Properties

Label 1344.4.a.ba.1.1
Level $1344$
Weight $4$
Character 1344.1
Self dual yes
Analytic conductor $79.299$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1344.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(79.2985670477\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 21)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1344.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.00000 q^{3} +18.0000 q^{5} +7.00000 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q+3.00000 q^{3} +18.0000 q^{5} +7.00000 q^{7} +9.00000 q^{9} +36.0000 q^{11} +34.0000 q^{13} +54.0000 q^{15} +42.0000 q^{17} +124.000 q^{19} +21.0000 q^{21} +199.000 q^{25} +27.0000 q^{27} -102.000 q^{29} -160.000 q^{31} +108.000 q^{33} +126.000 q^{35} -398.000 q^{37} +102.000 q^{39} -318.000 q^{41} +268.000 q^{43} +162.000 q^{45} +240.000 q^{47} +49.0000 q^{49} +126.000 q^{51} +498.000 q^{53} +648.000 q^{55} +372.000 q^{57} +132.000 q^{59} -398.000 q^{61} +63.0000 q^{63} +612.000 q^{65} -92.0000 q^{67} -720.000 q^{71} -502.000 q^{73} +597.000 q^{75} +252.000 q^{77} -1024.00 q^{79} +81.0000 q^{81} +204.000 q^{83} +756.000 q^{85} -306.000 q^{87} +354.000 q^{89} +238.000 q^{91} -480.000 q^{93} +2232.00 q^{95} -286.000 q^{97} +324.000 q^{99} +O(q^{100})\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 3.00000 0.577350
\(4\) 0 0
\(5\) 18.0000 1.60997 0.804984 0.593296i \(-0.202174\pi\)
0.804984 + 0.593296i \(0.202174\pi\)
\(6\) 0 0
\(7\) 7.00000 0.377964
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 36.0000 0.986764 0.493382 0.869813i \(-0.335760\pi\)
0.493382 + 0.869813i \(0.335760\pi\)
\(12\) 0 0
\(13\) 34.0000 0.725377 0.362689 0.931910i \(-0.381859\pi\)
0.362689 + 0.931910i \(0.381859\pi\)
\(14\) 0 0
\(15\) 54.0000 0.929516
\(16\) 0 0
\(17\) 42.0000 0.599206 0.299603 0.954064i \(-0.403146\pi\)
0.299603 + 0.954064i \(0.403146\pi\)
\(18\) 0 0
\(19\) 124.000 1.49724 0.748620 0.663000i \(-0.230717\pi\)
0.748620 + 0.663000i \(0.230717\pi\)
\(20\) 0 0
\(21\) 21.0000 0.218218
\(22\) 0 0
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 199.000 1.59200
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) −102.000 −0.653135 −0.326568 0.945174i \(-0.605892\pi\)
−0.326568 + 0.945174i \(0.605892\pi\)
\(30\) 0 0
\(31\) −160.000 −0.926995 −0.463498 0.886098i \(-0.653406\pi\)
−0.463498 + 0.886098i \(0.653406\pi\)
\(32\) 0 0
\(33\) 108.000 0.569709
\(34\) 0 0
\(35\) 126.000 0.608511
\(36\) 0 0
\(37\) −398.000 −1.76840 −0.884200 0.467109i \(-0.845296\pi\)
−0.884200 + 0.467109i \(0.845296\pi\)
\(38\) 0 0
\(39\) 102.000 0.418797
\(40\) 0 0
\(41\) −318.000 −1.21130 −0.605649 0.795732i \(-0.707087\pi\)
−0.605649 + 0.795732i \(0.707087\pi\)
\(42\) 0 0
\(43\) 268.000 0.950456 0.475228 0.879863i \(-0.342366\pi\)
0.475228 + 0.879863i \(0.342366\pi\)
\(44\) 0 0
\(45\) 162.000 0.536656
\(46\) 0 0
\(47\) 240.000 0.744843 0.372421 0.928064i \(-0.378528\pi\)
0.372421 + 0.928064i \(0.378528\pi\)
\(48\) 0 0
\(49\) 49.0000 0.142857
\(50\) 0 0
\(51\) 126.000 0.345952
\(52\) 0 0
\(53\) 498.000 1.29067 0.645335 0.763899i \(-0.276718\pi\)
0.645335 + 0.763899i \(0.276718\pi\)
\(54\) 0 0
\(55\) 648.000 1.58866
\(56\) 0 0
\(57\) 372.000 0.864432
\(58\) 0 0
\(59\) 132.000 0.291270 0.145635 0.989338i \(-0.453477\pi\)
0.145635 + 0.989338i \(0.453477\pi\)
\(60\) 0 0
\(61\) −398.000 −0.835388 −0.417694 0.908588i \(-0.637162\pi\)
−0.417694 + 0.908588i \(0.637162\pi\)
\(62\) 0 0
\(63\) 63.0000 0.125988
\(64\) 0 0
\(65\) 612.000 1.16783
\(66\) 0 0
\(67\) −92.0000 −0.167755 −0.0838775 0.996476i \(-0.526730\pi\)
−0.0838775 + 0.996476i \(0.526730\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −720.000 −1.20350 −0.601748 0.798686i \(-0.705529\pi\)
−0.601748 + 0.798686i \(0.705529\pi\)
\(72\) 0 0
\(73\) −502.000 −0.804858 −0.402429 0.915451i \(-0.631834\pi\)
−0.402429 + 0.915451i \(0.631834\pi\)
\(74\) 0 0
\(75\) 597.000 0.919142
\(76\) 0 0
\(77\) 252.000 0.372962
\(78\) 0 0
\(79\) −1024.00 −1.45834 −0.729171 0.684332i \(-0.760094\pi\)
−0.729171 + 0.684332i \(0.760094\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 204.000 0.269782 0.134891 0.990860i \(-0.456932\pi\)
0.134891 + 0.990860i \(0.456932\pi\)
\(84\) 0 0
\(85\) 756.000 0.964703
