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Properties

Label 8325.2.a.e.1.1
Level $8325$
Weight $2$
Character 8325.1
Self dual yes
Analytic conductor $66.475$
Analytic rank $1$
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] = [8325,2,Mod(1,8325)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8325, 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("8325.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8325 = 3^{2} \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8325.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: \(66.4754596827\)
Analytic rank: \(1\)
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 \(\chi\) \(=\) 8325.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000 q^{2} +2.00000 q^{4} +1.00000 q^{7} +5.00000 q^{11} +2.00000 q^{13} -2.00000 q^{14} -4.00000 q^{16} -10.0000 q^{22} +2.00000 q^{23} -4.00000 q^{26} +2.00000 q^{28} -6.00000 q^{29} -4.00000 q^{31} +8.00000 q^{32} +1.00000 q^{37} +9.00000 q^{41} -2.00000 q^{43} +10.0000 q^{44} -4.00000 q^{46} -9.00000 q^{47} -6.00000 q^{49} +4.00000 q^{52} +1.00000 q^{53} +12.0000 q^{58} -8.00000 q^{59} -8.00000 q^{61} +8.00000 q^{62} -8.00000 q^{64} -8.00000 q^{67} -9.00000 q^{71} +1.00000 q^{73} -2.00000 q^{74} +5.00000 q^{77} +4.00000 q^{79} -18.0000 q^{82} -15.0000 q^{83} +4.00000 q^{86} -4.00000 q^{89} +2.00000 q^{91} +4.00000 q^{92} +18.0000 q^{94} -4.00000 q^{97} +12.0000 q^{98} +O(q^{100})\)

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 \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(3\) 0 0
\(4\) 2.00000 1.00000
\(5\) 0 0
\(6\) 0 0
\(7\) 1.00000 0.377964 0.188982 − 0.981981i \(-0.439481\pi\)
0.188982 + 0.981981i \(0.439481\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 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 \(0.410544\pi\)
\(14\) −2.00000 −0.534522
\(15\) 0 0
\(16\) −4.00000 −1.00000
\(17\) 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
\(21\) 0 0
\(22\) −10.0000 −2.13201
\(23\) 2.00000 0.417029 0.208514 − 0.978019i \(-0.433137\pi\)
0.208514 + 0.978019i \(0.433137\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −4.00000 −0.784465
\(27\) 0 0
\(28\) 2.00000 0.377964
\(29\) −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 \(0.616954\pi\)
\(32\) 8.00000 1.41421
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 1.00000 0.164399
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 9.00000 1.40556 0.702782 − 0.711405i \(-0.251941\pi\)
0.702782 + 0.711405i \(0.251941\pi\)
\(42\) 0 0
\(43\) −2.00000 −0.304997 −0.152499 − 0.988304i \(-0.548732\pi\)
−0.152499 + 0.988304i \(0.548732\pi\)
\(44\) 10.0000 1.50756
\(45\) 0 0
\(46\) −4.00000 −0.589768
\(47\) −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
\(52\) 4.00000 0.554700
\(53\) 1.00000 0.137361 0.0686803 − 0.997639i \(-0.478121\pi\)
0.0686803 + 0.997639i \(0.478121\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 12.0000 1.57568
\(59\) −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 \(-0.671150\pi\)
−0.512148 + 0.858898i \(0.671150\pi\)
\(62\) 8.00000 1.01600
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) −8.00000 −0.977356 −0.488678 − 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −9.00000 −1.06810 −0.534052 − 0.845452i \(-0.679331\pi\)
−0.534052 + 0.845452i \(0.679331\pi\)
\(72\) 0 0
\(73\) 1.00000 0.117041 0.0585206 − 0.998286i \(-0.481362\pi\)
0.0585206 + 0.998286i \(0.481362\pi\)
\(74\) −2.00000 −0.232495
\(75\) 0 0
\(76\) 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
\(81\) 0 0
\(82\) −18.0000 −1.98777
\(83\) −15.0000 −1.64646 −0.823232 − 0.567705i \(-0.807831\pi\)
