Properties

Label 800.4.a.ba.1.1
Level $800$
Weight $4$
Character 800.1
Self dual yes
Analytic conductor $47.202$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [800,4,Mod(1,800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(800, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("800.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 800 = 2^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 800.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: \(47.2015280046\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.2106005.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 24x^{2} + 25x - 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-4.54854\) of defining polynomial
Character \(\chi\) \(=\) 800.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-10.0971 q^{3} -10.5923 q^{7} +74.9510 q^{9} -38.9027 q^{11} -68.9510 q^{13} +65.9510 q^{17} +49.4950 q^{19} +106.951 q^{21} -164.524 q^{23} -484.164 q^{27} -170.853 q^{29} -166.505 q^{31} +392.804 q^{33} -384.853 q^{37} +696.203 q^{39} -22.8038 q^{41} +136.709 q^{43} -307.562 q^{47} -230.804 q^{49} -665.912 q^{51} +222.000 q^{53} -499.755 q^{57} -522.765 q^{59} +393.049 q^{61} -793.901 q^{63} -476.845 q^{67} +1661.22 q^{69} +4.26352 q^{71} -601.166 q^{73} +412.068 q^{77} +1075.93 q^{79} +2864.97 q^{81} +1132.74 q^{83} +1725.11 q^{87} +479.755 q^{89} +730.347 q^{91} +1681.22 q^{93} +635.510 q^{97} -2915.80 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 96 q^{9} - 72 q^{13} + 60 q^{17} + 224 q^{21} - 72 q^{29} + 756 q^{33} - 928 q^{37} + 724 q^{41} - 108 q^{49} + 888 q^{53} - 980 q^{57} + 1776 q^{61} + 3384 q^{69} + 1060 q^{73} - 2224 q^{77} + 7180 q^{81}+ \cdots + 504 q^{97}+O(q^{100}) \) 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
\(3\) −10.0971 −1.94318 −0.971592 0.236664i \(-0.923946\pi\)
−0.971592 + 0.236664i \(0.923946\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −10.5923 −0.571929 −0.285964 0.958240i \(-0.592314\pi\)
−0.285964 + 0.958240i \(0.592314\pi\)
\(8\) 0 0
\(9\) 74.9510 2.77596
\(10\) 0 0
\(11\) −38.9027 −1.06633 −0.533164 0.846012i \(-0.678997\pi\)
−0.533164 + 0.846012i \(0.678997\pi\)
\(12\) 0 0
\(13\) −68.9510 −1.47104 −0.735521 0.677502i \(-0.763063\pi\)
−0.735521 + 0.677502i \(0.763063\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 65.9510 0.940909 0.470455 0.882424i \(-0.344090\pi\)
0.470455 + 0.882424i \(0.344090\pi\)
\(18\) 0 0
\(19\) 49.4950 0.597628 0.298814 0.954311i \(-0.403409\pi\)
0.298814 + 0.954311i \(0.403409\pi\)
\(20\) 0 0
\(21\) 106.951 1.11136
\(22\) 0 0
\(23\) −164.524 −1.49155 −0.745776 0.666197i \(-0.767921\pi\)
−0.745776 + 0.666197i \(0.767921\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −484.164 −3.45102
\(28\) 0 0
\(29\) −170.853 −1.09402 −0.547010 0.837126i \(-0.684234\pi\)
−0.547010 + 0.837126i \(0.684234\pi\)
\(30\) 0 0
\(31\) −166.505 −0.964684 −0.482342 0.875983i \(-0.660214\pi\)
−0.482342 + 0.875983i \(0.660214\pi\)
\(32\) 0 0
\(33\) 392.804 2.07207
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −384.853 −1.70998 −0.854992 0.518641i \(-0.826438\pi\)
−0.854992 + 0.518641i \(0.826438\pi\)
\(38\) 0 0
\(39\) 696.203 2.85851
\(40\) 0 0
\(41\) −22.8038 −0.0868624 −0.0434312 0.999056i \(-0.513829\pi\)
−0.0434312 + 0.999056i \(0.513829\pi\)
\(42\) 0 0
\(43\) 136.709 0.484836 0.242418 0.970172i \(-0.422059\pi\)
0.242418 + 0.970172i \(0.422059\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −307.562 −0.954523 −0.477261 0.878761i \(-0.658370\pi\)
−0.477261 + 0.878761i \(0.658370\pi\)
\(48\) 0 0
\(49\) −230.804 −0.672897
\(50\) 0 0
\(51\) −665.912 −1.82836
\(52\) 0 0
\(53\) 222.000 0.575359 0.287680 0.957727i \(-0.407116\pi\)
0.287680 + 0.957727i \(0.407116\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −499.755 −1.16130
\(58\) 0 0
\(59\) −522.765 −1.15353 −0.576765 0.816910i \(-0.695685\pi\)
−0.576765 + 0.816910i \(0.695685\pi\)
\(60\) 0 0
\(61\) 393.049 0.824996 0.412498 0.910958i \(-0.364656\pi\)
0.412498 + 0.910958i \(0.364656\pi\)
\(62\) 0 0
\(63\) −793.901 −1.58765
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −476.845 −0.869492 −0.434746 0.900553i \(-0.643162\pi\)
