Properties

Label 8004.2.a.g.1.7
Level $8004$
Weight $2$
Character 8004.1
Self dual yes
Analytic conductor $63.912$
Analytic rank $1$
Dimension $12$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8004.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(63.9122617778\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 3 x^{11} - 31 x^{10} + 80 x^{9} + 347 x^{8} - 697 x^{7} - 1714 x^{6} + 2146 x^{5} + 3304 x^{4} - 1156 x^{3} - 616 x^{2} + 136 x + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(-0.0891245\) of defining polynomial
Character \(\chi\) \(=\) 8004.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +0.0891245 q^{5} +2.52085 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +0.0891245 q^{5} +2.52085 q^{7} +1.00000 q^{9} -1.55149 q^{11} +6.31044 q^{13} -0.0891245 q^{15} -6.02488 q^{17} +4.11930 q^{19} -2.52085 q^{21} +1.00000 q^{23} -4.99206 q^{25} -1.00000 q^{27} -1.00000 q^{29} -6.28492 q^{31} +1.55149 q^{33} +0.224669 q^{35} -3.14158 q^{37} -6.31044 q^{39} -3.88280 q^{41} +1.91638 q^{43} +0.0891245 q^{45} -12.5261 q^{47} -0.645334 q^{49} +6.02488 q^{51} -3.50409 q^{53} -0.138276 q^{55} -4.11930 q^{57} -9.16270 q^{59} +10.3666 q^{61} +2.52085 q^{63} +0.562415 q^{65} -1.85364 q^{67} -1.00000 q^{69} +2.06730 q^{71} -0.852008 q^{73} +4.99206 q^{75} -3.91107 q^{77} -14.6715 q^{79} +1.00000 q^{81} +1.87772 q^{83} -0.536965 q^{85} +1.00000 q^{87} -11.8205 q^{89} +15.9077 q^{91} +6.28492 q^{93} +0.367131 q^{95} -5.06804 q^{97} -1.55149 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 12q^{3} - 3q^{5} + 4q^{7} + 12q^{9} + O(q^{10}) \) \( 12q - 12q^{3} - 3q^{5} + 4q^{7} + 12q^{9} - 5q^{11} - 6q^{13} + 3q^{15} - 7q^{17} - 3q^{19} - 4q^{21} + 12q^{23} + 11q^{25} - 12q^{27} - 12q^{29} + 2q^{31} + 5q^{33} - 9q^{35} - 20q^{37} + 6q^{39} - 3q^{41} + 5q^{43} - 3q^{45} - 2q^{49} + 7q^{51} - 3q^{53} + 19q^{55} + 3q^{57} - 20q^{59} - 17q^{61} + 4q^{63} - 4q^{65} - 9q^{67} - 12q^{69} + 7q^{71} - 9q^{73} - 11q^{75} - 34q^{77} + 14q^{79} + 12q^{81} + 5q^{83} - 12q^{85} + 12q^{87} - 22q^{89} - 3q^{91} - 2q^{93} - 27q^{95} + 17q^{97} - 5q^{99} + O(q^{100}) \)

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
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 0.0891245 0.0398577 0.0199289 0.999801i \(-0.493656\pi\)
0.0199289 + 0.999801i \(0.493656\pi\)
\(6\) 0 0
\(7\) 2.52085 0.952790 0.476395 0.879231i \(-0.341943\pi\)
0.476395 + 0.879231i \(0.341943\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.55149 −0.467792 −0.233896 0.972262i \(-0.575148\pi\)
−0.233896 + 0.972262i \(0.575148\pi\)
\(12\) 0 0
\(13\) 6.31044 1.75020 0.875101 0.483941i \(-0.160795\pi\)
0.875101 + 0.483941i \(0.160795\pi\)
\(14\) 0 0
\(15\) −0.0891245 −0.0230119
\(16\) 0 0
\(17\) −6.02488 −1.46125 −0.730624 0.682780i \(-0.760771\pi\)
−0.730624 + 0.682780i \(0.760771\pi\)
\(18\) 0 0
\(19\) 4.11930 0.945033 0.472516 0.881322i \(-0.343346\pi\)
0.472516 + 0.881322i \(0.343346\pi\)
\(20\) 0 0
\(21\) −2.52085 −0.550094
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) −4.99206 −0.998411
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −1.00000 −0.185695
\(30\) 0 0
\(31\) −6.28492 −1.12881 −0.564403 0.825500i \(-0.690893\pi\)
−0.564403 + 0.825500i \(0.690893\pi\)
\(32\) 0 0
\(33\) 1.55149 0.270080
\(34\) 0 0
\(35\) 0.224669 0.0379760
\(36\) 0 0
\(37\) −3.14158 −0.516472 −0.258236 0.966082i \(-0.583141\pi\)
−0.258236 + 0.966082i \(0.583141\pi\)
\(38\) 0 0
\(39\) −6.31044 −1.01048
\(40\) 0 0
\(41\) −3.88280 −0.606392 −0.303196 0.952928i \(-0.598054\pi\)
−0.303196 + 0.952928i \(0.598054\pi\)
\(42\) 0 0
\(43\) 1.91638 0.292245 0.146122 0.989267i \(-0.453321\pi\)
0.146122 + 0.989267i \(0.453321\pi\)
\(44\) 0 0
\(45\) 0.0891245 0.0132859
\(46\) 0 0
\(47\) −12.5261 −1.82712 −0.913562 0.406700i \(-0.866680\pi\)
−0.913562 + 0.406700i \(0.866680\pi\)
\(48\) 0 0
\(49\) −0.645334 −0.0921905
\(50\) 0 0
\(51\) 6.02488 0.843652
\(52\) 0 0
\(53\) −3.50409 −0.481324 −0.240662 0.970609i \(-0.577365\pi\)
−0.240662 + 0.970609i \(0.577365\pi\)
\(54\) 0 0
\(55\) −0.138276 −0.0186451
\(56\) 0 0
\(57\) −4.11930 −0.545615
\(58\) 0 0
\(59\) −9.16270 −1.19288 −0.596441 0.802657i \(-0.703419\pi\)
−0.596441 + 0.802657i \(0.703419\pi\)
\(60\) 0 0
