Properties

Label 4016.2.a.j.1.7
Level $4016$
Weight $2$
Character 4016.1
Self dual yes
Analytic conductor $32.068$
Analytic rank $1$
Dimension $14$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4016,2,Mod(1,4016)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4016, 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("4016.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4016 = 2^{4} \cdot 251 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4016.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: \(32.0679214517\)
Analytic rank: \(1\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 3 x^{13} - 27 x^{12} + 79 x^{11} + 274 x^{10} - 747 x^{9} - 1422 x^{8} + 3287 x^{7} + 4161 x^{6} - 6861 x^{5} - 6676 x^{4} + 5599 x^{3} + 4627 x^{2} - 359 x - 196 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1004)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(0.224635\) of defining polynomial
Character \(\chi\) \(=\) 4016.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.224635 q^{3} -2.42875 q^{5} +2.24830 q^{7} -2.94954 q^{9} +O(q^{10})\) \(q-0.224635 q^{3} -2.42875 q^{5} +2.24830 q^{7} -2.94954 q^{9} -1.64215 q^{11} +6.22381 q^{13} +0.545582 q^{15} -1.62535 q^{17} -7.83395 q^{19} -0.505047 q^{21} +6.98629 q^{23} +0.898832 q^{25} +1.33647 q^{27} +8.44758 q^{29} -7.71452 q^{31} +0.368883 q^{33} -5.46056 q^{35} +9.24659 q^{37} -1.39808 q^{39} +8.41207 q^{41} -3.97586 q^{43} +7.16370 q^{45} +0.735589 q^{47} -1.94514 q^{49} +0.365111 q^{51} -10.9903 q^{53} +3.98836 q^{55} +1.75978 q^{57} -8.63639 q^{59} -8.71829 q^{61} -6.63145 q^{63} -15.1161 q^{65} -4.17803 q^{67} -1.56936 q^{69} +4.49162 q^{71} -12.2570 q^{73} -0.201909 q^{75} -3.69204 q^{77} -3.66788 q^{79} +8.54840 q^{81} -1.17546 q^{83} +3.94758 q^{85} -1.89762 q^{87} +17.2264 q^{89} +13.9930 q^{91} +1.73295 q^{93} +19.0267 q^{95} +9.08104 q^{97} +4.84357 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 3 q^{3} - 2 q^{5} - 8 q^{7} + 21 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 3 q^{3} - 2 q^{5} - 8 q^{7} + 21 q^{9} - 9 q^{11} - q^{13} - 14 q^{15} - 27 q^{19} - 3 q^{21} - 13 q^{23} + 26 q^{25} - 15 q^{27} - 25 q^{31} + 16 q^{33} - 21 q^{35} - q^{37} - 33 q^{39} + 10 q^{41} - 35 q^{43} - 4 q^{45} - 6 q^{47} + 36 q^{49} - 48 q^{51} - q^{53} - 41 q^{55} + 14 q^{57} - 30 q^{59} + 3 q^{61} - 31 q^{63} + 7 q^{65} - 22 q^{67} - 17 q^{69} - 6 q^{71} + 5 q^{73} - 4 q^{75} - 14 q^{77} - 56 q^{79} + 26 q^{81} + 28 q^{83} - 23 q^{85} - 11 q^{87} - 24 q^{89} - 38 q^{91} - 55 q^{93} + 4 q^{95} + 6 q^{97} - 12 q^{99}+O(q^{100}) \) 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
\(3\) −0.224635 −0.129693 −0.0648465 0.997895i \(-0.520656\pi\)
−0.0648465 + 0.997895i \(0.520656\pi\)
\(4\) 0 0
\(5\) −2.42875 −1.08617 −0.543085 0.839678i \(-0.682744\pi\)
−0.543085 + 0.839678i \(0.682744\pi\)
\(6\) 0 0
\(7\) 2.24830 0.849778 0.424889 0.905245i \(-0.360313\pi\)
0.424889 + 0.905245i \(0.360313\pi\)
\(8\) 0 0
\(9\) −2.94954 −0.983180
\(10\) 0 0
\(11\) −1.64215 −0.495125 −0.247563 0.968872i \(-0.579630\pi\)
−0.247563 + 0.968872i \(0.579630\pi\)
\(12\) 0 0
\(13\) 6.22381 1.72617 0.863087 0.505055i \(-0.168528\pi\)
0.863087 + 0.505055i \(0.168528\pi\)
\(14\) 0 0
\(15\) 0.545582 0.140869
\(16\) 0 0
\(17\) −1.62535 −0.394206 −0.197103 0.980383i \(-0.563153\pi\)
−0.197103 + 0.980383i \(0.563153\pi\)
\(18\) 0 0
\(19\) −7.83395 −1.79723 −0.898615 0.438738i \(-0.855426\pi\)
−0.898615 + 0.438738i \(0.855426\pi\)
\(20\) 0 0
\(21\) −0.505047 −0.110210
\(22\) 0 0
\(23\) 6.98629 1.45674 0.728371 0.685183i \(-0.240278\pi\)
0.728371 + 0.685183i \(0.240278\pi\)
\(24\) 0 0
\(25\) 0.898832 0.179766
\(26\) 0 0
\(27\) 1.33647 0.257204
\(28\) 0 0
\(29\) 8.44758 1.56868 0.784338 0.620334i \(-0.213003\pi\)
0.784338 + 0.620334i \(0.213003\pi\)
\(30\) 0 0
\(31\) −7.71452 −1.38557 −0.692785 0.721145i \(-0.743616\pi\)
−0.692785 + 0.721145i \(0.743616\pi\)
\(32\) 0 0
\(33\) 0.368883 0.0642143
\(34\) 0 0
\(35\) −5.46056 −0.923004
\(36\) 0 0
\(37\) 9.24659 1.52013 0.760065 0.649847i \(-0.225167\pi\)
0.760065 + 0.649847i \(0.225167\pi\)
\(38\) 0 0
\(39\) −1.39808 −0.223873
\(40\) 0 0
\(41\) 8.41207 1.31375 0.656873 0.754002i \(-0.271879\pi\)
0.656873 + 0.754002i \(0.271879\pi\)
\(42\) 0 0
\(43\) −3.97586 −0.606313 −0.303157 0.952941i \(-0.598041\pi\)
