Properties

Label 6384.2.a.bv.1.1
Level $6384$
Weight $2$
Character 6384.1
Self dual yes
Analytic conductor $50.976$
Analytic rank $1$
Dimension $3$
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] = [6384,2,Mod(1,6384)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6384, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("6384.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6384 = 2^{4} \cdot 3 \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6384.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: \(50.9764966504\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.316.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 3192)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.34292\) of defining polynomial
Character \(\chi\) \(=\) 6384.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +1.00000 q^{7} +1.00000 q^{9} -2.68585 q^{11} -6.97858 q^{13} +2.29273 q^{17} +1.00000 q^{19} -1.00000 q^{21} +4.39312 q^{23} -5.00000 q^{25} -1.00000 q^{27} +0.978577 q^{29} +7.66442 q^{31} +2.68585 q^{33} +5.66442 q^{37} +6.97858 q^{39} +7.37169 q^{41} +9.37169 q^{43} +4.68585 q^{47} +1.00000 q^{49} -2.29273 q^{51} -12.9786 q^{53} -1.00000 q^{57} +0.585462 q^{59} -6.58546 q^{61} +1.00000 q^{63} -12.0575 q^{67} -4.39312 q^{69} +1.60688 q^{71} -7.37169 q^{73} +5.00000 q^{75} -2.68585 q^{77} -14.3503 q^{79} +1.00000 q^{81} -8.35027 q^{83} -0.978577 q^{87} -15.9572 q^{89} -6.97858 q^{91} -7.66442 q^{93} +5.27131 q^{97} -2.68585 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} + 3 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{3} + 3 q^{7} + 3 q^{9} + 4 q^{11} - 6 q^{13} + 4 q^{17} + 3 q^{19} - 3 q^{21} + 4 q^{23} - 15 q^{25} - 3 q^{27} - 12 q^{29} - 4 q^{31} - 4 q^{33} - 10 q^{37} + 6 q^{39} - 2 q^{41} + 4 q^{43} + 2 q^{47} + 3 q^{49} - 4 q^{51} - 24 q^{53} - 3 q^{57} - 4 q^{59} - 14 q^{61} + 3 q^{63} - 4 q^{69} + 14 q^{71} + 2 q^{73} + 15 q^{75} + 4 q^{77} - 4 q^{79} + 3 q^{81} + 14 q^{83} + 12 q^{87} - 18 q^{89} - 6 q^{91} + 4 q^{93} - 2 q^{97} + 4 q^{99}+O(q^{100}) \) 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
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −2.68585 −0.809813 −0.404907 0.914358i \(-0.632696\pi\)
−0.404907 + 0.914358i \(0.632696\pi\)
\(12\) 0 0
\(13\) −6.97858 −1.93551 −0.967755 0.251895i \(-0.918946\pi\)
−0.967755 + 0.251895i \(0.918946\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.29273 0.556069 0.278034 0.960571i \(-0.410317\pi\)
0.278034 + 0.960571i \(0.410317\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) 4.39312 0.916028 0.458014 0.888945i \(-0.348561\pi\)
0.458014 + 0.888945i \(0.348561\pi\)
\(24\) 0 0
\(25\) −5.00000 −1.00000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 0.978577 0.181717 0.0908586 0.995864i \(-0.471039\pi\)
0.0908586 + 0.995864i \(0.471039\pi\)
\(30\) 0 0
\(31\) 7.66442 1.37657 0.688286 0.725440i \(-0.258364\pi\)
0.688286 + 0.725440i \(0.258364\pi\)
\(32\) 0 0
\(33\) 2.68585 0.467546
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 5.66442 0.931225 0.465613 0.884989i \(-0.345834\pi\)
0.465613 + 0.884989i \(0.345834\pi\)
\(38\) 0 0
\(39\) 6.97858 1.11747
\(40\) 0 0
\(41\) 7.37169 1.15126 0.575632 0.817709i \(-0.304756\pi\)
0.575632 + 0.817709i \(0.304756\pi\)
\(42\) 0 0
\(43\) 9.37169 1.42917 0.714585 0.699549i \(-0.246616\pi\)
0.714585 + 0.699549i \(0.246616\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 4.68585 0.683501 0.341750 0.939791i \(-0.388980\pi\)
0.341750 + 0.939791i \(0.388980\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −2.29273 −0.321047
\(52\) 0 0
\(53\) −12.9786 −1.78274 −0.891372 0.453272i \(-0.850257\pi\)
−0.891372 + 0.453272i \(0.850257\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −1.00000 −0.132453
\(58\) 0 0
\(59\) 0.585462 0.0762207 0.0381103 0.999274i \(-0.487866\pi\)
0.0381103 + 0.999274i \(0.487866\pi\)
\(60\) 0 0
\(61\) −6.58546 −0.843182 −0.421591 0.906786i \(-0.638528\pi\)
−0.421591 + 0.906786i \(0.638528\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −12.0575 −1.47306 −0.736531 0.676403i \(-0.763538\pi\)
