Properties

Label 2205.2.a.l.1.1
Level $2205$
Weight $2$
Character 2205.1
Self dual yes
Analytic conductor $17.607$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2205.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(17.6070136457\)
Analytic rank: \(0\)
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 \(\chi\) \(=\) 2205.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.00000 q^{2} +2.00000 q^{4} +1.00000 q^{5} +O(q^{10})\) \(q+2.00000 q^{2} +2.00000 q^{4} +1.00000 q^{5} +2.00000 q^{10} -1.00000 q^{11} +3.00000 q^{13} -4.00000 q^{16} +3.00000 q^{17} +6.00000 q^{19} +2.00000 q^{20} -2.00000 q^{22} +4.00000 q^{23} +1.00000 q^{25} +6.00000 q^{26} +1.00000 q^{29} +6.00000 q^{31} -8.00000 q^{32} +6.00000 q^{34} +12.0000 q^{38} -6.00000 q^{41} -6.00000 q^{43} -2.00000 q^{44} +8.00000 q^{46} +9.00000 q^{47} +2.00000 q^{50} +6.00000 q^{52} +10.0000 q^{53} -1.00000 q^{55} +2.00000 q^{58} +6.00000 q^{59} +12.0000 q^{62} -8.00000 q^{64} +3.00000 q^{65} -14.0000 q^{67} +6.00000 q^{68} +8.00000 q^{71} +6.00000 q^{73} +12.0000 q^{76} -1.00000 q^{79} -4.00000 q^{80} -12.0000 q^{82} -12.0000 q^{83} +3.00000 q^{85} -12.0000 q^{86} -12.0000 q^{89} +8.00000 q^{92} +18.0000 q^{94} +6.00000 q^{95} -15.0000 q^{97} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(3\) 0 0
\(4\) 2.00000 1.00000
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0 0
\(10\) 2.00000 0.632456
\(11\) −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
\(15\) 0 0
\(16\) −4.00000 −1.00000
\(17\) 3.00000 0.727607 0.363803 0.931476i \(-0.381478\pi\)
0.363803 + 0.931476i \(0.381478\pi\)
\(18\) 0 0
\(19\) 6.00000 1.37649 0.688247 0.725476i \(-0.258380\pi\)
0.688247 + 0.725476i \(0.258380\pi\)
\(20\) 2.00000 0.447214
\(21\) 0 0
\(22\) −2.00000 −0.426401
\(23\) 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
\(26\) 6.00000 1.17670
\(27\) 0 0
\(28\) 0 0
\(29\) 1.00000 0.185695 0.0928477 0.995680i \(-0.470403\pi\)
0.0928477 + 0.995680i \(0.470403\pi\)
\(30\) 0 0
\(31\) 6.00000 1.07763 0.538816 0.842424i \(-0.318872\pi\)
0.538816 + 0.842424i \(0.318872\pi\)
\(32\) −8.00000 −1.41421
\(33\) 0 0
\(34\) 6.00000 1.02899
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 12.0000 1.94666
\(39\) 0 0
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) −6.00000 −0.914991 −0.457496 0.889212i \(-0.651253\pi\)
−0.457496 + 0.889212i \(0.651253\pi\)
\(44\) −2.00000 −0.301511
\(45\) 0 0
\(46\) 8.00000 1.17954
\(47\) 9.00000 1.31278 0.656392 0.754420i \(-0.272082\pi\)
0.656392 + 0.754420i \(0.272082\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 2.00000 0.282843
\(51\) 0 0
\(52\) 6.00000 0.832050
\(53\) 10.0000 1.37361 0.686803 0.726844i \(-0.259014\pi\)
0.686803 + 0.726844i \(0.259014\pi\)
\(54\) 0 0
\(55\) −1.00000 −0.134840
\(56\) 0 0
\(57\) 0 0
\(58\) 2.00000 0.262613
\(59\) 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 \(0.5\pi\)
\(62\) 12.0000 1.52400
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) 3.00000 0.372104
\(66\) 0 0
\(67\) −14.0000 −1.71037 −0.855186 0.518321i \(-0.826557\pi\)
−0.855186 + 0.518321i \(0.826557\pi\)
\(68\) 6.00000 0.727607
\(69\) 0 0
\(70\) 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
\(75\) 0 0
\(76\) 12.0000 1.37649
\(77\) 0 0
\(78\) 0 0
\(79\) −1.00000 −0.112509 −0.0562544 0.998416i \(-0.517916\pi\)
−0.0562544 + 0.998416i \(0.517916\pi\)
\(80\) −4.00000 −0.447214
\(81\) 0 0
\(82\) −12.0000 −1.32518
\(83\) −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
\(86\) −12.0000 −1.29399
\(87\) 0 0
\(88\) 0 0
\(89\) −12.0000 −1.27200 −0.635999 0.771690i \(-0.719412\pi\)