\(86\) 0 0
\(87\) −306.000 −0.377088
\(88\) 0 0
\(89\) 354.000 0.421617 0.210809 0.977527i \(-0.432390\pi\)
0.210809 + 0.977527i \(0.432390\pi\)
\(90\) 0 0
\(91\) 238.000 0.274167
\(92\) 0 0
\(93\) −480.000 −0.535201
\(94\) 0 0
\(95\) 2232.00 2.41051
\(96\) 0 0
\(97\) −286.000 −0.299370 −0.149685 0.988734i \(-0.547826\pi\)
−0.149685 + 0.988734i \(0.547826\pi\)
\(98\) 0 0
\(99\) 324.000 0.328921
\(100\) 0 0
\(101\) −414.000 −0.407867 −0.203933 0.978985i \(-0.565373\pi\)
−0.203933 + 0.978985i \(0.565373\pi\)
\(102\) 0 0
\(103\) 56.0000 0.0535713 0.0267857 0.999641i \(-0.491473\pi\)
0.0267857 + 0.999641i \(0.491473\pi\)
\(104\) 0 0
\(105\) 378.000 0.351324
\(106\) 0 0
\(107\) −12.0000 −0.0108419 −0.00542095 0.999985i \(-0.501726\pi\)
−0.00542095 + 0.999985i \(0.501726\pi\)
\(108\) 0 0
\(109\) −1478.00 −1.29878 −0.649389 0.760457i \(-0.724975\pi\)
−0.649389 + 0.760457i \(0.724975\pi\)
\(110\) 0 0
\(111\) −1194.00 −1.02099
\(112\) 0 0
\(113\) 402.000 0.334664 0.167332 0.985901i \(-0.446485\pi\)
0.167332 + 0.985901i \(0.446485\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 306.000 0.241792
\(118\) 0 0
\(119\) 294.000 0.226478
\(120\) 0 0
\(121\) −35.0000 −0.0262960
\(122\) 0 0
\(123\) −954.000 −0.699344
\(124\) 0 0
\(125\) 1332.00 0.953102
\(126\) 0 0
\(127\) 1280.00 0.894344 0.447172 0.894448i \(-0.352431\pi\)
0.447172 + 0.894448i \(0.352431\pi\)
\(128\) 0 0
\(129\) 804.000 0.548746
\(130\) 0 0
\(131\) −1764.00 −1.17650 −0.588250 0.808679i \(-0.700183\pi\)
−0.588250 + 0.808679i \(0.700183\pi\)
\(132\) 0 0
\(133\) 868.000 0.565903
\(134\) 0 0
\(135\) 486.000 0.309839
\(136\) 0 0
\(137\) −2358.00 −1.47049 −0.735246 0.677800i \(-0.762934\pi\)
−0.735246 + 0.677800i \(0.762934\pi\)
\(138\) 0 0
\(139\) 52.0000 0.0317308 0.0158654 0.999874i \(-0.494950\pi\)
0.0158654 + 0.999874i \(0.494950\pi\)
\(140\) 0 0
\(141\) 720.000 0.430035
\(142\) 0 0
\(143\) 1224.00 0.715776
\(144\) 0 0
\(145\) −1836.00 −1.05153
\(146\) 0 0
\(147\) 147.000 0.0824786
\(148\) 0 0
\(149\) 1746.00 0.959986 0.479993 0.877272i \(-0.340639\pi\)
0.479993 + 0.877272i \(0.340639\pi\)
\(150\) 0 0
\(151\) −232.000 −0.125032 −0.0625162 0.998044i \(-0.519913\pi\)
−0.0625162 + 0.998044i \(0.519913\pi\)
\(152\) 0 0
\(153\) 378.000 0.199735
\(154\) 0 0
\(155\) −2880.00 −1.49243
\(156\) 0 0
\(157\) −1694.00 −0.861120 −0.430560 0.902562i \(-0.641684\pi\)
−0.430560 + 0.902562i \(0.641684\pi\)
\(158\) 0 0
\(159\) 1494.00 0.745169
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 2932.00 1.40891 0.704454 0.709750i \(-0.251192\pi\)
0.704454 + 0.709750i \(0.251192\pi\)
\(164\) 0 0
\(165\) 1944.00 0.917213
\(166\) 0 0
\(167\) 1176.00 0.544920 0.272460 0.962167i \(-0.412163\pi\)
0.272460 + 0.962167i \(0.412163\pi\)
\(168\) 0 0
\(169\) −1041.00 −0.473828
\(170\) 0 0
\(171\) 1116.00 0.499080
\(172\) 0 0
\(173\) −870.000 −0.382340 −0.191170 0.981557i \(-0.561228\pi\)
−0.191170 + 0.981557i \(0.561228\pi\)
\(174\) 0 0
\(175\) 1393.00 0.601719
\(176\) 0 0
\(177\) 396.000 0.168165
\(178\) 0 0
\(179\) 2316.00 0.967072 0.483536 0.875324i \(-0.339352\pi\)
0.483536 + 0.875324i \(0.339352\pi\)
\(180\) 0 0
\(181\) 106.000 0.0435299 0.0217650 0.999763i \(-0.493071\pi\)
0.0217650 + 0.999763i \(0.493071\pi\)
\(182\) 0 0
\(183\) −1194.00 −0.482312
\(184\) 0 0
\(185\) −7164.00 −2.84707
\(186\) 0 0
\(187\) 1512.00 0.591275
\(188\) 0 0
\(189\) 189.000 0.0727393
\(190\) 0 0
\(191\) −1128.00 −0.427326 −0.213663 0.976907i \(-0.568539\pi\)
−0.213663 + 0.976907i \(0.568539\pi\)
\(192\) 0 0
\(193\) 4034.00 1.50453 0.752263 0.658862i \(-0.228962\pi\)
0.752263 + 0.658862i \(0.228962\pi\)
\(194\) 0 0
\(195\) 1836.00 0.674250
\(196\) 0 0
\(197\) 1314.00 0.475221 0.237611 0.971360i \(-0.423636\pi\)
0.237611 + 0.971360i \(0.423636\pi\)
\(198\) 0 0
\(199\) 5096.00 1.81531 0.907653 0.419722i \(-0.137872\pi\)
0.907653 + 0.419722i \(0.137872\pi\)
\(200\) 0 0
\(201\) −276.000 −0.0968534
\(202\) 0 0
\(203\) −714.000 −0.246862
\(204\) 0 0
\(205\) −5724.00 −1.95015
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 4464.00 1.47742
\(210\) 0 0
\(211\) 3076.00 1.00360 0.501802 0.864982i \(-0.332670\pi\)
0.501802 + 0.864982i \(0.332670\pi\)
\(212\) 0 0
\(213\) −2160.00 −0.694839
