Properties

Label 768.4.a.q.1.1
Level $768$
Weight $4$
Character 768.1
Self dual yes
Analytic conductor $45.313$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 768.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(45.3134668844\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.1436.1
Defining polynomial: \( x^{3} - 11x - 12 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 24)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.48361\) of defining polynomial
Character \(\chi\) \(=\) 768.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.00000 q^{3} -18.5422 q^{5} +9.32669 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q-3.00000 q^{3} -18.5422 q^{5} +9.32669 q^{7} +9.00000 q^{9} -39.7378 q^{11} +32.9533 q^{13} +55.6266 q^{15} +90.5998 q^{17} +72.5998 q^{19} -27.9801 q^{21} +45.3466 q^{23} +218.813 q^{25} -27.0000 q^{27} -143.364 q^{29} -90.4865 q^{31} +119.213 q^{33} -172.937 q^{35} -1.77977 q^{37} -98.8599 q^{39} -195.827 q^{41} +407.027 q^{43} -166.880 q^{45} +278.467 q^{47} -256.013 q^{49} -271.799 q^{51} -241.303 q^{53} +736.826 q^{55} -217.799 q^{57} -149.724 q^{59} -508.314 q^{61} +83.9402 q^{63} -611.027 q^{65} +950.026 q^{67} -136.040 q^{69} -803.559 q^{71} -449.786 q^{73} -656.440 q^{75} -370.622 q^{77} +157.220 q^{79} +81.0000 q^{81} -175.063 q^{83} -1679.92 q^{85} +430.093 q^{87} -127.200 q^{89} +307.345 q^{91} +271.459 q^{93} -1346.16 q^{95} +158.826 q^{97} -357.640 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 9 q^{3} - 10 q^{5} + 14 q^{7} + 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 9 q^{3} - 10 q^{5} + 14 q^{7} + 27 q^{9} - 52 q^{13} + 30 q^{15} + 26 q^{17} - 28 q^{19} - 42 q^{21} + 164 q^{23} + 53 q^{25} - 81 q^{27} - 174 q^{29} + 318 q^{31} + 92 q^{35} - 296 q^{37} + 156 q^{39} - 118 q^{41} + 260 q^{43} - 90 q^{45} + 204 q^{47} + 327 q^{49} - 78 q^{51} - 1086 q^{53} + 512 q^{55} + 84 q^{57} - 196 q^{59} - 1536 q^{61} + 126 q^{63} - 872 q^{65} + 660 q^{67} - 492 q^{69} - 852 q^{71} - 478 q^{73} - 159 q^{75} - 2304 q^{77} + 22 q^{79} + 243 q^{81} - 1136 q^{83} - 2732 q^{85} + 522 q^{87} + 110 q^{89} + 632 q^{91} - 954 q^{93} - 2552 q^{95} - 1222 q^{97}+O(q^{100}) \) 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
\(3\) −3.00000 −0.577350
\(4\) 0 0
\(5\) −18.5422 −1.65846 −0.829232 0.558904i \(-0.811222\pi\)
−0.829232 + 0.558904i \(0.811222\pi\)
\(6\) 0 0
\(7\) 9.32669 0.503594 0.251797 0.967780i \(-0.418978\pi\)
0.251797 + 0.967780i \(0.418978\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −39.7378 −1.08922 −0.544609 0.838690i \(-0.683322\pi\)
−0.544609 + 0.838690i \(0.683322\pi\)
\(12\) 0 0
\(13\) 32.9533 0.703046 0.351523 0.936179i \(-0.385664\pi\)
0.351523 + 0.936179i \(0.385664\pi\)
\(14\) 0 0
\(15\) 55.6266 0.957515
\(16\) 0 0
\(17\) 90.5998 1.29257 0.646285 0.763096i \(-0.276322\pi\)
0.646285 + 0.763096i \(0.276322\pi\)
\(18\) 0 0
\(19\) 72.5998 0.876607 0.438304 0.898827i \(-0.355579\pi\)
0.438304 + 0.898827i \(0.355579\pi\)
\(20\) 0 0
\(21\) −27.9801 −0.290750
\(22\) 0 0
\(23\) 45.3466 0.411105 0.205553 0.978646i \(-0.434101\pi\)
0.205553 + 0.978646i \(0.434101\pi\)
\(24\) 0 0
\(25\) 218.813 1.75051
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) −143.364 −0.918003 −0.459002 0.888435i \(-0.651793\pi\)
−0.459002 + 0.888435i \(0.651793\pi\)
\(30\) 0 0
\(31\) −90.4865 −0.524253 −0.262127 0.965033i \(-0.584424\pi\)
−0.262127 + 0.965033i \(0.584424\pi\)
\(32\) 0 0
\(33\) 119.213 0.628860
\(34\) 0 0
\(35\) −172.937 −0.835193
\(36\) 0 0
\(37\) −1.77977 −0.00790792 −0.00395396 0.999992i \(-0.501259\pi\)
−0.00395396 + 0.999992i \(0.501259\pi\)
\(38\) 0 0
\(39\) −98.8599 −0.405904
\(40\) 0 0
\(41\) −195.827 −0.745927 −0.372964 0.927846i \(-0.621658\pi\)
−0.372964 + 0.927846i \(0.621658\pi\)
\(42\) 0 0
\(43\) 407.027 1.44351 0.721755 0.692148i \(-0.243336\pi\)
0.721755 + 0.692148i \(0.243336\pi\)
\(44\) 0 0
\(45\) −166.880 −0.552822
\(46\) 0 0
\(47\) 278.467 0.864224 0.432112 0.901820i \(-0.357769\pi\)
0.432112 + 0.901820i \(0.357769\pi\)
\(48\) 0 0
\(49\) −256.013 −0.746393
\(50\) 0 0
\(51\) −271.799 −0.746265
\(52\) 0 0
\(53\) −241.303 −0.625386 −0.312693 0.949854i \(-0.601231\pi\)
−0.312693 + 0.949854i \(0.601231\pi\)
\(54\) 0 0
\(55\) 736.826 1.80643
\(56\) 0 0
\(57\) −217.799 −0.506109
\(58\) 0 0
\(59\) −149.724 −0.330380 −0.165190 0.986262i \(-0.552824\pi\)
