Properties

Label 4140.2.f.d.829.2
Level $4140$
Weight $2$
Character 4140.829
Analytic conductor $33.058$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4140,2,Mod(829,4140)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4140, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("4140.829");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4140.f (of order \(2\), degree \(1\), 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: \(33.0580664368\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 829.2
Character \(\chi\) \(=\) 4140.829
Dual form 4140.2.f.d.829.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-2.23083 + 0.152950i) q^{5} +4.09652i q^{7} +O(q^{10})\) \(q+(-2.23083 + 0.152950i) q^{5} +4.09652i q^{7} +0.777594 q^{11} +1.67412i q^{13} -3.55545i q^{17} +4.61421 q^{19} +1.00000i q^{23} +(4.95321 - 0.682413i) q^{25} +8.24889 q^{29} +6.43498 q^{31} +(-0.626564 - 9.13864i) q^{35} -1.05292i q^{37} -5.98753 q^{41} -3.37477i q^{43} +1.24747i q^{47} -9.78146 q^{49} -6.08565i q^{53} +(-1.73468 + 0.118933i) q^{55} -2.37044 q^{59} -2.12459 q^{61} +(-0.256057 - 3.73468i) q^{65} +14.7691i q^{67} +11.4600 q^{71} +15.7443i q^{73} +3.18543i q^{77} +1.63206 q^{79} +17.0898i q^{83} +(0.543808 + 7.93162i) q^{85} +5.52443 q^{89} -6.85807 q^{91} +(-10.2935 + 0.705745i) q^{95} +8.78620i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q+O(q^{10}) \) Copy content Toggle raw display \( 24 q + 8 q^{19} - 8 q^{25} - 12 q^{31} - 28 q^{49} - 16 q^{55} - 16 q^{61} + 8 q^{79} + 12 q^{85} - 16 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4140\mathbb{Z}\right)^\times\).

\(n\) \(461\) \(1657\) \(2071\) \(3961\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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
\(5\) −2.23083 + 0.152950i −0.997658 + 0.0684015i
\(6\) 0 0
\(7\) 4.09652i 1.54834i 0.632979 + 0.774169i \(0.281832\pi\)
−0.632979 + 0.774169i \(0.718168\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0.777594 0.234453 0.117227 0.993105i \(-0.462600\pi\)
0.117227 + 0.993105i \(0.462600\pi\)
\(12\) 0 0
\(13\) 1.67412i 0.464318i 0.972678 + 0.232159i \(0.0745789\pi\)
−0.972678 + 0.232159i \(0.925421\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.55545i 0.862324i −0.902274 0.431162i \(-0.858104\pi\)
0.902274 0.431162i \(-0.141896\pi\)
\(18\) 0 0
\(19\) 4.61421 1.05857 0.529286 0.848444i \(-0.322460\pi\)
0.529286 + 0.848444i \(0.322460\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.00000i 0.208514i
\(24\) 0 0
\(25\) 4.95321 0.682413i 0.990642 0.136483i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 8.24889 1.53178 0.765890 0.642972i \(-0.222299\pi\)
0.765890 + 0.642972i \(0.222299\pi\)
\(30\) 0 0
\(31\) 6.43498 1.15576 0.577878 0.816123i \(-0.303881\pi\)
0.577878 + 0.816123i \(0.303881\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −0.626564 9.13864i −0.105909 1.54471i
\(36\) 0 0
\(37\) 1.05292i 0.173099i −0.996248 0.0865495i \(-0.972416\pi\)
0.996248 0.0865495i \(-0.0275841\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −5.98753 −0.935095 −0.467548 0.883968i \(-0.654862\pi\)
−0.467548 + 0.883968i \(0.654862\pi\)
\(42\) 0 0
\(43\) 3.37477i 0.514648i −0.966325 0.257324i \(-0.917159\pi\)
0.966325 0.257324i \(-0.0828408\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 1.24747i 0.181962i 0.995853 + 0.0909809i \(0.0290002\pi\)
−0.995853 + 0.0909809i \(0.971000\pi\)
\(48\) 0 0
\(49\) −9.78146 −1.39735
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 6.08565i 0.835928i −0.908463 0.417964i \(-0.862744\pi\)
0.908463 0.417964i \(-0.137256\pi\)
\(54\) 0 0
\(55\) −1.73468 + 0.118933i −0.233904 + 0.0160370i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −2.37044 −0.308605 −0.154302 0.988024i \(-0.549313\pi\)
−0.154302 + 0.988024i \(0.549313\pi\)
\(60\) 0 0
\(61\) −2.12459 −0.272026 −0.136013 0.990707i \(-0.543429\pi\)
−0.136013 + 0.990707i \(0.543429\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −0.256057 3.73468i −0.0317600 0.463230i
\(66\) 0 0
\(67\) 14.7691i 1.80433i 0.431391 + 0.902165i \(0.358023\pi\)
−0.431391 + 0.902165i \(0.641977\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 11.4600 1.36005 0.680026 0.733188i \(-0.261968\pi\)
