Properties

Label 116.4.a.a.1.2
Level $116$
Weight $4$
Character 116.1
Self dual yes
Analytic conductor $6.844$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [116,4,Mod(1,116)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(116, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("116.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 116 = 2^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 116.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(6.84422156067\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.30278\) of defining polynomial
Character \(\chi\) \(=\) 116.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.60555 q^{3} -12.2111 q^{5} -24.4222 q^{7} -14.0000 q^{9} +O(q^{10})\) \(q+3.60555 q^{3} -12.2111 q^{5} -24.4222 q^{7} -14.0000 q^{9} -19.6056 q^{11} -5.36669 q^{13} -44.0278 q^{15} +57.3221 q^{17} +27.2666 q^{19} -88.0555 q^{21} -54.0555 q^{23} +24.1110 q^{25} -147.828 q^{27} -29.0000 q^{29} -0.816654 q^{31} -70.6888 q^{33} +298.222 q^{35} -245.066 q^{37} -19.3499 q^{39} +55.2111 q^{41} +87.5500 q^{43} +170.955 q^{45} +474.672 q^{47} +253.444 q^{49} +206.678 q^{51} -211.700 q^{53} +239.405 q^{55} +98.3112 q^{57} +614.389 q^{59} -673.766 q^{61} +341.911 q^{63} +65.5332 q^{65} +999.532 q^{67} -194.900 q^{69} -650.522 q^{71} -1041.73 q^{73} +86.9335 q^{75} +478.811 q^{77} -1004.92 q^{79} -155.000 q^{81} -438.500 q^{83} -699.966 q^{85} -104.561 q^{87} +1361.32 q^{89} +131.066 q^{91} -2.94449 q^{93} -332.955 q^{95} -796.567 q^{97} +274.478 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 10 q^{5} - 20 q^{7} - 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 10 q^{5} - 20 q^{7} - 28 q^{9} - 32 q^{11} - 54 q^{13} - 52 q^{15} - 44 q^{17} - 32 q^{19} - 104 q^{21} - 36 q^{23} - 96 q^{25} - 58 q^{29} + 20 q^{31} - 26 q^{33} + 308 q^{35} - 144 q^{37} + 156 q^{39} + 96 q^{41} + 240 q^{43} + 140 q^{45} + 596 q^{47} - 70 q^{49} + 572 q^{51} - 34 q^{53} + 212 q^{55} + 312 q^{57} + 724 q^{59} - 612 q^{61} + 280 q^{63} - 42 q^{65} + 528 q^{67} - 260 q^{69} - 104 q^{71} - 872 q^{73} + 520 q^{75} + 424 q^{77} - 820 q^{79} - 310 q^{81} - 228 q^{83} - 924 q^{85} - 32 q^{89} - 84 q^{91} - 78 q^{93} - 464 q^{95} - 1896 q^{97} + 448 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 3.60555 0.693889 0.346944 0.937886i \(-0.387219\pi\)
0.346944 + 0.937886i \(0.387219\pi\)
\(4\) 0 0
\(5\) −12.2111 −1.09219 −0.546097 0.837722i \(-0.683887\pi\)
−0.546097 + 0.837722i \(0.683887\pi\)
\(6\) 0 0
\(7\) −24.4222 −1.31868 −0.659338 0.751847i \(-0.729163\pi\)
−0.659338 + 0.751847i \(0.729163\pi\)
\(8\) 0 0
\(9\) −14.0000 −0.518519
\(10\) 0 0
\(11\) −19.6056 −0.537391 −0.268695 0.963225i \(-0.586592\pi\)
−0.268695 + 0.963225i \(0.586592\pi\)
\(12\) 0 0
\(13\) −5.36669 −0.114496 −0.0572482 0.998360i \(-0.518233\pi\)
−0.0572482 + 0.998360i \(0.518233\pi\)
\(14\) 0 0
\(15\) −44.0278 −0.757861
\(16\) 0 0
\(17\) 57.3221 0.817803 0.408902 0.912578i \(-0.365912\pi\)
0.408902 + 0.912578i \(0.365912\pi\)
\(18\) 0 0
\(19\) 27.2666 0.329231 0.164616 0.986358i \(-0.447362\pi\)
0.164616 + 0.986358i \(0.447362\pi\)
\(20\) 0 0
\(21\) −88.0555 −0.915014
\(22\) 0 0
\(23\) −54.0555 −0.490059 −0.245029 0.969516i \(-0.578798\pi\)
−0.245029 + 0.969516i \(0.578798\pi\)
\(24\) 0 0
\(25\) 24.1110 0.192888
\(26\) 0 0
\(27\) −147.828 −1.05368
\(28\) 0 0
\(29\) −29.0000 −0.185695
\(30\) 0 0
\(31\) −0.816654 −0.00473146 −0.00236573 0.999997i \(-0.500753\pi\)
−0.00236573 + 0.999997i \(0.500753\pi\)
\(32\) 0 0
\(33\) −70.6888 −0.372889
\(34\) 0 0
\(35\) 298.222 1.44025
\(36\) 0 0
\(37\) −245.066 −1.08888 −0.544442 0.838799i \(-0.683258\pi\)
−0.544442 + 0.838799i \(0.683258\pi\)
\(38\) 0 0
\(39\) −19.3499 −0.0794477
\(40\) 0 0
\(41\) 55.2111 0.210306 0.105153 0.994456i \(-0.466467\pi\)
0.105153 + 0.994456i \(0.466467\pi\)
\(42\) 0 0
\(43\) 87.5500 0.310494 0.155247 0.987876i \(-0.450383\pi\)
0.155247 + 0.987876i \(0.450383\pi\)
\(44\) 0 0
\(45\) 170.955 0.566323
\(46\) 0 0
\(47\) 474.672 1.47315 0.736575 0.676356i \(-0.236442\pi\)
0.736575 + 0.676356i \(0.236442\pi\)
\(48\) 0 0
\(49\) 253.444 0.738904
\(50\) 0 0
\(51\) 206.678 0.567465
\(52\) 0 0
\(53\) −211.700 −0.548664 −0.274332 0.961635i \(-0.588457\pi\)
−0.274332 + 0.961635i \(0.588457\pi\)
\(54\) 0 0