−0.434746 + 0.900553i \(0.643162\pi\)
\(68\) 0 0
\(69\) 1661.22 2.89836
\(70\) 0 0
\(71\) 4.26352 0.00712657 0.00356328 0.999994i \(-0.498866\pi\)
0.00356328 + 0.999994i \(0.498866\pi\)
\(72\) 0 0
\(73\) −601.166 −0.963852 −0.481926 0.876212i \(-0.660063\pi\)
−0.481926 + 0.876212i \(0.660063\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 412.068 0.609864
\(78\) 0 0
\(79\) 1075.93 1.53230 0.766150 0.642662i \(-0.222170\pi\)
0.766150 + 0.642662i \(0.222170\pi\)
\(80\) 0 0
\(81\) 2864.97 3.93000
\(82\) 0 0
\(83\) 1132.74 1.49801 0.749005 0.662564i \(-0.230532\pi\)
0.749005 + 0.662564i \(0.230532\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 1725.11 2.12588
\(88\) 0 0
\(89\) 479.755 0.571392 0.285696 0.958320i \(-0.407775\pi\)
0.285696 + 0.958320i \(0.407775\pi\)
\(90\) 0 0
\(91\) 730.347 0.841332
\(92\) 0 0
\(93\) 1681.22 1.87456
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 635.510 0.665219 0.332609 0.943065i \(-0.392071\pi\)
0.332609 + 0.943065i \(0.392071\pi\)
\(98\) 0 0
\(99\) −2915.80 −2.96009
\(100\) 0 0
\(101\) −256.264 −0.252468 −0.126234 0.992001i \(-0.540289\pi\)
−0.126234 + 0.992001i \(0.540289\pi\)
\(102\) 0 0
\(103\) −81.6825 −0.0781400 −0.0390700 0.999236i \(-0.512440\pi\)
−0.0390700 + 0.999236i \(0.512440\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 742.944 0.671244 0.335622 0.941997i \(-0.391053\pi\)
0.335622 + 0.941997i \(0.391053\pi\)
\(108\) 0 0
\(109\) −495.872 −0.435742 −0.217871 0.975978i \(-0.569911\pi\)
−0.217871 + 0.975978i \(0.569911\pi\)
\(110\) 0 0
\(111\) 3885.89 3.32281
\(112\) 0 0
\(113\) 1878.61 1.56393 0.781967 0.623320i \(-0.214217\pi\)
0.781967 + 0.623320i \(0.214217\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −5167.94 −4.08356
\(118\) 0 0
\(119\) −698.570 −0.538133
\(120\) 0 0
\(121\) 182.422 0.137057
\(122\) 0 0
\(123\) 230.252 0.168790
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 2113.24 1.47653 0.738265 0.674511i \(-0.235645\pi\)
0.738265 + 0.674511i \(0.235645\pi\)
\(128\) 0 0
\(129\) −1380.36 −0.942125
\(130\) 0 0
\(131\) −1105.42 −0.737263 −0.368631 0.929576i \(-0.620173\pi\)
−0.368631 + 0.929576i \(0.620173\pi\)
\(132\) 0 0
\(133\) −524.264 −0.341801
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −1151.92 −0.718360 −0.359180 0.933268i \(-0.616944\pi\)
−0.359180 + 0.933268i \(0.616944\pi\)
\(138\) 0 0
\(139\) 354.304 0.216199 0.108099 0.994140i \(-0.465524\pi\)
0.108099 + 0.994140i \(0.465524\pi\)
\(140\) 0 0
\(141\) 3105.48 1.85481
\(142\) 0 0
\(143\) 2682.38 1.56861
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 2330.44 1.30756
\(148\) 0 0
\(149\) 715.638 0.393472 0.196736 0.980457i \(-0.436966\pi\)
0.196736 + 0.980457i \(0.436966\pi\)
\(150\) 0 0
\(151\) −2799.37 −1.50867 −0.754335 0.656490i \(-0.772041\pi\)
−0.754335 + 0.656490i \(0.772041\pi\)
\(152\) 0 0
\(153\) 4943.09 2.61193
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −845.275 −0.429683 −0.214842 0.976649i \(-0.568924\pi\)
−0.214842 + 0.976649i \(0.568924\pi\)
\(158\) 0 0
\(159\) −2241.55 −1.11803
\(160\) 0 0
\(161\) 1742.69 0.853062
\(162\) 0 0
\(163\) −539.964 −0.259468 −0.129734 0.991549i \(-0.541412\pi\)
−0.129734 + 0.991549i \(0.541412\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 1209.20 0.560302 0.280151 0.959956i \(-0.409615\pi\)
0.280151 + 0.959956i \(0.409615\pi\)
\(168\) 0 0
\(169\) 2557.23 1.16397
\(170\) 0 0
\(171\) 3709.70 1.65899
\(172\) 0 0
\(173\) −4061.65 −1.78498 −0.892489 0.451069i \(-0.851043\pi\)
−0.892489 + 0.451069i \(0.851043\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 5278.40 2.24152
\(178\) 0 0
\(179\) 3113.67 1.30015 0.650074 0.759871i \(-0.274738\pi\)
0.650074 + 0.759871i \(0.274738\pi\)
\(180\) 0 0
\(181\) 473.215 0.194330 0.0971652 0.995268i \(-0.469022\pi\)
0.0971652 + 0.995268i \(0.469022\pi\)
\(182\) 0 0
\(183\) −3968.65 −1.60312
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −2565.67 −1.00332
\(188\) 0 0
\(189\) 5128.40 1.97374
\(190\) 0 0
\(191\) −2250.68 −0.852637 −0.426318 0.904573i \(-0.640190\pi\)
−0.426318 + 0.904573i \(0.640190\pi\)
\(192\) 0 0
\(193\) −553.461 −0.206419 −0.103210 0.994660i \(-0.532911\pi\)
