Properties

Label 3584.2.a.k.1.6
Level $3584$
Weight $2$
Character 3584.1
Self dual yes
Analytic conductor $28.618$
Analytic rank $0$
Dimension $6$
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] = [3584,2,Mod(1,3584)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3584, 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("3584.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3584 = 2^{9} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3584.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: \(28.6183840844\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.18857984.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 9x^{4} + 16x^{3} + 6x^{2} - 12x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(-2.79169\) of defining polynomial
Character \(\chi\) \(=\) 3584.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.22299 q^{3} -0.725063 q^{5} -1.00000 q^{7} +7.38766 q^{9} +O(q^{10})\) \(q+3.22299 q^{3} -0.725063 q^{5} -1.00000 q^{7} +7.38766 q^{9} +2.21848 q^{11} +1.75046 q^{13} -2.33687 q^{15} +1.69102 q^{17} +1.80241 q^{19} -3.22299 q^{21} +8.47013 q^{23} -4.47428 q^{25} +14.1414 q^{27} +0.704174 q^{29} -6.28756 q^{31} +7.15015 q^{33} +0.725063 q^{35} -6.75496 q^{37} +5.64171 q^{39} +8.72453 q^{41} -7.33448 q^{43} -5.35652 q^{45} -10.1709 q^{47} +1.00000 q^{49} +5.45013 q^{51} +13.2440 q^{53} -1.60854 q^{55} +5.80913 q^{57} +8.86761 q^{59} -10.4417 q^{61} -7.38766 q^{63} -1.26919 q^{65} -1.71940 q^{67} +27.2992 q^{69} +1.47808 q^{71} +8.52472 q^{73} -14.4206 q^{75} -2.21848 q^{77} +10.6421 q^{79} +23.4146 q^{81} -10.8440 q^{83} -1.22609 q^{85} +2.26954 q^{87} -2.13326 q^{89} -1.75046 q^{91} -20.2648 q^{93} -1.30686 q^{95} -5.73933 q^{97} +16.3894 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 4 q^{3} - 6 q^{7} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 4 q^{3} - 6 q^{7} + 10 q^{9} + 8 q^{11} + 8 q^{15} + 20 q^{19} - 4 q^{21} + 2 q^{25} + 16 q^{27} - 4 q^{29} + 8 q^{31} + 4 q^{33} - 20 q^{37} - 4 q^{41} + 24 q^{43} - 8 q^{47} + 6 q^{49} + 24 q^{51} - 4 q^{53} - 16 q^{55} - 4 q^{57} + 12 q^{59} + 8 q^{61} - 10 q^{63} - 8 q^{65} + 16 q^{67} + 40 q^{69} + 8 q^{71} + 16 q^{73} + 28 q^{75} - 8 q^{77} + 24 q^{79} + 10 q^{81} + 12 q^{83} + 8 q^{87} + 16 q^{89} - 24 q^{93} + 16 q^{95} + 16 q^{97} + 40 q^{99}+O(q^{100}) \) 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
\(3\) 3.22299 1.86079 0.930397 0.366554i \(-0.119462\pi\)
0.930397 + 0.366554i \(0.119462\pi\)
\(4\) 0 0
\(5\) −0.725063 −0.324258 −0.162129 0.986770i \(-0.551836\pi\)
−0.162129 + 0.986770i \(0.551836\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 7.38766 2.46255
\(10\) 0 0
\(11\) 2.21848 0.668898 0.334449 0.942414i \(-0.391450\pi\)
0.334449 + 0.942414i \(0.391450\pi\)
\(12\) 0 0
\(13\) 1.75046 0.485490 0.242745 0.970090i \(-0.421952\pi\)
0.242745 + 0.970090i \(0.421952\pi\)
\(14\) 0 0
\(15\) −2.33687 −0.603378
\(16\) 0 0
\(17\) 1.69102 0.410132 0.205066 0.978748i \(-0.434259\pi\)
0.205066 + 0.978748i \(0.434259\pi\)
\(18\) 0 0
\(19\) 1.80241 0.413500 0.206750 0.978394i \(-0.433711\pi\)
0.206750 + 0.978394i \(0.433711\pi\)
\(20\) 0 0
\(21\) −3.22299 −0.703314
\(22\) 0 0
\(23\) 8.47013 1.76615 0.883073 0.469236i \(-0.155471\pi\)
0.883073 + 0.469236i \(0.155471\pi\)
\(24\) 0 0
\(25\) −4.47428 −0.894857
\(26\) 0 0
\(27\) 14.1414 2.72151
\(28\) 0 0
\(29\) 0.704174 0.130762 0.0653809 0.997860i \(-0.479174\pi\)
0.0653809 + 0.997860i \(0.479174\pi\)
\(30\) 0 0
\(31\) −6.28756 −1.12928 −0.564640 0.825337i \(-0.690985\pi\)
−0.564640 + 0.825337i \(0.690985\pi\)
\(32\) 0 0
\(33\) 7.15015 1.24468
\(34\) 0 0
\(35\) 0.725063 0.122558
\(36\) 0 0
\(37\) −6.75496 −1.11051 −0.555255 0.831680i \(-0.687379\pi\)
−0.555255 + 0.831680i \(0.687379\pi\)
\(38\) 0 0
\(39\) 5.64171 0.903396
\(40\) 0 0
\(41\) 8.72453 1.36254 0.681272 0.732031i \(-0.261427\pi\)
0.681272 + 0.732031i \(0.261427\pi\)
\(42\) 0 0
\(43\) −7.33448 −1.11850 −0.559249 0.829000i \(-0.688910\pi\)
−0.559249 + 0.829000i \(0.688910\pi\)
\(44\) 0 0
\(45\) −5.35652 −0.798503
\(46\) 0 0
\(47\) −10.1709 −1.48358 −0.741791 0.670631i \(-0.766023\pi\)
−0.741791 + 0.670631i \(0.766023\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 5.45013 0.763170
\(52\) 0 0
\(53\) 13.2440 1.81920 0.909600 0.415485i \(-0.136388\pi\)
