Properties

Label 2366.2.a.h.1.1
Level $2366$
Weight $2$
Character 2366.1
Self dual yes
Analytic conductor $18.893$
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] = [2366,2,Mod(1,2366)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2366, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("2366.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2366 = 2 \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2366.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: \(18.8926051182\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 182)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2366.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +3.00000 q^{3} +1.00000 q^{4} +4.00000 q^{5} -3.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +6.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +3.00000 q^{3} +1.00000 q^{4} +4.00000 q^{5} -3.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +6.00000 q^{9} -4.00000 q^{10} -1.00000 q^{11} +3.00000 q^{12} -1.00000 q^{14} +12.0000 q^{15} +1.00000 q^{16} -6.00000 q^{18} +6.00000 q^{19} +4.00000 q^{20} +3.00000 q^{21} +1.00000 q^{22} -7.00000 q^{23} -3.00000 q^{24} +11.0000 q^{25} +9.00000 q^{27} +1.00000 q^{28} -4.00000 q^{29} -12.0000 q^{30} -7.00000 q^{31} -1.00000 q^{32} -3.00000 q^{33} +4.00000 q^{35} +6.00000 q^{36} -9.00000 q^{37} -6.00000 q^{38} -4.00000 q^{40} +3.00000 q^{41} -3.00000 q^{42} +4.00000 q^{43} -1.00000 q^{44} +24.0000 q^{45} +7.00000 q^{46} -7.00000 q^{47} +3.00000 q^{48} +1.00000 q^{49} -11.0000 q^{50} -9.00000 q^{54} -4.00000 q^{55} -1.00000 q^{56} +18.0000 q^{57} +4.00000 q^{58} +10.0000 q^{59} +12.0000 q^{60} +1.00000 q^{61} +7.00000 q^{62} +6.00000 q^{63} +1.00000 q^{64} +3.00000 q^{66} -1.00000 q^{67} -21.0000 q^{69} -4.00000 q^{70} -16.0000 q^{71} -6.00000 q^{72} -5.00000 q^{73} +9.00000 q^{74} +33.0000 q^{75} +6.00000 q^{76} -1.00000 q^{77} +11.0000 q^{79} +4.00000 q^{80} +9.00000 q^{81} -3.00000 q^{82} +3.00000 q^{84} -4.00000 q^{86} -12.0000 q^{87} +1.00000 q^{88} +6.00000 q^{89} -24.0000 q^{90} -7.00000 q^{92} -21.0000 q^{93} +7.00000 q^{94} +24.0000 q^{95} -3.00000 q^{96} +1.00000 q^{97} -1.00000 q^{98} -6.00000 q^{99} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 3.00000 1.73205 0.866025 0.500000i \(-0.166667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(4\) 1.00000 0.500000
\(5\) 4.00000 1.78885 0.894427 0.447214i \(-0.147584\pi\)
0.894427 + 0.447214i \(0.147584\pi\)
\(6\) −3.00000 −1.22474
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) 6.00000 2.00000
\(10\) −4.00000 −1.26491
\(11\) −1.00000 −0.301511 −0.150756 0.988571i \(-0.548171\pi\)
−0.150756 + 0.988571i \(0.548171\pi\)
\(12\) 3.00000 0.866025
\(13\) 0 0
\(14\) −1.00000 −0.267261
\(15\) 12.0000 3.09839
\(16\) 1.00000 0.250000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) −6.00000 −1.41421
\(19\) 6.00000 1.37649 0.688247 0.725476i \(-0.258380\pi\)
0.688247 + 0.725476i \(0.258380\pi\)
\(20\) 4.00000 0.894427
\(21\) 3.00000 0.654654
\(22\) 1.00000 0.213201
\(23\) −7.00000 −1.45960 −0.729800 0.683660i \(-0.760387\pi\)
−0.729800 + 0.683660i \(0.760387\pi\)
\(24\) −3.00000 −0.612372
\(25\) 11.0000 2.20000
\(26\) 0 0
\(27\) 9.00000 1.73205
\(28\) 1.00000 0.188982
\(29\) −4.00000 −0.742781 −0.371391 0.928477i \(-0.621119\pi\)
−0.371391 + 0.928477i \(0.621119\pi\)
\(30\) −12.0000 −2.19089
\(31\) −7.00000 −1.25724 −0.628619 0.777714i \(-0.716379\pi\)
−0.628619 + 0.777714i \(0.716379\pi\)
\(32\) −1.00000 −0.176777
\(33\) −3.00000 −0.522233
\(34\) 0 0
\(35\) 4.00000 0.676123
\(36\) 6.00000 1.00000
\(37\) −9.00000 −1.47959 −0.739795 0.672832i \(-0.765078\pi\)
−0.739795 + 0.672832i \(0.765078\pi\)
\(38\) −6.00000 −0.973329
\(39\) 0 0
\(40\) −4.00000 −0.632456
\(41\) 3.00000 0.468521 0.234261 0.972174i \(-0.424733\pi\)
0.234261 + 0.972174i \(0.424733\pi\)
\(42\) −3.00000 −0.462910
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) −1.00000 −0.150756
\(45\) 24.0000 3.57771
\(46\) 7.00000 1.03209
\(47\) −7.00000 −1.02105 −0.510527 0.859861i \(-0.670550\pi\)
−0.510527 + 0.859861i \(0.670550\pi\)
\(48\) 3.00000 0.433013
\(49\) 1.00000 0.142857
\(50\) −11.0000 −1.55563
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) −9.00000 −1.22474
\(55\) −4.00000 −0.539360
\(56\) −1.00000 −0.133631
\(57\) 18.0000 2.38416
\(58\) 4.00000 0.525226
\(59\) 10.0000 1.30189 0.650945 0.759125i \(-0.274373\pi\)
0.650945 + 0.759125i \(0.274373\pi\)
\(60\) 12.0000 1.54919
\(61\) 1.00000 0.128037 0.0640184 0.997949i \(-0.479608\pi\)
0.0640184 + 0.997949i \(0.479608\pi\)
\(62\) 7.00000 0.889001
\(63\) 6.00000 0.755929
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 3.00000 0.369274
\(67\) −1.00000 −0.122169 −0.0610847 0.998133i \(-0.519456\pi\)
−0.0610847 + 0.998133i \(0.519456\pi\)
\(68\) 0 0
\(69\) −21.0000 −2.52810
