Properties

Label 784.4.a.z.1.2
Level $784$
Weight $4$
Character 784.1
Self dual yes
Analytic conductor $46.257$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 784.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(46.2574974445\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{22}) \)
Defining polynomial: \(x^{2} - 22\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 98)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(4.69042\) of defining polynomial
Character \(\chi\) \(=\) 784.1

$q$-expansion

\(f(q)\) \(=\) \(q+9.38083 q^{3} +9.38083 q^{5} +61.0000 q^{9} +O(q^{10})\) \(q+9.38083 q^{3} +9.38083 q^{5} +61.0000 q^{9} -20.0000 q^{11} +65.6658 q^{13} +88.0000 q^{15} +56.2850 q^{17} -9.38083 q^{19} -48.0000 q^{23} -37.0000 q^{25} +318.948 q^{27} -166.000 q^{29} +206.378 q^{31} -187.617 q^{33} -78.0000 q^{37} +616.000 q^{39} +393.995 q^{41} -436.000 q^{43} +572.231 q^{45} -206.378 q^{47} +528.000 q^{51} +62.0000 q^{53} -187.617 q^{55} -88.0000 q^{57} +666.039 q^{59} +272.044 q^{61} +616.000 q^{65} -580.000 q^{67} -450.280 q^{69} +544.000 q^{71} -600.373 q^{73} -347.091 q^{75} +680.000 q^{79} +1345.00 q^{81} -196.997 q^{83} +528.000 q^{85} -1557.22 q^{87} -1500.93 q^{89} +1936.00 q^{93} -88.0000 q^{95} -656.658 q^{97} -1220.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 122q^{9} + O(q^{10}) \) \( 2q + 122q^{9} - 40q^{11} + 176q^{15} - 96q^{23} - 74q^{25} - 332q^{29} - 156q^{37} + 1232q^{39} - 872q^{43} + 1056q^{51} + 124q^{53} - 176q^{57} + 1232q^{65} - 1160q^{67} + 1088q^{71} + 1360q^{79} + 2690q^{81} + 1056q^{85} + 3872q^{93} - 176q^{95} - 2440q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 9.38083 1.80534 0.902671 0.430331i \(-0.141603\pi\)
0.902671 + 0.430331i \(0.141603\pi\)
\(4\) 0 0
\(5\) 9.38083 0.839047 0.419524 0.907744i \(-0.362197\pi\)
0.419524 + 0.907744i \(0.362197\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 61.0000 2.25926
\(10\) 0 0
\(11\) −20.0000 −0.548202 −0.274101 0.961701i \(-0.588380\pi\)
−0.274101 + 0.961701i \(0.588380\pi\)
\(12\) 0 0
\(13\) 65.6658 1.40096 0.700478 0.713674i \(-0.252970\pi\)
0.700478 + 0.713674i \(0.252970\pi\)
\(14\) 0 0
\(15\) 88.0000 1.51477
\(16\) 0 0
\(17\) 56.2850 0.803007 0.401503 0.915858i \(-0.368488\pi\)
0.401503 + 0.915858i \(0.368488\pi\)
\(18\) 0 0
\(19\) −9.38083 −0.113269 −0.0566345 0.998395i \(-0.518037\pi\)
−0.0566345 + 0.998395i \(0.518037\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −48.0000 −0.435161 −0.217580 0.976042i \(-0.569816\pi\)
−0.217580 + 0.976042i \(0.569816\pi\)
\(24\) 0 0
\(25\) −37.0000 −0.296000
\(26\) 0 0
\(27\) 318.948 2.27339
\(28\) 0 0
\(29\) −166.000 −1.06295 −0.531473 0.847075i \(-0.678361\pi\)
−0.531473 + 0.847075i \(0.678361\pi\)
\(30\) 0 0
\(31\) 206.378 1.19570 0.597849 0.801609i \(-0.296022\pi\)
0.597849 + 0.801609i \(0.296022\pi\)
\(32\) 0 0
\(33\) −187.617 −0.989693
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −78.0000 −0.346571 −0.173285 0.984872i \(-0.555438\pi\)
−0.173285 + 0.984872i \(0.555438\pi\)
\(38\) 0 0
\(39\) 616.000 2.52920
\(40\) 0 0
\(41\) 393.995 1.50077 0.750386 0.661000i \(-0.229868\pi\)
0.750386 + 0.661000i \(0.229868\pi\)
\(42\) 0 0
\(43\) −436.000 −1.54626 −0.773132 0.634245i \(-0.781311\pi\)
−0.773132 + 0.634245i \(0.781311\pi\)
\(44\) 0 0
\(45\) 572.231 1.89562
\(46\) 0 0
\(47\) −206.378 −0.640497 −0.320249 0.947334i \(-0.603766\pi\)
−0.320249 + 0.947334i \(0.603766\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 528.000 1.44970
\(52\) 0 0
\(53\) 62.0000 0.160686 0.0803430 0.996767i \(-0.474398\pi\)
0.0803430 + 0.996767i \(0.474398\pi\)
\(54\) 0 0
\(55\) −187.617 −0.459968
\(56\) 0 0
\(57\) −88.0000 −0.204489
\(58\) 0 0
\(59\) 666.039 1.46968 0.734838 0.678243i \(-0.237258\pi\)
0.734838 + 0.678243i \(0.237258\pi\)
\(60\) 0 0
\(61\) 272.044 0.571011 0.285506 0.958377i \(-0.407838\pi\)
0.285506 + 0.958377i \(0.407838\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 616.000 1.17547
\(66\) 0 0
\(67\) −580.000 −1.05759 −0.528793 0.848751i \(-0.677355\pi\)
−0.528793 + 0.848751i \(0.677355\pi\)
\(68\) 0 0
\(69\) −450.280 −0.785613
\(70\) 0 0
\(71\) 544.000 0.909309 0.454654 0.890668i \(-0.349763\pi\)
