Properties

Label 336.4.a.o.1.1
Level $336$
Weight $4$
Character 336.1
Self dual yes
Analytic conductor $19.825$
Analytic rank $0$
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] = [336,4,Mod(1,336)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(336, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 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("336.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 336.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: \(19.8246417619\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{337}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 84 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 168)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(9.67878\) of defining polynomial
Character \(\chi\) \(=\) 336.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.00000 q^{3} -21.3576 q^{5} +7.00000 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q+3.00000 q^{3} -21.3576 q^{5} +7.00000 q^{7} +9.00000 q^{9} -31.3576 q^{11} +84.7151 q^{13} -64.0727 q^{15} -16.0727 q^{17} -20.0000 q^{19} +21.0000 q^{21} +80.7878 q^{23} +331.145 q^{25} +27.0000 q^{27} +102.000 q^{29} +245.721 q^{31} -94.0727 q^{33} -149.503 q^{35} +215.285 q^{37} +254.145 q^{39} +150.933 q^{41} -441.721 q^{43} -192.218 q^{45} +206.424 q^{47} +49.0000 q^{49} -48.2180 q^{51} +426.436 q^{53} +669.721 q^{55} -60.0000 q^{57} -363.006 q^{59} -343.721 q^{61} +63.0000 q^{63} -1809.31 q^{65} -69.2849 q^{67} +242.363 q^{69} +468.933 q^{71} +747.576 q^{73} +993.436 q^{75} -219.503 q^{77} -1298.74 q^{79} +81.0000 q^{81} +1293.16 q^{83} +343.273 q^{85} +306.000 q^{87} -563.346 q^{89} +593.006 q^{91} +737.163 q^{93} +427.151 q^{95} +1485.03 q^{97} -282.218 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{3} - 6 q^{5} + 14 q^{7} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 6 q^{3} - 6 q^{5} + 14 q^{7} + 18 q^{9} - 26 q^{11} + 96 q^{13} - 18 q^{15} + 78 q^{17} - 40 q^{19} + 42 q^{21} - 22 q^{23} + 442 q^{25} + 54 q^{27} + 204 q^{29} - 96 q^{31} - 78 q^{33} - 42 q^{35} + 504 q^{37} + 288 q^{39} - 102 q^{41} - 296 q^{43} - 54 q^{45} + 780 q^{47} + 98 q^{49} + 234 q^{51} + 192 q^{53} + 752 q^{55} - 120 q^{57} - 212 q^{59} - 100 q^{61} + 126 q^{63} - 1636 q^{65} - 212 q^{67} - 66 q^{69} + 534 q^{71} + 1128 q^{73} + 1326 q^{75} - 182 q^{77} - 468 q^{79} + 162 q^{81} + 824 q^{83} + 1788 q^{85} + 612 q^{87} - 2118 q^{89} + 672 q^{91} - 288 q^{93} + 120 q^{95} + 400 q^{97} - 234 q^{99}+O(q^{100}) \) 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
\(3\) 3.00000 0.577350
\(4\) 0 0
\(5\) −21.3576 −1.91028 −0.955139 0.296158i \(-0.904295\pi\)
−0.955139 + 0.296158i \(0.904295\pi\)
\(6\) 0 0
\(7\) 7.00000 0.377964
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −31.3576 −0.859515 −0.429757 0.902944i \(-0.641401\pi\)
−0.429757 + 0.902944i \(0.641401\pi\)
\(12\) 0 0
\(13\) 84.7151 1.80737 0.903683 0.428203i \(-0.140853\pi\)
0.903683 + 0.428203i \(0.140853\pi\)
\(14\) 0 0
\(15\) −64.0727 −1.10290
\(16\) 0 0
\(17\) −16.0727 −0.229306 −0.114653 0.993406i \(-0.536576\pi\)
−0.114653 + 0.993406i \(0.536576\pi\)
\(18\) 0 0
\(19\) −20.0000 −0.241490 −0.120745 0.992684i \(-0.538528\pi\)
−0.120745 + 0.992684i \(0.538528\pi\)
\(20\) 0 0
\(21\) 21.0000 0.218218
\(22\) 0 0
\(23\) 80.7878 0.732410 0.366205 0.930534i \(-0.380657\pi\)
0.366205 + 0.930534i \(0.380657\pi\)
\(24\) 0 0
\(25\) 331.145 2.64916
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) 102.000 0.653135 0.326568 0.945174i \(-0.394108\pi\)
0.326568 + 0.945174i \(0.394108\pi\)
\(30\) 0 0
\(31\) 245.721 1.42364 0.711819 0.702363i \(-0.247872\pi\)
0.711819 + 0.702363i \(0.247872\pi\)
\(32\) 0 0
\(33\) −94.0727 −0.496241
\(34\) 0 0
\(35\) −149.503 −0.722017
\(36\) 0 0
\(37\) 215.285 0.956557 0.478279 0.878208i \(-0.341261\pi\)
0.478279 + 0.878208i \(0.341261\pi\)
\(38\) 0 0
\(39\) 254.145 1.04348
\(40\) 0 0
\(41\) 150.933 0.574922 0.287461 0.957792i \(-0.407189\pi\)
0.287461 + 0.957792i \(0.407189\pi\)
\(42\) 0 0
\(43\) −441.721 −1.56655 −0.783277 0.621673i \(-0.786453\pi\)
−0.783277 + 0.621673i \(0.786453\pi\)
\(44\) 0 0
\(45\) −192.218 −0.636759
\(46\) 0 0
\(47\) 206.424 0.640640 0.320320 0.947309i \(-0.396210\pi\)
0.320320 + 0.947309i \(0.396210\pi\)
\(48\) 0 0
\(49\) 49.0000 0.142857
\(50\) 0 0
