Properties

Label 3136.2.a.bm.1.2
Level $3136$
Weight $2$
Character 3136.1
Self dual yes
Analytic conductor $25.041$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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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: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 98)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 3136.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.41421 q^{3} +2.82843 q^{5} -1.00000 q^{9} +O(q^{10})\) \(q+1.41421 q^{3} +2.82843 q^{5} -1.00000 q^{9} -2.00000 q^{11} +4.00000 q^{15} +1.41421 q^{17} +7.07107 q^{19} +4.00000 q^{23} +3.00000 q^{25} -5.65685 q^{27} -2.00000 q^{29} +8.48528 q^{31} -2.82843 q^{33} -10.0000 q^{37} +9.89949 q^{41} +2.00000 q^{43} -2.82843 q^{45} +2.82843 q^{47} +2.00000 q^{51} +2.00000 q^{53} -5.65685 q^{55} +10.0000 q^{57} +1.41421 q^{59} +2.82843 q^{61} +12.0000 q^{67} +5.65685 q^{69} +12.0000 q^{71} +1.41421 q^{73} +4.24264 q^{75} +4.00000 q^{79} -5.00000 q^{81} -9.89949 q^{83} +4.00000 q^{85} -2.82843 q^{87} +7.07107 q^{89} +12.0000 q^{93} +20.0000 q^{95} -9.89949 q^{97} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{9} + O(q^{10}) \) \( 2 q - 2 q^{9} - 4 q^{11} + 8 q^{15} + 8 q^{23} + 6 q^{25} - 4 q^{29} - 20 q^{37} + 4 q^{43} + 4 q^{51} + 4 q^{53} + 20 q^{57} + 24 q^{67} + 24 q^{71} + 8 q^{79} - 10 q^{81} + 8 q^{85} + 24 q^{93} + 40 q^{95} + 4 q^{99} + O(q^{100}) \)

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
\(3\) 1.41421 0.816497 0.408248 0.912871i \(-0.366140\pi\)
0.408248 + 0.912871i \(0.366140\pi\)
\(4\) 0 0
\(5\) 2.82843 1.26491 0.632456 0.774597i \(-0.282047\pi\)
0.632456 + 0.774597i \(0.282047\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 4.00000 1.03280
\(16\) 0 0
\(17\) 1.41421 0.342997 0.171499 0.985184i \(-0.445139\pi\)
0.171499 + 0.985184i \(0.445139\pi\)
\(18\) 0 0
\(19\) 7.07107 1.62221 0.811107 0.584898i \(-0.198865\pi\)
0.811107 + 0.584898i \(0.198865\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 0 0
\(25\) 3.00000 0.600000
\(26\) 0 0
\(27\) −5.65685 −1.08866
\(28\) 0 0
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) 8.48528 1.52400 0.762001 0.647576i \(-0.224217\pi\)
0.762001 + 0.647576i \(0.224217\pi\)
\(32\) 0 0
\(33\) −2.82843 −0.492366
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −10.0000 −1.64399 −0.821995 0.569495i \(-0.807139\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 9.89949 1.54604 0.773021 0.634381i \(-0.218745\pi\)
0.773021 + 0.634381i \(0.218745\pi\)
\(42\) 0 0
\(43\) 2.00000 0.304997 0.152499 0.988304i \(-0.451268\pi\)
0.152499 + 0.988304i \(0.451268\pi\)
\(44\) 0 0
\(45\) −2.82843 −0.421637
\(46\) 0 0
\(47\) 2.82843 0.412568 0.206284 0.978492i \(-0.433863\pi\)
0.206284 + 0.978492i \(0.433863\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 2.00000 0.280056
\(52\) 0 0
\(53\) 2.00000 0.274721 0.137361 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) 0 0
\(55\) −5.65685 −0.762770
\(56\) 0 0
\(57\) 10.0000 1.32453
\(58\) 0 0
\(59\) 1.41421 0.184115 0.0920575 0.995754i \(-0.470656\pi\)
0.0920575 + 0.995754i \(0.470656\pi\)
\(60\) 0 0
\(61\) 2.82843 0.362143 0.181071 0.983470i \(-0.442043\pi\)
0.181071 + 0.983470i \(0.442043\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 12.0000 1.46603 0.733017 0.680211i \(-0.238112\pi\)
0.733017 + 0.680211i \(0.238112\pi\)
\(68\) 0 0
\(69\) 5.65685 0.681005
\(70\) 0 0
\(71\) 12.0000 1.42414 0.712069 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 0 0
\(73\) 1.41421 0.165521 0.0827606 0.996569i \(-0.473626\pi\)
0.0827606 + 0.996569i \(0.473626\pi\)
\(74\) 0 0
\(75\) 4.24264 0.489898
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 4.00000 0.450035 0.225018 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 0 0
\(81\) −5.00000 −0.555556
\(82\) 0 0
\(83\) −9.89949 −1.08661 −0.543305 0.839535i \(-0.682827\pi\)
−0.543305 + 0.839535i \(0.682827\pi\)
\(84\) 0 0
\(85\) 4.00000 0.433861
\(86\) 0 0
\(87\) −2.82843 −0.303239
\(88\) 0 0
\(89\) 7.07107 0.749532 0.374766 0.927119i \(-0.377723\pi\)
