Properties

Label 576.3.e.f.449.2
Level $576$
Weight $3$
Character 576.449
Analytic conductor $15.695$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 576 = 2^{6} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 576.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(15.6948632272\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Defining polynomial: \(x^{2} + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 18)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 449.2
Root \(1.41421i\) of defining polynomial
Character \(\chi\) \(=\) 576.449
Dual form 576.3.e.f.449.1

$q$-expansion

\(f(q)\) \(=\) \(q+4.24264i q^{5} +4.00000 q^{7} +O(q^{10})\) \(q+4.24264i q^{5} +4.00000 q^{7} +16.9706i q^{11} -8.00000 q^{13} -12.7279i q^{17} -16.0000 q^{19} +16.9706i q^{23} +7.00000 q^{25} -4.24264i q^{29} -44.0000 q^{31} +16.9706i q^{35} +34.0000 q^{37} +46.6690i q^{41} -40.0000 q^{43} +84.8528i q^{47} -33.0000 q^{49} -38.1838i q^{53} -72.0000 q^{55} +33.9411i q^{59} -50.0000 q^{61} -33.9411i q^{65} +8.00000 q^{67} +50.9117i q^{71} -16.0000 q^{73} +67.8823i q^{77} +76.0000 q^{79} +118.794i q^{83} +54.0000 q^{85} +12.7279i q^{89} -32.0000 q^{91} -67.8823i q^{95} +176.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 8 q^{7} + O(q^{10}) \) \( 2 q + 8 q^{7} - 16 q^{13} - 32 q^{19} + 14 q^{25} - 88 q^{31} + 68 q^{37} - 80 q^{43} - 66 q^{49} - 144 q^{55} - 100 q^{61} + 16 q^{67} - 32 q^{73} + 152 q^{79} + 108 q^{85} - 64 q^{91} + 352 q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/576\mathbb{Z}\right)^\times\).

\(n\) \(65\) \(127\) \(325\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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 0.848528i 0.905539 + 0.424264i \(0.139467\pi\)
−0.905539 + 0.424264i \(0.860533\pi\)
\(6\) 0 0
\(7\) 4.00000 0.571429 0.285714 0.958315i \(-0.407769\pi\)
0.285714 + 0.958315i \(0.407769\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 16.9706i 1.54278i 0.636364 + 0.771389i \(0.280438\pi\)
−0.636364 + 0.771389i \(0.719562\pi\)
\(12\) 0 0
\(13\) −8.00000 −0.615385 −0.307692 0.951486i \(-0.599557\pi\)
−0.307692 + 0.951486i \(0.599557\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 12.7279i − 0.748701i −0.927287 0.374351i \(-0.877866\pi\)
0.927287 0.374351i \(-0.122134\pi\)
\(18\) 0 0
\(19\) −16.0000 −0.842105 −0.421053 0.907036i \(-0.638339\pi\)
−0.421053 + 0.907036i \(0.638339\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 16.9706i 0.737851i 0.929459 + 0.368925i \(0.120274\pi\)
−0.929459 + 0.368925i \(0.879726\pi\)
\(24\) 0 0
\(25\) 7.00000 0.280000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 4.24264i − 0.146298i −0.997321 0.0731490i \(-0.976695\pi\)
0.997321 0.0731490i \(-0.0233049\pi\)
\(30\) 0 0
\(31\) −44.0000 −1.41935 −0.709677 0.704527i \(-0.751159\pi\)
−0.709677 + 0.704527i \(0.751159\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 16.9706i 0.484873i
\(36\) 0 0
\(37\) 34.0000 0.918919 0.459459 0.888199i \(-0.348043\pi\)
0.459459 + 0.888199i \(0.348043\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 46.6690i 1.13827i 0.822244 + 0.569135i \(0.192722\pi\)
−0.822244 + 0.569135i \(0.807278\pi\)
\(42\) 0 0
\(43\) −40.0000 −0.930233 −0.465116 0.885250i \(-0.653987\pi\)
−0.465116 + 0.885250i \(0.653987\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 84.8528i 1.80538i 0.430293 + 0.902690i \(0.358410\pi\)
−0.430293 + 0.902690i \(0.641590\pi\)
\(48\) 0 0
\(49\) −33.0000 −0.673469
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 38.1838i − 0.720448i −0.932866 0.360224i \(-0.882700\pi\)
0.932866 0.360224i \(-0.117300\pi\)
\(54\) 0 0
\(55\) −72.0000 −1.30909
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 33.9411i 0.575273i 0.957740 + 0.287637i \(0.0928695\pi\)
−0.957740 + 0.287637i \(0.907130\pi\)
\(60\) 0 0
\(61\) −50.0000 −0.819672 −0.409836 0.912159i \(-0.634414\pi\)
−0.409836 + 0.912159i \(0.634414\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) − 33.9411i − 0.522171i
\(66\) 0 0
\(67\) 8.00000 0.119403 0.0597015 0.998216i \(-0.480985\pi\)
0.0597015 + 0.998216i \(0.480985\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 50.9117i 0.717066i 0.933517 + 0.358533i \(0.116723\pi\)
−0.933517 + 0.358533i \(0.883277\pi\)
\(72\) 0 0
\(73\) −16.0000 −0.219178 −0.109589 0.993977i \(-0.534953\pi\)
−0.109589 + 0.993977i \(0.534953\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 67.8823i 0.881588i
\(78\) 0 0
\(79\) 76.0000 0.962025 0.481013 0.876714i \(-0.340269\pi\)
