Properties

Label 1856.4.a.y.1.3
Level $1856$
Weight $4$
Character 1856.1
Self dual yes
Analytic conductor $109.508$
Analytic rank $1$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1856 = 2^{6} \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1856.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(109.507544971\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.13458092.1
Defining polynomial: \( x^{5} - x^{4} - 14x^{3} + 18x^{2} + 20x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 29)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(0.328194\) of defining polynomial
Character \(\chi\) \(=\) 1856.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.84328 q^{3} -18.3339 q^{5} -16.8583 q^{7} -23.6023 q^{9} +O(q^{10})\) \(q-1.84328 q^{3} -18.3339 q^{5} -16.8583 q^{7} -23.6023 q^{9} -52.4385 q^{11} +87.5580 q^{13} +33.7945 q^{15} +15.4072 q^{17} -67.0156 q^{19} +31.0745 q^{21} +132.679 q^{23} +211.133 q^{25} +93.2741 q^{27} +29.0000 q^{29} +90.2221 q^{31} +96.6587 q^{33} +309.078 q^{35} -11.1247 q^{37} -161.394 q^{39} -18.8392 q^{41} +147.756 q^{43} +432.723 q^{45} +21.0963 q^{47} -58.7983 q^{49} -28.3997 q^{51} +290.454 q^{53} +961.404 q^{55} +123.528 q^{57} +337.343 q^{59} -84.0147 q^{61} +397.895 q^{63} -1605.28 q^{65} -330.821 q^{67} -244.564 q^{69} +492.420 q^{71} -347.053 q^{73} -389.176 q^{75} +884.023 q^{77} -986.297 q^{79} +465.333 q^{81} -594.382 q^{83} -282.475 q^{85} -53.4550 q^{87} +1387.04 q^{89} -1476.08 q^{91} -166.304 q^{93} +1228.66 q^{95} -334.003 q^{97} +1237.67 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 8 q^{3} - 10 q^{5} + 40 q^{7} + 33 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 8 q^{3} - 10 q^{5} + 40 q^{7} + 33 q^{9} - 12 q^{11} - 14 q^{13} - 74 q^{15} + 66 q^{17} - 214 q^{19} + 164 q^{23} + 207 q^{25} - 362 q^{27} + 145 q^{29} + 420 q^{31} - 576 q^{33} + 52 q^{35} - 378 q^{37} - 374 q^{39} - 1158 q^{41} + 204 q^{43} + 1506 q^{45} + 248 q^{47} - 283 q^{49} - 228 q^{51} + 554 q^{53} + 546 q^{55} + 44 q^{57} - 440 q^{59} - 618 q^{61} + 804 q^{63} - 1656 q^{65} - 1164 q^{67} + 1968 q^{69} - 692 q^{71} - 1950 q^{73} - 3074 q^{75} + 1616 q^{77} + 272 q^{79} + 1801 q^{81} - 512 q^{83} + 1628 q^{85} - 232 q^{87} + 866 q^{89} - 2580 q^{91} + 40 q^{93} + 2244 q^{95} + 1562 q^{97} + 238 q^{99}+O(q^{100}) \) 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
\(3\) −1.84328 −0.354739 −0.177369 0.984144i \(-0.556759\pi\)
−0.177369 + 0.984144i \(0.556759\pi\)
\(4\) 0 0
\(5\) −18.3339 −1.63984 −0.819918 0.572481i \(-0.805981\pi\)
−0.819918 + 0.572481i \(0.805981\pi\)
\(6\) 0 0
\(7\) −16.8583 −0.910262 −0.455131 0.890425i \(-0.650408\pi\)
−0.455131 + 0.890425i \(0.650408\pi\)
\(8\) 0 0
\(9\) −23.6023 −0.874160
\(10\) 0 0
\(11\) −52.4385 −1.43735 −0.718673 0.695348i \(-0.755250\pi\)
−0.718673 + 0.695348i \(0.755250\pi\)
\(12\) 0 0
\(13\) 87.5580 1.86802 0.934008 0.357252i \(-0.116286\pi\)
0.934008 + 0.357252i \(0.116286\pi\)
\(14\) 0 0
\(15\) 33.7945 0.581713
\(16\) 0 0
\(17\) 15.4072 0.219812 0.109906 0.993942i \(-0.464945\pi\)
0.109906 + 0.993942i \(0.464945\pi\)
\(18\) 0 0
\(19\) −67.0156 −0.809181 −0.404591 0.914498i \(-0.632586\pi\)
−0.404591 + 0.914498i \(0.632586\pi\)
\(20\) 0 0
\(21\) 31.0745 0.322905
\(22\) 0 0
\(23\) 132.679 1.20285 0.601423 0.798931i \(-0.294601\pi\)
0.601423 + 0.798931i \(0.294601\pi\)
\(24\) 0 0
\(25\) 211.133 1.68906
\(26\) 0 0
\(27\) 93.2741 0.664837
\(28\) 0 0
\(29\) 29.0000 0.185695
\(30\) 0 0
\(31\) 90.2221 0.522721 0.261361 0.965241i \(-0.415829\pi\)
0.261361 + 0.965241i \(0.415829\pi\)
\(32\) 0 0
\(33\) 96.6587 0.509882
\(34\) 0 0
\(35\) 309.078 1.49268
\(36\) 0 0
\(37\) −11.1247 −0.0494296 −0.0247148 0.999695i \(-0.507868\pi\)
−0.0247148 + 0.999695i \(0.507868\pi\)
\(38\) 0 0
\(39\) −161.394 −0.662658
\(40\) 0 0
\(41\) −18.8392 −0.0717608 −0.0358804 0.999356i \(-0.511424\pi\)
−0.0358804 + 0.999356i \(0.511424\pi\)
\(42\) 0 0
\(43\) 147.756 0.524013 0.262007 0.965066i \(-0.415616\pi\)
0.262007 + 0.965066i \(0.415616\pi\)
\(44\) 0 0
\(45\) 432.723 1.43348
\(46\) 0 0
\(47\) 21.0963 0.0654726 0.0327363 0.999464i \(-0.489578\pi\)
0.0327363 + 0.999464i \(0.489578\pi\)
\(48\) 0 0
\(49\) −58.7983 −0.171424
\(50\) 0 0
\(51\) −28.3997 −0.0779757
\(52\) 0 0
\(53\) 290.454 0.752772 0.376386 0.926463i \(-0.377167\pi\)
