Properties

Label 4896.2.f.c.2449.1
Level $4896$
Weight $2$
Character 4896.2449
Analytic conductor $39.095$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4896,2,Mod(2449,4896)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4896, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 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("4896.2449");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4896 = 2^{5} \cdot 3^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4896.f (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: \(39.0947568296\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.1649659456.5
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} - 2x^{5} + 4x^{4} - 4x^{3} - 8x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 136)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2449.1
Root \(1.40014 - 0.199044i\) of defining polynomial
Character \(\chi\) \(=\) 4896.2449
Dual form 4896.2.f.c.2449.8

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.30391i q^{5} -3.45790 q^{7} +O(q^{10})\) \(q-3.30391i q^{5} -3.45790 q^{7} -3.09665i q^{11} -2.50773i q^{13} +1.00000 q^{17} -4.43508i q^{19} +4.14265 q^{23} -5.91579 q^{25} -7.15862i q^{29} +6.22515 q^{31} +11.4246i q^{35} -6.74411i q^{37} +6.99830 q^{41} -2.22951i q^{43} -3.31525 q^{47} +4.95705 q^{49} -6.19330i q^{53} -10.2310 q^{55} -8.83732i q^{59} +14.3707i q^{61} -8.28530 q^{65} +6.44195i q^{67} -5.66069 q^{71} -15.4486 q^{73} +10.7079i q^{77} +8.14265 q^{79} -13.8528i q^{83} -3.30391i q^{85} -7.32655 q^{89} +8.67146i q^{91} -14.6531 q^{95} +8.36780 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{7} + 8 q^{17} + 12 q^{23} - 8 q^{25} + 12 q^{31} - 28 q^{47} - 8 q^{49} - 44 q^{55} - 24 q^{65} + 8 q^{73} + 44 q^{79} - 8 q^{89} - 16 q^{95} + 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4896\mathbb{Z}\right)^\times\).

\(n\) \(613\) \(2143\) \(3809\) \(4321\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) − 3.30391i − 1.47755i −0.673951 0.738776i \(-0.735404\pi\)
0.673951 0.738776i \(-0.264596\pi\)
\(6\) 0 0
\(7\) −3.45790 −1.30696 −0.653481 0.756943i \(-0.726692\pi\)
−0.653481 + 0.756943i \(0.726692\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 3.09665i − 0.933675i −0.884343 0.466838i \(-0.845393\pi\)
0.884343 0.466838i \(-0.154607\pi\)
\(12\) 0 0
\(13\) − 2.50773i − 0.695519i −0.937584 0.347759i \(-0.886943\pi\)
0.937584 0.347759i \(-0.113057\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.00000 0.242536
\(18\) 0 0
\(19\) − 4.43508i − 1.01748i −0.860921 0.508739i \(-0.830112\pi\)
0.860921 0.508739i \(-0.169888\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.14265 0.863802 0.431901 0.901921i \(-0.357843\pi\)
0.431901 + 0.901921i \(0.357843\pi\)
\(24\) 0 0
\(25\) −5.91579 −1.18316
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 7.15862i − 1.32932i −0.747145 0.664661i \(-0.768576\pi\)
0.747145 0.664661i \(-0.231424\pi\)
\(30\) 0 0
\(31\) 6.22515 1.11807 0.559035 0.829144i \(-0.311172\pi\)
0.559035 + 0.829144i \(0.311172\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 11.4246i 1.93110i
\(36\) 0 0
\(37\) − 6.74411i − 1.10872i −0.832276 0.554362i \(-0.812962\pi\)
0.832276 0.554362i \(-0.187038\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 6.99830 1.09295 0.546475 0.837475i \(-0.315969\pi\)
0.546475 + 0.837475i \(0.315969\pi\)
\(42\) 0 0
\(43\) − 2.22951i − 0.339998i −0.985444 0.169999i \(-0.945624\pi\)
0.985444 0.169999i \(-0.0543764\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −3.31525 −0.483579 −0.241789 0.970329i \(-0.577734\pi\)
−0.241789 + 0.970329i \(0.577734\pi\)
\(48\) 0 0
\(49\) 4.95705 0.708149
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 6.19330i − 0.850715i −0.905025 0.425358i \(-0.860148\pi\)
0.905025 0.425358i \(-0.139852\pi\)
\(54\) 0 0
\(55\) −10.2310 −1.37955
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 8.83732i − 1.15052i −0.817970 0.575261i \(-0.804901\pi\)
0.817970 0.575261i \(-0.195099\pi\)
\(60\) 0 0
\(61\) 14.3707i 1.83999i 0.391936 + 0.919993i \(0.371806\pi\)
−0.391936 + 0.919993i \(0.628194\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −8.28530 −1.02766
\(66\) 0 0
