Properties

Label 153.4.a.g.1.2
Level $153$
Weight $4$
Character 153.1
Self dual yes
Analytic conductor $9.027$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 153 = 3^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 153.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(9.02729223088\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.2636.1
Defining polynomial: \( x^{3} - 14x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
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 \(\chi\) \(=\) 153.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.36122 q^{2} -6.14708 q^{4} -3.03171 q^{5} -7.94049 q^{7} +19.2573 q^{8} +O(q^{10})\) \(q-1.36122 q^{2} -6.14708 q^{4} -3.03171 q^{5} -7.94049 q^{7} +19.2573 q^{8} +4.12682 q^{10} -27.6161 q^{11} +58.1117 q^{13} +10.8088 q^{14} +22.9632 q^{16} +17.0000 q^{17} +89.1688 q^{19} +18.6361 q^{20} +37.5916 q^{22} +115.269 q^{23} -115.809 q^{25} -79.1029 q^{26} +48.8108 q^{28} +128.558 q^{29} +273.460 q^{31} -185.316 q^{32} -23.1408 q^{34} +24.0732 q^{35} -132.351 q^{37} -121.379 q^{38} -58.3825 q^{40} +470.559 q^{41} +352.642 q^{43} +169.758 q^{44} -156.907 q^{46} -152.598 q^{47} -279.949 q^{49} +157.641 q^{50} -357.217 q^{52} -527.614 q^{53} +83.7239 q^{55} -152.912 q^{56} -174.995 q^{58} +292.020 q^{59} -53.8962 q^{61} -372.239 q^{62} +68.5514 q^{64} -176.178 q^{65} +52.9572 q^{67} -104.500 q^{68} -32.7690 q^{70} -788.400 q^{71} +295.780 q^{73} +180.159 q^{74} -548.127 q^{76} +219.285 q^{77} -720.325 q^{79} -69.6175 q^{80} -640.535 q^{82} +116.051 q^{83} -51.5390 q^{85} -480.024 q^{86} -531.812 q^{88} +813.329 q^{89} -461.435 q^{91} -708.569 q^{92} +207.720 q^{94} -270.334 q^{95} +794.693 q^{97} +381.072 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{2} + 25 q^{4} + 8 q^{5} + 22 q^{7} + 39 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - q^{2} + 25 q^{4} + 8 q^{5} + 22 q^{7} + 39 q^{8} - 56 q^{10} + 28 q^{11} + 30 q^{13} - 92 q^{14} + 137 q^{16} + 51 q^{17} + 80 q^{19} + 168 q^{20} + 286 q^{22} - 142 q^{23} - 223 q^{25} - 26 q^{26} + 476 q^{28} + 456 q^{29} + 230 q^{31} + 71 q^{32} - 17 q^{34} + 332 q^{35} + 356 q^{37} - 724 q^{38} - 424 q^{40} + 294 q^{41} + 556 q^{43} + 1122 q^{44} - 704 q^{46} - 640 q^{47} - 269 q^{49} - 547 q^{50} - 774 q^{52} - 302 q^{53} + 76 q^{55} - 684 q^{56} - 1304 q^{58} - 636 q^{59} - 84 q^{61} - 508 q^{62} - 919 q^{64} - 408 q^{65} + 1008 q^{67} + 425 q^{68} - 1504 q^{70} + 402 q^{71} + 838 q^{73} - 836 q^{74} - 908 q^{76} + 504 q^{77} - 594 q^{79} + 40 q^{80} + 358 q^{82} + 2396 q^{83} + 136 q^{85} + 1264 q^{86} + 1838 q^{88} + 170 q^{89} - 1016 q^{91} - 4896 q^{92} - 2016 q^{94} + 472 q^{95} - 270 q^{97} - 2857 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.36122 −0.481264 −0.240632 0.970616i \(-0.577355\pi\)
−0.240632 + 0.970616i \(0.577355\pi\)
\(3\) 0 0
\(4\) −6.14708 −0.768385
\(5\) −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 \(0.568771\pi\)
\(8\) 19.2573 0.851061
\(9\) 0 0
\(10\) 4.12682 0.130502
\(11\) −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 \(0.287175\pi\)
\(14\) 10.8088 0.206340
\(15\) 0 0
\(16\) 22.9632 0.358799
\(17\) 17.0000 0.242536
\(18\) 0 0
\(19\) 89.1688 1.07667 0.538335 0.842731i \(-0.319054\pi\)
0.538335 + 0.842731i \(0.319054\pi\)
\(20\) 18.6361 0.208358
\(21\) 0 0
\(22\) 37.5916 0.364298
\(23\) 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
\(26\) −79.1029 −0.596668
\(27\) 0 0
\(28\) 48.8108 0.329442
\(29\) 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 \(0.208949\pi\)
\(32\) −185.316 −1.02374
\(33\) 0 0
\(34\) −23.1408 −0.116724
\(35\) 24.0732 0.116260
\(36\) 0 0
\(37\) −132.351 −0.588063 −0.294031 0.955796i \(-0.594997\pi\)
−0.294031 + 0.955796i \(0.594997\pi\)
\(38\) −121.379 −0.518163
\(39\) 0 0
\(40\) −58.3825 −0.230777
\(41\) 470.559 1.79241 0.896207 0.443636i \(-0.146312\pi\)
0.896207 + 0.443636i \(0.146312\pi\)
\(42\) 0 0
