Properties

Label 378.4.a.e.1.1
Level $378$
Weight $4$
Character 378.1
Self dual yes
Analytic conductor $22.303$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [378,4,Mod(1,378)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(378, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("378.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 378.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: \(22.3027219822\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 378.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000 q^{2} +4.00000 q^{4} +9.00000 q^{5} +7.00000 q^{7} -8.00000 q^{8} +O(q^{10})\) \(q-2.00000 q^{2} +4.00000 q^{4} +9.00000 q^{5} +7.00000 q^{7} -8.00000 q^{8} -18.0000 q^{10} +45.0000 q^{11} -16.0000 q^{13} -14.0000 q^{14} +16.0000 q^{16} +66.0000 q^{17} +11.0000 q^{19} +36.0000 q^{20} -90.0000 q^{22} +27.0000 q^{23} -44.0000 q^{25} +32.0000 q^{26} +28.0000 q^{28} -12.0000 q^{29} -169.000 q^{31} -32.0000 q^{32} -132.000 q^{34} +63.0000 q^{35} +209.000 q^{37} -22.0000 q^{38} -72.0000 q^{40} +291.000 q^{41} -394.000 q^{43} +180.000 q^{44} -54.0000 q^{46} +174.000 q^{47} +49.0000 q^{49} +88.0000 q^{50} -64.0000 q^{52} +228.000 q^{53} +405.000 q^{55} -56.0000 q^{56} +24.0000 q^{58} +474.000 q^{59} -232.000 q^{61} +338.000 q^{62} +64.0000 q^{64} -144.000 q^{65} +992.000 q^{67} +264.000 q^{68} -126.000 q^{70} -153.000 q^{71} +686.000 q^{73} -418.000 q^{74} +44.0000 q^{76} +315.000 q^{77} +1046.00 q^{79} +144.000 q^{80} -582.000 q^{82} +708.000 q^{83} +594.000 q^{85} +788.000 q^{86} -360.000 q^{88} +195.000 q^{89} -112.000 q^{91} +108.000 q^{92} -348.000 q^{94} +99.0000 q^{95} -88.0000 q^{97} -98.0000 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −0.707107
\(3\) 0 0
\(4\) 4.00000 0.500000
\(5\) 9.00000 0.804984 0.402492 0.915423i \(-0.368144\pi\)
0.402492 + 0.915423i \(0.368144\pi\)
\(6\) 0 0
\(7\) 7.00000 0.377964
\(8\) −8.00000 −0.353553
\(9\) 0 0
\(10\) −18.0000 −0.569210
\(11\) 45.0000 1.23346 0.616728 0.787177i \(-0.288458\pi\)
0.616728 + 0.787177i \(0.288458\pi\)
\(12\) 0 0
\(13\) −16.0000 −0.341354 −0.170677 0.985327i \(-0.554595\pi\)
−0.170677 + 0.985327i \(0.554595\pi\)
\(14\) −14.0000 −0.267261
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) 66.0000 0.941609 0.470804 0.882238i \(-0.343964\pi\)
0.470804 + 0.882238i \(0.343964\pi\)
\(18\) 0 0
\(19\) 11.0000 0.132820 0.0664098 0.997792i \(-0.478846\pi\)
0.0664098 + 0.997792i \(0.478846\pi\)
\(20\) 36.0000 0.402492
\(21\) 0 0
\(22\) −90.0000 −0.872185
\(23\) 27.0000 0.244778 0.122389 0.992482i \(-0.460944\pi\)
0.122389 + 0.992482i \(0.460944\pi\)
\(24\) 0 0
\(25\) −44.0000 −0.352000
\(26\) 32.0000 0.241374
\(27\) 0 0
\(28\) 28.0000 0.188982
\(29\) −12.0000 −0.0768395 −0.0384197 0.999262i \(-0.512232\pi\)
−0.0384197 + 0.999262i \(0.512232\pi\)
\(30\) 0 0
\(31\) −169.000 −0.979139 −0.489569 0.871964i \(-0.662846\pi\)
−0.489569 + 0.871964i \(0.662846\pi\)
\(32\) −32.0000 −0.176777
\(33\) 0 0
\(34\) −132.000 −0.665818
\(35\) 63.0000 0.304256
\(36\) 0 0
\(37\) 209.000 0.928632 0.464316 0.885670i \(-0.346300\pi\)
0.464316 + 0.885670i \(0.346300\pi\)
\(38\) −22.0000 −0.0939177
\(39\) 0 0
\(40\) −72.0000 −0.284605
\(41\) 291.000 1.10845 0.554226 0.832366i \(-0.313014\pi\)
0.554226 + 0.832366i \(0.313014\pi\)
\(42\) 0 0
\(43\) −394.000 −1.39731 −0.698656 0.715458i \(-0.746218\pi\)
−0.698656 + 0.715458i \(0.746218\pi\)
\(44\) 180.000 0.616728
\(45\) 0 0
\(46\) −54.0000 −0.173084
\(47\) 174.000 0.540011 0.270005 0.962859i \(-0.412975\pi\)
0.270005 + 0.962859i \(0.412975\pi\)
\(48\) 0 0
\(49\) 49.0000 0.142857
\(50\) 88.0000 0.248902
\(51\) 0 0
\(52\) −64.0000 −0.170677
\(53\) 228.000 0.590910 0.295455 0.955357i \(-0.404529\pi\)
0.295455 + 0.955357i \(0.404529\pi\)
\(54\) 0 0
\(55\) 405.000 0.992913
\(56\) −56.0000 −0.133631
\(57\) 0 0
\(58\) 24.0000 0.0543337
\(59\) 474.000 1.04592 0.522962 0.852356i \(-0.324827\pi\)
0.522962 + 0.852356i \(0.324827\pi\)
\(60\) 0 0
\(61\) −232.000 −0.486960 −0.243480 0.969906i \(-0.578289\pi\)
−0.243480 + 0.969906i \(0.578289\pi\)
\(62\) 338.000 0.692356
\(63\) 0 0
\(64\) 64.0000 0.125000
\(65\) −144.000 −0.274785
\(66\) 0 0
\(67\) 992.000 1.80884 0.904419 0.426646i \(-0.140305\pi\)
0.904419 + 0.426646i \(0.140305\pi\)
\(68\) 264.000 0.470804
\(69\) 0 0
\(70\) −126.000 −0.215141
\(71\) −153.000 −0.255743 −0.127872 0.991791i \(-0.540815\pi\)
−0.127872 + 0.991791i \(0.540815\pi\)
\(72\) 0 0
\(73\) 686.000 1.09987 0.549933 0.835209i \(-0.314653\pi\)
