Properties

Label 912.4.a.h.1.1
Level $912$
Weight $4$
Character 912.1
Self dual yes
Analytic conductor $53.810$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [912,4,Mod(1,912)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(912, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("912.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 912.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: \(53.8097419252\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 228)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 912.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.00000 q^{3} +4.00000 q^{5} +12.0000 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q+3.00000 q^{3} +4.00000 q^{5} +12.0000 q^{7} +9.00000 q^{9} -40.0000 q^{11} -40.0000 q^{13} +12.0000 q^{15} -66.0000 q^{17} +19.0000 q^{19} +36.0000 q^{21} +98.0000 q^{23} -109.000 q^{25} +27.0000 q^{27} -130.000 q^{29} -262.000 q^{31} -120.000 q^{33} +48.0000 q^{35} -296.000 q^{37} -120.000 q^{39} -442.000 q^{41} +164.000 q^{43} +36.0000 q^{45} +542.000 q^{47} -199.000 q^{49} -198.000 q^{51} +334.000 q^{53} -160.000 q^{55} +57.0000 q^{57} -60.0000 q^{59} +614.000 q^{61} +108.000 q^{63} -160.000 q^{65} +294.000 q^{69} -400.000 q^{71} +318.000 q^{73} -327.000 q^{75} -480.000 q^{77} -1154.00 q^{79} +81.0000 q^{81} +636.000 q^{83} -264.000 q^{85} -390.000 q^{87} -630.000 q^{89} -480.000 q^{91} -786.000 q^{93} +76.0000 q^{95} +1006.00 q^{97} -360.000 q^{99} +O(q^{100})\)

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
\(3\) 3.00000 0.577350
\(4\) 0 0
\(5\) 4.00000 0.357771 0.178885 0.983870i \(-0.442751\pi\)
0.178885 + 0.983870i \(0.442751\pi\)
\(6\) 0 0
\(7\) 12.0000 0.647939 0.323970 0.946068i \(-0.394982\pi\)
0.323970 + 0.946068i \(0.394982\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −40.0000 −1.09640 −0.548202 0.836346i \(-0.684688\pi\)
−0.548202 + 0.836346i \(0.684688\pi\)
\(12\) 0 0
\(13\) −40.0000 −0.853385 −0.426692 0.904397i \(-0.640321\pi\)
−0.426692 + 0.904397i \(0.640321\pi\)
\(14\) 0 0
\(15\) 12.0000 0.206559
\(16\) 0 0
\(17\) −66.0000 −0.941609 −0.470804 0.882238i \(-0.656036\pi\)
−0.470804 + 0.882238i \(0.656036\pi\)
\(18\) 0 0
\(19\) 19.0000 0.229416
\(20\) 0 0
\(21\) 36.0000 0.374088
\(22\) 0 0
\(23\) 98.0000 0.888453 0.444226 0.895915i \(-0.353479\pi\)
0.444226 + 0.895915i \(0.353479\pi\)
\(24\) 0 0
\(25\) −109.000 −0.872000
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) −130.000 −0.832427 −0.416214 0.909267i \(-0.636643\pi\)
−0.416214 + 0.909267i \(0.636643\pi\)
\(30\) 0 0
\(31\) −262.000 −1.51795 −0.758977 0.651117i \(-0.774301\pi\)
−0.758977 + 0.651117i \(0.774301\pi\)
\(32\) 0 0
\(33\) −120.000 −0.633010
\(34\) 0 0
\(35\) 48.0000 0.231814
\(36\) 0 0
\(37\) −296.000 −1.31519 −0.657596 0.753371i \(-0.728427\pi\)
−0.657596 + 0.753371i \(0.728427\pi\)
\(38\) 0 0
\(39\) −120.000 −0.492702
\(40\) 0 0
\(41\) −442.000 −1.68363 −0.841815 0.539767i \(-0.818512\pi\)
−0.841815 + 0.539767i \(0.818512\pi\)
\(42\) 0 0
\(43\) 164.000 0.581622 0.290811 0.956780i \(-0.406075\pi\)
0.290811 + 0.956780i \(0.406075\pi\)
\(44\) 0 0
\(45\) 36.0000 0.119257
\(46\) 0 0
\(47\) 542.000 1.68210 0.841051 0.540955i \(-0.181937\pi\)
0.841051 + 0.540955i \(0.181937\pi\)
\(48\) 0 0
\(49\) −199.000 −0.580175
\(50\) 0 0
\(51\) −198.000 −0.543638
\(52\) 0 0
\(53\) 334.000 0.865631 0.432815 0.901483i \(-0.357520\pi\)
0.432815 + 0.901483i \(0.357520\pi\)
\(54\) 0 0
\(55\) −160.000 −0.392262
\(56\) 0 0
\(57\) 57.0000 0.132453
\(58\) 0 0
\(59\) −60.0000 −0.132396 −0.0661978 0.997807i \(-0.521087\pi\)
−0.0661978 + 0.997807i \(0.521087\pi\)
\(60\) 0 0
\(61\) 614.000 1.28876 0.644382 0.764703i \(-0.277115\pi\)
0.644382 + 0.764703i \(0.277115\pi\)
\(62\) 0 0
\(63\) 108.000 0.215980
\(64\) 0 0
\(65\) −160.000 −0.305316
\(66\) 0 0
\(67\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 0 0
\(69\) 294.000 0.512948
\(70\) 0 0
\(71\) −400.000 −0.668609 −0.334305 0.942465i \(-0.608501\pi\)
−0.334305 + 0.942465i \(0.608501\pi\)
\(72\) 0 0
\(73\) 318.000 0.509850 0.254925 0.966961i \(-0.417949\pi\)
0.254925 + 0.966961i \(0.417949\pi\)
\(74\) 0 0
