Properties

Label 576.2.k.b.433.4
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 433.4
Root \(0.500000 + 2.10607i\) of defining polynomial
Character \(\chi\) \(=\) 576.433
Dual form 576.2.k.b.145.4

$q$-expansion

\(f(q)\) \(=\) \(q+(2.68554 + 2.68554i) q^{5} -2.15894i q^{7} +O(q^{10})\) \(q+(2.68554 + 2.68554i) q^{5} -2.15894i q^{7} +(1.79793 + 1.79793i) q^{11} +(1.38372 - 1.38372i) q^{13} +0.224777 q^{17} +(-0.158942 + 0.158942i) q^{19} +2.82843i q^{23} +9.42429i q^{25} +(1.85712 - 1.85712i) q^{29} -1.84106 q^{31} +(5.79793 - 5.79793i) q^{35} +(-3.66949 - 3.66949i) q^{37} +5.88163i q^{41} +(7.75481 + 7.75481i) q^{43} -2.82843 q^{47} +2.33897 q^{49} +(-7.51397 - 7.51397i) q^{53} +9.65685i q^{55} +(4.00000 + 4.00000i) q^{59} +(5.98737 - 5.98737i) q^{61} +7.43208 q^{65} +(10.4243 - 10.4243i) q^{67} -4.31788i q^{71} -5.97474i q^{73} +(3.88163 - 3.88163i) q^{77} -15.0075 q^{79} +(-10.1158 + 10.1158i) q^{83} +(0.603650 + 0.603650i) q^{85} +1.42847i q^{89} +(-2.98737 - 2.98737i) q^{91} -0.853690 q^{95} -16.3990 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q - 8q^{11} + 8q^{19} + 16q^{29} - 24q^{31} + 24q^{35} - 16q^{37} + 8q^{43} - 8q^{49} - 16q^{53} + 32q^{59} + 16q^{61} + 16q^{65} + 16q^{67} - 16q^{77} + 24q^{79} - 40q^{83} - 16q^{85} + 8q^{91} - 48q^{95} + O(q^{100}) \)

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
\(5\) 2.68554 + 2.68554i 1.20101 + 1.20101i 0.973859 + 0.227153i \(0.0729416\pi\)
0.227153 + 0.973859i \(0.427058\pi\)
\(6\) 0 0
\(7\) 2.15894i 0.816003i −0.912981 0.408002i \(-0.866226\pi\)
0.912981 0.408002i \(-0.133774\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.79793 + 1.79793i 0.542097 + 0.542097i 0.924143 0.382046i \(-0.124780\pi\)
−0.382046 + 0.924143i \(0.624780\pi\)
\(12\) 0 0
\(13\) 1.38372 1.38372i 0.383775 0.383775i −0.488685 0.872460i \(-0.662523\pi\)
0.872460 + 0.488685i \(0.162523\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0.224777 0.0545165 0.0272583 0.999628i \(-0.491322\pi\)
0.0272583 + 0.999628i \(0.491322\pi\)
\(18\) 0 0
\(19\) −0.158942 + 0.158942i −0.0364637 + 0.0364637i −0.725104 0.688640i \(-0.758208\pi\)
0.688640 + 0.725104i \(0.258208\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2.82843i 0.589768i 0.955533 + 0.294884i \(0.0952810\pi\)
−0.955533 + 0.294884i \(0.904719\pi\)
\(24\) 0 0
\(25\) 9.42429i 1.88486i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 1.85712 1.85712i 0.344858 0.344858i −0.513332 0.858190i \(-0.671589\pi\)
0.858190 + 0.513332i \(0.171589\pi\)
\(30\) 0 0
\(31\) −1.84106 −0.330664 −0.165332 0.986238i \(-0.552870\pi\)
−0.165332 + 0.986238i \(0.552870\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 5.79793 5.79793i 0.980029 0.980029i
\(36\) 0 0
\(37\) −3.66949 3.66949i −0.603260 0.603260i 0.337916 0.941176i \(-0.390278\pi\)
−0.941176 + 0.337916i \(0.890278\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 5.88163i 0.918557i 0.888292 + 0.459278i \(0.151892\pi\)
−0.888292 + 0.459278i \(0.848108\pi\)
\(42\) 0 0
\(43\) 7.75481 + 7.75481i 1.18260 + 1.18260i 0.979069 + 0.203528i \(0.0652407\pi\)
0.203528 + 0.979069i \(0.434759\pi\)
\(44\) 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
\(49\) 2.33897 0.334139
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −7.51397 7.51397i −1.03212 1.03212i −0.999467 0.0326567i \(-0.989603\pi\)
−0.0326567 0.999467i \(-0.510397\pi\)
\(54\) 0 0
\(55\) 9.65685i 1.30213i
\(56\) 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
\(61\) 5.98737 5.98737i 0.766604 0.766604i −0.210903 0.977507i \(-0.567640\pi\)