−0.736531 + 0.676403i \(0.763538\pi\)
\(68\) 0 0
\(69\) −4.39312 −0.528869
\(70\) 0 0
\(71\) 1.60688 0.190702 0.0953511 0.995444i \(-0.469603\pi\)
0.0953511 + 0.995444i \(0.469603\pi\)
\(72\) 0 0
\(73\) −7.37169 −0.862791 −0.431396 0.902163i \(-0.641979\pi\)
−0.431396 + 0.902163i \(0.641979\pi\)
\(74\) 0 0
\(75\) 5.00000 0.577350
\(76\) 0 0
\(77\) −2.68585 −0.306081
\(78\) 0 0
\(79\) −14.3503 −1.61453 −0.807266 0.590188i \(-0.799054\pi\)
−0.807266 + 0.590188i \(0.799054\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −8.35027 −0.916561 −0.458281 0.888808i \(-0.651535\pi\)
−0.458281 + 0.888808i \(0.651535\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −0.978577 −0.104914
\(88\) 0 0
\(89\) −15.9572 −1.69145 −0.845727 0.533615i \(-0.820833\pi\)
−0.845727 + 0.533615i \(0.820833\pi\)
\(90\) 0 0
\(91\) −6.97858 −0.731554
\(92\) 0 0
\(93\) −7.66442 −0.794764
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 5.27131 0.535220 0.267610 0.963527i \(-0.413766\pi\)
0.267610 + 0.963527i \(0.413766\pi\)
\(98\) 0 0
\(99\) −2.68585 −0.269938
\(100\) 0 0
\(101\) −8.00000 −0.796030 −0.398015 0.917379i \(-0.630301\pi\)
−0.398015 + 0.917379i \(0.630301\pi\)
\(102\) 0 0
\(103\) 11.0790 1.09164 0.545821 0.837902i \(-0.316218\pi\)
0.545821 + 0.837902i \(0.316218\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 7.56404 0.731243 0.365622 0.930764i \(-0.380856\pi\)
0.365622 + 0.930764i \(0.380856\pi\)
\(108\) 0 0
\(109\) −4.87819 −0.467246 −0.233623 0.972327i \(-0.575058\pi\)
−0.233623 + 0.972327i \(0.575058\pi\)
\(110\) 0 0
\(111\) −5.66442 −0.537643
\(112\) 0 0
\(113\) 10.3503 0.973671 0.486836 0.873494i \(-0.338151\pi\)
0.486836 + 0.873494i \(0.338151\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −6.97858 −0.645170
\(118\) 0 0
\(119\) 2.29273 0.210174
\(120\) 0 0
\(121\) −3.78623 −0.344203
\(122\) 0 0
\(123\) −7.37169 −0.664683
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 0.393115 0.0348833 0.0174417 0.999848i \(-0.494448\pi\)
0.0174417 + 0.999848i \(0.494448\pi\)
\(128\) 0 0
\(129\) −9.37169 −0.825132
\(130\) 0 0
\(131\) 1.80765 0.157935 0.0789677 0.996877i \(-0.474838\pi\)
0.0789677 + 0.996877i \(0.474838\pi\)
\(132\) 0 0
\(133\) 1.00000 0.0867110
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −14.7862 −1.26327 −0.631636 0.775265i \(-0.717616\pi\)
−0.631636 + 0.775265i \(0.717616\pi\)
\(138\) 0 0
\(139\) 13.3717 1.13417 0.567086 0.823659i \(-0.308071\pi\)
0.567086 + 0.823659i \(0.308071\pi\)
\(140\) 0 0
\(141\) −4.68585 −0.394619
\(142\) 0 0
\(143\) 18.7434 1.56740
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −1.00000 −0.0824786
\(148\) 0 0
\(149\) 5.07896 0.416085 0.208042 0.978120i \(-0.433291\pi\)
0.208042 + 0.978120i \(0.433291\pi\)
\(150\) 0 0
\(151\) −9.56404 −0.778310 −0.389155 0.921172i \(-0.627233\pi\)
−0.389155 + 0.921172i \(0.627233\pi\)
\(152\) 0 0
\(153\) 2.29273 0.185356
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 15.9572 1.27352 0.636760 0.771062i \(-0.280274\pi\)
0.636760 + 0.771062i \(0.280274\pi\)
\(158\) 0 0
\(159\) 12.9786 1.02927
\(160\) 0 0
\(161\) 4.39312 0.346226
\(162\) 0 0
\(163\) 1.37169 0.107439 0.0537196 0.998556i \(-0.482892\pi\)
0.0537196 + 0.998556i \(0.482892\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −9.37169 −0.725203 −0.362602 0.931944i \(-0.618111\pi\)
−0.362602 + 0.931944i \(0.618111\pi\)
\(168\) 0 0
\(169\) 35.7005 2.74620
\(170\) 0 0
\(171\) 1.00000 0.0764719
\(172\) 0 0
\(173\) −18.7862 −1.42829 −0.714145 0.699997i \(-0.753184\pi\)
−0.714145 + 0.699997i \(0.753184\pi\)
\(174\) 0 0
\(175\) −5.00000 −0.377964
\(176\) 0 0
\(177\) −0.585462 −0.0440060
\(178\) 0 0
\(179\) 4.35027 0.325154 0.162577 0.986696i \(-0.448019\pi\)
0.162577 + 0.986696i \(0.448019\pi\)
\(180\) 0 0
\(181\) −10.9786 −0.816031 −0.408016 0.912975i \(-0.633779\pi\)
−0.408016 + 0.912975i \(0.633779\pi\)
\(182\) 0 0
\(183\) 6.58546 0.486811
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −6.15792 −0.450312
\(188\) 0 0
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) −14.3503 −1.03835 −0.519175 0.854668i \(-0.673761\pi\)
