Properties

Label 2475.4.a.n.1.2
Level $2475$
Weight $4$
Character 2475.1
Self dual yes
Analytic conductor $146.030$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2475.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(146.029727264\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 165)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.56155\) of defining polynomial
Character \(\chi\) \(=\) 2475.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.56155 q^{2} -5.56155 q^{4} +10.2462 q^{7} -21.1771 q^{8} +O(q^{10})\) \(q+1.56155 q^{2} -5.56155 q^{4} +10.2462 q^{7} -21.1771 q^{8} +11.0000 q^{11} +40.8769 q^{13} +16.0000 q^{14} +11.4233 q^{16} -98.7083 q^{17} -39.6458 q^{19} +17.1771 q^{22} +61.6932 q^{23} +63.8314 q^{26} -56.9848 q^{28} +149.093 q^{29} +54.7386 q^{31} +187.255 q^{32} -154.138 q^{34} -44.8939 q^{37} -61.9091 q^{38} -336.479 q^{41} +2.36745 q^{43} -61.1771 q^{44} +96.3371 q^{46} -333.295 q^{47} -238.015 q^{49} -227.339 q^{52} +640.064 q^{53} -216.985 q^{56} +232.816 q^{58} +370.773 q^{59} -714.405 q^{61} +85.4773 q^{62} +201.022 q^{64} +404.985 q^{67} +548.972 q^{68} -939.292 q^{71} +362.570 q^{73} -70.1042 q^{74} +220.492 q^{76} +112.708 q^{77} +951.835 q^{79} -525.430 q^{82} +735.221 q^{83} +3.69690 q^{86} -232.948 q^{88} -385.879 q^{89} +418.833 q^{91} -343.110 q^{92} -520.458 q^{94} +966.345 q^{97} -371.673 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{2} - 7q^{4} + 4q^{7} + 3q^{8} + O(q^{10}) \) \( 2q - q^{2} - 7q^{4} + 4q^{7} + 3q^{8} + 22q^{11} + 90q^{13} + 32q^{14} - 39q^{16} - 16q^{17} - 170q^{19} - 11q^{22} - 124q^{23} - 62q^{26} - 48q^{28} + 158q^{29} + 60q^{31} + 123q^{32} - 366q^{34} + 372q^{37} + 272q^{38} - 38q^{41} + 516q^{43} - 77q^{44} + 572q^{46} + 224q^{47} - 542q^{49} - 298q^{52} + 472q^{53} - 368q^{56} + 210q^{58} - 248q^{59} + 72q^{61} + 72q^{62} + 769q^{64} + 744q^{67} + 430q^{68} - 2060q^{71} + 486q^{73} - 1138q^{74} + 408q^{76} + 44q^{77} + 642q^{79} - 1290q^{82} - 286q^{83} - 1312q^{86} + 33q^{88} - 244q^{89} + 112q^{91} - 76q^{92} - 1948q^{94} + 168q^{97} + 407q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.56155 0.552092 0.276046 0.961144i \(-0.410976\pi\)
0.276046 + 0.961144i \(0.410976\pi\)
\(3\) 0 0
\(4\) −5.56155 −0.695194
\(5\) 0 0
\(6\) 0 0
\(7\) 10.2462 0.553243 0.276622 0.960979i \(-0.410785\pi\)
0.276622 + 0.960979i \(0.410785\pi\)
\(8\) −21.1771 −0.935904
\(9\) 0 0
\(10\) 0 0
\(11\) 11.0000 0.301511
\(12\) 0 0
\(13\) 40.8769 0.872093 0.436047 0.899924i \(-0.356378\pi\)
0.436047 + 0.899924i \(0.356378\pi\)
\(14\) 16.0000 0.305441
\(15\) 0 0
\(16\) 11.4233 0.178489
\(17\) −98.7083 −1.40825 −0.704126 0.710075i \(-0.748661\pi\)
−0.704126 + 0.710075i \(0.748661\pi\)
\(18\) 0 0
\(19\) −39.6458 −0.478704 −0.239352 0.970933i \(-0.576935\pi\)
−0.239352 + 0.970933i \(0.576935\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 17.1771 0.166462
\(23\) 61.6932 0.559301 0.279650 0.960102i \(-0.409781\pi\)
0.279650 + 0.960102i \(0.409781\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 63.8314 0.481476
\(27\) 0 0
\(28\) −56.9848 −0.384612
\(29\) 149.093 0.954684 0.477342 0.878718i \(-0.341600\pi\)
0.477342 + 0.878718i \(0.341600\pi\)
\(30\) 0 0
\(31\) 54.7386 0.317140 0.158570 0.987348i \(-0.449312\pi\)
0.158570 + 0.987348i \(0.449312\pi\)
\(32\) 187.255 1.03445
\(33\) 0 0
\(34\) −154.138 −0.777485
\(35\) 0 0
\(36\) 0 0
\(37\) −44.8939 −0.199473 −0.0997367 0.995014i \(-0.531800\pi\)
−0.0997367 + 0.995014i \(0.531800\pi\)
\(38\) −61.9091 −0.264289
\(39\) 0 0
\(40\) 0 0
\(41\) −336.479 −1.28169 −0.640844 0.767671i \(-0.721415\pi\)
−0.640844 + 0.767671i \(0.721415\pi\)
\(42\) 0 0
\(43\) 2.36745 0.00839611 0.00419806 0.999991i \(-0.498664\pi\)
0.00419806 + 0.999991i \(0.498664\pi\)
\(44\) −61.1771 −0.209609
\(45\) 0 0
\(46\) 96.3371 0.308786
\(47\) −333.295 −1.03439 −0.517193 0.855869i \(-0.673023\pi\)
−0.517193 + 0.855869i \(0.673023\pi\)
\(48\) 0 0
\(49\) −238.015 −0.693922
\(50\) 0 0
\(51\) 0 0
\(52\) −227.339 −0.606274
\(53\) 640.064 1.65886 0.829430 0.558610i \(-0.188665\pi\)
0.829430 + 0.558610i \(0.188665\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −216.985 −0.517782
\(57\) 0 0
\(58\) 232.816 0.527074
\(59\) 370.773 0.818144 0.409072 0.912502i \(-0.365853\pi\)
0.409072 + 0.912502i \(0.365853\pi\)
\(60\) 0 0
\(61\) −714.405 −1.49951 −0.749756 0.661715i \(-0.769829\pi\)