0.977507 + 0.210903i \(0.0676404\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 7.43208 0.921836
\(66\) 0 0
\(67\) 10.4243 10.4243i 1.27353 1.27353i 0.329307 0.944223i \(-0.393185\pi\)
0.944223 0.329307i \(-0.106815\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 4.31788i 0.512438i −0.966619 0.256219i \(-0.917523\pi\)
0.966619 0.256219i \(-0.0824769\pi\)
\(72\) 0 0
\(73\) 5.97474i 0.699290i −0.936882 0.349645i \(-0.886302\pi\)
0.936882 0.349645i \(-0.113698\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 3.88163 3.88163i 0.442353 0.442353i
\(78\) 0 0
\(79\) −15.0075 −1.68848 −0.844239 0.535966i \(-0.819947\pi\)
−0.844239 + 0.535966i \(0.819947\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −10.1158 + 10.1158i −1.11036 + 1.11036i −0.117253 + 0.993102i \(0.537409\pi\)
−0.993102 + 0.117253i \(0.962591\pi\)
\(84\) 0 0
\(85\) 0.603650 + 0.603650i 0.0654750 + 0.0654750i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 1.42847i 0.151417i 0.997130 + 0.0757086i \(0.0241219\pi\)
−0.997130 + 0.0757086i \(0.975878\pi\)
\(90\) 0 0
\(91\) −2.98737 2.98737i −0.313161 0.313161i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −0.853690 −0.0875867
\(96\) 0 0
\(97\) −16.3990 −1.66507 −0.832535 0.553973i \(-0.813111\pi\)
−0.832535 + 0.553973i \(0.813111\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −0.0818942 0.0818942i −0.00814878 0.00814878i 0.703021 0.711169i \(-0.251834\pi\)
−0.711169 + 0.703021i \(0.751834\pi\)
\(102\) 0 0
\(103\) 13.3507i 1.31548i −0.753245 0.657740i \(-0.771512\pi\)
0.753245 0.657740i \(-0.228488\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −7.27798 7.27798i −0.703589 0.703589i 0.261590 0.965179i \(-0.415753\pi\)
−0.965179 + 0.261590i \(0.915753\pi\)
\(108\) 0 0
\(109\) −7.04057 + 7.04057i −0.674365 + 0.674365i −0.958719 0.284355i \(-0.908221\pi\)
0.284355 + 0.958719i \(0.408221\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −18.8486 −1.77313 −0.886563 0.462608i \(-0.846914\pi\)
−0.886563 + 0.462608i \(0.846914\pi\)
\(114\) 0 0
\(115\) −7.59587 + 7.59587i −0.708318 + 0.708318i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0.485281i 0.0444857i
\(120\) 0 0
\(121\) 4.53488i 0.412261i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −11.8816 + 11.8816i −1.06273 + 1.06273i
\(126\) 0 0
\(127\) 3.81580 0.338597 0.169299 0.985565i \(-0.445850\pi\)
0.169299 + 0.985565i \(0.445850\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −0.767438 + 0.767438i −0.0670514 + 0.0670514i −0.739837 0.672786i \(-0.765098\pi\)
0.672786 + 0.739837i \(0.265098\pi\)
\(132\) 0 0
\(133\) 0.343146 + 0.343146i 0.0297545 + 0.0297545i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 5.31010i 0.453672i −0.973933 0.226836i \(-0.927162\pi\)
0.973933 0.226836i \(-0.0728382\pi\)
\(138\) 0 0
\(139\) −8.76744 8.76744i −0.743644 0.743644i 0.229633 0.973277i \(-0.426247\pi\)
−0.973277 + 0.229633i \(0.926247\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 4.97567 0.416086
\(144\) 0 0
\(145\) 9.97474 0.828357
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 1.02869 + 1.02869i 0.0842735 + 0.0842735i 0.747987 0.663713i \(-0.231021\pi\)
−0.663713 + 0.747987i \(0.731021\pi\)
\(150\) 0 0
\(151\) 2.03696i 0.165766i 0.996559 + 0.0828829i \(0.0264127\pi\)
−0.996559 + 0.0828829i \(0.973587\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −4.94424 4.94424i −0.397131 0.397131i
\(156\) 0 0
\(157\) 6.09378 6.09378i 0.486336 0.486336i −0.420812 0.907148i \(-0.638255\pi\)
0.907148 + 0.420812i \(0.138255\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 6.10641 0.481252
\(162\) 0 0
