Properties

Label 8896.2.a.bd.1.3
Level $8896$
Weight $2$
Character 8896.1
Self dual yes
Analytic conductor $71.035$
Analytic rank $1$
Dimension $7$
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] = [8896,2,Mod(1,8896)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8896, 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("8896.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8896 = 2^{6} \cdot 139 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8896.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(71.0349176381\)
Analytic rank: \(1\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - x^{6} - 11x^{5} + 8x^{4} + 35x^{3} - 10x^{2} - 32x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 139)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-0.308806\) of defining polynomial
Character \(\chi\) \(=\) 8896.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.39811 q^{3} -2.83261 q^{5} -4.16776 q^{7} -1.04529 q^{9} +O(q^{10})\) \(q-1.39811 q^{3} -2.83261 q^{5} -4.16776 q^{7} -1.04529 q^{9} +1.93164 q^{11} -4.12937 q^{13} +3.96031 q^{15} +1.03363 q^{17} -7.34157 q^{19} +5.82699 q^{21} +4.09697 q^{23} +3.02371 q^{25} +5.65576 q^{27} -4.49067 q^{29} +1.28386 q^{31} -2.70065 q^{33} +11.8056 q^{35} +7.77849 q^{37} +5.77331 q^{39} +11.5603 q^{41} -7.87867 q^{43} +2.96089 q^{45} +6.84669 q^{47} +10.3702 q^{49} -1.44513 q^{51} -3.68473 q^{53} -5.47160 q^{55} +10.2643 q^{57} +1.24534 q^{59} -0.281096 q^{61} +4.35649 q^{63} +11.6969 q^{65} -8.51902 q^{67} -5.72803 q^{69} +6.40133 q^{71} -5.92608 q^{73} -4.22748 q^{75} -8.05062 q^{77} +8.65375 q^{79} -4.77152 q^{81} +15.8366 q^{83} -2.92789 q^{85} +6.27846 q^{87} -3.23844 q^{89} +17.2102 q^{91} -1.79497 q^{93} +20.7958 q^{95} -7.27325 q^{97} -2.01912 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q - 2 q^{3} - 11 q^{5} + 5 q^{7} + 13 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q - 2 q^{3} - 11 q^{5} + 5 q^{7} + 13 q^{9} + 2 q^{11} - 6 q^{13} + 3 q^{15} + 5 q^{17} - 10 q^{19} + 5 q^{21} + q^{23} + 14 q^{25} - 11 q^{27} - 30 q^{29} + 20 q^{31} - 20 q^{33} - 7 q^{35} - 6 q^{37} - 11 q^{39} + 19 q^{41} - 12 q^{43} - 27 q^{45} + 3 q^{47} - 8 q^{49} + 23 q^{51} - 38 q^{53} - 7 q^{55} - 19 q^{57} - 14 q^{59} - 4 q^{61} + 18 q^{63} + 10 q^{65} + 9 q^{67} - 9 q^{69} - 24 q^{71} - 5 q^{73} - 21 q^{75} + 13 q^{77} - 8 q^{79} + 39 q^{81} - 9 q^{83} + 22 q^{85} + 25 q^{87} + 10 q^{89} + 7 q^{91} + 15 q^{93} + 21 q^{95} - 5 q^{97} - 31 q^{99}+O(q^{100}) \) 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
\(3\) −1.39811 −0.807200 −0.403600 0.914936i \(-0.632241\pi\)
−0.403600 + 0.914936i \(0.632241\pi\)
\(4\) 0 0
\(5\) −2.83261 −1.26678 −0.633392 0.773831i \(-0.718338\pi\)
−0.633392 + 0.773831i \(0.718338\pi\)
\(6\) 0 0
\(7\) −4.16776 −1.57526 −0.787632 0.616146i \(-0.788693\pi\)
−0.787632 + 0.616146i \(0.788693\pi\)
\(8\) 0 0
\(9\) −1.04529 −0.348428
\(10\) 0 0
\(11\) 1.93164 0.582412 0.291206 0.956660i \(-0.405943\pi\)
0.291206 + 0.956660i \(0.405943\pi\)
\(12\) 0 0
\(13\) −4.12937 −1.14528 −0.572640 0.819807i \(-0.694081\pi\)
−0.572640 + 0.819807i \(0.694081\pi\)
\(14\) 0 0
\(15\) 3.96031 1.02255
\(16\) 0 0
\(17\) 1.03363 0.250693 0.125347 0.992113i \(-0.459996\pi\)
0.125347 + 0.992113i \(0.459996\pi\)
\(18\) 0 0
\(19\) −7.34157 −1.68427 −0.842135 0.539266i \(-0.818702\pi\)
−0.842135 + 0.539266i \(0.818702\pi\)
\(20\) 0 0
\(21\) 5.82699 1.27155
\(22\) 0 0
\(23\) 4.09697 0.854278 0.427139 0.904186i \(-0.359521\pi\)
0.427139 + 0.904186i \(0.359521\pi\)
\(24\) 0 0
\(25\) 3.02371 0.604741
\(26\) 0 0
\(27\) 5.65576 1.08845
\(28\) 0 0
\(29\) −4.49067 −0.833897 −0.416948 0.908930i \(-0.636900\pi\)
−0.416948 + 0.908930i \(0.636900\pi\)
\(30\) 0 0
\(31\) 1.28386 0.230587 0.115294 0.993331i \(-0.463219\pi\)
0.115294 + 0.993331i \(0.463219\pi\)
\(32\) 0 0
\(33\) −2.70065 −0.470123
\(34\) 0 0
\(35\) 11.8056 1.99552
\(36\) 0 0
\(37\) 7.77849 1.27878 0.639388 0.768884i \(-0.279188\pi\)
0.639388 + 0.768884i \(0.279188\pi\)
\(38\) 0 0
\(39\) 5.77331 0.924470
\(40\) 0 0
\(41\) 11.5603 1.80541 0.902706 0.430258i \(-0.141577\pi\)
0.902706 + 0.430258i \(0.141577\pi\)
\(42\) 0 0
\(43\) −7.87867 −1.20149 −0.600743 0.799442i \(-0.705129\pi\)
−0.600743 + 0.799442i \(0.705129\pi\)
\(44\) 0 0
\(45\) 2.96089 0.441383
\(46\) 0 0
