Properties

Label 1003.2.a.d.1.1
Level 1003
Weight 2
Character 1003.1
Self dual yes
Analytic conductor 8.009
Analytic rank 1
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 1003 = 17 \cdot 59 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1003.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(8.00899532273\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(0\) of \(x\)
Character \(\chi\) \(=\) 1003.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.00000 q^{2} +2.00000 q^{4} -2.00000 q^{5} -2.00000 q^{7} -3.00000 q^{9} +O(q^{10})\) \(q+2.00000 q^{2} +2.00000 q^{4} -2.00000 q^{5} -2.00000 q^{7} -3.00000 q^{9} -4.00000 q^{10} -3.00000 q^{11} +4.00000 q^{13} -4.00000 q^{14} -4.00000 q^{16} +1.00000 q^{17} -6.00000 q^{18} -1.00000 q^{19} -4.00000 q^{20} -6.00000 q^{22} +1.00000 q^{23} -1.00000 q^{25} +8.00000 q^{26} -4.00000 q^{28} -6.00000 q^{29} -8.00000 q^{31} -8.00000 q^{32} +2.00000 q^{34} +4.00000 q^{35} -6.00000 q^{36} -2.00000 q^{37} -2.00000 q^{38} +4.00000 q^{43} -6.00000 q^{44} +6.00000 q^{45} +2.00000 q^{46} +12.0000 q^{47} -3.00000 q^{49} -2.00000 q^{50} +8.00000 q^{52} -3.00000 q^{53} +6.00000 q^{55} -12.0000 q^{58} +1.00000 q^{59} +5.00000 q^{61} -16.0000 q^{62} +6.00000 q^{63} -8.00000 q^{64} -8.00000 q^{65} +10.0000 q^{67} +2.00000 q^{68} +8.00000 q^{70} -4.00000 q^{71} +15.0000 q^{73} -4.00000 q^{74} -2.00000 q^{76} +6.00000 q^{77} +6.00000 q^{79} +8.00000 q^{80} +9.00000 q^{81} +2.00000 q^{83} -2.00000 q^{85} +8.00000 q^{86} -14.0000 q^{89} +12.0000 q^{90} -8.00000 q^{91} +2.00000 q^{92} +24.0000 q^{94} +2.00000 q^{95} -19.0000 q^{97} -6.00000 q^{98} +9.00000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(3\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(4\) 2.00000 1.00000
\(5\) −2.00000 −0.894427 −0.447214 0.894427i \(-0.647584\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) 0 0
\(7\) −2.00000 −0.755929 −0.377964 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) 0 0
\(9\) −3.00000 −1.00000
\(10\) −4.00000 −1.26491
\(11\) −3.00000 −0.904534 −0.452267 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) 0 0
\(13\) 4.00000 1.10940 0.554700 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) −4.00000 −1.06904
\(15\) 0 0
\(16\) −4.00000 −1.00000
\(17\) 1.00000 0.242536
\(18\) −6.00000 −1.41421
\(19\) −1.00000 −0.229416 −0.114708 0.993399i \(-0.536593\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) −4.00000 −0.894427
\(21\) 0 0
\(22\) −6.00000 −1.27920
\(23\) 1.00000 0.208514 0.104257 0.994550i \(-0.466753\pi\)
0.104257 + 0.994550i \(0.466753\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 8.00000 1.56893
\(27\) 0 0
\(28\) −4.00000 −0.755929
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) −8.00000 −1.43684 −0.718421 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) −8.00000 −1.41421
\(33\) 0 0
\(34\) 2.00000 0.342997
\(35\) 4.00000 0.676123
\(36\) −6.00000 −1.00000
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) −2.00000 −0.324443
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) −6.00000 −0.904534
\(45\) 6.00000 0.894427
\(46\) 2.00000 0.294884
\(47\) 12.0000 1.75038 0.875190 0.483779i \(-0.160736\pi\)
0.875190 + 0.483779i \(0.160736\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) −2.00000 −0.282843
\(51\) 0 0
\(52\) 8.00000 1.10940
\(53\) −3.00000 −0.412082 −0.206041 0.978543i \(-0.566058\pi\)
−0.206041 + 0.978543i \(0.566058\pi\)
\(54\) 0 0
\(55\) 6.00000 0.809040
\(56\) 0 0
\(57\) 0 0
\(58\) −12.0000 −1.57568
\(59\) 1.00000 0.130189
\(60\) 0 0
\(61\) 5.00000 0.640184 0.320092 0.947386i \(-0.396286\pi\)
0.320092 + 0.947386i \(0.396286\pi\)
\(62\) −16.0000 −2.03200
\(63\) 6.00000 0.755929
\(64\) −8.00000 −1.00000
\(65\) −8.00000 −0.992278
\(66\) 0 0
\(67\) 10.0000 1.22169 0.610847 0.791748i \(-0.290829\pi\)
0.610847 + 0.791748i \(0.290829\pi\)
\(68\) 2.00000 0.242536
\(69\) 0 0
\(70\) 8.00000 0.956183
\(71\) −4.00000 −0.474713 −0.237356 0.971423i \(-0.576281\pi\)
−0.237356 + 0.971423i \(0.576281\pi\)
\(72\) 0 0
\(73\) 15.0000 1.75562 0.877809 0.479012i \(-0.159005\pi\)
0.877809 + 0.479012i \(0.159005\pi\)
\(74\) −4.00000 −0.464991
\(75\) 0 0
\(76\) −2.00000 −0.229416
\(77\) 6.00000 0.683763
\(78\) 0 0
\(79\) 6.00000 0.675053 0.337526 0.941316i \(-0.390410\pi\)
0.337526 + 0.941316i \(0.390410\pi\)
\(80\) 8.00000 0.894427
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) 2.00000 0.219529 0.109764 0.993958i \(-0.464990\pi\)