−0.519175 + 0.854668i \(0.673761\pi\)
\(192\) 0 0
\(193\) −13.3288 −0.959431 −0.479716 0.877424i \(-0.659260\pi\)
−0.479716 + 0.877424i \(0.659260\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −26.4507 −1.88453 −0.942266 0.334867i \(-0.891309\pi\)
−0.942266 + 0.334867i \(0.891309\pi\)
\(198\) 0 0
\(199\) −4.78623 −0.339287 −0.169643 0.985506i \(-0.554262\pi\)
−0.169643 + 0.985506i \(0.554262\pi\)
\(200\) 0 0
\(201\) 12.0575 0.850473
\(202\) 0 0
\(203\) 0.978577 0.0686827
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 4.39312 0.305343
\(208\) 0 0
\(209\) −2.68585 −0.185784
\(210\) 0 0
\(211\) −18.6858 −1.28639 −0.643193 0.765704i \(-0.722391\pi\)
−0.643193 + 0.765704i \(0.722391\pi\)
\(212\) 0 0
\(213\) −1.60688 −0.110102
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 7.66442 0.520295
\(218\) 0 0
\(219\) 7.37169 0.498133
\(220\) 0 0
\(221\) −16.0000 −1.07628
\(222\) 0 0
\(223\) 22.2070 1.48709 0.743547 0.668684i \(-0.233142\pi\)
0.743547 + 0.668684i \(0.233142\pi\)
\(224\) 0 0
\(225\) −5.00000 −0.333333
\(226\) 0 0
\(227\) −2.62831 −0.174447 −0.0872235 0.996189i \(-0.527799\pi\)
−0.0872235 + 0.996189i \(0.527799\pi\)
\(228\) 0 0
\(229\) 11.9572 0.790151 0.395075 0.918649i \(-0.370718\pi\)
0.395075 + 0.918649i \(0.370718\pi\)
\(230\) 0 0
\(231\) 2.68585 0.176716
\(232\) 0 0
\(233\) −6.58546 −0.431428 −0.215714 0.976457i \(-0.569208\pi\)
−0.215714 + 0.976457i \(0.569208\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 14.3503 0.932150
\(238\) 0 0
\(239\) −12.9786 −0.839514 −0.419757 0.907636i \(-0.637885\pi\)
−0.419757 + 0.907636i \(0.637885\pi\)
\(240\) 0 0
\(241\) 6.72869 0.433433 0.216717 0.976235i \(-0.430465\pi\)
0.216717 + 0.976235i \(0.430465\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −6.97858 −0.444036
\(248\) 0 0
\(249\) 8.35027 0.529177
\(250\) 0 0
\(251\) 1.80765 0.114098 0.0570490 0.998371i \(-0.481831\pi\)
0.0570490 + 0.998371i \(0.481831\pi\)
\(252\) 0 0
\(253\) −11.7992 −0.741811
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −15.3717 −0.958860 −0.479430 0.877580i \(-0.659157\pi\)
−0.479430 + 0.877580i \(0.659157\pi\)
\(258\) 0 0
\(259\) 5.66442 0.351970
\(260\) 0 0
\(261\) 0.978577 0.0605724
\(262\) 0 0
\(263\) −18.3503 −1.13153 −0.565763 0.824568i \(-0.691418\pi\)
−0.565763 + 0.824568i \(0.691418\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 15.9572 0.976562
\(268\) 0 0
\(269\) −10.5855 −0.645407 −0.322704 0.946500i \(-0.604592\pi\)
−0.322704 + 0.946500i \(0.604592\pi\)
\(270\) 0 0
\(271\) 8.00000 0.485965 0.242983 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) 0 0
\(273\) 6.97858 0.422363
\(274\) 0 0
\(275\) 13.4292 0.809813
\(276\) 0 0
\(277\) 1.21377 0.0729283 0.0364642 0.999335i \(-0.488391\pi\)
0.0364642 + 0.999335i \(0.488391\pi\)
\(278\) 0 0
\(279\) 7.66442 0.458857
\(280\) 0 0
\(281\) −25.5640 −1.52502 −0.762511 0.646975i \(-0.776034\pi\)
−0.762511 + 0.646975i \(0.776034\pi\)
\(282\) 0 0
\(283\) −22.5426 −1.34002 −0.670010 0.742352i \(-0.733710\pi\)
−0.670010 + 0.742352i \(0.733710\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 7.37169 0.435137
\(288\) 0 0
\(289\) −11.7434 −0.690787
\(290\) 0 0
\(291\) −5.27131 −0.309010
\(292\) 0 0
\(293\) −19.2860 −1.12670 −0.563350 0.826218i \(-0.690488\pi\)
−0.563350 + 0.826218i \(0.690488\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 2.68585 0.155849
\(298\) 0 0
\(299\) −30.6577 −1.77298
\(300\) 0 0
\(301\) 9.37169 0.540175
\(302\) 0 0
\(303\) 8.00000 0.459588
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −4.78623 −0.273165 −0.136582 0.990629i \(-0.543612\pi\)
−0.136582 + 0.990629i \(0.543612\pi\)
\(308\) 0 0
\(309\) −11.0790 −0.630260
\(310\) 0 0
\(311\) −19.4292 −1.10173 −0.550865 0.834594i \(-0.685702\pi\)
−0.550865 + 0.834594i \(0.685702\pi\)
\(312\) 0 0
\(313\) 25.3288 1.43167 0.715836 0.698269i \(-0.246046\pi\)
0.715836 + 0.698269i \(0.246046\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 29.6791 1.66695 0.833473 0.552561i \(-0.186349\pi\)
0.833473 + 0.552561i \(0.186349\pi\)
\(318\) 0 0
\(319\) −2.62831 −0.147157
\(320\) 0 0
