Properties

Label 144.4.c.b.143.4
Level $144$
Weight $4$
Character 144.143
Analytic conductor $8.496$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [144,4,Mod(143,144)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(144, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("144.143");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 144 = 2^{4} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 144.c (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(8.49627504083\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 143.4
Root \(-1.22474 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 144.143
Dual form 144.4.c.b.143.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+4.24264i q^{5} +13.8564i q^{7} +O(q^{10})\) \(q+4.24264i q^{5} +13.8564i q^{7} -58.7878 q^{11} +28.0000 q^{13} +97.5807i q^{17} +110.851i q^{19} -176.363 q^{23} +107.000 q^{25} -4.24264i q^{29} +124.708i q^{31} -58.7878 q^{35} -118.000 q^{37} +190.919i q^{41} -249.415i q^{43} +293.939 q^{47} +151.000 q^{49} -700.036i q^{53} -249.415i q^{55} +587.878 q^{59} -802.000 q^{61} +118.794i q^{65} -138.564i q^{67} +646.665 q^{71} +496.000 q^{73} -814.587i q^{77} -1011.52i q^{79} +58.7878 q^{83} -414.000 q^{85} +1124.30i q^{89} +387.979i q^{91} -470.302 q^{95} -152.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 112 q^{13} + 428 q^{25} - 472 q^{37} + 604 q^{49} - 3208 q^{61} + 1984 q^{73} - 1656 q^{85} - 608 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/144\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(65\) \(127\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 4.24264i 0.379473i 0.981835 + 0.189737i \(0.0607634\pi\)
−0.981835 + 0.189737i \(0.939237\pi\)
\(6\) 0 0
\(7\) 13.8564i 0.748176i 0.927393 + 0.374088i \(0.122044\pi\)
−0.927393 + 0.374088i \(0.877956\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −58.7878 −1.61138 −0.805690 0.592338i \(-0.798205\pi\)
−0.805690 + 0.592338i \(0.798205\pi\)
\(12\) 0 0
\(13\) 28.0000 0.597369 0.298685 0.954352i \(-0.403452\pi\)
0.298685 + 0.954352i \(0.403452\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 97.5807i 1.39216i 0.717962 + 0.696082i \(0.245075\pi\)
−0.717962 + 0.696082i \(0.754925\pi\)
\(18\) 0 0
\(19\) 110.851i 1.33847i 0.743049 + 0.669237i \(0.233379\pi\)
−0.743049 + 0.669237i \(0.766621\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −176.363 −1.59888 −0.799441 0.600745i \(-0.794871\pi\)
−0.799441 + 0.600745i \(0.794871\pi\)
\(24\) 0 0
\(25\) 107.000 0.856000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 4.24264i − 0.0271668i −0.999908 0.0135834i \(-0.995676\pi\)
0.999908 0.0135834i \(-0.00432387\pi\)
\(30\) 0 0
\(31\) 124.708i 0.722521i 0.932465 + 0.361261i \(0.117654\pi\)
−0.932465 + 0.361261i \(0.882346\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −58.7878 −0.283913
\(36\) 0 0
\(37\) −118.000 −0.524299 −0.262150 0.965027i \(-0.584431\pi\)
−0.262150 + 0.965027i \(0.584431\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 190.919i 0.727232i 0.931549 + 0.363616i \(0.118458\pi\)
−0.931549 + 0.363616i \(0.881542\pi\)
\(42\) 0 0
\(43\) − 249.415i − 0.884546i −0.896880 0.442273i \(-0.854172\pi\)
0.896880 0.442273i \(-0.145828\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 293.939 0.912242 0.456121 0.889918i \(-0.349238\pi\)
0.456121 + 0.889918i \(0.349238\pi\)
\(48\) 0 0
\(49\) 151.000 0.440233
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 700.036i − 1.81429i −0.420820 0.907144i \(-0.638257\pi\)
0.420820 0.907144i \(-0.361743\pi\)
\(54\) 0 0
\(55\) − 249.415i − 0.611476i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 587.878 1.29721 0.648603 0.761127i \(-0.275354\pi\)
0.648603 + 0.761127i \(0.275354\pi\)
\(60\) 0 0
\(61\) −802.000 −1.68337 −0.841685 0.539969i \(-0.818436\pi\)
−0.841685 + 0.539969i \(0.818436\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 118.794i 0.226686i
\(66\) 0 0
\(67\) − 138.564i − 0.252661i −0.991988 0.126331i \(-0.959680\pi\)
0.991988 0.126331i \(-0.0403200\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 646.665 1.08092 0.540458 0.841371i \(-0.318251\pi\)
0.540458 + 0.841371i \(0.318251\pi\)
\(72\) 0 0
