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Properties

Label 4650.2.a.bs.1.1
Level $4650$
Weight $2$
Character 4650.1
Self dual yes
Analytic conductor $37.130$
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] = [4650,2,Mod(1,4650)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4650, 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("4650.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4650.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: \(37.1304369399\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 930)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 4650.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -6.00000 q^{11} +1.00000 q^{12} -2.00000 q^{13} +2.00000 q^{14} +1.00000 q^{16} +6.00000 q^{17} +1.00000 q^{18} +8.00000 q^{19} +2.00000 q^{21} -6.00000 q^{22} +4.00000 q^{23} +1.00000 q^{24} -2.00000 q^{26} +1.00000 q^{27} +2.00000 q^{28} -4.00000 q^{29} -1.00000 q^{31} +1.00000 q^{32} -6.00000 q^{33} +6.00000 q^{34} +1.00000 q^{36} +2.00000 q^{37} +8.00000 q^{38} -2.00000 q^{39} +2.00000 q^{41} +2.00000 q^{42} -4.00000 q^{43} -6.00000 q^{44} +4.00000 q^{46} +8.00000 q^{47} +1.00000 q^{48} -3.00000 q^{49} +6.00000 q^{51} -2.00000 q^{52} +10.0000 q^{53} +1.00000 q^{54} +2.00000 q^{56} +8.00000 q^{57} -4.00000 q^{58} +6.00000 q^{59} -14.0000 q^{61} -1.00000 q^{62} +2.00000 q^{63} +1.00000 q^{64} -6.00000 q^{66} +4.00000 q^{67} +6.00000 q^{68} +4.00000 q^{69} +16.0000 q^{71} +1.00000 q^{72} -8.00000 q^{73} +2.00000 q^{74} +8.00000 q^{76} -12.0000 q^{77} -2.00000 q^{78} +1.00000 q^{81} +2.00000 q^{82} -4.00000 q^{83} +2.00000 q^{84} -4.00000 q^{86} -4.00000 q^{87} -6.00000 q^{88} +10.0000 q^{89} -4.00000 q^{91} +4.00000 q^{92} -1.00000 q^{93} +8.00000 q^{94} +1.00000 q^{96} +16.0000 q^{97} -3.00000 q^{98} -6.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 1.00000 0.408248
\(7\) 2.00000 0.755929 0.377964 − 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −6.00000 −1.80907 −0.904534 − 0.426401i \(-0.859781\pi\)
−0.904534 + 0.426401i \(0.859781\pi\)
\(12\) 1.00000 0.288675
\(13\) −2.00000 −0.554700 −0.277350 − 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 2.00000 0.534522
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 6.00000 1.45521 0.727607 − 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 1.00000 0.235702
\(19\) 8.00000 1.83533 0.917663 − 0.397360i \(-0.130073\pi\)
0.917663 + 0.397360i \(0.130073\pi\)
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) −6.00000 −1.27920
\(23\) 4.00000 0.834058 0.417029 − 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) −2.00000 −0.392232
\(27\) 1.00000 0.192450
\(28\) 2.00000 0.377964
\(29\) −4.00000 −0.742781 −0.371391 − 0.928477i \(-0.621119\pi\)
−0.371391 + 0.928477i \(0.621119\pi\)
\(30\) 0 0
\(31\) −1.00000 −0.179605
\(32\) 1.00000 0.176777
\(33\) −6.00000 −1.04447
\(34\) 6.00000 1.02899
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 2.00000 0.328798 0.164399 − 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 8.00000 1.29777
\(39\) −2.00000 −0.320256
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 − 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 2.00000 0.308607
\(43\) −4.00000 −0.609994 −0.304997 − 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) −6.00000 −0.904534
\(45\) 0 0
\(46\) 4.00000 0.589768
\(47\) 8.00000 1.16692 0.583460 − 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) 1.00000 0.144338
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) 6.00000 0.840168
\(52\) −2.00000 −0.277350
\(53\) 10.0000 1.37361 0.686803 − 0.726844i \(-0.259014\pi\)
0.686803 + 0.726844i \(0.259014\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) 2.00000 0.267261
\(57\) 8.00000 1.05963
\(58\) −4.00000 −0.525226
\(59\) 6.00000 0.781133 0.390567 − 0.920575i \(-0.372279\pi\)
0.390567 + 0.920575i \(0.372279\pi\)
\(60\) 0 0
\(61\) −14.0000 −1.79252 −0.896258 − 0.443533i \(-0.853725\pi\)
−0.896258 + 0.443533i \(0.853725\pi\)
\(62\) −1.00000 −0.127000
\(63\) 2.00000 0.251976
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −6.00000 −0.738549
\(67\) 4.00000 0.488678 0.244339 − 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 6.00000 0.727607
\(69\) 4.00000 0.481543
\(70\) 0 0
\(71\) 16.0000 1.89885 0.949425 − 0.313993i \(-0.101667\pi\)
0.949425 + 0.313993i \(0.101667\pi\)
\(72\) 1.00000 0.117851
\(73\) −8.00000 −0.936329 −0.468165 − 0.883641i \(-0.655085\pi\)
−0.468165 + 0.883641i \(0.655085\pi\)
\(74\) 2.00000 0.232495
\(75\) 0 0
\(76\) 8.00000 0.917663
\(77\) −12.0000 −1.36753
\(78\) −2.00000 −0.226455
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 2.00000 0.220863
