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Properties

Label 6018.2.a.l.1.1
Level $6018$
Weight $2$
Character 6018.1
Self dual yes
Analytic conductor $48.054$
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] = [6018,2,Mod(1,6018)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6018, 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("6018.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6018 = 2 \cdot 3 \cdot 17 \cdot 59 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6018.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: \(48.0539719364\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 6018.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{6} -4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{6} -4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{10} +4.00000 q^{11} +1.00000 q^{12} -2.00000 q^{13} -4.00000 q^{14} +2.00000 q^{15} +1.00000 q^{16} +1.00000 q^{17} +1.00000 q^{18} -4.00000 q^{19} +2.00000 q^{20} -4.00000 q^{21} +4.00000 q^{22} +4.00000 q^{23} +1.00000 q^{24} -1.00000 q^{25} -2.00000 q^{26} +1.00000 q^{27} -4.00000 q^{28} +10.0000 q^{29} +2.00000 q^{30} +4.00000 q^{31} +1.00000 q^{32} +4.00000 q^{33} +1.00000 q^{34} -8.00000 q^{35} +1.00000 q^{36} +2.00000 q^{37} -4.00000 q^{38} -2.00000 q^{39} +2.00000 q^{40} -6.00000 q^{41} -4.00000 q^{42} +12.0000 q^{43} +4.00000 q^{44} +2.00000 q^{45} +4.00000 q^{46} -8.00000 q^{47} +1.00000 q^{48} +9.00000 q^{49} -1.00000 q^{50} +1.00000 q^{51} -2.00000 q^{52} +6.00000 q^{53} +1.00000 q^{54} +8.00000 q^{55} -4.00000 q^{56} -4.00000 q^{57} +10.0000 q^{58} -1.00000 q^{59} +2.00000 q^{60} +10.0000 q^{61} +4.00000 q^{62} -4.00000 q^{63} +1.00000 q^{64} -4.00000 q^{65} +4.00000 q^{66} -4.00000 q^{67} +1.00000 q^{68} +4.00000 q^{69} -8.00000 q^{70} -12.0000 q^{71} +1.00000 q^{72} +10.0000 q^{73} +2.00000 q^{74} -1.00000 q^{75} -4.00000 q^{76} -16.0000 q^{77} -2.00000 q^{78} +12.0000 q^{79} +2.00000 q^{80} +1.00000 q^{81} -6.00000 q^{82} +12.0000 q^{83} -4.00000 q^{84} +2.00000 q^{85} +12.0000 q^{86} +10.0000 q^{87} +4.00000 q^{88} +10.0000 q^{89} +2.00000 q^{90} +8.00000 q^{91} +4.00000 q^{92} +4.00000 q^{93} -8.00000 q^{94} -8.00000 q^{95} +1.00000 q^{96} +10.0000 q^{97} +9.00000 q^{98} +4.00000 q^{99} +O(q^{100})\)

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
\(5\) 2.00000 0.894427 0.447214 − 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 1.00000 0.408248
\(7\) −4.00000 −1.51186 −0.755929 − 0.654654i \(-0.772814\pi\)
−0.755929 + 0.654654i \(0.772814\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 2.00000 0.632456
\(11\) 4.00000 1.20605 0.603023 − 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 1.00000 0.288675
\(13\) −2.00000 −0.554700 −0.277350 − 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) −4.00000 −1.06904
\(15\) 2.00000 0.516398
\(16\) 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 \(0.651732\pi\)
\(20\) 2.00000 0.447214
\(21\) −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
\(25\) −1.00000 −0.200000
\(26\) −2.00000 −0.392232
\(27\) 1.00000 0.192450
\(28\) −4.00000 −0.755929
\(29\) 10.0000 1.85695 0.928477 − 0.371391i \(-0.121119\pi\)
0.928477 + 0.371391i \(0.121119\pi\)
\(30\) 2.00000 0.365148
\(31\) 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
\(35\) −8.00000 −1.35225
\(36\) 1.00000 0.166667
\(37\) 2.00000 0.328798 0.164399 − 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) −4.00000 −0.648886
\(39\) −2.00000 −0.320256
\(40\) 2.00000 0.316228
\(41\) −6.00000 −0.937043 −0.468521 − 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) −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
\(45\) 2.00000 0.298142
\(46\) 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
\(50\) −1.00000 −0.141421
\(51\) 1.00000 0.140028
\(52\) −2.00000 −0.277350
\(53\) 6.00000 0.824163 0.412082 − 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 1.00000 0.136083
\(55\) 8.00000 1.07872
\(56\) −4.00000 −0.534522
\(57\) −4.00000 −0.529813
\(58\) 10.0000 1.31306
\(59\) −1.00000 −0.130189
\(60\) 2.00000 0.258199
\(61\) 10.0000 1.28037 0.640184 − 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 4.00000 0.508001
\(63\) −4.00000 −0.503953
\(64\) 1.00000 0.125000
\(65\) −4.00000 −0.496139
\(66\) 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
\(70\) −8.00000 −0.956183
\(71\) −12.0000 −1.42414 −0.712069 − 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 1.00000 0.117851
\(73\) 10.0000 1.17041 0.585206 − 0.810885i \(-0.301014\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) 2.00000 0.232495
\(75\) −1.00000 −0.115470
\(76\) −4.00000 −0.458831
\(77\) −16.0000 −1.82337
\(78\) −2.00000 −0.226455
\(79\) 12.0000 1.35011 0.675053 − 0.737769i \(-0.264121\pi\)
0.675053 + 0.737769i \(0.264121\pi\)
\(80\) 2.00000 0.223607
\(81\) 1.00000 0.111111
\(82\) −6.00000 −0.662589