0.909600 + 0.415485i \(0.136388\pi\)
\(54\) 0 0
\(55\) −1.60854 −0.216896
\(56\) 0 0
\(57\) 5.80913 0.769438
\(58\) 0 0
\(59\) 8.86761 1.15446 0.577232 0.816580i \(-0.304133\pi\)
0.577232 + 0.816580i \(0.304133\pi\)
\(60\) 0 0
\(61\) −10.4417 −1.33692 −0.668462 0.743746i \(-0.733047\pi\)
−0.668462 + 0.743746i \(0.733047\pi\)
\(62\) 0 0
\(63\) −7.38766 −0.930758
\(64\) 0 0
\(65\) −1.26919 −0.157424
\(66\) 0 0
\(67\) −1.71940 −0.210058 −0.105029 0.994469i \(-0.533494\pi\)
−0.105029 + 0.994469i \(0.533494\pi\)
\(68\) 0 0
\(69\) 27.2992 3.28643
\(70\) 0 0
\(71\) 1.47808 0.175416 0.0877079 0.996146i \(-0.472046\pi\)
0.0877079 + 0.996146i \(0.472046\pi\)
\(72\) 0 0
\(73\) 8.52472 0.997743 0.498872 0.866676i \(-0.333748\pi\)
0.498872 + 0.866676i \(0.333748\pi\)
\(74\) 0 0
\(75\) −14.4206 −1.66514
\(76\) 0 0
\(77\) −2.21848 −0.252820
\(78\) 0 0
\(79\) 10.6421 1.19733 0.598663 0.801001i \(-0.295699\pi\)
0.598663 + 0.801001i \(0.295699\pi\)
\(80\) 0 0
\(81\) 23.4146 2.60162
\(82\) 0 0
\(83\) −10.8440 −1.19029 −0.595144 0.803619i \(-0.702905\pi\)
−0.595144 + 0.803619i \(0.702905\pi\)
\(84\) 0 0
\(85\) −1.22609 −0.132989
\(86\) 0 0
\(87\) 2.26954 0.243321
\(88\) 0 0
\(89\) −2.13326 −0.226125 −0.113063 0.993588i \(-0.536066\pi\)
−0.113063 + 0.993588i \(0.536066\pi\)
\(90\) 0 0
\(91\) −1.75046 −0.183498
\(92\) 0 0
\(93\) −20.2648 −2.10136
\(94\) 0 0
\(95\) −1.30686 −0.134081
\(96\) 0 0
\(97\) −5.73933 −0.582740 −0.291370 0.956610i \(-0.594111\pi\)
−0.291370 + 0.956610i \(0.594111\pi\)
\(98\) 0 0
\(99\) 16.3894 1.64720
\(100\) 0 0
\(101\) 7.40731 0.737055 0.368528 0.929617i \(-0.379862\pi\)
0.368528 + 0.929617i \(0.379862\pi\)
\(102\) 0 0
\(103\) 16.6714 1.64268 0.821342 0.570436i \(-0.193226\pi\)
0.821342 + 0.570436i \(0.193226\pi\)
\(104\) 0 0
\(105\) 2.33687 0.228055
\(106\) 0 0
\(107\) −12.4492 −1.20351 −0.601755 0.798680i \(-0.705532\pi\)
−0.601755 + 0.798680i \(0.705532\pi\)
\(108\) 0 0
\(109\) 2.53093 0.242419 0.121210 0.992627i \(-0.461323\pi\)
0.121210 + 0.992627i \(0.461323\pi\)
\(110\) 0 0
\(111\) −21.7712 −2.06643
\(112\) 0 0
\(113\) 1.16568 0.109658 0.0548291 0.998496i \(-0.482539\pi\)
0.0548291 + 0.998496i \(0.482539\pi\)
\(114\) 0 0
\(115\) −6.14139 −0.572687
\(116\) 0 0
\(117\) 12.9318 1.19554
\(118\) 0 0
\(119\) −1.69102 −0.155015
\(120\) 0 0
\(121\) −6.07833 −0.552575
\(122\) 0 0
\(123\) 28.1191 2.53541
\(124\) 0 0
\(125\) 6.86946 0.614423
\(126\) 0 0
\(127\) −0.248418 −0.0220435 −0.0110218 0.999939i \(-0.503508\pi\)
−0.0110218 + 0.999939i \(0.503508\pi\)
\(128\) 0 0
\(129\) −23.6389 −2.08129
\(130\) 0 0
\(131\) 18.0997 1.58138 0.790690 0.612216i \(-0.209722\pi\)
0.790690 + 0.612216i \(0.209722\pi\)
\(132\) 0 0
\(133\) −1.80241 −0.156288
\(134\) 0 0
\(135\) −10.2534 −0.882472
\(136\) 0 0
\(137\) −17.7633 −1.51762 −0.758812 0.651310i \(-0.774220\pi\)
−0.758812 + 0.651310i \(0.774220\pi\)
\(138\) 0 0
\(139\) 12.7027 1.07743 0.538713 0.842490i \(-0.318911\pi\)
0.538713 + 0.842490i \(0.318911\pi\)
\(140\) 0 0
\(141\) −32.7808 −2.76064
\(142\) 0 0
\(143\) 3.88336 0.324743
\(144\) 0 0
\(145\) −0.510571 −0.0424006
\(146\) 0 0
\(147\) 3.22299 0.265828
\(148\) 0 0
\(149\) −7.98106 −0.653834 −0.326917 0.945053i \(-0.606010\pi\)
−0.326917 + 0.945053i \(0.606010\pi\)
\(150\) 0 0
\(151\) 18.6055 1.51409 0.757047 0.653361i \(-0.226641\pi\)
0.757047 + 0.653361i \(0.226641\pi\)
\(152\) 0 0
\(153\) 12.4927 1.00997
\(154\) 0 0
\(155\) 4.55888 0.366178
\(156\) 0 0
\(157\) 12.3166 0.982970 0.491485 0.870886i \(-0.336454\pi\)
0.491485 + 0.870886i \(0.336454\pi\)
\(158\) 0 0
\(159\) 42.6852 3.38516
\(160\) 0 0
\(161\) −8.47013 −0.667540
\(162\) 0 0
\(163\) 24.8444 1.94596 0.972982 0.230882i \(-0.0741610\pi\)
0.972982 + 0.230882i \(0.0741610\pi\)
\(164\) 0 0
\(165\) −5.18431 −0.403598
\(166\) 0 0
\(167\) −10.4042 −0.805101 −0.402551 0.915398i \(-0.631876\pi\)
−0.402551 + 0.915398i \(0.631876\pi\)
\(168\) 0 0
\(169\) −9.93590 −0.764300
\(170\) 0 0
\(171\) 13.3156 1.01827
\(172\) 0 0
\(173\) 13.1678 1.00113 0.500565 0.865699i \(-0.333126\pi\)
0.500565 + 0.865699i \(0.333126\pi\)
\(174\) 0 0
\(175\) 4.47428 0.338224
\(176\) 0 0
\(177\) 28.5802 2.14822
\(178\) 0 0
\(179\) −1.59431 −0.119164 −0.0595821 0.998223i \(-0.518977\pi\)
−0.0595821 + 0.998223i \(0.518977\pi\)
\(180\) 0 0
