Properties

Label 4840.2.a.bh.1.1
Level $4840$
Weight $2$
Character 4840.1
Self dual yes
Analytic conductor $38.648$
Analytic rank $0$
Dimension $8$
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] = [4840,2,Mod(1,4840)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4840, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("4840.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4840 = 2^{3} \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4840.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: \(38.6475945783\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} - 21x^{6} + 15x^{5} + 140x^{4} - 60x^{3} - 295x^{2} + 50x + 100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 440)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.90500\) of defining polynomial
Character \(\chi\) \(=\) 4840.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.90500 q^{3} +1.00000 q^{5} +2.18309 q^{7} +5.43902 q^{9} +O(q^{10})\) \(q-2.90500 q^{3} +1.00000 q^{5} +2.18309 q^{7} +5.43902 q^{9} -6.16092 q^{13} -2.90500 q^{15} -7.12818 q^{17} -6.35983 q^{19} -6.34188 q^{21} +2.17735 q^{23} +1.00000 q^{25} -7.08534 q^{27} +3.19299 q^{29} +4.39424 q^{31} +2.18309 q^{35} +5.29453 q^{37} +17.8975 q^{39} -3.28629 q^{41} +6.83874 q^{43} +5.43902 q^{45} -4.30445 q^{47} -2.23411 q^{49} +20.7074 q^{51} -10.6160 q^{53} +18.4753 q^{57} +13.7906 q^{59} -7.35286 q^{61} +11.8739 q^{63} -6.16092 q^{65} -7.67154 q^{67} -6.32521 q^{69} +5.74133 q^{71} +4.89575 q^{73} -2.90500 q^{75} +8.11407 q^{79} +4.26586 q^{81} -4.22384 q^{83} -7.12818 q^{85} -9.27563 q^{87} +16.8037 q^{89} -13.4499 q^{91} -12.7653 q^{93} -6.35983 q^{95} -3.27563 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + q^{3} + 8 q^{5} + 6 q^{7} + 19 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + q^{3} + 8 q^{5} + 6 q^{7} + 19 q^{9} - 12 q^{13} + q^{15} - 2 q^{17} + 6 q^{19} - 6 q^{21} + 10 q^{23} + 8 q^{25} + 13 q^{27} - 8 q^{29} + 19 q^{31} + 6 q^{35} + 12 q^{37} + 21 q^{39} + 3 q^{41} + 8 q^{43} + 19 q^{45} + 10 q^{47} + 22 q^{49} + 7 q^{51} + 28 q^{53} - 25 q^{57} + 25 q^{59} - 10 q^{61} + 64 q^{63} - 12 q^{65} - 2 q^{67} + 18 q^{69} + 25 q^{71} - 38 q^{73} + q^{75} + 38 q^{79} + 32 q^{81} + 28 q^{83} - 2 q^{85} - 15 q^{87} + 12 q^{89} + 8 q^{91} - 15 q^{93} + 6 q^{95} + 21 q^{97}+O(q^{100}) \) 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
\(3\) −2.90500 −1.67720 −0.838601 0.544746i \(-0.816626\pi\)
−0.838601 + 0.544746i \(0.816626\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 2.18309 0.825131 0.412566 0.910928i \(-0.364633\pi\)
0.412566 + 0.910928i \(0.364633\pi\)
\(8\) 0 0
\(9\) 5.43902 1.81301
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) −6.16092 −1.70873 −0.854366 0.519671i \(-0.826055\pi\)
−0.854366 + 0.519671i \(0.826055\pi\)
\(14\) 0 0
\(15\) −2.90500 −0.750067
\(16\) 0 0
\(17\) −7.12818 −1.72884 −0.864419 0.502772i \(-0.832314\pi\)
−0.864419 + 0.502772i \(0.832314\pi\)
\(18\) 0 0
\(19\) −6.35983 −1.45905 −0.729523 0.683956i \(-0.760258\pi\)
−0.729523 + 0.683956i \(0.760258\pi\)
\(20\) 0 0
\(21\) −6.34188 −1.38391
\(22\) 0 0
\(23\) 2.17735 0.454010 0.227005 0.973894i \(-0.427107\pi\)
0.227005 + 0.973894i \(0.427107\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −7.08534 −1.36357
\(28\) 0 0
\(29\) 3.19299 0.592924 0.296462 0.955045i \(-0.404193\pi\)
0.296462 + 0.955045i \(0.404193\pi\)
\(30\) 0 0
\(31\) 4.39424 0.789229 0.394614 0.918847i \(-0.370878\pi\)
0.394614 + 0.918847i \(0.370878\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2.18309 0.369010
\(36\) 0 0
\(37\) 5.29453 0.870415 0.435207 0.900330i \(-0.356675\pi\)
0.435207 + 0.900330i \(0.356675\pi\)
\(38\) 0 0
\(39\) 17.8975 2.86589
\(40\) 0 0
\(41\) −3.28629 −0.513232 −0.256616 0.966513i \(-0.582608\pi\)
−0.256616 + 0.966513i \(0.582608\pi\)
\(42\) 0 0
\(43\) 6.83874 1.04290 0.521449 0.853282i \(-0.325392\pi\)
0.521449 + 0.853282i \(0.325392\pi\)
\(44\) 0 0
\(45\) 5.43902 0.810801
\(46\) 0 0
\(47\) −4.30445 −0.627868 −0.313934 0.949445i \(-0.601647\pi\)
−0.313934 + 0.949445i \(0.601647\pi\)
\(48\) 0 0
\(49\) −2.23411 −0.319159
\(50\) 0 0
\(51\) 20.7074 2.89961
\(52\) 0 0
\(53\) −10.6160 −1.45822 −0.729110 0.684397i \(-0.760066\pi\)
