Properties

Label 208.10.a.a.1.1
Level $208$
Weight $10$
Character 208.1
Self dual yes
Analytic conductor $107.127$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [208,10,Mod(1,208)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(208, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("208.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 208 = 2^{4} \cdot 13 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 208.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: \(107.127453922\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 26)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 208.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-192.000 q^{3} -1310.00 q^{5} +5810.00 q^{7} +17181.0 q^{9} +O(q^{10})\) \(q-192.000 q^{3} -1310.00 q^{5} +5810.00 q^{7} +17181.0 q^{9} +4498.00 q^{11} -28561.0 q^{13} +251520. q^{15} -237498. q^{17} +913014. q^{19} -1.11552e6 q^{21} -201544. q^{23} -237025. q^{25} +480384. q^{27} +1.27683e6 q^{29} -4.16377e6 q^{31} -863616. q^{33} -7.61110e6 q^{35} -1.84427e7 q^{37} +5.48371e6 q^{39} -2.26017e7 q^{41} -1.17263e7 q^{43} -2.25071e7 q^{45} -5.92915e7 q^{47} -6.59751e6 q^{49} +4.55996e7 q^{51} +1.08159e8 q^{53} -5.89238e6 q^{55} -1.75299e8 q^{57} +1.49202e7 q^{59} -5.70037e7 q^{61} +9.98216e7 q^{63} +3.74149e7 q^{65} -2.20740e7 q^{67} +3.86964e7 q^{69} -4.44162e7 q^{71} +2.65795e8 q^{73} +4.55088e7 q^{75} +2.61334e7 q^{77} -4.76755e8 q^{79} -4.30407e8 q^{81} +5.05316e8 q^{83} +3.11122e8 q^{85} -2.45152e8 q^{87} +8.90841e8 q^{89} -1.65939e8 q^{91} +7.99444e8 q^{93} -1.19605e9 q^{95} -8.02777e8 q^{97} +7.72801e7 q^{99} +O(q^{100})\)

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
\(3\) −192.000 −1.36853 −0.684267 0.729232i \(-0.739878\pi\)
−0.684267 + 0.729232i \(0.739878\pi\)
\(4\) 0 0
\(5\) −1310.00 −0.937360 −0.468680 0.883368i \(-0.655270\pi\)
−0.468680 + 0.883368i \(0.655270\pi\)
\(6\) 0 0
\(7\) 5810.00 0.914608 0.457304 0.889310i \(-0.348815\pi\)
0.457304 + 0.889310i \(0.348815\pi\)
\(8\) 0 0
\(9\) 17181.0 0.872885
\(10\) 0 0
\(11\) 4498.00 0.0926302 0.0463151 0.998927i \(-0.485252\pi\)
0.0463151 + 0.998927i \(0.485252\pi\)
\(12\) 0 0
\(13\) −28561.0 −0.277350
\(14\) 0 0
\(15\) 251520. 1.28281
\(16\) 0 0
\(17\) −237498. −0.689668 −0.344834 0.938664i \(-0.612065\pi\)
−0.344834 + 0.938664i \(0.612065\pi\)
\(18\) 0 0
\(19\) 913014. 1.60726 0.803630 0.595129i \(-0.202899\pi\)
0.803630 + 0.595129i \(0.202899\pi\)
\(20\) 0 0
\(21\) −1.11552e6 −1.25167
\(22\) 0 0
\(23\) −201544. −0.150174 −0.0750870 0.997177i \(-0.523923\pi\)
−0.0750870 + 0.997177i \(0.523923\pi\)
\(24\) 0 0
\(25\) −237025. −0.121357
\(26\) 0 0
\(27\) 480384. 0.173961
\(28\) 0 0
\(29\) 1.27683e6 0.335230 0.167615 0.985852i \(-0.446393\pi\)
0.167615 + 0.985852i \(0.446393\pi\)
\(30\) 0 0
\(31\) −4.16377e6 −0.809765 −0.404883 0.914369i \(-0.632688\pi\)
−0.404883 + 0.914369i \(0.632688\pi\)
\(32\) 0 0
\(33\) −863616. −0.126768
\(34\) 0 0
\(35\) −7.61110e6 −0.857317
\(36\) 0 0
\(37\) −1.84427e7 −1.61777 −0.808883 0.587969i \(-0.799928\pi\)
−0.808883 + 0.587969i \(0.799928\pi\)
\(38\) 0 0
\(39\) 5.48371e6 0.379563
\(40\) 0 0
\(41\) −2.26017e7 −1.24915 −0.624573 0.780966i \(-0.714727\pi\)
−0.624573 + 0.780966i \(0.714727\pi\)
\(42\) 0 0
\(43\) −1.17263e7 −0.523062 −0.261531 0.965195i \(-0.584227\pi\)
−0.261531 + 0.965195i \(0.584227\pi\)
\(44\) 0 0
\(45\) −2.25071e7 −0.818207
\(46\) 0 0
\(47\) −5.92915e7 −1.77236 −0.886181 0.463340i \(-0.846651\pi\)
−0.886181 + 0.463340i \(0.846651\pi\)
\(48\) 0 0
\(49\) −6.59751e6 −0.163492
\(50\) 0 0
\(51\) 4.55996e7 0.943834
\(52\) 0 0
\(53\) 1.08159e8 1.88287 0.941434 0.337196i \(-0.109479\pi\)
0.941434 + 0.337196i \(0.109479\pi\)
\(54\) 0 0
\(55\) −5.89238e6 −0.0868278
\(56\) 0 0
\(57\) −1.75299e8 −2.19959
\(58\) 0 0
\(59\) 1.49202e7 0.160302 0.0801511 0.996783i \(-0.474460\pi\)
0.0801511 + 0.996783i \(0.474460\pi\)
\(60\) 0 0
\(61\) −5.70037e7 −0.527132 −0.263566 0.964641i \(-0.584899\pi\)
−0.263566 + 0.964641i \(0.584899\pi\)
\(62\) 0 0
\(63\) 9.98216e7 0.798348
\(64\) 0 0
\(65\) 3.74149e7 0.259977
\(66\) 0 0
\(67\) −2.20740e7 −0.133827 −0.0669136 0.997759i \(-0.521315\pi\)
−0.0669136 + 0.997759i \(0.521315\pi\)
\(68\) 0 0
\(69\) 3.86964e7 0.205518
\(70\) 0 0
\(71\) −4.44162e7 −0.207434 −0.103717 0.994607i \(-0.533074\pi\)
−0.103717 + 0.994607i \(0.533074\pi\)
