Properties

Label 6016.2.a.n.1.8
Level $6016$
Weight $2$
Character 6016.1
Self dual yes
Analytic conductor $48.038$
Analytic rank $0$
Dimension $13$
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] = [6016,2,Mod(1,6016)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6016, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6016.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6016 = 2^{7} \cdot 47 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6016.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: \(48.0380018560\)
Analytic rank: \(0\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{13} - 4 x^{12} - 22 x^{11} + 96 x^{10} + 163 x^{9} - 840 x^{8} - 438 x^{7} + 3358 x^{6} + 5 x^{5} - 6230 x^{4} + 1488 x^{3} + 4720 x^{2} - 1440 x - 832 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(-0.319482\) of defining polynomial
Character \(\chi\) \(=\) 6016.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.319482 q^{3} -2.22340 q^{5} +0.484368 q^{7} -2.89793 q^{9} +O(q^{10})\) \(q+0.319482 q^{3} -2.22340 q^{5} +0.484368 q^{7} -2.89793 q^{9} +3.07431 q^{11} +5.65430 q^{13} -0.710336 q^{15} +1.35477 q^{17} +2.21910 q^{19} +0.154747 q^{21} +1.53410 q^{23} -0.0565035 q^{25} -1.88428 q^{27} +3.86238 q^{29} +5.33192 q^{31} +0.982186 q^{33} -1.07694 q^{35} -6.61936 q^{37} +1.80645 q^{39} -0.811395 q^{41} -0.404417 q^{43} +6.44325 q^{45} -1.00000 q^{47} -6.76539 q^{49} +0.432825 q^{51} -9.23122 q^{53} -6.83541 q^{55} +0.708963 q^{57} -1.47489 q^{59} -2.46683 q^{61} -1.40366 q^{63} -12.5718 q^{65} -13.7368 q^{67} +0.490119 q^{69} +11.9263 q^{71} -3.68046 q^{73} -0.0180519 q^{75} +1.48910 q^{77} +15.1950 q^{79} +8.09180 q^{81} -5.17022 q^{83} -3.01219 q^{85} +1.23396 q^{87} +13.6589 q^{89} +2.73876 q^{91} +1.70345 q^{93} -4.93394 q^{95} +5.17590 q^{97} -8.90913 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13 q - 4 q^{3} + 6 q^{5} - 2 q^{7} + 21 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 13 q - 4 q^{3} + 6 q^{5} - 2 q^{7} + 21 q^{9} - 10 q^{11} + 4 q^{13} + 14 q^{15} + 10 q^{17} - 8 q^{19} + 10 q^{21} + 18 q^{23} + 23 q^{25} - 16 q^{27} + 14 q^{29} + 4 q^{31} + 14 q^{33} - 14 q^{35} + 16 q^{37} + 12 q^{39} + 10 q^{41} - 12 q^{43} + 10 q^{45} - 13 q^{47} + 9 q^{49} - 22 q^{51} + 26 q^{53} - 2 q^{55} + 20 q^{57} - 30 q^{59} + 18 q^{61} + 12 q^{63} - 4 q^{65} - 4 q^{67} + 2 q^{69} + 36 q^{71} + 10 q^{73} - 38 q^{75} + 42 q^{77} + 21 q^{81} - 12 q^{83} + 4 q^{85} + 6 q^{87} + 50 q^{89} + 4 q^{91} + 52 q^{93} + 8 q^{95} - 10 q^{97} - 34 q^{99}+O(q^{100}) \) 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
\(3\) 0.319482 0.184453 0.0922266 0.995738i \(-0.470602\pi\)
0.0922266 + 0.995738i \(0.470602\pi\)
\(4\) 0 0
\(5\) −2.22340 −0.994334 −0.497167 0.867655i \(-0.665626\pi\)
−0.497167 + 0.867655i \(0.665626\pi\)
\(6\) 0 0
\(7\) 0.484368 0.183074 0.0915369 0.995802i \(-0.470822\pi\)
0.0915369 + 0.995802i \(0.470822\pi\)
\(8\) 0 0
\(9\) −2.89793 −0.965977
\(10\) 0 0
\(11\) 3.07431 0.926938 0.463469 0.886113i \(-0.346604\pi\)
0.463469 + 0.886113i \(0.346604\pi\)
\(12\) 0 0
\(13\) 5.65430 1.56822 0.784111 0.620621i \(-0.213119\pi\)
0.784111 + 0.620621i \(0.213119\pi\)
\(14\) 0 0
\(15\) −0.710336 −0.183408
\(16\) 0 0
\(17\) 1.35477 0.328580 0.164290 0.986412i \(-0.447467\pi\)
0.164290 + 0.986412i \(0.447467\pi\)
\(18\) 0 0
\(19\) 2.21910 0.509097 0.254548 0.967060i \(-0.418073\pi\)
0.254548 + 0.967060i \(0.418073\pi\)
\(20\) 0 0
\(21\) 0.154747 0.0337686
\(22\) 0 0
\(23\) 1.53410 0.319883 0.159941 0.987127i \(-0.448869\pi\)
0.159941 + 0.987127i \(0.448869\pi\)
\(24\) 0 0
\(25\) −0.0565035 −0.0113007
\(26\) 0 0
\(27\) −1.88428 −0.362631
\(28\) 0 0
\(29\) 3.86238 0.717226 0.358613 0.933486i \(-0.383250\pi\)
0.358613 + 0.933486i \(0.383250\pi\)
\(30\) 0 0
\(31\) 5.33192 0.957641 0.478820 0.877913i \(-0.341065\pi\)
0.478820 + 0.877913i \(0.341065\pi\)
\(32\) 0 0
\(33\) 0.982186 0.170977
\(34\) 0 0
\(35\) −1.07694 −0.182036
\(36\) 0 0
\(37\) −6.61936 −1.08822 −0.544108 0.839015i \(-0.683132\pi\)
−0.544108 + 0.839015i \(0.683132\pi\)
\(38\) 0 0
\(39\) 1.80645 0.289264
\(40\) 0 0
\(41\) −0.811395 −0.126719 −0.0633593 0.997991i \(-0.520181\pi\)
−0.0633593 + 0.997991i \(0.520181\pi\)
\(42\) 0 0
\(43\) −0.404417 −0.0616731 −0.0308365 0.999524i \(-0.509817\pi\)
−0.0308365 + 0.999524i \(0.509817\pi\)