0.680026 + 0.733188i \(0.261968\pi\)
\(72\) 0 0
\(73\) 15.7443i 1.84273i 0.388698 + 0.921365i \(0.372925\pi\)
−0.388698 + 0.921365i \(0.627075\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 3.18543i 0.363013i
\(78\) 0 0
\(79\) 1.63206 0.183621 0.0918104 0.995777i \(-0.470735\pi\)
0.0918104 + 0.995777i \(0.470735\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 17.0898i 1.87585i 0.346844 + 0.937923i \(0.387253\pi\)
−0.346844 + 0.937923i \(0.612747\pi\)
\(84\) 0 0
\(85\) 0.543808 + 7.93162i 0.0589843 + 0.860305i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 5.52443 0.585588 0.292794 0.956176i \(-0.405415\pi\)
0.292794 + 0.956176i \(0.405415\pi\)
\(90\) 0 0
\(91\) −6.85807 −0.718921
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −10.2935 + 0.705745i −1.05609 + 0.0724079i
\(96\) 0 0
\(97\) 8.78620i 0.892103i 0.895007 + 0.446052i \(0.147170\pi\)
−0.895007 + 0.446052i \(0.852830\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −9.88643 −0.983736 −0.491868 0.870670i \(-0.663686\pi\)
−0.491868 + 0.870670i \(0.663686\pi\)
\(102\) 0 0
\(103\) 3.75550i 0.370040i 0.982735 + 0.185020i \(0.0592350\pi\)
−0.982735 + 0.185020i \(0.940765\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 2.92219i 0.282499i 0.989974 + 0.141249i \(0.0451120\pi\)
−0.989974 + 0.141249i \(0.954888\pi\)
\(108\) 0 0
\(109\) −11.9690 −1.14642 −0.573210 0.819409i \(-0.694302\pi\)
−0.573210 + 0.819409i \(0.694302\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 16.0939i 1.51399i −0.653424 0.756993i \(-0.726668\pi\)
0.653424 0.756993i \(-0.273332\pi\)
\(114\) 0 0
\(115\) −0.152950 2.23083i −0.0142627 0.208026i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 14.5650 1.33517
\(120\) 0 0
\(121\) −10.3953 −0.945032
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −10.9454 + 2.27994i −0.978987 + 0.203924i
\(126\) 0 0
\(127\) 9.16468i 0.813234i −0.913599 0.406617i \(-0.866708\pi\)
0.913599 0.406617i \(-0.133292\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −0.843290 −0.0736786 −0.0368393 0.999321i \(-0.511729\pi\)
−0.0368393 + 0.999321i \(0.511729\pi\)
\(132\) 0 0
\(133\) 18.9022i 1.63903i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 2.39231i 0.204389i −0.994764 0.102195i \(-0.967414\pi\)
0.994764 0.102195i \(-0.0325864\pi\)
\(138\) 0 0
\(139\) −10.7194 −0.909209 −0.454605 0.890693i \(-0.650219\pi\)
−0.454605 + 0.890693i \(0.650219\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 1.30179i 0.108861i
\(144\) 0 0
\(145\) −18.4019 + 1.26167i −1.52819 + 0.104776i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 17.3312 1.41983 0.709913 0.704289i \(-0.248734\pi\)
0.709913 + 0.704289i \(0.248734\pi\)
\(150\) 0 0
\(151\) 20.0120 1.62855 0.814275 0.580479i \(-0.197135\pi\)
0.814275 + 0.580479i \(0.197135\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −14.3554 + 0.984233i −1.15305 + 0.0790555i
\(156\) 0 0
\(157\) 7.92120i 0.632181i −0.948729 0.316090i \(-0.897630\pi\)
0.948729 0.316090i \(-0.102370\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −4.09652 −0.322851
\(162\) 0 0
\(163\) 11.8908i 0.931363i 0.884952 + 0.465681i \(0.154191\pi\)
−0.884952 + 0.465681i \(0.845809\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 15.5289i 1.20166i 0.799375 + 0.600832i \(0.205164\pi\)
−0.799375 + 0.600832i \(0.794836\pi\)
\(168\) 0 0
\(169\) 10.1973 0.784409
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 15.9023i 1.20903i 0.796595 + 0.604513i \(0.206632\pi\)
−0.796595 + 0.604513i \(0.793368\pi\)
\(174\) 0 0
\(175\) 2.79552 + 20.2909i 0.211321 + 1.53385i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −10.1510 −0.758723 −0.379362 0.925248i \(-0.623856\pi\)
−0.379362 + 0.925248i \(0.623856\pi\)
\(180\) 0 0
\(181\) −12.4562 −0.925861 −0.462930 0.886395i \(-0.653202\pi\)
−0.462930 + 0.886395i \(0.653202\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0.161045 + 2.34889i 0.0118402 + 0.172694i
\(186\) 0 0
\(187\) 2.76470i 0.202175i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −10.3633 −0.749860 −0.374930 0.927053i \(-0.622333\pi\)
−0.374930 + 0.927053i \(0.622333\pi\)
\(192\) 0 0
\(193\) 14.6888i 1.05732i 0.848833 + 0.528660i \(0.177305\pi\)
