Properties

Label 1856.4.a.p.1.2
Level $1856$
Weight $4$
Character 1856.1
Self dual yes
Analytic conductor $109.508$
Analytic rank $1$
Dimension $3$
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: \(3\)
Coefficient field: 3.3.229.1
Defining polynomial: \( x^{3} - 4x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 232)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.11491\) of defining polynomial
Character \(\chi\) \(=\) 1856.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.92622 q^{3} -14.3315 q^{5} +16.8804 q^{7} -18.4372 q^{9} +O(q^{10})\) \(q-2.92622 q^{3} -14.3315 q^{5} +16.8804 q^{7} -18.4372 q^{9} -16.5613 q^{11} -55.3675 q^{13} +41.9372 q^{15} +6.28120 q^{17} +119.008 q^{19} -49.3957 q^{21} +22.5433 q^{23} +80.3924 q^{25} +132.959 q^{27} +29.0000 q^{29} +228.987 q^{31} +48.4621 q^{33} -241.921 q^{35} -257.186 q^{37} +162.017 q^{39} +382.208 q^{41} +170.123 q^{43} +264.234 q^{45} -172.889 q^{47} -58.0532 q^{49} -18.3802 q^{51} +69.2381 q^{53} +237.349 q^{55} -348.244 q^{57} +43.6860 q^{59} -684.892 q^{61} -311.227 q^{63} +793.500 q^{65} +528.585 q^{67} -65.9667 q^{69} -488.054 q^{71} +80.3863 q^{73} -235.246 q^{75} -279.561 q^{77} -741.898 q^{79} +108.737 q^{81} +1360.29 q^{83} -90.0191 q^{85} -84.8604 q^{87} +957.563 q^{89} -934.623 q^{91} -670.068 q^{93} -1705.57 q^{95} -1111.59 q^{97} +305.345 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 6 q^{3} - 4 q^{5} + 16 q^{7} + 45 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 6 q^{3} - 4 q^{5} + 16 q^{7} + 45 q^{9} + 2 q^{11} - 28 q^{13} + 136 q^{15} - 66 q^{17} + 66 q^{19} - 472 q^{21} + 176 q^{23} - 9 q^{25} - 228 q^{27} + 87 q^{29} - 190 q^{31} + 154 q^{33} - 660 q^{35} - 442 q^{37} - 656 q^{39} + 1162 q^{41} - 30 q^{43} + 254 q^{45} - 738 q^{47} + 851 q^{49} + 576 q^{51} - 312 q^{53} + 464 q^{55} + 684 q^{57} - 44 q^{59} - 54 q^{61} + 964 q^{63} + 178 q^{65} + 116 q^{67} - 812 q^{69} - 1200 q^{71} - 1118 q^{73} + 1038 q^{75} - 792 q^{77} - 2262 q^{79} + 15 q^{81} + 1804 q^{83} + 8 q^{85} - 174 q^{87} + 1578 q^{89} + 1972 q^{91} + 706 q^{93} - 1052 q^{95} + 1450 q^{97} + 482 q^{99}+O(q^{100}) \) 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
\(3\) −2.92622 −0.563151 −0.281576 0.959539i \(-0.590857\pi\)
−0.281576 + 0.959539i \(0.590857\pi\)
\(4\) 0 0
\(5\) −14.3315 −1.28185 −0.640925 0.767603i \(-0.721449\pi\)
−0.640925 + 0.767603i \(0.721449\pi\)
\(6\) 0 0
\(7\) 16.8804 0.911454 0.455727 0.890120i \(-0.349379\pi\)
0.455727 + 0.890120i \(0.349379\pi\)
\(8\) 0 0
\(9\) −18.4372 −0.682860
\(10\) 0 0
\(11\) −16.5613 −0.453948 −0.226974 0.973901i \(-0.572883\pi\)
−0.226974 + 0.973901i \(0.572883\pi\)
\(12\) 0 0
\(13\) −55.3675 −1.18124 −0.590622 0.806948i \(-0.701118\pi\)
−0.590622 + 0.806948i \(0.701118\pi\)
\(14\) 0 0
\(15\) 41.9372 0.721876
\(16\) 0 0
\(17\) 6.28120 0.0896126 0.0448063 0.998996i \(-0.485733\pi\)
0.0448063 + 0.998996i \(0.485733\pi\)
\(18\) 0 0
\(19\) 119.008 1.43696 0.718482 0.695545i \(-0.244837\pi\)
0.718482 + 0.695545i \(0.244837\pi\)
\(20\) 0 0
\(21\) −49.3957 −0.513287
\(22\) 0 0
\(23\) 22.5433 0.204374 0.102187 0.994765i \(-0.467416\pi\)
0.102187 + 0.994765i \(0.467416\pi\)
\(24\) 0 0
\(25\) 80.3924 0.643139
\(26\) 0 0
\(27\) 132.959 0.947705
\(28\) 0 0
\(29\) 29.0000 0.185695
\(30\) 0 0
\(31\) 228.987 1.32669 0.663344 0.748314i \(-0.269136\pi\)
0.663344 + 0.748314i \(0.269136\pi\)
\(32\) 0 0
\(33\) 48.4621 0.255642
\(34\) 0 0
\(35\) −241.921 −1.16835
\(36\) 0 0
\(37\) −257.186 −1.14273 −0.571366 0.820696i \(-0.693586\pi\)
−0.571366 + 0.820696i \(0.693586\pi\)
\(38\) 0 0
\(39\) 162.017 0.665219
\(40\) 0 0
\(41\) 382.208 1.45588 0.727938 0.685643i \(-0.240479\pi\)
0.727938 + 0.685643i \(0.240479\pi\)
\(42\) 0 0
\(43\) 170.123 0.603337 0.301669 0.953413i \(-0.402456\pi\)
0.301669 + 0.953413i \(0.402456\pi\)
\(44\) 0 0
\(45\) 264.234 0.875325
\(46\) 0 0
\(47\) −172.889 −0.536564 −0.268282 0.963340i \(-0.586456\pi\)
−0.268282 + 0.963340i \(0.586456\pi\)
\(48\) 0 0
\(49\) −58.0532 −0.169251
\(50\) 0 0
\(51\) −18.3802 −0.0504654
\(52\) 0 0
\(53\) 69.2381 0.179445 0.0897224 0.995967i \(-0.471402\pi\)
0.0897224 + 0.995967i \(0.471402\pi\)
\(54\) 0 0
\(55\) 237.349 0.581893
\(56\) 0 0
\(57\) −348.244 −0.809229