\(73\) 496.000 0.795238 0.397619 0.917551i \(-0.369837\pi\)
0.397619 + 0.917551i \(0.369837\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 814.587i − 1.20559i
\(78\) 0 0
\(79\) − 1011.52i − 1.44056i −0.693681 0.720282i \(-0.744012\pi\)
0.693681 0.720282i \(-0.255988\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 58.7878 0.0777445 0.0388723 0.999244i \(-0.487623\pi\)
0.0388723 + 0.999244i \(0.487623\pi\)
\(84\) 0 0
\(85\) −414.000 −0.528289
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 1124.30i 1.33905i 0.742789 + 0.669525i \(0.233503\pi\)
−0.742789 + 0.669525i \(0.766497\pi\)
\(90\) 0 0
\(91\) 387.979i 0.446937i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −470.302 −0.507915
\(96\) 0 0
\(97\) −152.000 −0.159106 −0.0795529 0.996831i \(-0.525349\pi\)
−0.0795529 + 0.996831i \(0.525349\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) − 929.138i − 0.915373i −0.889114 0.457687i \(-0.848678\pi\)
0.889114 0.457687i \(-0.151322\pi\)
\(102\) 0 0
\(103\) 1759.76i 1.68344i 0.539912 + 0.841722i \(0.318458\pi\)
−0.539912 + 0.841722i \(0.681542\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −940.604 −0.849828 −0.424914 0.905234i \(-0.639696\pi\)
−0.424914 + 0.905234i \(0.639696\pi\)
\(108\) 0 0
\(109\) 628.000 0.551849 0.275924 0.961179i \(-0.411016\pi\)
0.275924 + 0.961179i \(0.411016\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 1243.09i 1.03487i 0.855722 + 0.517435i \(0.173113\pi\)
−0.855722 + 0.517435i \(0.826887\pi\)
\(114\) 0 0
\(115\) − 748.246i − 0.606733i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −1352.12 −1.04158
\(120\) 0 0
\(121\) 2125.00 1.59654
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 984.293i 0.704302i
\(126\) 0 0
\(127\) 1621.20i 1.13274i 0.824151 + 0.566371i \(0.191653\pi\)
−0.824151 + 0.566371i \(0.808347\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −823.029 −0.548919 −0.274459 0.961599i \(-0.588499\pi\)
−0.274459 + 0.961599i \(0.588499\pi\)
\(132\) 0 0
\(133\) −1536.00 −1.00141
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 1607.96i 1.00275i 0.865229 + 0.501377i \(0.167173\pi\)
−0.865229 + 0.501377i \(0.832827\pi\)
\(138\) 0 0
\(139\) 2133.89i 1.30211i 0.759029 + 0.651057i \(0.225674\pi\)
−0.759029 + 0.651057i \(0.774326\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −1646.06 −0.962589
\(144\) 0 0
\(145\) 18.0000 0.0103091
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) − 818.830i − 0.450209i −0.974335 0.225104i \(-0.927728\pi\)
0.974335 0.225104i \(-0.0722723\pi\)
\(150\) 0 0
\(151\) − 3256.26i − 1.75490i −0.479666 0.877451i \(-0.659242\pi\)
0.479666 0.877451i \(-0.340758\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −529.090 −0.274178
\(156\) 0 0
\(157\) 1730.00 0.879421 0.439710 0.898140i \(-0.355081\pi\)
0.439710 + 0.898140i \(0.355081\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) − 2443.76i − 1.19624i
\(162\) 0 0
\(163\) 1635.06i 0.785690i 0.919605 + 0.392845i \(0.128509\pi\)
−0.919605 + 0.392845i \(0.871491\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 1998.78 0.926171 0.463085 0.886314i \(-0.346742\pi\)
0.463085 + 0.886314i \(0.346742\pi\)
\(168\) 0 0
\(169\) −1413.00 −0.643150
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 2159.50i 0.949041i 0.880245 + 0.474520i \(0.157378\pi\)
−0.880245 + 0.474520i \(0.842622\pi\)
\(174\) 0 0
\(175\) 1482.64i 0.640438i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 2351.51 0.981900 0.490950 0.871188i \(-0.336650\pi\)
0.490950 + 0.871188i \(0.336650\pi\)
\(180\) 0 0
\(181\) −620.000 −0.254609 −0.127305 0.991864i \(-0.540633\pi\)
−0.127305 + 0.991864i \(0.540633\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) − 500.632i − 0.198958i
\(186\) 0 0
\(187\) − 5736.55i − 2.24331i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 3644.84 1.38079 0.690396 0.723431i \(-0.257436\pi\)
0.690396 + 0.723431i \(0.257436\pi\)
\(192\) 0 0
\(193\) −946.000 −0.352822 −0.176411 0.984317i \(-0.556449\pi\)
−0.176411 + 0.984317i \(0.556449\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 1913.43i − 0.692012i −0.938232 0.346006i \(-0.887538\pi\)
0.938232 0.346006i \(-0.112462\pi\)
\(198\) 0 0
\(199\) − 734.390i − 0.261605i −0.991408 0.130803i \(-0.958245\pi\)