0.481013 + 0.876714i \(0.340269\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 118.794i 1.43125i 0.698484 + 0.715626i \(0.253859\pi\)
−0.698484 + 0.715626i \(0.746141\pi\)
\(84\) 0 0
\(85\) 54.0000 0.635294
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 12.7279i 0.143010i 0.997440 + 0.0715052i \(0.0227802\pi\)
−0.997440 + 0.0715052i \(0.977220\pi\)
\(90\) 0 0
\(91\) −32.0000 −0.351648
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) − 67.8823i − 0.714550i
\(96\) 0 0
\(97\) 176.000 1.81443 0.907216 0.420664i \(-0.138203\pi\)
0.907216 + 0.420664i \(0.138203\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) − 29.6985i − 0.294044i −0.989133 0.147022i \(-0.953031\pi\)
0.989133 0.147022i \(-0.0469689\pi\)
\(102\) 0 0
\(103\) 28.0000 0.271845 0.135922 0.990719i \(-0.456600\pi\)
0.135922 + 0.990719i \(0.456600\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) −56.0000 −0.513761 −0.256881 0.966443i \(-0.582695\pi\)
−0.256881 + 0.966443i \(0.582695\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) − 156.978i − 1.38918i −0.719404 0.694592i \(-0.755585\pi\)
0.719404 0.694592i \(-0.244415\pi\)
\(114\) 0 0
\(115\) −72.0000 −0.626087
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) − 50.9117i − 0.427829i
\(120\) 0 0
\(121\) −167.000 −1.38017
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 135.765i 1.08612i
\(126\) 0 0
\(127\) −92.0000 −0.724409 −0.362205 0.932099i \(-0.617976\pi\)
−0.362205 + 0.932099i \(0.617976\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) − 169.706i − 1.29546i −0.761869 0.647731i \(-0.775718\pi\)
0.761869 0.647731i \(-0.224282\pi\)
\(132\) 0 0
\(133\) −64.0000 −0.481203
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 156.978i 1.14582i 0.819617 + 0.572911i \(0.194186\pi\)
−0.819617 + 0.572911i \(0.805814\pi\)
\(138\) 0 0
\(139\) 152.000 1.09353 0.546763 0.837288i \(-0.315860\pi\)
0.546763 + 0.837288i \(0.315860\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) − 135.765i − 0.949402i
\(144\) 0 0
\(145\) 18.0000 0.124138
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) − 275.772i − 1.85082i −0.378972 0.925408i \(-0.623722\pi\)
0.378972 0.925408i \(-0.376278\pi\)
\(150\) 0 0
\(151\) 148.000 0.980132 0.490066 0.871685i \(-0.336973\pi\)
0.490066 + 0.871685i \(0.336973\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) − 186.676i − 1.20436i
\(156\) 0 0
\(157\) 82.0000 0.522293 0.261146 0.965299i \(-0.415899\pi\)
0.261146 + 0.965299i \(0.415899\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 67.8823i 0.421629i
\(162\) 0 0
\(163\) 56.0000 0.343558 0.171779 0.985135i \(-0.445048\pi\)
0.171779 + 0.985135i \(0.445048\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 33.9411i − 0.203240i −0.994823 0.101620i \(-0.967597\pi\)
0.994823 0.101620i \(-0.0324026\pi\)
\(168\) 0 0
\(169\) −105.000 −0.621302
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 173.948i 1.00548i 0.864437 + 0.502741i \(0.167675\pi\)
−0.864437 + 0.502741i \(0.832325\pi\)
\(174\) 0 0
\(175\) 28.0000 0.160000
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) − 203.647i − 1.13769i −0.822444 0.568846i \(-0.807390\pi\)
0.822444 0.568846i \(-0.192610\pi\)
\(180\) 0 0
\(181\) 232.000 1.28177 0.640884 0.767638i \(-0.278568\pi\)
0.640884 + 0.767638i \(0.278568\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 144.250i 0.779729i
\(186\) 0 0
\(187\) 216.000 1.15508
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 33.9411i 0.177702i 0.996045 + 0.0888511i \(0.0283195\pi\)
−0.996045 + 0.0888511i \(0.971680\pi\)
\(192\) 0 0
\(193\) 206.000 1.06736 0.533679 0.845687i \(-0.320809\pi\)
0.533679 + 0.845687i \(0.320809\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 165.463i − 0.839914i −0.907544 0.419957i \(-0.862045\pi\)
0.907544 0.419957i \(-0.137955\pi\)
\(198\) 0 0
\(199\) −20.0000 −0.100503 −0.0502513 0.998737i \(-0.516002\pi\)
−0.0502513 + 0.998737i \(0.516002\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) − 16.9706i − 0.0835988i
\(204\) 0 0
\(205\) −198.000 −0.965854
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) − 271.529i − 1.29918i
\(210\) 0 0
\(211\) 296.000 1.40284 0.701422 0.712746i \(-0.252549\pi\)
0.701422 + 0.712746i \(0.252549\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) − 169.706i − 0.789328i
\(216\) 0 0
\(217\) −176.000 −0.811060
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 101.823i 0.460739i
\(222\) 0 0
\(223\) 436.000 1.95516 0.977578 0.210571i \(-0.0675325\pi\)