\(214\) 0 0
\(215\) 4824.00 1.53020
\(216\) 0 0
\(217\) −1120.00 −0.350371
\(218\) 0 0
\(219\) −1506.00 −0.464685
\(220\) 0 0
\(221\) 1428.00 0.434650
\(222\) 0 0
\(223\) −1888.00 −0.566950 −0.283475 0.958980i \(-0.591487\pi\)
−0.283475 + 0.958980i \(0.591487\pi\)
\(224\) 0 0
\(225\) 1791.00 0.530667
\(226\) 0 0
\(227\) 4716.00 1.37891 0.689454 0.724330i \(-0.257851\pi\)
0.689454 + 0.724330i \(0.257851\pi\)
\(228\) 0 0
\(229\) 1690.00 0.487678 0.243839 0.969816i \(-0.421593\pi\)
0.243839 + 0.969816i \(0.421593\pi\)
\(230\) 0 0
\(231\) 756.000 0.215330
\(232\) 0 0
\(233\) 138.000 0.0388012 0.0194006 0.999812i \(-0.493824\pi\)
0.0194006 + 0.999812i \(0.493824\pi\)
\(234\) 0 0
\(235\) 4320.00 1.19917
\(236\) 0 0
\(237\) −3072.00 −0.841974
\(238\) 0 0
\(239\) 1896.00 0.513147 0.256573 0.966525i \(-0.417406\pi\)
0.256573 + 0.966525i \(0.417406\pi\)
\(240\) 0 0
\(241\) −3598.00 −0.961691 −0.480846 0.876805i \(-0.659670\pi\)
−0.480846 + 0.876805i \(0.659670\pi\)
\(242\) 0 0
\(243\) 243.000 0.0641500
\(244\) 0 0
\(245\) 882.000 0.229996
\(246\) 0 0
\(247\) 4216.00 1.08606
\(248\) 0 0
\(249\) 612.000 0.155759
\(250\) 0 0
\(251\) 3060.00 0.769504 0.384752 0.923020i \(-0.374287\pi\)
0.384752 + 0.923020i \(0.374287\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 2268.00 0.556971
\(256\) 0 0
\(257\) −6822.00 −1.65582 −0.827908 0.560864i \(-0.810469\pi\)
−0.827908 + 0.560864i \(0.810469\pi\)
\(258\) 0 0
\(259\) −2786.00 −0.668392
\(260\) 0 0
\(261\) −918.000 −0.217712
\(262\) 0 0
\(263\) 2592.00 0.607717 0.303858 0.952717i \(-0.401725\pi\)
0.303858 + 0.952717i \(0.401725\pi\)
\(264\) 0 0
\(265\) 8964.00 2.07794
\(266\) 0 0
\(267\) 1062.00 0.243421
\(268\) 0 0
\(269\) −8214.00 −1.86177 −0.930886 0.365311i \(-0.880963\pi\)
−0.930886 + 0.365311i \(0.880963\pi\)
\(270\) 0 0
\(271\) −5344.00 −1.19788 −0.598939 0.800795i \(-0.704411\pi\)
−0.598939 + 0.800795i \(0.704411\pi\)
\(272\) 0 0
\(273\) 714.000 0.158290
\(274\) 0 0
\(275\) 7164.00 1.57093
\(276\) 0 0
\(277\) 6514.00 1.41295 0.706477 0.707736i \(-0.250283\pi\)
0.706477 + 0.707736i \(0.250283\pi\)
\(278\) 0 0
\(279\) −1440.00 −0.308998
\(280\) 0 0
\(281\) 6618.00 1.40497 0.702485 0.711698i \(-0.252074\pi\)
0.702485 + 0.711698i \(0.252074\pi\)
\(282\) 0 0
\(283\) −3260.00 −0.684759 −0.342380 0.939562i \(-0.611233\pi\)
−0.342380 + 0.939562i \(0.611233\pi\)
\(284\) 0 0
\(285\) 6696.00 1.39171
\(286\) 0 0
\(287\) −2226.00 −0.457828
\(288\) 0 0
\(289\) −3149.00 −0.640953
\(290\) 0 0
\(291\) −858.000 −0.172841
\(292\) 0 0
\(293\) −5118.00 −1.02047 −0.510233 0.860036i \(-0.670441\pi\)
−0.510233 + 0.860036i \(0.670441\pi\)
\(294\) 0 0
\(295\) 2376.00 0.468936
\(296\) 0 0
\(297\) 972.000 0.189903
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 1876.00 0.359239
\(302\) 0 0
\(303\) −1242.00 −0.235482
\(304\) 0 0
\(305\) −7164.00 −1.34495
\(306\) 0 0
\(307\) −452.000 −0.0840293 −0.0420147 0.999117i \(-0.513378\pi\)
−0.0420147 + 0.999117i \(0.513378\pi\)
\(308\) 0 0
\(309\) 168.000 0.0309294
\(310\) 0 0
\(311\) 5016.00 0.914570 0.457285 0.889320i \(-0.348822\pi\)
0.457285 + 0.889320i \(0.348822\pi\)
\(312\) 0 0
\(313\) 5402.00 0.975524 0.487762 0.872977i \(-0.337813\pi\)
0.487762 + 0.872977i \(0.337813\pi\)
\(314\) 0 0
\(315\) 1134.00 0.202837
\(316\) 0 0
\(317\) −10086.0 −1.78702 −0.893511 0.449041i \(-0.851766\pi\)
−0.893511 + 0.449041i \(0.851766\pi\)
\(318\) 0 0
\(319\) −3672.00 −0.644491
\(320\) 0 0
\(321\) −36.0000 −0.00625958
\(322\) 0 0
\(323\) 5208.00 0.897154
\(324\) 0 0
\(325\) 6766.00 1.15480
\(326\) 0 0
\(327\) −4434.00 −0.749849
\(328\) 0 0
\(329\) 1680.00 0.281524
\(330\) 0 0
\(331\) 8044.00 1.33577 0.667883 0.744267i \(-0.267201\pi\)
0.667883 + 0.744267i \(0.267201\pi\)
\(332\) 0 0
\(333\) −3582.00 −0.589467
\(334\) 0 0
\(335\) −1656.00 −0.270080
\(336\) 0 0
\(337\) 4178.00 0.675342 0.337671 0.941264i \(-0.390361\pi\)
0.337671 + 0.941264i \(0.390361\pi\)
\(338\) 0 0
\(339\) 1206.00 0.193218
\(340\) 0 0
\(341\) −5760.00 −0.914726
\(342\) 0 0
\(343\) 343.000 0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −156.000 −0.0241341 −0.0120670 0.999927i \(-0.503841\pi\)
−0.0120670 + 0.999927i \(0.503841\pi\)
\(348\) 0 0