\(83\) −4.00000 −0.439057 −0.219529 − 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 2.00000 0.218218
\(85\) 0 0
\(86\) −4.00000 −0.431331
\(87\) −4.00000 −0.428845
\(88\) −6.00000 −0.639602
\(89\) 10.0000 1.06000 0.529999 − 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) −4.00000 −0.419314
\(92\) 4.00000 0.417029
\(93\) −1.00000 −0.103695
\(94\) 8.00000 0.825137
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) 16.0000 1.62455 0.812277 − 0.583272i \(-0.198228\pi\)
0.812277 + 0.583272i \(0.198228\pi\)
\(98\) −3.00000 −0.303046
\(99\) −6.00000 −0.603023
\(100\) 0 0
\(101\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 6.00000 0.594089
\(103\) −6.00000 −0.591198 −0.295599 − 0.955312i \(-0.595519\pi\)
−0.295599 + 0.955312i \(0.595519\pi\)
\(104\) −2.00000 −0.196116
\(105\) 0 0
\(106\) 10.0000 0.971286
\(107\) −4.00000 −0.386695 −0.193347 − 0.981130i \(-0.561934\pi\)
−0.193347 + 0.981130i \(0.561934\pi\)
\(108\) 1.00000 0.0962250
\(109\) 2.00000 0.191565 0.0957826 − 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 0 0
\(111\) 2.00000 0.189832
\(112\) 2.00000 0.188982
\(113\) 18.0000 1.69330 0.846649 − 0.532152i \(-0.178617\pi\)
0.846649 + 0.532152i \(0.178617\pi\)
\(114\) 8.00000 0.749269
\(115\) 0 0
\(116\) −4.00000 −0.371391
\(117\) −2.00000 −0.184900
\(118\) 6.00000 0.552345
\(119\) 12.0000 1.10004
\(120\) 0 0
\(121\) 25.0000 2.27273
\(122\) −14.0000 −1.26750
\(123\) 2.00000 0.180334
\(124\) −1.00000 −0.0898027
\(125\) 0 0
\(126\) 2.00000 0.178174
\(127\) −2.00000 −0.177471 −0.0887357 − 0.996055i \(-0.528283\pi\)
−0.0887357 + 0.996055i \(0.528283\pi\)
\(128\) 1.00000 0.0883883
\(129\) −4.00000 −0.352180
\(130\) 0 0
\(131\) 10.0000 0.873704 0.436852 − 0.899533i \(-0.356093\pi\)
0.436852 + 0.899533i \(0.356093\pi\)
\(132\) −6.00000 −0.522233
\(133\) 16.0000 1.38738
\(134\) 4.00000 0.345547
\(135\) 0 0
\(136\) 6.00000 0.514496
\(137\) −14.0000 −1.19610 −0.598050 − 0.801459i \(-0.704058\pi\)
−0.598050 + 0.801459i \(0.704058\pi\)
\(138\) 4.00000 0.340503
\(139\) −20.0000 −1.69638 −0.848189 − 0.529694i \(-0.822307\pi\)
−0.848189 + 0.529694i \(0.822307\pi\)
\(140\) 0 0
\(141\) 8.00000 0.673722
\(142\) 16.0000 1.34269
\(143\) 12.0000 1.00349
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −8.00000 −0.662085
\(147\) −3.00000 −0.247436
\(148\) 2.00000 0.164399
\(149\) −12.0000 −0.983078 −0.491539 − 0.870855i \(-0.663566\pi\)
−0.491539 + 0.870855i \(0.663566\pi\)
\(150\) 0 0
\(151\) 8.00000 0.651031 0.325515 − 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) 8.00000 0.648886
\(153\) 6.00000 0.485071
\(154\) −12.0000 −0.966988
\(155\) 0 0
\(156\) −2.00000 −0.160128
\(157\) −6.00000 −0.478852 −0.239426 − 0.970915i \(-0.576959\pi\)
−0.239426 + 0.970915i \(0.576959\pi\)
\(158\) 0 0
\(159\) 10.0000 0.793052
\(160\) 0 0
\(161\) 8.00000 0.630488
\(162\) 1.00000 0.0785674
\(163\) −20.0000 −1.56652 −0.783260 − 0.621694i \(-0.786445\pi\)
−0.783260 + 0.621694i \(0.786445\pi\)
\(164\) 2.00000 0.156174
\(165\) 0 0
\(166\) −4.00000 −0.310460
\(167\) −12.0000 −0.928588 −0.464294 − 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) 2.00000 0.154303
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 8.00000 0.611775
\(172\) −4.00000 −0.304997
\(173\) 14.0000 1.06440 0.532200 − 0.846619i \(-0.321365\pi\)
0.532200 + 0.846619i \(0.321365\pi\)
\(174\) −4.00000 −0.303239
\(175\) 0 0
\(176\) −6.00000 −0.452267
\(177\) 6.00000 0.450988
\(178\) 10.0000 0.749532
\(179\) 18.0000 1.34538 0.672692 − 0.739923i \(-0.265138\pi\)
0.672692 + 0.739923i \(0.265138\pi\)
\(180\) 0 0
\(181\) 10.0000 0.743294 0.371647 − 0.928374i \(-0.378793\pi\)
0.371647 + 0.928374i \(0.378793\pi\)
\(182\) −4.00000 −0.296500
\(183\) −14.0000 −1.03491
\(184\) 4.00000 0.294884
\(185\) 0 0
\(186\) −1.00000 −0.0733236
\(187\) −36.0000 −2.63258
\(188\) 8.00000 0.583460
\(189\) 2.00000 0.145479
\(190\) 0 0
\(191\) 8.00000 0.578860 0.289430 − 0.957199i \(-0.406534\pi\)
0.289430 + 0.957199i \(0.406534\pi\)
\(192\) 1.00000 0.0721688
\(193\) −20.0000 −1.43963 −0.719816 − 0.694165i \(-0.755774\pi\)
−0.719816 + 0.694165i \(0.755774\pi\)
\(194\) 16.0000 1.14873
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) 18.0000 1.28245 0.641223 − 0.767354i \(-0.278427\pi\)
0.641223 + 0.767354i \(0.278427\pi\)
\(198\) −6.00000 −0.426401
\(199\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 0 0
\(201\) 4.00000 0.282138
\(202\) 0 0
\(203\) −8.00000 −0.561490
\(204\) 6.00000 0.420084
\(205\) 0 0
\(206\) −6.00000 −0.418040
\(207\) 4.00000 0.278019
\(208\) −2.00000 −0.138675
\(209\) −48.0000 −3.32023
\(210\) 0 0
\(211\) −20.0000 −1.37686 −0.688428 − 0.725304i \(-0.741699\pi\)