\(72\) 0 0
\(73\) 2.65795e8 1.09545 0.547726 0.836658i \(-0.315494\pi\)
0.547726 + 0.836658i \(0.315494\pi\)
\(74\) 0 0
\(75\) 4.55088e7 0.166081
\(76\) 0 0
\(77\) 2.61334e7 0.0847203
\(78\) 0 0
\(79\) −4.76755e8 −1.37713 −0.688563 0.725176i \(-0.741758\pi\)
−0.688563 + 0.725176i \(0.741758\pi\)
\(80\) 0 0
\(81\) −4.30407e8 −1.11096
\(82\) 0 0
\(83\) 5.05316e8 1.16872 0.584361 0.811494i \(-0.301345\pi\)
0.584361 + 0.811494i \(0.301345\pi\)
\(84\) 0 0
\(85\) 3.11122e8 0.646467
\(86\) 0 0
\(87\) −2.45152e8 −0.458774
\(88\) 0 0
\(89\) 8.90841e8 1.50503 0.752515 0.658575i \(-0.228841\pi\)
0.752515 + 0.658575i \(0.228841\pi\)
\(90\) 0 0
\(91\) −1.65939e8 −0.253667
\(92\) 0 0
\(93\) 7.99444e8 1.10819
\(94\) 0 0
\(95\) −1.19605e9 −1.50658
\(96\) 0 0
\(97\) −8.02777e8 −0.920708 −0.460354 0.887735i \(-0.652278\pi\)
−0.460354 + 0.887735i \(0.652278\pi\)
\(98\) 0 0
\(99\) 7.72801e7 0.0808555
\(100\) 0 0
\(101\) 1.19998e9 1.14743 0.573717 0.819053i \(-0.305501\pi\)
0.573717 + 0.819053i \(0.305501\pi\)
\(102\) 0 0
\(103\) 9.58027e8 0.838707 0.419353 0.907823i \(-0.362257\pi\)
0.419353 + 0.907823i \(0.362257\pi\)
\(104\) 0 0
\(105\) 1.46133e9 1.17327
\(106\) 0 0
\(107\) 2.39051e9 1.76304 0.881521 0.472145i \(-0.156520\pi\)
0.881521 + 0.472145i \(0.156520\pi\)
\(108\) 0 0
\(109\) −1.70171e9 −1.15469 −0.577346 0.816499i \(-0.695912\pi\)
−0.577346 + 0.816499i \(0.695912\pi\)
\(110\) 0 0
\(111\) 3.54099e9 2.21397
\(112\) 0 0
\(113\) −1.40793e9 −0.812320 −0.406160 0.913802i \(-0.633132\pi\)
−0.406160 + 0.913802i \(0.633132\pi\)
\(114\) 0 0
\(115\) 2.64023e8 0.140767
\(116\) 0 0
\(117\) −4.90707e8 −0.242095
\(118\) 0 0
\(119\) −1.37986e9 −0.630775
\(120\) 0 0
\(121\) −2.33772e9 −0.991420
\(122\) 0 0
\(123\) 4.33952e9 1.70950
\(124\) 0 0
\(125\) 2.86910e9 1.05111
\(126\) 0 0
\(127\) 3.31210e9 1.12976 0.564881 0.825172i \(-0.308922\pi\)
0.564881 + 0.825172i \(0.308922\pi\)
\(128\) 0 0
\(129\) 2.25145e9 0.715828
\(130\) 0 0
\(131\) −3.06389e9 −0.908977 −0.454489 0.890753i \(-0.650178\pi\)
−0.454489 + 0.890753i \(0.650178\pi\)
\(132\) 0 0
\(133\) 5.30461e9 1.47001
\(134\) 0 0
\(135\) −6.29303e8 −0.163064
\(136\) 0 0
\(137\) 5.62781e8 0.136489 0.0682444 0.997669i \(-0.478260\pi\)
0.0682444 + 0.997669i \(0.478260\pi\)
\(138\) 0 0
\(139\) 4.60597e8 0.104654 0.0523268 0.998630i \(-0.483336\pi\)
0.0523268 + 0.998630i \(0.483336\pi\)
\(140\) 0 0
\(141\) 1.13840e10 2.42554
\(142\) 0 0
\(143\) −1.28467e8 −0.0256910
\(144\) 0 0
\(145\) −1.67265e9 −0.314232
\(146\) 0 0
\(147\) 1.26672e9 0.223745
\(148\) 0 0
\(149\) −6.01717e8 −0.100012 −0.0500062 0.998749i \(-0.515924\pi\)
−0.0500062 + 0.998749i \(0.515924\pi\)
\(150\) 0 0
\(151\) 1.26695e10 1.98318 0.991589 0.129428i \(-0.0413142\pi\)
0.991589 + 0.129428i \(0.0413142\pi\)
\(152\) 0 0
\(153\) −4.08045e9 −0.602001
\(154\) 0 0
\(155\) 5.45454e9 0.759041
\(156\) 0 0
\(157\) −2.00733e8 −0.0263676 −0.0131838 0.999913i \(-0.504197\pi\)
−0.0131838 + 0.999913i \(0.504197\pi\)
\(158\) 0 0
\(159\) −2.07665e10 −2.57677
\(160\) 0 0
\(161\) −1.17097e9 −0.137350
\(162\) 0 0
\(163\) −6.32491e9 −0.701795 −0.350898 0.936414i \(-0.614123\pi\)
−0.350898 + 0.936414i \(0.614123\pi\)
\(164\) 0 0
\(165\) 1.13134e9 0.118827
\(166\) 0 0
\(167\) −1.51400e10 −1.50627 −0.753134 0.657867i \(-0.771459\pi\)
−0.753134 + 0.657867i \(0.771459\pi\)
\(168\) 0 0
\(169\) 8.15731e8 0.0769231
\(170\) 0 0
\(171\) 1.56865e10 1.40295
\(172\) 0 0
\(173\) −1.63483e9 −0.138760 −0.0693802 0.997590i \(-0.522102\pi\)
−0.0693802 + 0.997590i \(0.522102\pi\)
\(174\) 0 0
\(175\) −1.37712e9 −0.110994
\(176\) 0 0
\(177\) −2.86467e9 −0.219379
\(178\) 0 0
\(179\) 4.12980e9 0.300670 0.150335 0.988635i \(-0.451965\pi\)
0.150335 + 0.988635i \(0.451965\pi\)
\(180\) 0 0
\(181\) −2.13092e10 −1.47575 −0.737875 0.674937i \(-0.764171\pi\)
−0.737875 + 0.674937i \(0.764171\pi\)
\(182\) 0 0
\(183\) 1.09447e10 0.721398
\(184\) 0 0
\(185\) 2.41599e10 1.51643
\(186\) 0 0
\(187\) −1.06827e9 −0.0638840
\(188\) 0 0
\(189\) 2.79103e9 0.159106
\(190\) 0 0
\(191\) 3.08641e10 1.67804 0.839021 0.544099i \(-0.183128\pi\)
0.839021 + 0.544099i \(0.183128\pi\)
\(192\) 0 0
\(193\) −4.54917e9 −0.236007 −0.118003 0.993013i \(-0.537649\pi\)
−0.118003 + 0.993013i \(0.537649\pi\)
\(194\) 0 0
\(195\) −7.18366e9 −0.355787
\(196\) 0 0
\(197\) 2.26076e10 1.06944 0.534720 0.845030i \(-0.320417\pi\)
