Properties

Label 768.4.a.q.1.2
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 1.2
Root \(3.76644\) of defining polynomial
Character \(\chi\) \(=\) 768.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.00000 q^{3} -0.612661 q^{5} -22.7441 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q-3.00000 q^{3} -0.612661 q^{5} -22.7441 q^{7} +9.00000 q^{9} +60.2630 q^{11} -52.9062 q^{13} +1.83798 q^{15} +47.1643 q^{17} +29.1643 q^{19} +68.2324 q^{21} +109.488 q^{23} -124.625 q^{25} -27.0000 q^{27} +10.4250 q^{29} +220.881 q^{31} -180.789 q^{33} +13.9345 q^{35} -408.348 q^{37} +158.718 q^{39} +360.742 q^{41} -236.414 q^{43} -5.51395 q^{45} -129.113 q^{47} +174.296 q^{49} -141.493 q^{51} -117.819 q^{53} -36.9208 q^{55} -87.4928 q^{57} -262.854 q^{59} -273.465 q^{61} -204.697 q^{63} +32.4135 q^{65} +89.4077 q^{67} -328.465 q^{69} -350.521 q^{71} -532.610 q^{73} +373.874 q^{75} -1370.63 q^{77} +166.561 q^{79} +81.0000 q^{81} -361.934 q^{83} -28.8957 q^{85} -31.2750 q^{87} -40.3285 q^{89} +1203.31 q^{91} -662.642 q^{93} -17.8678 q^{95} -614.921 q^{97} +542.367 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 9 q^{3} - 10 q^{5} + 14 q^{7} + 27 q^{9}+O(q^{10}) \) 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
\(5\) −0.612661 −0.0547981 −0.0273990 0.999625i \(-0.508722\pi\)
−0.0273990 + 0.999625i \(0.508722\pi\)
\(6\) 0 0
\(7\) −22.7441 −1.22807 −0.614034 0.789279i \(-0.710454\pi\)
−0.614034 + 0.789279i \(0.710454\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 60.2630 1.65182 0.825908 0.563805i \(-0.190663\pi\)
0.825908 + 0.563805i \(0.190663\pi\)
\(12\) 0 0
\(13\) −52.9062 −1.12873 −0.564367 0.825524i \(-0.690880\pi\)
−0.564367 + 0.825524i \(0.690880\pi\)
\(14\) 0 0
\(15\) 1.83798 0.0316377
\(16\) 0 0
\(17\) 47.1643 0.672883 0.336442 0.941704i \(-0.390777\pi\)
0.336442 + 0.941704i \(0.390777\pi\)
\(18\) 0 0
\(19\) 29.1643 0.352144 0.176072 0.984377i \(-0.443661\pi\)
0.176072 + 0.984377i \(0.443661\pi\)
\(20\) 0 0
\(21\) 68.2324 0.709026
\(22\) 0 0
\(23\) 109.488 0.992604 0.496302 0.868150i \(-0.334691\pi\)
0.496302 + 0.868150i \(0.334691\pi\)
\(24\) 0 0
\(25\) −124.625 −0.996997
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) 10.4250 0.0667542 0.0333771 0.999443i \(-0.489374\pi\)
0.0333771 + 0.999443i \(0.489374\pi\)
\(30\) 0 0
\(31\) 220.881 1.27972 0.639860 0.768492i \(-0.278992\pi\)
0.639860 + 0.768492i \(0.278992\pi\)
\(32\) 0 0
\(33\) −180.789 −0.953676
\(34\) 0 0
\(35\) 13.9345 0.0672958
\(36\) 0 0
\(37\) −408.348 −1.81438 −0.907188 0.420725i \(-0.861776\pi\)
−0.907188 + 0.420725i \(0.861776\pi\)
\(38\) 0 0
\(39\) 158.718 0.651674
\(40\) 0 0
\(41\) 360.742 1.37411 0.687054 0.726606i \(-0.258903\pi\)
0.687054 + 0.726606i \(0.258903\pi\)
\(42\) 0 0
\(43\) −236.414 −0.838436 −0.419218 0.907886i \(-0.637696\pi\)
−0.419218 + 0.907886i \(0.637696\pi\)
\(44\) 0 0
\(45\) −5.51395 −0.0182660
\(46\) 0 0
\(47\) −129.113 −0.400703 −0.200352 0.979724i \(-0.564208\pi\)
−0.200352 + 0.979724i \(0.564208\pi\)
\(48\) 0 0
\(49\) 174.296 0.508152
\(50\) 0 0
\(51\) −141.493 −0.388489
\(52\) 0 0
\(53\) −117.819 −0.305353 −0.152677 0.988276i \(-0.548789\pi\)
−0.152677 + 0.988276i \(0.548789\pi\)
\(54\) 0 0
\(55\) −36.9208 −0.0905163
\(56\) 0 0
\(57\) −87.4928 −0.203311
\(58\) 0 0
\(59\) −262.854 −0.580012 −0.290006 0.957025i \(-0.593657\pi\)
−0.290006 + 0.957025i \(0.593657\pi\)
\(60\) 0 0
\(61\) −273.465 −0.573993 −0.286996 0.957932i \(-0.592657\pi\)
−0.286996 + 0.957932i \(0.592657\pi\)
\(62\) 0 0
\(63\) −204.697 −0.409356
\(64\) 0 0
\(65\) 32.4135 0.0618524
\(66\) 0 0
\(67\) 89.4077 0.163028 0.0815141 0.996672i \(-0.474024\pi\)
0.0815141 + 0.996672i \(0.474024\pi\)
\(68\) 0 0
\(69\) −328.465 −0.573080
\(70\) 0 0
\(71\) −350.521 −0.585904 −0.292952 0.956127i \(-0.594638\pi\)
−0.292952 + 0.956127i \(0.594638\pi\)
\(72\) 0 0
\(73\) −532.610 −0.853936 −0.426968 0.904267i \(-0.640418\pi\)
−0.426968 + 0.904267i \(0.640418\pi\)
\(74\) 0 0
\(75\) 373.874 0.575617
\(76\) 0 0
\(77\) −1370.63 −2.02854
\(78\) 0 0
\(79\) 166.561 0.237210 0.118605 0.992942i \(-0.462158\pi\)
0.118605 + 0.992942i \(0.462158\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) −361.934 −0.478644 −0.239322 0.970940i \(-0.576925\pi\)
