Properties

Label 1344.4.a.i.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

\(f(q)\) \(=\) \(q-3.00000 q^{3} +4.00000 q^{5} +7.00000 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q-3.00000 q^{3} +4.00000 q^{5} +7.00000 q^{7} +9.00000 q^{9} +62.0000 q^{11} +62.0000 q^{13} -12.0000 q^{15} +84.0000 q^{17} +100.000 q^{19} -21.0000 q^{21} +42.0000 q^{23} -109.000 q^{25} -27.0000 q^{27} +10.0000 q^{29} +48.0000 q^{31} -186.000 q^{33} +28.0000 q^{35} +246.000 q^{37} -186.000 q^{39} -248.000 q^{41} +68.0000 q^{43} +36.0000 q^{45} -324.000 q^{47} +49.0000 q^{49} -252.000 q^{51} -258.000 q^{53} +248.000 q^{55} -300.000 q^{57} +120.000 q^{59} -622.000 q^{61} +63.0000 q^{63} +248.000 q^{65} +904.000 q^{67} -126.000 q^{69} +678.000 q^{71} -642.000 q^{73} +327.000 q^{75} +434.000 q^{77} -740.000 q^{79} +81.0000 q^{81} +468.000 q^{83} +336.000 q^{85} -30.0000 q^{87} +200.000 q^{89} +434.000 q^{91} -144.000 q^{93} +400.000 q^{95} -1266.00 q^{97} +558.000 q^{99} +O(q^{100})\)

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
\(5\) 4.00000 0.357771 0.178885 0.983870i \(-0.442751\pi\)
0.178885 + 0.983870i \(0.442751\pi\)
\(6\) 0 0
\(7\) 7.00000 0.377964
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 62.0000 1.69943 0.849714 0.527244i \(-0.176775\pi\)
0.849714 + 0.527244i \(0.176775\pi\)
\(12\) 0 0
\(13\) 62.0000 1.32275 0.661373 0.750057i \(-0.269974\pi\)
0.661373 + 0.750057i \(0.269974\pi\)
\(14\) 0 0
\(15\) −12.0000 −0.206559
\(16\) 0 0
\(17\) 84.0000 1.19841 0.599206 0.800595i \(-0.295483\pi\)
0.599206 + 0.800595i \(0.295483\pi\)
\(18\) 0 0
\(19\) 100.000 1.20745 0.603726 0.797192i \(-0.293682\pi\)
0.603726 + 0.797192i \(0.293682\pi\)
\(20\) 0 0
\(21\) −21.0000 −0.218218
\(22\) 0 0
\(23\) 42.0000 0.380765 0.190383 0.981710i \(-0.439027\pi\)
0.190383 + 0.981710i \(0.439027\pi\)
\(24\) 0 0
\(25\) −109.000 −0.872000
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) 10.0000 0.0640329 0.0320164 0.999487i \(-0.489807\pi\)
0.0320164 + 0.999487i \(0.489807\pi\)
\(30\) 0 0
\(31\) 48.0000 0.278099 0.139049 0.990285i \(-0.455595\pi\)
0.139049 + 0.990285i \(0.455595\pi\)
\(32\) 0 0
\(33\) −186.000 −0.981165
\(34\) 0 0
\(35\) 28.0000 0.135225
\(36\) 0 0
\(37\) 246.000 1.09303 0.546516 0.837449i \(-0.315954\pi\)
0.546516 + 0.837449i \(0.315954\pi\)
\(38\) 0 0
\(39\) −186.000 −0.763688
\(40\) 0 0
\(41\) −248.000 −0.944661 −0.472330 0.881422i \(-0.656587\pi\)
−0.472330 + 0.881422i \(0.656587\pi\)
\(42\) 0 0
\(43\) 68.0000 0.241161 0.120580 0.992704i \(-0.461524\pi\)
0.120580 + 0.992704i \(0.461524\pi\)
\(44\) 0 0
\(45\) 36.0000 0.119257
\(46\) 0 0
\(47\) −324.000 −1.00554 −0.502769 0.864421i \(-0.667685\pi\)
−0.502769 + 0.864421i \(0.667685\pi\)
\(48\) 0 0
\(49\) 49.0000 0.142857
\(50\) 0 0
\(51\) −252.000 −0.691903
\(52\) 0 0
\(53\) −258.000 −0.668661 −0.334330 0.942456i \(-0.608510\pi\)
−0.334330 + 0.942456i \(0.608510\pi\)
\(54\) 0 0
\(55\) 248.000 0.608006
\(56\) 0 0
\(57\) −300.000 −0.697122
\(58\) 0 0
\(59\) 120.000 0.264791 0.132396 0.991197i \(-0.457733\pi\)
0.132396 + 0.991197i \(0.457733\pi\)
\(60\) 0 0
\(61\) −622.000 −1.30556 −0.652778 0.757549i \(-0.726397\pi\)
−0.652778 + 0.757549i \(0.726397\pi\)
\(62\) 0 0
\(63\) 63.0000 0.125988
\(64\) 0 0
\(65\) 248.000 0.473240
\(66\) 0 0
\(67\) 904.000 1.64838 0.824188 0.566316i \(-0.191632\pi\)
0.824188 + 0.566316i \(0.191632\pi\)
\(68\) 0 0
\(69\) −126.000 −0.219835
\(70\) 0 0
\(71\) 678.000 1.13329 0.566646 0.823961i \(-0.308241\pi\)
0.566646 + 0.823961i \(0.308241\pi\)
\(72\) 0 0
\(73\) −642.000 −1.02932 −0.514660 0.857394i \(-0.672082\pi\)
−0.514660 + 0.857394i \(0.672082\pi\)
\(74\) 0 0
\(75\) 327.000 0.503449
\(76\) 0 0
\(77\) 434.000 0.642323
\(78\) 0 0
\(79\) −740.000 −1.05388 −0.526940 0.849903i \(-0.676661\pi\)
−0.526940 + 0.849903i \(0.676661\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 468.000 0.618912 0.309456 0.950914i \(-0.399853\pi\)
0.309456 + 0.950914i \(0.399853\pi\)
\(84\) 0 0
\(85\) 336.000 0.428757
\(86\) 0 0
\(87\) −30.0000 −0.0369694
\(88\) 0 0
\(89\) 200.000 0.238202 0.119101 0.992882i \(-0.461999\pi\)
0.119101 + 0.992882i \(0.461999\pi\)
\(90\) 0 0
\(91\) 434.000 0.499951
\(92\) 0 0
\(93\) −144.000 −0.160560
