Properties

Label 6240.2.a.bh.1.2
Level $6240$
Weight $2$
Character 6240.1
Self dual yes
Analytic conductor $49.827$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6240 = 2^{5} \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6240.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(49.8266508613\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 6240.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} -1.00000 q^{5} +2.82843 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -1.00000 q^{5} +2.82843 q^{7} +1.00000 q^{9} +2.82843 q^{11} +1.00000 q^{13} +1.00000 q^{15} -3.65685 q^{17} -1.17157 q^{19} -2.82843 q^{21} +4.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} -7.65685 q^{29} -6.82843 q^{31} -2.82843 q^{33} -2.82843 q^{35} -2.00000 q^{37} -1.00000 q^{39} -2.00000 q^{41} -9.65685 q^{43} -1.00000 q^{45} -6.82843 q^{47} +1.00000 q^{49} +3.65685 q^{51} -2.00000 q^{53} -2.82843 q^{55} +1.17157 q^{57} -8.48528 q^{59} +6.00000 q^{61} +2.82843 q^{63} -1.00000 q^{65} +2.82843 q^{67} -4.00000 q^{69} +5.17157 q^{71} +15.6569 q^{73} -1.00000 q^{75} +8.00000 q^{77} +2.34315 q^{79} +1.00000 q^{81} -1.17157 q^{83} +3.65685 q^{85} +7.65685 q^{87} +17.3137 q^{89} +2.82843 q^{91} +6.82843 q^{93} +1.17157 q^{95} -3.65685 q^{97} +2.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 2 q^{5} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} - 2 q^{5} + 2 q^{9} + 2 q^{13} + 2 q^{15} + 4 q^{17} - 8 q^{19} + 8 q^{23} + 2 q^{25} - 2 q^{27} - 4 q^{29} - 8 q^{31} - 4 q^{37} - 2 q^{39} - 4 q^{41} - 8 q^{43} - 2 q^{45} - 8 q^{47} + 2 q^{49} - 4 q^{51} - 4 q^{53} + 8 q^{57} + 12 q^{61} - 2 q^{65} - 8 q^{69} + 16 q^{71} + 20 q^{73} - 2 q^{75} + 16 q^{77} + 16 q^{79} + 2 q^{81} - 8 q^{83} - 4 q^{85} + 4 q^{87} + 12 q^{89} + 8 q^{93} + 8 q^{95} + 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 2.82843 1.06904 0.534522 0.845154i \(-0.320491\pi\)
0.534522 + 0.845154i \(0.320491\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.82843 0.852803 0.426401 0.904534i \(-0.359781\pi\)
0.426401 + 0.904534i \(0.359781\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) −3.65685 −0.886917 −0.443459 0.896295i \(-0.646249\pi\)
−0.443459 + 0.896295i \(0.646249\pi\)
\(18\) 0 0
\(19\) −1.17157 −0.268777 −0.134389 0.990929i \(-0.542907\pi\)
−0.134389 + 0.990929i \(0.542907\pi\)
\(20\) 0 0
\(21\) −2.82843 −0.617213
\(22\) 0 0
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −7.65685 −1.42184 −0.710921 0.703272i \(-0.751722\pi\)
−0.710921 + 0.703272i \(0.751722\pi\)
\(30\) 0 0
\(31\) −6.82843 −1.22642 −0.613211 0.789919i \(-0.710122\pi\)
−0.613211 + 0.789919i \(0.710122\pi\)
\(32\) 0 0
\(33\) −2.82843 −0.492366
\(34\) 0 0
\(35\) −2.82843 −0.478091
\(36\) 0 0
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 0 0
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) −9.65685 −1.47266 −0.736328 0.676625i \(-0.763442\pi\)
−0.736328 + 0.676625i \(0.763442\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) −6.82843 −0.996028 −0.498014 0.867169i \(-0.665937\pi\)
−0.498014 + 0.867169i \(0.665937\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 3.65685 0.512062
\(52\) 0 0
\(53\) −2.00000 −0.274721 −0.137361 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 0 0
\(55\) −2.82843 −0.381385
\(56\) 0 0
\(57\) 1.17157 0.155179
\(58\) 0 0
\(59\) −8.48528 −1.10469 −0.552345 0.833616i \(-0.686267\pi\)
−0.552345 + 0.833616i \(0.686267\pi\)
\(60\) 0 0
\(61\) 6.00000 0.768221 0.384111 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) 0 0
\(63\) 2.82843 0.356348
\(64\) 0 0
\(65\) −1.00000 −0.124035
\(66\) 0 0
\(67\) 2.82843 0.345547 0.172774 0.984962i \(-0.444727\pi\)
0.172774 + 0.984962i \(0.444727\pi\)
\(68\) 0 0
\(69\) −4.00000 −0.481543
\(70\) 0 0
\(71\) 5.17157 0.613753 0.306876 0.951749i \(-0.400716\pi\)
0.306876 + 0.951749i \(0.400716\pi\)
\(72\) 0 0
\(73\) 15.6569 1.83250 0.916248 0.400611i \(-0.131202\pi\)
0.916248 + 0.400611i \(0.131202\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) 8.00000 0.911685
\(78\) 0 0
\(79\) 2.34315 0.263624 0.131812 0.991275i \(-0.457920\pi\)
