Properties

Label 1638.4.a.k.1.1
Level $1638$
Weight $4$
Character 1638.1
Self dual yes
Analytic conductor $96.645$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1638.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(96.6451285894\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1638.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.00000 q^{2} +4.00000 q^{4} +3.00000 q^{5} -7.00000 q^{7} +8.00000 q^{8} +O(q^{10})\) \(q+2.00000 q^{2} +4.00000 q^{4} +3.00000 q^{5} -7.00000 q^{7} +8.00000 q^{8} +6.00000 q^{10} +23.0000 q^{11} -13.0000 q^{13} -14.0000 q^{14} +16.0000 q^{16} -133.000 q^{17} +103.000 q^{19} +12.0000 q^{20} +46.0000 q^{22} -113.000 q^{23} -116.000 q^{25} -26.0000 q^{26} -28.0000 q^{28} +141.000 q^{29} -90.0000 q^{31} +32.0000 q^{32} -266.000 q^{34} -21.0000 q^{35} -233.000 q^{37} +206.000 q^{38} +24.0000 q^{40} +230.000 q^{41} -337.000 q^{43} +92.0000 q^{44} -226.000 q^{46} -96.0000 q^{47} +49.0000 q^{49} -232.000 q^{50} -52.0000 q^{52} +134.000 q^{53} +69.0000 q^{55} -56.0000 q^{56} +282.000 q^{58} -838.000 q^{59} +295.000 q^{61} -180.000 q^{62} +64.0000 q^{64} -39.0000 q^{65} +468.000 q^{67} -532.000 q^{68} -42.0000 q^{70} -54.0000 q^{71} -553.000 q^{73} -466.000 q^{74} +412.000 q^{76} -161.000 q^{77} -694.000 q^{79} +48.0000 q^{80} +460.000 q^{82} +1010.00 q^{83} -399.000 q^{85} -674.000 q^{86} +184.000 q^{88} -390.000 q^{89} +91.0000 q^{91} -452.000 q^{92} -192.000 q^{94} +309.000 q^{95} -102.000 q^{97} +98.0000 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000 0.707107
\(3\) 0 0
\(4\) 4.00000 0.500000
\(5\) 3.00000 0.268328 0.134164 0.990959i \(-0.457165\pi\)
0.134164 + 0.990959i \(0.457165\pi\)
\(6\) 0 0
\(7\) −7.00000 −0.377964
\(8\) 8.00000 0.353553
\(9\) 0 0
\(10\) 6.00000 0.189737
\(11\) 23.0000 0.630433 0.315216 0.949020i \(-0.397923\pi\)
0.315216 + 0.949020i \(0.397923\pi\)
\(12\) 0 0
\(13\) −13.0000 −0.277350
\(14\) −14.0000 −0.267261
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) −133.000 −1.89748 −0.948742 0.316051i \(-0.897643\pi\)
−0.948742 + 0.316051i \(0.897643\pi\)
\(18\) 0 0
\(19\) 103.000 1.24367 0.621837 0.783146i \(-0.286386\pi\)
0.621837 + 0.783146i \(0.286386\pi\)
\(20\) 12.0000 0.134164
\(21\) 0 0
\(22\) 46.0000 0.445783
\(23\) −113.000 −1.02444 −0.512220 0.858854i \(-0.671177\pi\)
−0.512220 + 0.858854i \(0.671177\pi\)
\(24\) 0 0
\(25\) −116.000 −0.928000
\(26\) −26.0000 −0.196116
\(27\) 0 0
\(28\) −28.0000 −0.188982
\(29\) 141.000 0.902864 0.451432 0.892306i \(-0.350913\pi\)
0.451432 + 0.892306i \(0.350913\pi\)
\(30\) 0 0
\(31\) −90.0000 −0.521435 −0.260717 0.965415i \(-0.583959\pi\)
−0.260717 + 0.965415i \(0.583959\pi\)
\(32\) 32.0000 0.176777
\(33\) 0 0
\(34\) −266.000 −1.34172
\(35\) −21.0000 −0.101419
\(36\) 0 0
\(37\) −233.000 −1.03527 −0.517635 0.855602i \(-0.673187\pi\)
−0.517635 + 0.855602i \(0.673187\pi\)
\(38\) 206.000 0.879411
\(39\) 0 0
\(40\) 24.0000 0.0948683
\(41\) 230.000 0.876097 0.438048 0.898951i \(-0.355670\pi\)
0.438048 + 0.898951i \(0.355670\pi\)
\(42\) 0 0
\(43\) −337.000 −1.19516 −0.597582 0.801808i \(-0.703872\pi\)
−0.597582 + 0.801808i \(0.703872\pi\)
\(44\) 92.0000 0.315216
\(45\) 0 0
\(46\) −226.000 −0.724389
\(47\) −96.0000 −0.297937 −0.148969 0.988842i \(-0.547595\pi\)
−0.148969 + 0.988842i \(0.547595\pi\)
\(48\) 0 0
\(49\) 49.0000 0.142857
\(50\) −232.000 −0.656195
\(51\) 0 0
\(52\) −52.0000 −0.138675
\(53\) 134.000 0.347289 0.173644 0.984808i \(-0.444446\pi\)
0.173644 + 0.984808i \(0.444446\pi\)
\(54\) 0 0
\(55\) 69.0000 0.169163
\(56\) −56.0000 −0.133631
\(57\) 0 0
\(58\) 282.000 0.638421
\(59\) −838.000 −1.84912 −0.924562 0.381032i \(-0.875569\pi\)
−0.924562 + 0.381032i \(0.875569\pi\)
\(60\) 0 0
\(61\) 295.000 0.619195 0.309597 0.950868i \(-0.399806\pi\)
0.309597 + 0.950868i \(0.399806\pi\)
\(62\) −180.000 −0.368710
\(63\) 0 0
\(64\) 64.0000 0.125000
\(65\) −39.0000 −0.0744208
\(66\) 0 0
\(67\) 468.000 0.853363 0.426681 0.904402i \(-0.359683\pi\)
0.426681 + 0.904402i \(0.359683\pi\)
\(68\) −532.000 −0.948742
\(69\) 0 0
\(70\) −42.0000 −0.0717137
\(71\) −54.0000 −0.0902623 −0.0451311 0.998981i \(-0.514371\pi\)
−0.0451311 + 0.998981i \(0.514371\pi\)
\(72\) 0 0
\(73\) −553.000 −0.886627 −0.443313 0.896367i \(-0.646197\pi\)
−0.443313 + 0.896367i \(0.646197\pi\)
\(74\) −466.000 −0.732046
\(75\) 0 0
\(76\) 412.000 0.621837
\(77\) −161.000 −0.238281
\(78\) 0 0
\(79\) −694.000 −0.988368 −0.494184 0.869357i \(-0.664533\pi\)