−0.823232 + 0.567705i \(0.807831\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 4.00000 0.431331
\(87\) 0 0
\(88\) 0 0
\(89\) −4.00000 −0.423999 −0.212000 − 0.977270i \(-0.567998\pi\)
−0.212000 + 0.977270i \(0.567998\pi\)
\(90\) 0 0
\(91\) 2.00000 0.209657
\(92\) 4.00000 0.417029
\(93\) 0 0
\(94\) 18.0000 1.85656
\(95\) 0 0
\(96\) 0 0
\(97\) −4.00000 −0.406138 −0.203069 − 0.979164i \(-0.565092\pi\)
−0.203069 + 0.979164i \(0.565092\pi\)
\(98\) 12.0000 1.21218
\(99\) 0 0
\(100\) 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 \(-0.847074\pi\)
−0.886796 + 0.462160i \(0.847074\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −2.00000 −0.194257
\(107\) −12.0000 −1.16008 −0.580042 − 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 0 0
\(109\) −16.0000 −1.53252 −0.766261 − 0.642529i \(-0.777885\pi\)
−0.766261 + 0.642529i \(0.777885\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −4.00000 −0.377964
\(113\) −18.0000 −1.69330 −0.846649 − 0.532152i \(-0.821383\pi\)
−0.846649 + 0.532152i \(0.821383\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −12.0000 −1.11417
\(117\) 0 0
\(118\) 16.0000 1.47292
\(119\) 0 0
\(120\) 0 0
\(121\) 14.0000 1.27273
\(122\) 16.0000 1.44857
\(123\) 0 0
\(124\) −8.00000 −0.718421
\(125\) 0 0
\(126\) 0 0
\(127\) −1.00000 −0.0887357 −0.0443678 − 0.999015i \(-0.514127\pi\)
−0.0443678 + 0.999015i \(0.514127\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 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
\(134\) 16.0000 1.38219
\(135\) 0 0
\(136\) 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 \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 18.0000 1.51053
\(143\) 10.0000 0.836242
\(144\) 0 0
\(145\) 0 0
\(146\) −2.00000 −0.165521
\(147\) 0 0
\(148\) 2.00000 0.164399
\(149\) 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
\(154\) −10.0000 −0.805823
\(155\) 0 0
\(156\) 0 0
\(157\) −23.0000 −1.83560 −0.917800 − 0.397043i \(-0.870036\pi\)
−0.917800 + 0.397043i \(0.870036\pi\)
\(158\) −8.00000 −0.636446
\(159\) 0 0
\(160\) 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 \(0.250976\pi\)
\(164\) 18.0000 1.40556
\(165\) 0 0
\(166\) 30.0000 2.32845
\(167\) −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
\(172\) −4.00000 −0.304997
\(173\) 9.00000 0.684257 0.342129 − 0.939653i \(-0.388852\pi\)
0.342129 + 0.939653i \(0.388852\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −20.0000 −1.50756
\(177\) 0 0
\(178\) 8.00000 0.599625
\(179\) −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 \(-0.440505\pi\)
0.185824 + 0.982583i \(0.440505\pi\)
\(182\) −4.00000 −0.296500
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) −18.0000 −1.31278
\(189\) 0 0
\(190\) 0 0
\(191\) 4.00000 0.289430 0.144715 − 0.989473i \(-0.453773\pi\)
0.144715 + 0.989473i \(0.453773\pi\)
\(192\) 0 0
\(193\) 26.0000 1.87152 0.935760 − 0.352636i \(-0.114715\pi\)
0.935760 + 0.352636i \(0.114715\pi\)
\(194\) 8.00000 0.574367
\(195\) 0 0
\(196\) −12.0000 −0.857143
\(197\) 3.00000 0.213741 0.106871 − 0.994273i \(-0.465917\pi\)
0.106871 + 0.994273i \(0.465917\pi\)
\(198\) 0 0
\(199\) 2.00000 0.141776 0.0708881 − 0.997484i \(-0.477417\pi\)
0.0708881 + 0.997484i \(0.477417\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 6.00000 0.422159
\(203\) −6.00000 −0.421117
\(204\) 0 0
\(205\) 0 0
\(206\) 36.0000 2.50824
\(207\) 0 0
\(208\) −8.00000 −0.554700
\(209\) 0 0
\(210\) 0 0
\(211\) −13.0000 −0.894957 −0.447478 − 0.894295i \(-0.647678\pi\)
−0.447478 + 0.894295i \(0.647678\pi\)
\(212\) 2.00000 0.137361
\(213\) 0 0
\(214\) 24.0000 1.64061
\(215\) 0 0
\(216\) 0 0
\(217\) −4.00000 −0.271538
\(218\) 32.0000 2.16731
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 17.0000 1.13840 0.569202 − 0.822198i \(-0.307252\pi\)