0.991408 0.130803i \(-0.0417555\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 58.7878 0.0203256
\(204\) 0 0
\(205\) −810.000 −0.275965
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) − 6516.70i − 2.15679i
\(210\) 0 0
\(211\) − 360.267i − 0.117544i −0.998271 0.0587720i \(-0.981282\pi\)
0.998271 0.0587720i \(-0.0187185\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 1058.18 0.335662
\(216\) 0 0
\(217\) −1728.00 −0.540573
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 2732.26i 0.831637i
\(222\) 0 0
\(223\) − 3865.94i − 1.16091i −0.814293 0.580454i \(-0.802875\pi\)
0.814293 0.580454i \(-0.197125\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 2763.02 0.807878 0.403939 0.914786i \(-0.367641\pi\)
0.403939 + 0.914786i \(0.367641\pi\)
\(228\) 0 0
\(229\) 700.000 0.201997 0.100998 0.994887i \(-0.467796\pi\)
0.100998 + 0.994887i \(0.467796\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 1964.34i 0.552311i 0.961113 + 0.276155i \(0.0890604\pi\)
−0.961113 + 0.276155i \(0.910940\pi\)
\(234\) 0 0
\(235\) 1247.08i 0.346172i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −235.151 −0.0636429 −0.0318215 0.999494i \(-0.510131\pi\)
−0.0318215 + 0.999494i \(0.510131\pi\)
\(240\) 0 0
\(241\) −6488.00 −1.73414 −0.867072 0.498182i \(-0.834001\pi\)
−0.867072 + 0.498182i \(0.834001\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 640.639i 0.167057i
\(246\) 0 0
\(247\) 3103.84i 0.799564i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 3586.05 0.901791 0.450896 0.892577i \(-0.351105\pi\)
0.450896 + 0.892577i \(0.351105\pi\)
\(252\) 0 0
\(253\) 10368.0 2.57641
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 5316.03i − 1.29029i −0.764060 0.645145i \(-0.776797\pi\)
0.764060 0.645145i \(-0.223203\pi\)
\(258\) 0 0
\(259\) − 1635.06i − 0.392268i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −5996.35 −1.40590 −0.702948 0.711241i \(-0.748134\pi\)
−0.702948 + 0.711241i \(0.748134\pi\)
\(264\) 0 0
\(265\) 2970.00 0.688474
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 6520.94i 1.47802i 0.673692 + 0.739012i \(0.264708\pi\)
−0.673692 + 0.739012i \(0.735292\pi\)
\(270\) 0 0
\(271\) 734.390i 0.164616i 0.996607 + 0.0823081i \(0.0262292\pi\)
−0.996607 + 0.0823081i \(0.973771\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −6290.29 −1.37934
\(276\) 0 0
\(277\) −4124.00 −0.894538 −0.447269 0.894399i \(-0.647603\pi\)
−0.447269 + 0.894399i \(0.647603\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 1294.01i 0.274711i 0.990522 + 0.137356i \(0.0438603\pi\)
−0.990522 + 0.137356i \(0.956140\pi\)
\(282\) 0 0
\(283\) 7482.46i 1.57168i 0.618428 + 0.785841i \(0.287770\pi\)
−0.618428 + 0.785841i \(0.712230\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −2645.45 −0.544097
\(288\) 0 0
\(289\) −4609.00 −0.938123
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 3542.60i 0.706352i 0.935557 + 0.353176i \(0.114898\pi\)
−0.935557 + 0.353176i \(0.885102\pi\)
\(294\) 0 0
\(295\) 2494.15i 0.492255i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −4938.17 −0.955123
\(300\) 0 0
\(301\) 3456.00 0.661796
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) − 3402.60i − 0.638794i
\(306\) 0 0
\(307\) 249.415i 0.0463677i 0.999731 + 0.0231839i \(0.00738031\pi\)
−0.999731 + 0.0231839i \(0.992620\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 2233.93 0.407315 0.203657 0.979042i \(-0.434717\pi\)
0.203657 + 0.979042i \(0.434717\pi\)
\(312\) 0 0
\(313\) 7946.00 1.43493 0.717467 0.696592i \(-0.245301\pi\)
0.717467 + 0.696592i \(0.245301\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 8964.70i 1.58835i 0.607688 + 0.794176i \(0.292097\pi\)
−0.607688 + 0.794176i \(0.707903\pi\)
\(318\) 0 0
\(319\) 249.415i 0.0437761i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −10816.9 −1.86338
\(324\) 0 0
\(325\) 2996.00 0.511348
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 4072.94i 0.682517i
\(330\) 0 0
\(331\) − 1856.76i − 0.308328i −0.988045 0.154164i \(-0.950732\pi\)
0.988045 0.154164i \(-0.0492685\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 587.878 0.0958782
\(336\) 0 0
