Properties

Label 37.2.a.a.1.1
Level $37$
Weight $2$
Character 37.1
Self dual yes
Analytic conductor $0.295$
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] = [37,2,Mod(1,37)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(37, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("37.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 37.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: \(0.295446487479\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 37.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000 q^{2} -3.00000 q^{3} +2.00000 q^{4} -2.00000 q^{5} +6.00000 q^{6} -1.00000 q^{7} +6.00000 q^{9} +O(q^{10})\) \(q-2.00000 q^{2} -3.00000 q^{3} +2.00000 q^{4} -2.00000 q^{5} +6.00000 q^{6} -1.00000 q^{7} +6.00000 q^{9} +4.00000 q^{10} -5.00000 q^{11} -6.00000 q^{12} -2.00000 q^{13} +2.00000 q^{14} +6.00000 q^{15} -4.00000 q^{16} -12.0000 q^{18} -4.00000 q^{20} +3.00000 q^{21} +10.0000 q^{22} +2.00000 q^{23} -1.00000 q^{25} +4.00000 q^{26} -9.00000 q^{27} -2.00000 q^{28} +6.00000 q^{29} -12.0000 q^{30} -4.00000 q^{31} +8.00000 q^{32} +15.0000 q^{33} +2.00000 q^{35} +12.0000 q^{36} -1.00000 q^{37} +6.00000 q^{39} -9.00000 q^{41} -6.00000 q^{42} +2.00000 q^{43} -10.0000 q^{44} -12.0000 q^{45} -4.00000 q^{46} -9.00000 q^{47} +12.0000 q^{48} -6.00000 q^{49} +2.00000 q^{50} -4.00000 q^{52} +1.00000 q^{53} +18.0000 q^{54} +10.0000 q^{55} -12.0000 q^{58} +8.00000 q^{59} +12.0000 q^{60} -8.00000 q^{61} +8.00000 q^{62} -6.00000 q^{63} -8.00000 q^{64} +4.00000 q^{65} -30.0000 q^{66} +8.00000 q^{67} -6.00000 q^{69} -4.00000 q^{70} +9.00000 q^{71} -1.00000 q^{73} +2.00000 q^{74} +3.00000 q^{75} +5.00000 q^{77} -12.0000 q^{78} +4.00000 q^{79} +8.00000 q^{80} +9.00000 q^{81} +18.0000 q^{82} -15.0000 q^{83} +6.00000 q^{84} -4.00000 q^{86} -18.0000 q^{87} +4.00000 q^{89} +24.0000 q^{90} +2.00000 q^{91} +4.00000 q^{92} +12.0000 q^{93} +18.0000 q^{94} -24.0000 q^{96} +4.00000 q^{97} +12.0000 q^{98} -30.0000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(3\) −3.00000 −1.73205 −0.866025 0.500000i \(-0.833333\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(4\) 2.00000 1.00000
\(5\) −2.00000 −0.894427 −0.447214 0.894427i \(-0.647584\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) 6.00000 2.44949
\(7\) −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
\(10\) 4.00000 1.26491
\(11\) −5.00000 −1.50756 −0.753778 0.657129i \(-0.771771\pi\)
−0.753778 + 0.657129i \(0.771771\pi\)
\(12\) −6.00000 −1.73205
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 2.00000 0.534522
\(15\) 6.00000 1.54919
\(16\) −4.00000 −1.00000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) −12.0000 −2.82843
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) −4.00000 −0.894427
\(21\) 3.00000 0.654654
\(22\) 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
\(25\) −1.00000 −0.200000
\(26\) 4.00000 0.784465
\(27\) −9.00000 −1.73205
\(28\) −2.00000 −0.377964
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) −12.0000 −2.19089
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 8.00000 1.41421
\(33\) 15.0000 2.61116
\(34\) 0 0
\(35\) 2.00000 0.338062
\(36\) 12.0000 2.00000
\(37\) −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 \(0.748059\pi\)
\(42\) −6.00000 −0.925820
\(43\) 2.00000 0.304997 0.152499 0.988304i \(-0.451268\pi\)
0.152499 + 0.988304i \(0.451268\pi\)
\(44\) −10.0000 −1.50756
\(45\) −12.0000 −1.78885
\(46\) −4.00000 −0.589768
\(47\) −9.00000 −1.31278 −0.656392 0.754420i \(-0.727918\pi\)
−0.656392 + 0.754420i \(0.727918\pi\)
\(48\) 12.0000 1.73205
\(49\) −6.00000 −0.857143
\(50\) 2.00000 0.282843
\(51\) 0 0
\(52\) −4.00000 −0.554700
\(53\) 1.00000 0.137361 0.0686803 0.997639i \(-0.478121\pi\)
0.0686803 + 0.997639i \(0.478121\pi\)
\(54\) 18.0000 2.44949
\(55\) 10.0000 1.34840
\(56\) 0 0
\(57\) 0 0
\(58\) −12.0000 −1.57568
\(59\) 8.00000 1.04151 0.520756 0.853706i \(-0.325650\pi\)
0.520756 + 0.853706i \(0.325650\pi\)
\(60\) 12.0000 1.54919
\(61\) −8.00000 −1.02430 −0.512148 0.858898i \(-0.671150\pi\)
−0.512148 + 0.858898i \(0.671150\pi\)
\(62\) 8.00000 1.01600
\(63\) −6.00000 −0.755929
\(64\) −8.00000 −1.00000
\(65\) 4.00000 0.496139
\(66\) −30.0000 −3.69274
\(67\) 8.00000 0.977356 0.488678 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) 0 0