−0.635999 + 0.771690i \(0.719412\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 8.00000 0.834058
\(93\) 0 0
\(94\) 18.0000 1.85656
\(95\) 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
\(99\) 0 0
\(100\) 2.00000 0.200000
\(101\) 18.0000 1.79107 0.895533 0.444994i \(-0.146794\pi\)
0.895533 + 0.444994i \(0.146794\pi\)
\(102\) 0 0
\(103\) −9.00000 −0.886796 −0.443398 0.896325i \(-0.646227\pi\)
−0.443398 + 0.896325i \(0.646227\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 20.0000 1.94257
\(107\) 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 \(0.755111\pi\)
\(110\) −2.00000 −0.190693
\(111\) 0 0
\(112\) 0 0
\(113\) −8.00000 −0.752577 −0.376288 0.926503i \(-0.622800\pi\)
−0.376288 + 0.926503i \(0.622800\pi\)
\(114\) 0 0
\(115\) 4.00000 0.373002
\(116\) 2.00000 0.185695
\(117\) 0 0
\(118\) 12.0000 1.10469
\(119\) 0 0
\(120\) 0 0
\(121\) −10.0000 −0.909091
\(122\) 0 0
\(123\) 0 0
\(124\) 12.0000 1.07763
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −2.00000 −0.177471 −0.0887357 0.996055i \(-0.528283\pi\)
−0.0887357 + 0.996055i \(0.528283\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 6.00000 0.526235
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −28.0000 −2.41883
\(135\) 0 0
\(136\) 0 0
\(137\) −8.00000 −0.683486 −0.341743 0.939793i \(-0.611017\pi\)
−0.341743 + 0.939793i \(0.611017\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 16.0000 1.34269
\(143\) −3.00000 −0.250873
\(144\) 0 0
\(145\) 1.00000 0.0830455
\(146\) 12.0000 0.993127
\(147\) 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
\(151\) 15.0000 1.22068 0.610341 0.792139i \(-0.291032\pi\)
0.610341 + 0.792139i \(0.291032\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 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 \(0.244931\pi\)
\(158\) −2.00000 −0.159111
\(159\) 0 0
\(160\) −8.00000 −0.632456
\(161\) 0 0
\(162\) 0 0
\(163\) 16.0000 1.25322 0.626608 0.779334i \(-0.284443\pi\)
0.626608 + 0.779334i \(0.284443\pi\)
\(164\) −12.0000 −0.937043
\(165\) 0 0
\(166\) −24.0000 −1.86276
\(167\) −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
\(170\) 6.00000 0.460179
\(171\) 0 0
\(172\) −12.0000 −0.914991
\(173\) 3.00000 0.228086 0.114043 0.993476i \(-0.463620\pi\)
0.114043 + 0.993476i \(0.463620\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 4.00000 0.301511
\(177\) 0 0
\(178\) −24.0000 −1.79888
\(179\) −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
\(188\) 18.0000 1.31278
\(189\) 0 0
\(190\) 12.0000 0.870572
\(191\) −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 \(0.357847\pi\)
\(194\) −30.0000 −2.15387
\(195\) 0 0
\(196\) 0 0
\(197\) 2.00000 0.142494 0.0712470 0.997459i \(-0.477302\pi\)
0.0712470 + 0.997459i \(0.477302\pi\)
\(198\) 0 0
\(199\) 6.00000 0.425329 0.212664 0.977125i \(-0.431786\pi\)
0.212664 + 0.977125i \(0.431786\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 36.0000 2.53295
\(203\) 0 0
\(204\) 0 0
\(205\) −6.00000 −0.419058
\(206\) −18.0000 −1.25412
\(207\) 0 0
\(208\) −12.0000 −0.832050
\(209\) −6.00000 −0.415029
\(210\) 0 0
\(211\) 15.0000 1.03264 0.516321 0.856395i \(-0.327301\pi\)
0.516321 + 0.856395i \(0.327301\pi\)
\(212\) 20.0000 1.37361
\(213\) 0 0
\(214\) 4.00000 0.273434
\(215\) −6.00000 −0.409197
\(216\) 0 0
\(217\) 0 0
\(218\) −30.0000 −2.03186
\(219\) 0 0
\(220\) −2.00000 −0.134840
\(221\) 9.00000 0.605406
\(222\) 0 0
\(223\) 3.00000 0.200895 0.100447 0.994942i \(-0.467973\pi\)
0.100447 + 0.994942i \(0.467973\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −16.0000 −1.06430