\(61\) 10.3666 1.32731 0.663656 0.748038i \(-0.269004\pi\)
0.663656 + 0.748038i \(0.269004\pi\)
\(62\) 0 0
\(63\) 2.52085 0.317597
\(64\) 0 0
\(65\) 0.562415 0.0697590
\(66\) 0 0
\(67\) −1.85364 −0.226458 −0.113229 0.993569i \(-0.536119\pi\)
−0.113229 + 0.993569i \(0.536119\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) 2.06730 0.245343 0.122672 0.992447i \(-0.460854\pi\)
0.122672 + 0.992447i \(0.460854\pi\)
\(72\) 0 0
\(73\) −0.852008 −0.0997200 −0.0498600 0.998756i \(-0.515878\pi\)
−0.0498600 + 0.998756i \(0.515878\pi\)
\(74\) 0 0
\(75\) 4.99206 0.576433
\(76\) 0 0
\(77\) −3.91107 −0.445708
\(78\) 0 0
\(79\) −14.6715 −1.65068 −0.825339 0.564638i \(-0.809016\pi\)
−0.825339 + 0.564638i \(0.809016\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 1.87772 0.206106 0.103053 0.994676i \(-0.467139\pi\)
0.103053 + 0.994676i \(0.467139\pi\)
\(84\) 0 0
\(85\) −0.536965 −0.0582420
\(86\) 0 0
\(87\) 1.00000 0.107211
\(88\) 0 0
\(89\) −11.8205 −1.25297 −0.626484 0.779434i \(-0.715507\pi\)
−0.626484 + 0.779434i \(0.715507\pi\)
\(90\) 0 0
\(91\) 15.9077 1.66757
\(92\) 0 0
\(93\) 6.28492 0.651716
\(94\) 0 0
\(95\) 0.367131 0.0376668
\(96\) 0 0
\(97\) −5.06804 −0.514581 −0.257291 0.966334i \(-0.582830\pi\)
−0.257291 + 0.966334i \(0.582830\pi\)
\(98\) 0 0
\(99\) −1.55149 −0.155931
\(100\) 0 0
\(101\) 2.45871 0.244651 0.122325 0.992490i \(-0.460965\pi\)
0.122325 + 0.992490i \(0.460965\pi\)
\(102\) 0 0
\(103\) 17.4216 1.71660 0.858299 0.513150i \(-0.171522\pi\)
0.858299 + 0.513150i \(0.171522\pi\)
\(104\) 0 0
\(105\) −0.224669 −0.0219255
\(106\) 0 0
\(107\) −7.09609 −0.686005 −0.343003 0.939334i \(-0.611444\pi\)
−0.343003 + 0.939334i \(0.611444\pi\)
\(108\) 0 0
\(109\) 2.52032 0.241403 0.120702 0.992689i \(-0.461486\pi\)
0.120702 + 0.992689i \(0.461486\pi\)
\(110\) 0 0
\(111\) 3.14158 0.298185
\(112\) 0 0
\(113\) 9.51675 0.895260 0.447630 0.894219i \(-0.352268\pi\)
0.447630 + 0.894219i \(0.352268\pi\)
\(114\) 0 0
\(115\) 0.0891245 0.00831091
\(116\) 0 0
\(117\) 6.31044 0.583400
\(118\) 0 0
\(119\) −15.1878 −1.39226
\(120\) 0 0
\(121\) −8.59288 −0.781171
\(122\) 0 0
\(123\) 3.88280 0.350101
\(124\) 0 0
\(125\) −0.890537 −0.0796521
\(126\) 0 0
\(127\) −6.81952 −0.605135 −0.302567 0.953128i \(-0.597844\pi\)
−0.302567 + 0.953128i \(0.597844\pi\)
\(128\) 0 0
\(129\) −1.91638 −0.168728
\(130\) 0 0
\(131\) 8.83290 0.771734 0.385867 0.922554i \(-0.373902\pi\)
0.385867 + 0.922554i \(0.373902\pi\)
\(132\) 0 0
\(133\) 10.3841 0.900418
\(134\) 0 0
\(135\) −0.0891245 −0.00767062
\(136\) 0 0
\(137\) −12.8582 −1.09855 −0.549273 0.835643i \(-0.685095\pi\)
−0.549273 + 0.835643i \(0.685095\pi\)
\(138\) 0 0
\(139\) 7.85086 0.665901 0.332951 0.942944i \(-0.391956\pi\)
0.332951 + 0.942944i \(0.391956\pi\)
\(140\) 0 0
\(141\) 12.5261 1.05489
\(142\) 0 0
\(143\) −9.79059 −0.818730
\(144\) 0 0
\(145\) −0.0891245 −0.00740139
\(146\) 0 0
\(147\) 0.645334 0.0532262
\(148\) 0 0
\(149\) −6.39581 −0.523965 −0.261983 0.965073i \(-0.584376\pi\)
−0.261983 + 0.965073i \(0.584376\pi\)
\(150\) 0 0
\(151\) 14.8499 1.20847 0.604234 0.796807i \(-0.293479\pi\)
0.604234 + 0.796807i \(0.293479\pi\)
\(152\) 0 0
\(153\) −6.02488 −0.487083
\(154\) 0 0
\(155\) −0.560141 −0.0449916
\(156\) 0 0
\(157\) −17.9017 −1.42871 −0.714357 0.699781i \(-0.753281\pi\)
−0.714357 + 0.699781i \(0.753281\pi\)
\(158\) 0 0
\(159\) 3.50409 0.277893
\(160\) 0 0
\(161\) 2.52085 0.198671
\(162\) 0 0
\(163\) −4.91880 −0.385270 −0.192635 0.981270i \(-0.561703\pi\)
−0.192635 + 0.981270i \(0.561703\pi\)
\(164\) 0 0
\(165\) 0.138276 0.0107648
\(166\) 0 0
\(167\) −3.61590 −0.279807 −0.139903 0.990165i \(-0.544679\pi\)
−0.139903 + 0.990165i \(0.544679\pi\)
\(168\) 0 0
\(169\) 26.8217 2.06320
\(170\) 0 0
\(171\) 4.11930 0.315011
\(172\) 0 0
\(173\) 4.94379 0.375869 0.187935 0.982182i \(-0.439821\pi\)
0.187935 + 0.982182i \(0.439821\pi\)
\(174\) 0 0
\(175\) −12.5842 −0.951277
\(176\) 0 0
\(177\) 9.16270 0.688710
\(178\) 0 0
\(179\) 4.60179 0.343954 0.171977 0.985101i \(-0.444985\pi\)
0.171977 + 0.985101i \(0.444985\pi\)
\(180\) 0 0
\(181\) 0.0656052 0.00487640 0.00243820 0.999997i \(-0.499224\pi\)
0.00243820 + 0.999997i \(0.499224\pi\)
\(182\) 0 0
\(183\) −10.3666 −0.766324
\(184\) 0 0