0.549933 + 0.835209i \(0.314653\pi\)
\(74\) −418.000 −0.656642
\(75\) 0 0
\(76\) 44.0000 0.0664098
\(77\) 315.000 0.466202
\(78\) 0 0
\(79\) 1046.00 1.48967 0.744837 0.667247i \(-0.232527\pi\)
0.744837 + 0.667247i \(0.232527\pi\)
\(80\) 144.000 0.201246
\(81\) 0 0
\(82\) −582.000 −0.783794
\(83\) 708.000 0.936302 0.468151 0.883648i \(-0.344920\pi\)
0.468151 + 0.883648i \(0.344920\pi\)
\(84\) 0 0
\(85\) 594.000 0.757981
\(86\) 788.000 0.988049
\(87\) 0 0
\(88\) −360.000 −0.436092
\(89\) 195.000 0.232247 0.116123 0.993235i \(-0.462953\pi\)
0.116123 + 0.993235i \(0.462953\pi\)
\(90\) 0 0
\(91\) −112.000 −0.129020
\(92\) 108.000 0.122389
\(93\) 0 0
\(94\) −348.000 −0.381845
\(95\) 99.0000 0.106918
\(96\) 0 0
\(97\) −88.0000 −0.0921139 −0.0460569 0.998939i \(-0.514666\pi\)
−0.0460569 + 0.998939i \(0.514666\pi\)
\(98\) −98.0000 −0.101015
\(99\) 0 0
\(100\) −176.000 −0.176000
\(101\) −1206.00 −1.18813 −0.594067 0.804416i \(-0.702479\pi\)
−0.594067 + 0.804416i \(0.702479\pi\)
\(102\) 0 0
\(103\) −1177.00 −1.12595 −0.562977 0.826473i \(-0.690344\pi\)
−0.562977 + 0.826473i \(0.690344\pi\)
\(104\) 128.000 0.120687
\(105\) 0 0
\(106\) −456.000 −0.417836
\(107\) −156.000 −0.140945 −0.0704724 0.997514i \(-0.522451\pi\)
−0.0704724 + 0.997514i \(0.522451\pi\)
\(108\) 0 0
\(109\) 11.0000 0.00966614 0.00483307 0.999988i \(-0.498462\pi\)
0.00483307 + 0.999988i \(0.498462\pi\)
\(110\) −810.000 −0.702095
\(111\) 0 0
\(112\) 112.000 0.0944911
\(113\) 1038.00 0.864131 0.432066 0.901842i \(-0.357785\pi\)
0.432066 + 0.901842i \(0.357785\pi\)
\(114\) 0 0
\(115\) 243.000 0.197042
\(116\) −48.0000 −0.0384197
\(117\) 0 0
\(118\) −948.000 −0.739580
\(119\) 462.000 0.355895
\(120\) 0 0
\(121\) 694.000 0.521412
\(122\) 464.000 0.344333
\(123\) 0 0
\(124\) −676.000 −0.489569
\(125\) −1521.00 −1.08834
\(126\) 0 0
\(127\) 2216.00 1.54833 0.774166 0.632982i \(-0.218169\pi\)
0.774166 + 0.632982i \(0.218169\pi\)
\(128\) −128.000 −0.0883883
\(129\) 0 0
\(130\) 288.000 0.194302
\(131\) 918.000 0.612260 0.306130 0.951990i \(-0.400966\pi\)
0.306130 + 0.951990i \(0.400966\pi\)
\(132\) 0 0
\(133\) 77.0000 0.0502011
\(134\) −1984.00 −1.27904
\(135\) 0 0
\(136\) −528.000 −0.332909
\(137\) −2448.00 −1.52662 −0.763309 0.646033i \(-0.776427\pi\)
−0.763309 + 0.646033i \(0.776427\pi\)
\(138\) 0 0
\(139\) 1244.00 0.759099 0.379549 0.925172i \(-0.376079\pi\)
0.379549 + 0.925172i \(0.376079\pi\)
\(140\) 252.000 0.152128
\(141\) 0 0
\(142\) 306.000 0.180838
\(143\) −720.000 −0.421045
\(144\) 0 0
\(145\) −108.000 −0.0618546
\(146\) −1372.00 −0.777723
\(147\) 0 0
\(148\) 836.000 0.464316
\(149\) −954.000 −0.524528 −0.262264 0.964996i \(-0.584469\pi\)
−0.262264 + 0.964996i \(0.584469\pi\)
\(150\) 0 0
\(151\) −1942.00 −1.04661 −0.523304 0.852146i \(-0.675301\pi\)
−0.523304 + 0.852146i \(0.675301\pi\)
\(152\) −88.0000 −0.0469588
\(153\) 0 0
\(154\) −630.000 −0.329655
\(155\) −1521.00 −0.788191
\(156\) 0 0
\(157\) −2032.00 −1.03294 −0.516469 0.856306i \(-0.672754\pi\)
−0.516469 + 0.856306i \(0.672754\pi\)
\(158\) −2092.00 −1.05336
\(159\) 0 0
\(160\) −288.000 −0.142302
\(161\) 189.000 0.0925173
\(162\) 0 0
\(163\) 2252.00 1.08215 0.541074 0.840975i \(-0.318018\pi\)
0.541074 + 0.840975i \(0.318018\pi\)
\(164\) 1164.00 0.554226
\(165\) 0 0
\(166\) −1416.00 −0.662066
\(167\) 4170.00 1.93224 0.966121 0.258091i \(-0.0830934\pi\)
0.966121 + 0.258091i \(0.0830934\pi\)
\(168\) 0 0
\(169\) −1941.00 −0.883477
\(170\) −1188.00 −0.535973
\(171\) 0 0
\(172\) −1576.00 −0.698656
\(173\) −4353.00 −1.91302 −0.956510 0.291700i \(-0.905779\pi\)
−0.956510 + 0.291700i \(0.905779\pi\)
\(174\) 0 0
\(175\) −308.000 −0.133043
\(176\) 720.000 0.308364
\(177\) 0 0
\(178\) −390.000 −0.164223
\(179\) −3984.00 −1.66357 −0.831783 0.555102i \(-0.812679\pi\)
−0.831783 + 0.555102i \(0.812679\pi\)
\(180\) 0 0
\(181\) 650.000 0.266929 0.133464 0.991054i \(-0.457390\pi\)
0.133464 + 0.991054i \(0.457390\pi\)
\(182\) 224.000 0.0912307
\(183\) 0 0
\(184\) −216.000 −0.0865420
\(185\) 1881.00 0.747534
\(186\) 0 0
\(187\) 2970.00 1.16143
\(188\) 696.000 0.270005
\(189\) 0 0
\(190\) −198.000 −0.0756023
\(191\) −2967.00 −1.12400 −0.562002 0.827136i \(-0.689969\pi\)
−0.562002 + 0.827136i \(0.689969\pi\)
\(192\) 0 0
\(193\) −2842.00 −1.05996 −0.529978 0.848011i \(-0.677800\pi\)
−0.529978 + 0.848011i \(0.677800\pi\)
\(194\) 176.000 0.0651343
\(195\) 0 0
\(196\) 196.000 0.0714286
\(197\) 2286.00 0.826755 0.413378 0.910560i \(-0.364349\pi\)