−0.239322 + 0.970940i \(0.576925\pi\)
\(84\) 0 0
\(85\) −28.8957 −0.0368727
\(86\) 0 0
\(87\) −31.2750 −0.0385405
\(88\) 0 0
\(89\) −40.3285 −0.0480316 −0.0240158 0.999712i \(-0.507645\pi\)
−0.0240158 + 0.999712i \(0.507645\pi\)
\(90\) 0 0
\(91\) 1203.31 1.38616
\(92\) 0 0
\(93\) −662.642 −0.738846
\(94\) 0 0
\(95\) −17.8678 −0.0192968
\(96\) 0 0
\(97\) −614.921 −0.643667 −0.321834 0.946796i \(-0.604299\pi\)
−0.321834 + 0.946796i \(0.604299\pi\)
\(98\) 0 0
\(99\) 542.367 0.550605
\(100\) 0 0
\(101\) −1664.99 −1.64033 −0.820163 0.572130i \(-0.806117\pi\)
−0.820163 + 0.572130i \(0.806117\pi\)
\(102\) 0 0
\(103\) 396.858 0.379647 0.189823 0.981818i \(-0.439208\pi\)
0.189823 + 0.981818i \(0.439208\pi\)
\(104\) 0 0
\(105\) −41.8034 −0.0388532
\(106\) 0 0
\(107\) 350.630 0.316791 0.158396 0.987376i \(-0.449368\pi\)
0.158396 + 0.987376i \(0.449368\pi\)
\(108\) 0 0
\(109\) −597.009 −0.524615 −0.262308 0.964984i \(-0.584484\pi\)
−0.262308 + 0.964984i \(0.584484\pi\)
\(110\) 0 0
\(111\) 1225.04 1.04753
\(112\) 0 0
\(113\) 496.422 0.413270 0.206635 0.978418i \(-0.433749\pi\)
0.206635 + 0.978418i \(0.433749\pi\)
\(114\) 0 0
\(115\) −67.0792 −0.0543928
\(116\) 0 0
\(117\) −476.155 −0.376244
\(118\) 0 0
\(119\) −1072.71 −0.826346
\(120\) 0 0
\(121\) 2300.63 1.72849
\(122\) 0 0
\(123\) −1082.23 −0.793342
\(124\) 0 0
\(125\) 152.935 0.109432
\(126\) 0 0
\(127\) −1799.85 −1.25756 −0.628782 0.777581i \(-0.716446\pi\)
−0.628782 + 0.777581i \(0.716446\pi\)
\(128\) 0 0
\(129\) 709.241 0.484071
\(130\) 0 0
\(131\) 1121.45 0.747949 0.373974 0.927439i \(-0.377995\pi\)
0.373974 + 0.927439i \(0.377995\pi\)
\(132\) 0 0
\(133\) −663.316 −0.432457
\(134\) 0 0
\(135\) 16.5419 0.0105459
\(136\) 0 0
\(137\) −2449.55 −1.52759 −0.763793 0.645461i \(-0.776665\pi\)
−0.763793 + 0.645461i \(0.776665\pi\)
\(138\) 0 0
\(139\) −2457.56 −1.49962 −0.749811 0.661652i \(-0.769856\pi\)
−0.749811 + 0.661652i \(0.769856\pi\)
\(140\) 0 0
\(141\) 387.339 0.231346
\(142\) 0 0
\(143\) −3188.28 −1.86446
\(144\) 0 0
\(145\) −6.38698 −0.00365800
\(146\) 0 0
\(147\) −522.888 −0.293382
\(148\) 0 0
\(149\) −2084.96 −1.14635 −0.573177 0.819432i \(-0.694289\pi\)
−0.573177 + 0.819432i \(0.694289\pi\)
\(150\) 0 0
\(151\) 1057.80 0.570084 0.285042 0.958515i \(-0.407992\pi\)
0.285042 + 0.958515i \(0.407992\pi\)
\(152\) 0 0
\(153\) 424.478 0.224294
\(154\) 0 0
\(155\) −135.325 −0.0701262
\(156\) 0 0
\(157\) −3193.01 −1.62312 −0.811559 0.584270i \(-0.801381\pi\)
−0.811559 + 0.584270i \(0.801381\pi\)
\(158\) 0 0
\(159\) 353.458 0.176296
\(160\) 0 0
\(161\) −2490.22 −1.21899
\(162\) 0 0
\(163\) −846.854 −0.406937 −0.203469 0.979081i \(-0.565221\pi\)
−0.203469 + 0.979081i \(0.565221\pi\)
\(164\) 0 0
\(165\) 110.762 0.0522596
\(166\) 0 0
\(167\) −2630.15 −1.21873 −0.609363 0.792892i \(-0.708575\pi\)
−0.609363 + 0.792892i \(0.708575\pi\)
\(168\) 0 0
\(169\) 602.062 0.274038
\(170\) 0 0
\(171\) 262.478 0.117381
\(172\) 0 0
\(173\) −429.843 −0.188904 −0.0944519 0.995529i \(-0.530110\pi\)
−0.0944519 + 0.995529i \(0.530110\pi\)
\(174\) 0 0
\(175\) 2834.48 1.22438
\(176\) 0 0
\(177\) 788.563 0.334870
\(178\) 0 0
\(179\) −1516.30 −0.633149 −0.316574 0.948568i \(-0.602533\pi\)
−0.316574 + 0.948568i \(0.602533\pi\)
\(180\) 0 0
\(181\) −3380.20 −1.38811 −0.694056 0.719921i \(-0.744178\pi\)
−0.694056 + 0.719921i \(0.744178\pi\)
\(182\) 0 0
\(183\) 820.394 0.331395
\(184\) 0 0
\(185\) 250.179 0.0994244
\(186\) 0 0
\(187\) 2842.26 1.11148
\(188\) 0 0
\(189\) 614.092 0.236342
\(190\) 0 0
\(191\) 2799.71 1.06063 0.530314 0.847801i \(-0.322074\pi\)
0.530314 + 0.847801i \(0.322074\pi\)
\(192\) 0 0
\(193\) 624.106 0.232768 0.116384 0.993204i \(-0.462870\pi\)
0.116384 + 0.993204i \(0.462870\pi\)
\(194\) 0 0
\(195\) −97.2406 −0.0357105
\(196\) 0 0
\(197\) 4779.25 1.72846 0.864232 0.503094i \(-0.167805\pi\)
0.864232 + 0.503094i \(0.167805\pi\)
\(198\) 0 0
\(199\) 2615.92 0.931846 0.465923 0.884825i \(-0.345722\pi\)
0.465923 + 0.884825i \(0.345722\pi\)
\(200\) 0 0
\(201\) −268.223 −0.0941244
\(202\) 0 0
\(203\) −237.107 −0.0819787
\(204\) 0 0
\(205\) −221.013 −0.0752985
\(206\) 0 0
\(207\) 985.395 0.330868
\(208\) 0 0
\(209\) 1757.52 0.581677
\(210\) 0 0