\(185\) −0.279992 −0.0205854
\(186\) 0 0
\(187\) 9.34754 0.683560
\(188\) 0 0
\(189\) −2.52085 −0.183365
\(190\) 0 0
\(191\) −1.40793 −0.101874 −0.0509372 0.998702i \(-0.516221\pi\)
−0.0509372 + 0.998702i \(0.516221\pi\)
\(192\) 0 0
\(193\) 12.5383 0.902530 0.451265 0.892390i \(-0.350973\pi\)
0.451265 + 0.892390i \(0.350973\pi\)
\(194\) 0 0
\(195\) −0.562415 −0.0402754
\(196\) 0 0
\(197\) 4.95060 0.352715 0.176358 0.984326i \(-0.443568\pi\)
0.176358 + 0.984326i \(0.443568\pi\)
\(198\) 0 0
\(199\) −5.40788 −0.383354 −0.191677 0.981458i \(-0.561393\pi\)
−0.191677 + 0.981458i \(0.561393\pi\)
\(200\) 0 0
\(201\) 1.85364 0.130745
\(202\) 0 0
\(203\) −2.52085 −0.176929
\(204\) 0 0
\(205\) −0.346053 −0.0241694
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) −6.39106 −0.442079
\(210\) 0 0
\(211\) −7.75222 −0.533685 −0.266843 0.963740i \(-0.585980\pi\)
−0.266843 + 0.963740i \(0.585980\pi\)
\(212\) 0 0
\(213\) −2.06730 −0.141649
\(214\) 0 0
\(215\) 0.170796 0.0116482
\(216\) 0 0
\(217\) −15.8433 −1.07551
\(218\) 0 0
\(219\) 0.852008 0.0575734
\(220\) 0 0
\(221\) −38.0196 −2.55748
\(222\) 0 0
\(223\) 19.2273 1.28755 0.643777 0.765214i \(-0.277367\pi\)
0.643777 + 0.765214i \(0.277367\pi\)
\(224\) 0 0
\(225\) −4.99206 −0.332804
\(226\) 0 0
\(227\) −24.0362 −1.59534 −0.797668 0.603096i \(-0.793933\pi\)
−0.797668 + 0.603096i \(0.793933\pi\)
\(228\) 0 0
\(229\) −1.05473 −0.0696985 −0.0348492 0.999393i \(-0.511095\pi\)
−0.0348492 + 0.999393i \(0.511095\pi\)
\(230\) 0 0
\(231\) 3.91107 0.257329
\(232\) 0 0
\(233\) −5.04352 −0.330412 −0.165206 0.986259i \(-0.552829\pi\)
−0.165206 + 0.986259i \(0.552829\pi\)
\(234\) 0 0
\(235\) −1.11639 −0.0728250
\(236\) 0 0
\(237\) 14.6715 0.953019
\(238\) 0 0
\(239\) 8.65130 0.559606 0.279803 0.960057i \(-0.409731\pi\)
0.279803 + 0.960057i \(0.409731\pi\)
\(240\) 0 0
\(241\) −25.9981 −1.67469 −0.837344 0.546677i \(-0.815893\pi\)
−0.837344 + 0.546677i \(0.815893\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −0.0575151 −0.00367450
\(246\) 0 0
\(247\) 25.9946 1.65400
\(248\) 0 0
\(249\) −1.87772 −0.118995
\(250\) 0 0
\(251\) 5.75346 0.363155 0.181577 0.983377i \(-0.441880\pi\)
0.181577 + 0.983377i \(0.441880\pi\)
\(252\) 0 0
\(253\) −1.55149 −0.0975414
\(254\) 0 0
\(255\) 0.536965 0.0336260
\(256\) 0 0
\(257\) 4.56911 0.285013 0.142507 0.989794i \(-0.454484\pi\)
0.142507 + 0.989794i \(0.454484\pi\)
\(258\) 0 0
\(259\) −7.91944 −0.492090
\(260\) 0 0
\(261\) −1.00000 −0.0618984
\(262\) 0 0
\(263\) 12.3698 0.762752 0.381376 0.924420i \(-0.375450\pi\)
0.381376 + 0.924420i \(0.375450\pi\)
\(264\) 0 0
\(265\) −0.312301 −0.0191845
\(266\) 0 0
\(267\) 11.8205 0.723401
\(268\) 0 0
\(269\) −1.92180 −0.117174 −0.0585870 0.998282i \(-0.518660\pi\)
−0.0585870 + 0.998282i \(0.518660\pi\)
\(270\) 0 0
\(271\) −9.32469 −0.566435 −0.283217 0.959056i \(-0.591402\pi\)
−0.283217 + 0.959056i \(0.591402\pi\)
\(272\) 0 0
\(273\) −15.9077 −0.962775
\(274\) 0 0
\(275\) 7.74513 0.467049
\(276\) 0 0
\(277\) −19.6085 −1.17816 −0.589079 0.808075i \(-0.700509\pi\)
−0.589079 + 0.808075i \(0.700509\pi\)
\(278\) 0 0
\(279\) −6.28492 −0.376268
\(280\) 0 0
\(281\) 12.2808 0.732610 0.366305 0.930495i \(-0.380623\pi\)
0.366305 + 0.930495i \(0.380623\pi\)
\(282\) 0 0
\(283\) 12.0243 0.714770 0.357385 0.933957i \(-0.383668\pi\)
0.357385 + 0.933957i \(0.383668\pi\)
\(284\) 0 0
\(285\) −0.367131 −0.0217470
\(286\) 0 0
\(287\) −9.78795 −0.577764
\(288\) 0 0
\(289\) 19.2992 1.13525
\(290\) 0 0
\(291\) 5.06804 0.297094
\(292\) 0 0
\(293\) 15.8746 0.927406 0.463703 0.885991i \(-0.346520\pi\)
0.463703 + 0.885991i \(0.346520\pi\)
\(294\) 0 0
\(295\) −0.816621 −0.0475455
\(296\) 0 0
\(297\) 1.55149 0.0900266
\(298\) 0 0
\(299\) 6.31044 0.364942
\(300\) 0 0
\(301\) 4.83089 0.278448
\(302\) 0 0
\(303\) −2.45871 −0.141249
\(304\) 0 0
\(305\) 0.923922 0.0529036
\(306\) 0 0
\(307\) 6.32225 0.360830 0.180415 0.983591i \(-0.442256\pi\)
0.180415 + 0.983591i \(0.442256\pi\)
\(308\) 0 0
\(309\) −17.4216 −0.991078
\(310\) 0 0
\(311\) 0.581953 0.0329995 0.0164998 0.999864i \(-0.494748\pi\)
0.0164998 + 0.999864i \(0.494748\pi\)
\(312\) 0 0
\(313\) −32.7136 −1.84908 −0.924542 0.381081i \(-0.875552\pi\)
−0.924542 + 0.381081i \(0.875552\pi\)
\(314\) 0 0
