Properties

Label 1152.2.k.f.865.4
Level $1152$
Weight $2$
Character 1152.865
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 865.4
Root \(0.500000 + 0.0297061i\) of defining polynomial
Character \(\chi\) \(=\) 1152.865
Dual form 1152.2.k.f.289.4

$q$-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 \(\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times\).

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

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 0.980207 0.197977i \(-0.0634373\pi\)
−0.197977 + 0.980207i \(0.563437\pi\)
\(6\) 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 0.632091 0.774894i \(-0.282197\pi\)
−0.774894 + 0.632091i \(0.782197\pi\)
\(12\) 0 0
\(13\) −2.88784 + 2.88784i −0.800943 + 0.800943i −0.983243 0.182300i \(-0.941646\pi\)
0.182300 + 0.983243i \(0.441646\pi\)
\(14\) 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 \(0.514866\pi\)
−0.998910 + 0.0466864i \(0.985134\pi\)
\(20\) 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 −0.932635 0.360821i \(-0.882496\pi\)
0.360821 + 0.932635i \(0.382496\pi\)
\(30\) 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 0.894916 0.446235i \(-0.147235\pi\)
−0.446235 + 0.894916i \(0.647235\pi\)
\(38\) 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 0.676775 0.736190i \(-0.263377\pi\)
−0.736190 + 0.676775i \(0.763377\pi\)
\(44\) 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 0.507562 0.861615i \(-0.330547\pi\)
−0.861615 + 0.507562i \(0.830547\pi\)
\(54\) 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 \(-0.370036\pi\)
−0.917800 + 0.397044i \(0.870036\pi\)
\(60\) 0 0
\(61\) 4.38607 4.38607i 0.561579 0.561579i −0.368177 0.929756i \(-0.620018\pi\)
0.929756 + 0.368177i \(0.120018\pi\)
\(62\) 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 −0.824589 0.565733i \(-0.808593\pi\)
0.565733 + 0.824589i \(0.308593\pi\)
\(68\) 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 −0.741445 0.671013i \(-0.765859\pi\)
0.671013 + 0.741445i \(0.265859\pi\)
\(84\) 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 0.974244 0.225498i \(-0.0724010\pi\)
−0.225498 + 0.974244i \(0.572401\pi\)
\(102\) 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 \(0.161151\pi\)
0.484918 + 0.874560i \(0.338849\pi\)
\(108\) 0 0
\(109\) −2.76901 + 2.76901i −0.265224 + 0.265224i −0.827172 0.561949i \(-0.810052\pi\)
0.561949 + 0.827172i \(0.310052\pi\)
\(110\) 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 −0.522659 0.852542i \(-0.675060\pi\)
0.852542 + 0.522659i \(0.175060\pi\)
\(132\) 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 0.999999 0.00119925i \(-0.000381735\pi\)
−0.00119925 + 0.999999i \(0.500382\pi\)
\(140\) 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 0.304640 0.952467i \(-0.401464\pi\)
−0.952467 + 0.304640i \(0.901464\pi\)
\(150\) 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 −0.570054 0.821607i \(-0.693078\pi\)
0.821607 + 0.570054i \(0.193078\pi\)
\(158\) 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 −0.457852 0.889029i \(-0.651381\pi\)
0.889029 + 0.457852i \(0.151381\pi\)
\(164\) 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 −0.521985 0.852955i \(-0.674808\pi\)
0.852955 + 0.521985i \(0.174808\pi\)
\(174\) 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 −0.00713130 0.999975i \(-0.502270\pi\)
0.999975 + 0.00713130i \(0.00226998\pi\)
\(180\) 0 0
\(181\) −6.34628 6.34628i −0.471715 0.471715i 0.430754 0.902469i \(-0.358248\pi\)
−0.902469 + 0.430754i \(0.858248\pi\)
\(182\) 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 0.749830 0.661631i \(-0.230135\pi\)
−0.661631 + 0.749830i \(0.730135\pi\)
\(198\) 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 −0.546588 0.837402i \(-0.684073\pi\)
0.837402 + 0.546588i \(0.184073\pi\)
\(212\) 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 −0.685492 0.728081i \(-0.740413\pi\)
0.728081 + 0.685492i \(0.240413\pi\)
\(228\) 0 0
\(229\) −5.34275 5.34275i −0.353059 0.353059i 0.508188 0.861246i \(-0.330316\pi\)
−0.861246 + 0.508188i \(0.830316\pi\)
\(230\) 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 0.944465 0.328611i \(-0.106581\pi\)
−0.328611 + 0.944465i \(0.606581\pi\)
\(252\) 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 \(-0.401996\pi\)
0.952976 0.303046i \(-0.0980037\pi\)
\(270\) 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 0.241951 0.970289i \(-0.422213\pi\)
−0.970289 + 0.241951i \(0.922213\pi\)
\(278\) 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 0.827738 0.561115i \(-0.189628\pi\)
−0.561115 + 0.827738i \(0.689628\pi\)
\(284\) 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 0.425463 0.904976i \(-0.360111\pi\)
−0.904976 + 0.425463i \(0.860111\pi\)
\(294\) 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 −0.342738 0.939431i \(-0.611354\pi\)
0.939431 + 0.342738i \(0.111354\pi\)
\(308\) 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 −0.870964 0.491346i \(-0.836505\pi\)
0.491346 + 0.870964i \(0.336505\pi\)
\(318\) 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 \(-0.973120\pi\)
−0.0843464 0.996436i \(-0.526880\pi\)
