Properties

Label 3920.2.a.a.1.1
Level $3920$
Weight $2$
Character 3920.1
Self dual yes
Analytic conductor $31.301$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3920 = 2^{4} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3920.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(31.3013575923\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 245)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 3920.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.00000 q^{3} -1.00000 q^{5} +6.00000 q^{9} +O(q^{10})\) \(q-3.00000 q^{3} -1.00000 q^{5} +6.00000 q^{9} -1.00000 q^{11} +3.00000 q^{13} +3.00000 q^{15} -3.00000 q^{17} -6.00000 q^{19} +4.00000 q^{23} +1.00000 q^{25} -9.00000 q^{27} -1.00000 q^{29} -6.00000 q^{31} +3.00000 q^{33} -9.00000 q^{39} +6.00000 q^{41} +6.00000 q^{43} -6.00000 q^{45} +9.00000 q^{47} +9.00000 q^{51} -10.0000 q^{53} +1.00000 q^{55} +18.0000 q^{57} +6.00000 q^{59} -3.00000 q^{65} +14.0000 q^{67} -12.0000 q^{69} +8.00000 q^{71} +6.00000 q^{73} -3.00000 q^{75} +1.00000 q^{79} +9.00000 q^{81} -12.0000 q^{83} +3.00000 q^{85} +3.00000 q^{87} +12.0000 q^{89} +18.0000 q^{93} +6.00000 q^{95} -15.0000 q^{97} -6.00000 q^{99} +O(q^{100})\)

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
\(3\) −3.00000 −1.73205 −0.866025 0.500000i \(-0.833333\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 6.00000 2.00000
\(10\) 0 0
\(11\) −1.00000 −0.301511 −0.150756 0.988571i \(-0.548171\pi\)
−0.150756 + 0.988571i \(0.548171\pi\)
\(12\) 0 0
\(13\) 3.00000 0.832050 0.416025 0.909353i \(-0.363423\pi\)
0.416025 + 0.909353i \(0.363423\pi\)
\(14\) 0 0
\(15\) 3.00000 0.774597
\(16\) 0 0
\(17\) −3.00000 −0.727607 −0.363803 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) 0 0
\(19\) −6.00000 −1.37649 −0.688247 0.725476i \(-0.741620\pi\)
−0.688247 + 0.725476i \(0.741620\pi\)
\(20\) 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
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −9.00000 −1.73205
\(28\) 0 0
\(29\) −1.00000 −0.185695 −0.0928477 0.995680i \(-0.529597\pi\)
−0.0928477 + 0.995680i \(0.529597\pi\)
\(30\) 0 0
\(31\) −6.00000 −1.07763 −0.538816 0.842424i \(-0.681128\pi\)
−0.538816 + 0.842424i \(0.681128\pi\)
\(32\) 0 0
\(33\) 3.00000 0.522233
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 0 0
\(39\) −9.00000 −1.44115
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) 6.00000 0.914991 0.457496 0.889212i \(-0.348747\pi\)
0.457496 + 0.889212i \(0.348747\pi\)
\(44\) 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
\(51\) 9.00000 1.26025
\(52\) 0 0
\(53\) −10.0000 −1.37361 −0.686803 0.726844i \(-0.740986\pi\)
−0.686803 + 0.726844i \(0.740986\pi\)
\(54\) 0 0
\(55\) 1.00000 0.134840
\(56\) 0 0
\(57\) 18.0000 2.38416
\(58\) 0 0
\(59\) 6.00000 0.781133 0.390567 0.920575i \(-0.372279\pi\)
0.390567 + 0.920575i \(0.372279\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −3.00000 −0.372104
\(66\) 0 0
\(67\) 14.0000 1.71037 0.855186 0.518321i \(-0.173443\pi\)
0.855186 + 0.518321i \(0.173443\pi\)
\(68\) 0 0
\(69\) −12.0000 −1.44463
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 0 0
\(73\) 6.00000 0.702247 0.351123 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) 0 0
\(75\) −3.00000 −0.346410
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 1.00000 0.112509 0.0562544 0.998416i \(-0.482084\pi\)
0.0562544 + 0.998416i \(0.482084\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) −12.0000 −1.31717 −0.658586 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 0 0
\(85\) 3.00000 0.325396
\(86\) 0 0
\(87\) 3.00000 0.321634
\(88\) 0 0
\(89\) 12.0000 1.27200 0.635999 0.771690i \(-0.280588\pi\)
0.635999 + 0.771690i \(0.280588\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 18.0000 1.86651
\(94\) 0 0
\(95\) 6.00000 0.615587
\(96\) 0 0
\(97\) −15.0000 −1.52302 −0.761510 0.648154i \(-0.775541\pi\)
−0.761510 + 0.648154i \(0.775541\pi\)
