Properties

Label 882.5.b.c.197.3
Level $882$
Weight $5$
Character 882.197
Analytic conductor $91.172$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 882.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(91.1723074400\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-53})\)
Defining polynomial: \(x^{4} - 52 x^{2} + 729\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 197.3
Root \(5.14782 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 882.197
Dual form 882.5.b.c.197.2

$q$-expansion

\(f(q)\) \(=\) \(q+2.82843i q^{2} -8.00000 q^{4} -29.1204i q^{5} -22.6274i q^{8} +O(q^{10})\) \(q+2.82843i q^{2} -8.00000 q^{4} -29.1204i q^{5} -22.6274i q^{8} +82.3650 q^{10} -1.41421i q^{11} -82.3650 q^{13} +64.0000 q^{16} -262.084i q^{17} -453.008 q^{19} +232.964i q^{20} +4.00000 q^{22} +63.6396i q^{23} -223.000 q^{25} -232.964i q^{26} -502.046i q^{29} -1276.66 q^{31} +181.019i q^{32} +741.285 q^{34} +1560.00 q^{37} -1281.30i q^{38} -658.920 q^{40} +1310.42i q^{41} +328.000 q^{43} +11.3137i q^{44} -180.000 q^{46} -4135.10i q^{47} -630.739i q^{50} +658.920 q^{52} +4012.12i q^{53} -41.1825 q^{55} +1420.00 q^{58} +757.131i q^{59} +1194.29 q^{61} -3610.93i q^{62} -512.000 q^{64} +2398.51i q^{65} -1130.00 q^{67} +2096.67i q^{68} +422.850i q^{71} -3006.32 q^{73} +4412.35i q^{74} +3624.06 q^{76} +8650.00 q^{79} -1863.71i q^{80} -3706.43 q^{82} +2970.28i q^{83} -7632.00 q^{85} +927.724i q^{86} -32.0000 q^{88} +13308.0i q^{89} -509.117i q^{92} +11695.8 q^{94} +13191.8i q^{95} -13137.2 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 32 q^{4} + O(q^{10}) \) \( 4 q - 32 q^{4} + 256 q^{16} + 16 q^{22} - 892 q^{25} + 6240 q^{37} + 1312 q^{43} - 720 q^{46} + 5680 q^{58} - 2048 q^{64} - 4520 q^{67} + 34600 q^{79} - 30528 q^{85} - 128 q^{88} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/882\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(785\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.82843i 0.707107i
\(3\) 0 0
\(4\) −8.00000 −0.500000
\(5\) − 29.1204i − 1.16482i −0.812896 0.582409i \(-0.802110\pi\)
0.812896 0.582409i \(-0.197890\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) − 22.6274i − 0.353553i
\(9\) 0 0
\(10\) 82.3650 0.823650
\(11\) − 1.41421i − 0.0116877i −0.999983 0.00584386i \(-0.998140\pi\)
0.999983 0.00584386i \(-0.00186017\pi\)
\(12\) 0 0
\(13\) −82.3650 −0.487367 −0.243684 0.969855i \(-0.578356\pi\)
−0.243684 + 0.969855i \(0.578356\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 64.0000 0.250000
\(17\) − 262.084i − 0.906865i −0.891291 0.453432i \(-0.850199\pi\)
0.891291 0.453432i \(-0.149801\pi\)
\(18\) 0 0
\(19\) −453.008 −1.25487 −0.627435 0.778669i \(-0.715895\pi\)
−0.627435 + 0.778669i \(0.715895\pi\)
\(20\) 232.964i 0.582409i
\(21\) 0 0
\(22\) 4.00000 0.00826446
\(23\) 63.6396i 0.120302i 0.998189 + 0.0601509i \(0.0191582\pi\)
−0.998189 + 0.0601509i \(0.980842\pi\)
\(24\) 0 0
\(25\) −223.000 −0.356800
\(26\) − 232.964i − 0.344621i
\(27\) 0 0
\(28\) 0 0
\(29\) − 502.046i − 0.596963i −0.954415 0.298481i \(-0.903520\pi\)
0.954415 0.298481i \(-0.0964801\pi\)
\(30\) 0 0
\(31\) −1276.66 −1.32847 −0.664234 0.747525i \(-0.731242\pi\)
−0.664234 + 0.747525i \(0.731242\pi\)
\(32\) 181.019i 0.176777i
\(33\) 0 0
\(34\) 741.285 0.641250
\(35\) 0 0
\(36\) 0 0
\(37\) 1560.00 1.13952 0.569759 0.821812i \(-0.307037\pi\)
0.569759 + 0.821812i \(0.307037\pi\)
\(38\) − 1281.30i − 0.887326i
\(39\) 0 0
\(40\) −658.920 −0.411825
\(41\) 1310.42i 0.779548i 0.920911 + 0.389774i \(0.127447\pi\)
−0.920911 + 0.389774i \(0.872553\pi\)
\(42\) 0 0
\(43\) 328.000 0.177393 0.0886966 0.996059i \(-0.471730\pi\)
0.0886966 + 0.996059i \(0.471730\pi\)
\(44\) 11.3137i 0.00584386i
\(45\) 0 0
\(46\) −180.000 −0.0850662
\(47\) − 4135.10i − 1.87193i −0.352088 0.935967i \(-0.614528\pi\)
0.352088 0.935967i \(-0.385472\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) − 630.739i − 0.252296i
\(51\) 0 0
\(52\) 658.920 0.243684
\(53\) 4012.12i 1.42831i 0.699987 + 0.714155i \(0.253189\pi\)
−0.699987 + 0.714155i \(0.746811\pi\)
\(54\) 0 0
\(55\) −41.1825 −0.0136141
\(56\) 0 0
\(57\) 0 0
\(58\) 1420.00 0.422117
\(59\) 757.131i 0.217504i 0.994069 + 0.108752i \(0.0346854\pi\)
−0.994069 + 0.108752i \(0.965315\pi\)
\(60\) 0 0
\(61\) 1194.29 0.320960 0.160480 0.987039i \(-0.448696\pi\)
0.160480 + 0.987039i \(0.448696\pi\)
\(62\) − 3610.93i − 0.939369i
\(63\) 0 0
\(64\) −512.000 −0.125000
\(65\) 2398.51i 0.567694i
\(66\) 0 0
\(67\) −1130.00 −0.251726 −0.125863 0.992048i \(-0.540170\pi\)
−0.125863 + 0.992048i \(0.540170\pi\)
