Properties

Label 288.4.d.d.145.3
Level $288$
Weight $4$
Character 288.145
Analytic conductor $16.993$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [288,4,Mod(145,288)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(288, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("288.145");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 288 = 2^{5} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 288.d (of order \(2\), degree \(1\), not 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: \(16.9925500817\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.8248384.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + x^{4} - 12x^{3} + 4x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{15}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 24)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 145.3
Root \(1.88322 + 0.673417i\) of defining polynomial
Character \(\chi\) \(=\) 288.145
Dual form 288.4.d.d.145.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.612661i q^{5} +22.7441 q^{7} +O(q^{10})\) \(q-0.612661i q^{5} +22.7441 q^{7} +60.2630i q^{11} -52.9062i q^{13} -47.1643 q^{17} +29.1643i q^{19} +109.488 q^{23} +124.625 q^{25} -10.4250i q^{29} +220.881 q^{31} -13.9345i q^{35} +408.348i q^{37} +360.742 q^{41} +236.414i q^{43} +129.113 q^{47} +174.296 q^{49} -117.819i q^{53} +36.9208 q^{55} -262.854i q^{59} -273.465i q^{61} -32.4135 q^{65} +89.4077i q^{67} -350.521 q^{71} +532.610 q^{73} +1370.63i q^{77} +166.561 q^{79} +361.934i q^{83} +28.8957i q^{85} -40.3285 q^{89} -1203.31i q^{91} +17.8678 q^{95} -614.921 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 28 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 28 q^{7} - 52 q^{17} + 328 q^{23} - 106 q^{25} + 636 q^{31} - 236 q^{41} - 408 q^{47} + 654 q^{49} - 1024 q^{55} + 1744 q^{65} - 1704 q^{71} + 956 q^{73} + 44 q^{79} + 220 q^{89} + 5104 q^{95} - 2444 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/288\mathbb{Z}\right)^\times\).

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

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
\(5\) − 0.612661i − 0.0547981i −0.999625 0.0273990i \(-0.991278\pi\)
0.999625 0.0273990i \(-0.00872248\pi\)
\(6\) 0 0
\(7\) 22.7441 1.22807 0.614034 0.789279i \(-0.289546\pi\)
0.614034 + 0.789279i \(0.289546\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 60.2630i 1.65182i 0.563805 + 0.825908i \(0.309337\pi\)
−0.563805 + 0.825908i \(0.690663\pi\)
\(12\) 0 0
\(13\) − 52.9062i − 1.12873i −0.825524 0.564367i \(-0.809120\pi\)
0.825524 0.564367i \(-0.190880\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −47.1643 −0.672883 −0.336442 0.941704i \(-0.609223\pi\)
−0.336442 + 0.941704i \(0.609223\pi\)
\(18\) 0 0
\(19\) 29.1643i 0.352144i 0.984377 + 0.176072i \(0.0563392\pi\)
−0.984377 + 0.176072i \(0.943661\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 109.488 0.992604 0.496302 0.868150i \(-0.334691\pi\)
0.496302 + 0.868150i \(0.334691\pi\)
\(24\) 0 0
\(25\) 124.625 0.996997
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 10.4250i − 0.0667542i −0.999443 0.0333771i \(-0.989374\pi\)
0.999443 0.0333771i \(-0.0106262\pi\)
\(30\) 0 0
\(31\) 220.881 1.27972 0.639860 0.768492i \(-0.278992\pi\)
0.639860 + 0.768492i \(0.278992\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 13.9345i − 0.0672958i
\(36\) 0 0
\(37\) 408.348i 1.81438i 0.420725 + 0.907188i \(0.361776\pi\)
−0.420725 + 0.907188i \(0.638224\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 360.742 1.37411 0.687054 0.726606i \(-0.258903\pi\)
0.687054 + 0.726606i \(0.258903\pi\)
\(42\) 0 0
\(43\) 236.414i 0.838436i 0.907886 + 0.419218i \(0.137696\pi\)
−0.907886 + 0.419218i \(0.862304\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 129.113 0.400703 0.200352 0.979724i \(-0.435792\pi\)
0.200352 + 0.979724i \(0.435792\pi\)
\(48\) 0 0
\(49\) 174.296 0.508152
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 117.819i − 0.305353i −0.988276 0.152677i \(-0.951211\pi\)
0.988276 0.152677i \(-0.0487893\pi\)
\(54\) 0 0
\(55\) 36.9208 0.0905163
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 262.854i − 0.580012i −0.957025 0.290006i \(-0.906343\pi\)
0.957025 0.290006i \(-0.0936574\pi\)
\(60\) 0 0