−0.165190 + 0.986262i \(0.552824\pi\)
\(60\) 0 0
\(61\) −508.314 −1.06693 −0.533466 0.845821i \(-0.679111\pi\)
−0.533466 + 0.845821i \(0.679111\pi\)
\(62\) 0 0
\(63\) 83.9402 0.167865
\(64\) 0 0
\(65\) −611.027 −1.16598
\(66\) 0 0
\(67\) 950.026 1.73230 0.866150 0.499784i \(-0.166587\pi\)
0.866150 + 0.499784i \(0.166587\pi\)
\(68\) 0 0
\(69\) −136.040 −0.237352
\(70\) 0 0
\(71\) −803.559 −1.34317 −0.671584 0.740929i \(-0.734386\pi\)
−0.671584 + 0.740929i \(0.734386\pi\)
\(72\) 0 0
\(73\) −449.786 −0.721143 −0.360571 0.932732i \(-0.617418\pi\)
−0.360571 + 0.932732i \(0.617418\pi\)
\(74\) 0 0
\(75\) −656.440 −1.01065
\(76\) 0 0
\(77\) −370.622 −0.548524
\(78\) 0 0
\(79\) 157.220 0.223906 0.111953 0.993713i \(-0.464289\pi\)
0.111953 + 0.993713i \(0.464289\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) −175.063 −0.231513 −0.115757 0.993278i \(-0.536929\pi\)
−0.115757 + 0.993278i \(0.536929\pi\)
\(84\) 0 0
\(85\) −1679.92 −2.14368
\(86\) 0 0
\(87\) 430.093 0.530009
\(88\) 0 0
\(89\) −127.200 −0.151496 −0.0757479 0.997127i \(-0.524134\pi\)
−0.0757479 + 0.997127i \(0.524134\pi\)
\(90\) 0 0
\(91\) 307.345 0.354050
\(92\) 0 0
\(93\) 271.459 0.302678
\(94\) 0 0
\(95\) −1346.16 −1.45382
\(96\) 0 0
\(97\) 158.826 0.166251 0.0831254 0.996539i \(-0.473510\pi\)
0.0831254 + 0.996539i \(0.473510\pi\)
\(98\) 0 0
\(99\) −357.640 −0.363073
\(100\) 0 0
\(101\) −1366.26 −1.34602 −0.673010 0.739633i \(-0.734999\pi\)
−0.673010 + 0.739633i \(0.734999\pi\)
\(102\) 0 0
\(103\) −1741.30 −1.66578 −0.832889 0.553440i \(-0.813315\pi\)
−0.832889 + 0.553440i \(0.813315\pi\)
\(104\) 0 0
\(105\) 518.812 0.482199
\(106\) 0 0
\(107\) −649.378 −0.586708 −0.293354 0.956004i \(-0.594771\pi\)
−0.293354 + 0.956004i \(0.594771\pi\)
\(108\) 0 0
\(109\) −1413.18 −1.24182 −0.620908 0.783883i \(-0.713236\pi\)
−0.620908 + 0.783883i \(0.713236\pi\)
\(110\) 0 0
\(111\) 5.33932 0.00456564
\(112\) 0 0
\(113\) 1096.43 0.912771 0.456386 0.889782i \(-0.349144\pi\)
0.456386 + 0.889782i \(0.349144\pi\)
\(114\) 0 0
\(115\) −840.826 −0.681804
\(116\) 0 0
\(117\) 296.580 0.234349
\(118\) 0 0
\(119\) 844.997 0.650930
\(120\) 0 0
\(121\) 248.092 0.186395
\(122\) 0 0
\(123\) 587.481 0.430661
\(124\) 0 0
\(125\) −1739.50 −1.24469
\(126\) 0 0
\(127\) 737.794 0.515501 0.257751 0.966211i \(-0.417019\pi\)
0.257751 + 0.966211i \(0.417019\pi\)
\(128\) 0 0
\(129\) −1221.08 −0.833411
\(130\) 0 0
\(131\) 147.698 0.0985074 0.0492537 0.998786i \(-0.484316\pi\)
0.0492537 + 0.998786i \(0.484316\pi\)
\(132\) 0 0
\(133\) 677.116 0.441454
\(134\) 0 0
\(135\) 500.639 0.319172
\(136\) 0 0
\(137\) 1880.79 1.17289 0.586447 0.809988i \(-0.300526\pi\)
0.586447 + 0.809988i \(0.300526\pi\)
\(138\) 0 0
\(139\) 629.333 0.384024 0.192012 0.981393i \(-0.438499\pi\)
0.192012 + 0.981393i \(0.438499\pi\)
\(140\) 0 0
\(141\) −835.400 −0.498960
\(142\) 0 0
\(143\) −1309.49 −0.765770
\(144\) 0 0
\(145\) 2658.29 1.52248
\(146\) 0 0
\(147\) 768.038 0.430930
\(148\) 0 0
\(149\) 429.457 0.236124 0.118062 0.993006i \(-0.462332\pi\)
0.118062 + 0.993006i \(0.462332\pi\)
\(150\) 0 0
\(151\) −27.3124 −0.0147196 −0.00735978 0.999973i \(-0.502343\pi\)
−0.00735978 + 0.999973i \(0.502343\pi\)
\(152\) 0 0
\(153\) 815.398 0.430857
\(154\) 0 0
\(155\) 1677.82 0.869456
\(156\) 0 0
\(157\) −1251.08 −0.635970 −0.317985 0.948096i \(-0.603006\pi\)
−0.317985 + 0.948096i \(0.603006\pi\)
\(158\) 0 0
\(159\) 723.908 0.361067
\(160\) 0 0
\(161\) 422.934 0.207030
\(162\) 0 0
\(163\) 127.884 0.0614517 0.0307258 0.999528i \(-0.490218\pi\)
0.0307258 + 0.999528i \(0.490218\pi\)
\(164\) 0 0
\(165\) −2210.48 −1.04294
\(166\) 0 0
\(167\) −2079.65 −0.963642 −0.481821 0.876270i \(-0.660024\pi\)
−0.481821 + 0.876270i \(0.660024\pi\)
\(168\) 0 0
\(169\) −1111.08 −0.505726
\(170\) 0 0
\(171\) 653.398 0.292202
\(172\) 0 0
\(173\) −685.140 −0.301099 −0.150550 0.988602i \(-0.548104\pi\)
−0.150550 + 0.988602i \(0.548104\pi\)
\(174\) 0 0
\(175\) 2040.80 0.881544
\(176\) 0 0
\(177\) 449.172 0.190745
\(178\) 0 0
\(179\) −429.423 −0.179310 −0.0896552 0.995973i \(-0.528576\pi\)
−0.0896552 + 0.995973i \(0.528576\pi\)
\(180\) 0 0
\(181\) −2842.85 −1.16745 −0.583723 0.811953i \(-0.698404\pi\)
−0.583723 + 0.811953i \(0.698404\pi\)
\(182\) 0 0
\(183\) 1524.94 0.615994
\(184\) 0 0
\(185\) 33.0009 0.0131150
\(186\) 0 0
\(187\) −3600.24 −1.40789
