Properties

Label 4900.2.e.s.2549.6
Level $4900$
Weight $2$
Character 4900.2549
Analytic conductor $39.127$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4900,2,Mod(2549,4900)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4900, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4900.2549");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4900 = 2^{2} \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4900.e (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(39.1266969904\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.4227136.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 9x^{4} + 22x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 700)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2549.6
Root \(0.713538i\) of defining polynomial
Character \(\chi\) \(=\) 4900.2549
Dual form 4900.2.e.s.2549.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.20440i q^{3} -7.26819 q^{9} +O(q^{10})\) \(q+3.20440i q^{3} -7.26819 q^{9} +4.20440 q^{11} -0.204402i q^{13} -5.06379i q^{17} +1.06379 q^{19} +2.14061i q^{23} -13.6770i q^{27} +7.47259 q^{29} +8.47259 q^{31} +13.4726i q^{33} +10.6132i q^{37} +0.654985 q^{39} -10.5494 q^{41} +8.26819i q^{43} +3.26819i q^{47} +16.2264 q^{51} +5.67699i q^{53} +3.40880i q^{57} +1.20440 q^{59} -1.65498 q^{61} +12.4088i q^{67} -6.85939 q^{69} -0.591197 q^{71} +4.00000i q^{73} -6.54942 q^{79} +22.0220 q^{81} -3.88139i q^{83} +23.9452i q^{87} -9.26819 q^{89} +27.1496i q^{93} +1.33198i q^{97} -30.5584 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 16 q^{9} + 8 q^{11} - 4 q^{19} + 6 q^{31} + 28 q^{39} - 22 q^{41} - 6 q^{51} - 10 q^{59} - 34 q^{61} - 48 q^{69} - 38 q^{71} + 2 q^{79} + 46 q^{81} - 28 q^{89} - 42 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4900\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(1177\) \(2451\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) 3.20440i 1.85006i 0.379892 + 0.925031i \(0.375961\pi\)
−0.379892 + 0.925031i \(0.624039\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −7.26819 −2.42273
\(10\) 0 0
\(11\) 4.20440 1.26767 0.633837 0.773466i \(-0.281479\pi\)
0.633837 + 0.773466i \(0.281479\pi\)
\(12\) 0 0
\(13\) − 0.204402i − 0.0566908i −0.999598 0.0283454i \(-0.990976\pi\)
0.999598 0.0283454i \(-0.00902383\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 5.06379i − 1.22815i −0.789248 0.614074i \(-0.789529\pi\)
0.789248 0.614074i \(-0.210471\pi\)
\(18\) 0 0
\(19\) 1.06379 0.244050 0.122025 0.992527i \(-0.461061\pi\)
0.122025 + 0.992527i \(0.461061\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2.14061i 0.446349i 0.974779 + 0.223174i \(0.0716419\pi\)
−0.974779 + 0.223174i \(0.928358\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 13.6770i − 2.63214i
\(28\) 0 0
\(29\) 7.47259 1.38763 0.693813 0.720156i \(-0.255930\pi\)
0.693813 + 0.720156i \(0.255930\pi\)
\(30\) 0 0
\(31\) 8.47259 1.52172 0.760861 0.648915i \(-0.224777\pi\)
0.760861 + 0.648915i \(0.224777\pi\)
\(32\) 0 0
\(33\) 13.4726i 2.34528i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 10.6132i 1.74480i 0.488793 + 0.872400i \(0.337437\pi\)
−0.488793 + 0.872400i \(0.662563\pi\)
\(38\) 0 0
\(39\) 0.654985 0.104881
\(40\) 0 0
\(41\) −10.5494 −1.64754 −0.823771 0.566923i \(-0.808134\pi\)
−0.823771 + 0.566923i \(0.808134\pi\)
\(42\) 0 0
\(43\) 8.26819i 1.26089i 0.776235 + 0.630444i \(0.217127\pi\)
−0.776235 + 0.630444i \(0.782873\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 3.26819i 0.476714i 0.971178 + 0.238357i \(0.0766089\pi\)
−0.971178 + 0.238357i \(0.923391\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 16.2264 2.27215
\(52\) 0 0
\(53\) 5.67699i 0.779795i 0.920858 + 0.389897i \(0.127490\pi\)
−0.920858 + 0.389897i \(0.872510\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 3.40880i 0.451507i
\(58\) 0 0
\(59\) 1.20440 0.156800 0.0783999 0.996922i \(-0.475019\pi\)
0.0783999 + 0.996922i \(0.475019\pi\)
\(60\) 0 0
\(61\) −1.65498 −0.211899 −0.105950 0.994372i \(-0.533788\pi\)
−0.105950 + 0.994372i \(0.533788\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 12.4088i 1.51598i 0.652268 + 0.757988i \(0.273818\pi\)