\(315\) 0.224669 0.0126587
\(316\) 0 0
\(317\) −34.5562 −1.94087 −0.970434 0.241366i \(-0.922405\pi\)
−0.970434 + 0.241366i \(0.922405\pi\)
\(318\) 0 0
\(319\) 1.55149 0.0868668
\(320\) 0 0
\(321\) 7.09609 0.396065
\(322\) 0 0
\(323\) −24.8183 −1.38093
\(324\) 0 0
\(325\) −31.5021 −1.74742
\(326\) 0 0
\(327\) −2.52032 −0.139374
\(328\) 0 0
\(329\) −31.5765 −1.74087
\(330\) 0 0
\(331\) 27.7031 1.52270 0.761351 0.648340i \(-0.224536\pi\)
0.761351 + 0.648340i \(0.224536\pi\)
\(332\) 0 0
\(333\) −3.14158 −0.172157
\(334\) 0 0
\(335\) −0.165205 −0.00902609
\(336\) 0 0
\(337\) 15.6495 0.852482 0.426241 0.904610i \(-0.359838\pi\)
0.426241 + 0.904610i \(0.359838\pi\)
\(338\) 0 0
\(339\) −9.51675 −0.516879
\(340\) 0 0
\(341\) 9.75100 0.528046
\(342\) 0 0
\(343\) −19.2727 −1.04063
\(344\) 0 0
\(345\) −0.0891245 −0.00479830
\(346\) 0 0
\(347\) 21.9827 1.18010 0.590048 0.807368i \(-0.299109\pi\)
0.590048 + 0.807368i \(0.299109\pi\)
\(348\) 0 0
\(349\) −7.13201 −0.381768 −0.190884 0.981613i \(-0.561135\pi\)
−0.190884 + 0.981613i \(0.561135\pi\)
\(350\) 0 0
\(351\) −6.31044 −0.336826
\(352\) 0 0
\(353\) 27.1766 1.44647 0.723233 0.690604i \(-0.242655\pi\)
0.723233 + 0.690604i \(0.242655\pi\)
\(354\) 0 0
\(355\) 0.184247 0.00977882
\(356\) 0 0
\(357\) 15.1878 0.803823
\(358\) 0 0
\(359\) 15.8778 0.837996 0.418998 0.907987i \(-0.362381\pi\)
0.418998 + 0.907987i \(0.362381\pi\)
\(360\) 0 0
\(361\) −2.03135 −0.106913
\(362\) 0 0
\(363\) 8.59288 0.451009
\(364\) 0 0
\(365\) −0.0759348 −0.00397461
\(366\) 0 0
\(367\) −8.03489 −0.419418 −0.209709 0.977764i \(-0.567252\pi\)
−0.209709 + 0.977764i \(0.567252\pi\)
\(368\) 0 0
\(369\) −3.88280 −0.202131
\(370\) 0 0
\(371\) −8.83328 −0.458601
\(372\) 0 0
\(373\) −29.2738 −1.51574 −0.757869 0.652406i \(-0.773760\pi\)
−0.757869 + 0.652406i \(0.773760\pi\)
\(374\) 0 0
\(375\) 0.890537 0.0459872
\(376\) 0 0
\(377\) −6.31044 −0.325004
\(378\) 0 0
\(379\) −18.0054 −0.924877 −0.462439 0.886651i \(-0.653025\pi\)
−0.462439 + 0.886651i \(0.653025\pi\)
\(380\) 0 0
\(381\) 6.81952 0.349375
\(382\) 0 0
\(383\) 8.65577 0.442289 0.221145 0.975241i \(-0.429021\pi\)
0.221145 + 0.975241i \(0.429021\pi\)
\(384\) 0 0
\(385\) −0.348572 −0.0177649
\(386\) 0 0
\(387\) 1.91638 0.0974149
\(388\) 0 0
\(389\) −13.1237 −0.665399 −0.332699 0.943033i \(-0.607959\pi\)
−0.332699 + 0.943033i \(0.607959\pi\)
\(390\) 0 0
\(391\) −6.02488 −0.304691
\(392\) 0 0
\(393\) −8.83290 −0.445561
\(394\) 0 0
\(395\) −1.30759 −0.0657922
\(396\) 0 0
\(397\) −25.4689 −1.27825 −0.639123 0.769104i \(-0.720703\pi\)
−0.639123 + 0.769104i \(0.720703\pi\)
\(398\) 0 0
\(399\) −10.3841 −0.519857
\(400\) 0 0
\(401\) −12.4628 −0.622363 −0.311181 0.950351i \(-0.600725\pi\)
−0.311181 + 0.950351i \(0.600725\pi\)
\(402\) 0 0
\(403\) −39.6606 −1.97564
\(404\) 0 0
\(405\) 0.0891245 0.00442863
\(406\) 0 0
\(407\) 4.87413 0.241602
\(408\) 0 0
\(409\) −16.1131 −0.796741 −0.398371 0.917225i \(-0.630424\pi\)
−0.398371 + 0.917225i \(0.630424\pi\)
\(410\) 0 0
\(411\) 12.8582 0.634246
\(412\) 0 0
\(413\) −23.0978 −1.13657
\(414\) 0 0
\(415\) 0.167351 0.00821492
\(416\) 0 0
\(417\) −7.85086 −0.384458
\(418\) 0 0
\(419\) −14.2081 −0.694109 −0.347055 0.937845i \(-0.612818\pi\)
−0.347055 + 0.937845i \(0.612818\pi\)
\(420\) 0 0
\(421\) 14.8848 0.725439 0.362720 0.931898i \(-0.381848\pi\)
0.362720 + 0.931898i \(0.381848\pi\)
\(422\) 0 0
\(423\) −12.5261 −0.609041
\(424\) 0 0
\(425\) 30.0765 1.45893
\(426\) 0 0
\(427\) 26.1327 1.26465
\(428\) 0 0
\(429\) 9.79059 0.472694
\(430\) 0 0
\(431\) 16.6692 0.802925 0.401462 0.915876i \(-0.368502\pi\)
0.401462 + 0.915876i \(0.368502\pi\)
\(432\) 0 0
\(433\) −9.60301 −0.461491 −0.230746 0.973014i \(-0.574117\pi\)
−0.230746 + 0.973014i \(0.574117\pi\)
\(434\) 0 0
\(435\) 0.0891245 0.00427319
\(436\) 0 0
\(437\) 4.11930 0.197053
\(438\) 0 0
\(439\) −25.0898 −1.19747 −0.598736 0.800947i \(-0.704330\pi\)
−0.598736 + 0.800947i \(0.704330\pi\)
\(440\) 0 0
\(441\) −0.645334 −0.0307302
\(442\) 0 0
\(443\) −26.7780 −1.27226 −0.636132 0.771581i \(-0.719466\pi\)
−0.636132 + 0.771581i \(0.719466\pi\)
\(444\) 0 0
\(445\) −1.05349 −0.0499404
\(446\) 0 0
\(447\) 6.39581 0.302511
\(448\) 0 0
\(449\) 21.0330 0.992610 0.496305 0.868148i \(-0.334690\pi\)
