Properties

Label 1176.4.a.bc.1.4
Level $1176$
Weight $4$
Character 1176.1
Self dual yes
Analytic conductor $69.386$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1176 = 2^{3} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1176.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(69.3862461668\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.145408.2
Defining polynomial: \(x^{4} - 24 x^{2} + 142\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-3.25358\) of defining polynomial
Character \(\chi\) \(=\) 1176.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.00000 q^{3} +11.4452 q^{5} +9.00000 q^{9} +O(q^{10})\) \(q+3.00000 q^{3} +11.4452 q^{5} +9.00000 q^{9} -52.5886 q^{11} -5.48395 q^{13} +34.3355 q^{15} +85.3636 q^{17} -110.620 q^{19} -209.332 q^{23} +5.99160 q^{25} +27.0000 q^{27} -132.843 q^{29} -49.3139 q^{31} -157.766 q^{33} +160.351 q^{37} -16.4518 q^{39} +138.120 q^{41} -365.073 q^{43} +103.006 q^{45} +131.370 q^{47} +256.091 q^{51} -561.227 q^{53} -601.885 q^{55} -331.859 q^{57} -436.354 q^{59} -291.961 q^{61} -62.7646 q^{65} -593.683 q^{67} -627.995 q^{69} +775.073 q^{71} +330.907 q^{73} +17.9748 q^{75} +243.473 q^{79} +81.0000 q^{81} +332.477 q^{83} +976.999 q^{85} -398.529 q^{87} -979.115 q^{89} -147.942 q^{93} -1266.06 q^{95} +466.790 q^{97} -473.298 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 12q^{3} - 8q^{5} + 36q^{9} + O(q^{10}) \) \( 4q + 12q^{3} - 8q^{5} + 36q^{9} - 40q^{11} - 48q^{13} - 24q^{15} + 152q^{17} - 224q^{19} - 8q^{23} - 28q^{25} + 108q^{27} - 144q^{29} - 400q^{31} - 120q^{33} - 304q^{37} - 144q^{39} + 152q^{41} + 160q^{43} - 72q^{45} - 544q^{47} + 456q^{51} - 1320q^{53} - 16q^{55} - 672q^{57} - 1040q^{59} - 896q^{61} - 648q^{65} - 416q^{67} - 24q^{69} + 248q^{71} + 752q^{73} - 84q^{75} + 864q^{79} + 324q^{81} - 1456q^{83} - 1608q^{85} - 432q^{87} - 2936q^{89} - 1200q^{93} - 80q^{95} - 144q^{97} - 360q^{99} + O(q^{100}) \)

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
\(5\) 11.4452 1.02369 0.511843 0.859079i \(-0.328963\pi\)
0.511843 + 0.859079i \(0.328963\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −52.5886 −1.44146 −0.720730 0.693216i \(-0.756193\pi\)
−0.720730 + 0.693216i \(0.756193\pi\)
\(12\) 0 0
\(13\) −5.48395 −0.116998 −0.0584990 0.998287i \(-0.518631\pi\)
−0.0584990 + 0.998287i \(0.518631\pi\)
\(14\) 0 0
\(15\) 34.3355 0.591025
\(16\) 0 0
\(17\) 85.3636 1.21786 0.608932 0.793222i \(-0.291598\pi\)
0.608932 + 0.793222i \(0.291598\pi\)
\(18\) 0 0
\(19\) −110.620 −1.33568 −0.667839 0.744306i \(-0.732780\pi\)
−0.667839 + 0.744306i \(0.732780\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −209.332 −1.89777 −0.948884 0.315624i \(-0.897786\pi\)
−0.948884 + 0.315624i \(0.897786\pi\)
\(24\) 0 0
\(25\) 5.99160 0.0479328
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) −132.843 −0.850632 −0.425316 0.905045i \(-0.639837\pi\)
−0.425316 + 0.905045i \(0.639837\pi\)
\(30\) 0 0
\(31\) −49.3139 −0.285711 −0.142856 0.989744i \(-0.545628\pi\)
−0.142856 + 0.989744i \(0.545628\pi\)
\(32\) 0 0
\(33\) −157.766 −0.832228
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 160.351 0.712474 0.356237 0.934396i \(-0.384060\pi\)
0.356237 + 0.934396i \(0.384060\pi\)
\(38\) 0 0
\(39\) −16.4518 −0.0675488
\(40\) 0 0
\(41\) 138.120 0.526116 0.263058 0.964780i \(-0.415269\pi\)
0.263058 + 0.964780i \(0.415269\pi\)
\(42\) 0 0
\(43\) −365.073 −1.29472 −0.647362 0.762183i \(-0.724128\pi\)
−0.647362 + 0.762183i \(0.724128\pi\)
\(44\) 0 0
\(45\) 103.006 0.341229
\(46\) 0 0
\(47\) 131.370 0.407708 0.203854 0.979001i \(-0.434653\pi\)
0.203854 + 0.979001i \(0.434653\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 256.091 0.703135
\(52\) 0 0
\(53\) −561.227 −1.45454 −0.727268 0.686353i \(-0.759210\pi\)
−0.727268 + 0.686353i \(0.759210\pi\)
\(54\) 0 0
\(55\) −601.885 −1.47560
\(56\) 0 0
\(57\) −331.859 −0.771154
\(58\) 0 0
\(59\) −436.354 −0.962854 −0.481427 0.876486i \(-0.659881\pi\)
−0.481427 + 0.876486i \(0.659881\pi\)
\(60\) 0 0
\(61\) −291.961 −0.612816 −0.306408 0.951900i \(-0.599127\pi\)
−0.306408 + 0.951900i \(0.599127\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −62.7646 −0.119769