\(98\) 0 0
\(99\) −6.00000 −0.603023
\(100\) 0 0
\(101\) −18.0000 −1.79107 −0.895533 0.444994i \(-0.853206\pi\)
−0.895533 + 0.444994i \(0.853206\pi\)
\(102\) 0 0
\(103\) 9.00000 0.886796 0.443398 0.896325i \(-0.353773\pi\)
0.443398 + 0.896325i \(0.353773\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 2.00000 0.193347 0.0966736 0.995316i \(-0.469180\pi\)
0.0966736 + 0.995316i \(0.469180\pi\)
\(108\) 0 0
\(109\) −15.0000 −1.43674 −0.718370 0.695662i \(-0.755111\pi\)
−0.718370 + 0.695662i \(0.755111\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 8.00000 0.752577 0.376288 0.926503i \(-0.377200\pi\)
0.376288 + 0.926503i \(0.377200\pi\)
\(114\) 0 0
\(115\) −4.00000 −0.373002
\(116\) 0 0
\(117\) 18.0000 1.66410
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −10.0000 −0.909091
\(122\) 0 0
\(123\) −18.0000 −1.62301
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 2.00000 0.177471 0.0887357 0.996055i \(-0.471717\pi\)
0.0887357 + 0.996055i \(0.471717\pi\)
\(128\) 0 0
\(129\) −18.0000 −1.58481
\(130\) 0 0
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 9.00000 0.774597
\(136\) 0 0
\(137\) 8.00000 0.683486 0.341743 0.939793i \(-0.388983\pi\)
0.341743 + 0.939793i \(0.388983\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) −27.0000 −2.27381
\(142\) 0 0
\(143\) −3.00000 −0.250873
\(144\) 0 0
\(145\) 1.00000 0.0830455
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 10.0000 0.819232 0.409616 0.912258i \(-0.365663\pi\)
0.409616 + 0.912258i \(0.365663\pi\)
\(150\) 0 0
\(151\) −15.0000 −1.22068 −0.610341 0.792139i \(-0.708968\pi\)
−0.610341 + 0.792139i \(0.708968\pi\)
\(152\) 0 0
\(153\) −18.0000 −1.45521
\(154\) 0 0
\(155\) 6.00000 0.481932
\(156\) 0 0
\(157\) 18.0000 1.43656 0.718278 0.695756i \(-0.244931\pi\)
0.718278 + 0.695756i \(0.244931\pi\)
\(158\) 0 0
\(159\) 30.0000 2.37915
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −16.0000 −1.25322 −0.626608 0.779334i \(-0.715557\pi\)
−0.626608 + 0.779334i \(0.715557\pi\)
\(164\) 0 0
\(165\) −3.00000 −0.233550
\(166\) 0 0
\(167\) −3.00000 −0.232147 −0.116073 0.993241i \(-0.537031\pi\)
−0.116073 + 0.993241i \(0.537031\pi\)
\(168\) 0 0
\(169\) −4.00000 −0.307692
\(170\) 0 0
\(171\) −36.0000 −2.75299
\(172\) 0 0
\(173\) −3.00000 −0.228086 −0.114043 0.993476i \(-0.536380\pi\)
−0.114043 + 0.993476i \(0.536380\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −18.0000 −1.35296
\(178\) 0 0
\(179\) −4.00000 −0.298974 −0.149487 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) 0 0
\(181\) −6.00000 −0.445976 −0.222988 0.974821i \(-0.571581\pi\)
−0.222988 + 0.974821i \(0.571581\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 3.00000 0.219382
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −17.0000 −1.23008 −0.615038 0.788497i \(-0.710860\pi\)
−0.615038 + 0.788497i \(0.710860\pi\)
\(192\) 0 0
\(193\) 12.0000 0.863779 0.431889 0.901927i \(-0.357847\pi\)
0.431889 + 0.901927i \(0.357847\pi\)
\(194\) 0 0
\(195\) 9.00000 0.644503
\(196\) 0 0
\(197\) −2.00000 −0.142494 −0.0712470 0.997459i \(-0.522698\pi\)
−0.0712470 + 0.997459i \(0.522698\pi\)
\(198\) 0 0
\(199\) −6.00000 −0.425329 −0.212664 0.977125i \(-0.568214\pi\)
−0.212664 + 0.977125i \(0.568214\pi\)
\(200\) 0 0
\(201\) −42.0000 −2.96245
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −6.00000 −0.419058
\(206\) 0 0
\(207\) 24.0000 1.66812
\(208\) 0 0
\(209\) 6.00000 0.415029
\(210\) 0 0
\(211\) −15.0000 −1.03264 −0.516321 0.856395i \(-0.672699\pi\)
−0.516321 + 0.856395i \(0.672699\pi\)
\(212\) 0 0
\(213\) −24.0000 −1.64445
\(214\) 0 0
\(215\) −6.00000 −0.409197
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −18.0000 −1.21633
\(220\) 0 0
\(221\) −9.00000 −0.605406
\(222\) 0 0
\(223\) −3.00000 −0.200895 −0.100447 0.994942i \(-0.532027\pi\)
−0.100447 + 0.994942i \(0.532027\pi\)
\(224\) 0 0
\(225\) 6.00000 0.400000
\(226\) 0 0