\(55\) 239.405 0.586935
\(56\) 0 0
\(57\) 98.3112 0.228450
\(58\) 0 0
\(59\) 614.389 1.35570 0.677852 0.735198i \(-0.262911\pi\)
0.677852 + 0.735198i \(0.262911\pi\)
\(60\) 0 0
\(61\) −673.766 −1.41421 −0.707106 0.707108i \(-0.750000\pi\)
−0.707106 + 0.707108i \(0.750000\pi\)
\(62\) 0 0
\(63\) 341.911 0.683757
\(64\) 0 0
\(65\) 65.5332 0.125052
\(66\) 0 0
\(67\) 999.532 1.82257 0.911286 0.411774i \(-0.135091\pi\)
0.911286 + 0.411774i \(0.135091\pi\)
\(68\) 0 0
\(69\) −194.900 −0.340046
\(70\) 0 0
\(71\) −650.522 −1.08736 −0.543681 0.839292i \(-0.682970\pi\)
−0.543681 + 0.839292i \(0.682970\pi\)
\(72\) 0 0
\(73\) −1041.73 −1.67021 −0.835107 0.550088i \(-0.814594\pi\)
−0.835107 + 0.550088i \(0.814594\pi\)
\(74\) 0 0
\(75\) 86.9335 0.133843
\(76\) 0 0
\(77\) 478.811 0.708644
\(78\) 0 0
\(79\) −1004.92 −1.43116 −0.715582 0.698529i \(-0.753838\pi\)
−0.715582 + 0.698529i \(0.753838\pi\)
\(80\) 0 0
\(81\) −155.000 −0.212620
\(82\) 0 0
\(83\) −438.500 −0.579899 −0.289949 0.957042i \(-0.593638\pi\)
−0.289949 + 0.957042i \(0.593638\pi\)
\(84\) 0 0
\(85\) −699.966 −0.893200
\(86\) 0 0
\(87\) −104.561 −0.128852
\(88\) 0 0
\(89\) 1361.32 1.62134 0.810672 0.585500i \(-0.199102\pi\)
0.810672 + 0.585500i \(0.199102\pi\)
\(90\) 0 0
\(91\) 131.066 0.150983
\(92\) 0 0
\(93\) −2.94449 −0.00328311
\(94\) 0 0
\(95\) −332.955 −0.359584
\(96\) 0 0
\(97\) −796.567 −0.833805 −0.416903 0.908951i \(-0.636884\pi\)
−0.416903 + 0.908951i \(0.636884\pi\)
\(98\) 0 0
\(99\) 274.478 0.278647
\(100\) 0 0
\(101\) −1234.80 −1.21651 −0.608253 0.793743i \(-0.708129\pi\)
−0.608253 + 0.793743i \(0.708129\pi\)
\(102\) 0 0
\(103\) 349.610 0.334447 0.167224 0.985919i \(-0.446520\pi\)
0.167224 + 0.985919i \(0.446520\pi\)
\(104\) 0 0
\(105\) 1075.25 0.999373
\(106\) 0 0
\(107\) −276.244 −0.249584 −0.124792 0.992183i \(-0.539826\pi\)
−0.124792 + 0.992183i \(0.539826\pi\)
\(108\) 0 0
\(109\) −598.300 −0.525750 −0.262875 0.964830i \(-0.584671\pi\)
−0.262875 + 0.964830i \(0.584671\pi\)
\(110\) 0 0
\(111\) −883.600 −0.755564
\(112\) 0 0
\(113\) −977.532 −0.813792 −0.406896 0.913474i \(-0.633389\pi\)
−0.406896 + 0.913474i \(0.633389\pi\)
\(114\) 0 0
\(115\) 660.077 0.535239
\(116\) 0 0
\(117\) 75.1337 0.0593685
\(118\) 0 0
\(119\) −1399.93 −1.07842
\(120\) 0 0
\(121\) −946.622 −0.711211
\(122\) 0 0
\(123\) 199.066 0.145929
\(124\) 0 0
\(125\) 1231.97 0.881523
\(126\) 0 0
\(127\) 143.822 0.100489 0.0502446 0.998737i \(-0.484000\pi\)
0.0502446 + 0.998737i \(0.484000\pi\)
\(128\) 0 0
\(129\) 315.666 0.215448
\(130\) 0 0
\(131\) 591.334 0.394390 0.197195 0.980364i \(-0.436817\pi\)
0.197195 + 0.980364i \(0.436817\pi\)
\(132\) 0 0
\(133\) −665.911 −0.434149
\(134\) 0 0
\(135\) 1805.14 1.15083
\(136\) 0 0
\(137\) −1166.15 −0.727235 −0.363618 0.931548i \(-0.618458\pi\)
−0.363618 + 0.931548i \(0.618458\pi\)
\(138\) 0 0
\(139\) −1738.48 −1.06083 −0.530416 0.847738i \(-0.677964\pi\)
−0.530416 + 0.847738i \(0.677964\pi\)
\(140\) 0 0
\(141\) 1711.45 1.02220
\(142\) 0 0
\(143\) 105.217 0.0615293
\(144\) 0 0
\(145\) 354.122 0.202815
\(146\) 0 0
\(147\) 913.806 0.512717
\(148\) 0 0
\(149\) 2783.57 1.53046 0.765230 0.643757i \(-0.222625\pi\)
0.765230 + 0.643757i \(0.222625\pi\)
\(150\) 0 0
\(151\) 2731.24 1.47196 0.735978 0.677006i \(-0.236723\pi\)
0.735978 + 0.677006i \(0.236723\pi\)
\(152\) 0 0
\(153\) −802.510 −0.424046
\(154\) 0 0
\(155\) 9.97224 0.00516768
\(156\) 0 0
\(157\) −606.676 −0.308395 −0.154198 0.988040i \(-0.549279\pi\)
−0.154198 + 0.988040i \(0.549279\pi\)
\(158\) 0 0
\(159\) −763.294 −0.380712
\(160\) 0 0
\(161\) 1320.15 0.646228
\(162\) 0 0
\(163\) −1132.29 −0.544098 −0.272049 0.962283i \(-0.587701\pi\)
−0.272049 + 0.962283i \(0.587701\pi\)
\(164\) 0 0
\(165\) 863.188 0.407267
\(166\) 0 0
\(167\) −922.787 −0.427589 −0.213795 0.976879i \(-0.568582\pi\)
−0.213795 + 0.976879i \(0.568582\pi\)
\(168\) 0 0
\(169\) −2168.20 −0.986891
\(170\) 0 0
\(171\) −381.733 −0.170712
\(172\) 0 0
\(173\) 1454.47 0.639197 0.319599 0.947553i \(-0.396452\pi\)
0.319599 + 0.947553i \(0.396452\pi\)
\(174\) 0 0
\(175\) −588.844 −0.254357
\(176\) 0 0
\(177\) 2215.21 0.940708
\(178\) 0 0
\(179\) 2862.23 1.19516 0.597578 0.801811i \(-0.296130\pi\)
0.597578 + 0.801811i \(0.296130\pi\)
\(180\) 0 0
