Properties

Label 3240.2.a.r.1.1
Level $3240$
Weight $2$
Character 3240.1
Self dual yes
Analytic conductor $25.872$
Analytic rank $1$
Dimension $3$
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] = [3240,2,Mod(1,3240)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3240, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3240.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3240 = 2^{3} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3240.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: \(25.8715302549\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.564.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 5x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 360)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(0.571993\) of defining polynomial
Character \(\chi\) \(=\) 3240.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{5} -5.24482 q^{7} +O(q^{10})\) \(q+1.00000 q^{5} -5.24482 q^{7} +2.67282 q^{11} +3.81681 q^{13} +3.52884 q^{17} -4.67282 q^{19} -4.95684 q^{23} +1.00000 q^{25} -1.85601 q^{29} -8.67282 q^{31} -5.24482 q^{35} -2.67282 q^{37} -3.67282 q^{41} +3.52884 q^{43} +9.26329 q^{47} +20.5081 q^{49} -2.85601 q^{53} +2.67282 q^{55} +4.20166 q^{59} -7.96080 q^{61} +3.81681 q^{65} -0.859966 q^{67} -15.1625 q^{71} +6.28797 q^{73} -14.0185 q^{77} +5.63362 q^{79} -3.89917 q^{83} +3.52884 q^{85} -11.0000 q^{89} -20.0185 q^{91} -4.67282 q^{95} -3.83528 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{5} - 5 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{5} - 5 q^{7} - 2 q^{11} + 2 q^{17} - 4 q^{19} - 7 q^{23} + 3 q^{25} - 7 q^{29} - 16 q^{31} - 5 q^{35} + 2 q^{37} - q^{41} + 2 q^{43} - 13 q^{47} + 10 q^{49} - 10 q^{53} - 2 q^{55} - 6 q^{59} - 11 q^{61} + q^{67} - 14 q^{71} + 16 q^{73} - 12 q^{77} - 6 q^{79} - 21 q^{83} + 2 q^{85} - 33 q^{89} - 30 q^{91} - 4 q^{95} + 30 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −5.24482 −1.98235 −0.991177 0.132543i \(-0.957686\pi\)
−0.991177 + 0.132543i \(0.957686\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.67282 0.805887 0.402943 0.915225i \(-0.367987\pi\)
0.402943 + 0.915225i \(0.367987\pi\)
\(12\) 0 0
\(13\) 3.81681 1.05859 0.529296 0.848437i \(-0.322456\pi\)
0.529296 + 0.848437i \(0.322456\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.52884 0.855869 0.427934 0.903810i \(-0.359241\pi\)
0.427934 + 0.903810i \(0.359241\pi\)
\(18\) 0 0
\(19\) −4.67282 −1.07202 −0.536010 0.844212i \(-0.680069\pi\)
−0.536010 + 0.844212i \(0.680069\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −4.95684 −1.03357 −0.516787 0.856114i \(-0.672872\pi\)
−0.516787 + 0.856114i \(0.672872\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −1.85601 −0.344653 −0.172327 0.985040i \(-0.555128\pi\)
−0.172327 + 0.985040i \(0.555128\pi\)
\(30\) 0 0
\(31\) −8.67282 −1.55769 −0.778843 0.627219i \(-0.784193\pi\)
−0.778843 + 0.627219i \(0.784193\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −5.24482 −0.886536
\(36\) 0 0
\(37\) −2.67282 −0.439410 −0.219705 0.975566i \(-0.570509\pi\)
−0.219705 + 0.975566i \(0.570509\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −3.67282 −0.573599 −0.286799 0.957991i \(-0.592591\pi\)
−0.286799 + 0.957991i \(0.592591\pi\)
\(42\) 0 0
\(43\) 3.52884 0.538143 0.269071 0.963120i \(-0.413283\pi\)
0.269071 + 0.963120i \(0.413283\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 9.26329 1.35119 0.675595 0.737273i \(-0.263887\pi\)
0.675595 + 0.737273i \(0.263887\pi\)
\(48\) 0 0
\(49\) 20.5081 2.92973
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −2.85601 −0.392304 −0.196152 0.980574i \(-0.562845\pi\)
−0.196152 + 0.980574i \(0.562845\pi\)
\(54\) 0 0
\(55\) 2.67282 0.360403
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 4.20166 0.547010 0.273505 0.961871i \(-0.411817\pi\)
0.273505 + 0.961871i \(0.411817\pi\)
\(60\) 0 0
\(61\) −7.96080 −1.01928 −0.509638 0.860389i \(-0.670221\pi\)
−0.509638 + 0.860389i \(0.670221\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 3.81681 0.473417
\(66\) 0 0
\(67\) −0.859966 −0.105062 −0.0525308 0.998619i \(-0.516729\pi\)