0.496305 + 0.868148i \(0.334690\pi\)
\(450\) 0 0
\(451\) 6.02413 0.283665
\(452\) 0 0
\(453\) −14.8499 −0.697709
\(454\) 0 0
\(455\) 1.41776 0.0664657
\(456\) 0 0
\(457\) −6.28056 −0.293792 −0.146896 0.989152i \(-0.546928\pi\)
−0.146896 + 0.989152i \(0.546928\pi\)
\(458\) 0 0
\(459\) 6.02488 0.281217
\(460\) 0 0
\(461\) −17.1246 −0.797570 −0.398785 0.917044i \(-0.630568\pi\)
−0.398785 + 0.917044i \(0.630568\pi\)
\(462\) 0 0
\(463\) −17.9716 −0.835210 −0.417605 0.908629i \(-0.637130\pi\)
−0.417605 + 0.908629i \(0.637130\pi\)
\(464\) 0 0
\(465\) 0.560141 0.0259759
\(466\) 0 0
\(467\) 28.0902 1.29986 0.649931 0.759993i \(-0.274798\pi\)
0.649931 + 0.759993i \(0.274798\pi\)
\(468\) 0 0
\(469\) −4.67273 −0.215767
\(470\) 0 0
\(471\) 17.9017 0.824869
\(472\) 0 0
\(473\) −2.97324 −0.136710
\(474\) 0 0
\(475\) −20.5638 −0.943532
\(476\) 0 0
\(477\) −3.50409 −0.160441
\(478\) 0 0
\(479\) 26.5967 1.21523 0.607617 0.794230i \(-0.292125\pi\)
0.607617 + 0.794230i \(0.292125\pi\)
\(480\) 0 0
\(481\) −19.8247 −0.903931
\(482\) 0 0
\(483\) −2.52085 −0.114702
\(484\) 0 0
\(485\) −0.451687 −0.0205100
\(486\) 0 0
\(487\) −10.5066 −0.476101 −0.238050 0.971253i \(-0.576508\pi\)
−0.238050 + 0.971253i \(0.576508\pi\)
\(488\) 0 0
\(489\) 4.91880 0.222436
\(490\) 0 0
\(491\) 31.5247 1.42269 0.711344 0.702844i \(-0.248087\pi\)
0.711344 + 0.702844i \(0.248087\pi\)
\(492\) 0 0
\(493\) 6.02488 0.271347
\(494\) 0 0
\(495\) −0.138276 −0.00621504
\(496\) 0 0
\(497\) 5.21134 0.233761
\(498\) 0 0
\(499\) −6.46966 −0.289622 −0.144811 0.989459i \(-0.546257\pi\)
−0.144811 + 0.989459i \(0.546257\pi\)
\(500\) 0 0
\(501\) 3.61590 0.161546
\(502\) 0 0
\(503\) −39.1789 −1.74690 −0.873451 0.486912i \(-0.838123\pi\)
−0.873451 + 0.486912i \(0.838123\pi\)
\(504\) 0 0
\(505\) 0.219131 0.00975121
\(506\) 0 0
\(507\) −26.8217 −1.19119
\(508\) 0 0
\(509\) −42.2035 −1.87064 −0.935318 0.353808i \(-0.884887\pi\)
−0.935318 + 0.353808i \(0.884887\pi\)
\(510\) 0 0
\(511\) −2.14778 −0.0950122
\(512\) 0 0
\(513\) −4.11930 −0.181872
\(514\) 0 0
\(515\) 1.55269 0.0684197
\(516\) 0 0
\(517\) 19.4342 0.854714
\(518\) 0 0
\(519\) −4.94379 −0.217008
\(520\) 0 0
\(521\) 34.0529 1.49188 0.745942 0.666011i \(-0.232000\pi\)
0.745942 + 0.666011i \(0.232000\pi\)
\(522\) 0 0
\(523\) 9.94115 0.434696 0.217348 0.976094i \(-0.430259\pi\)
0.217348 + 0.976094i \(0.430259\pi\)
\(524\) 0 0
\(525\) 12.5842 0.549220
\(526\) 0 0
\(527\) 37.8659 1.64946
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −9.16270 −0.397627
\(532\) 0 0
\(533\) −24.5022 −1.06131
\(534\) 0 0
\(535\) −0.632436 −0.0273426
\(536\) 0 0
\(537\) −4.60179 −0.198582
\(538\) 0 0
\(539\) 1.00123 0.0431260
\(540\) 0 0
\(541\) 34.1745 1.46928 0.734638 0.678459i \(-0.237352\pi\)
0.734638 + 0.678459i \(0.237352\pi\)
\(542\) 0 0
\(543\) −0.0656052 −0.00281539
\(544\) 0 0
\(545\) 0.224623 0.00962177
\(546\) 0 0
\(547\) 38.3745 1.64078 0.820388 0.571808i \(-0.193758\pi\)
0.820388 + 0.571808i \(0.193758\pi\)
\(548\) 0 0
\(549\) 10.3666 0.442438
\(550\) 0 0
\(551\) −4.11930 −0.175488
\(552\) 0 0
\(553\) −36.9847 −1.57275
\(554\) 0 0
\(555\) 0.279992 0.0118850
\(556\) 0 0
\(557\) −30.0871 −1.27483 −0.637415 0.770521i \(-0.719996\pi\)
−0.637415 + 0.770521i \(0.719996\pi\)
\(558\) 0 0
\(559\) 12.0932 0.511487
\(560\) 0 0
\(561\) −9.34754 −0.394654
\(562\) 0 0
\(563\) −27.9924 −1.17974 −0.589870 0.807498i \(-0.700821\pi\)
−0.589870 + 0.807498i \(0.700821\pi\)
\(564\) 0 0
\(565\) 0.848176 0.0356830
\(566\) 0 0
\(567\) 2.52085 0.105866
\(568\) 0 0
\(569\) −38.7969 −1.62645 −0.813225 0.581950i \(-0.802290\pi\)
−0.813225 + 0.581950i \(0.802290\pi\)
\(570\) 0 0
\(571\) 1.50405 0.0629424 0.0314712 0.999505i \(-0.489981\pi\)
0.0314712 + 0.999505i \(0.489981\pi\)
\(572\) 0 0
\(573\) 1.40793 0.0588172
\(574\) 0 0
\(575\) −4.99206 −0.208183
\(576\) 0 0
\(577\) 29.7062 1.23669 0.618343 0.785908i \(-0.287804\pi\)
0.618343 + 0.785908i \(0.287804\pi\)
\(578\) 0 0
\(579\) −12.5383 −0.521076
\(580\) 0 0
\(581\) 4.73344 0.196376
\(582\) 0 0
\(583\) 5.43657 0.225160
\(584\) 0 0
\(585\) 0.562415 0.0232530
\(586\) 0 0
\(587\) 19.9946 0.825264 0.412632 0.910898i \(-0.364609\pi\)
0.412632 + 0.910898i \(0.364609\pi\)
\(588\) 0 0