−0.652268 + 0.757988i \(0.726182\pi\)
\(68\) 0 0
\(69\) −6.85939 −0.825773
\(70\) 0 0
\(71\) −0.591197 −0.0701622 −0.0350811 0.999384i \(-0.511169\pi\)
−0.0350811 + 0.999384i \(0.511169\pi\)
\(72\) 0 0
\(73\) 4.00000i 0.468165i 0.972217 + 0.234082i \(0.0752085\pi\)
−0.972217 + 0.234082i \(0.924791\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −6.54942 −0.736867 −0.368433 0.929654i \(-0.620106\pi\)
−0.368433 + 0.929654i \(0.620106\pi\)
\(80\) 0 0
\(81\) 22.0220 2.44689
\(82\) 0 0
\(83\) − 3.88139i − 0.426038i −0.977048 0.213019i \(-0.931670\pi\)
0.977048 0.213019i \(-0.0683297\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 23.9452i 2.56719i
\(88\) 0 0
\(89\) −9.26819 −0.982426 −0.491213 0.871039i \(-0.663446\pi\)
−0.491213 + 0.871039i \(0.663446\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 27.1496i 2.81528i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1.33198i 0.135242i 0.997711 + 0.0676209i \(0.0215408\pi\)
−0.997711 + 0.0676209i \(0.978459\pi\)
\(98\) 0 0
\(99\) −30.5584 −3.07123
\(100\) 0 0
\(101\) −6.19136 −0.616064 −0.308032 0.951376i \(-0.599670\pi\)
−0.308032 + 0.951376i \(0.599670\pi\)
\(102\) 0 0
\(103\) − 9.74078i − 0.959788i −0.877327 0.479894i \(-0.840675\pi\)
0.877327 0.479894i \(-0.159325\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 9.85939i − 0.953143i −0.879136 0.476571i \(-0.841879\pi\)
0.879136 0.476571i \(-0.158121\pi\)
\(108\) 0 0
\(109\) 4.07683 0.390489 0.195245 0.980755i \(-0.437450\pi\)
0.195245 + 0.980755i \(0.437450\pi\)
\(110\) 0 0
\(111\) −34.0090 −3.22799
\(112\) 0 0
\(113\) 2.21744i 0.208599i 0.994546 + 0.104300i \(0.0332601\pi\)
−0.994546 + 0.104300i \(0.966740\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 1.48563i 0.137346i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 6.67699 0.606999
\(122\) 0 0
\(123\) − 33.8046i − 3.04806i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 6.54942i − 0.581167i −0.956850 0.290583i \(-0.906151\pi\)
0.956850 0.290583i \(-0.0938494\pi\)
\(128\) 0 0
\(129\) −26.4946 −2.33272
\(130\) 0 0
\(131\) 17.0090 1.48608 0.743040 0.669247i \(-0.233383\pi\)
0.743040 + 0.669247i \(0.233383\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 17.0858i 1.45974i 0.683587 + 0.729869i \(0.260419\pi\)
−0.683587 + 0.729869i \(0.739581\pi\)
\(138\) 0 0
\(139\) 0.740780 0.0628322 0.0314161 0.999506i \(-0.489998\pi\)
0.0314161 + 0.999506i \(0.489998\pi\)
\(140\) 0 0
\(141\) −10.4726 −0.881951
\(142\) 0 0
\(143\) − 0.859386i − 0.0718655i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 16.4178 1.34500 0.672498 0.740099i \(-0.265221\pi\)
0.672498 + 0.740099i \(0.265221\pi\)
\(150\) 0 0
\(151\) 0.527409 0.0429199 0.0214600 0.999770i \(-0.493169\pi\)
0.0214600 + 0.999770i \(0.493169\pi\)
\(152\) 0 0
\(153\) 36.8046i 2.97547i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 9.48563i − 0.757036i −0.925594 0.378518i \(-0.876434\pi\)
0.925594 0.378518i \(-0.123566\pi\)
\(158\) 0 0
\(159\) −18.1914 −1.44267
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 8.85939i 0.693921i 0.937880 + 0.346960i \(0.112786\pi\)
−0.937880 + 0.346960i \(0.887214\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 8.33198i 0.644748i 0.946612 + 0.322374i \(0.104481\pi\)
−0.946612 + 0.322374i \(0.895519\pi\)
\(168\) 0 0
\(169\) 12.9582 0.996786
\(170\) 0 0
\(171\) −7.73181 −0.591266
\(172\) 0 0
\(173\) 18.3450i 1.39475i 0.716709 + 0.697373i \(0.245648\pi\)
−0.716709 + 0.697373i \(0.754352\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 3.85939i 0.290089i
\(178\) 0 0
\(179\) 4.28123 0.319994 0.159997 0.987118i \(-0.448852\pi\)
0.159997 + 0.987118i \(0.448852\pi\)
\(180\) 0 0
\(181\) −2.93621 −0.218247 −0.109123 0.994028i \(-0.534804\pi\)
−0.109123 + 0.994028i \(0.534804\pi\)
\(182\) 0 0
\(183\) − 5.30324i − 0.392026i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 21.2902i − 1.55689i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −11.9452 −0.864323 −0.432162 0.901796i \(-0.642249\pi\)
−0.432162 + 0.901796i \(0.642249\pi\)
\(192\) 0 0