\(44\) 0 0
\(45\) 6.44325 0.960503
\(46\) 0 0
\(47\) −1.00000 −0.145865
\(48\) 0 0
\(49\) −6.76539 −0.966484
\(50\) 0 0
\(51\) 0.432825 0.0606077
\(52\) 0 0
\(53\) −9.23122 −1.26801 −0.634003 0.773331i \(-0.718589\pi\)
−0.634003 + 0.773331i \(0.718589\pi\)
\(54\) 0 0
\(55\) −6.83541 −0.921686
\(56\) 0 0
\(57\) 0.708963 0.0939045
\(58\) 0 0
\(59\) −1.47489 −0.192014 −0.0960069 0.995381i \(-0.530607\pi\)
−0.0960069 + 0.995381i \(0.530607\pi\)
\(60\) 0 0
\(61\) −2.46683 −0.315845 −0.157923 0.987451i \(-0.550480\pi\)
−0.157923 + 0.987451i \(0.550480\pi\)
\(62\) 0 0
\(63\) −1.40366 −0.176845
\(64\) 0 0
\(65\) −12.5718 −1.55934
\(66\) 0 0
\(67\) −13.7368 −1.67822 −0.839108 0.543965i \(-0.816923\pi\)
−0.839108 + 0.543965i \(0.816923\pi\)
\(68\) 0 0
\(69\) 0.490119 0.0590034
\(70\) 0 0
\(71\) 11.9263 1.41539 0.707694 0.706519i \(-0.249736\pi\)
0.707694 + 0.706519i \(0.249736\pi\)
\(72\) 0 0
\(73\) −3.68046 −0.430766 −0.215383 0.976530i \(-0.569100\pi\)
−0.215383 + 0.976530i \(0.569100\pi\)
\(74\) 0 0
\(75\) −0.0180519 −0.00208445
\(76\) 0 0
\(77\) 1.48910 0.169698
\(78\) 0 0
\(79\) 15.1950 1.70957 0.854787 0.518980i \(-0.173688\pi\)
0.854787 + 0.518980i \(0.173688\pi\)
\(80\) 0 0
\(81\) 8.09180 0.899089
\(82\) 0 0
\(83\) −5.17022 −0.567505 −0.283753 0.958898i \(-0.591579\pi\)
−0.283753 + 0.958898i \(0.591579\pi\)
\(84\) 0 0
\(85\) −3.01219 −0.326718
\(86\) 0 0
\(87\) 1.23396 0.132295
\(88\) 0 0
\(89\) 13.6589 1.44784 0.723922 0.689882i \(-0.242337\pi\)
0.723922 + 0.689882i \(0.242337\pi\)
\(90\) 0 0
\(91\) 2.73876 0.287100
\(92\) 0 0
\(93\) 1.70345 0.176640
\(94\) 0 0
\(95\) −4.93394 −0.506212
\(96\) 0 0
\(97\) 5.17590 0.525533 0.262766 0.964859i \(-0.415365\pi\)
0.262766 + 0.964859i \(0.415365\pi\)
\(98\) 0 0
\(99\) −8.90913 −0.895401
\(100\) 0 0
\(101\) 9.62759 0.957981 0.478990 0.877820i \(-0.341003\pi\)
0.478990 + 0.877820i \(0.341003\pi\)
\(102\) 0 0
\(103\) 15.9258 1.56922 0.784608 0.619992i \(-0.212864\pi\)
0.784608 + 0.619992i \(0.212864\pi\)
\(104\) 0 0
\(105\) −0.344064 −0.0335772
\(106\) 0 0
\(107\) −14.9979 −1.44990 −0.724949 0.688803i \(-0.758137\pi\)
−0.724949 + 0.688803i \(0.758137\pi\)
\(108\) 0 0
\(109\) −3.54676 −0.339718 −0.169859 0.985468i \(-0.554331\pi\)
−0.169859 + 0.985468i \(0.554331\pi\)
\(110\) 0 0
\(111\) −2.11477 −0.200725
\(112\) 0 0
\(113\) 20.4175 1.92072 0.960358 0.278769i \(-0.0899264\pi\)
0.960358 + 0.278769i \(0.0899264\pi\)
\(114\) 0 0
\(115\) −3.41092 −0.318070
\(116\) 0 0
\(117\) −16.3858 −1.51487
\(118\) 0 0
\(119\) 0.656208 0.0601545
\(120\) 0 0
\(121\) −1.54864 −0.140785
\(122\) 0 0
\(123\) −0.259226 −0.0233736
\(124\) 0 0
\(125\) 11.2426 1.00557
\(126\) 0 0
\(127\) 3.36589 0.298674 0.149337 0.988786i \(-0.452286\pi\)
0.149337 + 0.988786i \(0.452286\pi\)
\(128\) 0 0
\(129\) −0.129204 −0.0113758
\(130\) 0 0
\(131\) 18.2049 1.59057 0.795285 0.606236i \(-0.207321\pi\)
0.795285 + 0.606236i \(0.207321\pi\)
\(132\) 0 0
\(133\) 1.07486 0.0932023
\(134\) 0 0
\(135\) 4.18951 0.360576
\(136\) 0 0
\(137\) 2.24716 0.191988 0.0959939 0.995382i \(-0.469397\pi\)
0.0959939 + 0.995382i \(0.469397\pi\)
\(138\) 0 0
\(139\) −5.07267 −0.430258 −0.215129 0.976586i \(-0.569017\pi\)
−0.215129 + 0.976586i \(0.569017\pi\)
\(140\) 0 0
\(141\) −0.319482 −0.0269053
\(142\) 0 0
\(143\) 17.3831 1.45364
\(144\) 0 0
\(145\) −8.58761 −0.713162
\(146\) 0 0
\(147\) −2.16142 −0.178271
\(148\) 0 0
\(149\) 0.0464165 0.00380259 0.00190129 0.999998i \(-0.499395\pi\)
0.00190129 + 0.999998i \(0.499395\pi\)
\(150\) 0 0
\(151\) −8.31165 −0.676392 −0.338196 0.941076i \(-0.609817\pi\)
−0.338196 + 0.941076i \(0.609817\pi\)
\(152\) 0 0
\(153\) −3.92603 −0.317401
\(154\) 0 0
\(155\) −11.8550 −0.952214
\(156\) 0 0
\(157\) 17.4540 1.39298 0.696489 0.717568i \(-0.254745\pi\)
0.696489 + 0.717568i \(0.254745\pi\)
\(158\) 0 0
\(159\) −2.94921 −0.233888
\(160\) 0 0
\(161\) 0.743071 0.0585622
\(162\) 0 0
\(163\) −23.1395 −1.81243 −0.906213 0.422821i \(-0.861040\pi\)
−0.906213 + 0.422821i \(0.861040\pi\)
\(164\) 0 0
\(165\) −2.18379 −0.170008
\(166\) 0 0
\(167\) −15.2533 −1.18034 −0.590168 0.807281i \(-0.700938\pi\)
−0.590168 + 0.807281i \(0.700938\pi\)
\(168\) 0 0
\(169\) 18.9712 1.45932
\(170\) 0 0
\(171\) −6.43080 −0.491776
\(172\) 0 0
\(173\) 22.7991 1.73339 0.866693 0.498843i \(-0.166241\pi\)