\(75\) −327.000 −0.503449
\(76\) 0 0
\(77\) −480.000 −0.710404
\(78\) 0 0
\(79\) −1154.00 −1.64348 −0.821741 0.569861i \(-0.806997\pi\)
−0.821741 + 0.569861i \(0.806997\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 636.000 0.841085 0.420543 0.907273i \(-0.361840\pi\)
0.420543 + 0.907273i \(0.361840\pi\)
\(84\) 0 0
\(85\) −264.000 −0.336880
\(86\) 0 0
\(87\) −390.000 −0.480602
\(88\) 0 0
\(89\) −630.000 −0.750336 −0.375168 0.926957i \(-0.622415\pi\)
−0.375168 + 0.926957i \(0.622415\pi\)
\(90\) 0 0
\(91\) −480.000 −0.552941
\(92\) 0 0
\(93\) −786.000 −0.876391
\(94\) 0 0
\(95\) 76.0000 0.0820783
\(96\) 0 0
\(97\) 1006.00 1.05303 0.526515 0.850166i \(-0.323499\pi\)
0.526515 + 0.850166i \(0.323499\pi\)
\(98\) 0 0
\(99\) −360.000 −0.365468
\(100\) 0 0
\(101\) −140.000 −0.137926 −0.0689630 0.997619i \(-0.521969\pi\)
−0.0689630 + 0.997619i \(0.521969\pi\)
\(102\) 0 0
\(103\) −66.0000 −0.0631376 −0.0315688 0.999502i \(-0.510050\pi\)
−0.0315688 + 0.999502i \(0.510050\pi\)
\(104\) 0 0
\(105\) 144.000 0.133838
\(106\) 0 0
\(107\) −1012.00 −0.914334 −0.457167 0.889381i \(-0.651136\pi\)
−0.457167 + 0.889381i \(0.651136\pi\)
\(108\) 0 0
\(109\) −588.000 −0.516699 −0.258349 0.966052i \(-0.583179\pi\)
−0.258349 + 0.966052i \(0.583179\pi\)
\(110\) 0 0
\(111\) −888.000 −0.759326
\(112\) 0 0
\(113\) 622.000 0.517813 0.258906 0.965902i \(-0.416638\pi\)
0.258906 + 0.965902i \(0.416638\pi\)
\(114\) 0 0
\(115\) 392.000 0.317863
\(116\) 0 0
\(117\) −360.000 −0.284462
\(118\) 0 0
\(119\) −792.000 −0.610105
\(120\) 0 0
\(121\) 269.000 0.202104
\(122\) 0 0
\(123\) −1326.00 −0.972044
\(124\) 0 0
\(125\) −936.000 −0.669747
\(126\) 0 0
\(127\) −198.000 −0.138344 −0.0691719 0.997605i \(-0.522036\pi\)
−0.0691719 + 0.997605i \(0.522036\pi\)
\(128\) 0 0
\(129\) 492.000 0.335800
\(130\) 0 0
\(131\) 1524.00 1.01643 0.508216 0.861230i \(-0.330305\pi\)
0.508216 + 0.861230i \(0.330305\pi\)
\(132\) 0 0
\(133\) 228.000 0.148647
\(134\) 0 0
\(135\) 108.000 0.0688530
\(136\) 0 0
\(137\) −1546.00 −0.964115 −0.482057 0.876140i \(-0.660110\pi\)
−0.482057 + 0.876140i \(0.660110\pi\)
\(138\) 0 0
\(139\) −2508.00 −1.53040 −0.765201 0.643792i \(-0.777360\pi\)
−0.765201 + 0.643792i \(0.777360\pi\)
\(140\) 0 0
\(141\) 1626.00 0.971162
\(142\) 0 0
\(143\) 1600.00 0.935655
\(144\) 0 0
\(145\) −520.000 −0.297818
\(146\) 0 0
\(147\) −597.000 −0.334964
\(148\) 0 0
\(149\) 936.000 0.514632 0.257316 0.966327i \(-0.417162\pi\)
0.257316 + 0.966327i \(0.417162\pi\)
\(150\) 0 0
\(151\) −98.0000 −0.0528154 −0.0264077 0.999651i \(-0.508407\pi\)
−0.0264077 + 0.999651i \(0.508407\pi\)
\(152\) 0 0
\(153\) −594.000 −0.313870
\(154\) 0 0
\(155\) −1048.00 −0.543080
\(156\) 0 0
\(157\) −926.000 −0.470719 −0.235359 0.971908i \(-0.575627\pi\)
−0.235359 + 0.971908i \(0.575627\pi\)
\(158\) 0 0
\(159\) 1002.00 0.499772
\(160\) 0 0
\(161\) 1176.00 0.575663
\(162\) 0 0
\(163\) −1268.00 −0.609309 −0.304655 0.952463i \(-0.598541\pi\)
−0.304655 + 0.952463i \(0.598541\pi\)
\(164\) 0 0
\(165\) −480.000 −0.226472
\(166\) 0 0
\(167\) 612.000 0.283581 0.141790 0.989897i \(-0.454714\pi\)
0.141790 + 0.989897i \(0.454714\pi\)
\(168\) 0 0
\(169\) −597.000 −0.271734
\(170\) 0 0
\(171\) 171.000 0.0764719
\(172\) 0 0
\(173\) −3294.00 −1.44762 −0.723810 0.690000i \(-0.757611\pi\)
−0.723810 + 0.690000i \(0.757611\pi\)
\(174\) 0 0
\(175\) −1308.00 −0.565003
\(176\) 0 0
\(177\) −180.000 −0.0764386
\(178\) 0 0
\(179\) 3684.00 1.53830 0.769148 0.639070i \(-0.220681\pi\)
0.769148 + 0.639070i \(0.220681\pi\)
\(180\) 0 0
\(181\) −3404.00 −1.39789 −0.698943 0.715177i \(-0.746346\pi\)
−0.698943 + 0.715177i \(0.746346\pi\)
\(182\) 0 0
\(183\) 1842.00 0.744069
\(184\) 0 0
\(185\) −1184.00 −0.470537
\(186\) 0 0
\(187\) 2640.00 1.03238
\(188\) 0 0
\(189\) 324.000 0.124696
\(190\) 0 0
\(191\) −2534.00 −0.959968 −0.479984 0.877277i \(-0.659357\pi\)
−0.479984 + 0.877277i \(0.659357\pi\)
\(192\) 0 0
\(193\) 5118.00 1.90882 0.954409 0.298503i \(-0.0964874\pi\)
0.954409 + 0.298503i \(0.0964874\pi\)
\(194\) 0 0
\(195\) −480.000 −0.176274
\(196\) 0 0
\(197\) 328.000 0.118625 0.0593123 0.998239i \(-0.481109\pi\)
0.0593123 + 0.998239i \(0.481109\pi\)
