Properties

Label 4400.2.b.b.4049.1
Level $4400$
Weight $2$
Character 4400.4049
Analytic conductor $35.134$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4400 = 2^{4} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4400.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(35.1341768894\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 88)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 4049.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 4400.4049
Dual form 4400.2.b.b.4049.2

$q$-expansion

\(f(q)\) \(=\) \(q-3.00000i q^{3} +2.00000i q^{7} -6.00000 q^{9} +O(q^{10})\) \(q-3.00000i q^{3} +2.00000i q^{7} -6.00000 q^{9} +1.00000 q^{11} -6.00000i q^{17} +4.00000 q^{19} +6.00000 q^{21} +1.00000i q^{23} +9.00000i q^{27} +8.00000 q^{29} +7.00000 q^{31} -3.00000i q^{33} -1.00000i q^{37} +4.00000 q^{41} +6.00000i q^{43} +8.00000i q^{47} +3.00000 q^{49} -18.0000 q^{51} -2.00000i q^{53} -12.0000i q^{57} -1.00000 q^{59} +4.00000 q^{61} -12.0000i q^{63} +5.00000i q^{67} +3.00000 q^{69} -3.00000 q^{71} -16.0000i q^{73} +2.00000i q^{77} +2.00000 q^{79} +9.00000 q^{81} -2.00000i q^{83} -24.0000i q^{87} -15.0000 q^{89} -21.0000i q^{93} -7.00000i q^{97} -6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 12 q^{9} + 2 q^{11} + 8 q^{19} + 12 q^{21} + 16 q^{29} + 14 q^{31} + 8 q^{41} + 6 q^{49} - 36 q^{51} - 2 q^{59} + 8 q^{61} + 6 q^{69} - 6 q^{71} + 4 q^{79} + 18 q^{81} - 30 q^{89} - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4400\mathbb{Z}\right)^\times\).

\(n\) \(177\) \(1201\) \(2751\) \(3301\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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
\(3\) − 3.00000i − 1.73205i −0.500000 0.866025i \(-0.666667\pi\)
0.500000 0.866025i \(-0.333333\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 2.00000i 0.755929i 0.925820 + 0.377964i \(0.123376\pi\)
−0.925820 + 0.377964i \(0.876624\pi\)
\(8\) 0 0
\(9\) −6.00000 −2.00000
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 6.00000i − 1.45521i −0.685994 0.727607i \(-0.740633\pi\)
0.685994 0.727607i \(-0.259367\pi\)
\(18\) 0 0
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) 6.00000 1.30931
\(22\) 0 0
\(23\) 1.00000i 0.208514i 0.994550 + 0.104257i \(0.0332465\pi\)
−0.994550 + 0.104257i \(0.966753\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 9.00000i 1.73205i
\(28\) 0 0
\(29\) 8.00000 1.48556 0.742781 0.669534i \(-0.233506\pi\)
0.742781 + 0.669534i \(0.233506\pi\)
\(30\) 0 0
\(31\) 7.00000 1.25724 0.628619 0.777714i \(-0.283621\pi\)
0.628619 + 0.777714i \(0.283621\pi\)
\(32\) 0 0
\(33\) − 3.00000i − 0.522233i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 1.00000i − 0.164399i −0.996616 0.0821995i \(-0.973806\pi\)
0.996616 0.0821995i \(-0.0261945\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 4.00000 0.624695 0.312348 0.949968i \(-0.398885\pi\)
0.312348 + 0.949968i \(0.398885\pi\)
\(42\) 0 0
\(43\) 6.00000i 0.914991i 0.889212 + 0.457496i \(0.151253\pi\)
−0.889212 + 0.457496i \(0.848747\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 8.00000i 1.16692i 0.812142 + 0.583460i \(0.198301\pi\)
−0.812142 + 0.583460i \(0.801699\pi\)
\(48\) 0 0
\(49\) 3.00000 0.428571
\(50\) 0 0
\(51\) −18.0000 −2.52050
\(52\) 0 0
\(53\) − 2.00000i − 0.274721i −0.990521 0.137361i \(-0.956138\pi\)
0.990521 0.137361i \(-0.0438619\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 12.0000i − 1.58944i
\(58\) 0 0
\(59\) −1.00000 −0.130189 −0.0650945 0.997879i \(-0.520735\pi\)
−0.0650945 + 0.997879i \(0.520735\pi\)
\(60\) 0 0
\(61\) 4.00000 0.512148 0.256074 0.966657i \(-0.417571\pi\)
0.256074 + 0.966657i \(0.417571\pi\)
\(62\) 0 0
\(63\) − 12.0000i − 1.51186i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 5.00000i 0.610847i 0.952217 + 0.305424i \(0.0987981\pi\)
−0.952217 + 0.305424i \(0.901202\pi\)
\(68\) 0 0
\(69\) 3.00000 0.361158
\(70\) 0 0
\(71\) −3.00000 −0.356034 −0.178017 0.984027i \(-0.556968\pi\)
−0.178017 + 0.984027i \(0.556968\pi\)
\(72\) 0 0
\(73\) − 16.0000i − 1.87266i −0.351123 0.936329i \(-0.614200\pi\)
0.351123 0.936329i \(-0.385800\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 2.00000i 0.227921i
\(78\) 0 0