0.413378 + 0.910560i \(0.364349\pi\)
\(198\) 0 0
\(199\) −1645.00 −0.585985 −0.292992 0.956115i \(-0.594651\pi\)
−0.292992 + 0.956115i \(0.594651\pi\)
\(200\) 352.000 0.124451
\(201\) 0 0
\(202\) 2412.00 0.840137
\(203\) −84.0000 −0.0290426
\(204\) 0 0
\(205\) 2619.00 0.892287
\(206\) 2354.00 0.796170
\(207\) 0 0
\(208\) −256.000 −0.0853385
\(209\) 495.000 0.163827
\(210\) 0 0
\(211\) −2878.00 −0.939003 −0.469502 0.882932i \(-0.655566\pi\)
−0.469502 + 0.882932i \(0.655566\pi\)
\(212\) 912.000 0.295455
\(213\) 0 0
\(214\) 312.000 0.0996630
\(215\) −3546.00 −1.12481
\(216\) 0 0
\(217\) −1183.00 −0.370080
\(218\) −22.0000 −0.00683499
\(219\) 0 0
\(220\) 1620.00 0.496456
\(221\) −1056.00 −0.321422
\(222\) 0 0
\(223\) 4349.00 1.30597 0.652983 0.757372i \(-0.273517\pi\)
0.652983 + 0.757372i \(0.273517\pi\)
\(224\) −224.000 −0.0668153
\(225\) 0 0
\(226\) −2076.00 −0.611033
\(227\) −4266.00 −1.24733 −0.623666 0.781691i \(-0.714357\pi\)
−0.623666 + 0.781691i \(0.714357\pi\)
\(228\) 0 0
\(229\) −1060.00 −0.305881 −0.152941 0.988235i \(-0.548874\pi\)
−0.152941 + 0.988235i \(0.548874\pi\)
\(230\) −486.000 −0.139330
\(231\) 0 0
\(232\) 96.0000 0.0271668
\(233\) 2814.00 0.791207 0.395604 0.918421i \(-0.370535\pi\)
0.395604 + 0.918421i \(0.370535\pi\)
\(234\) 0 0
\(235\) 1566.00 0.434700
\(236\) 1896.00 0.522962
\(237\) 0 0
\(238\) −924.000 −0.251656
\(239\) 1200.00 0.324776 0.162388 0.986727i \(-0.448080\pi\)
0.162388 + 0.986727i \(0.448080\pi\)
\(240\) 0 0
\(241\) −5650.00 −1.51016 −0.755080 0.655633i \(-0.772402\pi\)
−0.755080 + 0.655633i \(0.772402\pi\)
\(242\) −1388.00 −0.368694
\(243\) 0 0
\(244\) −928.000 −0.243480
\(245\) 441.000 0.114998
\(246\) 0 0
\(247\) −176.000 −0.0453385
\(248\) 1352.00 0.346178
\(249\) 0 0
\(250\) 3042.00 0.769572
\(251\) 2160.00 0.543179 0.271590 0.962413i \(-0.412451\pi\)
0.271590 + 0.962413i \(0.412451\pi\)
\(252\) 0 0
\(253\) 1215.00 0.301923
\(254\) −4432.00 −1.09484
\(255\) 0 0
\(256\) 256.000 0.0625000
\(257\) −2457.00 −0.596356 −0.298178 0.954510i \(-0.596379\pi\)
−0.298178 + 0.954510i \(0.596379\pi\)
\(258\) 0 0
\(259\) 1463.00 0.350990
\(260\) −576.000 −0.137392
\(261\) 0 0
\(262\) −1836.00 −0.432933
\(263\) −2889.00 −0.677351 −0.338676 0.940903i \(-0.609979\pi\)
−0.338676 + 0.940903i \(0.609979\pi\)
\(264\) 0 0
\(265\) 2052.00 0.475673
\(266\) −154.000 −0.0354975
\(267\) 0 0
\(268\) 3968.00 0.904419
\(269\) 7221.00 1.63670 0.818350 0.574721i \(-0.194889\pi\)
0.818350 + 0.574721i \(0.194889\pi\)
\(270\) 0 0
\(271\) 2504.00 0.561281 0.280641 0.959813i \(-0.409453\pi\)
0.280641 + 0.959813i \(0.409453\pi\)
\(272\) 1056.00 0.235402
\(273\) 0 0
\(274\) 4896.00 1.07948
\(275\) −1980.00 −0.434176
\(276\) 0 0
\(277\) −2455.00 −0.532515 −0.266257 0.963902i \(-0.585787\pi\)
−0.266257 + 0.963902i \(0.585787\pi\)
\(278\) −2488.00 −0.536764
\(279\) 0 0
\(280\) −504.000 −0.107571
\(281\) 8070.00 1.71322 0.856612 0.515961i \(-0.172565\pi\)
0.856612 + 0.515961i \(0.172565\pi\)
\(282\) 0 0
\(283\) 1244.00 0.261301 0.130650 0.991429i \(-0.458293\pi\)
0.130650 + 0.991429i \(0.458293\pi\)
\(284\) −612.000 −0.127872
\(285\) 0 0
\(286\) 1440.00 0.297724
\(287\) 2037.00 0.418956
\(288\) 0 0
\(289\) −557.000 −0.113373
\(290\) 216.000 0.0437378
\(291\) 0 0
\(292\) 2744.00 0.549933
\(293\) 3450.00 0.687888 0.343944 0.938990i \(-0.388237\pi\)
0.343944 + 0.938990i \(0.388237\pi\)
\(294\) 0 0
\(295\) 4266.00 0.841953
\(296\) −1672.00 −0.328321
\(297\) 0 0
\(298\) 1908.00 0.370898
\(299\) −432.000 −0.0835559
\(300\) 0 0
\(301\) −2758.00 −0.528134
\(302\) 3884.00 0.740063
\(303\) 0 0
\(304\) 176.000 0.0332049
\(305\) −2088.00 −0.391995
\(306\) 0 0
\(307\) −2761.00 −0.513285 −0.256643 0.966506i \(-0.582616\pi\)
−0.256643 + 0.966506i \(0.582616\pi\)
\(308\) 1260.00 0.233101
\(309\) 0 0
\(310\) 3042.00 0.557335
\(311\) 7008.00 1.27777 0.638886 0.769301i \(-0.279395\pi\)
0.638886 + 0.769301i \(0.279395\pi\)
\(312\) 0 0
\(313\) −160.000 −0.0288937 −0.0144469 0.999896i \(-0.504599\pi\)
−0.0144469 + 0.999896i \(0.504599\pi\)
\(314\) 4064.00 0.730397
\(315\) 0 0
\(316\) 4184.00 0.744837
\(317\) 750.000 0.132884 0.0664420 0.997790i \(-0.478835\pi\)
0.0664420 + 0.997790i \(0.478835\pi\)
\(318\) 0 0
\(319\) −540.000 −0.0947780
\(320\) 576.000 0.100623
\(321\) 0 0
\(322\) −378.000 −0.0654196
\(323\) 726.000 0.125064
\(324\) 0 0
\(325\) 704.000 0.120157
\(326\) −4504.00 −0.765195
\(327\) 0 0
\(328\) −2328.00 −0.391897
\(329\) 1218.00 0.204105
