Properties

Label 272.4.a.h.1.2
Level $272$
Weight $4$
Character 272.1
Self dual yes
Analytic conductor $16.049$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 272 = 2^{4} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 272.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(16.0485195216\)
Analytic rank: \(1\)
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 \(\chi\) \(=\) 272.1

$q$-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 \(-0.456710\pi\)
0.135582 + 0.990766i \(0.456710\pi\)
\(6\) 0 0
\(7\) 7.94049 0.428746 0.214373 0.976752i \(-0.431229\pi\)
0.214373 + 0.976752i \(0.431229\pi\)
\(8\) 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 \(-0.287175\pi\)
0.619896 + 0.784684i \(0.287175\pi\)
\(14\) 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 \(-0.324997\pi\)
0.522507 + 0.852635i \(0.324997\pi\)
\(24\) 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 \(-0.635028\pi\)
−0.411596 + 0.911367i \(0.635028\pi\)
\(30\) 0 0
\(31\) −273.460 −1.58435 −0.792174 0.610295i \(-0.791051\pi\)
−0.792174 + 0.610295i \(0.791051\pi\)
\(32\) 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 \(-0.594997\pi\)
−0.294031 + 0.955796i \(0.594997\pi\)
\(38\) 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 \(-0.576097\pi\)
−0.236795 + 0.971560i \(0.576097\pi\)
\(48\) 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 \(-0.260365\pi\)
0.683711 + 0.729753i \(0.260365\pi\)
\(54\) 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 \(-0.518014\pi\)
−0.0565632 + 0.998399i \(0.518014\pi\)
\(62\) 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 \(-0.728984\pi\)
−0.658915 + 0.752218i \(0.728984\pi\)
\(72\) 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 \(-0.328560\pi\)
0.512930 + 0.858430i \(0.328560\pi\)
\(80\) 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 \(-0.458249\pi\)
0.130790 + 0.991410i \(0.458249\pi\)
\(102\) 0 0
\(103\) −523.107 −0.500420 −0.250210 0.968192i \(-0.580500\pi\)
−0.250210 + 0.968192i \(0.580500\pi\)
\(104\) 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 \(-0.206320\pi\)
0.797188 + 0.603732i \(0.206320\pi\)
\(110\) 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 \(-0.876062\pi\)
−0.925151 + 0.379600i \(0.876062\pi\)
\(128\) 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 \(-0.525055\pi\)
−0.0786316 + 0.996904i \(0.525055\pi\)
\(150\) 0 0
\(151\) 669.626 0.360883 0.180442 0.983586i \(-0.442247\pi\)
0.180442 + 0.983586i \(0.442247\pi\)
\(152\) 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 \(-0.441352\pi\)
0.183206 + 0.983074i \(0.441352\pi\)
\(158\) 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 \(-0.271834\pi\)
0.656979 + 0.753909i \(0.271834\pi\)
\(168\) 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 \(-0.512405\pi\)
−0.0389621 + 0.999241i \(0.512405\pi\)
\(174\) 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 \(-0.750597\pi\)
−0.708432 + 0.705779i \(0.750597\pi\)
\(182\) 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 \(-0.470360\pi\)
0.0929839 + 0.995668i \(0.470360\pi\)
\(192\) 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 \(-0.420720\pi\)
0.246498 + 0.969143i \(0.420720\pi\)
\(198\) 0 0
\(199\) −3737.46 −1.33137 −0.665683 0.746235i \(-0.731860\pi\)
−0.665683 + 0.746235i \(0.731860\pi\)
\(200\) 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 \(-0.533735\pi\)
−0.105785 + 0.994389i \(0.533735\pi\)
\(224\) 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 \(-0.691181\pi\)
−0.565149 + 0.824989i \(0.691181\pi\)
\(230\) 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 \(-0.514405\pi\)
−0.0452393 + 0.998976i \(0.514405\pi\)
\(240\) 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 \(-0.652580\pi\)
−0.461198 + 0.887297i \(0.652580\pi\)