\(79\) 2.00000 0.225018 0.112509 0.993651i \(-0.464111\pi\)
0.112509 + 0.993651i \(0.464111\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) − 2.00000i − 0.219529i −0.993958 0.109764i \(-0.964990\pi\)
0.993958 0.109764i \(-0.0350096\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 24.0000i − 2.57307i
\(88\) 0 0
\(89\) −15.0000 −1.59000 −0.794998 0.606612i \(-0.792528\pi\)
−0.794998 + 0.606612i \(0.792528\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) − 21.0000i − 2.17760i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 7.00000i − 0.710742i −0.934725 0.355371i \(-0.884354\pi\)
0.934725 0.355371i \(-0.115646\pi\)
\(98\) 0 0
\(99\) −6.00000 −0.603023
\(100\) 0 0
\(101\) −10.0000 −0.995037 −0.497519 0.867453i \(-0.665755\pi\)
−0.497519 + 0.867453i \(0.665755\pi\)
\(102\) 0 0
\(103\) − 16.0000i − 1.57653i −0.615338 0.788263i \(-0.710980\pi\)
0.615338 0.788263i \(-0.289020\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 2.00000i − 0.193347i −0.995316 0.0966736i \(-0.969180\pi\)
0.995316 0.0966736i \(-0.0308203\pi\)
\(108\) 0 0
\(109\) 14.0000 1.34096 0.670478 0.741929i \(-0.266089\pi\)
0.670478 + 0.741929i \(0.266089\pi\)
\(110\) 0 0
\(111\) −3.00000 −0.284747
\(112\) 0 0
\(113\) 7.00000i 0.658505i 0.944242 + 0.329252i \(0.106797\pi\)
−0.944242 + 0.329252i \(0.893203\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 12.0000 1.10004
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) − 12.0000i − 1.08200i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 4.00000i − 0.354943i −0.984126 0.177471i \(-0.943208\pi\)
0.984126 0.177471i \(-0.0567917\pi\)
\(128\) 0 0
\(129\) 18.0000 1.58481
\(130\) 0 0
\(131\) 2.00000 0.174741 0.0873704 0.996176i \(-0.472154\pi\)
0.0873704 + 0.996176i \(0.472154\pi\)
\(132\) 0 0
\(133\) 8.00000i 0.693688i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 15.0000i − 1.28154i −0.767734 0.640768i \(-0.778616\pi\)
0.767734 0.640768i \(-0.221384\pi\)
\(138\) 0 0
\(139\) −22.0000 −1.86602 −0.933008 0.359856i \(-0.882826\pi\)
−0.933008 + 0.359856i \(0.882826\pi\)
\(140\) 0 0
\(141\) 24.0000 2.02116
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) − 9.00000i − 0.742307i
\(148\) 0 0
\(149\) −18.0000 −1.47462 −0.737309 0.675556i \(-0.763904\pi\)
−0.737309 + 0.675556i \(0.763904\pi\)
\(150\) 0 0
\(151\) 18.0000 1.46482 0.732410 0.680864i \(-0.238396\pi\)
0.732410 + 0.680864i \(0.238396\pi\)
\(152\) 0 0
\(153\) 36.0000i 2.91043i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 11.0000i − 0.877896i −0.898513 0.438948i \(-0.855351\pi\)
0.898513 0.438948i \(-0.144649\pi\)
\(158\) 0 0
\(159\) −6.00000 −0.475831
\(160\) 0 0
\(161\) −2.00000 −0.157622
\(162\) 0 0
\(163\) − 4.00000i − 0.313304i −0.987654 0.156652i \(-0.949930\pi\)
0.987654 0.156652i \(-0.0500701\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 16.0000i − 1.23812i −0.785345 0.619059i \(-0.787514\pi\)
0.785345 0.619059i \(-0.212486\pi\)
\(168\) 0 0
\(169\) 13.0000 1.00000
\(170\) 0 0
\(171\) −24.0000 −1.83533
\(172\) 0 0
\(173\) − 18.0000i − 1.36851i −0.729241 0.684257i \(-0.760127\pi\)
0.729241 0.684257i \(-0.239873\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 3.00000i 0.225494i
\(178\) 0 0
\(179\) −5.00000 −0.373718 −0.186859 0.982387i \(-0.559831\pi\)
−0.186859 + 0.982387i \(0.559831\pi\)
\(180\) 0 0
\(181\) −5.00000 −0.371647 −0.185824 0.982583i \(-0.559495\pi\)
−0.185824 + 0.982583i \(0.559495\pi\)
\(182\) 0 0
\(183\) − 12.0000i − 0.887066i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 6.00000i − 0.438763i
\(188\) 0 0
\(189\) −18.0000 −1.30931
\(190\) 0 0
\(191\) 9.00000 0.651217 0.325609 0.945505i \(-0.394431\pi\)
0.325609 + 0.945505i \(0.394431\pi\)
\(192\) 0 0
\(193\) − 4.00000i − 0.287926i −0.989583 0.143963i \(-0.954015\pi\)
0.989583 0.143963i \(-0.0459847\pi\)
\(194\) 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
\(199\) 8.00000 0.567105 0.283552 0.958957i \(-0.408487\pi\)
0.283552 + 0.958957i \(0.408487\pi\)
\(200\) 0 0
\(201\) 15.0000 1.05802
\(202\) 0 0
\(203\) 16.0000i 1.12298i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 6.00000i − 0.417029i
\(208\) 0 0
\(209\) 4.00000 0.276686
\(210\) 0 0