\(181\) −3740.03 −1.53588 −0.767940 0.640522i \(-0.778718\pi\)
−0.767940 + 0.640522i \(0.778718\pi\)
\(182\) 0 0
\(183\) −2429.30 −0.981306
\(184\) 0 0
\(185\) 2992.53 1.18927
\(186\) 0 0
\(187\) −1123.83 −0.439480
\(188\) 0 0
\(189\) 3610.28 1.38947
\(190\) 0 0
\(191\) 210.466 0.0797319 0.0398659 0.999205i \(-0.487307\pi\)
0.0398659 + 0.999205i \(0.487307\pi\)
\(192\) 0 0
\(193\) 4151.21 1.54824 0.774120 0.633038i \(-0.218192\pi\)
0.774120 + 0.633038i \(0.218192\pi\)
\(194\) 0 0
\(195\) 236.283 0.0867723
\(196\) 0 0
\(197\) 3838.13 1.38810 0.694050 0.719927i \(-0.255825\pi\)
0.694050 + 0.719927i \(0.255825\pi\)
\(198\) 0 0
\(199\) −3706.40 −1.32030 −0.660150 0.751134i \(-0.729507\pi\)
−0.660150 + 0.751134i \(0.729507\pi\)
\(200\) 0 0
\(201\) 3603.87 1.26466
\(202\) 0 0
\(203\) 708.244 0.244872
\(204\) 0 0
\(205\) −674.188 −0.229694
\(206\) 0 0
\(207\) 756.777 0.254105
\(208\) 0 0
\(209\) −534.577 −0.176926
\(210\) 0 0
\(211\) −267.272 −0.0872028 −0.0436014 0.999049i \(-0.513883\pi\)
−0.0436014 + 0.999049i \(0.513883\pi\)
\(212\) 0 0
\(213\) −2345.49 −0.754508
\(214\) 0 0
\(215\) −1069.08 −0.339120
\(216\) 0 0
\(217\) 19.9445 0.00623926
\(218\) 0 0
\(219\) −3756.02 −1.15894
\(220\) 0 0
\(221\) −307.630 −0.0936355
\(222\) 0 0
\(223\) −5263.48 −1.58058 −0.790288 0.612735i \(-0.790069\pi\)
−0.790288 + 0.612735i \(0.790069\pi\)
\(224\) 0 0
\(225\) −337.554 −0.100016
\(226\) 0 0
\(227\) 2314.81 0.676825 0.338412 0.940998i \(-0.390110\pi\)
0.338412 + 0.940998i \(0.390110\pi\)
\(228\) 0 0
\(229\) −3344.50 −0.965112 −0.482556 0.875865i \(-0.660292\pi\)
−0.482556 + 0.875865i \(0.660292\pi\)
\(230\) 0 0
\(231\) 1726.38 0.491720
\(232\) 0 0
\(233\) −5063.82 −1.42378 −0.711892 0.702289i \(-0.752161\pi\)
−0.711892 + 0.702289i \(0.752161\pi\)
\(234\) 0 0
\(235\) −5796.27 −1.60897
\(236\) 0 0
\(237\) −3623.28 −0.993068
\(238\) 0 0
\(239\) −1854.88 −0.502018 −0.251009 0.967985i \(-0.580762\pi\)
−0.251009 + 0.967985i \(0.580762\pi\)
\(240\) 0 0
\(241\) −2981.31 −0.796859 −0.398430 0.917199i \(-0.630445\pi\)
−0.398430 + 0.917199i \(0.630445\pi\)
\(242\) 0 0
\(243\) 3432.48 0.906148
\(244\) 0 0
\(245\) −3094.83 −0.807027
\(246\) 0 0
\(247\) −146.332 −0.0376958
\(248\) 0 0
\(249\) −1581.03 −0.402385
\(250\) 0 0
\(251\) 1639.16 0.412203 0.206102 0.978531i \(-0.433922\pi\)
0.206102 + 0.978531i \(0.433922\pi\)
\(252\) 0 0
\(253\) 1059.79 0.263353
\(254\) 0 0
\(255\) −2523.76 −0.619781
\(256\) 0 0
\(257\) −972.955 −0.236153 −0.118076 0.993005i \(-0.537673\pi\)
−0.118076 + 0.993005i \(0.537673\pi\)
\(258\) 0 0
\(259\) 5985.06 1.43588
\(260\) 0 0
\(261\) 406.000 0.0962865
\(262\) 0 0
\(263\) −3991.31 −0.935796 −0.467898 0.883782i \(-0.654989\pi\)
−0.467898 + 0.883782i \(0.654989\pi\)
\(264\) 0 0
\(265\) 2585.09 0.599248
\(266\) 0 0
\(267\) 4908.31 1.12503
\(268\) 0 0
\(269\) 1932.74 0.438072 0.219036 0.975717i \(-0.429709\pi\)
0.219036 + 0.975717i \(0.429709\pi\)
\(270\) 0 0
\(271\) 2679.57 0.600635 0.300317 0.953839i \(-0.402907\pi\)
0.300317 + 0.953839i \(0.402907\pi\)
\(272\) 0 0
\(273\) 472.567 0.104766
\(274\) 0 0
\(275\) −472.710 −0.103656
\(276\) 0 0
\(277\) 4898.39 1.06251 0.531256 0.847211i \(-0.321720\pi\)
0.531256 + 0.847211i \(0.321720\pi\)
\(278\) 0 0
\(279\) 11.4332 0.00245335
\(280\) 0 0
\(281\) −4851.20 −1.02989 −0.514943 0.857224i \(-0.672187\pi\)
−0.514943 + 0.857224i \(0.672187\pi\)
\(282\) 0 0
\(283\) 6486.83 1.36255 0.681276 0.732027i \(-0.261426\pi\)
0.681276 + 0.732027i \(0.261426\pi\)
\(284\) 0 0
\(285\) −1200.49 −0.249511
\(286\) 0 0
\(287\) −1348.38 −0.277325
\(288\) 0 0
\(289\) −1627.17 −0.331198
\(290\) 0 0
\(291\) −2872.06 −0.578568
\(292\) 0 0
\(293\) −1673.60 −0.333696 −0.166848 0.985983i \(-0.553359\pi\)
−0.166848 + 0.985983i \(0.553359\pi\)
\(294\) 0 0
\(295\) −7502.36 −1.48069
\(296\) 0 0
\(297\) 2898.24 0.566239
\(298\) 0 0
\(299\) 290.099 0.0561100
\(300\) 0 0
\(301\) −2138.16 −0.409441
\(302\) 0 0
\(303\) −4452.13 −0.844120
\(304\) 0 0
\(305\) 8227.43 1.54459
\(306\) 0 0
\(307\) 4619.62 0.858813 0.429406 0.903111i \(-0.358723\pi\)
0.429406 + 0.903111i \(0.358723\pi\)
\(308\) 0 0
\(309\) 1260.54 0.232069
\(310\) 0 0
\(311\) 3307.00 0.602967 0.301483 0.953471i \(-0.402518\pi\)
0.301483 + 0.953471i \(0.402518\pi\)
\(312\) 0 0
