Properties

Label 2880.3.e.j.2431.2
Level $2880$
Weight $3$
Character 2880.2431
Analytic conductor $78.474$
Analytic rank $0$
Dimension $8$
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] = [2880,3,Mod(2431,2880)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2880, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2880.2431");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2880 = 2^{6} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2880.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: \(78.4743161358\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.85100625.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} - 2x^{6} + x^{5} + 3x^{4} + 2x^{3} - 8x^{2} - 8x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{19} \)
Twist minimal: no (minimal twist has level 60)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2431.2
Root \(1.04064 - 0.957636i\) of defining polynomial
Character \(\chi\) \(=\) 2880.2431
Dual form 2880.3.e.j.2431.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.23607 q^{5} -5.46770i q^{7} +O(q^{10})\) \(q-2.23607 q^{5} -5.46770i q^{7} -11.0403i q^{11} -10.1242 q^{13} +24.4146 q^{17} +23.7757i q^{19} -37.2526i q^{23} +5.00000 q^{25} -25.7726 q^{29} +4.83647i q^{31} +12.2261i q^{35} -35.6493 q^{37} +9.30410 q^{41} -70.0287i q^{43} -38.0223i q^{47} +19.1043 q^{49} +55.7762 q^{53} +24.6869i q^{55} -55.5411i q^{59} +82.2412 q^{61} +22.6384 q^{65} +104.493i q^{67} +76.7471i q^{71} -93.5215 q^{73} -60.3651 q^{77} +49.3762i q^{79} -72.3768i q^{83} -54.5927 q^{85} -115.691 q^{89} +55.3560i q^{91} -53.1641i q^{95} -72.9589 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 16 q^{13} + 40 q^{25} + 64 q^{29} + 112 q^{37} + 16 q^{41} - 56 q^{49} + 352 q^{53} + 176 q^{61} + 80 q^{65} - 240 q^{73} - 288 q^{77} - 160 q^{85} - 80 q^{89} + 432 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2880\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(641\) \(901\) \(2431\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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
\(5\) −2.23607 −0.447214
\(6\) 0 0
\(7\) − 5.46770i − 0.781100i −0.920582 0.390550i \(-0.872285\pi\)
0.920582 0.390550i \(-0.127715\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 11.0403i − 1.00366i −0.864965 0.501832i \(-0.832659\pi\)
0.864965 0.501832i \(-0.167341\pi\)
\(12\) 0 0
\(13\) −10.1242 −0.778784 −0.389392 0.921072i \(-0.627315\pi\)
−0.389392 + 0.921072i \(0.627315\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 24.4146 1.43615 0.718077 0.695964i \(-0.245023\pi\)
0.718077 + 0.695964i \(0.245023\pi\)
\(18\) 0 0
\(19\) 23.7757i 1.25135i 0.780082 + 0.625677i \(0.215177\pi\)
−0.780082 + 0.625677i \(0.784823\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 37.2526i − 1.61968i −0.586653 0.809838i \(-0.699555\pi\)
0.586653 0.809838i \(-0.300445\pi\)
\(24\) 0 0
\(25\) 5.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −25.7726 −0.888712 −0.444356 0.895850i \(-0.646567\pi\)
−0.444356 + 0.895850i \(0.646567\pi\)
\(30\) 0 0
\(31\) 4.83647i 0.156015i 0.996953 + 0.0780076i \(0.0248558\pi\)
−0.996953 + 0.0780076i \(0.975144\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 12.2261i 0.349319i
\(36\) 0 0
\(37\) −35.6493 −0.963495 −0.481747 0.876310i \(-0.659998\pi\)
−0.481747 + 0.876310i \(0.659998\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 9.30410 0.226929 0.113465 0.993542i \(-0.463805\pi\)
0.113465 + 0.993542i \(0.463805\pi\)
\(42\) 0 0
\(43\) − 70.0287i − 1.62857i −0.580462 0.814287i \(-0.697128\pi\)
0.580462 0.814287i \(-0.302872\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 38.0223i − 0.808984i −0.914542 0.404492i \(-0.867448\pi\)
0.914542 0.404492i \(-0.132552\pi\)
\(48\) 0 0
\(49\) 19.1043 0.389883
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 55.7762 1.05238 0.526191 0.850366i \(-0.323620\pi\)
0.526191 + 0.850366i \(0.323620\pi\)
\(54\) 0 0
\(55\) 24.6869i 0.448853i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 55.5411i − 0.941374i −0.882300 0.470687i \(-0.844006\pi\)
0.882300 0.470687i \(-0.155994\pi\)
\(60\) 0 0
\(61\) 82.2412 1.34822 0.674108 0.738633i \(-0.264528\pi\)
0.674108 + 0.738633i \(0.264528\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 22.6384 0.348283