\(181\) 5.93385 0.441060 0.220530 0.975380i \(-0.429221\pi\)
0.220530 + 0.975380i \(0.429221\pi\)
\(182\) 0 0
\(183\) −33.6535 −2.48774
\(184\) 0 0
\(185\) 4.89778 0.360092
\(186\) 0 0
\(187\) 3.75149 0.274336
\(188\) 0 0
\(189\) −14.1414 −1.02863
\(190\) 0 0
\(191\) −16.2217 −1.17376 −0.586881 0.809673i \(-0.699644\pi\)
−0.586881 + 0.809673i \(0.699644\pi\)
\(192\) 0 0
\(193\) 0.805434 0.0579764 0.0289882 0.999580i \(-0.490771\pi\)
0.0289882 + 0.999580i \(0.490771\pi\)
\(194\) 0 0
\(195\) −4.09060 −0.292934
\(196\) 0 0
\(197\) −9.93905 −0.708128 −0.354064 0.935221i \(-0.615200\pi\)
−0.354064 + 0.935221i \(0.615200\pi\)
\(198\) 0 0
\(199\) −19.4280 −1.37722 −0.688608 0.725134i \(-0.741778\pi\)
−0.688608 + 0.725134i \(0.741778\pi\)
\(200\) 0 0
\(201\) −5.54161 −0.390875
\(202\) 0 0
\(203\) −0.704174 −0.0494233
\(204\) 0 0
\(205\) −6.32584 −0.441816
\(206\) 0 0
\(207\) 62.5745 4.34923
\(208\) 0 0
\(209\) 3.99861 0.276590
\(210\) 0 0
\(211\) 18.4252 1.26844 0.634222 0.773151i \(-0.281321\pi\)
0.634222 + 0.773151i \(0.281321\pi\)
\(212\) 0 0
\(213\) 4.76384 0.326413
\(214\) 0 0
\(215\) 5.31796 0.362682
\(216\) 0 0
\(217\) 6.28756 0.426828
\(218\) 0 0
\(219\) 27.4751 1.85659
\(220\) 0 0
\(221\) 2.96005 0.199115
\(222\) 0 0
\(223\) −3.60025 −0.241090 −0.120545 0.992708i \(-0.538464\pi\)
−0.120545 + 0.992708i \(0.538464\pi\)
\(224\) 0 0
\(225\) −33.0545 −2.20363
\(226\) 0 0
\(227\) −10.9011 −0.723531 −0.361765 0.932269i \(-0.617826\pi\)
−0.361765 + 0.932269i \(0.617826\pi\)
\(228\) 0 0
\(229\) 13.3147 0.879863 0.439932 0.898031i \(-0.355003\pi\)
0.439932 + 0.898031i \(0.355003\pi\)
\(230\) 0 0
\(231\) −7.15015 −0.470446
\(232\) 0 0
\(233\) −22.3624 −1.46501 −0.732506 0.680760i \(-0.761650\pi\)
−0.732506 + 0.680760i \(0.761650\pi\)
\(234\) 0 0
\(235\) 7.37457 0.481064
\(236\) 0 0
\(237\) 34.2992 2.22798
\(238\) 0 0
\(239\) −18.1505 −1.17406 −0.587029 0.809566i \(-0.699703\pi\)
−0.587029 + 0.809566i \(0.699703\pi\)
\(240\) 0 0
\(241\) −19.7278 −1.27078 −0.635388 0.772193i \(-0.719160\pi\)
−0.635388 + 0.772193i \(0.719160\pi\)
\(242\) 0 0
\(243\) 33.0407 2.11956
\(244\) 0 0
\(245\) −0.725063 −0.0463226
\(246\) 0 0
\(247\) 3.15503 0.200750
\(248\) 0 0
\(249\) −34.9502 −2.21488
\(250\) 0 0
\(251\) −7.38794 −0.466322 −0.233161 0.972438i \(-0.574907\pi\)
−0.233161 + 0.972438i \(0.574907\pi\)
\(252\) 0 0
\(253\) 18.7909 1.18137
\(254\) 0 0
\(255\) −3.95169 −0.247464
\(256\) 0 0
\(257\) −8.13863 −0.507674 −0.253837 0.967247i \(-0.581693\pi\)
−0.253837 + 0.967247i \(0.581693\pi\)
\(258\) 0 0
\(259\) 6.75496 0.419733
\(260\) 0 0
\(261\) 5.20220 0.322008
\(262\) 0 0
\(263\) 18.0689 1.11418 0.557088 0.830453i \(-0.311919\pi\)
0.557088 + 0.830453i \(0.311919\pi\)
\(264\) 0 0
\(265\) −9.60272 −0.589891
\(266\) 0 0
\(267\) −6.87548 −0.420773
\(268\) 0 0
\(269\) −28.5831 −1.74274 −0.871371 0.490624i \(-0.836769\pi\)
−0.871371 + 0.490624i \(0.836769\pi\)
\(270\) 0 0
\(271\) −19.6073 −1.19106 −0.595528 0.803335i \(-0.703057\pi\)
−0.595528 + 0.803335i \(0.703057\pi\)
\(272\) 0 0
\(273\) −5.64171 −0.341452
\(274\) 0 0
\(275\) −9.92613 −0.598568
\(276\) 0 0
\(277\) −23.1932 −1.39354 −0.696772 0.717293i \(-0.745381\pi\)
−0.696772 + 0.717293i \(0.745381\pi\)
\(278\) 0 0
\(279\) −46.4504 −2.78091
\(280\) 0 0
\(281\) −9.51739 −0.567760 −0.283880 0.958860i \(-0.591622\pi\)
−0.283880 + 0.958860i \(0.591622\pi\)
\(282\) 0 0
\(283\) 5.09510 0.302872 0.151436 0.988467i \(-0.451610\pi\)
0.151436 + 0.988467i \(0.451610\pi\)
\(284\) 0 0
\(285\) −4.21199 −0.249497
\(286\) 0 0
\(287\) −8.72453 −0.514993
\(288\) 0 0
\(289\) −14.1405 −0.831792
\(290\) 0 0
\(291\) −18.4978 −1.08436
\(292\) 0 0
\(293\) −2.71074 −0.158363 −0.0791815 0.996860i \(-0.525231\pi\)
−0.0791815 + 0.996860i \(0.525231\pi\)
\(294\) 0 0
\(295\) −6.42958 −0.374344
\(296\) 0 0
\(297\) 31.3724 1.82041
\(298\) 0 0
\(299\) 14.8266 0.857445
\(300\) 0 0
\(301\) 7.33448 0.422752
\(302\) 0 0
\(303\) 23.8737 1.37151
\(304\) 0 0
\(305\) 7.57091 0.433509
\(306\) 0 0
\(307\) −1.04975 −0.0599121 −0.0299561 0.999551i \(-0.509537\pi\)
−0.0299561 + 0.999551i \(0.509537\pi\)
\(308\) 0 0
\(309\) 53.7318 3.05670
\(310\) 0 0
\(311\) −1.41330 −0.0801410 −0.0400705 0.999197i \(-0.512758\pi\)
−0.0400705 + 0.999197i \(0.512758\pi\)
\(312\) 0 0
\(313\) −26.5463 −1.50049 −0.750244 0.661161i \(-0.770064\pi\)