\(51\) −48.2180 −0.132390
\(52\) 0 0
\(53\) 426.436 1.10520 0.552599 0.833447i \(-0.313636\pi\)
0.552599 + 0.833447i \(0.313636\pi\)
\(54\) 0 0
\(55\) 669.721 1.64191
\(56\) 0 0
\(57\) −60.0000 −0.139424
\(58\) 0 0
\(59\) −363.006 −0.801006 −0.400503 0.916296i \(-0.631165\pi\)
−0.400503 + 0.916296i \(0.631165\pi\)
\(60\) 0 0
\(61\) −343.721 −0.721458 −0.360729 0.932671i \(-0.617472\pi\)
−0.360729 + 0.932671i \(0.617472\pi\)
\(62\) 0 0
\(63\) 63.0000 0.125988
\(64\) 0 0
\(65\) −1809.31 −3.45257
\(66\) 0 0
\(67\) −69.2849 −0.126336 −0.0631679 0.998003i \(-0.520120\pi\)
−0.0631679 + 0.998003i \(0.520120\pi\)
\(68\) 0 0
\(69\) 242.363 0.422857
\(70\) 0 0
\(71\) 468.933 0.783833 0.391916 0.920001i \(-0.371812\pi\)
0.391916 + 0.920001i \(0.371812\pi\)
\(72\) 0 0
\(73\) 747.576 1.19859 0.599295 0.800528i \(-0.295448\pi\)
0.599295 + 0.800528i \(0.295448\pi\)
\(74\) 0 0
\(75\) 993.436 1.52949
\(76\) 0 0
\(77\) −219.503 −0.324866
\(78\) 0 0
\(79\) −1298.74 −1.84961 −0.924807 0.380437i \(-0.875774\pi\)
−0.924807 + 0.380437i \(0.875774\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 1293.16 1.71016 0.855079 0.518498i \(-0.173509\pi\)
0.855079 + 0.518498i \(0.173509\pi\)
\(84\) 0 0
\(85\) 343.273 0.438038
\(86\) 0 0
\(87\) 306.000 0.377088
\(88\) 0 0
\(89\) −563.346 −0.670950 −0.335475 0.942049i \(-0.608897\pi\)
−0.335475 + 0.942049i \(0.608897\pi\)
\(90\) 0 0
\(91\) 593.006 0.683120
\(92\) 0 0
\(93\) 737.163 0.821938
\(94\) 0 0
\(95\) 427.151 0.461314
\(96\) 0 0
\(97\) 1485.03 1.55445 0.777226 0.629221i \(-0.216626\pi\)
0.777226 + 0.629221i \(0.216626\pi\)
\(98\) 0 0
\(99\) −282.218 −0.286505
\(100\) 0 0
\(101\) 1526.25 1.50364 0.751821 0.659367i \(-0.229176\pi\)
0.751821 + 0.659367i \(0.229176\pi\)
\(102\) 0 0
\(103\) 297.988 0.285065 0.142532 0.989790i \(-0.454476\pi\)
0.142532 + 0.989790i \(0.454476\pi\)
\(104\) 0 0
\(105\) −448.509 −0.416857
\(106\) 0 0
\(107\) −259.916 −0.234832 −0.117416 0.993083i \(-0.537461\pi\)
−0.117416 + 0.993083i \(0.537461\pi\)
\(108\) 0 0
\(109\) −1436.33 −1.26216 −0.631078 0.775719i \(-0.717387\pi\)
−0.631078 + 0.775719i \(0.717387\pi\)
\(110\) 0 0
\(111\) 645.855 0.552269
\(112\) 0 0
\(113\) 121.128 0.100838 0.0504192 0.998728i \(-0.483944\pi\)
0.0504192 + 0.998728i \(0.483944\pi\)
\(114\) 0 0
\(115\) −1725.43 −1.39911
\(116\) 0 0
\(117\) 762.436 0.602455
\(118\) 0 0
\(119\) −112.509 −0.0866694
\(120\) 0 0
\(121\) −347.703 −0.261235
\(122\) 0 0
\(123\) 452.799 0.331931
\(124\) 0 0
\(125\) −4402.76 −3.15036
\(126\) 0 0
\(127\) −496.413 −0.346847 −0.173423 0.984847i \(-0.555483\pi\)
−0.173423 + 0.984847i \(0.555483\pi\)
\(128\) 0 0
\(129\) −1325.16 −0.904450
\(130\) 0 0
\(131\) −2674.59 −1.78382 −0.891909 0.452214i \(-0.850634\pi\)
−0.891909 + 0.452214i \(0.850634\pi\)
\(132\) 0 0
\(133\) −140.000 −0.0912747
\(134\) 0 0
\(135\) −576.654 −0.367633
\(136\) 0 0
\(137\) 2370.46 1.47826 0.739131 0.673561i \(-0.235236\pi\)
0.739131 + 0.673561i \(0.235236\pi\)
\(138\) 0 0
\(139\) 95.4419 0.0582394 0.0291197 0.999576i \(-0.490730\pi\)
0.0291197 + 0.999576i \(0.490730\pi\)
\(140\) 0 0
\(141\) 619.273 0.369874
\(142\) 0 0
\(143\) −2656.46 −1.55346
\(144\) 0 0
\(145\) −2178.47 −1.24767
\(146\) 0 0
\(147\) 147.000 0.0824786
\(148\) 0 0
\(149\) −2261.56 −1.24345 −0.621727 0.783234i \(-0.713569\pi\)
−0.621727 + 0.783234i \(0.713569\pi\)
\(150\) 0 0
\(151\) −126.884 −0.0683818 −0.0341909 0.999415i \(-0.510885\pi\)
−0.0341909 + 0.999415i \(0.510885\pi\)
\(152\) 0 0
\(153\) −144.654 −0.0764352
\(154\) 0 0
\(155\) −5248.00 −2.71955
\(156\) 0 0
\(157\) 2977.73 1.51369 0.756844 0.653596i \(-0.226740\pi\)
0.756844 + 0.653596i \(0.226740\pi\)
\(158\) 0 0
\(159\) 1279.31 0.638086
\(160\) 0 0
\(161\) 565.515 0.276825
\(162\) 0 0
\(163\) 303.029 0.145614 0.0728070 0.997346i \(-0.476804\pi\)
0.0728070 + 0.997346i \(0.476804\pi\)
\(164\) 0 0
\(165\) 2009.16 0.947958
\(166\) 0 0
\(167\) −3547.90 −1.64398 −0.821991 0.569501i \(-0.807136\pi\)
−0.821991 + 0.569501i \(0.807136\pi\)
\(168\) 0 0
\(169\) 4979.65 2.26657
\(170\) 0 0
\(171\) −180.000 −0.0804967
\(172\) 0 0
\(173\) 283.102 0.124415 0.0622076 0.998063i \(-0.480186\pi\)
0.0622076 + 0.998063i \(0.480186\pi\)
\(174\) 0 0
\(175\) 2318.02 1.00129
\(176\) 0 0
\(177\) −1089.02 −0.462461
\(178\) 0 0
\(179\) −1611.50 −0.672902 −0.336451 0.941701i \(-0.609227\pi\)
