↑

Properties

Label 2550.2.a.n.1.1
Level $2550$
Weight $2$
Character 2550.1
Self dual yes
Analytic conductor $20.362$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2550,2,Mod(1,2550)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2550, 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("2550.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2550 = 2 \cdot 3 \cdot 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2550.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: \(20.3618525154\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 510)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2550.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +4.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +4.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -4.00000 q^{11} +1.00000 q^{12} +2.00000 q^{13} -4.00000 q^{14} +1.00000 q^{16} -1.00000 q^{17} -1.00000 q^{18} -4.00000 q^{19} +4.00000 q^{21} +4.00000 q^{22} +4.00000 q^{23} -1.00000 q^{24} -2.00000 q^{26} +1.00000 q^{27} +4.00000 q^{28} +2.00000 q^{29} +4.00000 q^{31} -1.00000 q^{32} -4.00000 q^{33} +1.00000 q^{34} +1.00000 q^{36} +6.00000 q^{37} +4.00000 q^{38} +2.00000 q^{39} +2.00000 q^{41} -4.00000 q^{42} +12.0000 q^{43} -4.00000 q^{44} -4.00000 q^{46} -8.00000 q^{47} +1.00000 q^{48} +9.00000 q^{49} -1.00000 q^{51} +2.00000 q^{52} +2.00000 q^{53} -1.00000 q^{54} -4.00000 q^{56} -4.00000 q^{57} -2.00000 q^{58} +12.0000 q^{59} +2.00000 q^{61} -4.00000 q^{62} +4.00000 q^{63} +1.00000 q^{64} +4.00000 q^{66} -4.00000 q^{67} -1.00000 q^{68} +4.00000 q^{69} -4.00000 q^{71} -1.00000 q^{72} +14.0000 q^{73} -6.00000 q^{74} -4.00000 q^{76} -16.0000 q^{77} -2.00000 q^{78} -12.0000 q^{79} +1.00000 q^{81} -2.00000 q^{82} +4.00000 q^{83} +4.00000 q^{84} -12.0000 q^{86} +2.00000 q^{87} +4.00000 q^{88} +10.0000 q^{89} +8.00000 q^{91} +4.00000 q^{92} +4.00000 q^{93} +8.00000 q^{94} -1.00000 q^{96} -18.0000 q^{97} -9.00000 q^{98} -4.00000 q^{99} +O(q^{100})\)

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
\(7\) 4.00000 1.51186 0.755929 − 0.654654i \(-0.227186\pi\)
0.755929 + 0.654654i \(0.227186\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −4.00000 −1.20605 −0.603023 − 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 1.00000 0.288675
\(13\) 2.00000 0.554700 0.277350 − 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) −4.00000 −1.06904
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −1.00000 −0.242536
\(18\) −1.00000 −0.235702
\(19\) −4.00000 −0.917663 −0.458831 − 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 0 0
\(21\) 4.00000 0.872872
\(22\) 4.00000 0.852803
\(23\) 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
\(28\) 4.00000 0.755929
\(29\) 2.00000 0.371391 0.185695 − 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 − 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) −1.00000 −0.176777
\(33\) −4.00000 −0.696311
\(34\) 1.00000 0.171499
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 6.00000 0.986394 0.493197 − 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) 4.00000 0.648886
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 − 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) −4.00000 −0.617213
\(43\) 12.0000 1.82998 0.914991 − 0.403473i \(-0.132197\pi\)
0.914991 + 0.403473i \(0.132197\pi\)
\(44\) −4.00000 −0.603023
\(45\) 0 0
\(46\) −4.00000 −0.589768
\(47\) −8.00000 −1.16692 −0.583460 − 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 1.00000 0.144338
\(49\) 9.00000 1.28571
\(50\) 0 0
\(51\) −1.00000 −0.140028
\(52\) 2.00000 0.277350
\(53\) 2.00000 0.274721 0.137361 − 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) −4.00000 −0.534522
\(57\) −4.00000 −0.529813
\(58\) −2.00000 −0.262613
\(59\) 12.0000 1.56227 0.781133 − 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 − 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) −4.00000 −0.508001
\(63\) 4.00000 0.503953
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 4.00000 0.492366
\(67\) −4.00000 −0.488678 −0.244339 − 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) −1.00000 −0.121268
\(69\) 4.00000 0.481543
\(70\) 0 0
\(71\) −4.00000 −0.474713 −0.237356 − 0.971423i \(-0.576281\pi\)
−0.237356 + 0.971423i \(0.576281\pi\)
\(72\) −1.00000 −0.117851
\(73\) 14.0000 1.63858 0.819288 − 0.573382i \(-0.194369\pi\)
0.819288 + 0.573382i \(0.194369\pi\)
\(74\) −6.00000 −0.697486
\(75\) 0 0
\(76\) −4.00000 −0.458831
\(77\) −16.0000 −1.82337
\(78\) −2.00000 −0.226455
\(79\) −12.0000 −1.35011 −0.675053 − 0.737769i \(-0.735879\pi\)
−0.675053 + 0.737769i \(0.735879\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −2.00000 −0.220863
