Properties

Label 4900.2.e.i.2549.2
Level $4900$
Weight $2$
Character 4900.2549
Analytic conductor $39.127$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4900 = 2^{2} \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4900.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(39.1266969904\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 28)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2549.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 4900.2549
Dual form 4900.2.e.i.2549.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{3} +2.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{3} +2.00000 q^{9} -3.00000 q^{11} +2.00000i q^{13} -3.00000i q^{17} +1.00000 q^{19} +3.00000i q^{23} +5.00000i q^{27} +6.00000 q^{29} -7.00000 q^{31} -3.00000i q^{33} +1.00000i q^{37} -2.00000 q^{39} +6.00000 q^{41} -4.00000i q^{43} +9.00000i q^{47} +3.00000 q^{51} +3.00000i q^{53} +1.00000i q^{57} -9.00000 q^{59} -1.00000 q^{61} +7.00000i q^{67} -3.00000 q^{69} -1.00000i q^{73} +13.0000 q^{79} +1.00000 q^{81} +12.0000i q^{83} +6.00000i q^{87} -15.0000 q^{89} -7.00000i q^{93} +10.0000i q^{97} -6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{9} + O(q^{10}) \) \( 2q + 4q^{9} - 6q^{11} + 2q^{19} + 12q^{29} - 14q^{31} - 4q^{39} + 12q^{41} + 6q^{51} - 18q^{59} - 2q^{61} - 6q^{69} + 26q^{79} + 2q^{81} - 30q^{89} - 12q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4900\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(1177\) \(2451\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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 0.957427 + 0.288675i \(0.0932147\pi\)
−0.957427 + 0.288675i \(0.906785\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 2.00000 0.666667
\(10\) 0 0
\(11\) −3.00000 −0.904534 −0.452267 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) 0 0
\(13\) 2.00000i 0.554700i 0.960769 + 0.277350i \(0.0894562\pi\)
−0.960769 + 0.277350i \(0.910544\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 3.00000i − 0.727607i −0.931476 0.363803i \(-0.881478\pi\)
0.931476 0.363803i \(-0.118522\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416 0.114708 0.993399i \(-0.463407\pi\)
0.114708 + 0.993399i \(0.463407\pi\)
\(20\) 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
\(25\) 0 0
\(26\) 0 0
\(27\) 5.00000i 0.962250i
\(28\) 0 0
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) −7.00000 −1.25724 −0.628619 0.777714i \(-0.716379\pi\)
−0.628619 + 0.777714i \(0.716379\pi\)
\(32\) 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 \(0.0261945\pi\)
−0.996616 + 0.0821995i \(0.973806\pi\)
\(38\) 0 0
\(39\) −2.00000 −0.320256
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) − 4.00000i − 0.609994i −0.952353 0.304997i \(-0.901344\pi\)
0.952353 0.304997i \(-0.0986555\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 9.00000i 1.31278i 0.754420 + 0.656392i \(0.227918\pi\)
−0.754420 + 0.656392i \(0.772082\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 3.00000 0.420084
\(52\) 0 0
\(53\) 3.00000i 0.412082i 0.978543 + 0.206041i \(0.0660580\pi\)
−0.978543 + 0.206041i \(0.933942\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 1.00000i 0.132453i
\(58\) 0 0
\(59\) −9.00000 −1.17170 −0.585850 0.810419i \(-0.699239\pi\)
−0.585850 + 0.810419i \(0.699239\pi\)
\(60\) 0 0
\(61\) −1.00000 −0.128037 −0.0640184 0.997949i \(-0.520392\pi\)
−0.0640184 + 0.997949i \(0.520392\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 7.00000i 0.855186i 0.903971 + 0.427593i \(0.140638\pi\)
−0.903971 + 0.427593i \(0.859362\pi\)
\(68\) 0 0
\(69\) −3.00000 −0.361158
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) − 1.00000i − 0.117041i −0.998286 0.0585206i \(-0.981362\pi\)
0.998286 0.0585206i \(-0.0186383\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 13.0000 1.46261 0.731307 0.682048i \(-0.238911\pi\)
0.731307 + 0.682048i \(0.238911\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 12.0000i 1.31717i 0.752506 + 0.658586i \(0.228845\pi\)
−0.752506 + 0.658586i \(0.771155\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 6.00000i 0.643268i
\(88\) 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