−0.688428 + 0.725304i \(0.741699\pi\)
\(212\) 10.0000 0.686803
\(213\) 16.0000 1.09630
\(214\) −4.00000 −0.273434
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) −2.00000 −0.135769
\(218\) 2.00000 0.135457
\(219\) −8.00000 −0.540590
\(220\) 0 0
\(221\) −12.0000 −0.807207
\(222\) 2.00000 0.134231
\(223\) 14.0000 0.937509 0.468755 − 0.883328i \(-0.344703\pi\)
0.468755 + 0.883328i \(0.344703\pi\)
\(224\) 2.00000 0.133631
\(225\) 0 0
\(226\) 18.0000 1.19734
\(227\) −12.0000 −0.796468 −0.398234 − 0.917284i \(-0.630377\pi\)
−0.398234 + 0.917284i \(0.630377\pi\)
\(228\) 8.00000 0.529813
\(229\) 6.00000 0.396491 0.198246 − 0.980152i \(-0.436476\pi\)
0.198246 + 0.980152i \(0.436476\pi\)
\(230\) 0 0
\(231\) −12.0000 −0.789542
\(232\) −4.00000 −0.262613
\(233\) −18.0000 −1.17922 −0.589610 − 0.807688i \(-0.700718\pi\)
−0.589610 + 0.807688i \(0.700718\pi\)
\(234\) −2.00000 −0.130744
\(235\) 0 0
\(236\) 6.00000 0.390567
\(237\) 0 0
\(238\) 12.0000 0.777844
\(239\) −20.0000 −1.29369 −0.646846 − 0.762620i \(-0.723912\pi\)
−0.646846 + 0.762620i \(0.723912\pi\)
\(240\) 0 0
\(241\) −18.0000 −1.15948 −0.579741 − 0.814801i \(-0.696846\pi\)
−0.579741 + 0.814801i \(0.696846\pi\)
\(242\) 25.0000 1.60706
\(243\) 1.00000 0.0641500
\(244\) −14.0000 −0.896258
\(245\) 0 0
\(246\) 2.00000 0.127515
\(247\) −16.0000 −1.01806
\(248\) −1.00000 −0.0635001
\(249\) −4.00000 −0.253490
\(250\) 0 0
\(251\) 6.00000 0.378717 0.189358 − 0.981908i \(-0.439359\pi\)
0.189358 + 0.981908i \(0.439359\pi\)
\(252\) 2.00000 0.125988
\(253\) −24.0000 −1.50887
\(254\) −2.00000 −0.125491
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −14.0000 −0.873296 −0.436648 − 0.899632i \(-0.643834\pi\)
−0.436648 + 0.899632i \(0.643834\pi\)
\(258\) −4.00000 −0.249029
\(259\) 4.00000 0.248548
\(260\) 0 0
\(261\) −4.00000 −0.247594
\(262\) 10.0000 0.617802
\(263\) −12.0000 −0.739952 −0.369976 − 0.929041i \(-0.620634\pi\)
−0.369976 + 0.929041i \(0.620634\pi\)
\(264\) −6.00000 −0.369274
\(265\) 0 0
\(266\) 16.0000 0.981023
\(267\) 10.0000 0.611990
\(268\) 4.00000 0.244339
\(269\) −4.00000 −0.243884 −0.121942 − 0.992537i \(-0.538912\pi\)
−0.121942 + 0.992537i \(0.538912\pi\)
\(270\) 0 0
\(271\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) 6.00000 0.363803
\(273\) −4.00000 −0.242091
\(274\) −14.0000 −0.845771
\(275\) 0 0
\(276\) 4.00000 0.240772
\(277\) 10.0000 0.600842 0.300421 − 0.953807i \(-0.402873\pi\)
0.300421 + 0.953807i \(0.402873\pi\)
\(278\) −20.0000 −1.19952
\(279\) −1.00000 −0.0598684
\(280\) 0 0
\(281\) 22.0000 1.31241 0.656205 − 0.754583i \(-0.272161\pi\)
0.656205 + 0.754583i \(0.272161\pi\)
\(282\) 8.00000 0.476393
\(283\) −20.0000 −1.18888 −0.594438 − 0.804141i \(-0.702626\pi\)
−0.594438 + 0.804141i \(0.702626\pi\)
\(284\) 16.0000 0.949425
\(285\) 0 0
\(286\) 12.0000 0.709575
\(287\) 4.00000 0.236113
\(288\) 1.00000 0.0589256
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) 16.0000 0.937937
\(292\) −8.00000 −0.468165
\(293\) −6.00000 −0.350524 −0.175262 − 0.984522i \(-0.556077\pi\)
−0.175262 + 0.984522i \(0.556077\pi\)
\(294\) −3.00000 −0.174964
\(295\) 0 0
\(296\) 2.00000 0.116248
\(297\) −6.00000 −0.348155
\(298\) −12.0000 −0.695141
\(299\) −8.00000 −0.462652
\(300\) 0 0
\(301\) −8.00000 −0.461112
\(302\) 8.00000 0.460348
\(303\) 0 0
\(304\) 8.00000 0.458831
\(305\) 0 0
\(306\) 6.00000 0.342997
\(307\) −8.00000 −0.456584 −0.228292 − 0.973593i \(-0.573314\pi\)
−0.228292 + 0.973593i \(0.573314\pi\)
\(308\) −12.0000 −0.683763
\(309\) −6.00000 −0.341328
\(310\) 0 0
\(311\) −8.00000 −0.453638 −0.226819 − 0.973937i \(-0.572833\pi\)
−0.226819 + 0.973937i \(0.572833\pi\)
\(312\) −2.00000 −0.113228
\(313\) 24.0000 1.35656 0.678280 − 0.734803i \(-0.262726\pi\)
0.678280 + 0.734803i \(0.262726\pi\)
\(314\) −6.00000 −0.338600
\(315\) 0 0
\(316\) 0 0
\(317\) −18.0000 −1.01098 −0.505490 − 0.862832i \(-0.668688\pi\)
−0.505490 + 0.862832i \(0.668688\pi\)
\(318\) 10.0000 0.560772
\(319\) 24.0000 1.34374
\(320\) 0 0
\(321\) −4.00000 −0.223258
\(322\) 8.00000 0.445823
\(323\) 48.0000 2.67079
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −20.0000 −1.10770
\(327\) 2.00000 0.110600
\(328\) 2.00000 0.110432
\(329\) 16.0000 0.882109
\(330\) 0 0
\(331\) 8.00000 0.439720 0.219860 − 0.975531i \(-0.429440\pi\)
0.219860 + 0.975531i \(0.429440\pi\)
\(332\) −4.00000 −0.219529
\(333\) 2.00000 0.109599
\(334\) −12.0000 −0.656611
\(335\) 0 0
\(336\) 2.00000 0.109109
\(337\) −8.00000 −0.435788 −0.217894 − 0.975972i \(-0.569919\pi\)
−0.217894 + 0.975972i \(0.569919\pi\)
\(338\) −9.00000 −0.489535
\(339\) 18.0000 0.977626
\(340\) 0 0
\(341\) 6.00000 0.324918
