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Properties

Label 72.6.a.d.1.1
Level $72$
Weight $6$
Character 72.1
Self dual yes
Analytic conductor $11.548$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [72,6,Mod(1,72)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(72, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("72.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 72 = 2^{3} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 72.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: \(11.5476350265\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 72.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+16.0000 q^{5} +12.0000 q^{7} +O(q^{10})\) \(q+16.0000 q^{5} +12.0000 q^{7} +448.000 q^{11} -206.000 q^{13} +1952.00 q^{17} +1064.00 q^{19} +3712.00 q^{23} -2869.00 q^{25} +4080.00 q^{29} +5324.00 q^{31} +192.000 q^{35} -9690.00 q^{37} -9120.00 q^{41} +16552.0 q^{43} -14208.0 q^{47} -16663.0 q^{49} -21776.0 q^{53} +7168.00 q^{55} -31616.0 q^{59} -13154.0 q^{61} -3296.00 q^{65} +27056.0 q^{67} +9728.00 q^{71} +9046.00 q^{73} +5376.00 q^{77} -58292.0 q^{79} +86336.0 q^{83} +31232.0 q^{85} +75072.0 q^{89} -2472.00 q^{91} +17024.0 q^{95} +76046.0 q^{97} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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
\(5\) 16.0000 0.286217 0.143108 − 0.989707i \(-0.454290\pi\)
0.143108 + 0.989707i \(0.454290\pi\)
\(6\) 0 0
\(7\) 12.0000 0.0925627 0.0462814 − 0.998928i \(-0.485263\pi\)
0.0462814 + 0.998928i \(0.485263\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 448.000 1.11634 0.558170 − 0.829727i \(-0.311504\pi\)
0.558170 + 0.829727i \(0.311504\pi\)
\(12\) 0 0
\(13\) −206.000 −0.338072 −0.169036 − 0.985610i \(-0.554065\pi\)
−0.169036 + 0.985610i \(0.554065\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1952.00 1.63816 0.819082 − 0.573676i \(-0.194483\pi\)
0.819082 + 0.573676i \(0.194483\pi\)
\(18\) 0 0
\(19\) 1064.00 0.676173 0.338086 − 0.941115i \(-0.390220\pi\)
0.338086 + 0.941115i \(0.390220\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3712.00 1.46315 0.731574 − 0.681762i \(-0.238786\pi\)
0.731574 + 0.681762i \(0.238786\pi\)
\(24\) 0 0
\(25\) −2869.00 −0.918080
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 4080.00 0.900876 0.450438 − 0.892808i \(-0.351268\pi\)
0.450438 + 0.892808i \(0.351268\pi\)
\(30\) 0 0
\(31\) 5324.00 0.995025 0.497512 − 0.867457i \(-0.334247\pi\)
0.497512 + 0.867457i \(0.334247\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 192.000 0.0264930
\(36\) 0 0
\(37\) −9690.00 −1.16364 −0.581821 − 0.813317i \(-0.697660\pi\)
−0.581821 + 0.813317i \(0.697660\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −9120.00 −0.847296 −0.423648 − 0.905827i \(-0.639251\pi\)
−0.423648 + 0.905827i \(0.639251\pi\)
\(42\) 0 0
\(43\) 16552.0 1.36515 0.682573 − 0.730817i \(-0.260861\pi\)
0.682573 + 0.730817i \(0.260861\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −14208.0 −0.938185 −0.469092 − 0.883149i \(-0.655419\pi\)
−0.469092 + 0.883149i \(0.655419\pi\)
\(48\) 0 0
\(49\) −16663.0 −0.991432
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −21776.0 −1.06485 −0.532425 − 0.846477i \(-0.678719\pi\)
−0.532425 + 0.846477i \(0.678719\pi\)
\(54\) 0 0
\(55\) 7168.00 0.319515
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −31616.0 −1.18243 −0.591217 − 0.806513i \(-0.701352\pi\)
−0.591217 + 0.806513i \(0.701352\pi\)
\(60\) 0 0
\(61\) −13154.0 −0.452619 −0.226310 − 0.974055i \(-0.572666\pi\)
−0.226310 + 0.974055i \(0.572666\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −3296.00 −0.0967618
\(66\) 0 0
\(67\) 27056.0 0.736337 0.368168 − 0.929759i \(-0.379985\pi\)
0.368168 + 0.929759i \(0.379985\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 9728.00 0.229022 0.114511 − 0.993422i \(-0.463470\pi\)
0.114511 + 0.993422i \(0.463470\pi\)
\(72\) 0 0
\(73\) 9046.00 0.198678 0.0993389 − 0.995054i \(-0.468327\pi\)
0.0993389 + 0.995054i \(0.468327\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 5376.00 0.103331
\(78\) 0 0
\(79\) −58292.0 −1.05085 −0.525426 − 0.850840i \(-0.676094\pi\)