\(211\) −20.0000 −1.37686 −0.688428 0.725304i \(-0.741699\pi\)
−0.688428 + 0.725304i \(0.741699\pi\)
\(212\) 0 0
\(213\) 9.00000i 0.616670i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 14.0000i 0.950382i
\(218\) 0 0
\(219\) −48.0000 −3.24354
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 29.0000i 1.94198i 0.239113 + 0.970992i \(0.423143\pi\)
−0.239113 + 0.970992i \(0.576857\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 18.0000i − 1.19470i −0.801980 0.597351i \(-0.796220\pi\)
0.801980 0.597351i \(-0.203780\pi\)
\(228\) 0 0
\(229\) 21.0000 1.38772 0.693860 0.720110i \(-0.255909\pi\)
0.693860 + 0.720110i \(0.255909\pi\)
\(230\) 0 0
\(231\) 6.00000 0.394771
\(232\) 0 0
\(233\) 16.0000i 1.04819i 0.851658 + 0.524097i \(0.175597\pi\)
−0.851658 + 0.524097i \(0.824403\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 6.00000i − 0.389742i
\(238\) 0 0
\(239\) −2.00000 −0.129369 −0.0646846 0.997906i \(-0.520604\pi\)
−0.0646846 + 0.997906i \(0.520604\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −6.00000 −0.380235
\(250\) 0 0
\(251\) 13.0000 0.820553 0.410276 0.911961i \(-0.365432\pi\)
0.410276 + 0.911961i \(0.365432\pi\)
\(252\) 0 0
\(253\) 1.00000i 0.0628695i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 10.0000i − 0.623783i −0.950118 0.311891i \(-0.899037\pi\)
0.950118 0.311891i \(-0.100963\pi\)
\(258\) 0 0
\(259\) 2.00000 0.124274
\(260\) 0 0
\(261\) −48.0000 −2.97113
\(262\) 0 0
\(263\) 14.0000i 0.863277i 0.902047 + 0.431638i \(0.142064\pi\)
−0.902047 + 0.431638i \(0.857936\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 45.0000i 2.75396i
\(268\) 0 0
\(269\) −26.0000 −1.58525 −0.792624 0.609711i \(-0.791286\pi\)
−0.792624 + 0.609711i \(0.791286\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 2.00000i − 0.120168i −0.998193 0.0600842i \(-0.980863\pi\)
0.998193 0.0600842i \(-0.0191369\pi\)
\(278\) 0 0
\(279\) −42.0000 −2.51447
\(280\) 0 0
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 0 0
\(283\) 4.00000i 0.237775i 0.992908 + 0.118888i \(0.0379328\pi\)
−0.992908 + 0.118888i \(0.962067\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 8.00000i 0.472225i
\(288\) 0 0
\(289\) −19.0000 −1.11765
\(290\) 0 0
\(291\) −21.0000 −1.23104
\(292\) 0 0
\(293\) − 12.0000i − 0.701047i −0.936554 0.350524i \(-0.886004\pi\)
0.936554 0.350524i \(-0.113996\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 9.00000i 0.522233i
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −12.0000 −0.691669
\(302\) 0 0
\(303\) 30.0000i 1.72345i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 4.00000i 0.228292i 0.993464 + 0.114146i \(0.0364132\pi\)
−0.993464 + 0.114146i \(0.963587\pi\)
\(308\) 0 0
\(309\) −48.0000 −2.73062
\(310\) 0 0
\(311\) 12.0000 0.680458 0.340229 0.940343i \(-0.389495\pi\)
0.340229 + 0.940343i \(0.389495\pi\)
\(312\) 0 0
\(313\) 9.00000i 0.508710i 0.967111 + 0.254355i \(0.0818632\pi\)
−0.967111 + 0.254355i \(0.918137\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 15.0000i − 0.842484i −0.906948 0.421242i \(-0.861594\pi\)
0.906948 0.421242i \(-0.138406\pi\)
\(318\) 0 0
\(319\) 8.00000 0.447914
\(320\) 0 0
\(321\) −6.00000 −0.334887
\(322\) 0 0
\(323\) − 24.0000i − 1.33540i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 42.0000i − 2.32261i
\(328\) 0 0
\(329\) −16.0000 −0.882109
\(330\) 0 0
\(331\) 35.0000 1.92377 0.961887 0.273447i \(-0.0881639\pi\)
0.961887 + 0.273447i \(0.0881639\pi\)
\(332\) 0 0
\(333\) 6.00000i 0.328798i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 10.0000i − 0.544735i −0.962193 0.272367i \(-0.912193\pi\)
0.962193 0.272367i \(-0.0878066\pi\)
\(338\) 0 0
\(339\) 21.0000 1.14056
\(340\) 0 0
\(341\) 7.00000 0.379071
\(342\) 0 0
\(343\) 20.0000i 1.07990i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 32.0000i 1.71785i 0.512101 + 0.858925i \(0.328867\pi\)
−0.512101 + 0.858925i \(0.671133\pi\)
\(348\) 0 0
\(349\) 14.0000 0.749403 0.374701 0.927146i \(-0.377745\pi\)
0.374701 + 0.927146i \(0.377745\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 3.00000i − 0.159674i −0.996808 0.0798369i \(-0.974560\pi\)
0.996808 0.0798369i \(-0.0254400\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) − 36.0000i − 1.90532i