−0.336451 + 0.941701i \(0.609227\pi\)
\(180\) 0 0
\(181\) −2794.99 −1.14779 −0.573896 0.818928i \(-0.694569\pi\)
−0.573896 + 0.818928i \(0.694569\pi\)
\(182\) 0 0
\(183\) −1031.16 −0.416534
\(184\) 0 0
\(185\) −4597.96 −1.82729
\(186\) 0 0
\(187\) 504.000 0.197092
\(188\) 0 0
\(189\) 189.000 0.0727393
\(190\) 0 0
\(191\) 2786.99 1.05581 0.527905 0.849304i \(-0.322978\pi\)
0.527905 + 0.849304i \(0.322978\pi\)
\(192\) 0 0
\(193\) −1751.87 −0.653379 −0.326689 0.945132i \(-0.605933\pi\)
−0.326689 + 0.945132i \(0.605933\pi\)
\(194\) 0 0
\(195\) −5427.92 −1.99334
\(196\) 0 0
\(197\) 964.715 0.348899 0.174450 0.984666i \(-0.444185\pi\)
0.174450 + 0.984666i \(0.444185\pi\)
\(198\) 0 0
\(199\) 5284.36 1.88240 0.941202 0.337844i \(-0.109698\pi\)
0.941202 + 0.337844i \(0.109698\pi\)
\(200\) 0 0
\(201\) −207.855 −0.0729400
\(202\) 0 0
\(203\) 714.000 0.246862
\(204\) 0 0
\(205\) −3223.56 −1.09826
\(206\) 0 0
\(207\) 727.090 0.244137
\(208\) 0 0
\(209\) 627.151 0.207564
\(210\) 0 0
\(211\) 1998.04 0.651898 0.325949 0.945387i \(-0.394316\pi\)
0.325949 + 0.945387i \(0.394316\pi\)
\(212\) 0 0
\(213\) 1406.80 0.452546
\(214\) 0 0
\(215\) 9434.08 2.99255
\(216\) 0 0
\(217\) 1720.05 0.538085
\(218\) 0 0
\(219\) 2242.73 0.692007
\(220\) 0 0
\(221\) −1361.60 −0.414439
\(222\) 0 0
\(223\) 3969.41 1.19198 0.595989 0.802993i \(-0.296760\pi\)
0.595989 + 0.802993i \(0.296760\pi\)
\(224\) 0 0
\(225\) 2980.31 0.883054
\(226\) 0 0
\(227\) 2175.32 0.636040 0.318020 0.948084i \(-0.396982\pi\)
0.318020 + 0.948084i \(0.396982\pi\)
\(228\) 0 0
\(229\) 2567.33 0.740847 0.370424 0.928863i \(-0.379213\pi\)
0.370424 + 0.928863i \(0.379213\pi\)
\(230\) 0 0
\(231\) −658.509 −0.187561
\(232\) 0 0
\(233\) −2114.68 −0.594581 −0.297290 0.954787i \(-0.596083\pi\)
−0.297290 + 0.954787i \(0.596083\pi\)
\(234\) 0 0
\(235\) −4408.72 −1.22380
\(236\) 0 0
\(237\) −3896.22 −1.06788
\(238\) 0 0
\(239\) −5418.12 −1.46640 −0.733199 0.680014i \(-0.761974\pi\)
−0.733199 + 0.680014i \(0.761974\pi\)
\(240\) 0 0
\(241\) 376.378 0.100600 0.0503000 0.998734i \(-0.483982\pi\)
0.0503000 + 0.998734i \(0.483982\pi\)
\(242\) 0 0
\(243\) 243.000 0.0641500
\(244\) 0 0
\(245\) −1046.52 −0.272897
\(246\) 0 0
\(247\) −1694.30 −0.436461
\(248\) 0 0
\(249\) 3879.49 0.987360
\(250\) 0 0
\(251\) −1833.94 −0.461183 −0.230592 0.973051i \(-0.574066\pi\)
−0.230592 + 0.973051i \(0.574066\pi\)
\(252\) 0 0
\(253\) −2533.31 −0.629517
\(254\) 0 0
\(255\) 1029.82 0.252901
\(256\) 0 0
\(257\) 4606.35 1.11804 0.559020 0.829154i \(-0.311177\pi\)
0.559020 + 0.829154i \(0.311177\pi\)
\(258\) 0 0
\(259\) 1506.99 0.361545
\(260\) 0 0
\(261\) 918.000 0.217712
\(262\) 0 0
\(263\) 3565.97 0.836074 0.418037 0.908430i \(-0.362718\pi\)
0.418037 + 0.908430i \(0.362718\pi\)
\(264\) 0 0
\(265\) −9107.63 −2.11124
\(266\) 0 0
\(267\) −1690.04 −0.387373
\(268\) 0 0
\(269\) 926.084 0.209905 0.104952 0.994477i \(-0.466531\pi\)
0.104952 + 0.994477i \(0.466531\pi\)
\(270\) 0 0
\(271\) 4720.24 1.05806 0.529030 0.848603i \(-0.322556\pi\)
0.529030 + 0.848603i \(0.322556\pi\)
\(272\) 0 0
\(273\) 1779.02 0.394399
\(274\) 0 0
\(275\) −10383.9 −2.27699
\(276\) 0 0
\(277\) −5273.15 −1.14380 −0.571900 0.820323i \(-0.693793\pi\)
−0.571900 + 0.820323i \(0.693793\pi\)
\(278\) 0 0
\(279\) 2211.49 0.474546
\(280\) 0 0
\(281\) 4130.22 0.876826 0.438413 0.898774i \(-0.355541\pi\)
0.438413 + 0.898774i \(0.355541\pi\)
\(282\) 0 0
\(283\) −3161.77 −0.664126 −0.332063 0.943257i \(-0.607745\pi\)
−0.332063 + 0.943257i \(0.607745\pi\)
\(284\) 0 0
\(285\) 1281.45 0.266340
\(286\) 0 0
\(287\) 1056.53 0.217300
\(288\) 0 0
\(289\) −4654.67 −0.947419
\(290\) 0 0
\(291\) 4455.09 0.897463
\(292\) 0 0
\(293\) 3546.23 0.707075 0.353537 0.935420i \(-0.384979\pi\)
0.353537 + 0.935420i \(0.384979\pi\)
\(294\) 0 0
\(295\) 7752.92 1.53014
\(296\) 0 0
\(297\) −846.654 −0.165414
\(298\) 0 0
\(299\) 6843.95 1.32373
\(300\) 0 0
\(301\) −3092.05 −0.592102
\(302\) 0 0
\(303\) 4578.76 0.868128
\(304\) 0 0
\(305\) 7341.04 1.37819
\(306\) 0 0
\(307\) 596.966 0.110979 0.0554896 0.998459i \(-0.482328\pi\)
0.0554896 + 0.998459i \(0.482328\pi\)
\(308\) 0 0
\(309\) 893.965 0.164582
\(310\) 0 0
\(311\) 3234.94 0.589828 0.294914 0.955524i \(-0.404709\pi\)
0.294914 + 0.955524i \(0.404709\pi\)
\(312\) 0 0
