Properties

Label 1152.2.k.c.289.2
Level $1152$
Weight $2$
Character 1152.289
Analytic conductor $9.199$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1152 = 2^{7} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1152.k (of order \(4\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.19876631285\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.18939904.2
Defining polynomial: \(x^{8} - 4 x^{7} + 14 x^{6} - 28 x^{5} + 43 x^{4} - 44 x^{3} + 30 x^{2} - 12 x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 48)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 289.2
Root \(0.500000 + 1.44392i\) of defining polynomial
Character \(\chi\) \(=\) 1152.289
Dual form 1152.2.k.c.865.2

$q$-expansion

\(f(q)\) \(=\) \(q+(-0.334904 + 0.334904i) q^{5} -4.55765i q^{7} +O(q^{10})\) \(q+(-0.334904 + 0.334904i) q^{5} -4.55765i q^{7} +(-2.47363 + 2.47363i) q^{11} +(0.0594122 + 0.0594122i) q^{13} -3.61706 q^{17} +(-2.55765 - 2.55765i) q^{19} -2.82843i q^{23} +4.77568i q^{25} +(-5.16333 - 5.16333i) q^{29} -0.557647 q^{31} +(1.52637 + 1.52637i) q^{35} +(-4.38607 + 4.38607i) q^{37} +9.27391i q^{41} +(1.61040 - 1.61040i) q^{43} -2.82843 q^{47} -13.7721 q^{49} +(-0.493523 + 0.493523i) q^{53} -1.65685i q^{55} +(4.00000 - 4.00000i) q^{59} +(-2.72922 - 2.72922i) q^{61} -0.0397948 q^{65} +(-3.77568 - 3.77568i) q^{67} -9.11529i q^{71} -0.541560i q^{73} +(11.2739 + 11.2739i) q^{77} -10.9937 q^{79} +(-10.6417 - 10.6417i) q^{83} +(1.21137 - 1.21137i) q^{85} -14.6533i q^{89} +(0.270780 - 0.270780i) q^{91} +1.71313 q^{95} +4.31724 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q - 8q^{11} + 8q^{19} - 16q^{29} + 24q^{31} + 24q^{35} + 16q^{37} + 8q^{43} - 8q^{49} + 16q^{53} + 32q^{59} - 16q^{61} + 16q^{65} + 16q^{67} + 16q^{77} - 24q^{79} - 40q^{83} + 16q^{85} + 8q^{91} + 48q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(641\) \(901\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{3}{4}\right)\)

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
\(5\) −0.334904 + 0.334904i −0.149774 + 0.149774i −0.778017 0.628243i \(-0.783774\pi\)
0.628243 + 0.778017i \(0.283774\pi\)
\(6\) 0 0
\(7\) 4.55765i 1.72263i −0.508072 0.861314i \(-0.669642\pi\)
0.508072 0.861314i \(-0.330358\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −2.47363 + 2.47363i −0.745826 + 0.745826i −0.973692 0.227866i \(-0.926825\pi\)
0.227866 + 0.973692i \(0.426825\pi\)
\(12\) 0 0
\(13\) 0.0594122 + 0.0594122i 0.0164780 + 0.0164780i 0.715298 0.698820i \(-0.246291\pi\)
−0.698820 + 0.715298i \(0.746291\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −3.61706 −0.877266 −0.438633 0.898666i \(-0.644537\pi\)
−0.438633 + 0.898666i \(0.644537\pi\)
\(18\) 0 0
\(19\) −2.55765 2.55765i −0.586765 0.586765i 0.349989 0.936754i \(-0.386185\pi\)
−0.936754 + 0.349989i \(0.886185\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2.82843i 0.589768i −0.955533 0.294884i \(-0.904719\pi\)
0.955533 0.294884i \(-0.0952810\pi\)
\(24\) 0 0
\(25\) 4.77568i 0.955136i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −5.16333 5.16333i −0.958807 0.958807i 0.0403780 0.999184i \(-0.487144\pi\)
−0.999184 + 0.0403780i \(0.987144\pi\)
\(30\) 0 0
\(31\) −0.557647 −0.100156 −0.0500782 0.998745i \(-0.515947\pi\)
−0.0500782 + 0.998745i \(0.515947\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 1.52637 + 1.52637i 0.258004 + 0.258004i
\(36\) 0 0
\(37\) −4.38607 + 4.38607i −0.721066 + 0.721066i −0.968822 0.247756i \(-0.920307\pi\)
0.247756 + 0.968822i \(0.420307\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 9.27391i 1.44834i 0.689620 + 0.724171i \(0.257777\pi\)
−0.689620 + 0.724171i \(0.742223\pi\)
\(42\) 0 0
\(43\) 1.61040 1.61040i 0.245583 0.245583i −0.573572 0.819155i \(-0.694443\pi\)