\(342\) 8.00000 0.432590
\(343\) −20.0000 −1.07990
\(344\) −4.00000 −0.215666
\(345\) 0 0
\(346\) 14.0000 0.752645
\(347\) 4.00000 0.214731 0.107366 − 0.994220i \(-0.465758\pi\)
0.107366 + 0.994220i \(0.465758\pi\)
\(348\) −4.00000 −0.214423
\(349\) 2.00000 0.107058 0.0535288 − 0.998566i \(-0.482953\pi\)
0.0535288 + 0.998566i \(0.482953\pi\)
\(350\) 0 0
\(351\) −2.00000 −0.106752
\(352\) −6.00000 −0.319801
\(353\) −18.0000 −0.958043 −0.479022 − 0.877803i \(-0.659008\pi\)
−0.479022 + 0.877803i \(0.659008\pi\)
\(354\) 6.00000 0.318896
\(355\) 0 0
\(356\) 10.0000 0.529999
\(357\) 12.0000 0.635107
\(358\) 18.0000 0.951330
\(359\) 12.0000 0.633336 0.316668 − 0.948536i \(-0.397436\pi\)
0.316668 + 0.948536i \(0.397436\pi\)
\(360\) 0 0
\(361\) 45.0000 2.36842
\(362\) 10.0000 0.525588
\(363\) 25.0000 1.31216
\(364\) −4.00000 −0.209657
\(365\) 0 0
\(366\) −14.0000 −0.731792
\(367\) −26.0000 −1.35719 −0.678594 − 0.734513i \(-0.737411\pi\)
−0.678594 + 0.734513i \(0.737411\pi\)
\(368\) 4.00000 0.208514
\(369\) 2.00000 0.104116
\(370\) 0 0
\(371\) 20.0000 1.03835
\(372\) −1.00000 −0.0518476
\(373\) −22.0000 −1.13912 −0.569558 − 0.821951i \(-0.692886\pi\)
−0.569558 + 0.821951i \(0.692886\pi\)
\(374\) −36.0000 −1.86152
\(375\) 0 0
\(376\) 8.00000 0.412568
\(377\) 8.00000 0.412021
\(378\) 2.00000 0.102869
\(379\) 36.0000 1.84920 0.924598 − 0.380945i \(-0.124401\pi\)
0.924598 + 0.380945i \(0.124401\pi\)
\(380\) 0 0
\(381\) −2.00000 −0.102463
\(382\) 8.00000 0.409316
\(383\) −16.0000 −0.817562 −0.408781 − 0.912633i \(-0.634046\pi\)
−0.408781 + 0.912633i \(0.634046\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −20.0000 −1.01797
\(387\) −4.00000 −0.203331
\(388\) 16.0000 0.812277
\(389\) −16.0000 −0.811232 −0.405616 − 0.914044i \(-0.632943\pi\)
−0.405616 + 0.914044i \(0.632943\pi\)
\(390\) 0 0
\(391\) 24.0000 1.21373
\(392\) −3.00000 −0.151523
\(393\) 10.0000 0.504433
\(394\) 18.0000 0.906827
\(395\) 0 0
\(396\) −6.00000 −0.301511
\(397\) 22.0000 1.10415 0.552074 − 0.833795i \(-0.313837\pi\)
0.552074 + 0.833795i \(0.313837\pi\)
\(398\) 0 0
\(399\) 16.0000 0.801002
\(400\) 0 0
\(401\) 2.00000 0.0998752 0.0499376 − 0.998752i \(-0.484098\pi\)
0.0499376 + 0.998752i \(0.484098\pi\)
\(402\) 4.00000 0.199502
\(403\) 2.00000 0.0996271
\(404\) 0 0
\(405\) 0 0
\(406\) −8.00000 −0.397033
\(407\) −12.0000 −0.594818
\(408\) 6.00000 0.297044
\(409\) 22.0000 1.08783 0.543915 − 0.839140i \(-0.316941\pi\)
0.543915 + 0.839140i \(0.316941\pi\)
\(410\) 0 0
\(411\) −14.0000 −0.690569
\(412\) −6.00000 −0.295599
\(413\) 12.0000 0.590481
\(414\) 4.00000 0.196589
\(415\) 0 0
\(416\) −2.00000 −0.0980581
\(417\) −20.0000 −0.979404
\(418\) −48.0000 −2.34776
\(419\) −6.00000 −0.293119 −0.146560 − 0.989202i \(-0.546820\pi\)
−0.146560 + 0.989202i \(0.546820\pi\)
\(420\) 0 0
\(421\) −2.00000 −0.0974740 −0.0487370 − 0.998812i \(-0.515520\pi\)
−0.0487370 + 0.998812i \(0.515520\pi\)
\(422\) −20.0000 −0.973585
\(423\) 8.00000 0.388973
\(424\) 10.0000 0.485643
\(425\) 0 0
\(426\) 16.0000 0.775203
\(427\) −28.0000 −1.35501
\(428\) −4.00000 −0.193347
\(429\) 12.0000 0.579365
\(430\) 0 0
\(431\) 12.0000 0.578020 0.289010 − 0.957326i \(-0.406674\pi\)
0.289010 + 0.957326i \(0.406674\pi\)
\(432\) 1.00000 0.0481125
\(433\) 8.00000 0.384455 0.192228 − 0.981350i \(-0.438429\pi\)
0.192228 + 0.981350i \(0.438429\pi\)
\(434\) −2.00000 −0.0960031
\(435\) 0 0
\(436\) 2.00000 0.0957826
\(437\) 32.0000 1.53077
\(438\) −8.00000 −0.382255
\(439\) 32.0000 1.52728 0.763638 − 0.645644i \(-0.223411\pi\)
0.763638 + 0.645644i \(0.223411\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) −12.0000 −0.570782
\(443\) 4.00000 0.190046 0.0950229 − 0.995475i \(-0.469708\pi\)
0.0950229 + 0.995475i \(0.469708\pi\)
\(444\) 2.00000 0.0949158
\(445\) 0 0
\(446\) 14.0000 0.662919
\(447\) −12.0000 −0.567581
\(448\) 2.00000 0.0944911
\(449\) 26.0000 1.22702 0.613508 − 0.789689i \(-0.289758\pi\)
0.613508 + 0.789689i \(0.289758\pi\)
\(450\) 0 0
\(451\) −12.0000 −0.565058
\(452\) 18.0000 0.846649
\(453\) 8.00000 0.375873
\(454\) −12.0000 −0.563188
\(455\) 0 0
\(456\) 8.00000 0.374634
\(457\) −12.0000 −0.561336 −0.280668 − 0.959805i \(-0.590556\pi\)
−0.280668 + 0.959805i \(0.590556\pi\)
\(458\) 6.00000 0.280362
\(459\) 6.00000 0.280056
\(460\) 0 0
\(461\) −32.0000 −1.49039 −0.745194 − 0.666847i \(-0.767643\pi\)
−0.745194 + 0.666847i \(0.767643\pi\)
\(462\) −12.0000 −0.558291
\(463\) 26.0000 1.20832 0.604161 − 0.796862i \(-0.293508\pi\)
0.604161 + 0.796862i \(0.293508\pi\)
\(464\) −4.00000 −0.185695
\(465\) 0 0
\(466\) −18.0000 −0.833834
\(467\) −4.00000 −0.185098 −0.0925490 − 0.995708i \(-0.529501\pi\)
