Properties

Label 768.4.a.r.1.2
Level $768$
Weight $4$
Character 768.1
Self dual yes
Analytic conductor $45.313$
Analytic rank $0$
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: \(0\)
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 \(-0.491278\pi\)
0.0273990 + 0.999625i \(0.491278\pi\)
\(6\) 0 0
\(7\) 22.7441 1.22807 0.614034 0.789279i \(-0.289546\pi\)
0.614034 + 0.789279i \(0.289546\pi\)
\(8\) 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 \(-0.309120\pi\)
0.564367 + 0.825524i \(0.309120\pi\)
\(14\) 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 \(-0.665309\pi\)
−0.496302 + 0.868150i \(0.665309\pi\)
\(24\) 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 \(-0.510626\pi\)
−0.0333771 + 0.999443i \(0.510626\pi\)
\(30\) 0 0
\(31\) −220.881 −1.27972 −0.639860 0.768492i \(-0.721008\pi\)
−0.639860 + 0.768492i \(0.721008\pi\)
\(32\) 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 \(-0.138224\pi\)
0.907188 + 0.420725i \(0.138224\pi\)
\(38\) 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 \(-0.435792\pi\)
0.200352 + 0.979724i \(0.435792\pi\)
\(48\) 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 \(-0.451211\pi\)
0.152677 + 0.988276i \(0.451211\pi\)
\(54\) 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 \(-0.407343\pi\)
0.286996 + 0.957932i \(0.407343\pi\)
\(62\) 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 \(-0.405362\pi\)
0.292952 + 0.956127i \(0.405362\pi\)
\(72\) 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 \(-0.537842\pi\)
−0.118605 + 0.992942i \(0.537842\pi\)
\(80\) 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 \(-0.193883\pi\)
0.820163 + 0.572130i \(0.193883\pi\)
\(102\) 0 0
\(103\) −396.858 −0.379647 −0.189823 0.981818i \(-0.560792\pi\)
−0.189823 + 0.981818i \(0.560792\pi\)
\(104\) 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 \(-0.415516\pi\)
0.262308 + 0.964984i \(0.415516\pi\)
\(110\) 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 \(-0.283554\pi\)
0.628782 + 0.777581i \(0.283554\pi\)
\(128\) 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 \(-0.305711\pi\)
0.573177 + 0.819432i \(0.305711\pi\)
\(150\) 0 0
\(151\) −1057.80 −0.570084 −0.285042 0.958515i \(-0.592008\pi\)
−0.285042 + 0.958515i \(0.592008\pi\)
\(152\) 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 \(-0.198619\pi\)
0.811559 + 0.584270i \(0.198619\pi\)
\(158\) 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 \(-0.291425\pi\)
0.609363 + 0.792892i \(0.291425\pi\)
\(168\) 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 \(-0.469890\pi\)
0.0944519 + 0.995529i \(0.469890\pi\)
\(174\) 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 \(-0.255822\pi\)
0.694056 + 0.719921i \(0.255822\pi\)
\(182\) 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 \(-0.677926\pi\)
−0.530314 + 0.847801i \(0.677926\pi\)
\(192\) 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 \(-0.832195\pi\)
−0.864232 + 0.503094i \(0.832195\pi\)
\(198\) 0 0
\(199\) −2615.92 −0.931846 −0.465923 0.884825i \(-0.654278\pi\)
−0.465923 + 0.884825i \(0.654278\pi\)
\(200\) 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 \(-0.669738\pi\)
−0.508333 + 0.861161i \(0.669738\pi\)
\(224\) 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 \(-0.561705\pi\)
−0.192640 + 0.981270i \(0.561705\pi\)
\(230\) 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 \(-0.0917146\pi\)
0.958777 + 0.284160i \(0.0917146\pi\)
\(240\) 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 \(-0.300551\pi\)
0.586383 + 0.810034i \(0.300551\pi\)
\(264\) 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 \(-0.743839\pi\)
−0.693288 + 0.720661i \(0.743839\pi\)
\(270\) 0 0
\(271\) −3956.12 −0.886780 −0.443390 0.896329i \(-0.646224\pi\)