−0.494184 + 0.869357i \(0.664533\pi\)
\(80\) 48.0000 0.0670820
\(81\) 0 0
\(82\) 460.000 0.619494
\(83\) 1010.00 1.33569 0.667843 0.744302i \(-0.267218\pi\)
0.667843 + 0.744302i \(0.267218\pi\)
\(84\) 0 0
\(85\) −399.000 −0.509149
\(86\) −674.000 −0.845108
\(87\) 0 0
\(88\) 184.000 0.222892
\(89\) −390.000 −0.464493 −0.232247 0.972657i \(-0.574608\pi\)
−0.232247 + 0.972657i \(0.574608\pi\)
\(90\) 0 0
\(91\) 91.0000 0.104828
\(92\) −452.000 −0.512220
\(93\) 0 0
\(94\) −192.000 −0.210673
\(95\) 309.000 0.333713
\(96\) 0 0
\(97\) −102.000 −0.106768 −0.0533842 0.998574i \(-0.517001\pi\)
−0.0533842 + 0.998574i \(0.517001\pi\)
\(98\) 98.0000 0.101015
\(99\) 0 0
\(100\) −464.000 −0.464000
\(101\) −1590.00 −1.56644 −0.783222 0.621742i \(-0.786425\pi\)
−0.783222 + 0.621742i \(0.786425\pi\)
\(102\) 0 0
\(103\) 1679.00 1.60618 0.803091 0.595856i \(-0.203187\pi\)
0.803091 + 0.595856i \(0.203187\pi\)
\(104\) −104.000 −0.0980581
\(105\) 0 0
\(106\) 268.000 0.245570
\(107\) −30.0000 −0.0271048 −0.0135524 0.999908i \(-0.504314\pi\)
−0.0135524 + 0.999908i \(0.504314\pi\)
\(108\) 0 0
\(109\) −1423.00 −1.25045 −0.625223 0.780446i \(-0.714992\pi\)
−0.625223 + 0.780446i \(0.714992\pi\)
\(110\) 138.000 0.119616
\(111\) 0 0
\(112\) −112.000 −0.0944911
\(113\) −844.000 −0.702627 −0.351313 0.936258i \(-0.614265\pi\)
−0.351313 + 0.936258i \(0.614265\pi\)
\(114\) 0 0
\(115\) −339.000 −0.274886
\(116\) 564.000 0.451432
\(117\) 0 0
\(118\) −1676.00 −1.30753
\(119\) 931.000 0.717182
\(120\) 0 0
\(121\) −802.000 −0.602554
\(122\) 590.000 0.437837
\(123\) 0 0
\(124\) −360.000 −0.260717
\(125\) −723.000 −0.517337
\(126\) 0 0
\(127\) 328.000 0.229176 0.114588 0.993413i \(-0.463445\pi\)
0.114588 + 0.993413i \(0.463445\pi\)
\(128\) 128.000 0.0883883
\(129\) 0 0
\(130\) −78.0000 −0.0526235
\(131\) 267.000 0.178076 0.0890378 0.996028i \(-0.471621\pi\)
0.0890378 + 0.996028i \(0.471621\pi\)
\(132\) 0 0
\(133\) −721.000 −0.470065
\(134\) 936.000 0.603419
\(135\) 0 0
\(136\) −1064.00 −0.670862
\(137\) −1503.00 −0.937299 −0.468649 0.883384i \(-0.655259\pi\)
−0.468649 + 0.883384i \(0.655259\pi\)
\(138\) 0 0
\(139\) 958.000 0.584579 0.292290 0.956330i \(-0.405583\pi\)
0.292290 + 0.956330i \(0.405583\pi\)
\(140\) −84.0000 −0.0507093
\(141\) 0 0
\(142\) −108.000 −0.0638251
\(143\) −299.000 −0.174851
\(144\) 0 0
\(145\) 423.000 0.242264
\(146\) −1106.00 −0.626940
\(147\) 0 0
\(148\) −932.000 −0.517635
\(149\) −1864.00 −1.02486 −0.512432 0.858728i \(-0.671255\pi\)
−0.512432 + 0.858728i \(0.671255\pi\)
\(150\) 0 0
\(151\) −929.000 −0.500669 −0.250334 0.968159i \(-0.580541\pi\)
−0.250334 + 0.968159i \(0.580541\pi\)
\(152\) 824.000 0.439705
\(153\) 0 0
\(154\) −322.000 −0.168490
\(155\) −270.000 −0.139916
\(156\) 0 0
\(157\) −85.0000 −0.0432085 −0.0216043 0.999767i \(-0.506877\pi\)
−0.0216043 + 0.999767i \(0.506877\pi\)
\(158\) −1388.00 −0.698882
\(159\) 0 0
\(160\) 96.0000 0.0474342
\(161\) 791.000 0.387202
\(162\) 0 0
\(163\) −3002.00 −1.44254 −0.721272 0.692652i \(-0.756442\pi\)
−0.721272 + 0.692652i \(0.756442\pi\)
\(164\) 920.000 0.438048
\(165\) 0 0
\(166\) 2020.00 0.944472
\(167\) −1897.00 −0.879008 −0.439504 0.898241i \(-0.644846\pi\)
−0.439504 + 0.898241i \(0.644846\pi\)
\(168\) 0 0
\(169\) 169.000 0.0769231
\(170\) −798.000 −0.360022
\(171\) 0 0
\(172\) −1348.00 −0.597582
\(173\) −2222.00 −0.976506 −0.488253 0.872702i \(-0.662366\pi\)
−0.488253 + 0.872702i \(0.662366\pi\)
\(174\) 0 0
\(175\) 812.000 0.350751
\(176\) 368.000 0.157608
\(177\) 0 0
\(178\) −780.000 −0.328446
\(179\) −20.0000 −0.00835123 −0.00417562 0.999991i \(-0.501329\pi\)
−0.00417562 + 0.999991i \(0.501329\pi\)
\(180\) 0 0
\(181\) 4750.00 1.95063 0.975317 0.220810i \(-0.0708700\pi\)
0.975317 + 0.220810i \(0.0708700\pi\)
\(182\) 182.000 0.0741249
\(183\) 0 0
\(184\) −904.000 −0.362194
\(185\) −699.000 −0.277792
\(186\) 0 0
\(187\) −3059.00 −1.19624
\(188\) −384.000 −0.148969
\(189\) 0 0
\(190\) 618.000 0.235971
\(191\) −909.000 −0.344361 −0.172180 0.985065i \(-0.555081\pi\)
−0.172180 + 0.985065i \(0.555081\pi\)
\(192\) 0 0
\(193\) 442.000 0.164849 0.0824245 0.996597i \(-0.473734\pi\)
0.0824245 + 0.996597i \(0.473734\pi\)
\(194\) −204.000 −0.0754966
\(195\) 0 0
\(196\) 196.000 0.0714286
\(197\) −3286.00 −1.18842 −0.594208 0.804312i \(-0.702534\pi\)
−0.594208 + 0.804312i \(0.702534\pi\)
\(198\) 0 0
\(199\) 3993.00 1.42239 0.711197 0.702993i \(-0.248154\pi\)
0.711197 + 0.702993i \(0.248154\pi\)
