Properties

Label 3920.2.a.ce.1.2
Level $3920$
Weight $2$
Character 3920.1
Self dual yes
Analytic conductor $31.301$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3920 = 2^{4} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3920.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(31.3013575923\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.16448.2
Defining polynomial: \(x^{4} - 2 x^{3} - 7 x^{2} + 8 x + 14\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1960)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.18398\) of defining polynomial
Character \(\chi\) \(=\) 3920.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.18398 q^{3} -1.00000 q^{5} -1.59819 q^{9} +O(q^{10})\) \(q-1.18398 q^{3} -1.00000 q^{5} -1.59819 q^{9} -0.230234 q^{11} +2.27259 q^{13} +1.18398 q^{15} -6.53278 q^{17} -0.260186 q^{19} +8.87079 q^{23} +1.00000 q^{25} +5.44417 q^{27} +5.42662 q^{29} +4.87079 q^{31} +0.272593 q^{33} -1.15403 q^{37} -2.69070 q^{39} -4.43337 q^{41} -4.17723 q^{43} +1.59819 q^{45} +0.923793 q^{47} +7.73467 q^{51} -2.13487 q^{53} +0.230234 q^{55} +0.308055 q^{57} -5.07107 q^{59} -8.88833 q^{61} -2.27259 q^{65} -8.15968 q^{67} -10.5028 q^{69} +9.06556 q^{71} -6.00955 q^{73} -1.18398 q^{75} -0.112912 q^{79} -1.65120 q^{81} +8.72065 q^{83} +6.53278 q^{85} -6.42501 q^{87} -15.9170 q^{89} -5.76691 q^{93} +0.260186 q^{95} -4.29565 q^{97} +0.367959 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{3} - 4q^{5} + 6q^{9} + O(q^{10}) \) \( 4q + 2q^{3} - 4q^{5} + 6q^{9} - 2q^{11} - 10q^{13} - 2q^{15} - 6q^{17} + 4q^{23} + 4q^{25} + 14q^{27} - 2q^{29} - 12q^{31} - 18q^{33} - 14q^{39} - 12q^{41} + 8q^{43} - 6q^{45} - 2q^{47} - 2q^{51} - 4q^{53} + 2q^{55} - 8q^{57} + 8q^{59} - 20q^{61} + 10q^{65} + 8q^{67} - 24q^{69} - 4q^{71} - 16q^{73} + 2q^{75} - 22q^{79} - 20q^{81} + 36q^{83} + 6q^{85} - 18q^{87} - 40q^{89} - 32q^{93} - 26q^{97} - 12q^{99} + O(q^{100}) \)

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
\(3\) −1.18398 −0.683571 −0.341785 0.939778i \(-0.611032\pi\)
−0.341785 + 0.939778i \(0.611032\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −1.59819 −0.532731
\(10\) 0 0
\(11\) −0.230234 −0.0694182 −0.0347091 0.999397i \(-0.511050\pi\)
−0.0347091 + 0.999397i \(0.511050\pi\)
\(12\) 0 0
\(13\) 2.27259 0.630304 0.315152 0.949041i \(-0.397945\pi\)
0.315152 + 0.949041i \(0.397945\pi\)
\(14\) 0 0
\(15\) 1.18398 0.305702
\(16\) 0 0
\(17\) −6.53278 −1.58443 −0.792216 0.610241i \(-0.791073\pi\)
−0.792216 + 0.610241i \(0.791073\pi\)
\(18\) 0 0
\(19\) −0.260186 −0.0596908 −0.0298454 0.999555i \(-0.509501\pi\)
−0.0298454 + 0.999555i \(0.509501\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 8.87079 1.84969 0.924843 0.380348i \(-0.124196\pi\)
0.924843 + 0.380348i \(0.124196\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 5.44417 1.04773
\(28\) 0 0
\(29\) 5.42662 1.00770 0.503849 0.863792i \(-0.331917\pi\)
0.503849 + 0.863792i \(0.331917\pi\)
\(30\) 0 0
\(31\) 4.87079 0.874819 0.437409 0.899262i \(-0.355896\pi\)
0.437409 + 0.899262i \(0.355896\pi\)
\(32\) 0 0
\(33\) 0.272593 0.0474523
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −1.15403 −0.189721 −0.0948605 0.995491i \(-0.530240\pi\)
−0.0948605 + 0.995491i \(0.530240\pi\)
\(38\) 0 0
\(39\) −2.69070 −0.430857
\(40\) 0 0
\(41\) −4.43337 −0.692377 −0.346188 0.938165i \(-0.612524\pi\)
−0.346188 + 0.938165i \(0.612524\pi\)
\(42\) 0 0
\(43\) −4.17723 −0.637021 −0.318511 0.947919i \(-0.603183\pi\)
−0.318511 + 0.947919i \(0.603183\pi\)
\(44\) 0 0
\(45\) 1.59819 0.238245
\(46\) 0 0
\(47\) 0.923793 0.134749 0.0673745 0.997728i \(-0.478538\pi\)
0.0673745 + 0.997728i \(0.478538\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 7.73467 1.08307
\(52\) 0 0
\(53\) −2.13487 −0.293247 −0.146623 0.989192i \(-0.546841\pi\)
−0.146623 + 0.989192i \(0.546841\pi\)
\(54\) 0 0
\(55\) 0.230234 0.0310448
\(56\) 0 0
\(57\) 0.308055 0.0408029
\(58\) 0 0
\(59\) −5.07107 −0.660197 −0.330098 0.943947i \(-0.607082\pi\)
−0.330098 + 0.943947i \(0.607082\pi\)
\(60\) 0 0
\(61\) −8.88833 −1.13803 −0.569017 0.822326i \(-0.692676\pi\)
−0.569017 + 0.822326i \(0.692676\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −2.27259 −0.281880