−0.525426 + 0.850840i \(0.676094\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 86336.0 1.37561 0.687807 − 0.725893i \(-0.258573\pi\)
0.687807 + 0.725893i \(0.258573\pi\)
\(84\) 0 0
\(85\) 31232.0 0.468870
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 75072.0 1.00462 0.502311 − 0.864687i \(-0.332483\pi\)
0.502311 + 0.864687i \(0.332483\pi\)
\(90\) 0 0
\(91\) −2472.00 −0.0312928
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 17024.0 0.193532
\(96\) 0 0
\(97\) 76046.0 0.820629 0.410315 − 0.911944i \(-0.365419\pi\)
0.410315 + 0.911944i \(0.365419\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 134064. 1.30770 0.653851 − 0.756623i \(-0.273152\pi\)
0.653851 + 0.756623i \(0.273152\pi\)
\(102\) 0 0
\(103\) −176444. −1.63875 −0.819377 − 0.573255i \(-0.805681\pi\)
−0.819377 + 0.573255i \(0.805681\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 58752.0 0.496093 0.248047 − 0.968748i \(-0.420211\pi\)
0.248047 + 0.968748i \(0.420211\pi\)
\(108\) 0 0
\(109\) 125362. 1.01065 0.505324 − 0.862930i \(-0.331373\pi\)
0.505324 + 0.862930i \(0.331373\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −101568. −0.748274 −0.374137 − 0.927373i \(-0.622061\pi\)
−0.374137 + 0.927373i \(0.622061\pi\)
\(114\) 0 0
\(115\) 59392.0 0.418778
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 23424.0 0.151633
\(120\) 0 0
\(121\) 39653.0 0.246214
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −95904.0 −0.548987
\(126\) 0 0
\(127\) −312572. −1.71965 −0.859826 − 0.510586i \(-0.829428\pi\)
−0.859826 + 0.510586i \(0.829428\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −328192. −1.67090 −0.835449 − 0.549569i \(-0.814792\pi\)
−0.835449 + 0.549569i \(0.814792\pi\)
\(132\) 0 0
\(133\) 12768.0 0.0625884
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −16992.0 −0.0773469 −0.0386735 − 0.999252i \(-0.512313\pi\)
−0.0386735 + 0.999252i \(0.512313\pi\)
\(138\) 0 0
\(139\) 319152. 1.40107 0.700536 − 0.713617i \(-0.252944\pi\)
0.700536 + 0.713617i \(0.252944\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −92288.0 −0.377403
\(144\) 0 0
\(145\) 65280.0 0.257846
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −514480. −1.89847 −0.949233 − 0.314574i \(-0.898138\pi\)
−0.949233 + 0.314574i \(0.898138\pi\)
\(150\) 0 0
\(151\) −210292. −0.750551 −0.375276 − 0.926913i \(-0.622452\pi\)
−0.375276 + 0.926913i \(0.622452\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 85184.0 0.284793
\(156\) 0 0
\(157\) −198626. −0.643113 −0.321556 − 0.946890i \(-0.604206\pi\)
−0.321556 + 0.946890i \(0.604206\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 44544.0 0.135433
\(162\) 0 0
\(163\) 445224. 1.31253 0.656265 − 0.754530i \(-0.272135\pi\)
0.656265 + 0.754530i \(0.272135\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −324480. −0.900320 −0.450160 − 0.892948i \(-0.648633\pi\)
−0.450160 + 0.892948i \(0.648633\pi\)
\(168\) 0 0
\(169\) −328857. −0.885708
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 605712. 1.53869 0.769345 − 0.638834i \(-0.220583\pi\)
0.769345 + 0.638834i \(0.220583\pi\)
\(174\) 0 0
\(175\) −34428.0 −0.0849800
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 157824. 0.368163 0.184082 − 0.982911i \(-0.441069\pi\)
0.184082 + 0.982911i \(0.441069\pi\)
\(180\) 0 0
\(181\) −216918. −0.492152 −0.246076 − 0.969251i \(-0.579141\pi\)
−0.246076 + 0.969251i \(0.579141\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −155040. −0.333054
\(186\) 0 0
\(187\) 874496. 1.82875
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 917376. 1.81955 0.909775 − 0.415102i \(-0.136254\pi\)
0.909775 + 0.415102i \(0.136254\pi\)
\(192\) 0 0
\(193\) −368638. −0.712372 −0.356186 − 0.934415i \(-0.615923\pi\)
−0.356186 + 0.934415i \(0.615923\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 509808. 0.935925 0.467963 − 0.883748i \(-0.344988\pi\)
0.467963 + 0.883748i \(0.344988\pi\)
\(198\) 0 0
\(199\) −513452. −0.919109 −0.459555 − 0.888149i \(-0.651991\pi\)
−0.459555 + 0.888149i \(0.651991\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 48960.0 0.0833876
