Properties

Label 5376.2.c.bh.2689.2
Level $5376$
Weight $2$
Character 5376.2689
Analytic conductor $42.928$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 5376 = 2^{8} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5376.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(42.9275761266\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 672)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2689.2
Root \(0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 5376.2689
Dual form 5376.2.c.bh.2689.3

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{3} +3.46410i q^{5} -1.00000 q^{7} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{3} +3.46410i q^{5} -1.00000 q^{7} -1.00000 q^{9} +5.46410i q^{11} +2.00000i q^{13} +3.46410 q^{15} +7.46410 q^{17} +6.92820i q^{19} +1.00000i q^{21} +5.46410 q^{23} -7.00000 q^{25} +1.00000i q^{27} +8.92820i q^{29} -2.92820 q^{31} +5.46410 q^{33} -3.46410i q^{35} +2.00000i q^{37} +2.00000 q^{39} -4.53590 q^{41} -8.00000i q^{43} -3.46410i q^{45} +2.92820 q^{47} +1.00000 q^{49} -7.46410i q^{51} +2.00000i q^{53} -18.9282 q^{55} +6.92820 q^{57} -14.9282i q^{59} +4.92820i q^{61} +1.00000 q^{63} -6.92820 q^{65} -10.9282i q^{67} -5.46410i q^{69} +2.53590 q^{71} +0.928203 q^{73} +7.00000i q^{75} -5.46410i q^{77} -2.92820 q^{79} +1.00000 q^{81} +4.00000i q^{83} +25.8564i q^{85} +8.92820 q^{87} +3.46410 q^{89} -2.00000i q^{91} +2.92820i q^{93} -24.0000 q^{95} +4.92820 q^{97} -5.46410i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{7} - 4q^{9} + O(q^{10}) \) \( 4q - 4q^{7} - 4q^{9} + 16q^{17} + 8q^{23} - 28q^{25} + 16q^{31} + 8q^{33} + 8q^{39} - 32q^{41} - 16q^{47} + 4q^{49} - 48q^{55} + 4q^{63} + 24q^{71} - 24q^{73} + 16q^{79} + 4q^{81} + 8q^{87} - 96q^{95} - 8q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5376\mathbb{Z}\right)^\times\).

\(n\) \(1793\) \(2815\) \(4609\) \(5125\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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
\(3\) − 1.00000i − 0.577350i
\(4\) 0 0
\(5\) 3.46410i 1.54919i 0.632456 + 0.774597i \(0.282047\pi\)
−0.632456 + 0.774597i \(0.717953\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 5.46410i 1.64749i 0.566961 + 0.823744i \(0.308119\pi\)
−0.566961 + 0.823744i \(0.691881\pi\)
\(12\) 0 0
\(13\) 2.00000i 0.554700i 0.960769 + 0.277350i \(0.0894562\pi\)
−0.960769 + 0.277350i \(0.910544\pi\)
\(14\) 0 0
\(15\) 3.46410 0.894427
\(16\) 0 0
\(17\) 7.46410 1.81031 0.905155 0.425081i \(-0.139754\pi\)
0.905155 + 0.425081i \(0.139754\pi\)
\(18\) 0 0
\(19\) 6.92820i 1.58944i 0.606977 + 0.794719i \(0.292382\pi\)
−0.606977 + 0.794719i \(0.707618\pi\)
\(20\) 0 0
\(21\) 1.00000i 0.218218i
\(22\) 0 0
\(23\) 5.46410 1.13934 0.569672 0.821872i \(-0.307070\pi\)
0.569672 + 0.821872i \(0.307070\pi\)
\(24\) 0 0
\(25\) −7.00000 −1.40000
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) 8.92820i 1.65793i 0.559304 + 0.828963i \(0.311069\pi\)
−0.559304 + 0.828963i \(0.688931\pi\)
\(30\) 0 0
\(31\) −2.92820 −0.525921 −0.262960 0.964807i \(-0.584699\pi\)
−0.262960 + 0.964807i \(0.584699\pi\)
\(32\) 0 0
\(33\) 5.46410 0.951178
\(34\) 0 0
\(35\) − 3.46410i − 0.585540i
\(36\) 0 0
\(37\) 2.00000i 0.328798i 0.986394 + 0.164399i \(0.0525685\pi\)
−0.986394 + 0.164399i \(0.947432\pi\)
\(38\) 0 0
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) −4.53590 −0.708388 −0.354194 0.935172i \(-0.615245\pi\)
−0.354194 + 0.935172i \(0.615245\pi\)
\(42\) 0 0
\(43\) − 8.00000i − 1.21999i −0.792406 0.609994i \(-0.791172\pi\)
0.792406 0.609994i \(-0.208828\pi\)
\(44\) 0 0
\(45\) − 3.46410i − 0.516398i
\(46\) 0 0
\(47\) 2.92820 0.427122 0.213561 0.976930i \(-0.431494\pi\)
0.213561 + 0.976930i \(0.431494\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) − 7.46410i − 1.04518i
\(52\) 0 0
\(53\) 2.00000i 0.274721i 0.990521 + 0.137361i \(0.0438619\pi\)
−0.990521 + 0.137361i \(0.956138\pi\)
\(54\) 0 0
\(55\) −18.9282 −2.55228
\(56\) 0 0
\(57\) 6.92820 0.917663
\(58\) 0 0
\(59\) − 14.9282i − 1.94349i −0.236040 0.971743i \(-0.575850\pi\)
0.236040 0.971743i \(-0.424150\pi\)
\(60\) 0 0
\(61\) 4.92820i 0.630992i 0.948927 + 0.315496i \(0.102171\pi\)
−0.948927 + 0.315496i \(0.897829\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 0 0
\(65\) −6.92820 −0.859338
\(66\) 0 0
\(67\) − 10.9282i − 1.33509i −0.744568 0.667546i \(-0.767345\pi\)
0.744568 0.667546i \(-0.232655\pi\)
\(68\) 0 0