\(227\) −3.00000 −0.199117 −0.0995585 0.995032i \(-0.531743\pi\)
−0.0995585 + 0.995032i \(0.531743\pi\)
\(228\) 0 0
\(229\) −6.00000 −0.396491 −0.198246 0.980152i \(-0.563524\pi\)
−0.198246 + 0.980152i \(0.563524\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −8.00000 −0.524097 −0.262049 0.965055i \(-0.584398\pi\)
−0.262049 + 0.965055i \(0.584398\pi\)
\(234\) 0 0
\(235\) −9.00000 −0.587095
\(236\) 0 0
\(237\) −3.00000 −0.194871
\(238\) 0 0
\(239\) −7.00000 −0.452792 −0.226396 0.974035i \(-0.572694\pi\)
−0.226396 + 0.974035i \(0.572694\pi\)
\(240\) 0 0
\(241\) −24.0000 −1.54598 −0.772988 0.634421i \(-0.781239\pi\)
−0.772988 + 0.634421i \(0.781239\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −18.0000 −1.14531
\(248\) 0 0
\(249\) 36.0000 2.28141
\(250\) 0 0
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 0 0
\(253\) −4.00000 −0.251478
\(254\) 0 0
\(255\) −9.00000 −0.563602
\(256\) 0 0
\(257\) 18.0000 1.12281 0.561405 0.827541i \(-0.310261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −6.00000 −0.371391
\(262\) 0 0
\(263\) 10.0000 0.616626 0.308313 0.951285i \(-0.400236\pi\)
0.308313 + 0.951285i \(0.400236\pi\)
\(264\) 0 0
\(265\) 10.0000 0.614295
\(266\) 0 0
\(267\) −36.0000 −2.20316
\(268\) 0 0
\(269\) 24.0000 1.46331 0.731653 0.681677i \(-0.238749\pi\)
0.731653 + 0.681677i \(0.238749\pi\)
\(270\) 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
\(275\) −1.00000 −0.0603023
\(276\) 0 0
\(277\) −28.0000 −1.68236 −0.841178 0.540758i \(-0.818138\pi\)
−0.841178 + 0.540758i \(0.818138\pi\)
\(278\) 0 0
\(279\) −36.0000 −2.15526
\(280\) 0 0
\(281\) −5.00000 −0.298275 −0.149137 0.988816i \(-0.547650\pi\)
−0.149137 + 0.988816i \(0.547650\pi\)
\(282\) 0 0
\(283\) −21.0000 −1.24832 −0.624160 0.781296i \(-0.714559\pi\)
−0.624160 + 0.781296i \(0.714559\pi\)
\(284\) 0 0
\(285\) −18.0000 −1.06623
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) 45.0000 2.63795
\(292\) 0 0
\(293\) 9.00000 0.525786 0.262893 0.964825i \(-0.415323\pi\)
0.262893 + 0.964825i \(0.415323\pi\)
\(294\) 0 0
\(295\) −6.00000 −0.349334
\(296\) 0 0
\(297\) 9.00000 0.522233
\(298\) 0 0
\(299\) 12.0000 0.693978
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 54.0000 3.10222
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 3.00000 0.171219 0.0856095 0.996329i \(-0.472716\pi\)
0.0856095 + 0.996329i \(0.472716\pi\)
\(308\) 0 0
\(309\) −27.0000 −1.53598
\(310\) 0 0
\(311\) 6.00000 0.340229 0.170114 0.985424i \(-0.445586\pi\)
0.170114 + 0.985424i \(0.445586\pi\)
\(312\) 0 0
\(313\) 3.00000 0.169570 0.0847850 0.996399i \(-0.472980\pi\)
0.0847850 + 0.996399i \(0.472980\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −22.0000 −1.23564 −0.617822 0.786318i \(-0.711985\pi\)
−0.617822 + 0.786318i \(0.711985\pi\)
\(318\) 0 0
\(319\) 1.00000 0.0559893
\(320\) 0 0
\(321\) −6.00000 −0.334887
\(322\) 0 0
\(323\) 18.0000 1.00155
\(324\) 0 0
\(325\) 3.00000 0.166410
\(326\) 0 0
\(327\) 45.0000 2.48851
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −12.0000 −0.659580 −0.329790 0.944054i \(-0.606978\pi\)
−0.329790 + 0.944054i \(0.606978\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −14.0000 −0.764902
\(336\) 0 0
\(337\) −24.0000 −1.30736 −0.653682 0.756770i \(-0.726776\pi\)
−0.653682 + 0.756770i \(0.726776\pi\)
\(338\) 0 0
\(339\) −24.0000 −1.30350
\(340\) 0 0
\(341\) 6.00000 0.324918
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 12.0000 0.646058
\(346\) 0 0
\(347\) −16.0000 −0.858925 −0.429463 0.903085i \(-0.641297\pi\)
−0.429463 + 0.903085i \(0.641297\pi\)
\(348\) 0 0
\(349\) −30.0000 −1.60586 −0.802932 0.596071i \(-0.796728\pi\)
−0.802932 + 0.596071i \(0.796728\pi\)
\(350\) 0 0
\(351\) −27.0000 −1.44115
\(352\) 0 0
\(353\) −3.00000 −0.159674 −0.0798369 0.996808i \(-0.525440\pi\)
−0.0798369 + 0.996808i \(0.525440\pi\)
\(354\) 0 0
\(355\) −8.00000 −0.424596
