Properties

Label 1216.4.a.l.1.1
Level $1216$
Weight $4$
Character 1216.1
Self dual yes
Analytic conductor $71.746$
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] = [1216,4,Mod(1,1216)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1216, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1216.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1216 = 2^{6} \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1216.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: \(71.7463225670\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{177}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 44 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 38)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-6.15207\) of defining polynomial
Character \(\chi\) \(=\) 1216.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-6.15207 q^{3} -18.3041 q^{5} -21.8479 q^{7} +10.8479 q^{9} +O(q^{10})\) \(q-6.15207 q^{3} -18.3041 q^{5} -21.8479 q^{7} +10.8479 q^{9} -8.30413 q^{11} -53.0645 q^{13} +112.608 q^{15} +74.2810 q^{17} -19.0000 q^{19} +134.410 q^{21} +163.977 q^{23} +210.041 q^{25} +99.3686 q^{27} +232.410 q^{29} -98.4331 q^{31} +51.0876 q^{33} +399.908 q^{35} -296.433 q^{37} +326.456 q^{39} -434.912 q^{41} -171.299 q^{43} -198.562 q^{45} +366.083 q^{47} +134.332 q^{49} -456.982 q^{51} -138.631 q^{53} +152.000 q^{55} +116.889 q^{57} +572.797 q^{59} +632.691 q^{61} -237.005 q^{63} +971.299 q^{65} -183.461 q^{67} -1008.80 q^{69} -56.6545 q^{71} +68.1521 q^{73} -1292.19 q^{75} +181.428 q^{77} +332.820 q^{79} -904.217 q^{81} +1152.91 q^{83} -1359.65 q^{85} -1429.80 q^{87} -368.479 q^{89} +1159.35 q^{91} +605.567 q^{93} +347.779 q^{95} -426.443 q^{97} -90.0827 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} - 10 q^{5} - 57 q^{7} + 35 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{3} - 10 q^{5} - 57 q^{7} + 35 q^{9} + 10 q^{11} - 13 q^{13} + 172 q^{15} - 51 q^{17} - 38 q^{19} - 117 q^{21} + 155 q^{23} + 154 q^{25} + 79 q^{27} + 79 q^{29} + 16 q^{31} + 182 q^{33} + 108 q^{35} - 380 q^{37} + 613 q^{39} - 790 q^{41} + 296 q^{43} + 2 q^{45} + 200 q^{47} + 1027 q^{49} - 1353 q^{51} - 397 q^{53} + 304 q^{55} - 19 q^{57} + 201 q^{59} + 680 q^{61} - 1086 q^{63} + 1304 q^{65} - 939 q^{67} - 1073 q^{69} - 406 q^{71} + 123 q^{73} - 1693 q^{75} - 462 q^{77} - 106 q^{79} - 1702 q^{81} + 2226 q^{83} - 2400 q^{85} - 2527 q^{87} - 870 q^{89} - 249 q^{91} + 1424 q^{93} + 190 q^{95} - 1864 q^{97} + 352 q^{99}+O(q^{100}) \) 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
\(3\) −6.15207 −1.18397 −0.591983 0.805950i \(-0.701655\pi\)
−0.591983 + 0.805950i \(0.701655\pi\)
\(4\) 0 0
\(5\) −18.3041 −1.63717 −0.818586 0.574384i \(-0.805242\pi\)
−0.818586 + 0.574384i \(0.805242\pi\)
\(6\) 0 0
\(7\) −21.8479 −1.17968 −0.589839 0.807521i \(-0.700809\pi\)
−0.589839 + 0.807521i \(0.700809\pi\)
\(8\) 0 0
\(9\) 10.8479 0.401775
\(10\) 0 0
\(11\) −8.30413 −0.227617 −0.113809 0.993503i \(-0.536305\pi\)
−0.113809 + 0.993503i \(0.536305\pi\)
\(12\) 0 0
\(13\) −53.0645 −1.13211 −0.566055 0.824367i \(-0.691531\pi\)
−0.566055 + 0.824367i \(0.691531\pi\)
\(14\) 0 0
\(15\) 112.608 1.93836
\(16\) 0 0
\(17\) 74.2810 1.05975 0.529876 0.848075i \(-0.322238\pi\)
0.529876 + 0.848075i \(0.322238\pi\)
\(18\) 0 0
\(19\) −19.0000 −0.229416
\(20\) 0 0
\(21\) 134.410 1.39670
\(22\) 0 0
\(23\) 163.977 1.48659 0.743294 0.668964i \(-0.233262\pi\)
0.743294 + 0.668964i \(0.233262\pi\)
\(24\) 0 0
\(25\) 210.041 1.68033
\(26\) 0 0
\(27\) 99.3686 0.708278
\(28\) 0 0
\(29\) 232.410 1.48819 0.744094 0.668075i \(-0.232882\pi\)
0.744094 + 0.668075i \(0.232882\pi\)
\(30\) 0 0
\(31\) −98.4331 −0.570294 −0.285147 0.958484i \(-0.592042\pi\)
−0.285147 + 0.958484i \(0.592042\pi\)
\(32\) 0 0
\(33\) 51.0876 0.269491
\(34\) 0 0
\(35\) 399.908 1.93133
\(36\) 0 0
\(37\) −296.433 −1.31712 −0.658558 0.752530i \(-0.728833\pi\)
−0.658558 + 0.752530i \(0.728833\pi\)
\(38\) 0 0
\(39\) 326.456 1.34038
\(40\) 0 0
\(41\) −434.912 −1.65663 −0.828316 0.560261i \(-0.810701\pi\)
−0.828316 + 0.560261i \(0.810701\pi\)
\(42\) 0 0
\(43\) −171.299 −0.607509 −0.303755 0.952750i \(-0.598240\pi\)
−0.303755 + 0.952750i \(0.598240\pi\)
\(44\) 0 0
\(45\) −198.562 −0.657775
\(46\) 0 0
\(47\) 366.083 1.13614 0.568071 0.822980i \(-0.307690\pi\)
0.568071 + 0.822980i \(0.307690\pi\)