\(69\) −6.00000 −0.722315
\(70\) −4.00000 −0.478091
\(71\) 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 \(0.518638\pi\)
\(74\) 2.00000 0.232495
\(75\) 3.00000 0.346410
\(76\) 0 0
\(77\) 5.00000 0.569803
\(78\) −12.0000 −1.35873
\(79\) 4.00000 0.450035 0.225018 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 8.00000 0.894427
\(81\) 9.00000 1.00000
\(82\) 18.0000 1.98777
\(83\) −15.0000 −1.64646 −0.823232 0.567705i \(-0.807831\pi\)
−0.823232 + 0.567705i \(0.807831\pi\)
\(84\) 6.00000 0.654654
\(85\) 0 0
\(86\) −4.00000 −0.431331
\(87\) −18.0000 −1.92980
\(88\) 0 0
\(89\) 4.00000 0.423999 0.212000 0.977270i \(-0.432002\pi\)
0.212000 + 0.977270i \(0.432002\pi\)
\(90\) 24.0000 2.52982
\(91\) 2.00000 0.209657
\(92\) 4.00000 0.417029
\(93\) 12.0000 1.24434
\(94\) 18.0000 1.85656
\(95\) 0 0
\(96\) −24.0000 −2.44949
\(97\) 4.00000 0.406138 0.203069 0.979164i \(-0.434908\pi\)
0.203069 + 0.979164i \(0.434908\pi\)
\(98\) 12.0000 1.21218
\(99\) −30.0000 −3.01511
\(100\) −2.00000 −0.200000
\(101\) 3.00000 0.298511 0.149256 0.988799i \(-0.452312\pi\)
0.149256 + 0.988799i \(0.452312\pi\)
\(102\) 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
\(106\) −2.00000 −0.194257
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) −18.0000 −1.73205
\(109\) −16.0000 −1.53252 −0.766261 0.642529i \(-0.777885\pi\)
−0.766261 + 0.642529i \(0.777885\pi\)
\(110\) −20.0000 −1.90693
\(111\) 3.00000 0.284747
\(112\) 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
\(115\) −4.00000 −0.373002
\(116\) 12.0000 1.11417
\(117\) −12.0000 −1.10940
\(118\) −16.0000 −1.47292
\(119\) 0 0
\(120\) 0 0
\(121\) 14.0000 1.27273
\(122\) 16.0000 1.44857
\(123\) 27.0000 2.43451
\(124\) −8.00000 −0.718421
\(125\) 12.0000 1.07331
\(126\) 12.0000 1.06904
\(127\) 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
\(130\) −8.00000 −0.701646
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 30.0000 2.61116
\(133\) 0 0
\(134\) −16.0000 −1.38219
\(135\) 18.0000 1.54919
\(136\) 0 0
\(137\) −6.00000 −0.512615 −0.256307 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) 12.0000 1.02151
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 4.00000 0.338062
\(141\) 27.0000 2.27381
\(142\) −18.0000 −1.51053
\(143\) 10.0000 0.836242
\(144\) −24.0000 −2.00000
\(145\) −12.0000 −0.996546
\(146\) 2.00000 0.165521
\(147\) 18.0000 1.48461
\(148\) −2.00000 −0.164399
\(149\) −5.00000 −0.409616 −0.204808 0.978802i \(-0.565657\pi\)
−0.204808 + 0.978802i \(0.565657\pi\)
\(150\) −6.00000 −0.489898
\(151\) 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
\(155\) 8.00000 0.642575
\(156\) 12.0000 0.960769
\(157\) 23.0000 1.83560 0.917800 0.397043i \(-0.129964\pi\)
0.917800 + 0.397043i \(0.129964\pi\)
\(158\) −8.00000 −0.636446
\(159\) −3.00000 −0.237915
\(160\) −16.0000 −1.26491
\(161\) −2.00000 −0.157622
\(162\) −18.0000 −1.41421
\(163\) −18.0000 −1.40987 −0.704934 0.709273i \(-0.749024\pi\)
−0.704934 + 0.709273i \(0.749024\pi\)
\(164\) −18.0000 −1.40556
\(165\) −30.0000 −2.33550
\(166\) 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 \(0.388852\pi\)
\(174\) 36.0000 2.72915
\(175\) 1.00000 0.0755929
\(176\) 20.0000 1.50756
\(177\) −24.0000 −1.80395
\(178\) −8.00000 −0.599625
\(179\) 18.0000 1.34538 0.672692 0.739923i \(-0.265138\pi\)
0.672692 + 0.739923i \(0.265138\pi\)
\(180\) −24.0000 −1.78885
\(181\) 5.00000 0.371647 0.185824 0.982583i \(-0.440505\pi\)
0.185824 + 0.982583i \(0.440505\pi\)
\(182\) −4.00000 −0.296500
\(183\) 24.0000 1.77413
\(184\) 0 0
\(185\) 2.00000 0.147043
\(186\) −24.0000 −1.75977
\(187\) 0 0
\(188\) −18.0000 −1.31278
\(189\) 9.00000 0.654654
\(190\) 0 0
\(191\) −4.00000 −0.289430 −0.144715 0.989473i \(-0.546227\pi\)
−0.144715 + 0.989473i \(0.546227\pi\)
\(192\) 24.0000 1.73205
\(193\) −26.0000 −1.87152 −0.935760 0.352636i \(-0.885285\pi\)
−0.935760 + 0.352636i \(0.885285\pi\)
\(194\) −8.00000 −0.574367
\(195\) −12.0000 −0.859338
\(196\) −12.0000 −0.857143
\(197\) 3.00000 0.213741 0.106871 0.994273i \(-0.465917\pi\)
0.106871 + 0.994273i \(0.465917\pi\)
\(198\) 60.0000 4.26401
\(199\) 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
\(202\) −6.00000 −0.422159
\(203\) −6.00000 −0.421117
\(204\) 0 0
\(205\) 18.0000 1.25717
\(206\) −36.0000 −2.50824
