Properties

Label 168.4.a.h.1.2
Level $168$
Weight $4$
Character 168.1
Self dual yes
Analytic conductor $9.912$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 168.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(9.91232088096\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{177}) \)
Defining polynomial: \(x^{2} - x - 44\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-6.15207\) of defining polynomial
Character \(\chi\) \(=\) 168.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.00000 q^{3} +20.3041 q^{5} +7.00000 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q+3.00000 q^{3} +20.3041 q^{5} +7.00000 q^{7} +9.00000 q^{9} -30.9124 q^{11} +50.6083 q^{13} +60.9124 q^{15} -102.737 q^{17} -61.2165 q^{19} +21.0000 q^{21} +148.129 q^{23} +287.258 q^{25} +27.0000 q^{27} +159.217 q^{29} -121.217 q^{31} -92.7372 q^{33} +142.129 q^{35} -357.908 q^{37} +151.825 q^{39} +466.737 q^{41} -185.567 q^{43} +182.737 q^{45} -131.392 q^{47} +49.0000 q^{49} -308.212 q^{51} +200.175 q^{53} -627.650 q^{55} -183.650 q^{57} -591.908 q^{59} +70.5158 q^{61} +63.0000 q^{63} +1027.56 q^{65} -643.041 q^{67} +444.387 q^{69} -522.397 q^{71} -576.175 q^{73} +861.774 q^{75} -216.387 q^{77} +280.774 q^{79} +81.0000 q^{81} -557.732 q^{83} -2085.99 q^{85} +477.650 q^{87} -1228.89 q^{89} +354.258 q^{91} -363.650 q^{93} -1242.95 q^{95} +65.0413 q^{97} -278.212 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 6q^{3} + 14q^{5} + 14q^{7} + 18q^{9} + O(q^{10}) \) \( 2q + 6q^{3} + 14q^{5} + 14q^{7} + 18q^{9} + 18q^{11} + 48q^{13} + 42q^{15} + 34q^{17} - 16q^{19} + 42q^{21} + 110q^{23} + 202q^{25} + 54q^{27} + 212q^{29} - 136q^{31} + 54q^{33} + 98q^{35} - 24q^{37} + 144q^{39} + 694q^{41} - 584q^{43} + 126q^{45} - 316q^{47} + 98q^{49} + 102q^{51} + 560q^{53} - 936q^{55} - 48q^{57} - 492q^{59} - 604q^{61} + 126q^{63} + 1044q^{65} - 1020q^{67} + 330q^{69} - 1710q^{71} - 1312q^{73} + 606q^{75} + 126q^{77} - 556q^{79} + 162q^{81} - 264q^{83} - 2948q^{85} + 636q^{87} + 70q^{89} + 336q^{91} - 408q^{93} - 1528q^{95} - 136q^{97} + 162q^{99} + O(q^{100}) \)

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
\(3\) 3.00000 0.577350
\(4\) 0 0
\(5\) 20.3041 1.81606 0.908029 0.418908i \(-0.137587\pi\)
0.908029 + 0.418908i \(0.137587\pi\)
\(6\) 0 0
\(7\) 7.00000 0.377964
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −30.9124 −0.847313 −0.423656 0.905823i \(-0.639254\pi\)
−0.423656 + 0.905823i \(0.639254\pi\)
\(12\) 0 0
\(13\) 50.6083 1.07971 0.539854 0.841759i \(-0.318479\pi\)
0.539854 + 0.841759i \(0.318479\pi\)
\(14\) 0 0
\(15\) 60.9124 1.04850
\(16\) 0 0
\(17\) −102.737 −1.46573 −0.732866 0.680373i \(-0.761818\pi\)
−0.732866 + 0.680373i \(0.761818\pi\)
\(18\) 0 0
\(19\) −61.2165 −0.739160 −0.369580 0.929199i \(-0.620498\pi\)
−0.369580 + 0.929199i \(0.620498\pi\)
\(20\) 0 0
\(21\) 21.0000 0.218218
\(22\) 0 0
\(23\) 148.129 1.34291 0.671457 0.741044i \(-0.265669\pi\)
0.671457 + 0.741044i \(0.265669\pi\)
\(24\) 0 0
\(25\) 287.258 2.29806
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) 159.217 1.01951 0.509755 0.860320i \(-0.329736\pi\)
0.509755 + 0.860320i \(0.329736\pi\)
\(30\) 0 0
\(31\) −121.217 −0.702295 −0.351147 0.936320i \(-0.614208\pi\)
−0.351147 + 0.936320i \(0.614208\pi\)
\(32\) 0 0
\(33\) −92.7372 −0.489196
\(34\) 0 0
\(35\) 142.129 0.686405
\(36\) 0 0
\(37\) −357.908 −1.59026 −0.795130 0.606439i \(-0.792598\pi\)
−0.795130 + 0.606439i \(0.792598\pi\)
\(38\) 0 0
\(39\) 151.825 0.623370
\(40\) 0 0
\(41\) 466.737 1.77786 0.888928 0.458047i \(-0.151451\pi\)
0.888928 + 0.458047i \(0.151451\pi\)
\(42\) 0 0
\(43\) −185.567 −0.658109 −0.329055 0.944311i \(-0.606730\pi\)
−0.329055 + 0.944311i \(0.606730\pi\)
\(44\) 0 0
\(45\) 182.737 0.605352
\(46\) 0 0
\(47\) −131.392 −0.407776 −0.203888 0.978994i \(-0.565358\pi\)
−0.203888 + 0.978994i \(0.565358\pi\)
\(48\) 0 0
\(49\) 49.0000 0.142857
\(50\) 0 0
\(51\) −308.212 −0.846240
\(52\) 0 0
\(53\) 200.175 0.518796 0.259398 0.965771i \(-0.416476\pi\)
0.259398 + 0.965771i \(0.416476\pi\)
\(54\) 0 0
\(55\) −627.650 −1.53877
\(56\) 0 0
\(57\) −183.650 −0.426754
\(58\) 0 0
\(59\) −591.908 −1.30610 −0.653049 0.757316i \(-0.726510\pi\)
−0.653049 + 0.757316i \(0.726510\pi\)
\(60\) 0 0
