Properties

Label 5390.2.a.h.1.1
Level $5390$
Weight $2$
Character 5390.1
Self dual yes
Analytic conductor $43.039$
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] = [5390,2,Mod(1,5390)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5390, 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("5390.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5390 = 2 \cdot 5 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5390.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: \(43.0393666895\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 110)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 5390.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -1.00000 q^{8} -2.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -1.00000 q^{8} -2.00000 q^{9} -1.00000 q^{10} +1.00000 q^{11} -1.00000 q^{12} -2.00000 q^{13} -1.00000 q^{15} +1.00000 q^{16} -3.00000 q^{17} +2.00000 q^{18} +7.00000 q^{19} +1.00000 q^{20} -1.00000 q^{22} -6.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} +2.00000 q^{26} +5.00000 q^{27} -3.00000 q^{29} +1.00000 q^{30} +7.00000 q^{31} -1.00000 q^{32} -1.00000 q^{33} +3.00000 q^{34} -2.00000 q^{36} -7.00000 q^{37} -7.00000 q^{38} +2.00000 q^{39} -1.00000 q^{40} -6.00000 q^{41} +8.00000 q^{43} +1.00000 q^{44} -2.00000 q^{45} +6.00000 q^{46} -6.00000 q^{47} -1.00000 q^{48} -1.00000 q^{50} +3.00000 q^{51} -2.00000 q^{52} -3.00000 q^{53} -5.00000 q^{54} +1.00000 q^{55} -7.00000 q^{57} +3.00000 q^{58} +6.00000 q^{59} -1.00000 q^{60} +1.00000 q^{61} -7.00000 q^{62} +1.00000 q^{64} -2.00000 q^{65} +1.00000 q^{66} +8.00000 q^{67} -3.00000 q^{68} +6.00000 q^{69} +3.00000 q^{71} +2.00000 q^{72} -2.00000 q^{73} +7.00000 q^{74} -1.00000 q^{75} +7.00000 q^{76} -2.00000 q^{78} -10.0000 q^{79} +1.00000 q^{80} +1.00000 q^{81} +6.00000 q^{82} +6.00000 q^{83} -3.00000 q^{85} -8.00000 q^{86} +3.00000 q^{87} -1.00000 q^{88} -9.00000 q^{89} +2.00000 q^{90} -6.00000 q^{92} -7.00000 q^{93} +6.00000 q^{94} +7.00000 q^{95} +1.00000 q^{96} +4.00000 q^{97} -2.00000 q^{99} +O(q^{100})\)

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
\(6\) 1.00000 0.408248
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) −2.00000 −0.666667
\(10\) −1.00000 −0.316228
\(11\) 1.00000 0.301511
\(12\) −1.00000 −0.288675
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) −3.00000 −0.727607 −0.363803 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) 2.00000 0.471405
\(19\) 7.00000 1.60591 0.802955 0.596040i \(-0.203260\pi\)
0.802955 + 0.596040i \(0.203260\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) −1.00000 −0.213201
\(23\) −6.00000 −1.25109 −0.625543 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) 1.00000 0.204124
\(25\) 1.00000 0.200000
\(26\) 2.00000 0.392232
\(27\) 5.00000 0.962250
\(28\) 0 0
\(29\) −3.00000 −0.557086 −0.278543 0.960424i \(-0.589851\pi\)
−0.278543 + 0.960424i \(0.589851\pi\)
\(30\) 1.00000 0.182574
\(31\) 7.00000 1.25724 0.628619 0.777714i \(-0.283621\pi\)
0.628619 + 0.777714i \(0.283621\pi\)
\(32\) −1.00000 −0.176777
\(33\) −1.00000 −0.174078
\(34\) 3.00000 0.514496
\(35\) 0 0
\(36\) −2.00000 −0.333333
\(37\) −7.00000 −1.15079 −0.575396 0.817875i \(-0.695152\pi\)
−0.575396 + 0.817875i \(0.695152\pi\)
\(38\) −7.00000 −1.13555
\(39\) 2.00000 0.320256
\(40\) −1.00000 −0.158114
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) 8.00000 1.21999 0.609994 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) 1.00000 0.150756
\(45\) −2.00000 −0.298142
\(46\) 6.00000 0.884652
\(47\) −6.00000 −0.875190 −0.437595 0.899172i \(-0.644170\pi\)
−0.437595 + 0.899172i \(0.644170\pi\)
\(48\) −1.00000 −0.144338
\(49\) 0 0
\(50\) −1.00000 −0.141421
\(51\) 3.00000 0.420084
\(52\) −2.00000 −0.277350
\(53\) −3.00000 −0.412082 −0.206041 0.978543i \(-0.566058\pi\)
−0.206041 + 0.978543i \(0.566058\pi\)
\(54\) −5.00000 −0.680414
\(55\) 1.00000 0.134840
\(56\) 0 0
\(57\) −7.00000 −0.927173
\(58\) 3.00000 0.393919
\(59\) 6.00000 0.781133 0.390567 0.920575i \(-0.372279\pi\)
0.390567 + 0.920575i \(0.372279\pi\)
\(60\) −1.00000 −0.129099
\(61\) 1.00000 0.128037 0.0640184 0.997949i \(-0.479608\pi\)
0.0640184 + 0.997949i \(0.479608\pi\)
\(62\) −7.00000 −0.889001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −2.00000 −0.248069
\(66\) 1.00000 0.123091
\(67\) 8.00000 0.977356 0.488678 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) −3.00000 −0.363803
\(69\) 6.00000 0.722315
\(70\) 0 0
\(71\) 3.00000 0.356034 0.178017 0.984027i \(-0.443032\pi\)
0.178017 + 0.984027i \(0.443032\pi\)
\(72\) 2.00000 0.235702
\(73\) −2.00000 −0.234082 −0.117041 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) 7.00000 0.813733
\(75\) −1.00000 −0.115470
\(76\) 7.00000 0.802955
\(77\) 0 0
\(78\) −2.00000 −0.226455
\(79\) −10.0000 −1.12509 −0.562544 0.826767i \(-0.690177\pi\)