\(58\) 0 0
\(59\) 43.6860 0.0963971 0.0481986 0.998838i \(-0.484652\pi\)
0.0481986 + 0.998838i \(0.484652\pi\)
\(60\) 0 0
\(61\) −684.892 −1.43756 −0.718782 0.695235i \(-0.755300\pi\)
−0.718782 + 0.695235i \(0.755300\pi\)
\(62\) 0 0
\(63\) −311.227 −0.622396
\(64\) 0 0
\(65\) 793.500 1.51418
\(66\) 0 0
\(67\) 528.585 0.963835 0.481917 0.876217i \(-0.339941\pi\)
0.481917 + 0.876217i \(0.339941\pi\)
\(68\) 0 0
\(69\) −65.9667 −0.115094
\(70\) 0 0
\(71\) −488.054 −0.815794 −0.407897 0.913028i \(-0.633738\pi\)
−0.407897 + 0.913028i \(0.633738\pi\)
\(72\) 0 0
\(73\) 80.3863 0.128884 0.0644418 0.997921i \(-0.479473\pi\)
0.0644418 + 0.997921i \(0.479473\pi\)
\(74\) 0 0
\(75\) −235.246 −0.362185
\(76\) 0 0
\(77\) −279.561 −0.413753
\(78\) 0 0
\(79\) −741.898 −1.05658 −0.528292 0.849063i \(-0.677167\pi\)
−0.528292 + 0.849063i \(0.677167\pi\)
\(80\) 0 0
\(81\) 108.737 0.149159
\(82\) 0 0
\(83\) 1360.29 1.79893 0.899466 0.436991i \(-0.143956\pi\)
0.899466 + 0.436991i \(0.143956\pi\)
\(84\) 0 0
\(85\) −90.0191 −0.114870
\(86\) 0 0
\(87\) −84.8604 −0.104575
\(88\) 0 0
\(89\) 957.563 1.14047 0.570233 0.821483i \(-0.306853\pi\)
0.570233 + 0.821483i \(0.306853\pi\)
\(90\) 0 0
\(91\) −934.623 −1.07665
\(92\) 0 0
\(93\) −670.068 −0.747127
\(94\) 0 0
\(95\) −1705.57 −1.84197
\(96\) 0 0
\(97\) −1111.59 −1.16355 −0.581776 0.813349i \(-0.697642\pi\)
−0.581776 + 0.813349i \(0.697642\pi\)
\(98\) 0 0
\(99\) 305.345 0.309983
\(100\) 0 0
\(101\) 1691.56 1.66650 0.833249 0.552898i \(-0.186478\pi\)
0.833249 + 0.552898i \(0.186478\pi\)
\(102\) 0 0
\(103\) −439.312 −0.420259 −0.210130 0.977674i \(-0.567389\pi\)
−0.210130 + 0.977674i \(0.567389\pi\)
\(104\) 0 0
\(105\) 707.915 0.657957
\(106\) 0 0
\(107\) 861.531 0.778387 0.389193 0.921156i \(-0.372754\pi\)
0.389193 + 0.921156i \(0.372754\pi\)
\(108\) 0 0
\(109\) 459.780 0.404027 0.202013 0.979383i \(-0.435252\pi\)
0.202013 + 0.979383i \(0.435252\pi\)
\(110\) 0 0
\(111\) 752.582 0.643531
\(112\) 0 0
\(113\) −9.56230 −0.00796058 −0.00398029 0.999992i \(-0.501267\pi\)
−0.00398029 + 0.999992i \(0.501267\pi\)
\(114\) 0 0
\(115\) −323.080 −0.261977
\(116\) 0 0
\(117\) 1020.82 0.806625
\(118\) 0 0
\(119\) 106.029 0.0816777
\(120\) 0 0
\(121\) −1056.72 −0.793931
\(122\) 0 0
\(123\) −1118.43 −0.819879
\(124\) 0 0
\(125\) 639.294 0.457442
\(126\) 0 0
\(127\) 905.522 0.632694 0.316347 0.948644i \(-0.397544\pi\)
0.316347 + 0.948644i \(0.397544\pi\)
\(128\) 0 0
\(129\) −497.817 −0.339770
\(130\) 0 0
\(131\) −1039.43 −0.693250 −0.346625 0.938004i \(-0.612672\pi\)
−0.346625 + 0.938004i \(0.612672\pi\)
\(132\) 0 0
\(133\) 2008.90 1.30973
\(134\) 0 0
\(135\) −1905.51 −1.21482
\(136\) 0 0
\(137\) 1146.58 0.715028 0.357514 0.933908i \(-0.383624\pi\)
0.357514 + 0.933908i \(0.383624\pi\)
\(138\) 0 0
\(139\) 573.043 0.349675 0.174838 0.984597i \(-0.444060\pi\)
0.174838 + 0.984597i \(0.444060\pi\)
\(140\) 0 0
\(141\) 505.912 0.302167
\(142\) 0 0
\(143\) 916.959 0.536224
\(144\) 0 0
\(145\) −415.614 −0.238034
\(146\) 0 0
\(147\) 169.877 0.0953142
\(148\) 0 0
\(149\) −2324.32 −1.27796 −0.638979 0.769224i \(-0.720643\pi\)
−0.638979 + 0.769224i \(0.720643\pi\)
\(150\) 0 0
\(151\) −3188.68 −1.71848 −0.859242 0.511570i \(-0.829064\pi\)
−0.859242 + 0.511570i \(0.829064\pi\)
\(152\) 0 0
\(153\) −115.808 −0.0611929
\(154\) 0 0
\(155\) −3281.74 −1.70062
\(156\) 0 0
\(157\) 2960.71 1.50503 0.752516 0.658574i \(-0.228840\pi\)
0.752516 + 0.658574i \(0.228840\pi\)
\(158\) 0 0
\(159\) −202.606 −0.101055
\(160\) 0 0
\(161\) 380.540 0.186278
\(162\) 0 0
\(163\) −3417.14 −1.64203 −0.821015 0.570907i \(-0.806592\pi\)
−0.821015 + 0.570907i \(0.806592\pi\)
\(164\) 0 0
\(165\) −694.536 −0.327694
\(166\) 0 0
\(167\) −677.988 −0.314158 −0.157079 0.987586i \(-0.550208\pi\)
−0.157079 + 0.987586i \(0.550208\pi\)
\(168\) 0 0
\(169\) 868.556 0.395337
\(170\) 0 0
\(171\) −2194.18 −0.981246
\(172\) 0 0
\(173\) 773.271 0.339831 0.169915 0.985459i \(-0.445651\pi\)
0.169915 + 0.985459i \(0.445651\pi\)
\(174\) 0 0
\(175\) 1357.05 0.586192
\(176\) 0 0
\(177\) −127.835 −0.0542862
\(178\) 0 0
\(179\) −1413.92 −0.590399 −0.295199 0.955436i \(-0.595386\pi\)
−0.295199 + 0.955436i \(0.595386\pi\)
\(180\) 0 0
\(181\) −2160.51 −0.887234 −0.443617 0.896216i \(-0.646305\pi\)