\(204\) 0 0
\(205\) −145920. −0.242510
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 476672. 0.754838
\(210\) 0 0
\(211\) 1.02904e6 1.59120 0.795602 − 0.605819i \(-0.207154\pi\)
0.795602 + 0.605819i \(0.207154\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 264832. 0.390728
\(216\) 0 0
\(217\) 63888.0 0.0921022
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −402112. −0.553817
\(222\) 0 0
\(223\) −502892. −0.677193 −0.338597 − 0.940932i \(-0.609952\pi\)
−0.338597 + 0.940932i \(0.609952\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 649920. 0.837135 0.418567 − 0.908186i \(-0.362532\pi\)
0.418567 + 0.908186i \(0.362532\pi\)
\(228\) 0 0
\(229\) −547142. −0.689464 −0.344732 − 0.938701i \(-0.612030\pi\)
−0.344732 + 0.938701i \(0.612030\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 349888. 0.422220 0.211110 − 0.977462i \(-0.432292\pi\)
0.211110 + 0.977462i \(0.432292\pi\)
\(234\) 0 0
\(235\) −227328. −0.268524
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −1.68947e6 −1.91318 −0.956591 − 0.291434i \(-0.905868\pi\)
−0.956591 + 0.291434i \(0.905868\pi\)
\(240\) 0 0
\(241\) 375630. 0.416598 0.208299 − 0.978065i \(-0.433207\pi\)
0.208299 + 0.978065i \(0.433207\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −266608. −0.283764
\(246\) 0 0
\(247\) −219184. −0.228595
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −414016. −0.414794 −0.207397 − 0.978257i \(-0.566499\pi\)
−0.207397 + 0.978257i \(0.566499\pi\)
\(252\) 0 0
\(253\) 1.66298e6 1.63337
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −1.23789e6 −1.16909 −0.584546 − 0.811361i \(-0.698727\pi\)
−0.584546 + 0.811361i \(0.698727\pi\)
\(258\) 0 0
\(259\) −116280. −0.107710
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 117760. 0.104980 0.0524902 − 0.998621i \(-0.483284\pi\)
0.0524902 + 0.998621i \(0.483284\pi\)
\(264\) 0 0
\(265\) −348416. −0.304778
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −1.26549e6 −1.06629 −0.533147 − 0.846022i \(-0.678991\pi\)
−0.533147 + 0.846022i \(0.678991\pi\)
\(270\) 0 0
\(271\) −1.58206e6 −1.30858 −0.654289 − 0.756244i \(-0.727032\pi\)
−0.654289 + 0.756244i \(0.727032\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −1.28531e6 −1.02489
\(276\) 0 0
\(277\) 40234.0 0.0315060 0.0157530 − 0.999876i \(-0.494985\pi\)
0.0157530 + 0.999876i \(0.494985\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 50048.0 0.0378112 0.0189056 − 0.999821i \(-0.493982\pi\)
0.0189056 + 0.999821i \(0.493982\pi\)
\(282\) 0 0
\(283\) −387424. −0.287555 −0.143777 − 0.989610i \(-0.545925\pi\)
−0.143777 + 0.989610i \(0.545925\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −109440. −0.0784280
\(288\) 0 0
\(289\) 2.39045e6 1.68358
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 327408. 0.222803 0.111401 − 0.993776i \(-0.464466\pi\)
0.111401 + 0.993776i \(0.464466\pi\)
\(294\) 0 0
\(295\) −505856. −0.338432
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −764672. −0.494649
\(300\) 0 0
\(301\) 198624. 0.126362
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −210464. −0.129547
\(306\) 0 0
\(307\) −2.02555e6 −1.22658 −0.613292 − 0.789856i \(-0.710155\pi\)
−0.613292 + 0.789856i \(0.710155\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 1.70112e6 0.997319 0.498659 − 0.866798i \(-0.333826\pi\)
0.498659 + 0.866798i \(0.333826\pi\)
\(312\) 0 0
\(313\) −231990. −0.133847 −0.0669235 − 0.997758i \(-0.521318\pi\)
−0.0669235 + 0.997758i \(0.521318\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 1.32696e6 0.741668 0.370834 − 0.928699i \(-0.379072\pi\)
0.370834 + 0.928699i \(0.379072\pi\)
\(318\) 0 0
\(319\) 1.82784e6 1.00568
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 2.07693e6 1.10768
\(324\) 0 0
\(325\) 591014. 0.310377
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −170496. −0.0868409
\(330\) 0 0
\(331\) −200032. −0.100353 −0.0501764 − 0.998740i \(-0.515978\pi\)
−0.0501764 + 0.998740i \(0.515978\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 432896. 0.210752
\(336\) 0 0
\(337\) −548322. −0.263003 −0.131502 − 0.991316i \(-0.541980\pi\)
