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Properties

Label 2368.2.a.q.1.1
Level $2368$
Weight $2$
Character 2368.1
Self dual yes
Analytic conductor $18.909$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2368,2,Mod(1,2368)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2368, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2368.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2368 = 2^{6} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2368.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(18.9085751986\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 37)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2368.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.00000 q^{3} +2.00000 q^{5} -1.00000 q^{7} +6.00000 q^{9} +5.00000 q^{11} +2.00000 q^{13} +6.00000 q^{15} -3.00000 q^{21} +2.00000 q^{23} -1.00000 q^{25} +9.00000 q^{27} -6.00000 q^{29} -4.00000 q^{31} +15.0000 q^{33} -2.00000 q^{35} +1.00000 q^{37} +6.00000 q^{39} -9.00000 q^{41} -2.00000 q^{43} +12.0000 q^{45} -9.00000 q^{47} -6.00000 q^{49} -1.00000 q^{53} +10.0000 q^{55} -8.00000 q^{59} +8.00000 q^{61} -6.00000 q^{63} +4.00000 q^{65} -8.00000 q^{67} +6.00000 q^{69} +9.00000 q^{71} -1.00000 q^{73} -3.00000 q^{75} -5.00000 q^{77} +4.00000 q^{79} +9.00000 q^{81} +15.0000 q^{83} -18.0000 q^{87} +4.00000 q^{89} -2.00000 q^{91} -12.0000 q^{93} +4.00000 q^{97} +30.0000 q^{99} +O(q^{100})\)

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 \(-0.166667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(4\) 0 0
\(5\) 2.00000 0.894427 0.447214 − 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964 −0.188982 − 0.981981i \(-0.560519\pi\)
−0.188982 + 0.981981i \(0.560519\pi\)
\(8\) 0 0
\(9\) 6.00000 2.00000
\(10\) 0 0
\(11\) 5.00000 1.50756 0.753778 − 0.657129i \(-0.228229\pi\)
0.753778 + 0.657129i \(0.228229\pi\)
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 − 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 0 0
\(15\) 6.00000 1.54919
\(16\) 0 0
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) −3.00000 −0.654654
\(22\) 0 0
\(23\) 2.00000 0.417029 0.208514 − 0.978019i \(-0.433137\pi\)
0.208514 + 0.978019i \(0.433137\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 9.00000 1.73205
\(28\) 0 0
\(29\) −6.00000 −1.11417 −0.557086 − 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 − 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 0 0
\(33\) 15.0000 2.61116
\(34\) 0 0
\(35\) −2.00000 −0.338062
\(36\) 0 0
\(37\) 1.00000 0.164399
\(38\) 0 0
\(39\) 6.00000 0.960769
\(40\) 0 0
\(41\) −9.00000 −1.40556 −0.702782 − 0.711405i \(-0.748059\pi\)
−0.702782 + 0.711405i \(0.748059\pi\)
\(42\) 0 0
\(43\) −2.00000 −0.304997 −0.152499 − 0.988304i \(-0.548732\pi\)
−0.152499 + 0.988304i \(0.548732\pi\)
\(44\) 0 0
\(45\) 12.0000 1.78885
\(46\) 0 0
\(47\) −9.00000 −1.31278 −0.656392 − 0.754420i \(-0.727918\pi\)
−0.656392 + 0.754420i \(0.727918\pi\)
\(48\) 0 0
\(49\) −6.00000 −0.857143
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −1.00000 −0.137361 −0.0686803 − 0.997639i \(-0.521879\pi\)
−0.0686803 + 0.997639i \(0.521879\pi\)
\(54\) 0 0
\(55\) 10.0000 1.34840
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −8.00000 −1.04151 −0.520756 − 0.853706i \(-0.674350\pi\)
−0.520756 + 0.853706i \(0.674350\pi\)
\(60\) 0 0
\(61\) 8.00000 1.02430 0.512148 − 0.858898i \(-0.328850\pi\)
0.512148 + 0.858898i \(0.328850\pi\)
\(62\) 0 0
\(63\) −6.00000 −0.755929
\(64\) 0 0
\(65\) 4.00000 0.496139
\(66\) 0 0
\(67\) −8.00000 −0.977356 −0.488678 − 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) 0 0
\(69\) 6.00000 0.722315
\(70\) 0 0
\(71\) 9.00000 1.06810 0.534052 − 0.845452i \(-0.320669\pi\)
0.534052 + 0.845452i \(0.320669\pi\)
\(72\) 0 0
\(73\) −1.00000 −0.117041 −0.0585206 − 0.998286i \(-0.518638\pi\)
−0.0585206 + 0.998286i \(0.518638\pi\)
\(74\) 0 0
\(75\) −3.00000 −0.346410
\(76\) 0 0
\(77\) −5.00000 −0.569803
\(78\) 0 0
\(79\) 4.00000 0.450035 0.225018 − 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) 15.0000 1.64646 0.823232 − 0.567705i \(-0.192169\pi\)