\(200\) −928.000 −0.328098
\(201\) 0 0
\(202\) −3180.00 −1.10764
\(203\) −987.000 −0.341250
\(204\) 0 0
\(205\) 690.000 0.235081
\(206\) 3358.00 1.13574
\(207\) 0 0
\(208\) −208.000 −0.0693375
\(209\) 2369.00 0.784053
\(210\) 0 0
\(211\) 3445.00 1.12400 0.561999 0.827138i \(-0.310032\pi\)
0.561999 + 0.827138i \(0.310032\pi\)
\(212\) 536.000 0.173644
\(213\) 0 0
\(214\) −60.0000 −0.0191660
\(215\) −1011.00 −0.320696
\(216\) 0 0
\(217\) 630.000 0.197084
\(218\) −2846.00 −0.884199
\(219\) 0 0
\(220\) 276.000 0.0845814
\(221\) 1729.00 0.526268
\(222\) 0 0
\(223\) −1420.00 −0.426414 −0.213207 0.977007i \(-0.568391\pi\)
−0.213207 + 0.977007i \(0.568391\pi\)
\(224\) −224.000 −0.0668153
\(225\) 0 0
\(226\) −1688.00 −0.496832
\(227\) 510.000 0.149118 0.0745592 0.997217i \(-0.476245\pi\)
0.0745592 + 0.997217i \(0.476245\pi\)
\(228\) 0 0
\(229\) −1746.00 −0.503838 −0.251919 0.967748i \(-0.581062\pi\)
−0.251919 + 0.967748i \(0.581062\pi\)
\(230\) −678.000 −0.194374
\(231\) 0 0
\(232\) 1128.00 0.319210
\(233\) −1020.00 −0.286792 −0.143396 0.989665i \(-0.545802\pi\)
−0.143396 + 0.989665i \(0.545802\pi\)
\(234\) 0 0
\(235\) −288.000 −0.0799449
\(236\) −3352.00 −0.924562
\(237\) 0 0
\(238\) 1862.00 0.507124
\(239\) 2824.00 0.764307 0.382154 0.924099i \(-0.375183\pi\)
0.382154 + 0.924099i \(0.375183\pi\)
\(240\) 0 0
\(241\) 2538.00 0.678369 0.339185 0.940720i \(-0.389849\pi\)
0.339185 + 0.940720i \(0.389849\pi\)
\(242\) −1604.00 −0.426070
\(243\) 0 0
\(244\) 1180.00 0.309597
\(245\) 147.000 0.0383326
\(246\) 0 0
\(247\) −1339.00 −0.344933
\(248\) −720.000 −0.184355
\(249\) 0 0
\(250\) −1446.00 −0.365812
\(251\) −4725.00 −1.18820 −0.594102 0.804389i \(-0.702493\pi\)
−0.594102 + 0.804389i \(0.702493\pi\)
\(252\) 0 0
\(253\) −2599.00 −0.645841
\(254\) 656.000 0.162052
\(255\) 0 0
\(256\) 256.000 0.0625000
\(257\) 2642.00 0.641258 0.320629 0.947205i \(-0.396106\pi\)
0.320629 + 0.947205i \(0.396106\pi\)
\(258\) 0 0
\(259\) 1631.00 0.391295
\(260\) −156.000 −0.0372104
\(261\) 0 0
\(262\) 534.000 0.125918
\(263\) 5152.00 1.20793 0.603966 0.797010i \(-0.293586\pi\)
0.603966 + 0.797010i \(0.293586\pi\)
\(264\) 0 0
\(265\) 402.000 0.0931874
\(266\) −1442.00 −0.332386
\(267\) 0 0
\(268\) 1872.00 0.426681
\(269\) −518.000 −0.117409 −0.0587045 0.998275i \(-0.518697\pi\)
−0.0587045 + 0.998275i \(0.518697\pi\)
\(270\) 0 0
\(271\) −3642.00 −0.816368 −0.408184 0.912900i \(-0.633838\pi\)
−0.408184 + 0.912900i \(0.633838\pi\)
\(272\) −2128.00 −0.474371
\(273\) 0 0
\(274\) −3006.00 −0.662770
\(275\) −2668.00 −0.585042
\(276\) 0 0
\(277\) 4876.00 1.05766 0.528828 0.848729i \(-0.322632\pi\)
0.528828 + 0.848729i \(0.322632\pi\)
\(278\) 1916.00 0.413360
\(279\) 0 0
\(280\) −168.000 −0.0358569
\(281\) −2202.00 −0.467474 −0.233737 0.972300i \(-0.575096\pi\)
−0.233737 + 0.972300i \(0.575096\pi\)
\(282\) 0 0
\(283\) −4036.00 −0.847757 −0.423879 0.905719i \(-0.639332\pi\)
−0.423879 + 0.905719i \(0.639332\pi\)
\(284\) −216.000 −0.0451311
\(285\) 0 0
\(286\) −598.000 −0.123638
\(287\) −1610.00 −0.331133
\(288\) 0 0
\(289\) 12776.0 2.60045
\(290\) 846.000 0.171306
\(291\) 0 0
\(292\) −2212.00 −0.443313
\(293\) −5346.00 −1.06593 −0.532964 0.846138i \(-0.678922\pi\)
−0.532964 + 0.846138i \(0.678922\pi\)
\(294\) 0 0
\(295\) −2514.00 −0.496172
\(296\) −1864.00 −0.366023
\(297\) 0 0
\(298\) −3728.00 −0.724689
\(299\) 1469.00 0.284129
\(300\) 0 0
\(301\) 2359.00 0.451729
\(302\) −1858.00 −0.354026
\(303\) 0 0
\(304\) 1648.00 0.310919
\(305\) 885.000 0.166147
\(306\) 0 0
\(307\) −1556.00 −0.289269 −0.144635 0.989485i \(-0.546201\pi\)
−0.144635 + 0.989485i \(0.546201\pi\)
\(308\) −644.000 −0.119141
\(309\) 0 0
\(310\) −540.000 −0.0989353
\(311\) −3884.00 −0.708172 −0.354086 0.935213i \(-0.615208\pi\)
−0.354086 + 0.935213i \(0.615208\pi\)
\(312\) 0 0
\(313\) −5022.00 −0.906902 −0.453451 0.891281i \(-0.649807\pi\)
−0.453451 + 0.891281i \(0.649807\pi\)
\(314\) −170.000 −0.0305530
\(315\) 0 0
\(316\) −2776.00 −0.494184
\(317\) −5654.00 −1.00177 −0.500884 0.865515i \(-0.666992\pi\)
−0.500884 + 0.865515i \(0.666992\pi\)
\(318\) 0 0
\(319\) 3243.00 0.569195
\(320\) 192.000 0.0335410
\(321\) 0 0
\(322\) 1582.00 0.273793
\(323\) −13699.0 −2.35985
\(324\) 0 0
\(325\) 1508.00 0.257381
\(326\) −6004.00 −1.02003
\(327\) 0 0
\(328\) 1840.00 0.309747
\(329\) 672.000 0.112610
\(330\) 0 0
\(331\) 3214.00 0.533708 0.266854 0.963737i \(-0.414016\pi\)
0.266854 + 0.963737i \(0.414016\pi\)
