Properties

Label 144.7.e.a.17.2
Level $144$
Weight $7$
Character 144.17
Analytic conductor $33.128$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [144,7,Mod(17,144)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(144, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("144.17");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 144 = 2^{4} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 144.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(33.1277880413\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 9)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 17.2
Root \(1.41421i\) of defining polynomial
Character \(\chi\) \(=\) 144.17
Dual form 144.7.e.a.17.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+63.6396i q^{5} -524.000 q^{7} +O(q^{10})\) \(q+63.6396i q^{5} -524.000 q^{7} +865.499i q^{11} +344.000 q^{13} -7140.36i q^{17} +2320.00 q^{19} -5753.02i q^{23} +11575.0 q^{25} +23152.1i q^{29} +10564.0 q^{31} -33347.2i q^{35} -24082.0 q^{37} -108836. i q^{41} +90952.0 q^{43} -128959. i q^{47} +156927. q^{49} -196685. i q^{53} -55080.0 q^{55} -39812.9i q^{59} +251138. q^{61} +21892.0i q^{65} +216088. q^{67} -53915.5i q^{71} -308176. q^{73} -453521. i q^{77} +540124. q^{79} -932346. i q^{83} +454410. q^{85} +223413. i q^{89} -180256. q^{91} +147644. i q^{95} -37168.0 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 1048 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 1048 q^{7} + 688 q^{13} + 4640 q^{19} + 23150 q^{25} + 21128 q^{31} - 48164 q^{37} + 181904 q^{43} + 313854 q^{49} - 110160 q^{55} + 502276 q^{61} + 432176 q^{67} - 616352 q^{73} + 1080248 q^{79} + 908820 q^{85} - 360512 q^{91} - 74336 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/144\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(65\) \(127\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 63.6396i 0.509117i 0.967057 + 0.254558i \(0.0819301\pi\)
−0.967057 + 0.254558i \(0.918070\pi\)
\(6\) 0 0
\(7\) −524.000 −1.52770 −0.763848 0.645396i \(-0.776692\pi\)
−0.763848 + 0.645396i \(0.776692\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 865.499i 0.650262i 0.945669 + 0.325131i \(0.105408\pi\)
−0.945669 + 0.325131i \(0.894592\pi\)
\(12\) 0 0
\(13\) 344.000 0.156577 0.0782886 0.996931i \(-0.475054\pi\)
0.0782886 + 0.996931i \(0.475054\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 7140.36i − 1.45336i −0.686975 0.726681i \(-0.741062\pi\)
0.686975 0.726681i \(-0.258938\pi\)
\(18\) 0 0
\(19\) 2320.00 0.338242 0.169121 0.985595i \(-0.445907\pi\)
0.169121 + 0.985595i \(0.445907\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 5753.02i − 0.472838i −0.971651 0.236419i \(-0.924026\pi\)
0.971651 0.236419i \(-0.0759738\pi\)
\(24\) 0 0
\(25\) 11575.0 0.740800
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 23152.1i 0.949284i 0.880179 + 0.474642i \(0.157422\pi\)
−0.880179 + 0.474642i \(0.842578\pi\)
\(30\) 0 0
\(31\) 10564.0 0.354604 0.177302 0.984157i \(-0.443263\pi\)
0.177302 + 0.984157i \(0.443263\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 33347.2i − 0.777776i
\(36\) 0 0
\(37\) −24082.0 −0.475431 −0.237715 0.971335i \(-0.576399\pi\)
−0.237715 + 0.971335i \(0.576399\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 108836.i − 1.57915i −0.613655 0.789574i \(-0.710302\pi\)
0.613655 0.789574i \(-0.289698\pi\)
\(42\) 0 0
\(43\) 90952.0 1.14395 0.571975 0.820271i \(-0.306178\pi\)
0.571975 + 0.820271i \(0.306178\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 128959.i − 1.24211i −0.783768 0.621054i \(-0.786705\pi\)
0.783768 0.621054i \(-0.213295\pi\)
\(48\) 0 0
\(49\) 156927. 1.33386
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 196685.i − 1.32112i −0.750773 0.660561i \(-0.770319\pi\)
0.750773 0.660561i \(-0.229681\pi\)
\(54\) 0 0
\(55\) −55080.0 −0.331059
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 39812.9i − 0.193851i −0.995292 0.0969255i \(-0.969099\pi\)
0.995292 0.0969255i \(-0.0309009\pi\)
\(60\) 0 0
\(61\) 251138. 1.10643 0.553214 0.833039i \(-0.313401\pi\)
0.553214 + 0.833039i \(0.313401\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 21892.0i 0.0797161i
\(66\) 0 0
\(67\) 216088. 0.718466 0.359233 0.933248i \(-0.383038\pi\)
0.359233 + 0.933248i \(0.383038\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 53915.5i − 0.150639i −0.997159 0.0753197i \(-0.976002\pi\)