\(207\) 12.0000 0.834058
\(208\) 8.00000 0.554700
\(209\) 0 0
\(210\) 12.0000 0.828079
\(211\) −13.0000 −0.894957 −0.447478 0.894295i \(-0.647678\pi\)
−0.447478 + 0.894295i \(0.647678\pi\)
\(212\) 2.00000 0.137361
\(213\) −27.0000 −1.85001
\(214\) 24.0000 1.64061
\(215\) −4.00000 −0.272798
\(216\) 0 0
\(217\) 4.00000 0.271538
\(218\) 32.0000 2.16731
\(219\) 3.00000 0.202721
\(220\) 20.0000 1.34840
\(221\) 0 0
\(222\) −6.00000 −0.402694
\(223\) −17.0000 −1.13840 −0.569202 0.822198i \(-0.692748\pi\)
−0.569202 + 0.822198i \(0.692748\pi\)
\(224\) −8.00000 −0.534522
\(225\) −6.00000 −0.400000
\(226\) 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 \(0.425707\pi\)
\(230\) 8.00000 0.527504
\(231\) −15.0000 −0.986928
\(232\) 0 0
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 24.0000 1.56893
\(235\) 18.0000 1.17419
\(236\) 16.0000 1.04151
\(237\) −12.0000 −0.779484
\(238\) 0 0
\(239\) −6.00000 −0.388108 −0.194054 0.980991i \(-0.562164\pi\)
−0.194054 + 0.980991i \(0.562164\pi\)
\(240\) −24.0000 −1.54919
\(241\) 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
\(245\) 12.0000 0.766652
\(246\) −54.0000 −3.44291
\(247\) 0 0
\(248\) 0 0
\(249\) 45.0000 2.85176
\(250\) −24.0000 −1.51789
\(251\) −2.00000 −0.126239 −0.0631194 0.998006i \(-0.520105\pi\)
−0.0631194 + 0.998006i \(0.520105\pi\)
\(252\) −12.0000 −0.755929
\(253\) −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 \(0.5\pi\)
\(258\) 12.0000 0.747087
\(259\) 1.00000 0.0621370
\(260\) 8.00000 0.496139
\(261\) 36.0000 2.22834
\(262\) 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
\(265\) −2.00000 −0.122859
\(266\) 0 0
\(267\) −12.0000 −0.734388
\(268\) 16.0000 0.977356
\(269\) −6.00000 −0.365826 −0.182913 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) −36.0000 −2.19089
\(271\) −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
\(274\) 12.0000 0.724947
\(275\) 5.00000 0.301511
\(276\) −12.0000 −0.722315
\(277\) 12.0000 0.721010 0.360505 0.932757i \(-0.382604\pi\)
0.360505 + 0.932757i \(0.382604\pi\)
\(278\) −8.00000 −0.479808
\(279\) −24.0000 −1.43684
\(280\) 0 0
\(281\) 12.0000 0.715860 0.357930 0.933748i \(-0.383483\pi\)
0.357930 + 0.933748i \(0.383483\pi\)
\(282\) −54.0000 −3.21565
\(283\) 4.00000 0.237775 0.118888 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) 18.0000 1.06810
\(285\) 0 0
\(286\) −20.0000 −1.18262
\(287\) 9.00000 0.531253
\(288\) 48.0000 2.82843
\(289\) −17.0000 −1.00000
\(290\) 24.0000 1.40933
\(291\) −12.0000 −0.703452
\(292\) −2.00000 −0.117041
\(293\) −2.00000 −0.116841 −0.0584206 0.998292i \(-0.518606\pi\)
−0.0584206 + 0.998292i \(0.518606\pi\)
\(294\) −36.0000 −2.09956
\(295\) −16.0000 −0.931556
\(296\) 0 0
\(297\) 45.0000 2.61116
\(298\) 10.0000 0.579284
\(299\) −4.00000 −0.231326
\(300\) 6.00000 0.346410
\(301\) −2.00000 −0.115278
\(302\) −32.0000 −1.84139
\(303\) −9.00000 −0.517036
\(304\) 0 0
\(305\) 16.0000 0.916157
\(306\) 0 0
\(307\) −17.0000 −0.970241 −0.485121 0.874447i \(-0.661224\pi\)
−0.485121 + 0.874447i \(0.661224\pi\)
\(308\) 10.0000 0.569803
\(309\) −54.0000 −3.07195
\(310\) −16.0000 −0.908739
\(311\) 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 \(0.286419\pi\)
\(314\) −46.0000 −2.59593
\(315\) 12.0000 0.676123
\(316\) 8.00000 0.450035
\(317\) 22.0000 1.23564 0.617822 0.786318i \(-0.288015\pi\)
0.617822 + 0.786318i \(0.288015\pi\)
\(318\) 6.00000 0.336463
\(319\) −30.0000 −1.67968
\(320\) 16.0000 0.894427
\(321\) 36.0000 2.00932
\(322\) 4.00000 0.222911
\(323\) 0 0
\(324\) 18.0000 1.00000
\(325\) 2.00000 0.110940
\(326\) 36.0000 1.99386
\(327\) 48.0000 2.65441
\(328\) 0 0
\(329\) 9.00000 0.496186
\(330\) 60.0000 3.30289
\(331\) −2.00000 −0.109930 −0.0549650 0.998488i \(-0.517505\pi\)
−0.0549650 + 0.998488i \(0.517505\pi\)
\(332\) −30.0000 −1.64646
\(333\) −6.00000 −0.328798
\(334\) 24.0000 1.31322
\(335\) −16.0000 −0.874173
\(336\) −12.0000 −0.654654
\(337\) −25.0000 −1.36184 −0.680918 0.732359i \(-0.738419\pi\)
−0.680918 + 0.732359i \(0.738419\pi\)
\(338\) 18.0000 0.979071
\(339\) 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
\(346\) −18.0000 −0.967686
\(347\) −10.0000 −0.536828 −0.268414 0.963304i \(-0.586500\pi\)
−0.268414 + 0.963304i \(0.586500\pi\)
\(348\) −36.0000 −1.92980
\(349\) 6.00000 0.321173 0.160586 0.987022i \(-0.448662\pi\)