−0.562544 + 0.826767i \(0.690177\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) 6.00000 0.662589
\(83\) 6.00000 0.658586 0.329293 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) 0 0
\(85\) −3.00000 −0.325396
\(86\) −8.00000 −0.862662
\(87\) 3.00000 0.321634
\(88\) −1.00000 −0.106600
\(89\) −9.00000 −0.953998 −0.476999 0.878904i \(-0.658275\pi\)
−0.476999 + 0.878904i \(0.658275\pi\)
\(90\) 2.00000 0.210819
\(91\) 0 0
\(92\) −6.00000 −0.625543
\(93\) −7.00000 −0.725866
\(94\) 6.00000 0.618853
\(95\) 7.00000 0.718185
\(96\) 1.00000 0.102062
\(97\) 4.00000 0.406138 0.203069 0.979164i \(-0.434908\pi\)
0.203069 + 0.979164i \(0.434908\pi\)
\(98\) 0 0
\(99\) −2.00000 −0.201008
\(100\) 1.00000 0.100000
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) −3.00000 −0.297044
\(103\) 4.00000 0.394132 0.197066 0.980390i \(-0.436859\pi\)
0.197066 + 0.980390i \(0.436859\pi\)
\(104\) 2.00000 0.196116
\(105\) 0 0
\(106\) 3.00000 0.291386
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 5.00000 0.481125
\(109\) 14.0000 1.34096 0.670478 0.741929i \(-0.266089\pi\)
0.670478 + 0.741929i \(0.266089\pi\)
\(110\) −1.00000 −0.0953463
\(111\) 7.00000 0.664411
\(112\) 0 0
\(113\) 12.0000 1.12887 0.564433 0.825479i \(-0.309095\pi\)
0.564433 + 0.825479i \(0.309095\pi\)
\(114\) 7.00000 0.655610
\(115\) −6.00000 −0.559503
\(116\) −3.00000 −0.278543
\(117\) 4.00000 0.369800
\(118\) −6.00000 −0.552345
\(119\) 0 0
\(120\) 1.00000 0.0912871
\(121\) 1.00000 0.0909091
\(122\) −1.00000 −0.0905357
\(123\) 6.00000 0.541002
\(124\) 7.00000 0.628619
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −16.0000 −1.41977 −0.709885 0.704317i \(-0.751253\pi\)
−0.709885 + 0.704317i \(0.751253\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −8.00000 −0.704361
\(130\) 2.00000 0.175412
\(131\) 3.00000 0.262111 0.131056 0.991375i \(-0.458163\pi\)
0.131056 + 0.991375i \(0.458163\pi\)
\(132\) −1.00000 −0.0870388
\(133\) 0 0
\(134\) −8.00000 −0.691095
\(135\) 5.00000 0.430331
\(136\) 3.00000 0.257248
\(137\) 12.0000 1.02523 0.512615 0.858619i \(-0.328677\pi\)
0.512615 + 0.858619i \(0.328677\pi\)
\(138\) −6.00000 −0.510754
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) 6.00000 0.505291
\(142\) −3.00000 −0.251754
\(143\) −2.00000 −0.167248
\(144\) −2.00000 −0.166667
\(145\) −3.00000 −0.249136
\(146\) 2.00000 0.165521
\(147\) 0 0
\(148\) −7.00000 −0.575396
\(149\) 15.0000 1.22885 0.614424 0.788976i \(-0.289388\pi\)
0.614424 + 0.788976i \(0.289388\pi\)
\(150\) 1.00000 0.0816497
\(151\) 2.00000 0.162758 0.0813788 0.996683i \(-0.474068\pi\)
0.0813788 + 0.996683i \(0.474068\pi\)
\(152\) −7.00000 −0.567775
\(153\) 6.00000 0.485071
\(154\) 0 0
\(155\) 7.00000 0.562254
\(156\) 2.00000 0.160128
\(157\) 7.00000 0.558661 0.279330 0.960195i \(-0.409888\pi\)
0.279330 + 0.960195i \(0.409888\pi\)
\(158\) 10.0000 0.795557
\(159\) 3.00000 0.237915
\(160\) −1.00000 −0.0790569
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) 5.00000 0.391630 0.195815 0.980641i \(-0.437265\pi\)
0.195815 + 0.980641i \(0.437265\pi\)
\(164\) −6.00000 −0.468521
\(165\) −1.00000 −0.0778499
\(166\) −6.00000 −0.465690
\(167\) −21.0000 −1.62503 −0.812514 0.582941i \(-0.801902\pi\)
−0.812514 + 0.582941i \(0.801902\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 3.00000 0.230089
\(171\) −14.0000 −1.07061
\(172\) 8.00000 0.609994
\(173\) −18.0000 −1.36851 −0.684257 0.729241i \(-0.739873\pi\)
−0.684257 + 0.729241i \(0.739873\pi\)
\(174\) −3.00000 −0.227429
\(175\) 0 0
\(176\) 1.00000 0.0753778
\(177\) −6.00000 −0.450988
\(178\) 9.00000 0.674579
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) −2.00000 −0.149071
\(181\) 22.0000 1.63525 0.817624 0.575753i \(-0.195291\pi\)
0.817624 + 0.575753i \(0.195291\pi\)
\(182\) 0 0
\(183\) −1.00000 −0.0739221
\(184\) 6.00000 0.442326
\(185\) −7.00000 −0.514650
\(186\) 7.00000 0.513265
\(187\) −3.00000 −0.219382
\(188\) −6.00000 −0.437595
\(189\) 0 0
\(190\) −7.00000 −0.507833
\(191\) −12.0000 −0.868290 −0.434145 0.900843i \(-0.642949\pi\)
−0.434145 + 0.900843i \(0.642949\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 23.0000 1.65558 0.827788 0.561041i \(-0.189599\pi\)
0.827788 + 0.561041i \(0.189599\pi\)
\(194\) −4.00000 −0.287183
\(195\) 2.00000 0.143223
\(196\) 0 0
\(197\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 2.00000 0.142134
\(199\) −11.0000 −0.779769 −0.389885 0.920864i \(-0.627485\pi\)
−0.389885 + 0.920864i \(0.627485\pi\)
\(200\) −1.00000 −0.0707107
\(201\) −8.00000 −0.564276
\(202\) −6.00000 −0.422159
\(203\) 0 0
\(204\) 3.00000 0.210042
\(205\) −6.00000 −0.419058
\(206\) −4.00000 −0.278693
\(207\) 12.0000 0.834058
\(208\) −2.00000 −0.138675
\(209\) 7.00000 0.484200
\(210\) 0 0
\(211\) 5.00000 0.344214 0.172107 0.985078i \(-0.444942\pi\)
0.172107 + 0.985078i \(0.444942\pi\)
\(212\) −3.00000 −0.206041
\(213\) −3.00000 −0.205557
\(214\) 0 0
