Properties

Label 3024.2.k.f.1889.4
Level $3024$
Weight $2$
Character 3024.1889
Analytic conductor $24.147$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 3024 = 2^{4} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3024.k (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(24.1467615712\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: no (minimal twist has level 378)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1889.4
Root \(-0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 3024.1889
Dual form 3024.2.k.f.1889.3

$q$-expansion

\(f(q)\) \(=\) \(q+3.46410 q^{5} +(-2.00000 + 1.73205i) q^{7} +O(q^{10})\) \(q+3.46410 q^{5} +(-2.00000 + 1.73205i) q^{7} -6.00000i q^{11} +1.73205i q^{13} -1.73205 q^{17} +6.92820i q^{19} +3.00000i q^{23} +7.00000 q^{25} +3.00000i q^{29} +5.19615i q^{31} +(-6.92820 + 6.00000i) q^{35} -2.00000 q^{37} +6.92820 q^{41} +11.0000 q^{43} +6.92820 q^{47} +(1.00000 - 6.92820i) q^{49} +3.00000i q^{53} -20.7846i q^{55} +8.66025 q^{59} +13.8564i q^{61} +6.00000i q^{65} +7.00000 q^{67} -3.00000i q^{71} -6.92820i q^{73} +(10.3923 + 12.0000i) q^{77} -8.00000 q^{79} +3.46410 q^{83} -6.00000 q^{85} +5.19615 q^{89} +(-3.00000 - 3.46410i) q^{91} +24.0000i q^{95} +6.92820i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 8q^{7} + O(q^{10}) \) \( 4q - 8q^{7} + 28q^{25} - 8q^{37} + 44q^{43} + 4q^{49} + 28q^{67} - 32q^{79} - 24q^{85} - 12q^{91} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3024\mathbb{Z}\right)^\times\).

\(n\) \(757\) \(785\) \(1135\) \(2593\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 3.46410 1.54919 0.774597 0.632456i \(-0.217953\pi\)
0.774597 + 0.632456i \(0.217953\pi\)
\(6\) 0 0
\(7\) −2.00000 + 1.73205i −0.755929 + 0.654654i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 6.00000i 1.80907i −0.426401 0.904534i \(-0.640219\pi\)
0.426401 0.904534i \(-0.359781\pi\)
\(12\) 0 0
\(13\) 1.73205i 0.480384i 0.970725 + 0.240192i \(0.0772105\pi\)
−0.970725 + 0.240192i \(0.922790\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.73205 −0.420084 −0.210042 0.977692i \(-0.567360\pi\)
−0.210042 + 0.977692i \(0.567360\pi\)
\(18\) 0 0
\(19\) 6.92820i 1.58944i 0.606977 + 0.794719i \(0.292382\pi\)
−0.606977 + 0.794719i \(0.707618\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.00000i 0.625543i 0.949828 + 0.312772i \(0.101257\pi\)
−0.949828 + 0.312772i \(0.898743\pi\)
\(24\) 0 0
\(25\) 7.00000 1.40000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 3.00000i 0.557086i 0.960424 + 0.278543i \(0.0898515\pi\)
−0.960424 + 0.278543i \(0.910149\pi\)
\(30\) 0 0
\(31\) 5.19615i 0.933257i 0.884454 + 0.466628i \(0.154531\pi\)
−0.884454 + 0.466628i \(0.845469\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −6.92820 + 6.00000i −1.17108 + 1.01419i
\(36\) 0 0
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 6.92820 1.08200 0.541002 0.841021i \(-0.318045\pi\)
0.541002 + 0.841021i \(0.318045\pi\)
\(42\) 0 0
\(43\) 11.0000 1.67748 0.838742 0.544529i \(-0.183292\pi\)
0.838742 + 0.544529i \(0.183292\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 6.92820 1.01058 0.505291 0.862949i \(-0.331385\pi\)
0.505291 + 0.862949i \(0.331385\pi\)
\(48\) 0 0
\(49\) 1.00000 6.92820i 0.142857 0.989743i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 3.00000i 0.412082i 0.978543 + 0.206041i \(0.0660580\pi\)
−0.978543 + 0.206041i \(0.933942\pi\)
\(54\) 0 0
\(55\) 20.7846i 2.80260i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 8.66025 1.12747 0.563735 0.825956i \(-0.309364\pi\)
0.563735 + 0.825956i \(0.309364\pi\)
\(60\) 0 0
\(61\) 13.8564i 1.77413i 0.461644 + 0.887066i \(0.347260\pi\)
−0.461644 + 0.887066i \(0.652740\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 6.00000i 0.744208i