\(188\) 0 0
\(189\) −251.821 −0.0969167
\(190\) 0 0
\(191\) −2546.78 −0.964808 −0.482404 0.875949i \(-0.660236\pi\)
−0.482404 + 0.875949i \(0.660236\pi\)
\(192\) 0 0
\(193\) −3579.97 −1.33519 −0.667596 0.744524i \(-0.732677\pi\)
−0.667596 + 0.744524i \(0.732677\pi\)
\(194\) 0 0
\(195\) 1833.08 0.673177
\(196\) 0 0
\(197\) −3872.58 −1.40056 −0.700280 0.713869i \(-0.746941\pi\)
−0.700280 + 0.713869i \(0.746941\pi\)
\(198\) 0 0
\(199\) 5558.64 1.98011 0.990054 0.140686i \(-0.0449307\pi\)
0.990054 + 0.140686i \(0.0449307\pi\)
\(200\) 0 0
\(201\) −2850.08 −1.00014
\(202\) 0 0
\(203\) −1337.12 −0.462301
\(204\) 0 0
\(205\) 3631.06 1.23709
\(206\) 0 0
\(207\) 408.120 0.137035
\(208\) 0 0
\(209\) −2884.96 −0.954816
\(210\) 0 0
\(211\) −4658.93 −1.52007 −0.760034 0.649884i \(-0.774818\pi\)
−0.760034 + 0.649884i \(0.774818\pi\)
\(212\) 0 0
\(213\) 2410.68 0.775478
\(214\) 0 0
\(215\) −7547.17 −2.39401
\(216\) 0 0
\(217\) −843.940 −0.264011
\(218\) 0 0
\(219\) 1349.36 0.416352
\(220\) 0 0
\(221\) 2985.56 0.908736
\(222\) 0 0
\(223\) 1545.42 0.464077 0.232038 0.972707i \(-0.425460\pi\)
0.232038 + 0.972707i \(0.425460\pi\)
\(224\) 0 0
\(225\) 1969.32 0.583502
\(226\) 0 0
\(227\) 6545.76 1.91391 0.956954 0.290240i \(-0.0937351\pi\)
0.956954 + 0.290240i \(0.0937351\pi\)
\(228\) 0 0
\(229\) 5463.48 1.57658 0.788291 0.615303i \(-0.210966\pi\)
0.788291 + 0.615303i \(0.210966\pi\)
\(230\) 0 0
\(231\) 1111.87 0.316690
\(232\) 0 0
\(233\) −4722.40 −1.32779 −0.663894 0.747827i \(-0.731097\pi\)
−0.663894 + 0.747827i \(0.731097\pi\)
\(234\) 0 0
\(235\) −5163.38 −1.43328
\(236\) 0 0
\(237\) −471.659 −0.129272
\(238\) 0 0
\(239\) 1054.38 0.285363 0.142682 0.989769i \(-0.454427\pi\)
0.142682 + 0.989769i \(0.454427\pi\)
\(240\) 0 0
\(241\) −3134.40 −0.837777 −0.418888 0.908038i \(-0.637580\pi\)
−0.418888 + 0.908038i \(0.637580\pi\)
\(242\) 0 0
\(243\) −243.000 −0.0641500
\(244\) 0 0
\(245\) 4747.04 1.23787
\(246\) 0 0
\(247\) 2392.40 0.616295
\(248\) 0 0
\(249\) 525.188 0.133664
\(250\) 0 0
\(251\) 4881.91 1.22766 0.613831 0.789437i \(-0.289628\pi\)
0.613831 + 0.789437i \(0.289628\pi\)
\(252\) 0 0
\(253\) −1801.97 −0.447783
\(254\) 0 0
\(255\) 5039.76 1.23765
\(256\) 0 0
\(257\) −540.458 −0.131178 −0.0655892 0.997847i \(-0.520893\pi\)
−0.0655892 + 0.997847i \(0.520893\pi\)
\(258\) 0 0
\(259\) −16.5994 −0.00398238
\(260\) 0 0
\(261\) −1290.28 −0.306001
\(262\) 0 0
\(263\) 4800.12 1.12543 0.562715 0.826651i \(-0.309757\pi\)
0.562715 + 0.826651i \(0.309757\pi\)
\(264\) 0 0
\(265\) 4474.28 1.03718
\(266\) 0 0
\(267\) 381.599 0.0874662
\(268\) 0 0
\(269\) −3321.28 −0.752795 −0.376397 0.926458i \(-0.622837\pi\)
−0.376397 + 0.926458i \(0.622837\pi\)
\(270\) 0 0
\(271\) −5274.04 −1.18220 −0.591098 0.806600i \(-0.701305\pi\)
−0.591098 + 0.806600i \(0.701305\pi\)
\(272\) 0 0
\(273\) −922.036 −0.204411
\(274\) 0 0
\(275\) −8695.15 −1.90668
\(276\) 0 0
\(277\) −3190.24 −0.691996 −0.345998 0.938235i \(-0.612460\pi\)
−0.345998 + 0.938235i \(0.612460\pi\)
\(278\) 0 0
\(279\) −814.378 −0.174751
\(280\) 0 0
\(281\) 545.619 0.115832 0.0579162 0.998321i \(-0.481554\pi\)
0.0579162 + 0.998321i \(0.481554\pi\)
\(282\) 0 0
\(283\) 5927.74 1.24511 0.622557 0.782574i \(-0.286094\pi\)
0.622557 + 0.782574i \(0.286094\pi\)
\(284\) 0 0
\(285\) 4038.48 0.839365
\(286\) 0 0
\(287\) −1826.42 −0.375645
\(288\) 0 0
\(289\) 3295.33 0.670736
\(290\) 0 0
\(291\) −476.478 −0.0959850
\(292\) 0 0
\(293\) 5406.01 1.07789 0.538946 0.842340i \(-0.318823\pi\)
0.538946 + 0.842340i \(0.318823\pi\)
\(294\) 0 0
\(295\) 2776.21 0.547923
\(296\) 0 0
\(297\) 1072.92 0.209620
\(298\) 0 0
\(299\) 1494.32 0.289026
\(300\) 0 0
\(301\) 3796.21 0.726944
\(302\) 0 0
\(303\) 4098.78 0.777125
\(304\) 0 0
\(305\) 9425.25 1.76947
\(306\) 0 0
\(307\) −1558.56 −0.289745 −0.144873 0.989450i \(-0.546277\pi\)
−0.144873 + 0.989450i \(0.546277\pi\)
\(308\) 0 0
\(309\) 5223.89 0.961738
\(310\) 0 0
\(311\) −8348.21 −1.52213 −0.761067 0.648673i \(-0.775324\pi\)
−0.761067 + 0.648673i \(0.775324\pi\)
\(312\) 0 0
\(313\) 5213.09 0.941410 0.470705 0.882291i \(-0.344000\pi\)
0.470705 + 0.882291i \(0.344000\pi\)
\(314\) 0 0
\(315\) −1556.44 −0.278398
\(316\) 0 0
\(317\) 9070.57 1.60711 0.803555 0.595230i \(-0.202939\pi\)
0.803555 + 0.595230i \(0.202939\pi\)
\(318\) 0 0
\(319\) 5696.98 0.999905
\(320\) 0 0
\(321\) 1948.13 0.338736