\(83\) 4.00000 0.439057 0.219529 − 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 4.00000 0.436436
\(85\) 0 0
\(86\) −12.0000 −1.29399
\(87\) 2.00000 0.214423
\(88\) 4.00000 0.426401
\(89\) 10.0000 1.06000 0.529999 − 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) 8.00000 0.838628
\(92\) 4.00000 0.417029
\(93\) 4.00000 0.414781
\(94\) 8.00000 0.825137
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) −18.0000 −1.82762 −0.913812 − 0.406138i \(-0.866875\pi\)
−0.913812 + 0.406138i \(0.866875\pi\)
\(98\) −9.00000 −0.909137
\(99\) −4.00000 −0.402015
\(100\) 0 0
\(101\) −10.0000 −0.995037 −0.497519 − 0.867453i \(-0.665755\pi\)
−0.497519 + 0.867453i \(0.665755\pi\)
\(102\) 1.00000 0.0990148
\(103\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) −2.00000 −0.196116
\(105\) 0 0
\(106\) −2.00000 −0.194257
\(107\) −12.0000 −1.16008 −0.580042 − 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 1.00000 0.0962250
\(109\) 10.0000 0.957826 0.478913 − 0.877862i \(-0.341031\pi\)
0.478913 + 0.877862i \(0.341031\pi\)
\(110\) 0 0
\(111\) 6.00000 0.569495
\(112\) 4.00000 0.377964
\(113\) 6.00000 0.564433 0.282216 − 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 4.00000 0.374634
\(115\) 0 0
\(116\) 2.00000 0.185695
\(117\) 2.00000 0.184900
\(118\) −12.0000 −1.10469
\(119\) −4.00000 −0.366679
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) −2.00000 −0.181071
\(123\) 2.00000 0.180334
\(124\) 4.00000 0.359211
\(125\) 0 0
\(126\) −4.00000 −0.356348
\(127\) 16.0000 1.41977 0.709885 − 0.704317i \(-0.248747\pi\)
0.709885 + 0.704317i \(0.248747\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 12.0000 1.05654
\(130\) 0 0
\(131\) 20.0000 1.74741 0.873704 − 0.486458i \(-0.161711\pi\)
0.873704 + 0.486458i \(0.161711\pi\)
\(132\) −4.00000 −0.348155
\(133\) −16.0000 −1.38738
\(134\) 4.00000 0.345547
\(135\) 0 0
\(136\) 1.00000 0.0857493
\(137\) 6.00000 0.512615 0.256307 − 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) −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
\(142\) 4.00000 0.335673
\(143\) −8.00000 −0.668994
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −14.0000 −1.15865
\(147\) 9.00000 0.742307
\(148\) 6.00000 0.493197
\(149\) −2.00000 −0.163846 −0.0819232 − 0.996639i \(-0.526106\pi\)
−0.0819232 + 0.996639i \(0.526106\pi\)
\(150\) 0 0
\(151\) −8.00000 −0.651031 −0.325515 − 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) 4.00000 0.324443
\(153\) −1.00000 −0.0808452
\(154\) 16.0000 1.28932
\(155\) 0 0
\(156\) 2.00000 0.160128
\(157\) 10.0000 0.798087 0.399043 − 0.916932i \(-0.369342\pi\)
0.399043 + 0.916932i \(0.369342\pi\)
\(158\) 12.0000 0.954669
\(159\) 2.00000 0.158610
\(160\) 0 0
\(161\) 16.0000 1.26098
\(162\) −1.00000 −0.0785674
\(163\) 20.0000 1.56652 0.783260 − 0.621694i \(-0.213555\pi\)
0.783260 + 0.621694i \(0.213555\pi\)
\(164\) 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 \(0.653692\pi\)
\(168\) −4.00000 −0.308607
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) −4.00000 −0.305888
\(172\) 12.0000 0.914991
\(173\) 14.0000 1.06440 0.532200 − 0.846619i \(-0.321365\pi\)
0.532200 + 0.846619i \(0.321365\pi\)
\(174\) −2.00000 −0.151620
\(175\) 0 0
\(176\) −4.00000 −0.301511
\(177\) 12.0000 0.901975
\(178\) −10.0000 −0.749532
\(179\) 12.0000 0.896922 0.448461 − 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) −22.0000 −1.63525 −0.817624 − 0.575753i \(-0.804709\pi\)
−0.817624 + 0.575753i \(0.804709\pi\)
\(182\) −8.00000 −0.592999
\(183\) 2.00000 0.147844
\(184\) −4.00000 −0.294884
\(185\) 0 0
\(186\) −4.00000 −0.293294
\(187\) 4.00000 0.292509
\(188\) −8.00000 −0.583460
\(189\) 4.00000 0.290957
\(190\) 0 0
\(191\) −8.00000 −0.578860 −0.289430 − 0.957199i \(-0.593466\pi\)
−0.289430 + 0.957199i \(0.593466\pi\)
\(192\) 1.00000 0.0721688
\(193\) −18.0000 −1.29567 −0.647834 − 0.761781i \(-0.724325\pi\)
−0.647834 + 0.761781i \(0.724325\pi\)
\(194\) 18.0000 1.29232
\(195\) 0 0
\(196\) 9.00000 0.642857
\(197\) 14.0000 0.997459 0.498729 − 0.866758i \(-0.333800\pi\)
0.498729 + 0.866758i \(0.333800\pi\)
\(198\) 4.00000 0.284268
\(199\) 4.00000 0.283552 0.141776 − 0.989899i \(-0.454719\pi\)
0.141776 + 0.989899i \(0.454719\pi\)
\(200\) 0 0
\(201\) −4.00000 −0.282138
\(202\) 10.0000 0.703598
\(203\) 8.00000 0.561490
\(204\) −1.00000 −0.0700140
\(205\) 0 0
\(206\) 0 0
\(207\) 4.00000 0.278019
\(208\) 2.00000 0.138675
\(209\) 16.0000 1.10674
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 − 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) 2.00000 0.137361
\(213\) −4.00000 −0.274075
\(214\) 12.0000 0.820303