\(93\) − 7.00000i − 0.725866i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 10.0000i 1.01535i 0.861550 + 0.507673i \(0.169494\pi\)
−0.861550 + 0.507673i \(0.830506\pi\)
\(98\) 0 0
\(99\) −6.00000 −0.603023
\(100\) 0 0
\(101\) 15.0000 1.49256 0.746278 0.665635i \(-0.231839\pi\)
0.746278 + 0.665635i \(0.231839\pi\)
\(102\) 0 0
\(103\) 11.0000i 1.08386i 0.840423 + 0.541931i \(0.182307\pi\)
−0.840423 + 0.541931i \(0.817693\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 15.0000i − 1.45010i −0.688694 0.725052i \(-0.741816\pi\)
0.688694 0.725052i \(-0.258184\pi\)
\(108\) 0 0
\(109\) 1.00000 0.0957826 0.0478913 0.998853i \(-0.484750\pi\)
0.0478913 + 0.998853i \(0.484750\pi\)
\(110\) 0 0
\(111\) −1.00000 −0.0949158
\(112\) 0 0
\(113\) 6.00000i 0.564433i 0.959351 + 0.282216i \(0.0910696\pi\)
−0.959351 + 0.282216i \(0.908930\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 4.00000i 0.369800i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) 0 0
\(123\) 6.00000i 0.541002i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 8.00000i − 0.709885i −0.934888 0.354943i \(-0.884500\pi\)
0.934888 0.354943i \(-0.115500\pi\)
\(128\) 0 0
\(129\) 4.00000 0.352180
\(130\) 0 0
\(131\) 3.00000 0.262111 0.131056 0.991375i \(-0.458163\pi\)
0.131056 + 0.991375i \(0.458163\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 21.0000i 1.79415i 0.441877 + 0.897076i \(0.354313\pi\)
−0.441877 + 0.897076i \(0.645687\pi\)
\(138\) 0 0
\(139\) −20.0000 −1.69638 −0.848189 0.529694i \(-0.822307\pi\)
−0.848189 + 0.529694i \(0.822307\pi\)
\(140\) 0 0
\(141\) −9.00000 −0.757937
\(142\) 0 0
\(143\) − 6.00000i − 0.501745i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −3.00000 −0.245770 −0.122885 0.992421i \(-0.539215\pi\)
−0.122885 + 0.992421i \(0.539215\pi\)
\(150\) 0 0
\(151\) 17.0000 1.38344 0.691720 0.722166i \(-0.256853\pi\)
0.691720 + 0.722166i \(0.256853\pi\)
\(152\) 0 0
\(153\) − 6.00000i − 0.485071i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 13.0000i 1.03751i 0.854922 + 0.518756i \(0.173605\pi\)
−0.854922 + 0.518756i \(0.826395\pi\)
\(158\) 0 0
\(159\) −3.00000 −0.237915
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 11.0000i 0.861586i 0.902451 + 0.430793i \(0.141766\pi\)
−0.902451 + 0.430793i \(0.858234\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 12.0000i 0.928588i 0.885681 + 0.464294i \(0.153692\pi\)
−0.885681 + 0.464294i \(0.846308\pi\)
\(168\) 0 0
\(169\) 9.00000 0.692308
\(170\) 0 0
\(171\) 2.00000 0.152944
\(172\) 0 0
\(173\) − 9.00000i − 0.684257i −0.939653 0.342129i \(-0.888852\pi\)
0.939653 0.342129i \(-0.111148\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 9.00000i − 0.676481i
\(178\) 0 0
\(179\) −21.0000 −1.56961 −0.784807 0.619740i \(-0.787238\pi\)
−0.784807 + 0.619740i \(0.787238\pi\)
\(180\) 0 0
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 0 0
\(183\) − 1.00000i − 0.0739221i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 9.00000i 0.658145i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −9.00000 −0.651217 −0.325609 0.945505i \(-0.605569\pi\)
−0.325609 + 0.945505i \(0.605569\pi\)
\(192\) 0 0
\(193\) 11.0000i 0.791797i 0.918294 + 0.395899i \(0.129567\pi\)
−0.918294 + 0.395899i \(0.870433\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 18.0000i − 1.28245i −0.767354 0.641223i \(-0.778427\pi\)
0.767354 0.641223i \(-0.221573\pi\)
\(198\) 0 0
\(199\) 7.00000 0.496217 0.248108 0.968732i \(-0.420191\pi\)
0.248108 + 0.968732i \(0.420191\pi\)
\(200\) 0 0
\(201\) −7.00000 −0.493742
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 6.00000i 0.417029i
\(208\) 0 0
\(209\) −3.00000 −0.207514
\(210\) 0 0
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 1.00000 0.0675737
\(220\) 0 0
\(221\) 6.00000 0.403604
\(222\) 0 0
\(223\) 8.00000i 0.535720i 0.963458 + 0.267860i \(0.0863164\pi\)
−0.963458 + 0.267860i \(0.913684\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 3.00000i 0.199117i 0.995032 + 0.0995585i \(0.0317430\pi\)
−0.995032 + 0.0995585i \(0.968257\pi\)
\(228\) 0 0
\(229\) −11.0000 −0.726900 −0.363450 0.931614i \(-0.618401\pi\)
−0.363450 + 0.931614i \(0.618401\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 21.0000i − 1.37576i −0.725826 0.687878i \(-0.758542\pi\)