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −32.0000 −1.68890 −0.844448 0.535638i \(-0.820071\pi\)
−0.844448 + 0.535638i \(0.820071\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) 0 0
\(363\) 30.0000 1.57459
\(364\) 0 0
\(365\) −6.00000 −0.314054
\(366\) 0 0
\(367\) 33.0000 1.72259 0.861293 0.508109i \(-0.169655\pi\)
0.861293 + 0.508109i \(0.169655\pi\)
\(368\) 0 0
\(369\) 36.0000 1.87409
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 4.00000 0.207112 0.103556 0.994624i \(-0.466978\pi\)
0.103556 + 0.994624i \(0.466978\pi\)
\(374\) 0 0
\(375\) 3.00000 0.154919
\(376\) 0 0
\(377\) −3.00000 −0.154508
\(378\) 0 0
\(379\) 24.0000 1.23280 0.616399 0.787434i \(-0.288591\pi\)
0.616399 + 0.787434i \(0.288591\pi\)
\(380\) 0 0
\(381\) −6.00000 −0.307389
\(382\) 0 0
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 36.0000 1.82998
\(388\) 0 0
\(389\) 13.0000 0.659126 0.329563 0.944134i \(-0.393099\pi\)
0.329563 + 0.944134i \(0.393099\pi\)
\(390\) 0 0
\(391\) −12.0000 −0.606866
\(392\) 0 0
\(393\) 36.0000 1.81596
\(394\) 0 0
\(395\) −1.00000 −0.0503155
\(396\) 0 0
\(397\) 9.00000 0.451697 0.225849 0.974162i \(-0.427485\pi\)
0.225849 + 0.974162i \(0.427485\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −19.0000 −0.948815 −0.474407 0.880305i \(-0.657338\pi\)
−0.474407 + 0.880305i \(0.657338\pi\)
\(402\) 0 0
\(403\) −18.0000 −0.896644
\(404\) 0 0
\(405\) −9.00000 −0.447214
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 30.0000 1.48340 0.741702 0.670729i \(-0.234019\pi\)
0.741702 + 0.670729i \(0.234019\pi\)
\(410\) 0 0
\(411\) −24.0000 −1.18383
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 12.0000 0.589057
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 6.00000 0.293119 0.146560 0.989202i \(-0.453180\pi\)
0.146560 + 0.989202i \(0.453180\pi\)
\(420\) 0 0
\(421\) 1.00000 0.0487370 0.0243685 0.999703i \(-0.492242\pi\)
0.0243685 + 0.999703i \(0.492242\pi\)
\(422\) 0 0
\(423\) 54.0000 2.62557
\(424\) 0 0
\(425\) −3.00000 −0.145521
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 9.00000 0.434524
\(430\) 0 0
\(431\) −5.00000 −0.240842 −0.120421 0.992723i \(-0.538424\pi\)
−0.120421 + 0.992723i \(0.538424\pi\)
\(432\) 0 0
\(433\) −30.0000 −1.44171 −0.720854 0.693087i \(-0.756250\pi\)
−0.720854 + 0.693087i \(0.756250\pi\)
\(434\) 0 0
\(435\) −3.00000 −0.143839
\(436\) 0 0
\(437\) −24.0000 −1.14808
\(438\) 0 0
\(439\) −12.0000 −0.572729 −0.286364 0.958121i \(-0.592447\pi\)
−0.286364 + 0.958121i \(0.592447\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −34.0000 −1.61539 −0.807694 0.589601i \(-0.799285\pi\)
−0.807694 + 0.589601i \(0.799285\pi\)
\(444\) 0 0
\(445\) −12.0000 −0.568855
\(446\) 0 0
\(447\) −30.0000 −1.41895
\(448\) 0 0
\(449\) 23.0000 1.08544 0.542719 0.839915i \(-0.317395\pi\)
0.542719 + 0.839915i \(0.317395\pi\)
\(450\) 0 0
\(451\) −6.00000 −0.282529
\(452\) 0 0
\(453\) 45.0000 2.11428
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 6.00000 0.280668 0.140334 0.990104i \(-0.455182\pi\)
0.140334 + 0.990104i \(0.455182\pi\)
\(458\) 0 0
\(459\) 27.0000 1.26025
\(460\) 0 0
\(461\) 36.0000 1.67669 0.838344 0.545142i \(-0.183524\pi\)
0.838344 + 0.545142i \(0.183524\pi\)
\(462\) 0 0
\(463\) 4.00000 0.185896 0.0929479 0.995671i \(-0.470371\pi\)
0.0929479 + 0.995671i \(0.470371\pi\)
\(464\) 0 0
\(465\) −18.0000 −0.834730
\(466\) 0 0
\(467\) 15.0000 0.694117 0.347059 0.937843i \(-0.387180\pi\)
0.347059 + 0.937843i \(0.387180\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −54.0000 −2.48819
\(472\) 0 0
\(473\) −6.00000 −0.275880
\(474\) 0 0
\(475\) −6.00000 −0.275299
\(476\) 0 0
\(477\) −60.0000 −2.74721
\(478\) 0 0
\(479\) 24.0000 1.09659 0.548294 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 15.0000 0.681115
\(486\) 0 0
\(487\) −36.0000 −1.63132 −0.815658 0.578535i \(-0.803625\pi\)
−0.815658 + 0.578535i \(0.803625\pi\)
\(488\) 0 0
\(489\) 48.0000 2.17064
\(490\) 0 0
