Properties

Label 7168.2.a.bf.1.5
Level $7168$
Weight $2$
Character 7168.1
Self dual yes
Analytic conductor $57.237$
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] = [7168,2,Mod(1,7168)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7168, 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("7168.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7168 = 2^{10} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7168.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: \(57.2367681689\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.8.9433055232.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} - 6x^{6} + 32x^{5} + 9x^{4} - 76x^{3} - 4x^{2} + 48x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 1792)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-1.21236\) of defining polynomial
Character \(\chi\) \(=\) 7168.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.463900 q^{3} -1.98268 q^{5} +1.00000 q^{7} -2.78480 q^{9} +O(q^{10})\) \(q+0.463900 q^{3} -1.98268 q^{5} +1.00000 q^{7} -2.78480 q^{9} +0.628034 q^{11} -3.19033 q^{13} -0.919765 q^{15} -4.85578 q^{17} +0.161226 q^{19} +0.463900 q^{21} +3.20529 q^{23} -1.06898 q^{25} -2.68357 q^{27} -1.41010 q^{29} -5.34435 q^{31} +0.291345 q^{33} -1.98268 q^{35} -2.87754 q^{37} -1.47999 q^{39} -9.57673 q^{41} +9.71222 q^{43} +5.52136 q^{45} +9.70703 q^{47} +1.00000 q^{49} -2.25259 q^{51} +10.8106 q^{53} -1.24519 q^{55} +0.0747927 q^{57} -2.12508 q^{59} -2.46295 q^{61} -2.78480 q^{63} +6.32541 q^{65} +12.6783 q^{67} +1.48693 q^{69} +7.18356 q^{71} -9.04029 q^{73} -0.495897 q^{75} +0.628034 q^{77} +9.58806 q^{79} +7.10949 q^{81} -7.49880 q^{83} +9.62746 q^{85} -0.654145 q^{87} -2.49938 q^{89} -3.19033 q^{91} -2.47924 q^{93} -0.319660 q^{95} +5.89073 q^{97} -1.74895 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{5} + 8 q^{7} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{5} + 8 q^{7} + 12 q^{9} + 12 q^{11} + 20 q^{13} + 4 q^{17} + 4 q^{19} + 8 q^{23} + 12 q^{25} - 12 q^{27} + 8 q^{29} - 4 q^{31} + 8 q^{33} + 8 q^{35} + 8 q^{37} - 16 q^{39} - 12 q^{41} - 4 q^{43} + 52 q^{45} + 20 q^{47} + 8 q^{49} + 32 q^{51} + 40 q^{53} + 24 q^{55} - 4 q^{57} + 4 q^{59} - 8 q^{61} + 12 q^{63} + 36 q^{65} + 28 q^{67} + 4 q^{69} - 16 q^{71} + 16 q^{73} - 28 q^{75} + 12 q^{77} + 20 q^{81} - 8 q^{83} + 16 q^{85} + 20 q^{87} + 16 q^{89} + 20 q^{91} + 16 q^{93} - 40 q^{95} - 36 q^{97} - 4 q^{99}+O(q^{100}) \) 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
\(3\) 0.463900 0.267833 0.133916 0.990993i \(-0.457245\pi\)
0.133916 + 0.990993i \(0.457245\pi\)
\(4\) 0 0
\(5\) −1.98268 −0.886682 −0.443341 0.896353i \(-0.646207\pi\)
−0.443341 + 0.896353i \(0.646207\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) −2.78480 −0.928266
\(10\) 0 0
\(11\) 0.628034 0.189359 0.0946797 0.995508i \(-0.469817\pi\)
0.0946797 + 0.995508i \(0.469817\pi\)
\(12\) 0 0
\(13\) −3.19033 −0.884839 −0.442419 0.896808i \(-0.645880\pi\)
−0.442419 + 0.896808i \(0.645880\pi\)
\(14\) 0 0
\(15\) −0.919765 −0.237482
\(16\) 0 0
\(17\) −4.85578 −1.17770 −0.588849 0.808243i \(-0.700419\pi\)
−0.588849 + 0.808243i \(0.700419\pi\)
\(18\) 0 0
\(19\) 0.161226 0.0369878 0.0184939 0.999829i \(-0.494113\pi\)
0.0184939 + 0.999829i \(0.494113\pi\)
\(20\) 0 0
\(21\) 0.463900 0.101231
\(22\) 0 0
\(23\) 3.20529 0.668350 0.334175 0.942511i \(-0.391542\pi\)
0.334175 + 0.942511i \(0.391542\pi\)
\(24\) 0 0
\(25\) −1.06898 −0.213795
\(26\) 0 0
\(27\) −2.68357 −0.516452
\(28\) 0 0
\(29\) −1.41010 −0.261849 −0.130924 0.991392i \(-0.541795\pi\)
−0.130924 + 0.991392i \(0.541795\pi\)
\(30\) 0 0
\(31\) −5.34435 −0.959874 −0.479937 0.877303i \(-0.659340\pi\)
−0.479937 + 0.877303i \(0.659340\pi\)
\(32\) 0 0
\(33\) 0.291345 0.0507167
\(34\) 0 0
\(35\) −1.98268 −0.335134
\(36\) 0 0
\(37\) −2.87754 −0.473064 −0.236532 0.971624i \(-0.576011\pi\)
−0.236532 + 0.971624i \(0.576011\pi\)
\(38\) 0 0
\(39\) −1.47999 −0.236989
\(40\) 0 0
\(41\) −9.57673 −1.49563 −0.747817 0.663905i \(-0.768898\pi\)
−0.747817 + 0.663905i \(0.768898\pi\)
\(42\) 0 0
\(43\) 9.71222 1.48110 0.740550 0.672001i \(-0.234565\pi\)
0.740550 + 0.672001i \(0.234565\pi\)
\(44\) 0 0
\(45\) 5.52136 0.823076
\(46\) 0 0
\(47\) 9.70703 1.41592 0.707958 0.706255i \(-0.249617\pi\)
0.707958 + 0.706255i \(0.249617\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −2.25259 −0.315426