0.569202 + 0.822198i \(0.307252\pi\)
\(224\) 8.00000 0.534522
\(225\) 0 0
\(226\) 36.0000 2.39468
\(227\) −16.0000 −1.06196 −0.530979 − 0.847385i \(-0.678176\pi\)
−0.530979 + 0.847385i \(0.678176\pi\)
\(228\) 0 0
\(229\) 7.00000 0.462573 0.231287 − 0.972886i \(-0.425707\pi\)
0.231287 + 0.972886i \(0.425707\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 6.00000 0.393073 0.196537 − 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −16.0000 −1.04151
\(237\) 0 0
\(238\) 0 0
\(239\) 6.00000 0.388108 0.194054 − 0.980991i \(-0.437836\pi\)
0.194054 + 0.980991i \(0.437836\pi\)
\(240\) 0 0
\(241\) 14.0000 0.901819 0.450910 − 0.892570i \(-0.351100\pi\)
0.450910 + 0.892570i \(0.351100\pi\)
\(242\) −28.0000 −1.79991
\(243\) 0 0
\(244\) −16.0000 −1.02430
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 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
\(254\) 2.00000 0.125491
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(258\) 0 0
\(259\) 1.00000 0.0621370
\(260\) 0 0
\(261\) 0 0
\(262\) −24.0000 −1.48272
\(263\) 19.0000 1.17159 0.585795 − 0.810459i \(-0.300782\pi\)
0.585795 + 0.810459i \(0.300782\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) −16.0000 −0.977356
\(269\) 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
\(273\) 0 0
\(274\) 12.0000 0.724947
\(275\) 0 0
\(276\) 0 0
\(277\) −12.0000 −0.721010 −0.360505 − 0.932757i \(-0.617396\pi\)
−0.360505 + 0.932757i \(0.617396\pi\)
\(278\) −8.00000 −0.479808
\(279\) 0 0
\(280\) 0 0
\(281\) −12.0000 −0.715860 −0.357930 − 0.933748i \(-0.616517\pi\)
−0.357930 + 0.933748i \(0.616517\pi\)
\(282\) 0 0
\(283\) −4.00000 −0.237775 −0.118888 − 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) −18.0000 −1.06810
\(285\) 0 0
\(286\) −20.0000 −1.18262
\(287\) 9.00000 0.531253
\(288\) 0 0
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 2.00000 0.117041
\(293\) −2.00000 −0.116841 −0.0584206 − 0.998292i \(-0.518606\pi\)
−0.0584206 + 0.998292i \(0.518606\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) −10.0000 −0.579284
\(299\) 4.00000 0.231326
\(300\) 0 0
\(301\) −2.00000 −0.115278
\(302\) −32.0000 −1.84139
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 17.0000 0.970241 0.485121 − 0.874447i \(-0.338776\pi\)
0.485121 + 0.874447i \(0.338776\pi\)
\(308\) 10.0000 0.569803
\(309\) 0 0
\(310\) 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 \(-0.713581\pi\)
−0.621757 + 0.783210i \(0.713581\pi\)
\(314\) 46.0000 2.59593
\(315\) 0 0
\(316\) 8.00000 0.450035
\(317\) 22.0000 1.23564 0.617822 − 0.786318i \(-0.288015\pi\)
0.617822 + 0.786318i \(0.288015\pi\)
\(318\) 0 0
\(319\) −30.0000 −1.67968
\(320\) 0 0
\(321\) 0 0
\(322\) −4.00000 −0.222911
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) −36.0000 −1.99386
\(327\) 0 0
\(328\) 0 0
\(329\) −9.00000 −0.496186
\(330\) 0 0
\(331\) −2.00000 −0.109930 −0.0549650 − 0.998488i \(-0.517505\pi\)
−0.0549650 + 0.998488i \(0.517505\pi\)
\(332\) −30.0000 −1.64646
\(333\) 0 0
\(334\) 24.0000 1.31322
\(335\) 0 0
\(336\) 0 0
\(337\) 25.0000 1.36184 0.680918 − 0.732359i \(-0.261581\pi\)
0.680918 + 0.732359i \(0.261581\pi\)
\(338\) 18.0000 0.979071
\(339\) 0 0
\(340\) 0 0
\(341\) −20.0000 −1.08306
\(342\) 0 0
\(343\) −13.0000 −0.701934
\(344\) 0 0
\(345\) 0 0
\(346\) −18.0000 −0.967686
\(347\) −10.0000 −0.536828 −0.268414 − 0.963304i \(-0.586500\pi\)
−0.268414 + 0.963304i \(0.586500\pi\)
\(348\) 0 0
\(349\) 6.00000 0.321173 0.160586 − 0.987022i \(-0.448662\pi\)
0.160586 + 0.987022i \(0.448662\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 40.0000 2.13201
\(353\) 8.00000 0.425797 0.212899 − 0.977074i \(-0.431710\pi\)