0.823232 + 0.567705i \(0.192169\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −18.0000 −1.92980
\(88\) 0 0
\(89\) 4.00000 0.423999 0.212000 − 0.977270i \(-0.432002\pi\)
0.212000 + 0.977270i \(0.432002\pi\)
\(90\) 0 0
\(91\) −2.00000 −0.209657
\(92\) 0 0
\(93\) −12.0000 −1.24434
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 4.00000 0.406138 0.203069 − 0.979164i \(-0.434908\pi\)
0.203069 + 0.979164i \(0.434908\pi\)
\(98\) 0 0
\(99\) 30.0000 3.01511
\(100\) 0 0
\(101\) −3.00000 −0.298511 −0.149256 − 0.988799i \(-0.547688\pi\)
−0.149256 + 0.988799i \(0.547688\pi\)
\(102\) 0 0
\(103\) 18.0000 1.77359 0.886796 − 0.462160i \(-0.152926\pi\)
0.886796 + 0.462160i \(0.152926\pi\)
\(104\) 0 0
\(105\) −6.00000 −0.585540
\(106\) 0 0
\(107\) 12.0000 1.16008 0.580042 − 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 0 0
\(109\) 16.0000 1.53252 0.766261 − 0.642529i \(-0.222115\pi\)
0.766261 + 0.642529i \(0.222115\pi\)
\(110\) 0 0
\(111\) 3.00000 0.284747
\(112\) 0 0
\(113\) −18.0000 −1.69330 −0.846649 − 0.532152i \(-0.821383\pi\)
−0.846649 + 0.532152i \(0.821383\pi\)
\(114\) 0 0
\(115\) 4.00000 0.373002
\(116\) 0 0
\(117\) 12.0000 1.10940
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 14.0000 1.27273
\(122\) 0 0
\(123\) −27.0000 −2.43451
\(124\) 0 0
\(125\) −12.0000 −1.07331
\(126\) 0 0
\(127\) 1.00000 0.0887357 0.0443678 − 0.999015i \(-0.485873\pi\)
0.0443678 + 0.999015i \(0.485873\pi\)
\(128\) 0 0
\(129\) −6.00000 −0.528271
\(130\) 0 0
\(131\) 12.0000 1.04844 0.524222 − 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 18.0000 1.54919
\(136\) 0 0
\(137\) −6.00000 −0.512615 −0.256307 − 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.339276 −0.169638 − 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 0 0
\(141\) −27.0000 −2.27381
\(142\) 0 0
\(143\) 10.0000 0.836242
\(144\) 0 0
\(145\) −12.0000 −0.996546
\(146\) 0 0
\(147\) −18.0000 −1.48461
\(148\) 0 0
\(149\) 5.00000 0.409616 0.204808 − 0.978802i \(-0.434343\pi\)
0.204808 + 0.978802i \(0.434343\pi\)
\(150\) 0 0
\(151\) 16.0000 1.30206 0.651031 − 0.759051i \(-0.274337\pi\)
0.651031 + 0.759051i \(0.274337\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −8.00000 −0.642575
\(156\) 0 0
\(157\) −23.0000 −1.83560 −0.917800 − 0.397043i \(-0.870036\pi\)
−0.917800 + 0.397043i \(0.870036\pi\)
\(158\) 0 0
\(159\) −3.00000 −0.237915
\(160\) 0 0
\(161\) −2.00000 −0.157622
\(162\) 0 0
\(163\) 18.0000 1.40987 0.704934 − 0.709273i \(-0.250976\pi\)
0.704934 + 0.709273i \(0.250976\pi\)
\(164\) 0 0
\(165\) 30.0000 2.33550
\(166\) 0 0
\(167\) −12.0000 −0.928588 −0.464294 − 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −9.00000 −0.684257 −0.342129 − 0.939653i \(-0.611148\pi\)
−0.342129 + 0.939653i \(0.611148\pi\)
\(174\) 0 0
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) −24.0000 −1.80395
\(178\) 0 0
\(179\) −18.0000 −1.34538 −0.672692 − 0.739923i \(-0.734862\pi\)
−0.672692 + 0.739923i \(0.734862\pi\)
\(180\) 0 0
\(181\) −5.00000 −0.371647 −0.185824 − 0.982583i \(-0.559495\pi\)
−0.185824 + 0.982583i \(0.559495\pi\)
\(182\) 0 0
\(183\) 24.0000 1.77413
\(184\) 0 0
\(185\) 2.00000 0.147043
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −9.00000 −0.654654
\(190\) 0 0
\(191\) −4.00000 −0.289430 −0.144715 − 0.989473i \(-0.546227\pi\)
−0.144715 + 0.989473i \(0.546227\pi\)
\(192\) 0 0
\(193\) −26.0000 −1.87152 −0.935760 − 0.352636i \(-0.885285\pi\)
−0.935760 + 0.352636i \(0.885285\pi\)
\(194\) 0 0
\(195\) 12.0000 0.859338
\(196\) 0 0
\(197\) −3.00000 −0.213741 −0.106871 − 0.994273i \(-0.534083\pi\)
−0.106871 + 0.994273i \(0.534083\pi\)
\(198\) 0 0
\(199\) 2.00000 0.141776 0.0708881 − 0.997484i \(-0.477417\pi\)
0.0708881 + 0.997484i \(0.477417\pi\)
\(200\) 0 0
\(201\) −24.0000 −1.69283
\(202\) 0 0
\(203\) 6.00000 0.421117
\(204\) 0 0
\(205\) −18.0000 −1.25717
\(206\) 0 0
\(207\) 12.0000 0.834058
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 13.0000 0.894957 0.447478 − 0.894295i \(-0.352322\pi\)
0.447478 + 0.894295i \(0.352322\pi\)
\(212\) 0 0
\(213\) 27.0000 1.85001
\(214\) 0 0
\(215\) −4.00000 −0.272798