\(67\) 6.44195i 0.787009i 0.919323 + 0.393505i \(0.128738\pi\)
−0.919323 + 0.393505i \(0.871262\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −5.66069 −0.671800 −0.335900 0.941898i \(-0.609040\pi\)
−0.335900 + 0.941898i \(0.609040\pi\)
\(72\) 0 0
\(73\) −15.4486 −1.80812 −0.904061 0.427403i \(-0.859429\pi\)
−0.904061 + 0.427403i \(0.859429\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 10.7079i 1.22028i
\(78\) 0 0
\(79\) 8.14265 0.916120 0.458060 0.888921i \(-0.348545\pi\)
0.458060 + 0.888921i \(0.348545\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 13.8528i − 1.52054i −0.649607 0.760270i \(-0.725067\pi\)
0.649607 0.760270i \(-0.274933\pi\)
\(84\) 0 0
\(85\) − 3.30391i − 0.358359i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −7.32655 −0.776613 −0.388306 0.921530i \(-0.626940\pi\)
−0.388306 + 0.921530i \(0.626940\pi\)
\(90\) 0 0
\(91\) 8.67146i 0.909016i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −14.6531 −1.50338
\(96\) 0 0
\(97\) 8.36780 0.849622 0.424811 0.905282i \(-0.360341\pi\)
0.424811 + 0.905282i \(0.360341\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 7.95480i 0.791532i 0.918351 + 0.395766i \(0.129521\pi\)
−0.918351 + 0.395766i \(0.870479\pi\)
\(102\) 0 0
\(103\) 7.91409 0.779798 0.389899 0.920858i \(-0.372510\pi\)
0.389899 + 0.920858i \(0.372510\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 13.3605i 1.29161i 0.763504 + 0.645803i \(0.223477\pi\)
−0.763504 + 0.645803i \(0.776523\pi\)
\(108\) 0 0
\(109\) − 2.12606i − 0.203640i −0.994803 0.101820i \(-0.967533\pi\)
0.994803 0.101820i \(-0.0324665\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −2.28530 −0.214983 −0.107491 0.994206i \(-0.534282\pi\)
−0.107491 + 0.994206i \(0.534282\pi\)
\(114\) 0 0
\(115\) − 13.6869i − 1.27631i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −3.45790 −0.316985
\(120\) 0 0
\(121\) 1.41076 0.128251
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 3.02569i 0.270626i
\(126\) 0 0
\(127\) −2.36780 −0.210109 −0.105054 0.994466i \(-0.533502\pi\)
−0.105054 + 0.994466i \(0.533502\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) − 2.73893i − 0.239301i −0.992816 0.119651i \(-0.961823\pi\)
0.992816 0.119651i \(-0.0381774\pi\)
\(132\) 0 0
\(133\) 15.3361i 1.32981i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −22.7044 −1.93977 −0.969885 0.243564i \(-0.921683\pi\)
−0.969885 + 0.243564i \(0.921683\pi\)
\(138\) 0 0
\(139\) − 0.502570i − 0.0426275i −0.999773 0.0213137i \(-0.993215\pi\)
0.999773 0.0213137i \(-0.00678488\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −7.76556 −0.649388
\(144\) 0 0
\(145\) −23.6514 −1.96414
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 7.21213i 0.590840i 0.955367 + 0.295420i \(0.0954596\pi\)
−0.955367 + 0.295420i \(0.904540\pi\)
\(150\) 0 0
\(151\) −3.16671 −0.257704 −0.128852 0.991664i \(-0.541129\pi\)
−0.128852 + 0.991664i \(0.541129\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) − 20.5673i − 1.65201i
\(156\) 0 0
\(157\) 3.16761i 0.252803i 0.991979 + 0.126401i \(0.0403427\pi\)
−0.991979 + 0.126401i \(0.959657\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −14.3248 −1.12896
\(162\) 0 0
\(163\) 1.66331i 0.130281i 0.997876 + 0.0651404i \(0.0207495\pi\)
−0.997876 + 0.0651404i \(0.979250\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −8.57648 −0.663668 −0.331834 0.943338i \(-0.607667\pi\)
−0.331834 + 0.943338i \(0.607667\pi\)
\(168\) 0 0
\(169\) 6.71130 0.516254
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 4.48175i 0.340741i 0.985380 + 0.170371i \(0.0544965\pi\)
−0.985380 + 0.170371i \(0.945504\pi\)
\(174\) 0 0
\(175\) 20.4562 1.54634
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 3.60606i 0.269530i 0.990878 + 0.134765i \(0.0430279\pi\)
−0.990878 + 0.134765i \(0.956972\pi\)
\(180\) 0 0
\(181\) 17.5384i 1.30362i 0.758384 + 0.651808i \(0.225989\pi\)
−0.758384 + 0.651808i \(0.774011\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −22.2819 −1.63820
\(186\) 0 0
\(187\) − 3.09665i − 0.226449i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 8.76386 0.634130 0.317065 0.948404i \(-0.397303\pi\)
0.317065 + 0.948404i \(0.397303\pi\)
\(192\) 0 0