\(47\) 6.84669 0.998692 0.499346 0.866403i \(-0.333574\pi\)
0.499346 + 0.866403i \(0.333574\pi\)
\(48\) 0 0
\(49\) 10.3702 1.48145
\(50\) 0 0
\(51\) −1.44513 −0.202359
\(52\) 0 0
\(53\) −3.68473 −0.506136 −0.253068 0.967449i \(-0.581440\pi\)
−0.253068 + 0.967449i \(0.581440\pi\)
\(54\) 0 0
\(55\) −5.47160 −0.737791
\(56\) 0 0
\(57\) 10.2643 1.35954
\(58\) 0 0
\(59\) 1.24534 0.162129 0.0810647 0.996709i \(-0.474168\pi\)
0.0810647 + 0.996709i \(0.474168\pi\)
\(60\) 0 0
\(61\) −0.281096 −0.0359907 −0.0179953 0.999838i \(-0.505728\pi\)
−0.0179953 + 0.999838i \(0.505728\pi\)
\(62\) 0 0
\(63\) 4.35649 0.548867
\(64\) 0 0
\(65\) 11.6969 1.45082
\(66\) 0 0
\(67\) −8.51902 −1.04076 −0.520382 0.853934i \(-0.674210\pi\)
−0.520382 + 0.853934i \(0.674210\pi\)
\(68\) 0 0
\(69\) −5.72803 −0.689573
\(70\) 0 0
\(71\) 6.40133 0.759698 0.379849 0.925048i \(-0.375976\pi\)
0.379849 + 0.925048i \(0.375976\pi\)
\(72\) 0 0
\(73\) −5.92608 −0.693595 −0.346798 0.937940i \(-0.612731\pi\)
−0.346798 + 0.937940i \(0.612731\pi\)
\(74\) 0 0
\(75\) −4.22748 −0.488147
\(76\) 0 0
\(77\) −8.05062 −0.917453
\(78\) 0 0
\(79\) 8.65375 0.973623 0.486811 0.873507i \(-0.338160\pi\)
0.486811 + 0.873507i \(0.338160\pi\)
\(80\) 0 0
\(81\) −4.77152 −0.530169
\(82\) 0 0
\(83\) 15.8366 1.73829 0.869146 0.494555i \(-0.164669\pi\)
0.869146 + 0.494555i \(0.164669\pi\)
\(84\) 0 0
\(85\) −2.92789 −0.317574
\(86\) 0 0
\(87\) 6.27846 0.673121
\(88\) 0 0
\(89\) −3.23844 −0.343274 −0.171637 0.985160i \(-0.554906\pi\)
−0.171637 + 0.985160i \(0.554906\pi\)
\(90\) 0 0
\(91\) 17.2102 1.80412
\(92\) 0 0
\(93\) −1.79497 −0.186130
\(94\) 0 0
\(95\) 20.7958 2.13361
\(96\) 0 0
\(97\) −7.27325 −0.738487 −0.369243 0.929333i \(-0.620383\pi\)
−0.369243 + 0.929333i \(0.620383\pi\)
\(98\) 0 0
\(99\) −2.01912 −0.202929
\(100\) 0 0
\(101\) 13.0456 1.29809 0.649044 0.760751i \(-0.275169\pi\)
0.649044 + 0.760751i \(0.275169\pi\)
\(102\) 0 0
\(103\) 11.9048 1.17302 0.586510 0.809942i \(-0.300502\pi\)
0.586510 + 0.809942i \(0.300502\pi\)
\(104\) 0 0
\(105\) −16.5056 −1.61078
\(106\) 0 0
\(107\) 16.1058 1.55700 0.778502 0.627642i \(-0.215980\pi\)
0.778502 + 0.627642i \(0.215980\pi\)
\(108\) 0 0
\(109\) −13.1464 −1.25919 −0.629597 0.776922i \(-0.716780\pi\)
−0.629597 + 0.776922i \(0.716780\pi\)
\(110\) 0 0
\(111\) −10.8752 −1.03223
\(112\) 0 0
\(113\) −14.7595 −1.38846 −0.694228 0.719755i \(-0.744254\pi\)
−0.694228 + 0.719755i \(0.744254\pi\)
\(114\) 0 0
\(115\) −11.6052 −1.08219
\(116\) 0 0
\(117\) 4.31636 0.399048
\(118\) 0 0
\(119\) −4.30793 −0.394908
\(120\) 0 0
\(121\) −7.26876 −0.660796
\(122\) 0 0
\(123\) −16.1626 −1.45733
\(124\) 0 0
\(125\) 5.59808 0.500707
\(126\) 0 0
\(127\) −11.0142 −0.977350 −0.488675 0.872466i \(-0.662520\pi\)
−0.488675 + 0.872466i \(0.662520\pi\)
\(128\) 0 0
\(129\) 11.0153 0.969840
\(130\) 0 0
\(131\) −5.25402 −0.459046 −0.229523 0.973303i \(-0.573717\pi\)
−0.229523 + 0.973303i \(0.573717\pi\)
\(132\) 0 0
\(133\) 30.5979 2.65317
\(134\) 0 0
\(135\) −16.0206 −1.37883
\(136\) 0 0
\(137\) −1.68115 −0.143630 −0.0718151 0.997418i \(-0.522879\pi\)
−0.0718151 + 0.997418i \(0.522879\pi\)
\(138\) 0 0
\(139\) 1.00000 0.0848189
\(140\) 0 0
\(141\) −9.57243 −0.806144
\(142\) 0 0
\(143\) −7.97646 −0.667025
\(144\) 0 0
\(145\) 12.7203 1.05637
\(146\) 0 0
\(147\) −14.4987 −1.19583
\(148\) 0 0
\(149\) −13.9257 −1.14084 −0.570421 0.821353i \(-0.693220\pi\)
−0.570421 + 0.821353i \(0.693220\pi\)
\(150\) 0 0
\(151\) −8.63888 −0.703022 −0.351511 0.936184i \(-0.614332\pi\)
−0.351511 + 0.936184i \(0.614332\pi\)
\(152\) 0 0
\(153\) −1.08044 −0.0873486
\(154\) 0 0
\(155\) −3.63667 −0.292104
\(156\) 0 0
\(157\) −10.4362 −0.832900 −0.416450 0.909159i \(-0.636726\pi\)
−0.416450 + 0.909159i \(0.636726\pi\)
\(158\) 0 0
\(159\) 5.15166 0.408553
\(160\) 0 0
\(161\) −17.0752 −1.34571
\(162\) 0 0
\(163\) −3.73406 −0.292474 −0.146237 0.989250i \(-0.546716\pi\)
−0.146237 + 0.989250i \(0.546716\pi\)
\(164\) 0 0
\(165\) 7.64991 0.595544
\(166\) 0 0
\(167\) 19.7503 1.52832 0.764161 0.645025i \(-0.223153\pi\)
0.764161 + 0.645025i \(0.223153\pi\)
\(168\) 0 0
\(169\) 4.05166 0.311666
\(170\) 0 0
\(171\) 7.67403 0.586848
\(172\) 0 0
\(173\) 7.51428 0.571300 0.285650 0.958334i \(-0.407790\pi\)
0.285650 + 0.958334i \(0.407790\pi\)
\(174\) 0 0
\(175\) −12.6021 −0.952627