−0.443617 + 0.896216i \(0.646305\pi\)
\(182\) 0 0
\(183\) 2004.15 0.809567
\(184\) 0 0
\(185\) 3685.86 1.46481
\(186\) 0 0
\(187\) −104.025 −0.0406795
\(188\) 0 0
\(189\) 2244.40 0.863790
\(190\) 0 0
\(191\) 1586.39 0.600981 0.300490 0.953785i \(-0.402850\pi\)
0.300490 + 0.953785i \(0.402850\pi\)
\(192\) 0 0
\(193\) −1970.84 −0.735047 −0.367523 0.930014i \(-0.619794\pi\)
−0.367523 + 0.930014i \(0.619794\pi\)
\(194\) 0 0
\(195\) −2321.96 −0.852711
\(196\) 0 0
\(197\) −4803.57 −1.73726 −0.868629 0.495463i \(-0.834998\pi\)
−0.868629 + 0.495463i \(0.834998\pi\)
\(198\) 0 0
\(199\) −2692.85 −0.959252 −0.479626 0.877473i \(-0.659228\pi\)
−0.479626 + 0.877473i \(0.659228\pi\)
\(200\) 0 0
\(201\) −1546.76 −0.542785
\(202\) 0 0
\(203\) 489.531 0.169253
\(204\) 0 0
\(205\) −5477.63 −1.86621
\(206\) 0 0
\(207\) −415.636 −0.139559
\(208\) 0 0
\(209\) −1970.93 −0.652308
\(210\) 0 0
\(211\) 703.260 0.229452 0.114726 0.993397i \(-0.463401\pi\)
0.114726 + 0.993397i \(0.463401\pi\)
\(212\) 0 0
\(213\) 1428.15 0.459415
\(214\) 0 0
\(215\) −2438.12 −0.773388
\(216\) 0 0
\(217\) 3865.39 1.20922
\(218\) 0 0
\(219\) −235.228 −0.0725810
\(220\) 0 0
\(221\) −347.774 −0.105854
\(222\) 0 0
\(223\) −3965.95 −1.19094 −0.595469 0.803378i \(-0.703034\pi\)
−0.595469 + 0.803378i \(0.703034\pi\)
\(224\) 0 0
\(225\) −1482.21 −0.439174
\(226\) 0 0
\(227\) −3976.09 −1.16256 −0.581282 0.813702i \(-0.697449\pi\)
−0.581282 + 0.813702i \(0.697449\pi\)
\(228\) 0 0
\(229\) 611.004 0.176316 0.0881578 0.996107i \(-0.471902\pi\)
0.0881578 + 0.996107i \(0.471902\pi\)
\(230\) 0 0
\(231\) 818.058 0.233006
\(232\) 0 0
\(233\) −1051.90 −0.295761 −0.147880 0.989005i \(-0.547245\pi\)
−0.147880 + 0.989005i \(0.547245\pi\)
\(234\) 0 0
\(235\) 2477.77 0.687794
\(236\) 0 0
\(237\) 2170.96 0.595017
\(238\) 0 0
\(239\) 2085.97 0.564561 0.282280 0.959332i \(-0.408909\pi\)
0.282280 + 0.959332i \(0.408909\pi\)
\(240\) 0 0
\(241\) −6064.55 −1.62096 −0.810481 0.585765i \(-0.800794\pi\)
−0.810481 + 0.585765i \(0.800794\pi\)
\(242\) 0 0
\(243\) −3908.09 −1.03170
\(244\) 0 0
\(245\) 831.991 0.216955
\(246\) 0 0
\(247\) −6589.18 −1.69741
\(248\) 0 0
\(249\) −3980.51 −1.01307
\(250\) 0 0
\(251\) −5169.26 −1.29992 −0.649962 0.759967i \(-0.725215\pi\)
−0.649962 + 0.759967i \(0.725215\pi\)
\(252\) 0 0
\(253\) −373.347 −0.0927753
\(254\) 0 0
\(255\) 263.416 0.0646891
\(256\) 0 0
\(257\) −7238.84 −1.75699 −0.878495 0.477751i \(-0.841452\pi\)
−0.878495 + 0.477751i \(0.841452\pi\)
\(258\) 0 0
\(259\) −4341.39 −1.04155
\(260\) 0 0
\(261\) −534.680 −0.126804
\(262\) 0 0
\(263\) 4855.39 1.13839 0.569195 0.822203i \(-0.307255\pi\)
0.569195 + 0.822203i \(0.307255\pi\)
\(264\) 0 0
\(265\) −992.286 −0.230021
\(266\) 0 0
\(267\) −2802.04 −0.642255
\(268\) 0 0
\(269\) −5559.70 −1.26015 −0.630076 0.776533i \(-0.716976\pi\)
−0.630076 + 0.776533i \(0.716976\pi\)
\(270\) 0 0
\(271\) 2534.36 0.568087 0.284044 0.958811i \(-0.408324\pi\)
0.284044 + 0.958811i \(0.408324\pi\)
\(272\) 0 0
\(273\) 2734.91 0.606317
\(274\) 0 0
\(275\) −1331.41 −0.291952
\(276\) 0 0
\(277\) −7540.72 −1.63566 −0.817831 0.575459i \(-0.804823\pi\)
−0.817831 + 0.575459i \(0.804823\pi\)
\(278\) 0 0
\(279\) −4221.89 −0.905943
\(280\) 0 0
\(281\) 5426.97 1.15212 0.576060 0.817407i \(-0.304589\pi\)
0.576060 + 0.817407i \(0.304589\pi\)
\(282\) 0 0
\(283\) −1297.75 −0.272590 −0.136295 0.990668i \(-0.543520\pi\)
−0.136295 + 0.990668i \(0.543520\pi\)
\(284\) 0 0
\(285\) 4990.87 1.03731
\(286\) 0 0
\(287\) 6451.82 1.32696
\(288\) 0 0
\(289\) −4873.55 −0.991970
\(290\) 0 0
\(291\) 3252.75 0.655256
\(292\) 0 0
\(293\) 6201.84 1.23657 0.618285 0.785954i \(-0.287828\pi\)
0.618285 + 0.785954i \(0.287828\pi\)
\(294\) 0 0
\(295\) −626.086 −0.123567
\(296\) 0 0
\(297\) −2201.98 −0.430209
\(298\) 0 0
\(299\) −1248.17 −0.241416
\(300\) 0 0
\(301\) 2871.74 0.549914
\(302\) 0 0
\(303\) −4949.87 −0.938491
\(304\) 0 0
\(305\) 9815.55 1.84274
\(306\) 0 0
\(307\) 9181.06 1.70681 0.853406 0.521247i \(-0.174533\pi\)
0.853406 + 0.521247i \(0.174533\pi\)
\(308\) 0 0
\(309\) 1285.52 0.236670
\(310\) 0 0
\(311\) 3753.29 0.684339 0.342170 0.939638i \(-0.388838\pi\)
0.342170 + 0.939638i \(0.388838\pi\)
\(312\) 0 0
\(313\) 2115.29 0.381991 0.190996 0.981591i \(-0.438828\pi\)