\(216\) 0 0
\(217\) 4.00000 0.271538
\(218\) 0 0
\(219\) −3.00000 −0.202721
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −17.0000 −1.13840 −0.569202 − 0.822198i \(-0.692748\pi\)
−0.569202 + 0.822198i \(0.692748\pi\)
\(224\) 0 0
\(225\) −6.00000 −0.400000
\(226\) 0 0
\(227\) 16.0000 1.06196 0.530979 − 0.847385i \(-0.321824\pi\)
0.530979 + 0.847385i \(0.321824\pi\)
\(228\) 0 0
\(229\) −7.00000 −0.462573 −0.231287 − 0.972886i \(-0.574293\pi\)
−0.231287 + 0.972886i \(0.574293\pi\)
\(230\) 0 0
\(231\) −15.0000 −0.986928
\(232\) 0 0
\(233\) 6.00000 0.393073 0.196537 − 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 0 0
\(235\) −18.0000 −1.17419
\(236\) 0 0
\(237\) 12.0000 0.779484
\(238\) 0 0
\(239\) −6.00000 −0.388108 −0.194054 − 0.980991i \(-0.562164\pi\)
−0.194054 + 0.980991i \(0.562164\pi\)
\(240\) 0 0
\(241\) 14.0000 0.901819 0.450910 − 0.892570i \(-0.351100\pi\)
0.450910 + 0.892570i \(0.351100\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −12.0000 −0.766652
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 45.0000 2.85176
\(250\) 0 0
\(251\) 2.00000 0.126239 0.0631194 − 0.998006i \(-0.479895\pi\)
0.0631194 + 0.998006i \(0.479895\pi\)
\(252\) 0 0
\(253\) 10.0000 0.628695
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(258\) 0 0
\(259\) −1.00000 −0.0621370
\(260\) 0 0
\(261\) −36.0000 −2.22834
\(262\) 0 0
\(263\) 19.0000 1.17159 0.585795 − 0.810459i \(-0.300782\pi\)
0.585795 + 0.810459i \(0.300782\pi\)
\(264\) 0 0
\(265\) −2.00000 −0.122859
\(266\) 0 0
\(267\) 12.0000 0.734388
\(268\) 0 0
\(269\) 6.00000 0.365826 0.182913 − 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) 0 0
\(271\) −31.0000 −1.88312 −0.941558 − 0.336851i \(-0.890638\pi\)
−0.941558 + 0.336851i \(0.890638\pi\)
\(272\) 0 0
\(273\) −6.00000 −0.363137
\(274\) 0 0
\(275\) −5.00000 −0.301511
\(276\) 0 0
\(277\) −12.0000 −0.721010 −0.360505 − 0.932757i \(-0.617396\pi\)
−0.360505 + 0.932757i \(0.617396\pi\)
\(278\) 0 0
\(279\) −24.0000 −1.43684
\(280\) 0 0
\(281\) 12.0000 0.715860 0.357930 − 0.933748i \(-0.383483\pi\)
0.357930 + 0.933748i \(0.383483\pi\)
\(282\) 0 0
\(283\) −4.00000 −0.237775 −0.118888 − 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 9.00000 0.531253
\(288\) 0 0
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) 12.0000 0.703452
\(292\) 0 0
\(293\) 2.00000 0.116841 0.0584206 − 0.998292i \(-0.481394\pi\)
0.0584206 + 0.998292i \(0.481394\pi\)
\(294\) 0 0
\(295\) −16.0000 −0.931556
\(296\) 0 0
\(297\) 45.0000 2.61116
\(298\) 0 0
\(299\) 4.00000 0.231326
\(300\) 0 0
\(301\) 2.00000 0.115278
\(302\) 0 0
\(303\) −9.00000 −0.517036
\(304\) 0 0
\(305\) 16.0000 0.916157
\(306\) 0 0
\(307\) 17.0000 0.970241 0.485121 − 0.874447i \(-0.338776\pi\)
0.485121 + 0.874447i \(0.338776\pi\)
\(308\) 0 0
\(309\) 54.0000 3.07195
\(310\) 0 0
\(311\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 22.0000 1.24351 0.621757 − 0.783210i \(-0.286419\pi\)
0.621757 + 0.783210i \(0.286419\pi\)
\(314\) 0 0
\(315\) −12.0000 −0.676123
\(316\) 0 0
\(317\) −22.0000 −1.23564 −0.617822 − 0.786318i \(-0.711985\pi\)
−0.617822 + 0.786318i \(0.711985\pi\)
\(318\) 0 0
\(319\) −30.0000 −1.67968
\(320\) 0 0
\(321\) 36.0000 2.00932
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −2.00000 −0.110940
\(326\) 0 0
\(327\) 48.0000 2.65441
\(328\) 0 0
\(329\) 9.00000 0.496186
\(330\) 0 0
\(331\) 2.00000 0.109930 0.0549650 − 0.998488i \(-0.482495\pi\)
0.0549650 + 0.998488i \(0.482495\pi\)
\(332\) 0 0
\(333\) 6.00000 0.328798
\(334\) 0 0
\(335\) −16.0000 −0.874173
\(336\) 0 0
\(337\) −25.0000 −1.36184 −0.680918 − 0.732359i \(-0.738419\pi\)
−0.680918 + 0.732359i \(0.738419\pi\)
\(338\) 0 0
\(339\) −54.0000 −2.93288
\(340\) 0 0
\(341\) −20.0000 −1.08306
\(342\) 0 0
\(343\) 13.0000 0.701934
\(344\) 0 0
\(345\) 12.0000 0.646058
\(346\) 0 0
\(347\) 10.0000 0.536828 0.268414 − 0.963304i \(-0.413500\pi\)
0.268414 + 0.963304i \(0.413500\pi\)
\(348\) 0 0
\(349\) −6.00000 −0.321173 −0.160586 − 0.987022i \(-0.551338\pi\)
−0.160586 + 0.987022i \(0.551338\pi\)
\(350\) 0 0
\(351\) 18.0000 0.960769
\(352\) 0 0