−0.303157 + 0.952941i \(0.598041\pi\)
\(44\) 0 0
\(45\) 7.16370 1.06790
\(46\) 0 0
\(47\) 0.735589 0.107297 0.0536484 0.998560i \(-0.482915\pi\)
0.0536484 + 0.998560i \(0.482915\pi\)
\(48\) 0 0
\(49\) −1.94514 −0.277877
\(50\) 0 0
\(51\) 0.365111 0.0511258
\(52\) 0 0
\(53\) −10.9903 −1.50963 −0.754817 0.655936i \(-0.772274\pi\)
−0.754817 + 0.655936i \(0.772274\pi\)
\(54\) 0 0
\(55\) 3.98836 0.537791
\(56\) 0 0
\(57\) 1.75978 0.233088
\(58\) 0 0
\(59\) −8.63639 −1.12436 −0.562181 0.827014i \(-0.690038\pi\)
−0.562181 + 0.827014i \(0.690038\pi\)
\(60\) 0 0
\(61\) −8.71829 −1.11626 −0.558131 0.829753i \(-0.688481\pi\)
−0.558131 + 0.829753i \(0.688481\pi\)
\(62\) 0 0
\(63\) −6.63145 −0.835485
\(64\) 0 0
\(65\) −15.1161 −1.87492
\(66\) 0 0
\(67\) −4.17803 −0.510428 −0.255214 0.966885i \(-0.582146\pi\)
−0.255214 + 0.966885i \(0.582146\pi\)
\(68\) 0 0
\(69\) −1.56936 −0.188929
\(70\) 0 0
\(71\) 4.49162 0.533058 0.266529 0.963827i \(-0.414123\pi\)
0.266529 + 0.963827i \(0.414123\pi\)
\(72\) 0 0
\(73\) −12.2570 −1.43458 −0.717288 0.696777i \(-0.754617\pi\)
−0.717288 + 0.696777i \(0.754617\pi\)
\(74\) 0 0
\(75\) −0.201909 −0.0233144
\(76\) 0 0
\(77\) −3.69204 −0.420747
\(78\) 0 0
\(79\) −3.66788 −0.412669 −0.206334 0.978482i \(-0.566153\pi\)
−0.206334 + 0.978482i \(0.566153\pi\)
\(80\) 0 0
\(81\) 8.54840 0.949822
\(82\) 0 0
\(83\) −1.17546 −0.129023 −0.0645116 0.997917i \(-0.520549\pi\)
−0.0645116 + 0.997917i \(0.520549\pi\)
\(84\) 0 0
\(85\) 3.94758 0.428175
\(86\) 0 0
\(87\) −1.89762 −0.203446
\(88\) 0 0
\(89\) 17.2264 1.82599 0.912996 0.407968i \(-0.133762\pi\)
0.912996 + 0.407968i \(0.133762\pi\)
\(90\) 0 0
\(91\) 13.9930 1.46687
\(92\) 0 0
\(93\) 1.73295 0.179699
\(94\) 0 0
\(95\) 19.0267 1.95210
\(96\) 0 0
\(97\) 9.08104 0.922040 0.461020 0.887390i \(-0.347484\pi\)
0.461020 + 0.887390i \(0.347484\pi\)
\(98\) 0 0
\(99\) 4.84357 0.486797
\(100\) 0 0
\(101\) 1.31035 0.130385 0.0651924 0.997873i \(-0.479234\pi\)
0.0651924 + 0.997873i \(0.479234\pi\)
\(102\) 0 0
\(103\) 5.17455 0.509863 0.254932 0.966959i \(-0.417947\pi\)
0.254932 + 0.966959i \(0.417947\pi\)
\(104\) 0 0
\(105\) 1.22663 0.119707
\(106\) 0 0
\(107\) −15.5548 −1.50374 −0.751869 0.659313i \(-0.770847\pi\)
−0.751869 + 0.659313i \(0.770847\pi\)
\(108\) 0 0
\(109\) −7.94967 −0.761441 −0.380720 0.924690i \(-0.624324\pi\)
−0.380720 + 0.924690i \(0.624324\pi\)
\(110\) 0 0
\(111\) −2.07710 −0.197150
\(112\) 0 0
\(113\) −4.80047 −0.451590 −0.225795 0.974175i \(-0.572498\pi\)
−0.225795 + 0.974175i \(0.572498\pi\)
\(114\) 0 0
\(115\) −16.9680 −1.58227
\(116\) 0 0
\(117\) −18.3574 −1.69714
\(118\) 0 0
\(119\) −3.65429 −0.334988
\(120\) 0 0
\(121\) −8.30336 −0.754851
\(122\) 0 0
\(123\) −1.88964 −0.170383
\(124\) 0 0
\(125\) 9.96072 0.890914
\(126\) 0 0
\(127\) −17.1925 −1.52559 −0.762795 0.646641i \(-0.776173\pi\)
−0.762795 + 0.646641i \(0.776173\pi\)
\(128\) 0 0
\(129\) 0.893117 0.0786345
\(130\) 0 0
\(131\) 4.30291 0.375947 0.187974 0.982174i \(-0.439808\pi\)
0.187974 + 0.982174i \(0.439808\pi\)
\(132\) 0 0
\(133\) −17.6131 −1.52725
\(134\) 0 0
\(135\) −3.24596 −0.279368
\(136\) 0 0
\(137\) 0.00645063 0.000551114 0 0.000275557 1.00000i \(-0.499912\pi\)
0.000275557 1.00000i \(0.499912\pi\)
\(138\) 0 0
\(139\) 0.181781 0.0154185 0.00770923 0.999970i \(-0.497546\pi\)
0.00770923 + 0.999970i \(0.497546\pi\)
\(140\) 0 0
\(141\) −0.165239 −0.0139156
\(142\) 0 0
\(143\) −10.2204 −0.854673
\(144\) 0 0
\(145\) −20.5171 −1.70385
\(146\) 0 0
\(147\) 0.436946 0.0360387
\(148\) 0 0
\(149\) −23.6239 −1.93535 −0.967673 0.252209i \(-0.918843\pi\)
−0.967673 + 0.252209i \(0.918843\pi\)
\(150\) 0 0
\(151\) −2.45736 −0.199977 −0.0999886 0.994989i \(-0.531881\pi\)
−0.0999886 + 0.994989i \(0.531881\pi\)
\(152\) 0 0
\(153\) 4.79405 0.387576
\(154\) 0 0
\(155\) 18.7367 1.50496
\(156\) 0 0
\(157\) 12.7494 1.01751 0.508755 0.860912i \(-0.330106\pi\)
0.508755 + 0.860912i \(0.330106\pi\)
\(158\) 0 0
\(159\) 2.46880 0.195789
\(160\) 0 0
\(161\) 15.7073 1.23791
\(162\) 0 0
\(163\) −18.8119 −1.47346 −0.736730 0.676187i \(-0.763631\pi\)
−0.736730 + 0.676187i \(0.763631\pi\)
\(164\) 0 0
\(165\) −0.895925 −0.0697476
\(166\) 0 0
\(167\) −5.91688 −0.457862 −0.228931 0.973443i \(-0.573523\pi\)
−0.228931 + 0.973443i \(0.573523\pi\)
\(168\) 0 0
\(169\) 25.7358 1.97968