\(68\) 2096.67i 0.453432i
\(69\) 0 0
\(70\) 0 0
\(71\) 422.850i 0.0838821i 0.999120 + 0.0419411i \(0.0133542\pi\)
−0.999120 + 0.0419411i \(0.986646\pi\)
\(72\) 0 0
\(73\) −3006.32 −0.564144 −0.282072 0.959393i \(-0.591022\pi\)
−0.282072 + 0.959393i \(0.591022\pi\)
\(74\) 4412.35i 0.805761i
\(75\) 0 0
\(76\) 3624.06 0.627435
\(77\) 0 0
\(78\) 0 0
\(79\) 8650.00 1.38600 0.692998 0.720940i \(-0.256289\pi\)
0.692998 + 0.720940i \(0.256289\pi\)
\(80\) − 1863.71i − 0.291204i
\(81\) 0 0
\(82\) −3706.43 −0.551224
\(83\) 2970.28i 0.431163i 0.976486 + 0.215582i \(0.0691648\pi\)
−0.976486 + 0.215582i \(0.930835\pi\)
\(84\) 0 0
\(85\) −7632.00 −1.05633
\(86\) 927.724i 0.125436i
\(87\) 0 0
\(88\) −32.0000 −0.00413223
\(89\) 13308.0i 1.68010i 0.542512 + 0.840048i \(0.317473\pi\)
−0.542512 + 0.840048i \(0.682527\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) − 509.117i − 0.0601509i
\(93\) 0 0
\(94\) 11695.8 1.32366
\(95\) 13191.8i 1.46169i
\(96\) 0 0
\(97\) −13137.2 −1.39624 −0.698120 0.715981i \(-0.745980\pi\)
−0.698120 + 0.715981i \(0.745980\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 1784.00 0.178400
\(101\) 12026.7i 1.17898i 0.807777 + 0.589488i \(0.200671\pi\)
−0.807777 + 0.589488i \(0.799329\pi\)
\(102\) 0 0
\(103\) 10913.4 1.02869 0.514345 0.857583i \(-0.328035\pi\)
0.514345 + 0.857583i \(0.328035\pi\)
\(104\) 1863.71i 0.172310i
\(105\) 0 0
\(106\) −11348.0 −1.00997
\(107\) 12703.9i 1.10961i 0.831982 + 0.554803i \(0.187206\pi\)
−0.831982 + 0.554803i \(0.812794\pi\)
\(108\) 0 0
\(109\) −9752.00 −0.820806 −0.410403 0.911904i \(-0.634612\pi\)
−0.410403 + 0.911904i \(0.634612\pi\)
\(110\) − 116.482i − 0.00962659i
\(111\) 0 0
\(112\) 0 0
\(113\) 11303.8i 0.885254i 0.896706 + 0.442627i \(0.145953\pi\)
−0.896706 + 0.442627i \(0.854047\pi\)
\(114\) 0 0
\(115\) 1853.21 0.140130
\(116\) 4016.37i 0.298481i
\(117\) 0 0
\(118\) −2141.49 −0.153799
\(119\) 0 0
\(120\) 0 0
\(121\) 14639.0 0.999863
\(122\) 3377.97i 0.226953i
\(123\) 0 0
\(124\) 10213.3 0.664234
\(125\) − 11706.4i − 0.749211i
\(126\) 0 0
\(127\) −30922.0 −1.91717 −0.958584 0.284810i \(-0.908069\pi\)
−0.958584 + 0.284810i \(0.908069\pi\)
\(128\) − 1448.15i − 0.0883883i
\(129\) 0 0
\(130\) −6784.00 −0.401420
\(131\) − 5940.57i − 0.346167i −0.984907 0.173083i \(-0.944627\pi\)
0.984907 0.173083i \(-0.0553730\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) − 3196.12i − 0.177997i
\(135\) 0 0
\(136\) −5930.28 −0.320625
\(137\) 18929.2i 1.00854i 0.863547 + 0.504269i \(0.168238\pi\)
−0.863547 + 0.504269i \(0.831762\pi\)
\(138\) 0 0
\(139\) 10872.2 0.562713 0.281357 0.959603i \(-0.409216\pi\)
0.281357 + 0.959603i \(0.409216\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −1196.00 −0.0593136
\(143\) 116.482i 0.00569621i
\(144\) 0 0
\(145\) −14619.8 −0.695353
\(146\) − 8503.17i − 0.398910i
\(147\) 0 0
\(148\) −12480.0 −0.569759
\(149\) 7448.66i 0.335510i 0.985829 + 0.167755i \(0.0536518\pi\)
−0.985829 + 0.167755i \(0.946348\pi\)
\(150\) 0 0
\(151\) −35792.0 −1.56976 −0.784878 0.619651i \(-0.787274\pi\)
−0.784878 + 0.619651i \(0.787274\pi\)
\(152\) 10250.4i 0.443663i
\(153\) 0 0
\(154\) 0 0
\(155\) 37176.8i 1.54742i
\(156\) 0 0
\(157\) −2676.86 −0.108599 −0.0542996 0.998525i \(-0.517293\pi\)
−0.0542996 + 0.998525i \(0.517293\pi\)
\(158\) 24465.9i 0.980047i
\(159\) 0 0
\(160\) 5271.36 0.205913
\(161\) 0 0
\(162\) 0 0
\(163\) 19254.0 0.724679 0.362340 0.932046i \(-0.381978\pi\)
0.362340 + 0.932046i \(0.381978\pi\)
\(164\) − 10483.4i − 0.389774i
\(165\) 0 0
\(166\) −8401.23 −0.304879
\(167\) 12521.8i 0.448987i 0.974476 + 0.224493i \(0.0720727\pi\)
−0.974476 + 0.224493i \(0.927927\pi\)
\(168\) 0 0
\(169\) −21777.0 −0.762473
\(170\) − 21586.6i − 0.746940i
\(171\) 0 0
\(172\) −2624.00 −0.0886966
\(173\) − 23558.4i − 0.787144i −0.919294 0.393572i \(-0.871239\pi\)
0.919294 0.393572i \(-0.128761\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) − 90.5097i − 0.00292193i
\(177\) 0 0
\(178\) −37640.8 −1.18801
\(179\) 25511.0i 0.796199i 0.917342 + 0.398099i \(0.130330\pi\)
−0.917342 + 0.398099i \(0.869670\pi\)
\(180\) 0 0
\(181\) 38876.3 1.18666 0.593332 0.804958i \(-0.297812\pi\)
0.593332 + 0.804958i \(0.297812\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 1440.00 0.0425331
\(185\) − 45427.9i − 1.32733i
\(186\) 0 0
\(187\) −370.643 −0.0105992
\(188\) 33080.8i 0.935967i
\(189\) 0 0
\(190\) −37312.0 −1.03357
\(191\) − 34352.7i − 0.941659i −0.882224 0.470829i \(-0.843955\pi\)
0.882224 0.470829i \(-0.156045\pi\)
\(192\) 0 0
