Properties

Label 1856.4.a.l.1.1
Level $1856$
Weight $4$
Character 1856.1
Self dual yes
Analytic conductor $109.508$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1856 = 2^{6} \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1856.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(109.507544971\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
Defining polynomial: \( x^{2} - 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 58)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.44949\) of defining polynomial
Character \(\chi\) \(=\) 1856.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.44949 q^{3} +19.6969 q^{5} +11.5959 q^{7} -24.8990 q^{9} +O(q^{10})\) \(q-1.44949 q^{3} +19.6969 q^{5} +11.5959 q^{7} -24.8990 q^{9} +37.6515 q^{11} +44.5959 q^{13} -28.5505 q^{15} -61.1918 q^{17} -63.7980 q^{19} -16.8082 q^{21} +177.060 q^{23} +262.969 q^{25} +75.2270 q^{27} -29.0000 q^{29} -233.994 q^{31} -54.5755 q^{33} +228.404 q^{35} -10.2020 q^{37} -64.6413 q^{39} +347.959 q^{41} +194.823 q^{43} -490.434 q^{45} -14.5005 q^{47} -208.535 q^{49} +88.6969 q^{51} +606.373 q^{53} +741.620 q^{55} +92.4745 q^{57} +702.372 q^{59} -543.394 q^{61} -288.727 q^{63} +878.403 q^{65} +407.010 q^{67} -256.647 q^{69} +314.717 q^{71} -859.110 q^{73} -381.171 q^{75} +436.604 q^{77} +725.266 q^{79} +563.232 q^{81} +820.919 q^{83} -1205.29 q^{85} +42.0352 q^{87} -648.363 q^{89} +517.131 q^{91} +339.172 q^{93} -1256.62 q^{95} -60.9490 q^{97} -937.485 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 10 q^{5} - 16 q^{7} - 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} + 10 q^{5} - 16 q^{7} - 40 q^{9} + 90 q^{11} + 50 q^{13} - 62 q^{15} - 44 q^{17} - 108 q^{19} - 112 q^{21} - 28 q^{23} + 232 q^{25} - 70 q^{27} - 58 q^{29} + 66 q^{31} + 126 q^{33} + 496 q^{35} - 40 q^{37} - 46 q^{39} + 304 q^{41} + 130 q^{43} - 344 q^{45} - 514 q^{47} + 210 q^{49} + 148 q^{51} + 958 q^{53} + 234 q^{55} - 60 q^{57} + 180 q^{59} - 1028 q^{61} + 128 q^{63} + 826 q^{65} + 912 q^{67} - 964 q^{69} + 796 q^{71} - 856 q^{73} - 488 q^{75} - 1008 q^{77} - 318 q^{79} + 470 q^{81} + 1828 q^{83} - 1372 q^{85} - 58 q^{87} - 944 q^{89} + 368 q^{91} + 1374 q^{93} - 828 q^{95} + 368 q^{97} - 1728 q^{99}+O(q^{100}) \) 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
\(3\) −1.44949 −0.278954 −0.139477 0.990225i \(-0.544542\pi\)
−0.139477 + 0.990225i \(0.544542\pi\)
\(4\) 0 0
\(5\) 19.6969 1.76175 0.880874 0.473351i \(-0.156956\pi\)
0.880874 + 0.473351i \(0.156956\pi\)
\(6\) 0 0
\(7\) 11.5959 0.626121 0.313060 0.949733i \(-0.398646\pi\)
0.313060 + 0.949733i \(0.398646\pi\)
\(8\) 0 0
\(9\) −24.8990 −0.922184
\(10\) 0 0
\(11\) 37.6515 1.03203 0.516017 0.856579i \(-0.327414\pi\)
0.516017 + 0.856579i \(0.327414\pi\)
\(12\) 0 0
\(13\) 44.5959 0.951437 0.475719 0.879598i \(-0.342188\pi\)
0.475719 + 0.879598i \(0.342188\pi\)
\(14\) 0 0
\(15\) −28.5505 −0.491447
\(16\) 0 0
\(17\) −61.1918 −0.873012 −0.436506 0.899701i \(-0.643784\pi\)
−0.436506 + 0.899701i \(0.643784\pi\)
\(18\) 0 0
\(19\) −63.7980 −0.770329 −0.385165 0.922848i \(-0.625855\pi\)
−0.385165 + 0.922848i \(0.625855\pi\)
\(20\) 0 0
\(21\) −16.8082 −0.174659
\(22\) 0 0
\(23\) 177.060 1.60520 0.802600 0.596517i \(-0.203449\pi\)
0.802600 + 0.596517i \(0.203449\pi\)
\(24\) 0 0
\(25\) 262.969 2.10376
\(26\) 0 0
\(27\) 75.2270 0.536202
\(28\) 0 0
\(29\) −29.0000 −0.185695
\(30\) 0 0
\(31\) −233.994 −1.35570 −0.677849 0.735201i \(-0.737088\pi\)
−0.677849 + 0.735201i \(0.737088\pi\)
\(32\) 0 0
\(33\) −54.5755 −0.287890
\(34\) 0 0
\(35\) 228.404 1.10307
\(36\) 0 0
\(37\) −10.2020 −0.0453299 −0.0226649 0.999743i \(-0.507215\pi\)
−0.0226649 + 0.999743i \(0.507215\pi\)
\(38\) 0 0
\(39\) −64.6413 −0.265408
\(40\) 0 0
\(41\) 347.959 1.32542 0.662708 0.748877i \(-0.269407\pi\)
0.662708 + 0.748877i \(0.269407\pi\)
\(42\) 0 0
\(43\) 194.823 0.690935 0.345468 0.938431i \(-0.387720\pi\)
0.345468 + 0.938431i \(0.387720\pi\)
\(44\) 0 0
\(45\) −490.434 −1.62466
\(46\) 0 0
\(47\) −14.5005 −0.0450025 −0.0225013 0.999747i \(-0.507163\pi\)
−0.0225013 + 0.999747i \(0.507163\pi\)
\(48\) 0 0
\(49\) −208.535 −0.607973
\(50\) 0 0
\(51\) 88.6969 0.243531
\(52\) 0 0
\(53\) 606.373 1.57154 0.785772 0.618517i \(-0.212266\pi\)
0.785772 + 0.618517i \(0.212266\pi\)
\(54\) 0 0