\(170\) 0 0
\(171\) 23.1065 1.76700
\(172\) 0 0
\(173\) −13.0221 −0.990053 −0.495027 0.868878i \(-0.664842\pi\)
−0.495027 + 0.868878i \(0.664842\pi\)
\(174\) 0 0
\(175\) 2.02085 0.152762
\(176\) 0 0
\(177\) 1.94003 0.145822
\(178\) 0 0
\(179\) −11.3014 −0.844710 −0.422355 0.906431i \(-0.638796\pi\)
−0.422355 + 0.906431i \(0.638796\pi\)
\(180\) 0 0
\(181\) −10.6066 −0.788379 −0.394190 0.919029i \(-0.628975\pi\)
−0.394190 + 0.919029i \(0.628975\pi\)
\(182\) 0 0
\(183\) 1.95843 0.144771
\(184\) 0 0
\(185\) −22.4577 −1.65112
\(186\) 0 0
\(187\) 2.66907 0.195182
\(188\) 0 0
\(189\) 3.00479 0.218567
\(190\) 0 0
\(191\) 10.1524 0.734599 0.367300 0.930103i \(-0.380282\pi\)
0.367300 + 0.930103i \(0.380282\pi\)
\(192\) 0 0
\(193\) −9.19678 −0.661999 −0.330999 0.943631i \(-0.607386\pi\)
−0.330999 + 0.943631i \(0.607386\pi\)
\(194\) 0 0
\(195\) 3.39560 0.243164
\(196\) 0 0
\(197\) −20.1222 −1.43365 −0.716825 0.697253i \(-0.754405\pi\)
−0.716825 + 0.697253i \(0.754405\pi\)
\(198\) 0 0
\(199\) 12.2239 0.866531 0.433266 0.901266i \(-0.357361\pi\)
0.433266 + 0.901266i \(0.357361\pi\)
\(200\) 0 0
\(201\) 0.938532 0.0661989
\(202\) 0 0
\(203\) 18.9927 1.33303
\(204\) 0 0
\(205\) −20.4308 −1.42695
\(206\) 0 0
\(207\) −20.6063 −1.43224
\(208\) 0 0
\(209\) 12.8645 0.889854
\(210\) 0 0
\(211\) −24.4289 −1.68175 −0.840877 0.541227i \(-0.817960\pi\)
−0.840877 + 0.541227i \(0.817960\pi\)
\(212\) 0 0
\(213\) −1.00897 −0.0691338
\(214\) 0 0
\(215\) 9.65638 0.658560
\(216\) 0 0
\(217\) −17.3446 −1.17743
\(218\) 0 0
\(219\) 2.75335 0.186054
\(220\) 0 0
\(221\) −10.1159 −0.680469
\(222\) 0 0
\(223\) −19.9845 −1.33826 −0.669130 0.743145i \(-0.733333\pi\)
−0.669130 + 0.743145i \(0.733333\pi\)
\(224\) 0 0
\(225\) −2.65114 −0.176743
\(226\) 0 0
\(227\) −3.68676 −0.244699 −0.122349 0.992487i \(-0.539043\pi\)
−0.122349 + 0.992487i \(0.539043\pi\)
\(228\) 0 0
\(229\) 1.96570 0.129897 0.0649485 0.997889i \(-0.479312\pi\)
0.0649485 + 0.997889i \(0.479312\pi\)
\(230\) 0 0
\(231\) 0.829360 0.0545679
\(232\) 0 0
\(233\) −7.53560 −0.493674 −0.246837 0.969057i \(-0.579391\pi\)
−0.246837 + 0.969057i \(0.579391\pi\)
\(234\) 0 0
\(235\) −1.78656 −0.116543
\(236\) 0 0
\(237\) 0.823933 0.0535202
\(238\) 0 0
\(239\) 7.87974 0.509698 0.254849 0.966981i \(-0.417974\pi\)
0.254849 + 0.966981i \(0.417974\pi\)
\(240\) 0 0
\(241\) 7.20915 0.464382 0.232191 0.972670i \(-0.425411\pi\)
0.232191 + 0.972670i \(0.425411\pi\)
\(242\) 0 0
\(243\) −5.92969 −0.380390
\(244\) 0 0
\(245\) 4.72426 0.301822
\(246\) 0 0
\(247\) −48.7570 −3.10233
\(248\) 0 0
\(249\) 0.264049 0.0167334
\(250\) 0 0
\(251\) 1.00000 0.0631194
\(252\) 0 0
\(253\) −11.4725 −0.721270
\(254\) 0 0
\(255\) −0.886764 −0.0555313
\(256\) 0 0
\(257\) 25.9988 1.62176 0.810879 0.585213i \(-0.198989\pi\)
0.810879 + 0.585213i \(0.198989\pi\)
\(258\) 0 0
\(259\) 20.7891 1.29177
\(260\) 0 0
\(261\) −24.9165 −1.54229
\(262\) 0 0
\(263\) 31.1491 1.92074 0.960369 0.278731i \(-0.0899139\pi\)
0.960369 + 0.278731i \(0.0899139\pi\)
\(264\) 0 0
\(265\) 26.6927 1.63972
\(266\) 0 0
\(267\) −3.86964 −0.236818
\(268\) 0 0
\(269\) 3.41438 0.208178 0.104089 0.994568i \(-0.466807\pi\)
0.104089 + 0.994568i \(0.466807\pi\)
\(270\) 0 0
\(271\) −22.8379 −1.38730 −0.693652 0.720310i \(-0.743999\pi\)
−0.693652 + 0.720310i \(0.743999\pi\)
\(272\) 0 0
\(273\) −3.14331 −0.190242
\(274\) 0 0
\(275\) −1.47601 −0.0890069
\(276\) 0 0
\(277\) −4.04269 −0.242902 −0.121451 0.992597i \(-0.538755\pi\)
−0.121451 + 0.992597i \(0.538755\pi\)
\(278\) 0 0
\(279\) 22.7543 1.36226
\(280\) 0 0
\(281\) −2.35718 −0.140618 −0.0703088 0.997525i \(-0.522398\pi\)
−0.0703088 + 0.997525i \(0.522398\pi\)
\(282\) 0 0
\(283\) −20.1634 −1.19859 −0.599296 0.800528i \(-0.704553\pi\)
−0.599296 + 0.800528i \(0.704553\pi\)
\(284\) 0 0
\(285\) −4.27406 −0.253173
\(286\) 0 0
\(287\) 18.9129 1.11639
\(288\) 0 0
\(289\) −14.3582 −0.844601
\(290\) 0 0
\(291\) −2.03992 −0.119582
\(292\) 0 0
\(293\) −12.9511 −0.756610 −0.378305 0.925681i \(-0.623493\pi\)
−0.378305 + 0.925681i \(0.623493\pi\)
\(294\) 0 0
\(295\) 20.9757 1.22125
\(296\) 0 0
\(297\) −2.19468 −0.127348
\(298\) 0 0
\(299\) 43.4814 2.51459
\(300\) 0 0
\(301\) −8.93894 −0.515232
\(302\) 0 0
\(303\) −0.294350 −0.0169100
\(304\) 0 0
\(305\) 21.1746 1.21245
\(306\) 0 0