\(66\) 0 0
\(67\) 104.493i 1.55960i 0.626026 + 0.779802i \(0.284680\pi\)
−0.626026 + 0.779802i \(0.715320\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 76.7471i 1.08094i 0.841362 + 0.540472i \(0.181754\pi\)
−0.841362 + 0.540472i \(0.818246\pi\)
\(72\) 0 0
\(73\) −93.5215 −1.28112 −0.640558 0.767910i \(-0.721297\pi\)
−0.640558 + 0.767910i \(0.721297\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −60.3651 −0.783963
\(78\) 0 0
\(79\) 49.3762i 0.625016i 0.949915 + 0.312508i \(0.101169\pi\)
−0.949915 + 0.312508i \(0.898831\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 72.3768i − 0.872010i −0.899944 0.436005i \(-0.856393\pi\)
0.899944 0.436005i \(-0.143607\pi\)
\(84\) 0 0
\(85\) −54.5927 −0.642267
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −115.691 −1.29990 −0.649950 0.759977i \(-0.725210\pi\)
−0.649950 + 0.759977i \(0.725210\pi\)
\(90\) 0 0
\(91\) 55.3560i 0.608308i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) − 53.1641i − 0.559623i
\(96\) 0 0
\(97\) −72.9589 −0.752154 −0.376077 0.926588i \(-0.622727\pi\)
−0.376077 + 0.926588i \(0.622727\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 29.4092 0.291180 0.145590 0.989345i \(-0.453492\pi\)
0.145590 + 0.989345i \(0.453492\pi\)
\(102\) 0 0
\(103\) 28.1884i 0.273673i 0.990594 + 0.136837i \(0.0436935\pi\)
−0.990594 + 0.136837i \(0.956306\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 4.50700i 0.0421215i 0.999778 + 0.0210607i \(0.00670434\pi\)
−0.999778 + 0.0210607i \(0.993296\pi\)
\(108\) 0 0
\(109\) −193.315 −1.77353 −0.886767 0.462217i \(-0.847054\pi\)
−0.886767 + 0.462217i \(0.847054\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −75.5727 −0.668785 −0.334392 0.942434i \(-0.608531\pi\)
−0.334392 + 0.942434i \(0.608531\pi\)
\(114\) 0 0
\(115\) 83.2992i 0.724341i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) − 133.492i − 1.12178i
\(120\) 0 0
\(121\) −0.888544 −0.00734334
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −11.1803 −0.0894427
\(126\) 0 0
\(127\) 131.306i 1.03390i 0.856015 + 0.516951i \(0.172933\pi\)
−0.856015 + 0.516951i \(0.827067\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 75.7533i 0.578270i 0.957288 + 0.289135i \(0.0933676\pi\)
−0.957288 + 0.289135i \(0.906632\pi\)
\(132\) 0 0
\(133\) 129.999 0.977433
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −66.7927 −0.487538 −0.243769 0.969833i \(-0.578384\pi\)
−0.243769 + 0.969833i \(0.578384\pi\)
\(138\) 0 0
\(139\) 38.1214i 0.274255i 0.990553 + 0.137127i \(0.0437869\pi\)
−0.990553 + 0.137127i \(0.956213\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 111.774i 0.781638i
\(144\) 0 0
\(145\) 57.6294 0.397444
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −126.717 −0.850449 −0.425225 0.905088i \(-0.639805\pi\)
−0.425225 + 0.905088i \(0.639805\pi\)
\(150\) 0 0
\(151\) 68.4403i 0.453247i 0.973982 + 0.226623i \(0.0727687\pi\)
−0.973982 + 0.226623i \(0.927231\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) − 10.8147i − 0.0697721i
\(156\) 0 0
\(157\) −25.5777 −0.162915 −0.0814577 0.996677i \(-0.525958\pi\)
−0.0814577 + 0.996677i \(0.525958\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −203.686 −1.26513
\(162\) 0 0
\(163\) 63.4771i 0.389430i 0.980860 + 0.194715i \(0.0623782\pi\)
−0.980860 + 0.194715i \(0.937622\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 12.3771i 0.0741144i 0.999313 + 0.0370572i \(0.0117984\pi\)
−0.999313 + 0.0370572i \(0.988202\pi\)
\(168\) 0 0
\(169\) −66.5008 −0.393496
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 59.3729 0.343196 0.171598 0.985167i \(-0.445107\pi\)
0.171598 + 0.985167i \(0.445107\pi\)
\(174\) 0 0
\(175\) − 27.3385i − 0.156220i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) − 252.782i − 1.41219i −0.708118 0.706094i \(-0.750455\pi\)
0.708118 0.706094i \(-0.249545\pi\)
\(180\) 0 0
\(181\) −125.373 −0.692670 −0.346335 0.938111i \(-0.612574\pi\)
−0.346335 + 0.938111i \(0.612574\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 79.7143 0.430888
\(186\) 0 0
\(187\) − 269.545i − 1.44142i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) − 97.4640i − 0.510283i −0.966904 0.255141i \(-0.917878\pi\)
0.966904 0.255141i \(-0.0821220\pi\)
\(192\) 0 0
\(193\) −342.376 −1.77397 −0.886985 0.461798i \(-0.847204\pi\)