0.160586 + 0.987022i \(0.448662\pi\)
\(350\) −2.00000 −0.106904
\(351\) 18.0000 0.960769
\(352\) −40.0000 −2.13201
\(353\) 8.00000 0.425797 0.212899 0.977074i \(-0.431710\pi\)
0.212899 + 0.977074i \(0.431710\pi\)
\(354\) 48.0000 2.55117
\(355\) −18.0000 −0.955341
\(356\) 8.00000 0.423999
\(357\) 0 0
\(358\) −36.0000 −1.90266
\(359\) −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
\(362\) −10.0000 −0.525588
\(363\) −42.0000 −2.20443
\(364\) 4.00000 0.209657
\(365\) 2.00000 0.104685
\(366\) −48.0000 −2.50900
\(367\) 8.00000 0.417597 0.208798 0.977959i \(-0.433045\pi\)
0.208798 + 0.977959i \(0.433045\pi\)
\(368\) −8.00000 −0.417029
\(369\) −54.0000 −2.81113
\(370\) −4.00000 −0.207950
\(371\) −1.00000 −0.0519174
\(372\) 24.0000 1.24434
\(373\) −19.0000 −0.983783 −0.491891 0.870657i \(-0.663694\pi\)
−0.491891 + 0.870657i \(0.663694\pi\)
\(374\) 0 0
\(375\) −36.0000 −1.85903
\(376\) 0 0
\(377\) −12.0000 −0.618031
\(378\) −18.0000 −0.925820
\(379\) 15.0000 0.770498 0.385249 0.922813i \(-0.374116\pi\)
0.385249 + 0.922813i \(0.374116\pi\)
\(380\) 0 0
\(381\) −3.00000 −0.153695
\(382\) 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
\(385\) −10.0000 −0.509647
\(386\) 52.0000 2.64673
\(387\) 12.0000 0.609994
\(388\) 8.00000 0.406138
\(389\) 4.00000 0.202808 0.101404 0.994845i \(-0.467667\pi\)
0.101404 + 0.994845i \(0.467667\pi\)
\(390\) 24.0000 1.21529
\(391\) 0 0
\(392\) 0 0
\(393\) 36.0000 1.81596
\(394\) −6.00000 −0.302276
\(395\) −8.00000 −0.402524
\(396\) −60.0000 −3.01511
\(397\) −5.00000 −0.250943 −0.125471 0.992097i \(-0.540044\pi\)
−0.125471 + 0.992097i \(0.540044\pi\)
\(398\) −4.00000 −0.200502
\(399\) 0 0
\(400\) 4.00000 0.200000
\(401\) 18.0000 0.898877 0.449439 0.893311i \(-0.351624\pi\)
0.449439 + 0.893311i \(0.351624\pi\)
\(402\) 48.0000 2.39402
\(403\) 8.00000 0.398508
\(404\) 6.00000 0.298511
\(405\) −18.0000 −0.894427
\(406\) 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 \(0.335363\pi\)
\(410\) −36.0000 −1.77791
\(411\) 18.0000 0.887875
\(412\) 36.0000 1.77359
\(413\) −8.00000 −0.393654
\(414\) −24.0000 −1.17954
\(415\) 30.0000 1.47264
\(416\) −16.0000 −0.784465
\(417\) −12.0000 −0.587643
\(418\) 0 0
\(419\) 7.00000 0.341972 0.170986 0.985273i \(-0.445305\pi\)
0.170986 + 0.985273i \(0.445305\pi\)
\(420\) −12.0000 −0.585540
\(421\) −24.0000 −1.16969 −0.584844 0.811146i \(-0.698844\pi\)
−0.584844 + 0.811146i \(0.698844\pi\)
\(422\) 26.0000 1.26566
\(423\) −54.0000 −2.62557
\(424\) 0 0
\(425\) 0 0
\(426\) 54.0000 2.61631
\(427\) 8.00000 0.387147
\(428\) −24.0000 −1.16008
\(429\) −30.0000 −1.44841
\(430\) 8.00000 0.385794
\(431\) −30.0000 −1.44505 −0.722525 0.691345i \(-0.757018\pi\)
−0.722525 + 0.691345i \(0.757018\pi\)
\(432\) 36.0000 1.73205
\(433\) 9.00000 0.432512 0.216256 0.976337i \(-0.430615\pi\)
0.216256 + 0.976337i \(0.430615\pi\)
\(434\) −8.00000 −0.384012
\(435\) 36.0000 1.72607
\(436\) −32.0000 −1.53252
\(437\) 0 0
\(438\) −6.00000 −0.286691
\(439\) 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 \(-0.492438\pi\)
0.0237557 + 0.999718i \(0.492438\pi\)
\(444\) 6.00000 0.284747
\(445\) −8.00000 −0.379236
\(446\) 34.0000 1.60995
\(447\) 15.0000 0.709476
\(448\) 8.00000 0.377964
\(449\) 36.0000 1.69895 0.849473 0.527633i \(-0.176920\pi\)
0.849473 + 0.527633i \(0.176920\pi\)
\(450\) 12.0000 0.565685
\(451\) 45.0000 2.11897
\(452\) −36.0000 −1.69330
\(453\) −48.0000 −2.25524
\(454\) 32.0000 1.50183
\(455\) −4.00000 −0.187523
\(456\) 0 0
\(457\) 18.0000 0.842004 0.421002 0.907060i \(-0.361678\pi\)
0.421002 + 0.907060i \(0.361678\pi\)
\(458\) −14.0000 −0.654177
\(459\) 0 0
\(460\) −8.00000 −0.373002
\(461\) 30.0000 1.39724 0.698620 0.715493i \(-0.253798\pi\)
0.698620 + 0.715493i \(0.253798\pi\)
\(462\) 30.0000 1.39573
\(463\) −22.0000 −1.02243 −0.511213 0.859454i \(-0.670804\pi\)
−0.511213 + 0.859454i \(0.670804\pi\)
\(464\) −24.0000 −1.11417
\(465\) −24.0000 −1.11297
\(466\) −12.0000 −0.555889
\(467\) −2.00000 −0.0925490 −0.0462745 0.998929i \(-0.514735\pi\)
−0.0462745 + 0.998929i \(0.514735\pi\)
\(468\) −24.0000 −1.10940
\(469\) −8.00000 −0.369406
\(470\) −36.0000 −1.66056
\(471\) −69.0000 −3.17935
\(472\) 0 0
\(473\) −10.0000 −0.459800
\(474\) 24.0000 1.10236
\(475\) 0 0
\(476\) 0 0