0.212899 + 0.977074i \(0.431710\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −8.00000 −0.423999
\(357\) 0 0
\(358\) 36.0000 1.90266
\(359\) 15.0000 0.791670 0.395835 − 0.918322i \(-0.370455\pi\)
0.395835 + 0.918322i \(0.370455\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) −10.0000 −0.525588
\(363\) 0 0
\(364\) 4.00000 0.209657
\(365\) 0 0
\(366\) 0 0
\(367\) −8.00000 −0.417597 −0.208798 − 0.977959i \(-0.566955\pi\)
−0.208798 + 0.977959i \(0.566955\pi\)
\(368\) −8.00000 −0.417029
\(369\) 0 0
\(370\) 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
\(375\) 0 0
\(376\) 0 0
\(377\) −12.0000 −0.618031
\(378\) 0 0
\(379\) 15.0000 0.770498 0.385249 − 0.922813i \(-0.374116\pi\)
0.385249 + 0.922813i \(0.374116\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −8.00000 −0.409316
\(383\) 20.0000 1.02195 0.510976 − 0.859595i \(-0.329284\pi\)
0.510976 + 0.859595i \(0.329284\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −52.0000 −2.64673
\(387\) 0 0
\(388\) −8.00000 −0.406138
\(389\) −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
\(393\) 0 0
\(394\) −6.00000 −0.302276
\(395\) 0 0
\(396\) 0 0
\(397\) 5.00000 0.250943 0.125471 − 0.992097i \(-0.459956\pi\)
0.125471 + 0.992097i \(0.459956\pi\)
\(398\) −4.00000 −0.200502
\(399\) 0 0
\(400\) 0 0
\(401\) −18.0000 −0.898877 −0.449439 − 0.893311i \(-0.648376\pi\)
−0.449439 + 0.893311i \(0.648376\pi\)
\(402\) 0 0
\(403\) −8.00000 −0.398508
\(404\) −6.00000 −0.298511
\(405\) 0 0
\(406\) 12.0000 0.595550
\(407\) 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
\(411\) 0 0
\(412\) −36.0000 −1.77359
\(413\) −8.00000 −0.393654
\(414\) 0 0
\(415\) 0 0
\(416\) 16.0000 0.784465
\(417\) 0 0
\(418\) 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 \(-0.698844\pi\)
−0.584844 + 0.811146i \(0.698844\pi\)
\(422\) 26.0000 1.26566
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −8.00000 −0.387147
\(428\) −24.0000 −1.16008
\(429\) 0 0
\(430\) 0 0
\(431\) 30.0000 1.44505 0.722525 − 0.691345i \(-0.242982\pi\)
0.722525 + 0.691345i \(0.242982\pi\)
\(432\) 0 0
\(433\) −9.00000 −0.432512 −0.216256 − 0.976337i \(-0.569385\pi\)
−0.216256 + 0.976337i \(0.569385\pi\)
\(434\) 8.00000 0.384012
\(435\) 0 0
\(436\) −32.0000 −1.53252
\(437\) 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
\(441\) 0 0
\(442\) 0 0
\(443\) 1.00000 0.0475114 0.0237557 − 0.999718i \(-0.492438\pi\)
0.0237557 + 0.999718i \(0.492438\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −34.0000 −1.60995
\(447\) 0 0
\(448\) −8.00000 −0.377964
\(449\) −36.0000 −1.69895 −0.849473 − 0.527633i \(-0.823080\pi\)
−0.849473 + 0.527633i \(0.823080\pi\)
\(450\) 0 0
\(451\) 45.0000 2.11897
\(452\) −36.0000 −1.69330
\(453\) 0 0
\(454\) 32.0000 1.50183
\(455\) 0 0
\(456\) 0 0
\(457\) −18.0000 −0.842004 −0.421002 − 0.907060i \(-0.638322\pi\)
−0.421002 + 0.907060i \(0.638322\pi\)
\(458\) −14.0000 −0.654177
\(459\) 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 \(-0.329196\pi\)
0.511213 + 0.859454i \(0.329196\pi\)
\(464\) 24.0000 1.11417
\(465\) 0 0
\(466\) −12.0000 −0.555889
\(467\) −2.00000 −0.0925490 −0.0462745 − 0.998929i \(-0.514735\pi\)
−0.0462745 + 0.998929i \(0.514735\pi\)
\(468\) 0 0
\(469\) −8.00000 −0.369406
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −10.0000 −0.459800
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) −12.0000 −0.548867
\(479\) −14.0000 −0.639676 −0.319838 − 0.947472i \(-0.603629\pi\)
−0.319838 + 0.947472i \(0.603629\pi\)
\(480\) 0 0
\(481\) 2.00000 0.0911922
\(482\) −28.0000 −1.27537
\(483\) 0 0
\(484\) 28.0000 1.27273
\(485\) 0 0
\(486\) 0 0
\(487\) 24.0000 1.08754 0.543772 − 0.839233i \(-0.316996\pi\)