0.190996 + 0.981591i \(0.438828\pi\)
\(314\) 0 0
\(315\) 4460.36 0.797818
\(316\) 0 0
\(317\) 4739.89 0.839807 0.419903 0.907569i \(-0.362064\pi\)
0.419903 + 0.907569i \(0.362064\pi\)
\(318\) 0 0
\(319\) −480.279 −0.0842961
\(320\) 0 0
\(321\) −2521.03 −0.438350
\(322\) 0 0
\(323\) 747.513 0.128770
\(324\) 0 0
\(325\) −4451.12 −0.759705
\(326\) 0 0
\(327\) −1345.42 −0.227528
\(328\) 0 0
\(329\) −2918.43 −0.489053
\(330\) 0 0
\(331\) 6958.68 1.15554 0.577770 0.816200i \(-0.303923\pi\)
0.577770 + 0.816200i \(0.303923\pi\)
\(332\) 0 0
\(333\) 4741.79 0.780326
\(334\) 0 0
\(335\) −7575.42 −1.23549
\(336\) 0 0
\(337\) 2997.55 0.484530 0.242265 0.970210i \(-0.422110\pi\)
0.242265 + 0.970210i \(0.422110\pi\)
\(338\) 0 0
\(339\) 27.9814 0.00448301
\(340\) 0 0
\(341\) −3792.34 −0.602248
\(342\) 0 0
\(343\) −6769.93 −1.06572
\(344\) 0 0
\(345\) 945.404 0.147533
\(346\) 0 0
\(347\) −7788.46 −1.20492 −0.602459 0.798150i \(-0.705812\pi\)
−0.602459 + 0.798150i \(0.705812\pi\)
\(348\) 0 0
\(349\) 6404.16 0.982254 0.491127 0.871088i \(-0.336585\pi\)
0.491127 + 0.871088i \(0.336585\pi\)
\(350\) 0 0
\(351\) −7361.62 −1.11947
\(352\) 0 0
\(353\) −5869.46 −0.884986 −0.442493 0.896772i \(-0.645906\pi\)
−0.442493 + 0.896772i \(0.645906\pi\)
\(354\) 0 0
\(355\) 6994.55 1.04572
\(356\) 0 0
\(357\) −310.264 −0.0459969
\(358\) 0 0
\(359\) −2761.48 −0.405975 −0.202988 0.979181i \(-0.565065\pi\)
−0.202988 + 0.979181i \(0.565065\pi\)
\(360\) 0 0
\(361\) 7303.93 1.06487
\(362\) 0 0
\(363\) 3092.20 0.447103
\(364\) 0 0
\(365\) −1152.06 −0.165209
\(366\) 0 0
\(367\) 3039.36 0.432298 0.216149 0.976360i \(-0.430650\pi\)
0.216149 + 0.976360i \(0.430650\pi\)
\(368\) 0 0
\(369\) −7046.86 −0.994160
\(370\) 0 0
\(371\) 1168.76 0.163556
\(372\) 0 0
\(373\) −2051.57 −0.284789 −0.142394 0.989810i \(-0.545480\pi\)
−0.142394 + 0.989810i \(0.545480\pi\)
\(374\) 0 0
\(375\) −1870.72 −0.257609
\(376\) 0 0
\(377\) −1605.66 −0.219351
\(378\) 0 0
\(379\) 1852.42 0.251062 0.125531 0.992090i \(-0.459937\pi\)
0.125531 + 0.992090i \(0.459937\pi\)
\(380\) 0 0
\(381\) −2649.76 −0.356302
\(382\) 0 0
\(383\) 5723.43 0.763587 0.381793 0.924248i \(-0.375307\pi\)
0.381793 + 0.924248i \(0.375307\pi\)
\(384\) 0 0
\(385\) 4006.54 0.530369
\(386\) 0 0
\(387\) −3136.59 −0.411995
\(388\) 0 0
\(389\) 4946.70 0.644750 0.322375 0.946612i \(-0.395519\pi\)
0.322375 + 0.946612i \(0.395519\pi\)
\(390\) 0 0
\(391\) 141.599 0.0183145
\(392\) 0 0
\(393\) 3041.61 0.390405
\(394\) 0 0
\(395\) 10632.5 1.35438
\(396\) 0 0
\(397\) −9275.78 −1.17264 −0.586320 0.810080i \(-0.699424\pi\)
−0.586320 + 0.810080i \(0.699424\pi\)
\(398\) 0 0
\(399\) −5878.49 −0.737575
\(400\) 0 0
\(401\) 5051.01 0.629016 0.314508 0.949255i \(-0.398161\pi\)
0.314508 + 0.949255i \(0.398161\pi\)
\(402\) 0 0
\(403\) −12678.4 −1.56714
\(404\) 0 0
\(405\) −1558.36 −0.191199
\(406\) 0 0
\(407\) 4259.34 0.518741
\(408\) 0 0
\(409\) 5049.42 0.610459 0.305229 0.952279i \(-0.401267\pi\)
0.305229 + 0.952279i \(0.401267\pi\)
\(410\) 0 0
\(411\) −3355.14 −0.402669
\(412\) 0 0
\(413\) 737.435 0.0878616
\(414\) 0 0
\(415\) −19495.0 −2.30596
\(416\) 0 0
\(417\) −1676.85 −0.196920
\(418\) 0 0
\(419\) −5332.54 −0.621746 −0.310873 0.950451i \(-0.600621\pi\)
−0.310873 + 0.950451i \(0.600621\pi\)
\(420\) 0 0
\(421\) −6195.89 −0.717266 −0.358633 0.933479i \(-0.616757\pi\)
−0.358633 + 0.933479i \(0.616757\pi\)
\(422\) 0 0
\(423\) 3187.60 0.366398
\(424\) 0 0
\(425\) 504.961 0.0576334
\(426\) 0 0
\(427\) −11561.2 −1.31027
\(428\) 0 0
\(429\) −2683.22 −0.301975
\(430\) 0 0
\(431\) 1399.50 0.156408 0.0782038 0.996937i \(-0.475082\pi\)
0.0782038 + 0.996937i \(0.475082\pi\)
\(432\) 0 0
\(433\) 1689.45 0.187505 0.0937526 0.995596i \(-0.470114\pi\)
0.0937526 + 0.995596i \(0.470114\pi\)
\(434\) 0 0
\(435\) 1216.18 0.134049
\(436\) 0 0
\(437\) 2682.84 0.293679
\(438\) 0 0
\(439\) −12352.8 −1.34298 −0.671491 0.741013i \(-0.734346\pi\)
−0.671491 + 0.741013i \(0.734346\pi\)
\(440\) 0 0
\(441\) 1070.34 0.115575
\(442\) 0 0
\(443\) 13427.1 1.44005 0.720023 0.693950i \(-0.244131\pi\)
0.720023 + 0.693950i \(0.244131\pi\)
\(444\) 0 0
\(445\) −13723.3 −1.46191
\(446\) 0 0
\(447\) 6801.47 0.719684
\(448\) 0 0
\(449\) 2912.34 0.306107 0.153053 0.988218i \(-0.451089\pi\)