\(52\) 0 0
\(53\) 10.8106 1.48495 0.742476 0.669872i \(-0.233651\pi\)
0.742476 + 0.669872i \(0.233651\pi\)
\(54\) 0 0
\(55\) −1.24519 −0.167902
\(56\) 0 0
\(57\) 0.0747927 0.00990653
\(58\) 0 0
\(59\) −2.12508 −0.276662 −0.138331 0.990386i \(-0.544174\pi\)
−0.138331 + 0.990386i \(0.544174\pi\)
\(60\) 0 0
\(61\) −2.46295 −0.315348 −0.157674 0.987491i \(-0.550399\pi\)
−0.157674 + 0.987491i \(0.550399\pi\)
\(62\) 0 0
\(63\) −2.78480 −0.350851
\(64\) 0 0
\(65\) 6.32541 0.784571
\(66\) 0 0
\(67\) 12.6783 1.54890 0.774450 0.632635i \(-0.218027\pi\)
0.774450 + 0.632635i \(0.218027\pi\)
\(68\) 0 0
\(69\) 1.48693 0.179006
\(70\) 0 0
\(71\) 7.18356 0.852532 0.426266 0.904598i \(-0.359829\pi\)
0.426266 + 0.904598i \(0.359829\pi\)
\(72\) 0 0
\(73\) −9.04029 −1.05809 −0.529043 0.848595i \(-0.677449\pi\)
−0.529043 + 0.848595i \(0.677449\pi\)
\(74\) 0 0
\(75\) −0.495897 −0.0572613
\(76\) 0 0
\(77\) 0.628034 0.0715712
\(78\) 0 0
\(79\) 9.58806 1.07874 0.539370 0.842069i \(-0.318662\pi\)
0.539370 + 0.842069i \(0.318662\pi\)
\(80\) 0 0
\(81\) 7.10949 0.789943
\(82\) 0 0
\(83\) −7.49880 −0.823100 −0.411550 0.911387i \(-0.635012\pi\)
−0.411550 + 0.911387i \(0.635012\pi\)
\(84\) 0 0
\(85\) 9.62746 1.04424
\(86\) 0 0
\(87\) −0.654145 −0.0701317
\(88\) 0 0
\(89\) −2.49938 −0.264934 −0.132467 0.991187i \(-0.542290\pi\)
−0.132467 + 0.991187i \(0.542290\pi\)
\(90\) 0 0
\(91\) −3.19033 −0.334438
\(92\) 0 0
\(93\) −2.47924 −0.257086
\(94\) 0 0
\(95\) −0.319660 −0.0327964
\(96\) 0 0
\(97\) 5.89073 0.598113 0.299057 0.954235i \(-0.403328\pi\)
0.299057 + 0.954235i \(0.403328\pi\)
\(98\) 0 0
\(99\) −1.74895 −0.175776
\(100\) 0 0
\(101\) 9.90655 0.985738 0.492869 0.870103i \(-0.335948\pi\)
0.492869 + 0.870103i \(0.335948\pi\)
\(102\) 0 0
\(103\) −8.10887 −0.798991 −0.399496 0.916735i \(-0.630815\pi\)
−0.399496 + 0.916735i \(0.630815\pi\)
\(104\) 0 0
\(105\) −0.919765 −0.0897599
\(106\) 0 0
\(107\) −3.17303 −0.306749 −0.153374 0.988168i \(-0.549014\pi\)
−0.153374 + 0.988168i \(0.549014\pi\)
\(108\) 0 0
\(109\) −0.423434 −0.0405576 −0.0202788 0.999794i \(-0.506455\pi\)
−0.0202788 + 0.999794i \(0.506455\pi\)
\(110\) 0 0
\(111\) −1.33489 −0.126702
\(112\) 0 0
\(113\) −4.67432 −0.439723 −0.219862 0.975531i \(-0.570561\pi\)
−0.219862 + 0.975531i \(0.570561\pi\)
\(114\) 0 0
\(115\) −6.35507 −0.592614
\(116\) 0 0
\(117\) 8.88442 0.821365
\(118\) 0 0
\(119\) −4.85578 −0.445128
\(120\) 0 0
\(121\) −10.6056 −0.964143
\(122\) 0 0
\(123\) −4.44264 −0.400579
\(124\) 0 0
\(125\) 12.0328 1.07625
\(126\) 0 0
\(127\) −13.7602 −1.22102 −0.610509 0.792009i \(-0.709035\pi\)
−0.610509 + 0.792009i \(0.709035\pi\)
\(128\) 0 0
\(129\) 4.50550 0.396687
\(130\) 0 0
\(131\) 9.78426 0.854855 0.427427 0.904050i \(-0.359420\pi\)
0.427427 + 0.904050i \(0.359420\pi\)
\(132\) 0 0
\(133\) 0.161226 0.0139801
\(134\) 0 0
\(135\) 5.32066 0.457929
\(136\) 0 0
\(137\) −7.10937 −0.607395 −0.303697 0.952769i \(-0.598221\pi\)
−0.303697 + 0.952769i \(0.598221\pi\)
\(138\) 0 0
\(139\) 15.2977 1.29754 0.648768 0.760987i \(-0.275285\pi\)
0.648768 + 0.760987i \(0.275285\pi\)
\(140\) 0 0
\(141\) 4.50309 0.379228
\(142\) 0 0
\(143\) −2.00364 −0.167553
\(144\) 0 0
\(145\) 2.79578 0.232177
\(146\) 0 0
\(147\) 0.463900 0.0382618
\(148\) 0 0
\(149\) 21.6738 1.77559 0.887794 0.460241i \(-0.152237\pi\)
0.887794 + 0.460241i \(0.152237\pi\)
\(150\) 0 0
\(151\) 21.5070 1.75022 0.875109 0.483925i \(-0.160789\pi\)
0.875109 + 0.483925i \(0.160789\pi\)
\(152\) 0 0
\(153\) 13.5224 1.09322
\(154\) 0 0
\(155\) 10.5961 0.851103
\(156\) 0 0
\(157\) 10.9266 0.872036 0.436018 0.899938i \(-0.356388\pi\)
0.436018 + 0.899938i \(0.356388\pi\)
\(158\) 0 0
\(159\) 5.01504 0.397719
\(160\) 0 0
\(161\) 3.20529 0.252612
\(162\) 0 0
\(163\) −23.9477 −1.87573 −0.937864 0.347003i \(-0.887199\pi\)
−0.937864 + 0.347003i \(0.887199\pi\)
\(164\) 0 0
\(165\) −0.577644 −0.0449695
\(166\) 0 0
\(167\) −8.57678 −0.663691 −0.331846 0.943334i \(-0.607671\pi\)
−0.331846 + 0.943334i \(0.607671\pi\)
\(168\) 0 0
\(169\) −2.82179 −0.217061
\(170\) 0 0
\(171\) −0.448981 −0.0343345
\(172\) 0 0
\(173\) 7.69377 0.584947 0.292473 0.956274i \(-0.405522\pi\)
0.292473 + 0.956274i \(0.405522\pi\)
\(174\) 0 0
\(175\) −1.06898 −0.0808069
\(176\) 0 0
\(177\) −0.985825 −0.0740991
\(178\) 0 0
\(179\) −11.7768 −0.880242 −0.440121 0.897939i \(-0.645064\pi\)
−0.440121 + 0.897939i \(0.645064\pi\)