0.725826 0.687878i \(-0.241458\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 13.0000i 0.844441i
\(238\) 0 0
\(239\) 12.0000 0.776215 0.388108 0.921614i \(-0.373129\pi\)
0.388108 + 0.921614i \(0.373129\pi\)
\(240\) 0 0
\(241\) −1.00000 −0.0644157 −0.0322078 0.999481i \(-0.510254\pi\)
−0.0322078 + 0.999481i \(0.510254\pi\)
\(242\) 0 0
\(243\) 16.0000i 1.02640i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 2.00000i 0.127257i
\(248\) 0 0
\(249\) −12.0000 −0.760469
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) − 9.00000i − 0.565825i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 3.00000i − 0.187135i −0.995613 0.0935674i \(-0.970173\pi\)
0.995613 0.0935674i \(-0.0298271\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 12.0000 0.742781
\(262\) 0 0
\(263\) − 3.00000i − 0.184988i −0.995713 0.0924940i \(-0.970516\pi\)
0.995713 0.0924940i \(-0.0294839\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 15.0000i − 0.917985i
\(268\) 0 0
\(269\) −3.00000 −0.182913 −0.0914566 0.995809i \(-0.529152\pi\)
−0.0914566 + 0.995809i \(0.529152\pi\)
\(270\) 0 0
\(271\) 11.0000 0.668202 0.334101 0.942537i \(-0.391567\pi\)
0.334101 + 0.942537i \(0.391567\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 13.0000i 0.781094i 0.920583 + 0.390547i \(0.127714\pi\)
−0.920583 + 0.390547i \(0.872286\pi\)
\(278\) 0 0
\(279\) −14.0000 −0.838158
\(280\) 0 0
\(281\) 30.0000 1.78965 0.894825 0.446417i \(-0.147300\pi\)
0.894825 + 0.446417i \(0.147300\pi\)
\(282\) 0 0
\(283\) 29.0000i 1.72387i 0.507018 + 0.861936i \(0.330748\pi\)
−0.507018 + 0.861936i \(0.669252\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 8.00000 0.470588
\(290\) 0 0
\(291\) −10.0000 −0.586210
\(292\) 0 0
\(293\) 6.00000i 0.350524i 0.984522 + 0.175262i \(0.0560772\pi\)
−0.984522 + 0.175262i \(0.943923\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) − 15.0000i − 0.870388i
\(298\) 0 0
\(299\) −6.00000 −0.346989
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 15.0000i 0.861727i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 28.0000i 1.59804i 0.601302 + 0.799022i \(0.294649\pi\)
−0.601302 + 0.799022i \(0.705351\pi\)
\(308\) 0 0
\(309\) −11.0000 −0.625768
\(310\) 0 0
\(311\) −27.0000 −1.53103 −0.765515 0.643418i \(-0.777516\pi\)
−0.765515 + 0.643418i \(0.777516\pi\)
\(312\) 0 0
\(313\) 23.0000i 1.30004i 0.759918 + 0.650018i \(0.225239\pi\)
−0.759918 + 0.650018i \(0.774761\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 9.00000i 0.505490i 0.967533 + 0.252745i \(0.0813334\pi\)
−0.967533 + 0.252745i \(0.918667\pi\)
\(318\) 0 0
\(319\) −18.0000 −1.00781
\(320\) 0 0
\(321\) 15.0000 0.837218
\(322\) 0 0
\(323\) − 3.00000i − 0.166924i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 1.00000i 0.0553001i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −13.0000 −0.714545 −0.357272 0.934000i \(-0.616293\pi\)
−0.357272 + 0.934000i \(0.616293\pi\)
\(332\) 0 0
\(333\) 2.00000i 0.109599i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 34.0000i 1.85210i 0.377403 + 0.926049i \(0.376817\pi\)
−0.377403 + 0.926049i \(0.623183\pi\)
\(338\) 0 0
\(339\) −6.00000 −0.325875
\(340\) 0 0
\(341\) 21.0000 1.13721
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 9.00000i − 0.483145i −0.970383 0.241573i \(-0.922337\pi\)
0.970383 0.241573i \(-0.0776632\pi\)
\(348\) 0 0
\(349\) −26.0000 −1.39175 −0.695874 0.718164i \(-0.744983\pi\)
−0.695874 + 0.718164i \(0.744983\pi\)
\(350\) 0 0
\(351\) −10.0000 −0.533761
\(352\) 0 0
\(353\) − 21.0000i − 1.11772i −0.829263 0.558859i \(-0.811239\pi\)
0.829263 0.558859i \(-0.188761\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −15.0000 −0.791670 −0.395835 0.918322i \(-0.629545\pi\)
−0.395835 + 0.918322i \(0.629545\pi\)
\(360\) 0 0
\(361\) −18.0000 −0.947368
\(362\) 0 0
\(363\) − 2.00000i − 0.104973i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 5.00000i − 0.260998i −0.991448 0.130499i \(-0.958342\pi\)
0.991448 0.130499i \(-0.0416579\pi\)
\(368\) 0 0
\(369\) 12.0000 0.624695
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) − 25.0000i − 1.29445i −0.762299 0.647225i \(-0.775929\pi\)