\(193\) 7.08580i 0.510047i 0.966935 + 0.255023i \(0.0820832\pi\)
−0.966935 + 0.255023i \(0.917917\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 5.48563i 0.390835i 0.980720 + 0.195417i \(0.0626062\pi\)
−0.980720 + 0.195417i \(0.937394\pi\)
\(198\) 0 0
\(199\) 16.0858 1.14029 0.570146 0.821543i \(-0.306887\pi\)
0.570146 + 0.821543i \(0.306887\pi\)
\(200\) 0 0
\(201\) −39.7628 −2.80465
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 15.5584i − 1.08138i
\(208\) 0 0
\(209\) 4.47259 0.309376
\(210\) 0 0
\(211\) 16.6002 1.14280 0.571401 0.820671i \(-0.306400\pi\)
0.571401 + 0.820671i \(0.306400\pi\)
\(212\) 0 0
\(213\) − 1.89443i − 0.129804i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −12.8176 −0.866134
\(220\) 0 0
\(221\) −1.03505 −0.0696247
\(222\) 0 0
\(223\) 15.4856i 1.03699i 0.855079 + 0.518497i \(0.173508\pi\)
−0.855079 + 0.518497i \(0.826492\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 3.78256i − 0.251057i −0.992090 0.125529i \(-0.959937\pi\)
0.992090 0.125529i \(-0.0400627\pi\)
\(228\) 0 0
\(229\) −11.2044 −0.740408 −0.370204 0.928951i \(-0.620712\pi\)
−0.370204 + 0.928951i \(0.620712\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 11.9452i − 0.782555i −0.920273 0.391277i \(-0.872033\pi\)
0.920273 0.391277i \(-0.127967\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 20.9870i − 1.36325i
\(238\) 0 0
\(239\) −15.3450 −0.992587 −0.496293 0.868155i \(-0.665306\pi\)
−0.496293 + 0.868155i \(0.665306\pi\)
\(240\) 0 0
\(241\) 14.3958 0.927313 0.463656 0.886015i \(-0.346537\pi\)
0.463656 + 0.886015i \(0.346537\pi\)
\(242\) 0 0
\(243\) 29.5364i 1.89476i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 0.217440i − 0.0138354i
\(248\) 0 0
\(249\) 12.4375 0.788197
\(250\) 0 0
\(251\) 19.4178 1.22564 0.612819 0.790223i \(-0.290035\pi\)
0.612819 + 0.790223i \(0.290035\pi\)
\(252\) 0 0
\(253\) 9.00000i 0.565825i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 13.8724i − 0.865338i −0.901553 0.432669i \(-0.857572\pi\)
0.901553 0.432669i \(-0.142428\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −54.3122 −3.36184
\(262\) 0 0
\(263\) 3.45955i 0.213325i 0.994295 + 0.106663i \(0.0340165\pi\)
−0.994295 + 0.106663i \(0.965984\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 29.6990i − 1.81755i
\(268\) 0 0
\(269\) 25.6640 1.56476 0.782379 0.622802i \(-0.214006\pi\)
0.782379 + 0.622802i \(0.214006\pi\)
\(270\) 0 0
\(271\) −19.6770 −1.19529 −0.597646 0.801760i \(-0.703897\pi\)
−0.597646 + 0.801760i \(0.703897\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 18.1496i − 1.09050i −0.838273 0.545251i \(-0.816434\pi\)
0.838273 0.545251i \(-0.183566\pi\)
\(278\) 0 0
\(279\) −61.5804 −3.68672
\(280\) 0 0
\(281\) −18.5364 −1.10579 −0.552894 0.833252i \(-0.686476\pi\)
−0.552894 + 0.833252i \(0.686476\pi\)
\(282\) 0 0
\(283\) 28.5584i 1.69762i 0.528698 + 0.848810i \(0.322680\pi\)
−0.528698 + 0.848810i \(0.677320\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −8.64195 −0.508350
\(290\) 0 0
\(291\) −4.26819 −0.250206
\(292\) 0 0
\(293\) 25.5494i 1.49261i 0.665603 + 0.746306i \(0.268175\pi\)
−0.665603 + 0.746306i \(0.731825\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) − 57.5036i − 3.33670i
\(298\) 0 0
\(299\) 0.437545 0.0253039
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) − 19.8396i − 1.13976i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 10.1145i − 0.577267i −0.957440 0.288634i \(-0.906799\pi\)
0.957440 0.288634i \(-0.0932009\pi\)
\(308\) 0 0
\(309\) 31.2134 1.77567
\(310\) 0 0
\(311\) −34.1716 −1.93769 −0.968847 0.247662i \(-0.920338\pi\)
−0.968847 + 0.247662i \(0.920338\pi\)
\(312\) 0 0
\(313\) − 8.61320i − 0.486847i −0.969920 0.243424i \(-0.921730\pi\)
0.969920 0.243424i \(-0.0782705\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 7.81761i − 0.439081i −0.975603 0.219540i \(-0.929544\pi\)
0.975603 0.219540i \(-0.0704557\pi\)
\(318\) 0 0
\(319\) 31.4178 1.75906
\(320\) 0 0
\(321\) 31.5934 1.76337
\(322\) 0 0
\(323\) − 5.38680i − 0.299729i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 13.0638i 0.722429i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −22.0988 −1.21466 −0.607331 0.794449i \(-0.707760\pi\)