0.543772 + 0.839233i \(0.316996\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 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
\(495\) 0 0
\(496\) 16.0000 0.718421
\(497\) −9.00000 −0.403705
\(498\) 0 0
\(499\) 12.0000 0.537194 0.268597 − 0.963253i \(-0.413440\pi\)
0.268597 + 0.963253i \(0.413440\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −4.00000 −0.178529
\(503\) 16.0000 0.713405 0.356702 − 0.934218i \(-0.383901\pi\)
0.356702 + 0.934218i \(0.383901\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −20.0000 −0.889108
\(507\) 0 0
\(508\) −2.00000 −0.0887357
\(509\) 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
\(512\) −32.0000 −1.41421
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −45.0000 −1.97910
\(518\) −2.00000 −0.0878750
\(519\) 0 0
\(520\) 0 0
\(521\) 33.0000 1.44576 0.722878 − 0.690976i \(-0.242819\pi\)
0.722878 + 0.690976i \(0.242819\pi\)
\(522\) 0 0
\(523\) 22.0000 0.961993 0.480996 − 0.876723i \(-0.340275\pi\)
0.480996 + 0.876723i \(0.340275\pi\)
\(524\) 24.0000 1.04844
\(525\) 0 0
\(526\) −38.0000 −1.65688
\(527\) 0 0
\(528\) 0 0
\(529\) −19.0000 −0.826087
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 18.0000 0.779667
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) −12.0000 −0.517357
\(539\) −30.0000 −1.29219
\(540\) 0 0
\(541\) 20.0000 0.859867 0.429934 − 0.902861i \(-0.358537\pi\)
0.429934 + 0.902861i \(0.358537\pi\)
\(542\) 62.0000 2.66313
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −8.00000 −0.342055 −0.171028 − 0.985266i \(-0.554709\pi\)
−0.171028 + 0.985266i \(0.554709\pi\)
\(548\) −12.0000 −0.512615
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 4.00000 0.170097
\(554\) 24.0000 1.01966
\(555\) 0 0
\(556\) 8.00000 0.339276
\(557\) −18.0000 −0.762684 −0.381342 − 0.924434i \(-0.624538\pi\)
−0.381342 + 0.924434i \(0.624538\pi\)
\(558\) 0 0
\(559\) −4.00000 −0.169182
\(560\) 0 0
\(561\) 0 0
\(562\) 24.0000 1.01238
\(563\) −30.0000 −1.26435 −0.632175 − 0.774826i \(-0.717837\pi\)
−0.632175 + 0.774826i \(0.717837\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 8.00000 0.336265
\(567\) 0 0
\(568\) 0 0
\(569\) 24.0000 1.00613 0.503066 − 0.864248i \(-0.332205\pi\)
0.503066 + 0.864248i \(0.332205\pi\)
\(570\) 0 0
\(571\) 7.00000 0.292941 0.146470 − 0.989215i \(-0.453209\pi\)
0.146470 + 0.989215i \(0.453209\pi\)
\(572\) 20.0000 0.836242
\(573\) 0 0
\(574\) −18.0000 −0.751305
\(575\) 0 0
\(576\) 0 0
\(577\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(578\) 34.0000 1.41421
\(579\) 0 0
\(580\) 0 0
\(581\) −15.0000 −0.622305
\(582\) 0 0
\(583\) 5.00000 0.207079
\(584\) 0 0
\(585\) 0 0
\(586\) 4.00000 0.165238
\(587\) −32.0000 −1.32078 −0.660391 − 0.750922i \(-0.729609\pi\)
−0.660391 + 0.750922i \(0.729609\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) −4.00000 −0.164399
\(593\) −5.00000 −0.205325 −0.102663 − 0.994716i \(-0.532736\pi\)
−0.102663 + 0.994716i \(0.532736\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 10.0000 0.409616
\(597\) 0 0
\(598\) −8.00000 −0.327144
\(599\) −1.00000 −0.0408589 −0.0204294 − 0.999791i \(-0.506503\pi\)
−0.0204294 + 0.999791i \(0.506503\pi\)
\(600\) 0 0
\(601\) −22.0000 −0.897399 −0.448699 − 0.893683i \(-0.648113\pi\)
−0.448699 + 0.893683i \(0.648113\pi\)
\(602\) 4.00000 0.163028
\(603\) 0 0
\(604\) 32.0000 1.30206
\(605\) 0 0
\(606\) 0 0
\(607\) 32.0000 1.29884 0.649420 − 0.760430i \(-0.275012\pi\)
0.649420 + 0.760430i \(0.275012\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 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 \(0.597962\pi\)
\(614\) −34.0000 −1.37213
\(615\) 0 0