\(477\) 6.00000 0.274721
\(478\) 12.0000 0.548867
\(479\) 14.0000 0.639676 0.319838 0.947472i \(-0.396371\pi\)
0.319838 + 0.947472i \(0.396371\pi\)
\(480\) 48.0000 2.19089
\(481\) 2.00000 0.0911922
\(482\) −28.0000 −1.27537
\(483\) 6.00000 0.273009
\(484\) 28.0000 1.27273
\(485\) −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
\(490\) −24.0000 −1.08421
\(491\) −28.0000 −1.26362 −0.631811 0.775122i \(-0.717688\pi\)
−0.631811 + 0.775122i \(0.717688\pi\)
\(492\) 54.0000 2.43451
\(493\) 0 0
\(494\) 0 0
\(495\) 60.0000 2.69680
\(496\) 16.0000 0.718421
\(497\) −9.00000 −0.403705
\(498\) −90.0000 −4.03300
\(499\) 12.0000 0.537194 0.268597 0.963253i \(-0.413440\pi\)
0.268597 + 0.963253i \(0.413440\pi\)
\(500\) 24.0000 1.07331
\(501\) 36.0000 1.60836
\(502\) 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
\(505\) −6.00000 −0.266996
\(506\) 20.0000 0.889108
\(507\) 27.0000 1.19911
\(508\) 2.00000 0.0887357
\(509\) −31.0000 −1.37405 −0.687025 0.726633i \(-0.741084\pi\)
−0.687025 + 0.726633i \(0.741084\pi\)
\(510\) 0 0
\(511\) 1.00000 0.0442374
\(512\) −32.0000 −1.41421
\(513\) 0 0
\(514\) 0 0
\(515\) −36.0000 −1.58635
\(516\) −12.0000 −0.528271
\(517\) 45.0000 1.97910
\(518\) −2.00000 −0.0878750
\(519\) −27.0000 −1.18517
\(520\) 0 0
\(521\) −33.0000 −1.44576 −0.722878 0.690976i \(-0.757181\pi\)
−0.722878 + 0.690976i \(0.757181\pi\)
\(522\) −72.0000 −3.15135
\(523\) −22.0000 −0.961993 −0.480996 0.876723i \(-0.659725\pi\)
−0.480996 + 0.876723i \(0.659725\pi\)
\(524\) −24.0000 −1.04844
\(525\) −3.00000 −0.130931
\(526\) −38.0000 −1.65688
\(527\) 0 0
\(528\) −60.0000 −2.61116
\(529\) −19.0000 −0.826087
\(530\) 4.00000 0.173749
\(531\) 48.0000 2.08302
\(532\) 0 0
\(533\) 18.0000 0.779667
\(534\) 24.0000 1.03858
\(535\) 24.0000 1.03761
\(536\) 0 0
\(537\) −54.0000 −2.33027
\(538\) 12.0000 0.517357
\(539\) 30.0000 1.29219
\(540\) 36.0000 1.54919
\(541\) 20.0000 0.859867 0.429934 0.902861i \(-0.358537\pi\)
0.429934 + 0.902861i \(0.358537\pi\)
\(542\) 62.0000 2.66313
\(543\) −15.0000 −0.643712
\(544\) 0 0
\(545\) 32.0000 1.37073
\(546\) 12.0000 0.513553
\(547\) 8.00000 0.342055 0.171028 0.985266i \(-0.445291\pi\)
0.171028 + 0.985266i \(0.445291\pi\)
\(548\) −12.0000 −0.512615
\(549\) −48.0000 −2.04859
\(550\) −10.0000 −0.426401
\(551\) 0 0
\(552\) 0 0
\(553\) −4.00000 −0.170097
\(554\) −24.0000 −1.01966
\(555\) −6.00000 −0.254686
\(556\) 8.00000 0.339276
\(557\) −18.0000 −0.762684 −0.381342 0.924434i \(-0.624538\pi\)
−0.381342 + 0.924434i \(0.624538\pi\)
\(558\) 48.0000 2.03200
\(559\) −4.00000 −0.169182
\(560\) −8.00000 −0.338062
\(561\) 0 0
\(562\) −24.0000 −1.01238
\(563\) −30.0000 −1.26435 −0.632175 0.774826i \(-0.717837\pi\)
−0.632175 + 0.774826i \(0.717837\pi\)
\(564\) 54.0000 2.27381
\(565\) 36.0000 1.51453
\(566\) −8.00000 −0.336265
\(567\) −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 \(-0.453209\pi\)
0.146470 + 0.989215i \(0.453209\pi\)
\(572\) 20.0000 0.836242
\(573\) 12.0000 0.501307
\(574\) −18.0000 −0.751305
\(575\) −2.00000 −0.0834058
\(576\) −48.0000 −2.00000
\(577\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(578\) 34.0000 1.41421
\(579\) 78.0000 3.24157
\(580\) −24.0000 −0.996546
\(581\) 15.0000 0.622305
\(582\) 24.0000 0.994832
\(583\) −5.00000 −0.207079
\(584\) 0 0
\(585\) 24.0000 0.992278
\(586\) 4.00000 0.165238
\(587\) −32.0000 −1.32078 −0.660391 0.750922i \(-0.729609\pi\)
−0.660391 + 0.750922i \(0.729609\pi\)
\(588\) 36.0000 1.48461
\(589\) 0 0
\(590\) 32.0000 1.31742
\(591\) −9.00000 −0.370211
\(592\) 4.00000 0.164399
\(593\) −5.00000 −0.205325 −0.102663 0.994716i \(-0.532736\pi\)
−0.102663 + 0.994716i \(0.532736\pi\)
\(594\) −90.0000 −3.69274
\(595\) 0 0
\(596\) −10.0000 −0.409616
\(597\) −6.00000 −0.245564
\(598\) 8.00000 0.327144
\(599\) 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 \(0.648113\pi\)
\(602\) 4.00000 0.163028
\(603\) 48.0000 1.95471
\(604\) 32.0000 1.30206
\(605\) −28.0000 −1.13836
\(606\) 18.0000 0.731200
\(607\) −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
\(610\) −32.0000 −1.29564
\(611\) 18.0000 0.728202
\(612\) 0 0
\(613\) 15.0000 0.605844 0.302922 0.953015i \(-0.402038\pi\)
0.302922 + 0.953015i \(0.402038\pi\)
\(614\) 34.0000 1.37213
\(615\) −54.0000 −2.17749
\(616\) 0 0