−0.131502 + 0.991316i \(0.541980\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 2.38515e6 1.11079
\(342\) 0 0
\(343\) −401640. −0.184332
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 1.53734e6 0.685405 0.342703 − 0.939444i \(-0.388658\pi\)
0.342703 + 0.939444i \(0.388658\pi\)
\(348\) 0 0
\(349\) 94494.0 0.0415280 0.0207640 − 0.999784i \(-0.493390\pi\)
0.0207640 + 0.999784i \(0.493390\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 3.29658e6 1.40808 0.704038 − 0.710162i \(-0.251378\pi\)
0.704038 + 0.710162i \(0.251378\pi\)
\(354\) 0 0
\(355\) 155648. 0.0655500
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −1.29638e6 −0.530881 −0.265441 − 0.964127i \(-0.585517\pi\)
−0.265441 + 0.964127i \(0.585517\pi\)
\(360\) 0 0
\(361\) −1.34400e6 −0.542790
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 144736. 0.0568649
\(366\) 0 0
\(367\) 3.57512e6 1.38556 0.692779 − 0.721150i \(-0.256386\pi\)
0.692779 + 0.721150i \(0.256386\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −261312. −0.0985654
\(372\) 0 0
\(373\) −3.61969e6 −1.34710 −0.673549 − 0.739142i \(-0.735231\pi\)
−0.673549 + 0.739142i \(0.735231\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −840480. −0.304561
\(378\) 0 0
\(379\) −2.57135e6 −0.919525 −0.459762 − 0.888042i \(-0.652065\pi\)
−0.459762 + 0.888042i \(0.652065\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 3.76358e6 1.31101 0.655503 − 0.755193i \(-0.272457\pi\)
0.655503 + 0.755193i \(0.272457\pi\)
\(384\) 0 0
\(385\) 86016.0 0.0295752
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −5.87077e6 −1.96707 −0.983537 − 0.180704i \(-0.942162\pi\)
−0.983537 + 0.180704i \(0.942162\pi\)
\(390\) 0 0
\(391\) 7.24582e6 2.39688
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −932672. −0.300771
\(396\) 0 0
\(397\) 2.18334e6 0.695257 0.347629 − 0.937632i \(-0.386987\pi\)
0.347629 + 0.937632i \(0.386987\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 1.93891e6 0.602140 0.301070 − 0.953602i \(-0.402656\pi\)
0.301070 + 0.953602i \(0.402656\pi\)
\(402\) 0 0
\(403\) −1.09674e6 −0.336390
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −4.34112e6 −1.29902
\(408\) 0 0
\(409\) −5.91993e6 −1.74988 −0.874940 − 0.484231i \(-0.839099\pi\)
−0.874940 + 0.484231i \(0.839099\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −379392. −0.109449
\(414\) 0 0
\(415\) 1.38138e6 0.393724
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −4.10400e6 −1.14202 −0.571008 − 0.820944i \(-0.693448\pi\)
−0.571008 + 0.820944i \(0.693448\pi\)
\(420\) 0 0
\(421\) −5.31463e6 −1.46140 −0.730698 − 0.682701i \(-0.760805\pi\)
−0.730698 + 0.682701i \(0.760805\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −5.60029e6 −1.50397
\(426\) 0 0
\(427\) −157848. −0.0418957
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −2.64026e6 −0.684625 −0.342313 − 0.939586i \(-0.611210\pi\)
−0.342313 + 0.939586i \(0.611210\pi\)
\(432\) 0 0
\(433\) 5.35405e6 1.37234 0.686172 − 0.727440i \(-0.259290\pi\)
0.686172 + 0.727440i \(0.259290\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 3.94957e6 0.989341
\(438\) 0 0
\(439\) 3.53711e6 0.875966 0.437983 − 0.898983i \(-0.355693\pi\)
0.437983 + 0.898983i \(0.355693\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 762176. 0.184521 0.0922605 − 0.995735i \(-0.470591\pi\)
0.0922605 + 0.995735i \(0.470591\pi\)
\(444\) 0 0
\(445\) 1.20115e6 0.287540
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −6.85155e6 −1.60388 −0.801942 − 0.597402i \(-0.796200\pi\)
−0.801942 + 0.597402i \(0.796200\pi\)
\(450\) 0 0
\(451\) −4.08576e6 −0.945870
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −39552.0 −0.00895653
\(456\) 0 0
\(457\) −3.67249e6 −0.822565 −0.411282 − 0.911508i \(-0.634919\pi\)
−0.411282 + 0.911508i \(0.634919\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −1.54616e6 −0.338846 −0.169423 − 0.985543i \(-0.554190\pi\)
−0.169423 + 0.985543i \(0.554190\pi\)
\(462\) 0 0
\(463\) 3.13562e6 0.679784 0.339892 − 0.940464i \(-0.389609\pi\)
0.339892 + 0.940464i \(0.389609\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 1.23744e6 0.262562 0.131281 − 0.991345i \(-0.458091\pi\)