\(313\) 2307.12 0.416632 0.208316 0.978062i \(-0.433202\pi\)
0.208316 + 0.978062i \(0.433202\pi\)
\(314\) 0 0
\(315\) −1345.53 −0.240672
\(316\) 0 0
\(317\) 5190.60 0.919663 0.459831 0.888006i \(-0.347910\pi\)
0.459831 + 0.888006i \(0.347910\pi\)
\(318\) 0 0
\(319\) −3198.47 −0.561379
\(320\) 0 0
\(321\) −779.747 −0.135580
\(322\) 0 0
\(323\) 321.454 0.0553751
\(324\) 0 0
\(325\) 28053.0 4.78800
\(326\) 0 0
\(327\) −4308.98 −0.728706
\(328\) 0 0
\(329\) 1444.97 0.242139
\(330\) 0 0
\(331\) −1651.08 −0.274174 −0.137087 0.990559i \(-0.543774\pi\)
−0.137087 + 0.990559i \(0.543774\pi\)
\(332\) 0 0
\(333\) 1937.56 0.318852
\(334\) 0 0
\(335\) 1479.76 0.241336
\(336\) 0 0
\(337\) −5633.73 −0.910650 −0.455325 0.890325i \(-0.650477\pi\)
−0.455325 + 0.890325i \(0.650477\pi\)
\(338\) 0 0
\(339\) 363.384 0.0582191
\(340\) 0 0
\(341\) −7705.21 −1.22364
\(342\) 0 0
\(343\) 343.000 0.0539949
\(344\) 0 0
\(345\) −5176.29 −0.807774
\(346\) 0 0
\(347\) 2701.17 0.417886 0.208943 0.977928i \(-0.432998\pi\)
0.208943 + 0.977928i \(0.432998\pi\)
\(348\) 0 0
\(349\) 5855.26 0.898065 0.449032 0.893515i \(-0.351769\pi\)
0.449032 + 0.893515i \(0.351769\pi\)
\(350\) 0 0
\(351\) 2287.31 0.347828
\(352\) 0 0
\(353\) 7167.35 1.08068 0.540339 0.841447i \(-0.318296\pi\)
0.540339 + 0.841447i \(0.318296\pi\)
\(354\) 0 0
\(355\) −10015.3 −1.49734
\(356\) 0 0
\(357\) −337.526 −0.0500386
\(358\) 0 0
\(359\) −5899.16 −0.867258 −0.433629 0.901091i \(-0.642767\pi\)
−0.433629 + 0.901091i \(0.642767\pi\)
\(360\) 0 0
\(361\) −6459.00 −0.941682
\(362\) 0 0
\(363\) −1043.11 −0.150824
\(364\) 0 0
\(365\) −15966.4 −2.28964
\(366\) 0 0
\(367\) −10663.5 −1.51670 −0.758349 0.651848i \(-0.773994\pi\)
−0.758349 + 0.651848i \(0.773994\pi\)
\(368\) 0 0
\(369\) 1358.40 0.191641
\(370\) 0 0
\(371\) 2985.05 0.417726
\(372\) 0 0
\(373\) 3563.67 0.494692 0.247346 0.968927i \(-0.420442\pi\)
0.247346 + 0.968927i \(0.420442\pi\)
\(374\) 0 0
\(375\) −13208.3 −1.81886
\(376\) 0 0
\(377\) 8640.94 1.18045
\(378\) 0 0
\(379\) −2702.04 −0.366212 −0.183106 0.983093i \(-0.558615\pi\)
−0.183106 + 0.983093i \(0.558615\pi\)
\(380\) 0 0
\(381\) −1489.24 −0.200252
\(382\) 0 0
\(383\) −6244.41 −0.833092 −0.416546 0.909115i \(-0.636760\pi\)
−0.416546 + 0.909115i \(0.636760\pi\)
\(384\) 0 0
\(385\) 4688.05 0.620584
\(386\) 0 0
\(387\) −3975.49 −0.522185
\(388\) 0 0
\(389\) 3753.93 0.489285 0.244642 0.969613i \(-0.421329\pi\)
0.244642 + 0.969613i \(0.421329\pi\)
\(390\) 0 0
\(391\) −1298.48 −0.167946
\(392\) 0 0
\(393\) −8023.78 −1.02989
\(394\) 0 0
\(395\) 27737.9 3.53328
\(396\) 0 0
\(397\) 5717.71 0.722830 0.361415 0.932405i \(-0.382294\pi\)
0.361415 + 0.932405i \(0.382294\pi\)
\(398\) 0 0
\(399\) −420.000 −0.0526975
\(400\) 0 0
\(401\) 622.680 0.0775440 0.0387720 0.999248i \(-0.487655\pi\)
0.0387720 + 0.999248i \(0.487655\pi\)
\(402\) 0 0
\(403\) 20816.3 2.57303
\(404\) 0 0
\(405\) −1729.96 −0.212253
\(406\) 0 0
\(407\) −6750.81 −0.822175
\(408\) 0 0
\(409\) −6462.06 −0.781243 −0.390622 0.920551i \(-0.627740\pi\)
−0.390622 + 0.920551i \(0.627740\pi\)
\(410\) 0 0
\(411\) 7111.38 0.853475
\(412\) 0 0
\(413\) −2541.04 −0.302752
\(414\) 0 0
\(415\) −27618.8 −3.26688
\(416\) 0 0
\(417\) 286.326 0.0336245
\(418\) 0 0
\(419\) −1418.38 −0.165375 −0.0826877 0.996576i \(-0.526350\pi\)
−0.0826877 + 0.996576i \(0.526350\pi\)
\(420\) 0 0
\(421\) −5773.61 −0.668381 −0.334191 0.942505i \(-0.608463\pi\)
−0.334191 + 0.942505i \(0.608463\pi\)
\(422\) 0 0
\(423\) 1857.82 0.213547
\(424\) 0 0
\(425\) −5322.39 −0.607468
\(426\) 0 0
\(427\) −2406.05 −0.272686
\(428\) 0 0
\(429\) −7969.38 −0.896889
\(430\) 0 0
\(431\) 14770.4 1.65073 0.825366 0.564598i \(-0.190969\pi\)
0.825366 + 0.564598i \(0.190969\pi\)
\(432\) 0 0
\(433\) −2353.06 −0.261156 −0.130578 0.991438i \(-0.541683\pi\)
−0.130578 + 0.991438i \(0.541683\pi\)
\(434\) 0 0
\(435\) −6535.41 −0.720343
\(436\) 0 0
\(437\) −1615.76 −0.176870
\(438\) 0 0
\(439\) 8437.95 0.917361 0.458681 0.888601i \(-0.348322\pi\)
0.458681 + 0.888601i \(0.348322\pi\)
\(440\) 0 0
\(441\) 441.000 0.0476190
\(442\) 0 0
\(443\) −5577.32 −0.598163 −0.299082 0.954228i \(-0.596680\pi\)
−0.299082 + 0.954228i \(0.596680\pi\)
\(444\) 0 0
\(445\) 12031.7 1.28170
\(446\) 0 0
\(447\) −6784.69 −0.717908
\(448\) 0 0
\(449\) 2837.09 0.298198 0.149099 0.988822i \(-0.452363\pi\)
