Properties

Label 1856.4.a.n.1.1
Level $1856$
Weight $4$
Character 1856.1
Self dual yes
Analytic conductor $109.508$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1856 = 2^{6} \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1856.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(109.507544971\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 29)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 1856.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.757359 q^{3} +10.6569 q^{5} -22.1421 q^{7} -26.4264 q^{9} +O(q^{10})\) \(q+0.757359 q^{3} +10.6569 q^{5} -22.1421 q^{7} -26.4264 q^{9} -39.3259 q^{11} -23.7696 q^{13} +8.07107 q^{15} +4.54416 q^{17} +155.255 q^{19} -16.7696 q^{21} -41.8823 q^{23} -11.4315 q^{25} -40.4630 q^{27} -29.0000 q^{29} -57.9045 q^{31} -29.7838 q^{33} -235.966 q^{35} -235.196 q^{37} -18.0021 q^{39} -175.161 q^{41} +402.831 q^{43} -281.622 q^{45} +227.742 q^{47} +147.274 q^{49} +3.44156 q^{51} -673.534 q^{53} -419.090 q^{55} +117.584 q^{57} +800.725 q^{59} +222.270 q^{61} +585.137 q^{63} -253.309 q^{65} +524.479 q^{67} -31.7199 q^{69} -281.917 q^{71} +1229.10 q^{73} -8.65772 q^{75} +870.759 q^{77} +611.247 q^{79} +682.868 q^{81} -515.490 q^{83} +48.4264 q^{85} -21.9634 q^{87} -358.219 q^{89} +526.309 q^{91} -43.8545 q^{93} +1654.53 q^{95} +829.415 q^{97} +1039.24 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 10 q^{3} + 10 q^{5} - 16 q^{7} + 32 q^{9} + O(q^{10}) \) \( 2 q + 10 q^{3} + 10 q^{5} - 16 q^{7} + 32 q^{9} + 26 q^{11} + 26 q^{13} + 2 q^{15} + 60 q^{17} + 220 q^{19} + 40 q^{21} + 52 q^{23} - 136 q^{25} + 250 q^{27} - 58 q^{29} - 294 q^{31} + 574 q^{33} - 240 q^{35} - 312 q^{37} + 442 q^{39} + 40 q^{41} + 322 q^{43} - 320 q^{45} - 130 q^{47} - 158 q^{49} + 516 q^{51} - 1002 q^{53} - 462 q^{55} + 716 q^{57} + 900 q^{59} + 948 q^{61} + 944 q^{63} - 286 q^{65} - 320 q^{67} + 836 q^{69} - 660 q^{71} + 648 q^{73} - 1160 q^{75} + 1272 q^{77} + 258 q^{79} + 1790 q^{81} - 1212 q^{83} + 12 q^{85} - 290 q^{87} + 760 q^{89} + 832 q^{91} - 2226 q^{93} + 1612 q^{95} + 24 q^{97} + 4856 q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0.757359 0.145754 0.0728769 0.997341i \(-0.476782\pi\)
0.0728769 + 0.997341i \(0.476782\pi\)
\(4\) 0 0
\(5\) 10.6569 0.953178 0.476589 0.879126i \(-0.341873\pi\)
0.476589 + 0.879126i \(0.341873\pi\)
\(6\) 0 0
\(7\) −22.1421 −1.19556 −0.597781 0.801659i \(-0.703951\pi\)
−0.597781 + 0.801659i \(0.703951\pi\)
\(8\) 0 0
\(9\) −26.4264 −0.978756
\(10\) 0 0
\(11\) −39.3259 −1.07793 −0.538964 0.842329i \(-0.681184\pi\)
−0.538964 + 0.842329i \(0.681184\pi\)
\(12\) 0 0
\(13\) −23.7696 −0.507114 −0.253557 0.967320i \(-0.581601\pi\)
−0.253557 + 0.967320i \(0.581601\pi\)
\(14\) 0 0
\(15\) 8.07107 0.138929
\(16\) 0 0
\(17\) 4.54416 0.0648306 0.0324153 0.999474i \(-0.489680\pi\)
0.0324153 + 0.999474i \(0.489680\pi\)
\(18\) 0 0
\(19\) 155.255 1.87463 0.937313 0.348488i \(-0.113305\pi\)
0.937313 + 0.348488i \(0.113305\pi\)
\(20\) 0 0
\(21\) −16.7696 −0.174258
\(22\) 0 0
\(23\) −41.8823 −0.379698 −0.189849 0.981813i \(-0.560800\pi\)
−0.189849 + 0.981813i \(0.560800\pi\)
\(24\) 0 0
\(25\) −11.4315 −0.0914517
\(26\) 0 0
\(27\) −40.4630 −0.288411
\(28\) 0 0
\(29\) −29.0000 −0.185695
\(30\) 0 0
\(31\) −57.9045 −0.335483 −0.167741 0.985831i \(-0.553647\pi\)
−0.167741 + 0.985831i \(0.553647\pi\)
\(32\) 0 0
\(33\) −29.7838 −0.157112
\(34\) 0 0
\(35\) −235.966 −1.13958
\(36\) 0 0
\(37\) −235.196 −1.04503 −0.522513 0.852631i \(-0.675005\pi\)
−0.522513 + 0.852631i \(0.675005\pi\)
\(38\) 0 0
\(39\) −18.0021 −0.0739139
\(40\) 0 0
\(41\) −175.161 −0.667210 −0.333605 0.942713i \(-0.608265\pi\)
−0.333605 + 0.942713i \(0.608265\pi\)
\(42\) 0 0
\(43\) 402.831 1.42863 0.714315 0.699824i \(-0.246738\pi\)
0.714315 + 0.699824i \(0.246738\pi\)
\(44\) 0 0
\(45\) −281.622 −0.932929
\(46\) 0 0
\(47\) 227.742 0.706800 0.353400 0.935472i \(-0.385025\pi\)
0.353400 + 0.935472i \(0.385025\pi\)
\(48\) 0 0
\(49\) 147.274 0.429371
\(50\) 0 0
\(51\) 3.44156 0.00944931
\(52\) 0 0
\(53\) −673.534 −1.74560 −0.872802 0.488074i \(-0.837699\pi\)
−0.872802 + 0.488074i \(0.837699\pi\)
\(54\) 0 0
\(55\) −419.090 −1.02746
\(56\) 0 0
\(57\) 117.584 0.273234
\(58\) 0 0
\(59\) 800.725 1.76687 0.883437 0.468551i \(-0.155224\pi\)