−0.750244 + 0.661161i \(0.770064\pi\)
\(314\) 0 0
\(315\) 5.35652 0.301806
\(316\) 0 0
\(317\) −24.5350 −1.37802 −0.689011 0.724751i \(-0.741955\pi\)
−0.689011 + 0.724751i \(0.741955\pi\)
\(318\) 0 0
\(319\) 1.56220 0.0874663
\(320\) 0 0
\(321\) −40.1237 −2.23949
\(322\) 0 0
\(323\) 3.04790 0.169589
\(324\) 0 0
\(325\) −7.83204 −0.434444
\(326\) 0 0
\(327\) 8.15716 0.451092
\(328\) 0 0
\(329\) 10.1709 0.560741
\(330\) 0 0
\(331\) 21.6727 1.19124 0.595619 0.803267i \(-0.296907\pi\)
0.595619 + 0.803267i \(0.296907\pi\)
\(332\) 0 0
\(333\) −49.9034 −2.73469
\(334\) 0 0
\(335\) 1.24667 0.0681131
\(336\) 0 0
\(337\) −8.42780 −0.459092 −0.229546 0.973298i \(-0.573724\pi\)
−0.229546 + 0.973298i \(0.573724\pi\)
\(338\) 0 0
\(339\) 3.75698 0.204051
\(340\) 0 0
\(341\) −13.9489 −0.755373
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) −19.7936 −1.06565
\(346\) 0 0
\(347\) −32.4126 −1.74000 −0.869999 0.493053i \(-0.835881\pi\)
−0.869999 + 0.493053i \(0.835881\pi\)
\(348\) 0 0
\(349\) −27.3672 −1.46493 −0.732467 0.680803i \(-0.761631\pi\)
−0.732467 + 0.680803i \(0.761631\pi\)
\(350\) 0 0
\(351\) 24.7539 1.32127
\(352\) 0 0
\(353\) −8.81238 −0.469035 −0.234518 0.972112i \(-0.575351\pi\)
−0.234518 + 0.972112i \(0.575351\pi\)
\(354\) 0 0
\(355\) −1.07170 −0.0568800
\(356\) 0 0
\(357\) −5.45013 −0.288451
\(358\) 0 0
\(359\) 4.54142 0.239687 0.119844 0.992793i \(-0.461761\pi\)
0.119844 + 0.992793i \(0.461761\pi\)
\(360\) 0 0
\(361\) −15.7513 −0.829018
\(362\) 0 0
\(363\) −19.5904 −1.02823
\(364\) 0 0
\(365\) −6.18096 −0.323526
\(366\) 0 0
\(367\) −15.7472 −0.821996 −0.410998 0.911636i \(-0.634820\pi\)
−0.410998 + 0.911636i \(0.634820\pi\)
\(368\) 0 0
\(369\) 64.4539 3.35534
\(370\) 0 0
\(371\) −13.2440 −0.687593
\(372\) 0 0
\(373\) 10.3575 0.536289 0.268144 0.963379i \(-0.413590\pi\)
0.268144 + 0.963379i \(0.413590\pi\)
\(374\) 0 0
\(375\) 22.1402 1.14331
\(376\) 0 0
\(377\) 1.23263 0.0634835
\(378\) 0 0
\(379\) −17.7152 −0.909967 −0.454983 0.890500i \(-0.650355\pi\)
−0.454983 + 0.890500i \(0.650355\pi\)
\(380\) 0 0
\(381\) −0.800648 −0.0410184
\(382\) 0 0
\(383\) 27.1645 1.38804 0.694020 0.719956i \(-0.255838\pi\)
0.694020 + 0.719956i \(0.255838\pi\)
\(384\) 0 0
\(385\) 1.60854 0.0819789
\(386\) 0 0
\(387\) −54.1846 −2.75436
\(388\) 0 0
\(389\) 20.7629 1.05272 0.526361 0.850261i \(-0.323556\pi\)
0.526361 + 0.850261i \(0.323556\pi\)
\(390\) 0 0
\(391\) 14.3231 0.724352
\(392\) 0 0
\(393\) 58.3352 2.94262
\(394\) 0 0
\(395\) −7.71617 −0.388243
\(396\) 0 0
\(397\) 31.0965 1.56069 0.780345 0.625350i \(-0.215044\pi\)
0.780345 + 0.625350i \(0.215044\pi\)
\(398\) 0 0
\(399\) −5.80913 −0.290820
\(400\) 0 0
\(401\) −21.6415 −1.08072 −0.540362 0.841433i \(-0.681713\pi\)
−0.540362 + 0.841433i \(0.681713\pi\)
\(402\) 0 0
\(403\) −11.0061 −0.548254
\(404\) 0 0
\(405\) −16.9770 −0.843596
\(406\) 0 0
\(407\) −14.9858 −0.742818
\(408\) 0 0
\(409\) −18.1632 −0.898111 −0.449056 0.893504i \(-0.648239\pi\)
−0.449056 + 0.893504i \(0.648239\pi\)
\(410\) 0 0
\(411\) −57.2510 −2.82398
\(412\) 0 0
\(413\) −8.86761 −0.436346
\(414\) 0 0
\(415\) 7.86262 0.385961
\(416\) 0 0
\(417\) 40.9405 2.00487
\(418\) 0 0
\(419\) 24.1668 1.18062 0.590312 0.807175i \(-0.299005\pi\)
0.590312 + 0.807175i \(0.299005\pi\)
\(420\) 0 0
\(421\) −23.7262 −1.15635 −0.578173 0.815914i \(-0.696234\pi\)
−0.578173 + 0.815914i \(0.696234\pi\)
\(422\) 0 0
\(423\) −75.1394 −3.65340
\(424\) 0 0
\(425\) −7.56608 −0.367009
\(426\) 0 0
\(427\) 10.4417 0.505310
\(428\) 0 0
\(429\) 12.5160 0.604280
\(430\) 0 0
\(431\) 21.8343 1.05172 0.525861 0.850571i \(-0.323743\pi\)
0.525861 + 0.850571i \(0.323743\pi\)
\(432\) 0 0
\(433\) −35.6444 −1.71296 −0.856480 0.516181i \(-0.827353\pi\)
−0.856480 + 0.516181i \(0.827353\pi\)
\(434\) 0 0
\(435\) −1.64556 −0.0788987
\(436\) 0 0
\(437\) 15.2666 0.730301
\(438\) 0 0
\(439\) 0.0557759 0.00266204 0.00133102 0.999999i \(-0.499576\pi\)
0.00133102 + 0.999999i \(0.499576\pi\)
\(440\) 0 0
\(441\) 7.38766 0.351793
\(442\) 0 0
\(443\) −14.7541 −0.700991 −0.350495 0.936564i \(-0.613987\pi\)
−0.350495 + 0.936564i \(0.613987\pi\)
\(444\) 0 0
\(445\) 1.54675 0.0733230
\(446\) 0 0
\(447\) −25.7229 −1.21665
\(448\) 0 0
\(449\) −11.0183 −0.519986 −0.259993 0.965611i \(-0.583720\pi\)
−0.259993 + 0.965611i \(0.583720\pi\)
