Properties

Label 118.2.a.c.1.1
Level $118$
Weight $2$
Character 118.1
Self dual yes
Analytic conductor $0.942$
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] = [118,2,Mod(1,118)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(118, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("118.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 118 = 2 \cdot 59 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 118.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(0.942234743851\)
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 \(\chi\) \(=\) 118.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +3.00000 q^{7} +1.00000 q^{8} -2.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +3.00000 q^{7} +1.00000 q^{8} -2.00000 q^{9} +1.00000 q^{10} +2.00000 q^{11} -1.00000 q^{12} -6.00000 q^{13} +3.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} -2.00000 q^{17} -2.00000 q^{18} -5.00000 q^{19} +1.00000 q^{20} -3.00000 q^{21} +2.00000 q^{22} +4.00000 q^{23} -1.00000 q^{24} -4.00000 q^{25} -6.00000 q^{26} +5.00000 q^{27} +3.00000 q^{28} -5.00000 q^{29} -1.00000 q^{30} +2.00000 q^{31} +1.00000 q^{32} -2.00000 q^{33} -2.00000 q^{34} +3.00000 q^{35} -2.00000 q^{36} +8.00000 q^{37} -5.00000 q^{38} +6.00000 q^{39} +1.00000 q^{40} +7.00000 q^{41} -3.00000 q^{42} -6.00000 q^{43} +2.00000 q^{44} -2.00000 q^{45} +4.00000 q^{46} -2.00000 q^{47} -1.00000 q^{48} +2.00000 q^{49} -4.00000 q^{50} +2.00000 q^{51} -6.00000 q^{52} +9.00000 q^{53} +5.00000 q^{54} +2.00000 q^{55} +3.00000 q^{56} +5.00000 q^{57} -5.00000 q^{58} -1.00000 q^{59} -1.00000 q^{60} -8.00000 q^{61} +2.00000 q^{62} -6.00000 q^{63} +1.00000 q^{64} -6.00000 q^{65} -2.00000 q^{66} -2.00000 q^{67} -2.00000 q^{68} -4.00000 q^{69} +3.00000 q^{70} +12.0000 q^{71} -2.00000 q^{72} +4.00000 q^{73} +8.00000 q^{74} +4.00000 q^{75} -5.00000 q^{76} +6.00000 q^{77} +6.00000 q^{78} +5.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} +7.00000 q^{82} +14.0000 q^{83} -3.00000 q^{84} -2.00000 q^{85} -6.00000 q^{86} +5.00000 q^{87} +2.00000 q^{88} -2.00000 q^{90} -18.0000 q^{91} +4.00000 q^{92} -2.00000 q^{93} -2.00000 q^{94} -5.00000 q^{95} -1.00000 q^{96} +8.00000 q^{97} +2.00000 q^{98} -4.00000 q^{99} +O(q^{100})\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −1.00000 −0.577350 −0.288675 0.957427i \(-0.593215\pi\)
−0.288675 + 0.957427i \(0.593215\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214 0.223607 0.974679i \(-0.428217\pi\)
0.223607 + 0.974679i \(0.428217\pi\)
\(6\) −1.00000 −0.408248
\(7\) 3.00000 1.13389 0.566947 0.823754i \(-0.308125\pi\)
0.566947 + 0.823754i \(0.308125\pi\)
\(8\) 1.00000 0.353553
\(9\) −2.00000 −0.666667
\(10\) 1.00000 0.316228
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) −1.00000 −0.288675
\(13\) −6.00000 −1.66410 −0.832050 0.554700i \(-0.812833\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) 3.00000 0.801784
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) −2.00000 −0.471405
\(19\) −5.00000 −1.14708 −0.573539 0.819178i \(-0.694430\pi\)
−0.573539 + 0.819178i \(0.694430\pi\)
\(20\) 1.00000 0.223607
\(21\) −3.00000 −0.654654
\(22\) 2.00000 0.426401
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) −1.00000 −0.204124
\(25\) −4.00000 −0.800000
\(26\) −6.00000 −1.17670
\(27\) 5.00000 0.962250
\(28\) 3.00000 0.566947
\(29\) −5.00000 −0.928477 −0.464238 0.885710i \(-0.653672\pi\)
−0.464238 + 0.885710i \(0.653672\pi\)
\(30\) −1.00000 −0.182574
\(31\) 2.00000 0.359211 0.179605 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) 1.00000 0.176777
\(33\) −2.00000 −0.348155
\(34\) −2.00000 −0.342997
\(35\) 3.00000 0.507093
\(36\) −2.00000 −0.333333
\(37\) 8.00000 1.31519 0.657596 0.753371i \(-0.271573\pi\)
0.657596 + 0.753371i \(0.271573\pi\)
\(38\) −5.00000 −0.811107
\(39\) 6.00000 0.960769
\(40\) 1.00000 0.158114
\(41\) 7.00000 1.09322 0.546608 0.837389i \(-0.315919\pi\)
0.546608 + 0.837389i \(0.315919\pi\)
\(42\) −3.00000 −0.462910
\(43\) −6.00000 −0.914991 −0.457496 0.889212i \(-0.651253\pi\)
−0.457496 + 0.889212i \(0.651253\pi\)
\(44\) 2.00000 0.301511
\(45\) −2.00000 −0.298142
\(46\) 4.00000 0.589768
\(47\) −2.00000 −0.291730 −0.145865 0.989305i \(-0.546597\pi\)
−0.145865 + 0.989305i \(0.546597\pi\)
\(48\) −1.00000 −0.144338
\(49\) 2.00000 0.285714
\(50\) −4.00000 −0.565685
\(51\) 2.00000 0.280056
\(52\) −6.00000 −0.832050
\(53\) 9.00000 1.23625 0.618123 0.786082i \(-0.287894\pi\)
0.618123 + 0.786082i \(0.287894\pi\)
\(54\) 5.00000 0.680414
\(55\) 2.00000 0.269680
\(56\) 3.00000 0.400892
\(57\) 5.00000 0.662266
\(58\) −5.00000 −0.656532
\(59\) −1.00000 −0.130189
\(60\) −1.00000 −0.129099
\(61\) −8.00000 −1.02430 −0.512148 0.858898i \(-0.671150\pi\)
−0.512148 + 0.858898i \(0.671150\pi\)
\(62\) 2.00000 0.254000
\(63\) −6.00000 −0.755929
\(64\) 1.00000 0.125000
\(65\) −6.00000 −0.744208
\(66\) −2.00000 −0.246183
\(67\) −2.00000 −0.244339 −0.122169 0.992509i \(-0.538985\pi\)
−0.122169 + 0.992509i \(0.538985\pi\)
\(68\) −2.00000 −0.242536
\(69\) −4.00000 −0.481543
\(70\) 3.00000 0.358569
