Properties

Label 7605.2.a.d.1.1
Level $7605$
Weight $2$
Character 7605.1
Self dual yes
Analytic conductor $60.726$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [7605,2,Mod(1,7605)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7605, 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("7605.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7605 = 3^{2} \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7605.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: \(60.7262307372\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 195)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 7605.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{4} -1.00000 q^{5} +2.00000 q^{7} +3.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{4} -1.00000 q^{5} +2.00000 q^{7} +3.00000 q^{8} +1.00000 q^{10} -2.00000 q^{14} -1.00000 q^{16} +2.00000 q^{17} +2.00000 q^{19} +1.00000 q^{20} -8.00000 q^{23} +1.00000 q^{25} -2.00000 q^{28} -2.00000 q^{29} -2.00000 q^{31} -5.00000 q^{32} -2.00000 q^{34} -2.00000 q^{35} +8.00000 q^{37} -2.00000 q^{38} -3.00000 q^{40} +2.00000 q^{41} +4.00000 q^{43} +8.00000 q^{46} -4.00000 q^{47} -3.00000 q^{49} -1.00000 q^{50} +6.00000 q^{53} +6.00000 q^{56} +2.00000 q^{58} -12.0000 q^{59} +10.0000 q^{61} +2.00000 q^{62} +7.00000 q^{64} -6.00000 q^{67} -2.00000 q^{68} +2.00000 q^{70} -8.00000 q^{71} -16.0000 q^{73} -8.00000 q^{74} -2.00000 q^{76} -8.00000 q^{79} +1.00000 q^{80} -2.00000 q^{82} -12.0000 q^{83} -2.00000 q^{85} -4.00000 q^{86} +6.00000 q^{89} +8.00000 q^{92} +4.00000 q^{94} -2.00000 q^{95} +16.0000 q^{97} +3.00000 q^{98} +O(q^{100})\)

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 −0.353553 0.935414i \(-0.615027\pi\)
−0.353553 + 0.935414i \(0.615027\pi\)
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 2.00000 0.755929 0.377964 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) 3.00000 1.06066
\(9\) 0 0
\(10\) 1.00000 0.316228
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) −2.00000 −0.534522
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) 0 0
\(23\) −8.00000 −1.66812 −0.834058 0.551677i \(-0.813988\pi\)
−0.834058 + 0.551677i \(0.813988\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) −2.00000 −0.377964
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) −2.00000 −0.359211 −0.179605 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) −5.00000 −0.883883
\(33\) 0 0
\(34\) −2.00000 −0.342997
\(35\) −2.00000 −0.338062
\(36\) 0 0
\(37\) 8.00000 1.31519 0.657596 0.753371i \(-0.271573\pi\)
0.657596 + 0.753371i \(0.271573\pi\)
\(38\) −2.00000 −0.324443
\(39\) 0 0
\(40\) −3.00000 −0.474342
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 8.00000 1.17954
\(47\) −4.00000 −0.583460 −0.291730 0.956501i \(-0.594231\pi\)
−0.291730 + 0.956501i \(0.594231\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) 0 0
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 6.00000 0.801784
\(57\) 0 0
\(58\) 2.00000 0.262613
\(59\) −12.0000 −1.56227 −0.781133 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) 0 0
\(61\) 10.0000 1.28037 0.640184 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 2.00000 0.254000
\(63\) 0 0