−0.607331 + 0.794449i \(0.707760\pi\)
\(332\) 0 0
\(333\) − 77.1388i − 4.22718i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 1.29020i 0.0702815i 0.999382 + 0.0351408i \(0.0111880\pi\)
−0.999382 + 0.0351408i \(0.988812\pi\)
\(338\) 0 0
\(339\) −7.10557 −0.385921
\(340\) 0 0
\(341\) 35.6222 1.92905
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 9.45955i 0.507815i 0.967228 + 0.253908i \(0.0817159\pi\)
−0.967228 + 0.253908i \(0.918284\pi\)
\(348\) 0 0
\(349\) 15.5494 0.832341 0.416171 0.909287i \(-0.363372\pi\)
0.416171 + 0.909287i \(0.363372\pi\)
\(350\) 0 0
\(351\) −2.79560 −0.149218
\(352\) 0 0
\(353\) − 15.1365i − 0.805637i −0.915280 0.402818i \(-0.868031\pi\)
0.915280 0.402818i \(-0.131969\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −10.3189 −0.544613 −0.272306 0.962211i \(-0.587786\pi\)
−0.272306 + 0.962211i \(0.587786\pi\)
\(360\) 0 0
\(361\) −17.8684 −0.940440
\(362\) 0 0
\(363\) 21.3958i 1.12299i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 17.9362i 0.936263i 0.883659 + 0.468131i \(0.155073\pi\)
−0.883659 + 0.468131i \(0.844927\pi\)
\(368\) 0 0
\(369\) 76.6752 3.99155
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) − 16.1914i − 0.838357i −0.907904 0.419179i \(-0.862318\pi\)
0.907904 0.419179i \(-0.137682\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 1.52741i − 0.0786656i
\(378\) 0 0
\(379\) 12.6002 0.647227 0.323614 0.946189i \(-0.395102\pi\)
0.323614 + 0.946189i \(0.395102\pi\)
\(380\) 0 0
\(381\) 20.9870 1.07519
\(382\) 0 0
\(383\) − 15.1757i − 0.775440i −0.921777 0.387720i \(-0.873263\pi\)
0.921777 0.387720i \(-0.126737\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 60.0948i − 3.05479i
\(388\) 0 0
\(389\) 29.2394 1.48250 0.741249 0.671230i \(-0.234234\pi\)
0.741249 + 0.671230i \(0.234234\pi\)
\(390\) 0 0
\(391\) 10.8396 0.548183
\(392\) 0 0
\(393\) 54.5036i 2.74934i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 11.2682i 0.565534i 0.959189 + 0.282767i \(0.0912524\pi\)
−0.959189 + 0.282767i \(0.908748\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −18.0220 −0.899976 −0.449988 0.893035i \(-0.648572\pi\)
−0.449988 + 0.893035i \(0.648572\pi\)
\(402\) 0 0
\(403\) − 1.73181i − 0.0862676i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 44.6222i 2.21184i
\(408\) 0 0
\(409\) −15.7915 −0.780841 −0.390420 0.920637i \(-0.627670\pi\)
−0.390420 + 0.920637i \(0.627670\pi\)
\(410\) 0 0
\(411\) −54.7497 −2.70061
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 2.37376i 0.116243i
\(418\) 0 0
\(419\) −13.4726 −0.658179 −0.329090 0.944299i \(-0.606742\pi\)
−0.329090 + 0.944299i \(0.606742\pi\)
\(420\) 0 0
\(421\) −26.8396 −1.30808 −0.654041 0.756459i \(-0.726928\pi\)
−0.654041 + 0.756459i \(0.726928\pi\)
\(422\) 0 0
\(423\) − 23.7538i − 1.15495i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 2.75382 0.132956
\(430\) 0 0
\(431\) 16.0507 0.773137 0.386569 0.922261i \(-0.373660\pi\)
0.386569 + 0.922261i \(0.373660\pi\)
\(432\) 0 0
\(433\) − 0.910136i − 0.0437383i −0.999761 0.0218692i \(-0.993038\pi\)
0.999761 0.0218692i \(-0.00696173\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 2.27716i 0.108931i
\(438\) 0 0
\(439\) −37.9034 −1.80903 −0.904515 0.426441i \(-0.859767\pi\)
−0.904515 + 0.426441i \(0.859767\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 6.88139i − 0.326945i −0.986548 0.163472i \(-0.947731\pi\)
0.986548 0.163472i \(-0.0522695\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 52.6091i 2.48833i
\(448\) 0 0
\(449\) −6.43754 −0.303807 −0.151903 0.988395i \(-0.548540\pi\)
−0.151903 + 0.988395i \(0.548540\pi\)
\(450\) 0 0
\(451\) −44.3540 −2.08855
\(452\) 0 0
\(453\) 1.69003i 0.0794046i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 9.29427i 0.434767i 0.976086 + 0.217384i \(0.0697523\pi\)
−0.976086 + 0.217384i \(0.930248\pi\)
\(458\) 0 0
\(459\) −69.2574 −3.23266
\(460\) 0 0
\(461\) −9.29020 −0.432688 −0.216344 0.976317i \(-0.569413\pi\)
−0.216344 + 0.976317i \(0.569413\pi\)
\(462\) 0 0
\(463\) − 29.6091i − 1.37605i −0.725685 0.688027i \(-0.758477\pi\)