\(332\) 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 0.518317 0.855188i \(-0.326559\pi\)
−0.855188 + 0.518317i \(0.826559\pi\)
\(348\) 0 0
\(349\) 4.74255 4.74255i 0.253863 0.253863i −0.568690 0.822552i \(-0.692549\pi\)
0.822552 + 0.568690i \(0.192549\pi\)
\(350\) 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 0.991382 0.131004i \(-0.0418200\pi\)
−0.131004 + 0.991382i \(0.541820\pi\)
\(374\) 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 0.878075 0.478523i \(-0.158828\pi\)
−0.478523 + 0.878075i \(0.658828\pi\)
\(380\) 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 \(-0.923486\pi\)
−0.238069 0.971248i \(-0.576514\pi\)
\(390\) 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 −0.400847 0.916145i \(-0.631284\pi\)
0.916145 + 0.400847i \(0.131284\pi\)
\(398\) 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 −0.852442 0.522822i \(-0.824879\pi\)
0.522822 + 0.852442i \(0.324879\pi\)
\(420\) 0 0
\(421\) 17.2239 + 17.2239i 0.839443 + 0.839443i 0.988785 0.149343i \(-0.0477158\pi\)
−0.149343 + 0.988785i \(0.547716\pi\)
\(422\) 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 0.227997 0.973662i \(-0.426782\pi\)
−0.973662 + 0.227997i \(0.926782\pi\)
\(444\) 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 −0.745429 0.666585i \(-0.767755\pi\)
0.666585 + 0.745429i \(0.267755\pi\)
\(462\) 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 −0.189883 0.981807i \(-0.560811\pi\)
0.981807 + 0.189883i \(0.0608108\pi\)
\(468\) 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 0.524772 0.851243i \(-0.324151\pi\)
−0.851243 + 0.524772i \(0.824151\pi\)
\(492\) 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 −0.646735 0.762715i \(-0.723866\pi\)
0.762715 + 0.646735i \(0.223866\pi\)
\(500\) 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 −0.0961393 0.995368i \(-0.530649\pi\)
0.995368 + 0.0961393i \(0.0306494\pi\)
\(510\) 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 0.569210 0.822192i \(-0.307249\pi\)
−0.822192 + 0.569210i \(0.807249\pi\)
\(524\) 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 −0.798152 0.602457i \(-0.794189\pi\)
0.602457 + 0.798152i \(0.294189\pi\)
\(542\) 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 \(0.566594\pi\)
−0.978195 + 0.207689i \(0.933406\pi\)
\(548\) 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 −0.731569 0.681767i \(-0.761212\pi\)
0.681767 + 0.731569i \(0.261212\pi\)
\(558\) 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 \(-0.422746\pi\)
0.970692 0.240326i \(-0.0772544\pi\)
\(564\) 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 0.325122 0.945672i \(-0.394595\pi\)
−0.945672 + 0.325122i \(0.894595\pi\)
\(572\) 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 \(0.115665\pi\)
0.355429 + 0.934703i \(0.384335\pi\)
\(588\) 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 0.886677 0.462389i \(-0.153007\pi\)
−0.462389 + 0.886677i \(0.653007\pi\)
\(614\) 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 0.901215 0.433372i \(-0.142676\pi\)
−0.433372 + 0.901215i \(0.642676\pi\)
\(620\) 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 −0.837378 0.546625i \(-0.815912\pi\)
0.546625 + 0.837378i \(0.315912\pi\)
\(644\) 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 −0.743870 0.668324i \(-0.767012\pi\)
0.668324 + 0.743870i \(0.267012\pi\)
\(654\) 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 −0.992371 0.123290i \(-0.960656\pi\)
0.123290 + 0.992371i \(0.460656\pi\)
\(660\) 0 0
\(661\) −10.7033 10.7033i −0.416311 0.416311i 0.467619 0.883930i \(-0.345112\pi\)
−0.883930 + 0.467619i \(0.845112\pi\)
\(662\) 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 \(0.0100120\pi\)
0.0314484 + 0.999505i \(0.489988\pi\)
\(678\) 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 0.415766 0.909472i \(-0.363513\pi\)
−0.909472 + 0.415766i \(0.863513\pi\)
\(684\) 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 −0.985202 0.171399i \(-0.945171\pi\)
0.171399 + 0.985202i \(0.445171\pi\)
\(692\) 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 −0.386014 0.922493i \(-0.626148\pi\)
0.922493 + 0.386014i \(0.126148\pi\)
\(702\) 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 \(0.0371626\pi\)
0.116485 + 0.993192i \(0.462837\pi\)
\(710\) 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 \(0.597833\pi\)
−0.953138 + 0.302536i \(0.902167\pi\)
\(734\) 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 −0.370855 0.928691i \(-0.620935\pi\)
0.928691 + 0.370855i \(0.120935\pi\)
\(740\) 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 0.928497 0.371340i \(-0.121101\pi\)
−0.371340 + 0.928497i \(0.621101\pi\)
\(758\) 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 \(-0.900190\pi\)
−0.308450 0.951240i \(-0.599810\pi\)
\(774\) 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 −0.131873 0.991267i \(-0.542099\pi\)
0.991267 + 0.131873i \(0.0420992\pi\)
\(788\) 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 \(0.506160\pi\)
−0.999813 + 0.0193524i \(0.993840\pi\)
\(798\) 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