\(66\) 0 0
\(67\) −593.683 −1.08254 −0.541268 0.840850i \(-0.682056\pi\)
−0.541268 + 0.840850i \(0.682056\pi\)
\(68\) 0 0
\(69\) −627.995 −1.09568
\(70\) 0 0
\(71\) 775.073 1.29555 0.647776 0.761831i \(-0.275699\pi\)
0.647776 + 0.761831i \(0.275699\pi\)
\(72\) 0 0
\(73\) 330.907 0.530544 0.265272 0.964174i \(-0.414538\pi\)
0.265272 + 0.964174i \(0.414538\pi\)
\(74\) 0 0
\(75\) 17.9748 0.0276740
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 243.473 0.346745 0.173373 0.984856i \(-0.444533\pi\)
0.173373 + 0.984856i \(0.444533\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 332.477 0.439688 0.219844 0.975535i \(-0.429445\pi\)
0.219844 + 0.975535i \(0.429445\pi\)
\(84\) 0 0
\(85\) 976.999 1.24671
\(86\) 0 0
\(87\) −398.529 −0.491113
\(88\) 0 0
\(89\) −979.115 −1.16614 −0.583068 0.812424i \(-0.698148\pi\)
−0.583068 + 0.812424i \(0.698148\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −147.942 −0.164955
\(94\) 0 0
\(95\) −1266.06 −1.36731
\(96\) 0 0
\(97\) 466.790 0.488612 0.244306 0.969698i \(-0.421440\pi\)
0.244306 + 0.969698i \(0.421440\pi\)
\(98\) 0 0
\(99\) −473.298 −0.480487
\(100\) 0 0
\(101\) 866.429 0.853593 0.426796 0.904348i \(-0.359642\pi\)
0.426796 + 0.904348i \(0.359642\pi\)
\(102\) 0 0
\(103\) 926.755 0.886562 0.443281 0.896383i \(-0.353814\pi\)
0.443281 + 0.896383i \(0.353814\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1046.38 0.945395 0.472698 0.881225i \(-0.343280\pi\)
0.472698 + 0.881225i \(0.343280\pi\)
\(108\) 0 0
\(109\) −486.397 −0.427416 −0.213708 0.976898i \(-0.568554\pi\)
−0.213708 + 0.976898i \(0.568554\pi\)
\(110\) 0 0
\(111\) 481.053 0.411347
\(112\) 0 0
\(113\) 1955.03 1.62755 0.813777 0.581178i \(-0.197408\pi\)
0.813777 + 0.581178i \(0.197408\pi\)
\(114\) 0 0
\(115\) −2395.83 −1.94272
\(116\) 0 0
\(117\) −49.3555 −0.0389993
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 1434.56 1.07781
\(122\) 0 0
\(123\) 414.360 0.303753
\(124\) 0 0
\(125\) −1362.07 −0.974618
\(126\) 0 0
\(127\) 1255.53 0.877246 0.438623 0.898671i \(-0.355466\pi\)
0.438623 + 0.898671i \(0.355466\pi\)
\(128\) 0 0
\(129\) −1095.22 −0.747509
\(130\) 0 0
\(131\) −1029.41 −0.686561 −0.343281 0.939233i \(-0.611538\pi\)
−0.343281 + 0.939233i \(0.611538\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 309.019 0.197008
\(136\) 0 0
\(137\) −1931.36 −1.20443 −0.602217 0.798332i \(-0.705716\pi\)
−0.602217 + 0.798332i \(0.705716\pi\)
\(138\) 0 0
\(139\) −2926.04 −1.78549 −0.892747 0.450559i \(-0.851225\pi\)
−0.892747 + 0.450559i \(0.851225\pi\)
\(140\) 0 0
\(141\) 394.110 0.235390
\(142\) 0 0
\(143\) 288.393 0.168648
\(144\) 0 0
\(145\) −1520.41 −0.870780
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 131.809 0.0724712 0.0362356 0.999343i \(-0.488463\pi\)
0.0362356 + 0.999343i \(0.488463\pi\)
\(150\) 0 0
\(151\) 177.956 0.0959066 0.0479533 0.998850i \(-0.484730\pi\)
0.0479533 + 0.998850i \(0.484730\pi\)
\(152\) 0 0
\(153\) 768.272 0.405955
\(154\) 0 0
\(155\) −564.406 −0.292479
\(156\) 0 0
\(157\) −3607.47 −1.83381 −0.916903 0.399111i \(-0.869319\pi\)
−0.916903 + 0.399111i \(0.869319\pi\)
\(158\) 0 0
\(159\) −1683.68 −0.839777
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 1540.88 0.740438 0.370219 0.928945i \(-0.379283\pi\)
0.370219 + 0.928945i \(0.379283\pi\)
\(164\) 0 0
\(165\) −1805.66 −0.851940
\(166\) 0 0
\(167\) 761.798 0.352992 0.176496 0.984301i \(-0.443524\pi\)
0.176496 + 0.984301i \(0.443524\pi\)
\(168\) 0 0
\(169\) −2166.93 −0.986311
\(170\) 0 0
\(171\) −995.576 −0.445226
\(172\) 0 0
\(173\) −2691.32 −1.18276 −0.591379 0.806394i \(-0.701416\pi\)
−0.591379 + 0.806394i \(0.701416\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −1309.06 −0.555904
\(178\) 0 0
\(179\) 1123.25 0.469027 0.234513 0.972113i \(-0.424650\pi\)
0.234513 + 0.972113i \(0.424650\pi\)
\(180\) 0 0
\(181\) 4221.85 1.73374 0.866872 0.498531i \(-0.166127\pi\)
0.866872 + 0.498531i \(0.166127\pi\)
\(182\) 0 0
\(183\) −875.882 −0.353809
\(184\) 0 0
\(185\) 1835.24 0.729350
\(186\) 0 0
\(187\) −4489.15 −1.75550
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 3293.81 1.24781 0.623904 0.781501i \(-0.285545\pi\)