0.725685 0.688027i \(-0.241523\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 26.0638i − 1.20609i −0.797708 0.603044i \(-0.793954\pi\)
0.797708 0.603044i \(-0.206046\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 30.3958 1.40056
\(472\) 0 0
\(473\) 34.7628i 1.59839i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) − 41.2615i − 1.88923i
\(478\) 0 0
\(479\) −1.10557 −0.0505147 −0.0252573 0.999681i \(-0.508041\pi\)
−0.0252573 + 0.999681i \(0.508041\pi\)
\(480\) 0 0
\(481\) 2.16936 0.0989141
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 5.19136i − 0.235243i −0.993058 0.117622i \(-0.962473\pi\)
0.993058 0.117622i \(-0.0375270\pi\)
\(488\) 0 0
\(489\) −28.3890 −1.28380
\(490\) 0 0
\(491\) −28.6640 −1.29359 −0.646793 0.762666i \(-0.723890\pi\)
−0.646793 + 0.762666i \(0.723890\pi\)
\(492\) 0 0
\(493\) − 37.8396i − 1.70421i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 26.0728 1.16718 0.583588 0.812050i \(-0.301648\pi\)
0.583588 + 0.812050i \(0.301648\pi\)
\(500\) 0 0
\(501\) −26.6990 −1.19282
\(502\) 0 0
\(503\) − 8.80864i − 0.392758i −0.980528 0.196379i \(-0.937082\pi\)
0.980528 0.196379i \(-0.0629182\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 41.5233i 1.84412i
\(508\) 0 0
\(509\) −22.0179 −0.975928 −0.487964 0.872864i \(-0.662260\pi\)
−0.487964 + 0.872864i \(0.662260\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) − 14.5494i − 0.642372i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 13.7408i 0.604319i
\(518\) 0 0
\(519\) −58.7848 −2.58037
\(520\) 0 0
\(521\) 15.0988 0.661492 0.330746 0.943720i \(-0.392700\pi\)
0.330746 + 0.943720i \(0.392700\pi\)
\(522\) 0 0
\(523\) 4.59120i 0.200759i 0.994949 + 0.100380i \(0.0320057\pi\)
−0.994949 + 0.100380i \(0.967994\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 42.9034i − 1.86890i
\(528\) 0 0
\(529\) 18.4178 0.800773
\(530\) 0 0
\(531\) −8.75382 −0.379883
\(532\) 0 0
\(533\) 2.15632i 0.0934005i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 13.7188i 0.592009i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 13.6770 0.588020 0.294010 0.955802i \(-0.405010\pi\)
0.294010 + 0.955802i \(0.405010\pi\)
\(542\) 0 0
\(543\) − 9.40880i − 0.403770i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 11.3581i 0.485635i 0.970072 + 0.242818i \(0.0780717\pi\)
−0.970072 + 0.242818i \(0.921928\pi\)
\(548\) 0 0
\(549\) 12.0287 0.513374
\(550\) 0 0
\(551\) 7.94925 0.338649
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 16.3958i − 0.694711i −0.937734 0.347355i \(-0.887080\pi\)
0.937734 0.347355i \(-0.112920\pi\)
\(558\) 0 0
\(559\) 1.69003 0.0714807
\(560\) 0 0
\(561\) 68.2223 2.88035
\(562\) 0 0
\(563\) 4.87242i 0.205348i 0.994715 + 0.102674i \(0.0327399\pi\)
−0.994715 + 0.102674i \(0.967260\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 37.8553 1.58698 0.793489 0.608585i \(-0.208263\pi\)
0.793489 + 0.608585i \(0.208263\pi\)
\(570\) 0 0
\(571\) −41.7408 −1.74680 −0.873399 0.487006i \(-0.838089\pi\)
−0.873399 + 0.487006i \(0.838089\pi\)
\(572\) 0 0
\(573\) − 38.2772i − 1.59905i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 43.0530i 1.79232i 0.443732 + 0.896160i \(0.353654\pi\)
−0.443732 + 0.896160i \(0.646346\pi\)
\(578\) 0 0
\(579\) −22.7057 −0.943618
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 23.8684i 0.988526i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 5.67699i − 0.234315i −0.993113 0.117157i \(-0.962622\pi\)
0.993113 0.117157i \(-0.0373782\pi\)
\(588\) 0 0
\(589\) 9.01304 0.371376
\(590\) 0 0
\(591\) −17.5782 −0.723069
\(592\) 0 0
\(593\) − 8.94518i − 0.367335i −0.982988 0.183667i \(-0.941203\pi\)
0.982988 0.183667i \(-0.0587969\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 51.5453i 2.10961i
\(598\) 0 0
\(599\) 25.2264 1.03072 0.515362 0.856973i \(-0.327658\pi\)
0.515362 + 0.856973i \(0.327658\pi\)
\(600\) 0 0
\(601\) −15.3409 −0.625770 −0.312885 0.949791i \(-0.601295\pi\)
−0.312885 + 0.949791i \(0.601295\pi\)
\(602\) 0 0
\(603\) − 90.1895i − 3.67280i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 20.9362i 0.849775i 0.905246 + 0.424887i \(0.139686\pi\)