\(61\) 70.5158 0.148010 0.0740051 0.997258i \(-0.476422\pi\)
0.0740051 + 0.997258i \(0.476422\pi\)
\(62\) 0 0
\(63\) 63.0000 0.125988
\(64\) 0 0
\(65\) 1027.56 1.96081
\(66\) 0 0
\(67\) −643.041 −1.17254 −0.586269 0.810117i \(-0.699404\pi\)
−0.586269 + 0.810117i \(0.699404\pi\)
\(68\) 0 0
\(69\) 444.387 0.775332
\(70\) 0 0
\(71\) −522.397 −0.873198 −0.436599 0.899656i \(-0.643817\pi\)
−0.436599 + 0.899656i \(0.643817\pi\)
\(72\) 0 0
\(73\) −576.175 −0.923784 −0.461892 0.886936i \(-0.652829\pi\)
−0.461892 + 0.886936i \(0.652829\pi\)
\(74\) 0 0
\(75\) 861.774 1.32679
\(76\) 0 0
\(77\) −216.387 −0.320254
\(78\) 0 0
\(79\) 280.774 0.399867 0.199934 0.979809i \(-0.435927\pi\)
0.199934 + 0.979809i \(0.435927\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) −557.732 −0.737579 −0.368790 0.929513i \(-0.620228\pi\)
−0.368790 + 0.929513i \(0.620228\pi\)
\(84\) 0 0
\(85\) −2085.99 −2.66185
\(86\) 0 0
\(87\) 477.650 0.588614
\(88\) 0 0
\(89\) −1228.89 −1.46362 −0.731811 0.681508i \(-0.761325\pi\)
−0.731811 + 0.681508i \(0.761325\pi\)
\(90\) 0 0
\(91\) 354.258 0.408091
\(92\) 0 0
\(93\) −363.650 −0.405470
\(94\) 0 0
\(95\) −1242.95 −1.34236
\(96\) 0 0
\(97\) 65.0413 0.0680819 0.0340410 0.999420i \(-0.489162\pi\)
0.0340410 + 0.999420i \(0.489162\pi\)
\(98\) 0 0
\(99\) −278.212 −0.282438
\(100\) 0 0
\(101\) 303.511 0.299014 0.149507 0.988761i \(-0.452231\pi\)
0.149507 + 0.988761i \(0.452231\pi\)
\(102\) 0 0
\(103\) 174.268 0.166710 0.0833549 0.996520i \(-0.473436\pi\)
0.0833549 + 0.996520i \(0.473436\pi\)
\(104\) 0 0
\(105\) 426.387 0.396296
\(106\) 0 0
\(107\) 1945.42 1.75767 0.878835 0.477126i \(-0.158321\pi\)
0.878835 + 0.477126i \(0.158321\pi\)
\(108\) 0 0
\(109\) −1102.87 −0.969132 −0.484566 0.874755i \(-0.661023\pi\)
−0.484566 + 0.874755i \(0.661023\pi\)
\(110\) 0 0
\(111\) −1073.72 −0.918137
\(112\) 0 0
\(113\) 80.0827 0.0666685 0.0333343 0.999444i \(-0.489387\pi\)
0.0333343 + 0.999444i \(0.489387\pi\)
\(114\) 0 0
\(115\) 3007.63 2.43881
\(116\) 0 0
\(117\) 455.474 0.359903
\(118\) 0 0
\(119\) −719.160 −0.553994
\(120\) 0 0
\(121\) −375.423 −0.282061
\(122\) 0 0
\(123\) 1400.21 1.02645
\(124\) 0 0
\(125\) 3294.51 2.35736
\(126\) 0 0
\(127\) −1981.12 −1.38422 −0.692112 0.721791i \(-0.743319\pi\)
−0.692112 + 0.721791i \(0.743319\pi\)
\(128\) 0 0
\(129\) −556.701 −0.379959
\(130\) 0 0
\(131\) 2854.25 1.90364 0.951820 0.306658i \(-0.0992107\pi\)
0.951820 + 0.306658i \(0.0992107\pi\)
\(132\) 0 0
\(133\) −428.516 −0.279376
\(134\) 0 0
\(135\) 548.212 0.349500
\(136\) 0 0
\(137\) −535.474 −0.333932 −0.166966 0.985963i \(-0.553397\pi\)
−0.166966 + 0.985963i \(0.553397\pi\)
\(138\) 0 0
\(139\) 2326.43 1.41961 0.709804 0.704399i \(-0.248784\pi\)
0.709804 + 0.704399i \(0.248784\pi\)
\(140\) 0 0
\(141\) −394.175 −0.235429
\(142\) 0 0
\(143\) −1564.42 −0.914851
\(144\) 0 0
\(145\) 3232.75 1.85149
\(146\) 0 0
\(147\) 147.000 0.0824786
\(148\) 0 0
\(149\) 2061.04 1.13320 0.566601 0.823992i \(-0.308258\pi\)
0.566601 + 0.823992i \(0.308258\pi\)
\(150\) 0 0
\(151\) −1113.03 −0.599849 −0.299925 0.953963i \(-0.596962\pi\)
−0.299925 + 0.953963i \(0.596962\pi\)
\(152\) 0 0
\(153\) −924.635 −0.488577
\(154\) 0 0
\(155\) −2461.20 −1.27541
\(156\) 0 0
\(157\) −1537.30 −0.781464 −0.390732 0.920505i \(-0.627778\pi\)
−0.390732 + 0.920505i \(0.627778\pi\)
\(158\) 0 0
\(159\) 600.526 0.299527
\(160\) 0 0
\(161\) 1036.90 0.507574
\(162\) 0 0
\(163\) −2043.52 −0.981967 −0.490984 0.871169i \(-0.663363\pi\)
−0.490984 + 0.871169i \(0.663363\pi\)
\(164\) 0 0
\(165\) −1882.95 −0.888408
\(166\) 0 0
\(167\) 2513.44 1.16464 0.582322 0.812958i \(-0.302144\pi\)
0.582322 + 0.812958i \(0.302144\pi\)
\(168\) 0 0
\(169\) 364.197 0.165770
\(170\) 0 0
\(171\) −550.949 −0.246387
\(172\) 0 0
\(173\) 1591.00 0.699197 0.349599 0.936900i \(-0.386318\pi\)
0.349599 + 0.936900i \(0.386318\pi\)
\(174\) 0 0
\(175\) 2010.81 0.868586
\(176\) 0 0
\(177\) −1775.72 −0.754076
\(178\) 0 0
\(179\) 2552.27 1.06573 0.532866 0.846200i \(-0.321115\pi\)
0.532866 + 0.846200i \(0.321115\pi\)
\(180\) 0 0
\(181\) −3361.33 −1.38036 −0.690182 0.723636i \(-0.742470\pi\)
−0.690182 + 0.723636i \(0.742470\pi\)
\(182\) 0 0
\(183\) 211.547 0.0854537
\(184\) 0 0
\(185\) −7267.00 −2.88800
\(186\) 0 0
\(187\) 3175.85 1.24193
