Properties

Label 1856.4.a.bg.1.8
Level $1856$
Weight $4$
Character 1856.1
Self dual yes
Analytic conductor $109.508$
Analytic rank $1$
Dimension $9$
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: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Defining polynomial: \( x^{9} - 4x^{8} - 138x^{7} + 394x^{6} + 5872x^{5} - 10822x^{4} - 85158x^{3} + 30654x^{2} + 439999x + 396802 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: no (minimal twist has level 928)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(6.50540\) of defining polynomial
Character \(\chi\) \(=\) 1856.1

$q$-expansion

\(f(q)\) \(=\) \(q+6.50540 q^{3} +1.74434 q^{5} -26.0581 q^{7} +15.3202 q^{9} +O(q^{10})\) \(q+6.50540 q^{3} +1.74434 q^{5} -26.0581 q^{7} +15.3202 q^{9} +34.5346 q^{11} -43.9865 q^{13} +11.3476 q^{15} +134.214 q^{17} -39.5628 q^{19} -169.519 q^{21} +22.5760 q^{23} -121.957 q^{25} -75.9817 q^{27} +29.0000 q^{29} +90.1691 q^{31} +224.661 q^{33} -45.4542 q^{35} -213.354 q^{37} -286.150 q^{39} -328.864 q^{41} +2.01802 q^{43} +26.7236 q^{45} -16.3670 q^{47} +336.027 q^{49} +873.113 q^{51} -149.430 q^{53} +60.2400 q^{55} -257.372 q^{57} -292.706 q^{59} +111.495 q^{61} -399.216 q^{63} -76.7273 q^{65} +159.934 q^{67} +146.866 q^{69} -129.341 q^{71} +742.501 q^{73} -793.381 q^{75} -899.907 q^{77} -1064.85 q^{79} -907.937 q^{81} -496.664 q^{83} +234.114 q^{85} +188.657 q^{87} +676.600 q^{89} +1146.21 q^{91} +586.586 q^{93} -69.0110 q^{95} -103.230 q^{97} +529.077 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 4 q^{3} - 10 q^{5} - 12 q^{7} + 49 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q + 4 q^{3} - 10 q^{5} - 12 q^{7} + 49 q^{9} + 64 q^{11} - 70 q^{13} - 170 q^{15} - 66 q^{17} + 42 q^{19} - 76 q^{21} - 40 q^{23} + 111 q^{25} + 322 q^{27} + 261 q^{29} + 64 q^{31} - 52 q^{33} + 496 q^{35} + 54 q^{37} - 590 q^{39} - 378 q^{41} - 32 q^{43} - 1046 q^{45} - 1164 q^{47} - 351 q^{49} + 376 q^{51} - 278 q^{53} - 614 q^{55} + 28 q^{57} + 640 q^{59} - 1054 q^{61} - 1660 q^{63} - 708 q^{65} + 1184 q^{67} - 188 q^{69} - 1988 q^{71} - 750 q^{73} + 3126 q^{75} - 1260 q^{77} - 2916 q^{79} + 293 q^{81} + 2832 q^{83} - 56 q^{85} + 116 q^{87} - 370 q^{89} + 3016 q^{91} + 1696 q^{93} - 4412 q^{95} - 2234 q^{97} + 4118 q^{99}+O(q^{100}) \) 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
\(3\) 6.50540 1.25196 0.625982 0.779837i \(-0.284698\pi\)
0.625982 + 0.779837i \(0.284698\pi\)
\(4\) 0 0
\(5\) 1.74434 0.156018 0.0780092 0.996953i \(-0.475144\pi\)
0.0780092 + 0.996953i \(0.475144\pi\)
\(6\) 0 0
\(7\) −26.0581 −1.40701 −0.703504 0.710692i \(-0.748382\pi\)
−0.703504 + 0.710692i \(0.748382\pi\)
\(8\) 0 0
\(9\) 15.3202 0.567415
\(10\) 0 0
\(11\) 34.5346 0.946597 0.473298 0.880902i \(-0.343063\pi\)
0.473298 + 0.880902i \(0.343063\pi\)
\(12\) 0 0
\(13\) −43.9865 −0.938435 −0.469218 0.883083i \(-0.655464\pi\)
−0.469218 + 0.883083i \(0.655464\pi\)
\(14\) 0 0
\(15\) 11.3476 0.195329
\(16\) 0 0
\(17\) 134.214 1.91480 0.957400 0.288766i \(-0.0932448\pi\)
0.957400 + 0.288766i \(0.0932448\pi\)
\(18\) 0 0
\(19\) −39.5628 −0.477702 −0.238851 0.971056i \(-0.576771\pi\)
−0.238851 + 0.971056i \(0.576771\pi\)
\(20\) 0 0
\(21\) −169.519 −1.76152
\(22\) 0 0
\(23\) 22.5760 0.204671 0.102335 0.994750i \(-0.467368\pi\)
0.102335 + 0.994750i \(0.467368\pi\)
\(24\) 0 0
\(25\) −121.957 −0.975658
\(26\) 0 0
\(27\) −75.9817 −0.541581
\(28\) 0 0
\(29\) 29.0000 0.185695
\(30\) 0 0
\(31\) 90.1691 0.522414 0.261207 0.965283i \(-0.415879\pi\)
0.261207 + 0.965283i \(0.415879\pi\)
\(32\) 0 0
\(33\) 224.661 1.18511
\(34\) 0 0
\(35\) −45.4542 −0.219519
\(36\) 0 0
\(37\) −213.354 −0.947976 −0.473988 0.880531i \(-0.657186\pi\)
−0.473988 + 0.880531i \(0.657186\pi\)
\(38\) 0 0
\(39\) −286.150 −1.17489
\(40\) 0 0
\(41\) −328.864 −1.25268 −0.626340 0.779550i \(-0.715448\pi\)
−0.626340 + 0.779550i \(0.715448\pi\)
\(42\) 0 0
\(43\) 2.01802 0.00715686 0.00357843 0.999994i \(-0.498861\pi\)
0.00357843 + 0.999994i \(0.498861\pi\)
\(44\) 0 0
\(45\) 26.7236 0.0885272
\(46\) 0 0
\(47\) −16.3670 −0.0507952 −0.0253976 0.999677i \(-0.508085\pi\)
−0.0253976 + 0.999677i \(0.508085\pi\)
\(48\) 0 0
\(49\) 336.027 0.979670
\(50\) 0 0
\(51\) 873.113 2.39726
\(52\) 0 0
\(53\) −149.430 −0.387280 −0.193640 0.981073i \(-0.562029\pi\)