0.534720 + 0.845030i \(0.320417\pi\)
\(198\) 0 0
\(199\) −1.05027e10 −0.474749 −0.237375 0.971418i \(-0.576287\pi\)
−0.237375 + 0.971418i \(0.576287\pi\)
\(200\) 0 0
\(201\) 4.23821e9 0.183147
\(202\) 0 0
\(203\) 7.41841e9 0.306604
\(204\) 0 0
\(205\) 2.96082e10 1.17090
\(206\) 0 0
\(207\) −3.46273e9 −0.131085
\(208\) 0 0
\(209\) 4.10674e9 0.148881
\(210\) 0 0
\(211\) 5.66420e9 0.196729 0.0983643 0.995150i \(-0.468639\pi\)
0.0983643 + 0.995150i \(0.468639\pi\)
\(212\) 0 0
\(213\) 8.52792e9 0.283880
\(214\) 0 0
\(215\) 1.53615e10 0.490297
\(216\) 0 0
\(217\) −2.41915e10 −0.740618
\(218\) 0 0
\(219\) −5.10326e10 −1.49916
\(220\) 0 0
\(221\) 6.78318e9 0.191279
\(222\) 0 0
\(223\) 3.19607e10 0.865454 0.432727 0.901525i \(-0.357551\pi\)
0.432727 + 0.901525i \(0.357551\pi\)
\(224\) 0 0
\(225\) −4.07233e9 −0.105931
\(226\) 0 0
\(227\) 5.07782e10 1.26929 0.634645 0.772804i \(-0.281146\pi\)
0.634645 + 0.772804i \(0.281146\pi\)
\(228\) 0 0
\(229\) 5.99836e10 1.44136 0.720681 0.693267i \(-0.243829\pi\)
0.720681 + 0.693267i \(0.243829\pi\)
\(230\) 0 0
\(231\) −5.01761e9 −0.115943
\(232\) 0 0
\(233\) 4.77228e10 1.06078 0.530389 0.847755i \(-0.322046\pi\)
0.530389 + 0.847755i \(0.322046\pi\)
\(234\) 0 0
\(235\) 7.76719e10 1.66134
\(236\) 0 0
\(237\) 9.15371e10 1.88464
\(238\) 0 0
\(239\) 8.71569e10 1.72787 0.863936 0.503602i \(-0.167992\pi\)
0.863936 + 0.503602i \(0.167992\pi\)
\(240\) 0 0
\(241\) −1.04205e11 −1.98981 −0.994903 0.100835i \(-0.967849\pi\)
−0.994903 + 0.100835i \(0.967849\pi\)
\(242\) 0 0
\(243\) 7.31828e10 1.34642
\(244\) 0 0
\(245\) 8.64273e9 0.153251
\(246\) 0 0
\(247\) −2.60766e10 −0.445774
\(248\) 0 0
\(249\) −9.70206e10 −1.59944
\(250\) 0 0
\(251\) −2.82027e9 −0.0448496 −0.0224248 0.999749i \(-0.507139\pi\)
−0.0224248 + 0.999749i \(0.507139\pi\)
\(252\) 0 0
\(253\) −9.06545e8 −0.0139106
\(254\) 0 0
\(255\) −5.97355e10 −0.884711
\(256\) 0 0
\(257\) −1.41573e10 −0.202433 −0.101216 0.994864i \(-0.532273\pi\)
−0.101216 + 0.994864i \(0.532273\pi\)
\(258\) 0 0
\(259\) −1.07152e11 −1.47962
\(260\) 0 0
\(261\) 2.19373e10 0.292618
\(262\) 0 0
\(263\) 3.58497e10 0.462045 0.231023 0.972948i \(-0.425793\pi\)
0.231023 + 0.972948i \(0.425793\pi\)
\(264\) 0 0
\(265\) −1.41688e11 −1.76493
\(266\) 0 0
\(267\) −1.71041e11 −2.05968
\(268\) 0 0
\(269\) −7.14394e10 −0.831864 −0.415932 0.909396i \(-0.636545\pi\)
−0.415932 + 0.909396i \(0.636545\pi\)
\(270\) 0 0
\(271\) 6.79344e9 0.0765117 0.0382558 0.999268i \(-0.487820\pi\)
0.0382558 + 0.999268i \(0.487820\pi\)
\(272\) 0 0
\(273\) 3.18604e10 0.347151
\(274\) 0 0
\(275\) −1.06614e9 −0.0112413
\(276\) 0 0
\(277\) −6.93103e10 −0.707357 −0.353679 0.935367i \(-0.615069\pi\)
−0.353679 + 0.935367i \(0.615069\pi\)
\(278\) 0 0
\(279\) −7.15377e10 −0.706832
\(280\) 0 0
\(281\) 3.10369e10 0.296961 0.148480 0.988915i \(-0.452562\pi\)
0.148480 + 0.988915i \(0.452562\pi\)
\(282\) 0 0
\(283\) −1.35312e11 −1.25400 −0.627001 0.779018i \(-0.715718\pi\)
−0.627001 + 0.779018i \(0.715718\pi\)
\(284\) 0 0
\(285\) 2.29641e11 2.06181
\(286\) 0 0
\(287\) −1.31316e11 −1.14248
\(288\) 0 0
\(289\) −6.21826e10 −0.524359
\(290\) 0 0
\(291\) 1.54133e11 1.26002
\(292\) 0 0
\(293\) −7.55078e10 −0.598532 −0.299266 0.954170i \(-0.596742\pi\)
−0.299266 + 0.954170i \(0.596742\pi\)
\(294\) 0 0
\(295\) −1.95454e10 −0.150261
\(296\) 0 0
\(297\) 2.16077e9 0.0161140
\(298\) 0 0
\(299\) 5.75630e9 0.0416508
\(300\) 0 0
\(301\) −6.81298e10 −0.478397
\(302\) 0 0
\(303\) −2.30396e11 −1.57030
\(304\) 0 0
\(305\) 7.46749e10 0.494112
\(306\) 0 0
\(307\) −1.42760e10 −0.0917241 −0.0458620 0.998948i \(-0.514603\pi\)
−0.0458620 + 0.998948i \(0.514603\pi\)
\(308\) 0 0
\(309\) −1.83941e11 −1.14780
\(310\) 0 0
\(311\) −3.58426e9 −0.0217259 −0.0108630 0.999941i \(-0.503458\pi\)
−0.0108630 + 0.999941i \(0.503458\pi\)
\(312\) 0 0
\(313\) 2.79830e11 1.64795 0.823977 0.566623i \(-0.191750\pi\)
0.823977 + 0.566623i \(0.191750\pi\)
\(314\) 0 0
\(315\) −1.30766e11 −0.748339
\(316\) 0 0
\(317\) 2.40148e10 0.133571 0.0667855 0.997767i \(-0.478726\pi\)
0.0667855 + 0.997767i \(0.478726\pi\)
\(318\) 0 0
\(319\) 5.74320e9 0.0310524
\(320\) 0 0
\(321\) −4.58977e11 −2.41278
\(322\) 0 0
\(323\) −2.16839e11 −1.10848
\(324\) 0 0
\(325\) 6.76967e9 0.0336583
\(326\) 0 0
\(327\) 3.26728e11 1.58024
\(328\) 0 0
\(329\) −3.44484e11 −1.62102
\(330\) 0 0