\(66\) 0 0
\(67\) −8.15968 −0.996864 −0.498432 0.866929i \(-0.666091\pi\)
−0.498432 + 0.866929i \(0.666091\pi\)
\(68\) 0 0
\(69\) −10.5028 −1.26439
\(70\) 0 0
\(71\) 9.06556 1.07588 0.537942 0.842982i \(-0.319202\pi\)
0.537942 + 0.842982i \(0.319202\pi\)
\(72\) 0 0
\(73\) −6.00955 −0.703365 −0.351682 0.936119i \(-0.614390\pi\)
−0.351682 + 0.936119i \(0.614390\pi\)
\(74\) 0 0
\(75\) −1.18398 −0.136714
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −0.112912 −0.0127035 −0.00635177 0.999980i \(-0.502022\pi\)
−0.00635177 + 0.999980i \(0.502022\pi\)
\(80\) 0 0
\(81\) −1.65120 −0.183467
\(82\) 0 0
\(83\) 8.72065 0.957216 0.478608 0.878029i \(-0.341141\pi\)
0.478608 + 0.878029i \(0.341141\pi\)
\(84\) 0 0
\(85\) 6.53278 0.708579
\(86\) 0 0
\(87\) −6.42501 −0.688833
\(88\) 0 0
\(89\) −15.9170 −1.68720 −0.843601 0.536970i \(-0.819569\pi\)
−0.843601 + 0.536970i \(0.819569\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −5.76691 −0.598001
\(94\) 0 0
\(95\) 0.260186 0.0266945
\(96\) 0 0
\(97\) −4.29565 −0.436157 −0.218079 0.975931i \(-0.569979\pi\)
−0.218079 + 0.975931i \(0.569979\pi\)
\(98\) 0 0
\(99\) 0.367959 0.0369812
\(100\) 0 0
\(101\) −7.38712 −0.735046 −0.367523 0.930014i \(-0.619794\pi\)
−0.367523 + 0.930014i \(0.619794\pi\)
\(102\) 0 0
\(103\) −10.5806 −1.04254 −0.521271 0.853391i \(-0.674542\pi\)
−0.521271 + 0.853391i \(0.674542\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 13.4661 1.30182 0.650910 0.759155i \(-0.274388\pi\)
0.650910 + 0.759155i \(0.274388\pi\)
\(108\) 0 0
\(109\) 8.50568 0.814697 0.407348 0.913273i \(-0.366453\pi\)
0.407348 + 0.913273i \(0.366453\pi\)
\(110\) 0 0
\(111\) 1.36634 0.129688
\(112\) 0 0
\(113\) 18.3968 1.73063 0.865313 0.501232i \(-0.167120\pi\)
0.865313 + 0.501232i \(0.167120\pi\)
\(114\) 0 0
\(115\) −8.87079 −0.827205
\(116\) 0 0
\(117\) −3.63204 −0.335782
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −10.9470 −0.995181
\(122\) 0 0
\(123\) 5.24902 0.473288
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −8.41032 −0.746295 −0.373147 0.927772i \(-0.621721\pi\)
−0.373147 + 0.927772i \(0.621721\pi\)
\(128\) 0 0
\(129\) 4.94575 0.435449
\(130\) 0 0
\(131\) −1.78217 −0.155709 −0.0778546 0.996965i \(-0.524807\pi\)
−0.0778546 + 0.996965i \(0.524807\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −5.44417 −0.468559
\(136\) 0 0
\(137\) −11.3137 −0.966595 −0.483298 0.875456i \(-0.660561\pi\)
−0.483298 + 0.875456i \(0.660561\pi\)
\(138\) 0 0
\(139\) 15.4390 1.30952 0.654761 0.755836i \(-0.272769\pi\)
0.654761 + 0.755836i \(0.272769\pi\)
\(140\) 0 0
\(141\) −1.09375 −0.0921105
\(142\) 0 0
\(143\) −0.523229 −0.0437546
\(144\) 0 0
\(145\) −5.42662 −0.450656
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 23.2811 1.90726 0.953631 0.300978i \(-0.0973130\pi\)
0.953631 + 0.300978i \(0.0973130\pi\)
\(150\) 0 0
\(151\) −22.3398 −1.81798 −0.908992 0.416813i \(-0.863147\pi\)
−0.908992 + 0.416813i \(0.863147\pi\)
\(152\) 0 0
\(153\) 10.4406 0.844076
\(154\) 0 0
\(155\) −4.87079 −0.391231
\(156\) 0 0
\(157\) 3.52603 0.281407 0.140704 0.990052i \(-0.455063\pi\)
0.140704 + 0.990052i \(0.455063\pi\)
\(158\) 0 0
\(159\) 2.52764 0.200455
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 16.2522 1.27297 0.636485 0.771289i \(-0.280388\pi\)
0.636485 + 0.771289i \(0.280388\pi\)
\(164\) 0 0
\(165\) −0.272593 −0.0212213
\(166\) 0 0
\(167\) −3.99714 −0.309308 −0.154654 0.987969i \(-0.549426\pi\)
−0.154654 + 0.987969i \(0.549426\pi\)
\(168\) 0 0
\(169\) −7.83532 −0.602717
\(170\) 0 0
\(171\) 0.415828 0.0317991
\(172\) 0 0
\(173\) −17.6214 −1.33973 −0.669865 0.742483i \(-0.733648\pi\)
−0.669865 + 0.742483i \(0.733648\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 6.00404 0.451291
\(178\) 0 0
\(179\) −25.2811 −1.88960 −0.944799 0.327650i \(-0.893743\pi\)
−0.944799 + 0.327650i \(0.893743\pi\)
\(180\) 0 0
\(181\) −18.4950 −1.37473 −0.687363 0.726315i \(-0.741232\pi\)
−0.687363 + 0.726315i \(0.741232\pi\)
\(182\) 0 0
\(183\) 10.5236 0.777927
\(184\) 0 0
\(185\) 1.15403 0.0848458
\(186\) 0 0
\(187\) 1.50407 0.109988
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −19.8658 −1.43744 −0.718719 0.695301i \(-0.755271\pi\)
−0.718719 + 0.695301i \(0.755271\pi\)