\(193\) −65824.0 −1.76713 −0.883567 0.468304i \(-0.844865\pi\)
−0.883567 + 0.468304i \(0.844865\pi\)
\(194\) − 37157.7i − 0.987291i
\(195\) 0 0
\(196\) 0 0
\(197\) 64831.8i 1.67053i 0.549844 + 0.835267i \(0.314687\pi\)
−0.549844 + 0.835267i \(0.685313\pi\)
\(198\) 0 0
\(199\) 46536.2 1.17513 0.587564 0.809178i \(-0.300087\pi\)
0.587564 + 0.809178i \(0.300087\pi\)
\(200\) 5045.91i 0.126148i
\(201\) 0 0
\(202\) −34016.8 −0.833662
\(203\) 0 0
\(204\) 0 0
\(205\) 38160.0 0.908031
\(206\) 30867.7i 0.727393i
\(207\) 0 0
\(208\) −5271.36 −0.121842
\(209\) 640.650i 0.0146666i
\(210\) 0 0
\(211\) 8504.00 0.191011 0.0955055 0.995429i \(-0.469553\pi\)
0.0955055 + 0.995429i \(0.469553\pi\)
\(212\) − 32097.0i − 0.714155i
\(213\) 0 0
\(214\) −35932.0 −0.784610
\(215\) − 9551.50i − 0.206631i
\(216\) 0 0
\(217\) 0 0
\(218\) − 27582.8i − 0.580398i
\(219\) 0 0
\(220\) 329.460 0.00680703
\(221\) 21586.6i 0.441976i
\(222\) 0 0
\(223\) −56749.5 −1.14118 −0.570588 0.821237i \(-0.693285\pi\)
−0.570588 + 0.821237i \(0.693285\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −31972.0 −0.625969
\(227\) − 55969.5i − 1.08617i −0.839676 0.543087i \(-0.817255\pi\)
0.839676 0.543087i \(-0.182745\pi\)
\(228\) 0 0
\(229\) −84588.9 −1.61303 −0.806515 0.591214i \(-0.798649\pi\)
−0.806515 + 0.591214i \(0.798649\pi\)
\(230\) 5241.68i 0.0990866i
\(231\) 0 0
\(232\) −11360.0 −0.211058
\(233\) − 35469.9i − 0.653353i −0.945136 0.326677i \(-0.894071\pi\)
0.945136 0.326677i \(-0.105929\pi\)
\(234\) 0 0
\(235\) −120416. −2.18046
\(236\) − 6057.05i − 0.108752i
\(237\) 0 0
\(238\) 0 0
\(239\) 57330.8i 1.00367i 0.864963 + 0.501836i \(0.167342\pi\)
−0.864963 + 0.501836i \(0.832658\pi\)
\(240\) 0 0
\(241\) −85783.2 −1.47696 −0.738479 0.674277i \(-0.764456\pi\)
−0.738479 + 0.674277i \(0.764456\pi\)
\(242\) 41405.3i 0.707010i
\(243\) 0 0
\(244\) −9554.34 −0.160480
\(245\) 0 0
\(246\) 0 0
\(247\) 37312.0 0.611582
\(248\) 28887.5i 0.469685i
\(249\) 0 0
\(250\) 33110.7 0.529772
\(251\) − 25276.5i − 0.401209i −0.979672 0.200604i \(-0.935709\pi\)
0.979672 0.200604i \(-0.0642905\pi\)
\(252\) 0 0
\(253\) 90.0000 0.00140605
\(254\) − 87460.6i − 1.35564i
\(255\) 0 0
\(256\) 4096.00 0.0625000
\(257\) 87215.7i 1.32047i 0.751059 + 0.660235i \(0.229543\pi\)
−0.751059 + 0.660235i \(0.770457\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) − 19188.0i − 0.283847i
\(261\) 0 0
\(262\) 16802.5 0.244777
\(263\) − 60747.5i − 0.878248i −0.898426 0.439124i \(-0.855289\pi\)
0.898426 0.439124i \(-0.144711\pi\)
\(264\) 0 0
\(265\) 116835. 1.66372
\(266\) 0 0
\(267\) 0 0
\(268\) 9040.00 0.125863
\(269\) 109871.i 1.51838i 0.650870 + 0.759189i \(0.274404\pi\)
−0.650870 + 0.759189i \(0.725596\pi\)
\(270\) 0 0
\(271\) 115188. 1.56844 0.784218 0.620485i \(-0.213064\pi\)
0.784218 + 0.620485i \(0.213064\pi\)
\(272\) − 16773.4i − 0.226716i
\(273\) 0 0
\(274\) −53540.0 −0.713144
\(275\) 315.370i 0.00417018i
\(276\) 0 0
\(277\) −109310. −1.42462 −0.712312 0.701863i \(-0.752352\pi\)
−0.712312 + 0.701863i \(0.752352\pi\)
\(278\) 30751.2i 0.397898i
\(279\) 0 0
\(280\) 0 0
\(281\) 103576.i 1.31173i 0.754878 + 0.655865i \(0.227696\pi\)
−0.754878 + 0.655865i \(0.772304\pi\)
\(282\) 0 0
\(283\) 144015. 1.79819 0.899095 0.437753i \(-0.144226\pi\)
0.899095 + 0.437753i \(0.144226\pi\)
\(284\) − 3382.80i − 0.0419411i
\(285\) 0 0
\(286\) −329.460 −0.00402783
\(287\) 0 0
\(288\) 0 0
\(289\) 14833.0 0.177596
\(290\) − 41351.0i − 0.491689i
\(291\) 0 0
\(292\) 24050.6 0.282072
\(293\) − 42370.2i − 0.493544i −0.969074 0.246772i \(-0.920630\pi\)
0.969074 0.246772i \(-0.0793698\pi\)
\(294\) 0 0
\(295\) 22048.0 0.253352
\(296\) − 35298.8i − 0.402880i
\(297\) 0 0
\(298\) −21068.0 −0.237242
\(299\) − 5241.68i − 0.0586311i
\(300\) 0 0
\(301\) 0 0
\(302\) − 101235.i − 1.10998i
\(303\) 0 0
\(304\) −28992.5 −0.313717
\(305\) − 34778.3i − 0.373860i
\(306\) 0 0
\(307\) −64533.0 −0.684708 −0.342354 0.939571i \(-0.611224\pi\)
−0.342354 + 0.939571i \(0.611224\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −105152. −1.09419
\(311\) 33838.0i 0.349851i 0.984582 + 0.174926i \(0.0559685\pi\)
−0.984582 + 0.174926i \(0.944032\pi\)
\(312\) 0 0
\(313\) 53619.6 0.547312 0.273656 0.961828i \(-0.411767\pi\)
0.273656 + 0.961828i \(0.411767\pi\)
\(314\) − 7571.31i − 0.0767913i
\(315\) 0 0
\(316\) −69200.0 −0.692998
\(317\) − 25426.1i − 0.253024i −0.991965 0.126512i \(-0.959622\pi\)
0.991965 0.126512i \(-0.0403783\pi\)
\(318\) 0 0
\(319\) −710.000 −0.00697713
\(320\) 14909.7i 0.145602i
\(321\) 0 0
\(322\) 0 0
\(323\) 118726.i 1.13800i