\(349\) 12418.0 1.90464 0.952321 0.305097i \(-0.0986888\pi\)
0.952321 + 0.305097i \(0.0986888\pi\)
\(350\) 0 0
\(351\) 918.000 0.139599
\(352\) 0 0
\(353\) −7830.00 −1.18059 −0.590296 0.807187i \(-0.700989\pi\)
−0.590296 + 0.807187i \(0.700989\pi\)
\(354\) 0 0
\(355\) −12960.0 −1.93759
\(356\) 0 0
\(357\) 882.000 0.130757
\(358\) 0 0
\(359\) −9312.00 −1.36899 −0.684497 0.729016i \(-0.739978\pi\)
−0.684497 + 0.729016i \(0.739978\pi\)
\(360\) 0 0
\(361\) 8517.00 1.24173
\(362\) 0 0
\(363\) −105.000 −0.0151820
\(364\) 0 0
\(365\) −9036.00 −1.29580
\(366\) 0 0
\(367\) −3760.00 −0.534797 −0.267398 0.963586i \(-0.586164\pi\)
−0.267398 + 0.963586i \(0.586164\pi\)
\(368\) 0 0
\(369\) −2862.00 −0.403766
\(370\) 0 0
\(371\) 3486.00 0.487828
\(372\) 0 0
\(373\) −5870.00 −0.814845 −0.407422 0.913240i \(-0.633572\pi\)
−0.407422 + 0.913240i \(0.633572\pi\)
\(374\) 0 0
\(375\) 3996.00 0.550273
\(376\) 0 0
\(377\) −3468.00 −0.473769
\(378\) 0 0
\(379\) 1852.00 0.251005 0.125502 0.992093i \(-0.459946\pi\)
0.125502 + 0.992093i \(0.459946\pi\)
\(380\) 0 0
\(381\) 3840.00 0.516350
\(382\) 0 0
\(383\) 2160.00 0.288175 0.144087 0.989565i \(-0.453975\pi\)
0.144087 + 0.989565i \(0.453975\pi\)
\(384\) 0 0
\(385\) 4536.00 0.600457
\(386\) 0 0
\(387\) 2412.00 0.316819
\(388\) 0 0
\(389\) 6786.00 0.884483 0.442241 0.896896i \(-0.354183\pi\)
0.442241 + 0.896896i \(0.354183\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) −5292.00 −0.679252
\(394\) 0 0
\(395\) −18432.0 −2.34788
\(396\) 0 0
\(397\) 6514.00 0.823497 0.411748 0.911298i \(-0.364918\pi\)
0.411748 + 0.911298i \(0.364918\pi\)
\(398\) 0 0
\(399\) 2604.00 0.326724
\(400\) 0 0
\(401\) 3330.00 0.414694 0.207347 0.978267i \(-0.433517\pi\)
0.207347 + 0.978267i \(0.433517\pi\)
\(402\) 0 0
\(403\) −5440.00 −0.672421
\(404\) 0 0
\(405\) 1458.00 0.178885
\(406\) 0 0
\(407\) −14328.0 −1.74499
\(408\) 0 0
\(409\) −5398.00 −0.652601 −0.326301 0.945266i \(-0.605802\pi\)
−0.326301 + 0.945266i \(0.605802\pi\)
\(410\) 0 0
\(411\) −7074.00 −0.848990
\(412\) 0 0
\(413\) 924.000 0.110090
\(414\) 0 0
\(415\) 3672.00 0.434341
\(416\) 0 0
\(417\) 156.000 0.0183198
\(418\) 0 0
\(419\) −13092.0 −1.52646 −0.763229 0.646128i \(-0.776387\pi\)
−0.763229 + 0.646128i \(0.776387\pi\)
\(420\) 0 0
\(421\) 322.000 0.0372763 0.0186381 0.999826i \(-0.494067\pi\)
0.0186381 + 0.999826i \(0.494067\pi\)
\(422\) 0 0
\(423\) 2160.00 0.248281
\(424\) 0 0
\(425\) 8358.00 0.953935
\(426\) 0 0
\(427\) −2786.00 −0.315747
\(428\) 0 0
\(429\) 3672.00 0.413254
\(430\) 0 0
\(431\) 2616.00 0.292363 0.146181 0.989258i \(-0.453302\pi\)
0.146181 + 0.989258i \(0.453302\pi\)
\(432\) 0 0
\(433\) 4322.00 0.479681 0.239841 0.970812i \(-0.422905\pi\)
0.239841 + 0.970812i \(0.422905\pi\)
\(434\) 0 0
\(435\) −5508.00 −0.607100
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −9016.00 −0.980205 −0.490103 0.871665i \(-0.663041\pi\)
−0.490103 + 0.871665i \(0.663041\pi\)
\(440\) 0 0
\(441\) 441.000 0.0476190
\(442\) 0 0
\(443\) 5268.00 0.564989 0.282495 0.959269i \(-0.408838\pi\)
0.282495 + 0.959269i \(0.408838\pi\)
\(444\) 0 0
\(445\) 6372.00 0.678790
\(446\) 0 0
\(447\) 5238.00 0.554248
\(448\) 0 0
\(449\) −5310.00 −0.558117 −0.279058 0.960274i \(-0.590022\pi\)
−0.279058 + 0.960274i \(0.590022\pi\)
\(450\) 0 0
\(451\) −11448.0 −1.19527
\(452\) 0 0
\(453\) −696.000 −0.0721875
\(454\) 0 0
\(455\) 4284.00 0.441400
\(456\) 0 0
\(457\) 15770.0 1.61420 0.807100 0.590415i \(-0.201036\pi\)
0.807100 + 0.590415i \(0.201036\pi\)
\(458\) 0 0
\(459\) 1134.00 0.115317
\(460\) 0 0
\(461\) 5370.00 0.542529 0.271264 0.962505i \(-0.412558\pi\)
0.271264 + 0.962505i \(0.412558\pi\)
\(462\) 0 0
\(463\) −3328.00 −0.334050 −0.167025 0.985953i \(-0.553416\pi\)
−0.167025 + 0.985953i \(0.553416\pi\)
\(464\) 0 0
\(465\) −8640.00 −0.861657
\(466\) 0 0
\(467\) −4548.00 −0.450656 −0.225328 0.974283i \(-0.572345\pi\)
−0.225328 + 0.974283i \(0.572345\pi\)
\(468\) 0 0
\(469\) −644.000 −0.0634055
\(470\) 0 0
\(471\) −5082.00 −0.497168
\(472\) 0 0
\(473\) 9648.00 0.937876
\(474\) 0 0
\(475\) 24676.0 2.38361
\(476\) 0 0
\(477\) 4482.00 0.430224
\(478\) 0 0
\(479\) −8064.00 −0.769214 −0.384607 0.923080i \(-0.625663\pi\)
−0.384607 + 0.923080i \(0.625663\pi\)
\(480\) 0 0
\(481\) −13532.0 −1.28276