\(617\) 17.0000 0.684394 0.342197 0.939628i \(-0.388829\pi\)
0.342197 + 0.939628i \(0.388829\pi\)
\(618\) 108.000 4.34440
\(619\) −1.00000 −0.0401934 −0.0200967 0.999798i \(-0.506397\pi\)
−0.0200967 + 0.999798i \(0.506397\pi\)
\(620\) 16.0000 0.642575
\(621\) −18.0000 −0.722315
\(622\) 0 0
\(623\) −4.00000 −0.160257
\(624\) −24.0000 −0.960769
\(625\) −19.0000 −0.760000
\(626\) −44.0000 −1.75859
\(627\) 0 0
\(628\) 46.0000 1.83560
\(629\) 0 0
\(630\) −24.0000 −0.956183
\(631\) −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
\(634\) −44.0000 −1.74746
\(635\) −2.00000 −0.0793676
\(636\) −6.00000 −0.237915
\(637\) 12.0000 0.475457
\(638\) 60.0000 2.37542
\(639\) 54.0000 2.13621
\(640\) 0 0
\(641\) −1.00000 −0.0394976 −0.0197488 0.999805i \(-0.506287\pi\)
−0.0197488 + 0.999805i \(0.506287\pi\)
\(642\) −72.0000 −2.84161
\(643\) 14.0000 0.552106 0.276053 0.961142i \(-0.410973\pi\)
0.276053 + 0.961142i \(0.410973\pi\)
\(644\) −4.00000 −0.157622
\(645\) 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
\(650\) −4.00000 −0.156893
\(651\) −12.0000 −0.470317
\(652\) −36.0000 −1.40987
\(653\) −24.0000 −0.939193 −0.469596 0.882881i \(-0.655601\pi\)
−0.469596 + 0.882881i \(0.655601\pi\)
\(654\) −96.0000 −3.75390
\(655\) 24.0000 0.937758
\(656\) 36.0000 1.40556
\(657\) −6.00000 −0.234082
\(658\) −18.0000 −0.701713
\(659\) −15.0000 −0.584317 −0.292159 0.956370i \(-0.594373\pi\)
−0.292159 + 0.956370i \(0.594373\pi\)
\(660\) −60.0000 −2.33550
\(661\) −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
\(666\) 12.0000 0.464991
\(667\) 12.0000 0.464642
\(668\) −24.0000 −0.928588
\(669\) 51.0000 1.97177
\(670\) 32.0000 1.23627
\(671\) 40.0000 1.54418
\(672\) 24.0000 0.925820
\(673\) 27.0000 1.04077 0.520387 0.853931i \(-0.325788\pi\)
0.520387 + 0.853931i \(0.325788\pi\)
\(674\) 50.0000 1.92593
\(675\) 9.00000 0.346410
\(676\) −18.0000 −0.692308
\(677\) −11.0000 −0.422764 −0.211382 0.977403i \(-0.567796\pi\)
−0.211382 + 0.977403i \(0.567796\pi\)
\(678\) −108.000 −4.14772
\(679\) −4.00000 −0.153506
\(680\) 0 0
\(681\) 48.0000 1.83936
\(682\) −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
\(685\) 12.0000 0.458496
\(686\) −26.0000 −0.992685
\(687\) −21.0000 −0.801200
\(688\) −8.00000 −0.304997
\(689\) −2.00000 −0.0761939
\(690\) −24.0000 −0.913664
\(691\) −20.0000 −0.760836 −0.380418 0.924815i \(-0.624220\pi\)
−0.380418 + 0.924815i \(0.624220\pi\)
\(692\) 18.0000 0.684257
\(693\) 30.0000 1.13961
\(694\) 20.0000 0.759190
\(695\) −8.00000 −0.303457
\(696\) 0 0
\(697\) 0 0
\(698\) −12.0000 −0.454207
\(699\) −18.0000 −0.680823
\(700\) 2.00000 0.0755929
\(701\) −12.0000 −0.453234 −0.226617 0.973984i \(-0.572767\pi\)
−0.226617 + 0.973984i \(0.572767\pi\)
\(702\) −36.0000 −1.35873
\(703\) 0 0
\(704\) 40.0000 1.50756
\(705\) −54.0000 −2.03376
\(706\) −16.0000 −0.602168
\(707\) −3.00000 −0.112827
\(708\) −48.0000 −1.80395
\(709\) 40.0000 1.50223 0.751116 0.660171i \(-0.229516\pi\)
0.751116 + 0.660171i \(0.229516\pi\)
\(710\) 36.0000 1.35106
\(711\) 24.0000 0.900070
\(712\) 0 0
\(713\) −8.00000 −0.299602
\(714\) 0 0
\(715\) −20.0000 −0.747958
\(716\) 36.0000 1.34538
\(717\) 18.0000 0.672222
\(718\) 30.0000 1.11959
\(719\) 39.0000 1.45445 0.727227 0.686397i \(-0.240809\pi\)
0.727227 + 0.686397i \(0.240809\pi\)
\(720\) 48.0000 1.78885
\(721\) −18.0000 −0.670355
\(722\) 38.0000 1.41421
\(723\) −42.0000 −1.56200
\(724\) 10.0000 0.371647
\(725\) −6.00000 −0.222834
\(726\) 84.0000 3.11753
\(727\) 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
\(730\) −4.00000 −0.148047
\(731\) 0 0
\(732\) 48.0000 1.77413
\(733\) 7.00000 0.258551 0.129275 0.991609i \(-0.458735\pi\)
0.129275 + 0.991609i \(0.458735\pi\)
\(734\) −16.0000 −0.590571
\(735\) −36.0000 −1.32788
\(736\) 16.0000 0.589768
\(737\) −40.0000 −1.47342
\(738\) 108.000 3.97553
\(739\) −9.00000 −0.331070 −0.165535 0.986204i \(-0.552935\pi\)
−0.165535 + 0.986204i \(0.552935\pi\)
\(740\) 4.00000 0.147043
\(741\) 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
\(745\) 10.0000 0.366372
\(746\) 38.0000 1.39128
\(747\) −90.0000 −3.29293
\(748\) 0 0
\(749\) 12.0000 0.438470
\(750\) 72.0000 2.62907
\(751\) 25.0000 0.912263 0.456131 0.889912i \(-0.349235\pi\)
0.456131 + 0.889912i \(0.349235\pi\)
\(752\) 36.0000 1.31278
\(753\) 6.00000 0.218652