\(193\) −18.4847 −1.33056 −0.665278 0.746595i \(-0.731687\pi\)
−0.665278 + 0.746595i \(0.731687\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 3.87744i 0.276256i 0.990414 + 0.138128i \(0.0441085\pi\)
−0.990414 + 0.138128i \(0.955891\pi\)
\(198\) 0 0
\(199\) −13.1409 −0.931537 −0.465769 0.884907i \(-0.654222\pi\)
−0.465769 + 0.884907i \(0.654222\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 24.7538i 1.73737i
\(204\) 0 0
\(205\) − 23.1217i − 1.61489i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −13.7339 −0.949994
\(210\) 0 0
\(211\) 13.5024i 0.929543i 0.885431 + 0.464771i \(0.153863\pi\)
−0.885431 + 0.464771i \(0.846137\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −7.36610 −0.502364
\(216\) 0 0
\(217\) −21.5259 −1.46128
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) − 2.50773i − 0.168688i
\(222\) 0 0
\(223\) 7.85147 0.525773 0.262887 0.964827i \(-0.415326\pi\)
0.262887 + 0.964827i \(0.415326\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 21.1858i 1.40615i 0.711115 + 0.703076i \(0.248191\pi\)
−0.711115 + 0.703076i \(0.751809\pi\)
\(228\) 0 0
\(229\) 13.6745i 0.903633i 0.892111 + 0.451817i \(0.149224\pi\)
−0.892111 + 0.451817i \(0.850776\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 18.1994 1.19228 0.596141 0.802880i \(-0.296700\pi\)
0.596141 + 0.802880i \(0.296700\pi\)
\(234\) 0 0
\(235\) 10.9533i 0.714512i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 2.93227 0.189673 0.0948364 0.995493i \(-0.469767\pi\)
0.0948364 + 0.995493i \(0.469767\pi\)
\(240\) 0 0
\(241\) 27.3180 1.75971 0.879853 0.475247i \(-0.157641\pi\)
0.879853 + 0.475247i \(0.157641\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) − 16.3776i − 1.04633i
\(246\) 0 0
\(247\) −11.1220 −0.707675
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) − 18.8286i − 1.18845i −0.804300 0.594224i \(-0.797459\pi\)
0.804300 0.594224i \(-0.202541\pi\)
\(252\) 0 0
\(253\) − 12.8283i − 0.806510i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −7.58924 −0.473404 −0.236702 0.971582i \(-0.576066\pi\)
−0.236702 + 0.971582i \(0.576066\pi\)
\(258\) 0 0
\(259\) 23.3204i 1.44906i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 30.0310 1.85179 0.925895 0.377782i \(-0.123313\pi\)
0.925895 + 0.377782i \(0.123313\pi\)
\(264\) 0 0
\(265\) −20.4621 −1.25698
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 3.24478i 0.197837i 0.995096 + 0.0989187i \(0.0315384\pi\)
−0.995096 + 0.0989187i \(0.968462\pi\)
\(270\) 0 0
\(271\) 11.6453 0.707400 0.353700 0.935359i \(-0.384923\pi\)
0.353700 + 0.935359i \(0.384923\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 18.3191i 1.10469i
\(276\) 0 0
\(277\) − 16.9340i − 1.01747i −0.860924 0.508734i \(-0.830114\pi\)
0.860924 0.508734i \(-0.169886\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −7.38127 −0.440330 −0.220165 0.975463i \(-0.570660\pi\)
−0.220165 + 0.975463i \(0.570660\pi\)
\(282\) 0 0
\(283\) − 19.3209i − 1.14851i −0.818678 0.574253i \(-0.805293\pi\)
0.818678 0.574253i \(-0.194707\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −24.1994 −1.42844
\(288\) 0 0
\(289\) 1.00000 0.0588235
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 5.71966i − 0.334146i −0.985945 0.167073i \(-0.946568\pi\)
0.985945 0.167073i \(-0.0534316\pi\)
\(294\) 0 0
\(295\) −29.1977 −1.69995
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) − 10.3886i − 0.600790i
\(300\) 0 0
\(301\) 7.70943i 0.444364i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 47.4796 2.71867
\(306\) 0 0
\(307\) 0.222648i 0.0127072i 0.999980 + 0.00635359i \(0.00202242\pi\)
−0.999980 + 0.00635359i \(0.997978\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 3.88560 0.220332 0.110166 0.993913i \(-0.464862\pi\)
0.110166 + 0.993913i \(0.464862\pi\)
\(312\) 0 0
\(313\) −20.4022 −1.15320 −0.576600 0.817027i \(-0.695621\pi\)
−0.576600 + 0.817027i \(0.695621\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 6.32960i 0.355506i 0.984075 + 0.177753i \(0.0568827\pi\)
−0.984075 + 0.177753i \(0.943117\pi\)
\(318\) 0 0
\(319\) −22.1677 −1.24115
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 4.43508i − 0.246775i
\(324\) 0 0
