Properties

Label 1176.4.a.s.1.2
Level $1176$
Weight $4$
Character 1176.1
Self dual yes
Analytic conductor $69.386$
Analytic rank $0$
Dimension $2$
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] = [1176,4,Mod(1,1176)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1176, 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("1176.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1176 = 2^{3} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1176.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: \(69.3862461668\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{137}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 34 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-5.35235\) of defining polynomial
Character \(\chi\) \(=\) 1176.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.00000 q^{3} +20.7047 q^{5} +9.00000 q^{9} +O(q^{10})\) \(q-3.00000 q^{3} +20.7047 q^{5} +9.00000 q^{9} -6.11410 q^{11} -22.8188 q^{13} -62.1141 q^{15} +8.70470 q^{17} -75.0470 q^{19} +178.570 q^{23} +303.685 q^{25} -27.0000 q^{27} -122.228 q^{29} +102.591 q^{31} +18.3423 q^{33} +182.228 q^{37} +68.4564 q^{39} +488.208 q^{41} -388.000 q^{43} +186.342 q^{45} +254.550 q^{47} -26.1141 q^{51} +610.228 q^{53} -126.591 q^{55} +225.141 q^{57} -31.2752 q^{59} -587.503 q^{61} -472.456 q^{65} -689.597 q^{67} -535.711 q^{69} -563.940 q^{71} +275.315 q^{73} -911.054 q^{75} -219.315 q^{79} +81.0000 q^{81} +1447.28 q^{83} +180.228 q^{85} +366.685 q^{87} +1018.57 q^{89} -307.772 q^{93} -1553.83 q^{95} +792.497 q^{97} -55.0269 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{3} + 18 q^{5} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 6 q^{3} + 18 q^{5} + 18 q^{9} + 58 q^{11} + 48 q^{13} - 54 q^{15} - 6 q^{17} + 84 q^{19} + 6 q^{23} + 186 q^{25} - 54 q^{27} - 104 q^{29} + 252 q^{31} - 174 q^{33} + 224 q^{37} - 144 q^{39} + 438 q^{41} - 776 q^{43} + 162 q^{45} - 240 q^{47} + 18 q^{51} + 1080 q^{53} - 300 q^{55} - 252 q^{57} + 312 q^{59} - 660 q^{61} - 664 q^{65} - 396 q^{67} - 18 q^{69} + 66 q^{71} + 972 q^{73} - 558 q^{75} - 860 q^{79} + 162 q^{81} + 2520 q^{83} + 220 q^{85} + 312 q^{87} + 1686 q^{89} - 756 q^{93} - 1984 q^{95} + 2100 q^{97} + 522 q^{99}+O(q^{100}) \) 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
\(3\) −3.00000 −0.577350
\(4\) 0 0
\(5\) 20.7047 1.85188 0.925942 0.377665i \(-0.123273\pi\)
0.925942 + 0.377665i \(0.123273\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −6.11410 −0.167588 −0.0837941 0.996483i \(-0.526704\pi\)
−0.0837941 + 0.996483i \(0.526704\pi\)
\(12\) 0 0
\(13\) −22.8188 −0.486830 −0.243415 0.969922i \(-0.578268\pi\)
−0.243415 + 0.969922i \(0.578268\pi\)
\(14\) 0 0
\(15\) −62.1141 −1.06919
\(16\) 0 0
\(17\) 8.70470 0.124188 0.0620941 0.998070i \(-0.480222\pi\)
0.0620941 + 0.998070i \(0.480222\pi\)
\(18\) 0 0
\(19\) −75.0470 −0.906156 −0.453078 0.891471i \(-0.649674\pi\)
−0.453078 + 0.891471i \(0.649674\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 178.570 1.61889 0.809446 0.587194i \(-0.199767\pi\)
0.809446 + 0.587194i \(0.199767\pi\)
\(24\) 0 0
\(25\) 303.685 2.42948
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) −122.228 −0.782662 −0.391331 0.920250i \(-0.627985\pi\)
−0.391331 + 0.920250i \(0.627985\pi\)
\(30\) 0 0
\(31\) 102.591 0.594381 0.297191 0.954818i \(-0.403950\pi\)
0.297191 + 0.954818i \(0.403950\pi\)
\(32\) 0 0
\(33\) 18.3423 0.0967571
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 182.228 0.809679 0.404840 0.914388i \(-0.367327\pi\)
0.404840 + 0.914388i \(0.367327\pi\)
\(38\) 0 0
\(39\) 68.4564 0.281072
\(40\) 0 0
\(41\) 488.208 1.85964 0.929821 0.368013i \(-0.119962\pi\)
0.929821 + 0.368013i \(0.119962\pi\)
\(42\) 0 0
\(43\) −388.000 −1.37603 −0.688017 0.725695i \(-0.741518\pi\)
−0.688017 + 0.725695i \(0.741518\pi\)
\(44\) 0 0
\(45\) 186.342 0.617295
\(46\) 0 0
\(47\) 254.550 0.790000 0.395000 0.918681i \(-0.370745\pi\)
0.395000 + 0.918681i \(0.370745\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −26.1141 −0.0717001
\(52\) 0 0
\(53\) 610.228 1.58153 0.790767 0.612117i \(-0.209682\pi\)
0.790767 + 0.612117i \(0.209682\pi\)
\(54\) 0 0
\(55\) −126.591 −0.310354
\(56\) 0 0