\(754\) 24.0000 0.874028
\(755\) −32.0000 −1.16460
\(756\) 18.0000 0.654654
\(757\) −50.0000 −1.81728 −0.908640 0.417579i \(-0.862879\pi\)
−0.908640 + 0.417579i \(0.862879\pi\)
\(758\) −30.0000 −1.08965
\(759\) 30.0000 1.08893
\(760\) 0 0
\(761\) −35.0000 −1.26875 −0.634375 0.773026i \(-0.718742\pi\)
−0.634375 + 0.773026i \(0.718742\pi\)
\(762\) 6.00000 0.217357
\(763\) 16.0000 0.579239
\(764\) −8.00000 −0.289430
\(765\) 0 0
\(766\) −40.0000 −1.44526
\(767\) −16.0000 −0.577727
\(768\) −48.0000 −1.73205
\(769\) 26.0000 0.937584 0.468792 0.883309i \(-0.344689\pi\)
0.468792 + 0.883309i \(0.344689\pi\)
\(770\) 20.0000 0.720750
\(771\) 0 0
\(772\) −52.0000 −1.87152
\(773\) −9.00000 −0.323708 −0.161854 0.986815i \(-0.551747\pi\)
−0.161854 + 0.986815i \(0.551747\pi\)
\(774\) −24.0000 −0.862662
\(775\) 4.00000 0.143684
\(776\) 0 0
\(777\) −3.00000 −0.107624
\(778\) −8.00000 −0.286814
\(779\) 0 0
\(780\) −24.0000 −0.859338
\(781\) −45.0000 −1.61023
\(782\) 0 0
\(783\) −54.0000 −1.92980
\(784\) 24.0000 0.857143
\(785\) −46.0000 −1.64181
\(786\) −72.0000 −2.56815
\(787\) −5.00000 −0.178231 −0.0891154 0.996021i \(-0.528404\pi\)
−0.0891154 + 0.996021i \(0.528404\pi\)
\(788\) 6.00000 0.213741
\(789\) −57.0000 −2.02925
\(790\) 16.0000 0.569254
\(791\) 18.0000 0.640006
\(792\) 0 0
\(793\) 16.0000 0.568177
\(794\) 10.0000 0.354887
\(795\) 6.00000 0.212798
\(796\) 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
\(800\) −8.00000 −0.282843
\(801\) 24.0000 0.847998
\(802\) −36.0000 −1.27120
\(803\) 5.00000 0.176446
\(804\) −48.0000 −1.69283
\(805\) 4.00000 0.140981
\(806\) −16.0000 −0.563576
\(807\) 18.0000 0.633630
\(808\) 0 0
\(809\) 2.00000 0.0703163 0.0351581 0.999382i \(-0.488807\pi\)
0.0351581 + 0.999382i \(0.488807\pi\)
\(810\) 36.0000 1.26491
\(811\) 47.0000 1.65039 0.825197 0.564846i \(-0.191064\pi\)
0.825197 + 0.564846i \(0.191064\pi\)
\(812\) −12.0000 −0.421117
\(813\) 93.0000 3.26165
\(814\) −10.0000 −0.350500
\(815\) 36.0000 1.26102
\(816\) 0 0
\(817\) 0 0
\(818\) −40.0000 −1.39857
\(819\) 12.0000 0.419314
\(820\) 36.0000 1.25717
\(821\) −47.0000 −1.64031 −0.820156 0.572140i \(-0.806113\pi\)
−0.820156 + 0.572140i \(0.806113\pi\)
\(822\) −36.0000 −1.25564
\(823\) −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
\(826\) 16.0000 0.556711
\(827\) 22.0000 0.765015 0.382507 0.923952i \(-0.375061\pi\)
0.382507 + 0.923952i \(0.375061\pi\)
\(828\) 24.0000 0.834058
\(829\) −4.00000 −0.138926 −0.0694629 0.997585i \(-0.522129\pi\)
−0.0694629 + 0.997585i \(0.522129\pi\)
\(830\) −60.0000 −2.08263
\(831\) −36.0000 −1.24883
\(832\) 16.0000 0.554700
\(833\) 0 0
\(834\) 24.0000 0.831052
\(835\) 24.0000 0.830554
\(836\) 0 0
\(837\) 36.0000 1.24434
\(838\) −14.0000 −0.483622
\(839\) 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
\(842\) 48.0000 1.65419
\(843\) −36.0000 −1.23991
\(844\) −26.0000 −0.894957
\(845\) 18.0000 0.619219
\(846\) 108.000 3.71312
\(847\) −14.0000 −0.481046
\(848\) −4.00000 −0.137361
\(849\) −12.0000 −0.411839
\(850\) 0 0
\(851\) −2.00000 −0.0685591
\(852\) −54.0000 −1.85001
\(853\) 26.0000 0.890223 0.445112 0.895475i \(-0.353164\pi\)
0.445112 + 0.895475i \(0.353164\pi\)
\(854\) −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 \(0.805929\pi\)
\(858\) 60.0000 2.04837
\(859\) −20.0000 −0.682391 −0.341196 0.939992i \(-0.610832\pi\)
−0.341196 + 0.939992i \(0.610832\pi\)
\(860\) −8.00000 −0.272798
\(861\) −27.0000 −0.920158
\(862\) 60.0000 2.04361
\(863\) −24.0000 −0.816970 −0.408485 0.912765i \(-0.633943\pi\)
−0.408485 + 0.912765i \(0.633943\pi\)
\(864\) −72.0000 −2.44949
\(865\) −18.0000 −0.612018
\(866\) −18.0000 −0.611665
\(867\) 51.0000 1.73205
\(868\) 8.00000 0.271538
\(869\) −20.0000 −0.678454
\(870\) −72.0000 −2.44103
\(871\) −16.0000 −0.542139
\(872\) 0 0
\(873\) 24.0000 0.812277
\(874\) 0 0
\(875\) −12.0000 −0.405674
\(876\) 6.00000 0.202721
\(877\) 50.0000 1.68838 0.844190 0.536044i \(-0.180082\pi\)
0.844190 + 0.536044i \(0.180082\pi\)
\(878\) −56.0000 −1.88991
\(879\) 6.00000 0.202375
\(880\) −40.0000 −1.34840
\(881\) −14.0000 −0.471672 −0.235836 0.971793i \(-0.575783\pi\)
−0.235836 + 0.971793i \(0.575783\pi\)
\(882\) 72.0000 2.42437
\(883\) 48.0000 1.61533 0.807664 0.589643i \(-0.200731\pi\)
0.807664 + 0.589643i \(0.200731\pi\)