\(331\) −3.73009e11 −1.70802 −0.854010 0.520257i \(-0.825836\pi\)
−0.854010 + 0.520257i \(0.825836\pi\)
\(332\) 0 0
\(333\) −3.16863e11 −1.41212
\(334\) 0 0
\(335\) 2.89170e10 0.125444
\(336\) 0 0
\(337\) 1.91157e11 0.807340 0.403670 0.914905i \(-0.367734\pi\)
0.403670 + 0.914905i \(0.367734\pi\)
\(338\) 0 0
\(339\) 2.70322e11 1.11169
\(340\) 0 0
\(341\) −1.87286e10 −0.0750087
\(342\) 0 0
\(343\) −2.72786e11 −1.06414
\(344\) 0 0
\(345\) −5.06923e10 −0.192644
\(346\) 0 0
\(347\) −8.60398e10 −0.318579 −0.159289 0.987232i \(-0.550920\pi\)
−0.159289 + 0.987232i \(0.550920\pi\)
\(348\) 0 0
\(349\) −1.33612e11 −0.482094 −0.241047 0.970513i \(-0.577491\pi\)
−0.241047 + 0.970513i \(0.577491\pi\)
\(350\) 0 0
\(351\) −1.37202e10 −0.0482481
\(352\) 0 0
\(353\) −6.23799e10 −0.213825 −0.106912 0.994268i \(-0.534096\pi\)
−0.106912 + 0.994268i \(0.534096\pi\)
\(354\) 0 0
\(355\) 5.81853e10 0.194440
\(356\) 0 0
\(357\) 2.64934e11 0.863238
\(358\) 0 0
\(359\) 3.82739e11 1.21612 0.608062 0.793890i \(-0.291947\pi\)
0.608062 + 0.793890i \(0.291947\pi\)
\(360\) 0 0
\(361\) 5.10907e11 1.58329
\(362\) 0 0
\(363\) 4.48841e11 1.35679
\(364\) 0 0
\(365\) −3.48191e11 −1.02683
\(366\) 0 0
\(367\) −2.59802e11 −0.747560 −0.373780 0.927517i \(-0.621938\pi\)
−0.373780 + 0.927517i \(0.621938\pi\)
\(368\) 0 0
\(369\) −3.88319e11 −1.09036
\(370\) 0 0
\(371\) 6.28402e11 1.72209
\(372\) 0 0
\(373\) 4.70946e11 1.25974 0.629870 0.776700i \(-0.283108\pi\)
0.629870 + 0.776700i \(0.283108\pi\)
\(374\) 0 0
\(375\) −5.50867e11 −1.43849
\(376\) 0 0
\(377\) −3.64677e10 −0.0929762
\(378\) 0 0
\(379\) 3.60046e11 0.896358 0.448179 0.893944i \(-0.352073\pi\)
0.448179 + 0.893944i \(0.352073\pi\)
\(380\) 0 0
\(381\) −6.35924e11 −1.54612
\(382\) 0 0
\(383\) 9.59380e10 0.227822 0.113911 0.993491i \(-0.463662\pi\)
0.113911 + 0.993491i \(0.463662\pi\)
\(384\) 0 0
\(385\) −3.42347e10 −0.0794134
\(386\) 0 0
\(387\) −2.01470e11 −0.456573
\(388\) 0 0
\(389\) −4.60488e11 −1.01964 −0.509818 0.860282i \(-0.670287\pi\)
−0.509818 + 0.860282i \(0.670287\pi\)
\(390\) 0 0
\(391\) 4.78663e10 0.103570
\(392\) 0 0
\(393\) 5.88268e11 1.24397
\(394\) 0 0
\(395\) 6.24550e11 1.29086
\(396\) 0 0
\(397\) 2.90299e11 0.586528 0.293264 0.956032i \(-0.405259\pi\)
0.293264 + 0.956032i \(0.405259\pi\)
\(398\) 0 0
\(399\) −1.01849e12 −2.01176
\(400\) 0 0
\(401\) 6.85495e11 1.32390 0.661949 0.749549i \(-0.269730\pi\)
0.661949 + 0.749549i \(0.269730\pi\)
\(402\) 0 0
\(403\) 1.18921e11 0.224588
\(404\) 0 0
\(405\) 5.63834e11 1.04137
\(406\) 0 0
\(407\) −8.29551e10 −0.149854
\(408\) 0 0
\(409\) 1.00030e12 1.76756 0.883779 0.467905i \(-0.154991\pi\)
0.883779 + 0.467905i \(0.154991\pi\)
\(410\) 0 0
\(411\) −1.08054e11 −0.186790
\(412\) 0 0
\(413\) 8.66861e10 0.146614
\(414\) 0 0
\(415\) −6.61964e11 −1.09551
\(416\) 0 0
\(417\) −8.84346e10 −0.143222
\(418\) 0 0
\(419\) 8.64798e11 1.37073 0.685364 0.728200i \(-0.259643\pi\)
0.685364 + 0.728200i \(0.259643\pi\)
\(420\) 0 0
\(421\) −9.57784e10 −0.148593 −0.0742965 0.997236i \(-0.523671\pi\)
−0.0742965 + 0.997236i \(0.523671\pi\)
\(422\) 0 0
\(423\) −1.01869e12 −1.54707
\(424\) 0 0
\(425\) 5.62930e10 0.0836959
\(426\) 0 0
\(427\) −3.31192e11 −0.482119
\(428\) 0 0
\(429\) 2.46657e10 0.0351590
\(430\) 0 0
\(431\) 1.27185e11 0.177536 0.0887682 0.996052i \(-0.471707\pi\)
0.0887682 + 0.996052i \(0.471707\pi\)
\(432\) 0 0
\(433\) −1.55264e11 −0.212264 −0.106132 0.994352i \(-0.533847\pi\)
−0.106132 + 0.994352i \(0.533847\pi\)
\(434\) 0 0
\(435\) 3.21149e11 0.430036
\(436\) 0 0
\(437\) −1.84012e11 −0.241369
\(438\) 0 0
\(439\) 1.02610e12 1.31855 0.659277 0.751901i \(-0.270863\pi\)
0.659277 + 0.751901i \(0.270863\pi\)
\(440\) 0 0
\(441\) −1.13352e11 −0.142710
\(442\) 0 0
\(443\) −2.52039e11 −0.310922 −0.155461 0.987842i \(-0.549686\pi\)
−0.155461 + 0.987842i \(0.549686\pi\)
\(444\) 0 0
\(445\) −1.16700e12 −1.41075
\(446\) 0 0
\(447\) 1.15530e11 0.136870
\(448\) 0 0
\(449\) 7.66198e11 0.889678 0.444839 0.895611i \(-0.353261\pi\)
0.444839 + 0.895611i \(0.353261\pi\)
\(450\) 0 0
\(451\) −1.01662e11 −0.115709
\(452\) 0 0
\(453\) −2.43254e12 −2.71405
\(454\) 0 0
\(455\) 2.17381e11 0.237777
\(456\) 0 0
\(457\) 1.75683e12 1.88411 0.942057 0.335454i \(-0.108890\pi\)
0.942057 + 0.335454i \(0.108890\pi\)
\(458\) 0 0
\(459\) −1.14090e11 −0.119975
\(460\) 0 0