\(57\) 225.141 0.523169
\(58\) 0 0
\(59\) −31.2752 −0.0690116 −0.0345058 0.999404i \(-0.510986\pi\)
−0.0345058 + 0.999404i \(0.510986\pi\)
\(60\) 0 0
\(61\) −587.503 −1.23315 −0.616575 0.787296i \(-0.711480\pi\)
−0.616575 + 0.787296i \(0.711480\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −472.456 −0.901554
\(66\) 0 0
\(67\) −689.597 −1.25743 −0.628714 0.777636i \(-0.716419\pi\)
−0.628714 + 0.777636i \(0.716419\pi\)
\(68\) 0 0
\(69\) −535.711 −0.934668
\(70\) 0 0
\(71\) −563.940 −0.942638 −0.471319 0.881963i \(-0.656222\pi\)
−0.471319 + 0.881963i \(0.656222\pi\)
\(72\) 0 0
\(73\) 275.315 0.441414 0.220707 0.975340i \(-0.429164\pi\)
0.220707 + 0.975340i \(0.429164\pi\)
\(74\) 0 0
\(75\) −911.054 −1.40266
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −219.315 −0.312341 −0.156170 0.987730i \(-0.549915\pi\)
−0.156170 + 0.987730i \(0.549915\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 1447.28 1.91396 0.956982 0.290146i \(-0.0937038\pi\)
0.956982 + 0.290146i \(0.0937038\pi\)
\(84\) 0 0
\(85\) 180.228 0.229982
\(86\) 0 0
\(87\) 366.685 0.451870
\(88\) 0 0
\(89\) 1018.57 1.21313 0.606563 0.795035i \(-0.292548\pi\)
0.606563 + 0.795035i \(0.292548\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −307.772 −0.343166
\(94\) 0 0
\(95\) −1553.83 −1.67810
\(96\) 0 0
\(97\) 792.497 0.829545 0.414772 0.909925i \(-0.363861\pi\)
0.414772 + 0.909925i \(0.363861\pi\)
\(98\) 0 0
\(99\) −55.0269 −0.0558627
\(100\) 0 0
\(101\) 509.658 0.502107 0.251054 0.967973i \(-0.419223\pi\)
0.251054 + 0.967973i \(0.419223\pi\)
\(102\) 0 0
\(103\) −217.490 −0.208057 −0.104029 0.994574i \(-0.533173\pi\)
−0.104029 + 0.994574i \(0.533173\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −1897.31 −1.71420 −0.857102 0.515147i \(-0.827737\pi\)
−0.857102 + 0.515147i \(0.827737\pi\)
\(108\) 0 0
\(109\) 850.456 0.747330 0.373665 0.927564i \(-0.378101\pi\)
0.373665 + 0.927564i \(0.378101\pi\)
\(110\) 0 0
\(111\) −546.685 −0.467469
\(112\) 0 0
\(113\) 1527.48 1.27162 0.635809 0.771846i \(-0.280666\pi\)
0.635809 + 0.771846i \(0.280666\pi\)
\(114\) 0 0
\(115\) 3697.25 2.99800
\(116\) 0 0
\(117\) −205.369 −0.162277
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −1293.62 −0.971914
\(122\) 0 0
\(123\) −1464.62 −1.07366
\(124\) 0 0
\(125\) 3699.61 2.64723
\(126\) 0 0
\(127\) 1038.51 0.725613 0.362807 0.931864i \(-0.381819\pi\)
0.362807 + 0.931864i \(0.381819\pi\)
\(128\) 0 0
\(129\) 1164.00 0.794453
\(130\) 0 0
\(131\) −961.181 −0.641059 −0.320530 0.947238i \(-0.603861\pi\)
−0.320530 + 0.947238i \(0.603861\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −559.027 −0.356395
\(136\) 0 0
\(137\) −1690.00 −1.05392 −0.526958 0.849891i \(-0.676667\pi\)
−0.526958 + 0.849891i \(0.676667\pi\)
\(138\) 0 0
\(139\) −1732.30 −1.05706 −0.528530 0.848915i \(-0.677257\pi\)
−0.528530 + 0.848915i \(0.677257\pi\)
\(140\) 0 0
\(141\) −763.651 −0.456107
\(142\) 0 0
\(143\) 139.516 0.0815871
\(144\) 0 0
\(145\) −2530.70 −1.44940
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 2725.88 1.49874 0.749372 0.662150i \(-0.230356\pi\)
0.749372 + 0.662150i \(0.230356\pi\)
\(150\) 0 0
\(151\) 1129.83 0.608900 0.304450 0.952528i \(-0.401527\pi\)
0.304450 + 0.952528i \(0.401527\pi\)
\(152\) 0 0
\(153\) 78.3423 0.0413961
\(154\) 0 0
\(155\) 2124.11 1.10073
\(156\) 0 0
\(157\) −719.691 −0.365845 −0.182922 0.983127i \(-0.558556\pi\)
−0.182922 + 0.983127i \(0.558556\pi\)
\(158\) 0 0
\(159\) −1830.68 −0.913099
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 2469.25 1.18654 0.593271 0.805003i \(-0.297836\pi\)
0.593271 + 0.805003i \(0.297836\pi\)
\(164\) 0 0
\(165\) 379.772 0.179183
\(166\) 0 0
\(167\) 2419.03 1.12090 0.560451 0.828188i \(-0.310628\pi\)
0.560451 + 0.828188i \(0.310628\pi\)
\(168\) 0 0
\(169\) −1676.30 −0.762996
\(170\) 0 0
\(171\) −675.423 −0.302052
\(172\) 0 0
\(173\) 3559.16 1.56415 0.782075 0.623184i \(-0.214161\pi\)
0.782075 + 0.623184i \(0.214161\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 93.8256 0.0398439
\(178\) 0 0
\(179\) 3864.15 1.61352 0.806761 0.590878i \(-0.201218\pi\)
0.806761 + 0.590878i \(0.201218\pi\)
\(180\) 0 0
\(181\) 4100.46 1.68389 0.841946 0.539561i \(-0.181410\pi\)