−0.749756 + 0.661715i \(0.769829\pi\)
\(62\) 85.4773 0.175091
\(63\) 0 0
\(64\) 201.022 0.392621
\(65\) 0 0
\(66\) 0 0
\(67\) 404.985 0.738459 0.369230 0.929338i \(-0.379622\pi\)
0.369230 + 0.929338i \(0.379622\pi\)
\(68\) 548.972 0.979009
\(69\) 0 0
\(70\) 0 0
\(71\) −939.292 −1.57005 −0.785024 0.619465i \(-0.787349\pi\)
−0.785024 + 0.619465i \(0.787349\pi\)
\(72\) 0 0
\(73\) 362.570 0.581310 0.290655 0.956828i \(-0.406127\pi\)
0.290655 + 0.956828i \(0.406127\pi\)
\(74\) −70.1042 −0.110128
\(75\) 0 0
\(76\) 220.492 0.332792
\(77\) 112.708 0.166809
\(78\) 0 0
\(79\) 951.835 1.35557 0.677784 0.735261i \(-0.262941\pi\)
0.677784 + 0.735261i \(0.262941\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −525.430 −0.707610
\(83\) 735.221 0.972302 0.486151 0.873875i \(-0.338401\pi\)
0.486151 + 0.873875i \(0.338401\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 3.69690 0.00463543
\(87\) 0 0
\(88\) −232.948 −0.282186
\(89\) −385.879 −0.459585 −0.229793 0.973240i \(-0.573805\pi\)
−0.229793 + 0.973240i \(0.573805\pi\)
\(90\) 0 0
\(91\) 418.833 0.482480
\(92\) −343.110 −0.388823
\(93\) 0 0
\(94\) −520.458 −0.571076
\(95\) 0 0
\(96\) 0 0
\(97\) 966.345 1.01152 0.505760 0.862674i \(-0.331212\pi\)
0.505760 + 0.862674i \(0.331212\pi\)
\(98\) −371.673 −0.383109
\(99\) 0 0
\(100\) 0 0
\(101\) −348.600 −0.343436 −0.171718 0.985146i \(-0.554932\pi\)
−0.171718 + 0.985146i \(0.554932\pi\)
\(102\) 0 0
\(103\) 1536.38 1.46975 0.734873 0.678204i \(-0.237242\pi\)
0.734873 + 0.678204i \(0.237242\pi\)
\(104\) −865.653 −0.816195
\(105\) 0 0
\(106\) 999.494 0.915844
\(107\) −779.180 −0.703983 −0.351991 0.936003i \(-0.614495\pi\)
−0.351991 + 0.936003i \(0.614495\pi\)
\(108\) 0 0
\(109\) −1501.79 −1.31968 −0.659842 0.751404i \(-0.729377\pi\)
−0.659842 + 0.751404i \(0.729377\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 117.045 0.0987478
\(113\) 170.000 0.141524 0.0707622 0.997493i \(-0.477457\pi\)
0.0707622 + 0.997493i \(0.477457\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −829.187 −0.663691
\(117\) 0 0
\(118\) 578.981 0.451691
\(119\) −1011.39 −0.779106
\(120\) 0 0
\(121\) 121.000 0.0909091
\(122\) −1115.58 −0.827869
\(123\) 0 0
\(124\) −304.432 −0.220474
\(125\) 0 0
\(126\) 0 0
\(127\) 1739.82 1.21562 0.607811 0.794082i \(-0.292048\pi\)
0.607811 + 0.794082i \(0.292048\pi\)
\(128\) −1184.13 −0.817683
\(129\) 0 0
\(130\) 0 0
\(131\) −312.837 −0.208647 −0.104323 0.994543i \(-0.533268\pi\)
−0.104323 + 0.994543i \(0.533268\pi\)
\(132\) 0 0
\(133\) −406.220 −0.264840
\(134\) 632.405 0.407698
\(135\) 0 0
\(136\) 2090.35 1.31799
\(137\) −716.928 −0.447090 −0.223545 0.974694i \(-0.571763\pi\)
−0.223545 + 0.974694i \(0.571763\pi\)
\(138\) 0 0
\(139\) −876.483 −0.534837 −0.267418 0.963581i \(-0.586171\pi\)
−0.267418 + 0.963581i \(0.586171\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −1466.75 −0.866811
\(143\) 449.646 0.262946
\(144\) 0 0
\(145\) 0 0
\(146\) 566.172 0.320937
\(147\) 0 0
\(148\) 249.680 0.138673
\(149\) 2376.36 1.30657 0.653285 0.757112i \(-0.273390\pi\)
0.653285 + 0.757112i \(0.273390\pi\)
\(150\) 0 0
\(151\) −92.8466 −0.0500381 −0.0250190 0.999687i \(-0.507965\pi\)
−0.0250190 + 0.999687i \(0.507965\pi\)
\(152\) 839.583 0.448021
\(153\) 0 0
\(154\) 176.000 0.0920941
\(155\) 0 0
\(156\) 0 0
\(157\) 1881.24 0.956301 0.478150 0.878278i \(-0.341307\pi\)
0.478150 + 0.878278i \(0.341307\pi\)
\(158\) 1486.34 0.748398
\(159\) 0 0
\(160\) 0 0
\(161\) 632.121 0.309429
\(162\) 0 0
\(163\) 2465.49 1.18474 0.592369 0.805667i \(-0.298193\pi\)
0.592369 + 0.805667i \(0.298193\pi\)
\(164\) 1871.35 0.891022
\(165\) 0 0
\(166\) 1148.09 0.536800
\(167\) −1254.30 −0.581200 −0.290600 0.956845i \(-0.593855\pi\)
−0.290600 + 0.956845i \(0.593855\pi\)
\(168\) 0 0
\(169\) −526.080 −0.239454
\(170\) 0 0
\(171\) 0 0
\(172\) −13.1667 −0.00583693
\(173\) −1206.71 −0.530314 −0.265157 0.964205i \(-0.585424\pi\)
−0.265157 + 0.964205i \(0.585424\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 125.656 0.0538164
\(177\) 0 0
\(178\) −602.570 −0.253733
\(179\) 1442.29 0.602244 0.301122 0.953586i \(-0.402639\pi\)
0.301122 + 0.953586i \(0.402639\pi\)
\(180\) 0 0
\(181\) 4261.81 1.75015 0.875076 0.483985i \(-0.160811\pi\)
0.875076 + 0.483985i \(0.160811\pi\)
\(182\) 654.030 0.266373
\(183\) 0 0
\(184\) −1306.48 −0.523451
\(185\) 0 0
\(186\) 0 0
\(187\) −1085.79 −0.424604
\(188\) 1853.64 0.719099