\(70\) −4.00000 −0.478091
\(71\) −16.0000 −1.89885 −0.949425 0.313993i \(-0.898333\pi\)
−0.949425 + 0.313993i \(0.898333\pi\)
\(72\) −6.00000 −0.707107
\(73\) −5.00000 −0.585206 −0.292603 0.956234i \(-0.594521\pi\)
−0.292603 + 0.956234i \(0.594521\pi\)
\(74\) 9.00000 1.04623
\(75\) 33.0000 3.81051
\(76\) 6.00000 0.688247
\(77\) −1.00000 −0.113961
\(78\) 0 0
\(79\) 11.0000 1.23760 0.618798 0.785550i \(-0.287620\pi\)
0.618798 + 0.785550i \(0.287620\pi\)
\(80\) 4.00000 0.447214
\(81\) 9.00000 1.00000
\(82\) −3.00000 −0.331295
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 3.00000 0.327327
\(85\) 0 0
\(86\) −4.00000 −0.431331
\(87\) −12.0000 −1.28654
\(88\) 1.00000 0.106600
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) −24.0000 −2.52982
\(91\) 0 0
\(92\) −7.00000 −0.729800
\(93\) −21.0000 −2.17760
\(94\) 7.00000 0.721995
\(95\) 24.0000 2.46235
\(96\) −3.00000 −0.306186
\(97\) 1.00000 0.101535 0.0507673 0.998711i \(-0.483833\pi\)
0.0507673 + 0.998711i \(0.483833\pi\)
\(98\) −1.00000 −0.101015
\(99\) −6.00000 −0.603023
\(100\) 11.0000 1.10000
\(101\) −5.00000 −0.497519 −0.248759 0.968565i \(-0.580023\pi\)
−0.248759 + 0.968565i \(0.580023\pi\)
\(102\) 0 0
\(103\) −14.0000 −1.37946 −0.689730 0.724066i \(-0.742271\pi\)
−0.689730 + 0.724066i \(0.742271\pi\)
\(104\) 0 0
\(105\) 12.0000 1.17108
\(106\) 0 0
\(107\) −4.00000 −0.386695 −0.193347 0.981130i \(-0.561934\pi\)
−0.193347 + 0.981130i \(0.561934\pi\)
\(108\) 9.00000 0.866025
\(109\) 14.0000 1.34096 0.670478 0.741929i \(-0.266089\pi\)
0.670478 + 0.741929i \(0.266089\pi\)
\(110\) 4.00000 0.381385
\(111\) −27.0000 −2.56273
\(112\) 1.00000 0.0944911
\(113\) −7.00000 −0.658505 −0.329252 0.944242i \(-0.606797\pi\)
−0.329252 + 0.944242i \(0.606797\pi\)
\(114\) −18.0000 −1.68585
\(115\) −28.0000 −2.61101
\(116\) −4.00000 −0.371391
\(117\) 0 0
\(118\) −10.0000 −0.920575
\(119\) 0 0
\(120\) −12.0000 −1.09545
\(121\) −10.0000 −0.909091
\(122\) −1.00000 −0.0905357
\(123\) 9.00000 0.811503
\(124\) −7.00000 −0.628619
\(125\) 24.0000 2.14663
\(126\) −6.00000 −0.534522
\(127\) −11.0000 −0.976092 −0.488046 0.872818i \(-0.662290\pi\)
−0.488046 + 0.872818i \(0.662290\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 12.0000 1.05654
\(130\) 0 0
\(131\) 8.00000 0.698963 0.349482 0.936943i \(-0.386358\pi\)
0.349482 + 0.936943i \(0.386358\pi\)
\(132\) −3.00000 −0.261116
\(133\) 6.00000 0.520266
\(134\) 1.00000 0.0863868
\(135\) 36.0000 3.09839
\(136\) 0 0
\(137\) 6.00000 0.512615 0.256307 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) 21.0000 1.78764
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 4.00000 0.338062
\(141\) −21.0000 −1.76852
\(142\) 16.0000 1.34269
\(143\) 0 0
\(144\) 6.00000 0.500000
\(145\) −16.0000 −1.32873
\(146\) 5.00000 0.413803
\(147\) 3.00000 0.247436
\(148\) −9.00000 −0.739795
\(149\) −9.00000 −0.737309 −0.368654 0.929567i \(-0.620181\pi\)
−0.368654 + 0.929567i \(0.620181\pi\)
\(150\) −33.0000 −2.69444
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) −6.00000 −0.486664
\(153\) 0 0
\(154\) 1.00000 0.0805823
\(155\) −28.0000 −2.24901
\(156\) 0 0
\(157\) 5.00000 0.399043 0.199522 0.979893i \(-0.436061\pi\)
0.199522 + 0.979893i \(0.436061\pi\)
\(158\) −11.0000 −0.875113
\(159\) 0 0
\(160\) −4.00000 −0.316228
\(161\) −7.00000 −0.551677
\(162\) −9.00000 −0.707107
\(163\) 4.00000 0.313304 0.156652 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) 3.00000 0.234261
\(165\) −12.0000 −0.934199
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) −3.00000 −0.231455
\(169\) 0 0
\(170\) 0 0
\(171\) 36.0000 2.75299
\(172\) 4.00000 0.304997
\(173\) 2.00000 0.152057 0.0760286 0.997106i \(-0.475776\pi\)
0.0760286 + 0.997106i \(0.475776\pi\)
\(174\) 12.0000 0.909718
\(175\) 11.0000 0.831522
\(176\) −1.00000 −0.0753778
\(177\) 30.0000 2.25494
\(178\) −6.00000 −0.449719
\(179\) −6.00000 −0.448461 −0.224231 0.974536i \(-0.571987\pi\)
−0.224231 + 0.974536i \(0.571987\pi\)
\(180\) 24.0000 1.78885
\(181\) 15.0000 1.11494 0.557471 0.830197i \(-0.311772\pi\)
0.557471 + 0.830197i \(0.311772\pi\)
\(182\) 0 0
\(183\) 3.00000 0.221766
\(184\) 7.00000 0.516047
\(185\) −36.0000 −2.64677
\(186\) 21.0000 1.53979
\(187\) 0 0
\(188\) −7.00000 −0.510527
\(189\) 9.00000 0.654654
\(190\) −24.0000 −1.74114
\(191\) −8.00000 −0.578860 −0.289430 0.957199i \(-0.593466\pi\)
−0.289430 + 0.957199i \(0.593466\pi\)
\(192\) 3.00000 0.216506
\(193\) −20.0000 −1.43963 −0.719816 0.694165i \(-0.755774\pi\)
−0.719816 + 0.694165i \(0.755774\pi\)
\(194\) −1.00000 −0.0717958
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 27.0000 1.92367 0.961835 0.273629i \(-0.0882242\pi\)
0.961835 + 0.273629i \(0.0882242\pi\)
\(198\) 6.00000 0.426401
\(199\) −4.00000 −0.283552 −0.141776 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) −11.0000 −0.777817