\(94\) 0 0
\(95\) 400.000 0.431991
\(96\) 0 0
\(97\) −1266.00 −1.32518 −0.662592 0.748981i \(-0.730544\pi\)
−0.662592 + 0.748981i \(0.730544\pi\)
\(98\) 0 0
\(99\) 558.000 0.566476
\(100\) 0 0
\(101\) −232.000 −0.228563 −0.114281 0.993448i \(-0.536457\pi\)
−0.114281 + 0.993448i \(0.536457\pi\)
\(102\) 0 0
\(103\) 1792.00 1.71428 0.857141 0.515082i \(-0.172239\pi\)
0.857141 + 0.515082i \(0.172239\pi\)
\(104\) 0 0
\(105\) −84.0000 −0.0780720
\(106\) 0 0
\(107\) −1906.00 −1.72206 −0.861028 0.508558i \(-0.830179\pi\)
−0.861028 + 0.508558i \(0.830179\pi\)
\(108\) 0 0
\(109\) 90.0000 0.0790866 0.0395433 0.999218i \(-0.487410\pi\)
0.0395433 + 0.999218i \(0.487410\pi\)
\(110\) 0 0
\(111\) −738.000 −0.631062
\(112\) 0 0
\(113\) 458.000 0.381283 0.190642 0.981660i \(-0.438943\pi\)
0.190642 + 0.981660i \(0.438943\pi\)
\(114\) 0 0
\(115\) 168.000 0.136227
\(116\) 0 0
\(117\) 558.000 0.440916
\(118\) 0 0
\(119\) 588.000 0.452957
\(120\) 0 0
\(121\) 2513.00 1.88805
\(122\) 0 0
\(123\) 744.000 0.545400
\(124\) 0 0
\(125\) −936.000 −0.669747
\(126\) 0 0
\(127\) −804.000 −0.561760 −0.280880 0.959743i \(-0.590626\pi\)
−0.280880 + 0.959743i \(0.590626\pi\)
\(128\) 0 0
\(129\) −204.000 −0.139234
\(130\) 0 0
\(131\) 812.000 0.541563 0.270782 0.962641i \(-0.412718\pi\)
0.270782 + 0.962641i \(0.412718\pi\)
\(132\) 0 0
\(133\) 700.000 0.456374
\(134\) 0 0
\(135\) −108.000 −0.0688530
\(136\) 0 0
\(137\) 414.000 0.258178 0.129089 0.991633i \(-0.458795\pi\)
0.129089 + 0.991633i \(0.458795\pi\)
\(138\) 0 0
\(139\) −1620.00 −0.988537 −0.494268 0.869309i \(-0.664564\pi\)
−0.494268 + 0.869309i \(0.664564\pi\)
\(140\) 0 0
\(141\) 972.000 0.580547
\(142\) 0 0
\(143\) 3844.00 2.24791
\(144\) 0 0
\(145\) 40.0000 0.0229091
\(146\) 0 0
\(147\) −147.000 −0.0824786
\(148\) 0 0
\(149\) −2370.00 −1.30307 −0.651537 0.758617i \(-0.725875\pi\)
−0.651537 + 0.758617i \(0.725875\pi\)
\(150\) 0 0
\(151\) 568.000 0.306114 0.153057 0.988217i \(-0.451088\pi\)
0.153057 + 0.988217i \(0.451088\pi\)
\(152\) 0 0
\(153\) 756.000 0.399470
\(154\) 0 0
\(155\) 192.000 0.0994956
\(156\) 0 0
\(157\) 266.000 0.135217 0.0676086 0.997712i \(-0.478463\pi\)
0.0676086 + 0.997712i \(0.478463\pi\)
\(158\) 0 0
\(159\) 774.000 0.386052
\(160\) 0 0
\(161\) 294.000 0.143916
\(162\) 0 0
\(163\) −272.000 −0.130704 −0.0653518 0.997862i \(-0.520817\pi\)
−0.0653518 + 0.997862i \(0.520817\pi\)
\(164\) 0 0
\(165\) −744.000 −0.351032
\(166\) 0 0
\(167\) 1876.00 0.869277 0.434638 0.900605i \(-0.356876\pi\)
0.434638 + 0.900605i \(0.356876\pi\)
\(168\) 0 0
\(169\) 1647.00 0.749659
\(170\) 0 0
\(171\) 900.000 0.402484
\(172\) 0 0
\(173\) 152.000 0.0667997 0.0333998 0.999442i \(-0.489367\pi\)
0.0333998 + 0.999442i \(0.489367\pi\)
\(174\) 0 0
\(175\) −763.000 −0.329585
\(176\) 0 0
\(177\) −360.000 −0.152877
\(178\) 0 0
\(179\) 610.000 0.254713 0.127356 0.991857i \(-0.459351\pi\)
0.127356 + 0.991857i \(0.459351\pi\)
\(180\) 0 0
\(181\) −1042.00 −0.427907 −0.213954 0.976844i \(-0.568634\pi\)
−0.213954 + 0.976844i \(0.568634\pi\)
\(182\) 0 0
\(183\) 1866.00 0.753763
\(184\) 0 0
\(185\) 984.000 0.391055
\(186\) 0 0
\(187\) 5208.00 2.03661
\(188\) 0 0
\(189\) −189.000 −0.0727393
\(190\) 0 0
\(191\) 2038.00 0.772065 0.386033 0.922485i \(-0.373845\pi\)
0.386033 + 0.922485i \(0.373845\pi\)
\(192\) 0 0
\(193\) −2602.00 −0.970446 −0.485223 0.874390i \(-0.661262\pi\)
−0.485223 + 0.874390i \(0.661262\pi\)
\(194\) 0 0
\(195\) −744.000 −0.273225
\(196\) 0 0
\(197\) −2354.00 −0.851348 −0.425674 0.904877i \(-0.639963\pi\)
−0.425674 + 0.904877i \(0.639963\pi\)
\(198\) 0 0
\(199\) −1680.00 −0.598452 −0.299226 0.954182i \(-0.596729\pi\)
−0.299226 + 0.954182i \(0.596729\pi\)
\(200\) 0 0
\(201\) −2712.00 −0.951690
\(202\) 0 0
\(203\) 70.0000 0.0242022
\(204\) 0 0
\(205\) −992.000 −0.337972
\(206\) 0 0
\(207\) 378.000 0.126922
\(208\) 0 0
\(209\) 6200.00 2.05198
\(210\) 0 0
\(211\) −668.000 −0.217948 −0.108974 0.994045i \(-0.534757\pi\)
−0.108974 + 0.994045i \(0.534757\pi\)
\(212\) 0 0
\(213\) −2034.00 −0.654307
\(214\) 0 0
\(215\) 272.000 0.0862802
\(216\) 0 0
\(217\) 336.000 0.105111
\(218\) 0 0
\(219\) 1926.00 0.594279
\(220\) 0 0
\(221\) 5208.00 1.58519
\(222\) 0 0