\(189\) 0 0
\(190\) 0 0
\(191\) 852.223 0.322852 0.161426 0.986885i \(-0.448391\pi\)
0.161426 + 0.986885i \(0.448391\pi\)
\(192\) 0 0
\(193\) 2459.95 0.917468 0.458734 0.888574i \(-0.348303\pi\)
0.458734 + 0.888574i \(0.348303\pi\)
\(194\) 1509.00 0.558452
\(195\) 0 0
\(196\) 1323.73 0.482410
\(197\) 3477.06 1.25751 0.628756 0.777602i \(-0.283564\pi\)
0.628756 + 0.777602i \(0.283564\pi\)
\(198\) 0 0
\(199\) 3995.04 1.42312 0.711560 0.702626i \(-0.247989\pi\)
0.711560 + 0.702626i \(0.247989\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −544.358 −0.189608
\(203\) 1527.64 0.528173
\(204\) 0 0
\(205\) 0 0
\(206\) 2399.14 0.811436
\(207\) 0 0
\(208\) 466.949 0.155659
\(209\) −436.104 −0.144335
\(210\) 0 0
\(211\) 1046.13 0.341319 0.170660 0.985330i \(-0.445410\pi\)
0.170660 + 0.985330i \(0.445410\pi\)
\(212\) −3559.75 −1.15323
\(213\) 0 0
\(214\) −1216.73 −0.388664
\(215\) 0 0
\(216\) 0 0
\(217\) 560.864 0.175456
\(218\) −2345.13 −0.728587
\(219\) 0 0
\(220\) 0 0
\(221\) −4034.89 −1.22813
\(222\) 0 0
\(223\) 506.265 0.152027 0.0760135 0.997107i \(-0.475781\pi\)
0.0760135 + 0.997107i \(0.475781\pi\)
\(224\) 1918.65 0.572300
\(225\) 0 0
\(226\) 265.464 0.0781345
\(227\) −4286.29 −1.25326 −0.626632 0.779315i \(-0.715567\pi\)
−0.626632 + 0.779315i \(0.715567\pi\)
\(228\) 0 0
\(229\) 5709.37 1.64754 0.823769 0.566926i \(-0.191867\pi\)
0.823769 + 0.566926i \(0.191867\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −3157.35 −0.893492
\(233\) 2946.09 0.828348 0.414174 0.910198i \(-0.364071\pi\)
0.414174 + 0.910198i \(0.364071\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −2062.07 −0.568769
\(237\) 0 0
\(238\) −1579.33 −0.430139
\(239\) 2078.89 0.562646 0.281323 0.959613i \(-0.409227\pi\)
0.281323 + 0.959613i \(0.409227\pi\)
\(240\) 0 0
\(241\) 1853.37 0.495378 0.247689 0.968840i \(-0.420329\pi\)
0.247689 + 0.968840i \(0.420329\pi\)
\(242\) 188.948 0.0501902
\(243\) 0 0
\(244\) 3973.20 1.04245
\(245\) 0 0
\(246\) 0 0
\(247\) −1620.60 −0.417475
\(248\) −1159.20 −0.296813
\(249\) 0 0
\(250\) 0 0
\(251\) 2358.39 0.593068 0.296534 0.955022i \(-0.404169\pi\)
0.296534 + 0.955022i \(0.404169\pi\)
\(252\) 0 0
\(253\) 678.625 0.168635
\(254\) 2716.82 0.671135
\(255\) 0 0
\(256\) −3457.26 −0.844057
\(257\) 5519.25 1.33962 0.669809 0.742534i \(-0.266376\pi\)
0.669809 + 0.742534i \(0.266376\pi\)
\(258\) 0 0
\(259\) −459.993 −0.110357
\(260\) 0 0
\(261\) 0 0
\(262\) −488.512 −0.115192
\(263\) 2259.65 0.529795 0.264898 0.964277i \(-0.414662\pi\)
0.264898 + 0.964277i \(0.414662\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −634.333 −0.146216
\(267\) 0 0
\(268\) −2252.34 −0.513373
\(269\) −7039.53 −1.59557 −0.797783 0.602944i \(-0.793994\pi\)
−0.797783 + 0.602944i \(0.793994\pi\)
\(270\) 0 0
\(271\) 5155.08 1.15553 0.577765 0.816203i \(-0.303925\pi\)
0.577765 + 0.816203i \(0.303925\pi\)
\(272\) −1127.57 −0.251357
\(273\) 0 0
\(274\) −1119.52 −0.246835
\(275\) 0 0
\(276\) 0 0
\(277\) −9074.52 −1.96836 −0.984179 0.177175i \(-0.943304\pi\)
−0.984179 + 0.177175i \(0.943304\pi\)
\(278\) −1368.67 −0.295279
\(279\) 0 0
\(280\) 0 0
\(281\) 3407.79 0.723459 0.361729 0.932283i \(-0.382186\pi\)
0.361729 + 0.932283i \(0.382186\pi\)
\(282\) 0 0
\(283\) 8827.73 1.85425 0.927127 0.374746i \(-0.122270\pi\)
0.927127 + 0.374746i \(0.122270\pi\)
\(284\) 5223.92 1.09149
\(285\) 0 0
\(286\) 702.146 0.145170
\(287\) −3447.64 −0.709085
\(288\) 0 0
\(289\) 4830.33 0.983174
\(290\) 0 0
\(291\) 0 0
\(292\) −2016.45 −0.404123
\(293\) −4528.29 −0.902886 −0.451443 0.892300i \(-0.649091\pi\)
−0.451443 + 0.892300i \(0.649091\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 950.722 0.186688
\(297\) 0 0
\(298\) 3710.81 0.721347
\(299\) 2521.83 0.487762
\(300\) 0 0
\(301\) 24.2574 0.00464509
\(302\) −144.985 −0.0276256
\(303\) 0 0
\(304\) −452.886 −0.0854434
\(305\) 0 0
\(306\) 0 0
\(307\) −568.106 −0.105614 −0.0528071 0.998605i \(-0.516817\pi\)
−0.0528071 + 0.998605i \(0.516817\pi\)
\(308\) −626.833 −0.115965
\(309\) 0 0
\(310\) 0 0
\(311\) 6853.59 1.24962 0.624809 0.780778i \(-0.285177\pi\)
0.624809 + 0.780778i \(0.285177\pi\)
\(312\) 0 0
\(313\) 1138.92 0.205673 0.102837 0.994698i \(-0.467208\pi\)
0.102837 + 0.994698i \(0.467208\pi\)
\(314\) 2937.65 0.527966
\(315\) 0 0
\(316\) −5293.68 −0.942382
\(317\) 3207.48 0.568297 0.284148 0.958780i \(-0.408289\pi\)
0.284148 + 0.958780i \(0.408289\pi\)