\(48\) 0 0
\(49\) 134.332 0.391639
\(50\) 0 0
\(51\) −456.982 −1.25471
\(52\) 0 0
\(53\) −138.631 −0.359292 −0.179646 0.983731i \(-0.557495\pi\)
−0.179646 + 0.983731i \(0.557495\pi\)
\(54\) 0 0
\(55\) 152.000 0.372649
\(56\) 0 0
\(57\) 116.889 0.271620
\(58\) 0 0
\(59\) 572.797 1.26393 0.631964 0.774997i \(-0.282249\pi\)
0.631964 + 0.774997i \(0.282249\pi\)
\(60\) 0 0
\(61\) 632.691 1.32800 0.663998 0.747734i \(-0.268858\pi\)
0.663998 + 0.747734i \(0.268858\pi\)
\(62\) 0 0
\(63\) −237.005 −0.473965
\(64\) 0 0
\(65\) 971.299 1.85346
\(66\) 0 0
\(67\) −183.461 −0.334527 −0.167264 0.985912i \(-0.553493\pi\)
−0.167264 + 0.985912i \(0.553493\pi\)
\(68\) 0 0
\(69\) −1008.80 −1.76007
\(70\) 0 0
\(71\) −56.6545 −0.0946994 −0.0473497 0.998878i \(-0.515078\pi\)
−0.0473497 + 0.998878i \(0.515078\pi\)
\(72\) 0 0
\(73\) 68.1521 0.109268 0.0546342 0.998506i \(-0.482601\pi\)
0.0546342 + 0.998506i \(0.482601\pi\)
\(74\) 0 0
\(75\) −1292.19 −1.98945
\(76\) 0 0
\(77\) 181.428 0.268515
\(78\) 0 0
\(79\) 332.820 0.473989 0.236995 0.971511i \(-0.423838\pi\)
0.236995 + 0.971511i \(0.423838\pi\)
\(80\) 0 0
\(81\) −904.217 −1.24035
\(82\) 0 0
\(83\) 1152.91 1.52468 0.762341 0.647176i \(-0.224050\pi\)
0.762341 + 0.647176i \(0.224050\pi\)
\(84\) 0 0
\(85\) −1359.65 −1.73500
\(86\) 0 0
\(87\) −1429.80 −1.76196
\(88\) 0 0
\(89\) −368.479 −0.438862 −0.219431 0.975628i \(-0.570420\pi\)
−0.219431 + 0.975628i \(0.570420\pi\)
\(90\) 0 0
\(91\) 1159.35 1.33553
\(92\) 0 0
\(93\) 605.567 0.675208
\(94\) 0 0
\(95\) 347.779 0.375593
\(96\) 0 0
\(97\) −426.443 −0.446378 −0.223189 0.974775i \(-0.571647\pi\)
−0.223189 + 0.974775i \(0.571647\pi\)
\(98\) 0 0
\(99\) −90.0827 −0.0914510
\(100\) 0 0
\(101\) 403.124 0.397152 0.198576 0.980086i \(-0.436368\pi\)
0.198576 + 0.980086i \(0.436368\pi\)
\(102\) 0 0
\(103\) 1135.68 1.08642 0.543211 0.839596i \(-0.317208\pi\)
0.543211 + 0.839596i \(0.317208\pi\)
\(104\) 0 0
\(105\) −2460.26 −2.28663
\(106\) 0 0
\(107\) −380.096 −0.343414 −0.171707 0.985148i \(-0.554928\pi\)
−0.171707 + 0.985148i \(0.554928\pi\)
\(108\) 0 0
\(109\) −1180.74 −1.03756 −0.518782 0.854907i \(-0.673614\pi\)
−0.518782 + 0.854907i \(0.673614\pi\)
\(110\) 0 0
\(111\) 1823.68 1.55942
\(112\) 0 0
\(113\) −1132.51 −0.942807 −0.471404 0.881918i \(-0.656252\pi\)
−0.471404 + 0.881918i \(0.656252\pi\)
\(114\) 0 0
\(115\) −3001.45 −2.43380
\(116\) 0 0
\(117\) −575.640 −0.454854
\(118\) 0 0
\(119\) −1622.89 −1.25017
\(120\) 0 0
\(121\) −1262.04 −0.948190
\(122\) 0 0
\(123\) 2675.61 1.96140
\(124\) 0 0
\(125\) −1556.61 −1.11382
\(126\) 0 0
\(127\) 40.0462 0.0279806 0.0139903 0.999902i \(-0.495547\pi\)
0.0139903 + 0.999902i \(0.495547\pi\)
\(128\) 0 0
\(129\) 1053.84 0.719270
\(130\) 0 0
\(131\) −177.898 −0.118649 −0.0593244 0.998239i \(-0.518895\pi\)
−0.0593244 + 0.998239i \(0.518895\pi\)
\(132\) 0 0
\(133\) 415.111 0.270637
\(134\) 0 0
\(135\) −1818.86 −1.15957
\(136\) 0 0
\(137\) −24.7603 −0.0154410 −0.00772050 0.999970i \(-0.502458\pi\)
−0.00772050 + 0.999970i \(0.502458\pi\)
\(138\) 0 0
\(139\) 2867.21 1.74959 0.874796 0.484491i \(-0.160995\pi\)
0.874796 + 0.484491i \(0.160995\pi\)
\(140\) 0 0
\(141\) −2252.17 −1.34515
\(142\) 0 0
\(143\) 440.655 0.257688
\(144\) 0 0
\(145\) −4254.06 −2.43642
\(146\) 0 0
\(147\) −826.421 −0.463687
\(148\) 0 0
\(149\) −1949.35 −1.07179 −0.535895 0.844285i \(-0.680026\pi\)
−0.535895 + 0.844285i \(0.680026\pi\)
\(150\) 0 0
\(151\) −1120.99 −0.604136 −0.302068 0.953286i \(-0.597677\pi\)
−0.302068 + 0.953286i \(0.597677\pi\)
\(152\) 0 0
\(153\) 805.795 0.425782
\(154\) 0 0
\(155\) 1801.73 0.933669
\(156\) 0 0
\(157\) 2360.23 1.19979 0.599894 0.800079i \(-0.295209\pi\)
0.599894 + 0.800079i \(0.295209\pi\)
\(158\) 0 0
\(159\) 852.870 0.425390
\(160\) 0 0
\(161\) −3582.56 −1.75370
\(162\) 0 0
\(163\) 861.825 0.414131 0.207065 0.978327i \(-0.433609\pi\)
0.207065 + 0.978327i \(0.433609\pi\)
\(164\) 0 0
\(165\) −935.114 −0.441203
\(166\) 0 0
\(167\) −1686.51 −0.781472 −0.390736 0.920503i \(-0.627779\pi\)
−0.390736 + 0.920503i \(0.627779\pi\)
\(168\) 0 0
\(169\) 618.838 0.281674
\(170\) 0 0
\(171\) −206.111 −0.0921736
\(172\) 0 0
\(173\) 3191.44 1.40255 0.701273 0.712893i \(-0.252616\pi\)
0.701273 + 0.712893i \(0.252616\pi\)
\(174\) 0 0
\(175\) −4588.97 −1.98225
\(176\) 0 0
\(177\) −3523.88 −1.49645
\(178\) 0 0
