Properties

Label 4900.2.e.c.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: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 140)
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.c.2549.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.00000i q^{3} -6.00000 q^{9} +O(q^{10})\) \(q+3.00000i q^{3} -6.00000 q^{9} -2.00000 q^{11} +6.00000i q^{13} +2.00000i q^{17} +9.00000i q^{23} -9.00000i q^{27} -3.00000 q^{29} +2.00000 q^{31} -6.00000i q^{33} +8.00000i q^{37} -18.0000 q^{39} +5.00000 q^{41} -1.00000i q^{43} +8.00000i q^{47} -6.00000 q^{51} -4.00000i q^{53} +8.00000 q^{59} +7.00000 q^{61} -3.00000i q^{67} -27.0000 q^{69} +8.00000 q^{71} -14.0000i q^{73} -4.00000 q^{79} +9.00000 q^{81} +1.00000i q^{83} -9.00000i q^{87} -13.0000 q^{89} +6.00000i q^{93} -10.0000i q^{97} +12.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 12q^{9} + O(q^{10}) \) \( 2q - 12q^{9} - 4q^{11} - 6q^{29} + 4q^{31} - 36q^{39} + 10q^{41} - 12q^{51} + 16q^{59} + 14q^{61} - 54q^{69} + 16q^{71} - 8q^{79} + 18q^{81} - 26q^{89} + 24q^{99} + O(q^{100}) \)

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
\(3\) 3.00000i 1.73205i 0.500000 + 0.866025i \(0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −6.00000 −2.00000
\(10\) 0 0
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 0 0
\(13\) 6.00000i 1.66410i 0.554700 + 0.832050i \(0.312833\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.00000i 0.485071i 0.970143 + 0.242536i \(0.0779791\pi\)
−0.970143 + 0.242536i \(0.922021\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 9.00000i 1.87663i 0.345782 + 0.938315i \(0.387614\pi\)
−0.345782 + 0.938315i \(0.612386\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 9.00000i − 1.73205i
\(28\) 0 0
\(29\) −3.00000 −0.557086 −0.278543 0.960424i \(-0.589851\pi\)
−0.278543 + 0.960424i \(0.589851\pi\)
\(30\) 0 0
\(31\) 2.00000 0.359211 0.179605 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) 0 0
\(33\) − 6.00000i − 1.04447i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 8.00000i 1.31519i 0.753371 + 0.657596i \(0.228427\pi\)
−0.753371 + 0.657596i \(0.771573\pi\)
\(38\) 0 0
\(39\) −18.0000 −2.88231
\(40\) 0 0
\(41\) 5.00000 0.780869 0.390434 0.920631i \(-0.372325\pi\)
0.390434 + 0.920631i \(0.372325\pi\)
\(42\) 0 0
\(43\) − 1.00000i − 0.152499i −0.997089 0.0762493i \(-0.975706\pi\)
0.997089 0.0762493i \(-0.0242945\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 8.00000i 1.16692i 0.812142 + 0.583460i \(0.198301\pi\)
−0.812142 + 0.583460i \(0.801699\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −6.00000 −0.840168
\(52\) 0 0
\(53\) − 4.00000i − 0.549442i −0.961524 0.274721i \(-0.911414\pi\)
0.961524 0.274721i \(-0.0885855\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 8.00000 1.04151 0.520756 0.853706i \(-0.325650\pi\)
0.520756 + 0.853706i \(0.325650\pi\)
\(60\) 0 0
\(61\) 7.00000 0.896258 0.448129 0.893969i \(-0.352090\pi\)
0.448129 + 0.893969i \(0.352090\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 3.00000i − 0.366508i −0.983066 0.183254i \(-0.941337\pi\)
0.983066 0.183254i \(-0.0586631\pi\)
\(68\) 0 0
\(69\) −27.0000 −3.25042
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 0 0
\(73\) − 14.0000i − 1.63858i −0.573382 0.819288i \(-0.694369\pi\)
0.573382 0.819288i \(-0.305631\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) 1.00000i 0.109764i 0.998493 + 0.0548821i \(0.0174783\pi\)
−0.998493 + 0.0548821i \(0.982522\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 9.00000i − 0.964901i
\(88\) 0 0
\(89\) −13.0000 −1.37800 −0.688999 0.724763i \(-0.741949\pi\)
−0.688999 + 0.724763i \(0.741949\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 6.00000i 0.622171i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 10.0000i − 1.01535i −0.861550 0.507673i \(-0.830506\pi\)
0.861550 0.507673i \(-0.169494\pi\)