\(450\) 0 0
\(451\) 19.3552 0.911403
\(452\) 0 0
\(453\) 59.9653 2.81742
\(454\) 0 0
\(455\) 1.26919 0.0595007
\(456\) 0 0
\(457\) 30.5056 1.42699 0.713495 0.700660i \(-0.247111\pi\)
0.713495 + 0.700660i \(0.247111\pi\)
\(458\) 0 0
\(459\) 23.9133 1.11618
\(460\) 0 0
\(461\) −9.76890 −0.454983 −0.227492 0.973780i \(-0.573052\pi\)
−0.227492 + 0.973780i \(0.573052\pi\)
\(462\) 0 0
\(463\) −21.0301 −0.977352 −0.488676 0.872465i \(-0.662520\pi\)
−0.488676 + 0.872465i \(0.662520\pi\)
\(464\) 0 0
\(465\) 14.6932 0.681382
\(466\) 0 0
\(467\) −5.67899 −0.262792 −0.131396 0.991330i \(-0.541946\pi\)
−0.131396 + 0.991330i \(0.541946\pi\)
\(468\) 0 0
\(469\) 1.71940 0.0793945
\(470\) 0 0
\(471\) 39.6962 1.82910
\(472\) 0 0
\(473\) −16.2714 −0.748161
\(474\) 0 0
\(475\) −8.06447 −0.370023
\(476\) 0 0
\(477\) 97.8420 4.47988
\(478\) 0 0
\(479\) −30.3538 −1.38690 −0.693449 0.720505i \(-0.743910\pi\)
−0.693449 + 0.720505i \(0.743910\pi\)
\(480\) 0 0
\(481\) −11.8243 −0.539141
\(482\) 0 0
\(483\) −27.2992 −1.24215
\(484\) 0 0
\(485\) 4.16138 0.188958
\(486\) 0 0
\(487\) 12.6275 0.572206 0.286103 0.958199i \(-0.407640\pi\)
0.286103 + 0.958199i \(0.407640\pi\)
\(488\) 0 0
\(489\) 80.0732 3.62104
\(490\) 0 0
\(491\) 27.5915 1.24519 0.622594 0.782545i \(-0.286079\pi\)
0.622594 + 0.782545i \(0.286079\pi\)
\(492\) 0 0
\(493\) 1.19077 0.0536295
\(494\) 0 0
\(495\) −11.8834 −0.534118
\(496\) 0 0
\(497\) −1.47808 −0.0663009
\(498\) 0 0
\(499\) −8.82017 −0.394845 −0.197423 0.980318i \(-0.563257\pi\)
−0.197423 + 0.980318i \(0.563257\pi\)
\(500\) 0 0
\(501\) −33.5326 −1.49813
\(502\) 0 0
\(503\) 30.2244 1.34764 0.673819 0.738896i \(-0.264653\pi\)
0.673819 + 0.738896i \(0.264653\pi\)
\(504\) 0 0
\(505\) −5.37077 −0.238996
\(506\) 0 0
\(507\) −32.0233 −1.42220
\(508\) 0 0
\(509\) 6.45130 0.285949 0.142974 0.989726i \(-0.454333\pi\)
0.142974 + 0.989726i \(0.454333\pi\)
\(510\) 0 0
\(511\) −8.52472 −0.377111
\(512\) 0 0
\(513\) 25.4885 1.12535
\(514\) 0 0
\(515\) −12.0878 −0.532654
\(516\) 0 0
\(517\) −22.5640 −0.992366
\(518\) 0 0
\(519\) 42.4397 1.86290
\(520\) 0 0
\(521\) −29.9742 −1.31319 −0.656596 0.754242i \(-0.728004\pi\)
−0.656596 + 0.754242i \(0.728004\pi\)
\(522\) 0 0
\(523\) 7.07881 0.309535 0.154767 0.987951i \(-0.450537\pi\)
0.154767 + 0.987951i \(0.450537\pi\)
\(524\) 0 0
\(525\) 14.4206 0.629365
\(526\) 0 0
\(527\) −10.6324 −0.463153
\(528\) 0 0
\(529\) 48.7432 2.11927
\(530\) 0 0
\(531\) 65.5109 2.84293
\(532\) 0 0
\(533\) 15.2719 0.661501
\(534\) 0 0
\(535\) 9.02647 0.390248
\(536\) 0 0
\(537\) −5.13844 −0.221740
\(538\) 0 0
\(539\) 2.21848 0.0955569
\(540\) 0 0
\(541\) 21.6394 0.930351 0.465175 0.885218i \(-0.345991\pi\)
0.465175 + 0.885218i \(0.345991\pi\)
\(542\) 0 0
\(543\) 19.1247 0.820721
\(544\) 0 0
\(545\) −1.83508 −0.0786064
\(546\) 0 0
\(547\) −14.8782 −0.636145 −0.318073 0.948066i \(-0.603036\pi\)
−0.318073 + 0.948066i \(0.603036\pi\)
\(548\) 0 0
\(549\) −77.1399 −3.29225
\(550\) 0 0
\(551\) 1.26921 0.0540700
\(552\) 0 0
\(553\) −10.6421 −0.452546
\(554\) 0 0
\(555\) 15.7855 0.670056
\(556\) 0 0
\(557\) −18.1723 −0.769986 −0.384993 0.922920i \(-0.625796\pi\)
−0.384993 + 0.922920i \(0.625796\pi\)
\(558\) 0 0
\(559\) −12.8387 −0.543019
\(560\) 0 0
\(561\) 12.0910 0.510483
\(562\) 0 0
\(563\) 5.41059 0.228029 0.114015 0.993479i \(-0.463629\pi\)
0.114015 + 0.993479i \(0.463629\pi\)
\(564\) 0 0
\(565\) −0.845193 −0.0355575
\(566\) 0 0
\(567\) −23.4146 −0.983319
\(568\) 0 0
\(569\) 23.6309 0.990660 0.495330 0.868705i \(-0.335047\pi\)
0.495330 + 0.868705i \(0.335047\pi\)
\(570\) 0 0
\(571\) 15.9091 0.665773 0.332887 0.942967i \(-0.391977\pi\)
0.332887 + 0.942967i \(0.391977\pi\)
\(572\) 0 0
\(573\) −52.2824 −2.18413
\(574\) 0 0
\(575\) −37.8978 −1.58045
\(576\) 0 0
\(577\) 10.9906 0.457544 0.228772 0.973480i \(-0.426529\pi\)
0.228772 + 0.973480i \(0.426529\pi\)
\(578\) 0 0
\(579\) 2.59591 0.107882
\(580\) 0 0
\(581\) 10.8440 0.449887
\(582\) 0 0
\(583\) 29.3816 1.21686
\(584\) 0 0
\(585\) −9.37637 −0.387665
\(586\) 0 0
\(587\) −19.7045 −0.813291 −0.406646 0.913586i \(-0.633302\pi\)
−0.406646 + 0.913586i \(0.633302\pi\)
\(588\) 0 0
\(589\) −11.3327 −0.466957
\(590\) 0 0
\(591\) −32.0334 −1.31768
\(592\) 0 0
\(593\) 7.05691 0.289792 0.144896 0.989447i \(-0.453715\pi\)
0.144896 + 0.989447i \(0.453715\pi\)