\(332\) 4040.00 0.667843
\(333\) 0 0
\(334\) −3794.00 −0.621552
\(335\) 1404.00 0.228981
\(336\) 0 0
\(337\) −657.000 −0.106199 −0.0530995 0.998589i \(-0.516910\pi\)
−0.0530995 + 0.998589i \(0.516910\pi\)
\(338\) 338.000 0.0543928
\(339\) 0 0
\(340\) −1596.00 −0.254574
\(341\) −2070.00 −0.328730
\(342\) 0 0
\(343\) −343.000 −0.0539949
\(344\) −2696.00 −0.422554
\(345\) 0 0
\(346\) −4444.00 −0.690494
\(347\) −5336.00 −0.825509 −0.412754 0.910842i \(-0.635433\pi\)
−0.412754 + 0.910842i \(0.635433\pi\)
\(348\) 0 0
\(349\) 5816.00 0.892044 0.446022 0.895022i \(-0.352840\pi\)
0.446022 + 0.895022i \(0.352840\pi\)
\(350\) 1624.00 0.248018
\(351\) 0 0
\(352\) 736.000 0.111446
\(353\) 910.000 0.137208 0.0686040 0.997644i \(-0.478146\pi\)
0.0686040 + 0.997644i \(0.478146\pi\)
\(354\) 0 0
\(355\) −162.000 −0.0242199
\(356\) −1560.00 −0.232247
\(357\) 0 0
\(358\) −40.0000 −0.00590521
\(359\) −8274.00 −1.21639 −0.608196 0.793787i \(-0.708107\pi\)
−0.608196 + 0.793787i \(0.708107\pi\)
\(360\) 0 0
\(361\) 3750.00 0.546727
\(362\) 9500.00 1.37931
\(363\) 0 0
\(364\) 364.000 0.0524142
\(365\) −1659.00 −0.237907
\(366\) 0 0
\(367\) 5624.00 0.799919 0.399960 0.916533i \(-0.369024\pi\)
0.399960 + 0.916533i \(0.369024\pi\)
\(368\) −1808.00 −0.256110
\(369\) 0 0
\(370\) −1398.00 −0.196429
\(371\) −938.000 −0.131263
\(372\) 0 0
\(373\) 246.000 0.0341485 0.0170743 0.999854i \(-0.494565\pi\)
0.0170743 + 0.999854i \(0.494565\pi\)
\(374\) −6118.00 −0.845867
\(375\) 0 0
\(376\) −768.000 −0.105337
\(377\) −1833.00 −0.250409
\(378\) 0 0
\(379\) −802.000 −0.108696 −0.0543482 0.998522i \(-0.517308\pi\)
−0.0543482 + 0.998522i \(0.517308\pi\)
\(380\) 1236.00 0.166856
\(381\) 0 0
\(382\) −1818.00 −0.243500
\(383\) −2597.00 −0.346477 −0.173238 0.984880i \(-0.555423\pi\)
−0.173238 + 0.984880i \(0.555423\pi\)
\(384\) 0 0
\(385\) −483.000 −0.0639376
\(386\) 884.000 0.116566
\(387\) 0 0
\(388\) −408.000 −0.0533842
\(389\) −5986.00 −0.780211 −0.390106 0.920770i \(-0.627562\pi\)
−0.390106 + 0.920770i \(0.627562\pi\)
\(390\) 0 0
\(391\) 15029.0 1.94386
\(392\) 392.000 0.0505076
\(393\) 0 0
\(394\) −6572.00 −0.840336
\(395\) −2082.00 −0.265207
\(396\) 0 0
\(397\) 14524.0 1.83612 0.918059 0.396444i \(-0.129756\pi\)
0.918059 + 0.396444i \(0.129756\pi\)
\(398\) 7986.00 1.00578
\(399\) 0 0
\(400\) −1856.00 −0.232000
\(401\) 7334.00 0.913323 0.456661 0.889641i \(-0.349045\pi\)
0.456661 + 0.889641i \(0.349045\pi\)
\(402\) 0 0
\(403\) 1170.00 0.144620
\(404\) −6360.00 −0.783222
\(405\) 0 0
\(406\) −1974.00 −0.241300
\(407\) −5359.00 −0.652668
\(408\) 0 0
\(409\) 2981.00 0.360394 0.180197 0.983631i \(-0.442327\pi\)
0.180197 + 0.983631i \(0.442327\pi\)
\(410\) 1380.00 0.166228
\(411\) 0 0
\(412\) 6716.00 0.803091
\(413\) 5866.00 0.698903
\(414\) 0 0
\(415\) 3030.00 0.358402
\(416\) −416.000 −0.0490290
\(417\) 0 0
\(418\) 4738.00 0.554409
\(419\) 169.000 0.0197045 0.00985226 0.999951i \(-0.496864\pi\)
0.00985226 + 0.999951i \(0.496864\pi\)
\(420\) 0 0
\(421\) 9950.00 1.15186 0.575930 0.817499i \(-0.304640\pi\)
0.575930 + 0.817499i \(0.304640\pi\)
\(422\) 6890.00 0.794787
\(423\) 0 0
\(424\) 1072.00 0.122785
\(425\) 15428.0 1.76087
\(426\) 0 0
\(427\) −2065.00 −0.234034
\(428\) −120.000 −0.0135524
\(429\) 0 0
\(430\) −2022.00 −0.226766
\(431\) 7462.00 0.833949 0.416974 0.908918i \(-0.363090\pi\)
0.416974 + 0.908918i \(0.363090\pi\)
\(432\) 0 0
\(433\) −782.000 −0.0867910 −0.0433955 0.999058i \(-0.513818\pi\)
−0.0433955 + 0.999058i \(0.513818\pi\)
\(434\) 1260.00 0.139359
\(435\) 0 0
\(436\) −5692.00 −0.625223
\(437\) −11639.0 −1.27407
\(438\) 0 0
\(439\) 1079.00 0.117307 0.0586536 0.998278i \(-0.481319\pi\)
0.0586536 + 0.998278i \(0.481319\pi\)
\(440\) 552.000 0.0598081
\(441\) 0 0
\(442\) 3458.00 0.372127
\(443\) −4538.00 −0.486697 −0.243349 0.969939i \(-0.578246\pi\)
−0.243349 + 0.969939i \(0.578246\pi\)
\(444\) 0 0
\(445\) −1170.00 −0.124637
\(446\) −2840.00 −0.301520
\(447\) 0 0
\(448\) −448.000 −0.0472456
\(449\) 15773.0 1.65785 0.828924 0.559361i \(-0.188953\pi\)
0.828924 + 0.559361i \(0.188953\pi\)
\(450\) 0 0
\(451\) 5290.00 0.552320
\(452\) −3376.00 −0.351313
\(453\) 0 0
\(454\) 1020.00 0.105443
\(455\) 273.000 0.0281284
\(456\) 0 0
\(457\) −3112.00 −0.318541 −0.159270 0.987235i \(-0.550914\pi\)
−0.159270 + 0.987235i \(0.550914\pi\)
\(458\) −3492.00 −0.356267
\(459\) 0 0
\(460\) −1356.00 −0.137443
\(461\) 9305.00 0.940080 0.470040 0.882645i \(-0.344239\pi\)