−0.0525308 + 0.998619i \(0.516729\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −15.1625 −1.79945 −0.899726 0.436454i \(-0.856234\pi\)
−0.899726 + 0.436454i \(0.856234\pi\)
\(72\) 0 0
\(73\) 6.28797 0.735952 0.367976 0.929835i \(-0.380051\pi\)
0.367976 + 0.929835i \(0.380051\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −14.0185 −1.59755
\(78\) 0 0
\(79\) 5.63362 0.633832 0.316916 0.948454i \(-0.397353\pi\)
0.316916 + 0.948454i \(0.397353\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −3.89917 −0.427989 −0.213995 0.976835i \(-0.568648\pi\)
−0.213995 + 0.976835i \(0.568648\pi\)
\(84\) 0 0
\(85\) 3.52884 0.382756
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −11.0000 −1.16600 −0.582999 0.812473i \(-0.698121\pi\)
−0.582999 + 0.812473i \(0.698121\pi\)
\(90\) 0 0
\(91\) −20.0185 −2.09851
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −4.67282 −0.479422
\(96\) 0 0
\(97\) −3.83528 −0.389414 −0.194707 0.980861i \(-0.562376\pi\)
−0.194707 + 0.980861i \(0.562376\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −10.4896 −1.04376 −0.521879 0.853020i \(-0.674769\pi\)
−0.521879 + 0.853020i \(0.674769\pi\)
\(102\) 0 0
\(103\) 4.10478 0.404456 0.202228 0.979338i \(-0.435182\pi\)
0.202228 + 0.979338i \(0.435182\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −19.7345 −1.90780 −0.953901 0.300122i \(-0.902972\pi\)
−0.953901 + 0.300122i \(0.902972\pi\)
\(108\) 0 0
\(109\) −9.50811 −0.910711 −0.455356 0.890310i \(-0.650488\pi\)
−0.455356 + 0.890310i \(0.650488\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −9.43196 −0.887284 −0.443642 0.896204i \(-0.646314\pi\)
−0.443642 + 0.896204i \(0.646314\pi\)
\(114\) 0 0
\(115\) −4.95684 −0.462228
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −18.5081 −1.69664
\(120\) 0 0
\(121\) −3.85601 −0.350547
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 18.6089 1.65128 0.825638 0.564200i \(-0.190815\pi\)
0.825638 + 0.564200i \(0.190815\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −15.3456 −1.34076 −0.670378 0.742020i \(-0.733868\pi\)
−0.670378 + 0.742020i \(0.733868\pi\)
\(132\) 0 0
\(133\) 24.5081 2.12512
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0.489634 0.0418323 0.0209161 0.999781i \(-0.493342\pi\)
0.0209161 + 0.999781i \(0.493342\pi\)
\(138\) 0 0
\(139\) −10.6913 −0.906824 −0.453412 0.891301i \(-0.649793\pi\)
−0.453412 + 0.891301i \(0.649793\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 10.2017 0.853106
\(144\) 0 0
\(145\) −1.85601 −0.154134
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 3.87448 0.317410 0.158705 0.987326i \(-0.449268\pi\)
0.158705 + 0.987326i \(0.449268\pi\)
\(150\) 0 0
\(151\) −14.6728 −1.19406 −0.597029 0.802220i \(-0.703652\pi\)
−0.597029 + 0.802220i \(0.703652\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −8.67282 −0.696618
\(156\) 0 0
\(157\) −13.1809 −1.05195 −0.525976 0.850499i \(-0.676300\pi\)
−0.525976 + 0.850499i \(0.676300\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 25.9977 2.04891
\(162\) 0 0
\(163\) 20.8745 1.63502 0.817508 0.575917i \(-0.195355\pi\)
0.817508 + 0.575917i \(0.195355\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0.00395288 0.000305883 0 0.000152941 1.00000i \(-0.499951\pi\)
0.000152941 1.00000i \(0.499951\pi\)
\(168\) 0 0
\(169\) 1.56804 0.120618
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −20.6129 −1.56717 −0.783584 0.621285i \(-0.786611\pi\)
−0.783584 + 0.621285i \(0.786611\pi\)
\(174\) 0 0
\(175\) −5.24482 −0.396471
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 4.10478 0.306806 0.153403 0.988164i \(-0.450977\pi\)
0.153403 + 0.988164i \(0.450977\pi\)
\(180\) 0 0
\(181\) −4.43196 −0.329425 −0.164712 0.986342i \(-0.552670\pi\)
−0.164712 + 0.986342i \(0.552670\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −2.67282 −0.196510
\(186\) 0 0
\(187\) 9.43196 0.689733
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −17.1625 −1.24183 −0.620916 0.783877i \(-0.713239\pi\)
−0.620916 + 0.783877i \(0.713239\pi\)
\(192\) 0 0
\(193\) 5.89522 0.424347 0.212173 0.977232i \(-0.431946\pi\)