\(264\) 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 \(-0.373151\pi\)
0.388044 + 0.921641i \(0.373151\pi\)
\(270\) 0 0
\(271\) −549.034 −0.123068 −0.0615340 0.998105i \(-0.519599\pi\)
−0.0615340 + 0.998105i \(0.519599\pi\)
\(272\) 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 \(-0.309122\pi\)
0.564363 + 0.825527i \(0.309122\pi\)
\(278\) 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 \(-0.754950\pi\)
−0.718017 + 0.696025i \(0.754950\pi\)
\(294\) 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 \(-0.164011\pi\)
0.870167 + 0.492757i \(0.164011\pi\)
\(312\) 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 \(-0.262846\pi\)
0.678001 + 0.735061i \(0.262846\pi\)
\(318\) 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 \(-0.463410\pi\)
0.114697 + 0.993401i \(0.463410\pi\)
\(350\) 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 \(-0.381624\pi\)
0.363376 + 0.931643i \(0.381624\pi\)
\(360\) 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 \(-0.499663\pi\)
0.00105872 + 0.999999i \(0.499663\pi\)
\(368\) 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 \(-0.457384\pi\)
0.133483 + 0.991051i \(0.457384\pi\)
\(374\) 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 \(-0.292793\pi\)
0.605951 + 0.795502i \(0.292793\pi\)
\(384\) 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 \(-0.523749\pi\)
−0.0745396 + 0.997218i \(0.523749\pi\)
\(390\) 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 \(-0.266162\pi\)
0.670307 + 0.742084i \(0.266162\pi\)
\(398\) 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 \(-0.753775\pi\)
−0.715444 + 0.698671i \(0.753775\pi\)
\(422\) 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 \(-0.553084\pi\)
−0.165995 + 0.986127i \(0.553084\pi\)
\(432\) 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 \(-0.455824\pi\)
0.138339 + 0.990385i \(0.455824\pi\)
\(440\) 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 \(-0.619578\pi\)
−0.366892 + 0.930264i \(0.619578\pi\)
\(462\) 0 0
\(463\) −352.898 −0.0354224 −0.0177112 0.999843i \(-0.505638\pi\)
−0.0177112 + 0.999843i \(0.505638\pi\)
\(464\) 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 \(-0.341916\pi\)
0.476471 + 0.879190i \(0.341916\pi\)
\(480\) 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 \(-0.483260\pi\)
0.0525663 + 0.998617i \(0.483260\pi\)
\(488\) 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 \(-0.125288\pi\)
0.923533 + 0.383518i \(0.125288\pi\)
\(504\) 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 \(-0.327680\pi\)
0.515301 + 0.857009i \(0.327680\pi\)
\(510\) 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 \(-0.506912\pi\)
−0.0217145 + 0.999764i \(0.506912\pi\)
\(542\) 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 \(-0.559447\pi\)
−0.185675 + 0.982611i \(0.559447\pi\)
\(558\) 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 \(-0.655084\pi\)
−0.468164 + 0.883642i \(0.655084\pi\)
\(600\) 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 \(-0.501969\pi\)
−0.00618540 + 0.999981i \(0.501969\pi\)
\(608\) 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 \(-0.698254\pi\)
−0.583339 + 0.812228i \(0.698254\pi\)
\(614\) 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 \(-0.774191\pi\)
−0.758752 + 0.651379i \(0.774191\pi\)
\(632\) 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 \(-0.457529\pi\)
0.133030 + 0.991112i \(0.457529\pi\)
\(648\) 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 \(-0.573768\pi\)
−0.229679 + 0.973266i \(0.573768\pi\)
\(654\) 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 \(-0.871743\pi\)
−0.919916 + 0.392116i \(0.871743\pi\)
\(662\) 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 \(-0.502616\pi\)
−0.00821829 + 0.999966i \(0.502616\pi\)