\(201\) −3.00000 −0.211604
\(202\) 5.00000 0.351799
\(203\) −4.00000 −0.280745
\(204\) 0 0
\(205\) 12.0000 0.838116
\(206\) 14.0000 0.975426
\(207\) −42.0000 −2.91920
\(208\) 0 0
\(209\) −6.00000 −0.415029
\(210\) −12.0000 −0.828079
\(211\) −10.0000 −0.688428 −0.344214 0.938891i \(-0.611855\pi\)
−0.344214 + 0.938891i \(0.611855\pi\)
\(212\) 0 0
\(213\) −48.0000 −3.28891
\(214\) 4.00000 0.273434
\(215\) 16.0000 1.09119
\(216\) −9.00000 −0.612372
\(217\) −7.00000 −0.475191
\(218\) −14.0000 −0.948200
\(219\) −15.0000 −1.01361
\(220\) −4.00000 −0.269680
\(221\) 0 0
\(222\) 27.0000 1.81212
\(223\) −21.0000 −1.40626 −0.703132 0.711059i \(-0.748216\pi\)
−0.703132 + 0.711059i \(0.748216\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 66.0000 4.40000
\(226\) 7.00000 0.465633
\(227\) −24.0000 −1.59294 −0.796468 0.604681i \(-0.793301\pi\)
−0.796468 + 0.604681i \(0.793301\pi\)
\(228\) 18.0000 1.19208
\(229\) 24.0000 1.58596 0.792982 0.609245i \(-0.208527\pi\)
0.792982 + 0.609245i \(0.208527\pi\)
\(230\) 28.0000 1.84627
\(231\) −3.00000 −0.197386
\(232\) 4.00000 0.262613
\(233\) −5.00000 −0.327561 −0.163780 0.986497i \(-0.552369\pi\)
−0.163780 + 0.986497i \(0.552369\pi\)
\(234\) 0 0
\(235\) −28.0000 −1.82652
\(236\) 10.0000 0.650945
\(237\) 33.0000 2.14358
\(238\) 0 0
\(239\) −6.00000 −0.388108 −0.194054 0.980991i \(-0.562164\pi\)
−0.194054 + 0.980991i \(0.562164\pi\)
\(240\) 12.0000 0.774597
\(241\) 18.0000 1.15948 0.579741 0.814801i \(-0.303154\pi\)
0.579741 + 0.814801i \(0.303154\pi\)
\(242\) 10.0000 0.642824
\(243\) 0 0
\(244\) 1.00000 0.0640184
\(245\) 4.00000 0.255551
\(246\) −9.00000 −0.573819
\(247\) 0 0
\(248\) 7.00000 0.444500
\(249\) 0 0
\(250\) −24.0000 −1.51789
\(251\) 3.00000 0.189358 0.0946792 0.995508i \(-0.469817\pi\)
0.0946792 + 0.995508i \(0.469817\pi\)
\(252\) 6.00000 0.377964
\(253\) 7.00000 0.440086
\(254\) 11.0000 0.690201
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 24.0000 1.49708 0.748539 0.663090i \(-0.230755\pi\)
0.748539 + 0.663090i \(0.230755\pi\)
\(258\) −12.0000 −0.747087
\(259\) −9.00000 −0.559233
\(260\) 0 0
\(261\) −24.0000 −1.48556
\(262\) −8.00000 −0.494242
\(263\) 24.0000 1.47990 0.739952 0.672660i \(-0.234848\pi\)
0.739952 + 0.672660i \(0.234848\pi\)
\(264\) 3.00000 0.184637
\(265\) 0 0
\(266\) −6.00000 −0.367884
\(267\) 18.0000 1.10158
\(268\) −1.00000 −0.0610847
\(269\) 9.00000 0.548740 0.274370 0.961624i \(-0.411531\pi\)
0.274370 + 0.961624i \(0.411531\pi\)
\(270\) −36.0000 −2.19089
\(271\) 23.0000 1.39715 0.698575 0.715537i \(-0.253818\pi\)
0.698575 + 0.715537i \(0.253818\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) −6.00000 −0.362473
\(275\) −11.0000 −0.663325
\(276\) −21.0000 −1.26405
\(277\) −18.0000 −1.08152 −0.540758 0.841178i \(-0.681862\pi\)
−0.540758 + 0.841178i \(0.681862\pi\)
\(278\) −4.00000 −0.239904
\(279\) −42.0000 −2.51447
\(280\) −4.00000 −0.239046
\(281\) 8.00000 0.477240 0.238620 0.971113i \(-0.423305\pi\)
0.238620 + 0.971113i \(0.423305\pi\)
\(282\) 21.0000 1.25053
\(283\) 19.0000 1.12943 0.564716 0.825285i \(-0.308986\pi\)
0.564716 + 0.825285i \(0.308986\pi\)
\(284\) −16.0000 −0.949425
\(285\) 72.0000 4.26491
\(286\) 0 0
\(287\) 3.00000 0.177084
\(288\) −6.00000 −0.353553
\(289\) −17.0000 −1.00000
\(290\) 16.0000 0.939552
\(291\) 3.00000 0.175863
\(292\) −5.00000 −0.292603
\(293\) 26.0000 1.51894 0.759468 0.650545i \(-0.225459\pi\)
0.759468 + 0.650545i \(0.225459\pi\)
\(294\) −3.00000 −0.174964
\(295\) 40.0000 2.32889
\(296\) 9.00000 0.523114
\(297\) −9.00000 −0.522233
\(298\) 9.00000 0.521356
\(299\) 0 0
\(300\) 33.0000 1.90526
\(301\) 4.00000 0.230556
\(302\) 0 0
\(303\) −15.0000 −0.861727
\(304\) 6.00000 0.344124
\(305\) 4.00000 0.229039
\(306\) 0 0
\(307\) 4.00000 0.228292 0.114146 0.993464i \(-0.463587\pi\)
0.114146 + 0.993464i \(0.463587\pi\)
\(308\) −1.00000 −0.0569803
\(309\) −42.0000 −2.38930
\(310\) 28.0000 1.59029
\(311\) 30.0000 1.70114 0.850572 0.525859i \(-0.176256\pi\)
0.850572 + 0.525859i \(0.176256\pi\)
\(312\) 0 0
\(313\) 14.0000 0.791327 0.395663 0.918396i \(-0.370515\pi\)
0.395663 + 0.918396i \(0.370515\pi\)
\(314\) −5.00000 −0.282166
\(315\) 24.0000 1.35225
\(316\) 11.0000 0.618798
\(317\) −21.0000 −1.17948 −0.589739 0.807594i \(-0.700769\pi\)
−0.589739 + 0.807594i \(0.700769\pi\)
\(318\) 0 0
\(319\) 4.00000 0.223957
\(320\) 4.00000 0.223607
\(321\) −12.0000 −0.669775
\(322\) 7.00000 0.390095
\(323\) 0 0
\(324\) 9.00000 0.500000
\(325\) 0 0
\(326\) −4.00000 −0.221540
\(327\) 42.0000 2.32261
\(328\) −3.00000 −0.165647
\(329\) −7.00000 −0.385922
\(330\) 12.0000 0.660578
\(331\) 7.00000 0.384755 0.192377 0.981321i \(-0.438380\pi\)
0.192377 + 0.981321i \(0.438380\pi\)
\(332\) 0 0
\(333\) −54.0000 −2.95918
\(334\) 0 0
\(335\) −4.00000 −0.218543
\(336\) 3.00000 0.163663
\(337\) 17.0000 0.926049 0.463025 0.886345i \(-0.346764\pi\)