0.149099 + 0.988822i \(0.452363\pi\)
\(450\) 0 0
\(451\) −4732.90 −0.494154
\(452\) 0 0
\(453\) −380.651 −0.0394803
\(454\) 0 0
\(455\) −12665.2 −1.30495
\(456\) 0 0
\(457\) −4460.98 −0.456621 −0.228310 0.973588i \(-0.573320\pi\)
−0.228310 + 0.973588i \(0.573320\pi\)
\(458\) 0 0
\(459\) −433.962 −0.0441299
\(460\) 0 0
\(461\) −5008.00 −0.505956 −0.252978 0.967472i \(-0.581410\pi\)
−0.252978 + 0.967472i \(0.581410\pi\)
\(462\) 0 0
\(463\) −348.867 −0.0350178 −0.0175089 0.999847i \(-0.505574\pi\)
−0.0175089 + 0.999847i \(0.505574\pi\)
\(464\) 0 0
\(465\) −15744.0 −1.57013
\(466\) 0 0
\(467\) −9786.32 −0.969715 −0.484857 0.874593i \(-0.661129\pi\)
−0.484857 + 0.874593i \(0.661129\pi\)
\(468\) 0 0
\(469\) −484.994 −0.0477504
\(470\) 0 0
\(471\) 8933.20 0.873928
\(472\) 0 0
\(473\) 13851.3 1.34648
\(474\) 0 0
\(475\) −6622.91 −0.639747
\(476\) 0 0
\(477\) 3837.92 0.368399
\(478\) 0 0
\(479\) 13073.7 1.24708 0.623540 0.781792i \(-0.285694\pi\)
0.623540 + 0.781792i \(0.285694\pi\)
\(480\) 0 0
\(481\) 18237.9 1.72885
\(482\) 0 0
\(483\) 1696.54 0.159825
\(484\) 0 0
\(485\) −31716.6 −2.96944
\(486\) 0 0
\(487\) −1220.76 −0.113589 −0.0567944 0.998386i \(-0.518088\pi\)
−0.0567944 + 0.998386i \(0.518088\pi\)
\(488\) 0 0
\(489\) 909.088 0.0840703
\(490\) 0 0
\(491\) 6367.89 0.585293 0.292647 0.956221i \(-0.405464\pi\)
0.292647 + 0.956221i \(0.405464\pi\)
\(492\) 0 0
\(493\) −1639.41 −0.149768
\(494\) 0 0
\(495\) 6027.49 0.547304
\(496\) 0 0
\(497\) 3282.53 0.296261
\(498\) 0 0
\(499\) 11715.5 1.05101 0.525507 0.850789i \(-0.323876\pi\)
0.525507 + 0.850789i \(0.323876\pi\)
\(500\) 0 0
\(501\) −10643.7 −0.949153
\(502\) 0 0
\(503\) 13412.1 1.18890 0.594450 0.804133i \(-0.297370\pi\)
0.594450 + 0.804133i \(0.297370\pi\)
\(504\) 0 0
\(505\) −32597.0 −2.87237
\(506\) 0 0
\(507\) 14939.0 1.30860
\(508\) 0 0
\(509\) 10029.1 0.873345 0.436672 0.899621i \(-0.356157\pi\)
0.436672 + 0.899621i \(0.356157\pi\)
\(510\) 0 0
\(511\) 5233.03 0.453025
\(512\) 0 0
\(513\) −540.000 −0.0464748
\(514\) 0 0
\(515\) −6364.30 −0.544553
\(516\) 0 0
\(517\) −6472.97 −0.550640
\(518\) 0 0
\(519\) 849.306 0.0718312
\(520\) 0 0
\(521\) −14668.9 −1.23350 −0.616752 0.787158i \(-0.711552\pi\)
−0.616752 + 0.787158i \(0.711552\pi\)
\(522\) 0 0
\(523\) −4305.05 −0.359936 −0.179968 0.983672i \(-0.557599\pi\)
−0.179968 + 0.983672i \(0.557599\pi\)
\(524\) 0 0
\(525\) 6954.05 0.578095
\(526\) 0 0
\(527\) −3949.39 −0.326448
\(528\) 0 0
\(529\) −5640.33 −0.463576
\(530\) 0 0
\(531\) −3267.05 −0.267002
\(532\) 0 0
\(533\) 12786.3 1.03909
\(534\) 0 0
\(535\) 5551.16 0.448594
\(536\) 0 0
\(537\) −4834.51 −0.388500
\(538\) 0 0
\(539\) −1536.52 −0.122788
\(540\) 0 0
\(541\) 1876.27 0.149108 0.0745539 0.997217i \(-0.476247\pi\)
0.0745539 + 0.997217i \(0.476247\pi\)
\(542\) 0 0
\(543\) −8384.98 −0.662678
\(544\) 0 0
\(545\) 30676.4 2.41107
\(546\) 0 0
\(547\) 4753.99 0.371602 0.185801 0.982587i \(-0.440512\pi\)
0.185801 + 0.982587i \(0.440512\pi\)
\(548\) 0 0
\(549\) −3093.49 −0.240486
\(550\) 0 0
\(551\) −2040.00 −0.157726
\(552\) 0 0
\(553\) −9091.17 −0.699088
\(554\) 0 0
\(555\) −13793.9 −1.05499
\(556\) 0 0
\(557\) −18408.3 −1.40033 −0.700165 0.713981i \(-0.746890\pi\)
−0.700165 + 0.713981i \(0.746890\pi\)
\(558\) 0 0
\(559\) −37420.4 −2.83133
\(560\) 0 0
\(561\) 1512.00 0.113791
\(562\) 0 0
\(563\) 1220.61 0.0913723 0.0456861 0.998956i \(-0.485453\pi\)
0.0456861 + 0.998956i \(0.485453\pi\)
\(564\) 0 0
\(565\) −2587.00 −0.192630
\(566\) 0 0
\(567\) 567.000 0.0419961
\(568\) 0 0
\(569\) −12709.3 −0.936381 −0.468191 0.883627i \(-0.655094\pi\)
−0.468191 + 0.883627i \(0.655094\pi\)
\(570\) 0 0
\(571\) −6789.95 −0.497637 −0.248818 0.968550i \(-0.580042\pi\)
−0.248818 + 0.968550i \(0.580042\pi\)
\(572\) 0 0
\(573\) 8360.97 0.609572
\(574\) 0 0
\(575\) 26752.5 1.94027
\(576\) 0 0
\(577\) −4818.41 −0.347648 −0.173824 0.984777i \(-0.555612\pi\)
−0.173824 + 0.984777i \(0.555612\pi\)
\(578\) 0 0
\(579\) −5255.60 −0.377228
\(580\) 0 0
\(581\) 9052.14 0.646379
\(582\) 0 0
\(583\) −13372.0 −0.949934
\(584\) 0 0
\(585\) −16283.8 −1.15086
\(586\) 0 0
\(587\) −2086.69 −0.146724 −0.0733620 0.997305i \(-0.523373\pi\)
−0.0733620 + 0.997305i \(0.523373\pi\)
\(588\) 0 0
\(589\) −4914.42 −0.343795
\(590\) 0 0
\(591\) 2894.15 0.201437
\(592\) 0 0