\(43\) 352.642 1.25064 0.625318 0.780370i \(-0.284969\pi\)
0.625318 + 0.780370i \(0.284969\pi\)
\(44\) 169.758 0.581637
\(45\) 0 0
\(46\) −156.907 −0.502928
\(47\) −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
\(50\) 157.641 0.445877
\(51\) 0 0
\(52\) −357.217 −0.952637
\(53\) −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
\(56\) −152.912 −0.364889
\(57\) 0 0
\(58\) −174.995 −0.396173
\(59\) 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 \(0.518014\pi\)
\(62\) −372.239 −0.762490
\(63\) 0 0
\(64\) 68.5514 0.133889
\(65\) −176.178 −0.336187
\(66\) 0 0
\(67\) 52.9572 0.0965635 0.0482817 0.998834i \(-0.484625\pi\)
0.0482817 + 0.998834i \(0.484625\pi\)
\(68\) −104.500 −0.186361
\(69\) 0 0
\(70\) −32.7690 −0.0559520
\(71\) −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 \(0.423799\pi\)
\(74\) 180.159 0.283014
\(75\) 0 0
\(76\) −548.127 −0.827296
\(77\) 219.285 0.324544
\(78\) 0 0
\(79\) −720.325 −1.02586 −0.512930 0.858430i \(-0.671440\pi\)
−0.512930 + 0.858430i \(0.671440\pi\)
\(80\) −69.6175 −0.0972934
\(81\) 0 0
\(82\) −640.535 −0.862625
\(83\) 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
\(86\) −480.024 −0.601887
\(87\) 0 0
\(88\) −531.812 −0.644220
\(89\) 813.329 0.968682 0.484341 0.874879i \(-0.339059\pi\)
0.484341 + 0.874879i \(0.339059\pi\)
\(90\) 0 0
\(91\) −461.435 −0.531556
\(92\) −708.569 −0.802972
\(93\) 0 0
\(94\) 207.720 0.227922
\(95\) −270.334 −0.291954
\(96\) 0 0
\(97\) 794.693 0.831844 0.415922 0.909400i \(-0.363459\pi\)
0.415922 + 0.909400i \(0.363459\pi\)
\(98\) 381.072 0.392797
\(99\) 0 0
\(100\) 711.885 0.711885
\(101\) −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 \(0.419500\pi\)
\(104\) 1119.07 1.05514
\(105\) 0 0
\(106\) 718.199 0.658091
\(107\) 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 \(0.206320\pi\)
\(110\) −113.967 −0.0987846
\(111\) 0 0
\(112\) −182.339 −0.153834
\(113\) 707.339 0.588857 0.294429 0.955673i \(-0.404871\pi\)
0.294429 + 0.955673i \(0.404871\pi\)
\(114\) 0 0
\(115\) −349.463 −0.283370
\(116\) −790.253 −0.632527
\(117\) 0 0
\(118\) −397.503 −0.310112
\(119\) −134.988 −0.103986
\(120\) 0 0
\(121\) −568.350 −0.427010
\(122\) 73.3647 0.0544437
\(123\) 0 0
\(124\) −1680.98 −1.21739
\(125\) 730.061 0.522389
\(126\) 0 0
\(127\) 2648.18 1.85030 0.925151 0.379600i \(-0.123938\pi\)
0.925151 + 0.379600i \(0.123938\pi\)
\(128\) 1389.22 0.959302
\(129\) 0 0
\(130\) 239.817 0.161795
\(131\) 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
\(134\) −72.0865 −0.0464726
\(135\) 0 0
\(136\) 327.374 0.206413
\(137\) −3141.92 −1.95936 −0.979679 0.200570i \(-0.935721\pi\)
−0.979679 + 0.200570i \(0.935721\pi\)
\(138\) 0 0
\(139\) 1468.07 0.895830 0.447915 0.894076i \(-0.352167\pi\)
0.447915 + 0.894076i \(0.352167\pi\)
\(140\) −147.980 −0.0893327
\(141\) 0 0
\(142\) 1073.19 0.634224
\(143\) −1604.82 −0.938474
\(144\) 0 0
\(145\) −389.749 −0.223220
\(146\) −402.621 −0.228227
\(147\) 0 0
\(148\) 813.570 0.451858
\(149\) 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 \(0.557753\pi\)
\(152\) 1717.15 0.916311
\(153\) 0 0
\(154\) −298.496 −0.156191
\(155\) −829.049 −0.429618
\(156\) 0 0
\(157\) 720.809 0.366413 0.183206 0.983074i \(-0.441352\pi\)
0.183206 + 0.983074i \(0.441352\pi\)
\(158\) 980.522 0.493710
\(159\) 0 0
\(160\) 561.825 0.277601
\(161\) −915.294 −0.448045
\(162\) 0 0
\(163\) −676.599 −0.325125 −0.162562 0.986698i \(-0.551976\pi\)
−0.162562 + 0.986698i \(0.551976\pi\)
\(164\) −2892.56 −1.37726
\(165\) 0 0
\(166\) −157.971 −0.0738609
\(167\) 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
\(170\) 70.1560 0.0316513
\(171\) 0 0
\(172\) −2167.72 −0.960970
\(173\) 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
\(176\) −634.153 −0.271597
\(177\) 0 0
\(178\) −1107.12 −0.466192
\(179\) 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 \(0.750597\pi\)
\(182\) 628.116 0.255819
\(183\) 0 0
\(184\) 2219.78 0.889370