\(324\) 0 0
\(325\) 18367.4 0.173893
\(326\) 54458.5i 0.512426i
\(327\) 0 0
\(328\) 29651.4 0.275612
\(329\) 0 0
\(330\) 0 0
\(331\) 24392.0 0.222634 0.111317 0.993785i \(-0.464493\pi\)
0.111317 + 0.993785i \(0.464493\pi\)
\(332\) − 23762.3i − 0.215582i
\(333\) 0 0
\(334\) −35417.0 −0.317482
\(335\) 32906.1i 0.293215i
\(336\) 0 0
\(337\) 105296. 0.927154 0.463577 0.886057i \(-0.346566\pi\)
0.463577 + 0.886057i \(0.346566\pi\)
\(338\) − 61594.7i − 0.539150i
\(339\) 0 0
\(340\) 61056.0 0.528166
\(341\) 1805.47i 0.0155268i
\(342\) 0 0
\(343\) 0 0
\(344\) − 7421.79i − 0.0627180i
\(345\) 0 0
\(346\) 66633.3 0.556595
\(347\) − 39045.0i − 0.324270i −0.986769 0.162135i \(-0.948162\pi\)
0.986769 0.162135i \(-0.0518380\pi\)
\(348\) 0 0
\(349\) −188410. −1.54687 −0.773434 0.633877i \(-0.781463\pi\)
−0.773434 + 0.633877i \(0.781463\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 256.000 0.00206612
\(353\) 124140.i 0.996240i 0.867108 + 0.498120i \(0.165976\pi\)
−0.867108 + 0.498120i \(0.834024\pi\)
\(354\) 0 0
\(355\) 12313.6 0.0977074
\(356\) − 106464.i − 0.840048i
\(357\) 0 0
\(358\) −72156.0 −0.562997
\(359\) 202443.i 1.57078i 0.619003 + 0.785388i \(0.287537\pi\)
−0.619003 + 0.785388i \(0.712463\pi\)
\(360\) 0 0
\(361\) 74895.0 0.574696
\(362\) 109959.i 0.839098i
\(363\) 0 0
\(364\) 0 0
\(365\) 87545.5i 0.657125i
\(366\) 0 0
\(367\) −228728. −1.69819 −0.849096 0.528239i \(-0.822853\pi\)
−0.849096 + 0.528239i \(0.822853\pi\)
\(368\) 4072.94i 0.0300754i
\(369\) 0 0
\(370\) 128489. 0.938564
\(371\) 0 0
\(372\) 0 0
\(373\) −49874.0 −0.358473 −0.179237 0.983806i \(-0.557363\pi\)
−0.179237 + 0.983806i \(0.557363\pi\)
\(374\) − 1048.34i − 0.00749475i
\(375\) 0 0
\(376\) −93566.7 −0.661829
\(377\) 41351.0i 0.290940i
\(378\) 0 0
\(379\) 16632.0 0.115789 0.0578943 0.998323i \(-0.481561\pi\)
0.0578943 + 0.998323i \(0.481561\pi\)
\(380\) − 105534.i − 0.730847i
\(381\) 0 0
\(382\) 97164.0 0.665853
\(383\) − 128596.i − 0.876656i −0.898815 0.438328i \(-0.855571\pi\)
0.898815 0.438328i \(-0.144429\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) − 186178.i − 1.24955i
\(387\) 0 0
\(388\) 105098. 0.698120
\(389\) − 82370.9i − 0.544345i −0.962248 0.272173i \(-0.912258\pi\)
0.962248 0.272173i \(-0.0877422\pi\)
\(390\) 0 0
\(391\) 16678.9 0.109097
\(392\) 0 0
\(393\) 0 0
\(394\) −183372. −1.18125
\(395\) − 251892.i − 1.61443i
\(396\) 0 0
\(397\) −208013. −1.31980 −0.659902 0.751352i \(-0.729402\pi\)
−0.659902 + 0.751352i \(0.729402\pi\)
\(398\) 131624.i 0.830941i
\(399\) 0 0
\(400\) −14272.0 −0.0892000
\(401\) 274325.i 1.70599i 0.521919 + 0.852995i \(0.325216\pi\)
−0.521919 + 0.852995i \(0.674784\pi\)
\(402\) 0 0
\(403\) 105152. 0.647452
\(404\) − 96213.9i − 0.589488i
\(405\) 0 0
\(406\) 0 0
\(407\) − 2206.17i − 0.0133184i
\(408\) 0 0
\(409\) −88830.7 −0.531027 −0.265513 0.964107i \(-0.585541\pi\)
−0.265513 + 0.964107i \(0.585541\pi\)
\(410\) 107933.i 0.642075i
\(411\) 0 0
\(412\) −87306.9 −0.514345
\(413\) 0 0
\(414\) 0 0
\(415\) 86496.0 0.502227
\(416\) − 14909.7i − 0.0861551i
\(417\) 0 0
\(418\) −1812.03 −0.0103708
\(419\) 228071.i 1.29910i 0.760319 + 0.649550i \(0.225043\pi\)
−0.760319 + 0.649550i \(0.774957\pi\)
\(420\) 0 0
\(421\) 149694. 0.844579 0.422289 0.906461i \(-0.361227\pi\)
0.422289 + 0.906461i \(0.361227\pi\)
\(422\) 24052.9i 0.135065i
\(423\) 0 0
\(424\) 90784.0 0.504984
\(425\) 58444.7i 0.323569i
\(426\) 0 0
\(427\) 0 0
\(428\) − 101631.i − 0.554803i
\(429\) 0 0
\(430\) 27015.7 0.146110
\(431\) − 161643.i − 0.870168i −0.900390 0.435084i \(-0.856719\pi\)
0.900390 0.435084i \(-0.143281\pi\)
\(432\) 0 0
\(433\) 112099. 0.597895 0.298948 0.954269i \(-0.403364\pi\)
0.298948 + 0.954269i \(0.403364\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 78016.0 0.410403
\(437\) − 28829.2i − 0.150963i
\(438\) 0 0
\(439\) 185321. 0.961604 0.480802 0.876829i \(-0.340346\pi\)
0.480802 + 0.876829i \(0.340346\pi\)
\(440\) 931.854i 0.00481330i
\(441\) 0 0
\(442\) −61056.0 −0.312524
\(443\) 129337.i 0.659045i 0.944148 + 0.329522i \(0.106888\pi\)
−0.944148 + 0.329522i \(0.893112\pi\)
\(444\) 0 0
\(445\) 387536. 1.95701
\(446\) − 160512.i − 0.806933i
\(447\) 0 0
\(448\) 0 0
\(449\) 4323.25i 0.0214446i 0.999943 + 0.0107223i \(0.00341308\pi\)
−0.999943 + 0.0107223i \(0.996587\pi\)
\(450\) 0 0
\(451\) 1853.21 0.00911113
\(452\) − 90430.5i − 0.442627i
\(453\) 0 0
\(454\) 158306. 0.768041
\(455\) 0 0
\(456\) 0 0
\(457\) 133376. 0.638624 0.319312 0.947650i \(-0.396548\pi\)
0.319312 + 0.947650i \(0.396548\pi\)
\(458\) − 239254.i − 1.14058i