\(358\) 0 0
\(359\) 24.0000 1.26667 0.633336 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 0 0
\(363\) − 3.00000i − 0.157459i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 33.0000i − 1.72259i −0.508109 0.861293i \(-0.669655\pi\)
0.508109 0.861293i \(-0.330345\pi\)
\(368\) 0 0
\(369\) −24.0000 −1.24939
\(370\) 0 0
\(371\) 4.00000 0.207670
\(372\) 0 0
\(373\) 22.0000i 1.13912i 0.821951 + 0.569558i \(0.192886\pi\)
−0.821951 + 0.569558i \(0.807114\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 25.0000 1.28416 0.642082 0.766636i \(-0.278071\pi\)
0.642082 + 0.766636i \(0.278071\pi\)
\(380\) 0 0
\(381\) −12.0000 −0.614779
\(382\) 0 0
\(383\) 1.00000i 0.0510976i 0.999674 + 0.0255488i \(0.00813332\pi\)
−0.999674 + 0.0255488i \(0.991867\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 36.0000i − 1.82998i
\(388\) 0 0
\(389\) −13.0000 −0.659126 −0.329563 0.944134i \(-0.606901\pi\)
−0.329563 + 0.944134i \(0.606901\pi\)
\(390\) 0 0
\(391\) 6.00000 0.303433
\(392\) 0 0
\(393\) − 6.00000i − 0.302660i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 26.0000i − 1.30490i −0.757831 0.652451i \(-0.773741\pi\)
0.757831 0.652451i \(-0.226259\pi\)
\(398\) 0 0
\(399\) 24.0000 1.20150
\(400\) 0 0
\(401\) −6.00000 −0.299626 −0.149813 0.988714i \(-0.547867\pi\)
−0.149813 + 0.988714i \(0.547867\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 1.00000i − 0.0495682i
\(408\) 0 0
\(409\) −34.0000 −1.68119 −0.840596 0.541663i \(-0.817795\pi\)
−0.840596 + 0.541663i \(0.817795\pi\)
\(410\) 0 0
\(411\) −45.0000 −2.21969
\(412\) 0 0
\(413\) − 2.00000i − 0.0984136i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 66.0000i 3.23203i
\(418\) 0 0
\(419\) −28.0000 −1.36789 −0.683945 0.729534i \(-0.739737\pi\)
−0.683945 + 0.729534i \(0.739737\pi\)
\(420\) 0 0
\(421\) 30.0000 1.46211 0.731055 0.682318i \(-0.239028\pi\)
0.731055 + 0.682318i \(0.239028\pi\)
\(422\) 0 0
\(423\) − 48.0000i − 2.33384i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 8.00000i 0.387147i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 26.0000 1.25238 0.626188 0.779672i \(-0.284614\pi\)
0.626188 + 0.779672i \(0.284614\pi\)
\(432\) 0 0
\(433\) − 13.0000i − 0.624740i −0.949960 0.312370i \(-0.898877\pi\)
0.949960 0.312370i \(-0.101123\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 4.00000i 0.191346i
\(438\) 0 0
\(439\) −32.0000 −1.52728 −0.763638 0.645644i \(-0.776589\pi\)
−0.763638 + 0.645644i \(0.776589\pi\)
\(440\) 0 0
\(441\) −18.0000 −0.857143
\(442\) 0 0
\(443\) − 9.00000i − 0.427603i −0.976877 0.213801i \(-0.931415\pi\)
0.976877 0.213801i \(-0.0685846\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 54.0000i 2.55411i
\(448\) 0 0
\(449\) 21.0000 0.991051 0.495526 0.868593i \(-0.334975\pi\)
0.495526 + 0.868593i \(0.334975\pi\)
\(450\) 0 0
\(451\) 4.00000 0.188353
\(452\) 0 0
\(453\) − 54.0000i − 2.53714i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 12.0000i 0.561336i 0.959805 + 0.280668i \(0.0905560\pi\)
−0.959805 + 0.280668i \(0.909444\pi\)
\(458\) 0 0
\(459\) 54.0000 2.52050
\(460\) 0 0
\(461\) −28.0000 −1.30409 −0.652045 0.758180i \(-0.726089\pi\)
−0.652045 + 0.758180i \(0.726089\pi\)
\(462\) 0 0
\(463\) 27.0000i 1.25480i 0.778699 + 0.627398i \(0.215880\pi\)
−0.778699 + 0.627398i \(0.784120\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 33.0000i 1.52706i 0.645774 + 0.763529i \(0.276535\pi\)
−0.645774 + 0.763529i \(0.723465\pi\)
\(468\) 0 0
\(469\) −10.0000 −0.461757
\(470\) 0 0
\(471\) −33.0000 −1.52056
\(472\) 0 0
\(473\) 6.00000i 0.275880i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 12.0000i 0.549442i
\(478\) 0 0
\(479\) 8.00000 0.365529 0.182765 0.983157i \(-0.441495\pi\)
0.182765 + 0.983157i \(0.441495\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 6.00000i 0.273009i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 9.00000i − 0.407829i −0.978989 0.203914i \(-0.934634\pi\)
0.978989 0.203914i \(-0.0653664\pi\)
\(488\) 0 0
\(489\) −12.0000 −0.542659
\(490\) 0 0
\(491\) −8.00000 −0.361035 −0.180517 0.983572i \(-0.557777\pi\)
−0.180517 + 0.983572i \(0.557777\pi\)
\(492\) 0 0