\(69\) − 5.46410i − 0.657801i
\(70\) 0 0
\(71\) 2.53590 0.300956 0.150478 0.988613i \(-0.451919\pi\)
0.150478 + 0.988613i \(0.451919\pi\)
\(72\) 0 0
\(73\) 0.928203 0.108638 0.0543190 0.998524i \(-0.482701\pi\)
0.0543190 + 0.998524i \(0.482701\pi\)
\(74\) 0 0
\(75\) 7.00000i 0.808290i
\(76\) 0 0
\(77\) − 5.46410i − 0.622692i
\(78\) 0 0
\(79\) −2.92820 −0.329449 −0.164724 0.986340i \(-0.552673\pi\)
−0.164724 + 0.986340i \(0.552673\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 4.00000i 0.439057i 0.975606 + 0.219529i \(0.0704519\pi\)
−0.975606 + 0.219529i \(0.929548\pi\)
\(84\) 0 0
\(85\) 25.8564i 2.80452i
\(86\) 0 0
\(87\) 8.92820 0.957204
\(88\) 0 0
\(89\) 3.46410 0.367194 0.183597 0.983002i \(-0.441226\pi\)
0.183597 + 0.983002i \(0.441226\pi\)
\(90\) 0 0
\(91\) − 2.00000i − 0.209657i
\(92\) 0 0
\(93\) 2.92820i 0.303641i
\(94\) 0 0
\(95\) −24.0000 −2.46235
\(96\) 0 0
\(97\) 4.92820 0.500383 0.250192 0.968196i \(-0.419506\pi\)
0.250192 + 0.968196i \(0.419506\pi\)
\(98\) 0 0
\(99\) − 5.46410i − 0.549163i
\(100\) 0 0
\(101\) 8.53590i 0.849354i 0.905345 + 0.424677i \(0.139612\pi\)
−0.905345 + 0.424677i \(0.860388\pi\)
\(102\) 0 0
\(103\) 2.92820 0.288524 0.144262 0.989539i \(-0.453919\pi\)
0.144262 + 0.989539i \(0.453919\pi\)
\(104\) 0 0
\(105\) −3.46410 −0.338062
\(106\) 0 0
\(107\) − 8.39230i − 0.811315i −0.914025 0.405657i \(-0.867043\pi\)
0.914025 0.405657i \(-0.132957\pi\)
\(108\) 0 0
\(109\) − 2.00000i − 0.191565i −0.995402 0.0957826i \(-0.969465\pi\)
0.995402 0.0957826i \(-0.0305354\pi\)
\(110\) 0 0
\(111\) 2.00000 0.189832
\(112\) 0 0
\(113\) 7.85641 0.739069 0.369534 0.929217i \(-0.379517\pi\)
0.369534 + 0.929217i \(0.379517\pi\)
\(114\) 0 0
\(115\) 18.9282i 1.76506i
\(116\) 0 0
\(117\) − 2.00000i − 0.184900i
\(118\) 0 0
\(119\) −7.46410 −0.684233
\(120\) 0 0
\(121\) −18.8564 −1.71422
\(122\) 0 0
\(123\) 4.53590i 0.408988i
\(124\) 0 0
\(125\) − 6.92820i − 0.619677i
\(126\) 0 0
\(127\) 10.9282 0.969721 0.484861 0.874591i \(-0.338870\pi\)
0.484861 + 0.874591i \(0.338870\pi\)
\(128\) 0 0
\(129\) −8.00000 −0.704361
\(130\) 0 0
\(131\) − 12.0000i − 1.04844i −0.851581 0.524222i \(-0.824356\pi\)
0.851581 0.524222i \(-0.175644\pi\)
\(132\) 0 0
\(133\) − 6.92820i − 0.600751i
\(134\) 0 0
\(135\) −3.46410 −0.298142
\(136\) 0 0
\(137\) −4.92820 −0.421045 −0.210522 0.977589i \(-0.567516\pi\)
−0.210522 + 0.977589i \(0.567516\pi\)
\(138\) 0 0
\(139\) 9.85641i 0.836009i 0.908445 + 0.418005i \(0.137270\pi\)
−0.908445 + 0.418005i \(0.862730\pi\)
\(140\) 0 0
\(141\) − 2.92820i − 0.246599i
\(142\) 0 0
\(143\) −10.9282 −0.913862
\(144\) 0 0
\(145\) −30.9282 −2.56845
\(146\) 0 0
\(147\) − 1.00000i − 0.0824786i
\(148\) 0 0
\(149\) 18.0000i 1.47462i 0.675556 + 0.737309i \(0.263904\pi\)
−0.675556 + 0.737309i \(0.736096\pi\)
\(150\) 0 0
\(151\) −5.85641 −0.476588 −0.238294 0.971193i \(-0.576588\pi\)
−0.238294 + 0.971193i \(0.576588\pi\)
\(152\) 0 0
\(153\) −7.46410 −0.603437
\(154\) 0 0
\(155\) − 10.1436i − 0.814753i
\(156\) 0 0
\(157\) 12.9282i 1.03178i 0.856654 + 0.515891i \(0.172539\pi\)
−0.856654 + 0.515891i \(0.827461\pi\)
\(158\) 0 0
\(159\) 2.00000 0.158610
\(160\) 0 0
\(161\) −5.46410 −0.430632
\(162\) 0 0
\(163\) 2.92820i 0.229355i 0.993403 + 0.114677i \(0.0365834\pi\)
−0.993403 + 0.114677i \(0.963417\pi\)
\(164\) 0 0
\(165\) 18.9282i 1.47356i
\(166\) 0 0
\(167\) 2.92820 0.226591 0.113296 0.993561i \(-0.463859\pi\)
0.113296 + 0.993561i \(0.463859\pi\)
\(168\) 0 0
\(169\) 9.00000 0.692308
\(170\) 0 0
\(171\) − 6.92820i − 0.529813i
\(172\) 0 0
\(173\) 1.60770i 0.122231i 0.998131 + 0.0611154i \(0.0194658\pi\)
−0.998131 + 0.0611154i \(0.980534\pi\)
\(174\) 0 0
\(175\) 7.00000 0.529150
\(176\) 0 0
\(177\) −14.9282 −1.12207
\(178\) 0 0
\(179\) − 2.53590i − 0.189542i −0.995499 0.0947710i \(-0.969788\pi\)
0.995499 0.0947710i \(-0.0302119\pi\)
\(180\) 0 0
\(181\) 6.00000i 0.445976i 0.974821 + 0.222988i \(0.0715812\pi\)
−0.974821 + 0.222988i \(0.928419\pi\)
\(182\) 0 0
\(183\) 4.92820 0.364303
\(184\) 0 0
\(185\) −6.92820 −0.509372
\(186\) 0 0
\(187\) 40.7846i 2.98247i
\(188\) 0 0
\(189\) − 1.00000i − 0.0727393i
\(190\) 0 0
\(191\) −2.53590 −0.183491 −0.0917456 0.995782i \(-0.529245\pi\)
−0.0917456 + 0.995782i \(0.529245\pi\)
\(192\) 0 0
\(193\) −11.8564 −0.853443 −0.426721 0.904383i \(-0.640332\pi\)
−0.426721 + 0.904383i \(0.640332\pi\)
\(194\) 0 0
\(195\) 6.92820i 0.496139i
\(196\) 0 0