−0.729110 + 0.684397i \(0.760066\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 18.4753 2.44711
\(58\) 0 0
\(59\) 13.7906 1.79538 0.897689 0.440630i \(-0.145245\pi\)
0.897689 + 0.440630i \(0.145245\pi\)
\(60\) 0 0
\(61\) −7.35286 −0.941437 −0.470719 0.882283i \(-0.656005\pi\)
−0.470719 + 0.882283i \(0.656005\pi\)
\(62\) 0 0
\(63\) 11.8739 1.49597
\(64\) 0 0
\(65\) −6.16092 −0.764169
\(66\) 0 0
\(67\) −7.67154 −0.937228 −0.468614 0.883403i \(-0.655246\pi\)
−0.468614 + 0.883403i \(0.655246\pi\)
\(68\) 0 0
\(69\) −6.32521 −0.761466
\(70\) 0 0
\(71\) 5.74133 0.681370 0.340685 0.940177i \(-0.389341\pi\)
0.340685 + 0.940177i \(0.389341\pi\)
\(72\) 0 0
\(73\) 4.89575 0.573004 0.286502 0.958080i \(-0.407507\pi\)
0.286502 + 0.958080i \(0.407507\pi\)
\(74\) 0 0
\(75\) −2.90500 −0.335440
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 8.11407 0.912904 0.456452 0.889748i \(-0.349120\pi\)
0.456452 + 0.889748i \(0.349120\pi\)
\(80\) 0 0
\(81\) 4.26586 0.473984
\(82\) 0 0
\(83\) −4.22384 −0.463626 −0.231813 0.972760i \(-0.574466\pi\)
−0.231813 + 0.972760i \(0.574466\pi\)
\(84\) 0 0
\(85\) −7.12818 −0.773160
\(86\) 0 0
\(87\) −9.27563 −0.994452
\(88\) 0 0
\(89\) 16.8037 1.78119 0.890594 0.454800i \(-0.150289\pi\)
0.890594 + 0.454800i \(0.150289\pi\)
\(90\) 0 0
\(91\) −13.4499 −1.40993
\(92\) 0 0
\(93\) −12.7653 −1.32370
\(94\) 0 0
\(95\) −6.35983 −0.652505
\(96\) 0 0
\(97\) −3.27563 −0.332589 −0.166295 0.986076i \(-0.553180\pi\)
−0.166295 + 0.986076i \(0.553180\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 6.91048 0.687618 0.343809 0.939040i \(-0.388283\pi\)
0.343809 + 0.939040i \(0.388283\pi\)
\(102\) 0 0
\(103\) 7.58591 0.747462 0.373731 0.927537i \(-0.378078\pi\)
0.373731 + 0.927537i \(0.378078\pi\)
\(104\) 0 0
\(105\) −6.34188 −0.618904
\(106\) 0 0
\(107\) −2.80495 −0.271164 −0.135582 0.990766i \(-0.543290\pi\)
−0.135582 + 0.990766i \(0.543290\pi\)
\(108\) 0 0
\(109\) −8.02896 −0.769035 −0.384517 0.923118i \(-0.625632\pi\)
−0.384517 + 0.923118i \(0.625632\pi\)
\(110\) 0 0
\(111\) −15.3806 −1.45986
\(112\) 0 0
\(113\) 6.95944 0.654689 0.327344 0.944905i \(-0.393846\pi\)
0.327344 + 0.944905i \(0.393846\pi\)
\(114\) 0 0
\(115\) 2.17735 0.203039
\(116\) 0 0
\(117\) −33.5094 −3.09794
\(118\) 0 0
\(119\) −15.5615 −1.42652
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) 9.54667 0.860794
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 10.3872 0.921716 0.460858 0.887474i \(-0.347542\pi\)
0.460858 + 0.887474i \(0.347542\pi\)
\(128\) 0 0
\(129\) −19.8665 −1.74915
\(130\) 0 0
\(131\) −3.48613 −0.304585 −0.152292 0.988335i \(-0.548666\pi\)
−0.152292 + 0.988335i \(0.548666\pi\)
\(132\) 0 0
\(133\) −13.8841 −1.20390
\(134\) 0 0
\(135\) −7.08534 −0.609809
\(136\) 0 0
\(137\) −14.5471 −1.24284 −0.621422 0.783476i \(-0.713445\pi\)
−0.621422 + 0.783476i \(0.713445\pi\)
\(138\) 0 0
\(139\) 0.501893 0.0425700 0.0212850 0.999773i \(-0.493224\pi\)
0.0212850 + 0.999773i \(0.493224\pi\)
\(140\) 0 0
\(141\) 12.5044 1.05306
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 3.19299 0.265163
\(146\) 0 0
\(147\) 6.49009 0.535294
\(148\) 0 0
\(149\) −0.747620 −0.0612474 −0.0306237 0.999531i \(-0.509749\pi\)
−0.0306237 + 0.999531i \(0.509749\pi\)
\(150\) 0 0
\(151\) 3.27337 0.266383 0.133192 0.991090i \(-0.457477\pi\)
0.133192 + 0.991090i \(0.457477\pi\)
\(152\) 0 0
\(153\) −38.7703 −3.13439
\(154\) 0 0
\(155\) 4.39424 0.352954
\(156\) 0 0
\(157\) −15.0296 −1.19949 −0.599747 0.800190i \(-0.704732\pi\)
−0.599747 + 0.800190i \(0.704732\pi\)
\(158\) 0 0
\(159\) 30.8395 2.44573
\(160\) 0 0
\(161\) 4.75336 0.374617
\(162\) 0 0
\(163\) 14.4813 1.13426 0.567132 0.823627i \(-0.308053\pi\)
0.567132 + 0.823627i \(0.308053\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0.304676 0.0235766 0.0117883 0.999931i \(-0.496248\pi\)
0.0117883 + 0.999931i \(0.496248\pi\)
\(168\) 0 0
\(169\) 24.9570 1.91977
\(170\) 0 0
\(171\) −34.5912 −2.64526
\(172\) 0 0
\(173\) 9.71345 0.738500 0.369250 0.929330i \(-0.379615\pi\)
0.369250 + 0.929330i \(0.379615\pi\)
\(174\) 0 0
\(175\) 2.18309 0.165026
\(176\) 0 0
\(177\) −40.0615 −3.01121
\(178\) 0 0
\(179\) 9.40166 0.702713 0.351356 0.936242i \(-0.385721\pi\)
0.351356 + 0.936242i \(0.385721\pi\)
\(180\) 0 0