\(491\) −23.0000 −1.03798 −0.518988 0.854782i \(-0.673691\pi\)
−0.518988 + 0.854782i \(0.673691\pi\)
\(492\) 0 0
\(493\) 3.00000 0.135113
\(494\) 0 0
\(495\) 6.00000 0.269680
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 27.0000 1.20869 0.604343 0.796724i \(-0.293436\pi\)
0.604343 + 0.796724i \(0.293436\pi\)
\(500\) 0 0
\(501\) 9.00000 0.402090
\(502\) 0 0
\(503\) −9.00000 −0.401290 −0.200645 0.979664i \(-0.564304\pi\)
−0.200645 + 0.979664i \(0.564304\pi\)
\(504\) 0 0
\(505\) 18.0000 0.800989
\(506\) 0 0
\(507\) 12.0000 0.532939
\(508\) 0 0
\(509\) −24.0000 −1.06378 −0.531891 0.846813i \(-0.678518\pi\)
−0.531891 + 0.846813i \(0.678518\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 54.0000 2.38416
\(514\) 0 0
\(515\) −9.00000 −0.396587
\(516\) 0 0
\(517\) −9.00000 −0.395820
\(518\) 0 0
\(519\) 9.00000 0.395056
\(520\) 0 0
\(521\) −18.0000 −0.788594 −0.394297 0.918983i \(-0.629012\pi\)
−0.394297 + 0.918983i \(0.629012\pi\)
\(522\) 0 0
\(523\) −24.0000 −1.04945 −0.524723 0.851273i \(-0.675831\pi\)
−0.524723 + 0.851273i \(0.675831\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 18.0000 0.784092
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 36.0000 1.56227
\(532\) 0 0
\(533\) 18.0000 0.779667
\(534\) 0 0
\(535\) −2.00000 −0.0864675
\(536\) 0 0
\(537\) 12.0000 0.517838
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 11.0000 0.472927 0.236463 0.971640i \(-0.424012\pi\)
0.236463 + 0.971640i \(0.424012\pi\)
\(542\) 0 0
\(543\) 18.0000 0.772454
\(544\) 0 0
\(545\) 15.0000 0.642529
\(546\) 0 0
\(547\) −8.00000 −0.342055 −0.171028 0.985266i \(-0.554709\pi\)
−0.171028 + 0.985266i \(0.554709\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 6.00000 0.255609
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 4.00000 0.169485 0.0847427 0.996403i \(-0.472993\pi\)
0.0847427 + 0.996403i \(0.472993\pi\)
\(558\) 0 0
\(559\) 18.0000 0.761319
\(560\) 0 0
\(561\) −9.00000 −0.379980
\(562\) 0 0
\(563\) −12.0000 −0.505740 −0.252870 0.967500i \(-0.581374\pi\)
−0.252870 + 0.967500i \(0.581374\pi\)
\(564\) 0 0
\(565\) −8.00000 −0.336563
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −38.0000 −1.59304 −0.796521 0.604610i \(-0.793329\pi\)
−0.796521 + 0.604610i \(0.793329\pi\)
\(570\) 0 0
\(571\) −4.00000 −0.167395 −0.0836974 0.996491i \(-0.526673\pi\)
−0.0836974 + 0.996491i \(0.526673\pi\)
\(572\) 0 0
\(573\) 51.0000 2.13056
\(574\) 0 0
\(575\) 4.00000 0.166812
\(576\) 0 0
\(577\) 15.0000 0.624458 0.312229 0.950007i \(-0.398924\pi\)
0.312229 + 0.950007i \(0.398924\pi\)
\(578\) 0 0
\(579\) −36.0000 −1.49611
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 10.0000 0.414158
\(584\) 0 0
\(585\) −18.0000 −0.744208
\(586\) 0 0
\(587\) 12.0000 0.495293 0.247647 0.968850i \(-0.420343\pi\)
0.247647 + 0.968850i \(0.420343\pi\)
\(588\) 0 0
\(589\) 36.0000 1.48335
\(590\) 0 0
\(591\) 6.00000 0.246807
\(592\) 0 0
\(593\) −45.0000 −1.84793 −0.923964 0.382479i \(-0.875070\pi\)
−0.923964 + 0.382479i \(0.875070\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 18.0000 0.736691
\(598\) 0 0
\(599\) −17.0000 −0.694601 −0.347301 0.937754i \(-0.612902\pi\)
−0.347301 + 0.937754i \(0.612902\pi\)
\(600\) 0 0
\(601\) −30.0000 −1.22373 −0.611863 0.790964i \(-0.709580\pi\)
−0.611863 + 0.790964i \(0.709580\pi\)
\(602\) 0 0
\(603\) 84.0000 3.42074
\(604\) 0 0
\(605\) 10.0000 0.406558
\(606\) 0 0
\(607\) −21.0000 −0.852364 −0.426182 0.904638i \(-0.640142\pi\)
−0.426182 + 0.904638i \(0.640142\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 27.0000 1.09230
\(612\) 0 0
\(613\) −2.00000 −0.0807792 −0.0403896 0.999184i \(-0.512860\pi\)
−0.0403896 + 0.999184i \(0.512860\pi\)
\(614\) 0 0
\(615\) 18.0000 0.725830
\(616\) 0 0
\(617\) 4.00000 0.161034 0.0805170 0.996753i \(-0.474343\pi\)
0.0805170 + 0.996753i \(0.474343\pi\)
\(618\) 0 0
\(619\) 24.0000 0.964641 0.482321 0.875995i \(-0.339794\pi\)