\(98\) 0 0
\(99\) 12.0000 1.20605
\(100\) 0 0
\(101\) −3.00000 −0.298511 −0.149256 0.988799i \(-0.547688\pi\)
−0.149256 + 0.988799i \(0.547688\pi\)
\(102\) 0 0
\(103\) − 13.0000i − 1.28093i −0.767988 0.640464i \(-0.778742\pi\)
0.767988 0.640464i \(-0.221258\pi\)
\(104\) 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
\(109\) −9.00000 −0.862044 −0.431022 0.902342i \(-0.641847\pi\)
−0.431022 + 0.902342i \(0.641847\pi\)
\(110\) 0 0
\(111\) −24.0000 −2.27798
\(112\) 0 0
\(113\) 4.00000i 0.376288i 0.982141 + 0.188144i \(0.0602472\pi\)
−0.982141 + 0.188144i \(0.939753\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 36.0000i − 3.32820i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 0 0
\(123\) 15.0000i 1.35250i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 16.0000i 1.41977i 0.704317 + 0.709885i \(0.251253\pi\)
−0.704317 + 0.709885i \(0.748747\pi\)
\(128\) 0 0
\(129\) 3.00000 0.264135
\(130\) 0 0
\(131\) −4.00000 −0.349482 −0.174741 0.984614i \(-0.555909\pi\)
−0.174741 + 0.984614i \(0.555909\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 12.0000i − 1.02523i −0.858619 0.512615i \(-0.828677\pi\)
0.858619 0.512615i \(-0.171323\pi\)
\(138\) 0 0
\(139\) −10.0000 −0.848189 −0.424094 0.905618i \(-0.639408\pi\)
−0.424094 + 0.905618i \(0.639408\pi\)
\(140\) 0 0
\(141\) −24.0000 −2.02116
\(142\) 0 0
\(143\) − 12.0000i − 1.00349i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −9.00000 −0.737309 −0.368654 0.929567i \(-0.620181\pi\)
−0.368654 + 0.929567i \(0.620181\pi\)
\(150\) 0 0
\(151\) −10.0000 −0.813788 −0.406894 0.913475i \(-0.633388\pi\)
−0.406894 + 0.913475i \(0.633388\pi\)
\(152\) 0 0
\(153\) − 12.0000i − 0.970143i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 2.00000i − 0.159617i −0.996810 0.0798087i \(-0.974569\pi\)
0.996810 0.0798087i \(-0.0254309\pi\)
\(158\) 0 0
\(159\) 12.0000 0.951662
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) − 8.00000i − 0.626608i −0.949653 0.313304i \(-0.898564\pi\)
0.949653 0.313304i \(-0.101436\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 9.00000i − 0.696441i −0.937413 0.348220i \(-0.886786\pi\)
0.937413 0.348220i \(-0.113214\pi\)
\(168\) 0 0
\(169\) −23.0000 −1.76923
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 16.0000i 1.21646i 0.793762 + 0.608229i \(0.208120\pi\)
−0.793762 + 0.608229i \(0.791880\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 24.0000i 1.80395i
\(178\) 0 0
\(179\) 6.00000 0.448461 0.224231 0.974536i \(-0.428013\pi\)
0.224231 + 0.974536i \(0.428013\pi\)
\(180\) 0 0
\(181\) 1.00000 0.0743294 0.0371647 0.999309i \(-0.488167\pi\)
0.0371647 + 0.999309i \(0.488167\pi\)
\(182\) 0 0
\(183\) 21.0000i 1.55236i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 4.00000i − 0.292509i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 6.00000 0.434145 0.217072 0.976156i \(-0.430349\pi\)
0.217072 + 0.976156i \(0.430349\pi\)
\(192\) 0 0
\(193\) − 2.00000i − 0.143963i −0.997406 0.0719816i \(-0.977068\pi\)
0.997406 0.0719816i \(-0.0229323\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 14.0000i 0.997459i 0.866758 + 0.498729i \(0.166200\pi\)
−0.866758 + 0.498729i \(0.833800\pi\)
\(198\) 0 0
\(199\) −20.0000 −1.41776 −0.708881 0.705328i \(-0.750800\pi\)
−0.708881 + 0.705328i \(0.750800\pi\)
\(200\) 0 0
\(201\) 9.00000 0.634811
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 54.0000i − 3.75326i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) 0 0
\(213\) 24.0000i 1.64445i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 42.0000 2.83810
\(220\) 0 0
\(221\) −12.0000 −0.807207
\(222\) 0 0
\(223\) − 16.0000i − 1.07144i −0.844396 0.535720i \(-0.820040\pi\)
0.844396 0.535720i \(-0.179960\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 4.00000i − 0.265489i −0.991150 0.132745i \(-0.957621\pi\)
0.991150 0.132745i \(-0.0423790\pi\)
\(228\) 0 0