\(337\) −1952.00 −0.315526 −0.157763 0.987477i \(-0.550428\pi\)
−0.157763 + 0.987477i \(0.550428\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) − 7331.28i − 1.16426i
\(342\) 0 0
\(343\) 6845.06i 1.07755i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 5819.99 0.900384 0.450192 0.892932i \(-0.351356\pi\)
0.450192 + 0.892932i \(0.351356\pi\)
\(348\) 0 0
\(349\) −226.000 −0.0346633 −0.0173317 0.999850i \(-0.505517\pi\)
−0.0173317 + 0.999850i \(0.505517\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 1845.55i − 0.278268i −0.990274 0.139134i \(-0.955568\pi\)
0.990274 0.139134i \(-0.0444319\pi\)
\(354\) 0 0
\(355\) 2743.57i 0.410179i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 58.7878 0.00864262 0.00432131 0.999991i \(-0.498624\pi\)
0.00432131 + 0.999991i \(0.498624\pi\)
\(360\) 0 0
\(361\) −5429.00 −0.791515
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 2104.35i 0.301772i
\(366\) 0 0
\(367\) − 2480.30i − 0.352780i −0.984320 0.176390i \(-0.943558\pi\)
0.984320 0.176390i \(-0.0564421\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 9699.98 1.35741
\(372\) 0 0
\(373\) −5338.00 −0.740995 −0.370498 0.928833i \(-0.620813\pi\)
−0.370498 + 0.928833i \(0.620813\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 118.794i − 0.0162286i
\(378\) 0 0
\(379\) − 5376.29i − 0.728658i −0.931270 0.364329i \(-0.881298\pi\)
0.931270 0.364329i \(-0.118702\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −6349.08 −0.847057 −0.423528 0.905883i \(-0.639209\pi\)
−0.423528 + 0.905883i \(0.639209\pi\)
\(384\) 0 0
\(385\) 3456.00 0.457491
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) − 11238.8i − 1.46485i −0.680847 0.732426i \(-0.738388\pi\)
0.680847 0.732426i \(-0.261612\pi\)
\(390\) 0 0
\(391\) − 17209.7i − 2.22591i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 4291.51 0.546656
\(396\) 0 0
\(397\) −466.000 −0.0589115 −0.0294558 0.999566i \(-0.509377\pi\)
−0.0294558 + 0.999566i \(0.509377\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) − 7403.41i − 0.921966i −0.887409 0.460983i \(-0.847497\pi\)
0.887409 0.460983i \(-0.152503\pi\)
\(402\) 0 0
\(403\) 3491.81i 0.431612i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 6936.95 0.844845
\(408\) 0 0
\(409\) 4024.00 0.486489 0.243244 0.969965i \(-0.421788\pi\)
0.243244 + 0.969965i \(0.421788\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 8145.87i 0.970538i
\(414\) 0 0
\(415\) 249.415i 0.0295020i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 15108.5 1.76157 0.880783 0.473520i \(-0.157017\pi\)
0.880783 + 0.473520i \(0.157017\pi\)
\(420\) 0 0
\(421\) −8540.00 −0.988632 −0.494316 0.869282i \(-0.664581\pi\)
−0.494316 + 0.869282i \(0.664581\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 10441.1i 1.19169i
\(426\) 0 0
\(427\) − 11112.8i − 1.25946i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 16049.1 1.79363 0.896817 0.442403i \(-0.145874\pi\)
0.896817 + 0.442403i \(0.145874\pi\)
\(432\) 0 0
\(433\) −11266.0 −1.25037 −0.625184 0.780477i \(-0.714976\pi\)
−0.625184 + 0.780477i \(0.714976\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 19550.1i − 2.14006i
\(438\) 0 0
\(439\) 6110.68i 0.664343i 0.943219 + 0.332172i \(0.107781\pi\)
−0.943219 + 0.332172i \(0.892219\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −13580.0 −1.45644 −0.728221 0.685342i \(-0.759653\pi\)
−0.728221 + 0.685342i \(0.759653\pi\)
\(444\) 0 0
\(445\) −4770.00 −0.508134
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 547.301i 0.0575250i 0.999586 + 0.0287625i \(0.00915665\pi\)
−0.999586 + 0.0287625i \(0.990843\pi\)
\(450\) 0 0
\(451\) − 11223.7i − 1.17185i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −1646.06 −0.169601
\(456\) 0 0
\(457\) 15784.0 1.61563 0.807817 0.589434i \(-0.200649\pi\)
0.807817 + 0.589434i \(0.200649\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 18620.9i − 1.88127i −0.339424 0.940634i \(-0.610232\pi\)
0.339424 0.940634i \(-0.389768\pi\)
\(462\) 0 0
\(463\) 8244.56i 0.827554i 0.910378 + 0.413777i \(0.135791\pi\)
−0.910378 + 0.413777i \(0.864209\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 2057.57 0.203882 0.101941 0.994790i \(-0.467495\pi\)
0.101941 + 0.994790i \(0.467495\pi\)