\(227\) −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 \(0.563524\pi\)
\(230\) 8.00000 0.527504
\(231\) 0 0
\(232\) 0 0
\(233\) 8.00000 0.524097 0.262049 0.965055i \(-0.415602\pi\)
0.262049 + 0.965055i \(0.415602\pi\)
\(234\) 0 0
\(235\) 9.00000 0.587095
\(236\) 12.0000 0.781133
\(237\) 0 0
\(238\) 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 \(0.781239\pi\)
\(242\) −20.0000 −1.28565
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 18.0000 1.14531
\(248\) 0 0
\(249\) 0 0
\(250\) 2.00000 0.126491
\(251\) 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
\(254\) −4.00000 −0.250982
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) −18.0000 −1.12281 −0.561405 0.827541i \(-0.689739\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 6.00000 0.372104
\(261\) 0 0
\(262\) −24.0000 −1.48272
\(263\) 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
\(267\) 0 0
\(268\) −28.0000 −1.71037
\(269\) −24.0000 −1.46331 −0.731653 0.681677i \(-0.761251\pi\)
−0.731653 + 0.681677i \(0.761251\pi\)
\(270\) 0 0
\(271\) 24.0000 1.45790 0.728948 0.684569i \(-0.240010\pi\)
0.728948 + 0.684569i \(0.240010\pi\)
\(272\) −12.0000 −0.727607
\(273\) 0 0
\(274\) −16.0000 −0.966595
\(275\) −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
\(279\) 0 0
\(280\) 0 0
\(281\) 5.00000 0.298275 0.149137 0.988816i \(-0.452350\pi\)
0.149137 + 0.988816i \(0.452350\pi\)
\(282\) 0 0
\(283\) 21.0000 1.24832 0.624160 0.781296i \(-0.285441\pi\)
0.624160 + 0.781296i \(0.285441\pi\)
\(284\) 16.0000 0.949425
\(285\) 0 0
\(286\) −6.00000 −0.354787
\(287\) 0 0
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 2.00000 0.117444
\(291\) 0 0
\(292\) 12.0000 0.702247
\(293\) −9.00000 −0.525786 −0.262893 0.964825i \(-0.584677\pi\)
−0.262893 + 0.964825i \(0.584677\pi\)
\(294\) 0 0
\(295\) 6.00000 0.349334
\(296\) 0 0
\(297\) 0 0
\(298\) −20.0000 −1.15857
\(299\) 12.0000 0.693978
\(300\) 0 0
\(301\) 0 0
\(302\) 30.0000 1.72631
\(303\) 0 0
\(304\) −24.0000 −1.37649
\(305\) 0 0
\(306\) 0 0
\(307\) −3.00000 −0.171219 −0.0856095 0.996329i \(-0.527284\pi\)
−0.0856095 + 0.996329i \(0.527284\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 12.0000 0.681554
\(311\) 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 \(0.472980\pi\)
\(314\) 36.0000 2.03160
\(315\) 0 0
\(316\) −2.00000 −0.112509
\(317\) 22.0000 1.23564 0.617822 0.786318i \(-0.288015\pi\)
0.617822 + 0.786318i \(0.288015\pi\)
\(318\) 0 0
\(319\) −1.00000 −0.0559893
\(320\) −8.00000 −0.447214
\(321\) 0 0
\(322\) 0 0
\(323\) 18.0000 1.00155
\(324\) 0 0
\(325\) 3.00000 0.166410
\(326\) 32.0000 1.77232
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 12.0000 0.659580 0.329790 0.944054i \(-0.393022\pi\)
0.329790 + 0.944054i \(0.393022\pi\)
\(332\) −24.0000 −1.31717
\(333\) 0 0
\(334\) −6.00000 −0.328305
\(335\) −14.0000 −0.764902
\(336\) 0 0
\(337\) −24.0000 −1.30736 −0.653682 0.756770i \(-0.726776\pi\)
−0.653682 + 0.756770i \(0.726776\pi\)
\(338\) −8.00000 −0.435143
\(339\) 0 0
\(340\) 6.00000 0.325396
\(341\) −6.00000 −0.324918
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 6.00000 0.322562
\(347\) −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
\(351\) 0 0
\(352\) 8.00000 0.426401
\(353\) 3.00000 0.159674 0.0798369 0.996808i \(-0.474560\pi\)
0.0798369 + 0.996808i \(0.474560\pi\)
\(354\) 0 0
\(355\) 8.00000 0.424596
\(356\) −24.0000 −1.27200
\(357\) 0 0
\(358\) −8.00000 −0.422813
\(359\) −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
\(362\) −12.0000 −0.630706
\(363\) 0 0
\(364\) 0 0
\(365\) 6.00000 0.314054
\(366\) 0 0