−0.103210 + 0.994660i \(0.532911\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 385.744 0.139508 0.0697541 0.997564i \(-0.477779\pi\)
0.0697541 + 0.997564i \(0.477779\pi\)
\(198\) 0 0
\(199\) 559.494 0.199304 0.0996520 0.995022i \(-0.468227\pi\)
0.0996520 + 0.995022i \(0.468227\pi\)
\(200\) 0 0
\(201\) 4814.74 1.68958
\(202\) 0 0
\(203\) 1809.72 0.625702
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −12331.3 −4.14049
\(208\) 0 0
\(209\) −1925.49 −0.637268
\(210\) 0 0
\(211\) −3721.00 −1.21405 −0.607025 0.794683i \(-0.707637\pi\)
−0.607025 + 0.794683i \(0.707637\pi\)
\(212\) 0 0
\(213\) −43.0490 −0.0138482
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 1763.67 0.551731
\(218\) 0 0
\(219\) 6070.02 1.87294
\(220\) 0 0
\(221\) −4547.38 −1.38412
\(222\) 0 0
\(223\) 5506.41 1.65353 0.826763 0.562550i \(-0.190180\pi\)
0.826763 + 0.562550i \(0.190180\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 1139.27 0.333109 0.166555 0.986032i \(-0.446736\pi\)
0.166555 + 0.986032i \(0.446736\pi\)
\(228\) 0 0
\(229\) 1727.08 0.498378 0.249189 0.968455i \(-0.419836\pi\)
0.249189 + 0.968455i \(0.419836\pi\)
\(230\) 0 0
\(231\) −4160.68 −1.18508
\(232\) 0 0
\(233\) −1267.84 −0.356477 −0.178238 0.983987i \(-0.557040\pi\)
−0.178238 + 0.983987i \(0.557040\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −10863.8 −2.97754
\(238\) 0 0
\(239\) 493.995 0.133698 0.0668492 0.997763i \(-0.478705\pi\)
0.0668492 + 0.997763i \(0.478705\pi\)
\(240\) 0 0
\(241\) −332.676 −0.0889192 −0.0444596 0.999011i \(-0.514157\pi\)
−0.0444596 + 0.999011i \(0.514157\pi\)
\(242\) 0 0
\(243\) −15855.4 −4.18569
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −3412.73 −0.879136
\(248\) 0 0
\(249\) −11437.4 −2.91091
\(250\) 0 0
\(251\) 4365.75 1.09786 0.548932 0.835867i \(-0.315035\pi\)
0.548932 + 0.835867i \(0.315035\pi\)
\(252\) 0 0
\(253\) 6400.45 1.59048
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −3407.19 −0.826984 −0.413492 0.910508i \(-0.635691\pi\)
−0.413492 + 0.910508i \(0.635691\pi\)
\(258\) 0 0
\(259\) 4076.47 0.977989
\(260\) 0 0
\(261\) −12805.6 −3.03696
\(262\) 0 0
\(263\) −2388.20 −0.559934 −0.279967 0.960010i \(-0.590324\pi\)
−0.279967 + 0.960010i \(0.590324\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −4844.12 −1.11032
\(268\) 0 0
\(269\) −3800.33 −0.861377 −0.430688 0.902501i \(-0.641729\pi\)
−0.430688 + 0.902501i \(0.641729\pi\)
\(270\) 0 0
\(271\) −2270.39 −0.508917 −0.254459 0.967084i \(-0.581897\pi\)
−0.254459 + 0.967084i \(0.581897\pi\)
\(272\) 0 0
\(273\) −7374.37 −1.63486
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 7070.93 1.53376 0.766879 0.641792i \(-0.221809\pi\)
0.766879 + 0.641792i \(0.221809\pi\)
\(278\) 0 0
\(279\) −12479.7 −2.67793
\(280\) 0 0
\(281\) 1536.57 0.326206 0.163103 0.986609i \(-0.447850\pi\)
0.163103 + 0.986609i \(0.447850\pi\)
\(282\) 0 0
\(283\) 6013.32 1.26309 0.631545 0.775339i \(-0.282421\pi\)
0.631545 + 0.775339i \(0.282421\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 241.544 0.0496791
\(288\) 0 0
\(289\) −563.471 −0.114690
\(290\) 0 0
\(291\) −6416.79 −1.29264
\(292\) 0 0
\(293\) 4651.03 0.927358 0.463679 0.886003i \(-0.346529\pi\)
0.463679 + 0.886003i \(0.346529\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 18835.3 3.67992
\(298\) 0 0
\(299\) 11344.1 2.19414
\(300\) 0 0
\(301\) −1448.06 −0.277292
\(302\) 0 0
\(303\) 2587.52 0.490591
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 7261.88 1.35002 0.675012 0.737807i \(-0.264139\pi\)
0.675012 + 0.737807i \(0.264139\pi\)
\(308\) 0 0
\(309\) 824.755 0.151840
\(310\) 0 0
\(311\) −6759.67 −1.23249 −0.616247 0.787553i \(-0.711348\pi\)
−0.616247 + 0.787553i \(0.711348\pi\)
\(312\) 0 0
\(313\) 1865.55 0.336891 0.168446 0.985711i \(-0.446125\pi\)
0.168446 + 0.985711i \(0.446125\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −2759.41 −0.488908 −0.244454 0.969661i \(-0.578609\pi\)
−0.244454 + 0.969661i \(0.578609\pi\)
\(318\) 0 0
\(319\) 6646.64 1.16658
\(320\) 0 0
\(321\) −7501.57 −1.30435
\(322\) 0 0
\(323\) 3264.24 0.562314
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 5006.86 0.846727
\(328\) 0 0
\(329\) 3257.78 0.545919