\(321\) −7.56404 −0.422183
\(322\) 0 0
\(323\) 2.29273 0.127571
\(324\) 0 0
\(325\) 34.8929 1.93551
\(326\) 0 0
\(327\) 4.87819 0.269765
\(328\) 0 0
\(329\) 4.68585 0.258339
\(330\) 0 0
\(331\) 5.22846 0.287382 0.143691 0.989623i \(-0.454103\pi\)
0.143691 + 0.989623i \(0.454103\pi\)
\(332\) 0 0
\(333\) 5.66442 0.310408
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −9.32885 −0.508175 −0.254087 0.967181i \(-0.581775\pi\)
−0.254087 + 0.967181i \(0.581775\pi\)
\(338\) 0 0
\(339\) −10.3503 −0.562149
\(340\) 0 0
\(341\) −20.5855 −1.11477
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 19.3864 1.04072 0.520358 0.853948i \(-0.325799\pi\)
0.520358 + 0.853948i \(0.325799\pi\)
\(348\) 0 0
\(349\) 7.95715 0.425937 0.212968 0.977059i \(-0.431687\pi\)
0.212968 + 0.977059i \(0.431687\pi\)
\(350\) 0 0
\(351\) 6.97858 0.372489
\(352\) 0 0
\(353\) 33.6216 1.78950 0.894748 0.446571i \(-0.147355\pi\)
0.894748 + 0.446571i \(0.147355\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −2.29273 −0.121344
\(358\) 0 0
\(359\) −11.7220 −0.618661 −0.309331 0.950955i \(-0.600105\pi\)
−0.309331 + 0.950955i \(0.600105\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 3.78623 0.198726
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 11.2138 0.585354 0.292677 0.956211i \(-0.405454\pi\)
0.292677 + 0.956211i \(0.405454\pi\)
\(368\) 0 0
\(369\) 7.37169 0.383755
\(370\) 0 0
\(371\) −12.9786 −0.673814
\(372\) 0 0
\(373\) −18.4507 −0.955339 −0.477669 0.878540i \(-0.658518\pi\)
−0.477669 + 0.878540i \(0.658518\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −6.82908 −0.351715
\(378\) 0 0
\(379\) −11.9425 −0.613443 −0.306722 0.951799i \(-0.599232\pi\)
−0.306722 + 0.951799i \(0.599232\pi\)
\(380\) 0 0
\(381\) −0.393115 −0.0201399
\(382\) 0 0
\(383\) −3.21377 −0.164216 −0.0821080 0.996623i \(-0.526165\pi\)
−0.0821080 + 0.996623i \(0.526165\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 9.37169 0.476390
\(388\) 0 0
\(389\) 24.2070 1.22735 0.613673 0.789560i \(-0.289691\pi\)
0.613673 + 0.789560i \(0.289691\pi\)
\(390\) 0 0
\(391\) 10.0722 0.509375
\(392\) 0 0
\(393\) −1.80765 −0.0911840
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −6.20077 −0.311208 −0.155604 0.987820i \(-0.549732\pi\)
−0.155604 + 0.987820i \(0.549732\pi\)
\(398\) 0 0
\(399\) −1.00000 −0.0500626
\(400\) 0 0
\(401\) 5.17935 0.258644 0.129322 0.991603i \(-0.458720\pi\)
0.129322 + 0.991603i \(0.458720\pi\)
\(402\) 0 0
\(403\) −53.4868 −2.66437
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −15.2138 −0.754119
\(408\) 0 0
\(409\) −12.6002 −0.623038 −0.311519 0.950240i \(-0.600838\pi\)
−0.311519 + 0.950240i \(0.600838\pi\)
\(410\) 0 0
\(411\) 14.7862 0.729351
\(412\) 0 0
\(413\) 0.585462 0.0288087
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −13.3717 −0.654815
\(418\) 0 0
\(419\) −25.1365 −1.22800 −0.613999 0.789307i \(-0.710440\pi\)
−0.613999 + 0.789307i \(0.710440\pi\)
\(420\) 0 0
\(421\) −20.4935 −0.998792 −0.499396 0.866374i \(-0.666445\pi\)
−0.499396 + 0.866374i \(0.666445\pi\)
\(422\) 0 0
\(423\) 4.68585 0.227834
\(424\) 0 0
\(425\) −11.4637 −0.556069
\(426\) 0 0
\(427\) −6.58546 −0.318693
\(428\) 0 0
\(429\) −18.7434 −0.904939
\(430\) 0 0
\(431\) −34.3074 −1.65253 −0.826265 0.563281i \(-0.809539\pi\)
−0.826265 + 0.563281i \(0.809539\pi\)
\(432\) 0 0
\(433\) −34.5573 −1.66072 −0.830359 0.557229i \(-0.811865\pi\)
−0.830359 + 0.557229i \(0.811865\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 4.39312 0.210151
\(438\) 0 0
\(439\) 4.92104 0.234868 0.117434 0.993081i \(-0.462533\pi\)
0.117434 + 0.993081i \(0.462533\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 12.4422 0.591148 0.295574 0.955320i \(-0.404489\pi\)
0.295574 + 0.955320i \(0.404489\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −5.07896 −0.240227
\(448\) 0 0
\(449\) −27.6069 −1.30285 −0.651425 0.758713i \(-0.725828\pi\)
−0.651425 + 0.758713i \(0.725828\pi\)
\(450\) 0 0
\(451\) −19.7992 −0.932309
\(452\) 0 0
\(453\) 9.56404 0.449358
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 23.5725 1.10267 0.551337 0.834283i \(-0.314118\pi\)