\(325\) 14.8352i 0.822909i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 11.4638 0.632019
\(330\) 0 0
\(331\) − 20.8922i − 1.14834i −0.818736 0.574170i \(-0.805325\pi\)
0.818736 0.574170i \(-0.194675\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 21.2836 1.16285
\(336\) 0 0
\(337\) 8.82988 0.480994 0.240497 0.970650i \(-0.422690\pi\)
0.240497 + 0.970650i \(0.422690\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) − 19.2771i − 1.04391i
\(342\) 0 0
\(343\) 7.06433 0.381438
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 2.62301i − 0.140811i −0.997518 0.0704053i \(-0.977571\pi\)
0.997518 0.0704053i \(-0.0224292\pi\)
\(348\) 0 0
\(349\) 13.3575i 0.715013i 0.933911 + 0.357506i \(0.116373\pi\)
−0.933911 + 0.357506i \(0.883627\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −0.510209 −0.0271557 −0.0135779 0.999908i \(-0.504322\pi\)
−0.0135779 + 0.999908i \(0.504322\pi\)
\(354\) 0 0
\(355\) 18.7024i 0.992619i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −0.861534 −0.0454700 −0.0227350 0.999742i \(-0.507237\pi\)
−0.0227350 + 0.999742i \(0.507237\pi\)
\(360\) 0 0
\(361\) −0.669976 −0.0352619
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 51.0407i 2.67159i
\(366\) 0 0
\(367\) −3.07492 −0.160509 −0.0802547 0.996774i \(-0.525573\pi\)
−0.0802547 + 0.996774i \(0.525573\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 21.4158i 1.11185i
\(372\) 0 0
\(373\) − 8.12750i − 0.420826i −0.977613 0.210413i \(-0.932519\pi\)
0.977613 0.210413i \(-0.0674809\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −17.9519 −0.924568
\(378\) 0 0
\(379\) 9.01502i 0.463070i 0.972827 + 0.231535i \(0.0743748\pi\)
−0.972827 + 0.231535i \(0.925625\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −1.79721 −0.0918331 −0.0459165 0.998945i \(-0.514621\pi\)
−0.0459165 + 0.998945i \(0.514621\pi\)
\(384\) 0 0
\(385\) 35.3779 1.80302
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 18.7490i 0.950614i 0.879820 + 0.475307i \(0.157663\pi\)
−0.879820 + 0.475307i \(0.842337\pi\)
\(390\) 0 0
\(391\) 4.14265 0.209503
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) − 26.9025i − 1.35361i
\(396\) 0 0
\(397\) − 23.9280i − 1.20091i −0.799658 0.600456i \(-0.794986\pi\)
0.799658 0.600456i \(-0.205014\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −6.58236 −0.328708 −0.164354 0.986401i \(-0.552554\pi\)
−0.164354 + 0.986401i \(0.552554\pi\)
\(402\) 0 0
\(403\) − 15.6110i − 0.777639i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −20.8841 −1.03519
\(408\) 0 0
\(409\) 29.7717 1.47212 0.736058 0.676919i \(-0.236685\pi\)
0.736058 + 0.676919i \(0.236685\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 30.5585i 1.50369i
\(414\) 0 0
\(415\) −45.7683 −2.24668
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 24.2375i 1.18408i 0.805909 + 0.592040i \(0.201677\pi\)
−0.805909 + 0.592040i \(0.798323\pi\)
\(420\) 0 0
\(421\) − 19.8507i − 0.967462i −0.875217 0.483731i \(-0.839281\pi\)
0.875217 0.483731i \(-0.160719\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −5.91579 −0.286958
\(426\) 0 0
\(427\) − 49.6925i − 2.40479i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −1.84218 −0.0887346 −0.0443673 0.999015i \(-0.514127\pi\)
−0.0443673 + 0.999015i \(0.514127\pi\)
\(432\) 0 0
\(433\) 8.03955 0.386356 0.193178 0.981164i \(-0.438120\pi\)
0.193178 + 0.981164i \(0.438120\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 18.3730i − 0.878900i
\(438\) 0 0
\(439\) −14.4184 −0.688153 −0.344077 0.938942i \(-0.611808\pi\)
−0.344077 + 0.938942i \(0.611808\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 30.7835i − 1.46257i −0.682072 0.731285i \(-0.738921\pi\)
0.682072 0.731285i \(-0.261079\pi\)
\(444\) 0 0
\(445\) 24.2062i 1.14749i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −12.6305 −0.596070 −0.298035 0.954555i \(-0.596331\pi\)
−0.298035 + 0.954555i \(0.596331\pi\)
\(450\) 0 0
\(451\) − 21.6713i − 1.02046i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 28.6497 1.34312
\(456\) 0 0
\(457\) 20.3592 0.952364 0.476182 0.879347i \(-0.342020\pi\)
0.476182 + 0.879347i \(0.342020\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 33.5671i − 1.56338i −0.623669 0.781689i \(-0.714359\pi\)