0.841946 + 0.539561i \(0.181410\pi\)
\(182\) 0 0
\(183\) 1762.51 0.711959
\(184\) 0 0
\(185\) 3772.98 1.49943
\(186\) 0 0
\(187\) −53.2214 −0.0208125
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 1120.17 0.424359 0.212179 0.977231i \(-0.431944\pi\)
0.212179 + 0.977231i \(0.431944\pi\)
\(192\) 0 0
\(193\) −1418.44 −0.529025 −0.264512 0.964382i \(-0.585211\pi\)
−0.264512 + 0.964382i \(0.585211\pi\)
\(194\) 0 0
\(195\) 1417.37 0.520512
\(196\) 0 0
\(197\) −4611.01 −1.66762 −0.833809 0.552053i \(-0.813844\pi\)
−0.833809 + 0.552053i \(0.813844\pi\)
\(198\) 0 0
\(199\) 5049.83 1.79886 0.899429 0.437067i \(-0.143983\pi\)
0.899429 + 0.437067i \(0.143983\pi\)
\(200\) 0 0
\(201\) 2068.79 0.725977
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 10108.2 3.44384
\(206\) 0 0
\(207\) 1607.13 0.539631
\(208\) 0 0
\(209\) 458.845 0.151861
\(210\) 0 0
\(211\) −1712.11 −0.558608 −0.279304 0.960203i \(-0.590104\pi\)
−0.279304 + 0.960203i \(0.590104\pi\)
\(212\) 0 0
\(213\) 1691.82 0.544233
\(214\) 0 0
\(215\) −8033.42 −2.54826
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −825.946 −0.254851
\(220\) 0 0
\(221\) −198.631 −0.0604586
\(222\) 0 0
\(223\) −161.906 −0.0486190 −0.0243095 0.999704i \(-0.507739\pi\)
−0.0243095 + 0.999704i \(0.507739\pi\)
\(224\) 0 0
\(225\) 2733.16 0.809826
\(226\) 0 0
\(227\) 3947.76 1.15428 0.577141 0.816645i \(-0.304168\pi\)
0.577141 + 0.816645i \(0.304168\pi\)
\(228\) 0 0
\(229\) −3584.78 −1.03445 −0.517225 0.855850i \(-0.673035\pi\)
−0.517225 + 0.855850i \(0.673035\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −3448.85 −0.969706 −0.484853 0.874596i \(-0.661127\pi\)
−0.484853 + 0.874596i \(0.661127\pi\)
\(234\) 0 0
\(235\) 5270.39 1.46299
\(236\) 0 0
\(237\) 657.946 0.180330
\(238\) 0 0
\(239\) 3818.57 1.03348 0.516742 0.856141i \(-0.327145\pi\)
0.516742 + 0.856141i \(0.327145\pi\)
\(240\) 0 0
\(241\) −1314.54 −0.351356 −0.175678 0.984448i \(-0.556212\pi\)
−0.175678 + 0.984448i \(0.556212\pi\)
\(242\) 0 0
\(243\) −243.000 −0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 1712.48 0.441144
\(248\) 0 0
\(249\) −4341.83 −1.10503
\(250\) 0 0
\(251\) −4829.66 −1.21452 −0.607262 0.794501i \(-0.707732\pi\)
−0.607262 + 0.794501i \(0.707732\pi\)
\(252\) 0 0
\(253\) −1091.80 −0.271307
\(254\) 0 0
\(255\) −540.685 −0.132780
\(256\) 0 0
\(257\) −1479.93 −0.359203 −0.179602 0.983739i \(-0.557481\pi\)
−0.179602 + 0.983739i \(0.557481\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −1100.05 −0.260887
\(262\) 0 0
\(263\) −1191.01 −0.279244 −0.139622 0.990205i \(-0.544589\pi\)
−0.139622 + 0.990205i \(0.544589\pi\)
\(264\) 0 0
\(265\) 12634.6 2.92882
\(266\) 0 0
\(267\) −3055.71 −0.700399
\(268\) 0 0
\(269\) 558.772 0.126650 0.0633251 0.997993i \(-0.479830\pi\)
0.0633251 + 0.997993i \(0.479830\pi\)
\(270\) 0 0
\(271\) 5992.31 1.34320 0.671599 0.740914i \(-0.265608\pi\)
0.671599 + 0.740914i \(0.265608\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −1856.76 −0.407152
\(276\) 0 0
\(277\) 7884.03 1.71013 0.855064 0.518523i \(-0.173518\pi\)
0.855064 + 0.518523i \(0.173518\pi\)
\(278\) 0 0
\(279\) 923.315 0.198127
\(280\) 0 0
\(281\) −75.8256 −0.0160974 −0.00804871 0.999968i \(-0.502562\pi\)
−0.00804871 + 0.999968i \(0.502562\pi\)
\(282\) 0 0
\(283\) 803.369 0.168747 0.0843733 0.996434i \(-0.473111\pi\)
0.0843733 + 0.996434i \(0.473111\pi\)
\(284\) 0 0
\(285\) 4661.48 0.968849
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −4837.23 −0.984577
\(290\) 0 0
\(291\) −2377.49 −0.478938
\(292\) 0 0
\(293\) −2626.26 −0.523645 −0.261822 0.965116i \(-0.584323\pi\)
−0.261822 + 0.965116i \(0.584323\pi\)
\(294\) 0 0
\(295\) −647.544 −0.127802
\(296\) 0 0
\(297\) 165.081 0.0322524
\(298\) 0 0
\(299\) −4074.76 −0.788126
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −1528.97 −0.289892
\(304\) 0 0
\(305\) −12164.1 −2.28365
\(306\) 0 0
\(307\) −4865.84 −0.904587 −0.452293 0.891869i \(-0.649394\pi\)
−0.452293 + 0.891869i \(0.649394\pi\)
\(308\) 0 0
\(309\) 652.469 0.120122
\(310\) 0 0
\(311\) 6152.97 1.12187 0.560937 0.827859i \(-0.310441\pi\)
0.560937 + 0.827859i \(0.310441\pi\)
\(312\) 0 0
\(313\) 7892.78 1.42532 0.712662 0.701508i \(-0.247489\pi\)