\(229\) 14.0000 0.925146 0.462573 0.886581i \(-0.346926\pi\)
0.462573 + 0.886581i \(0.346926\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 18.0000i 1.17922i 0.807688 + 0.589610i \(0.200718\pi\)
−0.807688 + 0.589610i \(0.799282\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 12.0000i − 0.779484i
\(238\) 0 0
\(239\) 26.0000 1.68180 0.840900 0.541190i \(-0.182026\pi\)
0.840900 + 0.541190i \(0.182026\pi\)
\(240\) 0 0
\(241\) 10.0000 0.644157 0.322078 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −3.00000 −0.190117
\(250\) 0 0
\(251\) 30.0000 1.89358 0.946792 0.321847i \(-0.104304\pi\)
0.946792 + 0.321847i \(0.104304\pi\)
\(252\) 0 0
\(253\) − 18.0000i − 1.13165i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 8.00000i 0.499026i 0.968371 + 0.249513i \(0.0802706\pi\)
−0.968371 + 0.249513i \(0.919729\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 18.0000 1.11417
\(262\) 0 0
\(263\) − 17.0000i − 1.04826i −0.851637 0.524132i \(-0.824390\pi\)
0.851637 0.524132i \(-0.175610\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 39.0000i − 2.38676i
\(268\) 0 0
\(269\) −9.00000 −0.548740 −0.274370 0.961624i \(-0.588469\pi\)
−0.274370 + 0.961624i \(0.588469\pi\)
\(270\) 0 0
\(271\) −24.0000 −1.45790 −0.728948 0.684569i \(-0.759990\pi\)
−0.728948 + 0.684569i \(0.759990\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 18.0000i 1.08152i 0.841178 + 0.540758i \(0.181862\pi\)
−0.841178 + 0.540758i \(0.818138\pi\)
\(278\) 0 0
\(279\) −12.0000 −0.718421
\(280\) 0 0
\(281\) −22.0000 −1.31241 −0.656205 0.754583i \(-0.727839\pi\)
−0.656205 + 0.754583i \(0.727839\pi\)
\(282\) 0 0
\(283\) 4.00000i 0.237775i 0.992908 + 0.118888i \(0.0379328\pi\)
−0.992908 + 0.118888i \(0.962067\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 13.0000 0.764706
\(290\) 0 0
\(291\) 30.0000 1.75863
\(292\) 0 0
\(293\) − 4.00000i − 0.233682i −0.993151 0.116841i \(-0.962723\pi\)
0.993151 0.116841i \(-0.0372769\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 18.0000i 1.04447i
\(298\) 0 0
\(299\) −54.0000 −3.12290
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) − 9.00000i − 0.517036i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 1.00000i 0.0570730i 0.999593 + 0.0285365i \(0.00908469\pi\)
−0.999593 + 0.0285365i \(0.990915\pi\)
\(308\) 0 0
\(309\) 39.0000 2.21863
\(310\) 0 0
\(311\) 24.0000 1.36092 0.680458 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) 0 0
\(313\) 4.00000i 0.226093i 0.993590 + 0.113047i \(0.0360610\pi\)
−0.993590 + 0.113047i \(0.963939\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 10.0000i 0.561656i 0.959758 + 0.280828i \(0.0906090\pi\)
−0.959758 + 0.280828i \(0.909391\pi\)
\(318\) 0 0
\(319\) 6.00000 0.335936
\(320\) 0 0
\(321\) 45.0000 2.51166
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 27.0000i − 1.49310i
\(328\) 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
\(333\) − 48.0000i − 2.63038i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 28.0000i 1.52526i 0.646837 + 0.762629i \(0.276092\pi\)
−0.646837 + 0.762629i \(0.723908\pi\)
\(338\) 0 0
\(339\) −12.0000 −0.651751
\(340\) 0 0
\(341\) −4.00000 −0.216612
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 25.0000i 1.34207i 0.741426 + 0.671035i \(0.234150\pi\)
−0.741426 + 0.671035i \(0.765850\pi\)
\(348\) 0 0
\(349\) 17.0000 0.909989 0.454995 0.890494i \(-0.349641\pi\)
0.454995 + 0.890494i \(0.349641\pi\)
\(350\) 0 0
\(351\) 54.0000 2.88231
\(352\) 0 0
\(353\) 36.0000i 1.91609i 0.286623 + 0.958043i \(0.407467\pi\)
−0.286623 + 0.958043i \(0.592533\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 10.0000 0.527780 0.263890 0.964553i \(-0.414994\pi\)
0.263890 + 0.964553i \(0.414994\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 0 0
\(363\) − 21.0000i − 1.10221i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 1.00000i 0.0521996i 0.999659 + 0.0260998i \(0.00830876\pi\)