\(330\) 0 0
\(331\) 3188.00 0.529391 0.264695 0.964332i \(-0.414729\pi\)
0.264695 + 0.964332i \(0.414729\pi\)
\(332\) 2832.00 0.468151
\(333\) 0 0
\(334\) −8340.00 −1.36630
\(335\) 8928.00 1.45609
\(336\) 0 0
\(337\) −8917.00 −1.44136 −0.720682 0.693265i \(-0.756171\pi\)
−0.720682 + 0.693265i \(0.756171\pi\)
\(338\) 3882.00 0.624713
\(339\) 0 0
\(340\) 2376.00 0.378990
\(341\) −7605.00 −1.20772
\(342\) 0 0
\(343\) 343.000 0.0539949
\(344\) 3152.00 0.494025
\(345\) 0 0
\(346\) 8706.00 1.35271
\(347\) −7491.00 −1.15890 −0.579449 0.815008i \(-0.696732\pi\)
−0.579449 + 0.815008i \(0.696732\pi\)
\(348\) 0 0
\(349\) −7450.00 −1.14266 −0.571331 0.820719i \(-0.693573\pi\)
−0.571331 + 0.820719i \(0.693573\pi\)
\(350\) 616.000 0.0940760
\(351\) 0 0
\(352\) −1440.00 −0.218046
\(353\) 9225.00 1.39093 0.695463 0.718561i \(-0.255199\pi\)
0.695463 + 0.718561i \(0.255199\pi\)
\(354\) 0 0
\(355\) −1377.00 −0.205869
\(356\) 780.000 0.116123
\(357\) 0 0
\(358\) 7968.00 1.17632
\(359\) 3660.00 0.538071 0.269035 0.963130i \(-0.413295\pi\)
0.269035 + 0.963130i \(0.413295\pi\)
\(360\) 0 0
\(361\) −6738.00 −0.982359
\(362\) −1300.00 −0.188747
\(363\) 0 0
\(364\) −448.000 −0.0645098
\(365\) 6174.00 0.885375
\(366\) 0 0
\(367\) −10213.0 −1.45263 −0.726314 0.687363i \(-0.758768\pi\)
−0.726314 + 0.687363i \(0.758768\pi\)
\(368\) 432.000 0.0611944
\(369\) 0 0
\(370\) −3762.00 −0.528587
\(371\) 1596.00 0.223343
\(372\) 0 0
\(373\) −8629.00 −1.19784 −0.598918 0.800811i \(-0.704402\pi\)
−0.598918 + 0.800811i \(0.704402\pi\)
\(374\) −5940.00 −0.821257
\(375\) 0 0
\(376\) −1392.00 −0.190923
\(377\) 192.000 0.0262295
\(378\) 0 0
\(379\) 1226.00 0.166162 0.0830810 0.996543i \(-0.473524\pi\)
0.0830810 + 0.996543i \(0.473524\pi\)
\(380\) 396.000 0.0534589
\(381\) 0 0
\(382\) 5934.00 0.794790
\(383\) 10998.0 1.46729 0.733644 0.679534i \(-0.237818\pi\)
0.733644 + 0.679534i \(0.237818\pi\)
\(384\) 0 0
\(385\) 2835.00 0.375286
\(386\) 5684.00 0.749503
\(387\) 0 0
\(388\) −352.000 −0.0460569
\(389\) −2772.00 −0.361301 −0.180650 0.983547i \(-0.557820\pi\)
−0.180650 + 0.983547i \(0.557820\pi\)
\(390\) 0 0
\(391\) 1782.00 0.230485
\(392\) −392.000 −0.0505076
\(393\) 0 0
\(394\) −4572.00 −0.584604
\(395\) 9414.00 1.19916
\(396\) 0 0
\(397\) −6118.00 −0.773435 −0.386717 0.922198i \(-0.626391\pi\)
−0.386717 + 0.922198i \(0.626391\pi\)
\(398\) 3290.00 0.414354
\(399\) 0 0
\(400\) −704.000 −0.0880000
\(401\) −14076.0 −1.75292 −0.876461 0.481472i \(-0.840102\pi\)
−0.876461 + 0.481472i \(0.840102\pi\)
\(402\) 0 0
\(403\) 2704.00 0.334233
\(404\) −4824.00 −0.594067
\(405\) 0 0
\(406\) 168.000 0.0205362
\(407\) 9405.00 1.14543
\(408\) 0 0
\(409\) 542.000 0.0655261 0.0327631 0.999463i \(-0.489569\pi\)
0.0327631 + 0.999463i \(0.489569\pi\)
\(410\) −5238.00 −0.630942
\(411\) 0 0
\(412\) −4708.00 −0.562977
\(413\) 3318.00 0.395322
\(414\) 0 0
\(415\) 6372.00 0.753709
\(416\) 512.000 0.0603434
\(417\) 0 0
\(418\) −990.000 −0.115843
\(419\) −6090.00 −0.710062 −0.355031 0.934855i \(-0.615530\pi\)
−0.355031 + 0.934855i \(0.615530\pi\)
\(420\) 0 0
\(421\) −14335.0 −1.65949 −0.829745 0.558143i \(-0.811514\pi\)
−0.829745 + 0.558143i \(0.811514\pi\)
\(422\) 5756.00 0.663976
\(423\) 0 0
\(424\) −1824.00 −0.208918
\(425\) −2904.00 −0.331446
\(426\) 0 0
\(427\) −1624.00 −0.184054
\(428\) −624.000 −0.0704724
\(429\) 0 0
\(430\) 7092.00 0.795364
\(431\) −16755.0 −1.87253 −0.936264 0.351296i \(-0.885741\pi\)
−0.936264 + 0.351296i \(0.885741\pi\)
\(432\) 0 0
\(433\) 7436.00 0.825292 0.412646 0.910892i \(-0.364605\pi\)
0.412646 + 0.910892i \(0.364605\pi\)
\(434\) 2366.00 0.261686
\(435\) 0 0
\(436\) 44.0000 0.00483307
\(437\) 297.000 0.0325113
\(438\) 0 0
\(439\) 1352.00 0.146987 0.0734937 0.997296i \(-0.476585\pi\)
0.0734937 + 0.997296i \(0.476585\pi\)
\(440\) −3240.00 −0.351048
\(441\) 0 0
\(442\) 2112.00 0.227280
\(443\) 147.000 0.0157656 0.00788282 0.999969i \(-0.497491\pi\)
0.00788282 + 0.999969i \(0.497491\pi\)
\(444\) 0 0
\(445\) 1755.00 0.186955
\(446\) −8698.00 −0.923458
\(447\) 0 0
\(448\) 448.000 0.0472456
\(449\) −10638.0 −1.11813 −0.559063 0.829125i \(-0.688839\pi\)
−0.559063 + 0.829125i \(0.688839\pi\)
\(450\) 0 0
\(451\) 13095.0 1.36723
\(452\) 4152.00 0.432066
\(453\) 0 0
\(454\) 8532.00 0.881997
\(455\) −1008.00 −0.103859
\(456\) 0 0
\(457\) −3499.00 −0.358154 −0.179077 0.983835i \(-0.557311\pi\)
−0.179077 + 0.983835i \(0.557311\pi\)
\(458\) 2120.00 0.216291
\(459\) 0 0
\(460\) 972.000 0.0985212