\(459\) 0 0
\(460\) −14825.7 −0.0700648
\(461\) − 111735.i − 0.525760i −0.964828 0.262880i \(-0.915328\pi\)
0.964828 0.262880i \(-0.0846724\pi\)
\(462\) 0 0
\(463\) 324458. 1.51355 0.756775 0.653676i \(-0.226774\pi\)
0.756775 + 0.653676i \(0.226774\pi\)
\(464\) − 32130.9i − 0.149241i
\(465\) 0 0
\(466\) 100324. 0.461990
\(467\) − 330109.i − 1.51364i −0.653621 0.756822i \(-0.726751\pi\)
0.653621 0.756822i \(-0.273249\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) − 340588.i − 1.54182i
\(471\) 0 0
\(472\) 17131.9 0.0768993
\(473\) − 463.862i − 0.00207332i
\(474\) 0 0
\(475\) 101021. 0.447737
\(476\) 0 0
\(477\) 0 0
\(478\) −162156. −0.709704
\(479\) 142224.i 0.619873i 0.950757 + 0.309936i \(0.100308\pi\)
−0.950757 + 0.309936i \(0.899692\pi\)
\(480\) 0 0
\(481\) −128489. −0.555364
\(482\) − 242632.i − 1.04437i
\(483\) 0 0
\(484\) −117112. −0.499932
\(485\) 382562.i 1.62637i
\(486\) 0 0
\(487\) 35920.0 0.151453 0.0757266 0.997129i \(-0.475872\pi\)
0.0757266 + 0.997129i \(0.475872\pi\)
\(488\) − 27023.8i − 0.113477i
\(489\) 0 0
\(490\) 0 0
\(491\) − 347986.i − 1.44344i −0.692186 0.721719i \(-0.743352\pi\)
0.692186 0.721719i \(-0.256648\pi\)
\(492\) 0 0
\(493\) −131578. −0.541365
\(494\) 105534.i 0.432454i
\(495\) 0 0
\(496\) −81706.1 −0.332117
\(497\) 0 0
\(498\) 0 0
\(499\) 12120.0 0.0486745 0.0243373 0.999704i \(-0.492252\pi\)
0.0243373 + 0.999704i \(0.492252\pi\)
\(500\) 93651.3i 0.374605i
\(501\) 0 0
\(502\) 71492.9 0.283697
\(503\) − 222655.i − 0.880028i −0.897991 0.440014i \(-0.854974\pi\)
0.897991 0.440014i \(-0.145026\pi\)
\(504\) 0 0
\(505\) 350224. 1.37329
\(506\) 254.558i 0 0.000994229i
\(507\) 0 0
\(508\) 247376. 0.958584
\(509\) 232760.i 0.898405i 0.893430 + 0.449203i \(0.148292\pi\)
−0.893430 + 0.449203i \(0.851708\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 11585.2i 0.0441942i
\(513\) 0 0
\(514\) −246683. −0.933713
\(515\) − 317802.i − 1.19824i
\(516\) 0 0
\(517\) −5847.92 −0.0218786
\(518\) 0 0
\(519\) 0 0
\(520\) 54272.0 0.200710
\(521\) 37245.0i 0.137212i 0.997644 + 0.0686061i \(0.0218552\pi\)
−0.997644 + 0.0686061i \(0.978145\pi\)
\(522\) 0 0
\(523\) −112016. −0.409523 −0.204762 0.978812i \(-0.565642\pi\)
−0.204762 + 0.978812i \(0.565642\pi\)
\(524\) 47524.6i 0.173083i
\(525\) 0 0
\(526\) 171820. 0.621015
\(527\) 334592.i 1.20474i
\(528\) 0 0
\(529\) 275791. 0.985527
\(530\) 330459.i 1.17643i
\(531\) 0 0
\(532\) 0 0
\(533\) − 107933.i − 0.379926i
\(534\) 0 0
\(535\) 369943. 1.29249
\(536\) 25569.0i 0.0889987i
\(537\) 0 0
\(538\) −310763. −1.07366
\(539\) 0 0
\(540\) 0 0
\(541\) 128194. 0.437999 0.219000 0.975725i \(-0.429721\pi\)
0.219000 + 0.975725i \(0.429721\pi\)
\(542\) 325799.i 1.10905i
\(543\) 0 0
\(544\) 47442.3 0.160313
\(545\) 283983.i 0.956090i
\(546\) 0 0
\(547\) 186442. 0.623116 0.311558 0.950227i \(-0.399149\pi\)
0.311558 + 0.950227i \(0.399149\pi\)
\(548\) − 151434.i − 0.504269i
\(549\) 0 0
\(550\) −892.000 −0.00294876
\(551\) 227431.i 0.749110i
\(552\) 0 0
\(553\) 0 0
\(554\) − 309175.i − 1.00736i
\(555\) 0 0
\(556\) −86977.5 −0.281357
\(557\) − 14318.9i − 0.0461530i −0.999734 0.0230765i \(-0.992654\pi\)
0.999734 0.0230765i \(-0.00734613\pi\)
\(558\) 0 0
\(559\) −27015.7 −0.0864556
\(560\) 0 0
\(561\) 0 0
\(562\) −292956. −0.927534
\(563\) 66336.4i 0.209283i 0.994510 + 0.104642i \(0.0333696\pi\)
−0.994510 + 0.104642i \(0.966630\pi\)
\(564\) 0 0
\(565\) 329172. 1.03116
\(566\) 407337.i 1.27151i
\(567\) 0 0
\(568\) 9568.00 0.0296568
\(569\) − 208920.i − 0.645292i −0.946520 0.322646i \(-0.895428\pi\)
0.946520 0.322646i \(-0.104572\pi\)
\(570\) 0 0
\(571\) 515862. 1.58220 0.791100 0.611687i \(-0.209509\pi\)
0.791100 + 0.611687i \(0.209509\pi\)
\(572\) − 931.854i − 0.00284810i
\(573\) 0 0
\(574\) 0 0
\(575\) − 14191.6i − 0.0429237i
\(576\) 0 0
\(577\) 352440. 1.05860 0.529302 0.848433i \(-0.322454\pi\)
0.529302 + 0.848433i \(0.322454\pi\)
\(578\) 41954.1i 0.125579i
\(579\) 0 0
\(580\) 116958. 0.347676
\(581\) 0 0
\(582\) 0 0
\(583\) 5674.00 0.0166937
\(584\) 68025.3i 0.199455i
\(585\) 0 0
\(586\) 119841. 0.348988
\(587\) − 296621.i − 0.860846i −0.902627 0.430423i \(-0.858364\pi\)
0.902627 0.430423i \(-0.141636\pi\)
\(588\) 0 0
\(589\) 578336. 1.66705
\(590\) 62361.2i 0.179147i
\(591\) 0 0
\(592\) 99840.0 0.284879
\(593\) − 22626.6i − 0.0643442i −0.999482 0.0321721i \(-0.989758\pi\)
0.999482 0.0321721i \(-0.0102425\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) − 59589.3i − 0.167755i
\(597\) 0 0
\(598\) 14825.7 0.0414584