0.712662 + 0.701508i \(0.247489\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −3688.75 −0.653568 −0.326784 0.945099i \(-0.605965\pi\)
−0.326784 + 0.945099i \(0.605965\pi\)
\(318\) 0 0
\(319\) 747.315 0.131165
\(320\) 0 0
\(321\) 5691.93 0.989696
\(322\) 0 0
\(323\) −653.262 −0.112534
\(324\) 0 0
\(325\) −6929.72 −1.18274
\(326\) 0 0
\(327\) −2551.37 −0.431471
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −1981.80 −0.329092 −0.164546 0.986369i \(-0.552616\pi\)
−0.164546 + 0.986369i \(0.552616\pi\)
\(332\) 0 0
\(333\) 1640.05 0.269893
\(334\) 0 0
\(335\) −14277.9 −2.32861
\(336\) 0 0
\(337\) 1578.35 0.255128 0.127564 0.991830i \(-0.459284\pi\)
0.127564 + 0.991830i \(0.459284\pi\)
\(338\) 0 0
\(339\) −4582.43 −0.734169
\(340\) 0 0
\(341\) −627.249 −0.0996113
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −11091.7 −1.73090
\(346\) 0 0
\(347\) −3213.51 −0.497148 −0.248574 0.968613i \(-0.579962\pi\)
−0.248574 + 0.968613i \(0.579962\pi\)
\(348\) 0 0
\(349\) −725.920 −0.111340 −0.0556699 0.998449i \(-0.517729\pi\)
−0.0556699 + 0.998449i \(0.517729\pi\)
\(350\) 0 0
\(351\) 616.108 0.0936906
\(352\) 0 0
\(353\) −619.403 −0.0933923 −0.0466962 0.998909i \(-0.514869\pi\)
−0.0466962 + 0.998909i \(0.514869\pi\)
\(354\) 0 0
\(355\) −11676.2 −1.74566
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −7912.05 −1.16318 −0.581590 0.813482i \(-0.697569\pi\)
−0.581590 + 0.813482i \(0.697569\pi\)
\(360\) 0 0
\(361\) −1226.95 −0.178881
\(362\) 0 0
\(363\) 3880.85 0.561135
\(364\) 0 0
\(365\) 5700.32 0.817448
\(366\) 0 0
\(367\) 2916.08 0.414764 0.207382 0.978260i \(-0.433506\pi\)
0.207382 + 0.978260i \(0.433506\pi\)
\(368\) 0 0
\(369\) 4393.87 0.619880
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −53.1688 −0.00738063 −0.00369031 0.999993i \(-0.501175\pi\)
−0.00369031 + 0.999993i \(0.501175\pi\)
\(374\) 0 0
\(375\) −11098.8 −1.52838
\(376\) 0 0
\(377\) 2789.10 0.381024
\(378\) 0 0
\(379\) −12650.6 −1.71456 −0.857281 0.514848i \(-0.827848\pi\)
−0.857281 + 0.514848i \(0.827848\pi\)
\(380\) 0 0
\(381\) −3115.53 −0.418933
\(382\) 0 0
\(383\) −10401.5 −1.38770 −0.693852 0.720118i \(-0.744088\pi\)
−0.693852 + 0.720118i \(0.744088\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −3492.00 −0.458678
\(388\) 0 0
\(389\) 2665.16 0.347375 0.173687 0.984801i \(-0.444432\pi\)
0.173687 + 0.984801i \(0.444432\pi\)
\(390\) 0 0
\(391\) 1554.40 0.201047
\(392\) 0 0
\(393\) 2883.54 0.370116
\(394\) 0 0
\(395\) −4540.86 −0.578419
\(396\) 0 0
\(397\) 10880.9 1.37556 0.687779 0.725920i \(-0.258586\pi\)
0.687779 + 0.725920i \(0.258586\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −12599.0 −1.56899 −0.784493 0.620137i \(-0.787077\pi\)
−0.784493 + 0.620137i \(0.787077\pi\)
\(402\) 0 0
\(403\) −2340.99 −0.289363
\(404\) 0 0
\(405\) 1677.08 0.205765
\(406\) 0 0
\(407\) −1114.16 −0.135693
\(408\) 0 0
\(409\) −5740.47 −0.694005 −0.347002 0.937864i \(-0.612800\pi\)
−0.347002 + 0.937864i \(0.612800\pi\)
\(410\) 0 0
\(411\) 5070.00 0.608478
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 29965.4 3.54444
\(416\) 0 0
\(417\) 5196.89 0.610294
\(418\) 0 0
\(419\) −3433.66 −0.400347 −0.200174 0.979760i \(-0.564151\pi\)
−0.200174 + 0.979760i \(0.564151\pi\)
\(420\) 0 0
\(421\) −4793.88 −0.554963 −0.277481 0.960731i \(-0.589500\pi\)
−0.277481 + 0.960731i \(0.589500\pi\)
\(422\) 0 0
\(423\) 2290.95 0.263333
\(424\) 0 0
\(425\) 2643.48 0.301712
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −418.549 −0.0471043
\(430\) 0 0
\(431\) 5350.29 0.597945 0.298973 0.954262i \(-0.403356\pi\)
0.298973 + 0.954262i \(0.403356\pi\)
\(432\) 0 0
\(433\) −4440.03 −0.492781 −0.246390 0.969171i \(-0.579245\pi\)
−0.246390 + 0.969171i \(0.579245\pi\)
\(434\) 0 0
\(435\) 7592.09 0.836812
\(436\) 0 0
\(437\) −13401.2 −1.46697
\(438\) 0 0
\(439\) 10376.0 1.12806 0.564030 0.825754i \(-0.309250\pi\)
0.564030 + 0.825754i \(0.309250\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −4186.32 −0.448979 −0.224490 0.974476i \(-0.572072\pi\)
−0.224490 + 0.974476i \(0.572072\pi\)
\(444\) 0 0
\(445\) 21089.2 2.24657
\(446\) 0 0
\(447\) −8177.64 −0.865300
\(448\) 0 0
\(449\) 5216.36 0.548275 0.274137 0.961691i \(-0.411608\pi\)