\(185\) 401.248 0.159461
\(186\) 0 0
\(187\) −469.474 −0.183590
\(188\) 938.031 0.363899
\(189\) 0 0
\(190\) 367.984 0.140507
\(191\) 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 \(0.730200\pi\)
\(194\) −1081.75 −0.400337
\(195\) 0 0
\(196\) 1720.87 0.627138
\(197\) −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 \(0.268140\pi\)
\(200\) −2230.16 −0.788482
\(201\) 0 0
\(202\) 361.422 0.125889
\(203\) −1020.81 −0.352940
\(204\) 0 0
\(205\) −1426.60 −0.486038
\(206\) −712.064 −0.240834
\(207\) 0 0
\(208\) 1334.43 0.444836
\(209\) −2462.50 −0.814997
\(210\) 0 0
\(211\) −5266.12 −1.71817 −0.859087 0.511829i \(-0.828968\pi\)
−0.859087 + 0.511829i \(0.828968\pi\)
\(212\) 3243.28 1.05071
\(213\) 0 0
\(214\) −1342.22 −0.428748
\(215\) −1069.11 −0.339128
\(216\) 0 0
\(217\) −2171.40 −0.679283
\(218\) −2469.78 −0.767316
\(219\) 0 0
\(220\) −514.657 −0.157719
\(221\) 987.899 0.300694
\(222\) 0 0
\(223\) 704.546 0.211569 0.105785 0.994389i \(-0.466265\pi\)
0.105785 + 0.994389i \(0.466265\pi\)
\(224\) 1471.50 0.438923
\(225\) 0 0
\(226\) −962.845 −0.283396
\(227\) 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 \(0.691181\pi\)
\(230\) 475.696 0.136376
\(231\) 0 0
\(232\) 2475.67 0.700586
\(233\) 5192.74 1.46003 0.730017 0.683429i \(-0.239512\pi\)
0.730017 + 0.683429i \(0.239512\pi\)
\(234\) 0 0
\(235\) 462.632 0.128420
\(236\) −1795.07 −0.495123
\(237\) 0 0
\(238\) 183.749 0.0500448
\(239\) −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 \(0.582532\pi\)
\(242\) 773.651 0.205505
\(243\) 0 0
\(244\) 331.304 0.0869245
\(245\) 848.722 0.221318
\(246\) 0 0
\(247\) 5181.75 1.33485
\(248\) 5266.09 1.34838
\(249\) 0 0
\(250\) −993.775 −0.251407
\(251\) −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
\(254\) −3604.76 −0.890484
\(255\) 0 0
\(256\) −2439.44 −0.595567
\(257\) −5335.10 −1.29492 −0.647460 0.762099i \(-0.724169\pi\)
−0.647460 + 0.762099i \(0.724169\pi\)
\(258\) 0 0
\(259\) 1050.93 0.252130
\(260\) 1082.98 0.258321
\(261\) 0 0
\(262\) −2693.97 −0.635244
\(263\) −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
\(266\) 963.804 0.222160
\(267\) 0 0
\(268\) −325.532 −0.0741979
\(269\) −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 \(0.480401\pi\)
\(272\) 390.374 0.0870216
\(273\) 0 0
\(274\) 4276.85 0.942970
\(275\) 3198.19 0.701302
\(276\) 0 0
\(277\) 5203.65 1.12873 0.564363 0.825527i \(-0.309122\pi\)
0.564363 + 0.825527i \(0.309122\pi\)
\(278\) −1998.37 −0.431131
\(279\) 0 0
\(280\) 463.585 0.0989447
\(281\) 1986.73 0.421774 0.210887 0.977510i \(-0.432365\pi\)
0.210887 + 0.977510i \(0.432365\pi\)
\(282\) 0 0
\(283\) 753.696 0.158313 0.0791565 0.996862i \(-0.474777\pi\)
0.0791565 + 0.996862i \(0.474777\pi\)
\(284\) 4846.36 1.01260
\(285\) 0 0
\(286\) 2184.51 0.451654
\(287\) −3736.47 −0.768490
\(288\) 0 0
\(289\) 289.000 0.0588235
\(290\) 530.534 0.107428
\(291\) 0 0
\(292\) −1818.18 −0.364387
\(293\) 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
\(296\) −2548.72 −0.500477
\(297\) 0 0
\(298\) −389.345 −0.0756852
\(299\) 6698.50 1.29560
\(300\) 0 0
\(301\) −2800.15 −0.536205
\(302\) 911.509 0.173680
\(303\) 0 0
\(304\) 2047.60 0.386308
\(305\) 163.398 0.0306758
\(306\) 0 0
\(307\) 2425.71 0.450953 0.225477 0.974249i \(-0.427606\pi\)
0.225477 + 0.974249i \(0.427606\pi\)
\(308\) −1347.96 −0.249375
\(309\) 0 0
\(310\) 1128.52 0.206760
\(311\) 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 \(0.483081\pi\)
\(314\) −981.180 −0.176341
\(315\) 0 0
\(316\) 4427.89 0.788255
\(317\) −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
\(320\) −207.828 −0.0363060
\(321\) 0 0
\(322\) 1245.92 0.215628
\(323\) 1515.87 0.261131
\(324\) 0 0
\(325\) −6729.85 −1.14863
\(326\) 921.001 0.156471
\(327\) 0 0
\(328\) 9061.70 1.52545
\(329\) 1211.70 0.203050
\(330\) 0 0
\(331\) 752.266 0.124919 0.0624597 0.998047i \(-0.480106\pi\)
0.0624597 + 0.998047i \(0.480106\pi\)