0.623904 + 0.781501i \(0.285545\pi\)
\(192\) 0 0
\(193\) −4623.91 −1.72454 −0.862270 0.506448i \(-0.830958\pi\)
−0.862270 + 0.506448i \(0.830958\pi\)
\(194\) 0 0
\(195\) −188.294 −0.0691487
\(196\) 0 0
\(197\) 2644.18 0.956293 0.478147 0.878280i \(-0.341309\pi\)
0.478147 + 0.878280i \(0.341309\pi\)
\(198\) 0 0
\(199\) −1382.33 −0.492417 −0.246209 0.969217i \(-0.579185\pi\)
−0.246209 + 0.969217i \(0.579185\pi\)
\(200\) 0 0
\(201\) −1781.05 −0.625002
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 1580.81 0.538577
\(206\) 0 0
\(207\) −1883.99 −0.632590
\(208\) 0 0
\(209\) 5817.33 1.92533
\(210\) 0 0
\(211\) 4533.28 1.47907 0.739535 0.673118i \(-0.235045\pi\)
0.739535 + 0.673118i \(0.235045\pi\)
\(212\) 0 0
\(213\) 2325.22 0.747987
\(214\) 0 0
\(215\) −4178.32 −1.32539
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 992.721 0.306310
\(220\) 0 0
\(221\) −468.129 −0.142488
\(222\) 0 0
\(223\) −3052.46 −0.916628 −0.458314 0.888790i \(-0.651547\pi\)
−0.458314 + 0.888790i \(0.651547\pi\)
\(224\) 0 0
\(225\) 53.9244 0.0159776
\(226\) 0 0
\(227\) −5194.86 −1.51892 −0.759461 0.650553i \(-0.774537\pi\)
−0.759461 + 0.650553i \(0.774537\pi\)
\(228\) 0 0
\(229\) 5497.05 1.58627 0.793133 0.609048i \(-0.208448\pi\)
0.793133 + 0.609048i \(0.208448\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −5457.17 −1.53438 −0.767191 0.641419i \(-0.778346\pi\)
−0.767191 + 0.641419i \(0.778346\pi\)
\(234\) 0 0
\(235\) 1503.55 0.417365
\(236\) 0 0
\(237\) 730.420 0.200194
\(238\) 0 0
\(239\) −5622.17 −1.52162 −0.760812 0.648972i \(-0.775199\pi\)
−0.760812 + 0.648972i \(0.775199\pi\)
\(240\) 0 0
\(241\) 4083.02 1.09133 0.545665 0.838003i \(-0.316277\pi\)
0.545665 + 0.838003i \(0.316277\pi\)
\(242\) 0 0
\(243\) 243.000 0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 606.632 0.156271
\(248\) 0 0
\(249\) 997.430 0.253854
\(250\) 0 0
\(251\) −1751.99 −0.440576 −0.220288 0.975435i \(-0.570700\pi\)
−0.220288 + 0.975435i \(0.570700\pi\)
\(252\) 0 0
\(253\) 11008.5 2.73556
\(254\) 0 0
\(255\) 2931.00 0.719789
\(256\) 0 0
\(257\) −3825.26 −0.928457 −0.464228 0.885716i \(-0.653668\pi\)
−0.464228 + 0.885716i \(0.653668\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −1195.59 −0.283544
\(262\) 0 0
\(263\) −5272.57 −1.23620 −0.618099 0.786100i \(-0.712097\pi\)
−0.618099 + 0.786100i \(0.712097\pi\)
\(264\) 0 0
\(265\) −6423.33 −1.48899
\(266\) 0 0
\(267\) −2937.35 −0.673268
\(268\) 0 0
\(269\) −4535.04 −1.02790 −0.513952 0.857819i \(-0.671819\pi\)
−0.513952 + 0.857819i \(0.671819\pi\)
\(270\) 0 0
\(271\) −1439.10 −0.322580 −0.161290 0.986907i \(-0.551565\pi\)
−0.161290 + 0.986907i \(0.551565\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −315.090 −0.0690932
\(276\) 0 0
\(277\) −5984.85 −1.29818 −0.649088 0.760713i \(-0.724849\pi\)
−0.649088 + 0.760713i \(0.724849\pi\)
\(278\) 0 0
\(279\) −443.826 −0.0952371
\(280\) 0 0
\(281\) −2647.87 −0.562131 −0.281066 0.959689i \(-0.590688\pi\)
−0.281066 + 0.959689i \(0.590688\pi\)
\(282\) 0 0
\(283\) −400.822 −0.0841922 −0.0420961 0.999114i \(-0.513404\pi\)
−0.0420961 + 0.999114i \(0.513404\pi\)
\(284\) 0 0
\(285\) −3798.17 −0.789419
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 2373.94 0.483195
\(290\) 0 0
\(291\) 1400.37 0.282100
\(292\) 0 0
\(293\) −5307.12 −1.05817 −0.529087 0.848567i \(-0.677466\pi\)
−0.529087 + 0.848567i \(0.677466\pi\)
\(294\) 0 0
\(295\) −4994.14 −0.985660
\(296\) 0 0
\(297\) −1419.89 −0.277409
\(298\) 0 0
\(299\) 1147.96 0.222035
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 2599.29 0.492822
\(304\) 0 0
\(305\) −3341.54 −0.627331
\(306\) 0 0
\(307\) 4130.45 0.767873 0.383936 0.923360i \(-0.374568\pi\)
0.383936 + 0.923360i \(0.374568\pi\)
\(308\) 0 0
\(309\) 2780.26 0.511857
\(310\) 0 0
\(311\) −7653.87 −1.39553 −0.697767 0.716325i \(-0.745823\pi\)
−0.697767 + 0.716325i \(0.745823\pi\)
\(312\) 0 0
\(313\) 8546.27 1.54333 0.771667 0.636026i \(-0.219423\pi\)
0.771667 + 0.636026i \(0.219423\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 7643.65 1.35429 0.677145 0.735850i \(-0.263217\pi\)
0.677145 + 0.735850i \(0.263217\pi\)