0.212173 + 0.977232i \(0.431946\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −21.8353 −1.55570 −0.777850 0.628450i \(-0.783690\pi\)
−0.777850 + 0.628450i \(0.783690\pi\)
\(198\) 0 0
\(199\) −13.2488 −0.939180 −0.469590 0.882885i \(-0.655598\pi\)
−0.469590 + 0.882885i \(0.655598\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 9.73445 0.683225
\(204\) 0 0
\(205\) −3.67282 −0.256521
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −12.4896 −0.863926
\(210\) 0 0
\(211\) 0.384851 0.0264942 0.0132471 0.999912i \(-0.495783\pi\)
0.0132471 + 0.999912i \(0.495783\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 3.52884 0.240665
\(216\) 0 0
\(217\) 45.4874 3.08788
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 13.4689 0.906016
\(222\) 0 0
\(223\) −11.4280 −0.765276 −0.382638 0.923898i \(-0.624984\pi\)
−0.382638 + 0.923898i \(0.624984\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 17.9401 1.19072 0.595362 0.803458i \(-0.297009\pi\)
0.595362 + 0.803458i \(0.297009\pi\)
\(228\) 0 0
\(229\) −4.91369 −0.324706 −0.162353 0.986733i \(-0.551908\pi\)
−0.162353 + 0.986733i \(0.551908\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 4.59442 0.300990 0.150495 0.988611i \(-0.451913\pi\)
0.150495 + 0.988611i \(0.451913\pi\)
\(234\) 0 0
\(235\) 9.26329 0.604270
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 26.2386 1.69723 0.848617 0.529008i \(-0.177436\pi\)
0.848617 + 0.529008i \(0.177436\pi\)
\(240\) 0 0
\(241\) −0.895217 −0.0576660 −0.0288330 0.999584i \(-0.509179\pi\)
−0.0288330 + 0.999584i \(0.509179\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 20.5081 1.31021
\(246\) 0 0
\(247\) −17.8353 −1.13483
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 2.10478 0.132853 0.0664264 0.997791i \(-0.478840\pi\)
0.0664264 + 0.997791i \(0.478840\pi\)
\(252\) 0 0
\(253\) −13.2488 −0.832943
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −4.69129 −0.292635 −0.146317 0.989238i \(-0.546742\pi\)
−0.146317 + 0.989238i \(0.546742\pi\)
\(258\) 0 0
\(259\) 14.0185 0.871065
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 9.81681 0.605330 0.302665 0.953097i \(-0.402124\pi\)
0.302665 + 0.953097i \(0.402124\pi\)
\(264\) 0 0
\(265\) −2.85601 −0.175444
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 20.4504 1.24688 0.623442 0.781869i \(-0.285734\pi\)
0.623442 + 0.781869i \(0.285734\pi\)
\(270\) 0 0
\(271\) −5.03920 −0.306110 −0.153055 0.988218i \(-0.548911\pi\)
−0.153055 + 0.988218i \(0.548911\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 2.67282 0.161177
\(276\) 0 0
\(277\) 1.04711 0.0629147 0.0314573 0.999505i \(-0.489985\pi\)
0.0314573 + 0.999505i \(0.489985\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 1.75914 0.104941 0.0524706 0.998622i \(-0.483290\pi\)
0.0524706 + 0.998622i \(0.483290\pi\)
\(282\) 0 0
\(283\) 2.87844 0.171105 0.0855527 0.996334i \(-0.472734\pi\)
0.0855527 + 0.996334i \(0.472734\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 19.2633 1.13708
\(288\) 0 0
\(289\) −4.54731 −0.267489
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 24.9114 1.45534 0.727671 0.685927i \(-0.240603\pi\)
0.727671 + 0.685927i \(0.240603\pi\)
\(294\) 0 0
\(295\) 4.20166 0.244630
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −18.9193 −1.09413
\(300\) 0 0
\(301\) −18.5081 −1.06679
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −7.96080 −0.455834
\(306\) 0 0
\(307\) −1.32322 −0.0755203 −0.0377602 0.999287i \(-0.512022\pi\)
−0.0377602 + 0.999287i \(0.512022\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 4.95289 0.280853 0.140426 0.990091i \(-0.455153\pi\)
0.140426 + 0.990091i \(0.455153\pi\)
\(312\) 0 0
\(313\) 9.61515 0.543480 0.271740 0.962371i \(-0.412401\pi\)
0.271740 + 0.962371i \(0.412401\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −12.3064 −0.691199 −0.345599 0.938382i \(-0.612324\pi\)
−0.345599 + 0.938382i \(0.612324\pi\)
\(318\) 0 0
\(319\) −4.96080 −0.277751
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −16.4896 −0.917508
\(324\) 0 0
\(325\) 3.81681 0.211719
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −48.5843 −2.67854