0.374766 + 0.927119i \(0.377723\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 12.0000 1.24434
\(94\) 0 0
\(95\) 20.0000 2.05196
\(96\) 0 0
\(97\) −9.89949 −1.00514 −0.502571 0.864536i \(-0.667612\pi\)
−0.502571 + 0.864536i \(0.667612\pi\)
\(98\) 0 0
\(99\) 2.00000 0.201008
\(100\) 0 0
\(101\) 8.48528 0.844317 0.422159 0.906522i \(-0.361273\pi\)
0.422159 + 0.906522i \(0.361273\pi\)
\(102\) 0 0
\(103\) 2.82843 0.278693 0.139347 0.990244i \(-0.455500\pi\)
0.139347 + 0.990244i \(0.455500\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −4.00000 −0.386695 −0.193347 0.981130i \(-0.561934\pi\)
−0.193347 + 0.981130i \(0.561934\pi\)
\(108\) 0 0
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 0 0
\(111\) −14.1421 −1.34231
\(112\) 0 0
\(113\) −12.0000 −1.12887 −0.564433 0.825479i \(-0.690905\pi\)
−0.564433 + 0.825479i \(0.690905\pi\)
\(114\) 0 0
\(115\) 11.3137 1.05501
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 0 0
\(123\) 14.0000 1.26234
\(124\) 0 0
\(125\) −5.65685 −0.505964
\(126\) 0 0
\(127\) −16.0000 −1.41977 −0.709885 0.704317i \(-0.751253\pi\)
−0.709885 + 0.704317i \(0.751253\pi\)
\(128\) 0 0
\(129\) 2.82843 0.249029
\(130\) 0 0
\(131\) −12.7279 −1.11204 −0.556022 0.831168i \(-0.687673\pi\)
−0.556022 + 0.831168i \(0.687673\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −16.0000 −1.37706
\(136\) 0 0
\(137\) 12.0000 1.02523 0.512615 0.858619i \(-0.328677\pi\)
0.512615 + 0.858619i \(0.328677\pi\)
\(138\) 0 0
\(139\) −9.89949 −0.839664 −0.419832 0.907602i \(-0.637911\pi\)
−0.419832 + 0.907602i \(0.637911\pi\)
\(140\) 0 0
\(141\) 4.00000 0.336861
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −5.65685 −0.469776
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −10.0000 −0.819232 −0.409616 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) 0 0
\(151\) 16.0000 1.30206 0.651031 0.759051i \(-0.274337\pi\)
0.651031 + 0.759051i \(0.274337\pi\)
\(152\) 0 0
\(153\) −1.41421 −0.114332
\(154\) 0 0
\(155\) 24.0000 1.92773
\(156\) 0 0
\(157\) −11.3137 −0.902932 −0.451466 0.892288i \(-0.649099\pi\)
−0.451466 + 0.892288i \(0.649099\pi\)
\(158\) 0 0
\(159\) 2.82843 0.224309
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 10.0000 0.783260 0.391630 0.920123i \(-0.371911\pi\)
0.391630 + 0.920123i \(0.371911\pi\)
\(164\) 0 0
\(165\) −8.00000 −0.622799
\(166\) 0 0
\(167\) −19.7990 −1.53209 −0.766046 0.642786i \(-0.777779\pi\)
−0.766046 + 0.642786i \(0.777779\pi\)
\(168\) 0 0
\(169\) −13.0000 −1.00000
\(170\) 0 0
\(171\) −7.07107 −0.540738
\(172\) 0 0
\(173\) −16.9706 −1.29025 −0.645124 0.764078i \(-0.723194\pi\)
−0.645124 + 0.764078i \(0.723194\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 2.00000 0.150329
\(178\) 0 0
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(182\) 0 0
\(183\) 4.00000 0.295689
\(184\) 0 0
\(185\) −28.2843 −2.07950
\(186\) 0 0
\(187\) −2.82843 −0.206835
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 4.00000 0.289430 0.144715 0.989473i \(-0.453773\pi\)
0.144715 + 0.989473i \(0.453773\pi\)
\(192\) 0 0
\(193\) −16.0000 −1.15171 −0.575853 0.817554i \(-0.695330\pi\)
−0.575853 + 0.817554i \(0.695330\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −2.00000 −0.142494 −0.0712470 0.997459i \(-0.522698\pi\)
−0.0712470 + 0.997459i \(0.522698\pi\)
\(198\) 0 0
\(199\) 8.48528 0.601506 0.300753 0.953702i \(-0.402762\pi\)
0.300753 + 0.953702i \(0.402762\pi\)
\(200\) 0 0
\(201\) 16.9706 1.19701
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 28.0000 1.95560
\(206\) 0 0
\(207\) −4.00000 −0.278019
\(208\) 0 0
\(209\) −14.1421 −0.978232
\(210\) 0 0
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) 0 0
\(213\) 16.9706 1.16280
\(214\) 0 0
\(215\) 5.65685 0.385794
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 2.00000 0.135147
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 0 0
\(225\) −3.00000 −0.200000
\(226\) 0 0
\(227\) 21.2132 1.40797 0.703985 0.710215i \(-0.251402\pi\)
0.703985 + 0.710215i \(0.251402\pi\)
\(228\) 0 0
\(229\) −16.9706 −1.12145 −0.560723 0.828003i \(-0.689477\pi\)