\(188\) 0 0
\(189\) 189.000 0.0727393
\(190\) 0 0
\(191\) −3868.64 −1.46558 −0.732789 0.680456i \(-0.761782\pi\)
−0.732789 + 0.680456i \(0.761782\pi\)
\(192\) 0 0
\(193\) 2693.17 1.00445 0.502224 0.864738i \(-0.332515\pi\)
0.502224 + 0.864738i \(0.332515\pi\)
\(194\) 0 0
\(195\) 3082.67 1.13208
\(196\) 0 0
\(197\) −1064.86 −0.385116 −0.192558 0.981286i \(-0.561678\pi\)
−0.192558 + 0.981286i \(0.561678\pi\)
\(198\) 0 0
\(199\) 1625.92 0.579187 0.289594 0.957150i \(-0.406480\pi\)
0.289594 + 0.957150i \(0.406480\pi\)
\(200\) 0 0
\(201\) −1929.12 −0.676965
\(202\) 0 0
\(203\) 1114.52 0.385338
\(204\) 0 0
\(205\) 9476.70 3.22869
\(206\) 0 0
\(207\) 1333.16 0.447638
\(208\) 0 0
\(209\) 1892.35 0.626300
\(210\) 0 0
\(211\) 1138.04 0.371309 0.185654 0.982615i \(-0.440560\pi\)
0.185654 + 0.982615i \(0.440560\pi\)
\(212\) 0 0
\(213\) −1567.19 −0.504141
\(214\) 0 0
\(215\) −3767.78 −1.19516
\(216\) 0 0
\(217\) −848.516 −0.265442
\(218\) 0 0
\(219\) −1728.53 −0.533347
\(220\) 0 0
\(221\) −5199.35 −1.58256
\(222\) 0 0
\(223\) 3535.78 1.06176 0.530881 0.847446i \(-0.321861\pi\)
0.530881 + 0.847446i \(0.321861\pi\)
\(224\) 0 0
\(225\) 2585.32 0.766021
\(226\) 0 0
\(227\) −3055.21 −0.893309 −0.446655 0.894706i \(-0.647385\pi\)
−0.446655 + 0.894706i \(0.647385\pi\)
\(228\) 0 0
\(229\) −321.703 −0.0928329 −0.0464164 0.998922i \(-0.514780\pi\)
−0.0464164 + 0.998922i \(0.514780\pi\)
\(230\) 0 0
\(231\) −649.160 −0.184899
\(232\) 0 0
\(233\) −906.939 −0.255002 −0.127501 0.991838i \(-0.540696\pi\)
−0.127501 + 0.991838i \(0.540696\pi\)
\(234\) 0 0
\(235\) −2667.80 −0.740544
\(236\) 0 0
\(237\) 842.321 0.230863
\(238\) 0 0
\(239\) −4290.84 −1.16130 −0.580651 0.814152i \(-0.697202\pi\)
−0.580651 + 0.814152i \(0.697202\pi\)
\(240\) 0 0
\(241\) −6985.64 −1.86716 −0.933579 0.358373i \(-0.883332\pi\)
−0.933579 + 0.358373i \(0.883332\pi\)
\(242\) 0 0
\(243\) 243.000 0.0641500
\(244\) 0 0
\(245\) 994.903 0.259437
\(246\) 0 0
\(247\) −3098.06 −0.798077
\(248\) 0 0
\(249\) −1673.20 −0.425842
\(250\) 0 0
\(251\) −5666.63 −1.42500 −0.712499 0.701673i \(-0.752437\pi\)
−0.712499 + 0.701673i \(0.752437\pi\)
\(252\) 0 0
\(253\) −4579.02 −1.13787
\(254\) 0 0
\(255\) −6257.97 −1.53682
\(256\) 0 0
\(257\) 840.596 0.204027 0.102013 0.994783i \(-0.467472\pi\)
0.102013 + 0.994783i \(0.467472\pi\)
\(258\) 0 0
\(259\) −2505.35 −0.601062
\(260\) 0 0
\(261\) 1432.95 0.339836
\(262\) 0 0
\(263\) 3386.69 0.794039 0.397019 0.917810i \(-0.370045\pi\)
0.397019 + 0.917810i \(0.370045\pi\)
\(264\) 0 0
\(265\) 4064.38 0.942163
\(266\) 0 0
\(267\) −3686.68 −0.845023
\(268\) 0 0
\(269\) 7929.42 1.79727 0.898635 0.438698i \(-0.144560\pi\)
0.898635 + 0.438698i \(0.144560\pi\)
\(270\) 0 0
\(271\) 5080.25 1.13876 0.569379 0.822075i \(-0.307184\pi\)
0.569379 + 0.822075i \(0.307184\pi\)
\(272\) 0 0
\(273\) 1062.77 0.235612
\(274\) 0 0
\(275\) −8879.83 −1.94718
\(276\) 0 0
\(277\) −195.329 −0.0423688 −0.0211844 0.999776i \(-0.506744\pi\)
−0.0211844 + 0.999776i \(0.506744\pi\)
\(278\) 0 0
\(279\) −1090.95 −0.234098
\(280\) 0 0
\(281\) 4029.83 0.855514 0.427757 0.903894i \(-0.359304\pi\)
0.427757 + 0.903894i \(0.359304\pi\)
\(282\) 0 0
\(283\) 6927.28 1.45507 0.727534 0.686072i \(-0.240667\pi\)
0.727534 + 0.686072i \(0.240667\pi\)
\(284\) 0 0
\(285\) −3728.85 −0.775010
\(286\) 0 0
\(287\) 3267.16 0.671967
\(288\) 0 0
\(289\) 5641.93 1.14837
\(290\) 0 0
\(291\) 195.124 0.0393071
\(292\) 0 0
\(293\) −911.009 −0.181644 −0.0908221 0.995867i \(-0.528949\pi\)
−0.0908221 + 0.995867i \(0.528949\pi\)
\(294\) 0 0
\(295\) −12018.2 −2.37195
\(296\) 0 0
\(297\) −834.635 −0.163065
\(298\) 0 0
\(299\) 7496.55 1.44996
\(300\) 0 0
\(301\) −1298.97 −0.248742
\(302\) 0 0
\(303\) 910.533 0.172636
\(304\) 0 0
\(305\) 1431.76 0.268795
\(306\) 0 0
\(307\) 3310.72 0.615482 0.307741 0.951470i \(-0.400427\pi\)
0.307741 + 0.951470i \(0.400427\pi\)
\(308\) 0 0
\(309\) 522.803 0.0962499
\(310\) 0 0
\(311\) 4979.13 0.907847 0.453924 0.891041i \(-0.350024\pi\)
0.453924 + 0.891041i \(0.350024\pi\)
\(312\) 0 0
\(313\) −5116.19 −0.923911 −0.461955 0.886903i \(-0.652852\pi\)
−0.461955 + 0.886903i \(0.652852\pi\)
\(314\) 0 0
\(315\) 1279.16 0.228802
\(316\) 0 0
\(317\) 978.317 0.173337 0.0866684 0.996237i \(-0.472378\pi\)
