Properties

Label 1232.4.a.s.1.4
Level $1232$
Weight $4$
Character 1232.1
Self dual yes
Analytic conductor $72.690$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1232 = 2^{4} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1232.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(72.6903531271\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.522072.1
Defining polynomial: \( x^{4} - x^{3} - 12x^{2} + 5x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 77)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(0.555307\) of defining polynomial
Character \(\chi\) \(=\) 1232.1

$q$-expansion

\(f(q)\) \(=\) \(q+5.49244 q^{3} -16.0955 q^{5} +7.00000 q^{7} +3.16692 q^{9} +O(q^{10})\) \(q+5.49244 q^{3} -16.0955 q^{5} +7.00000 q^{7} +3.16692 q^{9} +11.0000 q^{11} +35.3712 q^{13} -88.4036 q^{15} +40.4757 q^{17} -118.159 q^{19} +38.4471 q^{21} +174.510 q^{23} +134.065 q^{25} -130.902 q^{27} -262.725 q^{29} +36.1894 q^{31} +60.4169 q^{33} -112.668 q^{35} +19.0464 q^{37} +194.274 q^{39} +156.996 q^{41} -287.182 q^{43} -50.9731 q^{45} -397.244 q^{47} +49.0000 q^{49} +222.311 q^{51} +272.483 q^{53} -177.050 q^{55} -648.984 q^{57} +507.466 q^{59} +35.5608 q^{61} +22.1684 q^{63} -569.317 q^{65} -979.229 q^{67} +958.484 q^{69} -750.404 q^{71} +395.594 q^{73} +736.344 q^{75} +77.0000 q^{77} +736.516 q^{79} -804.477 q^{81} -582.975 q^{83} -651.477 q^{85} -1443.00 q^{87} -806.201 q^{89} +247.598 q^{91} +198.768 q^{93} +1901.84 q^{95} -957.232 q^{97} +34.8361 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 14 q^{3} + 10 q^{5} + 28 q^{7} + 76 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 14 q^{3} + 10 q^{5} + 28 q^{7} + 76 q^{9} + 44 q^{11} + 58 q^{13} - 284 q^{15} + 4 q^{17} - 258 q^{19} - 98 q^{21} - 8 q^{23} + 80 q^{25} - 428 q^{27} - 396 q^{29} + 56 q^{31} - 154 q^{33} + 70 q^{35} + 84 q^{37} + 412 q^{39} + 52 q^{41} - 408 q^{43} + 826 q^{45} - 8 q^{47} + 196 q^{49} + 388 q^{51} + 624 q^{53} + 110 q^{55} + 48 q^{57} + 238 q^{59} - 162 q^{61} + 532 q^{63} - 32 q^{65} - 1340 q^{67} + 2416 q^{69} - 1788 q^{71} + 1456 q^{73} + 806 q^{75} + 308 q^{77} + 1324 q^{79} + 1444 q^{81} - 450 q^{83} - 1736 q^{85} - 588 q^{87} - 3072 q^{89} + 406 q^{91} - 1264 q^{93} - 24 q^{95} - 652 q^{97} + 836 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) 5.49244 1.05702 0.528510 0.848927i \(-0.322751\pi\)
0.528510 + 0.848927i \(0.322751\pi\)
\(4\) 0 0
\(5\) −16.0955 −1.43962 −0.719812 0.694169i \(-0.755772\pi\)
−0.719812 + 0.694169i \(0.755772\pi\)
\(6\) 0 0
\(7\) 7.00000 0.377964
\(8\) 0 0
\(9\) 3.16692 0.117293
\(10\) 0 0
\(11\) 11.0000 0.301511
\(12\) 0 0
\(13\) 35.3712 0.754631 0.377316 0.926085i \(-0.376847\pi\)
0.377316 + 0.926085i \(0.376847\pi\)
\(14\) 0 0
\(15\) −88.4036 −1.52171
\(16\) 0 0
\(17\) 40.4757 0.577459 0.288730 0.957411i \(-0.406767\pi\)
0.288730 + 0.957411i \(0.406767\pi\)
\(18\) 0 0
\(19\) −118.159 −1.42672 −0.713359 0.700799i \(-0.752827\pi\)
−0.713359 + 0.700799i \(0.752827\pi\)
\(20\) 0 0
\(21\) 38.4471 0.399516
\(22\) 0 0
\(23\) 174.510 1.58208 0.791039 0.611766i \(-0.209541\pi\)
0.791039 + 0.611766i \(0.209541\pi\)
\(24\) 0 0
\(25\) 134.065 1.07252
\(26\) 0 0
\(27\) −130.902 −0.933040
\(28\) 0 0
\(29\) −262.725 −1.68230 −0.841152 0.540799i \(-0.818122\pi\)
−0.841152 + 0.540799i \(0.818122\pi\)
\(30\) 0 0
\(31\) 36.1894 0.209671 0.104836 0.994490i \(-0.466568\pi\)
0.104836 + 0.994490i \(0.466568\pi\)
\(32\) 0 0
\(33\) 60.4169 0.318704
\(34\) 0 0
\(35\) −112.668 −0.544127
\(36\) 0 0
\(37\) 19.0464 0.0846274 0.0423137 0.999104i \(-0.486527\pi\)
0.0423137 + 0.999104i \(0.486527\pi\)
\(38\) 0 0
\(39\) 194.274 0.797661
\(40\) 0 0
\(41\) 156.996 0.598017 0.299008 0.954250i \(-0.403344\pi\)
0.299008 + 0.954250i \(0.403344\pi\)
\(42\) 0 0
\(43\) −287.182 −1.01849 −0.509243 0.860623i \(-0.670074\pi\)
−0.509243 + 0.860623i \(0.670074\pi\)
\(44\) 0 0
\(45\) −50.9731 −0.168858
\(46\) 0 0
\(47\) −397.244 −1.23285 −0.616425 0.787413i \(-0.711420\pi\)
−0.616425 + 0.787413i \(0.711420\pi\)
\(48\) 0 0
\(49\) 49.0000 0.142857
\(50\) 0 0
\(51\) 222.311 0.610386
\(52\) 0 0
\(53\) 272.483 0.706196 0.353098 0.935586i \(-0.385128\pi\)
0.353098 + 0.935586i \(0.385128\pi\)
\(54\) 0 0
\(55\) −177.050 −0.434063
\(56\) 0 0
\(57\) −648.984 −1.50807