−0.0925490 + 0.995708i \(0.529501\pi\)
\(468\) −2.00000 −0.0924500
\(469\) 8.00000 0.369406
\(470\) 0 0
\(471\) −6.00000 −0.276465
\(472\) 6.00000 0.276172
\(473\) 24.0000 1.10352
\(474\) 0 0
\(475\) 0 0
\(476\) 12.0000 0.550019
\(477\) 10.0000 0.457869
\(478\) −20.0000 −0.914779
\(479\) −40.0000 −1.82765 −0.913823 − 0.406112i \(-0.866884\pi\)
−0.913823 + 0.406112i \(0.866884\pi\)
\(480\) 0 0
\(481\) −4.00000 −0.182384
\(482\) −18.0000 −0.819878
\(483\) 8.00000 0.364013
\(484\) 25.0000 1.13636
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) −14.0000 −0.634401 −0.317200 − 0.948359i \(-0.602743\pi\)
−0.317200 + 0.948359i \(0.602743\pi\)
\(488\) −14.0000 −0.633750
\(489\) −20.0000 −0.904431
\(490\) 0 0
\(491\) 30.0000 1.35388 0.676941 − 0.736038i \(-0.263305\pi\)
0.676941 + 0.736038i \(0.263305\pi\)
\(492\) 2.00000 0.0901670
\(493\) −24.0000 −1.08091
\(494\) −16.0000 −0.719874
\(495\) 0 0
\(496\) −1.00000 −0.0449013
\(497\) 32.0000 1.43540
\(498\) −4.00000 −0.179244
\(499\) −32.0000 −1.43252 −0.716258 − 0.697835i \(-0.754147\pi\)
−0.716258 + 0.697835i \(0.754147\pi\)
\(500\) 0 0
\(501\) −12.0000 −0.536120
\(502\) 6.00000 0.267793
\(503\) −8.00000 −0.356702 −0.178351 − 0.983967i \(-0.557076\pi\)
−0.178351 + 0.983967i \(0.557076\pi\)
\(504\) 2.00000 0.0890871
\(505\) 0 0
\(506\) −24.0000 −1.06693
\(507\) −9.00000 −0.399704
\(508\) −2.00000 −0.0887357
\(509\) 20.0000 0.886484 0.443242 − 0.896402i \(-0.353828\pi\)
0.443242 + 0.896402i \(0.353828\pi\)
\(510\) 0 0
\(511\) −16.0000 −0.707798
\(512\) 1.00000 0.0441942
\(513\) 8.00000 0.353209
\(514\) −14.0000 −0.617514
\(515\) 0 0
\(516\) −4.00000 −0.176090
\(517\) −48.0000 −2.11104
\(518\) 4.00000 0.175750
\(519\) 14.0000 0.614532
\(520\) 0 0
\(521\) −42.0000 −1.84005 −0.920027 − 0.391856i \(-0.871833\pi\)
−0.920027 + 0.391856i \(0.871833\pi\)
\(522\) −4.00000 −0.175075
\(523\) −40.0000 −1.74908 −0.874539 − 0.484955i \(-0.838836\pi\)
−0.874539 + 0.484955i \(0.838836\pi\)
\(524\) 10.0000 0.436852
\(525\) 0 0
\(526\) −12.0000 −0.523225
\(527\) −6.00000 −0.261364
\(528\) −6.00000 −0.261116
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 6.00000 0.260378
\(532\) 16.0000 0.693688
\(533\) −4.00000 −0.173259
\(534\) 10.0000 0.432742
\(535\) 0 0
\(536\) 4.00000 0.172774
\(537\) 18.0000 0.776757
\(538\) −4.00000 −0.172452
\(539\) 18.0000 0.775315
\(540\) 0 0
\(541\) 18.0000 0.773880 0.386940 − 0.922105i \(-0.373532\pi\)
0.386940 + 0.922105i \(0.373532\pi\)
\(542\) 0 0
\(543\) 10.0000 0.429141
\(544\) 6.00000 0.257248
\(545\) 0 0
\(546\) −4.00000 −0.171184
\(547\) 8.00000 0.342055 0.171028 − 0.985266i \(-0.445291\pi\)
0.171028 + 0.985266i \(0.445291\pi\)
\(548\) −14.0000 −0.598050
\(549\) −14.0000 −0.597505
\(550\) 0 0
\(551\) −32.0000 −1.36325
\(552\) 4.00000 0.170251
\(553\) 0 0
\(554\) 10.0000 0.424859
\(555\) 0 0
\(556\) −20.0000 −0.848189
\(557\) −30.0000 −1.27114 −0.635570 − 0.772043i \(-0.719235\pi\)
−0.635570 + 0.772043i \(0.719235\pi\)
\(558\) −1.00000 −0.0423334
\(559\) 8.00000 0.338364
\(560\) 0 0
\(561\) −36.0000 −1.51992
\(562\) 22.0000 0.928014
\(563\) −36.0000 −1.51722 −0.758610 − 0.651546i \(-0.774121\pi\)
−0.758610 + 0.651546i \(0.774121\pi\)
\(564\) 8.00000 0.336861
\(565\) 0 0
\(566\) −20.0000 −0.840663
\(567\) 2.00000 0.0839921
\(568\) 16.0000 0.671345
\(569\) 18.0000 0.754599 0.377300 − 0.926091i \(-0.376853\pi\)
0.377300 + 0.926091i \(0.376853\pi\)
\(570\) 0 0
\(571\) −40.0000 −1.67395 −0.836974 − 0.547243i \(-0.815677\pi\)
−0.836974 + 0.547243i \(0.815677\pi\)
\(572\) 12.0000 0.501745
\(573\) 8.00000 0.334205
\(574\) 4.00000 0.166957
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 16.0000 0.666089 0.333044 − 0.942911i \(-0.391924\pi\)
0.333044 + 0.942911i \(0.391924\pi\)
\(578\) 19.0000 0.790296
\(579\) −20.0000 −0.831172
\(580\) 0 0
\(581\) −8.00000 −0.331896
\(582\) 16.0000 0.663221
\(583\) −60.0000 −2.48495
\(584\) −8.00000 −0.331042
\(585\) 0 0
\(586\) −6.00000 −0.247858
\(587\) −12.0000 −0.495293 −0.247647 − 0.968850i \(-0.579657\pi\)
−0.247647 + 0.968850i \(0.579657\pi\)
\(588\) −3.00000 −0.123718
\(589\) −8.00000 −0.329634
\(590\) 0 0
\(591\) 18.0000 0.740421
\(592\) 2.00000 0.0821995
\(593\) −30.0000 −1.23195 −0.615976 − 0.787765i \(-0.711238\pi\)
−0.615976 + 0.787765i \(0.711238\pi\)
\(594\) −6.00000 −0.246183
\(595\) 0 0
\(596\) −12.0000 −0.491539
\(597\) 0 0
\(598\) −8.00000 −0.327144
\(599\) −20.0000 −0.817178 −0.408589 − 0.912719i \(-0.633979\pi\)
−0.408589 + 0.912719i \(0.633979\pi\)
\(600\) 0 0
\(601\) 10.0000 0.407909 0.203954 − 0.978980i \(-0.434621\pi\)
0.203954 + 0.978980i \(0.434621\pi\)
\(602\) −8.00000 −0.326056
\(603\) 4.00000 0.162893
\(604\) 8.00000 0.325515