\(332\) −713.373 −0.117926
\(333\) 0 0
\(334\) −3859.98 −0.632361
\(335\) −160.551 −0.0261845
\(336\) 0 0
\(337\) −1968.57 −0.318204 −0.159102 0.987262i \(-0.550860\pi\)
−0.159102 + 0.987262i \(0.550860\pi\)
\(338\) −1606.20 −0.258479
\(339\) 0 0
\(340\) 316.814 0.0505343
\(341\) −7551.89 −1.19929
\(342\) 0 0
\(343\) 4946.52 0.778678
\(344\) 6790.93 1.06437
\(345\) 0 0
\(346\) −241.363 −0.0375022
\(347\) −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 \(0.463410\pi\)
\(350\) −1251.75 −0.191168
\(351\) 0 0
\(352\) 5117.72 0.774930
\(353\) −6482.49 −0.977417 −0.488708 0.872447i \(-0.662532\pi\)
−0.488708 + 0.872447i \(0.662532\pi\)
\(354\) 0 0
\(355\) 2390.20 0.357348
\(356\) −4999.59 −0.744320
\(357\) 0 0
\(358\) −1393.56 −0.205732
\(359\) 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
\(362\) 4696.50 0.681886
\(363\) 0 0
\(364\) 2836.48 0.408439
\(365\) −896.717 −0.128593
\(366\) 0 0
\(367\) −14.8871 −0.00211743 −0.00105872 0.999999i \(-0.500337\pi\)
−0.00105872 + 0.999999i \(0.500337\pi\)
\(368\) 2646.95 0.374950
\(369\) 0 0
\(370\) −546.188 −0.0767431
\(371\) 4189.51 0.586276
\(372\) 0 0
\(373\) 1923.18 0.266966 0.133483 0.991051i \(-0.457384\pi\)
0.133483 + 0.991051i \(0.457384\pi\)
\(374\) 639.058 0.0883554
\(375\) 0 0
\(376\) −2938.63 −0.403053
\(377\) 7470.70 1.02059
\(378\) 0 0
\(379\) −9592.87 −1.30014 −0.650069 0.759875i \(-0.725260\pi\)
−0.650069 + 0.759875i \(0.725260\pi\)
\(380\) 1661.76 0.224333
\(381\) 0 0
\(382\) −668.215 −0.0894996
\(383\) 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
\(386\) 4830.70 0.636985
\(387\) 0 0
\(388\) −4885.04 −0.639176
\(389\) 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
\(392\) −5391.06 −0.694616
\(393\) 0 0
\(394\) 1855.55 0.237262
\(395\) 2183.81 0.278176
\(396\) 0 0
\(397\) 10604.5 1.34061 0.670307 0.742084i \(-0.266162\pi\)
0.670307 + 0.742084i \(0.266162\pi\)
\(398\) −5087.51 −0.640739
\(399\) 0 0
\(400\) −2659.33 −0.332417
\(401\) −13785.4 −1.71674 −0.858368 0.513035i \(-0.828521\pi\)
−0.858368 + 0.513035i \(0.828521\pi\)
\(402\) 0 0
\(403\) 15891.2 1.96426
\(404\) 1632.13 0.200993
\(405\) 0 0
\(406\) 1389.55 0.169857
\(407\) 3655.01 0.445141
\(408\) 0 0
\(409\) −9505.94 −1.14924 −0.574619 0.818421i \(-0.694850\pi\)
−0.574619 + 0.818421i \(0.694850\pi\)
\(410\) 1941.91 0.233913
\(411\) 0 0
\(412\) −3215.58 −0.384515
\(413\) −2318.78 −0.276270
\(414\) 0 0
\(415\) −351.832 −0.0416162
\(416\) −10769.1 −1.26922
\(417\) 0 0
\(418\) 3352.00 0.392229
\(419\) −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 \(0.753775\pi\)
\(422\) 7168.36 0.826897
\(423\) 0 0
\(424\) −10160.4 −1.16376
\(425\) −1968.75 −0.224702
\(426\) 0 0
\(427\) 427.962 0.0485025
\(428\) −6061.25 −0.684537
\(429\) 0 0
\(430\) 1455.29 0.163210
\(431\) −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 \(0.389487\pi\)
\(434\) 2955.76 0.326915
\(435\) 0 0
\(436\) −11153.2 −1.22509
\(437\) 10278.4 1.12513
\(438\) 0 0
\(439\) −2544.91 −0.276679 −0.138339 0.990385i \(-0.544176\pi\)
−0.138339 + 0.990385i \(0.544176\pi\)
\(440\) 1612.30 0.174689
\(441\) 0 0
\(442\) −1344.75 −0.144713
\(443\) −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
\(446\) −959.043 −0.101821
\(447\) 0 0
\(448\) −544.331 −0.0574046
\(449\) −8855.74 −0.930798 −0.465399 0.885101i \(-0.654089\pi\)
−0.465399 + 0.885101i \(0.654089\pi\)
\(450\) 0 0
\(451\) −12995.0 −1.35679
\(452\) −4348.07 −0.452469
\(453\) 0 0
\(454\) −2928.35 −0.302718
\(455\) 1398.94 0.144139
\(456\) 0 0
\(457\) −7154.78 −0.732356 −0.366178 0.930545i \(-0.619334\pi\)
−0.366178 + 0.930545i \(0.619334\pi\)
\(458\) 5331.82 0.543973
\(459\) 0 0
\(460\) 2148.17 0.217737
\(461\) 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 \(0.494362\pi\)
\(464\) 2952.09 0.295360
\(465\) 0 0
\(466\) −7068.47 −0.702662
\(467\) −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
\(470\) −629.745 −0.0618042
\(471\) 0 0
\(472\) 5623.51 0.548396
