Properties

Label 1088.4.a.v.1.2
Level $1088$
Weight $4$
Character 1088.1
Self dual yes
Analytic conductor $64.194$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1088 = 2^{6} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1088.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.1940780862\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.2636.1
Defining polynomial: \( x^{3} - 14x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 17)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(3.87707\) of defining polynomial
Character \(\chi\) \(=\) 1088.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.15463 q^{3} -3.03171 q^{5} -7.94049 q^{7} -17.0483 q^{9} +O(q^{10})\) \(q-3.15463 q^{3} -3.03171 q^{5} -7.94049 q^{7} -17.0483 q^{9} -27.6161 q^{11} -58.1117 q^{13} +9.56391 q^{15} -17.0000 q^{17} -89.1688 q^{19} +25.0493 q^{21} -115.269 q^{23} -115.809 q^{25} +138.956 q^{27} +128.558 q^{29} +273.460 q^{31} +87.1187 q^{33} +24.0732 q^{35} +132.351 q^{37} +183.321 q^{39} -470.559 q^{41} -352.642 q^{43} +51.6854 q^{45} +152.598 q^{47} -279.949 q^{49} +53.6287 q^{51} -527.614 q^{53} +83.7239 q^{55} +281.295 q^{57} +292.020 q^{59} +53.8962 q^{61} +135.372 q^{63} +176.178 q^{65} -52.9572 q^{67} +363.632 q^{69} +788.400 q^{71} +295.780 q^{73} +365.334 q^{75} +219.285 q^{77} -720.325 q^{79} +21.9487 q^{81} +116.051 q^{83} +51.5390 q^{85} -405.552 q^{87} -813.329 q^{89} +461.435 q^{91} -862.664 q^{93} +270.334 q^{95} +794.693 q^{97} +470.808 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 4 q^{3} + 8 q^{5} + 22 q^{7} + 59 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 4 q^{3} + 8 q^{5} + 22 q^{7} + 59 q^{9} + 28 q^{11} - 30 q^{13} + 108 q^{15} - 51 q^{17} - 80 q^{19} + 192 q^{21} + 142 q^{23} - 223 q^{25} + 20 q^{27} + 456 q^{29} + 230 q^{31} - 332 q^{33} + 332 q^{35} - 356 q^{37} + 268 q^{39} - 294 q^{41} - 556 q^{43} + 384 q^{45} + 640 q^{47} - 269 q^{49} + 68 q^{51} - 302 q^{53} + 76 q^{55} - 720 q^{57} - 636 q^{59} + 84 q^{61} + 1122 q^{63} + 408 q^{65} - 1008 q^{67} - 576 q^{69} - 402 q^{71} + 838 q^{73} + 1548 q^{75} + 504 q^{77} - 594 q^{79} - 505 q^{81} + 2396 q^{83} - 136 q^{85} + 1428 q^{87} - 170 q^{89} + 1016 q^{91} - 632 q^{93} - 472 q^{95} - 270 q^{97} + 2920 q^{99}+O(q^{100}) \) 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
\(3\) −3.15463 −0.607109 −0.303555 0.952814i \(-0.598173\pi\)
−0.303555 + 0.952814i \(0.598173\pi\)
\(4\) 0 0
\(5\) −3.03171 −0.271164 −0.135582 0.990766i \(-0.543290\pi\)
−0.135582 + 0.990766i \(0.543290\pi\)
\(6\) 0 0
\(7\) −7.94049 −0.428746 −0.214373 0.976752i \(-0.568771\pi\)
−0.214373 + 0.976752i \(0.568771\pi\)
\(8\) 0 0
\(9\) −17.0483 −0.631419
\(10\) 0 0
\(11\) −27.6161 −0.756961 −0.378481 0.925609i \(-0.623553\pi\)
−0.378481 + 0.925609i \(0.623553\pi\)
\(12\) 0 0
\(13\) −58.1117 −1.23979 −0.619896 0.784684i \(-0.712825\pi\)
−0.619896 + 0.784684i \(0.712825\pi\)
\(14\) 0 0
\(15\) 9.56391 0.164626
\(16\) 0 0
\(17\) −17.0000 −0.242536
\(18\) 0 0
\(19\) −89.1688 −1.07667 −0.538335 0.842731i \(-0.680946\pi\)
−0.538335 + 0.842731i \(0.680946\pi\)
\(20\) 0 0
\(21\) 25.0493 0.260296
\(22\) 0 0
\(23\) −115.269 −1.04501 −0.522507 0.852635i \(-0.675003\pi\)
−0.522507 + 0.852635i \(0.675003\pi\)
\(24\) 0 0
\(25\) −115.809 −0.926470
\(26\) 0 0
\(27\) 138.956 0.990449
\(28\) 0 0
\(29\) 128.558 0.823191 0.411596 0.911367i \(-0.364972\pi\)
0.411596 + 0.911367i \(0.364972\pi\)
\(30\) 0 0
\(31\) 273.460 1.58435 0.792174 0.610295i \(-0.208949\pi\)
0.792174 + 0.610295i \(0.208949\pi\)
\(32\) 0 0
\(33\) 87.1187 0.459558
\(34\) 0 0
\(35\) 24.0732 0.116260
\(36\) 0 0
\(37\) 132.351 0.588063 0.294031 0.955796i \(-0.405003\pi\)
0.294031 + 0.955796i \(0.405003\pi\)
\(38\) 0 0
\(39\) 183.321 0.752689
\(40\) 0 0
\(41\) −470.559 −1.79241 −0.896207 0.443636i \(-0.853688\pi\)
−0.896207 + 0.443636i \(0.853688\pi\)
\(42\) 0 0
\(43\) −352.642 −1.25064 −0.625318 0.780370i \(-0.715031\pi\)
−0.625318 + 0.780370i \(0.715031\pi\)
\(44\) 0 0
\(45\) 51.6854 0.171218
\(46\) 0 0
\(47\) 152.598 0.473589 0.236795 0.971560i \(-0.423903\pi\)
0.236795 + 0.971560i \(0.423903\pi\)
\(48\) 0 0
\(49\) −279.949 −0.816177
\(50\) 0 0
\(51\) 53.6287 0.147246
\(52\) 0 0
\(53\) −527.614 −1.36742 −0.683711 0.729753i \(-0.739635\pi\)
−0.683711 + 0.729753i \(0.739635\pi\)
\(54\) 0 0
\(55\) 83.7239 0.205261