0.819155 + 0.573572i \(0.194443\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −2.82843 −0.412568 −0.206284 0.978492i \(-0.566137\pi\)
−0.206284 + 0.978492i \(0.566137\pi\)
\(48\) 0 0
\(49\) −13.7721 −1.96745
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −0.493523 + 0.493523i −0.0677906 + 0.0677906i −0.740189 0.672399i \(-0.765264\pi\)
0.672399 + 0.740189i \(0.265264\pi\)
\(54\) 0 0
\(55\) 1.65685i 0.223410i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 4.00000 4.00000i 0.520756 0.520756i −0.397044 0.917800i \(-0.629964\pi\)
0.917800 + 0.397044i \(0.129964\pi\)
\(60\) 0 0
\(61\) −2.72922 2.72922i −0.349441 0.349441i 0.510460 0.859901i \(-0.329475\pi\)
−0.859901 + 0.510460i \(0.829475\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −0.0397948 −0.00493593
\(66\) 0 0
\(67\) −3.77568 3.77568i −0.461273 0.461273i 0.437800 0.899072i \(-0.355758\pi\)
−0.899072 + 0.437800i \(0.855758\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 9.11529i 1.08179i −0.841091 0.540893i \(-0.818086\pi\)
0.841091 0.540893i \(-0.181914\pi\)
\(72\) 0 0
\(73\) 0.541560i 0.0633848i −0.999498 0.0316924i \(-0.989910\pi\)
0.999498 0.0316924i \(-0.0100897\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 11.2739 + 11.2739i 1.28478 + 1.28478i
\(78\) 0 0
\(79\) −10.9937 −1.23689 −0.618445 0.785828i \(-0.712237\pi\)
−0.618445 + 0.785828i \(0.712237\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −10.6417 10.6417i −1.16807 1.16807i −0.982660 0.185415i \(-0.940637\pi\)
−0.185415 0.982660i \(-0.559363\pi\)
\(84\) 0 0
\(85\) 1.21137 1.21137i 0.131391 0.131391i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 14.6533i 1.55325i −0.629964 0.776625i \(-0.716930\pi\)
0.629964 0.776625i \(-0.283070\pi\)
\(90\) 0 0
\(91\) 0.270780 0.270780i 0.0283854 0.0283854i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 1.71313 0.175764
\(96\) 0 0
\(97\) 4.31724 0.438349 0.219175 0.975686i \(-0.429664\pi\)
0.219175 + 0.975686i \(0.429664\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −0.453728 + 0.453728i −0.0451477 + 0.0451477i −0.729320 0.684173i \(-0.760164\pi\)
0.684173 + 0.729320i \(0.260164\pi\)
\(102\) 0 0
\(103\) 1.33686i 0.131724i 0.997829 + 0.0658622i \(0.0209798\pi\)
−0.997829 + 0.0658622i \(0.979020\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 6.06255 6.06255i 0.586088 0.586088i −0.350481 0.936570i \(-0.613982\pi\)
0.936570 + 0.350481i \(0.113982\pi\)
\(108\) 0 0
\(109\) −5.71627 5.71627i −0.547519 0.547519i 0.378203 0.925722i \(-0.376542\pi\)
−0.925722 + 0.378203i \(0.876542\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 9.55136 0.898516 0.449258 0.893402i \(-0.351688\pi\)
0.449258 + 0.893402i \(0.351688\pi\)
\(114\) 0 0
\(115\) 0.947252 + 0.947252i 0.0883317 + 0.0883317i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 16.4853i 1.51120i
\(120\) 0 0
\(121\) 1.23765i 0.112514i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −3.27391 3.27391i −0.292828 0.292828i
\(126\) 0 0
\(127\) 5.09921 0.452481 0.226241 0.974071i \(-0.427356\pi\)
0.226241 + 0.974071i \(0.427356\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 2.11882 + 2.11882i 0.185123 + 0.185123i 0.793584 0.608461i \(-0.208213\pi\)
−0.608461 + 0.793584i \(0.708213\pi\)
\(132\) 0 0
\(133\) −11.6569 + 11.6569i −1.01078 + 1.01078i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 3.37941i 0.288723i 0.989525 + 0.144361i \(0.0461127\pi\)
−0.989525 + 0.144361i \(0.953887\pi\)
\(138\) 0 0
\(139\) −5.88118 + 5.88118i −0.498835 + 0.498835i −0.911075 0.412240i \(-0.864746\pi\)
0.412240 + 0.911075i \(0.364746\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −0.293927 −0.0245794
\(144\) 0 0