\(884\) 0 0
\(885\) 48.0000 1.61350
\(886\) −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
\(890\) 16.0000 0.536321
\(891\) −45.0000 −1.50756
\(892\) −34.0000 −1.13840
\(893\) 0 0
\(894\) −30.0000 −1.00335
\(895\) −36.0000 −1.20335
\(896\) 0 0
\(897\) 12.0000 0.400668
\(898\) −72.0000 −2.40267
\(899\) −24.0000 −0.800445
\(900\) −12.0000 −0.400000
\(901\) 0 0
\(902\) −90.0000 −2.99667
\(903\) 6.00000 0.199667
\(904\) 0 0
\(905\) −10.0000 −0.332411
\(906\) 96.0000 3.18939
\(907\) 52.0000 1.72663 0.863316 0.504664i \(-0.168384\pi\)
0.863316 + 0.504664i \(0.168384\pi\)
\(908\) −32.0000 −1.06196
\(909\) 18.0000 0.597022
\(910\) 8.00000 0.265197
\(911\) 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
\(914\) −36.0000 −1.19077
\(915\) −48.0000 −1.58683
\(916\) 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
\(921\) 51.0000 1.68051
\(922\) −60.0000 −1.97599
\(923\) −18.0000 −0.592477
\(924\) −30.0000 −0.986928
\(925\) 1.00000 0.0328798
\(926\) 44.0000 1.44593
\(927\) 108.000 3.54719
\(928\) 48.0000 1.57568
\(929\) 18.0000 0.590561 0.295280 0.955411i \(-0.404587\pi\)
0.295280 + 0.955411i \(0.404587\pi\)
\(930\) 48.0000 1.57398
\(931\) 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 \(-0.293425\pi\)
0.604369 + 0.796705i \(0.293425\pi\)
\(938\) 16.0000 0.522419
\(939\) −66.0000 −2.15383
\(940\) 36.0000 1.17419
\(941\) −10.0000 −0.325991 −0.162995 0.986627i \(-0.552116\pi\)
−0.162995 + 0.986627i \(0.552116\pi\)
\(942\) 138.000 4.49628
\(943\) −18.0000 −0.586161
\(944\) −32.0000 −1.04151
\(945\) −18.0000 −0.585540
\(946\) 20.0000 0.650256
\(947\) 12.0000 0.389948 0.194974 0.980808i \(-0.437538\pi\)
0.194974 + 0.980808i \(0.437538\pi\)
\(948\) −24.0000 −0.779484
\(949\) 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 \(0.0493785\pi\)
\(954\) −12.0000 −0.388514
\(955\) 8.00000 0.258874
\(956\) −12.0000 −0.388108
\(957\) 90.0000 2.90929
\(958\) −28.0000 −0.904639
\(959\) 6.00000 0.193750
\(960\) −48.0000 −1.54919
\(961\) −15.0000 −0.483871
\(962\) −4.00000 −0.128965
\(963\) −72.0000 −2.32017
\(964\) 28.0000 0.901819
\(965\) 52.0000 1.67394
\(966\) −12.0000 −0.386094
\(967\) −14.0000 −0.450210 −0.225105 0.974335i \(-0.572272\pi\)
−0.225105 + 0.974335i \(0.572272\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 16.0000 0.513729
\(971\) −8.00000 −0.256732 −0.128366 0.991727i \(-0.540973\pi\)
−0.128366 + 0.991727i \(0.540973\pi\)
\(972\) 0 0
\(973\) −4.00000 −0.128234
\(974\) 48.0000 1.53802
\(975\) −6.00000 −0.192154
\(976\) 32.0000 1.02430
\(977\) 28.0000 0.895799 0.447900 0.894084i \(-0.352172\pi\)
0.447900 + 0.894084i \(0.352172\pi\)
\(978\) −108.000 −3.45346
\(979\) −20.0000 −0.639203
\(980\) 24.0000 0.766652
\(981\) −96.0000 −3.06504
\(982\) 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
\(985\) −6.00000 −0.191176
\(986\) 0 0
\(987\) −27.0000 −0.859419
\(988\) 0 0
\(989\) 4.00000 0.127193
\(990\) −120.000 −3.81385
\(991\) −18.0000 −0.571789 −0.285894 0.958261i \(-0.592291\pi\)
−0.285894 + 0.958261i \(0.592291\pi\)
\(992\) −32.0000 −1.01600
\(993\) 6.00000 0.190404
\(994\) 18.0000 0.570925
\(995\) −4.00000 −0.126809
\(996\) 90.0000 2.85176
\(997\) −42.0000 −1.33015 −0.665077 0.746775i \(-0.731601\pi\)
−0.665077 + 0.746775i \(0.731601\pi\)
\(998\) −24.0000 −0.759707
\(999\) 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 37.2.a.a.1.1 1
3.2 odd 2 333.2.a.d.1.1 1
4.3 odd 2 592.2.a.e.1.1 1
5.2 odd 4 925.2.b.b.149.1 2
5.3 odd 4 925.2.b.b.149.2 2
5.4 even 2 925.2.a.e.1.1 1
7.6 odd 2 1813.2.a.a.1.1 1
8.3 odd 2 2368.2.a.b.1.1 1
8.5 even 2 2368.2.a.q.1.1 1
11.10 odd 2 4477.2.a.b.1.1 1
12.11 even 2 5328.2.a.r.1.1 1
13.12 even 2 6253.2.a.c.1.1 1
15.14 odd 2 8325.2.a.e.1.1 1
37.6 odd 4 1369.2.b.c.1368.2 2
37.31 odd 4 1369.2.b.c.1368.1 2
37.36 even 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 1.1 even 1 trivial
333.2.a.d.1.1 1 3.2 odd 2
592.2.a.e.1.1 1 4.3 odd 2
925.2.a.e.1.1 1 5.4 even 2
925.2.b.b.149.1 2 5.2 odd 4
925.2.b.b.149.2 2 5.3 odd 4
1369.2.a.e.1.1 1 37.36 even 2
1369.2.b.c.1368.1 2 37.31 odd 4
1369.2.b.c.1368.2 2 37.6 odd 4
1813.2.a.a.1.1 1 7.6 odd 2
2368.2.a.b.1.1 1 8.3 odd 2
2368.2.a.q.1.1 1 8.5 even 2
4477.2.a.b.1.1 1 11.10 odd 2
5328.2.a.r.1.1 1 12.11 even 2
6253.2.a.c.1.1 1 13.12 even 2
8325.2.a.e.1.1 1 15.14 odd 2