\(353\) 8.00000 0.425797 0.212899 − 0.977074i \(-0.431710\pi\)
0.212899 + 0.977074i \(0.431710\pi\)
\(354\) 0 0
\(355\) 18.0000 0.955341
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −15.0000 −0.791670 −0.395835 − 0.918322i \(-0.629545\pi\)
−0.395835 + 0.918322i \(0.629545\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 0 0
\(363\) 42.0000 2.20443
\(364\) 0 0
\(365\) −2.00000 −0.104685
\(366\) 0 0
\(367\) 8.00000 0.417597 0.208798 − 0.977959i \(-0.433045\pi\)
0.208798 + 0.977959i \(0.433045\pi\)
\(368\) 0 0
\(369\) −54.0000 −2.81113
\(370\) 0 0
\(371\) 1.00000 0.0519174
\(372\) 0 0
\(373\) 19.0000 0.983783 0.491891 − 0.870657i \(-0.336306\pi\)
0.491891 + 0.870657i \(0.336306\pi\)
\(374\) 0 0
\(375\) −36.0000 −1.85903
\(376\) 0 0
\(377\) −12.0000 −0.618031
\(378\) 0 0
\(379\) −15.0000 −0.770498 −0.385249 − 0.922813i \(-0.625884\pi\)
−0.385249 + 0.922813i \(0.625884\pi\)
\(380\) 0 0
\(381\) 3.00000 0.153695
\(382\) 0 0
\(383\) 20.0000 1.02195 0.510976 − 0.859595i \(-0.329284\pi\)
0.510976 + 0.859595i \(0.329284\pi\)
\(384\) 0 0
\(385\) −10.0000 −0.509647
\(386\) 0 0
\(387\) −12.0000 −0.609994
\(388\) 0 0
\(389\) −4.00000 −0.202808 −0.101404 − 0.994845i \(-0.532333\pi\)
−0.101404 + 0.994845i \(0.532333\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 36.0000 1.81596
\(394\) 0 0
\(395\) 8.00000 0.402524
\(396\) 0 0
\(397\) 5.00000 0.250943 0.125471 − 0.992097i \(-0.459956\pi\)
0.125471 + 0.992097i \(0.459956\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 18.0000 0.898877 0.449439 − 0.893311i \(-0.351624\pi\)
0.449439 + 0.893311i \(0.351624\pi\)
\(402\) 0 0
\(403\) −8.00000 −0.398508
\(404\) 0 0
\(405\) 18.0000 0.894427
\(406\) 0 0
\(407\) 5.00000 0.247841
\(408\) 0 0
\(409\) 20.0000 0.988936 0.494468 − 0.869196i \(-0.335363\pi\)
0.494468 + 0.869196i \(0.335363\pi\)
\(410\) 0 0
\(411\) −18.0000 −0.887875
\(412\) 0 0
\(413\) 8.00000 0.393654
\(414\) 0 0
\(415\) 30.0000 1.47264
\(416\) 0 0
\(417\) −12.0000 −0.587643
\(418\) 0 0
\(419\) −7.00000 −0.341972 −0.170986 − 0.985273i \(-0.554695\pi\)
−0.170986 + 0.985273i \(0.554695\pi\)
\(420\) 0 0
\(421\) 24.0000 1.16969 0.584844 − 0.811146i \(-0.301156\pi\)
0.584844 + 0.811146i \(0.301156\pi\)
\(422\) 0 0
\(423\) −54.0000 −2.62557
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −8.00000 −0.387147
\(428\) 0 0
\(429\) 30.0000 1.44841
\(430\) 0 0
\(431\) −30.0000 −1.44505 −0.722525 − 0.691345i \(-0.757018\pi\)
−0.722525 + 0.691345i \(0.757018\pi\)
\(432\) 0 0
\(433\) 9.00000 0.432512 0.216256 − 0.976337i \(-0.430615\pi\)
0.216256 + 0.976337i \(0.430615\pi\)
\(434\) 0 0
\(435\) −36.0000 −1.72607
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 28.0000 1.33637 0.668184 − 0.743996i \(-0.267072\pi\)
0.668184 + 0.743996i \(0.267072\pi\)
\(440\) 0 0
\(441\) −36.0000 −1.71429
\(442\) 0 0
\(443\) −1.00000 −0.0475114 −0.0237557 − 0.999718i \(-0.507562\pi\)
−0.0237557 + 0.999718i \(0.507562\pi\)
\(444\) 0 0
\(445\) 8.00000 0.379236
\(446\) 0 0
\(447\) 15.0000 0.709476
\(448\) 0 0
\(449\) 36.0000 1.69895 0.849473 − 0.527633i \(-0.176920\pi\)
0.849473 + 0.527633i \(0.176920\pi\)
\(450\) 0 0
\(451\) −45.0000 −2.11897
\(452\) 0 0
\(453\) 48.0000 2.25524
\(454\) 0 0
\(455\) −4.00000 −0.187523
\(456\) 0 0
\(457\) 18.0000 0.842004 0.421002 − 0.907060i \(-0.361678\pi\)
0.421002 + 0.907060i \(0.361678\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −30.0000 −1.39724 −0.698620 − 0.715493i \(-0.746202\pi\)
−0.698620 + 0.715493i \(0.746202\pi\)
\(462\) 0 0
\(463\) −22.0000 −1.02243 −0.511213 − 0.859454i \(-0.670804\pi\)
−0.511213 + 0.859454i \(0.670804\pi\)
\(464\) 0 0
\(465\) −24.0000 −1.11297
\(466\) 0 0
\(467\) 2.00000 0.0925490 0.0462745 − 0.998929i \(-0.485265\pi\)
0.0462745 + 0.998929i \(0.485265\pi\)
\(468\) 0 0
\(469\) 8.00000 0.369406
\(470\) 0 0
\(471\) −69.0000 −3.17935
\(472\) 0 0
\(473\) −10.0000 −0.459800
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −6.00000 −0.274721
\(478\) 0 0
\(479\) 14.0000 0.639676 0.319838 − 0.947472i \(-0.396371\pi\)
0.319838 + 0.947472i \(0.396371\pi\)
\(480\) 0 0