\(192\) 0 0
\(193\) −11.4335 −0.823002 −0.411501 0.911409i \(-0.634995\pi\)
−0.411501 + 0.911409i \(0.634995\pi\)
\(194\) 0 0
\(195\) 2.69070 0.192685
\(196\) 0 0
\(197\) −4.56273 −0.325081 −0.162541 0.986702i \(-0.551969\pi\)
−0.162541 + 0.986702i \(0.551969\pi\)
\(198\) 0 0
\(199\) −14.4429 −1.02383 −0.511916 0.859036i \(-0.671064\pi\)
−0.511916 + 0.859036i \(0.671064\pi\)
\(200\) 0 0
\(201\) 9.66089 0.681427
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 4.43337 0.309640
\(206\) 0 0
\(207\) −14.1772 −0.985385
\(208\) 0 0
\(209\) 0.0599037 0.00414363
\(210\) 0 0
\(211\) −6.63651 −0.456876 −0.228438 0.973558i \(-0.573362\pi\)
−0.228438 + 0.973558i \(0.573362\pi\)
\(212\) 0 0
\(213\) −10.7334 −0.735443
\(214\) 0 0
\(215\) 4.17723 0.284884
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 7.11518 0.480800
\(220\) 0 0
\(221\) −14.8463 −0.998673
\(222\) 0 0
\(223\) −20.3325 −1.36156 −0.680782 0.732486i \(-0.738360\pi\)
−0.680782 + 0.732486i \(0.738360\pi\)
\(224\) 0 0
\(225\) −1.59819 −0.106546
\(226\) 0 0
\(227\) 23.9312 1.58837 0.794185 0.607676i \(-0.207898\pi\)
0.794185 + 0.607676i \(0.207898\pi\)
\(228\) 0 0
\(229\) −2.59534 −0.171505 −0.0857523 0.996316i \(-0.527329\pi\)
−0.0857523 + 0.996316i \(0.527329\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −15.1638 −0.993412 −0.496706 0.867919i \(-0.665457\pi\)
−0.496706 + 0.867919i \(0.665457\pi\)
\(234\) 0 0
\(235\) −0.923793 −0.0602616
\(236\) 0 0
\(237\) 0.133685 0.00868377
\(238\) 0 0
\(239\) −10.5656 −0.683431 −0.341716 0.939803i \(-0.611008\pi\)
−0.341716 + 0.939803i \(0.611008\pi\)
\(240\) 0 0
\(241\) −19.6762 −1.26745 −0.633726 0.773557i \(-0.718475\pi\)
−0.633726 + 0.773557i \(0.718475\pi\)
\(242\) 0 0
\(243\) −14.3775 −0.922318
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −0.591297 −0.0376233
\(248\) 0 0
\(249\) −10.3251 −0.654325
\(250\) 0 0
\(251\) −25.0498 −1.58113 −0.790564 0.612380i \(-0.790212\pi\)
−0.790564 + 0.612380i \(0.790212\pi\)
\(252\) 0 0
\(253\) −2.04236 −0.128402
\(254\) 0 0
\(255\) −7.73467 −0.484364
\(256\) 0 0
\(257\) −1.71500 −0.106979 −0.0534894 0.998568i \(-0.517034\pi\)
−0.0534894 + 0.998568i \(0.517034\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −8.67279 −0.536832
\(262\) 0 0
\(263\) 22.1173 1.36381 0.681906 0.731440i \(-0.261151\pi\)
0.681906 + 0.731440i \(0.261151\pi\)
\(264\) 0 0
\(265\) 2.13487 0.131144
\(266\) 0 0
\(267\) 18.8454 1.15332
\(268\) 0 0
\(269\) 31.9261 1.94657 0.973283 0.229608i \(-0.0737443\pi\)
0.973283 + 0.229608i \(0.0737443\pi\)
\(270\) 0 0
\(271\) −25.7767 −1.56582 −0.782910 0.622135i \(-0.786266\pi\)
−0.782910 + 0.622135i \(0.786266\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −0.230234 −0.0138836
\(276\) 0 0
\(277\) −8.01755 −0.481728 −0.240864 0.970559i \(-0.577431\pi\)
−0.240864 + 0.970559i \(0.577431\pi\)
\(278\) 0 0
\(279\) −7.78445 −0.466043
\(280\) 0 0
\(281\) −3.13207 −0.186844 −0.0934218 0.995627i \(-0.529781\pi\)
−0.0934218 + 0.995627i \(0.529781\pi\)
\(282\) 0 0
\(283\) 3.17047 0.188465 0.0942325 0.995550i \(-0.469960\pi\)
0.0942325 + 0.995550i \(0.469960\pi\)
\(284\) 0 0
\(285\) −0.308055 −0.0182476
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 25.6772 1.51042
\(290\) 0 0
\(291\) 5.08596 0.298144
\(292\) 0 0
\(293\) 17.5264 1.02390 0.511952 0.859014i \(-0.328923\pi\)
0.511952 + 0.859014i \(0.328923\pi\)
\(294\) 0 0
\(295\) 5.07107 0.295249
\(296\) 0 0
\(297\) −1.25343 −0.0727316
\(298\) 0 0
\(299\) 20.1597 1.16586
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 8.74620 0.502456
\(304\) 0 0
\(305\) 8.88833 0.508944
\(306\) 0 0
\(307\) 2.11063 0.120460 0.0602300 0.998185i \(-0.480817\pi\)
0.0602300 + 0.998185i \(0.480817\pi\)
\(308\) 0 0
\(309\) 12.5273 0.712651
\(310\) 0 0
\(311\) 20.8149 1.18031 0.590153 0.807291i \(-0.299067\pi\)
0.590153 + 0.807291i \(0.299067\pi\)
\(312\) 0 0
\(313\) 21.3860 1.20881 0.604405 0.796678i \(-0.293411\pi\)
0.604405 + 0.796678i \(0.293411\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 15.4909 0.870058 0.435029 0.900417i \(-0.356738\pi\)
0.435029 + 0.900417i \(0.356738\pi\)
\(318\) 0 0
\(319\) −1.24939 −0.0699526
\(320\) 0 0
\(321\) −15.9436 −0.889886
\(322\) 0 0
