.59 even 2
2025.2.a.o.1.3 3 20.19 odd 2
2025.2.b.l.649.1 6 60.23 odd 4
2025.2.b.l.649.6 6 60.47 odd 4
2025.2.b.m.649.1 6 20.7 even 4
2025.2.b.m.649.6 6 20.3 even 4
2160.2.q.k.721.1 6 9.4 even 3
2160.2.q.k.1441.1 6 9.7 even 3
6480.2.a.bs.1.3 3 1.1 even 1 trivial
6480.2.a.bv.1.3 3 3.2 odd 2