Properties

Label 3136.2.a.s.1.1
Level $3136$
Weight $2$
Character 3136.1
Self dual yes
Analytic conductor $25.041$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3136 = 2^{6} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3136.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(25.0410860739\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 28)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 3136.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} -3.00000 q^{5} -2.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -3.00000 q^{5} -2.00000 q^{9} -3.00000 q^{11} -2.00000 q^{13} -3.00000 q^{15} +3.00000 q^{17} -1.00000 q^{19} -3.00000 q^{23} +4.00000 q^{25} -5.00000 q^{27} +6.00000 q^{29} +7.00000 q^{31} -3.00000 q^{33} +1.00000 q^{37} -2.00000 q^{39} +6.00000 q^{41} -4.00000 q^{43} +6.00000 q^{45} +9.00000 q^{47} +3.00000 q^{51} -3.00000 q^{53} +9.00000 q^{55} -1.00000 q^{57} +9.00000 q^{59} +1.00000 q^{61} +6.00000 q^{65} -7.00000 q^{67} -3.00000 q^{69} -1.00000 q^{73} +4.00000 q^{75} +13.0000 q^{79} +1.00000 q^{81} +12.0000 q^{83} -9.00000 q^{85} +6.00000 q^{87} +15.0000 q^{89} +7.00000 q^{93} +3.00000 q^{95} -10.0000 q^{97} +6.00000 q^{99} +O(q^{100})\)

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
\(3\) 1.00000 0.577350 0.288675 0.957427i \(-0.406785\pi\)
0.288675 + 0.957427i \(0.406785\pi\)
\(4\) 0 0
\(5\) −3.00000 −1.34164 −0.670820 0.741620i \(-0.734058\pi\)
−0.670820 + 0.741620i \(0.734058\pi\)
\(6\) 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
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) −3.00000 −0.774597
\(16\) 0 0
\(17\) 3.00000 0.727607 0.363803 0.931476i \(-0.381478\pi\)
0.363803 + 0.931476i \(0.381478\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416 −0.114708 0.993399i \(-0.536593\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −3.00000 −0.625543 −0.312772 0.949828i \(-0.601257\pi\)
−0.312772 + 0.949828i \(0.601257\pi\)
\(24\) 0 0
\(25\) 4.00000 0.800000
\(26\) 0 0
\(27\) −5.00000 −0.962250
\(28\) 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 \(-0.283621\pi\)
0.628619 + 0.777714i \(0.283621\pi\)
\(32\) 0 0
\(33\) −3.00000 −0.522233
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 1.00000 0.164399 0.0821995 0.996616i \(-0.473806\pi\)
0.0821995 + 0.996616i \(0.473806\pi\)
\(38\) 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
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) 6.00000 0.894427
\(46\) 0 0
\(47\) 9.00000 1.31278 0.656392 0.754420i \(-0.272082\pi\)
0.656392 + 0.754420i \(0.272082\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 3.00000 0.420084
\(52\) 0 0
\(53\) −3.00000 −0.412082 −0.206041 0.978543i \(-0.566058\pi\)
−0.206041 + 0.978543i \(0.566058\pi\)
\(54\) 0 0
\(55\) 9.00000 1.21356
\(56\) 0 0
\(57\) −1.00000 −0.132453
\(58\) 0 0
\(59\) 9.00000 1.17170 0.585850 0.810419i \(-0.300761\pi\)
0.585850 + 0.810419i \(0.300761\pi\)
\(60\) 0 0
\(61\) 1.00000 0.128037 0.0640184 0.997949i \(-0.479608\pi\)
0.0640184 + 0.997949i \(0.479608\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 6.00000 0.744208
\(66\) 0 0
\(67\) −7.00000 −0.855186 −0.427593 0.903971i \(-0.640638\pi\)
−0.427593 + 0.903971i \(0.640638\pi\)
\(68\) 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
\(73\) −1.00000 −0.117041 −0.0585206 0.998286i \(-0.518638\pi\)
−0.0585206 + 0.998286i \(0.518638\pi\)
\(74\) 0 0
\(75\) 4.00000 0.461880
\(76\) 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
\(83\) 12.0000 1.31717 0.658586 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) 0 0
\(85\) −9.00000 −0.976187
\(86\) 0 0
\(87\) 6.00000 0.643268
\(88\) 0 0
\(89\) 15.0000 1.59000 0.794998 0.606612i \(-0.207472\pi\)
0.794998 + 0.606612i \(0.207472\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 7.00000 0.725866