−0.848833 + 0.528660i \(0.822695\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 19.2517i 1.37163i −0.727777 0.685814i \(-0.759447\pi\)
0.727777 0.685814i \(-0.240553\pi\)
\(198\) 0 0
\(199\) 8.57495 0.607862 0.303931 0.952694i \(-0.401701\pi\)
0.303931 + 0.952694i \(0.401701\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 33.7917i 2.37171i
\(204\) 0 0
\(205\) 13.3572 0.915796i 0.932905 0.0639619i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 3.58798 0.248186
\(210\) 0 0
\(211\) −16.0719 −1.10644 −0.553218 0.833036i \(-0.686600\pi\)
−0.553218 + 0.833036i \(0.686600\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0.516173 + 7.52855i 0.0352027 + 0.513443i
\(216\) 0 0
\(217\) 26.3610i 1.78950i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 5.95226 0.400392
\(222\) 0 0
\(223\) 15.4350i 1.03360i −0.856105 0.516802i \(-0.827122\pi\)
0.856105 0.516802i \(-0.172878\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 19.5993i 1.30085i −0.759571 0.650424i \(-0.774591\pi\)
0.759571 0.650424i \(-0.225409\pi\)
\(228\) 0 0
\(229\) 17.7275 1.17147 0.585734 0.810504i \(-0.300806\pi\)
0.585734 + 0.810504i \(0.300806\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 23.4197i 1.53428i 0.641482 + 0.767138i \(0.278320\pi\)
−0.641482 + 0.767138i \(0.721680\pi\)
\(234\) 0 0
\(235\) −0.190801 2.78289i −0.0124465 0.181536i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 7.79898 0.504474 0.252237 0.967666i \(-0.418834\pi\)
0.252237 + 0.967666i \(0.418834\pi\)
\(240\) 0 0
\(241\) −16.1742 −1.04187 −0.520936 0.853595i \(-0.674417\pi\)
−0.520936 + 0.853595i \(0.674417\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 21.8208 1.49608i 1.39408 0.0955810i
\(246\) 0 0
\(247\) 7.72474i 0.491513i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 7.46275 0.471045 0.235522 0.971869i \(-0.424320\pi\)
0.235522 + 0.971869i \(0.424320\pi\)
\(252\) 0 0
\(253\) 0.777594i 0.0488869i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 11.8277i 0.737794i 0.929470 + 0.368897i \(0.120264\pi\)
−0.929470 + 0.368897i \(0.879736\pi\)
\(258\) 0 0
\(259\) 4.31330 0.268016
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0.466006i 0.0287352i 0.999897 + 0.0143676i \(0.00457350\pi\)
−0.999897 + 0.0143676i \(0.995426\pi\)
\(264\) 0 0
\(265\) 0.930803 + 13.5761i 0.0571788 + 0.833971i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 11.5971 0.707089 0.353545 0.935418i \(-0.384976\pi\)
0.353545 + 0.935418i \(0.384976\pi\)
\(270\) 0 0
\(271\) 16.0761 0.976553 0.488276 0.872689i \(-0.337626\pi\)
0.488276 + 0.872689i \(0.337626\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 3.85159 0.530640i 0.232259 0.0319988i
\(276\) 0 0
\(277\) 17.6670i 1.06151i 0.847526 + 0.530754i \(0.178091\pi\)
−0.847526 + 0.530754i \(0.821909\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −18.9009 −1.12753 −0.563767 0.825934i \(-0.690648\pi\)
−0.563767 + 0.825934i \(0.690648\pi\)
\(282\) 0 0
\(283\) 0.0766236i 0.00455480i −0.999997 0.00227740i \(-0.999275\pi\)
0.999997 0.00227740i \(-0.000724919\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 24.5280i 1.44784i
\(288\) 0 0
\(289\) 4.35875 0.256397
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 1.75829i 0.102720i −0.998680 0.0513602i \(-0.983644\pi\)
0.998680 0.0513602i \(-0.0163557\pi\)
\(294\) 0 0
\(295\) 5.28805 0.362560i 0.307882 0.0211090i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −1.67412 −0.0968169
\(300\) 0 0
\(301\) 13.8248 0.796849
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 4.73960 0.324957i 0.271389 0.0186070i
\(306\) 0 0
\(307\) 2.55583i 0.145869i −0.997337 0.0729345i \(-0.976764\pi\)
0.997337 0.0729345i \(-0.0232364\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −20.4792 −1.16127 −0.580636 0.814163i \(-0.697196\pi\)
−0.580636 + 0.814163i \(0.697196\pi\)
\(312\) 0 0
\(313\) 2.11052i 0.119294i −0.998220 0.0596468i \(-0.981003\pi\)
0.998220 0.0596468i \(-0.0189974\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 25.3095i 1.42152i −0.703433 0.710761i \(-0.748351\pi\)
0.703433 0.710761i \(-0.251649\pi\)
\(318\) 0 0
\(319\) 6.41428 0.359131
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 16.4056i 0.912832i
\(324\) 0 0
\(325\) 1.14244 + 8.29228i 0.0633713 + 0.459973i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −5.11027 −0.281739