\(197\) − 19.8564i − 1.41471i −0.706858 0.707355i \(-0.749888\pi\)
0.706858 0.707355i \(-0.250112\pi\)
\(198\) 0 0
\(199\) −21.8564 −1.54936 −0.774680 0.632354i \(-0.782089\pi\)
−0.774680 + 0.632354i \(0.782089\pi\)
\(200\) 0 0
\(201\) −10.9282 −0.770816
\(202\) 0 0
\(203\) − 8.92820i − 0.626637i
\(204\) 0 0
\(205\) − 15.7128i − 1.09743i
\(206\) 0 0
\(207\) −5.46410 −0.379781
\(208\) 0 0
\(209\) −37.8564 −2.61858
\(210\) 0 0
\(211\) 16.0000i 1.10149i 0.834675 + 0.550743i \(0.185655\pi\)
−0.834675 + 0.550743i \(0.814345\pi\)
\(212\) 0 0
\(213\) − 2.53590i − 0.173757i
\(214\) 0 0
\(215\) 27.7128 1.89000
\(216\) 0 0
\(217\) 2.92820 0.198779
\(218\) 0 0
\(219\) − 0.928203i − 0.0627222i
\(220\) 0 0
\(221\) 14.9282i 1.00418i
\(222\) 0 0
\(223\) 24.0000 1.60716 0.803579 0.595198i \(-0.202926\pi\)
0.803579 + 0.595198i \(0.202926\pi\)
\(224\) 0 0
\(225\) 7.00000 0.466667
\(226\) 0 0
\(227\) 9.07180i 0.602116i 0.953606 + 0.301058i \(0.0973398\pi\)
−0.953606 + 0.301058i \(0.902660\pi\)
\(228\) 0 0
\(229\) 0.143594i 0.00948893i 0.999989 + 0.00474446i \(0.00151022\pi\)
−0.999989 + 0.00474446i \(0.998490\pi\)
\(230\) 0 0
\(231\) −5.46410 −0.359511
\(232\) 0 0
\(233\) 0.928203 0.0608086 0.0304043 0.999538i \(-0.490321\pi\)
0.0304043 + 0.999538i \(0.490321\pi\)
\(234\) 0 0
\(235\) 10.1436i 0.661695i
\(236\) 0 0
\(237\) 2.92820i 0.190207i
\(238\) 0 0
\(239\) −21.4641 −1.38840 −0.694199 0.719783i \(-0.744241\pi\)
−0.694199 + 0.719783i \(0.744241\pi\)
\(240\) 0 0
\(241\) −16.9282 −1.09044 −0.545221 0.838293i \(-0.683554\pi\)
−0.545221 + 0.838293i \(0.683554\pi\)
\(242\) 0 0
\(243\) − 1.00000i − 0.0641500i
\(244\) 0 0
\(245\) 3.46410i 0.221313i
\(246\) 0 0
\(247\) −13.8564 −0.881662
\(248\) 0 0
\(249\) 4.00000 0.253490
\(250\) 0 0
\(251\) − 22.9282i − 1.44722i −0.690212 0.723608i \(-0.742483\pi\)
0.690212 0.723608i \(-0.257517\pi\)
\(252\) 0 0
\(253\) 29.8564i 1.87706i
\(254\) 0 0
\(255\) 25.8564 1.61919
\(256\) 0 0
\(257\) 4.53590 0.282942 0.141471 0.989942i \(-0.454817\pi\)
0.141471 + 0.989942i \(0.454817\pi\)
\(258\) 0 0
\(259\) − 2.00000i − 0.124274i
\(260\) 0 0
\(261\) − 8.92820i − 0.552642i
\(262\) 0 0
\(263\) −8.39230 −0.517492 −0.258746 0.965945i \(-0.583309\pi\)
−0.258746 + 0.965945i \(0.583309\pi\)
\(264\) 0 0
\(265\) −6.92820 −0.425596
\(266\) 0 0
\(267\) − 3.46410i − 0.212000i
\(268\) 0 0
\(269\) − 3.46410i − 0.211210i −0.994408 0.105605i \(-0.966322\pi\)
0.994408 0.105605i \(-0.0336779\pi\)
\(270\) 0 0
\(271\) 10.9282 0.663841 0.331921 0.943307i \(-0.392303\pi\)
0.331921 + 0.943307i \(0.392303\pi\)
\(272\) 0 0
\(273\) −2.00000 −0.121046
\(274\) 0 0
\(275\) − 38.2487i − 2.30648i
\(276\) 0 0
\(277\) − 27.8564i − 1.67373i −0.547410 0.836865i \(-0.684386\pi\)
0.547410 0.836865i \(-0.315614\pi\)
\(278\) 0 0
\(279\) 2.92820 0.175307
\(280\) 0 0
\(281\) 0.928203 0.0553720 0.0276860 0.999617i \(-0.491186\pi\)
0.0276860 + 0.999617i \(0.491186\pi\)
\(282\) 0 0
\(283\) − 1.07180i − 0.0637117i −0.999492 0.0318559i \(-0.989858\pi\)
0.999492 0.0318559i \(-0.0101417\pi\)
\(284\) 0 0
\(285\) 24.0000i 1.42164i
\(286\) 0 0
\(287\) 4.53590 0.267746
\(288\) 0 0
\(289\) 38.7128 2.27722
\(290\) 0 0
\(291\) − 4.92820i − 0.288896i
\(292\) 0 0
\(293\) − 23.4641i − 1.37079i −0.728173 0.685394i \(-0.759630\pi\)
0.728173 0.685394i \(-0.240370\pi\)
\(294\) 0 0
\(295\) 51.7128 3.01084
\(296\) 0 0
\(297\) −5.46410 −0.317059
\(298\) 0 0
\(299\) 10.9282i 0.631994i
\(300\) 0 0
\(301\) 8.00000i 0.461112i
\(302\) 0 0
\(303\) 8.53590 0.490375
\(304\) 0 0
\(305\) −17.0718 −0.977528
\(306\) 0 0
\(307\) − 9.07180i − 0.517755i −0.965910 0.258877i \(-0.916647\pi\)
0.965910 0.258877i \(-0.0833526\pi\)
\(308\) 0 0
\(309\) − 2.92820i − 0.166580i
\(310\) 0 0
\(311\) 5.07180 0.287595 0.143798 0.989607i \(-0.454069\pi\)
0.143798 + 0.989607i \(0.454069\pi\)
\(312\) 0 0
\(313\) 14.0000 0.791327 0.395663 0.918396i \(-0.370515\pi\)
0.395663 + 0.918396i \(0.370515\pi\)
\(314\) 0 0
\(315\) 3.46410i 0.195180i
\(316\) 0 0
\(317\) 27.8564i 1.56457i 0.622920 + 0.782286i \(0.285946\pi\)
−0.622920 + 0.782286i \(0.714054\pi\)
\(318\) 0 0
\(319\) −48.7846 −2.73141
\(320\) 0 0
\(321\) −8.39230 −0.468413
\(322\) 0 0
\(323\) 51.7128i 2.87738i
\(324\) 0 0
\(325\) − 14.0000i − 0.776580i
\(326\) 0 0
\(327\) −2.00000 −0.110600
\(328\) 0 0
\(329\) −2.92820 −0.161437
\(330\) 0 0
\(331\) 24.0000i 1.31916i 0.751635 + 0.659580i \(0.229266\pi\)