0.883437 + 0.468551i \(0.155224\pi\)
\(60\) 0 0
\(61\) 222.270 0.466537 0.233268 0.972412i \(-0.425058\pi\)
0.233268 + 0.972412i \(0.425058\pi\)
\(62\) 0 0
\(63\) 585.137 1.17016
\(64\) 0 0
\(65\) −253.309 −0.483370
\(66\) 0 0
\(67\) 524.479 0.956349 0.478174 0.878265i \(-0.341299\pi\)
0.478174 + 0.878265i \(0.341299\pi\)
\(68\) 0 0
\(69\) −31.7199 −0.0553424
\(70\) 0 0
\(71\) −281.917 −0.471230 −0.235615 0.971846i \(-0.575711\pi\)
−0.235615 + 0.971846i \(0.575711\pi\)
\(72\) 0 0
\(73\) 1229.10 1.97061 0.985307 0.170790i \(-0.0546320\pi\)
0.985307 + 0.170790i \(0.0546320\pi\)
\(74\) 0 0
\(75\) −8.65772 −0.0133294
\(76\) 0 0
\(77\) 870.759 1.28873
\(78\) 0 0
\(79\) 611.247 0.870514 0.435257 0.900306i \(-0.356657\pi\)
0.435257 + 0.900306i \(0.356657\pi\)
\(80\) 0 0
\(81\) 682.868 0.936719
\(82\) 0 0
\(83\) −515.490 −0.681716 −0.340858 0.940115i \(-0.610718\pi\)
−0.340858 + 0.940115i \(0.610718\pi\)
\(84\) 0 0
\(85\) 48.4264 0.0617951
\(86\) 0 0
\(87\) −21.9634 −0.0270658
\(88\) 0 0
\(89\) −358.219 −0.426643 −0.213321 0.976982i \(-0.568428\pi\)
−0.213321 + 0.976982i \(0.568428\pi\)
\(90\) 0 0
\(91\) 526.309 0.606287
\(92\) 0 0
\(93\) −43.8545 −0.0488979
\(94\) 0 0
\(95\) 1654.53 1.78685
\(96\) 0 0
\(97\) 829.415 0.868189 0.434095 0.900867i \(-0.357068\pi\)
0.434095 + 0.900867i \(0.357068\pi\)
\(98\) 0 0
\(99\) 1039.24 1.05503
\(100\) 0 0
\(101\) 978.010 0.963521 0.481761 0.876303i \(-0.339997\pi\)
0.481761 + 0.876303i \(0.339997\pi\)
\(102\) 0 0
\(103\) −1217.33 −1.16453 −0.582266 0.812998i \(-0.697834\pi\)
−0.582266 + 0.812998i \(0.697834\pi\)
\(104\) 0 0
\(105\) −178.711 −0.166099
\(106\) 0 0
\(107\) 707.044 0.638808 0.319404 0.947619i \(-0.396517\pi\)
0.319404 + 0.947619i \(0.396517\pi\)
\(108\) 0 0
\(109\) 1496.14 1.31471 0.657357 0.753580i \(-0.271674\pi\)
0.657357 + 0.753580i \(0.271674\pi\)
\(110\) 0 0
\(111\) −178.128 −0.152317
\(112\) 0 0
\(113\) −1067.23 −0.888469 −0.444234 0.895911i \(-0.646524\pi\)
−0.444234 + 0.895911i \(0.646524\pi\)
\(114\) 0 0
\(115\) −446.333 −0.361920
\(116\) 0 0
\(117\) 628.144 0.496341
\(118\) 0 0
\(119\) −100.617 −0.0775090
\(120\) 0 0
\(121\) 215.527 0.161928
\(122\) 0 0
\(123\) −132.660 −0.0972485
\(124\) 0 0
\(125\) −1453.93 −1.04035
\(126\) 0 0
\(127\) −1179.58 −0.824177 −0.412088 0.911144i \(-0.635201\pi\)
−0.412088 + 0.911144i \(0.635201\pi\)
\(128\) 0 0
\(129\) 305.087 0.208228
\(130\) 0 0
\(131\) 2357.47 1.57231 0.786156 0.618028i \(-0.212068\pi\)
0.786156 + 0.618028i \(0.212068\pi\)
\(132\) 0 0
\(133\) −3437.67 −2.24123
\(134\) 0 0
\(135\) −431.208 −0.274907
\(136\) 0 0
\(137\) 722.489 0.450558 0.225279 0.974294i \(-0.427671\pi\)
0.225279 + 0.974294i \(0.427671\pi\)
\(138\) 0 0
\(139\) −1398.24 −0.853219 −0.426610 0.904436i \(-0.640292\pi\)
−0.426610 + 0.904436i \(0.640292\pi\)
\(140\) 0 0
\(141\) 172.483 0.103019
\(142\) 0 0
\(143\) 934.759 0.546633
\(144\) 0 0
\(145\) −309.049 −0.177001
\(146\) 0 0
\(147\) 111.539 0.0625824
\(148\) 0 0
\(149\) −2830.63 −1.55634 −0.778168 0.628056i \(-0.783851\pi\)
−0.778168 + 0.628056i \(0.783851\pi\)
\(150\) 0 0
\(151\) 1705.58 0.919194 0.459597 0.888128i \(-0.347994\pi\)
0.459597 + 0.888128i \(0.347994\pi\)
\(152\) 0 0
\(153\) −120.086 −0.0634533
\(154\) 0 0
\(155\) −617.080 −0.319775
\(156\) 0 0
\(157\) 2670.84 1.35768 0.678841 0.734286i \(-0.262483\pi\)
0.678841 + 0.734286i \(0.262483\pi\)
\(158\) 0 0
\(159\) −510.107 −0.254429
\(160\) 0 0
\(161\) 927.362 0.453953
\(162\) 0 0
\(163\) 2151.92 1.03406 0.517028 0.855968i \(-0.327038\pi\)
0.517028 + 0.855968i \(0.327038\pi\)
\(164\) 0 0
\(165\) −317.402 −0.149756
\(166\) 0 0
\(167\) 999.387 0.463083 0.231542 0.972825i \(-0.425623\pi\)
0.231542 + 0.972825i \(0.425623\pi\)
\(168\) 0 0
\(169\) −1632.01 −0.742835
\(170\) 0 0
\(171\) −4102.83 −1.83480
\(172\) 0 0
\(173\) 2534.30 1.11375 0.556877 0.830595i \(-0.311999\pi\)
0.556877 + 0.830595i \(0.311999\pi\)
\(174\) 0 0
\(175\) 253.117 0.109336
\(176\) 0 0
\(177\) 606.437 0.257529
\(178\) 0 0
\(179\) −3550.27 −1.48245 −0.741227 0.671254i \(-0.765756\pi\)
−0.741227 + 0.671254i \(0.765756\pi\)
\(180\) 0 0
\(181\) 3034.68 1.24622 0.623110 0.782135i \(-0.285869\pi\)
0.623110 + 0.782135i \(0.285869\pi\)
\(182\) 0 0
\(183\) 168.338 0.0679996
\(184\) 0 0
\(185\) −2506.45 −0.996096