\(678\) 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 \(-0.276423\pi\)
0.646043 + 0.763301i \(0.276423\pi\)
\(702\) 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 \(-0.565453\pi\)
−0.204181 + 0.978933i \(0.565453\pi\)
\(710\) 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 \(-0.399468\pi\)
0.310605 + 0.950539i \(0.399468\pi\)
\(720\) 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 \(-0.342675\pi\)
0.474372 + 0.880324i \(0.342675\pi\)
\(728\) 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 \(-0.702386\pi\)
−0.593833 + 0.804588i \(0.702386\pi\)
\(734\) 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 \(-0.740325\pi\)
−0.685291 + 0.728269i \(0.740325\pi\)
\(744\) 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 \(-0.493695\pi\)
0.0198056 + 0.999804i \(0.493695\pi\)
\(752\) 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 \(-0.602954\pi\)
−0.317829 + 0.948148i \(0.602954\pi\)
\(758\) 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 \(-0.142823\pi\)
0.901016 + 0.433786i \(0.142823\pi\)
\(774\) 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 \(-0.614734\pi\)
−0.352693 + 0.935739i \(0.614734\pi\)
\(798\) 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 \(-0.591079\pi\)
−0.282244 + 0.959343i \(0.591079\pi\)
\(822\) 0 0
\(823\) −28934.0 −1.22549 −0.612745 0.790281i \(-0.709935\pi\)
−0.612745 + 0.790281i \(0.709935\pi\)
\(824\) 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 \(-0.387559\pi\)
0.345943 + 0.938256i \(0.387559\pi\)
\(830\) 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 \(-0.499424\pi\)
0.00181022 + 0.999998i \(0.499424\pi\)
\(840\) 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 \(-0.447615\pi\)
0.163830 + 0.986489i \(0.447615\pi\)
\(854\) 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 \(-0.694670\pi\)
−0.574157 + 0.818746i \(0.694670\pi\)
\(864\) 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 \(-0.228143\pi\)
0.753957 + 0.656924i \(0.228143\pi\)
\(878\) 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 \(-0.580975\pi\)
−0.251654 + 0.967817i \(0.580975\pi\)
\(888\) 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 \(-0.609537\pi\)
−0.337369 + 0.941372i \(0.609537\pi\)
\(912\) 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 \(-0.708601\pi\)
−0.609429 + 0.792841i \(0.708601\pi\)
\(920\) 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 \(-0.361188\pi\)
0.422400 + 0.906410i \(0.361188\pi\)
\(942\) 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 \(-0.635104\pi\)
−0.411813 + 0.911269i \(0.635104\pi\)
\(968\) 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 \(-0.441252\pi\)
0.183516 + 0.983017i \(0.441252\pi\)
\(984\) 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 \(-0.343788\pi\)
0.471290 + 0.881978i \(0.343788\pi\)
\(992\) 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 \(-0.837216\pi\)
−0.872060 + 0.489398i \(0.837216\pi\)
\(998\) 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 272.4.a.h.1.2 3
3.2 odd 2 2448.4.a.bi.1.1 3
4.3 odd 2 17.4.a.b.1.2 3
8.3 odd 2 1088.4.a.v.1.2 3
8.5 even 2 1088.4.a.x.1.2 3
12.11 even 2 153.4.a.g.1.2 3
20.3 even 4 425.4.b.f.324.3 6
20.7 even 4 425.4.b.f.324.4 6
20.19 odd 2 425.4.a.g.1.2 3
28.27 even 2 833.4.a.d.1.2 3
44.43 even 2 2057.4.a.e.1.2 3
68.47 odd 4 289.4.b.b.288.4 6
68.55 odd 4 289.4.b.b.288.3 6
68.67 odd 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 4.3 odd 2
153.4.a.g.1.2 3 12.11 even 2
272.4.a.h.1.2 3 1.1 even 1 trivial
289.4.a.b.1.2 3 68.67 odd 2
289.4.b.b.288.3 6 68.55 odd 4
289.4.b.b.288.4 6 68.47 odd 4
425.4.a.g.1.2 3 20.19 odd 2
425.4.b.f.324.3 6 20.3 even 4
425.4.b.f.324.4 6 20.7 even 4
833.4.a.d.1.2 3 28.27 even 2
1088.4.a.v.1.2 3 8.3 odd 2
1088.4.a.x.1.2 3 8.5 even 2
2057.4.a.e.1.2 3 44.43 even 2
2448.4.a.bi.1.1 3 3.2 odd 2