−0.886985 + 0.461798i \(0.847204\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 74.4829 0.378086 0.189043 0.981969i \(-0.439461\pi\)
0.189043 + 0.981969i \(0.439461\pi\)
\(198\) 0 0
\(199\) − 178.027i − 0.894606i −0.894382 0.447303i \(-0.852385\pi\)
0.894382 0.447303i \(-0.147615\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 140.917i 0.694173i
\(204\) 0 0
\(205\) −20.8046 −0.101486
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 262.491 1.25594
\(210\) 0 0
\(211\) − 185.893i − 0.881008i −0.897751 0.440504i \(-0.854800\pi\)
0.897751 0.440504i \(-0.145200\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 156.589i 0.728321i
\(216\) 0 0
\(217\) 26.4444 0.121863
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −247.178 −1.11845
\(222\) 0 0
\(223\) − 202.724i − 0.909074i −0.890728 0.454537i \(-0.849805\pi\)
0.890728 0.454537i \(-0.150195\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 51.2708i − 0.225863i −0.993603 0.112931i \(-0.963976\pi\)
0.993603 0.112931i \(-0.0360240\pi\)
\(228\) 0 0
\(229\) −337.056 −1.47186 −0.735930 0.677058i \(-0.763255\pi\)
−0.735930 + 0.677058i \(0.763255\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 80.2851 0.344571 0.172286 0.985047i \(-0.444885\pi\)
0.172286 + 0.985047i \(0.444885\pi\)
\(234\) 0 0
\(235\) 85.0203i 0.361789i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) − 330.808i − 1.38413i −0.721834 0.692066i \(-0.756701\pi\)
0.721834 0.692066i \(-0.243299\pi\)
\(240\) 0 0
\(241\) −359.914 −1.49342 −0.746710 0.665150i \(-0.768368\pi\)
−0.746710 + 0.665150i \(0.768368\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −42.7184 −0.174361
\(246\) 0 0
\(247\) − 240.710i − 0.974534i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 312.213i 1.24388i 0.783067 + 0.621938i \(0.213654\pi\)
−0.783067 + 0.621938i \(0.786346\pi\)
\(252\) 0 0
\(253\) −411.280 −1.62561
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −80.2592 −0.312293 −0.156146 0.987734i \(-0.549907\pi\)
−0.156146 + 0.987734i \(0.549907\pi\)
\(258\) 0 0
\(259\) 194.920i 0.752586i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 487.967i 1.85539i 0.373342 + 0.927694i \(0.378212\pi\)
−0.373342 + 0.927694i \(0.621788\pi\)
\(264\) 0 0
\(265\) −124.719 −0.470639
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −309.553 −1.15076 −0.575378 0.817888i \(-0.695145\pi\)
−0.575378 + 0.817888i \(0.695145\pi\)
\(270\) 0 0
\(271\) 48.9693i 0.180698i 0.995910 + 0.0903492i \(0.0287983\pi\)
−0.995910 + 0.0903492i \(0.971202\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 55.2016i − 0.200733i
\(276\) 0 0
\(277\) 199.644 0.720736 0.360368 0.932810i \(-0.382651\pi\)
0.360368 + 0.932810i \(0.382651\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −61.1598 −0.217650 −0.108825 0.994061i \(-0.534709\pi\)
−0.108825 + 0.994061i \(0.534709\pi\)
\(282\) 0 0
\(283\) − 432.506i − 1.52829i −0.645044 0.764145i \(-0.723161\pi\)
0.645044 0.764145i \(-0.276839\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 50.8720i − 0.177254i
\(288\) 0 0
\(289\) 307.073 1.06254
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −283.234 −0.966668 −0.483334 0.875436i \(-0.660574\pi\)
−0.483334 + 0.875436i \(0.660574\pi\)
\(294\) 0 0
\(295\) 124.194i 0.420995i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 377.152i 1.26138i
\(300\) 0 0
\(301\) −382.896 −1.27208
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −183.897 −0.602940
\(306\) 0 0
\(307\) 100.077i 0.325983i 0.986627 + 0.162992i \(0.0521144\pi\)
−0.986627 + 0.162992i \(0.947886\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 404.185i 1.29963i 0.760092 + 0.649815i \(0.225154\pi\)
−0.760092 + 0.649815i \(0.774846\pi\)
\(312\) 0 0
\(313\) −128.579 −0.410795 −0.205398 0.978679i \(-0.565849\pi\)
−0.205398 + 0.978679i \(0.565849\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 85.9315 0.271077 0.135539 0.990772i \(-0.456724\pi\)
0.135539 + 0.990772i \(0.456724\pi\)
\(318\) 0 0
\(319\) 284.538i 0.891969i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 580.475i 1.79714i
\(324\) 0 0
\(325\) −50.6209 −0.155757
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −207.894 −0.631898
\(330\) 0 0
\(331\) − 183.391i − 0.554052i −0.960862 0.277026i \(-0.910651\pi\)
