Properties

Label 2160.4.a.bs.1.2
Level $2160$
Weight $4$
Character 2160.1
Self dual yes
Analytic conductor $127.444$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2160 = 2^{4} \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2160.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(127.444125612\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.1257.1
Defining polynomial: \( x^{3} - x^{2} - 8x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3}\cdot 3 \)
Twist minimal: no (minimal twist has level 1080)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.14974\) of defining polynomial
Character \(\chi\) \(=\) 2160.1

$q$-expansion

\(f(q)\) \(=\) \(q+5.00000 q^{5} +3.85875 q^{7} +O(q^{10})\) \(q+5.00000 q^{5} +3.85875 q^{7} +42.2783 q^{11} +4.96700 q^{13} +25.8876 q^{17} -28.9958 q^{19} -191.469 q^{23} +25.0000 q^{25} -287.595 q^{29} -52.6915 q^{31} +19.2937 q^{35} -225.550 q^{37} -73.9907 q^{41} +275.370 q^{43} -192.271 q^{47} -328.110 q^{49} -275.204 q^{53} +211.392 q^{55} +497.084 q^{59} +44.0473 q^{61} +24.8350 q^{65} -761.667 q^{67} +264.284 q^{71} +728.781 q^{73} +163.141 q^{77} -664.347 q^{79} +1491.33 q^{83} +129.438 q^{85} -106.783 q^{89} +19.1664 q^{91} -144.979 q^{95} -924.560 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 15 q^{5} + 10 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 15 q^{5} + 10 q^{7} - 28 q^{11} - 78 q^{13} + 11 q^{17} + 71 q^{19} + 25 q^{23} + 75 q^{25} - 118 q^{29} + 107 q^{31} + 50 q^{35} - 410 q^{37} - 592 q^{41} - 52 q^{43} + 580 q^{47} + 479 q^{49} - 169 q^{53} - 140 q^{55} - 234 q^{59} - 673 q^{61} - 390 q^{65} - 386 q^{67} - 16 q^{71} - 892 q^{73} - 1800 q^{77} - 1263 q^{79} + 1815 q^{83} + 55 q^{85} - 1800 q^{89} - 1284 q^{91} + 355 q^{95} - 840 q^{97}+O(q^{100}) \) 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
\(3\) 0 0
\(4\) 0 0
\(5\) 5.00000 0.447214
\(6\) 0 0
\(7\) 3.85875 0.208353 0.104176 0.994559i \(-0.466779\pi\)
0.104176 + 0.994559i \(0.466779\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 42.2783 1.15885 0.579427 0.815024i \(-0.303276\pi\)
0.579427 + 0.815024i \(0.303276\pi\)
\(12\) 0 0
\(13\) 4.96700 0.105969 0.0529845 0.998595i \(-0.483127\pi\)
0.0529845 + 0.998595i \(0.483127\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 25.8876 0.369333 0.184666 0.982801i \(-0.440880\pi\)
0.184666 + 0.982801i \(0.440880\pi\)
\(18\) 0 0
\(19\) −28.9958 −0.350110 −0.175055 0.984559i \(-0.556010\pi\)
−0.175055 + 0.984559i \(0.556010\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −191.469 −1.73583 −0.867914 0.496715i \(-0.834539\pi\)
−0.867914 + 0.496715i \(0.834539\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −287.595 −1.84155 −0.920776 0.390092i \(-0.872443\pi\)
−0.920776 + 0.390092i \(0.872443\pi\)
\(30\) 0 0
\(31\) −52.6915 −0.305280 −0.152640 0.988282i \(-0.548777\pi\)
−0.152640 + 0.988282i \(0.548777\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 19.2937 0.0931782
\(36\) 0 0
\(37\) −225.550 −1.00217 −0.501084 0.865398i \(-0.667065\pi\)
−0.501084 + 0.865398i \(0.667065\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −73.9907 −0.281839 −0.140920 0.990021i \(-0.545006\pi\)
−0.140920 + 0.990021i \(0.545006\pi\)
\(42\) 0 0
\(43\) 275.370 0.976593 0.488297 0.872678i \(-0.337618\pi\)
0.488297 + 0.872678i \(0.337618\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −192.271 −0.596715 −0.298357 0.954454i \(-0.596439\pi\)
−0.298357 + 0.954454i \(0.596439\pi\)
\(48\) 0 0
\(49\) −328.110 −0.956589
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −275.204 −0.713249 −0.356624 0.934248i \(-0.616072\pi\)
−0.356624 + 0.934248i \(0.616072\pi\)
\(54\) 0 0
\(55\) 211.392 0.518255
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 497.084 1.09686 0.548431 0.836196i \(-0.315225\pi\)
0.548431 + 0.836196i \(0.315225\pi\)
\(60\) 0 0
\(61\) 44.0473 0.0924538 0.0462269 0.998931i \(-0.485280\pi\)
0.0462269 + 0.998931i \(0.485280\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 24.8350 0.0473908
\(66\) 0 0