\(83\) 12.0000 1.31717 0.658586 − 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) −4.00000 −0.436436
\(85\) 2.00000 0.216930
\(86\) 12.0000 1.29399
\(87\) 10.0000 1.07211
\(88\) 4.00000 0.426401
\(89\) 10.0000 1.06000 0.529999 − 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 2.00000 0.210819
\(91\) 8.00000 0.838628
\(92\) 4.00000 0.417029
\(93\) 4.00000 0.414781
\(94\) −8.00000 −0.825137
\(95\) −8.00000 −0.820783
\(96\) 1.00000 0.102062
\(97\) 10.0000 1.01535 0.507673 − 0.861550i \(-0.330506\pi\)
0.507673 + 0.861550i \(0.330506\pi\)
\(98\) 9.00000 0.909137
\(99\) 4.00000 0.402015
\(100\) −1.00000 −0.100000
\(101\) −18.0000 −1.79107 −0.895533 − 0.444994i \(-0.853206\pi\)
−0.895533 + 0.444994i \(0.853206\pi\)
\(102\) 1.00000 0.0990148
\(103\) 8.00000 0.788263 0.394132 − 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) −2.00000 −0.196116
\(105\) −8.00000 −0.780720
\(106\) 6.00000 0.582772
\(107\) −12.0000 −1.16008 −0.580042 − 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 1.00000 0.0962250
\(109\) 2.00000 0.191565 0.0957826 − 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 8.00000 0.762770
\(111\) 2.00000 0.189832
\(112\) −4.00000 −0.377964
\(113\) 10.0000 0.940721 0.470360 − 0.882474i \(-0.344124\pi\)
0.470360 + 0.882474i \(0.344124\pi\)
\(114\) −4.00000 −0.374634
\(115\) 8.00000 0.746004
\(116\) 10.0000 0.928477
\(117\) −2.00000 −0.184900
\(118\) −1.00000 −0.0920575
\(119\) −4.00000 −0.366679
\(120\) 2.00000 0.182574
\(121\) 5.00000 0.454545
\(122\) 10.0000 0.905357
\(123\) −6.00000 −0.541002
\(124\) 4.00000 0.359211
\(125\) −12.0000 −1.07331
\(126\) −4.00000 −0.356348
\(127\) −16.0000 −1.41977 −0.709885 − 0.704317i \(-0.751253\pi\)
−0.709885 + 0.704317i \(0.751253\pi\)
\(128\) 1.00000 0.0883883
\(129\) 12.0000 1.05654
\(130\) −4.00000 −0.350823
\(131\) −4.00000 −0.349482 −0.174741 − 0.984614i \(-0.555909\pi\)
−0.174741 + 0.984614i \(0.555909\pi\)
\(132\) 4.00000 0.348155
\(133\) 16.0000 1.38738
\(134\) −4.00000 −0.345547
\(135\) 2.00000 0.172133
\(136\) 1.00000 0.0857493
\(137\) −6.00000 −0.512615 −0.256307 − 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) 4.00000 0.340503
\(139\) 4.00000 0.339276 0.169638 − 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) −8.00000 −0.676123
\(141\) −8.00000 −0.673722
\(142\) −12.0000 −1.00702
\(143\) −8.00000 −0.668994
\(144\) 1.00000 0.0833333
\(145\) 20.0000 1.66091
\(146\) 10.0000 0.827606
\(147\) 9.00000 0.742307
\(148\) 2.00000 0.164399
\(149\) 6.00000 0.491539 0.245770 − 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) −1.00000 −0.0816497
\(151\) −16.0000 −1.30206 −0.651031 − 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) −4.00000 −0.324443
\(153\) 1.00000 0.0808452
\(154\) −16.0000 −1.28932
\(155\) 8.00000 0.642575
\(156\) −2.00000 −0.160128
\(157\) −10.0000 −0.798087 −0.399043 − 0.916932i \(-0.630658\pi\)
−0.399043 + 0.916932i \(0.630658\pi\)
\(158\) 12.0000 0.954669
\(159\) 6.00000 0.475831
\(160\) 2.00000 0.158114
\(161\) −16.0000 −1.26098
\(162\) 1.00000 0.0785674
\(163\) 4.00000 0.313304 0.156652 − 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) −6.00000 −0.468521
\(165\) 8.00000 0.622799
\(166\) 12.0000 0.931381
\(167\) 12.0000 0.928588 0.464294 − 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) −4.00000 −0.308607
\(169\) −9.00000 −0.692308
\(170\) 2.00000 0.153393
\(171\) −4.00000 −0.305888
\(172\) 12.0000 0.914991
\(173\) 2.00000 0.152057 0.0760286 − 0.997106i \(-0.475776\pi\)
0.0760286 + 0.997106i \(0.475776\pi\)
\(174\) 10.0000 0.758098
\(175\) 4.00000 0.302372
\(176\) 4.00000 0.301511
\(177\) −1.00000 −0.0751646
\(178\) 10.0000 0.749532
\(179\) 20.0000 1.49487 0.747435 − 0.664335i \(-0.231285\pi\)
0.747435 + 0.664335i \(0.231285\pi\)
\(180\) 2.00000 0.149071
\(181\) −6.00000 −0.445976 −0.222988 − 0.974821i \(-0.571581\pi\)
−0.222988 + 0.974821i \(0.571581\pi\)
\(182\) 8.00000 0.592999
\(183\) 10.0000 0.739221
\(184\) 4.00000 0.294884
\(185\) 4.00000 0.294086
\(186\) 4.00000 0.293294
\(187\) 4.00000 0.292509
\(188\) −8.00000 −0.583460
\(189\) −4.00000 −0.290957
\(190\) −8.00000 −0.580381
\(191\) 24.0000 1.73658 0.868290 − 0.496058i \(-0.165220\pi\)
0.868290 + 0.496058i \(0.165220\pi\)
\(192\) 1.00000 0.0721688
\(193\) −6.00000 −0.431889 −0.215945 − 0.976406i \(-0.569283\pi\)
−0.215945 + 0.976406i \(0.569283\pi\)
\(194\) 10.0000 0.717958
\(195\) −4.00000 −0.286446
\(196\) 9.00000 0.642857
\(197\) 26.0000 1.85242 0.926212 − 0.377004i \(-0.123046\pi\)
0.926212 + 0.377004i \(0.123046\pi\)
\(198\) 4.00000 0.284268
\(199\) −20.0000 −1.41776 −0.708881 − 0.705328i \(-0.750800\pi\)
−0.708881 + 0.705328i \(0.750800\pi\)
\(200\) −1.00000 −0.0707107
\(201\) −4.00000 −0.282138
\(202\) −18.0000 −1.26648
\(203\) −40.0000 −2.80745
\(204\) 1.00000 0.0700140
\(205\) −12.0000 −0.838116
\(206\) 8.00000 0.557386
\(207\) 4.00000 0.278019
\(208\) −2.00000 −0.138675
\(209\) −16.0000 −1.10674