\(318\) 0 0
\(319\) 1640.02 0.287848
\(320\) 0 0
\(321\) 0 0
\(322\) 987.091 0.170834
\(323\) 3913.37 0.674136
\(324\) 0 0
\(325\) 0 0
\(326\) 3850.00 0.654085
\(327\) 0 0
\(328\) 7125.65 1.19954
\(329\) −3415.02 −0.572267
\(330\) 0 0
\(331\) −9135.12 −1.51695 −0.758477 0.651700i \(-0.774056\pi\)
−0.758477 + 0.651700i \(0.774056\pi\)
\(332\) −4088.97 −0.675938
\(333\) 0 0
\(334\) −1958.65 −0.320876
\(335\) 0 0
\(336\) 0 0
\(337\) 3470.05 0.560907 0.280453 0.959868i \(-0.409515\pi\)
0.280453 + 0.959868i \(0.409515\pi\)
\(338\) −821.501 −0.132200
\(339\) 0 0
\(340\) 0 0
\(341\) 602.125 0.0956214
\(342\) 0 0
\(343\) −5953.20 −0.937151
\(344\) −50.1357 −0.00785795
\(345\) 0 0
\(346\) −1884.34 −0.292782
\(347\) −89.3315 −0.0138201 −0.00691004 0.999976i \(-0.502200\pi\)
−0.00691004 + 0.999976i \(0.502200\pi\)
\(348\) 0 0
\(349\) −149.375 −0.0229107 −0.0114554 0.999934i \(-0.503646\pi\)
−0.0114554 + 0.999934i \(0.503646\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 2059.80 0.311897
\(353\) 7867.64 1.18627 0.593133 0.805104i \(-0.297891\pi\)
0.593133 + 0.805104i \(0.297891\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 2146.09 0.319501
\(357\) 0 0
\(358\) 2252.21 0.332494
\(359\) −4974.22 −0.731279 −0.365639 0.930757i \(-0.619150\pi\)
−0.365639 + 0.930757i \(0.619150\pi\)
\(360\) 0 0
\(361\) −5287.21 −0.770842
\(362\) 6655.04 0.966246
\(363\) 0 0
\(364\) −2329.36 −0.335417
\(365\) 0 0
\(366\) 0 0
\(367\) 13266.7 1.88696 0.943479 0.331433i \(-0.107532\pi\)
0.943479 + 0.331433i \(0.107532\pi\)
\(368\) 704.739 0.0998290
\(369\) 0 0
\(370\) 0 0
\(371\) 6558.23 0.917754
\(372\) 0 0
\(373\) 4632.77 0.643099 0.321549 0.946893i \(-0.395796\pi\)
0.321549 + 0.946893i \(0.395796\pi\)
\(374\) −1695.52 −0.234421
\(375\) 0 0
\(376\) 7058.22 0.968085
\(377\) 6094.45 0.832573
\(378\) 0 0
\(379\) 6503.31 0.881406 0.440703 0.897653i \(-0.354729\pi\)
0.440703 + 0.897653i \(0.354729\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 1330.79 0.178244
\(383\) 12734.5 1.69897 0.849484 0.527614i \(-0.176913\pi\)
0.849484 + 0.527614i \(0.176913\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 3841.35 0.506527
\(387\) 0 0
\(388\) −5374.38 −0.703203
\(389\) −12024.6 −1.56728 −0.783639 0.621216i \(-0.786639\pi\)
−0.783639 + 0.621216i \(0.786639\pi\)
\(390\) 0 0
\(391\) −6089.63 −0.787636
\(392\) 5040.47 0.649444
\(393\) 0 0
\(394\) 5429.61 0.694263
\(395\) 0 0
\(396\) 0 0
\(397\) 5223.65 0.660371 0.330186 0.943916i \(-0.392889\pi\)
0.330186 + 0.943916i \(0.392889\pi\)
\(398\) 6238.46 0.785693
\(399\) 0 0
\(400\) 0 0
\(401\) −9648.18 −1.20151 −0.600757 0.799432i \(-0.705134\pi\)
−0.600757 + 0.799432i \(0.705134\pi\)
\(402\) 0 0
\(403\) 2237.55 0.276576
\(404\) 1938.76 0.238755
\(405\) 0 0
\(406\) 2385.48 0.291600
\(407\) −493.833 −0.0601435
\(408\) 0 0
\(409\) −2010.47 −0.243060 −0.121530 0.992588i \(-0.538780\pi\)
−0.121530 + 0.992588i \(0.538780\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −8544.65 −1.02176
\(413\) 3799.02 0.452633
\(414\) 0 0
\(415\) 0 0
\(416\) 7654.39 0.902133
\(417\) 0 0
\(418\) −681.000 −0.0796861
\(419\) −4435.27 −0.517129 −0.258565 0.965994i \(-0.583250\pi\)
−0.258565 + 0.965994i \(0.583250\pi\)
\(420\) 0 0
\(421\) 15217.9 1.76170 0.880852 0.473392i \(-0.156971\pi\)
0.880852 + 0.473392i \(0.156971\pi\)
\(422\) 1633.58 0.188440
\(423\) 0 0
\(424\) −13554.7 −1.55253
\(425\) 0 0
\(426\) 0 0
\(427\) −7319.95 −0.829595
\(428\) 4333.45 0.489405
\(429\) 0 0
\(430\) 0 0
\(431\) 5622.11 0.628324 0.314162 0.949369i \(-0.398277\pi\)
0.314162 + 0.949369i \(0.398277\pi\)
\(432\) 0 0
\(433\) 14306.3 1.58780 0.793898 0.608051i \(-0.208049\pi\)
0.793898 + 0.608051i \(0.208049\pi\)
\(434\) 875.818 0.0968678
\(435\) 0 0
\(436\) 8352.29 0.917436
\(437\) −2445.88 −0.267740
\(438\) 0 0
\(439\) 4384.20 0.476643 0.238322 0.971186i \(-0.423403\pi\)
0.238322 + 0.971186i \(0.423403\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −6300.69 −0.678039
\(443\) −10090.0 −1.08214 −0.541071 0.840977i \(-0.681981\pi\)
−0.541071 + 0.840977i \(0.681981\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 790.560 0.0839330
\(447\) 0 0
\(448\) 2059.71 0.217215
\(449\) −9582.52 −1.00719 −0.503594 0.863941i \(-0.667989\pi\)
−0.503594 + 0.863941i \(0.667989\pi\)
\(450\) 0 0
\(451\) −3701.27 −0.386444
\(452\) −945.464 −0.0983869
\(453\) 0 0
\(454\) −6693.27 −0.691918