0.109764 + 0.993958i \(0.464990\pi\)
\(84\) 0 0
\(85\) −2.00000 −0.216930
\(86\) 8.00000 0.862662
\(87\) 0 0
\(88\) 0 0
\(89\) −14.0000 −1.48400 −0.741999 0.670402i \(-0.766122\pi\)
−0.741999 + 0.670402i \(0.766122\pi\)
\(90\) 12.0000 1.26491
\(91\) −8.00000 −0.838628
\(92\) 2.00000 0.208514
\(93\) 0 0
\(94\) 24.0000 2.47541
\(95\) 2.00000 0.205196
\(96\) 0 0
\(97\) −19.0000 −1.92916 −0.964579 0.263795i \(-0.915026\pi\)
−0.964579 + 0.263795i \(0.915026\pi\)
\(98\) −6.00000 −0.606092
\(99\) 9.00000 0.904534
\(100\) −2.00000 −0.200000
\(101\) −4.00000 −0.398015 −0.199007 0.979998i \(-0.563772\pi\)
−0.199007 + 0.979998i \(0.563772\pi\)
\(102\) 0 0
\(103\) −10.0000 −0.985329 −0.492665 0.870219i \(-0.663977\pi\)
−0.492665 + 0.870219i \(0.663977\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) −10.0000 −0.966736 −0.483368 0.875417i \(-0.660587\pi\)
−0.483368 + 0.875417i \(0.660587\pi\)
\(108\) 0 0
\(109\) 3.00000 0.287348 0.143674 0.989625i \(-0.454108\pi\)
0.143674 + 0.989625i \(0.454108\pi\)
\(110\) 12.0000 1.14416
\(111\) 0 0
\(112\) 8.00000 0.755929
\(113\) −2.00000 −0.188144 −0.0940721 0.995565i \(-0.529988\pi\)
−0.0940721 + 0.995565i \(0.529988\pi\)
\(114\) 0 0
\(115\) −2.00000 −0.186501
\(116\) −12.0000 −1.11417
\(117\) −12.0000 −1.10940
\(118\) 2.00000 0.184115
\(119\) −2.00000 −0.183340
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) 10.0000 0.905357
\(123\) 0 0
\(124\) −16.0000 −1.43684
\(125\) 12.0000 1.07331
\(126\) 12.0000 1.06904
\(127\) 20.0000 1.77471 0.887357 0.461084i \(-0.152539\pi\)
0.887357 + 0.461084i \(0.152539\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) −16.0000 −1.40329
\(131\) −20.0000 −1.74741 −0.873704 0.486458i \(-0.838289\pi\)
−0.873704 + 0.486458i \(0.838289\pi\)
\(132\) 0 0
\(133\) 2.00000 0.173422
\(134\) 20.0000 1.72774
\(135\) 0 0
\(136\) 0 0
\(137\) 9.00000 0.768922 0.384461 0.923141i \(-0.374387\pi\)
0.384461 + 0.923141i \(0.374387\pi\)
\(138\) 0 0
\(139\) −8.00000 −0.678551 −0.339276 0.940687i \(-0.610182\pi\)
−0.339276 + 0.940687i \(0.610182\pi\)
\(140\) 8.00000 0.676123
\(141\) 0 0
\(142\) −8.00000 −0.671345
\(143\) −12.0000 −1.00349
\(144\) 12.0000 1.00000
\(145\) 12.0000 0.996546
\(146\) 30.0000 2.48282
\(147\) 0 0
\(148\) −4.00000 −0.328798
\(149\) −22.0000 −1.80231 −0.901155 0.433497i \(-0.857280\pi\)
−0.901155 + 0.433497i \(0.857280\pi\)
\(150\) 0 0
\(151\) −16.0000 −1.30206 −0.651031 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) 0 0
\(153\) −3.00000 −0.242536
\(154\) 12.0000 0.966988
\(155\) 16.0000 1.28515
\(156\) 0 0
\(157\) 10.0000 0.798087 0.399043 0.916932i \(-0.369342\pi\)
0.399043 + 0.916932i \(0.369342\pi\)
\(158\) 12.0000 0.954669
\(159\) 0 0
\(160\) 16.0000 1.26491
\(161\) −2.00000 −0.157622
\(162\) 18.0000 1.41421
\(163\) 12.0000 0.939913 0.469956 0.882690i \(-0.344270\pi\)
0.469956 + 0.882690i \(0.344270\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 4.00000 0.310460
\(167\) −22.0000 −1.70241 −0.851206 0.524832i \(-0.824128\pi\)
−0.851206 + 0.524832i \(0.824128\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) −4.00000 −0.306786
\(171\) 3.00000 0.229416
\(172\) 8.00000 0.609994
\(173\) −5.00000 −0.380143 −0.190071 0.981770i \(-0.560872\pi\)
−0.190071 + 0.981770i \(0.560872\pi\)
\(174\) 0 0
\(175\) 2.00000 0.151186
\(176\) 12.0000 0.904534
\(177\) 0 0
\(178\) −28.0000 −2.09869
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 12.0000 0.894427
\(181\) −8.00000 −0.594635 −0.297318 0.954779i \(-0.596092\pi\)
−0.297318 + 0.954779i \(0.596092\pi\)
\(182\) −16.0000 −1.18600
\(183\) 0 0
\(184\) 0 0
\(185\) 4.00000 0.294086
\(186\) 0 0
\(187\) −3.00000 −0.219382
\(188\) 24.0000 1.75038
\(189\) 0 0
\(190\) 4.00000 0.290191
\(191\) 8.00000 0.578860 0.289430 0.957199i \(-0.406534\pi\)
0.289430 + 0.957199i \(0.406534\pi\)
\(192\) 0 0
\(193\) 2.00000 0.143963 0.0719816 0.997406i \(-0.477068\pi\)
0.0719816 + 0.997406i \(0.477068\pi\)
\(194\) −38.0000 −2.72824
\(195\) 0 0
\(196\) −6.00000 −0.428571
\(197\) −12.0000 −0.854965 −0.427482 0.904024i \(-0.640599\pi\)
−0.427482 + 0.904024i \(0.640599\pi\)
\(198\) 18.0000 1.27920
\(199\) −20.0000 −1.41776 −0.708881 0.705328i \(-0.750800\pi\)
−0.708881 + 0.705328i \(0.750800\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −8.00000 −0.562878
\(203\) 12.0000 0.842235
\(204\) 0 0
\(205\) 0 0