\(330\) 0 0
\(331\) 1.54731 0.0850478 0.0425239 0.999095i \(-0.486460\pi\)
0.0425239 + 0.999095i \(0.486460\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −0.859966 −0.0469850
\(336\) 0 0
\(337\) 32.3170 1.76042 0.880210 0.474585i \(-0.157402\pi\)
0.880210 + 0.474585i \(0.157402\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −23.1809 −1.25532
\(342\) 0 0
\(343\) −70.8475 −3.82541
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −25.0841 −1.34658 −0.673291 0.739377i \(-0.735120\pi\)
−0.673291 + 0.739377i \(0.735120\pi\)
\(348\) 0 0
\(349\) 20.7490 1.11067 0.555333 0.831628i \(-0.312591\pi\)
0.555333 + 0.831628i \(0.312591\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −18.1153 −0.964183 −0.482091 0.876121i \(-0.660123\pi\)
−0.482091 + 0.876121i \(0.660123\pi\)
\(354\) 0 0
\(355\) −15.1625 −0.804740
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 25.3720 1.33908 0.669542 0.742774i \(-0.266490\pi\)
0.669542 + 0.742774i \(0.266490\pi\)
\(360\) 0 0
\(361\) 2.83528 0.149225
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 6.28797 0.329128
\(366\) 0 0
\(367\) 6.18319 0.322760 0.161380 0.986892i \(-0.448406\pi\)
0.161380 + 0.986892i \(0.448406\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 14.9793 0.777685
\(372\) 0 0
\(373\) −16.8145 −0.870624 −0.435312 0.900280i \(-0.643362\pi\)
−0.435312 + 0.900280i \(0.643362\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −7.08405 −0.364847
\(378\) 0 0
\(379\) 22.0000 1.13006 0.565032 0.825069i \(-0.308864\pi\)
0.565032 + 0.825069i \(0.308864\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −6.79608 −0.347263 −0.173632 0.984811i \(-0.555550\pi\)
−0.173632 + 0.984811i \(0.555550\pi\)
\(384\) 0 0
\(385\) −14.0185 −0.714447
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −8.61515 −0.436805 −0.218403 0.975859i \(-0.570085\pi\)
−0.218403 + 0.975859i \(0.570085\pi\)
\(390\) 0 0
\(391\) −17.4919 −0.884603
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 5.63362 0.283458
\(396\) 0 0
\(397\) 22.4033 1.12439 0.562195 0.827005i \(-0.309957\pi\)
0.562195 + 0.827005i \(0.309957\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 13.9137 0.694816 0.347408 0.937714i \(-0.387062\pi\)
0.347408 + 0.937714i \(0.387062\pi\)
\(402\) 0 0
\(403\) −33.1025 −1.64895
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −7.14399 −0.354114
\(408\) 0 0
\(409\) 30.6129 1.51371 0.756855 0.653583i \(-0.226735\pi\)
0.756855 + 0.653583i \(0.226735\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −22.0369 −1.08437
\(414\) 0 0
\(415\) −3.89917 −0.191403
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −13.7305 −0.670779 −0.335389 0.942080i \(-0.608868\pi\)
−0.335389 + 0.942080i \(0.608868\pi\)
\(420\) 0 0
\(421\) 15.9216 0.775971 0.387985 0.921665i \(-0.373171\pi\)
0.387985 + 0.921665i \(0.373171\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 3.52884 0.171174
\(426\) 0 0
\(427\) 41.7529 2.02057
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −25.2672 −1.21708 −0.608540 0.793523i \(-0.708245\pi\)
−0.608540 + 0.793523i \(0.708245\pi\)
\(432\) 0 0
\(433\) 9.24086 0.444088 0.222044 0.975037i \(-0.428727\pi\)
0.222044 + 0.975037i \(0.428727\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 23.1625 1.10801
\(438\) 0 0
\(439\) −17.4320 −0.831982 −0.415991 0.909369i \(-0.636565\pi\)
−0.415991 + 0.909369i \(0.636565\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −3.62967 −0.172451 −0.0862254 0.996276i \(-0.527481\pi\)
−0.0862254 + 0.996276i \(0.527481\pi\)
\(444\) 0 0
\(445\) −11.0000 −0.521450
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −39.6785 −1.87254 −0.936271 0.351278i \(-0.885747\pi\)
−0.936271 + 0.351278i \(0.885747\pi\)
\(450\) 0 0
\(451\) −9.81681 −0.462256
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −20.0185 −0.938480
\(456\) 0 0
\(457\) 5.69356 0.266333 0.133167 0.991094i \(-0.457485\pi\)
0.133167 + 0.991094i \(0.457485\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 17.3170 0.806534 0.403267 0.915082i \(-0.367875\pi\)
0.403267 + 0.915082i \(0.367875\pi\)