\(599\) − 146528.i − 0.408383i −0.978931 0.204191i \(-0.934544\pi\)
0.978931 0.204191i \(-0.0654565\pi\)
\(600\) 0 0
\(601\) −136479. −0.377847 −0.188924 0.981992i \(-0.560500\pi\)
−0.188924 + 0.981992i \(0.560500\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 286336. 0.784878
\(605\) − 426294.i − 1.16466i
\(606\) 0 0
\(607\) 79317.5 0.215274 0.107637 0.994190i \(-0.465672\pi\)
0.107637 + 0.994190i \(0.465672\pi\)
\(608\) − 82003.2i − 0.221832i
\(609\) 0 0
\(610\) 98368.0 0.264359
\(611\) 340588.i 0.912319i
\(612\) 0 0
\(613\) 13742.0 0.0365703 0.0182852 0.999833i \(-0.494179\pi\)
0.0182852 + 0.999833i \(0.494179\pi\)
\(614\) − 182527.i − 0.484161i
\(615\) 0 0
\(616\) 0 0
\(617\) 125015.i 0.328392i 0.986428 + 0.164196i \(0.0525029\pi\)
−0.986428 + 0.164196i \(0.947497\pi\)
\(618\) 0 0
\(619\) 318341. 0.830828 0.415414 0.909632i \(-0.363637\pi\)
0.415414 + 0.909632i \(0.363637\pi\)
\(620\) − 297415.i − 0.773712i
\(621\) 0 0
\(622\) −95708.2 −0.247382
\(623\) 0 0
\(624\) 0 0
\(625\) −480271. −1.22949
\(626\) 151659.i 0.387008i
\(627\) 0 0
\(628\) 21414.9 0.0542996
\(629\) − 408851.i − 1.03339i
\(630\) 0 0
\(631\) −12618.0 −0.0316907 −0.0158453 0.999874i \(-0.505044\pi\)
−0.0158453 + 0.999874i \(0.505044\pi\)
\(632\) − 195727.i − 0.490024i
\(633\) 0 0
\(634\) 71916.0 0.178915
\(635\) 900462.i 2.23315i
\(636\) 0 0
\(637\) 0 0
\(638\) − 2008.18i − 0.00493358i
\(639\) 0 0
\(640\) −42170.9 −0.102956
\(641\) − 88644.3i − 0.215742i −0.994165 0.107871i \(-0.965597\pi\)
0.994165 0.107871i \(-0.0344034\pi\)
\(642\) 0 0
\(643\) −195329. −0.472437 −0.236219 0.971700i \(-0.575908\pi\)
−0.236219 + 0.971700i \(0.575908\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −335808. −0.804685
\(647\) 441524.i 1.05474i 0.849635 + 0.527371i \(0.176822\pi\)
−0.849635 + 0.527371i \(0.823178\pi\)
\(648\) 0 0
\(649\) 1070.75 0.00254212
\(650\) 51950.9i 0.122961i
\(651\) 0 0
\(652\) −154032. −0.362340
\(653\) − 346158.i − 0.811799i −0.913918 0.405900i \(-0.866958\pi\)
0.913918 0.405900i \(-0.133042\pi\)
\(654\) 0 0
\(655\) −172992. −0.403221
\(656\) 83866.9i 0.194887i
\(657\) 0 0
\(658\) 0 0
\(659\) − 255677.i − 0.588737i −0.955692 0.294368i \(-0.904891\pi\)
0.955692 0.294368i \(-0.0951092\pi\)
\(660\) 0 0
\(661\) −124742. −0.285502 −0.142751 0.989759i \(-0.545595\pi\)
−0.142751 + 0.989759i \(0.545595\pi\)
\(662\) 68991.0i 0.157426i
\(663\) 0 0
\(664\) 67209.9 0.152439
\(665\) 0 0
\(666\) 0 0
\(667\) 31950.0 0.0718157
\(668\) − 100174.i − 0.224493i
\(669\) 0 0
\(670\) −93072.5 −0.207335
\(671\) − 1688.99i − 0.00375129i
\(672\) 0 0
\(673\) −168784. −0.372650 −0.186325 0.982488i \(-0.559658\pi\)
−0.186325 + 0.982488i \(0.559658\pi\)
\(674\) 297822.i 0.655597i
\(675\) 0 0
\(676\) 174216. 0.381237
\(677\) − 709287.i − 1.54755i −0.633461 0.773774i \(-0.718366\pi\)
0.633461 0.773774i \(-0.281634\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 172692.i 0.373470i
\(681\) 0 0
\(682\) −5106.63 −0.0109791
\(683\) − 746025.i − 1.59923i −0.600511 0.799616i \(-0.705036\pi\)
0.600511 0.799616i \(-0.294964\pi\)
\(684\) 0 0
\(685\) 551228. 1.17476
\(686\) 0 0
\(687\) 0 0
\(688\) 20992.0 0.0443483
\(689\) − 330459.i − 0.696112i
\(690\) 0 0
\(691\) −901815. −1.88869 −0.944346 0.328953i \(-0.893304\pi\)
−0.944346 + 0.328953i \(0.893304\pi\)
\(692\) 188467.i 0.393572i
\(693\) 0 0
\(694\) 110436. 0.229293
\(695\) − 316603.i − 0.655458i
\(696\) 0 0
\(697\) 343440. 0.706944
\(698\) − 532904.i − 1.09380i
\(699\) 0 0
\(700\) 0 0
\(701\) 571146.i 1.16228i 0.813803 + 0.581140i \(0.197393\pi\)
−0.813803 + 0.581140i \(0.802607\pi\)
\(702\) 0 0
\(703\) −706692. −1.42995
\(704\) 724.077i 0.00146096i
\(705\) 0 0
\(706\) −351122. −0.704448
\(707\) 0 0
\(708\) 0 0
\(709\) −403688. −0.803070 −0.401535 0.915844i \(-0.631523\pi\)
−0.401535 + 0.915844i \(0.631523\pi\)
\(710\) 34828.0i 0.0690896i
\(711\) 0 0
\(712\) 301127. 0.594004
\(713\) − 81246.0i − 0.159817i
\(714\) 0 0
\(715\) 3392.00 0.00663504
\(716\) − 204088.i − 0.398099i
\(717\) 0 0
\(718\) −572596. −1.11071
\(719\) − 365170.i − 0.706379i −0.935552 0.353189i \(-0.885097\pi\)
0.935552 0.353189i \(-0.114903\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 211835.i 0.406372i
\(723\) 0 0
\(724\) −311010. −0.593332
\(725\) 111956.i 0.212996i
\(726\) 0 0
\(727\) 941968. 1.78224 0.891122 0.453764i \(-0.149919\pi\)
0.891122 + 0.453764i \(0.149919\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −247616. −0.464658
\(731\) − 85963.5i − 0.160872i
\(732\) 0 0
\(733\) 418909. 0.779671 0.389835 0.920885i \(-0.372532\pi\)