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −5148.00 −0.481977
\(486\) 0 0
\(487\) 16616.0 1.54608 0.773042 0.634355i \(-0.218734\pi\)
0.773042 + 0.634355i \(0.218734\pi\)
\(488\) 0 0
\(489\) 8796.00 0.813433
\(490\) 0 0
\(491\) 7140.00 0.656260 0.328130 0.944633i \(-0.393582\pi\)
0.328130 + 0.944633i \(0.393582\pi\)
\(492\) 0 0
\(493\) −4284.00 −0.391362
\(494\) 0 0
\(495\) 5832.00 0.529553
\(496\) 0 0
\(497\) −5040.00 −0.454879
\(498\) 0 0
\(499\) 9124.00 0.818530 0.409265 0.912416i \(-0.365785\pi\)
0.409265 + 0.912416i \(0.365785\pi\)
\(500\) 0 0
\(501\) 3528.00 0.314610
\(502\) 0 0
\(503\) −6552.00 −0.580794 −0.290397 0.956906i \(-0.593787\pi\)
−0.290397 + 0.956906i \(0.593787\pi\)
\(504\) 0 0
\(505\) −7452.00 −0.656653
\(506\) 0 0
\(507\) −3123.00 −0.273565
\(508\) 0 0
\(509\) −2790.00 −0.242956 −0.121478 0.992594i \(-0.538763\pi\)
−0.121478 + 0.992594i \(0.538763\pi\)
\(510\) 0 0
\(511\) −3514.00 −0.304208
\(512\) 0 0
\(513\) 3348.00 0.288144
\(514\) 0 0
\(515\) 1008.00 0.0862481
\(516\) 0 0
\(517\) 8640.00 0.734984
\(518\) 0 0
\(519\) −2610.00 −0.220744
\(520\) 0 0
\(521\) −14862.0 −1.24974 −0.624871 0.780728i \(-0.714849\pi\)
−0.624871 + 0.780728i \(0.714849\pi\)
\(522\) 0 0
\(523\) −17660.0 −1.47652 −0.738258 0.674518i \(-0.764351\pi\)
−0.738258 + 0.674518i \(0.764351\pi\)
\(524\) 0 0
\(525\) 4179.00 0.347403
\(526\) 0 0
\(527\) −6720.00 −0.555461
\(528\) 0 0
\(529\) −12167.0 −1.00000
\(530\) 0 0
\(531\) 1188.00 0.0970900
\(532\) 0 0
\(533\) −10812.0 −0.878649
\(534\) 0 0
\(535\) −216.000 −0.0174551
\(536\) 0 0
\(537\) 6948.00 0.558340
\(538\) 0 0
\(539\) 1764.00 0.140966
\(540\) 0 0
\(541\) 19834.0 1.57621 0.788106 0.615540i \(-0.211062\pi\)
0.788106 + 0.615540i \(0.211062\pi\)
\(542\) 0 0
\(543\) 318.000 0.0251320
\(544\) 0 0
\(545\) −26604.0 −2.09099
\(546\) 0 0
\(547\) −20972.0 −1.63930 −0.819651 0.572863i \(-0.805833\pi\)
−0.819651 + 0.572863i \(0.805833\pi\)
\(548\) 0 0
\(549\) −3582.00 −0.278463
\(550\) 0 0
\(551\) −12648.0 −0.977900
\(552\) 0 0
\(553\) −7168.00 −0.551201
\(554\) 0 0
\(555\) −21492.0 −1.64376
\(556\) 0 0
\(557\) −21174.0 −1.61072 −0.805360 0.592786i \(-0.798028\pi\)
−0.805360 + 0.592786i \(0.798028\pi\)
\(558\) 0 0
\(559\) 9112.00 0.689439
\(560\) 0 0
\(561\) 4536.00 0.341373
\(562\) 0 0
\(563\) 17772.0 1.33037 0.665187 0.746677i \(-0.268352\pi\)
0.665187 + 0.746677i \(0.268352\pi\)
\(564\) 0 0
\(565\) 7236.00 0.538798
\(566\) 0 0
\(567\) 567.000 0.0419961
\(568\) 0 0
\(569\) 8250.00 0.607835 0.303917 0.952698i \(-0.401705\pi\)
0.303917 + 0.952698i \(0.401705\pi\)
\(570\) 0 0
\(571\) −20756.0 −1.52121 −0.760606 0.649214i \(-0.775098\pi\)
−0.760606 + 0.649214i \(0.775098\pi\)
\(572\) 0 0
\(573\) −3384.00 −0.246717
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 2.00000 0.000144300 0 7.21500e−5 1.00000i \(-0.499977\pi\)
7.21500e−5 1.00000i \(0.499977\pi\)
\(578\) 0 0
\(579\) 12102.0 0.868639
\(580\) 0 0
\(581\) 1428.00 0.101968
\(582\) 0 0
\(583\) 17928.0 1.27359
\(584\) 0 0
\(585\) 5508.00 0.389278
\(586\) 0 0
\(587\) −26364.0 −1.85376 −0.926881 0.375354i \(-0.877521\pi\)
−0.926881 + 0.375354i \(0.877521\pi\)
\(588\) 0 0
\(589\) −19840.0 −1.38793
\(590\) 0 0
\(591\) 3942.00 0.274369
\(592\) 0 0
\(593\) 2298.00 0.159136 0.0795679 0.996829i \(-0.474646\pi\)
0.0795679 + 0.996829i \(0.474646\pi\)
\(594\) 0 0
\(595\) 5292.00 0.364623
\(596\) 0 0
\(597\) 15288.0 1.04807
\(598\) 0 0
\(599\) 3072.00 0.209547 0.104773 0.994496i \(-0.466588\pi\)
0.104773 + 0.994496i \(0.466588\pi\)
\(600\) 0 0
\(601\) 24554.0 1.66652 0.833260 0.552881i \(-0.186472\pi\)
0.833260 + 0.552881i \(0.186472\pi\)
\(602\) 0 0
\(603\) −828.000 −0.0559184
\(604\) 0 0
\(605\) −630.000 −0.0423358
\(606\) 0 0
\(607\) 16832.0 1.12552 0.562759 0.826621i \(-0.309740\pi\)
0.562759 + 0.826621i \(0.309740\pi\)
\(608\) 0 0
\(609\) −2142.00 −0.142526
\(610\) 0 0
\(611\) 8160.00 0.540292
\(612\) 0 0
\(613\) 2482.00 0.163535 0.0817676 0.996651i \(-0.473943\pi\)
0.0817676 + 0.996651i \(0.473943\pi\)
\(614\) 0 0
\(615\) −17172.0 −1.12592
\(616\) 0 0
\(617\) −15798.0 −1.03080 −0.515400 0.856950i \(-0.672357\pi\)
−0.515400 + 0.856950i \(0.672357\pi\)
\(618\) 0 0
\(619\) 15460.0 1.00386 0.501930 0.864908i \(-0.332623\pi\)