\(223\) 1832.00 0.550134 0.275067 0.961425i \(-0.411300\pi\)
0.275067 + 0.961425i \(0.411300\pi\)
\(224\) 0 0
\(225\) −981.000 −0.290667
\(226\) 0 0
\(227\) 4944.00 1.44557 0.722786 0.691072i \(-0.242861\pi\)
0.722786 + 0.691072i \(0.242861\pi\)
\(228\) 0 0
\(229\) 5470.00 1.57846 0.789231 0.614096i \(-0.210479\pi\)
0.789231 + 0.614096i \(0.210479\pi\)
\(230\) 0 0
\(231\) −1302.00 −0.370846
\(232\) 0 0
\(233\) −2802.00 −0.787833 −0.393917 0.919146i \(-0.628880\pi\)
−0.393917 + 0.919146i \(0.628880\pi\)
\(234\) 0 0
\(235\) −1296.00 −0.359752
\(236\) 0 0
\(237\) 2220.00 0.608458
\(238\) 0 0
\(239\) 1170.00 0.316657 0.158328 0.987386i \(-0.449390\pi\)
0.158328 + 0.987386i \(0.449390\pi\)
\(240\) 0 0
\(241\) −2338.00 −0.624912 −0.312456 0.949932i \(-0.601152\pi\)
−0.312456 + 0.949932i \(0.601152\pi\)
\(242\) 0 0
\(243\) −243.000 −0.0641500
\(244\) 0 0
\(245\) 196.000 0.0511101
\(246\) 0 0
\(247\) 6200.00 1.59715
\(248\) 0 0
\(249\) −1404.00 −0.357329
\(250\) 0 0
\(251\) 2792.00 0.702109 0.351055 0.936355i \(-0.385823\pi\)
0.351055 + 0.936355i \(0.385823\pi\)
\(252\) 0 0
\(253\) 2604.00 0.647083
\(254\) 0 0
\(255\) −1008.00 −0.247543
\(256\) 0 0
\(257\) 7024.00 1.70484 0.852422 0.522854i \(-0.175133\pi\)
0.852422 + 0.522854i \(0.175133\pi\)
\(258\) 0 0
\(259\) 1722.00 0.413127
\(260\) 0 0
\(261\) 90.0000 0.0213443
\(262\) 0 0
\(263\) −2438.00 −0.571610 −0.285805 0.958288i \(-0.592261\pi\)
−0.285805 + 0.958288i \(0.592261\pi\)
\(264\) 0 0
\(265\) −1032.00 −0.239227
\(266\) 0 0
\(267\) −600.000 −0.137526
\(268\) 0 0
\(269\) 6780.00 1.53674 0.768372 0.640004i \(-0.221067\pi\)
0.768372 + 0.640004i \(0.221067\pi\)
\(270\) 0 0
\(271\) 1928.00 0.432168 0.216084 0.976375i \(-0.430671\pi\)
0.216084 + 0.976375i \(0.430671\pi\)
\(272\) 0 0
\(273\) −1302.00 −0.288647
\(274\) 0 0
\(275\) −6758.00 −1.48190
\(276\) 0 0
\(277\) −5554.00 −1.20472 −0.602360 0.798224i \(-0.705773\pi\)
−0.602360 + 0.798224i \(0.705773\pi\)
\(278\) 0 0
\(279\) 432.000 0.0926995
\(280\) 0 0
\(281\) 1942.00 0.412278 0.206139 0.978523i \(-0.433910\pi\)
0.206139 + 0.978523i \(0.433910\pi\)
\(282\) 0 0
\(283\) 4828.00 1.01412 0.507058 0.861912i \(-0.330733\pi\)
0.507058 + 0.861912i \(0.330733\pi\)
\(284\) 0 0
\(285\) −1200.00 −0.249410
\(286\) 0 0
\(287\) −1736.00 −0.357048
\(288\) 0 0
\(289\) 2143.00 0.436190
\(290\) 0 0
\(291\) 3798.00 0.765095
\(292\) 0 0
\(293\) 6152.00 1.22663 0.613317 0.789837i \(-0.289835\pi\)
0.613317 + 0.789837i \(0.289835\pi\)
\(294\) 0 0
\(295\) 480.000 0.0947345
\(296\) 0 0
\(297\) −1674.00 −0.327055
\(298\) 0 0
\(299\) 2604.00 0.503656
\(300\) 0 0
\(301\) 476.000 0.0911501
\(302\) 0 0
\(303\) 696.000 0.131961
\(304\) 0 0
\(305\) −2488.00 −0.467090
\(306\) 0 0
\(307\) 5884.00 1.09387 0.546934 0.837176i \(-0.315795\pi\)
0.546934 + 0.837176i \(0.315795\pi\)
\(308\) 0 0
\(309\) −5376.00 −0.989741
\(310\) 0 0
\(311\) −9132.00 −1.66504 −0.832521 0.553993i \(-0.813103\pi\)
−0.832521 + 0.553993i \(0.813103\pi\)
\(312\) 0 0
\(313\) −9382.00 −1.69426 −0.847128 0.531389i \(-0.821670\pi\)
−0.847128 + 0.531389i \(0.821670\pi\)
\(314\) 0 0
\(315\) 252.000 0.0450749
\(316\) 0 0
\(317\) −3114.00 −0.551734 −0.275867 0.961196i \(-0.588965\pi\)
−0.275867 + 0.961196i \(0.588965\pi\)
\(318\) 0 0
\(319\) 620.000 0.108819
\(320\) 0 0
\(321\) 5718.00 0.994229
\(322\) 0 0
\(323\) 8400.00 1.44702
\(324\) 0 0
\(325\) −6758.00 −1.15344
\(326\) 0 0
\(327\) −270.000 −0.0456607
\(328\) 0 0
\(329\) −2268.00 −0.380057
\(330\) 0 0
\(331\) 1532.00 0.254400 0.127200 0.991877i \(-0.459401\pi\)
0.127200 + 0.991877i \(0.459401\pi\)
\(332\) 0 0
\(333\) 2214.00 0.364344
\(334\) 0 0
\(335\) 3616.00 0.589741
\(336\) 0 0
\(337\) −4166.00 −0.673402 −0.336701 0.941612i \(-0.609311\pi\)
−0.336701 + 0.941612i \(0.609311\pi\)
\(338\) 0 0
\(339\) −1374.00 −0.220134
\(340\) 0 0
\(341\) 2976.00 0.472608
\(342\) 0 0
\(343\) 343.000 0.0539949
\(344\) 0 0
\(345\) −504.000 −0.0786506
\(346\) 0 0
\(347\) −11366.0 −1.75838 −0.879191 0.476469i \(-0.841917\pi\)
−0.879191 + 0.476469i \(0.841917\pi\)
\(348\) 0 0
\(349\) −9310.00 −1.42795 −0.713973 0.700174i \(-0.753106\pi\)
−0.713973 + 0.700174i \(0.753106\pi\)
\(350\) 0 0
\(351\) −1674.00 −0.254563
\(352\) 0 0