\(145\) 3.45844 0.287208
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −9.99176 + 9.99176i −0.818557 + 0.818557i −0.985899 0.167342i \(-0.946482\pi\)
0.167342 + 0.985899i \(0.446482\pi\)
\(150\) 0 0
\(151\) 9.97685i 0.811905i 0.913894 + 0.405952i \(0.133060\pi\)
−0.913894 + 0.405952i \(0.866940\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0.186758 0.186758i 0.0150008 0.0150008i
\(156\) 0 0
\(157\) 16.1618 + 16.1618i 1.28985 + 1.28985i 0.934877 + 0.354971i \(0.115509\pi\)
0.354971 + 0.934877i \(0.384491\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −12.8910 −1.01595
\(162\) 0 0
\(163\) 7.50490 + 7.50490i 0.587829 + 0.587829i 0.937043 0.349214i \(-0.113551\pi\)
−0.349214 + 0.937043i \(0.613551\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 5.83822i 0.451775i 0.974153 + 0.225888i \(0.0725282\pi\)
−0.974153 + 0.225888i \(0.927472\pi\)
\(168\) 0 0
\(169\) 12.9929i 0.999457i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −3.62530 3.62530i −0.275627 0.275627i 0.555734 0.831360i \(-0.312437\pi\)
−0.831360 + 0.555734i \(0.812437\pi\)
\(174\) 0 0
\(175\) 21.7659 1.64534
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 9.28334 + 9.28334i 0.693869 + 0.693869i 0.963081 0.269212i \(-0.0867632\pi\)
−0.269212 + 0.963081i \(0.586763\pi\)
\(180\) 0 0
\(181\) 10.8316 10.8316i 0.805104 0.805104i −0.178785 0.983888i \(-0.557217\pi\)
0.983888 + 0.178785i \(0.0572165\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 2.93783i 0.215993i
\(186\) 0 0
\(187\) 8.94725 8.94725i 0.654288 0.654288i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 8.63001 0.624446 0.312223 0.950009i \(-0.398926\pi\)
0.312223 + 0.950009i \(0.398926\pi\)
\(192\) 0 0
\(193\) 11.4514 0.824288 0.412144 0.911119i \(-0.364780\pi\)
0.412144 + 0.911119i \(0.364780\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 7.48999 7.48999i 0.533640 0.533640i −0.388014 0.921654i \(-0.626839\pi\)
0.921654 + 0.388014i \(0.126839\pi\)
\(198\) 0 0
\(199\) 3.68000i 0.260868i −0.991457 0.130434i \(-0.958363\pi\)
0.991457 0.130434i \(-0.0416371\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −23.5326 + 23.5326i −1.65167 + 1.65167i
\(204\) 0 0
\(205\) −3.10587 3.10587i −0.216923 0.216923i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 12.6533 0.875249
\(210\) 0 0
\(211\) −10.1188 10.1188i −0.696609 0.696609i 0.267069 0.963677i \(-0.413945\pi\)
−0.963677 + 0.267069i \(0.913945\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 1.07866i 0.0735637i
\(216\) 0 0
\(217\) 2.54156i 0.172532i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −0.214897 0.214897i −0.0144556 0.0144556i
\(222\) 0 0
\(223\) −4.86156 −0.325554 −0.162777 0.986663i \(-0.552045\pi\)
−0.162777 + 0.986663i \(0.552045\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 10.6417 + 10.6417i 0.706312 + 0.706312i 0.965758 0.259445i \(-0.0835398\pi\)
−0.259445 + 0.965758i \(0.583540\pi\)
\(228\) 0 0
\(229\) 20.1712 20.1712i 1.33295 1.33295i 0.430229 0.902720i \(-0.358433\pi\)
0.902720 0.430229i \(-0.141567\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 13.5702i 0.889014i −0.895775 0.444507i \(-0.853379\pi\)
0.895775 0.444507i \(-0.146621\pi\)
\(234\) 0 0
\(235\) 0.947252 0.947252i 0.0617919 0.0617919i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −29.3629 −1.89933 −0.949665 0.313267i \(-0.898576\pi\)
−0.949665 + 0.313267i \(0.898576\pi\)
\(240\) 0 0
\(241\) 24.0063 1.54638 0.773190 0.634175i \(-0.218660\pi\)
0.773190 + 0.634175i \(0.218660\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 4.61235 4.61235i 0.294672 0.294672i
\(246\) 0 0
\(247\) 0.303911i 0.0193374i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 15.7570 15.7570i 0.994571 0.994571i −0.00541463 0.999985i \(-0.501724\pi\)