\(593\) 22800.0 1.57889 0.789447 0.613818i \(-0.210367\pi\)
0.789447 + 0.613818i \(0.210367\pi\)
\(594\) 0 0
\(595\) 2402.91 0.165563
\(596\) 0 0
\(597\) 15853.1 1.08681
\(598\) 0 0
\(599\) −21433.3 −1.46200 −0.731001 0.682376i \(-0.760947\pi\)
−0.731001 + 0.682376i \(0.760947\pi\)
\(600\) 0 0
\(601\) −9749.85 −0.661738 −0.330869 0.943677i \(-0.607342\pi\)
−0.330869 + 0.943677i \(0.607342\pi\)
\(602\) 0 0
\(603\) −623.564 −0.0421119
\(604\) 0 0
\(605\) 7426.10 0.499031
\(606\) 0 0
\(607\) −22837.3 −1.52708 −0.763541 0.645760i \(-0.776541\pi\)
−0.763541 + 0.645760i \(0.776541\pi\)
\(608\) 0 0
\(609\) 2142.00 0.142526
\(610\) 0 0
\(611\) 17487.3 1.15787
\(612\) 0 0
\(613\) −280.302 −0.0184687 −0.00923434 0.999957i \(-0.502939\pi\)
−0.00923434 + 0.999957i \(0.502939\pi\)
\(614\) 0 0
\(615\) −9670.69 −0.634081
\(616\) 0 0
\(617\) −13271.3 −0.865934 −0.432967 0.901410i \(-0.642533\pi\)
−0.432967 + 0.901410i \(0.642533\pi\)
\(618\) 0 0
\(619\) −30464.9 −1.97817 −0.989085 0.147349i \(-0.952926\pi\)
−0.989085 + 0.147349i \(0.952926\pi\)
\(620\) 0 0
\(621\) 2181.27 0.140952
\(622\) 0 0
\(623\) −3943.42 −0.253595
\(624\) 0 0
\(625\) 52639.1 3.36890
\(626\) 0 0
\(627\) 1881.45 0.119837
\(628\) 0 0
\(629\) −3460.20 −0.219344
\(630\) 0 0
\(631\) 8564.10 0.540303 0.270152 0.962818i \(-0.412926\pi\)
0.270152 + 0.962818i \(0.412926\pi\)
\(632\) 0 0
\(633\) 5994.11 0.376373
\(634\) 0 0
\(635\) 10602.2 0.662573
\(636\) 0 0
\(637\) 4151.04 0.258195
\(638\) 0 0
\(639\) 4220.40 0.261278
\(640\) 0 0
\(641\) −3170.52 −0.195363 −0.0976817 0.995218i \(-0.531143\pi\)
−0.0976817 + 0.995218i \(0.531143\pi\)
\(642\) 0 0
\(643\) 22499.6 1.37993 0.689966 0.723842i \(-0.257626\pi\)
0.689966 + 0.723842i \(0.257626\pi\)
\(644\) 0 0
\(645\) 28302.2 1.72775
\(646\) 0 0
\(647\) −24475.4 −1.48722 −0.743608 0.668615i \(-0.766887\pi\)
−0.743608 + 0.668615i \(0.766887\pi\)
\(648\) 0 0
\(649\) 11383.0 0.688476
\(650\) 0 0
\(651\) 5160.14 0.310663
\(652\) 0 0
\(653\) −6047.57 −0.362419 −0.181210 0.983445i \(-0.558001\pi\)
−0.181210 + 0.983445i \(0.558001\pi\)
\(654\) 0 0
\(655\) 57122.8 3.40759
\(656\) 0 0
\(657\) 6728.18 0.399530
\(658\) 0 0
\(659\) −5415.60 −0.320124 −0.160062 0.987107i \(-0.551169\pi\)
−0.160062 + 0.987107i \(0.551169\pi\)
\(660\) 0 0
\(661\) −21652.6 −1.27411 −0.637056 0.770817i \(-0.719848\pi\)
−0.637056 + 0.770817i \(0.719848\pi\)
\(662\) 0 0
\(663\) −4084.80 −0.239277
\(664\) 0 0
\(665\) 2990.06 0.174360
\(666\) 0 0
\(667\) 8240.36 0.478363
\(668\) 0 0
\(669\) 11908.2 0.688189
\(670\) 0 0
\(671\) 10778.3 0.620104
\(672\) 0 0
\(673\) 525.611 0.0301052 0.0150526 0.999887i \(-0.495208\pi\)
0.0150526 + 0.999887i \(0.495208\pi\)
\(674\) 0 0
\(675\) 8940.92 0.509832
\(676\) 0 0
\(677\) 1169.54 0.0663947 0.0331974 0.999449i \(-0.489431\pi\)
0.0331974 + 0.999449i \(0.489431\pi\)
\(678\) 0 0
\(679\) 10395.2 0.587528
\(680\) 0 0
\(681\) 6525.96 0.367218
\(682\) 0 0
\(683\) 11840.7 0.663356 0.331678 0.943393i \(-0.392385\pi\)
0.331678 + 0.943393i \(0.392385\pi\)
\(684\) 0 0
\(685\) −50627.2 −2.82389
\(686\) 0 0
\(687\) 7701.99 0.427728
\(688\) 0 0
\(689\) 36125.6 1.99750
\(690\) 0 0
\(691\) 267.476 0.0147254 0.00736271 0.999973i \(-0.497656\pi\)
0.00736271 + 0.999973i \(0.497656\pi\)
\(692\) 0 0
\(693\) −1975.53 −0.108289
\(694\) 0 0
\(695\) −2038.41 −0.111253
\(696\) 0 0
\(697\) −2425.90 −0.131833
\(698\) 0 0
\(699\) −6344.04 −0.343281
\(700\) 0 0
\(701\) 9700.72 0.522669 0.261335 0.965248i \(-0.415837\pi\)
0.261335 + 0.965248i \(0.415837\pi\)
\(702\) 0 0
\(703\) −4305.70 −0.230999
\(704\) 0 0
\(705\) −13226.2 −0.706562
\(706\) 0 0
\(707\) 10683.8 0.568323
\(708\) 0 0
\(709\) 33319.9 1.76496 0.882479 0.470353i \(-0.155873\pi\)
0.882479 + 0.470353i \(0.155873\pi\)
\(710\) 0 0
\(711\) −11688.6 −0.616538
\(712\) 0 0
\(713\) 19851.3 1.04269
\(714\) 0 0
\(715\) 56735.5 2.96753
\(716\) 0 0
\(717\) −16254.4 −0.846625
\(718\) 0 0
\(719\) −34321.8 −1.78023 −0.890117 0.455733i \(-0.849377\pi\)
−0.890117 + 0.455733i \(0.849377\pi\)
\(720\) 0 0
\(721\) 2085.92 0.107744
\(722\) 0 0
\(723\) 1129.13 0.0580815
\(724\) 0 0
\(725\) 33776.8 1.73026
\(726\) 0 0
\(727\) −27768.3 −1.41660 −0.708301 0.705910i \(-0.750538\pi\)
−0.708301 + 0.705910i \(0.750538\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 7099.64 0.359220
\(732\) 0 0