\(58\) 0 0
\(59\) 507.466 1.11977 0.559885 0.828570i \(-0.310845\pi\)
0.559885 + 0.828570i \(0.310845\pi\)
\(60\) 0 0
\(61\) 35.5608 0.0746409 0.0373205 0.999303i \(-0.488118\pi\)
0.0373205 + 0.999303i \(0.488118\pi\)
\(62\) 0 0
\(63\) 22.1684 0.0443326
\(64\) 0 0
\(65\) −569.317 −1.08639
\(66\) 0 0
\(67\) −979.229 −1.78555 −0.892775 0.450502i \(-0.851245\pi\)
−0.892775 + 0.450502i \(0.851245\pi\)
\(68\) 0 0
\(69\) 958.484 1.67229
\(70\) 0 0
\(71\) −750.404 −1.25432 −0.627159 0.778891i \(-0.715782\pi\)
−0.627159 + 0.778891i \(0.715782\pi\)
\(72\) 0 0
\(73\) 395.594 0.634257 0.317129 0.948383i \(-0.397281\pi\)
0.317129 + 0.948383i \(0.397281\pi\)
\(74\) 0 0
\(75\) 736.344 1.13368
\(76\) 0 0
\(77\) 77.0000 0.113961
\(78\) 0 0
\(79\) 736.516 1.04892 0.524459 0.851436i \(-0.324268\pi\)
0.524459 + 0.851436i \(0.324268\pi\)
\(80\) 0 0
\(81\) −804.477 −1.10354
\(82\) 0 0
\(83\) −582.975 −0.770962 −0.385481 0.922716i \(-0.625964\pi\)
−0.385481 + 0.922716i \(0.625964\pi\)
\(84\) 0 0
\(85\) −651.477 −0.831325
\(86\) 0 0
\(87\) −1443.00 −1.77823
\(88\) 0 0
\(89\) −806.201 −0.960192 −0.480096 0.877216i \(-0.659398\pi\)
−0.480096 + 0.877216i \(0.659398\pi\)
\(90\) 0 0
\(91\) 247.598 0.285224
\(92\) 0 0
\(93\) 198.768 0.221627
\(94\) 0 0
\(95\) 1901.84 2.05394
\(96\) 0 0
\(97\) −957.232 −1.00198 −0.500991 0.865453i \(-0.667031\pi\)
−0.500991 + 0.865453i \(0.667031\pi\)
\(98\) 0 0
\(99\) 34.8361 0.0353652
\(100\) 0 0
\(101\) −996.143 −0.981386 −0.490693 0.871333i \(-0.663256\pi\)
−0.490693 + 0.871333i \(0.663256\pi\)
\(102\) 0 0
\(103\) −1338.55 −1.28050 −0.640248 0.768169i \(-0.721168\pi\)
−0.640248 + 0.768169i \(0.721168\pi\)
\(104\) 0 0
\(105\) −618.825 −0.575154
\(106\) 0 0
\(107\) −1449.25 −1.30939 −0.654693 0.755895i \(-0.727202\pi\)
−0.654693 + 0.755895i \(0.727202\pi\)
\(108\) 0 0
\(109\) −654.535 −0.575166 −0.287583 0.957756i \(-0.592852\pi\)
−0.287583 + 0.957756i \(0.592852\pi\)
\(110\) 0 0
\(111\) 104.611 0.0894529
\(112\) 0 0
\(113\) −1160.63 −0.966223 −0.483111 0.875559i \(-0.660493\pi\)
−0.483111 + 0.875559i \(0.660493\pi\)
\(114\) 0 0
\(115\) −2808.82 −2.27760
\(116\) 0 0
\(117\) 112.018 0.0885131
\(118\) 0 0
\(119\) 283.330 0.218259
\(120\) 0 0
\(121\) 121.000 0.0909091
\(122\) 0 0
\(123\) 862.293 0.632116
\(124\) 0 0
\(125\) −145.906 −0.104402
\(126\) 0 0
\(127\) 1055.41 0.737420 0.368710 0.929545i \(-0.379800\pi\)
0.368710 + 0.929545i \(0.379800\pi\)
\(128\) 0 0
\(129\) −1577.33 −1.07656
\(130\) 0 0
\(131\) −2657.40 −1.77235 −0.886175 0.463351i \(-0.846647\pi\)
−0.886175 + 0.463351i \(0.846647\pi\)
\(132\) 0 0
\(133\) −827.116 −0.539249
\(134\) 0 0
\(135\) 2106.93 1.34323
\(136\) 0 0
\(137\) −147.314 −0.0918676 −0.0459338 0.998944i \(-0.514626\pi\)
−0.0459338 + 0.998944i \(0.514626\pi\)
\(138\) 0 0
\(139\) −902.634 −0.550794 −0.275397 0.961331i \(-0.588809\pi\)
−0.275397 + 0.961331i \(0.588809\pi\)
\(140\) 0 0
\(141\) −2181.84 −1.30315
\(142\) 0 0
\(143\) 389.083 0.227530
\(144\) 0 0
\(145\) 4228.69 2.42189
\(146\) 0 0
\(147\) 269.130 0.151003
\(148\) 0 0
\(149\) 1212.63 0.666727 0.333363 0.942798i \(-0.391816\pi\)
0.333363 + 0.942798i \(0.391816\pi\)
\(150\) 0 0
\(151\) −2565.33 −1.38254 −0.691270 0.722597i \(-0.742948\pi\)
−0.691270 + 0.722597i \(0.742948\pi\)
\(152\) 0 0
\(153\) 128.183 0.0677320
\(154\) 0 0
\(155\) −582.486 −0.301848
\(156\) 0 0
\(157\) 702.237 0.356972 0.178486 0.983942i \(-0.442880\pi\)
0.178486 + 0.983942i \(0.442880\pi\)
\(158\) 0 0
\(159\) 1496.60 0.746464
\(160\) 0 0
\(161\) 1221.57 0.597969
\(162\) 0 0
\(163\) 1146.27 0.550814 0.275407 0.961328i \(-0.411187\pi\)
0.275407 + 0.961328i \(0.411187\pi\)
\(164\) 0 0
\(165\) −972.439 −0.458814
\(166\) 0 0
\(167\) 3255.04 1.50828 0.754140 0.656714i \(-0.228054\pi\)
0.754140 + 0.656714i \(0.228054\pi\)
\(168\) 0 0
\(169\) −945.879 −0.430532
\(170\) 0 0
\(171\) −374.201 −0.167344
\(172\) 0 0
\(173\) 4024.24 1.76854 0.884269 0.466977i \(-0.154657\pi\)
0.884269 + 0.466977i \(0.154657\pi\)
\(174\) 0 0
\(175\) 938.455 0.405374
\(176\) 0 0
\(177\) 2787.23 1.18362
\(178\) 0 0
\(179\) −706.090 −0.294836 −0.147418 0.989074i \(-0.547096\pi\)
−0.147418 + 0.989074i \(0.547096\pi\)
\(180\) 0 0
\(181\) −1268.90 −0.521087 −0.260544 0.965462i \(-0.583902\pi\)