\(186\) 0 0
\(187\) −178.703 −0.0698827
\(188\) 0 0
\(189\) 895.937 0.344814
\(190\) 0 0
\(191\) 2224.00 0.842529 0.421265 0.906938i \(-0.361586\pi\)
0.421265 + 0.906938i \(0.361586\pi\)
\(192\) 0 0
\(193\) −632.830 −0.236021 −0.118011 0.993012i \(-0.537652\pi\)
−0.118011 + 0.993012i \(0.537652\pi\)
\(194\) 0 0
\(195\) −191.846 −0.0704531
\(196\) 0 0
\(197\) −1369.80 −0.495404 −0.247702 0.968836i \(-0.579675\pi\)
−0.247702 + 0.968836i \(0.579675\pi\)
\(198\) 0 0
\(199\) 1416.38 0.504544 0.252272 0.967656i \(-0.418822\pi\)
0.252272 + 0.967656i \(0.418822\pi\)
\(200\) 0 0
\(201\) 397.219 0.139391
\(202\) 0 0
\(203\) 642.122 0.222010
\(204\) 0 0
\(205\) −1866.67 −0.635970
\(206\) 0 0
\(207\) 1106.80 0.371632
\(208\) 0 0
\(209\) −6105.54 −2.02071
\(210\) 0 0
\(211\) 896.432 0.292478 0.146239 0.989249i \(-0.453283\pi\)
0.146239 + 0.989249i \(0.453283\pi\)
\(212\) 0 0
\(213\) −213.512 −0.0686837
\(214\) 0 0
\(215\) 4292.91 1.36174
\(216\) 0 0
\(217\) 1282.13 0.401091
\(218\) 0 0
\(219\) 930.868 0.287225
\(220\) 0 0
\(221\) −108.013 −0.0328765
\(222\) 0 0
\(223\) −2268.94 −0.681344 −0.340672 0.940182i \(-0.610655\pi\)
−0.340672 + 0.940182i \(0.610655\pi\)
\(224\) 0 0
\(225\) 302.092 0.0895088
\(226\) 0 0
\(227\) 2078.09 0.607610 0.303805 0.952734i \(-0.401743\pi\)
0.303805 + 0.952734i \(0.401743\pi\)
\(228\) 0 0
\(229\) 3715.05 1.07204 0.536020 0.844205i \(-0.319927\pi\)
0.536020 + 0.844205i \(0.319927\pi\)
\(230\) 0 0
\(231\) 659.478 0.187837
\(232\) 0 0
\(233\) 2521.35 0.708923 0.354461 0.935071i \(-0.384664\pi\)
0.354461 + 0.935071i \(0.384664\pi\)
\(234\) 0 0
\(235\) 2427.02 0.673707
\(236\) 0 0
\(237\) 462.933 0.126881
\(238\) 0 0
\(239\) 3940.04 1.06636 0.533179 0.846002i \(-0.320997\pi\)
0.533179 + 0.846002i \(0.320997\pi\)
\(240\) 0 0
\(241\) −1973.06 −0.527369 −0.263684 0.964609i \(-0.584938\pi\)
−0.263684 + 0.964609i \(0.584938\pi\)
\(242\) 0 0
\(243\) 1609.68 0.424942
\(244\) 0 0
\(245\) 1569.48 0.409267
\(246\) 0 0
\(247\) −3690.34 −0.950650
\(248\) 0 0
\(249\) −390.411 −0.0993627
\(250\) 0 0
\(251\) 1236.65 0.310982 0.155491 0.987837i \(-0.450304\pi\)
0.155491 + 0.987837i \(0.450304\pi\)
\(252\) 0 0
\(253\) 1647.06 0.409287
\(254\) 0 0
\(255\) 36.6762 0.00900687
\(256\) 0 0
\(257\) 2918.10 0.708272 0.354136 0.935194i \(-0.384775\pi\)
0.354136 + 0.935194i \(0.384775\pi\)
\(258\) 0 0
\(259\) 5207.74 1.24939
\(260\) 0 0
\(261\) 766.366 0.181750
\(262\) 0 0
\(263\) −310.789 −0.0728673 −0.0364336 0.999336i \(-0.511600\pi\)
−0.0364336 + 0.999336i \(0.511600\pi\)
\(264\) 0 0
\(265\) −7177.75 −1.66387
\(266\) 0 0
\(267\) −271.301 −0.0621848
\(268\) 0 0
\(269\) 1839.98 0.417047 0.208523 0.978017i \(-0.433134\pi\)
0.208523 + 0.978017i \(0.433134\pi\)
\(270\) 0 0
\(271\) 5187.35 1.16277 0.581383 0.813630i \(-0.302512\pi\)
0.581383 + 0.813630i \(0.302512\pi\)
\(272\) 0 0
\(273\) 398.605 0.0883687
\(274\) 0 0
\(275\) 449.552 0.0985783
\(276\) 0 0
\(277\) −8666.41 −1.87983 −0.939917 0.341403i \(-0.889098\pi\)
−0.939917 + 0.341403i \(0.889098\pi\)
\(278\) 0 0
\(279\) 1530.21 0.328356
\(280\) 0 0
\(281\) 7856.96 1.66800 0.833998 0.551767i \(-0.186046\pi\)
0.833998 + 0.551767i \(0.186046\pi\)
\(282\) 0 0
\(283\) 3054.71 0.641638 0.320819 0.947141i \(-0.396042\pi\)
0.320819 + 0.947141i \(0.396042\pi\)
\(284\) 0 0
\(285\) 1253.07 0.260441
\(286\) 0 0
\(287\) 3878.45 0.797692
\(288\) 0 0
\(289\) −4892.35 −0.995797
\(290\) 0 0
\(291\) 628.166 0.126542
\(292\) 0 0
\(293\) −2847.83 −0.567822 −0.283911 0.958851i \(-0.591632\pi\)
−0.283911 + 0.958851i \(0.591632\pi\)
\(294\) 0 0
\(295\) 8533.21 1.68414
\(296\) 0 0
\(297\) 1591.24 0.310887
\(298\) 0 0
\(299\) 995.522 0.192550
\(300\) 0 0
\(301\) −8919.53 −1.70802
\(302\) 0 0
\(303\) 740.705 0.140437
\(304\) 0 0
\(305\) 2368.70 0.444693
\(306\) 0 0
\(307\) 2470.79 0.459334 0.229667 0.973269i \(-0.426236\pi\)
0.229667 + 0.973269i \(0.426236\pi\)
\(308\) 0 0
\(309\) −921.955 −0.169735
\(310\) 0 0
\(311\) −5653.29 −1.03077 −0.515384 0.856959i \(-0.672351\pi\)
−0.515384 + 0.856959i \(0.672351\pi\)
\(312\) 0 0
\(313\) −8289.72 −1.49701 −0.748503 0.663132i \(-0.769227\pi\)
−0.748503 + 0.663132i \(0.769227\pi\)
\(314\) 0 0
\(315\) 6235.72 1.11537
\(316\) 0 0
\(317\) −2577.06 −0.456600 −0.228300 0.973591i \(-0.573317\pi\)