0.551337 + 0.834283i \(0.314118\pi\)
\(458\) 0 0
\(459\) −2.29273 −0.107016
\(460\) 0 0
\(461\) −3.41454 −0.159031 −0.0795154 0.996834i \(-0.525337\pi\)
−0.0795154 + 0.996834i \(0.525337\pi\)
\(462\) 0 0
\(463\) −34.7434 −1.61466 −0.807331 0.590099i \(-0.799089\pi\)
−0.807331 + 0.590099i \(0.799089\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 16.3503 0.756600 0.378300 0.925683i \(-0.376509\pi\)
0.378300 + 0.925683i \(0.376509\pi\)
\(468\) 0 0
\(469\) −12.0575 −0.556765
\(470\) 0 0
\(471\) −15.9572 −0.735267
\(472\) 0 0
\(473\) −25.1709 −1.15736
\(474\) 0 0
\(475\) −5.00000 −0.229416
\(476\) 0 0
\(477\) −12.9786 −0.594248
\(478\) 0 0
\(479\) 12.6002 0.575716 0.287858 0.957673i \(-0.407057\pi\)
0.287858 + 0.957673i \(0.407057\pi\)
\(480\) 0 0
\(481\) −39.5296 −1.80240
\(482\) 0 0
\(483\) −4.39312 −0.199894
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 25.7648 1.16751 0.583757 0.811928i \(-0.301582\pi\)
0.583757 + 0.811928i \(0.301582\pi\)
\(488\) 0 0
\(489\) −1.37169 −0.0620301
\(490\) 0 0
\(491\) 20.0575 0.905184 0.452592 0.891718i \(-0.350499\pi\)
0.452592 + 0.891718i \(0.350499\pi\)
\(492\) 0 0
\(493\) 2.24361 0.101047
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 1.60688 0.0720786
\(498\) 0 0
\(499\) −29.2860 −1.31102 −0.655511 0.755186i \(-0.727547\pi\)
−0.655511 + 0.755186i \(0.727547\pi\)
\(500\) 0 0
\(501\) 9.37169 0.418696
\(502\) 0 0
\(503\) −19.4292 −0.866307 −0.433153 0.901320i \(-0.642599\pi\)
−0.433153 + 0.901320i \(0.642599\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −35.7005 −1.58552
\(508\) 0 0
\(509\) 20.0722 0.889686 0.444843 0.895609i \(-0.353259\pi\)
0.444843 + 0.895609i \(0.353259\pi\)
\(510\) 0 0
\(511\) −7.37169 −0.326104
\(512\) 0 0
\(513\) −1.00000 −0.0441511
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −12.5855 −0.553508
\(518\) 0 0
\(519\) 18.7862 0.824624
\(520\) 0 0
\(521\) −39.4868 −1.72995 −0.864973 0.501818i \(-0.832665\pi\)
−0.864973 + 0.501818i \(0.832665\pi\)
\(522\) 0 0
\(523\) 31.1281 1.36114 0.680568 0.732685i \(-0.261733\pi\)
0.680568 + 0.732685i \(0.261733\pi\)
\(524\) 0 0
\(525\) 5.00000 0.218218
\(526\) 0 0
\(527\) 17.5725 0.765468
\(528\) 0 0
\(529\) −3.70054 −0.160893
\(530\) 0 0
\(531\) 0.585462 0.0254069
\(532\) 0 0
\(533\) −51.4439 −2.22828
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −4.35027 −0.187728
\(538\) 0 0
\(539\) −2.68585 −0.115688
\(540\) 0 0
\(541\) 22.7862 0.979657 0.489828 0.871819i \(-0.337059\pi\)
0.489828 + 0.871819i \(0.337059\pi\)
\(542\) 0 0
\(543\) 10.9786 0.471136
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 2.30115 0.0983902 0.0491951 0.998789i \(-0.484334\pi\)
0.0491951 + 0.998789i \(0.484334\pi\)
\(548\) 0 0
\(549\) −6.58546 −0.281061
\(550\) 0 0
\(551\) 0.978577 0.0416888
\(552\) 0 0
\(553\) −14.3503 −0.610236
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 27.7367 1.17524 0.587620 0.809137i \(-0.300065\pi\)
0.587620 + 0.809137i \(0.300065\pi\)
\(558\) 0 0
\(559\) −65.4011 −2.76617
\(560\) 0 0
\(561\) 6.15792 0.259988
\(562\) 0 0
\(563\) −22.5426 −0.950058 −0.475029 0.879970i \(-0.657562\pi\)
−0.475029 + 0.879970i \(0.657562\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) −6.43596 −0.269810 −0.134905 0.990859i \(-0.543073\pi\)
−0.134905 + 0.990859i \(0.543073\pi\)
\(570\) 0 0
\(571\) 7.91431 0.331204 0.165602 0.986193i \(-0.447043\pi\)
0.165602 + 0.986193i \(0.447043\pi\)
\(572\) 0 0
\(573\) 14.3503 0.599491
\(574\) 0 0
\(575\) −21.9656 −0.916028
\(576\) 0 0
\(577\) 28.7434 1.19660 0.598301 0.801271i \(-0.295843\pi\)
0.598301 + 0.801271i \(0.295843\pi\)
\(578\) 0 0
\(579\) 13.3288 0.553928
\(580\) 0 0
\(581\) −8.35027 −0.346428
\(582\) 0 0
\(583\) 34.8585 1.44369
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 38.1067 1.57283 0.786415 0.617699i \(-0.211935\pi\)
0.786415 + 0.617699i \(0.211935\pi\)
\(588\) 0 0
\(589\) 7.66442 0.315807
\(590\) 0 0
\(591\) 26.4507 1.08803
\(592\) 0 0
\(593\) −20.4507 −0.839808 −0.419904 0.907569i \(-0.637936\pi\)
−0.419904 + 0.907569i \(0.637936\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 4.78623 0.195887