0.131281 + 0.991345i \(0.458091\pi\)
\(468\) 0 0
\(469\) 324672. 0.0681574
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 7.41530e6 1.52397
\(474\) 0 0
\(475\) −3.05262e6 −0.620781
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −4.25114e6 −0.846577 −0.423288 − 0.905995i \(-0.639124\pi\)
−0.423288 + 0.905995i \(0.639124\pi\)
\(480\) 0 0
\(481\) 1.99614e6 0.393395
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 1.21674e6 0.234878
\(486\) 0 0
\(487\) −909188. −0.173713 −0.0868563 − 0.996221i \(-0.527682\pi\)
−0.0868563 + 0.996221i \(0.527682\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 7.21613e6 1.35083 0.675415 − 0.737438i \(-0.263965\pi\)
0.675415 + 0.737438i \(0.263965\pi\)
\(492\) 0 0
\(493\) 7.96416e6 1.47578
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 116736. 0.0211989
\(498\) 0 0
\(499\) −4.55664e6 −0.819206 −0.409603 − 0.912264i \(-0.634333\pi\)
−0.409603 + 0.912264i \(0.634333\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 7.10630e6 1.25234 0.626172 − 0.779685i \(-0.284621\pi\)
0.626172 + 0.779685i \(0.284621\pi\)
\(504\) 0 0
\(505\) 2.14502e6 0.374286
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 663216. 0.113465 0.0567323 − 0.998389i \(-0.481932\pi\)
0.0567323 + 0.998389i \(0.481932\pi\)
\(510\) 0 0
\(511\) 108552. 0.0183902
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −2.82310e6 −0.469039
\(516\) 0 0
\(517\) −6.36518e6 −1.04733
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 9.09872e6 1.46854 0.734270 − 0.678857i \(-0.237524\pi\)
0.734270 + 0.678857i \(0.237524\pi\)
\(522\) 0 0
\(523\) −9.95055e6 −1.59072 −0.795359 − 0.606139i \(-0.792717\pi\)
−0.795359 + 0.606139i \(0.792717\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 1.03924e7 1.63001
\(528\) 0 0
\(529\) 7.34260e6 1.14080
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 1.87872e6 0.286447
\(534\) 0 0
\(535\) 940032. 0.141990
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −7.46502e6 −1.10677
\(540\) 0 0
\(541\) 8.84781e6 1.29970 0.649849 − 0.760063i \(-0.274832\pi\)
0.649849 + 0.760063i \(0.274832\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 2.00579e6 0.289264
\(546\) 0 0
\(547\) −534472. −0.0763760 −0.0381880 − 0.999271i \(-0.512159\pi\)
−0.0381880 + 0.999271i \(0.512159\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 4.34112e6 0.609148
\(552\) 0 0
\(553\) −699504. −0.0972697
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −631664. −0.0862677 −0.0431338 − 0.999069i \(-0.513734\pi\)
−0.0431338 + 0.999069i \(0.513734\pi\)
\(558\) 0 0
\(559\) −3.40971e6 −0.461518
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 1.11349e7 1.48053 0.740263 − 0.672318i \(-0.234701\pi\)
0.740263 + 0.672318i \(0.234701\pi\)
\(564\) 0 0
\(565\) −1.62509e6 −0.214169
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −8.25296e6 −1.06863 −0.534317 − 0.845284i \(-0.679431\pi\)
−0.534317 + 0.845284i \(0.679431\pi\)
\(570\) 0 0
\(571\) 2.46494e6 0.316386 0.158193 − 0.987408i \(-0.449433\pi\)
0.158193 + 0.987408i \(0.449433\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −1.06497e7 −1.34329
\(576\) 0 0
\(577\) 6.94767e6 0.868759 0.434380 − 0.900730i \(-0.356968\pi\)
0.434380 + 0.900730i \(0.356968\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 1.03603e6 0.127331
\(582\) 0 0
\(583\) −9.75565e6 −1.18873
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −8.57754e6 −1.02747 −0.513733 − 0.857950i \(-0.671738\pi\)
−0.513733 + 0.857950i \(0.671738\pi\)
\(588\) 0 0
\(589\) 5.66474e6 0.672808
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 2.86125e6 0.334133 0.167066 − 0.985946i \(-0.446571\pi\)
0.167066 + 0.985946i \(0.446571\pi\)
\(594\) 0 0
\(595\) 374784. 0.0433999
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −1.31401e7 −1.49634 −0.748172 − 0.663505i \(-0.769068\pi\)
−0.748172 + 0.663505i \(0.769068\pi\)
\(600\) 0 0
\(601\) −1.49112e7 −1.68394 −0.841972 − 0.539522i \(-0.818605\pi\)
−0.841972 + 0.539522i \(0.818605\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 634448. 0.0704705