0.997159 0.0753197i \(-0.0239977\pi\)
\(72\) 0 0
\(73\) −308176. −0.792192 −0.396096 0.918209i \(-0.629635\pi\)
−0.396096 + 0.918209i \(0.629635\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 453521.i − 0.993403i
\(78\) 0 0
\(79\) 540124. 1.09550 0.547750 0.836642i \(-0.315485\pi\)
0.547750 + 0.836642i \(0.315485\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 932346.i − 1.63058i −0.579051 0.815291i \(-0.696577\pi\)
0.579051 0.815291i \(-0.303423\pi\)
\(84\) 0 0
\(85\) 454410. 0.739931
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 223413.i 0.316912i 0.987366 + 0.158456i \(0.0506516\pi\)
−0.987366 + 0.158456i \(0.949348\pi\)
\(90\) 0 0
\(91\) −180256. −0.239202
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 147644.i 0.172205i
\(96\) 0 0
\(97\) −37168.0 −0.0407243 −0.0203622 0.999793i \(-0.506482\pi\)
−0.0203622 + 0.999793i \(0.506482\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 559787.i 0.543323i 0.962393 + 0.271662i \(0.0875732\pi\)
−0.962393 + 0.271662i \(0.912427\pi\)
\(102\) 0 0
\(103\) −1.46018e6 −1.33627 −0.668136 0.744039i \(-0.732907\pi\)
−0.668136 + 0.744039i \(0.732907\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1.29031e6i 1.05327i 0.850090 + 0.526637i \(0.176547\pi\)
−0.850090 + 0.526637i \(0.823453\pi\)
\(108\) 0 0
\(109\) −1.43548e6 −1.10845 −0.554227 0.832366i \(-0.686986\pi\)
−0.554227 + 0.832366i \(0.686986\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) − 186426.i − 0.129202i −0.997911 0.0646012i \(-0.979422\pi\)
0.997911 0.0646012i \(-0.0205775\pi\)
\(114\) 0 0
\(115\) 366120. 0.240730
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 3.74155e6i 2.22030i
\(120\) 0 0
\(121\) 1.02247e6 0.577159
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1.73100e6i 0.886271i
\(126\) 0 0
\(127\) 127060. 0.0620294 0.0310147 0.999519i \(-0.490126\pi\)
0.0310147 + 0.999519i \(0.490126\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) − 2.62348e6i − 1.16698i −0.812120 0.583490i \(-0.801687\pi\)
0.812120 0.583490i \(-0.198313\pi\)
\(132\) 0 0
\(133\) −1.21568e6 −0.516731
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 202310.i 0.0786785i 0.999226 + 0.0393393i \(0.0125253\pi\)
−0.999226 + 0.0393393i \(0.987475\pi\)
\(138\) 0 0
\(139\) −2.02642e6 −0.754546 −0.377273 0.926102i \(-0.623138\pi\)
−0.377273 + 0.926102i \(0.623138\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 297732.i 0.101816i
\(144\) 0 0
\(145\) −1.47339e6 −0.483297
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 4.25534e6i 1.28640i 0.765699 + 0.643199i \(0.222393\pi\)
−0.765699 + 0.643199i \(0.777607\pi\)
\(150\) 0 0
\(151\) −3.74035e6 −1.08638 −0.543189 0.839610i \(-0.682783\pi\)
−0.543189 + 0.839610i \(0.682783\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 672289.i 0.180535i
\(156\) 0 0
\(157\) −2.38813e6 −0.617105 −0.308552 0.951207i \(-0.599845\pi\)
−0.308552 + 0.951207i \(0.599845\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 3.01458e6i 0.722353i
\(162\) 0 0
\(163\) 6.74519e6 1.55751 0.778756 0.627327i \(-0.215851\pi\)
0.778756 + 0.627327i \(0.215851\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 6.61699e6i 1.42073i 0.703834 + 0.710364i \(0.251470\pi\)
−0.703834 + 0.710364i \(0.748530\pi\)
\(168\) 0 0
\(169\) −4.70847e6 −0.975484
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 5.58077e6i − 1.07784i −0.842356 0.538922i \(-0.818832\pi\)
0.842356 0.538922i \(-0.181168\pi\)
\(174\) 0 0
\(175\) −6.06530e6 −1.13172
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) − 4.87103e6i − 0.849301i −0.905357 0.424650i \(-0.860397\pi\)
0.905357 0.424650i \(-0.139603\pi\)
\(180\) 0 0
\(181\) 8.47546e6 1.42931 0.714657 0.699475i \(-0.246583\pi\)
0.714657 + 0.699475i \(0.246583\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) − 1.53257e6i − 0.242050i
\(186\) 0 0
\(187\) 6.17998e6 0.945066
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) − 97037.7i − 0.0139264i −0.999976 0.00696322i \(-0.997784\pi\)
0.999976 0.00696322i \(-0.00221648\pi\)
\(192\) 0 0
\(193\) 7.49473e6 1.04252 0.521260 0.853398i \(-0.325462\pi\)
0.521260 + 0.853398i \(0.325462\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 9.84656e6i − 1.28791i −0.765063 0.643956i \(-0.777292\pi\)
0.765063 0.643956i \(-0.222708\pi\)