\(55\) 741.620 1.81818
\(56\) 0 0
\(57\) 92.4745 0.214887
\(58\) 0 0
\(59\) 702.372 1.54985 0.774925 0.632054i \(-0.217788\pi\)
0.774925 + 0.632054i \(0.217788\pi\)
\(60\) 0 0
\(61\) −543.394 −1.14056 −0.570282 0.821449i \(-0.693166\pi\)
−0.570282 + 0.821449i \(0.693166\pi\)
\(62\) 0 0
\(63\) −288.727 −0.577399
\(64\) 0 0
\(65\) 878.403 1.67619
\(66\) 0 0
\(67\) 407.010 0.742152 0.371076 0.928602i \(-0.378989\pi\)
0.371076 + 0.928602i \(0.378989\pi\)
\(68\) 0 0
\(69\) −256.647 −0.447778
\(70\) 0 0
\(71\) 314.717 0.526057 0.263029 0.964788i \(-0.415279\pi\)
0.263029 + 0.964788i \(0.415279\pi\)
\(72\) 0 0
\(73\) −859.110 −1.37741 −0.688707 0.725040i \(-0.741821\pi\)
−0.688707 + 0.725040i \(0.741821\pi\)
\(74\) 0 0
\(75\) −381.171 −0.586852
\(76\) 0 0
\(77\) 436.604 0.646177
\(78\) 0 0
\(79\) 725.266 1.03290 0.516448 0.856319i \(-0.327254\pi\)
0.516448 + 0.856319i \(0.327254\pi\)
\(80\) 0 0
\(81\) 563.232 0.772609
\(82\) 0 0
\(83\) 820.919 1.08563 0.542817 0.839851i \(-0.317358\pi\)
0.542817 + 0.839851i \(0.317358\pi\)
\(84\) 0 0
\(85\) −1205.29 −1.53803
\(86\) 0 0
\(87\) 42.0352 0.0518005
\(88\) 0 0
\(89\) −648.363 −0.772206 −0.386103 0.922456i \(-0.626179\pi\)
−0.386103 + 0.922456i \(0.626179\pi\)
\(90\) 0 0
\(91\) 517.131 0.595714
\(92\) 0 0
\(93\) 339.172 0.378178
\(94\) 0 0
\(95\) −1256.62 −1.35713
\(96\) 0 0
\(97\) −60.9490 −0.0637983 −0.0318991 0.999491i \(-0.510156\pi\)
−0.0318991 + 0.999491i \(0.510156\pi\)
\(98\) 0 0
\(99\) −937.485 −0.951725
\(100\) 0 0
\(101\) −1100.16 −1.08386 −0.541930 0.840423i \(-0.682306\pi\)
−0.541930 + 0.840423i \(0.682306\pi\)
\(102\) 0 0
\(103\) 747.930 0.715492 0.357746 0.933819i \(-0.383545\pi\)
0.357746 + 0.933819i \(0.383545\pi\)
\(104\) 0 0
\(105\) −331.069 −0.307705
\(106\) 0 0
\(107\) −176.059 −0.159068 −0.0795340 0.996832i \(-0.525343\pi\)
−0.0795340 + 0.996832i \(0.525343\pi\)
\(108\) 0 0
\(109\) −173.856 −0.152774 −0.0763871 0.997078i \(-0.524338\pi\)
−0.0763871 + 0.997078i \(0.524338\pi\)
\(110\) 0 0
\(111\) 14.7878 0.0126450
\(112\) 0 0
\(113\) −1707.65 −1.42161 −0.710807 0.703387i \(-0.751670\pi\)
−0.710807 + 0.703387i \(0.751670\pi\)
\(114\) 0 0
\(115\) 3487.54 2.82796
\(116\) 0 0
\(117\) −1110.39 −0.877400
\(118\) 0 0
\(119\) −709.576 −0.546611
\(120\) 0 0
\(121\) 86.6378 0.0650922
\(122\) 0 0
\(123\) −504.363 −0.369731
\(124\) 0 0
\(125\) 2717.57 1.94454
\(126\) 0 0
\(127\) −881.473 −0.615891 −0.307945 0.951404i \(-0.599641\pi\)
−0.307945 + 0.951404i \(0.599641\pi\)
\(128\) 0 0
\(129\) −282.394 −0.192739
\(130\) 0 0
\(131\) −2025.49 −1.35090 −0.675451 0.737405i \(-0.736051\pi\)
−0.675451 + 0.737405i \(0.736051\pi\)
\(132\) 0 0
\(133\) −739.796 −0.482319
\(134\) 0 0
\(135\) 1481.74 0.944652
\(136\) 0 0
\(137\) 1594.32 0.994248 0.497124 0.867679i \(-0.334389\pi\)
0.497124 + 0.867679i \(0.334389\pi\)
\(138\) 0 0
\(139\) 2855.40 1.74239 0.871195 0.490938i \(-0.163346\pi\)
0.871195 + 0.490938i \(0.163346\pi\)
\(140\) 0 0
\(141\) 21.0183 0.0125536
\(142\) 0 0
\(143\) 1679.10 0.981915
\(144\) 0 0
\(145\) −571.211 −0.327148
\(146\) 0 0
\(147\) 302.269 0.169597
\(148\) 0 0
\(149\) 18.3755 0.0101032 0.00505162 0.999987i \(-0.498392\pi\)
0.00505162 + 0.999987i \(0.498392\pi\)
\(150\) 0 0
\(151\) −778.806 −0.419724 −0.209862 0.977731i \(-0.567301\pi\)
−0.209862 + 0.977731i \(0.567301\pi\)
\(152\) 0 0
\(153\) 1523.61 0.805078
\(154\) 0 0
\(155\) −4608.97 −2.38840
\(156\) 0 0
\(157\) 511.896 0.260215 0.130107 0.991500i \(-0.458468\pi\)
0.130107 + 0.991500i \(0.458468\pi\)
\(158\) 0 0
\(159\) −878.932 −0.438389
\(160\) 0 0
\(161\) 2053.18 1.00505
\(162\) 0 0
\(163\) 1451.59 0.697529 0.348764 0.937210i \(-0.386601\pi\)
0.348764 + 0.937210i \(0.386601\pi\)
\(164\) 0 0
\(165\) −1074.97 −0.507190
\(166\) 0 0
\(167\) 600.374 0.278194 0.139097 0.990279i \(-0.455580\pi\)
0.139097 + 0.990279i \(0.455580\pi\)
\(168\) 0 0
\(169\) −208.204 −0.0947675
\(170\) 0 0
\(171\) 1588.50 0.710386
\(172\) 0 0
\(173\) 2574.20 1.13129 0.565644 0.824650i \(-0.308628\pi\)
0.565644 + 0.824650i \(0.308628\pi\)
\(174\) 0 0
\(175\) 3049.37 1.31720
\(176\) 0 0
\(177\) −1018.08 −0.432337
\(178\) 0 0
\(179\) −3828.72 −1.59872 −0.799362 0.600850i \(-0.794829\pi\)
−0.799362 + 0.600850i \(0.794829\pi\)
\(180\) 0 0
\(181\) −2075.64 −0.852381 −0.426190 0.904633i \(-0.640145\pi\)