\(353\) −8572.00 −1.29247 −0.646234 0.763139i \(-0.723657\pi\)
−0.646234 + 0.763139i \(0.723657\pi\)
\(354\) 0 0
\(355\) 2712.00 0.405459
\(356\) 0 0
\(357\) −1764.00 −0.261515
\(358\) 0 0
\(359\) 4790.00 0.704196 0.352098 0.935963i \(-0.385468\pi\)
0.352098 + 0.935963i \(0.385468\pi\)
\(360\) 0 0
\(361\) 3141.00 0.457938
\(362\) 0 0
\(363\) −7539.00 −1.09007
\(364\) 0 0
\(365\) −2568.00 −0.368261
\(366\) 0 0
\(367\) −5424.00 −0.771473 −0.385736 0.922609i \(-0.626053\pi\)
−0.385736 + 0.922609i \(0.626053\pi\)
\(368\) 0 0
\(369\) −2232.00 −0.314887
\(370\) 0 0
\(371\) −1806.00 −0.252730
\(372\) 0 0
\(373\) −1838.00 −0.255142 −0.127571 0.991829i \(-0.540718\pi\)
−0.127571 + 0.991829i \(0.540718\pi\)
\(374\) 0 0
\(375\) 2808.00 0.386679
\(376\) 0 0
\(377\) 620.000 0.0846993
\(378\) 0 0
\(379\) −4260.00 −0.577365 −0.288683 0.957425i \(-0.593217\pi\)
−0.288683 + 0.957425i \(0.593217\pi\)
\(380\) 0 0
\(381\) 2412.00 0.324332
\(382\) 0 0
\(383\) −9048.00 −1.20713 −0.603566 0.797313i \(-0.706254\pi\)
−0.603566 + 0.797313i \(0.706254\pi\)
\(384\) 0 0
\(385\) 1736.00 0.229805
\(386\) 0 0
\(387\) 612.000 0.0803868
\(388\) 0 0
\(389\) 11490.0 1.49760 0.748800 0.662796i \(-0.230631\pi\)
0.748800 + 0.662796i \(0.230631\pi\)
\(390\) 0 0
\(391\) 3528.00 0.456314
\(392\) 0 0
\(393\) −2436.00 −0.312672
\(394\) 0 0
\(395\) −2960.00 −0.377048
\(396\) 0 0
\(397\) 1866.00 0.235899 0.117949 0.993020i \(-0.462368\pi\)
0.117949 + 0.993020i \(0.462368\pi\)
\(398\) 0 0
\(399\) −2100.00 −0.263487
\(400\) 0 0
\(401\) 13662.0 1.70137 0.850683 0.525679i \(-0.176189\pi\)
0.850683 + 0.525679i \(0.176189\pi\)
\(402\) 0 0
\(403\) 2976.00 0.367854
\(404\) 0 0
\(405\) 324.000 0.0397523
\(406\) 0 0
\(407\) 15252.0 1.85753
\(408\) 0 0
\(409\) −13210.0 −1.59705 −0.798524 0.601963i \(-0.794385\pi\)
−0.798524 + 0.601963i \(0.794385\pi\)
\(410\) 0 0
\(411\) −1242.00 −0.149059
\(412\) 0 0
\(413\) 840.000 0.100082
\(414\) 0 0
\(415\) 1872.00 0.221429
\(416\) 0 0
\(417\) 4860.00 0.570732
\(418\) 0 0
\(419\) 6960.00 0.811499 0.405750 0.913984i \(-0.367010\pi\)
0.405750 + 0.913984i \(0.367010\pi\)
\(420\) 0 0
\(421\) −8162.00 −0.944873 −0.472437 0.881365i \(-0.656625\pi\)
−0.472437 + 0.881365i \(0.656625\pi\)
\(422\) 0 0
\(423\) −2916.00 −0.335179
\(424\) 0 0
\(425\) −9156.00 −1.04501
\(426\) 0 0
\(427\) −4354.00 −0.493454
\(428\) 0 0
\(429\) −11532.0 −1.29783
\(430\) 0 0
\(431\) −16602.0 −1.85543 −0.927715 0.373290i \(-0.878230\pi\)
−0.927715 + 0.373290i \(0.878230\pi\)
\(432\) 0 0
\(433\) 7738.00 0.858810 0.429405 0.903112i \(-0.358723\pi\)
0.429405 + 0.903112i \(0.358723\pi\)
\(434\) 0 0
\(435\) −120.000 −0.0132266
\(436\) 0 0
\(437\) 4200.00 0.459756
\(438\) 0 0
\(439\) 840.000 0.0913235 0.0456617 0.998957i \(-0.485460\pi\)
0.0456617 + 0.998957i \(0.485460\pi\)
\(440\) 0 0
\(441\) 441.000 0.0476190
\(442\) 0 0
\(443\) 6618.00 0.709776 0.354888 0.934909i \(-0.384519\pi\)
0.354888 + 0.934909i \(0.384519\pi\)
\(444\) 0 0
\(445\) 800.000 0.0852217
\(446\) 0 0
\(447\) 7110.00 0.752330
\(448\) 0 0
\(449\) 3090.00 0.324780 0.162390 0.986727i \(-0.448080\pi\)
0.162390 + 0.986727i \(0.448080\pi\)
\(450\) 0 0
\(451\) −15376.0 −1.60538
\(452\) 0 0
\(453\) −1704.00 −0.176735
\(454\) 0 0
\(455\) 1736.00 0.178868
\(456\) 0 0
\(457\) 5914.00 0.605351 0.302675 0.953094i \(-0.402120\pi\)
0.302675 + 0.953094i \(0.402120\pi\)
\(458\) 0 0
\(459\) −2268.00 −0.230634
\(460\) 0 0
\(461\) 15968.0 1.61324 0.806620 0.591070i \(-0.201294\pi\)
0.806620 + 0.591070i \(0.201294\pi\)
\(462\) 0 0
\(463\) 1172.00 0.117640 0.0588202 0.998269i \(-0.481266\pi\)
0.0588202 + 0.998269i \(0.481266\pi\)
\(464\) 0 0
\(465\) −576.000 −0.0574438
\(466\) 0 0
\(467\) 5304.00 0.525567 0.262784 0.964855i \(-0.415359\pi\)
0.262784 + 0.964855i \(0.415359\pi\)
\(468\) 0 0
\(469\) 6328.00 0.623027
\(470\) 0 0
\(471\) −798.000 −0.0780677
\(472\) 0 0
\(473\) 4216.00 0.409835
\(474\) 0 0
\(475\) −10900.0 −1.05290
\(476\) 0 0
\(477\) −2322.00 −0.222887
\(478\) 0 0
\(479\) −5740.00 −0.547531 −0.273765 0.961796i \(-0.588269\pi\)
−0.273765 + 0.961796i \(0.588269\pi\)
\(480\) 0 0
\(481\) 15252.0 1.44580
\(482\) 0 0
\(483\) −882.000 −0.0830898
\(484\) 0 0
\(485\) −5064.00 −0.474112
\(486\) 0 0