0.999985 + 0.00541463i \(0.00172354\pi\)
\(252\) 0 0
\(253\) 6.99647 + 6.99647i 0.439864 + 0.439864i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −8.66038 −0.540220 −0.270110 0.962829i \(-0.587060\pi\)
−0.270110 + 0.962829i \(0.587060\pi\)
\(258\) 0 0
\(259\) 19.9902 + 19.9902i 1.24213 + 1.24213i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 13.3208i 0.821394i −0.911772 0.410697i \(-0.865285\pi\)
0.911772 0.410697i \(-0.134715\pi\)
\(264\) 0 0
\(265\) 0.330566i 0.0203065i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −11.6714 11.6714i −0.711616 0.711616i 0.255257 0.966873i \(-0.417840\pi\)
−0.966873 + 0.255257i \(0.917840\pi\)
\(270\) 0 0
\(271\) −21.9769 −1.33500 −0.667499 0.744610i \(-0.732635\pi\)
−0.667499 + 0.744610i \(0.732635\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −11.8132 11.8132i −0.712365 0.712365i
\(276\) 0 0
\(277\) 10.9504 10.9504i 0.657945 0.657945i −0.296949 0.954893i \(-0.595969\pi\)
0.954893 + 0.296949i \(0.0959690\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 22.8910i 1.36556i 0.730624 + 0.682780i \(0.239229\pi\)
−0.730624 + 0.682780i \(0.760771\pi\)
\(282\) 0 0
\(283\) −4.48528 + 4.48528i −0.266622 + 0.266622i −0.827738 0.561115i \(-0.810372\pi\)
0.561115 + 0.827738i \(0.310372\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 42.2672 2.49496
\(288\) 0 0
\(289\) −3.91688 −0.230405
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 21.6221 21.6221i 1.26318 1.26318i 0.313636 0.949543i \(-0.398453\pi\)
0.949543 0.313636i \(-0.101547\pi\)
\(294\) 0 0
\(295\) 2.67923i 0.155991i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0.168043 0.168043i 0.00971818 0.00971818i
\(300\) 0 0
\(301\) −7.33962 7.33962i −0.423048 0.423048i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 1.82805 0.104674
\(306\) 0 0
\(307\) 12.1118 + 12.1118i 0.691255 + 0.691255i 0.962508 0.271253i \(-0.0874380\pi\)
−0.271253 + 0.962508i \(0.587438\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 26.8651i 1.52338i 0.647943 + 0.761689i \(0.275630\pi\)
−0.647943 + 0.761689i \(0.724370\pi\)
\(312\) 0 0
\(313\) 19.6890i 1.11289i 0.830885 + 0.556445i \(0.187835\pi\)
−0.830885 + 0.556445i \(0.812165\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 21.3447 + 21.3447i 1.19884 + 1.19884i 0.974515 + 0.224323i \(0.0720171\pi\)
0.224323 + 0.974515i \(0.427983\pi\)
\(318\) 0 0
\(319\) 25.5443 1.43021
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 9.25116 + 9.25116i 0.514748 + 0.514748i
\(324\) 0 0
\(325\) −0.283734 + 0.283734i −0.0157387 + 0.0157387i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 12.8910i 0.710702i
\(330\) 0 0
\(331\) −14.6926 + 14.6926i −0.807576 + 0.807576i −0.984266 0.176690i \(-0.943461\pi\)
0.176690 + 0.984266i \(0.443461\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 2.52898 0.138173
\(336\) 0 0
\(337\) −23.0098 −1.25342 −0.626712 0.779251i \(-0.715600\pi\)
−0.626712 + 0.779251i \(0.715600\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 1.37941 1.37941i 0.0746993 0.0746993i
\(342\) 0 0
\(343\) 30.8651i 1.66656i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −10.9026 + 10.9026i −0.585284 + 0.585284i −0.936350 0.351067i \(-0.885819\pi\)
0.351067 + 0.936350i \(0.385819\pi\)
\(348\) 0 0
\(349\) −20.0563 20.0563i −1.07359 1.07359i −0.997068 0.0765186i \(-0.975620\pi\)
−0.0765186 0.997068i \(-0.524380\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 12.2117 0.649965 0.324983 0.945720i \(-0.394642\pi\)
0.324983 + 0.945720i \(0.394642\pi\)
\(354\) 0 0
\(355\) 3.05275 + 3.05275i 0.162023 + 0.162023i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 33.4780i 1.76690i −0.468522 0.883452i \(-0.655214\pi\)