−0.260544 + 0.965462i \(0.583902\pi\)
\(182\) 0 0
\(183\) 195.316 0.0788970
\(184\) 0 0
\(185\) −306.562 −0.121832
\(186\) 0 0
\(187\) 445.233 0.174110
\(188\) 0 0
\(189\) −916.313 −0.352656
\(190\) 0 0
\(191\) −4864.58 −1.84287 −0.921436 0.388529i \(-0.872983\pi\)
−0.921436 + 0.388529i \(0.872983\pi\)
\(192\) 0 0
\(193\) −2675.49 −0.997855 −0.498928 0.866644i \(-0.666273\pi\)
−0.498928 + 0.866644i \(0.666273\pi\)
\(194\) 0 0
\(195\) −3126.94 −1.14833
\(196\) 0 0
\(197\) 1627.73 0.588684 0.294342 0.955700i \(-0.404899\pi\)
0.294342 + 0.955700i \(0.404899\pi\)
\(198\) 0 0
\(199\) 2254.07 0.802947 0.401474 0.915871i \(-0.368498\pi\)
0.401474 + 0.915871i \(0.368498\pi\)
\(200\) 0 0
\(201\) −5378.36 −1.88736
\(202\) 0 0
\(203\) −1839.08 −0.635851
\(204\) 0 0
\(205\) −2526.93 −0.860920
\(206\) 0 0
\(207\) 552.657 0.185567
\(208\) 0 0
\(209\) −1299.75 −0.430172
\(210\) 0 0
\(211\) −3112.07 −1.01537 −0.507687 0.861542i \(-0.669499\pi\)
−0.507687 + 0.861542i \(0.669499\pi\)
\(212\) 0 0
\(213\) −4121.55 −1.32584
\(214\) 0 0
\(215\) 4622.34 1.46624
\(216\) 0 0
\(217\) 253.326 0.0792482
\(218\) 0 0
\(219\) 2172.78 0.670423
\(220\) 0 0
\(221\) 1431.67 0.435769
\(222\) 0 0
\(223\) 3558.38 1.06855 0.534275 0.845311i \(-0.320585\pi\)
0.534275 + 0.845311i \(0.320585\pi\)
\(224\) 0 0
\(225\) 424.572 0.125799
\(226\) 0 0
\(227\) 2330.51 0.681416 0.340708 0.940169i \(-0.389333\pi\)
0.340708 + 0.940169i \(0.389333\pi\)
\(228\) 0 0
\(229\) 676.106 0.195102 0.0975509 0.995231i \(-0.468899\pi\)
0.0975509 + 0.995231i \(0.468899\pi\)
\(230\) 0 0
\(231\) 422.918 0.120459
\(232\) 0 0
\(233\) 1620.20 0.455548 0.227774 0.973714i \(-0.426855\pi\)
0.227774 + 0.973714i \(0.426855\pi\)
\(234\) 0 0
\(235\) 6393.84 1.77484
\(236\) 0 0
\(237\) 4045.27 1.10873
\(238\) 0 0
\(239\) 2001.39 0.541669 0.270835 0.962626i \(-0.412700\pi\)
0.270835 + 0.962626i \(0.412700\pi\)
\(240\) 0 0
\(241\) −1586.39 −0.424018 −0.212009 0.977268i \(-0.568001\pi\)
−0.212009 + 0.977268i \(0.568001\pi\)
\(242\) 0 0
\(243\) −884.196 −0.233420
\(244\) 0 0
\(245\) −788.679 −0.205661
\(246\) 0 0
\(247\) −4179.44 −1.07665
\(248\) 0 0
\(249\) −3201.96 −0.814923
\(250\) 0 0
\(251\) −2612.23 −0.656902 −0.328451 0.944521i \(-0.606527\pi\)
−0.328451 + 0.944521i \(0.606527\pi\)
\(252\) 0 0
\(253\) 1919.61 0.477014
\(254\) 0 0
\(255\) −3578.20 −0.878727
\(256\) 0 0
\(257\) 4198.23 1.01898 0.509491 0.860476i \(-0.329834\pi\)
0.509491 + 0.860476i \(0.329834\pi\)
\(258\) 0 0
\(259\) 133.325 0.0319861
\(260\) 0 0
\(261\) −832.028 −0.197323
\(262\) 0 0
\(263\) 5170.31 1.21222 0.606112 0.795380i \(-0.292728\pi\)
0.606112 + 0.795380i \(0.292728\pi\)
\(264\) 0 0
\(265\) −4385.74 −1.01666
\(266\) 0 0
\(267\) −4428.01 −1.01494
\(268\) 0 0
\(269\) 1648.20 0.373578 0.186789 0.982400i \(-0.440192\pi\)
0.186789 + 0.982400i \(0.440192\pi\)
\(270\) 0 0
\(271\) −2562.79 −0.574459 −0.287230 0.957862i \(-0.592734\pi\)
−0.287230 + 0.957862i \(0.592734\pi\)
\(272\) 0 0
\(273\) 1359.92 0.301487
\(274\) 0 0
\(275\) 1474.72 0.323377
\(276\) 0 0
\(277\) 2762.94 0.599310 0.299655 0.954048i \(-0.403128\pi\)
0.299655 + 0.954048i \(0.403128\pi\)
\(278\) 0 0
\(279\) 114.609 0.0245930
\(280\) 0 0
\(281\) 6453.68 1.37009 0.685044 0.728502i \(-0.259783\pi\)
0.685044 + 0.728502i \(0.259783\pi\)
\(282\) 0 0
\(283\) −4540.78 −0.953787 −0.476893 0.878961i \(-0.658237\pi\)
−0.476893 + 0.878961i \(0.658237\pi\)
\(284\) 0 0
\(285\) 10445.7 2.17106
\(286\) 0 0
\(287\) 1098.97 0.226029
\(288\) 0 0
\(289\) −3274.72 −0.666541
\(290\) 0 0
\(291\) −5257.54 −1.05912
\(292\) 0 0
\(293\) 1916.34 0.382094 0.191047 0.981581i \(-0.438812\pi\)
0.191047 + 0.981581i \(0.438812\pi\)
\(294\) 0 0
\(295\) −8167.92 −1.61205
\(296\) 0 0
\(297\) −1439.92 −0.281322
\(298\) 0 0
\(299\) 6172.61 1.19388
\(300\) 0 0
\(301\) −2010.28 −0.384951
\(302\) 0 0
\(303\) −5471.26 −1.03735
\(304\) 0 0
\(305\) −572.369 −0.107455
\(306\) 0 0
\(307\) −6876.30 −1.27834 −0.639171 0.769064i \(-0.720722\pi\)
−0.639171 + 0.769064i \(0.720722\pi\)
\(308\) 0 0
\(309\) −7351.89 −1.35351
\(310\) 0 0
\(311\) −8469.32 −1.54422 −0.772108 0.635492i \(-0.780797\pi\)
−0.772108 + 0.635492i \(0.780797\pi\)
\(312\) 0 0
\(313\) −2882.28 −0.520498 −0.260249 0.965542i \(-0.583805\pi\)
−0.260249 + 0.965542i \(0.583805\pi\)