0.0866684 + 0.996237i \(0.472378\pi\)
\(318\) 0 0
\(319\) −4921.77 −0.863843
\(320\) 0 0
\(321\) 5836.26 1.01479
\(322\) 0 0
\(323\) 6289.22 1.08341
\(324\) 0 0
\(325\) 14537.6 2.48124
\(326\) 0 0
\(327\) −3308.60 −0.559529
\(328\) 0 0
\(329\) −919.742 −0.154125
\(330\) 0 0
\(331\) 9092.36 1.50985 0.754926 0.655810i \(-0.227673\pi\)
0.754926 + 0.655810i \(0.227673\pi\)
\(332\) 0 0
\(333\) −3221.17 −0.530087
\(334\) 0 0
\(335\) −13056.4 −2.12939
\(336\) 0 0
\(337\) −621.406 −0.100445 −0.0502227 0.998738i \(-0.515993\pi\)
−0.0502227 + 0.998738i \(0.515993\pi\)
\(338\) 0 0
\(339\) 240.248 0.0384911
\(340\) 0 0
\(341\) 3747.09 0.595063
\(342\) 0 0
\(343\) 343.000 0.0539949
\(344\) 0 0
\(345\) 9022.89 1.40805
\(346\) 0 0
\(347\) 4276.85 0.661652 0.330826 0.943692i \(-0.392673\pi\)
0.330826 + 0.943692i \(0.392673\pi\)
\(348\) 0 0
\(349\) −7497.41 −1.14993 −0.574967 0.818177i \(-0.694985\pi\)
−0.574967 + 0.818177i \(0.694985\pi\)
\(350\) 0 0
\(351\) 1366.42 0.207790
\(352\) 0 0
\(353\) −2295.77 −0.346151 −0.173076 0.984909i \(-0.555371\pi\)
−0.173076 + 0.984909i \(0.555371\pi\)
\(354\) 0 0
\(355\) −10606.8 −1.58578
\(356\) 0 0
\(357\) −2157.48 −0.319849
\(358\) 0 0
\(359\) 681.808 0.100235 0.0501176 0.998743i \(-0.484040\pi\)
0.0501176 + 0.998743i \(0.484040\pi\)
\(360\) 0 0
\(361\) −3111.54 −0.453643
\(362\) 0 0
\(363\) −1126.27 −0.162848
\(364\) 0 0
\(365\) −11698.7 −1.67764
\(366\) 0 0
\(367\) 8497.77 1.20867 0.604333 0.796732i \(-0.293440\pi\)
0.604333 + 0.796732i \(0.293440\pi\)
\(368\) 0 0
\(369\) 4200.63 0.592619
\(370\) 0 0
\(371\) 1401.23 0.196086
\(372\) 0 0
\(373\) 12306.1 1.70828 0.854138 0.520046i \(-0.174085\pi\)
0.854138 + 0.520046i \(0.174085\pi\)
\(374\) 0 0
\(375\) 9883.52 1.36102
\(376\) 0 0
\(377\) 8057.67 1.10077
\(378\) 0 0
\(379\) 7064.19 0.957423 0.478711 0.877972i \(-0.341104\pi\)
0.478711 + 0.877972i \(0.341104\pi\)
\(380\) 0 0
\(381\) −5943.37 −0.799182
\(382\) 0 0
\(383\) 3082.91 0.411304 0.205652 0.978625i \(-0.434069\pi\)
0.205652 + 0.978625i \(0.434069\pi\)
\(384\) 0 0
\(385\) −4393.55 −0.581600
\(386\) 0 0
\(387\) −1670.10 −0.219370
\(388\) 0 0
\(389\) 9990.11 1.30210 0.651052 0.759033i \(-0.274328\pi\)
0.651052 + 0.759033i \(0.274328\pi\)
\(390\) 0 0
\(391\) −15218.4 −1.96835
\(392\) 0 0
\(393\) 8562.74 1.09907
\(394\) 0 0
\(395\) 5700.87 0.726182
\(396\) 0 0
\(397\) 8327.91 1.05281 0.526405 0.850234i \(-0.323540\pi\)
0.526405 + 0.850234i \(0.323540\pi\)
\(398\) 0 0
\(399\) −1285.55 −0.161298
\(400\) 0 0
\(401\) −9715.82 −1.20994 −0.604969 0.796249i \(-0.706815\pi\)
−0.604969 + 0.796249i \(0.706815\pi\)
\(402\) 0 0
\(403\) −6134.56 −0.758273
\(404\) 0 0
\(405\) 1644.63 0.201784
\(406\) 0 0
\(407\) 11063.8 1.34745
\(408\) 0 0
\(409\) 13341.8 1.61299 0.806493 0.591244i \(-0.201363\pi\)
0.806493 + 0.591244i \(0.201363\pi\)
\(410\) 0 0
\(411\) −1606.42 −0.192796
\(412\) 0 0
\(413\) −4143.35 −0.493659
\(414\) 0 0
\(415\) −11324.3 −1.33949
\(416\) 0 0
\(417\) 6979.30 0.819611
\(418\) 0 0
\(419\) 4281.34 0.499181 0.249591 0.968351i \(-0.419704\pi\)
0.249591 + 0.968351i \(0.419704\pi\)
\(420\) 0 0
\(421\) −3296.60 −0.381631 −0.190815 0.981626i \(-0.561113\pi\)
−0.190815 + 0.981626i \(0.561113\pi\)
\(422\) 0 0
\(423\) −1182.53 −0.135925
\(424\) 0 0
\(425\) −29512.1 −3.36834
\(426\) 0 0
\(427\) 493.610 0.0559426
\(428\) 0 0
\(429\) −4693.27 −0.528189
\(430\) 0 0
\(431\) 4122.73 0.460754 0.230377 0.973101i \(-0.426004\pi\)
0.230377 + 0.973101i \(0.426004\pi\)
\(432\) 0 0
\(433\) 1070.60 0.118822 0.0594110 0.998234i \(-0.481078\pi\)
0.0594110 + 0.998234i \(0.481078\pi\)
\(434\) 0 0
\(435\) 9698.26 1.06896
\(436\) 0 0
\(437\) −9067.94 −0.992628
\(438\) 0 0
\(439\) 6.22847 0.000677150 0 0.000338575 1.00000i \(-0.499892\pi\)
0.000338575 1.00000i \(0.499892\pi\)
\(440\) 0 0
\(441\) 441.000 0.0476190
\(442\) 0 0
\(443\) −6835.85 −0.733140 −0.366570 0.930390i \(-0.619468\pi\)
−0.366570 + 0.930390i \(0.619468\pi\)
\(444\) 0 0
\(445\) −24951.6 −2.65802
\(446\) 0 0
\(447\) 6183.12 0.654254
\(448\) 0 0
\(449\) 5590.94 0.587646 0.293823 0.955860i \(-0.405072\pi\)
0.293823 + 0.955860i \(0.405072\pi\)
\(450\) 0 0
\(451\) −14428.0 −1.50640
\(452\) 0 0
\(453\) −3339.09 −0.346323
\(454\) 0 0
\(455\) 7192.90 0.741117
\(456\) 0 0