\(461\) 4119.00 0.416141 0.208070 0.978114i \(-0.433282\pi\)
0.208070 + 0.978114i \(0.433282\pi\)
\(462\) 0 0
\(463\) −13318.0 −1.33680 −0.668402 0.743801i \(-0.733021\pi\)
−0.668402 + 0.743801i \(0.733021\pi\)
\(464\) −192.000 −0.0192099
\(465\) 0 0
\(466\) −5628.00 −0.559468
\(467\) −3306.00 −0.327588 −0.163794 0.986495i \(-0.552373\pi\)
−0.163794 + 0.986495i \(0.552373\pi\)
\(468\) 0 0
\(469\) 6944.00 0.683676
\(470\) −3132.00 −0.307380
\(471\) 0 0
\(472\) −3792.00 −0.369790
\(473\) −17730.0 −1.72352
\(474\) 0 0
\(475\) −484.000 −0.0467525
\(476\) 1848.00 0.177947
\(477\) 0 0
\(478\) −2400.00 −0.229652
\(479\) 2376.00 0.226643 0.113322 0.993558i \(-0.463851\pi\)
0.113322 + 0.993558i \(0.463851\pi\)
\(480\) 0 0
\(481\) −3344.00 −0.316992
\(482\) 11300.0 1.06784
\(483\) 0 0
\(484\) 2776.00 0.260706
\(485\) −792.000 −0.0741502
\(486\) 0 0
\(487\) 14186.0 1.31998 0.659989 0.751276i \(-0.270561\pi\)
0.659989 + 0.751276i \(0.270561\pi\)
\(488\) 1856.00 0.172166
\(489\) 0 0
\(490\) −882.000 −0.0813157
\(491\) −10599.0 −0.974188 −0.487094 0.873350i \(-0.661943\pi\)
−0.487094 + 0.873350i \(0.661943\pi\)
\(492\) 0 0
\(493\) −792.000 −0.0723527
\(494\) 352.000 0.0320592
\(495\) 0 0
\(496\) −2704.00 −0.244785
\(497\) −1071.00 −0.0966618
\(498\) 0 0
\(499\) −3022.00 −0.271109 −0.135554 0.990770i \(-0.543282\pi\)
−0.135554 + 0.990770i \(0.543282\pi\)
\(500\) −6084.00 −0.544170
\(501\) 0 0
\(502\) −4320.00 −0.384086
\(503\) 8244.00 0.730779 0.365389 0.930855i \(-0.380936\pi\)
0.365389 + 0.930855i \(0.380936\pi\)
\(504\) 0 0
\(505\) −10854.0 −0.956429
\(506\) −2430.00 −0.213491
\(507\) 0 0
\(508\) 8864.00 0.774166
\(509\) −8406.00 −0.732003 −0.366001 0.930614i \(-0.619273\pi\)
−0.366001 + 0.930614i \(0.619273\pi\)
\(510\) 0 0
\(511\) 4802.00 0.415710
\(512\) −512.000 −0.0441942
\(513\) 0 0
\(514\) 4914.00 0.421687
\(515\) −10593.0 −0.906375
\(516\) 0 0
\(517\) 7830.00 0.666079
\(518\) −2926.00 −0.248187
\(519\) 0 0
\(520\) 1152.00 0.0971510
\(521\) 7005.00 0.589049 0.294525 0.955644i \(-0.404839\pi\)
0.294525 + 0.955644i \(0.404839\pi\)
\(522\) 0 0
\(523\) 18695.0 1.56305 0.781525 0.623874i \(-0.214442\pi\)
0.781525 + 0.623874i \(0.214442\pi\)
\(524\) 3672.00 0.306130
\(525\) 0 0
\(526\) 5778.00 0.478960
\(527\) −11154.0 −0.921966
\(528\) 0 0
\(529\) −11438.0 −0.940084
\(530\) −4104.00 −0.336352
\(531\) 0 0
\(532\) 308.000 0.0251006
\(533\) −4656.00 −0.378375
\(534\) 0 0
\(535\) −1404.00 −0.113458
\(536\) −7936.00 −0.639521
\(537\) 0 0
\(538\) −14442.0 −1.15732
\(539\) 2205.00 0.176208
\(540\) 0 0
\(541\) 6113.00 0.485801 0.242901 0.970051i \(-0.421901\pi\)
0.242901 + 0.970051i \(0.421901\pi\)
\(542\) −5008.00 −0.396886
\(543\) 0 0
\(544\) −2112.00 −0.166455
\(545\) 99.0000 0.00778109
\(546\) 0 0
\(547\) 2036.00 0.159146 0.0795732 0.996829i \(-0.474644\pi\)
0.0795732 + 0.996829i \(0.474644\pi\)
\(548\) −9792.00 −0.763309
\(549\) 0 0
\(550\) 3960.00 0.307009
\(551\) −132.000 −0.0102058
\(552\) 0 0
\(553\) 7322.00 0.563044
\(554\) 4910.00 0.376545
\(555\) 0 0
\(556\) 4976.00 0.379549
\(557\) 11262.0 0.856708 0.428354 0.903611i \(-0.359094\pi\)
0.428354 + 0.903611i \(0.359094\pi\)
\(558\) 0 0
\(559\) 6304.00 0.476978
\(560\) 1008.00 0.0760639
\(561\) 0 0
\(562\) −16140.0 −1.21143
\(563\) −11028.0 −0.825532 −0.412766 0.910837i \(-0.635437\pi\)
−0.412766 + 0.910837i \(0.635437\pi\)
\(564\) 0 0
\(565\) 9342.00 0.695612
\(566\) −2488.00 −0.184768
\(567\) 0 0
\(568\) 1224.00 0.0904188
\(569\) 10686.0 0.787312 0.393656 0.919258i \(-0.371210\pi\)
0.393656 + 0.919258i \(0.371210\pi\)
\(570\) 0 0
\(571\) 2018.00 0.147900 0.0739498 0.997262i \(-0.476440\pi\)
0.0739498 + 0.997262i \(0.476440\pi\)
\(572\) −2880.00 −0.210522
\(573\) 0 0
\(574\) −4074.00 −0.296246
\(575\) −1188.00 −0.0861618
\(576\) 0 0
\(577\) −10006.0 −0.721933 −0.360966 0.932579i \(-0.617553\pi\)
−0.360966 + 0.932579i \(0.617553\pi\)
\(578\) 1114.00 0.0801666
\(579\) 0 0
\(580\) −432.000 −0.0309273
\(581\) 4956.00 0.353889
\(582\) 0 0
\(583\) 10260.0 0.728861
\(584\) −5488.00 −0.388861
\(585\) 0 0
\(586\) −6900.00 −0.486410
\(587\) 13254.0 0.931944 0.465972 0.884799i \(-0.345705\pi\)
0.465972 + 0.884799i \(0.345705\pi\)
\(588\) 0 0
\(589\) −1859.00 −0.130049
\(590\) −8532.00 −0.595351
\(591\) 0 0
\(592\) 3344.00 0.232158
\(593\) −23187.0 −1.60569 −0.802847 0.596186i \(-0.796682\pi\)
−0.802847 + 0.596186i \(0.796682\pi\)
\(594\) 0 0
\(595\) 4158.00 0.286490
\(596\) −3816.00 −0.262264
\(597\) 0 0