0.482321 + 0.875995i \(0.339794\pi\)
\(620\) 0 0
\(621\) −36.0000 −1.44463
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −18.0000 −0.718851
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 11.0000 0.437903 0.218952 0.975736i \(-0.429736\pi\)
0.218952 + 0.975736i \(0.429736\pi\)
\(632\) 0 0
\(633\) 45.0000 1.78859
\(634\) 0 0
\(635\) −2.00000 −0.0793676
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 48.0000 1.89885
\(640\) 0 0
\(641\) −2.00000 −0.0789953 −0.0394976 0.999220i \(-0.512576\pi\)
−0.0394976 + 0.999220i \(0.512576\pi\)
\(642\) 0 0
\(643\) −27.0000 −1.06478 −0.532388 0.846500i \(-0.678705\pi\)
−0.532388 + 0.846500i \(0.678705\pi\)
\(644\) 0 0
\(645\) 18.0000 0.708749
\(646\) 0 0
\(647\) −12.0000 −0.471769 −0.235884 0.971781i \(-0.575799\pi\)
−0.235884 + 0.971781i \(0.575799\pi\)
\(648\) 0 0
\(649\) −6.00000 −0.235521
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 14.0000 0.547862 0.273931 0.961749i \(-0.411676\pi\)
0.273931 + 0.961749i \(0.411676\pi\)
\(654\) 0 0
\(655\) 12.0000 0.468879
\(656\) 0 0
\(657\) 36.0000 1.40449
\(658\) 0 0
\(659\) 19.0000 0.740135 0.370067 0.929005i \(-0.379335\pi\)
0.370067 + 0.929005i \(0.379335\pi\)
\(660\) 0 0
\(661\) 12.0000 0.466746 0.233373 0.972387i \(-0.425024\pi\)
0.233373 + 0.972387i \(0.425024\pi\)
\(662\) 0 0
\(663\) 27.0000 1.04859
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −4.00000 −0.154881
\(668\) 0 0
\(669\) 9.00000 0.347960
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 12.0000 0.462566 0.231283 0.972887i \(-0.425708\pi\)
0.231283 + 0.972887i \(0.425708\pi\)
\(674\) 0 0
\(675\) −9.00000 −0.346410
\(676\) 0 0
\(677\) −9.00000 −0.345898 −0.172949 0.984931i \(-0.555330\pi\)
−0.172949 + 0.984931i \(0.555330\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 9.00000 0.344881
\(682\) 0 0
\(683\) −8.00000 −0.306111 −0.153056 0.988218i \(-0.548911\pi\)
−0.153056 + 0.988218i \(0.548911\pi\)
\(684\) 0 0
\(685\) −8.00000 −0.305664
\(686\) 0 0
\(687\) 18.0000 0.686743
\(688\) 0 0
\(689\) −30.0000 −1.14291
\(690\) 0 0
\(691\) −36.0000 −1.36950 −0.684752 0.728776i \(-0.740090\pi\)
−0.684752 + 0.728776i \(0.740090\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −18.0000 −0.681799
\(698\) 0 0
\(699\) 24.0000 0.907763
\(700\) 0 0
\(701\) 47.0000 1.77517 0.887583 0.460648i \(-0.152383\pi\)
0.887583 + 0.460648i \(0.152383\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 27.0000 1.01688
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −33.0000 −1.23934 −0.619671 0.784862i \(-0.712734\pi\)
−0.619671 + 0.784862i \(0.712734\pi\)
\(710\) 0 0
\(711\) 6.00000 0.225018
\(712\) 0 0
\(713\) −24.0000 −0.898807
\(714\) 0 0
\(715\) 3.00000 0.112194
\(716\) 0 0
\(717\) 21.0000 0.784259
\(718\) 0 0
\(719\) 36.0000 1.34257 0.671287 0.741198i \(-0.265742\pi\)
0.671287 + 0.741198i \(0.265742\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 72.0000 2.67771
\(724\) 0 0
\(725\) −1.00000 −0.0371391
\(726\) 0 0
\(727\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) −18.0000 −0.665754
\(732\) 0 0
\(733\) 15.0000 0.554038 0.277019 0.960864i \(-0.410654\pi\)
0.277019 + 0.960864i \(0.410654\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −14.0000 −0.515697
\(738\) 0 0
\(739\) 43.0000 1.58178 0.790890 0.611958i \(-0.209618\pi\)
0.790890 + 0.611958i \(0.209618\pi\)
\(740\) 0 0
\(741\) 54.0000 1.98374
\(742\) 0 0
\(743\) 22.0000 0.807102 0.403551 0.914957i \(-0.367776\pi\)
0.403551 + 0.914957i \(0.367776\pi\)
\(744\) 0 0
\(745\) −10.0000 −0.366372
\(746\) 0 0
\(747\) −72.0000 −2.63434
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −15.0000 −0.547358 −0.273679 0.961821i \(-0.588241\pi\)
−0.273679 + 0.961821i \(0.588241\pi\)
\(752\) 0 0
\(753\) −36.0000 −1.31191
\(754\) 0 0
\(755\) 15.0000 0.545906
\(756\) 0 0
\(757\) 48.0000 1.74459 0.872295 0.488980i \(-0.162631\pi\)
0.872295 + 0.488980i \(0.162631\pi\)
\(758\) 0 0