\(457\) −7615.65 −0.779530 −0.389765 0.920914i \(-0.627444\pi\)
−0.389765 + 0.920914i \(0.627444\pi\)
\(458\) 0 0
\(459\) −2773.90 −0.282080
\(460\) 0 0
\(461\) 12191.7 1.23172 0.615860 0.787855i \(-0.288808\pi\)
0.615860 + 0.787855i \(0.288808\pi\)
\(462\) 0 0
\(463\) 4210.28 0.422610 0.211305 0.977420i \(-0.432229\pi\)
0.211305 + 0.977420i \(0.432229\pi\)
\(464\) 0 0
\(465\) −7383.59 −0.736357
\(466\) 0 0
\(467\) 16128.2 1.59812 0.799062 0.601248i \(-0.205330\pi\)
0.799062 + 0.601248i \(0.205330\pi\)
\(468\) 0 0
\(469\) −4501.29 −0.443177
\(470\) 0 0
\(471\) −4611.90 −0.451178
\(472\) 0 0
\(473\) 5736.32 0.557624
\(474\) 0 0
\(475\) −17584.9 −1.69864
\(476\) 0 0
\(477\) 1801.58 0.172932
\(478\) 0 0
\(479\) −3736.70 −0.356438 −0.178219 0.983991i \(-0.557034\pi\)
−0.178219 + 0.983991i \(0.557034\pi\)
\(480\) 0 0
\(481\) −18113.1 −1.71702
\(482\) 0 0
\(483\) 3110.71 0.293048
\(484\) 0 0
\(485\) 1320.61 0.123641
\(486\) 0 0
\(487\) 15812.5 1.47132 0.735661 0.677350i \(-0.236872\pi\)
0.735661 + 0.677350i \(0.236872\pi\)
\(488\) 0 0
\(489\) −6130.55 −0.566939
\(490\) 0 0
\(491\) 2564.56 0.235717 0.117858 0.993030i \(-0.462397\pi\)
0.117858 + 0.993030i \(0.462397\pi\)
\(492\) 0 0
\(493\) −16357.5 −1.49433
\(494\) 0 0
\(495\) −5648.85 −0.512923
\(496\) 0 0
\(497\) −3656.78 −0.330038
\(498\) 0 0
\(499\) −8511.36 −0.763569 −0.381784 0.924251i \(-0.624690\pi\)
−0.381784 + 0.924251i \(0.624690\pi\)
\(500\) 0 0
\(501\) 7540.31 0.672407
\(502\) 0 0
\(503\) 6486.89 0.575022 0.287511 0.957777i \(-0.407172\pi\)
0.287511 + 0.957777i \(0.407172\pi\)
\(504\) 0 0
\(505\) 6162.53 0.543027
\(506\) 0 0
\(507\) 1092.59 0.0957074
\(508\) 0 0
\(509\) −13980.3 −1.21742 −0.608710 0.793393i \(-0.708313\pi\)
−0.608710 + 0.793393i \(0.708313\pi\)
\(510\) 0 0
\(511\) −4033.23 −0.349157
\(512\) 0 0
\(513\) −1652.85 −0.142251
\(514\) 0 0
\(515\) 3538.35 0.302754
\(516\) 0 0
\(517\) 4061.63 0.345513
\(518\) 0 0
\(519\) 4772.99 0.403682
\(520\) 0 0
\(521\) −10684.8 −0.898485 −0.449242 0.893410i \(-0.648306\pi\)
−0.449242 + 0.893410i \(0.648306\pi\)
\(522\) 0 0
\(523\) 2500.29 0.209044 0.104522 0.994523i \(-0.466669\pi\)
0.104522 + 0.994523i \(0.466669\pi\)
\(524\) 0 0
\(525\) 6032.42 0.501478
\(526\) 0 0
\(527\) 12453.4 1.02938
\(528\) 0 0
\(529\) 9775.18 0.803418
\(530\) 0 0
\(531\) −5327.17 −0.435366
\(532\) 0 0
\(533\) 23620.8 1.91957
\(534\) 0 0
\(535\) 39500.0 3.19203
\(536\) 0 0
\(537\) 7656.82 0.615300
\(538\) 0 0
\(539\) −1514.71 −0.121045
\(540\) 0 0
\(541\) −8780.94 −0.697823 −0.348911 0.937156i \(-0.613449\pi\)
−0.348911 + 0.937156i \(0.613449\pi\)
\(542\) 0 0
\(543\) −10084.0 −0.796954
\(544\) 0 0
\(545\) −22392.7 −1.76000
\(546\) 0 0
\(547\) −10965.6 −0.857140 −0.428570 0.903509i \(-0.640982\pi\)
−0.428570 + 0.903509i \(0.640982\pi\)
\(548\) 0 0
\(549\) 634.642 0.0493367
\(550\) 0 0
\(551\) −9746.69 −0.753580
\(552\) 0 0
\(553\) 1965.42 0.151136
\(554\) 0 0
\(555\) −21801.0 −1.66739
\(556\) 0 0
\(557\) 3156.31 0.240103 0.120051 0.992768i \(-0.461694\pi\)
0.120051 + 0.992768i \(0.461694\pi\)
\(558\) 0 0
\(559\) −9391.22 −0.710566
\(560\) 0 0
\(561\) 9527.56 0.717030
\(562\) 0 0
\(563\) −16805.3 −1.25801 −0.629003 0.777403i \(-0.716536\pi\)
−0.629003 + 0.777403i \(0.716536\pi\)
\(564\) 0 0
\(565\) 1626.01 0.121074
\(566\) 0 0
\(567\) 567.000 0.0419961
\(568\) 0 0
\(569\) −16899.6 −1.24511 −0.622557 0.782575i \(-0.713906\pi\)
−0.622557 + 0.782575i \(0.713906\pi\)
\(570\) 0 0
\(571\) −7998.44 −0.586207 −0.293104 0.956081i \(-0.594688\pi\)
−0.293104 + 0.956081i \(0.594688\pi\)
\(572\) 0 0
\(573\) −11605.9 −0.846152
\(574\) 0 0
\(575\) 42551.2 3.08610
\(576\) 0 0
\(577\) 23385.2 1.68724 0.843619 0.536942i \(-0.180421\pi\)
0.843619 + 0.536942i \(0.180421\pi\)
\(578\) 0 0
\(579\) 8079.50 0.579918
\(580\) 0 0
\(581\) −3904.13 −0.278779
\(582\) 0 0
\(583\) −6187.90 −0.439582
\(584\) 0 0
\(585\) 9248.01 0.653604
\(586\) 0 0
\(587\) −7583.02 −0.533194 −0.266597 0.963808i \(-0.585899\pi\)
−0.266597 + 0.963808i \(0.585899\pi\)
\(588\) 0 0
\(589\) 7420.46 0.519108
\(590\) 0 0
\(591\) −3194.57 −0.222347
\(592\) 0 0
\(593\) 10593.1 0.733567 0.366783 0.930306i \(-0.380459\pi\)
0.366783 + 0.930306i \(0.380459\pi\)
\(594\) 0 0
\(595\) −14601.9 −1.00609
\(596\) 0 0