−0.905246 + 0.424887i \(0.860314\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0.668023 0.0270253
\(612\) 0 0
\(613\) − 5.31894i − 0.214830i −0.994214 0.107415i \(-0.965743\pi\)
0.994214 0.107415i \(-0.0342573\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 28.6262i 1.15245i 0.817291 + 0.576225i \(0.195475\pi\)
−0.817291 + 0.576225i \(0.804525\pi\)
\(618\) 0 0
\(619\) −21.5625 −0.866668 −0.433334 0.901233i \(-0.642663\pi\)
−0.433334 + 0.901233i \(0.642663\pi\)
\(620\) 0 0
\(621\) 29.2772 1.17485
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 14.3320i 0.572364i
\(628\) 0 0
\(629\) 53.7430 2.14287
\(630\) 0 0
\(631\) −25.1365 −1.00067 −0.500335 0.865832i \(-0.666790\pi\)
−0.500335 + 0.865832i \(0.666790\pi\)
\(632\) 0 0
\(633\) 53.1936i 2.11426i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 4.29693 0.169984
\(640\) 0 0
\(641\) 44.1078 1.74215 0.871077 0.491147i \(-0.163422\pi\)
0.871077 + 0.491147i \(0.163422\pi\)
\(642\) 0 0
\(643\) 16.4816i 0.649969i 0.945719 + 0.324985i \(0.105359\pi\)
−0.945719 + 0.324985i \(0.894641\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 31.4398i − 1.23603i −0.786168 0.618013i \(-0.787938\pi\)
0.786168 0.618013i \(-0.212062\pi\)
\(648\) 0 0
\(649\) 5.06379 0.198771
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 40.8773i 1.59965i 0.600231 + 0.799827i \(0.295075\pi\)
−0.600231 + 0.799827i \(0.704925\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) − 29.0728i − 1.13424i
\(658\) 0 0
\(659\) 46.6860 1.81863 0.909313 0.416112i \(-0.136608\pi\)
0.909313 + 0.416112i \(0.136608\pi\)
\(660\) 0 0
\(661\) −0.127575 −0.00496211 −0.00248106 0.999997i \(-0.500790\pi\)
−0.00248106 + 0.999997i \(0.500790\pi\)
\(662\) 0 0
\(663\) − 3.31670i − 0.128810i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 15.9959i 0.619365i
\(668\) 0 0
\(669\) −49.6222 −1.91850
\(670\) 0 0
\(671\) −6.95822 −0.268619
\(672\) 0 0
\(673\) 48.1936i 1.85773i 0.370423 + 0.928863i \(0.379213\pi\)
−0.370423 + 0.928863i \(0.620787\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 46.4178i − 1.78398i −0.452055 0.891990i \(-0.649309\pi\)
0.452055 0.891990i \(-0.350691\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 12.1208 0.464472
\(682\) 0 0
\(683\) 43.5494i 1.66637i 0.552993 + 0.833186i \(0.313486\pi\)
−0.552993 + 0.833186i \(0.686514\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) − 35.9034i − 1.36980i
\(688\) 0 0
\(689\) 1.16039 0.0442072
\(690\) 0 0
\(691\) 26.0948 0.992692 0.496346 0.868125i \(-0.334675\pi\)
0.496346 + 0.868125i \(0.334675\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 53.4200i 2.02343i
\(698\) 0 0
\(699\) 38.2772 1.44778
\(700\) 0 0
\(701\) 21.7277 0.820645 0.410323 0.911940i \(-0.365416\pi\)
0.410323 + 0.911940i \(0.365416\pi\)
\(702\) 0 0
\(703\) 11.2902i 0.425818i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −17.0768 −0.641334 −0.320667 0.947192i \(-0.603907\pi\)
−0.320667 + 0.947192i \(0.603907\pi\)
\(710\) 0 0
\(711\) 47.6024 1.78523
\(712\) 0 0
\(713\) 18.1365i 0.679219i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 49.1716i − 1.83635i
\(718\) 0 0
\(719\) 27.2462 1.01611 0.508056 0.861324i \(-0.330364\pi\)
0.508056 + 0.861324i \(0.330364\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 46.1298i 1.71559i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 19.9232i 0.738910i 0.929249 + 0.369455i \(0.120456\pi\)
−0.929249 + 0.369455i \(0.879544\pi\)
\(728\) 0 0
\(729\) −28.5804 −1.05853
\(730\) 0 0
\(731\) 41.8684 1.54856
\(732\) 0 0
\(733\) − 31.1626i − 1.15102i −0.817796 0.575509i \(-0.804804\pi\)
0.817796 0.575509i \(-0.195196\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 52.1716i 1.92177i
\(738\) 0 0
\(739\) −1.90117 −0.0699355 −0.0349678 0.999388i \(-0.511133\pi\)
−0.0349678 + 0.999388i \(0.511133\pi\)
\(740\) 0 0
\(741\) 0.696765 0.0255963
\(742\) 0 0
\(743\) 36.0660i 1.32313i 0.749886 + 0.661567i \(0.230108\pi\)
−0.749886 + 0.661567i \(0.769892\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 28.2107i 1.03218i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −0.769522 −0.0280803 −0.0140401 0.999901i \(-0.504469\pi\)