\(181\) −13.1767 −0.979420 −0.489710 0.871885i \(-0.662897\pi\)
−0.489710 + 0.871885i \(0.662897\pi\)
\(182\) 0 0
\(183\) 21.3600 1.57898
\(184\) 0 0
\(185\) 5.29453 0.389261
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −15.4679 −1.12513
\(190\) 0 0
\(191\) −7.35752 −0.532372 −0.266186 0.963922i \(-0.585764\pi\)
−0.266186 + 0.963922i \(0.585764\pi\)
\(192\) 0 0
\(193\) 18.4749 1.32985 0.664927 0.746908i \(-0.268463\pi\)
0.664927 + 0.746908i \(0.268463\pi\)
\(194\) 0 0
\(195\) 17.8975 1.28166
\(196\) 0 0
\(197\) −9.58209 −0.682696 −0.341348 0.939937i \(-0.610883\pi\)
−0.341348 + 0.939937i \(0.610883\pi\)
\(198\) 0 0
\(199\) 13.3287 0.944844 0.472422 0.881372i \(-0.343380\pi\)
0.472422 + 0.881372i \(0.343380\pi\)
\(200\) 0 0
\(201\) 22.2858 1.57192
\(202\) 0 0
\(203\) 6.97059 0.489240
\(204\) 0 0
\(205\) −3.28629 −0.229524
\(206\) 0 0
\(207\) 11.8427 0.823122
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −10.0841 −0.694221 −0.347111 0.937824i \(-0.612837\pi\)
−0.347111 + 0.937824i \(0.612837\pi\)
\(212\) 0 0
\(213\) −16.6785 −1.14279
\(214\) 0 0
\(215\) 6.83874 0.466398
\(216\) 0 0
\(217\) 9.59303 0.651217
\(218\) 0 0
\(219\) −14.2221 −0.961044
\(220\) 0 0
\(221\) 43.9162 2.95412
\(222\) 0 0
\(223\) 10.3500 0.693084 0.346542 0.938034i \(-0.387356\pi\)
0.346542 + 0.938034i \(0.387356\pi\)
\(224\) 0 0
\(225\) 5.43902 0.362601
\(226\) 0 0
\(227\) −10.9354 −0.725805 −0.362902 0.931827i \(-0.618214\pi\)
−0.362902 + 0.931827i \(0.618214\pi\)
\(228\) 0 0
\(229\) 22.3845 1.47921 0.739606 0.673040i \(-0.235012\pi\)
0.739606 + 0.673040i \(0.235012\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 15.5657 1.01974 0.509871 0.860251i \(-0.329693\pi\)
0.509871 + 0.860251i \(0.329693\pi\)
\(234\) 0 0
\(235\) −4.30445 −0.280791
\(236\) 0 0
\(237\) −23.5714 −1.53112
\(238\) 0 0
\(239\) 24.4294 1.58021 0.790103 0.612974i \(-0.210027\pi\)
0.790103 + 0.612974i \(0.210027\pi\)
\(240\) 0 0
\(241\) 5.75661 0.370816 0.185408 0.982662i \(-0.440639\pi\)
0.185408 + 0.982662i \(0.440639\pi\)
\(242\) 0 0
\(243\) 8.86372 0.568608
\(244\) 0 0
\(245\) −2.23411 −0.142732
\(246\) 0 0
\(247\) 39.1825 2.49312
\(248\) 0 0
\(249\) 12.2702 0.777595
\(250\) 0 0
\(251\) 12.0240 0.758950 0.379475 0.925202i \(-0.376105\pi\)
0.379475 + 0.925202i \(0.376105\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 20.7074 1.29675
\(256\) 0 0
\(257\) −19.5364 −1.21865 −0.609324 0.792922i \(-0.708559\pi\)
−0.609324 + 0.792922i \(0.708559\pi\)
\(258\) 0 0
\(259\) 11.5584 0.718206
\(260\) 0 0
\(261\) 17.3667 1.07497
\(262\) 0 0
\(263\) 24.6213 1.51821 0.759106 0.650967i \(-0.225636\pi\)
0.759106 + 0.650967i \(0.225636\pi\)
\(264\) 0 0
\(265\) −10.6160 −0.652135
\(266\) 0 0
\(267\) −48.8147 −2.98741
\(268\) 0 0
\(269\) 1.54495 0.0941974 0.0470987 0.998890i \(-0.485002\pi\)
0.0470987 + 0.998890i \(0.485002\pi\)
\(270\) 0 0
\(271\) −22.5647 −1.37071 −0.685355 0.728209i \(-0.740353\pi\)
−0.685355 + 0.728209i \(0.740353\pi\)
\(272\) 0 0
\(273\) 39.0718 2.36473
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 13.8189 0.830298 0.415149 0.909754i \(-0.363730\pi\)
0.415149 + 0.909754i \(0.363730\pi\)
\(278\) 0 0
\(279\) 23.9003 1.43088
\(280\) 0 0
\(281\) 8.70719 0.519427 0.259714 0.965686i \(-0.416372\pi\)
0.259714 + 0.965686i \(0.416372\pi\)
\(282\) 0 0
\(283\) −1.52467 −0.0906325 −0.0453162 0.998973i \(-0.514430\pi\)
−0.0453162 + 0.998973i \(0.514430\pi\)
\(284\) 0 0
\(285\) 18.4753 1.09438
\(286\) 0 0
\(287\) −7.17427 −0.423484
\(288\) 0 0
\(289\) 33.8110 1.98888
\(290\) 0 0
\(291\) 9.51569 0.557820
\(292\) 0 0
\(293\) −23.7371 −1.38674 −0.693368 0.720584i \(-0.743874\pi\)
−0.693368 + 0.720584i \(0.743874\pi\)
\(294\) 0 0
\(295\) 13.7906 0.802917
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −13.4145 −0.775781
\(300\) 0 0
\(301\) 14.9296 0.860527
\(302\) 0 0
\(303\) −20.0749 −1.15327
\(304\) 0 0
\(305\) −7.35286 −0.421024
\(306\) 0 0
\(307\) 17.7127 1.01092 0.505458 0.862851i \(-0.331324\pi\)
0.505458 + 0.862851i \(0.331324\pi\)
\(308\) 0 0
\(309\) −22.0371 −1.25364
\(310\) 0 0
\(311\) 6.78493 0.384738 0.192369 0.981323i \(-0.438383\pi\)
0.192369 + 0.981323i \(0.438383\pi\)
\(312\) 0 0
\(313\) 23.8056 1.34557 0.672786 0.739838i \(-0.265098\pi\)