0.131812 + 0.991275i \(0.457920\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −1.17157 −0.128597 −0.0642984 0.997931i \(-0.520481\pi\)
−0.0642984 + 0.997931i \(0.520481\pi\)
\(84\) 0 0
\(85\) 3.65685 0.396642
\(86\) 0 0
\(87\) 7.65685 0.820901
\(88\) 0 0
\(89\) 17.3137 1.83525 0.917625 0.397448i \(-0.130104\pi\)
0.917625 + 0.397448i \(0.130104\pi\)
\(90\) 0 0
\(91\) 2.82843 0.296500
\(92\) 0 0
\(93\) 6.82843 0.708075
\(94\) 0 0
\(95\) 1.17157 0.120201
\(96\) 0 0
\(97\) −3.65685 −0.371297 −0.185649 0.982616i \(-0.559439\pi\)
−0.185649 + 0.982616i \(0.559439\pi\)
\(98\) 0 0
\(99\) 2.82843 0.284268
\(100\) 0 0
\(101\) −7.65685 −0.761885 −0.380943 0.924599i \(-0.624401\pi\)
−0.380943 + 0.924599i \(0.624401\pi\)
\(102\) 0 0
\(103\) −9.65685 −0.951518 −0.475759 0.879576i \(-0.657827\pi\)
−0.475759 + 0.879576i \(0.657827\pi\)
\(104\) 0 0
\(105\) 2.82843 0.276026
\(106\) 0 0
\(107\) −15.3137 −1.48043 −0.740216 0.672369i \(-0.765277\pi\)
−0.740216 + 0.672369i \(0.765277\pi\)
\(108\) 0 0
\(109\) −4.34315 −0.415998 −0.207999 0.978129i \(-0.566695\pi\)
−0.207999 + 0.978129i \(0.566695\pi\)
\(110\) 0 0
\(111\) 2.00000 0.189832
\(112\) 0 0
\(113\) −0.343146 −0.0322804 −0.0161402 0.999870i \(-0.505138\pi\)
−0.0161402 + 0.999870i \(0.505138\pi\)
\(114\) 0 0
\(115\) −4.00000 −0.373002
\(116\) 0 0
\(117\) 1.00000 0.0924500
\(118\) 0 0
\(119\) −10.3431 −0.948155
\(120\) 0 0
\(121\) −3.00000 −0.272727
\(122\) 0 0
\(123\) 2.00000 0.180334
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 1.65685 0.147022 0.0735110 0.997294i \(-0.476580\pi\)
0.0735110 + 0.997294i \(0.476580\pi\)
\(128\) 0 0
\(129\) 9.65685 0.850239
\(130\) 0 0
\(131\) −16.0000 −1.39793 −0.698963 0.715158i \(-0.746355\pi\)
−0.698963 + 0.715158i \(0.746355\pi\)
\(132\) 0 0
\(133\) −3.31371 −0.287335
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) −18.0000 −1.53784 −0.768922 0.639343i \(-0.779207\pi\)
−0.768922 + 0.639343i \(0.779207\pi\)
\(138\) 0 0
\(139\) 16.9706 1.43942 0.719712 0.694273i \(-0.244274\pi\)
0.719712 + 0.694273i \(0.244274\pi\)
\(140\) 0 0
\(141\) 6.82843 0.575057
\(142\) 0 0
\(143\) 2.82843 0.236525
\(144\) 0 0
\(145\) 7.65685 0.635867
\(146\) 0 0
\(147\) −1.00000 −0.0824786
\(148\) 0 0
\(149\) −9.31371 −0.763009 −0.381504 0.924367i \(-0.624594\pi\)
−0.381504 + 0.924367i \(0.624594\pi\)
\(150\) 0 0
\(151\) 6.82843 0.555690 0.277845 0.960626i \(-0.410380\pi\)
0.277845 + 0.960626i \(0.410380\pi\)
\(152\) 0 0
\(153\) −3.65685 −0.295639
\(154\) 0 0
\(155\) 6.82843 0.548472
\(156\) 0 0
\(157\) 14.0000 1.11732 0.558661 0.829396i \(-0.311315\pi\)
0.558661 + 0.829396i \(0.311315\pi\)
\(158\) 0 0
\(159\) 2.00000 0.158610
\(160\) 0 0
\(161\) 11.3137 0.891645
\(162\) 0 0
\(163\) −22.1421 −1.73431 −0.867153 0.498042i \(-0.834053\pi\)
−0.867153 + 0.498042i \(0.834053\pi\)
\(164\) 0 0
\(165\) 2.82843 0.220193
\(166\) 0 0
\(167\) 4.48528 0.347081 0.173541 0.984827i \(-0.444479\pi\)
0.173541 + 0.984827i \(0.444479\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) −1.17157 −0.0895924
\(172\) 0 0
\(173\) 20.6274 1.56827 0.784137 0.620588i \(-0.213106\pi\)
0.784137 + 0.620588i \(0.213106\pi\)
\(174\) 0 0
\(175\) 2.82843 0.213809
\(176\) 0 0
\(177\) 8.48528 0.637793
\(178\) 0 0
\(179\) −13.6569 −1.02076 −0.510381 0.859949i \(-0.670495\pi\)
−0.510381 + 0.859949i \(0.670495\pi\)
\(180\) 0 0
\(181\) −5.31371 −0.394965 −0.197482 0.980306i \(-0.563277\pi\)
−0.197482 + 0.980306i \(0.563277\pi\)
\(182\) 0 0
\(183\) −6.00000 −0.443533
\(184\) 0 0
\(185\) 2.00000 0.147043
\(186\) 0 0
\(187\) −10.3431 −0.756366
\(188\) 0 0
\(189\) −2.82843 −0.205738
\(190\) 0 0
\(191\) −5.65685 −0.409316 −0.204658 0.978834i \(-0.565608\pi\)
−0.204658 + 0.978834i \(0.565608\pi\)
\(192\) 0 0
\(193\) 15.6569 1.12701 0.563503 0.826114i \(-0.309454\pi\)
0.563503 + 0.826114i \(0.309454\pi\)
\(194\) 0 0
\(195\) 1.00000 0.0716115
\(196\) 0 0
\(197\) −22.0000 −1.56744 −0.783718 0.621117i \(-0.786679\pi\)
−0.783718 + 0.621117i \(0.786679\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 0 0
\(201\) −2.82843 −0.199502
\(202\) 0 0