\(597\) 4877.75 0.334394
\(598\) 0 0
\(599\) −22125.1 −1.50919 −0.754597 0.656188i \(-0.772168\pi\)
−0.754597 + 0.656188i \(0.772168\pi\)
\(600\) 0 0
\(601\) −19841.6 −1.34669 −0.673343 0.739331i \(-0.735142\pi\)
−0.673343 + 0.739331i \(0.735142\pi\)
\(602\) 0 0
\(603\) −5787.37 −0.390846
\(604\) 0 0
\(605\) −7622.64 −0.512239
\(606\) 0 0
\(607\) 15507.5 1.03695 0.518475 0.855093i \(-0.326500\pi\)
0.518475 + 0.855093i \(0.326500\pi\)
\(608\) 0 0
\(609\) 3343.55 0.222475
\(610\) 0 0
\(611\) −6649.51 −0.440279
\(612\) 0 0
\(613\) −14042.8 −0.925255 −0.462628 0.886553i \(-0.653093\pi\)
−0.462628 + 0.886553i \(0.653093\pi\)
\(614\) 0 0
\(615\) 28430.1 1.86408
\(616\) 0 0
\(617\) 13175.9 0.859708 0.429854 0.902898i \(-0.358565\pi\)
0.429854 + 0.902898i \(0.358565\pi\)
\(618\) 0 0
\(619\) 10828.3 0.703109 0.351555 0.936167i \(-0.385653\pi\)
0.351555 + 0.936167i \(0.385653\pi\)
\(620\) 0 0
\(621\) 3999.48 0.258444
\(622\) 0 0
\(623\) −8602.25 −0.553197
\(624\) 0 0
\(625\) 30984.9 1.98303
\(626\) 0 0
\(627\) 5677.05 0.361594
\(628\) 0 0
\(629\) 36770.4 2.33089
\(630\) 0 0
\(631\) −23922.2 −1.50923 −0.754617 0.656166i \(-0.772177\pi\)
−0.754617 + 0.656166i \(0.772177\pi\)
\(632\) 0 0
\(633\) 3414.13 0.214375
\(634\) 0 0
\(635\) −40225.0 −2.51383
\(636\) 0 0
\(637\) 2479.81 0.154244
\(638\) 0 0
\(639\) −4701.57 −0.291066
\(640\) 0 0
\(641\) −21962.6 −1.35331 −0.676654 0.736301i \(-0.736571\pi\)
−0.676654 + 0.736301i \(0.736571\pi\)
\(642\) 0 0
\(643\) 17927.1 1.09950 0.549749 0.835330i \(-0.314724\pi\)
0.549749 + 0.835330i \(0.314724\pi\)
\(644\) 0 0
\(645\) −11303.3 −0.690028
\(646\) 0 0
\(647\) 850.926 0.0517054 0.0258527 0.999666i \(-0.491770\pi\)
0.0258527 + 0.999666i \(0.491770\pi\)
\(648\) 0 0
\(649\) 18297.3 1.10667
\(650\) 0 0
\(651\) −2545.55 −0.153253
\(652\) 0 0
\(653\) 18674.9 1.11915 0.559575 0.828780i \(-0.310964\pi\)
0.559575 + 0.828780i \(0.310964\pi\)
\(654\) 0 0
\(655\) 57953.0 3.45712
\(656\) 0 0
\(657\) −5185.58 −0.307928
\(658\) 0 0
\(659\) −777.121 −0.0459368 −0.0229684 0.999736i \(-0.507312\pi\)
−0.0229684 + 0.999736i \(0.507312\pi\)
\(660\) 0 0
\(661\) 21963.4 1.29240 0.646201 0.763167i \(-0.276357\pi\)
0.646201 + 0.763167i \(0.276357\pi\)
\(662\) 0 0
\(663\) −15598.1 −0.913693
\(664\) 0 0
\(665\) −8700.64 −0.507363
\(666\) 0 0
\(667\) 23584.6 1.36911
\(668\) 0 0
\(669\) 10607.3 0.613009
\(670\) 0 0
\(671\) −2179.81 −0.125411
\(672\) 0 0
\(673\) 29380.0 1.68279 0.841393 0.540423i \(-0.181736\pi\)
0.841393 + 0.540423i \(0.181736\pi\)
\(674\) 0 0
\(675\) 7755.96 0.442262
\(676\) 0 0
\(677\) 7898.58 0.448400 0.224200 0.974543i \(-0.428023\pi\)
0.224200 + 0.974543i \(0.428023\pi\)
\(678\) 0 0
\(679\) 455.289 0.0257326
\(680\) 0 0
\(681\) −9165.62 −0.515752
\(682\) 0 0
\(683\) 3097.04 0.173507 0.0867533 0.996230i \(-0.472351\pi\)
0.0867533 + 0.996230i \(0.472351\pi\)
\(684\) 0 0
\(685\) −10872.3 −0.606439
\(686\) 0 0
\(687\) −965.109 −0.0535971
\(688\) 0 0
\(689\) 10130.5 0.560148
\(690\) 0 0
\(691\) 26926.3 1.48238 0.741189 0.671296i \(-0.234262\pi\)
0.741189 + 0.671296i \(0.234262\pi\)
\(692\) 0 0
\(693\) −1947.48 −0.106751
\(694\) 0 0
\(695\) 47236.2 2.57809
\(696\) 0 0
\(697\) −47951.3 −2.60586
\(698\) 0 0
\(699\) −2720.82 −0.147226
\(700\) 0 0
\(701\) −16209.3 −0.873347 −0.436674 0.899620i \(-0.643844\pi\)
−0.436674 + 0.899620i \(0.643844\pi\)
\(702\) 0 0
\(703\) 21909.9 1.17546
\(704\) 0 0
\(705\) −8003.39 −0.427553
\(706\) 0 0
\(707\) 2124.58 0.113017
\(708\) 0 0
\(709\) −8277.60 −0.438465 −0.219233 0.975673i \(-0.570355\pi\)
−0.219233 + 0.975673i \(0.570355\pi\)
\(710\) 0 0
\(711\) 2526.96 0.133289
\(712\) 0 0
\(713\) −17955.7 −0.943121
\(714\) 0 0
\(715\) −31764.3 −1.66142
\(716\) 0 0
\(717\) −12872.5 −0.670478
\(718\) 0 0
\(719\) −2334.31 −0.121078 −0.0605389 0.998166i \(-0.519282\pi\)
−0.0605389 + 0.998166i \(0.519282\pi\)
\(720\) 0 0
\(721\) 1219.87 0.0630104
\(722\) 0 0
\(723\) −20956.9 −1.07800
\(724\) 0 0
\(725\) 45736.2 2.34290
\(726\) 0 0
\(727\) −18885.8 −0.963462 −0.481731 0.876319i \(-0.659992\pi\)
−0.481731 + 0.876319i \(0.659992\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 19064.6 0.964611
\(732\) 0 0
\(733\) 3071.96 0.154796 0.0773980 0.997000i \(-0.475339\pi\)
0.0773980 + 0.997000i \(0.475339\pi\)
