Properties

Label 2790.2.a.bk.1.1
Level $2790$
Weight $2$
Character 2790.1
Self dual yes
Analytic conductor $22.278$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2790 = 2 \cdot 3^{2} \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2790.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(22.2782621639\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.17428.1
Defining polynomial: \( x^{4} - x^{3} - 6x^{2} + 4x + 6 \) Copy content Toggle raw display
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 1.1
Root \(2.36865\) of defining polynomial
Character \(\chi\) \(=\) 2790.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.81471 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.81471 q^{7} +1.00000 q^{8} +1.00000 q^{10} +3.81471 q^{11} -2.73730 q^{13} -1.81471 q^{14} +1.00000 q^{16} +0.483701 q^{17} +2.92259 q^{19} +1.00000 q^{20} +3.81471 q^{22} -1.40629 q^{23} +1.00000 q^{25} -2.73730 q^{26} -1.81471 q^{28} +9.47460 q^{29} +1.00000 q^{31} +1.00000 q^{32} +0.483701 q^{34} -1.81471 q^{35} -3.70470 q^{37} +2.92259 q^{38} +1.00000 q^{40} +9.47460 q^{41} +11.5194 q^{43} +3.81471 q^{44} -1.40629 q^{46} -10.4420 q^{47} -3.70683 q^{49} +1.00000 q^{50} -2.73730 q^{52} -8.55201 q^{53} +3.81471 q^{55} -1.81471 q^{56} +9.47460 q^{58} +13.3341 q^{59} +10.7751 q^{61} +1.00000 q^{62} +1.00000 q^{64} -2.73730 q^{65} +9.54988 q^{67} +0.483701 q^{68} -1.81471 q^{70} -5.65989 q^{71} +8.70683 q^{73} -3.70470 q^{74} +2.92259 q^{76} -6.92259 q^{77} -9.36459 q^{79} +1.00000 q^{80} +9.47460 q^{82} +2.81258 q^{83} +0.483701 q^{85} +11.5194 q^{86} +3.81471 q^{88} -12.4115 q^{89} +4.96740 q^{91} -1.40629 q^{92} -10.4420 q^{94} +2.92259 q^{95} +0.154821 q^{97} -3.70683 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} + 4 q^{4} + 4 q^{5} + 5 q^{7} + 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{2} + 4 q^{4} + 4 q^{5} + 5 q^{7} + 4 q^{8} + 4 q^{10} + 3 q^{11} + 6 q^{13} + 5 q^{14} + 4 q^{16} + 7 q^{19} + 4 q^{20} + 3 q^{22} + q^{23} + 4 q^{25} + 6 q^{26} + 5 q^{28} + 4 q^{29} + 4 q^{31} + 4 q^{32} + 5 q^{35} + 6 q^{37} + 7 q^{38} + 4 q^{40} + 4 q^{41} + 13 q^{43} + 3 q^{44} + q^{46} - 4 q^{47} + 5 q^{49} + 4 q^{50} + 6 q^{52} - 5 q^{53} + 3 q^{55} + 5 q^{56} + 4 q^{58} + 8 q^{59} - 4 q^{61} + 4 q^{62} + 4 q^{64} + 6 q^{65} + 8 q^{67} + 5 q^{70} - q^{71} + 15 q^{73} + 6 q^{74} + 7 q^{76} - 23 q^{77} + 5 q^{79} + 4 q^{80} + 4 q^{82} - 2 q^{83} + 13 q^{86} + 3 q^{88} - 9 q^{89} + 16 q^{91} + q^{92} - 4 q^{94} + 7 q^{95} + 10 q^{97} + 5 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −1.81471 −0.685896 −0.342948 0.939354i \(-0.611425\pi\)
−0.342948 + 0.939354i \(0.611425\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 1.00000 0.316228
\(11\) 3.81471 1.15018 0.575089 0.818091i \(-0.304967\pi\)
0.575089 + 0.818091i \(0.304967\pi\)
\(12\) 0 0
\(13\) −2.73730 −0.759190 −0.379595 0.925153i \(-0.623937\pi\)
−0.379595 + 0.925153i \(0.623937\pi\)
\(14\) −1.81471 −0.485002
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 0.483701 0.117315 0.0586574 0.998278i \(-0.481318\pi\)
0.0586574 + 0.998278i \(0.481318\pi\)
\(18\) 0 0
\(19\) 2.92259 0.670488 0.335244 0.942131i \(-0.391181\pi\)
0.335244 + 0.942131i \(0.391181\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) 3.81471 0.813299
\(23\) −1.40629 −0.293232 −0.146616 0.989193i \(-0.546838\pi\)
−0.146616 + 0.989193i \(0.546838\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −2.73730 −0.536828
\(27\) 0 0
\(28\) −1.81471 −0.342948
\(29\) 9.47460 1.75939 0.879694 0.475540i \(-0.157747\pi\)
0.879694 + 0.475540i \(0.157747\pi\)
\(30\) 0 0
\(31\) 1.00000 0.179605
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 0.483701 0.0829540
\(35\) −1.81471 −0.306742
\(36\) 0 0
\(37\) −3.70470 −0.609049 −0.304525 0.952504i \(-0.598498\pi\)
−0.304525 + 0.952504i \(0.598498\pi\)
\(38\) 2.92259 0.474107
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) 9.47460 1.47968 0.739842 0.672781i \(-0.234900\pi\)
0.739842 + 0.672781i \(0.234900\pi\)
\(42\) 0 0
\(43\) 11.5194 1.75669 0.878347 0.478024i \(-0.158647\pi\)
0.878347 + 0.478024i \(0.158647\pi\)
\(44\) 3.81471 0.575089
\(45\) 0 0
\(46\) −1.40629 −0.207346
\(47\) −10.4420 −1.52312 −0.761561 0.648093i \(-0.775567\pi\)
−0.761561 + 0.648093i \(0.775567\pi\)
\(48\) 0 0
\(49\) −3.70683 −0.529547
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) −2.73730 −0.379595
\(53\) −8.55201 −1.17471 −0.587354 0.809330i \(-0.699830\pi\)
−0.587354 + 0.809330i \(0.699830\pi\)
\(54\) 0 0
\(55\) 3.81471 0.514375
\(56\) −1.81471 −0.242501
\(57\) 0 0