0.376386 + 0.926463i \(0.377167\pi\)
\(54\) 0 0
\(55\) 961.404 2.35701
\(56\) 0 0
\(57\) 123.528 0.287048
\(58\) 0 0
\(59\) 337.343 0.744379 0.372190 0.928157i \(-0.378607\pi\)
0.372190 + 0.928157i \(0.378607\pi\)
\(60\) 0 0
\(61\) −84.0147 −0.176344 −0.0881720 0.996105i \(-0.528103\pi\)
−0.0881720 + 0.996105i \(0.528103\pi\)
\(62\) 0 0
\(63\) 397.895 0.795715
\(64\) 0 0
\(65\) −1605.28 −3.06324
\(66\) 0 0
\(67\) −330.821 −0.603228 −0.301614 0.953430i \(-0.597525\pi\)
−0.301614 + 0.953430i \(0.597525\pi\)
\(68\) 0 0
\(69\) −244.564 −0.426696
\(70\) 0 0
\(71\) 492.420 0.823092 0.411546 0.911389i \(-0.364989\pi\)
0.411546 + 0.911389i \(0.364989\pi\)
\(72\) 0 0
\(73\) −347.053 −0.556431 −0.278216 0.960519i \(-0.589743\pi\)
−0.278216 + 0.960519i \(0.589743\pi\)
\(74\) 0 0
\(75\) −389.176 −0.599176
\(76\) 0 0
\(77\) 884.023 1.30836
\(78\) 0 0
\(79\) −986.297 −1.40465 −0.702324 0.711858i \(-0.747854\pi\)
−0.702324 + 0.711858i \(0.747854\pi\)
\(80\) 0 0
\(81\) 465.333 0.638317
\(82\) 0 0
\(83\) −594.382 −0.786048 −0.393024 0.919528i \(-0.628571\pi\)
−0.393024 + 0.919528i \(0.628571\pi\)
\(84\) 0 0
\(85\) −282.475 −0.360455
\(86\) 0 0
\(87\) −53.4550 −0.0658733
\(88\) 0 0
\(89\) 1387.04 1.65197 0.825987 0.563689i \(-0.190618\pi\)
0.825987 + 0.563689i \(0.190618\pi\)
\(90\) 0 0
\(91\) −1476.08 −1.70038
\(92\) 0 0
\(93\) −166.304 −0.185430
\(94\) 0 0
\(95\) 1228.66 1.32692
\(96\) 0 0
\(97\) −334.003 −0.349617 −0.174808 0.984602i \(-0.555931\pi\)
−0.174808 + 0.984602i \(0.555931\pi\)
\(98\) 0 0
\(99\) 1237.67 1.25647
\(100\) 0 0
\(101\) 245.919 0.242276 0.121138 0.992636i \(-0.461346\pi\)
0.121138 + 0.992636i \(0.461346\pi\)
\(102\) 0 0
\(103\) −531.298 −0.508255 −0.254128 0.967171i \(-0.581788\pi\)
−0.254128 + 0.967171i \(0.581788\pi\)
\(104\) 0 0
\(105\) −569.717 −0.529511
\(106\) 0 0
\(107\) 429.030 0.387625 0.193812 0.981039i \(-0.437915\pi\)
0.193812 + 0.981039i \(0.437915\pi\)
\(108\) 0 0
\(109\) 967.263 0.849972 0.424986 0.905200i \(-0.360279\pi\)
0.424986 + 0.905200i \(0.360279\pi\)
\(110\) 0 0
\(111\) 20.5060 0.0175346
\(112\) 0 0
\(113\) −1705.23 −1.41960 −0.709798 0.704405i \(-0.751214\pi\)
−0.709798 + 0.704405i \(0.751214\pi\)
\(114\) 0 0
\(115\) −2432.52 −1.97247
\(116\) 0 0
\(117\) −2066.57 −1.63295
\(118\) 0 0
\(119\) −259.739 −0.200086
\(120\) 0 0
\(121\) 1418.80 1.06596
\(122\) 0 0
\(123\) 34.7259 0.0254563
\(124\) 0 0
\(125\) −1579.15 −1.12995
\(126\) 0 0
\(127\) −2670.28 −1.86574 −0.932870 0.360213i \(-0.882704\pi\)
−0.932870 + 0.360213i \(0.882704\pi\)
\(128\) 0 0
\(129\) −272.355 −0.185888
\(130\) 0 0
\(131\) −879.993 −0.586911 −0.293455 0.955973i \(-0.594805\pi\)
−0.293455 + 0.955973i \(0.594805\pi\)
\(132\) 0 0
\(133\) 1129.77 0.736567
\(134\) 0 0
\(135\) −1710.08 −1.09022
\(136\) 0 0
\(137\) 2064.15 1.28724 0.643620 0.765345i \(-0.277432\pi\)
0.643620 + 0.765345i \(0.277432\pi\)
\(138\) 0 0
\(139\) −605.130 −0.369255 −0.184628 0.982809i \(-0.559108\pi\)
−0.184628 + 0.982809i \(0.559108\pi\)
\(140\) 0 0
\(141\) −38.8863 −0.0232257
\(142\) 0 0
\(143\) −4591.41 −2.68499
\(144\) 0 0
\(145\) −531.684 −0.304510
\(146\) 0 0
\(147\) 108.381 0.0608106
\(148\) 0 0
\(149\) 775.322 0.426287 0.213144 0.977021i \(-0.431630\pi\)
0.213144 + 0.977021i \(0.431630\pi\)
\(150\) 0 0
\(151\) 427.925 0.230623 0.115311 0.993329i \(-0.463213\pi\)
0.115311 + 0.993329i \(0.463213\pi\)
\(152\) 0 0
\(153\) −363.646 −0.192151
\(154\) 0 0
\(155\) −1654.12 −0.857177
\(156\) 0 0
\(157\) 1680.93 0.854474 0.427237 0.904140i \(-0.359487\pi\)
0.427237 + 0.904140i \(0.359487\pi\)
\(158\) 0 0
\(159\) −535.387 −0.267037
\(160\) 0 0
\(161\) −2236.74 −1.09490
\(162\) 0 0
\(163\) −2038.68 −0.979645 −0.489822 0.871822i \(-0.662938\pi\)
−0.489822 + 0.871822i \(0.662938\pi\)
\(164\) 0 0
\(165\) −1772.13 −0.836123
\(166\) 0 0
\(167\) 2543.12 1.17840 0.589199 0.807988i \(-0.299443\pi\)
0.589199 + 0.807988i \(0.299443\pi\)
\(168\) 0 0
\(169\) 5469.40 2.48949
\(170\) 0 0
\(171\) 1581.73 0.707354
\(172\) 0 0
\(173\) 306.031 0.134492 0.0672460 0.997736i \(-0.478579\pi\)
0.0672460 + 0.997736i \(0.478579\pi\)
\(174\) 0 0
\(175\) −3559.34 −1.53749
\(176\) 0 0
\(177\) −621.817 −0.264060
\(178\) 0 0
\(179\) 478.797 0.199927 0.0999635 0.994991i \(-0.468127\pi\)
0.0999635 + 0.994991i \(0.468127\pi\)