−0.999659 + 0.0260998i \(0.991691\pi\)
\(368\) 0 0
\(369\) −30.0000 −1.56174
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) − 32.0000i − 1.65690i −0.560065 0.828449i \(-0.689224\pi\)
0.560065 0.828449i \(-0.310776\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 18.0000i − 0.927047i
\(378\) 0 0
\(379\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(380\) 0 0
\(381\) −48.0000 −2.45911
\(382\) 0 0
\(383\) 9.00000i 0.459879i 0.973205 + 0.229939i \(0.0738528\pi\)
−0.973205 + 0.229939i \(0.926147\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 6.00000i 0.304997i
\(388\) 0 0
\(389\) 10.0000 0.507020 0.253510 0.967333i \(-0.418415\pi\)
0.253510 + 0.967333i \(0.418415\pi\)
\(390\) 0 0
\(391\) −18.0000 −0.910299
\(392\) 0 0
\(393\) − 12.0000i − 0.605320i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 10.0000i 0.501886i 0.968002 + 0.250943i \(0.0807406\pi\)
−0.968002 + 0.250943i \(0.919259\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −29.0000 −1.44819 −0.724095 0.689700i \(-0.757743\pi\)
−0.724095 + 0.689700i \(0.757743\pi\)
\(402\) 0 0
\(403\) 12.0000i 0.597763i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 16.0000i − 0.793091i
\(408\) 0 0
\(409\) −3.00000 −0.148340 −0.0741702 0.997246i \(-0.523631\pi\)
−0.0741702 + 0.997246i \(0.523631\pi\)
\(410\) 0 0
\(411\) 36.0000 1.77575
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 30.0000i − 1.46911i
\(418\) 0 0
\(419\) 24.0000 1.17248 0.586238 0.810139i \(-0.300608\pi\)
0.586238 + 0.810139i \(0.300608\pi\)
\(420\) 0 0
\(421\) −1.00000 −0.0487370 −0.0243685 0.999703i \(-0.507758\pi\)
−0.0243685 + 0.999703i \(0.507758\pi\)
\(422\) 0 0
\(423\) − 48.0000i − 2.33384i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 36.0000 1.73810
\(430\) 0 0
\(431\) 6.00000 0.289010 0.144505 0.989504i \(-0.453841\pi\)
0.144505 + 0.989504i \(0.453841\pi\)
\(432\) 0 0
\(433\) 4.00000i 0.192228i 0.995370 + 0.0961139i \(0.0306413\pi\)
−0.995370 + 0.0961139i \(0.969359\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −4.00000 −0.190910 −0.0954548 0.995434i \(-0.530431\pi\)
−0.0954548 + 0.995434i \(0.530431\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 37.0000i − 1.75792i −0.476893 0.878962i \(-0.658237\pi\)
0.476893 0.878962i \(-0.341763\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 27.0000i − 1.27706i
\(448\) 0 0
\(449\) −15.0000 −0.707894 −0.353947 0.935266i \(-0.615161\pi\)
−0.353947 + 0.935266i \(0.615161\pi\)
\(450\) 0 0
\(451\) −10.0000 −0.470882
\(452\) 0 0
\(453\) − 30.0000i − 1.40952i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 28.0000i 1.30978i 0.755722 + 0.654892i \(0.227286\pi\)
−0.755722 + 0.654892i \(0.772714\pi\)
\(458\) 0 0
\(459\) 18.0000 0.840168
\(460\) 0 0
\(461\) −2.00000 −0.0931493 −0.0465746 0.998915i \(-0.514831\pi\)
−0.0465746 + 0.998915i \(0.514831\pi\)
\(462\) 0 0
\(463\) − 17.0000i − 0.790057i −0.918669 0.395029i \(-0.870735\pi\)
0.918669 0.395029i \(-0.129265\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 5.00000i 0.231372i 0.993286 + 0.115686i \(0.0369067\pi\)
−0.993286 + 0.115686i \(0.963093\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 6.00000 0.276465
\(472\) 0 0
\(473\) 2.00000i 0.0919601i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 24.0000i 1.09888i
\(478\) 0 0
\(479\) 30.0000 1.37073 0.685367 0.728197i \(-0.259642\pi\)
0.685367 + 0.728197i \(0.259642\pi\)
\(480\) 0 0
\(481\) −48.0000 −2.18861
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 32.0000i 1.45006i 0.688718 + 0.725029i \(0.258174\pi\)
−0.688718 + 0.725029i \(0.741826\pi\)
\(488\) 0 0
\(489\) 24.0000 1.08532
\(490\) 0 0
\(491\) 6.00000 0.270776 0.135388 0.990793i \(-0.456772\pi\)
0.135388 + 0.990793i \(0.456772\pi\)
\(492\) 0 0