0.454654 + 0.890668i \(0.349763\pi\)
\(72\) 0 0
\(73\) −600.373 −0.962580 −0.481290 0.876561i \(-0.659832\pi\)
−0.481290 + 0.876561i \(0.659832\pi\)
\(74\) 0 0
\(75\) −347.091 −0.534381
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 680.000 0.968430 0.484215 0.874949i \(-0.339105\pi\)
0.484215 + 0.874949i \(0.339105\pi\)
\(80\) 0 0
\(81\) 1345.00 1.84499
\(82\) 0 0
\(83\) −196.997 −0.260521 −0.130261 0.991480i \(-0.541581\pi\)
−0.130261 + 0.991480i \(0.541581\pi\)
\(84\) 0 0
\(85\) 528.000 0.673760
\(86\) 0 0
\(87\) −1557.22 −1.91898
\(88\) 0 0
\(89\) −1500.93 −1.78762 −0.893812 0.448441i \(-0.851979\pi\)
−0.893812 + 0.448441i \(0.851979\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 1936.00 2.15864
\(94\) 0 0
\(95\) −88.0000 −0.0950380
\(96\) 0 0
\(97\) −656.658 −0.687356 −0.343678 0.939088i \(-0.611673\pi\)
−0.343678 + 0.939088i \(0.611673\pi\)
\(98\) 0 0
\(99\) −1220.00 −1.23853
\(100\) 0 0
\(101\) 121.951 0.120144 0.0600721 0.998194i \(-0.480867\pi\)
0.0600721 + 0.998194i \(0.480867\pi\)
\(102\) 0 0
\(103\) 1369.60 1.31020 0.655101 0.755541i \(-0.272626\pi\)
0.655101 + 0.755541i \(0.272626\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 260.000 0.234908 0.117454 0.993078i \(-0.462527\pi\)
0.117454 + 0.993078i \(0.462527\pi\)
\(108\) 0 0
\(109\) 1882.00 1.65379 0.826894 0.562358i \(-0.190106\pi\)
0.826894 + 0.562358i \(0.190106\pi\)
\(110\) 0 0
\(111\) −731.705 −0.625679
\(112\) 0 0
\(113\) −1286.00 −1.07059 −0.535295 0.844665i \(-0.679800\pi\)
−0.535295 + 0.844665i \(0.679800\pi\)
\(114\) 0 0
\(115\) −450.280 −0.365120
\(116\) 0 0
\(117\) 4005.62 3.16512
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −931.000 −0.699474
\(122\) 0 0
\(123\) 3696.00 2.70941
\(124\) 0 0
\(125\) −1519.69 −1.08741
\(126\) 0 0
\(127\) −2312.00 −1.61541 −0.807704 0.589588i \(-0.799290\pi\)
−0.807704 + 0.589588i \(0.799290\pi\)
\(128\) 0 0
\(129\) −4090.04 −2.79154
\(130\) 0 0
\(131\) 253.282 0.168927 0.0844633 0.996427i \(-0.473082\pi\)
0.0844633 + 0.996427i \(0.473082\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 2992.00 1.90748
\(136\) 0 0
\(137\) −1114.00 −0.694711 −0.347356 0.937733i \(-0.612920\pi\)
−0.347356 + 0.937733i \(0.612920\pi\)
\(138\) 0 0
\(139\) −1378.98 −0.841466 −0.420733 0.907185i \(-0.638227\pi\)
−0.420733 + 0.907185i \(0.638227\pi\)
\(140\) 0 0
\(141\) −1936.00 −1.15632
\(142\) 0 0
\(143\) −1313.32 −0.768007
\(144\) 0 0
\(145\) −1557.22 −0.891862
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −946.000 −0.520130 −0.260065 0.965591i \(-0.583744\pi\)
−0.260065 + 0.965591i \(0.583744\pi\)
\(150\) 0 0
\(151\) −832.000 −0.448392 −0.224196 0.974544i \(-0.571976\pi\)
−0.224196 + 0.974544i \(0.571976\pi\)
\(152\) 0 0
\(153\) 3433.38 1.81420
\(154\) 0 0
\(155\) 1936.00 1.00325
\(156\) 0 0
\(157\) 2879.92 1.46396 0.731982 0.681324i \(-0.238596\pi\)
0.731982 + 0.681324i \(0.238596\pi\)
\(158\) 0 0
\(159\) 581.612 0.290093
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −636.000 −0.305616 −0.152808 0.988256i \(-0.548832\pi\)
−0.152808 + 0.988256i \(0.548832\pi\)
\(164\) 0 0
\(165\) −1760.00 −0.830399
\(166\) 0 0
\(167\) 656.658 0.304274 0.152137 0.988359i \(-0.451385\pi\)
0.152137 + 0.988359i \(0.451385\pi\)
\(168\) 0 0
\(169\) 2115.00 0.962676
\(170\) 0 0
\(171\) −572.231 −0.255904
\(172\) 0 0
\(173\) 666.039 0.292705 0.146353 0.989232i \(-0.453247\pi\)
0.146353 + 0.989232i \(0.453247\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 6248.00 2.65327
\(178\) 0 0
\(179\) 3228.00 1.34789 0.673944 0.738782i \(-0.264599\pi\)
0.673944 + 0.738782i \(0.264599\pi\)
\(180\) 0 0
\(181\) −2823.63 −1.15955 −0.579776 0.814776i \(-0.696860\pi\)
−0.579776 + 0.814776i \(0.696860\pi\)
\(182\) 0 0
\(183\) 2552.00 1.03087
\(184\) 0 0
\(185\) −731.705 −0.290789
\(186\) 0 0
\(187\) −1125.70 −0.440210
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 2136.00 0.809191 0.404596 0.914496i \(-0.367412\pi\)
0.404596 + 0.914496i \(0.367412\pi\)
\(192\) 0 0
\(193\) 1658.00 0.618370 0.309185 0.951002i \(-0.399944\pi\)
0.309185 + 0.951002i \(0.399944\pi\)
\(194\) 0 0
\(195\) 5778.59 2.12212
\(196\) 0 0
\(197\) −978.000 −0.353704 −0.176852 0.984237i \(-0.556591\pi\)