0.153053 + 0.988218i \(0.451089\pi\)
\(450\) 0 0
\(451\) −6329.88 −0.660892
\(452\) 0 0
\(453\) 9330.78 0.967767
\(454\) 0 0
\(455\) 13394.6 1.38010
\(456\) 0 0
\(457\) 5906.26 0.604558 0.302279 0.953219i \(-0.402253\pi\)
0.302279 + 0.953219i \(0.402253\pi\)
\(458\) 0 0
\(459\) 835.144 0.0849263
\(460\) 0 0
\(461\) −2847.28 −0.287659 −0.143830 0.989602i \(-0.545942\pi\)
−0.143830 + 0.989602i \(0.545942\pi\)
\(462\) 0 0
\(463\) 8561.02 0.859319 0.429659 0.902991i \(-0.358634\pi\)
0.429659 + 0.902991i \(0.358634\pi\)
\(464\) 0 0
\(465\) 9603.09 0.957704
\(466\) 0 0
\(467\) −14998.9 −1.48622 −0.743112 0.669167i \(-0.766651\pi\)
−0.743112 + 0.669167i \(0.766651\pi\)
\(468\) 0 0
\(469\) 8922.71 0.878491
\(470\) 0 0
\(471\) −8663.68 −0.847561
\(472\) 0 0
\(473\) −2817.46 −0.273884
\(474\) 0 0
\(475\) 9567.35 0.924169
\(476\) 0 0
\(477\) −1276.56 −0.122536
\(478\) 0 0
\(479\) −8422.27 −0.803389 −0.401695 0.915774i \(-0.631579\pi\)
−0.401695 + 0.915774i \(0.631579\pi\)
\(480\) 0 0
\(481\) 14239.7 1.34984
\(482\) 0 0
\(483\) −1113.54 −0.104903
\(484\) 0 0
\(485\) 15930.7 1.49150
\(486\) 0 0
\(487\) −1773.91 −0.165058 −0.0825291 0.996589i \(-0.526300\pi\)
−0.0825291 + 0.996589i \(0.526300\pi\)
\(488\) 0 0
\(489\) 9999.30 0.924711
\(490\) 0 0
\(491\) 2157.20 0.198275 0.0991375 0.995074i \(-0.468392\pi\)
0.0991375 + 0.995074i \(0.468392\pi\)
\(492\) 0 0
\(493\) 182.155 0.0166406
\(494\) 0 0
\(495\) −4376.06 −0.397352
\(496\) 0 0
\(497\) −8238.53 −0.743558
\(498\) 0 0
\(499\) −4303.31 −0.386058 −0.193029 0.981193i \(-0.561831\pi\)
−0.193029 + 0.981193i \(0.561831\pi\)
\(500\) 0 0
\(501\) 1983.94 0.176918
\(502\) 0 0
\(503\) 16020.5 1.42011 0.710057 0.704144i \(-0.248669\pi\)
0.710057 + 0.704144i \(0.248669\pi\)
\(504\) 0 0
\(505\) −24242.6 −2.13620
\(506\) 0 0
\(507\) −2541.59 −0.222635
\(508\) 0 0
\(509\) −18269.6 −1.59094 −0.795469 0.605994i \(-0.792776\pi\)
−0.795469 + 0.605994i \(0.792776\pi\)
\(510\) 0 0
\(511\) 1356.95 0.117472
\(512\) 0 0
\(513\) 15823.2 1.36182
\(514\) 0 0
\(515\) 6296.01 0.538709
\(516\) 0 0
\(517\) 2863.28 0.243572
\(518\) 0 0
\(519\) −2262.76 −0.191376
\(520\) 0 0
\(521\) 13128.7 1.10399 0.551996 0.833847i \(-0.313866\pi\)
0.551996 + 0.833847i \(0.313866\pi\)
\(522\) 0 0
\(523\) −8295.67 −0.693583 −0.346792 0.937942i \(-0.612729\pi\)
−0.346792 + 0.937942i \(0.612729\pi\)
\(524\) 0 0
\(525\) −3971.04 −0.330115
\(526\) 0 0
\(527\) 1438.31 0.118888
\(528\) 0 0
\(529\) −11658.8 −0.958231
\(530\) 0 0
\(531\) −805.449 −0.0658258
\(532\) 0 0
\(533\) −21161.9 −1.71974
\(534\) 0 0
\(535\) −12347.1 −0.997775
\(536\) 0 0
\(537\) 4137.44 0.332484
\(538\) 0 0
\(539\) 961.439 0.0768314
\(540\) 0 0
\(541\) −23563.6 −1.87261 −0.936303 0.351194i \(-0.885776\pi\)
−0.936303 + 0.351194i \(0.885776\pi\)
\(542\) 0 0
\(543\) 6322.13 0.499647
\(544\) 0 0
\(545\) −6589.34 −0.517902
\(546\) 0 0
\(547\) −17151.4 −1.34066 −0.670330 0.742063i \(-0.733848\pi\)
−0.670330 + 0.742063i \(0.733848\pi\)
\(548\) 0 0
\(549\) 12627.5 0.981656
\(550\) 0 0
\(551\) 3451.23 0.266838
\(552\) 0 0
\(553\) −12523.5 −0.963027
\(554\) 0 0
\(555\) −10785.6 −0.824910
\(556\) 0 0
\(557\) −18572.6 −1.41283 −0.706415 0.707798i \(-0.749689\pi\)
−0.706415 + 0.707798i \(0.749689\pi\)
\(558\) 0 0
\(559\) −9419.27 −0.712688
\(560\) 0 0
\(561\) 304.400 0.0229087
\(562\) 0 0
\(563\) −7835.40 −0.586541 −0.293270 0.956030i \(-0.594744\pi\)
−0.293270 + 0.956030i \(0.594744\pi\)
\(564\) 0 0
\(565\) 137.042 0.0102043
\(566\) 0 0
\(567\) 1835.52 0.135951
\(568\) 0 0
\(569\) 5891.09 0.434038 0.217019 0.976167i \(-0.430367\pi\)
0.217019 + 0.976167i \(0.430367\pi\)
\(570\) 0 0
\(571\) 20083.8 1.47195 0.735973 0.677010i \(-0.236725\pi\)
0.735973 + 0.677010i \(0.236725\pi\)
\(572\) 0 0
\(573\) −4642.13 −0.338443
\(574\) 0 0
\(575\) 1812.31 0.131441
\(576\) 0 0
\(577\) −21008.9 −1.51579 −0.757896 0.652375i \(-0.773773\pi\)
−0.757896 + 0.652375i \(0.773773\pi\)
\(578\) 0 0
\(579\) 5767.11 0.413943
\(580\) 0 0
\(581\) 22962.2 1.63964
\(582\) 0 0
\(583\) −1146.67 −0.0814587
\(584\) 0 0
\(585\) −14629.9 −1.03397
\(586\) 0 0
\(587\) 20222.3 1.42192 0.710958 0.703235i \(-0.248262\pi\)
0.710958 + 0.703235i \(0.248262\pi\)
\(588\) 0 0
\(589\) 27251.4 1.90640
\(590\) 0 0
\(591\) 14056.3 0.978339