\(598\) 864.000 0.0590829
\(599\) −27255.0 −1.85911 −0.929557 0.368679i \(-0.879810\pi\)
−0.929557 + 0.368679i \(0.879810\pi\)
\(600\) 0 0
\(601\) −21976.0 −1.49155 −0.745773 0.666200i \(-0.767920\pi\)
−0.745773 + 0.666200i \(0.767920\pi\)
\(602\) 5516.00 0.373447
\(603\) 0 0
\(604\) −7768.00 −0.523304
\(605\) 6246.00 0.419729
\(606\) 0 0
\(607\) 13052.0 0.872758 0.436379 0.899763i \(-0.356261\pi\)
0.436379 + 0.899763i \(0.356261\pi\)
\(608\) −352.000 −0.0234794
\(609\) 0 0
\(610\) 4176.00 0.277182
\(611\) −2784.00 −0.184335
\(612\) 0 0
\(613\) −2923.00 −0.192592 −0.0962960 0.995353i \(-0.530700\pi\)
−0.0962960 + 0.995353i \(0.530700\pi\)
\(614\) 5522.00 0.362948
\(615\) 0 0
\(616\) −2520.00 −0.164827
\(617\) −3714.00 −0.242334 −0.121167 0.992632i \(-0.538664\pi\)
−0.121167 + 0.992632i \(0.538664\pi\)
\(618\) 0 0
\(619\) 17705.0 1.14963 0.574817 0.818282i \(-0.305073\pi\)
0.574817 + 0.818282i \(0.305073\pi\)
\(620\) −6084.00 −0.394096
\(621\) 0 0
\(622\) −14016.0 −0.903522
\(623\) 1365.00 0.0877810
\(624\) 0 0
\(625\) −8189.00 −0.524096
\(626\) 320.000 0.0204309
\(627\) 0 0
\(628\) −8128.00 −0.516469
\(629\) 13794.0 0.874408
\(630\) 0 0
\(631\) −19978.0 −1.26040 −0.630199 0.776433i \(-0.717027\pi\)
−0.630199 + 0.776433i \(0.717027\pi\)
\(632\) −8368.00 −0.526679
\(633\) 0 0
\(634\) −1500.00 −0.0939631
\(635\) 19944.0 1.24638
\(636\) 0 0
\(637\) −784.000 −0.0487649
\(638\) 1080.00 0.0670182
\(639\) 0 0
\(640\) −1152.00 −0.0711512
\(641\) −11502.0 −0.708739 −0.354369 0.935105i \(-0.615304\pi\)
−0.354369 + 0.935105i \(0.615304\pi\)
\(642\) 0 0
\(643\) 31439.0 1.92820 0.964100 0.265538i \(-0.0855496\pi\)
0.964100 + 0.265538i \(0.0855496\pi\)
\(644\) 756.000 0.0462587
\(645\) 0 0
\(646\) −1452.00 −0.0884337
\(647\) 10650.0 0.647132 0.323566 0.946206i \(-0.395118\pi\)
0.323566 + 0.946206i \(0.395118\pi\)
\(648\) 0 0
\(649\) 21330.0 1.29010
\(650\) −1408.00 −0.0849635
\(651\) 0 0
\(652\) 9008.00 0.541074
\(653\) −9774.00 −0.585737 −0.292868 0.956153i \(-0.594610\pi\)
−0.292868 + 0.956153i \(0.594610\pi\)
\(654\) 0 0
\(655\) 8262.00 0.492860
\(656\) 4656.00 0.277113
\(657\) 0 0
\(658\) −2436.00 −0.144324
\(659\) 13857.0 0.819108 0.409554 0.912286i \(-0.365684\pi\)
0.409554 + 0.912286i \(0.365684\pi\)
\(660\) 0 0
\(661\) 3908.00 0.229960 0.114980 0.993368i \(-0.463320\pi\)
0.114980 + 0.993368i \(0.463320\pi\)
\(662\) −6376.00 −0.374336
\(663\) 0 0
\(664\) −5664.00 −0.331033
\(665\) 693.000 0.0404111
\(666\) 0 0
\(667\) −324.000 −0.0188086
\(668\) 16680.0 0.966121
\(669\) 0 0
\(670\) −17856.0 −1.02961
\(671\) −10440.0 −0.600643
\(672\) 0 0
\(673\) 26570.0 1.52184 0.760920 0.648846i \(-0.224748\pi\)
0.760920 + 0.648846i \(0.224748\pi\)
\(674\) 17834.0 1.01920
\(675\) 0 0
\(676\) −7764.00 −0.441739
\(677\) −1725.00 −0.0979278 −0.0489639 0.998801i \(-0.515592\pi\)
−0.0489639 + 0.998801i \(0.515592\pi\)
\(678\) 0 0
\(679\) −616.000 −0.0348158
\(680\) −4752.00 −0.267987
\(681\) 0 0
\(682\) 15210.0 0.853990
\(683\) 31071.0 1.74070 0.870350 0.492433i \(-0.163892\pi\)
0.870350 + 0.492433i \(0.163892\pi\)
\(684\) 0 0
\(685\) −22032.0 −1.22890
\(686\) −686.000 −0.0381802
\(687\) 0 0
\(688\) −6304.00 −0.349328
\(689\) −3648.00 −0.201709
\(690\) 0 0
\(691\) 19964.0 1.09908 0.549541 0.835466i \(-0.314802\pi\)
0.549541 + 0.835466i \(0.314802\pi\)
\(692\) −17412.0 −0.956510
\(693\) 0 0
\(694\) 14982.0 0.819465
\(695\) 11196.0 0.611063
\(696\) 0 0
\(697\) 19206.0 1.04373
\(698\) 14900.0 0.807985
\(699\) 0 0
\(700\) −1232.00 −0.0665217
\(701\) −25884.0 −1.39462 −0.697308 0.716772i \(-0.745619\pi\)
−0.697308 + 0.716772i \(0.745619\pi\)
\(702\) 0 0
\(703\) 2299.00 0.123341
\(704\) 2880.00 0.154182
\(705\) 0 0
\(706\) −18450.0 −0.983534
\(707\) −8442.00 −0.449072
\(708\) 0 0
\(709\) −23821.0 −1.26180 −0.630900 0.775864i \(-0.717314\pi\)
−0.630900 + 0.775864i \(0.717314\pi\)
\(710\) 2754.00 0.145572
\(711\) 0 0
\(712\) −1560.00 −0.0821116
\(713\) −4563.00 −0.239671
\(714\) 0 0
\(715\) −6480.00 −0.338935
\(716\) −15936.0 −0.831783
\(717\) 0 0
\(718\) −7320.00 −0.380474
\(719\) −28254.0 −1.46550 −0.732751 0.680497i \(-0.761764\pi\)
−0.732751 + 0.680497i \(0.761764\pi\)
\(720\) 0 0
\(721\) −8239.00 −0.425571
\(722\) 13476.0 0.694633
\(723\) 0 0
\(724\) 2600.00 0.133464
\(725\) 528.000 0.0270475
\(726\) 0 0
\(727\) −17152.0 −0.875010 −0.437505 0.899216i \(-0.644138\pi\)
−0.437505 + 0.899216i \(0.644138\pi\)
\(728\) 896.000 0.0456153
\(729\) 0 0
\(730\) −12348.0 −0.626055
