Properties

Label 576.2.k.b.433.1
Level $576$
Weight $2$
Character 576.433
Analytic conductor $4.599$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 576 = 2^{6} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 576.k (of order \(4\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.59938315643\)
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 433.1
Root \(0.500000 + 0.0297061i\) of defining polynomial
Character \(\chi\) \(=\) 576.433
Dual form 576.2.k.b.145.1

$q$-expansion

\(f(q)\) \(=\) \(q+(-1.74912 - 1.74912i) q^{5} +2.55765i q^{7} +O(q^{10})\) \(q+(-1.74912 - 1.74912i) q^{5} +2.55765i q^{7} +(0.473626 + 0.473626i) q^{11} +(2.88784 - 2.88784i) q^{13} +6.44549 q^{17} +(4.55765 - 4.55765i) q^{19} -2.82843i q^{23} +1.11882i q^{25} +(3.07931 - 3.07931i) q^{29} -6.55765 q^{31} +(4.47363 - 4.47363i) q^{35} +(-2.72922 - 2.72922i) q^{37} +0.788632i q^{41} +(0.389604 + 0.389604i) q^{43} +2.82843 q^{47} +0.458440 q^{49} +(2.57754 + 2.57754i) q^{53} -1.65685i q^{55} +(4.00000 + 4.00000i) q^{59} +(-4.38607 + 4.38607i) q^{61} -10.1023 q^{65} +(2.11882 - 2.11882i) q^{67} +5.11529i q^{71} +14.7721i q^{73} +(-1.21137 + 1.21137i) q^{77} +6.32000 q^{79} +(0.641669 - 0.641669i) q^{83} +(-11.2739 - 11.2739i) q^{85} +6.31724i q^{89} +(7.38607 + 7.38607i) q^{91} -15.9437 q^{95} +12.6533 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q - 8q^{11} + 8q^{19} + 16q^{29} - 24q^{31} + 24q^{35} - 16q^{37} + 8q^{43} - 8q^{49} - 16q^{53} + 32q^{59} + 16q^{61} + 16q^{65} + 16q^{67} - 16q^{77} + 24q^{79} - 40q^{83} - 16q^{85} + 8q^{91} - 48q^{95} + O(q^{100}) \)

Character values

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

\(n\) \(65\) \(127\) \(325\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{1}{4}\right)\)

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
\(5\) −1.74912 1.74912i −0.782229 0.782229i 0.197977 0.980207i \(-0.436563\pi\)
−0.980207 + 0.197977i \(0.936563\pi\)
\(6\) 0 0
\(7\) 2.55765i 0.966700i 0.875427 + 0.483350i \(0.160580\pi\)
−0.875427 + 0.483350i \(0.839420\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0.473626 + 0.473626i 0.142804 + 0.142804i 0.774894 0.632091i \(-0.217803\pi\)
−0.632091 + 0.774894i \(0.717803\pi\)
\(12\) 0 0
\(13\) 2.88784 2.88784i 0.800943 0.800943i −0.182300 0.983243i \(-0.558354\pi\)
0.983243 + 0.182300i \(0.0583543\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 6.44549 1.56326 0.781630 0.623742i \(-0.214389\pi\)
0.781630 + 0.623742i \(0.214389\pi\)
\(18\) 0 0
\(19\) 4.55765 4.55765i 1.04560 1.04560i 0.0466864 0.998910i \(-0.485134\pi\)
0.998910 0.0466864i \(-0.0148661\pi\)
\(20\) 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
\(25\) 1.11882i 0.223765i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 3.07931 3.07931i 0.571813 0.571813i −0.360821 0.932635i \(-0.617504\pi\)
0.932635 + 0.360821i \(0.117504\pi\)
\(30\) 0 0
\(31\) −6.55765 −1.17779 −0.588894 0.808210i \(-0.700437\pi\)
−0.588894 + 0.808210i \(0.700437\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 4.47363 4.47363i 0.756181 0.756181i
\(36\) 0 0
\(37\) −2.72922 2.72922i −0.448681 0.448681i 0.446235 0.894916i \(-0.352765\pi\)
−0.894916 + 0.446235i \(0.852765\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0.788632i 0.123164i 0.998102 + 0.0615818i \(0.0196145\pi\)
−0.998102 + 0.0615818i \(0.980385\pi\)
\(42\) 0 0
\(43\) 0.389604 + 0.389604i 0.0594141 + 0.0594141i 0.736190 0.676775i \(-0.236623\pi\)
−0.676775 + 0.736190i \(0.736623\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2.82843 0.412568 0.206284 0.978492i \(-0.433863\pi\)
0.206284 + 0.978492i \(0.433863\pi\)
\(48\) 0 0