\(64\) 7.00000 0.875000
\(65\) 0 0
\(66\) 0 0
\(67\) −6.00000 −0.733017 −0.366508 0.930415i \(-0.619447\pi\)
−0.366508 + 0.930415i \(0.619447\pi\)
\(68\) −2.00000 −0.242536
\(69\) 0 0
\(70\) 2.00000 0.239046
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 0 0
\(73\) −16.0000 −1.87266 −0.936329 0.351123i \(-0.885800\pi\)
−0.936329 + 0.351123i \(0.885800\pi\)
\(74\) −8.00000 −0.929981
\(75\) 0 0
\(76\) −2.00000 −0.229416
\(77\) 0 0
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 1.00000 0.111803
\(81\) 0 0
\(82\) −2.00000 −0.220863
\(83\) −12.0000 −1.31717 −0.658586 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 0 0
\(85\) −2.00000 −0.216930
\(86\) −4.00000 −0.431331
\(87\) 0 0
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 8.00000 0.834058
\(93\) 0 0
\(94\) 4.00000 0.412568
\(95\) −2.00000 −0.205196
\(96\) 0 0
\(97\) 16.0000 1.62455 0.812277 0.583272i \(-0.198228\pi\)
0.812277 + 0.583272i \(0.198228\pi\)
\(98\) 3.00000 0.303046
\(99\) 0 0
\(100\) −1.00000 −0.100000
\(101\) −2.00000 −0.199007 −0.0995037 0.995037i \(-0.531726\pi\)
−0.0995037 + 0.995037i \(0.531726\pi\)
\(102\) 0 0
\(103\) −16.0000 −1.57653 −0.788263 0.615338i \(-0.789020\pi\)
−0.788263 + 0.615338i \(0.789020\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) −8.00000 −0.773389 −0.386695 0.922208i \(-0.626383\pi\)
−0.386695 + 0.922208i \(0.626383\pi\)
\(108\) 0 0
\(109\) 16.0000 1.53252 0.766261 0.642529i \(-0.222115\pi\)
0.766261 + 0.642529i \(0.222115\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −2.00000 −0.188982
\(113\) 14.0000 1.31701 0.658505 0.752577i \(-0.271189\pi\)
0.658505 + 0.752577i \(0.271189\pi\)
\(114\) 0 0
\(115\) 8.00000 0.746004
\(116\) 2.00000 0.185695
\(117\) 0 0
\(118\) 12.0000 1.10469
\(119\) 4.00000 0.366679
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) −10.0000 −0.905357
\(123\) 0 0
\(124\) 2.00000 0.179605
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) 3.00000 0.265165
\(129\) 0 0
\(130\) 0 0
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 0 0
\(133\) 4.00000 0.346844
\(134\) 6.00000 0.518321
\(135\) 0 0
\(136\) 6.00000 0.514496
\(137\) −6.00000 −0.512615 −0.256307 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) 0 0
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 2.00000 0.169031
\(141\) 0 0
\(142\) 8.00000 0.671345
\(143\) 0 0
\(144\) 0 0
\(145\) 2.00000 0.166091
\(146\) 16.0000 1.32417
\(147\) 0 0
\(148\) −8.00000 −0.657596
\(149\) 14.0000 1.14692 0.573462 0.819232i \(-0.305600\pi\)
0.573462 + 0.819232i \(0.305600\pi\)
\(150\) 0 0
\(151\) −14.0000 −1.13930 −0.569652 0.821886i \(-0.692922\pi\)
−0.569652 + 0.821886i \(0.692922\pi\)
\(152\) 6.00000 0.486664
\(153\) 0 0
\(154\) 0 0
\(155\) 2.00000 0.160644
\(156\) 0 0
\(157\) 14.0000 1.11732 0.558661 0.829396i \(-0.311315\pi\)
0.558661 + 0.829396i \(0.311315\pi\)
\(158\) 8.00000 0.636446
\(159\) 0 0
\(160\) 5.00000 0.395285
\(161\) −16.0000 −1.26098
\(162\) 0 0
\(163\) 18.0000 1.40987 0.704934 0.709273i \(-0.250976\pi\)
0.704934 + 0.709273i \(0.250976\pi\)
\(164\) −2.00000 −0.156174
\(165\) 0 0
\(166\) 12.0000 0.931381
\(167\) −20.0000 −1.54765 −0.773823 0.633402i \(-0.781658\pi\)