0.960862 0.277026i \(-0.0893488\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) − 233.654i − 0.697476i
\(336\) 0 0
\(337\) 168.130 0.498901 0.249451 0.968388i \(-0.419750\pi\)
0.249451 + 0.968388i \(0.419750\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 53.3962 0.156587
\(342\) 0 0
\(343\) − 372.374i − 1.08564i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 137.414i − 0.396006i −0.980201 0.198003i \(-0.936554\pi\)
0.980201 0.198003i \(-0.0634455\pi\)
\(348\) 0 0
\(349\) 13.4893 0.0386513 0.0193256 0.999813i \(-0.493848\pi\)
0.0193256 + 0.999813i \(0.493848\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −243.547 −0.689935 −0.344968 0.938615i \(-0.612110\pi\)
−0.344968 + 0.938615i \(0.612110\pi\)
\(354\) 0 0
\(355\) − 171.612i − 0.483413i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) − 17.9166i − 0.0499068i −0.999689 0.0249534i \(-0.992056\pi\)
0.999689 0.0249534i \(-0.00794374\pi\)
\(360\) 0 0
\(361\) −204.285 −0.565887
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 209.120 0.572933
\(366\) 0 0
\(367\) − 238.417i − 0.649637i −0.945776 0.324818i \(-0.894697\pi\)
0.945776 0.324818i \(-0.105303\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) − 304.968i − 0.822016i
\(372\) 0 0
\(373\) 181.271 0.485981 0.242990 0.970029i \(-0.421872\pi\)
0.242990 + 0.970029i \(0.421872\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 260.927 0.692114
\(378\) 0 0
\(379\) 306.206i 0.807931i 0.914774 + 0.403965i \(0.132368\pi\)
−0.914774 + 0.403965i \(0.867632\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) − 144.027i − 0.376050i −0.982164 0.188025i \(-0.939791\pi\)
0.982164 0.188025i \(-0.0602086\pi\)
\(384\) 0 0
\(385\) 134.981 0.350599
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 14.0099 0.0360152 0.0180076 0.999838i \(-0.494268\pi\)
0.0180076 + 0.999838i \(0.494268\pi\)
\(390\) 0 0
\(391\) − 909.506i − 2.32610i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) − 110.409i − 0.279515i
\(396\) 0 0
\(397\) −39.1084 −0.0985098 −0.0492549 0.998786i \(-0.515685\pi\)
−0.0492549 + 0.998786i \(0.515685\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 121.067 0.301913 0.150957 0.988540i \(-0.451765\pi\)
0.150957 + 0.988540i \(0.451765\pi\)
\(402\) 0 0
\(403\) − 48.9653i − 0.121502i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 393.579i 0.967026i
\(408\) 0 0
\(409\) −541.795 −1.32468 −0.662342 0.749202i \(-0.730437\pi\)
−0.662342 + 0.749202i \(0.730437\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −303.682 −0.735307
\(414\) 0 0
\(415\) 161.839i 0.389975i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) − 687.825i − 1.64159i −0.571224 0.820794i \(-0.693531\pi\)
0.571224 0.820794i \(-0.306469\pi\)
\(420\) 0 0
\(421\) 454.396 1.07932 0.539662 0.841882i \(-0.318552\pi\)
0.539662 + 0.841882i \(0.318552\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 122.073 0.287231
\(426\) 0 0
\(427\) − 449.670i − 1.05309i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 466.145i 1.08154i 0.841169 + 0.540772i \(0.181868\pi\)
−0.841169 + 0.540772i \(0.818132\pi\)
\(432\) 0 0
\(433\) 457.094 1.05565 0.527823 0.849355i \(-0.323009\pi\)
0.527823 + 0.849355i \(0.323009\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 885.706 2.02679
\(438\) 0 0
\(439\) 777.467i 1.77100i 0.464644 + 0.885498i \(0.346182\pi\)
−0.464644 + 0.885498i \(0.653818\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 247.484i 0.558654i 0.960196 + 0.279327i \(0.0901114\pi\)
−0.960196 + 0.279327i \(0.909889\pi\)
\(444\) 0 0
\(445\) 258.693 0.581333
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −412.508 −0.918726 −0.459363 0.888249i \(-0.651922\pi\)
−0.459363 + 0.888249i \(0.651922\pi\)
\(450\) 0 0
\(451\) − 102.720i − 0.227761i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) − 123.780i − 0.272044i
\(456\) 0 0
\(457\) −745.400 −1.63107 −0.815537 0.578706i \(-0.803558\pi\)
−0.815537 + 0.578706i \(0.803558\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 81.6151 0.177039 0.0885196 0.996074i \(-0.471786\pi\)
0.0885196 + 0.996074i \(0.471786\pi\)
\(462\) 0 0
\(463\) 292.248i 0.631205i 0.948891 + 0.315603i \(0.102207\pi\)