−0.176852 + 0.984237i \(0.556591\pi\)
\(198\) 0 0
\(199\) 4934.32 1.75771 0.878855 0.477088i \(-0.158308\pi\)
0.878855 + 0.477088i \(0.158308\pi\)
\(200\) 0 0
\(201\) −5440.88 −1.90930
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 3696.00 1.25922
\(206\) 0 0
\(207\) −2928.00 −0.983140
\(208\) 0 0
\(209\) 187.617 0.0620943
\(210\) 0 0
\(211\) −1556.00 −0.507675 −0.253838 0.967247i \(-0.581693\pi\)
−0.253838 + 0.967247i \(0.581693\pi\)
\(212\) 0 0
\(213\) 5103.17 1.64161
\(214\) 0 0
\(215\) −4090.04 −1.29739
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −5632.00 −1.73779
\(220\) 0 0
\(221\) 3696.00 1.12498
\(222\) 0 0
\(223\) −2889.30 −0.867630 −0.433815 0.901002i \(-0.642833\pi\)
−0.433815 + 0.901002i \(0.642833\pi\)
\(224\) 0 0
\(225\) −2257.00 −0.668741
\(226\) 0 0
\(227\) 1979.36 0.578742 0.289371 0.957217i \(-0.406554\pi\)
0.289371 + 0.957217i \(0.406554\pi\)
\(228\) 0 0
\(229\) 2767.35 0.798565 0.399282 0.916828i \(-0.369259\pi\)
0.399282 + 0.916828i \(0.369259\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −6490.00 −1.82478 −0.912391 0.409321i \(-0.865766\pi\)
−0.912391 + 0.409321i \(0.865766\pi\)
\(234\) 0 0
\(235\) −1936.00 −0.537407
\(236\) 0 0
\(237\) 6378.97 1.74835
\(238\) 0 0
\(239\) 4296.00 1.16270 0.581350 0.813654i \(-0.302525\pi\)
0.581350 + 0.813654i \(0.302525\pi\)
\(240\) 0 0
\(241\) 4521.56 1.20854 0.604272 0.796778i \(-0.293464\pi\)
0.604272 + 0.796778i \(0.293464\pi\)
\(242\) 0 0
\(243\) 4005.62 1.05745
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −616.000 −0.158685
\(248\) 0 0
\(249\) −1848.00 −0.470330
\(250\) 0 0
\(251\) −5581.59 −1.40361 −0.701807 0.712367i \(-0.747623\pi\)
−0.701807 + 0.712367i \(0.747623\pi\)
\(252\) 0 0
\(253\) 960.000 0.238556
\(254\) 0 0
\(255\) 4953.08 1.21637
\(256\) 0 0
\(257\) −1500.93 −0.364302 −0.182151 0.983271i \(-0.558306\pi\)
−0.182151 + 0.983271i \(0.558306\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −10126.0 −2.40147
\(262\) 0 0
\(263\) 400.000 0.0937835 0.0468917 0.998900i \(-0.485068\pi\)
0.0468917 + 0.998900i \(0.485068\pi\)
\(264\) 0 0
\(265\) 581.612 0.134823
\(266\) 0 0
\(267\) −14080.0 −3.22727
\(268\) 0 0
\(269\) −272.044 −0.0616610 −0.0308305 0.999525i \(-0.509815\pi\)
−0.0308305 + 0.999525i \(0.509815\pi\)
\(270\) 0 0
\(271\) −6904.29 −1.54762 −0.773812 0.633416i \(-0.781652\pi\)
−0.773812 + 0.633416i \(0.781652\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 740.000 0.162268
\(276\) 0 0
\(277\) −6770.00 −1.46848 −0.734242 0.678888i \(-0.762462\pi\)
−0.734242 + 0.678888i \(0.762462\pi\)
\(278\) 0 0
\(279\) 12589.1 2.70139
\(280\) 0 0
\(281\) 1878.00 0.398691 0.199345 0.979929i \(-0.436118\pi\)
0.199345 + 0.979929i \(0.436118\pi\)
\(282\) 0 0
\(283\) −384.614 −0.0807878 −0.0403939 0.999184i \(-0.512861\pi\)
−0.0403939 + 0.999184i \(0.512861\pi\)
\(284\) 0 0
\(285\) −825.513 −0.171576
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −1745.00 −0.355180
\(290\) 0 0
\(291\) −6160.00 −1.24091
\(292\) 0 0
\(293\) −3742.95 −0.746299 −0.373149 0.927771i \(-0.621722\pi\)
−0.373149 + 0.927771i \(0.621722\pi\)
\(294\) 0 0
\(295\) 6248.00 1.23313
\(296\) 0 0
\(297\) −6378.97 −1.24628
\(298\) 0 0
\(299\) −3151.96 −0.609641
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 1144.00 0.216901
\(304\) 0 0
\(305\) 2552.00 0.479105
\(306\) 0 0
\(307\) −722.324 −0.134284 −0.0671420 0.997743i \(-0.521388\pi\)
−0.0671420 + 0.997743i \(0.521388\pi\)
\(308\) 0 0
\(309\) 12848.0 2.36536
\(310\) 0 0
\(311\) −7279.53 −1.32728 −0.663640 0.748052i \(-0.730989\pi\)
−0.663640 + 0.748052i \(0.730989\pi\)
\(312\) 0 0
\(313\) 1519.69 0.274435 0.137218 0.990541i \(-0.456184\pi\)
0.137218 + 0.990541i \(0.456184\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 2358.00 0.417787 0.208893 0.977938i \(-0.433014\pi\)
0.208893 + 0.977938i \(0.433014\pi\)
\(318\) 0 0
\(319\) 3320.00 0.582709
\(320\) 0 0
\(321\) 2439.02 0.424089
\(322\) 0 0
\(323\) −528.000 −0.0909557
\(324\) 0 0
\(325\) −2429.64 −0.414683
\(326\) 0 0
\(327\) 17654.7 2.98565
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −2372.00 −0.393888 −0.196944 0.980415i \(-0.563102\pi\)
−0.196944 + 0.980415i \(0.563102\pi\)
\(332\) 0 0
\(333\) −4758.00 −0.782993