0.762299 0.647225i \(-0.224071\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 12.0000i 0.618031i
\(378\) 0 0
\(379\) −8.00000 −0.410932 −0.205466 0.978664i \(-0.565871\pi\)
−0.205466 + 0.978664i \(0.565871\pi\)
\(380\) 0 0
\(381\) 8.00000 0.409852
\(382\) 0 0
\(383\) − 33.0000i − 1.68622i −0.537740 0.843111i \(-0.680722\pi\)
0.537740 0.843111i \(-0.319278\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 8.00000i − 0.406663i
\(388\) 0 0
\(389\) −15.0000 −0.760530 −0.380265 0.924878i \(-0.624167\pi\)
−0.380265 + 0.924878i \(0.624167\pi\)
\(390\) 0 0
\(391\) 9.00000 0.455150
\(392\) 0 0
\(393\) 3.00000i 0.151330i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 37.0000i 1.85698i 0.371361 + 0.928488i \(0.378891\pi\)
−0.371361 + 0.928488i \(0.621109\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 3.00000 0.149813 0.0749064 0.997191i \(-0.476134\pi\)
0.0749064 + 0.997191i \(0.476134\pi\)
\(402\) 0 0
\(403\) − 14.0000i − 0.697390i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 3.00000i − 0.148704i
\(408\) 0 0
\(409\) −11.0000 −0.543915 −0.271957 0.962309i \(-0.587671\pi\)
−0.271957 + 0.962309i \(0.587671\pi\)
\(410\) 0 0
\(411\) −21.0000 −1.03585
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 20.0000i − 0.979404i
\(418\) 0 0
\(419\) 12.0000 0.586238 0.293119 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 0 0
\(421\) −22.0000 −1.07221 −0.536107 0.844150i \(-0.680106\pi\)
−0.536107 + 0.844150i \(0.680106\pi\)
\(422\) 0 0
\(423\) 18.0000i 0.875190i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 6.00000 0.289683
\(430\) 0 0
\(431\) −15.0000 −0.722525 −0.361262 0.932464i \(-0.617654\pi\)
−0.361262 + 0.932464i \(0.617654\pi\)
\(432\) 0 0
\(433\) − 10.0000i − 0.480569i −0.970702 0.240285i \(-0.922759\pi\)
0.970702 0.240285i \(-0.0772408\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 3.00000i 0.143509i
\(438\) 0 0
\(439\) 1.00000 0.0477274 0.0238637 0.999715i \(-0.492403\pi\)
0.0238637 + 0.999715i \(0.492403\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 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
\(447\) − 3.00000i − 0.141895i
\(448\) 0 0
\(449\) 18.0000 0.849473 0.424736 0.905317i \(-0.360367\pi\)
0.424736 + 0.905317i \(0.360367\pi\)
\(450\) 0 0
\(451\) −18.0000 −0.847587
\(452\) 0 0
\(453\) 17.0000i 0.798730i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 23.0000i − 1.07589i −0.842978 0.537947i \(-0.819200\pi\)
0.842978 0.537947i \(-0.180800\pi\)
\(458\) 0 0
\(459\) 15.0000 0.700140
\(460\) 0 0
\(461\) −6.00000 −0.279448 −0.139724 0.990190i \(-0.544622\pi\)
−0.139724 + 0.990190i \(0.544622\pi\)
\(462\) 0 0
\(463\) − 16.0000i − 0.743583i −0.928316 0.371792i \(-0.878744\pi\)
0.928316 0.371792i \(-0.121256\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 21.0000i 0.971764i 0.874024 + 0.485882i \(0.161502\pi\)
−0.874024 + 0.485882i \(0.838498\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −13.0000 −0.599008
\(472\) 0 0
\(473\) 12.0000i 0.551761i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 6.00000i 0.274721i
\(478\) 0 0
\(479\) 3.00000 0.137073 0.0685367 0.997649i \(-0.478167\pi\)
0.0685367 + 0.997649i \(0.478167\pi\)
\(480\) 0 0
\(481\) −2.00000 −0.0911922
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 19.0000i 0.860972i 0.902597 + 0.430486i \(0.141658\pi\)
−0.902597 + 0.430486i \(0.858342\pi\)
\(488\) 0 0
\(489\) −11.0000 −0.497437
\(490\) 0 0
\(491\) 24.0000 1.08310 0.541552 0.840667i \(-0.317837\pi\)
0.541552 + 0.840667i \(0.317837\pi\)
\(492\) 0 0
\(493\) − 18.0000i − 0.810679i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −11.0000 −0.492428 −0.246214 0.969216i \(-0.579187\pi\)
−0.246214 + 0.969216i \(0.579187\pi\)
\(500\) 0 0
\(501\) −12.0000 −0.536120
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 9.00000i 0.399704i
\(508\) 0 0
\(509\) −3.00000 −0.132973 −0.0664863 0.997787i \(-0.521179\pi\)
−0.0664863 + 0.997787i \(0.521179\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 5.00000i 0.220755i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) − 27.0000i − 1.18746i
\(518\) 0 0
\(519\) 9.00000 0.395056
\(520\) 0 0
\(521\) 39.0000 1.70862 0.854311 0.519763i \(-0.173980\pi\)