\(179\) −1229.49 −0.513389 −0.256695 0.966493i \(-0.582633\pi\)
−0.256695 + 0.966493i \(0.582633\pi\)
\(180\) 0 0
\(181\) 3108.95 1.27672 0.638360 0.769738i \(-0.279613\pi\)
0.638360 + 0.769738i \(0.279613\pi\)
\(182\) 0 0
\(183\) −3892.36 −1.57230
\(184\) 0 0
\(185\) 5425.95 2.15635
\(186\) 0 0
\(187\) −616.840 −0.241218
\(188\) 0 0
\(189\) −2171.00 −0.835539
\(190\) 0 0
\(191\) 1415.48 0.536233 0.268117 0.963386i \(-0.413599\pi\)
0.268117 + 0.963386i \(0.413599\pi\)
\(192\) 0 0
\(193\) 1443.40 0.538333 0.269167 0.963094i \(-0.413252\pi\)
0.269167 + 0.963094i \(0.413252\pi\)
\(194\) 0 0
\(195\) −5975.50 −2.19443
\(196\) 0 0
\(197\) −5271.92 −1.90664 −0.953322 0.301954i \(-0.902361\pi\)
−0.953322 + 0.301954i \(0.902361\pi\)
\(198\) 0 0
\(199\) 2510.19 0.894183 0.447091 0.894488i \(-0.352460\pi\)
0.447091 + 0.894488i \(0.352460\pi\)
\(200\) 0 0
\(201\) 1128.67 0.396069
\(202\) 0 0
\(203\) −5077.68 −1.75558
\(204\) 0 0
\(205\) 7960.70 2.71219
\(206\) 0 0
\(207\) 1778.81 0.597275
\(208\) 0 0
\(209\) 157.779 0.0522190
\(210\) 0 0
\(211\) 1854.44 0.605046 0.302523 0.953142i \(-0.402171\pi\)
0.302523 + 0.953142i \(0.402171\pi\)
\(212\) 0 0
\(213\) 348.542 0.112121
\(214\) 0 0
\(215\) 3135.48 0.994597
\(216\) 0 0
\(217\) 2150.56 0.672763
\(218\) 0 0
\(219\) −419.276 −0.129370
\(220\) 0 0
\(221\) −3941.68 −1.19976
\(222\) 0 0
\(223\) −1880.34 −0.564649 −0.282325 0.959319i \(-0.591105\pi\)
−0.282325 + 0.959319i \(0.591105\pi\)
\(224\) 0 0
\(225\) 2278.51 0.675115
\(226\) 0 0
\(227\) 1799.23 0.526075 0.263038 0.964786i \(-0.415276\pi\)
0.263038 + 0.964786i \(0.415276\pi\)
\(228\) 0 0
\(229\) −4835.34 −1.39532 −0.697660 0.716429i \(-0.745776\pi\)
−0.697660 + 0.716429i \(0.745776\pi\)
\(230\) 0 0
\(231\) −1116.16 −0.317913
\(232\) 0 0
\(233\) 865.299 0.243295 0.121647 0.992573i \(-0.461182\pi\)
0.121647 + 0.992573i \(0.461182\pi\)
\(234\) 0 0
\(235\) −6700.83 −1.86006
\(236\) 0 0
\(237\) −2047.53 −0.561187
\(238\) 0 0
\(239\) 4764.27 1.28943 0.644717 0.764421i \(-0.276975\pi\)
0.644717 + 0.764421i \(0.276975\pi\)
\(240\) 0 0
\(241\) 615.336 0.164470 0.0822350 0.996613i \(-0.473794\pi\)
0.0822350 + 0.996613i \(0.473794\pi\)
\(242\) 0 0
\(243\) 2879.85 0.760257
\(244\) 0 0
\(245\) −2458.83 −0.641180
\(246\) 0 0
\(247\) 1008.22 0.259724
\(248\) 0 0
\(249\) −7092.79 −1.80517
\(250\) 0 0
\(251\) −1658.08 −0.416959 −0.208480 0.978027i \(-0.566852\pi\)
−0.208480 + 0.978027i \(0.566852\pi\)
\(252\) 0 0
\(253\) −1361.69 −0.338373
\(254\) 0 0
\(255\) 8364.66 2.05418
\(256\) 0 0
\(257\) 3446.12 0.836432 0.418216 0.908348i \(-0.362656\pi\)
0.418216 + 0.908348i \(0.362656\pi\)
\(258\) 0 0
\(259\) 6476.45 1.55377
\(260\) 0 0
\(261\) 2521.17 0.597917
\(262\) 0 0
\(263\) −5755.80 −1.34950 −0.674748 0.738048i \(-0.735748\pi\)
−0.674748 + 0.738048i \(0.735748\pi\)
\(264\) 0 0
\(265\) 2537.53 0.588223
\(266\) 0 0
\(267\) 2266.91 0.519598
\(268\) 0 0
\(269\) −2257.28 −0.511631 −0.255816 0.966726i \(-0.582344\pi\)
−0.255816 + 0.966726i \(0.582344\pi\)
\(270\) 0 0
\(271\) −7012.13 −1.57180 −0.785898 0.618357i \(-0.787799\pi\)
−0.785898 + 0.618357i \(0.787799\pi\)
\(272\) 0 0
\(273\) −7132.39 −1.58122
\(274\) 0 0
\(275\) −1744.21 −0.382472
\(276\) 0 0
\(277\) 372.810 0.0808664 0.0404332 0.999182i \(-0.487126\pi\)
0.0404332 + 0.999182i \(0.487126\pi\)
\(278\) 0 0
\(279\) −1067.80 −0.229130
\(280\) 0 0
\(281\) −1888.96 −0.401017 −0.200508 0.979692i \(-0.564259\pi\)
−0.200508 + 0.979692i \(0.564259\pi\)
\(282\) 0 0
\(283\) −3884.43 −0.815920 −0.407960 0.913000i \(-0.633760\pi\)
−0.407960 + 0.913000i \(0.633760\pi\)
\(284\) 0 0
\(285\) −2139.56 −0.444689
\(286\) 0 0
\(287\) 9501.94 1.95429
\(288\) 0 0
\(289\) 604.668 0.123075
\(290\) 0 0
\(291\) 2623.51 0.528497
\(292\) 0 0
\(293\) 1273.01 0.253823 0.126911 0.991914i \(-0.459494\pi\)
0.126911 + 0.991914i \(0.459494\pi\)
\(294\) 0 0
\(295\) −10484.5 −2.06927
\(296\) 0 0
\(297\) −825.170 −0.161216
\(298\) 0 0
\(299\) −8701.35 −1.68298
\(300\) 0 0
\(301\) 3742.53 0.716665
\(302\) 0 0
\(303\) −2480.05 −0.470214
\(304\) 0 0
\(305\) −11580.9 −2.17416
\(306\) 0 0
\(307\) 819.153 0.152285 0.0761426 0.997097i \(-0.475740\pi\)
0.0761426 + 0.997097i \(0.475740\pi\)
\(308\) 0 0
\(309\) −6986.76 −1.28629
\(310\) 0 0
\(311\) −2104.67 −0.383745 −0.191872 0.981420i \(-0.561456\pi\)
−0.191872 + 0.981420i \(0.561456\pi\)