\(330\) 0 0
\(331\) 16.2710 0.894336 0.447168 0.894450i \(-0.352433\pi\)
0.447168 + 0.894450i \(0.352433\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −2.25894 32.9473i −0.123419 1.80010i
\(336\) 0 0
\(337\) 35.1877i 1.91680i 0.285434 + 0.958398i \(0.407862\pi\)
−0.285434 + 0.958398i \(0.592138\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 5.00380 0.270971
\(342\) 0 0
\(343\) 11.3943i 0.615235i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 1.66067i 0.0891493i 0.999006 + 0.0445747i \(0.0141933\pi\)
−0.999006 + 0.0445747i \(0.985807\pi\)
\(348\) 0 0
\(349\) 5.43378 0.290863 0.145432 0.989368i \(-0.453543\pi\)
0.145432 + 0.989368i \(0.453543\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 32.4056i 1.72478i 0.506248 + 0.862388i \(0.331032\pi\)
−0.506248 + 0.862388i \(0.668968\pi\)
\(354\) 0 0
\(355\) −25.5653 + 1.75281i −1.35687 + 0.0930296i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −26.9119 −1.42036 −0.710178 0.704023i \(-0.751385\pi\)
−0.710178 + 0.704023i \(0.751385\pi\)
\(360\) 0 0
\(361\) 2.29090 0.120573
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −2.40810 35.1229i −0.126046 1.83841i
\(366\) 0 0
\(367\) 0.0525023i 0.00274060i 0.999999 + 0.00137030i \(0.000436179\pi\)
−0.999999 + 0.00137030i \(0.999564\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 24.9300 1.29430
\(372\) 0 0
\(373\) 0.561607i 0.0290789i −0.999894 0.0145395i \(-0.995372\pi\)
0.999894 0.0145395i \(-0.00462822\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 13.8096i 0.711232i
\(378\) 0 0
\(379\) −20.3025 −1.04287 −0.521434 0.853292i \(-0.674603\pi\)
−0.521434 + 0.853292i \(0.674603\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 22.7628i 1.16313i 0.813501 + 0.581564i \(0.197559\pi\)
−0.813501 + 0.581564i \(0.802441\pi\)
\(384\) 0 0
\(385\) −0.487212 7.10615i −0.0248306 0.362163i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −31.0468 −1.57414 −0.787068 0.616866i \(-0.788402\pi\)
−0.787068 + 0.616866i \(0.788402\pi\)
\(390\) 0 0
\(391\) 3.55545 0.179807
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −3.64085 + 0.249624i −0.183191 + 0.0125599i
\(396\) 0 0
\(397\) 0.111698i 0.00560595i −0.999996 0.00280297i \(-0.999108\pi\)
0.999996 0.00280297i \(-0.000892215\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −18.8391 −0.940778 −0.470389 0.882459i \(-0.655886\pi\)
−0.470389 + 0.882459i \(0.655886\pi\)
\(402\) 0 0
\(403\) 10.7729i 0.536638i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0.818744i 0.0405836i
\(408\) 0 0
\(409\) 26.8758 1.32892 0.664461 0.747323i \(-0.268661\pi\)
0.664461 + 0.747323i \(0.268661\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 9.71055i 0.477825i
\(414\) 0 0
\(415\) −2.61389 38.1244i −0.128311 1.87145i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 33.8796 1.65513 0.827564 0.561372i \(-0.189726\pi\)
0.827564 + 0.561372i \(0.189726\pi\)
\(420\) 0 0
\(421\) 25.0252 1.21965 0.609827 0.792535i \(-0.291239\pi\)
0.609827 + 0.792535i \(0.291239\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −2.42629 17.6109i −0.117692 0.854255i
\(426\) 0 0
\(427\) 8.70342i 0.421188i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 18.3854 0.885595 0.442798 0.896622i \(-0.353986\pi\)
0.442798 + 0.896622i \(0.353986\pi\)
\(432\) 0 0
\(433\) 26.6774i 1.28203i 0.767527 + 0.641016i \(0.221487\pi\)
−0.767527 + 0.641016i \(0.778513\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 4.61421i 0.220727i
\(438\) 0 0
\(439\) 18.3341 0.875037 0.437519 0.899209i \(-0.355857\pi\)
0.437519 + 0.899209i \(0.355857\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 26.7077i 1.26892i 0.772955 + 0.634461i \(0.218778\pi\)
−0.772955 + 0.634461i \(0.781222\pi\)
\(444\) 0 0
\(445\) −12.3241 + 0.844964i −0.584217 + 0.0400551i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 4.44716 0.209875 0.104937 0.994479i \(-0.466536\pi\)
0.104937 + 0.994479i \(0.466536\pi\)
\(450\) 0 0
\(451\) −4.65587 −0.219236
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 15.2992 1.04894i 0.717237 0.0491753i
\(456\) 0 0
\(457\) 16.6048i 0.776739i −0.921504 0.388370i \(-0.873038\pi\)