\(330\) 0 0
\(331\) 5123.99 0.850876 0.425438 0.904988i \(-0.360120\pi\)
0.425438 + 0.904988i \(0.360120\pi\)
\(332\) 0 0
\(333\) −28845.1 −4.74685
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −1475.31 −0.238473 −0.119236 0.992866i \(-0.538045\pi\)
−0.119236 + 0.992866i \(0.538045\pi\)
\(338\) 0 0
\(339\) −18968.4 −3.03901
\(340\) 0 0
\(341\) 6477.50 1.02867
\(342\) 0 0
\(343\) 6077.88 0.956778
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 3478.67 0.538170 0.269085 0.963116i \(-0.413279\pi\)
0.269085 + 0.963116i \(0.413279\pi\)
\(348\) 0 0
\(349\) 11392.3 1.74732 0.873660 0.486537i \(-0.161740\pi\)
0.873660 + 0.486537i \(0.161740\pi\)
\(350\) 0 0
\(351\) 33383.6 5.07660
\(352\) 0 0
\(353\) 2735.39 0.412437 0.206218 0.978506i \(-0.433884\pi\)
0.206218 + 0.978506i \(0.433884\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 7053.52 1.04569
\(358\) 0 0
\(359\) −2390.41 −0.351424 −0.175712 0.984442i \(-0.556223\pi\)
−0.175712 + 0.984442i \(0.556223\pi\)
\(360\) 0 0
\(361\) −4409.25 −0.642841
\(362\) 0 0
\(363\) −1841.93 −0.266326
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 9209.84 1.30994 0.654972 0.755653i \(-0.272680\pi\)
0.654972 + 0.755653i \(0.272680\pi\)
\(368\) 0 0
\(369\) −1709.17 −0.241127
\(370\) 0 0
\(371\) −2351.48 −0.329065
\(372\) 0 0
\(373\) −13768.4 −1.91127 −0.955634 0.294556i \(-0.904828\pi\)
−0.955634 + 0.294556i \(0.904828\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 11780.5 1.60935
\(378\) 0 0
\(379\) −8666.50 −1.17459 −0.587293 0.809374i \(-0.699807\pi\)
−0.587293 + 0.809374i \(0.699807\pi\)
\(380\) 0 0
\(381\) −21337.5 −2.86917
\(382\) 0 0
\(383\) 7434.12 0.991816 0.495908 0.868375i \(-0.334835\pi\)
0.495908 + 0.868375i \(0.334835\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 10246.5 1.34589
\(388\) 0 0
\(389\) −7323.06 −0.954483 −0.477241 0.878772i \(-0.658363\pi\)
−0.477241 + 0.878772i \(0.658363\pi\)
\(390\) 0 0
\(391\) −10850.5 −1.40342
\(392\) 0 0
\(393\) 11161.6 1.43264
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −5185.92 −0.655602 −0.327801 0.944747i \(-0.606307\pi\)
−0.327801 + 0.944747i \(0.606307\pi\)
\(398\) 0 0
\(399\) 5293.54 0.664181
\(400\) 0 0
\(401\) −9044.81 −1.12637 −0.563187 0.826329i \(-0.690425\pi\)
−0.563187 + 0.826329i \(0.690425\pi\)
\(402\) 0 0
\(403\) 11480.7 1.41909
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 14971.8 1.82341
\(408\) 0 0
\(409\) −2311.41 −0.279442 −0.139721 0.990191i \(-0.544621\pi\)
−0.139721 + 0.990191i \(0.544621\pi\)
\(410\) 0 0
\(411\) 11631.0 1.39590
\(412\) 0 0
\(413\) 5537.27 0.659737
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −3577.43 −0.420114
\(418\) 0 0
\(419\) −330.860 −0.0385765 −0.0192882 0.999814i \(-0.506140\pi\)
−0.0192882 + 0.999814i \(0.506140\pi\)
\(420\) 0 0
\(421\) 1933.39 0.223819 0.111909 0.993718i \(-0.464303\pi\)
0.111909 + 0.993718i \(0.464303\pi\)
\(422\) 0 0
\(423\) −23052.1 −2.64972
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −4163.28 −0.471839
\(428\) 0 0
\(429\) −27084.2 −3.04811
\(430\) 0 0
\(431\) −8820.32 −0.985753 −0.492877 0.870099i \(-0.664055\pi\)
−0.492877 + 0.870099i \(0.664055\pi\)
\(432\) 0 0
\(433\) −11269.7 −1.25078 −0.625390 0.780312i \(-0.715060\pi\)
−0.625390 + 0.780312i \(0.715060\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −8143.13 −0.891393
\(438\) 0 0
\(439\) −2361.79 −0.256770 −0.128385 0.991724i \(-0.540979\pi\)
−0.128385 + 0.991724i \(0.540979\pi\)
\(440\) 0 0
\(441\) −17299.0 −1.86794
\(442\) 0 0
\(443\) 15029.6 1.61192 0.805959 0.591971i \(-0.201650\pi\)
0.805959 + 0.591971i \(0.201650\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −7225.85 −0.764588
\(448\) 0 0
\(449\) −12661.3 −1.33079 −0.665396 0.746491i \(-0.731737\pi\)
−0.665396 + 0.746491i \(0.731737\pi\)
\(450\) 0 0
\(451\) 887.131 0.0926239
\(452\) 0 0
\(453\) 28265.4 2.93162
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 1620.47 0.165870 0.0829348 0.996555i \(-0.473571\pi\)
0.0829348 + 0.996555i \(0.473571\pi\)
\(458\) 0 0
\(459\) −31931.1 −3.24710
\(460\) 0 0
\(461\) 6893.84 0.696482 0.348241 0.937405i \(-0.386779\pi\)
0.348241 + 0.937405i \(0.386779\pi\)