\(203\) −21.6569 −1.52001
\(204\) 0 0
\(205\) 2.00000 0.139686
\(206\) 0 0
\(207\) 4.00000 0.278019
\(208\) 0 0
\(209\) −3.31371 −0.229214
\(210\) 0 0
\(211\) 5.65685 0.389434 0.194717 0.980859i \(-0.437621\pi\)
0.194717 + 0.980859i \(0.437621\pi\)
\(212\) 0 0
\(213\) −5.17157 −0.354350
\(214\) 0 0
\(215\) 9.65685 0.658592
\(216\) 0 0
\(217\) −19.3137 −1.31110
\(218\) 0 0
\(219\) −15.6569 −1.05799
\(220\) 0 0
\(221\) −3.65685 −0.245987
\(222\) 0 0
\(223\) 5.17157 0.346314 0.173157 0.984894i \(-0.444603\pi\)
0.173157 + 0.984894i \(0.444603\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −17.1716 −1.13972 −0.569859 0.821743i \(-0.693002\pi\)
−0.569859 + 0.821743i \(0.693002\pi\)
\(228\) 0 0
\(229\) 8.34315 0.551331 0.275665 0.961254i \(-0.411102\pi\)
0.275665 + 0.961254i \(0.411102\pi\)
\(230\) 0 0
\(231\) −8.00000 −0.526361
\(232\) 0 0
\(233\) 10.9706 0.718705 0.359353 0.933202i \(-0.382997\pi\)
0.359353 + 0.933202i \(0.382997\pi\)
\(234\) 0 0
\(235\) 6.82843 0.445437
\(236\) 0 0
\(237\) −2.34315 −0.152204
\(238\) 0 0
\(239\) −7.51472 −0.486087 −0.243043 0.970015i \(-0.578146\pi\)
−0.243043 + 0.970015i \(0.578146\pi\)
\(240\) 0 0
\(241\) 21.3137 1.37294 0.686468 0.727160i \(-0.259160\pi\)
0.686468 + 0.727160i \(0.259160\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −1.00000 −0.0638877
\(246\) 0 0
\(247\) −1.17157 −0.0745454
\(248\) 0 0
\(249\) 1.17157 0.0742454
\(250\) 0 0
\(251\) −8.97056 −0.566217 −0.283108 0.959088i \(-0.591366\pi\)
−0.283108 + 0.959088i \(0.591366\pi\)
\(252\) 0 0
\(253\) 11.3137 0.711287
\(254\) 0 0
\(255\) −3.65685 −0.229001
\(256\) 0 0
\(257\) −14.9706 −0.933838 −0.466919 0.884300i \(-0.654636\pi\)
−0.466919 + 0.884300i \(0.654636\pi\)
\(258\) 0 0
\(259\) −5.65685 −0.351500
\(260\) 0 0
\(261\) −7.65685 −0.473947
\(262\) 0 0
\(263\) 1.65685 0.102166 0.0510830 0.998694i \(-0.483733\pi\)
0.0510830 + 0.998694i \(0.483733\pi\)
\(264\) 0 0
\(265\) 2.00000 0.122859
\(266\) 0 0
\(267\) −17.3137 −1.05958
\(268\) 0 0
\(269\) 3.65685 0.222962 0.111481 0.993767i \(-0.464441\pi\)
0.111481 + 0.993767i \(0.464441\pi\)
\(270\) 0 0
\(271\) −12.4853 −0.758427 −0.379213 0.925309i \(-0.623805\pi\)
−0.379213 + 0.925309i \(0.623805\pi\)
\(272\) 0 0
\(273\) −2.82843 −0.171184
\(274\) 0 0
\(275\) 2.82843 0.170561
\(276\) 0 0
\(277\) −6.68629 −0.401740 −0.200870 0.979618i \(-0.564377\pi\)
−0.200870 + 0.979618i \(0.564377\pi\)
\(278\) 0 0
\(279\) −6.82843 −0.408807
\(280\) 0 0
\(281\) 20.6274 1.23053 0.615264 0.788321i \(-0.289049\pi\)
0.615264 + 0.788321i \(0.289049\pi\)
\(282\) 0 0
\(283\) −6.34315 −0.377061 −0.188530 0.982067i \(-0.560372\pi\)
−0.188530 + 0.982067i \(0.560372\pi\)
\(284\) 0 0
\(285\) −1.17157 −0.0693980
\(286\) 0 0
\(287\) −5.65685 −0.333914
\(288\) 0 0
\(289\) −3.62742 −0.213377
\(290\) 0 0
\(291\) 3.65685 0.214369
\(292\) 0 0
\(293\) 13.3137 0.777795 0.388898 0.921281i \(-0.372856\pi\)
0.388898 + 0.921281i \(0.372856\pi\)
\(294\) 0 0
\(295\) 8.48528 0.494032
\(296\) 0 0
\(297\) −2.82843 −0.164122
\(298\) 0 0
\(299\) 4.00000 0.231326
\(300\) 0 0
\(301\) −27.3137 −1.57434
\(302\) 0 0
\(303\) 7.65685 0.439875
\(304\) 0 0
\(305\) −6.00000 −0.343559
\(306\) 0 0
\(307\) −26.8284 −1.53118 −0.765590 0.643329i \(-0.777553\pi\)
−0.765590 + 0.643329i \(0.777553\pi\)
\(308\) 0 0
\(309\) 9.65685 0.549359
\(310\) 0 0
\(311\) 27.3137 1.54882 0.774409 0.632685i \(-0.218047\pi\)
0.774409 + 0.632685i \(0.218047\pi\)
\(312\) 0 0
\(313\) −14.0000 −0.791327 −0.395663 0.918396i \(-0.629485\pi\)
−0.395663 + 0.918396i \(0.629485\pi\)
\(314\) 0 0
\(315\) −2.82843 −0.159364
\(316\) 0 0
\(317\) −25.3137 −1.42176 −0.710880 0.703314i \(-0.751703\pi\)
−0.710880 + 0.703314i \(0.751703\pi\)
\(318\) 0 0
\(319\) −21.6569 −1.21255
\(320\) 0 0
\(321\) 15.3137 0.854728
\(322\) 0 0
\(323\) 4.28427 0.238383
\(324\) 0 0
\(325\) 1.00000 0.0554700
\(326\) 0 0
\(327\) 4.34315 0.240177
\(328\) 0 0
\(329\) −19.3137 −1.06480
\(330\) 0 0
\(331\) 10.1421 0.557462 0.278731 0.960369i \(-0.410086\pi\)
0.278731 + 0.960369i \(0.410086\pi\)
\(332\) 0 0
\(333\) −2.00000 −0.109599
\(334\) 0 0