\(493\) − 48.0000i − 2.16181i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 6.00000i − 0.269137i
\(498\) 0 0
\(499\) −20.0000 −0.895323 −0.447661 0.894203i \(-0.647743\pi\)
−0.447661 + 0.894203i \(0.647743\pi\)
\(500\) 0 0
\(501\) −48.0000 −2.14448
\(502\) 0 0
\(503\) − 30.0000i − 1.33763i −0.743427 0.668817i \(-0.766801\pi\)
0.743427 0.668817i \(-0.233199\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 39.0000i − 1.73205i
\(508\) 0 0
\(509\) 13.0000 0.576215 0.288107 0.957598i \(-0.406974\pi\)
0.288107 + 0.957598i \(0.406974\pi\)
\(510\) 0 0
\(511\) 32.0000 1.41560
\(512\) 0 0
\(513\) 36.0000i 1.58944i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 8.00000i 0.351840i
\(518\) 0 0
\(519\) −54.0000 −2.37034
\(520\) 0 0
\(521\) 37.0000 1.62100 0.810500 0.585739i \(-0.199196\pi\)
0.810500 + 0.585739i \(0.199196\pi\)
\(522\) 0 0
\(523\) 44.0000i 1.92399i 0.273075 + 0.961993i \(0.411959\pi\)
−0.273075 + 0.961993i \(0.588041\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 42.0000i − 1.82955i
\(528\) 0 0
\(529\) 22.0000 0.956522
\(530\) 0 0
\(531\) 6.00000 0.260378
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 15.0000i 0.647298i
\(538\) 0 0
\(539\) 3.00000 0.129219
\(540\) 0 0
\(541\) 12.0000 0.515920 0.257960 0.966156i \(-0.416950\pi\)
0.257960 + 0.966156i \(0.416950\pi\)
\(542\) 0 0
\(543\) 15.0000i 0.643712i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) 0 0
\(549\) −24.0000 −1.02430
\(550\) 0 0
\(551\) 32.0000 1.36325
\(552\) 0 0
\(553\) 4.00000i 0.170097i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 42.0000i 1.77960i 0.456354 + 0.889799i \(0.349155\pi\)
−0.456354 + 0.889799i \(0.650845\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) −18.0000 −0.759961
\(562\) 0 0
\(563\) − 12.0000i − 0.505740i −0.967500 0.252870i \(-0.918626\pi\)
0.967500 0.252870i \(-0.0813744\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 18.0000i 0.755929i
\(568\) 0 0
\(569\) 24.0000 1.00613 0.503066 0.864248i \(-0.332205\pi\)
0.503066 + 0.864248i \(0.332205\pi\)
\(570\) 0 0
\(571\) 4.00000 0.167395 0.0836974 0.996491i \(-0.473327\pi\)
0.0836974 + 0.996491i \(0.473327\pi\)
\(572\) 0 0
\(573\) − 27.0000i − 1.12794i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 23.0000i − 0.957503i −0.877951 0.478751i \(-0.841090\pi\)
0.877951 0.478751i \(-0.158910\pi\)
\(578\) 0 0
\(579\) −12.0000 −0.498703
\(580\) 0 0
\(581\) 4.00000 0.165948
\(582\) 0 0
\(583\) − 2.00000i − 0.0828315i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 12.0000i − 0.495293i −0.968850 0.247647i \(-0.920343\pi\)
0.968850 0.247647i \(-0.0796572\pi\)
\(588\) 0 0
\(589\) 28.0000 1.15372
\(590\) 0 0
\(591\) 18.0000 0.740421
\(592\) 0 0
\(593\) 4.00000i 0.164260i 0.996622 + 0.0821302i \(0.0261723\pi\)
−0.996622 + 0.0821302i \(0.973828\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) − 24.0000i − 0.982255i
\(598\) 0 0
\(599\) 48.0000 1.96123 0.980613 0.195952i \(-0.0627798\pi\)
0.980613 + 0.195952i \(0.0627798\pi\)
\(600\) 0 0
\(601\) 38.0000 1.55005 0.775026 0.631929i \(-0.217737\pi\)
0.775026 + 0.631929i \(0.217737\pi\)
\(602\) 0 0
\(603\) − 30.0000i − 1.22169i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 18.0000i 0.730597i 0.930890 + 0.365299i \(0.119033\pi\)
−0.930890 + 0.365299i \(0.880967\pi\)
\(608\) 0 0
\(609\) 48.0000 1.94506
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 16.0000i 0.646234i 0.946359 + 0.323117i \(0.104731\pi\)
−0.946359 + 0.323117i \(0.895269\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 18.0000i 0.724653i 0.932051 + 0.362326i \(0.118017\pi\)
−0.932051 + 0.362326i \(0.881983\pi\)
\(618\) 0 0
\(619\) −3.00000 −0.120580 −0.0602901 0.998181i \(-0.519203\pi\)
−0.0602901 + 0.998181i \(0.519203\pi\)
\(620\) 0 0
\(621\) −9.00000 −0.361158
\(622\) 0 0
\(623\) − 30.0000i − 1.20192i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) − 12.0000i − 0.479234i
\(628\) 0 0
\(629\) −6.00000 −0.239236
\(630\) 0 0
\(631\) −9.00000 −0.358284 −0.179142 0.983823i \(-0.557332\pi\)
−0.179142 + 0.983823i \(0.557332\pi\)
\(632\) 0 0
\(633\) 60.0000i 2.38479i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 18.0000 0.712069
\(640\) 0 0