\(94\) 0 0
\(95\) 3.00000 0.307794
\(96\) 0 0
\(97\) −10.0000 −1.01535 −0.507673 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) 0 0
\(99\) 6.00000 0.603023
\(100\) 0 0
\(101\) −15.0000 −1.49256 −0.746278 0.665635i \(-0.768161\pi\)
−0.746278 + 0.665635i \(0.768161\pi\)
\(102\) 0 0
\(103\) −11.0000 −1.08386 −0.541931 0.840423i \(-0.682307\pi\)
−0.541931 + 0.840423i \(0.682307\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 15.0000 1.45010 0.725052 0.688694i \(-0.241816\pi\)
0.725052 + 0.688694i \(0.241816\pi\)
\(108\) 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
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 0 0
\(115\) 9.00000 0.839254
\(116\) 0 0
\(117\) 4.00000 0.369800
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) 0 0
\(123\) 6.00000 0.541002
\(124\) 0 0
\(125\) 3.00000 0.268328
\(126\) 0 0
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) 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
\(135\) 15.0000 1.29099
\(136\) 0 0
\(137\) −21.0000 −1.79415 −0.897076 0.441877i \(-0.854313\pi\)
−0.897076 + 0.441877i \(0.854313\pi\)
\(138\) 0 0
\(139\) 20.0000 1.69638 0.848189 0.529694i \(-0.177693\pi\)
0.848189 + 0.529694i \(0.177693\pi\)
\(140\) 0 0
\(141\) 9.00000 0.757937
\(142\) 0 0
\(143\) 6.00000 0.501745
\(144\) 0 0
\(145\) −18.0000 −1.49482
\(146\) 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 \(-0.743147\pi\)
−0.691720 + 0.722166i \(0.743147\pi\)
\(152\) 0 0
\(153\) −6.00000 −0.485071
\(154\) 0 0
\(155\) −21.0000 −1.68676
\(156\) 0 0
\(157\) 13.0000 1.03751 0.518756 0.854922i \(-0.326395\pi\)
0.518756 + 0.854922i \(0.326395\pi\)
\(158\) 0 0
\(159\) −3.00000 −0.237915
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 11.0000 0.861586 0.430793 0.902451i \(-0.358234\pi\)
0.430793 + 0.902451i \(0.358234\pi\)
\(164\) 0 0
\(165\) 9.00000 0.700649
\(166\) 0 0
\(167\) 12.0000 0.928588 0.464294 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 2.00000 0.152944
\(172\) 0 0
\(173\) 9.00000 0.684257 0.342129 0.939653i \(-0.388852\pi\)
0.342129 + 0.939653i \(0.388852\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 9.00000 0.676481
\(178\) 0 0
\(179\) 21.0000 1.56961 0.784807 0.619740i \(-0.212762\pi\)
0.784807 + 0.619740i \(0.212762\pi\)
\(180\) 0 0
\(181\) 10.0000 0.743294 0.371647 0.928374i \(-0.378793\pi\)
0.371647 + 0.928374i \(0.378793\pi\)
\(182\) 0 0
\(183\) 1.00000 0.0739221
\(184\) 0 0
\(185\) −3.00000 −0.220564
\(186\) 0 0
\(187\) −9.00000 −0.658145
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 9.00000 0.651217 0.325609 0.945505i \(-0.394431\pi\)
0.325609 + 0.945505i \(0.394431\pi\)
\(192\) 0 0
\(193\) 11.0000 0.791797 0.395899 0.918294i \(-0.370433\pi\)
0.395899 + 0.918294i \(0.370433\pi\)
\(194\) 0 0
\(195\) 6.00000 0.429669
\(196\) 0 0
\(197\) −18.0000 −1.28245 −0.641223 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) 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
\(205\) −18.0000 −1.25717
\(206\) 0 0
\(207\) 6.00000 0.417029
\(208\) 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
\(215\) 12.0000 0.818393
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −1.00000 −0.0675737
\(220\) 0 0
\(221\) −6.00000 −0.403604
\(222\) 0 0
\(223\) −8.00000 −0.535720 −0.267860 0.963458i \(-0.586316\pi\)
−0.267860 + 0.963458i \(0.586316\pi\)
\(224\) 0 0
\(225\) −8.00000 −0.533333
\(226\) 0 0
\(227\) −3.00000 −0.199117 −0.0995585 0.995032i \(-0.531743\pi\)
−0.0995585 + 0.995032i \(0.531743\pi\)
\(228\) 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
\(233\) −21.0000 −1.37576 −0.687878 0.725826i \(-0.741458\pi\)
−0.687878 + 0.725826i \(0.741458\pi\)
\(234\) 0 0
\(235\) −27.0000 −1.76129
\(236\) 0 0
\(237\) 13.0000 0.844441
\(238\) 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
\(243\) 16.0000 1.02640