−0.426190 + 0.904633i \(0.640145\pi\)
\(182\) 0 0
\(183\) 787.644 0.318166
\(184\) 0 0
\(185\) −200.949 −0.0798598
\(186\) 0 0
\(187\) −2303.97 −0.900977
\(188\) 0 0
\(189\) 872.327 0.335727
\(190\) 0 0
\(191\) 4070.08 1.54189 0.770944 0.636902i \(-0.219785\pi\)
0.770944 + 0.636902i \(0.219785\pi\)
\(192\) 0 0
\(193\) 2373.24 0.885129 0.442565 0.896737i \(-0.354069\pi\)
0.442565 + 0.896737i \(0.354069\pi\)
\(194\) 0 0
\(195\) −1273.24 −0.467581
\(196\) 0 0
\(197\) 3108.58 1.12425 0.562125 0.827052i \(-0.309984\pi\)
0.562125 + 0.827052i \(0.309984\pi\)
\(198\) 0 0
\(199\) −4048.30 −1.44209 −0.721046 0.692887i \(-0.756338\pi\)
−0.721046 + 0.692887i \(0.756338\pi\)
\(200\) 0 0
\(201\) −589.957 −0.207027
\(202\) 0 0
\(203\) −336.282 −0.116268
\(204\) 0 0
\(205\) 6853.73 2.33505
\(206\) 0 0
\(207\) −4408.62 −1.48029
\(208\) 0 0
\(209\) −2402.09 −0.795005
\(210\) 0 0
\(211\) −3591.66 −1.17185 −0.585924 0.810366i \(-0.699268\pi\)
−0.585924 + 0.810366i \(0.699268\pi\)
\(212\) 0 0
\(213\) −456.180 −0.146746
\(214\) 0 0
\(215\) 3837.42 1.21725
\(216\) 0 0
\(217\) −2713.38 −0.848830
\(218\) 0 0
\(219\) 1245.27 0.384236
\(220\) 0 0
\(221\) −2728.91 −0.830616
\(222\) 0 0
\(223\) −772.085 −0.231850 −0.115925 0.993258i \(-0.536983\pi\)
−0.115925 + 0.993258i \(0.536983\pi\)
\(224\) 0 0
\(225\) −6547.67 −1.94005
\(226\) 0 0
\(227\) 4435.16 1.29679 0.648397 0.761303i \(-0.275440\pi\)
0.648397 + 0.761303i \(0.275440\pi\)
\(228\) 0 0
\(229\) −2213.12 −0.638634 −0.319317 0.947648i \(-0.603454\pi\)
−0.319317 + 0.947648i \(0.603454\pi\)
\(230\) 0 0
\(231\) −632.853 −0.180254
\(232\) 0 0
\(233\) −2980.41 −0.837998 −0.418999 0.907987i \(-0.637619\pi\)
−0.418999 + 0.907987i \(0.637619\pi\)
\(234\) 0 0
\(235\) −285.616 −0.0792831
\(236\) 0 0
\(237\) −1051.27 −0.288131
\(238\) 0 0
\(239\) −557.093 −0.150775 −0.0753877 0.997154i \(-0.524019\pi\)
−0.0753877 + 0.997154i \(0.524019\pi\)
\(240\) 0 0
\(241\) −4168.65 −1.11422 −0.557109 0.830439i \(-0.688089\pi\)
−0.557109 + 0.830439i \(0.688089\pi\)
\(242\) 0 0
\(243\) −2847.53 −0.751724
\(244\) 0 0
\(245\) −4107.49 −1.07109
\(246\) 0 0
\(247\) −2845.13 −0.732920
\(248\) 0 0
\(249\) −1189.91 −0.302842
\(250\) 0 0
\(251\) 7132.56 1.79364 0.896819 0.442398i \(-0.145872\pi\)
0.896819 + 0.442398i \(0.145872\pi\)
\(252\) 0 0
\(253\) 6666.59 1.65662
\(254\) 0 0
\(255\) 1747.06 0.429039
\(256\) 0 0
\(257\) 5564.76 1.35066 0.675331 0.737514i \(-0.264001\pi\)
0.675331 + 0.737514i \(0.264001\pi\)
\(258\) 0 0
\(259\) −118.302 −0.0283820
\(260\) 0 0
\(261\) 722.070 0.171245
\(262\) 0 0
\(263\) 7188.72 1.68546 0.842729 0.538338i \(-0.180947\pi\)
0.842729 + 0.538338i \(0.180947\pi\)
\(264\) 0 0
\(265\) 11943.7 2.76866
\(266\) 0 0
\(267\) 939.796 0.215410
\(268\) 0 0
\(269\) −3184.38 −0.721765 −0.360883 0.932611i \(-0.617524\pi\)
−0.360883 + 0.932611i \(0.617524\pi\)
\(270\) 0 0
\(271\) 1732.34 0.388311 0.194155 0.980971i \(-0.437803\pi\)
0.194155 + 0.980971i \(0.437803\pi\)
\(272\) 0 0
\(273\) −749.576 −0.166177
\(274\) 0 0
\(275\) 9901.20 2.17114
\(276\) 0 0
\(277\) −6061.31 −1.31476 −0.657381 0.753558i \(-0.728336\pi\)
−0.657381 + 0.753558i \(0.728336\pi\)
\(278\) 0 0
\(279\) 5826.22 1.25020
\(280\) 0 0
\(281\) 6183.75 1.31278 0.656391 0.754421i \(-0.272082\pi\)
0.656391 + 0.754421i \(0.272082\pi\)
\(282\) 0 0
\(283\) 9076.75 1.90656 0.953281 0.302086i \(-0.0976829\pi\)
0.953281 + 0.302086i \(0.0976829\pi\)
\(284\) 0 0
\(285\) 1821.46 0.378576
\(286\) 0 0
\(287\) 4034.91 0.829871
\(288\) 0 0
\(289\) −1168.56 −0.237850
\(290\) 0 0
\(291\) 88.3449 0.0177968
\(292\) 0 0
\(293\) 3636.66 0.725105 0.362553 0.931963i \(-0.381905\pi\)
0.362553 + 0.931963i \(0.381905\pi\)
\(294\) 0 0
\(295\) 13834.6 2.73044
\(296\) 0 0
\(297\) 2832.41 0.553378
\(298\) 0 0
\(299\) 7896.16 1.52725
\(300\) 0 0
\(301\) 2259.15 0.432609
\(302\) 0 0
\(303\) 1594.67 0.302348
\(304\) 0 0
\(305\) −10703.2 −2.00939
\(306\) 0 0
\(307\) 4619.06 0.858708 0.429354 0.903136i \(-0.358741\pi\)
0.429354 + 0.903136i \(0.358741\pi\)
\(308\) 0 0
\(309\) −1084.12 −0.199590
\(310\) 0 0
\(311\) 8094.07 1.47580 0.737898 0.674912i \(-0.235818\pi\)
0.737898 + 0.674912i \(0.235818\pi\)
\(312\) 0 0
\(313\) 6901.79 1.24637 0.623183 0.782076i \(-0.285839\pi\)
0.623183 + 0.782076i \(0.285839\pi\)