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) 16.0000 1.08615
\(218\) −10.0000 −0.677285
\(219\) 14.0000 0.946032
\(220\) 0 0
\(221\) −2.00000 −0.134535
\(222\) −6.00000 −0.402694
\(223\) 8.00000 0.535720 0.267860 − 0.963458i \(-0.413684\pi\)
0.267860 + 0.963458i \(0.413684\pi\)
\(224\) −4.00000 −0.267261
\(225\) 0 0
\(226\) −6.00000 −0.399114
\(227\) 20.0000 1.32745 0.663723 − 0.747978i \(-0.268975\pi\)
0.663723 + 0.747978i \(0.268975\pi\)
\(228\) −4.00000 −0.264906
\(229\) 6.00000 0.396491 0.198246 − 0.980152i \(-0.436476\pi\)
0.198246 + 0.980152i \(0.436476\pi\)
\(230\) 0 0
\(231\) −16.0000 −1.05272
\(232\) −2.00000 −0.131306
\(233\) −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
\(236\) 12.0000 0.781133
\(237\) −12.0000 −0.779484
\(238\) 4.00000 0.259281
\(239\) −24.0000 −1.55243 −0.776215 − 0.630468i \(-0.782863\pi\)
−0.776215 + 0.630468i \(0.782863\pi\)
\(240\) 0 0
\(241\) 10.0000 0.644157 0.322078 − 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) −5.00000 −0.321412
\(243\) 1.00000 0.0641500
\(244\) 2.00000 0.128037
\(245\) 0 0
\(246\) −2.00000 −0.127515
\(247\) −8.00000 −0.509028
\(248\) −4.00000 −0.254000
\(249\) 4.00000 0.253490
\(250\) 0 0
\(251\) −28.0000 −1.76734 −0.883672 − 0.468106i \(-0.844936\pi\)
−0.883672 + 0.468106i \(0.844936\pi\)
\(252\) 4.00000 0.251976
\(253\) −16.0000 −1.00591
\(254\) −16.0000 −1.00393
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −18.0000 −1.12281 −0.561405 − 0.827541i \(-0.689739\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(258\) −12.0000 −0.747087
\(259\) 24.0000 1.49129
\(260\) 0 0
\(261\) 2.00000 0.123797
\(262\) −20.0000 −1.23560
\(263\) 16.0000 0.986602 0.493301 − 0.869859i \(-0.335790\pi\)
0.493301 + 0.869859i \(0.335790\pi\)
\(264\) 4.00000 0.246183
\(265\) 0 0
\(266\) 16.0000 0.981023
\(267\) 10.0000 0.611990
\(268\) −4.00000 −0.244339
\(269\) −30.0000 −1.82913 −0.914566 − 0.404436i \(-0.867468\pi\)
−0.914566 + 0.404436i \(0.867468\pi\)
\(270\) 0 0
\(271\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) −1.00000 −0.0606339
\(273\) 8.00000 0.484182
\(274\) −6.00000 −0.362473
\(275\) 0 0
\(276\) 4.00000 0.240772
\(277\) −2.00000 −0.120168 −0.0600842 − 0.998193i \(-0.519137\pi\)
−0.0600842 + 0.998193i \(0.519137\pi\)
\(278\) 20.0000 1.19952
\(279\) 4.00000 0.239474
\(280\) 0 0
\(281\) 10.0000 0.596550 0.298275 − 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) 8.00000 0.476393
\(283\) 20.0000 1.18888 0.594438 − 0.804141i \(-0.297374\pi\)
0.594438 + 0.804141i \(0.297374\pi\)
\(284\) −4.00000 −0.237356
\(285\) 0 0
\(286\) 8.00000 0.473050
\(287\) 8.00000 0.472225
\(288\) −1.00000 −0.0589256
\(289\) 1.00000 0.0588235
\(290\) 0 0
\(291\) −18.0000 −1.05518
\(292\) 14.0000 0.819288
\(293\) 26.0000 1.51894 0.759468 − 0.650545i \(-0.225459\pi\)
0.759468 + 0.650545i \(0.225459\pi\)
\(294\) −9.00000 −0.524891
\(295\) 0 0
\(296\) −6.00000 −0.348743
\(297\) −4.00000 −0.232104
\(298\) 2.00000 0.115857
\(299\) 8.00000 0.462652
\(300\) 0 0
\(301\) 48.0000 2.76667
\(302\) 8.00000 0.460348
\(303\) −10.0000 −0.574485
\(304\) −4.00000 −0.229416
\(305\) 0 0
\(306\) 1.00000 0.0571662
\(307\) 4.00000 0.228292 0.114146 − 0.993464i \(-0.463587\pi\)
0.114146 + 0.993464i \(0.463587\pi\)
\(308\) −16.0000 −0.911685
\(309\) 0 0
\(310\) 0 0
\(311\) 20.0000 1.13410 0.567048 − 0.823685i \(-0.308085\pi\)
0.567048 + 0.823685i \(0.308085\pi\)
\(312\) −2.00000 −0.113228
\(313\) −10.0000 −0.565233 −0.282617 − 0.959233i \(-0.591202\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) −10.0000 −0.564333
\(315\) 0 0
\(316\) −12.0000 −0.675053
\(317\) −10.0000 −0.561656 −0.280828 − 0.959758i \(-0.590609\pi\)
−0.280828 + 0.959758i \(0.590609\pi\)
\(318\) −2.00000 −0.112154
\(319\) −8.00000 −0.447914
\(320\) 0 0
\(321\) −12.0000 −0.669775
\(322\) −16.0000 −0.891645
\(323\) 4.00000 0.222566
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −20.0000 −1.10770
\(327\) 10.0000 0.553001
\(328\) −2.00000 −0.110432
\(329\) −32.0000 −1.76422
\(330\) 0 0
\(331\) 28.0000 1.53902 0.769510 − 0.638635i \(-0.220501\pi\)
0.769510 + 0.638635i \(0.220501\pi\)
\(332\) 4.00000 0.219529
\(333\) 6.00000 0.328798
\(334\) 12.0000 0.656611
\(335\) 0 0
\(336\) 4.00000 0.218218
\(337\) 22.0000 1.19842 0.599208 − 0.800593i \(-0.295482\pi\)
0.599208 + 0.800593i \(0.295482\pi\)
\(338\) 9.00000 0.489535
\(339\) 6.00000 0.325875
\(340\) 0 0
\(341\) −16.0000 −0.866449
\(342\) 4.00000 0.216295
\(343\) 8.00000 0.431959
\(344\) −12.0000 −0.646997
\(345\) 0 0
\(346\) −14.0000 −0.752645
\(347\) −20.0000 −1.07366 −0.536828 − 0.843692i \(-0.680378\pi\)
−0.536828 + 0.843692i \(0.680378\pi\)
\(348\) 2.00000 0.107211