\(49\) 0.458440 0.0654915
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 2.57754 + 2.57754i 0.354053 + 0.354053i 0.861615 0.507562i \(-0.169453\pi\)
−0.507562 + 0.861615i \(0.669453\pi\)
\(54\) 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 0.917800 0.397044i \(-0.129964\pi\)
−0.397044 + 0.917800i \(0.629964\pi\)
\(60\) 0 0
\(61\) −4.38607 + 4.38607i −0.561579 + 0.561579i −0.929756 0.368177i \(-0.879982\pi\)
0.368177 + 0.929756i \(0.379982\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −10.1023 −1.25304
\(66\) 0 0
\(67\) 2.11882 2.11882i 0.258856 0.258856i −0.565733 0.824589i \(-0.691407\pi\)
0.824589 + 0.565733i \(0.191407\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 5.11529i 0.607074i 0.952820 + 0.303537i \(0.0981676\pi\)
−0.952820 + 0.303537i \(0.901832\pi\)
\(72\) 0 0
\(73\) 14.7721i 1.72895i 0.502676 + 0.864475i \(0.332349\pi\)
−0.502676 + 0.864475i \(0.667651\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −1.21137 + 1.21137i −0.138048 + 0.138048i
\(78\) 0 0
\(79\) 6.32000 0.711055 0.355528 0.934666i \(-0.384301\pi\)
0.355528 + 0.934666i \(0.384301\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0.641669 0.641669i 0.0704323 0.0704323i −0.671013 0.741445i \(-0.734141\pi\)
0.741445 + 0.671013i \(0.234141\pi\)
\(84\) 0 0
\(85\) −11.2739 11.2739i −1.22283 1.22283i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 6.31724i 0.669626i 0.942285 + 0.334813i \(0.108673\pi\)
−0.942285 + 0.334813i \(0.891327\pi\)
\(90\) 0 0
\(91\) 7.38607 + 7.38607i 0.774271 + 0.774271i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −15.9437 −1.63579
\(96\) 0 0
\(97\) 12.6533 1.28475 0.642375 0.766390i \(-0.277949\pi\)
0.642375 + 0.766390i \(0.277949\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −7.52480 7.52480i −0.748745 0.748745i 0.225498 0.974244i \(-0.427599\pi\)
−0.974244 + 0.225498i \(0.927599\pi\)
\(102\) 0 0
\(103\) 3.33686i 0.328790i −0.986395 0.164395i \(-0.947433\pi\)
0.986395 0.164395i \(-0.0525672\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −14.0625 14.0625i −1.35948 1.35948i −0.874560 0.484918i \(-0.838849\pi\)
−0.484918 0.874560i \(-0.661151\pi\)
\(108\) 0 0
\(109\) 2.76901 2.76901i 0.265224 0.265224i −0.561949 0.827172i \(-0.689948\pi\)
0.827172 + 0.561949i \(0.189948\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −2.23765 −0.210500 −0.105250 0.994446i \(-0.533564\pi\)
−0.105250 + 0.994446i \(0.533564\pi\)
\(114\) 0 0
\(115\) −4.94725 + 4.94725i −0.461334 + 0.461334i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 16.4853i 1.51120i
\(120\) 0 0
\(121\) 10.5514i 0.959214i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −6.78863 + 6.78863i −0.607194 + 0.607194i
\(126\) 0 0
\(127\) −12.2145 −1.08386 −0.541931 0.840423i \(-0.682307\pi\)
−0.541931 + 0.840423i \(0.682307\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −3.77568 + 3.77568i −0.329883 + 0.329883i −0.852542 0.522659i \(-0.824940\pi\)
0.522659 + 0.852542i \(0.324940\pi\)
\(132\) 0 0
\(133\) 11.6569 + 11.6569i 1.01078 + 1.01078i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 5.10587i 0.436224i −0.975924 0.218112i \(-0.930010\pi\)
0.975924 0.218112i \(-0.0699898\pi\)
\(138\) 0 0
\(139\) −11.7757 11.7757i −0.998800 0.998800i 0.00119925 0.999999i \(-0.499618\pi\)
−0.999999 + 0.00119925i \(0.999618\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 2.73551 0.228755
\(144\) 0 0
\(145\) −10.7721 −0.894578
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 7.90774 + 7.90774i 0.647827 + 0.647827i 0.952467 0.304640i \(-0.0985363\pi\)
−0.304640 + 0.952467i \(0.598536\pi\)
\(150\) 0 0
\(151\) 14.6506i 1.19225i 0.802893 + 0.596123i \(0.203293\pi\)