\(211\) −1745.78 −0.569595 −0.284798 0.958588i \(-0.591926\pi\)
−0.284798 + 0.958588i \(0.591926\pi\)
\(212\) 0 0
\(213\) 1051.56 0.338272
\(214\) 0 0
\(215\) 144.841 0.0459447
\(216\) 0 0
\(217\) −5023.74 −1.57158
\(218\) 0 0
\(219\) 1597.83 0.493020
\(220\) 0 0
\(221\) −2495.28 −0.759505
\(222\) 0 0
\(223\) 3385.60 1.01667 0.508333 0.861161i \(-0.330262\pi\)
0.508333 + 0.861161i \(0.330262\pi\)
\(224\) 0 0
\(225\) −1121.62 −0.332332
\(226\) 0 0
\(227\) 3847.72 1.12503 0.562515 0.826787i \(-0.309834\pi\)
0.562515 + 0.826787i \(0.309834\pi\)
\(228\) 0 0
\(229\) 1335.15 0.385279 0.192640 0.981270i \(-0.438295\pi\)
0.192640 + 0.981270i \(0.438295\pi\)
\(230\) 0 0
\(231\) 4111.89 1.17118
\(232\) 0 0
\(233\) 5146.38 1.44700 0.723499 0.690325i \(-0.242532\pi\)
0.723499 + 0.690325i \(0.242532\pi\)
\(234\) 0 0
\(235\) 79.1025 0.0219578
\(236\) 0 0
\(237\) −499.683 −0.136953
\(238\) 0 0
\(239\) −7085.07 −1.91755 −0.958777 0.284160i \(-0.908285\pi\)
−0.958777 + 0.284160i \(0.908285\pi\)
\(240\) 0 0
\(241\) 2538.40 0.678476 0.339238 0.940701i \(-0.389831\pi\)
0.339238 + 0.940701i \(0.389831\pi\)
\(242\) 0 0
\(243\) −243.000 −0.0641500
\(244\) 0 0
\(245\) −106.784 −0.0278458
\(246\) 0 0
\(247\) −1542.97 −0.397477
\(248\) 0 0
\(249\) 1085.80 0.276345
\(250\) 0 0
\(251\) 1696.99 0.426746 0.213373 0.976971i \(-0.431555\pi\)
0.213373 + 0.976971i \(0.431555\pi\)
\(252\) 0 0
\(253\) 6598.09 1.63960
\(254\) 0 0
\(255\) 86.6871 0.0212885
\(256\) 0 0
\(257\) −382.902 −0.0929369 −0.0464685 0.998920i \(-0.514797\pi\)
−0.0464685 + 0.998920i \(0.514797\pi\)
\(258\) 0 0
\(259\) 9287.52 2.22818
\(260\) 0 0
\(261\) 93.8249 0.0222514
\(262\) 0 0
\(263\) −5002.02 −1.17277 −0.586383 0.810034i \(-0.699449\pi\)
−0.586383 + 0.810034i \(0.699449\pi\)
\(264\) 0 0
\(265\) 72.1833 0.0167328
\(266\) 0 0
\(267\) 120.986 0.0277311
\(268\) 0 0
\(269\) 6117.47 1.38658 0.693288 0.720661i \(-0.256161\pi\)
0.693288 + 0.720661i \(0.256161\pi\)
\(270\) 0 0
\(271\) 3956.12 0.886780 0.443390 0.896329i \(-0.353776\pi\)
0.443390 + 0.896329i \(0.353776\pi\)
\(272\) 0 0
\(273\) −3609.92 −0.800301
\(274\) 0 0
\(275\) −7510.25 −1.64686
\(276\) 0 0
\(277\) 4842.17 1.05032 0.525158 0.851004i \(-0.324006\pi\)
0.525158 + 0.851004i \(0.324006\pi\)
\(278\) 0 0
\(279\) 1987.92 0.426573
\(280\) 0 0
\(281\) −1878.68 −0.398835 −0.199417 0.979915i \(-0.563905\pi\)
−0.199417 + 0.979915i \(0.563905\pi\)
\(282\) 0 0
\(283\) −5724.87 −1.20250 −0.601251 0.799060i \(-0.705331\pi\)
−0.601251 + 0.799060i \(0.705331\pi\)
\(284\) 0 0
\(285\) 53.6034 0.0111410
\(286\) 0 0
\(287\) −8204.77 −1.68750
\(288\) 0 0
\(289\) −2688.53 −0.547228
\(290\) 0 0
\(291\) 1844.76 0.371622
\(292\) 0 0
\(293\) −5088.75 −1.01464 −0.507318 0.861759i \(-0.669363\pi\)
−0.507318 + 0.861759i \(0.669363\pi\)
\(294\) 0 0
\(295\) 161.041 0.0317836
\(296\) 0 0
\(297\) −1627.10 −0.317892
\(298\) 0 0
\(299\) −5792.61 −1.12038
\(300\) 0 0
\(301\) 5377.02 1.02966
\(302\) 0 0
\(303\) 4994.98 0.947043
\(304\) 0 0
\(305\) 167.541 0.0314537
\(306\) 0 0
\(307\) −7219.21 −1.34209 −0.671046 0.741416i \(-0.734155\pi\)
−0.671046 + 0.741416i \(0.734155\pi\)
\(308\) 0 0
\(309\) −1190.58 −0.219189
\(310\) 0 0
\(311\) −1537.06 −0.280252 −0.140126 0.990134i \(-0.544751\pi\)
−0.140126 + 0.990134i \(0.544751\pi\)
\(312\) 0 0
\(313\) 2200.93 0.397456 0.198728 0.980055i \(-0.436319\pi\)
0.198728 + 0.980055i \(0.436319\pi\)
\(314\) 0 0
\(315\) 125.410 0.0224319
\(316\) 0 0
\(317\) 2840.41 0.503260 0.251630 0.967824i \(-0.419033\pi\)
0.251630 + 0.967824i \(0.419033\pi\)
\(318\) 0 0
\(319\) 628.240 0.110266
\(320\) 0 0
\(321\) −1051.89 −0.182899
\(322\) 0 0
\(323\) 1375.51 0.236952
\(324\) 0 0
\(325\) 6593.41 1.12534
\(326\) 0 0
\(327\) 1791.03 0.302887
\(328\) 0 0
\(329\) 2936.56 0.492091
\(330\) 0 0
\(331\) 2118.52 0.351795 0.175898 0.984408i \(-0.443717\pi\)
0.175898 + 0.984408i \(0.443717\pi\)
\(332\) 0 0
\(333\) −3675.13 −0.604792
\(334\) 0 0
\(335\) −54.7766 −0.00893364
\(336\) 0 0
\(337\) −659.599 −0.106619 −0.0533096 0.998578i \(-0.516977\pi\)
−0.0533096 + 0.998578i \(0.516977\pi\)
\(338\) 0 0
\(339\) −1489.27 −0.238601
\(340\) 0 0
\(341\) 13310.9 2.11386
\(342\) 0 0
\(343\) 3837.03 0.604023
\(344\) 0 0
\(345\) 201.238 0.0314037