0.463025 + 0.886345i \(0.346764\pi\)
\(338\) 0 0
\(339\) −21.0000 −1.14056
\(340\) 0 0
\(341\) 7.00000 0.379071
\(342\) −36.0000 −1.94666
\(343\) 1.00000 0.0539949
\(344\) −4.00000 −0.215666
\(345\) −84.0000 −4.52241
\(346\) −2.00000 −0.107521
\(347\) −24.0000 −1.28839 −0.644194 0.764862i \(-0.722807\pi\)
−0.644194 + 0.764862i \(0.722807\pi\)
\(348\) −12.0000 −0.643268
\(349\) 26.0000 1.39175 0.695874 0.718164i \(-0.255017\pi\)
0.695874 + 0.718164i \(0.255017\pi\)
\(350\) −11.0000 −0.587975
\(351\) 0 0
\(352\) 1.00000 0.0533002
\(353\) −25.0000 −1.33062 −0.665308 0.746569i \(-0.731700\pi\)
−0.665308 + 0.746569i \(0.731700\pi\)
\(354\) −30.0000 −1.59448
\(355\) −64.0000 −3.39677
\(356\) 6.00000 0.317999
\(357\) 0 0
\(358\) 6.00000 0.317110
\(359\) −12.0000 −0.633336 −0.316668 0.948536i \(-0.602564\pi\)
−0.316668 + 0.948536i \(0.602564\pi\)
\(360\) −24.0000 −1.26491
\(361\) 17.0000 0.894737
\(362\) −15.0000 −0.788382
\(363\) −30.0000 −1.57459
\(364\) 0 0
\(365\) −20.0000 −1.04685
\(366\) −3.00000 −0.156813
\(367\) 32.0000 1.67039 0.835193 0.549957i \(-0.185356\pi\)
0.835193 + 0.549957i \(0.185356\pi\)
\(368\) −7.00000 −0.364900
\(369\) 18.0000 0.937043
\(370\) 36.0000 1.87155
\(371\) 0 0
\(372\) −21.0000 −1.08880
\(373\) −16.0000 −0.828449 −0.414224 0.910175i \(-0.635947\pi\)
−0.414224 + 0.910175i \(0.635947\pi\)
\(374\) 0 0
\(375\) 72.0000 3.71806
\(376\) 7.00000 0.360997
\(377\) 0 0
\(378\) −9.00000 −0.462910
\(379\) 8.00000 0.410932 0.205466 0.978664i \(-0.434129\pi\)
0.205466 + 0.978664i \(0.434129\pi\)
\(380\) 24.0000 1.23117
\(381\) −33.0000 −1.69064
\(382\) 8.00000 0.409316
\(383\) 15.0000 0.766464 0.383232 0.923652i \(-0.374811\pi\)
0.383232 + 0.923652i \(0.374811\pi\)
\(384\) −3.00000 −0.153093
\(385\) −4.00000 −0.203859
\(386\) 20.0000 1.01797
\(387\) 24.0000 1.21999
\(388\) 1.00000 0.0507673
\(389\) −36.0000 −1.82527 −0.912636 0.408773i \(-0.865957\pi\)
−0.912636 + 0.408773i \(0.865957\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −1.00000 −0.0505076
\(393\) 24.0000 1.21064
\(394\) −27.0000 −1.36024
\(395\) 44.0000 2.21388
\(396\) −6.00000 −0.301511
\(397\) 28.0000 1.40528 0.702640 0.711546i \(-0.252005\pi\)
0.702640 + 0.711546i \(0.252005\pi\)
\(398\) 4.00000 0.200502
\(399\) 18.0000 0.901127
\(400\) 11.0000 0.550000
\(401\) −12.0000 −0.599251 −0.299626 0.954057i \(-0.596862\pi\)
−0.299626 + 0.954057i \(0.596862\pi\)
\(402\) 3.00000 0.149626
\(403\) 0 0
\(404\) −5.00000 −0.248759
\(405\) 36.0000 1.78885
\(406\) 4.00000 0.198517
\(407\) 9.00000 0.446113
\(408\) 0 0
\(409\) 26.0000 1.28562 0.642809 0.766027i \(-0.277769\pi\)
0.642809 + 0.766027i \(0.277769\pi\)
\(410\) −12.0000 −0.592638
\(411\) 18.0000 0.887875
\(412\) −14.0000 −0.689730
\(413\) 10.0000 0.492068
\(414\) 42.0000 2.06419
\(415\) 0 0
\(416\) 0 0
\(417\) 12.0000 0.587643
\(418\) 6.00000 0.293470
\(419\) 15.0000 0.732798 0.366399 0.930458i \(-0.380591\pi\)
0.366399 + 0.930458i \(0.380591\pi\)
\(420\) 12.0000 0.585540
\(421\) −19.0000 −0.926003 −0.463002 0.886357i \(-0.653228\pi\)
−0.463002 + 0.886357i \(0.653228\pi\)
\(422\) 10.0000 0.486792
\(423\) −42.0000 −2.04211
\(424\) 0 0
\(425\) 0 0
\(426\) 48.0000 2.32561
\(427\) 1.00000 0.0483934
\(428\) −4.00000 −0.193347
\(429\) 0 0
\(430\) −16.0000 −0.771589
\(431\) −18.0000 −0.867029 −0.433515 0.901146i \(-0.642727\pi\)
−0.433515 + 0.901146i \(0.642727\pi\)
\(432\) 9.00000 0.433013
\(433\) 34.0000 1.63394 0.816968 0.576683i \(-0.195653\pi\)
0.816968 + 0.576683i \(0.195653\pi\)
\(434\) 7.00000 0.336011
\(435\) −48.0000 −2.30142
\(436\) 14.0000 0.670478
\(437\) −42.0000 −2.00913
\(438\) 15.0000 0.716728
\(439\) 2.00000 0.0954548 0.0477274 0.998860i \(-0.484802\pi\)
0.0477274 + 0.998860i \(0.484802\pi\)
\(440\) 4.00000 0.190693
\(441\) 6.00000 0.285714
\(442\) 0 0
\(443\) 12.0000 0.570137 0.285069 0.958507i \(-0.407984\pi\)
0.285069 + 0.958507i \(0.407984\pi\)
\(444\) −27.0000 −1.28136
\(445\) 24.0000 1.13771
\(446\) 21.0000 0.994379
\(447\) −27.0000 −1.27706
\(448\) 1.00000 0.0472456
\(449\) 2.00000 0.0943858 0.0471929 0.998886i \(-0.484972\pi\)
0.0471929 + 0.998886i \(0.484972\pi\)
\(450\) −66.0000 −3.11127
\(451\) −3.00000 −0.141264
\(452\) −7.00000 −0.329252
\(453\) 0 0
\(454\) 24.0000 1.12638
\(455\) 0 0
\(456\) −18.0000 −0.842927
\(457\) 26.0000 1.21623 0.608114 0.793849i \(-0.291926\pi\)
0.608114 + 0.793849i \(0.291926\pi\)
\(458\) −24.0000 −1.12145
\(459\) 0 0
\(460\) −28.0000 −1.30551
\(461\) −12.0000 −0.558896 −0.279448 0.960161i \(-0.590151\pi\)
−0.279448 + 0.960161i \(0.590151\pi\)
\(462\) 3.00000 0.139573
\(463\) −16.0000 −0.743583 −0.371792 0.928316i \(-0.621256\pi\)
−0.371792 + 0.928316i \(0.621256\pi\)
\(464\) −4.00000 −0.185695
\(465\) −84.0000 −3.89541
\(466\) 5.00000 0.231621
\(467\) −4.00000 −0.185098 −0.0925490 0.995708i \(-0.529501\pi\)