0.470040 + 0.882645i \(0.344239\pi\)
\(462\) 0 0
\(463\) 13763.0 1.38147 0.690735 0.723108i \(-0.257287\pi\)
0.690735 + 0.723108i \(0.257287\pi\)
\(464\) 2256.00 0.225716
\(465\) 0 0
\(466\) −2040.00 −0.202792
\(467\) −14625.0 −1.44917 −0.724587 0.689183i \(-0.757969\pi\)
−0.724587 + 0.689183i \(0.757969\pi\)
\(468\) 0 0
\(469\) −3276.00 −0.322541
\(470\) −576.000 −0.0565296
\(471\) 0 0
\(472\) −6704.00 −0.653764
\(473\) −7751.00 −0.753470
\(474\) 0 0
\(475\) −11948.0 −1.15413
\(476\) 3724.00 0.358591
\(477\) 0 0
\(478\) 5648.00 0.540447
\(479\) 6797.00 0.648357 0.324178 0.945996i \(-0.394912\pi\)
0.324178 + 0.945996i \(0.394912\pi\)
\(480\) 0 0
\(481\) 3029.00 0.287132
\(482\) 5076.00 0.479679
\(483\) 0 0
\(484\) −3208.00 −0.301277
\(485\) −306.000 −0.0286490
\(486\) 0 0
\(487\) −13400.0 −1.24684 −0.623421 0.781886i \(-0.714258\pi\)
−0.623421 + 0.781886i \(0.714258\pi\)
\(488\) 2360.00 0.218918
\(489\) 0 0
\(490\) 294.000 0.0271052
\(491\) −3176.00 −0.291916 −0.145958 0.989291i \(-0.546626\pi\)
−0.145958 + 0.989291i \(0.546626\pi\)
\(492\) 0 0
\(493\) −18753.0 −1.71317
\(494\) −2678.00 −0.243905
\(495\) 0 0
\(496\) −1440.00 −0.130359
\(497\) 378.000 0.0341159
\(498\) 0 0
\(499\) −12164.0 −1.09125 −0.545627 0.838028i \(-0.683708\pi\)
−0.545627 + 0.838028i \(0.683708\pi\)
\(500\) −2892.00 −0.258668
\(501\) 0 0
\(502\) −9450.00 −0.840188
\(503\) −14808.0 −1.31264 −0.656318 0.754484i \(-0.727887\pi\)
−0.656318 + 0.754484i \(0.727887\pi\)
\(504\) 0 0
\(505\) −4770.00 −0.420321
\(506\) −5198.00 −0.456678
\(507\) 0 0
\(508\) 1312.00 0.114588
\(509\) 10977.0 0.955888 0.477944 0.878390i \(-0.341382\pi\)
0.477944 + 0.878390i \(0.341382\pi\)
\(510\) 0 0
\(511\) 3871.00 0.335113
\(512\) 512.000 0.0441942
\(513\) 0 0
\(514\) 5284.00 0.453438
\(515\) 5037.00 0.430984
\(516\) 0 0
\(517\) −2208.00 −0.187829
\(518\) 3262.00 0.276687
\(519\) 0 0
\(520\) −312.000 −0.0263117
\(521\) 14885.0 1.25168 0.625838 0.779953i \(-0.284757\pi\)
0.625838 + 0.779953i \(0.284757\pi\)
\(522\) 0 0
\(523\) −1054.00 −0.0881228 −0.0440614 0.999029i \(-0.514030\pi\)
−0.0440614 + 0.999029i \(0.514030\pi\)
\(524\) 1068.00 0.0890378
\(525\) 0 0
\(526\) 10304.0 0.854136
\(527\) 11970.0 0.989414
\(528\) 0 0
\(529\) 602.000 0.0494781
\(530\) 804.000 0.0658934
\(531\) 0 0
\(532\) −2884.00 −0.235032
\(533\) −2990.00 −0.242986
\(534\) 0 0
\(535\) −90.0000 −0.00727297
\(536\) 3744.00 0.301709
\(537\) 0 0
\(538\) −1036.00 −0.0830207
\(539\) 1127.00 0.0900618
\(540\) 0 0
\(541\) 21697.0 1.72426 0.862132 0.506684i \(-0.169129\pi\)
0.862132 + 0.506684i \(0.169129\pi\)
\(542\) −7284.00 −0.577259
\(543\) 0 0
\(544\) −4256.00 −0.335431
\(545\) −4269.00 −0.335530
\(546\) 0 0
\(547\) 23836.0 1.86317 0.931585 0.363524i \(-0.118427\pi\)
0.931585 + 0.363524i \(0.118427\pi\)
\(548\) −6012.00 −0.468649
\(549\) 0 0
\(550\) −5336.00 −0.413687
\(551\) 14523.0 1.12287
\(552\) 0 0
\(553\) 4858.00 0.373568
\(554\) 9752.00 0.747875
\(555\) 0 0
\(556\) 3832.00 0.292290
\(557\) −4902.00 −0.372898 −0.186449 0.982465i \(-0.559698\pi\)
−0.186449 + 0.982465i \(0.559698\pi\)
\(558\) 0 0
\(559\) 4381.00 0.331479
\(560\) −336.000 −0.0253546
\(561\) 0 0
\(562\) −4404.00 −0.330554
\(563\) 3147.00 0.235578 0.117789 0.993039i \(-0.462419\pi\)
0.117789 + 0.993039i \(0.462419\pi\)
\(564\) 0 0
\(565\) −2532.00 −0.188535
\(566\) −8072.00 −0.599455
\(567\) 0 0
\(568\) −432.000 −0.0319125
\(569\) 15394.0 1.13418 0.567091 0.823655i \(-0.308069\pi\)
0.567091 + 0.823655i \(0.308069\pi\)
\(570\) 0 0
\(571\) −11588.0 −0.849287 −0.424643 0.905361i \(-0.639601\pi\)
−0.424643 + 0.905361i \(0.639601\pi\)
\(572\) −1196.00 −0.0874253
\(573\) 0 0
\(574\) −3220.00 −0.234147
\(575\) 13108.0 0.950681
\(576\) 0 0
\(577\) 25474.0 1.83795 0.918974 0.394317i \(-0.129019\pi\)
0.918974 + 0.394317i \(0.129019\pi\)
\(578\) 25552.0 1.83879
\(579\) 0 0
\(580\) 1692.00 0.121132
\(581\) −7070.00 −0.504842
\(582\) 0 0
\(583\) 3082.00 0.218942
\(584\) −4424.00 −0.313470
\(585\) 0 0
\(586\) −10692.0 −0.753724
\(587\) 10122.0 0.711720 0.355860 0.934539i \(-0.384188\pi\)
0.355860 + 0.934539i \(0.384188\pi\)
\(588\) 0 0
\(589\) −9270.00 −0.648495
\(590\) −5028.00 −0.350847
\(591\) 0 0
\(592\) −3728.00 −0.258817
\(593\) 19524.0 1.35203 0.676016 0.736887i \(-0.263705\pi\)
0.676016 + 0.736887i \(0.263705\pi\)
\(594\) 0 0
\(595\) 2793.00 0.192440
\(596\) −7456.00 −0.512432
\(597\) 0 0
\(598\) 2938.00 0.200909
\(599\) 18157.0 1.23852 0.619261 0.785185i \(-0.287432\pi\)