0.921504 0.388370i \(-0.126962\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 9.58567 0.446449 0.223225 0.974767i \(-0.428342\pi\)
0.223225 + 0.974767i \(0.428342\pi\)
\(462\) 0 0
\(463\) 5.87507i 0.273038i 0.990637 + 0.136519i \(0.0435914\pi\)
−0.990637 + 0.136519i \(0.956409\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 37.0199i 1.71308i −0.516082 0.856539i \(-0.672610\pi\)
0.516082 0.856539i \(-0.327390\pi\)
\(468\) 0 0
\(469\) −60.5018 −2.79371
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 2.62420i 0.120661i
\(474\) 0 0
\(475\) 22.8551 3.14879i 1.04867 0.144477i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −23.1392 −1.05726 −0.528628 0.848854i \(-0.677293\pi\)
−0.528628 + 0.848854i \(0.677293\pi\)
\(480\) 0 0
\(481\) 1.76271 0.0803729
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −1.34385 19.6005i −0.0610212 0.890014i
\(486\) 0 0
\(487\) 3.00217i 0.136041i 0.997684 + 0.0680206i \(0.0216684\pi\)
−0.997684 + 0.0680206i \(0.978332\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 23.4243 1.05713 0.528563 0.848894i \(-0.322731\pi\)
0.528563 + 0.848894i \(0.322731\pi\)
\(492\) 0 0
\(493\) 29.3285i 1.32089i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 46.9461i 2.10582i
\(498\) 0 0
\(499\) −27.5583 −1.23368 −0.616839 0.787089i \(-0.711587\pi\)
−0.616839 + 0.787089i \(0.711587\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 14.5757i 0.649898i −0.945732 0.324949i \(-0.894653\pi\)
0.945732 0.324949i \(-0.105347\pi\)
\(504\) 0 0
\(505\) 22.0549 1.51213i 0.981432 0.0672890i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 5.03925 0.223361 0.111680 0.993744i \(-0.464377\pi\)
0.111680 + 0.993744i \(0.464377\pi\)
\(510\) 0 0
\(511\) −64.4968 −2.85317
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −0.574405 8.37788i −0.0253113 0.369173i
\(516\) 0 0
\(517\) 0.970023i 0.0426616i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 4.06479 0.178082 0.0890409 0.996028i \(-0.471620\pi\)
0.0890409 + 0.996028i \(0.471620\pi\)
\(522\) 0 0
\(523\) 17.3203i 0.757362i −0.925527 0.378681i \(-0.876378\pi\)
0.925527 0.378681i \(-0.123622\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 22.8793i 0.996637i
\(528\) 0 0
\(529\) −1.00000 −0.0434783
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 10.0239i 0.434181i
\(534\) 0 0
\(535\) −0.446950 6.51892i −0.0193234 0.281837i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −7.60600 −0.327614
\(540\) 0 0
\(541\) 12.2414 0.526298 0.263149 0.964755i \(-0.415239\pi\)
0.263149 + 0.964755i \(0.415239\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 26.7008 1.83066i 1.14373 0.0784168i
\(546\) 0 0
\(547\) 40.5381i 1.73328i −0.498931 0.866642i \(-0.666274\pi\)
0.498931 0.866642i \(-0.333726\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 38.0621 1.62150
\(552\) 0 0
\(553\) 6.68576i 0.284307i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 23.7396i 1.00588i −0.864322 0.502940i \(-0.832252\pi\)
0.864322 0.502940i \(-0.167748\pi\)
\(558\) 0 0
\(559\) 5.64978 0.238960
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 10.4297i 0.439558i 0.975550 + 0.219779i \(0.0705336\pi\)
−0.975550 + 0.219779i \(0.929466\pi\)
\(564\) 0 0
\(565\) 2.46157 + 35.9027i 0.103559 + 1.51044i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −8.80513 −0.369130 −0.184565 0.982820i \(-0.559088\pi\)
−0.184565 + 0.982820i \(0.559088\pi\)
\(570\) 0 0
\(571\) −37.1688 −1.55546 −0.777732 0.628596i \(-0.783630\pi\)
−0.777732 + 0.628596i \(0.783630\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0.682413 + 4.95321i 0.0284586 + 0.206563i
\(576\) 0 0
\(577\) 1.71439i 0.0713710i 0.999363 + 0.0356855i \(0.0113615\pi\)
−0.999363 + 0.0356855i \(0.988639\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −70.0085 −2.90444
\(582\) 0 0
\(583\) 4.73216i 0.195986i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 22.4610i 0.927064i −0.886080 0.463532i \(-0.846582\pi\)
0.886080 0.463532i \(-0.153418\pi\)
\(588\) 0 0
\(589\) 29.6923 1.22345
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 8.76392i 0.359891i 0.983677 + 0.179945i \(0.0575921\pi\)
−0.983677 + 0.179945i \(0.942408\pi\)
\(594\) 0 0
\(595\) −32.4920 + 2.22772i −1.33204 + 0.0913276i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −41.7472 −1.70575 −0.852873 0.522119i \(-0.825142\pi\)