−0.0925490 + 0.995708i \(0.529501\pi\)
\(468\) 0 0
\(469\) −1.00000 −0.0461757
\(470\) 28.0000 1.29154
\(471\) 15.0000 0.691164
\(472\) −10.0000 −0.460287
\(473\) −4.00000 −0.183920
\(474\) −33.0000 −1.51574
\(475\) 66.0000 3.02829
\(476\) 0 0
\(477\) 0 0
\(478\) 6.00000 0.274434
\(479\) −24.0000 −1.09659 −0.548294 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) −12.0000 −0.547723
\(481\) 0 0
\(482\) −18.0000 −0.819878
\(483\) −21.0000 −0.955533
\(484\) −10.0000 −0.454545
\(485\) 4.00000 0.181631
\(486\) 0 0
\(487\) −38.0000 −1.72194 −0.860972 0.508652i \(-0.830144\pi\)
−0.860972 + 0.508652i \(0.830144\pi\)
\(488\) −1.00000 −0.0452679
\(489\) 12.0000 0.542659
\(490\) −4.00000 −0.180702
\(491\) 16.0000 0.722070 0.361035 0.932552i \(-0.382424\pi\)
0.361035 + 0.932552i \(0.382424\pi\)
\(492\) 9.00000 0.405751
\(493\) 0 0
\(494\) 0 0
\(495\) −24.0000 −1.07872
\(496\) −7.00000 −0.314309
\(497\) −16.0000 −0.717698
\(498\) 0 0
\(499\) 5.00000 0.223831 0.111915 0.993718i \(-0.464301\pi\)
0.111915 + 0.993718i \(0.464301\pi\)
\(500\) 24.0000 1.07331
\(501\) 0 0
\(502\) −3.00000 −0.133897
\(503\) −20.0000 −0.891756 −0.445878 0.895094i \(-0.647108\pi\)
−0.445878 + 0.895094i \(0.647108\pi\)
\(504\) −6.00000 −0.267261
\(505\) −20.0000 −0.889988
\(506\) −7.00000 −0.311188
\(507\) 0 0
\(508\) −11.0000 −0.488046
\(509\) −10.0000 −0.443242 −0.221621 0.975133i \(-0.571135\pi\)
−0.221621 + 0.975133i \(0.571135\pi\)
\(510\) 0 0
\(511\) −5.00000 −0.221187
\(512\) −1.00000 −0.0441942
\(513\) 54.0000 2.38416
\(514\) −24.0000 −1.05859
\(515\) −56.0000 −2.46765
\(516\) 12.0000 0.528271
\(517\) 7.00000 0.307860
\(518\) 9.00000 0.395437
\(519\) 6.00000 0.263371
\(520\) 0 0
\(521\) −18.0000 −0.788594 −0.394297 0.918983i \(-0.629012\pi\)
−0.394297 + 0.918983i \(0.629012\pi\)
\(522\) 24.0000 1.05045
\(523\) 27.0000 1.18063 0.590314 0.807174i \(-0.299004\pi\)
0.590314 + 0.807174i \(0.299004\pi\)
\(524\) 8.00000 0.349482
\(525\) 33.0000 1.44024
\(526\) −24.0000 −1.04645
\(527\) 0 0
\(528\) −3.00000 −0.130558
\(529\) 26.0000 1.13043
\(530\) 0 0
\(531\) 60.0000 2.60378
\(532\) 6.00000 0.260133
\(533\) 0 0
\(534\) −18.0000 −0.778936
\(535\) −16.0000 −0.691740
\(536\) 1.00000 0.0431934
\(537\) −18.0000 −0.776757
\(538\) −9.00000 −0.388018
\(539\) −1.00000 −0.0430730
\(540\) 36.0000 1.54919
\(541\) 10.0000 0.429934 0.214967 0.976621i \(-0.431036\pi\)
0.214967 + 0.976621i \(0.431036\pi\)
\(542\) −23.0000 −0.987935
\(543\) 45.0000 1.93113
\(544\) 0 0
\(545\) 56.0000 2.39878
\(546\) 0 0
\(547\) 28.0000 1.19719 0.598597 0.801050i \(-0.295725\pi\)
0.598597 + 0.801050i \(0.295725\pi\)
\(548\) 6.00000 0.256307
\(549\) 6.00000 0.256074
\(550\) 11.0000 0.469042
\(551\) −24.0000 −1.02243
\(552\) 21.0000 0.893819
\(553\) 11.0000 0.467768
\(554\) 18.0000 0.764747
\(555\) −108.000 −4.58434
\(556\) 4.00000 0.169638
\(557\) −9.00000 −0.381342 −0.190671 0.981654i \(-0.561066\pi\)
−0.190671 + 0.981654i \(0.561066\pi\)
\(558\) 42.0000 1.77800
\(559\) 0 0
\(560\) 4.00000 0.169031
\(561\) 0 0
\(562\) −8.00000 −0.337460
\(563\) 11.0000 0.463595 0.231797 0.972764i \(-0.425539\pi\)
0.231797 + 0.972764i \(0.425539\pi\)
\(564\) −21.0000 −0.884260
\(565\) −28.0000 −1.17797
\(566\) −19.0000 −0.798630
\(567\) 9.00000 0.377964
\(568\) 16.0000 0.671345
\(569\) −35.0000 −1.46728 −0.733638 0.679540i \(-0.762179\pi\)
−0.733638 + 0.679540i \(0.762179\pi\)
\(570\) −72.0000 −3.01575
\(571\) 6.00000 0.251092 0.125546 0.992088i \(-0.459932\pi\)
0.125546 + 0.992088i \(0.459932\pi\)
\(572\) 0 0
\(573\) −24.0000 −1.00261
\(574\) −3.00000 −0.125218
\(575\) −77.0000 −3.21112
\(576\) 6.00000 0.250000
\(577\) −30.0000 −1.24892 −0.624458 0.781058i \(-0.714680\pi\)
−0.624458 + 0.781058i \(0.714680\pi\)
\(578\) 17.0000 0.707107
\(579\) −60.0000 −2.49351
\(580\) −16.0000 −0.664364
\(581\) 0 0
\(582\) −3.00000 −0.124354
\(583\) 0 0
\(584\) 5.00000 0.206901
\(585\) 0 0
\(586\) −26.0000 −1.07405
\(587\) −14.0000 −0.577842 −0.288921 0.957353i \(-0.593296\pi\)
−0.288921 + 0.957353i \(0.593296\pi\)
\(588\) 3.00000 0.123718
\(589\) −42.0000 −1.73058
\(590\) −40.0000 −1.64677
\(591\) 81.0000 3.33189
\(592\) −9.00000 −0.369898
\(593\) −14.0000 −0.574911 −0.287456 0.957794i \(-0.592809\pi\)
−0.287456 + 0.957794i \(0.592809\pi\)
\(594\) 9.00000 0.369274
\(595\) 0 0
\(596\) −9.00000 −0.368654
\(597\) −12.0000 −0.491127
\(598\) 0 0
\(599\) −29.0000 −1.18491 −0.592454 0.805604i \(-0.701841\pi\)
−0.592454 + 0.805604i \(0.701841\pi\)
\(600\) −33.0000 −1.34722
\(601\) −26.0000 −1.06056 −0.530281 0.847822i \(-0.677914\pi\)
−0.530281 + 0.847822i \(0.677914\pi\)
\(602\) −4.00000 −0.163028
\(603\) −6.00000 −0.244339
\(604\) 0 0
\(605\) −40.0000 −1.62623
\(606\) 15.0000 0.609333
\(607\) 6.00000 0.243532 0.121766 0.992559i \(-0.461144\pi\)
0.121766 + 0.992559i \(0.461144\pi\)