−0.802893 + 0.596123i \(0.796707\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 11.4701 + 11.4701i 0.921300 + 0.921300i
\(156\) 0 0
\(157\) −3.15196 + 3.15196i −0.251553 + 0.251553i −0.821607 0.570054i \(-0.806922\pi\)
0.570054 + 0.821607i \(0.306922\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 7.23412 0.570128
\(162\) 0 0
\(163\) −5.50490 + 5.50490i −0.431177 + 0.431177i −0.889029 0.457852i \(-0.848619\pi\)
0.457852 + 0.889029i \(0.348619\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 20.1814i 1.56168i −0.624730 0.780841i \(-0.714791\pi\)
0.624730 0.780841i \(-0.285209\pi\)
\(168\) 0 0
\(169\) 3.67923i 0.283018i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −4.35322 + 4.35322i −0.330969 + 0.330969i −0.852955 0.521985i \(-0.825192\pi\)
0.521985 + 0.852955i \(0.325192\pi\)
\(174\) 0 0
\(175\) −2.86156 −0.216313
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −13.2833 + 13.2833i −0.992843 + 0.992843i −0.999975 0.00713130i \(-0.997730\pi\)
0.00713130 + 0.999975i \(0.497730\pi\)
\(180\) 0 0
\(181\) 6.34628 + 6.34628i 0.471715 + 0.471715i 0.902469 0.430754i \(-0.141752\pi\)
−0.430754 + 0.902469i \(0.641752\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 9.54745i 0.701943i
\(186\) 0 0
\(187\) 3.05275 + 3.05275i 0.223239 + 0.223239i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 5.60058 0.405243 0.202622 0.979257i \(-0.435054\pi\)
0.202622 + 0.979257i \(0.435054\pi\)
\(192\) 0 0
\(193\) −19.4514 −1.40014 −0.700071 0.714074i \(-0.746848\pi\)
−0.700071 + 0.714074i \(0.746848\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −1.23793 1.23793i −0.0881988 0.0881988i 0.661631 0.749830i \(-0.269865\pi\)
−0.749830 + 0.661631i \(0.769865\pi\)
\(198\) 0 0
\(199\) 0.993710i 0.0704422i 0.999380 + 0.0352211i \(0.0112135\pi\)
−0.999380 + 0.0352211i \(0.988786\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 7.87579 + 7.87579i 0.552772 + 0.552772i
\(204\) 0 0
\(205\) 1.37941 1.37941i 0.0963422 0.0963422i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 4.31724 0.298630
\(210\) 0 0
\(211\) −4.22432 + 4.22432i −0.290814 + 0.290814i −0.837402 0.546588i \(-0.815927\pi\)
0.546588 + 0.837402i \(0.315927\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 1.36293i 0.0929509i
\(216\) 0 0
\(217\) 16.7721i 1.13857i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 18.6135 18.6135i 1.25208 1.25208i
\(222\) 0 0
\(223\) 23.7659 1.59148 0.795740 0.605639i \(-0.207082\pi\)
0.795740 + 0.605639i \(0.207082\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −0.641669 + 0.641669i −0.0425891 + 0.0425891i −0.728081 0.685492i \(-0.759587\pi\)
0.685492 + 0.728081i \(0.259587\pi\)
\(228\) 0 0
\(229\) 5.34275 + 5.34275i 0.353059 + 0.353059i 0.861246 0.508188i \(-0.169684\pi\)
−0.508188 + 0.861246i \(0.669684\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 23.2271i 1.52166i −0.648954 0.760828i \(-0.724793\pi\)
0.648954 0.760828i \(-0.275207\pi\)
\(234\) 0 0
\(235\) −4.94725 4.94725i −0.322723 0.322723i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 26.9213 1.74140 0.870698 0.491817i \(-0.163667\pi\)
0.870698 + 0.491817i \(0.163667\pi\)
\(240\) 0 0
\(241\) −10.3494 −0.666664 −0.333332 0.942809i \(-0.608173\pi\)
−0.333332 + 0.942809i \(0.608173\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −0.801866 0.801866i −0.0512293 0.0512293i
\(246\) 0 0
\(247\) 26.3235i 1.67492i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −9.75696 9.75696i −0.615854 0.615854i 0.328611 0.944465i \(-0.393419\pi\)
−0.944465 + 0.328611i \(0.893419\pi\)
\(252\) 0 0