\(198\) 0 0
\(199\) 1348.00 0.480187 0.240093 0.970750i \(-0.422822\pi\)
0.240093 + 0.970750i \(0.422822\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −1560.00 −0.539362
\(204\) 0 0
\(205\) −1768.00 −0.602354
\(206\) 0 0
\(207\) 882.000 0.296151
\(208\) 0 0
\(209\) −760.000 −0.251533
\(210\) 0 0
\(211\) 2196.00 0.716488 0.358244 0.933628i \(-0.383376\pi\)
0.358244 + 0.933628i \(0.383376\pi\)
\(212\) 0 0
\(213\) −1200.00 −0.386022
\(214\) 0 0
\(215\) 656.000 0.208088
\(216\) 0 0
\(217\) −3144.00 −0.983542
\(218\) 0 0
\(219\) 954.000 0.294362
\(220\) 0 0
\(221\) 2640.00 0.803555
\(222\) 0 0
\(223\) −2942.00 −0.883457 −0.441728 0.897149i \(-0.645635\pi\)
−0.441728 + 0.897149i \(0.645635\pi\)
\(224\) 0 0
\(225\) −981.000 −0.290667
\(226\) 0 0
\(227\) −1340.00 −0.391801 −0.195901 0.980624i \(-0.562763\pi\)
−0.195901 + 0.980624i \(0.562763\pi\)
\(228\) 0 0
\(229\) −6466.00 −1.86587 −0.932937 0.360039i \(-0.882763\pi\)
−0.932937 + 0.360039i \(0.882763\pi\)
\(230\) 0 0
\(231\) −1440.00 −0.410152
\(232\) 0 0
\(233\) −294.000 −0.0826634 −0.0413317 0.999145i \(-0.513160\pi\)
−0.0413317 + 0.999145i \(0.513160\pi\)
\(234\) 0 0
\(235\) 2168.00 0.601807
\(236\) 0 0
\(237\) −3462.00 −0.948865
\(238\) 0 0
\(239\) −3546.00 −0.959714 −0.479857 0.877347i \(-0.659311\pi\)
−0.479857 + 0.877347i \(0.659311\pi\)
\(240\) 0 0
\(241\) −1514.00 −0.404669 −0.202335 0.979316i \(-0.564853\pi\)
−0.202335 + 0.979316i \(0.564853\pi\)
\(242\) 0 0
\(243\) 243.000 0.0641500
\(244\) 0 0
\(245\) −796.000 −0.207570
\(246\) 0 0
\(247\) −760.000 −0.195780
\(248\) 0 0
\(249\) 1908.00 0.485601
\(250\) 0 0
\(251\) 7364.00 1.85184 0.925919 0.377721i \(-0.123292\pi\)
0.925919 + 0.377721i \(0.123292\pi\)
\(252\) 0 0
\(253\) −3920.00 −0.974104
\(254\) 0 0
\(255\) −792.000 −0.194498
\(256\) 0 0
\(257\) 2002.00 0.485920 0.242960 0.970036i \(-0.421882\pi\)
0.242960 + 0.970036i \(0.421882\pi\)
\(258\) 0 0
\(259\) −3552.00 −0.852164
\(260\) 0 0
\(261\) −1170.00 −0.277476
\(262\) 0 0
\(263\) 582.000 0.136455 0.0682275 0.997670i \(-0.478266\pi\)
0.0682275 + 0.997670i \(0.478266\pi\)
\(264\) 0 0
\(265\) 1336.00 0.309697
\(266\) 0 0
\(267\) −1890.00 −0.433206
\(268\) 0 0
\(269\) 6090.00 1.38035 0.690174 0.723643i \(-0.257534\pi\)
0.690174 + 0.723643i \(0.257534\pi\)
\(270\) 0 0
\(271\) −5452.00 −1.22209 −0.611043 0.791597i \(-0.709250\pi\)
−0.611043 + 0.791597i \(0.709250\pi\)
\(272\) 0 0
\(273\) −1440.00 −0.319241
\(274\) 0 0
\(275\) 4360.00 0.956065
\(276\) 0 0
\(277\) 4894.00 1.06156 0.530780 0.847510i \(-0.321899\pi\)
0.530780 + 0.847510i \(0.321899\pi\)
\(278\) 0 0
\(279\) −2358.00 −0.505985
\(280\) 0 0
\(281\) 4958.00 1.05256 0.526280 0.850311i \(-0.323586\pi\)
0.526280 + 0.850311i \(0.323586\pi\)
\(282\) 0 0
\(283\) 1308.00 0.274744 0.137372 0.990520i \(-0.456134\pi\)
0.137372 + 0.990520i \(0.456134\pi\)
\(284\) 0 0
\(285\) 228.000 0.0473879
\(286\) 0 0
\(287\) −5304.00 −1.09089
\(288\) 0 0
\(289\) −557.000 −0.113373
\(290\) 0 0
\(291\) 3018.00 0.607967
\(292\) 0 0
\(293\) 7722.00 1.53967 0.769836 0.638241i \(-0.220338\pi\)
0.769836 + 0.638241i \(0.220338\pi\)
\(294\) 0 0
\(295\) −240.000 −0.0473673
\(296\) 0 0
\(297\) −1080.00 −0.211003
\(298\) 0 0
\(299\) −3920.00 −0.758192
\(300\) 0 0
\(301\) 1968.00 0.376856
\(302\) 0 0
\(303\) −420.000 −0.0796316
\(304\) 0 0
\(305\) 2456.00 0.461082
\(306\) 0 0
\(307\) 3912.00 0.727263 0.363631 0.931543i \(-0.381537\pi\)
0.363631 + 0.931543i \(0.381537\pi\)
\(308\) 0 0
\(309\) −198.000 −0.0364525
\(310\) 0 0
\(311\) −1994.00 −0.363567 −0.181784 0.983339i \(-0.558187\pi\)
−0.181784 + 0.983339i \(0.558187\pi\)
\(312\) 0 0
\(313\) −3226.00 −0.582570 −0.291285 0.956636i \(-0.594083\pi\)
−0.291285 + 0.956636i \(0.594083\pi\)
\(314\) 0 0
\(315\) 432.000 0.0772712
\(316\) 0 0
\(317\) 498.000 0.0882349 0.0441175 0.999026i \(-0.485952\pi\)
0.0441175 + 0.999026i \(0.485952\pi\)
\(318\) 0 0
\(319\) 5200.00 0.912677
\(320\) 0 0
\(321\) −3036.00 −0.527891
\(322\) 0 0
\(323\) −1254.00 −0.216020
\(324\) 0 0
\(325\) 4360.00 0.744152
\(326\) 0 0
\(327\) −1764.00 −0.298316
\(328\) 0 0
\(329\) 6504.00 1.08990
\(330\) 0 0