\(66\) 0 0
\(67\) 7.00000 0.855186 0.427593 0.903971i \(-0.359362\pi\)
0.427593 + 0.903971i \(0.359362\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 3.00000i 0.356034i −0.984027 0.178017i \(-0.943032\pi\)
0.984027 0.178017i \(-0.0569683\pi\)
\(72\) 0 0
\(73\) 6.92820i 0.810885i −0.914121 0.405442i \(-0.867117\pi\)
0.914121 0.405442i \(-0.132883\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 10.3923 + 12.0000i 1.18431 + 1.36753i
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 3.46410 0.380235 0.190117 0.981761i \(-0.439113\pi\)
0.190117 + 0.981761i \(0.439113\pi\)
\(84\) 0 0
\(85\) −6.00000 −0.650791
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 5.19615 0.550791 0.275396 0.961331i \(-0.411191\pi\)
0.275396 + 0.961331i \(0.411191\pi\)
\(90\) 0 0
\(91\) −3.00000 3.46410i −0.314485 0.363137i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 24.0000i 2.46235i
\(96\) 0 0
\(97\) 6.92820i 0.703452i 0.936103 + 0.351726i \(0.114405\pi\)
−0.936103 + 0.351726i \(0.885595\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) 8.66025i 0.853320i −0.904412 0.426660i \(-0.859690\pi\)
0.904412 0.426660i \(-0.140310\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) 4.00000 0.383131 0.191565 0.981480i \(-0.438644\pi\)
0.191565 + 0.981480i \(0.438644\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 6.00000i 0.564433i −0.959351 0.282216i \(-0.908930\pi\)
0.959351 0.282216i \(-0.0910696\pi\)
\(114\) 0 0
\(115\) 10.3923i 0.969087i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 3.46410 3.00000i 0.317554 0.275010i
\(120\) 0 0
\(121\) −25.0000 −2.27273
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 6.92820 0.619677
\(126\) 0 0
\(127\) −14.0000 −1.24230 −0.621150 0.783692i \(-0.713334\pi\)
−0.621150 + 0.783692i \(0.713334\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 5.19615 0.453990 0.226995 0.973896i \(-0.427110\pi\)
0.226995 + 0.973896i \(0.427110\pi\)
\(132\) 0 0
\(133\) −12.0000 13.8564i −1.04053 1.20150i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) 3.46410i 0.293821i −0.989150 0.146911i \(-0.953067\pi\)
0.989150 0.146911i \(-0.0469330\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 10.3923 0.869048
\(144\) 0 0
\(145\) 10.3923i 0.863034i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 15.0000i 1.22885i 0.788976 + 0.614424i \(0.210612\pi\)
−0.788976 + 0.614424i \(0.789388\pi\)
\(150\) 0 0
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 18.0000i 1.44579i
\(156\) 0 0
\(157\) 15.5885i 1.24409i −0.782980 0.622047i \(-0.786301\pi\)
0.782980 0.622047i \(-0.213699\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −5.19615 6.00000i −0.409514 0.472866i
\(162\) 0 0
\(163\) −11.0000 −0.861586 −0.430793 0.902451i \(-0.641766\pi\)
−0.430793 + 0.902451i \(0.641766\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 17.3205 1.34030 0.670151 0.742225i \(-0.266230\pi\)
0.670151 + 0.742225i \(0.266230\pi\)
\(168\) 0 0
\(169\) 10.0000 0.769231
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −20.7846 −1.58022 −0.790112 0.612962i \(-0.789978\pi\)
−0.790112 + 0.612962i \(0.789978\pi\)
\(174\) 0 0
\(175\) −14.0000 + 12.1244i −1.05830 + 0.916515i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 6.00000i 0.448461i −0.974536 0.224231i \(-0.928013\pi\)
0.974536 0.224231i \(-0.0719869\pi\)
\(180\) 0 0
\(181\) 1.73205i 0.128742i 0.997926 + 0.0643712i \(0.0205042\pi\)
−0.997926 + 0.0643712i \(0.979496\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −6.92820 −0.509372
\(186\) 0 0
\(187\) 10.3923i 0.759961i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 24.0000i 1.73658i 0.496058 + 0.868290i \(0.334780\pi\)
−0.496058 + 0.868290i \(0.665220\pi\)
\(192\) 0 0