\(312\) 0 0
\(313\) 2395.47 0.432587 0.216293 0.976328i \(-0.430603\pi\)
0.216293 + 0.976328i \(0.430603\pi\)
\(314\) 0 0
\(315\) 4338.17 0.775962
\(316\) 0 0
\(317\) −2158.23 −0.382393 −0.191196 0.981552i \(-0.561237\pi\)
−0.191196 + 0.981552i \(0.561237\pi\)
\(318\) 0 0
\(319\) −1929.96 −0.338737
\(320\) 0 0
\(321\) 2338.38 0.406590
\(322\) 0 0
\(323\) −1411.34 −0.243124
\(324\) 0 0
\(325\) −11145.7 −1.90232
\(326\) 0 0
\(327\) 7264.00 1.22844
\(328\) 0 0
\(329\) −7998.15 −1.34028
\(330\) 0 0
\(331\) 517.782 0.0859815 0.0429908 0.999075i \(-0.486311\pi\)
0.0429908 + 0.999075i \(0.486311\pi\)
\(332\) 0 0
\(333\) −3215.69 −0.529185
\(334\) 0 0
\(335\) 3358.10 0.547679
\(336\) 0 0
\(337\) 6503.63 1.05126 0.525631 0.850713i \(-0.323829\pi\)
0.525631 + 0.850713i \(0.323829\pi\)
\(338\) 0 0
\(339\) 6967.25 1.11625
\(340\) 0 0
\(341\) 817.402 0.129809
\(342\) 0 0
\(343\) 4558.96 0.717670
\(344\) 0 0
\(345\) 18465.2 2.88154
\(346\) 0 0
\(347\) 6058.33 0.937257 0.468628 0.883395i \(-0.344748\pi\)
0.468628 + 0.883395i \(0.344748\pi\)
\(348\) 0 0
\(349\) 10955.1 1.68027 0.840135 0.542377i \(-0.182476\pi\)
0.840135 + 0.542377i \(0.182476\pi\)
\(350\) 0 0
\(351\) −5272.94 −0.801849
\(352\) 0 0
\(353\) −1806.43 −0.272369 −0.136185 0.990683i \(-0.543484\pi\)
−0.136185 + 0.990683i \(0.543484\pi\)
\(354\) 0 0
\(355\) 1037.01 0.155039
\(356\) 0 0
\(357\) 9984.11 1.48015
\(358\) 0 0
\(359\) 7964.50 1.17089 0.585446 0.810711i \(-0.300919\pi\)
0.585446 + 0.810711i \(0.300919\pi\)
\(360\) 0 0
\(361\) 361.000 0.0526316
\(362\) 0 0
\(363\) 7764.16 1.12263
\(364\) 0 0
\(365\) −1247.46 −0.178891
\(366\) 0 0
\(367\) 7311.58 1.03995 0.519974 0.854182i \(-0.325941\pi\)
0.519974 + 0.854182i \(0.325941\pi\)
\(368\) 0 0
\(369\) −4717.90 −0.665594
\(370\) 0 0
\(371\) 3028.81 0.423849
\(372\) 0 0
\(373\) −5518.38 −0.766035 −0.383017 0.923741i \(-0.625115\pi\)
−0.383017 + 0.923741i \(0.625115\pi\)
\(374\) 0 0
\(375\) 9576.36 1.31872
\(376\) 0 0
\(377\) −12332.7 −1.68479
\(378\) 0 0
\(379\) −1139.97 −0.154502 −0.0772512 0.997012i \(-0.524614\pi\)
−0.0772512 + 0.997012i \(0.524614\pi\)
\(380\) 0 0
\(381\) −246.367 −0.0331280
\(382\) 0 0
\(383\) 10409.5 1.38877 0.694385 0.719604i \(-0.255677\pi\)
0.694385 + 0.719604i \(0.255677\pi\)
\(384\) 0 0
\(385\) −3320.89 −0.439605
\(386\) 0 0
\(387\) −1858.24 −0.244082
\(388\) 0 0
\(389\) −10471.2 −1.36481 −0.682404 0.730975i \(-0.739066\pi\)
−0.682404 + 0.730975i \(0.739066\pi\)
\(390\) 0 0
\(391\) 12180.4 1.57542
\(392\) 0 0
\(393\) 1094.44 0.140476
\(394\) 0 0
\(395\) −6091.98 −0.776002
\(396\) 0 0
\(397\) 9588.68 1.21220 0.606098 0.795390i \(-0.292734\pi\)
0.606098 + 0.795390i \(0.292734\pi\)
\(398\) 0 0
\(399\) −2553.79 −0.320424
\(400\) 0 0
\(401\) 8549.30 1.06467 0.532334 0.846535i \(-0.321315\pi\)
0.532334 + 0.846535i \(0.321315\pi\)
\(402\) 0 0
\(403\) 5223.30 0.645635
\(404\) 0 0
\(405\) 16550.9 2.03067
\(406\) 0 0
\(407\) 2461.62 0.299798
\(408\) 0 0
\(409\) 266.960 0.0322746 0.0161373 0.999870i \(-0.494863\pi\)
0.0161373 + 0.999870i \(0.494863\pi\)
\(410\) 0 0
\(411\) 152.327 0.0182816
\(412\) 0 0
\(413\) −12514.4 −1.49103
\(414\) 0 0
\(415\) −21103.1 −2.49617
\(416\) 0 0
\(417\) −17639.2 −2.07146
\(418\) 0 0
\(419\) 8643.83 1.00783 0.503913 0.863755i \(-0.331893\pi\)
0.503913 + 0.863755i \(0.331893\pi\)
\(420\) 0 0
\(421\) 16801.3 1.94500 0.972499 0.232905i \(-0.0748232\pi\)
0.972499 + 0.232905i \(0.0748232\pi\)
\(422\) 0 0
\(423\) 3971.24 0.456474
\(424\) 0 0
\(425\) 15602.1 1.78073
\(426\) 0 0
\(427\) −13823.0 −1.56661
\(428\) 0 0
\(429\) −2710.94 −0.305094
\(430\) 0 0
\(431\) −12053.5 −1.34710 −0.673548 0.739144i \(-0.735231\pi\)
−0.673548 + 0.739144i \(0.735231\pi\)
\(432\) 0 0
\(433\) 9034.61 1.00271 0.501357 0.865240i \(-0.332834\pi\)
0.501357 + 0.865240i \(0.332834\pi\)
\(434\) 0 0
\(435\) 26171.3 2.88464
\(436\) 0 0
\(437\) −3115.56 −0.341047
\(438\) 0 0
\(439\) −3008.87 −0.327120 −0.163560 0.986533i \(-0.552298\pi\)
−0.163560 + 0.986533i \(0.552298\pi\)
\(440\) 0 0
\(441\) 1457.23 0.157351
\(442\) 0 0
\(443\) −229.594 −0.0246237 −0.0123119 0.999924i \(-0.503919\pi\)
−0.0123119 + 0.999924i \(0.503919\pi\)
\(444\) 0 0
\(445\) 6744.70 0.718493
\(446\) 0 0
\(447\) 11992.5 1.26896
\(448\) 0 0
\(449\) −7559.44 −0.794548 −0.397274 0.917700i \(-0.630044\pi\)