−0.193640 + 0.981073i \(0.562029\pi\)
\(54\) 0 0
\(55\) 60.2400 0.147686
\(56\) 0 0
\(57\) −257.372 −0.598066
\(58\) 0 0
\(59\) −292.706 −0.645882 −0.322941 0.946419i \(-0.604672\pi\)
−0.322941 + 0.946419i \(0.604672\pi\)
\(60\) 0 0
\(61\) 111.495 0.234024 0.117012 0.993131i \(-0.462668\pi\)
0.117012 + 0.993131i \(0.462668\pi\)
\(62\) 0 0
\(63\) −399.216 −0.798358
\(64\) 0 0
\(65\) −76.7273 −0.146413
\(66\) 0 0
\(67\) 159.934 0.291628 0.145814 0.989312i \(-0.453420\pi\)
0.145814 + 0.989312i \(0.453420\pi\)
\(68\) 0 0
\(69\) 146.866 0.256241
\(70\) 0 0
\(71\) −129.341 −0.216196 −0.108098 0.994140i \(-0.534476\pi\)
−0.108098 + 0.994140i \(0.534476\pi\)
\(72\) 0 0
\(73\) 742.501 1.19045 0.595227 0.803558i \(-0.297062\pi\)
0.595227 + 0.803558i \(0.297062\pi\)
\(74\) 0 0
\(75\) −793.381 −1.22149
\(76\) 0 0
\(77\) −899.907 −1.33187
\(78\) 0 0
\(79\) −1064.85 −1.51651 −0.758257 0.651956i \(-0.773949\pi\)
−0.758257 + 0.651956i \(0.773949\pi\)
\(80\) 0 0
\(81\) −907.937 −1.24546
\(82\) 0 0
\(83\) −496.664 −0.656818 −0.328409 0.944536i \(-0.606513\pi\)
−0.328409 + 0.944536i \(0.606513\pi\)
\(84\) 0 0
\(85\) 234.114 0.298744
\(86\) 0 0
\(87\) 188.657 0.232484
\(88\) 0 0
\(89\) 676.600 0.805836 0.402918 0.915236i \(-0.367996\pi\)
0.402918 + 0.915236i \(0.367996\pi\)
\(90\) 0 0
\(91\) 1146.21 1.32039
\(92\) 0 0
\(93\) 586.586 0.654044
\(94\) 0 0
\(95\) −69.0110 −0.0745303
\(96\) 0 0
\(97\) −103.230 −0.108056 −0.0540278 0.998539i \(-0.517206\pi\)
−0.0540278 + 0.998539i \(0.517206\pi\)
\(98\) 0 0
\(99\) 529.077 0.537113
\(100\) 0 0
\(101\) −1396.34 −1.37566 −0.687828 0.725874i \(-0.741436\pi\)
−0.687828 + 0.725874i \(0.741436\pi\)
\(102\) 0 0
\(103\) −1828.75 −1.74944 −0.874719 0.484630i \(-0.838954\pi\)
−0.874719 + 0.484630i \(0.838954\pi\)
\(104\) 0 0
\(105\) −295.698 −0.274830
\(106\) 0 0
\(107\) 1116.86 1.00907 0.504536 0.863390i \(-0.331663\pi\)
0.504536 + 0.863390i \(0.331663\pi\)
\(108\) 0 0
\(109\) 598.302 0.525751 0.262876 0.964830i \(-0.415329\pi\)
0.262876 + 0.964830i \(0.415329\pi\)
\(110\) 0 0
\(111\) −1387.95 −1.18683
\(112\) 0 0
\(113\) −2090.66 −1.74046 −0.870232 0.492643i \(-0.836031\pi\)
−0.870232 + 0.492643i \(0.836031\pi\)
\(114\) 0 0
\(115\) 39.3803 0.0319324
\(116\) 0 0
\(117\) −673.883 −0.532483
\(118\) 0 0
\(119\) −3497.36 −2.69414
\(120\) 0 0
\(121\) −138.364 −0.103955
\(122\) 0 0
\(123\) −2139.39 −1.56831
\(124\) 0 0
\(125\) −430.777 −0.308239
\(126\) 0 0
\(127\) 1767.83 1.23519 0.617597 0.786495i \(-0.288106\pi\)
0.617597 + 0.786495i \(0.288106\pi\)
\(128\) 0 0
\(129\) 13.1280 0.00896013
\(130\) 0 0
\(131\) −1823.75 −1.21635 −0.608174 0.793804i \(-0.708098\pi\)
−0.608174 + 0.793804i \(0.708098\pi\)
\(132\) 0 0
\(133\) 1030.93 0.672130
\(134\) 0 0
\(135\) −132.538 −0.0844965
\(136\) 0 0
\(137\) −2417.52 −1.50761 −0.753806 0.657097i \(-0.771784\pi\)
−0.753806 + 0.657097i \(0.771784\pi\)
\(138\) 0 0
\(139\) −1787.82 −1.09094 −0.545470 0.838130i \(-0.683649\pi\)
−0.545470 + 0.838130i \(0.683649\pi\)
\(140\) 0 0
\(141\) −106.474 −0.0635938
\(142\) 0 0
\(143\) −1519.05 −0.888320
\(144\) 0 0
\(145\) 50.5858 0.0289719
\(146\) 0 0
\(147\) 2185.99 1.22651
\(148\) 0 0
\(149\) 348.880 0.191821 0.0959106 0.995390i \(-0.469424\pi\)
0.0959106 + 0.995390i \(0.469424\pi\)
\(150\) 0 0
\(151\) −1024.52 −0.552147 −0.276073 0.961137i \(-0.589033\pi\)
−0.276073 + 0.961137i \(0.589033\pi\)
\(152\) 0 0
\(153\) 2056.18 1.08649
\(154\) 0 0
\(155\) 157.285 0.0815062
\(156\) 0 0
\(157\) 920.398 0.467871 0.233935 0.972252i \(-0.424840\pi\)
0.233935 + 0.972252i \(0.424840\pi\)
\(158\) 0 0
\(159\) −972.104 −0.484861
\(160\) 0 0
\(161\) −588.290 −0.287974
\(162\) 0 0
\(163\) 1214.70 0.583696 0.291848 0.956465i \(-0.405730\pi\)
0.291848 + 0.956465i \(0.405730\pi\)
\(164\) 0 0
\(165\) 391.885 0.184898
\(166\) 0 0
\(167\) −3724.63 −1.72587 −0.862935 0.505315i \(-0.831376\pi\)
−0.862935 + 0.505315i \(0.831376\pi\)
\(168\) 0 0
\(169\) −262.188 −0.119339
\(170\) 0 0
\(171\) −606.111 −0.271056
\(172\) 0 0
\(173\) −1251.03 −0.549792 −0.274896 0.961474i \(-0.588643\pi\)
−0.274896 + 0.961474i \(0.588643\pi\)
\(174\) 0 0
\(175\) 3177.98 1.37276
\(176\) 0 0
\(177\) −1904.17 −0.808622
\(178\) 0 0
\(179\) 3162.67 1.32061 0.660305 0.750998i \(-0.270427\pi\)