\(314\) 0 0
\(315\) −5687.03 −1.01723
\(316\) 0 0
\(317\) 6119.75 1.08429 0.542144 0.840286i \(-0.317613\pi\)
0.542144 + 0.840286i \(0.317613\pi\)
\(318\) 0 0
\(319\) −1091.89 −0.191644
\(320\) 0 0
\(321\) 255.196 0.0443727
\(322\) 0 0
\(323\) 3903.91 0.672506
\(324\) 0 0
\(325\) 11727.4 2.00159
\(326\) 0 0
\(327\) 252.003 0.0426171
\(328\) 0 0
\(329\) −168.147 −0.0281770
\(330\) 0 0
\(331\) −198.924 −0.0330328 −0.0165164 0.999864i \(-0.505258\pi\)
−0.0165164 + 0.999864i \(0.505258\pi\)
\(332\) 0 0
\(333\) 254.020 0.0418025
\(334\) 0 0
\(335\) 8016.85 1.30749
\(336\) 0 0
\(337\) 2102.43 0.339841 0.169921 0.985458i \(-0.445649\pi\)
0.169921 + 0.985458i \(0.445649\pi\)
\(338\) 0 0
\(339\) 2475.23 0.396566
\(340\) 0 0
\(341\) −8810.25 −1.39912
\(342\) 0 0
\(343\) −6395.55 −1.00679
\(344\) 0 0
\(345\) −5055.16 −0.788871
\(346\) 0 0
\(347\) −6543.83 −1.01237 −0.506183 0.862426i \(-0.668944\pi\)
−0.506183 + 0.862426i \(0.668944\pi\)
\(348\) 0 0
\(349\) −793.427 −0.121694 −0.0608469 0.998147i \(-0.519380\pi\)
−0.0608469 + 0.998147i \(0.519380\pi\)
\(350\) 0 0
\(351\) 3354.82 0.510162
\(352\) 0 0
\(353\) 7378.84 1.11257 0.556284 0.830992i \(-0.312227\pi\)
0.556284 + 0.830992i \(0.312227\pi\)
\(354\) 0 0
\(355\) 6198.97 0.926780
\(356\) 0 0
\(357\) 1028.52 0.152479
\(358\) 0 0
\(359\) −7142.97 −1.05012 −0.525058 0.851066i \(-0.675956\pi\)
−0.525058 + 0.851066i \(0.675956\pi\)
\(360\) 0 0
\(361\) −2788.82 −0.406593
\(362\) 0 0
\(363\) −125.581 −0.0181578
\(364\) 0 0
\(365\) −16921.8 −2.42666
\(366\) 0 0
\(367\) −1806.47 −0.256940 −0.128470 0.991713i \(-0.541007\pi\)
−0.128470 + 0.991713i \(0.541007\pi\)
\(368\) 0 0
\(369\) −8663.83 −1.22228
\(370\) 0 0
\(371\) 7031.46 0.983976
\(372\) 0 0
\(373\) −9396.05 −1.30431 −0.652157 0.758084i \(-0.726136\pi\)
−0.652157 + 0.758084i \(0.726136\pi\)
\(374\) 0 0
\(375\) −3939.10 −0.542437
\(376\) 0 0
\(377\) −1293.28 −0.176677
\(378\) 0 0
\(379\) 421.482 0.0571242 0.0285621 0.999592i \(-0.490907\pi\)
0.0285621 + 0.999592i \(0.490907\pi\)
\(380\) 0 0
\(381\) 1277.69 0.171805
\(382\) 0 0
\(383\) 3189.84 0.425570 0.212785 0.977099i \(-0.431747\pi\)
0.212785 + 0.977099i \(0.431747\pi\)
\(384\) 0 0
\(385\) 8599.76 1.13840
\(386\) 0 0
\(387\) −4850.89 −0.637170
\(388\) 0 0
\(389\) 4110.96 0.535820 0.267910 0.963444i \(-0.413667\pi\)
0.267910 + 0.963444i \(0.413667\pi\)
\(390\) 0 0
\(391\) −10834.6 −1.40136
\(392\) 0 0
\(393\) 2935.93 0.376840
\(394\) 0 0
\(395\) 14285.5 1.81970
\(396\) 0 0
\(397\) 827.505 0.104613 0.0523064 0.998631i \(-0.483343\pi\)
0.0523064 + 0.998631i \(0.483343\pi\)
\(398\) 0 0
\(399\) 1072.33 0.134545
\(400\) 0 0
\(401\) 675.145 0.0840776 0.0420388 0.999116i \(-0.486615\pi\)
0.0420388 + 0.999116i \(0.486615\pi\)
\(402\) 0 0
\(403\) −10435.2 −1.28986
\(404\) 0 0
\(405\) 11093.9 1.36114
\(406\) 0 0
\(407\) −384.122 −0.0467819
\(408\) 0 0
\(409\) 764.847 0.0924676 0.0462338 0.998931i \(-0.485278\pi\)
0.0462338 + 0.998931i \(0.485278\pi\)
\(410\) 0 0
\(411\) −2310.95 −0.277350
\(412\) 0 0
\(413\) 8144.65 0.970393
\(414\) 0 0
\(415\) 16169.6 1.91261
\(416\) 0 0
\(417\) −4138.88 −0.486047
\(418\) 0 0
\(419\) 2573.33 0.300037 0.150019 0.988683i \(-0.452067\pi\)
0.150019 + 0.988683i \(0.452067\pi\)
\(420\) 0 0
\(421\) −14287.4 −1.65398 −0.826992 0.562213i \(-0.809950\pi\)
−0.826992 + 0.562213i \(0.809950\pi\)
\(422\) 0 0
\(423\) 361.048 0.0415006
\(424\) 0 0
\(425\) −16091.6 −1.83660
\(426\) 0 0
\(427\) −6301.15 −0.714131
\(428\) 0 0
\(429\) −2433.84 −0.273909
\(430\) 0 0
\(431\) −11283.2 −1.26100 −0.630500 0.776189i \(-0.717150\pi\)
−0.630500 + 0.776189i \(0.717150\pi\)
\(432\) 0 0
\(433\) 8069.64 0.895617 0.447809 0.894129i \(-0.352205\pi\)
0.447809 + 0.894129i \(0.352205\pi\)
\(434\) 0 0
\(435\) 827.965 0.0912595
\(436\) 0 0
\(437\) −11296.1 −1.23653
\(438\) 0 0
\(439\) −7819.42 −0.850116 −0.425058 0.905166i \(-0.639746\pi\)
−0.425058 + 0.905166i \(0.639746\pi\)
\(440\) 0 0
\(441\) 5192.30 0.560663
\(442\) 0 0
\(443\) 10153.4 1.08894 0.544472 0.838779i \(-0.316730\pi\)
0.544472 + 0.838779i \(0.316730\pi\)
\(444\) 0 0
\(445\) −12770.8 −1.36043
\(446\) 0 0
\(447\) −26.6351 −0.00281834
\(448\) 0 0
\(449\) 13608.3 1.43033 0.715163 0.698958i \(-0.246353\pi\)
0.715163 + 0.698958i \(0.246353\pi\)
\(450\) 0 0