\(313\) 6739.95 1.21714 0.608570 0.793501i \(-0.291744\pi\)
0.608570 + 0.793501i \(0.291744\pi\)
\(314\) 0 0
\(315\) −4175.11 −0.746796
\(316\) 0 0
\(317\) 9278.23 1.64390 0.821952 0.569557i \(-0.192885\pi\)
0.821952 + 0.569557i \(0.192885\pi\)
\(318\) 0 0
\(319\) 568.561 0.0997909
\(320\) 0 0
\(321\) −996.012 −0.173184
\(322\) 0 0
\(323\) 1562.98 0.269246
\(324\) 0 0
\(325\) −129.396 −0.0220850
\(326\) 0 0
\(327\) −2157.20 −0.364812
\(328\) 0 0
\(329\) −11592.5 −1.94261
\(330\) 0 0
\(331\) 8424.51 1.39895 0.699475 0.714657i \(-0.253417\pi\)
0.699475 + 0.714657i \(0.253417\pi\)
\(332\) 0 0
\(333\) 3430.93 0.564606
\(334\) 0 0
\(335\) −12205.4 −1.99060
\(336\) 0 0
\(337\) 7069.41 1.14272 0.571358 0.820701i \(-0.306417\pi\)
0.571358 + 0.820701i \(0.306417\pi\)
\(338\) 0 0
\(339\) −3524.54 −0.564681
\(340\) 0 0
\(341\) 16.0109 0.00254264
\(342\) 0 0
\(343\) 2187.15 0.344301
\(344\) 0 0
\(345\) 2379.94 0.371397
\(346\) 0 0
\(347\) 10677.5 1.65186 0.825931 0.563772i \(-0.190650\pi\)
0.825931 + 0.563772i \(0.190650\pi\)
\(348\) 0 0
\(349\) 8612.96 1.32103 0.660517 0.750811i \(-0.270337\pi\)
0.660517 + 0.750811i \(0.270337\pi\)
\(350\) 0 0
\(351\) 793.345 0.120643
\(352\) 0 0
\(353\) −9586.96 −1.44550 −0.722751 0.691109i \(-0.757123\pi\)
−0.722751 + 0.691109i \(0.757123\pi\)
\(354\) 0 0
\(355\) 7943.58 1.18761
\(356\) 0 0
\(357\) −5047.53 −0.748301
\(358\) 0 0
\(359\) −10525.2 −1.54736 −0.773678 0.633579i \(-0.781585\pi\)
−0.773678 + 0.633579i \(0.781585\pi\)
\(360\) 0 0
\(361\) −6115.53 −0.891607
\(362\) 0 0
\(363\) −3413.10 −0.493502
\(364\) 0 0
\(365\) 12720.7 1.82420
\(366\) 0 0
\(367\) 8807.65 1.25274 0.626370 0.779526i \(-0.284540\pi\)
0.626370 + 0.779526i \(0.284540\pi\)
\(368\) 0 0
\(369\) −772.955 −0.109047
\(370\) 0 0
\(371\) 5170.18 0.723510
\(372\) 0 0
\(373\) −3134.23 −0.435079 −0.217540 0.976052i \(-0.569803\pi\)
−0.217540 + 0.976052i \(0.569803\pi\)
\(374\) 0 0
\(375\) 4441.92 0.611679
\(376\) 0 0
\(377\) 155.634 0.0212614
\(378\) 0 0
\(379\) −7830.92 −1.06134 −0.530670 0.847579i \(-0.678059\pi\)
−0.530670 + 0.847579i \(0.678059\pi\)
\(380\) 0 0
\(381\) 518.557 0.0697282
\(382\) 0 0
\(383\) 13034.5 1.73899 0.869496 0.493939i \(-0.164444\pi\)
0.869496 + 0.493939i \(0.164444\pi\)
\(384\) 0 0
\(385\) −5846.81 −0.773976
\(386\) 0 0
\(387\) −1225.70 −0.160997
\(388\) 0 0
\(389\) −5012.18 −0.653284 −0.326642 0.945148i \(-0.605917\pi\)
−0.326642 + 0.945148i \(0.605917\pi\)
\(390\) 0 0
\(391\) −3098.58 −0.400772
\(392\) 0 0
\(393\) 2132.08 0.273663
\(394\) 0 0
\(395\) 12271.1 1.56311
\(396\) 0 0
\(397\) 2200.61 0.278200 0.139100 0.990278i \(-0.455579\pi\)
0.139100 + 0.990278i \(0.455579\pi\)
\(398\) 0 0
\(399\) −2400.98 −0.301251
\(400\) 0 0
\(401\) −9028.68 −1.12437 −0.562183 0.827013i \(-0.690038\pi\)
−0.562183 + 0.827013i \(0.690038\pi\)
\(402\) 0 0
\(403\) 4.38273 0.000541735 0
\(404\) 0 0
\(405\) 1892.72 0.232222
\(406\) 0 0
\(407\) 4804.66 0.585156
\(408\) 0 0
\(409\) −5274.47 −0.637667 −0.318833 0.947811i \(-0.603291\pi\)
−0.318833 + 0.947811i \(0.603291\pi\)
\(410\) 0 0
\(411\) −4204.63 −0.504620
\(412\) 0 0
\(413\) −15004.7 −1.78773
\(414\) 0 0
\(415\) 5354.56 0.633362
\(416\) 0 0
\(417\) −6268.16 −0.736099
\(418\) 0 0
\(419\) 2450.27 0.285688 0.142844 0.989745i \(-0.454375\pi\)
0.142844 + 0.989745i \(0.454375\pi\)
\(420\) 0 0
\(421\) −2060.09 −0.238487 −0.119243 0.992865i \(-0.538047\pi\)
−0.119243 + 0.992865i \(0.538047\pi\)
\(422\) 0 0
\(423\) −6645.41 −0.763855
\(424\) 0 0
\(425\) 1382.10 0.157745
\(426\) 0 0
\(427\) 16454.9 1.86489
\(428\) 0 0
\(429\) 379.365 0.0426945
\(430\) 0 0
\(431\) −10108.0 −1.12966 −0.564830 0.825207i \(-0.691058\pi\)
−0.564830 + 0.825207i \(0.691058\pi\)
\(432\) 0 0
\(433\) −11457.1 −1.27158 −0.635791 0.771861i \(-0.719326\pi\)
−0.635791 + 0.771861i \(0.719326\pi\)
\(434\) 0 0
\(435\) 1276.80 0.140731
\(436\) 0 0
\(437\) −1473.91 −0.161343
\(438\) 0 0
\(439\) −14239.8 −1.54813 −0.774064 0.633107i \(-0.781779\pi\)
−0.774064 + 0.633107i \(0.781779\pi\)
\(440\) 0 0
\(441\) −3548.22 −0.383135
\(442\) 0 0
\(443\) −3220.82 −0.345431 −0.172716 0.984972i \(-0.555254\pi\)
−0.172716 + 0.984972i \(0.555254\pi\)
\(444\) 0 0
\(445\) −16623.2 −1.77082
\(446\) 0 0
\(447\) 10036.3 1.06197
\(448\) 0 0
\(449\) 3189.72 0.335262 0.167631 0.985850i \(-0.446388\pi\)