\(468\) 0 0
\(469\) 1920.00 0.189035
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 14662.6i 1.42534i
\(474\) 0 0
\(475\) 11861.1i 1.14573i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 2175.15 0.207484 0.103742 0.994604i \(-0.466918\pi\)
0.103742 + 0.994604i \(0.466918\pi\)
\(480\) 0 0
\(481\) −3304.00 −0.313200
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) − 644.881i − 0.0603764i
\(486\) 0 0
\(487\) − 11708.7i − 1.08947i −0.838609 0.544733i \(-0.816631\pi\)
0.838609 0.544733i \(-0.183369\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −1646.06 −0.151294 −0.0756472 0.997135i \(-0.524102\pi\)
−0.0756472 + 0.997135i \(0.524102\pi\)
\(492\) 0 0
\(493\) 414.000 0.0378207
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 8960.46i 0.808715i
\(498\) 0 0
\(499\) − 2992.98i − 0.268506i −0.990947 0.134253i \(-0.957137\pi\)
0.990947 0.134253i \(-0.0428634\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −9112.10 −0.807731 −0.403865 0.914818i \(-0.632334\pi\)
−0.403865 + 0.914818i \(0.632334\pi\)
\(504\) 0 0
\(505\) 3942.00 0.347360
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 12384.3i 1.07843i 0.842167 + 0.539217i \(0.181280\pi\)
−0.842167 + 0.539217i \(0.818720\pi\)
\(510\) 0 0
\(511\) 6872.78i 0.594978i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −7466.04 −0.638822
\(516\) 0 0
\(517\) −17280.0 −1.46997
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 3882.02i 0.326438i 0.986590 + 0.163219i \(0.0521877\pi\)
−0.986590 + 0.163219i \(0.947812\pi\)
\(522\) 0 0
\(523\) 15574.6i 1.30216i 0.759009 + 0.651080i \(0.225684\pi\)
−0.759009 + 0.651080i \(0.774316\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −12169.1 −1.00587
\(528\) 0 0
\(529\) 18937.0 1.55642
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 5345.73i 0.434426i
\(534\) 0 0
\(535\) − 3990.65i − 0.322487i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −8876.95 −0.709383
\(540\) 0 0
\(541\) 11188.0 0.889112 0.444556 0.895751i \(-0.353361\pi\)
0.444556 + 0.895751i \(0.353361\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 2664.38i 0.209412i
\(546\) 0 0
\(547\) 21588.3i 1.68747i 0.536757 + 0.843737i \(0.319649\pi\)
−0.536757 + 0.843737i \(0.680351\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 470.302 0.0363621
\(552\) 0 0
\(553\) 14016.0 1.07780
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 16346.9i − 1.24352i −0.783208 0.621760i \(-0.786418\pi\)
0.783208 0.621760i \(-0.213582\pi\)
\(558\) 0 0
\(559\) − 6983.63i − 0.528401i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −21692.7 −1.62387 −0.811934 0.583749i \(-0.801585\pi\)
−0.811934 + 0.583749i \(0.801585\pi\)
\(564\) 0 0
\(565\) −5274.00 −0.392706
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 9414.42i 0.693626i 0.937934 + 0.346813i \(0.112736\pi\)
−0.937934 + 0.346813i \(0.887264\pi\)
\(570\) 0 0
\(571\) − 10364.6i − 0.759623i −0.925064 0.379811i \(-0.875989\pi\)
0.925064 0.379811i \(-0.124011\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −18870.9 −1.36864
\(576\) 0 0
\(577\) −9406.00 −0.678643 −0.339321 0.940670i \(-0.610197\pi\)
−0.339321 + 0.940670i \(0.610197\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 814.587i 0.0581665i
\(582\) 0 0
\(583\) 41153.5i 2.92351i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 21986.6 1.54597 0.772985 0.634424i \(-0.218763\pi\)
0.772985 + 0.634424i \(0.218763\pi\)
\(588\) 0 0
\(589\) −13824.0 −0.967076
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 13419.5i 0.929295i 0.885496 + 0.464647i \(0.153819\pi\)
−0.885496 + 0.464647i \(0.846181\pi\)
\(594\) 0 0
\(595\) − 5736.55i − 0.395253i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 6878.17 0.469172 0.234586 0.972095i \(-0.424626\pi\)
0.234586 + 0.972095i \(0.424626\pi\)
\(600\) 0 0
\(601\) 6086.00 0.413067 0.206533 0.978440i \(-0.433782\pi\)
0.206533 + 0.978440i \(0.433782\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 9015.61i 0.605846i
\(606\) 0 0
\(607\) − 7967.43i − 0.532765i −0.963867 0.266382i \(-0.914172\pi\)
0.963867 0.266382i \(-0.0858284\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 8230.29 0.544946
\(612\) 0 0