\(487\) −8944.00 −0.832220 −0.416110 0.909314i \(-0.636607\pi\)
−0.416110 + 0.909314i \(0.636607\pi\)
\(488\) 0 0
\(489\) 816.000 0.0754617
\(490\) 0 0
\(491\) −5558.00 −0.510853 −0.255427 0.966828i \(-0.582216\pi\)
−0.255427 + 0.966828i \(0.582216\pi\)
\(492\) 0 0
\(493\) 840.000 0.0767377
\(494\) 0 0
\(495\) 2232.00 0.202669
\(496\) 0 0
\(497\) 4746.00 0.428344
\(498\) 0 0
\(499\) −19820.0 −1.77809 −0.889043 0.457823i \(-0.848629\pi\)
−0.889043 + 0.457823i \(0.848629\pi\)
\(500\) 0 0
\(501\) −5628.00 −0.501877
\(502\) 0 0
\(503\) −1848.00 −0.163814 −0.0819068 0.996640i \(-0.526101\pi\)
−0.0819068 + 0.996640i \(0.526101\pi\)
\(504\) 0 0
\(505\) −928.000 −0.0817732
\(506\) 0 0
\(507\) −4941.00 −0.432816
\(508\) 0 0
\(509\) −340.000 −0.0296075 −0.0148038 0.999890i \(-0.504712\pi\)
−0.0148038 + 0.999890i \(0.504712\pi\)
\(510\) 0 0
\(511\) −4494.00 −0.389047
\(512\) 0 0
\(513\) −2700.00 −0.232374
\(514\) 0 0
\(515\) 7168.00 0.613320
\(516\) 0 0
\(517\) −20088.0 −1.70884
\(518\) 0 0
\(519\) −456.000 −0.0385668
\(520\) 0 0
\(521\) 10212.0 0.858725 0.429363 0.903132i \(-0.358738\pi\)
0.429363 + 0.903132i \(0.358738\pi\)
\(522\) 0 0
\(523\) −9332.00 −0.780229 −0.390115 0.920766i \(-0.627565\pi\)
−0.390115 + 0.920766i \(0.627565\pi\)
\(524\) 0 0
\(525\) 2289.00 0.190286
\(526\) 0 0
\(527\) 4032.00 0.333276
\(528\) 0 0
\(529\) −10403.0 −0.855018
\(530\) 0 0
\(531\) 1080.00 0.0882637
\(532\) 0 0
\(533\) −15376.0 −1.24955
\(534\) 0 0
\(535\) −7624.00 −0.616101
\(536\) 0 0
\(537\) −1830.00 −0.147058
\(538\) 0 0
\(539\) 3038.00 0.242775
\(540\) 0 0
\(541\) 8998.00 0.715073 0.357536 0.933899i \(-0.383617\pi\)
0.357536 + 0.933899i \(0.383617\pi\)
\(542\) 0 0
\(543\) 3126.00 0.247052
\(544\) 0 0
\(545\) 360.000 0.0282949
\(546\) 0 0
\(547\) −3416.00 −0.267016 −0.133508 0.991048i \(-0.542624\pi\)
−0.133508 + 0.991048i \(0.542624\pi\)
\(548\) 0 0
\(549\) −5598.00 −0.435185
\(550\) 0 0
\(551\) 1000.00 0.0773166
\(552\) 0 0
\(553\) −5180.00 −0.398329
\(554\) 0 0
\(555\) −2952.00 −0.225776
\(556\) 0 0
\(557\) 526.000 0.0400132 0.0200066 0.999800i \(-0.493631\pi\)
0.0200066 + 0.999800i \(0.493631\pi\)
\(558\) 0 0
\(559\) 4216.00 0.318994
\(560\) 0 0
\(561\) −15624.0 −1.17584
\(562\) 0 0
\(563\) −6712.00 −0.502446 −0.251223 0.967929i \(-0.580833\pi\)
−0.251223 + 0.967929i \(0.580833\pi\)
\(564\) 0 0
\(565\) 1832.00 0.136412
\(566\) 0 0
\(567\) 567.000 0.0419961
\(568\) 0 0
\(569\) 4190.00 0.308706 0.154353 0.988016i \(-0.450671\pi\)
0.154353 + 0.988016i \(0.450671\pi\)
\(570\) 0 0
\(571\) 3032.00 0.222216 0.111108 0.993808i \(-0.464560\pi\)
0.111108 + 0.993808i \(0.464560\pi\)
\(572\) 0 0
\(573\) −6114.00 −0.445752
\(574\) 0 0
\(575\) −4578.00 −0.332027
\(576\) 0 0
\(577\) 5434.00 0.392063 0.196032 0.980598i \(-0.437195\pi\)
0.196032 + 0.980598i \(0.437195\pi\)
\(578\) 0 0
\(579\) 7806.00 0.560287
\(580\) 0 0
\(581\) 3276.00 0.233927
\(582\) 0 0
\(583\) −15996.0 −1.13634
\(584\) 0 0
\(585\) 2232.00 0.157747
\(586\) 0 0
\(587\) 464.000 0.0326258 0.0163129 0.999867i \(-0.494807\pi\)
0.0163129 + 0.999867i \(0.494807\pi\)
\(588\) 0 0
\(589\) 4800.00 0.335790
\(590\) 0 0
\(591\) 7062.00 0.491526
\(592\) 0 0
\(593\) 11748.0 0.813546 0.406773 0.913529i \(-0.366654\pi\)
0.406773 + 0.913529i \(0.366654\pi\)
\(594\) 0 0
\(595\) 2352.00 0.162055
\(596\) 0 0
\(597\) 5040.00 0.345517
\(598\) 0 0
\(599\) −7650.00 −0.521821 −0.260910 0.965363i \(-0.584023\pi\)
−0.260910 + 0.965363i \(0.584023\pi\)
\(600\) 0 0
\(601\) −22878.0 −1.55277 −0.776384 0.630261i \(-0.782948\pi\)
−0.776384 + 0.630261i \(0.782948\pi\)
\(602\) 0 0
\(603\) 8136.00 0.549459
\(604\) 0 0
\(605\) 10052.0 0.675491
\(606\) 0 0
\(607\) −704.000 −0.0470749 −0.0235375 0.999723i \(-0.507493\pi\)
−0.0235375 + 0.999723i \(0.507493\pi\)
\(608\) 0 0
\(609\) −210.000 −0.0139731
\(610\) 0 0
\(611\) −20088.0 −1.33007
\(612\) 0 0
\(613\) −24958.0 −1.64444 −0.822222 0.569167i \(-0.807266\pi\)
−0.822222 + 0.569167i \(0.807266\pi\)
\(614\) 0 0
\(615\) 2976.00 0.195128
\(616\) 0 0
\(617\) −8826.00 −0.575886 −0.287943 0.957648i \(-0.592971\pi\)
−0.287943 + 0.957648i \(0.592971\pi\)
\(618\) 0 0
\(619\) 21220.0 1.37787 0.688937 0.724821i \(-0.258078\pi\)
0.688937 + 0.724821i \(0.258078\pi\)
\(620\) 0 0