0.660305 + 0.750998i \(0.270427\pi\)
\(180\) 0 0
\(181\) 2922.50 1.20015 0.600076 0.799943i \(-0.295137\pi\)
0.600076 + 0.799943i \(0.295137\pi\)
\(182\) 0 0
\(183\) 725.319 0.292990
\(184\) 0 0
\(185\) −372.161 −0.147902
\(186\) 0 0
\(187\) 4635.01 1.81254
\(188\) 0 0
\(189\) 1979.94 0.762008
\(190\) 0 0
\(191\) 220.312 0.0834620 0.0417310 0.999129i \(-0.486713\pi\)
0.0417310 + 0.999129i \(0.486713\pi\)
\(192\) 0 0
\(193\) −3256.89 −1.21470 −0.607348 0.794436i \(-0.707767\pi\)
−0.607348 + 0.794436i \(0.707767\pi\)
\(194\) 0 0
\(195\) −499.142 −0.183304
\(196\) 0 0
\(197\) −3771.73 −1.36409 −0.682043 0.731312i \(-0.738908\pi\)
−0.682043 + 0.731312i \(0.738908\pi\)
\(198\) 0 0
\(199\) 2497.30 0.889593 0.444797 0.895632i \(-0.353276\pi\)
0.444797 + 0.895632i \(0.353276\pi\)
\(200\) 0 0
\(201\) 1040.44 0.365108
\(202\) 0 0
\(203\) −755.686 −0.261275
\(204\) 0 0
\(205\) −573.650 −0.195441
\(206\) 0 0
\(207\) 345.870 0.116133
\(208\) 0 0
\(209\) −1366.29 −0.452191
\(210\) 0 0
\(211\) −913.180 −0.297943 −0.148971 0.988842i \(-0.547596\pi\)
−0.148971 + 0.988842i \(0.547596\pi\)
\(212\) 0 0
\(213\) −841.415 −0.270670
\(214\) 0 0
\(215\) 3.52011 0.00111660
\(216\) 0 0
\(217\) −2349.64 −0.735041
\(218\) 0 0
\(219\) 4830.26 1.49041
\(220\) 0 0
\(221\) −5903.59 −1.79692
\(222\) 0 0
\(223\) 4122.81 1.23805 0.619023 0.785373i \(-0.287529\pi\)
0.619023 + 0.785373i \(0.287529\pi\)
\(224\) 0 0
\(225\) −1868.41 −0.553604
\(226\) 0 0
\(227\) 3632.84 1.06220 0.531102 0.847308i \(-0.321778\pi\)
0.531102 + 0.847308i \(0.321778\pi\)
\(228\) 0 0
\(229\) −6428.20 −1.85497 −0.927484 0.373864i \(-0.878033\pi\)
−0.927484 + 0.373864i \(0.878033\pi\)
\(230\) 0 0
\(231\) −5854.25 −1.66745
\(232\) 0 0
\(233\) −3844.90 −1.08106 −0.540532 0.841324i \(-0.681777\pi\)
−0.540532 + 0.841324i \(0.681777\pi\)
\(234\) 0 0
\(235\) −28.5496 −0.00792498
\(236\) 0 0
\(237\) −6927.25 −1.89862
\(238\) 0 0
\(239\) 2434.71 0.658947 0.329473 0.944165i \(-0.393129\pi\)
0.329473 + 0.944165i \(0.393129\pi\)
\(240\) 0 0
\(241\) −4450.57 −1.18957 −0.594785 0.803885i \(-0.702763\pi\)
−0.594785 + 0.803885i \(0.702763\pi\)
\(242\) 0 0
\(243\) −3854.99 −1.01769
\(244\) 0 0
\(245\) 586.144 0.152846
\(246\) 0 0
\(247\) 1740.23 0.448293
\(248\) 0 0
\(249\) −3231.00 −0.822313
\(250\) 0 0
\(251\) 3603.45 0.906166 0.453083 0.891468i \(-0.350324\pi\)
0.453083 + 0.891468i \(0.350324\pi\)
\(252\) 0 0
\(253\) 779.654 0.193741
\(254\) 0 0
\(255\) 1523.01 0.374017
\(256\) 0 0
\(257\) 3169.32 0.769248 0.384624 0.923073i \(-0.374331\pi\)
0.384624 + 0.923073i \(0.374331\pi\)
\(258\) 0 0
\(259\) 5559.60 1.33381
\(260\) 0 0
\(261\) 444.286 0.105366
\(262\) 0 0
\(263\) 901.230 0.211301 0.105651 0.994403i \(-0.466307\pi\)
0.105651 + 0.994403i \(0.466307\pi\)
\(264\) 0 0
\(265\) −260.657 −0.0604228
\(266\) 0 0
\(267\) 4401.55 1.00888
\(268\) 0 0
\(269\) −3218.85 −0.729578 −0.364789 0.931090i \(-0.618859\pi\)
−0.364789 + 0.931090i \(0.618859\pi\)
\(270\) 0 0
\(271\) 8364.23 1.87487 0.937437 0.348154i \(-0.113191\pi\)
0.937437 + 0.348154i \(0.113191\pi\)
\(272\) 0 0
\(273\) 7456.53 1.65308
\(274\) 0 0
\(275\) −4211.74 −0.923555
\(276\) 0 0
\(277\) −2507.41 −0.543882 −0.271941 0.962314i \(-0.587666\pi\)
−0.271941 + 0.962314i \(0.587666\pi\)
\(278\) 0 0
\(279\) 1381.41 0.296426
\(280\) 0 0
\(281\) 7813.51 1.65877 0.829386 0.558676i \(-0.188690\pi\)
0.829386 + 0.558676i \(0.188690\pi\)
\(282\) 0 0
\(283\) 7382.99 1.55079 0.775394 0.631478i \(-0.217551\pi\)
0.775394 + 0.631478i \(0.217551\pi\)
\(284\) 0 0
\(285\) −448.944 −0.0933093
\(286\) 0 0
\(287\) 8569.58 1.76253
\(288\) 0 0
\(289\) 13100.3 2.66646
\(290\) 0 0
\(291\) −671.551 −0.135282
\(292\) 0 0
\(293\) 4876.95 0.972405 0.486203 0.873846i \(-0.338382\pi\)
0.486203 + 0.873846i \(0.338382\pi\)
\(294\) 0 0
\(295\) −510.578 −0.100770
\(296\) 0 0
\(297\) −2623.99 −0.512658
\(298\) 0 0
\(299\) −993.041 −0.192070
\(300\) 0 0
\(301\) −52.5858 −0.0100698
\(302\) 0 0
\(303\) −9083.76 −1.72227
\(304\) 0 0
\(305\) 194.485 0.0365120
\(306\) 0 0
\(307\) −6722.28 −1.24971 −0.624854 0.780741i \(-0.714842\pi\)
−0.624854 + 0.780741i \(0.714842\pi\)
\(308\) 0 0
\(309\) −11896.8 −2.19024
\(310\) 0 0
\(311\) −1684.02 −0.307049 −0.153524 0.988145i \(-0.549062\pi\)
−0.153524 + 0.988145i \(0.549062\pi\)