\(335\) −2.82843 −0.154533
\(336\) 0 0
\(337\) −6.00000 −0.326841 −0.163420 0.986557i \(-0.552253\pi\)
−0.163420 + 0.986557i \(0.552253\pi\)
\(338\) 0 0
\(339\) 0.343146 0.0186371
\(340\) 0 0
\(341\) −19.3137 −1.04590
\(342\) 0 0
\(343\) −16.9706 −0.916324
\(344\) 0 0
\(345\) 4.00000 0.215353
\(346\) 0 0
\(347\) −34.6274 −1.85890 −0.929449 0.368952i \(-0.879717\pi\)
−0.929449 + 0.368952i \(0.879717\pi\)
\(348\) 0 0
\(349\) 30.9706 1.65782 0.828908 0.559385i \(-0.188963\pi\)
0.828908 + 0.559385i \(0.188963\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) 0 0
\(353\) −5.31371 −0.282820 −0.141410 0.989951i \(-0.545164\pi\)
−0.141410 + 0.989951i \(0.545164\pi\)
\(354\) 0 0
\(355\) −5.17157 −0.274479
\(356\) 0 0
\(357\) 10.3431 0.547417
\(358\) 0 0
\(359\) −0.485281 −0.0256122 −0.0128061 0.999918i \(-0.504076\pi\)
−0.0128061 + 0.999918i \(0.504076\pi\)
\(360\) 0 0
\(361\) −17.6274 −0.927759
\(362\) 0 0
\(363\) 3.00000 0.157459
\(364\) 0 0
\(365\) −15.6569 −0.819517
\(366\) 0 0
\(367\) −31.3137 −1.63456 −0.817281 0.576239i \(-0.804520\pi\)
−0.817281 + 0.576239i \(0.804520\pi\)
\(368\) 0 0
\(369\) −2.00000 −0.104116
\(370\) 0 0
\(371\) −5.65685 −0.293689
\(372\) 0 0
\(373\) −16.6274 −0.860935 −0.430468 0.902606i \(-0.641651\pi\)
−0.430468 + 0.902606i \(0.641651\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) −7.65685 −0.394348
\(378\) 0 0
\(379\) 10.1421 0.520967 0.260483 0.965478i \(-0.416118\pi\)
0.260483 + 0.965478i \(0.416118\pi\)
\(380\) 0 0
\(381\) −1.65685 −0.0848832
\(382\) 0 0
\(383\) −9.17157 −0.468645 −0.234323 0.972159i \(-0.575287\pi\)
−0.234323 + 0.972159i \(0.575287\pi\)
\(384\) 0 0
\(385\) −8.00000 −0.407718
\(386\) 0 0
\(387\) −9.65685 −0.490885
\(388\) 0 0
\(389\) −12.3431 −0.625822 −0.312911 0.949782i \(-0.601304\pi\)
−0.312911 + 0.949782i \(0.601304\pi\)
\(390\) 0 0
\(391\) −14.6274 −0.739740
\(392\) 0 0
\(393\) 16.0000 0.807093
\(394\) 0 0
\(395\) −2.34315 −0.117896
\(396\) 0 0
\(397\) 28.6274 1.43677 0.718384 0.695646i \(-0.244882\pi\)
0.718384 + 0.695646i \(0.244882\pi\)
\(398\) 0 0
\(399\) 3.31371 0.165893
\(400\) 0 0
\(401\) −29.3137 −1.46386 −0.731928 0.681382i \(-0.761379\pi\)
−0.731928 + 0.681382i \(0.761379\pi\)
\(402\) 0 0
\(403\) −6.82843 −0.340148
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) −5.65685 −0.280400
\(408\) 0 0
\(409\) 24.6274 1.21775 0.608874 0.793267i \(-0.291622\pi\)
0.608874 + 0.793267i \(0.291622\pi\)
\(410\) 0 0
\(411\) 18.0000 0.887875
\(412\) 0 0
\(413\) −24.0000 −1.18096
\(414\) 0 0
\(415\) 1.17157 0.0575103
\(416\) 0 0
\(417\) −16.9706 −0.831052
\(418\) 0 0
\(419\) −30.6274 −1.49625 −0.748124 0.663559i \(-0.769045\pi\)
−0.748124 + 0.663559i \(0.769045\pi\)
\(420\) 0 0
\(421\) 14.9706 0.729621 0.364810 0.931082i \(-0.381134\pi\)
0.364810 + 0.931082i \(0.381134\pi\)
\(422\) 0 0
\(423\) −6.82843 −0.332009
\(424\) 0 0
\(425\) −3.65685 −0.177383
\(426\) 0 0
\(427\) 16.9706 0.821263
\(428\) 0 0
\(429\) −2.82843 −0.136558
\(430\) 0 0
\(431\) 1.85786 0.0894902 0.0447451 0.998998i \(-0.485752\pi\)
0.0447451 + 0.998998i \(0.485752\pi\)
\(432\) 0 0
\(433\) 13.3137 0.639816 0.319908 0.947449i \(-0.396348\pi\)
0.319908 + 0.947449i \(0.396348\pi\)
\(434\) 0 0
\(435\) −7.65685 −0.367118
\(436\) 0 0
\(437\) −4.68629 −0.224176
\(438\) 0 0
\(439\) 30.6274 1.46177 0.730883 0.682502i \(-0.239108\pi\)
0.730883 + 0.682502i \(0.239108\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 26.6274 1.26511 0.632553 0.774517i \(-0.282007\pi\)
0.632553 + 0.774517i \(0.282007\pi\)
\(444\) 0 0
\(445\) −17.3137 −0.820748
\(446\) 0 0
\(447\) 9.31371 0.440523
\(448\) 0 0
\(449\) −8.62742 −0.407153 −0.203576 0.979059i \(-0.565257\pi\)
−0.203576 + 0.979059i \(0.565257\pi\)
\(450\) 0 0
\(451\) −5.65685 −0.266371
\(452\) 0 0
\(453\) −6.82843 −0.320827
\(454\) 0 0
\(455\) −2.82843 −0.132599
\(456\) 0 0
\(457\) −34.2843 −1.60375 −0.801875 0.597491i \(-0.796164\pi\)
−0.801875 + 0.597491i \(0.796164\pi\)
\(458\) 0 0
\(459\) 3.65685 0.170687
\(460\) 0 0
\(461\) 13.3137 0.620081 0.310041 0.950723i \(-0.399657\pi\)
0.310041 + 0.950723i \(0.399657\pi\)
\(462\) 0 0