−0.751635 + 0.659580i \(0.770734\pi\)
\(332\) 0 0
\(333\) − 2.00000i − 0.109599i
\(334\) 0 0
\(335\) 37.8564 2.06832
\(336\) 0 0
\(337\) −19.8564 −1.08165 −0.540824 0.841136i \(-0.681887\pi\)
−0.540824 + 0.841136i \(0.681887\pi\)
\(338\) 0 0
\(339\) − 7.85641i − 0.426701i
\(340\) 0 0
\(341\) − 16.0000i − 0.866449i
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 18.9282 1.01906
\(346\) 0 0
\(347\) − 11.3205i − 0.607717i −0.952717 0.303858i \(-0.901725\pi\)
0.952717 0.303858i \(-0.0982750\pi\)
\(348\) 0 0
\(349\) − 30.7846i − 1.64786i −0.566690 0.823931i \(-0.691776\pi\)
0.566690 0.823931i \(-0.308224\pi\)
\(350\) 0 0
\(351\) −2.00000 −0.106752
\(352\) 0 0
\(353\) 15.4641 0.823071 0.411536 0.911394i \(-0.364993\pi\)
0.411536 + 0.911394i \(0.364993\pi\)
\(354\) 0 0
\(355\) 8.78461i 0.466239i
\(356\) 0 0
\(357\) 7.46410i 0.395042i
\(358\) 0 0
\(359\) −27.3205 −1.44192 −0.720961 0.692976i \(-0.756299\pi\)
−0.720961 + 0.692976i \(0.756299\pi\)
\(360\) 0 0
\(361\) −29.0000 −1.52632
\(362\) 0 0
\(363\) 18.8564i 0.989705i
\(364\) 0 0
\(365\) 3.21539i 0.168301i
\(366\) 0 0
\(367\) −8.00000 −0.417597 −0.208798 0.977959i \(-0.566955\pi\)
−0.208798 + 0.977959i \(0.566955\pi\)
\(368\) 0 0
\(369\) 4.53590 0.236129
\(370\) 0 0
\(371\) − 2.00000i − 0.103835i
\(372\) 0 0
\(373\) − 30.0000i − 1.55334i −0.629907 0.776671i \(-0.716907\pi\)
0.629907 0.776671i \(-0.283093\pi\)
\(374\) 0 0
\(375\) −6.92820 −0.357771
\(376\) 0 0
\(377\) −17.8564 −0.919652
\(378\) 0 0
\(379\) − 8.00000i − 0.410932i −0.978664 0.205466i \(-0.934129\pi\)
0.978664 0.205466i \(-0.0658711\pi\)
\(380\) 0 0
\(381\) − 10.9282i − 0.559869i
\(382\) 0 0
\(383\) 19.7128 1.00728 0.503639 0.863914i \(-0.331994\pi\)
0.503639 + 0.863914i \(0.331994\pi\)
\(384\) 0 0
\(385\) 18.9282 0.964671
\(386\) 0 0
\(387\) 8.00000i 0.406663i
\(388\) 0 0
\(389\) 26.7846i 1.35803i 0.734123 + 0.679017i \(0.237594\pi\)
−0.734123 + 0.679017i \(0.762406\pi\)
\(390\) 0 0
\(391\) 40.7846 2.06257
\(392\) 0 0
\(393\) −12.0000 −0.605320
\(394\) 0 0
\(395\) − 10.1436i − 0.510380i
\(396\) 0 0
\(397\) − 30.7846i − 1.54504i −0.634993 0.772518i \(-0.718997\pi\)
0.634993 0.772518i \(-0.281003\pi\)
\(398\) 0 0
\(399\) −6.92820 −0.346844
\(400\) 0 0
\(401\) −8.92820 −0.445853 −0.222927 0.974835i \(-0.571561\pi\)
−0.222927 + 0.974835i \(0.571561\pi\)
\(402\) 0 0
\(403\) − 5.85641i − 0.291728i
\(404\) 0 0
\(405\) 3.46410i 0.172133i
\(406\) 0 0
\(407\) −10.9282 −0.541691
\(408\) 0 0
\(409\) 8.92820 0.441471 0.220736 0.975334i \(-0.429154\pi\)
0.220736 + 0.975334i \(0.429154\pi\)
\(410\) 0 0
\(411\) 4.92820i 0.243090i
\(412\) 0 0
\(413\) 14.9282i 0.734569i
\(414\) 0 0
\(415\) −13.8564 −0.680184
\(416\) 0 0
\(417\) 9.85641 0.482670
\(418\) 0 0
\(419\) − 30.9282i − 1.51094i −0.655182 0.755471i \(-0.727408\pi\)
0.655182 0.755471i \(-0.272592\pi\)
\(420\) 0 0
\(421\) 31.8564i 1.55259i 0.630372 + 0.776293i \(0.282902\pi\)
−0.630372 + 0.776293i \(0.717098\pi\)
\(422\) 0 0
\(423\) −2.92820 −0.142374
\(424\) 0 0
\(425\) −52.2487 −2.53443
\(426\) 0 0
\(427\) − 4.92820i − 0.238492i
\(428\) 0 0
\(429\) 10.9282i 0.527619i
\(430\) 0 0
\(431\) −38.2487 −1.84238 −0.921188 0.389118i \(-0.872780\pi\)
−0.921188 + 0.389118i \(0.872780\pi\)
\(432\) 0 0
\(433\) −0.143594 −0.00690067 −0.00345033 0.999994i \(-0.501098\pi\)
−0.00345033 + 0.999994i \(0.501098\pi\)
\(434\) 0 0
\(435\) 30.9282i 1.48289i
\(436\) 0 0
\(437\) 37.8564i 1.81092i
\(438\) 0 0
\(439\) −13.8564 −0.661330 −0.330665 0.943748i \(-0.607273\pi\)
−0.330665 + 0.943748i \(0.607273\pi\)
\(440\) 0 0
\(441\) −1.00000 −0.0476190
\(442\) 0 0
\(443\) − 7.60770i − 0.361453i −0.983533 0.180726i \(-0.942155\pi\)
0.983533 0.180726i \(-0.0578448\pi\)
\(444\) 0 0
\(445\) 12.0000i 0.568855i
\(446\) 0 0
\(447\) 18.0000 0.851371
\(448\) 0 0
\(449\) −19.8564 −0.937082 −0.468541 0.883442i \(-0.655220\pi\)
−0.468541 + 0.883442i \(0.655220\pi\)
\(450\) 0 0
\(451\) − 24.7846i − 1.16706i
\(452\) 0 0
\(453\) 5.85641i 0.275158i
\(454\) 0 0
\(455\) 6.92820 0.324799
\(456\) 0 0
\(457\) 17.7128 0.828570 0.414285 0.910147i \(-0.364032\pi\)
0.414285 + 0.910147i \(0.364032\pi\)
\(458\) 0 0
\(459\) 7.46410i 0.348394i
\(460\) 0 0
\(461\) 35.1769i 1.63835i 0.573542 + 0.819176i \(0.305569\pi\)
−0.573542 + 0.819176i \(0.694431\pi\)
\(462\) 0 0
\(463\) 8.78461 0.408255 0.204128 0.978944i \(-0.434564\pi\)
0.204128 + 0.978944i \(0.434564\pi\)
\(464\) 0 0
\(465\) −10.1436 −0.470398