\(163\) −3.43692 + 3.43692i −0.269201 + 0.269201i −0.828778 0.559577i \(-0.810963\pi\)
0.559577 + 0.828778i \(0.310963\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 21.7023i 1.67937i −0.543072 0.839686i \(-0.682739\pi\)
0.543072 0.839686i \(-0.317261\pi\)
\(168\) 0 0
\(169\) 9.17064i 0.705434i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 8.74653 8.74653i 0.664987 0.664987i −0.291565 0.956551i \(-0.594176\pi\)
0.956551 + 0.291565i \(0.0941758\pi\)
\(174\) 0 0
\(175\) 20.3465 1.53805
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 8.23163 8.23163i 0.615261 0.615261i −0.329051 0.944312i \(-0.606729\pi\)
0.944312 + 0.329051i \(0.106729\pi\)
\(180\) 0 0
\(181\) 6.72269 + 6.72269i 0.499694 + 0.499694i 0.911343 0.411649i \(-0.135047\pi\)
−0.411649 + 0.911343i \(0.635047\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 19.7091i 1.44904i
\(186\) 0 0
\(187\) 0.404135 + 0.404135i 0.0295533 + 0.0295533i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −20.8032 −1.50526 −0.752632 0.658441i \(-0.771216\pi\)
−0.752632 + 0.658441i \(0.771216\pi\)
\(192\) 0 0
\(193\) 14.1454 1.01821 0.509103 0.860705i \(-0.329977\pi\)
0.509103 + 0.860705i \(0.329977\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 2.42865 + 2.42865i 0.173034 + 0.173034i 0.788311 0.615277i \(-0.210956\pi\)
−0.615277 + 0.788311i \(0.710956\pi\)
\(198\) 0 0
\(199\) 0.306182i 0.0217047i −0.999941 0.0108523i \(-0.996546\pi\)
0.999941 0.0108523i \(-0.00345447\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −4.00941 4.00941i −0.281405 0.281405i
\(204\) 0 0
\(205\) −15.7954 + 15.7954i −1.10320 + 1.10320i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −0.571533 −0.0395337
\(210\) 0 0
\(211\) −7.23256 + 7.23256i −0.497910 + 0.497910i −0.910787 0.412877i \(-0.864524\pi\)
0.412877 + 0.910787i \(0.364524\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 41.6517i 2.84063i
\(216\) 0 0
\(217\) 3.97474i 0.269823i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0.311029 0.311029i 0.0209221 0.0209221i
\(222\) 0 0
\(223\) 1.71908 0.115118 0.0575591 0.998342i \(-0.481668\pi\)
0.0575591 + 0.998342i \(0.481668\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 10.1158 10.1158i 0.671410 0.671410i −0.286631 0.958041i \(-0.592535\pi\)
0.958041 + 0.286631i \(0.0925353\pi\)
\(228\) 0 0
\(229\) −12.0195 12.0195i −0.794270 0.794270i 0.187915 0.982185i \(-0.439827\pi\)
−0.982185 + 0.187915i \(0.939827\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 13.3779i 0.876418i 0.898873 + 0.438209i \(0.144387\pi\)
−0.898873 + 0.438209i \(0.855613\pi\)
\(234\) 0 0
\(235\) −7.59587 7.59587i −0.495500 0.495500i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 13.3675 0.864670 0.432335 0.901713i \(-0.357690\pi\)
0.432335 + 0.901713i \(0.357690\pi\)
\(240\) 0 0
\(241\) 0.211474 0.0136222 0.00681112 0.999977i \(-0.497832\pi\)
0.00681112 + 0.999977i \(0.497832\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 6.28141 + 6.28141i 0.401305 + 0.401305i
\(246\) 0 0
\(247\) 0.439861i 0.0279877i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 10.4337 + 10.4337i 0.658569 + 0.658569i 0.955041 0.296472i \(-0.0958102\pi\)
−0.296472 + 0.955041i \(0.595810\pi\)
\(252\) 0 0
\(253\) −5.08532 + 5.08532i −0.319711 + 0.319711i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0.742176 0.0462957 0.0231478 0.999732i \(-0.492631\pi\)
0.0231478 + 0.999732i \(0.492631\pi\)
\(258\) 0 0
\(259\) −7.92221 + 7.92221i −0.492262 + 0.492262i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 5.48435i 0.338180i 0.985601 + 0.169090i \(0.0540828\pi\)
−0.985601 + 0.169090i \(0.945917\pi\)