\(58\) 9.47460 1.24408
\(59\) 13.3341 1.73595 0.867977 0.496604i \(-0.165420\pi\)
0.867977 + 0.496604i \(0.165420\pi\)
\(60\) 0 0
\(61\) 10.7751 1.37961 0.689807 0.723993i \(-0.257695\pi\)
0.689807 + 0.723993i \(0.257695\pi\)
\(62\) 1.00000 0.127000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −2.73730 −0.339520
\(66\) 0 0
\(67\) 9.54988 1.16670 0.583352 0.812220i \(-0.301741\pi\)
0.583352 + 0.812220i \(0.301741\pi\)
\(68\) 0.483701 0.0586574
\(69\) 0 0
\(70\) −1.81471 −0.216899
\(71\) −5.65989 −0.671705 −0.335853 0.941915i \(-0.609024\pi\)
−0.335853 + 0.941915i \(0.609024\pi\)
\(72\) 0 0
\(73\) 8.70683 1.01906 0.509529 0.860454i \(-0.329820\pi\)
0.509529 + 0.860454i \(0.329820\pi\)
\(74\) −3.70470 −0.430663
\(75\) 0 0
\(76\) 2.92259 0.335244
\(77\) −6.92259 −0.788902
\(78\) 0 0
\(79\) −9.36459 −1.05360 −0.526799 0.849990i \(-0.676608\pi\)
−0.526799 + 0.849990i \(0.676608\pi\)
\(80\) 1.00000 0.111803
\(81\) 0 0
\(82\) 9.47460 1.04629
\(83\) 2.81258 0.308721 0.154360 0.988015i \(-0.450668\pi\)
0.154360 + 0.988015i \(0.450668\pi\)
\(84\) 0 0
\(85\) 0.483701 0.0524647
\(86\) 11.5194 1.24217
\(87\) 0 0
\(88\) 3.81471 0.406649
\(89\) −12.4115 −1.31562 −0.657810 0.753184i \(-0.728517\pi\)
−0.657810 + 0.753184i \(0.728517\pi\)
\(90\) 0 0
\(91\) 4.96740 0.520725
\(92\) −1.40629 −0.146616
\(93\) 0 0
\(94\) −10.4420 −1.07701
\(95\) 2.92259 0.299851
\(96\) 0 0
\(97\) 0.154821 0.0157197 0.00785987 0.999969i \(-0.497498\pi\)
0.00785987 + 0.999969i \(0.497498\pi\)
\(98\) −3.70683 −0.374446
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −1.88999 −0.188061 −0.0940306 0.995569i \(-0.529975\pi\)
−0.0940306 + 0.995569i \(0.529975\pi\)
\(102\) 0 0
\(103\) −5.33412 −0.525586 −0.262793 0.964852i \(-0.584644\pi\)
−0.262793 + 0.964852i \(0.584644\pi\)
\(104\) −2.73730 −0.268414
\(105\) 0 0
\(106\) −8.55201 −0.830644
\(107\) 3.51941 0.340234 0.170117 0.985424i \(-0.445585\pi\)
0.170117 + 0.985424i \(0.445585\pi\)
\(108\) 0 0
\(109\) 20.0714 1.92249 0.961247 0.275690i \(-0.0889063\pi\)
0.961247 + 0.275690i \(0.0889063\pi\)
\(110\) 3.81471 0.363718
\(111\) 0 0
\(112\) −1.81471 −0.171474
\(113\) 3.73517 0.351375 0.175688 0.984446i \(-0.443785\pi\)
0.175688 + 0.984446i \(0.443785\pi\)
\(114\) 0 0
\(115\) −1.40629 −0.131137
\(116\) 9.47460 0.879694
\(117\) 0 0
\(118\) 13.3341 1.22751
\(119\) −0.877777 −0.0804657
\(120\) 0 0
\(121\) 3.55201 0.322910
\(122\) 10.7751 0.975535
\(123\) 0 0
\(124\) 1.00000 0.0898027
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −12.2119 −1.08363 −0.541815 0.840498i \(-0.682263\pi\)
−0.541815 + 0.840498i \(0.682263\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) −2.73730 −0.240077
\(131\) 9.55414 0.834749 0.417374 0.908735i \(-0.362950\pi\)
0.417374 + 0.908735i \(0.362950\pi\)
\(132\) 0 0
\(133\) −5.30365 −0.459885
\(134\) 9.54988 0.824984
\(135\) 0 0
\(136\) 0.483701 0.0414770
\(137\) −12.9300 −1.10468 −0.552340 0.833619i \(-0.686265\pi\)
−0.552340 + 0.833619i \(0.686265\pi\)
\(138\) 0 0
\(139\) 10.1131 0.857784 0.428892 0.903356i \(-0.358904\pi\)
0.428892 + 0.903356i \(0.358904\pi\)
\(140\) −1.81471 −0.153371
\(141\) 0 0
\(142\) −5.65989 −0.474967
\(143\) −10.4420 −0.873204
\(144\) 0 0
\(145\) 9.47460 0.786822
\(146\) 8.70683 0.720582
\(147\) 0 0
\(148\) −3.70470 −0.304525
\(149\) −18.9940 −1.55605 −0.778025 0.628233i \(-0.783778\pi\)
−0.778025 + 0.628233i \(0.783778\pi\)
\(150\) 0 0
\(151\) −4.96740 −0.404241 −0.202121 0.979361i \(-0.564783\pi\)
−0.202121 + 0.979361i \(0.564783\pi\)
\(152\) 2.92259 0.237053
\(153\) 0 0
\(154\) −6.92259 −0.557838
\(155\) 1.00000 0.0803219
\(156\) 0 0
\(157\) 5.73517 0.457716 0.228858 0.973460i \(-0.426501\pi\)
0.228858 + 0.973460i \(0.426501\pi\)
\(158\) −9.36459 −0.745007
\(159\) 0 0
\(160\) 1.00000 0.0790569
\(161\) 2.55201 0.201126
\(162\) 0 0
\(163\) −1.17930 −0.0923698 −0.0461849 0.998933i \(-0.514706\pi\)
−0.0461849 + 0.998933i \(0.514706\pi\)
\(164\) 9.47460 0.739842
\(165\) 0 0
\(166\) 2.81258 0.218299
\(167\) 21.7691 1.68455 0.842273 0.539050i \(-0.181217\pi\)
0.842273 + 0.539050i \(0.181217\pi\)
\(168\) 0 0
\(169\) −5.50720 −0.423630
\(170\) 0.483701 0.0370982
\(171\) 0 0
\(172\) 11.5194 0.878347
\(173\) −18.2872 −1.39035 −0.695174 0.718841i \(-0.744673\pi\)
−0.695174 + 0.718841i \(0.744673\pi\)
\(174\) 0 0
\(175\) −1.81471 −0.137179
\(176\) 3.81471 0.287545
\(177\) 0 0
\(178\) −12.4115 −0.930284
\(179\) −5.92046 −0.442516 −0.221258 0.975215i \(-0.571016\pi\)
−0.221258 + 0.975215i \(0.571016\pi\)
\(180\) 0 0
\(181\) 12.8809 0.957429 0.478714 0.877971i \(-0.341103\pi\)