\(734\) 0 0
\(735\) 2984.71 0.149786
\(736\) 0 0
\(737\) 19878.0 0.993506
\(738\) 0 0
\(739\) −27388.4 −1.36333 −0.681663 0.731666i \(-0.738743\pi\)
−0.681663 + 0.731666i \(0.738743\pi\)
\(740\) 0 0
\(741\) −9294.19 −0.460770
\(742\) 0 0
\(743\) −30541.6 −1.50803 −0.754013 0.656859i \(-0.771885\pi\)
−0.754013 + 0.656859i \(0.771885\pi\)
\(744\) 0 0
\(745\) 41847.7 2.05796
\(746\) 0 0
\(747\) −5019.59 −0.245860
\(748\) 0 0
\(749\) 13617.9 0.664337
\(750\) 0 0
\(751\) −36868.0 −1.79139 −0.895695 0.444670i \(-0.853321\pi\)
−0.895695 + 0.444670i \(0.853321\pi\)
\(752\) 0 0
\(753\) −16999.9 −0.822723
\(754\) 0 0
\(755\) −22599.1 −1.08936
\(756\) 0 0
\(757\) 1804.07 0.0866184 0.0433092 0.999062i \(-0.486210\pi\)
0.0433092 + 0.999062i \(0.486210\pi\)
\(758\) 0 0
\(759\) −13737.1 −0.656948
\(760\) 0 0
\(761\) −17838.8 −0.849746 −0.424873 0.905253i \(-0.639681\pi\)
−0.424873 + 0.905253i \(0.639681\pi\)
\(762\) 0 0
\(763\) −7720.06 −0.366298
\(764\) 0 0
\(765\) −18773.9 −0.887284
\(766\) 0 0
\(767\) −29955.4 −1.41021
\(768\) 0 0
\(769\) 6804.58 0.319089 0.159544 0.987191i \(-0.448997\pi\)
0.159544 + 0.987191i \(0.448997\pi\)
\(770\) 0 0
\(771\) 2521.79 0.117795
\(772\) 0 0
\(773\) −23600.9 −1.09814 −0.549072 0.835775i \(-0.685019\pi\)
−0.549072 + 0.835775i \(0.685019\pi\)
\(774\) 0 0
\(775\) −34820.4 −1.61392
\(776\) 0 0
\(777\) −7516.06 −0.347023
\(778\) 0 0
\(779\) −28572.0 −1.31412
\(780\) 0 0
\(781\) 16148.5 0.739872
\(782\) 0 0
\(783\) 4298.85 0.196205
\(784\) 0 0
\(785\) −31213.5 −1.41918
\(786\) 0 0
\(787\) 9183.98 0.415977 0.207988 0.978131i \(-0.433308\pi\)
0.207988 + 0.978131i \(0.433308\pi\)
\(788\) 0 0
\(789\) 10160.1 0.458438
\(790\) 0 0
\(791\) 560.579 0.0251983
\(792\) 0 0
\(793\) 3568.68 0.159808
\(794\) 0 0
\(795\) 12193.2 0.543958
\(796\) 0 0
\(797\) 11076.9 0.492302 0.246151 0.969231i \(-0.420834\pi\)
0.246151 + 0.969231i \(0.420834\pi\)
\(798\) 0 0
\(799\) 13498.8 0.597690
\(800\) 0 0
\(801\) −11060.0 −0.487874
\(802\) 0 0
\(803\) 17811.0 0.782734
\(804\) 0 0
\(805\) 21053.4 0.921783
\(806\) 0 0
\(807\) 23788.3 1.03765
\(808\) 0 0
\(809\) 32258.3 1.40190 0.700952 0.713209i \(-0.252759\pi\)
0.700952 + 0.713209i \(0.252759\pi\)
\(810\) 0 0
\(811\) 17693.9 0.766114 0.383057 0.923725i \(-0.374871\pi\)
0.383057 + 0.923725i \(0.374871\pi\)
\(812\) 0 0
\(813\) 15240.8 0.657462
\(814\) 0 0
\(815\) −41491.9 −1.78331
\(816\) 0 0
\(817\) 11359.8 0.486448
\(818\) 0 0
\(819\) 3188.32 0.136030
\(820\) 0 0
\(821\) −18430.7 −0.783480 −0.391740 0.920076i \(-0.628127\pi\)
−0.391740 + 0.920076i \(0.628127\pi\)
\(822\) 0 0
\(823\) −32285.9 −1.36746 −0.683729 0.729736i \(-0.739643\pi\)
−0.683729 + 0.729736i \(0.739643\pi\)
\(824\) 0 0
\(825\) −26639.5 −1.12420
\(826\) 0 0
\(827\) 36266.1 1.52491 0.762453 0.647044i \(-0.223995\pi\)
0.762453 + 0.647044i \(0.223995\pi\)
\(828\) 0 0
\(829\) −31156.2 −1.30531 −0.652654 0.757656i \(-0.726345\pi\)
−0.652654 + 0.757656i \(0.726345\pi\)
\(830\) 0 0
\(831\) −585.986 −0.0244616
\(832\) 0 0
\(833\) −5034.12 −0.209390
\(834\) 0 0
\(835\) 51033.1 2.11506
\(836\) 0 0
\(837\) −3272.85 −0.135157
\(838\) 0 0
\(839\) −85.5051 −0.00351843 −0.00175922 0.999998i \(-0.500560\pi\)
−0.00175922 + 0.999998i \(0.500560\pi\)
\(840\) 0 0
\(841\) 960.906 0.0393992
\(842\) 0 0
\(843\) 12089.5 0.493931
\(844\) 0 0
\(845\) 7394.70 0.301048
\(846\) 0 0
\(847\) −2627.96 −0.106609
\(848\) 0 0
\(849\) 20781.8 0.840083
\(850\) 0 0
\(851\) −53016.5 −2.13558
\(852\) 0 0
\(853\) 1148.37 0.0460956 0.0230478 0.999734i \(-0.492663\pi\)
0.0230478 + 0.999734i \(0.492663\pi\)
\(854\) 0 0
\(855\) −11186.5 −0.447452
\(856\) 0 0
\(857\) −11674.6 −0.465340 −0.232670 0.972556i \(-0.574746\pi\)
−0.232670 + 0.972556i \(0.574746\pi\)
\(858\) 0 0
\(859\) −47503.1 −1.88683 −0.943413 0.331619i \(-0.892405\pi\)
−0.943413 + 0.331619i \(0.892405\pi\)
\(860\) 0 0
\(861\) 9801.48 0.387960
\(862\) 0 0
\(863\) −10361.9 −0.408717 −0.204359 0.978896i \(-0.565511\pi\)
−0.204359 + 0.978896i \(0.565511\pi\)
\(864\) 0 0
\(865\) 32303.8 1.26978
\(866\) 0 0
\(867\) 16925.8 0.663011
\(868\) 0 0
\(869\) −8679.39 −0.338813
\(870\) 0 0
\(871\) −32543.2 −1.26600
\(872\) 0 0
\(873\) 585.372 0.0226940
\(874\) 0 0
\(875\) 23061.5 0.890997
\(876\) 0 0