\(312\) 0 0
\(313\) −9547.82 −1.72420 −0.862100 0.506738i \(-0.830851\pi\)
−0.862100 + 0.506738i \(0.830851\pi\)
\(314\) 0 0
\(315\) −696.368 −0.124558
\(316\) 0 0
\(317\) 4080.15 0.722915 0.361457 0.932389i \(-0.382279\pi\)
0.361457 + 0.932389i \(0.382279\pi\)
\(318\) 0 0
\(319\) 1001.50 0.175779
\(320\) 0 0
\(321\) 7265.61 1.26332
\(322\) 0 0
\(323\) −5309.87 −0.914704
\(324\) 0 0
\(325\) 5364.47 0.915592
\(326\) 0 0
\(327\) 3892.19 0.658222
\(328\) 0 0
\(329\) 426.494 0.0714692
\(330\) 0 0
\(331\) 2013.41 0.334341 0.167171 0.985928i \(-0.446537\pi\)
0.167171 + 0.985928i \(0.446537\pi\)
\(332\) 0 0
\(333\) −3268.62 −0.537896
\(334\) 0 0
\(335\) 278.980 0.0454993
\(336\) 0 0
\(337\) −8373.60 −1.35353 −0.676764 0.736200i \(-0.736618\pi\)
−0.676764 + 0.736200i \(0.736618\pi\)
\(338\) 0 0
\(339\) −13600.5 −2.17900
\(340\) 0 0
\(341\) 3113.95 0.494516
\(342\) 0 0
\(343\) 181.708 0.0286044
\(344\) 0 0
\(345\) 256.184 0.0399783
\(346\) 0 0
\(347\) −6584.86 −1.01871 −0.509357 0.860555i \(-0.670117\pi\)
−0.509357 + 0.860555i \(0.670117\pi\)
\(348\) 0 0
\(349\) 2267.59 0.347798 0.173899 0.984764i \(-0.444363\pi\)
0.173899 + 0.984764i \(0.444363\pi\)
\(350\) 0 0
\(351\) 3342.17 0.508238
\(352\) 0 0
\(353\) 1025.12 0.154565 0.0772826 0.997009i \(-0.475376\pi\)
0.0772826 + 0.997009i \(0.475376\pi\)
\(354\) 0 0
\(355\) −225.614 −0.0337306
\(356\) 0 0
\(357\) −22751.7 −3.37296
\(358\) 0 0
\(359\) −1688.16 −0.248183 −0.124091 0.992271i \(-0.539602\pi\)
−0.124091 + 0.992271i \(0.539602\pi\)
\(360\) 0 0
\(361\) −5293.78 −0.771801
\(362\) 0 0
\(363\) −900.112 −0.130148
\(364\) 0 0
\(365\) 1295.17 0.185733
\(366\) 0 0
\(367\) −192.611 −0.0273957 −0.0136978 0.999906i \(-0.504360\pi\)
−0.0136978 + 0.999906i \(0.504360\pi\)
\(368\) 0 0
\(369\) −5038.27 −0.710790
\(370\) 0 0
\(371\) 3893.88 0.544906
\(372\) 0 0
\(373\) 6947.18 0.964374 0.482187 0.876068i \(-0.339843\pi\)
0.482187 + 0.876068i \(0.339843\pi\)
\(374\) 0 0
\(375\) −2802.38 −0.385904
\(376\) 0 0
\(377\) −1275.61 −0.174263
\(378\) 0 0
\(379\) 7500.31 1.01653 0.508265 0.861200i \(-0.330287\pi\)
0.508265 + 0.861200i \(0.330287\pi\)
\(380\) 0 0
\(381\) 11500.4 1.54642
\(382\) 0 0
\(383\) 7259.99 0.968586 0.484293 0.874906i \(-0.339077\pi\)
0.484293 + 0.874906i \(0.339077\pi\)
\(384\) 0 0
\(385\) −1569.74 −0.207796
\(386\) 0 0
\(387\) 30.9165 0.00406091
\(388\) 0 0
\(389\) 3912.76 0.509987 0.254994 0.966943i \(-0.417927\pi\)
0.254994 + 0.966943i \(0.417927\pi\)
\(390\) 0 0
\(391\) 3030.01 0.391904
\(392\) 0 0
\(393\) −11864.2 −1.52283
\(394\) 0 0
\(395\) −1857.45 −0.236604
\(396\) 0 0
\(397\) −3532.50 −0.446577 −0.223288 0.974752i \(-0.571679\pi\)
−0.223288 + 0.974752i \(0.571679\pi\)
\(398\) 0 0
\(399\) 6706.64 0.841483
\(400\) 0 0
\(401\) −9853.53 −1.22709 −0.613543 0.789661i \(-0.710256\pi\)
−0.613543 + 0.789661i \(0.710256\pi\)
\(402\) 0 0
\(403\) −3966.22 −0.490252
\(404\) 0 0
\(405\) −1583.75 −0.194314
\(406\) 0 0
\(407\) −7368.07 −0.897351
\(408\) 0 0
\(409\) −8972.45 −1.08474 −0.542370 0.840139i \(-0.682473\pi\)
−0.542370 + 0.840139i \(0.682473\pi\)
\(410\) 0 0
\(411\) −15726.9 −1.88748
\(412\) 0 0
\(413\) 7627.37 0.908761
\(414\) 0 0
\(415\) −866.350 −0.102476
\(416\) 0 0
\(417\) −11630.5 −1.36582
\(418\) 0 0
\(419\) −12190.9 −1.42139 −0.710697 0.703498i \(-0.751620\pi\)
−0.710697 + 0.703498i \(0.751620\pi\)
\(420\) 0 0
\(421\) −4044.75 −0.468241 −0.234120 0.972208i \(-0.575221\pi\)
−0.234120 + 0.972208i \(0.575221\pi\)
\(422\) 0 0
\(423\) −250.746 −0.0288220
\(424\) 0 0
\(425\) −16368.3 −1.86819
\(426\) 0 0
\(427\) −2905.35 −0.329273
\(428\) 0 0
\(429\) −9882.06 −1.11214
\(430\) 0 0
\(431\) −14523.1 −1.62309 −0.811545 0.584290i \(-0.801373\pi\)
−0.811545 + 0.584290i \(0.801373\pi\)
\(432\) 0 0
\(433\) 5205.13 0.577697 0.288848 0.957375i \(-0.406728\pi\)
0.288848 + 0.957375i \(0.406728\pi\)
\(434\) 0 0
\(435\) 329.081 0.0362718
\(436\) 0 0
\(437\) −893.173 −0.0977717
\(438\) 0 0
\(439\) 12554.6 1.36492 0.682458 0.730925i \(-0.260911\pi\)
0.682458 + 0.730925i \(0.260911\pi\)
\(440\) 0 0
\(441\) 5148.00 0.555880
\(442\) 0 0
\(443\) 15143.8 1.62416 0.812082 0.583544i \(-0.198334\pi\)
0.812082 + 0.583544i \(0.198334\pi\)
\(444\) 0 0
\(445\) 1180.22 0.125725
\(446\) 0 0