0.478714 + 0.877971i \(0.341103\pi\)
\(182\) 4.96740 0.368208
\(183\) 0 0
\(184\) −1.40629 −0.103673
\(185\) −3.70470 −0.272375
\(186\) 0 0
\(187\) 1.84518 0.134933
\(188\) −10.4420 −0.761561
\(189\) 0 0
\(190\) 2.92259 0.212027
\(191\) −3.55414 −0.257168 −0.128584 0.991699i \(-0.541043\pi\)
−0.128584 + 0.991699i \(0.541043\pi\)
\(192\) 0 0
\(193\) 17.2588 1.24232 0.621159 0.783684i \(-0.286662\pi\)
0.621159 + 0.783684i \(0.286662\pi\)
\(194\) 0.154821 0.0111155
\(195\) 0 0
\(196\) −3.70683 −0.264774
\(197\) −8.52540 −0.607410 −0.303705 0.952766i \(-0.598224\pi\)
−0.303705 + 0.952766i \(0.598224\pi\)
\(198\) 0 0
\(199\) 20.0266 1.41965 0.709824 0.704379i \(-0.248774\pi\)
0.709824 + 0.704379i \(0.248774\pi\)
\(200\) 1.00000 0.0707107
\(201\) 0 0
\(202\) −1.88999 −0.132979
\(203\) −17.1936 −1.20676
\(204\) 0 0
\(205\) 9.47460 0.661735
\(206\) −5.33412 −0.371646
\(207\) 0 0
\(208\) −2.73730 −0.189798
\(209\) 11.1488 0.771181
\(210\) 0 0
\(211\) −26.4686 −1.82217 −0.911087 0.412214i \(-0.864756\pi\)
−0.911087 + 0.412214i \(0.864756\pi\)
\(212\) −8.55201 −0.587354
\(213\) 0 0
\(214\) 3.51941 0.240582
\(215\) 11.5194 0.785617
\(216\) 0 0
\(217\) −1.81471 −0.123191
\(218\) 20.0714 1.35941
\(219\) 0 0
\(220\) 3.81471 0.257188
\(221\) −1.32403 −0.0890642
\(222\) 0 0
\(223\) 24.3625 1.63143 0.815715 0.578453i \(-0.196344\pi\)
0.815715 + 0.578453i \(0.196344\pi\)
\(224\) −1.81471 −0.121250
\(225\) 0 0
\(226\) 3.73517 0.248460
\(227\) −13.6742 −0.907591 −0.453795 0.891106i \(-0.649930\pi\)
−0.453795 + 0.891106i \(0.649930\pi\)
\(228\) 0 0
\(229\) −8.97051 −0.592788 −0.296394 0.955066i \(-0.595784\pi\)
−0.296394 + 0.955066i \(0.595784\pi\)
\(230\) −1.40629 −0.0927280
\(231\) 0 0
\(232\) 9.47460 0.622038
\(233\) 25.0654 1.64209 0.821045 0.570863i \(-0.193391\pi\)
0.821045 + 0.570863i \(0.193391\pi\)
\(234\) 0 0
\(235\) −10.4420 −0.681161
\(236\) 13.3341 0.867977
\(237\) 0 0
\(238\) −0.877777 −0.0568978
\(239\) −12.4463 −0.805081 −0.402541 0.915402i \(-0.631873\pi\)
−0.402541 + 0.915402i \(0.631873\pi\)
\(240\) 0 0
\(241\) −9.03260 −0.581841 −0.290920 0.956747i \(-0.593961\pi\)
−0.290920 + 0.956747i \(0.593961\pi\)
\(242\) 3.55201 0.228332
\(243\) 0 0
\(244\) 10.7751 0.689807
\(245\) −3.70683 −0.236821
\(246\) 0 0
\(247\) −8.00000 −0.509028
\(248\) 1.00000 0.0635001
\(249\) 0 0
\(250\) 1.00000 0.0632456
\(251\) 5.92046 0.373696 0.186848 0.982389i \(-0.440173\pi\)
0.186848 + 0.982389i \(0.440173\pi\)
\(252\) 0 0
\(253\) −5.36459 −0.337269
\(254\) −12.2119 −0.766243
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −31.8718 −1.98811 −0.994054 0.108892i \(-0.965270\pi\)
−0.994054 + 0.108892i \(0.965270\pi\)
\(258\) 0 0
\(259\) 6.72296 0.417744
\(260\) −2.73730 −0.169760
\(261\) 0 0
\(262\) 9.55414 0.590257
\(263\) −1.82594 −0.112592 −0.0562962 0.998414i \(-0.517929\pi\)
−0.0562962 + 0.998414i \(0.517929\pi\)
\(264\) 0 0
\(265\) −8.55201 −0.525346
\(266\) −5.30365 −0.325188
\(267\) 0 0
\(268\) 9.54988 0.583352
\(269\) −13.8452 −0.844155 −0.422078 0.906560i \(-0.638699\pi\)
−0.422078 + 0.906560i \(0.638699\pi\)
\(270\) 0 0
\(271\) −25.5012 −1.54909 −0.774544 0.632520i \(-0.782021\pi\)
−0.774544 + 0.632520i \(0.782021\pi\)
\(272\) 0.483701 0.0293287
\(273\) 0 0
\(274\) −12.9300 −0.781127
\(275\) 3.81471 0.230036
\(276\) 0 0
\(277\) −17.9961 −1.08128 −0.540642 0.841253i \(-0.681818\pi\)
−0.540642 + 0.841253i \(0.681818\pi\)
\(278\) 10.1131 0.606545
\(279\) 0 0
\(280\) −1.81471 −0.108450
\(281\) −18.9492 −1.13041 −0.565207 0.824949i \(-0.691204\pi\)
−0.565207 + 0.824949i \(0.691204\pi\)
\(282\) 0 0
\(283\) 19.7047 1.17132 0.585661 0.810556i \(-0.300835\pi\)
0.585661 + 0.810556i \(0.300835\pi\)
\(284\) −5.65989 −0.335853
\(285\) 0 0
\(286\) −10.4420 −0.617448
\(287\) −17.1936 −1.01491
\(288\) 0 0
\(289\) −16.7660 −0.986237
\(290\) 9.47460 0.556368
\(291\) 0 0
\(292\) 8.70683 0.509529
\(293\) −12.6620 −0.739723 −0.369861 0.929087i \(-0.620595\pi\)
−0.369861 + 0.929087i \(0.620595\pi\)
\(294\) 0 0
\(295\) 13.3341 0.776342
\(296\) −3.70470 −0.215331
\(297\) 0 0
\(298\) −18.9940 −1.10029
\(299\) 3.84944 0.222619
\(300\) 0 0
\(301\) −20.9044 −1.20491
\(302\) −4.96740 −0.285842
\(303\) 0 0
\(304\) 2.92259 0.167622
\(305\) 10.7751 0.616983
\(306\) 0 0
\(307\) 14.8130 0.845421 0.422711 0.906265i \(-0.361079\pi\)
0.422711 + 0.906265i \(0.361079\pi\)
\(308\) −6.92259 −0.394451
\(309\) 0 0
\(310\) 1.00000 0.0567962
\(311\) 25.0287 1.41925 0.709625 0.704580i \(-0.248865\pi\)
0.709625 + 0.704580i \(0.248865\pi\)
\(312\) 0 0
\(313\) −20.1366 −1.13819 −0.569094 0.822272i \(-0.692706\pi\)