\(621\) −1134.00 −0.0732783
\(622\) 0 0
\(623\) 1400.00 0.0900318
\(624\) 0 0
\(625\) 9881.00 0.632384
\(626\) 0 0
\(627\) −18600.0 −1.18471
\(628\) 0 0
\(629\) 20664.0 1.30990
\(630\) 0 0
\(631\) 3268.00 0.206176 0.103088 0.994672i \(-0.467128\pi\)
0.103088 + 0.994672i \(0.467128\pi\)
\(632\) 0 0
\(633\) 2004.00 0.125832
\(634\) 0 0
\(635\) −3216.00 −0.200981
\(636\) 0 0
\(637\) 3038.00 0.188964
\(638\) 0 0
\(639\) 6102.00 0.377764
\(640\) 0 0
\(641\) 13062.0 0.804864 0.402432 0.915450i \(-0.368165\pi\)
0.402432 + 0.915450i \(0.368165\pi\)
\(642\) 0 0
\(643\) −28012.0 −1.71802 −0.859009 0.511961i \(-0.828919\pi\)
−0.859009 + 0.511961i \(0.828919\pi\)
\(644\) 0 0
\(645\) −816.000 −0.0498139
\(646\) 0 0
\(647\) −3844.00 −0.233575 −0.116788 0.993157i \(-0.537260\pi\)
−0.116788 + 0.993157i \(0.537260\pi\)
\(648\) 0 0
\(649\) 7440.00 0.449993
\(650\) 0 0
\(651\) −1008.00 −0.0606861
\(652\) 0 0
\(653\) 28482.0 1.70687 0.853436 0.521198i \(-0.174515\pi\)
0.853436 + 0.521198i \(0.174515\pi\)
\(654\) 0 0
\(655\) 3248.00 0.193756
\(656\) 0 0
\(657\) −5778.00 −0.343107
\(658\) 0 0
\(659\) −9330.00 −0.551510 −0.275755 0.961228i \(-0.588928\pi\)
−0.275755 + 0.961228i \(0.588928\pi\)
\(660\) 0 0
\(661\) −8782.00 −0.516763 −0.258381 0.966043i \(-0.583189\pi\)
−0.258381 + 0.966043i \(0.583189\pi\)
\(662\) 0 0
\(663\) −15624.0 −0.915212
\(664\) 0 0
\(665\) 2800.00 0.163277
\(666\) 0 0
\(667\) 420.000 0.0243815
\(668\) 0 0
\(669\) −5496.00 −0.317620
\(670\) 0 0
\(671\) −38564.0 −2.21870
\(672\) 0 0
\(673\) −10562.0 −0.604956 −0.302478 0.953156i \(-0.597814\pi\)
−0.302478 + 0.953156i \(0.597814\pi\)
\(674\) 0 0
\(675\) 2943.00 0.167816
\(676\) 0 0
\(677\) 26016.0 1.47692 0.738461 0.674296i \(-0.235553\pi\)
0.738461 + 0.674296i \(0.235553\pi\)
\(678\) 0 0
\(679\) −8862.00 −0.500872
\(680\) 0 0
\(681\) −14832.0 −0.834601
\(682\) 0 0
\(683\) 8898.00 0.498496 0.249248 0.968440i \(-0.419817\pi\)
0.249248 + 0.968440i \(0.419817\pi\)
\(684\) 0 0
\(685\) 1656.00 0.0923686
\(686\) 0 0
\(687\) −16410.0 −0.911325
\(688\) 0 0
\(689\) −15996.0 −0.884469
\(690\) 0 0
\(691\) 30572.0 1.68309 0.841544 0.540189i \(-0.181647\pi\)
0.841544 + 0.540189i \(0.181647\pi\)
\(692\) 0 0
\(693\) 3906.00 0.214108
\(694\) 0 0
\(695\) −6480.00 −0.353670
\(696\) 0 0
\(697\) −20832.0 −1.13209
\(698\) 0 0
\(699\) 8406.00 0.454856
\(700\) 0 0
\(701\) 30618.0 1.64968 0.824840 0.565366i \(-0.191265\pi\)
0.824840 + 0.565366i \(0.191265\pi\)
\(702\) 0 0
\(703\) 24600.0 1.31978
\(704\) 0 0
\(705\) 3888.00 0.207703
\(706\) 0 0
\(707\) −1624.00 −0.0863887
\(708\) 0 0
\(709\) 8130.00 0.430647 0.215323 0.976543i \(-0.430919\pi\)
0.215323 + 0.976543i \(0.430919\pi\)
\(710\) 0 0
\(711\) −6660.00 −0.351293
\(712\) 0 0
\(713\) 2016.00 0.105890
\(714\) 0 0
\(715\) 15376.0 0.804237
\(716\) 0 0
\(717\) −3510.00 −0.182822
\(718\) 0 0
\(719\) 27840.0 1.44403 0.722014 0.691878i \(-0.243216\pi\)
0.722014 + 0.691878i \(0.243216\pi\)
\(720\) 0 0
\(721\) 12544.0 0.647938
\(722\) 0 0
\(723\) 7014.00 0.360793
\(724\) 0 0
\(725\) −1090.00 −0.0558367
\(726\) 0 0
\(727\) −14624.0 −0.746044 −0.373022 0.927822i \(-0.621678\pi\)
−0.373022 + 0.927822i \(0.621678\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 5712.00 0.289010
\(732\) 0 0
\(733\) 20862.0 1.05124 0.525618 0.850721i \(-0.323834\pi\)
0.525618 + 0.850721i \(0.323834\pi\)
\(734\) 0 0
\(735\) −588.000 −0.0295084
\(736\) 0 0
\(737\) 56048.0 2.80130
\(738\) 0 0
\(739\) −13920.0 −0.692903 −0.346452 0.938068i \(-0.612614\pi\)
−0.346452 + 0.938068i \(0.612614\pi\)
\(740\) 0 0
\(741\) −18600.0 −0.922116
\(742\) 0 0
\(743\) −25578.0 −1.26294 −0.631471 0.775400i \(-0.717548\pi\)
−0.631471 + 0.775400i \(0.717548\pi\)
\(744\) 0 0
\(745\) −9480.00 −0.466202
\(746\) 0 0
\(747\) 4212.00 0.206304
\(748\) 0 0
\(749\) −13342.0 −0.650876
\(750\) 0 0
\(751\) −33472.0 −1.62638 −0.813189 0.581999i \(-0.802271\pi\)
−0.813189 + 0.581999i \(0.802271\pi\)
\(752\) 0 0
\(753\) −8376.00 −0.405363
\(754\) 0 0
\(755\) 2272.00 0.109519
\(756\) 0 0
\(757\) −25934.0 −1.24516 −0.622581 0.782556i \(-0.713916\pi\)
−0.622581 + 0.782556i \(0.713916\pi\)
\(758\) 0 0
\(759\) −7812.00 −0.373594
\(760\) 0 0
\(761\) 26952.0 1.28385 0.641925 0.766768i \(-0.278136\pi\)