\(307\) −6.71716 −0.383369 −0.191684 0.981457i \(-0.561395\pi\)
−0.191684 + 0.981457i \(0.561395\pi\)
\(308\) 0 0
\(309\) −1.16238 −0.0661257
\(310\) 0 0
\(311\) −15.8087 −0.896429 −0.448214 0.893926i \(-0.647940\pi\)
−0.448214 + 0.893926i \(0.647940\pi\)
\(312\) 0 0
\(313\) −27.6720 −1.56411 −0.782057 0.623207i \(-0.785830\pi\)
−0.782057 + 0.623207i \(0.785830\pi\)
\(314\) 0 0
\(315\) 16.1061 0.907479
\(316\) 0 0
\(317\) −2.14088 −0.120244 −0.0601218 0.998191i \(-0.519149\pi\)
−0.0601218 + 0.998191i \(0.519149\pi\)
\(318\) 0 0
\(319\) −13.8721 −0.776691
\(320\) 0 0
\(321\) 3.49414 0.195024
\(322\) 0 0
\(323\) 12.7329 0.708479
\(324\) 0 0
\(325\) 5.59416 0.310308
\(326\) 0 0
\(327\) 1.78577 0.0987535
\(328\) 0 0
\(329\) 1.65383 0.0911784
\(330\) 0 0
\(331\) 13.0310 0.716246 0.358123 0.933674i \(-0.383417\pi\)
0.358123 + 0.933674i \(0.383417\pi\)
\(332\) 0 0
\(333\) −27.2732 −1.49456
\(334\) 0 0
\(335\) 10.1474 0.554412
\(336\) 0 0
\(337\) −2.70826 −0.147528 −0.0737642 0.997276i \(-0.523501\pi\)
−0.0737642 + 0.997276i \(0.523501\pi\)
\(338\) 0 0
\(339\) 1.07835 0.0585680
\(340\) 0 0
\(341\) 12.6684 0.686030
\(342\) 0 0
\(343\) −20.1114 −1.08591
\(344\) 0 0
\(345\) 3.81159 0.205209
\(346\) 0 0
\(347\) 26.4326 1.41898 0.709489 0.704716i \(-0.248926\pi\)
0.709489 + 0.704716i \(0.248926\pi\)
\(348\) 0 0
\(349\) −14.6941 −0.786559 −0.393280 0.919419i \(-0.628660\pi\)
−0.393280 + 0.919419i \(0.628660\pi\)
\(350\) 0 0
\(351\) 8.31796 0.443980
\(352\) 0 0
\(353\) −12.0164 −0.639566 −0.319783 0.947491i \(-0.603610\pi\)
−0.319783 + 0.947491i \(0.603610\pi\)
\(354\) 0 0
\(355\) −10.9090 −0.578992
\(356\) 0 0
\(357\) 0.820879 0.0434455
\(358\) 0 0
\(359\) −32.2776 −1.70354 −0.851772 0.523912i \(-0.824472\pi\)
−0.851772 + 0.523912i \(0.824472\pi\)
\(360\) 0 0
\(361\) 42.3707 2.23004
\(362\) 0 0
\(363\) 1.86522 0.0978988
\(364\) 0 0
\(365\) 29.7693 1.55819
\(366\) 0 0
\(367\) 27.7888 1.45056 0.725282 0.688452i \(-0.241709\pi\)
0.725282 + 0.688452i \(0.241709\pi\)
\(368\) 0 0
\(369\) −24.8117 −1.29165
\(370\) 0 0
\(371\) −24.7095 −1.28285
\(372\) 0 0
\(373\) 23.2716 1.20496 0.602478 0.798135i \(-0.294180\pi\)
0.602478 + 0.798135i \(0.294180\pi\)
\(374\) 0 0
\(375\) −2.23752 −0.115545
\(376\) 0 0
\(377\) 52.5761 2.70781
\(378\) 0 0
\(379\) −21.4700 −1.10284 −0.551420 0.834228i \(-0.685914\pi\)
−0.551420 + 0.834228i \(0.685914\pi\)
\(380\) 0 0
\(381\) 3.86204 0.197858
\(382\) 0 0
\(383\) 27.7055 1.41569 0.707843 0.706370i \(-0.249669\pi\)
0.707843 + 0.706370i \(0.249669\pi\)
\(384\) 0 0
\(385\) 8.96704 0.457003
\(386\) 0 0
\(387\) 11.7270 0.596115
\(388\) 0 0
\(389\) 22.2832 1.12980 0.564901 0.825158i \(-0.308914\pi\)
0.564901 + 0.825158i \(0.308914\pi\)
\(390\) 0 0
\(391\) −11.3552 −0.574257
\(392\) 0 0
\(393\) −0.966584 −0.0487577
\(394\) 0 0
\(395\) 8.90837 0.448229
\(396\) 0 0
\(397\) −2.54049 −0.127504 −0.0637518 0.997966i \(-0.520307\pi\)
−0.0637518 + 0.997966i \(0.520307\pi\)
\(398\) 0 0
\(399\) 3.95651 0.198073
\(400\) 0 0
\(401\) 1.42313 0.0710676 0.0355338 0.999368i \(-0.488687\pi\)
0.0355338 + 0.999368i \(0.488687\pi\)
\(402\) 0 0
\(403\) −48.0137 −2.39173
\(404\) 0 0
\(405\) −20.7619 −1.03167
\(406\) 0 0
\(407\) −15.1842 −0.752655
\(408\) 0 0
\(409\) 8.83345 0.436786 0.218393 0.975861i \(-0.429919\pi\)
0.218393 + 0.975861i \(0.429919\pi\)
\(410\) 0 0
\(411\) −0.00144903 −7.14756e−5 0
\(412\) 0 0
\(413\) −19.4172 −0.955459
\(414\) 0 0
\(415\) 2.85490 0.140141
\(416\) 0 0
\(417\) −0.0408343 −0.00199967
\(418\) 0 0
\(419\) −34.0729 −1.66457 −0.832286 0.554346i \(-0.812968\pi\)
−0.832286 + 0.554346i \(0.812968\pi\)
\(420\) 0 0
\(421\) 29.2308 1.42462 0.712311 0.701864i \(-0.247649\pi\)
0.712311 + 0.701864i \(0.247649\pi\)
\(422\) 0 0
\(423\) −2.16965 −0.105492
\(424\) 0 0
\(425\) −1.46092 −0.0708650
\(426\) 0 0
\(427\) −19.6013 −0.948576
\(428\) 0 0
\(429\) 2.29586 0.110845
\(430\) 0 0
\(431\) 4.82572 0.232447 0.116223 0.993223i \(-0.462921\pi\)
0.116223 + 0.993223i \(0.462921\pi\)
\(432\) 0 0
\(433\) −23.7354 −1.14065 −0.570326 0.821418i \(-0.693183\pi\)
−0.570326 + 0.821418i \(0.693183\pi\)
\(434\) 0 0
\(435\) 4.60884 0.220977
\(436\) 0 0
\(437\) −54.7302 −2.61810
\(438\) 0 0
\(439\) 30.8566 1.47271 0.736354 0.676597i \(-0.236546\pi\)
0.736354 + 0.676597i \(0.236546\pi\)
\(440\) 0 0
\(441\) 5.73727 0.273203