\(733\) −4358.45 −0.219622 −0.109811 0.993952i \(-0.535025\pi\)
−0.109811 + 0.993952i \(0.535025\pi\)
\(734\) 0 0
\(735\) −3139.56 −0.157557
\(736\) 0 0
\(737\) 2172.60 0.108587
\(738\) 0 0
\(739\) 34023.0 1.69358 0.846789 0.531928i \(-0.178532\pi\)
0.846789 + 0.531928i \(0.178532\pi\)
\(740\) 0 0
\(741\) −5082.91 −0.251991
\(742\) 0 0
\(743\) −19494.5 −0.962563 −0.481281 0.876566i \(-0.659828\pi\)
−0.481281 + 0.876566i \(0.659828\pi\)
\(744\) 0 0
\(745\) 48301.5 2.37534
\(746\) 0 0
\(747\) 11638.5 0.570052
\(748\) 0 0
\(749\) −1819.41 −0.0887580
\(750\) 0 0
\(751\) −3936.95 −0.191294 −0.0956468 0.995415i \(-0.530492\pi\)
−0.0956468 + 0.995415i \(0.530492\pi\)
\(752\) 0 0
\(753\) −5501.81 −0.266264
\(754\) 0 0
\(755\) 2709.93 0.130628
\(756\) 0 0
\(757\) 27339.3 1.31263 0.656316 0.754486i \(-0.272114\pi\)
0.656316 + 0.754486i \(0.272114\pi\)
\(758\) 0 0
\(759\) −7599.92 −0.363452
\(760\) 0 0
\(761\) 1917.52 0.0913402 0.0456701 0.998957i \(-0.485458\pi\)
0.0456701 + 0.998957i \(0.485458\pi\)
\(762\) 0 0
\(763\) −10054.3 −0.477050
\(764\) 0 0
\(765\) 3089.46 0.146013
\(766\) 0 0
\(767\) −30752.1 −1.44771
\(768\) 0 0
\(769\) −9055.78 −0.424655 −0.212328 0.977199i \(-0.568104\pi\)
−0.212328 + 0.977199i \(0.568104\pi\)
\(770\) 0 0
\(771\) 13819.1 0.645501
\(772\) 0 0
\(773\) −29066.5 −1.35246 −0.676228 0.736693i \(-0.736386\pi\)
−0.676228 + 0.736693i \(0.736386\pi\)
\(774\) 0 0
\(775\) 81369.4 3.77145
\(776\) 0 0
\(777\) 4520.98 0.208738
\(778\) 0 0
\(779\) −3018.66 −0.138838
\(780\) 0 0
\(781\) −14704.6 −0.673716
\(782\) 0 0
\(783\) 2754.00 0.125696
\(784\) 0 0
\(785\) −63597.1 −2.89156
\(786\) 0 0
\(787\) −15939.6 −0.721962 −0.360981 0.932573i \(-0.617558\pi\)
−0.360981 + 0.932573i \(0.617558\pi\)
\(788\) 0 0
\(789\) 10697.9 0.482707
\(790\) 0 0
\(791\) 847.895 0.0381134
\(792\) 0 0
\(793\) −29118.4 −1.30394
\(794\) 0 0
\(795\) −27322.9 −1.21892
\(796\) 0 0
\(797\) 16655.0 0.740212 0.370106 0.928989i \(-0.379321\pi\)
0.370106 + 0.928989i \(0.379321\pi\)
\(798\) 0 0
\(799\) −3317.79 −0.146902
\(800\) 0 0
\(801\) −5070.11 −0.223650
\(802\) 0 0
\(803\) −23442.1 −1.03021
\(804\) 0 0
\(805\) −12078.0 −0.528812
\(806\) 0 0
\(807\) 2778.25 0.121189
\(808\) 0 0
\(809\) −3173.02 −0.137896 −0.0689478 0.997620i \(-0.521964\pi\)
−0.0689478 + 0.997620i \(0.521964\pi\)
\(810\) 0 0
\(811\) −2879.83 −0.124691 −0.0623455 0.998055i \(-0.519858\pi\)
−0.0623455 + 0.998055i \(0.519858\pi\)
\(812\) 0 0
\(813\) 14160.7 0.610872
\(814\) 0 0
\(815\) −6471.96 −0.278163
\(816\) 0 0
\(817\) 8834.42 0.378307
\(818\) 0 0
\(819\) 5337.05 0.227707
\(820\) 0 0
\(821\) −16714.5 −0.710522 −0.355261 0.934767i \(-0.615608\pi\)
−0.355261 + 0.934767i \(0.615608\pi\)
\(822\) 0 0
\(823\) −37949.4 −1.60733 −0.803665 0.595082i \(-0.797119\pi\)
−0.803665 + 0.595082i \(0.797119\pi\)
\(824\) 0 0
\(825\) −31151.7 −1.31462
\(826\) 0 0
\(827\) −17817.1 −0.749165 −0.374583 0.927194i \(-0.622214\pi\)
−0.374583 + 0.927194i \(0.622214\pi\)
\(828\) 0 0
\(829\) −22637.8 −0.948425 −0.474213 0.880410i \(-0.657267\pi\)
−0.474213 + 0.880410i \(0.657267\pi\)
\(830\) 0 0
\(831\) −15819.4 −0.660373
\(832\) 0 0
\(833\) −787.561 −0.0327580
\(834\) 0 0
\(835\) 75774.5 3.14046
\(836\) 0 0
\(837\) 6634.47 0.273979
\(838\) 0 0
\(839\) 43907.0 1.80672 0.903359 0.428884i \(-0.141093\pi\)
0.903359 + 0.428884i \(0.141093\pi\)
\(840\) 0 0
\(841\) −13985.0 −0.573414
\(842\) 0 0
\(843\) 12390.6 0.506235
\(844\) 0 0
\(845\) −106353. −4.32978
\(846\) 0 0
\(847\) −2433.92 −0.0987375
\(848\) 0 0
\(849\) −9485.30 −0.383433
\(850\) 0 0
\(851\) 17392.4 0.700592
\(852\) 0 0
\(853\) 41480.0 1.66500 0.832502 0.554022i \(-0.186908\pi\)
0.832502 + 0.554022i \(0.186908\pi\)
\(854\) 0 0
\(855\) 3844.36 0.153771
\(856\) 0 0
\(857\) −34365.9 −1.36980 −0.684899 0.728638i \(-0.740154\pi\)
−0.684899 + 0.728638i \(0.740154\pi\)
\(858\) 0 0
\(859\) −13207.6 −0.524606 −0.262303 0.964986i \(-0.584482\pi\)
−0.262303 + 0.964986i \(0.584482\pi\)
\(860\) 0 0
\(861\) 3169.60 0.125458
\(862\) 0 0
\(863\) 5894.31 0.232497 0.116248 0.993220i \(-0.462913\pi\)
0.116248 + 0.993220i \(0.462913\pi\)
\(864\) 0 0
\(865\) −6046.36 −0.237668
\(866\) 0 0
\(867\) −13964.0 −0.546993
\(868\) 0 0
\(869\) 40725.3 1.58977
\(870\) 0 0
\(871\) −5869.48 −0.228335
\(872\) 0 0
\(873\) 13365.3 0.518151
\(874\) 0 0