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 2.00000 0.127257
\(248\) 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
\(253\) 9.00000 0.565825
\(254\) 0 0
\(255\) −9.00000 −0.563602
\(256\) 0 0
\(257\) 3.00000 0.187135 0.0935674 0.995613i \(-0.470173\pi\)
0.0935674 + 0.995613i \(0.470173\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −12.0000 −0.742781
\(262\) 0 0
\(263\) 3.00000 0.184988 0.0924940 0.995713i \(-0.470516\pi\)
0.0924940 + 0.995713i \(0.470516\pi\)
\(264\) 0 0
\(265\) 9.00000 0.552866
\(266\) 0 0
\(267\) 15.0000 0.917985
\(268\) 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 \(-0.608433\pi\)
−0.334101 + 0.942537i \(0.608433\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −12.0000 −0.723627
\(276\) 0 0
\(277\) 13.0000 0.781094 0.390547 0.920583i \(-0.372286\pi\)
0.390547 + 0.920583i \(0.372286\pi\)
\(278\) 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
\(283\) 29.0000 1.72387 0.861936 0.507018i \(-0.169252\pi\)
0.861936 + 0.507018i \(0.169252\pi\)
\(284\) 0 0
\(285\) 3.00000 0.177705
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) −10.0000 −0.586210
\(292\) 0 0
\(293\) −6.00000 −0.350524 −0.175262 0.984522i \(-0.556077\pi\)
−0.175262 + 0.984522i \(0.556077\pi\)
\(294\) 0 0
\(295\) −27.0000 −1.57200
\(296\) 0 0
\(297\) 15.0000 0.870388
\(298\) 0 0
\(299\) 6.00000 0.346989
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −15.0000 −0.861727
\(304\) 0 0
\(305\) −3.00000 −0.171780
\(306\) 0 0
\(307\) −28.0000 −1.59804 −0.799022 0.601302i \(-0.794649\pi\)
−0.799022 + 0.601302i \(0.794649\pi\)
\(308\) 0 0
\(309\) −11.0000 −0.625768
\(310\) 0 0
\(311\) 27.0000 1.53103 0.765515 0.643418i \(-0.222484\pi\)
0.765515 + 0.643418i \(0.222484\pi\)
\(312\) 0 0
\(313\) 23.0000 1.30004 0.650018 0.759918i \(-0.274761\pi\)
0.650018 + 0.759918i \(0.274761\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 9.00000 0.505490 0.252745 0.967533i \(-0.418667\pi\)
0.252745 + 0.967533i \(0.418667\pi\)
\(318\) 0 0
\(319\) −18.0000 −1.00781
\(320\) 0 0
\(321\) 15.0000 0.837218
\(322\) 0 0
\(323\) −3.00000 −0.166924
\(324\) 0 0
\(325\) −8.00000 −0.443760
\(326\) 0 0
\(327\) 1.00000 0.0553001
\(328\) 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
\(333\) −2.00000 −0.109599
\(334\) 0 0
\(335\) 21.0000 1.14735
\(336\) 0 0
\(337\) −34.0000 −1.85210 −0.926049 0.377403i \(-0.876817\pi\)
−0.926049 + 0.377403i \(0.876817\pi\)
\(338\) 0 0
\(339\) 6.00000 0.325875
\(340\) 0 0
\(341\) −21.0000 −1.13721
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 9.00000 0.484544
\(346\) 0 0
\(347\) 9.00000 0.483145 0.241573 0.970383i \(-0.422337\pi\)
0.241573 + 0.970383i \(0.422337\pi\)
\(348\) 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
\(353\) −21.0000 −1.11772 −0.558859 0.829263i \(-0.688761\pi\)
−0.558859 + 0.829263i \(0.688761\pi\)
\(354\) 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
\(363\) −2.00000 −0.104973
\(364\) 0 0
\(365\) 3.00000 0.157027
\(366\) 0 0
\(367\) −5.00000 −0.260998 −0.130499 0.991448i \(-0.541658\pi\)
−0.130499 + 0.991448i \(0.541658\pi\)
\(368\) 0 0
\(369\) −12.0000 −0.624695
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 25.0000 1.29445 0.647225 0.762299i \(-0.275929\pi\)
0.647225 + 0.762299i \(0.275929\pi\)
\(374\) 0 0
\(375\) 3.00000 0.154919
\(376\) 0 0
\(377\) −12.0000 −0.618031
\(378\) 0 0
\(379\) 8.00000 0.410932 0.205466 0.978664i \(-0.434129\pi\)
0.205466 + 0.978664i \(0.434129\pi\)
\(380\) 0 0
\(381\) −8.00000 −0.409852
\(382\) 0 0
\(383\) 33.0000 1.68622 0.843111 0.537740i \(-0.180722\pi\)
0.843111 + 0.537740i \(0.180722\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 8.00000 0.406663
\(388\) 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
\(393\) 3.00000 0.151330
\(394\) 0 0
\(395\) −39.0000 −1.96230