−0.773823 + 0.633402i \(0.781658\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 2.00000 0.153393
\(171\) 0 0
\(172\) −4.00000 −0.304997
\(173\) 10.0000 0.760286 0.380143 0.924928i \(-0.375875\pi\)
0.380143 + 0.924928i \(0.375875\pi\)
\(174\) 0 0
\(175\) 2.00000 0.151186
\(176\) 0 0
\(177\) 0 0
\(178\) −6.00000 −0.449719
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) −6.00000 −0.445976 −0.222988 0.974821i \(-0.571581\pi\)
−0.222988 + 0.974821i \(0.571581\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −24.0000 −1.76930
\(185\) −8.00000 −0.588172
\(186\) 0 0
\(187\) 0 0
\(188\) 4.00000 0.291730
\(189\) 0 0
\(190\) 2.00000 0.145095
\(191\) 20.0000 1.44715 0.723575 0.690246i \(-0.242498\pi\)
0.723575 + 0.690246i \(0.242498\pi\)
\(192\) 0 0
\(193\) −4.00000 −0.287926 −0.143963 0.989583i \(-0.545985\pi\)
−0.143963 + 0.989583i \(0.545985\pi\)
\(194\) −16.0000 −1.14873
\(195\) 0 0
\(196\) 3.00000 0.214286
\(197\) −22.0000 −1.56744 −0.783718 0.621117i \(-0.786679\pi\)
−0.783718 + 0.621117i \(0.786679\pi\)
\(198\) 0 0
\(199\) 24.0000 1.70131 0.850657 0.525720i \(-0.176204\pi\)
0.850657 + 0.525720i \(0.176204\pi\)
\(200\) 3.00000 0.212132
\(201\) 0 0
\(202\) 2.00000 0.140720
\(203\) −4.00000 −0.280745
\(204\) 0 0
\(205\) −2.00000 −0.139686
\(206\) 16.0000 1.11477
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) −6.00000 −0.412082
\(213\) 0 0
\(214\) 8.00000 0.546869
\(215\) −4.00000 −0.272798
\(216\) 0 0
\(217\) −4.00000 −0.271538
\(218\) −16.0000 −1.08366
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −22.0000 −1.47323 −0.736614 0.676313i \(-0.763577\pi\)
−0.736614 + 0.676313i \(0.763577\pi\)
\(224\) −10.0000 −0.668153
\(225\) 0 0
\(226\) −14.0000 −0.931266
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) −8.00000 −0.528655 −0.264327 0.964433i \(-0.585150\pi\)
−0.264327 + 0.964433i \(0.585150\pi\)
\(230\) −8.00000 −0.527504
\(231\) 0 0
\(232\) −6.00000 −0.393919
\(233\) 14.0000 0.917170 0.458585 0.888650i \(-0.348356\pi\)
0.458585 + 0.888650i \(0.348356\pi\)
\(234\) 0 0
\(235\) 4.00000 0.260931
\(236\) 12.0000 0.781133
\(237\) 0 0
\(238\) −4.00000 −0.259281
\(239\) −8.00000 −0.517477 −0.258738 0.965947i \(-0.583307\pi\)
−0.258738 + 0.965947i \(0.583307\pi\)
\(240\) 0 0
\(241\) −12.0000 −0.772988 −0.386494 0.922292i \(-0.626314\pi\)
−0.386494 + 0.922292i \(0.626314\pi\)
\(242\) 11.0000 0.707107
\(243\) 0 0
\(244\) −10.0000 −0.640184
\(245\) 3.00000 0.191663
\(246\) 0 0
\(247\) 0 0
\(248\) −6.00000 −0.381000
\(249\) 0 0
\(250\) 1.00000 0.0632456
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) −17.0000 −1.06250
\(257\) 30.0000 1.87135 0.935674 0.352865i \(-0.114792\pi\)
0.935674 + 0.352865i \(0.114792\pi\)
\(258\) 0 0
\(259\) 16.0000 0.994192
\(260\) 0 0
\(261\) 0 0
\(262\) −12.0000 −0.741362
\(263\) −12.0000 −0.739952 −0.369976 0.929041i \(-0.620634\pi\)
−0.369976 + 0.929041i \(0.620634\pi\)
\(264\) 0 0
\(265\) −6.00000 −0.368577
\(266\) −4.00000 −0.245256
\(267\) 0 0
\(268\) 6.00000 0.366508
\(269\) −10.0000 −0.609711 −0.304855 0.952399i \(-0.598608\pi\)
−0.304855 + 0.952399i \(0.598608\pi\)
\(270\) 0 0
\(271\) 14.0000 0.850439 0.425220 0.905090i \(-0.360197\pi\)
0.425220 + 0.905090i \(0.360197\pi\)