\(71\) 12.0000 1.42414 0.712069 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) −2.00000 −0.235702
\(73\) 4.00000 0.468165 0.234082 0.972217i \(-0.424791\pi\)
0.234082 + 0.972217i \(0.424791\pi\)
\(74\) 8.00000 0.929981
\(75\) 4.00000 0.461880
\(76\) −5.00000 −0.573539
\(77\) 6.00000 0.683763
\(78\) 6.00000 0.679366
\(79\) 5.00000 0.562544 0.281272 0.959628i \(-0.409244\pi\)
0.281272 + 0.959628i \(0.409244\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) 7.00000 0.773021
\(83\) 14.0000 1.53670 0.768350 0.640030i \(-0.221078\pi\)
0.768350 + 0.640030i \(0.221078\pi\)
\(84\) −3.00000 −0.327327
\(85\) −2.00000 −0.216930
\(86\) −6.00000 −0.646997
\(87\) 5.00000 0.536056
\(88\) 2.00000 0.213201
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) −2.00000 −0.210819
\(91\) −18.0000 −1.88691
\(92\) 4.00000 0.417029
\(93\) −2.00000 −0.207390
\(94\) −2.00000 −0.206284
\(95\) −5.00000 −0.512989
\(96\) −1.00000 −0.102062
\(97\) 8.00000 0.812277 0.406138 0.913812i \(-0.366875\pi\)
0.406138 + 0.913812i \(0.366875\pi\)
\(98\) 2.00000 0.202031
\(99\) −4.00000 −0.402015
\(100\) −4.00000 −0.400000
\(101\) −18.0000 −1.79107 −0.895533 0.444994i \(-0.853206\pi\)
−0.895533 + 0.444994i \(0.853206\pi\)
\(102\) 2.00000 0.198030
\(103\) −16.0000 −1.57653 −0.788263 0.615338i \(-0.789020\pi\)
−0.788263 + 0.615338i \(0.789020\pi\)
\(104\) −6.00000 −0.588348
\(105\) −3.00000 −0.292770
\(106\) 9.00000 0.874157
\(107\) 3.00000 0.290021 0.145010 0.989430i \(-0.453678\pi\)
0.145010 + 0.989430i \(0.453678\pi\)
\(108\) 5.00000 0.481125
\(109\) −20.0000 −1.91565 −0.957826 0.287348i \(-0.907226\pi\)
−0.957826 + 0.287348i \(0.907226\pi\)
\(110\) 2.00000 0.190693
\(111\) −8.00000 −0.759326
\(112\) 3.00000 0.283473
\(113\) 14.0000 1.31701 0.658505 0.752577i \(-0.271189\pi\)
0.658505 + 0.752577i \(0.271189\pi\)
\(114\) 5.00000 0.468293
\(115\) 4.00000 0.373002
\(116\) −5.00000 −0.464238
\(117\) 12.0000 1.10940
\(118\) −1.00000 −0.0920575
\(119\) −6.00000 −0.550019
\(120\) −1.00000 −0.0912871
\(121\) −7.00000 −0.636364
\(122\) −8.00000 −0.724286
\(123\) −7.00000 −0.631169
\(124\) 2.00000 0.179605
\(125\) −9.00000 −0.804984
\(126\) −6.00000 −0.534522
\(127\) 13.0000 1.15356 0.576782 0.816898i \(-0.304308\pi\)
0.576782 + 0.816898i \(0.304308\pi\)
\(128\) 1.00000 0.0883883
\(129\) 6.00000 0.528271
\(130\) −6.00000 −0.526235
\(131\) −18.0000 −1.57267 −0.786334 0.617802i \(-0.788023\pi\)
−0.786334 + 0.617802i \(0.788023\pi\)
\(132\) −2.00000 −0.174078
\(133\) −15.0000 −1.30066
\(134\) −2.00000 −0.172774
\(135\) 5.00000 0.430331
\(136\) −2.00000 −0.171499
\(137\) 3.00000 0.256307 0.128154 0.991754i \(-0.459095\pi\)
0.128154 + 0.991754i \(0.459095\pi\)
\(138\) −4.00000 −0.340503
\(139\) 20.0000 1.69638 0.848189 0.529694i \(-0.177693\pi\)
0.848189 + 0.529694i \(0.177693\pi\)
\(140\) 3.00000 0.253546
\(141\) 2.00000 0.168430
\(142\) 12.0000 1.00702
\(143\) −12.0000 −1.00349
\(144\) −2.00000 −0.166667
\(145\) −5.00000 −0.415227
\(146\) 4.00000 0.331042
\(147\) −2.00000 −0.164957
\(148\) 8.00000 0.657596
\(149\) 10.0000 0.819232 0.409616 0.912258i \(-0.365663\pi\)
0.409616 + 0.912258i \(0.365663\pi\)
\(150\) 4.00000 0.326599
\(151\) 2.00000 0.162758 0.0813788 0.996683i \(-0.474068\pi\)
0.0813788 + 0.996683i \(0.474068\pi\)
\(152\) −5.00000 −0.405554
\(153\) 4.00000 0.323381
\(154\) 6.00000 0.483494
\(155\) 2.00000 0.160644
\(156\) 6.00000 0.480384
\(157\) −12.0000 −0.957704 −0.478852 0.877896i \(-0.658947\pi\)
−0.478852 + 0.877896i \(0.658947\pi\)
\(158\) 5.00000 0.397779
\(159\) −9.00000 −0.713746
\(160\) 1.00000 0.0790569
\(161\) 12.0000 0.945732
\(162\) 1.00000 0.0785674
\(163\) 4.00000 0.313304 0.156652 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) 7.00000 0.546608
\(165\) −2.00000 −0.155700
\(166\) 14.0000 1.08661
\(167\) 3.00000 0.232147 0.116073 0.993241i \(-0.462969\pi\)
0.116073 + 0.993241i \(0.462969\pi\)
\(168\) −3.00000 −0.231455
\(169\) 23.0000 1.76923
\(170\) −2.00000 −0.153393
\(171\) 10.0000 0.764719
\(172\) −6.00000 −0.457496
\(173\) 4.00000 0.304114 0.152057 0.988372i \(-0.451410\pi\)
0.152057 + 0.988372i \(0.451410\pi\)
\(174\) 5.00000 0.379049
\(175\) −12.0000 −0.907115
\(176\) 2.00000 0.150756
\(177\) 1.00000 0.0751646
\(178\) 0 0
\(179\) 20.0000 1.49487 0.747435 0.664335i \(-0.231285\pi\)
0.747435 + 0.664335i \(0.231285\pi\)
\(180\) −2.00000 −0.149071
\(181\) −13.0000 −0.966282 −0.483141 0.875542i \(-0.660504\pi\)
−0.483141 + 0.875542i \(0.660504\pi\)
\(182\) −18.0000 −1.33425
\(183\) 8.00000 0.591377
\(184\) 4.00000 0.294884
\(185\) 8.00000 0.588172
\(186\) −2.00000 −0.146647
\(187\) −4.00000 −0.292509
\(188\) −2.00000 −0.145865
\(189\) 15.0000 1.09109
\(190\) −5.00000 −0.362738
\(191\) −18.0000 −1.30243 −0.651217 0.758891i \(-0.725741\pi\)
−0.651217 + 0.758891i \(0.725741\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 19.0000 1.36765 0.683825 0.729646i \(-0.260315\pi\)
0.683825 + 0.729646i \(0.260315\pi\)
\(194\) 8.00000 0.574367
\(195\) 6.00000 0.429669
\(196\) 2.00000 0.142857
\(197\) 18.0000 1.28245 0.641223 0.767354i \(-0.278427\pi\)
0.641223 + 0.767354i \(0.278427\pi\)