\(589\) −25.8895 −1.06676
\(590\) 0 0
\(591\) −4.95060 −0.203640
\(592\) 0 0
\(593\) 23.6830 0.972543 0.486272 0.873808i \(-0.338357\pi\)
0.486272 + 0.873808i \(0.338357\pi\)
\(594\) 0 0
\(595\) −1.35361 −0.0554924
\(596\) 0 0
\(597\) 5.40788 0.221330
\(598\) 0 0
\(599\) −30.3527 −1.24018 −0.620088 0.784532i \(-0.712903\pi\)
−0.620088 + 0.784532i \(0.712903\pi\)
\(600\) 0 0
\(601\) −38.9978 −1.59075 −0.795377 0.606114i \(-0.792727\pi\)
−0.795377 + 0.606114i \(0.792727\pi\)
\(602\) 0 0
\(603\) −1.85364 −0.0754859
\(604\) 0 0
\(605\) −0.765836 −0.0311357
\(606\) 0 0
\(607\) −44.0987 −1.78991 −0.894956 0.446154i \(-0.852793\pi\)
−0.894956 + 0.446154i \(0.852793\pi\)
\(608\) 0 0
\(609\) 2.52085 0.102150
\(610\) 0 0
\(611\) −79.0454 −3.19783
\(612\) 0 0
\(613\) −29.1904 −1.17899 −0.589494 0.807773i \(-0.700673\pi\)
−0.589494 + 0.807773i \(0.700673\pi\)
\(614\) 0 0
\(615\) 0.346053 0.0139542
\(616\) 0 0
\(617\) 18.9272 0.761980 0.380990 0.924579i \(-0.375583\pi\)
0.380990 + 0.924579i \(0.375583\pi\)
\(618\) 0 0
\(619\) −2.81697 −0.113223 −0.0566117 0.998396i \(-0.518030\pi\)
−0.0566117 + 0.998396i \(0.518030\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 0 0
\(623\) −29.7976 −1.19382
\(624\) 0 0
\(625\) 24.8809 0.995237
\(626\) 0 0
\(627\) 6.39106 0.255234
\(628\) 0 0
\(629\) 18.9276 0.754694
\(630\) 0 0
\(631\) 13.5049 0.537622 0.268811 0.963193i \(-0.413369\pi\)
0.268811 + 0.963193i \(0.413369\pi\)
\(632\) 0 0
\(633\) 7.75222 0.308123
\(634\) 0 0
\(635\) −0.607787 −0.0241193
\(636\) 0 0
\(637\) −4.07234 −0.161352
\(638\) 0 0
\(639\) 2.06730 0.0817811
\(640\) 0 0
\(641\) −33.2190 −1.31207 −0.656037 0.754729i \(-0.727768\pi\)
−0.656037 + 0.754729i \(0.727768\pi\)
\(642\) 0 0
\(643\) 46.8166 1.84627 0.923133 0.384481i \(-0.125619\pi\)
0.923133 + 0.384481i \(0.125619\pi\)
\(644\) 0 0
\(645\) −0.170796 −0.00672510
\(646\) 0 0
\(647\) −31.6817 −1.24554 −0.622769 0.782406i \(-0.713992\pi\)
−0.622769 + 0.782406i \(0.713992\pi\)
\(648\) 0 0
\(649\) 14.2158 0.558020
\(650\) 0 0
\(651\) 15.8433 0.620949
\(652\) 0 0
\(653\) 8.43461 0.330072 0.165036 0.986288i \(-0.447226\pi\)
0.165036 + 0.986288i \(0.447226\pi\)
\(654\) 0 0
\(655\) 0.787228 0.0307595
\(656\) 0 0
\(657\) −0.852008 −0.0332400
\(658\) 0 0
\(659\) 23.9452 0.932771 0.466385 0.884582i \(-0.345556\pi\)
0.466385 + 0.884582i \(0.345556\pi\)
\(660\) 0 0
\(661\) −16.1122 −0.626690 −0.313345 0.949639i \(-0.601450\pi\)
−0.313345 + 0.949639i \(0.601450\pi\)
\(662\) 0 0
\(663\) 38.0196 1.47656
\(664\) 0 0
\(665\) 0.925481 0.0358886
\(666\) 0 0
\(667\) −1.00000 −0.0387202
\(668\) 0 0
\(669\) −19.2273 −0.743369
\(670\) 0 0
\(671\) −16.0837 −0.620906
\(672\) 0 0
\(673\) 26.6979 1.02913 0.514564 0.857452i \(-0.327954\pi\)
0.514564 + 0.857452i \(0.327954\pi\)
\(674\) 0 0
\(675\) 4.99206 0.192144
\(676\) 0 0
\(677\) 32.2930 1.24112 0.620560 0.784159i \(-0.286906\pi\)
0.620560 + 0.784159i \(0.286906\pi\)
\(678\) 0 0
\(679\) −12.7757 −0.490288
\(680\) 0 0
\(681\) 24.0362 0.921068
\(682\) 0 0
\(683\) −22.2531 −0.851490 −0.425745 0.904843i \(-0.639988\pi\)
−0.425745 + 0.904843i \(0.639988\pi\)
\(684\) 0 0
\(685\) −1.14598 −0.0437855
\(686\) 0 0
\(687\) 1.05473 0.0402404
\(688\) 0 0
\(689\) −22.1124 −0.842414
\(690\) 0 0
\(691\) 11.3411 0.431435 0.215718 0.976456i \(-0.430791\pi\)
0.215718 + 0.976456i \(0.430791\pi\)
\(692\) 0 0
\(693\) −3.91107 −0.148569
\(694\) 0 0
\(695\) 0.699704 0.0265413
\(696\) 0 0
\(697\) 23.3934 0.886089
\(698\) 0 0
\(699\) 5.04352 0.190763
\(700\) 0 0
\(701\) −10.9018 −0.411754 −0.205877 0.978578i \(-0.566005\pi\)
−0.205877 + 0.978578i \(0.566005\pi\)
\(702\) 0 0
\(703\) −12.9411 −0.488083
\(704\) 0 0
\(705\) 1.11639 0.0420455
\(706\) 0 0
\(707\) 6.19802 0.233101
\(708\) 0 0
\(709\) 11.6697 0.438266 0.219133 0.975695i \(-0.429677\pi\)
0.219133 + 0.975695i \(0.429677\pi\)
\(710\) 0 0
\(711\) −14.6715 −0.550226
\(712\) 0 0
\(713\) −6.28492 −0.235372
\(714\) 0 0
\(715\) −0.872582 −0.0326327
\(716\) 0 0
\(717\) −8.65130 −0.323089
\(718\) 0 0
\(719\) −9.96159 −0.371505 −0.185752 0.982597i \(-0.559472\pi\)
−0.185752 + 0.982597i \(0.559472\pi\)
\(720\) 0 0
\(721\) 43.9171 1.63556
\(722\) 0 0
\(723\) 25.9981 0.966881
\(724\) 0 0
\(725\) 4.99206 0.185400
\(726\) 0 0