\(462\) 0 0
\(463\) 25.7674 1.19751 0.598757 0.800931i \(-0.295661\pi\)
0.598757 + 0.800931i \(0.295661\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −1.04711 −0.0484544 −0.0242272 0.999706i \(-0.507713\pi\)
−0.0242272 + 0.999706i \(0.507713\pi\)
\(468\) 0 0
\(469\) 4.51037 0.208269
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 9.43196 0.433682
\(474\) 0 0
\(475\) −4.67282 −0.214404
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 22.5759 1.03152 0.515761 0.856733i \(-0.327509\pi\)
0.515761 + 0.856733i \(0.327509\pi\)
\(480\) 0 0
\(481\) −10.2017 −0.465156
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −3.83528 −0.174151
\(486\) 0 0
\(487\) 0.923855 0.0418638 0.0209319 0.999781i \(-0.493337\pi\)
0.0209319 + 0.999781i \(0.493337\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 8.08631 0.364930 0.182465 0.983212i \(-0.441592\pi\)
0.182465 + 0.983212i \(0.441592\pi\)
\(492\) 0 0
\(493\) −6.54957 −0.294978
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 79.5243 3.56715
\(498\) 0 0
\(499\) −12.6834 −0.567786 −0.283893 0.958856i \(-0.591626\pi\)
−0.283893 + 0.958856i \(0.591626\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 4.93837 0.220191 0.110096 0.993921i \(-0.464884\pi\)
0.110096 + 0.993921i \(0.464884\pi\)
\(504\) 0 0
\(505\) −10.4896 −0.466783
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 15.2096 0.674152 0.337076 0.941477i \(-0.390562\pi\)
0.337076 + 0.941477i \(0.390562\pi\)
\(510\) 0 0
\(511\) −32.9793 −1.45892
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 4.10478 0.180878
\(516\) 0 0
\(517\) 24.7591 1.08891
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 22.1809 0.971764 0.485882 0.874024i \(-0.338498\pi\)
0.485882 + 0.874024i \(0.338498\pi\)
\(522\) 0 0
\(523\) −9.46495 −0.413873 −0.206937 0.978354i \(-0.566349\pi\)
−0.206937 + 0.978354i \(0.566349\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −30.6050 −1.33317
\(528\) 0 0
\(529\) 1.57030 0.0682740
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −14.0185 −0.607207
\(534\) 0 0
\(535\) −19.7345 −0.853195
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 54.8145 2.36103
\(540\) 0 0
\(541\) −2.42405 −0.104218 −0.0521091 0.998641i \(-0.516594\pi\)
−0.0521091 + 0.998641i \(0.516594\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −9.50811 −0.407282
\(546\) 0 0
\(547\) −15.2818 −0.653401 −0.326700 0.945128i \(-0.605937\pi\)
−0.326700 + 0.945128i \(0.605937\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 8.67282 0.369475
\(552\) 0 0
\(553\) −29.5473 −1.25648
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 38.9793 1.65160 0.825802 0.563960i \(-0.190723\pi\)
0.825802 + 0.563960i \(0.190723\pi\)
\(558\) 0 0
\(559\) 13.4689 0.569674
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −4.60894 −0.194243 −0.0971217 0.995273i \(-0.530964\pi\)
−0.0971217 + 0.995273i \(0.530964\pi\)
\(564\) 0 0
\(565\) −9.43196 −0.396806
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 11.3826 0.477183 0.238591 0.971120i \(-0.423314\pi\)
0.238591 + 0.971120i \(0.423314\pi\)
\(570\) 0 0
\(571\) 45.2258 1.89264 0.946320 0.323231i \(-0.104769\pi\)
0.946320 + 0.323231i \(0.104769\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −4.95684 −0.206715
\(576\) 0 0
\(577\) 5.14399 0.214147 0.107073 0.994251i \(-0.465852\pi\)
0.107073 + 0.994251i \(0.465852\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 20.4504 0.848427
\(582\) 0 0
\(583\) −7.63362 −0.316152
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −43.2818 −1.78643 −0.893215 0.449630i \(-0.851556\pi\)
−0.893215 + 0.449630i \(0.851556\pi\)
\(588\) 0 0
\(589\) 40.5266 1.66987
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 33.3641 1.37010 0.685050 0.728496i \(-0.259780\pi\)
0.685050 + 0.728496i \(0.259780\pi\)
\(594\) 0 0
\(595\) −18.5081 −0.758758
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 40.1153 1.63907 0.819534 0.573030i \(-0.194232\pi\)
0.819534 + 0.573030i \(0.194232\pi\)
\(600\) 0 0
\(601\) −5.96306 −0.243238 −0.121619 0.992577i \(-0.538809\pi\)