\(367\) −33.0000 −1.72259 −0.861293 0.508109i \(-0.830345\pi\)
−0.861293 + 0.508109i \(0.830345\pi\)
\(368\) −16.0000 −0.834058
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 4.00000 0.207112 0.103556 0.994624i \(-0.466978\pi\)
0.103556 + 0.994624i \(0.466978\pi\)
\(374\) −6.00000 −0.310253
\(375\) 0 0
\(376\) 0 0
\(377\) 3.00000 0.154508
\(378\) 0 0
\(379\) −24.0000 −1.23280 −0.616399 0.787434i \(-0.711409\pi\)
−0.616399 + 0.787434i \(0.711409\pi\)
\(380\) 12.0000 0.615587
\(381\) 0 0
\(382\) −34.0000 −1.73959
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 24.0000 1.22157
\(387\) 0 0
\(388\) −30.0000 −1.52302
\(389\) −13.0000 −0.659126 −0.329563 0.944134i \(-0.606901\pi\)
−0.329563 + 0.944134i \(0.606901\pi\)
\(390\) 0 0
\(391\) 12.0000 0.606866
\(392\) 0 0
\(393\) 0 0
\(394\) 4.00000 0.201517
\(395\) −1.00000 −0.0503155
\(396\) 0 0
\(397\) 9.00000 0.451697 0.225849 0.974162i \(-0.427485\pi\)
0.225849 + 0.974162i \(0.427485\pi\)
\(398\) 12.0000 0.601506
\(399\) 0 0
\(400\) −4.00000 −0.200000
\(401\) 19.0000 0.948815 0.474407 0.880305i \(-0.342662\pi\)
0.474407 + 0.880305i \(0.342662\pi\)
\(402\) 0 0
\(403\) 18.0000 0.896644
\(404\) 36.0000 1.79107
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 30.0000 1.48340 0.741702 0.670729i \(-0.234019\pi\)
0.741702 + 0.670729i \(0.234019\pi\)
\(410\) −12.0000 −0.592638
\(411\) 0 0
\(412\) −18.0000 −0.886796
\(413\) 0 0
\(414\) 0 0
\(415\) −12.0000 −0.589057
\(416\) −24.0000 −1.17670
\(417\) 0 0
\(418\) −12.0000 −0.586939
\(419\) 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 \(0.492242\pi\)
\(422\) 30.0000 1.46038
\(423\) 0 0
\(424\) 0 0
\(425\) 3.00000 0.145521
\(426\) 0 0
\(427\) 0 0
\(428\) 4.00000 0.193347
\(429\) 0 0
\(430\) −12.0000 −0.578691
\(431\) −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
\(435\) 0 0
\(436\) −30.0000 −1.43674
\(437\) 24.0000 1.14808
\(438\) 0 0
\(439\) 12.0000 0.572729 0.286364 0.958121i \(-0.407553\pi\)
0.286364 + 0.958121i \(0.407553\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 18.0000 0.856173
\(443\) −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
\(446\) 6.00000 0.284108
\(447\) 0 0
\(448\) 0 0
\(449\) −23.0000 −1.08544 −0.542719 0.839915i \(-0.682605\pi\)
−0.542719 + 0.839915i \(0.682605\pi\)
\(450\) 0 0
\(451\) 6.00000 0.282529
\(452\) −16.0000 −0.752577
\(453\) 0 0
\(454\) −6.00000 −0.281594
\(455\) 0 0
\(456\) 0 0
\(457\) 6.00000 0.280668 0.140334 0.990104i \(-0.455182\pi\)
0.140334 + 0.990104i \(0.455182\pi\)
\(458\) −12.0000 −0.560723
\(459\) 0 0
\(460\) 8.00000 0.373002
\(461\) −36.0000 −1.67669 −0.838344 0.545142i \(-0.816476\pi\)
−0.838344 + 0.545142i \(0.816476\pi\)
\(462\) 0 0
\(463\) −4.00000 −0.185896 −0.0929479 0.995671i \(-0.529629\pi\)
−0.0929479 + 0.995671i \(0.529629\pi\)
\(464\) −4.00000 −0.185695
\(465\) 0 0
\(466\) 16.0000 0.741186
\(467\) 15.0000 0.694117 0.347059 0.937843i \(-0.387180\pi\)
0.347059 + 0.937843i \(0.387180\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 18.0000 0.830278
\(471\) 0 0
\(472\) 0 0
\(473\) 6.00000 0.275880
\(474\) 0 0
\(475\) 6.00000 0.275299
\(476\) 0 0
\(477\) 0 0
\(478\) −14.0000 −0.640345
\(479\) 24.0000 1.09659 0.548294 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −48.0000 −2.18634
\(483\) 0 0
\(484\) −20.0000 −0.909091
\(485\) −15.0000 −0.681115
\(486\) 0 0
\(487\) 36.0000 1.63132 0.815658 0.578535i \(-0.196375\pi\)
0.815658 + 0.578535i \(0.196375\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 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
\(494\) 36.0000 1.61972
\(495\) 0 0
\(496\) −24.0000 −1.07763