\(396\) 0 0
\(397\) 37.0000 1.85698 0.928488 0.371361i \(-0.121109\pi\)
0.928488 + 0.371361i \(0.121109\pi\)
\(398\) 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
\(403\) −14.0000 −0.697390
\(404\) 0 0
\(405\) −3.00000 −0.149071
\(406\) 0 0
\(407\) −3.00000 −0.148704
\(408\) 0 0
\(409\) 11.0000 0.543915 0.271957 0.962309i \(-0.412329\pi\)
0.271957 + 0.962309i \(0.412329\pi\)
\(410\) 0 0
\(411\) −21.0000 −1.03585
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −36.0000 −1.76717
\(416\) 0 0
\(417\) 20.0000 0.979404
\(418\) 0 0
\(419\) −12.0000 −0.586238 −0.293119 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 0 0
\(421\) 22.0000 1.07221 0.536107 0.844150i \(-0.319894\pi\)
0.536107 + 0.844150i \(0.319894\pi\)
\(422\) 0 0
\(423\) −18.0000 −0.875190
\(424\) 0 0
\(425\) 12.0000 0.582086
\(426\) 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 \(-0.382346\pi\)
0.361262 + 0.932464i \(0.382346\pi\)
\(432\) 0 0
\(433\) −10.0000 −0.480569 −0.240285 0.970702i \(-0.577241\pi\)
−0.240285 + 0.970702i \(0.577241\pi\)
\(434\) 0 0
\(435\) −18.0000 −0.863034
\(436\) 0 0
\(437\) 3.00000 0.143509
\(438\) 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
\(443\) −9.00000 −0.427603 −0.213801 0.976877i \(-0.568585\pi\)
−0.213801 + 0.976877i \(0.568585\pi\)
\(444\) 0 0
\(445\) −45.0000 −2.13320
\(446\) 0 0
\(447\) −3.00000 −0.141895
\(448\) 0 0
\(449\) −18.0000 −0.849473 −0.424736 0.905317i \(-0.639633\pi\)
−0.424736 + 0.905317i \(0.639633\pi\)
\(450\) 0 0
\(451\) −18.0000 −0.847587
\(452\) 0 0
\(453\) −17.0000 −0.798730
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 23.0000 1.07589 0.537947 0.842978i \(-0.319200\pi\)
0.537947 + 0.842978i \(0.319200\pi\)
\(458\) 0 0
\(459\) −15.0000 −0.700140
\(460\) 0 0
\(461\) 6.00000 0.279448 0.139724 0.990190i \(-0.455378\pi\)
0.139724 + 0.990190i \(0.455378\pi\)
\(462\) 0 0
\(463\) 16.0000 0.743583 0.371792 0.928316i \(-0.378744\pi\)
0.371792 + 0.928316i \(0.378744\pi\)
\(464\) 0 0
\(465\) −21.0000 −0.973852
\(466\) 0 0
\(467\) −21.0000 −0.971764 −0.485882 0.874024i \(-0.661502\pi\)
−0.485882 + 0.874024i \(0.661502\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 13.0000 0.599008
\(472\) 0 0
\(473\) 12.0000 0.551761
\(474\) 0 0
\(475\) −4.00000 −0.183533
\(476\) 0 0
\(477\) 6.00000 0.274721
\(478\) 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
\(485\) 30.0000 1.36223
\(486\) 0 0
\(487\) 19.0000 0.860972 0.430486 0.902597i \(-0.358342\pi\)
0.430486 + 0.902597i \(0.358342\pi\)
\(488\) 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
\(493\) 18.0000 0.810679
\(494\) 0 0
\(495\) −18.0000 −0.809040
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 11.0000 0.492428 0.246214 0.969216i \(-0.420813\pi\)
0.246214 + 0.969216i \(0.420813\pi\)
\(500\) 0 0
\(501\) 12.0000 0.536120
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 45.0000 2.00247
\(506\) 0 0
\(507\) −9.00000 −0.399704
\(508\) 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
\(513\) 5.00000 0.220755
\(514\) 0 0
\(515\) 33.0000 1.45415
\(516\) 0 0
\(517\) −27.0000 −1.18746
\(518\) 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
\(523\) −1.00000 −0.0437269 −0.0218635 0.999761i \(-0.506960\pi\)
−0.0218635 + 0.999761i \(0.506960\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 21.0000 0.914774
\(528\) 0 0
\(529\) −14.0000 −0.608696
\(530\) 0 0
\(531\) −18.0000 −0.781133
\(532\) 0 0
\(533\) −12.0000 −0.519778
\(534\) 0 0
\(535\) −45.0000 −1.94552
\(536\) 0 0
\(537\) 21.0000 0.906217
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −35.0000 −1.50477 −0.752384 0.658725i \(-0.771096\pi\)
−0.752384 + 0.658725i \(0.771096\pi\)
\(542\) 0 0
\(543\) 10.0000 0.429141
\(544\) 0 0
\(545\) −3.00000 −0.128506
\(546\) 0 0
\(547\) 8.00000 0.342055 0.171028 0.985266i \(-0.445291\pi\)