\(349\) 22.0000 1.17763 0.588817 − 0.808267i \(-0.299594\pi\)
0.588817 + 0.808267i \(0.299594\pi\)
\(350\) 0 0
\(351\) 2.00000 0.106752
\(352\) 4.00000 0.213201
\(353\) −2.00000 −0.106449 −0.0532246 − 0.998583i \(-0.516950\pi\)
−0.0532246 + 0.998583i \(0.516950\pi\)
\(354\) −12.0000 −0.637793
\(355\) 0 0
\(356\) 10.0000 0.529999
\(357\) −4.00000 −0.211702
\(358\) −12.0000 −0.634220
\(359\) −24.0000 −1.26667 −0.633336 − 0.773877i \(-0.718315\pi\)
−0.633336 + 0.773877i \(0.718315\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 22.0000 1.15629
\(363\) 5.00000 0.262432
\(364\) 8.00000 0.419314
\(365\) 0 0
\(366\) −2.00000 −0.104542
\(367\) −28.0000 −1.46159 −0.730794 − 0.682598i \(-0.760850\pi\)
−0.730794 + 0.682598i \(0.760850\pi\)
\(368\) 4.00000 0.208514
\(369\) 2.00000 0.104116
\(370\) 0 0
\(371\) 8.00000 0.415339
\(372\) 4.00000 0.207390
\(373\) −38.0000 −1.96757 −0.983783 − 0.179364i \(-0.942596\pi\)
−0.983783 + 0.179364i \(0.942596\pi\)
\(374\) −4.00000 −0.206835
\(375\) 0 0
\(376\) 8.00000 0.412568
\(377\) 4.00000 0.206010
\(378\) −4.00000 −0.205738
\(379\) 20.0000 1.02733 0.513665 − 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) 0 0
\(381\) 16.0000 0.819705
\(382\) 8.00000 0.409316
\(383\) −8.00000 −0.408781 −0.204390 − 0.978889i \(-0.565521\pi\)
−0.204390 + 0.978889i \(0.565521\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 18.0000 0.916176
\(387\) 12.0000 0.609994
\(388\) −18.0000 −0.913812
\(389\) −18.0000 −0.912636 −0.456318 − 0.889817i \(-0.650832\pi\)
−0.456318 + 0.889817i \(0.650832\pi\)
\(390\) 0 0
\(391\) −4.00000 −0.202289
\(392\) −9.00000 −0.454569
\(393\) 20.0000 1.00887
\(394\) −14.0000 −0.705310
\(395\) 0 0
\(396\) −4.00000 −0.201008
\(397\) 30.0000 1.50566 0.752828 − 0.658217i \(-0.228689\pi\)
0.752828 + 0.658217i \(0.228689\pi\)
\(398\) −4.00000 −0.200502
\(399\) −16.0000 −0.801002
\(400\) 0 0
\(401\) −6.00000 −0.299626 −0.149813 − 0.988714i \(-0.547867\pi\)
−0.149813 + 0.988714i \(0.547867\pi\)
\(402\) 4.00000 0.199502
\(403\) 8.00000 0.398508
\(404\) −10.0000 −0.497519
\(405\) 0 0
\(406\) −8.00000 −0.397033
\(407\) −24.0000 −1.18964
\(408\) 1.00000 0.0495074
\(409\) 10.0000 0.494468 0.247234 − 0.968956i \(-0.420478\pi\)
0.247234 + 0.968956i \(0.420478\pi\)
\(410\) 0 0
\(411\) 6.00000 0.295958
\(412\) 0 0
\(413\) 48.0000 2.36193
\(414\) −4.00000 −0.196589
\(415\) 0 0
\(416\) −2.00000 −0.0980581
\(417\) −20.0000 −0.979404
\(418\) −16.0000 −0.782586
\(419\) 12.0000 0.586238 0.293119 − 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 0 0
\(421\) −18.0000 −0.877266 −0.438633 − 0.898666i \(-0.644537\pi\)
−0.438633 + 0.898666i \(0.644537\pi\)
\(422\) −4.00000 −0.194717
\(423\) −8.00000 −0.388973
\(424\) −2.00000 −0.0971286
\(425\) 0 0
\(426\) 4.00000 0.193801
\(427\) 8.00000 0.387147
\(428\) −12.0000 −0.580042
\(429\) −8.00000 −0.386244
\(430\) 0 0
\(431\) −4.00000 −0.192673 −0.0963366 − 0.995349i \(-0.530713\pi\)
−0.0963366 + 0.995349i \(0.530713\pi\)
\(432\) 1.00000 0.0481125
\(433\) 14.0000 0.672797 0.336399 − 0.941720i \(-0.390791\pi\)
0.336399 + 0.941720i \(0.390791\pi\)
\(434\) −16.0000 −0.768025
\(435\) 0 0
\(436\) 10.0000 0.478913
\(437\) −16.0000 −0.765384
\(438\) −14.0000 −0.668946
\(439\) −12.0000 −0.572729 −0.286364 − 0.958121i \(-0.592447\pi\)
−0.286364 + 0.958121i \(0.592447\pi\)
\(440\) 0 0
\(441\) 9.00000 0.428571
\(442\) 2.00000 0.0951303
\(443\) 36.0000 1.71041 0.855206 − 0.518289i \(-0.173431\pi\)
0.855206 + 0.518289i \(0.173431\pi\)
\(444\) 6.00000 0.284747
\(445\) 0 0
\(446\) −8.00000 −0.378811
\(447\) −2.00000 −0.0945968
\(448\) 4.00000 0.188982
\(449\) −6.00000 −0.283158 −0.141579 − 0.989927i \(-0.545218\pi\)
−0.141579 + 0.989927i \(0.545218\pi\)
\(450\) 0 0
\(451\) −8.00000 −0.376705
\(452\) 6.00000 0.282216
\(453\) −8.00000 −0.375873
\(454\) −20.0000 −0.938647
\(455\) 0 0
\(456\) 4.00000 0.187317
\(457\) −10.0000 −0.467780 −0.233890 − 0.972263i \(-0.575146\pi\)
−0.233890 + 0.972263i \(0.575146\pi\)
\(458\) −6.00000 −0.280362
\(459\) −1.00000 −0.0466760
\(460\) 0 0
\(461\) 14.0000 0.652045 0.326023 − 0.945362i \(-0.394291\pi\)
0.326023 + 0.945362i \(0.394291\pi\)
\(462\) 16.0000 0.744387
\(463\) −40.0000 −1.85896 −0.929479 − 0.368875i \(-0.879743\pi\)
−0.929479 + 0.368875i \(0.879743\pi\)
\(464\) 2.00000 0.0928477
\(465\) 0 0
\(466\) 18.0000 0.833834
\(467\) −36.0000 −1.66588 −0.832941 − 0.553362i \(-0.813345\pi\)
−0.832941 + 0.553362i \(0.813345\pi\)
\(468\) 2.00000 0.0924500
\(469\) −16.0000 −0.738811
\(470\) 0 0
\(471\) 10.0000 0.460776
\(472\) −12.0000 −0.552345
\(473\) −48.0000 −2.20704
\(474\) 12.0000 0.551178
\(475\) 0 0
\(476\) −4.00000 −0.183340