\(473\) −9738.60 −0.946683
\(474\) 0 0
\(475\) −10326.5 −0.997502
\(476\) 829.783 0.0799014
\(477\) 0 0
\(478\) 455.063 0.0435441
\(479\) 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
\(482\) 2611.44 0.246779
\(483\) 0 0
\(484\) 3493.69 0.328108
\(485\) −2409.27 −0.225566
\(486\) 0 0
\(487\) −1129.88 −0.105133 −0.0525663 0.998617i \(-0.516740\pi\)
−0.0525663 + 0.998617i \(0.516740\pi\)
\(488\) −1037.90 −0.0962774
\(489\) 0 0
\(490\) −1155.30 −0.106512
\(491\) −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
\(494\) −7053.51 −0.642414
\(495\) 0 0
\(496\) 6279.49 0.568463
\(497\) 6260.28 0.565014
\(498\) 0 0
\(499\) 17329.1 1.55462 0.777310 0.629118i \(-0.216584\pi\)
0.777310 + 0.629118i \(0.216584\pi\)
\(500\) −4487.74 −0.401396
\(501\) 0 0
\(502\) 10475.6 0.931369
\(503\) 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
\(506\) 4333.16 0.380697
\(507\) 0 0
\(508\) −16278.6 −1.42174
\(509\) −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
\(512\) −7793.12 −0.672676
\(513\) 0 0
\(514\) 7262.26 0.623199
\(515\) −1585.91 −0.135696
\(516\) 0 0
\(517\) 4214.16 0.358489
\(518\) −1430.55 −0.121341
\(519\) 0 0
\(520\) −3392.71 −0.286115
\(521\) −7686.37 −0.646346 −0.323173 0.946340i \(-0.604750\pi\)
−0.323173 + 0.946340i \(0.604750\pi\)
\(522\) 0 0
\(523\) 11476.4 0.959518 0.479759 0.877400i \(-0.340724\pi\)
0.479759 + 0.877400i \(0.340724\pi\)
\(524\) −12165.6 −1.01423
\(525\) 0 0
\(526\) 5355.25 0.443916
\(527\) 4648.81 0.384261
\(528\) 0 0
\(529\) 1120.01 0.0920535
\(530\) −2177.37 −0.178451
\(531\) 0 0
\(532\) 4352.40 0.354700
\(533\) 27345.0 2.22222
\(534\) 0 0
\(535\) −2989.38 −0.241574
\(536\) 1019.81 0.0821814
\(537\) 0 0
\(538\) 4660.88 0.373504
\(539\) 7731.09 0.617814
\(540\) 0 0
\(541\) −546.481 −0.0434289 −0.0217145 0.999764i \(-0.506912\pi\)
−0.0217145 + 0.999764i \(0.506912\pi\)
\(542\) −747.357 −0.0592283
\(543\) 0 0
\(544\) −3150.38 −0.248293
\(545\) −5500.69 −0.432337
\(546\) 0 0
\(547\) 8397.33 0.656388 0.328194 0.944610i \(-0.393560\pi\)
0.328194 + 0.944610i \(0.393560\pi\)
\(548\) 19313.6 1.50554
\(549\) 0 0
\(550\) −4353.44 −0.337512
\(551\) 11463.3 0.886305
\(552\) 0 0
\(553\) 5719.73 0.439833
\(554\) −7083.32 −0.543215
\(555\) 0 0
\(556\) −9024.36 −0.688342
\(557\) 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
\(560\) 552.797 0.0417142
\(561\) 0 0
\(562\) −2704.38 −0.202985
\(563\) −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
\(566\) −1025.95 −0.0761904
\(567\) 0 0
\(568\) −15182.5 −1.12155
\(569\) 23946.9 1.76433 0.882167 0.470937i \(-0.156084\pi\)
0.882167 + 0.470937i \(0.156084\pi\)
\(570\) 0 0
\(571\) 1593.15 0.116763 0.0583813 0.998294i \(-0.481406\pi\)
0.0583813 + 0.998294i \(0.481406\pi\)
\(572\) 9864.95 0.721109
\(573\) 0 0
\(574\) 5086.16 0.369847
\(575\) −13349.2 −0.968174
\(576\) 0 0
\(577\) 12937.4 0.933435 0.466717 0.884406i \(-0.345436\pi\)
0.466717 + 0.884406i \(0.345436\pi\)
\(578\) −393.393 −0.0283097
\(579\) 0 0
\(580\) 2395.82 0.171519
\(581\) −921.499 −0.0658007
\(582\) 0 0
\(583\) 14570.6 1.03508
\(584\) 5695.92 0.403594
\(585\) 0 0
\(586\) −9803.82 −0.691112
\(587\) 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
\(590\) 1205.11 0.0840911
\(591\) 0 0
\(592\) −3039.19 −0.210997
\(593\) −4357.13 −0.301730 −0.150865 0.988554i \(-0.548206\pi\)
−0.150865 + 0.988554i \(0.548206\pi\)
\(594\) 0 0
\(595\) 409.245 0.0281973
\(596\) −1758.23 −0.120839
\(597\) 0 0
\(598\) −9118.14 −0.623526
\(599\) −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 \(0.472622\pi\)
\(602\) 3811.62 0.258057
\(603\) 0 0
\(604\) 4116.24 0.277297
\(605\) 1723.07 0.115790
\(606\) 0 0
\(607\) 185.004 0.0123708 0.00618540 0.999981i \(-0.498031\pi\)
0.00618540 + 0.999981i \(0.498031\pi\)
\(608\) −16524.4 −1.10223
\(609\) 0 0
\(610\) −222.420 −0.0147632
\(611\) −8867.73 −0.587152
\(612\) 0 0
\(613\) −17706.9 −1.16668 −0.583339 0.812228i \(-0.698254\pi\)