\(67\) −761.667 −1.38884 −0.694422 0.719568i \(-0.744340\pi\)
−0.694422 + 0.719568i \(0.744340\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 264.284 0.441757 0.220879 0.975301i \(-0.429108\pi\)
0.220879 + 0.975301i \(0.429108\pi\)
\(72\) 0 0
\(73\) 728.781 1.16846 0.584229 0.811589i \(-0.301397\pi\)
0.584229 + 0.811589i \(0.301397\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 163.141 0.241450
\(78\) 0 0
\(79\) −664.347 −0.946137 −0.473069 0.881026i \(-0.656854\pi\)
−0.473069 + 0.881026i \(0.656854\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 1491.33 1.97223 0.986113 0.166075i \(-0.0531094\pi\)
0.986113 + 0.166075i \(0.0531094\pi\)
\(84\) 0 0
\(85\) 129.438 0.165171
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −106.783 −0.127180 −0.0635899 0.997976i \(-0.520255\pi\)
−0.0635899 + 0.997976i \(0.520255\pi\)
\(90\) 0 0
\(91\) 19.1664 0.0220789
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −144.979 −0.156574
\(96\) 0 0
\(97\) −924.560 −0.967782 −0.483891 0.875128i \(-0.660777\pi\)
−0.483891 + 0.875128i \(0.660777\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −627.973 −0.618670 −0.309335 0.950953i \(-0.600106\pi\)
−0.309335 + 0.950953i \(0.600106\pi\)
\(102\) 0 0
\(103\) 224.878 0.215125 0.107562 0.994198i \(-0.465695\pi\)
0.107562 + 0.994198i \(0.465695\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −70.1076 −0.0633417 −0.0316708 0.999498i \(-0.510083\pi\)
−0.0316708 + 0.999498i \(0.510083\pi\)
\(108\) 0 0
\(109\) −235.022 −0.206523 −0.103261 0.994654i \(-0.532928\pi\)
−0.103261 + 0.994654i \(0.532928\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −1647.70 −1.37170 −0.685850 0.727743i \(-0.740569\pi\)
−0.685850 + 0.727743i \(0.740569\pi\)
\(114\) 0 0
\(115\) −957.344 −0.776286
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 99.8936 0.0769515
\(120\) 0 0
\(121\) 456.458 0.342943
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 125.000 0.0894427
\(126\) 0 0
\(127\) 1282.64 0.896189 0.448095 0.893986i \(-0.352103\pi\)
0.448095 + 0.893986i \(0.352103\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 1070.64 0.714066 0.357033 0.934092i \(-0.383788\pi\)
0.357033 + 0.934092i \(0.383788\pi\)
\(132\) 0 0
\(133\) −111.888 −0.0729465
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −2045.70 −1.27574 −0.637869 0.770145i \(-0.720184\pi\)
−0.637869 + 0.770145i \(0.720184\pi\)
\(138\) 0 0
\(139\) −784.256 −0.478559 −0.239280 0.970951i \(-0.576911\pi\)
−0.239280 + 0.970951i \(0.576911\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 209.997 0.122803
\(144\) 0 0
\(145\) −1437.97 −0.823567
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −1300.68 −0.715140 −0.357570 0.933886i \(-0.616395\pi\)
−0.357570 + 0.933886i \(0.616395\pi\)
\(150\) 0 0
\(151\) −1511.72 −0.814718 −0.407359 0.913268i \(-0.633550\pi\)
−0.407359 + 0.913268i \(0.633550\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −263.458 −0.136525
\(156\) 0 0
\(157\) 167.967 0.0853838 0.0426919 0.999088i \(-0.486407\pi\)
0.0426919 + 0.999088i \(0.486407\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −738.830 −0.361664
\(162\) 0 0
\(163\) 67.1490 0.0322670 0.0161335 0.999870i \(-0.494864\pi\)
0.0161335 + 0.999870i \(0.494864\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 2568.36 1.19009 0.595046 0.803692i \(-0.297134\pi\)
0.595046 + 0.803692i \(0.297134\pi\)
\(168\) 0 0
\(169\) −2172.33 −0.988771
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 3012.46 1.32389 0.661945 0.749553i \(-0.269731\pi\)
0.661945 + 0.749553i \(0.269731\pi\)
\(174\) 0 0
\(175\) 96.4687 0.0416705
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 2462.80 1.02837 0.514185 0.857679i \(-0.328095\pi\)
0.514185 + 0.857679i \(0.328095\pi\)
\(180\) 0 0
\(181\) −3691.10 −1.51578 −0.757892 0.652380i \(-0.773771\pi\)
−0.757892 + 0.652380i \(0.773771\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −1127.75 −0.448183
\(186\) 0 0
\(187\) 1094.48 0.428003
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 1969.02 0.745932 0.372966 0.927845i \(-0.378341\pi\)