\(56\) 0 0
\(57\) 281.295 0.653656
\(58\) 0 0
\(59\) 292.020 0.644368 0.322184 0.946677i \(-0.395583\pi\)
0.322184 + 0.946677i \(0.395583\pi\)
\(60\) 0 0
\(61\) 53.8962 0.113126 0.0565632 0.998399i \(-0.481986\pi\)
0.0565632 + 0.998399i \(0.481986\pi\)
\(62\) 0 0
\(63\) 135.372 0.270718
\(64\) 0 0
\(65\) 176.178 0.336187
\(66\) 0 0
\(67\) −52.9572 −0.0965635 −0.0482817 0.998834i \(-0.515375\pi\)
−0.0482817 + 0.998834i \(0.515375\pi\)
\(68\) 0 0
\(69\) 363.632 0.634437
\(70\) 0 0
\(71\) 788.400 1.31783 0.658915 0.752218i \(-0.271016\pi\)
0.658915 + 0.752218i \(0.271016\pi\)
\(72\) 0 0
\(73\) 295.780 0.474224 0.237112 0.971482i \(-0.423799\pi\)
0.237112 + 0.971482i \(0.423799\pi\)
\(74\) 0 0
\(75\) 365.334 0.562468
\(76\) 0 0
\(77\) 219.285 0.324544
\(78\) 0 0
\(79\) −720.325 −1.02586 −0.512930 0.858430i \(-0.671440\pi\)
−0.512930 + 0.858430i \(0.671440\pi\)
\(80\) 0 0
\(81\) 21.9487 0.0301079
\(82\) 0 0
\(83\) 116.051 0.153473 0.0767363 0.997051i \(-0.475550\pi\)
0.0767363 + 0.997051i \(0.475550\pi\)
\(84\) 0 0
\(85\) 51.5390 0.0657669
\(86\) 0 0
\(87\) −405.552 −0.499767
\(88\) 0 0
\(89\) −813.329 −0.968682 −0.484341 0.874879i \(-0.660941\pi\)
−0.484341 + 0.874879i \(0.660941\pi\)
\(90\) 0 0
\(91\) 461.435 0.531556
\(92\) 0 0
\(93\) −862.664 −0.961872
\(94\) 0 0
\(95\) 270.334 0.291954
\(96\) 0 0
\(97\) 794.693 0.831844 0.415922 0.909400i \(-0.363459\pi\)
0.415922 + 0.909400i \(0.363459\pi\)
\(98\) 0 0
\(99\) 470.808 0.477959
\(100\) 0 0
\(101\) −265.513 −0.261579 −0.130790 0.991410i \(-0.541751\pi\)
−0.130790 + 0.991410i \(0.541751\pi\)
\(102\) 0 0
\(103\) 523.107 0.500420 0.250210 0.968192i \(-0.419500\pi\)
0.250210 + 0.968192i \(0.419500\pi\)
\(104\) 0 0
\(105\) −75.9421 −0.0705828
\(106\) 0 0
\(107\) 986.039 0.890878 0.445439 0.895312i \(-0.353048\pi\)
0.445439 + 0.895312i \(0.353048\pi\)
\(108\) 0 0
\(109\) −1814.39 −1.59438 −0.797188 0.603732i \(-0.793680\pi\)
−0.797188 + 0.603732i \(0.793680\pi\)
\(110\) 0 0
\(111\) −417.518 −0.357018
\(112\) 0 0
\(113\) −707.339 −0.588857 −0.294429 0.955673i \(-0.595129\pi\)
−0.294429 + 0.955673i \(0.595129\pi\)
\(114\) 0 0
\(115\) 349.463 0.283370
\(116\) 0 0
\(117\) 990.706 0.782827
\(118\) 0 0
\(119\) 134.988 0.103986
\(120\) 0 0
\(121\) −568.350 −0.427010
\(122\) 0 0
\(123\) 1484.44 1.08819
\(124\) 0 0
\(125\) 730.061 0.522389
\(126\) 0 0
\(127\) 2648.18 1.85030 0.925151 0.379600i \(-0.123938\pi\)
0.925151 + 0.379600i \(0.123938\pi\)
\(128\) 0 0
\(129\) 1112.46 0.759273
\(130\) 0 0
\(131\) 1979.08 1.31995 0.659974 0.751289i \(-0.270567\pi\)
0.659974 + 0.751289i \(0.270567\pi\)
\(132\) 0 0
\(133\) 708.044 0.461618
\(134\) 0 0
\(135\) −421.274 −0.268574
\(136\) 0 0
\(137\) 3141.92 1.95936 0.979679 0.200570i \(-0.0642794\pi\)
0.979679 + 0.200570i \(0.0642794\pi\)
\(138\) 0 0
\(139\) −1468.07 −0.895830 −0.447915 0.894076i \(-0.647833\pi\)
−0.447915 + 0.894076i \(0.647833\pi\)
\(140\) 0 0
\(141\) −481.390 −0.287520
\(142\) 0 0
\(143\) 1604.82 0.938474
\(144\) 0 0
\(145\) −389.749 −0.223220
\(146\) 0 0
\(147\) 883.135 0.495508
\(148\) 0 0
\(149\) 286.027 0.157263 0.0786316 0.996904i \(-0.474945\pi\)
0.0786316 + 0.996904i \(0.474945\pi\)
\(150\) 0 0
\(151\) −669.626 −0.360883 −0.180442 0.983586i \(-0.557753\pi\)
−0.180442 + 0.983586i \(0.557753\pi\)
\(152\) 0 0
\(153\) 289.821 0.153141
\(154\) 0 0
\(155\) −829.049 −0.429618
\(156\) 0 0
\(157\) −720.809 −0.366413 −0.183206 0.983074i \(-0.558648\pi\)
−0.183206 + 0.983074i \(0.558648\pi\)
\(158\) 0 0
\(159\) 1664.43 0.830174
\(160\) 0 0
\(161\) 915.294 0.448045
\(162\) 0 0
\(163\) 676.599 0.325125 0.162562 0.986698i \(-0.448024\pi\)
0.162562 + 0.986698i \(0.448024\pi\)
\(164\) 0 0
\(165\) −264.118 −0.124616
\(166\) 0 0
\(167\) −2835.67 −1.31396 −0.656979 0.753909i \(-0.728166\pi\)
−0.656979 + 0.753909i \(0.728166\pi\)
\(168\) 0 0
\(169\) 1179.97 0.537083
\(170\) 0 0
\(171\) 1520.18 0.679829
\(172\) 0 0
\(173\) 177.314 0.0779243 0.0389621 0.999241i \(-0.487595\pi\)
0.0389621 + 0.999241i \(0.487595\pi\)
\(174\) 0 0
\(175\) 919.578 0.397220
\(176\) 0 0
\(177\) −921.214 −0.391202
\(178\) 0 0
\(179\) 1023.76 0.427483 0.213741 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(180\) 0 0
\(181\) 3450.21 1.41686 0.708432 0.705779i \(-0.249403\pi\)
0.708432 + 0.705779i \(0.249403\pi\)
\(182\) 0 0
\(183\) −170.023 −0.0686800
\(184\) 0 0