\(497\) 0 0
\(498\) 0 0
\(499\) −27.0000 −1.20869 −0.604343 0.796724i \(-0.706564\pi\)
−0.604343 + 0.796724i \(0.706564\pi\)
\(500\) 2.00000 0.0894427
\(501\) 0 0
\(502\) 24.0000 1.07117
\(503\) −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
\(506\) −8.00000 −0.355643
\(507\) 0 0
\(508\) −4.00000 −0.177471
\(509\) 24.0000 1.06378 0.531891 0.846813i \(-0.321482\pi\)
0.531891 + 0.846813i \(0.321482\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 32.0000 1.41421
\(513\) 0 0
\(514\) −36.0000 −1.58789
\(515\) −9.00000 −0.396587
\(516\) 0 0
\(517\) −9.00000 −0.395820
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 18.0000 0.788594 0.394297 0.918983i \(-0.370988\pi\)
0.394297 + 0.918983i \(0.370988\pi\)
\(522\) 0 0
\(523\) 24.0000 1.04945 0.524723 0.851273i \(-0.324169\pi\)
0.524723 + 0.851273i \(0.324169\pi\)
\(524\) −24.0000 −1.04844
\(525\) 0 0
\(526\) 20.0000 0.872041
\(527\) 18.0000 0.784092
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 20.0000 0.868744
\(531\) 0 0
\(532\) 0 0
\(533\) −18.0000 −0.779667
\(534\) 0 0
\(535\) 2.00000 0.0864675
\(536\) 0 0
\(537\) 0 0
\(538\) −48.0000 −2.06943
\(539\) 0 0
\(540\) 0 0
\(541\) 11.0000 0.472927 0.236463 0.971640i \(-0.424012\pi\)
0.236463 + 0.971640i \(0.424012\pi\)
\(542\) 48.0000 2.06178
\(543\) 0 0
\(544\) −24.0000 −1.02899
\(545\) −15.0000 −0.642529
\(546\) 0 0
\(547\) 8.00000 0.342055 0.171028 0.985266i \(-0.445291\pi\)
0.171028 + 0.985266i \(0.445291\pi\)
\(548\) −16.0000 −0.683486
\(549\) 0 0
\(550\) −2.00000 −0.0852803
\(551\) 6.00000 0.255609
\(552\) 0 0
\(553\) 0 0
\(554\) −56.0000 −2.37921
\(555\) 0 0
\(556\) 0 0
\(557\) −4.00000 −0.169485 −0.0847427 0.996403i \(-0.527007\pi\)
−0.0847427 + 0.996403i \(0.527007\pi\)
\(558\) 0 0
\(559\) −18.0000 −0.761319
\(560\) 0 0
\(561\) 0 0
\(562\) 10.0000 0.421825
\(563\) −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
\(566\) 42.0000 1.76539
\(567\) 0 0
\(568\) 0 0
\(569\) 38.0000 1.59304 0.796521 0.604610i \(-0.206671\pi\)
0.796521 + 0.604610i \(0.206671\pi\)
\(570\) 0 0
\(571\) 4.00000 0.167395 0.0836974 0.996491i \(-0.473327\pi\)
0.0836974 + 0.996491i \(0.473327\pi\)
\(572\) −6.00000 −0.250873
\(573\) 0 0
\(574\) 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 \(0.398924\pi\)
\(578\) −16.0000 −0.665512
\(579\) 0 0
\(580\) 2.00000 0.0830455
\(581\) 0 0
\(582\) 0 0
\(583\) −10.0000 −0.414158
\(584\) 0 0
\(585\) 0 0
\(586\) −18.0000 −0.743573
\(587\) 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
\(590\) 12.0000 0.494032
\(591\) 0 0
\(592\) 0 0
\(593\) 45.0000 1.84793 0.923964 0.382479i \(-0.124930\pi\)
0.923964 + 0.382479i \(0.124930\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −20.0000 −0.819232
\(597\) 0 0
\(598\) 24.0000 0.981433
\(599\) −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
\(603\) 0 0
\(604\) 30.0000 1.22068
\(605\) −10.0000 −0.406558
\(606\) 0 0
\(607\) 21.0000 0.852364 0.426182 0.904638i \(-0.359858\pi\)
0.426182 + 0.904638i \(0.359858\pi\)
\(608\) −48.0000 −1.94666
\(609\) 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 \(0.512860\pi\)
\(614\) −6.00000 −0.242140
\(615\) 0 0
\(616\) 0 0
\(617\) −4.00000 −0.161034 −0.0805170 0.996753i \(-0.525657\pi\)
−0.0805170 + 0.996753i \(0.525657\pi\)
\(618\) 0 0
\(619\) −24.0000 −0.964641 −0.482321 0.875995i \(-0.660206\pi\)
−0.482321 + 0.875995i \(0.660206\pi\)
\(620\) 12.0000 0.481932
\(621\) 0 0
\(622\) 12.0000 0.481156
\(623\) 0 0
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 6.00000 0.239808
\(627\) 0 0
\(628\) 36.0000 1.43656
\(629\) 0 0
\(630\) 0 0