\(346\) 0 0
\(347\) 8377.42 1.29603 0.648017 0.761626i \(-0.275599\pi\)
0.648017 + 0.761626i \(0.275599\pi\)
\(348\) 0 0
\(349\) 3254.18 0.499119 0.249559 0.968360i \(-0.419714\pi\)
0.249559 + 0.968360i \(0.419714\pi\)
\(350\) 0 0
\(351\) 1428.47 0.217225
\(352\) 0 0
\(353\) 11117.5 1.67627 0.838137 0.545459i \(-0.183645\pi\)
0.838137 + 0.545459i \(0.183645\pi\)
\(354\) 0 0
\(355\) 214.750 0.0321064
\(356\) 0 0
\(357\) 3218.13 0.477091
\(358\) 0 0
\(359\) −4756.56 −0.699281 −0.349640 0.936884i \(-0.613696\pi\)
−0.349640 + 0.936884i \(0.613696\pi\)
\(360\) 0 0
\(361\) −6008.45 −0.875994
\(362\) 0 0
\(363\) −6901.88 −0.997946
\(364\) 0 0
\(365\) 326.310 0.0467940
\(366\) 0 0
\(367\) −1837.40 −0.261339 −0.130670 0.991426i \(-0.541713\pi\)
−0.130670 + 0.991426i \(0.541713\pi\)
\(368\) 0 0
\(369\) 3246.68 0.458036
\(370\) 0 0
\(371\) 2679.70 0.374995
\(372\) 0 0
\(373\) 5598.07 0.777097 0.388549 0.921428i \(-0.372977\pi\)
0.388549 + 0.921428i \(0.372977\pi\)
\(374\) 0 0
\(375\) −458.806 −0.0631804
\(376\) 0 0
\(377\) −551.546 −0.0753476
\(378\) 0 0
\(379\) 3460.18 0.468965 0.234482 0.972120i \(-0.424661\pi\)
0.234482 + 0.972120i \(0.424661\pi\)
\(380\) 0 0
\(381\) 5399.55 0.726055
\(382\) 0 0
\(383\) −5059.63 −0.675027 −0.337513 0.941321i \(-0.609586\pi\)
−0.337513 + 0.941321i \(0.609586\pi\)
\(384\) 0 0
\(385\) 839.732 0.111160
\(386\) 0 0
\(387\) −2127.72 −0.279479
\(388\) 0 0
\(389\) 2192.22 0.285732 0.142866 0.989742i \(-0.454368\pi\)
0.142866 + 0.989742i \(0.454368\pi\)
\(390\) 0 0
\(391\) 5163.93 0.667906
\(392\) 0 0
\(393\) −3364.34 −0.431828
\(394\) 0 0
\(395\) −102.045 −0.0129986
\(396\) 0 0
\(397\) −5519.94 −0.697828 −0.348914 0.937155i \(-0.613449\pi\)
−0.348914 + 0.937155i \(0.613449\pi\)
\(398\) 0 0
\(399\) 1989.95 0.249679
\(400\) 0 0
\(401\) −7352.64 −0.915645 −0.457822 0.889044i \(-0.651370\pi\)
−0.457822 + 0.889044i \(0.651370\pi\)
\(402\) 0 0
\(403\) −11685.9 −1.44446
\(404\) 0 0
\(405\) −49.6256 −0.00608868
\(406\) 0 0
\(407\) −24608.2 −2.99702
\(408\) 0 0
\(409\) 11311.2 1.36749 0.683745 0.729721i \(-0.260350\pi\)
0.683745 + 0.729721i \(0.260350\pi\)
\(410\) 0 0
\(411\) 7348.66 0.881952
\(412\) 0 0
\(413\) 5978.40 0.712295
\(414\) 0 0
\(415\) 221.743 0.0262288
\(416\) 0 0
\(417\) 7372.68 0.865808
\(418\) 0 0
\(419\) 13042.2 1.52065 0.760325 0.649543i \(-0.225040\pi\)
0.760325 + 0.649543i \(0.225040\pi\)
\(420\) 0 0
\(421\) −4544.38 −0.526080 −0.263040 0.964785i \(-0.584725\pi\)
−0.263040 + 0.964785i \(0.584725\pi\)
\(422\) 0 0
\(423\) −1162.02 −0.133568
\(424\) 0 0
\(425\) −5877.83 −0.670863
\(426\) 0 0
\(427\) 6219.72 0.704903
\(428\) 0 0
\(429\) 9564.85 1.07645
\(430\) 0 0
\(431\) −7713.83 −0.862093 −0.431047 0.902330i \(-0.641856\pi\)
−0.431047 + 0.902330i \(0.641856\pi\)
\(432\) 0 0
\(433\) −15068.3 −1.67237 −0.836183 0.548451i \(-0.815218\pi\)
−0.836183 + 0.548451i \(0.815218\pi\)
\(434\) 0 0
\(435\) 19.1609 0.00211195
\(436\) 0 0
\(437\) 3193.14 0.349540
\(438\) 0 0
\(439\) 11004.7 1.19642 0.598208 0.801341i \(-0.295880\pi\)
0.598208 + 0.801341i \(0.295880\pi\)
\(440\) 0 0
\(441\) 1568.67 0.169384
\(442\) 0 0
\(443\) −2513.04 −0.269522 −0.134761 0.990878i \(-0.543027\pi\)
−0.134761 + 0.990878i \(0.543027\pi\)
\(444\) 0 0
\(445\) 24.7077 0.00263204
\(446\) 0 0
\(447\) 6254.89 0.661848
\(448\) 0 0
\(449\) −15752.7 −1.65571 −0.827855 0.560942i \(-0.810439\pi\)
−0.827855 + 0.560942i \(0.810439\pi\)
\(450\) 0 0
\(451\) 21739.4 2.26977
\(452\) 0 0
\(453\) −3173.40 −0.329138
\(454\) 0 0
\(455\) −737.218 −0.0759590
\(456\) 0 0
\(457\) 5257.06 0.538107 0.269053 0.963125i \(-0.413289\pi\)
0.269053 + 0.963125i \(0.413289\pi\)
\(458\) 0 0
\(459\) −1273.43 −0.129496
\(460\) 0 0
\(461\) 8066.31 0.814936 0.407468 0.913220i \(-0.366412\pi\)
0.407468 + 0.913220i \(0.366412\pi\)
\(462\) 0 0
\(463\) −5683.43 −0.570478 −0.285239 0.958456i \(-0.592073\pi\)
−0.285239 + 0.958456i \(0.592073\pi\)
\(464\) 0 0
\(465\) 405.975 0.0404874
\(466\) 0 0
\(467\) −11139.3 −1.10378 −0.551891 0.833916i \(-0.686094\pi\)
−0.551891 + 0.833916i \(0.686094\pi\)
\(468\) 0 0
\(469\) −2033.50 −0.200210
\(470\) 0 0
\(471\) 9579.02 0.937108
\(472\) 0 0
\(473\) −14247.0 −1.38494
\(474\) 0 0
\(475\) −3634.59 −0.351087