−0.560723 + 0.828003i \(0.689477\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 24.0000 1.57229 0.786146 0.618041i \(-0.212073\pi\)
0.786146 + 0.618041i \(0.212073\pi\)
\(234\) 0 0
\(235\) 8.00000 0.521862
\(236\) 0 0
\(237\) 5.65685 0.367452
\(238\) 0 0
\(239\) 12.0000 0.776215 0.388108 0.921614i \(-0.373129\pi\)
0.388108 + 0.921614i \(0.373129\pi\)
\(240\) 0 0
\(241\) 21.2132 1.36646 0.683231 0.730202i \(-0.260574\pi\)
0.683231 + 0.730202i \(0.260574\pi\)
\(242\) 0 0
\(243\) 9.89949 0.635053
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −14.0000 −0.887214
\(250\) 0 0
\(251\) −9.89949 −0.624851 −0.312425 0.949942i \(-0.601141\pi\)
−0.312425 + 0.949942i \(0.601141\pi\)
\(252\) 0 0
\(253\) −8.00000 −0.502956
\(254\) 0 0
\(255\) 5.65685 0.354246
\(256\) 0 0
\(257\) −12.7279 −0.793946 −0.396973 0.917830i \(-0.629939\pi\)
−0.396973 + 0.917830i \(0.629939\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 2.00000 0.123797
\(262\) 0 0
\(263\) −12.0000 −0.739952 −0.369976 0.929041i \(-0.620634\pi\)
−0.369976 + 0.929041i \(0.620634\pi\)
\(264\) 0 0
\(265\) 5.65685 0.347498
\(266\) 0 0
\(267\) 10.0000 0.611990
\(268\) 0 0
\(269\) −11.3137 −0.689809 −0.344904 0.938638i \(-0.612089\pi\)
−0.344904 + 0.938638i \(0.612089\pi\)
\(270\) 0 0
\(271\) 22.6274 1.37452 0.687259 0.726413i \(-0.258814\pi\)
0.687259 + 0.726413i \(0.258814\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −6.00000 −0.361814
\(276\) 0 0
\(277\) 2.00000 0.120168 0.0600842 0.998193i \(-0.480863\pi\)
0.0600842 + 0.998193i \(0.480863\pi\)
\(278\) 0 0
\(279\) −8.48528 −0.508001
\(280\) 0 0
\(281\) 16.0000 0.954480 0.477240 0.878773i \(-0.341637\pi\)
0.477240 + 0.878773i \(0.341637\pi\)
\(282\) 0 0
\(283\) 1.41421 0.0840663 0.0420331 0.999116i \(-0.486616\pi\)
0.0420331 + 0.999116i \(0.486616\pi\)
\(284\) 0 0
\(285\) 28.2843 1.67542
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −15.0000 −0.882353
\(290\) 0 0
\(291\) −14.0000 −0.820695
\(292\) 0 0
\(293\) −19.7990 −1.15667 −0.578335 0.815800i \(-0.696297\pi\)
−0.578335 + 0.815800i \(0.696297\pi\)
\(294\) 0 0
\(295\) 4.00000 0.232889
\(296\) 0 0
\(297\) 11.3137 0.656488
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 12.0000 0.689382
\(304\) 0 0
\(305\) 8.00000 0.458079
\(306\) 0 0
\(307\) 9.89949 0.564994 0.282497 0.959268i \(-0.408837\pi\)
0.282497 + 0.959268i \(0.408837\pi\)
\(308\) 0 0
\(309\) 4.00000 0.227552
\(310\) 0 0
\(311\) −11.3137 −0.641542 −0.320771 0.947157i \(-0.603942\pi\)
−0.320771 + 0.947157i \(0.603942\pi\)
\(312\) 0 0
\(313\) −12.7279 −0.719425 −0.359712 0.933063i \(-0.617125\pi\)
−0.359712 + 0.933063i \(0.617125\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −10.0000 −0.561656 −0.280828 0.959758i \(-0.590609\pi\)
−0.280828 + 0.959758i \(0.590609\pi\)
\(318\) 0 0
\(319\) 4.00000 0.223957
\(320\) 0 0
\(321\) −5.65685 −0.315735
\(322\) 0 0
\(323\) 10.0000 0.556415
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 2.82843 0.156412
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 10.0000 0.549650 0.274825 0.961494i \(-0.411380\pi\)
0.274825 + 0.961494i \(0.411380\pi\)
\(332\) 0 0
\(333\) 10.0000 0.547997
\(334\) 0 0
\(335\) 33.9411 1.85440
\(336\) 0 0
\(337\) 2.00000 0.108947 0.0544735 0.998515i \(-0.482652\pi\)
0.0544735 + 0.998515i \(0.482652\pi\)
\(338\) 0 0
\(339\) −16.9706 −0.921714
\(340\) 0 0
\(341\) −16.9706 −0.919007
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 16.0000 0.861411
\(346\) 0 0
\(347\) −30.0000 −1.61048 −0.805242 0.592946i \(-0.797965\pi\)
−0.805242 + 0.592946i \(0.797965\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 1.41421 0.0752710 0.0376355 0.999292i \(-0.488017\pi\)
0.0376355 + 0.999292i \(0.488017\pi\)
\(354\) 0 0
\(355\) 33.9411 1.80141
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 32.0000 1.68890 0.844448 0.535638i \(-0.179929\pi\)
0.844448 + 0.535638i \(0.179929\pi\)
\(360\) 0 0
\(361\) 31.0000 1.63158
\(362\) 0 0
\(363\) −9.89949 −0.519589
\(364\) 0 0
\(365\) 4.00000 0.209370
\(366\) 0 0
\(367\) 28.2843 1.47643 0.738213 0.674567i \(-0.235670\pi\)