\(253\) 1.33962 1.33962i 0.0842209 0.0842209i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −16.9965 −1.06021 −0.530105 0.847932i \(-0.677848\pi\)
−0.530105 + 0.847932i \(0.677848\pi\)
\(258\) 0 0
\(259\) 6.98038 6.98038i 0.433740 0.433740i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 29.9929i 1.84944i −0.380643 0.924722i \(-0.624297\pi\)
0.380643 0.924722i \(-0.375703\pi\)
\(264\) 0 0
\(265\) 9.01686i 0.553901i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −20.6003 + 20.6003i −1.25602 + 1.25602i −0.303046 + 0.952976i \(0.598004\pi\)
−0.952976 + 0.303046i \(0.901996\pi\)
\(270\) 0 0
\(271\) 26.6506 1.61891 0.809453 0.587184i \(-0.199764\pi\)
0.809453 + 0.587184i \(0.199764\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −0.529904 + 0.529904i −0.0319544 + 0.0319544i
\(276\) 0 0
\(277\) 12.1220 + 12.1220i 0.728338 + 0.728338i 0.970289 0.241951i \(-0.0777872\pi\)
−0.241951 + 0.970289i \(0.577787\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 2.76588i 0.164999i −0.996591 0.0824993i \(-0.973710\pi\)
0.996591 0.0824993i \(-0.0262902\pi\)
\(282\) 0 0
\(283\) −4.48528 4.48528i −0.266622 0.266622i 0.561115 0.827738i \(-0.310372\pi\)
−0.827738 + 0.561115i \(0.810372\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −2.01704 −0.119062
\(288\) 0 0
\(289\) 24.5443 1.44378
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 8.20793 + 8.20793i 0.479512 + 0.479512i 0.904976 0.425463i \(-0.139889\pi\)
−0.425463 + 0.904976i \(0.639889\pi\)
\(294\) 0 0
\(295\) 13.9929i 0.814700i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −8.16804 8.16804i −0.472370 0.472370i
\(300\) 0 0
\(301\) −0.996470 + 0.996470i −0.0574356 + 0.0574356i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 15.3435 0.878567
\(306\) 0 0
\(307\) −10.4549 + 10.4549i −0.596693 + 0.596693i −0.939431 0.342738i \(-0.888646\pi\)
0.342738 + 0.939431i \(0.388646\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 15.0761i 0.854885i 0.904043 + 0.427442i \(0.140585\pi\)
−0.904043 + 0.427442i \(0.859415\pi\)
\(312\) 0 0
\(313\) 23.0027i 1.30019i 0.759852 + 0.650096i \(0.225271\pi\)
−0.759852 + 0.650096i \(0.774729\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 6.75892 6.75892i 0.379618 0.379618i −0.491346 0.870964i \(-0.663495\pi\)
0.870964 + 0.491346i \(0.163495\pi\)
\(318\) 0 0
\(319\) 2.91688 0.163314
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 29.3763 29.3763i 1.63454 1.63454i
\(324\) 0 0
\(325\) 3.23099 + 3.23099i 0.179223 + 0.179223i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 7.23412i 0.398830i
\(330\) 0 0
\(331\) 19.6631 + 19.6631i 1.08078 + 1.08078i 0.996436 + 0.0843464i \(0.0268802\pi\)
0.0843464 + 0.996436i \(0.473120\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −7.41215 −0.404969
\(336\) 0 0
\(337\) 3.00980 0.163954 0.0819771 0.996634i \(-0.473877\pi\)
0.0819771 + 0.996634i \(0.473877\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −3.10587 3.10587i −0.168192 0.168192i
\(342\) 0 0
\(343\) 19.0761i 1.03001i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 6.27521 + 6.27521i 0.336871 + 0.336871i 0.855188 0.518317i \(-0.173441\pi\)
−0.518317 + 0.855188i \(0.673441\pi\)
\(348\) 0 0
\(349\) −4.74255 + 4.74255i −0.253863 + 0.253863i −0.822552 0.568690i \(-0.807451\pi\)
0.568690 + 0.822552i \(0.307451\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 8.75882 0.466185 0.233093 0.972455i \(-0.425116\pi\)
0.233093 + 0.972455i \(0.425116\pi\)
\(354\) 0 0
\(355\) 8.94725 8.94725i 0.474871 0.474871i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 32.7917i 1.73068i 0.501184 + 0.865341i \(0.332898\pi\)
−0.501184 + 0.865341i \(0.667102\pi\)
\(360\) 0 0