0.171028 + 0.985266i \(0.445291\pi\)
\(548\) 0 0
\(549\) −2.00000 −0.0853579
\(550\) 0 0
\(551\) −6.00000 −0.255609
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −3.00000 −0.127343
\(556\) 0 0
\(557\) 33.0000 1.39825 0.699127 0.714997i \(-0.253572\pi\)
0.699127 + 0.714997i \(0.253572\pi\)
\(558\) 0 0
\(559\) 8.00000 0.338364
\(560\) 0 0
\(561\) −9.00000 −0.379980
\(562\) 0 0
\(563\) 9.00000 0.379305 0.189652 0.981851i \(-0.439264\pi\)
0.189652 + 0.981851i \(0.439264\pi\)
\(564\) 0 0
\(565\) −18.0000 −0.757266
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −9.00000 −0.377300 −0.188650 0.982044i \(-0.560411\pi\)
−0.188650 + 0.982044i \(0.560411\pi\)
\(570\) 0 0
\(571\) 29.0000 1.21361 0.606806 0.794850i \(-0.292450\pi\)
0.606806 + 0.794850i \(0.292450\pi\)
\(572\) 0 0
\(573\) 9.00000 0.375980
\(574\) 0 0
\(575\) −12.0000 −0.500435
\(576\) 0 0
\(577\) −1.00000 −0.0416305 −0.0208153 0.999783i \(-0.506626\pi\)
−0.0208153 + 0.999783i \(0.506626\pi\)
\(578\) 0 0
\(579\) 11.0000 0.457144
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 9.00000 0.372742
\(584\) 0 0
\(585\) −12.0000 −0.496139
\(586\) 0 0
\(587\) 12.0000 0.495293 0.247647 0.968850i \(-0.420343\pi\)
0.247647 + 0.968850i \(0.420343\pi\)
\(588\) 0 0
\(589\) −7.00000 −0.288430
\(590\) 0 0
\(591\) −18.0000 −0.740421
\(592\) 0 0
\(593\) −21.0000 −0.862367 −0.431183 0.902264i \(-0.641904\pi\)
−0.431183 + 0.902264i \(0.641904\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 7.00000 0.286491
\(598\) 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
\(603\) 14.0000 0.570124
\(604\) 0 0
\(605\) 6.00000 0.243935
\(606\) 0 0
\(607\) −47.0000 −1.90767 −0.953836 0.300329i \(-0.902903\pi\)
−0.953836 + 0.300329i \(0.902903\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −18.0000 −0.728202
\(612\) 0 0
\(613\) 25.0000 1.00974 0.504870 0.863195i \(-0.331540\pi\)
0.504870 + 0.863195i \(0.331540\pi\)
\(614\) 0 0
\(615\) −18.0000 −0.725830
\(616\) 0 0
\(617\) 6.00000 0.241551 0.120775 0.992680i \(-0.461462\pi\)
0.120775 + 0.992680i \(0.461462\pi\)
\(618\) 0 0
\(619\) −31.0000 −1.24600 −0.622998 0.782224i \(-0.714085\pi\)
−0.622998 + 0.782224i \(0.714085\pi\)
\(620\) 0 0
\(621\) 15.0000 0.601929
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −29.0000 −1.16000
\(626\) 0 0
\(627\) 3.00000 0.119808
\(628\) 0 0
\(629\) 3.00000 0.119618
\(630\) 0 0
\(631\) 16.0000 0.636950 0.318475 0.947931i \(-0.396829\pi\)
0.318475 + 0.947931i \(0.396829\pi\)
\(632\) 0 0
\(633\) −4.00000 −0.158986
\(634\) 0 0
\(635\) 24.0000 0.952411
\(636\) 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
\(643\) 20.0000 0.788723 0.394362 0.918955i \(-0.370966\pi\)
0.394362 + 0.918955i \(0.370966\pi\)
\(644\) 0 0
\(645\) 12.0000 0.472500
\(646\) 0 0
\(647\) −21.0000 −0.825595 −0.412798 0.910823i \(-0.635448\pi\)
−0.412798 + 0.910823i \(0.635448\pi\)
\(648\) 0 0
\(649\) −27.0000 −1.05984
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −39.0000 −1.52619 −0.763094 0.646288i \(-0.776321\pi\)
−0.763094 + 0.646288i \(0.776321\pi\)
\(654\) 0 0
\(655\) −9.00000 −0.351659
\(656\) 0 0
\(657\) 2.00000 0.0780274
\(658\) 0 0
\(659\) 12.0000 0.467454 0.233727 0.972302i \(-0.424908\pi\)
0.233727 + 0.972302i \(0.424908\pi\)
\(660\) 0 0
\(661\) −11.0000 −0.427850 −0.213925 0.976850i \(-0.568625\pi\)
−0.213925 + 0.976850i \(0.568625\pi\)
\(662\) 0 0
\(663\) −6.00000 −0.233021
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −18.0000 −0.696963
\(668\) 0 0
\(669\) −8.00000 −0.309298
\(670\) 0 0
\(671\) −3.00000 −0.115814
\(672\) 0 0
\(673\) 14.0000 0.539660 0.269830 0.962908i \(-0.413032\pi\)
0.269830 + 0.962908i \(0.413032\pi\)
\(674\) 0 0
\(675\) −20.0000 −0.769800
\(676\) 0 0
\(677\) −27.0000 −1.03769 −0.518847 0.854867i \(-0.673639\pi\)