\(210\) −8.00000 −0.552052
\(211\) −20.0000 −1.37686 −0.688428 − 0.725304i \(-0.741699\pi\)
−0.688428 + 0.725304i \(0.741699\pi\)
\(212\) 6.00000 0.412082
\(213\) −12.0000 −0.822226
\(214\) −12.0000 −0.820303
\(215\) 24.0000 1.63679
\(216\) 1.00000 0.0680414
\(217\) −16.0000 −1.08615
\(218\) 2.00000 0.135457
\(219\) 10.0000 0.675737
\(220\) 8.00000 0.539360
\(221\) −2.00000 −0.134535
\(222\) 2.00000 0.134231
\(223\) 8.00000 0.535720 0.267860 − 0.963458i \(-0.413684\pi\)
0.267860 + 0.963458i \(0.413684\pi\)
\(224\) −4.00000 −0.267261
\(225\) −1.00000 −0.0666667
\(226\) 10.0000 0.665190
\(227\) −12.0000 −0.796468 −0.398234 − 0.917284i \(-0.630377\pi\)
−0.398234 + 0.917284i \(0.630377\pi\)
\(228\) −4.00000 −0.264906
\(229\) −2.00000 −0.132164 −0.0660819 − 0.997814i \(-0.521050\pi\)
−0.0660819 + 0.997814i \(0.521050\pi\)
\(230\) 8.00000 0.527504
\(231\) −16.0000 −1.05272
\(232\) 10.0000 0.656532
\(233\) 18.0000 1.17922 0.589610 − 0.807688i \(-0.299282\pi\)
0.589610 + 0.807688i \(0.299282\pi\)
\(234\) −2.00000 −0.130744
\(235\) −16.0000 −1.04372
\(236\) −1.00000 −0.0650945
\(237\) 12.0000 0.779484
\(238\) −4.00000 −0.259281
\(239\) −16.0000 −1.03495 −0.517477 − 0.855697i \(-0.673129\pi\)
−0.517477 + 0.855697i \(0.673129\pi\)
\(240\) 2.00000 0.129099
\(241\) 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
\(244\) 10.0000 0.640184
\(245\) 18.0000 1.14998
\(246\) −6.00000 −0.382546
\(247\) 8.00000 0.509028
\(248\) 4.00000 0.254000
\(249\) 12.0000 0.760469
\(250\) −12.0000 −0.758947
\(251\) 12.0000 0.757433 0.378717 − 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) −4.00000 −0.251976
\(253\) 16.0000 1.00591
\(254\) −16.0000 −1.00393
\(255\) 2.00000 0.125245
\(256\) 1.00000 0.0625000
\(257\) −30.0000 −1.87135 −0.935674 − 0.352865i \(-0.885208\pi\)
−0.935674 + 0.352865i \(0.885208\pi\)
\(258\) 12.0000 0.747087
\(259\) −8.00000 −0.497096
\(260\) −4.00000 −0.248069
\(261\) 10.0000 0.618984
\(262\) −4.00000 −0.247121
\(263\) −24.0000 −1.47990 −0.739952 − 0.672660i \(-0.765152\pi\)
−0.739952 + 0.672660i \(0.765152\pi\)
\(264\) 4.00000 0.246183
\(265\) 12.0000 0.737154
\(266\) 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 \(0.867468\pi\)
\(270\) 2.00000 0.121716
\(271\) −16.0000 −0.971931 −0.485965 − 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) 1.00000 0.0606339
\(273\) 8.00000 0.484182
\(274\) −6.00000 −0.362473
\(275\) −4.00000 −0.241209
\(276\) 4.00000 0.240772
\(277\) 18.0000 1.08152 0.540758 − 0.841178i \(-0.318138\pi\)
0.540758 + 0.841178i \(0.318138\pi\)
\(278\) 4.00000 0.239904
\(279\) 4.00000 0.239474
\(280\) −8.00000 −0.478091
\(281\) −6.00000 −0.357930 −0.178965 − 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) −8.00000 −0.476393
\(283\) 20.0000 1.18888 0.594438 − 0.804141i \(-0.297374\pi\)
0.594438 + 0.804141i \(0.297374\pi\)
\(284\) −12.0000 −0.712069
\(285\) −8.00000 −0.473879
\(286\) −8.00000 −0.473050
\(287\) 24.0000 1.41668
\(288\) 1.00000 0.0589256
\(289\) 1.00000 0.0588235
\(290\) 20.0000 1.17444
\(291\) 10.0000 0.586210
\(292\) 10.0000 0.585206
\(293\) −18.0000 −1.05157 −0.525786 − 0.850617i \(-0.676229\pi\)
−0.525786 + 0.850617i \(0.676229\pi\)
\(294\) 9.00000 0.524891
\(295\) −2.00000 −0.116445
\(296\) 2.00000 0.116248
\(297\) 4.00000 0.232104
\(298\) 6.00000 0.347571
\(299\) −8.00000 −0.462652
\(300\) −1.00000 −0.0577350
\(301\) −48.0000 −2.76667
\(302\) −16.0000 −0.920697
\(303\) −18.0000 −1.03407
\(304\) −4.00000 −0.229416
\(305\) 20.0000 1.14520
\(306\) 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
\(309\) 8.00000 0.455104
\(310\) 8.00000 0.454369
\(311\) 12.0000 0.680458 0.340229 − 0.940343i \(-0.389495\pi\)
0.340229 + 0.940343i \(0.389495\pi\)
\(312\) −2.00000 −0.113228
\(313\) 2.00000 0.113047 0.0565233 − 0.998401i \(-0.481998\pi\)
0.0565233 + 0.998401i \(0.481998\pi\)
\(314\) −10.0000 −0.564333
\(315\) −8.00000 −0.450749
\(316\) 12.0000 0.675053
\(317\) 18.0000 1.01098 0.505490 − 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) 6.00000 0.336463
\(319\) 40.0000 2.23957
\(320\) 2.00000 0.111803
\(321\) −12.0000 −0.669775
\(322\) −16.0000 −0.891645
\(323\) −4.00000 −0.222566
\(324\) 1.00000 0.0555556
\(325\) 2.00000 0.110940
\(326\) 4.00000 0.221540
\(327\) 2.00000 0.110600
\(328\) −6.00000 −0.331295
\(329\) 32.0000 1.76422
\(330\) 8.00000 0.440386
\(331\) 12.0000 0.659580 0.329790 − 0.944054i \(-0.393022\pi\)
0.329790 + 0.944054i \(0.393022\pi\)
\(332\) 12.0000 0.658586
\(333\) 2.00000 0.109599
\(334\) 12.0000 0.656611
\(335\) −8.00000 −0.437087
\(336\) −4.00000 −0.218218
\(337\) 34.0000 1.85210 0.926049 − 0.377403i \(-0.123183\pi\)
0.926049 + 0.377403i \(0.123183\pi\)
\(338\) −9.00000 −0.489535
\(339\) 10.0000 0.543125
\(340\) 2.00000 0.108465
\(341\) 16.0000 0.866449
\(342\) −4.00000 −0.216295
\(343\) −8.00000 −0.431959
\(344\) 12.0000 0.646997
\(345\) 8.00000 0.430706
\(346\) 2.00000 0.107521