\(451\) 13101.2 1.36787
\(452\) 0 0
\(453\) 1128.87 0.117084
\(454\) 0 0
\(455\) 10185.9 1.04950
\(456\) 0 0
\(457\) −1882.55 −0.192696 −0.0963479 0.995348i \(-0.530716\pi\)
−0.0963479 + 0.995348i \(0.530716\pi\)
\(458\) 0 0
\(459\) −4603.28 −0.468111
\(460\) 0 0
\(461\) 13947.2 1.40908 0.704542 0.709662i \(-0.251153\pi\)
0.704542 + 0.709662i \(0.251153\pi\)
\(462\) 0 0
\(463\) −12454.8 −1.25016 −0.625079 0.780562i \(-0.714933\pi\)
−0.625079 + 0.780562i \(0.714933\pi\)
\(464\) 0 0
\(465\) 6680.66 0.666254
\(466\) 0 0
\(467\) 6561.59 0.650180 0.325090 0.945683i \(-0.394605\pi\)
0.325090 + 0.945683i \(0.394605\pi\)
\(468\) 0 0
\(469\) 4719.66 0.464677
\(470\) 0 0
\(471\) −741.988 −0.0725881
\(472\) 0 0
\(473\) 7335.38 0.713068
\(474\) 0 0
\(475\) −16776.9 −1.62058
\(476\) 0 0
\(477\) −15098.1 −1.44925
\(478\) 0 0
\(479\) −17407.5 −1.66048 −0.830240 0.557406i \(-0.811797\pi\)
−0.830240 + 0.557406i \(0.811797\pi\)
\(480\) 0 0
\(481\) −454.969 −0.0431285
\(482\) 0 0
\(483\) −2976.06 −0.280363
\(484\) 0 0
\(485\) −1200.51 −0.112396
\(486\) 0 0
\(487\) −2544.79 −0.236787 −0.118393 0.992967i \(-0.537774\pi\)
−0.118393 + 0.992967i \(0.537774\pi\)
\(488\) 0 0
\(489\) −2104.06 −0.194579
\(490\) 0 0
\(491\) −2712.89 −0.249351 −0.124675 0.992198i \(-0.539789\pi\)
−0.124675 + 0.992198i \(0.539789\pi\)
\(492\) 0 0
\(493\) 1774.56 0.162114
\(494\) 0 0
\(495\) −18465.6 −1.67670
\(496\) 0 0
\(497\) 3649.44 0.329375
\(498\) 0 0
\(499\) −4583.79 −0.411220 −0.205610 0.978634i \(-0.565918\pi\)
−0.205610 + 0.978634i \(0.565918\pi\)
\(500\) 0 0
\(501\) −870.237 −0.0776034
\(502\) 0 0
\(503\) −1950.19 −0.172872 −0.0864359 0.996257i \(-0.527548\pi\)
−0.0864359 + 0.996257i \(0.527548\pi\)
\(504\) 0 0
\(505\) −21669.8 −1.90949
\(506\) 0 0
\(507\) 301.790 0.0264358
\(508\) 0 0
\(509\) 8853.80 0.770997 0.385499 0.922708i \(-0.374029\pi\)
0.385499 + 0.922708i \(0.374029\pi\)
\(510\) 0 0
\(511\) −9962.17 −0.862428
\(512\) 0 0
\(513\) −4799.33 −0.413052
\(514\) 0 0
\(515\) 14731.9 1.26052
\(516\) 0 0
\(517\) −545.967 −0.0464441
\(518\) 0 0
\(519\) −3731.28 −0.315578
\(520\) 0 0
\(521\) −19939.5 −1.67671 −0.838355 0.545125i \(-0.816482\pi\)
−0.838355 + 0.545125i \(0.816482\pi\)
\(522\) 0 0
\(523\) 12185.8 1.01883 0.509416 0.860520i \(-0.329862\pi\)
0.509416 + 0.860520i \(0.329862\pi\)
\(524\) 0 0
\(525\) −4420.03 −0.367440
\(526\) 0 0
\(527\) 14318.5 1.18354
\(528\) 0 0
\(529\) 19183.3 1.57667
\(530\) 0 0
\(531\) −17488.4 −1.42925
\(532\) 0 0
\(533\) 15517.6 1.26105
\(534\) 0 0
\(535\) −3467.83 −0.280238
\(536\) 0 0
\(537\) 5549.68 0.445971
\(538\) 0 0
\(539\) −7851.65 −0.627448
\(540\) 0 0
\(541\) −700.188 −0.0556440 −0.0278220 0.999613i \(-0.508857\pi\)
−0.0278220 + 0.999613i \(0.508857\pi\)
\(542\) 0 0
\(543\) 3008.62 0.237775
\(544\) 0 0
\(545\) −3424.43 −0.269150
\(546\) 0 0
\(547\) −10664.2 −0.833576 −0.416788 0.909004i \(-0.636844\pi\)
−0.416788 + 0.909004i \(0.636844\pi\)
\(548\) 0 0
\(549\) 13530.0 1.05181
\(550\) 0 0
\(551\) 1850.14 0.143047
\(552\) 0 0
\(553\) 8410.12 0.646718
\(554\) 0 0
\(555\) 291.273 0.0222772
\(556\) 0 0
\(557\) −8410.60 −0.639800 −0.319900 0.947451i \(-0.603649\pi\)
−0.319900 + 0.947451i \(0.603649\pi\)
\(558\) 0 0
\(559\) 8688.31 0.657382
\(560\) 0 0
\(561\) 3339.58 0.251332
\(562\) 0 0
\(563\) −8770.15 −0.656515 −0.328257 0.944588i \(-0.606461\pi\)
−0.328257 + 0.944588i \(0.606461\pi\)
\(564\) 0 0
\(565\) −33635.5 −2.50453
\(566\) 0 0
\(567\) 6531.19 0.483746
\(568\) 0 0
\(569\) −10493.4 −0.773125 −0.386562 0.922263i \(-0.626338\pi\)
−0.386562 + 0.922263i \(0.626338\pi\)
\(570\) 0 0
\(571\) 2517.50 0.184508 0.0922542 0.995735i \(-0.470593\pi\)
0.0922542 + 0.995735i \(0.470593\pi\)
\(572\) 0 0
\(573\) −5899.54 −0.430117
\(574\) 0 0
\(575\) 46561.4 3.37695
\(576\) 0 0
\(577\) 1394.88 0.100641 0.0503203 0.998733i \(-0.483976\pi\)
0.0503203 + 0.998733i \(0.483976\pi\)
\(578\) 0 0
\(579\) −3439.99 −0.246911
\(580\) 0 0
\(581\) 9519.31 0.679738
\(582\) 0 0
\(583\) 22830.9 1.62188
\(584\) 0 0
\(585\) −21871.3 −1.54576
\(586\) 0 0
\(587\) −24391.0 −1.71504 −0.857518 0.514455i \(-0.827994\pi\)
−0.857518 + 0.514455i \(0.827994\pi\)
\(588\) 0 0
\(589\) 14928.4 1.04433
\(590\) 0 0
\(591\) −4505.86 −0.313615
\(592\) 0 0
\(593\) −16089.4 −1.11418 −0.557092 0.830451i \(-0.688083\pi\)