−0.852873 + 0.522119i \(0.825142\pi\)
\(600\) 0 0
\(601\) −7.55678 −0.308247 −0.154124 0.988052i \(-0.549255\pi\)
−0.154124 + 0.988052i \(0.549255\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 23.1903 1.58997i 0.942818 0.0646416i
\(606\) 0 0
\(607\) 33.5791i 1.36293i −0.731849 0.681466i \(-0.761343\pi\)
0.731849 0.681466i \(-0.238657\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −2.08841 −0.0844881
\(612\) 0 0
\(613\) 8.48045i 0.342522i −0.985226 0.171261i \(-0.945216\pi\)
0.985226 0.171261i \(-0.0547842\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 20.4302i 0.822490i 0.911525 + 0.411245i \(0.134906\pi\)
−0.911525 + 0.411245i \(0.865094\pi\)
\(618\) 0 0
\(619\) −20.9158 −0.840677 −0.420338 0.907367i \(-0.638089\pi\)
−0.420338 + 0.907367i \(0.638089\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 22.6309i 0.906689i
\(624\) 0 0
\(625\) 24.0686 6.76027i 0.962745 0.270411i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −3.74361 −0.149267
\(630\) 0 0
\(631\) 12.8205 0.510375 0.255187 0.966892i \(-0.417863\pi\)
0.255187 + 0.966892i \(0.417863\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 1.40174 + 20.4449i 0.0556264 + 0.811329i
\(636\) 0 0
\(637\) 16.3753i 0.648815i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 8.35480 0.329995 0.164997 0.986294i \(-0.447238\pi\)
0.164997 + 0.986294i \(0.447238\pi\)
\(642\) 0 0
\(643\) 28.6155i 1.12849i −0.825609 0.564243i \(-0.809168\pi\)
0.825609 0.564243i \(-0.190832\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 47.9816i 1.88635i 0.332297 + 0.943175i \(0.392176\pi\)
−0.332297 + 0.943175i \(0.607824\pi\)
\(648\) 0 0
\(649\) −1.84324 −0.0723534
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 47.3883i 1.85445i 0.374510 + 0.927223i \(0.377811\pi\)
−0.374510 + 0.927223i \(0.622189\pi\)
\(654\) 0 0
\(655\) 1.88124 0.128982i 0.0735060 0.00503973i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −21.2054 −0.826046 −0.413023 0.910721i \(-0.635527\pi\)
−0.413023 + 0.910721i \(0.635527\pi\)
\(660\) 0 0
\(661\) 8.00823 0.311484 0.155742 0.987798i \(-0.450223\pi\)
0.155742 + 0.987798i \(0.450223\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −2.89110 42.1676i −0.112112 1.63519i
\(666\) 0 0
\(667\) 8.24889i 0.319398i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −1.65207 −0.0637773
\(672\) 0 0
\(673\) 21.5201i 0.829540i −0.909926 0.414770i \(-0.863862\pi\)
0.909926 0.414770i \(-0.136138\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 2.53930i 0.0975932i 0.998809 + 0.0487966i \(0.0155386\pi\)
−0.998809 + 0.0487966i \(0.984461\pi\)
\(678\) 0 0
\(679\) −35.9928 −1.38128
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 21.5117i 0.823123i 0.911382 + 0.411561i \(0.135016\pi\)
−0.911382 + 0.411561i \(0.864984\pi\)
\(684\) 0 0
\(685\) 0.365905 + 5.33685i 0.0139805 + 0.203910i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 10.1881 0.388136
\(690\) 0 0
\(691\) 16.2563 0.618420 0.309210 0.950994i \(-0.399935\pi\)
0.309210 + 0.950994i \(0.399935\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 23.9132 1.63954i 0.907080 0.0621913i
\(696\) 0 0
\(697\) 21.2884i 0.806355i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 26.7080 1.00875 0.504373 0.863486i \(-0.331724\pi\)
0.504373 + 0.863486i \(0.331724\pi\)
\(702\) 0 0
\(703\) 4.85839i 0.183238i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 40.4999i 1.52316i
\(708\) 0 0
\(709\) −14.7454 −0.553774 −0.276887 0.960902i \(-0.589303\pi\)
−0.276887 + 0.960902i \(0.589303\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 6.43498i 0.240992i
\(714\) 0 0
\(715\) −0.199109 2.90406i −0.00744624 0.108606i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −22.8495 −0.852143 −0.426071 0.904689i \(-0.640103\pi\)
−0.426071 + 0.904689i \(0.640103\pi\)
\(720\) 0 0
\(721\) −15.3845 −0.572947
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 40.8585 5.62915i 1.51745 0.209061i
\(726\) 0 0
\(727\) 16.4360i 0.609578i −0.952420 0.304789i \(-0.901414\pi\)
0.952420 0.304789i \(-0.0985860\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −11.9989 −0.443794
\(732\) 0 0
\(733\) 30.6694i 1.13280i 0.824130 + 0.566400i \(0.191664\pi\)