\(442\) 0 0
\(443\) −37.8298 −1.79735 −0.898674 0.438616i \(-0.855469\pi\)
−0.898674 + 0.438616i \(0.855469\pi\)
\(444\) 0 0
\(445\) −41.8386 −1.98334
\(446\) 0 0
\(447\) 5.30675 0.251001
\(448\) 0 0
\(449\) 38.1436 1.80011 0.900054 0.435779i \(-0.143527\pi\)
0.900054 + 0.435779i \(0.143527\pi\)
\(450\) 0 0
\(451\) −13.8138 −0.650469
\(452\) 0 0
\(453\) 0.552009 0.0259356
\(454\) 0 0
\(455\) −33.9855 −1.59327
\(456\) 0 0
\(457\) −20.1481 −0.942489 −0.471244 0.882003i \(-0.656195\pi\)
−0.471244 + 0.882003i \(0.656195\pi\)
\(458\) 0 0
\(459\) −2.17224 −0.101392
\(460\) 0 0
\(461\) −0.500312 −0.0233019 −0.0116509 0.999932i \(-0.503709\pi\)
−0.0116509 + 0.999932i \(0.503709\pi\)
\(462\) 0 0
\(463\) −4.86536 −0.226113 −0.113056 0.993589i \(-0.536064\pi\)
−0.113056 + 0.993589i \(0.536064\pi\)
\(464\) 0 0
\(465\) −4.20890 −0.195183
\(466\) 0 0
\(467\) 3.56899 0.165153 0.0825765 0.996585i \(-0.473685\pi\)
0.0825765 + 0.996585i \(0.473685\pi\)
\(468\) 0 0
\(469\) −9.39348 −0.433751
\(470\) 0 0
\(471\) −2.86395 −0.131964
\(472\) 0 0
\(473\) 6.52894 0.300201
\(474\) 0 0
\(475\) −7.04140 −0.323082
\(476\) 0 0
\(477\) 32.4163 1.48424
\(478\) 0 0
\(479\) −0.936786 −0.0428028 −0.0214014 0.999771i \(-0.506813\pi\)
−0.0214014 + 0.999771i \(0.506813\pi\)
\(480\) 0 0
\(481\) 57.5490 2.62401
\(482\) 0 0
\(483\) −3.52840 −0.160548
\(484\) 0 0
\(485\) −22.0556 −1.00149
\(486\) 0 0
\(487\) 19.2487 0.872242 0.436121 0.899888i \(-0.356352\pi\)
0.436121 + 0.899888i \(0.356352\pi\)
\(488\) 0 0
\(489\) 4.22580 0.191097
\(490\) 0 0
\(491\) 15.5938 0.703739 0.351870 0.936049i \(-0.385546\pi\)
0.351870 + 0.936049i \(0.385546\pi\)
\(492\) 0 0
\(493\) −13.7303 −0.618382
\(494\) 0 0
\(495\) −11.7638 −0.528745
\(496\) 0 0
\(497\) 10.0985 0.452981
\(498\) 0 0
\(499\) 36.3922 1.62914 0.814570 0.580066i \(-0.196973\pi\)
0.814570 + 0.580066i \(0.196973\pi\)
\(500\) 0 0
\(501\) 1.32914 0.0593814
\(502\) 0 0
\(503\) −9.65241 −0.430380 −0.215190 0.976572i \(-0.569037\pi\)
−0.215190 + 0.976572i \(0.569037\pi\)
\(504\) 0 0
\(505\) −3.18252 −0.141620
\(506\) 0 0
\(507\) −5.78116 −0.256750
\(508\) 0 0
\(509\) 16.1978 0.717955 0.358977 0.933346i \(-0.383126\pi\)
0.358977 + 0.933346i \(0.383126\pi\)
\(510\) 0 0
\(511\) −27.5575 −1.21907
\(512\) 0 0
\(513\) −10.4699 −0.462256
\(514\) 0 0
\(515\) −12.5677 −0.553799
\(516\) 0 0
\(517\) −1.20794 −0.0531253
\(518\) 0 0
\(519\) 2.92522 0.128403
\(520\) 0 0
\(521\) −14.2198 −0.622981 −0.311491 0.950249i \(-0.600828\pi\)
−0.311491 + 0.950249i \(0.600828\pi\)
\(522\) 0 0
\(523\) −3.66535 −0.160275 −0.0801373 0.996784i \(-0.525536\pi\)
−0.0801373 + 0.996784i \(0.525536\pi\)
\(524\) 0 0
\(525\) −0.453952 −0.0198121
\(526\) 0 0
\(527\) 12.5388 0.546200
\(528\) 0 0
\(529\) 25.8083 1.12210
\(530\) 0 0
\(531\) 25.4734 1.10545
\(532\) 0 0
\(533\) 52.3552 2.26775
\(534\) 0 0
\(535\) 37.7787 1.63332
\(536\) 0 0
\(537\) 2.53870 0.109553
\(538\) 0 0
\(539\) 3.19420 0.137584
\(540\) 0 0
\(541\) −17.4901 −0.751959 −0.375980 0.926628i \(-0.622694\pi\)
−0.375980 + 0.926628i \(0.622694\pi\)
\(542\) 0 0
\(543\) 2.38260 0.102247
\(544\) 0 0
\(545\) 19.3078 0.827054
\(546\) 0 0
\(547\) −26.1597 −1.11851 −0.559254 0.828996i \(-0.688912\pi\)
−0.559254 + 0.828996i \(0.688912\pi\)
\(548\) 0 0
\(549\) 25.7149 1.09749
\(550\) 0 0
\(551\) −66.1778 −2.81927
\(552\) 0 0
\(553\) −8.24650 −0.350677
\(554\) 0 0
\(555\) 5.04477 0.214139
\(556\) 0 0
\(557\) −19.8690 −0.841879 −0.420939 0.907089i \(-0.638299\pi\)
−0.420939 + 0.907089i \(0.638299\pi\)
\(558\) 0 0
\(559\) −24.7450 −1.04660
\(560\) 0 0
\(561\) −0.599565 −0.0253137
\(562\) 0 0
\(563\) 26.1613 1.10257 0.551284 0.834318i \(-0.314138\pi\)
0.551284 + 0.834318i \(0.314138\pi\)
\(564\) 0 0
\(565\) 11.6591 0.490504
\(566\) 0 0
\(567\) 19.2194 0.807138
\(568\) 0 0
\(569\) −20.8786 −0.875277 −0.437638 0.899151i \(-0.644185\pi\)
−0.437638 + 0.899151i \(0.644185\pi\)
\(570\) 0 0
\(571\) −43.1297 −1.80492 −0.902461 0.430772i \(-0.858241\pi\)
−0.902461 + 0.430772i \(0.858241\pi\)
\(572\) 0 0
\(573\) −2.28057 −0.0952723
\(574\) 0 0
\(575\) 6.27950 0.261873
\(576\) 0 0
\(577\) −17.6062 −0.732958 −0.366479 0.930426i \(-0.619437\pi\)
−0.366479 + 0.930426i \(0.619437\pi\)
\(578\) 0 0
\(579\) 2.06592 0.0858566
\(580\) 0 0
\(581\) −2.64278 −0.109641
\(582\) 0 0