\(331\) −3440.00 −0.571237 −0.285619 0.958343i \(-0.592199\pi\)
−0.285619 + 0.958343i \(0.592199\pi\)
\(332\) 0 0
\(333\) −2664.00 −0.438397
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −6014.00 −0.972117 −0.486059 0.873926i \(-0.661566\pi\)
−0.486059 + 0.873926i \(0.661566\pi\)
\(338\) 0 0
\(339\) 1866.00 0.298959
\(340\) 0 0
\(341\) 10480.0 1.66429
\(342\) 0 0
\(343\) −6504.00 −1.02386
\(344\) 0 0
\(345\) 1176.00 0.183518
\(346\) 0 0
\(347\) 10144.0 1.56933 0.784666 0.619918i \(-0.212834\pi\)
0.784666 + 0.619918i \(0.212834\pi\)
\(348\) 0 0
\(349\) 8818.00 1.35248 0.676242 0.736680i \(-0.263607\pi\)
0.676242 + 0.736680i \(0.263607\pi\)
\(350\) 0 0
\(351\) −1080.00 −0.164234
\(352\) 0 0
\(353\) 8010.00 1.20773 0.603866 0.797086i \(-0.293626\pi\)
0.603866 + 0.797086i \(0.293626\pi\)
\(354\) 0 0
\(355\) −1600.00 −0.239209
\(356\) 0 0
\(357\) −2376.00 −0.352244
\(358\) 0 0
\(359\) 3426.00 0.503670 0.251835 0.967770i \(-0.418966\pi\)
0.251835 + 0.967770i \(0.418966\pi\)
\(360\) 0 0
\(361\) 361.000 0.0526316
\(362\) 0 0
\(363\) 807.000 0.116685
\(364\) 0 0
\(365\) 1272.00 0.182410
\(366\) 0 0
\(367\) 8396.00 1.19419 0.597095 0.802171i \(-0.296322\pi\)
0.597095 + 0.802171i \(0.296322\pi\)
\(368\) 0 0
\(369\) −3978.00 −0.561210
\(370\) 0 0
\(371\) 4008.00 0.560876
\(372\) 0 0
\(373\) −2200.00 −0.305393 −0.152697 0.988273i \(-0.548796\pi\)
−0.152697 + 0.988273i \(0.548796\pi\)
\(374\) 0 0
\(375\) −2808.00 −0.386679
\(376\) 0 0
\(377\) 5200.00 0.710381
\(378\) 0 0
\(379\) −2304.00 −0.312265 −0.156133 0.987736i \(-0.549903\pi\)
−0.156133 + 0.987736i \(0.549903\pi\)
\(380\) 0 0
\(381\) −594.000 −0.0798728
\(382\) 0 0
\(383\) −10376.0 −1.38431 −0.692153 0.721751i \(-0.743337\pi\)
−0.692153 + 0.721751i \(0.743337\pi\)
\(384\) 0 0
\(385\) −1920.00 −0.254162
\(386\) 0 0
\(387\) 1476.00 0.193874
\(388\) 0 0
\(389\) −8436.00 −1.09954 −0.549771 0.835315i \(-0.685285\pi\)
−0.549771 + 0.835315i \(0.685285\pi\)
\(390\) 0 0
\(391\) −6468.00 −0.836575
\(392\) 0 0
\(393\) 4572.00 0.586837
\(394\) 0 0
\(395\) −4616.00 −0.587990
\(396\) 0 0
\(397\) −9142.00 −1.15573 −0.577864 0.816133i \(-0.696114\pi\)
−0.577864 + 0.816133i \(0.696114\pi\)
\(398\) 0 0
\(399\) 684.000 0.0858216
\(400\) 0 0
\(401\) 6350.00 0.790783 0.395391 0.918513i \(-0.370609\pi\)
0.395391 + 0.918513i \(0.370609\pi\)
\(402\) 0 0
\(403\) 10480.0 1.29540
\(404\) 0 0
\(405\) 324.000 0.0397523
\(406\) 0 0
\(407\) 11840.0 1.44198
\(408\) 0 0
\(409\) −10586.0 −1.27981 −0.639907 0.768452i \(-0.721027\pi\)
−0.639907 + 0.768452i \(0.721027\pi\)
\(410\) 0 0
\(411\) −4638.00 −0.556632
\(412\) 0 0
\(413\) −720.000 −0.0857842
\(414\) 0 0
\(415\) 2544.00 0.300916
\(416\) 0 0
\(417\) −7524.00 −0.883578
\(418\) 0 0
\(419\) 14856.0 1.73213 0.866066 0.499930i \(-0.166641\pi\)
0.866066 + 0.499930i \(0.166641\pi\)
\(420\) 0 0
\(421\) 11696.0 1.35399 0.676993 0.735989i \(-0.263283\pi\)
0.676993 + 0.735989i \(0.263283\pi\)
\(422\) 0 0
\(423\) 4878.00 0.560701
\(424\) 0 0
\(425\) 7194.00 0.821083
\(426\) 0 0
\(427\) 7368.00 0.835041
\(428\) 0 0
\(429\) 4800.00 0.540201
\(430\) 0 0
\(431\) −1356.00 −0.151546 −0.0757729 0.997125i \(-0.524142\pi\)
−0.0757729 + 0.997125i \(0.524142\pi\)
\(432\) 0 0
\(433\) 10862.0 1.20553 0.602765 0.797919i \(-0.294066\pi\)
0.602765 + 0.797919i \(0.294066\pi\)
\(434\) 0 0
\(435\) −1560.00 −0.171945
\(436\) 0 0
\(437\) 1862.00 0.203825
\(438\) 0 0
\(439\) −13838.0 −1.50445 −0.752223 0.658909i \(-0.771018\pi\)
−0.752223 + 0.658909i \(0.771018\pi\)
\(440\) 0 0
\(441\) −1791.00 −0.193392
\(442\) 0 0
\(443\) −1088.00 −0.116687 −0.0583436 0.998297i \(-0.518582\pi\)
−0.0583436 + 0.998297i \(0.518582\pi\)
\(444\) 0 0
\(445\) −2520.00 −0.268448
\(446\) 0 0
\(447\) 2808.00 0.297123
\(448\) 0 0
\(449\) −10078.0 −1.05927 −0.529633 0.848227i \(-0.677670\pi\)
−0.529633 + 0.848227i \(0.677670\pi\)
\(450\) 0 0
\(451\) 17680.0 1.84594
\(452\) 0 0
\(453\) −294.000 −0.0304930
\(454\) 0 0
\(455\) −1920.00 −0.197826
\(456\) 0 0
\(457\) 13634.0 1.39556 0.697781 0.716311i \(-0.254171\pi\)
0.697781 + 0.716311i \(0.254171\pi\)
\(458\) 0 0
\(459\) −1782.00 −0.181213
\(460\) 0 0