\(322\) 0 0
\(323\) 6577.53 1.13308
\(324\) 0 0
\(325\) 7210.61 1.23069
\(326\) 0 0
\(327\) 4239.54 0.716963
\(328\) 0 0
\(329\) 2597.17 0.435218
\(330\) 0 0
\(331\) −186.537 −0.0309758 −0.0154879 0.999880i \(-0.504930\pi\)
−0.0154879 + 0.999880i \(0.504930\pi\)
\(332\) 0 0
\(333\) −16.0180 −0.00263597
\(334\) 0 0
\(335\) −17615.6 −2.87296
\(336\) 0 0
\(337\) 829.350 0.134058 0.0670290 0.997751i \(-0.478648\pi\)
0.0670290 + 0.997751i \(0.478648\pi\)
\(338\) 0 0
\(339\) −3289.28 −0.526989
\(340\) 0 0
\(341\) 3595.73 0.571026
\(342\) 0 0
\(343\) −5586.81 −0.879473
\(344\) 0 0
\(345\) 2522.48 0.393640
\(346\) 0 0
\(347\) −12005.6 −1.85734 −0.928669 0.370908i \(-0.879046\pi\)
−0.928669 + 0.370908i \(0.879046\pi\)
\(348\) 0 0
\(349\) 77.8551 0.0119412 0.00597062 0.999982i \(-0.498099\pi\)
0.00597062 + 0.999982i \(0.498099\pi\)
\(350\) 0 0
\(351\) −889.739 −0.135301
\(352\) 0 0
\(353\) −4925.06 −0.742591 −0.371296 0.928515i \(-0.621086\pi\)
−0.371296 + 0.928515i \(0.621086\pi\)
\(354\) 0 0
\(355\) 14899.8 2.22760
\(356\) 0 0
\(357\) −2534.99 −0.375815
\(358\) 0 0
\(359\) −12260.2 −1.80242 −0.901209 0.433384i \(-0.857319\pi\)
−0.901209 + 0.433384i \(0.857319\pi\)
\(360\) 0 0
\(361\) −1588.27 −0.231560
\(362\) 0 0
\(363\) −744.275 −0.107615
\(364\) 0 0
\(365\) 8340.02 1.19599
\(366\) 0 0
\(367\) 8600.86 1.22333 0.611664 0.791118i \(-0.290500\pi\)
0.611664 + 0.791118i \(0.290500\pi\)
\(368\) 0 0
\(369\) −1762.44 −0.248642
\(370\) 0 0
\(371\) −2250.56 −0.314941
\(372\) 0 0
\(373\) −3996.47 −0.554770 −0.277385 0.960759i \(-0.589468\pi\)
−0.277385 + 0.960759i \(0.589468\pi\)
\(374\) 0 0
\(375\) 5218.51 0.718620
\(376\) 0 0
\(377\) −4724.33 −0.645399
\(378\) 0 0
\(379\) 10404.7 1.41017 0.705087 0.709121i \(-0.250908\pi\)
0.705087 + 0.709121i \(0.250908\pi\)
\(380\) 0 0
\(381\) −2213.38 −0.297625
\(382\) 0 0
\(383\) −6814.19 −0.909109 −0.454554 0.890719i \(-0.650202\pi\)
−0.454554 + 0.890719i \(0.650202\pi\)
\(384\) 0 0
\(385\) 6872.15 0.909707
\(386\) 0 0
\(387\) 3663.24 0.481170
\(388\) 0 0
\(389\) −779.329 −0.101577 −0.0507886 0.998709i \(-0.516173\pi\)
−0.0507886 + 0.998709i \(0.516173\pi\)
\(390\) 0 0
\(391\) 4108.39 0.531382
\(392\) 0 0
\(393\) −443.095 −0.0568733
\(394\) 0 0
\(395\) −2915.20 −0.371340
\(396\) 0 0
\(397\) 12514.1 1.58202 0.791011 0.611802i \(-0.209555\pi\)
0.791011 + 0.611802i \(0.209555\pi\)
\(398\) 0 0
\(399\) −2031.35 −0.254874
\(400\) 0 0
\(401\) −7949.68 −0.989995 −0.494998 0.868894i \(-0.664831\pi\)
−0.494998 + 0.868894i \(0.664831\pi\)
\(402\) 0 0
\(403\) −2981.83 −0.368574
\(404\) 0 0
\(405\) −1501.92 −0.184274
\(406\) 0 0
\(407\) 70.7243 0.00861345
\(408\) 0 0
\(409\) −15183.9 −1.83569 −0.917846 0.396937i \(-0.870073\pi\)
−0.917846 + 0.396937i \(0.870073\pi\)
\(410\) 0 0
\(411\) −5642.36 −0.677170
\(412\) 0 0
\(413\) −1396.43 −0.166377
\(414\) 0 0
\(415\) 3246.05 0.383957
\(416\) 0 0
\(417\) −1888.00 −0.221716
\(418\) 0 0
\(419\) −11767.2 −1.37199 −0.685996 0.727606i \(-0.740633\pi\)
−0.685996 + 0.727606i \(0.740633\pi\)
\(420\) 0 0
\(421\) 14459.8 1.67394 0.836970 0.547248i \(-0.184325\pi\)
0.836970 + 0.547248i \(0.184325\pi\)
\(422\) 0 0
\(423\) 2506.20 0.288075
\(424\) 0 0
\(425\) 19824.4 2.26265
\(426\) 0 0
\(427\) −4740.89 −0.537301
\(428\) 0 0
\(429\) 3928.47 0.442118
\(430\) 0 0
\(431\) 9248.50 1.03361 0.516804 0.856104i \(-0.327122\pi\)
0.516804 + 0.856104i \(0.327122\pi\)
\(432\) 0 0
\(433\) −3456.12 −0.383581 −0.191790 0.981436i \(-0.561429\pi\)
−0.191790 + 0.981436i \(0.561429\pi\)
\(434\) 0 0
\(435\) −7974.87 −0.879002
\(436\) 0 0
\(437\) 3292.16 0.360378
\(438\) 0 0
\(439\) 8075.68 0.877975 0.438988 0.898493i \(-0.355337\pi\)
0.438988 + 0.898493i \(0.355337\pi\)
\(440\) 0 0
\(441\) −2304.12 −0.248798
\(442\) 0 0
\(443\) 11447.7 1.22776 0.613878 0.789401i \(-0.289609\pi\)
0.613878 + 0.789401i \(0.289609\pi\)
\(444\) 0 0
\(445\) 2358.56 0.251251
\(446\) 0 0
\(447\) −1288.37 −0.136326
\(448\) 0 0
\(449\) −1010.18 −0.106177 −0.0530886 0.998590i \(-0.516907\pi\)
−0.0530886 + 0.998590i \(0.516907\pi\)
\(450\) 0 0
\(451\) 7781.73 0.812477
\(452\) 0 0
\(453\) 81.9373 0.00849835
\(454\) 0 0
\(455\) −5698.86 −0.587179
\(456\) 0 0
\(457\) −15949.2 −1.63254 −0.816272 0.577667i \(-0.803963\pi\)
−0.816272 + 0.577667i \(0.803963\pi\)
\(458\) 0 0
\(459\) −2446.19 −0.248755
\(460\) 0 0