0.672786 + 0.739838i \(0.265098\pi\)
\(314\) 0 0
\(315\) 11.8739 0.669017
\(316\) 0 0
\(317\) −1.84797 −0.103793 −0.0518963 0.998652i \(-0.516527\pi\)
−0.0518963 + 0.998652i \(0.516527\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 8.14836 0.454797
\(322\) 0 0
\(323\) 45.3341 2.52245
\(324\) 0 0
\(325\) −6.16092 −0.341747
\(326\) 0 0
\(327\) 23.3241 1.28983
\(328\) 0 0
\(329\) −9.39700 −0.518073
\(330\) 0 0
\(331\) 17.5654 0.965484 0.482742 0.875763i \(-0.339641\pi\)
0.482742 + 0.875763i \(0.339641\pi\)
\(332\) 0 0
\(333\) 28.7970 1.57807
\(334\) 0 0
\(335\) −7.67154 −0.419141
\(336\) 0 0
\(337\) −21.3539 −1.16322 −0.581610 0.813468i \(-0.697577\pi\)
−0.581610 + 0.813468i \(0.697577\pi\)
\(338\) 0 0
\(339\) −20.2172 −1.09805
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −20.1589 −1.08848
\(344\) 0 0
\(345\) −6.32521 −0.340538
\(346\) 0 0
\(347\) −5.36134 −0.287812 −0.143906 0.989591i \(-0.545966\pi\)
−0.143906 + 0.989591i \(0.545966\pi\)
\(348\) 0 0
\(349\) 11.5164 0.616456 0.308228 0.951312i \(-0.400264\pi\)
0.308228 + 0.951312i \(0.400264\pi\)
\(350\) 0 0
\(351\) 43.6523 2.32998
\(352\) 0 0
\(353\) 9.70208 0.516389 0.258195 0.966093i \(-0.416872\pi\)
0.258195 + 0.966093i \(0.416872\pi\)
\(354\) 0 0
\(355\) 5.74133 0.304718
\(356\) 0 0
\(357\) 45.2061 2.39256
\(358\) 0 0
\(359\) 32.5559 1.71823 0.859117 0.511780i \(-0.171014\pi\)
0.859117 + 0.511780i \(0.171014\pi\)
\(360\) 0 0
\(361\) 21.4475 1.12882
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 4.89575 0.256255
\(366\) 0 0
\(367\) −26.5031 −1.38345 −0.691725 0.722161i \(-0.743149\pi\)
−0.691725 + 0.722161i \(0.743149\pi\)
\(368\) 0 0
\(369\) −17.8742 −0.930493
\(370\) 0 0
\(371\) −23.1757 −1.20322
\(372\) 0 0
\(373\) −25.2185 −1.30577 −0.652883 0.757459i \(-0.726441\pi\)
−0.652883 + 0.757459i \(0.726441\pi\)
\(374\) 0 0
\(375\) −2.90500 −0.150013
\(376\) 0 0
\(377\) −19.6718 −1.01315
\(378\) 0 0
\(379\) −17.5353 −0.900729 −0.450364 0.892845i \(-0.648706\pi\)
−0.450364 + 0.892845i \(0.648706\pi\)
\(380\) 0 0
\(381\) −30.1748 −1.54590
\(382\) 0 0
\(383\) −13.2063 −0.674811 −0.337406 0.941359i \(-0.609549\pi\)
−0.337406 + 0.941359i \(0.609549\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 37.1960 1.89078
\(388\) 0 0
\(389\) 32.8461 1.66536 0.832681 0.553753i \(-0.186805\pi\)
0.832681 + 0.553753i \(0.186805\pi\)
\(390\) 0 0
\(391\) −15.5206 −0.784909
\(392\) 0 0
\(393\) 10.1272 0.510850
\(394\) 0 0
\(395\) 8.11407 0.408263
\(396\) 0 0
\(397\) 25.2290 1.26621 0.633104 0.774066i \(-0.281780\pi\)
0.633104 + 0.774066i \(0.281780\pi\)
\(398\) 0 0
\(399\) 40.3333 2.01919
\(400\) 0 0
\(401\) 17.0748 0.852676 0.426338 0.904564i \(-0.359803\pi\)
0.426338 + 0.904564i \(0.359803\pi\)
\(402\) 0 0
\(403\) −27.0726 −1.34858
\(404\) 0 0
\(405\) 4.26586 0.211972
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0.798538 0.0394851 0.0197426 0.999805i \(-0.493715\pi\)
0.0197426 + 0.999805i \(0.493715\pi\)
\(410\) 0 0
\(411\) 42.2593 2.08450
\(412\) 0 0
\(413\) 30.1060 1.48142
\(414\) 0 0
\(415\) −4.22384 −0.207340
\(416\) 0 0
\(417\) −1.45800 −0.0713984
\(418\) 0 0
\(419\) −3.76575 −0.183969 −0.0919844 0.995760i \(-0.529321\pi\)
−0.0919844 + 0.995760i \(0.529321\pi\)
\(420\) 0 0
\(421\) −16.9537 −0.826272 −0.413136 0.910669i \(-0.635567\pi\)
−0.413136 + 0.910669i \(0.635567\pi\)
\(422\) 0 0
\(423\) −23.4120 −1.13833
\(424\) 0 0
\(425\) −7.12818 −0.345768
\(426\) 0 0
\(427\) −16.0520 −0.776809
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −26.0432 −1.25446 −0.627228 0.778836i \(-0.715810\pi\)
−0.627228 + 0.778836i \(0.715810\pi\)
\(432\) 0 0
\(433\) −22.2935 −1.07136 −0.535678 0.844422i \(-0.679944\pi\)
−0.535678 + 0.844422i \(0.679944\pi\)
\(434\) 0 0
\(435\) −9.27563 −0.444733
\(436\) 0 0
\(437\) −13.8476 −0.662421
\(438\) 0 0
\(439\) −13.8298 −0.660060 −0.330030 0.943970i \(-0.607059\pi\)
−0.330030 + 0.943970i \(0.607059\pi\)
\(440\) 0 0
\(441\) −12.1514 −0.578637
\(442\) 0 0
\(443\) 15.0923 0.717057 0.358528 0.933519i \(-0.383279\pi\)
0.358528 + 0.933519i \(0.383279\pi\)
\(444\) 0 0
\(445\) 16.8037 0.796571
\(446\) 0 0
\(447\) 2.17183 0.102724
\(448\) 0 0
\(449\) 17.6708 0.833936 0.416968 0.908921i \(-0.363093\pi\)