\(461\) −1.13127e12 −1.16657 −0.583287 0.812266i \(-0.698234\pi\)
−0.583287 + 0.812266i \(0.698234\pi\)
\(462\) 0 0
\(463\) 2.71657e11 0.274730 0.137365 0.990521i \(-0.456137\pi\)
0.137365 + 0.990521i \(0.456137\pi\)
\(464\) 0 0
\(465\) −1.04727e12 −1.03877
\(466\) 0 0
\(467\) 9.54617e11 0.928759 0.464380 0.885636i \(-0.346277\pi\)
0.464380 + 0.885636i \(0.346277\pi\)
\(468\) 0 0
\(469\) −1.28250e11 −0.122399
\(470\) 0 0
\(471\) 3.85407e10 0.0360849
\(472\) 0 0
\(473\) −5.27449e10 −0.0484513
\(474\) 0 0
\(475\) −2.16407e11 −0.195052
\(476\) 0 0
\(477\) 1.85827e12 1.64353
\(478\) 0 0
\(479\) −1.43680e12 −1.24705 −0.623527 0.781802i \(-0.714301\pi\)
−0.623527 + 0.781802i \(0.714301\pi\)
\(480\) 0 0
\(481\) 5.26741e11 0.448688
\(482\) 0 0
\(483\) 2.24826e11 0.187969
\(484\) 0 0
\(485\) 1.05164e12 0.863035
\(486\) 0 0
\(487\) 1.22744e12 0.988826 0.494413 0.869227i \(-0.335383\pi\)
0.494413 + 0.869227i \(0.335383\pi\)
\(488\) 0 0
\(489\) 1.21438e12 0.960431
\(490\) 0 0
\(491\) 1.00389e12 0.779506 0.389753 0.920919i \(-0.372560\pi\)
0.389753 + 0.920919i \(0.372560\pi\)
\(492\) 0 0
\(493\) −3.03246e11 −0.231198
\(494\) 0 0
\(495\) −1.01237e11 −0.0757907
\(496\) 0 0
\(497\) −2.58058e11 −0.189721
\(498\) 0 0
\(499\) 7.58262e11 0.547478 0.273739 0.961804i \(-0.411740\pi\)
0.273739 + 0.961804i \(0.411740\pi\)
\(500\) 0 0
\(501\) 2.90688e12 2.06138
\(502\) 0 0
\(503\) −1.82032e12 −1.26792 −0.633959 0.773367i \(-0.718571\pi\)
−0.633959 + 0.773367i \(0.718571\pi\)
\(504\) 0 0
\(505\) −1.57197e12 −1.07556
\(506\) 0 0
\(507\) −1.56620e11 −0.105272
\(508\) 0 0
\(509\) 6.57012e11 0.433854 0.216927 0.976188i \(-0.430397\pi\)
0.216927 + 0.976188i \(0.430397\pi\)
\(510\) 0 0
\(511\) 1.54427e12 1.00191
\(512\) 0 0
\(513\) 4.38597e11 0.279600
\(514\) 0 0
\(515\) −1.25502e12 −0.786170
\(516\) 0 0
\(517\) −2.66693e11 −0.164174
\(518\) 0 0
\(519\) 3.13887e11 0.189898
\(520\) 0 0
\(521\) 3.17678e11 0.188894 0.0944468 0.995530i \(-0.469892\pi\)
0.0944468 + 0.995530i \(0.469892\pi\)
\(522\) 0 0
\(523\) 2.88365e12 1.68533 0.842666 0.538436i \(-0.180985\pi\)
0.842666 + 0.538436i \(0.180985\pi\)
\(524\) 0 0
\(525\) 2.64406e11 0.151899
\(526\) 0 0
\(527\) 9.88887e11 0.558469
\(528\) 0 0
\(529\) −1.76053e12 −0.977448
\(530\) 0 0
\(531\) 2.56343e11 0.139925
\(532\) 0 0
\(533\) 6.45526e11 0.346451
\(534\) 0 0
\(535\) −3.13156e12 −1.65260
\(536\) 0 0
\(537\) −7.92921e11 −0.411477
\(538\) 0 0
\(539\) −2.96756e10 −0.0151443
\(540\) 0 0
\(541\) 2.16753e12 1.08787 0.543937 0.839126i \(-0.316933\pi\)
0.543937 + 0.839126i \(0.316933\pi\)
\(542\) 0 0
\(543\) 4.09136e12 2.01961
\(544\) 0 0
\(545\) 2.22924e12 1.08236
\(546\) 0 0
\(547\) 9.14427e11 0.436723 0.218361 0.975868i \(-0.429929\pi\)
0.218361 + 0.975868i \(0.429929\pi\)
\(548\) 0 0
\(549\) −9.79381e11 −0.460125
\(550\) 0 0
\(551\) 1.16577e12 0.538803
\(552\) 0 0
\(553\) −2.76995e12 −1.25953
\(554\) 0 0
\(555\) −4.63870e12 −2.07528
\(556\) 0 0
\(557\) 1.75791e10 0.00773834 0.00386917 0.999993i \(-0.498768\pi\)
0.00386917 + 0.999993i \(0.498768\pi\)
\(558\) 0 0
\(559\) 3.34915e11 0.145071
\(560\) 0 0
\(561\) 2.05107e11 0.0874274
\(562\) 0 0
\(563\) 3.70644e12 1.55478 0.777390 0.629019i \(-0.216543\pi\)
0.777390 + 0.629019i \(0.216543\pi\)
\(564\) 0 0
\(565\) 1.84438e12 0.761436
\(566\) 0 0
\(567\) −2.50067e12 −1.01609
\(568\) 0 0
\(569\) −3.40051e12 −1.36000 −0.679999 0.733213i \(-0.738020\pi\)
−0.679999 + 0.733213i \(0.738020\pi\)
\(570\) 0 0
\(571\) 4.46270e12 1.75685 0.878427 0.477877i \(-0.158594\pi\)
0.878427 + 0.477877i \(0.158594\pi\)
\(572\) 0 0
\(573\) −5.92590e12 −2.29646
\(574\) 0 0
\(575\) 4.77710e10 0.0182246
\(576\) 0 0
\(577\) 3.47193e12 1.30401 0.652004 0.758215i \(-0.273929\pi\)
0.652004 + 0.758215i \(0.273929\pi\)
\(578\) 0 0
\(579\) 8.73441e11 0.322983
\(580\) 0 0
\(581\) 2.93588e12 1.06892
\(582\) 0 0
\(583\) 4.86498e11 0.174410
\(584\) 0 0
\(585\) 6.42826e11 0.226930
\(586\) 0 0
\(587\) −2.86266e12 −0.995171 −0.497586 0.867415i \(-0.665780\pi\)
−0.497586 + 0.867415i \(0.665780\pi\)
\(588\) 0 0
\(589\) −3.80158e12 −1.30150
\(590\) 0 0
\(591\) −4.34066e12 −1.46356
\(592\) 0 0
\(593\) −8.38217e11 −0.278362 −0.139181 0.990267i \(-0.544447\pi\)
−0.139181 + 0.990267i \(0.544447\pi\)
\(594\) 0 0
\(595\) 1.80762e12 0.591263
\(596\) 0 0
\(597\) 2.01653e12 0.649710
\(598\) 0 0
\(599\) −3.94489e12 −1.25203 −0.626013 0.779812i \(-0.715315\pi\)