\(583\) 18.0477 0.747458
\(584\) 0 0
\(585\) 44.5855 1.84338
\(586\) 0 0
\(587\) 45.5328 1.87934 0.939670 0.342083i \(-0.111132\pi\)
0.939670 + 0.342083i \(0.111132\pi\)
\(588\) 0 0
\(589\) 60.4351 2.49019
\(590\) 0 0
\(591\) 4.52015 0.185934
\(592\) 0 0
\(593\) 3.18374 0.130741 0.0653703 0.997861i \(-0.479177\pi\)
0.0653703 + 0.997861i \(0.479177\pi\)
\(594\) 0 0
\(595\) 8.87535 0.363854
\(596\) 0 0
\(597\) −2.74592 −0.112383
\(598\) 0 0
\(599\) −9.93843 −0.406073 −0.203037 0.979171i \(-0.565081\pi\)
−0.203037 + 0.979171i \(0.565081\pi\)
\(600\) 0 0
\(601\) 11.1826 0.456147 0.228074 0.973644i \(-0.426757\pi\)
0.228074 + 0.973644i \(0.426757\pi\)
\(602\) 0 0
\(603\) 12.3233 0.501843
\(604\) 0 0
\(605\) 20.1668 0.819897
\(606\) 0 0
\(607\) 18.8713 0.765961 0.382981 0.923756i \(-0.374898\pi\)
0.382981 + 0.923756i \(0.374898\pi\)
\(608\) 0 0
\(609\) −4.26642 −0.172884
\(610\) 0 0
\(611\) 4.57817 0.185213
\(612\) 0 0
\(613\) 42.1894 1.70401 0.852007 0.523531i \(-0.175386\pi\)
0.852007 + 0.523531i \(0.175386\pi\)
\(614\) 0 0
\(615\) 4.58948 0.185065
\(616\) 0 0
\(617\) 16.7840 0.675696 0.337848 0.941201i \(-0.390301\pi\)
0.337848 + 0.941201i \(0.390301\pi\)
\(618\) 0 0
\(619\) 31.7125 1.27463 0.637317 0.770602i \(-0.280044\pi\)
0.637317 + 0.770602i \(0.280044\pi\)
\(620\) 0 0
\(621\) 9.33699 0.374681
\(622\) 0 0
\(623\) 38.7301 1.55169
\(624\) 0 0
\(625\) −28.6863 −1.14745
\(626\) 0 0
\(627\) −2.88981 −0.115408
\(628\) 0 0
\(629\) −15.0290 −0.599245
\(630\) 0 0
\(631\) 6.58208 0.262028 0.131014 0.991381i \(-0.458177\pi\)
0.131014 + 0.991381i \(0.458177\pi\)
\(632\) 0 0
\(633\) 5.48757 0.218112
\(634\) 0 0
\(635\) 41.7564 1.65705
\(636\) 0 0
\(637\) −12.1062 −0.479665
\(638\) 0 0
\(639\) −13.2482 −0.524092
\(640\) 0 0
\(641\) −40.0105 −1.58032 −0.790161 0.612900i \(-0.790003\pi\)
−0.790161 + 0.612900i \(0.790003\pi\)
\(642\) 0 0
\(643\) 24.1750 0.953370 0.476685 0.879074i \(-0.341838\pi\)
0.476685 + 0.879074i \(0.341838\pi\)
\(644\) 0 0
\(645\) −2.16916 −0.0854105
\(646\) 0 0
\(647\) 12.5910 0.495004 0.247502 0.968887i \(-0.420390\pi\)
0.247502 + 0.968887i \(0.420390\pi\)
\(648\) 0 0
\(649\) 14.1822 0.556701
\(650\) 0 0
\(651\) 3.89619 0.152704
\(652\) 0 0
\(653\) 34.6001 1.35401 0.677003 0.735981i \(-0.263279\pi\)
0.677003 + 0.735981i \(0.263279\pi\)
\(654\) 0 0
\(655\) −10.4507 −0.408343
\(656\) 0 0
\(657\) 36.1526 1.41045
\(658\) 0 0
\(659\) −31.6427 −1.23263 −0.616313 0.787502i \(-0.711374\pi\)
−0.616313 + 0.787502i \(0.711374\pi\)
\(660\) 0 0
\(661\) 14.5221 0.564843 0.282422 0.959290i \(-0.408862\pi\)
0.282422 + 0.959290i \(0.408862\pi\)
\(662\) 0 0
\(663\) 2.27238 0.0882520
\(664\) 0 0
\(665\) 42.7778 1.65885
\(666\) 0 0
\(667\) 59.0172 2.28516
\(668\) 0 0
\(669\) 4.48921 0.173563
\(670\) 0 0
\(671\) 14.3167 0.552690
\(672\) 0 0
\(673\) −11.7338 −0.452304 −0.226152 0.974092i \(-0.572615\pi\)
−0.226152 + 0.974092i \(0.572615\pi\)
\(674\) 0 0
\(675\) 1.20126 0.0462367
\(676\) 0 0
\(677\) 2.50545 0.0962923 0.0481461 0.998840i \(-0.484669\pi\)
0.0481461 + 0.998840i \(0.484669\pi\)
\(678\) 0 0
\(679\) 20.4169 0.783529
\(680\) 0 0
\(681\) 0.828175 0.0317357
\(682\) 0 0
\(683\) −14.4053 −0.551203 −0.275602 0.961272i \(-0.588877\pi\)
−0.275602 + 0.961272i \(0.588877\pi\)
\(684\) 0 0
\(685\) −0.0156670 −0.000598604 0
\(686\) 0 0
\(687\) −0.441564 −0.0168467
\(688\) 0 0
\(689\) −68.4015 −2.60589
\(690\) 0 0
\(691\) −11.1265 −0.423271 −0.211635 0.977349i \(-0.567879\pi\)
−0.211635 + 0.977349i \(0.567879\pi\)
\(692\) 0 0
\(693\) 10.8898 0.413670
\(694\) 0 0
\(695\) −0.441501 −0.0167471
\(696\) 0 0
\(697\) −13.6726 −0.517887
\(698\) 0 0
\(699\) 1.69276 0.0640260
\(700\) 0 0
\(701\) 14.5012 0.547704 0.273852 0.961772i \(-0.411702\pi\)
0.273852 + 0.961772i \(0.411702\pi\)
\(702\) 0 0
\(703\) −72.4373 −2.73202
\(704\) 0 0
\(705\) 0.401324 0.0151147
\(706\) 0 0
\(707\) 2.94606 0.110798
\(708\) 0 0
\(709\) 19.8600 0.745859 0.372929 0.927860i \(-0.378353\pi\)
0.372929 + 0.927860i \(0.378353\pi\)
\(710\) 0 0
\(711\) 10.8186 0.405728
\(712\) 0 0
\(713\) −53.8959 −2.01842
\(714\) 0 0
\(715\) 24.8228 0.928320
\(716\) 0 0
\(717\) −1.77006 −0.0661042
\(718\) 0 0
\(719\) −6.68379 −0.249263 −0.124632 0.992203i \(-0.539775\pi\)
−0.124632 + 0.992203i \(0.539775\pi\)
\(720\) 0 0
\(721\) 11.6339 0.433271
\(722\) 0 0