\(193\) 13.0000 0.935760 0.467880 0.883792i \(-0.345018\pi\)
0.467880 + 0.883792i \(0.345018\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 6.00000i 0.427482i 0.976890 + 0.213741i \(0.0685649\pi\)
−0.976890 + 0.213741i \(0.931435\pi\)
\(198\) 0 0
\(199\) 8.66025i 0.613909i 0.951724 + 0.306955i \(0.0993100\pi\)
−0.951724 + 0.306955i \(0.900690\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −5.19615 6.00000i −0.364698 0.421117i
\(204\) 0 0
\(205\) 24.0000 1.67623
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 41.5692 2.87540
\(210\) 0 0
\(211\) −7.00000 −0.481900 −0.240950 0.970538i \(-0.577459\pi\)
−0.240950 + 0.970538i \(0.577459\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 38.1051 2.59875
\(216\) 0 0
\(217\) −9.00000 10.3923i −0.610960 0.705476i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 3.00000i 0.201802i
\(222\) 0 0
\(223\) 10.3923i 0.695920i −0.937509 0.347960i \(-0.886874\pi\)
0.937509 0.347960i \(-0.113126\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −25.9808 −1.72440 −0.862202 0.506565i \(-0.830915\pi\)
−0.862202 + 0.506565i \(0.830915\pi\)
\(228\) 0 0
\(229\) 20.7846i 1.37349i −0.726900 0.686743i \(-0.759040\pi\)
0.726900 0.686743i \(-0.240960\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 24.0000i 1.57229i −0.618041 0.786146i \(-0.712073\pi\)
0.618041 0.786146i \(-0.287927\pi\)
\(234\) 0 0
\(235\) 24.0000 1.56559
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) 13.8564i 0.892570i 0.894891 + 0.446285i \(0.147253\pi\)
−0.894891 + 0.446285i \(0.852747\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 3.46410 24.0000i 0.221313 1.53330i
\(246\) 0 0
\(247\) −12.0000 −0.763542
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −3.46410 −0.218652 −0.109326 0.994006i \(-0.534869\pi\)
−0.109326 + 0.994006i \(0.534869\pi\)
\(252\) 0 0
\(253\) 18.0000 1.13165
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −20.7846 −1.29651 −0.648254 0.761424i \(-0.724501\pi\)
−0.648254 + 0.761424i \(0.724501\pi\)
\(258\) 0 0
\(259\) 4.00000 3.46410i 0.248548 0.215249i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 9.00000i 0.554964i −0.960731 0.277482i \(-0.910500\pi\)
0.960731 0.277482i \(-0.0894999\pi\)
\(264\) 0 0
\(265\) 10.3923i 0.638394i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −3.46410 −0.211210 −0.105605 0.994408i \(-0.533678\pi\)
−0.105605 + 0.994408i \(0.533678\pi\)
\(270\) 0 0
\(271\) 12.1244i 0.736502i −0.929726 0.368251i \(-0.879957\pi\)
0.929726 0.368251i \(-0.120043\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 42.0000i 2.53270i
\(276\) 0 0
\(277\) 20.0000 1.20168 0.600842 0.799368i \(-0.294832\pi\)
0.600842 + 0.799368i \(0.294832\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 30.0000i 1.78965i 0.446417 + 0.894825i \(0.352700\pi\)
−0.446417 + 0.894825i \(0.647300\pi\)
\(282\) 0 0
\(283\) 20.7846i 1.23552i −0.786368 0.617758i \(-0.788041\pi\)
0.786368 0.617758i \(-0.211959\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −13.8564 + 12.0000i −0.817918 + 0.708338i
\(288\) 0 0
\(289\) −14.0000 −0.823529
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 17.3205 1.01187 0.505937 0.862570i \(-0.331147\pi\)
0.505937 + 0.862570i \(0.331147\pi\)
\(294\) 0 0
\(295\) 30.0000 1.74667
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −5.19615 −0.300501
\(300\) 0 0
\(301\) −22.0000 + 19.0526i −1.26806 + 1.09817i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 48.0000i 2.74847i
\(306\) 0 0
\(307\) 31.1769i 1.77936i 0.456584 + 0.889680i \(0.349073\pi\)
−0.456584 + 0.889680i \(0.650927\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −13.8564 −0.785725 −0.392862 0.919597i \(-0.628515\pi\)
−0.392862 + 0.919597i \(0.628515\pi\)
\(312\) 0 0