\(461\) 5848.00 0.590821 0.295411 0.955370i \(-0.404544\pi\)
0.295411 + 0.955370i \(0.404544\pi\)
\(462\) 0 0
\(463\) 10508.0 1.05475 0.527374 0.849633i \(-0.323177\pi\)
0.527374 + 0.849633i \(0.323177\pi\)
\(464\) 0 0
\(465\) −3144.00 −0.313547
\(466\) 0 0
\(467\) 17236.0 1.70789 0.853947 0.520359i \(-0.174202\pi\)
0.853947 + 0.520359i \(0.174202\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −2778.00 −0.271770
\(472\) 0 0
\(473\) −6560.00 −0.637694
\(474\) 0 0
\(475\) −2071.00 −0.200051
\(476\) 0 0
\(477\) 3006.00 0.288544
\(478\) 0 0
\(479\) 5890.00 0.561839 0.280920 0.959731i \(-0.409361\pi\)
0.280920 + 0.959731i \(0.409361\pi\)
\(480\) 0 0
\(481\) 11840.0 1.12236
\(482\) 0 0
\(483\) 3528.00 0.332359
\(484\) 0 0
\(485\) 4024.00 0.376743
\(486\) 0 0
\(487\) 7982.00 0.742708 0.371354 0.928491i \(-0.378894\pi\)
0.371354 + 0.928491i \(0.378894\pi\)
\(488\) 0 0
\(489\) −3804.00 −0.351785
\(490\) 0 0
\(491\) 2732.00 0.251107 0.125553 0.992087i \(-0.459929\pi\)
0.125553 + 0.992087i \(0.459929\pi\)
\(492\) 0 0
\(493\) 8580.00 0.783821
\(494\) 0 0
\(495\) −1440.00 −0.130754
\(496\) 0 0
\(497\) −4800.00 −0.433218
\(498\) 0 0
\(499\) −17548.0 −1.57426 −0.787131 0.616786i \(-0.788434\pi\)
−0.787131 + 0.616786i \(0.788434\pi\)
\(500\) 0 0
\(501\) 1836.00 0.163725
\(502\) 0 0
\(503\) −2082.00 −0.184556 −0.0922781 0.995733i \(-0.529415\pi\)
−0.0922781 + 0.995733i \(0.529415\pi\)
\(504\) 0 0
\(505\) −560.000 −0.0493459
\(506\) 0 0
\(507\) −1791.00 −0.156886
\(508\) 0 0
\(509\) −15654.0 −1.36317 −0.681583 0.731741i \(-0.738708\pi\)
−0.681583 + 0.731741i \(0.738708\pi\)
\(510\) 0 0
\(511\) 3816.00 0.330352
\(512\) 0 0
\(513\) 513.000 0.0441511
\(514\) 0 0
\(515\) −264.000 −0.0225888
\(516\) 0 0
\(517\) −21680.0 −1.84427
\(518\) 0 0
\(519\) −9882.00 −0.835784
\(520\) 0 0
\(521\) −16382.0 −1.37756 −0.688780 0.724971i \(-0.741853\pi\)
−0.688780 + 0.724971i \(0.741853\pi\)
\(522\) 0 0
\(523\) 19216.0 1.60661 0.803305 0.595568i \(-0.203073\pi\)
0.803305 + 0.595568i \(0.203073\pi\)
\(524\) 0 0
\(525\) −3924.00 −0.326205
\(526\) 0 0
\(527\) 17292.0 1.42932
\(528\) 0 0
\(529\) −2563.00 −0.210652
\(530\) 0 0
\(531\) −540.000 −0.0441318
\(532\) 0 0
\(533\) 17680.0 1.43678
\(534\) 0 0
\(535\) −4048.00 −0.327122
\(536\) 0 0
\(537\) 11052.0 0.888136
\(538\) 0 0
\(539\) 7960.00 0.636107
\(540\) 0 0
\(541\) −11302.0 −0.898172 −0.449086 0.893489i \(-0.648250\pi\)
−0.449086 + 0.893489i \(0.648250\pi\)
\(542\) 0 0
\(543\) −10212.0 −0.807070
\(544\) 0 0
\(545\) −2352.00 −0.184860
\(546\) 0 0
\(547\) −10120.0 −0.791042 −0.395521 0.918457i \(-0.629436\pi\)
−0.395521 + 0.918457i \(0.629436\pi\)
\(548\) 0 0
\(549\) 5526.00 0.429588
\(550\) 0 0
\(551\) −2470.00 −0.190972
\(552\) 0 0
\(553\) −13848.0 −1.06488
\(554\) 0 0
\(555\) −3552.00 −0.271665
\(556\) 0 0
\(557\) 14984.0 1.13984 0.569921 0.821699i \(-0.306974\pi\)
0.569921 + 0.821699i \(0.306974\pi\)
\(558\) 0 0
\(559\) −6560.00 −0.496348
\(560\) 0 0
\(561\) 7920.00 0.596048
\(562\) 0 0
\(563\) 13020.0 0.974649 0.487325 0.873221i \(-0.337973\pi\)
0.487325 + 0.873221i \(0.337973\pi\)
\(564\) 0 0
\(565\) 2488.00 0.185258
\(566\) 0 0
\(567\) 972.000 0.0719932
\(568\) 0 0
\(569\) 11546.0 0.850674 0.425337 0.905035i \(-0.360156\pi\)
0.425337 + 0.905035i \(0.360156\pi\)
\(570\) 0 0
\(571\) −2084.00 −0.152737 −0.0763684 0.997080i \(-0.524333\pi\)
−0.0763684 + 0.997080i \(0.524333\pi\)
\(572\) 0 0
\(573\) −7602.00 −0.554238
\(574\) 0 0
\(575\) −10682.0 −0.774731
\(576\) 0 0
\(577\) −9022.00 −0.650937 −0.325469 0.945553i \(-0.605522\pi\)
−0.325469 + 0.945553i \(0.605522\pi\)
\(578\) 0 0
\(579\) 15354.0 1.10206
\(580\) 0 0
\(581\) 7632.00 0.544972
\(582\) 0 0
\(583\) −13360.0 −0.949082
\(584\) 0 0
\(585\) −1440.00 −0.101772
\(586\) 0 0
\(587\) 25800.0 1.81411 0.907053 0.421017i \(-0.138327\pi\)
0.907053 + 0.421017i \(0.138327\pi\)
\(588\) 0 0
\(589\) −4978.00 −0.348243
\(590\) 0 0
\(591\) 984.000 0.0684879
\(592\) 0 0
\(593\) −19606.0 −1.35771 −0.678855 0.734272i \(-0.737523\pi\)
−0.678855 + 0.734272i \(0.737523\pi\)
\(594\) 0 0
\(595\) −3168.00 −0.218278
\(596\) 0 0
\(597\) 4044.00 0.277236
\(598\) 0 0
\(599\) 21300.0 1.45291 0.726456 0.687213i \(-0.241166\pi\)