\(61\) − 273.465i − 0.573993i −0.957932 0.286996i \(-0.907343\pi\)
0.957932 0.286996i \(-0.0926568\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −32.4135 −0.0618524
\(66\) 0 0
\(67\) 89.4077i 0.163028i 0.996672 + 0.0815141i \(0.0259756\pi\)
−0.996672 + 0.0815141i \(0.974024\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −350.521 −0.585904 −0.292952 0.956127i \(-0.594638\pi\)
−0.292952 + 0.956127i \(0.594638\pi\)
\(72\) 0 0
\(73\) 532.610 0.853936 0.426968 0.904267i \(-0.359582\pi\)
0.426968 + 0.904267i \(0.359582\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1370.63i 2.02854i
\(78\) 0 0
\(79\) 166.561 0.237210 0.118605 0.992942i \(-0.462158\pi\)
0.118605 + 0.992942i \(0.462158\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 361.934i 0.478644i 0.970940 + 0.239322i \(0.0769252\pi\)
−0.970940 + 0.239322i \(0.923075\pi\)
\(84\) 0 0
\(85\) 28.8957i 0.0368727i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −40.3285 −0.0480316 −0.0240158 0.999712i \(-0.507645\pi\)
−0.0240158 + 0.999712i \(0.507645\pi\)
\(90\) 0 0
\(91\) − 1203.31i − 1.38616i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 17.8678 0.0192968
\(96\) 0 0
\(97\) −614.921 −0.643667 −0.321834 0.946796i \(-0.604299\pi\)
−0.321834 + 0.946796i \(0.604299\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) − 1664.99i − 1.64033i −0.572130 0.820163i \(-0.693883\pi\)
0.572130 0.820163i \(-0.306117\pi\)
\(102\) 0 0
\(103\) −396.858 −0.379647 −0.189823 0.981818i \(-0.560792\pi\)
−0.189823 + 0.981818i \(0.560792\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 350.630i 0.316791i 0.987376 + 0.158396i \(0.0506321\pi\)
−0.987376 + 0.158396i \(0.949368\pi\)
\(108\) 0 0
\(109\) − 597.009i − 0.524615i −0.964984 0.262308i \(-0.915516\pi\)
0.964984 0.262308i \(-0.0844835\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −496.422 −0.413270 −0.206635 0.978418i \(-0.566251\pi\)
−0.206635 + 0.978418i \(0.566251\pi\)
\(114\) 0 0
\(115\) − 67.0792i − 0.0543928i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −1072.71 −0.826346
\(120\) 0 0
\(121\) −2300.63 −1.72849
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 152.935i − 0.109432i
\(126\) 0 0
\(127\) −1799.85 −1.25756 −0.628782 0.777581i \(-0.716446\pi\)
−0.628782 + 0.777581i \(0.716446\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) − 1121.45i − 0.747949i −0.927439 0.373974i \(-0.877995\pi\)
0.927439 0.373974i \(-0.122005\pi\)
\(132\) 0 0
\(133\) 663.316i 0.432457i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −2449.55 −1.52759 −0.763793 0.645461i \(-0.776665\pi\)
−0.763793 + 0.645461i \(0.776665\pi\)
\(138\) 0 0
\(139\) 2457.56i 1.49962i 0.661652 + 0.749811i \(0.269856\pi\)
−0.661652 + 0.749811i \(0.730144\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 3188.28 1.86446
\(144\) 0 0
\(145\) −6.38698 −0.00365800
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) − 2084.96i − 1.14635i −0.819432 0.573177i \(-0.805711\pi\)
0.819432 0.573177i \(-0.194289\pi\)
\(150\) 0 0
\(151\) −1057.80 −0.570084 −0.285042 0.958515i \(-0.592008\pi\)
−0.285042 + 0.958515i \(0.592008\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) − 135.325i − 0.0701262i
\(156\) 0 0
\(157\) − 3193.01i − 1.62312i −0.584270 0.811559i \(-0.698619\pi\)
0.584270 0.811559i \(-0.301381\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 2490.22 1.21899
\(162\) 0 0
\(163\) − 846.854i − 0.406937i −0.979081 0.203469i \(-0.934779\pi\)
0.979081 0.203469i \(-0.0652215\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −2630.15 −1.21873 −0.609363 0.792892i \(-0.708575\pi\)
−0.609363 + 0.792892i \(0.708575\pi\)
\(168\) 0 0
\(169\) −602.062 −0.274038
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 429.843i 0.188904i 0.995529 + 0.0944519i \(0.0301099\pi\)
−0.995529 + 0.0944519i \(0.969890\pi\)
\(174\) 0 0
\(175\) 2834.48 1.22438
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 1516.30i 0.633149i 0.948568 + 0.316574i \(0.102533\pi\)
−0.948568 + 0.316574i \(0.897467\pi\)
\(180\) 0 0
\(181\) 3380.20i 1.38811i 0.719921 + 0.694056i \(0.244178\pi\)
−0.719921 + 0.694056i \(0.755822\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 250.179 0.0994244
\(186\) 0 0