\(323\) 1.69974 0.0945760
\(324\) 0 0
\(325\) 2.27259 0.126061
\(326\) 0 0
\(327\) −10.0706 −0.556903
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 8.52360 0.468499 0.234250 0.972176i \(-0.424737\pi\)
0.234250 + 0.972176i \(0.424737\pi\)
\(332\) 0 0
\(333\) 1.84436 0.101070
\(334\) 0 0
\(335\) 8.15968 0.445811
\(336\) 0 0
\(337\) −18.1246 −0.987309 −0.493655 0.869658i \(-0.664339\pi\)
−0.493655 + 0.869658i \(0.664339\pi\)
\(338\) 0 0
\(339\) −21.7814 −1.18301
\(340\) 0 0
\(341\) −1.12142 −0.0607284
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 10.5028 0.565453
\(346\) 0 0
\(347\) −4.10226 −0.220221 −0.110110 0.993919i \(-0.535120\pi\)
−0.110110 + 0.993919i \(0.535120\pi\)
\(348\) 0 0
\(349\) 8.67850 0.464549 0.232275 0.972650i \(-0.425383\pi\)
0.232275 + 0.972650i \(0.425383\pi\)
\(350\) 0 0
\(351\) 12.3724 0.660388
\(352\) 0 0
\(353\) −30.1162 −1.60292 −0.801462 0.598045i \(-0.795944\pi\)
−0.801462 + 0.598045i \(0.795944\pi\)
\(354\) 0 0
\(355\) −9.06556 −0.481150
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −36.2843 −1.91501 −0.957505 0.288416i \(-0.906872\pi\)
−0.957505 + 0.288416i \(0.906872\pi\)
\(360\) 0 0
\(361\) −18.9323 −0.996437
\(362\) 0 0
\(363\) 12.9610 0.680277
\(364\) 0 0
\(365\) 6.00955 0.314554
\(366\) 0 0
\(367\) −10.7115 −0.559134 −0.279567 0.960126i \(-0.590191\pi\)
−0.279567 + 0.960126i \(0.590191\pi\)
\(368\) 0 0
\(369\) 7.08539 0.368850
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 5.64335 0.292202 0.146101 0.989270i \(-0.453328\pi\)
0.146101 + 0.989270i \(0.453328\pi\)
\(374\) 0 0
\(375\) 1.18398 0.0611404
\(376\) 0 0
\(377\) 12.3325 0.635156
\(378\) 0 0
\(379\) 1.59944 0.0821575 0.0410787 0.999156i \(-0.486921\pi\)
0.0410787 + 0.999156i \(0.486921\pi\)
\(380\) 0 0
\(381\) 9.95764 0.510145
\(382\) 0 0
\(383\) −28.8894 −1.47618 −0.738089 0.674704i \(-0.764271\pi\)
−0.738089 + 0.674704i \(0.764271\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 6.67601 0.339361
\(388\) 0 0
\(389\) −23.5226 −1.19265 −0.596323 0.802745i \(-0.703372\pi\)
−0.596323 + 0.802745i \(0.703372\pi\)
\(390\) 0 0
\(391\) −57.9509 −2.93070
\(392\) 0 0
\(393\) 2.11006 0.106438
\(394\) 0 0
\(395\) 0.112912 0.00568120
\(396\) 0 0
\(397\) 16.5806 0.832159 0.416079 0.909328i \(-0.363404\pi\)
0.416079 + 0.909328i \(0.363404\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −1.52161 −0.0759858 −0.0379929 0.999278i \(-0.512096\pi\)
−0.0379929 + 0.999278i \(0.512096\pi\)
\(402\) 0 0
\(403\) 11.0693 0.551402
\(404\) 0 0
\(405\) 1.65120 0.0820488
\(406\) 0 0
\(407\) 0.265697 0.0131701
\(408\) 0 0
\(409\) 5.42927 0.268460 0.134230 0.990950i \(-0.457144\pi\)
0.134230 + 0.990950i \(0.457144\pi\)
\(410\) 0 0
\(411\) 13.3952 0.660736
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −8.72065 −0.428080
\(416\) 0 0
\(417\) −18.2795 −0.895150
\(418\) 0 0
\(419\) 12.4199 0.606750 0.303375 0.952871i \(-0.401886\pi\)
0.303375 + 0.952871i \(0.401886\pi\)
\(420\) 0 0
\(421\) −20.6804 −1.00790 −0.503951 0.863732i \(-0.668121\pi\)
−0.503951 + 0.863732i \(0.668121\pi\)
\(422\) 0 0
\(423\) −1.47640 −0.0717850
\(424\) 0 0
\(425\) −6.53278 −0.316886
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0.619492 0.0299093
\(430\) 0 0
\(431\) 31.6152 1.52285 0.761424 0.648254i \(-0.224500\pi\)
0.761424 + 0.648254i \(0.224500\pi\)
\(432\) 0 0
\(433\) 31.1247 1.49576 0.747880 0.663834i \(-0.231072\pi\)
0.747880 + 0.663834i \(0.231072\pi\)
\(434\) 0 0
\(435\) 6.42501 0.308055
\(436\) 0 0
\(437\) −2.30805 −0.110409
\(438\) 0 0
\(439\) 0.244398 0.0116645 0.00583223 0.999983i \(-0.498144\pi\)
0.00583223 + 0.999983i \(0.498144\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 3.45481 0.164143 0.0820716 0.996626i \(-0.473846\pi\)
0.0820716 + 0.996626i \(0.473846\pi\)
\(444\) 0 0
\(445\) 15.9170 0.754540
\(446\) 0 0
\(447\) −27.5643 −1.30375
\(448\) 0 0
\(449\) −15.5329 −0.733044 −0.366522 0.930409i \(-0.619452\pi\)
−0.366522 + 0.930409i \(0.619452\pi\)
\(450\) 0 0
\(451\) 1.02071 0.0480636
\(452\) 0 0
\(453\) 26.4498 1.24272
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −10.5379 −0.492943 −0.246471 0.969150i \(-0.579271\pi\)
−0.246471 + 0.969150i \(0.579271\pi\)
\(458\) 0 0