\(598\) 0 0
\(599\) 2.30742 0.0942788 0.0471394 0.998888i \(-0.484990\pi\)
0.0471394 + 0.998888i \(0.484990\pi\)
\(600\) 0 0
\(601\) 39.3435 1.60486 0.802428 0.596749i \(-0.203541\pi\)
0.802428 + 0.596749i \(0.203541\pi\)
\(602\) 0 0
\(603\) −12.0575 −0.491021
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 29.7073 1.20578 0.602890 0.797824i \(-0.294016\pi\)
0.602890 + 0.797824i \(0.294016\pi\)
\(608\) 0 0
\(609\) −0.978577 −0.0396539
\(610\) 0 0
\(611\) −32.7005 −1.32292
\(612\) 0 0
\(613\) 4.62831 0.186936 0.0934678 0.995622i \(-0.470205\pi\)
0.0934678 + 0.995622i \(0.470205\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −31.2860 −1.25953 −0.629763 0.776787i \(-0.716848\pi\)
−0.629763 + 0.776787i \(0.716848\pi\)
\(618\) 0 0
\(619\) −40.1151 −1.61236 −0.806181 0.591670i \(-0.798469\pi\)
−0.806181 + 0.591670i \(0.798469\pi\)
\(620\) 0 0
\(621\) −4.39312 −0.176290
\(622\) 0 0
\(623\) −15.9572 −0.639310
\(624\) 0 0
\(625\) 25.0000 1.00000
\(626\) 0 0
\(627\) 2.68585 0.107262
\(628\) 0 0
\(629\) 12.9870 0.517826
\(630\) 0 0
\(631\) −33.9572 −1.35181 −0.675906 0.736987i \(-0.736248\pi\)
−0.675906 + 0.736987i \(0.736248\pi\)
\(632\) 0 0
\(633\) 18.6858 0.742696
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −6.97858 −0.276501
\(638\) 0 0
\(639\) 1.60688 0.0635674
\(640\) 0 0
\(641\) 47.1365 1.86178 0.930890 0.365300i \(-0.119034\pi\)
0.930890 + 0.365300i \(0.119034\pi\)
\(642\) 0 0
\(643\) 7.61531 0.300318 0.150159 0.988662i \(-0.452021\pi\)
0.150159 + 0.988662i \(0.452021\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −35.9290 −1.41251 −0.706257 0.707955i \(-0.749618\pi\)
−0.706257 + 0.707955i \(0.749618\pi\)
\(648\) 0 0
\(649\) −1.57246 −0.0617245
\(650\) 0 0
\(651\) −7.66442 −0.300392
\(652\) 0 0
\(653\) −27.7367 −1.08542 −0.542710 0.839920i \(-0.682602\pi\)
−0.542710 + 0.839920i \(0.682602\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −7.37169 −0.287597
\(658\) 0 0
\(659\) −6.39312 −0.249040 −0.124520 0.992217i \(-0.539739\pi\)
−0.124520 + 0.992217i \(0.539739\pi\)
\(660\) 0 0
\(661\) 17.1365 0.666533 0.333266 0.942833i \(-0.391849\pi\)
0.333266 + 0.942833i \(0.391849\pi\)
\(662\) 0 0
\(663\) 16.0000 0.621389
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 4.29900 0.166458
\(668\) 0 0
\(669\) −22.2070 −0.858574
\(670\) 0 0
\(671\) 17.6875 0.682820
\(672\) 0 0
\(673\) 39.8715 1.53693 0.768466 0.639891i \(-0.221020\pi\)
0.768466 + 0.639891i \(0.221020\pi\)
\(674\) 0 0
\(675\) 5.00000 0.192450
\(676\) 0 0
\(677\) −23.9572 −0.920748 −0.460374 0.887725i \(-0.652285\pi\)
−0.460374 + 0.887725i \(0.652285\pi\)
\(678\) 0 0
\(679\) 5.27131 0.202294
\(680\) 0 0
\(681\) 2.62831 0.100717
\(682\) 0 0
\(683\) 17.7220 0.678112 0.339056 0.940766i \(-0.389892\pi\)
0.339056 + 0.940766i \(0.389892\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −11.9572 −0.456194
\(688\) 0 0
\(689\) 90.5720 3.45052
\(690\) 0 0
\(691\) 1.25662 0.0478039 0.0239020 0.999714i \(-0.492391\pi\)
0.0239020 + 0.999714i \(0.492391\pi\)
\(692\) 0 0
\(693\) −2.68585 −0.102027
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 16.9013 0.640183
\(698\) 0 0
\(699\) 6.58546 0.249085
\(700\) 0 0
\(701\) −22.9210 −0.865716 −0.432858 0.901462i \(-0.642495\pi\)
−0.432858 + 0.901462i \(0.642495\pi\)
\(702\) 0 0
\(703\) 5.66442 0.213638
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −8.00000 −0.300871
\(708\) 0 0
\(709\) 7.75639 0.291297 0.145649 0.989336i \(-0.453473\pi\)
0.145649 + 0.989336i \(0.453473\pi\)
\(710\) 0 0
\(711\) −14.3503 −0.538177
\(712\) 0 0
\(713\) 33.6707 1.26098
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 12.9786 0.484694
\(718\) 0 0
\(719\) 34.3565 1.28128 0.640641 0.767840i \(-0.278669\pi\)
0.640641 + 0.767840i \(0.278669\pi\)
\(720\) 0 0
\(721\) 11.0790 0.412602
\(722\) 0 0
\(723\) −6.72869 −0.250243
\(724\) 0 0
\(725\) −4.89289 −0.181717
\(726\) 0 0
\(727\) −16.4998 −0.611943 −0.305971 0.952041i \(-0.598981\pi\)
−0.305971 + 0.952041i \(0.598981\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 21.4868 0.794717
\(732\) 0 0