0.501930 + 0.864908i \(0.332623\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 2478.00 0.159356
\(624\) 0 0
\(625\) −899.000 −0.0575360
\(626\) 0 0
\(627\) 13392.0 0.852990
\(628\) 0 0
\(629\) −16716.0 −1.05964
\(630\) 0 0
\(631\) −7720.00 −0.487050 −0.243525 0.969895i \(-0.578304\pi\)
−0.243525 + 0.969895i \(0.578304\pi\)
\(632\) 0 0
\(633\) 9228.00 0.579431
\(634\) 0 0
\(635\) 23040.0 1.43987
\(636\) 0 0
\(637\) 1666.00 0.103625
\(638\) 0 0
\(639\) −6480.00 −0.401166
\(640\) 0 0
\(641\) −17262.0 −1.06366 −0.531832 0.846850i \(-0.678496\pi\)
−0.531832 + 0.846850i \(0.678496\pi\)
\(642\) 0 0
\(643\) 12220.0 0.749471 0.374735 0.927132i \(-0.377734\pi\)
0.374735 + 0.927132i \(0.377734\pi\)
\(644\) 0 0
\(645\) 14472.0 0.883464
\(646\) 0 0
\(647\) 13560.0 0.823955 0.411977 0.911194i \(-0.364838\pi\)
0.411977 + 0.911194i \(0.364838\pi\)
\(648\) 0 0
\(649\) 4752.00 0.287415
\(650\) 0 0
\(651\) −3360.00 −0.202287
\(652\) 0 0
\(653\) −23094.0 −1.38398 −0.691989 0.721908i \(-0.743265\pi\)
−0.691989 + 0.721908i \(0.743265\pi\)
\(654\) 0 0
\(655\) −31752.0 −1.89413
\(656\) 0 0
\(657\) −4518.00 −0.268286
\(658\) 0 0
\(659\) −22548.0 −1.33285 −0.666423 0.745574i \(-0.732175\pi\)
−0.666423 + 0.745574i \(0.732175\pi\)
\(660\) 0 0
\(661\) −17462.0 −1.02752 −0.513762 0.857933i \(-0.671748\pi\)
−0.513762 + 0.857933i \(0.671748\pi\)
\(662\) 0 0
\(663\) 4284.00 0.250945
\(664\) 0 0
\(665\) 15624.0 0.911087
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) −5664.00 −0.327329
\(670\) 0 0
\(671\) −14328.0 −0.824331
\(672\) 0 0
\(673\) −22462.0 −1.28655 −0.643274 0.765636i \(-0.722424\pi\)
−0.643274 + 0.765636i \(0.722424\pi\)
\(674\) 0 0
\(675\) 5373.00 0.306381
\(676\) 0 0
\(677\) 25554.0 1.45069 0.725347 0.688383i \(-0.241679\pi\)
0.725347 + 0.688383i \(0.241679\pi\)
\(678\) 0 0
\(679\) −2002.00 −0.113151
\(680\) 0 0
\(681\) 14148.0 0.796112
\(682\) 0 0
\(683\) −9276.00 −0.519672 −0.259836 0.965653i \(-0.583669\pi\)
−0.259836 + 0.965653i \(0.583669\pi\)
\(684\) 0 0
\(685\) −42444.0 −2.36745
\(686\) 0 0
\(687\) 5070.00 0.281561
\(688\) 0 0
\(689\) 16932.0 0.936223
\(690\) 0 0
\(691\) −27380.0 −1.50736 −0.753679 0.657243i \(-0.771723\pi\)
−0.753679 + 0.657243i \(0.771723\pi\)
\(692\) 0 0
\(693\) 2268.00 0.124321
\(694\) 0 0
\(695\) 936.000 0.0510856
\(696\) 0 0
\(697\) −13356.0 −0.725817
\(698\) 0 0
\(699\) 414.000 0.0224019
\(700\) 0 0
\(701\) −25830.0 −1.39171 −0.695853 0.718184i \(-0.744973\pi\)
−0.695853 + 0.718184i \(0.744973\pi\)
\(702\) 0 0
\(703\) −49352.0 −2.64772
\(704\) 0 0
\(705\) 12960.0 0.692343
\(706\) 0 0
\(707\) −2898.00 −0.154159
\(708\) 0 0
\(709\) 6226.00 0.329792 0.164896 0.986311i \(-0.447271\pi\)
0.164896 + 0.986311i \(0.447271\pi\)
\(710\) 0 0
\(711\) −9216.00 −0.486114
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 22032.0 1.15238
\(716\) 0 0
\(717\) 5688.00 0.296265
\(718\) 0 0
\(719\) −15072.0 −0.781767 −0.390884 0.920440i \(-0.627831\pi\)
−0.390884 + 0.920440i \(0.627831\pi\)
\(720\) 0 0
\(721\) 392.000 0.0202480
\(722\) 0 0
\(723\) −10794.0 −0.555233
\(724\) 0 0
\(725\) −20298.0 −1.03979
\(726\) 0 0
\(727\) −32920.0 −1.67942 −0.839708 0.543038i \(-0.817274\pi\)
−0.839708 + 0.543038i \(0.817274\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 11256.0 0.569519
\(732\) 0 0
\(733\) 6946.00 0.350009 0.175004 0.984568i \(-0.444006\pi\)
0.175004 + 0.984568i \(0.444006\pi\)
\(734\) 0 0
\(735\) 2646.00 0.132788
\(736\) 0 0
\(737\) −3312.00 −0.165535
\(738\) 0 0
\(739\) 2356.00 0.117276 0.0586379 0.998279i \(-0.481324\pi\)
0.0586379 + 0.998279i \(0.481324\pi\)
\(740\) 0 0
\(741\) 12648.0 0.627039
\(742\) 0 0
\(743\) −23520.0 −1.16133 −0.580663 0.814144i \(-0.697207\pi\)
−0.580663 + 0.814144i \(0.697207\pi\)
\(744\) 0 0
\(745\) 31428.0 1.54555
\(746\) 0 0
\(747\) 1836.00 0.0899273
\(748\) 0 0
\(749\) −84.0000 −0.00409785
\(750\) 0 0
\(751\) 3008.00 0.146156 0.0730782 0.997326i \(-0.476718\pi\)
0.0730782 + 0.997326i \(0.476718\pi\)
\(752\) 0 0
\(753\) 9180.00 0.444273
\(754\) 0 0
\(755\) −4176.00 −0.201298
\(756\) 0 0
\(757\) 20770.0 0.997224 0.498612 0.866825i \(-0.333843\pi\)
0.498612 + 0.866825i \(0.333843\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 11538.0 0.549609 0.274804 0.961500i \(-0.411387\pi\)