\(459\) −35.5655 −1.66006
\(460\) 0 0
\(461\) −5.98972 −0.278969 −0.139485 0.990224i \(-0.544545\pi\)
−0.139485 + 0.990224i \(0.544545\pi\)
\(462\) 0 0
\(463\) 33.3601 1.55038 0.775188 0.631731i \(-0.217655\pi\)
0.775188 + 0.631731i \(0.217655\pi\)
\(464\) 0 0
\(465\) 5.76691 0.267434
\(466\) 0 0
\(467\) 19.7076 0.911958 0.455979 0.889991i \(-0.349289\pi\)
0.455979 + 0.889991i \(0.349289\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −4.17474 −0.192362
\(472\) 0 0
\(473\) 0.961740 0.0442209
\(474\) 0 0
\(475\) −0.260186 −0.0119382
\(476\) 0 0
\(477\) 3.41193 0.156222
\(478\) 0 0
\(479\) 0.325600 0.0148771 0.00743853 0.999972i \(-0.497632\pi\)
0.00743853 + 0.999972i \(0.497632\pi\)
\(480\) 0 0
\(481\) −2.62263 −0.119582
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 4.29565 0.195055
\(486\) 0 0
\(487\) 16.9204 0.766737 0.383369 0.923595i \(-0.374764\pi\)
0.383369 + 0.923595i \(0.374764\pi\)
\(488\) 0 0
\(489\) −19.2423 −0.870165
\(490\) 0 0
\(491\) −15.0203 −0.677859 −0.338929 0.940812i \(-0.610065\pi\)
−0.338929 + 0.940812i \(0.610065\pi\)
\(492\) 0 0
\(493\) −35.4509 −1.59663
\(494\) 0 0
\(495\) −0.367959 −0.0165385
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −41.0431 −1.83734 −0.918670 0.395025i \(-0.870736\pi\)
−0.918670 + 0.395025i \(0.870736\pi\)
\(500\) 0 0
\(501\) 4.73254 0.211434
\(502\) 0 0
\(503\) 8.29741 0.369963 0.184982 0.982742i \(-0.440777\pi\)
0.184982 + 0.982742i \(0.440777\pi\)
\(504\) 0 0
\(505\) 7.38712 0.328722
\(506\) 0 0
\(507\) 9.27686 0.412000
\(508\) 0 0
\(509\) −8.71266 −0.386182 −0.193091 0.981181i \(-0.561851\pi\)
−0.193091 + 0.981181i \(0.561851\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −1.41650 −0.0625398
\(514\) 0 0
\(515\) 10.5806 0.466239
\(516\) 0 0
\(517\) −0.212689 −0.00935404
\(518\) 0 0
\(519\) 20.8634 0.915800
\(520\) 0 0
\(521\) 27.1056 1.18752 0.593760 0.804642i \(-0.297643\pi\)
0.593760 + 0.804642i \(0.297643\pi\)
\(522\) 0 0
\(523\) −1.37186 −0.0599870 −0.0299935 0.999550i \(-0.509549\pi\)
−0.0299935 + 0.999550i \(0.509549\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −31.8198 −1.38609
\(528\) 0 0
\(529\) 55.6908 2.42134
\(530\) 0 0
\(531\) 8.10454 0.351707
\(532\) 0 0
\(533\) −10.0753 −0.436408
\(534\) 0 0
\(535\) −13.4661 −0.582191
\(536\) 0 0
\(537\) 29.9323 1.29167
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 8.56559 0.368263 0.184132 0.982902i \(-0.441053\pi\)
0.184132 + 0.982902i \(0.441053\pi\)
\(542\) 0 0
\(543\) 21.8977 0.939722
\(544\) 0 0
\(545\) −8.50568 −0.364343
\(546\) 0 0
\(547\) −44.2056 −1.89009 −0.945046 0.326936i \(-0.893984\pi\)
−0.945046 + 0.326936i \(0.893984\pi\)
\(548\) 0 0
\(549\) 14.2053 0.606266
\(550\) 0 0
\(551\) −1.41193 −0.0601503
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −1.36634 −0.0579981
\(556\) 0 0
\(557\) −13.3385 −0.565171 −0.282586 0.959242i \(-0.591192\pi\)
−0.282586 + 0.959242i \(0.591192\pi\)
\(558\) 0 0
\(559\) −9.49313 −0.401517
\(560\) 0 0
\(561\) −1.78079 −0.0751849
\(562\) 0 0
\(563\) 29.7320 1.25306 0.626528 0.779399i \(-0.284476\pi\)
0.626528 + 0.779399i \(0.284476\pi\)
\(564\) 0 0
\(565\) −18.3968 −0.773960
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 23.5859 0.988774 0.494387 0.869242i \(-0.335393\pi\)
0.494387 + 0.869242i \(0.335393\pi\)
\(570\) 0 0
\(571\) −32.6434 −1.36608 −0.683042 0.730379i \(-0.739343\pi\)
−0.683042 + 0.730379i \(0.739343\pi\)
\(572\) 0 0
\(573\) 23.5207 0.982591
\(574\) 0 0
\(575\) 8.87079 0.369937
\(576\) 0 0
\(577\) 28.8195 1.19977 0.599886 0.800085i \(-0.295212\pi\)
0.599886 + 0.800085i \(0.295212\pi\)
\(578\) 0 0
\(579\) 13.5370 0.562580
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0.491520 0.0203567
\(584\) 0 0
\(585\) 3.63204 0.150166
\(586\) 0 0
\(587\) −1.82205 −0.0752039 −0.0376019 0.999293i \(-0.511972\pi\)
−0.0376019 + 0.999293i \(0.511972\pi\)
\(588\) 0 0
\(589\) −1.26731 −0.0522186
\(590\) 0 0
\(591\) 5.40218 0.222216
\(592\) 0 0
\(593\) −30.9369 −1.27042 −0.635212 0.772338i \(-0.719087\pi\)
−0.635212 + 0.772338i \(0.719087\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 17.1001 0.699861
\(598\) 0 0
\(599\) 32.2912 1.31938 0.659691 0.751537i \(-0.270687\pi\)
0.659691 + 0.751537i \(0.270687\pi\)