−0.121619 + 0.992577i \(0.538809\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −3.85601 −0.156769
\(606\) 0 0
\(607\) −15.1400 −0.614515 −0.307258 0.951626i \(-0.599411\pi\)
−0.307258 + 0.951626i \(0.599411\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 35.3562 1.43036
\(612\) 0 0
\(613\) 28.5081 1.15143 0.575716 0.817650i \(-0.304723\pi\)
0.575716 + 0.817650i \(0.304723\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 17.4320 0.701784 0.350892 0.936416i \(-0.385878\pi\)
0.350892 + 0.936416i \(0.385878\pi\)
\(618\) 0 0
\(619\) 29.6521 1.19182 0.595909 0.803052i \(-0.296792\pi\)
0.595909 + 0.803052i \(0.296792\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 57.6930 2.31142
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −9.43196 −0.376077
\(630\) 0 0
\(631\) −40.1233 −1.59728 −0.798641 0.601808i \(-0.794447\pi\)
−0.798641 + 0.601808i \(0.794447\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 18.6089 0.738473
\(636\) 0 0
\(637\) 78.2755 3.10139
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 4.41349 0.174322 0.0871612 0.996194i \(-0.472220\pi\)
0.0871612 + 0.996194i \(0.472220\pi\)
\(642\) 0 0
\(643\) 19.8313 0.782071 0.391036 0.920376i \(-0.372117\pi\)
0.391036 + 0.920376i \(0.372117\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −42.6873 −1.67821 −0.839106 0.543967i \(-0.816921\pi\)
−0.839106 + 0.543967i \(0.816921\pi\)
\(648\) 0 0
\(649\) 11.2303 0.440828
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −37.3641 −1.46217 −0.731085 0.682286i \(-0.760986\pi\)
−0.731085 + 0.682286i \(0.760986\pi\)
\(654\) 0 0
\(655\) −15.3456 −0.599604
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −5.43196 −0.211599 −0.105800 0.994387i \(-0.533740\pi\)
−0.105800 + 0.994387i \(0.533740\pi\)
\(660\) 0 0
\(661\) 17.0946 0.664904 0.332452 0.943120i \(-0.392124\pi\)
0.332452 + 0.943120i \(0.392124\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 24.5081 0.950384
\(666\) 0 0
\(667\) 9.19997 0.356224
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −21.2778 −0.821421
\(672\) 0 0
\(673\) −39.4011 −1.51880 −0.759400 0.650624i \(-0.774507\pi\)
−0.759400 + 0.650624i \(0.774507\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 12.2695 0.471555 0.235778 0.971807i \(-0.424236\pi\)
0.235778 + 0.971807i \(0.424236\pi\)
\(678\) 0 0
\(679\) 20.1153 0.771956
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 11.8952 0.455158 0.227579 0.973760i \(-0.426919\pi\)
0.227579 + 0.973760i \(0.426919\pi\)
\(684\) 0 0
\(685\) 0.489634 0.0187080
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −10.9009 −0.415290
\(690\) 0 0
\(691\) −5.26724 −0.200375 −0.100188 0.994969i \(-0.531944\pi\)
−0.100188 + 0.994969i \(0.531944\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −10.6913 −0.405544
\(696\) 0 0
\(697\) −12.9608 −0.490925
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −22.1131 −0.835200 −0.417600 0.908631i \(-0.637129\pi\)
−0.417600 + 0.908631i \(0.637129\pi\)
\(702\) 0 0
\(703\) 12.4896 0.471055
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 55.0162 2.06910
\(708\) 0 0
\(709\) 17.4112 0.653892 0.326946 0.945043i \(-0.393980\pi\)
0.326946 + 0.945043i \(0.393980\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 42.9898 1.60998
\(714\) 0 0
\(715\) 10.2017 0.381520
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −19.4689 −0.726068 −0.363034 0.931776i \(-0.618259\pi\)
−0.363034 + 0.931776i \(0.618259\pi\)
\(720\) 0 0
\(721\) −21.5288 −0.801776
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −1.85601 −0.0689306
\(726\) 0 0
\(727\) −23.9255 −0.887349 −0.443675 0.896188i \(-0.646325\pi\)
−0.443675 + 0.896188i \(0.646325\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 12.4527 0.460579
\(732\) 0 0
\(733\) −4.11535 −0.152004 −0.0760019 0.997108i \(-0.524216\pi\)
−0.0760019 + 0.997108i \(0.524216\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −2.29854 −0.0846678
\(738\) 0 0
\(739\) 16.8145 0.618533 0.309267 0.950975i \(-0.399916\pi\)
0.309267 + 0.950975i \(0.399916\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 27.3602 1.00375 0.501874 0.864941i \(-0.332644\pi\)