\(455\) 0 0
\(456\) 0 0
\(457\) −9999.34 −1.02352 −0.511761 0.859128i \(-0.671007\pi\)
−0.511761 + 0.859128i \(0.671007\pi\)
\(458\) 8915.49 0.909593
\(459\) 0 0
\(460\) 0 0
\(461\) 11115.8 1.12302 0.561512 0.827468i \(-0.310220\pi\)
0.561512 + 0.827468i \(0.310220\pi\)
\(462\) 0 0
\(463\) 1567.16 0.157305 0.0786524 0.996902i \(-0.474938\pi\)
0.0786524 + 0.996902i \(0.474938\pi\)
\(464\) 1703.13 0.170401
\(465\) 0 0
\(466\) 4600.48 0.457325
\(467\) 12648.8 1.25335 0.626675 0.779281i \(-0.284415\pi\)
0.626675 + 0.779281i \(0.284415\pi\)
\(468\) 0 0
\(469\) 4149.56 0.408548
\(470\) 0 0
\(471\) 0 0
\(472\) −7851.88 −0.765704
\(473\) 26.0420 0.00253152
\(474\) 0 0
\(475\) 0 0
\(476\) 5624.88 0.541630
\(477\) 0 0
\(478\) 3246.30 0.310633
\(479\) 10719.2 1.02249 0.511247 0.859434i \(-0.329184\pi\)
0.511247 + 0.859434i \(0.329184\pi\)
\(480\) 0 0
\(481\) −1835.12 −0.173959
\(482\) 2894.14 0.273494
\(483\) 0 0
\(484\) −672.948 −0.0631995
\(485\) 0 0
\(486\) 0 0
\(487\) −7161.20 −0.666335 −0.333167 0.942868i \(-0.608117\pi\)
−0.333167 + 0.942868i \(0.608117\pi\)
\(488\) 15129.0 1.40340
\(489\) 0 0
\(490\) 0 0
\(491\) −14567.3 −1.33893 −0.669463 0.742845i \(-0.733476\pi\)
−0.669463 + 0.742845i \(0.733476\pi\)
\(492\) 0 0
\(493\) −14716.7 −1.34444
\(494\) −2530.65 −0.230485
\(495\) 0 0
\(496\) 625.295 0.0566060
\(497\) −9624.18 −0.868619
\(498\) 0 0
\(499\) −4638.99 −0.416172 −0.208086 0.978111i \(-0.566723\pi\)
−0.208086 + 0.978111i \(0.566723\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 3682.75 0.327428
\(503\) −12206.3 −1.08201 −0.541006 0.841019i \(-0.681956\pi\)
−0.541006 + 0.841019i \(0.681956\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 1059.71 0.0931024
\(507\) 0 0
\(508\) −9676.09 −0.845093
\(509\) −10018.6 −0.872427 −0.436214 0.899843i \(-0.643681\pi\)
−0.436214 + 0.899843i \(0.643681\pi\)
\(510\) 0 0
\(511\) 3714.97 0.321606
\(512\) 4074.36 0.351686
\(513\) 0 0
\(514\) 8618.61 0.739592
\(515\) 0 0
\(516\) 0 0
\(517\) −3666.25 −0.311879
\(518\) −718.303 −0.0609274
\(519\) 0 0
\(520\) 0 0
\(521\) −1054.72 −0.0886916 −0.0443458 0.999016i \(-0.514120\pi\)
−0.0443458 + 0.999016i \(0.514120\pi\)
\(522\) 0 0
\(523\) 16234.2 1.35730 0.678652 0.734460i \(-0.262564\pi\)
0.678652 + 0.734460i \(0.262564\pi\)
\(524\) 1739.86 0.145050
\(525\) 0 0
\(526\) 3528.57 0.292496
\(527\) −5403.16 −0.446613
\(528\) 0 0
\(529\) −8360.95 −0.687183
\(530\) 0 0
\(531\) 0 0
\(532\) 2259.21 0.184115
\(533\) −13754.2 −1.11775
\(534\) 0 0
\(535\) 0 0
\(536\) −8576.40 −0.691127
\(537\) 0 0
\(538\) −10992.6 −0.880900
\(539\) −2618.17 −0.209225
\(540\) 0 0
\(541\) 675.936 0.0537167 0.0268584 0.999639i \(-0.491450\pi\)
0.0268584 + 0.999639i \(0.491450\pi\)
\(542\) 8049.93 0.637960
\(543\) 0 0
\(544\) −18483.6 −1.45676
\(545\) 0 0
\(546\) 0 0
\(547\) −13058.2 −1.02071 −0.510355 0.859964i \(-0.670486\pi\)
−0.510355 + 0.859964i \(0.670486\pi\)
\(548\) 3987.23 0.310814
\(549\) 0 0
\(550\) 0 0
\(551\) −5910.91 −0.457011
\(552\) 0 0
\(553\) 9752.70 0.749959
\(554\) −14170.3 −1.08672
\(555\) 0 0
\(556\) 4874.61 0.371815
\(557\) −6710.48 −0.510471 −0.255236 0.966879i \(-0.582153\pi\)
−0.255236 + 0.966879i \(0.582153\pi\)
\(558\) 0 0
\(559\) 96.7741 0.00732219
\(560\) 0 0
\(561\) 0 0
\(562\) 5321.45 0.399416
\(563\) −20820.5 −1.55858 −0.779288 0.626666i \(-0.784419\pi\)
−0.779288 + 0.626666i \(0.784419\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 13785.0 1.02372
\(567\) 0 0
\(568\) 19891.5 1.46941
\(569\) −3251.08 −0.239530 −0.119765 0.992802i \(-0.538214\pi\)
−0.119765 + 0.992802i \(0.538214\pi\)
\(570\) 0 0
\(571\) −4637.50 −0.339883 −0.169941 0.985454i \(-0.554358\pi\)
−0.169941 + 0.985454i \(0.554358\pi\)
\(572\) −2500.73 −0.182798
\(573\) 0 0
\(574\) −5383.67 −0.391481
\(575\) 0 0
\(576\) 0 0
\(577\) −14462.4 −1.04346 −0.521730 0.853111i \(-0.674713\pi\)
−0.521730 + 0.853111i \(0.674713\pi\)
\(578\) 7542.82 0.542803
\(579\) 0 0
\(580\) 0 0
\(581\) 7533.23 0.537920
\(582\) 0 0
\(583\) 7040.71 0.500165
\(584\) −7678.18 −0.544050
\(585\) 0 0
\(586\) −7071.17 −0.498476
\(587\) −22759.7 −1.60033 −0.800166 0.599779i \(-0.795255\pi\)
−0.800166 + 0.599779i \(0.795255\pi\)
\(588\) 0 0
\(589\) −2170.16 −0.151816
\(590\) 0 0
\(591\) 0 0
\(592\) −512.836 −0.0356038
\(593\) −14956.4 −1.03573 −0.517864 0.855463i \(-0.673273\pi\)
−0.517864 + 0.855463i \(0.673273\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −13216.2 −0.908319