0.274137 + 0.961691i \(0.411608\pi\)
\(450\) 0 0
\(451\) −2984.95 −0.311654
\(452\) 0 0
\(453\) −3389.48 −0.351549
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 9469.65 0.969303 0.484652 0.874707i \(-0.338946\pi\)
0.484652 + 0.874707i \(0.338946\pi\)
\(458\) 0 0
\(459\) −235.027 −0.0239000
\(460\) 0 0
\(461\) −3455.07 −0.349064 −0.174532 0.984651i \(-0.555841\pi\)
−0.174532 + 0.984651i \(0.555841\pi\)
\(462\) 0 0
\(463\) −10881.9 −1.09228 −0.546139 0.837695i \(-0.683903\pi\)
−0.546139 + 0.837695i \(0.683903\pi\)
\(464\) 0 0
\(465\) −6372.32 −0.635504
\(466\) 0 0
\(467\) 12053.6 1.19437 0.597186 0.802102i \(-0.296285\pi\)
0.597186 + 0.802102i \(0.296285\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 2159.07 0.211221
\(472\) 0 0
\(473\) 2372.27 0.230607
\(474\) 0 0
\(475\) −22790.6 −2.20148
\(476\) 0 0
\(477\) 5492.05 0.527178
\(478\) 0 0
\(479\) −7836.59 −0.747522 −0.373761 0.927525i \(-0.621932\pi\)
−0.373761 + 0.927525i \(0.621932\pi\)
\(480\) 0 0
\(481\) −4158.23 −0.394177
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 16408.4 1.53622
\(486\) 0 0
\(487\) 8799.54 0.818779 0.409390 0.912360i \(-0.365742\pi\)
0.409390 + 0.912360i \(0.365742\pi\)
\(488\) 0 0
\(489\) −7407.75 −0.685051
\(490\) 0 0
\(491\) −11382.9 −1.04623 −0.523117 0.852261i \(-0.675231\pi\)
−0.523117 + 0.852261i \(0.675231\pi\)
\(492\) 0 0
\(493\) −1063.96 −0.0971974
\(494\) 0 0
\(495\) −1139.32 −0.103451
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 17510.7 1.57092 0.785458 0.618915i \(-0.212427\pi\)
0.785458 + 0.618915i \(0.212427\pi\)
\(500\) 0 0
\(501\) −7257.10 −0.647153
\(502\) 0 0
\(503\) −14573.9 −1.29188 −0.645940 0.763388i \(-0.723535\pi\)
−0.645940 + 0.763388i \(0.723535\pi\)
\(504\) 0 0
\(505\) 10552.3 0.929845
\(506\) 0 0
\(507\) 5028.91 0.440516
\(508\) 0 0
\(509\) 17451.3 1.51968 0.759839 0.650112i \(-0.225278\pi\)
0.759839 + 0.650112i \(0.225278\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 2026.27 0.174390
\(514\) 0 0
\(515\) −4503.06 −0.385298
\(516\) 0 0
\(517\) −1556.35 −0.132395
\(518\) 0 0
\(519\) −10677.5 −0.903063
\(520\) 0 0
\(521\) −3210.52 −0.269972 −0.134986 0.990848i \(-0.543099\pi\)
−0.134986 + 0.990848i \(0.543099\pi\)
\(522\) 0 0
\(523\) −16647.8 −1.39188 −0.695942 0.718098i \(-0.745013\pi\)
−0.695942 + 0.718098i \(0.745013\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 893.020 0.0738151
\(528\) 0 0
\(529\) 19720.4 1.62081
\(530\) 0 0
\(531\) −281.477 −0.0230039
\(532\) 0 0
\(533\) −11140.3 −0.905330
\(534\) 0 0
\(535\) −39283.2 −3.17451
\(536\) 0 0
\(537\) −11592.5 −0.931568
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 10709.9 0.851118 0.425559 0.904931i \(-0.360078\pi\)
0.425559 + 0.904931i \(0.360078\pi\)
\(542\) 0 0
\(543\) −12301.4 −0.972196
\(544\) 0 0
\(545\) 17608.4 1.38397
\(546\) 0 0
\(547\) 1263.05 0.0987277 0.0493638 0.998781i \(-0.484281\pi\)
0.0493638 + 0.998781i \(0.484281\pi\)
\(548\) 0 0
\(549\) −5287.53 −0.411050
\(550\) 0 0
\(551\) 9172.86 0.709214
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −11318.9 −0.865698
\(556\) 0 0
\(557\) −24281.3 −1.84710 −0.923548 0.383483i \(-0.874725\pi\)
−0.923548 + 0.383483i \(0.874725\pi\)
\(558\) 0 0
\(559\) 8853.69 0.669895
\(560\) 0 0
\(561\) 159.664 0.0120161
\(562\) 0 0
\(563\) −5467.57 −0.409291 −0.204645 0.978836i \(-0.565604\pi\)
−0.204645 + 0.978836i \(0.565604\pi\)
\(564\) 0 0
\(565\) 31625.9 2.35489
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 6474.08 0.476991 0.238495 0.971144i \(-0.423346\pi\)
0.238495 + 0.971144i \(0.423346\pi\)
\(570\) 0 0
\(571\) 18112.8 1.32749 0.663747 0.747957i \(-0.268965\pi\)
0.663747 + 0.747957i \(0.268965\pi\)
\(572\) 0 0
\(573\) −3360.50 −0.245004
\(574\) 0 0
\(575\) 54229.1 3.93306
\(576\) 0 0
\(577\) 10175.8 0.734181 0.367090 0.930185i \(-0.380354\pi\)
0.367090 + 0.930185i \(0.380354\pi\)
\(578\) 0 0
\(579\) 4255.33 0.305433
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −3731.00 −0.265046
\(584\) 0 0
\(585\) −4252.11 −0.300518
\(586\) 0 0
\(587\) 11566.7 0.813304 0.406652 0.913583i \(-0.366696\pi\)
0.406652 + 0.913583i \(0.366696\pi\)
\(588\) 0 0
\(589\) −7699.12 −0.538602