\(731\) −26004.0 −1.31572
\(732\) 0 0
\(733\) 254.000 0.0127991 0.00639953 0.999980i \(-0.497963\pi\)
0.00639953 + 0.999980i \(0.497963\pi\)
\(734\) 20426.0 1.02716
\(735\) 0 0
\(736\) −864.000 −0.0432710
\(737\) 44640.0 2.23112
\(738\) 0 0
\(739\) −12382.0 −0.616345 −0.308173 0.951330i \(-0.599717\pi\)
−0.308173 + 0.951330i \(0.599717\pi\)
\(740\) 7524.00 0.373767
\(741\) 0 0
\(742\) −3192.00 −0.157927
\(743\) 34671.0 1.71192 0.855959 0.517043i \(-0.172967\pi\)
0.855959 + 0.517043i \(0.172967\pi\)
\(744\) 0 0
\(745\) −8586.00 −0.422237
\(746\) 17258.0 0.846998
\(747\) 0 0
\(748\) 11880.0 0.580716
\(749\) −1092.00 −0.0532721
\(750\) 0 0
\(751\) −28150.0 −1.36779 −0.683894 0.729582i \(-0.739715\pi\)
−0.683894 + 0.729582i \(0.739715\pi\)
\(752\) 2784.00 0.135003
\(753\) 0 0
\(754\) −384.000 −0.0185470
\(755\) −17478.0 −0.842503
\(756\) 0 0
\(757\) −26638.0 −1.27896 −0.639481 0.768807i \(-0.720851\pi\)
−0.639481 + 0.768807i \(0.720851\pi\)
\(758\) −2452.00 −0.117494
\(759\) 0 0
\(760\) −792.000 −0.0378011
\(761\) 26154.0 1.24584 0.622918 0.782287i \(-0.285947\pi\)
0.622918 + 0.782287i \(0.285947\pi\)
\(762\) 0 0
\(763\) 77.0000 0.00365346
\(764\) −11868.0 −0.562002
\(765\) 0 0
\(766\) −21996.0 −1.03753
\(767\) −7584.00 −0.357030
\(768\) 0 0
\(769\) 36416.0 1.70767 0.853833 0.520548i \(-0.174272\pi\)
0.853833 + 0.520548i \(0.174272\pi\)
\(770\) −5670.00 −0.265367
\(771\) 0 0
\(772\) −11368.0 −0.529978
\(773\) −16917.0 −0.787144 −0.393572 0.919294i \(-0.628761\pi\)
−0.393572 + 0.919294i \(0.628761\pi\)
\(774\) 0 0
\(775\) 7436.00 0.344657
\(776\) 704.000 0.0325672
\(777\) 0 0
\(778\) 5544.00 0.255478
\(779\) 3201.00 0.147224
\(780\) 0 0
\(781\) −6885.00 −0.315448
\(782\) −3564.00 −0.162977
\(783\) 0 0
\(784\) 784.000 0.0357143
\(785\) −18288.0 −0.831499
\(786\) 0 0
\(787\) −20032.0 −0.907324 −0.453662 0.891174i \(-0.649883\pi\)
−0.453662 + 0.891174i \(0.649883\pi\)
\(788\) 9144.00 0.413378
\(789\) 0 0
\(790\) −18828.0 −0.847937
\(791\) 7266.00 0.326611
\(792\) 0 0
\(793\) 3712.00 0.166226
\(794\) 12236.0 0.546901
\(795\) 0 0
\(796\) −6580.00 −0.292992
\(797\) 34095.0 1.51532 0.757658 0.652652i \(-0.226344\pi\)
0.757658 + 0.652652i \(0.226344\pi\)
\(798\) 0 0
\(799\) 11484.0 0.508479
\(800\) 1408.00 0.0622254
\(801\) 0 0
\(802\) 28152.0 1.23950
\(803\) 30870.0 1.35664
\(804\) 0 0
\(805\) 1701.00 0.0744750
\(806\) −5408.00 −0.236338
\(807\) 0 0
\(808\) 9648.00 0.420069
\(809\) 37164.0 1.61510 0.807550 0.589798i \(-0.200793\pi\)
0.807550 + 0.589798i \(0.200793\pi\)
\(810\) 0 0
\(811\) 17525.0 0.758799 0.379399 0.925233i \(-0.376131\pi\)
0.379399 + 0.925233i \(0.376131\pi\)
\(812\) −336.000 −0.0145213
\(813\) 0 0
\(814\) −18810.0 −0.809939
\(815\) 20268.0 0.871113
\(816\) 0 0
\(817\) −4334.00 −0.185591
\(818\) −1084.00 −0.0463340
\(819\) 0 0
\(820\) 10476.0 0.446144
\(821\) −2778.00 −0.118091 −0.0590456 0.998255i \(-0.518806\pi\)
−0.0590456 + 0.998255i \(0.518806\pi\)
\(822\) 0 0
\(823\) −5470.00 −0.231679 −0.115840 0.993268i \(-0.536956\pi\)
−0.115840 + 0.993268i \(0.536956\pi\)
\(824\) 9416.00 0.398085
\(825\) 0 0
\(826\) −6636.00 −0.279535
\(827\) 27711.0 1.16518 0.582591 0.812765i \(-0.302039\pi\)
0.582591 + 0.812765i \(0.302039\pi\)
\(828\) 0 0
\(829\) −6550.00 −0.274416 −0.137208 0.990542i \(-0.543813\pi\)
−0.137208 + 0.990542i \(0.543813\pi\)
\(830\) −12744.0 −0.532953
\(831\) 0 0
\(832\) −1024.00 −0.0426692
\(833\) 3234.00 0.134516
\(834\) 0 0
\(835\) 37530.0 1.55542
\(836\) 1980.00 0.0819136
\(837\) 0 0
\(838\) 12180.0 0.502090
\(839\) −7068.00 −0.290840 −0.145420 0.989370i \(-0.546453\pi\)
−0.145420 + 0.989370i \(0.546453\pi\)
\(840\) 0 0
\(841\) −24245.0 −0.994096
\(842\) 28670.0 1.17344
\(843\) 0 0
\(844\) −11512.0 −0.469502
\(845\) −17469.0 −0.711186
\(846\) 0 0
\(847\) 4858.00 0.197075
\(848\) 3648.00 0.147727
\(849\) 0 0
\(850\) 5808.00 0.234368
\(851\) 5643.00 0.227309
\(852\) 0 0
\(853\) 41528.0 1.66693 0.833465 0.552572i \(-0.186354\pi\)
0.833465 + 0.552572i \(0.186354\pi\)
\(854\) 3248.00 0.130146
\(855\) 0 0
\(856\) 1248.00 0.0498315
\(857\) 32751.0 1.30543 0.652715 0.757604i \(-0.273630\pi\)
0.652715 + 0.757604i \(0.273630\pi\)
\(858\) 0 0
\(859\) 15995.0 0.635323 0.317661 0.948204i \(-0.397102\pi\)
0.317661 + 0.948204i \(0.397102\pi\)
\(860\) −14184.0 −0.562407
\(861\) 0 0
\(862\) 33510.0 1.32408
\(863\) −15528.0 −0.612490 −0.306245 0.951953i \(-0.599073\pi\)
−0.306245 + 0.951953i \(0.599073\pi\)
\(864\) 0 0
\(865\) −39177.0 −1.53995