0.167631 + 0.985850i \(0.446388\pi\)
\(450\) 0 0
\(451\) −1082.44 −0.113016
\(452\) 0 0
\(453\) 9847.63 1.02137
\(454\) 0 0
\(455\) −1600.47 −0.164903
\(456\) 0 0
\(457\) 11209.7 1.14741 0.573704 0.819062i \(-0.305506\pi\)
0.573704 + 0.819062i \(0.305506\pi\)
\(458\) 0 0
\(459\) −8473.79 −0.861705
\(460\) 0 0
\(461\) −8038.44 −0.812121 −0.406060 0.913846i \(-0.633098\pi\)
−0.406060 + 0.913846i \(0.633098\pi\)
\(462\) 0 0
\(463\) 16556.0 1.66182 0.830909 0.556408i \(-0.187821\pi\)
0.830909 + 0.556408i \(0.187821\pi\)
\(464\) 0 0
\(465\) 35.9554 0.00358579
\(466\) 0 0
\(467\) −17211.9 −1.70550 −0.852752 0.522317i \(-0.825068\pi\)
−0.852752 + 0.522317i \(0.825068\pi\)
\(468\) 0 0
\(469\) −24410.8 −2.40338
\(470\) 0 0
\(471\) −2187.40 −0.213992
\(472\) 0 0
\(473\) −1716.47 −0.166857
\(474\) 0 0
\(475\) 657.426 0.0635048
\(476\) 0 0
\(477\) 2963.80 0.284493
\(478\) 0 0
\(479\) 6601.25 0.629685 0.314842 0.949144i \(-0.398048\pi\)
0.314842 + 0.949144i \(0.398048\pi\)
\(480\) 0 0
\(481\) 1315.20 0.124673
\(482\) 0 0
\(483\) 4759.89 0.448411
\(484\) 0 0
\(485\) 9726.96 0.910677
\(486\) 0 0
\(487\) 669.609 0.0623057 0.0311529 0.999515i \(-0.490082\pi\)
0.0311529 + 0.999515i \(0.490082\pi\)
\(488\) 0 0
\(489\) −4082.54 −0.377543
\(490\) 0 0
\(491\) −2822.98 −0.259469 −0.129735 0.991549i \(-0.541413\pi\)
−0.129735 + 0.991549i \(0.541413\pi\)
\(492\) 0 0
\(493\) −1662.34 −0.151862
\(494\) 0 0
\(495\) −3351.68 −0.304337
\(496\) 0 0
\(497\) 15887.2 1.43388
\(498\) 0 0
\(499\) 1879.69 0.168630 0.0843151 0.996439i \(-0.473130\pi\)
0.0843151 + 0.996439i \(0.473130\pi\)
\(500\) 0 0
\(501\) −3327.16 −0.296699
\(502\) 0 0
\(503\) −5616.84 −0.497897 −0.248949 0.968517i \(-0.580085\pi\)
−0.248949 + 0.968517i \(0.580085\pi\)
\(504\) 0 0
\(505\) 15078.3 1.32866
\(506\) 0 0
\(507\) −7817.55 −0.684792
\(508\) 0 0
\(509\) −15417.9 −1.34261 −0.671304 0.741182i \(-0.734266\pi\)
−0.671304 + 0.741182i \(0.734266\pi\)
\(510\) 0 0
\(511\) 25441.4 2.20247
\(512\) 0 0
\(513\) −4030.76 −0.346905
\(514\) 0 0
\(515\) −4269.12 −0.365282
\(516\) 0 0
\(517\) −9306.21 −0.791657
\(518\) 0 0
\(519\) 5244.16 0.443532
\(520\) 0 0
\(521\) 8475.84 0.712732 0.356366 0.934346i \(-0.384016\pi\)
0.356366 + 0.934346i \(0.384016\pi\)
\(522\) 0 0
\(523\) 15153.8 1.26698 0.633490 0.773750i \(-0.281622\pi\)
0.633490 + 0.773750i \(0.281622\pi\)
\(524\) 0 0
\(525\) −2123.11 −0.176495
\(526\) 0 0
\(527\) −46.8123 −0.00386941
\(528\) 0 0
\(529\) −9245.00 −0.759842
\(530\) 0 0
\(531\) −8601.44 −0.702958
\(532\) 0 0
\(533\) −296.301 −0.0240792
\(534\) 0 0
\(535\) 3373.24 0.272594
\(536\) 0 0
\(537\) 10319.9 0.829306
\(538\) 0 0
\(539\) −4968.91 −0.397080
\(540\) 0 0
\(541\) 21278.1 1.69098 0.845488 0.533994i \(-0.179310\pi\)
0.845488 + 0.533994i \(0.179310\pi\)
\(542\) 0 0
\(543\) −13484.9 −1.06573
\(544\) 0 0
\(545\) 7305.91 0.574221
\(546\) 0 0
\(547\) −13816.0 −1.07994 −0.539972 0.841683i \(-0.681565\pi\)
−0.539972 + 0.841683i \(0.681565\pi\)
\(548\) 0 0
\(549\) 9432.73 0.733295
\(550\) 0 0
\(551\) −790.732 −0.0611367
\(552\) 0 0
\(553\) 24542.3 1.88724
\(554\) 0 0
\(555\) 10789.7 0.825222
\(556\) 0 0
\(557\) −7735.09 −0.588414 −0.294207 0.955742i \(-0.595055\pi\)
−0.294207 + 0.955742i \(0.595055\pi\)
\(558\) 0 0
\(559\) −469.854 −0.0355505
\(560\) 0 0
\(561\) −4052.03 −0.304950
\(562\) 0 0
\(563\) −7911.82 −0.592262 −0.296131 0.955147i \(-0.595696\pi\)
−0.296131 + 0.955147i \(0.595696\pi\)
\(564\) 0 0
\(565\) 11936.7 0.888819
\(566\) 0 0
\(567\) 3785.44 0.280377
\(568\) 0 0
\(569\) 16436.7 1.21101 0.605503 0.795843i \(-0.292972\pi\)
0.605503 + 0.795843i \(0.292972\pi\)
\(570\) 0 0
\(571\) 10244.9 0.750854 0.375427 0.926852i \(-0.377496\pi\)
0.375427 + 0.926852i \(0.377496\pi\)
\(572\) 0 0
\(573\) 758.846 0.0553250
\(574\) 0 0
\(575\) −1303.33 −0.0945266
\(576\) 0 0
\(577\) 7544.04 0.544303 0.272151 0.962254i \(-0.412265\pi\)
0.272151 + 0.962254i \(0.412265\pi\)
\(578\) 0 0
\(579\) 14967.4 1.07431
\(580\) 0 0
\(581\) 10709.1 0.764698
\(582\) 0 0
\(583\) 4150.49 0.294847
\(584\) 0 0
\(585\) −917.465 −0.0648419
\(586\) 0 0
\(587\) −19532.4 −1.37341 −0.686703 0.726938i \(-0.740943\pi\)
−0.686703 + 0.726938i \(0.740943\pi\)
\(588\) 0 0
\(589\) −22.2674 −0.00155774
\(590\) 0 0
\(591\) 13838.6 0.963186