0.726456 + 0.687213i \(0.241166\pi\)
\(600\) 0 0
\(601\) 11818.0 0.802107 0.401053 0.916055i \(-0.368644\pi\)
0.401053 + 0.916055i \(0.368644\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 1076.00 0.0723068
\(606\) 0 0
\(607\) −28154.0 −1.88260 −0.941298 0.337577i \(-0.890393\pi\)
−0.941298 + 0.337577i \(0.890393\pi\)
\(608\) 0 0
\(609\) −4680.00 −0.311401
\(610\) 0 0
\(611\) −21680.0 −1.43548
\(612\) 0 0
\(613\) −5966.00 −0.393090 −0.196545 0.980495i \(-0.562972\pi\)
−0.196545 + 0.980495i \(0.562972\pi\)
\(614\) 0 0
\(615\) −5304.00 −0.347769
\(616\) 0 0
\(617\) −19278.0 −1.25787 −0.628933 0.777460i \(-0.716508\pi\)
−0.628933 + 0.777460i \(0.716508\pi\)
\(618\) 0 0
\(619\) −10492.0 −0.681275 −0.340637 0.940195i \(-0.610643\pi\)
−0.340637 + 0.940195i \(0.610643\pi\)
\(620\) 0 0
\(621\) 2646.00 0.170983
\(622\) 0 0
\(623\) −7560.00 −0.486172
\(624\) 0 0
\(625\) 9881.00 0.632384
\(626\) 0 0
\(627\) −2280.00 −0.145222
\(628\) 0 0
\(629\) 19536.0 1.23840
\(630\) 0 0
\(631\) −20768.0 −1.31024 −0.655120 0.755525i \(-0.727382\pi\)
−0.655120 + 0.755525i \(0.727382\pi\)
\(632\) 0 0
\(633\) 6588.00 0.413664
\(634\) 0 0
\(635\) −792.000 −0.0494954
\(636\) 0 0
\(637\) 7960.00 0.495113
\(638\) 0 0
\(639\) −3600.00 −0.222870
\(640\) 0 0
\(641\) −13110.0 −0.807822 −0.403911 0.914798i \(-0.632349\pi\)
−0.403911 + 0.914798i \(0.632349\pi\)
\(642\) 0 0
\(643\) 11612.0 0.712181 0.356091 0.934451i \(-0.384109\pi\)
0.356091 + 0.934451i \(0.384109\pi\)
\(644\) 0 0
\(645\) 1968.00 0.120139
\(646\) 0 0
\(647\) −12094.0 −0.734875 −0.367438 0.930048i \(-0.619765\pi\)
−0.367438 + 0.930048i \(0.619765\pi\)
\(648\) 0 0
\(649\) 2400.00 0.145159
\(650\) 0 0
\(651\) −9432.00 −0.567848
\(652\) 0 0
\(653\) 15072.0 0.903236 0.451618 0.892211i \(-0.350847\pi\)
0.451618 + 0.892211i \(0.350847\pi\)
\(654\) 0 0
\(655\) 6096.00 0.363650
\(656\) 0 0
\(657\) 2862.00 0.169950
\(658\) 0 0
\(659\) −11572.0 −0.684038 −0.342019 0.939693i \(-0.611111\pi\)
−0.342019 + 0.939693i \(0.611111\pi\)
\(660\) 0 0
\(661\) 26680.0 1.56994 0.784971 0.619532i \(-0.212678\pi\)
0.784971 + 0.619532i \(0.212678\pi\)
\(662\) 0 0
\(663\) 7920.00 0.463933
\(664\) 0 0
\(665\) 912.000 0.0531817
\(666\) 0 0
\(667\) −12740.0 −0.739572
\(668\) 0 0
\(669\) −8826.00 −0.510064
\(670\) 0 0
\(671\) −24560.0 −1.41301
\(672\) 0 0
\(673\) 5726.00 0.327966 0.163983 0.986463i \(-0.447566\pi\)
0.163983 + 0.986463i \(0.447566\pi\)
\(674\) 0 0
\(675\) −2943.00 −0.167816
\(676\) 0 0
\(677\) −7786.00 −0.442009 −0.221005 0.975273i \(-0.570934\pi\)
−0.221005 + 0.975273i \(0.570934\pi\)
\(678\) 0 0
\(679\) 12072.0 0.682299
\(680\) 0 0
\(681\) −4020.00 −0.226207
\(682\) 0 0
\(683\) 19372.0 1.08528 0.542642 0.839964i \(-0.317424\pi\)
0.542642 + 0.839964i \(0.317424\pi\)
\(684\) 0 0
\(685\) −6184.00 −0.344932
\(686\) 0 0
\(687\) −19398.0 −1.07726
\(688\) 0 0
\(689\) −13360.0 −0.738716
\(690\) 0 0
\(691\) −24996.0 −1.37611 −0.688055 0.725658i \(-0.741535\pi\)
−0.688055 + 0.725658i \(0.741535\pi\)
\(692\) 0 0
\(693\) −4320.00 −0.236801
\(694\) 0 0
\(695\) −10032.0 −0.547533
\(696\) 0 0
\(697\) 29172.0 1.58532
\(698\) 0 0
\(699\) −882.000 −0.0477258
\(700\) 0 0
\(701\) −14972.0 −0.806683 −0.403341 0.915050i \(-0.632151\pi\)
−0.403341 + 0.915050i \(0.632151\pi\)
\(702\) 0 0
\(703\) −5624.00 −0.301726
\(704\) 0 0
\(705\) 6504.00 0.347454
\(706\) 0 0
\(707\) −1680.00 −0.0893676
\(708\) 0 0
\(709\) −12258.0 −0.649307 −0.324654 0.945833i \(-0.605248\pi\)
−0.324654 + 0.945833i \(0.605248\pi\)
\(710\) 0 0
\(711\) −10386.0 −0.547828
\(712\) 0 0
\(713\) −25676.0 −1.34863
\(714\) 0 0
\(715\) 6400.00 0.334750
\(716\) 0 0
\(717\) −10638.0 −0.554091
\(718\) 0 0
\(719\) 19846.0 1.02939 0.514695 0.857374i \(-0.327905\pi\)
0.514695 + 0.857374i \(0.327905\pi\)
\(720\) 0 0
\(721\) −792.000 −0.0409093
\(722\) 0 0
\(723\) −4542.00 −0.233636
\(724\) 0 0
\(725\) 14170.0 0.725877
\(726\) 0 0
\(727\) 2744.00 0.139985 0.0699927 0.997548i \(-0.477702\pi\)
0.0699927 + 0.997548i \(0.477702\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −10824.0 −0.547661
\(732\) 0 0
\(733\) 834.000 0.0420252 0.0210126 0.999779i \(-0.493311\pi\)