0.738213 + 0.674567i \(0.235670\pi\)
\(368\) 0 0
\(369\) −9.89949 −0.515347
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −10.0000 −0.517780 −0.258890 0.965907i \(-0.583357\pi\)
−0.258890 + 0.965907i \(0.583357\pi\)
\(374\) 0 0
\(375\) −8.00000 −0.413118
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −26.0000 −1.33553 −0.667765 0.744372i \(-0.732749\pi\)
−0.667765 + 0.744372i \(0.732749\pi\)
\(380\) 0 0
\(381\) −22.6274 −1.15924
\(382\) 0 0
\(383\) −36.7696 −1.87884 −0.939418 0.342773i \(-0.888634\pi\)
−0.939418 + 0.342773i \(0.888634\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −2.00000 −0.101666
\(388\) 0 0
\(389\) −26.0000 −1.31825 −0.659126 0.752032i \(-0.729074\pi\)
−0.659126 + 0.752032i \(0.729074\pi\)
\(390\) 0 0
\(391\) 5.65685 0.286079
\(392\) 0 0
\(393\) −18.0000 −0.907980
\(394\) 0 0
\(395\) 11.3137 0.569254
\(396\) 0 0
\(397\) 22.6274 1.13564 0.567819 0.823154i \(-0.307787\pi\)
0.567819 + 0.823154i \(0.307787\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −18.0000 −0.898877 −0.449439 0.893311i \(-0.648376\pi\)
−0.449439 + 0.893311i \(0.648376\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −14.1421 −0.702728
\(406\) 0 0
\(407\) 20.0000 0.991363
\(408\) 0 0
\(409\) −38.1838 −1.88807 −0.944033 0.329851i \(-0.893001\pi\)
−0.944033 + 0.329851i \(0.893001\pi\)
\(410\) 0 0
\(411\) 16.9706 0.837096
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −28.0000 −1.37447
\(416\) 0 0
\(417\) −14.0000 −0.685583
\(418\) 0 0
\(419\) 9.89949 0.483622 0.241811 0.970323i \(-0.422259\pi\)
0.241811 + 0.970323i \(0.422259\pi\)
\(420\) 0 0
\(421\) −30.0000 −1.46211 −0.731055 0.682318i \(-0.760972\pi\)
−0.731055 + 0.682318i \(0.760972\pi\)
\(422\) 0 0
\(423\) −2.82843 −0.137523
\(424\) 0 0
\(425\) 4.24264 0.205798
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −12.0000 −0.578020 −0.289010 0.957326i \(-0.593326\pi\)
−0.289010 + 0.957326i \(0.593326\pi\)
\(432\) 0 0
\(433\) 29.6985 1.42722 0.713609 0.700544i \(-0.247059\pi\)
0.713609 + 0.700544i \(0.247059\pi\)
\(434\) 0 0
\(435\) −8.00000 −0.383571
\(436\) 0 0
\(437\) 28.2843 1.35302
\(438\) 0 0
\(439\) −16.9706 −0.809961 −0.404980 0.914325i \(-0.632722\pi\)
−0.404980 + 0.914325i \(0.632722\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −4.00000 −0.190046 −0.0950229 0.995475i \(-0.530292\pi\)
−0.0950229 + 0.995475i \(0.530292\pi\)
\(444\) 0 0
\(445\) 20.0000 0.948091
\(446\) 0 0
\(447\) −14.1421 −0.668900
\(448\) 0 0
\(449\) 30.0000 1.41579 0.707894 0.706319i \(-0.249646\pi\)
0.707894 + 0.706319i \(0.249646\pi\)
\(450\) 0 0
\(451\) −19.7990 −0.932298
\(452\) 0 0
\(453\) 22.6274 1.06313
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 24.0000 1.12267 0.561336 0.827588i \(-0.310287\pi\)
0.561336 + 0.827588i \(0.310287\pi\)
\(458\) 0 0
\(459\) −8.00000 −0.373408
\(460\) 0 0
\(461\) 39.5980 1.84426 0.922131 0.386878i \(-0.126447\pi\)
0.922131 + 0.386878i \(0.126447\pi\)
\(462\) 0 0
\(463\) −16.0000 −0.743583 −0.371792 0.928316i \(-0.621256\pi\)
−0.371792 + 0.928316i \(0.621256\pi\)
\(464\) 0 0
\(465\) 33.9411 1.57398
\(466\) 0 0
\(467\) −32.5269 −1.50517 −0.752583 0.658497i \(-0.771192\pi\)
−0.752583 + 0.658497i \(0.771192\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −16.0000 −0.737241
\(472\) 0 0
\(473\) −4.00000 −0.183920
\(474\) 0 0
\(475\) 21.2132 0.973329
\(476\) 0 0
\(477\) −2.00000 −0.0915737
\(478\) 0 0
\(479\) −31.1127 −1.42158 −0.710788 0.703407i \(-0.751661\pi\)
−0.710788 + 0.703407i \(0.751661\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −28.0000 −1.27141
\(486\) 0 0
\(487\) −12.0000 −0.543772 −0.271886 0.962329i \(-0.587647\pi\)
−0.271886 + 0.962329i \(0.587647\pi\)
\(488\) 0 0
\(489\) 14.1421 0.639529
\(490\) 0 0
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) 0 0
\(493\) −2.82843 −0.127386
\(494\) 0 0
\(495\) 5.65685 0.254257
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −4.00000 −0.179065 −0.0895323 0.995984i \(-0.528537\pi\)
−0.0895323 + 0.995984i \(0.528537\pi\)
\(500\) 0 0
\(501\) −28.0000 −1.25095
\(502\) 0 0