\(600\) 0 0
\(601\) 25.2829 1.03131 0.515655 0.856797i \(-0.327549\pi\)
0.515655 + 0.856797i \(0.327549\pi\)
\(602\) 0 0
\(603\) 13.0407 0.531060
\(604\) 0 0
\(605\) 10.9470 0.445059
\(606\) 0 0
\(607\) −9.28852 −0.377010 −0.188505 0.982072i \(-0.560364\pi\)
−0.188505 + 0.982072i \(0.560364\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 2.09941 0.0849329
\(612\) 0 0
\(613\) 40.0609 1.61805 0.809023 0.587777i \(-0.199997\pi\)
0.809023 + 0.587777i \(0.199997\pi\)
\(614\) 0 0
\(615\) −5.24902 −0.211661
\(616\) 0 0
\(617\) 27.9860 1.12667 0.563336 0.826228i \(-0.309518\pi\)
0.563336 + 0.826228i \(0.309518\pi\)
\(618\) 0 0
\(619\) −45.8048 −1.84105 −0.920525 0.390684i \(-0.872239\pi\)
−0.920525 + 0.390684i \(0.872239\pi\)
\(620\) 0 0
\(621\) 48.2940 1.93797
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −0.0709248 −0.00283246
\(628\) 0 0
\(629\) 7.53901 0.300600
\(630\) 0 0
\(631\) −22.5847 −0.899082 −0.449541 0.893260i \(-0.648412\pi\)
−0.449541 + 0.893260i \(0.648412\pi\)
\(632\) 0 0
\(633\) 7.85749 0.312307
\(634\) 0 0
\(635\) 8.41032 0.333753
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −14.4885 −0.573157
\(640\) 0 0
\(641\) −33.5125 −1.32366 −0.661832 0.749652i \(-0.730221\pi\)
−0.661832 + 0.749652i \(0.730221\pi\)
\(642\) 0 0
\(643\) 46.4980 1.83370 0.916851 0.399231i \(-0.130723\pi\)
0.916851 + 0.399231i \(0.130723\pi\)
\(644\) 0 0
\(645\) −4.94575 −0.194739
\(646\) 0 0
\(647\) −1.57565 −0.0619453 −0.0309726 0.999520i \(-0.509860\pi\)
−0.0309726 + 0.999520i \(0.509860\pi\)
\(648\) 0 0
\(649\) 1.16753 0.0458297
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 12.1311 0.474727 0.237364 0.971421i \(-0.423717\pi\)
0.237364 + 0.971421i \(0.423717\pi\)
\(654\) 0 0
\(655\) 1.78217 0.0696352
\(656\) 0 0
\(657\) 9.60442 0.374704
\(658\) 0 0
\(659\) −35.1682 −1.36996 −0.684979 0.728563i \(-0.740189\pi\)
−0.684979 + 0.728563i \(0.740189\pi\)
\(660\) 0 0
\(661\) 1.62425 0.0631759 0.0315880 0.999501i \(-0.489944\pi\)
0.0315880 + 0.999501i \(0.489944\pi\)
\(662\) 0 0
\(663\) 17.5778 0.682664
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 48.1384 1.86393
\(668\) 0 0
\(669\) 24.0733 0.930726
\(670\) 0 0
\(671\) 2.04640 0.0790003
\(672\) 0 0
\(673\) −24.3194 −0.937443 −0.468721 0.883346i \(-0.655285\pi\)
−0.468721 + 0.883346i \(0.655285\pi\)
\(674\) 0 0
\(675\) 5.44417 0.209546
\(676\) 0 0
\(677\) −12.1553 −0.467165 −0.233582 0.972337i \(-0.575045\pi\)
−0.233582 + 0.972337i \(0.575045\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −28.3341 −1.08576
\(682\) 0 0
\(683\) 32.5158 1.24418 0.622091 0.782945i \(-0.286283\pi\)
0.622091 + 0.782945i \(0.286283\pi\)
\(684\) 0 0
\(685\) 11.3137 0.432275
\(686\) 0 0
\(687\) 3.07282 0.117236
\(688\) 0 0
\(689\) −4.85169 −0.184834
\(690\) 0 0
\(691\) 20.3311 0.773432 0.386716 0.922199i \(-0.373609\pi\)
0.386716 + 0.922199i \(0.373609\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −15.4390 −0.585636
\(696\) 0 0
\(697\) 28.9622 1.09702
\(698\) 0 0
\(699\) 17.9536 0.679068
\(700\) 0 0
\(701\) 7.35321 0.277727 0.138863 0.990312i \(-0.455655\pi\)
0.138863 + 0.990312i \(0.455655\pi\)
\(702\) 0 0
\(703\) 0.300262 0.0113246
\(704\) 0 0
\(705\) 1.09375 0.0411931
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −11.6638 −0.438041 −0.219021 0.975720i \(-0.570286\pi\)
−0.219021 + 0.975720i \(0.570286\pi\)
\(710\) 0 0
\(711\) 0.180454 0.00676757
\(712\) 0 0
\(713\) 43.2077 1.61814
\(714\) 0 0
\(715\) 0.523229 0.0195676
\(716\) 0 0
\(717\) 12.5094 0.467173
\(718\) 0 0
\(719\) −5.68571 −0.212041 −0.106021 0.994364i \(-0.533811\pi\)
−0.106021 + 0.994364i \(0.533811\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 23.2962 0.866394
\(724\) 0 0
\(725\) 5.42662 0.201540
\(726\) 0 0
\(727\) 18.6873 0.693074 0.346537 0.938036i \(-0.387357\pi\)
0.346537 + 0.938036i \(0.387357\pi\)
\(728\) 0 0
\(729\) 21.9763 0.813936
\(730\) 0 0
\(731\) 27.2889 1.00932
\(732\) 0 0
\(733\) −1.83694 −0.0678488 −0.0339244 0.999424i \(-0.510801\pi\)
−0.0339244 + 0.999424i \(0.510801\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1.87864 0.0692005
\(738\) 0 0
\(739\) 24.5634 0.903579 0.451789 0.892125i \(-0.350786\pi\)