0.977578 + 0.210571i \(0.0675325\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 16.9706i 0.0747602i 0.999301 + 0.0373801i \(0.0119012\pi\)
−0.999301 + 0.0373801i \(0.988099\pi\)
\(228\) 0 0
\(229\) −8.00000 −0.0349345 −0.0174672 0.999847i \(-0.505560\pi\)
−0.0174672 + 0.999847i \(0.505560\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 12.7279i − 0.0546263i −0.999627 0.0273131i \(-0.991305\pi\)
0.999627 0.0273131i \(-0.00869512\pi\)
\(234\) 0 0
\(235\) −360.000 −1.53191
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) − 135.765i − 0.568052i −0.958817 0.284026i \(-0.908330\pi\)
0.958817 0.284026i \(-0.0916703\pi\)
\(240\) 0 0
\(241\) 32.0000 0.132780 0.0663900 0.997794i \(-0.478852\pi\)
0.0663900 + 0.997794i \(0.478852\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) − 140.007i − 0.571458i
\(246\) 0 0
\(247\) 128.000 0.518219
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 50.9117i 0.202835i 0.994844 + 0.101418i \(0.0323379\pi\)
−0.994844 + 0.101418i \(0.967662\pi\)
\(252\) 0 0
\(253\) −288.000 −1.13834
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 182.434i − 0.709858i −0.934893 0.354929i \(-0.884505\pi\)
0.934893 0.354929i \(-0.115495\pi\)
\(258\) 0 0
\(259\) 136.000 0.525097
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) − 373.352i − 1.41959i −0.704408 0.709795i \(-0.748787\pi\)
0.704408 0.709795i \(-0.251213\pi\)
\(264\) 0 0
\(265\) 162.000 0.611321
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 343.654i 1.27752i 0.769404 + 0.638762i \(0.220553\pi\)
−0.769404 + 0.638762i \(0.779447\pi\)
\(270\) 0 0
\(271\) −380.000 −1.40221 −0.701107 0.713056i \(-0.747310\pi\)
−0.701107 + 0.713056i \(0.747310\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 118.794i 0.431978i
\(276\) 0 0
\(277\) 328.000 1.18412 0.592058 0.805896i \(-0.298316\pi\)
0.592058 + 0.805896i \(0.298316\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 284.257i 1.01159i 0.862654 + 0.505795i \(0.168801\pi\)
−0.862654 + 0.505795i \(0.831199\pi\)
\(282\) 0 0
\(283\) −208.000 −0.734982 −0.367491 0.930027i \(-0.619783\pi\)
−0.367491 + 0.930027i \(0.619783\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 186.676i 0.650440i
\(288\) 0 0
\(289\) 127.000 0.439446
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 436.992i 1.49144i 0.666259 + 0.745720i \(0.267894\pi\)
−0.666259 + 0.745720i \(0.732106\pi\)
\(294\) 0 0
\(295\) −144.000 −0.488136
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) − 135.765i − 0.454062i
\(300\) 0 0
\(301\) −160.000 −0.531561
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) − 212.132i − 0.695515i
\(306\) 0 0
\(307\) −520.000 −1.69381 −0.846906 0.531743i \(-0.821537\pi\)
−0.846906 + 0.531743i \(0.821537\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 373.352i 1.20049i 0.799816 + 0.600245i \(0.204930\pi\)
−0.799816 + 0.600245i \(0.795070\pi\)
\(312\) 0 0
\(313\) −94.0000 −0.300319 −0.150160 0.988662i \(-0.547979\pi\)
−0.150160 + 0.988662i \(0.547979\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 335.169i − 1.05731i −0.848835 0.528657i \(-0.822696\pi\)
0.848835 0.528657i \(-0.177304\pi\)
\(318\) 0 0
\(319\) 72.0000 0.225705
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 203.647i 0.630485i
\(324\) 0 0
\(325\) −56.0000 −0.172308
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 339.411i 1.03165i
\(330\) 0 0
\(331\) 536.000 1.61934 0.809668 0.586889i \(-0.199647\pi\)
0.809668 + 0.586889i \(0.199647\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 33.9411i 0.101317i
\(336\) 0 0
\(337\) −208.000 −0.617211 −0.308605 0.951190i \(-0.599862\pi\)
−0.308605 + 0.951190i \(0.599862\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) − 746.705i − 2.18975i
\(342\) 0 0
\(343\) −328.000 −0.956268
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 288.500i 0.831411i 0.909499 + 0.415705i \(0.136465\pi\)
−0.909499 + 0.415705i \(0.863535\pi\)
\(348\) 0 0
\(349\) 238.000 0.681948 0.340974 0.940073i \(-0.389243\pi\)
0.340974 + 0.940073i \(0.389243\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 224.860i − 0.636997i −0.947923 0.318499i \(-0.896821\pi\)
0.947923 0.318499i \(-0.103179\pi\)
\(354\) 0 0
\(355\) −216.000 −0.608451
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 560.029i 1.55997i 0.625799 + 0.779984i \(0.284773\pi\)
−0.625799 + 0.779984i \(0.715227\pi\)
\(360\) 0 0
\(361\) −105.000 −0.290859
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) − 67.8823i − 0.185979i