\(877\) −26153.1 −1.00699 −0.503493 0.863999i \(-0.667952\pi\)
−0.503493 + 0.863999i \(0.667952\pi\)
\(878\) 0 0
\(879\) −2733.03 −0.104872
\(880\) 0 0
\(881\) 35711.7 1.36567 0.682837 0.730571i \(-0.260746\pi\)
0.682837 + 0.730571i \(0.260746\pi\)
\(882\) 0 0
\(883\) 5654.75 0.215513 0.107756 0.994177i \(-0.465633\pi\)
0.107756 + 0.994177i \(0.465633\pi\)
\(884\) 0 0
\(885\) −36054.5 −1.36945
\(886\) 0 0
\(887\) 8344.44 0.315873 0.157936 0.987449i \(-0.449516\pi\)
0.157936 + 0.987449i \(0.449516\pi\)
\(888\) 0 0
\(889\) −13867.9 −0.523187
\(890\) 0 0
\(891\) −2503.90 −0.0941459
\(892\) 0 0
\(893\) 8043.35 0.301411
\(894\) 0 0
\(895\) 51821.7 1.93543
\(896\) 0 0
\(897\) 22489.6 0.837132
\(898\) 0 0
\(899\) −19299.7 −0.715996
\(900\) 0 0
\(901\) −20565.4 −0.760415
\(902\) 0 0
\(903\) −3896.91 −0.143611
\(904\) 0 0
\(905\) −68249.0 −2.50682
\(906\) 0 0
\(907\) 52612.1 1.92608 0.963040 0.269358i \(-0.0868116\pi\)
0.963040 + 0.269358i \(0.0868116\pi\)
\(908\) 0 0
\(909\) 2731.60 0.0996715
\(910\) 0 0
\(911\) −24706.6 −0.898536 −0.449268 0.893397i \(-0.648315\pi\)
−0.449268 + 0.893397i \(0.648315\pi\)
\(912\) 0 0
\(913\) 17240.8 0.624960
\(914\) 0 0
\(915\) 4295.29 0.155189
\(916\) 0 0
\(917\) 19979.7 0.719508
\(918\) 0 0
\(919\) 42479.2 1.52477 0.762383 0.647127i \(-0.224030\pi\)
0.762383 + 0.647127i \(0.224030\pi\)
\(920\) 0 0
\(921\) 9932.17 0.355349
\(922\) 0 0
\(923\) −26437.6 −0.942799
\(924\) 0 0
\(925\) −102812. −3.65452
\(926\) 0 0
\(927\) 1568.41 0.0555699
\(928\) 0 0
\(929\) 9423.54 0.332805 0.166403 0.986058i \(-0.446785\pi\)
0.166403 + 0.986058i \(0.446785\pi\)
\(930\) 0 0
\(931\) −2999.61 −0.105594
\(932\) 0 0
\(933\) 14937.4 0.524146
\(934\) 0 0
\(935\) 64483.0 2.25542
\(936\) 0 0
\(937\) −33158.7 −1.15608 −0.578041 0.816008i \(-0.696183\pi\)
−0.578041 + 0.816008i \(0.696183\pi\)
\(938\) 0 0
\(939\) −15348.6 −0.533420
\(940\) 0 0
\(941\) 15005.1 0.519822 0.259911 0.965633i \(-0.416307\pi\)
0.259911 + 0.965633i \(0.416307\pi\)
\(942\) 0 0
\(943\) 69137.3 2.38751
\(944\) 0 0
\(945\) 3837.48 0.132099
\(946\) 0 0
\(947\) 7631.38 0.261865 0.130933 0.991391i \(-0.458203\pi\)
0.130933 + 0.991391i \(0.458203\pi\)
\(948\) 0 0
\(949\) −29159.2 −0.997417
\(950\) 0 0
\(951\) 2934.95 0.100076
\(952\) 0 0
\(953\) −42615.3 −1.44853 −0.724263 0.689524i \(-0.757820\pi\)
−0.724263 + 0.689524i \(0.757820\pi\)
\(954\) 0 0
\(955\) −78549.5 −2.66157
\(956\) 0 0
\(957\) −14765.3 −0.498740
\(958\) 0 0
\(959\) −3748.32 −0.126214
\(960\) 0 0
\(961\) −15097.6 −0.506782
\(962\) 0 0
\(963\) 17508.8 0.585890
\(964\) 0 0
\(965\) 54682.4 1.82413
\(966\) 0 0
\(967\) 34795.1 1.15712 0.578559 0.815640i \(-0.303615\pi\)
0.578559 + 0.815640i \(0.303615\pi\)
\(968\) 0 0
\(969\) 18867.6 0.625507
\(970\) 0 0
\(971\) 7714.99 0.254980 0.127490 0.991840i \(-0.459308\pi\)
0.127490 + 0.991840i \(0.459308\pi\)
\(972\) 0 0
\(973\) 16285.0 0.536561
\(974\) 0 0
\(975\) 43612.9 1.43254
\(976\) 0 0
\(977\) 22817.7 0.747188 0.373594 0.927592i \(-0.378125\pi\)
0.373594 + 0.927592i \(0.378125\pi\)
\(978\) 0 0
\(979\) 37988.0 1.24015
\(980\) 0 0
\(981\) −9925.80 −0.323044
\(982\) 0 0
\(983\) −10924.8 −0.354473 −0.177237 0.984168i \(-0.556716\pi\)
−0.177237 + 0.984168i \(0.556716\pi\)
\(984\) 0 0
\(985\) −21621.0 −0.699393
\(986\) 0 0
\(987\) −2759.23 −0.0889839
\(988\) 0 0
\(989\) −27487.8 −0.883784
\(990\) 0 0
\(991\) −49279.5 −1.57963 −0.789816 0.613343i \(-0.789824\pi\)
−0.789816 + 0.613343i \(0.789824\pi\)
\(992\) 0 0
\(993\) 27277.1 0.871714
\(994\) 0 0
\(995\) 33012.8 1.05184
\(996\) 0 0
\(997\) −49601.6 −1.57563 −0.787813 0.615914i \(-0.788787\pi\)
−0.787813 + 0.615914i \(0.788787\pi\)
\(998\) 0 0
\(999\) −9663.50 −0.306046
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 168.4.a.h.1.2 2
3.2 odd 2 504.4.a.j.1.1 2
4.3 odd 2 336.4.a.n.1.2 2
7.6 odd 2 1176.4.a.p.1.1 2
8.3 odd 2 1344.4.a.bl.1.1 2
8.5 even 2 1344.4.a.bd.1.1 2
12.11 even 2 1008.4.a.y.1.1 2
28.27 even 2 2352.4.a.bv.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
168.4.a.h.1.2 2 1.1 even 1 trivial
336.4.a.n.1.2 2 4.3 odd 2
504.4.a.j.1.1 2 3.2 odd 2
1008.4.a.y.1.1 2 12.11 even 2
1176.4.a.p.1.1 2 7.6 odd 2
1344.4.a.bd.1.1 2 8.5 even 2
1344.4.a.bl.1.1 2 8.3 odd 2
2352.4.a.bv.1.1 2 28.27 even 2