\(727\) 0.970079 0.0359782 0.0179891 0.999838i \(-0.494274\pi\)
0.0179891 + 0.999838i \(0.494274\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −11.5459 −0.427042
\(732\) 0 0
\(733\) −35.5445 −1.31287 −0.656433 0.754385i \(-0.727935\pi\)
−0.656433 + 0.754385i \(0.727935\pi\)
\(734\) 0 0
\(735\) 0.0575151 0.00212148
\(736\) 0 0
\(737\) 2.87590 0.105935
\(738\) 0 0
\(739\) −26.2939 −0.967236 −0.483618 0.875279i \(-0.660678\pi\)
−0.483618 + 0.875279i \(0.660678\pi\)
\(740\) 0 0
\(741\) −25.9946 −0.954936
\(742\) 0 0
\(743\) −10.2289 −0.375262 −0.187631 0.982240i \(-0.560081\pi\)
−0.187631 + 0.982240i \(0.560081\pi\)
\(744\) 0 0
\(745\) −0.570024 −0.0208840
\(746\) 0 0
\(747\) 1.87772 0.0687021
\(748\) 0 0
\(749\) −17.8882 −0.653619
\(750\) 0 0
\(751\) −9.74607 −0.355639 −0.177820 0.984063i \(-0.556904\pi\)
−0.177820 + 0.984063i \(0.556904\pi\)
\(752\) 0 0
\(753\) −5.75346 −0.209668
\(754\) 0 0
\(755\) 1.32349 0.0481668
\(756\) 0 0
\(757\) −1.26270 −0.0458936 −0.0229468 0.999737i \(-0.507305\pi\)
−0.0229468 + 0.999737i \(0.507305\pi\)
\(758\) 0 0
\(759\) 1.55149 0.0563155
\(760\) 0 0
\(761\) −15.3340 −0.555858 −0.277929 0.960602i \(-0.589648\pi\)
−0.277929 + 0.960602i \(0.589648\pi\)
\(762\) 0 0
\(763\) 6.35334 0.230007
\(764\) 0 0
\(765\) −0.536965 −0.0194140
\(766\) 0 0
\(767\) −57.8207 −2.08778
\(768\) 0 0
\(769\) 49.0640 1.76929 0.884647 0.466261i \(-0.154399\pi\)
0.884647 + 0.466261i \(0.154399\pi\)
\(770\) 0 0
\(771\) −4.56911 −0.164553
\(772\) 0 0
\(773\) −35.8777 −1.29043 −0.645216 0.764000i \(-0.723233\pi\)
−0.645216 + 0.764000i \(0.723233\pi\)
\(774\) 0 0
\(775\) 31.3747 1.12701
\(776\) 0 0
\(777\) 7.91944 0.284108
\(778\) 0 0
\(779\) −15.9944 −0.573060
\(780\) 0 0
\(781\) −3.20739 −0.114770
\(782\) 0 0
\(783\) 1.00000 0.0357371
\(784\) 0 0
\(785\) −1.59548 −0.0569453
\(786\) 0 0
\(787\) 30.6960 1.09419 0.547097 0.837069i \(-0.315733\pi\)
0.547097 + 0.837069i \(0.315733\pi\)
\(788\) 0 0
\(789\) −12.3698 −0.440375
\(790\) 0 0
\(791\) 23.9903 0.852995
\(792\) 0 0
\(793\) 65.4181 2.32306
\(794\) 0 0
\(795\) 0.312301 0.0110762
\(796\) 0 0
\(797\) −0.541641 −0.0191859 −0.00959296 0.999954i \(-0.503054\pi\)
−0.00959296 + 0.999954i \(0.503054\pi\)
\(798\) 0 0
\(799\) 75.4684 2.66988
\(800\) 0 0
\(801\) −11.8205 −0.417656
\(802\) 0 0
\(803\) 1.32188 0.0466482
\(804\) 0 0
\(805\) 0.224669 0.00791855
\(806\) 0 0
\(807\) 1.92180 0.0676505
\(808\) 0 0
\(809\) −49.2737 −1.73237 −0.866186 0.499722i \(-0.833436\pi\)
−0.866186 + 0.499722i \(0.833436\pi\)
\(810\) 0 0
\(811\) 21.8789 0.768274 0.384137 0.923276i \(-0.374499\pi\)
0.384137 + 0.923276i \(0.374499\pi\)
\(812\) 0 0
\(813\) 9.32469 0.327031
\(814\) 0 0
\(815\) −0.438386 −0.0153560
\(816\) 0 0
\(817\) 7.89414 0.276181
\(818\) 0 0
\(819\) 15.9077 0.555858
\(820\) 0 0
\(821\) −40.0618 −1.39817 −0.699083 0.715040i \(-0.746408\pi\)
−0.699083 + 0.715040i \(0.746408\pi\)
\(822\) 0 0
\(823\) 34.0723 1.18769 0.593844 0.804580i \(-0.297610\pi\)
0.593844 + 0.804580i \(0.297610\pi\)
\(824\) 0 0
\(825\) −7.74513 −0.269651
\(826\) 0 0
\(827\) −4.55925 −0.158541 −0.0792704 0.996853i \(-0.525259\pi\)
−0.0792704 + 0.996853i \(0.525259\pi\)
\(828\) 0 0
\(829\) −28.3451 −0.984465 −0.492233 0.870464i \(-0.663819\pi\)
−0.492233 + 0.870464i \(0.663819\pi\)
\(830\) 0 0
\(831\) 19.6085 0.680210
\(832\) 0 0
\(833\) 3.88806 0.134713
\(834\) 0 0
\(835\) −0.322265 −0.0111525
\(836\) 0 0
\(837\) 6.28492 0.217239
\(838\) 0 0
\(839\) −42.2492 −1.45860 −0.729302 0.684192i \(-0.760155\pi\)
−0.729302 + 0.684192i \(0.760155\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 0 0
\(843\) −12.2808 −0.422972
\(844\) 0 0
\(845\) 2.39047 0.0822346
\(846\) 0 0
\(847\) −21.6613 −0.744292
\(848\) 0 0
\(849\) −12.0243 −0.412673
\(850\) 0 0
\(851\) −3.14158 −0.107692
\(852\) 0 0
\(853\) −29.4475 −1.00826 −0.504131 0.863627i \(-0.668187\pi\)
−0.504131 + 0.863627i \(0.668187\pi\)
\(854\) 0 0
\(855\) 0.367131 0.0125556
\(856\) 0 0
\(857\) 12.9477 0.442284 0.221142 0.975242i \(-0.429022\pi\)
0.221142 + 0.975242i \(0.429022\pi\)
\(858\) 0 0
\(859\) 9.37239 0.319782 0.159891 0.987135i \(-0.448886\pi\)
0.159891 + 0.987135i \(0.448886\pi\)
\(860\) 0 0
\(861\) 9.78795 0.333572
\(862\) 0 0