0.468522 0.883452i \(-0.344786\pi\)
\(360\) 0 0
\(361\) 5.91688i 0.311415i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0.181370 + 0.181370i 0.00949337 + 0.00949337i
\(366\) 0 0
\(367\) −0.702379 −0.0366639 −0.0183319 0.999832i \(-0.505836\pi\)
−0.0183319 + 0.999832i \(0.505836\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 2.24930 + 2.24930i 0.116778 + 0.116778i
\(372\) 0 0
\(373\) −18.9598 + 18.9598i −0.981702 + 0.981702i −0.999836 0.0181339i \(-0.994227\pi\)
0.0181339 + 0.999836i \(0.494227\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0.613530i 0.0315984i
\(378\) 0 0
\(379\) 1.77844 1.77844i 0.0913523 0.0913523i −0.659954 0.751306i \(-0.729424\pi\)
0.751306 + 0.659954i \(0.229424\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −25.4880 −1.30238 −0.651188 0.758916i \(-0.725729\pi\)
−0.651188 + 0.758916i \(0.725729\pi\)
\(384\) 0 0
\(385\) −7.55136 −0.384853
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −11.7049 + 11.7049i −0.593462 + 0.593462i −0.938565 0.345103i \(-0.887844\pi\)
0.345103 + 0.938565i \(0.387844\pi\)
\(390\) 0 0
\(391\) 10.2306i 0.517383i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 3.68184 3.68184i 0.185253 0.185253i
\(396\) 0 0
\(397\) 9.04646 + 9.04646i 0.454029 + 0.454029i 0.896689 0.442661i \(-0.145965\pi\)
−0.442661 + 0.896689i \(0.645965\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 18.0853 0.903137 0.451568 0.892237i \(-0.350865\pi\)
0.451568 + 0.892237i \(0.350865\pi\)
\(402\) 0 0
\(403\) −0.0331311 0.0331311i −0.00165038 0.00165038i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 21.6990i 1.07558i
\(408\) 0 0
\(409\) 25.2271i 1.24740i −0.781665 0.623699i \(-0.785629\pi\)
0.781665 0.623699i \(-0.214371\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −18.2306 18.2306i −0.897069 0.897069i
\(414\) 0 0
\(415\) 7.12787 0.349894
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 7.25283 + 7.25283i 0.354324 + 0.354324i 0.861716 0.507392i \(-0.169390\pi\)
−0.507392 + 0.861716i \(0.669390\pi\)
\(420\) 0 0
\(421\) −2.39550 + 2.39550i −0.116749 + 0.116749i −0.763068 0.646318i \(-0.776308\pi\)
0.646318 + 0.763068i \(0.276308\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 17.2739i 0.837908i
\(426\) 0 0
\(427\) −12.4388 + 12.4388i −0.601957 + 0.601957i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 4.42454 0.213123 0.106561 0.994306i \(-0.466016\pi\)
0.106561 + 0.994306i \(0.466016\pi\)
\(432\) 0 0
\(433\) 7.31371 0.351474 0.175737 0.984437i \(-0.443769\pi\)
0.175737 + 0.984437i \(0.443769\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −7.23412 + 7.23412i −0.346055 + 0.346055i
\(438\) 0 0
\(439\) 29.6533i 1.41527i −0.706576 0.707637i \(-0.749761\pi\)
0.706576 0.707637i \(-0.250239\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 10.3056 10.3056i 0.489633 0.489633i −0.418557 0.908190i \(-0.637464\pi\)
0.908190 + 0.418557i \(0.137464\pi\)
\(444\) 0 0
\(445\) 4.90746 + 4.90746i 0.232636 + 0.232636i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 6.48844 0.306208 0.153104 0.988210i \(-0.451073\pi\)
0.153104 + 0.988210i \(0.451073\pi\)
\(450\) 0 0
\(451\) −22.9402 22.9402i −1.08021 1.08021i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0.181370i 0.00850278i
\(456\) 0 0
\(457\) 9.00353i 0.421167i −0.977576 0.210584i \(-0.932464\pi\)
0.977576 0.210584i \(-0.0675364\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 14.6218 + 14.6218i 0.681004 + 0.681004i 0.960226 0.279223i \(-0.0900767\pi\)
−0.279223 + 0.960226i \(0.590077\pi\)
\(462\) 0 0
\(463\) −18.6435 −0.866437 −0.433219 0.901289i \(-0.642622\pi\)