\(361\) 22.5443i 1.18654i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 25.8382 25.8382i 1.35243 1.35243i
\(366\) 0 0
\(367\) −20.6435 −1.07758 −0.538791 0.842439i \(-0.681119\pi\)
−0.538791 + 0.842439i \(0.681119\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −6.59245 + 6.59245i −0.342263 + 0.342263i
\(372\) 0 0
\(373\) −16.6167 16.6167i −0.860378 0.860378i 0.131004 0.991382i \(-0.458180\pi\)
−0.991382 + 0.131004i \(0.958180\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 17.7851i 0.915979i
\(378\) 0 0
\(379\) −7.77844 7.77844i −0.399552 0.399552i 0.478523 0.878075i \(-0.341172\pi\)
−0.878075 + 0.478523i \(0.841172\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −17.2037 −0.879070 −0.439535 0.898225i \(-0.644857\pi\)
−0.439535 + 0.898225i \(0.644857\pi\)
\(384\) 0 0
\(385\) 4.23765 0.215971
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 23.8515 + 23.8515i 1.20932 + 1.20932i 0.971248 + 0.238069i \(0.0765143\pi\)
0.238069 + 0.971248i \(0.423486\pi\)
\(390\) 0 0
\(391\) 18.2306i 0.921961i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −11.0544 11.0544i −0.556208 0.556208i
\(396\) 0 0
\(397\) −10.2673 + 10.2673i −0.515299 + 0.515299i −0.916145 0.400847i \(-0.868716\pi\)
0.400847 + 0.916145i \(0.368716\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −32.2274 −1.60936 −0.804681 0.593708i \(-0.797663\pi\)
−0.804681 + 0.593708i \(0.797663\pi\)
\(402\) 0 0
\(403\) −18.9374 + 18.9374i −0.943341 + 0.943341i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 2.58526i 0.128146i
\(408\) 0 0
\(409\) 11.5702i 0.572110i −0.958213 0.286055i \(-0.907656\pi\)
0.958213 0.286055i \(-0.0923440\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −10.2306 + 10.2306i −0.503414 + 0.503414i
\(414\) 0 0
\(415\) −2.24471 −0.110188
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 6.74717 6.74717i 0.329621 0.329621i −0.522822 0.852442i \(-0.675121\pi\)
0.852442 + 0.522822i \(0.175121\pi\)
\(420\) 0 0
\(421\) −17.2239 17.2239i −0.839443 0.839443i 0.149343 0.988785i \(-0.452284\pi\)
−0.988785 + 0.149343i \(0.952284\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 7.21137i 0.349803i
\(426\) 0 0
\(427\) −11.2180 11.2180i −0.542879 0.542879i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 40.7088 1.96087 0.980437 0.196832i \(-0.0630654\pi\)
0.980437 + 0.196832i \(0.0630654\pi\)
\(432\) 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
\(437\) −12.8910 12.8910i −0.616659 0.616659i
\(438\) 0 0
\(439\) 17.7122i 0.845356i 0.906280 + 0.422678i \(0.138910\pi\)
−0.906280 + 0.422678i \(0.861090\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 15.6944 + 15.6944i 0.745664 + 0.745664i 0.973662 0.227997i \(-0.0732178\pi\)
−0.227997 + 0.973662i \(0.573218\pi\)
\(444\) 0 0
\(445\) 11.0496 11.0496i 0.523801 0.523801i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 28.3400 1.33745 0.668723 0.743511i \(-0.266841\pi\)
0.668723 + 0.743511i \(0.266841\pi\)
\(450\) 0 0
\(451\) −0.373517 + 0.373517i −0.0175882 + 0.0175882i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 25.8382i 1.21131i
\(456\) 0 0
\(457\) 17.3396i 0.811113i 0.914070 + 0.405557i \(0.132922\pi\)
−0.914070 + 0.405557i \(0.867078\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 1.69284 1.69284i 0.0788434 0.0788434i −0.666585 0.745429i \(-0.732245\pi\)
0.745429 + 0.666585i \(0.232245\pi\)
\(462\) 0 0
\(463\) −2.70238 −0.125590 −0.0627951 0.998026i \(-0.520001\pi\)
−0.0627951 + 0.998026i \(0.520001\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −17.1136 + 17.1136i −0.791924 + 0.791924i −0.981807 0.189883i \(-0.939189\pi\)
0.189883 + 0.981807i \(0.439189\pi\)