0.372966 + 0.927845i \(0.378341\pi\)
\(192\) 0 0
\(193\) 519.119 0.193612 0.0968058 0.995303i \(-0.469137\pi\)
0.0968058 + 0.995303i \(0.469137\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −3286.33 −1.18853 −0.594267 0.804268i \(-0.702558\pi\)
−0.594267 + 0.804268i \(0.702558\pi\)
\(198\) 0 0
\(199\) −3626.47 −1.29183 −0.645914 0.763410i \(-0.723524\pi\)
−0.645914 + 0.763410i \(0.723524\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −1109.76 −0.383692
\(204\) 0 0
\(205\) −369.954 −0.126042
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −1225.90 −0.405727
\(210\) 0 0
\(211\) −1410.39 −0.460166 −0.230083 0.973171i \(-0.573900\pi\)
−0.230083 + 0.973171i \(0.573900\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 1376.85 0.436746
\(216\) 0 0
\(217\) −203.323 −0.0636059
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 128.584 0.0391379
\(222\) 0 0
\(223\) −4026.25 −1.20905 −0.604524 0.796587i \(-0.706637\pi\)
−0.604524 + 0.796587i \(0.706637\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 2288.33 0.669083 0.334542 0.942381i \(-0.391419\pi\)
0.334542 + 0.942381i \(0.391419\pi\)
\(228\) 0 0
\(229\) 2003.69 0.578200 0.289100 0.957299i \(-0.406644\pi\)
0.289100 + 0.957299i \(0.406644\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −2837.72 −0.797877 −0.398939 0.916978i \(-0.630621\pi\)
−0.398939 + 0.916978i \(0.630621\pi\)
\(234\) 0 0
\(235\) −961.355 −0.266859
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −1749.50 −0.473496 −0.236748 0.971571i \(-0.576082\pi\)
−0.236748 + 0.971571i \(0.576082\pi\)
\(240\) 0 0
\(241\) 3645.10 0.974279 0.487140 0.873324i \(-0.338040\pi\)
0.487140 + 0.873324i \(0.338040\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −1640.55 −0.427800
\(246\) 0 0
\(247\) −144.022 −0.0371009
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −1286.39 −0.323491 −0.161745 0.986833i \(-0.551712\pi\)
−0.161745 + 0.986833i \(0.551712\pi\)
\(252\) 0 0
\(253\) −8094.99 −2.01157
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 2860.83 0.694372 0.347186 0.937796i \(-0.387137\pi\)
0.347186 + 0.937796i \(0.387137\pi\)
\(258\) 0 0
\(259\) −870.341 −0.208805
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 4615.68 1.08219 0.541093 0.840963i \(-0.318011\pi\)
0.541093 + 0.840963i \(0.318011\pi\)
\(264\) 0 0
\(265\) −1376.02 −0.318974
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 3053.54 0.692111 0.346055 0.938214i \(-0.387521\pi\)
0.346055 + 0.938214i \(0.387521\pi\)
\(270\) 0 0
\(271\) 5867.51 1.31523 0.657613 0.753356i \(-0.271566\pi\)
0.657613 + 0.753356i \(0.271566\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 1056.96 0.231771
\(276\) 0 0
\(277\) −7256.41 −1.57399 −0.786995 0.616959i \(-0.788364\pi\)
−0.786995 + 0.616959i \(0.788364\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 1318.05 0.279816 0.139908 0.990165i \(-0.455319\pi\)
0.139908 + 0.990165i \(0.455319\pi\)
\(282\) 0 0
\(283\) −4810.74 −1.01049 −0.505245 0.862976i \(-0.668598\pi\)
−0.505245 + 0.862976i \(0.668598\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −285.511 −0.0587220
\(288\) 0 0
\(289\) −4242.83 −0.863593
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −1512.59 −0.301593 −0.150796 0.988565i \(-0.548184\pi\)
−0.150796 + 0.988565i \(0.548184\pi\)
\(294\) 0 0
\(295\) 2485.42 0.490531
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −951.026 −0.183944
\(300\) 0 0
\(301\) 1062.58 0.203476
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 220.237 0.0413466
\(306\) 0 0
\(307\) −3092.67 −0.574945 −0.287472 0.957789i \(-0.592815\pi\)
−0.287472 + 0.957789i \(0.592815\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 21.7251 0.00396115 0.00198058 0.999998i \(-0.499370\pi\)
0.00198058 + 0.999998i \(0.499370\pi\)
\(312\) 0 0
\(313\) 758.733 0.137016 0.0685082 0.997651i \(-0.478176\pi\)
0.0685082 + 0.997651i \(0.478176\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 1764.70 0.312668 0.156334 0.987704i \(-0.450032\pi\)
0.156334 + 0.987704i \(0.450032\pi\)