\(185\) −401.248 −0.159461
\(186\) 0 0
\(187\) 469.474 0.183590
\(188\) 0 0
\(189\) −1103.38 −0.424651
\(190\) 0 0
\(191\) −490.894 −0.185968 −0.0929839 0.995668i \(-0.529640\pi\)
−0.0929839 + 0.995668i \(0.529640\pi\)
\(192\) 0 0
\(193\) −3548.80 −1.32357 −0.661783 0.749696i \(-0.730200\pi\)
−0.661783 + 0.749696i \(0.730200\pi\)
\(194\) 0 0
\(195\) −555.775 −0.204102
\(196\) 0 0
\(197\) −1363.15 −0.492996 −0.246498 0.969143i \(-0.579280\pi\)
−0.246498 + 0.969143i \(0.579280\pi\)
\(198\) 0 0
\(199\) 3737.46 1.33137 0.665683 0.746235i \(-0.268140\pi\)
0.665683 + 0.746235i \(0.268140\pi\)
\(200\) 0 0
\(201\) 167.060 0.0586246
\(202\) 0 0
\(203\) −1020.81 −0.352940
\(204\) 0 0
\(205\) 1426.60 0.486038
\(206\) 0 0
\(207\) 1965.15 0.659841
\(208\) 0 0
\(209\) 2462.50 0.814997
\(210\) 0 0
\(211\) 5266.12 1.71817 0.859087 0.511829i \(-0.171032\pi\)
0.859087 + 0.511829i \(0.171032\pi\)
\(212\) 0 0
\(213\) −2487.11 −0.800066
\(214\) 0 0
\(215\) 1069.11 0.339128
\(216\) 0 0
\(217\) −2171.40 −0.679283
\(218\) 0 0
\(219\) −933.075 −0.287906
\(220\) 0 0
\(221\) 987.899 0.300694
\(222\) 0 0
\(223\) 704.546 0.211569 0.105785 0.994389i \(-0.466265\pi\)
0.105785 + 0.994389i \(0.466265\pi\)
\(224\) 0 0
\(225\) 1974.34 0.584990
\(226\) 0 0
\(227\) 2151.26 0.629006 0.314503 0.949256i \(-0.398162\pi\)
0.314503 + 0.949256i \(0.398162\pi\)
\(228\) 0 0
\(229\) 3916.94 1.13030 0.565149 0.824989i \(-0.308819\pi\)
0.565149 + 0.824989i \(0.308819\pi\)
\(230\) 0 0
\(231\) −691.764 −0.197034
\(232\) 0 0
\(233\) −5192.74 −1.46003 −0.730017 0.683429i \(-0.760488\pi\)
−0.730017 + 0.683429i \(0.760488\pi\)
\(234\) 0 0
\(235\) −462.632 −0.128420
\(236\) 0 0
\(237\) 2272.36 0.622809
\(238\) 0 0
\(239\) 334.305 0.0904786 0.0452393 0.998976i \(-0.485595\pi\)
0.0452393 + 0.998976i \(0.485595\pi\)
\(240\) 0 0
\(241\) −1918.45 −0.512773 −0.256386 0.966574i \(-0.582532\pi\)
−0.256386 + 0.966574i \(0.582532\pi\)
\(242\) 0 0
\(243\) −3821.06 −1.00873
\(244\) 0 0
\(245\) 848.722 0.221318
\(246\) 0 0
\(247\) 5181.75 1.33485
\(248\) 0 0
\(249\) −366.097 −0.0931746
\(250\) 0 0
\(251\) −7695.71 −1.93525 −0.967627 0.252385i \(-0.918785\pi\)
−0.967627 + 0.252385i \(0.918785\pi\)
\(252\) 0 0
\(253\) 3183.29 0.791035
\(254\) 0 0
\(255\) −162.587 −0.0399277
\(256\) 0 0
\(257\) 5335.10 1.29492 0.647460 0.762099i \(-0.275831\pi\)
0.647460 + 0.762099i \(0.275831\pi\)
\(258\) 0 0
\(259\) −1050.93 −0.252130
\(260\) 0 0
\(261\) −2191.69 −0.519778
\(262\) 0 0
\(263\) 3934.15 0.922396 0.461198 0.887297i \(-0.347420\pi\)
0.461198 + 0.887297i \(0.347420\pi\)
\(264\) 0 0
\(265\) 1599.57 0.370795
\(266\) 0 0
\(267\) 2565.75 0.588095
\(268\) 0 0
\(269\) −3424.04 −0.776088 −0.388044 0.921641i \(-0.626849\pi\)
−0.388044 + 0.921641i \(0.626849\pi\)
\(270\) 0 0
\(271\) 549.034 0.123068 0.0615340 0.998105i \(-0.480401\pi\)
0.0615340 + 0.998105i \(0.480401\pi\)
\(272\) 0 0
\(273\) −1455.66 −0.322712
\(274\) 0 0
\(275\) 3198.19 0.701302
\(276\) 0 0
\(277\) −5203.65 −1.12873 −0.564363 0.825527i \(-0.690878\pi\)
−0.564363 + 0.825527i \(0.690878\pi\)
\(278\) 0 0
\(279\) −4662.02 −1.00039
\(280\) 0 0
\(281\) −1986.73 −0.421774 −0.210887 0.977510i \(-0.567635\pi\)
−0.210887 + 0.977510i \(0.567635\pi\)
\(282\) 0 0
\(283\) −753.696 −0.158313 −0.0791565 0.996862i \(-0.525223\pi\)
−0.0791565 + 0.996862i \(0.525223\pi\)
\(284\) 0 0
\(285\) −852.803 −0.177248
\(286\) 0 0
\(287\) 3736.47 0.768490
\(288\) 0 0
\(289\) 289.000 0.0588235
\(290\) 0 0
\(291\) −2506.96 −0.505020
\(292\) 0 0
\(293\) 7202.22 1.43603 0.718017 0.696025i \(-0.245050\pi\)
0.718017 + 0.696025i \(0.245050\pi\)
\(294\) 0 0
\(295\) −885.318 −0.174729
\(296\) 0 0
\(297\) −3837.43 −0.749731
\(298\) 0 0
\(299\) 6698.50 1.29560
\(300\) 0 0
\(301\) 2800.15 0.536205
\(302\) 0 0
\(303\) 837.595 0.158807
\(304\) 0 0
\(305\) −163.398 −0.0306758
\(306\) 0 0
\(307\) −2425.71 −0.450953 −0.225477 0.974249i \(-0.572394\pi\)
−0.225477 + 0.974249i \(0.572394\pi\)
\(308\) 0 0
\(309\) −1650.21 −0.303809
\(310\) 0 0
\(311\) −9544.94 −1.74033 −0.870167 0.492757i \(-0.835989\pi\)
−0.870167 + 0.492757i \(0.835989\pi\)
\(312\) 0 0
\(313\) 588.379 0.106253 0.0531264 0.998588i \(-0.483081\pi\)
0.0531264 + 0.998588i \(0.483081\pi\)
\(314\) 0 0
\(315\) −410.407 −0.0734090
\(316\) 0 0
\(317\) −7653.31 −1.35600 −0.678001 0.735061i \(-0.737154\pi\)
−0.678001 + 0.735061i \(0.737154\pi\)