\(215\) 8.00000 0.545595
\(216\) −5.00000 −0.340207
\(217\) 0 0
\(218\) −14.0000 −0.948200
\(219\) 2.00000 0.135147
\(220\) 1.00000 0.0674200
\(221\) 6.00000 0.403604
\(222\) −7.00000 −0.469809
\(223\) −14.0000 −0.937509 −0.468755 0.883328i \(-0.655297\pi\)
−0.468755 + 0.883328i \(0.655297\pi\)
\(224\) 0 0
\(225\) −2.00000 −0.133333
\(226\) −12.0000 −0.798228
\(227\) 18.0000 1.19470 0.597351 0.801980i \(-0.296220\pi\)
0.597351 + 0.801980i \(0.296220\pi\)
\(228\) −7.00000 −0.463586
\(229\) −14.0000 −0.925146 −0.462573 0.886581i \(-0.653074\pi\)
−0.462573 + 0.886581i \(0.653074\pi\)
\(230\) 6.00000 0.395628
\(231\) 0 0
\(232\) 3.00000 0.196960
\(233\) 27.0000 1.76883 0.884414 0.466702i \(-0.154558\pi\)
0.884414 + 0.466702i \(0.154558\pi\)
\(234\) −4.00000 −0.261488
\(235\) −6.00000 −0.391397
\(236\) 6.00000 0.390567
\(237\) 10.0000 0.649570
\(238\) 0 0
\(239\) 6.00000 0.388108 0.194054 0.980991i \(-0.437836\pi\)
0.194054 + 0.980991i \(0.437836\pi\)
\(240\) −1.00000 −0.0645497
\(241\) 22.0000 1.41714 0.708572 0.705638i \(-0.249340\pi\)
0.708572 + 0.705638i \(0.249340\pi\)
\(242\) −1.00000 −0.0642824
\(243\) −16.0000 −1.02640
\(244\) 1.00000 0.0640184
\(245\) 0 0
\(246\) −6.00000 −0.382546
\(247\) −14.0000 −0.890799
\(248\) −7.00000 −0.444500
\(249\) −6.00000 −0.380235
\(250\) −1.00000 −0.0632456
\(251\) 30.0000 1.89358 0.946792 0.321847i \(-0.104304\pi\)
0.946792 + 0.321847i \(0.104304\pi\)
\(252\) 0 0
\(253\) −6.00000 −0.377217
\(254\) 16.0000 1.00393
\(255\) 3.00000 0.187867
\(256\) 1.00000 0.0625000
\(257\) −6.00000 −0.374270 −0.187135 0.982334i \(-0.559920\pi\)
−0.187135 + 0.982334i \(0.559920\pi\)
\(258\) 8.00000 0.498058
\(259\) 0 0
\(260\) −2.00000 −0.124035
\(261\) 6.00000 0.371391
\(262\) −3.00000 −0.185341
\(263\) −9.00000 −0.554964 −0.277482 0.960731i \(-0.589500\pi\)
−0.277482 + 0.960731i \(0.589500\pi\)
\(264\) 1.00000 0.0615457
\(265\) −3.00000 −0.184289
\(266\) 0 0
\(267\) 9.00000 0.550791
\(268\) 8.00000 0.488678
\(269\) 12.0000 0.731653 0.365826 0.930683i \(-0.380786\pi\)
0.365826 + 0.930683i \(0.380786\pi\)
\(270\) −5.00000 −0.304290
\(271\) −20.0000 −1.21491 −0.607457 0.794353i \(-0.707810\pi\)
−0.607457 + 0.794353i \(0.707810\pi\)
\(272\) −3.00000 −0.181902
\(273\) 0 0
\(274\) −12.0000 −0.724947
\(275\) 1.00000 0.0603023
\(276\) 6.00000 0.361158
\(277\) −4.00000 −0.240337 −0.120168 0.992754i \(-0.538343\pi\)
−0.120168 + 0.992754i \(0.538343\pi\)
\(278\) −4.00000 −0.239904
\(279\) −14.0000 −0.838158
\(280\) 0 0
\(281\) 30.0000 1.78965 0.894825 0.446417i \(-0.147300\pi\)
0.894825 + 0.446417i \(0.147300\pi\)
\(282\) −6.00000 −0.357295
\(283\) −2.00000 −0.118888 −0.0594438 0.998232i \(-0.518933\pi\)
−0.0594438 + 0.998232i \(0.518933\pi\)
\(284\) 3.00000 0.178017
\(285\) −7.00000 −0.414644
\(286\) 2.00000 0.118262
\(287\) 0 0
\(288\) 2.00000 0.117851
\(289\) −8.00000 −0.470588
\(290\) 3.00000 0.176166
\(291\) −4.00000 −0.234484
\(292\) −2.00000 −0.117041
\(293\) 6.00000 0.350524 0.175262 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) 0 0
\(295\) 6.00000 0.349334
\(296\) 7.00000 0.406867
\(297\) 5.00000 0.290129
\(298\) −15.0000 −0.868927
\(299\) 12.0000 0.693978
\(300\) −1.00000 −0.0577350
\(301\) 0 0
\(302\) −2.00000 −0.115087
\(303\) −6.00000 −0.344691
\(304\) 7.00000 0.401478
\(305\) 1.00000 0.0572598
\(306\) −6.00000 −0.342997
\(307\) 10.0000 0.570730 0.285365 0.958419i \(-0.407885\pi\)
0.285365 + 0.958419i \(0.407885\pi\)
\(308\) 0 0
\(309\) −4.00000 −0.227552
\(310\) −7.00000 −0.397573
\(311\) 15.0000 0.850572 0.425286 0.905059i \(-0.360174\pi\)
0.425286 + 0.905059i \(0.360174\pi\)
\(312\) −2.00000 −0.113228
\(313\) −14.0000 −0.791327 −0.395663 0.918396i \(-0.629485\pi\)
−0.395663 + 0.918396i \(0.629485\pi\)
\(314\) −7.00000 −0.395033
\(315\) 0 0
\(316\) −10.0000 −0.562544
\(317\) −21.0000 −1.17948 −0.589739 0.807594i \(-0.700769\pi\)
−0.589739 + 0.807594i \(0.700769\pi\)
\(318\) −3.00000 −0.168232
\(319\) −3.00000 −0.167968
\(320\) 1.00000 0.0559017
\(321\) 0 0
\(322\) 0 0
\(323\) −21.0000 −1.16847
\(324\) 1.00000 0.0555556
\(325\) −2.00000 −0.110940
\(326\) −5.00000 −0.276924
\(327\) −14.0000 −0.774202
\(328\) 6.00000 0.331295
\(329\) 0 0
\(330\) 1.00000 0.0550482
\(331\) 8.00000 0.439720 0.219860 0.975531i \(-0.429440\pi\)
0.219860 + 0.975531i \(0.429440\pi\)
\(332\) 6.00000 0.329293
\(333\) 14.0000 0.767195
\(334\) 21.0000 1.14907
\(335\) 8.00000 0.437087
\(336\) 0 0
\(337\) 5.00000 0.272367 0.136184 0.990684i \(-0.456516\pi\)
0.136184 + 0.990684i \(0.456516\pi\)
\(338\) 9.00000 0.489535
\(339\) −12.0000 −0.651751
\(340\) −3.00000 −0.162698
\(341\) 7.00000 0.379071
\(342\) 14.0000 0.757033
\(343\) 0 0
\(344\) −8.00000 −0.431331
\(345\) 6.00000 0.323029
\(346\) 18.0000 0.967686
\(347\) 30.0000 1.61048 0.805242 0.592946i \(-0.202035\pi\)
0.805242 + 0.592946i \(0.202035\pi\)
\(348\) 3.00000 0.160817