\(187\) − 2842.26i − 1.11148i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −2799.71 −1.06063 −0.530314 0.847801i \(-0.677926\pi\)
−0.530314 + 0.847801i \(0.677926\pi\)
\(192\) 0 0
\(193\) 624.106 0.232768 0.116384 0.993204i \(-0.462870\pi\)
0.116384 + 0.993204i \(0.462870\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 4779.25i 1.72846i 0.503094 + 0.864232i \(0.332195\pi\)
−0.503094 + 0.864232i \(0.667805\pi\)
\(198\) 0 0
\(199\) −2615.92 −0.931846 −0.465923 0.884825i \(-0.654278\pi\)
−0.465923 + 0.884825i \(0.654278\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) − 237.107i − 0.0819787i
\(204\) 0 0
\(205\) − 221.013i − 0.0752985i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −1757.52 −0.581677
\(210\) 0 0
\(211\) − 1745.78i − 0.569595i −0.958588 0.284798i \(-0.908074\pi\)
0.958588 0.284798i \(-0.0919264\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 144.841 0.0459447
\(216\) 0 0
\(217\) 5023.74 1.57158
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 2495.28i 0.759505i
\(222\) 0 0
\(223\) 3385.60 1.01667 0.508333 0.861161i \(-0.330262\pi\)
0.508333 + 0.861161i \(0.330262\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 3847.72i − 1.12503i −0.826787 0.562515i \(-0.809834\pi\)
0.826787 0.562515i \(-0.190166\pi\)
\(228\) 0 0
\(229\) − 1335.15i − 0.385279i −0.981270 0.192640i \(-0.938295\pi\)
0.981270 0.192640i \(-0.0617049\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 5146.38 1.44700 0.723499 0.690325i \(-0.242532\pi\)
0.723499 + 0.690325i \(0.242532\pi\)
\(234\) 0 0
\(235\) − 79.1025i − 0.0219578i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 7085.07 1.91755 0.958777 0.284160i \(-0.0917146\pi\)
0.958777 + 0.284160i \(0.0917146\pi\)
\(240\) 0 0
\(241\) 2538.40 0.678476 0.339238 0.940701i \(-0.389831\pi\)
0.339238 + 0.940701i \(0.389831\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) − 106.784i − 0.0278458i
\(246\) 0 0
\(247\) 1542.97 0.397477
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 1696.99i 0.426746i 0.976971 + 0.213373i \(0.0684449\pi\)
−0.976971 + 0.213373i \(0.931555\pi\)
\(252\) 0 0
\(253\) 6598.09i 1.63960i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 382.902 0.0929369 0.0464685 0.998920i \(-0.485203\pi\)
0.0464685 + 0.998920i \(0.485203\pi\)
\(258\) 0 0
\(259\) 9287.52i 2.22818i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −5002.02 −1.17277 −0.586383 0.810034i \(-0.699449\pi\)
−0.586383 + 0.810034i \(0.699449\pi\)
\(264\) 0 0
\(265\) −72.1833 −0.0167328
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) − 6117.47i − 1.38658i −0.720661 0.693288i \(-0.756161\pi\)
0.720661 0.693288i \(-0.243839\pi\)
\(270\) 0 0
\(271\) 3956.12 0.886780 0.443390 0.896329i \(-0.353776\pi\)
0.443390 + 0.896329i \(0.353776\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 7510.25i 1.64686i
\(276\) 0 0
\(277\) − 4842.17i − 1.05032i −0.851004 0.525158i \(-0.824006\pi\)
0.851004 0.525158i \(-0.175994\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −1878.68 −0.398835 −0.199417 0.979915i \(-0.563905\pi\)
−0.199417 + 0.979915i \(0.563905\pi\)
\(282\) 0 0
\(283\) 5724.87i 1.20250i 0.799060 + 0.601251i \(0.205331\pi\)
−0.799060 + 0.601251i \(0.794669\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 8204.77 1.68750
\(288\) 0 0
\(289\) −2688.53 −0.547228
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 5088.75i − 1.01464i −0.861759 0.507318i \(-0.830637\pi\)
0.861759 0.507318i \(-0.169363\pi\)
\(294\) 0 0
\(295\) −161.041 −0.0317836
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) − 5792.61i − 1.12038i
\(300\) 0 0
\(301\) 5377.02i 1.02966i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −167.541 −0.0314537
\(306\) 0 0
\(307\) − 7219.21i − 1.34209i −0.741416 0.671046i \(-0.765845\pi\)
0.741416 0.671046i \(-0.234155\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −1537.06 −0.280252 −0.140126 0.990134i \(-0.544751\pi\)
−0.140126 + 0.990134i \(0.544751\pi\)
\(312\) 0 0
\(313\) −2200.93 −0.397456 −0.198728 0.980055i \(-0.563681\pi\)
−0.198728 + 0.980055i \(0.563681\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 2840.41i − 0.503260i −0.967824 0.251630i \(-0.919033\pi\)
0.967824 0.251630i \(-0.0809665\pi\)
\(318\) 0 0
\(319\) 628.240 0.110266
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 1375.51i − 0.236952i