\(477\) 2.00000 0.0915737
\(478\) 24.0000 1.09773
\(479\) 20.0000 0.913823 0.456912 − 0.889512i \(-0.348956\pi\)
0.456912 + 0.889512i \(0.348956\pi\)
\(480\) 0 0
\(481\) 12.0000 0.547153
\(482\) −10.0000 −0.455488
\(483\) 16.0000 0.728025
\(484\) 5.00000 0.227273
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) −20.0000 −0.906287 −0.453143 − 0.891438i \(-0.649697\pi\)
−0.453143 + 0.891438i \(0.649697\pi\)
\(488\) −2.00000 −0.0905357
\(489\) 20.0000 0.904431
\(490\) 0 0
\(491\) −12.0000 −0.541552 −0.270776 − 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) 2.00000 0.0901670
\(493\) −2.00000 −0.0900755
\(494\) 8.00000 0.359937
\(495\) 0 0
\(496\) 4.00000 0.179605
\(497\) −16.0000 −0.717698
\(498\) −4.00000 −0.179244
\(499\) 4.00000 0.179065 0.0895323 − 0.995984i \(-0.471463\pi\)
0.0895323 + 0.995984i \(0.471463\pi\)
\(500\) 0 0
\(501\) −12.0000 −0.536120
\(502\) 28.0000 1.24970
\(503\) −44.0000 −1.96186 −0.980932 − 0.194354i \(-0.937739\pi\)
−0.980932 + 0.194354i \(0.937739\pi\)
\(504\) −4.00000 −0.178174
\(505\) 0 0
\(506\) 16.0000 0.711287
\(507\) −9.00000 −0.399704
\(508\) 16.0000 0.709885
\(509\) 38.0000 1.68432 0.842160 − 0.539227i \(-0.181284\pi\)
0.842160 + 0.539227i \(0.181284\pi\)
\(510\) 0 0
\(511\) 56.0000 2.47729
\(512\) −1.00000 −0.0441942
\(513\) −4.00000 −0.176604
\(514\) 18.0000 0.793946
\(515\) 0 0
\(516\) 12.0000 0.528271
\(517\) 32.0000 1.40736
\(518\) −24.0000 −1.05450
\(519\) 14.0000 0.614532
\(520\) 0 0
\(521\) 18.0000 0.788594 0.394297 − 0.918983i \(-0.370988\pi\)
0.394297 + 0.918983i \(0.370988\pi\)
\(522\) −2.00000 −0.0875376
\(523\) −28.0000 −1.22435 −0.612177 − 0.790721i \(-0.709706\pi\)
−0.612177 + 0.790721i \(0.709706\pi\)
\(524\) 20.0000 0.873704
\(525\) 0 0
\(526\) −16.0000 −0.697633
\(527\) −4.00000 −0.174243
\(528\) −4.00000 −0.174078
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 12.0000 0.520756
\(532\) −16.0000 −0.693688
\(533\) 4.00000 0.173259
\(534\) −10.0000 −0.432742
\(535\) 0 0
\(536\) 4.00000 0.172774
\(537\) 12.0000 0.517838
\(538\) 30.0000 1.29339
\(539\) −36.0000 −1.55063
\(540\) 0 0
\(541\) 18.0000 0.773880 0.386940 − 0.922105i \(-0.373532\pi\)
0.386940 + 0.922105i \(0.373532\pi\)
\(542\) 0 0
\(543\) −22.0000 −0.944110
\(544\) 1.00000 0.0428746
\(545\) 0 0
\(546\) −8.00000 −0.342368
\(547\) −20.0000 −0.855138 −0.427569 − 0.903983i \(-0.640630\pi\)
−0.427569 + 0.903983i \(0.640630\pi\)
\(548\) 6.00000 0.256307
\(549\) 2.00000 0.0853579
\(550\) 0 0
\(551\) −8.00000 −0.340811
\(552\) −4.00000 −0.170251
\(553\) −48.0000 −2.04117
\(554\) 2.00000 0.0849719
\(555\) 0 0
\(556\) −20.0000 −0.848189
\(557\) 26.0000 1.10166 0.550828 − 0.834619i \(-0.314312\pi\)
0.550828 + 0.834619i \(0.314312\pi\)
\(558\) −4.00000 −0.169334
\(559\) 24.0000 1.01509
\(560\) 0 0
\(561\) 4.00000 0.168880
\(562\) −10.0000 −0.421825
\(563\) 36.0000 1.51722 0.758610 − 0.651546i \(-0.225879\pi\)
0.758610 + 0.651546i \(0.225879\pi\)
\(564\) −8.00000 −0.336861
\(565\) 0 0
\(566\) −20.0000 −0.840663
\(567\) 4.00000 0.167984
\(568\) 4.00000 0.167836
\(569\) −6.00000 −0.251533 −0.125767 − 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) 0 0
\(571\) −44.0000 −1.84134 −0.920671 − 0.390339i \(-0.872358\pi\)
−0.920671 + 0.390339i \(0.872358\pi\)
\(572\) −8.00000 −0.334497
\(573\) −8.00000 −0.334205
\(574\) −8.00000 −0.333914
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) −18.0000 −0.749350 −0.374675 − 0.927156i \(-0.622246\pi\)
−0.374675 + 0.927156i \(0.622246\pi\)
\(578\) −1.00000 −0.0415945
\(579\) −18.0000 −0.748054
\(580\) 0 0
\(581\) 16.0000 0.663792
\(582\) 18.0000 0.746124
\(583\) −8.00000 −0.331326
\(584\) −14.0000 −0.579324
\(585\) 0 0
\(586\) −26.0000 −1.07405
\(587\) −44.0000 −1.81607 −0.908037 − 0.418890i \(-0.862419\pi\)
−0.908037 + 0.418890i \(0.862419\pi\)
\(588\) 9.00000 0.371154
\(589\) −16.0000 −0.659269
\(590\) 0 0
\(591\) 14.0000 0.575883
\(592\) 6.00000 0.246598
\(593\) −18.0000 −0.739171 −0.369586 − 0.929197i \(-0.620500\pi\)
−0.369586 + 0.929197i \(0.620500\pi\)
\(594\) 4.00000 0.164122
\(595\) 0 0
\(596\) −2.00000 −0.0819232
\(597\) 4.00000 0.163709
\(598\) −8.00000 −0.327144
\(599\) 24.0000 0.980613 0.490307 − 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) 0 0
\(601\) −6.00000 −0.244745 −0.122373 − 0.992484i \(-0.539050\pi\)
−0.122373 + 0.992484i \(0.539050\pi\)
\(602\) −48.0000 −1.95633
\(603\) −4.00000 −0.162893
\(604\) −8.00000 −0.325515
\(605\) 0 0
\(606\) 10.0000 0.406222
\(607\) −28.0000 −1.13648 −0.568242 − 0.822861i \(-0.692376\pi\)
−0.568242 + 0.822861i \(0.692376\pi\)
\(608\) 4.00000 0.162221
\(609\) 8.00000 0.324176
\(610\) 0 0
\(611\) −16.0000 −0.647291