\(463\) 6.14214 0.285449 0.142725 0.989762i \(-0.454414\pi\)
0.142725 + 0.989762i \(0.454414\pi\)
\(464\) 0 0
\(465\) −6.82843 −0.316661
\(466\) 0 0
\(467\) 4.97056 0.230010 0.115005 0.993365i \(-0.463312\pi\)
0.115005 + 0.993365i \(0.463312\pi\)
\(468\) 0 0
\(469\) 8.00000 0.369406
\(470\) 0 0
\(471\) −14.0000 −0.645086
\(472\) 0 0
\(473\) −27.3137 −1.25589
\(474\) 0 0
\(475\) −1.17157 −0.0537555
\(476\) 0 0
\(477\) −2.00000 −0.0915737
\(478\) 0 0
\(479\) 7.51472 0.343356 0.171678 0.985153i \(-0.445081\pi\)
0.171678 + 0.985153i \(0.445081\pi\)
\(480\) 0 0
\(481\) −2.00000 −0.0911922
\(482\) 0 0
\(483\) −11.3137 −0.514792
\(484\) 0 0
\(485\) 3.65685 0.166049
\(486\) 0 0
\(487\) −25.4558 −1.15351 −0.576757 0.816916i \(-0.695682\pi\)
−0.576757 + 0.816916i \(0.695682\pi\)
\(488\) 0 0
\(489\) 22.1421 1.00130
\(490\) 0 0
\(491\) 2.34315 0.105745 0.0528723 0.998601i \(-0.483162\pi\)
0.0528723 + 0.998601i \(0.483162\pi\)
\(492\) 0 0
\(493\) 28.0000 1.26106
\(494\) 0 0
\(495\) −2.82843 −0.127128
\(496\) 0 0
\(497\) 14.6274 0.656129
\(498\) 0 0
\(499\) −25.1716 −1.12683 −0.563417 0.826173i \(-0.690514\pi\)
−0.563417 + 0.826173i \(0.690514\pi\)
\(500\) 0 0
\(501\) −4.48528 −0.200388
\(502\) 0 0
\(503\) −9.65685 −0.430578 −0.215289 0.976550i \(-0.569069\pi\)
−0.215289 + 0.976550i \(0.569069\pi\)
\(504\) 0 0
\(505\) 7.65685 0.340726
\(506\) 0 0
\(507\) −1.00000 −0.0444116
\(508\) 0 0
\(509\) 6.68629 0.296365 0.148182 0.988960i \(-0.452658\pi\)
0.148182 + 0.988960i \(0.452658\pi\)
\(510\) 0 0
\(511\) 44.2843 1.95902
\(512\) 0 0
\(513\) 1.17157 0.0517262
\(514\) 0 0
\(515\) 9.65685 0.425532
\(516\) 0 0
\(517\) −19.3137 −0.849416
\(518\) 0 0
\(519\) −20.6274 −0.905443
\(520\) 0 0
\(521\) −44.6274 −1.95516 −0.977581 0.210558i \(-0.932472\pi\)
−0.977581 + 0.210558i \(0.932472\pi\)
\(522\) 0 0
\(523\) 15.3137 0.669622 0.334811 0.942285i \(-0.391328\pi\)
0.334811 + 0.942285i \(0.391328\pi\)
\(524\) 0 0
\(525\) −2.82843 −0.123443
\(526\) 0 0
\(527\) 24.9706 1.08773
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) −8.48528 −0.368230
\(532\) 0 0
\(533\) −2.00000 −0.0866296
\(534\) 0 0
\(535\) 15.3137 0.662069
\(536\) 0 0
\(537\) 13.6569 0.589337
\(538\) 0 0
\(539\) 2.82843 0.121829
\(540\) 0 0
\(541\) −10.9706 −0.471661 −0.235831 0.971794i \(-0.575781\pi\)
−0.235831 + 0.971794i \(0.575781\pi\)
\(542\) 0 0
\(543\) 5.31371 0.228033
\(544\) 0 0
\(545\) 4.34315 0.186040
\(546\) 0 0
\(547\) −20.0000 −0.855138 −0.427569 0.903983i \(-0.640630\pi\)
−0.427569 + 0.903983i \(0.640630\pi\)
\(548\) 0 0
\(549\) 6.00000 0.256074
\(550\) 0 0
\(551\) 8.97056 0.382159
\(552\) 0 0
\(553\) 6.62742 0.281826
\(554\) 0 0
\(555\) −2.00000 −0.0848953
\(556\) 0 0
\(557\) 27.9411 1.18390 0.591952 0.805973i \(-0.298358\pi\)
0.591952 + 0.805973i \(0.298358\pi\)
\(558\) 0 0
\(559\) −9.65685 −0.408441
\(560\) 0 0
\(561\) 10.3431 0.436688
\(562\) 0 0
\(563\) 36.9706 1.55812 0.779062 0.626947i \(-0.215696\pi\)
0.779062 + 0.626947i \(0.215696\pi\)
\(564\) 0 0
\(565\) 0.343146 0.0144363
\(566\) 0 0
\(567\) 2.82843 0.118783
\(568\) 0 0
\(569\) −2.68629 −0.112615 −0.0563076 0.998413i \(-0.517933\pi\)
−0.0563076 + 0.998413i \(0.517933\pi\)
\(570\) 0 0
\(571\) −14.6274 −0.612138 −0.306069 0.952009i \(-0.599014\pi\)
−0.306069 + 0.952009i \(0.599014\pi\)
\(572\) 0 0
\(573\) 5.65685 0.236318
\(574\) 0 0
\(575\) 4.00000 0.166812
\(576\) 0 0
\(577\) −5.02944 −0.209378 −0.104689 0.994505i \(-0.533385\pi\)
−0.104689 + 0.994505i \(0.533385\pi\)
\(578\) 0 0
\(579\) −15.6569 −0.650677
\(580\) 0 0
\(581\) −3.31371 −0.137476
\(582\) 0 0
\(583\) −5.65685 −0.234283
\(584\) 0 0
\(585\) −1.00000 −0.0413449
\(586\) 0 0
\(587\) 2.14214 0.0884154 0.0442077 0.999022i \(-0.485924\pi\)
0.0442077 + 0.999022i \(0.485924\pi\)
\(588\) 0 0
\(589\) 8.00000 0.329634
\(590\) 0 0
\(591\) 22.0000 0.904959
\(592\) 0 0
\(593\) −29.3137 −1.20377 −0.601885 0.798583i \(-0.705583\pi\)
−0.601885 + 0.798583i \(0.705583\pi\)
\(594\) 0 0
\(595\) 10.3431 0.424028
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −26.3431 −1.07635 −0.538176 0.842833i \(-0.680886\pi\)