0.623669 0.781689i \(-0.285641\pi\)
\(462\) 0 0
\(463\) 21.8316 1.01460 0.507300 0.861770i \(-0.330644\pi\)
0.507300 + 0.861770i \(0.330644\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 23.2705i 1.07683i 0.842680 + 0.538415i \(0.180977\pi\)
−0.842680 + 0.538415i \(0.819023\pi\)
\(468\) 0 0
\(469\) − 22.2756i − 1.02859i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −6.90402 −0.317447
\(474\) 0 0
\(475\) 26.2370i 1.20384i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −43.5363 −1.98923 −0.994613 0.103662i \(-0.966944\pi\)
−0.994613 + 0.103662i \(0.966944\pi\)
\(480\) 0 0
\(481\) −16.9124 −0.771139
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) − 27.6464i − 1.25536i
\(486\) 0 0
\(487\) −20.2912 −0.919481 −0.459741 0.888053i \(-0.652058\pi\)
−0.459741 + 0.888053i \(0.652058\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 39.6117i 1.78765i 0.448416 + 0.893825i \(0.351988\pi\)
−0.448416 + 0.893825i \(0.648012\pi\)
\(492\) 0 0
\(493\) − 7.15862i − 0.322408i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 19.5741 0.878017
\(498\) 0 0
\(499\) 9.69556i 0.434033i 0.976168 + 0.217016i \(0.0696325\pi\)
−0.976168 + 0.217016i \(0.930367\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −39.2861 −1.75168 −0.875840 0.482602i \(-0.839692\pi\)
−0.875840 + 0.482602i \(0.839692\pi\)
\(504\) 0 0
\(505\) 26.2819 1.16953
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) − 7.64374i − 0.338803i −0.985547 0.169401i \(-0.945817\pi\)
0.985547 0.169401i \(-0.0541834\pi\)
\(510\) 0 0
\(511\) 53.4197 2.36315
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 26.1474i − 1.15219i
\(516\) 0 0
\(517\) 10.2662i 0.451505i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −4.66487 −0.204372 −0.102186 0.994765i \(-0.532584\pi\)
−0.102186 + 0.994765i \(0.532584\pi\)
\(522\) 0 0
\(523\) 34.7381i 1.51899i 0.650511 + 0.759496i \(0.274555\pi\)
−0.650511 + 0.759496i \(0.725445\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 6.22515 0.271172
\(528\) 0 0
\(529\) −5.83846 −0.253846
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) − 17.5498i − 0.760168i
\(534\) 0 0
\(535\) 44.1417 1.90841
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) − 15.3502i − 0.661181i
\(540\) 0 0
\(541\) − 24.8504i − 1.06840i −0.845358 0.534200i \(-0.820613\pi\)
0.845358 0.534200i \(-0.179387\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −7.02431 −0.300888
\(546\) 0 0
\(547\) − 7.32619i − 0.313245i −0.987659 0.156623i \(-0.949939\pi\)
0.987659 0.156623i \(-0.0500606\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −31.7491 −1.35256
\(552\) 0 0
\(553\) −28.1564 −1.19733
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 36.3646i − 1.54082i −0.637551 0.770408i \(-0.720053\pi\)
0.637551 0.770408i \(-0.279947\pi\)
\(558\) 0 0
\(559\) −5.59101 −0.236475
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 20.9579i − 0.883270i −0.897195 0.441635i \(-0.854399\pi\)
0.897195 0.441635i \(-0.145601\pi\)
\(564\) 0 0
\(565\) 7.55041i 0.317648i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 17.3214 0.726150 0.363075 0.931760i \(-0.381727\pi\)
0.363075 + 0.931760i \(0.381727\pi\)
\(570\) 0 0
\(571\) − 41.5633i − 1.73937i −0.493606 0.869686i \(-0.664322\pi\)
0.493606 0.869686i \(-0.335678\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −24.5070 −1.02201
\(576\) 0 0
\(577\) −3.88631 −0.161789 −0.0808946 0.996723i \(-0.525778\pi\)
−0.0808946 + 0.996723i \(0.525778\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 47.9015i 1.98729i
\(582\) 0 0
\(583\) −19.1785 −0.794292
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 14.7386i − 0.608327i −0.952620 0.304163i \(-0.901623\pi\)
0.952620 0.304163i \(-0.0983769\pi\)
\(588\) 0 0
\(589\) − 27.6091i − 1.13761i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 40.2819 1.65418 0.827090 0.562070i \(-0.189995\pi\)
0.827090 + 0.562070i \(0.189995\pi\)
\(594\) 0 0
\(595\) 11.4246i 0.468361i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 15.4982 0.633238 0.316619 0.948553i \(-0.397452\pi\)
0.316619 + 0.948553i \(0.397452\pi\)
\(600\) 0 0
\(601\) 31.9519 1.30334 0.651672 0.758501i \(-0.274068\pi\)