\(334\) 0 0
\(335\) −5440.88 −0.887365
\(336\) 0 0
\(337\) −250.000 −0.0404106 −0.0202053 0.999796i \(-0.506432\pi\)
−0.0202053 + 0.999796i \(0.506432\pi\)
\(338\) 0 0
\(339\) −12063.7 −1.93278
\(340\) 0 0
\(341\) −4127.57 −0.655485
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −4224.00 −0.659167
\(346\) 0 0
\(347\) −9540.00 −1.47589 −0.737945 0.674861i \(-0.764204\pi\)
−0.737945 + 0.674861i \(0.764204\pi\)
\(348\) 0 0
\(349\) −5712.93 −0.876235 −0.438117 0.898918i \(-0.644355\pi\)
−0.438117 + 0.898918i \(0.644355\pi\)
\(350\) 0 0
\(351\) 20944.0 3.18492
\(352\) 0 0
\(353\) 4390.23 0.661950 0.330975 0.943640i \(-0.392622\pi\)
0.330975 + 0.943640i \(0.392622\pi\)
\(354\) 0 0
\(355\) 5103.17 0.762953
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −1840.00 −0.270506 −0.135253 0.990811i \(-0.543185\pi\)
−0.135253 + 0.990811i \(0.543185\pi\)
\(360\) 0 0
\(361\) −6771.00 −0.987170
\(362\) 0 0
\(363\) −8733.55 −1.26279
\(364\) 0 0
\(365\) −5632.00 −0.807650
\(366\) 0 0
\(367\) 2964.34 0.421628 0.210814 0.977526i \(-0.432389\pi\)
0.210814 + 0.977526i \(0.432389\pi\)
\(368\) 0 0
\(369\) 24033.7 3.39063
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 3982.00 0.552762 0.276381 0.961048i \(-0.410865\pi\)
0.276381 + 0.961048i \(0.410865\pi\)
\(374\) 0 0
\(375\) −14256.0 −1.96314
\(376\) 0 0
\(377\) −10900.5 −1.48914
\(378\) 0 0
\(379\) −2676.00 −0.362683 −0.181342 0.983420i \(-0.558044\pi\)
−0.181342 + 0.983420i \(0.558044\pi\)
\(380\) 0 0
\(381\) −21688.5 −2.91636
\(382\) 0 0
\(383\) −7035.62 −0.938652 −0.469326 0.883025i \(-0.655503\pi\)
−0.469326 + 0.883025i \(0.655503\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −26596.0 −3.49341
\(388\) 0 0
\(389\) 8658.00 1.12848 0.564239 0.825611i \(-0.309170\pi\)
0.564239 + 0.825611i \(0.309170\pi\)
\(390\) 0 0
\(391\) −2701.68 −0.349437
\(392\) 0 0
\(393\) 2376.00 0.304970
\(394\) 0 0
\(395\) 6378.97 0.812558
\(396\) 0 0
\(397\) −9052.50 −1.14441 −0.572207 0.820109i \(-0.693912\pi\)
−0.572207 + 0.820109i \(0.693912\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −5706.00 −0.710584 −0.355292 0.934755i \(-0.615619\pi\)
−0.355292 + 0.934755i \(0.615619\pi\)
\(402\) 0 0
\(403\) 13552.0 1.67512
\(404\) 0 0
\(405\) 12617.2 1.54804
\(406\) 0 0
\(407\) 1560.00 0.189991
\(408\) 0 0
\(409\) 2420.25 0.292601 0.146301 0.989240i \(-0.453263\pi\)
0.146301 + 0.989240i \(0.453263\pi\)
\(410\) 0 0
\(411\) −10450.2 −1.25419
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −1848.00 −0.218590
\(416\) 0 0
\(417\) −12936.0 −1.51913
\(418\) 0 0
\(419\) 1510.31 0.176095 0.0880473 0.996116i \(-0.471937\pi\)
0.0880473 + 0.996116i \(0.471937\pi\)
\(420\) 0 0
\(421\) −16770.0 −1.94138 −0.970689 0.240341i \(-0.922741\pi\)
−0.970689 + 0.240341i \(0.922741\pi\)
\(422\) 0 0
\(423\) −12589.1 −1.44705
\(424\) 0 0
\(425\) −2082.54 −0.237690
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −12320.0 −1.38652
\(430\) 0 0
\(431\) −1336.00 −0.149311 −0.0746553 0.997209i \(-0.523786\pi\)
−0.0746553 + 0.997209i \(0.523786\pi\)
\(432\) 0 0
\(433\) −11163.2 −1.23896 −0.619479 0.785013i \(-0.712656\pi\)
−0.619479 + 0.785013i \(0.712656\pi\)
\(434\) 0 0
\(435\) −14608.0 −1.61011
\(436\) 0 0
\(437\) 450.280 0.0492902
\(438\) 0 0
\(439\) 3602.24 0.391630 0.195815 0.980641i \(-0.437265\pi\)
0.195815 + 0.980641i \(0.437265\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −6348.00 −0.680818 −0.340409 0.940277i \(-0.610566\pi\)
−0.340409 + 0.940277i \(0.610566\pi\)
\(444\) 0 0
\(445\) −14080.0 −1.49990
\(446\) 0 0
\(447\) −8874.27 −0.939012
\(448\) 0 0
\(449\) 7170.00 0.753615 0.376808 0.926292i \(-0.377022\pi\)
0.376808 + 0.926292i \(0.377022\pi\)
\(450\) 0 0
\(451\) −7879.90 −0.822727
\(452\) 0 0
\(453\) −7804.85 −0.809501
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 6866.00 0.702796 0.351398 0.936226i \(-0.385706\pi\)
0.351398 + 0.936226i \(0.385706\pi\)
\(458\) 0 0
\(459\) 17952.0 1.82555
\(460\) 0 0
\(461\) 1378.98 0.139318 0.0696590 0.997571i \(-0.477809\pi\)
0.0696590 + 0.997571i \(0.477809\pi\)
\(462\) 0 0
\(463\) −2648.00 −0.265795 −0.132897 0.991130i \(-0.542428\pi\)
−0.132897 + 0.991130i \(0.542428\pi\)
\(464\) 0 0
\(465\) 18161.3 1.81120