\(318\) 0 0
\(319\) −12159.0 −2.13409
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −750.632 −0.129307
\(324\) 0 0
\(325\) 124.175 0.0211938
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −741.925 −0.124327
\(330\) 0 0
\(331\) −10098.1 −1.67687 −0.838433 0.545005i \(-0.816528\pi\)
−0.838433 + 0.545005i \(0.816528\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −3808.34 −0.621109
\(336\) 0 0
\(337\) −4485.28 −0.725012 −0.362506 0.931981i \(-0.618079\pi\)
−0.362506 + 0.931981i \(0.618079\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −2227.71 −0.353775
\(342\) 0 0
\(343\) −2589.64 −0.407661
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −585.137 −0.0905239 −0.0452620 0.998975i \(-0.514412\pi\)
−0.0452620 + 0.998975i \(0.514412\pi\)
\(348\) 0 0
\(349\) 3461.97 0.530989 0.265494 0.964112i \(-0.414465\pi\)
0.265494 + 0.964112i \(0.414465\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 6797.01 1.02484 0.512420 0.858735i \(-0.328749\pi\)
0.512420 + 0.858735i \(0.328749\pi\)
\(354\) 0 0
\(355\) 1321.42 0.197560
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −4622.13 −0.679518 −0.339759 0.940513i \(-0.610345\pi\)
−0.339759 + 0.940513i \(0.610345\pi\)
\(360\) 0 0
\(361\) −6018.24 −0.877423
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 3643.91 0.522550
\(366\) 0 0
\(367\) 147.843 0.0210281 0.0105141 0.999945i \(-0.496653\pi\)
0.0105141 + 0.999945i \(0.496653\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −1061.94 −0.148607
\(372\) 0 0
\(373\) 8840.27 1.22716 0.613582 0.789631i \(-0.289728\pi\)
0.613582 + 0.789631i \(0.289728\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −1428.48 −0.195148
\(378\) 0 0
\(379\) 6063.51 0.821799 0.410899 0.911681i \(-0.365215\pi\)
0.410899 + 0.911681i \(0.365215\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 496.589 0.0662520 0.0331260 0.999451i \(-0.489454\pi\)
0.0331260 + 0.999451i \(0.489454\pi\)
\(384\) 0 0
\(385\) 815.707 0.107980
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 3432.37 0.447373 0.223686 0.974661i \(-0.428191\pi\)
0.223686 + 0.974661i \(0.428191\pi\)
\(390\) 0 0
\(391\) −4956.66 −0.641098
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −3321.73 −0.423125
\(396\) 0 0
\(397\) −15311.8 −1.93571 −0.967856 0.251506i \(-0.919074\pi\)
−0.967856 + 0.251506i \(0.919074\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −7088.63 −0.882766 −0.441383 0.897319i \(-0.645512\pi\)
−0.441383 + 0.897319i \(0.645512\pi\)
\(402\) 0 0
\(403\) −261.719 −0.0323502
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −9535.89 −1.16137
\(408\) 0 0
\(409\) 9991.23 1.20791 0.603954 0.797019i \(-0.293591\pi\)
0.603954 + 0.797019i \(0.293591\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 1918.12 0.228534
\(414\) 0 0
\(415\) 7456.65 0.882006
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −8172.83 −0.952909 −0.476455 0.879199i \(-0.658078\pi\)
−0.476455 + 0.879199i \(0.658078\pi\)
\(420\) 0 0
\(421\) −15475.1 −1.79147 −0.895734 0.444590i \(-0.853350\pi\)
−0.895734 + 0.444590i \(0.853350\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 647.189 0.0738666
\(426\) 0 0
\(427\) 169.968 0.0192630
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −3426.00 −0.382888 −0.191444 0.981504i \(-0.561317\pi\)
−0.191444 + 0.981504i \(0.561317\pi\)
\(432\) 0 0
\(433\) −1806.95 −0.200546 −0.100273 0.994960i \(-0.531972\pi\)
−0.100273 + 0.994960i \(0.531972\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 5551.80 0.607731
\(438\) 0 0
\(439\) −7974.34 −0.866957 −0.433479 0.901164i \(-0.642714\pi\)
−0.433479 + 0.901164i \(0.642714\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −3566.86 −0.382544 −0.191272 0.981537i \(-0.561261\pi\)
−0.191272 + 0.981537i \(0.561261\pi\)
\(444\) 0 0
\(445\) −533.916 −0.0568765
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −13687.2 −1.43862 −0.719308 0.694692i \(-0.755541\pi\)
−0.719308 + 0.694692i \(0.755541\pi\)
\(450\) 0 0
\(451\) −3128.21 −0.326611
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 95.8320 0.00987401