\(324\) 0 0
\(325\) − 6593.41i − 1.12534i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 2936.56 0.492091
\(330\) 0 0
\(331\) − 2118.52i − 0.351795i −0.984408 0.175898i \(-0.943717\pi\)
0.984408 0.175898i \(-0.0562828\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 54.7766 0.00893364
\(336\) 0 0
\(337\) −659.599 −0.106619 −0.0533096 0.998578i \(-0.516977\pi\)
−0.0533096 + 0.998578i \(0.516977\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 13310.9i 2.11386i
\(342\) 0 0
\(343\) −3837.03 −0.604023
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 8377.42i 1.29603i 0.761626 + 0.648017i \(0.224401\pi\)
−0.761626 + 0.648017i \(0.775599\pi\)
\(348\) 0 0
\(349\) 3254.18i 0.499119i 0.968360 + 0.249559i \(0.0802857\pi\)
−0.968360 + 0.249559i \(0.919714\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −11117.5 −1.67627 −0.838137 0.545459i \(-0.816355\pi\)
−0.838137 + 0.545459i \(0.816355\pi\)
\(354\) 0 0
\(355\) 214.750i 0.0321064i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −4756.56 −0.699281 −0.349640 0.936884i \(-0.613696\pi\)
−0.349640 + 0.936884i \(0.613696\pi\)
\(360\) 0 0
\(361\) 6008.45 0.875994
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) − 326.310i − 0.0467940i
\(366\) 0 0
\(367\) −1837.40 −0.261339 −0.130670 0.991426i \(-0.541713\pi\)
−0.130670 + 0.991426i \(0.541713\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) − 2679.70i − 0.374995i
\(372\) 0 0
\(373\) − 5598.07i − 0.777097i −0.921428 0.388549i \(-0.872977\pi\)
0.921428 0.388549i \(-0.127023\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −551.546 −0.0753476
\(378\) 0 0
\(379\) − 3460.18i − 0.468965i −0.972120 0.234482i \(-0.924661\pi\)
0.972120 0.234482i \(-0.0753395\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 5059.63 0.675027 0.337513 0.941321i \(-0.390414\pi\)
0.337513 + 0.941321i \(0.390414\pi\)
\(384\) 0 0
\(385\) 839.732 0.111160
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 2192.22i 0.285732i 0.989742 + 0.142866i \(0.0456318\pi\)
−0.989742 + 0.142866i \(0.954368\pi\)
\(390\) 0 0
\(391\) −5163.93 −0.667906
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) − 102.045i − 0.0129986i
\(396\) 0 0
\(397\) − 5519.94i − 0.697828i −0.937155 0.348914i \(-0.886551\pi\)
0.937155 0.348914i \(-0.113449\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 7352.64 0.915645 0.457822 0.889044i \(-0.348630\pi\)
0.457822 + 0.889044i \(0.348630\pi\)
\(402\) 0 0
\(403\) − 11685.9i − 1.44446i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −24608.2 −2.99702
\(408\) 0 0
\(409\) −11311.2 −1.36749 −0.683745 0.729721i \(-0.739650\pi\)
−0.683745 + 0.729721i \(0.739650\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) − 5978.40i − 0.712295i
\(414\) 0 0
\(415\) 221.743 0.0262288
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) − 13042.2i − 1.52065i −0.649543 0.760325i \(-0.725040\pi\)
0.649543 0.760325i \(-0.274960\pi\)
\(420\) 0 0
\(421\) 4544.38i 0.526080i 0.964785 + 0.263040i \(0.0847251\pi\)
−0.964785 + 0.263040i \(0.915275\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −5877.83 −0.670863
\(426\) 0 0
\(427\) − 6219.72i − 0.704903i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 7713.83 0.862093 0.431047 0.902330i \(-0.358144\pi\)
0.431047 + 0.902330i \(0.358144\pi\)
\(432\) 0 0
\(433\) −15068.3 −1.67237 −0.836183 0.548451i \(-0.815218\pi\)
−0.836183 + 0.548451i \(0.815218\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 3193.14i 0.349540i
\(438\) 0 0
\(439\) −11004.7 −1.19642 −0.598208 0.801341i \(-0.704120\pi\)
−0.598208 + 0.801341i \(0.704120\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 2513.04i − 0.269522i −0.990878 0.134761i \(-0.956973\pi\)
0.990878 0.134761i \(-0.0430266\pi\)
\(444\) 0 0
\(445\) 24.7077i 0.00263204i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 15752.7 1.65571 0.827855 0.560942i \(-0.189561\pi\)
0.827855 + 0.560942i \(0.189561\pi\)
\(450\) 0 0
\(451\) 21739.4i 2.26977i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −737.218 −0.0759590
\(456\) 0 0
\(457\) −5257.06 −0.538107 −0.269053 0.963125i \(-0.586711\pi\)
−0.269053 + 0.963125i \(0.586711\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 8066.31i − 0.814936i −0.913220 0.407468i \(-0.866412\pi\)