0.619261 + 0.785185i \(0.287432\pi\)
\(600\) 0 0
\(601\) 7438.00 0.504829 0.252415 0.967619i \(-0.418775\pi\)
0.252415 + 0.967619i \(0.418775\pi\)
\(602\) 4718.00 0.319421
\(603\) 0 0
\(604\) −3716.00 −0.250334
\(605\) −2406.00 −0.161682
\(606\) 0 0
\(607\) 9451.00 0.631967 0.315984 0.948765i \(-0.397665\pi\)
0.315984 + 0.948765i \(0.397665\pi\)
\(608\) 3296.00 0.219853
\(609\) 0 0
\(610\) 1770.00 0.117484
\(611\) 1248.00 0.0826329
\(612\) 0 0
\(613\) −12631.0 −0.832237 −0.416119 0.909310i \(-0.636610\pi\)
−0.416119 + 0.909310i \(0.636610\pi\)
\(614\) −3112.00 −0.204544
\(615\) 0 0
\(616\) −1288.00 −0.0842451
\(617\) −16065.0 −1.04822 −0.524111 0.851650i \(-0.675602\pi\)
−0.524111 + 0.851650i \(0.675602\pi\)
\(618\) 0 0
\(619\) −21773.0 −1.41378 −0.706891 0.707323i \(-0.749903\pi\)
−0.706891 + 0.707323i \(0.749903\pi\)
\(620\) −1080.00 −0.0699578
\(621\) 0 0
\(622\) −7768.00 −0.500753
\(623\) 2730.00 0.175562
\(624\) 0 0
\(625\) 12331.0 0.789184
\(626\) −10044.0 −0.641276
\(627\) 0 0
\(628\) −340.000 −0.0216043
\(629\) 30989.0 1.96441
\(630\) 0 0
\(631\) 15575.0 0.982616 0.491308 0.870986i \(-0.336519\pi\)
0.491308 + 0.870986i \(0.336519\pi\)
\(632\) −5552.00 −0.349441
\(633\) 0 0
\(634\) −11308.0 −0.708357
\(635\) 984.000 0.0614943
\(636\) 0 0
\(637\) −637.000 −0.0396214
\(638\) 6486.00 0.402482
\(639\) 0 0
\(640\) 384.000 0.0237171
\(641\) −8814.00 −0.543108 −0.271554 0.962423i \(-0.587538\pi\)
−0.271554 + 0.962423i \(0.587538\pi\)
\(642\) 0 0
\(643\) 10629.0 0.651892 0.325946 0.945388i \(-0.394317\pi\)
0.325946 + 0.945388i \(0.394317\pi\)
\(644\) 3164.00 0.193601
\(645\) 0 0
\(646\) −27398.0 −1.66867
\(647\) 18624.0 1.13166 0.565831 0.824521i \(-0.308556\pi\)
0.565831 + 0.824521i \(0.308556\pi\)
\(648\) 0 0
\(649\) −19274.0 −1.16575
\(650\) 3016.00 0.181996
\(651\) 0 0
\(652\) −12008.0 −0.721272
\(653\) −29561.0 −1.77153 −0.885767 0.464131i \(-0.846367\pi\)
−0.885767 + 0.464131i \(0.846367\pi\)
\(654\) 0 0
\(655\) 801.000 0.0477827
\(656\) 3680.00 0.219024
\(657\) 0 0
\(658\) 1344.00 0.0796270
\(659\) −18432.0 −1.08954 −0.544771 0.838585i \(-0.683384\pi\)
−0.544771 + 0.838585i \(0.683384\pi\)
\(660\) 0 0
\(661\) 18956.0 1.11544 0.557718 0.830031i \(-0.311677\pi\)
0.557718 + 0.830031i \(0.311677\pi\)
\(662\) 6428.00 0.377389
\(663\) 0 0
\(664\) 8080.00 0.472236
\(665\) −2163.00 −0.126132
\(666\) 0 0
\(667\) −15933.0 −0.924930
\(668\) −7588.00 −0.439504
\(669\) 0 0
\(670\) 2808.00 0.161914
\(671\) 6785.00 0.390361
\(672\) 0 0
\(673\) 32299.0 1.84998 0.924989 0.379994i \(-0.124074\pi\)
0.924989 + 0.379994i \(0.124074\pi\)
\(674\) −1314.00 −0.0750940
\(675\) 0 0
\(676\) 676.000 0.0384615
\(677\) 5786.00 0.328470 0.164235 0.986421i \(-0.447484\pi\)
0.164235 + 0.986421i \(0.447484\pi\)
\(678\) 0 0
\(679\) 714.000 0.0403546
\(680\) −3192.00 −0.180011
\(681\) 0 0
\(682\) −4140.00 −0.232447
\(683\) 17761.0 0.995030 0.497515 0.867455i \(-0.334246\pi\)
0.497515 + 0.867455i \(0.334246\pi\)
\(684\) 0 0
\(685\) −4509.00 −0.251504
\(686\) −686.000 −0.0381802
\(687\) 0 0
\(688\) −5392.00 −0.298791
\(689\) −1742.00 −0.0963206
\(690\) 0 0
\(691\) −592.000 −0.0325915 −0.0162958 0.999867i \(-0.505187\pi\)
−0.0162958 + 0.999867i \(0.505187\pi\)
\(692\) −8888.00 −0.488253
\(693\) 0 0
\(694\) −10672.0 −0.583723
\(695\) 2874.00 0.156859
\(696\) 0 0
\(697\) −30590.0 −1.66238
\(698\) 11632.0 0.630770
\(699\) 0 0
\(700\) 3248.00 0.175376
\(701\) 3498.00 0.188470 0.0942351 0.995550i \(-0.469959\pi\)
0.0942351 + 0.995550i \(0.469959\pi\)
\(702\) 0 0
\(703\) −23999.0 −1.28754
\(704\) 1472.00 0.0788041
\(705\) 0 0
\(706\) 1820.00 0.0970207
\(707\) 11130.0 0.592060
\(708\) 0 0
\(709\) −19378.0 −1.02645 −0.513227 0.858253i \(-0.671550\pi\)
−0.513227 + 0.858253i \(0.671550\pi\)
\(710\) −324.000 −0.0171261
\(711\) 0 0
\(712\) −3120.00 −0.164223
\(713\) 10170.0 0.534179
\(714\) 0 0
\(715\) −897.000 −0.0469173
\(716\) −80.0000 −0.00417562
\(717\) 0 0
\(718\) −16548.0 −0.860120
\(719\) −2856.00 −0.148137 −0.0740687 0.997253i \(-0.523598\pi\)
−0.0740687 + 0.997253i \(0.523598\pi\)
\(720\) 0 0
\(721\) −11753.0 −0.607080
\(722\) 7500.00 0.386594
\(723\) 0 0
\(724\) 19000.0 0.975317
\(725\) −16356.0 −0.837857
\(726\) 0 0
\(727\) 19699.0 1.00495 0.502473 0.864593i \(-0.332424\pi\)
0.502473 + 0.864593i \(0.332424\pi\)
\(728\) 728.000 0.0370625
\(729\) 0 0
\(730\) −3318.00 −0.168226
\(731\) 44821.0 2.26780
\(732\) 0 0