\(447\) 2269.60 0.240153
\(448\) 0 0
\(449\) 2918.35 0.306738 0.153369 0.988169i \(-0.450988\pi\)
0.153369 + 0.988169i \(0.450988\pi\)
\(450\) 0 0
\(451\) −11357.2 −1.18578
\(452\) 0 0
\(453\) −6664.91 −0.691268
\(454\) 0 0
\(455\) 1999.37 0.206004
\(456\) 0 0
\(457\) −9572.94 −0.979876 −0.489938 0.871757i \(-0.662981\pi\)
−0.489938 + 0.871757i \(0.662981\pi\)
\(458\) 0 0
\(459\) −10197.8 −1.03702
\(460\) 0 0
\(461\) 15933.7 1.60977 0.804885 0.593430i \(-0.202227\pi\)
0.804885 + 0.593430i \(0.202227\pi\)
\(462\) 0 0
\(463\) 104.965 0.0105360 0.00526798 0.999986i \(-0.498323\pi\)
0.00526798 + 0.999986i \(0.498323\pi\)
\(464\) 0 0
\(465\) 1023.20 0.102043
\(466\) 0 0
\(467\) 138.716 0.0137452 0.00687259 0.999976i \(-0.497812\pi\)
0.00687259 + 0.999976i \(0.497812\pi\)
\(468\) 0 0
\(469\) −4167.59 −0.410323
\(470\) 0 0
\(471\) 5987.55 0.585758
\(472\) 0 0
\(473\) 69.6914 0.00677466
\(474\) 0 0
\(475\) 4824.98 0.466074
\(476\) 0 0
\(477\) −2289.31 −0.219749
\(478\) 0 0
\(479\) 9798.67 0.934682 0.467341 0.884077i \(-0.345212\pi\)
0.467341 + 0.884077i \(0.345212\pi\)
\(480\) 0 0
\(481\) 9384.68 0.889614
\(482\) 0 0
\(483\) −3827.06 −0.360533
\(484\) 0 0
\(485\) −180.068 −0.0168587
\(486\) 0 0
\(487\) 8385.14 0.780220 0.390110 0.920768i \(-0.372437\pi\)
0.390110 + 0.920768i \(0.372437\pi\)
\(488\) 0 0
\(489\) 7902.09 0.730766
\(490\) 0 0
\(491\) 3106.61 0.285539 0.142769 0.989756i \(-0.454399\pi\)
0.142769 + 0.989756i \(0.454399\pi\)
\(492\) 0 0
\(493\) 3892.20 0.355569
\(494\) 0 0
\(495\) 922.889 0.0837996
\(496\) 0 0
\(497\) 3370.39 0.304190
\(498\) 0 0
\(499\) 54.8571 0.00492132 0.00246066 0.999997i \(-0.499217\pi\)
0.00246066 + 0.999997i \(0.499217\pi\)
\(500\) 0 0
\(501\) −24230.2 −2.16073
\(502\) 0 0
\(503\) −16425.8 −1.45604 −0.728020 0.685556i \(-0.759559\pi\)
−0.728020 + 0.685556i \(0.759559\pi\)
\(504\) 0 0
\(505\) −2435.69 −0.214627
\(506\) 0 0
\(507\) −1705.64 −0.149408
\(508\) 0 0
\(509\) −564.244 −0.0491349 −0.0245675 0.999698i \(-0.507821\pi\)
−0.0245675 + 0.999698i \(0.507821\pi\)
\(510\) 0 0
\(511\) −19348.2 −1.67498
\(512\) 0 0
\(513\) 3006.05 0.258714
\(514\) 0 0
\(515\) −3189.96 −0.272945
\(516\) 0 0
\(517\) −565.228 −0.0480826
\(518\) 0 0
\(519\) −8138.45 −0.688320
\(520\) 0 0
\(521\) 11436.1 0.961657 0.480828 0.876815i \(-0.340336\pi\)
0.480828 + 0.876815i \(0.340336\pi\)
\(522\) 0 0
\(523\) −9988.34 −0.835104 −0.417552 0.908653i \(-0.637112\pi\)
−0.417552 + 0.908653i \(0.637112\pi\)
\(524\) 0 0
\(525\) 20674.0 1.71865
\(526\) 0 0
\(527\) 12101.9 1.00032
\(528\) 0 0
\(529\) −11657.3 −0.958110
\(530\) 0 0
\(531\) −4484.32 −0.366484
\(532\) 0 0
\(533\) 14465.6 1.17556
\(534\) 0 0
\(535\) 1948.18 0.157434
\(536\) 0 0
\(537\) 20574.4 1.65336
\(538\) 0 0
\(539\) 11604.5 0.927352
\(540\) 0 0
\(541\) −1900.85 −0.151061 −0.0755303 0.997144i \(-0.524065\pi\)
−0.0755303 + 0.997144i \(0.524065\pi\)
\(542\) 0 0
\(543\) 19012.0 1.50255
\(544\) 0 0
\(545\) 1043.64 0.0820269
\(546\) 0 0
\(547\) −9310.74 −0.727785 −0.363892 0.931441i \(-0.618552\pi\)
−0.363892 + 0.931441i \(0.618552\pi\)
\(548\) 0 0
\(549\) 1708.13 0.132789
\(550\) 0 0
\(551\) −1147.32 −0.0887070
\(552\) 0 0
\(553\) 27747.9 2.13375
\(554\) 0 0
\(555\) −2421.05 −0.185168
\(556\) 0 0
\(557\) 10861.6 0.826250 0.413125 0.910674i \(-0.364437\pi\)
0.413125 + 0.910674i \(0.364437\pi\)
\(558\) 0 0
\(559\) −88.7655 −0.00671625
\(560\) 0 0
\(561\) 30152.6 2.26924
\(562\) 0 0
\(563\) 246.144 0.0184258 0.00921290 0.999958i \(-0.497067\pi\)
0.00921290 + 0.999958i \(0.497067\pi\)
\(564\) 0 0
\(565\) −3646.81 −0.271544
\(566\) 0 0
\(567\) 23659.1 1.75236
\(568\) 0 0
\(569\) 1180.19 0.0869527 0.0434763 0.999054i \(-0.486157\pi\)
0.0434763 + 0.999054i \(0.486157\pi\)
\(570\) 0 0
\(571\) 6841.42 0.501409 0.250704 0.968064i \(-0.419338\pi\)
0.250704 + 0.968064i \(0.419338\pi\)
\(572\) 0 0
\(573\) 1433.22 0.104492
\(574\) 0 0
\(575\) −2753.31 −0.199689
\(576\) 0 0
\(577\) −6422.86 −0.463409 −0.231705 0.972786i \(-0.574430\pi\)
−0.231705 + 0.972786i \(0.574430\pi\)
\(578\) 0 0
\(579\) −21187.4 −1.52076
\(580\) 0 0
\(581\) 12942.1 0.924149
\(582\) 0 0
\(583\) −5160.51 −0.366598
\(584\) 0 0
\(585\) −1175.48 −0.0830771
\(586\) 0 0
\(587\) 16445.0 1.15632 0.578158 0.815925i \(-0.303772\pi\)
0.578158 + 0.815925i \(0.303772\pi\)