\(176\) 0 0
\(177\) −1.74112 −0.130871
\(178\) 0 0
\(179\) 24.2625 1.81346 0.906732 0.421708i \(-0.138569\pi\)
0.906732 + 0.421708i \(0.138569\pi\)
\(180\) 0 0
\(181\) 3.06020 0.227463 0.113732 0.993512i \(-0.463720\pi\)
0.113732 + 0.993512i \(0.463720\pi\)
\(182\) 0 0
\(183\) 0.393004 0.0290517
\(184\) 0 0
\(185\) −22.0335 −1.61993
\(186\) 0 0
\(187\) 1.99661 0.146007
\(188\) 0 0
\(189\) −23.5718 −1.71460
\(190\) 0 0
\(191\) −8.83544 −0.639310 −0.319655 0.947534i \(-0.603567\pi\)
−0.319655 + 0.947534i \(0.603567\pi\)
\(192\) 0 0
\(193\) 14.9309 1.07475 0.537374 0.843344i \(-0.319416\pi\)
0.537374 + 0.843344i \(0.319416\pi\)
\(194\) 0 0
\(195\) −16.3536 −1.17110
\(196\) 0 0
\(197\) 11.8981 0.847702 0.423851 0.905732i \(-0.360678\pi\)
0.423851 + 0.905732i \(0.360678\pi\)
\(198\) 0 0
\(199\) 4.40723 0.312420 0.156210 0.987724i \(-0.450072\pi\)
0.156210 + 0.987724i \(0.450072\pi\)
\(200\) 0 0
\(201\) 11.9105 0.840104
\(202\) 0 0
\(203\) 18.7160 1.31361
\(204\) 0 0
\(205\) −32.7458 −2.28707
\(206\) 0 0
\(207\) −4.28251 −0.297655
\(208\) 0 0
\(209\) −14.1813 −0.980940
\(210\) 0 0
\(211\) 21.4218 1.47474 0.737368 0.675492i \(-0.236069\pi\)
0.737368 + 0.675492i \(0.236069\pi\)
\(212\) 0 0
\(213\) −8.94977 −0.613228
\(214\) 0 0
\(215\) 22.3173 1.52202
\(216\) 0 0
\(217\) −5.35079 −0.363236
\(218\) 0 0
\(219\) 8.28532 0.559870
\(220\) 0 0
\(221\) −4.26825 −0.287114
\(222\) 0 0
\(223\) 7.93581 0.531421 0.265711 0.964053i \(-0.414393\pi\)
0.265711 + 0.964053i \(0.414393\pi\)
\(224\) 0 0
\(225\) −3.16064 −0.210709
\(226\) 0 0
\(227\) 15.9354 1.05767 0.528836 0.848724i \(-0.322629\pi\)
0.528836 + 0.848724i \(0.322629\pi\)
\(228\) 0 0
\(229\) −13.4712 −0.890203 −0.445102 0.895480i \(-0.646832\pi\)
−0.445102 + 0.895480i \(0.646832\pi\)
\(230\) 0 0
\(231\) 11.2557 0.740568
\(232\) 0 0
\(233\) 18.4315 1.20749 0.603745 0.797177i \(-0.293674\pi\)
0.603745 + 0.797177i \(0.293674\pi\)
\(234\) 0 0
\(235\) −19.3940 −1.26513
\(236\) 0 0
\(237\) −12.0989 −0.785908
\(238\) 0 0
\(239\) −0.889441 −0.0575331 −0.0287666 0.999586i \(-0.509158\pi\)
−0.0287666 + 0.999586i \(0.509158\pi\)
\(240\) 0 0
\(241\) −11.7554 −0.757233 −0.378616 0.925554i \(-0.623600\pi\)
−0.378616 + 0.925554i \(0.623600\pi\)
\(242\) 0 0
\(243\) −10.2962 −0.660499
\(244\) 0 0
\(245\) −29.3747 −1.87668
\(246\) 0 0
\(247\) 30.3160 1.92896
\(248\) 0 0
\(249\) −22.1413 −1.40315
\(250\) 0 0
\(251\) −14.4551 −0.912397 −0.456199 0.889878i \(-0.650789\pi\)
−0.456199 + 0.889878i \(0.650789\pi\)
\(252\) 0 0
\(253\) 7.91389 0.497542
\(254\) 0 0
\(255\) 4.09351 0.256346
\(256\) 0 0
\(257\) 17.2989 1.07907 0.539537 0.841962i \(-0.318599\pi\)
0.539537 + 0.841962i \(0.318599\pi\)
\(258\) 0 0
\(259\) −32.4189 −2.01441
\(260\) 0 0
\(261\) 4.69403 0.290553
\(262\) 0 0
\(263\) −4.00876 −0.247191 −0.123595 0.992333i \(-0.539442\pi\)
−0.123595 + 0.992333i \(0.539442\pi\)
\(264\) 0 0
\(265\) 10.4374 0.641165
\(266\) 0 0
\(267\) 4.52770 0.277091
\(268\) 0 0
\(269\) −28.0301 −1.70902 −0.854511 0.519433i \(-0.826143\pi\)
−0.854511 + 0.519433i \(0.826143\pi\)
\(270\) 0 0
\(271\) −16.3058 −0.990507 −0.495253 0.868749i \(-0.664925\pi\)
−0.495253 + 0.868749i \(0.664925\pi\)
\(272\) 0 0
\(273\) −24.0617 −1.45628
\(274\) 0 0
\(275\) 5.84072 0.352209
\(276\) 0 0
\(277\) 26.3241 1.58166 0.790831 0.612035i \(-0.209649\pi\)
0.790831 + 0.612035i \(0.209649\pi\)
\(278\) 0 0
\(279\) −1.34199 −0.0803431
\(280\) 0 0
\(281\) −7.58167 −0.452285 −0.226142 0.974094i \(-0.572611\pi\)
−0.226142 + 0.974094i \(0.572611\pi\)
\(282\) 0 0
\(283\) −14.2221 −0.845417 −0.422709 0.906266i \(-0.638921\pi\)
−0.422709 + 0.906266i \(0.638921\pi\)
\(284\) 0 0
\(285\) −29.0749 −1.72225
\(286\) 0 0
\(287\) −48.1804 −2.84400
\(288\) 0 0
\(289\) −15.9316 −0.937153
\(290\) 0 0
\(291\) 10.1688 0.596107
\(292\) 0 0
\(293\) 4.68983 0.273983 0.136991 0.990572i \(-0.456257\pi\)
0.136991 + 0.990572i \(0.456257\pi\)
\(294\) 0 0
\(295\) −3.52757 −0.205383
\(296\) 0 0
\(297\) 10.9249 0.633927
\(298\) 0 0
\(299\) −16.9179 −0.978388
\(300\) 0 0
\(301\) 32.8364 1.89266
\(302\) 0 0
\(303\) −18.2392 −1.04782
\(304\) 0 0
\(305\) 0.796238 0.0455924
\(306\) 0 0
\(307\) 10.6916 0.610199 0.305100 0.952320i \(-0.401310\pi\)
0.305100 + 0.952320i \(0.401310\pi\)
\(308\) 0 0
\(309\) −16.6443 −0.946861
\(310\) 0 0