0.0210126 + 0.999779i \(0.493311\pi\)
\(734\) 0 0
\(735\) −2388.00 −0.119840
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 30892.0 1.53773 0.768864 0.639413i \(-0.220822\pi\)
0.768864 + 0.639413i \(0.220822\pi\)
\(740\) 0 0
\(741\) −2280.00 −0.113034
\(742\) 0 0
\(743\) −4636.00 −0.228908 −0.114454 0.993429i \(-0.536512\pi\)
−0.114454 + 0.993429i \(0.536512\pi\)
\(744\) 0 0
\(745\) 3744.00 0.184120
\(746\) 0 0
\(747\) 5724.00 0.280362
\(748\) 0 0
\(749\) −12144.0 −0.592433
\(750\) 0 0
\(751\) −17666.0 −0.858377 −0.429189 0.903215i \(-0.641200\pi\)
−0.429189 + 0.903215i \(0.641200\pi\)
\(752\) 0 0
\(753\) 22092.0 1.06916
\(754\) 0 0
\(755\) −392.000 −0.0188958
\(756\) 0 0
\(757\) −27374.0 −1.31430 −0.657150 0.753760i \(-0.728238\pi\)
−0.657150 + 0.753760i \(0.728238\pi\)
\(758\) 0 0
\(759\) −11760.0 −0.562399
\(760\) 0 0
\(761\) 22350.0 1.06463 0.532317 0.846545i \(-0.321321\pi\)
0.532317 + 0.846545i \(0.321321\pi\)
\(762\) 0 0
\(763\) −7056.00 −0.334789
\(764\) 0 0
\(765\) −2376.00 −0.112293
\(766\) 0 0
\(767\) 2400.00 0.112984
\(768\) 0 0
\(769\) −37586.0 −1.76253 −0.881265 0.472622i \(-0.843308\pi\)
−0.881265 + 0.472622i \(0.843308\pi\)
\(770\) 0 0
\(771\) 6006.00 0.280546
\(772\) 0 0
\(773\) −27462.0 −1.27780 −0.638900 0.769290i \(-0.720610\pi\)
−0.638900 + 0.769290i \(0.720610\pi\)
\(774\) 0 0
\(775\) 28558.0 1.32366
\(776\) 0 0
\(777\) −10656.0 −0.491997
\(778\) 0 0
\(779\) −8398.00 −0.386251
\(780\) 0 0
\(781\) 16000.0 0.733067
\(782\) 0 0
\(783\) −3510.00 −0.160201
\(784\) 0 0
\(785\) −3704.00 −0.168409
\(786\) 0 0
\(787\) −6244.00 −0.282814 −0.141407 0.989952i \(-0.545163\pi\)
−0.141407 + 0.989952i \(0.545163\pi\)
\(788\) 0 0
\(789\) 1746.00 0.0787823
\(790\) 0 0
\(791\) 7464.00 0.335511
\(792\) 0 0
\(793\) −24560.0 −1.09981
\(794\) 0 0
\(795\) 4008.00 0.178804
\(796\) 0 0
\(797\) 1362.00 0.0605326 0.0302663 0.999542i \(-0.490364\pi\)
0.0302663 + 0.999542i \(0.490364\pi\)
\(798\) 0 0
\(799\) −35772.0 −1.58388
\(800\) 0 0
\(801\) −5670.00 −0.250112
\(802\) 0 0
\(803\) −12720.0 −0.559003
\(804\) 0 0
\(805\) 4704.00 0.205956
\(806\) 0 0
\(807\) 18270.0 0.796945
\(808\) 0 0
\(809\) 18222.0 0.791905 0.395953 0.918271i \(-0.370414\pi\)
0.395953 + 0.918271i \(0.370414\pi\)
\(810\) 0 0
\(811\) 12216.0 0.528929 0.264465 0.964395i \(-0.414805\pi\)
0.264465 + 0.964395i \(0.414805\pi\)
\(812\) 0 0
\(813\) −16356.0 −0.705572
\(814\) 0 0
\(815\) −5072.00 −0.217993
\(816\) 0 0
\(817\) 3116.00 0.133433
\(818\) 0 0
\(819\) −4320.00 −0.184314
\(820\) 0 0
\(821\) −29268.0 −1.24417 −0.622083 0.782951i \(-0.713713\pi\)
−0.622083 + 0.782951i \(0.713713\pi\)
\(822\) 0 0
\(823\) −19568.0 −0.828794 −0.414397 0.910096i \(-0.636008\pi\)
−0.414397 + 0.910096i \(0.636008\pi\)
\(824\) 0 0
\(825\) 13080.0 0.551984
\(826\) 0 0
\(827\) −14308.0 −0.601618 −0.300809 0.953684i \(-0.597257\pi\)
−0.300809 + 0.953684i \(0.597257\pi\)
\(828\) 0 0
\(829\) −15036.0 −0.629942 −0.314971 0.949101i \(-0.601995\pi\)
−0.314971 + 0.949101i \(0.601995\pi\)
\(830\) 0 0
\(831\) 14682.0 0.612892
\(832\) 0 0
\(833\) 13134.0 0.546298
\(834\) 0 0
\(835\) 2448.00 0.101457
\(836\) 0 0
\(837\) −7074.00 −0.292130
\(838\) 0 0
\(839\) 9092.00 0.374125 0.187062 0.982348i \(-0.440103\pi\)
0.187062 + 0.982348i \(0.440103\pi\)
\(840\) 0 0
\(841\) −7489.00 −0.307065
\(842\) 0 0
\(843\) 14874.0 0.607696
\(844\) 0 0
\(845\) −2388.00 −0.0972186
\(846\) 0 0
\(847\) 3228.00 0.130951
\(848\) 0 0
\(849\) 3924.00 0.158623
\(850\) 0 0
\(851\) −29008.0 −1.16849
\(852\) 0 0
\(853\) −16242.0 −0.651953 −0.325976 0.945378i \(-0.605693\pi\)
−0.325976 + 0.945378i \(0.605693\pi\)
\(854\) 0 0
\(855\) 684.000 0.0273594
\(856\) 0 0
\(857\) −9966.00 −0.397237 −0.198618 0.980077i \(-0.563645\pi\)
−0.198618 + 0.980077i \(0.563645\pi\)
\(858\) 0 0
\(859\) 33788.0 1.34206 0.671031 0.741429i \(-0.265852\pi\)
0.671031 + 0.741429i \(0.265852\pi\)
\(860\) 0 0
\(861\) −15912.0 −0.629825
\(862\) 0 0
\(863\) −1524.00 −0.0601131 −0.0300565 0.999548i \(-0.509569\pi\)
−0.0300565 + 0.999548i \(0.509569\pi\)
\(864\) 0 0
\(865\) −13176.0 −0.517916
\(866\) 0 0
\(867\) −1671.00 −0.0654558