\(723\) −1.61943 −0.0602271
\(724\) 0 0
\(725\) 7.59295 0.281995
\(726\) 0 0
\(727\) 13.6007 0.504421 0.252210 0.967672i \(-0.418842\pi\)
0.252210 + 0.967672i \(0.418842\pi\)
\(728\) 0 0
\(729\) −24.3132 −0.900488
\(730\) 0 0
\(731\) 6.46218 0.239012
\(732\) 0 0
\(733\) 0.562255 0.0207674 0.0103837 0.999946i \(-0.496695\pi\)
0.0103837 + 0.999946i \(0.496695\pi\)
\(734\) 0 0
\(735\) −1.06123 −0.0391442
\(736\) 0 0
\(737\) 6.86094 0.252726
\(738\) 0 0
\(739\) −30.9406 −1.13817 −0.569085 0.822279i \(-0.692702\pi\)
−0.569085 + 0.822279i \(0.692702\pi\)
\(740\) 0 0
\(741\) 10.9525 0.402351
\(742\) 0 0
\(743\) −10.2863 −0.377369 −0.188685 0.982038i \(-0.560422\pi\)
−0.188685 + 0.982038i \(0.560422\pi\)
\(744\) 0 0
\(745\) 57.3766 2.10212
\(746\) 0 0
\(747\) 3.46706 0.126853
\(748\) 0 0
\(749\) −34.9718 −1.27784
\(750\) 0 0
\(751\) 23.1131 0.843410 0.421705 0.906733i \(-0.361432\pi\)
0.421705 + 0.906733i \(0.361432\pi\)
\(752\) 0 0
\(753\) −0.224635 −0.00818615
\(754\) 0 0
\(755\) 5.96832 0.217209
\(756\) 0 0
\(757\) 40.0512 1.45569 0.727843 0.685743i \(-0.240523\pi\)
0.727843 + 0.685743i \(0.240523\pi\)
\(758\) 0 0
\(759\) 2.57712 0.0935436
\(760\) 0 0
\(761\) 31.9526 1.15828 0.579141 0.815228i \(-0.303388\pi\)
0.579141 + 0.815228i \(0.303388\pi\)
\(762\) 0 0
\(763\) −17.8733 −0.647055
\(764\) 0 0
\(765\) −11.6435 −0.420973
\(766\) 0 0
\(767\) −53.7513 −1.94085
\(768\) 0 0
\(769\) −11.2993 −0.407463 −0.203732 0.979027i \(-0.565307\pi\)
−0.203732 + 0.979027i \(0.565307\pi\)
\(770\) 0 0
\(771\) −5.84023 −0.210331
\(772\) 0 0
\(773\) 4.44963 0.160042 0.0800210 0.996793i \(-0.474501\pi\)
0.0800210 + 0.996793i \(0.474501\pi\)
\(774\) 0 0
\(775\) −6.93406 −0.249079
\(776\) 0 0
\(777\) −4.66996 −0.167534
\(778\) 0 0
\(779\) −65.8997 −2.36110
\(780\) 0 0
\(781\) −7.37590 −0.263930
\(782\) 0 0
\(783\) 11.2900 0.403470
\(784\) 0 0
\(785\) −30.9650 −1.10519
\(786\) 0 0
\(787\) 12.2070 0.435133 0.217566 0.976046i \(-0.430188\pi\)
0.217566 + 0.976046i \(0.430188\pi\)
\(788\) 0 0
\(789\) −6.99718 −0.249106
\(790\) 0 0
\(791\) −10.7929 −0.383751
\(792\) 0 0
\(793\) −54.2610 −1.92686
\(794\) 0 0
\(795\) −5.99611 −0.212660
\(796\) 0 0
\(797\) −27.5856 −0.977131 −0.488565 0.872527i \(-0.662480\pi\)
−0.488565 + 0.872527i \(0.662480\pi\)
\(798\) 0 0
\(799\) −1.19559 −0.0422970
\(800\) 0 0
\(801\) −50.8099 −1.79528
\(802\) 0 0
\(803\) 20.1278 0.710295
\(804\) 0 0
\(805\) −38.1491 −1.34458
\(806\) 0 0
\(807\) −0.766988 −0.0269993
\(808\) 0 0
\(809\) −17.3522 −0.610071 −0.305036 0.952341i \(-0.598668\pi\)
−0.305036 + 0.952341i \(0.598668\pi\)
\(810\) 0 0
\(811\) 22.4430 0.788081 0.394041 0.919093i \(-0.371077\pi\)
0.394041 + 0.919093i \(0.371077\pi\)
\(812\) 0 0
\(813\) 5.13019 0.179924
\(814\) 0 0
\(815\) 45.6894 1.60043
\(816\) 0 0
\(817\) 31.1467 1.08968
\(818\) 0 0
\(819\) −41.2729 −1.44219
\(820\) 0 0
\(821\) 2.18088 0.0761133 0.0380567 0.999276i \(-0.487883\pi\)
0.0380567 + 0.999276i \(0.487883\pi\)
\(822\) 0 0
\(823\) 6.22243 0.216900 0.108450 0.994102i \(-0.465411\pi\)
0.108450 + 0.994102i \(0.465411\pi\)
\(824\) 0 0
\(825\) 0.331564 0.0115436
\(826\) 0 0
\(827\) −17.5574 −0.610530 −0.305265 0.952267i \(-0.598745\pi\)
−0.305265 + 0.952267i \(0.598745\pi\)
\(828\) 0 0
\(829\) −0.230530 −0.00800664 −0.00400332 0.999992i \(-0.501274\pi\)
−0.00400332 + 0.999992i \(0.501274\pi\)
\(830\) 0 0
\(831\) 0.908130 0.0315027
\(832\) 0 0
\(833\) 3.16154 0.109541
\(834\) 0 0
\(835\) 14.3706 0.497316
\(836\) 0 0
\(837\) −10.3103 −0.356374
\(838\) 0 0
\(839\) 42.3974 1.46372 0.731860 0.681455i \(-0.238652\pi\)
0.731860 + 0.681455i \(0.238652\pi\)
\(840\) 0 0
\(841\) 42.3615 1.46074
\(842\) 0 0
\(843\) 0.529504 0.0182371
\(844\) 0 0
\(845\) −62.5059 −2.15027
\(846\) 0 0
\(847\) −18.6685 −0.641456
\(848\) 0 0
\(849\) 4.52941 0.155449
\(850\) 0 0
\(851\) 64.5994 2.21444
\(852\) 0 0
\(853\) −27.0749 −0.927026 −0.463513 0.886090i \(-0.653411\pi\)
−0.463513 + 0.886090i \(0.653411\pi\)
\(854\) 0 0
\(855\) −56.1200 −1.91926
\(856\) 0 0
\(857\) 10.5979 0.362017 0.181008 0.983482i \(-0.442064\pi\)
0.181008 + 0.983482i \(0.442064\pi\)
\(858\) 0 0
\(859\) 57.4152 1.95898 0.979491 0.201489i \(-0.0645781\pi\)
0.979491 + 0.201489i \(0.0645781\pi\)
\(860\) 0 0
\(861\) −4.24849 −0.144788
\(862\) 0 0
\(863\) −31.6763 −1.07827 −0.539136 0.842219i \(-0.681249\pi\)