\(180\) 0 0
\(181\) −8.82678 −0.656090 −0.328045 0.944662i \(-0.606390\pi\)
−0.328045 + 0.944662i \(0.606390\pi\)
\(182\) 0 0
\(183\) −1.14256 −0.0844605
\(184\) 0 0
\(185\) 5.70524 0.419457
\(186\) 0 0
\(187\) −3.04960 −0.223008
\(188\) 0 0
\(189\) −2.68357 −0.195201
\(190\) 0 0
\(191\) 1.57521 0.113979 0.0569893 0.998375i \(-0.481850\pi\)
0.0569893 + 0.998375i \(0.481850\pi\)
\(192\) 0 0
\(193\) 0.649369 0.0467426 0.0233713 0.999727i \(-0.492560\pi\)
0.0233713 + 0.999727i \(0.492560\pi\)
\(194\) 0 0
\(195\) 2.93436 0.210134
\(196\) 0 0
\(197\) 1.65273 0.117752 0.0588760 0.998265i \(-0.481248\pi\)
0.0588760 + 0.998265i \(0.481248\pi\)
\(198\) 0 0
\(199\) −4.09013 −0.289942 −0.144971 0.989436i \(-0.546309\pi\)
−0.144971 + 0.989436i \(0.546309\pi\)
\(200\) 0 0
\(201\) 5.88146 0.414846
\(202\) 0 0
\(203\) −1.41010 −0.0989696
\(204\) 0 0
\(205\) 18.9876 1.32615
\(206\) 0 0
\(207\) −8.92609 −0.620406
\(208\) 0 0
\(209\) 0.101255 0.00700398
\(210\) 0 0
\(211\) 26.9295 1.85391 0.926953 0.375177i \(-0.122418\pi\)
0.926953 + 0.375177i \(0.122418\pi\)
\(212\) 0 0
\(213\) 3.33245 0.228336
\(214\) 0 0
\(215\) −19.2562 −1.31326
\(216\) 0 0
\(217\) −5.34435 −0.362798
\(218\) 0 0
\(219\) −4.19379 −0.283390
\(220\) 0 0
\(221\) 15.4915 1.04207
\(222\) 0 0
\(223\) 26.0121 1.74190 0.870950 0.491372i \(-0.163504\pi\)
0.870950 + 0.491372i \(0.163504\pi\)
\(224\) 0 0
\(225\) 2.97688 0.198459
\(226\) 0 0
\(227\) 0.256838 0.0170470 0.00852348 0.999964i \(-0.497287\pi\)
0.00852348 + 0.999964i \(0.497287\pi\)
\(228\) 0 0
\(229\) 0.961093 0.0635108 0.0317554 0.999496i \(-0.489890\pi\)
0.0317554 + 0.999496i \(0.489890\pi\)
\(230\) 0 0
\(231\) 0.291345 0.0191691
\(232\) 0 0
\(233\) 19.6676 1.28847 0.644235 0.764828i \(-0.277176\pi\)
0.644235 + 0.764828i \(0.277176\pi\)
\(234\) 0 0
\(235\) −19.2459 −1.25547
\(236\) 0 0
\(237\) 4.44790 0.288922
\(238\) 0 0
\(239\) 9.44846 0.611170 0.305585 0.952165i \(-0.401148\pi\)
0.305585 + 0.952165i \(0.401148\pi\)
\(240\) 0 0
\(241\) 22.8233 1.47018 0.735088 0.677971i \(-0.237141\pi\)
0.735088 + 0.677971i \(0.237141\pi\)
\(242\) 0 0
\(243\) 11.3488 0.728025
\(244\) 0 0
\(245\) −1.98268 −0.126669
\(246\) 0 0
\(247\) −0.514364 −0.0327282
\(248\) 0 0
\(249\) −3.47869 −0.220453
\(250\) 0 0
\(251\) 17.3482 1.09501 0.547504 0.836803i \(-0.315578\pi\)
0.547504 + 0.836803i \(0.315578\pi\)
\(252\) 0 0
\(253\) 2.01303 0.126558
\(254\) 0 0
\(255\) 4.46618 0.279683
\(256\) 0 0
\(257\) 7.71213 0.481070 0.240535 0.970641i \(-0.422677\pi\)
0.240535 + 0.970641i \(0.422677\pi\)
\(258\) 0 0
\(259\) −2.87754 −0.178801
\(260\) 0 0
\(261\) 3.92684 0.243065
\(262\) 0 0
\(263\) 9.24972 0.570362 0.285181 0.958474i \(-0.407946\pi\)
0.285181 + 0.958474i \(0.407946\pi\)
\(264\) 0 0
\(265\) −21.4340 −1.31668
\(266\) 0 0
\(267\) −1.15946 −0.0709578
\(268\) 0 0
\(269\) −11.8311 −0.721356 −0.360678 0.932690i \(-0.617455\pi\)
−0.360678 + 0.932690i \(0.617455\pi\)
\(270\) 0 0
\(271\) 1.15740 0.0703071 0.0351535 0.999382i \(-0.488808\pi\)
0.0351535 + 0.999382i \(0.488808\pi\)
\(272\) 0 0
\(273\) −1.47999 −0.0895733
\(274\) 0 0
\(275\) −0.671353 −0.0404841
\(276\) 0 0
\(277\) −23.6068 −1.41839 −0.709197 0.705011i \(-0.750942\pi\)
−0.709197 + 0.705011i \(0.750942\pi\)
\(278\) 0 0
\(279\) 14.8829 0.891018
\(280\) 0 0
\(281\) −29.3656 −1.75180 −0.875902 0.482490i \(-0.839732\pi\)
−0.875902 + 0.482490i \(0.839732\pi\)
\(282\) 0 0
\(283\) −18.4557 −1.09708 −0.548539 0.836125i \(-0.684816\pi\)
−0.548539 + 0.836125i \(0.684816\pi\)
\(284\) 0 0
\(285\) −0.148290 −0.00878394
\(286\) 0 0
\(287\) −9.57673 −0.565296
\(288\) 0 0
\(289\) 6.57857 0.386975
\(290\) 0 0
\(291\) 2.73271 0.160194
\(292\) 0 0
\(293\) 27.4972 1.60641 0.803203 0.595705i \(-0.203127\pi\)
0.803203 + 0.595705i \(0.203127\pi\)
\(294\) 0 0
\(295\) 4.21336 0.245311
\(296\) 0 0
\(297\) −1.68537 −0.0977952
\(298\) 0 0
\(299\) −10.2259 −0.591382
\(300\) 0 0
\(301\) 9.71222 0.559803
\(302\) 0 0
\(303\) 4.59564 0.264013
\(304\) 0 0
\(305\) 4.88324 0.279613
\(306\) 0 0
\(307\) −27.6567 −1.57845 −0.789226 0.614103i \(-0.789518\pi\)
−0.789226 + 0.614103i \(0.789518\pi\)
\(308\) 0 0
\(309\) −3.76170 −0.213996
\(310\) 0 0
\(311\) −24.5927 −1.39453 −0.697263 0.716815i \(-0.745599\pi\)
−0.697263 + 0.716815i \(0.745599\pi\)
\(312\) 0 0
\(313\) −18.0125 −1.01812 −0.509062 0.860730i \(-0.670008\pi\)
−0.509062 + 0.860730i \(0.670008\pi\)