\(349\) 10.0000 0.535288 0.267644 0.963518i \(-0.413755\pi\)
0.267644 + 0.963518i \(0.413755\pi\)
\(350\) 0 0
\(351\) −10.0000 −0.533761
\(352\) −1.00000 −0.0533002
\(353\) −6.00000 −0.319348 −0.159674 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(354\) 6.00000 0.318896
\(355\) 3.00000 0.159223
\(356\) −9.00000 −0.476999
\(357\) 0 0
\(358\) 12.0000 0.634220
\(359\) −12.0000 −0.633336 −0.316668 0.948536i \(-0.602564\pi\)
−0.316668 + 0.948536i \(0.602564\pi\)
\(360\) 2.00000 0.105409
\(361\) 30.0000 1.57895
\(362\) −22.0000 −1.15629
\(363\) −1.00000 −0.0524864
\(364\) 0 0
\(365\) −2.00000 −0.104685
\(366\) 1.00000 0.0522708
\(367\) 4.00000 0.208798 0.104399 0.994535i \(-0.466708\pi\)
0.104399 + 0.994535i \(0.466708\pi\)
\(368\) −6.00000 −0.312772
\(369\) 12.0000 0.624695
\(370\) 7.00000 0.363913
\(371\) 0 0
\(372\) −7.00000 −0.362933
\(373\) 2.00000 0.103556 0.0517780 0.998659i \(-0.483511\pi\)
0.0517780 + 0.998659i \(0.483511\pi\)
\(374\) 3.00000 0.155126
\(375\) −1.00000 −0.0516398
\(376\) 6.00000 0.309426
\(377\) 6.00000 0.309016
\(378\) 0 0
\(379\) 26.0000 1.33553 0.667765 0.744372i \(-0.267251\pi\)
0.667765 + 0.744372i \(0.267251\pi\)
\(380\) 7.00000 0.359092
\(381\) 16.0000 0.819705
\(382\) 12.0000 0.613973
\(383\) 18.0000 0.919757 0.459879 0.887982i \(-0.347893\pi\)
0.459879 + 0.887982i \(0.347893\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −23.0000 −1.17067
\(387\) −16.0000 −0.813326
\(388\) 4.00000 0.203069
\(389\) 18.0000 0.912636 0.456318 0.889817i \(-0.349168\pi\)
0.456318 + 0.889817i \(0.349168\pi\)
\(390\) −2.00000 −0.101274
\(391\) 18.0000 0.910299
\(392\) 0 0
\(393\) −3.00000 −0.151330
\(394\) 0 0
\(395\) −10.0000 −0.503155
\(396\) −2.00000 −0.100504
\(397\) 22.0000 1.10415 0.552074 0.833795i \(-0.313837\pi\)
0.552074 + 0.833795i \(0.313837\pi\)
\(398\) 11.0000 0.551380
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 3.00000 0.149813 0.0749064 0.997191i \(-0.476134\pi\)
0.0749064 + 0.997191i \(0.476134\pi\)
\(402\) 8.00000 0.399004
\(403\) −14.0000 −0.697390
\(404\) 6.00000 0.298511
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) −7.00000 −0.346977
\(408\) −3.00000 −0.148522
\(409\) 4.00000 0.197787 0.0988936 0.995098i \(-0.468470\pi\)
0.0988936 + 0.995098i \(0.468470\pi\)
\(410\) 6.00000 0.296319
\(411\) −12.0000 −0.591916
\(412\) 4.00000 0.197066
\(413\) 0 0
\(414\) −12.0000 −0.589768
\(415\) 6.00000 0.294528
\(416\) 2.00000 0.0980581
\(417\) −4.00000 −0.195881
\(418\) −7.00000 −0.342381
\(419\) −24.0000 −1.17248 −0.586238 0.810139i \(-0.699392\pi\)
−0.586238 + 0.810139i \(0.699392\pi\)
\(420\) 0 0
\(421\) 8.00000 0.389896 0.194948 0.980814i \(-0.437546\pi\)
0.194948 + 0.980814i \(0.437546\pi\)
\(422\) −5.00000 −0.243396
\(423\) 12.0000 0.583460
\(424\) 3.00000 0.145693
\(425\) −3.00000 −0.145521
\(426\) 3.00000 0.145350
\(427\) 0 0
\(428\) 0 0
\(429\) 2.00000 0.0965609
\(430\) −8.00000 −0.385794
\(431\) −24.0000 −1.15604 −0.578020 0.816023i \(-0.696174\pi\)
−0.578020 + 0.816023i \(0.696174\pi\)
\(432\) 5.00000 0.240563
\(433\) 28.0000 1.34559 0.672797 0.739827i \(-0.265093\pi\)
0.672797 + 0.739827i \(0.265093\pi\)
\(434\) 0 0
\(435\) 3.00000 0.143839
\(436\) 14.0000 0.670478
\(437\) −42.0000 −2.00913
\(438\) −2.00000 −0.0955637
\(439\) 16.0000 0.763638 0.381819 0.924237i \(-0.375298\pi\)
0.381819 + 0.924237i \(0.375298\pi\)
\(440\) −1.00000 −0.0476731
\(441\) 0 0
\(442\) −6.00000 −0.285391
\(443\) 12.0000 0.570137 0.285069 0.958507i \(-0.407984\pi\)
0.285069 + 0.958507i \(0.407984\pi\)
\(444\) 7.00000 0.332205
\(445\) −9.00000 −0.426641
\(446\) 14.0000 0.662919
\(447\) −15.0000 −0.709476
\(448\) 0 0
\(449\) −30.0000 −1.41579 −0.707894 0.706319i \(-0.750354\pi\)
−0.707894 + 0.706319i \(0.750354\pi\)
\(450\) 2.00000 0.0942809
\(451\) −6.00000 −0.282529
\(452\) 12.0000 0.564433
\(453\) −2.00000 −0.0939682
\(454\) −18.0000 −0.844782
\(455\) 0 0
\(456\) 7.00000 0.327805
\(457\) 17.0000 0.795226 0.397613 0.917553i \(-0.369839\pi\)
0.397613 + 0.917553i \(0.369839\pi\)
\(458\) 14.0000 0.654177
\(459\) −15.0000 −0.700140
\(460\) −6.00000 −0.279751
\(461\) −3.00000 −0.139724 −0.0698620 0.997557i \(-0.522256\pi\)
−0.0698620 + 0.997557i \(0.522256\pi\)
\(462\) 0 0
\(463\) −22.0000 −1.02243 −0.511213 0.859454i \(-0.670804\pi\)
−0.511213 + 0.859454i \(0.670804\pi\)
\(464\) −3.00000 −0.139272
\(465\) −7.00000 −0.324617
\(466\) −27.0000 −1.25075
\(467\) 39.0000 1.80470 0.902352 0.430999i \(-0.141839\pi\)
0.902352 + 0.430999i \(0.141839\pi\)
\(468\) 4.00000 0.184900
\(469\) 0 0
\(470\) 6.00000 0.276759
\(471\) −7.00000 −0.322543
\(472\) −6.00000 −0.276172
\(473\) 8.00000 0.367840
\(474\) −10.0000 −0.459315
\(475\) 7.00000 0.321182
\(476\) 0 0
\(477\) 6.00000 0.274721
\(478\) −6.00000 −0.274434
\(479\) 36.0000 1.64488 0.822441 0.568850i \(-0.192612\pi\)
0.822441 + 0.568850i \(0.192612\pi\)