\(314\) 0 0
\(315\) −356.812 −0.0638224
\(316\) 0 0
\(317\) 9795.31 1.73552 0.867759 0.496985i \(-0.165560\pi\)
0.867759 + 0.496985i \(0.165560\pi\)
\(318\) 0 0
\(319\) −2889.98 −0.507234
\(320\) 0 0
\(321\) −7959.92 −1.38405
\(322\) 0 0
\(323\) −4782.59 −0.823871
\(324\) 0 0
\(325\) 4742.04 0.809357
\(326\) 0 0
\(327\) −3595.00 −0.607963
\(328\) 0 0
\(329\) −2780.71 −0.465974
\(330\) 0 0
\(331\) −1813.70 −0.301178 −0.150589 0.988596i \(-0.548117\pi\)
−0.150589 + 0.988596i \(0.548117\pi\)
\(332\) 0 0
\(333\) 60.3184 0.00992621
\(334\) 0 0
\(335\) 15761.2 2.57052
\(336\) 0 0
\(337\) −11964.2 −1.93393 −0.966964 0.254913i \(-0.917953\pi\)
−0.966964 + 0.254913i \(0.917953\pi\)
\(338\) 0 0
\(339\) −6374.71 −1.02132
\(340\) 0 0
\(341\) 398.083 0.0632182
\(342\) 0 0
\(343\) 343.000 0.0539949
\(344\) 0 0
\(345\) −15427.3 −2.40747
\(346\) 0 0
\(347\) 2283.89 0.353330 0.176665 0.984271i \(-0.443469\pi\)
0.176665 + 0.984271i \(0.443469\pi\)
\(348\) 0 0
\(349\) −2472.48 −0.379223 −0.189612 0.981859i \(-0.560723\pi\)
−0.189612 + 0.981859i \(0.560723\pi\)
\(350\) 0 0
\(351\) −4630.15 −0.704101
\(352\) 0 0
\(353\) 10190.0 1.53642 0.768211 0.640197i \(-0.221147\pi\)
0.768211 + 0.640197i \(0.221147\pi\)
\(354\) 0 0
\(355\) 12078.1 1.80575
\(356\) 0 0
\(357\) 1556.17 0.230704
\(358\) 0 0
\(359\) 43.4294 0.00638473 0.00319236 0.999995i \(-0.498984\pi\)
0.00319236 + 0.999995i \(0.498984\pi\)
\(360\) 0 0
\(361\) 7102.66 1.03552
\(362\) 0 0
\(363\) 664.585 0.0960928
\(364\) 0 0
\(365\) −6367.28 −0.913093
\(366\) 0 0
\(367\) −8775.78 −1.24821 −0.624104 0.781342i \(-0.714536\pi\)
−0.624104 + 0.781342i \(0.714536\pi\)
\(368\) 0 0
\(369\) 497.194 0.0701433
\(370\) 0 0
\(371\) 1907.38 0.266917
\(372\) 0 0
\(373\) −2751.89 −0.382004 −0.191002 0.981590i \(-0.561174\pi\)
−0.191002 + 0.981590i \(0.561174\pi\)
\(374\) 0 0
\(375\) −801.379 −0.110355
\(376\) 0 0
\(377\) −9292.90 −1.26952
\(378\) 0 0
\(379\) −2605.59 −0.353141 −0.176570 0.984288i \(-0.556500\pi\)
−0.176570 + 0.984288i \(0.556500\pi\)
\(380\) 0 0
\(381\) 5796.77 0.779468
\(382\) 0 0
\(383\) 14360.5 1.91590 0.957949 0.286940i \(-0.0926380\pi\)
0.957949 + 0.286940i \(0.0926380\pi\)
\(384\) 0 0
\(385\) −1239.35 −0.164060
\(386\) 0 0
\(387\) −909.482 −0.119461
\(388\) 0 0
\(389\) −10607.9 −1.38262 −0.691311 0.722557i \(-0.742967\pi\)
−0.691311 + 0.722557i \(0.742967\pi\)
\(390\) 0 0
\(391\) 7063.40 0.913585
\(392\) 0 0
\(393\) −14595.6 −1.87341
\(394\) 0 0
\(395\) −11854.6 −1.51005
\(396\) 0 0
\(397\) 7362.35 0.930745 0.465373 0.885115i \(-0.345920\pi\)
0.465373 + 0.885115i \(0.345920\pi\)
\(398\) 0 0
\(399\) −4542.89 −0.569997
\(400\) 0 0
\(401\) −264.514 −0.0329406 −0.0164703 0.999864i \(-0.505243\pi\)
−0.0164703 + 0.999864i \(0.505243\pi\)
\(402\) 0 0
\(403\) 1280.06 0.158224
\(404\) 0 0
\(405\) 12948.5 1.58868
\(406\) 0 0
\(407\) 209.511 0.0255161
\(408\) 0 0
\(409\) −1244.03 −0.150400 −0.0751999 0.997168i \(-0.523959\pi\)
−0.0751999 + 0.997168i \(0.523959\pi\)
\(410\) 0 0
\(411\) −809.112 −0.0971060
\(412\) 0 0
\(413\) 3552.26 0.423233
\(414\) 0 0
\(415\) 9383.27 1.10990
\(416\) 0 0
\(417\) −4957.66 −0.582201
\(418\) 0 0
\(419\) −3974.76 −0.463436 −0.231718 0.972783i \(-0.574435\pi\)
−0.231718 + 0.972783i \(0.574435\pi\)
\(420\) 0 0
\(421\) −14910.2 −1.72608 −0.863041 0.505134i \(-0.831443\pi\)
−0.863041 + 0.505134i \(0.831443\pi\)
\(422\) 0 0
\(423\) −1258.04 −0.144605
\(424\) 0 0
\(425\) 5426.38 0.619336
\(426\) 0 0
\(427\) 248.926 0.0282116
\(428\) 0 0
\(429\) 2137.02 0.240504
\(430\) 0 0
\(431\) 10622.4 1.18715 0.593574 0.804779i \(-0.297716\pi\)
0.593574 + 0.804779i \(0.297716\pi\)
\(432\) 0 0
\(433\) 17440.5 1.93565 0.967823 0.251630i \(-0.0809667\pi\)
0.967823 + 0.251630i \(0.0809667\pi\)
\(434\) 0 0
\(435\) 23225.8 2.55999
\(436\) 0 0
\(437\) −20620.0 −2.25718
\(438\) 0 0
\(439\) −13627.1 −1.48151 −0.740756 0.671774i \(-0.765533\pi\)
−0.740756 + 0.671774i \(0.765533\pi\)
\(440\) 0 0
\(441\) 155.179 0.0167562
\(442\) 0 0
\(443\) −2135.29 −0.229008 −0.114504 0.993423i \(-0.536528\pi\)
−0.114504 + 0.993423i \(0.536528\pi\)
\(444\) 0 0
\(445\) 12976.2 1.38232
\(446\) 0 0
\(447\) 6660.28 0.704744
\(448\) 0 0
\(449\) 17780.8 1.86889 0.934443 0.356113i \(-0.115898\pi\)
0.934443 + 0.356113i \(0.115898\pi\)