−0.824130 + 0.566400i \(0.808336\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 11.4843i 0.423031i
\(738\) 0 0
\(739\) 6.64541 0.244455 0.122228 0.992502i \(-0.460996\pi\)
0.122228 + 0.992502i \(0.460996\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 1.54318i 0.0566137i 0.999599 + 0.0283068i \(0.00901155\pi\)
−0.999599 + 0.0283068i \(0.990988\pi\)
\(744\) 0 0
\(745\) −38.6629 + 2.65081i −1.41650 + 0.0971182i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −11.9708 −0.437404
\(750\) 0 0
\(751\) 50.2481 1.83358 0.916790 0.399369i \(-0.130771\pi\)
0.916790 + 0.399369i \(0.130771\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −44.6433 + 3.06084i −1.62474 + 0.111395i
\(756\) 0 0
\(757\) 35.2879i 1.28256i −0.767307 0.641280i \(-0.778404\pi\)
0.767307 0.641280i \(-0.221596\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 9.39667 0.340629 0.170315 0.985390i \(-0.445522\pi\)
0.170315 + 0.985390i \(0.445522\pi\)
\(762\) 0 0
\(763\) 49.0311i 1.77505i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 3.96840i 0.143291i
\(768\) 0 0
\(769\) −13.6412 −0.491913 −0.245957 0.969281i \(-0.579102\pi\)
−0.245957 + 0.969281i \(0.579102\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 39.3829i 1.41650i 0.705960 + 0.708252i \(0.250516\pi\)
−0.705960 + 0.708252i \(0.749484\pi\)
\(774\) 0 0
\(775\) 31.8738 4.39131i 1.14494 0.157741i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −27.6277 −0.989865
\(780\) 0 0
\(781\) 8.91123 0.318869
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 1.21155 + 17.6709i 0.0432421 + 0.630700i
\(786\) 0 0
\(787\) 21.3591i 0.761369i −0.924705 0.380685i \(-0.875688\pi\)
0.924705 0.380685i \(-0.124312\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 65.9289 2.34416
\(792\) 0 0
\(793\) 3.55682i 0.126306i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 3.83989i 0.136016i 0.997685 + 0.0680080i \(0.0216643\pi\)
−0.997685 + 0.0680080i \(0.978336\pi\)
\(798\) 0 0
\(799\) 4.43531 0.156910
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 12.2427i 0.432034i
\(804\) 0 0
\(805\) 9.13864 0.626564i 0.322095 0.0220835i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −24.8509 −0.873710 −0.436855 0.899532i \(-0.643908\pi\)
−0.436855 + 0.899532i \(0.643908\pi\)
\(810\) 0 0
\(811\) 27.9097 0.980043 0.490021 0.871710i \(-0.336989\pi\)
0.490021 + 0.871710i \(0.336989\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −1.81871 26.5265i −0.0637066 0.929181i
\(816\) 0 0
\(817\) 15.5719i 0.544792i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −40.2831 −1.40589 −0.702945 0.711245i \(-0.748132\pi\)
−0.702945 + 0.711245i \(0.748132\pi\)
\(822\) 0 0
\(823\) 4.67595i 0.162993i 0.996674 + 0.0814967i \(0.0259700\pi\)
−0.996674 + 0.0814967i \(0.974030\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 13.4452i 0.467537i −0.972292 0.233768i \(-0.924894\pi\)
0.972292 0.233768i \(-0.0751057\pi\)
\(828\) 0 0
\(829\) 40.5096 1.40695 0.703477 0.710718i \(-0.251630\pi\)
0.703477 + 0.710718i \(0.251630\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 34.7775i 1.20497i
\(834\) 0 0
\(835\) −2.37515 34.6424i −0.0821956 1.19885i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −35.7585 −1.23452 −0.617260 0.786759i \(-0.711757\pi\)
−0.617260 + 0.786759i \(0.711757\pi\)
\(840\) 0 0
\(841\) 39.0441 1.34635
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −22.7485 + 1.55968i −0.782572 + 0.0536548i
\(846\) 0 0
\(847\) 42.5847i 1.46323i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 1.05292 0.0360936
\(852\) 0 0
\(853\) 5.42183i 0.185640i −0.995683 0.0928200i \(-0.970412\pi\)
0.995683 0.0928200i \(-0.0295881\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 40.5420i 1.38489i 0.721471 + 0.692444i \(0.243466\pi\)
−0.721471 + 0.692444i \(0.756534\pi\)
\(858\) 0 0
\(859\) 27.0671 0.923517 0.461758 0.887006i \(-0.347219\pi\)
0.461758 + 0.887006i \(0.347219\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 21.6380i 0.736566i −0.929714 0.368283i \(-0.879946\pi\)
0.929714 0.368283i \(-0.120054\pi\)
\(864\) 0 0
\(865\) −2.43226 35.4752i −0.0826992 1.20619i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 1.26908 0.0430505
\(870\) 0 0
\(871\) −24.7252 −0.837782