\(366\) 0 0
\(367\) −284.000 −0.773842 −0.386921 0.922113i \(-0.626461\pi\)
−0.386921 + 0.922113i \(0.626461\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) − 152.735i − 0.411685i
\(372\) 0 0
\(373\) 190.000 0.509383 0.254692 0.967022i \(-0.418026\pi\)
0.254692 + 0.967022i \(0.418026\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 33.9411i 0.0900295i
\(378\) 0 0
\(379\) −160.000 −0.422164 −0.211082 0.977468i \(-0.567699\pi\)
−0.211082 + 0.977468i \(0.567699\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 271.529i 0.708953i 0.935065 + 0.354477i \(0.115341\pi\)
−0.935065 + 0.354477i \(0.884659\pi\)
\(384\) 0 0
\(385\) −288.000 −0.748052
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 403.051i 1.03612i 0.855344 + 0.518060i \(0.173346\pi\)
−0.855344 + 0.518060i \(0.826654\pi\)
\(390\) 0 0
\(391\) 216.000 0.552430
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 322.441i 0.816306i
\(396\) 0 0
\(397\) −146.000 −0.367758 −0.183879 0.982949i \(-0.558865\pi\)
−0.183879 + 0.982949i \(0.558865\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 326.683i 0.814672i 0.913278 + 0.407336i \(0.133542\pi\)
−0.913278 + 0.407336i \(0.866458\pi\)
\(402\) 0 0
\(403\) 352.000 0.873449
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 576.999i 1.41769i
\(408\) 0 0
\(409\) 368.000 0.899756 0.449878 0.893090i \(-0.351468\pi\)
0.449878 + 0.893090i \(0.351468\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 135.765i 0.328728i
\(414\) 0 0
\(415\) −504.000 −1.21446
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 390.323i 0.931558i 0.884901 + 0.465779i \(0.154226\pi\)
−0.884901 + 0.465779i \(0.845774\pi\)
\(420\) 0 0
\(421\) 40.0000 0.0950119 0.0475059 0.998871i \(-0.484873\pi\)
0.0475059 + 0.998871i \(0.484873\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) − 89.0955i − 0.209636i
\(426\) 0 0
\(427\) −200.000 −0.468384
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 152.735i 0.354374i 0.984177 + 0.177187i \(0.0566997\pi\)
−0.984177 + 0.177187i \(0.943300\pi\)
\(432\) 0 0
\(433\) 542.000 1.25173 0.625866 0.779931i \(-0.284746\pi\)
0.625866 + 0.779931i \(0.284746\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 271.529i − 0.621348i
\(438\) 0 0
\(439\) 4.00000 0.00911162 0.00455581 0.999990i \(-0.498550\pi\)
0.00455581 + 0.999990i \(0.498550\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 322.441i 0.727857i 0.931427 + 0.363929i \(0.118565\pi\)
−0.931427 + 0.363929i \(0.881435\pi\)
\(444\) 0 0
\(445\) −54.0000 −0.121348
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 216.375i 0.481904i 0.970537 + 0.240952i \(0.0774596\pi\)
−0.970537 + 0.240952i \(0.922540\pi\)
\(450\) 0 0
\(451\) −792.000 −1.75610
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) − 135.765i − 0.298384i
\(456\) 0 0
\(457\) −400.000 −0.875274 −0.437637 0.899152i \(-0.644184\pi\)
−0.437637 + 0.899152i \(0.644184\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 301.227i 0.653422i 0.945124 + 0.326711i \(0.105940\pi\)
−0.945124 + 0.326711i \(0.894060\pi\)
\(462\) 0 0
\(463\) 604.000 1.30454 0.652268 0.757989i \(-0.273818\pi\)
0.652268 + 0.757989i \(0.273818\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 356.382i 0.763130i 0.924342 + 0.381565i \(0.124615\pi\)
−0.924342 + 0.381565i \(0.875385\pi\)
\(468\) 0 0
\(469\) 32.0000 0.0682303
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) − 678.823i − 1.43514i
\(474\) 0 0
\(475\) −112.000 −0.235789
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) − 526.087i − 1.09830i −0.835723 0.549152i \(-0.814951\pi\)
0.835723 0.549152i \(-0.185049\pi\)
\(480\) 0 0
\(481\) −272.000 −0.565489
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 746.705i 1.53960i
\(486\) 0 0
\(487\) −596.000 −1.22382 −0.611910 0.790928i \(-0.709598\pi\)
−0.611910 + 0.790928i \(0.709598\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) − 271.529i − 0.553012i −0.961012 0.276506i \(-0.910823\pi\)
0.961012 0.276506i \(-0.0891766\pi\)
\(492\) 0 0
\(493\) −54.0000 −0.109533
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 203.647i 0.409752i
\(498\) 0 0
\(499\) 224.000 0.448898 0.224449 0.974486i \(-0.427942\pi\)
0.224449 + 0.974486i \(0.427942\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) − 865.499i − 1.72067i −0.509726 0.860337i \(-0.670253\pi\)
0.509726 0.860337i \(-0.329747\pi\)
\(504\) 0 0
\(505\) 126.000 0.249505
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) − 479.418i − 0.941883i −0.882164 0.470941i \(-0.843914\pi\)