\(480\) 1.00000 0.0456435
\(481\) 14.0000 0.638345
\(482\) −22.0000 −1.00207
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) 4.00000 0.181631
\(486\) 16.0000 0.725775
\(487\) 8.00000 0.362515 0.181257 0.983436i \(-0.441983\pi\)
0.181257 + 0.983436i \(0.441983\pi\)
\(488\) −1.00000 −0.0452679
\(489\) −5.00000 −0.226108
\(490\) 0 0
\(491\) 33.0000 1.48927 0.744635 0.667472i \(-0.232624\pi\)
0.744635 + 0.667472i \(0.232624\pi\)
\(492\) 6.00000 0.270501
\(493\) 9.00000 0.405340
\(494\) 14.0000 0.629890
\(495\) −2.00000 −0.0898933
\(496\) 7.00000 0.314309
\(497\) 0 0
\(498\) 6.00000 0.268866
\(499\) 44.0000 1.96971 0.984855 0.173379i \(-0.0554684\pi\)
0.984855 + 0.173379i \(0.0554684\pi\)
\(500\) 1.00000 0.0447214
\(501\) 21.0000 0.938211
\(502\) −30.0000 −1.33897
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 6.00000 0.266996
\(506\) 6.00000 0.266733
\(507\) 9.00000 0.399704
\(508\) −16.0000 −0.709885
\(509\) 12.0000 0.531891 0.265945 0.963988i \(-0.414316\pi\)
0.265945 + 0.963988i \(0.414316\pi\)
\(510\) −3.00000 −0.132842
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 35.0000 1.54529
\(514\) 6.00000 0.264649
\(515\) 4.00000 0.176261
\(516\) −8.00000 −0.352180
\(517\) −6.00000 −0.263880
\(518\) 0 0
\(519\) 18.0000 0.790112
\(520\) 2.00000 0.0877058
\(521\) 18.0000 0.788594 0.394297 0.918983i \(-0.370988\pi\)
0.394297 + 0.918983i \(0.370988\pi\)
\(522\) −6.00000 −0.262613
\(523\) 16.0000 0.699631 0.349816 0.936819i \(-0.386244\pi\)
0.349816 + 0.936819i \(0.386244\pi\)
\(524\) 3.00000 0.131056
\(525\) 0 0
\(526\) 9.00000 0.392419
\(527\) −21.0000 −0.914774
\(528\) −1.00000 −0.0435194
\(529\) 13.0000 0.565217
\(530\) 3.00000 0.130312
\(531\) −12.0000 −0.520756
\(532\) 0 0
\(533\) 12.0000 0.519778
\(534\) −9.00000 −0.389468
\(535\) 0 0
\(536\) −8.00000 −0.345547
\(537\) 12.0000 0.517838
\(538\) −12.0000 −0.517357
\(539\) 0 0
\(540\) 5.00000 0.215166
\(541\) 41.0000 1.76273 0.881364 0.472438i \(-0.156626\pi\)
0.881364 + 0.472438i \(0.156626\pi\)
\(542\) 20.0000 0.859074
\(543\) −22.0000 −0.944110
\(544\) 3.00000 0.128624
\(545\) 14.0000 0.599694
\(546\) 0 0
\(547\) 8.00000 0.342055 0.171028 0.985266i \(-0.445291\pi\)
0.171028 + 0.985266i \(0.445291\pi\)
\(548\) 12.0000 0.512615
\(549\) −2.00000 −0.0853579
\(550\) −1.00000 −0.0426401
\(551\) −21.0000 −0.894630
\(552\) −6.00000 −0.255377
\(553\) 0 0
\(554\) 4.00000 0.169944
\(555\) 7.00000 0.297133
\(556\) 4.00000 0.169638
\(557\) −6.00000 −0.254228 −0.127114 0.991888i \(-0.540571\pi\)
−0.127114 + 0.991888i \(0.540571\pi\)
\(558\) 14.0000 0.592667
\(559\) −16.0000 −0.676728
\(560\) 0 0
\(561\) 3.00000 0.126660
\(562\) −30.0000 −1.26547
\(563\) 6.00000 0.252870 0.126435 0.991975i \(-0.459647\pi\)
0.126435 + 0.991975i \(0.459647\pi\)
\(564\) 6.00000 0.252646
\(565\) 12.0000 0.504844
\(566\) 2.00000 0.0840663
\(567\) 0 0
\(568\) −3.00000 −0.125877
\(569\) 24.0000 1.00613 0.503066 0.864248i \(-0.332205\pi\)
0.503066 + 0.864248i \(0.332205\pi\)
\(570\) 7.00000 0.293198
\(571\) −25.0000 −1.04622 −0.523109 0.852266i \(-0.675228\pi\)
−0.523109 + 0.852266i \(0.675228\pi\)
\(572\) −2.00000 −0.0836242
\(573\) 12.0000 0.501307
\(574\) 0 0
\(575\) −6.00000 −0.250217
\(576\) −2.00000 −0.0833333
\(577\) −38.0000 −1.58196 −0.790980 0.611842i \(-0.790429\pi\)
−0.790980 + 0.611842i \(0.790429\pi\)
\(578\) 8.00000 0.332756
\(579\) −23.0000 −0.955847
\(580\) −3.00000 −0.124568
\(581\) 0 0
\(582\) 4.00000 0.165805
\(583\) −3.00000 −0.124247
\(584\) 2.00000 0.0827606
\(585\) 4.00000 0.165380
\(586\) −6.00000 −0.247858
\(587\) −3.00000 −0.123823 −0.0619116 0.998082i \(-0.519720\pi\)
−0.0619116 + 0.998082i \(0.519720\pi\)
\(588\) 0 0
\(589\) 49.0000 2.01901
\(590\) −6.00000 −0.247016
\(591\) 0 0
\(592\) −7.00000 −0.287698
\(593\) 6.00000 0.246390 0.123195 0.992382i \(-0.460686\pi\)
0.123195 + 0.992382i \(0.460686\pi\)
\(594\) −5.00000 −0.205152
\(595\) 0 0
\(596\) 15.0000 0.614424
\(597\) 11.0000 0.450200
\(598\) −12.0000 −0.490716
\(599\) 9.00000 0.367730 0.183865 0.982952i \(-0.441139\pi\)
0.183865 + 0.982952i \(0.441139\pi\)
\(600\) 1.00000 0.0408248
\(601\) 22.0000 0.897399 0.448699 0.893683i \(-0.351887\pi\)
0.448699 + 0.893683i \(0.351887\pi\)
\(602\) 0 0
\(603\) −16.0000 −0.651570
\(604\) 2.00000 0.0813788
\(605\) 1.00000 0.0406558
\(606\) 6.00000 0.243733
\(607\) 49.0000 1.98885 0.994424 0.105453i \(-0.0336291\pi\)
0.994424 + 0.105453i \(0.0336291\pi\)
\(608\) −7.00000 −0.283887
\(609\) 0 0
\(610\) −1.00000 −0.0404888
\(611\) 12.0000 0.485468
\(612\) 6.00000 0.242536
\(613\) 2.00000 0.0807792 0.0403896 0.999184i \(-0.487140\pi\)
0.0403896 + 0.999184i \(0.487140\pi\)
\(614\) −10.0000 −0.403567
\(615\) 6.00000 0.241943
\(616\) 0 0
\(617\) −48.0000 −1.93241 −0.966204 0.257780i \(-0.917009\pi\)
−0.966204 + 0.257780i \(0.917009\pi\)
\(618\) 4.00000 0.160904