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −9.33983 44.8380i −0.315744 1.51580i
\(876\) 0 0
\(877\) 27.5945i 0.931799i 0.884838 + 0.465899i \(0.154269\pi\)
−0.884838 + 0.465899i \(0.845731\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 31.8296 1.07237 0.536183 0.844102i \(-0.319866\pi\)
0.536183 + 0.844102i \(0.319866\pi\)
\(882\) 0 0
\(883\) 29.9009i 1.00625i −0.864215 0.503123i \(-0.832184\pi\)
0.864215 0.503123i \(-0.167816\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 33.1746i 1.11389i −0.830548 0.556947i \(-0.811973\pi\)
0.830548 0.556947i \(-0.188027\pi\)
\(888\) 0 0
\(889\) 37.5433 1.25916
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 5.75607i 0.192620i
\(894\) 0 0
\(895\) 22.6452 1.55260i 0.756946 0.0518978i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 53.0814 1.77036
\(900\) 0 0
\(901\) −21.6373 −0.720841
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 27.7876 1.90518i 0.923692 0.0633303i
\(906\) 0 0
\(907\) 45.3211i 1.50486i −0.658671 0.752431i \(-0.728881\pi\)
0.658671 0.752431i \(-0.271119\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 23.8181 0.789128 0.394564 0.918869i \(-0.370896\pi\)
0.394564 + 0.918869i \(0.370896\pi\)
\(912\) 0 0
\(913\) 13.2889i 0.439798i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 3.45455i 0.114079i
\(918\) 0 0
\(919\) 47.6196 1.57083 0.785413 0.618972i \(-0.212450\pi\)
0.785413 + 0.618972i \(0.212450\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 19.1854i 0.631496i
\(924\) 0 0
\(925\) −0.718526 5.21534i −0.0236250 0.171479i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 10.1069 0.331595 0.165798 0.986160i \(-0.446980\pi\)
0.165798 + 0.986160i \(0.446980\pi\)
\(930\) 0 0
\(931\) −45.1337 −1.47920
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0.422862 + 6.16757i 0.0138291 + 0.201701i
\(936\) 0 0
\(937\) 26.1597i 0.854601i 0.904110 + 0.427300i \(0.140535\pi\)
−0.904110 + 0.427300i \(0.859465\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −12.5128 −0.407905 −0.203952 0.978981i \(-0.565379\pi\)
−0.203952 + 0.978981i \(0.565379\pi\)
\(942\) 0 0
\(943\) 5.98753i 0.194981i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 1.43126i 0.0465097i −0.999730 0.0232549i \(-0.992597\pi\)
0.999730 0.0232549i \(-0.00740292\pi\)
\(948\) 0 0
\(949\) −26.3579 −0.855612
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 49.3959i 1.60009i 0.599940 + 0.800045i \(0.295191\pi\)
−0.599940 + 0.800045i \(0.704809\pi\)
\(954\) 0 0
\(955\) 23.1187 1.58507i 0.748103 0.0512915i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 9.80016 0.316464
\(960\) 0 0
\(961\) 10.4090 0.335773
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −2.24665 32.7682i −0.0723223 1.05484i
\(966\) 0 0
\(967\) 15.4504i 0.496852i 0.968651 + 0.248426i \(0.0799132\pi\)
−0.968651 + 0.248426i \(0.920087\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −26.3962 −0.847095 −0.423548 0.905874i \(-0.639215\pi\)
−0.423548 + 0.905874i \(0.639215\pi\)
\(972\) 0 0
\(973\) 43.9123i 1.40776i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 52.0950i 1.66667i −0.552770 0.833334i \(-0.686429\pi\)
0.552770 0.833334i \(-0.313571\pi\)
\(978\) 0 0
\(979\) 4.29576 0.137293
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 14.7660i 0.470962i 0.971879 + 0.235481i \(0.0756666\pi\)
−0.971879 + 0.235481i \(0.924333\pi\)
\(984\) 0 0
\(985\) 2.94456 + 42.9473i 0.0938214 + 1.36841i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 3.37477 0.107312
\(990\) 0 0
\(991\) 46.5090 1.47741 0.738703 0.674031i \(-0.235439\pi\)
0.738703 + 0.674031i \(0.235439\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −19.1293 + 1.31154i −0.606438 + 0.0415787i
\(996\) 0 0
\(997\) 53.1892i 1.68452i −0.539071 0.842260i \(-0.681225\pi\)
0.539071 0.842260i \(-0.318775\pi\)
\(998\) 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 4140.2.f.d.829.2 yes 24
3.2 odd 2 inner 4140.2.f.d.829.23 yes 24
5.4 even 2 inner 4140.2.f.d.829.1 24
15.14 odd 2 inner 4140.2.f.d.829.24 yes 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4140.2.f.d.829.1 24 5.4 even 2 inner
4140.2.f.d.829.2 yes 24 1.1 even 1 trivial
4140.2.f.d.829.23 yes 24 3.2 odd 2 inner
4140.2.f.d.829.24 yes 24 15.14 odd 2 inner