\(206\) −20.0000 −1.39347
\(207\) −3.00000 −0.208514
\(208\) −16.0000 −1.10940
\(209\) 3.00000 0.207514
\(210\) 0 0
\(211\) −1.00000 −0.0688428 −0.0344214 0.999407i \(-0.510959\pi\)
−0.0344214 + 0.999407i \(0.510959\pi\)
\(212\) −6.00000 −0.412082
\(213\) 0 0
\(214\) −20.0000 −1.36717
\(215\) −8.00000 −0.545595
\(216\) 0 0
\(217\) 16.0000 1.08615
\(218\) 6.00000 0.406371
\(219\) 0 0
\(220\) 12.0000 0.809040
\(221\) 4.00000 0.269069
\(222\) 0 0
\(223\) 21.0000 1.40626 0.703132 0.711059i \(-0.251784\pi\)
0.703132 + 0.711059i \(0.251784\pi\)
\(224\) 16.0000 1.06904
\(225\) 3.00000 0.200000
\(226\) −4.00000 −0.266076
\(227\) 12.0000 0.796468 0.398234 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) 0 0
\(229\) 6.00000 0.396491 0.198246 0.980152i \(-0.436476\pi\)
0.198246 + 0.980152i \(0.436476\pi\)
\(230\) −4.00000 −0.263752
\(231\) 0 0
\(232\) 0 0
\(233\) −3.00000 −0.196537 −0.0982683 0.995160i \(-0.531330\pi\)
−0.0982683 + 0.995160i \(0.531330\pi\)
\(234\) −24.0000 −1.56893
\(235\) −24.0000 −1.56559
\(236\) 2.00000 0.130189
\(237\) 0 0
\(238\) −4.00000 −0.259281
\(239\) −7.00000 −0.452792 −0.226396 0.974035i \(-0.572694\pi\)
−0.226396 + 0.974035i \(0.572694\pi\)
\(240\) 0 0
\(241\) 12.0000 0.772988 0.386494 0.922292i \(-0.373686\pi\)
0.386494 + 0.922292i \(0.373686\pi\)
\(242\) −4.00000 −0.257130
\(243\) 0 0
\(244\) 10.0000 0.640184
\(245\) 6.00000 0.383326
\(246\) 0 0
\(247\) −4.00000 −0.254514
\(248\) 0 0
\(249\) 0 0
\(250\) 24.0000 1.51789
\(251\) −21.0000 −1.32551 −0.662754 0.748837i \(-0.730613\pi\)
−0.662754 + 0.748837i \(0.730613\pi\)
\(252\) 12.0000 0.755929
\(253\) −3.00000 −0.188608
\(254\) 40.0000 2.50982
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) −15.0000 −0.935674 −0.467837 0.883815i \(-0.654967\pi\)
−0.467837 + 0.883815i \(0.654967\pi\)
\(258\) 0 0
\(259\) 4.00000 0.248548
\(260\) −16.0000 −0.992278
\(261\) 18.0000 1.11417
\(262\) −40.0000 −2.47121
\(263\) 23.0000 1.41824 0.709120 0.705087i \(-0.249092\pi\)
0.709120 + 0.705087i \(0.249092\pi\)
\(264\) 0 0
\(265\) 6.00000 0.368577
\(266\) 4.00000 0.245256
\(267\) 0 0
\(268\) 20.0000 1.22169
\(269\) −17.0000 −1.03651 −0.518254 0.855227i \(-0.673418\pi\)
−0.518254 + 0.855227i \(0.673418\pi\)
\(270\) 0 0
\(271\) −25.0000 −1.51864 −0.759321 0.650716i \(-0.774469\pi\)
−0.759321 + 0.650716i \(0.774469\pi\)
\(272\) −4.00000 −0.242536
\(273\) 0 0
\(274\) 18.0000 1.08742
\(275\) 3.00000 0.180907
\(276\) 0 0
\(277\) 8.00000 0.480673 0.240337 0.970690i \(-0.422742\pi\)
0.240337 + 0.970690i \(0.422742\pi\)
\(278\) −16.0000 −0.959616
\(279\) 24.0000 1.43684
\(280\) 0 0
\(281\) −27.0000 −1.61068 −0.805342 0.592810i \(-0.798019\pi\)
−0.805342 + 0.592810i \(0.798019\pi\)
\(282\) 0 0
\(283\) −4.00000 −0.237775 −0.118888 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) −8.00000 −0.474713
\(285\) 0 0
\(286\) −24.0000 −1.41915
\(287\) 0 0
\(288\) 24.0000 1.41421
\(289\) 1.00000 0.0588235
\(290\) 24.0000 1.40933
\(291\) 0 0
\(292\) 30.0000 1.75562
\(293\) 9.00000 0.525786 0.262893 0.964825i \(-0.415323\pi\)
0.262893 + 0.964825i \(0.415323\pi\)
\(294\) 0 0
\(295\) −2.00000 −0.116445
\(296\) 0 0
\(297\) 0 0
\(298\) −44.0000 −2.54885
\(299\) 4.00000 0.231326
\(300\) 0 0
\(301\) −8.00000 −0.461112
\(302\) −32.0000 −1.84139
\(303\) 0 0
\(304\) 4.00000 0.229416
\(305\) −10.0000 −0.572598
\(306\) −6.00000 −0.342997
\(307\) −15.0000 −0.856095 −0.428048 0.903756i \(-0.640798\pi\)
−0.428048 + 0.903756i \(0.640798\pi\)
\(308\) 12.0000 0.683763
\(309\) 0 0
\(310\) 32.0000 1.81748
\(311\) 10.0000 0.567048 0.283524 0.958965i \(-0.408496\pi\)
0.283524 + 0.958965i \(0.408496\pi\)
\(312\) 0 0
\(313\) −10.0000 −0.565233 −0.282617 0.959233i \(-0.591202\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) 20.0000 1.12867
\(315\) −12.0000 −0.676123
\(316\) 12.0000 0.675053
\(317\) −28.0000 −1.57264 −0.786318 0.617822i \(-0.788015\pi\)
−0.786318 + 0.617822i \(0.788015\pi\)
\(318\) 0 0
\(319\) 18.0000 1.00781
\(320\) 16.0000 0.894427
\(321\) 0 0
\(322\) −4.00000 −0.222911
\(323\) −1.00000 −0.0556415
\(324\) 18.0000 1.00000
\(325\) −4.00000 −0.221880
\(326\) 24.0000 1.32924
\(327\) 0 0
\(328\) 0 0
\(329\) −24.0000 −1.32316
\(330\) 0 0
\(331\) −8.00000 −0.439720 −0.219860 0.975531i \(-0.570560\pi\)
−0.219860 + 0.975531i \(0.570560\pi\)
\(332\) 4.00000 0.219529
\(333\) 6.00000 0.328798
\(334\) −44.0000 −2.40757
\(335\) −20.0000 −1.09272
\(336\) 0 0