\(733\) 18776.0 0.946122 0.473061 0.881030i \(-0.343149\pi\)
0.473061 + 0.881030i \(0.343149\pi\)
\(734\) 11248.0 0.565628
\(735\) 0 0
\(736\) −3616.00 −0.181097
\(737\) 10764.0 0.537988
\(738\) 0 0
\(739\) −26158.0 −1.30208 −0.651040 0.759043i \(-0.725667\pi\)
−0.651040 + 0.759043i \(0.725667\pi\)
\(740\) −2796.00 −0.138896
\(741\) 0 0
\(742\) −1876.00 −0.0928169
\(743\) −33700.0 −1.66397 −0.831987 0.554795i \(-0.812797\pi\)
−0.831987 + 0.554795i \(0.812797\pi\)
\(744\) 0 0
\(745\) −5592.00 −0.275000
\(746\) 492.000 0.0241466
\(747\) 0 0
\(748\) −12236.0 −0.598118
\(749\) 210.000 0.0102446
\(750\) 0 0
\(751\) −14416.0 −0.700462 −0.350231 0.936663i \(-0.613897\pi\)
−0.350231 + 0.936663i \(0.613897\pi\)
\(752\) −1536.00 −0.0744843
\(753\) 0 0
\(754\) −3666.00 −0.177066
\(755\) −2787.00 −0.134343
\(756\) 0 0
\(757\) −27724.0 −1.33110 −0.665552 0.746351i \(-0.731804\pi\)
−0.665552 + 0.746351i \(0.731804\pi\)
\(758\) −1604.00 −0.0768600
\(759\) 0 0
\(760\) 2472.00 0.117985
\(761\) 24476.0 1.16591 0.582953 0.812506i \(-0.301897\pi\)
0.582953 + 0.812506i \(0.301897\pi\)
\(762\) 0 0
\(763\) 9961.00 0.472624
\(764\) −3636.00 −0.172180
\(765\) 0 0
\(766\) −5194.00 −0.244996
\(767\) 10894.0 0.512855
\(768\) 0 0
\(769\) 27869.0 1.30687 0.653434 0.756983i \(-0.273328\pi\)
0.653434 + 0.756983i \(0.273328\pi\)
\(770\) −966.000 −0.0452107
\(771\) 0 0
\(772\) 1768.00 0.0824245
\(773\) −15407.0 −0.716884 −0.358442 0.933552i \(-0.616692\pi\)
−0.358442 + 0.933552i \(0.616692\pi\)
\(774\) 0 0
\(775\) 10440.0 0.483891
\(776\) −816.000 −0.0377483
\(777\) 0 0
\(778\) −11972.0 −0.551693
\(779\) 23690.0 1.08958
\(780\) 0 0
\(781\) −1242.00 −0.0569043
\(782\) 30058.0 1.37452
\(783\) 0 0
\(784\) 784.000 0.0357143
\(785\) −255.000 −0.0115941
\(786\) 0 0
\(787\) 20675.0 0.936447 0.468224 0.883610i \(-0.344894\pi\)
0.468224 + 0.883610i \(0.344894\pi\)
\(788\) −13144.0 −0.594208
\(789\) 0 0
\(790\) −4164.00 −0.187530
\(791\) 5908.00 0.265568
\(792\) 0 0
\(793\) −3835.00 −0.171734
\(794\) 29048.0 1.29833
\(795\) 0 0
\(796\) 15972.0 0.711197
\(797\) 37704.0 1.67571 0.837857 0.545890i \(-0.183808\pi\)
0.837857 + 0.545890i \(0.183808\pi\)
\(798\) 0 0
\(799\) 12768.0 0.565331
\(800\) −3712.00 −0.164049
\(801\) 0 0
\(802\) 14668.0 0.645817
\(803\) −12719.0 −0.558959
\(804\) 0 0
\(805\) 2373.00 0.103897
\(806\) 2340.00 0.102262
\(807\) 0 0
\(808\) −12720.0 −0.553822
\(809\) −31714.0 −1.37825 −0.689125 0.724642i \(-0.742005\pi\)
−0.689125 + 0.724642i \(0.742005\pi\)
\(810\) 0 0
\(811\) −2493.00 −0.107942 −0.0539711 0.998542i \(-0.517188\pi\)
−0.0539711 + 0.998542i \(0.517188\pi\)
\(812\) −3948.00 −0.170625
\(813\) 0 0
\(814\) −10718.0 −0.461506
\(815\) −9006.00 −0.387075
\(816\) 0 0
\(817\) −34711.0 −1.48639
\(818\) 5962.00 0.254837
\(819\) 0 0
\(820\) 2760.00 0.117541
\(821\) −17932.0 −0.762279 −0.381140 0.924518i \(-0.624468\pi\)
−0.381140 + 0.924518i \(0.624468\pi\)
\(822\) 0 0
\(823\) 22.0000 0.000931800 0 0.000465900 1.00000i \(-0.499852\pi\)
0.000465900 1.00000i \(0.499852\pi\)
\(824\) 13432.0 0.567871
\(825\) 0 0
\(826\) 11732.0 0.494199
\(827\) −27225.0 −1.14475 −0.572374 0.819993i \(-0.693977\pi\)
−0.572374 + 0.819993i \(0.693977\pi\)
\(828\) 0 0
\(829\) 40759.0 1.70762 0.853811 0.520583i \(-0.174285\pi\)
0.853811 + 0.520583i \(0.174285\pi\)
\(830\) 6060.00 0.253429
\(831\) 0 0
\(832\) −832.000 −0.0346688
\(833\) −6517.00 −0.271069
\(834\) 0 0
\(835\) −5691.00 −0.235862
\(836\) 9476.00 0.392027
\(837\) 0 0
\(838\) 338.000 0.0139332
\(839\) −3680.00 −0.151428 −0.0757138 0.997130i \(-0.524124\pi\)
−0.0757138 + 0.997130i \(0.524124\pi\)
\(840\) 0 0
\(841\) −4508.00 −0.184837
\(842\) 19900.0 0.814488
\(843\) 0 0
\(844\) 13780.0 0.561999
\(845\) 507.000 0.0206406
\(846\) 0 0
\(847\) 5614.00 0.227744
\(848\) 2144.00 0.0868222
\(849\) 0 0
\(850\) 30856.0 1.24512
\(851\) 26329.0 1.06057
\(852\) 0 0
\(853\) −12112.0 −0.486175 −0.243087 0.970004i \(-0.578160\pi\)
−0.243087 + 0.970004i \(0.578160\pi\)
\(854\) −4130.00 −0.165487
\(855\) 0 0
\(856\) −240.000 −0.00958298
\(857\) −21322.0 −0.849878 −0.424939 0.905222i \(-0.639704\pi\)
−0.424939 + 0.905222i \(0.639704\pi\)
\(858\) 0 0
\(859\) −43598.0 −1.73172 −0.865858 0.500289i \(-0.833227\pi\)
−0.865858 + 0.500289i \(0.833227\pi\)
\(860\) −4044.00 −0.160348
\(861\) 0 0
\(862\) 14924.0 0.589691
\(863\) 37560.0 1.48153 0.740763 0.671766i \(-0.234464\pi\)
0.740763 + 0.671766i \(0.234464\pi\)
\(864\) 0 0
\(865\) −6666.00 −0.262024