\(198\) −4.00000 −0.284268
\(199\) −5.00000 −0.354441 −0.177220 0.984171i \(-0.556711\pi\)
−0.177220 + 0.984171i \(0.556711\pi\)
\(200\) −4.00000 −0.282843
\(201\) 2.00000 0.141069
\(202\) −18.0000 −1.26648
\(203\) −15.0000 −1.05279
\(204\) 2.00000 0.140028
\(205\) 7.00000 0.488901
\(206\) −16.0000 −1.11477
\(207\) −8.00000 −0.556038
\(208\) −6.00000 −0.416025
\(209\) −10.0000 −0.691714
\(210\) −3.00000 −0.207020
\(211\) −8.00000 −0.550743 −0.275371 0.961338i \(-0.588801\pi\)
−0.275371 + 0.961338i \(0.588801\pi\)
\(212\) 9.00000 0.618123
\(213\) −12.0000 −0.822226
\(214\) 3.00000 0.205076
\(215\) −6.00000 −0.409197
\(216\) 5.00000 0.340207
\(217\) 6.00000 0.407307
\(218\) −20.0000 −1.35457
\(219\) −4.00000 −0.270295
\(220\) 2.00000 0.134840
\(221\) 12.0000 0.807207
\(222\) −8.00000 −0.536925
\(223\) 24.0000 1.60716 0.803579 0.595198i \(-0.202926\pi\)
0.803579 + 0.595198i \(0.202926\pi\)
\(224\) 3.00000 0.200446
\(225\) 8.00000 0.533333
\(226\) 14.0000 0.931266
\(227\) 8.00000 0.530979 0.265489 0.964114i \(-0.414466\pi\)
0.265489 + 0.964114i \(0.414466\pi\)
\(228\) 5.00000 0.331133
\(229\) −10.0000 −0.660819 −0.330409 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) 4.00000 0.263752
\(231\) −6.00000 −0.394771
\(232\) −5.00000 −0.328266
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 12.0000 0.784465
\(235\) −2.00000 −0.130466
\(236\) −1.00000 −0.0650945
\(237\) −5.00000 −0.324785
\(238\) −6.00000 −0.388922
\(239\) −25.0000 −1.61712 −0.808558 0.588417i \(-0.799751\pi\)
−0.808558 + 0.588417i \(0.799751\pi\)
\(240\) −1.00000 −0.0645497
\(241\) 17.0000 1.09507 0.547533 0.836784i \(-0.315567\pi\)
0.547533 + 0.836784i \(0.315567\pi\)
\(242\) −7.00000 −0.449977
\(243\) −16.0000 −1.02640
\(244\) −8.00000 −0.512148
\(245\) 2.00000 0.127775
\(246\) −7.00000 −0.446304
\(247\) 30.0000 1.90885
\(248\) 2.00000 0.127000
\(249\) −14.0000 −0.887214
\(250\) −9.00000 −0.569210
\(251\) −23.0000 −1.45175 −0.725874 0.687828i \(-0.758564\pi\)
−0.725874 + 0.687828i \(0.758564\pi\)
\(252\) −6.00000 −0.377964
\(253\) 8.00000 0.502956
\(254\) 13.0000 0.815693
\(255\) 2.00000 0.125245
\(256\) 1.00000 0.0625000
\(257\) −27.0000 −1.68421 −0.842107 0.539311i \(-0.818685\pi\)
−0.842107 + 0.539311i \(0.818685\pi\)
\(258\) 6.00000 0.373544
\(259\) 24.0000 1.49129
\(260\) −6.00000 −0.372104
\(261\) 10.0000 0.618984
\(262\) −18.0000 −1.11204
\(263\) −21.0000 −1.29492 −0.647458 0.762101i \(-0.724168\pi\)
−0.647458 + 0.762101i \(0.724168\pi\)
\(264\) −2.00000 −0.123091
\(265\) 9.00000 0.552866
\(266\) −15.0000 −0.919709
\(267\) 0 0
\(268\) −2.00000 −0.122169
\(269\) −10.0000 −0.609711 −0.304855 0.952399i \(-0.598608\pi\)
−0.304855 + 0.952399i \(0.598608\pi\)
\(270\) 5.00000 0.304290
\(271\) 17.0000 1.03268 0.516338 0.856385i \(-0.327295\pi\)
0.516338 + 0.856385i \(0.327295\pi\)
\(272\) −2.00000 −0.121268
\(273\) 18.0000 1.08941
\(274\) 3.00000 0.181237
\(275\) −8.00000 −0.482418
\(276\) −4.00000 −0.240772
\(277\) −27.0000 −1.62227 −0.811136 0.584857i \(-0.801151\pi\)
−0.811136 + 0.584857i \(0.801151\pi\)
\(278\) 20.0000 1.19952
\(279\) −4.00000 −0.239474
\(280\) 3.00000 0.179284
\(281\) −13.0000 −0.775515 −0.387757 0.921761i \(-0.626750\pi\)
−0.387757 + 0.921761i \(0.626750\pi\)
\(282\) 2.00000 0.119098
\(283\) −16.0000 −0.951101 −0.475551 0.879688i \(-0.657751\pi\)
−0.475551 + 0.879688i \(0.657751\pi\)
\(284\) 12.0000 0.712069
\(285\) 5.00000 0.296174
\(286\) −12.0000 −0.709575
\(287\) 21.0000 1.23959
\(288\) −2.00000 −0.117851
\(289\) −13.0000 −0.764706
\(290\) −5.00000 −0.293610
\(291\) −8.00000 −0.468968
\(292\) 4.00000 0.234082
\(293\) 19.0000 1.10999 0.554996 0.831853i \(-0.312720\pi\)
0.554996 + 0.831853i \(0.312720\pi\)
\(294\) −2.00000 −0.116642
\(295\) −1.00000 −0.0582223
\(296\) 8.00000 0.464991
\(297\) 10.0000 0.580259
\(298\) 10.0000 0.579284
\(299\) −24.0000 −1.38796
\(300\) 4.00000 0.230940
\(301\) −18.0000 −1.03750
\(302\) 2.00000 0.115087
\(303\) 18.0000 1.03407
\(304\) −5.00000 −0.286770
\(305\) −8.00000 −0.458079
\(306\) 4.00000 0.228665
\(307\) 13.0000 0.741949 0.370975 0.928643i \(-0.379024\pi\)
0.370975 + 0.928643i \(0.379024\pi\)
\(308\) 6.00000 0.341882
\(309\) 16.0000 0.910208
\(310\) 2.00000 0.113592
\(311\) 7.00000 0.396934 0.198467 0.980108i \(-0.436404\pi\)
0.198467 + 0.980108i \(0.436404\pi\)
\(312\) 6.00000 0.339683
\(313\) 24.0000 1.35656 0.678280 0.734803i \(-0.262726\pi\)
0.678280 + 0.734803i \(0.262726\pi\)
\(314\) −12.0000 −0.677199
\(315\) −6.00000 −0.338062
\(316\) 5.00000 0.281272
\(317\) −2.00000 −0.112331 −0.0561656 0.998421i \(-0.517887\pi\)
−0.0561656 + 0.998421i \(0.517887\pi\)
\(318\) −9.00000 −0.504695
\(319\) −10.0000 −0.559893
\(320\) 1.00000 0.0559017
\(321\) −3.00000 −0.167444
\(322\) 12.0000 0.668734
\(323\) 10.0000 0.556415
\(324\) 1.00000 0.0555556
\(325\) 24.0000 1.33128
\(326\) 4.00000 0.221540
\(327\) 20.0000 1.10600
\(328\) 7.00000 0.386510
\(329\) −6.00000 −0.330791
\(330\) −2.00000 −0.110096
\(331\) 27.0000 1.48405 0.742027 0.670370i \(-0.233865\pi\)
0.742027 + 0.670370i \(0.233865\pi\)
\(332\) 14.0000 0.768350
\(333\) −16.0000 −0.876795