\(481\) 2.00000 0.0911922
\(482\) 0 0
\(483\) −6.00000 −0.273009
\(484\) 0 0
\(485\) 8.00000 0.363261
\(486\) 0 0
\(487\) −24.0000 −1.08754 −0.543772 − 0.839233i \(-0.683004\pi\)
−0.543772 + 0.839233i \(0.683004\pi\)
\(488\) 0 0
\(489\) 54.0000 2.44196
\(490\) 0 0
\(491\) 28.0000 1.26362 0.631811 − 0.775122i \(-0.282312\pi\)
0.631811 + 0.775122i \(0.282312\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 60.0000 2.69680
\(496\) 0 0
\(497\) −9.00000 −0.403705
\(498\) 0 0
\(499\) −12.0000 −0.537194 −0.268597 − 0.963253i \(-0.586560\pi\)
−0.268597 + 0.963253i \(0.586560\pi\)
\(500\) 0 0
\(501\) −36.0000 −1.60836
\(502\) 0 0
\(503\) 16.0000 0.713405 0.356702 − 0.934218i \(-0.383901\pi\)
0.356702 + 0.934218i \(0.383901\pi\)
\(504\) 0 0
\(505\) −6.00000 −0.266996
\(506\) 0 0
\(507\) −27.0000 −1.19911
\(508\) 0 0
\(509\) 31.0000 1.37405 0.687025 − 0.726633i \(-0.258916\pi\)
0.687025 + 0.726633i \(0.258916\pi\)
\(510\) 0 0
\(511\) 1.00000 0.0442374
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 36.0000 1.58635
\(516\) 0 0
\(517\) −45.0000 −1.97910
\(518\) 0 0
\(519\) −27.0000 −1.18517
\(520\) 0 0
\(521\) −33.0000 −1.44576 −0.722878 − 0.690976i \(-0.757181\pi\)
−0.722878 + 0.690976i \(0.757181\pi\)
\(522\) 0 0
\(523\) 22.0000 0.961993 0.480996 − 0.876723i \(-0.340275\pi\)
0.480996 + 0.876723i \(0.340275\pi\)
\(524\) 0 0
\(525\) 3.00000 0.130931
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −19.0000 −0.826087
\(530\) 0 0
\(531\) −48.0000 −2.08302
\(532\) 0 0
\(533\) −18.0000 −0.779667
\(534\) 0 0
\(535\) 24.0000 1.03761
\(536\) 0 0
\(537\) −54.0000 −2.33027
\(538\) 0 0
\(539\) −30.0000 −1.29219
\(540\) 0 0
\(541\) −20.0000 −0.859867 −0.429934 − 0.902861i \(-0.641463\pi\)
−0.429934 + 0.902861i \(0.641463\pi\)
\(542\) 0 0
\(543\) −15.0000 −0.643712
\(544\) 0 0
\(545\) 32.0000 1.37073
\(546\) 0 0
\(547\) −8.00000 −0.342055 −0.171028 − 0.985266i \(-0.554709\pi\)
−0.171028 + 0.985266i \(0.554709\pi\)
\(548\) 0 0
\(549\) 48.0000 2.04859
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −4.00000 −0.170097
\(554\) 0 0
\(555\) 6.00000 0.254686
\(556\) 0 0
\(557\) 18.0000 0.762684 0.381342 − 0.924434i \(-0.375462\pi\)
0.381342 + 0.924434i \(0.375462\pi\)
\(558\) 0 0
\(559\) −4.00000 −0.169182
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 30.0000 1.26435 0.632175 − 0.774826i \(-0.282163\pi\)
0.632175 + 0.774826i \(0.282163\pi\)
\(564\) 0 0
\(565\) −36.0000 −1.51453
\(566\) 0 0
\(567\) −9.00000 −0.377964
\(568\) 0 0
\(569\) −24.0000 −1.00613 −0.503066 − 0.864248i \(-0.667795\pi\)
−0.503066 + 0.864248i \(0.667795\pi\)
\(570\) 0 0
\(571\) −7.00000 −0.292941 −0.146470 − 0.989215i \(-0.546791\pi\)
−0.146470 + 0.989215i \(0.546791\pi\)
\(572\) 0 0
\(573\) −12.0000 −0.501307
\(574\) 0 0
\(575\) −2.00000 −0.0834058
\(576\) 0 0
\(577\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(578\) 0 0
\(579\) −78.0000 −3.24157
\(580\) 0 0
\(581\) −15.0000 −0.622305
\(582\) 0 0
\(583\) −5.00000 −0.207079
\(584\) 0 0
\(585\) 24.0000 0.992278
\(586\) 0 0
\(587\) 32.0000 1.32078 0.660391 − 0.750922i \(-0.270391\pi\)
0.660391 + 0.750922i \(0.270391\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) −9.00000 −0.370211
\(592\) 0 0
\(593\) −5.00000 −0.205325 −0.102663 − 0.994716i \(-0.532736\pi\)
−0.102663 + 0.994716i \(0.532736\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 6.00000 0.245564
\(598\) 0 0
\(599\) 1.00000 0.0408589 0.0204294 − 0.999791i \(-0.493497\pi\)
0.0204294 + 0.999791i \(0.493497\pi\)
\(600\) 0 0
\(601\) −22.0000 −0.897399 −0.448699 − 0.893683i \(-0.648113\pi\)
−0.448699 + 0.893683i \(0.648113\pi\)
\(602\) 0 0
\(603\) −48.0000 −1.95471
\(604\) 0 0
\(605\) 28.0000 1.13836
\(606\) 0 0
\(607\) −32.0000 −1.29884 −0.649420 − 0.760430i \(-0.724988\pi\)
−0.649420 + 0.760430i \(0.724988\pi\)
\(608\) 0 0
\(609\) 18.0000 0.729397
\(610\) 0 0
\(611\) −18.0000 −0.728202
\(612\) 0 0
\(613\) −15.0000 −0.605844 −0.302922 − 0.953015i \(-0.597962\pi\)
−0.302922 + 0.953015i \(0.597962\pi\)
\(614\) 0 0
\(615\) −54.0000 −2.17749
\(616\) 0 0
\(617\) 17.0000 0.684394 0.342197 − 0.939628i \(-0.388829\pi\)