−0.948891 + 0.315603i \(0.897793\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 51.4163i − 0.110099i −0.998484 0.0550495i \(-0.982468\pi\)
0.998484 0.0550495i \(-0.0175317\pi\)
\(468\) 0 0
\(469\) 571.339 1.21821
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −773.139 −1.63454
\(474\) 0 0
\(475\) 118.879i 0.250271i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) − 122.593i − 0.255935i −0.991778 0.127967i \(-0.959155\pi\)
0.991778 0.127967i \(-0.0408453\pi\)
\(480\) 0 0
\(481\) 360.920 0.750354
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 163.141 0.336373
\(486\) 0 0
\(487\) 65.9859i 0.135495i 0.997703 + 0.0677474i \(0.0215812\pi\)
−0.997703 + 0.0677474i \(0.978419\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 361.163i 0.735567i 0.929911 + 0.367783i \(0.119883\pi\)
−0.929911 + 0.367783i \(0.880117\pi\)
\(492\) 0 0
\(493\) −629.229 −1.27633
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 419.630 0.844326
\(498\) 0 0
\(499\) 711.138i 1.42513i 0.701608 + 0.712564i \(0.252466\pi\)
−0.701608 + 0.712564i \(0.747534\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) − 353.756i − 0.703292i −0.936133 0.351646i \(-0.885622\pi\)
0.936133 0.351646i \(-0.114378\pi\)
\(504\) 0 0
\(505\) −65.7610 −0.130220
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 478.049 0.939192 0.469596 0.882881i \(-0.344400\pi\)
0.469596 + 0.882881i \(0.344400\pi\)
\(510\) 0 0
\(511\) 511.348i 1.00068i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 63.0311i − 0.122390i
\(516\) 0 0
\(517\) −419.778 −0.811949
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 35.7365 0.0685921 0.0342960 0.999412i \(-0.489081\pi\)
0.0342960 + 0.999412i \(0.489081\pi\)
\(522\) 0 0
\(523\) − 733.562i − 1.40260i −0.712864 0.701302i \(-0.752602\pi\)
0.712864 0.701302i \(-0.247398\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 118.081i 0.224062i
\(528\) 0 0
\(529\) −858.753 −1.62335
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −94.1965 −0.176729
\(534\) 0 0
\(535\) − 10.0780i − 0.0188373i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) − 210.917i − 0.391312i
\(540\) 0 0
\(541\) −608.939 −1.12558 −0.562790 0.826600i \(-0.690272\pi\)
−0.562790 + 0.826600i \(0.690272\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 432.266 0.793148
\(546\) 0 0
\(547\) 78.5868i 0.143669i 0.997417 + 0.0718344i \(0.0228853\pi\)
−0.997417 + 0.0718344i \(0.977115\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) − 612.763i − 1.11209i
\(552\) 0 0
\(553\) 269.974 0.488200
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −928.488 −1.66694 −0.833472 0.552561i \(-0.813651\pi\)
−0.833472 + 0.552561i \(0.813651\pi\)
\(558\) 0 0
\(559\) 708.984i 1.26831i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 447.978i − 0.795697i −0.917451 0.397849i \(-0.869757\pi\)
0.917451 0.397849i \(-0.130243\pi\)
\(564\) 0 0
\(565\) 168.986 0.299090
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −571.441 −1.00429 −0.502145 0.864783i \(-0.667456\pi\)
−0.502145 + 0.864783i \(0.667456\pi\)
\(570\) 0 0
\(571\) 990.801i 1.73520i 0.497260 + 0.867602i \(0.334340\pi\)
−0.497260 + 0.867602i \(0.665660\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) − 186.263i − 0.323935i
\(576\) 0 0
\(577\) 826.638 1.43265 0.716324 0.697768i \(-0.245823\pi\)
0.716324 + 0.697768i \(0.245823\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −395.735 −0.681127
\(582\) 0 0
\(583\) − 615.787i − 1.05624i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 900.009i 1.53323i 0.642104 + 0.766617i \(0.278062\pi\)
−0.642104 + 0.766617i \(0.721938\pi\)
\(588\) 0 0
\(589\) −114.991 −0.195230
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 704.088 1.18733 0.593666 0.804711i \(-0.297680\pi\)
0.593666 + 0.804711i \(0.297680\pi\)
\(594\) 0 0
\(595\) 298.497i 0.501675i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 376.098i 0.627876i 0.949444 + 0.313938i \(0.101648\pi\)
−0.949444 + 0.313938i \(0.898352\pi\)
\(600\) 0 0
\(601\) 430.191 0.715791 0.357896 0.933762i \(-0.383494\pi\)
0.357896 + 0.933762i \(0.383494\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 1.98684 0.00328404
\(606\) 0 0
\(607\) − 93.4019i − 0.153875i −0.997036 0.0769373i \(-0.975486\pi\)
0.997036 0.0769373i \(-0.0245141\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 384.944i 0.630024i