\(594\) 0 0
\(595\) 1.22609 0.0502650
\(596\) 0 0
\(597\) −62.6163 −2.56272
\(598\) 0 0
\(599\) 6.03378 0.246534 0.123267 0.992374i \(-0.460663\pi\)
0.123267 + 0.992374i \(0.460663\pi\)
\(600\) 0 0
\(601\) 26.3735 1.07580 0.537900 0.843009i \(-0.319218\pi\)
0.537900 + 0.843009i \(0.319218\pi\)
\(602\) 0 0
\(603\) −12.7023 −0.517280
\(604\) 0 0
\(605\) 4.40717 0.179177
\(606\) 0 0
\(607\) 7.38950 0.299930 0.149965 0.988691i \(-0.452084\pi\)
0.149965 + 0.988691i \(0.452084\pi\)
\(608\) 0 0
\(609\) −2.26954 −0.0919666
\(610\) 0 0
\(611\) −17.8038 −0.720264
\(612\) 0 0
\(613\) −38.9532 −1.57330 −0.786652 0.617397i \(-0.788187\pi\)
−0.786652 + 0.617397i \(0.788187\pi\)
\(614\) 0 0
\(615\) −20.3881 −0.822128
\(616\) 0 0
\(617\) −12.2102 −0.491563 −0.245781 0.969325i \(-0.579045\pi\)
−0.245781 + 0.969325i \(0.579045\pi\)
\(618\) 0 0
\(619\) 47.9001 1.92527 0.962634 0.270804i \(-0.0872896\pi\)
0.962634 + 0.270804i \(0.0872896\pi\)
\(620\) 0 0
\(621\) 119.779 4.80658
\(622\) 0 0
\(623\) 2.13326 0.0854674
\(624\) 0 0
\(625\) 17.3906 0.695625
\(626\) 0 0
\(627\) 12.8875 0.514676
\(628\) 0 0
\(629\) −11.4228 −0.455455
\(630\) 0 0
\(631\) 13.8111 0.549810 0.274905 0.961471i \(-0.411354\pi\)
0.274905 + 0.961471i \(0.411354\pi\)
\(632\) 0 0
\(633\) 59.3843 2.36031
\(634\) 0 0
\(635\) 0.180119 0.00714779
\(636\) 0 0
\(637\) 1.75046 0.0693557
\(638\) 0 0
\(639\) 10.9196 0.431971
\(640\) 0 0
\(641\) 5.42340 0.214212 0.107106 0.994248i \(-0.465842\pi\)
0.107106 + 0.994248i \(0.465842\pi\)
\(642\) 0 0
\(643\) 29.1749 1.15054 0.575272 0.817962i \(-0.304896\pi\)
0.575272 + 0.817962i \(0.304896\pi\)
\(644\) 0 0
\(645\) 17.1397 0.674876
\(646\) 0 0
\(647\) 9.72718 0.382415 0.191207 0.981550i \(-0.438760\pi\)
0.191207 + 0.981550i \(0.438760\pi\)
\(648\) 0 0
\(649\) 19.6726 0.772219
\(650\) 0 0
\(651\) 20.2648 0.794238
\(652\) 0 0
\(653\) 22.5984 0.884346 0.442173 0.896930i \(-0.354208\pi\)
0.442173 + 0.896930i \(0.354208\pi\)
\(654\) 0 0
\(655\) −13.1235 −0.512776
\(656\) 0 0
\(657\) 62.9777 2.45700
\(658\) 0 0
\(659\) −21.0888 −0.821502 −0.410751 0.911748i \(-0.634733\pi\)
−0.410751 + 0.911748i \(0.634733\pi\)
\(660\) 0 0
\(661\) −21.3835 −0.831722 −0.415861 0.909428i \(-0.636520\pi\)
−0.415861 + 0.909428i \(0.636520\pi\)
\(662\) 0 0
\(663\) 9.54022 0.370511
\(664\) 0 0
\(665\) 1.30686 0.0506778
\(666\) 0 0
\(667\) 5.96445 0.230944
\(668\) 0 0
\(669\) −11.6036 −0.448619
\(670\) 0 0
\(671\) −23.1648 −0.894267
\(672\) 0 0
\(673\) −9.99170 −0.385152 −0.192576 0.981282i \(-0.561684\pi\)
−0.192576 + 0.981282i \(0.561684\pi\)
\(674\) 0 0
\(675\) −63.2726 −2.43536
\(676\) 0 0
\(677\) 4.11871 0.158295 0.0791475 0.996863i \(-0.474780\pi\)
0.0791475 + 0.996863i \(0.474780\pi\)
\(678\) 0 0
\(679\) 5.73933 0.220255
\(680\) 0 0
\(681\) −35.1341 −1.34634
\(682\) 0 0
\(683\) −2.28787 −0.0875429 −0.0437714 0.999042i \(-0.513937\pi\)
−0.0437714 + 0.999042i \(0.513937\pi\)
\(684\) 0 0
\(685\) 12.8795 0.492102
\(686\) 0 0
\(687\) 42.9133 1.63724
\(688\) 0 0
\(689\) 23.1830 0.883203
\(690\) 0 0
\(691\) 9.67099 0.367902 0.183951 0.982935i \(-0.441111\pi\)
0.183951 + 0.982935i \(0.441111\pi\)
\(692\) 0 0
\(693\) −16.3894 −0.622582
\(694\) 0 0
\(695\) −9.21023 −0.349364
\(696\) 0 0
\(697\) 14.7533 0.558822
\(698\) 0 0
\(699\) −72.0739 −2.72609
\(700\) 0 0
\(701\) 51.4462 1.94310 0.971549 0.236840i \(-0.0761116\pi\)
0.971549 + 0.236840i \(0.0761116\pi\)
\(702\) 0 0
\(703\) −12.1752 −0.459196
\(704\) 0 0
\(705\) 23.7682 0.895161
\(706\) 0 0
\(707\) −7.40731 −0.278581
\(708\) 0 0
\(709\) −9.65849 −0.362732 −0.181366 0.983416i \(-0.558052\pi\)
−0.181366 + 0.983416i \(0.558052\pi\)
\(710\) 0 0
\(711\) 78.6199 2.94848
\(712\) 0 0
\(713\) −53.2565 −1.99447
\(714\) 0 0
\(715\) −2.81569 −0.105301
\(716\) 0 0
\(717\) −58.4989 −2.18468
\(718\) 0 0
\(719\) 19.5159 0.727820 0.363910 0.931434i \(-0.381441\pi\)
0.363910 + 0.931434i \(0.381441\pi\)
\(720\) 0 0
\(721\) −16.6714 −0.620876
\(722\) 0 0
\(723\) −63.5824 −2.36465
\(724\) 0 0
\(725\) −3.15067 −0.117013
\(726\) 0 0
\(727\) −15.5192 −0.575575 −0.287787 0.957694i \(-0.592920\pi\)
−0.287787 + 0.957694i \(0.592920\pi\)
\(728\) 0 0
\(729\) 36.2462 1.34245
\(730\) 0 0
\(731\) −12.4027 −0.458731
\(732\) 0 0
\(733\) 3.95395 0.146042 0.0730212 0.997330i \(-0.476736\pi\)