\(616\) 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 \(-0.506397\pi\)
−0.0200967 + 0.999798i \(0.506397\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −4.00000 −0.160257
\(624\) 0 0
\(625\) 0 0
\(626\) 44.0000 1.75859
\(627\) 0 0
\(628\) −46.0000 −1.83560
\(629\) 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
\(633\) 0 0
\(634\) −44.0000 −1.74746
\(635\) 0 0
\(636\) 0 0
\(637\) −12.0000 −0.475457
\(638\) 60.0000 2.37542
\(639\) 0 0
\(640\) 0 0
\(641\) 1.00000 0.0394976 0.0197488 − 0.999805i \(-0.493713\pi\)
0.0197488 + 0.999805i \(0.493713\pi\)
\(642\) 0 0
\(643\) −14.0000 −0.552106 −0.276053 − 0.961142i \(-0.589027\pi\)
−0.276053 + 0.961142i \(0.589027\pi\)
\(644\) 4.00000 0.157622
\(645\) 0 0
\(646\) 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
\(651\) 0 0
\(652\) 36.0000 1.40987
\(653\) −24.0000 −0.939193 −0.469596 − 0.882881i \(-0.655601\pi\)
−0.469596 + 0.882881i \(0.655601\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −36.0000 −1.40556
\(657\) 0 0
\(658\) 18.0000 0.701713
\(659\) 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 \(-0.683295\pi\)
−0.544537 + 0.838737i \(0.683295\pi\)
\(662\) 4.00000 0.155464
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −12.0000 −0.464642
\(668\) −24.0000 −0.928588
\(669\) 0 0
\(670\) 0 0
\(671\) −40.0000 −1.54418
\(672\) 0 0
\(673\) −27.0000 −1.04077 −0.520387 − 0.853931i \(-0.674212\pi\)
−0.520387 + 0.853931i \(0.674212\pi\)
\(674\) −50.0000 −1.92593
\(675\) 0 0
\(676\) −18.0000 −0.692308
\(677\) −11.0000 −0.422764 −0.211382 − 0.977403i \(-0.567796\pi\)
−0.211382 + 0.977403i \(0.567796\pi\)
\(678\) 0 0
\(679\) −4.00000 −0.153506
\(680\) 0 0
\(681\) 0 0
\(682\) 40.0000 1.53168
\(683\) 18.0000 0.688751 0.344375 − 0.938832i \(-0.388091\pi\)
0.344375 + 0.938832i \(0.388091\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 26.0000 0.992685
\(687\) 0 0
\(688\) 8.00000 0.304997
\(689\) 2.00000 0.0761939
\(690\) 0 0
\(691\) −20.0000 −0.760836 −0.380418 − 0.924815i \(-0.624220\pi\)
−0.380418 + 0.924815i \(0.624220\pi\)
\(692\) 18.0000 0.684257
\(693\) 0 0
\(694\) 20.0000 0.759190
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) −12.0000 −0.454207
\(699\) 0 0
\(700\) 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
\(704\) −40.0000 −1.50756
\(705\) 0 0
\(706\) −16.0000 −0.602168
\(707\) −3.00000 −0.112827
\(708\) 0 0
\(709\) 40.0000 1.50223 0.751116 − 0.660171i \(-0.229516\pi\)
0.751116 + 0.660171i \(0.229516\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −8.00000 −0.299602
\(714\) 0 0
\(715\) 0 0
\(716\) −36.0000 −1.34538
\(717\) 0 0
\(718\) −30.0000 −1.11959
\(719\) −39.0000 −1.45445 −0.727227 − 0.686397i \(-0.759191\pi\)
−0.727227 + 0.686397i \(0.759191\pi\)
\(720\) 0 0
\(721\) −18.0000 −0.670355
\(722\) 38.0000 1.41421
\(723\) 0 0
\(724\) 10.0000 0.371647
\(725\) 0 0
\(726\) 0 0
\(727\) −16.0000 −0.593407 −0.296704 − 0.954970i \(-0.595887\pi\)
−0.296704 + 0.954970i \(0.595887\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −7.00000 −0.258551 −0.129275 − 0.991609i \(-0.541265\pi\)
−0.129275 + 0.991609i \(0.541265\pi\)
\(734\) 16.0000 0.590571
\(735\) 0 0
\(736\) 16.0000 0.589768
\(737\) −40.0000 −1.47342
\(738\) 0 0
\(739\) −9.00000 −0.331070 −0.165535 − 0.986204i \(-0.552935\pi\)
−0.165535 + 0.986204i \(0.552935\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −2.00000 −0.0734223
\(743\) 21.0000 0.770415 0.385208 − 0.922830i \(-0.374130\pi\)
0.385208 + 0.922830i \(0.374130\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −38.0000 −1.39128
\(747\) 0 0
\(748\) 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 \(0.349235\pi\)
\(752\) 36.0000 1.31278
\(753\) 0 0
\(754\) 24.0000 0.874028