\(264\) 0 0
\(265\) 40.3582i 2.47918i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −14.4741 + 14.4741i −0.882500 + 0.882500i −0.993788 0.111289i \(-0.964502\pi\)
0.111289 + 0.993788i \(0.464502\pi\)
\(270\) 0 0
\(271\) 14.0370 0.852685 0.426342 0.904562i \(-0.359802\pi\)
0.426342 + 0.904562i \(0.359802\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −16.9442 + 16.9442i −1.02178 + 1.02178i
\(276\) 0 0
\(277\) 9.49013 + 9.49013i 0.570207 + 0.570207i 0.932186 0.361980i \(-0.117899\pi\)
−0.361980 + 0.932186i \(0.617899\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 3.89359i 0.232272i −0.993233 0.116136i \(-0.962949\pi\)
0.993233 0.116136i \(-0.0370509\pi\)
\(282\) 0 0
\(283\) 12.4853 + 12.4853i 0.742173 + 0.742173i 0.972996 0.230823i \(-0.0741418\pi\)
−0.230823 + 0.972996i \(0.574142\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 12.6981 0.749545
\(288\) 0 0
\(289\) −16.9495 −0.997028
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 11.1553 + 11.1553i 0.651697 + 0.651697i 0.953402 0.301704i \(-0.0975556\pi\)
−0.301704 + 0.953402i \(0.597556\pi\)
\(294\) 0 0
\(295\) 21.4844i 1.25087i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 3.91375 + 3.91375i 0.226338 + 0.226338i
\(300\) 0 0
\(301\) 16.7422 16.7422i 0.965003 0.965003i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 32.1587 1.84140
\(306\) 0 0
\(307\) 5.40320 5.40320i 0.308377 0.308377i −0.535903 0.844280i \(-0.680029\pi\)
0.844280 + 0.535903i \(0.180029\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 24.1623i 1.37012i −0.728488 0.685059i \(-0.759776\pi\)
0.728488 0.685059i \(-0.240224\pi\)
\(312\) 0 0
\(313\) 16.6105i 0.938881i −0.882964 0.469441i \(-0.844456\pi\)
0.882964 0.469441i \(-0.155544\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −1.81170 + 1.81170i −0.101755 + 0.101755i −0.756152 0.654397i \(-0.772923\pi\)
0.654397 + 0.756152i \(0.272923\pi\)
\(318\) 0 0
\(319\) 6.67794 0.373893
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −0.0357265 + 0.0357265i −0.00198788 + 0.00198788i
\(324\) 0 0
\(325\) 13.0406 + 13.0406i 0.723361 + 0.723361i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 6.10641i 0.336657i
\(330\) 0 0
\(331\) −13.5252 13.5252i −0.743411 0.743411i 0.229822 0.973233i \(-0.426186\pi\)
−0.973233 + 0.229822i \(0.926186\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 55.9898 3.05905
\(336\) 0 0
\(337\) −1.12615 −0.0613454 −0.0306727 0.999529i \(-0.509765\pi\)
−0.0306727 + 0.999529i \(0.509765\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −3.31010 3.31010i −0.179252 0.179252i
\(342\) 0 0
\(343\) 20.1623i 1.08866i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 20.7938 + 20.7938i 1.11627 + 1.11627i 0.992284 + 0.123983i \(0.0395669\pi\)
0.123983 + 0.992284i \(0.460433\pi\)
\(348\) 0 0
\(349\) −19.2855 + 19.2855i −1.03233 + 1.03233i −0.0328700 + 0.999460i \(0.510465\pi\)
−0.999460 + 0.0328700i \(0.989535\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −25.5908 −1.36206 −0.681029 0.732256i \(-0.738467\pi\)
−0.681029 + 0.732256i \(0.738467\pi\)
\(354\) 0 0
\(355\) 11.5959 11.5959i 0.615445 0.615445i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 3.77296i 0.199129i 0.995031 + 0.0995645i \(0.0317450\pi\)
−0.995031 + 0.0995645i \(0.968255\pi\)
\(360\) 0 0
\(361\) 18.9495i 0.997341i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 16.0454 16.0454i 0.839856 0.839856i
\(366\) 0 0
\(367\) 27.4474 1.43274 0.716371 0.697720i \(-0.245802\pi\)
0.716371 + 0.697720i \(0.245802\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −16.2222 + 16.2222i −0.842216 + 0.842216i