0.866693 + 0.498843i \(0.166241\pi\)
\(174\) 0 0
\(175\) −0.0273685 −0.00206886
\(176\) 0 0
\(177\) −0.471200 −0.0354175
\(178\) 0 0
\(179\) 3.28783 0.245744 0.122872 0.992423i \(-0.460790\pi\)
0.122872 + 0.992423i \(0.460790\pi\)
\(180\) 0 0
\(181\) 20.7882 1.54517 0.772586 0.634910i \(-0.218963\pi\)
0.772586 + 0.634910i \(0.218963\pi\)
\(182\) 0 0
\(183\) −0.788108 −0.0582587
\(184\) 0 0
\(185\) 14.7175 1.08205
\(186\) 0 0
\(187\) 4.16498 0.304574
\(188\) 0 0
\(189\) −0.912687 −0.0663882
\(190\) 0 0
\(191\) 0.671922 0.0486186 0.0243093 0.999704i \(-0.492261\pi\)
0.0243093 + 0.999704i \(0.492261\pi\)
\(192\) 0 0
\(193\) −4.70443 −0.338632 −0.169316 0.985562i \(-0.554156\pi\)
−0.169316 + 0.985562i \(0.554156\pi\)
\(194\) 0 0
\(195\) −4.01646 −0.287624
\(196\) 0 0
\(197\) 16.1862 1.15322 0.576610 0.817020i \(-0.304375\pi\)
0.576610 + 0.817020i \(0.304375\pi\)
\(198\) 0 0
\(199\) 19.0137 1.34785 0.673923 0.738801i \(-0.264608\pi\)
0.673923 + 0.738801i \(0.264608\pi\)
\(200\) 0 0
\(201\) −4.38866 −0.309552
\(202\) 0 0
\(203\) 1.87081 0.131305
\(204\) 0 0
\(205\) 1.80405 0.126001
\(206\) 0 0
\(207\) −4.44573 −0.308999
\(208\) 0 0
\(209\) 6.82220 0.471901
\(210\) 0 0
\(211\) 17.2828 1.18980 0.594898 0.803801i \(-0.297192\pi\)
0.594898 + 0.803801i \(0.297192\pi\)
\(212\) 0 0
\(213\) 3.81023 0.261073
\(214\) 0 0
\(215\) 0.899181 0.0613236
\(216\) 0 0
\(217\) 2.58261 0.175319
\(218\) 0 0
\(219\) −1.17584 −0.0794561
\(220\) 0 0
\(221\) 7.66029 0.515287
\(222\) 0 0
\(223\) 5.52092 0.369708 0.184854 0.982766i \(-0.440819\pi\)
0.184854 + 0.982766i \(0.440819\pi\)
\(224\) 0 0
\(225\) 0.163743 0.0109162
\(226\) 0 0
\(227\) 10.7768 0.715284 0.357642 0.933859i \(-0.383581\pi\)
0.357642 + 0.933859i \(0.383581\pi\)
\(228\) 0 0
\(229\) 1.62920 0.107661 0.0538303 0.998550i \(-0.482857\pi\)
0.0538303 + 0.998550i \(0.482857\pi\)
\(230\) 0 0
\(231\) 0.475740 0.0313014
\(232\) 0 0
\(233\) 20.9418 1.37194 0.685970 0.727630i \(-0.259378\pi\)
0.685970 + 0.727630i \(0.259378\pi\)
\(234\) 0 0
\(235\) 2.22340 0.145038
\(236\) 0 0
\(237\) 4.85454 0.315336
\(238\) 0 0
\(239\) 5.74445 0.371578 0.185789 0.982590i \(-0.440516\pi\)
0.185789 + 0.982590i \(0.440516\pi\)
\(240\) 0 0
\(241\) 6.31475 0.406769 0.203384 0.979099i \(-0.434806\pi\)
0.203384 + 0.979099i \(0.434806\pi\)
\(242\) 0 0
\(243\) 8.23804 0.528470
\(244\) 0 0
\(245\) 15.0421 0.961007
\(246\) 0 0
\(247\) 12.5475 0.798376
\(248\) 0 0
\(249\) −1.65179 −0.104678
\(250\) 0 0
\(251\) −1.66570 −0.105138 −0.0525690 0.998617i \(-0.516741\pi\)
−0.0525690 + 0.998617i \(0.516741\pi\)
\(252\) 0 0
\(253\) 4.71631 0.296512
\(254\) 0 0
\(255\) −0.962343 −0.0602642
\(256\) 0 0
\(257\) 6.19750 0.386589 0.193295 0.981141i \(-0.438083\pi\)
0.193295 + 0.981141i \(0.438083\pi\)
\(258\) 0 0
\(259\) −3.20620 −0.199224
\(260\) 0 0
\(261\) −11.1929 −0.692824
\(262\) 0 0
\(263\) −17.3028 −1.06694 −0.533469 0.845819i \(-0.679112\pi\)
−0.533469 + 0.845819i \(0.679112\pi\)
\(264\) 0 0
\(265\) 20.5247 1.26082
\(266\) 0 0
\(267\) 4.36379 0.267060
\(268\) 0 0
\(269\) −6.30797 −0.384604 −0.192302 0.981336i \(-0.561595\pi\)
−0.192302 + 0.981336i \(0.561595\pi\)
\(270\) 0 0
\(271\) −10.6993 −0.649938 −0.324969 0.945725i \(-0.605354\pi\)
−0.324969 + 0.945725i \(0.605354\pi\)
\(272\) 0 0
\(273\) 0.874986 0.0529566
\(274\) 0 0
\(275\) −0.173709 −0.0104751
\(276\) 0 0
\(277\) 3.81187 0.229033 0.114517 0.993421i \(-0.463468\pi\)
0.114517 + 0.993421i \(0.463468\pi\)
\(278\) 0 0
\(279\) −15.4515 −0.925059
\(280\) 0 0
\(281\) 15.0178 0.895889 0.447944 0.894061i \(-0.352156\pi\)
0.447944 + 0.894061i \(0.352156\pi\)
\(282\) 0 0
\(283\) −26.5823 −1.58015 −0.790076 0.613009i \(-0.789959\pi\)
−0.790076 + 0.613009i \(0.789959\pi\)
\(284\) 0 0
\(285\) −1.57631 −0.0933724
\(286\) 0 0
\(287\) −0.393014 −0.0231989
\(288\) 0 0
\(289\) −15.1646 −0.892035
\(290\) 0 0
\(291\) 1.65361 0.0969362
\(292\) 0 0
\(293\) 1.92391 0.112396 0.0561979 0.998420i \(-0.482102\pi\)
0.0561979 + 0.998420i \(0.482102\pi\)
\(294\) 0 0
\(295\) 3.27926 0.190926
\(296\) 0 0
\(297\) −5.79287 −0.336136
\(298\) 0 0
\(299\) 8.67429 0.501647
\(300\) 0 0
\(301\) −0.195887 −0.0112907
\(302\) 0 0
\(303\) 3.07584 0.176703
\(304\) 0 0
\(305\) 5.48474 0.314056
\(306\) 0 0
\(307\) 5.02012 0.286513 0.143257 0.989686i \(-0.454243\pi\)
0.143257 + 0.989686i \(0.454243\pi\)