\(337\) 13.0000 0.708155 0.354078 0.935216i \(-0.384795\pi\)
0.354078 + 0.935216i \(0.384795\pi\)
\(338\) 6.00000 0.326357
\(339\) 0 0
\(340\) −4.00000 −0.216930
\(341\) 24.0000 1.29967
\(342\) 6.00000 0.324443
\(343\) 20.0000 1.07990
\(344\) 0 0
\(345\) 0 0
\(346\) −10.0000 −0.537603
\(347\) 5.00000 0.268414 0.134207 0.990953i \(-0.457151\pi\)
0.134207 + 0.990953i \(0.457151\pi\)
\(348\) 0 0
\(349\) 6.00000 0.321173 0.160586 0.987022i \(-0.448662\pi\)
0.160586 + 0.987022i \(0.448662\pi\)
\(350\) 4.00000 0.213809
\(351\) 0 0
\(352\) 24.0000 1.27920
\(353\) −6.00000 −0.319348 −0.159674 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(354\) 0 0
\(355\) 8.00000 0.424596
\(356\) −28.0000 −1.48400
\(357\) 0 0
\(358\) 24.0000 1.26844
\(359\) 15.0000 0.791670 0.395835 0.918322i \(-0.370455\pi\)
0.395835 + 0.918322i \(0.370455\pi\)
\(360\) 0 0
\(361\) −18.0000 −0.947368
\(362\) −16.0000 −0.840941
\(363\) 0 0
\(364\) −16.0000 −0.838628
\(365\) −30.0000 −1.57027
\(366\) 0 0
\(367\) −5.00000 −0.260998 −0.130499 0.991448i \(-0.541658\pi\)
−0.130499 + 0.991448i \(0.541658\pi\)
\(368\) −4.00000 −0.208514
\(369\) 0 0
\(370\) 8.00000 0.415900
\(371\) 6.00000 0.311504
\(372\) 0 0
\(373\) −3.00000 −0.155334 −0.0776671 0.996979i \(-0.524747\pi\)
−0.0776671 + 0.996979i \(0.524747\pi\)
\(374\) −6.00000 −0.310253
\(375\) 0 0
\(376\) 0 0
\(377\) −24.0000 −1.23606
\(378\) 0 0
\(379\) 28.0000 1.43826 0.719132 0.694874i \(-0.244540\pi\)
0.719132 + 0.694874i \(0.244540\pi\)
\(380\) 4.00000 0.205196
\(381\) 0 0
\(382\) 16.0000 0.818631
\(383\) −3.00000 −0.153293 −0.0766464 0.997058i \(-0.524421\pi\)
−0.0766464 + 0.997058i \(0.524421\pi\)
\(384\) 0 0
\(385\) −12.0000 −0.611577
\(386\) 4.00000 0.203595
\(387\) −12.0000 −0.609994
\(388\) −38.0000 −1.92916
\(389\) −6.00000 −0.304212 −0.152106 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 0 0
\(391\) 1.00000 0.0505722
\(392\) 0 0
\(393\) 0 0
\(394\) −24.0000 −1.20910
\(395\) −12.0000 −0.603786
\(396\) 18.0000 0.904534
\(397\) −9.00000 −0.451697 −0.225849 0.974162i \(-0.572515\pi\)
−0.225849 + 0.974162i \(0.572515\pi\)
\(398\) −40.0000 −2.00502
\(399\) 0 0
\(400\) 4.00000 0.200000
\(401\) 7.00000 0.349563 0.174782 0.984607i \(-0.444078\pi\)
0.174782 + 0.984607i \(0.444078\pi\)
\(402\) 0 0
\(403\) −32.0000 −1.59403
\(404\) −8.00000 −0.398015
\(405\) −18.0000 −0.894427
\(406\) 24.0000 1.19110
\(407\) 6.00000 0.297409
\(408\) 0 0
\(409\) 20.0000 0.988936 0.494468 0.869196i \(-0.335363\pi\)
0.494468 + 0.869196i \(0.335363\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −20.0000 −0.985329
\(413\) −2.00000 −0.0984136
\(414\) −6.00000 −0.294884
\(415\) −4.00000 −0.196352
\(416\) −32.0000 −1.56893
\(417\) 0 0
\(418\) 6.00000 0.293470
\(419\) −7.00000 −0.341972 −0.170986 0.985273i \(-0.554695\pi\)
−0.170986 + 0.985273i \(0.554695\pi\)
\(420\) 0 0
\(421\) 10.0000 0.487370 0.243685 0.969854i \(-0.421644\pi\)
0.243685 + 0.969854i \(0.421644\pi\)
\(422\) −2.00000 −0.0973585
\(423\) −36.0000 −1.75038
\(424\) 0 0
\(425\) −1.00000 −0.0485071
\(426\) 0 0
\(427\) −10.0000 −0.483934
\(428\) −20.0000 −0.966736
\(429\) 0 0
\(430\) −16.0000 −0.771589
\(431\) −15.0000 −0.722525 −0.361262 0.932464i \(-0.617654\pi\)
−0.361262 + 0.932464i \(0.617654\pi\)
\(432\) 0 0
\(433\) 23.0000 1.10531 0.552655 0.833410i \(-0.313615\pi\)
0.552655 + 0.833410i \(0.313615\pi\)
\(434\) 32.0000 1.53605
\(435\) 0 0
\(436\) 6.00000 0.287348
\(437\) −1.00000 −0.0478365
\(438\) 0 0
\(439\) 40.0000 1.90910 0.954548 0.298057i \(-0.0963387\pi\)
0.954548 + 0.298057i \(0.0963387\pi\)
\(440\) 0 0
\(441\) 9.00000 0.428571
\(442\) 8.00000 0.380521
\(443\) 16.0000 0.760183 0.380091 0.924949i \(-0.375893\pi\)
0.380091 + 0.924949i \(0.375893\pi\)
\(444\) 0 0
\(445\) 28.0000 1.32733
\(446\) 42.0000 1.98876
\(447\) 0 0
\(448\) 16.0000 0.755929
\(449\) 28.0000 1.32140 0.660701 0.750649i \(-0.270259\pi\)
0.660701 + 0.750649i \(0.270259\pi\)
\(450\) 6.00000 0.282843
\(451\) 0 0
\(452\) −4.00000 −0.188144
\(453\) 0 0
\(454\) 24.0000 1.12638
\(455\) 16.0000 0.750092
\(456\) 0 0
\(457\) −42.0000 −1.96468 −0.982339 0.187112i \(-0.940087\pi\)
−0.982339 + 0.187112i \(0.940087\pi\)
\(458\) 12.0000 0.560723
\(459\) 0 0
\(460\) −4.00000 −0.186501
\(461\) 13.0000 0.605470 0.302735 0.953075i \(-0.402100\pi\)
0.302735 + 0.953075i \(0.402100\pi\)
\(462\) 0 0
\(463\) 14.0000 0.650635 0.325318 0.945605i \(-0.394529\pi\)