\(612\) 0 0
\(613\) 156.506 0.255312 0.127656 0.991818i \(-0.459255\pi\)
0.127656 + 0.991818i \(0.459255\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 553.493 0.897072 0.448536 0.893765i \(-0.351946\pi\)
0.448536 + 0.893765i \(0.351946\pi\)
\(618\) 0 0
\(619\) 14.4398i 0.0233276i 0.999932 + 0.0116638i \(0.00371278\pi\)
−0.999932 + 0.0116638i \(0.996287\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 632.564i 1.01535i
\(624\) 0 0
\(625\) 25.0000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −870.364 −1.38373
\(630\) 0 0
\(631\) − 352.389i − 0.558460i −0.960224 0.279230i \(-0.909921\pi\)
0.960224 0.279230i \(-0.0900793\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) − 293.608i − 0.462375i
\(636\) 0 0
\(637\) −193.415 −0.303634
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 545.742 0.851391 0.425696 0.904866i \(-0.360029\pi\)
0.425696 + 0.904866i \(0.360029\pi\)
\(642\) 0 0
\(643\) 757.447i 1.17799i 0.808137 + 0.588995i \(0.200476\pi\)
−0.808137 + 0.588995i \(0.799524\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 1161.36i − 1.79500i −0.441016 0.897499i \(-0.645382\pi\)
0.441016 0.897499i \(-0.354618\pi\)
\(648\) 0 0
\(649\) −613.191 −0.944824
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −621.231 −0.951348 −0.475674 0.879622i \(-0.657796\pi\)
−0.475674 + 0.879622i \(0.657796\pi\)
\(654\) 0 0
\(655\) − 169.390i − 0.258610i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) − 736.047i − 1.11692i −0.829533 0.558458i \(-0.811393\pi\)
0.829533 0.558458i \(-0.188607\pi\)
\(660\) 0 0
\(661\) −383.845 −0.580704 −0.290352 0.956920i \(-0.593772\pi\)
−0.290352 + 0.956920i \(0.593772\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −290.686 −0.437121
\(666\) 0 0
\(667\) 960.096i 1.43942i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) − 907.968i − 1.35316i
\(672\) 0 0
\(673\) 984.464 1.46280 0.731400 0.681949i \(-0.238867\pi\)
0.731400 + 0.681949i \(0.238867\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 673.154 0.994319 0.497160 0.867659i \(-0.334376\pi\)
0.497160 + 0.867659i \(0.334376\pi\)
\(678\) 0 0
\(679\) 398.918i 0.587507i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) − 291.192i − 0.426343i −0.977015 0.213171i \(-0.931621\pi\)
0.977015 0.213171i \(-0.0683793\pi\)
\(684\) 0 0
\(685\) 149.353 0.218034
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −564.689 −0.819578
\(690\) 0 0
\(691\) − 943.693i − 1.36569i −0.730563 0.682846i \(-0.760742\pi\)
0.730563 0.682846i \(-0.239258\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) − 85.2420i − 0.122650i
\(696\) 0 0
\(697\) 227.156 0.325905
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 885.681 1.26345 0.631727 0.775191i \(-0.282346\pi\)
0.631727 + 0.775191i \(0.282346\pi\)
\(702\) 0 0
\(703\) − 847.588i − 1.20567i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 160.801i − 0.227441i
\(708\) 0 0
\(709\) 286.183 0.403644 0.201822 0.979422i \(-0.435314\pi\)
0.201822 + 0.979422i \(0.435314\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 180.171 0.252694
\(714\) 0 0
\(715\) − 249.935i − 0.349559i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) − 666.163i − 0.926513i −0.886224 0.463257i \(-0.846681\pi\)
0.886224 0.463257i \(-0.153319\pi\)
\(720\) 0 0
\(721\) 154.125 0.213766
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −128.863 −0.177742
\(726\) 0 0
\(727\) 856.270i 1.17781i 0.808201 + 0.588907i \(0.200441\pi\)
−0.808201 + 0.588907i \(0.799559\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) − 1709.72i − 2.33888i
\(732\) 0 0
\(733\) 769.487 1.04978 0.524889 0.851171i \(-0.324107\pi\)
0.524889 + 0.851171i \(0.324107\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1153.64 1.56532
\(738\) 0 0
\(739\) − 1156.70i − 1.56522i −0.622511 0.782611i \(-0.713887\pi\)
0.622511 0.782611i \(-0.286113\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 426.794i 0.574421i 0.957868 + 0.287210i \(0.0927278\pi\)
−0.957868 + 0.287210i \(0.907272\pi\)
\(744\) 0 0
\(745\) 283.348 0.380332
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 24.6429 0.0329011
\(750\) 0 0
\(751\) 1222.03i 1.62721i 0.581420 + 0.813604i \(0.302497\pi\)