−0.569094 + 0.822272i \(0.692706\pi\)
\(314\) 5.73517 0.323654
\(315\) 0 0
\(316\) −9.36459 −0.526799
\(317\) −21.2588 −1.19402 −0.597008 0.802236i \(-0.703644\pi\)
−0.597008 + 0.802236i \(0.703644\pi\)
\(318\) 0 0
\(319\) 36.1428 2.02361
\(320\) 1.00000 0.0559017
\(321\) 0 0
\(322\) 2.55201 0.142218
\(323\) 1.41366 0.0786581
\(324\) 0 0
\(325\) −2.73730 −0.151838
\(326\) −1.17930 −0.0653153
\(327\) 0 0
\(328\) 9.47460 0.523147
\(329\) 18.9492 1.04470
\(330\) 0 0
\(331\) 16.0949 0.884656 0.442328 0.896853i \(-0.354153\pi\)
0.442328 + 0.896853i \(0.354153\pi\)
\(332\) 2.81258 0.154360
\(333\) 0 0
\(334\) 21.7691 1.19115
\(335\) 9.54988 0.521766
\(336\) 0 0
\(337\) 2.07142 0.112837 0.0564187 0.998407i \(-0.482032\pi\)
0.0564187 + 0.998407i \(0.482032\pi\)
\(338\) −5.50720 −0.299552
\(339\) 0 0
\(340\) 0.483701 0.0262324
\(341\) 3.81471 0.206578
\(342\) 0 0
\(343\) 19.4298 1.04911
\(344\) 11.5194 0.621085
\(345\) 0 0
\(346\) −18.2872 −0.983125
\(347\) 4.22624 0.226876 0.113438 0.993545i \(-0.463814\pi\)
0.113438 + 0.993545i \(0.463814\pi\)
\(348\) 0 0
\(349\) 13.5685 0.726304 0.363152 0.931730i \(-0.381701\pi\)
0.363152 + 0.931730i \(0.381701\pi\)
\(350\) −1.81471 −0.0970003
\(351\) 0 0
\(352\) 3.81471 0.203325
\(353\) 7.59198 0.404080 0.202040 0.979377i \(-0.435243\pi\)
0.202040 + 0.979377i \(0.435243\pi\)
\(354\) 0 0
\(355\) −5.65989 −0.300396
\(356\) −12.4115 −0.657810
\(357\) 0 0
\(358\) −5.92046 −0.312906
\(359\) −26.7030 −1.40933 −0.704664 0.709541i \(-0.748902\pi\)
−0.704664 + 0.709541i \(0.748902\pi\)
\(360\) 0 0
\(361\) −10.4585 −0.550446
\(362\) 12.8809 0.677004
\(363\) 0 0
\(364\) 4.96740 0.260363
\(365\) 8.70683 0.455736
\(366\) 0 0
\(367\) −29.8371 −1.55748 −0.778741 0.627346i \(-0.784141\pi\)
−0.778741 + 0.627346i \(0.784141\pi\)
\(368\) −1.40629 −0.0733079
\(369\) 0 0
\(370\) −3.70470 −0.192598
\(371\) 15.5194 0.805728
\(372\) 0 0
\(373\) 30.4686 1.57760 0.788802 0.614647i \(-0.210702\pi\)
0.788802 + 0.614647i \(0.210702\pi\)
\(374\) 1.84518 0.0954119
\(375\) 0 0
\(376\) −10.4420 −0.538505
\(377\) −25.9348 −1.33571
\(378\) 0 0
\(379\) 6.86165 0.352459 0.176230 0.984349i \(-0.443610\pi\)
0.176230 + 0.984349i \(0.443610\pi\)
\(380\) 2.92259 0.149926
\(381\) 0 0
\(382\) −3.55414 −0.181845
\(383\) −9.20666 −0.470438 −0.235219 0.971942i \(-0.575581\pi\)
−0.235219 + 0.971942i \(0.575581\pi\)
\(384\) 0 0
\(385\) −6.92259 −0.352808
\(386\) 17.2588 0.878452
\(387\) 0 0
\(388\) 0.154821 0.00785987
\(389\) −12.0000 −0.608424 −0.304212 0.952604i \(-0.598393\pi\)
−0.304212 + 0.952604i \(0.598393\pi\)
\(390\) 0 0
\(391\) −0.680224 −0.0344004
\(392\) −3.70683 −0.187223
\(393\) 0 0
\(394\) −8.52540 −0.429504
\(395\) −9.36459 −0.471184
\(396\) 0 0
\(397\) 9.58461 0.481038 0.240519 0.970644i \(-0.422682\pi\)
0.240519 + 0.970644i \(0.422682\pi\)
\(398\) 20.0266 1.00384
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −30.7701 −1.53659 −0.768293 0.640098i \(-0.778894\pi\)
−0.768293 + 0.640098i \(0.778894\pi\)
\(402\) 0 0
\(403\) −2.73730 −0.136355
\(404\) −1.88999 −0.0940306
\(405\) 0 0
\(406\) −17.1936 −0.853306
\(407\) −14.1324 −0.700515
\(408\) 0 0
\(409\) −35.4669 −1.75372 −0.876862 0.480742i \(-0.840367\pi\)
−0.876862 + 0.480742i \(0.840367\pi\)
\(410\) 9.47460 0.467917
\(411\) 0 0
\(412\) −5.33412 −0.262793
\(413\) −24.1976 −1.19068
\(414\) 0 0
\(415\) 2.81258 0.138064
\(416\) −2.73730 −0.134207
\(417\) 0 0
\(418\) 11.1488 0.545307
\(419\) 5.61508 0.274314 0.137157 0.990549i \(-0.456203\pi\)
0.137157 + 0.990549i \(0.456203\pi\)
\(420\) 0 0
\(421\) 15.9880 0.779208 0.389604 0.920982i \(-0.372612\pi\)
0.389604 + 0.920982i \(0.372612\pi\)
\(422\) −26.4686 −1.28847
\(423\) 0 0
\(424\) −8.55201 −0.415322
\(425\) 0.483701 0.0234629
\(426\) 0 0
\(427\) −19.5537 −0.946272
\(428\) 3.51941 0.170117
\(429\) 0 0
\(430\) 11.5194 0.555515
\(431\) 23.7047 1.14182 0.570908 0.821014i \(-0.306591\pi\)
0.570908 + 0.821014i \(0.306591\pi\)
\(432\) 0 0
\(433\) 34.6948 1.66733 0.833664 0.552272i \(-0.186239\pi\)
0.833664 + 0.552272i \(0.186239\pi\)
\(434\) −1.81471 −0.0871088
\(435\) 0 0
\(436\) 20.0714 0.961247
\(437\) −4.11001 −0.196608
\(438\) 0 0
\(439\) 7.31978 0.349354 0.174677 0.984626i \(-0.444112\pi\)
0.174677 + 0.984626i \(0.444112\pi\)
\(440\) 3.81471 0.181859
\(441\) 0 0
\(442\) −1.32403 −0.0629779
\(443\) 23.8823 1.13468 0.567340 0.823483i \(-0.307972\pi\)
0.567340 + 0.823483i \(0.307972\pi\)
\(444\) 0 0
\(445\) −12.4115 −0.588363
\(446\) 24.3625 1.15360
\(447\) 0 0
\(448\) −1.81471 −0.0857370