0.913220 0.407468i \(-0.133588\pi\)
\(462\) 0 0
\(463\) −5683.43 −0.570478 −0.285239 0.958456i \(-0.592073\pi\)
−0.285239 + 0.958456i \(0.592073\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 11139.3i 1.10378i 0.833916 + 0.551891i \(0.186094\pi\)
−0.833916 + 0.551891i \(0.813906\pi\)
\(468\) 0 0
\(469\) 2033.50i 0.200210i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −14247.0 −1.38494
\(474\) 0 0
\(475\) 3634.59i 0.351087i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 3477.35 0.331699 0.165850 0.986151i \(-0.446963\pi\)
0.165850 + 0.986151i \(0.446963\pi\)
\(480\) 0 0
\(481\) 21604.1 2.04795
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 376.738i 0.0352717i
\(486\) 0 0
\(487\) 478.797 0.0445510 0.0222755 0.999752i \(-0.492909\pi\)
0.0222755 + 0.999752i \(0.492909\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) − 16601.8i − 1.52592i −0.646444 0.762961i \(-0.723745\pi\)
0.646444 0.762961i \(-0.276255\pi\)
\(492\) 0 0
\(493\) 491.687i 0.0449178i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −7972.29 −0.719530
\(498\) 0 0
\(499\) − 9482.20i − 0.850664i −0.905037 0.425332i \(-0.860157\pi\)
0.905037 0.425332i \(-0.139843\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 16561.2 1.46805 0.734023 0.679124i \(-0.237640\pi\)
0.734023 + 0.679124i \(0.237640\pi\)
\(504\) 0 0
\(505\) −1020.08 −0.0898867
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 4197.35i 0.365509i 0.983159 + 0.182755i \(0.0585014\pi\)
−0.983159 + 0.182755i \(0.941499\pi\)
\(510\) 0 0
\(511\) 12113.8 1.04869
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 243.140i 0.0208039i
\(516\) 0 0
\(517\) 7780.73i 0.661888i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 15755.5 1.32488 0.662440 0.749115i \(-0.269521\pi\)
0.662440 + 0.749115i \(0.269521\pi\)
\(522\) 0 0
\(523\) 11555.1i 0.966098i 0.875593 + 0.483049i \(0.160471\pi\)
−0.875593 + 0.483049i \(0.839529\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −10417.7 −0.861102
\(528\) 0 0
\(529\) −179.314 −0.0147378
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) − 19085.5i − 1.55100i
\(534\) 0 0
\(535\) 214.817 0.0173595
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 10503.6i 0.839373i
\(540\) 0 0
\(541\) − 7475.65i − 0.594091i −0.954863 0.297045i \(-0.903999\pi\)
0.954863 0.297045i \(-0.0960013\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −365.764 −0.0287479
\(546\) 0 0
\(547\) − 6028.08i − 0.471192i −0.971851 0.235596i \(-0.924296\pi\)
0.971851 0.235596i \(-0.0757043\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 304.037 0.0235071
\(552\) 0 0
\(553\) 3788.29 0.291310
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 19381.5i 1.47436i 0.675696 + 0.737181i \(0.263843\pi\)
−0.675696 + 0.737181i \(0.736157\pi\)
\(558\) 0 0
\(559\) 12507.7 0.946370
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 20565.0i 1.53946i 0.638372 + 0.769728i \(0.279608\pi\)
−0.638372 + 0.769728i \(0.720392\pi\)
\(564\) 0 0
\(565\) 304.139i 0.0226464i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −15252.9 −1.12379 −0.561895 0.827209i \(-0.689927\pi\)
−0.561895 + 0.827209i \(0.689927\pi\)
\(570\) 0 0
\(571\) − 16492.8i − 1.20876i −0.796697 0.604379i \(-0.793421\pi\)
0.796697 0.604379i \(-0.206579\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 13644.9 0.989623
\(576\) 0 0
\(577\) −10298.2 −0.743016 −0.371508 0.928430i \(-0.621159\pi\)
−0.371508 + 0.928430i \(0.621159\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 8231.89i 0.587808i
\(582\) 0 0
\(583\) 7100.14 0.504387
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 13104.8i 0.921453i 0.887542 + 0.460727i \(0.152411\pi\)
−0.887542 + 0.460727i \(0.847589\pi\)
\(588\) 0 0
\(589\) 6441.82i 0.450646i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −4163.34 −0.288310 −0.144155 0.989555i \(-0.546046\pi\)
−0.144155 + 0.989555i \(0.546046\pi\)
\(594\) 0 0
\(595\) 657.208i 0.0452822i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −5718.60 −0.390076 −0.195038 0.980796i \(-0.562483\pi\)
−0.195038 + 0.980796i \(0.562483\pi\)
\(600\) 0 0
\(601\) 17473.0 1.18592 0.592959 0.805233i \(-0.297960\pi\)