\(318\) 0 0
\(319\) −3550.26 −0.623124
\(320\) 0 0
\(321\) −3110.59 −0.540860
\(322\) 0 0
\(323\) 1515.87 0.261131
\(324\) 0 0
\(325\) 6729.85 1.14863
\(326\) 0 0
\(327\) 5723.73 0.967960
\(328\) 0 0
\(329\) −1211.70 −0.203050
\(330\) 0 0
\(331\) −752.266 −0.124919 −0.0624597 0.998047i \(-0.519894\pi\)
−0.0624597 + 0.998047i \(0.519894\pi\)
\(332\) 0 0
\(333\) −2256.36 −0.371314
\(334\) 0 0
\(335\) 160.551 0.0261845
\(336\) 0 0
\(337\) −1968.57 −0.318204 −0.159102 0.987262i \(-0.550860\pi\)
−0.159102 + 0.987262i \(0.550860\pi\)
\(338\) 0 0
\(339\) 2231.39 0.357501
\(340\) 0 0
\(341\) −7551.89 −1.19929
\(342\) 0 0
\(343\) 4946.52 0.778678
\(344\) 0 0
\(345\) −1102.43 −0.172037
\(346\) 0 0
\(347\) −3983.10 −0.616207 −0.308104 0.951353i \(-0.599694\pi\)
−0.308104 + 0.951353i \(0.599694\pi\)
\(348\) 0 0
\(349\) −1495.61 −0.229393 −0.114697 0.993401i \(-0.536590\pi\)
−0.114697 + 0.993401i \(0.536590\pi\)
\(350\) 0 0
\(351\) −8074.98 −1.22795
\(352\) 0 0
\(353\) 6482.49 0.977417 0.488708 0.872447i \(-0.337468\pi\)
0.488708 + 0.872447i \(0.337468\pi\)
\(354\) 0 0
\(355\) −2390.20 −0.357348
\(356\) 0 0
\(357\) −425.838 −0.0631309
\(358\) 0 0
\(359\) −4943.42 −0.726751 −0.363376 0.931643i \(-0.618376\pi\)
−0.363376 + 0.931643i \(0.618376\pi\)
\(360\) 0 0
\(361\) 1092.08 0.159218
\(362\) 0 0
\(363\) 1792.94 0.259242
\(364\) 0 0
\(365\) −896.717 −0.128593
\(366\) 0 0
\(367\) −14.8871 −0.00211743 −0.00105872 0.999999i \(-0.500337\pi\)
−0.00105872 + 0.999999i \(0.500337\pi\)
\(368\) 0 0
\(369\) 8022.23 1.13176
\(370\) 0 0
\(371\) 4189.51 0.586276
\(372\) 0 0
\(373\) −1923.18 −0.266966 −0.133483 0.991051i \(-0.542616\pi\)
−0.133483 + 0.991051i \(0.542616\pi\)
\(374\) 0 0
\(375\) −2303.07 −0.317147
\(376\) 0 0
\(377\) −7470.70 −1.02059
\(378\) 0 0
\(379\) 9592.87 1.30014 0.650069 0.759875i \(-0.274740\pi\)
0.650069 + 0.759875i \(0.274740\pi\)
\(380\) 0 0
\(381\) −8354.04 −1.12333
\(382\) 0 0
\(383\) −9083.77 −1.21190 −0.605951 0.795502i \(-0.707207\pi\)
−0.605951 + 0.795502i \(0.707207\pi\)
\(384\) 0 0
\(385\) −664.809 −0.0880046
\(386\) 0 0
\(387\) 6011.95 0.789675
\(388\) 0 0
\(389\) 1143.78 0.149079 0.0745396 0.997218i \(-0.476251\pi\)
0.0745396 + 0.997218i \(0.476251\pi\)
\(390\) 0 0
\(391\) 1959.58 0.253453
\(392\) 0 0
\(393\) −6243.27 −0.801352
\(394\) 0 0
\(395\) 2183.81 0.278176
\(396\) 0 0
\(397\) −10604.5 −1.34061 −0.670307 0.742084i \(-0.733838\pi\)
−0.670307 + 0.742084i \(0.733838\pi\)
\(398\) 0 0
\(399\) −2233.62 −0.280252
\(400\) 0 0
\(401\) 13785.4 1.71674 0.858368 0.513035i \(-0.171479\pi\)
0.858368 + 0.513035i \(0.171479\pi\)
\(402\) 0 0
\(403\) −15891.2 −1.96426
\(404\) 0 0
\(405\) −66.5420 −0.00816419
\(406\) 0 0
\(407\) −3655.01 −0.445141
\(408\) 0 0
\(409\) −9505.94 −1.14924 −0.574619 0.818421i \(-0.694850\pi\)
−0.574619 + 0.818421i \(0.694850\pi\)
\(410\) 0 0
\(411\) −9911.59 −1.18954
\(412\) 0 0
\(413\) −2318.78 −0.276270
\(414\) 0 0
\(415\) −351.832 −0.0416162
\(416\) 0 0
\(417\) 4631.23 0.543866
\(418\) 0 0
\(419\) −9680.86 −1.12874 −0.564369 0.825523i \(-0.690880\pi\)
−0.564369 + 0.825523i \(0.690880\pi\)
\(420\) 0 0
\(421\) 12360.3 1.43089 0.715444 0.698671i \(-0.246225\pi\)
0.715444 + 0.698671i \(0.246225\pi\)
\(422\) 0 0
\(423\) −2601.54 −0.299033
\(424\) 0 0
\(425\) 1968.75 0.224702
\(426\) 0 0
\(427\) −427.962 −0.0485025
\(428\) 0 0
\(429\) −5062.61 −0.569756
\(430\) 0 0
\(431\) 2970.58 0.331990 0.165995 0.986127i \(-0.446916\pi\)
0.165995 + 0.986127i \(0.446916\pi\)
\(432\) 0 0
\(433\) 6131.50 0.680510 0.340255 0.940333i \(-0.389487\pi\)
0.340255 + 0.940333i \(0.389487\pi\)
\(434\) 0 0
\(435\) 1229.51 0.135519
\(436\) 0 0
\(437\) 10278.4 1.12513
\(438\) 0 0
\(439\) −2544.91 −0.276679 −0.138339 0.990385i \(-0.544176\pi\)
−0.138339 + 0.990385i \(0.544176\pi\)
\(440\) 0 0
\(441\) 4772.65 0.515349
\(442\) 0 0
\(443\) −8529.82 −0.914817 −0.457408 0.889257i \(-0.651222\pi\)
−0.457408 + 0.889257i \(0.651222\pi\)
\(444\) 0 0
\(445\) 2465.77 0.262672
\(446\) 0 0
\(447\) −902.308 −0.0954759
\(448\) 0 0
\(449\) 8855.74 0.930798 0.465399 0.885101i \(-0.345911\pi\)
0.465399 + 0.885101i \(0.345911\pi\)
\(450\) 0 0
\(451\) 12995.0 1.35679
\(452\) 0 0
\(453\) 2112.42 0.219095
\(454\) 0 0
\(455\) −1398.94 −0.144139
\(456\) 0 0
\(457\) −7154.78 −0.732356 −0.366178 0.930545i \(-0.619334\pi\)