\(461\) −6737.61 −0.680698 −0.340349 0.940299i \(-0.610545\pi\)
−0.340349 + 0.940299i \(0.610545\pi\)
\(462\) 0 0
\(463\) 2602.69 0.261246 0.130623 0.991432i \(-0.458302\pi\)
0.130623 + 0.991432i \(0.458302\pi\)
\(464\) 0 0
\(465\) −5033.45 −0.501980
\(466\) 0 0
\(467\) −9326.18 −0.924120 −0.462060 0.886849i \(-0.652890\pi\)
−0.462060 + 0.886849i \(0.652890\pi\)
\(468\) 0 0
\(469\) 8860.60 0.872376
\(470\) 0 0
\(471\) 3753.25 0.367178
\(472\) 0 0
\(473\) −16174.3 −1.57230
\(474\) 0 0
\(475\) 15885.8 1.53451
\(476\) 0 0
\(477\) −2171.72 −0.208462
\(478\) 0 0
\(479\) −11472.0 −1.09430 −0.547149 0.837035i \(-0.684287\pi\)
−0.547149 + 0.837035i \(0.684287\pi\)
\(480\) 0 0
\(481\) −58.6494 −0.00555963
\(482\) 0 0
\(483\) −1268.80 −0.119529
\(484\) 0 0
\(485\) −2944.98 −0.275721
\(486\) 0 0
\(487\) −15048.0 −1.40018 −0.700090 0.714054i \(-0.746857\pi\)
−0.700090 + 0.714054i \(0.746857\pi\)
\(488\) 0 0
\(489\) −383.651 −0.0354792
\(490\) 0 0
\(491\) −14373.5 −1.32112 −0.660558 0.750775i \(-0.729680\pi\)
−0.660558 + 0.750775i \(0.729680\pi\)
\(492\) 0 0
\(493\) −12988.8 −1.18658
\(494\) 0 0
\(495\) 6631.43 0.602143
\(496\) 0 0
\(497\) −7494.55 −0.676411
\(498\) 0 0
\(499\) −9167.08 −0.822395 −0.411197 0.911546i \(-0.634889\pi\)
−0.411197 + 0.911546i \(0.634889\pi\)
\(500\) 0 0
\(501\) 6238.95 0.556359
\(502\) 0 0
\(503\) 7690.02 0.681672 0.340836 0.940123i \(-0.389290\pi\)
0.340836 + 0.940123i \(0.389290\pi\)
\(504\) 0 0
\(505\) 25333.5 2.23233
\(506\) 0 0
\(507\) 3333.24 0.291981
\(508\) 0 0
\(509\) −6854.21 −0.596872 −0.298436 0.954430i \(-0.596465\pi\)
−0.298436 + 0.954430i \(0.596465\pi\)
\(510\) 0 0
\(511\) −4195.01 −0.363163
\(512\) 0 0
\(513\) −1960.19 −0.168703
\(514\) 0 0
\(515\) 32287.5 2.76264
\(516\) 0 0
\(517\) −11065.6 −0.941328
\(518\) 0 0
\(519\) 2055.42 0.173840
\(520\) 0 0
\(521\) −1641.63 −0.138044 −0.0690221 0.997615i \(-0.521988\pi\)
−0.0690221 + 0.997615i \(0.521988\pi\)
\(522\) 0 0
\(523\) −1976.34 −0.165238 −0.0826188 0.996581i \(-0.526328\pi\)
−0.0826188 + 0.996581i \(0.526328\pi\)
\(524\) 0 0
\(525\) −6122.41 −0.508960
\(526\) 0 0
\(527\) −8198.06 −0.677634
\(528\) 0 0
\(529\) −10110.7 −0.830992
\(530\) 0 0
\(531\) −1347.52 −0.110127
\(532\) 0 0
\(533\) −6453.14 −0.524421
\(534\) 0 0
\(535\) 12040.9 0.973034
\(536\) 0 0
\(537\) 1288.27 0.103525
\(538\) 0 0
\(539\) 10173.4 0.812984
\(540\) 0 0
\(541\) 2892.17 0.229841 0.114921 0.993375i \(-0.463339\pi\)
0.114921 + 0.993375i \(0.463339\pi\)
\(542\) 0 0
\(543\) 8528.56 0.674025
\(544\) 0 0
\(545\) 26203.5 2.05951
\(546\) 0 0
\(547\) 7033.62 0.549791 0.274896 0.961474i \(-0.411357\pi\)
0.274896 + 0.961474i \(0.411357\pi\)
\(548\) 0 0
\(549\) −4574.82 −0.355644
\(550\) 0 0
\(551\) −10408.2 −0.804728
\(552\) 0 0
\(553\) 1466.34 0.112758
\(554\) 0 0
\(555\) −99.0028 −0.00757196
\(556\) 0 0
\(557\) 8702.92 0.662037 0.331019 0.943624i \(-0.392608\pi\)
0.331019 + 0.943624i \(0.392608\pi\)
\(558\) 0 0
\(559\) 13412.9 1.01485
\(560\) 0 0
\(561\) 10800.7 0.812845
\(562\) 0 0
\(563\) −14119.7 −1.05697 −0.528484 0.848943i \(-0.677239\pi\)
−0.528484 + 0.848943i \(0.677239\pi\)
\(564\) 0 0
\(565\) −20330.2 −1.51380
\(566\) 0 0
\(567\) 755.462 0.0559549
\(568\) 0 0
\(569\) −383.132 −0.0282280 −0.0141140 0.999900i \(-0.504493\pi\)
−0.0141140 + 0.999900i \(0.504493\pi\)
\(570\) 0 0
\(571\) −21917.8 −1.60636 −0.803178 0.595739i \(-0.796859\pi\)
−0.803178 + 0.595739i \(0.796859\pi\)
\(572\) 0 0
\(573\) 7640.33 0.557032
\(574\) 0 0
\(575\) 9922.44 0.719642
\(576\) 0 0
\(577\) 1570.50 0.113311 0.0566556 0.998394i \(-0.481956\pi\)
0.0566556 + 0.998394i \(0.481956\pi\)
\(578\) 0 0
\(579\) 10739.9 0.770873
\(580\) 0 0
\(581\) −1632.76 −0.116589
\(582\) 0 0
\(583\) 9588.84 0.681182
\(584\) 0 0
\(585\) −5499.24 −0.388659
\(586\) 0 0
\(587\) 14387.7 1.01166 0.505830 0.862633i \(-0.331186\pi\)
0.505830 + 0.862633i \(0.331186\pi\)
\(588\) 0 0
\(589\) −6569.30 −0.459564
\(590\) 0 0
\(591\) 11617.8 0.808613
\(592\) 0 0
\(593\) 17903.0 1.23978 0.619889 0.784690i \(-0.287178\pi\)
0.619889 + 0.784690i \(0.287178\pi\)
\(594\) 0 0
\(595\) −15668.1 −1.07955
\(596\) 0 0
\(597\) −16675.9 −1.14322
\(598\) 0 0
\(599\) 9474.88 0.646299 0.323150 0.946348i \(-0.395258\pi\)
0.323150 + 0.946348i \(0.395258\pi\)
\(600\) 0 0
\(601\) 16945.6 1.15012 0.575061 0.818110i \(-0.304978\pi\)
0.575061 + 0.818110i \(0.304978\pi\)