\(347\) 12.0000 0.644194 0.322097 − 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) 10.0000 0.536056
\(349\) 30.0000 1.60586 0.802932 − 0.596071i \(-0.203272\pi\)
0.802932 + 0.596071i \(0.203272\pi\)
\(350\) 4.00000 0.213809
\(351\) −2.00000 −0.106752
\(352\) 4.00000 0.213201
\(353\) −30.0000 −1.59674 −0.798369 − 0.602168i \(-0.794304\pi\)
−0.798369 + 0.602168i \(0.794304\pi\)
\(354\) −1.00000 −0.0531494
\(355\) −24.0000 −1.27379
\(356\) 10.0000 0.529999
\(357\) −4.00000 −0.211702
\(358\) 20.0000 1.05703
\(359\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 2.00000 0.105409
\(361\) −3.00000 −0.157895
\(362\) −6.00000 −0.315353
\(363\) 5.00000 0.262432
\(364\) 8.00000 0.419314
\(365\) 20.0000 1.04685
\(366\) 10.0000 0.522708
\(367\) 4.00000 0.208798 0.104399 − 0.994535i \(-0.466708\pi\)
0.104399 + 0.994535i \(0.466708\pi\)
\(368\) 4.00000 0.208514
\(369\) −6.00000 −0.312348
\(370\) 4.00000 0.207950
\(371\) −24.0000 −1.24602
\(372\) 4.00000 0.207390
\(373\) −18.0000 −0.932005 −0.466002 − 0.884783i \(-0.654306\pi\)
−0.466002 + 0.884783i \(0.654306\pi\)
\(374\) 4.00000 0.206835
\(375\) −12.0000 −0.619677
\(376\) −8.00000 −0.412568
\(377\) −20.0000 −1.03005
\(378\) −4.00000 −0.205738
\(379\) −36.0000 −1.84920 −0.924598 − 0.380945i \(-0.875599\pi\)
−0.924598 + 0.380945i \(0.875599\pi\)
\(380\) −8.00000 −0.410391
\(381\) −16.0000 −0.819705
\(382\) 24.0000 1.22795
\(383\) −32.0000 −1.63512 −0.817562 − 0.575841i \(-0.804675\pi\)
−0.817562 + 0.575841i \(0.804675\pi\)
\(384\) 1.00000 0.0510310
\(385\) −32.0000 −1.63087
\(386\) −6.00000 −0.305392
\(387\) 12.0000 0.609994
\(388\) 10.0000 0.507673
\(389\) 30.0000 1.52106 0.760530 − 0.649303i \(-0.224939\pi\)
0.760530 + 0.649303i \(0.224939\pi\)
\(390\) −4.00000 −0.202548
\(391\) 4.00000 0.202289
\(392\) 9.00000 0.454569
\(393\) −4.00000 −0.201773
\(394\) 26.0000 1.30986
\(395\) 24.0000 1.20757
\(396\) 4.00000 0.201008
\(397\) −38.0000 −1.90717 −0.953583 − 0.301131i \(-0.902636\pi\)
−0.953583 + 0.301131i \(0.902636\pi\)
\(398\) −20.0000 −1.00251
\(399\) 16.0000 0.801002
\(400\) −1.00000 −0.0500000
\(401\) 2.00000 0.0998752 0.0499376 − 0.998752i \(-0.484098\pi\)
0.0499376 + 0.998752i \(0.484098\pi\)
\(402\) −4.00000 −0.199502
\(403\) −8.00000 −0.398508
\(404\) −18.0000 −0.895533
\(405\) 2.00000 0.0993808
\(406\) −40.0000 −1.98517
\(407\) 8.00000 0.396545
\(408\) 1.00000 0.0495074
\(409\) −6.00000 −0.296681 −0.148340 − 0.988936i \(-0.547393\pi\)
−0.148340 + 0.988936i \(0.547393\pi\)
\(410\) −12.0000 −0.592638
\(411\) −6.00000 −0.295958
\(412\) 8.00000 0.394132
\(413\) 4.00000 0.196827
\(414\) 4.00000 0.196589
\(415\) 24.0000 1.17811
\(416\) −2.00000 −0.0980581
\(417\) 4.00000 0.195881
\(418\) −16.0000 −0.782586
\(419\) −12.0000 −0.586238 −0.293119 − 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) −8.00000 −0.390360
\(421\) −10.0000 −0.487370 −0.243685 − 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) −20.0000 −0.973585
\(423\) −8.00000 −0.388973
\(424\) 6.00000 0.291386
\(425\) −1.00000 −0.0485071
\(426\) −12.0000 −0.581402
\(427\) −40.0000 −1.93574
\(428\) −12.0000 −0.580042
\(429\) −8.00000 −0.386244
\(430\) 24.0000 1.15738
\(431\) 12.0000 0.578020 0.289010 − 0.957326i \(-0.406674\pi\)
0.289010 + 0.957326i \(0.406674\pi\)
\(432\) 1.00000 0.0481125
\(433\) −30.0000 −1.44171 −0.720854 − 0.693087i \(-0.756250\pi\)
−0.720854 + 0.693087i \(0.756250\pi\)
\(434\) −16.0000 −0.768025
\(435\) 20.0000 0.958927
\(436\) 2.00000 0.0957826
\(437\) −16.0000 −0.765384
\(438\) 10.0000 0.477818
\(439\) −20.0000 −0.954548 −0.477274 − 0.878755i \(-0.658375\pi\)
−0.477274 + 0.878755i \(0.658375\pi\)
\(440\) 8.00000 0.381385
\(441\) 9.00000 0.428571
\(442\) −2.00000 −0.0951303
\(443\) −36.0000 −1.71041 −0.855206 − 0.518289i \(-0.826569\pi\)
−0.855206 + 0.518289i \(0.826569\pi\)
\(444\) 2.00000 0.0949158
\(445\) 20.0000 0.948091
\(446\) 8.00000 0.378811
\(447\) 6.00000 0.283790
\(448\) −4.00000 −0.188982
\(449\) −30.0000 −1.41579 −0.707894 − 0.706319i \(-0.750354\pi\)
−0.707894 + 0.706319i \(0.750354\pi\)
\(450\) −1.00000 −0.0471405
\(451\) −24.0000 −1.13012
\(452\) 10.0000 0.470360
\(453\) −16.0000 −0.751746
\(454\) −12.0000 −0.563188
\(455\) 16.0000 0.750092
\(456\) −4.00000 −0.187317
\(457\) 10.0000 0.467780 0.233890 − 0.972263i \(-0.424854\pi\)
0.233890 + 0.972263i \(0.424854\pi\)
\(458\) −2.00000 −0.0934539
\(459\) 1.00000 0.0466760
\(460\) 8.00000 0.373002
\(461\) 14.0000 0.652045 0.326023 − 0.945362i \(-0.394291\pi\)
0.326023 + 0.945362i \(0.394291\pi\)
\(462\) −16.0000 −0.744387
\(463\) 16.0000 0.743583 0.371792 − 0.928316i \(-0.378744\pi\)
0.371792 + 0.928316i \(0.378744\pi\)
\(464\) 10.0000 0.464238
\(465\) 8.00000 0.370991
\(466\) 18.0000 0.833834
\(467\) −12.0000 −0.555294 −0.277647 − 0.960683i \(-0.589555\pi\)
−0.277647 + 0.960683i \(0.589555\pi\)
\(468\) −2.00000 −0.0924500
\(469\) 16.0000 0.738811
\(470\) −16.0000 −0.738025