\(605\) 0 0
\(606\) 0 0
\(607\) 6.00000 0.243532 0.121766 − 0.992559i \(-0.461144\pi\)
0.121766 + 0.992559i \(0.461144\pi\)
\(608\) 8.00000 0.324443
\(609\) −8.00000 −0.324176
\(610\) 0 0
\(611\) −16.0000 −0.647291
\(612\) 6.00000 0.242536
\(613\) −30.0000 −1.21169 −0.605844 − 0.795583i \(-0.707165\pi\)
−0.605844 + 0.795583i \(0.707165\pi\)
\(614\) −8.00000 −0.322854
\(615\) 0 0
\(616\) −12.0000 −0.483494
\(617\) 22.0000 0.885687 0.442843 − 0.896599i \(-0.353970\pi\)
0.442843 + 0.896599i \(0.353970\pi\)
\(618\) −6.00000 −0.241355
\(619\) −12.0000 −0.482321 −0.241160 − 0.970485i \(-0.577528\pi\)
−0.241160 + 0.970485i \(0.577528\pi\)
\(620\) 0 0
\(621\) 4.00000 0.160514
\(622\) −8.00000 −0.320771
\(623\) 20.0000 0.801283
\(624\) −2.00000 −0.0800641
\(625\) 0 0
\(626\) 24.0000 0.959233
\(627\) −48.0000 −1.91694
\(628\) −6.00000 −0.239426
\(629\) 12.0000 0.478471
\(630\) 0 0
\(631\) −8.00000 −0.318475 −0.159237 − 0.987240i \(-0.550904\pi\)
−0.159237 + 0.987240i \(0.550904\pi\)
\(632\) 0 0
\(633\) −20.0000 −0.794929
\(634\) −18.0000 −0.714871
\(635\) 0 0
\(636\) 10.0000 0.396526
\(637\) 6.00000 0.237729
\(638\) 24.0000 0.950169
\(639\) 16.0000 0.632950
\(640\) 0 0
\(641\) 14.0000 0.552967 0.276483 − 0.961019i \(-0.410831\pi\)
0.276483 + 0.961019i \(0.410831\pi\)
\(642\) −4.00000 −0.157867
\(643\) −32.0000 −1.26196 −0.630978 − 0.775800i \(-0.717346\pi\)
−0.630978 + 0.775800i \(0.717346\pi\)
\(644\) 8.00000 0.315244
\(645\) 0 0
\(646\) 48.0000 1.88853
\(647\) 32.0000 1.25805 0.629025 − 0.777385i \(-0.283454\pi\)
0.629025 + 0.777385i \(0.283454\pi\)
\(648\) 1.00000 0.0392837
\(649\) −36.0000 −1.41312
\(650\) 0 0
\(651\) −2.00000 −0.0783862
\(652\) −20.0000 −0.783260
\(653\) −42.0000 −1.64359 −0.821794 − 0.569785i \(-0.807026\pi\)
−0.821794 + 0.569785i \(0.807026\pi\)
\(654\) 2.00000 0.0782062
\(655\) 0 0
\(656\) 2.00000 0.0780869
\(657\) −8.00000 −0.312110
\(658\) 16.0000 0.623745
\(659\) 18.0000 0.701180 0.350590 − 0.936529i \(-0.385981\pi\)
0.350590 + 0.936529i \(0.385981\pi\)
\(660\) 0 0
\(661\) 18.0000 0.700119 0.350059 − 0.936727i \(-0.386161\pi\)
0.350059 + 0.936727i \(0.386161\pi\)
\(662\) 8.00000 0.310929
\(663\) −12.0000 −0.466041
\(664\) −4.00000 −0.155230
\(665\) 0 0
\(666\) 2.00000 0.0774984
\(667\) −16.0000 −0.619522
\(668\) −12.0000 −0.464294
\(669\) 14.0000 0.541271
\(670\) 0 0
\(671\) 84.0000 3.24278
\(672\) 2.00000 0.0771517
\(673\) 24.0000 0.925132 0.462566 − 0.886585i \(-0.346929\pi\)
0.462566 + 0.886585i \(0.346929\pi\)
\(674\) −8.00000 −0.308148
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) −10.0000 −0.384331 −0.192166 − 0.981363i \(-0.561551\pi\)
−0.192166 + 0.981363i \(0.561551\pi\)
\(678\) 18.0000 0.691286
\(679\) 32.0000 1.22805
\(680\) 0 0
\(681\) −12.0000 −0.459841
\(682\) 6.00000 0.229752
\(683\) 4.00000 0.153056 0.0765279 − 0.997067i \(-0.475617\pi\)
0.0765279 + 0.997067i \(0.475617\pi\)
\(684\) 8.00000 0.305888
\(685\) 0 0
\(686\) −20.0000 −0.763604
\(687\) 6.00000 0.228914
\(688\) −4.00000 −0.152499
\(689\) −20.0000 −0.761939
\(690\) 0 0
\(691\) −32.0000 −1.21734 −0.608669 − 0.793424i \(-0.708296\pi\)
−0.608669 + 0.793424i \(0.708296\pi\)
\(692\) 14.0000 0.532200
\(693\) −12.0000 −0.455842
\(694\) 4.00000 0.151838
\(695\) 0 0
\(696\) −4.00000 −0.151620
\(697\) 12.0000 0.454532
\(698\) 2.00000 0.0757011
\(699\) −18.0000 −0.680823
\(700\) 0 0
\(701\) 16.0000 0.604312 0.302156 − 0.953259i \(-0.402294\pi\)
0.302156 + 0.953259i \(0.402294\pi\)
\(702\) −2.00000 −0.0754851
\(703\) 16.0000 0.603451
\(704\) −6.00000 −0.226134
\(705\) 0 0
\(706\) −18.0000 −0.677439
\(707\) 0 0
\(708\) 6.00000 0.225494
\(709\) −10.0000 −0.375558 −0.187779 − 0.982211i \(-0.560129\pi\)
−0.187779 + 0.982211i \(0.560129\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 10.0000 0.374766
\(713\) −4.00000 −0.149801
\(714\) 12.0000 0.449089
\(715\) 0 0
\(716\) 18.0000 0.672692
\(717\) −20.0000 −0.746914
\(718\) 12.0000 0.447836
\(719\) −8.00000 −0.298350 −0.149175 − 0.988811i \(-0.547662\pi\)
−0.149175 + 0.988811i \(0.547662\pi\)
\(720\) 0 0
\(721\) −12.0000 −0.446903
\(722\) 45.0000 1.67473
\(723\) −18.0000 −0.669427
\(724\) 10.0000 0.371647
\(725\) 0 0
\(726\) 25.0000 0.927837
\(727\) 38.0000 1.40934 0.704671 − 0.709534i \(-0.251095\pi\)
0.704671 + 0.709534i \(0.251095\pi\)
\(728\) −4.00000 −0.148250
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −24.0000 −0.887672
\(732\) −14.0000 −0.517455
\(733\) 22.0000 0.812589 0.406294 − 0.913742i \(-0.366821\pi\)
0.406294 + 0.913742i \(0.366821\pi\)
\(734\) −26.0000 −0.959678
\(735\) 0 0
\(736\) 4.00000 0.147442
\(737\) −24.0000 −0.884051
\(738\) 2.00000 0.0736210