\(468\) 0 0
\(469\) 5.41921 + 5.41921i 0.250236 + 0.250236i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0.369053i 0.0169691i
\(474\) 0 0
\(475\) 5.09921 + 5.09921i 0.233968 + 0.233968i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −22.2251 −1.01549 −0.507745 0.861508i \(-0.669521\pi\)
−0.507745 + 0.861508i \(0.669521\pi\)
\(480\) 0 0
\(481\) −15.7631 −0.718735
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −22.1322 22.1322i −1.00497 1.00497i
\(486\) 0 0
\(487\) 13.9839i 0.633672i −0.948480 0.316836i \(-0.897380\pi\)
0.948480 0.316836i \(-0.102620\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 7.23412 + 7.23412i 0.326471 + 0.326471i 0.851243 0.524772i \(-0.175849\pi\)
−0.524772 + 0.851243i \(0.675849\pi\)
\(492\) 0 0
\(493\) 19.8476 19.8476i 0.893893 0.893893i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −13.0831 −0.586858
\(498\) 0 0
\(499\) −2.59078 + 2.59078i −0.115979 + 0.115979i −0.762715 0.646735i \(-0.776134\pi\)
0.646735 + 0.762715i \(0.276134\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 39.6443i 1.76765i 0.467817 + 0.883825i \(0.345041\pi\)
−0.467817 + 0.883825i \(0.654959\pi\)
\(504\) 0 0
\(505\) 26.3235i 1.17138i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −20.2875 + 20.2875i −0.899229 + 0.899229i −0.995368 0.0961393i \(-0.969351\pi\)
0.0961393 + 0.995368i \(0.469351\pi\)
\(510\) 0 0
\(511\) −37.7819 −1.67137
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −5.83655 + 5.83655i −0.257189 + 0.257189i
\(516\) 0 0
\(517\) 1.33962 + 1.33962i 0.0589162 + 0.0589162i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 23.1784i 1.01546i 0.861515 + 0.507732i \(0.169516\pi\)
−0.861515 + 0.507732i \(0.830484\pi\)
\(522\) 0 0
\(523\) 5.78550 + 5.78550i 0.252982 + 0.252982i 0.822192 0.569210i \(-0.192751\pi\)
−0.569210 + 0.822192i \(0.692751\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −42.2672 −1.84119
\(528\) 0 0
\(529\) 15.0000 0.652174
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 2.27744 + 2.27744i 0.0986470 + 0.0986470i
\(534\) 0 0
\(535\) 49.1941i 2.12685i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0.217129 + 0.217129i 0.00935241 + 0.00935241i
\(540\) 0 0
\(541\) 4.55175 4.55175i 0.195695 0.195695i −0.602457 0.798152i \(-0.705811\pi\)
0.798152 + 0.602457i \(0.205811\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −9.68667 −0.414931
\(546\) 0 0
\(547\) 27.7355 27.7355i 1.18588 1.18588i 0.207689 0.978195i \(-0.433406\pi\)
0.978195 0.207689i \(-0.0665942\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 28.0688i 1.19577i
\(552\) 0 0
\(553\) 16.1643i 0.687377i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 1.17538 1.17538i 0.0498026 0.0498026i −0.681767 0.731569i \(-0.738788\pi\)
0.731569 + 0.681767i \(0.238788\pi\)
\(558\) 0 0
\(559\) 2.25023 0.0951745
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −28.7346 + 28.7346i −1.21102 + 1.21102i −0.240326 + 0.970692i \(0.577254\pi\)
−0.970692 + 0.240326i \(0.922746\pi\)
\(564\) 0 0
\(565\) 3.91391 + 3.91391i 0.164659 + 0.164659i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 27.0004i 1.13191i 0.824435 + 0.565957i \(0.191493\pi\)
−0.824435 + 0.565957i \(0.808507\pi\)
\(570\) 0 0
\(571\) 14.8284 + 14.8284i 0.620550 + 0.620550i 0.945672 0.325122i \(-0.105405\pi\)
−0.325122 + 0.945672i \(0.605405\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 3.16451 0.131969
\(576\) 0 0
\(577\) −37.6372 −1.56686 −0.783429 0.621481i \(-0.786531\pi\)
−0.783429 + 0.621481i \(0.786531\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 1.64116 + 1.64116i 0.0680869 + 0.0680869i
\(582\) 0 0