\(863\) 19.2209 0.654286 0.327143 0.944975i \(-0.393914\pi\)
0.327143 + 0.944975i \(0.393914\pi\)
\(864\) 0 0
\(865\) 0.440613 0.0149813
\(866\) 0 0
\(867\) −19.2992 −0.655434
\(868\) 0 0
\(869\) 22.7628 0.772174
\(870\) 0 0
\(871\) −11.6973 −0.396347
\(872\) 0 0
\(873\) −5.06804 −0.171527
\(874\) 0 0
\(875\) −2.24491 −0.0758917
\(876\) 0 0
\(877\) −42.0093 −1.41855 −0.709277 0.704930i \(-0.750978\pi\)
−0.709277 + 0.704930i \(0.750978\pi\)
\(878\) 0 0
\(879\) −15.8746 −0.535438
\(880\) 0 0
\(881\) 24.8647 0.837714 0.418857 0.908052i \(-0.362431\pi\)
0.418857 + 0.908052i \(0.362431\pi\)
\(882\) 0 0
\(883\) −3.82093 −0.128585 −0.0642923 0.997931i \(-0.520479\pi\)
−0.0642923 + 0.997931i \(0.520479\pi\)
\(884\) 0 0
\(885\) 0.816621 0.0274504
\(886\) 0 0
\(887\) 28.7748 0.966163 0.483081 0.875576i \(-0.339518\pi\)
0.483081 + 0.875576i \(0.339518\pi\)
\(888\) 0 0
\(889\) −17.1910 −0.576567
\(890\) 0 0
\(891\) −1.55149 −0.0519769
\(892\) 0 0
\(893\) −51.5989 −1.72669
\(894\) 0 0
\(895\) 0.410132 0.0137092
\(896\) 0 0
\(897\) −6.31044 −0.210699
\(898\) 0 0
\(899\) 6.28492 0.209614
\(900\) 0 0
\(901\) 21.1117 0.703334
\(902\) 0 0
\(903\) −4.83089 −0.160762
\(904\) 0 0
\(905\) 0.00584703 0.000194362 0
\(906\) 0 0
\(907\) −54.9930 −1.82601 −0.913006 0.407947i \(-0.866245\pi\)
−0.913006 + 0.407947i \(0.866245\pi\)
\(908\) 0 0
\(909\) 2.45871 0.0815502
\(910\) 0 0
\(911\) 25.6668 0.850378 0.425189 0.905105i \(-0.360208\pi\)
0.425189 + 0.905105i \(0.360208\pi\)
\(912\) 0 0
\(913\) −2.91326 −0.0964148
\(914\) 0 0
\(915\) −0.923922 −0.0305439
\(916\) 0 0
\(917\) 22.2664 0.735300
\(918\) 0 0
\(919\) 29.0602 0.958607 0.479303 0.877649i \(-0.340889\pi\)
0.479303 + 0.877649i \(0.340889\pi\)
\(920\) 0 0
\(921\) −6.32225 −0.208325
\(922\) 0 0
\(923\) 13.0456 0.429400
\(924\) 0 0
\(925\) 15.6829 0.515652
\(926\) 0 0
\(927\) 17.4216 0.572199
\(928\) 0 0
\(929\) 56.6942 1.86008 0.930039 0.367462i \(-0.119773\pi\)
0.930039 + 0.367462i \(0.119773\pi\)
\(930\) 0 0
\(931\) −2.65832 −0.0871231
\(932\) 0 0
\(933\) −0.581953 −0.0190523
\(934\) 0 0
\(935\) 0.833095 0.0272451
\(936\) 0 0
\(937\) −24.8864 −0.813003 −0.406502 0.913650i \(-0.633251\pi\)
−0.406502 + 0.913650i \(0.633251\pi\)
\(938\) 0 0
\(939\) 32.7136 1.06757
\(940\) 0 0
\(941\) −23.5747 −0.768513 −0.384257 0.923226i \(-0.625542\pi\)
−0.384257 + 0.923226i \(0.625542\pi\)
\(942\) 0 0
\(943\) −3.88280 −0.126441
\(944\) 0 0
\(945\) −0.224669 −0.00730849
\(946\) 0 0
\(947\) −34.4993 −1.12108 −0.560539 0.828128i \(-0.689406\pi\)
−0.560539 + 0.828128i \(0.689406\pi\)
\(948\) 0 0
\(949\) −5.37655 −0.174530
\(950\) 0 0
\(951\) 34.5562 1.12056
\(952\) 0 0
\(953\) 20.7822 0.673201 0.336601 0.941647i \(-0.390723\pi\)
0.336601 + 0.941647i \(0.390723\pi\)
\(954\) 0 0
\(955\) −0.125481 −0.00406048
\(956\) 0 0
\(957\) −1.55149 −0.0501526
\(958\) 0 0
\(959\) −32.4134 −1.04668
\(960\) 0 0
\(961\) 8.50026 0.274202
\(962\) 0 0
\(963\) −7.09609 −0.228668
\(964\) 0 0
\(965\) 1.11747 0.0359728
\(966\) 0 0
\(967\) 6.46309 0.207839 0.103920 0.994586i \(-0.466862\pi\)
0.103920 + 0.994586i \(0.466862\pi\)
\(968\) 0 0
\(969\) 24.8183 0.797279
\(970\) 0 0
\(971\) 35.0870 1.12600 0.562998 0.826459i \(-0.309648\pi\)
0.562998 + 0.826459i \(0.309648\pi\)
\(972\) 0 0
\(973\) 19.7908 0.634464
\(974\) 0 0
\(975\) 31.5021 1.00887
\(976\) 0 0
\(977\) 20.3200 0.650094 0.325047 0.945698i \(-0.394620\pi\)
0.325047 + 0.945698i \(0.394620\pi\)
\(978\) 0 0
\(979\) 18.3394 0.586128
\(980\) 0 0
\(981\) 2.52032 0.0804677
\(982\) 0 0
\(983\) −12.7194 −0.405687 −0.202843 0.979211i \(-0.565018\pi\)
−0.202843 + 0.979211i \(0.565018\pi\)
\(984\) 0 0
\(985\) 0.441220 0.0140584
\(986\) 0 0
\(987\) 31.5765 1.00509
\(988\) 0 0
\(989\) 1.91638 0.0609373
\(990\) 0 0
\(991\) −8.66921 −0.275387 −0.137693 0.990475i \(-0.543969\pi\)
−0.137693 + 0.990475i \(0.543969\pi\)
\(992\) 0 0
\(993\) −27.7031 −0.879132
\(994\) 0 0
\(995\) −0.481974 −0.0152796
\(996\) 0 0
\(997\) −33.1550 −1.05003 −0.525014 0.851094i \(-0.675940\pi\)
−0.525014 + 0.851094i \(0.675940\pi\)
\(998\) 0 0
\(999\) 3.14158 0.0993952
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 8004.2.a.g.1.7 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8004.2.a.g.1.7 12 1.1 even 1 trivial