\(471\) −10.0000 −0.460776
\(472\) −1.00000 −0.0460287
\(473\) 48.0000 2.20704
\(474\) 12.0000 0.551178
\(475\) 4.00000 0.183533
\(476\) −4.00000 −0.183340
\(477\) 6.00000 0.274721
\(478\) −16.0000 −0.731823
\(479\) −36.0000 −1.64488 −0.822441 − 0.568850i \(-0.807388\pi\)
−0.822441 + 0.568850i \(0.807388\pi\)
\(480\) 2.00000 0.0912871
\(481\) −4.00000 −0.182384
\(482\) 10.0000 0.455488
\(483\) −16.0000 −0.728025
\(484\) 5.00000 0.227273
\(485\) 20.0000 0.908153
\(486\) 1.00000 0.0453609
\(487\) 4.00000 0.181257 0.0906287 − 0.995885i \(-0.471112\pi\)
0.0906287 + 0.995885i \(0.471112\pi\)
\(488\) 10.0000 0.452679
\(489\) 4.00000 0.180886
\(490\) 18.0000 0.813157
\(491\) −20.0000 −0.902587 −0.451294 − 0.892375i \(-0.649037\pi\)
−0.451294 + 0.892375i \(0.649037\pi\)
\(492\) −6.00000 −0.270501
\(493\) 10.0000 0.450377
\(494\) 8.00000 0.359937
\(495\) 8.00000 0.359573
\(496\) 4.00000 0.179605
\(497\) 48.0000 2.15309
\(498\) 12.0000 0.537733
\(499\) −20.0000 −0.895323 −0.447661 − 0.894203i \(-0.647743\pi\)
−0.447661 + 0.894203i \(0.647743\pi\)
\(500\) −12.0000 −0.536656
\(501\) 12.0000 0.536120
\(502\) 12.0000 0.535586
\(503\) −12.0000 −0.535054 −0.267527 − 0.963550i \(-0.586206\pi\)
−0.267527 + 0.963550i \(0.586206\pi\)
\(504\) −4.00000 −0.178174
\(505\) −36.0000 −1.60198
\(506\) 16.0000 0.711287
\(507\) −9.00000 −0.399704
\(508\) −16.0000 −0.709885
\(509\) −34.0000 −1.50702 −0.753512 − 0.657434i \(-0.771642\pi\)
−0.753512 + 0.657434i \(0.771642\pi\)
\(510\) 2.00000 0.0885615
\(511\) −40.0000 −1.76950
\(512\) 1.00000 0.0441942
\(513\) −4.00000 −0.176604
\(514\) −30.0000 −1.32324
\(515\) 16.0000 0.705044
\(516\) 12.0000 0.528271
\(517\) −32.0000 −1.40736
\(518\) −8.00000 −0.351500
\(519\) 2.00000 0.0877903
\(520\) −4.00000 −0.175412
\(521\) −6.00000 −0.262865 −0.131432 − 0.991325i \(-0.541958\pi\)
−0.131432 + 0.991325i \(0.541958\pi\)
\(522\) 10.0000 0.437688
\(523\) 36.0000 1.57417 0.787085 − 0.616844i \(-0.211589\pi\)
0.787085 + 0.616844i \(0.211589\pi\)
\(524\) −4.00000 −0.174741
\(525\) 4.00000 0.174574
\(526\) −24.0000 −1.04645
\(527\) 4.00000 0.174243
\(528\) 4.00000 0.174078
\(529\) −7.00000 −0.304348
\(530\) 12.0000 0.521247
\(531\) −1.00000 −0.0433963
\(532\) 16.0000 0.693688
\(533\) 12.0000 0.519778
\(534\) 10.0000 0.432742
\(535\) −24.0000 −1.03761
\(536\) −4.00000 −0.172774
\(537\) 20.0000 0.863064
\(538\) −30.0000 −1.29339
\(539\) 36.0000 1.55063
\(540\) 2.00000 0.0860663
\(541\) 10.0000 0.429934 0.214967 − 0.976621i \(-0.431036\pi\)
0.214967 + 0.976621i \(0.431036\pi\)
\(542\) −16.0000 −0.687259
\(543\) −6.00000 −0.257485
\(544\) 1.00000 0.0428746
\(545\) 4.00000 0.171341
\(546\) 8.00000 0.342368
\(547\) −36.0000 −1.53925 −0.769624 − 0.638497i \(-0.779557\pi\)
−0.769624 + 0.638497i \(0.779557\pi\)
\(548\) −6.00000 −0.256307
\(549\) 10.0000 0.426790
\(550\) −4.00000 −0.170561
\(551\) −40.0000 −1.70406
\(552\) 4.00000 0.170251
\(553\) −48.0000 −2.04117
\(554\) 18.0000 0.764747
\(555\) 4.00000 0.169791
\(556\) 4.00000 0.169638
\(557\) 30.0000 1.27114 0.635570 − 0.772043i \(-0.280765\pi\)
0.635570 + 0.772043i \(0.280765\pi\)
\(558\) 4.00000 0.169334
\(559\) −24.0000 −1.01509
\(560\) −8.00000 −0.338062
\(561\) 4.00000 0.168880
\(562\) −6.00000 −0.253095
\(563\) 28.0000 1.18006 0.590030 − 0.807382i \(-0.299116\pi\)
0.590030 + 0.807382i \(0.299116\pi\)
\(564\) −8.00000 −0.336861
\(565\) 20.0000 0.841406
\(566\) 20.0000 0.840663
\(567\) −4.00000 −0.167984
\(568\) −12.0000 −0.503509
\(569\) 42.0000 1.76073 0.880366 − 0.474295i \(-0.157297\pi\)
0.880366 + 0.474295i \(0.157297\pi\)
\(570\) −8.00000 −0.335083
\(571\) −4.00000 −0.167395 −0.0836974 − 0.996491i \(-0.526673\pi\)
−0.0836974 + 0.996491i \(0.526673\pi\)
\(572\) −8.00000 −0.334497
\(573\) 24.0000 1.00261
\(574\) 24.0000 1.00174
\(575\) −4.00000 −0.166812
\(576\) 1.00000 0.0416667
\(577\) 34.0000 1.41544 0.707719 − 0.706494i \(-0.249724\pi\)
0.707719 + 0.706494i \(0.249724\pi\)
\(578\) 1.00000 0.0415945
\(579\) −6.00000 −0.249351
\(580\) 20.0000 0.830455
\(581\) −48.0000 −1.99138
\(582\) 10.0000 0.414513
\(583\) 24.0000 0.993978
\(584\) 10.0000 0.413803
\(585\) −4.00000 −0.165380
\(586\) −18.0000 −0.743573
\(587\) −20.0000 −0.825488 −0.412744 − 0.910847i \(-0.635430\pi\)
−0.412744 + 0.910847i \(0.635430\pi\)
\(588\) 9.00000 0.371154
\(589\) −16.0000 −0.659269
\(590\) −2.00000 −0.0823387
\(591\) 26.0000 1.06950
\(592\) 2.00000 0.0821995
\(593\) 34.0000 1.39621 0.698106 − 0.715994i \(-0.254026\pi\)
0.698106 + 0.715994i \(0.254026\pi\)
\(594\) 4.00000 0.164122
\(595\) −8.00000 −0.327968
\(596\) 6.00000 0.245770
\(597\) −20.0000 −0.818546
\(598\) −8.00000 −0.327144
\(599\) 16.0000 0.653742 0.326871 − 0.945069i \(-0.394006\pi\)
0.326871 + 0.945069i \(0.394006\pi\)
\(600\) −1.00000 −0.0408248
\(601\) 42.0000 1.71322 0.856608 − 0.515968i \(-0.172568\pi\)
0.856608 + 0.515968i \(0.172568\pi\)
\(602\) −48.0000 −1.95633