\(318\) 0 0
\(319\) 6986.03 1.22615
\(320\) 0 0
\(321\) 3139.14 0.545824
\(322\) 0 0
\(323\) −9442.88 −1.62667
\(324\) 0 0
\(325\) −32.8576 −0.00560804
\(326\) 0 0
\(327\) −1459.19 −0.246769
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 7550.18 1.25376 0.626881 0.779115i \(-0.284331\pi\)
0.626881 + 0.779115i \(0.284331\pi\)
\(332\) 0 0
\(333\) 1443.16 0.237491
\(334\) 0 0
\(335\) −6794.79 −1.10818
\(336\) 0 0
\(337\) 9391.92 1.51813 0.759066 0.651014i \(-0.225656\pi\)
0.759066 + 0.651014i \(0.225656\pi\)
\(338\) 0 0
\(339\) 5865.08 0.939668
\(340\) 0 0
\(341\) 2593.35 0.411841
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −7187.50 −1.12163
\(346\) 0 0
\(347\) 1005.66 0.155580 0.0777902 0.996970i \(-0.475214\pi\)
0.0777902 + 0.996970i \(0.475214\pi\)
\(348\) 0 0
\(349\) −3109.75 −0.476966 −0.238483 0.971147i \(-0.576650\pi\)
−0.238483 + 0.971147i \(0.576650\pi\)
\(350\) 0 0
\(351\) −148.067 −0.0225163
\(352\) 0 0
\(353\) 7449.10 1.12316 0.561580 0.827423i \(-0.310194\pi\)
0.561580 + 0.827423i \(0.310194\pi\)
\(354\) 0 0
\(355\) 8870.83 1.32624
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −6427.61 −0.944948 −0.472474 0.881344i \(-0.656639\pi\)
−0.472474 + 0.881344i \(0.656639\pi\)
\(360\) 0 0
\(361\) 5377.69 0.784034
\(362\) 0 0
\(363\) 4303.69 0.622273
\(364\) 0 0
\(365\) 3787.28 0.543111
\(366\) 0 0
\(367\) −10767.3 −1.53147 −0.765734 0.643157i \(-0.777624\pi\)
−0.765734 + 0.643157i \(0.777624\pi\)
\(368\) 0 0
\(369\) 1243.08 0.175372
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −3601.03 −0.499877 −0.249938 0.968262i \(-0.580410\pi\)
−0.249938 + 0.968262i \(0.580410\pi\)
\(374\) 0 0
\(375\) −4086.21 −0.562696
\(376\) 0 0
\(377\) 728.504 0.0995222
\(378\) 0 0
\(379\) 6832.02 0.925956 0.462978 0.886370i \(-0.346781\pi\)
0.462978 + 0.886370i \(0.346781\pi\)
\(380\) 0 0
\(381\) 3766.59 0.506478
\(382\) 0 0
\(383\) 3339.87 0.445586 0.222793 0.974866i \(-0.428483\pi\)
0.222793 + 0.974866i \(0.428483\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −3285.66 −0.431575
\(388\) 0 0
\(389\) 4701.31 0.612765 0.306383 0.951908i \(-0.400881\pi\)
0.306383 + 0.951908i \(0.400881\pi\)
\(390\) 0 0
\(391\) −17869.3 −2.31123
\(392\) 0 0
\(393\) −3088.22 −0.396386
\(394\) 0 0
\(395\) 2786.59 0.354958
\(396\) 0 0
\(397\) 11715.8 1.48110 0.740552 0.671999i \(-0.234564\pi\)
0.740552 + 0.671999i \(0.234564\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −1643.72 −0.204697 −0.102349 0.994749i \(-0.532636\pi\)
−0.102349 + 0.994749i \(0.532636\pi\)
\(402\) 0 0
\(403\) 270.435 0.0334276
\(404\) 0 0
\(405\) 927.058 0.113743
\(406\) 0 0
\(407\) −8432.64 −1.02700
\(408\) 0 0
\(409\) 12206.6 1.47574 0.737869 0.674944i \(-0.235832\pi\)
0.737869 + 0.674944i \(0.235832\pi\)
\(410\) 0 0
\(411\) −5794.09 −0.695380
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 3805.25 0.450102
\(416\) 0 0
\(417\) −8778.12 −1.03086
\(418\) 0 0
\(419\) 11189.5 1.30464 0.652319 0.757945i \(-0.273796\pi\)
0.652319 + 0.757945i \(0.273796\pi\)
\(420\) 0 0
\(421\) 2718.12 0.314663 0.157331 0.987546i \(-0.449711\pi\)
0.157331 + 0.987546i \(0.449711\pi\)
\(422\) 0 0
\(423\) 1182.33 0.135903
\(424\) 0 0
\(425\) 511.464 0.0583757
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 865.179 0.0973689
\(430\) 0 0
\(431\) 11085.2 1.23888 0.619439 0.785045i \(-0.287360\pi\)
0.619439 + 0.785045i \(0.287360\pi\)
\(432\) 0 0
\(433\) 10880.6 1.20759 0.603795 0.797140i \(-0.293655\pi\)
0.603795 + 0.797140i \(0.293655\pi\)
\(434\) 0 0
\(435\) −4561.23 −0.502745
\(436\) 0 0
\(437\) 23156.2 2.53481
\(438\) 0 0
\(439\) 14902.3 1.62015 0.810075 0.586326i \(-0.199426\pi\)
0.810075 + 0.586326i \(0.199426\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −12504.3 −1.34108 −0.670538 0.741875i \(-0.733937\pi\)
−0.670538 + 0.741875i \(0.733937\pi\)
\(444\) 0 0
\(445\) −11206.1 −1.19376
\(446\) 0 0
\(447\) 395.427 0.0418413
\(448\) 0 0
\(449\) −1612.21 −0.169454 −0.0847268 0.996404i \(-0.527002\pi\)
−0.0847268 + 0.996404i \(0.527002\pi\)
\(450\) 0 0
\(451\) −7263.55 −0.758375
\(452\) 0 0
\(453\) 533.869 0.0553717