−0.433219 + 0.901289i \(0.642622\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −23.5138 23.5138i −1.08809 1.08809i −0.995725 0.0923633i \(-0.970558\pi\)
−0.0923633 0.995725i \(-0.529442\pi\)
\(468\) 0 0
\(469\) −17.2082 + 17.2082i −0.794601 + 0.794601i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 7.96703i 0.366325i
\(474\) 0 0
\(475\) 12.2145 12.2145i 0.560440 0.560440i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 1.08864 0.0497412 0.0248706 0.999691i \(-0.492083\pi\)
0.0248706 + 0.999691i \(0.492083\pi\)
\(480\) 0 0
\(481\) −0.521173 −0.0237634
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −1.44586 + 1.44586i −0.0656531 + 0.0656531i
\(486\) 0 0
\(487\) 35.3298i 1.60095i −0.599369 0.800473i \(-0.704582\pi\)
0.599369 0.800473i \(-0.295418\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −12.8910 + 12.8910i −0.581761 + 0.581761i −0.935387 0.353626i \(-0.884949\pi\)
0.353626 + 0.935387i \(0.384949\pi\)
\(492\) 0 0
\(493\) 18.6761 + 18.6761i 0.841128 + 0.841128i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −41.5443 −1.86352
\(498\) 0 0
\(499\) −14.3798 14.3798i −0.643728 0.643728i 0.307742 0.951470i \(-0.400427\pi\)
−0.951470 + 0.307742i \(0.900427\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 30.2969i 1.35087i 0.737420 + 0.675435i \(0.236044\pi\)
−0.737420 + 0.675435i \(0.763956\pi\)
\(504\) 0 0
\(505\) 0.303911i 0.0135239i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 10.5825 + 10.5825i 0.469063 + 0.469063i 0.901611 0.432548i \(-0.142385\pi\)
−0.432548 + 0.901611i \(0.642385\pi\)
\(510\) 0 0
\(511\) −2.46824 −0.109188
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −0.447718 0.447718i −0.0197288 0.0197288i
\(516\) 0 0
\(517\) 6.99647 6.99647i 0.307704 0.307704i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 24.9049i 1.09110i −0.838078 0.545551i \(-0.816320\pi\)
0.838078 0.545551i \(-0.183680\pi\)
\(522\) 0 0
\(523\) 12.9008 12.9008i 0.564112 0.564112i −0.366361 0.930473i \(-0.619396\pi\)
0.930473 + 0.366361i \(0.119396\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 2.01704 0.0878638
\(528\) 0 0
\(529\) 15.0000 0.652174
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −0.550984 + 0.550984i −0.0238657 + 0.0238657i
\(534\) 0 0
\(535\) 4.06074i 0.175561i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 34.0671 34.0671i 1.46738 1.46738i
\(540\) 0 0
\(541\) −18.2767 18.2767i −0.785776 0.785776i 0.195023 0.980799i \(-0.437522\pi\)
−0.980799 + 0.195023i \(0.937522\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 3.82880 0.164008
\(546\) 0 0
\(547\) −13.7355 13.7355i −0.587287 0.587287i 0.349609 0.936896i \(-0.386315\pi\)
−0.936896 + 0.349609i \(0.886315\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 26.4120i 1.12519i
\(552\) 0 0
\(553\) 50.1055i 2.13070i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −27.5525 27.5525i −1.16744 1.16744i −0.982808 0.184631i \(-0.940891\pi\)
−0.184631 0.982808i \(-0.559109\pi\)
\(558\) 0 0
\(559\) 0.191354 0.00809342
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −19.8928 19.8928i −0.838383 0.838383i 0.150263 0.988646i \(-0.451988\pi\)
−0.988646 + 0.150263i \(0.951988\pi\)
\(564\) 0 0
\(565\) −3.19879 + 3.19879i −0.134574 + 0.134574i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 13.4849i 0.565317i −0.959221 0.282658i \(-0.908784\pi\)
0.959221 0.282658i \(-0.0912163\pi\)
\(570\) 0 0
\(571\) 14.8284 14.8284i 0.620550 0.620550i −0.325122 0.945672i \(-0.605405\pi\)
0.945672 + 0.325122i \(0.105405\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 13.5077 0.563308
\(576\) 0 0
\(577\) −11.6176 −0.483648 −0.241824 0.970320i \(-0.577746\pi\)
−0.241824 + 0.970320i \(0.577746\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −48.5010 + 48.5010i −2.01216 + 2.01216i