\(272\) −2.00000 −0.121268
\(273\) 0 0
\(274\) 6.00000 0.362473
\(275\) 0 0
\(276\) 0 0
\(277\) 2.00000 0.120168 0.0600842 0.998193i \(-0.480863\pi\)
0.0600842 + 0.998193i \(0.480863\pi\)
\(278\) −4.00000 −0.239904
\(279\) 0 0
\(280\) −6.00000 −0.358569
\(281\) −6.00000 −0.357930 −0.178965 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) 0 0
\(283\) 4.00000 0.237775 0.118888 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) 8.00000 0.474713
\(285\) 0 0
\(286\) 0 0
\(287\) 4.00000 0.236113
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) −2.00000 −0.117444
\(291\) 0 0
\(292\) 16.0000 0.936329
\(293\) −18.0000 −1.05157 −0.525786 0.850617i \(-0.676229\pi\)
−0.525786 + 0.850617i \(0.676229\pi\)
\(294\) 0 0
\(295\) 12.0000 0.698667
\(296\) 24.0000 1.39497
\(297\) 0 0
\(298\) −14.0000 −0.810998
\(299\) 0 0
\(300\) 0 0
\(301\) 8.00000 0.461112
\(302\) 14.0000 0.805609
\(303\) 0 0
\(304\) −2.00000 −0.114708
\(305\) −10.0000 −0.572598
\(306\) 0 0
\(307\) −26.0000 −1.48390 −0.741949 0.670456i \(-0.766098\pi\)
−0.741949 + 0.670456i \(0.766098\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −2.00000 −0.113592
\(311\) 4.00000 0.226819 0.113410 0.993548i \(-0.463823\pi\)
0.113410 + 0.993548i \(0.463823\pi\)
\(312\) 0 0
\(313\) −10.0000 −0.565233 −0.282617 0.959233i \(-0.591202\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) −14.0000 −0.790066
\(315\) 0 0
\(316\) 8.00000 0.450035
\(317\) 26.0000 1.46031 0.730153 0.683284i \(-0.239449\pi\)
0.730153 + 0.683284i \(0.239449\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −7.00000 −0.391312
\(321\) 0 0
\(322\) 16.0000 0.891645
\(323\) 4.00000 0.222566
\(324\) 0 0
\(325\) 0 0
\(326\) −18.0000 −0.996928
\(327\) 0 0
\(328\) 6.00000 0.331295
\(329\) −8.00000 −0.441054
\(330\) 0 0
\(331\) −26.0000 −1.42909 −0.714545 0.699590i \(-0.753366\pi\)
−0.714545 + 0.699590i \(0.753366\pi\)
\(332\) 12.0000 0.658586
\(333\) 0 0
\(334\) 20.0000 1.09435
\(335\) 6.00000 0.327815
\(336\) 0 0
\(337\) −6.00000 −0.326841 −0.163420 0.986557i \(-0.552253\pi\)
−0.163420 + 0.986557i \(0.552253\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 2.00000 0.108465
\(341\) 0 0
\(342\) 0 0
\(343\) −20.0000 −1.07990
\(344\) 12.0000 0.646997
\(345\) 0 0
\(346\) −10.0000 −0.537603
\(347\) 20.0000 1.07366 0.536828 0.843692i \(-0.319622\pi\)
0.536828 + 0.843692i \(0.319622\pi\)
\(348\) 0 0
\(349\) 20.0000 1.07058 0.535288 0.844670i \(-0.320203\pi\)
0.535288 + 0.844670i \(0.320203\pi\)
\(350\) −2.00000 −0.106904
\(351\) 0 0
\(352\) 0 0
\(353\) −30.0000 −1.59674 −0.798369 0.602168i \(-0.794304\pi\)
−0.798369 + 0.602168i \(0.794304\pi\)
\(354\) 0 0
\(355\) 8.00000 0.424596
\(356\) −6.00000 −0.317999
\(357\) 0 0
\(358\) −12.0000 −0.634220
\(359\) 4.00000 0.211112 0.105556 0.994413i \(-0.466338\pi\)
0.105556 + 0.994413i \(0.466338\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 6.00000 0.315353
\(363\) 0 0
\(364\) 0 0
\(365\) 16.0000 0.837478
\(366\) 0 0
\(367\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(368\) 8.00000 0.417029
\(369\) 0 0
\(370\) 8.00000 0.415900
\(371\) 12.0000 0.623009
\(372\) 0 0
\(373\) −38.0000 −1.96757 −0.983783 0.179364i \(-0.942596\pi\)