\(606\) 0 0
\(607\) 3.54612e6 0.390644 0.195322 − 0.980739i \(-0.437425\pi\)
0.195322 + 0.980739i \(0.437425\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 2.92685e6 0.317174
\(612\) 0 0
\(613\) −5.91126e6 −0.635373 −0.317686 − 0.948196i \(-0.602906\pi\)
−0.317686 + 0.948196i \(0.602906\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −9.93754e6 −1.05091 −0.525455 − 0.850821i \(-0.676105\pi\)
−0.525455 + 0.850821i \(0.676105\pi\)
\(618\) 0 0
\(619\) 1.08121e7 1.13418 0.567090 − 0.823656i \(-0.308069\pi\)
0.567090 + 0.823656i \(0.308069\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 900864. 0.0929906
\(624\) 0 0
\(625\) 7.43116e6 0.760951
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −1.89149e7 −1.90624
\(630\) 0 0
\(631\) 2.01958e6 0.201924 0.100962 − 0.994890i \(-0.467808\pi\)
0.100962 + 0.994890i \(0.467808\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −5.00115e6 −0.492193
\(636\) 0 0
\(637\) 3.43258e6 0.335175
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −9.12598e6 −0.877273 −0.438636 − 0.898665i \(-0.644538\pi\)
−0.438636 + 0.898665i \(0.644538\pi\)
\(642\) 0 0
\(643\) 2.46943e6 0.235543 0.117771 − 0.993041i \(-0.462425\pi\)
0.117771 + 0.993041i \(0.462425\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 6.90982e6 0.648943 0.324471 − 0.945896i \(-0.394814\pi\)
0.324471 + 0.945896i \(0.394814\pi\)
\(648\) 0 0
\(649\) −1.41640e7 −1.32000
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 9.45902e6 0.868087 0.434044 − 0.900892i \(-0.357086\pi\)
0.434044 + 0.900892i \(0.357086\pi\)
\(654\) 0 0
\(655\) −5.25107e6 −0.478239
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −1.07267e7 −0.962168 −0.481084 − 0.876675i \(-0.659757\pi\)
−0.481084 + 0.876675i \(0.659757\pi\)
\(660\) 0 0
\(661\) −1.72413e7 −1.53485 −0.767427 − 0.641137i \(-0.778463\pi\)
−0.767427 + 0.641137i \(0.778463\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 204288. 0.0179138
\(666\) 0 0
\(667\) 1.51450e7 1.31812
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −5.89299e6 −0.505277
\(672\) 0 0
\(673\) 1.33530e7 1.13643 0.568213 − 0.822881i \(-0.307635\pi\)
0.568213 + 0.822881i \(0.307635\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 8.75694e6 0.734312 0.367156 − 0.930159i \(-0.380331\pi\)
0.367156 + 0.930159i \(0.380331\pi\)
\(678\) 0 0
\(679\) 912552. 0.0759597
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 6.40403e6 0.525293 0.262647 − 0.964892i \(-0.415405\pi\)
0.262647 + 0.964892i \(0.415405\pi\)
\(684\) 0 0
\(685\) −271872. −0.0221380
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 4.48586e6 0.359996
\(690\) 0 0
\(691\) 2.30918e7 1.83976 0.919881 − 0.392197i \(-0.128285\pi\)
0.919881 + 0.392197i \(0.128285\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 5.10643e6 0.401010
\(696\) 0 0
\(697\) −1.78022e7 −1.38801
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 2.09669e6 0.161153 0.0805766 − 0.996748i \(-0.474324\pi\)
0.0805766 + 0.996748i \(0.474324\pi\)
\(702\) 0 0
\(703\) −1.03102e7 −0.786823
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 1.60877e6 0.121044
\(708\) 0 0
\(709\) −1.70186e7 −1.27147 −0.635737 − 0.771906i \(-0.719304\pi\)
−0.635737 + 0.771906i \(0.719304\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 1.97627e7 1.45587
\(714\) 0 0
\(715\) −1.47661e6 −0.108019
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 7.78624e6 0.561701 0.280851 − 0.959751i \(-0.409383\pi\)
0.280851 + 0.959751i \(0.409383\pi\)
\(720\) 0 0
\(721\) −2.11733e6 −0.151688
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −1.17055e7 −0.827077
\(726\) 0 0
\(727\) −5.22445e6 −0.366610 −0.183305 − 0.983056i \(-0.558680\pi\)
−0.183305 + 0.983056i \(0.558680\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 3.23095e7 2.23633
\(732\) 0 0
\(733\) 2.51692e7 1.73025 0.865126 − 0.501555i \(-0.167239\pi\)
0.865126 + 0.501555i \(0.167239\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1.21211e7 0.822002
\(738\) 0 0
\(739\) 2.22776e7 1.50057 0.750285 − 0.661114i \(-0.229916\pi\)