\(334\) 3.00000 0.164153
\(335\) −2.00000 −0.109272
\(336\) −3.00000 −0.163663
\(337\) −32.0000 −1.74315 −0.871576 0.490261i \(-0.836901\pi\)
−0.871576 + 0.490261i \(0.836901\pi\)
\(338\) 23.0000 1.25104
\(339\) −14.0000 −0.760376
\(340\) −2.00000 −0.108465
\(341\) 4.00000 0.216612
\(342\) 10.0000 0.540738
\(343\) −15.0000 −0.809924
\(344\) −6.00000 −0.323498
\(345\) −4.00000 −0.215353
\(346\) 4.00000 0.215041
\(347\) −22.0000 −1.18102 −0.590511 0.807030i \(-0.701074\pi\)
−0.590511 + 0.807030i \(0.701074\pi\)
\(348\) 5.00000 0.268028
\(349\) 10.0000 0.535288 0.267644 0.963518i \(-0.413755\pi\)
0.267644 + 0.963518i \(0.413755\pi\)
\(350\) −12.0000 −0.641427
\(351\) −30.0000 −1.60128
\(352\) 2.00000 0.106600
\(353\) 4.00000 0.212899 0.106449 0.994318i \(-0.466052\pi\)
0.106449 + 0.994318i \(0.466052\pi\)
\(354\) 1.00000 0.0531494
\(355\) 12.0000 0.636894
\(356\) 0 0
\(357\) 6.00000 0.317554
\(358\) 20.0000 1.05703
\(359\) −15.0000 −0.791670 −0.395835 0.918322i \(-0.629545\pi\)
−0.395835 + 0.918322i \(0.629545\pi\)
\(360\) −2.00000 −0.105409
\(361\) 6.00000 0.315789
\(362\) −13.0000 −0.683265
\(363\) 7.00000 0.367405
\(364\) −18.0000 −0.943456
\(365\) 4.00000 0.209370
\(366\) 8.00000 0.418167
\(367\) −2.00000 −0.104399 −0.0521996 0.998637i \(-0.516623\pi\)
−0.0521996 + 0.998637i \(0.516623\pi\)
\(368\) 4.00000 0.208514
\(369\) −14.0000 −0.728811
\(370\) 8.00000 0.415900
\(371\) 27.0000 1.40177
\(372\) −2.00000 −0.103695
\(373\) −26.0000 −1.34623 −0.673114 0.739538i \(-0.735044\pi\)
−0.673114 + 0.739538i \(0.735044\pi\)
\(374\) −4.00000 −0.206835
\(375\) 9.00000 0.464758
\(376\) −2.00000 −0.103142
\(377\) 30.0000 1.54508
\(378\) 15.0000 0.771517
\(379\) 5.00000 0.256833 0.128416 0.991720i \(-0.459011\pi\)
0.128416 + 0.991720i \(0.459011\pi\)
\(380\) −5.00000 −0.256495
\(381\) −13.0000 −0.666010
\(382\) −18.0000 −0.920960
\(383\) −16.0000 −0.817562 −0.408781 0.912633i \(-0.634046\pi\)
−0.408781 + 0.912633i \(0.634046\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 6.00000 0.305788
\(386\) 19.0000 0.967075
\(387\) 12.0000 0.609994
\(388\) 8.00000 0.406138
\(389\) −10.0000 −0.507020 −0.253510 0.967333i \(-0.581585\pi\)
−0.253510 + 0.967333i \(0.581585\pi\)
\(390\) 6.00000 0.303822
\(391\) −8.00000 −0.404577
\(392\) 2.00000 0.101015
\(393\) 18.0000 0.907980
\(394\) 18.0000 0.906827
\(395\) 5.00000 0.251577
\(396\) −4.00000 −0.201008
\(397\) −2.00000 −0.100377 −0.0501886 0.998740i \(-0.515982\pi\)
−0.0501886 + 0.998740i \(0.515982\pi\)
\(398\) −5.00000 −0.250627
\(399\) 15.0000 0.750939
\(400\) −4.00000 −0.200000
\(401\) 12.0000 0.599251 0.299626 0.954057i \(-0.403138\pi\)
0.299626 + 0.954057i \(0.403138\pi\)
\(402\) 2.00000 0.0997509
\(403\) −12.0000 −0.597763
\(404\) −18.0000 −0.895533
\(405\) 1.00000 0.0496904
\(406\) −15.0000 −0.744438
\(407\) 16.0000 0.793091
\(408\) 2.00000 0.0990148
\(409\) −20.0000 −0.988936 −0.494468 0.869196i \(-0.664637\pi\)
−0.494468 + 0.869196i \(0.664637\pi\)
\(410\) 7.00000 0.345705
\(411\) −3.00000 −0.147979
\(412\) −16.0000 −0.788263
\(413\) −3.00000 −0.147620
\(414\) −8.00000 −0.393179
\(415\) 14.0000 0.687233
\(416\) −6.00000 −0.294174
\(417\) −20.0000 −0.979404
\(418\) −10.0000 −0.489116
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) −3.00000 −0.146385
\(421\) −8.00000 −0.389896 −0.194948 0.980814i \(-0.562454\pi\)
−0.194948 + 0.980814i \(0.562454\pi\)
\(422\) −8.00000 −0.389434
\(423\) 4.00000 0.194487
\(424\) 9.00000 0.437079
\(425\) 8.00000 0.388057
\(426\) −12.0000 −0.581402
\(427\) −24.0000 −1.16144
\(428\) 3.00000 0.145010
\(429\) 12.0000 0.579365
\(430\) −6.00000 −0.289346
\(431\) −18.0000 −0.867029 −0.433515 0.901146i \(-0.642727\pi\)
−0.433515 + 0.901146i \(0.642727\pi\)
\(432\) 5.00000 0.240563
\(433\) 19.0000 0.913082 0.456541 0.889702i \(-0.349088\pi\)
0.456541 + 0.889702i \(0.349088\pi\)
\(434\) 6.00000 0.288009
\(435\) 5.00000 0.239732
\(436\) −20.0000 −0.957826
\(437\) −20.0000 −0.956730
\(438\) −4.00000 −0.191127
\(439\) 20.0000 0.954548 0.477274 0.878755i \(-0.341625\pi\)
0.477274 + 0.878755i \(0.341625\pi\)
\(440\) 2.00000 0.0953463
\(441\) −4.00000 −0.190476
\(442\) 12.0000 0.570782
\(443\) 24.0000 1.14027 0.570137 0.821549i \(-0.306890\pi\)
0.570137 + 0.821549i \(0.306890\pi\)
\(444\) −8.00000 −0.379663
\(445\) 0 0
\(446\) 24.0000 1.13643
\(447\) −10.0000 −0.472984
\(448\) 3.00000 0.141737
\(449\) 5.00000 0.235965 0.117982 0.993016i \(-0.462357\pi\)
0.117982 + 0.993016i \(0.462357\pi\)
\(450\) 8.00000 0.377124
\(451\) 14.0000 0.659234
\(452\) 14.0000 0.658505
\(453\) −2.00000 −0.0939682
\(454\) 8.00000 0.375459
\(455\) −18.0000 −0.843853
\(456\) 5.00000 0.234146
\(457\) −22.0000 −1.02912 −0.514558 0.857455i \(-0.672044\pi\)
−0.514558 + 0.857455i \(0.672044\pi\)
\(458\) −10.0000 −0.467269
\(459\) −10.0000 −0.466760
\(460\) 4.00000 0.186501
\(461\) −18.0000 −0.838344 −0.419172 0.907907i \(-0.637680\pi\)
−0.419172 + 0.907907i \(0.637680\pi\)
\(462\) −6.00000 −0.279145
\(463\) −36.0000 −1.67306 −0.836531 0.547920i \(-0.815420\pi\)
−0.836531 + 0.547920i \(0.815420\pi\)
\(464\) −5.00000 −0.232119