−0.397274 + 0.917700i \(0.630044\pi\)
\(450\) 0 0
\(451\) 3611.57 0.377078
\(452\) 0 0
\(453\) 6896.38 0.715276
\(454\) 0 0
\(455\) −21220.9 −2.18648
\(456\) 0 0
\(457\) 11556.4 1.18290 0.591449 0.806343i \(-0.298556\pi\)
0.591449 + 0.806343i \(0.298556\pi\)
\(458\) 0 0
\(459\) 7381.20 0.750599
\(460\) 0 0
\(461\) −9191.58 −0.928622 −0.464311 0.885672i \(-0.653698\pi\)
−0.464311 + 0.885672i \(0.653698\pi\)
\(462\) 0 0
\(463\) −1356.03 −0.136113 −0.0680564 0.997681i \(-0.521680\pi\)
−0.0680564 + 0.997681i \(0.521680\pi\)
\(464\) 0 0
\(465\) −11084.4 −1.10543
\(466\) 0 0
\(467\) −14808.9 −1.46740 −0.733700 0.679473i \(-0.762208\pi\)
−0.733700 + 0.679473i \(0.762208\pi\)
\(468\) 0 0
\(469\) 4008.25 0.394635
\(470\) 0 0
\(471\) −14520.3 −1.42051
\(472\) 0 0
\(473\) 1422.49 0.138280
\(474\) 0 0
\(475\) −3990.79 −0.385494
\(476\) 0 0
\(477\) −1503.86 −0.144355
\(478\) 0 0
\(479\) −9834.66 −0.938115 −0.469058 0.883168i \(-0.655406\pi\)
−0.469058 + 0.883168i \(0.655406\pi\)
\(480\) 0 0
\(481\) 15730.1 1.49112
\(482\) 0 0
\(483\) 22040.1 2.07632
\(484\) 0 0
\(485\) 7805.67 0.730798
\(486\) 0 0
\(487\) −3687.82 −0.343144 −0.171572 0.985172i \(-0.554885\pi\)
−0.171572 + 0.985172i \(0.554885\pi\)
\(488\) 0 0
\(489\) −5302.00 −0.490317
\(490\) 0 0
\(491\) −11197.4 −1.02919 −0.514593 0.857435i \(-0.672057\pi\)
−0.514593 + 0.857435i \(0.672057\pi\)
\(492\) 0 0
\(493\) 17263.6 1.57711
\(494\) 0 0
\(495\) 1648.89 0.149721
\(496\) 0 0
\(497\) 1237.78 0.111715
\(498\) 0 0
\(499\) −12101.6 −1.08566 −0.542829 0.839843i \(-0.682647\pi\)
−0.542829 + 0.839843i \(0.682647\pi\)
\(500\) 0 0
\(501\) 10375.5 0.925236
\(502\) 0 0
\(503\) 7266.58 0.644136 0.322068 0.946716i \(-0.395622\pi\)
0.322068 + 0.946716i \(0.395622\pi\)
\(504\) 0 0
\(505\) −7378.84 −0.650206
\(506\) 0 0
\(507\) −3807.13 −0.333493
\(508\) 0 0
\(509\) −2564.99 −0.223362 −0.111681 0.993744i \(-0.535624\pi\)
−0.111681 + 0.993744i \(0.535624\pi\)
\(510\) 0 0
\(511\) −1488.98 −0.128902
\(512\) 0 0
\(513\) −1888.00 −0.162490
\(514\) 0 0
\(515\) −20787.6 −1.77866
\(516\) 0 0
\(517\) −3040.00 −0.258606
\(518\) 0 0
\(519\) −19633.9 −1.66057
\(520\) 0 0
\(521\) 11254.6 0.946401 0.473201 0.880955i \(-0.343099\pi\)
0.473201 + 0.880955i \(0.343099\pi\)
\(522\) 0 0
\(523\) −18357.1 −1.53480 −0.767401 0.641168i \(-0.778450\pi\)
−0.767401 + 0.641168i \(0.778450\pi\)
\(524\) 0 0
\(525\) 28231.6 2.34691
\(526\) 0 0
\(527\) −7311.71 −0.604370
\(528\) 0 0
\(529\) 14721.4 1.20995
\(530\) 0 0
\(531\) 6213.66 0.507815
\(532\) 0 0
\(533\) 23078.4 1.87549
\(534\) 0 0
\(535\) 6957.33 0.562227
\(536\) 0 0
\(537\) 7563.93 0.607836
\(538\) 0 0
\(539\) −1115.51 −0.0891438
\(540\) 0 0
\(541\) −11381.8 −0.904510 −0.452255 0.891889i \(-0.649380\pi\)
−0.452255 + 0.891889i \(0.649380\pi\)
\(542\) 0 0
\(543\) −19126.5 −1.51159
\(544\) 0 0
\(545\) 21612.4 1.69867
\(546\) 0 0
\(547\) −8996.84 −0.703248 −0.351624 0.936141i \(-0.614371\pi\)
−0.351624 + 0.936141i \(0.614371\pi\)
\(548\) 0 0
\(549\) 6863.39 0.533556
\(550\) 0 0
\(551\) −4415.79 −0.341414
\(552\) 0 0
\(553\) −7271.43 −0.559155
\(554\) 0 0
\(555\) −33380.8 −2.55304
\(556\) 0 0
\(557\) −10759.8 −0.818507 −0.409253 0.912421i \(-0.634211\pi\)
−0.409253 + 0.912421i \(0.634211\pi\)
\(558\) 0 0
\(559\) 9089.90 0.687767
\(560\) 0 0
\(561\) 3794.84 0.285594
\(562\) 0 0
\(563\) 25381.8 1.90003 0.950015 0.312205i \(-0.101068\pi\)
0.950015 + 0.312205i \(0.101068\pi\)
\(564\) 0 0
\(565\) 20729.5 1.54354
\(566\) 0 0
\(567\) 19755.3 1.46322
\(568\) 0 0
\(569\) −5546.00 −0.408612 −0.204306 0.978907i \(-0.565494\pi\)
−0.204306 + 0.978907i \(0.565494\pi\)
\(570\) 0 0
\(571\) 7714.96 0.565431 0.282715 0.959204i \(-0.408765\pi\)
0.282715 + 0.959204i \(0.408765\pi\)
\(572\) 0 0
\(573\) −8708.13 −0.634882
\(574\) 0 0
\(575\) 34441.9 2.49796
\(576\) 0 0
\(577\) 21335.1 1.53933 0.769665 0.638448i \(-0.220423\pi\)
0.769665 + 0.638448i \(0.220423\pi\)
\(578\) 0 0
\(579\) −8879.90 −0.637368
\(580\) 0 0
\(581\) −25188.8 −1.79863
\(582\) 0 0
\(583\) 1151.21 0.0817811
\(584\) 0 0
\(585\) 10536.6 0.744674
\(586\) 0 0
\(587\) 12370.9 0.869846 0.434923 0.900468i \(-0.356776\pi\)
0.434923 + 0.900468i \(0.356776\pi\)
\(588\) 0 0
\(589\) 1870.23 0.130834
\(590\) 0 0
\(591\) 32433.2 2.25740
\(592\) 0 0
\(593\) 12982.5 0.899034 0.449517 0.893272i \(-0.351596\pi\)