0.416968 + 0.908921i \(0.363093\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −9.50914 −0.446778
\(454\) 0 0
\(455\) −13.4499 −0.630539
\(456\) 0 0
\(457\) 34.2329 1.60135 0.800674 0.599100i \(-0.204475\pi\)
0.800674 + 0.599100i \(0.204475\pi\)
\(458\) 0 0
\(459\) 50.5056 2.35740
\(460\) 0 0
\(461\) 10.1143 0.471069 0.235534 0.971866i \(-0.424316\pi\)
0.235534 + 0.971866i \(0.424316\pi\)
\(462\) 0 0
\(463\) −11.2505 −0.522857 −0.261429 0.965223i \(-0.584194\pi\)
−0.261429 + 0.965223i \(0.584194\pi\)
\(464\) 0 0
\(465\) −12.7653 −0.591975
\(466\) 0 0
\(467\) 34.3894 1.59135 0.795676 0.605722i \(-0.207116\pi\)
0.795676 + 0.605722i \(0.207116\pi\)
\(468\) 0 0
\(469\) −16.7477 −0.773336
\(470\) 0 0
\(471\) 43.6610 2.01179
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −6.35983 −0.291809
\(476\) 0 0
\(477\) −57.7406 −2.64376
\(478\) 0 0
\(479\) 25.6708 1.17293 0.586465 0.809974i \(-0.300519\pi\)
0.586465 + 0.809974i \(0.300519\pi\)
\(480\) 0 0
\(481\) −32.6192 −1.48731
\(482\) 0 0
\(483\) −13.8085 −0.628309
\(484\) 0 0
\(485\) −3.27563 −0.148739
\(486\) 0 0
\(487\) −19.3029 −0.874698 −0.437349 0.899292i \(-0.644083\pi\)
−0.437349 + 0.899292i \(0.644083\pi\)
\(488\) 0 0
\(489\) −42.0682 −1.90239
\(490\) 0 0
\(491\) 17.1628 0.774544 0.387272 0.921965i \(-0.373417\pi\)
0.387272 + 0.921965i \(0.373417\pi\)
\(492\) 0 0
\(493\) −22.7602 −1.02507
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 12.5338 0.562220
\(498\) 0 0
\(499\) −41.2377 −1.84605 −0.923027 0.384734i \(-0.874293\pi\)
−0.923027 + 0.384734i \(0.874293\pi\)
\(500\) 0 0
\(501\) −0.885084 −0.0395427
\(502\) 0 0
\(503\) −26.7780 −1.19397 −0.596986 0.802252i \(-0.703635\pi\)
−0.596986 + 0.802252i \(0.703635\pi\)
\(504\) 0 0
\(505\) 6.91048 0.307512
\(506\) 0 0
\(507\) −72.5000 −3.21984
\(508\) 0 0
\(509\) −25.0638 −1.11093 −0.555467 0.831539i \(-0.687460\pi\)
−0.555467 + 0.831539i \(0.687460\pi\)
\(510\) 0 0
\(511\) 10.6879 0.472804
\(512\) 0 0
\(513\) 45.0616 1.98952
\(514\) 0 0
\(515\) 7.58591 0.334275
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −28.2176 −1.23861
\(520\) 0 0
\(521\) 0.959798 0.0420495 0.0210248 0.999779i \(-0.493307\pi\)
0.0210248 + 0.999779i \(0.493307\pi\)
\(522\) 0 0
\(523\) 2.81911 0.123271 0.0616354 0.998099i \(-0.480368\pi\)
0.0616354 + 0.998099i \(0.480368\pi\)
\(524\) 0 0
\(525\) −6.34188 −0.276782
\(526\) 0 0
\(527\) −31.3229 −1.36445
\(528\) 0 0
\(529\) −18.2591 −0.793875
\(530\) 0 0
\(531\) 75.0071 3.25503
\(532\) 0 0
\(533\) 20.2466 0.876977
\(534\) 0 0
\(535\) −2.80495 −0.121268
\(536\) 0 0
\(537\) −27.3118 −1.17859
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 24.3158 1.04542 0.522709 0.852511i \(-0.324921\pi\)
0.522709 + 0.852511i \(0.324921\pi\)
\(542\) 0 0
\(543\) 38.2784 1.64268
\(544\) 0 0
\(545\) −8.02896 −0.343923
\(546\) 0 0
\(547\) 43.2096 1.84751 0.923754 0.382985i \(-0.125104\pi\)
0.923754 + 0.382985i \(0.125104\pi\)
\(548\) 0 0
\(549\) −39.9923 −1.70683
\(550\) 0 0
\(551\) −20.3069 −0.865103
\(552\) 0 0
\(553\) 17.7137 0.753265
\(554\) 0 0
\(555\) −15.3806 −0.652870
\(556\) 0 0
\(557\) −8.39276 −0.355613 −0.177806 0.984065i \(-0.556900\pi\)
−0.177806 + 0.984065i \(0.556900\pi\)
\(558\) 0 0
\(559\) −42.1330 −1.78203
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 6.15829 0.259541 0.129771 0.991544i \(-0.458576\pi\)
0.129771 + 0.991544i \(0.458576\pi\)
\(564\) 0 0
\(565\) 6.95944 0.292786
\(566\) 0 0
\(567\) 9.31275 0.391099
\(568\) 0 0
\(569\) −4.08468 −0.171239 −0.0856194 0.996328i \(-0.527287\pi\)
−0.0856194 + 0.996328i \(0.527287\pi\)
\(570\) 0 0
\(571\) −30.8820 −1.29237 −0.646185 0.763181i \(-0.723637\pi\)
−0.646185 + 0.763181i \(0.723637\pi\)
\(572\) 0 0
\(573\) 21.3736 0.892895
\(574\) 0 0
\(575\) 2.17735 0.0908019
\(576\) 0 0
\(577\) −14.4553 −0.601783 −0.300891 0.953658i \(-0.597284\pi\)
−0.300891 + 0.953658i \(0.597284\pi\)
\(578\) 0 0
\(579\) −53.6696 −2.23043
\(580\) 0 0
\(581\) −9.22102 −0.382552
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −33.5094 −1.38544
\(586\) 0 0
\(587\) 30.8993 1.27535 0.637676 0.770305i \(-0.279896\pi\)
0.637676 + 0.770305i \(0.279896\pi\)
\(588\) 0 0
\(589\) −27.9466 −1.15152
\(590\) 0 0
\(591\) 27.8360 1.14502
\(592\) 0 0
\(593\) 29.6191 1.21631 0.608156 0.793818i \(-0.291910\pi\)