\(314\) 0 0
\(315\) 5.52136 0.311094
\(316\) 0 0
\(317\) 13.7247 0.770856 0.385428 0.922738i \(-0.374054\pi\)
0.385428 + 0.922738i \(0.374054\pi\)
\(318\) 0 0
\(319\) −0.885591 −0.0495836
\(320\) 0 0
\(321\) −1.47197 −0.0821573
\(322\) 0 0
\(323\) −0.782877 −0.0435604
\(324\) 0 0
\(325\) 3.41038 0.189174
\(326\) 0 0
\(327\) −0.196431 −0.0108627
\(328\) 0 0
\(329\) 9.70703 0.535166
\(330\) 0 0
\(331\) 2.12614 0.116863 0.0584317 0.998291i \(-0.481390\pi\)
0.0584317 + 0.998291i \(0.481390\pi\)
\(332\) 0 0
\(333\) 8.01335 0.439129
\(334\) 0 0
\(335\) −25.1370 −1.37338
\(336\) 0 0
\(337\) 17.7244 0.965512 0.482756 0.875755i \(-0.339636\pi\)
0.482756 + 0.875755i \(0.339636\pi\)
\(338\) 0 0
\(339\) −2.16842 −0.117772
\(340\) 0 0
\(341\) −3.35644 −0.181761
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) −2.94812 −0.158721
\(346\) 0 0
\(347\) −6.95932 −0.373596 −0.186798 0.982398i \(-0.559811\pi\)
−0.186798 + 0.982398i \(0.559811\pi\)
\(348\) 0 0
\(349\) 35.2104 1.88477 0.942386 0.334527i \(-0.108577\pi\)
0.942386 + 0.334527i \(0.108577\pi\)
\(350\) 0 0
\(351\) 8.56146 0.456977
\(352\) 0 0
\(353\) −5.71306 −0.304075 −0.152038 0.988375i \(-0.548584\pi\)
−0.152038 + 0.988375i \(0.548584\pi\)
\(354\) 0 0
\(355\) −14.2427 −0.755925
\(356\) 0 0
\(357\) −2.25259 −0.119220
\(358\) 0 0
\(359\) 25.2978 1.33516 0.667582 0.744536i \(-0.267329\pi\)
0.667582 + 0.744536i \(0.267329\pi\)
\(360\) 0 0
\(361\) −18.9740 −0.998632
\(362\) 0 0
\(363\) −4.91992 −0.258229
\(364\) 0 0
\(365\) 17.9240 0.938186
\(366\) 0 0
\(367\) −13.9944 −0.730499 −0.365250 0.930910i \(-0.619016\pi\)
−0.365250 + 0.930910i \(0.619016\pi\)
\(368\) 0 0
\(369\) 26.6692 1.38835
\(370\) 0 0
\(371\) 10.8106 0.561259
\(372\) 0 0
\(373\) −9.11095 −0.471747 −0.235874 0.971784i \(-0.575795\pi\)
−0.235874 + 0.971784i \(0.575795\pi\)
\(374\) 0 0
\(375\) 5.58203 0.288255
\(376\) 0 0
\(377\) 4.49869 0.231694
\(378\) 0 0
\(379\) 28.2898 1.45315 0.726576 0.687086i \(-0.241111\pi\)
0.726576 + 0.687086i \(0.241111\pi\)
\(380\) 0 0
\(381\) −6.38334 −0.327029
\(382\) 0 0
\(383\) −8.23256 −0.420664 −0.210332 0.977630i \(-0.567455\pi\)
−0.210332 + 0.977630i \(0.567455\pi\)
\(384\) 0 0
\(385\) −1.24519 −0.0634609
\(386\) 0 0
\(387\) −27.0466 −1.37485
\(388\) 0 0
\(389\) 32.2729 1.63630 0.818150 0.575004i \(-0.195000\pi\)
0.818150 + 0.575004i \(0.195000\pi\)
\(390\) 0 0
\(391\) −15.5642 −0.787115
\(392\) 0 0
\(393\) 4.53891 0.228958
\(394\) 0 0
\(395\) −19.0101 −0.956500
\(396\) 0 0
\(397\) 28.6434 1.43757 0.718786 0.695231i \(-0.244698\pi\)
0.718786 + 0.695231i \(0.244698\pi\)
\(398\) 0 0
\(399\) 0.0747927 0.00374432
\(400\) 0 0
\(401\) 20.9844 1.04791 0.523956 0.851745i \(-0.324456\pi\)
0.523956 + 0.851745i \(0.324456\pi\)
\(402\) 0 0
\(403\) 17.0502 0.849333
\(404\) 0 0
\(405\) −14.0958 −0.700428
\(406\) 0 0
\(407\) −1.80719 −0.0895791
\(408\) 0 0
\(409\) −22.9268 −1.13366 −0.566828 0.823836i \(-0.691830\pi\)
−0.566828 + 0.823836i \(0.691830\pi\)
\(410\) 0 0
\(411\) −3.29804 −0.162680
\(412\) 0 0
\(413\) −2.12508 −0.104568
\(414\) 0 0
\(415\) 14.8677 0.729828
\(416\) 0 0
\(417\) 7.09661 0.347522
\(418\) 0 0
\(419\) 39.4377 1.92666 0.963329 0.268323i \(-0.0864693\pi\)
0.963329 + 0.268323i \(0.0864693\pi\)
\(420\) 0 0
\(421\) 5.55909 0.270933 0.135467 0.990782i \(-0.456747\pi\)
0.135467 + 0.990782i \(0.456747\pi\)
\(422\) 0 0
\(423\) −27.0321 −1.31435
\(424\) 0 0
\(425\) 5.19071 0.251786
\(426\) 0 0
\(427\) −2.46295 −0.119190
\(428\) 0 0
\(429\) −0.929487 −0.0448761
\(430\) 0 0
\(431\) −30.8494 −1.48596 −0.742981 0.669313i \(-0.766589\pi\)
−0.742981 + 0.669313i \(0.766589\pi\)
\(432\) 0 0
\(433\) 22.5480 1.08359 0.541794 0.840511i \(-0.317745\pi\)
0.541794 + 0.840511i \(0.317745\pi\)
\(434\) 0 0
\(435\) 1.29696 0.0621845
\(436\) 0 0
\(437\) 0.516776 0.0247208
\(438\) 0 0
\(439\) −20.4983 −0.978329 −0.489164 0.872192i \(-0.662698\pi\)
−0.489164 + 0.872192i \(0.662698\pi\)
\(440\) 0 0
\(441\) −2.78480 −0.132609
\(442\) 0 0
\(443\) 22.4578 1.06700 0.533502 0.845799i \(-0.320876\pi\)
0.533502 + 0.845799i \(0.320876\pi\)
\(444\) 0 0
\(445\) 4.95547 0.234912
\(446\) 0 0
\(447\) 10.0545 0.475560
\(448\) 0 0
\(449\) 28.3273 1.33685 0.668423 0.743781i \(-0.266969\pi\)
0.668423 + 0.743781i \(0.266969\pi\)
\(450\) 0 0
\(451\) −6.01451 −0.283212
\(452\) 0 0
\(453\) 9.97711 0.468766
\(454\) 0 0
\(455\) 6.32541 0.296540