\(493\) − 6.00000i − 0.270226i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 6.00000 0.268597 0.134298 0.990941i \(-0.457122\pi\)
0.134298 + 0.990941i \(0.457122\pi\)
\(500\) 0 0
\(501\) 27.0000 1.20627
\(502\) 0 0
\(503\) − 27.0000i − 1.20387i −0.798545 0.601935i \(-0.794397\pi\)
0.798545 0.601935i \(-0.205603\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 69.0000i − 3.06440i
\(508\) 0 0
\(509\) 41.0000 1.81729 0.908647 0.417566i \(-0.137117\pi\)
0.908647 + 0.417566i \(0.137117\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) − 16.0000i − 0.703679i
\(518\) 0 0
\(519\) −48.0000 −2.10697
\(520\) 0 0
\(521\) 14.0000 0.613351 0.306676 0.951814i \(-0.400783\pi\)
0.306676 + 0.951814i \(0.400783\pi\)
\(522\) 0 0
\(523\) − 4.00000i − 0.174908i −0.996169 0.0874539i \(-0.972127\pi\)
0.996169 0.0874539i \(-0.0278730\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 4.00000i 0.174243i
\(528\) 0 0
\(529\) −58.0000 −2.52174
\(530\) 0 0
\(531\) −48.0000 −2.08302
\(532\) 0 0
\(533\) 30.0000i 1.29944i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 18.0000i 0.776757i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 33.0000 1.41878 0.709390 0.704816i \(-0.248970\pi\)
0.709390 + 0.704816i \(0.248970\pi\)
\(542\) 0 0
\(543\) 3.00000i 0.128742i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 15.0000i − 0.641354i −0.947189 0.320677i \(-0.896090\pi\)
0.947189 0.320677i \(-0.103910\pi\)
\(548\) 0 0
\(549\) −42.0000 −1.79252
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 2.00000i − 0.0847427i −0.999102 0.0423714i \(-0.986509\pi\)
0.999102 0.0423714i \(-0.0134913\pi\)
\(558\) 0 0
\(559\) 6.00000 0.253773
\(560\) 0 0
\(561\) 12.0000 0.506640
\(562\) 0 0
\(563\) 13.0000i 0.547885i 0.961746 + 0.273942i \(0.0883277\pi\)
−0.961746 + 0.273942i \(0.911672\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −30.0000 −1.25767 −0.628833 0.777541i \(-0.716467\pi\)
−0.628833 + 0.777541i \(0.716467\pi\)
\(570\) 0 0
\(571\) 24.0000 1.00437 0.502184 0.864761i \(-0.332530\pi\)
0.502184 + 0.864761i \(0.332530\pi\)
\(572\) 0 0
\(573\) 18.0000i 0.751961i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 22.0000i 0.915872i 0.888985 + 0.457936i \(0.151411\pi\)
−0.888985 + 0.457936i \(0.848589\pi\)
\(578\) 0 0
\(579\) 6.00000 0.249351
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 8.00000i 0.331326i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 20.0000i 0.825488i 0.910847 + 0.412744i \(0.135430\pi\)
−0.910847 + 0.412744i \(0.864570\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) −42.0000 −1.72765
\(592\) 0 0
\(593\) 36.0000i 1.47834i 0.673517 + 0.739171i \(0.264783\pi\)
−0.673517 + 0.739171i \(0.735217\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) − 60.0000i − 2.45564i
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) −6.00000 −0.244745 −0.122373 0.992484i \(-0.539050\pi\)
−0.122373 + 0.992484i \(0.539050\pi\)
\(602\) 0 0
\(603\) 18.0000i 0.733017i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 1.00000i 0.0405887i 0.999794 + 0.0202944i \(0.00646034\pi\)
−0.999794 + 0.0202944i \(0.993540\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −48.0000 −1.94187
\(612\) 0 0
\(613\) 6.00000i 0.242338i 0.992632 + 0.121169i \(0.0386643\pi\)
−0.992632 + 0.121169i \(0.961336\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 20.0000i 0.805170i 0.915383 + 0.402585i \(0.131888\pi\)
−0.915383 + 0.402585i \(0.868112\pi\)
\(618\) 0 0
\(619\) 10.0000 0.401934 0.200967 0.979598i \(-0.435592\pi\)
0.200967 + 0.979598i \(0.435592\pi\)
\(620\) 0 0
\(621\) 81.0000 3.25042
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −16.0000 −0.637962
\(630\) 0 0
\(631\) −34.0000 −1.35352 −0.676759 0.736204i \(-0.736616\pi\)
−0.676759 + 0.736204i \(0.736616\pi\)
\(632\) 0 0
\(633\) 12.0000i 0.476957i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −48.0000 −1.89885
\(640\) 0 0