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 1275.51 0.130560 0.0652800 0.997867i \(-0.479206\pi\)
0.0652800 + 0.997867i \(0.479206\pi\)
\(458\) 0 0
\(459\) 2304.82 0.234378
\(460\) 0 0
\(461\) 9267.92 0.936335 0.468167 0.883640i \(-0.344914\pi\)
0.468167 + 0.883640i \(0.344914\pi\)
\(462\) 0 0
\(463\) −12615.3 −1.26627 −0.633136 0.774041i \(-0.718233\pi\)
−0.633136 + 0.774041i \(0.718233\pi\)
\(464\) 0 0
\(465\) −1693.22 −0.168863
\(466\) 0 0
\(467\) −14407.1 −1.42758 −0.713790 0.700359i \(-0.753023\pi\)
−0.713790 + 0.700359i \(0.753023\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −10822.4 −1.05875
\(472\) 0 0
\(473\) 19198.7 1.86629
\(474\) 0 0
\(475\) −662.788 −0.0640228
\(476\) 0 0
\(477\) −5051.04 −0.484845
\(478\) 0 0
\(479\) −284.296 −0.0271186 −0.0135593 0.999908i \(-0.504316\pi\)
−0.0135593 + 0.999908i \(0.504316\pi\)
\(480\) 0 0
\(481\) −879.356 −0.0833580
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 5342.49 0.500185
\(486\) 0 0
\(487\) 10535.7 0.980323 0.490161 0.871632i \(-0.336938\pi\)
0.490161 + 0.871632i \(0.336938\pi\)
\(488\) 0 0
\(489\) 4622.65 0.427492
\(490\) 0 0
\(491\) −14230.4 −1.30796 −0.653982 0.756510i \(-0.726903\pi\)
−0.653982 + 0.756510i \(0.726903\pi\)
\(492\) 0 0
\(493\) −11340.0 −1.03596
\(494\) 0 0
\(495\) −5416.97 −0.491868
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −9796.09 −0.878824 −0.439412 0.898286i \(-0.644813\pi\)
−0.439412 + 0.898286i \(0.644813\pi\)
\(500\) 0 0
\(501\) 2285.39 0.203800
\(502\) 0 0
\(503\) 7285.16 0.645784 0.322892 0.946436i \(-0.395345\pi\)
0.322892 + 0.946436i \(0.395345\pi\)
\(504\) 0 0
\(505\) 9916.41 0.873811
\(506\) 0 0
\(507\) −6500.78 −0.569447
\(508\) 0 0
\(509\) −13269.6 −1.15553 −0.577764 0.816204i \(-0.696075\pi\)
−0.577764 + 0.816204i \(0.696075\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −2986.73 −0.257051
\(514\) 0 0
\(515\) 10606.9 0.907561
\(516\) 0 0
\(517\) −6908.56 −0.587695
\(518\) 0 0
\(519\) −8073.95 −0.682865
\(520\) 0 0
\(521\) 14711.8 1.23711 0.618556 0.785740i \(-0.287718\pi\)
0.618556 + 0.785740i \(0.287718\pi\)
\(522\) 0 0
\(523\) 11004.9 0.920093 0.460046 0.887895i \(-0.347833\pi\)
0.460046 + 0.887895i \(0.347833\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −4209.61 −0.347958
\(528\) 0 0
\(529\) 31652.8 2.60153
\(530\) 0 0
\(531\) −3927.18 −0.320951
\(532\) 0 0
\(533\) −757.443 −0.0615544
\(534\) 0 0
\(535\) 11976.0 0.967788
\(536\) 0 0
\(537\) 3369.75 0.270793
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −8884.91 −0.706085 −0.353042 0.935607i \(-0.614853\pi\)
−0.353042 + 0.935607i \(0.614853\pi\)
\(542\) 0 0
\(543\) 12665.5 1.00098
\(544\) 0 0
\(545\) −5566.89 −0.437540
\(546\) 0 0
\(547\) 13986.6 1.09328 0.546639 0.837368i \(-0.315907\pi\)
0.546639 + 0.837368i \(0.315907\pi\)
\(548\) 0 0
\(549\) −2627.65 −0.204272
\(550\) 0 0
\(551\) 14695.0 1.13617
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 5505.73 0.421090
\(556\) 0 0
\(557\) −11252.8 −0.856006 −0.428003 0.903777i \(-0.640783\pi\)
−0.428003 + 0.903777i \(0.640783\pi\)
\(558\) 0 0
\(559\) 2002.04 0.151480
\(560\) 0 0
\(561\) −13467.5 −1.01354
\(562\) 0 0
\(563\) −13559.4 −1.01503 −0.507514 0.861643i \(-0.669435\pi\)
−0.507514 + 0.861643i \(0.669435\pi\)
\(564\) 0 0
\(565\) 22375.6 1.66610
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −17208.6 −1.26788 −0.633940 0.773382i \(-0.718563\pi\)
−0.633940 + 0.773382i \(0.718563\pi\)
\(570\) 0 0
\(571\) 12910.3 0.946196 0.473098 0.881010i \(-0.343136\pi\)
0.473098 + 0.881010i \(0.343136\pi\)
\(572\) 0 0
\(573\) 9881.42 0.720423
\(574\) 0 0
\(575\) −1254.23 −0.0909654
\(576\) 0 0
\(577\) −16154.8 −1.16557 −0.582783 0.812628i \(-0.698036\pi\)
−0.582783 + 0.812628i \(0.698036\pi\)
\(578\) 0 0
\(579\) −13871.7 −0.995664
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 29514.1 2.09666
\(584\) 0 0
\(585\) −564.881 −0.0399230
\(586\) 0 0
\(587\) 3403.75 0.239332 0.119666 0.992814i \(-0.461818\pi\)
0.119666 + 0.992814i \(0.461818\pi\)
\(588\) 0 0
\(589\) 5455.09 0.381618
\(590\) 0 0
\(591\) 7932.53 0.552116
\(592\) 0 0
\(593\) 12884.9 0.892272 0.446136 0.894965i \(-0.352800\pi\)