0.501874 + 0.864941i \(0.332644\pi\)
\(744\) 0 0
\(745\) 3.87448 0.141950
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 103.504 3.78194
\(750\) 0 0
\(751\) 4.47116 0.163155 0.0815775 0.996667i \(-0.474004\pi\)
0.0815775 + 0.996667i \(0.474004\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −14.6728 −0.533999
\(756\) 0 0
\(757\) −17.7753 −0.646056 −0.323028 0.946389i \(-0.604701\pi\)
−0.323028 + 0.946389i \(0.604701\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −40.3905 −1.46415 −0.732077 0.681222i \(-0.761449\pi\)
−0.732077 + 0.681222i \(0.761449\pi\)
\(762\) 0 0
\(763\) 49.8683 1.80535
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 16.0369 0.579060
\(768\) 0 0
\(769\) −39.6992 −1.43159 −0.715795 0.698311i \(-0.753935\pi\)
−0.715795 + 0.698311i \(0.753935\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 8.97927 0.322962 0.161481 0.986876i \(-0.448373\pi\)
0.161481 + 0.986876i \(0.448373\pi\)
\(774\) 0 0
\(775\) −8.67282 −0.311537
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 17.1625 0.614909
\(780\) 0 0
\(781\) −40.5266 −1.45015
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −13.1809 −0.470448
\(786\) 0 0
\(787\) 35.8952 1.27953 0.639763 0.768572i \(-0.279032\pi\)
0.639763 + 0.768572i \(0.279032\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 49.4689 1.75891
\(792\) 0 0
\(793\) −30.3849 −1.07900
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 37.8458 1.34057 0.670284 0.742104i \(-0.266172\pi\)
0.670284 + 0.742104i \(0.266172\pi\)
\(798\) 0 0
\(799\) 32.6886 1.15644
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 16.8066 0.593094
\(804\) 0 0
\(805\) 25.9977 0.916300
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −27.8722 −0.979935 −0.489968 0.871741i \(-0.662991\pi\)
−0.489968 + 0.871741i \(0.662991\pi\)
\(810\) 0 0
\(811\) 48.3249 1.69692 0.848459 0.529262i \(-0.177531\pi\)
0.848459 + 0.529262i \(0.177531\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 20.8745 0.731201
\(816\) 0 0
\(817\) −16.4896 −0.576899
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −29.7098 −1.03688 −0.518439 0.855115i \(-0.673487\pi\)
−0.518439 + 0.855115i \(0.673487\pi\)
\(822\) 0 0
\(823\) −30.2056 −1.05290 −0.526451 0.850206i \(-0.676478\pi\)
−0.526451 + 0.850206i \(0.676478\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −1.14003 −0.0396429 −0.0198214 0.999804i \(-0.506310\pi\)
−0.0198214 + 0.999804i \(0.506310\pi\)
\(828\) 0 0
\(829\) −17.9608 −0.623804 −0.311902 0.950114i \(-0.600966\pi\)
−0.311902 + 0.950114i \(0.600966\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 72.3698 2.50746
\(834\) 0 0
\(835\) 0.00395288 0.000136795 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 1.21183 0.0418369 0.0209185 0.999781i \(-0.493341\pi\)
0.0209185 + 0.999781i \(0.493341\pi\)
\(840\) 0 0
\(841\) −25.5552 −0.881214
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 1.56804 0.0539422
\(846\) 0 0
\(847\) 20.2241 0.694908
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 13.2488 0.454162
\(852\) 0 0
\(853\) 35.3641 1.21084 0.605422 0.795905i \(-0.293004\pi\)
0.605422 + 0.795905i \(0.293004\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −41.8123 −1.42828 −0.714140 0.700003i \(-0.753182\pi\)
−0.714140 + 0.700003i \(0.753182\pi\)
\(858\) 0 0
\(859\) −11.7753 −0.401770 −0.200885 0.979615i \(-0.564382\pi\)
−0.200885 + 0.979615i \(0.564382\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −26.1193 −0.889111 −0.444556 0.895751i \(-0.646638\pi\)
−0.444556 + 0.895751i \(0.646638\pi\)
\(864\) 0 0
\(865\) −20.6129 −0.700859
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 15.0577 0.510797
\(870\) 0 0
\(871\) −3.28233 −0.111217
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −5.24482 −0.177307
\(876\) 0 0
\(877\) −52.6235 −1.77697 −0.888484 0.458908i \(-0.848241\pi\)
−0.888484 + 0.458908i \(0.848241\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −12.8089 −0.431543 −0.215771 0.976444i \(-0.569227\pi\)
−0.215771 + 0.976444i \(0.569227\pi\)
\(882\) 0 0