\(619\) −20.0000 −0.803868 −0.401934 0.915669i \(-0.631662\pi\)
−0.401934 + 0.915669i \(0.631662\pi\)
\(620\) 7.00000 0.281127
\(621\) −30.0000 −1.20386
\(622\) −15.0000 −0.601445
\(623\) 0 0
\(624\) 2.00000 0.0800641
\(625\) 1.00000 0.0400000
\(626\) 14.0000 0.559553
\(627\) −7.00000 −0.279553
\(628\) 7.00000 0.279330
\(629\) 21.0000 0.837325
\(630\) 0 0
\(631\) 35.0000 1.39333 0.696664 0.717398i \(-0.254667\pi\)
0.696664 + 0.717398i \(0.254667\pi\)
\(632\) 10.0000 0.397779
\(633\) −5.00000 −0.198732
\(634\) 21.0000 0.834017
\(635\) −16.0000 −0.634941
\(636\) 3.00000 0.118958
\(637\) 0 0
\(638\) 3.00000 0.118771
\(639\) −6.00000 −0.237356
\(640\) −1.00000 −0.0395285
\(641\) −33.0000 −1.30342 −0.651711 0.758468i \(-0.725948\pi\)
−0.651711 + 0.758468i \(0.725948\pi\)
\(642\) 0 0
\(643\) −5.00000 −0.197181 −0.0985904 0.995128i \(-0.531433\pi\)
−0.0985904 + 0.995128i \(0.531433\pi\)
\(644\) 0 0
\(645\) −8.00000 −0.315000
\(646\) 21.0000 0.826234
\(647\) −6.00000 −0.235884 −0.117942 0.993020i \(-0.537630\pi\)
−0.117942 + 0.993020i \(0.537630\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 6.00000 0.235521
\(650\) 2.00000 0.0784465
\(651\) 0 0
\(652\) 5.00000 0.195815
\(653\) −45.0000 −1.76099 −0.880493 0.474059i \(-0.842788\pi\)
−0.880493 + 0.474059i \(0.842788\pi\)
\(654\) 14.0000 0.547443
\(655\) 3.00000 0.117220
\(656\) −6.00000 −0.234261
\(657\) 4.00000 0.156055
\(658\) 0 0
\(659\) −21.0000 −0.818044 −0.409022 0.912525i \(-0.634130\pi\)
−0.409022 + 0.912525i \(0.634130\pi\)
\(660\) −1.00000 −0.0389249
\(661\) −14.0000 −0.544537 −0.272268 0.962221i \(-0.587774\pi\)
−0.272268 + 0.962221i \(0.587774\pi\)
\(662\) −8.00000 −0.310929
\(663\) −6.00000 −0.233021
\(664\) −6.00000 −0.232845
\(665\) 0 0
\(666\) −14.0000 −0.542489
\(667\) 18.0000 0.696963
\(668\) −21.0000 −0.812514
\(669\) 14.0000 0.541271
\(670\) −8.00000 −0.309067
\(671\) 1.00000 0.0386046
\(672\) 0 0
\(673\) −1.00000 −0.0385472 −0.0192736 0.999814i \(-0.506135\pi\)
−0.0192736 + 0.999814i \(0.506135\pi\)
\(674\) −5.00000 −0.192593
\(675\) 5.00000 0.192450
\(676\) −9.00000 −0.346154
\(677\) −12.0000 −0.461197 −0.230599 0.973049i \(-0.574068\pi\)
−0.230599 + 0.973049i \(0.574068\pi\)
\(678\) 12.0000 0.460857
\(679\) 0 0
\(680\) 3.00000 0.115045
\(681\) −18.0000 −0.689761
\(682\) −7.00000 −0.268044
\(683\) −9.00000 −0.344375 −0.172188 0.985064i \(-0.555084\pi\)
−0.172188 + 0.985064i \(0.555084\pi\)
\(684\) −14.0000 −0.535303
\(685\) 12.0000 0.458496
\(686\) 0 0
\(687\) 14.0000 0.534133
\(688\) 8.00000 0.304997
\(689\) 6.00000 0.228582
\(690\) −6.00000 −0.228416
\(691\) 10.0000 0.380418 0.190209 0.981744i \(-0.439083\pi\)
0.190209 + 0.981744i \(0.439083\pi\)
\(692\) −18.0000 −0.684257
\(693\) 0 0
\(694\) −30.0000 −1.13878
\(695\) 4.00000 0.151729
\(696\) −3.00000 −0.113715
\(697\) 18.0000 0.681799
\(698\) −10.0000 −0.378506
\(699\) −27.0000 −1.02123
\(700\) 0 0
\(701\) 39.0000 1.47301 0.736505 0.676432i \(-0.236475\pi\)
0.736505 + 0.676432i \(0.236475\pi\)
\(702\) 10.0000 0.377426
\(703\) −49.0000 −1.84807
\(704\) 1.00000 0.0376889
\(705\) 6.00000 0.225973
\(706\) 6.00000 0.225813
\(707\) 0 0
\(708\) −6.00000 −0.225494
\(709\) −34.0000 −1.27690 −0.638448 0.769665i \(-0.720423\pi\)
−0.638448 + 0.769665i \(0.720423\pi\)
\(710\) −3.00000 −0.112588
\(711\) 20.0000 0.750059
\(712\) 9.00000 0.337289
\(713\) −42.0000 −1.57291
\(714\) 0 0
\(715\) −2.00000 −0.0747958
\(716\) −12.0000 −0.448461
\(717\) −6.00000 −0.224074
\(718\) 12.0000 0.447836
\(719\) 51.0000 1.90198 0.950990 0.309223i \(-0.100069\pi\)
0.950990 + 0.309223i \(0.100069\pi\)
\(720\) −2.00000 −0.0745356
\(721\) 0 0
\(722\) −30.0000 −1.11648
\(723\) −22.0000 −0.818189
\(724\) 22.0000 0.817624
\(725\) −3.00000 −0.111417
\(726\) 1.00000 0.0371135
\(727\) 10.0000 0.370879 0.185440 0.982656i \(-0.440629\pi\)
0.185440 + 0.982656i \(0.440629\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 2.00000 0.0740233
\(731\) −24.0000 −0.887672
\(732\) −1.00000 −0.0369611
\(733\) 4.00000 0.147743 0.0738717 0.997268i \(-0.476464\pi\)
0.0738717 + 0.997268i \(0.476464\pi\)
\(734\) −4.00000 −0.147643
\(735\) 0 0
\(736\) 6.00000 0.221163
\(737\) 8.00000 0.294684
\(738\) −12.0000 −0.441726
\(739\) −16.0000 −0.588570 −0.294285 0.955718i \(-0.595081\pi\)
−0.294285 + 0.955718i \(0.595081\pi\)
\(740\) −7.00000 −0.257325
\(741\) 14.0000 0.514303
\(742\) 0 0
\(743\) −3.00000 −0.110059 −0.0550297 0.998485i \(-0.517525\pi\)
−0.0550297 + 0.998485i \(0.517525\pi\)
\(744\) 7.00000 0.256632
\(745\) 15.0000 0.549557
\(746\) −2.00000 −0.0732252
\(747\) −12.0000 −0.439057
\(748\) −3.00000 −0.109691
\(749\) 0 0
\(750\) 1.00000 0.0365148
\(751\) 5.00000 0.182453 0.0912263 0.995830i \(-0.470921\pi\)
0.0912263 + 0.995830i \(0.470921\pi\)
\(752\) −6.00000 −0.218797
\(753\) −30.0000 −1.09326
\(754\) −6.00000 −0.218507
\(755\) 2.00000 0.0727875
\(756\) 0 0