−0.539136 + 0.842219i \(0.681249\pi\)
\(864\) 0 0
\(865\) 31.6275 1.07537
\(866\) 0 0
\(867\) 3.22536 0.109539
\(868\) 0 0
\(869\) 6.02319 0.204323
\(870\) 0 0
\(871\) −26.0033 −0.881088
\(872\) 0 0
\(873\) −26.7849 −0.906531
\(874\) 0 0
\(875\) 22.3947 0.757079
\(876\) 0 0
\(877\) 23.1500 0.781719 0.390860 0.920450i \(-0.372178\pi\)
0.390860 + 0.920450i \(0.372178\pi\)
\(878\) 0 0
\(879\) 2.90926 0.0981270
\(880\) 0 0
\(881\) 9.26056 0.311996 0.155998 0.987757i \(-0.450141\pi\)
0.155998 + 0.987757i \(0.450141\pi\)
\(882\) 0 0
\(883\) 34.1105 1.14791 0.573954 0.818887i \(-0.305409\pi\)
0.573954 + 0.818887i \(0.305409\pi\)
\(884\) 0 0
\(885\) −4.71186 −0.158387
\(886\) 0 0
\(887\) −36.0360 −1.20997 −0.604986 0.796236i \(-0.706821\pi\)
−0.604986 + 0.796236i \(0.706821\pi\)
\(888\) 0 0
\(889\) −38.6540 −1.29641
\(890\) 0 0
\(891\) −14.0377 −0.470281
\(892\) 0 0
\(893\) −5.76257 −0.192837
\(894\) 0 0
\(895\) 27.4484 0.917499
\(896\) 0 0
\(897\) −9.76742 −0.326125
\(898\) 0 0
\(899\) −65.1690 −2.17351
\(900\) 0 0
\(901\) 17.8631 0.595107
\(902\) 0 0
\(903\) 2.00800 0.0668219
\(904\) 0 0
\(905\) 25.7607 0.856314
\(906\) 0 0
\(907\) −41.7267 −1.38551 −0.692756 0.721172i \(-0.743604\pi\)
−0.692756 + 0.721172i \(0.743604\pi\)
\(908\) 0 0
\(909\) −3.86493 −0.128192
\(910\) 0 0
\(911\) 32.2971 1.07005 0.535026 0.844836i \(-0.320302\pi\)
0.535026 + 0.844836i \(0.320302\pi\)
\(912\) 0 0
\(913\) 1.93027 0.0638827
\(914\) 0 0
\(915\) −4.75654 −0.157246
\(916\) 0 0
\(917\) 9.67425 0.319472
\(918\) 0 0
\(919\) −37.3668 −1.23262 −0.616308 0.787505i \(-0.711372\pi\)
−0.616308 + 0.787505i \(0.711372\pi\)
\(920\) 0 0
\(921\) 1.50891 0.0497202
\(922\) 0 0
\(923\) 27.9550 0.920151
\(924\) 0 0
\(925\) 8.31113 0.273268
\(926\) 0 0
\(927\) −15.2625 −0.501287
\(928\) 0 0
\(929\) 2.07810 0.0681804 0.0340902 0.999419i \(-0.489147\pi\)
0.0340902 + 0.999419i \(0.489147\pi\)
\(930\) 0 0
\(931\) 15.2381 0.499410
\(932\) 0 0
\(933\) 3.55118 0.116260
\(934\) 0 0
\(935\) −6.48250 −0.212000
\(936\) 0 0
\(937\) −4.54512 −0.148483 −0.0742413 0.997240i \(-0.523654\pi\)
−0.0742413 + 0.997240i \(0.523654\pi\)
\(938\) 0 0
\(939\) 6.21610 0.202855
\(940\) 0 0
\(941\) −27.5658 −0.898620 −0.449310 0.893376i \(-0.648330\pi\)
−0.449310 + 0.893376i \(0.648330\pi\)
\(942\) 0 0
\(943\) 58.7692 1.91379
\(944\) 0 0
\(945\) −7.29790 −0.237401
\(946\) 0 0
\(947\) 40.4811 1.31546 0.657729 0.753255i \(-0.271517\pi\)
0.657729 + 0.753255i \(0.271517\pi\)
\(948\) 0 0
\(949\) −76.2854 −2.47633
\(950\) 0 0
\(951\) 0.480915 0.0155948
\(952\) 0 0
\(953\) −33.9206 −1.09880 −0.549398 0.835561i \(-0.685143\pi\)
−0.549398 + 0.835561i \(0.685143\pi\)
\(954\) 0 0
\(955\) −24.6576 −0.797900
\(956\) 0 0
\(957\) 3.11617 0.100731
\(958\) 0 0
\(959\) 0.0145030 0.000468325 0
\(960\) 0 0
\(961\) 28.5139 0.919802
\(962\) 0 0
\(963\) 45.8794 1.47844
\(964\) 0 0
\(965\) 22.3367 0.719044
\(966\) 0 0
\(967\) −15.0645 −0.484441 −0.242221 0.970221i \(-0.577876\pi\)
−0.242221 + 0.970221i \(0.577876\pi\)
\(968\) 0 0
\(969\) −2.86026 −0.0918848
\(970\) 0 0
\(971\) 3.61438 0.115991 0.0579956 0.998317i \(-0.481529\pi\)
0.0579956 + 0.998317i \(0.481529\pi\)
\(972\) 0 0
\(973\) 0.408699 0.0131023
\(974\) 0 0
\(975\) −1.25664 −0.0402448
\(976\) 0 0
\(977\) 40.6929 1.30188 0.650940 0.759129i \(-0.274375\pi\)
0.650940 + 0.759129i \(0.274375\pi\)
\(978\) 0 0
\(979\) −28.2882 −0.904095
\(980\) 0 0
\(981\) 23.4479 0.748633
\(982\) 0 0
\(983\) 17.5420 0.559504 0.279752 0.960072i \(-0.409748\pi\)
0.279752 + 0.960072i \(0.409748\pi\)
\(984\) 0 0
\(985\) 48.8719 1.55719
\(986\) 0 0
\(987\) −0.371507 −0.0118252
\(988\) 0 0
\(989\) −27.7765 −0.883242
\(990\) 0 0
\(991\) 47.1532 1.49787 0.748936 0.662643i \(-0.230565\pi\)
0.748936 + 0.662643i \(0.230565\pi\)
\(992\) 0 0
\(993\) −2.92721 −0.0928920
\(994\) 0 0
\(995\) −29.6889 −0.941201
\(996\) 0 0
\(997\) −53.5944 −1.69735 −0.848676 0.528912i \(-0.822600\pi\)
−0.848676 + 0.528912i \(0.822600\pi\)
\(998\) 0 0
\(999\) 12.3578 0.390984
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 4016.2.a.j.1.7 14
4.3 odd 2 1004.2.a.b.1.8 14
12.11 even 2 9036.2.a.m.1.11 14
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1004.2.a.b.1.8 14 4.3 odd 2
4016.2.a.j.1.7 14 1.1 even 1 trivial
9036.2.a.m.1.11 14 12.11 even 2