\(588\) 0 0
\(589\) −3567.34 −0.249558
\(590\) 0 0
\(591\) −24536.6 −1.70779
\(592\) 0 0
\(593\) −13958.9 −0.966652 −0.483326 0.875440i \(-0.660571\pi\)
−0.483326 + 0.875440i \(0.660571\pi\)
\(594\) 0 0
\(595\) −6100.58 −0.420335
\(596\) 0 0
\(597\) 16245.9 1.11374
\(598\) 0 0
\(599\) 1007.54 0.0687263 0.0343632 0.999409i \(-0.489060\pi\)
0.0343632 + 0.999409i \(0.489060\pi\)
\(600\) 0 0
\(601\) −3198.61 −0.217095 −0.108548 0.994091i \(-0.534620\pi\)
−0.108548 + 0.994091i \(0.534620\pi\)
\(602\) 0 0
\(603\) 2450.23 0.165474
\(604\) 0 0
\(605\) −241.353 −0.0162189
\(606\) 0 0
\(607\) 17511.7 1.17097 0.585483 0.810684i \(-0.300905\pi\)
0.585483 + 0.810684i \(0.300905\pi\)
\(608\) 0 0
\(609\) −4916.04 −0.327107
\(610\) 0 0
\(611\) 719.928 0.0476680
\(612\) 0 0
\(613\) 2559.74 0.168657 0.0843285 0.996438i \(-0.473125\pi\)
0.0843285 + 0.996438i \(0.473125\pi\)
\(614\) 0 0
\(615\) −3731.82 −0.244685
\(616\) 0 0
\(617\) −9033.40 −0.589418 −0.294709 0.955587i \(-0.595223\pi\)
−0.294709 + 0.955587i \(0.595223\pi\)
\(618\) 0 0
\(619\) 17108.3 1.11089 0.555445 0.831553i \(-0.312548\pi\)
0.555445 + 0.831553i \(0.312548\pi\)
\(620\) 0 0
\(621\) −1715.37 −0.110846
\(622\) 0 0
\(623\) −17630.9 −1.13382
\(624\) 0 0
\(625\) 14493.2 0.927567
\(626\) 0 0
\(627\) −8888.23 −0.566127
\(628\) 0 0
\(629\) −28635.0 −1.81518
\(630\) 0 0
\(631\) 10191.7 0.642987 0.321494 0.946912i \(-0.395815\pi\)
0.321494 + 0.946912i \(0.395815\pi\)
\(632\) 0 0
\(633\) −5940.60 −0.373014
\(634\) 0 0
\(635\) 3083.69 0.192713
\(636\) 0 0
\(637\) −14780.6 −0.919357
\(638\) 0 0
\(639\) −1981.53 −0.122673
\(640\) 0 0
\(641\) −15069.3 −0.928551 −0.464275 0.885691i \(-0.653685\pi\)
−0.464275 + 0.885691i \(0.653685\pi\)
\(642\) 0 0
\(643\) 25251.4 1.54871 0.774354 0.632752i \(-0.218075\pi\)
0.774354 + 0.632752i \(0.218075\pi\)
\(644\) 0 0
\(645\) 22.8997 0.00139794
\(646\) 0 0
\(647\) −13049.6 −0.792941 −0.396470 0.918047i \(-0.629765\pi\)
−0.396470 + 0.918047i \(0.629765\pi\)
\(648\) 0 0
\(649\) −10108.5 −0.611390
\(650\) 0 0
\(651\) −15285.3 −0.920245
\(652\) 0 0
\(653\) −1741.94 −0.104391 −0.0521955 0.998637i \(-0.516622\pi\)
−0.0521955 + 0.998637i \(0.516622\pi\)
\(654\) 0 0
\(655\) −3181.23 −0.189773
\(656\) 0 0
\(657\) 11375.3 0.675482
\(658\) 0 0
\(659\) 29549.0 1.74669 0.873343 0.487105i \(-0.161947\pi\)
0.873343 + 0.487105i \(0.161947\pi\)
\(660\) 0 0
\(661\) −18000.8 −1.05923 −0.529615 0.848238i \(-0.677663\pi\)
−0.529615 + 0.848238i \(0.677663\pi\)
\(662\) 0 0
\(663\) −38405.2 −2.24967
\(664\) 0 0
\(665\) 1798.30 0.104865
\(666\) 0 0
\(667\) 654.705 0.0380064
\(668\) 0 0
\(669\) 26820.6 1.54999
\(670\) 0 0
\(671\) 3850.43 0.221526
\(672\) 0 0
\(673\) 6142.86 0.351842 0.175921 0.984404i \(-0.443710\pi\)
0.175921 + 0.984404i \(0.443710\pi\)
\(674\) 0 0
\(675\) 9266.52 0.528398
\(676\) 0 0
\(677\) 26970.2 1.53109 0.765546 0.643381i \(-0.222469\pi\)
0.765546 + 0.643381i \(0.222469\pi\)
\(678\) 0 0
\(679\) 2689.98 0.152035
\(680\) 0 0
\(681\) 23633.1 1.32984
\(682\) 0 0
\(683\) −11702.2 −0.655598 −0.327799 0.944748i \(-0.606307\pi\)
−0.327799 + 0.944748i \(0.606307\pi\)
\(684\) 0 0
\(685\) −4216.98 −0.235215
\(686\) 0 0
\(687\) −41818.0 −2.32235
\(688\) 0 0
\(689\) 6572.92 0.363437
\(690\) 0 0
\(691\) −22034.0 −1.21305 −0.606523 0.795066i \(-0.707436\pi\)
−0.606523 + 0.795066i \(0.707436\pi\)
\(692\) 0 0
\(693\) −13786.8 −0.755723
\(694\) 0 0
\(695\) −3118.56 −0.170207
\(696\) 0 0
\(697\) −44138.0 −2.39863
\(698\) 0 0
\(699\) −25012.6 −1.35345
\(700\) 0 0
\(701\) −26465.4 −1.42594 −0.712969 0.701195i \(-0.752650\pi\)
−0.712969 + 0.701195i \(0.752650\pi\)
\(702\) 0 0
\(703\) 8440.87 0.452850
\(704\) 0 0
\(705\) −185.727 −0.00992180
\(706\) 0 0
\(707\) 36386.1 1.93556
\(708\) 0 0
\(709\) 3909.36 0.207079 0.103540 0.994625i \(-0.466983\pi\)
0.103540 + 0.994625i \(0.466983\pi\)
\(710\) 0 0
\(711\) −16313.7 −0.860493
\(712\) 0 0
\(713\) 2035.66 0.106923
\(714\) 0 0
\(715\) −2649.74 −0.138594
\(716\) 0 0
\(717\) 15838.8 0.824978
\(718\) 0 0
\(719\) −27137.8 −1.40761 −0.703803 0.710395i \(-0.748516\pi\)
−0.703803 + 0.710395i \(0.748516\pi\)
\(720\) 0 0
\(721\) 47653.9 2.46147
\(722\) 0 0
\(723\) −28952.7 −1.48930
\(724\) 0 0
\(725\) −3536.76 −0.181175
\(726\) 0 0