0.446136 + 0.894965i \(0.352800\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −4147.00 −0.284297
\(598\) 0 0
\(599\) −19002.4 −1.29619 −0.648095 0.761559i \(-0.724434\pi\)
−0.648095 + 0.761559i \(0.724434\pi\)
\(600\) 0 0
\(601\) 8159.94 0.553829 0.276914 0.960895i \(-0.410688\pi\)
0.276914 + 0.960895i \(0.410688\pi\)
\(602\) 0 0
\(603\) −5343.15 −0.360845
\(604\) 0 0
\(605\) 16418.8 1.10334
\(606\) 0 0
\(607\) 5304.02 0.354668 0.177334 0.984151i \(-0.443253\pi\)
0.177334 + 0.984151i \(0.443253\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −720.425 −0.0477009
\(612\) 0 0
\(613\) 14871.9 0.979886 0.489943 0.871754i \(-0.337018\pi\)
0.489943 + 0.871754i \(0.337018\pi\)
\(614\) 0 0
\(615\) 4742.42 0.310948
\(616\) 0 0
\(617\) −6214.11 −0.405463 −0.202732 0.979234i \(-0.564982\pi\)
−0.202732 + 0.979234i \(0.564982\pi\)
\(618\) 0 0
\(619\) 26716.2 1.73475 0.867377 0.497652i \(-0.165804\pi\)
0.867377 + 0.497652i \(0.165804\pi\)
\(620\) 0 0
\(621\) −5651.96 −0.365226
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −16338.1 −1.04564
\(626\) 0 0
\(627\) 17452.0 1.11159
\(628\) 0 0
\(629\) 13688.1 0.867697
\(630\) 0 0
\(631\) −1788.18 −0.112815 −0.0564074 0.998408i \(-0.517965\pi\)
−0.0564074 + 0.998408i \(0.517965\pi\)
\(632\) 0 0
\(633\) 13599.8 0.853942
\(634\) 0 0
\(635\) 14369.7 0.898025
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 6975.65 0.431851
\(640\) 0 0
\(641\) −4845.85 −0.298595 −0.149298 0.988792i \(-0.547701\pi\)
−0.149298 + 0.988792i \(0.547701\pi\)
\(642\) 0 0
\(643\) −8154.19 −0.500109 −0.250054 0.968232i \(-0.580449\pi\)
−0.250054 + 0.968232i \(0.580449\pi\)
\(644\) 0 0
\(645\) −12535.0 −0.765215
\(646\) 0 0
\(647\) 14514.1 0.881928 0.440964 0.897525i \(-0.354637\pi\)
0.440964 + 0.897525i \(0.354637\pi\)
\(648\) 0 0
\(649\) 22947.2 1.38792
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −14194.4 −0.850641 −0.425321 0.905043i \(-0.639839\pi\)
−0.425321 + 0.905043i \(0.639839\pi\)
\(654\) 0 0
\(655\) −11781.7 −0.702823
\(656\) 0 0
\(657\) 2978.16 0.176848
\(658\) 0 0
\(659\) 24762.7 1.46376 0.731879 0.681435i \(-0.238644\pi\)
0.731879 + 0.681435i \(0.238644\pi\)
\(660\) 0 0
\(661\) −23482.5 −1.38179 −0.690897 0.722954i \(-0.742784\pi\)
−0.690897 + 0.722954i \(0.742784\pi\)
\(662\) 0 0
\(663\) −1404.39 −0.0822653
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 27808.3 1.61430
\(668\) 0 0
\(669\) −9157.39 −0.529216
\(670\) 0 0
\(671\) 15353.8 0.883350
\(672\) 0 0
\(673\) 12943.0 0.741333 0.370666 0.928766i \(-0.379129\pi\)
0.370666 + 0.928766i \(0.379129\pi\)
\(674\) 0 0
\(675\) 161.773 0.00922467
\(676\) 0 0
\(677\) −25409.8 −1.44251 −0.721255 0.692669i \(-0.756435\pi\)
−0.721255 + 0.692669i \(0.756435\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −15584.6 −0.876950
\(682\) 0 0
\(683\) 5747.89 0.322016 0.161008 0.986953i \(-0.448526\pi\)
0.161008 + 0.986953i \(0.448526\pi\)
\(684\) 0 0
\(685\) −22104.7 −1.23296
\(686\) 0 0
\(687\) 16491.1 0.915831
\(688\) 0 0
\(689\) 3077.74 0.170178
\(690\) 0 0
\(691\) 30578.5 1.68344 0.841722 0.539911i \(-0.181542\pi\)
0.841722 + 0.539911i \(0.181542\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −33489.0 −1.82778
\(696\) 0 0
\(697\) 11790.4 0.640738
\(698\) 0 0
\(699\) −16371.5 −0.885876
\(700\) 0 0
\(701\) 11378.9 0.613088 0.306544 0.951857i \(-0.400827\pi\)
0.306544 + 0.951857i \(0.400827\pi\)
\(702\) 0 0
\(703\) −17738.0 −0.951636
\(704\) 0 0
\(705\) 4510.65 0.240966
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 16050.7 0.850208 0.425104 0.905145i \(-0.360238\pi\)
0.425104 + 0.905145i \(0.360238\pi\)
\(710\) 0 0
\(711\) 2191.26 0.115582
\(712\) 0 0
\(713\) 10323.0 0.542214
\(714\) 0 0
\(715\) 3300.70 0.172642
\(716\) 0 0
\(717\) −16866.5 −0.878510
\(718\) 0 0
\(719\) 10628.6 0.551296 0.275648 0.961259i \(-0.411108\pi\)
0.275648 + 0.961259i \(0.411108\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 12249.1 0.630080
\(724\) 0 0
\(725\) −795.943 −0.0407732
\(726\) 0 0
\(727\) −16912.8 −0.862810 −0.431405 0.902158i \(-0.641982\pi\)
−0.431405 + 0.902158i \(0.641982\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −31163.9 −1.57680
\(732\) 0 0