−0.366178 + 0.930545i \(0.619334\pi\)
\(458\) 0 0
\(459\) −2362.25 −0.240219
\(460\) 0 0
\(461\) 7263.06 0.733784 0.366892 0.930264i \(-0.380422\pi\)
0.366892 + 0.930264i \(0.380422\pi\)
\(462\) 0 0
\(463\) 352.898 0.0354224 0.0177112 0.999843i \(-0.494362\pi\)
0.0177112 + 0.999843i \(0.494362\pi\)
\(464\) 0 0
\(465\) 2615.34 0.260825
\(466\) 0 0
\(467\) −1483.02 −0.146951 −0.0734753 0.997297i \(-0.523409\pi\)
−0.0734753 + 0.997297i \(0.523409\pi\)
\(468\) 0 0
\(469\) 420.506 0.0414012
\(470\) 0 0
\(471\) 2273.89 0.222452
\(472\) 0 0
\(473\) 9738.60 0.946683
\(474\) 0 0
\(475\) 10326.5 0.997502
\(476\) 0 0
\(477\) 8994.92 0.863415
\(478\) 0 0
\(479\) −9990.10 −0.952942 −0.476471 0.879190i \(-0.658084\pi\)
−0.476471 + 0.879190i \(0.658084\pi\)
\(480\) 0 0
\(481\) −7691.13 −0.729075
\(482\) 0 0
\(483\) −2887.42 −0.272012
\(484\) 0 0
\(485\) −2409.27 −0.225566
\(486\) 0 0
\(487\) −1129.88 −0.105133 −0.0525663 0.998617i \(-0.516740\pi\)
−0.0525663 + 0.998617i \(0.516740\pi\)
\(488\) 0 0
\(489\) −2134.42 −0.197386
\(490\) 0 0
\(491\) −18774.9 −1.72566 −0.862832 0.505491i \(-0.831311\pi\)
−0.862832 + 0.505491i \(0.831311\pi\)
\(492\) 0 0
\(493\) −2185.48 −0.199653
\(494\) 0 0
\(495\) −1427.35 −0.129605
\(496\) 0 0
\(497\) −6260.28 −0.565014
\(498\) 0 0
\(499\) −17329.1 −1.55462 −0.777310 0.629118i \(-0.783416\pi\)
−0.777310 + 0.629118i \(0.783416\pi\)
\(500\) 0 0
\(501\) 8945.50 0.797716
\(502\) 0 0
\(503\) −20837.0 −1.84707 −0.923533 0.383518i \(-0.874712\pi\)
−0.923533 + 0.383518i \(0.874712\pi\)
\(504\) 0 0
\(505\) 804.957 0.0709309
\(506\) 0 0
\(507\) −3722.37 −0.326068
\(508\) 0 0
\(509\) −11835.0 −1.03060 −0.515301 0.857009i \(-0.672320\pi\)
−0.515301 + 0.857009i \(0.672320\pi\)
\(510\) 0 0
\(511\) −2348.63 −0.203322
\(512\) 0 0
\(513\) −12390.6 −1.06639
\(514\) 0 0
\(515\) −1585.91 −0.135696
\(516\) 0 0
\(517\) −4214.16 −0.358489
\(518\) 0 0
\(519\) −559.359 −0.0473086
\(520\) 0 0
\(521\) 7686.37 0.646346 0.323173 0.946340i \(-0.395250\pi\)
0.323173 + 0.946340i \(0.395250\pi\)
\(522\) 0 0
\(523\) −11476.4 −0.959518 −0.479759 0.877400i \(-0.659276\pi\)
−0.479759 + 0.877400i \(0.659276\pi\)
\(524\) 0 0
\(525\) −2900.93 −0.241156
\(526\) 0 0
\(527\) −4648.81 −0.384261
\(528\) 0 0
\(529\) 1120.01 0.0920535
\(530\) 0 0
\(531\) −4978.44 −0.406866
\(532\) 0 0
\(533\) 27345.0 2.22222
\(534\) 0 0
\(535\) −2989.38 −0.241574
\(536\) 0 0
\(537\) −3229.59 −0.259529
\(538\) 0 0
\(539\) 7731.09 0.617814
\(540\) 0 0
\(541\) 546.481 0.0434289 0.0217145 0.999764i \(-0.493088\pi\)
0.0217145 + 0.999764i \(0.493088\pi\)
\(542\) 0 0
\(543\) −10884.1 −0.860191
\(544\) 0 0
\(545\) 5500.69 0.432337
\(546\) 0 0
\(547\) −8397.33 −0.656388 −0.328194 0.944610i \(-0.606440\pi\)
−0.328194 + 0.944610i \(0.606440\pi\)
\(548\) 0 0
\(549\) −918.839 −0.0714301
\(550\) 0 0
\(551\) −11463.3 −0.886305
\(552\) 0 0
\(553\) 5719.73 0.439833
\(554\) 0 0
\(555\) 1265.79 0.0968105
\(556\) 0 0
\(557\) 4881.65 0.371350 0.185675 0.982611i \(-0.440553\pi\)
0.185675 + 0.982611i \(0.440553\pi\)
\(558\) 0 0
\(559\) 20492.6 1.55053
\(560\) 0 0
\(561\) −1481.02 −0.111459
\(562\) 0 0
\(563\) −7198.57 −0.538870 −0.269435 0.963019i \(-0.586837\pi\)
−0.269435 + 0.963019i \(0.586837\pi\)
\(564\) 0 0
\(565\) 2144.44 0.159677
\(566\) 0 0
\(567\) −174.283 −0.0129087
\(568\) 0 0
\(569\) −23946.9 −1.76433 −0.882167 0.470937i \(-0.843916\pi\)
−0.882167 + 0.470937i \(0.843916\pi\)
\(570\) 0 0
\(571\) −1593.15 −0.116763 −0.0583813 0.998294i \(-0.518594\pi\)
−0.0583813 + 0.998294i \(0.518594\pi\)
\(572\) 0 0
\(573\) 1548.59 0.112903
\(574\) 0 0
\(575\) 13349.2 0.968174
\(576\) 0 0
\(577\) 12937.4 0.933435 0.466717 0.884406i \(-0.345436\pi\)
0.466717 + 0.884406i \(0.345436\pi\)
\(578\) 0 0
\(579\) 11195.2 0.803549
\(580\) 0 0
\(581\) −921.499 −0.0658007
\(582\) 0 0
\(583\) 14570.6 1.03508
\(584\) 0 0
\(585\) −3003.53 −0.212275
\(586\) 0 0
\(587\) 12899.2 0.906998 0.453499 0.891257i \(-0.350176\pi\)
0.453499 + 0.891257i \(0.350176\pi\)
\(588\) 0 0
\(589\) −24384.1 −1.70582
\(590\) 0 0
\(591\) 4300.23 0.299302
\(592\) 0 0
\(593\) 4357.13 0.301730 0.150865 0.988554i \(-0.451794\pi\)
0.150865 + 0.988554i \(0.451794\pi\)
\(594\) 0 0
\(595\) −409.245 −0.0281973
\(596\) 0 0
\(597\) −11790.3 −0.808284
\(598\) 0 0
\(599\) 13726.8 0.936328 0.468164 0.883642i \(-0.344916\pi\)
0.468164 + 0.883642i \(0.344916\pi\)