0.854311 + 0.519763i \(0.173980\pi\)
\(522\) 0 0
\(523\) − 1.00000i − 0.0437269i −0.999761 0.0218635i \(-0.993040\pi\)
0.999761 0.0218635i \(-0.00695991\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 21.0000i 0.914774i
\(528\) 0 0
\(529\) 14.0000 0.608696
\(530\) 0 0
\(531\) −18.0000 −0.781133
\(532\) 0 0
\(533\) 12.0000i 0.519778i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 21.0000i − 0.906217i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 35.0000 1.50477 0.752384 0.658725i \(-0.228904\pi\)
0.752384 + 0.658725i \(0.228904\pi\)
\(542\) 0 0
\(543\) − 10.0000i − 0.429141i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 8.00000i − 0.342055i −0.985266 0.171028i \(-0.945291\pi\)
0.985266 0.171028i \(-0.0547087\pi\)
\(548\) 0 0
\(549\) −2.00000 −0.0853579
\(550\) 0 0
\(551\) 6.00000 0.255609
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 33.0000i 1.39825i 0.714997 + 0.699127i \(0.246428\pi\)
−0.714997 + 0.699127i \(0.753572\pi\)
\(558\) 0 0
\(559\) 8.00000 0.338364
\(560\) 0 0
\(561\) −9.00000 −0.379980
\(562\) 0 0
\(563\) 9.00000i 0.379305i 0.981851 + 0.189652i \(0.0607361\pi\)
−0.981851 + 0.189652i \(0.939264\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 9.00000 0.377300 0.188650 0.982044i \(-0.439589\pi\)
0.188650 + 0.982044i \(0.439589\pi\)
\(570\) 0 0
\(571\) 29.0000 1.21361 0.606806 0.794850i \(-0.292450\pi\)
0.606806 + 0.794850i \(0.292450\pi\)
\(572\) 0 0
\(573\) − 9.00000i − 0.375980i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 1.00000i 0.0416305i 0.999783 + 0.0208153i \(0.00662619\pi\)
−0.999783 + 0.0208153i \(0.993374\pi\)
\(578\) 0 0
\(579\) −11.0000 −0.457144
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) − 9.00000i − 0.372742i
\(584\) 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
\(589\) −7.00000 −0.288430
\(590\) 0 0
\(591\) 18.0000 0.740421
\(592\) 0 0
\(593\) − 21.0000i − 0.862367i −0.902264 0.431183i \(-0.858096\pi\)
0.902264 0.431183i \(-0.141904\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 7.00000i 0.286491i
\(598\) 0 0
\(599\) 27.0000 1.10319 0.551595 0.834112i \(-0.314019\pi\)
0.551595 + 0.834112i \(0.314019\pi\)
\(600\) 0 0
\(601\) 14.0000 0.571072 0.285536 0.958368i \(-0.407828\pi\)
0.285536 + 0.958368i \(0.407828\pi\)
\(602\) 0 0
\(603\) 14.0000i 0.570124i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 47.0000i − 1.90767i −0.300329 0.953836i \(-0.597097\pi\)
0.300329 0.953836i \(-0.402903\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −18.0000 −0.728202
\(612\) 0 0
\(613\) − 25.0000i − 1.00974i −0.863195 0.504870i \(-0.831540\pi\)
0.863195 0.504870i \(-0.168460\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 6.00000i − 0.241551i −0.992680 0.120775i \(-0.961462\pi\)
0.992680 0.120775i \(-0.0385381\pi\)
\(618\) 0 0
\(619\) 31.0000 1.24600 0.622998 0.782224i \(-0.285915\pi\)
0.622998 + 0.782224i \(0.285915\pi\)
\(620\) 0 0
\(621\) −15.0000 −0.601929
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) − 3.00000i − 0.119808i
\(628\) 0 0
\(629\) 3.00000 0.119618
\(630\) 0 0
\(631\) −16.0000 −0.636950 −0.318475 0.947931i \(-0.603171\pi\)
−0.318475 + 0.947931i \(0.603171\pi\)
\(632\) 0 0
\(633\) − 4.00000i − 0.158986i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 15.0000 0.592464 0.296232 0.955116i \(-0.404270\pi\)
0.296232 + 0.955116i \(0.404270\pi\)
\(642\) 0 0
\(643\) 20.0000i 0.788723i 0.918955 + 0.394362i \(0.129034\pi\)
−0.918955 + 0.394362i \(0.870966\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 21.0000i − 0.825595i −0.910823 0.412798i \(-0.864552\pi\)
0.910823 0.412798i \(-0.135448\pi\)
\(648\) 0 0
\(649\) 27.0000 1.05984
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 39.0000i 1.52619i 0.646288 + 0.763094i \(0.276321\pi\)
−0.646288 + 0.763094i \(0.723679\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) − 2.00000i − 0.0780274i
\(658\) 0 0
\(659\) −12.0000 −0.467454 −0.233727 0.972302i \(-0.575092\pi\)
−0.233727 + 0.972302i \(0.575092\pi\)
\(660\) 0 0
\(661\) 11.0000 0.427850 0.213925 0.976850i \(-0.431375\pi\)
0.213925 + 0.976850i \(0.431375\pi\)
\(662\) 0 0
\(663\) 6.00000i 0.233021i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 18.0000i 0.696963i