−0.583339 + 0.812228i \(0.698254\pi\)
\(614\) −3301.93 −0.217028
\(615\) 0 0
\(616\) 4222.84 0.276207
\(617\) 6183.89 0.403491 0.201746 0.979438i \(-0.435339\pi\)
0.201746 + 0.979438i \(0.435339\pi\)
\(618\) 0 0
\(619\) −1247.51 −0.0810046 −0.0405023 0.999179i \(-0.512896\pi\)
−0.0405023 + 0.999179i \(0.512896\pi\)
\(620\) 5096.23 0.330112
\(621\) 0 0
\(622\) −12992.8 −0.837561
\(623\) −6458.23 −0.415318
\(624\) 0 0
\(625\) 12262.8 0.784817
\(626\) −800.914 −0.0511357
\(627\) 0 0
\(628\) −4430.87 −0.281546
\(629\) −2249.96 −0.142626
\(630\) 0 0
\(631\) 24053.3 1.51750 0.758752 0.651379i \(-0.225809\pi\)
0.758752 + 0.651379i \(0.225809\pi\)
\(632\) −13871.5 −0.873069
\(633\) 0 0
\(634\) 10417.8 0.652596
\(635\) −8028.51 −0.501735
\(636\) 0 0
\(637\) −16268.3 −1.01189
\(638\) 4832.69 0.299887
\(639\) 0 0
\(640\) −4211.70 −0.260128
\(641\) 21286.8 1.31167 0.655834 0.754905i \(-0.272317\pi\)
0.655834 + 0.754905i \(0.272317\pi\)
\(642\) 0 0
\(643\) −1789.41 −0.109747 −0.0548736 0.998493i \(-0.517476\pi\)
−0.0548736 + 0.998493i \(0.517476\pi\)
\(644\) 5626.38 0.344271
\(645\) 0 0
\(646\) −2063.43 −0.125673
\(647\) 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
\(650\) 9160.81 0.552795
\(651\) 0 0
\(652\) 4159.11 0.249821
\(653\) 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
\(656\) 10805.5 0.643117
\(657\) 0 0
\(658\) −1649.39 −0.0977205
\(659\) 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 \(0.871743\pi\)
\(662\) −1024.00 −0.0601192
\(663\) 0 0
\(664\) 2234.82 0.130614
\(665\) 2146.58 0.125174
\(666\) 0 0
\(667\) 14818.7 0.860246
\(668\) −17431.1 −1.00962
\(669\) 0 0
\(670\) 218.545 0.0126017
\(671\) 1488.40 0.0856322
\(672\) 0 0
\(673\) 11723.0 0.671454 0.335727 0.941959i \(-0.391018\pi\)
0.335727 + 0.941959i \(0.391018\pi\)
\(674\) 2679.65 0.153140
\(675\) 0 0
\(676\) −7253.37 −0.412686
\(677\) 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
\(680\) −992.502 −0.0559716
\(681\) 0 0
\(682\) 10279.8 0.577175
\(683\) −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
\(686\) −6733.30 −0.374750
\(687\) 0 0
\(688\) 8097.77 0.448728
\(689\) −30660.5 −1.69532
\(690\) 0 0
\(691\) −16777.7 −0.923665 −0.461832 0.886967i \(-0.652808\pi\)
−0.461832 + 0.886967i \(0.652808\pi\)
\(692\) −1089.96 −0.0598758
\(693\) 0 0
\(694\) 5421.88 0.296559
\(695\) −4450.77 −0.242917
\(696\) 0 0
\(697\) 7999.50 0.434724
\(698\) −2035.86 −0.110399
\(699\) 0 0
\(700\) −5652.71 −0.305218
\(701\) −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
\(704\) −1893.12 −0.101349
\(705\) 0 0
\(706\) 8824.10 0.470396
\(707\) 2108.30 0.112151
\(708\) 0 0
\(709\) −7709.28 −0.408361 −0.204181 0.978933i \(-0.565453\pi\)
−0.204181 + 0.978933i \(0.565453\pi\)
\(710\) −3253.59 −0.171979
\(711\) 0 0
\(712\) 15662.5 0.824407
\(713\) 31521.5 1.65567
\(714\) 0 0
\(715\) 4865.34 0.254480
\(716\) −6293.13 −0.328471
\(717\) 0 0
\(718\) −6729.09 −0.349760
\(719\) 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
\(722\) −1486.56 −0.0766260
\(723\) 0 0
\(724\) 21208.7 1.08870
\(725\) −14888.1 −0.762662
\(726\) 0 0
\(727\) −18597.3 −0.948745 −0.474372 0.880324i \(-0.657325\pi\)
−0.474372 + 0.880324i \(0.657325\pi\)
\(728\) −8886.00 −0.452386
\(729\) 0 0
\(730\) 1220.63 0.0618870
\(731\) 5994.91 0.303324
\(732\) 0 0
\(733\) −23569.5 −1.18767 −0.593833 0.804588i \(-0.702386\pi\)
−0.593833 + 0.804588i \(0.702386\pi\)
\(734\) 20.2646 0.00101905
\(735\) 0 0
\(736\) −21361.3 −1.06982
\(737\) −1462.47 −0.0730948
\(738\) 0 0
\(739\) 10149.1 0.505199 0.252599 0.967571i \(-0.418715\pi\)
0.252599 + 0.967571i \(0.418715\pi\)
\(740\) −2466.50 −0.122528
\(741\) 0 0
\(742\) −5702.85 −0.282154
\(743\) −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
\(746\) −2617.87 −0.128481
\(747\) 0 0
\(748\) 2885.89 0.141068
\(749\) −7829.63 −0.381960
\(750\) 0 0
\(751\) −815.225 −0.0396112 −0.0198056 0.999804i \(-0.506305\pi\)