−0.983783 + 0.179364i \(0.942596\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −12.0000 −0.618853
\(377\) 0 0
\(378\) 0 0
\(379\) −30.0000 −1.54100 −0.770498 0.637442i \(-0.779993\pi\)
−0.770498 + 0.637442i \(0.779993\pi\)
\(380\) 2.00000 0.102598
\(381\) 0 0
\(382\) −20.0000 −1.02329
\(383\) −4.00000 −0.204390 −0.102195 0.994764i \(-0.532587\pi\)
−0.102195 + 0.994764i \(0.532587\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 4.00000 0.203595
\(387\) 0 0
\(388\) −16.0000 −0.812277
\(389\) −38.0000 −1.92668 −0.963338 0.268290i \(-0.913542\pi\)
−0.963338 + 0.268290i \(0.913542\pi\)
\(390\) 0 0
\(391\) −16.0000 −0.809155
\(392\) −9.00000 −0.454569
\(393\) 0 0
\(394\) 22.0000 1.10834
\(395\) 8.00000 0.402524
\(396\) 0 0
\(397\) −8.00000 −0.401508 −0.200754 0.979642i \(-0.564339\pi\)
−0.200754 + 0.979642i \(0.564339\pi\)
\(398\) −24.0000 −1.20301
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) −10.0000 −0.499376 −0.249688 0.968326i \(-0.580328\pi\)
−0.249688 + 0.968326i \(0.580328\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 2.00000 0.0995037
\(405\) 0 0
\(406\) 4.00000 0.198517
\(407\) 0 0
\(408\) 0 0
\(409\) −12.0000 −0.593362 −0.296681 0.954977i \(-0.595880\pi\)
−0.296681 + 0.954977i \(0.595880\pi\)
\(410\) 2.00000 0.0987730
\(411\) 0 0
\(412\) 16.0000 0.788263
\(413\) −24.0000 −1.18096
\(414\) 0 0
\(415\) 12.0000 0.589057
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 12.0000 0.586238 0.293119 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 0 0
\(421\) −40.0000 −1.94948 −0.974740 0.223341i \(-0.928304\pi\)
−0.974740 + 0.223341i \(0.928304\pi\)
\(422\) 12.0000 0.584151
\(423\) 0 0
\(424\) 18.0000 0.874157
\(425\) 2.00000 0.0970143
\(426\) 0 0
\(427\) 20.0000 0.967868
\(428\) 8.00000 0.386695
\(429\) 0 0
\(430\) 4.00000 0.192897
\(431\) −12.0000 −0.578020 −0.289010 0.957326i \(-0.593326\pi\)
−0.289010 + 0.957326i \(0.593326\pi\)
\(432\) 0 0
\(433\) −14.0000 −0.672797 −0.336399 0.941720i \(-0.609209\pi\)
−0.336399 + 0.941720i \(0.609209\pi\)
\(434\) 4.00000 0.192006
\(435\) 0 0
\(436\) −16.0000 −0.766261
\(437\) −16.0000 −0.765384
\(438\) 0 0
\(439\) −16.0000 −0.763638 −0.381819 0.924237i \(-0.624702\pi\)
−0.381819 + 0.924237i \(0.624702\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −24.0000 −1.14027 −0.570137 0.821549i \(-0.693110\pi\)
−0.570137 + 0.821549i \(0.693110\pi\)
\(444\) 0 0
\(445\) −6.00000 −0.284427
\(446\) 22.0000 1.04173
\(447\) 0 0
\(448\) 14.0000 0.661438
\(449\) 6.00000 0.283158 0.141579 0.989927i \(-0.454782\pi\)
0.141579 + 0.989927i \(0.454782\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −14.0000 −0.658505
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 8.00000 0.374224 0.187112 0.982339i \(-0.440087\pi\)
0.187112 + 0.982339i \(0.440087\pi\)
\(458\) 8.00000 0.373815
\(459\) 0 0
\(460\) −8.00000 −0.373002
\(461\) 34.0000 1.58354 0.791769 0.610821i \(-0.209160\pi\)
0.791769 + 0.610821i \(0.209160\pi\)
\(462\) 0 0
\(463\) −14.0000 −0.650635 −0.325318 0.945605i \(-0.605471\pi\)
−0.325318 + 0.945605i \(0.605471\pi\)
\(464\) 2.00000 0.0928477
\(465\) 0 0
\(466\) −14.0000 −0.648537
\(467\) 36.0000 1.66588 0.832941 0.553362i \(-0.186655\pi\)