0.750285 + 0.661114i \(0.229916\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −5.05126e6 −0.335682 −0.167841 − 0.985814i \(-0.553680\pi\)
−0.167841 + 0.985814i \(0.553680\pi\)
\(744\) 0 0
\(745\) −8.23168e6 −0.543373
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 705024. 0.0459197
\(750\) 0 0
\(751\) −1.02532e6 −0.0663373 −0.0331687 − 0.999450i \(-0.510560\pi\)
−0.0331687 + 0.999450i \(0.510560\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −3.36467e6 −0.214820
\(756\) 0 0
\(757\) −1.34483e7 −0.852957 −0.426478 − 0.904498i \(-0.640246\pi\)
−0.426478 + 0.904498i \(0.640246\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 2.99370e6 0.187390 0.0936949 − 0.995601i \(-0.470132\pi\)
0.0936949 + 0.995601i \(0.470132\pi\)
\(762\) 0 0
\(763\) 1.50434e6 0.0935483
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 6.51290e6 0.399747
\(768\) 0 0
\(769\) 1.40552e7 0.857078 0.428539 − 0.903523i \(-0.359028\pi\)
0.428539 + 0.903523i \(0.359028\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 2.66477e7 1.60403 0.802013 − 0.597306i \(-0.203762\pi\)
0.802013 + 0.597306i \(0.203762\pi\)
\(774\) 0 0
\(775\) −1.52746e7 −0.913512
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −9.70368e6 −0.572918
\(780\) 0 0
\(781\) 4.35814e6 0.255667
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −3.17802e6 −0.184070
\(786\) 0 0
\(787\) −5.55697e6 −0.319817 −0.159908 − 0.987132i \(-0.551120\pi\)
−0.159908 + 0.987132i \(0.551120\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −1.21882e6 −0.0692623
\(792\) 0 0
\(793\) 2.70972e6 0.153018
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −2.02297e7 −1.12809 −0.564044 − 0.825745i \(-0.690755\pi\)
−0.564044 + 0.825745i \(0.690755\pi\)
\(798\) 0 0
\(799\) −2.77340e7 −1.53690
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 4.05261e6 0.221792
\(804\) 0 0
\(805\) 712704. 0.0387632
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −2.72398e7 −1.46330 −0.731648 − 0.681683i \(-0.761249\pi\)
−0.731648 + 0.681683i \(0.761249\pi\)
\(810\) 0 0
\(811\) 2.80834e7 1.49933 0.749664 − 0.661818i \(-0.230215\pi\)
0.749664 + 0.661818i \(0.230215\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 7.12358e6 0.375668
\(816\) 0 0
\(817\) 1.76113e7 0.923075
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 1.73375e7 0.897692 0.448846 − 0.893609i \(-0.351835\pi\)
0.448846 + 0.893609i \(0.351835\pi\)
\(822\) 0 0
\(823\) −2.66547e7 −1.37175 −0.685874 − 0.727721i \(-0.740580\pi\)
−0.685874 + 0.727721i \(0.740580\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −1.94633e7 −0.989584 −0.494792 − 0.869011i \(-0.664756\pi\)
−0.494792 + 0.869011i \(0.664756\pi\)
\(828\) 0 0
\(829\) 1.28797e7 0.650907 0.325453 − 0.945558i \(-0.394483\pi\)
0.325453 + 0.945558i \(0.394483\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −3.25262e7 −1.62413
\(834\) 0 0
\(835\) −5.19168e6 −0.257687
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −1.21928e7 −0.597995 −0.298997 − 0.954254i \(-0.596652\pi\)
−0.298997 + 0.954254i \(0.596652\pi\)
\(840\) 0 0
\(841\) −3.86475e6 −0.188422
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −5.26171e6 −0.253504
\(846\) 0 0
\(847\) 475836. 0.0227902
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −3.59693e7 −1.70258
\(852\) 0 0
\(853\) 1.54977e7 0.729278 0.364639 − 0.931149i \(-0.381192\pi\)
0.364639 + 0.931149i \(0.381192\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 2.00403e7 0.932077 0.466039 − 0.884764i \(-0.345681\pi\)
0.466039 + 0.884764i \(0.345681\pi\)
\(858\) 0 0
\(859\) −2.25281e7 −1.04170 −0.520849 − 0.853649i \(-0.674385\pi\)
−0.520849 + 0.853649i \(0.674385\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 3.23699e6 0.147950 0.0739749 − 0.997260i \(-0.476432\pi\)
0.0739749 + 0.997260i \(0.476432\pi\)
\(864\) 0 0
\(865\) 9.69139e6 0.440399
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −2.61148e7 −1.17311
\(870\) 0 0
\(871\) −5.57354e6 −0.248935
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −1.15085e6 −0.0508157
\(876\) 0 0
\(877\) 2.62078e7 1.15062 0.575309 − 0.817936i \(-0.304882\pi\)