−0.518847 + 0.854867i \(0.673639\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −3.00000 −0.114960
\(682\) 0 0
\(683\) 21.0000 0.803543 0.401771 0.915740i \(-0.368395\pi\)
0.401771 + 0.915740i \(0.368395\pi\)
\(684\) 0 0
\(685\) 63.0000 2.40711
\(686\) 0 0
\(687\) −11.0000 −0.419676
\(688\) 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
\(695\) −60.0000 −2.27593
\(696\) 0 0
\(697\) 18.0000 0.681799
\(698\) 0 0
\(699\) −21.0000 −0.794293
\(700\) 0 0
\(701\) 18.0000 0.679851 0.339925 0.940452i \(-0.389598\pi\)
0.339925 + 0.940452i \(0.389598\pi\)
\(702\) 0 0
\(703\) −1.00000 −0.0377157
\(704\) 0 0
\(705\) −27.0000 −1.01688
\(706\) 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
\(713\) −21.0000 −0.786456
\(714\) 0 0
\(715\) −18.0000 −0.673162
\(716\) 0 0
\(717\) 12.0000 0.448148
\(718\) 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
\(723\) −1.00000 −0.0371904
\(724\) 0 0
\(725\) 24.0000 0.891338
\(726\) 0 0
\(727\) −32.0000 −1.18681 −0.593407 0.804902i \(-0.702218\pi\)
−0.593407 + 0.804902i \(0.702218\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) −12.0000 −0.443836
\(732\) 0 0
\(733\) 25.0000 0.923396 0.461698 0.887037i \(-0.347240\pi\)
0.461698 + 0.887037i \(0.347240\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 21.0000 0.773545
\(738\) 0 0
\(739\) −19.0000 −0.698926 −0.349463 0.936950i \(-0.613636\pi\)
−0.349463 + 0.936950i \(0.613636\pi\)
\(740\) 0 0
\(741\) 2.00000 0.0734718
\(742\) 0 0
\(743\) 48.0000 1.76095 0.880475 0.474093i \(-0.157224\pi\)
0.880475 + 0.474093i \(0.157224\pi\)
\(744\) 0 0
\(745\) 9.00000 0.329734
\(746\) 0 0
\(747\) −24.0000 −0.878114
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 25.0000 0.912263 0.456131 0.889912i \(-0.349235\pi\)
0.456131 + 0.889912i \(0.349235\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 51.0000 1.85608
\(756\) 0 0
\(757\) −2.00000 −0.0726912 −0.0363456 0.999339i \(-0.511572\pi\)
−0.0363456 + 0.999339i \(0.511572\pi\)
\(758\) 0 0
\(759\) 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
\(765\) 18.0000 0.650791
\(766\) 0 0
\(767\) −18.0000 −0.649942
\(768\) 0 0
\(769\) −34.0000 −1.22607 −0.613036 0.790055i \(-0.710052\pi\)
−0.613036 + 0.790055i \(0.710052\pi\)
\(770\) 0 0
\(771\) 3.00000 0.108042
\(772\) 0 0
\(773\) 33.0000 1.18693 0.593464 0.804861i \(-0.297760\pi\)
0.593464 + 0.804861i \(0.297760\pi\)
\(774\) 0 0
\(775\) 28.0000 1.00579
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −6.00000 −0.214972
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −30.0000 −1.07211
\(784\) 0 0
\(785\) −39.0000 −1.39197
\(786\) 0 0
\(787\) −31.0000 −1.10503 −0.552515 0.833503i \(-0.686332\pi\)
−0.552515 + 0.833503i \(0.686332\pi\)
\(788\) 0 0
\(789\) 3.00000 0.106803
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −2.00000 −0.0710221
\(794\) 0 0
\(795\) 9.00000 0.319197
\(796\) 0 0
\(797\) −42.0000 −1.48772 −0.743858 0.668338i \(-0.767006\pi\)
−0.743858 + 0.668338i \(0.767006\pi\)
\(798\) 0 0
\(799\) 27.0000 0.955191
\(800\) 0 0
\(801\) −30.0000 −1.06000
\(802\) 0 0
\(803\) 3.00000 0.105868
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −3.00000 −0.105605
\(808\) 0 0
\(809\) −33.0000 −1.16022 −0.580109 0.814539i \(-0.696990\pi\)
−0.580109 + 0.814539i \(0.696990\pi\)
\(810\) 0 0
\(811\) −52.0000 −1.82597 −0.912983 0.407997i \(-0.866228\pi\)
−0.912983 + 0.407997i \(0.866228\pi\)
\(812\) 0 0
\(813\) −11.0000 −0.385787
\(814\) 0 0
\(815\) −33.0000 −1.15594
\(816\) 0 0
\(817\) 4.00000 0.139942
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −27.0000 −0.942306 −0.471153 0.882051i \(-0.656162\pi\)
−0.471153 + 0.882051i \(0.656162\pi\)
\(822\) 0 0
\(823\) −5.00000 −0.174289 −0.0871445 0.996196i \(-0.527774\pi\)
−0.0871445 + 0.996196i \(0.527774\pi\)
\(824\) 0 0
\(825\) −12.0000 −0.417786
\(826\) 0 0