0.608156 + 0.793818i \(0.291910\pi\)
\(594\) 0 0
\(595\) −15.5615 −0.637958
\(596\) 0 0
\(597\) −38.7198 −1.58469
\(598\) 0 0
\(599\) −6.78157 −0.277087 −0.138544 0.990356i \(-0.544242\pi\)
−0.138544 + 0.990356i \(0.544242\pi\)
\(600\) 0 0
\(601\) 17.7765 0.725120 0.362560 0.931961i \(-0.381903\pi\)
0.362560 + 0.931961i \(0.381903\pi\)
\(602\) 0 0
\(603\) −41.7256 −1.69920
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 38.8955 1.57872 0.789360 0.613931i \(-0.210413\pi\)
0.789360 + 0.613931i \(0.210413\pi\)
\(608\) 0 0
\(609\) −20.2496 −0.820554
\(610\) 0 0
\(611\) 26.5194 1.07286
\(612\) 0 0
\(613\) −30.7554 −1.24220 −0.621100 0.783731i \(-0.713314\pi\)
−0.621100 + 0.783731i \(0.713314\pi\)
\(614\) 0 0
\(615\) 9.54667 0.384959
\(616\) 0 0
\(617\) 23.3088 0.938377 0.469188 0.883098i \(-0.344547\pi\)
0.469188 + 0.883098i \(0.344547\pi\)
\(618\) 0 0
\(619\) 34.6072 1.39098 0.695490 0.718535i \(-0.255187\pi\)
0.695490 + 0.718535i \(0.255187\pi\)
\(620\) 0 0
\(621\) −15.4273 −0.619076
\(622\) 0 0
\(623\) 36.6840 1.46971
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −37.7404 −1.50481
\(630\) 0 0
\(631\) 11.1871 0.445352 0.222676 0.974893i \(-0.428521\pi\)
0.222676 + 0.974893i \(0.428521\pi\)
\(632\) 0 0
\(633\) 29.2944 1.16435
\(634\) 0 0
\(635\) 10.3872 0.412204
\(636\) 0 0
\(637\) 13.7642 0.545357
\(638\) 0 0
\(639\) 31.2272 1.23533
\(640\) 0 0
\(641\) 34.7930 1.37424 0.687121 0.726543i \(-0.258874\pi\)
0.687121 + 0.726543i \(0.258874\pi\)
\(642\) 0 0
\(643\) 24.4540 0.964370 0.482185 0.876069i \(-0.339843\pi\)
0.482185 + 0.876069i \(0.339843\pi\)
\(644\) 0 0
\(645\) −19.8665 −0.782244
\(646\) 0 0
\(647\) 18.7808 0.738349 0.369175 0.929360i \(-0.379640\pi\)
0.369175 + 0.929360i \(0.379640\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −27.8677 −1.09222
\(652\) 0 0
\(653\) 39.9031 1.56153 0.780765 0.624825i \(-0.214830\pi\)
0.780765 + 0.624825i \(0.214830\pi\)
\(654\) 0 0
\(655\) −3.48613 −0.136214
\(656\) 0 0
\(657\) 26.6281 1.03886
\(658\) 0 0
\(659\) −17.6187 −0.686326 −0.343163 0.939276i \(-0.611498\pi\)
−0.343163 + 0.939276i \(0.611498\pi\)
\(660\) 0 0
\(661\) 42.4404 1.65074 0.825371 0.564591i \(-0.190966\pi\)
0.825371 + 0.564591i \(0.190966\pi\)
\(662\) 0 0
\(663\) −127.576 −4.95466
\(664\) 0 0
\(665\) −13.8841 −0.538402
\(666\) 0 0
\(667\) 6.95227 0.269193
\(668\) 0 0
\(669\) −30.0666 −1.16244
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −6.08633 −0.234611 −0.117305 0.993096i \(-0.537426\pi\)
−0.117305 + 0.993096i \(0.537426\pi\)
\(674\) 0 0
\(675\) −7.08534 −0.272715
\(676\) 0 0
\(677\) 10.4432 0.401366 0.200683 0.979656i \(-0.435684\pi\)
0.200683 + 0.979656i \(0.435684\pi\)
\(678\) 0 0
\(679\) −7.15099 −0.274430
\(680\) 0 0
\(681\) 31.7672 1.21732
\(682\) 0 0
\(683\) 28.7312 1.09937 0.549685 0.835372i \(-0.314748\pi\)
0.549685 + 0.835372i \(0.314748\pi\)
\(684\) 0 0
\(685\) −14.5471 −0.555816
\(686\) 0 0
\(687\) −65.0271 −2.48094
\(688\) 0 0
\(689\) 65.4043 2.49171
\(690\) 0 0
\(691\) −28.0822 −1.06830 −0.534149 0.845390i \(-0.679368\pi\)
−0.534149 + 0.845390i \(0.679368\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0.501893 0.0190379
\(696\) 0 0
\(697\) 23.4253 0.887296
\(698\) 0 0
\(699\) −45.2183 −1.71031
\(700\) 0 0
\(701\) −12.0683 −0.455815 −0.227908 0.973683i \(-0.573188\pi\)
−0.227908 + 0.973683i \(0.573188\pi\)
\(702\) 0 0
\(703\) −33.6723 −1.26998
\(704\) 0 0
\(705\) 12.5044 0.470943
\(706\) 0 0
\(707\) 15.0862 0.567375
\(708\) 0 0
\(709\) 31.9154 1.19861 0.599304 0.800521i \(-0.295444\pi\)
0.599304 + 0.800521i \(0.295444\pi\)
\(710\) 0 0
\(711\) 44.1325 1.65510
\(712\) 0 0
\(713\) 9.56781 0.358317
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −70.9674 −2.65032
\(718\) 0 0
\(719\) −2.65067 −0.0988531 −0.0494266 0.998778i \(-0.515739\pi\)
−0.0494266 + 0.998778i \(0.515739\pi\)
\(720\) 0 0
\(721\) 16.5607 0.616754
\(722\) 0 0
\(723\) −16.7229 −0.621933
\(724\) 0 0
\(725\) 3.19299 0.118585
\(726\) 0 0
\(727\) −1.73626 −0.0643944 −0.0321972 0.999482i \(-0.510250\pi\)
−0.0321972 + 0.999482i \(0.510250\pi\)
\(728\) 0 0
\(729\) −38.5467 −1.42765
\(730\) 0 0
\(731\) −48.7478 −1.80300
\(732\) 0 0
\(733\) −33.3955 −1.23349 −0.616745 0.787163i \(-0.711549\pi\)
−0.616745 + 0.787163i \(0.711549\pi\)