\(883\) 21.7529 0.732044 0.366022 0.930606i \(-0.380719\pi\)
0.366022 + 0.930606i \(0.380719\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −33.0471 −1.10961 −0.554807 0.831979i \(-0.687208\pi\)
−0.554807 + 0.831979i \(0.687208\pi\)
\(888\) 0 0
\(889\) −97.6005 −3.27341
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −43.2857 −1.44850
\(894\) 0 0
\(895\) 4.10478 0.137208
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 16.0969 0.536861
\(900\) 0 0
\(901\) −10.0784 −0.335760
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −4.43196 −0.147323
\(906\) 0 0
\(907\) 49.1849 1.63316 0.816579 0.577234i \(-0.195868\pi\)
0.816579 + 0.577234i \(0.195868\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −6.45269 −0.213787 −0.106894 0.994270i \(-0.534090\pi\)
−0.106894 + 0.994270i \(0.534090\pi\)
\(912\) 0 0
\(913\) −10.4218 −0.344911
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 80.4851 2.65785
\(918\) 0 0
\(919\) −19.5552 −0.645067 −0.322533 0.946558i \(-0.604534\pi\)
−0.322533 + 0.946558i \(0.604534\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −57.8722 −1.90489
\(924\) 0 0
\(925\) −2.67282 −0.0878819
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −51.1730 −1.67893 −0.839466 0.543412i \(-0.817132\pi\)
−0.839466 + 0.543412i \(0.817132\pi\)
\(930\) 0 0
\(931\) −95.8308 −3.14073
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 9.43196 0.308458
\(936\) 0 0
\(937\) −37.6627 −1.23039 −0.615193 0.788377i \(-0.710922\pi\)
−0.615193 + 0.788377i \(0.710922\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −25.8145 −0.841530 −0.420765 0.907170i \(-0.638238\pi\)
−0.420765 + 0.907170i \(0.638238\pi\)
\(942\) 0 0
\(943\) 18.2056 0.592856
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −1.83923 −0.0597671 −0.0298835 0.999553i \(-0.509514\pi\)
−0.0298835 + 0.999553i \(0.509514\pi\)
\(948\) 0 0
\(949\) 24.0000 0.779073
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 39.7674 1.28819 0.644097 0.764944i \(-0.277233\pi\)
0.644097 + 0.764944i \(0.277233\pi\)
\(954\) 0 0
\(955\) −17.1625 −0.555364
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −2.56804 −0.0829263
\(960\) 0 0
\(961\) 44.2179 1.42638
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 5.89522 0.189774
\(966\) 0 0
\(967\) 9.70541 0.312105 0.156053 0.987749i \(-0.450123\pi\)
0.156053 + 0.987749i \(0.450123\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 13.8247 0.443656 0.221828 0.975086i \(-0.428798\pi\)
0.221828 + 0.975086i \(0.428798\pi\)
\(972\) 0 0
\(973\) 56.0739 1.79765
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 35.9770 1.15101 0.575503 0.817800i \(-0.304806\pi\)
0.575503 + 0.817800i \(0.304806\pi\)
\(978\) 0 0
\(979\) −29.4011 −0.939662
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −22.5490 −0.719201 −0.359601 0.933106i \(-0.617087\pi\)
−0.359601 + 0.933106i \(0.617087\pi\)
\(984\) 0 0
\(985\) −21.8353 −0.695730
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −17.4919 −0.556210
\(990\) 0 0
\(991\) −13.8767 −0.440809 −0.220405 0.975409i \(-0.570738\pi\)
−0.220405 + 0.975409i \(0.570738\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −13.2488 −0.420014
\(996\) 0 0
\(997\) 35.5058 1.12448 0.562241 0.826974i \(-0.309939\pi\)
0.562241 + 0.826974i \(0.309939\pi\)
\(998\) 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 3240.2.a.r.1.1 3
3.2 odd 2 3240.2.a.q.1.1 3
4.3 odd 2 6480.2.a.bx.1.3 3
9.2 odd 6 1080.2.q.d.361.3 6
9.4 even 3 360.2.q.d.241.3 yes 6
9.5 odd 6 1080.2.q.d.721.3 6
9.7 even 3 360.2.q.d.121.3 6
12.11 even 2 6480.2.a.bu.1.3 3
36.7 odd 6 720.2.q.j.481.1 6
36.11 even 6 2160.2.q.j.1441.1 6
36.23 even 6 2160.2.q.j.721.1 6
36.31 odd 6 720.2.q.j.241.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
360.2.q.d.121.3 6 9.7 even 3
360.2.q.d.241.3 yes 6 9.4 even 3
720.2.q.j.241.1 6 36.31 odd 6
720.2.q.j.481.1 6 36.7 odd 6
1080.2.q.d.361.3 6 9.2 odd 6
1080.2.q.d.721.3 6 9.5 odd 6
2160.2.q.j.721.1 6 36.23 even 6
2160.2.q.j.1441.1 6 36.11 even 6
3240.2.a.q.1.1 3 3.2 odd 2
3240.2.a.r.1.1 3 1.1 even 1 trivial
6480.2.a.bu.1.3 3 12.11 even 2
6480.2.a.bx.1.3 3 4.3 odd 2