\(180\) 0 0
\(181\) 478.433 0.196473 0.0982367 0.995163i \(-0.468680\pi\)
0.0982367 + 0.995163i \(0.468680\pi\)
\(182\) 0 0
\(183\) 154.862 0.0625560
\(184\) 0 0
\(185\) 203.960 0.0810565
\(186\) 0 0
\(187\) −807.931 −0.315945
\(188\) 0 0
\(189\) −1572.44 −0.605176
\(190\) 0 0
\(191\) 833.106 0.315610 0.157805 0.987470i \(-0.449558\pi\)
0.157805 + 0.987470i \(0.449558\pi\)
\(192\) 0 0
\(193\) −1449.88 −0.540751 −0.270376 0.962755i \(-0.587148\pi\)
−0.270376 + 0.962755i \(0.587148\pi\)
\(194\) 0 0
\(195\) 2958.98 1.08665
\(196\) 0 0
\(197\) 1993.27 0.720886 0.360443 0.932781i \(-0.382626\pi\)
0.360443 + 0.932781i \(0.382626\pi\)
\(198\) 0 0
\(199\) −356.359 −0.126943 −0.0634714 0.997984i \(-0.520217\pi\)
−0.0634714 + 0.997984i \(0.520217\pi\)
\(200\) 0 0
\(201\) 609.795 0.213988
\(202\) 0 0
\(203\) −488.890 −0.169031
\(204\) 0 0
\(205\) 345.397 0.117676
\(206\) 0 0
\(207\) −3131.53 −1.05148
\(208\) 0 0
\(209\) 3514.20 1.16307
\(210\) 0 0
\(211\) −4131.66 −1.34803 −0.674017 0.738716i \(-0.735433\pi\)
−0.674017 + 0.738716i \(0.735433\pi\)
\(212\) 0 0
\(213\) −907.666 −0.291982
\(214\) 0 0
\(215\) −2708.95 −0.859296
\(216\) 0 0
\(217\) −1520.99 −0.475813
\(218\) 0 0
\(219\) 639.715 0.197388
\(220\) 0 0
\(221\) 1349.02 0.410612
\(222\) 0 0
\(223\) 1332.32 0.400086 0.200043 0.979787i \(-0.435892\pi\)
0.200043 + 0.979787i \(0.435892\pi\)
\(224\) 0 0
\(225\) −4983.22 −1.47651
\(226\) 0 0
\(227\) 1329.33 0.388681 0.194340 0.980934i \(-0.437743\pi\)
0.194340 + 0.980934i \(0.437743\pi\)
\(228\) 0 0
\(229\) −5455.47 −1.57427 −0.787135 0.616780i \(-0.788437\pi\)
−0.787135 + 0.616780i \(0.788437\pi\)
\(230\) 0 0
\(231\) −1629.50 −0.464126
\(232\) 0 0
\(233\) −591.158 −0.166215 −0.0831075 0.996541i \(-0.526484\pi\)
−0.0831075 + 0.996541i \(0.526484\pi\)
\(234\) 0 0
\(235\) −386.778 −0.107364
\(236\) 0 0
\(237\) 1818.02 0.498283
\(238\) 0 0
\(239\) 6946.01 1.87992 0.939959 0.341289i \(-0.110863\pi\)
0.939959 + 0.341289i \(0.110863\pi\)
\(240\) 0 0
\(241\) 7105.62 1.89923 0.949613 0.313426i \(-0.101477\pi\)
0.949613 + 0.313426i \(0.101477\pi\)
\(242\) 0 0
\(243\) −3376.14 −0.891273
\(244\) 0 0
\(245\) 1078.00 0.281106
\(246\) 0 0
\(247\) −5867.75 −1.51156
\(248\) 0 0
\(249\) 1095.61 0.278842
\(250\) 0 0
\(251\) 4874.53 1.22581 0.612904 0.790158i \(-0.290001\pi\)
0.612904 + 0.790158i \(0.290001\pi\)
\(252\) 0 0
\(253\) −6957.49 −1.72891
\(254\) 0 0
\(255\) 520.679 0.127867
\(256\) 0 0
\(257\) 2488.22 0.603934 0.301967 0.953318i \(-0.402357\pi\)
0.301967 + 0.953318i \(0.402357\pi\)
\(258\) 0 0
\(259\) 187.544 0.0449939
\(260\) 0 0
\(261\) −684.468 −0.162328
\(262\) 0 0
\(263\) −2812.52 −0.659419 −0.329709 0.944082i \(-0.606951\pi\)
−0.329709 + 0.944082i \(0.606951\pi\)
\(264\) 0 0
\(265\) −5325.16 −1.23442
\(266\) 0 0
\(267\) −2556.69 −0.586019
\(268\) 0 0
\(269\) 5554.63 1.25900 0.629501 0.777000i \(-0.283259\pi\)
0.629501 + 0.777000i \(0.283259\pi\)
\(270\) 0 0
\(271\) −3168.41 −0.710211 −0.355105 0.934826i \(-0.615555\pi\)
−0.355105 + 0.934826i \(0.615555\pi\)
\(272\) 0 0
\(273\) 2720.82 0.603192
\(274\) 0 0
\(275\) −11071.5 −2.42777
\(276\) 0 0
\(277\) −3965.64 −0.860189 −0.430095 0.902784i \(-0.641520\pi\)
−0.430095 + 0.902784i \(0.641520\pi\)
\(278\) 0 0
\(279\) −2129.45 −0.456942
\(280\) 0 0
\(281\) 1655.16 0.351383 0.175692 0.984445i \(-0.443784\pi\)
0.175692 + 0.984445i \(0.443784\pi\)
\(282\) 0 0
\(283\) −7786.09 −1.63546 −0.817730 0.575602i \(-0.804768\pi\)
−0.817730 + 0.575602i \(0.804768\pi\)
\(284\) 0 0
\(285\) −2264.76 −0.470711
\(286\) 0 0
\(287\) 317.597 0.0653211
\(288\) 0 0
\(289\) −4675.62 −0.951683
\(290\) 0 0
\(291\) 615.659 0.124023
\(292\) 0 0
\(293\) −8090.80 −1.61321 −0.806603 0.591094i \(-0.798696\pi\)
−0.806603 + 0.591094i \(0.798696\pi\)
\(294\) 0 0
\(295\) −6184.83 −1.22066
\(296\) 0 0
\(297\) −4891.16 −0.955601
\(298\) 0 0
\(299\) 11617.1 2.24694
\(300\) 0 0
\(301\) −2490.91 −0.476989
\(302\) 0 0
\(303\) −453.297 −0.0859446
\(304\) 0 0
\(305\) 1540.32 0.289175
\(306\) 0 0
\(307\) 6129.49 1.13951 0.569753 0.821816i \(-0.307039\pi\)
0.569753 + 0.821816i \(0.307039\pi\)
\(308\) 0 0
\(309\) 979.328 0.180298
\(310\) 0 0
\(311\) −8167.93 −1.48926 −0.744632 0.667476i \(-0.767375\pi\)
−0.744632 + 0.667476i \(0.767375\pi\)
\(312\) 0 0