\(608\) −6.00000 −0.243332
\(609\) −12.0000 −0.486265
\(610\) −4.00000 −0.161955
\(611\) 0 0
\(612\) 0 0
\(613\) 49.0000 1.97909 0.989546 0.144220i \(-0.0460672\pi\)
0.989546 + 0.144220i \(0.0460672\pi\)
\(614\) −4.00000 −0.161427
\(615\) 36.0000 1.45166
\(616\) 1.00000 0.0402911
\(617\) −22.0000 −0.885687 −0.442843 0.896599i \(-0.646030\pi\)
−0.442843 + 0.896599i \(0.646030\pi\)
\(618\) 42.0000 1.68949
\(619\) −22.0000 −0.884255 −0.442127 0.896952i \(-0.645776\pi\)
−0.442127 + 0.896952i \(0.645776\pi\)
\(620\) −28.0000 −1.12451
\(621\) −63.0000 −2.52810
\(622\) −30.0000 −1.20289
\(623\) 6.00000 0.240385
\(624\) 0 0
\(625\) 41.0000 1.64000
\(626\) −14.0000 −0.559553
\(627\) −18.0000 −0.718851
\(628\) 5.00000 0.199522
\(629\) 0 0
\(630\) −24.0000 −0.956183
\(631\) 2.00000 0.0796187 0.0398094 0.999207i \(-0.487325\pi\)
0.0398094 + 0.999207i \(0.487325\pi\)
\(632\) −11.0000 −0.437557
\(633\) −30.0000 −1.19239
\(634\) 21.0000 0.834017
\(635\) −44.0000 −1.74609
\(636\) 0 0
\(637\) 0 0
\(638\) −4.00000 −0.158362
\(639\) −96.0000 −3.79770
\(640\) −4.00000 −0.158114
\(641\) 7.00000 0.276483 0.138242 0.990399i \(-0.455855\pi\)
0.138242 + 0.990399i \(0.455855\pi\)
\(642\) 12.0000 0.473602
\(643\) 16.0000 0.630978 0.315489 0.948929i \(-0.397831\pi\)
0.315489 + 0.948929i \(0.397831\pi\)
\(644\) −7.00000 −0.275839
\(645\) 48.0000 1.89000
\(646\) 0 0
\(647\) −8.00000 −0.314512 −0.157256 0.987558i \(-0.550265\pi\)
−0.157256 + 0.987558i \(0.550265\pi\)
\(648\) −9.00000 −0.353553
\(649\) −10.0000 −0.392534
\(650\) 0 0
\(651\) −21.0000 −0.823055
\(652\) 4.00000 0.156652
\(653\) −36.0000 −1.40879 −0.704394 0.709809i \(-0.748781\pi\)
−0.704394 + 0.709809i \(0.748781\pi\)
\(654\) −42.0000 −1.64233
\(655\) 32.0000 1.25034
\(656\) 3.00000 0.117130
\(657\) −30.0000 −1.17041
\(658\) 7.00000 0.272888
\(659\) −18.0000 −0.701180 −0.350590 0.936529i \(-0.614019\pi\)
−0.350590 + 0.936529i \(0.614019\pi\)
\(660\) −12.0000 −0.467099
\(661\) 4.00000 0.155582 0.0777910 0.996970i \(-0.475213\pi\)
0.0777910 + 0.996970i \(0.475213\pi\)
\(662\) −7.00000 −0.272063
\(663\) 0 0
\(664\) 0 0
\(665\) 24.0000 0.930680
\(666\) 54.0000 2.09246
\(667\) 28.0000 1.08416
\(668\) 0 0
\(669\) −63.0000 −2.43572
\(670\) 4.00000 0.154533
\(671\) −1.00000 −0.0386046
\(672\) −3.00000 −0.115728
\(673\) 7.00000 0.269830 0.134915 0.990857i \(-0.456924\pi\)
0.134915 + 0.990857i \(0.456924\pi\)
\(674\) −17.0000 −0.654816
\(675\) 99.0000 3.81051
\(676\) 0 0
\(677\) 39.0000 1.49889 0.749446 0.662066i \(-0.230320\pi\)
0.749446 + 0.662066i \(0.230320\pi\)
\(678\) 21.0000 0.806500
\(679\) 1.00000 0.0383765
\(680\) 0 0
\(681\) −72.0000 −2.75905
\(682\) −7.00000 −0.268044
\(683\) −1.00000 −0.0382639 −0.0191320 0.999817i \(-0.506090\pi\)
−0.0191320 + 0.999817i \(0.506090\pi\)
\(684\) 36.0000 1.37649
\(685\) 24.0000 0.916993
\(686\) −1.00000 −0.0381802
\(687\) 72.0000 2.74697
\(688\) 4.00000 0.152499
\(689\) 0 0
\(690\) 84.0000 3.19783
\(691\) 32.0000 1.21734 0.608669 0.793424i \(-0.291704\pi\)
0.608669 + 0.793424i \(0.291704\pi\)
\(692\) 2.00000 0.0760286
\(693\) −6.00000 −0.227921
\(694\) 24.0000 0.911028
\(695\) 16.0000 0.606915
\(696\) 12.0000 0.454859
\(697\) 0 0
\(698\) −26.0000 −0.984115
\(699\) −15.0000 −0.567352
\(700\) 11.0000 0.415761
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) −54.0000 −2.03665
\(704\) −1.00000 −0.0376889
\(705\) −84.0000 −3.16362
\(706\) 25.0000 0.940887
\(707\) −5.00000 −0.188044
\(708\) 30.0000 1.12747
\(709\) 13.0000 0.488225 0.244113 0.969747i \(-0.421503\pi\)
0.244113 + 0.969747i \(0.421503\pi\)
\(710\) 64.0000 2.40188
\(711\) 66.0000 2.47519
\(712\) −6.00000 −0.224860
\(713\) 49.0000 1.83506
\(714\) 0 0
\(715\) 0 0
\(716\) −6.00000 −0.224231
\(717\) −18.0000 −0.672222
\(718\) 12.0000 0.447836
\(719\) 2.00000 0.0745874 0.0372937 0.999304i \(-0.488126\pi\)
0.0372937 + 0.999304i \(0.488126\pi\)
\(720\) 24.0000 0.894427
\(721\) −14.0000 −0.521387
\(722\) −17.0000 −0.632674
\(723\) 54.0000 2.00828
\(724\) 15.0000 0.557471
\(725\) −44.0000 −1.63412
\(726\) 30.0000 1.11340
\(727\) −26.0000 −0.964287 −0.482143 0.876092i \(-0.660142\pi\)
−0.482143 + 0.876092i \(0.660142\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 20.0000 0.740233
\(731\) 0 0
\(732\) 3.00000 0.110883
\(733\) 4.00000 0.147743 0.0738717 0.997268i \(-0.476464\pi\)
0.0738717 + 0.997268i \(0.476464\pi\)
\(734\) −32.0000 −1.18114
\(735\) 12.0000 0.442627
\(736\) 7.00000 0.258023
\(737\) 1.00000 0.0368355
\(738\) −18.0000 −0.662589
\(739\) −24.0000 −0.882854 −0.441427 0.897297i \(-0.645528\pi\)
−0.441427 + 0.897297i \(0.645528\pi\)
\(740\) −36.0000 −1.32339
\(741\) 0 0
\(742\) 0 0
\(743\) −16.0000 −0.586983 −0.293492 0.955962i \(-0.594817\pi\)
−0.293492 + 0.955962i \(0.594817\pi\)
\(744\) 21.0000 0.769897
\(745\) −36.0000 −1.31894
\(746\) 16.0000 0.585802