\(466\) 0 0
\(467\) 12335.8 1.22234 0.611170 0.791500i \(-0.290699\pi\)
0.611170 + 0.791500i \(0.290699\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 27016.0 2.64295
\(472\) 0 0
\(473\) 8720.00 0.847666
\(474\) 0 0
\(475\) 347.091 0.0335276
\(476\) 0 0
\(477\) 3782.00 0.363031
\(478\) 0 0
\(479\) 13339.5 1.27244 0.636221 0.771507i \(-0.280497\pi\)
0.636221 + 0.771507i \(0.280497\pi\)
\(480\) 0 0
\(481\) −5121.93 −0.485530
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −6160.00 −0.576724
\(486\) 0 0
\(487\) −13936.0 −1.29672 −0.648358 0.761336i \(-0.724544\pi\)
−0.648358 + 0.761336i \(0.724544\pi\)
\(488\) 0 0
\(489\) −5966.21 −0.551741
\(490\) 0 0
\(491\) 12276.0 1.12833 0.564163 0.825663i \(-0.309199\pi\)
0.564163 + 0.825663i \(0.309199\pi\)
\(492\) 0 0
\(493\) −9343.31 −0.853553
\(494\) 0 0
\(495\) −11444.6 −1.03919
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 2220.00 0.199160 0.0995800 0.995030i \(-0.468250\pi\)
0.0995800 + 0.995030i \(0.468250\pi\)
\(500\) 0 0
\(501\) 6160.00 0.549318
\(502\) 0 0
\(503\) 11294.5 1.00119 0.500594 0.865682i \(-0.333115\pi\)
0.500594 + 0.865682i \(0.333115\pi\)
\(504\) 0 0
\(505\) 1144.00 0.100807
\(506\) 0 0
\(507\) 19840.5 1.73796
\(508\) 0 0
\(509\) −15881.7 −1.38300 −0.691499 0.722377i \(-0.743049\pi\)
−0.691499 + 0.722377i \(0.743049\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −2992.00 −0.257505
\(514\) 0 0
\(515\) 12848.0 1.09932
\(516\) 0 0
\(517\) 4127.57 0.351122
\(518\) 0 0
\(519\) 6248.00 0.528433
\(520\) 0 0
\(521\) 11613.5 0.976575 0.488287 0.872683i \(-0.337622\pi\)
0.488287 + 0.872683i \(0.337622\pi\)
\(522\) 0 0
\(523\) −12617.2 −1.05490 −0.527450 0.849586i \(-0.676852\pi\)
−0.527450 + 0.849586i \(0.676852\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 11616.0 0.960154
\(528\) 0 0
\(529\) −9863.00 −0.810635
\(530\) 0 0
\(531\) 40628.4 3.32038
\(532\) 0 0
\(533\) 25872.0 2.10252
\(534\) 0 0
\(535\) 2439.02 0.197099
\(536\) 0 0
\(537\) 30281.3 2.43340
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 1798.00 0.142887 0.0714437 0.997445i \(-0.477239\pi\)
0.0714437 + 0.997445i \(0.477239\pi\)
\(542\) 0 0
\(543\) −26488.0 −2.09339
\(544\) 0 0
\(545\) 17654.7 1.38761
\(546\) 0 0
\(547\) −1276.00 −0.0997401 −0.0498700 0.998756i \(-0.515881\pi\)
−0.0498700 + 0.998756i \(0.515881\pi\)
\(548\) 0 0
\(549\) 16594.7 1.29006
\(550\) 0 0
\(551\) 1557.22 0.120399
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −6864.00 −0.524974
\(556\) 0 0
\(557\) 2694.00 0.204934 0.102467 0.994736i \(-0.467326\pi\)
0.102467 + 0.994736i \(0.467326\pi\)
\(558\) 0 0
\(559\) −28630.3 −2.16625
\(560\) 0 0
\(561\) −10560.0 −0.794730
\(562\) 0 0
\(563\) 15769.2 1.18045 0.590223 0.807240i \(-0.299040\pi\)
0.590223 + 0.807240i \(0.299040\pi\)
\(564\) 0 0
\(565\) −12063.7 −0.898276
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 12606.0 0.928772 0.464386 0.885633i \(-0.346275\pi\)
0.464386 + 0.885633i \(0.346275\pi\)
\(570\) 0 0
\(571\) −6852.00 −0.502184 −0.251092 0.967963i \(-0.580790\pi\)
−0.251092 + 0.967963i \(0.580790\pi\)
\(572\) 0 0
\(573\) 20037.5 1.46087
\(574\) 0 0
\(575\) 1776.00 0.128808
\(576\) 0 0
\(577\) 14371.4 1.03690 0.518449 0.855108i \(-0.326509\pi\)
0.518449 + 0.855108i \(0.326509\pi\)
\(578\) 0 0
\(579\) 15553.4 1.11637
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −1240.00 −0.0880884
\(584\) 0 0
\(585\) 37576.0 2.65569
\(586\) 0 0
\(587\) −18977.4 −1.33438 −0.667191 0.744887i \(-0.732503\pi\)
−0.667191 + 0.744887i \(0.732503\pi\)
\(588\) 0 0
\(589\) −1936.00 −0.135435
\(590\) 0 0
\(591\) −9174.45 −0.638556
\(592\) 0 0
\(593\) 8217.61 0.569067 0.284534 0.958666i \(-0.408161\pi\)
0.284534 + 0.958666i \(0.408161\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 46288.0 3.17327
\(598\) 0 0
\(599\) 19104.0 1.30312 0.651559 0.758598i \(-0.274115\pi\)
0.651559 + 0.758598i \(0.274115\pi\)
\(600\) 0 0
\(601\) 21538.4 1.46185 0.730923 0.682460i \(-0.239090\pi\)
0.730923 + 0.682460i \(0.239090\pi\)
\(602\) 0 0
\(603\) −35380.0 −2.38936
\(604\) 0 0
\(605\) −8733.55 −0.586892
\(606\) 0 0
\(607\) −13733.5 −0.918331 −0.459166 0.888351i \(-0.651852\pi\)