\(466\) 0 0
\(467\) 20.7846i 0.961797i 0.876776 + 0.480899i \(0.159689\pi\)
−0.876776 + 0.480899i \(0.840311\pi\)
\(468\) 0 0
\(469\) 10.9282i 0.504618i
\(470\) 0 0
\(471\) 12.9282 0.595700
\(472\) 0 0
\(473\) 43.7128 2.00992
\(474\) 0 0
\(475\) − 48.4974i − 2.22521i
\(476\) 0 0
\(477\) − 2.00000i − 0.0915737i
\(478\) 0 0
\(479\) 18.9282 0.864852 0.432426 0.901670i \(-0.357658\pi\)
0.432426 + 0.901670i \(0.357658\pi\)
\(480\) 0 0
\(481\) −4.00000 −0.182384
\(482\) 0 0
\(483\) 5.46410i 0.248625i
\(484\) 0 0
\(485\) 17.0718i 0.775190i
\(486\) 0 0
\(487\) 27.7128 1.25579 0.627894 0.778299i \(-0.283917\pi\)
0.627894 + 0.778299i \(0.283917\pi\)
\(488\) 0 0
\(489\) 2.92820 0.132418
\(490\) 0 0
\(491\) − 13.4641i − 0.607626i −0.952732 0.303813i \(-0.901740\pi\)
0.952732 0.303813i \(-0.0982599\pi\)
\(492\) 0 0
\(493\) 66.6410i 3.00136i
\(494\) 0 0
\(495\) 18.9282 0.850759
\(496\) 0 0
\(497\) −2.53590 −0.113751
\(498\) 0 0
\(499\) − 5.85641i − 0.262169i −0.991371 0.131084i \(-0.958154\pi\)
0.991371 0.131084i \(-0.0418459\pi\)
\(500\) 0 0
\(501\) − 2.92820i − 0.130822i
\(502\) 0 0
\(503\) −16.0000 −0.713405 −0.356702 0.934218i \(-0.616099\pi\)
−0.356702 + 0.934218i \(0.616099\pi\)
\(504\) 0 0
\(505\) −29.5692 −1.31581
\(506\) 0 0
\(507\) − 9.00000i − 0.399704i
\(508\) 0 0
\(509\) 8.24871i 0.365618i 0.983148 + 0.182809i \(0.0585189\pi\)
−0.983148 + 0.182809i \(0.941481\pi\)
\(510\) 0 0
\(511\) −0.928203 −0.0410613
\(512\) 0 0
\(513\) −6.92820 −0.305888
\(514\) 0 0
\(515\) 10.1436i 0.446980i
\(516\) 0 0
\(517\) 16.0000i 0.703679i
\(518\) 0 0
\(519\) 1.60770 0.0705700
\(520\) 0 0
\(521\) −9.60770 −0.420921 −0.210460 0.977602i \(-0.567496\pi\)
−0.210460 + 0.977602i \(0.567496\pi\)
\(522\) 0 0
\(523\) 28.0000i 1.22435i 0.790721 + 0.612177i \(0.209706\pi\)
−0.790721 + 0.612177i \(0.790294\pi\)
\(524\) 0 0
\(525\) − 7.00000i − 0.305505i
\(526\) 0 0
\(527\) −21.8564 −0.952080
\(528\) 0 0
\(529\) 6.85641 0.298105
\(530\) 0 0
\(531\) 14.9282i 0.647829i
\(532\) 0 0
\(533\) − 9.07180i − 0.392943i
\(534\) 0 0
\(535\) 29.0718 1.25688
\(536\) 0 0
\(537\) −2.53590 −0.109432
\(538\) 0 0
\(539\) 5.46410i 0.235356i
\(540\) 0 0
\(541\) 19.8564i 0.853694i 0.904324 + 0.426847i \(0.140376\pi\)
−0.904324 + 0.426847i \(0.859624\pi\)
\(542\) 0 0
\(543\) 6.00000 0.257485
\(544\) 0 0
\(545\) 6.92820 0.296772
\(546\) 0 0
\(547\) 2.92820i 0.125201i 0.998039 + 0.0626005i \(0.0199394\pi\)
−0.998039 + 0.0626005i \(0.980061\pi\)
\(548\) 0 0
\(549\) − 4.92820i − 0.210331i
\(550\) 0 0
\(551\) −61.8564 −2.63517
\(552\) 0 0
\(553\) 2.92820 0.124520
\(554\) 0 0
\(555\) 6.92820i 0.294086i
\(556\) 0 0
\(557\) − 31.8564i − 1.34980i −0.737910 0.674900i \(-0.764187\pi\)
0.737910 0.674900i \(-0.235813\pi\)
\(558\) 0 0
\(559\) 16.0000 0.676728
\(560\) 0 0
\(561\) 40.7846 1.72193
\(562\) 0 0
\(563\) − 46.9282i − 1.97779i −0.148623 0.988894i \(-0.547484\pi\)
0.148623 0.988894i \(-0.452516\pi\)
\(564\) 0 0
\(565\) 27.2154i 1.14496i
\(566\) 0 0
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) 11.0718 0.464154 0.232077 0.972697i \(-0.425448\pi\)
0.232077 + 0.972697i \(0.425448\pi\)
\(570\) 0 0
\(571\) − 24.7846i − 1.03720i −0.855016 0.518602i \(-0.826453\pi\)
0.855016 0.518602i \(-0.173547\pi\)
\(572\) 0 0
\(573\) 2.53590i 0.105939i
\(574\) 0 0
\(575\) −38.2487 −1.59508
\(576\) 0 0
\(577\) −3.85641 −0.160544 −0.0802722 0.996773i \(-0.525579\pi\)
−0.0802722 + 0.996773i \(0.525579\pi\)
\(578\) 0 0
\(579\) 11.8564i 0.492735i
\(580\) 0 0
\(581\) − 4.00000i − 0.165948i
\(582\) 0 0
\(583\) −10.9282 −0.452600
\(584\) 0 0
\(585\) 6.92820 0.286446
\(586\) 0 0
\(587\) 20.7846i 0.857873i 0.903335 + 0.428936i \(0.141112\pi\)
−0.903335 + 0.428936i \(0.858888\pi\)
\(588\) 0 0
\(589\) − 20.2872i − 0.835919i
\(590\) 0 0
\(591\) −19.8564 −0.816783
\(592\) 0 0
\(593\) 15.4641 0.635035 0.317517 0.948252i \(-0.397151\pi\)
0.317517 + 0.948252i \(0.397151\pi\)
\(594\) 0 0
\(595\) − 25.8564i − 1.06001i
\(596\) 0 0
\(597\) 21.8564i 0.894523i
\(598\) 0 0
\(599\) −7.60770 −0.310842 −0.155421 0.987848i \(-0.549673\pi\)
−0.155421 + 0.987848i \(0.549673\pi\)
\(600\) 0 0
\(601\) 39.5692 1.61406 0.807031 0.590509i \(-0.201073\pi\)
0.807031 + 0.590509i \(0.201073\pi\)
\(602\) 0 0
\(603\) 10.9282i 0.445031i
\(604\) 0 0
\(605\) − 65.3205i − 2.65566i
\(606\) 0 0
\(607\) 13.8564 0.562414 0.281207 0.959647i \(-0.409265\pi\)
0.281207 + 0.959647i \(0.409265\pi\)