\(747\) 0 0
\(748\) 0 0
\(749\) −4.00000 −0.146157
\(750\) −72.0000 −2.62907
\(751\) −27.0000 −0.985244 −0.492622 0.870243i \(-0.663961\pi\)
−0.492622 + 0.870243i \(0.663961\pi\)
\(752\) −7.00000 −0.255264
\(753\) 9.00000 0.327978
\(754\) 0 0
\(755\) 0 0
\(756\) 9.00000 0.327327
\(757\) 54.0000 1.96266 0.981332 0.192323i \(-0.0616021\pi\)
0.981332 + 0.192323i \(0.0616021\pi\)
\(758\) −8.00000 −0.290573
\(759\) 21.0000 0.762252
\(760\) −24.0000 −0.870572
\(761\) −45.0000 −1.63125 −0.815624 0.578582i \(-0.803606\pi\)
−0.815624 + 0.578582i \(0.803606\pi\)
\(762\) 33.0000 1.19546
\(763\) 14.0000 0.506834
\(764\) −8.00000 −0.289430
\(765\) 0 0
\(766\) −15.0000 −0.541972
\(767\) 0 0
\(768\) 3.00000 0.108253
\(769\) −21.0000 −0.757279 −0.378640 0.925544i \(-0.623608\pi\)
−0.378640 + 0.925544i \(0.623608\pi\)
\(770\) 4.00000 0.144150
\(771\) 72.0000 2.59302
\(772\) −20.0000 −0.719816
\(773\) −14.0000 −0.503545 −0.251773 0.967786i \(-0.581013\pi\)
−0.251773 + 0.967786i \(0.581013\pi\)
\(774\) −24.0000 −0.862662
\(775\) −77.0000 −2.76592
\(776\) −1.00000 −0.0358979
\(777\) −27.0000 −0.968620
\(778\) 36.0000 1.29066
\(779\) 18.0000 0.644917
\(780\) 0 0
\(781\) 16.0000 0.572525
\(782\) 0 0
\(783\) −36.0000 −1.28654
\(784\) 1.00000 0.0357143
\(785\) 20.0000 0.713831
\(786\) −24.0000 −0.856052
\(787\) −32.0000 −1.14068 −0.570338 0.821410i \(-0.693188\pi\)
−0.570338 + 0.821410i \(0.693188\pi\)
\(788\) 27.0000 0.961835
\(789\) 72.0000 2.56327
\(790\) −44.0000 −1.56545
\(791\) −7.00000 −0.248891
\(792\) 6.00000 0.213201
\(793\) 0 0
\(794\) −28.0000 −0.993683
\(795\) 0 0
\(796\) −4.00000 −0.141776
\(797\) 15.0000 0.531327 0.265664 0.964066i \(-0.414409\pi\)
0.265664 + 0.964066i \(0.414409\pi\)
\(798\) −18.0000 −0.637193
\(799\) 0 0
\(800\) −11.0000 −0.388909
\(801\) 36.0000 1.27200
\(802\) 12.0000 0.423735
\(803\) 5.00000 0.176446
\(804\) −3.00000 −0.105802
\(805\) −28.0000 −0.986870
\(806\) 0 0
\(807\) 27.0000 0.950445
\(808\) 5.00000 0.175899
\(809\) 6.00000 0.210949 0.105474 0.994422i \(-0.466364\pi\)
0.105474 + 0.994422i \(0.466364\pi\)
\(810\) −36.0000 −1.26491
\(811\) 28.0000 0.983213 0.491606 0.870817i \(-0.336410\pi\)
0.491606 + 0.870817i \(0.336410\pi\)
\(812\) −4.00000 −0.140372
\(813\) 69.0000 2.41994
\(814\) −9.00000 −0.315450
\(815\) 16.0000 0.560456
\(816\) 0 0
\(817\) 24.0000 0.839654
\(818\) −26.0000 −0.909069
\(819\) 0 0
\(820\) 12.0000 0.419058
\(821\) 46.0000 1.60541 0.802706 0.596376i \(-0.203393\pi\)
0.802706 + 0.596376i \(0.203393\pi\)
\(822\) −18.0000 −0.627822
\(823\) −3.00000 −0.104573 −0.0522867 0.998632i \(-0.516651\pi\)
−0.0522867 + 0.998632i \(0.516651\pi\)
\(824\) 14.0000 0.487713
\(825\) −33.0000 −1.14891
\(826\) −10.0000 −0.347945
\(827\) 12.0000 0.417281 0.208640 0.977992i \(-0.433096\pi\)
0.208640 + 0.977992i \(0.433096\pi\)
\(828\) −42.0000 −1.45960
\(829\) −14.0000 −0.486240 −0.243120 0.969996i \(-0.578171\pi\)
−0.243120 + 0.969996i \(0.578171\pi\)
\(830\) 0 0
\(831\) −54.0000 −1.87324
\(832\) 0 0
\(833\) 0 0
\(834\) −12.0000 −0.415526
\(835\) 0 0
\(836\) −6.00000 −0.207514
\(837\) −63.0000 −2.17760
\(838\) −15.0000 −0.518166
\(839\) −17.0000 −0.586905 −0.293453 0.955974i \(-0.594804\pi\)
−0.293453 + 0.955974i \(0.594804\pi\)
\(840\) −12.0000 −0.414039
\(841\) −13.0000 −0.448276
\(842\) 19.0000 0.654783
\(843\) 24.0000 0.826604
\(844\) −10.0000 −0.344214
\(845\) 0 0
\(846\) 42.0000 1.44399
\(847\) −10.0000 −0.343604
\(848\) 0 0
\(849\) 57.0000 1.95623
\(850\) 0 0
\(851\) 63.0000 2.15961
\(852\) −48.0000 −1.64445
\(853\) −16.0000 −0.547830 −0.273915 0.961754i \(-0.588319\pi\)
−0.273915 + 0.961754i \(0.588319\pi\)
\(854\) −1.00000 −0.0342193
\(855\) 144.000 4.92470
\(856\) 4.00000 0.136717
\(857\) 22.0000 0.751506 0.375753 0.926720i \(-0.377384\pi\)
0.375753 + 0.926720i \(0.377384\pi\)
\(858\) 0 0
\(859\) 5.00000 0.170598 0.0852989 0.996355i \(-0.472815\pi\)
0.0852989 + 0.996355i \(0.472815\pi\)
\(860\) 16.0000 0.545595
\(861\) 9.00000 0.306719
\(862\) 18.0000 0.613082
\(863\) −28.0000 −0.953131 −0.476566 0.879139i \(-0.658119\pi\)
−0.476566 + 0.879139i \(0.658119\pi\)
\(864\) −9.00000 −0.306186
\(865\) 8.00000 0.272008
\(866\) −34.0000 −1.15537
\(867\) −51.0000 −1.73205
\(868\) −7.00000 −0.237595
\(869\) −11.0000 −0.373149
\(870\) 48.0000 1.62735
\(871\) 0 0
\(872\) −14.0000 −0.474100
\(873\) 6.00000 0.203069
\(874\) 42.0000 1.42067
\(875\) 24.0000 0.811348
\(876\) −15.0000 −0.506803
\(877\) 45.0000 1.51954 0.759771 0.650191i \(-0.225311\pi\)
0.759771 + 0.650191i \(0.225311\pi\)
\(878\) −2.00000 −0.0674967
\(879\) 78.0000 2.63087
\(880\) −4.00000 −0.134840
\(881\) 18.0000 0.606435 0.303218 0.952921i \(-0.401939\pi\)
0.303218 + 0.952921i \(0.401939\pi\)
\(882\) −6.00000 −0.202031
\(883\) −14.0000 −0.471138 −0.235569 0.971858i \(-0.575695\pi\)
−0.235569 + 0.971858i \(0.575695\pi\)