0.389835 + 0.920885i \(0.372532\pi\)
\(734\) − 646940.i − 1.20080i
\(735\) 0 0
\(736\) −11520.0 −0.0212665
\(737\) 1598.06i 0.00294211i
\(738\) 0 0
\(739\) −1.04476e6 −1.91305 −0.956526 0.291647i \(-0.905797\pi\)
−0.956526 + 0.291647i \(0.905797\pi\)
\(740\) 363423.i 0.663665i
\(741\) 0 0
\(742\) 0 0
\(743\) − 232297.i − 0.420791i −0.977616 0.210396i \(-0.932525\pi\)
0.977616 0.210396i \(-0.0674752\pi\)
\(744\) 0 0
\(745\) 216908. 0.390808
\(746\) − 141065.i − 0.253479i
\(747\) 0 0
\(748\) 2965.14 0.00529959
\(749\) 0 0
\(750\) 0 0
\(751\) −465408. −0.825190 −0.412595 0.910915i \(-0.635378\pi\)
−0.412595 + 0.910915i \(0.635378\pi\)
\(752\) − 264647.i − 0.467984i
\(753\) 0 0
\(754\) −116958. −0.205726
\(755\) 1.04228e6i 1.82848i
\(756\) 0 0
\(757\) −496088. −0.865699 −0.432850 0.901466i \(-0.642492\pi\)
−0.432850 + 0.901466i \(0.642492\pi\)
\(758\) 47042.4i 0.0818750i
\(759\) 0 0
\(760\) 298496. 0.516787
\(761\) − 738407.i − 1.27505i −0.770431 0.637524i \(-0.779959\pi\)
0.770431 0.637524i \(-0.220041\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 274821.i 0.470829i
\(765\) 0 0
\(766\) 363724. 0.619890
\(767\) − 62361.2i − 0.106004i
\(768\) 0 0
\(769\) −527136. −0.891395 −0.445698 0.895184i \(-0.647044\pi\)
−0.445698 + 0.895184i \(0.647044\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 526592. 0.883567
\(773\) 100378.i 0.167989i 0.996466 + 0.0839944i \(0.0267678\pi\)
−0.996466 + 0.0839944i \(0.973232\pi\)
\(774\) 0 0
\(775\) 284695. 0.473998
\(776\) 297261.i 0.493645i
\(777\) 0 0
\(778\) 232980. 0.384910
\(779\) − 593630.i − 0.978230i
\(780\) 0 0
\(781\) 598.000 0.000980391 0
\(782\) 47175.1i 0.0771435i
\(783\) 0 0
\(784\) 0 0
\(785\) 77951.5i 0.126498i
\(786\) 0 0
\(787\) −715752. −1.15562 −0.577808 0.816173i \(-0.696092\pi\)
−0.577808 + 0.816173i \(0.696092\pi\)
\(788\) − 518654.i − 0.835267i
\(789\) 0 0
\(790\) 712458. 1.14158
\(791\) 0 0
\(792\) 0 0
\(793\) −98368.0 −0.156425
\(794\) − 588349.i − 0.933242i
\(795\) 0 0
\(796\) −372290. −0.587564
\(797\) 921924.i 1.45137i 0.688027 + 0.725686i \(0.258477\pi\)
−0.688027 + 0.725686i \(0.741523\pi\)
\(798\) 0 0
\(799\) −1.08374e6 −1.69759
\(800\) − 40367.3i − 0.0630739i
\(801\) 0 0
\(802\) −775908. −1.20632
\(803\) 4251.58i 0.00659356i
\(804\) 0 0
\(805\) 0 0
\(806\) 297415.i 0.457818i
\(807\) 0 0
\(808\) 272134. 0.416831
\(809\) 601594.i 0.919192i 0.888128 + 0.459596i \(0.152006\pi\)
−0.888128 + 0.459596i \(0.847994\pi\)
\(810\) 0 0
\(811\) −1.22221e6 −1.85826 −0.929129 0.369757i \(-0.879441\pi\)
−0.929129 + 0.369757i \(0.879441\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 6240.00 0.00941750
\(815\) − 560685.i − 0.844119i
\(816\) 0 0
\(817\) −148587. −0.222605
\(818\) − 251251.i − 0.375493i
\(819\) 0 0
\(820\) −305280. −0.454015
\(821\) − 911304.i − 1.35200i −0.736901 0.676000i \(-0.763712\pi\)
0.736901 0.676000i \(-0.236288\pi\)
\(822\) 0 0
\(823\) −207542. −0.306412 −0.153206 0.988194i \(-0.548960\pi\)
−0.153206 + 0.988194i \(0.548960\pi\)
\(824\) − 246941.i − 0.363697i
\(825\) 0 0
\(826\) 0 0
\(827\) 746785.i 1.09190i 0.837816 + 0.545952i \(0.183832\pi\)
−0.837816 + 0.545952i \(0.816168\pi\)
\(828\) 0 0
\(829\) 233093. 0.339172 0.169586 0.985515i \(-0.445757\pi\)
0.169586 + 0.985515i \(0.445757\pi\)
\(830\) 244648.i 0.355128i
\(831\) 0 0
\(832\) 42170.9 0.0609209
\(833\) 0 0
\(834\) 0 0
\(835\) 364640. 0.522988
\(836\) − 5125.20i − 0.00733328i
\(837\) 0 0
\(838\) −645083. −0.918602
\(839\) − 798482.i − 1.13434i −0.823602 0.567168i \(-0.808039\pi\)
0.823602 0.567168i \(-0.191961\pi\)
\(840\) 0 0
\(841\) 455231. 0.643635
\(842\) 423399.i 0.597207i
\(843\) 0 0
\(844\) −68032.0 −0.0955055
\(845\) 634156.i 0.888142i
\(846\) 0 0
\(847\) 0 0
\(848\) 256776.i 0.357078i
\(849\) 0 0
\(850\) −165307. −0.228798
\(851\) 99277.8i 0.137086i
\(852\) 0 0
\(853\) −432128. −0.593902 −0.296951 0.954893i \(-0.595970\pi\)
−0.296951 + 0.954893i \(0.595970\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 287456. 0.392305
\(857\) 466247.i 0.634826i 0.948287 + 0.317413i \(0.102814\pi\)
−0.948287 + 0.317413i \(0.897186\pi\)
\(858\) 0 0
\(859\) −610449. −0.827299 −0.413650 0.910436i \(-0.635746\pi\)
−0.413650 + 0.910436i \(0.635746\pi\)
\(860\) 76412.0i 0.103315i
\(861\) 0 0
\(862\) 457196. 0.615301
\(863\) − 198424.i − 0.266424i −0.991088 0.133212i \(-0.957471\pi\)
0.991088 0.133212i \(-0.0425291\pi\)
\(864\) 0 0
\(865\) −686032. −0.916879
\(866\) 317063.i 0.422776i
\(867\) 0 0
\(868\) 0 0
\(869\) − 12232.9i − 0.0161991i
\(870\) 0 0
\(871\) 93072.5 0.122683