\(592\) 0 0
\(593\) 18803.0 1.30210 0.651051 0.759034i \(-0.274328\pi\)
0.651051 + 0.759034i \(0.274328\pi\)
\(594\) 0 0
\(595\) −1519.55 −0.104699
\(596\) 0 0
\(597\) 7879.87 0.540204
\(598\) 0 0
\(599\) 10612.1 0.723872 0.361936 0.932203i \(-0.382116\pi\)
0.361936 + 0.932203i \(0.382116\pi\)
\(600\) 0 0
\(601\) 27709.6 1.88070 0.940350 0.340210i \(-0.110498\pi\)
0.940350 + 0.340210i \(0.110498\pi\)
\(602\) 0 0
\(603\) −9745.64 −0.658164
\(604\) 0 0
\(605\) 15144.4 1.01770
\(606\) 0 0
\(607\) −21945.0 −1.46741 −0.733707 0.679466i \(-0.762212\pi\)
−0.733707 + 0.679466i \(0.762212\pi\)
\(608\) 0 0
\(609\) −1432.47 −0.0953150
\(610\) 0 0
\(611\) 9572.44 0.633813
\(612\) 0 0
\(613\) 11777.0 0.775971 0.387985 0.921665i \(-0.373171\pi\)
0.387985 + 0.921665i \(0.373171\pi\)
\(614\) 0 0
\(615\) 16028.7 1.05096
\(616\) 0 0
\(617\) −27161.1 −1.77223 −0.886115 0.463465i \(-0.846606\pi\)
−0.886115 + 0.463465i \(0.846606\pi\)
\(618\) 0 0
\(619\) 10366.3 0.673116 0.336558 0.941663i \(-0.390737\pi\)
0.336558 + 0.941663i \(0.390737\pi\)
\(620\) 0 0
\(621\) 2997.35 0.193687
\(622\) 0 0
\(623\) 16164.0 1.03948
\(624\) 0 0
\(625\) −19211.1 −1.22951
\(626\) 0 0
\(627\) 5767.38 0.367348
\(628\) 0 0
\(629\) −1615.43 −0.102403
\(630\) 0 0
\(631\) −11208.4 −0.707132 −0.353566 0.935410i \(-0.615031\pi\)
−0.353566 + 0.935410i \(0.615031\pi\)
\(632\) 0 0
\(633\) −2057.89 −0.129216
\(634\) 0 0
\(635\) −12977.5 −0.811018
\(636\) 0 0
\(637\) 3214.26 0.199927
\(638\) 0 0
\(639\) 8998.36 0.557073
\(640\) 0 0
\(641\) 351.957 0.0216872 0.0108436 0.999941i \(-0.496548\pi\)
0.0108436 + 0.999941i \(0.496548\pi\)
\(642\) 0 0
\(643\) −27628.0 −1.69447 −0.847234 0.531220i \(-0.821734\pi\)
−0.847234 + 0.531220i \(0.821734\pi\)
\(644\) 0 0
\(645\) 7134.48 0.435534
\(646\) 0 0
\(647\) −10858.0 −0.659771 −0.329886 0.944021i \(-0.607010\pi\)
−0.329886 + 0.944021i \(0.607010\pi\)
\(648\) 0 0
\(649\) −723.498 −0.0437593
\(650\) 0 0
\(651\) −11311.0 −0.680972
\(652\) 0 0
\(653\) 9714.54 0.582174 0.291087 0.956697i \(-0.405983\pi\)
0.291087 + 0.956697i \(0.405983\pi\)
\(654\) 0 0
\(655\) 14896.7 0.888643
\(656\) 0 0
\(657\) −1482.10 −0.0880095
\(658\) 0 0
\(659\) −1918.79 −0.113422 −0.0567111 0.998391i \(-0.518061\pi\)
−0.0567111 + 0.998391i \(0.518061\pi\)
\(660\) 0 0
\(661\) 12106.3 0.712375 0.356188 0.934414i \(-0.384076\pi\)
0.356188 + 0.934414i \(0.384076\pi\)
\(662\) 0 0
\(663\) 1017.66 0.0596120
\(664\) 0 0
\(665\) −28790.6 −1.67887
\(666\) 0 0
\(667\) 653.756 0.0379513
\(668\) 0 0
\(669\) 11605.2 0.670679
\(670\) 0 0
\(671\) 11342.7 0.652580
\(672\) 0 0
\(673\) 16348.0 0.936355 0.468178 0.883634i \(-0.344911\pi\)
0.468178 + 0.883634i \(0.344911\pi\)
\(674\) 0 0
\(675\) 10688.9 0.609507
\(676\) 0 0
\(677\) 19298.2 1.09555 0.547776 0.836625i \(-0.315475\pi\)
0.547776 + 0.836625i \(0.315475\pi\)
\(678\) 0 0
\(679\) −18764.0 −1.06052
\(680\) 0 0
\(681\) 11634.9 0.654700
\(682\) 0 0
\(683\) −9938.38 −0.556781 −0.278390 0.960468i \(-0.589801\pi\)
−0.278390 + 0.960468i \(0.589801\pi\)
\(684\) 0 0
\(685\) −16432.2 −0.916558
\(686\) 0 0
\(687\) −1787.93 −0.0992924
\(688\) 0 0
\(689\) −3833.54 −0.211968
\(690\) 0 0
\(691\) 18610.4 1.02456 0.512282 0.858817i \(-0.328800\pi\)
0.512282 + 0.858817i \(0.328800\pi\)
\(692\) 0 0
\(693\) 5154.34 0.282536
\(694\) 0 0
\(695\) −8212.58 −0.448231
\(696\) 0 0
\(697\) 2400.73 0.130465
\(698\) 0 0
\(699\) 3078.09 0.166558
\(700\) 0 0
\(701\) −26877.8 −1.44816 −0.724080 0.689716i \(-0.757735\pi\)
−0.724080 + 0.689716i \(0.757735\pi\)
\(702\) 0 0
\(703\) −30607.2 −1.64206
\(704\) 0 0
\(705\) −7250.49 −0.387332
\(706\) 0 0
\(707\) 28554.1 1.51894
\(708\) 0 0
\(709\) 3141.28 0.166394 0.0831970 0.996533i \(-0.473487\pi\)
0.0831970 + 0.996533i \(0.473487\pi\)
\(710\) 0 0
\(711\) 13678.6 0.721499
\(712\) 0 0
\(713\) 5162.14 0.271141
\(714\) 0 0
\(715\) −13141.4 −0.687358
\(716\) 0 0
\(717\) −6104.00 −0.317933
\(718\) 0 0
\(719\) 27198.6 1.41076 0.705379 0.708830i \(-0.250777\pi\)
0.705379 + 0.708830i \(0.250777\pi\)
\(720\) 0 0
\(721\) −7415.75 −0.383047
\(722\) 0 0
\(723\) 17746.2 0.912847
\(724\) 0 0
\(725\) 2331.38 0.119428
\(726\) 0 0
\(727\) 15713.1 0.801603 0.400802 0.916165i \(-0.368732\pi\)
0.400802 + 0.916165i \(0.368732\pi\)