\(465\) −2.00000 −0.0927478
\(466\) −6.00000 −0.277945
\(467\) 28.0000 1.29569 0.647843 0.761774i \(-0.275671\pi\)
0.647843 + 0.761774i \(0.275671\pi\)
\(468\) 12.0000 0.554700
\(469\) −6.00000 −0.277054
\(470\) −2.00000 −0.0922531
\(471\) 12.0000 0.552931
\(472\) −1.00000 −0.0460287
\(473\) −12.0000 −0.551761
\(474\) −5.00000 −0.229658
\(475\) 20.0000 0.917663
\(476\) −6.00000 −0.275010
\(477\) −18.0000 −0.824163
\(478\) −25.0000 −1.14347
\(479\) 40.0000 1.82765 0.913823 0.406112i \(-0.133116\pi\)
0.913823 + 0.406112i \(0.133116\pi\)
\(480\) −1.00000 −0.0456435
\(481\) −48.0000 −2.18861
\(482\) 17.0000 0.774329
\(483\) −12.0000 −0.546019
\(484\) −7.00000 −0.318182
\(485\) 8.00000 0.363261
\(486\) −16.0000 −0.725775
\(487\) −17.0000 −0.770344 −0.385172 0.922845i \(-0.625858\pi\)
−0.385172 + 0.922845i \(0.625858\pi\)
\(488\) −8.00000 −0.362143
\(489\) −4.00000 −0.180886
\(490\) 2.00000 0.0903508
\(491\) 37.0000 1.66979 0.834893 0.550412i \(-0.185529\pi\)
0.834893 + 0.550412i \(0.185529\pi\)
\(492\) −7.00000 −0.315584
\(493\) 10.0000 0.450377
\(494\) 30.0000 1.34976
\(495\) −4.00000 −0.179787
\(496\) 2.00000 0.0898027
\(497\) 36.0000 1.61482
\(498\) −14.0000 −0.627355
\(499\) 15.0000 0.671492 0.335746 0.941953i \(-0.391012\pi\)
0.335746 + 0.941953i \(0.391012\pi\)
\(500\) −9.00000 −0.402492
\(501\) −3.00000 −0.134030
\(502\) −23.0000 −1.02654
\(503\) 24.0000 1.07011 0.535054 0.844818i \(-0.320291\pi\)
0.535054 + 0.844818i \(0.320291\pi\)
\(504\) −6.00000 −0.267261
\(505\) −18.0000 −0.800989
\(506\) 8.00000 0.355643
\(507\) −23.0000 −1.02147
\(508\) 13.0000 0.576782
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 2.00000 0.0885615
\(511\) 12.0000 0.530849
\(512\) 1.00000 0.0441942
\(513\) −25.0000 −1.10378
\(514\) −27.0000 −1.19092
\(515\) −16.0000 −0.705044
\(516\) 6.00000 0.264135
\(517\) −4.00000 −0.175920
\(518\) 24.0000 1.05450
\(519\) −4.00000 −0.175581
\(520\) −6.00000 −0.263117
\(521\) 22.0000 0.963837 0.481919 0.876216i \(-0.339940\pi\)
0.481919 + 0.876216i \(0.339940\pi\)
\(522\) 10.0000 0.437688
\(523\) −11.0000 −0.480996 −0.240498 0.970650i \(-0.577311\pi\)
−0.240498 + 0.970650i \(0.577311\pi\)
\(524\) −18.0000 −0.786334
\(525\) 12.0000 0.523723
\(526\) −21.0000 −0.915644
\(527\) −4.00000 −0.174243
\(528\) −2.00000 −0.0870388
\(529\) −7.00000 −0.304348
\(530\) 9.00000 0.390935
\(531\) 2.00000 0.0867926
\(532\) −15.0000 −0.650332
\(533\) −42.0000 −1.81922
\(534\) 0 0
\(535\) 3.00000 0.129701
\(536\) −2.00000 −0.0863868
\(537\) −20.0000 −0.863064
\(538\) −10.0000 −0.431131
\(539\) 4.00000 0.172292
\(540\) 5.00000 0.215166
\(541\) 32.0000 1.37579 0.687894 0.725811i \(-0.258536\pi\)
0.687894 + 0.725811i \(0.258536\pi\)
\(542\) 17.0000 0.730213
\(543\) 13.0000 0.557883
\(544\) −2.00000 −0.0857493
\(545\) −20.0000 −0.856706
\(546\) 18.0000 0.770329
\(547\) 28.0000 1.19719 0.598597 0.801050i \(-0.295725\pi\)
0.598597 + 0.801050i \(0.295725\pi\)
\(548\) 3.00000 0.128154
\(549\) 16.0000 0.682863
\(550\) −8.00000 −0.341121
\(551\) 25.0000 1.06504
\(552\) −4.00000 −0.170251
\(553\) 15.0000 0.637865
\(554\) −27.0000 −1.14712
\(555\) −8.00000 −0.339581
\(556\) 20.0000 0.848189
\(557\) 33.0000 1.39825 0.699127 0.714997i \(-0.253572\pi\)
0.699127 + 0.714997i \(0.253572\pi\)
\(558\) −4.00000 −0.169334
\(559\) 36.0000 1.52264
\(560\) 3.00000 0.126773
\(561\) 4.00000 0.168880
\(562\) −13.0000 −0.548372
\(563\) 4.00000 0.168580 0.0842900 0.996441i \(-0.473138\pi\)
0.0842900 + 0.996441i \(0.473138\pi\)
\(564\) 2.00000 0.0842152
\(565\) 14.0000 0.588984
\(566\) −16.0000 −0.672530
\(567\) 3.00000 0.125988
\(568\) 12.0000 0.503509
\(569\) −30.0000 −1.25767 −0.628833 0.777541i \(-0.716467\pi\)
−0.628833 + 0.777541i \(0.716467\pi\)
\(570\) 5.00000 0.209427
\(571\) 22.0000 0.920671 0.460336 0.887745i \(-0.347729\pi\)
0.460336 + 0.887745i \(0.347729\pi\)
\(572\) −12.0000 −0.501745
\(573\) 18.0000 0.751961
\(574\) 21.0000 0.876523
\(575\) −16.0000 −0.667246
\(576\) −2.00000 −0.0833333
\(577\) −7.00000 −0.291414 −0.145707 0.989328i \(-0.546546\pi\)
−0.145707 + 0.989328i \(0.546546\pi\)
\(578\) −13.0000 −0.540729
\(579\) −19.0000 −0.789613
\(580\) −5.00000 −0.207614
\(581\) 42.0000 1.74245
\(582\) −8.00000 −0.331611
\(583\) 18.0000 0.745484
\(584\) 4.00000 0.165521
\(585\) 12.0000 0.496139
\(586\) 19.0000 0.784883
\(587\) −12.0000 −0.495293 −0.247647 0.968850i \(-0.579657\pi\)
−0.247647 + 0.968850i \(0.579657\pi\)
\(588\) −2.00000 −0.0824786
\(589\) −10.0000 −0.412043
\(590\) −1.00000 −0.0411693
\(591\) −18.0000 −0.740421
\(592\) 8.00000 0.328798
\(593\) −11.0000 −0.451716 −0.225858 0.974160i \(-0.572519\pi\)
−0.225858 + 0.974160i \(0.572519\pi\)
\(594\) 10.0000 0.410305
\(595\) −6.00000 −0.245976
\(596\) 10.0000 0.409616
\(597\) 5.00000 0.204636
\(598\) −24.0000 −0.981433
\(599\) −15.0000 −0.612883 −0.306442 0.951889i \(-0.599138\pi\)
−0.306442 + 0.951889i \(0.599138\pi\)
\(600\) 4.00000 0.163299
\(601\) −8.00000 −0.326327 −0.163163 0.986599i \(-0.552170\pi\)
−0.163163 + 0.986599i \(0.552170\pi\)
\(602\) −18.0000 −0.733625
\(603\) 4.00000 0.162893