\(600\) 0 0
\(601\) 2531.41 0.171811 0.0859056 0.996303i \(-0.472622\pi\)
0.0859056 + 0.996303i \(0.472622\pi\)
\(602\) 0 0
\(603\) 902.830 0.0609720
\(604\) 0 0
\(605\) 1723.07 0.115790
\(606\) 0 0
\(607\) 185.004 0.0123708 0.00618540 0.999981i \(-0.498031\pi\)
0.00618540 + 0.999981i \(0.498031\pi\)
\(608\) 0 0
\(609\) 3220.28 0.214273
\(610\) 0 0
\(611\) −8867.73 −0.587152
\(612\) 0 0
\(613\) 17706.9 1.16668 0.583339 0.812228i \(-0.301746\pi\)
0.583339 + 0.812228i \(0.301746\pi\)
\(614\) 0 0
\(615\) −4500.39 −0.295078
\(616\) 0 0
\(617\) −6183.89 −0.403491 −0.201746 0.979438i \(-0.564661\pi\)
−0.201746 + 0.979438i \(0.564661\pi\)
\(618\) 0 0
\(619\) 1247.51 0.0810046 0.0405023 0.999179i \(-0.487104\pi\)
0.0405023 + 0.999179i \(0.487104\pi\)
\(620\) 0 0
\(621\) −16017.4 −1.03503
\(622\) 0 0
\(623\) 6458.23 0.415318
\(624\) 0 0
\(625\) 12262.8 0.784817
\(626\) 0 0
\(627\) −7768.27 −0.494792
\(628\) 0 0
\(629\) −2249.96 −0.142626
\(630\) 0 0
\(631\) 24053.3 1.51750 0.758752 0.651379i \(-0.225809\pi\)
0.758752 + 0.651379i \(0.225809\pi\)
\(632\) 0 0
\(633\) −16612.7 −1.04312
\(634\) 0 0
\(635\) −8028.51 −0.501735
\(636\) 0 0
\(637\) 16268.3 1.01189
\(638\) 0 0
\(639\) −13440.9 −0.832102
\(640\) 0 0
\(641\) −21286.8 −1.31167 −0.655834 0.754905i \(-0.727683\pi\)
−0.655834 + 0.754905i \(0.727683\pi\)
\(642\) 0 0
\(643\) 1789.41 0.109747 0.0548736 0.998493i \(-0.482524\pi\)
0.0548736 + 0.998493i \(0.482524\pi\)
\(644\) 0 0
\(645\) −3372.64 −0.205888
\(646\) 0 0
\(647\) −4378.61 −0.266060 −0.133030 0.991112i \(-0.542471\pi\)
−0.133030 + 0.991112i \(0.542471\pi\)
\(648\) 0 0
\(649\) −8064.45 −0.487762
\(650\) 0 0
\(651\) 6849.97 0.412399
\(652\) 0 0
\(653\) 7665.15 0.459358 0.229679 0.973266i \(-0.426232\pi\)
0.229679 + 0.973266i \(0.426232\pi\)
\(654\) 0 0
\(655\) −5999.99 −0.357922
\(656\) 0 0
\(657\) −5042.54 −0.299434
\(658\) 0 0
\(659\) 4710.22 0.278428 0.139214 0.990262i \(-0.455542\pi\)
0.139214 + 0.990262i \(0.455542\pi\)
\(660\) 0 0
\(661\) 31266.6 1.83983 0.919916 0.392116i \(-0.128257\pi\)
0.919916 + 0.392116i \(0.128257\pi\)
\(662\) 0 0
\(663\) −3116.46 −0.182554
\(664\) 0 0
\(665\) −2146.58 −0.125174
\(666\) 0 0
\(667\) −14818.7 −0.860246
\(668\) 0 0
\(669\) −2222.58 −0.128446
\(670\) 0 0
\(671\) −1488.40 −0.0856322
\(672\) 0 0
\(673\) 11723.0 0.671454 0.335727 0.941959i \(-0.391018\pi\)
0.335727 + 0.941959i \(0.391018\pi\)
\(674\) 0 0
\(675\) −16092.3 −0.917621
\(676\) 0 0
\(677\) 289.531 0.0164366 0.00821829 0.999966i \(-0.497384\pi\)
0.00821829 + 0.999966i \(0.497384\pi\)
\(678\) 0 0
\(679\) −6310.25 −0.356650
\(680\) 0 0
\(681\) −6786.45 −0.381875
\(682\) 0 0
\(683\) −1720.10 −0.0963660 −0.0481830 0.998839i \(-0.515343\pi\)
−0.0481830 + 0.998839i \(0.515343\pi\)
\(684\) 0 0
\(685\) −9525.37 −0.531308
\(686\) 0 0
\(687\) −12356.5 −0.686215
\(688\) 0 0
\(689\) 30660.5 1.69532
\(690\) 0 0
\(691\) 16777.7 0.923665 0.461832 0.886967i \(-0.347192\pi\)
0.461832 + 0.886967i \(0.347192\pi\)
\(692\) 0 0
\(693\) −3738.44 −0.204923
\(694\) 0 0
\(695\) 4450.77 0.242917
\(696\) 0 0
\(697\) 7999.50 0.434724
\(698\) 0 0
\(699\) 16381.2 0.886400
\(700\) 0 0
\(701\) −23981.1 −1.29209 −0.646043 0.763301i \(-0.723577\pi\)
−0.646043 + 0.763301i \(0.723577\pi\)
\(702\) 0 0
\(703\) −11801.6 −0.633150
\(704\) 0 0
\(705\) 1459.43 0.0779652
\(706\) 0 0
\(707\) 2108.30 0.112151
\(708\) 0 0
\(709\) 7709.28 0.408361 0.204181 0.978933i \(-0.434547\pi\)
0.204181 + 0.978933i \(0.434547\pi\)
\(710\) 0 0
\(711\) 12280.3 0.647747
\(712\) 0 0
\(713\) −31521.5 −1.65567
\(714\) 0 0
\(715\) −4865.34 −0.254480
\(716\) 0 0
\(717\) −1054.61 −0.0549304
\(718\) 0 0
\(719\) −11976.5 −0.621209 −0.310605 0.950539i \(-0.600532\pi\)
−0.310605 + 0.950539i \(0.600532\pi\)
\(720\) 0 0
\(721\) −4153.72 −0.214553
\(722\) 0 0
\(723\) 6052.00 0.311309
\(724\) 0 0
\(725\) −14888.1 −0.762662
\(726\) 0 0
\(727\) −18597.3 −0.948745 −0.474372 0.880324i \(-0.657325\pi\)
−0.474372 + 0.880324i \(0.657325\pi\)
\(728\) 0 0
\(729\) 11461.4 0.582300
\(730\) 0 0
\(731\) 5994.91 0.303324
\(732\) 0 0
\(733\) 23569.5 1.18767 0.593833 0.804588i \(-0.297614\pi\)
0.593833 + 0.804588i \(0.297614\pi\)
\(734\) 0 0
\(735\) −2677.41 −0.134364
\(736\) 0 0
\(737\) 1462.47 0.0730948
\(738\) 0 0
\(739\) −10149.1 −0.505199 −0.252599 0.967571i \(-0.581285\pi\)
−0.252599 + 0.967571i \(0.581285\pi\)