\(372\) 0 0
\(373\) 12.6231 + 12.6231i 0.653601 + 0.653601i 0.953858 0.300257i \(-0.0970725\pi\)
−0.300257 + 0.953858i \(0.597072\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 5.13946i 0.264695i
\(378\) 0 0
\(379\) 11.6686 + 11.6686i 0.599373 + 0.599373i 0.940146 0.340772i \(-0.110689\pi\)
−0.340772 + 0.940146i \(0.610689\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −17.1885 −0.878291 −0.439145 0.898416i \(-0.644719\pi\)
−0.439145 + 0.898416i \(0.644719\pi\)
\(384\) 0 0
\(385\) 20.8486 1.06254
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 1.88238 + 1.88238i 0.0954404 + 0.0954404i 0.753215 0.657774i \(-0.228502\pi\)
−0.657774 + 0.753215i \(0.728502\pi\)
\(390\) 0 0
\(391\) 0.635767i 0.0321521i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −40.3034 40.3034i −2.02788 2.02788i
\(396\) 0 0
\(397\) 8.41166 8.41166i 0.422169 0.422169i −0.463781 0.885950i \(-0.653507\pi\)
0.885950 + 0.463781i \(0.153507\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −1.12389 −0.0561242 −0.0280621 0.999606i \(-0.508934\pi\)
−0.0280621 + 0.999606i \(0.508934\pi\)
\(402\) 0 0
\(403\) −2.54751 + 2.54751i −0.126900 + 0.126900i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 13.1950i 0.654051i
\(408\) 0 0
\(409\) 13.7211i 0.678464i 0.940703 + 0.339232i \(0.110167\pi\)
−0.940703 + 0.339232i \(0.889833\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 8.63577 8.63577i 0.424938 0.424938i
\(414\) 0 0
\(415\) −54.3329 −2.66710
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −9.30755 + 9.30755i −0.454703 + 0.454703i −0.896912 0.442209i \(-0.854195\pi\)
0.442209 + 0.896912i \(0.354195\pi\)
\(420\) 0 0
\(421\) 8.44378 + 8.44378i 0.411525 + 0.411525i 0.882269 0.470745i \(-0.156015\pi\)
−0.470745 + 0.882269i \(0.656015\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 2.11837i 0.102756i
\(426\) 0 0
\(427\) −12.9264 12.9264i −0.625551 0.625551i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −30.6054 −1.47421 −0.737105 0.675778i \(-0.763808\pi\)
−0.737105 + 0.675778i \(0.763808\pi\)
\(432\) 0 0
\(433\) −15.3137 −0.735930 −0.367965 0.929840i \(-0.619945\pi\)
−0.367965 + 0.929840i \(0.619945\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −0.449555 0.449555i −0.0215051 0.0215051i
\(438\) 0 0
\(439\) 33.3676i 1.59255i 0.604936 + 0.796274i \(0.293199\pi\)
−0.604936 + 0.796274i \(0.706801\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 2.28832 + 2.28832i 0.108721 + 0.108721i 0.759375 0.650653i \(-0.225505\pi\)
−0.650653 + 0.759375i \(0.725505\pi\)
\(444\) 0 0
\(445\) −3.83621 + 3.83621i −0.181854 + 0.181854i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 27.4165 1.29387 0.646933 0.762547i \(-0.276052\pi\)
0.646933 + 0.762547i \(0.276052\pi\)
\(450\) 0 0
\(451\) −10.5748 + 10.5748i −0.497947 + 0.497947i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 16.0454i 0.752221i
\(456\) 0 0
\(457\) 10.9147i 0.510567i 0.966866 + 0.255284i \(0.0821688\pi\)
−0.966866 + 0.255284i \(0.917831\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −17.8319 + 17.8319i −0.830512 + 0.830512i −0.987587 0.157075i \(-0.949794\pi\)
0.157075 + 0.987587i \(0.449794\pi\)
\(462\) 0 0
\(463\) −22.4937 −1.04537 −0.522686 0.852525i \(-0.675070\pi\)
−0.522686 + 0.852525i \(0.675070\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 24.2171 24.2171i 1.12063 1.12063i 0.128989 0.991646i \(-0.458827\pi\)
0.991646 0.128989i \(-0.0411731\pi\)
\(468\) 0 0
\(469\) −22.5054 22.5054i −1.03920 1.03920i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 27.8852i 1.28216i
\(474\) 0 0