\(668\) 0 0
\(669\) −8.00000 −0.309298
\(670\) 0 0
\(671\) 3.00000 0.115814
\(672\) 0 0
\(673\) 14.0000i 0.539660i 0.962908 + 0.269830i \(0.0869676\pi\)
−0.962908 + 0.269830i \(0.913032\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 27.0000i − 1.03769i −0.854867 0.518847i \(-0.826361\pi\)
0.854867 0.518847i \(-0.173639\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −3.00000 −0.114960
\(682\) 0 0
\(683\) 21.0000i 0.803543i 0.915740 + 0.401771i \(0.131605\pi\)
−0.915740 + 0.401771i \(0.868395\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) − 11.0000i − 0.419676i
\(688\) 0 0
\(689\) −6.00000 −0.228582
\(690\) 0 0
\(691\) −13.0000 −0.494543 −0.247272 0.968946i \(-0.579534\pi\)
−0.247272 + 0.968946i \(0.579534\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 18.0000i − 0.681799i
\(698\) 0 0
\(699\) 21.0000 0.794293
\(700\) 0 0
\(701\) −18.0000 −0.679851 −0.339925 0.940452i \(-0.610402\pi\)
−0.339925 + 0.940452i \(0.610402\pi\)
\(702\) 0 0
\(703\) 1.00000i 0.0377157i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 1.00000 0.0375558 0.0187779 0.999824i \(-0.494022\pi\)
0.0187779 + 0.999824i \(0.494022\pi\)
\(710\) 0 0
\(711\) 26.0000 0.975076
\(712\) 0 0
\(713\) − 21.0000i − 0.786456i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 12.0000i 0.448148i
\(718\) 0 0
\(719\) 21.0000 0.783168 0.391584 0.920142i \(-0.371927\pi\)
0.391584 + 0.920142i \(0.371927\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) − 1.00000i − 0.0371904i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 32.0000i − 1.18681i −0.804902 0.593407i \(-0.797782\pi\)
0.804902 0.593407i \(-0.202218\pi\)
\(728\) 0 0
\(729\) −13.0000 −0.481481
\(730\) 0 0
\(731\) −12.0000 −0.443836
\(732\) 0 0
\(733\) − 25.0000i − 0.923396i −0.887037 0.461698i \(-0.847240\pi\)
0.887037 0.461698i \(-0.152760\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 21.0000i − 0.773545i
\(738\) 0 0
\(739\) 19.0000 0.698926 0.349463 0.936950i \(-0.386364\pi\)
0.349463 + 0.936950i \(0.386364\pi\)
\(740\) 0 0
\(741\) −2.00000 −0.0734718
\(742\) 0 0
\(743\) − 48.0000i − 1.76095i −0.474093 0.880475i \(-0.657224\pi\)
0.474093 0.880475i \(-0.342776\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 24.0000i 0.878114i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −25.0000 −0.912263 −0.456131 0.889912i \(-0.650765\pi\)
−0.456131 + 0.889912i \(0.650765\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 2.00000i − 0.0726912i −0.999339 0.0363456i \(-0.988428\pi\)
0.999339 0.0363456i \(-0.0115717\pi\)
\(758\) 0 0
\(759\) 9.00000 0.326679
\(760\) 0 0
\(761\) 3.00000 0.108750 0.0543750 0.998521i \(-0.482683\pi\)
0.0543750 + 0.998521i \(0.482683\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 18.0000i − 0.649942i
\(768\) 0 0
\(769\) 34.0000 1.22607 0.613036 0.790055i \(-0.289948\pi\)
0.613036 + 0.790055i \(0.289948\pi\)
\(770\) 0 0
\(771\) 3.00000 0.108042
\(772\) 0 0
\(773\) − 33.0000i − 1.18693i −0.804861 0.593464i \(-0.797760\pi\)
0.804861 0.593464i \(-0.202240\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 6.00000 0.214972
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 30.0000i 1.07211i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 31.0000i 1.10503i 0.833503 + 0.552515i \(0.186332\pi\)
−0.833503 + 0.552515i \(0.813668\pi\)
\(788\) 0 0
\(789\) 3.00000 0.106803
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) − 2.00000i − 0.0710221i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 42.0000i − 1.48772i −0.668338 0.743858i \(-0.732994\pi\)
0.668338 0.743858i \(-0.267006\pi\)
\(798\) 0 0
\(799\) 27.0000 0.955191
\(800\) 0 0
\(801\) −30.0000 −1.06000
\(802\) 0 0
\(803\) 3.00000i 0.105868i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 3.00000i − 0.105605i
\(808\) 0 0
\(809\) 33.0000 1.16022 0.580109 0.814539i \(-0.303010\pi\)
0.580109 + 0.814539i \(0.303010\pi\)
\(810\) 0 0
\(811\) −52.0000 −1.82597 −0.912983 0.407997i \(-0.866228\pi\)
−0.912983 + 0.407997i \(0.866228\pi\)
\(812\) 0 0
\(813\) 11.0000i 0.385787i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 4.00000i − 0.139942i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 27.0000 0.942306 0.471153 0.882051i \(-0.343838\pi\)