\(884\) 0 0
\(885\) 120.000 4.03376
\(886\) −12.0000 −0.403148
\(887\) −38.0000 −1.27592 −0.637958 0.770072i \(-0.720220\pi\)
−0.637958 + 0.770072i \(0.720220\pi\)
\(888\) 27.0000 0.906061
\(889\) −11.0000 −0.368928
\(890\) −24.0000 −0.804482
\(891\) −9.00000 −0.301511
\(892\) −21.0000 −0.703132
\(893\) −42.0000 −1.40548
\(894\) 27.0000 0.903015
\(895\) −24.0000 −0.802232
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) −2.00000 −0.0667409
\(899\) 28.0000 0.933852
\(900\) 66.0000 2.20000
\(901\) 0 0
\(902\) 3.00000 0.0998891
\(903\) 12.0000 0.399335
\(904\) 7.00000 0.232817
\(905\) 60.0000 1.99447
\(906\) 0 0
\(907\) −30.0000 −0.996134 −0.498067 0.867139i \(-0.665957\pi\)
−0.498067 + 0.867139i \(0.665957\pi\)
\(908\) −24.0000 −0.796468
\(909\) −30.0000 −0.995037
\(910\) 0 0
\(911\) 36.0000 1.19273 0.596367 0.802712i \(-0.296610\pi\)
0.596367 + 0.802712i \(0.296610\pi\)
\(912\) 18.0000 0.596040
\(913\) 0 0
\(914\) −26.0000 −0.860004
\(915\) 12.0000 0.396708
\(916\) 24.0000 0.792982
\(917\) 8.00000 0.264183
\(918\) 0 0
\(919\) 27.0000 0.890648 0.445324 0.895370i \(-0.353089\pi\)
0.445324 + 0.895370i \(0.353089\pi\)
\(920\) 28.0000 0.923133
\(921\) 12.0000 0.395413
\(922\) 12.0000 0.395199
\(923\) 0 0
\(924\) −3.00000 −0.0986928
\(925\) −99.0000 −3.25510
\(926\) 16.0000 0.525793
\(927\) −84.0000 −2.75892
\(928\) 4.00000 0.131306
\(929\) 15.0000 0.492134 0.246067 0.969253i \(-0.420862\pi\)
0.246067 + 0.969253i \(0.420862\pi\)
\(930\) 84.0000 2.75447
\(931\) 6.00000 0.196642
\(932\) −5.00000 −0.163780
\(933\) 90.0000 2.94647
\(934\) 4.00000 0.130884
\(935\) 0 0
\(936\) 0 0
\(937\) −36.0000 −1.17607 −0.588034 0.808836i \(-0.700098\pi\)
−0.588034 + 0.808836i \(0.700098\pi\)
\(938\) 1.00000 0.0326512
\(939\) 42.0000 1.37062
\(940\) −28.0000 −0.913259
\(941\) 50.0000 1.62995 0.814977 0.579494i \(-0.196750\pi\)
0.814977 + 0.579494i \(0.196750\pi\)
\(942\) −15.0000 −0.488726
\(943\) −21.0000 −0.683854
\(944\) 10.0000 0.325472
\(945\) 36.0000 1.17108
\(946\) 4.00000 0.130051
\(947\) −24.0000 −0.779895 −0.389948 0.920837i \(-0.627507\pi\)
−0.389948 + 0.920837i \(0.627507\pi\)
\(948\) 33.0000 1.07179
\(949\) 0 0
\(950\) −66.0000 −2.14132
\(951\) −63.0000 −2.04291
\(952\) 0 0
\(953\) −34.0000 −1.10137 −0.550684 0.834714i \(-0.685633\pi\)
−0.550684 + 0.834714i \(0.685633\pi\)
\(954\) 0 0
\(955\) −32.0000 −1.03550
\(956\) −6.00000 −0.194054
\(957\) 12.0000 0.387905
\(958\) 24.0000 0.775405
\(959\) 6.00000 0.193750
\(960\) 12.0000 0.387298
\(961\) 18.0000 0.580645
\(962\) 0 0
\(963\) −24.0000 −0.773389
\(964\) 18.0000 0.579741
\(965\) −80.0000 −2.57529
\(966\) 21.0000 0.675664
\(967\) 22.0000 0.707472 0.353736 0.935345i \(-0.384911\pi\)
0.353736 + 0.935345i \(0.384911\pi\)
\(968\) 10.0000 0.321412
\(969\) 0 0
\(970\) −4.00000 −0.128432
\(971\) −15.0000 −0.481373 −0.240686 0.970603i \(-0.577373\pi\)
−0.240686 + 0.970603i \(0.577373\pi\)
\(972\) 0 0
\(973\) 4.00000 0.128234
\(974\) 38.0000 1.21760
\(975\) 0 0
\(976\) 1.00000 0.0320092
\(977\) −42.0000 −1.34370 −0.671850 0.740688i \(-0.734500\pi\)
−0.671850 + 0.740688i \(0.734500\pi\)
\(978\) −12.0000 −0.383718
\(979\) −6.00000 −0.191761
\(980\) 4.00000 0.127775
\(981\) 84.0000 2.68191
\(982\) −16.0000 −0.510581
\(983\) 40.0000 1.27580 0.637901 0.770118i \(-0.279803\pi\)
0.637901 + 0.770118i \(0.279803\pi\)
\(984\) −9.00000 −0.286910
\(985\) 108.000 3.44117
\(986\) 0 0
\(987\) −21.0000 −0.668437
\(988\) 0 0
\(989\) −28.0000 −0.890348
\(990\) 24.0000 0.762770
\(991\) −57.0000 −1.81066 −0.905332 0.424704i \(-0.860378\pi\)
−0.905332 + 0.424704i \(0.860378\pi\)
\(992\) 7.00000 0.222250
\(993\) 21.0000 0.666415
\(994\) 16.0000 0.507489
\(995\) −16.0000 −0.507234
\(996\) 0 0
\(997\) 5.00000 0.158352 0.0791758 0.996861i \(-0.474771\pi\)
0.0791758 + 0.996861i \(0.474771\pi\)
\(998\) −5.00000 −0.158272
\(999\) −81.0000 −2.56273
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 2366.2.a.h.1.1 1
13.5 odd 4 2366.2.d.j.337.2 2
13.8 odd 4 2366.2.d.j.337.1 2
13.12 even 2 182.2.a.e.1.1 1
39.38 odd 2 1638.2.a.j.1.1 1
52.51 odd 2 1456.2.a.a.1.1 1
65.64 even 2 4550.2.a.a.1.1 1
91.12 odd 6 1274.2.f.k.1145.1 2
91.25 even 6 1274.2.f.b.79.1 2
91.38 odd 6 1274.2.f.k.79.1 2
91.51 even 6 1274.2.f.b.1145.1 2
91.90 odd 2 1274.2.a.h.1.1 1
104.51 odd 2 5824.2.a.bf.1.1 1
104.77 even 2 5824.2.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
182.2.a.e.1.1 1 13.12 even 2
1274.2.a.h.1.1 1 91.90 odd 2
1274.2.f.b.79.1 2 91.25 even 6
1274.2.f.b.1145.1 2 91.51 even 6
1274.2.f.k.79.1 2 91.38 odd 6
1274.2.f.k.1145.1 2 91.12 odd 6
1456.2.a.a.1.1 1 52.51 odd 2
1638.2.a.j.1.1 1 39.38 odd 2
2366.2.a.h.1.1 1 1.1 even 1 trivial
2366.2.d.j.337.1 2 13.8 odd 4
2366.2.d.j.337.2 2 13.5 odd 4
4550.2.a.a.1.1 1 65.64 even 2
5824.2.a.b.1.1 1 104.77 even 2
5824.2.a.bf.1.1 1 104.51 odd 2