\(308\) 0 0
\(309\) 5.08801 0.289447
\(310\) 0 0
\(311\) −20.4391 −1.15900 −0.579498 0.814974i \(-0.696751\pi\)
−0.579498 + 0.814974i \(0.696751\pi\)
\(312\) 0 0
\(313\) 9.35860 0.528979 0.264490 0.964389i \(-0.414797\pi\)
0.264490 + 0.964389i \(0.414797\pi\)
\(314\) 0 0
\(315\) 3.12090 0.175843
\(316\) 0 0
\(317\) −3.50828 −0.197045 −0.0985224 0.995135i \(-0.531412\pi\)
−0.0985224 + 0.995135i \(0.531412\pi\)
\(318\) 0 0
\(319\) 11.8741 0.664824
\(320\) 0 0
\(321\) −4.79155 −0.267438
\(322\) 0 0
\(323\) 3.00637 0.167279
\(324\) 0 0
\(325\) −0.319488 −0.0177220
\(326\) 0 0
\(327\) −1.13313 −0.0626620
\(328\) 0 0
\(329\) −0.484368 −0.0267041
\(330\) 0 0
\(331\) 6.33727 0.348328 0.174164 0.984717i \(-0.444278\pi\)
0.174164 + 0.984717i \(0.444278\pi\)
\(332\) 0 0
\(333\) 19.1824 1.05119
\(334\) 0 0
\(335\) 30.5423 1.66871
\(336\) 0 0
\(337\) −20.3821 −1.11028 −0.555142 0.831756i \(-0.687336\pi\)
−0.555142 + 0.831756i \(0.687336\pi\)
\(338\) 0 0
\(339\) 6.52303 0.354282
\(340\) 0 0
\(341\) 16.3919 0.887674
\(342\) 0 0
\(343\) −6.66751 −0.360012
\(344\) 0 0
\(345\) −1.08973 −0.0586691
\(346\) 0 0
\(347\) 17.1246 0.919299 0.459649 0.888100i \(-0.347975\pi\)
0.459649 + 0.888100i \(0.347975\pi\)
\(348\) 0 0
\(349\) 5.94264 0.318102 0.159051 0.987270i \(-0.449157\pi\)
0.159051 + 0.987270i \(0.449157\pi\)
\(350\) 0 0
\(351\) −10.6543 −0.568685
\(352\) 0 0
\(353\) 21.4991 1.14428 0.572141 0.820155i \(-0.306113\pi\)
0.572141 + 0.820155i \(0.306113\pi\)
\(354\) 0 0
\(355\) −26.5168 −1.40737
\(356\) 0 0
\(357\) 0.209647 0.0110957
\(358\) 0 0
\(359\) 33.4888 1.76747 0.883735 0.467987i \(-0.155021\pi\)
0.883735 + 0.467987i \(0.155021\pi\)
\(360\) 0 0
\(361\) −14.0756 −0.740821
\(362\) 0 0
\(363\) −0.494763 −0.0259683
\(364\) 0 0
\(365\) 8.18314 0.428325
\(366\) 0 0
\(367\) −15.5899 −0.813786 −0.406893 0.913476i \(-0.633388\pi\)
−0.406893 + 0.913476i \(0.633388\pi\)
\(368\) 0 0
\(369\) 2.35137 0.122407
\(370\) 0 0
\(371\) −4.47131 −0.232139
\(372\) 0 0
\(373\) −4.96106 −0.256874 −0.128437 0.991718i \(-0.540996\pi\)
−0.128437 + 0.991718i \(0.540996\pi\)
\(374\) 0 0
\(375\) 3.59182 0.185481
\(376\) 0 0
\(377\) 21.8391 1.12477
\(378\) 0 0
\(379\) −6.00049 −0.308224 −0.154112 0.988053i \(-0.549252\pi\)
−0.154112 + 0.988053i \(0.549252\pi\)
\(380\) 0 0
\(381\) 1.07534 0.0550914
\(382\) 0 0
\(383\) −0.332863 −0.0170085 −0.00850426 0.999964i \(-0.502707\pi\)
−0.00850426 + 0.999964i \(0.502707\pi\)
\(384\) 0 0
\(385\) −3.31085 −0.168737
\(386\) 0 0
\(387\) 1.17197 0.0595748
\(388\) 0 0
\(389\) −13.0435 −0.661333 −0.330666 0.943748i \(-0.607273\pi\)
−0.330666 + 0.943748i \(0.607273\pi\)
\(390\) 0 0
\(391\) 2.07836 0.105107
\(392\) 0 0
\(393\) 5.81614 0.293386
\(394\) 0 0
\(395\) −33.7846 −1.69989
\(396\) 0 0
\(397\) −16.7275 −0.839528 −0.419764 0.907633i \(-0.637887\pi\)
−0.419764 + 0.907633i \(0.637887\pi\)
\(398\) 0 0
\(399\) 0.343399 0.0171915
\(400\) 0 0
\(401\) −9.79357 −0.489068 −0.244534 0.969641i \(-0.578635\pi\)
−0.244534 + 0.969641i \(0.578635\pi\)
\(402\) 0 0
\(403\) 30.1483 1.50179
\(404\) 0 0
\(405\) −17.9913 −0.893994
\(406\) 0 0
\(407\) −20.3499 −1.00871
\(408\) 0 0
\(409\) −10.1041 −0.499614 −0.249807 0.968296i \(-0.580367\pi\)
−0.249807 + 0.968296i \(0.580367\pi\)
\(410\) 0 0
\(411\) 0.717927 0.0354127
\(412\) 0 0
\(413\) −0.714387 −0.0351527
\(414\) 0 0
\(415\) 11.4954 0.564289
\(416\) 0 0
\(417\) −1.62063 −0.0793625
\(418\) 0 0
\(419\) −11.5777 −0.565607 −0.282804 0.959178i \(-0.591264\pi\)
−0.282804 + 0.959178i \(0.591264\pi\)
\(420\) 0 0
\(421\) −35.4182 −1.72618 −0.863088 0.505053i \(-0.831473\pi\)
−0.863088 + 0.505053i \(0.831473\pi\)
\(422\) 0 0
\(423\) 2.89793 0.140902
\(424\) 0 0
\(425\) −0.0765494 −0.00371319
\(426\) 0 0
\(427\) −1.19485 −0.0578230
\(428\) 0 0
\(429\) 5.55358 0.268129
\(430\) 0 0
\(431\) −3.17684 −0.153023 −0.0765115 0.997069i \(-0.524378\pi\)
−0.0765115 + 0.997069i \(0.524378\pi\)
\(432\) 0 0
\(433\) 12.8991 0.619891 0.309945 0.950754i \(-0.399689\pi\)
0.309945 + 0.950754i \(0.399689\pi\)
\(434\) 0 0
\(435\) −2.74359 −0.131545
\(436\) 0 0
\(437\) 3.40433 0.162851
\(438\) 0 0
\(439\) 21.3953 1.02114 0.510571 0.859836i \(-0.329434\pi\)
0.510571 + 0.859836i \(0.329434\pi\)
\(440\) 0 0
\(441\) 19.6056 0.933601
\(442\) 0 0
\(443\) 22.6642 1.07681 0.538404 0.842687i \(-0.319028\pi\)