\(608\) 0 0
\(609\) −8.92820 −0.361789
\(610\) 0 0
\(611\) 5.85641i 0.236925i
\(612\) 0 0
\(613\) − 0.143594i − 0.00579969i −0.999996 0.00289984i \(-0.999077\pi\)
0.999996 0.00289984i \(-0.000923050\pi\)
\(614\) 0 0
\(615\) −15.7128 −0.633602
\(616\) 0 0
\(617\) 22.7846 0.917274 0.458637 0.888624i \(-0.348338\pi\)
0.458637 + 0.888624i \(0.348338\pi\)
\(618\) 0 0
\(619\) − 4.00000i − 0.160774i −0.996764 0.0803868i \(-0.974384\pi\)
0.996764 0.0803868i \(-0.0256155\pi\)
\(620\) 0 0
\(621\) 5.46410i 0.219267i
\(622\) 0 0
\(623\) −3.46410 −0.138786
\(624\) 0 0
\(625\) −11.0000 −0.440000
\(626\) 0 0
\(627\) 37.8564i 1.51184i
\(628\) 0 0
\(629\) 14.9282i 0.595226i
\(630\) 0 0
\(631\) 13.0718 0.520380 0.260190 0.965557i \(-0.416215\pi\)
0.260190 + 0.965557i \(0.416215\pi\)
\(632\) 0 0
\(633\) 16.0000 0.635943
\(634\) 0 0
\(635\) 37.8564i 1.50229i
\(636\) 0 0
\(637\) 2.00000i 0.0792429i
\(638\) 0 0
\(639\) −2.53590 −0.100319
\(640\) 0 0
\(641\) 34.7846 1.37391 0.686955 0.726700i \(-0.258947\pi\)
0.686955 + 0.726700i \(0.258947\pi\)
\(642\) 0 0
\(643\) − 28.7846i − 1.13515i −0.823320 0.567577i \(-0.807881\pi\)
0.823320 0.567577i \(-0.192119\pi\)
\(644\) 0 0
\(645\) − 27.7128i − 1.09119i
\(646\) 0 0
\(647\) −18.9282 −0.744144 −0.372072 0.928204i \(-0.621353\pi\)
−0.372072 + 0.928204i \(0.621353\pi\)
\(648\) 0 0
\(649\) 81.5692 3.20187
\(650\) 0 0
\(651\) − 2.92820i − 0.114765i
\(652\) 0 0
\(653\) − 7.07180i − 0.276741i −0.990381 0.138370i \(-0.955814\pi\)
0.990381 0.138370i \(-0.0441864\pi\)
\(654\) 0 0
\(655\) 41.5692 1.62424
\(656\) 0 0
\(657\) −0.928203 −0.0362127
\(658\) 0 0
\(659\) − 21.4641i − 0.836123i −0.908419 0.418061i \(-0.862710\pi\)
0.908419 0.418061i \(-0.137290\pi\)
\(660\) 0 0
\(661\) 14.7846i 0.575055i 0.957772 + 0.287527i \(0.0928332\pi\)
−0.957772 + 0.287527i \(0.907167\pi\)
\(662\) 0 0
\(663\) 14.9282 0.579763
\(664\) 0 0
\(665\) 24.0000 0.930680
\(666\) 0 0
\(667\) 48.7846i 1.88895i
\(668\) 0 0
\(669\) − 24.0000i − 0.927894i
\(670\) 0 0
\(671\) −26.9282 −1.03955
\(672\) 0 0
\(673\) 4.14359 0.159724 0.0798619 0.996806i \(-0.474552\pi\)
0.0798619 + 0.996806i \(0.474552\pi\)
\(674\) 0 0
\(675\) − 7.00000i − 0.269430i
\(676\) 0 0
\(677\) 22.3923i 0.860606i 0.902684 + 0.430303i \(0.141593\pi\)
−0.902684 + 0.430303i \(0.858407\pi\)
\(678\) 0 0
\(679\) −4.92820 −0.189127
\(680\) 0 0
\(681\) 9.07180 0.347632
\(682\) 0 0
\(683\) 30.2487i 1.15743i 0.815528 + 0.578717i \(0.196447\pi\)
−0.815528 + 0.578717i \(0.803553\pi\)
\(684\) 0 0
\(685\) − 17.0718i − 0.652280i
\(686\) 0 0
\(687\) 0.143594 0.00547844
\(688\) 0 0
\(689\) −4.00000 −0.152388
\(690\) 0 0
\(691\) 14.1436i 0.538048i 0.963134 + 0.269024i \(0.0867010\pi\)
−0.963134 + 0.269024i \(0.913299\pi\)
\(692\) 0 0
\(693\) 5.46410i 0.207564i
\(694\) 0 0
\(695\) −34.1436 −1.29514
\(696\) 0 0
\(697\) −33.8564 −1.28240
\(698\) 0 0
\(699\) − 0.928203i − 0.0351079i
\(700\) 0 0
\(701\) − 40.6410i − 1.53499i −0.641055 0.767495i \(-0.721503\pi\)
0.641055 0.767495i \(-0.278497\pi\)
\(702\) 0 0
\(703\) −13.8564 −0.522604
\(704\) 0 0
\(705\) 10.1436 0.382030
\(706\) 0 0
\(707\) − 8.53590i − 0.321025i
\(708\) 0 0
\(709\) 37.7128i 1.41633i 0.706045 + 0.708167i \(0.250478\pi\)
−0.706045 + 0.708167i \(0.749522\pi\)
\(710\) 0 0
\(711\) 2.92820 0.109816
\(712\) 0 0
\(713\) −16.0000 −0.599205
\(714\) 0 0
\(715\) − 37.8564i − 1.41575i
\(716\) 0 0
\(717\) 21.4641i 0.801592i
\(718\) 0 0
\(719\) −13.8564 −0.516757 −0.258378 0.966044i \(-0.583188\pi\)
−0.258378 + 0.966044i \(0.583188\pi\)
\(720\) 0 0
\(721\) −2.92820 −0.109052
\(722\) 0 0
\(723\) 16.9282i 0.629567i
\(724\) 0 0
\(725\) − 62.4974i − 2.32110i
\(726\) 0 0
\(727\) 34.9282 1.29542 0.647708 0.761889i \(-0.275728\pi\)
0.647708 + 0.761889i \(0.275728\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) − 59.7128i − 2.20856i
\(732\) 0 0
\(733\) − 49.7128i − 1.83618i −0.396367 0.918092i \(-0.629729\pi\)
0.396367 0.918092i \(-0.370271\pi\)
\(734\) 0 0
\(735\) 3.46410 0.127775
\(736\) 0 0
\(737\) 59.7128 2.19955
\(738\) 0 0
\(739\) 50.9282i 1.87342i 0.350101 + 0.936712i \(0.386147\pi\)
−0.350101 + 0.936712i \(0.613853\pi\)
\(740\) 0 0
\(741\) 13.8564i 0.509028i
\(742\) 0 0
\(743\) 32.3923 1.18836 0.594179 0.804333i \(-0.297477\pi\)
0.594179 + 0.804333i \(0.297477\pi\)
\(744\) 0 0
\(745\) −62.3538 −2.28447
\(746\) 0 0
\(747\) − 4.00000i − 0.146352i
\(748\) 0 0
\(749\) 8.39230i 0.306648i
\(750\) 0 0