0.592959 + 0.805233i \(0.297960\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 1409.50i 0.0947181i
\(606\) 0 0
\(607\) 5647.60 0.377643 0.188821 0.982011i \(-0.439533\pi\)
0.188821 + 0.982011i \(0.439533\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) − 6830.87i − 0.452287i
\(612\) 0 0
\(613\) − 16023.6i − 1.05577i −0.849317 0.527884i \(-0.822986\pi\)
0.849317 0.527884i \(-0.177014\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 21022.1 1.37167 0.685834 0.727758i \(-0.259438\pi\)
0.685834 + 0.727758i \(0.259438\pi\)
\(618\) 0 0
\(619\) 17824.2i 1.15737i 0.815550 + 0.578686i \(0.196434\pi\)
−0.815550 + 0.578686i \(0.803566\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −917.238 −0.0589861
\(624\) 0 0
\(625\) 15484.4 0.991001
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 19259.4i − 1.22086i
\(630\) 0 0
\(631\) 22339.0 1.40935 0.704677 0.709528i \(-0.251092\pi\)
0.704677 + 0.709528i \(0.251092\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 1102.70i 0.0689121i
\(636\) 0 0
\(637\) − 9221.34i − 0.573568i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 5268.43 0.324634 0.162317 0.986739i \(-0.448103\pi\)
0.162317 + 0.986739i \(0.448103\pi\)
\(642\) 0 0
\(643\) 21965.8i 1.34719i 0.739099 + 0.673597i \(0.235252\pi\)
−0.739099 + 0.673597i \(0.764748\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 3165.40 0.192341 0.0961706 0.995365i \(-0.469341\pi\)
0.0961706 + 0.995365i \(0.469341\pi\)
\(648\) 0 0
\(649\) 15840.4 0.958073
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 12094.7i 0.724814i 0.932020 + 0.362407i \(0.118045\pi\)
−0.932020 + 0.362407i \(0.881955\pi\)
\(654\) 0 0
\(655\) −687.067 −0.0409861
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) − 7523.17i − 0.444706i −0.974966 0.222353i \(-0.928626\pi\)
0.974966 0.222353i \(-0.0713737\pi\)
\(660\) 0 0
\(661\) 24141.1i 1.42054i 0.703928 + 0.710271i \(0.251428\pi\)
−0.703928 + 0.710271i \(0.748572\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 406.388 0.0236978
\(666\) 0 0
\(667\) − 1141.41i − 0.0662604i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 16479.8 0.948130
\(672\) 0 0
\(673\) 944.143 0.0540773 0.0270387 0.999634i \(-0.491392\pi\)
0.0270387 + 0.999634i \(0.491392\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 4450.51i − 0.252654i −0.991989 0.126327i \(-0.959681\pi\)
0.991989 0.126327i \(-0.0403189\pi\)
\(678\) 0 0
\(679\) −13985.8 −0.790468
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) − 1726.40i − 0.0967189i −0.998830 0.0483594i \(-0.984601\pi\)
0.998830 0.0483594i \(-0.0153993\pi\)
\(684\) 0 0
\(685\) 1500.75i 0.0837088i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −6233.37 −0.344662
\(690\) 0 0
\(691\) 683.143i 0.0376092i 0.999823 + 0.0188046i \(0.00598605\pi\)
−0.999823 + 0.0188046i \(0.994014\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 1505.65 0.0821764
\(696\) 0 0
\(697\) −17014.1 −0.924614
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 19533.2i 1.05244i 0.850349 + 0.526220i \(0.176391\pi\)
−0.850349 + 0.526220i \(0.823609\pi\)
\(702\) 0 0
\(703\) −11909.2 −0.638922
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 37868.8i − 2.01443i
\(708\) 0 0
\(709\) 26081.9i 1.38156i 0.723066 + 0.690779i \(0.242732\pi\)
−0.723066 + 0.690779i \(0.757268\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 24183.8 1.27025
\(714\) 0 0
\(715\) − 1953.34i − 0.102169i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −30077.3 −1.56007 −0.780036 0.625734i \(-0.784800\pi\)
−0.780036 + 0.625734i \(0.784800\pi\)
\(720\) 0 0
\(721\) −9026.21 −0.466232
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) − 1299.21i − 0.0665537i
\(726\) 0 0
\(727\) −23049.9 −1.17589 −0.587946 0.808900i \(-0.700063\pi\)
−0.587946 + 0.808900i \(0.700063\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) − 11150.3i − 0.564169i
\(732\) 0 0
\(733\) − 4444.57i − 0.223962i −0.993710 0.111981i \(-0.964280\pi\)
0.993710 0.111981i \(-0.0357195\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −5387.98 −0.269293
\(738\) 0 0
\(739\) − 28465.1i − 1.41692i −0.705749 0.708462i \(-0.749390\pi\)
0.705749 0.708462i \(-0.250610\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 4389.76 0.216749 0.108374 0.994110i \(-0.465435\pi\)