0.538404 + 0.842687i \(0.319028\pi\)
\(444\) 0 0
\(445\) −30.3692 −1.43964
\(446\) 0 0
\(447\) 0.0148292 0.000701399 0
\(448\) 0 0
\(449\) −11.1457 −0.525997 −0.262999 0.964796i \(-0.584711\pi\)
−0.262999 + 0.964796i \(0.584711\pi\)
\(450\) 0 0
\(451\) −2.49448 −0.117460
\(452\) 0 0
\(453\) −2.65542 −0.124763
\(454\) 0 0
\(455\) −6.08936 −0.285474
\(456\) 0 0
\(457\) 9.75212 0.456185 0.228092 0.973639i \(-0.426751\pi\)
0.228092 + 0.973639i \(0.426751\pi\)
\(458\) 0 0
\(459\) −2.55277 −0.119153
\(460\) 0 0
\(461\) 24.1185 1.12331 0.561656 0.827371i \(-0.310164\pi\)
0.561656 + 0.827371i \(0.310164\pi\)
\(462\) 0 0
\(463\) 14.8910 0.692042 0.346021 0.938227i \(-0.387533\pi\)
0.346021 + 0.938227i \(0.387533\pi\)
\(464\) 0 0
\(465\) −3.78745 −0.175639
\(466\) 0 0
\(467\) −31.2381 −1.44552 −0.722762 0.691097i \(-0.757128\pi\)
−0.722762 + 0.691097i \(0.757128\pi\)
\(468\) 0 0
\(469\) −6.65366 −0.307238
\(470\) 0 0
\(471\) 5.57623 0.256939
\(472\) 0 0
\(473\) −1.24330 −0.0571671
\(474\) 0 0
\(475\) −0.125387 −0.00575315
\(476\) 0 0
\(477\) 26.7514 1.22486
\(478\) 0 0
\(479\) −30.5583 −1.39624 −0.698122 0.715979i \(-0.745980\pi\)
−0.698122 + 0.715979i \(0.745980\pi\)
\(480\) 0 0
\(481\) −37.4279 −1.70656
\(482\) 0 0
\(483\) 0.237398 0.0108020
\(484\) 0 0
\(485\) −11.5081 −0.522555
\(486\) 0 0
\(487\) 32.0918 1.45422 0.727108 0.686523i \(-0.240864\pi\)
0.727108 + 0.686523i \(0.240864\pi\)
\(488\) 0 0
\(489\) −7.39266 −0.334308
\(490\) 0 0
\(491\) −24.3060 −1.09692 −0.548458 0.836178i \(-0.684785\pi\)
−0.548458 + 0.836178i \(0.684785\pi\)
\(492\) 0 0
\(493\) 5.23264 0.235666
\(494\) 0 0
\(495\) 19.8085 0.890327
\(496\) 0 0
\(497\) 5.77670 0.259121
\(498\) 0 0
\(499\) −33.4247 −1.49630 −0.748148 0.663532i \(-0.769057\pi\)
−0.748148 + 0.663532i \(0.769057\pi\)
\(500\) 0 0
\(501\) −4.87316 −0.217717
\(502\) 0 0
\(503\) −2.00080 −0.0892114 −0.0446057 0.999005i \(-0.514203\pi\)
−0.0446057 + 0.999005i \(0.514203\pi\)
\(504\) 0 0
\(505\) −21.4060 −0.952553
\(506\) 0 0
\(507\) 6.06095 0.269176
\(508\) 0 0
\(509\) 18.4667 0.818524 0.409262 0.912417i \(-0.365786\pi\)
0.409262 + 0.912417i \(0.365786\pi\)
\(510\) 0 0
\(511\) −1.78270 −0.0788620
\(512\) 0 0
\(513\) −4.18142 −0.184614
\(514\) 0 0
\(515\) −35.4094 −1.56032
\(516\) 0 0
\(517\) −3.07431 −0.135208
\(518\) 0 0
\(519\) 7.28392 0.319728
\(520\) 0 0
\(521\) 35.7052 1.56427 0.782136 0.623108i \(-0.214130\pi\)
0.782136 + 0.623108i \(0.214130\pi\)
\(522\) 0 0
\(523\) −31.2754 −1.36758 −0.683788 0.729681i \(-0.739669\pi\)
−0.683788 + 0.729681i \(0.739669\pi\)
\(524\) 0 0
\(525\) −0.00874375 −0.000381609 0
\(526\) 0 0
\(527\) 7.22353 0.314662
\(528\) 0 0
\(529\) −20.6465 −0.897675
\(530\) 0 0
\(531\) 4.27412 0.185481
\(532\) 0 0
\(533\) −4.58787 −0.198723
\(534\) 0 0
\(535\) 33.3462 1.44168
\(536\) 0 0
\(537\) 1.05040 0.0453282
\(538\) 0 0
\(539\) −20.7989 −0.895871
\(540\) 0 0
\(541\) 2.59469 0.111554 0.0557771 0.998443i \(-0.482236\pi\)
0.0557771 + 0.998443i \(0.482236\pi\)
\(542\) 0 0
\(543\) 6.64145 0.285012
\(544\) 0 0
\(545\) 7.88585 0.337793
\(546\) 0 0
\(547\) −2.38576 −0.102008 −0.0510039 0.998698i \(-0.516242\pi\)
−0.0510039 + 0.998698i \(0.516242\pi\)
\(548\) 0 0
\(549\) 7.14870 0.305099
\(550\) 0 0
\(551\) 8.57101 0.365137
\(552\) 0 0
\(553\) 7.35998 0.312978
\(554\) 0 0
\(555\) 4.70197 0.199587
\(556\) 0 0
\(557\) −15.6704 −0.663975 −0.331987 0.943284i \(-0.607719\pi\)
−0.331987 + 0.943284i \(0.607719\pi\)
\(558\) 0 0
\(559\) −2.28670 −0.0967171
\(560\) 0 0
\(561\) 1.33064 0.0561796
\(562\) 0 0
\(563\) −7.36541 −0.310415 −0.155208 0.987882i \(-0.549605\pi\)
−0.155208 + 0.987882i \(0.549605\pi\)
\(564\) 0 0
\(565\) −45.3962 −1.90983
\(566\) 0 0
\(567\) 3.91941 0.164600
\(568\) 0 0
\(569\) −10.3538 −0.434055 −0.217027 0.976166i \(-0.569636\pi\)
−0.217027 + 0.976166i \(0.569636\pi\)
\(570\) 0 0
\(571\) 35.0150 1.46533 0.732667 0.680587i \(-0.238275\pi\)
0.732667 + 0.680587i \(0.238275\pi\)
\(572\) 0 0
\(573\) 0.214667 0.00896785
\(574\) 0 0
\(575\) −0.0866823 −0.00361490
\(576\) 0 0
\(577\) −2.45390 −0.102157 −0.0510787 0.998695i \(-0.516266\pi\)
−0.0510787 + 0.998695i \(0.516266\pi\)
\(578\) 0 0
\(579\) −1.50298 −0.0624618
\(580\) 0 0
\(581\) −2.50429 −0.103895
\(582\) 0 0
\(583\) −28.3796 −1.17536
\(584\) 0 0
\(585\) 36.4321 1.50628
\(586\) 0 0