\(827\) −36.0000 −1.25184 −0.625921 0.779886i \(-0.715277\pi\)
−0.625921 + 0.779886i \(0.715277\pi\)
\(828\) 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
\(835\) −36.0000 −1.24583
\(836\) 0 0
\(837\) −35.0000 −1.20978
\(838\) 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
\(843\) 30.0000 1.03325
\(844\) 0 0
\(845\) 27.0000 0.928828
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 29.0000 0.995277
\(850\) 0 0
\(851\) −3.00000 −0.102839
\(852\) 0 0
\(853\) 22.0000 0.753266 0.376633 0.926363i \(-0.377082\pi\)
0.376633 + 0.926363i \(0.377082\pi\)
\(854\) 0 0
\(855\) −6.00000 −0.205196
\(856\) 0 0
\(857\) 15.0000 0.512390 0.256195 0.966625i \(-0.417531\pi\)
0.256195 + 0.966625i \(0.417531\pi\)
\(858\) 0 0
\(859\) 23.0000 0.784750 0.392375 0.919805i \(-0.371654\pi\)
0.392375 + 0.919805i \(0.371654\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −51.0000 −1.73606 −0.868030 0.496512i \(-0.834614\pi\)
−0.868030 + 0.496512i \(0.834614\pi\)
\(864\) 0 0
\(865\) −27.0000 −0.918028
\(866\) 0 0
\(867\) −8.00000 −0.271694
\(868\) 0 0
\(869\) −39.0000 −1.32298
\(870\) 0 0
\(871\) 14.0000 0.474372
\(872\) 0 0
\(873\) 20.0000 0.676897
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 13.0000 0.438979 0.219489 0.975615i \(-0.429561\pi\)
0.219489 + 0.975615i \(0.429561\pi\)
\(878\) 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
\(883\) 44.0000 1.48072 0.740359 0.672212i \(-0.234656\pi\)
0.740359 + 0.672212i \(0.234656\pi\)
\(884\) 0 0
\(885\) −27.0000 −0.907595
\(886\) 0 0
\(887\) −39.0000 −1.30949 −0.654746 0.755849i \(-0.727224\pi\)
−0.654746 + 0.755849i \(0.727224\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −3.00000 −0.100504
\(892\) 0 0
\(893\) −9.00000 −0.301174
\(894\) 0 0
\(895\) −63.0000 −2.10586
\(896\) 0 0
\(897\) 6.00000 0.200334
\(898\) 0 0
\(899\) 42.0000 1.40078
\(900\) 0 0
\(901\) −9.00000 −0.299833
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −30.0000 −0.997234
\(906\) 0 0
\(907\) 41.0000 1.36138 0.680691 0.732570i \(-0.261680\pi\)
0.680691 + 0.732570i \(0.261680\pi\)
\(908\) 0 0
\(909\) 30.0000 0.995037
\(910\) 0 0
\(911\) −48.0000 −1.59031 −0.795155 0.606406i \(-0.792611\pi\)
−0.795155 + 0.606406i \(0.792611\pi\)
\(912\) 0 0
\(913\) −36.0000 −1.19143
\(914\) 0 0
\(915\) −3.00000 −0.0991769
\(916\) 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
\(925\) 4.00000 0.131519
\(926\) 0 0
\(927\) 22.0000 0.722575
\(928\) 0 0
\(929\) −9.00000 −0.295280 −0.147640 0.989041i \(-0.547168\pi\)
−0.147640 + 0.989041i \(0.547168\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 27.0000 0.883940
\(934\) 0 0
\(935\) 27.0000 0.882994
\(936\) 0 0
\(937\) 26.0000 0.849383 0.424691 0.905338i \(-0.360383\pi\)
0.424691 + 0.905338i \(0.360383\pi\)
\(938\) 0 0
\(939\) 23.0000 0.750577
\(940\) 0 0
\(941\) −27.0000 −0.880175 −0.440087 0.897955i \(-0.645053\pi\)
−0.440087 + 0.897955i \(0.645053\pi\)
\(942\) 0 0
\(943\) −18.0000 −0.586161
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −45.0000 −1.46230 −0.731152 0.682215i \(-0.761017\pi\)
−0.731152 + 0.682215i \(0.761017\pi\)
\(948\) 0 0
\(949\) 2.00000 0.0649227
\(950\) 0 0
\(951\) 9.00000 0.291845
\(952\) 0 0
\(953\) −18.0000 −0.583077 −0.291539 0.956559i \(-0.594167\pi\)
−0.291539 + 0.956559i \(0.594167\pi\)
\(954\) 0 0
\(955\) −27.0000 −0.873699
\(956\) 0 0
\(957\) −18.0000 −0.581857
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 18.0000 0.580645
\(962\) 0 0
\(963\) −30.0000 −0.966736
\(964\) 0 0
\(965\) −33.0000 −1.06231
\(966\) 0 0
\(967\) 16.0000 0.514525 0.257263 0.966342i \(-0.417179\pi\)
0.257263 + 0.966342i \(0.417179\pi\)
\(968\) 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
\(975\) −8.00000 −0.256205
\(976\) 0 0
\(977\) 27.0000 0.863807 0.431903 0.901920i \(-0.357842\pi\)