\(613\) 20438.0 1.34663 0.673314 0.739357i \(-0.264870\pi\)
0.673314 + 0.739357i \(0.264870\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 3568.06i 0.232812i 0.993202 + 0.116406i \(0.0371373\pi\)
−0.993202 + 0.116406i \(0.962863\pi\)
\(618\) 0 0
\(619\) − 1884.47i − 0.122364i −0.998127 0.0611820i \(-0.980513\pi\)
0.998127 0.0611820i \(-0.0194870\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −15578.8 −1.00185
\(624\) 0 0
\(625\) 9199.00 0.588736
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 11514.5i − 0.729911i
\(630\) 0 0
\(631\) − 9491.64i − 0.598821i −0.954124 0.299411i \(-0.903210\pi\)
0.954124 0.299411i \(-0.0967900\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −6878.17 −0.429845
\(636\) 0 0
\(637\) 4228.00 0.262982
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) − 16363.9i − 1.00832i −0.863610 0.504161i \(-0.831802\pi\)
0.863610 0.504161i \(-0.168198\pi\)
\(642\) 0 0
\(643\) − 24193.3i − 1.48381i −0.670505 0.741905i \(-0.733922\pi\)
0.670505 0.741905i \(-0.266078\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 14520.6 0.882323 0.441161 0.897428i \(-0.354567\pi\)
0.441161 + 0.897428i \(0.354567\pi\)
\(648\) 0 0
\(649\) −34560.0 −2.09029
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 4187.49i − 0.250948i −0.992097 0.125474i \(-0.959955\pi\)
0.992097 0.125474i \(-0.0400452\pi\)
\(654\) 0 0
\(655\) − 3491.81i − 0.208300i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −4115.14 −0.243252 −0.121626 0.992576i \(-0.538811\pi\)
−0.121626 + 0.992576i \(0.538811\pi\)
\(660\) 0 0
\(661\) −22438.0 −1.32033 −0.660164 0.751121i \(-0.729513\pi\)
−0.660164 + 0.751121i \(0.729513\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) − 6516.70i − 0.380010i
\(666\) 0 0
\(667\) 748.246i 0.0434366i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 47147.8 2.71255
\(672\) 0 0
\(673\) 18962.0 1.08608 0.543040 0.839707i \(-0.317273\pi\)
0.543040 + 0.839707i \(0.317273\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 2278.30i 0.129338i 0.997907 + 0.0646692i \(0.0205992\pi\)
−0.997907 + 0.0646692i \(0.979401\pi\)
\(678\) 0 0
\(679\) − 2106.17i − 0.119039i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 11581.2 0.648817 0.324408 0.945917i \(-0.394835\pi\)
0.324408 + 0.945917i \(0.394835\pi\)
\(684\) 0 0
\(685\) −6822.00 −0.380519
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) − 19601.0i − 1.08380i
\(690\) 0 0
\(691\) 11722.5i 0.645363i 0.946508 + 0.322681i \(0.104584\pi\)
−0.946508 + 0.322681i \(0.895416\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −9053.31 −0.494118
\(696\) 0 0
\(697\) −18630.0 −1.01243
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 9601.10i 0.517302i 0.965971 + 0.258651i \(0.0832779\pi\)
−0.965971 + 0.258651i \(0.916722\pi\)
\(702\) 0 0
\(703\) − 13080.4i − 0.701762i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 12874.5 0.684860
\(708\) 0 0
\(709\) 14572.0 0.771880 0.385940 0.922524i \(-0.373877\pi\)
0.385940 + 0.922524i \(0.373877\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) − 21993.8i − 1.15523i
\(714\) 0 0
\(715\) − 6983.63i − 0.365277i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 9817.55 0.509225 0.254613 0.967043i \(-0.418052\pi\)
0.254613 + 0.967043i \(0.418052\pi\)
\(720\) 0 0
\(721\) −24384.0 −1.25951
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) − 453.963i − 0.0232548i
\(726\) 0 0
\(727\) − 33047.5i − 1.68592i −0.537975 0.842961i \(-0.680811\pi\)
0.537975 0.842961i \(-0.319189\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 24338.1 1.23143
\(732\) 0 0
\(733\) 28636.0 1.44297 0.721483 0.692432i \(-0.243461\pi\)
0.721483 + 0.692432i \(0.243461\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 8145.87i 0.407133i
\(738\) 0 0
\(739\) 1496.49i 0.0744917i 0.999306 + 0.0372458i \(0.0118585\pi\)
−0.999306 + 0.0372458i \(0.988142\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −13050.9 −0.644402 −0.322201 0.946671i \(-0.604423\pi\)
−0.322201 + 0.946671i \(0.604423\pi\)
\(744\) 0 0
\(745\) 3474.00 0.170842
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) − 13033.4i − 0.635821i
\(750\) 0 0
\(751\) − 8244.56i − 0.400597i −0.979735 0.200298i \(-0.935809\pi\)