\(604\) 2.00000 0.0813788
\(605\) −7.00000 −0.284590
\(606\) 18.0000 0.731200
\(607\) 3.00000 0.121766 0.0608831 0.998145i \(-0.480608\pi\)
0.0608831 + 0.998145i \(0.480608\pi\)
\(608\) −5.00000 −0.202777
\(609\) 15.0000 0.607831
\(610\) −8.00000 −0.323911
\(611\) 12.0000 0.485468
\(612\) 4.00000 0.161690
\(613\) −16.0000 −0.646234 −0.323117 0.946359i \(-0.604731\pi\)
−0.323117 + 0.946359i \(0.604731\pi\)
\(614\) 13.0000 0.524637
\(615\) −7.00000 −0.282267
\(616\) 6.00000 0.241747
\(617\) −7.00000 −0.281809 −0.140905 0.990023i \(-0.545001\pi\)
−0.140905 + 0.990023i \(0.545001\pi\)
\(618\) 16.0000 0.643614
\(619\) 5.00000 0.200967 0.100483 0.994939i \(-0.467961\pi\)
0.100483 + 0.994939i \(0.467961\pi\)
\(620\) 2.00000 0.0803219
\(621\) 20.0000 0.802572
\(622\) 7.00000 0.280674
\(623\) 0 0
\(624\) 6.00000 0.240192
\(625\) 11.0000 0.440000
\(626\) 24.0000 0.959233
\(627\) 10.0000 0.399362
\(628\) −12.0000 −0.478852
\(629\) −16.0000 −0.637962
\(630\) −6.00000 −0.239046
\(631\) 32.0000 1.27390 0.636950 0.770905i \(-0.280196\pi\)
0.636950 + 0.770905i \(0.280196\pi\)
\(632\) 5.00000 0.198889
\(633\) 8.00000 0.317971
\(634\) −2.00000 −0.0794301
\(635\) 13.0000 0.515889
\(636\) −9.00000 −0.356873
\(637\) −12.0000 −0.475457
\(638\) −10.0000 −0.395904
\(639\) −24.0000 −0.949425
\(640\) 1.00000 0.0395285
\(641\) 42.0000 1.65890 0.829450 0.558581i \(-0.188654\pi\)
0.829450 + 0.558581i \(0.188654\pi\)
\(642\) −3.00000 −0.118401
\(643\) 19.0000 0.749287 0.374643 0.927169i \(-0.377765\pi\)
0.374643 + 0.927169i \(0.377765\pi\)
\(644\) 12.0000 0.472866
\(645\) 6.00000 0.236250
\(646\) 10.0000 0.393445
\(647\) 23.0000 0.904223 0.452112 0.891961i \(-0.350671\pi\)
0.452112 + 0.891961i \(0.350671\pi\)
\(648\) 1.00000 0.0392837
\(649\) −2.00000 −0.0785069
\(650\) 24.0000 0.941357
\(651\) −6.00000 −0.235159
\(652\) 4.00000 0.156652
\(653\) 9.00000 0.352197 0.176099 0.984373i \(-0.443652\pi\)
0.176099 + 0.984373i \(0.443652\pi\)
\(654\) 20.0000 0.782062
\(655\) −18.0000 −0.703318
\(656\) 7.00000 0.273304
\(657\) −8.00000 −0.312110
\(658\) −6.00000 −0.233904
\(659\) −40.0000 −1.55818 −0.779089 0.626913i \(-0.784318\pi\)
−0.779089 + 0.626913i \(0.784318\pi\)
\(660\) −2.00000 −0.0778499
\(661\) 17.0000 0.661223 0.330612 0.943767i \(-0.392745\pi\)
0.330612 + 0.943767i \(0.392745\pi\)
\(662\) 27.0000 1.04938
\(663\) −12.0000 −0.466041
\(664\) 14.0000 0.543305
\(665\) −15.0000 −0.581675
\(666\) −16.0000 −0.619987
\(667\) −20.0000 −0.774403
\(668\) 3.00000 0.116073
\(669\) −24.0000 −0.927894
\(670\) −2.00000 −0.0772667
\(671\) −16.0000 −0.617673
\(672\) −3.00000 −0.115728
\(673\) −36.0000 −1.38770 −0.693849 0.720121i \(-0.744086\pi\)
−0.693849 + 0.720121i \(0.744086\pi\)
\(674\) −32.0000 −1.23259
\(675\) −20.0000 −0.769800
\(676\) 23.0000 0.884615
\(677\) 18.0000 0.691796 0.345898 0.938272i \(-0.387574\pi\)
0.345898 + 0.938272i \(0.387574\pi\)
\(678\) −14.0000 −0.537667
\(679\) 24.0000 0.921035
\(680\) −2.00000 −0.0766965
\(681\) −8.00000 −0.306561
\(682\) 4.00000 0.153168
\(683\) −26.0000 −0.994862 −0.497431 0.867503i \(-0.665723\pi\)
−0.497431 + 0.867503i \(0.665723\pi\)
\(684\) 10.0000 0.382360
\(685\) 3.00000 0.114624
\(686\) −15.0000 −0.572703
\(687\) 10.0000 0.381524
\(688\) −6.00000 −0.228748
\(689\) −54.0000 −2.05724
\(690\) −4.00000 −0.152277
\(691\) −8.00000 −0.304334 −0.152167 0.988355i \(-0.548625\pi\)
−0.152167 + 0.988355i \(0.548625\pi\)
\(692\) 4.00000 0.152057
\(693\) −12.0000 −0.455842
\(694\) −22.0000 −0.835109
\(695\) 20.0000 0.758643
\(696\) 5.00000 0.189525
\(697\) −14.0000 −0.530288
\(698\) 10.0000 0.378506
\(699\) 6.00000 0.226941
\(700\) −12.0000 −0.453557
\(701\) 12.0000 0.453234 0.226617 0.973984i \(-0.427233\pi\)
0.226617 + 0.973984i \(0.427233\pi\)
\(702\) −30.0000 −1.13228
\(703\) −40.0000 −1.50863
\(704\) 2.00000 0.0753778
\(705\) 2.00000 0.0753244
\(706\) 4.00000 0.150542
\(707\) −54.0000 −2.03088
\(708\) 1.00000 0.0375823
\(709\) −5.00000 −0.187779 −0.0938895 0.995583i \(-0.529930\pi\)
−0.0938895 + 0.995583i \(0.529930\pi\)
\(710\) 12.0000 0.450352
\(711\) −10.0000 −0.375029
\(712\) 0 0
\(713\) 8.00000 0.299602
\(714\) 6.00000 0.224544
\(715\) −12.0000 −0.448775
\(716\) 20.0000 0.747435
\(717\) 25.0000 0.933642
\(718\) −15.0000 −0.559795
\(719\) −20.0000 −0.745874 −0.372937 0.927857i \(-0.621649\pi\)
−0.372937 + 0.927857i \(0.621649\pi\)
\(720\) −2.00000 −0.0745356
\(721\) −48.0000 −1.78761
\(722\) 6.00000 0.223297
\(723\) −17.0000 −0.632237
\(724\) −13.0000 −0.483141
\(725\) 20.0000 0.742781
\(726\) 7.00000 0.259794
\(727\) −32.0000 −1.18681 −0.593407 0.804902i \(-0.702218\pi\)
−0.593407 + 0.804902i \(0.702218\pi\)
\(728\) −18.0000 −0.667124
\(729\) 13.0000 0.481481
\(730\) 4.00000 0.148047
\(731\) 12.0000 0.443836
\(732\) 8.00000 0.295689
\(733\) 14.0000 0.517102 0.258551 0.965998i \(-0.416755\pi\)
0.258551 + 0.965998i \(0.416755\pi\)
\(734\) −2.00000 −0.0738213
\(735\) −2.00000 −0.0737711
\(736\) 4.00000 0.147442
\(737\) −4.00000 −0.147342
\(738\) −14.0000 −0.515347
\(739\) 20.0000 0.735712 0.367856 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(740\) 8.00000 0.294086