\(311\) 21.0455 1.19338 0.596690 0.802472i \(-0.296482\pi\)
0.596690 + 0.802472i \(0.296482\pi\)
\(312\) 0 0
\(313\) −14.2033 −0.802819 −0.401409 0.915899i \(-0.631480\pi\)
−0.401409 + 0.915899i \(0.631480\pi\)
\(314\) 0 0
\(315\) −12.3403 −0.695295
\(316\) 0 0
\(317\) −2.85738 −0.160487 −0.0802433 0.996775i \(-0.525570\pi\)
−0.0802433 + 0.996775i \(0.525570\pi\)
\(318\) 0 0
\(319\) −8.67437 −0.485672
\(320\) 0 0
\(321\) −22.5177 −1.25681
\(322\) 0 0
\(323\) −7.58849 −0.422235
\(324\) 0 0
\(325\) −12.4860 −0.692598
\(326\) 0 0
\(327\) 18.3801 1.01642
\(328\) 0 0
\(329\) −28.5353 −1.57320
\(330\) 0 0
\(331\) 12.1172 0.666019 0.333010 0.942923i \(-0.391936\pi\)
0.333010 + 0.942923i \(0.391936\pi\)
\(332\) 0 0
\(333\) −8.13074 −0.445562
\(334\) 0 0
\(335\) 24.1311 1.31842
\(336\) 0 0
\(337\) −12.4929 −0.680532 −0.340266 0.940329i \(-0.610517\pi\)
−0.340266 + 0.940329i \(0.610517\pi\)
\(338\) 0 0
\(339\) 20.6354 1.12076
\(340\) 0 0
\(341\) 2.47995 0.134297
\(342\) 0 0
\(343\) −14.0461 −0.758418
\(344\) 0 0
\(345\) 16.2253 0.873540
\(346\) 0 0
\(347\) −11.1030 −0.596039 −0.298020 0.954560i \(-0.596326\pi\)
−0.298020 + 0.954560i \(0.596326\pi\)
\(348\) 0 0
\(349\) −16.3372 −0.874509 −0.437254 0.899338i \(-0.644049\pi\)
−0.437254 + 0.899338i \(0.644049\pi\)
\(350\) 0 0
\(351\) −23.3547 −1.24658
\(352\) 0 0
\(353\) −26.2175 −1.39542 −0.697709 0.716382i \(-0.745797\pi\)
−0.697709 + 0.716382i \(0.745797\pi\)
\(354\) 0 0
\(355\) −18.1325 −0.962373
\(356\) 0 0
\(357\) 6.02297 0.318769
\(358\) 0 0
\(359\) −18.6480 −0.984204 −0.492102 0.870537i \(-0.663771\pi\)
−0.492102 + 0.870537i \(0.663771\pi\)
\(360\) 0 0
\(361\) 34.8986 1.83677
\(362\) 0 0
\(363\) 10.1625 0.533394
\(364\) 0 0
\(365\) 16.7863 0.878636
\(366\) 0 0
\(367\) 3.92562 0.204916 0.102458 0.994737i \(-0.467329\pi\)
0.102458 + 0.994737i \(0.467329\pi\)
\(368\) 0 0
\(369\) −12.0838 −0.629057
\(370\) 0 0
\(371\) 15.3570 0.797297
\(372\) 0 0
\(373\) −14.3279 −0.741870 −0.370935 0.928659i \(-0.620963\pi\)
−0.370935 + 0.928659i \(0.620963\pi\)
\(374\) 0 0
\(375\) −7.82673 −0.404171
\(376\) 0 0
\(377\) 18.5436 0.955045
\(378\) 0 0
\(379\) 5.65835 0.290650 0.145325 0.989384i \(-0.453577\pi\)
0.145325 + 0.989384i \(0.453577\pi\)
\(380\) 0 0
\(381\) 15.3990 0.788917
\(382\) 0 0
\(383\) −4.23323 −0.216308 −0.108154 0.994134i \(-0.534494\pi\)
−0.108154 + 0.994134i \(0.534494\pi\)
\(384\) 0 0
\(385\) 22.8043 1.16221
\(386\) 0 0
\(387\) 8.23546 0.418632
\(388\) 0 0
\(389\) 5.21409 0.264365 0.132182 0.991225i \(-0.457802\pi\)
0.132182 + 0.991225i \(0.457802\pi\)
\(390\) 0 0
\(391\) 4.23477 0.214162
\(392\) 0 0
\(393\) 7.34570 0.370542
\(394\) 0 0
\(395\) −24.5127 −1.23337
\(396\) 0 0
\(397\) −9.63263 −0.483448 −0.241724 0.970345i \(-0.577713\pi\)
−0.241724 + 0.970345i \(0.577713\pi\)
\(398\) 0 0
\(399\) −42.7792 −2.14164
\(400\) 0 0
\(401\) 24.8722 1.24206 0.621030 0.783787i \(-0.286714\pi\)
0.621030 + 0.783787i \(0.286714\pi\)
\(402\) 0 0
\(403\) −5.30151 −0.264087
\(404\) 0 0
\(405\) 13.5159 0.671610
\(406\) 0 0
\(407\) 15.0253 0.744775
\(408\) 0 0
\(409\) −28.4449 −1.40651 −0.703255 0.710938i \(-0.748271\pi\)
−0.703255 + 0.710938i \(0.748271\pi\)
\(410\) 0 0
\(411\) 2.35043 0.115938
\(412\) 0 0
\(413\) −5.19027 −0.255397
\(414\) 0 0
\(415\) −44.8590 −2.20204
\(416\) 0 0
\(417\) −1.39811 −0.0684658
\(418\) 0 0
\(419\) −4.59075 −0.224273 −0.112136 0.993693i \(-0.535769\pi\)
−0.112136 + 0.993693i \(0.535769\pi\)
\(420\) 0 0
\(421\) −20.9586 −1.02146 −0.510729 0.859742i \(-0.670625\pi\)
−0.510729 + 0.859742i \(0.670625\pi\)
\(422\) 0 0
\(423\) −7.15674 −0.347973
\(424\) 0 0
\(425\) 3.12541 0.151604
\(426\) 0 0
\(427\) 1.17154 0.0566948
\(428\) 0 0
\(429\) 11.1520 0.538423
\(430\) 0 0
\(431\) −14.1727 −0.682676 −0.341338 0.939941i \(-0.610880\pi\)
−0.341338 + 0.939941i \(0.610880\pi\)
\(432\) 0 0
\(433\) 7.32714 0.352120 0.176060 0.984379i \(-0.443665\pi\)
0.176060 + 0.984379i \(0.443665\pi\)
\(434\) 0 0
\(435\) −17.7844 −0.852699
\(436\) 0 0
\(437\) −30.0782 −1.43884
\(438\) 0 0
\(439\) −3.07088 −0.146565 −0.0732825 0.997311i \(-0.523347\pi\)
−0.0732825 + 0.997311i \(0.523347\pi\)
\(440\) 0 0
\(441\) −10.8398 −0.516181
\(442\) 0 0
\(443\) 14.9982 0.712586 0.356293 0.934374i \(-0.384040\pi\)
0.356293 + 0.934374i \(0.384040\pi\)
\(444\) 0 0