\(456\) 0 0
\(457\) 0.979734 0.0458300 0.0229150 0.999737i \(-0.492705\pi\)
0.0229150 + 0.999737i \(0.492705\pi\)
\(458\) 0 0
\(459\) 13.0308 0.608226
\(460\) 0 0
\(461\) −36.4648 −1.69834 −0.849169 0.528122i \(-0.822896\pi\)
−0.849169 + 0.528122i \(0.822896\pi\)
\(462\) 0 0
\(463\) 1.56380 0.0726760 0.0363380 0.999340i \(-0.488431\pi\)
0.0363380 + 0.999340i \(0.488431\pi\)
\(464\) 0 0
\(465\) 4.91555 0.227953
\(466\) 0 0
\(467\) 40.5676 1.87724 0.938622 0.344947i \(-0.112103\pi\)
0.938622 + 0.344947i \(0.112103\pi\)
\(468\) 0 0
\(469\) 12.6783 0.585429
\(470\) 0 0
\(471\) 5.06884 0.233560
\(472\) 0 0
\(473\) 6.09961 0.280460
\(474\) 0 0
\(475\) −0.172346 −0.00790780
\(476\) 0 0
\(477\) −30.1054 −1.37843
\(478\) 0 0
\(479\) 25.8040 1.17902 0.589508 0.807762i \(-0.299322\pi\)
0.589508 + 0.807762i \(0.299322\pi\)
\(480\) 0 0
\(481\) 9.18029 0.418585
\(482\) 0 0
\(483\) 1.48693 0.0676579
\(484\) 0 0
\(485\) −11.6794 −0.530336
\(486\) 0 0
\(487\) 2.56961 0.116440 0.0582201 0.998304i \(-0.481457\pi\)
0.0582201 + 0.998304i \(0.481457\pi\)
\(488\) 0 0
\(489\) −11.1093 −0.502381
\(490\) 0 0
\(491\) 2.98579 0.134747 0.0673734 0.997728i \(-0.478538\pi\)
0.0673734 + 0.997728i \(0.478538\pi\)
\(492\) 0 0
\(493\) 6.84713 0.308379
\(494\) 0 0
\(495\) 3.46761 0.155857
\(496\) 0 0
\(497\) 7.18356 0.322227
\(498\) 0 0
\(499\) 7.50113 0.335797 0.167898 0.985804i \(-0.446302\pi\)
0.167898 + 0.985804i \(0.446302\pi\)
\(500\) 0 0
\(501\) −3.97877 −0.177758
\(502\) 0 0
\(503\) 37.3432 1.66505 0.832525 0.553987i \(-0.186894\pi\)
0.832525 + 0.553987i \(0.186894\pi\)
\(504\) 0 0
\(505\) −19.6415 −0.874036
\(506\) 0 0
\(507\) −1.30903 −0.0581359
\(508\) 0 0
\(509\) 30.7313 1.36214 0.681070 0.732218i \(-0.261515\pi\)
0.681070 + 0.732218i \(0.261515\pi\)
\(510\) 0 0
\(511\) −9.04029 −0.399919
\(512\) 0 0
\(513\) −0.432660 −0.0191024
\(514\) 0 0
\(515\) 16.0773 0.708451
\(516\) 0 0
\(517\) 6.09635 0.268117
\(518\) 0 0
\(519\) 3.56914 0.156668
\(520\) 0 0
\(521\) 4.96517 0.217528 0.108764 0.994068i \(-0.465311\pi\)
0.108764 + 0.994068i \(0.465311\pi\)
\(522\) 0 0
\(523\) 1.30523 0.0570737 0.0285369 0.999593i \(-0.490915\pi\)
0.0285369 + 0.999593i \(0.490915\pi\)
\(524\) 0 0
\(525\) −0.495897 −0.0216427
\(526\) 0 0
\(527\) 25.9510 1.13044
\(528\) 0 0
\(529\) −12.7261 −0.553309
\(530\) 0 0
\(531\) 5.91792 0.256816
\(532\) 0 0
\(533\) 30.5529 1.32339
\(534\) 0 0
\(535\) 6.29111 0.271989
\(536\) 0 0
\(537\) −5.46327 −0.235757
\(538\) 0 0
\(539\) 0.628034 0.0270514
\(540\) 0 0
\(541\) −28.4295 −1.22228 −0.611139 0.791523i \(-0.709288\pi\)
−0.611139 + 0.791523i \(0.709288\pi\)
\(542\) 0 0
\(543\) −4.09474 −0.175722
\(544\) 0 0
\(545\) 0.839535 0.0359617
\(546\) 0 0
\(547\) −23.6760 −1.01231 −0.506156 0.862442i \(-0.668934\pi\)
−0.506156 + 0.862442i \(0.668934\pi\)
\(548\) 0 0
\(549\) 6.85880 0.292727
\(550\) 0 0
\(551\) −0.227345 −0.00968521
\(552\) 0 0
\(553\) 9.58806 0.407726
\(554\) 0 0
\(555\) 2.64666 0.112344
\(556\) 0 0
\(557\) −31.3625 −1.32887 −0.664436 0.747345i \(-0.731328\pi\)
−0.664436 + 0.747345i \(0.731328\pi\)
\(558\) 0 0
\(559\) −30.9852 −1.31053
\(560\) 0 0
\(561\) −1.41471 −0.0597290
\(562\) 0 0
\(563\) −34.3186 −1.44636 −0.723178 0.690662i \(-0.757319\pi\)
−0.723178 + 0.690662i \(0.757319\pi\)
\(564\) 0 0
\(565\) 9.26769 0.389895
\(566\) 0 0
\(567\) 7.10949 0.298570
\(568\) 0 0
\(569\) −42.8086 −1.79463 −0.897315 0.441392i \(-0.854485\pi\)
−0.897315 + 0.441392i \(0.854485\pi\)
\(570\) 0 0
\(571\) 9.75677 0.408308 0.204154 0.978939i \(-0.434556\pi\)
0.204154 + 0.978939i \(0.434556\pi\)
\(572\) 0 0
\(573\) 0.730742 0.0305272
\(574\) 0 0
\(575\) −3.42638 −0.142890
\(576\) 0 0
\(577\) −2.38332 −0.0992189 −0.0496094 0.998769i \(-0.515798\pi\)
−0.0496094 + 0.998769i \(0.515798\pi\)
\(578\) 0 0
\(579\) 0.301242 0.0125192
\(580\) 0 0
\(581\) −7.49880 −0.311103
\(582\) 0 0
\(583\) 6.78944 0.281190
\(584\) 0 0
\(585\) −17.6150 −0.728290
\(586\) 0 0
\(587\) −3.83243 −0.158181 −0.0790906 0.996867i \(-0.525202\pi\)
−0.0790906 + 0.996867i \(0.525202\pi\)
\(588\) 0 0
\(589\) −0.861648 −0.0355036
\(590\) 0 0
\(591\) 0.766700 0.0315378
\(592\) 0 0
\(593\) −29.3573 −1.20556 −0.602780 0.797907i \(-0.705940\pi\)
−0.602780 + 0.797907i \(0.705940\pi\)
\(594\) 0 0
\(595\) 9.62746 0.394687
\(596\) 0 0
\(597\) −1.89741 −0.0776558
\(598\) 0 0
\(599\) −40.3716 −1.64954 −0.824770 0.565468i \(-0.808696\pi\)