\(313\) 1877.25 0.339005 0.169502 0.985530i \(-0.445784\pi\)
0.169502 + 0.985530i \(0.445784\pi\)
\(314\) 0 0
\(315\) −7294.97 −1.30484
\(316\) 0 0
\(317\) −1222.93 −0.216677 −0.108338 0.994114i \(-0.534553\pi\)
−0.108338 + 0.994114i \(0.534553\pi\)
\(318\) 0 0
\(319\) −1520.72 −0.266908
\(320\) 0 0
\(321\) −790.820 −0.137506
\(322\) 0 0
\(323\) −1032.52 −0.177867
\(324\) 0 0
\(325\) 18486.4 3.15520
\(326\) 0 0
\(327\) −1782.93 −0.301518
\(328\) 0 0
\(329\) −355.648 −0.0595972
\(330\) 0 0
\(331\) −3769.03 −0.625876 −0.312938 0.949774i \(-0.601313\pi\)
−0.312938 + 0.949774i \(0.601313\pi\)
\(332\) 0 0
\(333\) 262.570 0.0432094
\(334\) 0 0
\(335\) 6065.25 0.989195
\(336\) 0 0
\(337\) −10900.2 −1.76193 −0.880967 0.473179i \(-0.843107\pi\)
−0.880967 + 0.473179i \(0.843107\pi\)
\(338\) 0 0
\(339\) 3143.21 0.503586
\(340\) 0 0
\(341\) −4731.11 −0.751332
\(342\) 0 0
\(343\) 6773.63 1.06630
\(344\) 0 0
\(345\) 4483.82 0.699712
\(346\) 0 0
\(347\) 8542.30 1.32154 0.660771 0.750588i \(-0.270230\pi\)
0.660771 + 0.750588i \(0.270230\pi\)
\(348\) 0 0
\(349\) 993.823 0.152430 0.0762151 0.997091i \(-0.475716\pi\)
0.0762151 + 0.997091i \(0.475716\pi\)
\(350\) 0 0
\(351\) 8166.89 1.24193
\(352\) 0 0
\(353\) 8191.10 1.23504 0.617519 0.786556i \(-0.288138\pi\)
0.617519 + 0.786556i \(0.288138\pi\)
\(354\) 0 0
\(355\) −9027.99 −1.34974
\(356\) 0 0
\(357\) 478.771 0.0709783
\(358\) 0 0
\(359\) −4703.71 −0.691510 −0.345755 0.938325i \(-0.612377\pi\)
−0.345755 + 0.938325i \(0.612377\pi\)
\(360\) 0 0
\(361\) −2367.90 −0.345226
\(362\) 0 0
\(363\) −2615.24 −0.378139
\(364\) 0 0
\(365\) 6362.85 0.912456
\(366\) 0 0
\(367\) 9431.88 1.34153 0.670763 0.741672i \(-0.265967\pi\)
0.670763 + 0.741672i \(0.265967\pi\)
\(368\) 0 0
\(369\) 444.650 0.0627305
\(370\) 0 0
\(371\) −4896.55 −0.685219
\(372\) 0 0
\(373\) 8281.46 1.14959 0.574796 0.818297i \(-0.305081\pi\)
0.574796 + 0.818297i \(0.305081\pi\)
\(374\) 0 0
\(375\) 2910.81 0.400836
\(376\) 0 0
\(377\) 2539.18 0.346882
\(378\) 0 0
\(379\) −6875.50 −0.931848 −0.465924 0.884825i \(-0.654278\pi\)
−0.465924 + 0.884825i \(0.654278\pi\)
\(380\) 0 0
\(381\) 4922.06 0.661850
\(382\) 0 0
\(383\) −4826.61 −0.643938 −0.321969 0.946750i \(-0.604345\pi\)
−0.321969 + 0.946750i \(0.604345\pi\)
\(384\) 0 0
\(385\) −16207.6 −2.14550
\(386\) 0 0
\(387\) −3487.38 −0.458072
\(388\) 0 0
\(389\) −4970.57 −0.647861 −0.323930 0.946081i \(-0.605004\pi\)
−0.323930 + 0.946081i \(0.605004\pi\)
\(390\) 0 0
\(391\) 2044.21 0.264400
\(392\) 0 0
\(393\) 1622.07 0.208200
\(394\) 0 0
\(395\) 18082.7 2.30339
\(396\) 0 0
\(397\) 12288.8 1.55355 0.776774 0.629779i \(-0.216855\pi\)
0.776774 + 0.629779i \(0.216855\pi\)
\(398\) 0 0
\(399\) −2082.48 −0.261289
\(400\) 0 0
\(401\) −11971.7 −1.49086 −0.745432 0.666581i \(-0.767757\pi\)
−0.745432 + 0.666581i \(0.767757\pi\)
\(402\) 0 0
\(403\) 7899.66 0.976452
\(404\) 0 0
\(405\) −8531.38 −1.04673
\(406\) 0 0
\(407\) 583.365 0.0710475
\(408\) 0 0
\(409\) 11147.7 1.34772 0.673861 0.738858i \(-0.264635\pi\)
0.673861 + 0.738858i \(0.264635\pi\)
\(410\) 0 0
\(411\) −3804.79 −0.456634
\(412\) 0 0
\(413\) −5687.03 −0.677580
\(414\) 0 0
\(415\) 10897.4 1.28899
\(416\) 0 0
\(417\) 1115.42 0.130989
\(418\) 0 0
\(419\) −11557.3 −1.34752 −0.673762 0.738949i \(-0.735323\pi\)
−0.673762 + 0.738949i \(0.735323\pi\)
\(420\) 0 0
\(421\) 12874.4 1.49040 0.745201 0.666840i \(-0.232353\pi\)
0.745201 + 0.666840i \(0.232353\pi\)
\(422\) 0 0
\(423\) −497.922 −0.0572336
\(424\) 0 0
\(425\) 3252.97 0.371275
\(426\) 0 0
\(427\) 1416.34 0.160519
\(428\) 0 0
\(429\) 8463.24 0.952469
\(430\) 0 0
\(431\) −4088.31 −0.456907 −0.228454 0.973555i \(-0.573367\pi\)
−0.228454 + 0.973555i \(0.573367\pi\)
\(432\) 0 0
\(433\) −3865.90 −0.429060 −0.214530 0.976717i \(-0.568822\pi\)
−0.214530 + 0.976717i \(0.568822\pi\)
\(434\) 0 0
\(435\) 980.040 0.108021
\(436\) 0 0
\(437\) −8891.56 −0.973320
\(438\) 0 0
\(439\) 10662.4 1.15920 0.579600 0.814901i \(-0.303209\pi\)
0.579600 + 0.814901i \(0.303209\pi\)
\(440\) 0 0
\(441\) 1387.78 0.149852
\(442\) 0 0
\(443\) −10288.9 −1.10347 −0.551736 0.834019i \(-0.686034\pi\)
−0.551736 + 0.834019i \(0.686034\pi\)
\(444\) 0 0
\(445\) −25429.8 −2.70897
\(446\) 0 0
\(447\) −1429.13 −0.151221
\(448\) 0 0
\(449\) 12426.2 1.30608 0.653041 0.757323i \(-0.273493\pi\)