0.274804 + 0.961500i \(0.411387\pi\)
\(762\) 0 0
\(763\) −10346.0 −0.490892
\(764\) 0 0
\(765\) 6804.00 0.321568
\(766\) 0 0
\(767\) 4488.00 0.211281
\(768\) 0 0
\(769\) 8498.00 0.398499 0.199249 0.979949i \(-0.436150\pi\)
0.199249 + 0.979949i \(0.436150\pi\)
\(770\) 0 0
\(771\) −20466.0 −0.955986
\(772\) 0 0
\(773\) 32322.0 1.50393 0.751967 0.659200i \(-0.229105\pi\)
0.751967 + 0.659200i \(0.229105\pi\)
\(774\) 0 0
\(775\) −31840.0 −1.47578
\(776\) 0 0
\(777\) −8358.00 −0.385896
\(778\) 0 0
\(779\) −39432.0 −1.81360
\(780\) 0 0
\(781\) −25920.0 −1.18757
\(782\) 0 0
\(783\) −2754.00 −0.125696
\(784\) 0 0
\(785\) −30492.0 −1.38638
\(786\) 0 0
\(787\) −26228.0 −1.18796 −0.593982 0.804479i \(-0.702445\pi\)
−0.593982 + 0.804479i \(0.702445\pi\)
\(788\) 0 0
\(789\) 7776.00 0.350866
\(790\) 0 0
\(791\) 2814.00 0.126491
\(792\) 0 0
\(793\) −13532.0 −0.605972
\(794\) 0 0
\(795\) 26892.0 1.19970
\(796\) 0 0
\(797\) 43338.0 1.92611 0.963056 0.269302i \(-0.0867931\pi\)
0.963056 + 0.269302i \(0.0867931\pi\)
\(798\) 0 0
\(799\) 10080.0 0.446314
\(800\) 0 0
\(801\) 3186.00 0.140539
\(802\) 0 0
\(803\) −18072.0 −0.794206
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −24642.0 −1.07489
\(808\) 0 0
\(809\) −28902.0 −1.25604 −0.628022 0.778195i \(-0.716135\pi\)
−0.628022 + 0.778195i \(0.716135\pi\)
\(810\) 0 0
\(811\) −27164.0 −1.17615 −0.588075 0.808807i \(-0.700114\pi\)
−0.588075 + 0.808807i \(0.700114\pi\)
\(812\) 0 0
\(813\) −16032.0 −0.691595
\(814\) 0 0
\(815\) 52776.0 2.26830
\(816\) 0 0
\(817\) 33232.0 1.42306
\(818\) 0 0
\(819\) 2142.00 0.0913889
\(820\) 0 0
\(821\) 17202.0 0.731247 0.365624 0.930763i \(-0.380856\pi\)
0.365624 + 0.930763i \(0.380856\pi\)
\(822\) 0 0
\(823\) −5992.00 −0.253789 −0.126894 0.991916i \(-0.540501\pi\)
−0.126894 + 0.991916i \(0.540501\pi\)
\(824\) 0 0
\(825\) 21492.0 0.906976
\(826\) 0 0
\(827\) −25884.0 −1.08836 −0.544181 0.838968i \(-0.683159\pi\)
−0.544181 + 0.838968i \(0.683159\pi\)
\(828\) 0 0
\(829\) 1474.00 0.0617541 0.0308770 0.999523i \(-0.490170\pi\)
0.0308770 + 0.999523i \(0.490170\pi\)
\(830\) 0 0
\(831\) 19542.0 0.815770
\(832\) 0 0
\(833\) 2058.00 0.0856008
\(834\) 0 0
\(835\) 21168.0 0.877304
\(836\) 0 0
\(837\) −4320.00 −0.178400
\(838\) 0 0
\(839\) 33528.0 1.37964 0.689818 0.723983i \(-0.257690\pi\)
0.689818 + 0.723983i \(0.257690\pi\)
\(840\) 0 0
\(841\) −13985.0 −0.573414
\(842\) 0 0
\(843\) 19854.0 0.811160
\(844\) 0 0
\(845\) −18738.0 −0.762848
\(846\) 0 0
\(847\) −245.000 −0.00993896
\(848\) 0 0
\(849\) −9780.00 −0.395346
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −1190.00 −0.0477665 −0.0238832 0.999715i \(-0.507603\pi\)
−0.0238832 + 0.999715i \(0.507603\pi\)
\(854\) 0 0
\(855\) 20088.0 0.803503
\(856\) 0 0
\(857\) 34578.0 1.37825 0.689126 0.724642i \(-0.257995\pi\)
0.689126 + 0.724642i \(0.257995\pi\)
\(858\) 0 0
\(859\) 44404.0 1.76373 0.881865 0.471501i \(-0.156288\pi\)
0.881865 + 0.471501i \(0.156288\pi\)
\(860\) 0 0
\(861\) −6678.00 −0.264327
\(862\) 0 0
\(863\) −38328.0 −1.51182 −0.755910 0.654676i \(-0.772805\pi\)
−0.755910 + 0.654676i \(0.772805\pi\)
\(864\) 0 0
\(865\) −15660.0 −0.615556
\(866\) 0 0
\(867\) −9447.00 −0.370054
\(868\) 0 0
\(869\) −36864.0 −1.43904
\(870\) 0 0
\(871\) −3128.00 −0.121686
\(872\) 0 0
\(873\) −2574.00 −0.0997900
\(874\) 0 0
\(875\) 9324.00 0.360239
\(876\) 0 0
\(877\) 38842.0 1.49555 0.747777 0.663950i \(-0.231121\pi\)
0.747777 + 0.663950i \(0.231121\pi\)
\(878\) 0 0
\(879\) −15354.0 −0.589167
\(880\) 0 0
\(881\) −35046.0 −1.34022 −0.670108 0.742264i \(-0.733752\pi\)
−0.670108 + 0.742264i \(0.733752\pi\)
\(882\) 0 0
\(883\) −14204.0 −0.541339 −0.270670 0.962672i \(-0.587245\pi\)
−0.270670 + 0.962672i \(0.587245\pi\)
\(884\) 0 0
\(885\) 7128.00 0.270740
\(886\) 0 0
\(887\) −26136.0 −0.989359 −0.494679 0.869076i \(-0.664714\pi\)
−0.494679 + 0.869076i \(0.664714\pi\)
\(888\) 0 0
\(889\) 8960.00 0.338030
\(890\) 0 0
\(891\) 2916.00 0.109640
\(892\) 0 0
\(893\) 29760.0 1.11521
\(894\) 0 0
\(895\) 41688.0 1.55696
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 16320.0 0.605453
\(900\) 0 0
\(901\) 20916.0 0.773377
\(902\) 0 0
\(903\) 5628.00 0.207407