\(590\) 0 0
\(591\) 13833.0 0.962799
\(592\) 0 0
\(593\) 4736.73 0.328017 0.164009 0.986459i \(-0.447557\pi\)
0.164009 + 0.986459i \(0.447557\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −15149.5 −1.03857
\(598\) 0 0
\(599\) 7774.56 0.530317 0.265158 0.964205i \(-0.414576\pi\)
0.265158 + 0.964205i \(0.414576\pi\)
\(600\) 0 0
\(601\) −16662.3 −1.13089 −0.565447 0.824784i \(-0.691296\pi\)
−0.565447 + 0.824784i \(0.691296\pi\)
\(602\) 0 0
\(603\) −6206.38 −0.419143
\(604\) 0 0
\(605\) −26784.0 −1.79987
\(606\) 0 0
\(607\) −26523.4 −1.77356 −0.886781 0.462190i \(-0.847064\pi\)
−0.886781 + 0.462190i \(0.847064\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −5808.53 −0.384596
\(612\) 0 0
\(613\) −6963.91 −0.458841 −0.229421 0.973327i \(-0.573683\pi\)
−0.229421 + 0.973327i \(0.573683\pi\)
\(614\) 0 0
\(615\) −30324.6 −1.98830
\(616\) 0 0
\(617\) −22945.5 −1.49716 −0.748582 0.663042i \(-0.769265\pi\)
−0.748582 + 0.663042i \(0.769265\pi\)
\(618\) 0 0
\(619\) 20783.6 1.34954 0.674769 0.738029i \(-0.264243\pi\)
0.674769 + 0.738029i \(0.264243\pi\)
\(620\) 0 0
\(621\) −4821.40 −0.311556
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 38638.8 2.47288
\(626\) 0 0
\(627\) −1376.53 −0.0876770
\(628\) 0 0
\(629\) 1586.24 0.100553
\(630\) 0 0
\(631\) 909.976 0.0574098 0.0287049 0.999588i \(-0.490862\pi\)
0.0287049 + 0.999588i \(0.490862\pi\)
\(632\) 0 0
\(633\) 5136.32 0.322513
\(634\) 0 0
\(635\) 21502.0 1.34375
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −5075.46 −0.314213
\(640\) 0 0
\(641\) −25103.9 −1.54687 −0.773437 0.633874i \(-0.781464\pi\)
−0.773437 + 0.633874i \(0.781464\pi\)
\(642\) 0 0
\(643\) −23810.1 −1.46031 −0.730156 0.683281i \(-0.760552\pi\)
−0.730156 + 0.683281i \(0.760552\pi\)
\(644\) 0 0
\(645\) 24100.3 1.47124
\(646\) 0 0
\(647\) −10985.6 −0.667527 −0.333763 0.942657i \(-0.608319\pi\)
−0.333763 + 0.942657i \(0.608319\pi\)
\(648\) 0 0
\(649\) 191.220 0.0115655
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −24506.4 −1.46862 −0.734309 0.678815i \(-0.762494\pi\)
−0.734309 + 0.678815i \(0.762494\pi\)
\(654\) 0 0
\(655\) −19901.0 −1.18717
\(656\) 0 0
\(657\) 2477.84 0.147138
\(658\) 0 0
\(659\) 22113.2 1.30714 0.653571 0.756865i \(-0.273270\pi\)
0.653571 + 0.756865i \(0.273270\pi\)
\(660\) 0 0
\(661\) −23937.9 −1.40859 −0.704294 0.709909i \(-0.748736\pi\)
−0.704294 + 0.709909i \(0.748736\pi\)
\(662\) 0 0
\(663\) 595.892 0.0349058
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −21826.4 −1.26705
\(668\) 0 0
\(669\) 485.718 0.0280702
\(670\) 0 0
\(671\) 3592.05 0.206661
\(672\) 0 0
\(673\) −15407.1 −0.882468 −0.441234 0.897392i \(-0.645459\pi\)
−0.441234 + 0.897392i \(0.645459\pi\)
\(674\) 0 0
\(675\) −8199.48 −0.467553
\(676\) 0 0
\(677\) −5145.81 −0.292126 −0.146063 0.989275i \(-0.546660\pi\)
−0.146063 + 0.989275i \(0.546660\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −11843.3 −0.666425
\(682\) 0 0
\(683\) 18247.7 1.02230 0.511148 0.859492i \(-0.329220\pi\)
0.511148 + 0.859492i \(0.329220\pi\)
\(684\) 0 0
\(685\) −34990.9 −1.95173
\(686\) 0 0
\(687\) 10754.3 0.597240
\(688\) 0 0
\(689\) −13924.7 −0.769939
\(690\) 0 0
\(691\) −10359.8 −0.570338 −0.285169 0.958477i \(-0.592050\pi\)
−0.285169 + 0.958477i \(0.592050\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −35866.7 −1.95755
\(696\) 0 0
\(697\) 4249.70 0.230946
\(698\) 0 0
\(699\) 10346.5 0.559860
\(700\) 0 0
\(701\) −369.772 −0.0199231 −0.00996155 0.999950i \(-0.503171\pi\)
−0.00996155 + 0.999950i \(0.503171\pi\)
\(702\) 0 0
\(703\) −13675.7 −0.733696
\(704\) 0 0
\(705\) −15811.2 −0.844657
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −16229.0 −0.859652 −0.429826 0.902912i \(-0.641425\pi\)
−0.429826 + 0.902912i \(0.641425\pi\)
\(710\) 0 0
\(711\) −1973.84 −0.104114
\(712\) 0 0
\(713\) 18319.7 0.962239
\(714\) 0 0
\(715\) 2888.65 0.151090
\(716\) 0 0
\(717\) −11455.7 −0.596683
\(718\) 0 0
\(719\) 27233.9 1.41259 0.706294 0.707918i \(-0.250365\pi\)
0.706294 + 0.707918i \(0.250365\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 3943.61 0.202855
\(724\) 0 0
\(725\) −37118.8 −1.90146
\(726\) 0 0
\(727\) 27653.4 1.41074 0.705370 0.708840i \(-0.250781\pi\)