\(445\) 9.17325 0.434854
\(446\) 0 0
\(447\) 19.4697 0.920887
\(448\) 0 0
\(449\) −22.4064 −1.05742 −0.528711 0.848802i \(-0.677324\pi\)
−0.528711 + 0.848802i \(0.677324\pi\)
\(450\) 0 0
\(451\) 22.3303 1.05149
\(452\) 0 0
\(453\) 12.0781 0.567479
\(454\) 0 0
\(455\) −48.7498 −2.28543
\(456\) 0 0
\(457\) −12.9317 −0.604919 −0.302460 0.953162i \(-0.597808\pi\)
−0.302460 + 0.953162i \(0.597808\pi\)
\(458\) 0 0
\(459\) 5.84598 0.272867
\(460\) 0 0
\(461\) 39.7115 1.84955 0.924774 0.380517i \(-0.124254\pi\)
0.924774 + 0.380517i \(0.124254\pi\)
\(462\) 0 0
\(463\) −21.4987 −0.999130 −0.499565 0.866276i \(-0.666507\pi\)
−0.499565 + 0.866276i \(0.666507\pi\)
\(464\) 0 0
\(465\) 5.08447 0.235786
\(466\) 0 0
\(467\) −19.2688 −0.891655 −0.445827 0.895119i \(-0.647090\pi\)
−0.445827 + 0.895119i \(0.647090\pi\)
\(468\) 0 0
\(469\) 35.5052 1.63948
\(470\) 0 0
\(471\) 14.5910 0.672317
\(472\) 0 0
\(473\) −15.2188 −0.699761
\(474\) 0 0
\(475\) −22.1987 −1.01855
\(476\) 0 0
\(477\) 3.85159 0.176352
\(478\) 0 0
\(479\) 23.7465 1.08500 0.542502 0.840055i \(-0.317477\pi\)
0.542502 + 0.840055i \(0.317477\pi\)
\(480\) 0 0
\(481\) −32.1202 −1.46456
\(482\) 0 0
\(483\) 23.8730 1.08626
\(484\) 0 0
\(485\) 20.6023 0.935503
\(486\) 0 0
\(487\) 27.3504 1.23937 0.619683 0.784852i \(-0.287261\pi\)
0.619683 + 0.784852i \(0.287261\pi\)
\(488\) 0 0
\(489\) 5.22062 0.236085
\(490\) 0 0
\(491\) −29.0333 −1.31025 −0.655126 0.755519i \(-0.727385\pi\)
−0.655126 + 0.755519i \(0.727385\pi\)
\(492\) 0 0
\(493\) −4.64171 −0.209052
\(494\) 0 0
\(495\) 5.71938 0.257067
\(496\) 0 0
\(497\) −26.6792 −1.19672
\(498\) 0 0
\(499\) 26.6880 1.19472 0.597359 0.801974i \(-0.296217\pi\)
0.597359 + 0.801974i \(0.296217\pi\)
\(500\) 0 0
\(501\) −27.6131 −1.23366
\(502\) 0 0
\(503\) −7.41588 −0.330658 −0.165329 0.986238i \(-0.552869\pi\)
−0.165329 + 0.986238i \(0.552869\pi\)
\(504\) 0 0
\(505\) −36.9532 −1.64440
\(506\) 0 0
\(507\) −5.66466 −0.251577
\(508\) 0 0
\(509\) −36.1455 −1.60212 −0.801060 0.598583i \(-0.795730\pi\)
−0.801060 + 0.598583i \(0.795730\pi\)
\(510\) 0 0
\(511\) 24.6985 1.09260
\(512\) 0 0
\(513\) −41.5221 −1.83325
\(514\) 0 0
\(515\) −33.7219 −1.48596
\(516\) 0 0
\(517\) 13.2254 0.581651
\(518\) 0 0
\(519\) −10.5058 −0.461153
\(520\) 0 0
\(521\) 33.9636 1.48797 0.743986 0.668196i \(-0.232933\pi\)
0.743986 + 0.668196i \(0.232933\pi\)
\(522\) 0 0
\(523\) −11.7578 −0.514134 −0.257067 0.966394i \(-0.582756\pi\)
−0.257067 + 0.966394i \(0.582756\pi\)
\(524\) 0 0
\(525\) 17.6191 0.768960
\(526\) 0 0
\(527\) 1.32704 0.0578066
\(528\) 0 0
\(529\) −6.21480 −0.270209
\(530\) 0 0
\(531\) −1.30174 −0.0564905
\(532\) 0 0
\(533\) −47.7366 −2.06770
\(534\) 0 0
\(535\) −45.6214 −1.97239
\(536\) 0 0
\(537\) −33.9217 −1.46383
\(538\) 0 0
\(539\) 20.0315 0.862818
\(540\) 0 0
\(541\) −0.804094 −0.0345707 −0.0172854 0.999851i \(-0.505502\pi\)
−0.0172854 + 0.999851i \(0.505502\pi\)
\(542\) 0 0
\(543\) −4.27851 −0.183608
\(544\) 0 0
\(545\) 37.2386 1.59513
\(546\) 0 0
\(547\) −4.34094 −0.185605 −0.0928027 0.995685i \(-0.529583\pi\)
−0.0928027 + 0.995685i \(0.529583\pi\)
\(548\) 0 0
\(549\) 0.293826 0.0125402
\(550\) 0 0
\(551\) 32.9686 1.40451
\(552\) 0 0
\(553\) −36.0667 −1.53371
\(554\) 0 0
\(555\) 30.8053 1.30761
\(556\) 0 0
\(557\) 29.2321 1.23861 0.619303 0.785152i \(-0.287415\pi\)
0.619303 + 0.785152i \(0.287415\pi\)
\(558\) 0 0
\(559\) 32.5339 1.37604
\(560\) 0 0
\(561\) −2.79148 −0.117857
\(562\) 0 0
\(563\) 27.9325 1.17721 0.588606 0.808420i \(-0.299677\pi\)
0.588606 + 0.808420i \(0.299677\pi\)
\(564\) 0 0
\(565\) 41.8079 1.75887
\(566\) 0 0
\(567\) 19.8865 0.835156
\(568\) 0 0
\(569\) 21.5023 0.901424 0.450712 0.892669i \(-0.351170\pi\)
0.450712 + 0.892669i \(0.351170\pi\)
\(570\) 0 0
\(571\) 19.6384 0.821843 0.410922 0.911671i \(-0.365207\pi\)
0.410922 + 0.911671i \(0.365207\pi\)
\(572\) 0 0
\(573\) 12.3529 0.516051
\(574\) 0 0
\(575\) 12.3881 0.516617
\(576\) 0 0
\(577\) 14.7632 0.614599 0.307299 0.951613i \(-0.400575\pi\)
0.307299 + 0.951613i \(0.400575\pi\)
\(578\) 0 0
\(579\) −20.8750 −0.867536
\(580\) 0 0
\(581\) −66.0031 −2.73827
\(582\) 0 0
\(583\) −7.11757 −0.294780
\(584\) 0 0
\(585\) −12.2266 −0.505508
\(586\) 0 0
\(587\) −6.27328 −0.258926 −0.129463 0.991584i \(-0.541325\pi\)
−0.129463 + 0.991584i \(0.541325\pi\)
\(588\) 0 0