\(603\) −4.00000 −0.162893
\(604\) −16.0000 −0.651031
\(605\) 10.0000 0.406558
\(606\) −18.0000 −0.731200
\(607\) −4.00000 −0.162355 −0.0811775 − 0.996700i \(-0.525868\pi\)
−0.0811775 + 0.996700i \(0.525868\pi\)
\(608\) −4.00000 −0.162221
\(609\) −40.0000 −1.62088
\(610\) 20.0000 0.809776
\(611\) 16.0000 0.647291
\(612\) 1.00000 0.0404226
\(613\) −2.00000 −0.0807792 −0.0403896 − 0.999184i \(-0.512860\pi\)
−0.0403896 + 0.999184i \(0.512860\pi\)
\(614\) 4.00000 0.161427
\(615\) −12.0000 −0.483887
\(616\) −16.0000 −0.644658
\(617\) 18.0000 0.724653 0.362326 − 0.932051i \(-0.381983\pi\)
0.362326 + 0.932051i \(0.381983\pi\)
\(618\) 8.00000 0.321807
\(619\) −20.0000 −0.803868 −0.401934 − 0.915669i \(-0.631662\pi\)
−0.401934 + 0.915669i \(0.631662\pi\)
\(620\) 8.00000 0.321288
\(621\) 4.00000 0.160514
\(622\) 12.0000 0.481156
\(623\) −40.0000 −1.60257
\(624\) −2.00000 −0.0800641
\(625\) −19.0000 −0.760000
\(626\) 2.00000 0.0799361
\(627\) −16.0000 −0.638978
\(628\) −10.0000 −0.399043
\(629\) 2.00000 0.0797452
\(630\) −8.00000 −0.318728
\(631\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(632\) 12.0000 0.477334
\(633\) −20.0000 −0.794929
\(634\) 18.0000 0.714871
\(635\) −32.0000 −1.26988
\(636\) 6.00000 0.237915
\(637\) −18.0000 −0.713186
\(638\) 40.0000 1.58362
\(639\) −12.0000 −0.474713
\(640\) 2.00000 0.0790569
\(641\) 2.00000 0.0789953 0.0394976 − 0.999220i \(-0.487424\pi\)
0.0394976 + 0.999220i \(0.487424\pi\)
\(642\) −12.0000 −0.473602
\(643\) −44.0000 −1.73519 −0.867595 − 0.497271i \(-0.834335\pi\)
−0.867595 + 0.497271i \(0.834335\pi\)
\(644\) −16.0000 −0.630488
\(645\) 24.0000 0.944999
\(646\) −4.00000 −0.157378
\(647\) −24.0000 −0.943537 −0.471769 − 0.881722i \(-0.656384\pi\)
−0.471769 + 0.881722i \(0.656384\pi\)
\(648\) 1.00000 0.0392837
\(649\) −4.00000 −0.157014
\(650\) 2.00000 0.0784465
\(651\) −16.0000 −0.627089
\(652\) 4.00000 0.156652
\(653\) −22.0000 −0.860927 −0.430463 − 0.902608i \(-0.641650\pi\)
−0.430463 + 0.902608i \(0.641650\pi\)
\(654\) 2.00000 0.0782062
\(655\) −8.00000 −0.312586
\(656\) −6.00000 −0.234261
\(657\) 10.0000 0.390137
\(658\) 32.0000 1.24749
\(659\) −36.0000 −1.40236 −0.701180 − 0.712984i \(-0.747343\pi\)
−0.701180 + 0.712984i \(0.747343\pi\)
\(660\) 8.00000 0.311400
\(661\) −34.0000 −1.32245 −0.661223 − 0.750189i \(-0.729962\pi\)
−0.661223 + 0.750189i \(0.729962\pi\)
\(662\) 12.0000 0.466393
\(663\) −2.00000 −0.0776736
\(664\) 12.0000 0.465690
\(665\) 32.0000 1.24091
\(666\) 2.00000 0.0774984
\(667\) 40.0000 1.54881
\(668\) 12.0000 0.464294
\(669\) 8.00000 0.309298
\(670\) −8.00000 −0.309067
\(671\) 40.0000 1.54418
\(672\) −4.00000 −0.154303
\(673\) −6.00000 −0.231283 −0.115642 − 0.993291i \(-0.536892\pi\)
−0.115642 + 0.993291i \(0.536892\pi\)
\(674\) 34.0000 1.30963
\(675\) −1.00000 −0.0384900
\(676\) −9.00000 −0.346154
\(677\) 42.0000 1.61419 0.807096 − 0.590421i \(-0.201038\pi\)
0.807096 + 0.590421i \(0.201038\pi\)
\(678\) 10.0000 0.384048
\(679\) −40.0000 −1.53506
\(680\) 2.00000 0.0766965
\(681\) −12.0000 −0.459841
\(682\) 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
\(685\) −12.0000 −0.458496
\(686\) −8.00000 −0.305441
\(687\) −2.00000 −0.0763048
\(688\) 12.0000 0.457496
\(689\) −12.0000 −0.457164
\(690\) 8.00000 0.304555
\(691\) 28.0000 1.06517 0.532585 − 0.846376i \(-0.321221\pi\)
0.532585 + 0.846376i \(0.321221\pi\)
\(692\) 2.00000 0.0760286
\(693\) −16.0000 −0.607790
\(694\) 12.0000 0.455514
\(695\) 8.00000 0.303457
\(696\) 10.0000 0.379049
\(697\) −6.00000 −0.227266
\(698\) 30.0000 1.13552
\(699\) 18.0000 0.680823
\(700\) 4.00000 0.151186
\(701\) 22.0000 0.830929 0.415464 − 0.909610i \(-0.363619\pi\)
0.415464 + 0.909610i \(0.363619\pi\)
\(702\) −2.00000 −0.0754851
\(703\) −8.00000 −0.301726
\(704\) 4.00000 0.150756
\(705\) −16.0000 −0.602595
\(706\) −30.0000 −1.12906
\(707\) 72.0000 2.70784
\(708\) −1.00000 −0.0375823
\(709\) 34.0000 1.27690 0.638448 − 0.769665i \(-0.279577\pi\)
0.638448 + 0.769665i \(0.279577\pi\)
\(710\) −24.0000 −0.900704
\(711\) 12.0000 0.450035
\(712\) 10.0000 0.374766
\(713\) 16.0000 0.599205
\(714\) −4.00000 −0.149696
\(715\) −16.0000 −0.598366
\(716\) 20.0000 0.747435
\(717\) −16.0000 −0.597531
\(718\) 0 0
\(719\) 4.00000 0.149175 0.0745874 − 0.997214i \(-0.476236\pi\)
0.0745874 + 0.997214i \(0.476236\pi\)
\(720\) 2.00000 0.0745356
\(721\) −32.0000 −1.19174
\(722\) −3.00000 −0.111648
\(723\) 10.0000 0.371904
\(724\) −6.00000 −0.222988
\(725\) −10.0000 −0.371391
\(726\) 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
\(730\) 20.0000 0.740233
\(731\) 12.0000 0.443836
\(732\) 10.0000 0.369611
\(733\) 6.00000 0.221615 0.110808 − 0.993842i \(-0.464656\pi\)
0.110808 + 0.993842i \(0.464656\pi\)
\(734\) 4.00000 0.147643
\(735\) 18.0000 0.663940
\(736\) 4.00000 0.147442
\(737\) −16.0000 −0.589368
\(738\) −6.00000 −0.220863
\(739\) −4.00000 −0.147142 −0.0735712 − 0.997290i \(-0.523440\pi\)