\(733\) −32129.7 −1.61901 −0.809506 0.587111i \(-0.800265\pi\)
−0.809506 + 0.587111i \(0.800265\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 31221.0 1.56043
\(738\) 0 0
\(739\) 21286.3 1.05958 0.529791 0.848128i \(-0.322270\pi\)
0.529791 + 0.848128i \(0.322270\pi\)
\(740\) 0 0
\(741\) 1819.90 0.0902234
\(742\) 0 0
\(743\) 16441.2 0.811804 0.405902 0.913917i \(-0.366957\pi\)
0.405902 + 0.913917i \(0.366957\pi\)
\(744\) 0 0
\(745\) 1508.57 0.0741877
\(746\) 0 0
\(747\) 2992.29 0.146563
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 844.181 0.0410181 0.0205091 0.999790i \(-0.493471\pi\)
0.0205091 + 0.999790i \(0.493471\pi\)
\(752\) 0 0
\(753\) −5255.97 −0.254367
\(754\) 0 0
\(755\) 2036.74 0.0981782
\(756\) 0 0
\(757\) −13607.9 −0.653352 −0.326676 0.945136i \(-0.605929\pi\)
−0.326676 + 0.945136i \(0.605929\pi\)
\(758\) 0 0
\(759\) 33025.4 1.57938
\(760\) 0 0
\(761\) 23150.7 1.10277 0.551387 0.834249i \(-0.314099\pi\)
0.551387 + 0.834249i \(0.314099\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 8792.99 0.415570
\(766\) 0 0
\(767\) 2392.94 0.112652
\(768\) 0 0
\(769\) −39732.0 −1.86316 −0.931581 0.363534i \(-0.881570\pi\)
−0.931581 + 0.363534i \(0.881570\pi\)
\(770\) 0 0
\(771\) −11475.8 −0.536045
\(772\) 0 0
\(773\) −4151.21 −0.193155 −0.0965773 0.995325i \(-0.530790\pi\)
−0.0965773 + 0.995325i \(0.530790\pi\)
\(774\) 0 0
\(775\) −295.469 −0.0136949
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −15278.8 −0.702721
\(780\) 0 0
\(781\) −40760.0 −1.86749
\(782\) 0 0
\(783\) −3586.76 −0.163704
\(784\) 0 0
\(785\) −41288.1 −1.87724
\(786\) 0 0
\(787\) 1290.94 0.0584714 0.0292357 0.999573i \(-0.490693\pi\)
0.0292357 + 0.999573i \(0.490693\pi\)
\(788\) 0 0
\(789\) −15817.7 −0.713720
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 1601.10 0.0716981
\(794\) 0 0
\(795\) −19270.0 −0.859668
\(796\) 0 0
\(797\) 4469.79 0.198655 0.0993276 0.995055i \(-0.468331\pi\)
0.0993276 + 0.995055i \(0.468331\pi\)
\(798\) 0 0
\(799\) 11214.2 0.496533
\(800\) 0 0
\(801\) −8812.04 −0.388712
\(802\) 0 0
\(803\) −17401.9 −0.764759
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −13605.1 −0.593460
\(808\) 0 0
\(809\) 29777.8 1.29411 0.647054 0.762445i \(-0.276001\pi\)
0.647054 + 0.762445i \(0.276001\pi\)
\(810\) 0 0
\(811\) 4083.89 0.176825 0.0884124 0.996084i \(-0.471821\pi\)
0.0884124 + 0.996084i \(0.471821\pi\)
\(812\) 0 0
\(813\) −4317.31 −0.186242
\(814\) 0 0
\(815\) 17635.7 0.757975
\(816\) 0 0
\(817\) 40384.2 1.72933
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 16909.2 0.718799 0.359399 0.933184i \(-0.382982\pi\)
0.359399 + 0.933184i \(0.382982\pi\)
\(822\) 0 0
\(823\) −294.522 −0.0124743 −0.00623717 0.999981i \(-0.501985\pi\)
−0.00623717 + 0.999981i \(0.501985\pi\)
\(824\) 0 0
\(825\) −945.270 −0.0398910
\(826\) 0 0
\(827\) −43346.8 −1.82263 −0.911315 0.411711i \(-0.864931\pi\)
−0.911315 + 0.411711i \(0.864931\pi\)
\(828\) 0 0
\(829\) −3542.74 −0.148425 −0.0742126 0.997242i \(-0.523644\pi\)
−0.0742126 + 0.997242i \(0.523644\pi\)
\(830\) 0 0
\(831\) −17954.6 −0.749503
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 8718.89 0.361353
\(836\) 0 0
\(837\) −1331.48 −0.0549851
\(838\) 0 0
\(839\) −26513.0 −1.09098 −0.545489 0.838118i \(-0.683656\pi\)
−0.545489 + 0.838118i \(0.683656\pi\)
\(840\) 0 0
\(841\) −6741.72 −0.276425
\(842\) 0 0
\(843\) −7943.62 −0.324547
\(844\) 0 0
\(845\) −24800.8 −1.00967
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −1202.47 −0.0486084
\(850\) 0 0
\(851\) −33566.6 −1.35211
\(852\) 0 0
\(853\) 31158.3 1.25069 0.625345 0.780348i \(-0.284958\pi\)
0.625345 + 0.780348i \(0.284958\pi\)
\(854\) 0 0
\(855\) −11394.5 −0.455771
\(856\) 0 0
\(857\) 22611.4 0.901271 0.450635 0.892708i \(-0.351197\pi\)
0.450635 + 0.892708i \(0.351197\pi\)
\(858\) 0 0
\(859\) −17135.6 −0.680627 −0.340313 0.940312i \(-0.610533\pi\)
−0.340313 + 0.940312i \(0.610533\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −31801.8 −1.25440 −0.627199 0.778859i \(-0.715799\pi\)
−0.627199 + 0.778859i \(0.715799\pi\)
\(864\) 0 0
\(865\) −30802.5 −1.21077
\(866\) 0 0
\(867\) 7121.81 0.278973
\(868\) 0 0