\(868\) 0 0
\(869\) 46160.0 1.80192
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 9054.00 0.351010
\(874\) 0 0
\(875\) −11232.0 −0.433955
\(876\) 0 0
\(877\) 9688.00 0.373022 0.186511 0.982453i \(-0.440282\pi\)
0.186511 + 0.982453i \(0.440282\pi\)
\(878\) 0 0
\(879\) 23166.0 0.888930
\(880\) 0 0
\(881\) −23262.0 −0.889576 −0.444788 0.895636i \(-0.646721\pi\)
−0.444788 + 0.895636i \(0.646721\pi\)
\(882\) 0 0
\(883\) 39452.0 1.50358 0.751792 0.659400i \(-0.229190\pi\)
0.751792 + 0.659400i \(0.229190\pi\)
\(884\) 0 0
\(885\) −720.000 −0.0273475
\(886\) 0 0
\(887\) 22492.0 0.851418 0.425709 0.904860i \(-0.360025\pi\)
0.425709 + 0.904860i \(0.360025\pi\)
\(888\) 0 0
\(889\) −2376.00 −0.0896383
\(890\) 0 0
\(891\) −3240.00 −0.121823
\(892\) 0 0
\(893\) 10298.0 0.385901
\(894\) 0 0
\(895\) 14736.0 0.550358
\(896\) 0 0
\(897\) −11760.0 −0.437742
\(898\) 0 0
\(899\) 34060.0 1.26359
\(900\) 0 0
\(901\) −22044.0 −0.815086
\(902\) 0 0
\(903\) 5904.00 0.217578
\(904\) 0 0
\(905\) −13616.0 −0.500123
\(906\) 0 0
\(907\) −5504.00 −0.201496 −0.100748 0.994912i \(-0.532124\pi\)
−0.100748 + 0.994912i \(0.532124\pi\)
\(908\) 0 0
\(909\) −1260.00 −0.0459753
\(910\) 0 0
\(911\) −23412.0 −0.851454 −0.425727 0.904852i \(-0.639982\pi\)
−0.425727 + 0.904852i \(0.639982\pi\)
\(912\) 0 0
\(913\) −25440.0 −0.922170
\(914\) 0 0
\(915\) 7368.00 0.266206
\(916\) 0 0
\(917\) 18288.0 0.658586
\(918\) 0 0
\(919\) −11360.0 −0.407760 −0.203880 0.978996i \(-0.565355\pi\)
−0.203880 + 0.978996i \(0.565355\pi\)
\(920\) 0 0
\(921\) 11736.0 0.419885
\(922\) 0 0
\(923\) 16000.0 0.570581
\(924\) 0 0
\(925\) 32264.0 1.14685
\(926\) 0 0
\(927\) −594.000 −0.0210459
\(928\) 0 0
\(929\) 12766.0 0.450849 0.225425 0.974261i \(-0.427623\pi\)
0.225425 + 0.974261i \(0.427623\pi\)
\(930\) 0 0
\(931\) −3781.00 −0.133101
\(932\) 0 0
\(933\) −5982.00 −0.209906
\(934\) 0 0
\(935\) 10560.0 0.369357
\(936\) 0 0
\(937\) −17002.0 −0.592776 −0.296388 0.955068i \(-0.595782\pi\)
−0.296388 + 0.955068i \(0.595782\pi\)
\(938\) 0 0
\(939\) −9678.00 −0.336347
\(940\) 0 0
\(941\) −48046.0 −1.66446 −0.832229 0.554432i \(-0.812935\pi\)
−0.832229 + 0.554432i \(0.812935\pi\)
\(942\) 0 0
\(943\) −43316.0 −1.49583
\(944\) 0 0
\(945\) 1296.00 0.0446126
\(946\) 0 0
\(947\) −20964.0 −0.719365 −0.359682 0.933075i \(-0.617115\pi\)
−0.359682 + 0.933075i \(0.617115\pi\)
\(948\) 0 0
\(949\) −12720.0 −0.435099
\(950\) 0 0
\(951\) 1494.00 0.0509424
\(952\) 0 0
\(953\) 31270.0 1.06289 0.531445 0.847093i \(-0.321649\pi\)
0.531445 + 0.847093i \(0.321649\pi\)
\(954\) 0 0
\(955\) −10136.0 −0.343448
\(956\) 0 0
\(957\) 15600.0 0.526935
\(958\) 0 0
\(959\) −18552.0 −0.624688
\(960\) 0 0
\(961\) 38853.0 1.30419
\(962\) 0 0
\(963\) −9108.00 −0.304778
\(964\) 0 0
\(965\) 20472.0 0.682919
\(966\) 0 0
\(967\) 45780.0 1.52243 0.761213 0.648502i \(-0.224604\pi\)
0.761213 + 0.648502i \(0.224604\pi\)
\(968\) 0 0
\(969\) −3762.00 −0.124719
\(970\) 0 0
\(971\) −51220.0 −1.69282 −0.846410 0.532532i \(-0.821241\pi\)
−0.846410 + 0.532532i \(0.821241\pi\)
\(972\) 0 0
\(973\) −30096.0 −0.991607
\(974\) 0 0
\(975\) 13080.0 0.429636
\(976\) 0 0
\(977\) −53954.0 −1.76678 −0.883389 0.468641i \(-0.844744\pi\)
−0.883389 + 0.468641i \(0.844744\pi\)
\(978\) 0 0
\(979\) 25200.0 0.822672
\(980\) 0 0
\(981\) −5292.00 −0.172233
\(982\) 0 0
\(983\) −45552.0 −1.47801 −0.739005 0.673700i \(-0.764704\pi\)
−0.739005 + 0.673700i \(0.764704\pi\)
\(984\) 0 0
\(985\) 1312.00 0.0424404
\(986\) 0 0
\(987\) 19512.0 0.629254
\(988\) 0 0
\(989\) 16072.0 0.516744
\(990\) 0 0
\(991\) −34510.0 −1.10620 −0.553101 0.833114i \(-0.686556\pi\)
−0.553101 + 0.833114i \(0.686556\pi\)
\(992\) 0 0
\(993\) −10320.0 −0.329804
\(994\) 0 0
\(995\) 5392.00 0.171797
\(996\) 0 0
\(997\) −6902.00 −0.219246 −0.109623 0.993973i \(-0.534964\pi\)
−0.109623 + 0.993973i \(0.534964\pi\)
\(998\) 0 0
\(999\) −7992.00 −0.253109
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 912.4.a.h.1.1 1
4.3 odd 2 228.4.a.b.1.1 1
12.11 even 2 684.4.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
228.4.a.b.1.1 1 4.3 odd 2
684.4.a.b.1.1 1 12.11 even 2
912.4.a.h.1.1 1 1.1 even 1 trivial