\(740\) 0 0
\(741\) −16346.5 −0.810397
\(742\) 0 0
\(743\) 27758.0 1.37058 0.685291 0.728269i \(-0.259675\pi\)
0.685291 + 0.728269i \(0.259675\pi\)
\(744\) 0 0
\(745\) −867.148 −0.0426441
\(746\) 0 0
\(747\) −1978.47 −0.0969054
\(748\) 0 0
\(749\) −7829.63 −0.381960
\(750\) 0 0
\(751\) −815.225 −0.0396112 −0.0198056 0.999804i \(-0.506305\pi\)
−0.0198056 + 0.999804i \(0.506305\pi\)
\(752\) 0 0
\(753\) 24277.1 1.17491
\(754\) 0 0
\(755\) 2030.11 0.0978585
\(756\) 0 0
\(757\) 13239.4 0.635659 0.317829 0.948148i \(-0.397046\pi\)
0.317829 + 0.948148i \(0.397046\pi\)
\(758\) 0 0
\(759\) −10042.1 −0.480244
\(760\) 0 0
\(761\) −11028.2 −0.525324 −0.262662 0.964888i \(-0.584600\pi\)
−0.262662 + 0.964888i \(0.584600\pi\)
\(762\) 0 0
\(763\) 14407.1 0.683582
\(764\) 0 0
\(765\) −878.652 −0.0415265
\(766\) 0 0
\(767\) −16969.8 −0.798882
\(768\) 0 0
\(769\) −18921.2 −0.887277 −0.443639 0.896206i \(-0.646313\pi\)
−0.443639 + 0.896206i \(0.646313\pi\)
\(770\) 0 0
\(771\) −16830.3 −0.786158
\(772\) 0 0
\(773\) −38728.6 −1.80203 −0.901016 0.433786i \(-0.857177\pi\)
−0.901016 + 0.433786i \(0.857177\pi\)
\(774\) 0 0
\(775\) −31669.0 −1.46785
\(776\) 0 0
\(777\) 3315.29 0.153070
\(778\) 0 0
\(779\) 41959.2 1.92984
\(780\) 0 0
\(781\) −21772.5 −0.997545
\(782\) 0 0
\(783\) 17863.9 0.815329
\(784\) 0 0
\(785\) 2185.28 0.0993579
\(786\) 0 0
\(787\) 20587.3 0.932477 0.466239 0.884659i \(-0.345609\pi\)
0.466239 + 0.884659i \(0.345609\pi\)
\(788\) 0 0
\(789\) −12410.8 −0.559995
\(790\) 0 0
\(791\) 5616.62 0.252470
\(792\) 0 0
\(793\) −3132.00 −0.140253
\(794\) 0 0
\(795\) −5046.05 −0.225113
\(796\) 0 0
\(797\) 15871.4 0.705385 0.352693 0.935739i \(-0.385266\pi\)
0.352693 + 0.935739i \(0.385266\pi\)
\(798\) 0 0
\(799\) −2594.17 −0.114862
\(800\) 0 0
\(801\) 13865.9 0.611644
\(802\) 0 0
\(803\) −8168.28 −0.358969
\(804\) 0 0
\(805\) −2774.90 −0.121494
\(806\) 0 0
\(807\) 10801.6 0.471170
\(808\) 0 0
\(809\) 39667.1 1.72388 0.861942 0.507007i \(-0.169248\pi\)
0.861942 + 0.507007i \(0.169248\pi\)
\(810\) 0 0
\(811\) −8003.87 −0.346552 −0.173276 0.984873i \(-0.555435\pi\)
−0.173276 + 0.984873i \(0.555435\pi\)
\(812\) 0 0
\(813\) −1732.00 −0.0747157
\(814\) 0 0
\(815\) −2051.25 −0.0881621
\(816\) 0 0
\(817\) 31444.7 1.34652
\(818\) 0 0
\(819\) −7866.69 −0.335634
\(820\) 0 0
\(821\) 13279.1 0.564489 0.282244 0.959343i \(-0.408921\pi\)
0.282244 + 0.959343i \(0.408921\pi\)
\(822\) 0 0
\(823\) 28934.0 1.22549 0.612745 0.790281i \(-0.290065\pi\)
0.612745 + 0.790281i \(0.290065\pi\)
\(824\) 0 0
\(825\) −10089.1 −0.425767
\(826\) 0 0
\(827\) 13679.6 0.575193 0.287597 0.957752i \(-0.407144\pi\)
0.287597 + 0.957752i \(0.407144\pi\)
\(828\) 0 0
\(829\) −16514.5 −0.691886 −0.345943 0.938256i \(-0.612441\pi\)
−0.345943 + 0.938256i \(0.612441\pi\)
\(830\) 0 0
\(831\) 16415.6 0.685260
\(832\) 0 0
\(833\) 4759.13 0.197952
\(834\) 0 0
\(835\) 8596.93 0.356298
\(836\) 0 0
\(837\) 37998.9 1.56922
\(838\) 0 0
\(839\) −87.9839 −0.00362043 −0.00181022 0.999998i \(-0.500576\pi\)
−0.00181022 + 0.999998i \(0.500576\pi\)
\(840\) 0 0
\(841\) −7861.94 −0.322356
\(842\) 0 0
\(843\) 6267.41 0.256063
\(844\) 0 0
\(845\) −3577.33 −0.145638
\(846\) 0 0
\(847\) 4512.98 0.183079
\(848\) 0 0
\(849\) 2377.63 0.0961133
\(850\) 0 0
\(851\) −15256.0 −0.614534
\(852\) 0 0
\(853\) −8162.96 −0.327660 −0.163830 0.986489i \(-0.552385\pi\)
−0.163830 + 0.986489i \(0.552385\pi\)
\(854\) 0 0
\(855\) −4608.73 −0.184345
\(856\) 0 0
\(857\) 18724.9 0.746361 0.373181 0.927759i \(-0.378267\pi\)
0.373181 + 0.927759i \(0.378267\pi\)
\(858\) 0 0
\(859\) 46422.5 1.84391 0.921953 0.387301i \(-0.126593\pi\)
0.921953 + 0.387301i \(0.126593\pi\)
\(860\) 0 0
\(861\) −11787.2 −0.466557
\(862\) 0 0
\(863\) 29112.3 1.14831 0.574157 0.818746i \(-0.305330\pi\)
0.574157 + 0.818746i \(0.305330\pi\)
\(864\) 0 0
\(865\) −537.563 −0.0211303
\(866\) 0 0
\(867\) −911.688 −0.0357123
\(868\) 0 0
\(869\) 19892.6 0.776536
\(870\) 0 0
\(871\) 3077.43 0.119719
\(872\) 0 0
\(873\) −13548.2 −0.525241
\(874\) 0 0
\(875\) −5797.04 −0.223972
\(876\) 0 0
\(877\) −39163.0 −1.50791 −0.753957 0.656924i \(-0.771857\pi\)
−0.753957 + 0.656924i \(0.771857\pi\)
\(878\) 0 0
\(879\) −22720.3 −0.871830
\(880\) 0 0
\(881\) −35073.2 −1.34125 −0.670627 0.741795i \(-0.733975\pi\)
−0.670627 + 0.741795i \(0.733975\pi\)
\(882\) 0 0