0.108374 + 0.994110i \(0.465435\pi\)
\(744\) 0 0
\(745\) −1277.38 −0.0628180
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 7974.77i 0.389041i
\(750\) 0 0
\(751\) −14121.9 −0.686171 −0.343085 0.939304i \(-0.611472\pi\)
−0.343085 + 0.939304i \(0.611472\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 648.074i 0.0312395i
\(756\) 0 0
\(757\) 17006.3i 0.816516i 0.912866 + 0.408258i \(0.133864\pi\)
−0.912866 + 0.408258i \(0.866136\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 4603.38 0.219280 0.109640 0.993971i \(-0.465030\pi\)
0.109640 + 0.993971i \(0.465030\pi\)
\(762\) 0 0
\(763\) − 13578.5i − 0.644264i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −13906.6 −0.654679
\(768\) 0 0
\(769\) −12459.1 −0.584248 −0.292124 0.956380i \(-0.594362\pi\)
−0.292124 + 0.956380i \(0.594362\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 27068.1i − 1.25947i −0.776808 0.629737i \(-0.783163\pi\)
0.776808 0.629737i \(-0.216837\pi\)
\(774\) 0 0
\(775\) 27527.2 1.27588
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 10520.8i 0.483884i
\(780\) 0 0
\(781\) − 21123.4i − 0.967804i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −1956.23 −0.0889438
\(786\) 0 0
\(787\) − 6986.86i − 0.316461i −0.987402 0.158230i \(-0.949421\pi\)
0.987402 0.158230i \(-0.0505789\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −11290.7 −0.507523
\(792\) 0 0
\(793\) −14468.0 −0.647885
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 27271.5i − 1.21205i −0.795444 0.606027i \(-0.792762\pi\)
0.795444 0.606027i \(-0.207238\pi\)
\(798\) 0 0
\(799\) −6089.52 −0.269627
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 32096.7i 1.41054i
\(804\) 0 0
\(805\) − 1525.66i − 0.0667981i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −23785.9 −1.03371 −0.516853 0.856074i \(-0.672897\pi\)
−0.516853 + 0.856074i \(0.672897\pi\)
\(810\) 0 0
\(811\) 21703.5i 0.939718i 0.882741 + 0.469859i \(0.155695\pi\)
−0.882741 + 0.469859i \(0.844305\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −518.835 −0.0222994
\(816\) 0 0
\(817\) −6894.83 −0.295250
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 33240.4i 1.41303i 0.707698 + 0.706515i \(0.249734\pi\)
−0.707698 + 0.706515i \(0.750266\pi\)
\(822\) 0 0
\(823\) 17227.5 0.729665 0.364832 0.931073i \(-0.381126\pi\)
0.364832 + 0.931073i \(0.381126\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 21678.4i − 0.911528i −0.890101 0.455764i \(-0.849366\pi\)
0.890101 0.455764i \(-0.150634\pi\)
\(828\) 0 0
\(829\) − 34269.8i − 1.43575i −0.696170 0.717877i \(-0.745114\pi\)
0.696170 0.717877i \(-0.254886\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −8220.55 −0.341927
\(834\) 0 0
\(835\) 1611.39i 0.0667838i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 33444.1 1.37618 0.688092 0.725623i \(-0.258448\pi\)
0.688092 + 0.725623i \(0.258448\pi\)
\(840\) 0 0
\(841\) 24280.3 0.995544
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 368.860i 0.0150168i
\(846\) 0 0
\(847\) −52325.8 −2.12271
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 44709.3i 1.80096i
\(852\) 0 0
\(853\) − 37037.3i − 1.48667i −0.668917 0.743337i \(-0.733242\pi\)
0.668917 0.743337i \(-0.266758\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −35794.2 −1.42673 −0.713365 0.700793i \(-0.752830\pi\)
−0.713365 + 0.700793i \(0.752830\pi\)
\(858\) 0 0
\(859\) − 20582.1i − 0.817522i −0.912641 0.408761i \(-0.865961\pi\)
0.912641 0.408761i \(-0.134039\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −25677.7 −1.01284 −0.506420 0.862287i \(-0.669031\pi\)
−0.506420 + 0.862287i \(0.669031\pi\)
\(864\) 0 0
\(865\) 263.348 0.0103516
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 10037.5i 0.391827i
\(870\) 0 0
\(871\) 4730.22 0.184015
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) − 3478.38i − 0.134389i
\(876\) 0 0
\(877\) − 32783.0i − 1.26226i −0.775676 0.631131i \(-0.782591\pi\)
0.775676 0.631131i \(-0.217409\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −26615.7 −1.01783 −0.508914 0.860818i \(-0.669953\pi\)
−0.508914 + 0.860818i \(0.669953\pi\)
\(882\) 0 0
\(883\) 19203.4i 0.731875i 0.930639 + 0.365937i \(0.119252\pi\)
−0.930639 + 0.365937i \(0.880748\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 23198.0 0.878144 0.439072 0.898452i \(-0.355307\pi\)