\(456\) 0 0
\(457\) 1691.89 0.173180 0.0865902 0.996244i \(-0.472403\pi\)
0.0865902 + 0.996244i \(0.472403\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 17881.6 1.80657 0.903285 0.429041i \(-0.141149\pi\)
0.903285 + 0.429041i \(0.141149\pi\)
\(462\) 0 0
\(463\) −11459.6 −1.15027 −0.575135 0.818059i \(-0.695050\pi\)
−0.575135 + 0.818059i \(0.695050\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −18329.4 −1.81624 −0.908119 0.418711i \(-0.862482\pi\)
−0.908119 + 0.418711i \(0.862482\pi\)
\(468\) 0 0
\(469\) −2939.08 −0.289369
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 11642.2 1.13173
\(474\) 0 0
\(475\) −724.896 −0.0700221
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 15475.4 1.47618 0.738090 0.674703i \(-0.235728\pi\)
0.738090 + 0.674703i \(0.235728\pi\)
\(480\) 0 0
\(481\) −1120.31 −0.106199
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −4622.80 −0.432805
\(486\) 0 0
\(487\) 16320.0 1.51854 0.759270 0.650776i \(-0.225556\pi\)
0.759270 + 0.650776i \(0.225556\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −6138.33 −0.564193 −0.282097 0.959386i \(-0.591030\pi\)
−0.282097 + 0.959386i \(0.591030\pi\)
\(492\) 0 0
\(493\) −7445.13 −0.680146
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 1019.81 0.0920413
\(498\) 0 0
\(499\) −3272.20 −0.293555 −0.146777 0.989170i \(-0.546890\pi\)
−0.146777 + 0.989170i \(0.546890\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −4696.96 −0.416356 −0.208178 0.978091i \(-0.566753\pi\)
−0.208178 + 0.978091i \(0.566753\pi\)
\(504\) 0 0
\(505\) −3139.87 −0.276678
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −10011.5 −0.871815 −0.435907 0.899992i \(-0.643572\pi\)
−0.435907 + 0.899992i \(0.643572\pi\)
\(510\) 0 0
\(511\) 2812.18 0.243451
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 1124.39 0.0962067
\(516\) 0 0
\(517\) −8128.89 −0.691505
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 19483.2 1.63833 0.819167 0.573554i \(-0.194436\pi\)
0.819167 + 0.573554i \(0.194436\pi\)
\(522\) 0 0
\(523\) 5181.82 0.433241 0.216621 0.976256i \(-0.430497\pi\)
0.216621 + 0.976256i \(0.430497\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −1364.06 −0.112750
\(528\) 0 0
\(529\) 24493.3 2.01310
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −367.512 −0.0298663
\(534\) 0 0
\(535\) −350.538 −0.0283273
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −13871.9 −1.10855
\(540\) 0 0
\(541\) 3178.24 0.252576 0.126288 0.991994i \(-0.459694\pi\)
0.126288 + 0.991994i \(0.459694\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −1175.11 −0.0923599
\(546\) 0 0
\(547\) 13887.9 1.08556 0.542781 0.839874i \(-0.317371\pi\)
0.542781 + 0.839874i \(0.317371\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 8339.05 0.644747
\(552\) 0 0
\(553\) −2563.55 −0.197130
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 10574.5 0.804406 0.402203 0.915551i \(-0.368245\pi\)
0.402203 + 0.915551i \(0.368245\pi\)
\(558\) 0 0
\(559\) 1367.76 0.103489
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 7815.91 0.585083 0.292541 0.956253i \(-0.405499\pi\)
0.292541 + 0.956253i \(0.405499\pi\)
\(564\) 0 0
\(565\) −8238.48 −0.613443
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 20024.0 1.47531 0.737655 0.675178i \(-0.235933\pi\)
0.737655 + 0.675178i \(0.235933\pi\)
\(570\) 0 0
\(571\) −23886.5 −1.75065 −0.875323 0.483538i \(-0.839351\pi\)
−0.875323 + 0.483538i \(0.839351\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −4786.72 −0.347165
\(576\) 0 0
\(577\) −18777.2 −1.35478 −0.677389 0.735625i \(-0.736888\pi\)
−0.677389 + 0.735625i \(0.736888\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 5754.67 0.410919
\(582\) 0 0
\(583\) −11635.2 −0.826551
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 23209.8 1.63198 0.815988 0.578069i \(-0.196194\pi\)
0.815988 + 0.578069i \(0.196194\pi\)
\(588\) 0 0
\(589\) 1527.83 0.106882
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 4204.44 0.291157 0.145578 0.989347i \(-0.453496\pi\)