\(582\) 0 0
\(583\) 2.44158i 0.101120i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −17.0268 + 17.0268i −0.702773 + 0.702773i −0.965005 0.262232i \(-0.915541\pi\)
0.262232 + 0.965005i \(0.415541\pi\)
\(588\) 0 0
\(589\) 1.42627 + 1.42627i 0.0587682 + 0.0587682i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −41.5372 −1.70573 −0.852865 0.522132i \(-0.825137\pi\)
−0.852865 + 0.522132i \(0.825137\pi\)
\(594\) 0 0
\(595\) −5.52099 5.52099i −0.226338 0.226338i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 6.43160i 0.262788i 0.991330 + 0.131394i \(0.0419453\pi\)
−0.991330 + 0.131394i \(0.958055\pi\)
\(600\) 0 0
\(601\) 3.45844i 0.141073i 0.997509 + 0.0705364i \(0.0224711\pi\)
−0.997509 + 0.0705364i \(0.977529\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0.414494 + 0.414494i 0.0168516 + 0.0168516i
\(606\) 0 0
\(607\) −30.1019 −1.22180 −0.610900 0.791708i \(-0.709192\pi\)
−0.610900 + 0.791708i \(0.709192\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −0.168043 0.168043i −0.00679829 0.00679829i
\(612\) 0 0
\(613\) −2.50490 + 2.50490i −0.101172 + 0.101172i −0.755881 0.654709i \(-0.772791\pi\)
0.654709 + 0.755881i \(0.272791\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 22.9098i 0.922315i 0.887318 + 0.461157i \(0.152566\pi\)
−0.887318 + 0.461157i \(0.847434\pi\)
\(618\) 0 0
\(619\) 28.6104 28.6104i 1.14995 1.14995i 0.163386 0.986562i \(-0.447758\pi\)
0.986562 0.163386i \(-0.0522415\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −66.7847 −2.67567
\(624\) 0 0
\(625\) −21.6855 −0.867420
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 15.8647 15.8647i 0.632567 0.632567i
\(630\) 0 0
\(631\) 11.1851i 0.445270i −0.974902 0.222635i \(-0.928534\pi\)
0.974902 0.222635i \(-0.0714659\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −1.70774 + 1.70774i −0.0677698 + 0.0677698i
\(636\) 0 0
\(637\) −0.818234 0.818234i −0.0324196 0.0324196i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 6.69312 0.264362 0.132181 0.991226i \(-0.457802\pi\)
0.132181 + 0.991226i \(0.457802\pi\)
\(642\) 0 0
\(643\) 17.9410 + 17.9410i 0.707522 + 0.707522i 0.966014 0.258491i \(-0.0832253\pi\)
−0.258491 + 0.966014i \(0.583225\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 6.72999i 0.264583i 0.991211 + 0.132292i \(0.0422335\pi\)
−0.991211 + 0.132292i \(0.957766\pi\)
\(648\) 0 0
\(649\) 19.7890i 0.776786i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 26.1731 + 26.1731i 1.02423 + 1.02423i 0.999699 + 0.0245347i \(0.00781042\pi\)
0.0245347 + 0.999699i \(0.492190\pi\)
\(654\) 0 0
\(655\) −1.41921 −0.0554529
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 13.9741 + 13.9741i 0.544353 + 0.544353i 0.924802 0.380449i \(-0.124230\pi\)
−0.380449 + 0.924802i \(0.624230\pi\)
\(660\) 0 0
\(661\) −11.9241 + 11.9241i −0.463794 + 0.463794i −0.899897 0.436103i \(-0.856358\pi\)
0.436103 + 0.899897i \(0.356358\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 7.80785i 0.302776i
\(666\) 0 0
\(667\) −14.6041 + 14.6041i −0.565473 + 0.565473i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 13.5021 0.521244
\(672\) 0 0
\(673\) −37.3066 −1.43807 −0.719033 0.694976i \(-0.755415\pi\)
−0.719033 + 0.694976i \(0.755415\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 0.447461 0.447461i 0.0171973 0.0171973i −0.698456 0.715653i \(-0.746129\pi\)
0.715653 + 0.698456i \(0.246129\pi\)
\(678\) 0 0
\(679\) 19.6764i 0.755113i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −4.27521 + 4.27521i −0.163586 + 0.163586i −0.784153 0.620567i \(-0.786902\pi\)
0.620567 + 0.784153i \(0.286902\pi\)
\(684\) 0 0
\(685\) −1.13178 1.13178i −0.0432430 0.0432430i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −0.0586426 −0.00223410