\(476\) 0 0
\(477\) −1060.37 −0.101784
\(478\) 0 0
\(479\) −3477.35 −0.331699 −0.165850 0.986151i \(-0.553037\pi\)
−0.165850 + 0.986151i \(0.553037\pi\)
\(480\) 0 0
\(481\) 21604.1 2.04795
\(482\) 0 0
\(483\) 7470.65 0.703782
\(484\) 0 0
\(485\) 376.738 0.0352717
\(486\) 0 0
\(487\) −478.797 −0.0445510 −0.0222755 0.999752i \(-0.507091\pi\)
−0.0222755 + 0.999752i \(0.507091\pi\)
\(488\) 0 0
\(489\) 2540.56 0.234945
\(490\) 0 0
\(491\) −16601.8 −1.52592 −0.762961 0.646444i \(-0.776255\pi\)
−0.762961 + 0.646444i \(0.776255\pi\)
\(492\) 0 0
\(493\) 491.687 0.0449178
\(494\) 0 0
\(495\) −332.287 −0.0301721
\(496\) 0 0
\(497\) 7972.29 0.719530
\(498\) 0 0
\(499\) −9482.20 −0.850664 −0.425332 0.905037i \(-0.639843\pi\)
−0.425332 + 0.905037i \(0.639843\pi\)
\(500\) 0 0
\(501\) 7890.45 0.703631
\(502\) 0 0
\(503\) 16561.2 1.46805 0.734023 0.679124i \(-0.237640\pi\)
0.734023 + 0.679124i \(0.237640\pi\)
\(504\) 0 0
\(505\) 1020.08 0.0898867
\(506\) 0 0
\(507\) −1806.19 −0.158216
\(508\) 0 0
\(509\) −4197.35 −0.365509 −0.182755 0.983159i \(-0.558501\pi\)
−0.182755 + 0.983159i \(0.558501\pi\)
\(510\) 0 0
\(511\) 12113.8 1.04869
\(512\) 0 0
\(513\) −787.435 −0.0677702
\(514\) 0 0
\(515\) −243.140 −0.0208039
\(516\) 0 0
\(517\) −7780.73 −0.661888
\(518\) 0 0
\(519\) 1289.53 0.109064
\(520\) 0 0
\(521\) 15755.5 1.32488 0.662440 0.749115i \(-0.269521\pi\)
0.662440 + 0.749115i \(0.269521\pi\)
\(522\) 0 0
\(523\) −11555.1 −0.966098 −0.483049 0.875593i \(-0.660471\pi\)
−0.483049 + 0.875593i \(0.660471\pi\)
\(524\) 0 0
\(525\) −8503.44 −0.706897
\(526\) 0 0
\(527\) 10417.7 0.861102
\(528\) 0 0
\(529\) −179.314 −0.0147378
\(530\) 0 0
\(531\) −2365.69 −0.193337
\(532\) 0 0
\(533\) −19085.5 −1.55100
\(534\) 0 0
\(535\) −214.817 −0.0173595
\(536\) 0 0
\(537\) 4548.90 0.365549
\(538\) 0 0
\(539\) 10503.6 0.839373
\(540\) 0 0
\(541\) −7475.65 −0.594091 −0.297045 0.954863i \(-0.596001\pi\)
−0.297045 + 0.954863i \(0.596001\pi\)
\(542\) 0 0
\(543\) 10140.6 0.801427
\(544\) 0 0
\(545\) 365.764 0.0287479
\(546\) 0 0
\(547\) −6028.08 −0.471192 −0.235596 0.971851i \(-0.575704\pi\)
−0.235596 + 0.971851i \(0.575704\pi\)
\(548\) 0 0
\(549\) −2461.18 −0.191331
\(550\) 0 0
\(551\) 304.037 0.0235071
\(552\) 0 0
\(553\) −3788.29 −0.291310
\(554\) 0 0
\(555\) −750.536 −0.0574027
\(556\) 0 0
\(557\) −19381.5 −1.47436 −0.737181 0.675696i \(-0.763843\pi\)
−0.737181 + 0.675696i \(0.763843\pi\)
\(558\) 0 0
\(559\) 12507.7 0.946370
\(560\) 0 0
\(561\) −8526.77 −0.641712
\(562\) 0 0
\(563\) −20565.0 −1.53946 −0.769728 0.638372i \(-0.779608\pi\)
−0.769728 + 0.638372i \(0.779608\pi\)
\(564\) 0 0
\(565\) −304.139 −0.0226464
\(566\) 0 0
\(567\) −1842.28 −0.136452
\(568\) 0 0
\(569\) −15252.9 −1.12379 −0.561895 0.827209i \(-0.689927\pi\)
−0.561895 + 0.827209i \(0.689927\pi\)
\(570\) 0 0
\(571\) 16492.8 1.20876 0.604379 0.796697i \(-0.293421\pi\)
0.604379 + 0.796697i \(0.293421\pi\)
\(572\) 0 0
\(573\) −8399.13 −0.612354
\(574\) 0 0
\(575\) −13644.9 −0.989623
\(576\) 0 0
\(577\) −10298.2 −0.743016 −0.371508 0.928430i \(-0.621159\pi\)
−0.371508 + 0.928430i \(0.621159\pi\)
\(578\) 0 0
\(579\) −1872.32 −0.134388
\(580\) 0 0
\(581\) 8231.89 0.587808
\(582\) 0 0
\(583\) −7100.14 −0.504387
\(584\) 0 0
\(585\) 291.722 0.0206175
\(586\) 0 0
\(587\) 13104.8 0.921453 0.460727 0.887542i \(-0.347589\pi\)
0.460727 + 0.887542i \(0.347589\pi\)
\(588\) 0 0
\(589\) 6441.82 0.450646
\(590\) 0 0
\(591\) −14337.7 −0.997929
\(592\) 0 0
\(593\) 4163.34 0.288310 0.144155 0.989555i \(-0.453954\pi\)
0.144155 + 0.989555i \(0.453954\pi\)
\(594\) 0 0
\(595\) 657.208 0.0452822
\(596\) 0 0
\(597\) −7847.75 −0.538001
\(598\) 0 0
\(599\) −5718.60 −0.390076 −0.195038 0.980796i \(-0.562483\pi\)
−0.195038 + 0.980796i \(0.562483\pi\)
\(600\) 0 0
\(601\) −17473.0 −1.18592 −0.592959 0.805233i \(-0.702040\pi\)
−0.592959 + 0.805233i \(0.702040\pi\)
\(602\) 0 0
\(603\) 804.670 0.0543428
\(604\) 0 0
\(605\) −1409.50 −0.0947181
\(606\) 0 0
\(607\) 5647.60 0.377643 0.188821 0.982011i \(-0.439533\pi\)
0.188821 + 0.982011i \(0.439533\pi\)
\(608\) 0 0
\(609\) 711.322 0.0473304
\(610\) 0 0
\(611\) 6830.87 0.452287
\(612\) 0 0
\(613\) 16023.6 1.05577 0.527884 0.849317i \(-0.322986\pi\)