\(872\) 220663.i 0.290199i
\(873\) 0 0
\(874\) 81541.4 0.106747
\(875\) 0 0
\(876\) 0 0
\(877\) −33678.0 −0.0437872 −0.0218936 0.999760i \(-0.506970\pi\)
−0.0218936 + 0.999760i \(0.506970\pi\)
\(878\) 524168.i 0.679957i
\(879\) 0 0
\(880\) −2635.68 −0.00340351
\(881\) − 1.11930e6i − 1.44210i −0.692883 0.721050i \(-0.743660\pi\)
0.692883 0.721050i \(-0.256340\pi\)
\(882\) 0 0
\(883\) −1.28444e6 −1.64737 −0.823687 0.567044i \(-0.808087\pi\)
−0.823687 + 0.567044i \(0.808087\pi\)
\(884\) − 172692.i − 0.220988i
\(885\) 0 0
\(886\) −365820. −0.466015
\(887\) 111997.i 0.142351i 0.997464 + 0.0711754i \(0.0226750\pi\)
−0.997464 + 0.0711754i \(0.977325\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 1.09612e6i 1.38381i
\(891\) 0 0
\(892\) 453996. 0.570588
\(893\) 1.87323e6i 2.34903i
\(894\) 0 0
\(895\) 742891. 0.927426
\(896\) 0 0
\(897\) 0 0
\(898\) −12228.0 −0.0151636
\(899\) 640941.i 0.793046i
\(900\) 0 0
\(901\) 1.05151e6 1.29528
\(902\) 5241.68i 0.00644254i
\(903\) 0 0
\(904\) 255776. 0.312985
\(905\) − 1.13209e6i − 1.38225i
\(906\) 0 0
\(907\) −1.49953e6 −1.82280 −0.911402 0.411517i \(-0.864999\pi\)
−0.911402 + 0.411517i \(0.864999\pi\)
\(908\) 447756.i 0.543087i
\(909\) 0 0
\(910\) 0 0
\(911\) − 1.62993e6i − 1.96396i −0.188997 0.981978i \(-0.560524\pi\)
0.188997 0.981978i \(-0.439476\pi\)
\(912\) 0 0
\(913\) 4200.62 0.00503932
\(914\) 377244.i 0.451575i
\(915\) 0 0
\(916\) 676711. 0.806515
\(917\) 0 0
\(918\) 0 0
\(919\) 503520. 0.596191 0.298096 0.954536i \(-0.403649\pi\)
0.298096 + 0.954536i \(0.403649\pi\)
\(920\) − 41933.4i − 0.0495433i
\(921\) 0 0
\(922\) 316035. 0.371769
\(923\) − 34828.0i − 0.0408814i
\(924\) 0 0
\(925\) −347880. −0.406580
\(926\) 917706.i 1.07024i
\(927\) 0 0
\(928\) 90880.0 0.105529
\(929\) − 71898.4i − 0.0833082i −0.999132 0.0416541i \(-0.986737\pi\)
0.999132 0.0416541i \(-0.0132627\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 283759.i 0.326677i
\(933\) 0 0
\(934\) 933690. 1.07031
\(935\) 10793.3i 0.0123461i
\(936\) 0 0
\(937\) −679017. −0.773396 −0.386698 0.922206i \(-0.626384\pi\)
−0.386698 + 0.922206i \(0.626384\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 963328. 1.09023
\(941\) 364908.i 0.412102i 0.978541 + 0.206051i \(0.0660612\pi\)
−0.978541 + 0.206051i \(0.933939\pi\)
\(942\) 0 0
\(943\) −83394.6 −0.0937809
\(944\) 48456.4i 0.0543760i
\(945\) 0 0
\(946\) 1312.00 0.00146606
\(947\) − 186576.i − 0.208044i −0.994575 0.104022i \(-0.966829\pi\)
0.994575 0.104022i \(-0.0331712\pi\)
\(948\) 0 0
\(949\) 247616. 0.274945
\(950\) 285730.i 0.316598i
\(951\) 0 0
\(952\) 0 0
\(953\) 1.32111e6i 1.45463i 0.686302 + 0.727317i \(0.259233\pi\)
−0.686302 + 0.727317i \(0.740767\pi\)
\(954\) 0 0
\(955\) −1.00036e6 −1.09686
\(956\) − 458646.i − 0.501836i
\(957\) 0 0
\(958\) −402271. −0.438316
\(959\) 0 0
\(960\) 0 0
\(961\) 706335. 0.764828
\(962\) − 363423.i − 0.392701i
\(963\) 0 0
\(964\) 686266. 0.738479
\(965\) 1.91682e6i 2.05839i
\(966\) 0 0
\(967\) −688022. −0.735782 −0.367891 0.929869i \(-0.619920\pi\)
−0.367891 + 0.929869i \(0.619920\pi\)
\(968\) − 331243.i − 0.353505i
\(969\) 0 0
\(970\) −1.08205e6 −1.15001
\(971\) 485088.i 0.514496i 0.966345 + 0.257248i \(0.0828158\pi\)
−0.966345 + 0.257248i \(0.917184\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 101597.i 0.107094i
\(975\) 0 0
\(976\) 76434.8 0.0802401
\(977\) 1.70595e6i 1.78721i 0.448850 + 0.893607i \(0.351834\pi\)
−0.448850 + 0.893607i \(0.648166\pi\)
\(978\) 0 0
\(979\) 18820.4 0.0196365
\(980\) 0 0
\(981\) 0 0
\(982\) 984252. 1.02067
\(983\) 964527.i 0.998177i 0.866551 + 0.499088i \(0.166332\pi\)
−0.866551 + 0.499088i \(0.833668\pi\)
\(984\) 0 0
\(985\) 1.88793e6 1.94587
\(986\) − 372159.i − 0.382803i
\(987\) 0 0
\(988\) −298496. −0.305791
\(989\) 20873.8i 0.0213407i
\(990\) 0 0
\(991\) −639856. −0.651531 −0.325765 0.945451i \(-0.605622\pi\)
−0.325765 + 0.945451i \(0.605622\pi\)
\(992\) − 231100.i − 0.234842i
\(993\) 0 0
\(994\) 0 0
\(995\) − 1.35516e6i − 1.36881i
\(996\) 0 0
\(997\) 1.60838e6 1.61808 0.809039 0.587755i \(-0.199988\pi\)
0.809039 + 0.587755i \(0.199988\pi\)
\(998\) 34280.5i 0.0344181i
\(999\) 0 0
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 882.5.b.c.197.3 yes 4
3.2 odd 2 inner 882.5.b.c.197.2 yes 4
7.6 odd 2 inner 882.5.b.c.197.4 yes 4
21.20 even 2 inner 882.5.b.c.197.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
882.5.b.c.197.1 4 21.20 even 2 inner
882.5.b.c.197.2 yes 4 3.2 odd 2 inner
882.5.b.c.197.3 yes 4 1.1 even 1 trivial
882.5.b.c.197.4 yes 4 7.6 odd 2 inner