\(741\) −30.0000 −1.10208
\(742\) 27.0000 0.991201
\(743\) 24.0000 0.880475 0.440237 0.897881i \(-0.354894\pi\)
0.440237 + 0.897881i \(0.354894\pi\)
\(744\) −2.00000 −0.0733236
\(745\) 10.0000 0.366372
\(746\) −26.0000 −0.951928
\(747\) −28.0000 −1.02447
\(748\) −4.00000 −0.146254
\(749\) 9.00000 0.328853
\(750\) 9.00000 0.328634
\(751\) −18.0000 −0.656829 −0.328415 0.944534i \(-0.606514\pi\)
−0.328415 + 0.944534i \(0.606514\pi\)
\(752\) −2.00000 −0.0729325
\(753\) 23.0000 0.838167
\(754\) 30.0000 1.09254
\(755\) 2.00000 0.0727875
\(756\) 15.0000 0.545545
\(757\) 23.0000 0.835949 0.417975 0.908459i \(-0.362740\pi\)
0.417975 + 0.908459i \(0.362740\pi\)
\(758\) 5.00000 0.181608
\(759\) −8.00000 −0.290382
\(760\) −5.00000 −0.181369
\(761\) −43.0000 −1.55875 −0.779374 0.626559i \(-0.784463\pi\)
−0.779374 + 0.626559i \(0.784463\pi\)
\(762\) −13.0000 −0.470940
\(763\) −60.0000 −2.17215
\(764\) −18.0000 −0.651217
\(765\) 4.00000 0.144620
\(766\) −16.0000 −0.578103
\(767\) 6.00000 0.216647
\(768\) −1.00000 −0.0360844
\(769\) 30.0000 1.08183 0.540914 0.841078i \(-0.318079\pi\)
0.540914 + 0.841078i \(0.318079\pi\)
\(770\) 6.00000 0.216225
\(771\) 27.0000 0.972381
\(772\) 19.0000 0.683825
\(773\) 24.0000 0.863220 0.431610 0.902060i \(-0.357946\pi\)
0.431610 + 0.902060i \(0.357946\pi\)
\(774\) 12.0000 0.431331
\(775\) −8.00000 −0.287368
\(776\) 8.00000 0.287183
\(777\) −24.0000 −0.860995
\(778\) −10.0000 −0.358517
\(779\) −35.0000 −1.25401
\(780\) 6.00000 0.214834
\(781\) 24.0000 0.858788
\(782\) −8.00000 −0.286079
\(783\) −25.0000 −0.893427
\(784\) 2.00000 0.0714286
\(785\) −12.0000 −0.428298
\(786\) 18.0000 0.642039
\(787\) −12.0000 −0.427754 −0.213877 0.976861i \(-0.568609\pi\)
−0.213877 + 0.976861i \(0.568609\pi\)
\(788\) 18.0000 0.641223
\(789\) 21.0000 0.747620
\(790\) 5.00000 0.177892
\(791\) 42.0000 1.49335
\(792\) −4.00000 −0.142134
\(793\) 48.0000 1.70453
\(794\) −2.00000 −0.0709773
\(795\) −9.00000 −0.319197
\(796\) −5.00000 −0.177220
\(797\) 8.00000 0.283375 0.141687 0.989911i \(-0.454747\pi\)
0.141687 + 0.989911i \(0.454747\pi\)
\(798\) 15.0000 0.530994
\(799\) 4.00000 0.141510
\(800\) −4.00000 −0.141421
\(801\) 0 0
\(802\) 12.0000 0.423735
\(803\) 8.00000 0.282314
\(804\) 2.00000 0.0705346
\(805\) 12.0000 0.422944
\(806\) −12.0000 −0.422682
\(807\) 10.0000 0.352017
\(808\) −18.0000 −0.633238
\(809\) 30.0000 1.05474 0.527372 0.849635i \(-0.323177\pi\)
0.527372 + 0.849635i \(0.323177\pi\)
\(810\) 1.00000 0.0351364
\(811\) 42.0000 1.47482 0.737410 0.675446i \(-0.236049\pi\)
0.737410 + 0.675446i \(0.236049\pi\)
\(812\) −15.0000 −0.526397
\(813\) −17.0000 −0.596216
\(814\) 16.0000 0.560800
\(815\) 4.00000 0.140114
\(816\) 2.00000 0.0700140
\(817\) 30.0000 1.04957
\(818\) −20.0000 −0.699284
\(819\) 36.0000 1.25794
\(820\) 7.00000 0.244451
\(821\) −8.00000 −0.279202 −0.139601 0.990208i \(-0.544582\pi\)
−0.139601 + 0.990208i \(0.544582\pi\)
\(822\) −3.00000 −0.104637
\(823\) 4.00000 0.139431 0.0697156 0.997567i \(-0.477791\pi\)
0.0697156 + 0.997567i \(0.477791\pi\)
\(824\) −16.0000 −0.557386
\(825\) 8.00000 0.278524
\(826\) −3.00000 −0.104383
\(827\) −12.0000 −0.417281 −0.208640 0.977992i \(-0.566904\pi\)
−0.208640 + 0.977992i \(0.566904\pi\)
\(828\) −8.00000 −0.278019
\(829\) −5.00000 −0.173657 −0.0868286 0.996223i \(-0.527673\pi\)
−0.0868286 + 0.996223i \(0.527673\pi\)
\(830\) 14.0000 0.485947
\(831\) 27.0000 0.936620
\(832\) −6.00000 −0.208013
\(833\) −4.00000 −0.138592
\(834\) −20.0000 −0.692543
\(835\) 3.00000 0.103819
\(836\) −10.0000 −0.345857
\(837\) 10.0000 0.345651
\(838\) 0 0
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) −3.00000 −0.103510
\(841\) −4.00000 −0.137931
\(842\) −8.00000 −0.275698
\(843\) 13.0000 0.447744
\(844\) −8.00000 −0.275371
\(845\) 23.0000 0.791224
\(846\) 4.00000 0.137523
\(847\) −21.0000 −0.721569
\(848\) 9.00000 0.309061
\(849\) 16.0000 0.549119
\(850\) 8.00000 0.274398
\(851\) 32.0000 1.09695
\(852\) −12.0000 −0.411113
\(853\) −21.0000 −0.719026 −0.359513 0.933140i \(-0.617057\pi\)
−0.359513 + 0.933140i \(0.617057\pi\)
\(854\) −24.0000 −0.821263
\(855\) 10.0000 0.341993
\(856\) 3.00000 0.102538
\(857\) −52.0000 −1.77629 −0.888143 0.459567i \(-0.848005\pi\)
−0.888143 + 0.459567i \(0.848005\pi\)
\(858\) 12.0000 0.409673
\(859\) 20.0000 0.682391 0.341196 0.939992i \(-0.389168\pi\)
0.341196 + 0.939992i \(0.389168\pi\)
\(860\) −6.00000 −0.204598
\(861\) −21.0000 −0.715678
\(862\) −18.0000 −0.613082
\(863\) 4.00000 0.136162 0.0680808 0.997680i \(-0.478312\pi\)
0.0680808 + 0.997680i \(0.478312\pi\)
\(864\) 5.00000 0.170103
\(865\) 4.00000 0.136004
\(866\) 19.0000 0.645646
\(867\) 13.0000 0.441503
\(868\) 6.00000 0.203653
\(869\) 10.0000 0.339227
\(870\) 5.00000 0.169516
\(871\) 12.0000 0.406604
\(872\) −20.0000 −0.677285
\(873\) −16.0000 −0.541518
\(874\) −20.0000 −0.676510
\(875\) −27.0000 −0.912767
\(876\) −4.00000 −0.135147
\(877\) −37.0000 −1.24940 −0.624701 0.780864i \(-0.714779\pi\)
−0.624701 + 0.780864i \(0.714779\pi\)
\(878\) 20.0000 0.674967
\(879\) −19.0000 −0.640854
\(880\) 2.00000 0.0674200