\(475\) −1.49791 1.49791i −0.0687289 0.0687289i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −36.2362 −1.65568 −0.827838 0.560968i \(-0.810429\pi\)
−0.827838 + 0.560968i \(0.810429\pi\)
\(480\) 0 0
\(481\) −10.1551 −0.463032
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −44.0403 44.0403i −1.99977 1.99977i
\(486\) 0 0
\(487\) 16.8200i 0.762186i −0.924537 0.381093i \(-0.875548\pi\)
0.924537 0.381093i \(-0.124452\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 6.10641 + 6.10641i 0.275578 + 0.275578i 0.831341 0.555763i \(-0.187574\pi\)
−0.555763 + 0.831341i \(0.687574\pi\)
\(492\) 0 0
\(493\) 0.417438 0.417438i 0.0188005 0.0188005i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −9.32206 −0.418151
\(498\) 0 0
\(499\) 19.6770 19.6770i 0.880864 0.880864i −0.112758 0.993622i \(-0.535969\pi\)
0.993622 + 0.112758i \(0.0359686\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 25.7308i 1.14728i 0.819108 + 0.573639i \(0.194469\pi\)
−0.819108 + 0.573639i \(0.805531\pi\)
\(504\) 0 0
\(505\) 0.439861i 0.0195736i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −1.73514 + 1.73514i −0.0769087 + 0.0769087i −0.744515 0.667606i \(-0.767319\pi\)
0.667606 + 0.744515i \(0.267319\pi\)
\(510\) 0 0
\(511\) −12.8991 −0.570623
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 35.8538 35.8538i 1.57991 1.57991i
\(516\) 0 0
\(517\) −5.08532 5.08532i −0.223652 0.223652i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 33.5944i 1.47180i −0.677092 0.735898i \(-0.736760\pi\)
0.677092 0.735898i \(-0.263240\pi\)
\(522\) 0 0
\(523\) 21.8158 + 21.8158i 0.953938 + 0.953938i 0.998985 0.0450467i \(-0.0143437\pi\)
−0.0450467 + 0.998985i \(0.514344\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −0.413828 −0.0180266
\(528\) 0 0
\(529\) 15.0000 0.652174
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 8.13853 + 8.13853i 0.352519 + 0.352519i
\(534\) 0 0
\(535\) 39.0907i 1.69004i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 4.20531 + 4.20531i 0.181136 + 0.181136i
\(540\) 0 0
\(541\) 27.2112 27.2112i 1.16990 1.16990i 0.187669 0.982232i \(-0.439907\pi\)
0.982232 0.187669i \(-0.0600933\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −37.8155 −1.61984
\(546\) 0 0
\(547\) 6.80116 6.80116i 0.290796 0.290796i −0.546598 0.837395i \(-0.684078\pi\)
0.837395 + 0.546598i \(0.184078\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0.590346i 0.0251496i
\(552\) 0 0
\(553\) 32.4004i 1.37780i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 4.29337 4.29337i 0.181916 0.181916i −0.610274 0.792190i \(-0.708941\pi\)
0.792190 + 0.610274i \(0.208941\pi\)
\(558\) 0 0
\(559\) 21.4609 0.907701
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −10.0801 + 10.0801i −0.424825 + 0.424825i −0.886861 0.462036i \(-0.847119\pi\)
0.462036 + 0.886861i \(0.347119\pi\)
\(564\) 0 0
\(565\) −50.6187 50.6187i −2.12954 2.12954i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 32.5018i 1.36255i 0.732029 + 0.681274i \(0.238574\pi\)
−0.732029 + 0.681274i \(0.761426\pi\)
\(570\) 0 0
\(571\) 9.17157 + 9.17157i 0.383818 + 0.383818i 0.872476 0.488657i \(-0.162513\pi\)
−0.488657 + 0.872476i \(0.662513\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −26.6559 −1.11163
\(576\) 0 0
\(577\) 11.7536 0.489308 0.244654 0.969611i \(-0.421326\pi\)
0.244654 + 0.969611i \(0.421326\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 21.8395 + 21.8395i 0.906053 + 0.906053i
\(582\) 0 0
\(583\) 27.0192i 1.11902i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 6.46002 + 6.46002i 0.266634 + 0.266634i 0.827742 0.561109i \(-0.189625\pi\)