−0.0735712 + 0.997290i \(0.523440\pi\)
\(740\) 4.00000 0.147043
\(741\) 8.00000 0.293887
\(742\) −24.0000 −0.881068
\(743\) −20.0000 −0.733729 −0.366864 − 0.930274i \(-0.619569\pi\)
−0.366864 + 0.930274i \(0.619569\pi\)
\(744\) 4.00000 0.146647
\(745\) 12.0000 0.439646
\(746\) −18.0000 −0.659027
\(747\) 12.0000 0.439057
\(748\) 4.00000 0.146254
\(749\) 48.0000 1.75388
\(750\) −12.0000 −0.438178
\(751\) 44.0000 1.60558 0.802791 − 0.596260i \(-0.203347\pi\)
0.802791 + 0.596260i \(0.203347\pi\)
\(752\) −8.00000 −0.291730
\(753\) 12.0000 0.437304
\(754\) −20.0000 −0.728357
\(755\) −32.0000 −1.16460
\(756\) −4.00000 −0.145479
\(757\) −10.0000 −0.363456 −0.181728 − 0.983349i \(-0.558169\pi\)
−0.181728 + 0.983349i \(0.558169\pi\)
\(758\) −36.0000 −1.30758
\(759\) 16.0000 0.580763
\(760\) −8.00000 −0.290191
\(761\) 42.0000 1.52250 0.761249 − 0.648459i \(-0.224586\pi\)
0.761249 + 0.648459i \(0.224586\pi\)
\(762\) −16.0000 −0.579619
\(763\) −8.00000 −0.289619
\(764\) 24.0000 0.868290
\(765\) 2.00000 0.0723102
\(766\) −32.0000 −1.15621
\(767\) 2.00000 0.0722158
\(768\) 1.00000 0.0360844
\(769\) 34.0000 1.22607 0.613036 − 0.790055i \(-0.289948\pi\)
0.613036 + 0.790055i \(0.289948\pi\)
\(770\) −32.0000 −1.15320
\(771\) −30.0000 −1.08042
\(772\) −6.00000 −0.215945
\(773\) 38.0000 1.36677 0.683383 − 0.730061i \(-0.260508\pi\)
0.683383 + 0.730061i \(0.260508\pi\)
\(774\) 12.0000 0.431331
\(775\) −4.00000 −0.143684
\(776\) 10.0000 0.358979
\(777\) −8.00000 −0.286998
\(778\) 30.0000 1.07555
\(779\) 24.0000 0.859889
\(780\) −4.00000 −0.143223
\(781\) −48.0000 −1.71758
\(782\) 4.00000 0.143040
\(783\) 10.0000 0.357371
\(784\) 9.00000 0.321429
\(785\) −20.0000 −0.713831
\(786\) −4.00000 −0.142675
\(787\) 44.0000 1.56843 0.784215 − 0.620489i \(-0.213066\pi\)
0.784215 + 0.620489i \(0.213066\pi\)
\(788\) 26.0000 0.926212
\(789\) −24.0000 −0.854423
\(790\) 24.0000 0.853882
\(791\) −40.0000 −1.42224
\(792\) 4.00000 0.142134
\(793\) −20.0000 −0.710221
\(794\) −38.0000 −1.34857
\(795\) 12.0000 0.425596
\(796\) −20.0000 −0.708881
\(797\) 6.00000 0.212531 0.106265 − 0.994338i \(-0.466111\pi\)
0.106265 + 0.994338i \(0.466111\pi\)
\(798\) 16.0000 0.566394
\(799\) −8.00000 −0.283020
\(800\) −1.00000 −0.0353553
\(801\) 10.0000 0.353333
\(802\) 2.00000 0.0706225
\(803\) 40.0000 1.41157
\(804\) −4.00000 −0.141069
\(805\) −32.0000 −1.12785
\(806\) −8.00000 −0.281788
\(807\) −30.0000 −1.05605
\(808\) −18.0000 −0.633238
\(809\) 10.0000 0.351581 0.175791 − 0.984428i \(-0.443752\pi\)
0.175791 + 0.984428i \(0.443752\pi\)
\(810\) 2.00000 0.0702728
\(811\) −20.0000 −0.702295 −0.351147 − 0.936320i \(-0.614208\pi\)
−0.351147 + 0.936320i \(0.614208\pi\)
\(812\) −40.0000 −1.40372
\(813\) −16.0000 −0.561144
\(814\) 8.00000 0.280400
\(815\) 8.00000 0.280228
\(816\) 1.00000 0.0350070
\(817\) −48.0000 −1.67931
\(818\) −6.00000 −0.209785
\(819\) 8.00000 0.279543
\(820\) −12.0000 −0.419058
\(821\) −46.0000 −1.60541 −0.802706 − 0.596376i \(-0.796607\pi\)
−0.802706 + 0.596376i \(0.796607\pi\)
\(822\) −6.00000 −0.209274
\(823\) 44.0000 1.53374 0.766872 − 0.641800i \(-0.221812\pi\)
0.766872 + 0.641800i \(0.221812\pi\)
\(824\) 8.00000 0.278693
\(825\) −4.00000 −0.139262
\(826\) 4.00000 0.139178
\(827\) 36.0000 1.25184 0.625921 − 0.779886i \(-0.284723\pi\)
0.625921 + 0.779886i \(0.284723\pi\)
\(828\) 4.00000 0.139010
\(829\) −18.0000 −0.625166 −0.312583 − 0.949890i \(-0.601194\pi\)
−0.312583 + 0.949890i \(0.601194\pi\)
\(830\) 24.0000 0.833052
\(831\) 18.0000 0.624413
\(832\) −2.00000 −0.0693375
\(833\) 9.00000 0.311832
\(834\) 4.00000 0.138509
\(835\) 24.0000 0.830554
\(836\) −16.0000 −0.553372
\(837\) 4.00000 0.138260
\(838\) −12.0000 −0.414533
\(839\) −20.0000 −0.690477 −0.345238 − 0.938515i \(-0.612202\pi\)
−0.345238 + 0.938515i \(0.612202\pi\)
\(840\) −8.00000 −0.276026
\(841\) 71.0000 2.44828
\(842\) −10.0000 −0.344623
\(843\) −6.00000 −0.206651
\(844\) −20.0000 −0.688428
\(845\) −18.0000 −0.619219
\(846\) −8.00000 −0.275046
\(847\) −20.0000 −0.687208
\(848\) 6.00000 0.206041
\(849\) 20.0000 0.686398
\(850\) −1.00000 −0.0342997
\(851\) 8.00000 0.274236
\(852\) −12.0000 −0.411113
\(853\) −30.0000 −1.02718 −0.513590 − 0.858036i \(-0.671685\pi\)
−0.513590 + 0.858036i \(0.671685\pi\)
\(854\) −40.0000 −1.36877
\(855\) −8.00000 −0.273594
\(856\) −12.0000 −0.410152
\(857\) −22.0000 −0.751506 −0.375753 − 0.926720i \(-0.622616\pi\)
−0.375753 + 0.926720i \(0.622616\pi\)
\(858\) −8.00000 −0.273115
\(859\) 28.0000 0.955348 0.477674 − 0.878537i \(-0.341480\pi\)
0.477674 + 0.878537i \(0.341480\pi\)
\(860\) 24.0000 0.818393
\(861\) 24.0000 0.817918
\(862\) 12.0000 0.408722
\(863\) −16.0000 −0.544646 −0.272323 − 0.962206i \(-0.587792\pi\)
−0.272323 + 0.962206i \(0.587792\pi\)
\(864\) 1.00000 0.0340207
\(865\) 4.00000 0.136004
\(866\) −30.0000 −1.01944
\(867\) 1.00000 0.0339618
\(868\) −16.0000 −0.543075
\(869\) 48.0000 1.62829