0.575309 + 0.817936i \(0.304882\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 3.83398e6 0.166422 0.0832110 − 0.996532i \(-0.473482\pi\)
0.0832110 + 0.996532i \(0.473482\pi\)
\(882\) 0 0
\(883\) −1.40149e7 −0.604908 −0.302454 − 0.953164i \(-0.597806\pi\)
−0.302454 + 0.953164i \(0.597806\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −1.87702e7 −0.801049 −0.400525 − 0.916286i \(-0.631172\pi\)
−0.400525 + 0.916286i \(0.631172\pi\)
\(888\) 0 0
\(889\) −3.75086e6 −0.159176
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −1.51173e7 −0.634375
\(894\) 0 0
\(895\) 2.52518e6 0.105374
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 2.17219e7 0.896394
\(900\) 0 0
\(901\) −4.25068e7 −1.74440
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −3.47069e6 −0.140862
\(906\) 0 0
\(907\) 1.33250e7 0.537834 0.268917 − 0.963163i \(-0.413334\pi\)
0.268917 + 0.963163i \(0.413334\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −1.28695e7 −0.513767 −0.256883 − 0.966442i \(-0.582696\pi\)
−0.256883 + 0.966442i \(0.582696\pi\)
\(912\) 0 0
\(913\) 3.86785e7 1.53565
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −3.93830e6 −0.154663
\(918\) 0 0
\(919\) 5.50582e6 0.215047 0.107523 − 0.994203i \(-0.465708\pi\)
0.107523 + 0.994203i \(0.465708\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −2.00397e6 −0.0774259
\(924\) 0 0
\(925\) 2.78006e7 1.06832
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 3.57775e7 1.36010 0.680050 − 0.733166i \(-0.261958\pi\)
0.680050 + 0.733166i \(0.261958\pi\)
\(930\) 0 0
\(931\) −1.77294e7 −0.670379
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 1.39919e7 0.523418
\(936\) 0 0
\(937\) −4.47066e7 −1.66350 −0.831750 − 0.555150i \(-0.812661\pi\)
−0.831750 + 0.555150i \(0.812661\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 1.41830e7 0.522148 0.261074 − 0.965319i \(-0.415923\pi\)
0.261074 + 0.965319i \(0.415923\pi\)
\(942\) 0 0
\(943\) −3.38534e7 −1.23972
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 4.65412e7 1.68641 0.843204 − 0.537594i \(-0.180667\pi\)
0.843204 + 0.537594i \(0.180667\pi\)
\(948\) 0 0
\(949\) −1.86348e6 −0.0671674
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −1.61041e7 −0.574386 −0.287193 − 0.957873i \(-0.592722\pi\)
−0.287193 + 0.957873i \(0.592722\pi\)
\(954\) 0 0
\(955\) 1.46780e7 0.520786
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −203904. −0.00715944
\(960\) 0 0
\(961\) −284175. −0.00992607
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −5.89821e6 −0.203893
\(966\) 0 0
\(967\) 3.33946e7 1.14845 0.574223 − 0.818699i \(-0.305304\pi\)
0.574223 + 0.818699i \(0.305304\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 4.81171e6 0.163776 0.0818882 − 0.996642i \(-0.473905\pi\)
0.0818882 + 0.996642i \(0.473905\pi\)
\(972\) 0 0
\(973\) 3.82982e6 0.129687
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 1.45783e7 0.488618 0.244309 − 0.969697i \(-0.421439\pi\)
0.244309 + 0.969697i \(0.421439\pi\)
\(978\) 0 0
\(979\) 3.36323e7 1.12150
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 5.04445e7 1.66506 0.832531 − 0.553978i \(-0.186891\pi\)
0.832531 + 0.553978i \(0.186891\pi\)
\(984\) 0 0
\(985\) 8.15693e6 0.267877
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 6.14410e7 1.99741
\(990\) 0 0
\(991\) 5.74114e7 1.85701 0.928504 − 0.371322i \(-0.121095\pi\)
0.928504 + 0.371322i \(0.121095\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −8.21523e6 −0.263064
\(996\) 0 0
\(997\) −4.63275e7 −1.47605 −0.738025 − 0.674773i \(-0.764241\pi\)
−0.738025 + 0.674773i \(0.764241\pi\)
\(998\) 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 72.6.a.d.1.1 yes 1
3.2 odd 2 72.6.a.c.1.1 ✓ 1
4.3 odd 2 144.6.a.h.1.1 1
8.3 odd 2 576.6.a.m.1.1 1
8.5 even 2 576.6.a.n.1.1 1
12.11 even 2 144.6.a.e.1.1 1
24.5 odd 2 576.6.a.x.1.1 1
24.11 even 2 576.6.a.w.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
72.6.a.c.1.1 ✓ 1 3.2 odd 2
72.6.a.d.1.1 yes 1 1.1 even 1 trivial
144.6.a.e.1.1 1 12.11 even 2
144.6.a.h.1.1 1 4.3 odd 2
576.6.a.m.1.1 1 8.3 odd 2
576.6.a.n.1.1 1 8.5 even 2
576.6.a.w.1.1 1 24.11 even 2
576.6.a.x.1.1 1 24.5 odd 2