0.653041 + 0.757323i \(0.273493\pi\)
\(450\) 0 0
\(451\) 987.901 0.103145
\(452\) 0 0
\(453\) −788.784 −0.0818108
\(454\) 0 0
\(455\) 27062.3 2.78835
\(456\) 0 0
\(457\) −10657.8 −1.09092 −0.545462 0.838136i \(-0.683646\pi\)
−0.545462 + 0.838136i \(0.683646\pi\)
\(458\) 0 0
\(459\) 1437.09 0.146139
\(460\) 0 0
\(461\) −9819.21 −0.992031 −0.496016 0.868314i \(-0.665204\pi\)
−0.496016 + 0.868314i \(0.665204\pi\)
\(462\) 0 0
\(463\) −19210.5 −1.92827 −0.964135 0.265412i \(-0.914492\pi\)
−0.964135 + 0.265412i \(0.914492\pi\)
\(464\) 0 0
\(465\) 3049.01 0.304074
\(466\) 0 0
\(467\) 345.566 0.0342417 0.0171208 0.999853i \(-0.494550\pi\)
0.0171208 + 0.999853i \(0.494550\pi\)
\(468\) 0 0
\(469\) 5577.08 0.549095
\(470\) 0 0
\(471\) −3098.41 −0.303115
\(472\) 0 0
\(473\) −7748.10 −0.753188
\(474\) 0 0
\(475\) −14149.2 −1.36676
\(476\) 0 0
\(477\) −6855.39 −0.658043
\(478\) 0 0
\(479\) −253.709 −0.0242009 −0.0121005 0.999927i \(-0.503852\pi\)
−0.0121005 + 0.999927i \(0.503852\pi\)
\(480\) 0 0
\(481\) −974.060 −0.0923354
\(482\) 0 0
\(483\) 4122.93 0.388405
\(484\) 0 0
\(485\) 6123.58 0.573314
\(486\) 0 0
\(487\) −13255.1 −1.23336 −0.616680 0.787214i \(-0.711523\pi\)
−0.616680 + 0.787214i \(0.711523\pi\)
\(488\) 0 0
\(489\) 3757.86 0.347518
\(490\) 0 0
\(491\) 6454.57 0.593260 0.296630 0.954993i \(-0.404137\pi\)
0.296630 + 0.954993i \(0.404137\pi\)
\(492\) 0 0
\(493\) 446.809 0.0408180
\(494\) 0 0
\(495\) −22691.4 −2.06041
\(496\) 0 0
\(497\) −8301.36 −0.749229
\(498\) 0 0
\(499\) −8090.41 −0.725805 −0.362902 0.931827i \(-0.618214\pi\)
−0.362902 + 0.931827i \(0.618214\pi\)
\(500\) 0 0
\(501\) −4687.67 −0.418023
\(502\) 0 0
\(503\) −18897.4 −1.67513 −0.837567 0.546334i \(-0.816023\pi\)
−0.837567 + 0.546334i \(0.816023\pi\)
\(504\) 0 0
\(505\) −4508.66 −0.397293
\(506\) 0 0
\(507\) −10081.6 −0.883117
\(508\) 0 0
\(509\) −4265.15 −0.371413 −0.185707 0.982605i \(-0.559457\pi\)
−0.185707 + 0.982605i \(0.559457\pi\)
\(510\) 0 0
\(511\) 5850.72 0.506498
\(512\) 0 0
\(513\) −6250.82 −0.537974
\(514\) 0 0
\(515\) 9740.77 0.833455
\(516\) 0 0
\(517\) −1106.26 −0.0941068
\(518\) 0 0
\(519\) −564.100 −0.0477096
\(520\) 0 0
\(521\) −3324.96 −0.279595 −0.139798 0.990180i \(-0.544645\pi\)
−0.139798 + 0.990180i \(0.544645\pi\)
\(522\) 0 0
\(523\) −13017.6 −1.08838 −0.544188 0.838964i \(-0.683162\pi\)
−0.544188 + 0.838964i \(0.683162\pi\)
\(524\) 0 0
\(525\) 6560.84 0.545407
\(526\) 0 0
\(527\) 1390.07 0.114900
\(528\) 0 0
\(529\) 5436.69 0.446839
\(530\) 0 0
\(531\) −7962.09 −0.650707
\(532\) 0 0
\(533\) −1649.52 −0.134050
\(534\) 0 0
\(535\) −7865.80 −0.635641
\(536\) 0 0
\(537\) −882.555 −0.0709219
\(538\) 0 0
\(539\) 3083.29 0.246395
\(540\) 0 0
\(541\) 17906.8 1.42305 0.711527 0.702658i \(-0.248004\pi\)
0.711527 + 0.702658i \(0.248004\pi\)
\(542\) 0 0
\(543\) −881.885 −0.0696967
\(544\) 0 0
\(545\) −17733.7 −1.39382
\(546\) 0 0
\(547\) −1612.94 −0.126078 −0.0630389 0.998011i \(-0.520079\pi\)
−0.0630389 + 0.998011i \(0.520079\pi\)
\(548\) 0 0
\(549\) 1982.94 0.154153
\(550\) 0 0
\(551\) −1943.45 −0.150261
\(552\) 0 0
\(553\) 16627.3 1.27860
\(554\) 0 0
\(555\) −375.955 −0.0287539
\(556\) 0 0
\(557\) 7803.94 0.593651 0.296826 0.954932i \(-0.404072\pi\)
0.296826 + 0.954932i \(0.404072\pi\)
\(558\) 0 0
\(559\) 12937.2 0.978865
\(560\) 0 0
\(561\) 1489.24 0.112078
\(562\) 0 0
\(563\) 12329.5 0.922958 0.461479 0.887151i \(-0.347319\pi\)
0.461479 + 0.887151i \(0.347319\pi\)
\(564\) 0 0
\(565\) 31263.5 2.32790
\(566\) 0 0
\(567\) −7844.72 −0.581035
\(568\) 0 0
\(569\) 1554.46 0.114528 0.0572640 0.998359i \(-0.481762\pi\)
0.0572640 + 0.998359i \(0.481762\pi\)
\(570\) 0 0
\(571\) −15951.8 −1.16911 −0.584556 0.811353i \(-0.698731\pi\)
−0.584556 + 0.811353i \(0.698731\pi\)
\(572\) 0 0
\(573\) −1535.64 −0.111959
\(574\) 0 0
\(575\) 28012.9 2.03168
\(576\) 0 0
\(577\) 10491.7 0.756975 0.378488 0.925606i \(-0.376444\pi\)
0.378488 + 0.925606i \(0.376444\pi\)
\(578\) 0 0
\(579\) 2672.54 0.191825
\(580\) 0 0
\(581\) 10020.3 0.715509
\(582\) 0 0
\(583\) −15231.0 −1.08199
\(584\) 0 0
\(585\) 37888.4 2.67776
\(586\) 0 0
\(587\) −6437.47 −0.452645 −0.226323 0.974052i \(-0.572670\pi\)
−0.226323 + 0.974052i \(0.572670\pi\)
\(588\) 0 0
\(589\) −6046.29 −0.422976
\(590\) 0 0
\(591\) −3674.15 −0.255726
\(592\) 0 0