0.0730212 + 0.997330i \(0.476736\pi\)
\(734\) 0 0
\(735\) −2.33687 −0.0861968
\(736\) 0 0
\(737\) −3.81446 −0.140508
\(738\) 0 0
\(739\) 45.5632 1.67607 0.838034 0.545617i \(-0.183705\pi\)
0.838034 + 0.545617i \(0.183705\pi\)
\(740\) 0 0
\(741\) 10.1686 0.373554
\(742\) 0 0
\(743\) −22.6814 −0.832100 −0.416050 0.909342i \(-0.636586\pi\)
−0.416050 + 0.909342i \(0.636586\pi\)
\(744\) 0 0
\(745\) 5.78677 0.212011
\(746\) 0 0
\(747\) −80.1121 −2.93115
\(748\) 0 0
\(749\) 12.4492 0.454884
\(750\) 0 0
\(751\) −39.5036 −1.44151 −0.720753 0.693192i \(-0.756204\pi\)
−0.720753 + 0.693192i \(0.756204\pi\)
\(752\) 0 0
\(753\) −23.8112 −0.867730
\(754\) 0 0
\(755\) −13.4902 −0.490957
\(756\) 0 0
\(757\) 2.75169 0.100012 0.0500059 0.998749i \(-0.484076\pi\)
0.0500059 + 0.998749i \(0.484076\pi\)
\(758\) 0 0
\(759\) 60.5628 2.19829
\(760\) 0 0
\(761\) 15.0780 0.546576 0.273288 0.961932i \(-0.411889\pi\)
0.273288 + 0.961932i \(0.411889\pi\)
\(762\) 0 0
\(763\) −2.53093 −0.0916258
\(764\) 0 0
\(765\) −9.05797 −0.327491
\(766\) 0 0
\(767\) 15.5224 0.560480
\(768\) 0 0
\(769\) −23.1596 −0.835158 −0.417579 0.908641i \(-0.637121\pi\)
−0.417579 + 0.908641i \(0.637121\pi\)
\(770\) 0 0
\(771\) −26.2307 −0.944677
\(772\) 0 0
\(773\) 14.7575 0.530790 0.265395 0.964140i \(-0.414498\pi\)
0.265395 + 0.964140i \(0.414498\pi\)
\(774\) 0 0
\(775\) 28.1323 1.01054
\(776\) 0 0
\(777\) 21.7712 0.781037
\(778\) 0 0
\(779\) 15.7251 0.563412
\(780\) 0 0
\(781\) 3.27910 0.117335
\(782\) 0 0
\(783\) 9.95799 0.355870
\(784\) 0 0
\(785\) −8.93030 −0.318736
\(786\) 0 0
\(787\) −12.3963 −0.441879 −0.220940 0.975287i \(-0.570912\pi\)
−0.220940 + 0.975287i \(0.570912\pi\)
\(788\) 0 0
\(789\) 58.2359 2.07325
\(790\) 0 0
\(791\) −1.16568 −0.0414469
\(792\) 0 0
\(793\) −18.2778 −0.649063
\(794\) 0 0
\(795\) −30.9495 −1.09766
\(796\) 0 0
\(797\) −18.9843 −0.672457 −0.336229 0.941780i \(-0.609152\pi\)
−0.336229 + 0.941780i \(0.609152\pi\)
\(798\) 0 0
\(799\) −17.1992 −0.608464
\(800\) 0 0
\(801\) −15.7598 −0.556846
\(802\) 0 0
\(803\) 18.9120 0.667389
\(804\) 0 0
\(805\) 6.14139 0.216455
\(806\) 0 0
\(807\) −92.1230 −3.24288
\(808\) 0 0
\(809\) 6.90560 0.242788 0.121394 0.992604i \(-0.461264\pi\)
0.121394 + 0.992604i \(0.461264\pi\)
\(810\) 0 0
\(811\) −13.8566 −0.486572 −0.243286 0.969955i \(-0.578225\pi\)
−0.243286 + 0.969955i \(0.578225\pi\)
\(812\) 0 0
\(813\) −63.1940 −2.21631
\(814\) 0 0
\(815\) −18.0138 −0.630995
\(816\) 0 0
\(817\) −13.2197 −0.462499
\(818\) 0 0
\(819\) −12.9318 −0.451873
\(820\) 0 0
\(821\) 30.6552 1.06987 0.534936 0.844892i \(-0.320336\pi\)
0.534936 + 0.844892i \(0.320336\pi\)
\(822\) 0 0
\(823\) −22.9379 −0.799564 −0.399782 0.916610i \(-0.630914\pi\)
−0.399782 + 0.916610i \(0.630914\pi\)
\(824\) 0 0
\(825\) −31.9918 −1.11381
\(826\) 0 0
\(827\) −12.7310 −0.442701 −0.221350 0.975194i \(-0.571046\pi\)
−0.221350 + 0.975194i \(0.571046\pi\)
\(828\) 0 0
\(829\) 19.3666 0.672630 0.336315 0.941750i \(-0.390819\pi\)
0.336315 + 0.941750i \(0.390819\pi\)
\(830\) 0 0
\(831\) −74.7514 −2.59310
\(832\) 0 0
\(833\) 1.69102 0.0585902
\(834\) 0 0
\(835\) 7.54370 0.261061
\(836\) 0 0
\(837\) −88.9149 −3.07335
\(838\) 0 0
\(839\) 26.7513 0.923558 0.461779 0.886995i \(-0.347211\pi\)
0.461779 + 0.886995i \(0.347211\pi\)
\(840\) 0 0
\(841\) −28.5041 −0.982901
\(842\) 0 0
\(843\) −30.6744 −1.05648
\(844\) 0 0
\(845\) 7.20416 0.247830
\(846\) 0 0
\(847\) 6.07833 0.208854
\(848\) 0 0
\(849\) 16.4214 0.563582
\(850\) 0 0
\(851\) −57.2154 −1.96132
\(852\) 0 0
\(853\) −48.1783 −1.64959 −0.824796 0.565430i \(-0.808710\pi\)
−0.824796 + 0.565430i \(0.808710\pi\)
\(854\) 0 0
\(855\) −9.65462 −0.330181
\(856\) 0 0
\(857\) 26.2828 0.897802 0.448901 0.893581i \(-0.351816\pi\)
0.448901 + 0.893581i \(0.351816\pi\)
\(858\) 0 0
\(859\) −15.3098 −0.522363 −0.261181 0.965290i \(-0.584112\pi\)
−0.261181 + 0.965290i \(0.584112\pi\)
\(860\) 0 0
\(861\) −28.1191 −0.958296
\(862\) 0 0
\(863\) 25.5692 0.870386 0.435193 0.900337i \(-0.356680\pi\)
0.435193 + 0.900337i \(0.356680\pi\)
\(864\) 0 0
\(865\) −9.54750 −0.324625
\(866\) 0 0
\(867\) −45.5746 −1.54779
\(868\) 0 0
\(869\) 23.6092 0.800889
\(870\) 0 0
\(871\) −3.00974 −0.101981
\(872\) 0 0
\(873\) −42.4002 −1.43503
\(874\) 0 0
\(875\) −6.86946 −0.232230
\(876\) 0 0
\(877\) 47.6643 1.60951 0.804755 0.593607i \(-0.202297\pi\)