\(641\) −31.0000 −1.22443 −0.612213 0.790693i \(-0.709721\pi\)
−0.612213 + 0.790693i \(0.709721\pi\)
\(642\) 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
\(647\) 1.00000i 0.0393141i 0.999807 + 0.0196570i \(0.00625743\pi\)
−0.999807 + 0.0196570i \(0.993743\pi\)
\(648\) 0 0
\(649\) −16.0000 −0.628055
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 34.0000i 1.33052i 0.746611 + 0.665261i \(0.231680\pi\)
−0.746611 + 0.665261i \(0.768320\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 84.0000i 3.27715i
\(658\) 0 0
\(659\) −4.00000 −0.155818 −0.0779089 0.996960i \(-0.524824\pi\)
−0.0779089 + 0.996960i \(0.524824\pi\)
\(660\) 0 0
\(661\) 31.0000 1.20576 0.602880 0.797832i \(-0.294020\pi\)
0.602880 + 0.797832i \(0.294020\pi\)
\(662\) 0 0
\(663\) − 36.0000i − 1.39812i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 27.0000i − 1.04544i
\(668\) 0 0
\(669\) 48.0000 1.85579
\(670\) 0 0
\(671\) −14.0000 −0.540464
\(672\) 0 0
\(673\) − 16.0000i − 0.616755i −0.951264 0.308377i \(-0.900214\pi\)
0.951264 0.308377i \(-0.0997859\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 6.00000i 0.230599i 0.993331 + 0.115299i \(0.0367827\pi\)
−0.993331 + 0.115299i \(0.963217\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 12.0000 0.459841
\(682\) 0 0
\(683\) − 29.0000i − 1.10965i −0.831966 0.554827i \(-0.812784\pi\)
0.831966 0.554827i \(-0.187216\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 42.0000i 1.60240i
\(688\) 0 0
\(689\) 24.0000 0.914327
\(690\) 0 0
\(691\) −26.0000 −0.989087 −0.494543 0.869153i \(-0.664665\pi\)
−0.494543 + 0.869153i \(0.664665\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 10.0000i 0.378777i
\(698\) 0 0
\(699\) −54.0000 −2.04247
\(700\) 0 0
\(701\) −29.0000 −1.09531 −0.547657 0.836703i \(-0.684480\pi\)
−0.547657 + 0.836703i \(0.684480\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −31.0000 −1.16423 −0.582115 0.813107i \(-0.697775\pi\)
−0.582115 + 0.813107i \(0.697775\pi\)
\(710\) 0 0
\(711\) 24.0000 0.900070
\(712\) 0 0
\(713\) 18.0000i 0.674105i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 78.0000i 2.91296i
\(718\) 0 0
\(719\) 6.00000 0.223762 0.111881 0.993722i \(-0.464312\pi\)
0.111881 + 0.993722i \(0.464312\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 30.0000i 1.11571i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 3.00000i 0.111264i 0.998451 + 0.0556319i \(0.0177173\pi\)
−0.998451 + 0.0556319i \(0.982283\pi\)
\(728\) 0 0
\(729\) 27.0000 1.00000
\(730\) 0 0
\(731\) 2.00000 0.0739727
\(732\) 0 0
\(733\) − 34.0000i − 1.25582i −0.778287 0.627909i \(-0.783911\pi\)
0.778287 0.627909i \(-0.216089\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 6.00000i 0.221013i
\(738\) 0 0
\(739\) −10.0000 −0.367856 −0.183928 0.982940i \(-0.558881\pi\)
−0.183928 + 0.982940i \(0.558881\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 3.00000i 0.110059i 0.998485 + 0.0550297i \(0.0175253\pi\)
−0.998485 + 0.0550297i \(0.982475\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) − 6.00000i − 0.219529i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 20.0000 0.729810 0.364905 0.931045i \(-0.381101\pi\)
0.364905 + 0.931045i \(0.381101\pi\)
\(752\) 0 0
\(753\) 90.0000i 3.27978i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 38.0000i − 1.38113i −0.723269 0.690567i \(-0.757361\pi\)
0.723269 0.690567i \(-0.242639\pi\)
\(758\) 0 0
\(759\) 54.0000 1.96008
\(760\) 0 0
\(761\) 30.0000 1.08750 0.543750 0.839248i \(-0.317004\pi\)
0.543750 + 0.839248i \(0.317004\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 48.0000i 1.73318i
\(768\) 0 0
\(769\) 2.00000 0.0721218 0.0360609 0.999350i \(-0.488519\pi\)
0.0360609 + 0.999350i \(0.488519\pi\)
\(770\) 0 0
\(771\) −24.0000 −0.864339
\(772\) 0 0
\(773\) 18.0000i 0.647415i 0.946157 + 0.323708i \(0.104929\pi\)