−0.443390 + 0.896329i \(0.646224\pi\)
\(272\) 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 \(-0.675994\pi\)
−0.525158 + 0.851004i \(0.675994\pi\)
\(278\) 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 \(-0.330637\pi\)
0.507318 + 0.861759i \(0.330637\pi\)
\(294\) 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 \(-0.455249\pi\)
0.140126 + 0.990134i \(0.455249\pi\)
\(312\) 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 \(-0.580967\pi\)
−0.251630 + 0.967824i \(0.580967\pi\)
\(318\) 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 \(-0.580286\pi\)
−0.249559 + 0.968360i \(0.580286\pi\)
\(350\) 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 \(-0.386304\pi\)
0.349640 + 0.936884i \(0.386304\pi\)
\(360\) 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 \(-0.458287\pi\)
0.130670 + 0.991426i \(0.458287\pi\)
\(368\) 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 \(-0.627023\pi\)
−0.388549 + 0.921428i \(0.627023\pi\)
\(374\) 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 \(-0.390414\pi\)
0.337513 + 0.941321i \(0.390414\pi\)
\(384\) 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 \(-0.545632\pi\)
−0.142866 + 0.989742i \(0.545632\pi\)
\(390\) 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 \(-0.386551\pi\)
0.348914 + 0.937155i \(0.386551\pi\)
\(398\) 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 \(-0.415275\pi\)
0.263040 + 0.964785i \(0.415275\pi\)
\(422\) 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 \(-0.358144\pi\)
0.431047 + 0.902330i \(0.358144\pi\)
\(432\) 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 \(-0.704120\pi\)
−0.598208 + 0.801341i \(0.704120\pi\)
\(440\) 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 \(-0.633588\pi\)
−0.407468 + 0.913220i \(0.633588\pi\)
\(462\) 0 0
\(463\) 5683.43 0.570478 0.285239 0.958456i \(-0.407927\pi\)
0.285239 + 0.958456i \(0.407927\pi\)
\(464\) 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 \(-0.446963\pi\)
0.165850 + 0.986151i \(0.446963\pi\)
\(480\) 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 \(-0.492909\pi\)
0.0222755 + 0.999752i \(0.492909\pi\)
\(488\) 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 \(-0.762360\pi\)
−0.734023 + 0.679124i \(0.762360\pi\)
\(504\) 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 \(-0.441499\pi\)
0.182755 + 0.983159i \(0.441499\pi\)
\(510\) 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 \(-0.403999\pi\)
0.297045 + 0.954863i \(0.403999\pi\)
\(542\) 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 \(-0.236157\pi\)
0.737181 + 0.675696i \(0.236157\pi\)
\(558\) 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 \(-0.437517\pi\)
0.195038 + 0.980796i \(0.437517\pi\)
\(600\) 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 \(-0.560467\pi\)
−0.188821 + 0.982011i \(0.560467\pi\)
\(608\) 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 \(-0.677014\pi\)
−0.527884 + 0.849317i \(0.677014\pi\)
\(614\) 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 \(-0.251092\pi\)
0.704677 + 0.709528i \(0.251092\pi\)
\(632\) 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 \(-0.530659\pi\)
−0.0961706 + 0.995365i \(0.530659\pi\)
\(648\) 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 \(-0.381955\pi\)
0.362407 + 0.932020i \(0.381955\pi\)
\(654\) 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 \(-0.248572\pi\)
0.710271 + 0.703928i \(0.248572\pi\)
\(662\) 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 \(-0.459681\pi\)
0.126327 + 0.991989i \(0.459681\pi\)
\(678\) 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 \(-0.323609\pi\)
0.526220 + 0.850349i \(0.323609\pi\)
\(702\) 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 \(-0.257268\pi\)