0.325318 + 0.945605i \(0.394529\pi\)
\(464\) 24.0000 1.11417
\(465\) 0 0
\(466\) −6.00000 −0.277945
\(467\) 6.00000 0.277647 0.138823 0.990317i \(-0.455668\pi\)
0.138823 + 0.990317i \(0.455668\pi\)
\(468\) −24.0000 −1.10940
\(469\) −20.0000 −0.923514
\(470\) −48.0000 −2.21407
\(471\) 0 0
\(472\) 0 0
\(473\) −12.0000 −0.551761
\(474\) 0 0
\(475\) 1.00000 0.0458831
\(476\) −4.00000 −0.183340
\(477\) 9.00000 0.412082
\(478\) −14.0000 −0.640345
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) −8.00000 −0.364769
\(482\) 24.0000 1.09317
\(483\) 0 0
\(484\) −4.00000 −0.181818
\(485\) 38.0000 1.72549
\(486\) 0 0
\(487\) −22.0000 −0.996915 −0.498458 0.866914i \(-0.666100\pi\)
−0.498458 + 0.866914i \(0.666100\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 12.0000 0.542105
\(491\) 17.0000 0.767199 0.383600 0.923499i \(-0.374684\pi\)
0.383600 + 0.923499i \(0.374684\pi\)
\(492\) 0 0
\(493\) −6.00000 −0.270226
\(494\) −8.00000 −0.359937
\(495\) −18.0000 −0.809040
\(496\) 32.0000 1.43684
\(497\) 8.00000 0.358849
\(498\) 0 0
\(499\) 20.0000 0.895323 0.447661 0.894203i \(-0.352257\pi\)
0.447661 + 0.894203i \(0.352257\pi\)
\(500\) 24.0000 1.07331
\(501\) 0 0
\(502\) −42.0000 −1.87455
\(503\) 27.0000 1.20387 0.601935 0.798545i \(-0.294397\pi\)
0.601935 + 0.798545i \(0.294397\pi\)
\(504\) 0 0
\(505\) 8.00000 0.355995
\(506\) −6.00000 −0.266733
\(507\) 0 0
\(508\) 40.0000 1.77471
\(509\) −26.0000 −1.15243 −0.576215 0.817298i \(-0.695471\pi\)
−0.576215 + 0.817298i \(0.695471\pi\)
\(510\) 0 0
\(511\) −30.0000 −1.32712
\(512\) 32.0000 1.41421
\(513\) 0 0
\(514\) −30.0000 −1.32324
\(515\) 20.0000 0.881305
\(516\) 0 0
\(517\) −36.0000 −1.58328
\(518\) 8.00000 0.351500
\(519\) 0 0
\(520\) 0 0
\(521\) 32.0000 1.40195 0.700973 0.713188i \(-0.252749\pi\)
0.700973 + 0.713188i \(0.252749\pi\)
\(522\) 36.0000 1.57568
\(523\) 9.00000 0.393543 0.196771 0.980449i \(-0.436954\pi\)
0.196771 + 0.980449i \(0.436954\pi\)
\(524\) −40.0000 −1.74741
\(525\) 0 0
\(526\) 46.0000 2.00570
\(527\) −8.00000 −0.348485
\(528\) 0 0
\(529\) −22.0000 −0.956522
\(530\) 12.0000 0.521247
\(531\) −3.00000 −0.130189
\(532\) 4.00000 0.173422
\(533\) 0 0
\(534\) 0 0
\(535\) 20.0000 0.864675
\(536\) 0 0
\(537\) 0 0
\(538\) −34.0000 −1.46584
\(539\) 9.00000 0.387657
\(540\) 0 0
\(541\) −42.0000 −1.80572 −0.902861 0.429934i \(-0.858537\pi\)
−0.902861 + 0.429934i \(0.858537\pi\)
\(542\) −50.0000 −2.14768
\(543\) 0 0
\(544\) −8.00000 −0.342997
\(545\) −6.00000 −0.257012
\(546\) 0 0
\(547\) 16.0000 0.684111 0.342055 0.939680i \(-0.388877\pi\)
0.342055 + 0.939680i \(0.388877\pi\)
\(548\) 18.0000 0.768922
\(549\) −15.0000 −0.640184
\(550\) 6.00000 0.255841
\(551\) 6.00000 0.255609
\(552\) 0 0
\(553\) −12.0000 −0.510292
\(554\) 16.0000 0.679775
\(555\) 0 0
\(556\) −16.0000 −0.678551
\(557\) −27.0000 −1.14403 −0.572013 0.820244i \(-0.693837\pi\)
−0.572013 + 0.820244i \(0.693837\pi\)
\(558\) 48.0000 2.03200
\(559\) 16.0000 0.676728
\(560\) −16.0000 −0.676123
\(561\) 0 0
\(562\) −54.0000 −2.27785
\(563\) 14.0000 0.590030 0.295015 0.955493i \(-0.404675\pi\)
0.295015 + 0.955493i \(0.404675\pi\)
\(564\) 0 0
\(565\) 4.00000 0.168281
\(566\) −8.00000 −0.336265
\(567\) −18.0000 −0.755929
\(568\) 0 0
\(569\) 4.00000 0.167689 0.0838444 0.996479i \(-0.473280\pi\)
0.0838444 + 0.996479i \(0.473280\pi\)
\(570\) 0 0
\(571\) −20.0000 −0.836974 −0.418487 0.908223i \(-0.637439\pi\)
−0.418487 + 0.908223i \(0.637439\pi\)
\(572\) −24.0000 −1.00349
\(573\) 0 0
\(574\) 0 0
\(575\) −1.00000 −0.0417029
\(576\) 24.0000 1.00000
\(577\) 43.0000 1.79011 0.895057 0.445952i \(-0.147135\pi\)
0.895057 + 0.445952i \(0.147135\pi\)
\(578\) 2.00000 0.0831890
\(579\) 0 0
\(580\) 24.0000 0.996546
\(581\) −4.00000 −0.165948
\(582\) 0 0
\(583\) 9.00000 0.372742
\(584\) 0 0
\(585\) 24.0000 0.992278
\(586\) 18.0000 0.743573
\(587\) 28.0000 1.15568 0.577842 0.816149i \(-0.303895\pi\)
0.577842 + 0.816149i \(0.303895\pi\)
\(588\) 0 0
\(589\) 8.00000 0.329634
\(590\) −4.00000 −0.164677
\(591\) 0 0
\(592\) 8.00000 0.328798
\(593\) −33.0000 −1.35515 −0.677574 0.735455i \(-0.736969\pi\)
−0.677574 + 0.735455i \(0.736969\pi\)
\(594\) 0 0
\(595\) 4.00000 0.163984
\(596\) −44.0000 −1.80231
\(597\) 0 0
\(598\) 8.00000 0.327144
\(599\) 12.0000 0.490307 0.245153 0.969484i \(-0.421162\pi\)
0.245153 + 0.969484i \(0.421162\pi\)
\(600\) 0 0
\(601\) −14.0000 −0.571072 −0.285536 0.958368i \(-0.592172\pi\)