0.705370 + 0.708840i \(0.250781\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −3377.42 −0.170887
\(732\) 0 0
\(733\) −20513.2 −1.03366 −0.516831 0.856088i \(-0.672888\pi\)
−0.516831 + 0.856088i \(0.672888\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 4216.27 0.210730
\(738\) 0 0
\(739\) −5032.87 −0.250524 −0.125262 0.992124i \(-0.539977\pi\)
−0.125262 + 0.992124i \(0.539977\pi\)
\(740\) 0 0
\(741\) −5137.45 −0.254695
\(742\) 0 0
\(743\) −6528.88 −0.322371 −0.161185 0.986924i \(-0.551532\pi\)
−0.161185 + 0.986924i \(0.551532\pi\)
\(744\) 0 0
\(745\) 56438.5 2.77550
\(746\) 0 0
\(747\) 13025.5 0.637988
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 11101.7 0.539423 0.269711 0.962941i \(-0.413072\pi\)
0.269711 + 0.962941i \(0.413072\pi\)
\(752\) 0 0
\(753\) 14489.0 0.701206
\(754\) 0 0
\(755\) 23392.7 1.12761
\(756\) 0 0
\(757\) −6375.85 −0.306122 −0.153061 0.988217i \(-0.548913\pi\)
−0.153061 + 0.988217i \(0.548913\pi\)
\(758\) 0 0
\(759\) 3275.39 0.156639
\(760\) 0 0
\(761\) −4038.93 −0.192393 −0.0961965 0.995362i \(-0.530668\pi\)
−0.0961965 + 0.995362i \(0.530668\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 1622.05 0.0766608
\(766\) 0 0
\(767\) 713.662 0.0335969
\(768\) 0 0
\(769\) 18864.4 0.884614 0.442307 0.896864i \(-0.354160\pi\)
0.442307 + 0.896864i \(0.354160\pi\)
\(770\) 0 0
\(771\) 4439.78 0.207386
\(772\) 0 0
\(773\) 10489.2 0.488060 0.244030 0.969768i \(-0.421530\pi\)
0.244030 + 0.969768i \(0.421530\pi\)
\(774\) 0 0
\(775\) 31155.2 1.44404
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −36638.6 −1.68513
\(780\) 0 0
\(781\) 3447.98 0.157975
\(782\) 0 0
\(783\) 3300.16 0.150623
\(784\) 0 0
\(785\) −14901.0 −0.677502
\(786\) 0 0
\(787\) −30948.6 −1.40178 −0.700888 0.713272i \(-0.747213\pi\)
−0.700888 + 0.713272i \(0.747213\pi\)
\(788\) 0 0
\(789\) 3573.04 0.161221
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 13406.1 0.600335
\(794\) 0 0
\(795\) −37903.8 −1.69095
\(796\) 0 0
\(797\) −38324.5 −1.70329 −0.851646 0.524117i \(-0.824395\pi\)
−0.851646 + 0.524117i \(0.824395\pi\)
\(798\) 0 0
\(799\) 2215.78 0.0981087
\(800\) 0 0
\(801\) 9167.13 0.404375
\(802\) 0 0
\(803\) −1683.31 −0.0739758
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −1676.31 −0.0731215
\(808\) 0 0
\(809\) 19368.1 0.841714 0.420857 0.907127i \(-0.361729\pi\)
0.420857 + 0.907127i \(0.361729\pi\)
\(810\) 0 0
\(811\) −5621.93 −0.243419 −0.121710 0.992566i \(-0.538838\pi\)
−0.121710 + 0.992566i \(0.538838\pi\)
\(812\) 0 0
\(813\) −17976.9 −0.775496
\(814\) 0 0
\(815\) 51125.1 2.19734
\(816\) 0 0
\(817\) 29118.2 1.24690
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 12512.3 0.531891 0.265945 0.963988i \(-0.414316\pi\)
0.265945 + 0.963988i \(0.414316\pi\)
\(822\) 0 0
\(823\) 26084.1 1.10478 0.552391 0.833585i \(-0.313716\pi\)
0.552391 + 0.833585i \(0.313716\pi\)
\(824\) 0 0
\(825\) 5570.27 0.235069
\(826\) 0 0
\(827\) −22210.4 −0.933895 −0.466947 0.884285i \(-0.654646\pi\)
−0.466947 + 0.884285i \(0.654646\pi\)
\(828\) 0 0
\(829\) 20945.2 0.877511 0.438755 0.898607i \(-0.355419\pi\)
0.438755 + 0.898607i \(0.355419\pi\)
\(830\) 0 0
\(831\) −23652.1 −0.987343
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 50085.4 2.07578
\(836\) 0 0
\(837\) −2769.95 −0.114389
\(838\) 0 0
\(839\) 1357.80 0.0558718 0.0279359 0.999610i \(-0.491107\pi\)
0.0279359 + 0.999610i \(0.491107\pi\)
\(840\) 0 0
\(841\) −9449.27 −0.387440
\(842\) 0 0
\(843\) 227.477 0.00929385
\(844\) 0 0
\(845\) −34707.3 −1.41298
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −2410.11 −0.0974259
\(850\) 0 0
\(851\) 32540.6 1.31078
\(852\) 0 0
\(853\) −24726.4 −0.992516 −0.496258 0.868175i \(-0.665293\pi\)
−0.496258 + 0.868175i \(0.665293\pi\)
\(854\) 0 0
\(855\) −13984.4 −0.559365
\(856\) 0 0
\(857\) 20843.2 0.830792 0.415396 0.909641i \(-0.363643\pi\)
0.415396 + 0.909641i \(0.363643\pi\)
\(858\) 0 0
\(859\) −46160.3 −1.83349 −0.916745 0.399473i \(-0.869193\pi\)
−0.916745 + 0.399473i \(0.869193\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 8635.84 0.340634 0.170317 0.985389i \(-0.445521\pi\)
0.170317 + 0.985389i \(0.445521\pi\)
\(864\) 0 0
\(865\) 73691.4 2.89663