\(602\) 0 0
\(603\) 8550.23 0.577433
\(604\) 0 0
\(605\) −4600.16 −0.309129
\(606\) 0 0
\(607\) −18736.2 −1.25285 −0.626423 0.779483i \(-0.715482\pi\)
−0.626423 + 0.779483i \(0.715482\pi\)
\(608\) 0 0
\(609\) 4011.35 0.266910
\(610\) 0 0
\(611\) 9176.39 0.607589
\(612\) 0 0
\(613\) −27850.0 −1.83499 −0.917497 0.397742i \(-0.869794\pi\)
−0.917497 + 0.397742i \(0.869794\pi\)
\(614\) 0 0
\(615\) −10893.2 −0.714237
\(616\) 0 0
\(617\) 12836.3 0.837551 0.418776 0.908090i \(-0.362459\pi\)
0.418776 + 0.908090i \(0.362459\pi\)
\(618\) 0 0
\(619\) −18030.3 −1.17076 −0.585378 0.810760i \(-0.699054\pi\)
−0.585378 + 0.810760i \(0.699054\pi\)
\(620\) 0 0
\(621\) −1224.36 −0.0791173
\(622\) 0 0
\(623\) −1186.35 −0.0762924
\(624\) 0 0
\(625\) 4902.56 0.313764
\(626\) 0 0
\(627\) 8654.87 0.551263
\(628\) 0 0
\(629\) −161.247 −0.0102215
\(630\) 0 0
\(631\) 16460.1 1.03846 0.519229 0.854635i \(-0.326219\pi\)
0.519229 + 0.854635i \(0.326219\pi\)
\(632\) 0 0
\(633\) 13976.8 0.877611
\(634\) 0 0
\(635\) −13680.3 −0.854940
\(636\) 0 0
\(637\) −8436.46 −0.524749
\(638\) 0 0
\(639\) −7232.03 −0.447723
\(640\) 0 0
\(641\) −19443.3 −1.19807 −0.599035 0.800723i \(-0.704449\pi\)
−0.599035 + 0.800723i \(0.704449\pi\)
\(642\) 0 0
\(643\) −7368.87 −0.451944 −0.225972 0.974134i \(-0.572556\pi\)
−0.225972 + 0.974134i \(0.572556\pi\)
\(644\) 0 0
\(645\) 22641.5 1.38218
\(646\) 0 0
\(647\) −11042.3 −0.670969 −0.335484 0.942046i \(-0.608900\pi\)
−0.335484 + 0.942046i \(0.608900\pi\)
\(648\) 0 0
\(649\) 5949.70 0.359856
\(650\) 0 0
\(651\) 2531.82 0.152427
\(652\) 0 0
\(653\) −14174.3 −0.849440 −0.424720 0.905325i \(-0.639627\pi\)
−0.424720 + 0.905325i \(0.639627\pi\)
\(654\) 0 0
\(655\) −2738.65 −0.163371
\(656\) 0 0
\(657\) −4048.07 −0.240381
\(658\) 0 0
\(659\) 1917.19 0.113328 0.0566640 0.998393i \(-0.481954\pi\)
0.0566640 + 0.998393i \(0.481954\pi\)
\(660\) 0 0
\(661\) 28084.2 1.65257 0.826284 0.563254i \(-0.190451\pi\)
0.826284 + 0.563254i \(0.190451\pi\)
\(662\) 0 0
\(663\) −8956.69 −0.524659
\(664\) 0 0
\(665\) −12555.2 −0.732136
\(666\) 0 0
\(667\) −6501.09 −0.377396
\(668\) 0 0
\(669\) −4636.27 −0.267935
\(670\) 0 0
\(671\) 20199.3 1.16212
\(672\) 0 0
\(673\) 1756.20 0.100589 0.0502947 0.998734i \(-0.483984\pi\)
0.0502947 + 0.998734i \(0.483984\pi\)
\(674\) 0 0
\(675\) −5907.96 −0.336885
\(676\) 0 0
\(677\) −10424.0 −0.591766 −0.295883 0.955224i \(-0.595614\pi\)
−0.295883 + 0.955224i \(0.595614\pi\)
\(678\) 0 0
\(679\) 1481.32 0.0837230
\(680\) 0 0
\(681\) −19637.3 −1.10500
\(682\) 0 0
\(683\) −5828.41 −0.326527 −0.163264 0.986582i \(-0.552202\pi\)
−0.163264 + 0.986582i \(0.552202\pi\)
\(684\) 0 0
\(685\) −34873.9 −1.94520
\(686\) 0 0
\(687\) −16390.4 −0.910240
\(688\) 0 0
\(689\) −7951.72 −0.439675
\(690\) 0 0
\(691\) −10673.5 −0.587611 −0.293806 0.955865i \(-0.594922\pi\)
−0.293806 + 0.955865i \(0.594922\pi\)
\(692\) 0 0
\(693\) −3335.60 −0.182841
\(694\) 0 0
\(695\) −11669.2 −0.636890
\(696\) 0 0
\(697\) −17741.9 −0.964163
\(698\) 0 0
\(699\) 14167.2 0.766599
\(700\) 0 0
\(701\) −14367.0 −0.774084 −0.387042 0.922062i \(-0.626503\pi\)
−0.387042 + 0.922062i \(0.626503\pi\)
\(702\) 0 0
\(703\) −129.211 −0.00693214
\(704\) 0 0
\(705\) 15490.1 0.827507
\(706\) 0 0
\(707\) −12742.7 −0.677848
\(708\) 0 0
\(709\) −25026.4 −1.32565 −0.662825 0.748774i \(-0.730643\pi\)
−0.662825 + 0.748774i \(0.730643\pi\)
\(710\) 0 0
\(711\) 1414.98 0.0746354
\(712\) 0 0
\(713\) −4103.26 −0.215523
\(714\) 0 0
\(715\) 24280.8 1.27000
\(716\) 0 0
\(717\) −3163.13 −0.164755
\(718\) 0 0
\(719\) −692.065 −0.0358966 −0.0179483 0.999839i \(-0.505713\pi\)
−0.0179483 + 0.999839i \(0.505713\pi\)
\(720\) 0 0
\(721\) −16240.6 −0.838876
\(722\) 0 0
\(723\) 9403.19 0.483691
\(724\) 0 0
\(725\) −31370.0 −1.60697
\(726\) 0 0
\(727\) −23929.5 −1.22076 −0.610382 0.792107i \(-0.708984\pi\)
−0.610382 + 0.792107i \(0.708984\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 36876.5 1.86584
\(732\) 0 0
\(733\) −5613.22 −0.282850 −0.141425 0.989949i \(-0.545168\pi\)
−0.141425 + 0.989949i \(0.545168\pi\)
\(734\) 0 0
\(735\) −14241.1 −0.714683
\(736\) 0 0
\(737\) −37751.9 −1.88685
\(738\) 0 0
\(739\) −4790.30 −0.238449 −0.119225 0.992867i \(-0.538041\pi\)
−0.119225 + 0.992867i \(0.538041\pi\)
\(740\) 0 0
\(741\) −7177.21 −0.355818
\(742\) 0 0
\(743\) 16695.0 0.824333 0.412166 0.911109i \(-0.364772\pi\)