\(597\) 0 0
\(598\) 3937.96 0.269290
\(599\) −2150.77 −0.146708 −0.0733539 0.997306i \(-0.523370\pi\)
−0.0733539 + 0.997306i \(0.523370\pi\)
\(600\) 0 0
\(601\) 27759.8 1.88410 0.942050 0.335472i \(-0.108896\pi\)
0.942050 + 0.335472i \(0.108896\pi\)
\(602\) 37.8792 0.00256452
\(603\) 0 0
\(604\) 516.371 0.0347862
\(605\) 0 0
\(606\) 0 0
\(607\) 10991.5 0.734974 0.367487 0.930029i \(-0.380218\pi\)
0.367487 + 0.930029i \(0.380218\pi\)
\(608\) −7423.87 −0.495194
\(609\) 0 0
\(610\) 0 0
\(611\) −13624.1 −0.902081
\(612\) 0 0
\(613\) −10646.1 −0.701457 −0.350728 0.936477i \(-0.614066\pi\)
−0.350728 + 0.936477i \(0.614066\pi\)
\(614\) −887.128 −0.0583088
\(615\) 0 0
\(616\) −2386.83 −0.156117
\(617\) 7199.92 0.469786 0.234893 0.972021i \(-0.424526\pi\)
0.234893 + 0.972021i \(0.424526\pi\)
\(618\) 0 0
\(619\) 12186.9 0.791332 0.395666 0.918395i \(-0.370514\pi\)
0.395666 + 0.918395i \(0.370514\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 10702.2 0.689905
\(623\) −3953.80 −0.254262
\(624\) 0 0
\(625\) 0 0
\(626\) 1778.49 0.113551
\(627\) 0 0
\(628\) −10462.6 −0.664814
\(629\) 4431.40 0.280909
\(630\) 0 0
\(631\) 7370.64 0.465009 0.232505 0.972595i \(-0.425308\pi\)
0.232505 + 0.972595i \(0.425308\pi\)
\(632\) −20157.1 −1.26868
\(633\) 0 0
\(634\) 5008.65 0.313752
\(635\) 0 0
\(636\) 0 0
\(637\) −9729.32 −0.605164
\(638\) 2560.98 0.158919
\(639\) 0 0
\(640\) 0 0
\(641\) 25014.9 1.54139 0.770693 0.637207i \(-0.219910\pi\)
0.770693 + 0.637207i \(0.219910\pi\)
\(642\) 0 0
\(643\) 21668.2 1.32894 0.664472 0.747313i \(-0.268657\pi\)
0.664472 + 0.747313i \(0.268657\pi\)
\(644\) −3515.58 −0.215113
\(645\) 0 0
\(646\) 6110.94 0.372185
\(647\) −27625.3 −1.67861 −0.839305 0.543661i \(-0.817038\pi\)
−0.839305 + 0.543661i \(0.817038\pi\)
\(648\) 0 0
\(649\) 4078.50 0.246680
\(650\) 0 0
\(651\) 0 0
\(652\) −13712.0 −0.823623
\(653\) −14314.0 −0.857810 −0.428905 0.903350i \(-0.641101\pi\)
−0.428905 + 0.903350i \(0.641101\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −3843.70 −0.228767
\(657\) 0 0
\(658\) −5332.73 −0.315944
\(659\) 28327.8 1.67450 0.837249 0.546822i \(-0.184163\pi\)
0.837249 + 0.546822i \(0.184163\pi\)
\(660\) 0 0
\(661\) −32190.9 −1.89422 −0.947112 0.320905i \(-0.896013\pi\)
−0.947112 + 0.320905i \(0.896013\pi\)
\(662\) −14265.0 −0.837498
\(663\) 0 0
\(664\) −15569.8 −0.909981
\(665\) 0 0
\(666\) 0 0
\(667\) 9198.01 0.533955
\(668\) 6975.84 0.404047
\(669\) 0 0
\(670\) 0 0
\(671\) −7858.46 −0.452120
\(672\) 0 0
\(673\) 6207.38 0.355538 0.177769 0.984072i \(-0.443112\pi\)
0.177769 + 0.984072i \(0.443112\pi\)
\(674\) 5418.66 0.309672
\(675\) 0 0
\(676\) 2925.82 0.166467
\(677\) 28831.1 1.63674 0.818368 0.574695i \(-0.194879\pi\)
0.818368 + 0.574695i \(0.194879\pi\)
\(678\) 0 0
\(679\) 9901.37 0.559617
\(680\) 0 0
\(681\) 0 0
\(682\) 940.250 0.0527918
\(683\) 3193.10 0.178888 0.0894441 0.995992i \(-0.471491\pi\)
0.0894441 + 0.995992i \(0.471491\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −9296.24 −0.517394
\(687\) 0 0
\(688\) 27.0441 0.00149861
\(689\) 26163.8 1.44668
\(690\) 0 0
\(691\) 7682.49 0.422946 0.211473 0.977384i \(-0.432174\pi\)
0.211473 + 0.977384i \(0.432174\pi\)
\(692\) 6711.17 0.368671
\(693\) 0 0
\(694\) −139.496 −0.00762996
\(695\) 0 0
\(696\) 0 0
\(697\) 33213.3 1.80494
\(698\) −233.256 −0.0126488
\(699\) 0 0
\(700\) 0 0
\(701\) −26551.6 −1.43058 −0.715292 0.698825i \(-0.753707\pi\)
−0.715292 + 0.698825i \(0.753707\pi\)
\(702\) 0 0
\(703\) 1779.86 0.0954887
\(704\) 2211.24 0.118380
\(705\) 0 0
\(706\) 12285.7 0.654928
\(707\) −3571.83 −0.190004
\(708\) 0 0
\(709\) −16304.6 −0.863655 −0.431828 0.901956i \(-0.642131\pi\)
−0.431828 + 0.901956i \(0.642131\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 8171.79 0.430127
\(713\) 3377.00 0.177377
\(714\) 0 0
\(715\) 0 0
\(716\) −8021.36 −0.418676
\(717\) 0 0
\(718\) −7767.50 −0.403733
\(719\) 3973.62 0.206107 0.103053 0.994676i \(-0.467139\pi\)
0.103053 + 0.994676i \(0.467139\pi\)
\(720\) 0 0
\(721\) 15742.1 0.813128
\(722\) −8256.25 −0.425576
\(723\) 0 0
\(724\) −23702.3 −1.21670
\(725\) 0 0
\(726\) 0 0
\(727\) 10780.4 0.549961 0.274980 0.961450i \(-0.411329\pi\)
0.274980 + 0.961450i \(0.411329\pi\)
\(728\) −8869.67 −0.451555
\(729\) 0 0
\(730\) 0 0
\(731\) −233.687 −0.0118238
\(732\) 0 0
\(733\) 9211.46 0.464165 0.232083 0.972696i \(-0.425446\pi\)