0.882164 0.470941i \(-0.156086\pi\)
\(510\) 0 0
\(511\) −64.0000 −0.125245
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 118.794i 0.230668i
\(516\) 0 0
\(517\) −1440.00 −2.78530
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) − 521.845i − 1.00162i −0.865557 0.500811i \(-0.833035\pi\)
0.865557 0.500811i \(-0.166965\pi\)
\(522\) 0 0
\(523\) −736.000 −1.40727 −0.703633 0.710564i \(-0.748440\pi\)
−0.703633 + 0.710564i \(0.748440\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 560.029i 1.06267i
\(528\) 0 0
\(529\) 241.000 0.455577
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) − 373.352i − 0.700474i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) − 560.029i − 1.03901i
\(540\) 0 0
\(541\) 808.000 1.49353 0.746765 0.665088i \(-0.231606\pi\)
0.746765 + 0.665088i \(0.231606\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) − 237.588i − 0.435941i
\(546\) 0 0
\(547\) 536.000 0.979890 0.489945 0.871753i \(-0.337017\pi\)
0.489945 + 0.871753i \(0.337017\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 67.8823i 0.123198i
\(552\) 0 0
\(553\) 304.000 0.549729
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 165.463i 0.297061i 0.988908 + 0.148531i \(0.0474543\pi\)
−0.988908 + 0.148531i \(0.952546\pi\)
\(558\) 0 0
\(559\) 320.000 0.572451
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 322.441i − 0.572719i −0.958122 0.286359i \(-0.907555\pi\)
0.958122 0.286359i \(-0.0924451\pi\)
\(564\) 0 0
\(565\) 666.000 1.17876
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 156.978i 0.275883i 0.990440 + 0.137942i \(0.0440487\pi\)
−0.990440 + 0.137942i \(0.955951\pi\)
\(570\) 0 0
\(571\) 368.000 0.644483 0.322242 0.946657i \(-0.395564\pi\)
0.322242 + 0.946657i \(0.395564\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 118.794i 0.206598i
\(576\) 0 0
\(577\) −142.000 −0.246101 −0.123050 0.992400i \(-0.539268\pi\)
−0.123050 + 0.992400i \(0.539268\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 475.176i 0.817858i
\(582\) 0 0
\(583\) 648.000 1.11149
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 373.352i 0.636035i 0.948085 + 0.318017i \(0.103017\pi\)
−0.948085 + 0.318017i \(0.896983\pi\)
\(588\) 0 0
\(589\) 704.000 1.19525
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 1107.33i 1.86733i 0.358142 + 0.933667i \(0.383410\pi\)
−0.358142 + 0.933667i \(0.616590\pi\)
\(594\) 0 0
\(595\) 216.000 0.363025
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) − 797.616i − 1.33158i −0.746139 0.665790i \(-0.768095\pi\)
0.746139 0.665790i \(-0.231905\pi\)
\(600\) 0 0
\(601\) 158.000 0.262895 0.131448 0.991323i \(-0.458037\pi\)
0.131448 + 0.991323i \(0.458037\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) − 708.521i − 1.17111i
\(606\) 0 0
\(607\) −332.000 −0.546952 −0.273476 0.961879i \(-0.588173\pi\)
−0.273476 + 0.961879i \(0.588173\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) − 678.823i − 1.11100i
\(612\) 0 0
\(613\) −578.000 −0.942904 −0.471452 0.881892i \(-0.656270\pi\)
−0.471452 + 0.881892i \(0.656270\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 55.1543i − 0.0893911i −0.999001 0.0446956i \(-0.985768\pi\)
0.999001 0.0446956i \(-0.0142318\pi\)
\(618\) 0 0
\(619\) 896.000 1.44750 0.723748 0.690064i \(-0.242418\pi\)
0.723748 + 0.690064i \(0.242418\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 50.9117i 0.0817202i
\(624\) 0 0
\(625\) −401.000 −0.641600
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 432.749i − 0.687996i
\(630\) 0 0
\(631\) −20.0000 −0.0316957 −0.0158479 0.999874i \(-0.505045\pi\)
−0.0158479 + 0.999874i \(0.505045\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) − 390.323i − 0.614682i
\(636\) 0 0
\(637\) 264.000 0.414443
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 258.801i 0.403746i 0.979412 + 0.201873i \(0.0647028\pi\)
−0.979412 + 0.201873i \(0.935297\pi\)
\(642\) 0 0
\(643\) 728.000 1.13219 0.566096 0.824339i \(-0.308453\pi\)
0.566096 + 0.824339i \(0.308453\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 458.205i 0.708200i 0.935208 + 0.354100i \(0.115213\pi\)
−0.935208 + 0.354100i \(0.884787\pi\)
\(648\) 0 0
\(649\) −576.000 −0.887519
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 301.227i − 0.461298i −0.973037 0.230649i \(-0.925915\pi\)
0.973037 0.230649i \(-0.0740849\pi\)
\(654\) 0 0
\(655\) 720.000 1.09924
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) − 1052.17i − 1.59662i −0.602244 0.798312i \(-0.705727\pi\)