0.471153 + 0.882051i \(0.343838\pi\)
\(822\) 0 0
\(823\) 5.00000i 0.174289i 0.996196 + 0.0871445i \(0.0277742\pi\)
−0.996196 + 0.0871445i \(0.972226\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 36.0000i 1.25184i 0.779886 + 0.625921i \(0.215277\pi\)
−0.779886 + 0.625921i \(0.784723\pi\)
\(828\) 0 0
\(829\) 1.00000 0.0347314 0.0173657 0.999849i \(-0.494472\pi\)
0.0173657 + 0.999849i \(0.494472\pi\)
\(830\) 0 0
\(831\) −13.0000 −0.450965
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) − 35.0000i − 1.20978i
\(838\) 0 0
\(839\) 24.0000 0.828572 0.414286 0.910147i \(-0.364031\pi\)
0.414286 + 0.910147i \(0.364031\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 0 0
\(843\) 30.0000i 1.03325i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −29.0000 −0.995277
\(850\) 0 0
\(851\) −3.00000 −0.102839
\(852\) 0 0
\(853\) − 22.0000i − 0.753266i −0.926363 0.376633i \(-0.877082\pi\)
0.926363 0.376633i \(-0.122918\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 15.0000i − 0.512390i −0.966625 0.256195i \(-0.917531\pi\)
0.966625 0.256195i \(-0.0824690\pi\)
\(858\) 0 0
\(859\) −23.0000 −0.784750 −0.392375 0.919805i \(-0.628346\pi\)
−0.392375 + 0.919805i \(0.628346\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 51.0000i 1.73606i 0.496512 + 0.868030i \(0.334614\pi\)
−0.496512 + 0.868030i \(0.665386\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 8.00000i 0.271694i
\(868\) 0 0
\(869\) −39.0000 −1.32298
\(870\) 0 0
\(871\) −14.0000 −0.474372
\(872\) 0 0
\(873\) 20.0000i 0.676897i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 13.0000i 0.438979i 0.975615 + 0.219489i \(0.0704391\pi\)
−0.975615 + 0.219489i \(0.929561\pi\)
\(878\) 0 0
\(879\) −6.00000 −0.202375
\(880\) 0 0
\(881\) 18.0000 0.606435 0.303218 0.952921i \(-0.401939\pi\)
0.303218 + 0.952921i \(0.401939\pi\)
\(882\) 0 0
\(883\) 44.0000i 1.48072i 0.672212 + 0.740359i \(0.265344\pi\)
−0.672212 + 0.740359i \(0.734656\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 39.0000i − 1.30949i −0.755849 0.654746i \(-0.772776\pi\)
0.755849 0.654746i \(-0.227224\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −3.00000 −0.100504
\(892\) 0 0
\(893\) 9.00000i 0.301174i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) − 6.00000i − 0.200334i
\(898\) 0 0
\(899\) −42.0000 −1.40078
\(900\) 0 0
\(901\) 9.00000 0.299833
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 41.0000i − 1.36138i −0.732570 0.680691i \(-0.761680\pi\)
0.732570 0.680691i \(-0.238320\pi\)
\(908\) 0 0
\(909\) 30.0000 0.995037
\(910\) 0 0
\(911\) 48.0000 1.59031 0.795155 0.606406i \(-0.207389\pi\)
0.795155 + 0.606406i \(0.207389\pi\)
\(912\) 0 0
\(913\) − 36.0000i − 1.19143i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 1.00000 0.0329870 0.0164935 0.999864i \(-0.494750\pi\)
0.0164935 + 0.999864i \(0.494750\pi\)
\(920\) 0 0
\(921\) −28.0000 −0.922631
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 22.0000i 0.722575i
\(928\) 0 0
\(929\) 9.00000 0.295280 0.147640 0.989041i \(-0.452832\pi\)
0.147640 + 0.989041i \(0.452832\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) − 27.0000i − 0.883940i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 26.0000i − 0.849383i −0.905338 0.424691i \(-0.860383\pi\)
0.905338 0.424691i \(-0.139617\pi\)
\(938\) 0 0
\(939\) −23.0000 −0.750577
\(940\) 0 0
\(941\) 27.0000 0.880175 0.440087 0.897955i \(-0.354947\pi\)
0.440087 + 0.897955i \(0.354947\pi\)
\(942\) 0 0
\(943\) 18.0000i 0.586161i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 45.0000i 1.46230i 0.682215 + 0.731152i \(0.261017\pi\)
−0.682215 + 0.731152i \(0.738983\pi\)
\(948\) 0 0
\(949\) 2.00000 0.0649227
\(950\) 0 0
\(951\) −9.00000 −0.291845
\(952\) 0 0
\(953\) − 18.0000i − 0.583077i −0.956559 0.291539i \(-0.905833\pi\)
0.956559 0.291539i \(-0.0941672\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) − 18.0000i − 0.581857i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 18.0000 0.580645
\(962\) 0 0
\(963\) − 30.0000i − 0.966736i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 16.0000i 0.514525i 0.966342 + 0.257263i \(0.0828206\pi\)
−0.966342 + 0.257263i \(0.917179\pi\)