\(751\) 27.7128 1.01125 0.505627 0.862752i \(-0.331261\pi\)
0.505627 + 0.862752i \(0.331261\pi\)
\(752\) 0 0
\(753\) −22.9282 −0.835550
\(754\) 0 0
\(755\) − 20.2872i − 0.738326i
\(756\) 0 0
\(757\) 26.0000i 0.944986i 0.881334 + 0.472493i \(0.156646\pi\)
−0.881334 + 0.472493i \(0.843354\pi\)
\(758\) 0 0
\(759\) 29.8564 1.08372
\(760\) 0 0
\(761\) −20.5359 −0.744426 −0.372213 0.928147i \(-0.621401\pi\)
−0.372213 + 0.928147i \(0.621401\pi\)
\(762\) 0 0
\(763\) 2.00000i 0.0724049i
\(764\) 0 0
\(765\) − 25.8564i − 0.934840i
\(766\) 0 0
\(767\) 29.8564 1.07805
\(768\) 0 0
\(769\) 31.8564 1.14877 0.574386 0.818585i \(-0.305241\pi\)
0.574386 + 0.818585i \(0.305241\pi\)
\(770\) 0 0
\(771\) − 4.53590i − 0.163356i
\(772\) 0 0
\(773\) − 29.3205i − 1.05459i −0.849684 0.527293i \(-0.823207\pi\)
0.849684 0.527293i \(-0.176793\pi\)
\(774\) 0 0
\(775\) 20.4974 0.736289
\(776\) 0 0
\(777\) −2.00000 −0.0717496
\(778\) 0 0
\(779\) − 31.4256i − 1.12594i
\(780\) 0 0
\(781\) 13.8564i 0.495821i
\(782\) 0 0
\(783\) −8.92820 −0.319068
\(784\) 0 0
\(785\) −44.7846 −1.59843
\(786\) 0 0
\(787\) 31.7128i 1.13044i 0.824940 + 0.565220i \(0.191209\pi\)
−0.824940 + 0.565220i \(0.808791\pi\)
\(788\) 0 0
\(789\) 8.39230i 0.298774i
\(790\) 0 0
\(791\) −7.85641 −0.279342
\(792\) 0 0
\(793\) −9.85641 −0.350011
\(794\) 0 0
\(795\) 6.92820i 0.245718i
\(796\) 0 0
\(797\) − 14.3923i − 0.509802i −0.966967 0.254901i \(-0.917957\pi\)
0.966967 0.254901i \(-0.0820428\pi\)
\(798\) 0 0
\(799\) 21.8564 0.773224
\(800\) 0 0
\(801\) −3.46410 −0.122398
\(802\) 0 0
\(803\) 5.07180i 0.178980i
\(804\) 0 0
\(805\) − 18.9282i − 0.667132i
\(806\) 0 0
\(807\) −3.46410 −0.121942
\(808\) 0 0
\(809\) 38.0000 1.33601 0.668004 0.744157i \(-0.267149\pi\)
0.668004 + 0.744157i \(0.267149\pi\)
\(810\) 0 0
\(811\) 45.5692i 1.60015i 0.599899 + 0.800076i \(0.295207\pi\)
−0.599899 + 0.800076i \(0.704793\pi\)
\(812\) 0 0
\(813\) − 10.9282i − 0.383269i
\(814\) 0 0
\(815\) −10.1436 −0.355315
\(816\) 0 0
\(817\) 55.4256 1.93910
\(818\) 0 0
\(819\) 2.00000i 0.0698857i
\(820\) 0 0
\(821\) 39.8564i 1.39100i 0.718527 + 0.695499i \(0.244817\pi\)
−0.718527 + 0.695499i \(0.755183\pi\)
\(822\) 0 0
\(823\) −34.9282 −1.21752 −0.608760 0.793354i \(-0.708333\pi\)
−0.608760 + 0.793354i \(0.708333\pi\)
\(824\) 0 0
\(825\) −38.2487 −1.33165
\(826\) 0 0
\(827\) 25.1769i 0.875487i 0.899100 + 0.437744i \(0.144222\pi\)
−0.899100 + 0.437744i \(0.855778\pi\)
\(828\) 0 0
\(829\) − 33.7128i − 1.17089i −0.810711 0.585447i \(-0.800919\pi\)
0.810711 0.585447i \(-0.199081\pi\)
\(830\) 0 0
\(831\) −27.8564 −0.966328
\(832\) 0 0
\(833\) 7.46410 0.258616
\(834\) 0 0
\(835\) 10.1436i 0.351034i
\(836\) 0 0
\(837\) − 2.92820i − 0.101214i
\(838\) 0 0
\(839\) 2.92820 0.101093 0.0505464 0.998722i \(-0.483904\pi\)
0.0505464 + 0.998722i \(0.483904\pi\)
\(840\) 0 0
\(841\) −50.7128 −1.74872
\(842\) 0 0
\(843\) − 0.928203i − 0.0319690i
\(844\) 0 0
\(845\) 31.1769i 1.07252i
\(846\) 0 0
\(847\) 18.8564 0.647914
\(848\) 0 0
\(849\) −1.07180 −0.0367840
\(850\) 0 0
\(851\) 10.9282i 0.374614i
\(852\) 0 0
\(853\) − 36.9282i − 1.26440i −0.774806 0.632199i \(-0.782153\pi\)
0.774806 0.632199i \(-0.217847\pi\)
\(854\) 0 0
\(855\) 24.0000 0.820783
\(856\) 0 0
\(857\) 7.17691 0.245159 0.122579 0.992459i \(-0.460883\pi\)
0.122579 + 0.992459i \(0.460883\pi\)
\(858\) 0 0
\(859\) − 25.0718i − 0.855439i −0.903912 0.427719i \(-0.859317\pi\)
0.903912 0.427719i \(-0.140683\pi\)
\(860\) 0 0
\(861\) − 4.53590i − 0.154583i
\(862\) 0 0
\(863\) 2.53590 0.0863230 0.0431615 0.999068i \(-0.486257\pi\)
0.0431615 + 0.999068i \(0.486257\pi\)
\(864\) 0 0
\(865\) −5.56922 −0.189359
\(866\) 0 0
\(867\) − 38.7128i − 1.31476i
\(868\) 0 0
\(869\) − 16.0000i − 0.542763i
\(870\) 0 0
\(871\) 21.8564 0.740576
\(872\) 0 0
\(873\) −4.92820 −0.166794
\(874\) 0 0
\(875\) 6.92820i 0.234216i
\(876\) 0 0
\(877\) 35.8564i 1.21078i 0.795927 + 0.605392i \(0.206984\pi\)
−0.795927 + 0.605392i \(0.793016\pi\)
\(878\) 0 0
\(879\) −23.4641 −0.791425
\(880\) 0 0
\(881\) 21.3205 0.718306 0.359153 0.933279i \(-0.383066\pi\)
0.359153 + 0.933279i \(0.383066\pi\)
\(882\) 0 0
\(883\) − 49.5692i − 1.66814i −0.551661 0.834069i \(-0.686006\pi\)
0.551661 0.834069i \(-0.313994\pi\)
\(884\) 0 0
\(885\) − 51.7128i − 1.73831i
\(886\) 0 0
\(887\) −13.0718 −0.438908 −0.219454 0.975623i \(-0.570428\pi\)
−0.219454 + 0.975623i \(0.570428\pi\)
\(888\) 0 0