\(875\) −30819.3 −1.19072
\(876\) 0 0
\(877\) −36195.4 −1.39365 −0.696825 0.717241i \(-0.745405\pi\)
−0.696825 + 0.717241i \(0.745405\pi\)
\(878\) 0 0
\(879\) 10638.7 0.408230
\(880\) 0 0
\(881\) −2389.46 −0.0913766 −0.0456883 0.998956i \(-0.514548\pi\)
−0.0456883 + 0.998956i \(0.514548\pi\)
\(882\) 0 0
\(883\) −7146.69 −0.272373 −0.136186 0.990683i \(-0.543485\pi\)
−0.136186 + 0.990683i \(0.543485\pi\)
\(884\) 0 0
\(885\) 23258.8 0.883429
\(886\) 0 0
\(887\) 13497.3 0.510930 0.255465 0.966818i \(-0.417771\pi\)
0.255465 + 0.966818i \(0.417771\pi\)
\(888\) 0 0
\(889\) −3474.89 −0.131096
\(890\) 0 0
\(891\) −2539.96 −0.0955016
\(892\) 0 0
\(893\) −4128.49 −0.154708
\(894\) 0 0
\(895\) 34417.8 1.28543
\(896\) 0 0
\(897\) 20531.8 0.764257
\(898\) 0 0
\(899\) 25063.5 0.929828
\(900\) 0 0
\(901\) −6853.97 −0.253428
\(902\) 0 0
\(903\) −9276.14 −0.341850
\(904\) 0 0
\(905\) 59694.3 2.19260
\(906\) 0 0
\(907\) −37635.1 −1.37779 −0.688894 0.724862i \(-0.741903\pi\)
−0.688894 + 0.724862i \(0.741903\pi\)
\(908\) 0 0
\(909\) 13736.3 0.501214
\(910\) 0 0
\(911\) −53207.9 −1.93508 −0.967540 0.252718i \(-0.918675\pi\)
−0.967540 + 0.252718i \(0.918675\pi\)
\(912\) 0 0
\(913\) −40550.4 −1.46991
\(914\) 0 0
\(915\) 22023.1 0.795696
\(916\) 0 0
\(917\) −18722.2 −0.674220
\(918\) 0 0
\(919\) −45909.8 −1.64790 −0.823952 0.566659i \(-0.808236\pi\)
−0.823952 + 0.566659i \(0.808236\pi\)
\(920\) 0 0
\(921\) 1790.90 0.0640739
\(922\) 0 0
\(923\) 39725.7 1.41667
\(924\) 0 0
\(925\) 71290.6 2.53408
\(926\) 0 0
\(927\) 2681.89 0.0950216
\(928\) 0 0
\(929\) 3855.67 0.136168 0.0680841 0.997680i \(-0.478311\pi\)
0.0680841 + 0.997680i \(0.478311\pi\)
\(930\) 0 0
\(931\) −980.000 −0.0344986
\(932\) 0 0
\(933\) 9704.81 0.340537
\(934\) 0 0
\(935\) −10764.2 −0.376500
\(936\) 0 0
\(937\) 5584.25 0.194695 0.0973476 0.995250i \(-0.468964\pi\)
0.0973476 + 0.995250i \(0.468964\pi\)
\(938\) 0 0
\(939\) 6921.35 0.240543
\(940\) 0 0
\(941\) −51277.2 −1.77640 −0.888199 0.459460i \(-0.848043\pi\)
−0.888199 + 0.459460i \(0.848043\pi\)
\(942\) 0 0
\(943\) 12193.6 0.421078
\(944\) 0 0
\(945\) −4036.58 −0.138952
\(946\) 0 0
\(947\) −48182.1 −1.65333 −0.826667 0.562692i \(-0.809766\pi\)
−0.826667 + 0.562692i \(0.809766\pi\)
\(948\) 0 0
\(949\) 63331.0 2.16629
\(950\) 0 0
\(951\) 15571.8 0.530968
\(952\) 0 0
\(953\) 6599.75 0.224330 0.112165 0.993690i \(-0.464221\pi\)
0.112165 + 0.993690i \(0.464221\pi\)
\(954\) 0 0
\(955\) −59523.3 −2.01689
\(956\) 0 0
\(957\) −9595.41 −0.324112
\(958\) 0 0
\(959\) 16593.2 0.558731
\(960\) 0 0
\(961\) 30587.8 1.02675
\(962\) 0 0
\(963\) −2339.24 −0.0782772
\(964\) 0 0
\(965\) 37415.6 1.24814
\(966\) 0 0
\(967\) −36957.9 −1.22905 −0.614523 0.788899i \(-0.710651\pi\)
−0.614523 + 0.788899i \(0.710651\pi\)
\(968\) 0 0
\(969\) 964.361 0.0319708
\(970\) 0 0
\(971\) 38023.4 1.25667 0.628336 0.777942i \(-0.283736\pi\)
0.628336 + 0.777942i \(0.283736\pi\)
\(972\) 0 0
\(973\) 668.093 0.0220124
\(974\) 0 0
\(975\) 84159.1 2.76436
\(976\) 0 0
\(977\) −34863.9 −1.14165 −0.570826 0.821071i \(-0.693377\pi\)
−0.570826 + 0.821071i \(0.693377\pi\)
\(978\) 0 0
\(979\) 17665.2 0.576691
\(980\) 0 0
\(981\) −12926.9 −0.420719
\(982\) 0 0
\(983\) −3499.55 −0.113549 −0.0567743 0.998387i \(-0.518082\pi\)
−0.0567743 + 0.998387i \(0.518082\pi\)
\(984\) 0 0
\(985\) −20604.0 −0.666494
\(986\) 0 0
\(987\) 4334.91 0.139799
\(988\) 0 0
\(989\) −35685.7 −1.14736
\(990\) 0 0
\(991\) 10944.5 0.350821 0.175410 0.984495i \(-0.443875\pi\)
0.175410 + 0.984495i \(0.443875\pi\)
\(992\) 0 0
\(993\) −4953.24 −0.158294
\(994\) 0 0
\(995\) −112861. −3.59592
\(996\) 0 0
\(997\) 46149.3 1.46596 0.732980 0.680250i \(-0.238129\pi\)
0.732980 + 0.680250i \(0.238129\pi\)
\(998\) 0 0
\(999\) 5812.69 0.184090
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 336.4.a.o.1.1 2
3.2 odd 2 1008.4.a.bg.1.2 2
4.3 odd 2 168.4.a.g.1.1 2
7.6 odd 2 2352.4.a.br.1.2 2
8.3 odd 2 1344.4.a.bq.1.2 2
8.5 even 2 1344.4.a.bi.1.2 2
12.11 even 2 504.4.a.n.1.2 2
28.27 even 2 1176.4.a.w.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
168.4.a.g.1.1 2 4.3 odd 2
336.4.a.o.1.1 2 1.1 even 1 trivial
504.4.a.n.1.2 2 12.11 even 2
1008.4.a.bg.1.2 2 3.2 odd 2
1176.4.a.w.1.2 2 28.27 even 2
1344.4.a.bi.1.2 2 8.5 even 2
1344.4.a.bq.1.2 2 8.3 odd 2
2352.4.a.br.1.2 2 7.6 odd 2