\(727\) 23333.1 1.19034 0.595169 0.803600i \(-0.297085\pi\)
0.595169 + 0.803600i \(0.297085\pi\)
\(728\) 0 0
\(729\) −563.931 −0.0286507
\(730\) 0 0
\(731\) 270.846 0.0137039
\(732\) 0 0
\(733\) −19651.7 −0.990248 −0.495124 0.868822i \(-0.664877\pi\)
−0.495124 + 0.868822i \(0.664877\pi\)
\(734\) 0 0
\(735\) 3813.10 0.191358
\(736\) 0 0
\(737\) 5523.26 0.276054
\(738\) 0 0
\(739\) 2012.48 0.100176 0.0500882 0.998745i \(-0.484050\pi\)
0.0500882 + 0.998745i \(0.484050\pi\)
\(740\) 0 0
\(741\) 11320.9 0.561246
\(742\) 0 0
\(743\) 14171.2 0.699720 0.349860 0.936802i \(-0.386229\pi\)
0.349860 + 0.936802i \(0.386229\pi\)
\(744\) 0 0
\(745\) 608.564 0.0299276
\(746\) 0 0
\(747\) −7609.00 −0.372689
\(748\) 0 0
\(749\) −29103.3 −1.41977
\(750\) 0 0
\(751\) 8500.52 0.413034 0.206517 0.978443i \(-0.433787\pi\)
0.206517 + 0.978443i \(0.433787\pi\)
\(752\) 0 0
\(753\) 23441.9 1.13449
\(754\) 0 0
\(755\) −1787.11 −0.0861450
\(756\) 0 0
\(757\) 15227.4 0.731109 0.365554 0.930790i \(-0.380879\pi\)
0.365554 + 0.930790i \(0.380879\pi\)
\(758\) 0 0
\(759\) 5071.96 0.242557
\(760\) 0 0
\(761\) −9243.82 −0.440326 −0.220163 0.975463i \(-0.570659\pi\)
−0.220163 + 0.975463i \(0.570659\pi\)
\(762\) 0 0
\(763\) −15590.6 −0.739736
\(764\) 0 0
\(765\) 3586.68 0.169512
\(766\) 0 0
\(767\) 12875.1 0.606119
\(768\) 0 0
\(769\) 418.701 0.0196343 0.00981714 0.999952i \(-0.496875\pi\)
0.00981714 + 0.999952i \(0.496875\pi\)
\(770\) 0 0
\(771\) 20617.7 0.963071
\(772\) 0 0
\(773\) 27488.2 1.27902 0.639508 0.768784i \(-0.279138\pi\)
0.639508 + 0.768784i \(0.279138\pi\)
\(774\) 0 0
\(775\) −10996.8 −0.509698
\(776\) 0 0
\(777\) 36167.4 1.66988
\(778\) 0 0
\(779\) 13010.8 0.598408
\(780\) 0 0
\(781\) −4466.73 −0.204651
\(782\) 0 0
\(783\) −2203.47 −0.100569
\(784\) 0 0
\(785\) 1605.48 0.0729965
\(786\) 0 0
\(787\) 7061.33 0.319834 0.159917 0.987130i \(-0.448877\pi\)
0.159917 + 0.987130i \(0.448877\pi\)
\(788\) 0 0
\(789\) 5862.86 0.264542
\(790\) 0 0
\(791\) 54478.6 2.44884
\(792\) 0 0
\(793\) −4904.27 −0.219616
\(794\) 0 0
\(795\) −1695.68 −0.0756472
\(796\) 0 0
\(797\) 33174.1 1.47439 0.737195 0.675680i \(-0.236150\pi\)
0.737195 + 0.675680i \(0.236150\pi\)
\(798\) 0 0
\(799\) −2196.68 −0.0972626
\(800\) 0 0
\(801\) 10365.7 0.457244
\(802\) 0 0
\(803\) 25641.9 1.12688
\(804\) 0 0
\(805\) −1026.18 −0.0449291
\(806\) 0 0
\(807\) −20939.9 −0.913406
\(808\) 0 0
\(809\) 10119.6 0.439787 0.219894 0.975524i \(-0.429429\pi\)
0.219894 + 0.975524i \(0.429429\pi\)
\(810\) 0 0
\(811\) −23079.6 −0.999303 −0.499652 0.866226i \(-0.666539\pi\)
−0.499652 + 0.866226i \(0.666539\pi\)
\(812\) 0 0
\(813\) 54412.7 2.34728
\(814\) 0 0
\(815\) 2118.84 0.0910672
\(816\) 0 0
\(817\) −79.8385 −0.00341884
\(818\) 0 0
\(819\) 17560.1 0.749207
\(820\) 0 0
\(821\) −10608.1 −0.450945 −0.225473 0.974250i \(-0.572393\pi\)
−0.225473 + 0.974250i \(0.572393\pi\)
\(822\) 0 0
\(823\) −6114.71 −0.258986 −0.129493 0.991580i \(-0.541335\pi\)
−0.129493 + 0.991580i \(0.541335\pi\)
\(824\) 0 0
\(825\) −27399.1 −1.15626
\(826\) 0 0
\(827\) 31584.8 1.32807 0.664033 0.747703i \(-0.268843\pi\)
0.664033 + 0.747703i \(0.268843\pi\)
\(828\) 0 0
\(829\) −24207.2 −1.01417 −0.507087 0.861895i \(-0.669278\pi\)
−0.507087 + 0.861895i \(0.669278\pi\)
\(830\) 0 0
\(831\) −16311.7 −0.680921
\(832\) 0 0
\(833\) 45099.4 1.87587
\(834\) 0 0
\(835\) −6497.01 −0.269267
\(836\) 0 0
\(837\) −6851.19 −0.282929
\(838\) 0 0
\(839\) −38899.3 −1.60066 −0.800329 0.599561i \(-0.795342\pi\)
−0.800329 + 0.599561i \(0.795342\pi\)
\(840\) 0 0
\(841\) 841.000 0.0344828
\(842\) 0 0
\(843\) 50830.0 2.07672
\(844\) 0 0
\(845\) −457.344 −0.0186191
\(846\) 0 0
\(847\) 3605.51 0.146265
\(848\) 0 0
\(849\) 48029.3 1.94153
\(850\) 0 0
\(851\) −4816.68 −0.194023
\(852\) 0 0
\(853\) 39006.0 1.56570 0.782848 0.622213i \(-0.213766\pi\)
0.782848 + 0.622213i \(0.213766\pi\)
\(854\) 0 0
\(855\) −1057.26 −0.0422896
\(856\) 0 0
\(857\) −11708.7 −0.466700 −0.233350 0.972393i \(-0.574969\pi\)
−0.233350 + 0.972393i \(0.574969\pi\)
\(858\) 0 0
\(859\) 39476.8 1.56802 0.784011 0.620747i \(-0.213171\pi\)
0.784011 + 0.620747i \(0.213171\pi\)
\(860\) 0 0
\(861\) 55748.6 2.20663
\(862\) 0 0
\(863\) −26810.7 −1.05753 −0.528764 0.848769i \(-0.677344\pi\)
−0.528764 + 0.848769i \(0.677344\pi\)