\(592\) 0 0
\(593\) −6854.78 −0.474691 −0.237346 0.971425i \(-0.576277\pi\)
−0.237346 + 0.971425i \(0.576277\pi\)
\(594\) 0 0
\(595\) 17094.7 1.17784
\(596\) 0 0
\(597\) −13363.6 −0.916141
\(598\) 0 0
\(599\) −15947.1 −1.08778 −0.543890 0.839156i \(-0.683049\pi\)
−0.543890 + 0.839156i \(0.683049\pi\)
\(600\) 0 0
\(601\) −10663.2 −0.723729 −0.361864 0.932231i \(-0.617860\pi\)
−0.361864 + 0.932231i \(0.617860\pi\)
\(602\) 0 0
\(603\) −13993.5 −0.945037
\(604\) 0 0
\(605\) 11559.3 0.776781
\(606\) 0 0
\(607\) 16197.4 1.08309 0.541543 0.840673i \(-0.317840\pi\)
0.541543 + 0.840673i \(0.317840\pi\)
\(608\) 0 0
\(609\) 2553.61 0.169914
\(610\) 0 0
\(611\) −2547.42 −0.168670
\(612\) 0 0
\(613\) 12176.2 0.802268 0.401134 0.916019i \(-0.368616\pi\)
0.401134 + 0.916019i \(0.368616\pi\)
\(614\) 0 0
\(615\) −2430.82 −0.159382
\(616\) 0 0
\(617\) 27706.3 1.80780 0.903900 0.427744i \(-0.140692\pi\)
0.903900 + 0.427744i \(0.140692\pi\)
\(618\) 0 0
\(619\) −23136.1 −1.50229 −0.751147 0.660136i \(-0.770499\pi\)
−0.751147 + 0.660136i \(0.770499\pi\)
\(620\) 0 0
\(621\) 7990.90 0.516367
\(622\) 0 0
\(623\) −33246.5 −2.13803
\(624\) 0 0
\(625\) −18057.5 −1.15568
\(626\) 0 0
\(627\) −1927.44 −0.122767
\(628\) 0 0
\(629\) −14047.7 −0.890492
\(630\) 0 0
\(631\) −425.446 −0.0268411 −0.0134205 0.999910i \(-0.504272\pi\)
−0.0134205 + 0.999910i \(0.504272\pi\)
\(632\) 0 0
\(633\) −963.665 −0.0605091
\(634\) 0 0
\(635\) −1756.22 −0.109754
\(636\) 0 0
\(637\) −1360.16 −0.0846018
\(638\) 0 0
\(639\) 9107.30 0.563817
\(640\) 0 0
\(641\) −20327.6 −1.25256 −0.626280 0.779598i \(-0.715423\pi\)
−0.626280 + 0.779598i \(0.715423\pi\)
\(642\) 0 0
\(643\) −13733.7 −0.842307 −0.421153 0.906989i \(-0.638375\pi\)
−0.421153 + 0.906989i \(0.638375\pi\)
\(644\) 0 0
\(645\) −3854.63 −0.235312
\(646\) 0 0
\(647\) −9986.00 −0.606786 −0.303393 0.952866i \(-0.598119\pi\)
−0.303393 + 0.952866i \(0.598119\pi\)
\(648\) 0 0
\(649\) −12045.4 −0.728543
\(650\) 0 0
\(651\) 71.9109 0.00432935
\(652\) 0 0
\(653\) 21134.3 1.26654 0.633268 0.773933i \(-0.281713\pi\)
0.633268 + 0.773933i \(0.281713\pi\)
\(654\) 0 0
\(655\) −7220.84 −0.430750
\(656\) 0 0
\(657\) 14584.3 0.866037
\(658\) 0 0
\(659\) 1452.86 0.0858806 0.0429403 0.999078i \(-0.486327\pi\)
0.0429403 + 0.999078i \(0.486327\pi\)
\(660\) 0 0
\(661\) −19476.0 −1.14604 −0.573018 0.819543i \(-0.694228\pi\)
−0.573018 + 0.819543i \(0.694228\pi\)
\(662\) 0 0
\(663\) −1109.18 −0.0649726
\(664\) 0 0
\(665\) 8131.51 0.474175
\(666\) 0 0
\(667\) 1567.61 0.0910016
\(668\) 0 0
\(669\) −18977.7 −1.09674
\(670\) 0 0
\(671\) 13209.6 0.759984
\(672\) 0 0
\(673\) 13453.9 0.770592 0.385296 0.922793i \(-0.374099\pi\)
0.385296 + 0.922793i \(0.374099\pi\)
\(674\) 0 0
\(675\) −3564.28 −0.203243
\(676\) 0 0
\(677\) 7227.43 0.410299 0.205150 0.978731i \(-0.434232\pi\)
0.205150 + 0.978731i \(0.434232\pi\)
\(678\) 0 0
\(679\) 19453.9 1.09952
\(680\) 0 0
\(681\) 8346.16 0.469641
\(682\) 0 0
\(683\) −27923.5 −1.56437 −0.782185 0.623047i \(-0.785895\pi\)
−0.782185 + 0.623047i \(0.785895\pi\)
\(684\) 0 0
\(685\) 14240.0 0.794282
\(686\) 0 0
\(687\) −12058.8 −0.669681
\(688\) 0 0
\(689\) 1136.13 0.0628200
\(690\) 0 0
\(691\) −8332.36 −0.458724 −0.229362 0.973341i \(-0.573664\pi\)
−0.229362 + 0.973341i \(0.573664\pi\)
\(692\) 0 0
\(693\) −6703.35 −0.367445
\(694\) 0 0
\(695\) 21228.7 1.15863
\(696\) 0 0
\(697\) 3164.82 0.171989
\(698\) 0 0
\(699\) −18257.8 −0.987947
\(700\) 0 0
\(701\) 14927.5 0.804288 0.402144 0.915576i \(-0.368265\pi\)
0.402144 + 0.915576i \(0.368265\pi\)
\(702\) 0 0
\(703\) −6682.13 −0.358494
\(704\) 0 0
\(705\) −20898.7 −1.11644
\(706\) 0 0
\(707\) 30156.5 1.60418
\(708\) 0 0
\(709\) 15555.3 0.823966 0.411983 0.911191i \(-0.364836\pi\)
0.411983 + 0.911191i \(0.364836\pi\)
\(710\) 0 0
\(711\) 14068.8 0.742085
\(712\) 0 0
\(713\) 44.1446 0.00231870
\(714\) 0 0
\(715\) −1284.82 −0.0672019
\(716\) 0 0
\(717\) −6687.86 −0.348344
\(718\) 0 0
\(719\) −7336.33 −0.380527 −0.190264 0.981733i \(-0.560934\pi\)
−0.190264 + 0.981733i \(0.560934\pi\)
\(720\) 0 0
\(721\) −8538.24 −0.441027
\(722\) 0 0
\(723\) −10749.3 −0.552932
\(724\) 0 0
\(725\) −699.220 −0.0358184
\(726\) 0 0
\(727\) −3280.35 −0.167347 −0.0836735 0.996493i \(-0.526665\pi\)