−0.228300 + 0.973591i \(0.573317\pi\)
\(318\) 0 0
\(319\) 1140.45 0.200166
\(320\) 0 0
\(321\) 535.486 0.0931088
\(322\) 0 0
\(323\) 705.502 0.121533
\(324\) 0 0
\(325\) 271.721 0.0463765
\(326\) 0 0
\(327\) 1133.11 0.191625
\(328\) 0 0
\(329\) −5042.70 −0.845024
\(330\) 0 0
\(331\) −7672.12 −1.27401 −0.637006 0.770859i \(-0.719827\pi\)
−0.637006 + 0.770859i \(0.719827\pi\)
\(332\) 0 0
\(333\) 6215.38 1.02283
\(334\) 0 0
\(335\) 5589.30 0.911570
\(336\) 0 0
\(337\) 3650.77 0.590120 0.295060 0.955479i \(-0.404660\pi\)
0.295060 + 0.955479i \(0.404660\pi\)
\(338\) 0 0
\(339\) −808.280 −0.129498
\(340\) 0 0
\(341\) 2277.15 0.361626
\(342\) 0 0
\(343\) 4333.79 0.682223
\(344\) 0 0
\(345\) −338.034 −0.0527512
\(346\) 0 0
\(347\) 8737.06 1.35167 0.675835 0.737053i \(-0.263783\pi\)
0.675835 + 0.737053i \(0.263783\pi\)
\(348\) 0 0
\(349\) 2143.83 0.328816 0.164408 0.986392i \(-0.447429\pi\)
0.164408 + 0.986392i \(0.447429\pi\)
\(350\) 0 0
\(351\) 961.787 0.146258
\(352\) 0 0
\(353\) −5111.17 −0.770651 −0.385326 0.922781i \(-0.625911\pi\)
−0.385326 + 0.922781i \(0.625911\pi\)
\(354\) 0 0
\(355\) −3004.35 −0.449167
\(356\) 0 0
\(357\) −76.2035 −0.0112972
\(358\) 0 0
\(359\) 4520.87 0.664630 0.332315 0.943168i \(-0.392170\pi\)
0.332315 + 0.943168i \(0.392170\pi\)
\(360\) 0 0
\(361\) 17245.1 2.51422
\(362\) 0 0
\(363\) 163.231 0.0236017
\(364\) 0 0
\(365\) 13098.3 1.87835
\(366\) 0 0
\(367\) −486.272 −0.0691641 −0.0345820 0.999402i \(-0.511010\pi\)
−0.0345820 + 0.999402i \(0.511010\pi\)
\(368\) 0 0
\(369\) 4628.89 0.653036
\(370\) 0 0
\(371\) 14913.5 2.08698
\(372\) 0 0
\(373\) −10053.1 −1.39552 −0.697758 0.716333i \(-0.745819\pi\)
−0.697758 + 0.716333i \(0.745819\pi\)
\(374\) 0 0
\(375\) −1101.15 −0.151635
\(376\) 0 0
\(377\) 689.317 0.0941688
\(378\) 0 0
\(379\) 11602.8 1.57255 0.786273 0.617879i \(-0.212008\pi\)
0.786273 + 0.617879i \(0.212008\pi\)
\(380\) 0 0
\(381\) −893.363 −0.120127
\(382\) 0 0
\(383\) −4161.38 −0.555187 −0.277593 0.960699i \(-0.589537\pi\)
−0.277593 + 0.960699i \(0.589537\pi\)
\(384\) 0 0
\(385\) 9279.56 1.22839
\(386\) 0 0
\(387\) −10645.4 −1.39828
\(388\) 0 0
\(389\) 6843.95 0.892036 0.446018 0.895024i \(-0.352842\pi\)
0.446018 + 0.895024i \(0.352842\pi\)
\(390\) 0 0
\(391\) −190.319 −0.0246160
\(392\) 0 0
\(393\) 1785.45 0.229171
\(394\) 0 0
\(395\) 6513.97 0.829755
\(396\) 0 0
\(397\) −5474.73 −0.692112 −0.346056 0.938214i \(-0.612479\pi\)
−0.346056 + 0.938214i \(0.612479\pi\)
\(398\) 0 0
\(399\) −2603.55 −0.326669
\(400\) 0 0
\(401\) 890.013 0.110836 0.0554178 0.998463i \(-0.482351\pi\)
0.0554178 + 0.998463i \(0.482351\pi\)
\(402\) 0 0
\(403\) 1376.37 0.170128
\(404\) 0 0
\(405\) 7277.22 0.892860
\(406\) 0 0
\(407\) 9249.29 1.12646
\(408\) 0 0
\(409\) 8107.81 0.980209 0.490104 0.871664i \(-0.336959\pi\)
0.490104 + 0.871664i \(0.336959\pi\)
\(410\) 0 0
\(411\) 547.184 0.0656706
\(412\) 0 0
\(413\) −17729.8 −2.11241
\(414\) 0 0
\(415\) −5493.51 −0.649797
\(416\) 0 0
\(417\) −1058.97 −0.124360
\(418\) 0 0
\(419\) −4237.59 −0.494080 −0.247040 0.969005i \(-0.579458\pi\)
−0.247040 + 0.969005i \(0.579458\pi\)
\(420\) 0 0
\(421\) 953.634 0.110397 0.0551987 0.998475i \(-0.482421\pi\)
0.0551987 + 0.998475i \(0.482421\pi\)
\(422\) 0 0
\(423\) −6018.41 −0.691785
\(424\) 0 0
\(425\) −51.9463 −0.00592886
\(426\) 0 0
\(427\) −4921.53 −0.557774
\(428\) 0 0
\(429\) 707.949 0.0796738
\(430\) 0 0
\(431\) −10098.8 −1.12864 −0.564318 0.825557i \(-0.690861\pi\)
−0.564318 + 0.825557i \(0.690861\pi\)
\(432\) 0 0
\(433\) −12833.1 −1.42430 −0.712148 0.702029i \(-0.752278\pi\)
−0.712148 + 0.702029i \(0.752278\pi\)
\(434\) 0 0
\(435\) −234.061 −0.0257985
\(436\) 0 0
\(437\) −6502.42 −0.711792
\(438\) 0 0
\(439\) −14434.8 −1.56932 −0.784662 0.619924i \(-0.787164\pi\)
−0.784662 + 0.619924i \(0.787164\pi\)
\(440\) 0 0
\(441\) −3891.93 −0.420249
\(442\) 0 0
\(443\) 7020.65 0.752959 0.376480 0.926425i \(-0.377134\pi\)
0.376480 + 0.926425i \(0.377134\pi\)
\(444\) 0 0
\(445\) −3817.49 −0.406666
\(446\) 0 0
\(447\) −2143.80 −0.226842
\(448\) 0 0
\(449\) 13210.5 1.38851 0.694254 0.719730i \(-0.255735\pi\)
0.694254 + 0.719730i \(0.255735\pi\)
\(450\) 0 0
\(451\) 6888.38 0.719205
\(452\) 0 0
\(453\) 1291.74 0.133976
\(454\) 0 0
\(455\) 5608.79 0.577900
\(456\) 0 0