\(462\) 0 0
\(463\) −12022.8 −1.20680 −0.603398 0.797440i \(-0.706187\pi\)
−0.603398 + 0.797440i \(0.706187\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −7349.60 −0.728263 −0.364131 0.931348i \(-0.618634\pi\)
−0.364131 + 0.931348i \(0.618634\pi\)
\(468\) 0 0
\(469\) 5050.87 0.497287
\(470\) 0 0
\(471\) 8534.81 0.834954
\(472\) 0 0
\(473\) −5318.36 −0.516994
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 16639.1 1.59718
\(478\) 0 0
\(479\) 12809.4 1.22187 0.610933 0.791682i \(-0.290794\pi\)
0.610933 + 0.791682i \(0.290794\pi\)
\(480\) 0 0
\(481\) 26536.0 2.51546
\(482\) 0 0
\(483\) −17596.0 −1.65766
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 15782.4 1.46852 0.734260 0.678868i \(-0.237529\pi\)
0.734260 + 0.678868i \(0.237529\pi\)
\(488\) 0 0
\(489\) 5452.06 0.504193
\(490\) 0 0
\(491\) −5972.10 −0.548914 −0.274457 0.961599i \(-0.588498\pi\)
−0.274457 + 0.961599i \(0.588498\pi\)
\(492\) 0 0
\(493\) −11267.9 −1.02937
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −45.1603 −0.00407589
\(498\) 0 0
\(499\) 14171.8 1.27138 0.635690 0.771944i \(-0.280716\pi\)
0.635690 + 0.771944i \(0.280716\pi\)
\(500\) 0 0
\(501\) −12209.4 −1.08877
\(502\) 0 0
\(503\) 5543.95 0.491436 0.245718 0.969341i \(-0.420976\pi\)
0.245718 + 0.969341i \(0.420976\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −25820.6 −2.26180
\(508\) 0 0
\(509\) 13228.9 1.15199 0.575994 0.817454i \(-0.304615\pi\)
0.575994 + 0.817454i \(0.304615\pi\)
\(510\) 0 0
\(511\) 6367.72 0.551255
\(512\) 0 0
\(513\) −23963.7 −2.06243
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 11965.0 1.01783
\(518\) 0 0
\(519\) 41010.8 3.46854
\(520\) 0 0
\(521\) −11092.8 −0.932795 −0.466398 0.884575i \(-0.654448\pi\)
−0.466398 + 0.884575i \(0.654448\pi\)
\(522\) 0 0
\(523\) 18550.2 1.55094 0.775470 0.631384i \(-0.217513\pi\)
0.775470 + 0.631384i \(0.217513\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −10981.2 −0.907680
\(528\) 0 0
\(529\) 14901.3 1.22473
\(530\) 0 0
\(531\) −39181.8 −3.20215
\(532\) 0 0
\(533\) 1572.35 0.127778
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −31438.9 −2.52643
\(538\) 0 0
\(539\) 8978.90 0.717530
\(540\) 0 0
\(541\) 8146.08 0.647370 0.323685 0.946165i \(-0.395078\pi\)
0.323685 + 0.946165i \(0.395078\pi\)
\(542\) 0 0
\(543\) −4778.09 −0.377620
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −18001.4 −1.40710 −0.703550 0.710645i \(-0.748403\pi\)
−0.703550 + 0.710645i \(0.748403\pi\)
\(548\) 0 0
\(549\) 29459.4 2.29016
\(550\) 0 0
\(551\) −8456.36 −0.653817
\(552\) 0 0
\(553\) −11396.5 −0.876366
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 5968.52 0.454029 0.227014 0.973891i \(-0.427104\pi\)
0.227014 + 0.973891i \(0.427104\pi\)
\(558\) 0 0
\(559\) −9426.22 −0.713214
\(560\) 0 0
\(561\) 25905.8 1.94963
\(562\) 0 0
\(563\) 6021.42 0.450751 0.225375 0.974272i \(-0.427639\pi\)
0.225375 + 0.974272i \(0.427639\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −30346.5 −2.24768
\(568\) 0 0
\(569\) 13499.2 0.994582 0.497291 0.867584i \(-0.334328\pi\)
0.497291 + 0.867584i \(0.334328\pi\)
\(570\) 0 0
\(571\) 2413.96 0.176920 0.0884599 0.996080i \(-0.471805\pi\)
0.0884599 + 0.996080i \(0.471805\pi\)
\(572\) 0 0
\(573\) 22725.3 1.65683
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 13549.7 0.977608 0.488804 0.872394i \(-0.337433\pi\)
0.488804 + 0.872394i \(0.337433\pi\)
\(578\) 0 0
\(579\) 5588.33 0.401111
\(580\) 0 0
\(581\) −11998.3 −0.856755
\(582\) 0 0
\(583\) −8636.41 −0.613522
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −19364.2 −1.36158 −0.680789 0.732479i \(-0.738363\pi\)
−0.680789 + 0.732479i \(0.738363\pi\)
\(588\) 0 0
\(589\) −8241.17 −0.576522
\(590\) 0 0
\(591\) −3894.89 −0.271090
\(592\) 0 0
\(593\) 21130.1 1.46325 0.731627 0.681705i \(-0.238761\pi\)
0.731627 + 0.681705i \(0.238761\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −5649.25 −0.387284
\(598\) 0 0
\(599\) −12938.1 −0.882532 −0.441266 0.897376i \(-0.645471\pi\)
−0.441266 + 0.897376i \(0.645471\pi\)
\(600\) 0 0
\(601\) −2173.64 −0.147528 −0.0737641 0.997276i \(-0.523501\pi\)
−0.0737641 + 0.997276i \(0.523501\pi\)
\(602\) 0 0
\(603\) −35740.0 −2.41367