\(449\) −17.6333 −0.832166 −0.416083 0.909327i \(-0.636597\pi\)
−0.416083 + 0.909327i \(0.636597\pi\)
\(450\) 0 0
\(451\) 36.1428 1.70190
\(452\) 3.73517 0.175688
\(453\) 0 0
\(454\) −13.6742 −0.641763
\(455\) 4.96740 0.232875
\(456\) 0 0
\(457\) −28.4463 −1.33066 −0.665330 0.746549i \(-0.731709\pi\)
−0.665330 + 0.746549i \(0.731709\pi\)
\(458\) −8.97051 −0.419165
\(459\) 0 0
\(460\) −1.40629 −0.0655686
\(461\) 5.62516 0.261990 0.130995 0.991383i \(-0.458183\pi\)
0.130995 + 0.991383i \(0.458183\pi\)
\(462\) 0 0
\(463\) −26.8801 −1.24923 −0.624613 0.780935i \(-0.714743\pi\)
−0.624613 + 0.780935i \(0.714743\pi\)
\(464\) 9.47460 0.439847
\(465\) 0 0
\(466\) 25.0654 1.16113
\(467\) 33.6174 1.55563 0.777815 0.628494i \(-0.216328\pi\)
0.777815 + 0.628494i \(0.216328\pi\)
\(468\) 0 0
\(469\) −17.3303 −0.800237
\(470\) −10.4420 −0.481654
\(471\) 0 0
\(472\) 13.3341 0.613753
\(473\) 43.9432 2.02051
\(474\) 0 0
\(475\) 2.92259 0.134098
\(476\) −0.877777 −0.0402328
\(477\) 0 0
\(478\) −12.4463 −0.569279
\(479\) −1.96953 −0.0899902 −0.0449951 0.998987i \(-0.514327\pi\)
−0.0449951 + 0.998987i \(0.514327\pi\)
\(480\) 0 0
\(481\) 10.1409 0.462384
\(482\) −9.03260 −0.411424
\(483\) 0 0
\(484\) 3.55201 0.161455
\(485\) 0.154821 0.00703008
\(486\) 0 0
\(487\) 23.1611 1.04953 0.524765 0.851247i \(-0.324153\pi\)
0.524765 + 0.851247i \(0.324153\pi\)
\(488\) 10.7751 0.487768
\(489\) 0 0
\(490\) −3.70683 −0.167457
\(491\) 22.0305 0.994221 0.497111 0.867687i \(-0.334394\pi\)
0.497111 + 0.867687i \(0.334394\pi\)
\(492\) 0 0
\(493\) 4.58287 0.206402
\(494\) −8.00000 −0.359937
\(495\) 0 0
\(496\) 1.00000 0.0449013
\(497\) 10.2711 0.460720
\(498\) 0 0
\(499\) −20.9257 −0.936763 −0.468382 0.883526i \(-0.655163\pi\)
−0.468382 + 0.883526i \(0.655163\pi\)
\(500\) 1.00000 0.0447214
\(501\) 0 0
\(502\) 5.92046 0.264243
\(503\) −16.4238 −0.732301 −0.366150 0.930556i \(-0.619324\pi\)
−0.366150 + 0.930556i \(0.619324\pi\)
\(504\) 0 0
\(505\) −1.88999 −0.0841035
\(506\) −5.36459 −0.238485
\(507\) 0 0
\(508\) −12.2119 −0.541815
\(509\) −13.4137 −0.594550 −0.297275 0.954792i \(-0.596078\pi\)
−0.297275 + 0.954792i \(0.596078\pi\)
\(510\) 0 0
\(511\) −15.8004 −0.698967
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −31.8718 −1.40580
\(515\) −5.33412 −0.235049
\(516\) 0 0
\(517\) −39.8332 −1.75186
\(518\) 6.72296 0.295390
\(519\) 0 0
\(520\) −2.73730 −0.120038
\(521\) −12.7334 −0.557862 −0.278931 0.960311i \(-0.589980\pi\)
−0.278931 + 0.960311i \(0.589980\pi\)
\(522\) 0 0
\(523\) 6.70257 0.293083 0.146541 0.989205i \(-0.453186\pi\)
0.146541 + 0.989205i \(0.453186\pi\)
\(524\) 9.55414 0.417374
\(525\) 0 0
\(526\) −1.82594 −0.0796148
\(527\) 0.483701 0.0210703
\(528\) 0 0
\(529\) −21.0223 −0.914015
\(530\) −8.55201 −0.371475
\(531\) 0 0
\(532\) −5.30365 −0.229942
\(533\) −25.9348 −1.12336
\(534\) 0 0
\(535\) 3.51941 0.152157
\(536\) 9.54988 0.412492
\(537\) 0 0
\(538\) −13.8452 −0.596908
\(539\) −14.1405 −0.609074
\(540\) 0 0
\(541\) 18.1366 0.779754 0.389877 0.920867i \(-0.372518\pi\)
0.389877 + 0.920867i \(0.372518\pi\)
\(542\) −25.5012 −1.09537
\(543\) 0 0
\(544\) 0.483701 0.0207385
\(545\) 20.0714 0.859765
\(546\) 0 0
\(547\) −2.45012 −0.104760 −0.0523798 0.998627i \(-0.516681\pi\)
−0.0523798 + 0.998627i \(0.516681\pi\)
\(548\) −12.9300 −0.552340
\(549\) 0 0
\(550\) 3.81471 0.162660
\(551\) 27.6904 1.17965
\(552\) 0 0
\(553\) 16.9940 0.722659
\(554\) −17.9961 −0.764583
\(555\) 0 0
\(556\) 10.1131 0.428892
\(557\) −21.5194 −0.911807 −0.455903 0.890029i \(-0.650684\pi\)
−0.455903 + 0.890029i \(0.650684\pi\)
\(558\) 0 0
\(559\) −31.5321 −1.33366
\(560\) −1.81471 −0.0766855
\(561\) 0 0
\(562\) −18.9492 −0.799324
\(563\) −18.5177 −0.780427 −0.390213 0.920724i \(-0.627599\pi\)
−0.390213 + 0.920724i \(0.627599\pi\)
\(564\) 0 0
\(565\) 3.73517 0.157140
\(566\) 19.7047 0.828250
\(567\) 0 0
\(568\) −5.65989 −0.237484
\(569\) −26.7030 −1.11945 −0.559723 0.828680i \(-0.689093\pi\)
−0.559723 + 0.828680i \(0.689093\pi\)
\(570\) 0 0
\(571\) −37.5982 −1.57344 −0.786718 0.617313i \(-0.788221\pi\)
−0.786718 + 0.617313i \(0.788221\pi\)
\(572\) −10.4420 −0.436602
\(573\) 0 0
\(574\) −17.1936 −0.717649
\(575\) −1.40629 −0.0586464
\(576\) 0 0
\(577\) −21.4789 −0.894176 −0.447088 0.894490i \(-0.647539\pi\)
−0.447088 + 0.894490i \(0.647539\pi\)
\(578\) −16.7660 −0.697375
\(579\) 0 0
\(580\) 9.47460 0.393411
\(581\) −5.10402 −0.211750
\(582\) 0 0
\(583\) −32.6234 −1.35112
\(584\) 8.70683 0.360291
\(585\) 0 0
\(586\) −12.6620 −0.523063
\(587\) −16.4238 −0.677882 −0.338941 0.940808i \(-0.610069\pi\)
−0.338941 + 0.940808i \(0.610069\pi\)