0.232083 + 0.972696i \(0.425446\pi\)
\(734\) 20716.6 1.04177
\(735\) 0 0
\(736\) 11552.3 0.578566
\(737\) 4454.83 0.222654
\(738\) 0 0
\(739\) 11084.7 0.551768 0.275884 0.961191i \(-0.411029\pi\)
0.275884 + 0.961191i \(0.411029\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 10241.0 0.506685
\(743\) −27420.4 −1.35391 −0.676955 0.736024i \(-0.736701\pi\)
−0.676955 + 0.736024i \(0.736701\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 7234.32 0.355050
\(747\) 0 0
\(748\) 6038.69 0.295182
\(749\) −7983.64 −0.389474
\(750\) 0 0
\(751\) −11290.8 −0.548614 −0.274307 0.961642i \(-0.588448\pi\)
−0.274307 + 0.961642i \(0.588448\pi\)
\(752\) −3807.33 −0.184626
\(753\) 0 0
\(754\) 9516.81 0.459657
\(755\) 0 0
\(756\) 0 0
\(757\) −3739.19 −0.179528 −0.0897642 0.995963i \(-0.528611\pi\)
−0.0897642 + 0.995963i \(0.528611\pi\)
\(758\) 10155.3 0.486617
\(759\) 0 0
\(760\) 0 0
\(761\) 15621.1 0.744107 0.372053 0.928211i \(-0.378654\pi\)
0.372053 + 0.928211i \(0.378654\pi\)
\(762\) 0 0
\(763\) −15387.7 −0.730106
\(764\) −4739.69 −0.224445
\(765\) 0 0
\(766\) 19885.7 0.937987
\(767\) 15156.0 0.713498
\(768\) 0 0
\(769\) 40241.7 1.88706 0.943531 0.331284i \(-0.107482\pi\)
0.943531 + 0.331284i \(0.107482\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −13681.2 −0.637818
\(773\) 22821.4 1.06187 0.530936 0.847412i \(-0.321840\pi\)
0.530936 + 0.847412i \(0.321840\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −20464.4 −0.946685
\(777\) 0 0
\(778\) −18777.0 −0.865282
\(779\) 13340.0 0.613549
\(780\) 0 0
\(781\) −10332.2 −0.473387
\(782\) −9509.28 −0.434848
\(783\) 0 0
\(784\) −2718.92 −0.123857
\(785\) 0 0
\(786\) 0 0
\(787\) 29454.3 1.33410 0.667048 0.745015i \(-0.267558\pi\)
0.667048 + 0.745015i \(0.267558\pi\)
\(788\) −19337.8 −0.874216
\(789\) 0 0
\(790\) 0 0
\(791\) 1741.86 0.0782974
\(792\) 0 0
\(793\) −29202.7 −1.30771
\(794\) 8157.00 0.364586
\(795\) 0 0
\(796\) −22218.6 −0.989344
\(797\) −27440.3 −1.21955 −0.609777 0.792573i \(-0.708741\pi\)
−0.609777 + 0.792573i \(0.708741\pi\)
\(798\) 0 0
\(799\) 32899.0 1.45668
\(800\) 0 0
\(801\) 0 0
\(802\) −15066.1 −0.663346
\(803\) 3988.27 0.175272
\(804\) 0 0
\(805\) 0 0
\(806\) 3494.05 0.152695
\(807\) 0 0
\(808\) 7382.34 0.321423
\(809\) 5060.18 0.219909 0.109954 0.993937i \(-0.464930\pi\)
0.109954 + 0.993937i \(0.464930\pi\)
\(810\) 0 0
\(811\) 30480.1 1.31973 0.659865 0.751384i \(-0.270613\pi\)
0.659865 + 0.751384i \(0.270613\pi\)
\(812\) −8496.03 −0.367183
\(813\) 0 0
\(814\) −771.146 −0.0332048
\(815\) 0 0
\(816\) 0 0
\(817\) −93.8596 −0.00401925
\(818\) −3139.46 −0.134192
\(819\) 0 0
\(820\) 0 0
\(821\) −37909.0 −1.61149 −0.805745 0.592263i \(-0.798235\pi\)
−0.805745 + 0.592263i \(0.798235\pi\)
\(822\) 0 0
\(823\) −23636.0 −1.00109 −0.500546 0.865710i \(-0.666867\pi\)
−0.500546 + 0.865710i \(0.666867\pi\)
\(824\) −32536.0 −1.37554
\(825\) 0 0
\(826\) 5932.36 0.249895
\(827\) −42634.3 −1.79267 −0.896336 0.443376i \(-0.853781\pi\)
−0.896336 + 0.443376i \(0.853781\pi\)
\(828\) 0 0
\(829\) −45152.5 −1.89169 −0.945845 0.324619i \(-0.894764\pi\)
−0.945845 + 0.324619i \(0.894764\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 8217.15 0.342402
\(833\) 23494.1 0.977217
\(834\) 0 0
\(835\) 0 0
\(836\) 2425.42 0.100341
\(837\) 0 0
\(838\) −6925.91 −0.285503
\(839\) −30431.5 −1.25222 −0.626110 0.779734i \(-0.715354\pi\)
−0.626110 + 0.779734i \(0.715354\pi\)
\(840\) 0 0
\(841\) −2160.34 −0.0885784
\(842\) 23763.6 0.972623
\(843\) 0 0
\(844\) −5818.09 −0.237283
\(845\) 0 0
\(846\) 0 0
\(847\) 1239.79 0.0502949
\(848\) 7311.64 0.296088
\(849\) 0 0
\(850\) 0 0
\(851\) −2769.65 −0.111566
\(852\) 0 0
\(853\) −10367.2 −0.416139 −0.208070 0.978114i \(-0.566718\pi\)
−0.208070 + 0.978114i \(0.566718\pi\)
\(854\) −11430.5 −0.458013
\(855\) 0 0
\(856\) 16500.8 0.658860
\(857\) 12947.1 0.516063 0.258032 0.966136i \(-0.416926\pi\)
0.258032 + 0.966136i \(0.416926\pi\)
\(858\) 0 0
\(859\) −20383.5 −0.809636 −0.404818 0.914397i \(-0.632665\pi\)
−0.404818 + 0.914397i \(0.632665\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 8779.22 0.346893
\(863\) 9056.42 0.357224 0.178612 0.983920i \(-0.442839\pi\)
0.178612 + 0.983920i \(0.442839\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 22340.0 0.876610
\(867\) 0 0
\(868\) −3119.27 −0.121976
\(869\) 10470.2 0.408719
\(870\) 0 0
\(871\) 16554.5 0.644005