0.602244 0.798312i \(-0.294273\pi\)
\(660\) 0 0
\(661\) −62.0000 −0.0937973 −0.0468986 0.998900i \(-0.514934\pi\)
−0.0468986 + 0.998900i \(0.514934\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) − 271.529i − 0.408314i
\(666\) 0 0
\(667\) 72.0000 0.107946
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) − 848.528i − 1.26457i
\(672\) 0 0
\(673\) −670.000 −0.995542 −0.497771 0.867308i \(-0.665848\pi\)
−0.497771 + 0.867308i \(0.665848\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 1294.01i 1.91138i 0.294372 + 0.955691i \(0.404889\pi\)
−0.294372 + 0.955691i \(0.595111\pi\)
\(678\) 0 0
\(679\) 704.000 1.03682
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) − 560.029i − 0.819954i −0.912096 0.409977i \(-0.865537\pi\)
0.912096 0.409977i \(-0.134463\pi\)
\(684\) 0 0
\(685\) −666.000 −0.972263
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 305.470i 0.443353i
\(690\) 0 0
\(691\) −40.0000 −0.0578871 −0.0289436 0.999581i \(-0.509214\pi\)
−0.0289436 + 0.999581i \(0.509214\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 644.881i 0.927887i
\(696\) 0 0
\(697\) 594.000 0.852224
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) − 954.594i − 1.36176i −0.732395 0.680880i \(-0.761597\pi\)
0.732395 0.680880i \(-0.238403\pi\)
\(702\) 0 0
\(703\) −544.000 −0.773826
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 118.794i − 0.168025i
\(708\) 0 0
\(709\) −968.000 −1.36530 −0.682652 0.730744i \(-0.739173\pi\)
−0.682652 + 0.730744i \(0.739173\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) − 746.705i − 1.04727i
\(714\) 0 0
\(715\) 576.000 0.805594
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) − 1170.97i − 1.62861i −0.580439 0.814304i \(-0.697119\pi\)
0.580439 0.814304i \(-0.302881\pi\)
\(720\) 0 0
\(721\) 112.000 0.155340
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) − 29.6985i − 0.0409634i
\(726\) 0 0
\(727\) 508.000 0.698762 0.349381 0.936981i \(-0.386392\pi\)
0.349381 + 0.936981i \(0.386392\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 509.117i 0.696466i
\(732\) 0 0
\(733\) 1144.00 1.56071 0.780355 0.625337i \(-0.215039\pi\)
0.780355 + 0.625337i \(0.215039\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 135.765i 0.184212i
\(738\) 0 0
\(739\) −304.000 −0.411367 −0.205683 0.978619i \(-0.565942\pi\)
−0.205683 + 0.978619i \(0.565942\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) − 848.528i − 1.14203i −0.820940 0.571015i \(-0.806550\pi\)
0.820940 0.571015i \(-0.193450\pi\)
\(744\) 0 0
\(745\) 1170.00 1.57047
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −188.000 −0.250333 −0.125166 0.992136i \(-0.539946\pi\)
−0.125166 + 0.992136i \(0.539946\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 627.911i 0.831670i
\(756\) 0 0
\(757\) 1240.00 1.63804 0.819022 0.573761i \(-0.194516\pi\)
0.819022 + 0.573761i \(0.194516\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) − 156.978i − 0.206278i −0.994667 0.103139i \(-0.967111\pi\)
0.994667 0.103139i \(-0.0328887\pi\)
\(762\) 0 0
\(763\) −224.000 −0.293578
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 271.529i − 0.354014i
\(768\) 0 0
\(769\) −910.000 −1.18336 −0.591678 0.806175i \(-0.701534\pi\)
−0.591678 + 0.806175i \(0.701534\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 1387.34i 1.79475i 0.441266 + 0.897376i \(0.354529\pi\)
−0.441266 + 0.897376i \(0.645471\pi\)
\(774\) 0 0
\(775\) −308.000 −0.397419
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 746.705i − 0.958543i
\(780\) 0 0
\(781\) −864.000 −1.10627
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 347.897i 0.443180i
\(786\) 0 0
\(787\) −1360.00 −1.72808 −0.864041 0.503422i \(-0.832074\pi\)
−0.864041 + 0.503422i \(0.832074\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) − 627.911i − 0.793819i
\(792\) 0 0
\(793\) 400.000 0.504414
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 106.066i 0.133082i 0.997784 + 0.0665408i \(0.0211963\pi\)
−0.997784 + 0.0665408i \(0.978804\pi\)
\(798\) 0 0
\(799\) 1080.00 1.35169
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) − 271.529i − 0.338143i
\(804\) 0 0
\(805\) −288.000 −0.357764
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) − 1107.33i − 1.36876i −0.729124 0.684381i \(-0.760072\pi\)
0.729124 0.684381i \(-0.239928\pi\)
\(810\) 0 0
\(811\) −160.000 −0.197287 −0.0986436 0.995123i \(-0.531450\pi\)
−0.0986436 + 0.995123i \(0.531450\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 237.588i 0.291519i