\(734\) 0 0
\(735\) 6.49009 0.239391
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 41.0369 1.50957 0.754784 0.655973i \(-0.227742\pi\)
0.754784 + 0.655973i \(0.227742\pi\)
\(740\) 0 0
\(741\) −113.825 −4.18147
\(742\) 0 0
\(743\) −41.8776 −1.53634 −0.768170 0.640246i \(-0.778833\pi\)
−0.768170 + 0.640246i \(0.778833\pi\)
\(744\) 0 0
\(745\) −0.747620 −0.0273907
\(746\) 0 0
\(747\) −22.9735 −0.840557
\(748\) 0 0
\(749\) −6.12345 −0.223746
\(750\) 0 0
\(751\) −18.6398 −0.680177 −0.340088 0.940394i \(-0.610457\pi\)
−0.340088 + 0.940394i \(0.610457\pi\)
\(752\) 0 0
\(753\) −34.9298 −1.27291
\(754\) 0 0
\(755\) 3.27337 0.119130
\(756\) 0 0
\(757\) −14.2930 −0.519488 −0.259744 0.965678i \(-0.583638\pi\)
−0.259744 + 0.965678i \(0.583638\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 15.7696 0.571649 0.285825 0.958282i \(-0.407733\pi\)
0.285825 + 0.958282i \(0.407733\pi\)
\(762\) 0 0
\(763\) −17.5279 −0.634554
\(764\) 0 0
\(765\) −38.7703 −1.40174
\(766\) 0 0
\(767\) −84.9626 −3.06782
\(768\) 0 0
\(769\) 16.5268 0.595971 0.297986 0.954570i \(-0.403685\pi\)
0.297986 + 0.954570i \(0.403685\pi\)
\(770\) 0 0
\(771\) 56.7532 2.04392
\(772\) 0 0
\(773\) 39.0636 1.40502 0.702511 0.711673i \(-0.252062\pi\)
0.702511 + 0.711673i \(0.252062\pi\)
\(774\) 0 0
\(775\) 4.39424 0.157846
\(776\) 0 0
\(777\) −33.5772 −1.20458
\(778\) 0 0
\(779\) 20.9003 0.748830
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −22.6234 −0.808496
\(784\) 0 0
\(785\) −15.0296 −0.536430
\(786\) 0 0
\(787\) −4.93078 −0.175763 −0.0878816 0.996131i \(-0.528010\pi\)
−0.0878816 + 0.996131i \(0.528010\pi\)
\(788\) 0 0
\(789\) −71.5247 −2.54635
\(790\) 0 0
\(791\) 15.1931 0.540204
\(792\) 0 0
\(793\) 45.3004 1.60866
\(794\) 0 0
\(795\) 30.8395 1.09376
\(796\) 0 0
\(797\) −50.4039 −1.78540 −0.892700 0.450652i \(-0.851191\pi\)
−0.892700 + 0.450652i \(0.851191\pi\)
\(798\) 0 0
\(799\) 30.6829 1.08548
\(800\) 0 0
\(801\) 91.3955 3.22930
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 4.75336 0.167534
\(806\) 0 0
\(807\) −4.48808 −0.157988
\(808\) 0 0
\(809\) 54.5875 1.91920 0.959598 0.281375i \(-0.0907905\pi\)
0.959598 + 0.281375i \(0.0907905\pi\)
\(810\) 0 0
\(811\) −13.6473 −0.479223 −0.239611 0.970869i \(-0.577020\pi\)
−0.239611 + 0.970869i \(0.577020\pi\)
\(812\) 0 0
\(813\) 65.5505 2.29896
\(814\) 0 0
\(815\) 14.4813 0.507258
\(816\) 0 0
\(817\) −43.4932 −1.52164
\(818\) 0 0
\(819\) −73.1540 −2.55621
\(820\) 0 0
\(821\) 37.3692 1.30419 0.652097 0.758136i \(-0.273889\pi\)
0.652097 + 0.758136i \(0.273889\pi\)
\(822\) 0 0
\(823\) −27.5161 −0.959150 −0.479575 0.877501i \(-0.659209\pi\)
−0.479575 + 0.877501i \(0.659209\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 38.9552 1.35460 0.677302 0.735705i \(-0.263149\pi\)
0.677302 + 0.735705i \(0.263149\pi\)
\(828\) 0 0
\(829\) 8.01799 0.278476 0.139238 0.990259i \(-0.455535\pi\)
0.139238 + 0.990259i \(0.455535\pi\)
\(830\) 0 0
\(831\) −40.1439 −1.39258
\(832\) 0 0
\(833\) 15.9252 0.551774
\(834\) 0 0
\(835\) 0.304676 0.0105438
\(836\) 0 0
\(837\) −31.1347 −1.07617
\(838\) 0 0
\(839\) −19.8626 −0.685733 −0.342866 0.939384i \(-0.611398\pi\)
−0.342866 + 0.939384i \(0.611398\pi\)
\(840\) 0 0
\(841\) −18.8048 −0.648442
\(842\) 0 0
\(843\) −25.2944 −0.871185
\(844\) 0 0
\(845\) 24.9570 0.858546
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 4.42918 0.152009
\(850\) 0 0
\(851\) 11.5281 0.395177
\(852\) 0 0
\(853\) −24.5518 −0.840637 −0.420318 0.907377i \(-0.638082\pi\)
−0.420318 + 0.907377i \(0.638082\pi\)
\(854\) 0 0
\(855\) −34.5912 −1.18300
\(856\) 0 0
\(857\) −9.17203 −0.313311 −0.156655 0.987653i \(-0.550071\pi\)
−0.156655 + 0.987653i \(0.550071\pi\)
\(858\) 0 0
\(859\) 2.83353 0.0966786 0.0483393 0.998831i \(-0.484607\pi\)
0.0483393 + 0.998831i \(0.484607\pi\)
\(860\) 0 0
\(861\) 20.8413 0.710268
\(862\) 0 0
\(863\) 40.7452 1.38698 0.693491 0.720466i \(-0.256072\pi\)
0.693491 + 0.720466i \(0.256072\pi\)
\(864\) 0 0
\(865\) 9.71345 0.330267
\(866\) 0 0
\(867\) −98.2209 −3.33576
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 47.2638 1.60147
\(872\) 0 0
\(873\) −17.8162 −0.602987
\(874\) 0 0
\(875\) 2.18309 0.0738020
\(876\) 0 0
\(877\) 2.84489 0.0960651 0.0480325 0.998846i \(-0.484705\pi\)