\(588\) 0 0
\(589\) 2.92259 0.120423
\(590\) 13.3341 0.548957
\(591\) 0 0
\(592\) −3.70470 −0.152262
\(593\) 3.09779 0.127211 0.0636056 0.997975i \(-0.479740\pi\)
0.0636056 + 0.997975i \(0.479740\pi\)
\(594\) 0 0
\(595\) −0.877777 −0.0359853
\(596\) −18.9940 −0.778025
\(597\) 0 0
\(598\) 3.84944 0.157415
\(599\) −4.97966 −0.203464 −0.101732 0.994812i \(-0.532438\pi\)
−0.101732 + 0.994812i \(0.532438\pi\)
\(600\) 0 0
\(601\) −43.8514 −1.78874 −0.894368 0.447332i \(-0.852374\pi\)
−0.894368 + 0.447332i \(0.852374\pi\)
\(602\) −20.9044 −0.851999
\(603\) 0 0
\(604\) −4.96740 −0.202121
\(605\) 3.55201 0.144410
\(606\) 0 0
\(607\) −39.1387 −1.58859 −0.794296 0.607531i \(-0.792160\pi\)
−0.794296 + 0.607531i \(0.792160\pi\)
\(608\) 2.92259 0.118527
\(609\) 0 0
\(610\) 10.7751 0.436273
\(611\) 28.5829 1.15634
\(612\) 0 0
\(613\) −29.9065 −1.20791 −0.603956 0.797017i \(-0.706410\pi\)
−0.603956 + 0.797017i \(0.706410\pi\)
\(614\) 14.8130 0.597803
\(615\) 0 0
\(616\) −6.92259 −0.278919
\(617\) −21.8066 −0.877900 −0.438950 0.898511i \(-0.644650\pi\)
−0.438950 + 0.898511i \(0.644650\pi\)
\(618\) 0 0
\(619\) −19.5120 −0.784255 −0.392128 0.919911i \(-0.628261\pi\)
−0.392128 + 0.919911i \(0.628261\pi\)
\(620\) 1.00000 0.0401610
\(621\) 0 0
\(622\) 25.0287 1.00356
\(623\) 22.5233 0.902378
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −20.1366 −0.804821
\(627\) 0 0
\(628\) 5.73517 0.228858
\(629\) −1.79197 −0.0714504
\(630\) 0 0
\(631\) −5.42553 −0.215987 −0.107993 0.994152i \(-0.534443\pi\)
−0.107993 + 0.994152i \(0.534443\pi\)
\(632\) −9.36459 −0.372503
\(633\) 0 0
\(634\) −21.2588 −0.844296
\(635\) −12.2119 −0.484614
\(636\) 0 0
\(637\) 10.1467 0.402027
\(638\) 36.1428 1.43091
\(639\) 0 0
\(640\) 1.00000 0.0395285
\(641\) 8.51532 0.336335 0.168167 0.985758i \(-0.446215\pi\)
0.168167 + 0.985758i \(0.446215\pi\)
\(642\) 0 0
\(643\) 19.5055 0.769220 0.384610 0.923079i \(-0.374336\pi\)
0.384610 + 0.923079i \(0.374336\pi\)
\(644\) 2.55201 0.100563
\(645\) 0 0
\(646\) 1.41366 0.0556197
\(647\) −18.0788 −0.710750 −0.355375 0.934724i \(-0.615647\pi\)
−0.355375 + 0.934724i \(0.615647\pi\)
\(648\) 0 0
\(649\) 50.8658 1.99666
\(650\) −2.73730 −0.107366
\(651\) 0 0
\(652\) −1.17930 −0.0461849
\(653\) −4.44200 −0.173829 −0.0869144 0.996216i \(-0.527701\pi\)
−0.0869144 + 0.996216i \(0.527701\pi\)
\(654\) 0 0
\(655\) 9.55414 0.373311
\(656\) 9.47460 0.369921
\(657\) 0 0
\(658\) 18.9492 0.738717
\(659\) 5.79433 0.225715 0.112857 0.993611i \(-0.464000\pi\)
0.112857 + 0.993611i \(0.464000\pi\)
\(660\) 0 0
\(661\) −9.91037 −0.385469 −0.192734 0.981251i \(-0.561736\pi\)
−0.192734 + 0.981251i \(0.561736\pi\)
\(662\) 16.0949 0.625547
\(663\) 0 0
\(664\) 2.81258 0.109149
\(665\) −5.30365 −0.205667
\(666\) 0 0
\(667\) −13.3240 −0.515909
\(668\) 21.7691 0.842273
\(669\) 0 0
\(670\) 9.54988 0.368944
\(671\) 41.1040 1.58680
\(672\) 0 0
\(673\) −28.2872 −1.09039 −0.545195 0.838309i \(-0.683545\pi\)
−0.545195 + 0.838309i \(0.683545\pi\)
\(674\) 2.07142 0.0797881
\(675\) 0 0
\(676\) −5.50720 −0.211815
\(677\) 48.2956 1.85615 0.928075 0.372394i \(-0.121463\pi\)
0.928075 + 0.372394i \(0.121463\pi\)
\(678\) 0 0
\(679\) −0.280956 −0.0107821
\(680\) 0.483701 0.0185491
\(681\) 0 0
\(682\) 3.81471 0.146073
\(683\) −18.2571 −0.698589 −0.349294 0.937013i \(-0.613579\pi\)
−0.349294 + 0.937013i \(0.613579\pi\)
\(684\) 0 0
\(685\) −12.9300 −0.494028
\(686\) 19.4298 0.741833
\(687\) 0 0
\(688\) 11.5194 0.439173
\(689\) 23.4094 0.891827
\(690\) 0 0
\(691\) −39.7926 −1.51378 −0.756892 0.653540i \(-0.773283\pi\)
−0.756892 + 0.653540i \(0.773283\pi\)
\(692\) −18.2872 −0.695174
\(693\) 0 0
\(694\) 4.22624 0.160426
\(695\) 10.1131 0.383613
\(696\) 0 0
\(697\) 4.58287 0.173589
\(698\) 13.5685 0.513575
\(699\) 0 0
\(700\) −1.81471 −0.0685896
\(701\) 37.5117 1.41680 0.708398 0.705813i \(-0.249418\pi\)
0.708398 + 0.705813i \(0.249418\pi\)
\(702\) 0 0
\(703\) −10.8273 −0.408360
\(704\) 3.81471 0.143772
\(705\) 0 0
\(706\) 7.59198 0.285728
\(707\) 3.42978 0.128990
\(708\) 0 0
\(709\) 1.16807 0.0438676 0.0219338 0.999759i \(-0.493018\pi\)
0.0219338 + 0.999759i \(0.493018\pi\)
\(710\) −5.65989 −0.212412
\(711\) 0 0
\(712\) −12.4115 −0.465142
\(713\) −1.40629 −0.0526660
\(714\) 0 0
\(715\) −10.4420 −0.390509
\(716\) −5.92046 −0.221258
\(717\) 0 0
\(718\) −26.7030 −0.996546
\(719\) 26.8658 1.00192 0.500962 0.865469i \(-0.332979\pi\)
0.500962 + 0.865469i \(0.332979\pi\)
\(720\) 0 0
\(721\) 9.67988 0.360497
\(722\) −10.4585 −0.389224
\(723\) 0 0
\(724\) 12.8809 0.478714
\(725\) 9.47460 0.351878