\(457\) −2811.18 −0.287749 −0.143875 0.989596i \(-0.545956\pi\)
−0.143875 + 0.989596i \(0.545956\pi\)
\(458\) 0 0
\(459\) −183.870 −0.0186979
\(460\) 0 0
\(461\) 9645.10 0.974441 0.487220 0.873279i \(-0.338011\pi\)
0.487220 + 0.873279i \(0.338011\pi\)
\(462\) 0 0
\(463\) 6923.23 0.694924 0.347462 0.937694i \(-0.387044\pi\)
0.347462 + 0.937694i \(0.387044\pi\)
\(464\) 0 0
\(465\) −467.352 −0.0466084
\(466\) 0 0
\(467\) 124.351 0.0123218 0.00616089 0.999981i \(-0.498039\pi\)
0.00616089 + 0.999981i \(0.498039\pi\)
\(468\) 0 0
\(469\) −11613.1 −1.14337
\(470\) 0 0
\(471\) 2022.78 0.197887
\(472\) 0 0
\(473\) −15841.7 −1.53996
\(474\) 0 0
\(475\) −1774.79 −0.171438
\(476\) 0 0
\(477\) 17799.1 1.70852
\(478\) 0 0
\(479\) −4461.48 −0.425575 −0.212788 0.977098i \(-0.568254\pi\)
−0.212788 + 0.977098i \(0.568254\pi\)
\(480\) 0 0
\(481\) 5590.50 0.529948
\(482\) 0 0
\(483\) 702.347 0.0661654
\(484\) 0 0
\(485\) 8838.96 0.827539
\(486\) 0 0
\(487\) −5660.72 −0.526718 −0.263359 0.964698i \(-0.584830\pi\)
−0.263359 + 0.964698i \(0.584830\pi\)
\(488\) 0 0
\(489\) 1629.77 0.150718
\(490\) 0 0
\(491\) −5587.50 −0.513565 −0.256782 0.966469i \(-0.582662\pi\)
−0.256782 + 0.966469i \(0.582662\pi\)
\(492\) 0 0
\(493\) −131.781 −0.0120387
\(494\) 0 0
\(495\) 11075.1 1.00563
\(496\) 0 0
\(497\) 6242.24 0.563386
\(498\) 0 0
\(499\) 4210.19 0.377703 0.188851 0.982006i \(-0.439524\pi\)
0.188851 + 0.982006i \(0.439524\pi\)
\(500\) 0 0
\(501\) 756.895 0.0674962
\(502\) 0 0
\(503\) 10796.7 0.957063 0.478532 0.878070i \(-0.341169\pi\)
0.478532 + 0.878070i \(0.341169\pi\)
\(504\) 0 0
\(505\) 10422.5 0.918407
\(506\) 0 0
\(507\) −1236.02 −0.108271
\(508\) 0 0
\(509\) −14983.6 −1.30479 −0.652395 0.757879i \(-0.726236\pi\)
−0.652395 + 0.757879i \(0.726236\pi\)
\(510\) 0 0
\(511\) −27214.8 −2.35599
\(512\) 0 0
\(513\) −6282.07 −0.540663
\(514\) 0 0
\(515\) −12972.9 −1.11001
\(516\) 0 0
\(517\) −8956.17 −0.761880
\(518\) 0 0
\(519\) 1919.38 0.162334
\(520\) 0 0
\(521\) −10770.3 −0.905671 −0.452836 0.891594i \(-0.649588\pi\)
−0.452836 + 0.891594i \(0.649588\pi\)
\(522\) 0 0
\(523\) 7538.93 0.630314 0.315157 0.949040i \(-0.397943\pi\)
0.315157 + 0.949040i \(0.397943\pi\)
\(524\) 0 0
\(525\) 191.700 0.0159362
\(526\) 0 0
\(527\) −263.127 −0.0217495
\(528\) 0 0
\(529\) −10412.9 −0.855829
\(530\) 0 0
\(531\) −21160.3 −1.72934
\(532\) 0 0
\(533\) 4163.51 0.338352
\(534\) 0 0
\(535\) 7534.86 0.608898
\(536\) 0 0
\(537\) −2688.83 −0.216073
\(538\) 0 0
\(539\) −5791.69 −0.462831
\(540\) 0 0
\(541\) −595.816 −0.0473496 −0.0236748 0.999720i \(-0.507537\pi\)
−0.0236748 + 0.999720i \(0.507537\pi\)
\(542\) 0 0
\(543\) 2298.34 0.181641
\(544\) 0 0
\(545\) 15944.1 1.25316
\(546\) 0 0
\(547\) −12703.4 −0.992978 −0.496489 0.868043i \(-0.665378\pi\)
−0.496489 + 0.868043i \(0.665378\pi\)
\(548\) 0 0
\(549\) −5873.80 −0.456626
\(550\) 0 0
\(551\) −4502.39 −0.348109
\(552\) 0 0
\(553\) −13534.3 −1.04075
\(554\) 0 0
\(555\) −1898.28 −0.145185
\(556\) 0 0
\(557\) 14973.0 1.13901 0.569503 0.821990i \(-0.307136\pi\)
0.569503 + 0.821990i \(0.307136\pi\)
\(558\) 0 0
\(559\) −9575.10 −0.724479
\(560\) 0 0
\(561\) −135.342 −0.0101857
\(562\) 0 0
\(563\) 10439.3 0.781466 0.390733 0.920504i \(-0.372222\pi\)
0.390733 + 0.920504i \(0.372222\pi\)
\(564\) 0 0
\(565\) −11373.4 −0.846869
\(566\) 0 0
\(567\) −15120.2 −1.11991
\(568\) 0 0
\(569\) −8859.72 −0.652757 −0.326379 0.945239i \(-0.605828\pi\)
−0.326379 + 0.945239i \(0.605828\pi\)
\(570\) 0 0
\(571\) 5078.17 0.372180 0.186090 0.982533i \(-0.440418\pi\)
0.186090 + 0.982533i \(0.440418\pi\)
\(572\) 0 0
\(573\) 1684.37 0.122802
\(574\) 0 0
\(575\) 478.775 0.0347240
\(576\) 0 0
\(577\) 20457.2 1.47599 0.737994 0.674808i \(-0.235773\pi\)
0.737994 + 0.674808i \(0.235773\pi\)
\(578\) 0 0
\(579\) −479.280 −0.0344010
\(580\) 0 0
\(581\) 11414.1 0.815034
\(582\) 0 0
\(583\) 26487.3 1.88164
\(584\) 0 0
\(585\) 6694.04 0.473102
\(586\) 0 0
\(587\) −4355.22 −0.306234 −0.153117 0.988208i \(-0.548931\pi\)
−0.153117 + 0.988208i \(0.548931\pi\)
\(588\) 0 0
\(589\) −8989.96 −0.628905
\(590\) 0 0
\(591\) −1037.43 −0.0722070
\(592\) 0 0
\(593\) 4342.49 0.300716 0.150358 0.988632i \(-0.451957\pi\)
0.150358 + 0.988632i \(0.451957\pi\)
\(594\) 0 0
\(595\) −1072.26 −0.0738799
\(596\) 0 0