\(739\) −24.0000 −0.882854 −0.441427 − 0.897297i \(-0.645528\pi\)
−0.441427 + 0.897297i \(0.645528\pi\)
\(740\) 0 0
\(741\) −16.0000 −0.587775
\(742\) 20.0000 0.734223
\(743\) 24.0000 0.880475 0.440237 − 0.897881i \(-0.354894\pi\)
0.440237 + 0.897881i \(0.354894\pi\)
\(744\) −1.00000 −0.0366618
\(745\) 0 0
\(746\) −22.0000 −0.805477
\(747\) −4.00000 −0.146352
\(748\) −36.0000 −1.31629
\(749\) −8.00000 −0.292314
\(750\) 0 0
\(751\) 40.0000 1.45962 0.729810 − 0.683650i \(-0.239608\pi\)
0.729810 + 0.683650i \(0.239608\pi\)
\(752\) 8.00000 0.291730
\(753\) 6.00000 0.218652
\(754\) 8.00000 0.291343
\(755\) 0 0
\(756\) 2.00000 0.0727393
\(757\) 34.0000 1.23575 0.617876 − 0.786276i \(-0.287994\pi\)
0.617876 + 0.786276i \(0.287994\pi\)
\(758\) 36.0000 1.30758
\(759\) −24.0000 −0.871145
\(760\) 0 0
\(761\) −54.0000 −1.95750 −0.978749 − 0.205061i \(-0.934261\pi\)
−0.978749 + 0.205061i \(0.934261\pi\)
\(762\) −2.00000 −0.0724524
\(763\) 4.00000 0.144810
\(764\) 8.00000 0.289430
\(765\) 0 0
\(766\) −16.0000 −0.578103
\(767\) −12.0000 −0.433295
\(768\) 1.00000 0.0360844
\(769\) −30.0000 −1.08183 −0.540914 − 0.841078i \(-0.681921\pi\)
−0.540914 + 0.841078i \(0.681921\pi\)
\(770\) 0 0
\(771\) −14.0000 −0.504198
\(772\) −20.0000 −0.719816
\(773\) −34.0000 −1.22290 −0.611448 − 0.791285i \(-0.709412\pi\)
−0.611448 + 0.791285i \(0.709412\pi\)
\(774\) −4.00000 −0.143777
\(775\) 0 0
\(776\) 16.0000 0.574367
\(777\) 4.00000 0.143499
\(778\) −16.0000 −0.573628
\(779\) 16.0000 0.573259
\(780\) 0 0
\(781\) −96.0000 −3.43515
\(782\) 24.0000 0.858238
\(783\) −4.00000 −0.142948
\(784\) −3.00000 −0.107143
\(785\) 0 0
\(786\) 10.0000 0.356688
\(787\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(788\) 18.0000 0.641223
\(789\) −12.0000 −0.427211
\(790\) 0 0
\(791\) 36.0000 1.28001
\(792\) −6.00000 −0.213201
\(793\) 28.0000 0.994309
\(794\) 22.0000 0.780751
\(795\) 0 0
\(796\) 0 0
\(797\) −50.0000 −1.77109 −0.885545 − 0.464553i \(-0.846215\pi\)
−0.885545 + 0.464553i \(0.846215\pi\)
\(798\) 16.0000 0.566394
\(799\) 48.0000 1.69812
\(800\) 0 0
\(801\) 10.0000 0.353333
\(802\) 2.00000 0.0706225
\(803\) 48.0000 1.69388
\(804\) 4.00000 0.141069
\(805\) 0 0
\(806\) 2.00000 0.0704470
\(807\) −4.00000 −0.140807
\(808\) 0 0
\(809\) −18.0000 −0.632846 −0.316423 − 0.948618i \(-0.602482\pi\)
−0.316423 + 0.948618i \(0.602482\pi\)
\(810\) 0 0
\(811\) −28.0000 −0.983213 −0.491606 − 0.870817i \(-0.663590\pi\)
−0.491606 + 0.870817i \(0.663590\pi\)
\(812\) −8.00000 −0.280745
\(813\) 0 0
\(814\) −12.0000 −0.420600
\(815\) 0 0
\(816\) 6.00000 0.210042
\(817\) −32.0000 −1.11954
\(818\) 22.0000 0.769212
\(819\) −4.00000 −0.139771
\(820\) 0 0
\(821\) 36.0000 1.25641 0.628204 − 0.778048i \(-0.283790\pi\)
0.628204 + 0.778048i \(0.283790\pi\)
\(822\) −14.0000 −0.488306
\(823\) −54.0000 −1.88232 −0.941161 − 0.337959i \(-0.890263\pi\)
−0.941161 + 0.337959i \(0.890263\pi\)
\(824\) −6.00000 −0.209020
\(825\) 0 0
\(826\) 12.0000 0.417533
\(827\) 4.00000 0.139094 0.0695468 − 0.997579i \(-0.477845\pi\)
0.0695468 + 0.997579i \(0.477845\pi\)
\(828\) 4.00000 0.139010
\(829\) 2.00000 0.0694629 0.0347314 − 0.999397i \(-0.488942\pi\)
0.0347314 + 0.999397i \(0.488942\pi\)
\(830\) 0 0
\(831\) 10.0000 0.346896
\(832\) −2.00000 −0.0693375
\(833\) −18.0000 −0.623663
\(834\) −20.0000 −0.692543
\(835\) 0 0
\(836\) −48.0000 −1.66011
\(837\) −1.00000 −0.0345651
\(838\) −6.00000 −0.207267
\(839\) −12.0000 −0.414286 −0.207143 − 0.978311i \(-0.566417\pi\)
−0.207143 + 0.978311i \(0.566417\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) −2.00000 −0.0689246
\(843\) 22.0000 0.757720
\(844\) −20.0000 −0.688428
\(845\) 0 0
\(846\) 8.00000 0.275046
\(847\) 50.0000 1.71802
\(848\) 10.0000 0.343401
\(849\) −20.0000 −0.686398
\(850\) 0 0
\(851\) 8.00000 0.274236
\(852\) 16.0000 0.548151
\(853\) 22.0000 0.753266 0.376633 − 0.926363i \(-0.377082\pi\)
0.376633 + 0.926363i \(0.377082\pi\)
\(854\) −28.0000 −0.958140
\(855\) 0 0
\(856\) −4.00000 −0.136717
\(857\) 46.0000 1.57133 0.785665 − 0.618652i \(-0.212321\pi\)
0.785665 + 0.618652i \(0.212321\pi\)
\(858\) 12.0000 0.409673
\(859\) 4.00000 0.136478 0.0682391 − 0.997669i \(-0.478262\pi\)
0.0682391 + 0.997669i \(0.478262\pi\)
\(860\) 0 0
\(861\) 4.00000 0.136320
\(862\) 12.0000 0.408722
\(863\) 44.0000 1.49778 0.748889 − 0.662696i \(-0.230588\pi\)
0.748889 + 0.662696i \(0.230588\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) 8.00000 0.271851
\(867\) 19.0000 0.645274
\(868\) −2.00000 −0.0678844
\(869\) 0 0
\(870\) 0 0
\(871\) −8.00000 −0.271070
\(872\) 2.00000 0.0677285
\(873\) 16.0000 0.541518
\(874\) 32.0000 1.08242
\(875\) 0 0
\(876\) −8.00000 −0.270295