\(733\) 3.57246 0.131952 0.0659759 0.997821i \(-0.478984\pi\)
0.0659759 + 0.997821i \(0.478984\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 32.3847 1.19291
\(738\) 0 0
\(739\) 6.74338 0.248059 0.124030 0.992279i \(-0.460418\pi\)
0.124030 + 0.992279i \(0.460418\pi\)
\(740\) 0 0
\(741\) 6.97858 0.256364
\(742\) 0 0
\(743\) −36.2646 −1.33042 −0.665209 0.746657i \(-0.731658\pi\)
−0.665209 + 0.746657i \(0.731658\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −8.35027 −0.305520
\(748\) 0 0
\(749\) 7.56404 0.276384
\(750\) 0 0
\(751\) −45.5934 −1.66373 −0.831864 0.554980i \(-0.812726\pi\)
−0.831864 + 0.554980i \(0.812726\pi\)
\(752\) 0 0
\(753\) −1.80765 −0.0658745
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 24.0722 0.874920 0.437460 0.899238i \(-0.355878\pi\)
0.437460 + 0.899238i \(0.355878\pi\)
\(758\) 0 0
\(759\) 11.7992 0.428285
\(760\) 0 0
\(761\) 40.3650 1.46323 0.731614 0.681719i \(-0.238767\pi\)
0.731614 + 0.681719i \(0.238767\pi\)
\(762\) 0 0
\(763\) −4.87819 −0.176602
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −4.08569 −0.147526
\(768\) 0 0
\(769\) 50.4998 1.82107 0.910534 0.413434i \(-0.135671\pi\)
0.910534 + 0.413434i \(0.135671\pi\)
\(770\) 0 0
\(771\) 15.3717 0.553598
\(772\) 0 0
\(773\) −27.2860 −0.981409 −0.490705 0.871326i \(-0.663261\pi\)
−0.490705 + 0.871326i \(0.663261\pi\)
\(774\) 0 0
\(775\) −38.3221 −1.37657
\(776\) 0 0
\(777\) −5.66442 −0.203210
\(778\) 0 0
\(779\) 7.37169 0.264118
\(780\) 0 0
\(781\) −4.31585 −0.154433
\(782\) 0 0
\(783\) −0.978577 −0.0349715
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 13.2860 0.473595 0.236797 0.971559i \(-0.423902\pi\)
0.236797 + 0.971559i \(0.423902\pi\)
\(788\) 0 0
\(789\) 18.3503 0.653287
\(790\) 0 0
\(791\) 10.3503 0.368013
\(792\) 0 0
\(793\) 45.9572 1.63199
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −26.7862 −0.948817 −0.474408 0.880305i \(-0.657338\pi\)
−0.474408 + 0.880305i \(0.657338\pi\)
\(798\) 0 0
\(799\) 10.7434 0.380074
\(800\) 0 0
\(801\) −15.9572 −0.563818
\(802\) 0 0
\(803\) 19.7992 0.698700
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 10.5855 0.372626
\(808\) 0 0
\(809\) 14.5855 0.512798 0.256399 0.966571i \(-0.417464\pi\)
0.256399 + 0.966571i \(0.417464\pi\)
\(810\) 0 0
\(811\) −2.04285 −0.0717340 −0.0358670 0.999357i \(-0.511419\pi\)
−0.0358670 + 0.999357i \(0.511419\pi\)
\(812\) 0 0
\(813\) −8.00000 −0.280572
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 9.37169 0.327874
\(818\) 0 0
\(819\) −6.97858 −0.243851
\(820\) 0 0
\(821\) 16.0920 0.561613 0.280807 0.959764i \(-0.409398\pi\)
0.280807 + 0.959764i \(0.409398\pi\)
\(822\) 0 0
\(823\) −6.82908 −0.238047 −0.119023 0.992891i \(-0.537976\pi\)
−0.119023 + 0.992891i \(0.537976\pi\)
\(824\) 0 0
\(825\) −13.4292 −0.467546
\(826\) 0 0
\(827\) −33.4355 −1.16267 −0.581333 0.813666i \(-0.697469\pi\)
−0.581333 + 0.813666i \(0.697469\pi\)
\(828\) 0 0
\(829\) 31.4783 1.09329 0.546644 0.837365i \(-0.315905\pi\)
0.546644 + 0.837365i \(0.315905\pi\)
\(830\) 0 0
\(831\) −1.21377 −0.0421052
\(832\) 0 0
\(833\) 2.29273 0.0794384
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −7.66442 −0.264921
\(838\) 0 0
\(839\) 18.0722 0.623923 0.311961 0.950095i \(-0.399014\pi\)
0.311961 + 0.950095i \(0.399014\pi\)
\(840\) 0 0
\(841\) −28.0424 −0.966979
\(842\) 0 0
\(843\) 25.5640 0.880472
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −3.78623 −0.130096
\(848\) 0 0
\(849\) 22.5426 0.773661
\(850\) 0 0
\(851\) 24.8845 0.853028
\(852\) 0 0
\(853\) −10.4998 −0.359505 −0.179753 0.983712i \(-0.557530\pi\)
−0.179753 + 0.983712i \(0.557530\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −22.6148 −0.772508 −0.386254 0.922392i \(-0.626231\pi\)
−0.386254 + 0.922392i \(0.626231\pi\)
\(858\) 0 0
\(859\) 16.9870 0.579589 0.289794 0.957089i \(-0.406413\pi\)
0.289794 + 0.957089i \(0.406413\pi\)
\(860\) 0 0
\(861\) −7.37169 −0.251227
\(862\) 0 0
\(863\) −35.9656 −1.22428 −0.612141 0.790748i \(-0.709692\pi\)
−0.612141 + 0.790748i \(0.709692\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 11.7434 0.398826