−0.285536 + 0.958368i \(0.592172\pi\)
\(602\) −16.0000 −0.652111
\(603\) −30.0000 −1.22169
\(604\) −32.0000 −1.30206
\(605\) 4.00000 0.162623
\(606\) 0 0
\(607\) −16.0000 −0.649420 −0.324710 0.945814i \(-0.605267\pi\)
−0.324710 + 0.945814i \(0.605267\pi\)
\(608\) 8.00000 0.324443
\(609\) 0 0
\(610\) −20.0000 −0.809776
\(611\) 48.0000 1.94187
\(612\) −6.00000 −0.242536
\(613\) −14.0000 −0.565455 −0.282727 0.959200i \(-0.591239\pi\)
−0.282727 + 0.959200i \(0.591239\pi\)
\(614\) −30.0000 −1.21070
\(615\) 0 0
\(616\) 0 0
\(617\) −18.0000 −0.724653 −0.362326 0.932051i \(-0.618017\pi\)
−0.362326 + 0.932051i \(0.618017\pi\)
\(618\) 0 0
\(619\) 24.0000 0.964641 0.482321 0.875995i \(-0.339794\pi\)
0.482321 + 0.875995i \(0.339794\pi\)
\(620\) 32.0000 1.28515
\(621\) 0 0
\(622\) 20.0000 0.801927
\(623\) 28.0000 1.12180
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) −20.0000 −0.799361
\(627\) 0 0
\(628\) 20.0000 0.798087
\(629\) −2.00000 −0.0797452
\(630\) −24.0000 −0.956183
\(631\) 13.0000 0.517522 0.258761 0.965941i \(-0.416686\pi\)
0.258761 + 0.965941i \(0.416686\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) −56.0000 −2.22404
\(635\) −40.0000 −1.58735
\(636\) 0 0
\(637\) −12.0000 −0.475457
\(638\) 36.0000 1.42525
\(639\) 12.0000 0.474713
\(640\) 0 0
\(641\) 24.0000 0.947943 0.473972 0.880540i \(-0.342820\pi\)
0.473972 + 0.880540i \(0.342820\pi\)
\(642\) 0 0
\(643\) 26.0000 1.02534 0.512670 0.858586i \(-0.328656\pi\)
0.512670 + 0.858586i \(0.328656\pi\)
\(644\) −4.00000 −0.157622
\(645\) 0 0
\(646\) −2.00000 −0.0786889
\(647\) −43.0000 −1.69050 −0.845252 0.534368i \(-0.820550\pi\)
−0.845252 + 0.534368i \(0.820550\pi\)
\(648\) 0 0
\(649\) −3.00000 −0.117760
\(650\) −8.00000 −0.313786
\(651\) 0 0
\(652\) 24.0000 0.939913
\(653\) 36.0000 1.40879 0.704394 0.709809i \(-0.251219\pi\)
0.704394 + 0.709809i \(0.251219\pi\)
\(654\) 0 0
\(655\) 40.0000 1.56293
\(656\) 0 0
\(657\) −45.0000 −1.75562
\(658\) −48.0000 −1.87123
\(659\) 10.0000 0.389545 0.194772 0.980848i \(-0.437603\pi\)
0.194772 + 0.980848i \(0.437603\pi\)
\(660\) 0 0
\(661\) −34.0000 −1.32245 −0.661223 0.750189i \(-0.729962\pi\)
−0.661223 + 0.750189i \(0.729962\pi\)
\(662\) −16.0000 −0.621858
\(663\) 0 0
\(664\) 0 0
\(665\) −4.00000 −0.155113
\(666\) 12.0000 0.464991
\(667\) −6.00000 −0.232321
\(668\) −44.0000 −1.70241
\(669\) 0 0
\(670\) −40.0000 −1.54533
\(671\) −15.0000 −0.579069
\(672\) 0 0
\(673\) 11.0000 0.424019 0.212009 0.977268i \(-0.431999\pi\)
0.212009 + 0.977268i \(0.431999\pi\)
\(674\) 26.0000 1.00148
\(675\) 0 0
\(676\) 6.00000 0.230769
\(677\) −42.0000 −1.61419 −0.807096 0.590421i \(-0.798962\pi\)
−0.807096 + 0.590421i \(0.798962\pi\)
\(678\) 0 0
\(679\) 38.0000 1.45831
\(680\) 0 0
\(681\) 0 0
\(682\) 48.0000 1.83801
\(683\) −4.00000 −0.153056 −0.0765279 0.997067i \(-0.524383\pi\)
−0.0765279 + 0.997067i \(0.524383\pi\)
\(684\) 6.00000 0.229416
\(685\) −18.0000 −0.687745
\(686\) 40.0000 1.52721
\(687\) 0 0
\(688\) −16.0000 −0.609994
\(689\) −12.0000 −0.457164
\(690\) 0 0
\(691\) −35.0000 −1.33146 −0.665731 0.746191i \(-0.731880\pi\)
−0.665731 + 0.746191i \(0.731880\pi\)
\(692\) −10.0000 −0.380143
\(693\) −18.0000 −0.683763
\(694\) 10.0000 0.379595
\(695\) 16.0000 0.606915
\(696\) 0 0
\(697\) 0 0
\(698\) 12.0000 0.454207
\(699\) 0 0
\(700\) 4.00000 0.151186
\(701\) −48.0000 −1.81293 −0.906467 0.422276i \(-0.861231\pi\)
−0.906467 + 0.422276i \(0.861231\pi\)
\(702\) 0 0
\(703\) 2.00000 0.0754314
\(704\) 24.0000 0.904534
\(705\) 0 0
\(706\) −12.0000 −0.451626
\(707\) 8.00000 0.300871
\(708\) 0 0
\(709\) 20.0000 0.751116 0.375558 0.926799i \(-0.377451\pi\)
0.375558 + 0.926799i \(0.377451\pi\)
\(710\) 16.0000 0.600469
\(711\) −18.0000 −0.675053
\(712\) 0 0
\(713\) −8.00000 −0.299602
\(714\) 0 0
\(715\) 24.0000 0.897549
\(716\) 24.0000 0.896922
\(717\) 0 0
\(718\) 30.0000 1.11959
\(719\) −8.00000 −0.298350 −0.149175 0.988811i \(-0.547662\pi\)
−0.149175 + 0.988811i \(0.547662\pi\)
\(720\) −24.0000 −0.894427
\(721\) 20.0000 0.744839
\(722\) −36.0000 −1.33978
\(723\) 0 0
\(724\) −16.0000 −0.594635
\(725\) 6.00000 0.222834
\(726\) 0 0
\(727\) 13.0000 0.482143 0.241072 0.970507i \(-0.422501\pi\)
0.241072 + 0.970507i \(0.422501\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) −60.0000 −2.22070
\(731\) 4.00000 0.147945
\(732\) 0 0
\(733\) 37.0000 1.36663 0.683313 0.730125i \(-0.260538\pi\)
0.683313 + 0.730125i \(0.260538\pi\)