\(313\) 13.8564i 0.783210i −0.920133 0.391605i \(-0.871920\pi\)
0.920133 0.391605i \(-0.128080\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 18.0000i 1.01098i −0.862832 0.505490i \(-0.831312\pi\)
0.862832 0.505490i \(-0.168688\pi\)
\(318\) 0 0
\(319\) 18.0000 1.00781
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 12.0000i 0.667698i
\(324\) 0 0
\(325\) 12.1244i 0.672538i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −13.8564 + 12.0000i −0.763928 + 0.661581i
\(330\) 0 0
\(331\) 17.0000 0.934405 0.467202 0.884150i \(-0.345262\pi\)
0.467202 + 0.884150i \(0.345262\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 24.2487 1.32485
\(336\) 0 0
\(337\) 13.0000 0.708155 0.354078 0.935216i \(-0.384795\pi\)
0.354078 + 0.935216i \(0.384795\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 31.1769 1.68832
\(342\) 0 0
\(343\) 10.0000 + 15.5885i 0.539949 + 0.841698i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 18.0000i 0.966291i 0.875540 + 0.483145i \(0.160506\pi\)
−0.875540 + 0.483145i \(0.839494\pi\)
\(348\) 0 0
\(349\) 19.0526i 1.01986i 0.860216 + 0.509930i \(0.170329\pi\)
−0.860216 + 0.509930i \(0.829671\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 8.66025 0.460939 0.230469 0.973080i \(-0.425974\pi\)
0.230469 + 0.973080i \(0.425974\pi\)
\(354\) 0 0
\(355\) 10.3923i 0.551566i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 27.0000i 1.42501i 0.701669 + 0.712503i \(0.252438\pi\)
−0.701669 + 0.712503i \(0.747562\pi\)
\(360\) 0 0
\(361\) −29.0000 −1.52632
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 24.0000i 1.25622i
\(366\) 0 0
\(367\) 19.0526i 0.994535i 0.867597 + 0.497268i \(0.165663\pi\)
−0.867597 + 0.497268i \(0.834337\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −5.19615 6.00000i −0.269771 0.311504i
\(372\) 0 0
\(373\) −4.00000 −0.207112 −0.103556 0.994624i \(-0.533022\pi\)
−0.103556 + 0.994624i \(0.533022\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −5.19615 −0.267615
\(378\) 0 0
\(379\) −16.0000 −0.821865 −0.410932 0.911666i \(-0.634797\pi\)
−0.410932 + 0.911666i \(0.634797\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 3.46410 0.177007 0.0885037 0.996076i \(-0.471792\pi\)
0.0885037 + 0.996076i \(0.471792\pi\)
\(384\) 0 0
\(385\) 36.0000 + 41.5692i 1.83473 + 2.11856i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 6.00000i 0.304212i −0.988364 0.152106i \(-0.951394\pi\)
0.988364 0.152106i \(-0.0486055\pi\)
\(390\) 0 0
\(391\) 5.19615i 0.262781i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −27.7128 −1.39438
\(396\) 0 0
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 18.0000i 0.898877i 0.893311 + 0.449439i \(0.148376\pi\)
−0.893311 + 0.449439i \(0.851624\pi\)
\(402\) 0 0
\(403\) −9.00000 −0.448322
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 12.0000i 0.594818i
\(408\) 0 0
\(409\) 3.46410i 0.171289i 0.996326 + 0.0856444i \(0.0272949\pi\)
−0.996326 + 0.0856444i \(0.972705\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −17.3205 + 15.0000i −0.852286 + 0.738102i
\(414\) 0 0
\(415\) 12.0000 0.589057
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −29.4449 −1.43848 −0.719238 0.694764i \(-0.755509\pi\)
−0.719238 + 0.694764i \(0.755509\pi\)
\(420\) 0 0
\(421\) −40.0000 −1.94948 −0.974740 0.223341i \(-0.928304\pi\)
−0.974740 + 0.223341i \(0.928304\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −12.1244 −0.588118
\(426\) 0 0
\(427\) −24.0000 27.7128i −1.16144 1.34112i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 12.0000i 0.578020i −0.957326 0.289010i \(-0.906674\pi\)
0.957326 0.289010i \(-0.0933260\pi\)
\(432\) 0 0
\(433\) 20.7846i 0.998845i −0.866359 0.499422i \(-0.833546\pi\)