\(583\) 2.44158i 0.101120i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −31.2574 31.2574i −1.29013 1.29013i −0.934703 0.355429i \(-0.884335\pi\)
−0.355429 0.934703i \(-0.615665\pi\)
\(588\) 0 0
\(589\) −29.8874 + 29.8874i −1.23149 + 1.23149i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 3.59611 0.147675 0.0738373 0.997270i \(-0.476475\pi\)
0.0738373 + 0.997270i \(0.476475\pi\)
\(594\) 0 0
\(595\) 28.8347 28.8347i 1.18211 1.18211i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 22.0296i 0.900104i −0.893002 0.450052i \(-0.851405\pi\)
0.893002 0.450052i \(-0.148595\pi\)
\(600\) 0 0
\(601\) 10.7721i 0.439405i 0.975567 + 0.219703i \(0.0705087\pi\)
−0.975567 + 0.219703i \(0.929491\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −18.4556 + 18.4556i −0.750325 + 0.750325i
\(606\) 0 0
\(607\) −5.47453 −0.222204 −0.111102 0.993809i \(-0.535438\pi\)
−0.111102 + 0.993809i \(0.535438\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 8.16804 8.16804i 0.330444 0.330444i
\(612\) 0 0
\(613\) −10.5049 10.5049i −0.424289 0.424289i 0.462389 0.886677i \(-0.346993\pi\)
−0.886677 + 0.462389i \(0.846993\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 22.2235i 0.894686i 0.894363 + 0.447343i \(0.147630\pi\)
−0.894363 + 0.447343i \(0.852370\pi\)
\(618\) 0 0
\(619\) −11.6398 11.6398i −0.467843 0.467843i 0.433372 0.901215i \(-0.357324\pi\)
−0.901215 + 0.433372i \(0.857324\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −16.1573 −0.647327
\(624\) 0 0
\(625\) 29.3424 1.17369
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −17.5912 17.5912i −0.701405 0.701405i
\(630\) 0 0
\(631\) 4.06977i 0.162015i −0.996713 0.0810075i \(-0.974186\pi\)
0.996713 0.0810075i \(-0.0258138\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 21.3646 + 21.3646i 0.847828 + 0.847828i
\(636\) 0 0
\(637\) 1.32390 1.32390i 0.0524549 0.0524549i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 8.41958 0.332553 0.166277 0.986079i \(-0.446826\pi\)
0.166277 + 0.986079i \(0.446826\pi\)
\(642\) 0 0
\(643\) 7.37275 7.37275i 0.290753 0.290753i −0.546625 0.837378i \(-0.684088\pi\)
0.837378 + 0.546625i \(0.184088\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 11.6132i 0.456560i 0.973595 + 0.228280i \(0.0733102\pi\)
−0.973595 + 0.228280i \(0.926690\pi\)
\(648\) 0 0
\(649\) 3.78901i 0.148731i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 1.93049 1.93049i 0.0755458 0.0755458i −0.668324 0.743870i \(-0.732988\pi\)
0.743870 + 0.668324i \(0.232988\pi\)
\(654\) 0 0
\(655\) 13.2082 0.516088
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 22.3102 22.3102i 0.869081 0.869081i −0.123290 0.992371i \(-0.539344\pi\)
0.992371 + 0.123290i \(0.0393444\pi\)
\(660\) 0 0
\(661\) 10.7033 + 10.7033i 0.416311 + 0.416311i 0.883930 0.467619i \(-0.154888\pi\)
−0.467619 + 0.883930i \(0.654888\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 40.7784i 1.58132i
\(666\) 0 0
\(667\) −8.70960 8.70960i −0.337237 0.337237i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −4.15472 −0.160391
\(672\) 0 0
\(673\) −20.6345 −0.795401 −0.397700 0.917515i \(-0.630192\pi\)
−0.397700 + 0.917515i \(0.630192\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −26.8246 26.8246i −1.03095 1.03095i −0.999505 0.0314484i \(-0.989988\pi\)
−0.0314484 0.999505i \(-0.510012\pi\)
\(678\) 0 0
\(679\) 32.3627i 1.24197i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 12.9026 + 12.9026i 0.493705 + 0.493705i 0.909472 0.415766i \(-0.136487\pi\)
−0.415766 + 0.909472i \(0.636487\pi\)
\(684\) 0 0
\(685\) −8.93077 + 8.93077i −0.341227 + 0.341227i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 14.8871 0.567152