\(631\) −11.0000 −0.437903 −0.218952 0.975736i \(-0.570264\pi\)
−0.218952 + 0.975736i \(0.570264\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 44.0000 1.74746
\(635\) −2.00000 −0.0793676
\(636\) 0 0
\(637\) 0 0
\(638\) −2.00000 −0.0791808
\(639\) 0 0
\(640\) 0 0
\(641\) 2.00000 0.0789953 0.0394976 0.999220i \(-0.487424\pi\)
0.0394976 + 0.999220i \(0.487424\pi\)
\(642\) 0 0
\(643\) 27.0000 1.06478 0.532388 0.846500i \(-0.321295\pi\)
0.532388 + 0.846500i \(0.321295\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 36.0000 1.41640
\(647\) −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
\(650\) 6.00000 0.235339
\(651\) 0 0
\(652\) 32.0000 1.25322
\(653\) −14.0000 −0.547862 −0.273931 0.961749i \(-0.588324\pi\)
−0.273931 + 0.961749i \(0.588324\pi\)
\(654\) 0 0
\(655\) −12.0000 −0.468879
\(656\) 24.0000 0.937043
\(657\) 0 0
\(658\) 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 \(0.425024\pi\)
\(662\) 24.0000 0.932786
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 4.00000 0.154881
\(668\) −6.00000 −0.232147
\(669\) 0 0
\(670\) −28.0000 −1.08173
\(671\) 0 0
\(672\) 0 0
\(673\) 12.0000 0.462566 0.231283 0.972887i \(-0.425708\pi\)
0.231283 + 0.972887i \(0.425708\pi\)
\(674\) −48.0000 −1.84889
\(675\) 0 0
\(676\) −8.00000 −0.307692
\(677\) 9.00000 0.345898 0.172949 0.984931i \(-0.444670\pi\)
0.172949 + 0.984931i \(0.444670\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) −12.0000 −0.459504
\(683\) −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
\(687\) 0 0
\(688\) 24.0000 0.914991
\(689\) 30.0000 1.14291
\(690\) 0 0
\(691\) 36.0000 1.36950 0.684752 0.728776i \(-0.259910\pi\)
0.684752 + 0.728776i \(0.259910\pi\)
\(692\) 6.00000 0.228086
\(693\) 0 0
\(694\) −32.0000 −1.21470
\(695\) 0 0
\(696\) 0 0
\(697\) −18.0000 −0.681799
\(698\) −60.0000 −2.27103
\(699\) 0 0
\(700\) 0 0
\(701\) −47.0000 −1.77517 −0.887583 0.460648i \(-0.847617\pi\)
−0.887583 + 0.460648i \(0.847617\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 8.00000 0.301511
\(705\) 0 0
\(706\) 6.00000 0.225813
\(707\) 0 0
\(708\) 0 0
\(709\) −33.0000 −1.23934 −0.619671 0.784862i \(-0.712734\pi\)
−0.619671 + 0.784862i \(0.712734\pi\)
\(710\) 16.0000 0.600469
\(711\) 0 0
\(712\) 0 0
\(713\) 24.0000 0.898807
\(714\) 0 0
\(715\) −3.00000 −0.112194
\(716\) −8.00000 −0.298974
\(717\) 0 0
\(718\) −64.0000 −2.38846
\(719\) 36.0000 1.34257 0.671287 0.741198i \(-0.265742\pi\)
0.671287 + 0.741198i \(0.265742\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 34.0000 1.26535
\(723\) 0 0
\(724\) −12.0000 −0.445976
\(725\) 1.00000 0.0371391
\(726\) 0 0
\(727\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 12.0000 0.444140
\(731\) −18.0000 −0.665754
\(732\) 0 0
\(733\) 15.0000 0.554038 0.277019 0.960864i \(-0.410654\pi\)
0.277019 + 0.960864i \(0.410654\pi\)
\(734\) −66.0000 −2.43610
\(735\) 0 0
\(736\) −32.0000 −1.17954
\(737\) 14.0000 0.515697
\(738\) 0 0
\(739\) −43.0000 −1.58178 −0.790890 0.611958i \(-0.790382\pi\)
−0.790890 + 0.611958i \(0.790382\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 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
\(746\) 8.00000 0.292901
\(747\) 0 0
\(748\) −6.00000 −0.219382
\(749\) 0 0
\(750\) 0 0
\(751\) 15.0000 0.547358 0.273679 0.961821i \(-0.411759\pi\)
0.273679 + 0.961821i \(0.411759\pi\)
\(752\) −36.0000 −1.31278
\(753\) 0 0
\(754\) 6.00000 0.218507
\(755\) 15.0000 0.545906
\(756\) 0 0
\(757\) 48.0000 1.74459 0.872295 0.488980i \(-0.162631\pi\)
0.872295 + 0.488980i \(0.162631\pi\)
\(758\) −48.0000 −1.74344
\(759\) 0 0
\(760\) 0 0
\(761\) −12.0000 −0.435000 −0.217500 0.976060i \(-0.569790\pi\)