\(641\) −9.00000 −0.355479 −0.177739 0.984078i \(-0.556878\pi\)
−0.177739 + 0.984078i \(0.556878\pi\)
\(642\) 0 0
\(643\) 7.00000i 0.276053i 0.990429 + 0.138027i \(0.0440759\pi\)
−0.990429 + 0.138027i \(0.955924\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 15.0000i − 0.589711i −0.955542 0.294855i \(-0.904729\pi\)
0.955542 0.294855i \(-0.0952715\pi\)
\(648\) 0 0
\(649\) −1.00000 −0.0392534
\(650\) 0 0
\(651\) 42.0000 1.64611
\(652\) 0 0
\(653\) − 11.0000i − 0.430463i −0.976563 0.215232i \(-0.930949\pi\)
0.976563 0.215232i \(-0.0690506\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 96.0000i 3.74532i
\(658\) 0 0
\(659\) 22.0000 0.856998 0.428499 0.903542i \(-0.359042\pi\)
0.428499 + 0.903542i \(0.359042\pi\)
\(660\) 0 0
\(661\) −7.00000 −0.272268 −0.136134 0.990690i \(-0.543468\pi\)
−0.136134 + 0.990690i \(0.543468\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 8.00000i 0.309761i
\(668\) 0 0
\(669\) 87.0000 3.36361
\(670\) 0 0
\(671\) 4.00000 0.154418
\(672\) 0 0
\(673\) 34.0000i 1.31060i 0.755367 + 0.655302i \(0.227459\pi\)
−0.755367 + 0.655302i \(0.772541\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 30.0000i − 1.15299i −0.817099 0.576497i \(-0.804419\pi\)
0.817099 0.576497i \(-0.195581\pi\)
\(678\) 0 0
\(679\) 14.0000 0.537271
\(680\) 0 0
\(681\) −54.0000 −2.06928
\(682\) 0 0
\(683\) 8.00000i 0.306111i 0.988218 + 0.153056i \(0.0489114\pi\)
−0.988218 + 0.153056i \(0.951089\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) − 63.0000i − 2.40360i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 29.0000 1.10321 0.551606 0.834105i \(-0.314015\pi\)
0.551606 + 0.834105i \(0.314015\pi\)
\(692\) 0 0
\(693\) − 12.0000i − 0.455842i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 24.0000i − 0.909065i
\(698\) 0 0
\(699\) 48.0000 1.81553
\(700\) 0 0
\(701\) −10.0000 −0.377695 −0.188847 0.982006i \(-0.560475\pi\)
−0.188847 + 0.982006i \(0.560475\pi\)
\(702\) 0 0
\(703\) − 4.00000i − 0.150863i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 20.0000i − 0.752177i
\(708\) 0 0
\(709\) −19.0000 −0.713560 −0.356780 0.934188i \(-0.616125\pi\)
−0.356780 + 0.934188i \(0.616125\pi\)
\(710\) 0 0
\(711\) −12.0000 −0.450035
\(712\) 0 0
\(713\) 7.00000i 0.262152i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 6.00000i 0.224074i
\(718\) 0 0
\(719\) −23.0000 −0.857755 −0.428878 0.903363i \(-0.641091\pi\)
−0.428878 + 0.903363i \(0.641091\pi\)
\(720\) 0 0
\(721\) 32.0000 1.19174
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 35.0000i 1.29808i 0.760755 + 0.649039i \(0.224829\pi\)
−0.760755 + 0.649039i \(0.775171\pi\)
\(728\) 0 0
\(729\) 27.0000 1.00000
\(730\) 0 0
\(731\) 36.0000 1.33151
\(732\) 0 0
\(733\) − 4.00000i − 0.147743i −0.997268 0.0738717i \(-0.976464\pi\)
0.997268 0.0738717i \(-0.0235355\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 5.00000i 0.184177i
\(738\) 0 0
\(739\) −26.0000 −0.956425 −0.478213 0.878244i \(-0.658715\pi\)
−0.478213 + 0.878244i \(0.658715\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) − 36.0000i − 1.32071i −0.750953 0.660356i \(-0.770405\pi\)
0.750953 0.660356i \(-0.229595\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 12.0000i 0.439057i
\(748\) 0 0
\(749\) 4.00000 0.146157
\(750\) 0 0
\(751\) −23.0000 −0.839282 −0.419641 0.907690i \(-0.637844\pi\)
−0.419641 + 0.907690i \(0.637844\pi\)
\(752\) 0 0
\(753\) − 39.0000i − 1.42124i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 10.0000i 0.363456i 0.983349 + 0.181728i \(0.0581691\pi\)
−0.983349 + 0.181728i \(0.941831\pi\)
\(758\) 0 0
\(759\) 3.00000 0.108893
\(760\) 0 0
\(761\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(762\) 0 0
\(763\) 28.0000i 1.01367i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −16.0000 −0.576975 −0.288487 0.957484i \(-0.593152\pi\)
−0.288487 + 0.957484i \(0.593152\pi\)
\(770\) 0 0
\(771\) −30.0000 −1.08042
\(772\) 0 0
\(773\) 6.00000i 0.215805i 0.994161 + 0.107903i \(0.0344134\pi\)
−0.994161 + 0.107903i \(0.965587\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 6.00000i − 0.215249i
\(778\) 0 0
\(779\) 16.0000 0.573259
\(780\) 0 0
\(781\) −3.00000 −0.107348
\(782\) 0 0