−0.626013 + 0.779812i \(0.715315\pi\)
\(600\) 0 0
\(601\) 4.99865e12 1.56285 0.781426 0.623998i \(-0.214492\pi\)
0.781426 + 0.623998i \(0.214492\pi\)
\(602\) 0 0
\(603\) −3.79254e11 −0.116816
\(604\) 0 0
\(605\) 3.06241e12 0.929317
\(606\) 0 0
\(607\) 3.95582e11 0.118274 0.0591368 0.998250i \(-0.481165\pi\)
0.0591368 + 0.998250i \(0.481165\pi\)
\(608\) 0 0
\(609\) −1.42433e12 −0.419599
\(610\) 0 0
\(611\) 1.69343e12 0.491565
\(612\) 0 0
\(613\) 5.03617e12 1.44055 0.720275 0.693688i \(-0.244016\pi\)
0.720275 + 0.693688i \(0.244016\pi\)
\(614\) 0 0
\(615\) −5.68477e12 −1.60242
\(616\) 0 0
\(617\) 4.06829e12 1.13013 0.565065 0.825046i \(-0.308851\pi\)
0.565065 + 0.825046i \(0.308851\pi\)
\(618\) 0 0
\(619\) 4.73136e12 1.29532 0.647662 0.761928i \(-0.275747\pi\)
0.647662 + 0.761928i \(0.275747\pi\)
\(620\) 0 0
\(621\) −9.68185e10 −0.0261244
\(622\) 0 0
\(623\) 5.17578e12 1.37651
\(624\) 0 0
\(625\) −3.29558e12 −0.863916
\(626\) 0 0
\(627\) −7.88493e11 −0.203748
\(628\) 0 0
\(629\) 4.38010e12 1.11572
\(630\) 0 0
\(631\) −3.45019e12 −0.866384 −0.433192 0.901302i \(-0.642613\pi\)
−0.433192 + 0.901302i \(0.642613\pi\)
\(632\) 0 0
\(633\) −1.08753e12 −0.269230
\(634\) 0 0
\(635\) −4.33886e12 −1.05899
\(636\) 0 0
\(637\) 1.88431e11 0.0453446
\(638\) 0 0
\(639\) −7.63116e11 −0.181066
\(640\) 0 0
\(641\) −3.87461e12 −0.906498 −0.453249 0.891384i \(-0.649735\pi\)
−0.453249 + 0.891384i \(0.649735\pi\)
\(642\) 0 0
\(643\) −2.77752e12 −0.640778 −0.320389 0.947286i \(-0.603814\pi\)
−0.320389 + 0.947286i \(0.603814\pi\)
\(644\) 0 0
\(645\) −2.94940e12 −0.670989
\(646\) 0 0
\(647\) −6.12025e12 −1.37309 −0.686546 0.727086i \(-0.740874\pi\)
−0.686546 + 0.727086i \(0.740874\pi\)
\(648\) 0 0
\(649\) 6.71109e10 0.0148488
\(650\) 0 0
\(651\) 4.64477e12 1.01356
\(652\) 0 0
\(653\) 2.50039e12 0.538143 0.269072 0.963120i \(-0.413283\pi\)
0.269072 + 0.963120i \(0.413283\pi\)
\(654\) 0 0
\(655\) 4.01370e12 0.852038
\(656\) 0 0
\(657\) 4.56662e12 0.956204
\(658\) 0 0
\(659\) 6.13676e12 1.26752 0.633760 0.773529i \(-0.281511\pi\)
0.633760 + 0.773529i \(0.281511\pi\)
\(660\) 0 0
\(661\) −6.28369e12 −1.28029 −0.640145 0.768254i \(-0.721126\pi\)
−0.640145 + 0.768254i \(0.721126\pi\)
\(662\) 0 0
\(663\) −1.30237e12 −0.261772
\(664\) 0 0
\(665\) −6.94904e12 −1.37793
\(666\) 0 0
\(667\) −2.57338e11 −0.0503429
\(668\) 0 0
\(669\) −6.13645e12 −1.18440
\(670\) 0 0
\(671\) −2.56403e11 −0.0488283
\(672\) 0 0
\(673\) −7.98616e12 −1.50062 −0.750309 0.661087i \(-0.770095\pi\)
−0.750309 + 0.661087i \(0.770095\pi\)
\(674\) 0 0
\(675\) −1.13863e11 −0.0211113
\(676\) 0 0
\(677\) −3.64428e12 −0.666749 −0.333374 0.942794i \(-0.608187\pi\)
−0.333374 + 0.942794i \(0.608187\pi\)
\(678\) 0 0
\(679\) −4.66413e12 −0.842087
\(680\) 0 0
\(681\) −9.74942e12 −1.73707
\(682\) 0 0
\(683\) 1.11273e12 0.195658 0.0978291 0.995203i \(-0.468810\pi\)
0.0978291 + 0.995203i \(0.468810\pi\)
\(684\) 0 0
\(685\) −7.37244e11 −0.127939
\(686\) 0 0
\(687\) −1.15169e13 −1.97255
\(688\) 0 0
\(689\) −3.08912e12 −0.522214
\(690\) 0 0
\(691\) 2.75811e12 0.460214 0.230107 0.973165i \(-0.426092\pi\)
0.230107 + 0.973165i \(0.426092\pi\)
\(692\) 0 0
\(693\) 4.48998e11 0.0739511
\(694\) 0 0
\(695\) −6.03382e11 −0.0980982
\(696\) 0 0
\(697\) 5.36785e12 0.861495
\(698\) 0 0
\(699\) −9.16278e12 −1.45171
\(700\) 0 0
\(701\) 8.08880e12 1.26518 0.632591 0.774486i \(-0.281991\pi\)
0.632591 + 0.774486i \(0.281991\pi\)
\(702\) 0 0
\(703\) −1.68384e13 −2.60017
\(704\) 0 0
\(705\) −1.49130e13 −2.27360
\(706\) 0 0
\(707\) 6.97189e12 1.04945
\(708\) 0 0
\(709\) 2.19552e11 0.0326310 0.0163155 0.999867i \(-0.494806\pi\)
0.0163155 + 0.999867i \(0.494806\pi\)
\(710\) 0 0
\(711\) −8.19114e12 −1.20207
\(712\) 0 0
\(713\) 8.39183e11 0.121606
\(714\) 0 0
\(715\) 1.68292e11 0.0240817
\(716\) 0 0
\(717\) −1.67341e13 −2.36465
\(718\) 0 0
\(719\) 8.58532e12 1.19805 0.599027 0.800729i \(-0.295554\pi\)
0.599027 + 0.800729i \(0.295554\pi\)
\(720\) 0 0
\(721\) 5.56614e12 0.767088
\(722\) 0 0
\(723\) 2.00073e13 2.72312
\(724\) 0 0
\(725\) −3.02642e11 −0.0406825
\(726\) 0 0
\(727\) −7.59563e11 −0.100846 −0.0504230 0.998728i \(-0.516057\pi\)
−0.0504230 + 0.998728i \(0.516057\pi\)
\(728\) 0 0
\(729\) −5.57939e12 −0.731666
\(730\) 0 0
\(731\) 2.78497e12 0.360739
\(732\) 0 0
\(733\) −7.83005e12 −1.00184 −0.500918 0.865495i \(-0.667004\pi\)