\(450\) 0 0
\(451\) 1726.96 0.180309
\(452\) 0 0
\(453\) −14089.9 −1.46137
\(454\) 0 0
\(455\) −3985.22 −0.410615
\(456\) 0 0
\(457\) 10357.0 1.06013 0.530064 0.847957i \(-0.322168\pi\)
0.530064 + 0.847957i \(0.322168\pi\)
\(458\) 0 0
\(459\) −5298.35 −0.538792
\(460\) 0 0
\(461\) −19679.5 −1.98821 −0.994106 0.108410i \(-0.965424\pi\)
−0.994106 + 0.108410i \(0.965424\pi\)
\(462\) 0 0
\(463\) 7171.43 0.719838 0.359919 0.932984i \(-0.382804\pi\)
0.359919 + 0.932984i \(0.382804\pi\)
\(464\) 0 0
\(465\) −3199.27 −0.319059
\(466\) 0 0
\(467\) 12192.8 1.20817 0.604085 0.796920i \(-0.293539\pi\)
0.604085 + 0.796920i \(0.293539\pi\)
\(468\) 0 0
\(469\) −6854.61 −0.674875
\(470\) 0 0
\(471\) 3856.99 0.377327
\(472\) 0 0
\(473\) −3159.00 −0.307085
\(474\) 0 0
\(475\) −15841.1 −1.53018
\(476\) 0 0
\(477\) 862.930 0.0828319
\(478\) 0 0
\(479\) −8475.11 −0.808429 −0.404215 0.914664i \(-0.632455\pi\)
−0.404215 + 0.914664i \(0.632455\pi\)
\(480\) 0 0
\(481\) 673.695 0.0638624
\(482\) 0 0
\(483\) 6709.39 0.632066
\(484\) 0 0
\(485\) 15407.1 1.44248
\(486\) 0 0
\(487\) −2735.29 −0.254513 −0.127257 0.991870i \(-0.540617\pi\)
−0.127257 + 0.991870i \(0.540617\pi\)
\(488\) 0 0
\(489\) 6295.81 0.582222
\(490\) 0 0
\(491\) 2233.82 0.205317 0.102659 0.994717i \(-0.467265\pi\)
0.102659 + 0.994717i \(0.467265\pi\)
\(492\) 0 0
\(493\) −10634.0 −0.971462
\(494\) 0 0
\(495\) −560.704 −0.0509126
\(496\) 0 0
\(497\) −5252.83 −0.474088
\(498\) 0 0
\(499\) 18100.3 1.62381 0.811904 0.583791i \(-0.198431\pi\)
0.811904 + 0.583791i \(0.198431\pi\)
\(500\) 0 0
\(501\) 17878.1 1.59428
\(502\) 0 0
\(503\) −6149.06 −0.545076 −0.272538 0.962145i \(-0.587863\pi\)
−0.272538 + 0.962145i \(0.587863\pi\)
\(504\) 0 0
\(505\) 16033.4 1.41283
\(506\) 0 0
\(507\) −5195.18 −0.455081
\(508\) 0 0
\(509\) 14193.9 1.23602 0.618008 0.786172i \(-0.287940\pi\)
0.618008 + 0.786172i \(0.287940\pi\)
\(510\) 0 0
\(511\) 2769.16 0.239727
\(512\) 0 0
\(513\) 15467.3 1.33118
\(514\) 0 0
\(515\) 21544.6 1.84343
\(516\) 0 0
\(517\) −4369.68 −0.371718
\(518\) 0 0
\(519\) 22102.9 1.86938
\(520\) 0 0
\(521\) 10371.8 0.872163 0.436082 0.899907i \(-0.356366\pi\)
0.436082 + 0.899907i \(0.356366\pi\)
\(522\) 0 0
\(523\) 11369.4 0.950569 0.475285 0.879832i \(-0.342345\pi\)
0.475285 + 0.879832i \(0.342345\pi\)
\(524\) 0 0
\(525\) 5154.41 0.428489
\(526\) 0 0
\(527\) 1464.79 0.121076
\(528\) 0 0
\(529\) 18286.6 1.50297
\(530\) 0 0
\(531\) 1607.10 0.131341
\(532\) 0 0
\(533\) 5553.14 0.451282
\(534\) 0 0
\(535\) 23326.4 1.88503
\(536\) 0 0
\(537\) −3878.16 −0.311648
\(538\) 0 0
\(539\) 539.000 0.0430730
\(540\) 0 0
\(541\) 5386.88 0.428096 0.214048 0.976823i \(-0.431335\pi\)
0.214048 + 0.976823i \(0.431335\pi\)
\(542\) 0 0
\(543\) −6969.38 −0.550800
\(544\) 0 0
\(545\) 10535.1 0.828024
\(546\) 0 0
\(547\) −13890.8 −1.08579 −0.542894 0.839801i \(-0.682672\pi\)
−0.542894 + 0.839801i \(0.682672\pi\)
\(548\) 0 0
\(549\) 112.618 0.00875487
\(550\) 0 0
\(551\) 31043.5 2.40017
\(552\) 0 0
\(553\) 5155.61 0.396454
\(554\) 0 0
\(555\) −1683.77 −0.128779
\(556\) 0 0
\(557\) 17498.3 1.33111 0.665553 0.746351i \(-0.268196\pi\)
0.665553 + 0.746351i \(0.268196\pi\)
\(558\) 0 0
\(559\) −10158.0 −0.768581
\(560\) 0 0
\(561\) 2445.42 0.184038
\(562\) 0 0
\(563\) −147.373 −0.0110320 −0.00551600 0.999985i \(-0.501756\pi\)
−0.00551600 + 0.999985i \(0.501756\pi\)
\(564\) 0 0
\(565\) 18681.0 1.39100
\(566\) 0 0
\(567\) −5631.34 −0.417097
\(568\) 0 0
\(569\) −14155.3 −1.04292 −0.521459 0.853276i \(-0.674612\pi\)
−0.521459 + 0.853276i \(0.674612\pi\)
\(570\) 0 0
\(571\) 248.361 0.0182025 0.00910123 0.999959i \(-0.497103\pi\)
0.00910123 + 0.999959i \(0.497103\pi\)
\(572\) 0 0
\(573\) −26718.4 −1.94795
\(574\) 0 0
\(575\) 23395.6 1.69681
\(576\) 0 0
\(577\) 19364.6 1.39715 0.698576 0.715536i \(-0.253817\pi\)
0.698576 + 0.715536i \(0.253817\pi\)
\(578\) 0 0
\(579\) −14695.0 −1.05475
\(580\) 0 0
\(581\) −4080.83 −0.291396
\(582\) 0 0
\(583\) 2997.31 0.212926
\(584\) 0 0
\(585\) −1802.98 −0.127426
\(586\) 0 0
\(587\) 9135.18 0.642332 0.321166 0.947023i \(-0.395925\pi\)
0.321166 + 0.947023i \(0.395925\pi\)
\(588\) 0 0
\(589\) −4276.12 −0.299141
\(590\) 0 0
\(591\) 8940.20 0.622252
\(592\) 0 0
\(593\) −12287.6 −0.850916 −0.425458 0.904978i \(-0.639887\pi\)