\(198\) 0 0
\(199\) −3.54170e6 −0.449420 −0.224710 0.974426i \(-0.572144\pi\)
−0.224710 + 0.974426i \(0.572144\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) − 1.21317e7i − 1.45022i
\(204\) 0 0
\(205\) 6.92631e6 0.803971
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 2.00796e6i 0.219946i
\(210\) 0 0
\(211\) 6.77298e6 0.720996 0.360498 0.932760i \(-0.382607\pi\)
0.360498 + 0.932760i \(0.382607\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 5.78815e6i 0.582404i
\(216\) 0 0
\(217\) −5.53554e6 −0.541727
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) − 2.45629e6i − 0.227563i
\(222\) 0 0
\(223\) 3.34186e6 0.301352 0.150676 0.988583i \(-0.451855\pi\)
0.150676 + 0.988583i \(0.451855\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 1.62013e7i − 1.38507i −0.721385 0.692535i \(-0.756494\pi\)
0.721385 0.692535i \(-0.243506\pi\)
\(228\) 0 0
\(229\) −1.66351e7 −1.38522 −0.692611 0.721311i \(-0.743540\pi\)
−0.692611 + 0.721311i \(0.743540\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 8.85600e6i − 0.700116i −0.936728 0.350058i \(-0.886162\pi\)
0.936728 0.350058i \(-0.113838\pi\)
\(234\) 0 0
\(235\) 8.20692e6 0.632378
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 1.12995e7i 0.827689i 0.910348 + 0.413845i \(0.135814\pi\)
−0.910348 + 0.413845i \(0.864186\pi\)
\(240\) 0 0
\(241\) 1.50090e7 1.07226 0.536129 0.844136i \(-0.319886\pi\)
0.536129 + 0.844136i \(0.319886\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 9.98677e6i 0.679089i
\(246\) 0 0
\(247\) 798080. 0.0529609
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) − 3.04076e7i − 1.92292i −0.274950 0.961458i \(-0.588661\pi\)
0.274950 0.961458i \(-0.411339\pi\)
\(252\) 0 0
\(253\) 4.97923e6 0.307469
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 1.97422e7i − 1.16305i −0.813530 0.581523i \(-0.802457\pi\)
0.813530 0.581523i \(-0.197543\pi\)
\(258\) 0 0
\(259\) 1.26190e7 0.726314
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) − 3.48531e6i − 0.191591i −0.995401 0.0957954i \(-0.969461\pi\)
0.995401 0.0957954i \(-0.0305394\pi\)
\(264\) 0 0
\(265\) 1.25169e7 0.672605
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 5.65391e6i 0.290464i 0.989398 + 0.145232i \(0.0463928\pi\)
−0.989398 + 0.145232i \(0.953607\pi\)
\(270\) 0 0
\(271\) −2.91893e7 −1.46662 −0.733308 0.679897i \(-0.762024\pi\)
−0.733308 + 0.679897i \(0.762024\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 1.00181e7i 0.481714i
\(276\) 0 0
\(277\) 2.29938e7 1.08186 0.540931 0.841067i \(-0.318072\pi\)
0.540931 + 0.841067i \(0.318072\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) − 1.25303e6i − 0.0564730i −0.999601 0.0282365i \(-0.991011\pi\)
0.999601 0.0282365i \(-0.00898916\pi\)
\(282\) 0 0
\(283\) 1.45129e7 0.640317 0.320159 0.947364i \(-0.396264\pi\)
0.320159 + 0.947364i \(0.396264\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 5.70303e7i 2.41246i
\(288\) 0 0
\(289\) −2.68472e7 −1.11226
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 1.12729e7i 0.448160i 0.974571 + 0.224080i \(0.0719377\pi\)
−0.974571 + 0.224080i \(0.928062\pi\)
\(294\) 0 0
\(295\) 2.53368e6 0.0986929
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) − 1.97904e6i − 0.0740356i
\(300\) 0 0
\(301\) −4.76588e7 −1.74761
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 1.59823e7i 0.563301i
\(306\) 0 0
\(307\) −4.51916e7 −1.56186 −0.780930 0.624618i \(-0.785255\pi\)
−0.780930 + 0.624618i \(0.785255\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 9.74134e6i 0.323845i 0.986803 + 0.161923i \(0.0517695\pi\)
−0.986803 + 0.161923i \(0.948230\pi\)
\(312\) 0 0
\(313\) −5.30265e6 −0.172926 −0.0864630 0.996255i \(-0.527556\pi\)
−0.0864630 + 0.996255i \(0.527556\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 1.05462e7i − 0.331068i −0.986204 0.165534i \(-0.947065\pi\)
0.986204 0.165534i \(-0.0529348\pi\)
\(318\) 0 0
\(319\) −2.00381e7 −0.617283
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 1.65656e7i − 0.491587i
\(324\) 0 0
\(325\) 3.98180e6 0.115992
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 6.75747e7i 1.89756i
\(330\) 0 0
\(331\) 3.81242e7 1.05128 0.525638 0.850708i \(-0.323826\pi\)
0.525638 + 0.850708i \(0.323826\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 1.37518e7i 0.365783i
\(336\) 0 0