\(864\) 0 0
\(865\) −2182.22 −0.0857777
\(866\) 0 0
\(867\) 85222.7 3.33831
\(868\) 0 0
\(869\) −36774.0 −1.43553
\(870\) 0 0
\(871\) −7034.95 −0.273674
\(872\) 0 0
\(873\) −1581.50 −0.0613125
\(874\) 0 0
\(875\) 11225.2 0.433694
\(876\) 0 0
\(877\) 36718.7 1.41380 0.706899 0.707314i \(-0.250093\pi\)
0.706899 + 0.707314i \(0.250093\pi\)
\(878\) 0 0
\(879\) 31726.5 1.21742
\(880\) 0 0
\(881\) −30253.6 −1.15695 −0.578473 0.815701i \(-0.696351\pi\)
−0.578473 + 0.815701i \(0.696351\pi\)
\(882\) 0 0
\(883\) 12267.1 0.467522 0.233761 0.972294i \(-0.424897\pi\)
0.233761 + 0.972294i \(0.424897\pi\)
\(884\) 0 0
\(885\) −3321.51 −0.126160
\(886\) 0 0
\(887\) −45007.2 −1.70371 −0.851856 0.523776i \(-0.824523\pi\)
−0.851856 + 0.523776i \(0.824523\pi\)
\(888\) 0 0
\(889\) −46066.4 −1.73793
\(890\) 0 0
\(891\) −31355.2 −1.17894
\(892\) 0 0
\(893\) 647.526 0.0242650
\(894\) 0 0
\(895\) 5516.77 0.206039
\(896\) 0 0
\(897\) −6460.13 −0.240465
\(898\) 0 0
\(899\) 2614.90 0.0970099
\(900\) 0 0
\(901\) −20055.6 −0.741564
\(902\) 0 0
\(903\) −342.092 −0.0126070
\(904\) 0 0
\(905\) 5097.83 0.187246
\(906\) 0 0
\(907\) 17604.7 0.644494 0.322247 0.946656i \(-0.395562\pi\)
0.322247 + 0.946656i \(0.395562\pi\)
\(908\) 0 0
\(909\) −21392.3 −0.780568
\(910\) 0 0
\(911\) 32031.3 1.16492 0.582461 0.812859i \(-0.302090\pi\)
0.582461 + 0.812859i \(0.302090\pi\)
\(912\) 0 0
\(913\) −17152.1 −0.621742
\(914\) 0 0
\(915\) 1265.20 0.0457118
\(916\) 0 0
\(917\) 47523.5 1.71141
\(918\) 0 0
\(919\) −11659.6 −0.418515 −0.209257 0.977861i \(-0.567105\pi\)
−0.209257 + 0.977861i \(0.567105\pi\)
\(920\) 0 0
\(921\) −43731.1 −1.56459
\(922\) 0 0
\(923\) 5689.26 0.202886
\(924\) 0 0
\(925\) 26020.0 0.924901
\(926\) 0 0
\(927\) −28016.9 −0.992658
\(928\) 0 0
\(929\) 25455.5 0.898997 0.449499 0.893281i \(-0.351603\pi\)
0.449499 + 0.893281i \(0.351603\pi\)
\(930\) 0 0
\(931\) −13294.2 −0.467990
\(932\) 0 0
\(933\) −10955.2 −0.384414
\(934\) 0 0
\(935\) 8085.03 0.282790
\(936\) 0 0
\(937\) 18209.7 0.634882 0.317441 0.948278i \(-0.397176\pi\)
0.317441 + 0.948278i \(0.397176\pi\)
\(938\) 0 0
\(939\) −62112.4 −2.15864
\(940\) 0 0
\(941\) 28483.5 0.986755 0.493377 0.869815i \(-0.335762\pi\)
0.493377 + 0.869815i \(0.335762\pi\)
\(942\) 0 0
\(943\) −7424.45 −0.256387
\(944\) 0 0
\(945\) 3453.69 0.118887
\(946\) 0 0
\(947\) 20823.1 0.714529 0.357264 0.934003i \(-0.383710\pi\)
0.357264 + 0.934003i \(0.383710\pi\)
\(948\) 0 0
\(949\) −32660.0 −1.11716
\(950\) 0 0
\(951\) 26543.0 0.905063
\(952\) 0 0
\(953\) −5514.74 −0.187450 −0.0937251 0.995598i \(-0.529877\pi\)
−0.0937251 + 0.995598i \(0.529877\pi\)
\(954\) 0 0
\(955\) 384.299 0.0130216
\(956\) 0 0
\(957\) 6515.17 0.220069
\(958\) 0 0
\(959\) 62996.1 2.12122
\(960\) 0 0
\(961\) −21660.5 −0.727083
\(962\) 0 0
\(963\) 17110.5 0.572563
\(964\) 0 0
\(965\) −5681.12 −0.189515
\(966\) 0 0
\(967\) 42258.1 1.40530 0.702652 0.711534i \(-0.251999\pi\)
0.702652 + 0.711534i \(0.251999\pi\)
\(968\) 0 0
\(969\) −34542.9 −1.14518
\(970\) 0 0
\(971\) −11158.8 −0.368798 −0.184399 0.982851i \(-0.559034\pi\)
−0.184399 + 0.982851i \(0.559034\pi\)
\(972\) 0 0
\(973\) 46587.2 1.53496
\(974\) 0 0
\(975\) 34898.0 1.14629
\(976\) 0 0
\(977\) −32314.4 −1.05817 −0.529083 0.848570i \(-0.677464\pi\)
−0.529083 + 0.848570i \(0.677464\pi\)
\(978\) 0 0
\(979\) 23366.1 0.762802
\(980\) 0 0
\(981\) 9166.11 0.298319
\(982\) 0 0
\(983\) −18702.9 −0.606847 −0.303424 0.952856i \(-0.598130\pi\)
−0.303424 + 0.952856i \(0.598130\pi\)
\(984\) 0 0
\(985\) −6579.18 −0.212822
\(986\) 0 0
\(987\) 2774.51 0.0894769
\(988\) 0 0
\(989\) 45.5589 0.00146480
\(990\) 0 0
\(991\) −11042.5 −0.353962 −0.176981 0.984214i \(-0.556633\pi\)
−0.176981 + 0.984214i \(0.556633\pi\)
\(992\) 0 0
\(993\) 13098.0 0.418584
\(994\) 0 0
\(995\) 4356.14 0.138793
\(996\) 0 0
\(997\) 26609.4 0.845265 0.422633 0.906301i \(-0.361106\pi\)
0.422633 + 0.906301i \(0.361106\pi\)
\(998\) 0 0
\(999\) 16211.0 0.513405
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.bg.1.8 9
4.3 odd 2 1856.4.a.bf.1.2 9
8.3 odd 2 928.4.a.e.1.8 yes 9
8.5 even 2 928.4.a.d.1.2 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
928.4.a.d.1.2 9 8.5 even 2
928.4.a.e.1.8 yes 9 8.3 odd 2
1856.4.a.bf.1.2 9 4.3 odd 2
1856.4.a.bg.1.8 9 1.1 even 1 trivial