\(881\) 2.00000 0.0673817 0.0336909 0.999432i \(-0.489274\pi\)
0.0336909 + 0.999432i \(0.489274\pi\)
\(882\) −4.00000 −0.134687
\(883\) 29.0000 0.975928 0.487964 0.872864i \(-0.337740\pi\)
0.487964 + 0.872864i \(0.337740\pi\)
\(884\) 12.0000 0.403604
\(885\) 1.00000 0.0336146
\(886\) 24.0000 0.806296
\(887\) 48.0000 1.61168 0.805841 0.592132i \(-0.201714\pi\)
0.805841 + 0.592132i \(0.201714\pi\)
\(888\) −8.00000 −0.268462
\(889\) 39.0000 1.30802
\(890\) 0 0
\(891\) 2.00000 0.0670025
\(892\) 24.0000 0.803579
\(893\) 10.0000 0.334637
\(894\) −10.0000 −0.334450
\(895\) 20.0000 0.668526
\(896\) 3.00000 0.100223
\(897\) 24.0000 0.801337
\(898\) 5.00000 0.166852
\(899\) −10.0000 −0.333519
\(900\) 8.00000 0.266667
\(901\) −18.0000 −0.599667
\(902\) 14.0000 0.466149
\(903\) 18.0000 0.599002
\(904\) 14.0000 0.465633
\(905\) −13.0000 −0.432135
\(906\) −2.00000 −0.0664455
\(907\) −37.0000 −1.22856 −0.614282 0.789086i \(-0.710554\pi\)
−0.614282 + 0.789086i \(0.710554\pi\)
\(908\) 8.00000 0.265489
\(909\) 36.0000 1.19404
\(910\) −18.0000 −0.596694
\(911\) −13.0000 −0.430709 −0.215355 0.976536i \(-0.569091\pi\)
−0.215355 + 0.976536i \(0.569091\pi\)
\(912\) 5.00000 0.165567
\(913\) 28.0000 0.926665
\(914\) −22.0000 −0.727695
\(915\) 8.00000 0.264472
\(916\) −10.0000 −0.330409
\(917\) −54.0000 −1.78324
\(918\) −10.0000 −0.330049
\(919\) −10.0000 −0.329870 −0.164935 0.986304i \(-0.552741\pi\)
−0.164935 + 0.986304i \(0.552741\pi\)
\(920\) 4.00000 0.131876
\(921\) −13.0000 −0.428365
\(922\) −18.0000 −0.592798
\(923\) −72.0000 −2.36991
\(924\) −6.00000 −0.197386
\(925\) −32.0000 −1.05215
\(926\) −36.0000 −1.18303
\(927\) 32.0000 1.05102
\(928\) −5.00000 −0.164133
\(929\) 50.0000 1.64045 0.820223 0.572043i \(-0.193849\pi\)
0.820223 + 0.572043i \(0.193849\pi\)
\(930\) −2.00000 −0.0655826
\(931\) −10.0000 −0.327737
\(932\) −6.00000 −0.196537
\(933\) −7.00000 −0.229170
\(934\) 28.0000 0.916188
\(935\) −4.00000 −0.130814
\(936\) 12.0000 0.392232
\(937\) 38.0000 1.24141 0.620703 0.784046i \(-0.286847\pi\)
0.620703 + 0.784046i \(0.286847\pi\)
\(938\) −6.00000 −0.195907
\(939\) −24.0000 −0.783210
\(940\) −2.00000 −0.0652328
\(941\) −8.00000 −0.260793 −0.130396 0.991462i \(-0.541625\pi\)
−0.130396 + 0.991462i \(0.541625\pi\)
\(942\) 12.0000 0.390981
\(943\) 28.0000 0.911805
\(944\) −1.00000 −0.0325472
\(945\) 15.0000 0.487950
\(946\) −12.0000 −0.390154
\(947\) 43.0000 1.39731 0.698656 0.715458i \(-0.253782\pi\)
0.698656 + 0.715458i \(0.253782\pi\)
\(948\) −5.00000 −0.162392
\(949\) −24.0000 −0.779073
\(950\) 20.0000 0.648886
\(951\) 2.00000 0.0648544
\(952\) −6.00000 −0.194461
\(953\) 34.0000 1.10137 0.550684 0.834714i \(-0.314367\pi\)
0.550684 + 0.834714i \(0.314367\pi\)
\(954\) −18.0000 −0.582772
\(955\) −18.0000 −0.582466
\(956\) −25.0000 −0.808558
\(957\) 10.0000 0.323254
\(958\) 40.0000 1.29234
\(959\) 9.00000 0.290625
\(960\) −1.00000 −0.0322749
\(961\) −27.0000 −0.870968
\(962\) −48.0000 −1.54758
\(963\) −6.00000 −0.193347
\(964\) 17.0000 0.547533
\(965\) 19.0000 0.611632
\(966\) −12.0000 −0.386094
\(967\) −42.0000 −1.35063 −0.675314 0.737530i \(-0.735992\pi\)
−0.675314 + 0.737530i \(0.735992\pi\)
\(968\) −7.00000 −0.224989
\(969\) −10.0000 −0.321246
\(970\) 8.00000 0.256865
\(971\) −3.00000 −0.0962746 −0.0481373 0.998841i \(-0.515328\pi\)
−0.0481373 + 0.998841i \(0.515328\pi\)
\(972\) −16.0000 −0.513200
\(973\) 60.0000 1.92351
\(974\) −17.0000 −0.544715
\(975\) −24.0000 −0.768615
\(976\) −8.00000 −0.256074
\(977\) −42.0000 −1.34370 −0.671850 0.740688i \(-0.734500\pi\)
−0.671850 + 0.740688i \(0.734500\pi\)
\(978\) −4.00000 −0.127906
\(979\) 0 0
\(980\) 2.00000 0.0638877
\(981\) 40.0000 1.27710
\(982\) 37.0000 1.18072
\(983\) −6.00000 −0.191370 −0.0956851 0.995412i \(-0.530504\pi\)
−0.0956851 + 0.995412i \(0.530504\pi\)
\(984\) −7.00000 −0.223152
\(985\) 18.0000 0.573528
\(986\) 10.0000 0.318465
\(987\) 6.00000 0.190982
\(988\) 30.0000 0.954427
\(989\) −24.0000 −0.763156
\(990\) −4.00000 −0.127128
\(991\) −18.0000 −0.571789 −0.285894 0.958261i \(-0.592291\pi\)
−0.285894 + 0.958261i \(0.592291\pi\)
\(992\) 2.00000 0.0635001
\(993\) −27.0000 −0.856819
\(994\) 36.0000 1.14185
\(995\) −5.00000 −0.158511
\(996\) −14.0000 −0.443607
\(997\) 23.0000 0.728417 0.364209 0.931317i \(-0.381339\pi\)
0.364209 + 0.931317i \(0.381339\pi\)
\(998\) 15.0000 0.474817
\(999\) 40.0000 1.26554
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 118.2.a.c.1.1 1
3.2 odd 2 1062.2.a.b.1.1 1
4.3 odd 2 944.2.a.j.1.1 1
5.2 odd 4 2950.2.c.d.1299.2 2
5.3 odd 4 2950.2.c.d.1299.1 2
5.4 even 2 2950.2.a.h.1.1 1
7.6 odd 2 5782.2.a.j.1.1 1
8.3 odd 2 3776.2.a.h.1.1 1
8.5 even 2 3776.2.a.o.1.1 1
12.11 even 2 8496.2.a.h.1.1 1
59.58 odd 2 6962.2.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
118.2.a.c.1.1 1 1.1 even 1 trivial
944.2.a.j.1.1 1 4.3 odd 2
1062.2.a.b.1.1 1 3.2 odd 2
2950.2.a.h.1.1 1 5.4 even 2
2950.2.c.d.1299.1 2 5.3 odd 4
2950.2.c.d.1299.2 2 5.2 odd 4
3776.2.a.h.1.1 1 8.3 odd 2
3776.2.a.o.1.1 1 8.5 even 2
5782.2.a.j.1.1 1 7.6 odd 2
6962.2.a.b.1.1 1 59.58 odd 2
8496.2.a.h.1.1 1 12.11 even 2