\(587\) −11.7076 −0.483223 −0.241612 0.970373i \(-0.577676\pi\)
−0.241612 + 0.970373i \(0.577676\pi\)
\(588\) 0 0
\(589\) 11.8321 0.487532
\(590\) 0 0
\(591\) 5.17121 0.212715
\(592\) 0 0
\(593\) 11.3762 0.467164 0.233582 0.972337i \(-0.424955\pi\)
0.233582 + 0.972337i \(0.424955\pi\)
\(594\) 0 0
\(595\) −1.45901 −0.0598136
\(596\) 0 0
\(597\) 6.07455 0.248615
\(598\) 0 0
\(599\) 20.1049 0.821462 0.410731 0.911757i \(-0.365273\pi\)
0.410731 + 0.911757i \(0.365273\pi\)
\(600\) 0 0
\(601\) 19.1573 0.781442 0.390721 0.920509i \(-0.372226\pi\)
0.390721 + 0.920509i \(0.372226\pi\)
\(602\) 0 0
\(603\) 39.8083 1.62112
\(604\) 0 0
\(605\) 3.44324 0.139988
\(606\) 0 0
\(607\) −7.28066 −0.295513 −0.147756 0.989024i \(-0.547205\pi\)
−0.147756 + 0.989024i \(0.547205\pi\)
\(608\) 0 0
\(609\) 0.597692 0.0242197
\(610\) 0 0
\(611\) −5.65430 −0.228749
\(612\) 0 0
\(613\) 46.2456 1.86784 0.933922 0.357478i \(-0.116363\pi\)
0.933922 + 0.357478i \(0.116363\pi\)
\(614\) 0 0
\(615\) 0.576363 0.0232412
\(616\) 0 0
\(617\) 27.3903 1.10269 0.551346 0.834276i \(-0.314114\pi\)
0.551346 + 0.834276i \(0.314114\pi\)
\(618\) 0 0
\(619\) −6.45368 −0.259395 −0.129698 0.991554i \(-0.541401\pi\)
−0.129698 + 0.991554i \(0.541401\pi\)
\(620\) 0 0
\(621\) −2.89069 −0.115999
\(622\) 0 0
\(623\) 6.61595 0.265062
\(624\) 0 0
\(625\) −24.7143 −0.988572
\(626\) 0 0
\(627\) 2.17957 0.0870437
\(628\) 0 0
\(629\) −8.96771 −0.357566
\(630\) 0 0
\(631\) −3.54519 −0.141132 −0.0705658 0.997507i \(-0.522480\pi\)
−0.0705658 + 0.997507i \(0.522480\pi\)
\(632\) 0 0
\(633\) 5.52154 0.219462
\(634\) 0 0
\(635\) −7.48371 −0.296982
\(636\) 0 0
\(637\) −38.2536 −1.51566
\(638\) 0 0
\(639\) −34.5615 −1.36723
\(640\) 0 0
\(641\) −13.8851 −0.548430 −0.274215 0.961668i \(-0.588418\pi\)
−0.274215 + 0.961668i \(0.588418\pi\)
\(642\) 0 0
\(643\) 26.5735 1.04796 0.523979 0.851731i \(-0.324447\pi\)
0.523979 + 0.851731i \(0.324447\pi\)
\(644\) 0 0
\(645\) 0.287272 0.0113113
\(646\) 0 0
\(647\) −25.2966 −0.994513 −0.497257 0.867604i \(-0.665659\pi\)
−0.497257 + 0.867604i \(0.665659\pi\)
\(648\) 0 0
\(649\) −4.53425 −0.177985
\(650\) 0 0
\(651\) 0.825098 0.0323381
\(652\) 0 0
\(653\) 36.8365 1.44152 0.720762 0.693183i \(-0.243792\pi\)
0.720762 + 0.693183i \(0.243792\pi\)
\(654\) 0 0
\(655\) −40.4767 −1.58156
\(656\) 0 0
\(657\) 10.6657 0.416110
\(658\) 0 0
\(659\) 35.2527 1.37325 0.686625 0.727011i \(-0.259091\pi\)
0.686625 + 0.727011i \(0.259091\pi\)
\(660\) 0 0
\(661\) 21.6035 0.840277 0.420139 0.907460i \(-0.361981\pi\)
0.420139 + 0.907460i \(0.361981\pi\)
\(662\) 0 0
\(663\) 2.44733 0.0950463
\(664\) 0 0
\(665\) −2.38984 −0.0926742
\(666\) 0 0
\(667\) 5.92529 0.229428
\(668\) 0 0
\(669\) 1.76384 0.0681938
\(670\) 0 0
\(671\) −7.58379 −0.292769
\(672\) 0 0
\(673\) 50.6048 1.95067 0.975336 0.220725i \(-0.0708422\pi\)
0.975336 + 0.220725i \(0.0708422\pi\)
\(674\) 0 0
\(675\) 0.106469 0.00409798
\(676\) 0 0
\(677\) −19.4981 −0.749372 −0.374686 0.927152i \(-0.622249\pi\)
−0.374686 + 0.927152i \(0.622249\pi\)
\(678\) 0 0
\(679\) 2.50704 0.0962113
\(680\) 0 0
\(681\) 3.44301 0.131936
\(682\) 0 0
\(683\) −15.7769 −0.603687 −0.301843 0.953358i \(-0.597602\pi\)
−0.301843 + 0.953358i \(0.597602\pi\)
\(684\) 0 0
\(685\) −4.99633 −0.190900
\(686\) 0 0
\(687\) 0.520501 0.0198583
\(688\) 0 0
\(689\) −52.1961 −1.98851
\(690\) 0 0
\(691\) −3.13237 −0.119161 −0.0595804 0.998224i \(-0.518976\pi\)
−0.0595804 + 0.998224i \(0.518976\pi\)
\(692\) 0 0
\(693\) −4.31530 −0.163925
\(694\) 0 0
\(695\) 11.2786 0.427820
\(696\) 0 0
\(697\) −1.09925 −0.0416372
\(698\) 0 0
\(699\) 6.69052 0.253059
\(700\) 0 0
\(701\) 21.3411 0.806041 0.403020 0.915191i \(-0.367960\pi\)
0.403020 + 0.915191i \(0.367960\pi\)
\(702\) 0 0
\(703\) −14.6890 −0.554007
\(704\) 0 0
\(705\) 0.710336 0.0267528
\(706\) 0 0
\(707\) 4.66330 0.175381
\(708\) 0 0
\(709\) −1.62825 −0.0611504 −0.0305752 0.999532i \(-0.509734\pi\)
−0.0305752 + 0.999532i \(0.509734\pi\)
\(710\) 0 0
\(711\) −44.0341 −1.65141
\(712\) 0 0
\(713\) 8.17972 0.306333
\(714\) 0 0
\(715\) −38.6495 −1.44541
\(716\) 0 0
\(717\) 1.83525 0.0685387
\(718\) 0 0
\(719\) 27.7237 1.03392 0.516959 0.856010i \(-0.327064\pi\)
0.516959 + 0.856010i \(0.327064\pi\)
\(720\) 0 0
\(721\) 7.71395 0.287283
\(722\) 0 0
\(723\) 2.01745 0.0750298
\(724\) 0 0
\(725\) −0.218238 −0.00810516
\(726\) 0 0