−0.459166 + 0.888351i \(0.651852\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −13552.0 −0.897308
\(612\) 0 0
\(613\) 28034.0 1.84712 0.923558 0.383458i \(-0.125267\pi\)
0.923558 + 0.383458i \(0.125267\pi\)
\(614\) 0 0
\(615\) 34671.6 2.27332
\(616\) 0 0
\(617\) −8258.00 −0.538824 −0.269412 0.963025i \(-0.586829\pi\)
−0.269412 + 0.963025i \(0.586829\pi\)
\(618\) 0 0
\(619\) 5131.31 0.333191 0.166595 0.986025i \(-0.446723\pi\)
0.166595 + 0.986025i \(0.446723\pi\)
\(620\) 0 0
\(621\) −15309.5 −0.989291
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −9631.00 −0.616384
\(626\) 0 0
\(627\) 1760.00 0.112101
\(628\) 0 0
\(629\) −4390.23 −0.278299
\(630\) 0 0
\(631\) −912.000 −0.0575375 −0.0287687 0.999586i \(-0.509159\pi\)
−0.0287687 + 0.999586i \(0.509159\pi\)
\(632\) 0 0
\(633\) −14596.6 −0.916527
\(634\) 0 0
\(635\) −21688.5 −1.35540
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 33184.0 2.05436
\(640\) 0 0
\(641\) −890.000 −0.0548407 −0.0274203 0.999624i \(-0.508729\pi\)
−0.0274203 + 0.999624i \(0.508729\pi\)
\(642\) 0 0
\(643\) 29352.6 1.80024 0.900120 0.435642i \(-0.143479\pi\)
0.900120 + 0.435642i \(0.143479\pi\)
\(644\) 0 0
\(645\) −38368.0 −2.34223
\(646\) 0 0
\(647\) −11876.1 −0.721637 −0.360818 0.932636i \(-0.617503\pi\)
−0.360818 + 0.932636i \(0.617503\pi\)
\(648\) 0 0
\(649\) −13320.8 −0.805680
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −21526.0 −1.29001 −0.645006 0.764178i \(-0.723145\pi\)
−0.645006 + 0.764178i \(0.723145\pi\)
\(654\) 0 0
\(655\) 2376.00 0.141737
\(656\) 0 0
\(657\) −36622.8 −2.17472
\(658\) 0 0
\(659\) −23452.0 −1.38628 −0.693141 0.720802i \(-0.743774\pi\)
−0.693141 + 0.720802i \(0.743774\pi\)
\(660\) 0 0
\(661\) −26669.7 −1.56934 −0.784668 0.619916i \(-0.787167\pi\)
−0.784668 + 0.619916i \(0.787167\pi\)
\(662\) 0 0
\(663\) 34671.6 2.03097
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 7968.00 0.462552
\(668\) 0 0
\(669\) −27104.0 −1.56637
\(670\) 0 0
\(671\) −5440.88 −0.313030
\(672\) 0 0
\(673\) −13858.0 −0.793739 −0.396870 0.917875i \(-0.629904\pi\)
−0.396870 + 0.917875i \(0.629904\pi\)
\(674\) 0 0
\(675\) −11801.1 −0.672924
\(676\) 0 0
\(677\) 32448.3 1.84208 0.921041 0.389466i \(-0.127340\pi\)
0.921041 + 0.389466i \(0.127340\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 18568.0 1.04483
\(682\) 0 0
\(683\) 27812.0 1.55812 0.779060 0.626949i \(-0.215696\pi\)
0.779060 + 0.626949i \(0.215696\pi\)
\(684\) 0 0
\(685\) −10450.2 −0.582895
\(686\) 0 0
\(687\) 25960.0 1.44168
\(688\) 0 0
\(689\) 4071.28 0.225114
\(690\) 0 0
\(691\) 1303.94 0.0717859 0.0358929 0.999356i \(-0.488572\pi\)
0.0358929 + 0.999356i \(0.488572\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −12936.0 −0.706029
\(696\) 0 0
\(697\) 22176.0 1.20513
\(698\) 0 0
\(699\) −60881.6 −3.29435
\(700\) 0 0
\(701\) 22906.0 1.23416 0.617081 0.786900i \(-0.288315\pi\)
0.617081 + 0.786900i \(0.288315\pi\)
\(702\) 0 0
\(703\) 731.705 0.0392557
\(704\) 0 0
\(705\) −18161.3 −0.970204
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −15086.0 −0.799107 −0.399553 0.916710i \(-0.630835\pi\)
−0.399553 + 0.916710i \(0.630835\pi\)
\(710\) 0 0
\(711\) 41480.0 2.18793
\(712\) 0 0
\(713\) −9906.16 −0.520321
\(714\) 0 0
\(715\) −12320.0 −0.644394
\(716\) 0 0
\(717\) 40300.1 2.09907
\(718\) 0 0
\(719\) 20544.0 1.06559 0.532797 0.846243i \(-0.321141\pi\)
0.532797 + 0.846243i \(0.321141\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 42416.0 2.18184
\(724\) 0 0
\(725\) 6142.00 0.314632
\(726\) 0 0
\(727\) 7223.24 0.368494 0.184247 0.982880i \(-0.441015\pi\)
0.184247 + 0.982880i \(0.441015\pi\)
\(728\) 0 0
\(729\) 1261.00 0.0640654
\(730\) 0 0
\(731\) −24540.3 −1.24166
\(732\) 0 0
\(733\) 29427.7 1.48286 0.741430 0.671031i \(-0.234148\pi\)
0.741430 + 0.671031i \(0.234148\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 11600.0 0.579771
\(738\) 0 0
\(739\) −32668.0 −1.62613 −0.813066 0.582171i \(-0.802203\pi\)
−0.813066 + 0.582171i \(0.802203\pi\)
\(740\) 0 0
\(741\) −5778.59 −0.286480
\(742\) 0 0
\(743\) 37056.0 1.82968 0.914840 0.403816i \(-0.132316\pi\)
0.914840 + 0.403816i \(0.132316\pi\)
\(744\) 0 0
\(745\) −8874.27 −0.436413
\(746\) 0 0
\(747\) −12016.8 −0.588586