\(866\) 0 0
\(867\) 14511.7 0.568446
\(868\) 0 0
\(869\) 1340.92 0.0523446
\(870\) 0 0
\(871\) 15735.8 0.612155
\(872\) 0 0
\(873\) 7132.47 0.276515
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −23282.3 −0.896453 −0.448226 0.893920i \(-0.647944\pi\)
−0.448226 + 0.893920i \(0.647944\pi\)
\(878\) 0 0
\(879\) 7878.79 0.302326
\(880\) 0 0
\(881\) 7097.62 0.271424 0.135712 0.990748i \(-0.456668\pi\)
0.135712 + 0.990748i \(0.456668\pi\)
\(882\) 0 0
\(883\) −35486.1 −1.35244 −0.676219 0.736700i \(-0.736383\pi\)
−0.676219 + 0.736700i \(0.736383\pi\)
\(884\) 0 0
\(885\) 1942.63 0.0737862
\(886\) 0 0
\(887\) −45062.5 −1.70581 −0.852903 0.522070i \(-0.825160\pi\)
−0.852903 + 0.522070i \(0.825160\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −495.242 −0.0186209
\(892\) 0 0
\(893\) −19103.2 −0.715863
\(894\) 0 0
\(895\) 80006.2 2.98806
\(896\) 0 0
\(897\) 12224.3 0.455025
\(898\) 0 0
\(899\) −12539.5 −0.465200
\(900\) 0 0
\(901\) 5311.85 0.196408
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 84898.7 3.11837
\(906\) 0 0
\(907\) 10899.4 0.399018 0.199509 0.979896i \(-0.436065\pi\)
0.199509 + 0.979896i \(0.436065\pi\)
\(908\) 0 0
\(909\) 4586.92 0.167369
\(910\) 0 0
\(911\) −10217.7 −0.371600 −0.185800 0.982588i \(-0.559488\pi\)
−0.185800 + 0.982588i \(0.559488\pi\)
\(912\) 0 0
\(913\) −8848.78 −0.320758
\(914\) 0 0
\(915\) 36492.2 1.31847
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 4396.64 0.157815 0.0789074 0.996882i \(-0.474857\pi\)
0.0789074 + 0.996882i \(0.474857\pi\)
\(920\) 0 0
\(921\) 14597.5 0.522263
\(922\) 0 0
\(923\) 12868.4 0.458905
\(924\) 0 0
\(925\) 55339.9 1.96710
\(926\) 0 0
\(927\) −1957.41 −0.0693525
\(928\) 0 0
\(929\) −42006.3 −1.48351 −0.741756 0.670670i \(-0.766007\pi\)
−0.741756 + 0.670670i \(0.766007\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −18458.9 −0.647714
\(934\) 0 0
\(935\) −1101.93 −0.0385423
\(936\) 0 0
\(937\) 39752.5 1.38597 0.692987 0.720950i \(-0.256294\pi\)
0.692987 + 0.720950i \(0.256294\pi\)
\(938\) 0 0
\(939\) −23678.3 −0.822911
\(940\) 0 0
\(941\) −13609.1 −0.471460 −0.235730 0.971819i \(-0.575748\pi\)
−0.235730 + 0.971819i \(0.575748\pi\)
\(942\) 0 0
\(943\) 87179.6 3.01056
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 30929.7 1.06133 0.530665 0.847581i \(-0.321942\pi\)
0.530665 + 0.847581i \(0.321942\pi\)
\(948\) 0 0
\(949\) −6282.37 −0.214894
\(950\) 0 0
\(951\) 11066.3 0.377337
\(952\) 0 0
\(953\) −26166.5 −0.889418 −0.444709 0.895675i \(-0.646693\pi\)
−0.444709 + 0.895675i \(0.646693\pi\)
\(954\) 0 0
\(955\) 23192.7 0.785863
\(956\) 0 0
\(957\) −2241.95 −0.0757281
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −19266.2 −0.646711
\(962\) 0 0
\(963\) −17075.8 −0.571401
\(964\) 0 0
\(965\) −29368.4 −0.979693
\(966\) 0 0
\(967\) −55883.6 −1.85842 −0.929212 0.369546i \(-0.879513\pi\)
−0.929212 + 0.369546i \(0.879513\pi\)
\(968\) 0 0
\(969\) 1959.78 0.0649715
\(970\) 0 0
\(971\) 10618.3 0.350933 0.175467 0.984485i \(-0.443857\pi\)
0.175467 + 0.984485i \(0.443857\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 20789.2 0.682857
\(976\) 0 0
\(977\) 11493.1 0.376352 0.188176 0.982135i \(-0.439743\pi\)
0.188176 + 0.982135i \(0.439743\pi\)
\(978\) 0 0
\(979\) −6227.64 −0.203306
\(980\) 0 0
\(981\) 7654.11 0.249110
\(982\) 0 0
\(983\) 25414.1 0.824602 0.412301 0.911048i \(-0.364725\pi\)
0.412301 + 0.911048i \(0.364725\pi\)
\(984\) 0 0
\(985\) −95469.5 −3.08824
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −69285.4 −2.22765
\(990\) 0 0
\(991\) −39966.1 −1.28109 −0.640547 0.767919i \(-0.721292\pi\)
−0.640547 + 0.767919i \(0.721292\pi\)
\(992\) 0 0
\(993\) 5945.40 0.190002
\(994\) 0 0
\(995\) 104555. 3.33128
\(996\) 0 0
\(997\) −16507.7 −0.524376 −0.262188 0.965017i \(-0.584444\pi\)
−0.262188 + 0.965017i \(0.584444\pi\)
\(998\) 0 0
\(999\) −4920.16 −0.155823
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 1176.4.a.s.1.2 2
4.3 odd 2 2352.4.a.ce.1.2 2
7.6 odd 2 1176.4.a.t.1.1 yes 2
28.27 even 2 2352.4.a.bm.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1176.4.a.s.1.2 2 1.1 even 1 trivial
1176.4.a.t.1.1 yes 2 7.6 odd 2
2352.4.a.bm.1.1 2 28.27 even 2
2352.4.a.ce.1.2 2 4.3 odd 2