\(727\) −1.03885 −0.0385289 −0.0192645 0.999814i \(-0.506132\pi\)
−0.0192645 + 0.999814i \(0.506132\pi\)
\(728\) 0 0
\(729\) −21.6435 −0.801611
\(730\) 0 0
\(731\) −0.547893 −0.0202646
\(732\) 0 0
\(733\) 14.7895 0.546264 0.273132 0.961977i \(-0.411940\pi\)
0.273132 + 0.961977i \(0.411940\pi\)
\(734\) 0 0
\(735\) 4.80570 0.177261
\(736\) 0 0
\(737\) −42.2311 −1.55560
\(738\) 0 0
\(739\) −51.0700 −1.87864 −0.939320 0.343042i \(-0.888543\pi\)
−0.939320 + 0.343042i \(0.888543\pi\)
\(740\) 0 0
\(741\) 4.00869 0.147263
\(742\) 0 0
\(743\) 50.3206 1.84609 0.923043 0.384698i \(-0.125694\pi\)
0.923043 + 0.384698i \(0.125694\pi\)
\(744\) 0 0
\(745\) −0.103202 −0.00378104
\(746\) 0 0
\(747\) 14.9829 0.548197
\(748\) 0 0
\(749\) −7.26448 −0.265438
\(750\) 0 0
\(751\) 20.6518 0.753593 0.376797 0.926296i \(-0.377026\pi\)
0.376797 + 0.926296i \(0.377026\pi\)
\(752\) 0 0
\(753\) −0.532161 −0.0193930
\(754\) 0 0
\(755\) 18.4801 0.672559
\(756\) 0 0
\(757\) −15.2835 −0.555490 −0.277745 0.960655i \(-0.589587\pi\)
−0.277745 + 0.960655i \(0.589587\pi\)
\(758\) 0 0
\(759\) 1.50678 0.0546925
\(760\) 0 0
\(761\) 17.4154 0.631307 0.315653 0.948875i \(-0.397776\pi\)
0.315653 + 0.948875i \(0.397776\pi\)
\(762\) 0 0
\(763\) −1.71793 −0.0621934
\(764\) 0 0
\(765\) 8.72913 0.315602
\(766\) 0 0
\(767\) −8.33945 −0.301120
\(768\) 0 0
\(769\) 20.3812 0.734964 0.367482 0.930031i \(-0.380220\pi\)
0.367482 + 0.930031i \(0.380220\pi\)
\(770\) 0 0
\(771\) 1.97999 0.0713076
\(772\) 0 0
\(773\) −27.7751 −0.999001 −0.499501 0.866314i \(-0.666483\pi\)
−0.499501 + 0.866314i \(0.666483\pi\)
\(774\) 0 0
\(775\) −0.301272 −0.0108220
\(776\) 0 0
\(777\) −1.02433 −0.0367475
\(778\) 0 0
\(779\) −1.80057 −0.0645120
\(780\) 0 0
\(781\) 36.6650 1.31198
\(782\) 0 0
\(783\) −7.27782 −0.260088
\(784\) 0 0
\(785\) −38.8071 −1.38508
\(786\) 0 0
\(787\) 21.2889 0.758869 0.379434 0.925219i \(-0.376119\pi\)
0.379434 + 0.925219i \(0.376119\pi\)
\(788\) 0 0
\(789\) −5.52795 −0.196800
\(790\) 0 0
\(791\) 9.88958 0.351633
\(792\) 0 0
\(793\) −13.9482 −0.495315
\(794\) 0 0
\(795\) 6.55727 0.232562
\(796\) 0 0
\(797\) −4.85844 −0.172095 −0.0860474 0.996291i \(-0.527424\pi\)
−0.0860474 + 0.996291i \(0.527424\pi\)
\(798\) 0 0
\(799\) −1.35477 −0.0479284
\(800\) 0 0
\(801\) −39.5827 −1.39858
\(802\) 0 0
\(803\) −11.3149 −0.399293
\(804\) 0 0
\(805\) −1.65214 −0.0582303
\(806\) 0 0
\(807\) −2.01528 −0.0709414
\(808\) 0 0
\(809\) 29.1902 1.02627 0.513137 0.858307i \(-0.328483\pi\)
0.513137 + 0.858307i \(0.328483\pi\)
\(810\) 0 0
\(811\) 27.0426 0.949593 0.474796 0.880096i \(-0.342522\pi\)
0.474796 + 0.880096i \(0.342522\pi\)
\(812\) 0 0
\(813\) −3.41825 −0.119883
\(814\) 0 0
\(815\) 51.4483 1.80216
\(816\) 0 0
\(817\) −0.897443 −0.0313976
\(818\) 0 0
\(819\) −7.93675 −0.277332
\(820\) 0 0
\(821\) −44.3932 −1.54933 −0.774667 0.632369i \(-0.782083\pi\)
−0.774667 + 0.632369i \(0.782083\pi\)
\(822\) 0 0
\(823\) −41.1924 −1.43588 −0.717939 0.696106i \(-0.754914\pi\)
−0.717939 + 0.696106i \(0.754914\pi\)
\(824\) 0 0
\(825\) −0.0554970 −0.00193216
\(826\) 0 0
\(827\) 16.2906 0.566478 0.283239 0.959049i \(-0.408591\pi\)
0.283239 + 0.959049i \(0.408591\pi\)
\(828\) 0 0
\(829\) −0.662232 −0.0230003 −0.0115001 0.999934i \(-0.503661\pi\)
−0.0115001 + 0.999934i \(0.503661\pi\)
\(830\) 0 0
\(831\) 1.21783 0.0422459
\(832\) 0 0
\(833\) −9.16555 −0.317568
\(834\) 0 0
\(835\) 33.9141 1.17365
\(836\) 0 0
\(837\) −10.0469 −0.347270
\(838\) 0 0
\(839\) 17.7298 0.612100 0.306050 0.952015i \(-0.400993\pi\)
0.306050 + 0.952015i \(0.400993\pi\)
\(840\) 0 0
\(841\) −14.0820 −0.485587
\(842\) 0 0
\(843\) 4.79793 0.165250
\(844\) 0 0
\(845\) −42.1804 −1.45105
\(846\) 0 0
\(847\) −0.750112 −0.0257741
\(848\) 0 0
\(849\) −8.49256 −0.291464
\(850\) 0 0
\(851\) −10.1548 −0.348101
\(852\) 0 0
\(853\) 43.0506 1.47403 0.737013 0.675879i \(-0.236236\pi\)
0.737013 + 0.675879i \(0.236236\pi\)
\(854\) 0 0
\(855\) 14.2982 0.488989
\(856\) 0 0
\(857\) 39.2426 1.34050 0.670250 0.742135i \(-0.266187\pi\)
0.670250 + 0.742135i \(0.266187\pi\)
\(858\) 0 0
\(859\) 20.7175 0.706871 0.353436 0.935459i \(-0.385013\pi\)
0.353436 + 0.935459i \(0.385013\pi\)
\(860\) 0 0
\(861\) −0.125561 −0.00427910
\(862\) 0 0
\(863\) 6.56027 0.223314 0.111657 0.993747i \(-0.464384\pi\)
0.111657 + 0.993747i \(0.464384\pi\)
\(864\) 0 0