−0.538176 + 0.842833i \(0.680886\pi\)
\(600\) 0 0
\(601\) −14.0000 −0.571072 −0.285536 0.958368i \(-0.592172\pi\)
−0.285536 + 0.958368i \(0.592172\pi\)
\(602\) 0 0
\(603\) 2.82843 0.115182
\(604\) 0 0
\(605\) 3.00000 0.121967
\(606\) 0 0
\(607\) −10.6274 −0.431354 −0.215677 0.976465i \(-0.569196\pi\)
−0.215677 + 0.976465i \(0.569196\pi\)
\(608\) 0 0
\(609\) 21.6569 0.877580
\(610\) 0 0
\(611\) −6.82843 −0.276249
\(612\) 0 0
\(613\) 41.3137 1.66864 0.834322 0.551277i \(-0.185859\pi\)
0.834322 + 0.551277i \(0.185859\pi\)
\(614\) 0 0
\(615\) −2.00000 −0.0806478
\(616\) 0 0
\(617\) 30.0000 1.20775 0.603877 0.797077i \(-0.293622\pi\)
0.603877 + 0.797077i \(0.293622\pi\)
\(618\) 0 0
\(619\) 15.7990 0.635015 0.317508 0.948256i \(-0.397154\pi\)
0.317508 + 0.948256i \(0.397154\pi\)
\(620\) 0 0
\(621\) −4.00000 −0.160514
\(622\) 0 0
\(623\) 48.9706 1.96196
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 3.31371 0.132337
\(628\) 0 0
\(629\) 7.31371 0.291617
\(630\) 0 0
\(631\) 34.1421 1.35918 0.679588 0.733594i \(-0.262158\pi\)
0.679588 + 0.733594i \(0.262158\pi\)
\(632\) 0 0
\(633\) −5.65685 −0.224840
\(634\) 0 0
\(635\) −1.65685 −0.0657503
\(636\) 0 0
\(637\) 1.00000 0.0396214
\(638\) 0 0
\(639\) 5.17157 0.204584
\(640\) 0 0
\(641\) −33.3137 −1.31581 −0.657906 0.753100i \(-0.728558\pi\)
−0.657906 + 0.753100i \(0.728558\pi\)
\(642\) 0 0
\(643\) −16.4853 −0.650116 −0.325058 0.945694i \(-0.605384\pi\)
−0.325058 + 0.945694i \(0.605384\pi\)
\(644\) 0 0
\(645\) −9.65685 −0.380238
\(646\) 0 0
\(647\) 4.00000 0.157256 0.0786281 0.996904i \(-0.474946\pi\)
0.0786281 + 0.996904i \(0.474946\pi\)
\(648\) 0 0
\(649\) −24.0000 −0.942082
\(650\) 0 0
\(651\) 19.3137 0.756964
\(652\) 0 0
\(653\) −21.3137 −0.834070 −0.417035 0.908890i \(-0.636931\pi\)
−0.417035 + 0.908890i \(0.636931\pi\)
\(654\) 0 0
\(655\) 16.0000 0.625172
\(656\) 0 0
\(657\) 15.6569 0.610832
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) 22.9706 0.893451 0.446726 0.894671i \(-0.352590\pi\)
0.446726 + 0.894671i \(0.352590\pi\)
\(662\) 0 0
\(663\) 3.65685 0.142020
\(664\) 0 0
\(665\) 3.31371 0.128500
\(666\) 0 0
\(667\) −30.6274 −1.18590
\(668\) 0 0
\(669\) −5.17157 −0.199945
\(670\) 0 0
\(671\) 16.9706 0.655141
\(672\) 0 0
\(673\) −17.3137 −0.667394 −0.333697 0.942680i \(-0.608296\pi\)
−0.333697 + 0.942680i \(0.608296\pi\)
\(674\) 0 0
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) 31.9411 1.22760 0.613799 0.789463i \(-0.289641\pi\)
0.613799 + 0.789463i \(0.289641\pi\)
\(678\) 0 0
\(679\) −10.3431 −0.396934
\(680\) 0 0
\(681\) 17.1716 0.658016
\(682\) 0 0
\(683\) −14.8284 −0.567394 −0.283697 0.958914i \(-0.591561\pi\)
−0.283697 + 0.958914i \(0.591561\pi\)
\(684\) 0 0
\(685\) 18.0000 0.687745
\(686\) 0 0
\(687\) −8.34315 −0.318311
\(688\) 0 0
\(689\) −2.00000 −0.0761939
\(690\) 0 0
\(691\) 8.20101 0.311981 0.155991 0.987759i \(-0.450143\pi\)
0.155991 + 0.987759i \(0.450143\pi\)
\(692\) 0 0
\(693\) 8.00000 0.303895
\(694\) 0 0
\(695\) −16.9706 −0.643730
\(696\) 0 0
\(697\) 7.31371 0.277026
\(698\) 0 0
\(699\) −10.9706 −0.414945
\(700\) 0 0
\(701\) −4.34315 −0.164038 −0.0820192 0.996631i \(-0.526137\pi\)
−0.0820192 + 0.996631i \(0.526137\pi\)
\(702\) 0 0
\(703\) 2.34315 0.0883734
\(704\) 0 0
\(705\) −6.82843 −0.257173
\(706\) 0 0
\(707\) −21.6569 −0.814490
\(708\) 0 0
\(709\) −23.6569 −0.888452 −0.444226 0.895915i \(-0.646521\pi\)
−0.444226 + 0.895915i \(0.646521\pi\)
\(710\) 0 0
\(711\) 2.34315 0.0878748
\(712\) 0 0
\(713\) −27.3137 −1.02291
\(714\) 0 0
\(715\) −2.82843 −0.105777
\(716\) 0 0
\(717\) 7.51472 0.280642
\(718\) 0 0
\(719\) 19.3137 0.720280 0.360140 0.932898i \(-0.382729\pi\)
0.360140 + 0.932898i \(0.382729\pi\)
\(720\) 0 0
\(721\) −27.3137 −1.01722
\(722\) 0 0
\(723\) −21.3137 −0.792665
\(724\) 0 0
\(725\) −7.65685 −0.284368
\(726\) 0 0
\(727\) 28.9706 1.07446 0.537229 0.843436i \(-0.319471\pi\)
0.537229 + 0.843436i \(0.319471\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 35.3137 1.30612
\(732\) 0 0
\(733\) 18.6863 0.690194 0.345097 0.938567i \(-0.387846\pi\)
0.345097 + 0.938567i \(0.387846\pi\)