\(503\) 39.5980 1.76559 0.882793 0.469762i \(-0.155660\pi\)
0.882793 + 0.469762i \(0.155660\pi\)
\(504\) 0 0
\(505\) 24.0000 1.06799
\(506\) 0 0
\(507\) −18.3848 −0.816497
\(508\) 0 0
\(509\) 22.6274 1.00294 0.501471 0.865174i \(-0.332792\pi\)
0.501471 + 0.865174i \(0.332792\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −40.0000 −1.76604
\(514\) 0 0
\(515\) 8.00000 0.352522
\(516\) 0 0
\(517\) −5.65685 −0.248788
\(518\) 0 0
\(519\) −24.0000 −1.05348
\(520\) 0 0
\(521\) 1.41421 0.0619578 0.0309789 0.999520i \(-0.490138\pi\)
0.0309789 + 0.999520i \(0.490138\pi\)
\(522\) 0 0
\(523\) −12.7279 −0.556553 −0.278277 0.960501i \(-0.589763\pi\)
−0.278277 + 0.960501i \(0.589763\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 12.0000 0.522728
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) −1.41421 −0.0613716
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −11.3137 −0.489134
\(536\) 0 0
\(537\) 16.9706 0.732334
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −10.0000 −0.429934 −0.214967 0.976621i \(-0.568964\pi\)
−0.214967 + 0.976621i \(0.568964\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 5.65685 0.242313
\(546\) 0 0
\(547\) −26.0000 −1.11168 −0.555840 0.831289i \(-0.687603\pi\)
−0.555840 + 0.831289i \(0.687603\pi\)
\(548\) 0 0
\(549\) −2.82843 −0.120714
\(550\) 0 0
\(551\) −14.1421 −0.602475
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −40.0000 −1.69791
\(556\) 0 0
\(557\) 30.0000 1.27114 0.635570 0.772043i \(-0.280765\pi\)
0.635570 + 0.772043i \(0.280765\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) −4.00000 −0.168880
\(562\) 0 0
\(563\) 1.41421 0.0596020 0.0298010 0.999556i \(-0.490513\pi\)
0.0298010 + 0.999556i \(0.490513\pi\)
\(564\) 0 0
\(565\) −33.9411 −1.42791
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 10.0000 0.419222 0.209611 0.977785i \(-0.432780\pi\)
0.209611 + 0.977785i \(0.432780\pi\)
\(570\) 0 0
\(571\) −2.00000 −0.0836974 −0.0418487 0.999124i \(-0.513325\pi\)
−0.0418487 + 0.999124i \(0.513325\pi\)
\(572\) 0 0
\(573\) 5.65685 0.236318
\(574\) 0 0
\(575\) 12.0000 0.500435
\(576\) 0 0
\(577\) 21.2132 0.883117 0.441559 0.897232i \(-0.354426\pi\)
0.441559 + 0.897232i \(0.354426\pi\)
\(578\) 0 0
\(579\) −22.6274 −0.940363
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −4.00000 −0.165663
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 29.6985 1.22579 0.612894 0.790165i \(-0.290005\pi\)
0.612894 + 0.790165i \(0.290005\pi\)
\(588\) 0 0
\(589\) 60.0000 2.47226
\(590\) 0 0
\(591\) −2.82843 −0.116346
\(592\) 0 0
\(593\) 7.07107 0.290374 0.145187 0.989404i \(-0.453622\pi\)
0.145187 + 0.989404i \(0.453622\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 12.0000 0.491127
\(598\) 0 0
\(599\) 16.0000 0.653742 0.326871 0.945069i \(-0.394006\pi\)
0.326871 + 0.945069i \(0.394006\pi\)
\(600\) 0 0
\(601\) −29.6985 −1.21143 −0.605713 0.795683i \(-0.707112\pi\)
−0.605713 + 0.795683i \(0.707112\pi\)
\(602\) 0 0
\(603\) −12.0000 −0.488678
\(604\) 0 0
\(605\) −19.7990 −0.804943
\(606\) 0 0
\(607\) −16.9706 −0.688814 −0.344407 0.938820i \(-0.611920\pi\)
−0.344407 + 0.938820i \(0.611920\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 30.0000 1.21169 0.605844 0.795583i \(-0.292835\pi\)
0.605844 + 0.795583i \(0.292835\pi\)
\(614\) 0 0
\(615\) 39.5980 1.59674
\(616\) 0 0
\(617\) −26.0000 −1.04672 −0.523360 0.852111i \(-0.675322\pi\)
−0.523360 + 0.852111i \(0.675322\pi\)
\(618\) 0 0
\(619\) −18.3848 −0.738947 −0.369473 0.929241i \(-0.620462\pi\)
−0.369473 + 0.929241i \(0.620462\pi\)
\(620\) 0 0
\(621\) −22.6274 −0.908007
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −31.0000 −1.24000
\(626\) 0 0
\(627\) −20.0000 −0.798723
\(628\) 0 0
\(629\) −14.1421 −0.563884
\(630\) 0 0
\(631\) −44.0000 −1.75161 −0.875806 0.482663i \(-0.839670\pi\)
−0.875806 + 0.482663i \(0.839670\pi\)
\(632\) 0 0
\(633\) −16.9706 −0.674519
\(634\) 0 0
\(635\) −45.2548 −1.79588
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −12.0000 −0.474713
\(640\) 0 0
\(641\) 26.0000 1.02694 0.513469 0.858108i \(-0.328360\pi\)