\(866\) −14872.0 −0.583569
\(867\) 0 0
\(868\) −4732.00 −0.185040
\(869\) 47070.0 1.83745
\(870\) 0 0
\(871\) −15872.0 −0.617454
\(872\) −88.0000 −0.00341750
\(873\) 0 0
\(874\) −594.000 −0.0229890
\(875\) −10647.0 −0.411353
\(876\) 0 0
\(877\) 40214.0 1.54838 0.774191 0.632953i \(-0.218157\pi\)
0.774191 + 0.632953i \(0.218157\pi\)
\(878\) −2704.00 −0.103936
\(879\) 0 0
\(880\) 6480.00 0.248228
\(881\) −24435.0 −0.934434 −0.467217 0.884143i \(-0.654743\pi\)
−0.467217 + 0.884143i \(0.654743\pi\)
\(882\) 0 0
\(883\) −18736.0 −0.714062 −0.357031 0.934093i \(-0.616211\pi\)
−0.357031 + 0.934093i \(0.616211\pi\)
\(884\) −4224.00 −0.160711
\(885\) 0 0
\(886\) −294.000 −0.0111480
\(887\) 23940.0 0.906231 0.453115 0.891452i \(-0.350313\pi\)
0.453115 + 0.891452i \(0.350313\pi\)
\(888\) 0 0
\(889\) 15512.0 0.585215
\(890\) −3510.00 −0.132197
\(891\) 0 0
\(892\) 17396.0 0.652983
\(893\) 1914.00 0.0717240
\(894\) 0 0
\(895\) −35856.0 −1.33914
\(896\) −896.000 −0.0334077
\(897\) 0 0
\(898\) 21276.0 0.790634
\(899\) 2028.00 0.0752365
\(900\) 0 0
\(901\) 15048.0 0.556406
\(902\) −26190.0 −0.966776
\(903\) 0 0
\(904\) −8304.00 −0.305517
\(905\) 5850.00 0.214874
\(906\) 0 0
\(907\) 31358.0 1.14799 0.573994 0.818859i \(-0.305393\pi\)
0.573994 + 0.818859i \(0.305393\pi\)
\(908\) −17064.0 −0.623666
\(909\) 0 0
\(910\) 2016.00 0.0734393
\(911\) −9408.00 −0.342153 −0.171076 0.985258i \(-0.554724\pi\)
−0.171076 + 0.985258i \(0.554724\pi\)
\(912\) 0 0
\(913\) 31860.0 1.15489
\(914\) 6998.00 0.253253
\(915\) 0 0
\(916\) −4240.00 −0.152941
\(917\) 6426.00 0.231412
\(918\) 0 0
\(919\) −6748.00 −0.242215 −0.121108 0.992639i \(-0.538645\pi\)
−0.121108 + 0.992639i \(0.538645\pi\)
\(920\) −1944.00 −0.0696650
\(921\) 0 0
\(922\) −8238.00 −0.294256
\(923\) 2448.00 0.0872989
\(924\) 0 0
\(925\) −9196.00 −0.326879
\(926\) 26636.0 0.945263
\(927\) 0 0
\(928\) 384.000 0.0135834
\(929\) 18246.0 0.644383 0.322192 0.946675i \(-0.395580\pi\)
0.322192 + 0.946675i \(0.395580\pi\)
\(930\) 0 0
\(931\) 539.000 0.0189742
\(932\) 11256.0 0.395604
\(933\) 0 0
\(934\) 6612.00 0.231639
\(935\) 26730.0 0.934935
\(936\) 0 0
\(937\) 19748.0 0.688516 0.344258 0.938875i \(-0.388131\pi\)
0.344258 + 0.938875i \(0.388131\pi\)
\(938\) −13888.0 −0.483432
\(939\) 0 0
\(940\) 6264.00 0.217350
\(941\) −55875.0 −1.93568 −0.967839 0.251571i \(-0.919053\pi\)
−0.967839 + 0.251571i \(0.919053\pi\)
\(942\) 0 0
\(943\) 7857.00 0.271325
\(944\) 7584.00 0.261481
\(945\) 0 0
\(946\) 35460.0 1.21871
\(947\) −37977.0 −1.30315 −0.651577 0.758583i \(-0.725892\pi\)
−0.651577 + 0.758583i \(0.725892\pi\)
\(948\) 0 0
\(949\) −10976.0 −0.375444
\(950\) 968.000 0.0330590
\(951\) 0 0
\(952\) −3696.00 −0.125828
\(953\) −28332.0 −0.963026 −0.481513 0.876439i \(-0.659913\pi\)
−0.481513 + 0.876439i \(0.659913\pi\)
\(954\) 0 0
\(955\) −26703.0 −0.904805
\(956\) 4800.00 0.162388
\(957\) 0 0
\(958\) −4752.00 −0.160261
\(959\) −17136.0 −0.577008
\(960\) 0 0
\(961\) −1230.00 −0.0412876
\(962\) 6688.00 0.224147
\(963\) 0 0
\(964\) −22600.0 −0.755080
\(965\) −25578.0 −0.853249
\(966\) 0 0
\(967\) −50902.0 −1.69276 −0.846380 0.532580i \(-0.821222\pi\)
−0.846380 + 0.532580i \(0.821222\pi\)
\(968\) −5552.00 −0.184347
\(969\) 0 0
\(970\) 1584.00 0.0524321
\(971\) −54444.0 −1.79937 −0.899686 0.436537i \(-0.856205\pi\)
−0.899686 + 0.436537i \(0.856205\pi\)
\(972\) 0 0
\(973\) 8708.00 0.286912
\(974\) −28372.0 −0.933365
\(975\) 0 0
\(976\) −3712.00 −0.121740
\(977\) 34836.0 1.14074 0.570370 0.821388i \(-0.306800\pi\)
0.570370 + 0.821388i \(0.306800\pi\)
\(978\) 0 0
\(979\) 8775.00 0.286466
\(980\) 1764.00 0.0574989
\(981\) 0 0
\(982\) 21198.0 0.688855
\(983\) −34974.0 −1.13479 −0.567394 0.823446i \(-0.692048\pi\)
−0.567394 + 0.823446i \(0.692048\pi\)
\(984\) 0 0
\(985\) 20574.0 0.665525
\(986\) 1584.00 0.0511611
\(987\) 0 0
\(988\) −704.000 −0.0226693
\(989\) −10638.0 −0.342031
\(990\) 0 0
\(991\) −36844.0 −1.18102 −0.590509 0.807031i \(-0.701073\pi\)
−0.590509 + 0.807031i \(0.701073\pi\)
\(992\) 5408.00 0.173089
\(993\) 0 0
\(994\) 2142.00 0.0683502
\(995\) −14805.0 −0.471709
\(996\) 0 0
\(997\) −45970.0 −1.46027 −0.730133 0.683305i \(-0.760542\pi\)
−0.730133 + 0.683305i \(0.760542\pi\)
\(998\) 6044.00 0.191703
\(999\) 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 378.4.a.e.1.1 1
3.2 odd 2 378.4.a.h.1.1 yes 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
378.4.a.e.1.1 1 1.1 even 1 trivial
378.4.a.h.1.1 yes 1 3.2 odd 2