\(612\) −1.00000 −0.0404226
\(613\) 2.00000 0.0807792 0.0403896 − 0.999184i \(-0.487140\pi\)
0.0403896 + 0.999184i \(0.487140\pi\)
\(614\) −4.00000 −0.161427
\(615\) 0 0
\(616\) 16.0000 0.644658
\(617\) 30.0000 1.20775 0.603877 − 0.797077i \(-0.293622\pi\)
0.603877 + 0.797077i \(0.293622\pi\)
\(618\) 0 0
\(619\) −28.0000 −1.12542 −0.562708 − 0.826656i \(-0.690240\pi\)
−0.562708 + 0.826656i \(0.690240\pi\)
\(620\) 0 0
\(621\) 4.00000 0.160514
\(622\) −20.0000 −0.801927
\(623\) 40.0000 1.60257
\(624\) 2.00000 0.0800641
\(625\) 0 0
\(626\) 10.0000 0.399680
\(627\) 16.0000 0.638978
\(628\) 10.0000 0.399043
\(629\) −6.00000 −0.239236
\(630\) 0 0
\(631\) −16.0000 −0.636950 −0.318475 − 0.947931i \(-0.603171\pi\)
−0.318475 + 0.947931i \(0.603171\pi\)
\(632\) 12.0000 0.477334
\(633\) 4.00000 0.158986
\(634\) 10.0000 0.397151
\(635\) 0 0
\(636\) 2.00000 0.0793052
\(637\) 18.0000 0.713186
\(638\) 8.00000 0.316723
\(639\) −4.00000 −0.158238
\(640\) 0 0
\(641\) −6.00000 −0.236986 −0.118493 − 0.992955i \(-0.537806\pi\)
−0.118493 + 0.992955i \(0.537806\pi\)
\(642\) 12.0000 0.473602
\(643\) −28.0000 −1.10421 −0.552106 − 0.833774i \(-0.686176\pi\)
−0.552106 + 0.833774i \(0.686176\pi\)
\(644\) 16.0000 0.630488
\(645\) 0 0
\(646\) −4.00000 −0.157378
\(647\) 16.0000 0.629025 0.314512 − 0.949253i \(-0.398159\pi\)
0.314512 + 0.949253i \(0.398159\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −48.0000 −1.88416
\(650\) 0 0
\(651\) 16.0000 0.627089
\(652\) 20.0000 0.783260
\(653\) 30.0000 1.17399 0.586995 − 0.809590i \(-0.300311\pi\)
0.586995 + 0.809590i \(0.300311\pi\)
\(654\) −10.0000 −0.391031
\(655\) 0 0
\(656\) 2.00000 0.0780869
\(657\) 14.0000 0.546192
\(658\) 32.0000 1.24749
\(659\) −28.0000 −1.09073 −0.545363 − 0.838200i \(-0.683608\pi\)
−0.545363 + 0.838200i \(0.683608\pi\)
\(660\) 0 0
\(661\) 14.0000 0.544537 0.272268 − 0.962221i \(-0.412226\pi\)
0.272268 + 0.962221i \(0.412226\pi\)
\(662\) −28.0000 −1.08825
\(663\) −2.00000 −0.0776736
\(664\) −4.00000 −0.155230
\(665\) 0 0
\(666\) −6.00000 −0.232495
\(667\) 8.00000 0.309761
\(668\) −12.0000 −0.464294
\(669\) 8.00000 0.309298
\(670\) 0 0
\(671\) −8.00000 −0.308837
\(672\) −4.00000 −0.154303
\(673\) −34.0000 −1.31060 −0.655302 − 0.755367i \(-0.727459\pi\)
−0.655302 + 0.755367i \(0.727459\pi\)
\(674\) −22.0000 −0.847408
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) −18.0000 −0.691796 −0.345898 − 0.938272i \(-0.612426\pi\)
−0.345898 + 0.938272i \(0.612426\pi\)
\(678\) −6.00000 −0.230429
\(679\) −72.0000 −2.76311
\(680\) 0 0
\(681\) 20.0000 0.766402
\(682\) 16.0000 0.612672
\(683\) −36.0000 −1.37750 −0.688751 − 0.724998i \(-0.741841\pi\)
−0.688751 + 0.724998i \(0.741841\pi\)
\(684\) −4.00000 −0.152944
\(685\) 0 0
\(686\) −8.00000 −0.305441
\(687\) 6.00000 0.228914
\(688\) 12.0000 0.457496
\(689\) 4.00000 0.152388
\(690\) 0 0
\(691\) −44.0000 −1.67384 −0.836919 − 0.547326i \(-0.815646\pi\)
−0.836919 + 0.547326i \(0.815646\pi\)
\(692\) 14.0000 0.532200
\(693\) −16.0000 −0.607790
\(694\) 20.0000 0.759190
\(695\) 0 0
\(696\) −2.00000 −0.0758098
\(697\) −2.00000 −0.0757554
\(698\) −22.0000 −0.832712
\(699\) −18.0000 −0.680823
\(700\) 0 0
\(701\) −18.0000 −0.679851 −0.339925 − 0.940452i \(-0.610402\pi\)
−0.339925 + 0.940452i \(0.610402\pi\)
\(702\) −2.00000 −0.0754851
\(703\) −24.0000 −0.905177
\(704\) −4.00000 −0.150756
\(705\) 0 0
\(706\) 2.00000 0.0752710
\(707\) −40.0000 −1.50435
\(708\) 12.0000 0.450988
\(709\) −14.0000 −0.525781 −0.262891 − 0.964826i \(-0.584676\pi\)
−0.262891 + 0.964826i \(0.584676\pi\)
\(710\) 0 0
\(711\) −12.0000 −0.450035
\(712\) −10.0000 −0.374766
\(713\) 16.0000 0.599205
\(714\) 4.00000 0.149696
\(715\) 0 0
\(716\) 12.0000 0.448461
\(717\) −24.0000 −0.896296
\(718\) 24.0000 0.895672
\(719\) −44.0000 −1.64092 −0.820462 − 0.571702i \(-0.806283\pi\)
−0.820462 + 0.571702i \(0.806283\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 3.00000 0.111648
\(723\) 10.0000 0.371904
\(724\) −22.0000 −0.817624
\(725\) 0 0
\(726\) −5.00000 −0.185567
\(727\) −8.00000 −0.296704 −0.148352 − 0.988935i \(-0.547397\pi\)
−0.148352 + 0.988935i \(0.547397\pi\)
\(728\) −8.00000 −0.296500
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −12.0000 −0.443836
\(732\) 2.00000 0.0739221
\(733\) 18.0000 0.664845 0.332423 − 0.943131i \(-0.392134\pi\)
0.332423 + 0.943131i \(0.392134\pi\)
\(734\) 28.0000 1.03350
\(735\) 0 0
\(736\) −4.00000 −0.147442
\(737\) 16.0000 0.589368
\(738\) −2.00000 −0.0736210
\(739\) 12.0000 0.441427 0.220714 − 0.975339i \(-0.429161\pi\)
0.220714 + 0.975339i \(0.429161\pi\)
\(740\) 0 0
\(741\) −8.00000 −0.293887
\(742\) −8.00000 −0.293689