\(759\) 12.0000 0.435572
\(760\) 0 0
\(761\) 12.0000 0.435000 0.217500 0.976060i \(-0.430210\pi\)
0.217500 + 0.976060i \(0.430210\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 18.0000 0.650791
\(766\) 0 0
\(767\) 18.0000 0.649942
\(768\) 0 0
\(769\) 42.0000 1.51456 0.757279 0.653091i \(-0.226528\pi\)
0.757279 + 0.653091i \(0.226528\pi\)
\(770\) 0 0
\(771\) −54.0000 −1.94476
\(772\) 0 0
\(773\) 3.00000 0.107903 0.0539513 0.998544i \(-0.482818\pi\)
0.0539513 + 0.998544i \(0.482818\pi\)
\(774\) 0 0
\(775\) −6.00000 −0.215526
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −36.0000 −1.28983
\(780\) 0 0
\(781\) −8.00000 −0.286263
\(782\) 0 0
\(783\) 9.00000 0.321634
\(784\) 0 0
\(785\) −18.0000 −0.642448
\(786\) 0 0
\(787\) −39.0000 −1.39020 −0.695100 0.718913i \(-0.744640\pi\)
−0.695100 + 0.718913i \(0.744640\pi\)
\(788\) 0 0
\(789\) −30.0000 −1.06803
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) −30.0000 −1.06399
\(796\) 0 0
\(797\) −27.0000 −0.956389 −0.478195 0.878254i \(-0.658709\pi\)
−0.478195 + 0.878254i \(0.658709\pi\)
\(798\) 0 0
\(799\) −27.0000 −0.955191
\(800\) 0 0
\(801\) 72.0000 2.54399
\(802\) 0 0
\(803\) −6.00000 −0.211735
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −72.0000 −2.53452
\(808\) 0 0
\(809\) −35.0000 −1.23053 −0.615267 0.788319i \(-0.710952\pi\)
−0.615267 + 0.788319i \(0.710952\pi\)
\(810\) 0 0
\(811\) −48.0000 −1.68551 −0.842754 0.538299i \(-0.819067\pi\)
−0.842754 + 0.538299i \(0.819067\pi\)
\(812\) 0 0
\(813\) 72.0000 2.52515
\(814\) 0 0
\(815\) 16.0000 0.560456
\(816\) 0 0
\(817\) −36.0000 −1.25948
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −23.0000 −0.802706 −0.401353 0.915924i \(-0.631460\pi\)
−0.401353 + 0.915924i \(0.631460\pi\)
\(822\) 0 0
\(823\) −50.0000 −1.74289 −0.871445 0.490493i \(-0.836817\pi\)
−0.871445 + 0.490493i \(0.836817\pi\)
\(824\) 0 0
\(825\) 3.00000 0.104447
\(826\) 0 0
\(827\) 16.0000 0.556375 0.278187 0.960527i \(-0.410266\pi\)
0.278187 + 0.960527i \(0.410266\pi\)
\(828\) 0 0
\(829\) 6.00000 0.208389 0.104194 0.994557i \(-0.466774\pi\)
0.104194 + 0.994557i \(0.466774\pi\)
\(830\) 0 0
\(831\) 84.0000 2.91393
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 3.00000 0.103819
\(836\) 0 0
\(837\) 54.0000 1.86651
\(838\) 0 0
\(839\) 24.0000 0.828572 0.414286 0.910147i \(-0.364031\pi\)
0.414286 + 0.910147i \(0.364031\pi\)
\(840\) 0 0
\(841\) −28.0000 −0.965517
\(842\) 0 0
\(843\) 15.0000 0.516627
\(844\) 0 0
\(845\) 4.00000 0.137604
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 63.0000 2.16215
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 30.0000 1.02718 0.513590 0.858036i \(-0.328315\pi\)
0.513590 + 0.858036i \(0.328315\pi\)
\(854\) 0 0
\(855\) 36.0000 1.23117
\(856\) 0 0
\(857\) 54.0000 1.84460 0.922302 0.386469i \(-0.126305\pi\)
0.922302 + 0.386469i \(0.126305\pi\)
\(858\) 0 0
\(859\) −12.0000 −0.409435 −0.204717 0.978821i \(-0.565628\pi\)
−0.204717 + 0.978821i \(0.565628\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 34.0000 1.15737 0.578687 0.815550i \(-0.303565\pi\)
0.578687 + 0.815550i \(0.303565\pi\)
\(864\) 0 0
\(865\) 3.00000 0.102003
\(866\) 0 0
\(867\) 24.0000 0.815083
\(868\) 0 0
\(869\) −1.00000 −0.0339227
\(870\) 0 0
\(871\) 42.0000 1.42312
\(872\) 0 0
\(873\) −90.0000 −3.04604
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −22.0000 −0.742887 −0.371444 0.928456i \(-0.621137\pi\)
−0.371444 + 0.928456i \(0.621137\pi\)
\(878\) 0 0
\(879\) −27.0000 −0.910687
\(880\) 0 0
\(881\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(884\) 0 0
\(885\) 18.0000 0.605063
\(886\) 0 0
\(887\) 24.0000 0.805841 0.402921 0.915235i \(-0.367995\pi\)
0.402921 + 0.915235i \(0.367995\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −9.00000 −0.301511
\(892\) 0 0
\(893\) −54.0000 −1.80704
\(894\) 0 0
\(895\) 4.00000 0.133705
\(896\) 0 0
\(897\) −36.0000 −1.20201
\(898\) 0 0