\(816\) 0 0
\(817\) 640.000 0.783354
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 436.992i − 0.532268i −0.963936 0.266134i \(-0.914254\pi\)
0.963936 0.266134i \(-0.0857464\pi\)
\(822\) 0 0
\(823\) −332.000 −0.403402 −0.201701 0.979447i \(-0.564647\pi\)
−0.201701 + 0.979447i \(0.564647\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 101.823i 0.123124i 0.998103 + 0.0615619i \(0.0196082\pi\)
−0.998103 + 0.0615619i \(0.980392\pi\)
\(828\) 0 0
\(829\) −632.000 −0.762364 −0.381182 0.924500i \(-0.624483\pi\)
−0.381182 + 0.924500i \(0.624483\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 420.021i 0.504227i
\(834\) 0 0
\(835\) 144.000 0.172455
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) − 729.734i − 0.869767i −0.900487 0.434883i \(-0.856790\pi\)
0.900487 0.434883i \(-0.143210\pi\)
\(840\) 0 0
\(841\) 823.000 0.978597
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) − 445.477i − 0.527192i
\(846\) 0 0
\(847\) −668.000 −0.788666
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 576.999i 0.678025i
\(852\) 0 0
\(853\) −446.000 −0.522860 −0.261430 0.965222i \(-0.584194\pi\)
−0.261430 + 0.965222i \(0.584194\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 428.507i − 0.500008i −0.968245 0.250004i \(-0.919568\pi\)
0.968245 0.250004i \(-0.0804319\pi\)
\(858\) 0 0
\(859\) 728.000 0.847497 0.423749 0.905780i \(-0.360714\pi\)
0.423749 + 0.905780i \(0.360714\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 916.410i 1.06189i 0.847407 + 0.530945i \(0.178163\pi\)
−0.847407 + 0.530945i \(0.821837\pi\)
\(864\) 0 0
\(865\) −738.000 −0.853179
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 1289.76i 1.48419i
\(870\) 0 0
\(871\) −64.0000 −0.0734788
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 543.058i 0.620638i
\(876\) 0 0
\(877\) 910.000 1.03763 0.518814 0.854887i \(-0.326374\pi\)
0.518814 + 0.854887i \(0.326374\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) − 929.138i − 1.05464i −0.849667 0.527320i \(-0.823197\pi\)
0.849667 0.527320i \(-0.176803\pi\)
\(882\) 0 0
\(883\) 1064.00 1.20498 0.602492 0.798125i \(-0.294175\pi\)
0.602492 + 0.798125i \(0.294175\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 1391.59i 1.56887i 0.620212 + 0.784434i \(0.287047\pi\)
−0.620212 + 0.784434i \(0.712953\pi\)
\(888\) 0 0
\(889\) −368.000 −0.413948
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 1357.65i − 1.52032i
\(894\) 0 0
\(895\) 864.000 0.965363
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 186.676i 0.207649i
\(900\) 0 0
\(901\) −486.000 −0.539401
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 984.293i 1.08762i
\(906\) 0 0
\(907\) −1768.00 −1.94928 −0.974642 0.223771i \(-0.928163\pi\)
−0.974642 + 0.223771i \(0.928163\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 237.588i 0.260799i 0.991462 + 0.130399i \(0.0416260\pi\)
−0.991462 + 0.130399i \(0.958374\pi\)
\(912\) 0 0
\(913\) −2016.00 −2.20811
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 678.823i − 0.740264i
\(918\) 0 0
\(919\) −380.000 −0.413493 −0.206746 0.978395i \(-0.566288\pi\)
−0.206746 + 0.978395i \(0.566288\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) − 407.294i − 0.441271i
\(924\) 0 0
\(925\) 238.000 0.257297
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 666.095i 0.717002i 0.933529 + 0.358501i \(0.116712\pi\)
−0.933529 + 0.358501i \(0.883288\pi\)
\(930\) 0 0
\(931\) 528.000 0.567132
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 916.410i 0.980118i
\(936\) 0 0
\(937\) −178.000 −0.189968 −0.0949840 0.995479i \(-0.530280\pi\)
−0.0949840 + 0.995479i \(0.530280\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 436.992i 0.464391i 0.972669 + 0.232196i \(0.0745909\pi\)
−0.972669 + 0.232196i \(0.925409\pi\)
\(942\) 0 0
\(943\) −792.000 −0.839873
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 1798.88i 1.89956i 0.312924 + 0.949778i \(0.398691\pi\)
−0.312924 + 0.949778i \(0.601309\pi\)
\(948\) 0 0
\(949\) 128.000 0.134879
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 1310.98i 1.37563i 0.725886 + 0.687815i \(0.241430\pi\)
−0.725886 + 0.687815i \(0.758570\pi\)
\(954\) 0 0
\(955\) −144.000 −0.150785
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 627.911i 0.654756i
\(960\) 0 0
\(961\) 975.000 1.01457
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 873.984i 0.905683i
\(966\) 0 0
\(967\) −1700.00 −1.75801 −0.879007 0.476808i \(-0.841794\pi\)