0.641925 + 0.766768i \(0.278136\pi\)
\(762\) 0 0
\(763\) 630.000 0.0298919
\(764\) 0 0
\(765\) 3024.00 0.142919
\(766\) 0 0
\(767\) 7440.00 0.350251
\(768\) 0 0
\(769\) 23450.0 1.09965 0.549824 0.835281i \(-0.314695\pi\)
0.549824 + 0.835281i \(0.314695\pi\)
\(770\) 0 0
\(771\) −21072.0 −0.984293
\(772\) 0 0
\(773\) −39568.0 −1.84109 −0.920545 0.390637i \(-0.872255\pi\)
−0.920545 + 0.390637i \(0.872255\pi\)
\(774\) 0 0
\(775\) −5232.00 −0.242502
\(776\) 0 0
\(777\) −5166.00 −0.238519
\(778\) 0 0
\(779\) −24800.0 −1.14063
\(780\) 0 0
\(781\) 42036.0 1.92595
\(782\) 0 0
\(783\) −270.000 −0.0123231
\(784\) 0 0
\(785\) 1064.00 0.0483768
\(786\) 0 0
\(787\) −12356.0 −0.559649 −0.279825 0.960051i \(-0.590276\pi\)
−0.279825 + 0.960051i \(0.590276\pi\)
\(788\) 0 0
\(789\) 7314.00 0.330019
\(790\) 0 0
\(791\) 3206.00 0.144112
\(792\) 0 0
\(793\) −38564.0 −1.72692
\(794\) 0 0
\(795\) 3096.00 0.138118
\(796\) 0 0
\(797\) 21736.0 0.966033 0.483017 0.875611i \(-0.339541\pi\)
0.483017 + 0.875611i \(0.339541\pi\)
\(798\) 0 0
\(799\) −27216.0 −1.20505
\(800\) 0 0
\(801\) 1800.00 0.0794006
\(802\) 0 0
\(803\) −39804.0 −1.74926
\(804\) 0 0
\(805\) 1176.00 0.0514889
\(806\) 0 0
\(807\) −20340.0 −0.887239
\(808\) 0 0
\(809\) −38310.0 −1.66490 −0.832452 0.554097i \(-0.813064\pi\)
−0.832452 + 0.554097i \(0.813064\pi\)
\(810\) 0 0
\(811\) 2132.00 0.0923115 0.0461558 0.998934i \(-0.485303\pi\)
0.0461558 + 0.998934i \(0.485303\pi\)
\(812\) 0 0
\(813\) −5784.00 −0.249513
\(814\) 0 0
\(815\) −1088.00 −0.0467619
\(816\) 0 0
\(817\) 6800.00 0.291190
\(818\) 0 0
\(819\) 3906.00 0.166650
\(820\) 0 0
\(821\) −5002.00 −0.212632 −0.106316 0.994332i \(-0.533906\pi\)
−0.106316 + 0.994332i \(0.533906\pi\)
\(822\) 0 0
\(823\) 3612.00 0.152985 0.0764923 0.997070i \(-0.475628\pi\)
0.0764923 + 0.997070i \(0.475628\pi\)
\(824\) 0 0
\(825\) 20274.0 0.855576
\(826\) 0 0
\(827\) −27666.0 −1.16329 −0.581645 0.813443i \(-0.697591\pi\)
−0.581645 + 0.813443i \(0.697591\pi\)
\(828\) 0 0
\(829\) −12890.0 −0.540034 −0.270017 0.962856i \(-0.587029\pi\)
−0.270017 + 0.962856i \(0.587029\pi\)
\(830\) 0 0
\(831\) 16662.0 0.695546
\(832\) 0 0
\(833\) 4116.00 0.171202
\(834\) 0 0
\(835\) 7504.00 0.311002
\(836\) 0 0
\(837\) −1296.00 −0.0535201
\(838\) 0 0
\(839\) 9340.00 0.384330 0.192165 0.981363i \(-0.438449\pi\)
0.192165 + 0.981363i \(0.438449\pi\)
\(840\) 0 0
\(841\) −24289.0 −0.995900
\(842\) 0 0
\(843\) −5826.00 −0.238029
\(844\) 0 0
\(845\) 6588.00 0.268206
\(846\) 0 0
\(847\) 17591.0 0.713617
\(848\) 0 0
\(849\) −14484.0 −0.585500
\(850\) 0 0
\(851\) 10332.0 0.416188
\(852\) 0 0
\(853\) 33082.0 1.32791 0.663954 0.747773i \(-0.268877\pi\)
0.663954 + 0.747773i \(0.268877\pi\)
\(854\) 0 0
\(855\) 3600.00 0.143997
\(856\) 0 0
\(857\) 7544.00 0.300698 0.150349 0.988633i \(-0.451960\pi\)
0.150349 + 0.988633i \(0.451960\pi\)
\(858\) 0 0
\(859\) 8180.00 0.324910 0.162455 0.986716i \(-0.448059\pi\)
0.162455 + 0.986716i \(0.448059\pi\)
\(860\) 0 0
\(861\) 5208.00 0.206142
\(862\) 0 0
\(863\) −10518.0 −0.414875 −0.207437 0.978248i \(-0.566512\pi\)
−0.207437 + 0.978248i \(0.566512\pi\)
\(864\) 0 0
\(865\) 608.000 0.0238990
\(866\) 0 0
\(867\) −6429.00 −0.251834
\(868\) 0 0
\(869\) −45880.0 −1.79099
\(870\) 0 0
\(871\) 56048.0 2.18038
\(872\) 0 0
\(873\) −11394.0 −0.441728
\(874\) 0 0
\(875\) −6552.00 −0.253141
\(876\) 0 0
\(877\) −14134.0 −0.544209 −0.272104 0.962268i \(-0.587720\pi\)
−0.272104 + 0.962268i \(0.587720\pi\)
\(878\) 0 0
\(879\) −18456.0 −0.708197
\(880\) 0 0
\(881\) 6492.00 0.248265 0.124132 0.992266i \(-0.460385\pi\)
0.124132 + 0.992266i \(0.460385\pi\)
\(882\) 0 0
\(883\) 38228.0 1.45694 0.728468 0.685080i \(-0.240233\pi\)
0.728468 + 0.685080i \(0.240233\pi\)
\(884\) 0 0
\(885\) −1440.00 −0.0546950
\(886\) 0 0
\(887\) 43076.0 1.63061 0.815305 0.579032i \(-0.196569\pi\)
0.815305 + 0.579032i \(0.196569\pi\)
\(888\) 0 0
\(889\) −5628.00 −0.212325
\(890\) 0 0
\(891\) 5022.00 0.188825
\(892\) 0 0
\(893\) −32400.0 −1.21414
\(894\) 0 0
\(895\) 2440.00 0.0911287
\(896\) 0 0
\(897\) −7812.00 −0.290786
\(898\) 0 0
\(899\) 480.000 0.0178074
\(900\) 0 0
\(901\) −21672.0 −0.801331
\(902\) 0 0
\(903\) −1428.00 −0.0526255
\(904\) 0 0