−0.0836735 + 0.996493i \(0.526665\pi\)
\(728\) 0 0
\(729\) 16561.0 0.841386
\(730\) 0 0
\(731\) 5018.55 0.253923
\(732\) 0 0
\(733\) −37704.0 −1.89990 −0.949952 0.312395i \(-0.898869\pi\)
−0.949952 + 0.312395i \(0.898869\pi\)
\(734\) 0 0
\(735\) −11158.6 −0.559987
\(736\) 0 0
\(737\) −19596.4 −0.979433
\(738\) 0 0
\(739\) 5442.77 0.270928 0.135464 0.990782i \(-0.456748\pi\)
0.135464 + 0.990782i \(0.456748\pi\)
\(740\) 0 0
\(741\) −527.606 −0.0261567
\(742\) 0 0
\(743\) −15637.1 −0.772097 −0.386049 0.922478i \(-0.626160\pi\)
−0.386049 + 0.922478i \(0.626160\pi\)
\(744\) 0 0
\(745\) −33990.4 −1.67156
\(746\) 0 0
\(747\) 6138.99 0.300688
\(748\) 0 0
\(749\) 6746.49 0.329120
\(750\) 0 0
\(751\) −29072.1 −1.41259 −0.706296 0.707917i \(-0.749635\pi\)
−0.706296 + 0.707917i \(0.749635\pi\)
\(752\) 0 0
\(753\) 5910.08 0.286023
\(754\) 0 0
\(755\) −33351.5 −1.60766
\(756\) 0 0
\(757\) −7801.68 −0.374580 −0.187290 0.982305i \(-0.559970\pi\)
−0.187290 + 0.982305i \(0.559970\pi\)
\(758\) 0 0
\(759\) 3821.12 0.182738
\(760\) 0 0
\(761\) −13233.8 −0.630386 −0.315193 0.949028i \(-0.602069\pi\)
−0.315193 + 0.949028i \(0.602069\pi\)
\(762\) 0 0
\(763\) 14611.8 0.693294
\(764\) 0 0
\(765\) 9799.53 0.463141
\(766\) 0 0
\(767\) −3297.23 −0.155223
\(768\) 0 0
\(769\) −35703.3 −1.67424 −0.837122 0.547017i \(-0.815763\pi\)
−0.837122 + 0.547017i \(0.815763\pi\)
\(770\) 0 0
\(771\) −3508.04 −0.163864
\(772\) 0 0
\(773\) 36775.8 1.71117 0.855585 0.517663i \(-0.173198\pi\)
0.855585 + 0.517663i \(0.173198\pi\)
\(774\) 0 0
\(775\) −19.6904 −0.000912643 0
\(776\) 0 0
\(777\) 21579.5 0.996343
\(778\) 0 0
\(779\) 1505.42 0.0692391
\(780\) 0 0
\(781\) 12753.8 0.584338
\(782\) 0 0
\(783\) 4287.00 0.195664
\(784\) 0 0
\(785\) 7408.19 0.336827
\(786\) 0 0
\(787\) −24987.9 −1.13179 −0.565897 0.824476i \(-0.691470\pi\)
−0.565897 + 0.824476i \(0.691470\pi\)
\(788\) 0 0
\(789\) −14390.9 −0.649339
\(790\) 0 0
\(791\) 23873.5 1.07313
\(792\) 0 0
\(793\) 3615.90 0.161922
\(794\) 0 0
\(795\) 9320.67 0.415811
\(796\) 0 0
\(797\) 12969.3 0.576409 0.288204 0.957569i \(-0.406942\pi\)
0.288204 + 0.957569i \(0.406942\pi\)
\(798\) 0 0
\(799\) 27209.2 1.20475
\(800\) 0 0
\(801\) −19058.5 −0.840697
\(802\) 0 0
\(803\) 20423.7 0.897557
\(804\) 0 0
\(805\) −16120.5 −0.705807
\(806\) 0 0
\(807\) 6968.59 0.303973
\(808\) 0 0
\(809\) 23818.8 1.03514 0.517568 0.855642i \(-0.326837\pi\)
0.517568 + 0.855642i \(0.326837\pi\)
\(810\) 0 0
\(811\) −11646.6 −0.504276 −0.252138 0.967691i \(-0.581134\pi\)
−0.252138 + 0.967691i \(0.581134\pi\)
\(812\) 0 0
\(813\) 9661.31 0.416774
\(814\) 0 0
\(815\) 13826.5 0.594261
\(816\) 0 0
\(817\) 2387.19 0.102224
\(818\) 0 0
\(819\) −1834.93 −0.0782877
\(820\) 0 0
\(821\) −4592.82 −0.195238 −0.0976190 0.995224i \(-0.531123\pi\)
−0.0976190 + 0.995224i \(0.531123\pi\)
\(822\) 0 0
\(823\) −4271.24 −0.180907 −0.0904533 0.995901i \(-0.528832\pi\)
−0.0904533 + 0.995901i \(0.528832\pi\)
\(824\) 0 0
\(825\) −1704.38 −0.0719259
\(826\) 0 0
\(827\) 20409.8 0.858185 0.429093 0.903260i \(-0.358833\pi\)
0.429093 + 0.903260i \(0.358833\pi\)
\(828\) 0 0
\(829\) 28910.2 1.21121 0.605606 0.795765i \(-0.292931\pi\)
0.605606 + 0.795765i \(0.292931\pi\)
\(830\) 0 0
\(831\) 17661.4 0.737265
\(832\) 0 0
\(833\) 14528.0 0.604278
\(834\) 0 0
\(835\) 11268.3 0.467011
\(836\) 0 0
\(837\) 120.724 0.00498546
\(838\) 0 0
\(839\) 15702.7 0.646148 0.323074 0.946374i \(-0.395284\pi\)
0.323074 + 0.946374i \(0.395284\pi\)
\(840\) 0 0
\(841\) 841.000 0.0344828
\(842\) 0 0
\(843\) −17491.2 −0.714627
\(844\) 0 0
\(845\) 26476.1 1.07788
\(846\) 0 0
\(847\) 23118.6 0.937857
\(848\) 0 0
\(849\) 23388.6 0.945459
\(850\) 0 0
\(851\) 13247.2 0.533617
\(852\) 0 0
\(853\) −22605.4 −0.907379 −0.453690 0.891160i \(-0.649893\pi\)
−0.453690 + 0.891160i \(0.649893\pi\)
\(854\) 0 0
\(855\) 4661.38 0.186451
\(856\) 0 0
\(857\) −16881.0 −0.672864 −0.336432 0.941708i \(-0.609220\pi\)
−0.336432 + 0.941708i \(0.609220\pi\)
\(858\) 0 0
\(859\) 16871.7 0.670147 0.335074 0.942192i \(-0.391239\pi\)
0.335074 + 0.942192i \(0.391239\pi\)
\(860\) 0 0
\(861\) −4861.64 −0.192432
\(862\) 0 0
\(863\) 17711.7 0.698624 0.349312 0.937006i \(-0.386415\pi\)
0.349312 + 0.937006i \(0.386415\pi\)
\(864\) 0 0
\(865\) −17760.7 −0.698128
\(866\) 0 0
\(867\) −5866.86 −0.229814