\(783\) 72.0000i 2.57307i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 24.0000i − 0.855508i −0.903895 0.427754i \(-0.859305\pi\)
0.903895 0.427754i \(-0.140695\pi\)
\(788\) 0 0
\(789\) 42.0000 1.49524
\(790\) 0 0
\(791\) −14.0000 −0.497783
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 1.00000i 0.0354218i 0.999843 + 0.0177109i \(0.00563785\pi\)
−0.999843 + 0.0177109i \(0.994362\pi\)
\(798\) 0 0
\(799\) 48.0000 1.69812
\(800\) 0 0
\(801\) 90.0000 3.17999
\(802\) 0 0
\(803\) − 16.0000i − 0.564628i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 78.0000i 2.74573i
\(808\) 0 0
\(809\) −12.0000 −0.421898 −0.210949 0.977497i \(-0.567655\pi\)
−0.210949 + 0.977497i \(0.567655\pi\)
\(810\) 0 0
\(811\) 26.0000 0.912983 0.456492 0.889728i \(-0.349106\pi\)
0.456492 + 0.889728i \(0.349106\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 24.0000i 0.839654i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −34.0000 −1.18661 −0.593304 0.804978i \(-0.702177\pi\)
−0.593304 + 0.804978i \(0.702177\pi\)
\(822\) 0 0
\(823\) − 31.0000i − 1.08059i −0.841475 0.540296i \(-0.818312\pi\)
0.841475 0.540296i \(-0.181688\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 16.0000i 0.556375i 0.960527 + 0.278187i \(0.0897336\pi\)
−0.960527 + 0.278187i \(0.910266\pi\)
\(828\) 0 0
\(829\) 35.0000 1.21560 0.607800 0.794090i \(-0.292052\pi\)
0.607800 + 0.794090i \(0.292052\pi\)
\(830\) 0 0
\(831\) −6.00000 −0.208138
\(832\) 0 0
\(833\) − 18.0000i − 0.623663i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 63.0000i 2.17760i
\(838\) 0 0
\(839\) 21.0000 0.725001 0.362500 0.931984i \(-0.381923\pi\)
0.362500 + 0.931984i \(0.381923\pi\)
\(840\) 0 0
\(841\) 35.0000 1.20690
\(842\) 0 0
\(843\) − 18.0000i − 0.619953i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 2.00000i 0.0687208i
\(848\) 0 0
\(849\) 12.0000 0.411839
\(850\) 0 0
\(851\) 1.00000 0.0342796
\(852\) 0 0
\(853\) 26.0000i 0.890223i 0.895475 + 0.445112i \(0.146836\pi\)
−0.895475 + 0.445112i \(0.853164\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 24.0000i − 0.819824i −0.912125 0.409912i \(-0.865559\pi\)
0.912125 0.409912i \(-0.134441\pi\)
\(858\) 0 0
\(859\) 11.0000 0.375315 0.187658 0.982235i \(-0.439910\pi\)
0.187658 + 0.982235i \(0.439910\pi\)
\(860\) 0 0
\(861\) 24.0000 0.817918
\(862\) 0 0
\(863\) − 16.0000i − 0.544646i −0.962206 0.272323i \(-0.912208\pi\)
0.962206 0.272323i \(-0.0877920\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 57.0000i 1.93582i
\(868\) 0 0
\(869\) 2.00000 0.0678454
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 42.0000i 1.42148i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 56.0000i 1.89099i 0.325643 + 0.945493i \(0.394419\pi\)
−0.325643 + 0.945493i \(0.605581\pi\)
\(878\) 0 0
\(879\) −36.0000 −1.21425
\(880\) 0 0
\(881\) −19.0000 −0.640126 −0.320063 0.947396i \(-0.603704\pi\)
−0.320063 + 0.947396i \(0.603704\pi\)
\(882\) 0 0
\(883\) − 28.0000i − 0.942275i −0.882060 0.471138i \(-0.843844\pi\)
0.882060 0.471138i \(-0.156156\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 34.0000i 1.14161i 0.821086 + 0.570804i \(0.193368\pi\)
−0.821086 + 0.570804i \(0.806632\pi\)
\(888\) 0 0
\(889\) 8.00000 0.268311
\(890\) 0 0
\(891\) 9.00000 0.301511
\(892\) 0 0
\(893\) 32.0000i 1.07084i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 56.0000 1.86770
\(900\) 0 0
\(901\) −12.0000 −0.399778
\(902\) 0 0
\(903\) 36.0000i 1.19800i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 28.0000i − 0.929725i −0.885383 0.464862i \(-0.846104\pi\)
0.885383 0.464862i \(-0.153896\pi\)
\(908\) 0 0
\(909\) 60.0000 1.99007
\(910\) 0 0
\(911\) −36.0000 −1.19273 −0.596367 0.802712i \(-0.703390\pi\)
−0.596367 + 0.802712i \(0.703390\pi\)
\(912\) 0 0
\(913\) − 2.00000i − 0.0661903i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 4.00000i 0.132092i
\(918\) 0 0
\(919\) −50.0000 −1.64935 −0.824674 0.565608i \(-0.808641\pi\)
−0.824674 + 0.565608i \(0.808641\pi\)
\(920\) 0 0
\(921\) 12.0000 0.395413
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 96.0000i 3.15305i
\(928\) 0 0
\(929\) 22.0000 0.721797 0.360898 0.932605i \(-0.382470\pi\)
0.360898 + 0.932605i \(0.382470\pi\)
\(930\) 0 0
\(931\) 12.0000 0.393284
\(932\) 0 0