\(337\) −1.22682e7 −0.320548 −0.160274 0.987073i \(-0.551238\pi\)
−0.160274 + 0.987073i \(0.551238\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 9.14313e6i 0.230585i
\(342\) 0 0
\(343\) −2.05817e7 −0.510033
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 1.65809e7i − 0.396843i −0.980117 0.198422i \(-0.936418\pi\)
0.980117 0.198422i \(-0.0635815\pi\)
\(348\) 0 0
\(349\) 4.81038e7 1.13163 0.565813 0.824534i \(-0.308562\pi\)
0.565813 + 0.824534i \(0.308562\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 1.47570e7i 0.335485i 0.985831 + 0.167742i \(0.0536477\pi\)
−0.985831 + 0.167742i \(0.946352\pi\)
\(354\) 0 0
\(355\) 3.43116e6 0.0766930
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) − 1.78509e7i − 0.385813i −0.981217 0.192907i \(-0.938209\pi\)
0.981217 0.192907i \(-0.0617914\pi\)
\(360\) 0 0
\(361\) −4.16635e7 −0.885593
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) − 1.96122e7i − 0.403318i
\(366\) 0 0
\(367\) −8.67940e6 −0.175587 −0.0877934 0.996139i \(-0.527982\pi\)
−0.0877934 + 0.996139i \(0.527982\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 1.03063e8i 2.01827i
\(372\) 0 0
\(373\) −7.94052e7 −1.53011 −0.765055 0.643965i \(-0.777288\pi\)
−0.765055 + 0.643965i \(0.777288\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 7.96432e6i 0.148636i
\(378\) 0 0
\(379\) 1.46346e7 0.268821 0.134410 0.990926i \(-0.457086\pi\)
0.134410 + 0.990926i \(0.457086\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 9.16736e6i 0.163173i 0.996666 + 0.0815865i \(0.0259987\pi\)
−0.996666 + 0.0815865i \(0.974001\pi\)
\(384\) 0 0
\(385\) 2.88619e7 0.505758
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 9.26668e6i 0.157426i 0.996897 + 0.0787128i \(0.0250810\pi\)
−0.996897 + 0.0787128i \(0.974919\pi\)
\(390\) 0 0
\(391\) −4.10787e7 −0.687205
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 3.43733e7i 0.557737i
\(396\) 0 0
\(397\) 7.25544e7 1.15956 0.579779 0.814774i \(-0.303139\pi\)
0.579779 + 0.814774i \(0.303139\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) − 7.37246e7i − 1.14335i −0.820480 0.571675i \(-0.806294\pi\)
0.820480 0.571675i \(-0.193706\pi\)
\(402\) 0 0
\(403\) 3.63402e6 0.0555228
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 2.08429e7i − 0.309155i
\(408\) 0 0
\(409\) 3.43558e7 0.502146 0.251073 0.967968i \(-0.419217\pi\)
0.251073 + 0.967968i \(0.419217\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 2.08620e7i 0.296146i
\(414\) 0 0
\(415\) 5.93341e7 0.830157
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) − 1.31347e8i − 1.78557i −0.450479 0.892787i \(-0.648747\pi\)
0.450479 0.892787i \(-0.351253\pi\)
\(420\) 0 0
\(421\) 2.36756e6 0.0317289 0.0158644 0.999874i \(-0.494950\pi\)
0.0158644 + 0.999874i \(0.494950\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) − 8.26497e7i − 1.07665i
\(426\) 0 0
\(427\) −1.31596e8 −1.69029
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) − 1.13049e8i − 1.41200i −0.708212 0.705999i \(-0.750498\pi\)
0.708212 0.705999i \(-0.249502\pi\)
\(432\) 0 0
\(433\) 4.50927e7 0.555447 0.277723 0.960661i \(-0.410420\pi\)
0.277723 + 0.960661i \(0.410420\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 1.33470e7i − 0.159934i
\(438\) 0 0
\(439\) 1.61605e8 1.91013 0.955064 0.296399i \(-0.0957858\pi\)
0.955064 + 0.296399i \(0.0957858\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 2.54011e7i − 0.292174i −0.989272 0.146087i \(-0.953332\pi\)
0.989272 0.146087i \(-0.0466679\pi\)
\(444\) 0 0
\(445\) −1.42179e7 −0.161345
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) − 9.43898e7i − 1.04276i −0.853323 0.521382i \(-0.825417\pi\)
0.853323 0.521382i \(-0.174583\pi\)
\(450\) 0 0
\(451\) 9.41978e7 1.02686
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) − 1.14714e7i − 0.121782i
\(456\) 0 0
\(457\) 4.68452e7 0.490813 0.245407 0.969420i \(-0.421078\pi\)
0.245407 + 0.969420i \(0.421078\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 2.33797e7i 0.238636i 0.992856 + 0.119318i \(0.0380708\pi\)
−0.992856 + 0.119318i \(0.961929\pi\)
\(462\) 0 0
\(463\) 4.98269e7 0.502019 0.251010 0.967985i \(-0.419237\pi\)
0.251010 + 0.967985i \(0.419237\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 9.52369e7i 0.935092i 0.883969 + 0.467546i \(0.154862\pi\)
−0.883969 + 0.467546i \(0.845138\pi\)
\(468\) 0 0
\(469\) −1.13230e8 −1.09760