\(872\) 31803.6 1.23510
\(873\) 0 0
\(874\) −3819.37 −0.147817
\(875\) 0 0
\(876\) 0 0
\(877\) 2867.88 0.110424 0.0552118 0.998475i \(-0.482417\pi\)
0.0552118 + 0.998475i \(0.482417\pi\)
\(878\) 6846.16 0.263151
\(879\) 0 0
\(880\) 0 0
\(881\) 11862.5 0.453640 0.226820 0.973937i \(-0.427167\pi\)
0.226820 + 0.973937i \(0.427167\pi\)
\(882\) 0 0
\(883\) −33463.8 −1.27537 −0.637683 0.770299i \(-0.720107\pi\)
−0.637683 + 0.770299i \(0.720107\pi\)
\(884\) 22440.3 0.853787
\(885\) 0 0
\(886\) −15756.0 −0.597442
\(887\) 2420.75 0.0916357 0.0458178 0.998950i \(-0.485411\pi\)
0.0458178 + 0.998950i \(0.485411\pi\)
\(888\) 0 0
\(889\) 17826.5 0.672534
\(890\) 0 0
\(891\) 0 0
\(892\) −2815.62 −0.105688
\(893\) 13213.8 0.495165
\(894\) 0 0
\(895\) 0 0
\(896\) −12132.9 −0.452378
\(897\) 0 0
\(898\) −14963.6 −0.556060
\(899\) 8161.14 0.302769
\(900\) 0 0
\(901\) −63179.7 −2.33609
\(902\) −5779.73 −0.213352
\(903\) 0 0
\(904\) −3600.10 −0.132453
\(905\) 0 0
\(906\) 0 0
\(907\) 38154.1 1.39679 0.698393 0.715714i \(-0.253899\pi\)
0.698393 + 0.715714i \(0.253899\pi\)
\(908\) 23838.4 0.871262
\(909\) 0 0
\(910\) 0 0
\(911\) 35758.0 1.30045 0.650227 0.759740i \(-0.274674\pi\)
0.650227 + 0.759740i \(0.274674\pi\)
\(912\) 0 0
\(913\) 8087.44 0.293160
\(914\) −15614.5 −0.565079
\(915\) 0 0
\(916\) −31753.0 −1.14536
\(917\) −3205.39 −0.115432
\(918\) 0 0
\(919\) 17387.5 0.624115 0.312058 0.950063i \(-0.398982\pi\)
0.312058 + 0.950063i \(0.398982\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 17357.9 0.620013
\(923\) −38395.3 −1.36923
\(924\) 0 0
\(925\) 0 0
\(926\) 2447.20 0.0868467
\(927\) 0 0
\(928\) 27918.3 0.987569
\(929\) −6955.93 −0.245658 −0.122829 0.992428i \(-0.539197\pi\)
−0.122829 + 0.992428i \(0.539197\pi\)
\(930\) 0 0
\(931\) 9436.31 0.332183
\(932\) −16384.9 −0.575863
\(933\) 0 0
\(934\) 19751.7 0.691965
\(935\) 0 0
\(936\) 0 0
\(937\) 16074.5 0.560438 0.280219 0.959936i \(-0.409593\pi\)
0.280219 + 0.959936i \(0.409593\pi\)
\(938\) 6479.76 0.225556
\(939\) 0 0
\(940\) 0 0
\(941\) 687.126 0.0238041 0.0119021 0.999929i \(-0.496211\pi\)
0.0119021 + 0.999929i \(0.496211\pi\)
\(942\) 0 0
\(943\) −20758.5 −0.716849
\(944\) 4235.44 0.146030
\(945\) 0 0
\(946\) 40.6659 0.00139763
\(947\) 35352.0 1.21308 0.606540 0.795053i \(-0.292557\pi\)
0.606540 + 0.795053i \(0.292557\pi\)
\(948\) 0 0
\(949\) 14820.7 0.506956
\(950\) 0 0
\(951\) 0 0
\(952\) 21418.2 0.729168
\(953\) −19390.7 −0.659103 −0.329552 0.944138i \(-0.606898\pi\)
−0.329552 + 0.944138i \(0.606898\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −11561.9 −0.391148
\(957\) 0 0
\(958\) 16738.7 0.564511
\(959\) −7345.80 −0.247349
\(960\) 0 0
\(961\) −26794.7 −0.899422
\(962\) −2865.64 −0.0960416
\(963\) 0 0
\(964\) −10307.6 −0.344384
\(965\) 0 0
\(966\) 0 0
\(967\) −28643.6 −0.952551 −0.476275 0.879296i \(-0.658013\pi\)
−0.476275 + 0.879296i \(0.658013\pi\)
\(968\) −2562.43 −0.0850821
\(969\) 0 0
\(970\) 0 0
\(971\) 19574.8 0.646946 0.323473 0.946237i \(-0.395149\pi\)
0.323473 + 0.946237i \(0.395149\pi\)
\(972\) 0 0
\(973\) −8980.63 −0.295895
\(974\) −11182.6 −0.367878
\(975\) 0 0
\(976\) −8160.86 −0.267646
\(977\) 50095.5 1.64043 0.820213 0.572058i \(-0.193855\pi\)
0.820213 + 0.572058i \(0.193855\pi\)
\(978\) 0 0
\(979\) −4244.67 −0.138570
\(980\) 0 0
\(981\) 0 0
\(982\) −22747.6 −0.739211
\(983\) 14445.4 0.468706 0.234353 0.972152i \(-0.424703\pi\)
0.234353 + 0.972152i \(0.424703\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −22980.9 −0.742253
\(987\) 0 0
\(988\) 9013.05 0.290226
\(989\) 146.056 0.00469595
\(990\) 0 0
\(991\) 29120.1 0.933430 0.466715 0.884408i \(-0.345437\pi\)
0.466715 + 0.884408i \(0.345437\pi\)
\(992\) 10250.1 0.328064
\(993\) 0 0
\(994\) −15028.7 −0.479558
\(995\) 0 0
\(996\) 0 0
\(997\) 9137.45 0.290257 0.145128 0.989413i \(-0.453640\pi\)
0.145128 + 0.989413i \(0.453640\pi\)
\(998\) −7244.03 −0.229765
\(999\) 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 2475.4.a.n.1.2 2
3.2 odd 2 825.4.a.m.1.1 2
5.4 even 2 495.4.a.d.1.1 2
15.2 even 4 825.4.c.j.199.2 4
15.8 even 4 825.4.c.j.199.3 4
15.14 odd 2 165.4.a.c.1.2 2
165.164 even 2 1815.4.a.n.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.4.a.c.1.2 2 15.14 odd 2
495.4.a.d.1.1 2 5.4 even 2
825.4.a.m.1.1 2 3.2 odd 2
825.4.c.j.199.2 4 15.2 even 4
825.4.c.j.199.3 4 15.8 even 4
1815.4.a.n.1.1 2 165.164 even 2
2475.4.a.n.1.2 2 1.1 even 1 trivial