0.439072 + 0.898452i \(0.355307\pi\)
\(888\) 0 0
\(889\) −40936.0 −1.54438
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 3765.48i 0.141105i
\(894\) 0 0
\(895\) 928.979 0.0346953
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 2302.68i − 0.0854266i
\(900\) 0 0
\(901\) 5556.86i 0.205467i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 2070.92 0.0760659
\(906\) 0 0
\(907\) − 16151.3i − 0.591285i −0.955299 0.295643i \(-0.904466\pi\)
0.955299 0.295643i \(-0.0955337\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −3487.34 −0.126829 −0.0634143 0.997987i \(-0.520199\pi\)
−0.0634143 + 0.997987i \(0.520199\pi\)
\(912\) 0 0
\(913\) −21811.2 −0.790632
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 25506.3i − 0.918532i
\(918\) 0 0
\(919\) −17055.5 −0.612198 −0.306099 0.952000i \(-0.599024\pi\)
−0.306099 + 0.952000i \(0.599024\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 18544.7i 0.661329i
\(924\) 0 0
\(925\) 50890.2i 1.80893i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 55339.3 1.95438 0.977192 0.212359i \(-0.0681146\pi\)
0.977192 + 0.212359i \(0.0681146\pi\)
\(930\) 0 0
\(931\) 5083.22i 0.178943i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −1741.34 −0.0609069
\(936\) 0 0
\(937\) −20457.6 −0.713255 −0.356627 0.934247i \(-0.616073\pi\)
−0.356627 + 0.934247i \(0.616073\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) − 55891.0i − 1.93623i −0.250502 0.968116i \(-0.580596\pi\)
0.250502 0.968116i \(-0.419404\pi\)
\(942\) 0 0
\(943\) 39497.0 1.36395
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 54727.0i − 1.87792i −0.344027 0.938960i \(-0.611791\pi\)
0.344027 0.938960i \(-0.388209\pi\)
\(948\) 0 0
\(949\) − 28178.4i − 0.963865i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −29958.8 −1.01832 −0.509160 0.860672i \(-0.670044\pi\)
−0.509160 + 0.860672i \(0.670044\pi\)
\(954\) 0 0
\(955\) 1715.27i 0.0581204i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −55713.0 −1.87598
\(960\) 0 0
\(961\) 18997.2 0.637682
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 382.365i − 0.0127552i
\(966\) 0 0
\(967\) −13498.5 −0.448896 −0.224448 0.974486i \(-0.572058\pi\)
−0.224448 + 0.974486i \(0.572058\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 9652.36i 0.319010i 0.987197 + 0.159505i \(0.0509899\pi\)
−0.987197 + 0.159505i \(0.949010\pi\)
\(972\) 0 0
\(973\) 55895.1i 1.84164i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −12169.8 −0.398511 −0.199256 0.979948i \(-0.563852\pi\)
−0.199256 + 0.979948i \(0.563852\pi\)
\(978\) 0 0
\(979\) − 2430.32i − 0.0793394i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 47545.2 1.54268 0.771341 0.636423i \(-0.219587\pi\)
0.771341 + 0.636423i \(0.219587\pi\)
\(984\) 0 0
\(985\) 2928.06 0.0947165
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 25884.5i 0.832234i
\(990\) 0 0
\(991\) 892.350 0.0286039 0.0143019 0.999898i \(-0.495447\pi\)
0.0143019 + 0.999898i \(0.495447\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 1602.67i 0.0510634i
\(996\) 0 0
\(997\) 4458.57i 0.141629i 0.997489 + 0.0708146i \(0.0225599\pi\)
−0.997489 + 0.0708146i \(0.977440\pi\)
\(998\) 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 288.4.d.d.145.3 6
3.2 odd 2 96.4.d.a.49.2 6
4.3 odd 2 72.4.d.d.37.2 6
8.3 odd 2 72.4.d.d.37.1 6
8.5 even 2 inner 288.4.d.d.145.4 6
12.11 even 2 24.4.d.a.13.5 6
16.3 odd 4 2304.4.a.bt.1.2 3
16.5 even 4 2304.4.a.bw.1.2 3
16.11 odd 4 2304.4.a.bv.1.2 3
16.13 even 4 2304.4.a.bu.1.2 3
24.5 odd 2 96.4.d.a.49.5 6
24.11 even 2 24.4.d.a.13.6 yes 6
48.5 odd 4 768.4.a.q.1.2 3
48.11 even 4 768.4.a.s.1.2 3
48.29 odd 4 768.4.a.t.1.2 3
48.35 even 4 768.4.a.r.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
24.4.d.a.13.5 6 12.11 even 2
24.4.d.a.13.6 yes 6 24.11 even 2
72.4.d.d.37.1 6 8.3 odd 2
72.4.d.d.37.2 6 4.3 odd 2
96.4.d.a.49.2 6 3.2 odd 2
96.4.d.a.49.5 6 24.5 odd 2
288.4.d.d.145.3 6 1.1 even 1 trivial
288.4.d.d.145.4 6 8.5 even 2 inner
768.4.a.q.1.2 3 48.5 odd 4
768.4.a.r.1.2 3 48.35 even 4
768.4.a.s.1.2 3 48.11 even 4
768.4.a.t.1.2 3 48.29 odd 4
2304.4.a.bt.1.2 3 16.3 odd 4
2304.4.a.bu.1.2 3 16.13 even 4
2304.4.a.bv.1.2 3 16.11 odd 4
2304.4.a.bw.1.2 3 16.5 even 4