0.979735 0.200298i \(-0.0641912\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 13815.1 0.665939
\(756\) 0 0
\(757\) 17740.0 0.851745 0.425873 0.904783i \(-0.359967\pi\)
0.425873 + 0.904783i \(0.359967\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) − 17254.8i − 0.821927i −0.911652 0.410964i \(-0.865192\pi\)
0.911652 0.410964i \(-0.134808\pi\)
\(762\) 0 0
\(763\) 8701.82i 0.412880i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 16460.6 0.774911
\(768\) 0 0
\(769\) −17038.0 −0.798967 −0.399484 0.916740i \(-0.630811\pi\)
−0.399484 + 0.916740i \(0.630811\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 38739.6i 1.80254i 0.433256 + 0.901271i \(0.357365\pi\)
−0.433256 + 0.901271i \(0.642635\pi\)
\(774\) 0 0
\(775\) 13343.7i 0.618478i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −21163.6 −0.973382
\(780\) 0 0
\(781\) −38016.0 −1.74177
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 7339.77i 0.333717i
\(786\) 0 0
\(787\) 3103.84i 0.140584i 0.997526 + 0.0702921i \(0.0223931\pi\)
−0.997526 + 0.0702921i \(0.977607\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −17224.8 −0.774265
\(792\) 0 0
\(793\) −22456.0 −1.00559
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 1098.84i − 0.0488370i −0.999702 0.0244185i \(-0.992227\pi\)
0.999702 0.0244185i \(-0.00777341\pi\)
\(798\) 0 0
\(799\) 28682.8i 1.26999i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −29158.7 −1.28143
\(804\) 0 0
\(805\) 10368.0 0.453943
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) − 38739.6i − 1.68357i −0.539811 0.841786i \(-0.681504\pi\)
0.539811 0.841786i \(-0.318496\pi\)
\(810\) 0 0
\(811\) − 34419.3i − 1.49029i −0.666902 0.745145i \(-0.732380\pi\)
0.666902 0.745145i \(-0.267620\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −6936.95 −0.298148
\(816\) 0 0
\(817\) 27648.0 1.18394
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 35387.9i 1.50432i 0.658982 + 0.752159i \(0.270987\pi\)
−0.658982 + 0.752159i \(0.729013\pi\)
\(822\) 0 0
\(823\) 11708.7i 0.495915i 0.968771 + 0.247958i \(0.0797594\pi\)
−0.968771 + 0.247958i \(0.920241\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −31392.7 −1.31999 −0.659994 0.751271i \(-0.729441\pi\)
−0.659994 + 0.751271i \(0.729441\pi\)
\(828\) 0 0
\(829\) 43444.0 1.82011 0.910056 0.414486i \(-0.136039\pi\)
0.910056 + 0.414486i \(0.136039\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 14734.7i 0.612877i
\(834\) 0 0
\(835\) 8480.12i 0.351457i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 15343.6 0.631371 0.315685 0.948864i \(-0.397766\pi\)
0.315685 + 0.948864i \(0.397766\pi\)
\(840\) 0 0
\(841\) 24371.0 0.999262
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) − 5994.85i − 0.244058i
\(846\) 0 0
\(847\) 29444.9i 1.19450i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 20810.9 0.838293
\(852\) 0 0
\(853\) −4870.00 −0.195481 −0.0977407 0.995212i \(-0.531162\pi\)
−0.0977407 + 0.995212i \(0.531162\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 9414.42i − 0.375251i −0.982241 0.187626i \(-0.939921\pi\)
0.982241 0.187626i \(-0.0600792\pi\)
\(858\) 0 0
\(859\) 16600.0i 0.659353i 0.944094 + 0.329676i \(0.106940\pi\)
−0.944094 + 0.329676i \(0.893060\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −36095.7 −1.42377 −0.711884 0.702297i \(-0.752158\pi\)
−0.711884 + 0.702297i \(0.752158\pi\)
\(864\) 0 0
\(865\) −9162.00 −0.360136
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 59464.9i 2.32130i
\(870\) 0 0
\(871\) − 3879.79i − 0.150932i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −13638.8 −0.526942
\(876\) 0 0
\(877\) −19954.0 −0.768300 −0.384150 0.923271i \(-0.625505\pi\)
−0.384150 + 0.923271i \(0.625505\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) − 8744.08i − 0.334388i −0.985924 0.167194i \(-0.946529\pi\)
0.985924 0.167194i \(-0.0534706\pi\)
\(882\) 0 0
\(883\) 30068.4i 1.14596i 0.819570 + 0.572980i \(0.194213\pi\)
−0.819570 + 0.572980i \(0.805787\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −28100.5 −1.06372 −0.531862 0.846831i \(-0.678508\pi\)
−0.531862 + 0.846831i \(0.678508\pi\)
\(888\) 0 0
\(889\) −22464.0 −0.847490
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 32583.5i 1.22101i