0.449517 + 0.893272i \(0.351596\pi\)
\(594\) 0 0
\(595\) 29705.5 2.04674
\(596\) 0 0
\(597\) −15442.8 −1.05868
\(598\) 0 0
\(599\) −13163.7 −0.897921 −0.448960 0.893552i \(-0.648206\pi\)
−0.448960 + 0.893552i \(0.648206\pi\)
\(600\) 0 0
\(601\) −29325.4 −1.99036 −0.995180 0.0980640i \(-0.968735\pi\)
−0.995180 + 0.0980640i \(0.968735\pi\)
\(602\) 0 0
\(603\) −1990.17 −0.134405
\(604\) 0 0
\(605\) 23100.6 1.55235
\(606\) 0 0
\(607\) 2170.11 0.145110 0.0725552 0.997364i \(-0.476885\pi\)
0.0725552 + 0.997364i \(0.476885\pi\)
\(608\) 0 0
\(609\) 31238.2 2.07855
\(610\) 0 0
\(611\) −19426.0 −1.28624
\(612\) 0 0
\(613\) 4917.96 0.324036 0.162018 0.986788i \(-0.448200\pi\)
0.162018 + 0.986788i \(0.448200\pi\)
\(614\) 0 0
\(615\) −48974.7 −3.21114
\(616\) 0 0
\(617\) 24763.5 1.61579 0.807894 0.589328i \(-0.200607\pi\)
0.807894 + 0.589328i \(0.200607\pi\)
\(618\) 0 0
\(619\) −27552.0 −1.78902 −0.894512 0.447043i \(-0.852477\pi\)
−0.894512 + 0.447043i \(0.852477\pi\)
\(620\) 0 0
\(621\) 16294.2 1.05292
\(622\) 0 0
\(623\) 8050.51 0.517716
\(624\) 0 0
\(625\) 2237.20 0.143181
\(626\) 0 0
\(627\) −970.664 −0.0618255
\(628\) 0 0
\(629\) −22019.3 −1.39582
\(630\) 0 0
\(631\) −377.839 −0.0238376 −0.0119188 0.999929i \(-0.503794\pi\)
−0.0119188 + 0.999929i \(0.503794\pi\)
\(632\) 0 0
\(633\) −11408.6 −0.716354
\(634\) 0 0
\(635\) −733.012 −0.0458090
\(636\) 0 0
\(637\) −7128.27 −0.443379
\(638\) 0 0
\(639\) −614.584 −0.0380479
\(640\) 0 0
\(641\) −14047.3 −0.865576 −0.432788 0.901496i \(-0.642470\pi\)
−0.432788 + 0.901496i \(0.642470\pi\)
\(642\) 0 0
\(643\) 5768.94 0.353818 0.176909 0.984227i \(-0.443390\pi\)
0.176909 + 0.984227i \(0.443390\pi\)
\(644\) 0 0
\(645\) −19289.7 −1.17757
\(646\) 0 0
\(647\) −4568.59 −0.277604 −0.138802 0.990320i \(-0.544325\pi\)
−0.138802 + 0.990320i \(0.544325\pi\)
\(648\) 0 0
\(649\) −4756.58 −0.287692
\(650\) 0 0
\(651\) −13230.4 −0.796528
\(652\) 0 0
\(653\) 16532.7 0.990774 0.495387 0.868672i \(-0.335026\pi\)
0.495387 + 0.868672i \(0.335026\pi\)
\(654\) 0 0
\(655\) 3256.26 0.194248
\(656\) 0 0
\(657\) 739.309 0.0439014
\(658\) 0 0
\(659\) −18630.0 −1.10125 −0.550624 0.834753i \(-0.685610\pi\)
−0.550624 + 0.834753i \(0.685610\pi\)
\(660\) 0 0
\(661\) −28851.3 −1.69771 −0.848854 0.528627i \(-0.822707\pi\)
−0.848854 + 0.528627i \(0.822707\pi\)
\(662\) 0 0
\(663\) 24249.5 1.42047
\(664\) 0 0
\(665\) −7598.24 −0.443079
\(666\) 0 0
\(667\) 38109.9 2.21232
\(668\) 0 0
\(669\) 11568.0 0.668525
\(670\) 0 0
\(671\) −5253.95 −0.302275
\(672\) 0 0
\(673\) −3873.41 −0.221856 −0.110928 0.993828i \(-0.535382\pi\)
−0.110928 + 0.993828i \(0.535382\pi\)
\(674\) 0 0
\(675\) 20871.5 1.19014
\(676\) 0 0
\(677\) −5025.24 −0.285282 −0.142641 0.989775i \(-0.545559\pi\)
−0.142641 + 0.989775i \(0.545559\pi\)
\(678\) 0 0
\(679\) 9316.90 0.526583
\(680\) 0 0
\(681\) −11069.0 −0.622855
\(682\) 0 0
\(683\) −14157.0 −0.793122 −0.396561 0.918008i \(-0.629796\pi\)
−0.396561 + 0.918008i \(0.629796\pi\)
\(684\) 0 0
\(685\) 453.217 0.0252796
\(686\) 0 0
\(687\) 29747.4 1.65201
\(688\) 0 0
\(689\) 7356.40 0.406758
\(690\) 0 0
\(691\) −2437.65 −0.134200 −0.0671002 0.997746i \(-0.521375\pi\)
−0.0671002 + 0.997746i \(0.521375\pi\)
\(692\) 0 0
\(693\) 1968.12 0.107883
\(694\) 0 0
\(695\) −52481.7 −2.86438
\(696\) 0 0
\(697\) −32305.7 −1.75562
\(698\) 0 0
\(699\) −5323.38 −0.288052
\(700\) 0 0
\(701\) −9496.96 −0.511691 −0.255845 0.966718i \(-0.582354\pi\)
−0.255845 + 0.966718i \(0.582354\pi\)
\(702\) 0 0
\(703\) 5632.23 0.302167
\(704\) 0 0
\(705\) 41223.9 2.20225
\(706\) 0 0
\(707\) −8807.43 −0.468511
\(708\) 0 0
\(709\) −6350.33 −0.336377 −0.168189 0.985755i \(-0.553792\pi\)
−0.168189 + 0.985755i \(0.553792\pi\)
\(710\) 0 0
\(711\) 3610.41 0.190437
\(712\) 0 0
\(713\) −16140.7 −0.847792
\(714\) 0 0
\(715\) −8065.80 −0.421879
\(716\) 0 0
\(717\) −29310.1 −1.52665
\(718\) 0 0
\(719\) 19722.6 1.02299 0.511496 0.859286i \(-0.329092\pi\)
0.511496 + 0.859286i \(0.329092\pi\)
\(720\) 0 0
\(721\) −24812.2 −1.28163
\(722\) 0 0
\(723\) −3785.59 −0.194727
\(724\) 0 0
\(725\) 48815.7 2.50065
\(726\) 0 0
\(727\) −28325.8 −1.44504 −0.722520 0.691350i \(-0.757016\pi\)
−0.722520 + 0.691350i \(0.757016\pi\)
\(728\) 0 0
\(729\) 6696.82 0.340234
\(730\) 0 0
\(731\) −12724.3 −0.643809
\(732\) 0 0
\(733\) −33981.8 −1.71234 −0.856170 0.516694i \(-0.827163\pi\)