\(597\) 1072.71 0.0735392
\(598\) 0 0
\(599\) −12761.8 −0.870506 −0.435253 0.900308i \(-0.643341\pi\)
−0.435253 + 0.900308i \(0.643341\pi\)
\(600\) 0 0
\(601\) −2804.87 −0.190371 −0.0951857 0.995460i \(-0.530344\pi\)
−0.0951857 + 0.995460i \(0.530344\pi\)
\(602\) 0 0
\(603\) −13860.1 −0.936032
\(604\) 0 0
\(605\) 2296.84 0.154346
\(606\) 0 0
\(607\) 12033.5 0.804654 0.402327 0.915496i \(-0.368202\pi\)
0.402327 + 0.915496i \(0.368202\pi\)
\(608\) 0 0
\(609\) 486.317 0.0323589
\(610\) 0 0
\(611\) −5413.33 −0.358429
\(612\) 0 0
\(613\) −7993.67 −0.526690 −0.263345 0.964702i \(-0.584826\pi\)
−0.263345 + 0.964702i \(0.584826\pi\)
\(614\) 0 0
\(615\) −1413.74 −0.0926951
\(616\) 0 0
\(617\) 9922.51 0.647432 0.323716 0.946154i \(-0.395068\pi\)
0.323716 + 0.946154i \(0.395068\pi\)
\(618\) 0 0
\(619\) 15401.4 1.00005 0.500027 0.866010i \(-0.333324\pi\)
0.500027 + 0.866010i \(0.333324\pi\)
\(620\) 0 0
\(621\) 1694.68 0.109509
\(622\) 0 0
\(623\) 7931.74 0.510078
\(624\) 0 0
\(625\) −14065.4 −0.900185
\(626\) 0 0
\(627\) −4624.08 −0.294527
\(628\) 0 0
\(629\) −1068.77 −0.0677497
\(630\) 0 0
\(631\) −3776.97 −0.238287 −0.119143 0.992877i \(-0.538015\pi\)
−0.119143 + 0.992877i \(0.538015\pi\)
\(632\) 0 0
\(633\) 678.921 0.0426298
\(634\) 0 0
\(635\) −12570.6 −0.785587
\(636\) 0 0
\(637\) −3500.64 −0.217740
\(638\) 0 0
\(639\) 7450.05 0.461220
\(640\) 0 0
\(641\) 464.125 0.0285988 0.0142994 0.999898i \(-0.495448\pi\)
0.0142994 + 0.999898i \(0.495448\pi\)
\(642\) 0 0
\(643\) −7607.61 −0.466586 −0.233293 0.972406i \(-0.574950\pi\)
−0.233293 + 0.972406i \(0.574950\pi\)
\(644\) 0 0
\(645\) 3251.27 0.198479
\(646\) 0 0
\(647\) 7776.85 0.472550 0.236275 0.971686i \(-0.424073\pi\)
0.236275 + 0.971686i \(0.424073\pi\)
\(648\) 0 0
\(649\) −31489.2 −1.90456
\(650\) 0 0
\(651\) 971.033 0.0584605
\(652\) 0 0
\(653\) 24486.1 1.46740 0.733702 0.679471i \(-0.237791\pi\)
0.733702 + 0.679471i \(0.237791\pi\)
\(654\) 0 0
\(655\) 25123.2 1.49869
\(656\) 0 0
\(657\) −32480.6 −1.92875
\(658\) 0 0
\(659\) −910.966 −0.0538486 −0.0269243 0.999637i \(-0.508571\pi\)
−0.0269243 + 0.999637i \(0.508571\pi\)
\(660\) 0 0
\(661\) −5826.64 −0.342859 −0.171430 0.985196i \(-0.554839\pi\)
−0.171430 + 0.985196i \(0.554839\pi\)
\(662\) 0 0
\(663\) −81.8043 −0.00479188
\(664\) 0 0
\(665\) −36634.8 −2.13629
\(666\) 0 0
\(667\) 1214.59 0.0705081
\(668\) 0 0
\(669\) −1718.40 −0.0993085
\(670\) 0 0
\(671\) −8740.97 −0.502893
\(672\) 0 0
\(673\) 27236.0 1.55999 0.779994 0.625788i \(-0.215222\pi\)
0.779994 + 0.625788i \(0.215222\pi\)
\(674\) 0 0
\(675\) 462.551 0.0263757
\(676\) 0 0
\(677\) −8989.70 −0.510343 −0.255172 0.966896i \(-0.582132\pi\)
−0.255172 + 0.966896i \(0.582132\pi\)
\(678\) 0 0
\(679\) −18365.0 −1.03798
\(680\) 0 0
\(681\) 1573.86 0.0885616
\(682\) 0 0
\(683\) −6934.65 −0.388502 −0.194251 0.980952i \(-0.562228\pi\)
−0.194251 + 0.980952i \(0.562228\pi\)
\(684\) 0 0
\(685\) 7699.46 0.429462
\(686\) 0 0
\(687\) 2813.63 0.156254
\(688\) 0 0
\(689\) 16009.6 0.885221
\(690\) 0 0
\(691\) −5234.54 −0.288178 −0.144089 0.989565i \(-0.546025\pi\)
−0.144089 + 0.989565i \(0.546025\pi\)
\(692\) 0 0
\(693\) −23011.0 −1.26135
\(694\) 0 0
\(695\) −14900.9 −0.813270
\(696\) 0 0
\(697\) −795.961 −0.0432556
\(698\) 0 0
\(699\) 1909.57 0.103328
\(700\) 0 0
\(701\) 33713.7 1.81647 0.908237 0.418457i \(-0.137429\pi\)
0.908237 + 0.418457i \(0.137429\pi\)
\(702\) 0 0
\(703\) −36515.3 −1.95903
\(704\) 0 0
\(705\) 1838.12 0.0981953
\(706\) 0 0
\(707\) −21655.2 −1.15195
\(708\) 0 0
\(709\) 12687.3 0.672049 0.336025 0.941853i \(-0.390917\pi\)
0.336025 + 0.941853i \(0.390917\pi\)
\(710\) 0 0
\(711\) −16153.1 −0.852021
\(712\) 0 0
\(713\) 2425.17 0.127382
\(714\) 0 0
\(715\) 9961.59 0.521038
\(716\) 0 0
\(717\) 2984.02 0.155426
\(718\) 0 0
\(719\) 30604.9 1.58744 0.793720 0.608283i \(-0.208141\pi\)
0.793720 + 0.608283i \(0.208141\pi\)
\(720\) 0 0
\(721\) 26954.2 1.39227
\(722\) 0 0
\(723\) −1494.31 −0.0768661
\(724\) 0 0
\(725\) 331.512 0.0169821
\(726\) 0 0
\(727\) 3727.96 0.190182 0.0950911 0.995469i \(-0.469686\pi\)
0.0950911 + 0.995469i \(0.469686\pi\)
\(728\) 0 0
\(729\) −17218.3 −0.874782
\(730\) 0 0
\(731\) 1830.52 0.0926189
\(732\) 0 0
\(733\) −1107.21 −0.0557921 −0.0278960 0.999611i \(-0.508881\pi\)