0.513469 + 0.858108i \(0.328360\pi\)
\(642\) 0 0
\(643\) −9.89949 −0.390398 −0.195199 0.980764i \(-0.562535\pi\)
−0.195199 + 0.980764i \(0.562535\pi\)
\(644\) 0 0
\(645\) 8.00000 0.315000
\(646\) 0 0
\(647\) 8.48528 0.333591 0.166795 0.985992i \(-0.446658\pi\)
0.166795 + 0.985992i \(0.446658\pi\)
\(648\) 0 0
\(649\) −2.82843 −0.111025
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 18.0000 0.704394 0.352197 0.935926i \(-0.385435\pi\)
0.352197 + 0.935926i \(0.385435\pi\)
\(654\) 0 0
\(655\) −36.0000 −1.40664
\(656\) 0 0
\(657\) −1.41421 −0.0551737
\(658\) 0 0
\(659\) 30.0000 1.16863 0.584317 0.811525i \(-0.301362\pi\)
0.584317 + 0.811525i \(0.301362\pi\)
\(660\) 0 0
\(661\) 8.48528 0.330039 0.165020 0.986290i \(-0.447231\pi\)
0.165020 + 0.986290i \(0.447231\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −8.00000 −0.309761
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −5.65685 −0.218380
\(672\) 0 0
\(673\) −12.0000 −0.462566 −0.231283 0.972887i \(-0.574292\pi\)
−0.231283 + 0.972887i \(0.574292\pi\)
\(674\) 0 0
\(675\) −16.9706 −0.653197
\(676\) 0 0
\(677\) −16.9706 −0.652232 −0.326116 0.945330i \(-0.605740\pi\)
−0.326116 + 0.945330i \(0.605740\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 30.0000 1.14960
\(682\) 0 0
\(683\) 12.0000 0.459167 0.229584 0.973289i \(-0.426264\pi\)
0.229584 + 0.973289i \(0.426264\pi\)
\(684\) 0 0
\(685\) 33.9411 1.29682
\(686\) 0 0
\(687\) −24.0000 −0.915657
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −12.7279 −0.484193 −0.242096 0.970252i \(-0.577835\pi\)
−0.242096 + 0.970252i \(0.577835\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −28.0000 −1.06210
\(696\) 0 0
\(697\) 14.0000 0.530288
\(698\) 0 0
\(699\) 33.9411 1.28377
\(700\) 0 0
\(701\) −30.0000 −1.13308 −0.566542 0.824033i \(-0.691719\pi\)
−0.566542 + 0.824033i \(0.691719\pi\)
\(702\) 0 0
\(703\) −70.7107 −2.66690
\(704\) 0 0
\(705\) 11.3137 0.426099
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −10.0000 −0.375558 −0.187779 0.982211i \(-0.560129\pi\)
−0.187779 + 0.982211i \(0.560129\pi\)
\(710\) 0 0
\(711\) −4.00000 −0.150012
\(712\) 0 0
\(713\) 33.9411 1.27111
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 16.9706 0.633777
\(718\) 0 0
\(719\) 2.82843 0.105483 0.0527413 0.998608i \(-0.483204\pi\)
0.0527413 + 0.998608i \(0.483204\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 30.0000 1.11571
\(724\) 0 0
\(725\) −6.00000 −0.222834
\(726\) 0 0
\(727\) −19.7990 −0.734304 −0.367152 0.930161i \(-0.619667\pi\)
−0.367152 + 0.930161i \(0.619667\pi\)
\(728\) 0 0
\(729\) 29.0000 1.07407
\(730\) 0 0
\(731\) 2.82843 0.104613
\(732\) 0 0
\(733\) 42.4264 1.56706 0.783528 0.621357i \(-0.213418\pi\)
0.783528 + 0.621357i \(0.213418\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −24.0000 −0.884051
\(738\) 0 0
\(739\) −30.0000 −1.10357 −0.551784 0.833987i \(-0.686053\pi\)
−0.551784 + 0.833987i \(0.686053\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −16.0000 −0.586983 −0.293492 0.955962i \(-0.594817\pi\)
−0.293492 + 0.955962i \(0.594817\pi\)
\(744\) 0 0
\(745\) −28.2843 −1.03626
\(746\) 0 0
\(747\) 9.89949 0.362204
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 4.00000 0.145962 0.0729810 0.997333i \(-0.476749\pi\)
0.0729810 + 0.997333i \(0.476749\pi\)
\(752\) 0 0
\(753\) −14.0000 −0.510188
\(754\) 0 0
\(755\) 45.2548 1.64699
\(756\) 0 0
\(757\) −2.00000 −0.0726912 −0.0363456 0.999339i \(-0.511572\pi\)
−0.0363456 + 0.999339i \(0.511572\pi\)
\(758\) 0 0
\(759\) −11.3137 −0.410662
\(760\) 0 0
\(761\) 7.07107 0.256326 0.128163 0.991753i \(-0.459092\pi\)
0.128163 + 0.991753i \(0.459092\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −4.00000 −0.144620
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −29.6985 −1.07095 −0.535477 0.844550i \(-0.679868\pi\)
−0.535477 + 0.844550i \(0.679868\pi\)
\(770\) 0 0
\(771\) −18.0000 −0.648254
\(772\) 0 0
\(773\) 48.0833 1.72943 0.864717 0.502259i \(-0.167498\pi\)
0.864717 + 0.502259i \(0.167498\pi\)
\(774\) 0 0
\(775\) 25.4558 0.914401
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 70.0000 2.50801