\(726\) 0 0
\(727\) −15.9086 −0.590017 −0.295009 0.955495i \(-0.595322\pi\)
−0.295009 + 0.955495i \(0.595322\pi\)
\(728\) 4.96740 0.184104
\(729\) 0 0
\(730\) 8.70683 0.322254
\(731\) 5.57195 0.206086
\(732\) 0 0
\(733\) −3.31978 −0.122619 −0.0613094 0.998119i \(-0.519528\pi\)
−0.0613094 + 0.998119i \(0.519528\pi\)
\(734\) −29.8371 −1.10131
\(735\) 0 0
\(736\) −1.40629 −0.0518365
\(737\) 36.4300 1.34192
\(738\) 0 0
\(739\) 31.8749 1.17254 0.586268 0.810117i \(-0.300596\pi\)
0.586268 + 0.810117i \(0.300596\pi\)
\(740\) −3.70470 −0.136188
\(741\) 0 0
\(742\) 15.5194 0.569735
\(743\) −8.80174 −0.322905 −0.161452 0.986881i \(-0.551618\pi\)
−0.161452 + 0.986881i \(0.551618\pi\)
\(744\) 0 0
\(745\) −18.9940 −0.695887
\(746\) 30.4686 1.11553
\(747\) 0 0
\(748\) 1.84518 0.0674664
\(749\) −6.38671 −0.233365
\(750\) 0 0
\(751\) 10.6682 0.389290 0.194645 0.980874i \(-0.437645\pi\)
0.194645 + 0.980874i \(0.437645\pi\)
\(752\) −10.4420 −0.380781
\(753\) 0 0
\(754\) −25.9348 −0.944490
\(755\) −4.96740 −0.180782
\(756\) 0 0
\(757\) −31.2507 −1.13583 −0.567913 0.823088i \(-0.692249\pi\)
−0.567913 + 0.823088i \(0.692249\pi\)
\(758\) 6.86165 0.249226
\(759\) 0 0
\(760\) 2.92259 0.106013
\(761\) 47.1407 1.70885 0.854425 0.519575i \(-0.173910\pi\)
0.854425 + 0.519575i \(0.173910\pi\)
\(762\) 0 0
\(763\) −36.4238 −1.31863
\(764\) −3.55414 −0.128584
\(765\) 0 0
\(766\) −9.20666 −0.332650
\(767\) −36.4995 −1.31792
\(768\) 0 0
\(769\) 44.5295 1.60578 0.802888 0.596130i \(-0.203296\pi\)
0.802888 + 0.596130i \(0.203296\pi\)
\(770\) −6.92259 −0.249473
\(771\) 0 0
\(772\) 17.2588 0.621159
\(773\) 47.5971 1.71195 0.855973 0.517020i \(-0.172959\pi\)
0.855973 + 0.517020i \(0.172959\pi\)
\(774\) 0 0
\(775\) 1.00000 0.0359211
\(776\) 0.154821 0.00555776
\(777\) 0 0
\(778\) −12.0000 −0.430221
\(779\) 27.6904 0.992110
\(780\) 0 0
\(781\) −21.5908 −0.772581
\(782\) −0.680224 −0.0243248
\(783\) 0 0
\(784\) −3.70683 −0.132387
\(785\) 5.73517 0.204697
\(786\) 0 0
\(787\) −27.7394 −0.988804 −0.494402 0.869233i \(-0.664613\pi\)
−0.494402 + 0.869233i \(0.664613\pi\)
\(788\) −8.52540 −0.303705
\(789\) 0 0
\(790\) −9.36459 −0.333177
\(791\) −6.77825 −0.241007
\(792\) 0 0
\(793\) −29.4948 −1.04739
\(794\) 9.58461 0.340145
\(795\) 0 0
\(796\) 20.0266 0.709824
\(797\) −17.8409 −0.631958 −0.315979 0.948766i \(-0.602333\pi\)
−0.315979 + 0.948766i \(0.602333\pi\)
\(798\) 0 0
\(799\) −5.05081 −0.178685
\(800\) 1.00000 0.0353553
\(801\) 0 0
\(802\) −30.7701 −1.08653
\(803\) 33.2140 1.17210
\(804\) 0 0
\(805\) 2.55201 0.0899465
\(806\) −2.73730 −0.0964172
\(807\) 0 0
\(808\) −1.88999 −0.0664897
\(809\) −35.0511 −1.23233 −0.616165 0.787617i \(-0.711315\pi\)
−0.616165 + 0.787617i \(0.711315\pi\)
\(810\) 0 0
\(811\) 41.5908 1.46045 0.730226 0.683206i \(-0.239415\pi\)
0.730226 + 0.683206i \(0.239415\pi\)
\(812\) −17.1936 −0.603379
\(813\) 0 0
\(814\) −14.1324 −0.495339
\(815\) −1.17930 −0.0413090
\(816\) 0 0
\(817\) 33.6665 1.17784
\(818\) −35.4669 −1.24007
\(819\) 0 0
\(820\) 9.47460 0.330867
\(821\) 5.10402 0.178131 0.0890657 0.996026i \(-0.471612\pi\)
0.0890657 + 0.996026i \(0.471612\pi\)
\(822\) 0 0
\(823\) 31.3811 1.09388 0.546938 0.837173i \(-0.315793\pi\)
0.546938 + 0.837173i \(0.315793\pi\)
\(824\) −5.33412 −0.185823
\(825\) 0 0
\(826\) −24.1976 −0.841941
\(827\) −11.3198 −0.393627 −0.196814 0.980441i \(-0.563059\pi\)
−0.196814 + 0.980441i \(0.563059\pi\)
\(828\) 0 0
\(829\) −14.7548 −0.512454 −0.256227 0.966617i \(-0.582479\pi\)
−0.256227 + 0.966617i \(0.582479\pi\)
\(830\) 2.81258 0.0976261
\(831\) 0 0
\(832\) −2.73730 −0.0948988
\(833\) −1.79300 −0.0621237
\(834\) 0 0
\(835\) 21.7691 0.753352
\(836\) 11.1488 0.385590
\(837\) 0 0
\(838\) 5.61508 0.193970
\(839\) −53.2121 −1.83709 −0.918543 0.395320i \(-0.870634\pi\)
−0.918543 + 0.395320i \(0.870634\pi\)
\(840\) 0 0
\(841\) 60.7680 2.09545
\(842\) 15.9880 0.550983
\(843\) 0 0
\(844\) −26.4686 −0.911087
\(845\) −5.50720 −0.189453
\(846\) 0 0
\(847\) −6.44586 −0.221482
\(848\) −8.55201 −0.293677
\(849\) 0 0
\(850\) 0.483701 0.0165908
\(851\) 5.20988 0.178593
\(852\) 0 0
\(853\) 3.97962 0.136259 0.0681297 0.997676i \(-0.478297\pi\)
0.0681297 + 0.997676i \(0.478297\pi\)
\(854\) −19.5537 −0.669115
\(855\) 0 0
\(856\) 3.51941 0.120291
\(857\) −1.38071 −0.0471643 −0.0235822 0.999722i \(-0.507507\pi\)
−0.0235822 + 0.999722i \(0.507507\pi\)
\(858\) 0 0
\(859\) 48.0262 1.63863 0.819317 0.573340i \(-0.194353\pi\)
0.819317 + 0.573340i \(0.194353\pi\)
\(860\) 11.5194 0.392809
\(861\) 0 0
\(862\) 23.7047 0.807385
\(863\) 40.6651 1.38426 0.692129 0.721774i \(-0.256673\pi\)