\(865\) −50.6915 −1.72356
\(866\) 0 0
\(867\) −4.84482 −0.164539
\(868\) 0 0
\(869\) 46.7141 1.58467
\(870\) 0 0
\(871\) −77.6720 −2.63182
\(872\) 0 0
\(873\) −14.9994 −0.507653
\(874\) 0 0
\(875\) 5.44556 0.184094
\(876\) 0 0
\(877\) −31.7489 −1.07208 −0.536042 0.844191i \(-0.680081\pi\)
−0.536042 + 0.844191i \(0.680081\pi\)
\(878\) 0 0
\(879\) 0.614654 0.0207318
\(880\) 0 0
\(881\) −59.0699 −1.99012 −0.995058 0.0992929i \(-0.968342\pi\)
−0.995058 + 0.0992929i \(0.968342\pi\)
\(882\) 0 0
\(883\) −35.7421 −1.20282 −0.601409 0.798941i \(-0.705394\pi\)
−0.601409 + 0.798941i \(0.705394\pi\)
\(884\) 0 0
\(885\) 1.04766 0.0352168
\(886\) 0 0
\(887\) 6.45708 0.216808 0.108404 0.994107i \(-0.465426\pi\)
0.108404 + 0.994107i \(0.465426\pi\)
\(888\) 0 0
\(889\) 1.63033 0.0546795
\(890\) 0 0
\(891\) 24.8767 0.833400
\(892\) 0 0
\(893\) −2.21910 −0.0742594
\(894\) 0 0
\(895\) −7.31015 −0.244351
\(896\) 0 0
\(897\) 2.77128 0.0925304
\(898\) 0 0
\(899\) 20.5939 0.686845
\(900\) 0 0
\(901\) −12.5062 −0.416641
\(902\) 0 0
\(903\) −0.0625824 −0.00208261
\(904\) 0 0
\(905\) −46.2203 −1.53642
\(906\) 0 0
\(907\) −8.85759 −0.294111 −0.147056 0.989128i \(-0.546980\pi\)
−0.147056 + 0.989128i \(0.546980\pi\)
\(908\) 0 0
\(909\) −27.9001 −0.925388
\(910\) 0 0
\(911\) 54.6382 1.81024 0.905122 0.425153i \(-0.139780\pi\)
0.905122 + 0.425153i \(0.139780\pi\)
\(912\) 0 0
\(913\) −15.8948 −0.526042
\(914\) 0 0
\(915\) 1.75228 0.0579285
\(916\) 0 0
\(917\) 8.81787 0.291192
\(918\) 0 0
\(919\) 10.8715 0.358617 0.179308 0.983793i \(-0.442614\pi\)
0.179308 + 0.983793i \(0.442614\pi\)
\(920\) 0 0
\(921\) 1.60384 0.0528483
\(922\) 0 0
\(923\) 67.4348 2.21964
\(924\) 0 0
\(925\) 0.374017 0.0122976
\(926\) 0 0
\(927\) −46.1519 −1.51583
\(928\) 0 0
\(929\) 23.4837 0.770475 0.385238 0.922817i \(-0.374120\pi\)
0.385238 + 0.922817i \(0.374120\pi\)
\(930\) 0 0
\(931\) −15.0131 −0.492034
\(932\) 0 0
\(933\) −6.52993 −0.213780
\(934\) 0 0
\(935\) −9.26041 −0.302848
\(936\) 0 0
\(937\) 6.18294 0.201988 0.100994 0.994887i \(-0.467798\pi\)
0.100994 + 0.994887i \(0.467798\pi\)
\(938\) 0 0
\(939\) 2.98991 0.0975719
\(940\) 0 0
\(941\) −8.11689 −0.264603 −0.132302 0.991210i \(-0.542237\pi\)
−0.132302 + 0.991210i \(0.542237\pi\)
\(942\) 0 0
\(943\) −1.24476 −0.0405351
\(944\) 0 0
\(945\) 2.02927 0.0660120
\(946\) 0 0
\(947\) 16.0127 0.520343 0.260172 0.965562i \(-0.416221\pi\)
0.260172 + 0.965562i \(0.416221\pi\)
\(948\) 0 0
\(949\) −20.8105 −0.675536
\(950\) 0 0
\(951\) −1.12083 −0.0363455
\(952\) 0 0
\(953\) 8.03372 0.260238 0.130119 0.991498i \(-0.458464\pi\)
0.130119 + 0.991498i \(0.458464\pi\)
\(954\) 0 0
\(955\) −1.49395 −0.0483431
\(956\) 0 0
\(957\) 3.79358 0.122629
\(958\) 0 0
\(959\) 1.08845 0.0351479
\(960\) 0 0
\(961\) −2.57065 −0.0829242
\(962\) 0 0
\(963\) 43.4628 1.40057
\(964\) 0 0
\(965\) 10.4598 0.336714
\(966\) 0 0
\(967\) −58.9584 −1.89597 −0.947986 0.318311i \(-0.896885\pi\)
−0.947986 + 0.318311i \(0.896885\pi\)
\(968\) 0 0
\(969\) 0.960483 0.0308552
\(970\) 0 0
\(971\) −53.8290 −1.72745 −0.863727 0.503961i \(-0.831876\pi\)
−0.863727 + 0.503961i \(0.831876\pi\)
\(972\) 0 0
\(973\) −2.45704 −0.0787690
\(974\) 0 0
\(975\) −0.102071 −0.00326888
\(976\) 0 0
\(977\) 31.0563 0.993578 0.496789 0.867871i \(-0.334512\pi\)
0.496789 + 0.867871i \(0.334512\pi\)
\(978\) 0 0
\(979\) 41.9918 1.34206
\(980\) 0 0
\(981\) 10.2783 0.328159
\(982\) 0 0
\(983\) 19.8434 0.632907 0.316454 0.948608i \(-0.397508\pi\)
0.316454 + 0.948608i \(0.397508\pi\)
\(984\) 0 0
\(985\) −35.9884 −1.14668
\(986\) 0 0
\(987\) −0.154747 −0.00492565
\(988\) 0 0
\(989\) −0.620418 −0.0197282
\(990\) 0 0
\(991\) −25.5528 −0.811711 −0.405856 0.913937i \(-0.633026\pi\)
−0.405856 + 0.913937i \(0.633026\pi\)
\(992\) 0 0
\(993\) 2.02465 0.0642502
\(994\) 0 0
\(995\) −42.2751 −1.34021
\(996\) 0 0
\(997\) −45.6882 −1.44696 −0.723481 0.690345i \(-0.757459\pi\)
−0.723481 + 0.690345i \(0.757459\pi\)
\(998\) 0 0
\(999\) 12.4728 0.394620
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 6016.2.a.n.1.8 yes 13
4.3 odd 2 6016.2.a.p.1.6 yes 13
8.3 odd 2 6016.2.a.m.1.8 13
8.5 even 2 6016.2.a.o.1.6 yes 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6016.2.a.m.1.8 13 8.3 odd 2
6016.2.a.n.1.8 yes 13 1.1 even 1 trivial
6016.2.a.o.1.6 yes 13 8.5 even 2
6016.2.a.p.1.6 yes 13 4.3 odd 2