−0.856170 + 0.516694i \(0.827163\pi\)
\(734\) 0 0
\(735\) 15126.9 0.759135
\(736\) 0 0
\(737\) 1523.49 0.0761443
\(738\) 0 0
\(739\) 9147.10 0.455320 0.227660 0.973741i \(-0.426893\pi\)
0.227660 + 0.973741i \(0.426893\pi\)
\(740\) 0 0
\(741\) −6202.67 −0.307504
\(742\) 0 0
\(743\) 34159.2 1.68665 0.843324 0.537405i \(-0.180596\pi\)
0.843324 + 0.537405i \(0.180596\pi\)
\(744\) 0 0
\(745\) 35681.1 1.75470
\(746\) 0 0
\(747\) 12506.7 0.612579
\(748\) 0 0
\(749\) 8304.31 0.405117
\(750\) 0 0
\(751\) −28361.8 −1.37808 −0.689038 0.724725i \(-0.741967\pi\)
−0.689038 + 0.724725i \(0.741967\pi\)
\(752\) 0 0
\(753\) 10200.6 0.493666
\(754\) 0 0
\(755\) 20518.7 0.989074
\(756\) 0 0
\(757\) −11464.9 −0.550462 −0.275231 0.961378i \(-0.588754\pi\)
−0.275231 + 0.961378i \(0.588754\pi\)
\(758\) 0 0
\(759\) 8377.18 0.400623
\(760\) 0 0
\(761\) 14289.3 0.680666 0.340333 0.940305i \(-0.389460\pi\)
0.340333 + 0.940305i \(0.389460\pi\)
\(762\) 0 0
\(763\) 25796.7 1.22399
\(764\) 0 0
\(765\) −14749.4 −0.697079
\(766\) 0 0
\(767\) −30395.2 −1.43091
\(768\) 0 0
\(769\) 100.811 0.00472734 0.00236367 0.999997i \(-0.499248\pi\)
0.00236367 + 0.999997i \(0.499248\pi\)
\(770\) 0 0
\(771\) −21200.8 −0.990307
\(772\) 0 0
\(773\) −52.1073 −0.00242454 −0.00121227 0.999999i \(-0.500386\pi\)
−0.00121227 + 0.999999i \(0.500386\pi\)
\(774\) 0 0
\(775\) −20675.0 −0.958282
\(776\) 0 0
\(777\) −39843.6 −1.83961
\(778\) 0 0
\(779\) 8263.34 0.380057
\(780\) 0 0
\(781\) 470.467 0.0215552
\(782\) 0 0
\(783\) 23094.3 1.05405
\(784\) 0 0
\(785\) −43201.9 −1.96426
\(786\) 0 0
\(787\) 15261.8 0.691266 0.345633 0.938370i \(-0.387664\pi\)
0.345633 + 0.938370i \(0.387664\pi\)
\(788\) 0 0
\(789\) 35410.0 1.59776
\(790\) 0 0
\(791\) 24742.9 1.11221
\(792\) 0 0
\(793\) −33573.4 −1.50344
\(794\) 0 0
\(795\) −15611.0 −0.696436
\(796\) 0 0
\(797\) −24432.3 −1.08587 −0.542934 0.839775i \(-0.682687\pi\)
−0.542934 + 0.839775i \(0.682687\pi\)
\(798\) 0 0
\(799\) 27193.0 1.20403
\(800\) 0 0
\(801\) −3997.24 −0.176324
\(802\) 0 0
\(803\) −565.944 −0.0248714
\(804\) 0 0
\(805\) 65575.6 2.87110
\(806\) 0 0
\(807\) 13886.9 0.605754
\(808\) 0 0
\(809\) 3635.86 0.158010 0.0790050 0.996874i \(-0.474826\pi\)
0.0790050 + 0.996874i \(0.474826\pi\)
\(810\) 0 0
\(811\) 1087.93 0.0471051 0.0235526 0.999723i \(-0.492502\pi\)
0.0235526 + 0.999723i \(0.492502\pi\)
\(812\) 0 0
\(813\) 43139.1 1.86095
\(814\) 0 0
\(815\) −15775.0 −0.678003
\(816\) 0 0
\(817\) 3254.69 0.139372
\(818\) 0 0
\(819\) 12576.5 0.536581
\(820\) 0 0
\(821\) −10953.7 −0.465636 −0.232818 0.972520i \(-0.574795\pi\)
−0.232818 + 0.972520i \(0.574795\pi\)
\(822\) 0 0
\(823\) −13502.5 −0.571893 −0.285947 0.958246i \(-0.592308\pi\)
−0.285947 + 0.958246i \(0.592308\pi\)
\(824\) 0 0
\(825\) 10730.5 0.452834
\(826\) 0 0
\(827\) 26812.4 1.12740 0.563699 0.825980i \(-0.309378\pi\)
0.563699 + 0.825980i \(0.309378\pi\)
\(828\) 0 0
\(829\) 4497.94 0.188444 0.0942218 0.995551i \(-0.469964\pi\)
0.0942218 + 0.995551i \(0.469964\pi\)
\(830\) 0 0
\(831\) −2293.55 −0.0957430
\(832\) 0 0
\(833\) 9978.33 0.415040
\(834\) 0 0
\(835\) 30870.0 1.27940
\(836\) 0 0
\(837\) −9781.16 −0.403926
\(838\) 0 0
\(839\) −30324.9 −1.24783 −0.623917 0.781491i \(-0.714460\pi\)
−0.623917 + 0.781491i \(0.714460\pi\)
\(840\) 0 0
\(841\) 29625.4 1.21470
\(842\) 0 0
\(843\) 11621.0 0.474790
\(844\) 0 0
\(845\) −11327.3 −0.461149
\(846\) 0 0
\(847\) 27573.0 1.11856
\(848\) 0 0
\(849\) 23897.3 0.966022
\(850\) 0 0
\(851\) −48608.2 −1.95801
\(852\) 0 0
\(853\) −17230.8 −0.691642 −0.345821 0.938300i \(-0.612400\pi\)
−0.345821 + 0.938300i \(0.612400\pi\)
\(854\) 0 0
\(855\) 3772.68 0.150904
\(856\) 0 0
\(857\) −6724.12 −0.268018 −0.134009 0.990980i \(-0.542785\pi\)
−0.134009 + 0.990980i \(0.542785\pi\)
\(858\) 0 0
\(859\) −36183.5 −1.43721 −0.718606 0.695417i \(-0.755219\pi\)
−0.718606 + 0.695417i \(0.755219\pi\)
\(860\) 0 0
\(861\) −58456.6 −2.31381
\(862\) 0 0
\(863\) −14350.8 −0.566056 −0.283028 0.959112i \(-0.591339\pi\)
−0.283028 + 0.959112i \(0.591339\pi\)
\(864\) 0 0
\(865\) −58416.5 −2.29621
\(866\) 0 0
\(867\) −3719.96 −0.145717
\(868\) 0 0
\(869\) −2763.78 −0.107888
\(870\) 0 0
\(871\) 9735.27 0.378722
\(872\) 0 0
\(873\) −4626.02 −0.179344
\(874\) 0 0
\(875\) 34008.7 1.31395
\(876\) 0 0
\(877\) 42978.6 1.65483 0.827414 0.561592i \(-0.189811\pi\)