−0.946157 + 0.323708i \(0.895071\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) −16.0000 −0.572525
\(782\) 0 0
\(783\) 27.0000i 0.964901i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 11.0000i − 0.392108i −0.980593 0.196054i \(-0.937187\pi\)
0.980593 0.196054i \(-0.0628127\pi\)
\(788\) 0 0
\(789\) 51.0000 1.81565
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 42.0000i 1.49146i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 36.0000i 1.27519i 0.770374 + 0.637593i \(0.220070\pi\)
−0.770374 + 0.637593i \(0.779930\pi\)
\(798\) 0 0
\(799\) −16.0000 −0.566039
\(800\) 0 0
\(801\) 78.0000 2.75599
\(802\) 0 0
\(803\) 28.0000i 0.988099i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 27.0000i − 0.950445i
\(808\) 0 0
\(809\) −45.0000 −1.58212 −0.791058 0.611741i \(-0.790469\pi\)
−0.791058 + 0.611741i \(0.790469\pi\)
\(810\) 0 0
\(811\) 50.0000 1.75574 0.877869 0.478901i \(-0.158965\pi\)
0.877869 + 0.478901i \(0.158965\pi\)
\(812\) 0 0
\(813\) − 72.0000i − 2.52515i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 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
\(823\) − 19.0000i − 0.662298i −0.943578 0.331149i \(-0.892564\pi\)
0.943578 0.331149i \(-0.107436\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 23.0000i 0.799788i 0.916561 + 0.399894i \(0.130953\pi\)
−0.916561 + 0.399894i \(0.869047\pi\)
\(828\) 0 0
\(829\) 10.0000 0.347314 0.173657 0.984806i \(-0.444442\pi\)
0.173657 + 0.984806i \(0.444442\pi\)
\(830\) 0 0
\(831\) −54.0000 −1.87324
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) − 18.0000i − 0.622171i
\(838\) 0 0
\(839\) −14.0000 −0.483334 −0.241667 0.970359i \(-0.577694\pi\)
−0.241667 + 0.970359i \(0.577694\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) 0 0
\(843\) − 66.0000i − 2.27316i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −12.0000 −0.411839
\(850\) 0 0
\(851\) −72.0000 −2.46813
\(852\) 0 0
\(853\) 40.0000i 1.36957i 0.728743 + 0.684787i \(0.240105\pi\)
−0.728743 + 0.684787i \(0.759895\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 54.0000i − 1.84460i −0.386469 0.922302i \(-0.626305\pi\)
0.386469 0.922302i \(-0.373695\pi\)
\(858\) 0 0
\(859\) 36.0000 1.22830 0.614152 0.789188i \(-0.289498\pi\)
0.614152 + 0.789188i \(0.289498\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) − 37.0000i − 1.25949i −0.776800 0.629747i \(-0.783158\pi\)
0.776800 0.629747i \(-0.216842\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 39.0000i 1.32451i
\(868\) 0 0
\(869\) 8.00000 0.271381
\(870\) 0 0
\(871\) 18.0000 0.609907
\(872\) 0 0
\(873\) 60.0000i 2.03069i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 32.0000i 1.08056i 0.841484 + 0.540282i \(0.181682\pi\)
−0.841484 + 0.540282i \(0.818318\pi\)
\(878\) 0 0
\(879\) 12.0000 0.404750
\(880\) 0 0
\(881\) −31.0000 −1.04442 −0.522208 0.852818i \(-0.674892\pi\)
−0.522208 + 0.852818i \(0.674892\pi\)
\(882\) 0 0
\(883\) 36.0000i 1.21150i 0.795656 + 0.605748i \(0.207126\pi\)
−0.795656 + 0.605748i \(0.792874\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 53.0000i 1.77957i 0.456384 + 0.889783i \(0.349144\pi\)
−0.456384 + 0.889783i \(0.650856\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −18.0000 −0.603023
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) − 162.000i − 5.40902i
\(898\) 0 0
\(899\) −6.00000 −0.200111
\(900\) 0 0
\(901\) 8.00000 0.266519
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 37.0000i − 1.22856i −0.789086 0.614282i \(-0.789446\pi\)
0.789086 0.614282i \(-0.210554\pi\)
\(908\) 0 0
\(909\) 18.0000 0.597022
\(910\) 0 0
\(911\) 54.0000 1.78910 0.894550 0.446968i \(-0.147496\pi\)
0.894550 + 0.446968i \(0.147496\pi\)
\(912\) 0 0
\(913\) − 2.00000i − 0.0661903i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −32.0000 −1.05558 −0.527791 0.849374i \(-0.676980\pi\)
−0.527791 + 0.849374i \(0.676980\pi\)
\(920\) 0 0
\(921\) −3.00000 −0.0988534