\(743\) −28.0000 −1.02722 −0.513610 − 0.858024i \(-0.671692\pi\)
−0.513610 + 0.858024i \(0.671692\pi\)
\(744\) −4.00000 −0.146647
\(745\) 0 0
\(746\) 38.0000 1.39128
\(747\) 4.00000 0.146352
\(748\) 4.00000 0.146254
\(749\) −48.0000 −1.75388
\(750\) 0 0
\(751\) 12.0000 0.437886 0.218943 − 0.975738i \(-0.429739\pi\)
0.218943 + 0.975738i \(0.429739\pi\)
\(752\) −8.00000 −0.291730
\(753\) −28.0000 −1.02038
\(754\) −4.00000 −0.145671
\(755\) 0 0
\(756\) 4.00000 0.145479
\(757\) 34.0000 1.23575 0.617876 − 0.786276i \(-0.287994\pi\)
0.617876 + 0.786276i \(0.287994\pi\)
\(758\) −20.0000 −0.726433
\(759\) −16.0000 −0.580763
\(760\) 0 0
\(761\) −6.00000 −0.217500 −0.108750 − 0.994069i \(-0.534685\pi\)
−0.108750 + 0.994069i \(0.534685\pi\)
\(762\) −16.0000 −0.579619
\(763\) 40.0000 1.44810
\(764\) −8.00000 −0.289430
\(765\) 0 0
\(766\) 8.00000 0.289052
\(767\) 24.0000 0.866590
\(768\) 1.00000 0.0360844
\(769\) −46.0000 −1.65880 −0.829401 − 0.558653i \(-0.811318\pi\)
−0.829401 + 0.558653i \(0.811318\pi\)
\(770\) 0 0
\(771\) −18.0000 −0.648254
\(772\) −18.0000 −0.647834
\(773\) −6.00000 −0.215805 −0.107903 − 0.994161i \(-0.534413\pi\)
−0.107903 + 0.994161i \(0.534413\pi\)
\(774\) −12.0000 −0.431331
\(775\) 0 0
\(776\) 18.0000 0.646162
\(777\) 24.0000 0.860995
\(778\) 18.0000 0.645331
\(779\) −8.00000 −0.286630
\(780\) 0 0
\(781\) 16.0000 0.572525
\(782\) 4.00000 0.143040
\(783\) 2.00000 0.0714742
\(784\) 9.00000 0.321429
\(785\) 0 0
\(786\) −20.0000 −0.713376
\(787\) −36.0000 −1.28326 −0.641631 − 0.767014i \(-0.721742\pi\)
−0.641631 + 0.767014i \(0.721742\pi\)
\(788\) 14.0000 0.498729
\(789\) 16.0000 0.569615
\(790\) 0 0
\(791\) 24.0000 0.853342
\(792\) 4.00000 0.142134
\(793\) 4.00000 0.142044
\(794\) −30.0000 −1.06466
\(795\) 0 0
\(796\) 4.00000 0.141776
\(797\) 26.0000 0.920967 0.460484 − 0.887668i \(-0.347676\pi\)
0.460484 + 0.887668i \(0.347676\pi\)
\(798\) 16.0000 0.566394
\(799\) 8.00000 0.283020
\(800\) 0 0
\(801\) 10.0000 0.353333
\(802\) 6.00000 0.211867
\(803\) −56.0000 −1.97620
\(804\) −4.00000 −0.141069
\(805\) 0 0
\(806\) −8.00000 −0.281788
\(807\) −30.0000 −1.05605
\(808\) 10.0000 0.351799
\(809\) 18.0000 0.632846 0.316423 − 0.948618i \(-0.397518\pi\)
0.316423 + 0.948618i \(0.397518\pi\)
\(810\) 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
\(814\) 24.0000 0.841200
\(815\) 0 0
\(816\) −1.00000 −0.0350070
\(817\) −48.0000 −1.67931
\(818\) −10.0000 −0.349642
\(819\) 8.00000 0.279543
\(820\) 0 0
\(821\) −30.0000 −1.04701 −0.523504 − 0.852023i \(-0.675375\pi\)
−0.523504 + 0.852023i \(0.675375\pi\)
\(822\) −6.00000 −0.209274
\(823\) 28.0000 0.976019 0.488009 − 0.872838i \(-0.337723\pi\)
0.488009 + 0.872838i \(0.337723\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) −48.0000 −1.67013
\(827\) −44.0000 −1.53003 −0.765015 − 0.644013i \(-0.777268\pi\)
−0.765015 + 0.644013i \(0.777268\pi\)
\(828\) 4.00000 0.139010
\(829\) −34.0000 −1.18087 −0.590434 − 0.807086i \(-0.701044\pi\)
−0.590434 + 0.807086i \(0.701044\pi\)
\(830\) 0 0
\(831\) −2.00000 −0.0693792
\(832\) 2.00000 0.0693375
\(833\) −9.00000 −0.311832
\(834\) 20.0000 0.692543
\(835\) 0 0
\(836\) 16.0000 0.553372
\(837\) 4.00000 0.138260
\(838\) −12.0000 −0.414533
\(839\) −36.0000 −1.24286 −0.621429 − 0.783470i \(-0.713448\pi\)
−0.621429 + 0.783470i \(0.713448\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 18.0000 0.620321
\(843\) 10.0000 0.344418
\(844\) 4.00000 0.137686
\(845\) 0 0
\(846\) 8.00000 0.275046
\(847\) 20.0000 0.687208
\(848\) 2.00000 0.0686803
\(849\) 20.0000 0.686398
\(850\) 0 0
\(851\) 24.0000 0.822709
\(852\) −4.00000 −0.137038
\(853\) 46.0000 1.57501 0.787505 − 0.616308i \(-0.211372\pi\)
0.787505 + 0.616308i \(0.211372\pi\)
\(854\) −8.00000 −0.273754
\(855\) 0 0
\(856\) 12.0000 0.410152
\(857\) 22.0000 0.751506 0.375753 − 0.926720i \(-0.377384\pi\)
0.375753 + 0.926720i \(0.377384\pi\)
\(858\) 8.00000 0.273115
\(859\) 28.0000 0.955348 0.477674 − 0.878537i \(-0.341480\pi\)
0.477674 + 0.878537i \(0.341480\pi\)
\(860\) 0 0
\(861\) 8.00000 0.272639
\(862\) 4.00000 0.136241
\(863\) −32.0000 −1.08929 −0.544646 − 0.838666i \(-0.683336\pi\)
−0.544646 + 0.838666i \(0.683336\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) −14.0000 −0.475739
\(867\) 1.00000 0.0339618
\(868\) 16.0000 0.543075
\(869\) 48.0000 1.62829
\(870\) 0 0
\(871\) −8.00000 −0.271070
\(872\) −10.0000 −0.338643
\(873\) −18.0000 −0.609208
\(874\) 16.0000 0.541208
\(875\) 0 0
\(876\) 14.0000 0.473016
\(877\) 22.0000 0.742887 0.371444 − 0.928456i \(-0.378863\pi\)
0.371444 + 0.928456i \(0.378863\pi\)
\(878\) 12.0000 0.404980
\(879\) 26.0000 0.876958
\(880\) 0 0