\(757\) 26.0000 0.944986 0.472493 0.881334i \(-0.343354\pi\)
0.472493 + 0.881334i \(0.343354\pi\)
\(758\) −26.0000 −0.944363
\(759\) 6.00000 0.217786
\(760\) −7.00000 −0.253917
\(761\) 30.0000 1.08750 0.543750 0.839248i \(-0.317004\pi\)
0.543750 + 0.839248i \(0.317004\pi\)
\(762\) −16.0000 −0.579619
\(763\) 0 0
\(764\) −12.0000 −0.434145
\(765\) 6.00000 0.216930
\(766\) −18.0000 −0.650366
\(767\) −12.0000 −0.433295
\(768\) −1.00000 −0.0360844
\(769\) 10.0000 0.360609 0.180305 0.983611i \(-0.442292\pi\)
0.180305 + 0.983611i \(0.442292\pi\)
\(770\) 0 0
\(771\) 6.00000 0.216085
\(772\) 23.0000 0.827788
\(773\) −9.00000 −0.323708 −0.161854 0.986815i \(-0.551747\pi\)
−0.161854 + 0.986815i \(0.551747\pi\)
\(774\) 16.0000 0.575108
\(775\) 7.00000 0.251447
\(776\) −4.00000 −0.143592
\(777\) 0 0
\(778\) −18.0000 −0.645331
\(779\) −42.0000 −1.50481
\(780\) 2.00000 0.0716115
\(781\) 3.00000 0.107348
\(782\) −18.0000 −0.643679
\(783\) −15.0000 −0.536056
\(784\) 0 0
\(785\) 7.00000 0.249841
\(786\) 3.00000 0.107006
\(787\) −8.00000 −0.285169 −0.142585 0.989783i \(-0.545541\pi\)
−0.142585 + 0.989783i \(0.545541\pi\)
\(788\) 0 0
\(789\) 9.00000 0.320408
\(790\) 10.0000 0.355784
\(791\) 0 0
\(792\) 2.00000 0.0710669
\(793\) −2.00000 −0.0710221
\(794\) −22.0000 −0.780751
\(795\) 3.00000 0.106399
\(796\) −11.0000 −0.389885
\(797\) −30.0000 −1.06265 −0.531327 0.847167i \(-0.678307\pi\)
−0.531327 + 0.847167i \(0.678307\pi\)
\(798\) 0 0
\(799\) 18.0000 0.636794
\(800\) −1.00000 −0.0353553
\(801\) 18.0000 0.635999
\(802\) −3.00000 −0.105934
\(803\) −2.00000 −0.0705785
\(804\) −8.00000 −0.282138
\(805\) 0 0
\(806\) 14.0000 0.493129
\(807\) −12.0000 −0.422420
\(808\) −6.00000 −0.211079
\(809\) −30.0000 −1.05474 −0.527372 0.849635i \(-0.676823\pi\)
−0.527372 + 0.849635i \(0.676823\pi\)
\(810\) −1.00000 −0.0351364
\(811\) 13.0000 0.456492 0.228246 0.973604i \(-0.426701\pi\)
0.228246 + 0.973604i \(0.426701\pi\)
\(812\) 0 0
\(813\) 20.0000 0.701431
\(814\) 7.00000 0.245350
\(815\) 5.00000 0.175142
\(816\) 3.00000 0.105021
\(817\) 56.0000 1.95919
\(818\) −4.00000 −0.139857
\(819\) 0 0
\(820\) −6.00000 −0.209529
\(821\) 30.0000 1.04701 0.523504 0.852023i \(-0.324625\pi\)
0.523504 + 0.852023i \(0.324625\pi\)
\(822\) 12.0000 0.418548
\(823\) −4.00000 −0.139431 −0.0697156 0.997567i \(-0.522209\pi\)
−0.0697156 + 0.997567i \(0.522209\pi\)
\(824\) −4.00000 −0.139347
\(825\) −1.00000 −0.0348155
\(826\) 0 0
\(827\) 12.0000 0.417281 0.208640 0.977992i \(-0.433096\pi\)
0.208640 + 0.977992i \(0.433096\pi\)
\(828\) 12.0000 0.417029
\(829\) 52.0000 1.80603 0.903017 0.429604i \(-0.141347\pi\)
0.903017 + 0.429604i \(0.141347\pi\)
\(830\) −6.00000 −0.208263
\(831\) 4.00000 0.138758
\(832\) −2.00000 −0.0693375
\(833\) 0 0
\(834\) 4.00000 0.138509
\(835\) −21.0000 −0.726735
\(836\) 7.00000 0.242100
\(837\) 35.0000 1.20978
\(838\) 24.0000 0.829066
\(839\) −48.0000 −1.65714 −0.828572 0.559883i \(-0.810846\pi\)
−0.828572 + 0.559883i \(0.810846\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) −8.00000 −0.275698
\(843\) −30.0000 −1.03325
\(844\) 5.00000 0.172107
\(845\) −9.00000 −0.309609
\(846\) −12.0000 −0.412568
\(847\) 0 0
\(848\) −3.00000 −0.103020
\(849\) 2.00000 0.0686398
\(850\) 3.00000 0.102899
\(851\) 42.0000 1.43974
\(852\) −3.00000 −0.102778
\(853\) 34.0000 1.16414 0.582069 0.813139i \(-0.302243\pi\)
0.582069 + 0.813139i \(0.302243\pi\)
\(854\) 0 0
\(855\) −14.0000 −0.478790
\(856\) 0 0
\(857\) −3.00000 −0.102478 −0.0512390 0.998686i \(-0.516317\pi\)
−0.0512390 + 0.998686i \(0.516317\pi\)
\(858\) −2.00000 −0.0682789
\(859\) −14.0000 −0.477674 −0.238837 0.971060i \(-0.576766\pi\)
−0.238837 + 0.971060i \(0.576766\pi\)
\(860\) 8.00000 0.272798
\(861\) 0 0
\(862\) 24.0000 0.817443
\(863\) −6.00000 −0.204242 −0.102121 0.994772i \(-0.532563\pi\)
−0.102121 + 0.994772i \(0.532563\pi\)
\(864\) −5.00000 −0.170103
\(865\) −18.0000 −0.612018
\(866\) −28.0000 −0.951479
\(867\) 8.00000 0.271694
\(868\) 0 0
\(869\) −10.0000 −0.339227
\(870\) −3.00000 −0.101710
\(871\) −16.0000 −0.542139
\(872\) −14.0000 −0.474100
\(873\) −8.00000 −0.270759
\(874\) 42.0000 1.42067
\(875\) 0 0
\(876\) 2.00000 0.0675737
\(877\) −22.0000 −0.742887 −0.371444 0.928456i \(-0.621137\pi\)
−0.371444 + 0.928456i \(0.621137\pi\)
\(878\) −16.0000 −0.539974
\(879\) −6.00000 −0.202375
\(880\) 1.00000 0.0337100
\(881\) −18.0000 −0.606435 −0.303218 0.952921i \(-0.598061\pi\)
−0.303218 + 0.952921i \(0.598061\pi\)
\(882\) 0 0
\(883\) −1.00000 −0.0336527 −0.0168263 0.999858i \(-0.505356\pi\)
−0.0168263 + 0.999858i \(0.505356\pi\)
\(884\) 6.00000 0.201802
\(885\) −6.00000 −0.201688
\(886\) −12.0000 −0.403148
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) −7.00000 −0.234905
\(889\) 0 0
\(890\) 9.00000 0.301681
\(891\) 1.00000 0.0335013
\(892\) −14.0000 −0.468755
\(893\) −42.0000 −1.40548