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 19608.0 0.952738 0.476369 0.879246i \(-0.341953\pi\)
0.476369 + 0.879246i \(0.341953\pi\)
\(752\) 0 0
\(753\) −52360.0 −2.53400
\(754\) 0 0
\(755\) −7804.85 −0.376222
\(756\) 0 0
\(757\) 19378.0 0.930390 0.465195 0.885208i \(-0.345984\pi\)
0.465195 + 0.885208i \(0.345984\pi\)
\(758\) 0 0
\(759\) 9005.60 0.430675
\(760\) 0 0
\(761\) −13977.4 −0.665810 −0.332905 0.942960i \(-0.608029\pi\)
−0.332905 + 0.942960i \(0.608029\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 32208.0 1.52220
\(766\) 0 0
\(767\) 43736.0 2.05895
\(768\) 0 0
\(769\) 8536.56 0.400307 0.200154 0.979765i \(-0.435856\pi\)
0.200154 + 0.979765i \(0.435856\pi\)
\(770\) 0 0
\(771\) −14080.0 −0.657690
\(772\) 0 0
\(773\) −29296.3 −1.36315 −0.681576 0.731748i \(-0.738705\pi\)
−0.681576 + 0.731748i \(0.738705\pi\)
\(774\) 0 0
\(775\) −7636.00 −0.353927
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −3696.00 −0.169991
\(780\) 0 0
\(781\) −10880.0 −0.498485
\(782\) 0 0
\(783\) −52945.4 −2.41649
\(784\) 0 0
\(785\) 27016.0 1.22833
\(786\) 0 0
\(787\) −13780.4 −0.624167 −0.312084 0.950055i \(-0.601027\pi\)
−0.312084 + 0.950055i \(0.601027\pi\)
\(788\) 0 0
\(789\) 3752.33 0.169311
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 17864.0 0.799961
\(794\) 0 0
\(795\) 5456.00 0.243402
\(796\) 0 0
\(797\) −34868.6 −1.54970 −0.774848 0.632148i \(-0.782174\pi\)
−0.774848 + 0.632148i \(0.782174\pi\)
\(798\) 0 0
\(799\) −11616.0 −0.514324
\(800\) 0 0
\(801\) −91556.9 −4.03871
\(802\) 0 0
\(803\) 12007.5 0.527689
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −2552.00 −0.111319
\(808\) 0 0
\(809\) 14034.0 0.609900 0.304950 0.952368i \(-0.401360\pi\)
0.304950 + 0.952368i \(0.401360\pi\)
\(810\) 0 0
\(811\) 6632.25 0.287164 0.143582 0.989638i \(-0.454138\pi\)
0.143582 + 0.989638i \(0.454138\pi\)
\(812\) 0 0
\(813\) −64768.0 −2.79399
\(814\) 0 0
\(815\) −5966.21 −0.256426
\(816\) 0 0
\(817\) 4090.04 0.175144
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 28622.0 1.21670 0.608352 0.793667i \(-0.291831\pi\)
0.608352 + 0.793667i \(0.291831\pi\)
\(822\) 0 0
\(823\) −24688.0 −1.04565 −0.522825 0.852440i \(-0.675122\pi\)
−0.522825 + 0.852440i \(0.675122\pi\)
\(824\) 0 0
\(825\) 6941.82 0.292949
\(826\) 0 0
\(827\) 30756.0 1.29322 0.646609 0.762822i \(-0.276187\pi\)
0.646609 + 0.762822i \(0.276187\pi\)
\(828\) 0 0
\(829\) 23236.3 0.973499 0.486750 0.873542i \(-0.338182\pi\)
0.486750 + 0.873542i \(0.338182\pi\)
\(830\) 0 0
\(831\) −63508.2 −2.65111
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 6160.00 0.255300
\(836\) 0 0
\(837\) 65824.0 2.71829
\(838\) 0 0
\(839\) 24033.7 0.988957 0.494479 0.869190i \(-0.335359\pi\)
0.494479 + 0.869190i \(0.335359\pi\)
\(840\) 0 0
\(841\) 3167.00 0.129854
\(842\) 0 0
\(843\) 17617.2 0.719773
\(844\) 0 0
\(845\) 19840.5 0.807731
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −3608.00 −0.145850
\(850\) 0 0
\(851\) 3744.00 0.150814
\(852\) 0 0
\(853\) −23574.0 −0.946260 −0.473130 0.880993i \(-0.656876\pi\)
−0.473130 + 0.880993i \(0.656876\pi\)
\(854\) 0 0
\(855\) −5368.00 −0.214715
\(856\) 0 0
\(857\) 24484.0 0.975912 0.487956 0.872868i \(-0.337743\pi\)
0.487956 + 0.872868i \(0.337743\pi\)
\(858\) 0 0
\(859\) 32954.9 1.30897 0.654485 0.756075i \(-0.272885\pi\)
0.654485 + 0.756075i \(0.272885\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −40872.0 −1.61217 −0.806083 0.591803i \(-0.798416\pi\)
−0.806083 + 0.591803i \(0.798416\pi\)
\(864\) 0 0
\(865\) 6248.00 0.245593
\(866\) 0 0
\(867\) −16369.6 −0.641222
\(868\) 0 0
\(869\) −13600.0 −0.530896
\(870\) 0 0
\(871\) −38086.2 −1.48163
\(872\) 0 0
\(873\) −40056.2 −1.55292
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −12006.0 −0.462273 −0.231137 0.972921i \(-0.574244\pi\)
−0.231137 + 0.972921i \(0.574244\pi\)
\(878\) 0 0
\(879\) −35112.0 −1.34732
\(880\) 0 0
\(881\) 35722.2 1.36607 0.683037 0.730383i \(-0.260659\pi\)
0.683037 + 0.730383i \(0.260659\pi\)
\(882\) 0 0
\(883\) −19588.0 −0.746533 −0.373267 0.927724i \(-0.621762\pi\)
−0.373267 + 0.927724i \(0.621762\pi\)
\(884\) 0 0
\(885\) 58611.4 2.22622
\(886\) 0 0
\(887\) 40243.8 1.52340 0.761699 0.647931i \(-0.224366\pi\)