\(589\) −9.42551 −0.388371
\(590\) 0 0
\(591\) −16.6348 −0.684265
\(592\) 0 0
\(593\) 40.8082 1.67579 0.837895 0.545831i \(-0.183786\pi\)
0.837895 + 0.545831i \(0.183786\pi\)
\(594\) 0 0
\(595\) 12.2027 0.500263
\(596\) 0 0
\(597\) −6.16179 −0.252185
\(598\) 0 0
\(599\) 10.3330 0.422193 0.211096 0.977465i \(-0.432297\pi\)
0.211096 + 0.977465i \(0.432297\pi\)
\(600\) 0 0
\(601\) 15.7883 0.644019 0.322009 0.946737i \(-0.395642\pi\)
0.322009 + 0.946737i \(0.395642\pi\)
\(602\) 0 0
\(603\) 8.90480 0.362632
\(604\) 0 0
\(605\) 20.5896 0.837086
\(606\) 0 0
\(607\) −12.8843 −0.522958 −0.261479 0.965209i \(-0.584210\pi\)
−0.261479 + 0.965209i \(0.584210\pi\)
\(608\) 0 0
\(609\) −26.1671 −1.06034
\(610\) 0 0
\(611\) −28.2725 −1.14378
\(612\) 0 0
\(613\) −6.99917 −0.282694 −0.141347 0.989960i \(-0.545143\pi\)
−0.141347 + 0.989960i \(0.545143\pi\)
\(614\) 0 0
\(615\) 45.7823 1.84612
\(616\) 0 0
\(617\) −8.16208 −0.328593 −0.164297 0.986411i \(-0.552535\pi\)
−0.164297 + 0.986411i \(0.552535\pi\)
\(618\) 0 0
\(619\) −12.6031 −0.506559 −0.253280 0.967393i \(-0.581509\pi\)
−0.253280 + 0.967393i \(0.581509\pi\)
\(620\) 0 0
\(621\) 23.1715 0.929840
\(622\) 0 0
\(623\) 13.4970 0.540747
\(624\) 0 0
\(625\) −30.9757 −1.23903
\(626\) 0 0
\(627\) 19.8270 0.791815
\(628\) 0 0
\(629\) 8.04011 0.320580
\(630\) 0 0
\(631\) −45.4769 −1.81041 −0.905203 0.424980i \(-0.860281\pi\)
−0.905203 + 0.424980i \(0.860281\pi\)
\(632\) 0 0
\(633\) −29.9500 −1.19041
\(634\) 0 0
\(635\) 31.1989 1.23809
\(636\) 0 0
\(637\) −42.8223 −1.69668
\(638\) 0 0
\(639\) −6.69122 −0.264700
\(640\) 0 0
\(641\) 12.1865 0.481339 0.240669 0.970607i \(-0.422633\pi\)
0.240669 + 0.970607i \(0.422633\pi\)
\(642\) 0 0
\(643\) 29.7340 1.17259 0.586296 0.810097i \(-0.300585\pi\)
0.586296 + 0.810097i \(0.300585\pi\)
\(644\) 0 0
\(645\) −31.2020 −1.22858
\(646\) 0 0
\(647\) 32.8228 1.29040 0.645199 0.764015i \(-0.276774\pi\)
0.645199 + 0.764015i \(0.276774\pi\)
\(648\) 0 0
\(649\) 2.40555 0.0944262
\(650\) 0 0
\(651\) 7.48100 0.293204
\(652\) 0 0
\(653\) −10.3258 −0.404078 −0.202039 0.979377i \(-0.564757\pi\)
−0.202039 + 0.979377i \(0.564757\pi\)
\(654\) 0 0
\(655\) 14.8826 0.581512
\(656\) 0 0
\(657\) 6.19445 0.241668
\(658\) 0 0
\(659\) 25.8076 1.00532 0.502661 0.864484i \(-0.332354\pi\)
0.502661 + 0.864484i \(0.332354\pi\)
\(660\) 0 0
\(661\) 13.3955 0.521025 0.260513 0.965470i \(-0.416108\pi\)
0.260513 + 0.965470i \(0.416108\pi\)
\(662\) 0 0
\(663\) 5.96749 0.231758
\(664\) 0 0
\(665\) −86.6719 −3.36099
\(666\) 0 0
\(667\) −18.3982 −0.712380
\(668\) 0 0
\(669\) −11.0952 −0.428963
\(670\) 0 0
\(671\) −0.542978 −0.0209614
\(672\) 0 0
\(673\) −6.82079 −0.262922 −0.131461 0.991321i \(-0.541967\pi\)
−0.131461 + 0.991321i \(0.541967\pi\)
\(674\) 0 0
\(675\) 17.1014 0.658232
\(676\) 0 0
\(677\) −10.5869 −0.406888 −0.203444 0.979087i \(-0.565213\pi\)
−0.203444 + 0.979087i \(0.565213\pi\)
\(678\) 0 0
\(679\) 30.3131 1.16331
\(680\) 0 0
\(681\) −22.2795 −0.853753
\(682\) 0 0
\(683\) 43.3107 1.65724 0.828620 0.559812i \(-0.189127\pi\)
0.828620 + 0.559812i \(0.189127\pi\)
\(684\) 0 0
\(685\) 4.76205 0.181948
\(686\) 0 0
\(687\) 18.8343 0.718572
\(688\) 0 0
\(689\) 15.2156 0.579667
\(690\) 0 0
\(691\) −27.5238 −1.04706 −0.523528 0.852009i \(-0.675384\pi\)
−0.523528 + 0.852009i \(0.675384\pi\)
\(692\) 0 0
\(693\) 8.41519 0.319667
\(694\) 0 0
\(695\) −2.83261 −0.107447
\(696\) 0 0
\(697\) 11.9491 0.452604
\(698\) 0 0
\(699\) −25.7694 −0.974686
\(700\) 0 0
\(701\) −51.4490 −1.94320 −0.971601 0.236625i \(-0.923959\pi\)
−0.971601 + 0.236625i \(0.923959\pi\)
\(702\) 0 0
\(703\) −57.1063 −2.15381
\(704\) 0 0
\(705\) 27.1150 1.02121
\(706\) 0 0
\(707\) −54.3710 −2.04483
\(708\) 0 0
\(709\) −16.8255 −0.631894 −0.315947 0.948777i \(-0.602322\pi\)
−0.315947 + 0.948777i \(0.602322\pi\)
\(710\) 0 0
\(711\) −9.04563 −0.339238
\(712\) 0 0
\(713\) 5.25992 0.196986
\(714\) 0 0
\(715\) 22.5942 0.844977
\(716\) 0 0
\(717\) 1.24354 0.0464407
\(718\) 0 0
\(719\) −26.2933 −0.980574 −0.490287 0.871561i \(-0.663108\pi\)
−0.490287 + 0.871561i \(0.663108\pi\)
\(720\) 0 0
\(721\) −49.6165 −1.84781
\(722\) 0 0
\(723\) 16.4354 0.611238
\(724\) 0 0
\(725\) −13.5785 −0.504292
\(726\) 0 0
\(727\) −23.5378 −0.872967 −0.436484 0.899712i \(-0.643776\pi\)
−0.436484 + 0.899712i \(0.643776\pi\)