0.827414 + 0.561592i \(0.189811\pi\)
\(878\) 0 0
\(879\) −7831.65 −0.300518
\(880\) 0 0
\(881\) −7298.75 −0.279116 −0.139558 0.990214i \(-0.544568\pi\)
−0.139558 + 0.990214i \(0.544568\pi\)
\(882\) 0 0
\(883\) 46722.1 1.78066 0.890331 0.455314i \(-0.150473\pi\)
0.890331 + 0.455314i \(0.150473\pi\)
\(884\) 0 0
\(885\) 64501.7 2.44994
\(886\) 0 0
\(887\) −35268.7 −1.33507 −0.667535 0.744578i \(-0.732651\pi\)
−0.667535 + 0.744578i \(0.732651\pi\)
\(888\) 0 0
\(889\) −874.928 −0.0330080
\(890\) 0 0
\(891\) 7508.74 0.282326
\(892\) 0 0
\(893\) −6955.57 −0.260649
\(894\) 0 0
\(895\) 22504.8 0.840506
\(896\) 0 0
\(897\) 53531.3 1.99259
\(898\) 0 0
\(899\) −22876.8 −0.848704
\(900\) 0 0
\(901\) −10297.7 −0.380761
\(902\) 0 0
\(903\) −23024.3 −0.848507
\(904\) 0 0
\(905\) −56906.6 −2.09021
\(906\) 0 0
\(907\) −46292.7 −1.69474 −0.847368 0.531007i \(-0.821814\pi\)
−0.847368 + 0.531007i \(0.821814\pi\)
\(908\) 0 0
\(909\) 4373.06 0.159566
\(910\) 0 0
\(911\) 42085.5 1.53058 0.765288 0.643688i \(-0.222597\pi\)
0.765288 + 0.643688i \(0.222597\pi\)
\(912\) 0 0
\(913\) −9573.94 −0.347044
\(914\) 0 0
\(915\) 71246.2 2.57413
\(916\) 0 0
\(917\) 3886.70 0.139967
\(918\) 0 0
\(919\) −21331.8 −0.765691 −0.382845 0.923812i \(-0.625056\pi\)
−0.382845 + 0.923812i \(0.625056\pi\)
\(920\) 0 0
\(921\) −5039.49 −0.180300
\(922\) 0 0
\(923\) 3006.34 0.107210
\(924\) 0 0
\(925\) −62263.2 −2.21319
\(926\) 0 0
\(927\) 12319.7 0.436498
\(928\) 0 0
\(929\) −34331.5 −1.21247 −0.606233 0.795287i \(-0.707320\pi\)
−0.606233 + 0.795287i \(0.707320\pi\)
\(930\) 0 0
\(931\) −2552.31 −0.0898481
\(932\) 0 0
\(933\) 12948.0 0.454341
\(934\) 0 0
\(935\) 11290.7 0.394915
\(936\) 0 0
\(937\) 13625.1 0.475041 0.237521 0.971383i \(-0.423665\pi\)
0.237521 + 0.971383i \(0.423665\pi\)
\(938\) 0 0
\(939\) −14737.1 −0.512168
\(940\) 0 0
\(941\) −7086.48 −0.245497 −0.122748 0.992438i \(-0.539171\pi\)
−0.122748 + 0.992438i \(0.539171\pi\)
\(942\) 0 0
\(943\) −71315.6 −2.46273
\(944\) 0 0
\(945\) 39738.3 1.36792
\(946\) 0 0
\(947\) 38735.9 1.32920 0.664598 0.747202i \(-0.268603\pi\)
0.664598 + 0.747202i \(0.268603\pi\)
\(948\) 0 0
\(949\) −3616.45 −0.123704
\(950\) 0 0
\(951\) 13277.6 0.452740
\(952\) 0 0
\(953\) 30964.3 1.05250 0.526249 0.850330i \(-0.323598\pi\)
0.526249 + 0.850330i \(0.323598\pi\)
\(954\) 0 0
\(955\) −25909.1 −0.877906
\(956\) 0 0
\(957\) 11873.3 0.401053
\(958\) 0 0
\(959\) 540.962 0.0182154
\(960\) 0 0
\(961\) −20101.9 −0.674765
\(962\) 0 0
\(963\) −4123.26 −0.137975
\(964\) 0 0
\(965\) −26420.2 −0.881344
\(966\) 0 0
\(967\) −10968.0 −0.364743 −0.182371 0.983230i \(-0.558377\pi\)
−0.182371 + 0.983230i \(0.558377\pi\)
\(968\) 0 0
\(969\) 8682.65 0.287850
\(970\) 0 0
\(971\) −30081.2 −0.994183 −0.497091 0.867698i \(-0.665599\pi\)
−0.497091 + 0.867698i \(0.665599\pi\)
\(972\) 0 0
\(973\) −62642.5 −2.06395
\(974\) 0 0
\(975\) 68569.3 2.25228
\(976\) 0 0
\(977\) 26628.1 0.871962 0.435981 0.899956i \(-0.356401\pi\)
0.435981 + 0.899956i \(0.356401\pi\)
\(978\) 0 0
\(979\) 3059.90 0.0998926
\(980\) 0 0
\(981\) −12808.6 −0.416867
\(982\) 0 0
\(983\) −25495.3 −0.827236 −0.413618 0.910450i \(-0.635735\pi\)
−0.413618 + 0.910450i \(0.635735\pi\)
\(984\) 0 0
\(985\) 96498.0 3.12150
\(986\) 0 0
\(987\) 49205.2 1.58685
\(988\) 0 0
\(989\) −28089.1 −0.903116
\(990\) 0 0
\(991\) 54385.4 1.74330 0.871649 0.490130i \(-0.163051\pi\)
0.871649 + 0.490130i \(0.163051\pi\)
\(992\) 0 0
\(993\) −3185.43 −0.101799
\(994\) 0 0
\(995\) −45946.8 −1.46393
\(996\) 0 0
\(997\) 36846.5 1.17045 0.585227 0.810870i \(-0.301006\pi\)
0.585227 + 0.810870i \(0.301006\pi\)
\(998\) 0 0
\(999\) −29456.1 −0.932884
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 1216.4.a.l.1.1 2
4.3 odd 2 1216.4.a.j.1.2 2
8.3 odd 2 38.4.a.b.1.1 2
8.5 even 2 304.4.a.d.1.2 2
24.11 even 2 342.4.a.k.1.1 2
40.3 even 4 950.4.b.g.799.3 4
40.19 odd 2 950.4.a.h.1.2 2
40.27 even 4 950.4.b.g.799.2 4
56.27 even 2 1862.4.a.b.1.2 2
152.75 even 2 722.4.a.i.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
38.4.a.b.1.1 2 8.3 odd 2
304.4.a.d.1.2 2 8.5 even 2
342.4.a.k.1.1 2 24.11 even 2
722.4.a.i.1.2 2 152.75 even 2
950.4.a.h.1.2 2 40.19 odd 2
950.4.b.g.799.2 4 40.27 even 4
950.4.b.g.799.3 4 40.3 even 4
1216.4.a.j.1.2 2 4.3 odd 2
1216.4.a.l.1.1 2 1.1 even 1 trivial
1862.4.a.b.1.2 2 56.27 even 2