\(755\) 0 0
\(756\) 0 0
\(757\) 50.0000 1.81728 0.908640 − 0.417579i \(-0.137121\pi\)
0.908640 + 0.417579i \(0.137121\pi\)
\(758\) −30.0000 −1.08965
\(759\) 0 0
\(760\) 0 0
\(761\) 35.0000 1.26875 0.634375 − 0.773026i \(-0.281258\pi\)
0.634375 + 0.773026i \(0.281258\pi\)
\(762\) 0 0
\(763\) −16.0000 −0.579239
\(764\) 8.00000 0.289430
\(765\) 0 0
\(766\) −40.0000 −1.44526
\(767\) −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
\(772\) 52.0000 1.87152
\(773\) −9.00000 −0.323708 −0.161854 − 0.986815i \(-0.551747\pi\)
−0.161854 + 0.986815i \(0.551747\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 8.00000 0.286814
\(779\) 0 0
\(780\) 0 0
\(781\) −45.0000 −1.61023
\(782\) 0 0
\(783\) 0 0
\(784\) 24.0000 0.857143
\(785\) 0 0
\(786\) 0 0
\(787\) 5.00000 0.178231 0.0891154 − 0.996021i \(-0.471596\pi\)
0.0891154 + 0.996021i \(0.471596\pi\)
\(788\) 6.00000 0.213741
\(789\) 0 0
\(790\) 0 0
\(791\) −18.0000 −0.640006
\(792\) 0 0
\(793\) −16.0000 −0.568177
\(794\) −10.0000 −0.354887
\(795\) 0 0
\(796\) 4.00000 0.141776
\(797\) 52.0000 1.84193 0.920967 − 0.389640i \(-0.127401\pi\)
0.920967 + 0.389640i \(0.127401\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 36.0000 1.27120
\(803\) 5.00000 0.176446
\(804\) 0 0
\(805\) 0 0
\(806\) 16.0000 0.563576
\(807\) 0 0
\(808\) 0 0
\(809\) −2.00000 −0.0703163 −0.0351581 − 0.999382i \(-0.511193\pi\)
−0.0351581 + 0.999382i \(0.511193\pi\)
\(810\) 0 0
\(811\) 47.0000 1.65039 0.825197 − 0.564846i \(-0.191064\pi\)
0.825197 + 0.564846i \(0.191064\pi\)
\(812\) −12.0000 −0.421117
\(813\) 0 0
\(814\) −10.0000 −0.350500
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) −40.0000 −1.39857
\(819\) 0 0
\(820\) 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 \(-0.410043\pi\)
0.278862 + 0.960331i \(0.410043\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 16.0000 0.556711
\(827\) 22.0000 0.765015 0.382507 − 0.923952i \(-0.375061\pi\)
0.382507 + 0.923952i \(0.375061\pi\)
\(828\) 0 0
\(829\) −4.00000 −0.138926 −0.0694629 − 0.997585i \(-0.522129\pi\)
−0.0694629 + 0.997585i \(0.522129\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −16.0000 −0.554700
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 14.0000 0.483622
\(839\) −44.0000 −1.51905 −0.759524 − 0.650479i \(-0.774568\pi\)
−0.759524 + 0.650479i \(0.774568\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 48.0000 1.65419
\(843\) 0 0
\(844\) −26.0000 −0.894957
\(845\) 0 0
\(846\) 0 0
\(847\) 14.0000 0.481046
\(848\) −4.00000 −0.137361
\(849\) 0 0
\(850\) 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 \(0.646836\pi\)
\(854\) 16.0000 0.547509
\(855\) 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 \(-0.610832\pi\)
−0.341196 + 0.939992i \(0.610832\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −60.0000 −2.04361
\(863\) −24.0000 −0.816970 −0.408485 − 0.912765i \(-0.633943\pi\)
−0.408485 + 0.912765i \(0.633943\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 18.0000 0.611665
\(867\) 0 0
\(868\) −8.00000 −0.271538
\(869\) 20.0000 0.678454
\(870\) 0 0
\(871\) −16.0000 −0.542139
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −50.0000 −1.68838 −0.844190 − 0.536044i \(-0.819918\pi\)
−0.844190 + 0.536044i \(0.819918\pi\)
\(878\) −56.0000 −1.88991
\(879\) 0 0
\(880\) 0 0
\(881\) 14.0000 0.471672 0.235836 − 0.971793i \(-0.424217\pi\)
0.235836 + 0.971793i \(0.424217\pi\)
\(882\) 0 0
\(883\) −48.0000 −1.61533 −0.807664 − 0.589643i \(-0.799269\pi\)
−0.807664 + 0.589643i \(0.799269\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −2.00000 −0.0671913
\(887\) 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
\(891\) 0 0
\(892\) 34.0000 1.13840