\(881\) −30.0000 −1.01073 −0.505363 − 0.862907i \(-0.668641\pi\)
−0.505363 + 0.862907i \(0.668641\pi\)
\(882\) −9.00000 −0.303046
\(883\) 28.0000 0.942275 0.471138 − 0.882060i \(-0.343844\pi\)
0.471138 + 0.882060i \(0.343844\pi\)
\(884\) −2.00000 −0.0672673
\(885\) 0 0
\(886\) −36.0000 −1.20944
\(887\) 12.0000 0.402921 0.201460 − 0.979497i \(-0.435431\pi\)
0.201460 + 0.979497i \(0.435431\pi\)
\(888\) −6.00000 −0.201347
\(889\) 64.0000 2.14649
\(890\) 0 0
\(891\) −4.00000 −0.134005
\(892\) 8.00000 0.267860
\(893\) 32.0000 1.07084
\(894\) 2.00000 0.0668900
\(895\) 0 0
\(896\) −4.00000 −0.133631
\(897\) 8.00000 0.267112
\(898\) 6.00000 0.200223
\(899\) 8.00000 0.266815
\(900\) 0 0
\(901\) −2.00000 −0.0666297
\(902\) 8.00000 0.266371
\(903\) 48.0000 1.59734
\(904\) −6.00000 −0.199557
\(905\) 0 0
\(906\) 8.00000 0.265782
\(907\) −28.0000 −0.929725 −0.464862 − 0.885383i \(-0.653896\pi\)
−0.464862 + 0.885383i \(0.653896\pi\)
\(908\) 20.0000 0.663723
\(909\) −10.0000 −0.331679
\(910\) 0 0
\(911\) 20.0000 0.662630 0.331315 − 0.943520i \(-0.392508\pi\)
0.331315 + 0.943520i \(0.392508\pi\)
\(912\) −4.00000 −0.132453
\(913\) −16.0000 −0.529523
\(914\) 10.0000 0.330771
\(915\) 0 0
\(916\) 6.00000 0.198246
\(917\) 80.0000 2.64183
\(918\) 1.00000 0.0330049
\(919\) −56.0000 −1.84727 −0.923635 − 0.383274i \(-0.874797\pi\)
−0.923635 + 0.383274i \(0.874797\pi\)
\(920\) 0 0
\(921\) 4.00000 0.131804
\(922\) −14.0000 −0.461065
\(923\) −8.00000 −0.263323
\(924\) −16.0000 −0.526361
\(925\) 0 0
\(926\) 40.0000 1.31448
\(927\) 0 0
\(928\) −2.00000 −0.0656532
\(929\) −6.00000 −0.196854 −0.0984268 − 0.995144i \(-0.531381\pi\)
−0.0984268 + 0.995144i \(0.531381\pi\)
\(930\) 0 0
\(931\) −36.0000 −1.17985
\(932\) −18.0000 −0.589610
\(933\) 20.0000 0.654771
\(934\) 36.0000 1.17796
\(935\) 0 0
\(936\) −2.00000 −0.0653720
\(937\) 22.0000 0.718709 0.359354 − 0.933201i \(-0.382997\pi\)
0.359354 + 0.933201i \(0.382997\pi\)
\(938\) 16.0000 0.522419
\(939\) −10.0000 −0.326338
\(940\) 0 0
\(941\) 10.0000 0.325991 0.162995 − 0.986627i \(-0.447884\pi\)
0.162995 + 0.986627i \(0.447884\pi\)
\(942\) −10.0000 −0.325818
\(943\) 8.00000 0.260516
\(944\) 12.0000 0.390567
\(945\) 0 0
\(946\) 48.0000 1.56061
\(947\) 28.0000 0.909878 0.454939 − 0.890523i \(-0.349661\pi\)
0.454939 + 0.890523i \(0.349661\pi\)
\(948\) −12.0000 −0.389742
\(949\) 28.0000 0.908918
\(950\) 0 0
\(951\) −10.0000 −0.324272
\(952\) 4.00000 0.129641
\(953\) 6.00000 0.194359 0.0971795 − 0.995267i \(-0.469018\pi\)
0.0971795 + 0.995267i \(0.469018\pi\)
\(954\) −2.00000 −0.0647524
\(955\) 0 0
\(956\) −24.0000 −0.776215
\(957\) −8.00000 −0.258603
\(958\) −20.0000 −0.646171
\(959\) 24.0000 0.775000
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) −12.0000 −0.386896
\(963\) −12.0000 −0.386695
\(964\) 10.0000 0.322078
\(965\) 0 0
\(966\) −16.0000 −0.514792
\(967\) 16.0000 0.514525 0.257263 − 0.966342i \(-0.417179\pi\)
0.257263 + 0.966342i \(0.417179\pi\)
\(968\) −5.00000 −0.160706
\(969\) 4.00000 0.128499
\(970\) 0 0
\(971\) 28.0000 0.898563 0.449281 − 0.893390i \(-0.351680\pi\)
0.449281 + 0.893390i \(0.351680\pi\)
\(972\) 1.00000 0.0320750
\(973\) −80.0000 −2.56468
\(974\) 20.0000 0.640841
\(975\) 0 0
\(976\) 2.00000 0.0640184
\(977\) 30.0000 0.959785 0.479893 − 0.877327i \(-0.340676\pi\)
0.479893 + 0.877327i \(0.340676\pi\)
\(978\) −20.0000 −0.639529
\(979\) −40.0000 −1.27841
\(980\) 0 0
\(981\) 10.0000 0.319275
\(982\) 12.0000 0.382935
\(983\) 28.0000 0.893061 0.446531 − 0.894768i \(-0.352659\pi\)
0.446531 + 0.894768i \(0.352659\pi\)
\(984\) −2.00000 −0.0637577
\(985\) 0 0
\(986\) 2.00000 0.0636930
\(987\) −32.0000 −1.01857
\(988\) −8.00000 −0.254514
\(989\) 48.0000 1.52631
\(990\) 0 0
\(991\) −4.00000 −0.127064 −0.0635321 − 0.997980i \(-0.520237\pi\)
−0.0635321 + 0.997980i \(0.520237\pi\)
\(992\) −4.00000 −0.127000
\(993\) 28.0000 0.888553
\(994\) 16.0000 0.507489
\(995\) 0 0
\(996\) 4.00000 0.126745
\(997\) 38.0000 1.20347 0.601736 − 0.798695i \(-0.294476\pi\)
0.601736 + 0.798695i \(0.294476\pi\)
\(998\) −4.00000 −0.126618
\(999\) 6.00000 0.189832
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 2550.2.a.n.1.1 1
3.2 odd 2 7650.2.a.cn.1.1 1
5.2 odd 4 2550.2.d.b.2449.1 2
5.3 odd 4 2550.2.d.b.2449.2 2
5.4 even 2 510.2.a.c.1.1 ✓ 1
15.14 odd 2 1530.2.a.d.1.1 1
20.19 odd 2 4080.2.a.x.1.1 1
85.84 even 2 8670.2.a.bb.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
510.2.a.c.1.1 ✓ 1 5.4 even 2
1530.2.a.d.1.1 1 15.14 odd 2
2550.2.a.n.1.1 1 1.1 even 1 trivial
2550.2.d.b.2449.1 2 5.2 odd 4
2550.2.d.b.2449.2 2 5.3 odd 4
4080.2.a.x.1.1 1 20.19 odd 2
7650.2.a.cn.1.1 1 3.2 odd 2
8670.2.a.bb.1.1 1 85.84 even 2