\(734\) −10.0000 −0.369107
\(735\) 0 0
\(736\) −8.00000 −0.294884
\(737\) −30.0000 −1.10506
\(738\) 0 0
\(739\) −26.0000 −0.956425 −0.478213 0.878244i \(-0.658715\pi\)
−0.478213 + 0.878244i \(0.658715\pi\)
\(740\) 8.00000 0.294086
\(741\) 0 0
\(742\) 12.0000 0.440534
\(743\) −12.0000 −0.440237 −0.220119 0.975473i \(-0.570644\pi\)
−0.220119 + 0.975473i \(0.570644\pi\)
\(744\) 0 0
\(745\) 44.0000 1.61204
\(746\) −6.00000 −0.219676
\(747\) −6.00000 −0.219529
\(748\) −6.00000 −0.219382
\(749\) 20.0000 0.730784
\(750\) 0 0
\(751\) 53.0000 1.93400 0.966999 0.254781i \(-0.0820034\pi\)
0.966999 + 0.254781i \(0.0820034\pi\)
\(752\) −48.0000 −1.75038
\(753\) 0 0
\(754\) −48.0000 −1.74806
\(755\) 32.0000 1.16460
\(756\) 0 0
\(757\) 1.00000 0.0363456 0.0181728 0.999835i \(-0.494215\pi\)
0.0181728 + 0.999835i \(0.494215\pi\)
\(758\) 56.0000 2.03401
\(759\) 0 0
\(760\) 0 0
\(761\) −25.0000 −0.906249 −0.453125 0.891447i \(-0.649691\pi\)
−0.453125 + 0.891447i \(0.649691\pi\)
\(762\) 0 0
\(763\) −6.00000 −0.217215
\(764\) 16.0000 0.578860
\(765\) 6.00000 0.216930
\(766\) −6.00000 −0.216789
\(767\) 4.00000 0.144432
\(768\) 0 0
\(769\) 40.0000 1.44244 0.721218 0.692708i \(-0.243582\pi\)
0.721218 + 0.692708i \(0.243582\pi\)
\(770\) −24.0000 −0.864900
\(771\) 0 0
\(772\) 4.00000 0.143963
\(773\) 4.00000 0.143870 0.0719350 0.997409i \(-0.477083\pi\)
0.0719350 + 0.997409i \(0.477083\pi\)
\(774\) −24.0000 −0.862662
\(775\) 8.00000 0.287368
\(776\) 0 0
\(777\) 0 0
\(778\) −12.0000 −0.430221
\(779\) 0 0
\(780\) 0 0
\(781\) 12.0000 0.429394
\(782\) 2.00000 0.0715199
\(783\) 0 0
\(784\) 12.0000 0.428571
\(785\) −20.0000 −0.713831
\(786\) 0 0
\(787\) −14.0000 −0.499046 −0.249523 0.968369i \(-0.580274\pi\)
−0.249523 + 0.968369i \(0.580274\pi\)
\(788\) −24.0000 −0.854965
\(789\) 0 0
\(790\) −24.0000 −0.853882
\(791\) 4.00000 0.142224
\(792\) 0 0
\(793\) 20.0000 0.710221
\(794\) −18.0000 −0.638796
\(795\) 0 0
\(796\) −40.0000 −1.41776
\(797\) −44.0000 −1.55856 −0.779280 0.626676i \(-0.784415\pi\)
−0.779280 + 0.626676i \(0.784415\pi\)
\(798\) 0 0
\(799\) 12.0000 0.424529
\(800\) 8.00000 0.282843
\(801\) 42.0000 1.48400
\(802\) 14.0000 0.494357
\(803\) −45.0000 −1.58802
\(804\) 0 0
\(805\) 4.00000 0.140981
\(806\) −64.0000 −2.25430
\(807\) 0 0
\(808\) 0 0
\(809\) 50.0000 1.75791 0.878953 0.476908i \(-0.158243\pi\)
0.878953 + 0.476908i \(0.158243\pi\)
\(810\) −36.0000 −1.26491
\(811\) 13.0000 0.456492 0.228246 0.973604i \(-0.426701\pi\)
0.228246 + 0.973604i \(0.426701\pi\)
\(812\) 24.0000 0.842235
\(813\) 0 0
\(814\) 12.0000 0.420600
\(815\) −24.0000 −0.840683
\(816\) 0 0
\(817\) −4.00000 −0.139942
\(818\) 40.0000 1.39857
\(819\) 24.0000 0.838628
\(820\) 0 0
\(821\) −27.0000 −0.942306 −0.471153 0.882051i \(-0.656162\pi\)
−0.471153 + 0.882051i \(0.656162\pi\)
\(822\) 0 0
\(823\) 29.0000 1.01088 0.505438 0.862863i \(-0.331331\pi\)
0.505438 + 0.862863i \(0.331331\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) −4.00000 −0.139178
\(827\) −18.0000 −0.625921 −0.312961 0.949766i \(-0.601321\pi\)
−0.312961 + 0.949766i \(0.601321\pi\)
\(828\) −6.00000 −0.208514
\(829\) 5.00000 0.173657 0.0868286 0.996223i \(-0.472327\pi\)
0.0868286 + 0.996223i \(0.472327\pi\)
\(830\) −8.00000 −0.277684
\(831\) 0 0
\(832\) −32.0000 −1.10940
\(833\) −3.00000 −0.103944
\(834\) 0 0
\(835\) 44.0000 1.52268
\(836\) 6.00000 0.207514
\(837\) 0 0
\(838\) −14.0000 −0.483622
\(839\) −31.0000 −1.07024 −0.535119 0.844776i \(-0.679733\pi\)
−0.535119 + 0.844776i \(0.679733\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 20.0000 0.689246
\(843\) 0 0
\(844\) −2.00000 −0.0688428
\(845\) −6.00000 −0.206406
\(846\) −72.0000 −2.47541
\(847\) 4.00000 0.137442
\(848\) 12.0000 0.412082
\(849\) 0 0
\(850\) −2.00000 −0.0685994
\(851\) −2.00000 −0.0685591
\(852\) 0 0
\(853\) −46.0000 −1.57501 −0.787505 0.616308i \(-0.788628\pi\)
−0.787505 + 0.616308i \(0.788628\pi\)
\(854\) −20.0000 −0.684386
\(855\) −6.00000 −0.205196
\(856\) 0 0
\(857\) −47.0000 −1.60549 −0.802745 0.596323i \(-0.796628\pi\)
−0.802745 + 0.596323i \(0.796628\pi\)
\(858\) 0 0
\(859\) −50.0000 −1.70598 −0.852989 0.521929i \(-0.825213\pi\)
−0.852989 + 0.521929i \(0.825213\pi\)
\(860\) −16.0000 −0.545595
\(861\) 0 0
\(862\) −30.0000 −1.02180
\(863\) 4.00000 0.136162 0.0680808 0.997680i \(-0.478312\pi\)
0.0680808 + 0.997680i \(0.478312\pi\)
\(864\) 0 0
\(865\) 10.0000 0.340010