−0.0278960 + 0.999611i \(0.508881\pi\)
\(734\) 0 0
\(735\) 1188.66 0.0596522
\(736\) 0 0
\(737\) −20625.6 −1.03087
\(738\) 0 0
\(739\) 29127.7 1.44991 0.724953 0.688798i \(-0.241861\pi\)
0.724953 + 0.688798i \(0.241861\pi\)
\(740\) 0 0
\(741\) −2794.91 −0.138561
\(742\) 0 0
\(743\) 12162.8 0.600552 0.300276 0.953852i \(-0.402921\pi\)
0.300276 + 0.953852i \(0.402921\pi\)
\(744\) 0 0
\(745\) −30165.6 −1.48347
\(746\) 0 0
\(747\) 13622.6 0.667233
\(748\) 0 0
\(749\) −15655.5 −0.763736
\(750\) 0 0
\(751\) −25067.6 −1.21802 −0.609008 0.793164i \(-0.708432\pi\)
−0.609008 + 0.793164i \(0.708432\pi\)
\(752\) 0 0
\(753\) 936.587 0.0453269
\(754\) 0 0
\(755\) 18176.1 0.876156
\(756\) 0 0
\(757\) 12020.6 0.577140 0.288570 0.957459i \(-0.406820\pi\)
0.288570 + 0.957459i \(0.406820\pi\)
\(758\) 0 0
\(759\) 1247.41 0.0596552
\(760\) 0 0
\(761\) 9255.42 0.440879 0.220439 0.975401i \(-0.429251\pi\)
0.220439 + 0.975401i \(0.429251\pi\)
\(762\) 0 0
\(763\) −33127.6 −1.57182
\(764\) 0 0
\(765\) −1279.74 −0.0604823
\(766\) 0 0
\(767\) −19032.9 −0.896007
\(768\) 0 0
\(769\) 28690.1 1.34537 0.672687 0.739928i \(-0.265140\pi\)
0.672687 + 0.739928i \(0.265140\pi\)
\(770\) 0 0
\(771\) 2210.05 0.103233
\(772\) 0 0
\(773\) 8984.36 0.418040 0.209020 0.977911i \(-0.432973\pi\)
0.209020 + 0.977911i \(0.432973\pi\)
\(774\) 0 0
\(775\) 661.933 0.0306804
\(776\) 0 0
\(777\) 3944.13 0.182104
\(778\) 0 0
\(779\) −27194.7 −1.25077
\(780\) 0 0
\(781\) 11086.6 0.507952
\(782\) 0 0
\(783\) 1173.43 0.0535566
\(784\) 0 0
\(785\) 28462.7 1.29411
\(786\) 0 0
\(787\) 5257.13 0.238115 0.119058 0.992887i \(-0.462013\pi\)
0.119058 + 0.992887i \(0.462013\pi\)
\(788\) 0 0
\(789\) −235.379 −0.0106207
\(790\) 0 0
\(791\) 23630.9 1.06222
\(792\) 0 0
\(793\) −5283.26 −0.236588
\(794\) 0 0
\(795\) −5436.14 −0.242516
\(796\) 0 0
\(797\) −7623.92 −0.338837 −0.169419 0.985544i \(-0.554189\pi\)
−0.169419 + 0.985544i \(0.554189\pi\)
\(798\) 0 0
\(799\) 1034.90 0.0458223
\(800\) 0 0
\(801\) 9466.45 0.417579
\(802\) 0 0
\(803\) −48335.3 −2.12418
\(804\) 0 0
\(805\) 9882.77 0.432698
\(806\) 0 0
\(807\) 1393.53 0.0607862
\(808\) 0 0
\(809\) −28238.7 −1.22722 −0.613609 0.789610i \(-0.710283\pi\)
−0.613609 + 0.789610i \(0.710283\pi\)
\(810\) 0 0
\(811\) −29851.2 −1.29250 −0.646251 0.763125i \(-0.723664\pi\)
−0.646251 + 0.763125i \(0.723664\pi\)
\(812\) 0 0
\(813\) 3928.69 0.169478
\(814\) 0 0
\(815\) 22932.7 0.985640
\(816\) 0 0
\(817\) 62541.4 2.67815
\(818\) 0 0
\(819\) −13908.4 −0.593407
\(820\) 0 0
\(821\) −2043.20 −0.0868552 −0.0434276 0.999057i \(-0.513828\pi\)
−0.0434276 + 0.999057i \(0.513828\pi\)
\(822\) 0 0
\(823\) 123.242 0.00521988 0.00260994 0.999997i \(-0.499169\pi\)
0.00260994 + 0.999997i \(0.499169\pi\)
\(824\) 0 0
\(825\) 340.473 0.0143682
\(826\) 0 0
\(827\) −34335.2 −1.44371 −0.721857 0.692043i \(-0.756711\pi\)
−0.721857 + 0.692043i \(0.756711\pi\)
\(828\) 0 0
\(829\) 14705.0 0.616073 0.308037 0.951375i \(-0.400328\pi\)
0.308037 + 0.951375i \(0.400328\pi\)
\(830\) 0 0
\(831\) −6563.58 −0.273993
\(832\) 0 0
\(833\) 669.237 0.0278364
\(834\) 0 0
\(835\) 10650.3 0.441401
\(836\) 0 0
\(837\) 2342.99 0.0967570
\(838\) 0 0
\(839\) −22204.3 −0.913679 −0.456840 0.889549i \(-0.651019\pi\)
−0.456840 + 0.889549i \(0.651019\pi\)
\(840\) 0 0
\(841\) 841.000 0.0344828
\(842\) 0 0
\(843\) 5950.54 0.243117
\(844\) 0 0
\(845\) −17392.1 −0.708054
\(846\) 0 0
\(847\) −4772.22 −0.193595
\(848\) 0 0
\(849\) 2313.51 0.0935212
\(850\) 0 0
\(851\) 9850.54 0.396794
\(852\) 0 0
\(853\) 40095.2 1.60942 0.804710 0.593669i \(-0.202321\pi\)
0.804710 + 0.593669i \(0.202321\pi\)
\(854\) 0 0
\(855\) −43723.2 −1.74889
\(856\) 0 0
\(857\) −12883.7 −0.513536 −0.256768 0.966473i \(-0.582658\pi\)
−0.256768 + 0.966473i \(0.582658\pi\)
\(858\) 0 0
\(859\) 34336.9 1.36386 0.681932 0.731416i \(-0.261140\pi\)
0.681932 + 0.731416i \(0.261140\pi\)
\(860\) 0 0
\(861\) 2937.38 0.116267
\(862\) 0 0
\(863\) 10864.2 0.428531 0.214266 0.976775i \(-0.431264\pi\)
0.214266 + 0.976775i \(0.431264\pi\)
\(864\) 0 0
\(865\) 27007.7 1.06161
\(866\) 0 0
\(867\) −3705.27 −0.145141
\(868\) 0 0
\(869\) −24037.8 −0.938352
\(870\) 0 0
\(871\) −12466.6 −0.484978
\(872\) 0 0
\(873\) −21918.5 −0.849745
\(874\) 0 0
\(875\) 32193.1 1.24380
\(876\) 0 0