\(905\) −4168.00 −0.153093
\(906\) 0 0
\(907\) −32236.0 −1.18013 −0.590065 0.807355i \(-0.700898\pi\)
−0.590065 + 0.807355i \(0.700898\pi\)
\(908\) 0 0
\(909\) −2088.00 −0.0761877
\(910\) 0 0
\(911\) 46518.0 1.69178 0.845889 0.533359i \(-0.179070\pi\)
0.845889 + 0.533359i \(0.179070\pi\)
\(912\) 0 0
\(913\) 29016.0 1.05180
\(914\) 0 0
\(915\) 7464.00 0.269675
\(916\) 0 0
\(917\) 5684.00 0.204692
\(918\) 0 0
\(919\) −17840.0 −0.640356 −0.320178 0.947357i \(-0.603743\pi\)
−0.320178 + 0.947357i \(0.603743\pi\)
\(920\) 0 0
\(921\) −17652.0 −0.631545
\(922\) 0 0
\(923\) 42036.0 1.49906
\(924\) 0 0
\(925\) −26814.0 −0.953123
\(926\) 0 0
\(927\) 16128.0 0.571427
\(928\) 0 0
\(929\) 7000.00 0.247215 0.123607 0.992331i \(-0.460554\pi\)
0.123607 + 0.992331i \(0.460554\pi\)
\(930\) 0 0
\(931\) 4900.00 0.172493
\(932\) 0 0
\(933\) 27396.0 0.961313
\(934\) 0 0
\(935\) 20832.0 0.728641
\(936\) 0 0
\(937\) 36114.0 1.25912 0.629559 0.776953i \(-0.283236\pi\)
0.629559 + 0.776953i \(0.283236\pi\)
\(938\) 0 0
\(939\) 28146.0 0.978179
\(940\) 0 0
\(941\) 4748.00 0.164485 0.0822425 0.996612i \(-0.473792\pi\)
0.0822425 + 0.996612i \(0.473792\pi\)
\(942\) 0 0
\(943\) −10416.0 −0.359694
\(944\) 0 0
\(945\) −756.000 −0.0260240
\(946\) 0 0
\(947\) 42694.0 1.46501 0.732507 0.680759i \(-0.238350\pi\)
0.732507 + 0.680759i \(0.238350\pi\)
\(948\) 0 0
\(949\) −39804.0 −1.36153
\(950\) 0 0
\(951\) 9342.00 0.318544
\(952\) 0 0
\(953\) −16742.0 −0.569073 −0.284537 0.958665i \(-0.591840\pi\)
−0.284537 + 0.958665i \(0.591840\pi\)
\(954\) 0 0
\(955\) 8152.00 0.276223
\(956\) 0 0
\(957\) −1860.00 −0.0628268
\(958\) 0 0
\(959\) 2898.00 0.0975822
\(960\) 0 0
\(961\) −27487.0 −0.922661
\(962\) 0 0
\(963\) −17154.0 −0.574019
\(964\) 0 0
\(965\) −10408.0 −0.347197
\(966\) 0 0
\(967\) 9956.00 0.331089 0.165545 0.986202i \(-0.447062\pi\)
0.165545 + 0.986202i \(0.447062\pi\)
\(968\) 0 0
\(969\) −25200.0 −0.835439
\(970\) 0 0
\(971\) −26388.0 −0.872123 −0.436061 0.899917i \(-0.643627\pi\)
−0.436061 + 0.899917i \(0.643627\pi\)
\(972\) 0 0
\(973\) −11340.0 −0.373632
\(974\) 0 0
\(975\) 20274.0 0.665936
\(976\) 0 0
\(977\) −786.000 −0.0257383 −0.0128692 0.999917i \(-0.504096\pi\)
−0.0128692 + 0.999917i \(0.504096\pi\)
\(978\) 0 0
\(979\) 12400.0 0.404807
\(980\) 0 0
\(981\) 810.000 0.0263622
\(982\) 0 0
\(983\) −51888.0 −1.68359 −0.841796 0.539796i \(-0.818501\pi\)
−0.841796 + 0.539796i \(0.818501\pi\)
\(984\) 0 0
\(985\) −9416.00 −0.304588
\(986\) 0 0
\(987\) 6804.00 0.219426
\(988\) 0 0
\(989\) 2856.00 0.0918256
\(990\) 0 0
\(991\) 51928.0 1.66453 0.832264 0.554379i \(-0.187044\pi\)
0.832264 + 0.554379i \(0.187044\pi\)
\(992\) 0 0
\(993\) −4596.00 −0.146878
\(994\) 0 0
\(995\) −6720.00 −0.214109
\(996\) 0 0
\(997\) 386.000 0.0122615 0.00613076 0.999981i \(-0.498049\pi\)
0.00613076 + 0.999981i \(0.498049\pi\)
\(998\) 0 0
\(999\) −6642.00 −0.210354
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.i.1.1 1
4.3 odd 2 1344.4.a.w.1.1 1
8.3 odd 2 21.4.a.b.1.1 1
8.5 even 2 336.4.a.h.1.1 1
24.5 odd 2 1008.4.a.m.1.1 1
24.11 even 2 63.4.a.a.1.1 1
40.3 even 4 525.4.d.b.274.1 2
40.19 odd 2 525.4.a.b.1.1 1
40.27 even 4 525.4.d.b.274.2 2
56.3 even 6 147.4.e.b.79.1 2
56.11 odd 6 147.4.e.c.79.1 2
56.13 odd 2 2352.4.a.l.1.1 1
56.19 even 6 147.4.e.b.67.1 2
56.27 even 2 147.4.a.g.1.1 1
56.51 odd 6 147.4.e.c.67.1 2
120.59 even 2 1575.4.a.k.1.1 1
168.11 even 6 441.4.e.m.226.1 2
168.59 odd 6 441.4.e.n.226.1 2
168.83 odd 2 441.4.a.b.1.1 1
168.107 even 6 441.4.e.m.361.1 2
168.131 odd 6 441.4.e.n.361.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.4.a.b.1.1 1 8.3 odd 2
63.4.a.a.1.1 1 24.11 even 2
147.4.a.g.1.1 1 56.27 even 2
147.4.e.b.67.1 2 56.19 even 6
147.4.e.b.79.1 2 56.3 even 6
147.4.e.c.67.1 2 56.51 odd 6
147.4.e.c.79.1 2 56.11 odd 6
336.4.a.h.1.1 1 8.5 even 2
441.4.a.b.1.1 1 168.83 odd 2
441.4.e.m.226.1 2 168.11 even 6
441.4.e.m.361.1 2 168.107 even 6
441.4.e.n.226.1 2 168.59 odd 6
441.4.e.n.361.1 2 168.131 odd 6
525.4.a.b.1.1 1 40.19 odd 2
525.4.d.b.274.1 2 40.3 even 4
525.4.d.b.274.2 2 40.27 even 4
1008.4.a.m.1.1 1 24.5 odd 2
1344.4.a.i.1.1 1 1.1 even 1 trivial
1344.4.a.w.1.1 1 4.3 odd 2
1575.4.a.k.1.1 1 120.59 even 2
2352.4.a.l.1.1 1 56.13 odd 2