−0.879007 + 0.476808i \(0.841794\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 458.205i 0.471890i 0.971766 + 0.235945i \(0.0758185\pi\)
−0.971766 + 0.235945i \(0.924181\pi\)
\(972\) 0 0
\(973\) 608.000 0.624872
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 759.433i 0.777311i 0.921383 + 0.388655i \(0.127060\pi\)
−0.921383 + 0.388655i \(0.872940\pi\)
\(978\) 0 0
\(979\) −216.000 −0.220633
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 1052.17i 1.07037i 0.844734 + 0.535186i \(0.179758\pi\)
−0.844734 + 0.535186i \(0.820242\pi\)
\(984\) 0 0
\(985\) 702.000 0.712690
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) − 678.823i − 0.686373i
\(990\) 0 0
\(991\) 772.000 0.779011 0.389506 0.921024i \(-0.372646\pi\)
0.389506 + 0.921024i \(0.372646\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) − 84.8528i − 0.0852792i
\(996\) 0 0
\(997\) −194.000 −0.194584 −0.0972919 0.995256i \(-0.531018\pi\)
−0.0972919 + 0.995256i \(0.531018\pi\)
\(998\) 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 576.3.e.f.449.2 2
3.2 odd 2 inner 576.3.e.f.449.1 2
4.3 odd 2 576.3.e.c.449.2 2
8.3 odd 2 18.3.b.a.17.2 yes 2
8.5 even 2 144.3.e.b.17.1 2
12.11 even 2 576.3.e.c.449.1 2
16.3 odd 4 2304.3.h.f.2177.3 4
16.5 even 4 2304.3.h.c.2177.2 4
16.11 odd 4 2304.3.h.f.2177.2 4
16.13 even 4 2304.3.h.c.2177.3 4
24.5 odd 2 144.3.e.b.17.2 2
24.11 even 2 18.3.b.a.17.1 2
40.3 even 4 450.3.b.b.449.4 4
40.13 odd 4 3600.3.c.b.449.1 4
40.19 odd 2 450.3.d.f.251.1 2
40.27 even 4 450.3.b.b.449.1 4
40.29 even 2 3600.3.l.d.1601.1 2
40.37 odd 4 3600.3.c.b.449.3 4
48.5 odd 4 2304.3.h.c.2177.4 4
48.11 even 4 2304.3.h.f.2177.4 4
48.29 odd 4 2304.3.h.c.2177.1 4
48.35 even 4 2304.3.h.f.2177.1 4
56.3 even 6 882.3.s.d.863.1 4
56.11 odd 6 882.3.s.b.863.1 4
56.19 even 6 882.3.s.d.557.2 4
56.27 even 2 882.3.b.a.197.2 2
56.51 odd 6 882.3.s.b.557.2 4
72.5 odd 6 1296.3.q.f.593.2 4
72.11 even 6 162.3.d.b.53.1 4
72.13 even 6 1296.3.q.f.593.1 4
72.29 odd 6 1296.3.q.f.1025.1 4
72.43 odd 6 162.3.d.b.53.2 4
72.59 even 6 162.3.d.b.107.2 4
72.61 even 6 1296.3.q.f.1025.2 4
72.67 odd 6 162.3.d.b.107.1 4
88.43 even 2 2178.3.c.d.485.1 2
104.51 odd 2 3042.3.c.e.1691.1 2
104.83 even 4 3042.3.d.a.3041.3 4
104.99 even 4 3042.3.d.a.3041.2 4
120.29 odd 2 3600.3.l.d.1601.2 2
120.53 even 4 3600.3.c.b.449.2 4
120.59 even 2 450.3.d.f.251.2 2
120.77 even 4 3600.3.c.b.449.4 4
120.83 odd 4 450.3.b.b.449.2 4
120.107 odd 4 450.3.b.b.449.3 4
168.11 even 6 882.3.s.b.863.2 4
168.59 odd 6 882.3.s.d.863.2 4
168.83 odd 2 882.3.b.a.197.1 2
168.107 even 6 882.3.s.b.557.1 4
168.131 odd 6 882.3.s.d.557.1 4
264.131 odd 2 2178.3.c.d.485.2 2
312.83 odd 4 3042.3.d.a.3041.1 4
312.155 even 2 3042.3.c.e.1691.2 2
312.203 odd 4 3042.3.d.a.3041.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
18.3.b.a.17.1 2 24.11 even 2
18.3.b.a.17.2 yes 2 8.3 odd 2
144.3.e.b.17.1 2 8.5 even 2
144.3.e.b.17.2 2 24.5 odd 2
162.3.d.b.53.1 4 72.11 even 6
162.3.d.b.53.2 4 72.43 odd 6
162.3.d.b.107.1 4 72.67 odd 6
162.3.d.b.107.2 4 72.59 even 6
450.3.b.b.449.1 4 40.27 even 4
450.3.b.b.449.2 4 120.83 odd 4
450.3.b.b.449.3 4 120.107 odd 4
450.3.b.b.449.4 4 40.3 even 4
450.3.d.f.251.1 2 40.19 odd 2
450.3.d.f.251.2 2 120.59 even 2
576.3.e.c.449.1 2 12.11 even 2
576.3.e.c.449.2 2 4.3 odd 2
576.3.e.f.449.1 2 3.2 odd 2 inner
576.3.e.f.449.2 2 1.1 even 1 trivial
882.3.b.a.197.1 2 168.83 odd 2
882.3.b.a.197.2 2 56.27 even 2
882.3.s.b.557.1 4 168.107 even 6
882.3.s.b.557.2 4 56.51 odd 6
882.3.s.b.863.1 4 56.11 odd 6
882.3.s.b.863.2 4 168.11 even 6
882.3.s.d.557.1 4 168.131 odd 6
882.3.s.d.557.2 4 56.19 even 6
882.3.s.d.863.1 4 56.3 even 6
882.3.s.d.863.2 4 168.59 odd 6
1296.3.q.f.593.1 4 72.13 even 6
1296.3.q.f.593.2 4 72.5 odd 6
1296.3.q.f.1025.1 4 72.29 odd 6
1296.3.q.f.1025.2 4 72.61 even 6
2178.3.c.d.485.1 2 88.43 even 2
2178.3.c.d.485.2 2 264.131 odd 2
2304.3.h.c.2177.1 4 48.29 odd 4
2304.3.h.c.2177.2 4 16.5 even 4
2304.3.h.c.2177.3 4 16.13 even 4
2304.3.h.c.2177.4 4 48.5 odd 4
2304.3.h.f.2177.1 4 48.35 even 4
2304.3.h.f.2177.2 4 16.11 odd 4
2304.3.h.f.2177.3 4 16.3 odd 4
2304.3.h.f.2177.4 4 48.11 even 4
3042.3.c.e.1691.1 2 104.51 odd 2
3042.3.c.e.1691.2 2 312.155 even 2
3042.3.d.a.3041.1 4 312.83 odd 4
3042.3.d.a.3041.2 4 104.99 even 4
3042.3.d.a.3041.3 4 104.83 even 4
3042.3.d.a.3041.4 4 312.203 odd 4
3600.3.c.b.449.1 4 40.13 odd 4
3600.3.c.b.449.2 4 120.53 even 4
3600.3.c.b.449.3 4 40.37 odd 4
3600.3.c.b.449.4 4 120.77 even 4
3600.3.l.d.1601.1 2 40.29 even 2
3600.3.l.d.1601.2 2 120.29 odd 2