−0.824770 + 0.565468i \(0.808696\pi\)
\(600\) 0 0
\(601\) 30.9931 1.26424 0.632118 0.774872i \(-0.282186\pi\)
0.632118 + 0.774872i \(0.282186\pi\)
\(602\) 0 0
\(603\) −35.3065 −1.43779
\(604\) 0 0
\(605\) 21.0275 0.854888
\(606\) 0 0
\(607\) −9.81876 −0.398531 −0.199265 0.979946i \(-0.563856\pi\)
−0.199265 + 0.979946i \(0.563856\pi\)
\(608\) 0 0
\(609\) −0.654145 −0.0265073
\(610\) 0 0
\(611\) −30.9686 −1.25286
\(612\) 0 0
\(613\) 7.14953 0.288767 0.144383 0.989522i \(-0.453880\pi\)
0.144383 + 0.989522i \(0.453880\pi\)
\(614\) 0 0
\(615\) 8.80834 0.355187
\(616\) 0 0
\(617\) 34.8190 1.40176 0.700881 0.713279i \(-0.252791\pi\)
0.700881 + 0.713279i \(0.252791\pi\)
\(618\) 0 0
\(619\) −48.3268 −1.94242 −0.971209 0.238231i \(-0.923433\pi\)
−0.971209 + 0.238231i \(0.923433\pi\)
\(620\) 0 0
\(621\) −8.60161 −0.345171
\(622\) 0 0
\(623\) −2.49938 −0.100135
\(624\) 0 0
\(625\) −18.5124 −0.740497
\(626\) 0 0
\(627\) 0.0469724 0.00187590
\(628\) 0 0
\(629\) 13.9727 0.557127
\(630\) 0 0
\(631\) −1.37799 −0.0548569 −0.0274285 0.999624i \(-0.508732\pi\)
−0.0274285 + 0.999624i \(0.508732\pi\)
\(632\) 0 0
\(633\) 12.4926 0.496537
\(634\) 0 0
\(635\) 27.2821 1.08266
\(636\) 0 0
\(637\) −3.19033 −0.126406
\(638\) 0 0
\(639\) −20.0048 −0.791376
\(640\) 0 0
\(641\) −1.24423 −0.0491440 −0.0245720 0.999698i \(-0.507822\pi\)
−0.0245720 + 0.999698i \(0.507822\pi\)
\(642\) 0 0
\(643\) 35.4647 1.39859 0.699296 0.714832i \(-0.253497\pi\)
0.699296 + 0.714832i \(0.253497\pi\)
\(644\) 0 0
\(645\) −8.93297 −0.351735
\(646\) 0 0
\(647\) −28.0343 −1.10214 −0.551071 0.834458i \(-0.685781\pi\)
−0.551071 + 0.834458i \(0.685781\pi\)
\(648\) 0 0
\(649\) −1.33462 −0.0523886
\(650\) 0 0
\(651\) −2.47924 −0.0971692
\(652\) 0 0
\(653\) 26.2484 1.02718 0.513589 0.858036i \(-0.328316\pi\)
0.513589 + 0.858036i \(0.328316\pi\)
\(654\) 0 0
\(655\) −19.3991 −0.757984
\(656\) 0 0
\(657\) 25.1754 0.982185
\(658\) 0 0
\(659\) −4.13485 −0.161071 −0.0805354 0.996752i \(-0.525663\pi\)
−0.0805354 + 0.996752i \(0.525663\pi\)
\(660\) 0 0
\(661\) −1.60137 −0.0622859 −0.0311430 0.999515i \(-0.509915\pi\)
−0.0311430 + 0.999515i \(0.509915\pi\)
\(662\) 0 0
\(663\) 7.18652 0.279101
\(664\) 0 0
\(665\) −0.319660 −0.0123959
\(666\) 0 0
\(667\) −4.51978 −0.175007
\(668\) 0 0
\(669\) 12.0670 0.466538
\(670\) 0 0
\(671\) −1.54681 −0.0597141
\(672\) 0 0
\(673\) −50.3603 −1.94125 −0.970623 0.240607i \(-0.922654\pi\)
−0.970623 + 0.240607i \(0.922654\pi\)
\(674\) 0 0
\(675\) 2.86867 0.110415
\(676\) 0 0
\(677\) 10.4285 0.400799 0.200399 0.979714i \(-0.435776\pi\)
0.200399 + 0.979714i \(0.435776\pi\)
\(678\) 0 0
\(679\) 5.89073 0.226065
\(680\) 0 0
\(681\) 0.119147 0.00456573
\(682\) 0 0
\(683\) 33.0316 1.26392 0.631960 0.775001i \(-0.282251\pi\)
0.631960 + 0.775001i \(0.282251\pi\)
\(684\) 0 0
\(685\) 14.0956 0.538566
\(686\) 0 0
\(687\) 0.445851 0.0170103
\(688\) 0 0
\(689\) −34.4895 −1.31394
\(690\) 0 0
\(691\) −17.1546 −0.652593 −0.326296 0.945267i \(-0.605801\pi\)
−0.326296 + 0.945267i \(0.605801\pi\)
\(692\) 0 0
\(693\) −1.74895 −0.0664371
\(694\) 0 0
\(695\) −30.3305 −1.15050
\(696\) 0 0
\(697\) 46.5025 1.76141
\(698\) 0 0
\(699\) 9.12382 0.345094
\(700\) 0 0
\(701\) 6.24526 0.235880 0.117940 0.993021i \(-0.462371\pi\)
0.117940 + 0.993021i \(0.462371\pi\)
\(702\) 0 0
\(703\) −0.463933 −0.0174976
\(704\) 0 0
\(705\) −8.92819 −0.336255
\(706\) 0 0
\(707\) 9.90655 0.372574
\(708\) 0 0
\(709\) −3.83146 −0.143893 −0.0719467 0.997408i \(-0.522921\pi\)
−0.0719467 + 0.997408i \(0.522921\pi\)
\(710\) 0 0
\(711\) −26.7008 −1.00136
\(712\) 0 0
\(713\) −17.1302 −0.641531
\(714\) 0 0
\(715\) 3.97258 0.148566
\(716\) 0 0
\(717\) 4.38314 0.163691
\(718\) 0 0
\(719\) 8.99485 0.335451 0.167726 0.985834i \(-0.446358\pi\)
0.167726 + 0.985834i \(0.446358\pi\)
\(720\) 0 0
\(721\) −8.10887 −0.301990
\(722\) 0 0
\(723\) 10.5877 0.393761
\(724\) 0 0
\(725\) 1.50736 0.0559820
\(726\) 0 0
\(727\) 33.6636 1.24851 0.624256 0.781219i \(-0.285402\pi\)
0.624256 + 0.781219i \(0.285402\pi\)
\(728\) 0 0
\(729\) −16.0638 −0.594954
\(730\) 0 0
\(731\) −47.1604 −1.74429
\(732\) 0 0
\(733\) −41.0889 −1.51765 −0.758827 0.651293i \(-0.774227\pi\)
−0.758827 + 0.651293i \(0.774227\pi\)
\(734\) 0 0
\(735\) −0.919765 −0.0339261
\(736\) 0 0
\(737\) 7.96241 0.293299
\(738\) 0 0
\(739\) 7.96250 0.292905 0.146453 0.989218i \(-0.453214\pi\)