\(899\) 6.00000 0.200111
\(900\) 0 0
\(901\) 30.0000 0.999445
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 6.00000 0.199447
\(906\) 0 0
\(907\) −24.0000 −0.796907 −0.398453 0.917189i \(-0.630453\pi\)
−0.398453 + 0.917189i \(0.630453\pi\)
\(908\) 0 0
\(909\) −108.000 −3.58213
\(910\) 0 0
\(911\) −40.0000 −1.32526 −0.662630 0.748947i \(-0.730560\pi\)
−0.662630 + 0.748947i \(0.730560\pi\)
\(912\) 0 0
\(913\) 12.0000 0.397142
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 9.00000 0.296883 0.148441 0.988921i \(-0.452574\pi\)
0.148441 + 0.988921i \(0.452574\pi\)
\(920\) 0 0
\(921\) −9.00000 −0.296560
\(922\) 0 0
\(923\) 24.0000 0.789970
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 54.0000 1.77359
\(928\) 0 0
\(929\) 42.0000 1.37798 0.688988 0.724773i \(-0.258055\pi\)
0.688988 + 0.724773i \(0.258055\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −18.0000 −0.589294
\(934\) 0 0
\(935\) −3.00000 −0.0981105
\(936\) 0 0
\(937\) −27.0000 −0.882052 −0.441026 0.897494i \(-0.645385\pi\)
−0.441026 + 0.897494i \(0.645385\pi\)
\(938\) 0 0
\(939\) −9.00000 −0.293704
\(940\) 0 0
\(941\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(942\) 0 0
\(943\) 24.0000 0.781548
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 2.00000 0.0649913 0.0324956 0.999472i \(-0.489654\pi\)
0.0324956 + 0.999472i \(0.489654\pi\)
\(948\) 0 0
\(949\) 18.0000 0.584305
\(950\) 0 0
\(951\) 66.0000 2.14020
\(952\) 0 0
\(953\) −38.0000 −1.23094 −0.615470 0.788160i \(-0.711034\pi\)
−0.615470 + 0.788160i \(0.711034\pi\)
\(954\) 0 0
\(955\) 17.0000 0.550107
\(956\) 0 0
\(957\) −3.00000 −0.0969762
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 5.00000 0.161290
\(962\) 0 0
\(963\) 12.0000 0.386695
\(964\) 0 0
\(965\) −12.0000 −0.386294
\(966\) 0 0
\(967\) 22.0000 0.707472 0.353736 0.935345i \(-0.384911\pi\)
0.353736 + 0.935345i \(0.384911\pi\)
\(968\) 0 0
\(969\) −54.0000 −1.73473
\(970\) 0 0
\(971\) 18.0000 0.577647 0.288824 0.957382i \(-0.406736\pi\)
0.288824 + 0.957382i \(0.406736\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −9.00000 −0.288231
\(976\) 0 0
\(977\) 34.0000 1.08776 0.543878 0.839164i \(-0.316955\pi\)
0.543878 + 0.839164i \(0.316955\pi\)
\(978\) 0 0
\(979\) −12.0000 −0.383522
\(980\) 0 0
\(981\) −90.0000 −2.87348
\(982\) 0 0
\(983\) −45.0000 −1.43528 −0.717639 0.696416i \(-0.754777\pi\)
−0.717639 + 0.696416i \(0.754777\pi\)
\(984\) 0 0
\(985\) 2.00000 0.0637253
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 24.0000 0.763156
\(990\) 0 0
\(991\) 36.0000 1.14358 0.571789 0.820401i \(-0.306250\pi\)
0.571789 + 0.820401i \(0.306250\pi\)
\(992\) 0 0
\(993\) 36.0000 1.14243
\(994\) 0 0
\(995\) 6.00000 0.190213
\(996\) 0 0
\(997\) −3.00000 −0.0950110 −0.0475055 0.998871i \(-0.515127\pi\)
−0.0475055 + 0.998871i \(0.515127\pi\)
\(998\) 0 0
\(999\) 0 0
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 3920.2.a.a.1.1 1
4.3 odd 2 245.2.a.b.1.1 yes 1
7.6 odd 2 3920.2.a.bj.1.1 1
12.11 even 2 2205.2.a.l.1.1 1
20.3 even 4 1225.2.b.a.99.2 2
20.7 even 4 1225.2.b.a.99.1 2
20.19 odd 2 1225.2.a.h.1.1 1
28.3 even 6 245.2.e.d.226.1 2
28.11 odd 6 245.2.e.c.226.1 2
28.19 even 6 245.2.e.d.116.1 2
28.23 odd 6 245.2.e.c.116.1 2
28.27 even 2 245.2.a.a.1.1 1
84.83 odd 2 2205.2.a.j.1.1 1
140.27 odd 4 1225.2.b.b.99.1 2
140.83 odd 4 1225.2.b.b.99.2 2
140.139 even 2 1225.2.a.j.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
245.2.a.a.1.1 1 28.27 even 2
245.2.a.b.1.1 yes 1 4.3 odd 2
245.2.e.c.116.1 2 28.23 odd 6
245.2.e.c.226.1 2 28.11 odd 6
245.2.e.d.116.1 2 28.19 even 6
245.2.e.d.226.1 2 28.3 even 6
1225.2.a.h.1.1 1 20.19 odd 2
1225.2.a.j.1.1 1 140.139 even 2
1225.2.b.a.99.1 2 20.7 even 4
1225.2.b.a.99.2 2 20.3 even 4
1225.2.b.b.99.1 2 140.27 odd 4
1225.2.b.b.99.2 2 140.83 odd 4
2205.2.a.j.1.1 1 84.83 odd 2
2205.2.a.l.1.1 1 12.11 even 2
3920.2.a.a.1.1 1 1.1 even 1 trivial
3920.2.a.bj.1.1 1 7.6 odd 2