\(728\) 0 0
\(729\) 28.7097 1.06332
\(730\) 0 0
\(731\) −8.14366 −0.301204
\(732\) 0 0
\(733\) 10.0422 0.370919 0.185459 0.982652i \(-0.440623\pi\)
0.185459 + 0.982652i \(0.440623\pi\)
\(734\) 0 0
\(735\) 41.0691 1.51486
\(736\) 0 0
\(737\) −16.4557 −0.606154
\(738\) 0 0
\(739\) 8.15975 0.300161 0.150081 0.988674i \(-0.452047\pi\)
0.150081 + 0.988674i \(0.452047\pi\)
\(740\) 0 0
\(741\) −42.3851 −1.55706
\(742\) 0 0
\(743\) −7.78206 −0.285496 −0.142748 0.989759i \(-0.545594\pi\)
−0.142748 + 0.989759i \(0.545594\pi\)
\(744\) 0 0
\(745\) 39.4463 1.44520
\(746\) 0 0
\(747\) −16.5538 −0.605670
\(748\) 0 0
\(749\) −67.1249 −2.45269
\(750\) 0 0
\(751\) 8.30887 0.303195 0.151598 0.988442i \(-0.451558\pi\)
0.151598 + 0.988442i \(0.451558\pi\)
\(752\) 0 0
\(753\) 20.2098 0.736487
\(754\) 0 0
\(755\) 24.4706 0.890577
\(756\) 0 0
\(757\) 38.8376 1.41158 0.705788 0.708424i \(-0.250593\pi\)
0.705788 + 0.708424i \(0.250593\pi\)
\(758\) 0 0
\(759\) −11.0645 −0.401616
\(760\) 0 0
\(761\) −25.3906 −0.920408 −0.460204 0.887813i \(-0.652224\pi\)
−0.460204 + 0.887813i \(0.652224\pi\)
\(762\) 0 0
\(763\) 54.7909 1.98356
\(764\) 0 0
\(765\) 3.06048 0.110652
\(766\) 0 0
\(767\) −5.14246 −0.185684
\(768\) 0 0
\(769\) 39.8098 1.43558 0.717788 0.696261i \(-0.245155\pi\)
0.717788 + 0.696261i \(0.245155\pi\)
\(770\) 0 0
\(771\) −24.1857 −0.871028
\(772\) 0 0
\(773\) −9.95735 −0.358141 −0.179070 0.983836i \(-0.557309\pi\)
−0.179070 + 0.983836i \(0.557309\pi\)
\(774\) 0 0
\(775\) 3.88200 0.139446
\(776\) 0 0
\(777\) 45.3252 1.62603
\(778\) 0 0
\(779\) −84.8706 −3.04080
\(780\) 0 0
\(781\) 12.3651 0.442458
\(782\) 0 0
\(783\) −25.3981 −0.907656
\(784\) 0 0
\(785\) 29.5618 1.05510
\(786\) 0 0
\(787\) 30.2211 1.07727 0.538633 0.842541i \(-0.318941\pi\)
0.538633 + 0.842541i \(0.318941\pi\)
\(788\) 0 0
\(789\) 5.60469 0.199532
\(790\) 0 0
\(791\) 61.5139 2.18718
\(792\) 0 0
\(793\) 1.16075 0.0412194
\(794\) 0 0
\(795\) −14.5927 −0.517548
\(796\) 0 0
\(797\) 10.6318 0.376597 0.188298 0.982112i \(-0.439703\pi\)
0.188298 + 0.982112i \(0.439703\pi\)
\(798\) 0 0
\(799\) 7.07697 0.250365
\(800\) 0 0
\(801\) 3.38509 0.119606
\(802\) 0 0
\(803\) −11.4471 −0.403959
\(804\) 0 0
\(805\) 48.3674 1.70473
\(806\) 0 0
\(807\) 39.1891 1.37952
\(808\) 0 0
\(809\) −15.9047 −0.559178 −0.279589 0.960120i \(-0.590198\pi\)
−0.279589 + 0.960120i \(0.590198\pi\)
\(810\) 0 0
\(811\) 53.3099 1.87196 0.935981 0.352049i \(-0.114515\pi\)
0.935981 + 0.352049i \(0.114515\pi\)
\(812\) 0 0
\(813\) 22.7973 0.799537
\(814\) 0 0
\(815\) 10.5771 0.370501
\(816\) 0 0
\(817\) 57.8418 2.02363
\(818\) 0 0
\(819\) −17.9895 −0.628606
\(820\) 0 0
\(821\) 16.7614 0.584975 0.292488 0.956269i \(-0.405517\pi\)
0.292488 + 0.956269i \(0.405517\pi\)
\(822\) 0 0
\(823\) −38.0271 −1.32554 −0.662770 0.748823i \(-0.730619\pi\)
−0.662770 + 0.748823i \(0.730619\pi\)
\(824\) 0 0
\(825\) −8.16598 −0.284303
\(826\) 0 0
\(827\) 40.6301 1.41285 0.706424 0.707789i \(-0.250307\pi\)
0.706424 + 0.707789i \(0.250307\pi\)
\(828\) 0 0
\(829\) 48.8098 1.69523 0.847617 0.530608i \(-0.178036\pi\)
0.847617 + 0.530608i \(0.178036\pi\)
\(830\) 0 0
\(831\) −36.8040 −1.27672
\(832\) 0 0
\(833\) 10.7190 0.371390
\(834\) 0 0
\(835\) −55.9449 −1.93605
\(836\) 0 0
\(837\) 7.26117 0.250983
\(838\) 0 0
\(839\) 13.9510 0.481641 0.240821 0.970570i \(-0.422583\pi\)
0.240821 + 0.970570i \(0.422583\pi\)
\(840\) 0 0
\(841\) −8.83388 −0.304617
\(842\) 0 0
\(843\) 10.6000 0.365084
\(844\) 0 0
\(845\) −11.4768 −0.394813
\(846\) 0 0
\(847\) 30.2944 1.04093
\(848\) 0 0
\(849\) 19.8841 0.682421
\(850\) 0 0
\(851\) 31.8683 1.09243
\(852\) 0 0
\(853\) 48.3304 1.65480 0.827401 0.561612i \(-0.189819\pi\)
0.827401 + 0.561612i \(0.189819\pi\)
\(854\) 0 0
\(855\) −21.7376 −0.743409
\(856\) 0 0
\(857\) −47.8068 −1.63305 −0.816525 0.577310i \(-0.804103\pi\)
−0.816525 + 0.577310i \(0.804103\pi\)
\(858\) 0 0
\(859\) 57.7110 1.96907 0.984537 0.175179i \(-0.0560503\pi\)
0.984537 + 0.175179i \(0.0560503\pi\)
\(860\) 0 0
\(861\) 67.3616 2.29568
\(862\) 0 0
\(863\) −22.5913 −0.769018 −0.384509 0.923121i \(-0.625629\pi\)
−0.384509 + 0.923121i \(0.625629\pi\)
\(864\) 0 0
\(865\) −21.2851 −0.723714
\(866\) 0 0
\(867\) 22.2741 0.756470
\(868\) 0 0
\(869\) 16.7160 0.567050
\(870\) 0 0
\(871\) 35.1781 1.19197
\(872\) 0 0