−0.425458 + 0.904978i \(0.639887\pi\)
\(594\) 0 0
\(595\) −4560.34 −0.314211
\(596\) 0 0
\(597\) 12380.3 0.848732
\(598\) 0 0
\(599\) −12351.9 −0.842549 −0.421275 0.906933i \(-0.638417\pi\)
−0.421275 + 0.906933i \(0.638417\pi\)
\(600\) 0 0
\(601\) 21624.2 1.46767 0.733836 0.679327i \(-0.237728\pi\)
0.733836 + 0.679327i \(0.237728\pi\)
\(602\) 0 0
\(603\) −3101.14 −0.209433
\(604\) 0 0
\(605\) −1947.56 −0.130875
\(606\) 0 0
\(607\) 2086.03 0.139488 0.0697442 0.997565i \(-0.477782\pi\)
0.0697442 + 0.997565i \(0.477782\pi\)
\(608\) 0 0
\(609\) −10101.0 −0.672108
\(610\) 0 0
\(611\) −14051.0 −0.930347
\(612\) 0 0
\(613\) −8338.46 −0.549408 −0.274704 0.961529i \(-0.588580\pi\)
−0.274704 + 0.961529i \(0.588580\pi\)
\(614\) 0 0
\(615\) −13879.0 −0.910011
\(616\) 0 0
\(617\) −4771.20 −0.311315 −0.155657 0.987811i \(-0.549750\pi\)
−0.155657 + 0.987811i \(0.549750\pi\)
\(618\) 0 0
\(619\) 16609.4 1.07850 0.539248 0.842147i \(-0.318708\pi\)
0.539248 + 0.842147i \(0.318708\pi\)
\(620\) 0 0
\(621\) −22843.6 −1.47614
\(622\) 0 0
\(623\) −5643.40 −0.362919
\(624\) 0 0
\(625\) −14409.7 −0.922221
\(626\) 0 0
\(627\) −7138.82 −0.454700
\(628\) 0 0
\(629\) 770.918 0.0488688
\(630\) 0 0
\(631\) −17254.9 −1.08860 −0.544299 0.838891i \(-0.683204\pi\)
−0.544299 + 0.838891i \(0.683204\pi\)
\(632\) 0 0
\(633\) −17092.9 −1.07327
\(634\) 0 0
\(635\) −16987.3 −1.06161
\(636\) 0 0
\(637\) 1733.19 0.107804
\(638\) 0 0
\(639\) −2376.47 −0.147123
\(640\) 0 0
\(641\) −7350.77 −0.452945 −0.226473 0.974018i \(-0.572719\pi\)
−0.226473 + 0.974018i \(0.572719\pi\)
\(642\) 0 0
\(643\) −10117.8 −0.620542 −0.310271 0.950648i \(-0.600420\pi\)
−0.310271 + 0.950648i \(0.600420\pi\)
\(644\) 0 0
\(645\) 25387.9 1.54984
\(646\) 0 0
\(647\) 24590.9 1.49423 0.747116 0.664693i \(-0.231438\pi\)
0.747116 + 0.664693i \(0.231438\pi\)
\(648\) 0 0
\(649\) 5582.13 0.337624
\(650\) 0 0
\(651\) 1391.38 0.0837670
\(652\) 0 0
\(653\) 2339.03 0.140173 0.0700867 0.997541i \(-0.477672\pi\)
0.0700867 + 0.997541i \(0.477672\pi\)
\(654\) 0 0
\(655\) 42772.1 2.55152
\(656\) 0 0
\(657\) 1252.81 0.0743940
\(658\) 0 0
\(659\) 15735.7 0.930162 0.465081 0.885268i \(-0.346025\pi\)
0.465081 + 0.885268i \(0.346025\pi\)
\(660\) 0 0
\(661\) 4846.75 0.285199 0.142600 0.989780i \(-0.454454\pi\)
0.142600 + 0.989780i \(0.454454\pi\)
\(662\) 0 0
\(663\) 7863.39 0.460617
\(664\) 0 0
\(665\) 13312.8 0.776316
\(666\) 0 0
\(667\) −45848.1 −2.66154
\(668\) 0 0
\(669\) 19544.2 1.12948
\(670\) 0 0
\(671\) 391.169 0.0225051
\(672\) 0 0
\(673\) 15716.8 0.900205 0.450103 0.892977i \(-0.351387\pi\)
0.450103 + 0.892977i \(0.351387\pi\)
\(674\) 0 0
\(675\) −17549.4 −1.00070
\(676\) 0 0
\(677\) −856.968 −0.0486499 −0.0243249 0.999704i \(-0.507744\pi\)
−0.0243249 + 0.999704i \(0.507744\pi\)
\(678\) 0 0
\(679\) −6700.63 −0.378713
\(680\) 0 0
\(681\) 12800.2 0.720271
\(682\) 0 0
\(683\) −24804.0 −1.38960 −0.694801 0.719202i \(-0.744508\pi\)
−0.694801 + 0.719202i \(0.744508\pi\)
\(684\) 0 0
\(685\) 2371.09 0.132255
\(686\) 0 0
\(687\) 3713.47 0.206227
\(688\) 0 0
\(689\) 9638.04 0.532917
\(690\) 0 0
\(691\) −3475.97 −0.191364 −0.0956818 0.995412i \(-0.530503\pi\)
−0.0956818 + 0.995412i \(0.530503\pi\)
\(692\) 0 0
\(693\) 243.852 0.0133668
\(694\) 0 0
\(695\) 14528.3 0.792937
\(696\) 0 0
\(697\) 6354.54 0.345330
\(698\) 0 0
\(699\) 8898.85 0.481524
\(700\) 0 0
\(701\) 17461.0 0.940790 0.470395 0.882456i \(-0.344111\pi\)
0.470395 + 0.882456i \(0.344111\pi\)
\(702\) 0 0
\(703\) −2250.52 −0.120739
\(704\) 0 0
\(705\) 35117.8 1.87605
\(706\) 0 0
\(707\) −6973.00 −0.370929
\(708\) 0 0
\(709\) −18405.7 −0.974950 −0.487475 0.873137i \(-0.662082\pi\)
−0.487475 + 0.873137i \(0.662082\pi\)
\(710\) 0 0
\(711\) 2332.48 0.123031
\(712\) 0 0
\(713\) 6315.39 0.331716
\(714\) 0 0
\(715\) −6262.49 −0.327558
\(716\) 0 0
\(717\) 10992.5 0.572555
\(718\) 0 0
\(719\) 892.380 0.0462867 0.0231434 0.999732i \(-0.492633\pi\)
0.0231434 + 0.999732i \(0.492633\pi\)
\(720\) 0 0
\(721\) −9369.83 −0.483982
\(722\) 0 0
\(723\) −8713.15 −0.448196
\(724\) 0 0
\(725\) −35222.2 −1.80431
\(726\) 0 0
\(727\) −27919.2 −1.42430 −0.712149 0.702028i \(-0.752278\pi\)
−0.712149 + 0.702028i \(0.752278\pi\)
\(728\) 0 0
\(729\) 16864.5 0.856805
\(730\) 0 0
\(731\) −11623.9 −0.588134