\(593\) 11240.6 0.778411 0.389205 0.921151i \(-0.372750\pi\)
0.389205 + 0.921151i \(0.372750\pi\)
\(594\) 0 0
\(595\) 4762.04 0.328108
\(596\) 0 0
\(597\) 656.869 0.0450316
\(598\) 0 0
\(599\) 15903.5 1.08481 0.542405 0.840117i \(-0.317514\pi\)
0.542405 + 0.840117i \(0.317514\pi\)
\(600\) 0 0
\(601\) 117.190 0.00795385 0.00397692 0.999992i \(-0.498734\pi\)
0.00397692 + 0.999992i \(0.498734\pi\)
\(602\) 0 0
\(603\) 7808.16 0.527318
\(604\) 0 0
\(605\) −26012.1 −1.74801
\(606\) 0 0
\(607\) −22047.8 −1.47429 −0.737143 0.675737i \(-0.763825\pi\)
−0.737143 + 0.675737i \(0.763825\pi\)
\(608\) 0 0
\(609\) 901.160 0.0599620
\(610\) 0 0
\(611\) 1847.15 0.122304
\(612\) 0 0
\(613\) 12719.2 0.838050 0.419025 0.907975i \(-0.362372\pi\)
0.419025 + 0.907975i \(0.362372\pi\)
\(614\) 0 0
\(615\) −636.662 −0.0417442
\(616\) 0 0
\(617\) 12736.9 0.831064 0.415532 0.909578i \(-0.363595\pi\)
0.415532 + 0.909578i \(0.363595\pi\)
\(618\) 0 0
\(619\) −28083.4 −1.82353 −0.911767 0.410709i \(-0.865281\pi\)
−0.911767 + 0.410709i \(0.865281\pi\)
\(620\) 0 0
\(621\) 12375.5 0.799697
\(622\) 0 0
\(623\) −23383.1 −1.50373
\(624\) 0 0
\(625\) 2560.44 0.163868
\(626\) 0 0
\(627\) −6477.64 −0.412587
\(628\) 0 0
\(629\) −171.401 −0.0108652
\(630\) 0 0
\(631\) 281.496 0.0177594 0.00887969 0.999961i \(-0.497173\pi\)
0.00887969 + 0.999961i \(0.497173\pi\)
\(632\) 0 0
\(633\) 7615.79 0.478200
\(634\) 0 0
\(635\) 48956.7 3.05951
\(636\) 0 0
\(637\) −5148.26 −0.320222
\(638\) 0 0
\(639\) −11622.3 −0.719514
\(640\) 0 0
\(641\) −8440.98 −0.520123 −0.260061 0.965592i \(-0.583743\pi\)
−0.260061 + 0.965592i \(0.583743\pi\)
\(642\) 0 0
\(643\) 1173.61 0.0719792 0.0359896 0.999352i \(-0.488542\pi\)
0.0359896 + 0.999352i \(0.488542\pi\)
\(644\) 0 0
\(645\) 4993.34 0.304825
\(646\) 0 0
\(647\) −10845.7 −0.659025 −0.329513 0.944151i \(-0.606884\pi\)
−0.329513 + 0.944151i \(0.606884\pi\)
\(648\) 0 0
\(649\) −17689.8 −1.06993
\(650\) 0 0
\(651\) 2803.60 0.168789
\(652\) 0 0
\(653\) 5282.40 0.316564 0.158282 0.987394i \(-0.449404\pi\)
0.158282 + 0.987394i \(0.449404\pi\)
\(654\) 0 0
\(655\) 16133.7 0.962437
\(656\) 0 0
\(657\) 8191.26 0.486410
\(658\) 0 0
\(659\) 19243.7 1.13752 0.568761 0.822503i \(-0.307423\pi\)
0.568761 + 0.822503i \(0.307423\pi\)
\(660\) 0 0
\(661\) −29196.9 −1.71804 −0.859021 0.511940i \(-0.828927\pi\)
−0.859021 + 0.511940i \(0.828927\pi\)
\(662\) 0 0
\(663\) −2486.62 −0.145660
\(664\) 0 0
\(665\) −20713.1 −1.20785
\(666\) 0 0
\(667\) 3847.69 0.223363
\(668\) 0 0
\(669\) −2455.84 −0.141926
\(670\) 0 0
\(671\) 4405.61 0.253467
\(672\) 0 0
\(673\) 19924.5 1.14121 0.570605 0.821224i \(-0.306709\pi\)
0.570605 + 0.821224i \(0.306709\pi\)
\(674\) 0 0
\(675\) 19693.2 1.12295
\(676\) 0 0
\(677\) 4980.43 0.282738 0.141369 0.989957i \(-0.454850\pi\)
0.141369 + 0.989957i \(0.454850\pi\)
\(678\) 0 0
\(679\) 5630.71 0.318243
\(680\) 0 0
\(681\) −2450.32 −0.137880
\(682\) 0 0
\(683\) −29295.8 −1.64125 −0.820624 0.571468i \(-0.806374\pi\)
−0.820624 + 0.571468i \(0.806374\pi\)
\(684\) 0 0
\(685\) −37843.9 −2.11086
\(686\) 0 0
\(687\) 10055.9 0.558455
\(688\) 0 0
\(689\) 25431.5 1.40619
\(690\) 0 0
\(691\) 32759.8 1.80353 0.901766 0.432225i \(-0.142271\pi\)
0.901766 + 0.432225i \(0.142271\pi\)
\(692\) 0 0
\(693\) −20865.0 −1.14372
\(694\) 0 0
\(695\) 11094.4 0.605518
\(696\) 0 0
\(697\) −290.260 −0.0157739
\(698\) 0 0
\(699\) 1089.67 0.0589629
\(700\) 0 0
\(701\) −27958.5 −1.50639 −0.753195 0.657797i \(-0.771488\pi\)
−0.753195 + 0.657797i \(0.771488\pi\)
\(702\) 0 0
\(703\) 745.532 0.0399975
\(704\) 0 0
\(705\) 712.939 0.0380863
\(706\) 0 0
\(707\) −4145.77 −0.220534
\(708\) 0 0
\(709\) 31863.5 1.68781 0.843906 0.536492i \(-0.180251\pi\)
0.843906 + 0.536492i \(0.180251\pi\)
\(710\) 0 0
\(711\) 23278.9 1.22789
\(712\) 0 0
\(713\) 11970.6 0.628753
\(714\) 0 0
\(715\) 84178.6 4.40294
\(716\) 0 0
\(717\) −12803.4 −0.666879
\(718\) 0 0
\(719\) 7944.76 0.412085 0.206043 0.978543i \(-0.433941\pi\)
0.206043 + 0.978543i \(0.433941\pi\)
\(720\) 0 0
\(721\) 8956.76 0.462645
\(722\) 0 0
\(723\) −13097.6 −0.673729
\(724\) 0 0
\(725\) 6122.85 0.313651
\(726\) 0 0
\(727\) 28640.3 1.46109 0.730543 0.682866i \(-0.239267\pi\)
0.730543 + 0.682866i \(0.239267\pi\)
\(728\) 0 0
\(729\) −6340.84 −0.322148
\(730\) 0 0
\(731\) 2276.51 0.115184
\(732\) 0 0