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 13712.9 0.916951 0.458475 0.888707i \(-0.348396\pi\)
0.458475 + 0.888707i \(0.348396\pi\)
\(608\) 0 0
\(609\) −18272.9 −1.21585
\(610\) 0 0
\(611\) 21206.7 1.40414
\(612\) 0 0
\(613\) 3666.83 0.241602 0.120801 0.992677i \(-0.461454\pi\)
0.120801 + 0.992677i \(0.461454\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −26402.5 −1.72273 −0.861367 0.507984i \(-0.830391\pi\)
−0.861367 + 0.507984i \(0.830391\pi\)
\(618\) 0 0
\(619\) 28554.8 1.85414 0.927072 0.374882i \(-0.122317\pi\)
0.927072 + 0.374882i \(0.122317\pi\)
\(620\) 0 0
\(621\) 79656.9 5.14737
\(622\) 0 0
\(623\) −5081.69 −0.326796
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 19441.8 1.23833
\(628\) 0 0
\(629\) −25381.4 −1.60894
\(630\) 0 0
\(631\) −29567.8 −1.86541 −0.932706 0.360637i \(-0.882559\pi\)
−0.932706 + 0.360637i \(0.882559\pi\)
\(632\) 0 0
\(633\) 37571.2 2.35912
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 15914.1 0.989861
\(638\) 0 0
\(639\) 319.555 0.0197831
\(640\) 0 0
\(641\) −21701.9 −1.33724 −0.668621 0.743604i \(-0.733115\pi\)
−0.668621 + 0.743604i \(0.733115\pi\)
\(642\) 0 0
\(643\) 6193.48 0.379855 0.189928 0.981798i \(-0.439175\pi\)
0.189928 + 0.981798i \(0.439175\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −2208.53 −0.134198 −0.0670991 0.997746i \(-0.521374\pi\)
−0.0670991 + 0.997746i \(0.521374\pi\)
\(648\) 0 0
\(649\) 20337.0 1.23004
\(650\) 0 0
\(651\) −17807.9 −1.07211
\(652\) 0 0
\(653\) −20505.9 −1.22888 −0.614438 0.788965i \(-0.710617\pi\)
−0.614438 + 0.788965i \(0.710617\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −45058.0 −2.67562
\(658\) 0 0
\(659\) −26030.4 −1.53870 −0.769348 0.638829i \(-0.779419\pi\)
−0.769348 + 0.638829i \(0.779419\pi\)
\(660\) 0 0
\(661\) −6729.32 −0.395976 −0.197988 0.980204i \(-0.563441\pi\)
−0.197988 + 0.980204i \(0.563441\pi\)
\(662\) 0 0
\(663\) 45915.3 2.68959
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 28109.5 1.63179
\(668\) 0 0
\(669\) −55598.6 −3.21311
\(670\) 0 0
\(671\) −15290.7 −0.879717
\(672\) 0 0
\(673\) 8716.54 0.499254 0.249627 0.968342i \(-0.419692\pi\)
0.249627 + 0.968342i \(0.419692\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −24843.1 −1.41034 −0.705170 0.709039i \(-0.749129\pi\)
−0.705170 + 0.709039i \(0.749129\pi\)
\(678\) 0 0
\(679\) −6731.49 −0.380458
\(680\) 0 0
\(681\) −11503.3 −0.647292
\(682\) 0 0
\(683\) −20079.1 −1.12490 −0.562449 0.826832i \(-0.690141\pi\)
−0.562449 + 0.826832i \(0.690141\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −17438.4 −0.968440
\(688\) 0 0
\(689\) −15307.1 −0.846378
\(690\) 0 0
\(691\) 16575.8 0.912550 0.456275 0.889839i \(-0.349183\pi\)
0.456275 + 0.889839i \(0.349183\pi\)
\(692\) 0 0
\(693\) 30884.9 1.69296
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −1503.93 −0.0817297
\(698\) 0 0
\(699\) 12801.5 0.692700
\(700\) 0 0
\(701\) 29052.6 1.56534 0.782668 0.622440i \(-0.213858\pi\)
0.782668 + 0.622440i \(0.213858\pi\)
\(702\) 0 0
\(703\) −19048.3 −1.02193
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 2714.42 0.144394
\(708\) 0 0
\(709\) 32100.2 1.70035 0.850175 0.526499i \(-0.176496\pi\)
0.850175 + 0.526499i \(0.176496\pi\)
\(710\) 0 0
\(711\) 80642.0 4.25360
\(712\) 0 0
\(713\) 27394.2 1.43888
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −4987.91 −0.259800
\(718\) 0 0
\(719\) −17819.4 −0.924274 −0.462137 0.886808i \(-0.652917\pi\)
−0.462137 + 0.886808i \(0.652917\pi\)
\(720\) 0 0
\(721\) 865.203 0.0446905
\(722\) 0 0
\(723\) 3359.05 0.172786
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −27816.4 −1.41906 −0.709529 0.704677i \(-0.751092\pi\)
−0.709529 + 0.704677i \(0.751092\pi\)
\(728\) 0 0
\(729\) 82738.8 4.20357
\(730\) 0 0
\(731\) 9016.10 0.456187
\(732\) 0 0
\(733\) −2522.56 −0.127112 −0.0635559 0.997978i \(-0.520244\pi\)
−0.0635559 + 0.997978i \(0.520244\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 18550.6 0.927164
\(738\) 0 0
\(739\) −3526.91 −0.175561 −0.0877805 0.996140i \(-0.527977\pi\)
−0.0877805 + 0.996140i \(0.527977\pi\)
\(740\) 0 0
\(741\) 34458.6 1.70832
\(742\) 0 0
\(743\) −8557.90 −0.422556 −0.211278 0.977426i \(-0.567763\pi\)