−0.557092 + 0.830451i \(0.688083\pi\)
\(594\) 0 0
\(595\) −13976.5 −0.962990
\(596\) 0 0
\(597\) 5867.97 0.402278
\(598\) 0 0
\(599\) −19228.2 −1.31159 −0.655796 0.754938i \(-0.727667\pi\)
−0.655796 + 0.754938i \(0.727667\pi\)
\(600\) 0 0
\(601\) −2345.94 −0.159223 −0.0796115 0.996826i \(-0.525368\pi\)
−0.0796115 + 0.996826i \(0.525368\pi\)
\(602\) 0 0
\(603\) −10134.1 −0.684401
\(604\) 0 0
\(605\) 1706.50 0.114676
\(606\) 0 0
\(607\) −4621.11 −0.309004 −0.154502 0.987993i \(-0.549377\pi\)
−0.154502 + 0.987993i \(0.549377\pi\)
\(608\) 0 0
\(609\) 487.437 0.0324334
\(610\) 0 0
\(611\) −646.664 −0.0428170
\(612\) 0 0
\(613\) 10464.8 0.689511 0.344755 0.938693i \(-0.387962\pi\)
0.344755 + 0.938693i \(0.387962\pi\)
\(614\) 0 0
\(615\) −9934.41 −0.651373
\(616\) 0 0
\(617\) −11873.3 −0.774718 −0.387359 0.921929i \(-0.626613\pi\)
−0.387359 + 0.921929i \(0.626613\pi\)
\(618\) 0 0
\(619\) −13102.2 −0.850762 −0.425381 0.905014i \(-0.639860\pi\)
−0.425381 + 0.905014i \(0.639860\pi\)
\(620\) 0 0
\(621\) 13319.7 0.860711
\(622\) 0 0
\(623\) −7518.37 −0.483494
\(624\) 0 0
\(625\) 20656.7 1.32203
\(626\) 0 0
\(627\) 3481.81 0.221770
\(628\) 0 0
\(629\) 624.282 0.0395735
\(630\) 0 0
\(631\) 12279.7 0.774719 0.387359 0.921929i \(-0.373387\pi\)
0.387359 + 0.921929i \(0.373387\pi\)
\(632\) 0 0
\(633\) 5206.07 0.326892
\(634\) 0 0
\(635\) −17362.3 −1.08504
\(636\) 0 0
\(637\) −9299.80 −0.578448
\(638\) 0 0
\(639\) −7836.14 −0.485122
\(640\) 0 0
\(641\) 17054.4 1.05087 0.525435 0.850834i \(-0.323903\pi\)
0.525435 + 0.850834i \(0.323903\pi\)
\(642\) 0 0
\(643\) 20697.3 1.26940 0.634698 0.772761i \(-0.281125\pi\)
0.634698 + 0.772761i \(0.281125\pi\)
\(644\) 0 0
\(645\) −5562.29 −0.339558
\(646\) 0 0
\(647\) −10613.6 −0.644924 −0.322462 0.946582i \(-0.604510\pi\)
−0.322462 + 0.946582i \(0.604510\pi\)
\(648\) 0 0
\(649\) 26445.4 1.59950
\(650\) 0 0
\(651\) 3933.02 0.236785
\(652\) 0 0
\(653\) 10019.4 0.600441 0.300220 0.953870i \(-0.402940\pi\)
0.300220 + 0.953870i \(0.402940\pi\)
\(654\) 0 0
\(655\) −39896.0 −2.37995
\(656\) 0 0
\(657\) 21391.0 1.27023
\(658\) 0 0
\(659\) −2400.70 −0.141909 −0.0709545 0.997480i \(-0.522605\pi\)
−0.0709545 + 0.997480i \(0.522605\pi\)
\(660\) 0 0
\(661\) 5037.05 0.296397 0.148199 0.988958i \(-0.452653\pi\)
0.148199 + 0.988958i \(0.452653\pi\)
\(662\) 0 0
\(663\) 3955.52 0.231704
\(664\) 0 0
\(665\) −14571.7 −0.849725
\(666\) 0 0
\(667\) −5134.75 −0.298078
\(668\) 0 0
\(669\) 1119.13 0.0646757
\(670\) 0 0
\(671\) −20459.6 −1.17710
\(672\) 0 0
\(673\) −20842.1 −1.19377 −0.596884 0.802328i \(-0.703595\pi\)
−0.596884 + 0.802328i \(0.703595\pi\)
\(674\) 0 0
\(675\) 19782.4 1.12804
\(676\) 0 0
\(677\) 23394.0 1.32807 0.664037 0.747700i \(-0.268842\pi\)
0.664037 + 0.747700i \(0.268842\pi\)
\(678\) 0 0
\(679\) −706.759 −0.0399454
\(680\) 0 0
\(681\) −6428.72 −0.361746
\(682\) 0 0
\(683\) 8567.11 0.479958 0.239979 0.970778i \(-0.422859\pi\)
0.239979 + 0.970778i \(0.422859\pi\)
\(684\) 0 0
\(685\) 31403.2 1.75161
\(686\) 0 0
\(687\) 3207.90 0.178150
\(688\) 0 0
\(689\) 27041.8 1.49522
\(690\) 0 0
\(691\) 11987.6 0.659959 0.329979 0.943988i \(-0.392958\pi\)
0.329979 + 0.943988i \(0.392958\pi\)
\(692\) 0 0
\(693\) −10871.0 −0.595895
\(694\) 0 0
\(695\) 56242.7 3.06965
\(696\) 0 0
\(697\) −21292.3 −1.15710
\(698\) 0 0
\(699\) 4320.08 0.233763
\(700\) 0 0
\(701\) −19429.6 −1.04686 −0.523428 0.852070i \(-0.675347\pi\)
−0.523428 + 0.852070i \(0.675347\pi\)
\(702\) 0 0
\(703\) 650.869 0.0349189
\(704\) 0 0
\(705\) 413.997 0.0221164
\(706\) 0 0
\(707\) −12757.4 −0.678628
\(708\) 0 0
\(709\) 804.014 0.0425887 0.0212944 0.999773i \(-0.493221\pi\)
0.0212944 + 0.999773i \(0.493221\pi\)
\(710\) 0 0
\(711\) −18058.4 −0.952521
\(712\) 0 0
\(713\) −41431.1 −2.17617
\(714\) 0 0
\(715\) 33073.2 1.72989
\(716\) 0 0
\(717\) 807.500 0.0420595
\(718\) 0 0
\(719\) 4975.94 0.258096 0.129048 0.991638i \(-0.458808\pi\)
0.129048 + 0.991638i \(0.458808\pi\)
\(720\) 0 0
\(721\) 8672.93 0.447984
\(722\) 0 0
\(723\) 6042.42 0.310816
\(724\) 0 0
\(725\) −7626.11 −0.390658
\(726\) 0 0
\(727\) −3761.32 −0.191884 −0.0959419 0.995387i \(-0.530586\pi\)
−0.0959419 + 0.995387i \(0.530586\pi\)
\(728\) 0 0
\(729\) −11079.8 −0.562912
\(730\) 0 0
\(731\) −11921.6 −0.603195
\(732\) 0 0