−0.217500 + 0.976060i \(0.569790\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −34.0000 −1.23008
\(765\) 0 0
\(766\) 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
\(771\) 0 0
\(772\) 24.0000 0.863779
\(773\) −3.00000 −0.107903 −0.0539513 0.998544i \(-0.517182\pi\)
−0.0539513 + 0.998544i \(0.517182\pi\)
\(774\) 0 0
\(775\) 6.00000 0.215526
\(776\) 0 0
\(777\) 0 0
\(778\) −26.0000 −0.932145
\(779\) −36.0000 −1.28983
\(780\) 0 0
\(781\) −8.00000 −0.286263
\(782\) 24.0000 0.858238
\(783\) 0 0
\(784\) 0 0
\(785\) 18.0000 0.642448
\(786\) 0 0
\(787\) 39.0000 1.39020 0.695100 0.718913i \(-0.255360\pi\)
0.695100 + 0.718913i \(0.255360\pi\)
\(788\) 4.00000 0.142494
\(789\) 0 0
\(790\) −2.00000 −0.0711568
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 18.0000 0.638796
\(795\) 0 0
\(796\) 12.0000 0.425329
\(797\) 27.0000 0.956389 0.478195 0.878254i \(-0.341291\pi\)
0.478195 + 0.878254i \(0.341291\pi\)
\(798\) 0 0
\(799\) 27.0000 0.955191
\(800\) −8.00000 −0.282843
\(801\) 0 0
\(802\) 38.0000 1.34183
\(803\) −6.00000 −0.211735
\(804\) 0 0
\(805\) 0 0
\(806\) 36.0000 1.26805
\(807\) 0 0
\(808\) 0 0
\(809\) 35.0000 1.23053 0.615267 0.788319i \(-0.289048\pi\)
0.615267 + 0.788319i \(0.289048\pi\)
\(810\) 0 0
\(811\) 48.0000 1.68551 0.842754 0.538299i \(-0.180933\pi\)
0.842754 + 0.538299i \(0.180933\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 16.0000 0.560456
\(816\) 0 0
\(817\) −36.0000 −1.25948
\(818\) 60.0000 2.09785
\(819\) 0 0
\(820\) −12.0000 −0.419058
\(821\) 23.0000 0.802706 0.401353 0.915924i \(-0.368540\pi\)
0.401353 + 0.915924i \(0.368540\pi\)
\(822\) 0 0
\(823\) 50.0000 1.74289 0.871445 0.490493i \(-0.163183\pi\)
0.871445 + 0.490493i \(0.163183\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 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 \(0.466774\pi\)
\(830\) −24.0000 −0.833052
\(831\) 0 0
\(832\) −24.0000 −0.832050
\(833\) 0 0
\(834\) 0 0
\(835\) −3.00000 −0.103819
\(836\) −12.0000 −0.415029
\(837\) 0 0
\(838\) 12.0000 0.414533
\(839\) 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
\(842\) 2.00000 0.0689246
\(843\) 0 0
\(844\) 30.0000 1.03264
\(845\) −4.00000 −0.137604
\(846\) 0 0
\(847\) 0 0
\(848\) −40.0000 −1.37361
\(849\) 0 0
\(850\) 6.00000 0.205798
\(851\) 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
\(855\) 0 0
\(856\) 0 0
\(857\) −54.0000 −1.84460 −0.922302 0.386469i \(-0.873695\pi\)
−0.922302 + 0.386469i \(0.873695\pi\)
\(858\) 0 0
\(859\) 12.0000 0.409435 0.204717 0.978821i \(-0.434372\pi\)
0.204717 + 0.978821i \(0.434372\pi\)
\(860\) −12.0000 −0.409197
\(861\) 0 0
\(862\) −10.0000 −0.340601
\(863\) 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
\(866\) −60.0000 −2.03888
\(867\) 0 0
\(868\) 0 0
\(869\) 1.00000 0.0339227
\(870\) 0 0
\(871\) −42.0000 −1.42312
\(872\) 0 0
\(873\) 0 0
\(874\) 48.0000 1.62362
\(875\) 0 0
\(876\) 0 0
\(877\) −22.0000 −0.742887 −0.371444 0.928456i \(-0.621137\pi\)
−0.371444 + 0.928456i \(0.621137\pi\)
\(878\) 24.0000 0.809961
\(879\) 0 0
\(880\) 4.00000 0.134840
\(881\) 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 \(0.5\pi\)
\(884\) 18.0000 0.605406
\(885\) 0 0
\(886\) −68.0000 −2.28450
\(887\) 24.0000 0.805841 0.402921 0.915235i \(-0.367995\pi\)
0.402921 + 0.915235i \(0.367995\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −24.0000 −0.804482
\(891\) 0 0
\(892\) 6.00000 0.200895
\(893\) 54.0000 1.80704
\(894\) 0 0
\(895\) −4.00000 −0.133705
\(896\) 0 0
\(897\) 0 0
\(898\) −46.0000 −1.53504
\(899\) 6.00000 0.200111
\(900\) 0 0
\(901\) 30.0000 0.999445