\(868\) 0 0
\(869\) 38.5426 1.30747
\(870\) 0 0
\(871\) 84.1445 2.85113
\(872\) 0 0
\(873\) 5.27131 0.178407
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −23.4208 −0.790865 −0.395432 0.918495i \(-0.629405\pi\)
−0.395432 + 0.918495i \(0.629405\pi\)
\(878\) 0 0
\(879\) 19.2860 0.650501
\(880\) 0 0
\(881\) −37.0361 −1.24778 −0.623889 0.781513i \(-0.714448\pi\)
−0.623889 + 0.781513i \(0.714448\pi\)
\(882\) 0 0
\(883\) 2.54262 0.0855658 0.0427829 0.999084i \(-0.486378\pi\)
0.0427829 + 0.999084i \(0.486378\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 1.37169 0.0460569 0.0230285 0.999735i \(-0.492669\pi\)
0.0230285 + 0.999735i \(0.492669\pi\)
\(888\) 0 0
\(889\) 0.393115 0.0131847
\(890\) 0 0
\(891\) −2.68585 −0.0899792
\(892\) 0 0
\(893\) 4.68585 0.156806
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 30.6577 1.02363
\(898\) 0 0
\(899\) 7.50023 0.250147
\(900\) 0 0
\(901\) −29.7564 −0.991329
\(902\) 0 0
\(903\) −9.37169 −0.311870
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −11.0705 −0.367591 −0.183796 0.982964i \(-0.558838\pi\)
−0.183796 + 0.982964i \(0.558838\pi\)
\(908\) 0 0
\(909\) −8.00000 −0.265343
\(910\) 0 0
\(911\) 37.0508 1.22755 0.613774 0.789482i \(-0.289651\pi\)
0.613774 + 0.789482i \(0.289651\pi\)
\(912\) 0 0
\(913\) 22.4275 0.742243
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 1.80765 0.0596940
\(918\) 0 0
\(919\) 29.4868 0.972679 0.486339 0.873770i \(-0.338332\pi\)
0.486339 + 0.873770i \(0.338332\pi\)
\(920\) 0 0
\(921\) 4.78623 0.157712
\(922\) 0 0
\(923\) −11.2138 −0.369106
\(924\) 0 0
\(925\) −28.3221 −0.931225
\(926\) 0 0
\(927\) 11.0790 0.363881
\(928\) 0 0
\(929\) −37.5359 −1.23151 −0.615756 0.787937i \(-0.711149\pi\)
−0.615756 + 0.787937i \(0.711149\pi\)
\(930\) 0 0
\(931\) 1.00000 0.0327737
\(932\) 0 0
\(933\) 19.4292 0.636084
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 2.78623 0.0910222 0.0455111 0.998964i \(-0.485508\pi\)
0.0455111 + 0.998964i \(0.485508\pi\)
\(938\) 0 0
\(939\) −25.3288 −0.826576
\(940\) 0 0
\(941\) −53.3288 −1.73847 −0.869235 0.494399i \(-0.835388\pi\)
−0.869235 + 0.494399i \(0.835388\pi\)
\(942\) 0 0
\(943\) 32.3847 1.05459
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 40.8438 1.32724 0.663622 0.748068i \(-0.269018\pi\)
0.663622 + 0.748068i \(0.269018\pi\)
\(948\) 0 0
\(949\) 51.4439 1.66994
\(950\) 0 0
\(951\) −29.6791 −0.962411
\(952\) 0 0
\(953\) 19.0508 0.617116 0.308558 0.951205i \(-0.400154\pi\)
0.308558 + 0.951205i \(0.400154\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 2.62831 0.0849611
\(958\) 0 0
\(959\) −14.7862 −0.477472
\(960\) 0 0
\(961\) 27.7434 0.894948
\(962\) 0 0
\(963\) 7.56404 0.243748
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 0.384694 0.0123709 0.00618546 0.999981i \(-0.498031\pi\)
0.00618546 + 0.999981i \(0.498031\pi\)
\(968\) 0 0
\(969\) −2.29273 −0.0736531
\(970\) 0 0
\(971\) −4.49977 −0.144405 −0.0722023 0.997390i \(-0.523003\pi\)
−0.0722023 + 0.997390i \(0.523003\pi\)
\(972\) 0 0
\(973\) 13.3717 0.428677
\(974\) 0 0
\(975\) −34.8929 −1.11747
\(976\) 0 0
\(977\) 35.4355 1.13368 0.566841 0.823827i \(-0.308165\pi\)
0.566841 + 0.823827i \(0.308165\pi\)
\(978\) 0 0
\(979\) 42.8585 1.36976
\(980\) 0 0
\(981\) −4.87819 −0.155749
\(982\) 0 0
\(983\) 1.17092 0.0373467 0.0186733 0.999826i \(-0.494056\pi\)
0.0186733 + 0.999826i \(0.494056\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −4.68585 −0.149152
\(988\) 0 0
\(989\) 41.1709 1.30916
\(990\) 0 0
\(991\) −25.7648 −0.818446 −0.409223 0.912434i \(-0.634200\pi\)
−0.409223 + 0.912434i \(0.634200\pi\)
\(992\) 0 0
\(993\) −5.22846 −0.165920
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −25.4439 −0.805817 −0.402909 0.915240i \(-0.632001\pi\)
−0.402909 + 0.915240i \(0.632001\pi\)
\(998\) 0 0
\(999\) −5.66442 −0.179214
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 6384.2.a.bv.1.1 3
4.3 odd 2 3192.2.a.v.1.3 3
12.11 even 2 9576.2.a.cc.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3192.2.a.v.1.3 3 4.3 odd 2
6384.2.a.bv.1.1 3 1.1 even 1 trivial
9576.2.a.cc.1.1 3 12.11 even 2