0.690779 + 0.723066i \(0.257268\pi\)
\(710\) 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 \(-0.784800\pi\)
−0.780036 + 0.625734i \(0.784800\pi\)
\(720\) 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 \(-0.700063\pi\)
−0.587946 + 0.808900i \(0.700063\pi\)
\(728\) 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 \(-0.464280\pi\)
0.111981 + 0.993710i \(0.464280\pi\)
\(734\) 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 \(-0.534565\pi\)
−0.108374 + 0.994110i \(0.534565\pi\)
\(744\) 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 \(-0.388528\pi\)
0.343085 + 0.939304i \(0.388528\pi\)
\(752\) 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 \(-0.366136\pi\)
0.408258 + 0.912866i \(0.366136\pi\)
\(758\) 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 \(-0.283163\pi\)
0.629737 + 0.776808i \(0.283163\pi\)
\(774\) 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 \(-0.707238\pi\)
−0.606027 + 0.795444i \(0.707238\pi\)
\(798\) 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 \(-0.749734\pi\)
−0.706515 + 0.707698i \(0.749734\pi\)
\(822\) 0 0
\(823\) 17227.5 0.729665 0.364832 0.931073i \(-0.381126\pi\)
0.364832 + 0.931073i \(0.381126\pi\)
\(824\) 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 \(-0.245114\pi\)
0.717877 + 0.696170i \(0.245114\pi\)
\(830\) 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 \(-0.741552\pi\)
−0.688092 + 0.725623i \(0.741552\pi\)
\(840\) 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 \(-0.766758\pi\)
−0.743337 + 0.668917i \(0.766758\pi\)
\(854\) 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 \(-0.669031\pi\)
−0.506420 + 0.862287i \(0.669031\pi\)
\(864\) 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 \(-0.282591\pi\)
0.631131 + 0.775676i \(0.282591\pi\)
\(878\) 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 \(-0.644693\pi\)
−0.439072 + 0.898452i \(0.644693\pi\)
\(888\) 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 \(-0.520199\pi\)
−0.0634143 + 0.997987i \(0.520199\pi\)
\(912\) 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 \(-0.599024\pi\)
−0.306099 + 0.952000i \(0.599024\pi\)
\(920\) 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 \(-0.919404\pi\)
−0.968116 + 0.250502i \(0.919404\pi\)
\(942\) 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 \(-0.572058\pi\)
−0.224448 + 0.974486i \(0.572058\pi\)
\(968\) 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 \(-0.780413\pi\)
−0.771341 + 0.636423i \(0.780413\pi\)
\(984\) 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 \(-0.504553\pi\)
−0.0143019 + 0.999898i \(0.504553\pi\)
\(992\) 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 \(-0.477440\pi\)
0.0708146 + 0.997489i \(0.477440\pi\)
\(998\) 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.r.1.2 3
3.2 odd 2 2304.4.a.bt.1.2 3
4.3 odd 2 768.4.a.t.1.2 3
8.3 odd 2 768.4.a.q.1.2 3
8.5 even 2 768.4.a.s.1.2 3
12.11 even 2 2304.4.a.bu.1.2 3
16.3 odd 4 96.4.d.a.49.5 6
16.5 even 4 24.4.d.a.13.5 6
16.11 odd 4 96.4.d.a.49.2 6
16.13 even 4 24.4.d.a.13.6 yes 6
24.5 odd 2 2304.4.a.bv.1.2 3
24.11 even 2 2304.4.a.bw.1.2 3
48.5 odd 4 72.4.d.d.37.2 6
48.11 even 4 288.4.d.d.145.3 6
48.29 odd 4 72.4.d.d.37.1 6
48.35 even 4 288.4.d.d.145.4 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
24.4.d.a.13.5 6 16.5 even 4
24.4.d.a.13.6 yes 6 16.13 even 4
72.4.d.d.37.1 6 48.29 odd 4
72.4.d.d.37.2 6 48.5 odd 4
96.4.d.a.49.2 6 16.11 odd 4
96.4.d.a.49.5 6 16.3 odd 4
288.4.d.d.145.3 6 48.11 even 4
288.4.d.d.145.4 6 48.35 even 4
768.4.a.q.1.2 3 8.3 odd 2
768.4.a.r.1.2 3 1.1 even 1 trivial
768.4.a.s.1.2 3 8.5 even 2
768.4.a.t.1.2 3 4.3 odd 2
2304.4.a.bt.1.2 3 3.2 odd 2
2304.4.a.bu.1.2 3 12.11 even 2
2304.4.a.bv.1.2 3 24.5 odd 2
2304.4.a.bw.1.2 3 24.11 even 2