\(870\) 20.0000 0.678064
\(871\) 8.00000 0.271070
\(872\) 2.00000 0.0677285
\(873\) 10.0000 0.338449
\(874\) −16.0000 −0.541208
\(875\) 48.0000 1.62270
\(876\) 10.0000 0.337869
\(877\) −38.0000 −1.28317 −0.641584 − 0.767052i \(-0.721723\pi\)
−0.641584 + 0.767052i \(0.721723\pi\)
\(878\) −20.0000 −0.674967
\(879\) −18.0000 −0.607125
\(880\) 8.00000 0.269680
\(881\) −54.0000 −1.81931 −0.909653 − 0.415369i \(-0.863653\pi\)
−0.909653 + 0.415369i \(0.863653\pi\)
\(882\) 9.00000 0.303046
\(883\) −20.0000 −0.673054 −0.336527 − 0.941674i \(-0.609252\pi\)
−0.336527 + 0.941674i \(0.609252\pi\)
\(884\) −2.00000 −0.0672673
\(885\) −2.00000 −0.0672293
\(886\) −36.0000 −1.20944
\(887\) 28.0000 0.940148 0.470074 − 0.882627i \(-0.344227\pi\)
0.470074 + 0.882627i \(0.344227\pi\)
\(888\) 2.00000 0.0671156
\(889\) 64.0000 2.14649
\(890\) 20.0000 0.670402
\(891\) 4.00000 0.134005
\(892\) 8.00000 0.267860
\(893\) 32.0000 1.07084
\(894\) 6.00000 0.200670
\(895\) 40.0000 1.33705
\(896\) −4.00000 −0.133631
\(897\) −8.00000 −0.267112
\(898\) −30.0000 −1.00111
\(899\) 40.0000 1.33407
\(900\) −1.00000 −0.0333333
\(901\) 6.00000 0.199889
\(902\) −24.0000 −0.799113
\(903\) −48.0000 −1.59734
\(904\) 10.0000 0.332595
\(905\) −12.0000 −0.398893
\(906\) −16.0000 −0.531564
\(907\) 52.0000 1.72663 0.863316 − 0.504664i \(-0.168384\pi\)
0.863316 + 0.504664i \(0.168384\pi\)
\(908\) −12.0000 −0.398234
\(909\) −18.0000 −0.597022
\(910\) 16.0000 0.530395
\(911\) −36.0000 −1.19273 −0.596367 − 0.802712i \(-0.703390\pi\)
−0.596367 + 0.802712i \(0.703390\pi\)
\(912\) −4.00000 −0.132453
\(913\) 48.0000 1.58857
\(914\) 10.0000 0.330771
\(915\) 20.0000 0.661180
\(916\) −2.00000 −0.0660819
\(917\) 16.0000 0.528367
\(918\) 1.00000 0.0330049
\(919\) 16.0000 0.527791 0.263896 − 0.964551i \(-0.414993\pi\)
0.263896 + 0.964551i \(0.414993\pi\)
\(920\) 8.00000 0.263752
\(921\) 4.00000 0.131804
\(922\) 14.0000 0.461065
\(923\) 24.0000 0.789970
\(924\) −16.0000 −0.526361
\(925\) −2.00000 −0.0657596
\(926\) 16.0000 0.525793
\(927\) 8.00000 0.262754
\(928\) 10.0000 0.328266
\(929\) 18.0000 0.590561 0.295280 − 0.955411i \(-0.404587\pi\)
0.295280 + 0.955411i \(0.404587\pi\)
\(930\) 8.00000 0.262330
\(931\) −36.0000 −1.17985
\(932\) 18.0000 0.589610
\(933\) 12.0000 0.392862
\(934\) −12.0000 −0.392652
\(935\) 8.00000 0.261628
\(936\) −2.00000 −0.0653720
\(937\) 26.0000 0.849383 0.424691 − 0.905338i \(-0.360383\pi\)
0.424691 + 0.905338i \(0.360383\pi\)
\(938\) 16.0000 0.522419
\(939\) 2.00000 0.0652675
\(940\) −16.0000 −0.521862
\(941\) −22.0000 −0.717180 −0.358590 − 0.933495i \(-0.616742\pi\)
−0.358590 + 0.933495i \(0.616742\pi\)
\(942\) −10.0000 −0.325818
\(943\) −24.0000 −0.781548
\(944\) −1.00000 −0.0325472
\(945\) −8.00000 −0.260240
\(946\) 48.0000 1.56061
\(947\) −52.0000 −1.68977 −0.844886 − 0.534946i \(-0.820332\pi\)
−0.844886 + 0.534946i \(0.820332\pi\)
\(948\) 12.0000 0.389742
\(949\) −20.0000 −0.649227
\(950\) 4.00000 0.129777
\(951\) 18.0000 0.583690
\(952\) −4.00000 −0.129641
\(953\) 58.0000 1.87880 0.939402 − 0.342817i \(-0.111381\pi\)
0.939402 + 0.342817i \(0.111381\pi\)
\(954\) 6.00000 0.194257
\(955\) 48.0000 1.55324
\(956\) −16.0000 −0.517477
\(957\) 40.0000 1.29302
\(958\) −36.0000 −1.16311
\(959\) 24.0000 0.775000
\(960\) 2.00000 0.0645497
\(961\) −15.0000 −0.483871
\(962\) −4.00000 −0.128965
\(963\) −12.0000 −0.386695
\(964\) 10.0000 0.322078
\(965\) −12.0000 −0.386294
\(966\) −16.0000 −0.514792
\(967\) 8.00000 0.257263 0.128631 − 0.991692i \(-0.458942\pi\)
0.128631 + 0.991692i \(0.458942\pi\)
\(968\) 5.00000 0.160706
\(969\) −4.00000 −0.128499
\(970\) 20.0000 0.642161
\(971\) −12.0000 −0.385098 −0.192549 − 0.981287i \(-0.561675\pi\)
−0.192549 + 0.981287i \(0.561675\pi\)
\(972\) 1.00000 0.0320750
\(973\) −16.0000 −0.512936
\(974\) 4.00000 0.128168
\(975\) 2.00000 0.0640513
\(976\) 10.0000 0.320092
\(977\) −30.0000 −0.959785 −0.479893 − 0.877327i \(-0.659324\pi\)
−0.479893 + 0.877327i \(0.659324\pi\)
\(978\) 4.00000 0.127906
\(979\) 40.0000 1.27841
\(980\) 18.0000 0.574989
\(981\) 2.00000 0.0638551
\(982\) −20.0000 −0.638226
\(983\) −52.0000 −1.65854 −0.829271 − 0.558846i \(-0.811244\pi\)
−0.829271 + 0.558846i \(0.811244\pi\)
\(984\) −6.00000 −0.191273
\(985\) 52.0000 1.65686
\(986\) 10.0000 0.318465
\(987\) 32.0000 1.01857
\(988\) 8.00000 0.254514
\(989\) 48.0000 1.52631
\(990\) 8.00000 0.254257
\(991\) −20.0000 −0.635321 −0.317660 − 0.948205i \(-0.602897\pi\)
−0.317660 + 0.948205i \(0.602897\pi\)
\(992\) 4.00000 0.127000
\(993\) 12.0000 0.380808
\(994\) 48.0000 1.52247
\(995\) −40.0000 −1.26809
\(996\) 12.0000 0.380235
\(997\) −38.0000 −1.20347 −0.601736 − 0.798695i \(-0.705524\pi\)
−0.601736 + 0.798695i \(0.705524\pi\)
\(998\) −20.0000 −0.633089
\(999\) 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 6018.2.a.l.1.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6018.2.a.l.1.1 ✓ 1 1.1 even 1 trivial