\(733\) −21439.2 −1.08032 −0.540161 0.841562i \(-0.681637\pi\)
−0.540161 + 0.841562i \(0.681637\pi\)
\(734\) 0 0
\(735\) 5953.77 0.298787
\(736\) 0 0
\(737\) 15324.6 0.765926
\(738\) 0 0
\(739\) −22000.3 −1.09512 −0.547561 0.836766i \(-0.684444\pi\)
−0.547561 + 0.836766i \(0.684444\pi\)
\(740\) 0 0
\(741\) 4123.98 0.204451
\(742\) 0 0
\(743\) −13371.8 −0.660246 −0.330123 0.943938i \(-0.607090\pi\)
−0.330123 + 0.943938i \(0.607090\pi\)
\(744\) 0 0
\(745\) 361.942 0.0177994
\(746\) 0 0
\(747\) −20440.1 −1.00115
\(748\) 0 0
\(749\) −2041.57 −0.0995958
\(750\) 0 0
\(751\) −17764.8 −0.863178 −0.431589 0.902070i \(-0.642047\pi\)
−0.431589 + 0.902070i \(0.642047\pi\)
\(752\) 0 0
\(753\) −10338.6 −0.500343
\(754\) 0 0
\(755\) −15340.1 −0.739448
\(756\) 0 0
\(757\) −33266.1 −1.59719 −0.798597 0.601867i \(-0.794424\pi\)
−0.798597 + 0.601867i \(0.794424\pi\)
\(758\) 0 0
\(759\) −9663.15 −0.462121
\(760\) 0 0
\(761\) −25327.3 −1.20646 −0.603228 0.797569i \(-0.706119\pi\)
−0.603228 + 0.797569i \(0.706119\pi\)
\(762\) 0 0
\(763\) −2016.02 −0.0956551
\(764\) 0 0
\(765\) 30010.5 1.41834
\(766\) 0 0
\(767\) 31322.9 1.47458
\(768\) 0 0
\(769\) −9587.28 −0.449579 −0.224789 0.974407i \(-0.572169\pi\)
−0.224789 + 0.974407i \(0.572169\pi\)
\(770\) 0 0
\(771\) −8066.07 −0.376773
\(772\) 0 0
\(773\) −2825.54 −0.131472 −0.0657359 0.997837i \(-0.520939\pi\)
−0.0657359 + 0.997837i \(0.520939\pi\)
\(774\) 0 0
\(775\) −61533.4 −2.85206
\(776\) 0 0
\(777\) 171.478 0.00791728
\(778\) 0 0
\(779\) −22199.1 −1.02101
\(780\) 0 0
\(781\) 11849.6 0.542909
\(782\) 0 0
\(783\) −2181.58 −0.0995702
\(784\) 0 0
\(785\) 10082.8 0.458433
\(786\) 0 0
\(787\) −2132.09 −0.0965702 −0.0482851 0.998834i \(-0.515376\pi\)
−0.0482851 + 0.998834i \(0.515376\pi\)
\(788\) 0 0
\(789\) −10420.0 −0.470166
\(790\) 0 0
\(791\) −19801.8 −0.890103
\(792\) 0 0
\(793\) −24233.1 −1.08518
\(794\) 0 0
\(795\) −17312.3 −0.772331
\(796\) 0 0
\(797\) −5443.62 −0.241936 −0.120968 0.992656i \(-0.538600\pi\)
−0.120968 + 0.992656i \(0.538600\pi\)
\(798\) 0 0
\(799\) 887.313 0.0392877
\(800\) 0 0
\(801\) 16143.6 0.712117
\(802\) 0 0
\(803\) −32346.8 −1.42154
\(804\) 0 0
\(805\) 40441.3 1.77064
\(806\) 0 0
\(807\) 4615.72 0.201340
\(808\) 0 0
\(809\) 41193.5 1.79022 0.895109 0.445847i \(-0.147098\pi\)
0.895109 + 0.445847i \(0.147098\pi\)
\(810\) 0 0
\(811\) 18225.9 0.789145 0.394572 0.918865i \(-0.370893\pi\)
0.394572 + 0.918865i \(0.370893\pi\)
\(812\) 0 0
\(813\) −2511.01 −0.108321
\(814\) 0 0
\(815\) 28591.8 1.22887
\(816\) 0 0
\(817\) −12429.3 −0.532248
\(818\) 0 0
\(819\) −12876.0 −0.549359
\(820\) 0 0
\(821\) −46061.7 −1.95806 −0.979029 0.203720i \(-0.934697\pi\)
−0.979029 + 0.203720i \(0.934697\pi\)
\(822\) 0 0
\(823\) 40166.3 1.70122 0.850612 0.525793i \(-0.176231\pi\)
0.850612 + 0.525793i \(0.176231\pi\)
\(824\) 0 0
\(825\) −14351.7 −0.605650
\(826\) 0 0
\(827\) 23994.5 1.00891 0.504457 0.863437i \(-0.331693\pi\)
0.504457 + 0.863437i \(0.331693\pi\)
\(828\) 0 0
\(829\) 13549.6 0.567668 0.283834 0.958873i \(-0.408394\pi\)
0.283834 + 0.958873i \(0.408394\pi\)
\(830\) 0 0
\(831\) 8785.81 0.366759
\(832\) 0 0
\(833\) 12760.6 0.530767
\(834\) 0 0
\(835\) 11825.5 0.490107
\(836\) 0 0
\(837\) −17602.7 −0.726928
\(838\) 0 0
\(839\) 34768.5 1.43068 0.715341 0.698776i \(-0.246271\pi\)
0.715341 + 0.698776i \(0.246271\pi\)
\(840\) 0 0
\(841\) 841.000 0.0344828
\(842\) 0 0
\(843\) −8963.29 −0.366207
\(844\) 0 0
\(845\) −4100.98 −0.166956
\(846\) 0 0
\(847\) 1004.64 0.0407556
\(848\) 0 0
\(849\) −13156.7 −0.531844
\(850\) 0 0
\(851\) −1806.38 −0.0727635
\(852\) 0 0
\(853\) −4691.07 −0.188299 −0.0941496 0.995558i \(-0.530013\pi\)
−0.0941496 + 0.995558i \(0.530013\pi\)
\(854\) 0 0
\(855\) 31288.7 1.25152
\(856\) 0 0
\(857\) −14362.7 −0.572485 −0.286242 0.958157i \(-0.592406\pi\)
−0.286242 + 0.958157i \(0.592406\pi\)
\(858\) 0 0
\(859\) −1392.43 −0.0553073 −0.0276537 0.999618i \(-0.508804\pi\)
−0.0276537 + 0.999618i \(0.508804\pi\)
\(860\) 0 0
\(861\) −5848.55 −0.231496
\(862\) 0 0
\(863\) 19162.2 0.755840 0.377920 0.925838i \(-0.376639\pi\)
0.377920 + 0.925838i \(0.376639\pi\)
\(864\) 0 0
\(865\) 50703.9 1.99304
\(866\) 0 0
\(867\) 1693.81 0.0663494
\(868\) 0 0
\(869\) 27307.4 1.06598
\(870\) 0 0
\(871\) 18151.0 0.706111
\(872\) 0 0