−0.561109 + 0.827742i \(0.689625\pi\)
\(588\) 0 0
\(589\) 0.292621 0.292621i 0.0120572 0.0120572i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −5.49270 −0.225558 −0.112779 0.993620i \(-0.535975\pi\)
−0.112779 + 0.993620i \(0.535975\pi\)
\(594\) 0 0
\(595\) 1.30324 1.30324i 0.0534278 0.0534278i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 36.4348i 1.48868i 0.667799 + 0.744342i \(0.267237\pi\)
−0.667799 + 0.744342i \(0.732763\pi\)
\(600\) 0 0
\(601\) 9.97474i 0.406878i −0.979088 0.203439i \(-0.934788\pi\)
0.979088 0.203439i \(-0.0652119\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 12.1786 12.1786i 0.495131 0.495131i
\(606\) 0 0
\(607\) −4.51900 −0.183421 −0.0917103 0.995786i \(-0.529233\pi\)
−0.0917103 + 0.995786i \(0.529233\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −3.91375 + 3.91375i −0.158333 + 0.158333i
\(612\) 0 0
\(613\) −8.43692 8.43692i −0.340764 0.340764i 0.515890 0.856655i \(-0.327461\pi\)
−0.856655 + 0.515890i \(0.827461\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 32.1201i 1.29311i −0.762869 0.646554i \(-0.776210\pi\)
0.762869 0.646554i \(-0.223790\pi\)
\(618\) 0 0
\(619\) −15.0412 15.0412i −0.604559 0.604559i 0.336960 0.941519i \(-0.390601\pi\)
−0.941519 + 0.336960i \(0.890601\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 3.08398 0.123557
\(624\) 0 0
\(625\) −16.6958 −0.667833
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −0.824818 0.824818i −0.0328876 0.0328876i
\(630\) 0 0
\(631\) 36.4685i 1.45179i 0.687807 + 0.725894i \(0.258574\pi\)
−0.687807 + 0.725894i \(0.741426\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 10.2475 + 10.2475i 0.406659 + 0.406659i
\(636\) 0 0
\(637\) 3.23648 3.23648i 0.128234 0.128234i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −14.0036 −0.553109 −0.276555 0.960998i \(-0.589193\pi\)
−0.276555 + 0.960998i \(0.589193\pi\)
\(642\) 0 0
\(643\) −16.6034 + 16.6034i −0.654774 + 0.654774i −0.954139 0.299365i \(-0.903225\pi\)
0.299365 + 0.954139i \(0.403225\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 12.1908i 0.479270i −0.970863 0.239635i \(-0.922972\pi\)
0.970863 0.239635i \(-0.0770277\pi\)
\(648\) 0 0
\(649\) 14.3835i 0.564600i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −0.983270 + 0.983270i −0.0384783 + 0.0384783i −0.726084 0.687606i \(-0.758662\pi\)
0.687606 + 0.726084i \(0.258662\pi\)
\(654\) 0 0
\(655\) −4.12198 −0.161059
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −18.0559 + 18.0559i −0.703357 + 0.703357i −0.965130 0.261772i \(-0.915693\pi\)
0.261772 + 0.965130i \(0.415693\pi\)
\(660\) 0 0
\(661\) −4.55890 4.55890i −0.177321 0.177321i 0.612866 0.790187i \(-0.290017\pi\)
−0.790187 + 0.612866i \(0.790017\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 1.84307i 0.0714710i
\(666\) 0 0
\(667\) 5.25272 + 5.25272i 0.203386 + 0.203386i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 21.5298 0.831148
\(672\) 0 0
\(673\) −10.8569 −0.418504 −0.209252 0.977862i \(-0.567103\pi\)
−0.209252 + 0.977862i \(0.567103\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −23.7066 23.7066i −0.911120 0.911120i 0.0852405 0.996360i \(-0.472834\pi\)
−0.996360 + 0.0852405i \(0.972834\pi\)
\(678\) 0 0
\(679\) 35.4045i 1.35870i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −17.8337 17.8337i −0.682386 0.682386i 0.278151 0.960537i \(-0.410278\pi\)
−0.960537 + 0.278151i \(0.910278\pi\)
\(684\) 0 0
\(685\) 14.2605 14.2605i 0.544866 0.544866i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −20.7945 −0.792205
\(690\) 0 0
\(691\) −10.8557 + 10.8557i −0.412970 + 0.412970i −0.882772 0.469802i \(-0.844325\pi\)