0.412166 + 0.911109i \(0.364772\pi\)
\(744\) 0 0
\(745\) −7963.07 −0.391603
\(746\) 0 0
\(747\) −1575.56 −0.0771711
\(748\) 0 0
\(749\) −6056.55 −0.295463
\(750\) 0 0
\(751\) −27366.8 −1.32973 −0.664866 0.746963i \(-0.731511\pi\)
−0.664866 + 0.746963i \(0.731511\pi\)
\(752\) 0 0
\(753\) −14645.7 −0.708791
\(754\) 0 0
\(755\) 506.433 0.0244119
\(756\) 0 0
\(757\) 5712.23 0.274260 0.137130 0.990553i \(-0.456212\pi\)
0.137130 + 0.990553i \(0.456212\pi\)
\(758\) 0 0
\(759\) 5405.92 0.258528
\(760\) 0 0
\(761\) −14015.9 −0.667643 −0.333822 0.942636i \(-0.608338\pi\)
−0.333822 + 0.942636i \(0.608338\pi\)
\(762\) 0 0
\(763\) −13180.3 −0.625372
\(764\) 0 0
\(765\) −15119.3 −0.714560
\(766\) 0 0
\(767\) −4933.90 −0.232272
\(768\) 0 0
\(769\) −3430.70 −0.160877 −0.0804384 0.996760i \(-0.525632\pi\)
−0.0804384 + 0.996760i \(0.525632\pi\)
\(770\) 0 0
\(771\) 1621.37 0.0757359
\(772\) 0 0
\(773\) 14821.8 0.689654 0.344827 0.938666i \(-0.387938\pi\)
0.344827 + 0.938666i \(0.387938\pi\)
\(774\) 0 0
\(775\) −19799.6 −0.917708
\(776\) 0 0
\(777\) 49.7982 0.00229923
\(778\) 0 0
\(779\) −14217.0 −0.653885
\(780\) 0 0
\(781\) 31931.7 1.46300
\(782\) 0 0
\(783\) 3870.84 0.176670
\(784\) 0 0
\(785\) 23197.8 1.05473
\(786\) 0 0
\(787\) −16917.4 −0.766253 −0.383126 0.923696i \(-0.625153\pi\)
−0.383126 + 0.923696i \(0.625153\pi\)
\(788\) 0 0
\(789\) −14400.4 −0.649768
\(790\) 0 0
\(791\) 10226.0 0.459666
\(792\) 0 0
\(793\) −16750.6 −0.750103
\(794\) 0 0
\(795\) −13422.9 −0.598817
\(796\) 0 0
\(797\) −23546.7 −1.04651 −0.523254 0.852177i \(-0.675282\pi\)
−0.523254 + 0.852177i \(0.675282\pi\)
\(798\) 0 0
\(799\) 25229.0 1.11707
\(800\) 0 0
\(801\) −1144.80 −0.0504986
\(802\) 0 0
\(803\) 17873.5 0.785482
\(804\) 0 0
\(805\) −7842.13 −0.343352
\(806\) 0 0
\(807\) 9963.83 0.434626
\(808\) 0 0
\(809\) 17647.6 0.766942 0.383471 0.923553i \(-0.374729\pi\)
0.383471 + 0.923553i \(0.374729\pi\)
\(810\) 0 0
\(811\) −11690.3 −0.506169 −0.253085 0.967444i \(-0.581445\pi\)
−0.253085 + 0.967444i \(0.581445\pi\)
\(812\) 0 0
\(813\) 15822.1 0.682541
\(814\) 0 0
\(815\) −2371.25 −0.101915
\(816\) 0 0
\(817\) 29550.0 1.26539
\(818\) 0 0
\(819\) 2766.11 0.118017
\(820\) 0 0
\(821\) −10576.3 −0.449594 −0.224797 0.974406i \(-0.572172\pi\)
−0.224797 + 0.974406i \(0.572172\pi\)
\(822\) 0 0
\(823\) −44624.4 −1.89005 −0.945023 0.327005i \(-0.893961\pi\)
−0.945023 + 0.327005i \(0.893961\pi\)
\(824\) 0 0
\(825\) 26085.5 1.10082
\(826\) 0 0
\(827\) −2532.98 −0.106506 −0.0532528 0.998581i \(-0.516959\pi\)
−0.0532528 + 0.998581i \(0.516959\pi\)
\(828\) 0 0
\(829\) 24029.6 1.00673 0.503366 0.864073i \(-0.332095\pi\)
0.503366 + 0.864073i \(0.332095\pi\)
\(830\) 0 0
\(831\) 9570.72 0.399524
\(832\) 0 0
\(833\) −23194.7 −0.964765
\(834\) 0 0
\(835\) 38561.3 1.59817
\(836\) 0 0
\(837\) 2443.13 0.100893
\(838\) 0 0
\(839\) 39117.5 1.60964 0.804819 0.593520i \(-0.202262\pi\)
0.804819 + 0.593520i \(0.202262\pi\)
\(840\) 0 0
\(841\) −3835.65 −0.157270
\(842\) 0 0
\(843\) −1636.86 −0.0668758
\(844\) 0 0
\(845\) 20601.9 0.838729
\(846\) 0 0
\(847\) 2313.87 0.0938674
\(848\) 0 0
\(849\) −17783.2 −0.718867
\(850\) 0 0
\(851\) −80.7068 −0.00325099
\(852\) 0 0
\(853\) 35436.1 1.42240 0.711201 0.702988i \(-0.248151\pi\)
0.711201 + 0.702988i \(0.248151\pi\)
\(854\) 0 0
\(855\) −12115.4 −0.484607
\(856\) 0 0
\(857\) −20451.8 −0.815191 −0.407596 0.913163i \(-0.633633\pi\)
−0.407596 + 0.913163i \(0.633633\pi\)
\(858\) 0 0
\(859\) −6477.74 −0.257297 −0.128648 0.991690i \(-0.541064\pi\)
−0.128648 + 0.991690i \(0.541064\pi\)
\(860\) 0 0
\(861\) 5479.25 0.216879
\(862\) 0 0
\(863\) 2068.34 0.0815843 0.0407922 0.999168i \(-0.487012\pi\)
0.0407922 + 0.999168i \(0.487012\pi\)
\(864\) 0 0
\(865\) 12704.0 0.499363
\(866\) 0 0
\(867\) −9885.98 −0.387250
\(868\) 0 0
\(869\) −6247.56 −0.243882
\(870\) 0 0
\(871\) 31306.5 1.21789
\(872\) 0 0
\(873\) 1429.43 0.0554170
\(874\) 0 0
\(875\) −16223.8 −0.626817
\(876\) 0 0
\(877\) −33095.0 −1.27427 −0.637137 0.770750i \(-0.719882\pi\)
−0.637137 + 0.770750i \(0.719882\pi\)
\(878\) 0 0
\(879\) −16218.0 −0.622321
\(880\) 0 0
\(881\) 33169.3 1.26845 0.634225 0.773149i \(-0.281319\pi\)
0.634225 + 0.773149i \(0.281319\pi\)
\(882\) 0 0
\(883\) 28990.4 1.10488 0.552438 0.833554i \(-0.313698\pi\)
0.552438 + 0.833554i \(0.313698\pi\)
\(884\) 0 0