\(734\) 0 0
\(735\) 1.00000 0.0368856
\(736\) 0 0
\(737\) 8.00000 0.294684
\(738\) 0 0
\(739\) −40.7696 −1.49973 −0.749866 0.661590i \(-0.769882\pi\)
−0.749866 + 0.661590i \(0.769882\pi\)
\(740\) 0 0
\(741\) 1.17157 0.0430388
\(742\) 0 0
\(743\) −23.7990 −0.873100 −0.436550 0.899680i \(-0.643800\pi\)
−0.436550 + 0.899680i \(0.643800\pi\)
\(744\) 0 0
\(745\) 9.31371 0.341228
\(746\) 0 0
\(747\) −1.17157 −0.0428656
\(748\) 0 0
\(749\) −43.3137 −1.58265
\(750\) 0 0
\(751\) 32.0000 1.16770 0.583848 0.811863i \(-0.301546\pi\)
0.583848 + 0.811863i \(0.301546\pi\)
\(752\) 0 0
\(753\) 8.97056 0.326905
\(754\) 0 0
\(755\) −6.82843 −0.248512
\(756\) 0 0
\(757\) 9.31371 0.338512 0.169256 0.985572i \(-0.445863\pi\)
0.169256 + 0.985572i \(0.445863\pi\)
\(758\) 0 0
\(759\) −11.3137 −0.410662
\(760\) 0 0
\(761\) 41.3137 1.49762 0.748810 0.662784i \(-0.230625\pi\)
0.748810 + 0.662784i \(0.230625\pi\)
\(762\) 0 0
\(763\) −12.2843 −0.444720
\(764\) 0 0
\(765\) 3.65685 0.132214
\(766\) 0 0
\(767\) −8.48528 −0.306386
\(768\) 0 0
\(769\) 2.00000 0.0721218 0.0360609 0.999350i \(-0.488519\pi\)
0.0360609 + 0.999350i \(0.488519\pi\)
\(770\) 0 0
\(771\) 14.9706 0.539152
\(772\) 0 0
\(773\) −9.31371 −0.334991 −0.167495 0.985873i \(-0.553568\pi\)
−0.167495 + 0.985873i \(0.553568\pi\)
\(774\) 0 0
\(775\) −6.82843 −0.245284
\(776\) 0 0
\(777\) 5.65685 0.202939
\(778\) 0 0
\(779\) 2.34315 0.0839519
\(780\) 0 0
\(781\) 14.6274 0.523410
\(782\) 0 0
\(783\) 7.65685 0.273634
\(784\) 0 0
\(785\) −14.0000 −0.499681
\(786\) 0 0
\(787\) −4.20101 −0.149750 −0.0748749 0.997193i \(-0.523856\pi\)
−0.0748749 + 0.997193i \(0.523856\pi\)
\(788\) 0 0
\(789\) −1.65685 −0.0589856
\(790\) 0 0
\(791\) −0.970563 −0.0345092
\(792\) 0 0
\(793\) 6.00000 0.213066
\(794\) 0 0
\(795\) −2.00000 −0.0709327
\(796\) 0 0
\(797\) −34.0000 −1.20434 −0.602171 0.798367i \(-0.705697\pi\)
−0.602171 + 0.798367i \(0.705697\pi\)
\(798\) 0 0
\(799\) 24.9706 0.883395
\(800\) 0 0
\(801\) 17.3137 0.611750
\(802\) 0 0
\(803\) 44.2843 1.56276
\(804\) 0 0
\(805\) −11.3137 −0.398756
\(806\) 0 0
\(807\) −3.65685 −0.128727
\(808\) 0 0
\(809\) −10.6863 −0.375710 −0.187855 0.982197i \(-0.560154\pi\)
−0.187855 + 0.982197i \(0.560154\pi\)
\(810\) 0 0
\(811\) −38.4264 −1.34933 −0.674667 0.738122i \(-0.735713\pi\)
−0.674667 + 0.738122i \(0.735713\pi\)
\(812\) 0 0
\(813\) 12.4853 0.437878
\(814\) 0 0
\(815\) 22.1421 0.775605
\(816\) 0 0
\(817\) 11.3137 0.395817
\(818\) 0 0
\(819\) 2.82843 0.0988332
\(820\) 0 0
\(821\) −9.31371 −0.325051 −0.162525 0.986704i \(-0.551964\pi\)
−0.162525 + 0.986704i \(0.551964\pi\)
\(822\) 0 0
\(823\) 22.3431 0.778833 0.389417 0.921062i \(-0.372677\pi\)
0.389417 + 0.921062i \(0.372677\pi\)
\(824\) 0 0
\(825\) −2.82843 −0.0984732
\(826\) 0 0
\(827\) 4.48528 0.155969 0.0779843 0.996955i \(-0.475152\pi\)
0.0779843 + 0.996955i \(0.475152\pi\)
\(828\) 0 0
\(829\) 12.6274 0.438568 0.219284 0.975661i \(-0.429628\pi\)
0.219284 + 0.975661i \(0.429628\pi\)
\(830\) 0 0
\(831\) 6.68629 0.231945
\(832\) 0 0
\(833\) −3.65685 −0.126702
\(834\) 0 0
\(835\) −4.48528 −0.155220
\(836\) 0 0
\(837\) 6.82843 0.236025
\(838\) 0 0
\(839\) −17.8579 −0.616522 −0.308261 0.951302i \(-0.599747\pi\)
−0.308261 + 0.951302i \(0.599747\pi\)
\(840\) 0 0
\(841\) 29.6274 1.02164
\(842\) 0 0
\(843\) −20.6274 −0.710446
\(844\) 0 0
\(845\) −1.00000 −0.0344010
\(846\) 0 0
\(847\) −8.48528 −0.291558
\(848\) 0 0
\(849\) 6.34315 0.217696
\(850\) 0 0
\(851\) −8.00000 −0.274236
\(852\) 0 0
\(853\) −47.2548 −1.61797 −0.808987 0.587826i \(-0.799984\pi\)
−0.808987 + 0.587826i \(0.799984\pi\)
\(854\) 0 0
\(855\) 1.17157 0.0400669
\(856\) 0 0
\(857\) −0.343146 −0.0117216 −0.00586082 0.999983i \(-0.501866\pi\)
−0.00586082 + 0.999983i \(0.501866\pi\)
\(858\) 0 0
\(859\) −18.3431 −0.625860 −0.312930 0.949776i \(-0.601311\pi\)
−0.312930 + 0.949776i \(0.601311\pi\)
\(860\) 0 0
\(861\) 5.65685 0.192785
\(862\) 0 0
\(863\) −27.1127 −0.922927 −0.461463 0.887159i \(-0.652675\pi\)
−0.461463 + 0.887159i \(0.652675\pi\)
\(864\) 0 0
\(865\) −20.6274 −0.701353
\(866\) 0 0
\(867\) 3.62742 0.123194