0.145578 + 0.989347i \(0.453496\pi\)
\(594\) 0 0
\(595\) 499.468 0.0344138
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −26962.9 −1.83919 −0.919593 0.392872i \(-0.871482\pi\)
−0.919593 + 0.392872i \(0.871482\pi\)
\(600\) 0 0
\(601\) 13595.2 0.922727 0.461363 0.887211i \(-0.347360\pi\)
0.461363 + 0.887211i \(0.347360\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 2282.29 0.153369
\(606\) 0 0
\(607\) −14337.5 −0.958720 −0.479360 0.877618i \(-0.659131\pi\)
−0.479360 + 0.877618i \(0.659131\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −955.010 −0.0632333
\(612\) 0 0
\(613\) 7963.62 0.524710 0.262355 0.964971i \(-0.415501\pi\)
0.262355 + 0.964971i \(0.415501\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 23594.2 1.53949 0.769745 0.638352i \(-0.220383\pi\)
0.769745 + 0.638352i \(0.220383\pi\)
\(618\) 0 0
\(619\) −12132.6 −0.787803 −0.393901 0.919153i \(-0.628875\pi\)
−0.393901 + 0.919153i \(0.628875\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −412.049 −0.0264982
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −5838.95 −0.370134
\(630\) 0 0
\(631\) 614.270 0.0387539 0.0193769 0.999812i \(-0.493832\pi\)
0.0193769 + 0.999812i \(0.493832\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 6413.21 0.400788
\(636\) 0 0
\(637\) −1629.72 −0.101369
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −2057.06 −0.126754 −0.0633769 0.997990i \(-0.520187\pi\)
−0.0633769 + 0.997990i \(0.520187\pi\)
\(642\) 0 0
\(643\) −17827.2 −1.09337 −0.546684 0.837339i \(-0.684110\pi\)
−0.546684 + 0.837339i \(0.684110\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 17499.4 1.06333 0.531664 0.846955i \(-0.321567\pi\)
0.531664 + 0.846955i \(0.321567\pi\)
\(648\) 0 0
\(649\) 21015.9 1.27110
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 6049.52 0.362536 0.181268 0.983434i \(-0.441980\pi\)
0.181268 + 0.983434i \(0.441980\pi\)
\(654\) 0 0
\(655\) 5353.22 0.319340
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −1436.32 −0.0849027 −0.0424514 0.999099i \(-0.513517\pi\)
−0.0424514 + 0.999099i \(0.513517\pi\)
\(660\) 0 0
\(661\) 9287.18 0.546489 0.273245 0.961945i \(-0.411903\pi\)
0.273245 + 0.961945i \(0.411903\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −559.438 −0.0326226
\(666\) 0 0
\(667\) 55065.4 3.19662
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 1862.25 0.107141
\(672\) 0 0
\(673\) 15014.9 0.860000 0.430000 0.902829i \(-0.358514\pi\)
0.430000 + 0.902829i \(0.358514\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −9147.19 −0.519284 −0.259642 0.965705i \(-0.583604\pi\)
−0.259642 + 0.965705i \(0.583604\pi\)
\(678\) 0 0
\(679\) −3567.64 −0.201640
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 10388.4 0.581992 0.290996 0.956724i \(-0.406013\pi\)
0.290996 + 0.956724i \(0.406013\pi\)
\(684\) 0 0
\(685\) −10228.5 −0.570527
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −1366.94 −0.0755823
\(690\) 0 0
\(691\) 25955.9 1.42895 0.714477 0.699659i \(-0.246665\pi\)
0.714477 + 0.699659i \(0.246665\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −3921.28 −0.214018
\(696\) 0 0
\(697\) −1915.44 −0.104093
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −31771.2 −1.71181 −0.855906 0.517131i \(-0.827000\pi\)
−0.855906 + 0.517131i \(0.827000\pi\)
\(702\) 0 0
\(703\) 6540.02 0.350870
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −2423.19 −0.128902
\(708\) 0 0
\(709\) −7002.47 −0.370921 −0.185461 0.982652i \(-0.559378\pi\)
−0.185461 + 0.982652i \(0.559378\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 10088.8 0.529913
\(714\) 0 0
\(715\) 1049.98 0.0549191
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −34634.3 −1.79644 −0.898221 0.439545i \(-0.855140\pi\)
−0.898221 + 0.439545i \(0.855140\pi\)
\(720\) 0 0
\(721\) 867.745 0.0448218
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −7189.87 −0.368310
\(726\) 0 0
\(727\) −10821.6 −0.552064 −0.276032 0.961148i \(-0.589020\pi\)
−0.276032 + 0.961148i \(0.589020\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 7128.66 0.360688