−0.0198056 + 0.999804i \(0.506305\pi\)
\(752\) −3504.13 −0.169924
\(753\) 0 0
\(754\) −10169.3 −0.491172
\(755\) 2030.11 0.0978585
\(756\) 0 0
\(757\) −13239.4 −0.635659 −0.317829 0.948148i \(-0.602954\pi\)
−0.317829 + 0.948148i \(0.602954\pi\)
\(758\) 13058.0 0.625710
\(759\) 0 0
\(760\) −5205.90 −0.248471
\(761\) 11028.2 0.525324 0.262662 0.964888i \(-0.415400\pi\)
0.262662 + 0.964888i \(0.415400\pi\)
\(762\) 0 0
\(763\) −14407.1 −0.683582
\(764\) −3017.56 −0.142895
\(765\) 0 0
\(766\) −12365.0 −0.583246
\(767\) 16969.8 0.798882
\(768\) 0 0
\(769\) −18921.2 −0.887277 −0.443639 0.896206i \(-0.646313\pi\)
−0.443639 + 0.896206i \(0.646313\pi\)
\(770\) 904.952 0.0423535
\(771\) 0 0
\(772\) 21814.7 1.01701
\(773\) −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
\(776\) 15303.6 0.707949
\(777\) 0 0
\(778\) −1556.93 −0.0717465
\(779\) 41959.2 1.92984
\(780\) 0 0
\(781\) 21772.5 0.997545
\(782\) −2667.42 −0.121978
\(783\) 0 0
\(784\) −6428.50 −0.292844
\(785\) −2185.28 −0.0993579
\(786\) 0 0
\(787\) −20587.3 −0.932477 −0.466239 0.884659i \(-0.654391\pi\)
−0.466239 + 0.884659i \(0.654391\pi\)
\(788\) 8379.37 0.378811
\(789\) 0 0
\(790\) −2972.66 −0.133876
\(791\) −5616.62 −0.252470
\(792\) 0 0
\(793\) −3132.00 −0.140253
\(794\) −14435.0 −0.645190
\(795\) 0 0
\(796\) −22974.5 −1.02300
\(797\) 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
\(800\) 21461.3 0.948463
\(801\) 0 0
\(802\) 18765.0 0.826204
\(803\) −8168.28 −0.358969
\(804\) 0 0
\(805\) 2774.90 0.121494
\(806\) −21631.4 −0.945329
\(807\) 0 0
\(808\) −5113.06 −0.222620
\(809\) −39667.1 −1.72388 −0.861942 0.507007i \(-0.830752\pi\)
−0.861942 + 0.507007i \(0.830752\pi\)
\(810\) 0 0
\(811\) 8003.87 0.346552 0.173276 0.984873i \(-0.444565\pi\)
0.173276 + 0.984873i \(0.444565\pi\)
\(812\) 6275.00 0.271194
\(813\) 0 0
\(814\) −4975.28 −0.214230
\(815\) 2051.25 0.0881621
\(816\) 0 0
\(817\) 31444.7 1.34652
\(818\) 12939.7 0.553088
\(819\) 0 0
\(820\) 8769.40 0.373464
\(821\) 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 \(0.290065\pi\)
\(824\) 10073.6 0.425887
\(825\) 0 0
\(826\) 3156.37 0.132959
\(827\) 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 \(0.387559\pi\)
\(830\) 478.921 0.0200284
\(831\) 0 0
\(832\) 3983.64 0.165995
\(833\) −4759.13 −0.197952
\(834\) 0 0
\(835\) −8596.93 −0.356298
\(836\) 15137.1 0.626231
\(837\) 0 0
\(838\) 13177.8 0.543221
\(839\) 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
\(842\) 16825.1 0.688635
\(843\) 0 0
\(844\) 32371.3 1.32022
\(845\) −3577.33 −0.145638
\(846\) 0 0
\(847\) 4512.98 0.183079
\(848\) −12115.7 −0.490630
\(849\) 0 0
\(850\) 2679.90 0.108141
\(851\) −15256.0 −0.614534
\(852\) 0 0
\(853\) 8162.96 0.327660 0.163830 0.986489i \(-0.447615\pi\)
0.163830 + 0.986489i \(0.447615\pi\)
\(854\) −582.551 −0.0233425
\(855\) 0 0
\(856\) 18988.4 0.758191
\(857\) −18724.9 −0.746361 −0.373181 0.927759i \(-0.621733\pi\)
−0.373181 + 0.927759i \(0.621733\pi\)
\(858\) 0 0
\(859\) −46422.5 −1.84391 −0.921953 0.387301i \(-0.873407\pi\)
−0.921953 + 0.387301i \(0.873407\pi\)
\(860\) 6571.88 0.260580
\(861\) 0 0
\(862\) 4043.61 0.159775
\(863\) −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
\(866\) −8346.33 −0.327505
\(867\) 0 0
\(868\) 13347.8 0.521950
\(869\) 19892.6 0.776536
\(870\) 0 0
\(871\) 3077.43 0.119719
\(872\) 34940.2 1.35691
\(873\) 0 0
\(874\) −13991.2 −0.541487
\(875\) −5797.04 −0.223972
\(876\) 0 0
\(877\) 39163.0 1.50791 0.753957 0.656924i \(-0.228143\pi\)
0.753957 + 0.656924i \(0.228143\pi\)
\(878\) 3464.19 0.133156
\(879\) 0 0
\(880\) 1922.57 0.0736473
\(881\) 35073.2 1.34125 0.670627 0.741795i \(-0.266025\pi\)
0.670627 + 0.741795i \(0.266025\pi\)
\(882\) 0 0
\(883\) −48775.7 −1.85893 −0.929463 0.368915i \(-0.879729\pi\)
−0.929463 + 0.368915i \(0.879729\pi\)
\(884\) −6072.69 −0.231048
\(885\) 0 0
\(886\) 11611.0 0.440269
\(887\) −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