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 7.87188e7i 0.743867i
\(474\) 0 0
\(475\) 2.68540e7 0.250569
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) − 1.97141e8i − 1.79378i −0.442251 0.896891i \(-0.645820\pi\)
0.442251 0.896891i \(-0.354180\pi\)
\(480\) 0 0
\(481\) −8.28421e6 −0.0744416
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) − 2.36536e6i − 0.0207334i
\(486\) 0 0
\(487\) −1.92602e6 −0.0166753 −0.00833765 0.999965i \(-0.502654\pi\)
−0.00833765 + 0.999965i \(0.502654\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 1.79722e8i 1.51830i 0.650916 + 0.759150i \(0.274385\pi\)
−0.650916 + 0.759150i \(0.725615\pi\)
\(492\) 0 0
\(493\) 1.65314e8 1.37965
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 2.82517e7i 0.230131i
\(498\) 0 0
\(499\) −1.54018e8 −1.23956 −0.619782 0.784774i \(-0.712779\pi\)
−0.619782 + 0.784774i \(0.712779\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) − 2.25142e7i − 0.176910i −0.996080 0.0884551i \(-0.971807\pi\)
0.996080 0.0884551i \(-0.0281930\pi\)
\(504\) 0 0
\(505\) −3.56246e7 −0.276615
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) − 2.16126e8i − 1.63891i −0.573147 0.819453i \(-0.694278\pi\)
0.573147 0.819453i \(-0.305722\pi\)
\(510\) 0 0
\(511\) 1.61484e8 1.21023
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 9.29253e7i − 0.680318i
\(516\) 0 0
\(517\) 1.11614e8 0.807695
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 8.35166e7i 0.590554i 0.955412 + 0.295277i \(0.0954120\pi\)
−0.955412 + 0.295277i \(0.904588\pi\)
\(522\) 0 0
\(523\) −2.08856e8 −1.45997 −0.729983 0.683466i \(-0.760472\pi\)
−0.729983 + 0.683466i \(0.760472\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 7.54308e7i − 0.515367i
\(528\) 0 0
\(529\) 1.14939e8 0.776424
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) − 3.74397e7i − 0.247258i
\(534\) 0 0
\(535\) −8.21146e7 −0.536240
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 1.35820e8i 0.867357i
\(540\) 0 0
\(541\) −1.21245e8 −0.765727 −0.382863 0.923805i \(-0.625062\pi\)
−0.382863 + 0.923805i \(0.625062\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) − 9.13534e7i − 0.564333i
\(546\) 0 0
\(547\) 1.33857e8 0.817861 0.408931 0.912565i \(-0.365902\pi\)
0.408931 + 0.912565i \(0.365902\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 5.37128e7i 0.321087i
\(552\) 0 0
\(553\) −2.83025e8 −1.67359
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 8.20694e7i − 0.474915i −0.971398 0.237457i \(-0.923686\pi\)
0.971398 0.237457i \(-0.0763140\pi\)
\(558\) 0 0
\(559\) 3.12875e7 0.179116
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 2.21977e8i 1.24389i 0.783059 + 0.621947i \(0.213658\pi\)
−0.783059 + 0.621947i \(0.786342\pi\)
\(564\) 0 0
\(565\) 1.18641e7 0.0657792
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 2.88087e8i 1.56382i 0.623391 + 0.781911i \(0.285755\pi\)
−0.623391 + 0.781911i \(0.714245\pi\)
\(570\) 0 0
\(571\) −1.83227e8 −0.984197 −0.492098 0.870540i \(-0.663770\pi\)
−0.492098 + 0.870540i \(0.663770\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) − 6.65912e7i − 0.350278i
\(576\) 0 0
\(577\) −2.07783e8 −1.08164 −0.540820 0.841139i \(-0.681886\pi\)
−0.540820 + 0.841139i \(0.681886\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 4.88549e8i 2.49104i
\(582\) 0 0
\(583\) 1.70230e8 0.859075
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 3.28908e8i 1.62615i 0.582161 + 0.813073i \(0.302207\pi\)
−0.582161 + 0.813073i \(0.697793\pi\)
\(588\) 0 0
\(589\) 2.45085e7 0.119942
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) − 1.97249e7i − 0.0945914i −0.998881 0.0472957i \(-0.984940\pi\)
0.998881 0.0472957i \(-0.0150603\pi\)
\(594\) 0 0
\(595\) −2.38111e8 −1.13039
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) − 9.38486e7i − 0.436664i −0.975875 0.218332i \(-0.929938\pi\)
0.975875 0.218332i \(-0.0700616\pi\)
\(600\) 0 0
\(601\) 1.53106e8 0.705293 0.352646 0.935757i \(-0.385282\pi\)
0.352646 + 0.935757i \(0.385282\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 6.50698e7i 0.293842i
\(606\) 0 0
\(607\) −1.08279e8 −0.484147 −0.242074 0.970258i \(-0.577828\pi\)
−0.242074 + 0.970258i \(0.577828\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) − 4.43620e7i − 0.194486i
\(612\) 0 0
\(613\) −2.60288e8 −1.12999 −0.564993 0.825096i \(-0.691121\pi\)