\(877\) −4818.40 −0.185525 −0.0927626 0.995688i \(-0.529570\pi\)
−0.0927626 + 0.995688i \(0.529570\pi\)
\(878\) 0 0
\(879\) −2156.83 −0.0827623
\(880\) 0 0
\(881\) −30195.7 −1.15473 −0.577366 0.816485i \(-0.695920\pi\)
−0.577366 + 0.816485i \(0.695920\pi\)
\(882\) 0 0
\(883\) −30734.0 −1.17133 −0.585663 0.810555i \(-0.699166\pi\)
−0.585663 + 0.810555i \(0.699166\pi\)
\(884\) 0 0
\(885\) 6462.71 0.245471
\(886\) 0 0
\(887\) −43409.3 −1.64323 −0.821614 0.570045i \(-0.806926\pi\)
−0.821614 + 0.570045i \(0.806926\pi\)
\(888\) 0 0
\(889\) 26118.3 0.985355
\(890\) 0 0
\(891\) −26854.4 −1.00972
\(892\) 0 0
\(893\) 35358.1 1.32499
\(894\) 0 0
\(895\) −37834.7 −1.41304
\(896\) 0 0
\(897\) 753.968 0.0280650
\(898\) 0 0
\(899\) 1679.23 0.0622976
\(900\) 0 0
\(901\) −3060.64 −0.113169
\(902\) 0 0
\(903\) −6755.29 −0.248950
\(904\) 0 0
\(905\) 32340.1 1.18787
\(906\) 0 0
\(907\) 7991.51 0.292562 0.146281 0.989243i \(-0.453270\pi\)
0.146281 + 0.989243i \(0.453270\pi\)
\(908\) 0 0
\(909\) −25845.3 −0.943052
\(910\) 0 0
\(911\) 24188.5 0.879694 0.439847 0.898073i \(-0.355033\pi\)
0.439847 + 0.898073i \(0.355033\pi\)
\(912\) 0 0
\(913\) 20272.1 0.734840
\(914\) 0 0
\(915\) 1793.96 0.0648157
\(916\) 0 0
\(917\) −52199.4 −1.87980
\(918\) 0 0
\(919\) 27421.3 0.984273 0.492136 0.870518i \(-0.336216\pi\)
0.492136 + 0.870518i \(0.336216\pi\)
\(920\) 0 0
\(921\) 1871.28 0.0669497
\(922\) 0 0
\(923\) 6701.03 0.238968
\(924\) 0 0
\(925\) 2688.63 0.0955694
\(926\) 0 0
\(927\) 32169.6 1.13979
\(928\) 0 0
\(929\) −9284.12 −0.327882 −0.163941 0.986470i \(-0.552421\pi\)
−0.163941 + 0.986470i \(0.552421\pi\)
\(930\) 0 0
\(931\) 22865.0 0.804910
\(932\) 0 0
\(933\) −4281.57 −0.150238
\(934\) 0 0
\(935\) −1904.41 −0.0666106
\(936\) 0 0
\(937\) −1891.00 −0.0659297 −0.0329649 0.999457i \(-0.510495\pi\)
−0.0329649 + 0.999457i \(0.510495\pi\)
\(938\) 0 0
\(939\) −6278.30 −0.218194
\(940\) 0 0
\(941\) −15163.9 −0.525322 −0.262661 0.964888i \(-0.584600\pi\)
−0.262661 + 0.964888i \(0.584600\pi\)
\(942\) 0 0
\(943\) 7336.16 0.253338
\(944\) 0 0
\(945\) 9547.87 0.328669
\(946\) 0 0
\(947\) 31251.8 1.07238 0.536192 0.844096i \(-0.319862\pi\)
0.536192 + 0.844096i \(0.319862\pi\)
\(948\) 0 0
\(949\) −29215.1 −0.999327
\(950\) 0 0
\(951\) −1951.76 −0.0665512
\(952\) 0 0
\(953\) −12521.4 −0.425612 −0.212806 0.977094i \(-0.568260\pi\)
−0.212806 + 0.977094i \(0.568260\pi\)
\(954\) 0 0
\(955\) 23700.9 0.803081
\(956\) 0 0
\(957\) 863.731 0.0291750
\(958\) 0 0
\(959\) −15997.5 −0.538670
\(960\) 0 0
\(961\) −26438.1 −0.887451
\(962\) 0 0
\(963\) −18684.6 −0.625237
\(964\) 0 0
\(965\) −6743.98 −0.224970
\(966\) 0 0
\(967\) −18348.5 −0.610183 −0.305091 0.952323i \(-0.598687\pi\)
−0.305091 + 0.952323i \(0.598687\pi\)
\(968\) 0 0
\(969\) 534.319 0.0177139
\(970\) 0 0
\(971\) −27188.0 −0.898564 −0.449282 0.893390i \(-0.648320\pi\)
−0.449282 + 0.893390i \(0.648320\pi\)
\(972\) 0 0
\(973\) 30960.1 1.02008
\(974\) 0 0
\(975\) 205.790 0.00675955
\(976\) 0 0
\(977\) 4733.67 0.155009 0.0775044 0.996992i \(-0.475305\pi\)
0.0775044 + 0.996992i \(0.475305\pi\)
\(978\) 0 0
\(979\) 14087.3 0.459890
\(980\) 0 0
\(981\) −39537.5 −1.28678
\(982\) 0 0
\(983\) −43971.5 −1.42673 −0.713363 0.700795i \(-0.752829\pi\)
−0.713363 + 0.700795i \(0.752829\pi\)
\(984\) 0 0
\(985\) −14597.8 −0.472208
\(986\) 0 0
\(987\) −3819.13 −0.123166
\(988\) 0 0
\(989\) −16871.4 −0.542448
\(990\) 0 0
\(991\) −14031.4 −0.449769 −0.224885 0.974385i \(-0.572201\pi\)
−0.224885 + 0.974385i \(0.572201\pi\)
\(992\) 0 0
\(993\) −5810.56 −0.185692
\(994\) 0 0
\(995\) 15094.1 0.480920
\(996\) 0 0
\(997\) −43177.4 −1.37156 −0.685779 0.727810i \(-0.740538\pi\)
−0.685779 + 0.727810i \(0.740538\pi\)
\(998\) 0 0
\(999\) 9516.73 0.301397
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1856.4.a.n.1.1 2
4.3 odd 2 1856.4.a.h.1.2 2
8.3 odd 2 464.4.a.f.1.1 2
8.5 even 2 29.4.a.a.1.1 2
24.5 odd 2 261.4.a.b.1.2 2
40.29 even 2 725.4.a.b.1.2 2
56.13 odd 2 1421.4.a.c.1.1 2
232.173 even 2 841.4.a.a.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
29.4.a.a.1.1 2 8.5 even 2
261.4.a.b.1.2 2 24.5 odd 2
464.4.a.f.1.1 2 8.3 odd 2
725.4.a.b.1.2 2 40.29 even 2
841.4.a.a.1.2 2 232.173 even 2
1421.4.a.c.1.1 2 56.13 odd 2
1856.4.a.h.1.2 2 4.3 odd 2
1856.4.a.n.1.1 2 1.1 even 1 trivial