\(933\) − 36.0000i − 1.17859i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 4.00000i 0.130674i 0.997863 + 0.0653372i \(0.0208123\pi\)
−0.997863 + 0.0653372i \(0.979188\pi\)
\(938\) 0 0
\(939\) 27.0000 0.881112
\(940\) 0 0
\(941\) 10.0000 0.325991 0.162995 0.986627i \(-0.447884\pi\)
0.162995 + 0.986627i \(0.447884\pi\)
\(942\) 0 0
\(943\) 4.00000i 0.130258i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 47.0000i − 1.52729i −0.645634 0.763647i \(-0.723407\pi\)
0.645634 0.763647i \(-0.276593\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) −45.0000 −1.45922
\(952\) 0 0
\(953\) 34.0000i 1.10137i 0.834714 + 0.550684i \(0.185633\pi\)
−0.834714 + 0.550684i \(0.814367\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) − 24.0000i − 0.775810i
\(958\) 0 0
\(959\) 30.0000 0.968751
\(960\) 0 0
\(961\) 18.0000 0.580645
\(962\) 0 0
\(963\) 12.0000i 0.386695i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 8.00000i 0.257263i 0.991692 + 0.128631i \(0.0410584\pi\)
−0.991692 + 0.128631i \(0.958942\pi\)
\(968\) 0 0
\(969\) −72.0000 −2.31297
\(970\) 0 0
\(971\) −53.0000 −1.70085 −0.850425 0.526096i \(-0.823655\pi\)
−0.850425 + 0.526096i \(0.823655\pi\)
\(972\) 0 0
\(973\) − 44.0000i − 1.41058i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 11.0000i − 0.351921i −0.984397 0.175961i \(-0.943697\pi\)
0.984397 0.175961i \(-0.0563031\pi\)
\(978\) 0 0
\(979\) −15.0000 −0.479402
\(980\) 0 0
\(981\) −84.0000 −2.68191
\(982\) 0 0
\(983\) 25.0000i 0.797376i 0.917087 + 0.398688i \(0.130534\pi\)
−0.917087 + 0.398688i \(0.869466\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 48.0000i 1.52786i
\(988\) 0 0
\(989\) −6.00000 −0.190789
\(990\) 0 0
\(991\) −16.0000 −0.508257 −0.254128 0.967170i \(-0.581789\pi\)
−0.254128 + 0.967170i \(0.581789\pi\)
\(992\) 0 0
\(993\) − 105.000i − 3.33207i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 2.00000i 0.0633406i 0.999498 + 0.0316703i \(0.0100827\pi\)
−0.999498 + 0.0316703i \(0.989917\pi\)
\(998\) 0 0
\(999\) 9.00000 0.284747
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 4400.2.b.b.4049.1 2
4.3 odd 2 2200.2.b.a.1849.2 2
5.2 odd 4 4400.2.a.a.1.1 1
5.3 odd 4 176.2.a.c.1.1 1
5.4 even 2 inner 4400.2.b.b.4049.2 2
15.8 even 4 1584.2.a.q.1.1 1
20.3 even 4 88.2.a.a.1.1 1
20.7 even 4 2200.2.a.k.1.1 1
20.19 odd 2 2200.2.b.a.1849.1 2
35.13 even 4 8624.2.a.c.1.1 1
40.3 even 4 704.2.a.l.1.1 1
40.13 odd 4 704.2.a.b.1.1 1
55.43 even 4 1936.2.a.l.1.1 1
60.23 odd 4 792.2.a.g.1.1 1
80.3 even 4 2816.2.c.i.1409.1 2
80.13 odd 4 2816.2.c.d.1409.2 2
80.43 even 4 2816.2.c.i.1409.2 2
80.53 odd 4 2816.2.c.d.1409.1 2
120.53 even 4 6336.2.a.k.1.1 1
120.83 odd 4 6336.2.a.h.1.1 1
140.83 odd 4 4312.2.a.l.1.1 1
220.3 even 20 968.2.i.j.9.1 4
220.43 odd 4 968.2.a.a.1.1 1
220.63 odd 20 968.2.i.i.9.1 4
220.83 odd 20 968.2.i.i.729.1 4
220.103 even 20 968.2.i.j.753.1 4
220.123 odd 20 968.2.i.i.81.1 4
220.163 even 20 968.2.i.j.81.1 4
220.183 odd 20 968.2.i.i.753.1 4
220.203 even 20 968.2.i.j.729.1 4
440.43 odd 4 7744.2.a.bk.1.1 1
440.373 even 4 7744.2.a.b.1.1 1
660.263 even 4 8712.2.a.x.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
88.2.a.a.1.1 1 20.3 even 4
176.2.a.c.1.1 1 5.3 odd 4
704.2.a.b.1.1 1 40.13 odd 4
704.2.a.l.1.1 1 40.3 even 4
792.2.a.g.1.1 1 60.23 odd 4
968.2.a.a.1.1 1 220.43 odd 4
968.2.i.i.9.1 4 220.63 odd 20
968.2.i.i.81.1 4 220.123 odd 20
968.2.i.i.729.1 4 220.83 odd 20
968.2.i.i.753.1 4 220.183 odd 20
968.2.i.j.9.1 4 220.3 even 20
968.2.i.j.81.1 4 220.163 even 20
968.2.i.j.729.1 4 220.203 even 20
968.2.i.j.753.1 4 220.103 even 20
1584.2.a.q.1.1 1 15.8 even 4
1936.2.a.l.1.1 1 55.43 even 4
2200.2.a.k.1.1 1 20.7 even 4
2200.2.b.a.1849.1 2 20.19 odd 2
2200.2.b.a.1849.2 2 4.3 odd 2
2816.2.c.d.1409.1 2 80.53 odd 4
2816.2.c.d.1409.2 2 80.13 odd 4
2816.2.c.i.1409.1 2 80.3 even 4
2816.2.c.i.1409.2 2 80.43 even 4
4312.2.a.l.1.1 1 140.83 odd 4
4400.2.a.a.1.1 1 5.2 odd 4
4400.2.b.b.4049.1 2 1.1 even 1 trivial
4400.2.b.b.4049.2 2 5.4 even 2 inner
6336.2.a.h.1.1 1 120.83 odd 4
6336.2.a.k.1.1 1 120.53 even 4
7744.2.a.b.1.1 1 440.373 even 4
7744.2.a.bk.1.1 1 440.43 odd 4
8624.2.a.c.1.1 1 35.13 even 4
8712.2.a.x.1.1 1 660.263 even 4