\(690\) 0 0
\(691\) 21.3923 21.3923i 0.813803 0.813803i −0.171399 0.985202i \(-0.554829\pi\)
0.985202 + 0.171399i \(0.0548286\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 41.1941i 1.56258i
\(696\) 0 0
\(697\) 5.08312i 0.192537i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −14.2040 + 14.2040i −0.536479 + 0.536479i −0.922493 0.386014i \(-0.873852\pi\)
0.386014 + 0.922493i \(0.373852\pi\)
\(702\) 0 0
\(703\) −24.8776 −0.938278
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 19.2458 19.2458i 0.723812 0.723812i
\(708\) 0 0
\(709\) −29.5474 29.5474i −1.10968 1.10968i −0.993192 0.116485i \(-0.962837\pi\)
−0.116485 0.993192i \(-0.537163\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 18.5478i 0.694622i
\(714\) 0 0
\(715\) −4.78473 4.78473i −0.178939 0.178939i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 28.3683 1.05796 0.528979 0.848635i \(-0.322575\pi\)
0.528979 + 0.848635i \(0.322575\pi\)
\(720\) 0 0
\(721\) 8.53450 0.317841
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 3.44521 + 3.44521i 0.127952 + 0.127952i
\(726\) 0 0
\(727\) 20.4843i 0.759722i −0.925044 0.379861i \(-0.875972\pi\)
0.925044 0.379861i \(-0.124028\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 2.51119 + 2.51119i 0.0928797 + 0.0928797i
\(732\) 0 0
\(733\) 33.9961 33.9961i 1.25567 1.25567i 0.302536 0.953138i \(-0.402167\pi\)
0.953138 0.302536i \(-0.0978331\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 2.00706 0.0739310
\(738\) 0 0
\(739\) −15.1645 + 15.1645i −0.557836 + 0.557836i −0.928691 0.370855i \(-0.879065\pi\)
0.370855 + 0.928691i \(0.379065\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 2.17431i 0.0797677i 0.999204 + 0.0398839i \(0.0126988\pi\)
−0.999204 + 0.0398839i \(0.987301\pi\)
\(744\) 0 0
\(745\) 27.6631i 1.01350i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 35.9670 35.9670i 1.31421 1.31421i
\(750\) 0 0
\(751\) −29.8980 −1.09099 −0.545497 0.838113i \(-0.683659\pi\)
−0.545497 + 0.838113i \(0.683659\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 25.6256 25.6256i 0.932610 0.932610i
\(756\) 0 0
\(757\) −15.3294 15.3294i −0.557157 0.557157i 0.371340 0.928497i \(-0.378899\pi\)
−0.928497 + 0.371340i \(0.878899\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 4.29449i 0.155675i −0.996966 0.0778375i \(-0.975198\pi\)
0.996966 0.0778375i \(-0.0248015\pi\)
\(762\) 0 0
\(763\) 7.08216 + 7.08216i 0.256392 + 0.256392i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 23.1027 0.834191
\(768\) 0 0
\(769\) 33.8819 1.22181 0.610907 0.791703i \(-0.290805\pi\)
0.610907 + 0.791703i \(0.290805\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 35.0230 + 35.0230i 1.25969 + 1.25969i 0.951240 + 0.308450i \(0.0998104\pi\)
0.308450 + 0.951240i \(0.400190\pi\)
\(774\) 0 0
\(775\) 7.33686i 0.263548i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 3.59431 + 3.59431i 0.128779 + 0.128779i
\(780\) 0 0
\(781\) −2.42274 + 2.42274i −0.0866923 + 0.0866923i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 11.0263 0.393545
\(786\) 0 0
\(787\) −24.1090 + 24.1090i −0.859393 + 0.859393i −0.991267 0.131873i \(-0.957901\pi\)
0.131873 + 0.991267i \(0.457901\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 5.72312i 0.203491i
\(792\) 0 0
\(793\) 25.3326i 0.899585i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 28.7722 28.7722i 1.01917 1.01917i 0.0193524 0.999813i \(-0.493840\pi\)
0.999813 0.0193524i \(-0.00616044\pi\)
\(798\) 0 0
\(799\) 18.2306 0.644952
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −6.99647 + 6.99647i −0.246900 + 0.246900i
\(804\) 0 0
\(805\) −12.6533 12.6533i −0.445971 0.445971i