0.866359 0.499422i \(-0.166454\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −20.7846 −0.994263
\(438\) 0 0
\(439\) 5.19615i 0.247999i 0.992282 + 0.123999i \(0.0395721\pi\)
−0.992282 + 0.123999i \(0.960428\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 18.0000i 0.855206i −0.903967 0.427603i \(-0.859358\pi\)
0.903967 0.427603i \(-0.140642\pi\)
\(444\) 0 0
\(445\) 18.0000 0.853282
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 6.00000i 0.283158i −0.989927 0.141579i \(-0.954782\pi\)
0.989927 0.141579i \(-0.0452178\pi\)
\(450\) 0 0
\(451\) 41.5692i 1.95742i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −10.3923 12.0000i −0.487199 0.562569i
\(456\) 0 0
\(457\) −17.0000 −0.795226 −0.397613 0.917553i \(-0.630161\pi\)
−0.397613 + 0.917553i \(0.630161\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 6.92820 0.322679 0.161339 0.986899i \(-0.448419\pi\)
0.161339 + 0.986899i \(0.448419\pi\)
\(462\) 0 0
\(463\) 4.00000 0.185896 0.0929479 0.995671i \(-0.470371\pi\)
0.0929479 + 0.995671i \(0.470371\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 31.1769 1.44270 0.721348 0.692573i \(-0.243523\pi\)
0.721348 + 0.692573i \(0.243523\pi\)
\(468\) 0 0
\(469\) −14.0000 + 12.1244i −0.646460 + 0.559851i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 66.0000i 3.03468i
\(474\) 0 0
\(475\) 48.4974i 2.22521i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 24.2487 1.10795 0.553976 0.832533i \(-0.313110\pi\)
0.553976 + 0.832533i \(0.313110\pi\)
\(480\) 0 0
\(481\) 3.46410i 0.157949i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 24.0000i 1.08978i
\(486\) 0 0
\(487\) −28.0000 −1.26880 −0.634401 0.773004i \(-0.718753\pi\)
−0.634401 + 0.773004i \(0.718753\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 18.0000i 0.812329i −0.913800 0.406164i \(-0.866866\pi\)
0.913800 0.406164i \(-0.133134\pi\)
\(492\) 0 0
\(493\) 5.19615i 0.234023i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 5.19615 + 6.00000i 0.233079 + 0.269137i
\(498\) 0 0
\(499\) −28.0000 −1.25345 −0.626726 0.779240i \(-0.715605\pi\)
−0.626726 + 0.779240i \(0.715605\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −17.3205 −0.772283 −0.386142 0.922440i \(-0.626192\pi\)
−0.386142 + 0.922440i \(0.626192\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −3.46410 −0.153544 −0.0767718 0.997049i \(-0.524461\pi\)
−0.0767718 + 0.997049i \(0.524461\pi\)
\(510\) 0 0
\(511\) 12.0000 + 13.8564i 0.530849 + 0.612971i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 30.0000i 1.32196i
\(516\) 0 0
\(517\) 41.5692i 1.82821i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 1.73205 0.0758825 0.0379413 0.999280i \(-0.487920\pi\)
0.0379413 + 0.999280i \(0.487920\pi\)
\(522\) 0 0
\(523\) 17.3205i 0.757373i −0.925525 0.378686i \(-0.876376\pi\)
0.925525 0.378686i \(-0.123624\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 9.00000i 0.392046i
\(528\) 0 0
\(529\) 14.0000 0.608696
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 12.0000i 0.519778i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −41.5692 6.00000i −1.79051 0.258438i
\(540\) 0 0
\(541\) 38.0000 1.63375 0.816874 0.576816i \(-0.195705\pi\)
0.816874 + 0.576816i \(0.195705\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 13.8564 0.593543
\(546\) 0 0
\(547\) −8.00000 −0.342055 −0.171028 0.985266i \(-0.554709\pi\)
−0.171028 + 0.985266i \(0.554709\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −20.7846 −0.885454
\(552\) 0 0
\(553\) 16.0000 13.8564i 0.680389 0.589234i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 39.0000i 1.65248i −0.563316 0.826242i \(-0.690475\pi\)
0.563316 0.826242i \(-0.309525\pi\)
\(558\) 0 0
\(559\) 19.0526i 0.805837i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −36.3731 −1.53294 −0.766471 0.642279i \(-0.777989\pi\)