\(902\) 12.0000 0.399556
\(903\) 0 0
\(904\) 0 0
\(905\) −6.00000 −0.199447
\(906\) 0 0
\(907\) 24.0000 0.796907 0.398453 0.917189i \(-0.369547\pi\)
0.398453 + 0.917189i \(0.369547\pi\)
\(908\) −6.00000 −0.199117
\(909\) 0 0
\(910\) 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
\(914\) 12.0000 0.396925
\(915\) 0 0
\(916\) −12.0000 −0.396491
\(917\) 0 0
\(918\) 0 0
\(919\) −9.00000 −0.296883 −0.148441 0.988921i \(-0.547426\pi\)
−0.148441 + 0.988921i \(0.547426\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −72.0000 −2.37119
\(923\) 24.0000 0.789970
\(924\) 0 0
\(925\) 0 0
\(926\) −8.00000 −0.262896
\(927\) 0 0
\(928\) −8.00000 −0.262613
\(929\) −42.0000 −1.37798 −0.688988 0.724773i \(-0.741945\pi\)
−0.688988 + 0.724773i \(0.741945\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 16.0000 0.524097
\(933\) 0 0
\(934\) 30.0000 0.981630
\(935\) −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
\(939\) 0 0
\(940\) 18.0000 0.587095
\(941\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(942\) 0 0
\(943\) −24.0000 −0.781548
\(944\) −24.0000 −0.781133
\(945\) 0 0
\(946\) 12.0000 0.390154
\(947\) 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
\(950\) 12.0000 0.389331
\(951\) 0 0
\(952\) 0 0
\(953\) 38.0000 1.23094 0.615470 0.788160i \(-0.288966\pi\)
0.615470 + 0.788160i \(0.288966\pi\)
\(954\) 0 0
\(955\) −17.0000 −0.550107
\(956\) −14.0000 −0.452792
\(957\) 0 0
\(958\) 48.0000 1.55081
\(959\) 0 0
\(960\) 0 0
\(961\) 5.00000 0.161290
\(962\) 0 0
\(963\) 0 0
\(964\) −48.0000 −1.54598
\(965\) 12.0000 0.386294
\(966\) 0 0
\(967\) −22.0000 −0.707472 −0.353736 0.935345i \(-0.615089\pi\)
−0.353736 + 0.935345i \(0.615089\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) −30.0000 −0.963242
\(971\) 18.0000 0.577647 0.288824 0.957382i \(-0.406736\pi\)
0.288824 + 0.957382i \(0.406736\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 72.0000 2.30703
\(975\) 0 0
\(976\) 0 0
\(977\) −34.0000 −1.08776 −0.543878 0.839164i \(-0.683045\pi\)
−0.543878 + 0.839164i \(0.683045\pi\)
\(978\) 0 0
\(979\) 12.0000 0.383522
\(980\) 0 0
\(981\) 0 0
\(982\) −46.0000 −1.46792
\(983\) −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
\(986\) 6.00000 0.191079
\(987\) 0 0
\(988\) 36.0000 1.14531
\(989\) −24.0000 −0.763156
\(990\) 0 0
\(991\) −36.0000 −1.14358 −0.571789 0.820401i \(-0.693750\pi\)
−0.571789 + 0.820401i \(0.693750\pi\)
\(992\) −48.0000 −1.52400
\(993\) 0 0
\(994\) 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 \(0.515127\pi\)
\(998\) −54.0000 −1.70934
\(999\) 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 2205.2.a.l.1.1 1
3.2 odd 2 245.2.a.b.1.1 yes 1
7.6 odd 2 2205.2.a.j.1.1 1
12.11 even 2 3920.2.a.a.1.1 1
15.2 even 4 1225.2.b.a.99.1 2
15.8 even 4 1225.2.b.a.99.2 2
15.14 odd 2 1225.2.a.h.1.1 1
21.2 odd 6 245.2.e.c.116.1 2
21.5 even 6 245.2.e.d.116.1 2
21.11 odd 6 245.2.e.c.226.1 2
21.17 even 6 245.2.e.d.226.1 2
21.20 even 2 245.2.a.a.1.1 1
84.83 odd 2 3920.2.a.bj.1.1 1
105.62 odd 4 1225.2.b.b.99.1 2
105.83 odd 4 1225.2.b.b.99.2 2
105.104 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 21.20 even 2
245.2.a.b.1.1 yes 1 3.2 odd 2
245.2.e.c.116.1 2 21.2 odd 6
245.2.e.c.226.1 2 21.11 odd 6
245.2.e.d.116.1 2 21.5 even 6
245.2.e.d.226.1 2 21.17 even 6
1225.2.a.h.1.1 1 15.14 odd 2
1225.2.a.j.1.1 1 105.104 even 2
1225.2.b.a.99.1 2 15.2 even 4
1225.2.b.a.99.2 2 15.8 even 4
1225.2.b.b.99.1 2 105.62 odd 4
1225.2.b.b.99.2 2 105.83 odd 4
2205.2.a.j.1.1 1 7.6 odd 2
2205.2.a.l.1.1 1 1.1 even 1 trivial
3920.2.a.a.1.1 1 12.11 even 2
3920.2.a.bj.1.1 1 84.83 odd 2