−0.211278 + 0.977426i \(0.567763\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 84900.3 4.15842
\(748\) 0 0
\(749\) −7869.47 −0.383904
\(750\) 0 0
\(751\) −806.886 −0.0392060 −0.0196030 0.999808i \(-0.506240\pi\)
−0.0196030 + 0.999808i \(0.506240\pi\)
\(752\) 0 0
\(753\) −44081.4 −2.13335
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 12898.6 0.619298 0.309649 0.950851i \(-0.399788\pi\)
0.309649 + 0.950851i \(0.399788\pi\)
\(758\) 0 0
\(759\) −64625.8 −3.09060
\(760\) 0 0
\(761\) 23315.9 1.11064 0.555322 0.831635i \(-0.312595\pi\)
0.555322 + 0.831635i \(0.312595\pi\)
\(762\) 0 0
\(763\) 5252.41 0.249214
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 36045.2 1.69689
\(768\) 0 0
\(769\) 41805.3 1.96039 0.980193 0.198045i \(-0.0634591\pi\)
0.980193 + 0.198045i \(0.0634591\pi\)
\(770\) 0 0
\(771\) 34402.7 1.60698
\(772\) 0 0
\(773\) 10587.9 0.492654 0.246327 0.969187i \(-0.420776\pi\)
0.246327 + 0.969187i \(0.420776\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −41160.4 −1.90041
\(778\) 0 0
\(779\) −1128.68 −0.0519114
\(780\) 0 0
\(781\) −165.862 −0.00759926
\(782\) 0 0
\(783\) 82720.9 3.77548
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −16085.8 −0.728584 −0.364292 0.931285i \(-0.618689\pi\)
−0.364292 + 0.931285i \(0.618689\pi\)
\(788\) 0 0
\(789\) 24113.8 1.08805
\(790\) 0 0
\(791\) −19898.7 −0.894459
\(792\) 0 0
\(793\) −27101.1 −1.21360
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −13813.3 −0.613919 −0.306959 0.951723i \(-0.599312\pi\)
−0.306959 + 0.951723i \(0.599312\pi\)
\(798\) 0 0
\(799\) −20284.0 −0.898119
\(800\) 0 0
\(801\) 35958.1 1.58616
\(802\) 0 0
\(803\) 23387.0 1.02778
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 38372.2 1.67381
\(808\) 0 0
\(809\) −14893.1 −0.647234 −0.323617 0.946188i \(-0.604899\pi\)
−0.323617 + 0.946188i \(0.604899\pi\)
\(810\) 0 0
\(811\) 24638.7 1.06681 0.533405 0.845860i \(-0.320912\pi\)
0.533405 + 0.845860i \(0.320912\pi\)
\(812\) 0 0
\(813\) 22924.3 0.988919
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 6766.42 0.289752
\(818\) 0 0
\(819\) 54740.2 2.33550
\(820\) 0 0
\(821\) −11953.6 −0.508142 −0.254071 0.967186i \(-0.581770\pi\)
−0.254071 + 0.967186i \(0.581770\pi\)
\(822\) 0 0
\(823\) 4320.27 0.182983 0.0914916 0.995806i \(-0.470837\pi\)
0.0914916 + 0.995806i \(0.470837\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 17980.9 0.756055 0.378028 0.925794i \(-0.376602\pi\)
0.378028 + 0.925794i \(0.376602\pi\)
\(828\) 0 0
\(829\) 29347.5 1.22953 0.614766 0.788710i \(-0.289250\pi\)
0.614766 + 0.788710i \(0.289250\pi\)
\(830\) 0 0
\(831\) −71395.7 −2.98037
\(832\) 0 0
\(833\) −15221.7 −0.633135
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 80615.9 3.32914
\(838\) 0 0
\(839\) 20819.5 0.856696 0.428348 0.903614i \(-0.359096\pi\)
0.428348 + 0.903614i \(0.359096\pi\)
\(840\) 0 0
\(841\) 4801.70 0.196880
\(842\) 0 0
\(843\) −15514.8 −0.633878
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −1932.27 −0.0783866
\(848\) 0 0
\(849\) −60716.9 −2.45442
\(850\) 0 0
\(851\) 63317.7 2.55053
\(852\) 0 0
\(853\) 16835.3 0.675766 0.337883 0.941188i \(-0.390289\pi\)
0.337883 + 0.941188i \(0.390289\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 2703.39 0.107755 0.0538775 0.998548i \(-0.482842\pi\)
0.0538775 + 0.998548i \(0.482842\pi\)
\(858\) 0 0
\(859\) −19064.6 −0.757247 −0.378624 0.925551i \(-0.623603\pi\)
−0.378624 + 0.925551i \(0.623603\pi\)
\(860\) 0 0
\(861\) −2438.89 −0.0965356
\(862\) 0 0
\(863\) −41930.0 −1.65390 −0.826949 0.562278i \(-0.809925\pi\)
−0.826949 + 0.562278i \(0.809925\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 5689.41 0.222863
\(868\) 0 0
\(869\) −41856.6 −1.63393
\(870\) 0 0
\(871\) 32878.9 1.27906
\(872\) 0 0
\(873\) 47632.1 1.84662
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 7701.66 0.296541 0.148270 0.988947i \(-0.452629\pi\)
0.148270 + 0.988947i \(0.452629\pi\)
\(878\) 0 0
\(879\) −46961.8 −1.80203
\(880\) 0 0
\(881\) 7799.94 0.298282 0.149141 0.988816i \(-0.452349\pi\)
0.149141 + 0.988816i \(0.452349\pi\)
\(882\) 0 0
\(883\) −40922.8 −1.55964 −0.779820 0.626003i \(-0.784690\pi\)