0.651672 + 0.758501i \(0.274068\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) − 4.66101i − 0.189497i
\(606\) 0 0
\(607\) 3.04667 0.123661 0.0618303 0.998087i \(-0.480306\pi\)
0.0618303 + 0.998087i \(0.480306\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 8.31374i 0.336338i
\(612\) 0 0
\(613\) 34.0750i 1.37628i 0.725580 + 0.688138i \(0.241572\pi\)
−0.725580 + 0.688138i \(0.758428\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −40.1395 −1.61595 −0.807977 0.589213i \(-0.799438\pi\)
−0.807977 + 0.589213i \(0.799438\pi\)
\(618\) 0 0
\(619\) − 9.21372i − 0.370331i −0.982707 0.185165i \(-0.940718\pi\)
0.982707 0.185165i \(-0.0592821\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 25.3344 1.01500
\(624\) 0 0
\(625\) −19.5824 −0.783295
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 6.74411i − 0.268905i
\(630\) 0 0
\(631\) −23.0526 −0.917708 −0.458854 0.888512i \(-0.651740\pi\)
−0.458854 + 0.888512i \(0.651740\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 7.82300i 0.310446i
\(636\) 0 0
\(637\) − 12.4309i − 0.492531i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −11.6322 −0.459444 −0.229722 0.973256i \(-0.573782\pi\)
−0.229722 + 0.973256i \(0.573782\pi\)
\(642\) 0 0
\(643\) − 21.1290i − 0.833247i −0.909079 0.416624i \(-0.863213\pi\)
0.909079 0.416624i \(-0.136787\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −29.5854 −1.16312 −0.581560 0.813503i \(-0.697558\pi\)
−0.581560 + 0.813503i \(0.697558\pi\)
\(648\) 0 0
\(649\) −27.3661 −1.07421
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 2.85861i − 0.111866i −0.998435 0.0559330i \(-0.982187\pi\)
0.998435 0.0559330i \(-0.0178133\pi\)
\(654\) 0 0
\(655\) −9.04915 −0.353580
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) − 0.654259i − 0.0254863i −0.999919 0.0127431i \(-0.995944\pi\)
0.999919 0.0127431i \(-0.00405638\pi\)
\(660\) 0 0
\(661\) − 16.3412i − 0.635599i −0.948158 0.317800i \(-0.897056\pi\)
0.948158 0.317800i \(-0.102944\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 50.6689 1.96486
\(666\) 0 0
\(667\) − 29.6556i − 1.14827i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 44.5012 1.71795
\(672\) 0 0
\(673\) 18.7921 0.724382 0.362191 0.932104i \(-0.382029\pi\)
0.362191 + 0.932104i \(0.382029\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 20.5706i − 0.790593i −0.918554 0.395296i \(-0.870642\pi\)
0.918554 0.395296i \(-0.129358\pi\)
\(678\) 0 0
\(679\) −28.9350 −1.11042
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 40.4925i 1.54940i 0.632327 + 0.774701i \(0.282100\pi\)
−0.632327 + 0.774701i \(0.717900\pi\)
\(684\) 0 0
\(685\) 75.0133i 2.86611i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −15.5311 −0.591688
\(690\) 0 0
\(691\) 9.29651i 0.353656i 0.984242 + 0.176828i \(0.0565836\pi\)
−0.984242 + 0.176828i \(0.943416\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −1.66044 −0.0629843
\(696\) 0 0
\(697\) 6.99830 0.265079
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 46.6793i 1.76305i 0.472134 + 0.881527i \(0.343484\pi\)
−0.472134 + 0.881527i \(0.656516\pi\)
\(702\) 0 0
\(703\) −29.9107 −1.12810
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 27.5069i − 1.03450i
\(708\) 0 0
\(709\) − 39.5821i − 1.48654i −0.668993 0.743269i \(-0.733274\pi\)
0.668993 0.743269i \(-0.266726\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 25.7886 0.965792
\(714\) 0 0
\(715\) 25.6567i 0.959505i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 38.3669 1.43084 0.715422 0.698693i \(-0.246235\pi\)
0.715422 + 0.698693i \(0.246235\pi\)
\(720\) 0 0
\(721\) −27.3661 −1.01917
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 42.3489i 1.57280i
\(726\) 0 0
\(727\) 17.3344 0.642899 0.321450 0.946927i \(-0.395830\pi\)
0.321450 + 0.946927i \(0.395830\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) − 2.22951i − 0.0824615i
\(732\) 0 0
\(733\) − 45.3211i − 1.67397i −0.547224 0.836986i \(-0.684315\pi\)
0.547224 0.836986i \(-0.315685\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 19.9485 0.734811
\(738\) 0 0
\(739\) 20.9602i 0.771035i 0.922701 + 0.385517i \(0.125977\pi\)
−0.922701 + 0.385517i \(0.874023\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −20.7354 −0.760707 −0.380353 0.924841i \(-0.624198\pi\)