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 72.0000 2.40267
\(899\) 24.0000 0.800445
\(900\) 0 0
\(901\) 0 0
\(902\) −90.0000 −2.99667
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −52.0000 −1.72663 −0.863316 − 0.504664i \(-0.831616\pi\)
−0.863316 + 0.504664i \(0.831616\pi\)
\(908\) −32.0000 −1.06196
\(909\) 0 0
\(910\) 0 0
\(911\) −26.0000 −0.861418 −0.430709 − 0.902491i \(-0.641737\pi\)
−0.430709 + 0.902491i \(0.641737\pi\)
\(912\) 0 0
\(913\) −75.0000 −2.48214
\(914\) 36.0000 1.19077
\(915\) 0 0
\(916\) 14.0000 0.462573
\(917\) 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
\(921\) 0 0
\(922\) 60.0000 1.97599
\(923\) −18.0000 −0.592477
\(924\) 0 0
\(925\) 0 0
\(926\) −44.0000 −1.44593
\(927\) 0 0
\(928\) −48.0000 −1.57568
\(929\) −18.0000 −0.590561 −0.295280 − 0.955411i \(-0.595413\pi\)
−0.295280 + 0.955411i \(0.595413\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 12.0000 0.393073
\(933\) 0 0
\(934\) 4.00000 0.130884
\(935\) 0 0
\(936\) 0 0
\(937\) −37.0000 −1.20874 −0.604369 − 0.796705i \(-0.706575\pi\)
−0.604369 + 0.796705i \(0.706575\pi\)
\(938\) 16.0000 0.522419
\(939\) 0 0
\(940\) 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
\(944\) 32.0000 1.04151
\(945\) 0 0
\(946\) 20.0000 0.650256
\(947\) 12.0000 0.389948 0.194974 − 0.980808i \(-0.437538\pi\)
0.194974 + 0.980808i \(0.437538\pi\)
\(948\) 0 0
\(949\) 2.00000 0.0649227
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 61.0000 1.97598 0.987992 − 0.154506i \(-0.0493785\pi\)
0.987992 + 0.154506i \(0.0493785\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 12.0000 0.388108
\(957\) 0 0
\(958\) 28.0000 0.904639
\(959\) −6.00000 −0.193750
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) −4.00000 −0.128965
\(963\) 0 0
\(964\) 28.0000 0.901819
\(965\) 0 0
\(966\) 0 0
\(967\) 14.0000 0.450210 0.225105 − 0.974335i \(-0.427728\pi\)
0.225105 + 0.974335i \(0.427728\pi\)
\(968\) 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
\(974\) −48.0000 −1.53802
\(975\) 0 0
\(976\) 32.0000 1.02430
\(977\) 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
\(981\) 0 0
\(982\) −56.0000 −1.78703
\(983\) 9.00000 0.287055 0.143528 − 0.989646i \(-0.454155\pi\)
0.143528 + 0.989646i \(0.454155\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 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 \(0.592291\pi\)
\(992\) −32.0000 −1.01600
\(993\) 0 0
\(994\) 18.0000 0.570925
\(995\) 0 0
\(996\) 0 0
\(997\) 42.0000 1.33015 0.665077 − 0.746775i \(-0.268399\pi\)
0.665077 + 0.746775i \(0.268399\pi\)
\(998\) −24.0000 −0.759707
\(999\) 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 8325.2.a.e.1.1 1
3.2 odd 2 925.2.a.e.1.1 1
5.4 even 2 333.2.a.d.1.1 1
15.2 even 4 925.2.b.b.149.2 2
15.8 even 4 925.2.b.b.149.1 2
15.14 odd 2 37.2.a.a.1.1 ✓ 1
20.19 odd 2 5328.2.a.r.1.1 1
60.59 even 2 592.2.a.e.1.1 1
105.104 even 2 1813.2.a.a.1.1 1
120.29 odd 2 2368.2.a.q.1.1 1
120.59 even 2 2368.2.a.b.1.1 1
165.164 even 2 4477.2.a.b.1.1 1
195.194 odd 2 6253.2.a.c.1.1 1
555.179 even 4 1369.2.b.c.1368.1 2
555.524 even 4 1369.2.b.c.1368.2 2
555.554 odd 2 1369.2.a.e.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
37.2.a.a.1.1 ✓ 1 15.14 odd 2
333.2.a.d.1.1 1 5.4 even 2
592.2.a.e.1.1 1 60.59 even 2
925.2.a.e.1.1 1 3.2 odd 2
925.2.b.b.149.1 2 15.8 even 4
925.2.b.b.149.2 2 15.2 even 4
1369.2.a.e.1.1 1 555.554 odd 2
1369.2.b.c.1368.1 2 555.179 even 4
1369.2.b.c.1368.2 2 555.524 even 4
1813.2.a.a.1.1 1 105.104 even 2
2368.2.a.b.1.1 1 120.59 even 2
2368.2.a.q.1.1 1 120.29 odd 2
4477.2.a.b.1.1 1 165.164 even 2
5328.2.a.r.1.1 1 20.19 odd 2
6253.2.a.c.1.1 1 195.194 odd 2
8325.2.a.e.1.1 1 1.1 even 1 trivial