\(732\) 0 0
\(733\) −5576.04 −0.280977 −0.140488 0.990082i \(-0.544867\pi\)
−0.140488 + 0.990082i \(0.544867\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −32202.0 −1.60947
\(738\) 0 0
\(739\) −29317.7 −1.45936 −0.729681 0.683788i \(-0.760331\pi\)
−0.729681 + 0.683788i \(0.760331\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −7521.08 −0.371361 −0.185681 0.982610i \(-0.559449\pi\)
−0.185681 + 0.982610i \(0.559449\pi\)
\(744\) 0 0
\(745\) −6503.40 −0.319820
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −270.528 −0.0131974
\(750\) 0 0
\(751\) 37300.7 1.81241 0.906206 0.422837i \(-0.138966\pi\)
0.906206 + 0.422837i \(0.138966\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −7558.62 −0.364353
\(756\) 0 0
\(757\) −26432.1 −1.26908 −0.634538 0.772891i \(-0.718810\pi\)
−0.634538 + 0.772891i \(0.718810\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −23440.4 −1.11658 −0.558288 0.829647i \(-0.688542\pi\)
−0.558288 + 0.829647i \(0.688542\pi\)
\(762\) 0 0
\(763\) −906.889 −0.0430296
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 2469.02 0.116233
\(768\) 0 0
\(769\) 34075.2 1.59790 0.798948 0.601401i \(-0.205390\pi\)
0.798948 + 0.601401i \(0.205390\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 14623.5 0.680427 0.340214 0.940348i \(-0.389501\pi\)
0.340214 + 0.940348i \(0.389501\pi\)
\(774\) 0 0
\(775\) −1317.29 −0.0610560
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 2145.42 0.0986749
\(780\) 0 0
\(781\) 11173.5 0.511932
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 839.837 0.0381848
\(786\) 0 0
\(787\) 33523.1 1.51839 0.759193 0.650865i \(-0.225594\pi\)
0.759193 + 0.650865i \(0.225594\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −6358.04 −0.285797
\(792\) 0 0
\(793\) 218.783 0.00979725
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −29447.5 −1.30876 −0.654381 0.756165i \(-0.727071\pi\)
−0.654381 + 0.756165i \(0.727071\pi\)
\(798\) 0 0
\(799\) −4977.43 −0.220386
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 30811.6 1.35407
\(804\) 0 0
\(805\) −3694.15 −0.161741
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −26891.1 −1.16866 −0.584328 0.811518i \(-0.698642\pi\)
−0.584328 + 0.811518i \(0.698642\pi\)
\(810\) 0 0
\(811\) −14912.6 −0.645687 −0.322844 0.946452i \(-0.604639\pi\)
−0.322844 + 0.946452i \(0.604639\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 335.745 0.0144302
\(816\) 0 0
\(817\) −7984.58 −0.341916
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 42644.2 1.81278 0.906391 0.422439i \(-0.138826\pi\)
0.906391 + 0.422439i \(0.138826\pi\)
\(822\) 0 0
\(823\) 25121.7 1.06402 0.532010 0.846738i \(-0.321437\pi\)
0.532010 + 0.846738i \(0.321437\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −4918.93 −0.206830 −0.103415 0.994638i \(-0.532977\pi\)
−0.103415 + 0.994638i \(0.532977\pi\)
\(828\) 0 0
\(829\) −22807.6 −0.955538 −0.477769 0.878486i \(-0.658554\pi\)
−0.477769 + 0.878486i \(0.658554\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −8493.97 −0.353300
\(834\) 0 0
\(835\) 12841.8 0.532225
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 25772.5 1.06051 0.530255 0.847839i \(-0.322096\pi\)
0.530255 + 0.847839i \(0.322096\pi\)
\(840\) 0 0
\(841\) 58321.7 2.39131
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −10861.6 −0.442192
\(846\) 0 0
\(847\) 1761.35 0.0714532
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 43185.9 1.73959
\(852\) 0 0
\(853\) 41636.0 1.67126 0.835632 0.549289i \(-0.185101\pi\)
0.835632 + 0.549289i \(0.185101\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 17941.6 0.715137 0.357569 0.933887i \(-0.383606\pi\)
0.357569 + 0.933887i \(0.383606\pi\)
\(858\) 0 0
\(859\) −23917.0 −0.949985 −0.474993 0.879990i \(-0.657549\pi\)
−0.474993 + 0.879990i \(0.657549\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 9786.53 0.386022 0.193011 0.981197i \(-0.438175\pi\)
0.193011 + 0.981197i \(0.438175\pi\)
\(864\) 0 0
\(865\) 15062.3 0.592061