\(690\) 0 0
\(691\) −20.0786 20.0786i −0.763827 0.763827i 0.213185 0.977012i \(-0.431616\pi\)
−0.977012 + 0.213185i \(0.931616\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 3.93926i 0.149425i
\(696\) 0 0
\(697\) 33.5443i 1.27058i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −10.4467 10.4467i −0.394565 0.394565i 0.481746 0.876311i \(-0.340003\pi\)
−0.876311 + 0.481746i \(0.840003\pi\)
\(702\) 0 0
\(703\) 22.4361 0.846192
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 2.06793 + 2.06793i 0.0777727 + 0.0777727i
\(708\) 0 0
\(709\) −16.0916 + 16.0916i −0.604332 + 0.604332i −0.941459 0.337127i \(-0.890545\pi\)
0.337127 + 0.941459i \(0.390545\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 1.57726i 0.0590690i
\(714\) 0 0
\(715\) 0.0984373 0.0984373i 0.00368135 0.00368135i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 30.9957 1.15594 0.577972 0.816057i \(-0.303844\pi\)
0.577972 + 0.816057i \(0.303844\pi\)
\(720\) 0 0
\(721\) 6.09292 0.226912
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 24.6584 24.6584i 0.915790 0.915790i
\(726\) 0 0
\(727\) 41.1117i 1.52475i 0.647135 + 0.762375i \(0.275967\pi\)
−0.647135 + 0.762375i \(0.724033\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −5.82490 + 5.82490i −0.215442 + 0.215442i
\(732\) 0 0
\(733\) −0.146061 0.146061i −0.00539490 0.00539490i 0.704404 0.709799i \(-0.251214\pi\)
−0.709799 + 0.704404i \(0.751214\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 18.6792 0.688058
\(738\) 0 0
\(739\) 1.50766 + 1.50766i 0.0554601 + 0.0554601i 0.734293 0.678833i \(-0.237514\pi\)
−0.678833 + 0.734293i \(0.737514\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 40.5175i 1.48644i −0.669046 0.743221i \(-0.733297\pi\)
0.669046 0.743221i \(-0.266703\pi\)
\(744\) 0 0
\(745\) 6.69256i 0.245196i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −27.6309 27.6309i −1.00961 1.00961i
\(750\) 0 0
\(751\) −12.5843 −0.459208 −0.229604 0.973284i \(-0.573743\pi\)
−0.229604 + 0.973284i \(0.573743\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −3.34129 3.34129i −0.121602 0.121602i
\(756\) 0 0
\(757\) 7.49900 7.49900i 0.272556 0.272556i −0.557572 0.830128i \(-0.688267\pi\)
0.830128 + 0.557572i \(0.188267\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 42.8182i 1.55216i 0.630635 + 0.776079i \(0.282794\pi\)
−0.630635 + 0.776079i \(0.717206\pi\)
\(762\) 0 0
\(763\) −26.0527 + 26.0527i −0.943172 + 0.943172i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0.475298 0.0171620
\(768\) 0 0
\(769\) 12.7455 0.459614 0.229807 0.973236i \(-0.426190\pi\)
0.229807 + 0.973236i \(0.426190\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −22.8765 + 22.8765i −0.822809 + 0.822809i −0.986510 0.163701i \(-0.947657\pi\)
0.163701 + 0.986510i \(0.447657\pi\)
\(774\) 0 0
\(775\) 2.66314i 0.0956630i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 23.7194 23.7194i 0.849836 0.849836i
\(780\) 0 0
\(781\) 22.5478 + 22.5478i 0.806825 + 0.806825i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −10.8253 −0.386370
\(786\) 0 0
\(787\) −5.20470 5.20470i −0.185528 0.185528i 0.608232 0.793759i \(-0.291879\pi\)
−0.793759 + 0.608232i \(0.791879\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 43.5317i 1.54781i
\(792\) 0 0
\(793\) 0.324298i 0.0115162i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 17.0149 + 17.0149i 0.602698 + 0.602698i 0.941028 0.338330i \(-0.109862\pi\)
−0.338330 + 0.941028i \(0.609862\pi\)
\(798\) 0 0
\(799\) 10.2306 0.361932
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 1.33962 + 1.33962i 0.0472740 + 0.0472740i
\(804\) 0 0
\(805\) 4.31724 4.31724i 0.152163 0.152163i