−0.779820 + 0.626003i \(0.784690\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −9632.80 −0.364642 −0.182321 0.983239i \(-0.558361\pi\)
−0.182321 + 0.983239i \(0.558361\pi\)
\(888\) 0 0
\(889\) −22384.0 −0.844470
\(890\) 0 0
\(891\) −111455. −4.19067
\(892\) 0 0
\(893\) −15222.8 −0.570449
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −114542. −4.26361
\(898\) 0 0
\(899\) 28447.9 1.05538
\(900\) 0 0
\(901\) 14641.1 0.541361
\(902\) 0 0
\(903\) 14621.2 0.538828
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −27797.0 −1.01762 −0.508812 0.860878i \(-0.669915\pi\)
−0.508812 + 0.860878i \(0.669915\pi\)
\(908\) 0 0
\(909\) −19207.3 −0.700841
\(910\) 0 0
\(911\) 20735.4 0.754110 0.377055 0.926191i \(-0.376937\pi\)
0.377055 + 0.926191i \(0.376937\pi\)
\(912\) 0 0
\(913\) −44066.9 −1.59737
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 11709.0 0.421662
\(918\) 0 0
\(919\) 6063.46 0.217644 0.108822 0.994061i \(-0.465292\pi\)
0.108822 + 0.994061i \(0.465292\pi\)
\(920\) 0 0
\(921\) −73323.7 −2.62334
\(922\) 0 0
\(923\) −293.973 −0.0104835
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −6122.18 −0.216914
\(928\) 0 0
\(929\) −694.043 −0.0245111 −0.0122556 0.999925i \(-0.503901\pi\)
−0.0122556 + 0.999925i \(0.503901\pi\)
\(930\) 0 0
\(931\) −11423.6 −0.402142
\(932\) 0 0
\(933\) 68252.9 2.39496
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 44470.5 1.55047 0.775234 0.631674i \(-0.217632\pi\)
0.775234 + 0.631674i \(0.217632\pi\)
\(938\) 0 0
\(939\) −18836.6 −0.654642
\(940\) 0 0
\(941\) −34075.3 −1.18047 −0.590235 0.807232i \(-0.700965\pi\)
−0.590235 + 0.807232i \(0.700965\pi\)
\(942\) 0 0
\(943\) 3751.79 0.129560
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −34495.9 −1.18370 −0.591851 0.806048i \(-0.701603\pi\)
−0.591851 + 0.806048i \(0.701603\pi\)
\(948\) 0 0
\(949\) 41451.0 1.41787
\(950\) 0 0
\(951\) 27862.0 0.950039
\(952\) 0 0
\(953\) 23975.9 0.814960 0.407480 0.913214i \(-0.366408\pi\)
0.407480 + 0.913214i \(0.366408\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −67111.7 −2.26689
\(958\) 0 0
\(959\) 12201.5 0.410851
\(960\) 0 0
\(961\) −2067.03 −0.0693845
\(962\) 0 0
\(963\) 55684.4 1.86335
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −27578.9 −0.917144 −0.458572 0.888657i \(-0.651639\pi\)
−0.458572 + 0.888657i \(0.651639\pi\)
\(968\) 0 0
\(969\) −32959.3 −1.09268
\(970\) 0 0
\(971\) −20834.3 −0.688574 −0.344287 0.938864i \(-0.611879\pi\)
−0.344287 + 0.938864i \(0.611879\pi\)
\(972\) 0 0
\(973\) −3752.88 −0.123650
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −41636.7 −1.36344 −0.681718 0.731615i \(-0.738767\pi\)
−0.681718 + 0.731615i \(0.738767\pi\)
\(978\) 0 0
\(979\) −18663.8 −0.609292
\(980\) 0 0
\(981\) −37166.1 −1.20960
\(982\) 0 0
\(983\) −18099.2 −0.587259 −0.293629 0.955919i \(-0.594863\pi\)
−0.293629 + 0.955919i \(0.594863\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −32894.1 −1.06082
\(988\) 0 0
\(989\) −22492.0 −0.723158
\(990\) 0 0
\(991\) −55904.9 −1.79201 −0.896003 0.444049i \(-0.853542\pi\)
−0.896003 + 0.444049i \(0.853542\pi\)
\(992\) 0 0
\(993\) −51737.3 −1.65341
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 15808.7 0.502174 0.251087 0.967965i \(-0.419212\pi\)
0.251087 + 0.967965i \(0.419212\pi\)
\(998\) 0 0
\(999\) 186332. 5.90119
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 800.4.a.ba.1.1 4
4.3 odd 2 inner 800.4.a.ba.1.4 yes 4
5.2 odd 4 800.4.c.o.449.7 8
5.3 odd 4 800.4.c.o.449.1 8
5.4 even 2 800.4.a.bb.1.4 yes 4
8.3 odd 2 1600.4.a.cx.1.1 4
8.5 even 2 1600.4.a.cx.1.4 4
20.3 even 4 800.4.c.o.449.8 8
20.7 even 4 800.4.c.o.449.2 8
20.19 odd 2 800.4.a.bb.1.1 yes 4
40.19 odd 2 1600.4.a.cw.1.4 4
40.29 even 2 1600.4.a.cw.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
800.4.a.ba.1.1 4 1.1 even 1 trivial
800.4.a.ba.1.4 yes 4 4.3 odd 2 inner
800.4.a.bb.1.1 yes 4 20.19 odd 2
800.4.a.bb.1.4 yes 4 5.4 even 2
800.4.c.o.449.1 8 5.3 odd 4
800.4.c.o.449.2 8 20.7 even 4
800.4.c.o.449.7 8 5.2 odd 4
800.4.c.o.449.8 8 20.3 even 4
1600.4.a.cw.1.1 4 40.29 even 2
1600.4.a.cw.1.4 4 40.19 odd 2
1600.4.a.cx.1.1 4 8.3 odd 2
1600.4.a.cx.1.4 4 8.5 even 2