\(968\) 0 0
\(969\) 3.00000 0.0963739
\(970\) 0 0
\(971\) 51.0000 1.63667 0.818334 0.574743i \(-0.194898\pi\)
0.818334 + 0.574743i \(0.194898\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 27.0000i − 0.863807i −0.901920 0.431903i \(-0.857842\pi\)
0.901920 0.431903i \(-0.142158\pi\)
\(978\) 0 0
\(979\) 45.0000 1.43821
\(980\) 0 0
\(981\) 2.00000 0.0638551
\(982\) 0 0
\(983\) 33.0000i 1.05254i 0.850319 + 0.526268i \(0.176409\pi\)
−0.850319 + 0.526268i \(0.823591\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 12.0000 0.381578
\(990\) 0 0
\(991\) −19.0000 −0.603555 −0.301777 0.953378i \(-0.597580\pi\)
−0.301777 + 0.953378i \(0.597580\pi\)
\(992\) 0 0
\(993\) − 13.0000i − 0.412543i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 59.0000i − 1.86855i −0.356555 0.934274i \(-0.616049\pi\)
0.356555 0.934274i \(-0.383951\pi\)
\(998\) 0 0
\(999\) −5.00000 −0.158193
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 4900.2.e.i.2549.2 2
5.2 odd 4 196.2.a.b.1.1 1
5.3 odd 4 4900.2.a.g.1.1 1
5.4 even 2 inner 4900.2.e.i.2549.1 2
7.2 even 3 700.2.r.b.249.1 4
7.4 even 3 700.2.r.b.149.2 4
7.6 odd 2 4900.2.e.h.2549.1 2
15.2 even 4 1764.2.a.a.1.1 1
20.7 even 4 784.2.a.d.1.1 1
35.2 odd 12 28.2.e.a.25.1 yes 2
35.4 even 6 700.2.r.b.149.1 4
35.9 even 6 700.2.r.b.249.2 4
35.12 even 12 196.2.e.a.165.1 2
35.13 even 4 4900.2.a.n.1.1 1
35.17 even 12 196.2.e.a.177.1 2
35.18 odd 12 700.2.i.c.401.1 2
35.23 odd 12 700.2.i.c.501.1 2
35.27 even 4 196.2.a.a.1.1 1
35.32 odd 12 28.2.e.a.9.1 2
35.34 odd 2 4900.2.e.h.2549.2 2
40.27 even 4 3136.2.a.s.1.1 1
40.37 odd 4 3136.2.a.h.1.1 1
60.47 odd 4 7056.2.a.f.1.1 1
105.2 even 12 252.2.k.c.109.1 2
105.17 odd 12 1764.2.k.b.1549.1 2
105.32 even 12 252.2.k.c.37.1 2
105.47 odd 12 1764.2.k.b.361.1 2
105.62 odd 4 1764.2.a.j.1.1 1
140.27 odd 4 784.2.a.g.1.1 1
140.47 odd 12 784.2.i.d.753.1 2
140.67 even 12 112.2.i.b.65.1 2
140.87 odd 12 784.2.i.d.177.1 2
140.107 even 12 112.2.i.b.81.1 2
280.27 odd 4 3136.2.a.k.1.1 1
280.37 odd 12 448.2.i.e.193.1 2
280.67 even 12 448.2.i.c.65.1 2
280.107 even 12 448.2.i.c.193.1 2
280.237 even 4 3136.2.a.v.1.1 1
280.277 odd 12 448.2.i.e.65.1 2
315.2 even 12 2268.2.l.a.109.1 2
315.32 even 12 2268.2.l.a.541.1 2
315.67 odd 12 2268.2.l.h.541.1 2
315.137 even 12 2268.2.i.h.2053.1 2
315.142 odd 12 2268.2.l.h.109.1 2
315.212 even 12 2268.2.i.h.865.1 2
315.247 odd 12 2268.2.i.a.865.1 2
315.277 odd 12 2268.2.i.a.2053.1 2
420.107 odd 12 1008.2.s.p.865.1 2
420.167 even 4 7056.2.a.bw.1.1 1
420.347 odd 12 1008.2.s.p.289.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
28.2.e.a.9.1 2 35.32 odd 12
28.2.e.a.25.1 yes 2 35.2 odd 12
112.2.i.b.65.1 2 140.67 even 12
112.2.i.b.81.1 2 140.107 even 12
196.2.a.a.1.1 1 35.27 even 4
196.2.a.b.1.1 1 5.2 odd 4
196.2.e.a.165.1 2 35.12 even 12
196.2.e.a.177.1 2 35.17 even 12
252.2.k.c.37.1 2 105.32 even 12
252.2.k.c.109.1 2 105.2 even 12
448.2.i.c.65.1 2 280.67 even 12
448.2.i.c.193.1 2 280.107 even 12
448.2.i.e.65.1 2 280.277 odd 12
448.2.i.e.193.1 2 280.37 odd 12
700.2.i.c.401.1 2 35.18 odd 12
700.2.i.c.501.1 2 35.23 odd 12
700.2.r.b.149.1 4 35.4 even 6
700.2.r.b.149.2 4 7.4 even 3
700.2.r.b.249.1 4 7.2 even 3
700.2.r.b.249.2 4 35.9 even 6
784.2.a.d.1.1 1 20.7 even 4
784.2.a.g.1.1 1 140.27 odd 4
784.2.i.d.177.1 2 140.87 odd 12
784.2.i.d.753.1 2 140.47 odd 12
1008.2.s.p.289.1 2 420.347 odd 12
1008.2.s.p.865.1 2 420.107 odd 12
1764.2.a.a.1.1 1 15.2 even 4
1764.2.a.j.1.1 1 105.62 odd 4
1764.2.k.b.361.1 2 105.47 odd 12
1764.2.k.b.1549.1 2 105.17 odd 12
2268.2.i.a.865.1 2 315.247 odd 12
2268.2.i.a.2053.1 2 315.277 odd 12
2268.2.i.h.865.1 2 315.212 even 12
2268.2.i.h.2053.1 2 315.137 even 12
2268.2.l.a.109.1 2 315.2 even 12
2268.2.l.a.541.1 2 315.32 even 12
2268.2.l.h.109.1 2 315.142 odd 12
2268.2.l.h.541.1 2 315.67 odd 12
3136.2.a.h.1.1 1 40.37 odd 4
3136.2.a.k.1.1 1 280.27 odd 4
3136.2.a.s.1.1 1 40.27 even 4
3136.2.a.v.1.1 1 280.237 even 4
4900.2.a.g.1.1 1 5.3 odd 4
4900.2.a.n.1.1 1 35.13 even 4
4900.2.e.h.2549.1 2 7.6 odd 2
4900.2.e.h.2549.2 2 35.34 odd 2
4900.2.e.i.2549.1 2 5.4 even 2 inner
4900.2.e.i.2549.2 2 1.1 even 1 trivial
7056.2.a.f.1.1 1 60.47 odd 4
7056.2.a.bw.1.1 1 420.167 even 4