\(869\) −12803.9 −0.499820
\(870\) 0 0
\(871\) 3255.72 0.126654
\(872\) 0 0
\(873\) 4201.11 0.162871
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −23693.0 −0.912265 −0.456133 0.889912i \(-0.650766\pi\)
−0.456133 + 0.889912i \(0.650766\pi\)
\(878\) 0 0
\(879\) −15921.4 −0.610938
\(880\) 0 0
\(881\) −35584.4 −1.36080 −0.680402 0.732839i \(-0.738195\pi\)
−0.680402 + 0.732839i \(0.738195\pi\)
\(882\) 0 0
\(883\) 31928.8 1.21686 0.608431 0.793607i \(-0.291799\pi\)
0.608431 + 0.793607i \(0.291799\pi\)
\(884\) 0 0
\(885\) −14982.4 −0.569071
\(886\) 0 0
\(887\) −24218.6 −0.916778 −0.458389 0.888752i \(-0.651573\pi\)
−0.458389 + 0.888752i \(0.651573\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −4259.68 −0.160162
\(892\) 0 0
\(893\) −14532.1 −0.544566
\(894\) 0 0
\(895\) 12855.8 0.480136
\(896\) 0 0
\(897\) 3443.89 0.128192
\(898\) 0 0
\(899\) 6551.02 0.243035
\(900\) 0 0
\(901\) −47908.3 −1.77143
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 48319.7 1.77481
\(906\) 0 0
\(907\) −37779.4 −1.38307 −0.691535 0.722343i \(-0.743065\pi\)
−0.691535 + 0.722343i \(0.743065\pi\)
\(908\) 0 0
\(909\) 7797.86 0.284531
\(910\) 0 0
\(911\) 33419.0 1.21539 0.607695 0.794170i \(-0.292094\pi\)
0.607695 + 0.794170i \(0.292094\pi\)
\(912\) 0 0
\(913\) −17484.5 −0.633792
\(914\) 0 0
\(915\) −10024.6 −0.362190
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 15771.4 0.566105 0.283053 0.959104i \(-0.408653\pi\)
0.283053 + 0.959104i \(0.408653\pi\)
\(920\) 0 0
\(921\) 12391.3 0.443332
\(922\) 0 0
\(923\) −4250.46 −0.151577
\(924\) 0 0
\(925\) 960.759 0.0341509
\(926\) 0 0
\(927\) 8340.79 0.295521
\(928\) 0 0
\(929\) −44216.4 −1.56157 −0.780783 0.624802i \(-0.785180\pi\)
−0.780783 + 0.624802i \(0.785180\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −22961.6 −0.805712
\(934\) 0 0
\(935\) −51379.1 −1.79709
\(936\) 0 0
\(937\) 8691.91 0.303044 0.151522 0.988454i \(-0.451583\pi\)
0.151522 + 0.988454i \(0.451583\pi\)
\(938\) 0 0
\(939\) 25638.8 0.891045
\(940\) 0 0
\(941\) −11824.1 −0.409622 −0.204811 0.978802i \(-0.565658\pi\)
−0.204811 + 0.978802i \(0.565658\pi\)
\(942\) 0 0
\(943\) −28912.9 −0.998446
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 4468.88 0.153346 0.0766732 0.997056i \(-0.475570\pi\)
0.0766732 + 0.997056i \(0.475570\pi\)
\(948\) 0 0
\(949\) −1814.68 −0.0620726
\(950\) 0 0
\(951\) 22930.9 0.781900
\(952\) 0 0
\(953\) 18172.0 0.617680 0.308840 0.951114i \(-0.400059\pi\)
0.308840 + 0.951114i \(0.400059\pi\)
\(954\) 0 0
\(955\) 37698.1 1.27736
\(956\) 0 0
\(957\) 20958.1 0.707920
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −27359.1 −0.918369
\(962\) 0 0
\(963\) 9417.41 0.315132
\(964\) 0 0
\(965\) −52921.4 −1.76539
\(966\) 0 0
\(967\) 53267.4 1.77142 0.885710 0.464239i \(-0.153672\pi\)
0.885710 + 0.464239i \(0.153672\pi\)
\(968\) 0 0
\(969\) −28328.6 −0.939161
\(970\) 0 0
\(971\) 14899.8 0.492439 0.246220 0.969214i \(-0.420812\pi\)
0.246220 + 0.969214i \(0.420812\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −98.5728 −0.00323780
\(976\) 0 0
\(977\) 50912.6 1.66718 0.833592 0.552380i \(-0.186280\pi\)
0.833592 + 0.552380i \(0.186280\pi\)
\(978\) 0 0
\(979\) 51490.3 1.68094
\(980\) 0 0
\(981\) −4377.57 −0.142472
\(982\) 0 0
\(983\) −49617.9 −1.60993 −0.804967 0.593320i \(-0.797817\pi\)
−0.804967 + 0.593320i \(0.797817\pi\)
\(984\) 0 0
\(985\) 30263.0 0.978944
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 76421.4 2.45709
\(990\) 0 0
\(991\) −16435.7 −0.526839 −0.263419 0.964681i \(-0.584850\pi\)
−0.263419 + 0.964681i \(0.584850\pi\)
\(992\) 0 0
\(993\) 22650.5 0.723860
\(994\) 0 0
\(995\) −15821.0 −0.504080
\(996\) 0 0
\(997\) 634.476 0.0201545 0.0100773 0.999949i \(-0.496792\pi\)
0.0100773 + 0.999949i \(0.496792\pi\)
\(998\) 0 0
\(999\) 4329.48 0.137116
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 1176.4.a.bc.1.4 yes 4
4.3 odd 2 2352.4.a.ck.1.4 4
7.6 odd 2 1176.4.a.bb.1.1 4
28.27 even 2 2352.4.a.cr.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1176.4.a.bb.1.1 4 7.6 odd 2
1176.4.a.bc.1.4 yes 4 1.1 even 1 trivial
2352.4.a.ck.1.4 4 4.3 odd 2
2352.4.a.cr.1.1 4 28.27 even 2