0.761699 + 0.647931i \(0.224366\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −26900.0 −1.01143
\(892\) 0 0
\(893\) 1936.00 0.0725485
\(894\) 0 0
\(895\) 30281.3 1.13094
\(896\) 0 0
\(897\) −29568.0 −1.10061
\(898\) 0 0
\(899\) −34258.8 −1.27096
\(900\) 0 0
\(901\) 3489.67 0.129032
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −26488.0 −0.972918
\(906\) 0 0
\(907\) −15868.0 −0.580913 −0.290457 0.956888i \(-0.593807\pi\)
−0.290457 + 0.956888i \(0.593807\pi\)
\(908\) 0 0
\(909\) 7439.00 0.271437
\(910\) 0 0
\(911\) −39832.0 −1.44862 −0.724310 0.689474i \(-0.757842\pi\)
−0.724310 + 0.689474i \(0.757842\pi\)
\(912\) 0 0
\(913\) 3939.95 0.142818
\(914\) 0 0
\(915\) 23939.9 0.864949
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 30528.0 1.09578 0.547892 0.836549i \(-0.315430\pi\)
0.547892 + 0.836549i \(0.315430\pi\)
\(920\) 0 0
\(921\) −6776.00 −0.242429
\(922\) 0 0
\(923\) 35722.2 1.27390
\(924\) 0 0
\(925\) 2886.00 0.102585
\(926\) 0 0
\(927\) 83545.7 2.96009
\(928\) 0 0
\(929\) −16604.1 −0.586396 −0.293198 0.956052i \(-0.594720\pi\)
−0.293198 + 0.956052i \(0.594720\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −68288.0 −2.39619
\(934\) 0 0
\(935\) −10560.0 −0.369357
\(936\) 0 0
\(937\) 29943.6 1.04399 0.521993 0.852950i \(-0.325189\pi\)
0.521993 + 0.852950i \(0.325189\pi\)
\(938\) 0 0
\(939\) 14256.0 0.495449
\(940\) 0 0
\(941\) 5375.22 0.186214 0.0931068 0.995656i \(-0.470320\pi\)
0.0931068 + 0.995656i \(0.470320\pi\)
\(942\) 0 0
\(943\) −18911.8 −0.653077
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −45212.0 −1.55142 −0.775709 0.631091i \(-0.782607\pi\)
−0.775709 + 0.631091i \(0.782607\pi\)
\(948\) 0 0
\(949\) −39424.0 −1.34853
\(950\) 0 0
\(951\) 22120.0 0.754248
\(952\) 0 0
\(953\) 34218.0 1.16310 0.581548 0.813512i \(-0.302447\pi\)
0.581548 + 0.813512i \(0.302447\pi\)
\(954\) 0 0
\(955\) 20037.5 0.678950
\(956\) 0 0
\(957\) 31144.4 1.05199
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 12801.0 0.429694
\(962\) 0 0
\(963\) 15860.0 0.530718
\(964\) 0 0
\(965\) 15553.4 0.518842
\(966\) 0 0
\(967\) −14464.0 −0.481004 −0.240502 0.970649i \(-0.577312\pi\)
−0.240502 + 0.970649i \(0.577312\pi\)
\(968\) 0 0
\(969\) −4953.08 −0.164206
\(970\) 0 0
\(971\) −37832.9 −1.25038 −0.625188 0.780474i \(-0.714978\pi\)
−0.625188 + 0.780474i \(0.714978\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −22792.0 −0.748644
\(976\) 0 0
\(977\) 42062.0 1.37736 0.688681 0.725065i \(-0.258190\pi\)
0.688681 + 0.725065i \(0.258190\pi\)
\(978\) 0 0
\(979\) 30018.7 0.979980
\(980\) 0 0
\(981\) 114802. 3.73634
\(982\) 0 0
\(983\) 43020.5 1.39587 0.697935 0.716161i \(-0.254102\pi\)
0.697935 + 0.716161i \(0.254102\pi\)
\(984\) 0 0
\(985\) −9174.45 −0.296774
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 20928.0 0.672873
\(990\) 0 0
\(991\) −21272.0 −0.681864 −0.340932 0.940088i \(-0.610743\pi\)
−0.340932 + 0.940088i \(0.610743\pi\)
\(992\) 0 0
\(993\) −22251.3 −0.711102
\(994\) 0 0
\(995\) 46288.0 1.47480
\(996\) 0 0
\(997\) 121.951 0.00387384 0.00193692 0.999998i \(-0.499383\pi\)
0.00193692 + 0.999998i \(0.499383\pi\)
\(998\) 0 0
\(999\) −24878.0 −0.787892
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 784.4.a.z.1.2 2
4.3 odd 2 98.4.a.h.1.1 2
7.6 odd 2 inner 784.4.a.z.1.1 2
12.11 even 2 882.4.a.w.1.1 2
20.19 odd 2 2450.4.a.bs.1.2 2
28.3 even 6 98.4.c.g.79.1 4
28.11 odd 6 98.4.c.g.79.2 4
28.19 even 6 98.4.c.g.67.1 4
28.23 odd 6 98.4.c.g.67.2 4
28.27 even 2 98.4.a.h.1.2 yes 2
84.11 even 6 882.4.g.bi.667.2 4
84.23 even 6 882.4.g.bi.361.2 4
84.47 odd 6 882.4.g.bi.361.1 4
84.59 odd 6 882.4.g.bi.667.1 4
84.83 odd 2 882.4.a.w.1.2 2
140.139 even 2 2450.4.a.bs.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
98.4.a.h.1.1 2 4.3 odd 2
98.4.a.h.1.2 yes 2 28.27 even 2
98.4.c.g.67.1 4 28.19 even 6
98.4.c.g.67.2 4 28.23 odd 6
98.4.c.g.79.1 4 28.3 even 6
98.4.c.g.79.2 4 28.11 odd 6
784.4.a.z.1.1 2 7.6 odd 2 inner
784.4.a.z.1.2 2 1.1 even 1 trivial
882.4.a.w.1.1 2 12.11 even 2
882.4.a.w.1.2 2 84.83 odd 2
882.4.g.bi.361.1 4 84.47 odd 6
882.4.g.bi.361.2 4 84.23 even 6
882.4.g.bi.667.1 4 84.59 odd 6
882.4.g.bi.667.2 4 84.11 even 6
2450.4.a.bs.1.1 2 140.139 even 2
2450.4.a.bs.1.2 2 20.19 odd 2