\(866\) 46.0000 1.56314
\(867\) 0 0
\(868\) 32.0000 1.08615
\(869\) −18.0000 −0.610608
\(870\) 0 0
\(871\) 40.0000 1.35535
\(872\) 0 0
\(873\) 57.0000 1.92916
\(874\) −2.00000 −0.0676510
\(875\) −24.0000 −0.811348
\(876\) 0 0
\(877\) −18.0000 −0.607817 −0.303908 0.952701i \(-0.598292\pi\)
−0.303908 + 0.952701i \(0.598292\pi\)
\(878\) 80.0000 2.69987
\(879\) 0 0
\(880\) −24.0000 −0.809040
\(881\) −19.0000 −0.640126 −0.320063 0.947396i \(-0.603704\pi\)
−0.320063 + 0.947396i \(0.603704\pi\)
\(882\) 18.0000 0.606092
\(883\) 45.0000 1.51437 0.757185 0.653200i \(-0.226574\pi\)
0.757185 + 0.653200i \(0.226574\pi\)
\(884\) 8.00000 0.269069
\(885\) 0 0
\(886\) 32.0000 1.07506
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 0 0
\(889\) −40.0000 −1.34156
\(890\) 56.0000 1.87712
\(891\) −27.0000 −0.904534
\(892\) 42.0000 1.40626
\(893\) −12.0000 −0.401565
\(894\) 0 0
\(895\) −24.0000 −0.802232
\(896\) 0 0
\(897\) 0 0
\(898\) 56.0000 1.86874
\(899\) 48.0000 1.60089
\(900\) 6.00000 0.200000
\(901\) −3.00000 −0.0999445
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 16.0000 0.531858
\(906\) 0 0
\(907\) 50.0000 1.66022 0.830111 0.557598i \(-0.188277\pi\)
0.830111 + 0.557598i \(0.188277\pi\)
\(908\) 24.0000 0.796468
\(909\) 12.0000 0.398015
\(910\) 32.0000 1.06079
\(911\) −22.0000 −0.728893 −0.364446 0.931224i \(-0.618742\pi\)
−0.364446 + 0.931224i \(0.618742\pi\)
\(912\) 0 0
\(913\) −6.00000 −0.198571
\(914\) −84.0000 −2.77847
\(915\) 0 0
\(916\) 12.0000 0.396491
\(917\) 40.0000 1.32092
\(918\) 0 0
\(919\) −48.0000 −1.58337 −0.791687 0.610927i \(-0.790797\pi\)
−0.791687 + 0.610927i \(0.790797\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 26.0000 0.856264
\(923\) −16.0000 −0.526646
\(924\) 0 0
\(925\) 2.00000 0.0657596
\(926\) 28.0000 0.920137
\(927\) 30.0000 0.985329
\(928\) 48.0000 1.57568
\(929\) −33.0000 −1.08269 −0.541347 0.840799i \(-0.682086\pi\)
−0.541347 + 0.840799i \(0.682086\pi\)
\(930\) 0 0
\(931\) 3.00000 0.0983210
\(932\) −6.00000 −0.196537
\(933\) 0 0
\(934\) 12.0000 0.392652
\(935\) 6.00000 0.196221
\(936\) 0 0
\(937\) 52.0000 1.69877 0.849383 0.527777i \(-0.176974\pi\)
0.849383 + 0.527777i \(0.176974\pi\)
\(938\) −40.0000 −1.30605
\(939\) 0 0
\(940\) −48.0000 −1.56559
\(941\) −42.0000 −1.36916 −0.684580 0.728937i \(-0.740015\pi\)
−0.684580 + 0.728937i \(0.740015\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) −4.00000 −0.130189
\(945\) 0 0
\(946\) −24.0000 −0.780307
\(947\) 32.0000 1.03986 0.519930 0.854209i \(-0.325958\pi\)
0.519930 + 0.854209i \(0.325958\pi\)
\(948\) 0 0
\(949\) 60.0000 1.94768
\(950\) 2.00000 0.0648886
\(951\) 0 0
\(952\) 0 0
\(953\) 59.0000 1.91120 0.955599 0.294671i \(-0.0952101\pi\)
0.955599 + 0.294671i \(0.0952101\pi\)
\(954\) 18.0000 0.582772
\(955\) −16.0000 −0.517748
\(956\) −14.0000 −0.452792
\(957\) 0 0
\(958\) 0 0
\(959\) −18.0000 −0.581250
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) −16.0000 −0.515861
\(963\) 30.0000 0.966736
\(964\) 24.0000 0.772988
\(965\) −4.00000 −0.128765
\(966\) 0 0
\(967\) 12.0000 0.385894 0.192947 0.981209i \(-0.438195\pi\)
0.192947 + 0.981209i \(0.438195\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 76.0000 2.44021
\(971\) −15.0000 −0.481373 −0.240686 0.970603i \(-0.577373\pi\)
−0.240686 + 0.970603i \(0.577373\pi\)
\(972\) 0 0
\(973\) 16.0000 0.512936
\(974\) −44.0000 −1.40985
\(975\) 0 0
\(976\) −20.0000 −0.640184
\(977\) 6.00000 0.191957 0.0959785 0.995383i \(-0.469402\pi\)
0.0959785 + 0.995383i \(0.469402\pi\)
\(978\) 0 0
\(979\) 42.0000 1.34233
\(980\) 12.0000 0.383326
\(981\) −9.00000 −0.287348
\(982\) 34.0000 1.08498
\(983\) 3.00000 0.0956851 0.0478426 0.998855i \(-0.484765\pi\)
0.0478426 + 0.998855i \(0.484765\pi\)
\(984\) 0 0
\(985\) 24.0000 0.764704
\(986\) −12.0000 −0.382158
\(987\) 0 0
\(988\) −8.00000 −0.254514
\(989\) 4.00000 0.127193
\(990\) −36.0000 −1.14416
\(991\) 9.00000 0.285894 0.142947 0.989730i \(-0.454342\pi\)
0.142947 + 0.989730i \(0.454342\pi\)
\(992\) 64.0000 2.03200
\(993\) 0 0
\(994\) 16.0000 0.507489
\(995\) 40.0000 1.26809
\(996\) 0 0
\(997\) 46.0000 1.45683 0.728417 0.685134i \(-0.240256\pi\)
0.728417 + 0.685134i \(0.240256\pi\)
\(998\) 40.0000 1.26618
\(999\) 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 1003.2.a.d.1.1 1
3.2 odd 2 9027.2.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1003.2.a.d.1.1 1 1.1 even 1 trivial
9027.2.a.a.1.1 1 3.2 odd 2