0.451789 + 0.892125i \(0.350786\pi\)
\(740\) 0 0
\(741\) 0.700083 0.0257182
\(742\) 0 0
\(743\) 27.1926 0.997601 0.498800 0.866717i \(-0.333774\pi\)
0.498800 + 0.866717i \(0.333774\pi\)
\(744\) 0 0
\(745\) −23.2811 −0.852954
\(746\) 0 0
\(747\) −13.9373 −0.509939
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −0.328457 −0.0119856 −0.00599278 0.999982i \(-0.501908\pi\)
−0.00599278 + 0.999982i \(0.501908\pi\)
\(752\) 0 0
\(753\) 29.6584 1.08081
\(754\) 0 0
\(755\) 22.3398 0.813027
\(756\) 0 0
\(757\) −48.6331 −1.76760 −0.883801 0.467864i \(-0.845024\pi\)
−0.883801 + 0.467864i \(0.845024\pi\)
\(758\) 0 0
\(759\) 2.41811 0.0877718
\(760\) 0 0
\(761\) 8.02253 0.290817 0.145408 0.989372i \(-0.453550\pi\)
0.145408 + 0.989372i \(0.453550\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −10.4406 −0.377482
\(766\) 0 0
\(767\) −11.5245 −0.416125
\(768\) 0 0
\(769\) −35.6370 −1.28510 −0.642552 0.766242i \(-0.722124\pi\)
−0.642552 + 0.766242i \(0.722124\pi\)
\(770\) 0 0
\(771\) 2.03053 0.0731276
\(772\) 0 0
\(773\) −49.5955 −1.78383 −0.891914 0.452206i \(-0.850637\pi\)
−0.891914 + 0.452206i \(0.850637\pi\)
\(774\) 0 0
\(775\) 4.87079 0.174964
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 1.15350 0.0413285
\(780\) 0 0
\(781\) −2.08720 −0.0746859
\(782\) 0 0
\(783\) 29.5434 1.05580
\(784\) 0 0
\(785\) −3.52603 −0.125849
\(786\) 0 0
\(787\) 21.4434 0.764376 0.382188 0.924085i \(-0.375171\pi\)
0.382188 + 0.924085i \(0.375171\pi\)
\(788\) 0 0
\(789\) −26.1865 −0.932262
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −20.1996 −0.717307
\(794\) 0 0
\(795\) −2.52764 −0.0896461
\(796\) 0 0
\(797\) −35.1429 −1.24482 −0.622412 0.782690i \(-0.713847\pi\)
−0.622412 + 0.782690i \(0.713847\pi\)
\(798\) 0 0
\(799\) −6.03494 −0.213501
\(800\) 0 0
\(801\) 25.4385 0.898825
\(802\) 0 0
\(803\) 1.38360 0.0488263
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −37.7998 −1.33062
\(808\) 0 0
\(809\) −2.32597 −0.0817768 −0.0408884 0.999164i \(-0.513019\pi\)
−0.0408884 + 0.999164i \(0.513019\pi\)
\(810\) 0 0
\(811\) −11.8045 −0.414512 −0.207256 0.978287i \(-0.566453\pi\)
−0.207256 + 0.978287i \(0.566453\pi\)
\(812\) 0 0
\(813\) 30.5190 1.07035
\(814\) 0 0
\(815\) −16.2522 −0.569289
\(816\) 0 0
\(817\) 1.08686 0.0380243
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −51.7790 −1.80710 −0.903550 0.428483i \(-0.859048\pi\)
−0.903550 + 0.428483i \(0.859048\pi\)
\(822\) 0 0
\(823\) 40.6259 1.41613 0.708064 0.706148i \(-0.249569\pi\)
0.708064 + 0.706148i \(0.249569\pi\)
\(824\) 0 0
\(825\) 0.272593 0.00949045
\(826\) 0 0
\(827\) −12.6201 −0.438846 −0.219423 0.975630i \(-0.570417\pi\)
−0.219423 + 0.975630i \(0.570417\pi\)
\(828\) 0 0
\(829\) 50.4905 1.75361 0.876803 0.480849i \(-0.159672\pi\)
0.876803 + 0.480849i \(0.159672\pi\)
\(830\) 0 0
\(831\) 9.49261 0.329295
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 3.99714 0.138327
\(836\) 0 0
\(837\) 26.5174 0.916574
\(838\) 0 0
\(839\) −4.41032 −0.152261 −0.0761305 0.997098i \(-0.524257\pi\)
−0.0761305 + 0.997098i \(0.524257\pi\)
\(840\) 0 0
\(841\) 0.448205 0.0154553
\(842\) 0 0
\(843\) 3.70831 0.127721
\(844\) 0 0
\(845\) 7.83532 0.269543
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −3.75378 −0.128829
\(850\) 0 0
\(851\) −10.2371 −0.350924
\(852\) 0 0
\(853\) −20.6785 −0.708018 −0.354009 0.935242i \(-0.615182\pi\)
−0.354009 + 0.935242i \(0.615182\pi\)
\(854\) 0 0
\(855\) −0.415828 −0.0142210
\(856\) 0 0
\(857\) 5.93055 0.202584 0.101292 0.994857i \(-0.467702\pi\)
0.101292 + 0.994857i \(0.467702\pi\)
\(858\) 0 0
\(859\) 31.3832 1.07078 0.535390 0.844605i \(-0.320165\pi\)
0.535390 + 0.844605i \(0.320165\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −26.1710 −0.890871 −0.445435 0.895314i \(-0.646951\pi\)
−0.445435 + 0.895314i \(0.646951\pi\)
\(864\) 0 0
\(865\) 17.6214 0.599145
\(866\) 0 0
\(867\) −30.4013 −1.03248
\(868\) 0 0
\(869\) 0.0259961 0.000881857 0
\(870\) 0 0
\(871\) −18.5436 −0.628327
\(872\) 0 0
\(873\) 6.86527 0.232354
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −52.2304 −1.76369 −0.881847 0.471536i \(-0.843700\pi\)
−0.881847 + 0.471536i \(0.843700\pi\)
\(878\) 0 0
\(879\) −20.7509 −0.699910
\(880\) 0 0