−0.581420 + 0.813604i \(0.697503\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) − 153.037i − 0.202698i
\(756\) 0 0
\(757\) 1312.95 1.73442 0.867209 0.497945i \(-0.165912\pi\)
0.867209 + 0.497945i \(0.165912\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 189.584 0.249124 0.124562 0.992212i \(-0.460247\pi\)
0.124562 + 0.992212i \(0.460247\pi\)
\(762\) 0 0
\(763\) 1056.99i 1.38531i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 562.308i 0.733127i
\(768\) 0 0
\(769\) 254.995 0.331594 0.165797 0.986160i \(-0.446980\pi\)
0.165797 + 0.986160i \(0.446980\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 23.2536 0.0300823 0.0150411 0.999887i \(-0.495212\pi\)
0.0150411 + 0.999887i \(0.495212\pi\)
\(774\) 0 0
\(775\) 24.1824i 0.0312030i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 221.212i 0.283969i
\(780\) 0 0
\(781\) 847.312 1.08491
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 57.1935 0.0728580
\(786\) 0 0
\(787\) 220.593i 0.280296i 0.990131 + 0.140148i \(0.0447579\pi\)
−0.990131 + 0.140148i \(0.955242\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 413.209i 0.522388i
\(792\) 0 0
\(793\) −832.625 −1.04997
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 1010.38 1.26773 0.633867 0.773442i \(-0.281467\pi\)
0.633867 + 0.773442i \(0.281467\pi\)
\(798\) 0 0
\(799\) − 928.298i − 1.16183i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 1032.51i 1.28581i
\(804\) 0 0
\(805\) 455.455 0.565783
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −1410.37 −1.74335 −0.871674 0.490086i \(-0.836965\pi\)
−0.871674 + 0.490086i \(0.836965\pi\)
\(810\) 0 0
\(811\) − 950.157i − 1.17159i −0.810460 0.585793i \(-0.800783\pi\)
0.810460 0.585793i \(-0.199217\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) − 141.939i − 0.174158i
\(816\) 0 0
\(817\) 1664.98 2.03792
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 77.3347 0.0941957 0.0470979 0.998890i \(-0.485003\pi\)
0.0470979 + 0.998890i \(0.485003\pi\)
\(822\) 0 0
\(823\) 1260.16i 1.53118i 0.643328 + 0.765591i \(0.277553\pi\)
−0.643328 + 0.765591i \(0.722447\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 438.047i 0.529681i 0.964292 + 0.264841i \(0.0853194\pi\)
−0.964292 + 0.264841i \(0.914681\pi\)
\(828\) 0 0
\(829\) −361.388 −0.435933 −0.217966 0.975956i \(-0.569942\pi\)
−0.217966 + 0.975956i \(0.569942\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 466.423 0.559931
\(834\) 0 0
\(835\) − 27.6760i − 0.0331450i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) − 785.017i − 0.935658i −0.883819 0.467829i \(-0.845036\pi\)
0.883819 0.467829i \(-0.154964\pi\)
\(840\) 0 0
\(841\) −176.771 −0.210192
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 148.700 0.175977
\(846\) 0 0
\(847\) 4.85829i 0.00573588i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 1328.03i 1.56055i
\(852\) 0 0
\(853\) −1113.79 −1.30573 −0.652865 0.757474i \(-0.726433\pi\)
−0.652865 + 0.757474i \(0.726433\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 306.591 0.357749 0.178875 0.983872i \(-0.442754\pi\)
0.178875 + 0.983872i \(0.442754\pi\)
\(858\) 0 0
\(859\) − 204.542i − 0.238116i −0.992887 0.119058i \(-0.962013\pi\)
0.992887 0.119058i \(-0.0379875\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 654.384i 0.758266i 0.925342 + 0.379133i \(0.123778\pi\)
−0.925342 + 0.379133i \(0.876222\pi\)
\(864\) 0 0
\(865\) −132.762 −0.153482
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 545.129 0.627306
\(870\) 0 0
\(871\) − 1057.91i − 1.21459i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 61.1307i 0.0698637i
\(876\) 0 0
\(877\) 604.453 0.689228 0.344614 0.938744i \(-0.388010\pi\)
0.344614 + 0.938744i \(0.388010\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −1436.81 −1.63089 −0.815445 0.578834i \(-0.803508\pi\)
−0.815445 + 0.578834i \(0.803508\pi\)
\(882\) 0 0
\(883\) − 120.993i − 0.137025i −0.997650 0.0685123i \(-0.978175\pi\)
0.997650 0.0685123i \(-0.0218252\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 286.448i 0.322941i 0.986878 + 0.161470i \(0.0516236\pi\)
−0.986878 + 0.161470i \(0.948376\pi\)
\(888\) 0 0
\(889\) 717.940 0.807582
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 904.007 1.01233
\(894\) 0 0
\(895\) 565.237i 0.631550i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 124.649i − 0.138652i