\(873\) 7.60262 0.257310
\(874\) 0 0
\(875\) −23.3314 −0.788746
\(876\) 0 0
\(877\) −8.20763 −0.277152 −0.138576 0.990352i \(-0.544252\pi\)
−0.138576 + 0.990352i \(0.544252\pi\)
\(878\) 0 0
\(879\) −6.55690 −0.221159
\(880\) 0 0
\(881\) 54.9341 1.85078 0.925388 0.379020i \(-0.123739\pi\)
0.925388 + 0.379020i \(0.123739\pi\)
\(882\) 0 0
\(883\) −42.9747 −1.44621 −0.723107 0.690736i \(-0.757286\pi\)
−0.723107 + 0.690736i \(0.757286\pi\)
\(884\) 0 0
\(885\) 4.93193 0.165785
\(886\) 0 0
\(887\) 16.0125 0.537647 0.268824 0.963189i \(-0.413365\pi\)
0.268824 + 0.963189i \(0.413365\pi\)
\(888\) 0 0
\(889\) 45.9044 1.53958
\(890\) 0 0
\(891\) −9.21688 −0.308777
\(892\) 0 0
\(893\) −50.2654 −1.68207
\(894\) 0 0
\(895\) −68.7263 −2.29727
\(896\) 0 0
\(897\) 23.6531 0.789754
\(898\) 0 0
\(899\) −5.76537 −0.192286
\(900\) 0 0
\(901\) −3.80866 −0.126885
\(902\) 0 0
\(903\) −45.9089 −1.52775
\(904\) 0 0
\(905\) −8.66838 −0.288147
\(906\) 0 0
\(907\) 17.4683 0.580026 0.290013 0.957023i \(-0.406340\pi\)
0.290013 + 0.957023i \(0.406340\pi\)
\(908\) 0 0
\(909\) −13.6364 −0.452291
\(910\) 0 0
\(911\) −16.6656 −0.552157 −0.276078 0.961135i \(-0.589035\pi\)
−0.276078 + 0.961135i \(0.589035\pi\)
\(912\) 0 0
\(913\) 30.5907 1.01240
\(914\) 0 0
\(915\) −1.11323 −0.0368022
\(916\) 0 0
\(917\) 21.8975 0.723118
\(918\) 0 0
\(919\) −18.5901 −0.613230 −0.306615 0.951834i \(-0.599196\pi\)
−0.306615 + 0.951834i \(0.599196\pi\)
\(920\) 0 0
\(921\) −14.9480 −0.492553
\(922\) 0 0
\(923\) −26.4334 −0.870067
\(924\) 0 0
\(925\) 23.5199 0.773329
\(926\) 0 0
\(927\) −12.4440 −0.408713
\(928\) 0 0
\(929\) 6.08296 0.199576 0.0997878 0.995009i \(-0.468184\pi\)
0.0997878 + 0.995009i \(0.468184\pi\)
\(930\) 0 0
\(931\) −76.1334 −2.49517
\(932\) 0 0
\(933\) −29.4239 −0.963296
\(934\) 0 0
\(935\) −5.65563 −0.184959
\(936\) 0 0
\(937\) −50.4851 −1.64928 −0.824639 0.565660i \(-0.808622\pi\)
−0.824639 + 0.565660i \(0.808622\pi\)
\(938\) 0 0
\(939\) 19.8578 0.648035
\(940\) 0 0
\(941\) 26.6303 0.868123 0.434061 0.900883i \(-0.357080\pi\)
0.434061 + 0.900883i \(0.357080\pi\)
\(942\) 0 0
\(943\) 47.3622 1.54232
\(944\) 0 0
\(945\) 66.7699 2.17202
\(946\) 0 0
\(947\) −41.7464 −1.35658 −0.678289 0.734796i \(-0.737278\pi\)
−0.678289 + 0.734796i \(0.737278\pi\)
\(948\) 0 0
\(949\) 24.4710 0.794361
\(950\) 0 0
\(951\) 3.99494 0.129545
\(952\) 0 0
\(953\) −45.7416 −1.48172 −0.740858 0.671662i \(-0.765581\pi\)
−0.740858 + 0.671662i \(0.765581\pi\)
\(954\) 0 0
\(955\) 25.0274 0.809867
\(956\) 0 0
\(957\) 12.1277 0.392034
\(958\) 0 0
\(959\) 7.00662 0.226255
\(960\) 0 0
\(961\) −29.3517 −0.946830
\(962\) 0 0
\(963\) −16.8351 −0.542504
\(964\) 0 0
\(965\) −42.2934 −1.36147
\(966\) 0 0
\(967\) −29.9836 −0.964207 −0.482104 0.876114i \(-0.660127\pi\)
−0.482104 + 0.876114i \(0.660127\pi\)
\(968\) 0 0
\(969\) 10.6096 0.340828
\(970\) 0 0
\(971\) −32.2248 −1.03414 −0.517071 0.855942i \(-0.672978\pi\)
−0.517071 + 0.855942i \(0.672978\pi\)
\(972\) 0 0
\(973\) −4.16776 −0.133612
\(974\) 0 0
\(975\) 17.4568 0.559065
\(976\) 0 0
\(977\) −1.45628 −0.0465907 −0.0232953 0.999729i \(-0.507416\pi\)
−0.0232953 + 0.999729i \(0.507416\pi\)
\(978\) 0 0
\(979\) −6.25551 −0.199927
\(980\) 0 0
\(981\) 13.7417 0.438739
\(982\) 0 0
\(983\) 35.1255 1.12033 0.560164 0.828382i \(-0.310738\pi\)
0.560164 + 0.828382i \(0.310738\pi\)
\(984\) 0 0
\(985\) −33.7026 −1.07385
\(986\) 0 0
\(987\) 39.8956 1.26989
\(988\) 0 0
\(989\) −32.2787 −1.02640
\(990\) 0 0
\(991\) 9.58149 0.304366 0.152183 0.988352i \(-0.451370\pi\)
0.152183 + 0.988352i \(0.451370\pi\)
\(992\) 0 0
\(993\) −16.9411 −0.537611
\(994\) 0 0
\(995\) −12.4840 −0.395769
\(996\) 0 0
\(997\) 16.5332 0.523613 0.261807 0.965120i \(-0.415682\pi\)
0.261807 + 0.965120i \(0.415682\pi\)
\(998\) 0 0
\(999\) 43.9933 1.39189
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 8896.2.a.bd.1.3 7
4.3 odd 2 8896.2.a.be.1.5 7
8.3 odd 2 139.2.a.c.1.4 7
8.5 even 2 2224.2.a.o.1.5 7
24.11 even 2 1251.2.a.k.1.4 7
40.19 odd 2 3475.2.a.e.1.4 7
56.27 even 2 6811.2.a.p.1.4 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
139.2.a.c.1.4 7 8.3 odd 2
1251.2.a.k.1.4 7 24.11 even 2
2224.2.a.o.1.5 7 8.5 even 2
3475.2.a.e.1.4 7 40.19 odd 2
6811.2.a.p.1.4 7 56.27 even 2
8896.2.a.bd.1.3 7 1.1 even 1 trivial
8896.2.a.be.1.5 7 4.3 odd 2