\(922\) 0 0
\(923\) 48.0000i 1.57994i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 78.0000i 2.56186i
\(928\) 0 0
\(929\) 5.00000 0.164045 0.0820223 0.996630i \(-0.473862\pi\)
0.0820223 + 0.996630i \(0.473862\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 72.0000i 2.35717i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 52.0000i − 1.69877i −0.527777 0.849383i \(-0.676974\pi\)
0.527777 0.849383i \(-0.323026\pi\)
\(938\) 0 0
\(939\) −12.0000 −0.391605
\(940\) 0 0
\(941\) 38.0000 1.23876 0.619382 0.785090i \(-0.287383\pi\)
0.619382 + 0.785090i \(0.287383\pi\)
\(942\) 0 0
\(943\) 45.0000i 1.46540i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 29.0000i 0.942373i 0.882034 + 0.471187i \(0.156174\pi\)
−0.882034 + 0.471187i \(0.843826\pi\)
\(948\) 0 0
\(949\) 84.0000 2.72676
\(950\) 0 0
\(951\) −30.0000 −0.972817
\(952\) 0 0
\(953\) 60.0000i 1.94359i 0.235826 + 0.971795i \(0.424220\pi\)
−0.235826 + 0.971795i \(0.575780\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 18.0000i 0.581857i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 0 0
\(963\) 90.0000i 2.90021i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 37.0000i 1.18984i 0.803785 + 0.594920i \(0.202816\pi\)
−0.803785 + 0.594920i \(0.797184\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −12.0000 −0.385098 −0.192549 0.981287i \(-0.561675\pi\)
−0.192549 + 0.981287i \(0.561675\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 6.00000i − 0.191957i −0.995383 0.0959785i \(-0.969402\pi\)
0.995383 0.0959785i \(-0.0305980\pi\)
\(978\) 0 0
\(979\) 26.0000 0.830964
\(980\) 0 0
\(981\) 54.0000 1.72409
\(982\) 0 0
\(983\) 19.0000i 0.606006i 0.952990 + 0.303003i \(0.0979892\pi\)
−0.952990 + 0.303003i \(0.902011\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 9.00000 0.286183
\(990\) 0 0
\(991\) −38.0000 −1.20711 −0.603555 0.797321i \(-0.706250\pi\)
−0.603555 + 0.797321i \(0.706250\pi\)
\(992\) 0 0
\(993\) 30.0000i 0.952021i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 40.0000i − 1.26681i −0.773819 0.633406i \(-0.781656\pi\)
0.773819 0.633406i \(-0.218344\pi\)
\(998\) 0 0
\(999\) 72.0000 2.27798
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.c.2549.2 2
5.2 odd 4 4900.2.a.v.1.1 1
5.3 odd 4 980.2.a.a.1.1 1
5.4 even 2 inner 4900.2.e.c.2549.1 2
7.2 even 3 700.2.r.c.249.1 4
7.4 even 3 700.2.r.c.149.2 4
7.6 odd 2 4900.2.e.b.2549.1 2
15.8 even 4 8820.2.a.w.1.1 1
20.3 even 4 3920.2.a.bi.1.1 1
35.2 odd 12 700.2.i.a.501.1 2
35.3 even 12 980.2.i.a.961.1 2
35.4 even 6 700.2.r.c.149.1 4
35.9 even 6 700.2.r.c.249.2 4
35.13 even 4 980.2.a.i.1.1 1
35.18 odd 12 140.2.i.b.121.1 yes 2
35.23 odd 12 140.2.i.b.81.1 2
35.27 even 4 4900.2.a.a.1.1 1
35.32 odd 12 700.2.i.a.401.1 2
35.33 even 12 980.2.i.a.361.1 2
35.34 odd 2 4900.2.e.b.2549.2 2
105.23 even 12 1260.2.s.b.361.1 2
105.53 even 12 1260.2.s.b.541.1 2
105.83 odd 4 8820.2.a.k.1.1 1
140.23 even 12 560.2.q.a.81.1 2
140.83 odd 4 3920.2.a.d.1.1 1
140.123 even 12 560.2.q.a.401.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
140.2.i.b.81.1 2 35.23 odd 12
140.2.i.b.121.1 yes 2 35.18 odd 12
560.2.q.a.81.1 2 140.23 even 12
560.2.q.a.401.1 2 140.123 even 12
700.2.i.a.401.1 2 35.32 odd 12
700.2.i.a.501.1 2 35.2 odd 12
700.2.r.c.149.1 4 35.4 even 6
700.2.r.c.149.2 4 7.4 even 3
700.2.r.c.249.1 4 7.2 even 3
700.2.r.c.249.2 4 35.9 even 6
980.2.a.a.1.1 1 5.3 odd 4
980.2.a.i.1.1 1 35.13 even 4
980.2.i.a.361.1 2 35.33 even 12
980.2.i.a.961.1 2 35.3 even 12
1260.2.s.b.361.1 2 105.23 even 12
1260.2.s.b.541.1 2 105.53 even 12
3920.2.a.d.1.1 1 140.83 odd 4
3920.2.a.bi.1.1 1 20.3 even 4
4900.2.a.a.1.1 1 35.27 even 4
4900.2.a.v.1.1 1 5.2 odd 4
4900.2.e.b.2549.1 2 7.6 odd 2
4900.2.e.b.2549.2 2 35.34 odd 2
4900.2.e.c.2549.1 2 5.4 even 2 inner
4900.2.e.c.2549.2 2 1.1 even 1 trivial
8820.2.a.k.1.1 1 105.83 odd 4
8820.2.a.w.1.1 1 15.8 even 4