0.527884 + 0.849317i \(0.322986\pi\)
\(614\) 0 0
\(615\) 663.038 0.0434736
\(616\) 0 0
\(617\) 21022.1 1.37167 0.685834 0.727758i \(-0.259438\pi\)
0.685834 + 0.727758i \(0.259438\pi\)
\(618\) 0 0
\(619\) −17824.2 −1.15737 −0.578686 0.815550i \(-0.696434\pi\)
−0.578686 + 0.815550i \(0.696434\pi\)
\(620\) 0 0
\(621\) −2956.18 −0.191027
\(622\) 0 0
\(623\) 917.238 0.0589861
\(624\) 0 0
\(625\) 15484.4 0.991001
\(626\) 0 0
\(627\) −5272.57 −0.335831
\(628\) 0 0
\(629\) −19259.4 −1.22086
\(630\) 0 0
\(631\) −22339.0 −1.40935 −0.704677 0.709528i \(-0.748908\pi\)
−0.704677 + 0.709528i \(0.748908\pi\)
\(632\) 0 0
\(633\) 5237.35 0.328856
\(634\) 0 0
\(635\) 1102.70 0.0689121
\(636\) 0 0
\(637\) −9221.34 −0.573568
\(638\) 0 0
\(639\) −3154.69 −0.195301
\(640\) 0 0
\(641\) −5268.43 −0.324634 −0.162317 0.986739i \(-0.551897\pi\)
−0.162317 + 0.986739i \(0.551897\pi\)
\(642\) 0 0
\(643\) 21965.8 1.34719 0.673597 0.739099i \(-0.264748\pi\)
0.673597 + 0.739099i \(0.264748\pi\)
\(644\) 0 0
\(645\) −434.524 −0.0265262
\(646\) 0 0
\(647\) 3165.40 0.192341 0.0961706 0.995365i \(-0.469341\pi\)
0.0961706 + 0.995365i \(0.469341\pi\)
\(648\) 0 0
\(649\) −15840.4 −0.958073
\(650\) 0 0
\(651\) 15071.2 0.907354
\(652\) 0 0
\(653\) −12094.7 −0.724814 −0.362407 0.932020i \(-0.618045\pi\)
−0.362407 + 0.932020i \(0.618045\pi\)
\(654\) 0 0
\(655\) −687.067 −0.0409861
\(656\) 0 0
\(657\) −4793.49 −0.284645
\(658\) 0 0
\(659\) 7523.17 0.444706 0.222353 0.974966i \(-0.428626\pi\)
0.222353 + 0.974966i \(0.428626\pi\)
\(660\) 0 0
\(661\) −24141.1 −1.42054 −0.710271 0.703928i \(-0.751428\pi\)
−0.710271 + 0.703928i \(0.751428\pi\)
\(662\) 0 0
\(663\) 7485.84 0.438501
\(664\) 0 0
\(665\) 406.388 0.0236978
\(666\) 0 0
\(667\) 1141.41 0.0662604
\(668\) 0 0
\(669\) −10156.8 −0.586972
\(670\) 0 0
\(671\) −16479.8 −0.948130
\(672\) 0 0
\(673\) 944.143 0.0540773 0.0270387 0.999634i \(-0.491392\pi\)
0.0270387 + 0.999634i \(0.491392\pi\)
\(674\) 0 0
\(675\) 3364.87 0.191872
\(676\) 0 0
\(677\) −4450.51 −0.252654 −0.126327 0.991989i \(-0.540319\pi\)
−0.126327 + 0.991989i \(0.540319\pi\)
\(678\) 0 0
\(679\) 13985.8 0.790468
\(680\) 0 0
\(681\) −11543.1 −0.649536
\(682\) 0 0
\(683\) −1726.40 −0.0967189 −0.0483594 0.998830i \(-0.515399\pi\)
−0.0483594 + 0.998830i \(0.515399\pi\)
\(684\) 0 0
\(685\) 1500.75 0.0837088
\(686\) 0 0
\(687\) −4005.44 −0.222441
\(688\) 0 0
\(689\) 6233.37 0.344662
\(690\) 0 0
\(691\) 683.143 0.0376092 0.0188046 0.999823i \(-0.494014\pi\)
0.0188046 + 0.999823i \(0.494014\pi\)
\(692\) 0 0
\(693\) −12335.7 −0.676181
\(694\) 0 0
\(695\) 1505.65 0.0821764
\(696\) 0 0
\(697\) 17014.1 0.924614
\(698\) 0 0
\(699\) −15439.1 −0.835425
\(700\) 0 0
\(701\) −19533.2 −1.05244 −0.526220 0.850349i \(-0.676391\pi\)
−0.526220 + 0.850349i \(0.676391\pi\)
\(702\) 0 0
\(703\) −11909.2 −0.638922
\(704\) 0 0
\(705\) −237.307 −0.0126773
\(706\) 0 0
\(707\) 37868.8 2.01443
\(708\) 0 0
\(709\) −26081.9 −1.38156 −0.690779 0.723066i \(-0.742732\pi\)
−0.690779 + 0.723066i \(0.742732\pi\)
\(710\) 0 0
\(711\) 1499.05 0.0790699
\(712\) 0 0
\(713\) 24183.8 1.27025
\(714\) 0 0
\(715\) 1953.34 0.102169
\(716\) 0 0
\(717\) 21255.2 1.10710
\(718\) 0 0
\(719\) 30077.3 1.56007 0.780036 0.625734i \(-0.215200\pi\)
0.780036 + 0.625734i \(0.215200\pi\)
\(720\) 0 0
\(721\) −9026.21 −0.466232
\(722\) 0 0
\(723\) −7615.20 −0.391718
\(724\) 0 0
\(725\) −1299.21 −0.0665537
\(726\) 0 0
\(727\) 23049.9 1.17589 0.587946 0.808900i \(-0.299937\pi\)
0.587946 + 0.808900i \(0.299937\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −11150.3 −0.564169
\(732\) 0 0
\(733\) −4444.57 −0.223962 −0.111981 0.993710i \(-0.535720\pi\)
−0.111981 + 0.993710i \(0.535720\pi\)
\(734\) 0 0
\(735\) 320.353 0.0160768
\(736\) 0 0
\(737\) 5387.98 0.269293
\(738\) 0 0
\(739\) −28465.1 −1.41692 −0.708462 0.705749i \(-0.750610\pi\)
−0.708462 + 0.705749i \(0.750610\pi\)
\(740\) 0 0
\(741\) 4628.91 0.229483
\(742\) 0 0
\(743\) 4389.76 0.216749 0.108374 0.994110i \(-0.465435\pi\)
0.108374 + 0.994110i \(0.465435\pi\)
\(744\) 0 0
\(745\) 1277.38 0.0628180
\(746\) 0 0
\(747\) −3257.41 −0.159548
\(748\) 0 0
\(749\) −7974.77 −0.389041
\(750\) 0 0
\(751\) −14121.9 −0.686171 −0.343085 0.939304i \(-0.611472\pi\)