0.342197 + 0.939628i \(0.388829\pi\)
\(618\) 0 0
\(619\) 1.00000 0.0401934 0.0200967 − 0.999798i \(-0.493603\pi\)
0.0200967 + 0.999798i \(0.493603\pi\)
\(620\) 0 0
\(621\) 18.0000 0.722315
\(622\) 0 0
\(623\) −4.00000 −0.160257
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −28.0000 −1.11466 −0.557331 − 0.830290i \(-0.688175\pi\)
−0.557331 + 0.830290i \(0.688175\pi\)
\(632\) 0 0
\(633\) 39.0000 1.55011
\(634\) 0 0
\(635\) 2.00000 0.0793676
\(636\) 0 0
\(637\) −12.0000 −0.475457
\(638\) 0 0
\(639\) 54.0000 2.13621
\(640\) 0 0
\(641\) −1.00000 −0.0394976 −0.0197488 − 0.999805i \(-0.506287\pi\)
−0.0197488 + 0.999805i \(0.506287\pi\)
\(642\) 0 0
\(643\) −14.0000 −0.552106 −0.276053 − 0.961142i \(-0.589027\pi\)
−0.276053 + 0.961142i \(0.589027\pi\)
\(644\) 0 0
\(645\) −12.0000 −0.472500
\(646\) 0 0
\(647\) −8.00000 −0.314512 −0.157256 − 0.987558i \(-0.550265\pi\)
−0.157256 + 0.987558i \(0.550265\pi\)
\(648\) 0 0
\(649\) −40.0000 −1.57014
\(650\) 0 0
\(651\) 12.0000 0.470317
\(652\) 0 0
\(653\) 24.0000 0.939193 0.469596 − 0.882881i \(-0.344399\pi\)
0.469596 + 0.882881i \(0.344399\pi\)
\(654\) 0 0
\(655\) 24.0000 0.937758
\(656\) 0 0
\(657\) −6.00000 −0.234082
\(658\) 0 0
\(659\) 15.0000 0.584317 0.292159 − 0.956370i \(-0.405627\pi\)
0.292159 + 0.956370i \(0.405627\pi\)
\(660\) 0 0
\(661\) 28.0000 1.08907 0.544537 − 0.838737i \(-0.316705\pi\)
0.544537 + 0.838737i \(0.316705\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −12.0000 −0.464642
\(668\) 0 0
\(669\) −51.0000 −1.97177
\(670\) 0 0
\(671\) 40.0000 1.54418
\(672\) 0 0
\(673\) 27.0000 1.04077 0.520387 − 0.853931i \(-0.325788\pi\)
0.520387 + 0.853931i \(0.325788\pi\)
\(674\) 0 0
\(675\) −9.00000 −0.346410
\(676\) 0 0
\(677\) 11.0000 0.422764 0.211382 − 0.977403i \(-0.432204\pi\)
0.211382 + 0.977403i \(0.432204\pi\)
\(678\) 0 0
\(679\) −4.00000 −0.153506
\(680\) 0 0
\(681\) 48.0000 1.83936
\(682\) 0 0
\(683\) −18.0000 −0.688751 −0.344375 − 0.938832i \(-0.611909\pi\)
−0.344375 + 0.938832i \(0.611909\pi\)
\(684\) 0 0
\(685\) −12.0000 −0.458496
\(686\) 0 0
\(687\) −21.0000 −0.801200
\(688\) 0 0
\(689\) −2.00000 −0.0761939
\(690\) 0 0
\(691\) 20.0000 0.760836 0.380418 − 0.924815i \(-0.375780\pi\)
0.380418 + 0.924815i \(0.375780\pi\)
\(692\) 0 0
\(693\) −30.0000 −1.13961
\(694\) 0 0
\(695\) −8.00000 −0.303457
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 18.0000 0.680823
\(700\) 0 0
\(701\) 12.0000 0.453234 0.226617 − 0.973984i \(-0.427233\pi\)
0.226617 + 0.973984i \(0.427233\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) −54.0000 −2.03376
\(706\) 0 0
\(707\) 3.00000 0.112827
\(708\) 0 0
\(709\) −40.0000 −1.50223 −0.751116 − 0.660171i \(-0.770484\pi\)
−0.751116 + 0.660171i \(0.770484\pi\)
\(710\) 0 0
\(711\) 24.0000 0.900070
\(712\) 0 0
\(713\) −8.00000 −0.299602
\(714\) 0 0
\(715\) 20.0000 0.747958
\(716\) 0 0
\(717\) −18.0000 −0.672222
\(718\) 0 0
\(719\) 39.0000 1.45445 0.727227 − 0.686397i \(-0.240809\pi\)
0.727227 + 0.686397i \(0.240809\pi\)
\(720\) 0 0
\(721\) −18.0000 −0.670355
\(722\) 0 0
\(723\) 42.0000 1.56200
\(724\) 0 0
\(725\) 6.00000 0.222834
\(726\) 0 0
\(727\) 16.0000 0.593407 0.296704 − 0.954970i \(-0.404113\pi\)
0.296704 + 0.954970i \(0.404113\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −7.00000 −0.258551 −0.129275 − 0.991609i \(-0.541265\pi\)
−0.129275 + 0.991609i \(0.541265\pi\)
\(734\) 0 0
\(735\) −36.0000 −1.32788
\(736\) 0 0
\(737\) −40.0000 −1.47342
\(738\) 0 0
\(739\) 9.00000 0.331070 0.165535 − 0.986204i \(-0.447065\pi\)
0.165535 + 0.986204i \(0.447065\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 21.0000 0.770415 0.385208 − 0.922830i \(-0.374130\pi\)
0.385208 + 0.922830i \(0.374130\pi\)
\(744\) 0 0
\(745\) 10.0000 0.366372
\(746\) 0 0
\(747\) 90.0000 3.29293
\(748\) 0 0
\(749\) −12.0000 −0.438470
\(750\) 0 0
\(751\) 25.0000 0.912263 0.456131 − 0.889912i \(-0.349235\pi\)
0.456131 + 0.889912i \(0.349235\pi\)
\(752\) 0 0
\(753\) 6.00000 0.218652
\(754\) 0 0
\(755\) 32.0000 1.16460
\(756\) 0 0
\(757\) 50.0000 1.81728 0.908640 − 0.417579i \(-0.137121\pi\)