\(890\) 3356.46 0.126415
\(891\) 0 0
\(892\) −4330.90 −0.162566
\(893\) −13607.0 −0.509899
\(894\) 0 0
\(895\) −3103.74 −0.115918
\(896\) −11031.1 −0.411297
\(897\) 0 0
\(898\) 12054.6 0.447960
\(899\) 35155.3 1.30422
\(900\) 0 0
\(901\) −8969.43 −0.331648
\(902\) 17689.1 0.652974
\(903\) 0 0
\(904\) 13621.4 0.501153
\(905\) 10460.0 0.384202
\(906\) 0 0
\(907\) −11675.0 −0.427410 −0.213705 0.976898i \(-0.568553\pi\)
−0.213705 + 0.976898i \(0.568553\pi\)
\(908\) −13224.0 −0.483319
\(909\) 0 0
\(910\) −1904.26 −0.0693689
\(911\) −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
\(914\) 9739.24 0.352457
\(915\) 0 0
\(916\) 24077.7 0.868504
\(917\) −15714.9 −0.565922
\(918\) 0 0
\(919\) 33956.8 1.21886 0.609429 0.792841i \(-0.291399\pi\)
0.609429 + 0.792841i \(0.291399\pi\)
\(920\) −6729.71 −0.241165
\(921\) 0 0
\(922\) −9886.63 −0.353144
\(923\) −45815.3 −1.63383
\(924\) 0 0
\(925\) 15327.4 0.544823
\(926\) −480.372 −0.0170475
\(927\) 0 0
\(928\) −23823.8 −0.842732
\(929\) 23695.3 0.836832 0.418416 0.908256i \(-0.362585\pi\)
0.418416 + 0.908256i \(0.362585\pi\)
\(930\) 0 0
\(931\) −24962.7 −0.878753
\(932\) −31920.2 −1.12187
\(933\) 0 0
\(934\) 2018.72 0.0707221
\(935\) 1423.31 0.0497830
\(936\) 0 0
\(937\) 7990.62 0.278593 0.139297 0.990251i \(-0.455516\pi\)
0.139297 + 0.990251i \(0.455516\pi\)
\(938\) 572.402 0.0199249
\(939\) 0 0
\(940\) −2843.83 −0.0986762
\(941\) −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
\(944\) 6705.69 0.231199
\(945\) 0 0
\(946\) 13256.4 0.455605
\(947\) 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
\(950\) 14056.7 0.480063
\(951\) 0 0
\(952\) −2599.51 −0.0884985
\(953\) 33546.9 1.14029 0.570143 0.821546i \(-0.306888\pi\)
0.570143 + 0.821546i \(0.306888\pi\)
\(954\) 0 0
\(955\) −1488.25 −0.0504277
\(956\) 2055.00 0.0695224
\(957\) 0 0
\(958\) −13598.7 −0.458617
\(959\) 24948.4 0.840067
\(960\) 0 0
\(961\) 44989.1 1.51016
\(962\) 10469.3 0.350878
\(963\) 0 0
\(964\) 11792.9 0.394007
\(965\) 10758.9 0.358903
\(966\) 0 0
\(967\) 24766.8 0.823625 0.411813 0.911269i \(-0.364896\pi\)
0.411813 + 0.911269i \(0.364896\pi\)
\(968\) −10944.9 −0.363411
\(969\) 0 0
\(970\) 3279.56 0.108557
\(971\) −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
\(974\) 1538.01 0.0505966
\(975\) 0 0
\(976\) −1237.63 −0.0405896
\(977\) 11320.4 0.370698 0.185349 0.982673i \(-0.440658\pi\)
0.185349 + 0.982673i \(0.440658\pi\)
\(978\) 0 0
\(979\) −22461.0 −0.733254
\(980\) −5217.16 −0.170057
\(981\) 0 0
\(982\) 25556.8 0.830500
\(983\) 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
\(986\) −2974.92 −0.0960860
\(987\) 0 0
\(988\) −31852.6 −1.02568
\(989\) 40648.8 1.30693
\(990\) 0 0
\(991\) −29405.5 −0.942580 −0.471290 0.881978i \(-0.656212\pi\)
−0.471290 + 0.881978i \(0.656212\pi\)
\(992\) −50676.5 −1.62196
\(993\) 0 0
\(994\) −8521.63 −0.271921
\(995\) −11330.9 −0.361018
\(996\) 0 0
\(997\) −54905.9 −1.74412 −0.872060 0.489398i \(-0.837216\pi\)
−0.872060 + 0.489398i \(0.837216\pi\)
\(998\) −23588.7 −0.748183
\(999\) 0 0
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 153.4.a.g.1.2 3
3.2 odd 2 17.4.a.b.1.2 3
4.3 odd 2 2448.4.a.bi.1.1 3
12.11 even 2 272.4.a.h.1.2 3
15.2 even 4 425.4.b.f.324.4 6
15.8 even 4 425.4.b.f.324.3 6
15.14 odd 2 425.4.a.g.1.2 3
21.20 even 2 833.4.a.d.1.2 3
24.5 odd 2 1088.4.a.v.1.2 3
24.11 even 2 1088.4.a.x.1.2 3
33.32 even 2 2057.4.a.e.1.2 3
51.38 odd 4 289.4.b.b.288.3 6
51.47 odd 4 289.4.b.b.288.4 6
51.50 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 3.2 odd 2
153.4.a.g.1.2 3 1.1 even 1 trivial
272.4.a.h.1.2 3 12.11 even 2
289.4.a.b.1.2 3 51.50 odd 2
289.4.b.b.288.3 6 51.38 odd 4
289.4.b.b.288.4 6 51.47 odd 4
425.4.a.g.1.2 3 15.14 odd 2
425.4.b.f.324.3 6 15.8 even 4
425.4.b.f.324.4 6 15.2 even 4
833.4.a.d.1.2 3 21.20 even 2
1088.4.a.v.1.2 3 24.5 odd 2
1088.4.a.x.1.2 3 24.11 even 2
2057.4.a.e.1.2 3 33.32 even 2
2448.4.a.bi.1.1 3 4.3 odd 2