0.146453 + 0.989218i \(0.453214\pi\)
\(740\) 0 0
\(741\) −0.238613 −0.00876568
\(742\) 0 0
\(743\) 27.7688 1.01874 0.509369 0.860548i \(-0.329879\pi\)
0.509369 + 0.860548i \(0.329879\pi\)
\(744\) 0 0
\(745\) −42.9723 −1.57438
\(746\) 0 0
\(747\) 20.8826 0.764056
\(748\) 0 0
\(749\) −3.17303 −0.115940
\(750\) 0 0
\(751\) 45.5138 1.66082 0.830410 0.557153i \(-0.188106\pi\)
0.830410 + 0.557153i \(0.188106\pi\)
\(752\) 0 0
\(753\) 8.04781 0.293279
\(754\) 0 0
\(755\) −42.6416 −1.55189
\(756\) 0 0
\(757\) 29.3644 1.06727 0.533633 0.845716i \(-0.320826\pi\)
0.533633 + 0.845716i \(0.320826\pi\)
\(758\) 0 0
\(759\) 0.933846 0.0338965
\(760\) 0 0
\(761\) 3.64393 0.132092 0.0660461 0.997817i \(-0.478962\pi\)
0.0660461 + 0.997817i \(0.478962\pi\)
\(762\) 0 0
\(763\) −0.423434 −0.0153293
\(764\) 0 0
\(765\) −26.8105 −0.969336
\(766\) 0 0
\(767\) 6.77971 0.244801
\(768\) 0 0
\(769\) −6.23588 −0.224872 −0.112436 0.993659i \(-0.535865\pi\)
−0.112436 + 0.993659i \(0.535865\pi\)
\(770\) 0 0
\(771\) 3.57766 0.128846
\(772\) 0 0
\(773\) 2.11122 0.0759352 0.0379676 0.999279i \(-0.487912\pi\)
0.0379676 + 0.999279i \(0.487912\pi\)
\(774\) 0 0
\(775\) 5.71298 0.205216
\(776\) 0 0
\(777\) −1.33489 −0.0478888
\(778\) 0 0
\(779\) −1.54402 −0.0553201
\(780\) 0 0
\(781\) 4.51152 0.161435
\(782\) 0 0
\(783\) 3.78410 0.135233
\(784\) 0 0
\(785\) −21.6639 −0.773219
\(786\) 0 0
\(787\) −12.4016 −0.442071 −0.221036 0.975266i \(-0.570944\pi\)
−0.221036 + 0.975266i \(0.570944\pi\)
\(788\) 0 0
\(789\) 4.29094 0.152762
\(790\) 0 0
\(791\) −4.67432 −0.166200
\(792\) 0 0
\(793\) 7.85761 0.279032
\(794\) 0 0
\(795\) −9.94323 −0.352650
\(796\) 0 0
\(797\) 32.1695 1.13950 0.569751 0.821817i \(-0.307040\pi\)
0.569751 + 0.821817i \(0.307040\pi\)
\(798\) 0 0
\(799\) −47.1352 −1.66752
\(800\) 0 0
\(801\) 6.96026 0.245929
\(802\) 0 0
\(803\) −5.67762 −0.200359
\(804\) 0 0
\(805\) −6.35507 −0.223987
\(806\) 0 0
\(807\) −5.48846 −0.193203
\(808\) 0 0
\(809\) −2.69445 −0.0947319 −0.0473660 0.998878i \(-0.515083\pi\)
−0.0473660 + 0.998878i \(0.515083\pi\)
\(810\) 0 0
\(811\) 43.1775 1.51617 0.758084 0.652157i \(-0.226136\pi\)
0.758084 + 0.652157i \(0.226136\pi\)
\(812\) 0 0
\(813\) 0.536918 0.0188305
\(814\) 0 0
\(815\) 47.4806 1.66317
\(816\) 0 0
\(817\) 1.56586 0.0547826
\(818\) 0 0
\(819\) 8.88442 0.310447
\(820\) 0 0
\(821\) 21.8510 0.762606 0.381303 0.924450i \(-0.375475\pi\)
0.381303 + 0.924450i \(0.375475\pi\)
\(822\) 0 0
\(823\) 22.4104 0.781178 0.390589 0.920565i \(-0.372271\pi\)
0.390589 + 0.920565i \(0.372271\pi\)
\(824\) 0 0
\(825\) −0.311441 −0.0108430
\(826\) 0 0
\(827\) 22.5736 0.784963 0.392481 0.919760i \(-0.371617\pi\)
0.392481 + 0.919760i \(0.371617\pi\)
\(828\) 0 0
\(829\) −9.73194 −0.338004 −0.169002 0.985616i \(-0.554054\pi\)
−0.169002 + 0.985616i \(0.554054\pi\)
\(830\) 0 0
\(831\) −10.9512 −0.379892
\(832\) 0 0
\(833\) −4.85578 −0.168243
\(834\) 0 0
\(835\) 17.0050 0.588483
\(836\) 0 0
\(837\) 14.3419 0.495729
\(838\) 0 0
\(839\) 24.1568 0.833986 0.416993 0.908910i \(-0.363084\pi\)
0.416993 + 0.908910i \(0.363084\pi\)
\(840\) 0 0
\(841\) −27.0116 −0.931435
\(842\) 0 0
\(843\) −13.6227 −0.469190
\(844\) 0 0
\(845\) 5.59470 0.192464
\(846\) 0 0
\(847\) −10.6056 −0.364412
\(848\) 0 0
\(849\) −8.56160 −0.293833
\(850\) 0 0
\(851\) −9.22334 −0.316172
\(852\) 0 0
\(853\) −30.3658 −1.03971 −0.519853 0.854256i \(-0.674013\pi\)
−0.519853 + 0.854256i \(0.674013\pi\)
\(854\) 0 0
\(855\) 0.890187 0.0304438
\(856\) 0 0
\(857\) 3.52721 0.120487 0.0602436 0.998184i \(-0.480812\pi\)
0.0602436 + 0.998184i \(0.480812\pi\)
\(858\) 0 0
\(859\) −55.0973 −1.87989 −0.939947 0.341319i \(-0.889126\pi\)
−0.939947 + 0.341319i \(0.889126\pi\)
\(860\) 0 0
\(861\) −4.44264 −0.151405
\(862\) 0 0
\(863\) −18.9798 −0.646081 −0.323040 0.946385i \(-0.604705\pi\)
−0.323040 + 0.946385i \(0.604705\pi\)
\(864\) 0 0
\(865\) −15.2543 −0.518662
\(866\) 0 0
\(867\) 3.05180 0.103645
\(868\) 0 0
\(869\) 6.02163 0.204270
\(870\) 0 0
\(871\) −40.4480 −1.37053
\(872\) 0 0
\(873\) −16.4045 −0.555208
\(874\) 0 0
\(875\) 12.0328 0.406784
\(876\) 0 0
\(877\) 9.25719 0.312593 0.156297 0.987710i \(-0.450044\pi\)
0.156297 + 0.987710i \(0.450044\pi\)
\(878\) 0 0
\(879\) 12.7560 0.430248
\(880\) 0 0
\(881\) −26.2370 −0.883946 −0.441973 0.897028i \(-0.645721\pi\)
−0.441973 + 0.897028i \(0.645721\pi\)
\(882\) 0 0