−0.343085 + 0.939304i \(0.611472\pi\)
\(752\) 0 0
\(753\) −5090.98 −0.246382
\(754\) 0 0
\(755\) −648.074 −0.0312395
\(756\) 0 0
\(757\) −17006.3 −0.816516 −0.408258 0.912866i \(-0.633864\pi\)
−0.408258 + 0.912866i \(0.633864\pi\)
\(758\) 0 0
\(759\) −19794.3 −0.946622
\(760\) 0 0
\(761\) 4603.38 0.219280 0.109640 0.993971i \(-0.465030\pi\)
0.109640 + 0.993971i \(0.465030\pi\)
\(762\) 0 0
\(763\) 13578.5 0.644264
\(764\) 0 0
\(765\) −260.061 −0.0122909
\(766\) 0 0
\(767\) 13906.6 0.654679
\(768\) 0 0
\(769\) −12459.1 −0.584248 −0.292124 0.956380i \(-0.594362\pi\)
−0.292124 + 0.956380i \(0.594362\pi\)
\(770\) 0 0
\(771\) 1148.71 0.0536571
\(772\) 0 0
\(773\) −27068.1 −1.25947 −0.629737 0.776808i \(-0.716837\pi\)
−0.629737 + 0.776808i \(0.716837\pi\)
\(774\) 0 0
\(775\) −27527.2 −1.27588
\(776\) 0 0
\(777\) −27862.6 −1.28644
\(778\) 0 0
\(779\) 10520.8 0.483884
\(780\) 0 0
\(781\) −21123.4 −0.967804
\(782\) 0 0
\(783\) −281.475 −0.0128468
\(784\) 0 0
\(785\) 1956.23 0.0889438
\(786\) 0 0
\(787\) −6986.86 −0.316461 −0.158230 0.987402i \(-0.550579\pi\)
−0.158230 + 0.987402i \(0.550579\pi\)
\(788\) 0 0
\(789\) 15006.1 0.677097
\(790\) 0 0
\(791\) −11290.7 −0.507523
\(792\) 0 0
\(793\) 14468.0 0.647885
\(794\) 0 0
\(795\) −216.550 −0.00966067
\(796\) 0 0
\(797\) 27271.5 1.21205 0.606027 0.795444i \(-0.292762\pi\)
0.606027 + 0.795444i \(0.292762\pi\)
\(798\) 0 0
\(799\) −6089.52 −0.269627
\(800\) 0 0
\(801\) −362.957 −0.0160105
\(802\) 0 0
\(803\) −32096.7 −1.41054
\(804\) 0 0
\(805\) 1525.66 0.0667981
\(806\) 0 0
\(807\) −18352.4 −0.800540
\(808\) 0 0
\(809\) −23785.9 −1.03371 −0.516853 0.856074i \(-0.672897\pi\)
−0.516853 + 0.856074i \(0.672897\pi\)
\(810\) 0 0
\(811\) −21703.5 −0.939718 −0.469859 0.882741i \(-0.655695\pi\)
−0.469859 + 0.882741i \(0.655695\pi\)
\(812\) 0 0
\(813\) −11868.4 −0.511982
\(814\) 0 0
\(815\) 518.835 0.0222994
\(816\) 0 0
\(817\) −6894.83 −0.295250
\(818\) 0 0
\(819\) 10829.7 0.462054
\(820\) 0 0
\(821\) 33240.4 1.41303 0.706515 0.707698i \(-0.250266\pi\)
0.706515 + 0.707698i \(0.250266\pi\)
\(822\) 0 0
\(823\) −17227.5 −0.729665 −0.364832 0.931073i \(-0.618874\pi\)
−0.364832 + 0.931073i \(0.618874\pi\)
\(824\) 0 0
\(825\) 22530.8 0.950812
\(826\) 0 0
\(827\) −21678.4 −0.911528 −0.455764 0.890101i \(-0.650634\pi\)
−0.455764 + 0.890101i \(0.650634\pi\)
\(828\) 0 0
\(829\) −34269.8 −1.43575 −0.717877 0.696170i \(-0.754886\pi\)
−0.717877 + 0.696170i \(0.754886\pi\)
\(830\) 0 0
\(831\) −14526.5 −0.606401
\(832\) 0 0
\(833\) 8220.55 0.341927
\(834\) 0 0
\(835\) 1611.39 0.0667838
\(836\) 0 0
\(837\) −5963.77 −0.246282
\(838\) 0 0
\(839\) 33444.1 1.37618 0.688092 0.725623i \(-0.258448\pi\)
0.688092 + 0.725623i \(0.258448\pi\)
\(840\) 0 0
\(841\) −24280.3 −0.995544
\(842\) 0 0
\(843\) 5636.04 0.230267
\(844\) 0 0
\(845\) −368.860 −0.0150168
\(846\) 0 0
\(847\) −52325.8 −2.12271
\(848\) 0 0
\(849\) 17174.6 0.694265
\(850\) 0 0
\(851\) −44709.3 −1.80096
\(852\) 0 0
\(853\) 37037.3 1.48667 0.743337 0.668917i \(-0.233242\pi\)
0.743337 + 0.668917i \(0.233242\pi\)
\(854\) 0 0
\(855\) −160.810 −0.00643227
\(856\) 0 0
\(857\) −35794.2 −1.42673 −0.713365 0.700793i \(-0.752830\pi\)
−0.713365 + 0.700793i \(0.752830\pi\)
\(858\) 0 0
\(859\) 20582.1 0.817522 0.408761 0.912641i \(-0.365961\pi\)
0.408761 + 0.912641i \(0.365961\pi\)
\(860\) 0 0
\(861\) 24614.3 0.974278
\(862\) 0 0
\(863\) 25677.7 1.01284 0.506420 0.862287i \(-0.330969\pi\)
0.506420 + 0.862287i \(0.330969\pi\)
\(864\) 0 0
\(865\) 263.348 0.0103516
\(866\) 0 0
\(867\) 8065.60 0.315942
\(868\) 0 0
\(869\) 10037.5 0.391827
\(870\) 0 0
\(871\) −4730.22 −0.184015
\(872\) 0 0
\(873\) −5534.29 −0.214556
\(874\) 0 0
\(875\) −3478.38 −0.134389
\(876\) 0 0
\(877\) −32783.0 −1.26226 −0.631131 0.775676i \(-0.717409\pi\)
−0.631131 + 0.775676i \(0.717409\pi\)
\(878\) 0 0
\(879\) 15266.3 0.585800
\(880\) 0 0
\(881\) 26615.7 1.01783 0.508914 0.860818i \(-0.330047\pi\)
0.508914 + 0.860818i \(0.330047\pi\)
\(882\) 0 0
\(883\) 19203.4 0.731875 0.365937 0.930639i \(-0.380748\pi\)
0.365937 + 0.930639i \(0.380748\pi\)
\(884\) 0 0
\(885\) −483.122 −0.0183503
\(886\) 0 0
\(887\) 23198.0 0.878144 0.439072 0.898452i \(-0.355307\pi\)
0.439072 + 0.898452i \(0.355307\pi\)