0.431903 + 0.901920i \(0.357842\pi\)
\(978\) 0 0
\(979\) −45.0000 −1.43821
\(980\) 0 0
\(981\) −2.00000 −0.0638551
\(982\) 0 0
\(983\) −33.0000 −1.05254 −0.526268 0.850319i \(-0.676409\pi\)
−0.526268 + 0.850319i \(0.676409\pi\)
\(984\) 0 0
\(985\) 54.0000 1.72058
\(986\) 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 \(-0.402420\pi\)
0.301777 + 0.953378i \(0.402420\pi\)
\(992\) 0 0
\(993\) −13.0000 −0.412543
\(994\) 0 0
\(995\) −21.0000 −0.665745
\(996\) 0 0
\(997\) −59.0000 −1.86855 −0.934274 0.356555i \(-0.883951\pi\)
−0.934274 + 0.356555i \(0.883951\pi\)
\(998\) 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 3136.2.a.s.1.1 1
4.3 odd 2 3136.2.a.h.1.1 1
7.2 even 3 448.2.i.c.193.1 2
7.4 even 3 448.2.i.c.65.1 2
7.6 odd 2 3136.2.a.k.1.1 1
8.3 odd 2 196.2.a.b.1.1 1
8.5 even 2 784.2.a.d.1.1 1
24.5 odd 2 7056.2.a.f.1.1 1
24.11 even 2 1764.2.a.a.1.1 1
28.11 odd 6 448.2.i.e.65.1 2
28.23 odd 6 448.2.i.e.193.1 2
28.27 even 2 3136.2.a.v.1.1 1
40.3 even 4 4900.2.e.i.2549.2 2
40.19 odd 2 4900.2.a.g.1.1 1
40.27 even 4 4900.2.e.i.2549.1 2
56.3 even 6 196.2.e.a.177.1 2
56.5 odd 6 784.2.i.d.753.1 2
56.11 odd 6 28.2.e.a.9.1 2
56.13 odd 2 784.2.a.g.1.1 1
56.19 even 6 196.2.e.a.165.1 2
56.27 even 2 196.2.a.a.1.1 1
56.37 even 6 112.2.i.b.81.1 2
56.45 odd 6 784.2.i.d.177.1 2
56.51 odd 6 28.2.e.a.25.1 yes 2
56.53 even 6 112.2.i.b.65.1 2
168.11 even 6 252.2.k.c.37.1 2
168.53 odd 6 1008.2.s.p.289.1 2
168.59 odd 6 1764.2.k.b.1549.1 2
168.83 odd 2 1764.2.a.j.1.1 1
168.107 even 6 252.2.k.c.109.1 2
168.125 even 2 7056.2.a.bw.1.1 1
168.131 odd 6 1764.2.k.b.361.1 2
168.149 odd 6 1008.2.s.p.865.1 2
280.27 odd 4 4900.2.e.h.2549.2 2
280.67 even 12 700.2.r.b.149.1 4
280.83 odd 4 4900.2.e.h.2549.1 2
280.107 even 12 700.2.r.b.249.2 4
280.123 even 12 700.2.r.b.149.2 4
280.139 even 2 4900.2.a.n.1.1 1
280.163 even 12 700.2.r.b.249.1 4
280.179 odd 6 700.2.i.c.401.1 2
280.219 odd 6 700.2.i.c.501.1 2
504.11 even 6 2268.2.i.h.2053.1 2
504.67 odd 6 2268.2.l.h.541.1 2
504.275 even 6 2268.2.i.h.865.1 2
504.331 odd 6 2268.2.l.h.109.1 2
504.347 even 6 2268.2.l.a.541.1 2
504.403 odd 6 2268.2.i.a.2053.1 2
504.443 even 6 2268.2.l.a.109.1 2
504.499 odd 6 2268.2.i.a.865.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
28.2.e.a.9.1 2 56.11 odd 6
28.2.e.a.25.1 yes 2 56.51 odd 6
112.2.i.b.65.1 2 56.53 even 6
112.2.i.b.81.1 2 56.37 even 6
196.2.a.a.1.1 1 56.27 even 2
196.2.a.b.1.1 1 8.3 odd 2
196.2.e.a.165.1 2 56.19 even 6
196.2.e.a.177.1 2 56.3 even 6
252.2.k.c.37.1 2 168.11 even 6
252.2.k.c.109.1 2 168.107 even 6
448.2.i.c.65.1 2 7.4 even 3
448.2.i.c.193.1 2 7.2 even 3
448.2.i.e.65.1 2 28.11 odd 6
448.2.i.e.193.1 2 28.23 odd 6
700.2.i.c.401.1 2 280.179 odd 6
700.2.i.c.501.1 2 280.219 odd 6
700.2.r.b.149.1 4 280.67 even 12
700.2.r.b.149.2 4 280.123 even 12
700.2.r.b.249.1 4 280.163 even 12
700.2.r.b.249.2 4 280.107 even 12
784.2.a.d.1.1 1 8.5 even 2
784.2.a.g.1.1 1 56.13 odd 2
784.2.i.d.177.1 2 56.45 odd 6
784.2.i.d.753.1 2 56.5 odd 6
1008.2.s.p.289.1 2 168.53 odd 6
1008.2.s.p.865.1 2 168.149 odd 6
1764.2.a.a.1.1 1 24.11 even 2
1764.2.a.j.1.1 1 168.83 odd 2
1764.2.k.b.361.1 2 168.131 odd 6
1764.2.k.b.1549.1 2 168.59 odd 6
2268.2.i.a.865.1 2 504.499 odd 6
2268.2.i.a.2053.1 2 504.403 odd 6
2268.2.i.h.865.1 2 504.275 even 6
2268.2.i.h.2053.1 2 504.11 even 6
2268.2.l.a.109.1 2 504.443 even 6
2268.2.l.a.541.1 2 504.347 even 6
2268.2.l.h.109.1 2 504.331 odd 6
2268.2.l.h.541.1 2 504.67 odd 6
3136.2.a.h.1.1 1 4.3 odd 2
3136.2.a.k.1.1 1 7.6 odd 2
3136.2.a.s.1.1 1 1.1 even 1 trivial
3136.2.a.v.1.1 1 28.27 even 2
4900.2.a.g.1.1 1 40.19 odd 2
4900.2.a.n.1.1 1 280.139 even 2
4900.2.e.h.2549.1 2 280.83 odd 4
4900.2.e.h.2549.2 2 280.27 odd 4
4900.2.e.i.2549.1 2 40.27 even 4
4900.2.e.i.2549.2 2 40.3 even 4
7056.2.a.f.1.1 1 24.5 odd 2
7056.2.a.bw.1.1 1 168.125 even 2