0.908640 + 0.417579i \(0.137121\pi\)
\(758\) 0 0
\(759\) 30.0000 1.08893
\(760\) 0 0
\(761\) −35.0000 −1.26875 −0.634375 − 0.773026i \(-0.718742\pi\)
−0.634375 + 0.773026i \(0.718742\pi\)
\(762\) 0 0
\(763\) −16.0000 −0.579239
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −16.0000 −0.577727
\(768\) 0 0
\(769\) 26.0000 0.937584 0.468792 − 0.883309i \(-0.344689\pi\)
0.468792 + 0.883309i \(0.344689\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 9.00000 0.323708 0.161854 − 0.986815i \(-0.448253\pi\)
0.161854 + 0.986815i \(0.448253\pi\)
\(774\) 0 0
\(775\) 4.00000 0.143684
\(776\) 0 0
\(777\) −3.00000 −0.107624
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 45.0000 1.61023
\(782\) 0 0
\(783\) −54.0000 −1.92980
\(784\) 0 0
\(785\) −46.0000 −1.64181
\(786\) 0 0
\(787\) 5.00000 0.178231 0.0891154 − 0.996021i \(-0.471596\pi\)
0.0891154 + 0.996021i \(0.471596\pi\)
\(788\) 0 0
\(789\) 57.0000 2.02925
\(790\) 0 0
\(791\) 18.0000 0.640006
\(792\) 0 0
\(793\) 16.0000 0.568177
\(794\) 0 0
\(795\) −6.00000 −0.212798
\(796\) 0 0
\(797\) −52.0000 −1.84193 −0.920967 − 0.389640i \(-0.872599\pi\)
−0.920967 + 0.389640i \(0.872599\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 24.0000 0.847998
\(802\) 0 0
\(803\) −5.00000 −0.176446
\(804\) 0 0
\(805\) −4.00000 −0.140981
\(806\) 0 0
\(807\) 18.0000 0.633630
\(808\) 0 0
\(809\) 2.00000 0.0703163 0.0351581 − 0.999382i \(-0.488807\pi\)
0.0351581 + 0.999382i \(0.488807\pi\)
\(810\) 0 0
\(811\) −47.0000 −1.65039 −0.825197 − 0.564846i \(-0.808936\pi\)
−0.825197 + 0.564846i \(0.808936\pi\)
\(812\) 0 0
\(813\) −93.0000 −3.26165
\(814\) 0 0
\(815\) 36.0000 1.26102
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) −12.0000 −0.419314
\(820\) 0 0
\(821\) 47.0000 1.64031 0.820156 − 0.572140i \(-0.193887\pi\)
0.820156 + 0.572140i \(0.193887\pi\)
\(822\) 0 0
\(823\) −16.0000 −0.557725 −0.278862 − 0.960331i \(-0.589957\pi\)
−0.278862 + 0.960331i \(0.589957\pi\)
\(824\) 0 0
\(825\) −15.0000 −0.522233
\(826\) 0 0
\(827\) −22.0000 −0.765015 −0.382507 − 0.923952i \(-0.624939\pi\)
−0.382507 + 0.923952i \(0.624939\pi\)
\(828\) 0 0
\(829\) 4.00000 0.138926 0.0694629 − 0.997585i \(-0.477871\pi\)
0.0694629 + 0.997585i \(0.477871\pi\)
\(830\) 0 0
\(831\) −36.0000 −1.24883
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −24.0000 −0.830554
\(836\) 0 0
\(837\) −36.0000 −1.24434
\(838\) 0 0
\(839\) 44.0000 1.51905 0.759524 − 0.650479i \(-0.225432\pi\)
0.759524 + 0.650479i \(0.225432\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 0 0
\(843\) 36.0000 1.23991
\(844\) 0 0
\(845\) −18.0000 −0.619219
\(846\) 0 0
\(847\) −14.0000 −0.481046
\(848\) 0 0
\(849\) −12.0000 −0.411839
\(850\) 0 0
\(851\) 2.00000 0.0685591
\(852\) 0 0
\(853\) −26.0000 −0.890223 −0.445112 − 0.895475i \(-0.646836\pi\)
−0.445112 + 0.895475i \(0.646836\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −48.0000 −1.63965 −0.819824 − 0.572615i \(-0.805929\pi\)
−0.819824 + 0.572615i \(0.805929\pi\)
\(858\) 0 0
\(859\) 20.0000 0.682391 0.341196 − 0.939992i \(-0.389168\pi\)
0.341196 + 0.939992i \(0.389168\pi\)
\(860\) 0 0
\(861\) 27.0000 0.920158
\(862\) 0 0
\(863\) −24.0000 −0.816970 −0.408485 − 0.912765i \(-0.633943\pi\)
−0.408485 + 0.912765i \(0.633943\pi\)
\(864\) 0 0
\(865\) −18.0000 −0.612018
\(866\) 0 0
\(867\) −51.0000 −1.73205
\(868\) 0 0
\(869\) 20.0000 0.678454
\(870\) 0 0
\(871\) −16.0000 −0.542139
\(872\) 0 0
\(873\) 24.0000 0.812277
\(874\) 0 0
\(875\) 12.0000 0.405674
\(876\) 0 0
\(877\) −50.0000 −1.68838 −0.844190 − 0.536044i \(-0.819918\pi\)
−0.844190 + 0.536044i \(0.819918\pi\)
\(878\) 0 0
\(879\) 6.00000 0.202375
\(880\) 0 0
\(881\) −14.0000 −0.471672 −0.235836 − 0.971793i \(-0.575783\pi\)
−0.235836 + 0.971793i \(0.575783\pi\)
\(882\) 0 0
\(883\) −48.0000 −1.61533 −0.807664 − 0.589643i \(-0.799269\pi\)
−0.807664 + 0.589643i \(0.799269\pi\)
\(884\) 0 0
\(885\) −48.0000 −1.61350
\(886\) 0 0
\(887\) 25.0000 0.839418 0.419709 − 0.907659i \(-0.362132\pi\)
0.419709 + 0.907659i \(0.362132\pi\)
\(888\) 0 0
\(889\) −1.00000 −0.0335389
\(890\) 0 0