\(868\) 0 0
\(869\) 19701.9 0.769093
\(870\) 0 0
\(871\) −5364.18 −0.208678
\(872\) 0 0
\(873\) 11151.9 0.432343
\(874\) 0 0
\(875\) −30087.3 −1.16244
\(876\) 0 0
\(877\) 42056.4 1.61932 0.809660 0.586899i \(-0.199652\pi\)
0.809660 + 0.586899i \(0.199652\pi\)
\(878\) 0 0
\(879\) −6034.26 −0.231548
\(880\) 0 0
\(881\) −44296.3 −1.69396 −0.846981 0.531622i \(-0.821583\pi\)
−0.846981 + 0.531622i \(0.821583\pi\)
\(882\) 0 0
\(883\) −32695.6 −1.24609 −0.623044 0.782187i \(-0.714104\pi\)
−0.623044 + 0.782187i \(0.714104\pi\)
\(884\) 0 0
\(885\) −27050.2 −1.02744
\(886\) 0 0
\(887\) 26911.3 1.01871 0.509353 0.860558i \(-0.329885\pi\)
0.509353 + 0.860558i \(0.329885\pi\)
\(888\) 0 0
\(889\) −3512.44 −0.132512
\(890\) 0 0
\(891\) 3038.86 0.114260
\(892\) 0 0
\(893\) 12942.7 0.485007
\(894\) 0 0
\(895\) −34951.0 −1.30534
\(896\) 0 0
\(897\) 1045.97 0.0389341
\(898\) 0 0
\(899\) 23.6830 0.000878611 0
\(900\) 0 0
\(901\) −12135.1 −0.448699
\(902\) 0 0
\(903\) −7709.26 −0.284107
\(904\) 0 0
\(905\) 45669.9 1.67748
\(906\) 0 0
\(907\) −23856.5 −0.873366 −0.436683 0.899615i \(-0.643847\pi\)
−0.436683 + 0.899615i \(0.643847\pi\)
\(908\) 0 0
\(909\) 17287.2 0.630781
\(910\) 0 0
\(911\) −15194.9 −0.552613 −0.276306 0.961070i \(-0.589110\pi\)
−0.276306 + 0.961070i \(0.589110\pi\)
\(912\) 0 0
\(913\) 8597.03 0.311632
\(914\) 0 0
\(915\) 29664.4 1.07178
\(916\) 0 0
\(917\) −14441.7 −0.520072
\(918\) 0 0
\(919\) 42629.9 1.53018 0.765088 0.643926i \(-0.222695\pi\)
0.765088 + 0.643926i \(0.222695\pi\)
\(920\) 0 0
\(921\) 16656.3 0.595920
\(922\) 0 0
\(923\) 3491.15 0.124499
\(924\) 0 0
\(925\) −5908.80 −0.210033
\(926\) 0 0
\(927\) −4894.54 −0.173417
\(928\) 0 0
\(929\) −24680.5 −0.871627 −0.435814 0.900037i \(-0.643539\pi\)
−0.435814 + 0.900037i \(0.643539\pi\)
\(930\) 0 0
\(931\) 6910.56 0.243270
\(932\) 0 0
\(933\) 11923.5 0.418392
\(934\) 0 0
\(935\) 13723.2 0.479997
\(936\) 0 0
\(937\) 15589.4 0.543525 0.271763 0.962364i \(-0.412393\pi\)
0.271763 + 0.962364i \(0.412393\pi\)
\(938\) 0 0
\(939\) 24301.2 0.844559
\(940\) 0 0
\(941\) −56872.4 −1.97023 −0.985116 0.171892i \(-0.945012\pi\)
−0.985116 + 0.171892i \(0.945012\pi\)
\(942\) 0 0
\(943\) −2984.46 −0.103062
\(944\) 0 0
\(945\) −44085.5 −1.51757
\(946\) 0 0
\(947\) −6452.91 −0.221427 −0.110714 0.993852i \(-0.535314\pi\)
−0.110714 + 0.993852i \(0.535314\pi\)
\(948\) 0 0
\(949\) 5590.66 0.191233
\(950\) 0 0
\(951\) 33453.1 1.14069
\(952\) 0 0
\(953\) 7140.93 0.242725 0.121363 0.992608i \(-0.461274\pi\)
0.121363 + 0.992608i \(0.461274\pi\)
\(954\) 0 0
\(955\) −2570.02 −0.0870827
\(956\) 0 0
\(957\) 2049.98 0.0692438
\(958\) 0 0
\(959\) 28480.0 0.958987
\(960\) 0 0
\(961\) −29790.3 −0.999978
\(962\) 0 0
\(963\) 3867.42 0.129414
\(964\) 0 0
\(965\) −50690.8 −1.69098
\(966\) 0 0
\(967\) −5957.77 −0.198127 −0.0990635 0.995081i \(-0.531585\pi\)
−0.0990635 + 0.995081i \(0.531585\pi\)
\(968\) 0 0
\(969\) 5635.41 0.186827
\(970\) 0 0
\(971\) −43221.7 −1.42848 −0.714238 0.699903i \(-0.753226\pi\)
−0.714238 + 0.699903i \(0.753226\pi\)
\(972\) 0 0
\(973\) 42457.4 1.39889
\(974\) 0 0
\(975\) −466.546 −0.0153245
\(976\) 0 0
\(977\) −28243.3 −0.924854 −0.462427 0.886657i \(-0.653021\pi\)
−0.462427 + 0.886657i \(0.653021\pi\)
\(978\) 0 0
\(979\) −26689.4 −0.871295
\(980\) 0 0
\(981\) 8376.20 0.272611
\(982\) 0 0
\(983\) −37893.9 −1.22953 −0.614765 0.788710i \(-0.710749\pi\)
−0.614765 + 0.788710i \(0.710749\pi\)
\(984\) 0 0
\(985\) −46867.8 −1.51607
\(986\) 0 0
\(987\) −41797.5 −1.34795
\(988\) 0 0
\(989\) −4732.56 −0.152160
\(990\) 0 0
\(991\) −29795.8 −0.955091 −0.477545 0.878607i \(-0.658473\pi\)
−0.477545 + 0.878607i \(0.658473\pi\)
\(992\) 0 0
\(993\) 30375.0 0.970716
\(994\) 0 0
\(995\) 45259.2 1.44202
\(996\) 0 0
\(997\) −43164.3 −1.37114 −0.685571 0.728006i \(-0.740447\pi\)
−0.685571 + 0.728006i \(0.740447\pi\)
\(998\) 0 0
\(999\) 36227.6 1.14734
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 116.4.a.a.1.2 2
3.2 odd 2 1044.4.a.d.1.2 2
4.3 odd 2 464.4.a.d.1.1 2
8.3 odd 2 1856.4.a.k.1.2 2
8.5 even 2 1856.4.a.j.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
116.4.a.a.1.2 2 1.1 even 1 trivial
464.4.a.d.1.1 2 4.3 odd 2
1044.4.a.d.1.2 2 3.2 odd 2
1856.4.a.j.1.1 2 8.5 even 2
1856.4.a.k.1.2 2 8.3 odd 2