\(904\) 0 0
\(905\) 1908.00 0.0700818
\(906\) 0 0
\(907\) 9052.00 0.331386 0.165693 0.986177i \(-0.447014\pi\)
0.165693 + 0.986177i \(0.447014\pi\)
\(908\) 0 0
\(909\) −3726.00 −0.135956
\(910\) 0 0
\(911\) 5016.00 0.182423 0.0912116 0.995832i \(-0.470926\pi\)
0.0912116 + 0.995832i \(0.470926\pi\)
\(912\) 0 0
\(913\) 7344.00 0.266211
\(914\) 0 0
\(915\) −21492.0 −0.776507
\(916\) 0 0
\(917\) −12348.0 −0.444675
\(918\) 0 0
\(919\) 44552.0 1.59917 0.799584 0.600555i \(-0.205054\pi\)
0.799584 + 0.600555i \(0.205054\pi\)
\(920\) 0 0
\(921\) −1356.00 −0.0485144
\(922\) 0 0
\(923\) −24480.0 −0.872989
\(924\) 0 0
\(925\) −79202.0 −2.81529
\(926\) 0 0
\(927\) 504.000 0.0178571
\(928\) 0 0
\(929\) 24234.0 0.855858 0.427929 0.903812i \(-0.359243\pi\)
0.427929 + 0.903812i \(0.359243\pi\)
\(930\) 0 0
\(931\) 6076.00 0.213891
\(932\) 0 0
\(933\) 15048.0 0.528027
\(934\) 0 0
\(935\) 27216.0 0.951934
\(936\) 0 0
\(937\) −13894.0 −0.484415 −0.242208 0.970224i \(-0.577872\pi\)
−0.242208 + 0.970224i \(0.577872\pi\)
\(938\) 0 0
\(939\) 16206.0 0.563219
\(940\) 0 0
\(941\) −46758.0 −1.61984 −0.809919 0.586542i \(-0.800489\pi\)
−0.809919 + 0.586542i \(0.800489\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 3402.00 0.117108
\(946\) 0 0
\(947\) −13812.0 −0.473949 −0.236974 0.971516i \(-0.576156\pi\)
−0.236974 + 0.971516i \(0.576156\pi\)
\(948\) 0 0
\(949\) −17068.0 −0.583826
\(950\) 0 0
\(951\) −30258.0 −1.03174
\(952\) 0 0
\(953\) −58518.0 −1.98907 −0.994535 0.104402i \(-0.966707\pi\)
−0.994535 + 0.104402i \(0.966707\pi\)
\(954\) 0 0
\(955\) −20304.0 −0.687981
\(956\) 0 0
\(957\) −11016.0 −0.372097
\(958\) 0 0
\(959\) −16506.0 −0.555794
\(960\) 0 0
\(961\) −4191.00 −0.140680
\(962\) 0 0
\(963\) −108.000 −0.00361397
\(964\) 0 0
\(965\) 72612.0 2.42224
\(966\) 0 0
\(967\) 19640.0 0.653133 0.326567 0.945174i \(-0.394108\pi\)
0.326567 + 0.945174i \(0.394108\pi\)
\(968\) 0 0
\(969\) 15624.0 0.517972
\(970\) 0 0
\(971\) 58308.0 1.92708 0.963539 0.267568i \(-0.0862200\pi\)
0.963539 + 0.267568i \(0.0862200\pi\)
\(972\) 0 0
\(973\) 364.000 0.0119931
\(974\) 0 0
\(975\) 20298.0 0.666724
\(976\) 0 0
\(977\) −23550.0 −0.771168 −0.385584 0.922673i \(-0.626000\pi\)
−0.385584 + 0.922673i \(0.626000\pi\)
\(978\) 0 0
\(979\) 12744.0 0.416037
\(980\) 0 0
\(981\) −13302.0 −0.432926
\(982\) 0 0
\(983\) 15768.0 0.511619 0.255809 0.966727i \(-0.417658\pi\)
0.255809 + 0.966727i \(0.417658\pi\)
\(984\) 0 0
\(985\) 23652.0 0.765092
\(986\) 0 0
\(987\) 5040.00 0.162538
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 35264.0 1.13037 0.565186 0.824964i \(-0.308805\pi\)
0.565186 + 0.824964i \(0.308805\pi\)
\(992\) 0 0
\(993\) 24132.0 0.771204
\(994\) 0 0
\(995\) 91728.0 2.92259
\(996\) 0 0
\(997\) 29338.0 0.931940 0.465970 0.884801i \(-0.345706\pi\)
0.465970 + 0.884801i \(0.345706\pi\)
\(998\) 0 0
\(999\) −10746.0 −0.340329
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 1344.4.a.ba.1.1 1
4.3 odd 2 1344.4.a.n.1.1 1
8.3 odd 2 336.4.a.f.1.1 1
8.5 even 2 21.4.a.a.1.1 1
24.5 odd 2 63.4.a.c.1.1 1
24.11 even 2 1008.4.a.v.1.1 1
40.13 odd 4 525.4.d.c.274.2 2
40.29 even 2 525.4.a.g.1.1 1
40.37 odd 4 525.4.d.c.274.1 2
56.5 odd 6 147.4.e.g.67.1 2
56.13 odd 2 147.4.a.c.1.1 1
56.27 even 2 2352.4.a.r.1.1 1
56.37 even 6 147.4.e.i.67.1 2
56.45 odd 6 147.4.e.g.79.1 2
56.53 even 6 147.4.e.i.79.1 2
120.29 odd 2 1575.4.a.b.1.1 1
168.5 even 6 441.4.e.d.361.1 2
168.53 odd 6 441.4.e.b.226.1 2
168.101 even 6 441.4.e.d.226.1 2
168.125 even 2 441.4.a.j.1.1 1
168.149 odd 6 441.4.e.b.361.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.4.a.a.1.1 1 8.5 even 2
63.4.a.c.1.1 1 24.5 odd 2
147.4.a.c.1.1 1 56.13 odd 2
147.4.e.g.67.1 2 56.5 odd 6
147.4.e.g.79.1 2 56.45 odd 6
147.4.e.i.67.1 2 56.37 even 6
147.4.e.i.79.1 2 56.53 even 6
336.4.a.f.1.1 1 8.3 odd 2
441.4.a.j.1.1 1 168.125 even 2
441.4.e.b.226.1 2 168.53 odd 6
441.4.e.b.361.1 2 168.149 odd 6
441.4.e.d.226.1 2 168.101 even 6
441.4.e.d.361.1 2 168.5 even 6
525.4.a.g.1.1 1 40.29 even 2
525.4.d.c.274.1 2 40.37 odd 4
525.4.d.c.274.2 2 40.13 odd 4
1008.4.a.v.1.1 1 24.11 even 2
1344.4.a.n.1.1 1 4.3 odd 2
1344.4.a.ba.1.1 1 1.1 even 1 trivial
1575.4.a.b.1.1 1 120.29 odd 2
2352.4.a.r.1.1 1 56.27 even 2