0.469802 + 0.882772i \(0.344325\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 47.0907i 1.78625i
\(696\) 0 0
\(697\) 1.32206i 0.0500765i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 6.08875 6.08875i 0.229969 0.229969i −0.582711 0.812680i \(-0.698008\pi\)
0.812680 + 0.582711i \(0.198008\pi\)
\(702\) 0 0
\(703\) 1.16647 0.0439942
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −0.176805 + 0.176805i −0.00664943 + 0.00664943i
\(708\) 0 0
\(709\) 22.8836 + 22.8836i 0.859413 + 0.859413i 0.991269 0.131856i \(-0.0420936\pi\)
−0.131856 + 0.991269i \(0.542094\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 5.20730i 0.195015i
\(714\) 0 0
\(715\) 13.3624 + 13.3624i 0.499724 + 0.499724i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −1.46744 −0.0547262 −0.0273631 0.999626i \(-0.508711\pi\)
−0.0273631 + 0.999626i \(0.508711\pi\)
\(720\) 0 0
\(721\) −28.8233 −1.07344
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 17.5020 + 17.5020i 0.650008 + 0.650008i
\(726\) 0 0
\(727\) 15.3928i 0.570889i −0.958395 0.285445i \(-0.907859\pi\)
0.958395 0.285445i \(-0.0921412\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 1.74311 + 1.74311i 0.0644711 + 0.0644711i
\(732\) 0 0
\(733\) −12.4185 + 12.4185i −0.458688 + 0.458688i −0.898225 0.439536i \(-0.855143\pi\)
0.439536 + 0.898225i \(0.355143\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 37.4844 1.38075
\(738\) 0 0
\(739\) 14.6559 14.6559i 0.539127 0.539127i −0.384146 0.923273i \(-0.625504\pi\)
0.923273 + 0.384146i \(0.125504\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 31.7821i 1.16597i −0.812482 0.582986i \(-0.801884\pi\)
0.812482 0.582986i \(-0.198116\pi\)
\(744\) 0 0
\(745\) 5.52518i 0.202427i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −15.7127 + 15.7127i −0.574131 + 0.574131i
\(750\) 0 0
\(751\) −29.7594 −1.08594 −0.542968 0.839753i \(-0.682699\pi\)
−0.542968 + 0.839753i \(0.682699\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −5.47036 + 5.47036i −0.199087 + 0.199087i
\(756\) 0 0
\(757\) 15.6355 + 15.6355i 0.568282 + 0.568282i 0.931647 0.363365i \(-0.118372\pi\)
−0.363365 + 0.931647i \(0.618372\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 4.55957i 0.165284i 0.996579 + 0.0826422i \(0.0263359\pi\)
−0.996579 + 0.0826422i \(0.973664\pi\)
\(762\) 0 0
\(763\) 15.2002 + 15.2002i 0.550284 + 0.550284i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 11.0698 0.399706
\(768\) 0 0
\(769\) 36.5794 1.31909 0.659543 0.751667i \(-0.270750\pi\)
0.659543 + 0.751667i \(0.270750\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 18.7108 + 18.7108i 0.672981 + 0.672981i 0.958402 0.285421i \(-0.0921335\pi\)
−0.285421 + 0.958402i \(0.592133\pi\)
\(774\) 0 0
\(775\) 17.3507i 0.623255i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −0.934836 0.934836i −0.0334940 0.0334940i
\(780\) 0 0
\(781\) 7.76326 7.76326i 0.277791 0.277791i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 32.7302 1.16819
\(786\) 0 0
\(787\) −13.3759 + 13.3759i −0.476801 + 0.476801i −0.904107 0.427306i \(-0.859463\pi\)
0.427306 + 0.904107i \(0.359463\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 40.6930i 1.44688i
\(792\) 0 0
\(793\) 16.5697i 0.588406i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 33.8043 33.8043i 1.19741 1.19741i 0.222471 0.974939i \(-0.428588\pi\)
0.974939 0.222471i \(-0.0714124\pi\)
\(798\) 0 0
\(799\) −0.635767 −0.0224918
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 10.7422 10.7422i 0.379083 0.379083i
\(804\) 0 0
\(805\) 16.3990 + 16.3990i 0.577990 + 0.577990i