0.0480325 + 0.998846i \(0.484705\pi\)
\(878\) 0 0
\(879\) 68.9562 2.32584
\(880\) 0 0
\(881\) −25.7884 −0.868834 −0.434417 0.900712i \(-0.643046\pi\)
−0.434417 + 0.900712i \(0.643046\pi\)
\(882\) 0 0
\(883\) 8.49346 0.285828 0.142914 0.989735i \(-0.454353\pi\)
0.142914 + 0.989735i \(0.454353\pi\)
\(884\) 0 0
\(885\) −40.0615 −1.34665
\(886\) 0 0
\(887\) −25.6413 −0.860951 −0.430475 0.902602i \(-0.641654\pi\)
−0.430475 + 0.902602i \(0.641654\pi\)
\(888\) 0 0
\(889\) 22.6762 0.760537
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 27.3756 0.916088
\(894\) 0 0
\(895\) 9.40166 0.314263
\(896\) 0 0
\(897\) 38.9691 1.30114
\(898\) 0 0
\(899\) 14.0308 0.467952
\(900\) 0 0
\(901\) 75.6728 2.52102
\(902\) 0 0
\(903\) −43.3704 −1.44328
\(904\) 0 0
\(905\) −13.1767 −0.438010
\(906\) 0 0
\(907\) −23.6343 −0.784763 −0.392382 0.919802i \(-0.628349\pi\)
−0.392382 + 0.919802i \(0.628349\pi\)
\(908\) 0 0
\(909\) 37.5862 1.24666
\(910\) 0 0
\(911\) −47.3905 −1.57012 −0.785058 0.619422i \(-0.787367\pi\)
−0.785058 + 0.619422i \(0.787367\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 21.3600 0.706141
\(916\) 0 0
\(917\) −7.61055 −0.251322
\(918\) 0 0
\(919\) 39.9139 1.31664 0.658319 0.752739i \(-0.271268\pi\)
0.658319 + 0.752739i \(0.271268\pi\)
\(920\) 0 0
\(921\) −51.4553 −1.69551
\(922\) 0 0
\(923\) −35.3719 −1.16428
\(924\) 0 0
\(925\) 5.29453 0.174083
\(926\) 0 0
\(927\) 41.2599 1.35515
\(928\) 0 0
\(929\) −10.7852 −0.353851 −0.176926 0.984224i \(-0.556615\pi\)
−0.176926 + 0.984224i \(0.556615\pi\)
\(930\) 0 0
\(931\) 14.2086 0.465667
\(932\) 0 0
\(933\) −19.7102 −0.645284
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 37.8842 1.23762 0.618812 0.785539i \(-0.287614\pi\)
0.618812 + 0.785539i \(0.287614\pi\)
\(938\) 0 0
\(939\) −69.1552 −2.25679
\(940\) 0 0
\(941\) −1.18812 −0.0387317 −0.0193659 0.999812i \(-0.506165\pi\)
−0.0193659 + 0.999812i \(0.506165\pi\)
\(942\) 0 0
\(943\) −7.15542 −0.233012
\(944\) 0 0
\(945\) −15.4679 −0.503172
\(946\) 0 0
\(947\) 4.75475 0.154508 0.0772542 0.997011i \(-0.475385\pi\)
0.0772542 + 0.997011i \(0.475385\pi\)
\(948\) 0 0
\(949\) −30.1623 −0.979111
\(950\) 0 0
\(951\) 5.36836 0.174081
\(952\) 0 0
\(953\) −1.92362 −0.0623120 −0.0311560 0.999515i \(-0.509919\pi\)
−0.0311560 + 0.999515i \(0.509919\pi\)
\(954\) 0 0
\(955\) −7.35752 −0.238084
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −31.7577 −1.02551
\(960\) 0 0
\(961\) −11.6907 −0.377118
\(962\) 0 0
\(963\) −15.2561 −0.491622
\(964\) 0 0
\(965\) 18.4749 0.594729
\(966\) 0 0
\(967\) 14.4688 0.465284 0.232642 0.972562i \(-0.425263\pi\)
0.232642 + 0.972562i \(0.425263\pi\)
\(968\) 0 0
\(969\) −131.695 −4.23067
\(970\) 0 0
\(971\) −13.7317 −0.440673 −0.220336 0.975424i \(-0.570715\pi\)
−0.220336 + 0.975424i \(0.570715\pi\)
\(972\) 0 0
\(973\) 1.09568 0.0351258
\(974\) 0 0
\(975\) 17.8975 0.573178
\(976\) 0 0
\(977\) −43.0472 −1.37720 −0.688602 0.725140i \(-0.741775\pi\)
−0.688602 + 0.725140i \(0.741775\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −43.6696 −1.39426
\(982\) 0 0
\(983\) 20.8242 0.664190 0.332095 0.943246i \(-0.392245\pi\)
0.332095 + 0.943246i \(0.392245\pi\)
\(984\) 0 0
\(985\) −9.58209 −0.305311
\(986\) 0 0
\(987\) 27.2983 0.868914
\(988\) 0 0
\(989\) 14.8904 0.473486
\(990\) 0 0
\(991\) 42.9441 1.36417 0.682083 0.731275i \(-0.261074\pi\)
0.682083 + 0.731275i \(0.261074\pi\)
\(992\) 0 0
\(993\) −51.0276 −1.61931
\(994\) 0 0
\(995\) 13.3287 0.422547
\(996\) 0 0
\(997\) 29.8397 0.945032 0.472516 0.881322i \(-0.343346\pi\)
0.472516 + 0.881322i \(0.343346\pi\)
\(998\) 0 0
\(999\) −37.5135 −1.18688
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 4840.2.a.bh.1.1 8
4.3 odd 2 9680.2.a.de.1.8 8
11.7 odd 10 440.2.y.d.401.4 yes 16
11.8 odd 10 440.2.y.d.361.4 16
11.10 odd 2 4840.2.a.bg.1.1 8
44.7 even 10 880.2.bo.k.401.1 16
44.19 even 10 880.2.bo.k.801.1 16
44.43 even 2 9680.2.a.df.1.8 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
440.2.y.d.361.4 16 11.8 odd 10
440.2.y.d.401.4 yes 16 11.7 odd 10
880.2.bo.k.401.1 16 44.7 even 10
880.2.bo.k.801.1 16 44.19 even 10
4840.2.a.bg.1.1 8 11.10 odd 2
4840.2.a.bh.1.1 8 1.1 even 1 trivial
9680.2.a.de.1.8 8 4.3 odd 2
9680.2.a.df.1.8 8 44.43 even 2