0.804755 + 0.593607i \(0.202297\pi\)
\(878\) 0 0
\(879\) −8.73668 −0.294681
\(880\) 0 0
\(881\) −57.2434 −1.92858 −0.964290 0.264850i \(-0.914678\pi\)
−0.964290 + 0.264850i \(0.914678\pi\)
\(882\) 0 0
\(883\) −25.8202 −0.868919 −0.434460 0.900691i \(-0.643061\pi\)
−0.434460 + 0.900691i \(0.643061\pi\)
\(884\) 0 0
\(885\) −20.7225 −0.696578
\(886\) 0 0
\(887\) 34.1175 1.14555 0.572777 0.819711i \(-0.305866\pi\)
0.572777 + 0.819711i \(0.305866\pi\)
\(888\) 0 0
\(889\) 0.248418 0.00833167
\(890\) 0 0
\(891\) 51.9448 1.74022
\(892\) 0 0
\(893\) −18.3321 −0.613461
\(894\) 0 0
\(895\) 1.15597 0.0386400
\(896\) 0 0
\(897\) 47.7860 1.59553
\(898\) 0 0
\(899\) −4.42754 −0.147667
\(900\) 0 0
\(901\) 22.3958 0.746111
\(902\) 0 0
\(903\) 23.6389 0.786655
\(904\) 0 0
\(905\) −4.30242 −0.143017
\(906\) 0 0
\(907\) 8.87794 0.294787 0.147394 0.989078i \(-0.452912\pi\)
0.147394 + 0.989078i \(0.452912\pi\)
\(908\) 0 0
\(909\) 54.7227 1.81504
\(910\) 0 0
\(911\) 4.33134 0.143504 0.0717519 0.997423i \(-0.477141\pi\)
0.0717519 + 0.997423i \(0.477141\pi\)
\(912\) 0 0
\(913\) −24.0573 −0.796182
\(914\) 0 0
\(915\) 24.4010 0.806671
\(916\) 0 0
\(917\) −18.0997 −0.597706
\(918\) 0 0
\(919\) 8.08794 0.266796 0.133398 0.991063i \(-0.457411\pi\)
0.133398 + 0.991063i \(0.457411\pi\)
\(920\) 0 0
\(921\) −3.38332 −0.111484
\(922\) 0 0
\(923\) 2.58732 0.0851626
\(924\) 0 0
\(925\) 30.2236 0.993746
\(926\) 0 0
\(927\) 123.163 4.04520
\(928\) 0 0
\(929\) −21.9875 −0.721387 −0.360694 0.932684i \(-0.617460\pi\)
−0.360694 + 0.932684i \(0.617460\pi\)
\(930\) 0 0
\(931\) 1.80241 0.0590714
\(932\) 0 0
\(933\) −4.55506 −0.149126
\(934\) 0 0
\(935\) −2.72007 −0.0889558
\(936\) 0 0
\(937\) 37.5410 1.22641 0.613205 0.789924i \(-0.289880\pi\)
0.613205 + 0.789924i \(0.289880\pi\)
\(938\) 0 0
\(939\) −85.5586 −2.79210
\(940\) 0 0
\(941\) −4.15697 −0.135513 −0.0677567 0.997702i \(-0.521584\pi\)
−0.0677567 + 0.997702i \(0.521584\pi\)
\(942\) 0 0
\(943\) 73.8980 2.40645
\(944\) 0 0
\(945\) 10.2534 0.333543
\(946\) 0 0
\(947\) −0.495081 −0.0160880 −0.00804399 0.999968i \(-0.502561\pi\)
−0.00804399 + 0.999968i \(0.502561\pi\)
\(948\) 0 0
\(949\) 14.9222 0.484394
\(950\) 0 0
\(951\) −79.0760 −2.56421
\(952\) 0 0
\(953\) 3.51461 0.113849 0.0569246 0.998378i \(-0.481871\pi\)
0.0569246 + 0.998378i \(0.481871\pi\)
\(954\) 0 0
\(955\) 11.7618 0.380602
\(956\) 0 0
\(957\) 5.03495 0.162757
\(958\) 0 0
\(959\) 17.7633 0.573608
\(960\) 0 0
\(961\) 8.53346 0.275273
\(962\) 0 0
\(963\) −91.9706 −2.96371
\(964\) 0 0
\(965\) −0.583991 −0.0187993
\(966\) 0 0
\(967\) −15.3243 −0.492795 −0.246398 0.969169i \(-0.579247\pi\)
−0.246398 + 0.969169i \(0.579247\pi\)
\(968\) 0 0
\(969\) 9.82334 0.315571
\(970\) 0 0
\(971\) −18.7915 −0.603049 −0.301524 0.953458i \(-0.597495\pi\)
−0.301524 + 0.953458i \(0.597495\pi\)
\(972\) 0 0
\(973\) −12.7027 −0.407229
\(974\) 0 0
\(975\) −25.2426 −0.808410
\(976\) 0 0
\(977\) 1.16324 0.0372154 0.0186077 0.999827i \(-0.494077\pi\)
0.0186077 + 0.999827i \(0.494077\pi\)
\(978\) 0 0
\(979\) −4.73261 −0.151255
\(980\) 0 0
\(981\) 18.6977 0.596970
\(982\) 0 0
\(983\) 0.537086 0.0171304 0.00856519 0.999963i \(-0.497274\pi\)
0.00856519 + 0.999963i \(0.497274\pi\)
\(984\) 0 0
\(985\) 7.20644 0.229616
\(986\) 0 0
\(987\) 32.7808 1.04342
\(988\) 0 0
\(989\) −62.1240 −1.97543
\(990\) 0 0
\(991\) 7.63276 0.242463 0.121231 0.992624i \(-0.461316\pi\)
0.121231 + 0.992624i \(0.461316\pi\)
\(992\) 0 0
\(993\) 69.8508 2.21665
\(994\) 0 0
\(995\) 14.0866 0.446574
\(996\) 0 0
\(997\) 33.0138 1.04556 0.522779 0.852468i \(-0.324895\pi\)
0.522779 + 0.852468i \(0.324895\pi\)
\(998\) 0 0
\(999\) −95.5245 −3.02226
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 3584.2.a.k.1.6 yes 6
4.3 odd 2 3584.2.a.f.1.1 yes 6
8.3 odd 2 3584.2.a.l.1.6 yes 6
8.5 even 2 3584.2.a.e.1.1 6
16.3 odd 4 3584.2.b.i.1793.1 12
16.5 even 4 3584.2.b.k.1793.1 12
16.11 odd 4 3584.2.b.i.1793.12 12
16.13 even 4 3584.2.b.k.1793.12 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3584.2.a.e.1.1 6 8.5 even 2
3584.2.a.f.1.1 yes 6 4.3 odd 2
3584.2.a.k.1.6 yes 6 1.1 even 1 trivial
3584.2.a.l.1.6 yes 6 8.3 odd 2
3584.2.b.i.1793.1 12 16.3 odd 4
3584.2.b.i.1793.12 12 16.11 odd 4
3584.2.b.k.1793.1 12 16.5 even 4
3584.2.b.k.1793.12 12 16.13 even 4