−0.380353 + 0.924841i \(0.624198\pi\)
\(744\) 0 0
\(745\) 23.8282 0.872997
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) − 46.1991i − 1.68808i
\(750\) 0 0
\(751\) 17.6725 0.644877 0.322439 0.946590i \(-0.395497\pi\)
0.322439 + 0.946590i \(0.395497\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 10.4625i 0.380770i
\(756\) 0 0
\(757\) − 22.0131i − 0.800081i −0.916498 0.400040i \(-0.868996\pi\)
0.916498 0.400040i \(-0.131004\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −18.9024 −0.685211 −0.342606 0.939479i \(-0.611309\pi\)
−0.342606 + 0.939479i \(0.611309\pi\)
\(762\) 0 0
\(763\) 7.35170i 0.266149i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −22.1616 −0.800209
\(768\) 0 0
\(769\) 44.8954 1.61897 0.809486 0.587140i \(-0.199746\pi\)
0.809486 + 0.587140i \(0.199746\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 14.2196i − 0.511445i −0.966750 0.255722i \(-0.917687\pi\)
0.966750 0.255722i \(-0.0823133\pi\)
\(774\) 0 0
\(775\) −36.8267 −1.32285
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 31.0380i − 1.11205i
\(780\) 0 0
\(781\) 17.5292i 0.627243i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 10.4655 0.373529
\(786\) 0 0
\(787\) 37.9008i 1.35102i 0.737353 + 0.675508i \(0.236076\pi\)
−0.737353 + 0.675508i \(0.763924\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 7.90232 0.280974
\(792\) 0 0
\(793\) 36.0379 1.27974
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 24.8276i − 0.879440i −0.898135 0.439720i \(-0.855078\pi\)
0.898135 0.439720i \(-0.144922\pi\)
\(798\) 0 0
\(799\) −3.31525 −0.117285
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 47.8389i 1.68820i
\(804\) 0 0
\(805\) 47.3279i 1.66809i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −6.87801 −0.241818 −0.120909 0.992664i \(-0.538581\pi\)
−0.120909 + 0.992664i \(0.538581\pi\)
\(810\) 0 0
\(811\) 5.35901i 0.188180i 0.995564 + 0.0940901i \(0.0299942\pi\)
−0.995564 + 0.0940901i \(0.970006\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 5.49543 0.192497
\(816\) 0 0
\(817\) −9.88808 −0.345940
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 2.17151i − 0.0757861i −0.999282 0.0378931i \(-0.987935\pi\)
0.999282 0.0378931i \(-0.0120646\pi\)
\(822\) 0 0
\(823\) −38.9210 −1.35670 −0.678350 0.734739i \(-0.737305\pi\)
−0.678350 + 0.734739i \(0.737305\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 22.5203i 0.783109i 0.920155 + 0.391554i \(0.128062\pi\)
−0.920155 + 0.391554i \(0.871938\pi\)
\(828\) 0 0
\(829\) − 11.4269i − 0.396873i −0.980114 0.198436i \(-0.936414\pi\)
0.980114 0.198436i \(-0.0635863\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 4.95705 0.171751
\(834\) 0 0
\(835\) 28.3359i 0.980604i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −28.1591 −0.972161 −0.486081 0.873914i \(-0.661574\pi\)
−0.486081 + 0.873914i \(0.661574\pi\)
\(840\) 0 0
\(841\) −22.2458 −0.767097
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) − 22.1735i − 0.762792i
\(846\) 0 0
\(847\) −4.87826 −0.167619
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) − 27.9385i − 0.957718i
\(852\) 0 0
\(853\) 0.516607i 0.0176883i 0.999961 + 0.00884415i \(0.00281522\pi\)
−0.999961 + 0.00884415i \(0.997185\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 7.72307 0.263815 0.131907 0.991262i \(-0.457890\pi\)
0.131907 + 0.991262i \(0.457890\pi\)
\(858\) 0 0
\(859\) 36.3045i 1.23869i 0.785118 + 0.619347i \(0.212602\pi\)
−0.785118 + 0.619347i \(0.787398\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 39.9299 1.35923 0.679615 0.733569i \(-0.262147\pi\)
0.679615 + 0.733569i \(0.262147\pi\)
\(864\) 0 0
\(865\) 14.8073 0.503462
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) − 25.2149i − 0.855358i
\(870\) 0 0
\(871\) 16.1547 0.547380
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) − 10.4625i − 0.353698i
\(876\) 0 0
\(877\) − 3.73210i − 0.126024i −0.998013 0.0630121i \(-0.979929\pi\)
0.998013 0.0630121i \(-0.0200707\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 26.3644 0.888239 0.444120 0.895968i \(-0.353517\pi\)
0.444120 + 0.895968i \(0.353517\pi\)
\(882\) 0 0