−0.500918 + 0.865495i \(0.667004\pi\)
\(734\) 0 0
\(735\) −1.65940e12 −0.209729
\(736\) 0 0
\(737\) −9.92889e10 −0.0123964
\(738\) 0 0
\(739\) −5.41643e12 −0.668056 −0.334028 0.942563i \(-0.608408\pi\)
−0.334028 + 0.942563i \(0.608408\pi\)
\(740\) 0 0
\(741\) 5.00671e12 0.610057
\(742\) 0 0
\(743\) −2.66408e12 −0.320699 −0.160349 0.987060i \(-0.551262\pi\)
−0.160349 + 0.987060i \(0.551262\pi\)
\(744\) 0 0
\(745\) 7.88249e11 0.0937476
\(746\) 0 0
\(747\) 8.68183e12 1.02016
\(748\) 0 0
\(749\) 1.38888e13 1.61249
\(750\) 0 0
\(751\) −5.82882e12 −0.668653 −0.334326 0.942457i \(-0.608509\pi\)
−0.334326 + 0.942457i \(0.608509\pi\)
\(752\) 0 0
\(753\) 5.41491e11 0.0613782
\(754\) 0 0
\(755\) −1.65970e13 −1.85895
\(756\) 0 0
\(757\) 5.67869e12 0.628517 0.314258 0.949338i \(-0.398244\pi\)
0.314258 + 0.949338i \(0.398244\pi\)
\(758\) 0 0
\(759\) 1.74057e11 0.0190372
\(760\) 0 0
\(761\) −1.12490e13 −1.21586 −0.607931 0.793990i \(-0.708000\pi\)
−0.607931 + 0.793990i \(0.708000\pi\)
\(762\) 0 0
\(763\) −9.88694e12 −1.05609
\(764\) 0 0
\(765\) 5.34539e12 0.564291
\(766\) 0 0
\(767\) −4.26135e11 −0.0444598
\(768\) 0 0
\(769\) 5.02943e12 0.518621 0.259311 0.965794i \(-0.416505\pi\)
0.259311 + 0.965794i \(0.416505\pi\)
\(770\) 0 0
\(771\) 2.71819e12 0.277036
\(772\) 0 0
\(773\) 1.22620e13 1.23524 0.617621 0.786476i \(-0.288097\pi\)
0.617621 + 0.786476i \(0.288097\pi\)
\(774\) 0 0
\(775\) 9.86918e11 0.0982705
\(776\) 0 0
\(777\) 2.05732e13 2.02491
\(778\) 0 0
\(779\) −2.06356e13 −2.00770
\(780\) 0 0
\(781\) −1.99784e11 −0.0192146
\(782\) 0 0
\(783\) 6.13371e11 0.0583170
\(784\) 0 0
\(785\) 2.62960e11 0.0247159
\(786\) 0 0
\(787\) 1.13978e13 1.05909 0.529547 0.848281i \(-0.322362\pi\)
0.529547 + 0.848281i \(0.322362\pi\)
\(788\) 0 0
\(789\) −6.88314e12 −0.632325
\(790\) 0 0
\(791\) −8.18005e12 −0.742954
\(792\) 0 0
\(793\) 1.62808e12 0.146200
\(794\) 0 0
\(795\) 2.72041e13 2.41536
\(796\) 0 0
\(797\) 9.66670e12 0.848625 0.424313 0.905516i \(-0.360516\pi\)
0.424313 + 0.905516i \(0.360516\pi\)
\(798\) 0 0
\(799\) 1.40816e13 1.22234
\(800\) 0 0
\(801\) 1.53055e13 1.31372
\(802\) 0 0
\(803\) 1.19554e12 0.101472
\(804\) 0 0
\(805\) 1.53397e12 0.128747
\(806\) 0 0
\(807\) 1.37164e13 1.13843
\(808\) 0 0
\(809\) 4.89988e12 0.402177 0.201088 0.979573i \(-0.435552\pi\)
0.201088 + 0.979573i \(0.435552\pi\)
\(810\) 0 0
\(811\) 9.97393e12 0.809604 0.404802 0.914404i \(-0.367340\pi\)
0.404802 + 0.914404i \(0.367340\pi\)
\(812\) 0 0
\(813\) −1.30434e12 −0.104709
\(814\) 0 0
\(815\) 8.28564e12 0.657835
\(816\) 0 0
\(817\) −1.07063e13 −0.840697
\(818\) 0 0
\(819\) −2.85101e12 −0.221422
\(820\) 0 0
\(821\) −6.20376e12 −0.476552 −0.238276 0.971197i \(-0.576582\pi\)
−0.238276 + 0.971197i \(0.576582\pi\)
\(822\) 0 0
\(823\) −2.05255e13 −1.55953 −0.779765 0.626072i \(-0.784661\pi\)
−0.779765 + 0.626072i \(0.784661\pi\)
\(824\) 0 0
\(825\) 2.04699e11 0.0153841
\(826\) 0 0
\(827\) 1.52447e13 1.13330 0.566649 0.823959i \(-0.308239\pi\)
0.566649 + 0.823959i \(0.308239\pi\)
\(828\) 0 0
\(829\) 1.59195e12 0.117067 0.0585334 0.998285i \(-0.481358\pi\)
0.0585334 + 0.998285i \(0.481358\pi\)
\(830\) 0 0
\(831\) 1.33076e13 0.968042
\(832\) 0 0
\(833\) 1.56689e12 0.112755
\(834\) 0 0
\(835\) 1.98334e13 1.41192
\(836\) 0 0
\(837\) −2.00021e12 −0.140867
\(838\) 0 0
\(839\) 5.56668e11 0.0387853 0.0193927 0.999812i \(-0.493827\pi\)
0.0193927 + 0.999812i \(0.493827\pi\)
\(840\) 0 0
\(841\) −1.28768e13 −0.887621
\(842\) 0 0
\(843\) −5.95908e12 −0.406401
\(844\) 0 0
\(845\) −1.06861e12 −0.0721046
\(846\) 0 0
\(847\) −1.35821e13 −0.906760
\(848\) 0 0
\(849\) 2.59800e13 1.71614
\(850\) 0 0
\(851\) 3.71701e12 0.242946
\(852\) 0 0
\(853\) −1.76959e13 −1.14446 −0.572231 0.820093i \(-0.693922\pi\)
−0.572231 + 0.820093i \(0.693922\pi\)
\(854\) 0 0
\(855\) −2.05493e13 −1.31507
\(856\) 0 0
\(857\) 1.34064e13 0.848983 0.424492 0.905432i \(-0.360453\pi\)
0.424492 + 0.905432i \(0.360453\pi\)
\(858\) 0 0
\(859\) −2.16215e13 −1.35493 −0.677466 0.735554i \(-0.736922\pi\)
−0.677466 + 0.735554i \(0.736922\pi\)
\(860\) 0 0
\(861\) 2.52126e13 1.56352
\(862\) 0 0
\(863\) 1.09637e13 0.672838 0.336419 0.941712i \(-0.390784\pi\)
0.336419 + 0.941712i \(0.390784\pi\)
\(864\) 0 0
\(865\) 2.14163e12 0.130068
\(866\) 0 0
\(867\) 1.19391e13 0.717603
\(868\) 0 0
\(869\) −2.14445e12 −0.127563