−0.564993 + 0.825096i \(0.691121\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 1.65376e8i 0.704073i 0.935986 + 0.352036i \(0.114511\pi\)
−0.935986 + 0.352036i \(0.885489\pi\)
\(618\) 0 0
\(619\) 1.36836e8 0.576935 0.288468 0.957490i \(-0.406854\pi\)
0.288468 + 0.957490i \(0.406854\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) − 1.17069e8i − 0.484146i
\(624\) 0 0
\(625\) 7.06994e7 0.289585
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 1.71954e8i 0.690973i
\(630\) 0 0
\(631\) −2.66941e8 −1.06249 −0.531247 0.847217i \(-0.678277\pi\)
−0.531247 + 0.847217i \(0.678277\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 8.08605e6i 0.0315802i
\(636\) 0 0
\(637\) 5.39829e7 0.208852
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) − 3.69866e8i − 1.40433i −0.712013 0.702167i \(-0.752216\pi\)
0.712013 0.702167i \(-0.247784\pi\)
\(642\) 0 0
\(643\) 9.29168e7 0.349511 0.174756 0.984612i \(-0.444086\pi\)
0.174756 + 0.984612i \(0.444086\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 9.21336e7i − 0.340177i −0.985429 0.170089i \(-0.945595\pi\)
0.985429 0.170089i \(-0.0544054\pi\)
\(648\) 0 0
\(649\) 3.44580e7 0.126054
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 2.20689e8i − 0.792576i −0.918126 0.396288i \(-0.870298\pi\)
0.918126 0.396288i \(-0.129702\pi\)
\(654\) 0 0
\(655\) 1.66957e8 0.594130
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 5.01619e8i 1.75274i 0.481639 + 0.876370i \(0.340042\pi\)
−0.481639 + 0.876370i \(0.659958\pi\)
\(660\) 0 0
\(661\) −2.78166e8 −0.963163 −0.481582 0.876401i \(-0.659937\pi\)
−0.481582 + 0.876401i \(0.659937\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) − 7.73654e7i − 0.263076i
\(666\) 0 0
\(667\) 1.33194e8 0.448858
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 2.17360e8i 0.719468i
\(672\) 0 0
\(673\) 5.34850e8 1.75464 0.877318 0.479910i \(-0.159331\pi\)
0.877318 + 0.479910i \(0.159331\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 1.41358e6i 0.00455568i 0.999997 + 0.00227784i \(0.000725059\pi\)
−0.999997 + 0.00227784i \(0.999275\pi\)
\(678\) 0 0
\(679\) 1.94760e7 0.0622144
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) − 2.41616e8i − 0.758340i −0.925327 0.379170i \(-0.876209\pi\)
0.925327 0.379170i \(-0.123791\pi\)
\(684\) 0 0
\(685\) −1.28750e7 −0.0400566
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) − 6.76595e7i − 0.206857i
\(690\) 0 0
\(691\) −5.30543e8 −1.60800 −0.804001 0.594628i \(-0.797299\pi\)
−0.804001 + 0.594628i \(0.797299\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) − 1.28961e8i − 0.384152i
\(696\) 0 0
\(697\) −7.77132e8 −2.29507
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) − 5.49256e7i − 0.159449i −0.996817 0.0797244i \(-0.974596\pi\)
0.996817 0.0797244i \(-0.0254040\pi\)
\(702\) 0 0
\(703\) −5.58702e7 −0.160811
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 2.93328e8i − 0.830034i
\(708\) 0 0
\(709\) 3.16706e7 0.0888622 0.0444311 0.999012i \(-0.485852\pi\)
0.0444311 + 0.999012i \(0.485852\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) − 6.07749e7i − 0.167670i
\(714\) 0 0
\(715\) −1.89475e7 −0.0518363
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 4.37723e8i 1.17764i 0.808264 + 0.588820i \(0.200407\pi\)
−0.808264 + 0.588820i \(0.799593\pi\)
\(720\) 0 0
\(721\) 7.65134e8 2.04142
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 2.67985e8i 0.703230i
\(726\) 0 0
\(727\) 4.64180e8 1.20805 0.604023 0.796967i \(-0.293564\pi\)
0.604023 + 0.796967i \(0.293564\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) − 6.49430e8i − 1.66257i
\(732\) 0 0
\(733\) 7.04886e8 1.78981 0.894905 0.446256i \(-0.147243\pi\)
0.894905 + 0.446256i \(0.147243\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1.87024e8i 0.467191i
\(738\) 0 0
\(739\) 2.83900e8 0.703448 0.351724 0.936104i \(-0.385596\pi\)
0.351724 + 0.936104i \(0.385596\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) − 5.01018e8i − 1.22148i −0.791830 0.610741i \(-0.790872\pi\)
0.791830 0.610741i \(-0.209128\pi\)
\(744\) 0 0
\(745\) −2.70808e8 −0.654927
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) − 6.76120e8i − 1.60908i
\(750\) 0 0
\(751\) 1.48249e8 0.350004 0.175002 0.984568i \(-0.444007\pi\)
0.175002 + 0.984568i \(0.444007\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) − 2.38034e8i − 0.553094i