\(732\) 0 0
\(733\) −4769.38 −0.240329 −0.120164 0.992754i \(-0.538342\pi\)
−0.120164 + 0.992754i \(0.538342\pi\)
\(734\) 0 0
\(735\) −4331.78 −0.217388
\(736\) 0 0
\(737\) −10771.5 −0.538364
\(738\) 0 0
\(739\) 5170.63 0.257381 0.128691 0.991685i \(-0.458923\pi\)
0.128691 + 0.991685i \(0.458923\pi\)
\(740\) 0 0
\(741\) −22955.3 −1.13804
\(742\) 0 0
\(743\) 29407.9 1.45205 0.726024 0.687669i \(-0.241366\pi\)
0.726024 + 0.687669i \(0.241366\pi\)
\(744\) 0 0
\(745\) −19517.8 −0.959836
\(746\) 0 0
\(747\) −1846.23 −0.0904286
\(748\) 0 0
\(749\) −10144.8 −0.494902
\(750\) 0 0
\(751\) 16956.5 0.823905 0.411952 0.911205i \(-0.364847\pi\)
0.411952 + 0.911205i \(0.364847\pi\)
\(752\) 0 0
\(753\) −14347.5 −0.694360
\(754\) 0 0
\(755\) 41290.3 1.99034
\(756\) 0 0
\(757\) −21322.0 −1.02373 −0.511864 0.859067i \(-0.671045\pi\)
−0.511864 + 0.859067i \(0.671045\pi\)
\(758\) 0 0
\(759\) 10543.3 0.504214
\(760\) 0 0
\(761\) −23548.0 −1.12170 −0.560851 0.827917i \(-0.689526\pi\)
−0.560851 + 0.827917i \(0.689526\pi\)
\(762\) 0 0
\(763\) −4581.75 −0.217392
\(764\) 0 0
\(765\) −2063.17 −0.0975087
\(766\) 0 0
\(767\) 17949.7 0.845014
\(768\) 0 0
\(769\) −17230.9 −0.808015 −0.404007 0.914756i \(-0.632383\pi\)
−0.404007 + 0.914756i \(0.632383\pi\)
\(770\) 0 0
\(771\) 23058.5 1.07709
\(772\) 0 0
\(773\) 12285.8 0.571655 0.285828 0.958281i \(-0.407732\pi\)
0.285828 + 0.958281i \(0.407732\pi\)
\(774\) 0 0
\(775\) 4851.73 0.224876
\(776\) 0 0
\(777\) 732.280 0.0338100
\(778\) 0 0
\(779\) −18550.6 −0.853202
\(780\) 0 0
\(781\) −8254.45 −0.378191
\(782\) 0 0
\(783\) 34391.2 1.56966
\(784\) 0 0
\(785\) −11302.8 −0.513906
\(786\) 0 0
\(787\) −663.152 −0.0300366 −0.0150183 0.999887i \(-0.504781\pi\)
−0.0150183 + 0.999887i \(0.504781\pi\)
\(788\) 0 0
\(789\) 28397.6 1.28135
\(790\) 0 0
\(791\) −8124.43 −0.365198
\(792\) 0 0
\(793\) 1257.83 0.0563264
\(794\) 0 0
\(795\) −24088.4 −1.07463
\(796\) 0 0
\(797\) 19216.3 0.854050 0.427025 0.904240i \(-0.359562\pi\)
0.427025 + 0.904240i \(0.359562\pi\)
\(798\) 0 0
\(799\) −16078.7 −0.711921
\(800\) 0 0
\(801\) −2553.17 −0.112624
\(802\) 0 0
\(803\) 4351.53 0.191236
\(804\) 0 0
\(805\) −19661.7 −0.860851
\(806\) 0 0
\(807\) 9052.65 0.394880
\(808\) 0 0
\(809\) −42881.8 −1.86359 −0.931794 0.362989i \(-0.881756\pi\)
−0.931794 + 0.362989i \(0.881756\pi\)
\(810\) 0 0
\(811\) 1205.73 0.0522058 0.0261029 0.999659i \(-0.491690\pi\)
0.0261029 + 0.999659i \(0.491690\pi\)
\(812\) 0 0
\(813\) −14076.0 −0.607215
\(814\) 0 0
\(815\) −18449.8 −0.792966
\(816\) 0 0
\(817\) 33933.3 1.45309
\(818\) 0 0
\(819\) 784.123 0.0334548
\(820\) 0 0
\(821\) −28577.6 −1.21482 −0.607408 0.794390i \(-0.707791\pi\)
−0.607408 + 0.794390i \(0.707791\pi\)
\(822\) 0 0
\(823\) −42524.8 −1.80112 −0.900561 0.434730i \(-0.856844\pi\)
−0.900561 + 0.434730i \(0.856844\pi\)
\(824\) 0 0
\(825\) 8099.79 0.341816
\(826\) 0 0
\(827\) −30768.7 −1.29375 −0.646876 0.762595i \(-0.723925\pi\)
−0.646876 + 0.762595i \(0.723925\pi\)
\(828\) 0 0
\(829\) −17583.3 −0.736661 −0.368330 0.929695i \(-0.620070\pi\)
−0.368330 + 0.929695i \(0.620070\pi\)
\(830\) 0 0
\(831\) 15175.3 0.633484
\(832\) 0 0
\(833\) 1983.31 0.0824942
\(834\) 0 0
\(835\) −52391.5 −2.17136
\(836\) 0 0
\(837\) −4737.25 −0.195631
\(838\) 0 0
\(839\) −19552.8 −0.804573 −0.402287 0.915514i \(-0.631784\pi\)
−0.402287 + 0.915514i \(0.631784\pi\)
\(840\) 0 0
\(841\) 44635.5 1.83015
\(842\) 0 0
\(843\) 35446.5 1.44821
\(844\) 0 0
\(845\) 15224.4 0.619805
\(846\) 0 0
\(847\) 847.000 0.0343604
\(848\) 0 0
\(849\) −24940.0 −1.00817
\(850\) 0 0
\(851\) 3323.78 0.133887
\(852\) 0 0
\(853\) −18524.1 −0.743557 −0.371779 0.928321i \(-0.621252\pi\)
−0.371779 + 0.928321i \(0.621252\pi\)
\(854\) 0 0
\(855\) 6022.95 0.240913
\(856\) 0 0
\(857\) 24439.0 0.974118 0.487059 0.873369i \(-0.338070\pi\)
0.487059 + 0.873369i \(0.338070\pi\)
\(858\) 0 0
\(859\) −11301.4 −0.448893 −0.224447 0.974486i \(-0.572057\pi\)
−0.224447 + 0.974486i \(0.572057\pi\)
\(860\) 0 0
\(861\) 6036.05 0.238918
\(862\) 0 0
\(863\) 26377.3 1.04043 0.520217 0.854034i \(-0.325851\pi\)
0.520217 + 0.854034i \(0.325851\pi\)
\(864\) 0 0
\(865\) −64772.1 −2.54603
\(866\) 0 0
\(867\) −17986.2 −0.704548
\(868\) 0 0
\(869\) 8101.68 0.316261
\(870\) 0 0
\(871\) −34636.5 −1.34743