\(894\) 15.0000 0.501675
\(895\) −12.0000 −0.401116
\(896\) 0 0
\(897\) −12.0000 −0.400668
\(898\) 30.0000 1.00111
\(899\) −21.0000 −0.700389
\(900\) −2.00000 −0.0666667
\(901\) 9.00000 0.299833
\(902\) 6.00000 0.199778
\(903\) 0 0
\(904\) −12.0000 −0.399114
\(905\) 22.0000 0.731305
\(906\) 2.00000 0.0664455
\(907\) 17.0000 0.564476 0.282238 0.959344i \(-0.408923\pi\)
0.282238 + 0.959344i \(0.408923\pi\)
\(908\) 18.0000 0.597351
\(909\) −12.0000 −0.398015
\(910\) 0 0
\(911\) 15.0000 0.496972 0.248486 0.968635i \(-0.420067\pi\)
0.248486 + 0.968635i \(0.420067\pi\)
\(912\) −7.00000 −0.231793
\(913\) 6.00000 0.198571
\(914\) −17.0000 −0.562310
\(915\) −1.00000 −0.0330590
\(916\) −14.0000 −0.462573
\(917\) 0 0
\(918\) 15.0000 0.495074
\(919\) −4.00000 −0.131948 −0.0659739 0.997821i \(-0.521015\pi\)
−0.0659739 + 0.997821i \(0.521015\pi\)
\(920\) 6.00000 0.197814
\(921\) −10.0000 −0.329511
\(922\) 3.00000 0.0987997
\(923\) −6.00000 −0.197492
\(924\) 0 0
\(925\) −7.00000 −0.230159
\(926\) 22.0000 0.722965
\(927\) −8.00000 −0.262754
\(928\) 3.00000 0.0984798
\(929\) 45.0000 1.47640 0.738201 0.674581i \(-0.235676\pi\)
0.738201 + 0.674581i \(0.235676\pi\)
\(930\) 7.00000 0.229539
\(931\) 0 0
\(932\) 27.0000 0.884414
\(933\) −15.0000 −0.491078
\(934\) −39.0000 −1.27612
\(935\) −3.00000 −0.0981105
\(936\) −4.00000 −0.130744
\(937\) −26.0000 −0.849383 −0.424691 0.905338i \(-0.639617\pi\)
−0.424691 + 0.905338i \(0.639617\pi\)
\(938\) 0 0
\(939\) 14.0000 0.456873
\(940\) −6.00000 −0.195698
\(941\) 15.0000 0.488986 0.244493 0.969651i \(-0.421378\pi\)
0.244493 + 0.969651i \(0.421378\pi\)
\(942\) 7.00000 0.228072
\(943\) 36.0000 1.17232
\(944\) 6.00000 0.195283
\(945\) 0 0
\(946\) −8.00000 −0.260102
\(947\) 3.00000 0.0974869 0.0487435 0.998811i \(-0.484478\pi\)
0.0487435 + 0.998811i \(0.484478\pi\)
\(948\) 10.0000 0.324785
\(949\) 4.00000 0.129845
\(950\) −7.00000 −0.227110
\(951\) 21.0000 0.680972
\(952\) 0 0
\(953\) −27.0000 −0.874616 −0.437308 0.899312i \(-0.644068\pi\)
−0.437308 + 0.899312i \(0.644068\pi\)
\(954\) −6.00000 −0.194257
\(955\) −12.0000 −0.388311
\(956\) 6.00000 0.194054
\(957\) 3.00000 0.0969762
\(958\) −36.0000 −1.16311
\(959\) 0 0
\(960\) −1.00000 −0.0322749
\(961\) 18.0000 0.580645
\(962\) −14.0000 −0.451378
\(963\) 0 0
\(964\) 22.0000 0.708572
\(965\) 23.0000 0.740396
\(966\) 0 0
\(967\) −13.0000 −0.418052 −0.209026 0.977910i \(-0.567029\pi\)
−0.209026 + 0.977910i \(0.567029\pi\)
\(968\) −1.00000 −0.0321412
\(969\) 21.0000 0.674617
\(970\) −4.00000 −0.128432
\(971\) 24.0000 0.770197 0.385098 0.922876i \(-0.374168\pi\)
0.385098 + 0.922876i \(0.374168\pi\)
\(972\) −16.0000 −0.513200
\(973\) 0 0
\(974\) −8.00000 −0.256337
\(975\) 2.00000 0.0640513
\(976\) 1.00000 0.0320092
\(977\) 12.0000 0.383914 0.191957 0.981403i \(-0.438517\pi\)
0.191957 + 0.981403i \(0.438517\pi\)
\(978\) 5.00000 0.159882
\(979\) −9.00000 −0.287641
\(980\) 0 0
\(981\) −28.0000 −0.893971
\(982\) −33.0000 −1.05307
\(983\) 6.00000 0.191370 0.0956851 0.995412i \(-0.469496\pi\)
0.0956851 + 0.995412i \(0.469496\pi\)
\(984\) −6.00000 −0.191273
\(985\) 0 0
\(986\) −9.00000 −0.286618
\(987\) 0 0
\(988\) −14.0000 −0.445399
\(989\) −48.0000 −1.52631
\(990\) 2.00000 0.0635642
\(991\) −40.0000 −1.27064 −0.635321 0.772248i \(-0.719132\pi\)
−0.635321 + 0.772248i \(0.719132\pi\)
\(992\) −7.00000 −0.222250
\(993\) −8.00000 −0.253872
\(994\) 0 0
\(995\) −11.0000 −0.348723
\(996\) −6.00000 −0.190117
\(997\) 28.0000 0.886769 0.443384 0.896332i \(-0.353778\pi\)
0.443384 + 0.896332i \(0.353778\pi\)
\(998\) −44.0000 −1.39280
\(999\) −35.0000 −1.10735
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 5390.2.a.h.1.1 1
7.6 odd 2 110.2.a.a.1.1 1
21.20 even 2 990.2.a.l.1.1 1
28.27 even 2 880.2.a.c.1.1 1
35.13 even 4 550.2.b.b.199.2 2
35.27 even 4 550.2.b.b.199.1 2
35.34 odd 2 550.2.a.i.1.1 1
56.13 odd 2 3520.2.a.l.1.1 1
56.27 even 2 3520.2.a.z.1.1 1
77.76 even 2 1210.2.a.k.1.1 1
84.83 odd 2 7920.2.a.s.1.1 1
105.62 odd 4 4950.2.c.a.199.2 2
105.83 odd 4 4950.2.c.a.199.1 2
105.104 even 2 4950.2.a.a.1.1 1
140.27 odd 4 4400.2.b.g.4049.2 2
140.83 odd 4 4400.2.b.g.4049.1 2
140.139 even 2 4400.2.a.w.1.1 1
308.307 odd 2 9680.2.a.j.1.1 1
385.384 even 2 6050.2.a.i.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
110.2.a.a.1.1 1 7.6 odd 2
550.2.a.i.1.1 1 35.34 odd 2
550.2.b.b.199.1 2 35.27 even 4
550.2.b.b.199.2 2 35.13 even 4
880.2.a.c.1.1 1 28.27 even 2
990.2.a.l.1.1 1 21.20 even 2
1210.2.a.k.1.1 1 77.76 even 2
3520.2.a.l.1.1 1 56.13 odd 2
3520.2.a.z.1.1 1 56.27 even 2
4400.2.a.w.1.1 1 140.139 even 2
4400.2.b.g.4049.1 2 140.83 odd 4
4400.2.b.g.4049.2 2 140.27 odd 4
4950.2.a.a.1.1 1 105.104 even 2
4950.2.c.a.199.1 2 105.83 odd 4
4950.2.c.a.199.2 2 105.62 odd 4
5390.2.a.h.1.1 1 1.1 even 1 trivial
6050.2.a.i.1.1 1 385.384 even 2
7920.2.a.s.1.1 1 84.83 odd 2
9680.2.a.j.1.1 1 308.307 odd 2