\(728\) 0 0
\(729\) 8500.05 0.431847
\(730\) 0 0
\(731\) 1068.58 0.0540666
\(732\) 0 0
\(733\) −10977.4 −0.553153 −0.276576 0.960992i \(-0.589200\pi\)
−0.276576 + 0.960992i \(0.589200\pi\)
\(734\) 0 0
\(735\) −2434.59 −0.122178
\(736\) 0 0
\(737\) −8754.07 −0.437531
\(738\) 0 0
\(739\) −12586.3 −0.626515 −0.313258 0.949668i \(-0.601420\pi\)
−0.313258 + 0.949668i \(0.601420\pi\)
\(740\) 0 0
\(741\) 19281.4 0.955897
\(742\) 0 0
\(743\) 21237.6 1.04863 0.524315 0.851524i \(-0.324321\pi\)
0.524315 + 0.851524i \(0.324321\pi\)
\(744\) 0 0
\(745\) 33311.0 1.63815
\(746\) 0 0
\(747\) −25080.0 −1.22842
\(748\) 0 0
\(749\) 14543.0 0.709464
\(750\) 0 0
\(751\) 31114.2 1.51181 0.755907 0.654679i \(-0.227196\pi\)
0.755907 + 0.654679i \(0.227196\pi\)
\(752\) 0 0
\(753\) 15126.4 0.732054
\(754\) 0 0
\(755\) 45698.6 2.20284
\(756\) 0 0
\(757\) −12838.7 −0.616423 −0.308211 0.951318i \(-0.599730\pi\)
−0.308211 + 0.951318i \(0.599730\pi\)
\(758\) 0 0
\(759\) 1092.50 0.0522466
\(760\) 0 0
\(761\) −11665.6 −0.555689 −0.277844 0.960626i \(-0.589620\pi\)
−0.277844 + 0.960626i \(0.589620\pi\)
\(762\) 0 0
\(763\) 7761.25 0.368252
\(764\) 0 0
\(765\) 1659.70 0.0784401
\(766\) 0 0
\(767\) −2418.78 −0.113869
\(768\) 0 0
\(769\) −33339.3 −1.56339 −0.781693 0.623663i \(-0.785644\pi\)
−0.781693 + 0.623663i \(0.785644\pi\)
\(770\) 0 0
\(771\) 21182.4 0.989452
\(772\) 0 0
\(773\) 17895.2 0.832660 0.416330 0.909214i \(-0.363316\pi\)
0.416330 + 0.909214i \(0.363316\pi\)
\(774\) 0 0
\(775\) 18408.9 0.853246
\(776\) 0 0
\(777\) 12703.9 0.586549
\(778\) 0 0
\(779\) 45485.9 2.09204
\(780\) 0 0
\(781\) 8082.82 0.370328
\(782\) 0 0
\(783\) 3855.82 0.175984
\(784\) 0 0
\(785\) −42431.4 −1.92923
\(786\) 0 0
\(787\) 15816.4 0.716384 0.358192 0.933648i \(-0.383393\pi\)
0.358192 + 0.933648i \(0.383393\pi\)
\(788\) 0 0
\(789\) −14208.0 −0.641086
\(790\) 0 0
\(791\) −161.415 −0.00725570
\(792\) 0 0
\(793\) 37920.7 1.69811
\(794\) 0 0
\(795\) 2903.65 0.129537
\(796\) 0 0
\(797\) −21146.2 −0.939822 −0.469911 0.882714i \(-0.655714\pi\)
−0.469911 + 0.882714i \(0.655714\pi\)
\(798\) 0 0
\(799\) −1085.95 −0.0480828
\(800\) 0 0
\(801\) −17654.8 −0.778779
\(802\) 0 0
\(803\) −1331.30 −0.0585065
\(804\) 0 0
\(805\) −5453.71 −0.238780
\(806\) 0 0
\(807\) 16268.9 0.709657
\(808\) 0 0
\(809\) −19190.2 −0.833982 −0.416991 0.908911i \(-0.636915\pi\)
−0.416991 + 0.908911i \(0.636915\pi\)
\(810\) 0 0
\(811\) 37890.9 1.64060 0.820302 0.571931i \(-0.193806\pi\)
0.820302 + 0.571931i \(0.193806\pi\)
\(812\) 0 0
\(813\) −7416.11 −0.319919
\(814\) 0 0
\(815\) 48972.8 2.10484
\(816\) 0 0
\(817\) 20246.0 0.866974
\(818\) 0 0
\(819\) 17231.9 0.735201
\(820\) 0 0
\(821\) 11006.2 0.467868 0.233934 0.972252i \(-0.424840\pi\)
0.233934 + 0.972252i \(0.424840\pi\)
\(822\) 0 0
\(823\) 42361.2 1.79419 0.897094 0.441839i \(-0.145674\pi\)
0.897094 + 0.441839i \(0.145674\pi\)
\(824\) 0 0
\(825\) 3895.99 0.164413
\(826\) 0 0
\(827\) 1786.13 0.0751028 0.0375514 0.999295i \(-0.488044\pi\)
0.0375514 + 0.999295i \(0.488044\pi\)
\(828\) 0 0
\(829\) −12070.8 −0.505712 −0.252856 0.967504i \(-0.581370\pi\)
−0.252856 + 0.967504i \(0.581370\pi\)
\(830\) 0 0
\(831\) 22065.8 0.921125
\(832\) 0 0
\(833\) −364.644 −0.0151671
\(834\) 0 0
\(835\) 9716.60 0.402703
\(836\) 0 0
\(837\) 30446.0 1.25731
\(838\) 0 0
\(839\) −39687.9 −1.63311 −0.816555 0.577267i \(-0.804119\pi\)
−0.816555 + 0.577267i \(0.804119\pi\)
\(840\) 0 0
\(841\) 841.000 0.0344828
\(842\) 0 0
\(843\) −15880.5 −0.648818
\(844\) 0 0
\(845\) −12447.7 −0.506763
\(846\) 0 0
\(847\) −17837.9 −0.723632
\(848\) 0 0
\(849\) 3797.50 0.153510
\(850\) 0 0
\(851\) −5797.82 −0.233545
\(852\) 0 0
\(853\) −36146.8 −1.45093 −0.725466 0.688258i \(-0.758376\pi\)
−0.725466 + 0.688258i \(0.758376\pi\)
\(854\) 0 0
\(855\) 31445.9 1.25781
\(856\) 0 0
\(857\) 1778.90 0.0709054 0.0354527 0.999371i \(-0.488713\pi\)
0.0354527 + 0.999371i \(0.488713\pi\)
\(858\) 0 0
\(859\) −37890.5 −1.50501 −0.752507 0.658584i \(-0.771156\pi\)
−0.752507 + 0.658584i \(0.771156\pi\)
\(860\) 0 0
\(861\) −18879.4 −0.747282
\(862\) 0 0
\(863\) −22754.4 −0.897529 −0.448764 0.893650i \(-0.648136\pi\)
−0.448764 + 0.893650i \(0.648136\pi\)
\(864\) 0 0
\(865\) −11082.1 −0.435612
\(866\) 0 0
\(867\) 14261.1 0.558629