\(866\) −1564.00 −0.0613705
\(867\) 0 0
\(868\) 2520.00 0.0985419
\(869\) −15962.0 −0.623100
\(870\) 0 0
\(871\) −6084.00 −0.236680
\(872\) −11384.0 −0.442100
\(873\) 0 0
\(874\) −23278.0 −0.900904
\(875\) 5061.00 0.195535
\(876\) 0 0
\(877\) 43230.0 1.66451 0.832254 0.554395i \(-0.187050\pi\)
0.832254 + 0.554395i \(0.187050\pi\)
\(878\) 2158.00 0.0829487
\(879\) 0 0
\(880\) 1104.00 0.0422907
\(881\) 25691.0 0.982465 0.491233 0.871028i \(-0.336547\pi\)
0.491233 + 0.871028i \(0.336547\pi\)
\(882\) 0 0
\(883\) −13789.0 −0.525523 −0.262761 0.964861i \(-0.584633\pi\)
−0.262761 + 0.964861i \(0.584633\pi\)
\(884\) 6916.00 0.263134
\(885\) 0 0
\(886\) −9076.00 −0.344147
\(887\) −10482.0 −0.396788 −0.198394 0.980122i \(-0.563573\pi\)
−0.198394 + 0.980122i \(0.563573\pi\)
\(888\) 0 0
\(889\) −2296.00 −0.0866202
\(890\) −2340.00 −0.0881314
\(891\) 0 0
\(892\) −5680.00 −0.213207
\(893\) −9888.00 −0.370537
\(894\) 0 0
\(895\) −60.0000 −0.00224087
\(896\) −896.000 −0.0334077
\(897\) 0 0
\(898\) 31546.0 1.17228
\(899\) −12690.0 −0.470784
\(900\) 0 0
\(901\) −17822.0 −0.658975
\(902\) 10580.0 0.390549
\(903\) 0 0
\(904\) −6752.00 −0.248416
\(905\) 14250.0 0.523410
\(906\) 0 0
\(907\) 5340.00 0.195493 0.0977463 0.995211i \(-0.468837\pi\)
0.0977463 + 0.995211i \(0.468837\pi\)
\(908\) 2040.00 0.0745592
\(909\) 0 0
\(910\) 546.000 0.0198898
\(911\) −21495.0 −0.781736 −0.390868 0.920447i \(-0.627825\pi\)
−0.390868 + 0.920447i \(0.627825\pi\)
\(912\) 0 0
\(913\) 23230.0 0.842060
\(914\) −6224.00 −0.225242
\(915\) 0 0
\(916\) −6984.00 −0.251919
\(917\) −1869.00 −0.0673062
\(918\) 0 0
\(919\) −52070.0 −1.86902 −0.934511 0.355935i \(-0.884163\pi\)
−0.934511 + 0.355935i \(0.884163\pi\)
\(920\) −2712.00 −0.0971869
\(921\) 0 0
\(922\) 18610.0 0.664737
\(923\) 702.000 0.0250342
\(924\) 0 0
\(925\) 27028.0 0.960730
\(926\) 27526.0 0.976847
\(927\) 0 0
\(928\) 4512.00 0.159605
\(929\) 4386.00 0.154898 0.0774489 0.996996i \(-0.475323\pi\)
0.0774489 + 0.996996i \(0.475323\pi\)
\(930\) 0 0
\(931\) 5047.00 0.177668
\(932\) −4080.00 −0.143396
\(933\) 0 0
\(934\) −29250.0 −1.02472
\(935\) −9177.00 −0.320984
\(936\) 0 0
\(937\) 33374.0 1.16359 0.581794 0.813337i \(-0.302351\pi\)
0.581794 + 0.813337i \(0.302351\pi\)
\(938\) −6552.00 −0.228071
\(939\) 0 0
\(940\) −1152.00 −0.0399724
\(941\) 22742.0 0.787851 0.393926 0.919142i \(-0.371117\pi\)
0.393926 + 0.919142i \(0.371117\pi\)
\(942\) 0 0
\(943\) −25990.0 −0.897509
\(944\) −13408.0 −0.462281
\(945\) 0 0
\(946\) −15502.0 −0.532784
\(947\) 45239.0 1.55234 0.776172 0.630521i \(-0.217159\pi\)
0.776172 + 0.630521i \(0.217159\pi\)
\(948\) 0 0
\(949\) 7189.00 0.245906
\(950\) −23896.0 −0.816093
\(951\) 0 0
\(952\) 7448.00 0.253562
\(953\) 46536.0 1.58179 0.790897 0.611950i \(-0.209615\pi\)
0.790897 + 0.611950i \(0.209615\pi\)
\(954\) 0 0
\(955\) −2727.00 −0.0924017
\(956\) 11296.0 0.382154
\(957\) 0 0
\(958\) 13594.0 0.458457
\(959\) 10521.0 0.354266
\(960\) 0 0
\(961\) −21691.0 −0.728106
\(962\) 6058.00 0.203033
\(963\) 0 0
\(964\) 10152.0 0.339185
\(965\) 1326.00 0.0442336
\(966\) 0 0
\(967\) −56219.0 −1.86958 −0.934789 0.355205i \(-0.884411\pi\)
−0.934789 + 0.355205i \(0.884411\pi\)
\(968\) −6416.00 −0.213035
\(969\) 0 0
\(970\) −612.000 −0.0202579
\(971\) 35376.0 1.16918 0.584588 0.811330i \(-0.301256\pi\)
0.584588 + 0.811330i \(0.301256\pi\)
\(972\) 0 0
\(973\) −6706.00 −0.220950
\(974\) −26800.0 −0.881650
\(975\) 0 0
\(976\) 4720.00 0.154799
\(977\) −37025.0 −1.21242 −0.606210 0.795304i \(-0.707311\pi\)
−0.606210 + 0.795304i \(0.707311\pi\)
\(978\) 0 0
\(979\) −8970.00 −0.292832
\(980\) 588.000 0.0191663
\(981\) 0 0
\(982\) −6352.00 −0.206416
\(983\) −47063.0 −1.52704 −0.763518 0.645786i \(-0.776530\pi\)
−0.763518 + 0.645786i \(0.776530\pi\)
\(984\) 0 0
\(985\) −9858.00 −0.318885
\(986\) −37506.0 −1.21139
\(987\) 0 0
\(988\) −5356.00 −0.172467
\(989\) 38081.0 1.22437
\(990\) 0 0
\(991\) −2138.00 −0.0685326 −0.0342663 0.999413i \(-0.510909\pi\)
−0.0342663 + 0.999413i \(0.510909\pi\)
\(992\) −2880.00 −0.0921775
\(993\) 0 0
\(994\) 756.000 0.0241236
\(995\) 11979.0 0.381668
\(996\) 0 0
\(997\) 42282.0 1.34311 0.671557 0.740953i \(-0.265626\pi\)
0.671557 + 0.740953i \(0.265626\pi\)
\(998\) −24328.0 −0.771633
\(999\) 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 1638.4.a.k.1.1 yes 1
3.2 odd 2 1638.4.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1638.4.a.c.1.1 1 3.2 odd 2
1638.4.a.k.1.1 yes 1 1.1 even 1 trivial