0.692129 + 0.721774i \(0.256673\pi\)
\(864\) 0 0
\(865\) −18.2872 −0.621783
\(866\) 34.6948 1.17898
\(867\) 0 0
\(868\) −1.81471 −0.0615953
\(869\) −35.7232 −1.21183
\(870\) 0 0
\(871\) −26.1409 −0.885750
\(872\) 20.0714 0.679704
\(873\) 0 0
\(874\) −4.11001 −0.139023
\(875\) −1.81471 −0.0613484
\(876\) 0 0
\(877\) 36.0819 1.21840 0.609200 0.793017i \(-0.291491\pi\)
0.609200 + 0.793017i \(0.291491\pi\)
\(878\) 7.31978 0.247030
\(879\) 0 0
\(880\) 3.81471 0.128594
\(881\) −16.3772 −0.551762 −0.275881 0.961192i \(-0.588970\pi\)
−0.275881 + 0.961192i \(0.588970\pi\)
\(882\) 0 0
\(883\) 20.2753 0.682319 0.341159 0.940006i \(-0.389180\pi\)
0.341159 + 0.940006i \(0.389180\pi\)
\(884\) −1.32403 −0.0445321
\(885\) 0 0
\(886\) 23.8823 0.802340
\(887\) 56.7397 1.90513 0.952566 0.304333i \(-0.0984336\pi\)
0.952566 + 0.304333i \(0.0984336\pi\)
\(888\) 0 0
\(889\) 22.1610 0.743258
\(890\) −12.4115 −0.416035
\(891\) 0 0
\(892\) 24.3625 0.815715
\(893\) −30.5177 −1.02124
\(894\) 0 0
\(895\) −5.92046 −0.197899
\(896\) −1.81471 −0.0606252
\(897\) 0 0
\(898\) −17.6333 −0.588430
\(899\) 9.47460 0.315996
\(900\) 0 0
\(901\) −4.13661 −0.137811
\(902\) 36.1428 1.20342
\(903\) 0 0
\(904\) 3.73517 0.124230
\(905\) 12.8809 0.428175
\(906\) 0 0
\(907\) 28.6582 0.951578 0.475789 0.879559i \(-0.342163\pi\)
0.475789 + 0.879559i \(0.342163\pi\)
\(908\) −13.6742 −0.453795
\(909\) 0 0
\(910\) 4.96740 0.164668
\(911\) −40.3543 −1.33700 −0.668499 0.743713i \(-0.733063\pi\)
−0.668499 + 0.743713i \(0.733063\pi\)
\(912\) 0 0
\(913\) 10.7292 0.355084
\(914\) −28.4463 −0.940919
\(915\) 0 0
\(916\) −8.97051 −0.296394
\(917\) −17.3380 −0.572551
\(918\) 0 0
\(919\) −1.31552 −0.0433949 −0.0216975 0.999765i \(-0.506907\pi\)
−0.0216975 + 0.999765i \(0.506907\pi\)
\(920\) −1.40629 −0.0463640
\(921\) 0 0
\(922\) 5.62516 0.185255
\(923\) 15.4928 0.509952
\(924\) 0 0
\(925\) −3.70470 −0.121810
\(926\) −26.8801 −0.883336
\(927\) 0 0
\(928\) 9.47460 0.311019
\(929\) 24.4500 0.802179 0.401089 0.916039i \(-0.368632\pi\)
0.401089 + 0.916039i \(0.368632\pi\)
\(930\) 0 0
\(931\) −10.8335 −0.355055
\(932\) 25.0654 0.821045
\(933\) 0 0
\(934\) 33.6174 1.10000
\(935\) 1.84518 0.0603438
\(936\) 0 0
\(937\) 10.3096 0.336801 0.168401 0.985719i \(-0.446140\pi\)
0.168401 + 0.985719i \(0.446140\pi\)
\(938\) −17.3303 −0.565853
\(939\) 0 0
\(940\) −10.4420 −0.340580
\(941\) −13.4137 −0.437273 −0.218636 0.975806i \(-0.570161\pi\)
−0.218636 + 0.975806i \(0.570161\pi\)
\(942\) 0 0
\(943\) −13.3240 −0.433890
\(944\) 13.3341 0.433989
\(945\) 0 0
\(946\) 43.9432 1.42872
\(947\) −9.92085 −0.322384 −0.161192 0.986923i \(-0.551534\pi\)
−0.161192 + 0.986923i \(0.551534\pi\)
\(948\) 0 0
\(949\) −23.8332 −0.773658
\(950\) 2.92259 0.0948213
\(951\) 0 0
\(952\) −0.877777 −0.0284489
\(953\) 14.5264 0.470557 0.235279 0.971928i \(-0.424400\pi\)
0.235279 + 0.971928i \(0.424400\pi\)
\(954\) 0 0
\(955\) −3.55414 −0.115009
\(956\) −12.4463 −0.402541
\(957\) 0 0
\(958\) −1.96953 −0.0636327
\(959\) 23.4641 0.757696
\(960\) 0 0
\(961\) 1.00000 0.0322581
\(962\) 10.1409 0.326955
\(963\) 0 0
\(964\) −9.03260 −0.290920
\(965\) 17.2588 0.555582
\(966\) 0 0
\(967\) −29.4952 −0.948501 −0.474250 0.880390i \(-0.657281\pi\)
−0.474250 + 0.880390i \(0.657281\pi\)
\(968\) 3.55201 0.114166
\(969\) 0 0
\(970\) 0.154821 0.00497102
\(971\) 17.8638 0.573276 0.286638 0.958039i \(-0.407462\pi\)
0.286638 + 0.958039i \(0.407462\pi\)
\(972\) 0 0
\(973\) −18.3524 −0.588350
\(974\) 23.1611 0.742129
\(975\) 0 0
\(976\) 10.7751 0.344904
\(977\) −20.2914 −0.649181 −0.324590 0.945855i \(-0.605226\pi\)
−0.324590 + 0.945855i \(0.605226\pi\)
\(978\) 0 0
\(979\) −47.3464 −1.51320
\(980\) −3.70683 −0.118410
\(981\) 0 0
\(982\) 22.0305 0.703021
\(983\) −23.0666 −0.735709 −0.367855 0.929883i \(-0.619908\pi\)
−0.367855 + 0.929883i \(0.619908\pi\)
\(984\) 0 0
\(985\) −8.52540 −0.271642
\(986\) 4.58287 0.145948
\(987\) 0 0
\(988\) −8.00000 −0.254514
\(989\) −16.1996 −0.515118
\(990\) 0 0
\(991\) 3.95945 0.125776 0.0628880 0.998021i \(-0.479969\pi\)
0.0628880 + 0.998021i \(0.479969\pi\)
\(992\) 1.00000 0.0317500
\(993\) 0 0
\(994\) 10.2711 0.325778
\(995\) 20.0266 0.634886
\(996\) 0 0
\(997\) −5.25458 −0.166414 −0.0832071 0.996532i \(-0.526516\pi\)
−0.0832071 + 0.996532i \(0.526516\pi\)
\(998\) −20.9257 −0.662391
\(999\) 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 2790.2.a.bk.1.1 yes 4
3.2 odd 2 2790.2.a.bj.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2790.2.a.bj.1.1 4 3.2 odd 2
2790.2.a.bk.1.1 yes 4 1.1 even 1 trivial