\(733\) −11852.7 −0.597258 −0.298629 0.954369i \(-0.596529\pi\)
−0.298629 + 0.954369i \(0.596529\pi\)
\(734\) 0 0
\(735\) −1987.06 −0.0997194
\(736\) 0 0
\(737\) 17347.8 0.867047
\(738\) 0 0
\(739\) 24052.5 1.19727 0.598636 0.801021i \(-0.295710\pi\)
0.598636 + 0.801021i \(0.295710\pi\)
\(740\) 0 0
\(741\) 10815.9 0.536210
\(742\) 0 0
\(743\) −12530.4 −0.618704 −0.309352 0.950948i \(-0.600112\pi\)
−0.309352 + 0.950948i \(0.600112\pi\)
\(744\) 0 0
\(745\) −14214.7 −0.699042
\(746\) 0 0
\(747\) 14028.8 0.687132
\(748\) 0 0
\(749\) −7232.71 −0.352840
\(750\) 0 0
\(751\) 30921.6 1.50246 0.751229 0.660042i \(-0.229461\pi\)
0.751229 + 0.660042i \(0.229461\pi\)
\(752\) 0 0
\(753\) −8985.11 −0.434841
\(754\) 0 0
\(755\) −7845.54 −0.378184
\(756\) 0 0
\(757\) 6257.40 0.300435 0.150217 0.988653i \(-0.452003\pi\)
0.150217 + 0.988653i \(0.452003\pi\)
\(758\) 0 0
\(759\) 12824.6 0.613310
\(760\) 0 0
\(761\) −20094.8 −0.957211 −0.478605 0.878030i \(-0.658858\pi\)
−0.478605 + 0.878030i \(0.658858\pi\)
\(762\) 0 0
\(763\) −16306.4 −0.773697
\(764\) 0 0
\(765\) 6667.06 0.315095
\(766\) 0 0
\(767\) 29537.1 1.39051
\(768\) 0 0
\(769\) −26647.8 −1.24960 −0.624801 0.780784i \(-0.714820\pi\)
−0.624801 + 0.780784i \(0.714820\pi\)
\(770\) 0 0
\(771\) −4586.48 −0.214239
\(772\) 0 0
\(773\) 18513.2 0.861414 0.430707 0.902492i \(-0.358264\pi\)
0.430707 + 0.902492i \(0.358264\pi\)
\(774\) 0 0
\(775\) 19048.8 0.882909
\(776\) 0 0
\(777\) −345.696 −0.0159611
\(778\) 0 0
\(779\) 1262.52 0.0580675
\(780\) 0 0
\(781\) −25821.8 −1.18307
\(782\) 0 0
\(783\) 2704.95 0.123457
\(784\) 0 0
\(785\) −30818.0 −1.40120
\(786\) 0 0
\(787\) 29524.5 1.33727 0.668636 0.743590i \(-0.266878\pi\)
0.668636 + 0.743590i \(0.266878\pi\)
\(788\) 0 0
\(789\) 5184.25 0.233921
\(790\) 0 0
\(791\) 28747.2 1.29220
\(792\) 0 0
\(793\) −7356.16 −0.329413
\(794\) 0 0
\(795\) 9815.74 0.437897
\(796\) 0 0
\(797\) −38789.4 −1.72395 −0.861976 0.506949i \(-0.830773\pi\)
−0.861976 + 0.506949i \(0.830773\pi\)
\(798\) 0 0
\(799\) 325.035 0.0143916
\(800\) 0 0
\(801\) −32737.3 −1.44409
\(802\) 0 0
\(803\) 18199.0 0.799785
\(804\) 0 0
\(805\) 41008.2 1.79546
\(806\) 0 0
\(807\) −10238.7 −0.446617
\(808\) 0 0
\(809\) −11552.0 −0.502037 −0.251018 0.967982i \(-0.580765\pi\)
−0.251018 + 0.967982i \(0.580765\pi\)
\(810\) 0 0
\(811\) −26939.1 −1.16641 −0.583205 0.812325i \(-0.698202\pi\)
−0.583205 + 0.812325i \(0.698202\pi\)
\(812\) 0 0
\(813\) 5840.25 0.251939
\(814\) 0 0
\(815\) 37377.1 1.60646
\(816\) 0 0
\(817\) −9901.96 −0.424022
\(818\) 0 0
\(819\) 34838.9 1.48641
\(820\) 0 0
\(821\) 7558.67 0.321315 0.160657 0.987010i \(-0.448639\pi\)
0.160657 + 0.987010i \(0.448639\pi\)
\(822\) 0 0
\(823\) −3201.90 −0.135615 −0.0678076 0.997698i \(-0.521600\pi\)
−0.0678076 + 0.997698i \(0.521600\pi\)
\(824\) 0 0
\(825\) 20407.8 0.861223
\(826\) 0 0
\(827\) 11479.1 0.482670 0.241335 0.970442i \(-0.422415\pi\)
0.241335 + 0.970442i \(0.422415\pi\)
\(828\) 0 0
\(829\) 1667.94 0.0698794 0.0349397 0.999389i \(-0.488876\pi\)
0.0349397 + 0.999389i \(0.488876\pi\)
\(830\) 0 0
\(831\) 7309.78 0.305142
\(832\) 0 0
\(833\) −905.917 −0.0376809
\(834\) 0 0
\(835\) −46625.3 −1.93238
\(836\) 0 0
\(837\) 8415.38 0.347525
\(838\) 0 0
\(839\) 15210.3 0.625884 0.312942 0.949772i \(-0.398685\pi\)
0.312942 + 0.949772i \(0.398685\pi\)
\(840\) 0 0
\(841\) 841.000 0.0344828
\(842\) 0 0
\(843\) −3050.92 −0.124649
\(844\) 0 0
\(845\) −100276. −4.08235
\(846\) 0 0
\(847\) −23918.5 −0.970306
\(848\) 0 0
\(849\) 14351.9 0.580161
\(850\) 0 0
\(851\) −1476.02 −0.0594563
\(852\) 0 0
\(853\) 18281.2 0.733805 0.366902 0.930259i \(-0.380418\pi\)
0.366902 + 0.930259i \(0.380418\pi\)
\(854\) 0 0
\(855\) −28999.2 −1.15994
\(856\) 0 0
\(857\) 585.415 0.0233342 0.0116671 0.999932i \(-0.496286\pi\)
0.0116671 + 0.999932i \(0.496286\pi\)
\(858\) 0 0
\(859\) 935.611 0.0371626 0.0185813 0.999827i \(-0.494085\pi\)
0.0185813 + 0.999827i \(0.494085\pi\)
\(860\) 0 0
\(861\) −585.419 −0.0231719
\(862\) 0 0
\(863\) −12110.7 −0.477696 −0.238848 0.971057i \(-0.576770\pi\)
−0.238848 + 0.971057i \(0.576770\pi\)
\(864\) 0 0
\(865\) −5610.75 −0.220545
\(866\) 0 0
\(867\) 8618.46 0.337599
\(868\) 0 0
\(869\) 51720.0 2.01896
\(870\) 0 0
\(871\) −28966.1 −1.12684
\(872\) 0 0
\(873\) 7883.24 0.305621
\(874\) 0 0