\(894\) 0 0
\(895\) 9976.61i 0.372605i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 529.090 0.0196286
\(900\) 0 0
\(901\) 68310.0 2.52579
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) − 2630.44i − 0.0966173i
\(906\) 0 0
\(907\) − 1136.23i − 0.0415962i −0.999784 0.0207981i \(-0.993379\pi\)
0.999784 0.0207981i \(-0.00662072\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −39387.8 −1.43247 −0.716233 0.697862i \(-0.754135\pi\)
−0.716233 + 0.697862i \(0.754135\pi\)
\(912\) 0 0
\(913\) −3456.00 −0.125276
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 11404.2i − 0.410688i
\(918\) 0 0
\(919\) − 5861.26i − 0.210386i −0.994452 0.105193i \(-0.966454\pi\)
0.994452 0.105193i \(-0.0335461\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 18106.6 0.645706
\(924\) 0 0
\(925\) −12626.0 −0.448800
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 6614.28i 0.233592i 0.993156 + 0.116796i \(0.0372624\pi\)
−0.993156 + 0.116796i \(0.962738\pi\)
\(930\) 0 0
\(931\) 16738.5i 0.589241i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 24338.1 0.851275
\(936\) 0 0
\(937\) −21610.0 −0.753434 −0.376717 0.926328i \(-0.622947\pi\)
−0.376717 + 0.926328i \(0.622947\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) − 50865.0i − 1.76212i −0.473007 0.881059i \(-0.656832\pi\)
0.473007 0.881059i \(-0.343168\pi\)
\(942\) 0 0
\(943\) − 33671.1i − 1.16276i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −22927.2 −0.786731 −0.393366 0.919382i \(-0.628689\pi\)
−0.393366 + 0.919382i \(0.628689\pi\)
\(948\) 0 0
\(949\) 13888.0 0.475051
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) − 8981.67i − 0.305294i −0.988281 0.152647i \(-0.951220\pi\)
0.988281 0.152647i \(-0.0487797\pi\)
\(954\) 0 0
\(955\) 15463.7i 0.523974i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −22280.6 −0.750236
\(960\) 0 0
\(961\) 14239.0 0.477963
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 4013.54i − 0.133886i
\(966\) 0 0
\(967\) 983.805i 0.0327167i 0.999866 + 0.0163583i \(0.00520725\pi\)
−0.999866 + 0.0163583i \(0.994793\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −12639.4 −0.417731 −0.208865 0.977944i \(-0.566977\pi\)
−0.208865 + 0.977944i \(0.566977\pi\)
\(972\) 0 0
\(973\) −29568.0 −0.974210
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 31501.6i 1.03155i 0.856724 + 0.515776i \(0.172496\pi\)
−0.856724 + 0.515776i \(0.827504\pi\)
\(978\) 0 0
\(979\) − 66095.1i − 2.15772i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 56318.7 1.82735 0.913676 0.406444i \(-0.133231\pi\)
0.913676 + 0.406444i \(0.133231\pi\)
\(984\) 0 0
\(985\) 8118.00 0.262600
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 43987.7i 1.41428i
\(990\) 0 0
\(991\) − 5473.28i − 0.175443i −0.996145 0.0877217i \(-0.972041\pi\)
0.996145 0.0877217i \(-0.0279586\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 3115.75 0.0992723
\(996\) 0 0
\(997\) −22858.0 −0.726098 −0.363049 0.931770i \(-0.618264\pi\)
−0.363049 + 0.931770i \(0.618264\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 144.4.c.b.143.4 yes 4
3.2 odd 2 inner 144.4.c.b.143.2 yes 4
4.3 odd 2 inner 144.4.c.b.143.3 yes 4
8.3 odd 2 576.4.c.c.575.1 4
8.5 even 2 576.4.c.c.575.2 4
12.11 even 2 inner 144.4.c.b.143.1 4
16.3 odd 4 2304.4.f.f.1151.8 8
16.5 even 4 2304.4.f.f.1151.1 8
16.11 odd 4 2304.4.f.f.1151.3 8
16.13 even 4 2304.4.f.f.1151.6 8
24.5 odd 2 576.4.c.c.575.4 4
24.11 even 2 576.4.c.c.575.3 4
48.5 odd 4 2304.4.f.f.1151.5 8
48.11 even 4 2304.4.f.f.1151.7 8
48.29 odd 4 2304.4.f.f.1151.2 8
48.35 even 4 2304.4.f.f.1151.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
144.4.c.b.143.1 4 12.11 even 2 inner
144.4.c.b.143.2 yes 4 3.2 odd 2 inner
144.4.c.b.143.3 yes 4 4.3 odd 2 inner
144.4.c.b.143.4 yes 4 1.1 even 1 trivial
576.4.c.c.575.1 4 8.3 odd 2
576.4.c.c.575.2 4 8.5 even 2
576.4.c.c.575.3 4 24.11 even 2
576.4.c.c.575.4 4 24.5 odd 2
2304.4.f.f.1151.1 8 16.5 even 4
2304.4.f.f.1151.2 8 48.29 odd 4
2304.4.f.f.1151.3 8 16.11 odd 4
2304.4.f.f.1151.4 8 48.35 even 4
2304.4.f.f.1151.5 8 48.5 odd 4
2304.4.f.f.1151.6 8 16.13 even 4
2304.4.f.f.1151.7 8 48.11 even 4
2304.4.f.f.1151.8 8 16.3 odd 4