−0.0140401 + 0.999901i \(0.504469\pi\)
\(752\) 0 0
\(753\) 62.2223i 2.26751i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 14.9542i 0.543518i 0.962365 + 0.271759i \(0.0876053\pi\)
−0.962365 + 0.271759i \(0.912395\pi\)
\(758\) 0 0
\(759\) −28.8396 −1.04681
\(760\) 0 0
\(761\) −19.1716 −0.694970 −0.347485 0.937686i \(-0.612964\pi\)
−0.347485 + 0.937686i \(0.612964\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 0.246182i − 0.00888910i
\(768\) 0 0
\(769\) 36.3189 1.30969 0.654847 0.755761i \(-0.272733\pi\)
0.654847 + 0.755761i \(0.272733\pi\)
\(770\) 0 0
\(771\) 44.4528 1.60093
\(772\) 0 0
\(773\) 47.4685i 1.70732i 0.520827 + 0.853662i \(0.325624\pi\)
−0.520827 + 0.853662i \(0.674376\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −11.2223 −0.402082
\(780\) 0 0
\(781\) −2.48563 −0.0889428
\(782\) 0 0
\(783\) − 102.203i − 3.65242i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 7.88546i 0.281086i 0.990075 + 0.140543i \(0.0448849\pi\)
−0.990075 + 0.140543i \(0.955115\pi\)
\(788\) 0 0
\(789\) −11.0858 −0.394665
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0.338281i 0.0120127i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 33.6132i − 1.19064i −0.803488 0.595320i \(-0.797025\pi\)
0.803488 0.595320i \(-0.202975\pi\)
\(798\) 0 0
\(799\) 16.5494 0.585476
\(800\) 0 0
\(801\) 67.3630 2.38015
\(802\) 0 0
\(803\) 16.8176i 0.593480i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 82.2376i 2.89490i
\(808\) 0 0
\(809\) 9.19136 0.323151 0.161576 0.986860i \(-0.448342\pi\)
0.161576 + 0.986860i \(0.448342\pi\)
\(810\) 0 0
\(811\) −21.6483 −0.760173 −0.380086 0.924951i \(-0.624106\pi\)
−0.380086 + 0.924951i \(0.624106\pi\)
\(812\) 0 0
\(813\) − 63.0530i − 2.21136i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 8.79560i 0.307719i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 32.2615 1.12593 0.562966 0.826480i \(-0.309660\pi\)
0.562966 + 0.826480i \(0.309660\pi\)
\(822\) 0 0
\(823\) 11.4438i 0.398908i 0.979907 + 0.199454i \(0.0639168\pi\)
−0.979907 + 0.199454i \(0.936083\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 9.82658i − 0.341704i −0.985297 0.170852i \(-0.945348\pi\)
0.985297 0.170852i \(-0.0546519\pi\)
\(828\) 0 0
\(829\) 38.2574 1.32873 0.664367 0.747407i \(-0.268701\pi\)
0.664367 + 0.747407i \(0.268701\pi\)
\(830\) 0 0
\(831\) 58.1586 2.01750
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) − 115.880i − 4.00538i
\(838\) 0 0
\(839\) 30.6900 1.05954 0.529769 0.848142i \(-0.322279\pi\)
0.529769 + 0.848142i \(0.322279\pi\)
\(840\) 0 0
\(841\) 26.8396 0.925504
\(842\) 0 0
\(843\) − 59.3980i − 2.04578i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −91.5125 −3.14070
\(850\) 0 0
\(851\) −22.7188 −0.778789
\(852\) 0 0
\(853\) − 39.4569i − 1.35098i −0.737370 0.675489i \(-0.763933\pi\)
0.737370 0.675489i \(-0.236067\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 39.8594i − 1.36157i −0.732483 0.680785i \(-0.761639\pi\)
0.732483 0.680785i \(-0.238361\pi\)
\(858\) 0 0
\(859\) −19.5144 −0.665822 −0.332911 0.942958i \(-0.608031\pi\)
−0.332911 + 0.942958i \(0.608031\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) − 37.9321i − 1.29123i −0.763665 0.645613i \(-0.776602\pi\)
0.763665 0.645613i \(-0.223398\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 27.6923i − 0.940479i
\(868\) 0 0
\(869\) −27.5364 −0.934108
\(870\) 0 0
\(871\) 2.53638 0.0859419
\(872\) 0 0
\(873\) − 9.68106i − 0.327654i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 10.9452i − 0.369593i −0.982777 0.184796i \(-0.940837\pi\)
0.982777 0.184796i \(-0.0591625\pi\)
\(878\) 0 0
\(879\) −81.8706 −2.76143
\(880\) 0 0
\(881\) 16.2592 0.547787 0.273894 0.961760i \(-0.411688\pi\)
0.273894 + 0.961760i \(0.411688\pi\)
\(882\) 0 0
\(883\) 23.2511i 0.782461i 0.920293 + 0.391231i \(0.127951\pi\)
−0.920293 + 0.391231i \(0.872049\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 23.2394i − 0.780304i −0.920750 0.390152i \(-0.872422\pi\)