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −28087.5 −1.09644
\(870\) 0 0
\(871\) −3783.20 −0.147174
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 482.343 0.0186356
\(876\) 0 0
\(877\) −29186.8 −1.12379 −0.561897 0.827207i \(-0.689928\pi\)
−0.561897 + 0.827207i \(0.689928\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 18426.6 0.704664 0.352332 0.935875i \(-0.385389\pi\)
0.352332 + 0.935875i \(0.385389\pi\)
\(882\) 0 0
\(883\) −3942.63 −0.150261 −0.0751303 0.997174i \(-0.523937\pi\)
−0.0751303 + 0.997174i \(0.523937\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −436.876 −0.0165376 −0.00826881 0.999966i \(-0.502632\pi\)
−0.00826881 + 0.999966i \(0.502632\pi\)
\(888\) 0 0
\(889\) 4949.39 0.186723
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 5575.05 0.208916
\(894\) 0 0
\(895\) 12314.0 0.459901
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 15153.8 0.562189
\(900\) 0 0
\(901\) −7124.36 −0.263426
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −18455.5 −0.677879
\(906\) 0 0
\(907\) 43243.2 1.58309 0.791547 0.611108i \(-0.209276\pi\)
0.791547 + 0.611108i \(0.209276\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −14141.5 −0.514301 −0.257150 0.966371i \(-0.582784\pi\)
−0.257150 + 0.966371i \(0.582784\pi\)
\(912\) 0 0
\(913\) 63051.0 2.28552
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 4131.34 0.148778
\(918\) 0 0
\(919\) −1506.36 −0.0540697 −0.0270349 0.999634i \(-0.508607\pi\)
−0.0270349 + 0.999634i \(0.508607\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 1312.70 0.0468126
\(924\) 0 0
\(925\) −5638.76 −0.200434
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −32173.3 −1.13624 −0.568122 0.822944i \(-0.692330\pi\)
−0.568122 + 0.822944i \(0.692330\pi\)
\(930\) 0 0
\(931\) 9513.82 0.334912
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 5472.42 0.191409
\(936\) 0 0
\(937\) 34833.5 1.21447 0.607236 0.794522i \(-0.292278\pi\)
0.607236 + 0.794522i \(0.292278\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 3709.66 0.128514 0.0642569 0.997933i \(-0.479532\pi\)
0.0642569 + 0.997933i \(0.479532\pi\)
\(942\) 0 0
\(943\) 14166.9 0.489224
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 32085.0 1.10097 0.550487 0.834844i \(-0.314442\pi\)
0.550487 + 0.834844i \(0.314442\pi\)
\(948\) 0 0
\(949\) 3619.86 0.123820
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −35654.2 −1.21191 −0.605956 0.795498i \(-0.707209\pi\)
−0.605956 + 0.795498i \(0.707209\pi\)
\(954\) 0 0
\(955\) 9845.08 0.333591
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −7893.84 −0.265803
\(960\) 0 0
\(961\) −27014.6 −0.906804
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 2595.60 0.0865857
\(966\) 0 0
\(967\) 32345.4 1.07565 0.537827 0.843055i \(-0.319245\pi\)
0.537827 + 0.843055i \(0.319245\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −20137.4 −0.665542 −0.332771 0.943008i \(-0.607984\pi\)
−0.332771 + 0.943008i \(0.607984\pi\)
\(972\) 0 0
\(973\) −3026.25 −0.0997091
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 30087.1 0.985231 0.492615 0.870247i \(-0.336041\pi\)
0.492615 + 0.870247i \(0.336041\pi\)
\(978\) 0 0
\(979\) −4514.62 −0.147383
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 59417.5 1.92790 0.963950 0.266084i \(-0.0857299\pi\)
0.963950 + 0.266084i \(0.0857299\pi\)
\(984\) 0 0
\(985\) −16431.6 −0.531529
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −52724.8 −1.69520
\(990\) 0 0
\(991\) 20153.5 0.646011 0.323006 0.946397i \(-0.395307\pi\)
0.323006 + 0.946397i \(0.395307\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −18132.4 −0.577723
\(996\) 0 0
\(997\) −20772.3 −0.659846 −0.329923 0.944008i \(-0.607023\pi\)
−0.329923 + 0.944008i \(0.607023\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 2160.4.a.bs.1.2 3
3.2 odd 2 2160.4.a.bk.1.2 3
4.3 odd 2 1080.4.a.j.1.2 yes 3
12.11 even 2 1080.4.a.d.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1080.4.a.d.1.2 3 12.11 even 2
1080.4.a.j.1.2 yes 3 4.3 odd 2
2160.4.a.bk.1.2 3 3.2 odd 2
2160.4.a.bs.1.2 3 1.1 even 1 trivial