\(888\) 0 0
\(889\) 40936.0 1.54438
\(890\) 0 0
\(891\) 4881.30 0.183535
\(892\) 0 0
\(893\) −3765.48 −0.141105
\(894\) 0 0
\(895\) 928.979 0.0346953
\(896\) 0 0
\(897\) 17377.8 0.646854
\(898\) 0 0
\(899\) 2302.68 0.0854266
\(900\) 0 0
\(901\) −5556.86 −0.205467
\(902\) 0 0
\(903\) −16131.1 −0.594472
\(904\) 0 0
\(905\) 2070.92 0.0760659
\(906\) 0 0
\(907\) 16151.3 0.591285 0.295643 0.955299i \(-0.404466\pi\)
0.295643 + 0.955299i \(0.404466\pi\)
\(908\) 0 0
\(909\) −14984.9 −0.546776
\(910\) 0 0
\(911\) 3487.34 0.126829 0.0634143 0.997987i \(-0.479801\pi\)
0.0634143 + 0.997987i \(0.479801\pi\)
\(912\) 0 0
\(913\) −21811.2 −0.790632
\(914\) 0 0
\(915\) −502.624 −0.0181598
\(916\) 0 0
\(917\) −25506.3 −0.918532
\(918\) 0 0
\(919\) 17055.5 0.612198 0.306099 0.952000i \(-0.400976\pi\)
0.306099 + 0.952000i \(0.400976\pi\)
\(920\) 0 0
\(921\) 21657.6 0.774857
\(922\) 0 0
\(923\) 18544.7 0.661329
\(924\) 0 0
\(925\) 50890.2 1.80893
\(926\) 0 0
\(927\) 3571.73 0.126549
\(928\) 0 0
\(929\) −55339.3 −1.95438 −0.977192 0.212359i \(-0.931885\pi\)
−0.977192 + 0.212359i \(0.931885\pi\)
\(930\) 0 0
\(931\) 5083.22 0.178943
\(932\) 0 0
\(933\) 4611.17 0.161804
\(934\) 0 0
\(935\) −1741.34 −0.0609069
\(936\) 0 0
\(937\) 20457.6 0.713255 0.356627 0.934247i \(-0.383927\pi\)
0.356627 + 0.934247i \(0.383927\pi\)
\(938\) 0 0
\(939\) −6602.78 −0.229471
\(940\) 0 0
\(941\) 55891.0 1.93623 0.968116 0.250502i \(-0.0805958\pi\)
0.968116 + 0.250502i \(0.0805958\pi\)
\(942\) 0 0
\(943\) 39497.0 1.36395
\(944\) 0 0
\(945\) −376.230 −0.0129511
\(946\) 0 0
\(947\) 54727.0 1.87792 0.938960 0.344027i \(-0.111791\pi\)
0.938960 + 0.344027i \(0.111791\pi\)
\(948\) 0 0
\(949\) 28178.4 0.963865
\(950\) 0 0
\(951\) −8521.23 −0.290557
\(952\) 0 0
\(953\) −29958.8 −1.01832 −0.509160 0.860672i \(-0.670044\pi\)
−0.509160 + 0.860672i \(0.670044\pi\)
\(954\) 0 0
\(955\) −1715.27 −0.0581204
\(956\) 0 0
\(957\) −1884.72 −0.0636619
\(958\) 0 0
\(959\) 55713.0 1.87598
\(960\) 0 0
\(961\) 18997.2 0.637682
\(962\) 0 0
\(963\) 3155.67 0.105597
\(964\) 0 0
\(965\) −382.365 −0.0127552
\(966\) 0 0
\(967\) 13498.5 0.448896 0.224448 0.974486i \(-0.427942\pi\)
0.224448 + 0.974486i \(0.427942\pi\)
\(968\) 0 0
\(969\) −4126.53 −0.136804
\(970\) 0 0
\(971\) 9652.36 0.319010 0.159505 0.987197i \(-0.449010\pi\)
0.159505 + 0.987197i \(0.449010\pi\)
\(972\) 0 0
\(973\) 55895.1 1.84164
\(974\) 0 0
\(975\) −19780.2 −0.649717
\(976\) 0 0
\(977\) 12169.8 0.398511 0.199256 0.979948i \(-0.436148\pi\)
0.199256 + 0.979948i \(0.436148\pi\)
\(978\) 0 0
\(979\) −2430.32 −0.0793394
\(980\) 0 0
\(981\) −5373.08 −0.174872
\(982\) 0 0
\(983\) 47545.2 1.54268 0.771341 0.636423i \(-0.219587\pi\)
0.771341 + 0.636423i \(0.219587\pi\)
\(984\) 0 0
\(985\) −2928.06 −0.0947165
\(986\) 0 0
\(987\) −8809.69 −0.284109
\(988\) 0 0
\(989\) −25884.5 −0.832234
\(990\) 0 0
\(991\) 892.350 0.0286039 0.0143019 0.999898i \(-0.495447\pi\)
0.0143019 + 0.999898i \(0.495447\pi\)
\(992\) 0 0
\(993\) −6355.55 −0.203109
\(994\) 0 0
\(995\) −1602.67 −0.0510634
\(996\) 0 0
\(997\) −4458.57 −0.141629 −0.0708146 0.997489i \(-0.522560\pi\)
−0.0708146 + 0.997489i \(0.522560\pi\)
\(998\) 0 0
\(999\) 11025.4 0.349177
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.1.2 3
3.2 odd 2 2304.4.a.bw.1.2 3
4.3 odd 2 768.4.a.s.1.2 3
8.3 odd 2 768.4.a.r.1.2 3
8.5 even 2 768.4.a.t.1.2 3
12.11 even 2 2304.4.a.bv.1.2 3
16.3 odd 4 24.4.d.a.13.5 6
16.5 even 4 96.4.d.a.49.5 6
16.11 odd 4 24.4.d.a.13.6 yes 6
16.13 even 4 96.4.d.a.49.2 6
24.5 odd 2 2304.4.a.bu.1.2 3
24.11 even 2 2304.4.a.bt.1.2 3
48.5 odd 4 288.4.d.d.145.4 6
48.11 even 4 72.4.d.d.37.1 6
48.29 odd 4 288.4.d.d.145.3 6
48.35 even 4 72.4.d.d.37.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
24.4.d.a.13.5 6 16.3 odd 4
24.4.d.a.13.6 yes 6 16.11 odd 4
72.4.d.d.37.1 6 48.11 even 4
72.4.d.d.37.2 6 48.35 even 4
96.4.d.a.49.2 6 16.13 even 4
96.4.d.a.49.5 6 16.5 even 4
288.4.d.d.145.3 6 48.29 odd 4
288.4.d.d.145.4 6 48.5 odd 4
768.4.a.q.1.2 3 1.1 even 1 trivial
768.4.a.r.1.2 3 8.3 odd 2
768.4.a.s.1.2 3 4.3 odd 2
768.4.a.t.1.2 3 8.5 even 2
2304.4.a.bt.1.2 3 24.11 even 2
2304.4.a.bu.1.2 3 24.5 odd 2
2304.4.a.bv.1.2 3 12.11 even 2
2304.4.a.bw.1.2 3 3.2 odd 2