−0.766471 + 0.642279i \(0.777989\pi\)
\(564\) 0 0
\(565\) 20.7846i 0.874415i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 18.0000i 0.754599i 0.926091 + 0.377300i \(0.123147\pi\)
−0.926091 + 0.377300i \(0.876853\pi\)
\(570\) 0 0
\(571\) −41.0000 −1.71580 −0.857898 0.513820i \(-0.828230\pi\)
−0.857898 + 0.513820i \(0.828230\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 21.0000i 0.875761i
\(576\) 0 0
\(577\) 38.1051i 1.58634i −0.609002 0.793168i \(-0.708430\pi\)
0.609002 0.793168i \(-0.291570\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −6.92820 + 6.00000i −0.287430 + 0.248922i
\(582\) 0 0
\(583\) 18.0000 0.745484
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 25.9808 1.07234 0.536170 0.844110i \(-0.319870\pi\)
0.536170 + 0.844110i \(0.319870\pi\)
\(588\) 0 0
\(589\) −36.0000 −1.48335
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 6.92820 0.284507 0.142254 0.989830i \(-0.454565\pi\)
0.142254 + 0.989830i \(0.454565\pi\)
\(594\) 0 0
\(595\) 12.0000 10.3923i 0.491952 0.426043i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 9.00000i 0.367730i 0.982952 + 0.183865i \(0.0588609\pi\)
−0.982952 + 0.183865i \(0.941139\pi\)
\(600\) 0 0
\(601\) 31.1769i 1.27173i −0.771799 0.635866i \(-0.780643\pi\)
0.771799 0.635866i \(-0.219357\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −86.6025 −3.52089
\(606\) 0 0
\(607\) 1.73205i 0.0703018i −0.999382 0.0351509i \(-0.988809\pi\)
0.999382 0.0351509i \(-0.0111912\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 12.0000i 0.485468i
\(612\) 0 0
\(613\) −28.0000 −1.13091 −0.565455 0.824779i \(-0.691299\pi\)
−0.565455 + 0.824779i \(0.691299\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 18.0000i 0.724653i −0.932051 0.362326i \(-0.881983\pi\)
0.932051 0.362326i \(-0.118017\pi\)
\(618\) 0 0
\(619\) 38.1051i 1.53157i −0.643094 0.765787i \(-0.722350\pi\)
0.643094 0.765787i \(-0.277650\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −10.3923 + 9.00000i −0.416359 + 0.360577i
\(624\) 0 0
\(625\) −11.0000 −0.440000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 3.46410 0.138123
\(630\) 0 0
\(631\) −2.00000 −0.0796187 −0.0398094 0.999207i \(-0.512675\pi\)
−0.0398094 + 0.999207i \(0.512675\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −48.4974 −1.92456
\(636\) 0 0
\(637\) 12.0000 + 1.73205i 0.475457 + 0.0686264i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) 10.3923i 0.409832i 0.978780 + 0.204916i \(0.0656922\pi\)
−0.978780 + 0.204916i \(0.934308\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 10.3923 0.408564 0.204282 0.978912i \(-0.434514\pi\)
0.204282 + 0.978912i \(0.434514\pi\)
\(648\) 0 0
\(649\) 51.9615i 2.03967i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 27.0000i 1.05659i −0.849060 0.528296i \(-0.822831\pi\)
0.849060 0.528296i \(-0.177169\pi\)
\(654\) 0 0
\(655\) 18.0000 0.703318
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 48.0000i 1.86981i −0.354892 0.934907i \(-0.615482\pi\)
0.354892 0.934907i \(-0.384518\pi\)
\(660\) 0 0
\(661\) 48.4974i 1.88633i 0.332323 + 0.943166i \(0.392168\pi\)
−0.332323 + 0.943166i \(0.607832\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −41.5692 48.0000i −1.61199 1.86136i
\(666\) 0 0
\(667\) −9.00000 −0.348481
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 83.1384 3.20952
\(672\) 0 0
\(673\) 31.0000 1.19496 0.597481 0.801883i \(-0.296168\pi\)
0.597481 + 0.801883i \(0.296168\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 10.3923 0.399409 0.199704 0.979856i \(-0.436002\pi\)
0.199704 + 0.979856i \(0.436002\pi\)
\(678\) 0 0
\(679\) −12.0000 13.8564i −0.460518 0.531760i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 42.0000i 1.60709i −0.595247 0.803543i \(-0.702946\pi\)