\(891\) 45.0000 1.50756
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) −36.0000 −1.20335
\(896\) 0 0
\(897\) 12.0000 0.400668
\(898\) 0 0
\(899\) 24.0000 0.800445
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 6.00000 0.199667
\(904\) 0 0
\(905\) −10.0000 −0.332411
\(906\) 0 0
\(907\) −52.0000 −1.72663 −0.863316 − 0.504664i \(-0.831616\pi\)
−0.863316 + 0.504664i \(0.831616\pi\)
\(908\) 0 0
\(909\) −18.0000 −0.597022
\(910\) 0 0
\(911\) 26.0000 0.861418 0.430709 − 0.902491i \(-0.358263\pi\)
0.430709 + 0.902491i \(0.358263\pi\)
\(912\) 0 0
\(913\) 75.0000 2.48214
\(914\) 0 0
\(915\) 48.0000 1.58683
\(916\) 0 0
\(917\) −12.0000 −0.396275
\(918\) 0 0
\(919\) −58.0000 −1.91324 −0.956622 − 0.291333i \(-0.905901\pi\)
−0.956622 + 0.291333i \(0.905901\pi\)
\(920\) 0 0
\(921\) 51.0000 1.68051
\(922\) 0 0
\(923\) 18.0000 0.592477
\(924\) 0 0
\(925\) −1.00000 −0.0328798
\(926\) 0 0
\(927\) 108.000 3.54719
\(928\) 0 0
\(929\) 18.0000 0.590561 0.295280 − 0.955411i \(-0.404587\pi\)
0.295280 + 0.955411i \(0.404587\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 37.0000 1.20874 0.604369 − 0.796705i \(-0.293425\pi\)
0.604369 + 0.796705i \(0.293425\pi\)
\(938\) 0 0
\(939\) 66.0000 2.15383
\(940\) 0 0
\(941\) 10.0000 0.325991 0.162995 − 0.986627i \(-0.447884\pi\)
0.162995 + 0.986627i \(0.447884\pi\)
\(942\) 0 0
\(943\) −18.0000 −0.586161
\(944\) 0 0
\(945\) −18.0000 −0.585540
\(946\) 0 0
\(947\) −12.0000 −0.389948 −0.194974 − 0.980808i \(-0.562462\pi\)
−0.194974 + 0.980808i \(0.562462\pi\)
\(948\) 0 0
\(949\) −2.00000 −0.0649227
\(950\) 0 0
\(951\) −66.0000 −2.14020
\(952\) 0 0
\(953\) 61.0000 1.97598 0.987992 − 0.154506i \(-0.0493785\pi\)
0.987992 + 0.154506i \(0.0493785\pi\)
\(954\) 0 0
\(955\) −8.00000 −0.258874
\(956\) 0 0
\(957\) −90.0000 −2.90929
\(958\) 0 0
\(959\) 6.00000 0.193750
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) 72.0000 2.32017
\(964\) 0 0
\(965\) −52.0000 −1.67394
\(966\) 0 0
\(967\) −14.0000 −0.450210 −0.225105 − 0.974335i \(-0.572272\pi\)
−0.225105 + 0.974335i \(0.572272\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 8.00000 0.256732 0.128366 − 0.991727i \(-0.459027\pi\)
0.128366 + 0.991727i \(0.459027\pi\)
\(972\) 0 0
\(973\) 4.00000 0.128234
\(974\) 0 0
\(975\) −6.00000 −0.192154
\(976\) 0 0
\(977\) 28.0000 0.895799 0.447900 − 0.894084i \(-0.352172\pi\)
0.447900 + 0.894084i \(0.352172\pi\)
\(978\) 0 0
\(979\) 20.0000 0.639203
\(980\) 0 0
\(981\) 96.0000 3.06504
\(982\) 0 0
\(983\) 9.00000 0.287055 0.143528 − 0.989646i \(-0.454155\pi\)
0.143528 + 0.989646i \(0.454155\pi\)
\(984\) 0 0
\(985\) −6.00000 −0.191176
\(986\) 0 0
\(987\) 27.0000 0.859419
\(988\) 0 0
\(989\) −4.00000 −0.127193
\(990\) 0 0
\(991\) −18.0000 −0.571789 −0.285894 − 0.958261i \(-0.592291\pi\)
−0.285894 + 0.958261i \(0.592291\pi\)
\(992\) 0 0
\(993\) 6.00000 0.190404
\(994\) 0 0
\(995\) 4.00000 0.126809
\(996\) 0 0
\(997\) 42.0000 1.33015 0.665077 − 0.746775i \(-0.268399\pi\)
0.665077 + 0.746775i \(0.268399\pi\)
\(998\) 0 0
\(999\) 9.00000 0.284747
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 2368.2.a.q.1.1 1
4.3 odd 2 2368.2.a.b.1.1 1
8.3 odd 2 592.2.a.e.1.1 1
8.5 even 2 37.2.a.a.1.1 ✓ 1
24.5 odd 2 333.2.a.d.1.1 1
24.11 even 2 5328.2.a.r.1.1 1
40.13 odd 4 925.2.b.b.149.2 2
40.29 even 2 925.2.a.e.1.1 1
40.37 odd 4 925.2.b.b.149.1 2
56.13 odd 2 1813.2.a.a.1.1 1
88.21 odd 2 4477.2.a.b.1.1 1
104.77 even 2 6253.2.a.c.1.1 1
120.29 odd 2 8325.2.a.e.1.1 1
296.117 odd 4 1369.2.b.c.1368.2 2
296.221 even 2 1369.2.a.e.1.1 1
296.253 odd 4 1369.2.b.c.1368.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
37.2.a.a.1.1 ✓ 1 8.5 even 2
333.2.a.d.1.1 1 24.5 odd 2
592.2.a.e.1.1 1 8.3 odd 2
925.2.a.e.1.1 1 40.29 even 2
925.2.b.b.149.1 2 40.37 odd 4
925.2.b.b.149.2 2 40.13 odd 4
1369.2.a.e.1.1 1 296.221 even 2
1369.2.b.c.1368.1 2 296.253 odd 4
1369.2.b.c.1368.2 2 296.117 odd 4
1813.2.a.a.1.1 1 56.13 odd 2
2368.2.a.b.1.1 1 4.3 odd 2
2368.2.a.q.1.1 1 1.1 even 1 trivial
4477.2.a.b.1.1 1 88.21 odd 2
5328.2.a.r.1.1 1 24.11 even 2
6253.2.a.c.1.1 1 104.77 even 2
8325.2.a.e.1.1 1 120.29 odd 2