\(872\) 0 0
\(873\) −3031.47 −0.117526
\(874\) 0 0
\(875\) −1021.34 −0.0394601
\(876\) 0 0
\(877\) −28425.0 −1.09446 −0.547232 0.836981i \(-0.684319\pi\)
−0.547232 + 0.836981i \(0.684319\pi\)
\(878\) 0 0
\(879\) 10525.4 0.403882
\(880\) 0 0
\(881\) 16897.1 0.646171 0.323086 0.946370i \(-0.395280\pi\)
0.323086 + 0.946370i \(0.395280\pi\)
\(882\) 0 0
\(883\) 25538.9 0.973331 0.486665 0.873588i \(-0.338213\pi\)
0.486665 + 0.873588i \(0.338213\pi\)
\(884\) 0 0
\(885\) −44861.8 −1.70397
\(886\) 0 0
\(887\) −6478.48 −0.245238 −0.122619 0.992454i \(-0.539129\pi\)
−0.122619 + 0.992454i \(0.539129\pi\)
\(888\) 0 0
\(889\) 7387.85 0.278718
\(890\) 0 0
\(891\) −8849.25 −0.332728
\(892\) 0 0
\(893\) 46938.1 1.75893
\(894\) 0 0
\(895\) 11364.9 0.424453
\(896\) 0 0
\(897\) 33902.7 1.26196
\(898\) 0 0
\(899\) −9507.86 −0.352731
\(900\) 0 0
\(901\) 11028.9 0.407799
\(902\) 0 0
\(903\) −11041.3 −0.406902
\(904\) 0 0
\(905\) 20423.6 0.750171
\(906\) 0 0
\(907\) 9356.17 0.342521 0.171260 0.985226i \(-0.445216\pi\)
0.171260 + 0.985226i \(0.445216\pi\)
\(908\) 0 0
\(909\) −3154.70 −0.115110
\(910\) 0 0
\(911\) 16574.0 0.602768 0.301384 0.953503i \(-0.402551\pi\)
0.301384 + 0.953503i \(0.402551\pi\)
\(912\) 0 0
\(913\) −6412.73 −0.232454
\(914\) 0 0
\(915\) −3143.70 −0.113582
\(916\) 0 0
\(917\) −18601.8 −0.669885
\(918\) 0 0
\(919\) −8214.92 −0.294870 −0.147435 0.989072i \(-0.547102\pi\)
−0.147435 + 0.989072i \(0.547102\pi\)
\(920\) 0 0
\(921\) −37767.7 −1.35123
\(922\) 0 0
\(923\) −26542.7 −0.946548
\(924\) 0 0
\(925\) 2553.46 0.0907646
\(926\) 0 0
\(927\) −4239.07 −0.150193
\(928\) 0 0
\(929\) 42653.9 1.50638 0.753192 0.657801i \(-0.228513\pi\)
0.753192 + 0.657801i \(0.228513\pi\)
\(930\) 0 0
\(931\) −5789.81 −0.203817
\(932\) 0 0
\(933\) −46517.2 −1.63227
\(934\) 0 0
\(935\) −7166.25 −0.250654
\(936\) 0 0
\(937\) −18484.8 −0.644473 −0.322237 0.946659i \(-0.604435\pi\)
−0.322237 + 0.946659i \(0.604435\pi\)
\(938\) 0 0
\(939\) −15830.7 −0.550177
\(940\) 0 0
\(941\) −7183.03 −0.248842 −0.124421 0.992230i \(-0.539707\pi\)
−0.124421 + 0.992230i \(0.539707\pi\)
\(942\) 0 0
\(943\) 27397.4 0.946109
\(944\) 0 0
\(945\) 14748.5 0.507692
\(946\) 0 0
\(947\) −41443.3 −1.42210 −0.711049 0.703143i \(-0.751779\pi\)
−0.711049 + 0.703143i \(0.751779\pi\)
\(948\) 0 0
\(949\) 13992.6 0.478630
\(950\) 0 0
\(951\) 53800.2 1.83448
\(952\) 0 0
\(953\) 7981.30 0.271290 0.135645 0.990757i \(-0.456689\pi\)
0.135645 + 0.990757i \(0.456689\pi\)
\(954\) 0 0
\(955\) 78297.8 2.65305
\(956\) 0 0
\(957\) −15873.0 −0.536157
\(958\) 0 0
\(959\) −1031.20 −0.0347227
\(960\) 0 0
\(961\) −28481.3 −0.956038
\(962\) 0 0
\(963\) −4589.65 −0.153582
\(964\) 0 0
\(965\) 43063.3 1.43654
\(966\) 0 0
\(967\) 18745.7 0.623394 0.311697 0.950182i \(-0.399103\pi\)
0.311697 + 0.950182i \(0.399103\pi\)
\(968\) 0 0
\(969\) −26268.1 −0.870849
\(970\) 0 0
\(971\) −3096.87 −0.102351 −0.0511757 0.998690i \(-0.516297\pi\)
−0.0511757 + 0.998690i \(0.516297\pi\)
\(972\) 0 0
\(973\) −6318.44 −0.208181
\(974\) 0 0
\(975\) 26045.4 0.855507
\(976\) 0 0
\(977\) 19960.2 0.653618 0.326809 0.945090i \(-0.394027\pi\)
0.326809 + 0.945090i \(0.394027\pi\)
\(978\) 0 0
\(979\) −8868.21 −0.289509
\(980\) 0 0
\(981\) −2072.86 −0.0674631
\(982\) 0 0
\(983\) 33434.5 1.08484 0.542419 0.840108i \(-0.317509\pi\)
0.542419 + 0.840108i \(0.317509\pi\)
\(984\) 0 0
\(985\) −26199.1 −0.847485
\(986\) 0 0
\(987\) −15272.9 −0.492544
\(988\) 0 0
\(989\) −50116.1 −1.61132
\(990\) 0 0
\(991\) −24855.2 −0.796722 −0.398361 0.917229i \(-0.630421\pi\)
−0.398361 + 0.917229i \(0.630421\pi\)
\(992\) 0 0
\(993\) −9961.63 −0.318351
\(994\) 0 0
\(995\) −36280.3 −1.15594
\(996\) 0 0
\(997\) −7810.38 −0.248102 −0.124051 0.992276i \(-0.539589\pi\)
−0.124051 + 0.992276i \(0.539589\pi\)
\(998\) 0 0
\(999\) −2493.21 −0.0789607
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 1232.4.a.s.1.4 4
4.3 odd 2 77.4.a.d.1.2 4
12.11 even 2 693.4.a.l.1.3 4
20.19 odd 2 1925.4.a.p.1.3 4
28.27 even 2 539.4.a.g.1.2 4
44.43 even 2 847.4.a.d.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
77.4.a.d.1.2 4 4.3 odd 2
539.4.a.g.1.2 4 28.27 even 2
693.4.a.l.1.3 4 12.11 even 2
847.4.a.d.1.3 4 44.43 even 2
1232.4.a.s.1.4 4 1.1 even 1 trivial
1925.4.a.p.1.3 4 20.19 odd 2