0.595247 0.803543i \(-0.297054\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −5.19615 −0.197958
\(690\) 0 0
\(691\) 10.3923i 0.395342i −0.980268 0.197671i \(-0.936662\pi\)
0.980268 0.197671i \(-0.0633378\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 12.0000i 0.455186i
\(696\) 0 0
\(697\) −12.0000 −0.454532
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 30.0000i 1.13308i −0.824033 0.566542i \(-0.808281\pi\)
0.824033 0.566542i \(-0.191719\pi\)
\(702\) 0 0
\(703\) 13.8564i 0.522604i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 10.0000 0.375558 0.187779 0.982211i \(-0.439871\pi\)
0.187779 + 0.982211i \(0.439871\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −15.5885 −0.583792
\(714\) 0 0
\(715\) 36.0000 1.34632
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −6.92820 −0.258378 −0.129189 0.991620i \(-0.541237\pi\)
−0.129189 + 0.991620i \(0.541237\pi\)
\(720\) 0 0
\(721\) 15.0000 + 17.3205i 0.558629 + 0.645049i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 21.0000i 0.779920i
\(726\) 0 0
\(727\) 25.9808i 0.963573i 0.876289 + 0.481787i \(0.160012\pi\)
−0.876289 + 0.481787i \(0.839988\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −19.0526 −0.704684
\(732\) 0 0
\(733\) 36.3731i 1.34347i 0.740792 + 0.671735i \(0.234451\pi\)
−0.740792 + 0.671735i \(0.765549\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 42.0000i 1.54709i
\(738\) 0 0
\(739\) −20.0000 −0.735712 −0.367856 0.929883i \(-0.619908\pi\)
−0.367856 + 0.929883i \(0.619908\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 9.00000i 0.330178i 0.986279 + 0.165089i \(0.0527911\pi\)
−0.986279 + 0.165089i \(0.947209\pi\)
\(744\) 0 0
\(745\) 51.9615i 1.90372i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 2.00000 0.0729810 0.0364905 0.999334i \(-0.488382\pi\)
0.0364905 + 0.999334i \(0.488382\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 27.7128 1.00857
\(756\) 0 0
\(757\) −2.00000 −0.0726912 −0.0363456 0.999339i \(-0.511572\pi\)
−0.0363456 + 0.999339i \(0.511572\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −19.0526 −0.690655 −0.345327 0.938482i \(-0.612232\pi\)
−0.345327 + 0.938482i \(0.612232\pi\)
\(762\) 0 0
\(763\) −8.00000 + 6.92820i −0.289619 + 0.250818i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 15.0000i 0.541619i
\(768\) 0 0
\(769\) 3.46410i 0.124919i 0.998048 + 0.0624593i \(0.0198944\pi\)
−0.998048 + 0.0624593i \(0.980106\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −38.1051 −1.37055 −0.685273 0.728286i \(-0.740317\pi\)
−0.685273 + 0.728286i \(0.740317\pi\)
\(774\) 0 0
\(775\) 36.3731i 1.30656i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 48.0000i 1.71978i
\(780\) 0 0
\(781\) −18.0000 −0.644091
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 54.0000i 1.92734i
\(786\) 0 0
\(787\) 3.46410i 0.123482i 0.998092 + 0.0617409i \(0.0196653\pi\)
−0.998092 + 0.0617409i \(0.980335\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 10.3923 + 12.0000i 0.369508 + 0.426671i
\(792\) 0 0
\(793\) −24.0000 −0.852265
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 31.1769 1.10434 0.552171 0.833731i \(-0.313799\pi\)
0.552171 + 0.833731i \(0.313799\pi\)
\(798\) 0 0
\(799\) −12.0000 −0.424529
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −41.5692 −1.46695
\(804\) 0 0
\(805\) −18.0000 20.7846i −0.634417 0.732561i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 24.0000i 0.843795i 0.906644 + 0.421898i \(0.138636\pi\)
−0.906644 + 0.421898i \(0.861364\pi\)
\(810\) 0 0
\(811\) 31.1769i 1.09477i −0.836881 0.547385i \(-0.815623\pi\)
0.836881 0.547385i \(-0.184377\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −38.1051 −1.33476