0.920750 0.390152i \(-0.127578\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 92.5894 3.10186
\(892\) 0 0
\(893\) 3.47666i 0.116342i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 1.40207i 0.0468137i
\(898\) 0 0
\(899\) 63.3122 2.11158
\(900\) 0 0
\(901\) 28.7471 0.957704
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 44.2264i 1.46851i 0.678872 + 0.734257i \(0.262469\pi\)
−0.678872 + 0.734257i \(0.737531\pi\)
\(908\) 0 0
\(909\) 45.0000 1.49256
\(910\) 0 0
\(911\) −27.4218 −0.908526 −0.454263 0.890868i \(-0.650097\pi\)
−0.454263 + 0.890868i \(0.650097\pi\)
\(912\) 0 0
\(913\) − 16.3189i − 0.540078i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 33.3163 1.09900 0.549501 0.835493i \(-0.314818\pi\)
0.549501 + 0.835493i \(0.314818\pi\)
\(920\) 0 0
\(921\) 32.4110 1.06798
\(922\) 0 0
\(923\) 0.120842i 0.00397755i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 70.7978i 2.32531i
\(928\) 0 0
\(929\) −13.2484 −0.434666 −0.217333 0.976097i \(-0.569736\pi\)
−0.217333 + 0.976097i \(0.569736\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) − 109.499i − 3.58485i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 10.1365i − 0.331146i −0.986197 0.165573i \(-0.947053\pi\)
0.986197 0.165573i \(-0.0529474\pi\)
\(938\) 0 0
\(939\) 27.6002 0.900697
\(940\) 0 0
\(941\) 32.8684 1.07148 0.535739 0.844384i \(-0.320033\pi\)
0.535739 + 0.844384i \(0.320033\pi\)
\(942\) 0 0
\(943\) − 22.5822i − 0.735379i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 22.3252i 0.725473i 0.931892 + 0.362736i \(0.118157\pi\)
−0.931892 + 0.362736i \(0.881843\pi\)
\(948\) 0 0
\(949\) 0.817606 0.0265406
\(950\) 0 0
\(951\) 25.0507 0.812326
\(952\) 0 0
\(953\) − 47.4685i − 1.53766i −0.639455 0.768828i \(-0.720840\pi\)
0.639455 0.768828i \(-0.279160\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 100.675i 3.25437i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 40.7848 1.31564
\(962\) 0 0
\(963\) 71.6599i 2.30921i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) − 21.3189i − 0.685571i −0.939414 0.342785i \(-0.888630\pi\)
0.939414 0.342785i \(-0.111370\pi\)
\(968\) 0 0
\(969\) 17.2615 0.554518
\(970\) 0 0
\(971\) 29.5625 0.948704 0.474352 0.880335i \(-0.342682\pi\)
0.474352 + 0.880335i \(0.342682\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 1.83961i 0.0588545i 0.999567 + 0.0294272i \(0.00936833\pi\)
−0.999567 + 0.0294272i \(0.990632\pi\)
\(978\) 0 0
\(979\) −38.9672 −1.24540
\(980\) 0 0
\(981\) −29.6311 −0.946050
\(982\) 0 0
\(983\) 11.3760i 0.362838i 0.983406 + 0.181419i \(0.0580690\pi\)
−0.983406 + 0.181419i \(0.941931\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −17.6990 −0.562795
\(990\) 0 0
\(991\) 7.79153 0.247506 0.123753 0.992313i \(-0.460507\pi\)
0.123753 + 0.992313i \(0.460507\pi\)
\(992\) 0 0
\(993\) − 70.8135i − 2.24720i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 11.4218i 0.361733i 0.983508 + 0.180867i \(0.0578903\pi\)
−0.983508 + 0.180867i \(0.942110\pi\)
\(998\) 0 0
\(999\) 145.157 4.59256
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 4900.2.e.s.2549.6 6
5.2 odd 4 4900.2.a.bd.1.3 3
5.3 odd 4 4900.2.a.bb.1.1 3
5.4 even 2 inner 4900.2.e.s.2549.1 6
7.3 odd 6 700.2.r.d.149.1 12
7.5 odd 6 700.2.r.d.249.6 12
7.6 odd 2 4900.2.e.t.2549.1 6
35.3 even 12 700.2.i.d.401.1 6
35.12 even 12 700.2.i.e.501.3 yes 6
35.13 even 4 4900.2.a.bc.1.3 3
35.17 even 12 700.2.i.e.401.3 yes 6
35.19 odd 6 700.2.r.d.249.1 12
35.24 odd 6 700.2.r.d.149.6 12
35.27 even 4 4900.2.a.ba.1.1 3
35.33 even 12 700.2.i.d.501.1 yes 6
35.34 odd 2 4900.2.e.t.2549.6 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
700.2.i.d.401.1 6 35.3 even 12
700.2.i.d.501.1 yes 6 35.33 even 12
700.2.i.e.401.3 yes 6 35.17 even 12
700.2.i.e.501.3 yes 6 35.12 even 12
700.2.r.d.149.1 12 7.3 odd 6
700.2.r.d.149.6 12 35.24 odd 6
700.2.r.d.249.1 12 35.19 odd 6
700.2.r.d.249.6 12 7.5 odd 6
4900.2.a.ba.1.1 3 35.27 even 4
4900.2.a.bb.1.1 3 5.3 odd 4
4900.2.a.bc.1.3 3 35.13 even 4
4900.2.a.bd.1.3 3 5.2 odd 4
4900.2.e.s.2549.1 6 5.4 even 2 inner
4900.2.e.s.2549.6 6 1.1 even 1 trivial
4900.2.e.t.2549.1 6 7.6 odd 2
4900.2.e.t.2549.6 6 35.34 odd 2