\(889\) −10.9282 −0.366520
\(890\) 0 0
\(891\) 5.46410i 0.183054i
\(892\) 0 0
\(893\) 20.2872i 0.678885i
\(894\) 0 0
\(895\) 8.78461 0.293637
\(896\) 0 0
\(897\) 10.9282 0.364882
\(898\) 0 0
\(899\) − 26.1436i − 0.871938i
\(900\) 0 0
\(901\) 14.9282i 0.497331i
\(902\) 0 0
\(903\) 8.00000 0.266223
\(904\) 0 0
\(905\) −20.7846 −0.690904
\(906\) 0 0
\(907\) 11.7128i 0.388918i 0.980911 + 0.194459i \(0.0622951\pi\)
−0.980911 + 0.194459i \(0.937705\pi\)
\(908\) 0 0
\(909\) − 8.53590i − 0.283118i
\(910\) 0 0
\(911\) 33.1769 1.09920 0.549600 0.835428i \(-0.314780\pi\)
0.549600 + 0.835428i \(0.314780\pi\)
\(912\) 0 0
\(913\) −21.8564 −0.723341
\(914\) 0 0
\(915\) 17.0718i 0.564376i
\(916\) 0 0
\(917\) 12.0000i 0.396275i
\(918\) 0 0
\(919\) −40.0000 −1.31948 −0.659739 0.751495i \(-0.729333\pi\)
−0.659739 + 0.751495i \(0.729333\pi\)
\(920\) 0 0
\(921\) −9.07180 −0.298926
\(922\) 0 0
\(923\) 5.07180i 0.166940i
\(924\) 0 0
\(925\) − 14.0000i − 0.460317i
\(926\) 0 0
\(927\) −2.92820 −0.0961748
\(928\) 0 0
\(929\) 34.3923 1.12837 0.564187 0.825647i \(-0.309190\pi\)
0.564187 + 0.825647i \(0.309190\pi\)
\(930\) 0 0
\(931\) 6.92820i 0.227063i
\(932\) 0 0
\(933\) − 5.07180i − 0.166043i
\(934\) 0 0
\(935\) −141.282 −4.62042
\(936\) 0 0
\(937\) −26.0000 −0.849383 −0.424691 0.905338i \(-0.639617\pi\)
−0.424691 + 0.905338i \(0.639617\pi\)
\(938\) 0 0
\(939\) − 14.0000i − 0.456873i
\(940\) 0 0
\(941\) − 43.4641i − 1.41689i −0.705766 0.708445i \(-0.749397\pi\)
0.705766 0.708445i \(-0.250603\pi\)
\(942\) 0 0
\(943\) −24.7846 −0.807098
\(944\) 0 0
\(945\) 3.46410 0.112687
\(946\) 0 0
\(947\) 22.2487i 0.722986i 0.932375 + 0.361493i \(0.117733\pi\)
−0.932375 + 0.361493i \(0.882267\pi\)
\(948\) 0 0
\(949\) 1.85641i 0.0602615i
\(950\) 0 0
\(951\) 27.8564 0.903306
\(952\) 0 0
\(953\) 22.0000 0.712650 0.356325 0.934362i \(-0.384030\pi\)
0.356325 + 0.934362i \(0.384030\pi\)
\(954\) 0 0
\(955\) − 8.78461i − 0.284263i
\(956\) 0 0
\(957\) 48.7846i 1.57698i
\(958\) 0 0
\(959\) 4.92820 0.159140
\(960\) 0 0
\(961\) −22.4256 −0.723407
\(962\) 0 0
\(963\) 8.39230i 0.270438i
\(964\) 0 0
\(965\) − 41.0718i − 1.32215i
\(966\) 0 0
\(967\) 21.0718 0.677623 0.338812 0.940854i \(-0.389975\pi\)
0.338812 + 0.940854i \(0.389975\pi\)
\(968\) 0 0
\(969\) 51.7128 1.66125
\(970\) 0 0
\(971\) − 25.8564i − 0.829772i −0.909873 0.414886i \(-0.863822\pi\)
0.909873 0.414886i \(-0.136178\pi\)
\(972\) 0 0
\(973\) − 9.85641i − 0.315982i
\(974\) 0 0
\(975\) −14.0000 −0.448359
\(976\) 0 0
\(977\) −58.4974 −1.87150 −0.935749 0.352666i \(-0.885275\pi\)
−0.935749 + 0.352666i \(0.885275\pi\)
\(978\) 0 0
\(979\) 18.9282i 0.604948i
\(980\) 0 0
\(981\) 2.00000i 0.0638551i
\(982\) 0 0
\(983\) 51.7128 1.64938 0.824691 0.565583i \(-0.191349\pi\)
0.824691 + 0.565583i \(0.191349\pi\)
\(984\) 0 0
\(985\) 68.7846 2.19166
\(986\) 0 0
\(987\) 2.92820i 0.0932057i
\(988\) 0 0
\(989\) − 43.7128i − 1.38999i
\(990\) 0 0
\(991\) 40.0000 1.27064 0.635321 0.772248i \(-0.280868\pi\)
0.635321 + 0.772248i \(0.280868\pi\)
\(992\) 0 0
\(993\) 24.0000 0.761617
\(994\) 0 0
\(995\) − 75.7128i − 2.40026i
\(996\) 0 0
\(997\) − 44.9282i − 1.42289i −0.702742 0.711445i \(-0.748041\pi\)
0.702742 0.711445i \(-0.251959\pi\)
\(998\) 0 0
\(999\) −2.00000 −0.0632772
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 \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 5376.2.c.bh.2689.2 4
4.3 odd 2 5376.2.c.bn.2689.4 4
8.3 odd 2 5376.2.c.bn.2689.1 4
8.5 even 2 inner 5376.2.c.bh.2689.3 4
16.3 odd 4 1344.2.a.u.1.2 2
16.5 even 4 672.2.a.i.1.1 2
16.11 odd 4 672.2.a.j.1.1 yes 2
16.13 even 4 1344.2.a.v.1.2 2
48.5 odd 4 2016.2.a.t.1.2 2
48.11 even 4 2016.2.a.s.1.2 2
48.29 odd 4 4032.2.a.bs.1.1 2
48.35 even 4 4032.2.a.br.1.1 2
112.13 odd 4 9408.2.a.do.1.1 2
112.27 even 4 4704.2.a.bm.1.2 2
112.69 odd 4 4704.2.a.bn.1.2 2
112.83 even 4 9408.2.a.dx.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
672.2.a.i.1.1 2 16.5 even 4
672.2.a.j.1.1 yes 2 16.11 odd 4
1344.2.a.u.1.2 2 16.3 odd 4
1344.2.a.v.1.2 2 16.13 even 4
2016.2.a.s.1.2 2 48.11 even 4
2016.2.a.t.1.2 2 48.5 odd 4
4032.2.a.br.1.1 2 48.35 even 4
4032.2.a.bs.1.1 2 48.29 odd 4
4704.2.a.bm.1.2 2 112.27 even 4
4704.2.a.bn.1.2 2 112.69 odd 4
5376.2.c.bh.2689.2 4 1.1 even 1 trivial
5376.2.c.bh.2689.3 4 8.5 even 2 inner
5376.2.c.bn.2689.1 4 8.3 odd 2
5376.2.c.bn.2689.4 4 4.3 odd 2
9408.2.a.do.1.1 2 112.13 odd 4
9408.2.a.dx.1.1 2 112.83 even 4