\(756\) 0 0
\(757\) 2.14422e8 0.494291 0.247145 0.968978i \(-0.420507\pi\)
0.247145 + 0.968978i \(0.420507\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 2.69760e7i 0.0612101i 0.999532 + 0.0306051i \(0.00974341\pi\)
−0.999532 + 0.0306051i \(0.990257\pi\)
\(762\) 0 0
\(763\) 7.52192e8 1.69338
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 1.36957e7i − 0.0303526i
\(768\) 0 0
\(769\) −4.90064e8 −1.07764 −0.538821 0.842421i \(-0.681130\pi\)
−0.538821 + 0.842421i \(0.681130\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 4.17032e8i − 0.902882i −0.892301 0.451441i \(-0.850910\pi\)
0.892301 0.451441i \(-0.149090\pi\)
\(774\) 0 0
\(775\) 1.22278e8 0.262690
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 2.52501e8i − 0.534134i
\(780\) 0 0
\(781\) 4.66638e7 0.0979550
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) − 1.51980e8i − 0.314179i
\(786\) 0 0
\(787\) 3.46111e8 0.710054 0.355027 0.934856i \(-0.384472\pi\)
0.355027 + 0.934856i \(0.384472\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 9.76872e7i 0.197382i
\(792\) 0 0
\(793\) 8.63915e7 0.173241
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 7.25306e8i − 1.43267i −0.697756 0.716335i \(-0.745818\pi\)
0.697756 0.716335i \(-0.254182\pi\)
\(798\) 0 0
\(799\) −9.20816e8 −1.80523
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) − 2.66726e8i − 0.515132i
\(804\) 0 0
\(805\) −1.91847e8 −0.367762
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 4.93161e8i 0.931415i 0.884939 + 0.465707i \(0.154200\pi\)
−0.884939 + 0.465707i \(0.845800\pi\)
\(810\) 0 0
\(811\) 5.34731e7 0.100247 0.0501237 0.998743i \(-0.484038\pi\)
0.0501237 + 0.998743i \(0.484038\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 4.29261e8i 0.792956i
\(816\) 0 0
\(817\) 2.11009e8 0.386931
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 1.07674e9i 1.94573i 0.231377 + 0.972864i \(0.425677\pi\)
−0.231377 + 0.972864i \(0.574323\pi\)
\(822\) 0 0
\(823\) −8.88441e7 −0.159378 −0.0796891 0.996820i \(-0.525393\pi\)
−0.0796891 + 0.996820i \(0.525393\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 9.82047e7i − 0.173626i −0.996225 0.0868132i \(-0.972332\pi\)
0.996225 0.0868132i \(-0.0276683\pi\)
\(828\) 0 0
\(829\) 2.25045e8 0.395008 0.197504 0.980302i \(-0.436717\pi\)
0.197504 + 0.980302i \(0.436717\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) − 1.12052e9i − 1.93858i
\(834\) 0 0
\(835\) −4.21103e8 −0.723317
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 7.84433e8i 1.32822i 0.747635 + 0.664110i \(0.231189\pi\)
−0.747635 + 0.664110i \(0.768811\pi\)
\(840\) 0 0
\(841\) 5.88040e7 0.0988597
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) − 2.99645e8i − 0.496635i
\(846\) 0 0
\(847\) −5.35776e8 −0.881724
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 1.38544e8i 0.224802i
\(852\) 0 0
\(853\) −7.97906e7 −0.128560 −0.0642798 0.997932i \(-0.520475\pi\)
−0.0642798 + 0.997932i \(0.520475\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 1.06107e9i 1.68579i 0.538080 + 0.842894i \(0.319150\pi\)
−0.538080 + 0.842894i \(0.680850\pi\)
\(858\) 0 0
\(859\) −3.34286e8 −0.527399 −0.263699 0.964605i \(-0.584943\pi\)
−0.263699 + 0.964605i \(0.584943\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 2.65292e7i 0.0412754i 0.999787 + 0.0206377i \(0.00656965\pi\)
−0.999787 + 0.0206377i \(0.993430\pi\)
\(864\) 0 0
\(865\) 3.55158e8 0.548749
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 4.67477e8i 0.712362i
\(870\) 0 0
\(871\) 7.43343e7 0.112495
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) − 9.07043e8i − 1.35395i
\(876\) 0 0
\(877\) −1.30791e8 −0.193901 −0.0969504 0.995289i \(-0.530909\pi\)
−0.0969504 + 0.995289i \(0.530909\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) − 9.18953e8i − 1.34390i −0.740598 0.671948i \(-0.765458\pi\)
0.740598 0.671948i \(-0.234542\pi\)
\(882\) 0 0
\(883\) 1.09112e9 1.58486 0.792432 0.609961i \(-0.208815\pi\)
0.792432 + 0.609961i \(0.208815\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 5.36427e8i 0.768670i 0.923194 + 0.384335i \(0.125569\pi\)
−0.923194 + 0.384335i \(0.874431\pi\)
\(888\) 0 0
\(889\) −6.65794e7 −0.0947621
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 2.99186e8i − 0.420133i
\(894\) 0 0
\(895\) 3.09990e8 0.432393