\(870\) 0 0
\(871\) 6.30456e11 0.0371170
\(872\) 0 0
\(873\) −1.37925e13 −0.803673
\(874\) 0 0
\(875\) 1.66695e13 0.961358
\(876\) 0 0
\(877\) 1.20955e13 0.690440 0.345220 0.938522i \(-0.387804\pi\)
0.345220 + 0.938522i \(0.387804\pi\)
\(878\) 0 0
\(879\) 1.44975e13 0.819112
\(880\) 0 0
\(881\) −3.33493e13 −1.86507 −0.932534 0.361083i \(-0.882407\pi\)
−0.932534 + 0.361083i \(0.882407\pi\)
\(882\) 0 0
\(883\) 1.01455e13 0.561628 0.280814 0.959762i \(-0.409396\pi\)
0.280814 + 0.959762i \(0.409396\pi\)
\(884\) 0 0
\(885\) 3.75272e12 0.205637
\(886\) 0 0
\(887\) −3.04816e13 −1.65341 −0.826707 0.562633i \(-0.809789\pi\)
−0.826707 + 0.562633i \(0.809789\pi\)
\(888\) 0 0
\(889\) 1.92433e13 1.03329
\(890\) 0 0
\(891\) −1.93597e12 −0.102908
\(892\) 0 0
\(893\) −5.41340e13 −2.84865
\(894\) 0 0
\(895\) −5.41003e12 −0.281836
\(896\) 0 0
\(897\) −1.10521e12 −0.0570005
\(898\) 0 0
\(899\) −5.31644e12 −0.271458
\(900\) 0 0
\(901\) −2.56875e13 −1.29855
\(902\) 0 0
\(903\) 1.30809e13 0.654702
\(904\) 0 0
\(905\) 2.79150e13 1.38331
\(906\) 0 0
\(907\) 2.08823e12 0.102458 0.0512289 0.998687i \(-0.483686\pi\)
0.0512289 + 0.998687i \(0.483686\pi\)
\(908\) 0 0
\(909\) 2.06169e13 1.00158
\(910\) 0 0
\(911\) 1.70747e13 0.821336 0.410668 0.911785i \(-0.365296\pi\)
0.410668 + 0.911785i \(0.365296\pi\)
\(912\) 0 0
\(913\) 2.27291e12 0.108259
\(914\) 0 0
\(915\) −1.43376e13 −0.676209
\(916\) 0 0
\(917\) −1.78012e13 −0.831358
\(918\) 0 0
\(919\) 3.86177e12 0.178594 0.0892970 0.996005i \(-0.471538\pi\)
0.0892970 + 0.996005i \(0.471538\pi\)
\(920\) 0 0
\(921\) 2.74099e12 0.125528
\(922\) 0 0
\(923\) 1.26857e12 0.0575318
\(924\) 0 0
\(925\) 4.37137e12 0.196327
\(926\) 0 0
\(927\) 1.64599e13 0.732095
\(928\) 0 0
\(929\) 2.72392e13 1.19984 0.599921 0.800059i \(-0.295199\pi\)
0.599921 + 0.800059i \(0.295199\pi\)
\(930\) 0 0
\(931\) −6.02362e12 −0.262775
\(932\) 0 0
\(933\) 6.88179e11 0.0297327
\(934\) 0 0
\(935\) 1.39943e12 0.0598823
\(936\) 0 0
\(937\) −1.33830e13 −0.567184 −0.283592 0.958945i \(-0.591526\pi\)
−0.283592 + 0.958945i \(0.591526\pi\)
\(938\) 0 0
\(939\) −5.37274e13 −2.25528
\(940\) 0 0
\(941\) 5.78614e12 0.240567 0.120283 0.992740i \(-0.461620\pi\)
0.120283 + 0.992740i \(0.461620\pi\)
\(942\) 0 0
\(943\) 4.55523e12 0.187589
\(944\) 0 0
\(945\) −3.65625e12 −0.149140
\(946\) 0 0
\(947\) 4.03191e13 1.62905 0.814527 0.580125i \(-0.196996\pi\)
0.814527 + 0.580125i \(0.196996\pi\)
\(948\) 0 0
\(949\) −7.59136e12 −0.303824
\(950\) 0 0
\(951\) −4.61084e12 −0.182797
\(952\) 0 0
\(953\) 1.46625e13 0.575824 0.287912 0.957657i \(-0.407039\pi\)
0.287912 + 0.957657i \(0.407039\pi\)
\(954\) 0 0
\(955\) −4.04319e13 −1.57293
\(956\) 0 0
\(957\) −1.10269e12 −0.0424963
\(958\) 0 0
\(959\) 3.26976e12 0.124834
\(960\) 0 0
\(961\) −9.10264e12 −0.344280
\(962\) 0 0
\(963\) 4.10713e13 1.53893
\(964\) 0 0
\(965\) 5.95942e12 0.221223
\(966\) 0 0
\(967\) −1.87662e13 −0.690172 −0.345086 0.938571i \(-0.612150\pi\)
−0.345086 + 0.938571i \(0.612150\pi\)
\(968\) 0 0
\(969\) 4.16331e13 1.51699
\(970\) 0 0
\(971\) −2.66964e13 −0.963755 −0.481877 0.876239i \(-0.660045\pi\)
−0.481877 + 0.876239i \(0.660045\pi\)
\(972\) 0 0
\(973\) 2.67607e12 0.0957171
\(974\) 0 0
\(975\) −1.29978e12 −0.0460626
\(976\) 0 0
\(977\) 1.44408e13 0.507068 0.253534 0.967327i \(-0.418407\pi\)
0.253534 + 0.967327i \(0.418407\pi\)
\(978\) 0 0
\(979\) 4.00700e12 0.139411
\(980\) 0 0
\(981\) −2.92371e13 −1.00791
\(982\) 0 0
\(983\) −3.96507e13 −1.35444 −0.677220 0.735781i \(-0.736815\pi\)
−0.677220 + 0.735781i \(0.736815\pi\)
\(984\) 0 0
\(985\) −2.96159e13 −1.00245
\(986\) 0 0
\(987\) 6.61409e13 2.21842
\(988\) 0 0
\(989\) 2.36337e12 0.0785503
\(990\) 0 0
\(991\) 4.04833e13 1.33335 0.666675 0.745348i \(-0.267717\pi\)
0.666675 + 0.745348i \(0.267717\pi\)
\(992\) 0 0
\(993\) 7.16177e13 2.33748
\(994\) 0 0
\(995\) 1.37586e13 0.445011
\(996\) 0 0
\(997\) −2.52148e13 −0.808217 −0.404109 0.914711i \(-0.632418\pi\)
−0.404109 + 0.914711i \(0.632418\pi\)
\(998\) 0 0
\(999\) −8.85956e12 −0.281428
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 208.10.a.a.1.1 1
4.3 odd 2 26.10.a.b.1.1 1
12.11 even 2 234.10.a.c.1.1 1
52.51 odd 2 338.10.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
26.10.a.b.1.1 1 4.3 odd 2
208.10.a.a.1.1 1 1.1 even 1 trivial
234.10.a.c.1.1 1 12.11 even 2
338.10.a.d.1.1 1 52.51 odd 2