\(883\) 48775.7 1.85893 0.929463 0.368915i \(-0.120271\pi\)
0.929463 + 0.368915i \(0.120271\pi\)
\(884\) 0 0
\(885\) 2792.85 0.106080
\(886\) 0 0
\(887\) 13296.0 0.503309 0.251654 0.967817i \(-0.419025\pi\)
0.251654 + 0.967817i \(0.419025\pi\)
\(888\) 0 0
\(889\) −21027.9 −0.793309
\(890\) 0 0
\(891\) −606.137 −0.0227905
\(892\) 0 0
\(893\) −13607.0 −0.509899
\(894\) 0 0
\(895\) −3103.74 −0.115918
\(896\) 0 0
\(897\) −21131.3 −0.786570
\(898\) 0 0
\(899\) 35155.3 1.30422
\(900\) 0 0
\(901\) 8969.43 0.331648
\(902\) 0 0
\(903\) −8833.44 −0.325535
\(904\) 0 0
\(905\) −10460.0 −0.384202
\(906\) 0 0
\(907\) 11675.0 0.427410 0.213705 0.976898i \(-0.431447\pi\)
0.213705 + 0.976898i \(0.431447\pi\)
\(908\) 0 0
\(909\) 4526.54 0.165166
\(910\) 0 0
\(911\) 18552.9 0.674738 0.337369 0.941372i \(-0.390463\pi\)
0.337369 + 0.941372i \(0.390463\pi\)
\(912\) 0 0
\(913\) −3204.87 −0.116173
\(914\) 0 0
\(915\) 515.459 0.0186235
\(916\) 0 0
\(917\) −15714.9 −0.565922
\(918\) 0 0
\(919\) 33956.8 1.21886 0.609429 0.792841i \(-0.291399\pi\)
0.609429 + 0.792841i \(0.291399\pi\)
\(920\) 0 0
\(921\) 7652.23 0.273778
\(922\) 0 0
\(923\) −45815.3 −1.63383
\(924\) 0 0
\(925\) −15327.4 −0.544823
\(926\) 0 0
\(927\) −8918.08 −0.315974
\(928\) 0 0
\(929\) −23695.3 −0.836832 −0.418416 0.908256i \(-0.637415\pi\)
−0.418416 + 0.908256i \(0.637415\pi\)
\(930\) 0 0
\(931\) 24962.7 0.878753
\(932\) 0 0
\(933\) 30110.8 1.05657
\(934\) 0 0
\(935\) −1423.31 −0.0497830
\(936\) 0 0
\(937\) 7990.62 0.278593 0.139297 0.990251i \(-0.455516\pi\)
0.139297 + 0.990251i \(0.455516\pi\)
\(938\) 0 0
\(939\) −1856.12 −0.0645071
\(940\) 0 0
\(941\) −24385.9 −0.844799 −0.422400 0.906410i \(-0.638812\pi\)
−0.422400 + 0.906410i \(0.638812\pi\)
\(942\) 0 0
\(943\) 54241.0 1.87310
\(944\) 0 0
\(945\) 3345.12 0.115150
\(946\) 0 0
\(947\) 1174.62 0.0403064 0.0201532 0.999797i \(-0.493585\pi\)
0.0201532 + 0.999797i \(0.493585\pi\)
\(948\) 0 0
\(949\) −17188.3 −0.587939
\(950\) 0 0
\(951\) 24143.4 0.823241
\(952\) 0 0
\(953\) −33546.9 −1.14029 −0.570143 0.821546i \(-0.693112\pi\)
−0.570143 + 0.821546i \(0.693112\pi\)
\(954\) 0 0
\(955\) 1488.25 0.0504277
\(956\) 0 0
\(957\) 11199.8 0.378304
\(958\) 0 0
\(959\) −24948.4 −0.840067
\(960\) 0 0
\(961\) 44989.1 1.51016
\(962\) 0 0
\(963\) −16810.3 −0.562517
\(964\) 0 0
\(965\) 10758.9 0.358903
\(966\) 0 0
\(967\) 24766.8 0.823625 0.411813 0.911269i \(-0.364896\pi\)
0.411813 + 0.911269i \(0.364896\pi\)
\(968\) 0 0
\(969\) −4782.01 −0.158535
\(970\) 0 0
\(971\) −42324.3 −1.39882 −0.699409 0.714721i \(-0.746553\pi\)
−0.699409 + 0.714721i \(0.746553\pi\)
\(972\) 0 0
\(973\) 11657.2 0.384083
\(974\) 0 0
\(975\) −21230.2 −0.697344
\(976\) 0 0
\(977\) −11320.4 −0.370698 −0.185349 0.982673i \(-0.559342\pi\)
−0.185349 + 0.982673i \(0.559342\pi\)
\(978\) 0 0
\(979\) 22461.0 0.733254
\(980\) 0 0
\(981\) 30932.2 1.00672
\(982\) 0 0
\(983\) −11311.9 −0.367032 −0.183516 0.983017i \(-0.558748\pi\)
−0.183516 + 0.983017i \(0.558748\pi\)
\(984\) 0 0
\(985\) 4132.66 0.133683
\(986\) 0 0
\(987\) 3822.47 0.123273
\(988\) 0 0
\(989\) 40648.8 1.30693
\(990\) 0 0
\(991\) −29405.5 −0.942580 −0.471290 0.881978i \(-0.656212\pi\)
−0.471290 + 0.881978i \(0.656212\pi\)
\(992\) 0 0
\(993\) 2373.12 0.0758397
\(994\) 0 0
\(995\) −11330.9 −0.361018
\(996\) 0 0
\(997\) 54905.9 1.74412 0.872060 0.489398i \(-0.162784\pi\)
0.872060 + 0.489398i \(0.162784\pi\)
\(998\) 0 0
\(999\) 18390.9 0.582446
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 1088.4.a.v.1.2 3
4.3 odd 2 1088.4.a.x.1.2 3
8.3 odd 2 272.4.a.h.1.2 3
8.5 even 2 17.4.a.b.1.2 3
24.5 odd 2 153.4.a.g.1.2 3
24.11 even 2 2448.4.a.bi.1.1 3
40.13 odd 4 425.4.b.f.324.3 6
40.29 even 2 425.4.a.g.1.2 3
40.37 odd 4 425.4.b.f.324.4 6
56.13 odd 2 833.4.a.d.1.2 3
88.21 odd 2 2057.4.a.e.1.2 3
136.13 even 4 289.4.b.b.288.4 6
136.21 even 4 289.4.b.b.288.3 6
136.101 even 2 289.4.a.b.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
17.4.a.b.1.2 3 8.5 even 2
153.4.a.g.1.2 3 24.5 odd 2
272.4.a.h.1.2 3 8.3 odd 2
289.4.a.b.1.2 3 136.101 even 2
289.4.b.b.288.3 6 136.21 even 4
289.4.b.b.288.4 6 136.13 even 4
425.4.a.g.1.2 3 40.29 even 2
425.4.b.f.324.3 6 40.13 odd 4
425.4.b.f.324.4 6 40.37 odd 4
833.4.a.d.1.2 3 56.13 odd 2
1088.4.a.v.1.2 3 1.1 even 1 trivial
1088.4.a.x.1.2 3 4.3 odd 2
2057.4.a.e.1.2 3 88.21 odd 2
2448.4.a.bi.1.1 3 24.11 even 2