\(877\) −30.0000 −1.01303 −0.506514 − 0.862232i \(-0.669066\pi\)
−0.506514 + 0.862232i \(0.669066\pi\)
\(878\) 32.0000 1.07995
\(879\) −6.00000 −0.202375
\(880\) 0 0
\(881\) 18.0000 0.606435 0.303218 − 0.952921i \(-0.401939\pi\)
0.303218 + 0.952921i \(0.401939\pi\)
\(882\) −3.00000 −0.101015
\(883\) 56.0000 1.88455 0.942275 − 0.334840i \(-0.108682\pi\)
0.942275 + 0.334840i \(0.108682\pi\)
\(884\) −12.0000 −0.403604
\(885\) 0 0
\(886\) 4.00000 0.134383
\(887\) 12.0000 0.402921 0.201460 − 0.979497i \(-0.435431\pi\)
0.201460 + 0.979497i \(0.435431\pi\)
\(888\) 2.00000 0.0671156
\(889\) −4.00000 −0.134156
\(890\) 0 0
\(891\) −6.00000 −0.201008
\(892\) 14.0000 0.468755
\(893\) 64.0000 2.14168
\(894\) −12.0000 −0.401340
\(895\) 0 0
\(896\) 2.00000 0.0668153
\(897\) −8.00000 −0.267112
\(898\) 26.0000 0.867631
\(899\) 4.00000 0.133407
\(900\) 0 0
\(901\) 60.0000 1.99889
\(902\) −12.0000 −0.399556
\(903\) −8.00000 −0.266223
\(904\) 18.0000 0.598671
\(905\) 0 0
\(906\) 8.00000 0.265782
\(907\) −24.0000 −0.796907 −0.398453 − 0.917189i \(-0.630453\pi\)
−0.398453 + 0.917189i \(0.630453\pi\)
\(908\) −12.0000 −0.398234
\(909\) 0 0
\(910\) 0 0
\(911\) −16.0000 −0.530104 −0.265052 − 0.964234i \(-0.585389\pi\)
−0.265052 + 0.964234i \(0.585389\pi\)
\(912\) 8.00000 0.264906
\(913\) 24.0000 0.794284
\(914\) −12.0000 −0.396925
\(915\) 0 0
\(916\) 6.00000 0.198246
\(917\) 20.0000 0.660458
\(918\) 6.00000 0.198030
\(919\) −16.0000 −0.527791 −0.263896 − 0.964551i \(-0.585007\pi\)
−0.263896 + 0.964551i \(0.585007\pi\)
\(920\) 0 0
\(921\) −8.00000 −0.263609
\(922\) −32.0000 −1.05386
\(923\) −32.0000 −1.05329
\(924\) −12.0000 −0.394771
\(925\) 0 0
\(926\) 26.0000 0.854413
\(927\) −6.00000 −0.197066
\(928\) −4.00000 −0.131306
\(929\) 50.0000 1.64045 0.820223 − 0.572043i \(-0.193849\pi\)
0.820223 + 0.572043i \(0.193849\pi\)
\(930\) 0 0
\(931\) −24.0000 −0.786568
\(932\) −18.0000 −0.589610
\(933\) −8.00000 −0.261908
\(934\) −4.00000 −0.130884
\(935\) 0 0
\(936\) −2.00000 −0.0653720
\(937\) −16.0000 −0.522697 −0.261349 − 0.965244i \(-0.584167\pi\)
−0.261349 + 0.965244i \(0.584167\pi\)
\(938\) 8.00000 0.261209
\(939\) 24.0000 0.783210
\(940\) 0 0
\(941\) 16.0000 0.521585 0.260793 − 0.965395i \(-0.416016\pi\)
0.260793 + 0.965395i \(0.416016\pi\)
\(942\) −6.00000 −0.195491
\(943\) 8.00000 0.260516
\(944\) 6.00000 0.195283
\(945\) 0 0
\(946\) 24.0000 0.780307
\(947\) 44.0000 1.42981 0.714904 − 0.699223i \(-0.246470\pi\)
0.714904 + 0.699223i \(0.246470\pi\)
\(948\) 0 0
\(949\) 16.0000 0.519382
\(950\) 0 0
\(951\) −18.0000 −0.583690
\(952\) 12.0000 0.388922
\(953\) −38.0000 −1.23094 −0.615470 − 0.788160i \(-0.711034\pi\)
−0.615470 + 0.788160i \(0.711034\pi\)
\(954\) 10.0000 0.323762
\(955\) 0 0
\(956\) −20.0000 −0.646846
\(957\) 24.0000 0.775810
\(958\) −40.0000 −1.29234
\(959\) −28.0000 −0.904167
\(960\) 0 0
\(961\) 1.00000 0.0322581
\(962\) −4.00000 −0.128965
\(963\) −4.00000 −0.128898
\(964\) −18.0000 −0.579741
\(965\) 0 0
\(966\) 8.00000 0.257396
\(967\) 14.0000 0.450210 0.225105 − 0.974335i \(-0.427728\pi\)
0.225105 + 0.974335i \(0.427728\pi\)
\(968\) 25.0000 0.803530
\(969\) 48.0000 1.54198
\(970\) 0 0
\(971\) −30.0000 −0.962746 −0.481373 − 0.876516i \(-0.659862\pi\)
−0.481373 + 0.876516i \(0.659862\pi\)
\(972\) 1.00000 0.0320750
\(973\) −40.0000 −1.28234
\(974\) −14.0000 −0.448589
\(975\) 0 0
\(976\) −14.0000 −0.448129
\(977\) 30.0000 0.959785 0.479893 − 0.877327i \(-0.340676\pi\)
0.479893 + 0.877327i \(0.340676\pi\)
\(978\) −20.0000 −0.639529
\(979\) −60.0000 −1.91761
\(980\) 0 0
\(981\) 2.00000 0.0638551
\(982\) 30.0000 0.957338
\(983\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) 2.00000 0.0637577
\(985\) 0 0
\(986\) −24.0000 −0.764316
\(987\) 16.0000 0.509286
\(988\) −16.0000 −0.509028
\(989\) −16.0000 −0.508770
\(990\) 0 0
\(991\) 16.0000 0.508257 0.254128 − 0.967170i \(-0.418211\pi\)
0.254128 + 0.967170i \(0.418211\pi\)
\(992\) −1.00000 −0.0317500
\(993\) 8.00000 0.253872
\(994\) 32.0000 1.01498
\(995\) 0 0
\(996\) −4.00000 −0.126745
\(997\) −30.0000 −0.950110 −0.475055 − 0.879956i \(-0.657572\pi\)
−0.475055 + 0.879956i \(0.657572\pi\)
\(998\) −32.0000 −1.01294
\(999\) 2.00000 0.0632772
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 4650.2.a.bs.1.1 1
5.2 odd 4 930.2.d.e.559.2 yes 2
5.3 odd 4 930.2.d.e.559.1 ✓ 2
5.4 even 2 4650.2.a.c.1.1 1
15.2 even 4 2790.2.d.c.559.1 2
15.8 even 4 2790.2.d.c.559.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
930.2.d.e.559.1 ✓ 2 5.3 odd 4
930.2.d.e.559.2 yes 2 5.2 odd 4
2790.2.d.c.559.1 2 15.2 even 4
2790.2.d.c.559.2 2 15.8 even 4
4650.2.a.c.1.1 1 5.4 even 2
4650.2.a.bs.1.1 1 1.1 even 1 trivial