\(875\) 26621.8 1.02855
\(876\) 0 0
\(877\) −5841.41 −0.224915 −0.112458 0.993657i \(-0.535872\pi\)
−0.112458 + 0.993657i \(0.535872\pi\)
\(878\) 0 0
\(879\) 14913.6 0.572267
\(880\) 0 0
\(881\) −47826.2 −1.82895 −0.914476 0.404641i \(-0.867397\pi\)
−0.914476 + 0.404641i \(0.867397\pi\)
\(882\) 0 0
\(883\) −17337.5 −0.660761 −0.330381 0.943848i \(-0.607177\pi\)
−0.330381 + 0.943848i \(0.607177\pi\)
\(884\) 0 0
\(885\) 11400.4 0.433015
\(886\) 0 0
\(887\) 35181.5 1.33177 0.665885 0.746055i \(-0.268054\pi\)
0.665885 + 0.746055i \(0.268054\pi\)
\(888\) 0 0
\(889\) 45016.3 1.69831
\(890\) 0 0
\(891\) −24401.4 −0.917482
\(892\) 0 0
\(893\) −1413.78 −0.0529792
\(894\) 0 0
\(895\) −8778.22 −0.327848
\(896\) 0 0
\(897\) −21413.5 −0.797075
\(898\) 0 0
\(899\) 2616.44 0.0970669
\(900\) 0 0
\(901\) 4475.08 0.165468
\(902\) 0 0
\(903\) 4591.44 0.169207
\(904\) 0 0
\(905\) −8771.56 −0.322184
\(906\) 0 0
\(907\) 28184.4 1.03180 0.515902 0.856648i \(-0.327457\pi\)
0.515902 + 0.856648i \(0.327457\pi\)
\(908\) 0 0
\(909\) −5804.26 −0.211788
\(910\) 0 0
\(911\) −34136.6 −1.24149 −0.620745 0.784013i \(-0.713170\pi\)
−0.620745 + 0.784013i \(0.713170\pi\)
\(912\) 0 0
\(913\) 31168.5 1.12982
\(914\) 0 0
\(915\) −2839.23 −0.102582
\(916\) 0 0
\(917\) 14835.2 0.534242
\(918\) 0 0
\(919\) 15512.0 0.556794 0.278397 0.960466i \(-0.410197\pi\)
0.278397 + 0.960466i \(0.410197\pi\)
\(920\) 0 0
\(921\) −11298.3 −0.404227
\(922\) 0 0
\(923\) 43115.3 1.53755
\(924\) 0 0
\(925\) −2348.80 −0.0834897
\(926\) 0 0
\(927\) 12539.9 0.444297
\(928\) 0 0
\(929\) −3100.72 −0.109506 −0.0547531 0.998500i \(-0.517437\pi\)
−0.0547531 + 0.998500i \(0.517437\pi\)
\(930\) 0 0
\(931\) 3940.40 0.138713
\(932\) 0 0
\(933\) 15055.8 0.528299
\(934\) 0 0
\(935\) 14812.5 0.518098
\(936\) 0 0
\(937\) −19638.8 −0.684708 −0.342354 0.939571i \(-0.611224\pi\)
−0.342354 + 0.939571i \(0.611224\pi\)
\(938\) 0 0
\(939\) −3460.29 −0.120258
\(940\) 0 0
\(941\) 50033.6 1.73332 0.866658 0.498903i \(-0.166264\pi\)
0.866658 + 0.498903i \(0.166264\pi\)
\(942\) 0 0
\(943\) −2499.57 −0.0863172
\(944\) 0 0
\(945\) 28829.0 0.992389
\(946\) 0 0
\(947\) −19758.4 −0.677994 −0.338997 0.940787i \(-0.610088\pi\)
−0.338997 + 0.940787i \(0.610088\pi\)
\(948\) 0 0
\(949\) −30387.3 −1.03942
\(950\) 0 0
\(951\) 2254.20 0.0768636
\(952\) 0 0
\(953\) 33843.4 1.15036 0.575180 0.818027i \(-0.304932\pi\)
0.575180 + 0.818027i \(0.304932\pi\)
\(954\) 0 0
\(955\) −15274.1 −0.517548
\(956\) 0 0
\(957\) 2803.10 0.0946828
\(958\) 0 0
\(959\) −34798.0 −1.17173
\(960\) 0 0
\(961\) −21651.0 −0.726762
\(962\) 0 0
\(963\) −10126.1 −0.338846
\(964\) 0 0
\(965\) 26582.1 0.886743
\(966\) 0 0
\(967\) 35236.9 1.17181 0.585906 0.810379i \(-0.300739\pi\)
0.585906 + 0.810379i \(0.300739\pi\)
\(968\) 0 0
\(969\) 1903.23 0.0630964
\(970\) 0 0
\(971\) 4505.40 0.148903 0.0744517 0.997225i \(-0.476279\pi\)
0.0744517 + 0.997225i \(0.476279\pi\)
\(972\) 0 0
\(973\) 10201.5 0.336119
\(974\) 0 0
\(975\) −34075.5 −1.11927
\(976\) 0 0
\(977\) −15167.4 −0.496671 −0.248336 0.968674i \(-0.579884\pi\)
−0.248336 + 0.968674i \(0.579884\pi\)
\(978\) 0 0
\(979\) −72734.2 −2.37446
\(980\) 0 0
\(981\) −22829.7 −0.743012
\(982\) 0 0
\(983\) 25342.5 0.822278 0.411139 0.911573i \(-0.365131\pi\)
0.411139 + 0.911573i \(0.365131\pi\)
\(984\) 0 0
\(985\) −36544.4 −1.18213
\(986\) 0 0
\(987\) 655.557 0.0211414
\(988\) 0 0
\(989\) 19604.1 0.630307
\(990\) 0 0
\(991\) 40576.1 1.30065 0.650324 0.759657i \(-0.274633\pi\)
0.650324 + 0.759657i \(0.274633\pi\)
\(992\) 0 0
\(993\) 6947.37 0.222022
\(994\) 0 0
\(995\) 6533.46 0.208166
\(996\) 0 0
\(997\) −7442.49 −0.236415 −0.118208 0.992989i \(-0.537715\pi\)
−0.118208 + 0.992989i \(0.537715\pi\)
\(998\) 0 0
\(999\) −1037.65 −0.0328627
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 1856.4.a.y.1.3 5
4.3 odd 2 1856.4.a.bb.1.3 5
8.3 odd 2 464.4.a.l.1.3 5
8.5 even 2 29.4.a.b.1.4 5
24.5 odd 2 261.4.a.f.1.2 5
40.29 even 2 725.4.a.c.1.2 5
56.13 odd 2 1421.4.a.e.1.4 5
232.173 even 2 841.4.a.b.1.2 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
29.4.a.b.1.4 5 8.5 even 2
261.4.a.f.1.2 5 24.5 odd 2
464.4.a.l.1.3 5 8.3 odd 2
725.4.a.c.1.2 5 40.29 even 2
841.4.a.b.1.2 5 232.173 even 2
1421.4.a.e.1.4 5 56.13 odd 2
1856.4.a.y.1.3 5 1.1 even 1 trivial
1856.4.a.bb.1.3 5 4.3 odd 2