\(780\) 0 0
\(781\) −24.0000 −0.858788
\(782\) 0 0
\(783\) 11.3137 0.404319
\(784\) 0 0
\(785\) −32.0000 −1.14213
\(786\) 0 0
\(787\) 1.41421 0.0504113 0.0252056 0.999682i \(-0.491976\pi\)
0.0252056 + 0.999682i \(0.491976\pi\)
\(788\) 0 0
\(789\) −16.9706 −0.604168
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 8.00000 0.283731
\(796\) 0 0
\(797\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(798\) 0 0
\(799\) 4.00000 0.141510
\(800\) 0 0
\(801\) −7.07107 −0.249844
\(802\) 0 0
\(803\) −2.82843 −0.0998130
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −16.0000 −0.563227
\(808\) 0 0
\(809\) −16.0000 −0.562530 −0.281265 0.959630i \(-0.590754\pi\)
−0.281265 + 0.959630i \(0.590754\pi\)
\(810\) 0 0
\(811\) 29.6985 1.04285 0.521427 0.853296i \(-0.325400\pi\)
0.521427 + 0.853296i \(0.325400\pi\)
\(812\) 0 0
\(813\) 32.0000 1.12229
\(814\) 0 0
\(815\) 28.2843 0.990755
\(816\) 0 0
\(817\) 14.1421 0.494771
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 18.0000 0.628204 0.314102 0.949389i \(-0.398297\pi\)
0.314102 + 0.949389i \(0.398297\pi\)
\(822\) 0 0
\(823\) −40.0000 −1.39431 −0.697156 0.716919i \(-0.745552\pi\)
−0.697156 + 0.716919i \(0.745552\pi\)
\(824\) 0 0
\(825\) −8.48528 −0.295420
\(826\) 0 0
\(827\) −12.0000 −0.417281 −0.208640 0.977992i \(-0.566904\pi\)
−0.208640 + 0.977992i \(0.566904\pi\)
\(828\) 0 0
\(829\) −31.1127 −1.08059 −0.540294 0.841476i \(-0.681687\pi\)
−0.540294 + 0.841476i \(0.681687\pi\)
\(830\) 0 0
\(831\) 2.82843 0.0981170
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −56.0000 −1.93796
\(836\) 0 0
\(837\) −48.0000 −1.65912
\(838\) 0 0
\(839\) 19.7990 0.683537 0.341769 0.939784i \(-0.388974\pi\)
0.341769 + 0.939784i \(0.388974\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 0 0
\(843\) 22.6274 0.779330
\(844\) 0 0
\(845\) −36.7696 −1.26491
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 2.00000 0.0686398
\(850\) 0 0
\(851\) −40.0000 −1.37118
\(852\) 0 0
\(853\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(854\) 0 0
\(855\) −20.0000 −0.683986
\(856\) 0 0
\(857\) −18.3848 −0.628012 −0.314006 0.949421i \(-0.601671\pi\)
−0.314006 + 0.949421i \(0.601671\pi\)
\(858\) 0 0
\(859\) 26.8701 0.916795 0.458397 0.888747i \(-0.348424\pi\)
0.458397 + 0.888747i \(0.348424\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 4.00000 0.136162 0.0680808 0.997680i \(-0.478312\pi\)
0.0680808 + 0.997680i \(0.478312\pi\)
\(864\) 0 0
\(865\) −48.0000 −1.63205
\(866\) 0 0
\(867\) −21.2132 −0.720438
\(868\) 0 0
\(869\) −8.00000 −0.271381
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 9.89949 0.335047
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 46.0000 1.55331 0.776655 0.629926i \(-0.216915\pi\)
0.776655 + 0.629926i \(0.216915\pi\)
\(878\) 0 0
\(879\) −28.0000 −0.944417
\(880\) 0 0
\(881\) −29.6985 −1.00057 −0.500284 0.865862i \(-0.666771\pi\)
−0.500284 + 0.865862i \(0.666771\pi\)
\(882\) 0 0
\(883\) 44.0000 1.48072 0.740359 0.672212i \(-0.234656\pi\)
0.740359 + 0.672212i \(0.234656\pi\)
\(884\) 0 0
\(885\) 5.65685 0.190153
\(886\) 0 0
\(887\) −36.7696 −1.23460 −0.617300 0.786728i \(-0.711774\pi\)
−0.617300 + 0.786728i \(0.711774\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 10.0000 0.335013
\(892\) 0 0
\(893\) 20.0000 0.669274
\(894\) 0 0
\(895\) 33.9411 1.13453
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −16.9706 −0.566000
\(900\) 0 0
\(901\) 2.82843 0.0942286
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −44.0000 −1.46100 −0.730498 0.682915i \(-0.760712\pi\)
−0.730498 + 0.682915i \(0.760712\pi\)
\(908\) 0 0
\(909\) −8.48528 −0.281439
\(910\) 0 0
\(911\) 40.0000 1.32526 0.662630 0.748947i \(-0.269440\pi\)
0.662630 + 0.748947i \(0.269440\pi\)
\(912\) 0 0
\(913\) 19.7990 0.655251
\(914\) 0 0
\(915\) 11.3137 0.374020
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 32.0000 1.05558 0.527791 0.849374i \(-0.323020\pi\)
0.527791 + 0.849374i \(0.323020\pi\)
\(920\) 0 0
\(921\) 14.0000 0.461316
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −30.0000 −0.986394