\(881\) −33.5930 −1.13178 −0.565888 0.824482i \(-0.691467\pi\)
−0.565888 + 0.824482i \(0.691467\pi\)
\(882\) 0 0
\(883\) −30.5923 −1.02951 −0.514757 0.857336i \(-0.672118\pi\)
−0.514757 + 0.857336i \(0.672118\pi\)
\(884\) 0 0
\(885\) −6.00404 −0.201824
\(886\) 0 0
\(887\) 49.0305 1.64628 0.823141 0.567837i \(-0.192220\pi\)
0.823141 + 0.567837i \(0.192220\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0.380163 0.0127359
\(892\) 0 0
\(893\) −0.240358 −0.00804328
\(894\) 0 0
\(895\) 25.2811 0.845054
\(896\) 0 0
\(897\) −23.8686 −0.796951
\(898\) 0 0
\(899\) 26.4319 0.881553
\(900\) 0 0
\(901\) 13.9466 0.464629
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 18.4950 0.614796
\(906\) 0 0
\(907\) 36.9013 1.22529 0.612644 0.790359i \(-0.290106\pi\)
0.612644 + 0.790359i \(0.290106\pi\)
\(908\) 0 0
\(909\) 11.8060 0.391582
\(910\) 0 0
\(911\) −34.6118 −1.14674 −0.573371 0.819296i \(-0.694364\pi\)
−0.573371 + 0.819296i \(0.694364\pi\)
\(912\) 0 0
\(913\) −2.00779 −0.0664483
\(914\) 0 0
\(915\) −10.5236 −0.347899
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 30.8170 1.01656 0.508279 0.861192i \(-0.330282\pi\)
0.508279 + 0.861192i \(0.330282\pi\)
\(920\) 0 0
\(921\) −2.49894 −0.0823429
\(922\) 0 0
\(923\) 20.6023 0.678134
\(924\) 0 0
\(925\) −1.15403 −0.0379442
\(926\) 0 0
\(927\) 16.9099 0.555395
\(928\) 0 0
\(929\) 17.3387 0.568865 0.284433 0.958696i \(-0.408195\pi\)
0.284433 + 0.958696i \(0.408195\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −24.6444 −0.806823
\(934\) 0 0
\(935\) −1.50407 −0.0491883
\(936\) 0 0
\(937\) −44.3736 −1.44962 −0.724811 0.688948i \(-0.758073\pi\)
−0.724811 + 0.688948i \(0.758073\pi\)
\(938\) 0 0
\(939\) −25.3206 −0.826307
\(940\) 0 0
\(941\) −1.65934 −0.0540929 −0.0270465 0.999634i \(-0.508610\pi\)
−0.0270465 + 0.999634i \(0.508610\pi\)
\(942\) 0 0
\(943\) −39.3275 −1.28068
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 37.4486 1.21692 0.608458 0.793586i \(-0.291789\pi\)
0.608458 + 0.793586i \(0.291789\pi\)
\(948\) 0 0
\(949\) −13.6573 −0.443333
\(950\) 0 0
\(951\) −18.3409 −0.594746
\(952\) 0 0
\(953\) 37.5875 1.21758 0.608789 0.793332i \(-0.291656\pi\)
0.608789 + 0.793332i \(0.291656\pi\)
\(954\) 0 0
\(955\) 19.8658 0.642842
\(956\) 0 0
\(957\) 1.47926 0.0478176
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −7.27545 −0.234692
\(962\) 0 0
\(963\) −21.5215 −0.693519
\(964\) 0 0
\(965\) 11.4335 0.368058
\(966\) 0 0
\(967\) 24.5757 0.790302 0.395151 0.918616i \(-0.370692\pi\)
0.395151 + 0.918616i \(0.370692\pi\)
\(968\) 0 0
\(969\) −2.01245 −0.0646494
\(970\) 0 0
\(971\) 48.6354 1.56078 0.780392 0.625290i \(-0.215019\pi\)
0.780392 + 0.625290i \(0.215019\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −2.69070 −0.0861714
\(976\) 0 0
\(977\) 2.33743 0.0747811 0.0373906 0.999301i \(-0.488095\pi\)
0.0373906 + 0.999301i \(0.488095\pi\)
\(978\) 0 0
\(979\) 3.66465 0.117123
\(980\) 0 0
\(981\) −13.5937 −0.434014
\(982\) 0 0
\(983\) 28.7331 0.916442 0.458221 0.888838i \(-0.348487\pi\)
0.458221 + 0.888838i \(0.348487\pi\)
\(984\) 0 0
\(985\) 4.56273 0.145381
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −37.0553 −1.17829
\(990\) 0 0
\(991\) 7.84787 0.249296 0.124648 0.992201i \(-0.460220\pi\)
0.124648 + 0.992201i \(0.460220\pi\)
\(992\) 0 0
\(993\) −10.0918 −0.320253
\(994\) 0 0
\(995\) 14.4429 0.457871
\(996\) 0 0
\(997\) −34.4523 −1.09112 −0.545558 0.838073i \(-0.683682\pi\)
−0.545558 + 0.838073i \(0.683682\pi\)
\(998\) 0 0
\(999\) −6.28272 −0.198776
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 3920.2.a.ce.1.2 4
4.3 odd 2 1960.2.a.x.1.3 4
7.6 odd 2 3920.2.a.cd.1.3 4
20.19 odd 2 9800.2.a.cs.1.2 4
28.3 even 6 1960.2.q.x.961.3 8
28.11 odd 6 1960.2.q.y.961.2 8
28.19 even 6 1960.2.q.x.361.3 8
28.23 odd 6 1960.2.q.y.361.2 8
28.27 even 2 1960.2.a.y.1.2 yes 4
140.139 even 2 9800.2.a.cl.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1960.2.a.x.1.3 4 4.3 odd 2
1960.2.a.y.1.2 yes 4 28.27 even 2
1960.2.q.x.361.3 8 28.19 even 6
1960.2.q.x.961.3 8 28.3 even 6
1960.2.q.y.361.2 8 28.23 odd 6
1960.2.q.y.961.2 8 28.11 odd 6
3920.2.a.cd.1.3 4 7.6 odd 2
3920.2.a.ce.1.2 4 1.1 even 1 trivial
9800.2.a.cl.1.3 4 140.139 even 2
9800.2.a.cs.1.2 4 20.19 odd 2