\(885\) −8328.64 −0.316344
\(886\) 0 0
\(887\) 458.565 0.0173586 0.00867931 0.999962i \(-0.497237\pi\)
0.00867931 + 0.999962i \(0.497237\pi\)
\(888\) 0 0
\(889\) 6881.18 0.259603
\(890\) 0 0
\(891\) −3218.76 −0.121024
\(892\) 0 0
\(893\) 20216.6 0.757585
\(894\) 0 0
\(895\) 7962.44 0.297380
\(896\) 0 0
\(897\) −4482.96 −0.166869
\(898\) 0 0
\(899\) 12972.5 0.481266
\(900\) 0 0
\(901\) −21862.0 −0.808355
\(902\) 0 0
\(903\) −11388.6 −0.419701
\(904\) 0 0
\(905\) 52712.8 1.93617
\(906\) 0 0
\(907\) 45653.8 1.67134 0.835672 0.549229i \(-0.185078\pi\)
0.835672 + 0.549229i \(0.185078\pi\)
\(908\) 0 0
\(909\) −12296.3 −0.448673
\(910\) 0 0
\(911\) 50760.6 1.84608 0.923038 0.384710i \(-0.125698\pi\)
0.923038 + 0.384710i \(0.125698\pi\)
\(912\) 0 0
\(913\) 6956.60 0.252169
\(914\) 0 0
\(915\) −28275.8 −1.02160
\(916\) 0 0
\(917\) 1377.54 0.0496078
\(918\) 0 0
\(919\) −25939.0 −0.931065 −0.465532 0.885031i \(-0.654137\pi\)
−0.465532 + 0.885031i \(0.654137\pi\)
\(920\) 0 0
\(921\) 4675.68 0.167284
\(922\) 0 0
\(923\) −26479.9 −0.944309
\(924\) 0 0
\(925\) −389.438 −0.0138429
\(926\) 0 0
\(927\) −15671.7 −0.555260
\(928\) 0 0
\(929\) 41850.4 1.47801 0.739003 0.673702i \(-0.235297\pi\)
0.739003 + 0.673702i \(0.235297\pi\)
\(930\) 0 0
\(931\) −18586.5 −0.654294
\(932\) 0 0
\(933\) 25044.6 0.878805
\(934\) 0 0
\(935\) 66756.3 2.33494
\(936\) 0 0
\(937\) −18888.1 −0.658534 −0.329267 0.944237i \(-0.606802\pi\)
−0.329267 + 0.944237i \(0.606802\pi\)
\(938\) 0 0
\(939\) −15639.3 −0.543523
\(940\) 0 0
\(941\) 21571.8 0.747313 0.373656 0.927567i \(-0.378104\pi\)
0.373656 + 0.927567i \(0.378104\pi\)
\(942\) 0 0
\(943\) −8880.09 −0.306655
\(944\) 0 0
\(945\) 4669.31 0.160733
\(946\) 0 0
\(947\) 2981.14 0.102296 0.0511479 0.998691i \(-0.483712\pi\)
0.0511479 + 0.998691i \(0.483712\pi\)
\(948\) 0 0
\(949\) −14821.9 −0.506997
\(950\) 0 0
\(951\) −27211.7 −0.927865
\(952\) 0 0
\(953\) 8353.84 0.283953 0.141977 0.989870i \(-0.454654\pi\)
0.141977 + 0.989870i \(0.454654\pi\)
\(954\) 0 0
\(955\) 47222.9 1.60010
\(956\) 0 0
\(957\) −17090.9 −0.577296
\(958\) 0 0
\(959\) 17541.5 0.590662
\(960\) 0 0
\(961\) −21603.2 −0.725159
\(962\) 0 0
\(963\) −5844.40 −0.195569
\(964\) 0 0
\(965\) 66380.6 2.21437
\(966\) 0 0
\(967\) −20156.9 −0.670323 −0.335162 0.942161i \(-0.608791\pi\)
−0.335162 + 0.942161i \(0.608791\pi\)
\(968\) 0 0
\(969\) −19732.6 −0.654182
\(970\) 0 0
\(971\) 32976.8 1.08988 0.544941 0.838474i \(-0.316552\pi\)
0.544941 + 0.838474i \(0.316552\pi\)
\(972\) 0 0
\(973\) 5869.60 0.193392
\(974\) 0 0
\(975\) −21631.8 −0.710537
\(976\) 0 0
\(977\) 2934.18 0.0960827 0.0480413 0.998845i \(-0.484702\pi\)
0.0480413 + 0.998845i \(0.484702\pi\)
\(978\) 0 0
\(979\) 5054.63 0.165012
\(980\) 0 0
\(981\) −12718.6 −0.413939
\(982\) 0 0
\(983\) 11965.0 0.388223 0.194111 0.980979i \(-0.437818\pi\)
0.194111 + 0.980979i \(0.437818\pi\)
\(984\) 0 0
\(985\) 71806.2 2.32278
\(986\) 0 0
\(987\) −7791.52 −0.251273
\(988\) 0 0
\(989\) 18457.3 0.593435
\(990\) 0 0
\(991\) −43262.0 −1.38674 −0.693371 0.720581i \(-0.743875\pi\)
−0.693371 + 0.720581i \(0.743875\pi\)
\(992\) 0 0
\(993\) 559.610 0.0178839
\(994\) 0 0
\(995\) −103069. −3.28394
\(996\) 0 0
\(997\) 35664.7 1.13291 0.566456 0.824092i \(-0.308314\pi\)
0.566456 + 0.824092i \(0.308314\pi\)
\(998\) 0 0
\(999\) 48.0539 0.00152188
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 768.4.a.q.1.1 3
3.2 odd 2 2304.4.a.bw.1.3 3
4.3 odd 2 768.4.a.s.1.1 3
8.3 odd 2 768.4.a.r.1.3 3
8.5 even 2 768.4.a.t.1.3 3
12.11 even 2 2304.4.a.bv.1.3 3
16.3 odd 4 24.4.d.a.13.1 6
16.5 even 4 96.4.d.a.49.4 6
16.11 odd 4 24.4.d.a.13.2 yes 6
16.13 even 4 96.4.d.a.49.3 6
24.5 odd 2 2304.4.a.bu.1.1 3
24.11 even 2 2304.4.a.bt.1.1 3
48.5 odd 4 288.4.d.d.145.6 6
48.11 even 4 72.4.d.d.37.5 6
48.29 odd 4 288.4.d.d.145.1 6
48.35 even 4 72.4.d.d.37.6 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
24.4.d.a.13.1 6 16.3 odd 4
24.4.d.a.13.2 yes 6 16.11 odd 4
72.4.d.d.37.5 6 48.11 even 4
72.4.d.d.37.6 6 48.35 even 4
96.4.d.a.49.3 6 16.13 even 4
96.4.d.a.49.4 6 16.5 even 4
288.4.d.d.145.1 6 48.29 odd 4
288.4.d.d.145.6 6 48.5 odd 4
768.4.a.q.1.1 3 1.1 even 1 trivial
768.4.a.r.1.3 3 8.3 odd 2
768.4.a.s.1.1 3 4.3 odd 2
768.4.a.t.1.3 3 8.5 even 2
2304.4.a.bt.1.1 3 24.11 even 2
2304.4.a.bu.1.1 3 24.5 odd 2
2304.4.a.bv.1.3 3 12.11 even 2
2304.4.a.bw.1.3 3 3.2 odd 2