\(883\) 27.3028 0.918814 0.459407 0.888226i \(-0.348062\pi\)
0.459407 + 0.888226i \(0.348062\pi\)
\(884\) 0 0
\(885\) 1.95458 0.0657024
\(886\) 0 0
\(887\) −3.40903 −0.114464 −0.0572321 0.998361i \(-0.518227\pi\)
−0.0572321 + 0.998361i \(0.518227\pi\)
\(888\) 0 0
\(889\) −13.7602 −0.461502
\(890\) 0 0
\(891\) 4.46500 0.149583
\(892\) 0 0
\(893\) 1.56502 0.0523715
\(894\) 0 0
\(895\) 23.3497 0.780494
\(896\) 0 0
\(897\) −4.74381 −0.158391
\(898\) 0 0
\(899\) 7.53607 0.251342
\(900\) 0 0
\(901\) −52.4940 −1.74883
\(902\) 0 0
\(903\) 4.50550 0.149934
\(904\) 0 0
\(905\) 17.5007 0.581743
\(906\) 0 0
\(907\) 25.3278 0.840997 0.420499 0.907293i \(-0.361855\pi\)
0.420499 + 0.907293i \(0.361855\pi\)
\(908\) 0 0
\(909\) −27.5877 −0.915027
\(910\) 0 0
\(911\) 23.6991 0.785186 0.392593 0.919712i \(-0.371578\pi\)
0.392593 + 0.919712i \(0.371578\pi\)
\(912\) 0 0
\(913\) −4.70950 −0.155862
\(914\) 0 0
\(915\) 2.26533 0.0748896
\(916\) 0 0
\(917\) 9.78426 0.323105
\(918\) 0 0
\(919\) −29.7662 −0.981897 −0.490949 0.871189i \(-0.663350\pi\)
−0.490949 + 0.871189i \(0.663350\pi\)
\(920\) 0 0
\(921\) −12.8299 −0.422761
\(922\) 0 0
\(923\) −22.9179 −0.754353
\(924\) 0 0
\(925\) 3.07601 0.101139
\(926\) 0 0
\(927\) 22.5816 0.741676
\(928\) 0 0
\(929\) 46.6355 1.53006 0.765031 0.643993i \(-0.222723\pi\)
0.765031 + 0.643993i \(0.222723\pi\)
\(930\) 0 0
\(931\) 0.161226 0.00528397
\(932\) 0 0
\(933\) −11.4086 −0.373500
\(934\) 0 0
\(935\) 6.04638 0.197738
\(936\) 0 0
\(937\) −13.5471 −0.442564 −0.221282 0.975210i \(-0.571024\pi\)
−0.221282 + 0.975210i \(0.571024\pi\)
\(938\) 0 0
\(939\) −8.35597 −0.272687
\(940\) 0 0
\(941\) −19.2235 −0.626668 −0.313334 0.949643i \(-0.601446\pi\)
−0.313334 + 0.949643i \(0.601446\pi\)
\(942\) 0 0
\(943\) −30.6962 −0.999606
\(944\) 0 0
\(945\) 5.32066 0.173081
\(946\) 0 0
\(947\) −40.6982 −1.32251 −0.661257 0.750159i \(-0.729977\pi\)
−0.661257 + 0.750159i \(0.729977\pi\)
\(948\) 0 0
\(949\) 28.8415 0.936236
\(950\) 0 0
\(951\) 6.36689 0.206460
\(952\) 0 0
\(953\) 30.3026 0.981598 0.490799 0.871273i \(-0.336705\pi\)
0.490799 + 0.871273i \(0.336705\pi\)
\(954\) 0 0
\(955\) −3.12315 −0.101063
\(956\) 0 0
\(957\) −0.410826 −0.0132801
\(958\) 0 0
\(959\) −7.10937 −0.229574
\(960\) 0 0
\(961\) −2.43792 −0.0786425
\(962\) 0 0
\(963\) 8.83625 0.284744
\(964\) 0 0
\(965\) −1.28749 −0.0414458
\(966\) 0 0
\(967\) 18.8733 0.606925 0.303462 0.952843i \(-0.401857\pi\)
0.303462 + 0.952843i \(0.401857\pi\)
\(968\) 0 0
\(969\) −0.363176 −0.0116669
\(970\) 0 0
\(971\) −40.4012 −1.29654 −0.648269 0.761412i \(-0.724507\pi\)
−0.648269 + 0.761412i \(0.724507\pi\)
\(972\) 0 0
\(973\) 15.2977 0.490422
\(974\) 0 0
\(975\) 1.58208 0.0506670
\(976\) 0 0
\(977\) −16.4828 −0.527331 −0.263666 0.964614i \(-0.584932\pi\)
−0.263666 + 0.964614i \(0.584932\pi\)
\(978\) 0 0
\(979\) −1.56970 −0.0501677
\(980\) 0 0
\(981\) 1.17918 0.0376482
\(982\) 0 0
\(983\) −33.4039 −1.06542 −0.532710 0.846298i \(-0.678826\pi\)
−0.532710 + 0.846298i \(0.678826\pi\)
\(984\) 0 0
\(985\) −3.27683 −0.104408
\(986\) 0 0
\(987\) 4.50309 0.143335
\(988\) 0 0
\(989\) 31.1305 0.989893
\(990\) 0 0
\(991\) −25.3648 −0.805740 −0.402870 0.915257i \(-0.631987\pi\)
−0.402870 + 0.915257i \(0.631987\pi\)
\(992\) 0 0
\(993\) 0.986318 0.0312998
\(994\) 0 0
\(995\) 8.10943 0.257086
\(996\) 0 0
\(997\) −15.7551 −0.498968 −0.249484 0.968379i \(-0.580261\pi\)
−0.249484 + 0.968379i \(0.580261\pi\)
\(998\) 0 0
\(999\) 7.72206 0.244315
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 7168.2.a.bf.1.5 8
4.3 odd 2 7168.2.a.be.1.4 8
8.3 odd 2 7168.2.a.ba.1.5 8
8.5 even 2 7168.2.a.bb.1.4 8
32.3 odd 8 1792.2.m.g.1345.4 yes 16
32.5 even 8 1792.2.m.h.449.4 yes 16
32.11 odd 8 1792.2.m.g.449.4 yes 16
32.13 even 8 1792.2.m.h.1345.4 yes 16
32.19 odd 8 1792.2.m.f.1345.5 yes 16
32.21 even 8 1792.2.m.e.449.5 16
32.27 odd 8 1792.2.m.f.449.5 yes 16
32.29 even 8 1792.2.m.e.1345.5 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1792.2.m.e.449.5 16 32.21 even 8
1792.2.m.e.1345.5 yes 16 32.29 even 8
1792.2.m.f.449.5 yes 16 32.27 odd 8
1792.2.m.f.1345.5 yes 16 32.19 odd 8
1792.2.m.g.449.4 yes 16 32.11 odd 8
1792.2.m.g.1345.4 yes 16 32.3 odd 8
1792.2.m.h.449.4 yes 16 32.5 even 8
1792.2.m.h.1345.4 yes 16 32.13 even 8
7168.2.a.ba.1.5 8 8.3 odd 2
7168.2.a.bb.1.4 8 8.5 even 2
7168.2.a.be.1.4 8 4.3 odd 2
7168.2.a.bf.1.5 8 1.1 even 1 trivial