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 2.44579e8i 0.336620i
\(900\) 0 0
\(901\) −1.40440e9 −1.92007
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 5.39375e8i 0.727688i
\(906\) 0 0
\(907\) 4.60985e8 0.617824 0.308912 0.951091i \(-0.400035\pi\)
0.308912 + 0.951091i \(0.400035\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 8.68400e8i 1.14859i 0.818649 + 0.574295i \(0.194724\pi\)
−0.818649 + 0.574295i \(0.805276\pi\)
\(912\) 0 0
\(913\) 8.06944e8 1.06031
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 1.37470e9i 1.78279i
\(918\) 0 0
\(919\) −1.28061e9 −1.64995 −0.824976 0.565168i \(-0.808811\pi\)
−0.824976 + 0.565168i \(0.808811\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) − 1.85469e7i − 0.0235867i
\(924\) 0 0
\(925\) −2.78749e8 −0.352199
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) − 3.23960e8i − 0.404058i −0.979380 0.202029i \(-0.935246\pi\)
0.979380 0.202029i \(-0.0647536\pi\)
\(930\) 0 0
\(931\) 3.64071e8 0.451166
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 3.93291e8i 0.481149i
\(936\) 0 0
\(937\) −1.33188e9 −1.61900 −0.809498 0.587123i \(-0.800261\pi\)
−0.809498 + 0.587123i \(0.800261\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 8.24869e8i 0.989957i 0.868905 + 0.494978i \(0.164824\pi\)
−0.868905 + 0.494978i \(0.835176\pi\)
\(942\) 0 0
\(943\) −6.26138e8 −0.746681
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 7.33660e8i − 0.863863i −0.901906 0.431931i \(-0.857832\pi\)
0.901906 0.431931i \(-0.142168\pi\)
\(948\) 0 0
\(949\) −1.06013e8 −0.124039
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 6.87744e8i 0.794600i 0.917689 + 0.397300i \(0.130053\pi\)
−0.917689 + 0.397300i \(0.869947\pi\)
\(954\) 0 0
\(955\) 6.17544e6 0.00709019
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) − 1.06011e8i − 0.120197i
\(960\) 0 0
\(961\) −7.75906e8 −0.874256
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 4.76962e8i 0.530764i
\(966\) 0 0
\(967\) −1.43001e9 −1.58147 −0.790733 0.612162i \(-0.790300\pi\)
−0.790733 + 0.612162i \(0.790300\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 1.05628e9i 1.15377i 0.816824 + 0.576887i \(0.195733\pi\)
−0.816824 + 0.576887i \(0.804267\pi\)
\(972\) 0 0
\(973\) 1.06185e9 1.15272
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 8.72371e8i 0.935443i 0.883876 + 0.467722i \(0.154925\pi\)
−0.883876 + 0.467722i \(0.845075\pi\)
\(978\) 0 0
\(979\) −1.93364e8 −0.206076
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 7.90860e8i 0.832605i 0.909226 + 0.416302i \(0.136674\pi\)
−0.909226 + 0.416302i \(0.863326\pi\)
\(984\) 0 0
\(985\) 6.26631e8 0.655697
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) − 5.23249e8i − 0.540903i
\(990\) 0 0
\(991\) −1.38796e8 −0.142612 −0.0713062 0.997454i \(-0.522717\pi\)
−0.0713062 + 0.997454i \(0.522717\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) − 2.25392e8i − 0.228807i
\(996\) 0 0
\(997\) −3.29036e8 −0.332015 −0.166008 0.986124i \(-0.553088\pi\)
−0.166008 + 0.986124i \(0.553088\pi\)
\(998\) 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 144.7.e.a.17.2 2
3.2 odd 2 inner 144.7.e.a.17.1 2
4.3 odd 2 9.7.b.a.8.2 yes 2
8.3 odd 2 576.7.e.l.449.1 2
8.5 even 2 576.7.e.a.449.1 2
12.11 even 2 9.7.b.a.8.1 2
20.3 even 4 225.7.d.a.224.3 4
20.7 even 4 225.7.d.a.224.2 4
20.19 odd 2 225.7.c.a.26.1 2
24.5 odd 2 576.7.e.a.449.2 2
24.11 even 2 576.7.e.l.449.2 2
28.27 even 2 441.7.b.a.197.2 2
36.7 odd 6 81.7.d.d.53.2 4
36.11 even 6 81.7.d.d.53.1 4
36.23 even 6 81.7.d.d.26.2 4
36.31 odd 6 81.7.d.d.26.1 4
60.23 odd 4 225.7.d.a.224.1 4
60.47 odd 4 225.7.d.a.224.4 4
60.59 even 2 225.7.c.a.26.2 2
84.83 odd 2 441.7.b.a.197.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
9.7.b.a.8.1 2 12.11 even 2
9.7.b.a.8.2 yes 2 4.3 odd 2
81.7.d.d.26.1 4 36.31 odd 6
81.7.d.d.26.2 4 36.23 even 6
81.7.d.d.53.1 4 36.11 even 6
81.7.d.d.53.2 4 36.7 odd 6
144.7.e.a.17.1 2 3.2 odd 2 inner
144.7.e.a.17.2 2 1.1 even 1 trivial
225.7.c.a.26.1 2 20.19 odd 2
225.7.c.a.26.2 2 60.59 even 2
225.7.d.a.224.1 4 60.23 odd 4
225.7.d.a.224.2 4 20.7 even 4
225.7.d.a.224.3 4 20.3 even 4
225.7.d.a.224.4 4 60.47 odd 4
441.7.b.a.197.1 2 84.83 odd 2
441.7.b.a.197.2 2 28.27 even 2
576.7.e.a.449.1 2 8.5 even 2
576.7.e.a.449.2 2 24.5 odd 2
576.7.e.l.449.1 2 8.3 odd 2
576.7.e.l.449.2 2 24.11 even 2