\(883\) 19.9236i 0.670483i 0.942132 + 0.335241i \(0.108818\pi\)
−0.942132 + 0.335241i \(0.891182\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 29.2662 0.982663 0.491331 0.870973i \(-0.336510\pi\)
0.491331 + 0.870973i \(0.336510\pi\)
\(888\) 0 0
\(889\) 8.18762 0.274604
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 14.7034i 0.492031i
\(894\) 0 0
\(895\) 11.9141 0.398244
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 44.5635i − 1.48628i
\(900\) 0 0
\(901\) − 6.19330i − 0.206329i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 57.9451 1.92616
\(906\) 0 0
\(907\) 54.0895i 1.79601i 0.439983 + 0.898006i \(0.354985\pi\)
−0.439983 + 0.898006i \(0.645015\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 51.8223 1.71695 0.858475 0.512856i \(-0.171413\pi\)
0.858475 + 0.512856i \(0.171413\pi\)
\(912\) 0 0
\(913\) −42.8972 −1.41969
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 9.47092i 0.312757i
\(918\) 0 0
\(919\) 19.5888 0.646174 0.323087 0.946369i \(-0.395279\pi\)
0.323087 + 0.946369i \(0.395279\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 14.1955i 0.467249i
\(924\) 0 0
\(925\) 39.8967i 1.31180i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 12.9090 0.423530 0.211765 0.977321i \(-0.432079\pi\)
0.211765 + 0.977321i \(0.432079\pi\)
\(930\) 0 0
\(931\) − 21.9849i − 0.720526i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −10.2310 −0.334591
\(936\) 0 0
\(937\) 4.57400 0.149426 0.0747130 0.997205i \(-0.476196\pi\)
0.0747130 + 0.997205i \(0.476196\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) − 27.2967i − 0.889846i −0.895569 0.444923i \(-0.853231\pi\)
0.895569 0.444923i \(-0.146769\pi\)
\(942\) 0 0
\(943\) 28.9915 0.944093
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 16.0284i − 0.520854i −0.965494 0.260427i \(-0.916137\pi\)
0.965494 0.260427i \(-0.0838634\pi\)
\(948\) 0 0
\(949\) 38.7409i 1.25758i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 34.0588 1.10327 0.551636 0.834085i \(-0.314004\pi\)
0.551636 + 0.834085i \(0.314004\pi\)
\(954\) 0 0
\(955\) − 28.9549i − 0.936960i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 78.5095 2.53520
\(960\) 0 0
\(961\) 7.75255 0.250082
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 61.0717i 1.96597i
\(966\) 0 0
\(967\) 22.8242 0.733978 0.366989 0.930225i \(-0.380389\pi\)
0.366989 + 0.930225i \(0.380389\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 5.49360i 0.176298i 0.996107 + 0.0881490i \(0.0280951\pi\)
−0.996107 + 0.0881490i \(0.971905\pi\)
\(972\) 0 0
\(973\) 1.73784i 0.0557125i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 24.2254 0.775039 0.387520 0.921861i \(-0.373332\pi\)
0.387520 + 0.921861i \(0.373332\pi\)
\(978\) 0 0
\(979\) 22.6878i 0.725104i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −5.00418 −0.159609 −0.0798043 0.996811i \(-0.525430\pi\)
−0.0798043 + 0.996811i \(0.525430\pi\)
\(984\) 0 0
\(985\) 12.8107 0.408182
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) − 9.23609i − 0.293691i
\(990\) 0 0
\(991\) 43.0455 1.36739 0.683693 0.729770i \(-0.260373\pi\)
0.683693 + 0.729770i \(0.260373\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 43.4164i 1.37639i
\(996\) 0 0
\(997\) 3.28681i 0.104094i 0.998645 + 0.0520471i \(0.0165746\pi\)
−0.998645 + 0.0520471i \(0.983425\pi\)
\(998\) 0 0
\(999\) 0 0
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 4896.2.f.c.2449.1 8
3.2 odd 2 544.2.c.a.273.6 8
4.3 odd 2 1224.2.f.d.613.8 8
8.3 odd 2 1224.2.f.d.613.7 8
8.5 even 2 inner 4896.2.f.c.2449.8 8
12.11 even 2 136.2.c.a.69.1 8
24.5 odd 2 544.2.c.a.273.3 8
24.11 even 2 136.2.c.a.69.2 yes 8
48.5 odd 4 4352.2.a.be.1.6 8
48.11 even 4 4352.2.a.bc.1.3 8
48.29 odd 4 4352.2.a.be.1.3 8
48.35 even 4 4352.2.a.bc.1.6 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
136.2.c.a.69.1 8 12.11 even 2
136.2.c.a.69.2 yes 8 24.11 even 2
544.2.c.a.273.3 8 24.5 odd 2
544.2.c.a.273.6 8 3.2 odd 2
1224.2.f.d.613.7 8 8.3 odd 2
1224.2.f.d.613.8 8 4.3 odd 2
4352.2.a.bc.1.3 8 48.11 even 4
4352.2.a.bc.1.6 8 48.35 even 4
4352.2.a.be.1.3 8 48.29 odd 4
4352.2.a.be.1.6 8 48.5 odd 4
4896.2.f.c.2449.1 8 1.1 even 1 trivial
4896.2.f.c.2449.8 8 8.5 even 2 inner