\(926\) 0 0
\(927\) −2.82843 −0.0928977
\(928\) 0 0
\(929\) −32.5269 −1.06717 −0.533587 0.845745i \(-0.679156\pi\)
−0.533587 + 0.845745i \(0.679156\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −16.0000 −0.523816
\(934\) 0 0
\(935\) −8.00000 −0.261628
\(936\) 0 0
\(937\) −9.89949 −0.323402 −0.161701 0.986840i \(-0.551698\pi\)
−0.161701 + 0.986840i \(0.551698\pi\)
\(938\) 0 0
\(939\) −18.0000 −0.587408
\(940\) 0 0
\(941\) −31.1127 −1.01424 −0.507122 0.861874i \(-0.669291\pi\)
−0.507122 + 0.861874i \(0.669291\pi\)
\(942\) 0 0
\(943\) 39.5980 1.28949
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −18.0000 −0.584921 −0.292461 0.956278i \(-0.594474\pi\)
−0.292461 + 0.956278i \(0.594474\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) −14.1421 −0.458590
\(952\) 0 0
\(953\) −26.0000 −0.842223 −0.421111 0.907009i \(-0.638360\pi\)
−0.421111 + 0.907009i \(0.638360\pi\)
\(954\) 0 0
\(955\) 11.3137 0.366103
\(956\) 0 0
\(957\) 5.65685 0.182860
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 41.0000 1.32258
\(962\) 0 0
\(963\) 4.00000 0.128898
\(964\) 0 0
\(965\) −45.2548 −1.45680
\(966\) 0 0
\(967\) 12.0000 0.385894 0.192947 0.981209i \(-0.438195\pi\)
0.192947 + 0.981209i \(0.438195\pi\)
\(968\) 0 0
\(969\) 14.1421 0.454311
\(970\) 0 0
\(971\) −32.5269 −1.04384 −0.521919 0.852995i \(-0.674784\pi\)
−0.521919 + 0.852995i \(0.674784\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 12.0000 0.383914 0.191957 0.981403i \(-0.438517\pi\)
0.191957 + 0.981403i \(0.438517\pi\)
\(978\) 0 0
\(979\) −14.1421 −0.451985
\(980\) 0 0
\(981\) −2.00000 −0.0638551
\(982\) 0 0
\(983\) 48.0833 1.53362 0.766809 0.641875i \(-0.221843\pi\)
0.766809 + 0.641875i \(0.221843\pi\)
\(984\) 0 0
\(985\) −5.65685 −0.180242
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 8.00000 0.254385
\(990\) 0 0
\(991\) 16.0000 0.508257 0.254128 0.967170i \(-0.418211\pi\)
0.254128 + 0.967170i \(0.418211\pi\)
\(992\) 0 0
\(993\) 14.1421 0.448787
\(994\) 0 0
\(995\) 24.0000 0.760851
\(996\) 0 0
\(997\) −31.1127 −0.985349 −0.492675 0.870214i \(-0.663981\pi\)
−0.492675 + 0.870214i \(0.663981\pi\)
\(998\) 0 0
\(999\) 56.5685 1.78975
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.bm.1.2 2
4.3 odd 2 3136.2.a.bn.1.1 2
7.6 odd 2 inner 3136.2.a.bm.1.1 2
8.3 odd 2 98.2.a.b.1.2 yes 2
8.5 even 2 784.2.a.l.1.1 2
24.5 odd 2 7056.2.a.cl.1.2 2
24.11 even 2 882.2.a.n.1.2 2
28.27 even 2 3136.2.a.bn.1.2 2
40.3 even 4 2450.2.c.v.99.2 4
40.19 odd 2 2450.2.a.bj.1.1 2
40.27 even 4 2450.2.c.v.99.3 4
56.3 even 6 98.2.c.c.79.2 4
56.5 odd 6 784.2.i.m.753.1 4
56.11 odd 6 98.2.c.c.79.1 4
56.13 odd 2 784.2.a.l.1.2 2
56.19 even 6 98.2.c.c.67.2 4
56.27 even 2 98.2.a.b.1.1 2
56.37 even 6 784.2.i.m.753.2 4
56.45 odd 6 784.2.i.m.177.1 4
56.51 odd 6 98.2.c.c.67.1 4
56.53 even 6 784.2.i.m.177.2 4
168.11 even 6 882.2.g.l.667.1 4
168.59 odd 6 882.2.g.l.667.2 4
168.83 odd 2 882.2.a.n.1.1 2
168.107 even 6 882.2.g.l.361.1 4
168.125 even 2 7056.2.a.cl.1.1 2
168.131 odd 6 882.2.g.l.361.2 4
280.27 odd 4 2450.2.c.v.99.4 4
280.83 odd 4 2450.2.c.v.99.1 4
280.139 even 2 2450.2.a.bj.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
98.2.a.b.1.1 2 56.27 even 2
98.2.a.b.1.2 yes 2 8.3 odd 2
98.2.c.c.67.1 4 56.51 odd 6
98.2.c.c.67.2 4 56.19 even 6
98.2.c.c.79.1 4 56.11 odd 6
98.2.c.c.79.2 4 56.3 even 6
784.2.a.l.1.1 2 8.5 even 2
784.2.a.l.1.2 2 56.13 odd 2
784.2.i.m.177.1 4 56.45 odd 6
784.2.i.m.177.2 4 56.53 even 6
784.2.i.m.753.1 4 56.5 odd 6
784.2.i.m.753.2 4 56.37 even 6
882.2.a.n.1.1 2 168.83 odd 2
882.2.a.n.1.2 2 24.11 even 2
882.2.g.l.361.1 4 168.107 even 6
882.2.g.l.361.2 4 168.131 odd 6
882.2.g.l.667.1 4 168.11 even 6
882.2.g.l.667.2 4 168.59 odd 6
2450.2.a.bj.1.1 2 40.19 odd 2
2450.2.a.bj.1.2 2 280.139 even 2
2450.2.c.v.99.1 4 280.83 odd 4
2450.2.c.v.99.2 4 40.3 even 4
2450.2.c.v.99.3 4 40.27 even 4
2450.2.c.v.99.4 4 280.27 odd 4
3136.2.a.bm.1.1 2 7.6 odd 2 inner
3136.2.a.bm.1.2 2 1.1 even 1 trivial
3136.2.a.bn.1.1 2 4.3 odd 2
3136.2.a.bn.1.2 2 28.27 even 2
7056.2.a.cl.1.1 2 168.125 even 2
7056.2.a.cl.1.2 2 24.5 odd 2