\(873\) 1517.57 0.0588338
\(874\) 0 0
\(875\) 31512.8 1.21752
\(876\) 0 0
\(877\) 6509.75 0.250649 0.125324 0.992116i \(-0.460003\pi\)
0.125324 + 0.992116i \(0.460003\pi\)
\(878\) 0 0
\(879\) −5271.30 −0.202271
\(880\) 0 0
\(881\) −589.496 −0.0225433 −0.0112716 0.999936i \(-0.503588\pi\)
−0.0112716 + 0.999936i \(0.503588\pi\)
\(882\) 0 0
\(883\) 3903.57 0.148772 0.0743859 0.997230i \(-0.476300\pi\)
0.0743859 + 0.997230i \(0.476300\pi\)
\(884\) 0 0
\(885\) −20053.1 −0.761669
\(886\) 0 0
\(887\) −24136.6 −0.913674 −0.456837 0.889550i \(-0.651018\pi\)
−0.456837 + 0.889550i \(0.651018\pi\)
\(888\) 0 0
\(889\) −10221.5 −0.385622
\(890\) 0 0
\(891\) 21206.5 0.797358
\(892\) 0 0
\(893\) 925.103 0.0346667
\(894\) 0 0
\(895\) −75414.0 −2.81655
\(896\) 0 0
\(897\) −11445.4 −0.426032
\(898\) 0 0
\(899\) 6785.84 0.251747
\(900\) 0 0
\(901\) −37105.1 −1.37198
\(902\) 0 0
\(903\) −3274.62 −0.120678
\(904\) 0 0
\(905\) −40883.7 −1.50168
\(906\) 0 0
\(907\) −22174.3 −0.811781 −0.405890 0.913922i \(-0.633038\pi\)
−0.405890 + 0.913922i \(0.633038\pi\)
\(908\) 0 0
\(909\) 27392.8 0.999519
\(910\) 0 0
\(911\) −46956.1 −1.70771 −0.853856 0.520509i \(-0.825742\pi\)
−0.853856 + 0.520509i \(0.825742\pi\)
\(912\) 0 0
\(913\) 30908.9 1.12041
\(914\) 0 0
\(915\) 15514.2 0.560528
\(916\) 0 0
\(917\) −23487.4 −0.845827
\(918\) 0 0
\(919\) −37166.4 −1.33407 −0.667033 0.745028i \(-0.732436\pi\)
−0.667033 + 0.745028i \(0.732436\pi\)
\(920\) 0 0
\(921\) −6695.27 −0.239540
\(922\) 0 0
\(923\) 14035.1 0.500511
\(924\) 0 0
\(925\) −2682.82 −0.0953629
\(926\) 0 0
\(927\) −18622.7 −0.659816
\(928\) 0 0
\(929\) 29571.7 1.04437 0.522183 0.852833i \(-0.325118\pi\)
0.522183 + 0.852833i \(0.325118\pi\)
\(930\) 0 0
\(931\) 13304.1 0.468339
\(932\) 0 0
\(933\) −11732.3 −0.411680
\(934\) 0 0
\(935\) −45381.1 −1.58729
\(936\) 0 0
\(937\) 8691.51 0.303030 0.151515 0.988455i \(-0.451585\pi\)
0.151515 + 0.988455i \(0.451585\pi\)
\(938\) 0 0
\(939\) −10004.1 −0.347679
\(940\) 0 0
\(941\) 2862.43 0.0991632 0.0495816 0.998770i \(-0.484211\pi\)
0.0495816 + 0.998770i \(0.484211\pi\)
\(942\) 0 0
\(943\) 61609.7 2.12756
\(944\) 0 0
\(945\) 17182.2 0.591466
\(946\) 0 0
\(947\) −36997.8 −1.26955 −0.634777 0.772695i \(-0.718908\pi\)
−0.634777 + 0.772695i \(0.718908\pi\)
\(948\) 0 0
\(949\) −38312.8 −1.31052
\(950\) 0 0
\(951\) −8870.51 −0.302467
\(952\) 0 0
\(953\) 20998.8 0.713764 0.356882 0.934150i \(-0.383840\pi\)
0.356882 + 0.934150i \(0.383840\pi\)
\(954\) 0 0
\(955\) 80168.1 2.71642
\(956\) 0 0
\(957\) 1582.69 0.0534599
\(958\) 0 0
\(959\) 18487.6 0.622519
\(960\) 0 0
\(961\) 24962.4 0.837917
\(962\) 0 0
\(963\) 4383.69 0.146690
\(964\) 0 0
\(965\) 46745.7 1.55937
\(966\) 0 0
\(967\) −30803.0 −1.02436 −0.512181 0.858878i \(-0.671162\pi\)
−0.512181 + 0.858878i \(0.671162\pi\)
\(968\) 0 0
\(969\) −5658.68 −0.187599
\(970\) 0 0
\(971\) 755.015 0.0249532 0.0124766 0.999922i \(-0.496028\pi\)
0.0124766 + 0.999922i \(0.496028\pi\)
\(972\) 0 0
\(973\) 33111.0 1.09095
\(974\) 0 0
\(975\) −16998.7 −0.558353
\(976\) 0 0
\(977\) −44011.7 −1.44121 −0.720603 0.693348i \(-0.756135\pi\)
−0.720603 + 0.693348i \(0.756135\pi\)
\(978\) 0 0
\(979\) −24411.9 −0.796943
\(980\) 0 0
\(981\) 4328.84 0.140886
\(982\) 0 0
\(983\) −36334.8 −1.17894 −0.589471 0.807790i \(-0.700664\pi\)
−0.589471 + 0.807790i \(0.700664\pi\)
\(984\) 0 0
\(985\) 61229.6 1.98065
\(986\) 0 0
\(987\) 243.727 0.00786010
\(988\) 0 0
\(989\) 34495.4 1.10909
\(990\) 0 0
\(991\) 9848.60 0.315692 0.157846 0.987464i \(-0.449545\pi\)
0.157846 + 0.987464i \(0.449545\pi\)
\(992\) 0 0
\(993\) 288.338 0.00921464
\(994\) 0 0
\(995\) −79739.1 −2.54060
\(996\) 0 0
\(997\) −3378.02 −0.107305 −0.0536525 0.998560i \(-0.517086\pi\)
−0.0536525 + 0.998560i \(0.517086\pi\)
\(998\) 0 0
\(999\) −767.469 −0.0243060
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 1856.4.a.l.1.1 2
4.3 odd 2 1856.4.a.i.1.2 2
8.3 odd 2 464.4.a.e.1.1 2
8.5 even 2 58.4.a.c.1.2 2
24.5 odd 2 522.4.a.j.1.2 2
40.29 even 2 1450.4.a.g.1.1 2
232.173 even 2 1682.4.a.c.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
58.4.a.c.1.2 2 8.5 even 2
464.4.a.e.1.1 2 8.3 odd 2
522.4.a.j.1.2 2 24.5 odd 2
1450.4.a.g.1.1 2 40.29 even 2
1682.4.a.c.1.1 2 232.173 even 2
1856.4.a.i.1.2 2 4.3 odd 2
1856.4.a.l.1.1 2 1.1 even 1 trivial