\(868\) 0 0
\(869\) 6.62742 0.224820
\(870\) 0 0
\(871\) 2.82843 0.0958376
\(872\) 0 0
\(873\) −3.65685 −0.123766
\(874\) 0 0
\(875\) −2.82843 −0.0956183
\(876\) 0 0
\(877\) 38.0000 1.28317 0.641584 0.767052i \(-0.278277\pi\)
0.641584 + 0.767052i \(0.278277\pi\)
\(878\) 0 0
\(879\) −13.3137 −0.449060
\(880\) 0 0
\(881\) 43.9411 1.48041 0.740207 0.672379i \(-0.234727\pi\)
0.740207 + 0.672379i \(0.234727\pi\)
\(882\) 0 0
\(883\) −24.2843 −0.817231 −0.408615 0.912707i \(-0.633988\pi\)
−0.408615 + 0.912707i \(0.633988\pi\)
\(884\) 0 0
\(885\) −8.48528 −0.285230
\(886\) 0 0
\(887\) 28.9706 0.972736 0.486368 0.873754i \(-0.338321\pi\)
0.486368 + 0.873754i \(0.338321\pi\)
\(888\) 0 0
\(889\) 4.68629 0.157173
\(890\) 0 0
\(891\) 2.82843 0.0947559
\(892\) 0 0
\(893\) 8.00000 0.267710
\(894\) 0 0
\(895\) 13.6569 0.456498
\(896\) 0 0
\(897\) −4.00000 −0.133556
\(898\) 0 0
\(899\) 52.2843 1.74378
\(900\) 0 0
\(901\) 7.31371 0.243655
\(902\) 0 0
\(903\) 27.3137 0.908943
\(904\) 0 0
\(905\) 5.31371 0.176634
\(906\) 0 0
\(907\) −3.02944 −0.100591 −0.0502954 0.998734i \(-0.516016\pi\)
−0.0502954 + 0.998734i \(0.516016\pi\)
\(908\) 0 0
\(909\) −7.65685 −0.253962
\(910\) 0 0
\(911\) 39.5980 1.31194 0.655970 0.754787i \(-0.272260\pi\)
0.655970 + 0.754787i \(0.272260\pi\)
\(912\) 0 0
\(913\) −3.31371 −0.109668
\(914\) 0 0
\(915\) 6.00000 0.198354
\(916\) 0 0
\(917\) −45.2548 −1.49445
\(918\) 0 0
\(919\) 20.2843 0.669116 0.334558 0.942375i \(-0.391413\pi\)
0.334558 + 0.942375i \(0.391413\pi\)
\(920\) 0 0
\(921\) 26.8284 0.884027
\(922\) 0 0
\(923\) 5.17157 0.170224
\(924\) 0 0
\(925\) −2.00000 −0.0657596
\(926\) 0 0
\(927\) −9.65685 −0.317173
\(928\) 0 0
\(929\) 28.6274 0.939235 0.469618 0.882870i \(-0.344392\pi\)
0.469618 + 0.882870i \(0.344392\pi\)
\(930\) 0 0
\(931\) −1.17157 −0.0383968
\(932\) 0 0
\(933\) −27.3137 −0.894211
\(934\) 0 0
\(935\) 10.3431 0.338257
\(936\) 0 0
\(937\) −44.6274 −1.45791 −0.728957 0.684559i \(-0.759995\pi\)
−0.728957 + 0.684559i \(0.759995\pi\)
\(938\) 0 0
\(939\) 14.0000 0.456873
\(940\) 0 0
\(941\) −12.6274 −0.411642 −0.205821 0.978590i \(-0.565986\pi\)
−0.205821 + 0.978590i \(0.565986\pi\)
\(942\) 0 0
\(943\) −8.00000 −0.260516
\(944\) 0 0
\(945\) 2.82843 0.0920087
\(946\) 0 0
\(947\) −46.8284 −1.52172 −0.760860 0.648916i \(-0.775222\pi\)
−0.760860 + 0.648916i \(0.775222\pi\)
\(948\) 0 0
\(949\) 15.6569 0.508243
\(950\) 0 0
\(951\) 25.3137 0.820853
\(952\) 0 0
\(953\) −11.6569 −0.377603 −0.188801 0.982015i \(-0.560460\pi\)
−0.188801 + 0.982015i \(0.560460\pi\)
\(954\) 0 0
\(955\) 5.65685 0.183052
\(956\) 0 0
\(957\) 21.6569 0.700067
\(958\) 0 0
\(959\) −50.9117 −1.64402
\(960\) 0 0
\(961\) 15.6274 0.504110
\(962\) 0 0
\(963\) −15.3137 −0.493477
\(964\) 0 0
\(965\) −15.6569 −0.504012
\(966\) 0 0
\(967\) −7.51472 −0.241657 −0.120829 0.992673i \(-0.538555\pi\)
−0.120829 + 0.992673i \(0.538555\pi\)
\(968\) 0 0
\(969\) −4.28427 −0.137631
\(970\) 0 0
\(971\) 12.6863 0.407122 0.203561 0.979062i \(-0.434748\pi\)
0.203561 + 0.979062i \(0.434748\pi\)
\(972\) 0 0
\(973\) 48.0000 1.53881
\(974\) 0 0
\(975\) −1.00000 −0.0320256
\(976\) 0 0
\(977\) −2.00000 −0.0639857 −0.0319928 0.999488i \(-0.510185\pi\)
−0.0319928 + 0.999488i \(0.510185\pi\)
\(978\) 0 0
\(979\) 48.9706 1.56511
\(980\) 0 0
\(981\) −4.34315 −0.138666
\(982\) 0 0
\(983\) −24.7696 −0.790026 −0.395013 0.918676i \(-0.629260\pi\)
−0.395013 + 0.918676i \(0.629260\pi\)
\(984\) 0 0
\(985\) 22.0000 0.700978
\(986\) 0 0
\(987\) 19.3137 0.614762
\(988\) 0 0
\(989\) −38.6274 −1.22828
\(990\) 0 0
\(991\) 24.9706 0.793216 0.396608 0.917988i \(-0.370187\pi\)
0.396608 + 0.917988i \(0.370187\pi\)
\(992\) 0 0
\(993\) −10.1421 −0.321851
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 2.68629 0.0850757 0.0425379 0.999095i \(-0.486456\pi\)
0.0425379 + 0.999095i \(0.486456\pi\)
\(998\) 0 0
\(999\) 2.00000 0.0632772
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 6240.2.a.bh.1.2 2
4.3 odd 2 6240.2.a.br.1.1 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6240.2.a.bh.1.2 2 1.1 even 1 trivial
6240.2.a.br.1.1 yes 2 4.3 odd 2