\(868\) 0 0
\(869\) 12286.8 0.479634
\(870\) 0 0
\(871\) −29266.4 −1.13852
\(872\) 0 0
\(873\) 20494.6 0.794543
\(874\) 0 0
\(875\) 10791.5 0.416937
\(876\) 0 0
\(877\) −28543.9 −1.09904 −0.549521 0.835480i \(-0.685190\pi\)
−0.549521 + 0.835480i \(0.685190\pi\)
\(878\) 0 0
\(879\) −18147.9 −0.696376
\(880\) 0 0
\(881\) 46848.4 1.79156 0.895779 0.444499i \(-0.146618\pi\)
0.895779 + 0.444499i \(0.146618\pi\)
\(882\) 0 0
\(883\) −7020.06 −0.267547 −0.133773 0.991012i \(-0.542709\pi\)
−0.133773 + 0.991012i \(0.542709\pi\)
\(884\) 0 0
\(885\) 1832.07 0.0695867
\(886\) 0 0
\(887\) 19200.5 0.726820 0.363410 0.931629i \(-0.381612\pi\)
0.363410 + 0.931629i \(0.381612\pi\)
\(888\) 0 0
\(889\) 15285.5 0.576671
\(890\) 0 0
\(891\) −1800.83 −0.0677104
\(892\) 0 0
\(893\) −20575.2 −0.771023
\(894\) 0 0
\(895\) 20263.6 0.756803
\(896\) 0 0
\(897\) 3652.41 0.135954
\(898\) 0 0
\(899\) 6640.63 0.246360
\(900\) 0 0
\(901\) 434.898 0.0160805
\(902\) 0 0
\(903\) −8403.34 −0.309685
\(904\) 0 0
\(905\) 30963.4 1.13730
\(906\) 0 0
\(907\) 36098.6 1.32154 0.660768 0.750590i \(-0.270231\pi\)
0.660768 + 0.750590i \(0.270231\pi\)
\(908\) 0 0
\(909\) −31187.6 −1.13799
\(910\) 0 0
\(911\) −19193.8 −0.698047 −0.349023 0.937114i \(-0.613487\pi\)
−0.349023 + 0.937114i \(0.613487\pi\)
\(912\) 0 0
\(913\) −22528.2 −0.816622
\(914\) 0 0
\(915\) −28722.5 −1.03774
\(916\) 0 0
\(917\) −17546.0 −0.631866
\(918\) 0 0
\(919\) 49242.2 1.76752 0.883760 0.467941i \(-0.155004\pi\)
0.883760 + 0.467941i \(0.155004\pi\)
\(920\) 0 0
\(921\) −26865.8 −0.961193
\(922\) 0 0
\(923\) 27022.3 0.963651
\(924\) 0 0
\(925\) −20675.8 −0.734936
\(926\) 0 0
\(927\) 8099.70 0.286978
\(928\) 0 0
\(929\) −43350.9 −1.53100 −0.765498 0.643438i \(-0.777507\pi\)
−0.765498 + 0.643438i \(0.777507\pi\)
\(930\) 0 0
\(931\) −6908.80 −0.243208
\(932\) 0 0
\(933\) −10983.0 −0.385387
\(934\) 0 0
\(935\) 1490.84 0.0521450
\(936\) 0 0
\(937\) 39167.8 1.36559 0.682794 0.730611i \(-0.260765\pi\)
0.682794 + 0.730611i \(0.260765\pi\)
\(938\) 0 0
\(939\) −6189.81 −0.215119
\(940\) 0 0
\(941\) 13329.9 0.461789 0.230895 0.972979i \(-0.425835\pi\)
0.230895 + 0.972979i \(0.425835\pi\)
\(942\) 0 0
\(943\) 8616.25 0.297544
\(944\) 0 0
\(945\) −32165.7 −1.10725
\(946\) 0 0
\(947\) 19115.3 0.655926 0.327963 0.944690i \(-0.393638\pi\)
0.327963 + 0.944690i \(0.393638\pi\)
\(948\) 0 0
\(949\) −4450.78 −0.152243
\(950\) 0 0
\(951\) −13870.0 −0.472938
\(952\) 0 0
\(953\) −29008.4 −0.986018 −0.493009 0.870024i \(-0.664103\pi\)
−0.493009 + 0.870024i \(0.664103\pi\)
\(954\) 0 0
\(955\) −22735.4 −0.770367
\(956\) 0 0
\(957\) 1405.40 0.0474715
\(958\) 0 0
\(959\) 19354.7 0.651715
\(960\) 0 0
\(961\) 22644.2 0.760103
\(962\) 0 0
\(963\) −15884.3 −0.531529
\(964\) 0 0
\(965\) 28245.1 0.942220
\(966\) 0 0
\(967\) 55491.5 1.84538 0.922691 0.385539i \(-0.125985\pi\)
0.922691 + 0.385539i \(0.125985\pi\)
\(968\) 0 0
\(969\) −2187.39 −0.0725171
\(970\) 0 0
\(971\) −39721.0 −1.31278 −0.656389 0.754422i \(-0.727917\pi\)
−0.656389 + 0.754422i \(0.727917\pi\)
\(972\) 0 0
\(973\) 9673.18 0.318713
\(974\) 0 0
\(975\) 13025.0 0.427829
\(976\) 0 0
\(977\) 4352.89 0.142540 0.0712699 0.997457i \(-0.477295\pi\)
0.0712699 + 0.997457i \(0.477295\pi\)
\(978\) 0 0
\(979\) −15858.5 −0.517712
\(980\) 0 0
\(981\) −8477.07 −0.275894
\(982\) 0 0
\(983\) −10317.1 −0.334754 −0.167377 0.985893i \(-0.553530\pi\)
−0.167377 + 0.985893i \(0.553530\pi\)
\(984\) 0 0
\(985\) 68842.4 2.22690
\(986\) 0 0
\(987\) 8539.98 0.275411
\(988\) 0 0
\(989\) 3835.13 0.123307
\(990\) 0 0
\(991\) −45439.0 −1.45653 −0.728264 0.685297i \(-0.759672\pi\)
−0.728264 + 0.685297i \(0.759672\pi\)
\(992\) 0 0
\(993\) −20362.6 −0.650744
\(994\) 0 0
\(995\) 38592.6 1.22962
\(996\) 0 0
\(997\) −31888.0 −1.01294 −0.506471 0.862257i \(-0.669050\pi\)
−0.506471 + 0.862257i \(0.669050\pi\)
\(998\) 0 0
\(999\) −34195.2 −1.08297
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.p.1.2 3
4.3 odd 2 1856.4.a.u.1.2 3
8.3 odd 2 464.4.a.g.1.2 3
8.5 even 2 232.4.a.b.1.2 3
24.5 odd 2 2088.4.a.b.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
232.4.a.b.1.2 3 8.5 even 2
464.4.a.g.1.2 3 8.3 odd 2
1856.4.a.p.1.2 3 1.1 even 1 trivial
1856.4.a.u.1.2 3 4.3 odd 2
2088.4.a.b.1.1 3 24.5 odd 2