\(900\) 0 0
\(901\) 1361.75 1.51138
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 280.343 0.309771
\(906\) 0 0
\(907\) − 234.706i − 0.258772i −0.991594 0.129386i \(-0.958699\pi\)
0.991594 0.129386i \(-0.0413006\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) − 491.244i − 0.539236i −0.962967 0.269618i \(-0.913103\pi\)
0.962967 0.269618i \(-0.0868974\pi\)
\(912\) 0 0
\(913\) −799.063 −0.875206
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 414.197 0.451687
\(918\) 0 0
\(919\) − 356.091i − 0.387477i −0.981053 0.193738i \(-0.937939\pi\)
0.981053 0.193738i \(-0.0620613\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) − 777.002i − 0.841822i
\(924\) 0 0
\(925\) −178.246 −0.192699
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 916.019 0.986027 0.493014 0.870022i \(-0.335895\pi\)
0.493014 + 0.870022i \(0.335895\pi\)
\(930\) 0 0
\(931\) 454.217i 0.487881i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 602.721i 0.644621i
\(936\) 0 0
\(937\) 143.818 0.153488 0.0767440 0.997051i \(-0.475548\pi\)
0.0767440 + 0.997051i \(0.475548\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −1488.04 −1.58133 −0.790667 0.612246i \(-0.790266\pi\)
−0.790667 + 0.612246i \(0.790266\pi\)
\(942\) 0 0
\(943\) − 346.602i − 0.367552i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 1095.51i − 1.15682i −0.815745 0.578411i \(-0.803673\pi\)
0.815745 0.578411i \(-0.196327\pi\)
\(948\) 0 0
\(949\) 946.829 0.997713
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −1277.86 −1.34089 −0.670443 0.741961i \(-0.733896\pi\)
−0.670443 + 0.741961i \(0.733896\pi\)
\(954\) 0 0
\(955\) 217.936i 0.228205i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 365.202i 0.380816i
\(960\) 0 0
\(961\) 937.609 0.975659
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 765.576 0.793343
\(966\) 0 0
\(967\) − 237.958i − 0.246079i −0.992402 0.123039i \(-0.960736\pi\)
0.992402 0.123039i \(-0.0392641\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) − 1602.10i − 1.64995i −0.565169 0.824975i \(-0.691189\pi\)
0.565169 0.824975i \(-0.308811\pi\)
\(972\) 0 0
\(973\) 208.436 0.214220
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −918.977 −0.940611 −0.470305 0.882504i \(-0.655856\pi\)
−0.470305 + 0.882504i \(0.655856\pi\)
\(978\) 0 0
\(979\) 1277.27i 1.30466i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) − 1162.30i − 1.18240i −0.806525 0.591200i \(-0.798654\pi\)
0.806525 0.591200i \(-0.201346\pi\)
\(984\) 0 0
\(985\) −166.549 −0.169085
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −2608.75 −2.63776
\(990\) 0 0
\(991\) − 491.614i − 0.496079i −0.968750 0.248040i \(-0.920214\pi\)
0.968750 0.248040i \(-0.0797863\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 398.080i 0.400080i
\(996\) 0 0
\(997\) 1262.51 1.26630 0.633152 0.774027i \(-0.281761\pi\)
0.633152 + 0.774027i \(0.281761\pi\)
\(998\) 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 2880.3.e.j.2431.2 8
3.2 odd 2 960.3.e.c.511.3 8
4.3 odd 2 inner 2880.3.e.j.2431.3 8
8.3 odd 2 180.3.c.b.91.2 8
8.5 even 2 180.3.c.b.91.1 8
12.11 even 2 960.3.e.c.511.8 8
24.5 odd 2 60.3.c.a.31.8 yes 8
24.11 even 2 60.3.c.a.31.7 8
40.3 even 4 900.3.f.f.199.12 16
40.13 odd 4 900.3.f.f.199.6 16
40.19 odd 2 900.3.c.u.451.7 8
40.27 even 4 900.3.f.f.199.5 16
40.29 even 2 900.3.c.u.451.8 8
40.37 odd 4 900.3.f.f.199.11 16
120.29 odd 2 300.3.c.d.151.1 8
120.53 even 4 300.3.f.b.199.11 16
120.59 even 2 300.3.c.d.151.2 8
120.77 even 4 300.3.f.b.199.6 16
120.83 odd 4 300.3.f.b.199.5 16
120.107 odd 4 300.3.f.b.199.12 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
60.3.c.a.31.7 8 24.11 even 2
60.3.c.a.31.8 yes 8 24.5 odd 2
180.3.c.b.91.1 8 8.5 even 2
180.3.c.b.91.2 8 8.3 odd 2
300.3.c.d.151.1 8 120.29 odd 2
300.3.c.d.151.2 8 120.59 even 2
300.3.f.b.199.5 16 120.83 odd 4
300.3.f.b.199.6 16 120.77 even 4
300.3.f.b.199.11 16 120.53 even 4
300.3.f.b.199.12 16 120.107 odd 4
900.3.c.u.451.7 8 40.19 odd 2
900.3.c.u.451.8 8 40.29 even 2
900.3.f.f.199.5 16 40.27 even 4
900.3.f.f.199.6 16 40.13 odd 4
900.3.f.f.199.11 16 40.37 odd 4
900.3.f.f.199.12 16 40.3 even 4
960.3.e.c.511.3 8 3.2 odd 2
960.3.e.c.511.8 8 12.11 even 2
2880.3.e.j.2431.2 8 1.1 even 1 trivial
2880.3.e.j.2431.3 8 4.3 odd 2 inner