0.832941 + 0.553362i \(0.186655\pi\)
\(468\) 0 0
\(469\) −12.0000 −0.554109
\(470\) −4.00000 −0.184506
\(471\) 0 0
\(472\) −36.0000 −1.65703
\(473\) 0 0
\(474\) 0 0
\(475\) 2.00000 0.0917663
\(476\) −4.00000 −0.183340
\(477\) 0 0
\(478\) 8.00000 0.365911
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 12.0000 0.546585
\(483\) 0 0
\(484\) 11.0000 0.500000
\(485\) −16.0000 −0.726523
\(486\) 0 0
\(487\) 18.0000 0.815658 0.407829 0.913058i \(-0.366286\pi\)
0.407829 + 0.913058i \(0.366286\pi\)
\(488\) 30.0000 1.35804
\(489\) 0 0
\(490\) −3.00000 −0.135526
\(491\) −40.0000 −1.80517 −0.902587 0.430507i \(-0.858335\pi\)
−0.902587 + 0.430507i \(0.858335\pi\)
\(492\) 0 0
\(493\) −4.00000 −0.180151
\(494\) 0 0
\(495\) 0 0
\(496\) 2.00000 0.0898027
\(497\) −16.0000 −0.717698
\(498\) 0 0
\(499\) −10.0000 −0.447661 −0.223831 0.974628i \(-0.571856\pi\)
−0.223831 + 0.974628i \(0.571856\pi\)
\(500\) 1.00000 0.0447214
\(501\) 0 0
\(502\) 0 0
\(503\) 4.00000 0.178351 0.0891756 0.996016i \(-0.471577\pi\)
0.0891756 + 0.996016i \(0.471577\pi\)
\(504\) 0 0
\(505\) 2.00000 0.0889988
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −6.00000 −0.265945 −0.132973 0.991120i \(-0.542452\pi\)
−0.132973 + 0.991120i \(0.542452\pi\)
\(510\) 0 0
\(511\) −32.0000 −1.41560
\(512\) 11.0000 0.486136
\(513\) 0 0
\(514\) −30.0000 −1.32324
\(515\) 16.0000 0.705044
\(516\) 0 0
\(517\) 0 0
\(518\) −16.0000 −0.703000
\(519\) 0 0
\(520\) 0 0
\(521\) −22.0000 −0.963837 −0.481919 0.876216i \(-0.660060\pi\)
−0.481919 + 0.876216i \(0.660060\pi\)
\(522\) 0 0
\(523\) −28.0000 −1.22435 −0.612177 0.790721i \(-0.709706\pi\)
−0.612177 + 0.790721i \(0.709706\pi\)
\(524\) −12.0000 −0.524222
\(525\) 0 0
\(526\) 12.0000 0.523225
\(527\) −4.00000 −0.174243
\(528\) 0 0
\(529\) 41.0000 1.78261
\(530\) 6.00000 0.260623
\(531\) 0 0
\(532\) −4.00000 −0.173422
\(533\) 0 0
\(534\) 0 0
\(535\) 8.00000 0.345870
\(536\) −18.0000 −0.777482
\(537\) 0 0
\(538\) 10.0000 0.431131
\(539\) 0 0
\(540\) 0 0
\(541\) 8.00000 0.343947 0.171973 0.985102i \(-0.444986\pi\)
0.171973 + 0.985102i \(0.444986\pi\)
\(542\) −14.0000 −0.601351
\(543\) 0 0
\(544\) −10.0000 −0.428746
\(545\) −16.0000 −0.685365
\(546\) 0 0
\(547\) −4.00000 −0.171028 −0.0855138 0.996337i \(-0.527253\pi\)
−0.0855138 + 0.996337i \(0.527253\pi\)
\(548\) 6.00000 0.256307
\(549\) 0 0
\(550\) 0 0
\(551\) −4.00000 −0.170406
\(552\) 0 0
\(553\) −16.0000 −0.680389
\(554\) −2.00000 −0.0849719
\(555\) 0 0
\(556\) −4.00000 −0.169638
\(557\) 6.00000 0.254228 0.127114 0.991888i \(-0.459429\pi\)
0.127114 + 0.991888i \(0.459429\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 2.00000 0.0845154
\(561\) 0 0
\(562\) 6.00000 0.253095
\(563\) 24.0000 1.01148 0.505740 0.862686i \(-0.331220\pi\)
0.505740 + 0.862686i \(0.331220\pi\)
\(564\) 0 0
\(565\) −14.0000 −0.588984
\(566\) −4.00000 −0.168133
\(567\) 0 0
\(568\) −24.0000 −1.00702
\(569\) −30.0000 −1.25767 −0.628833 0.777541i \(-0.716467\pi\)
−0.628833 + 0.777541i \(0.716467\pi\)
\(570\) 0 0
\(571\) 4.00000 0.167395 0.0836974 0.996491i \(-0.473327\pi\)
0.0836974 + 0.996491i \(0.473327\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −4.00000 −0.166957
\(575\) −8.00000 −0.333623
\(576\) 0 0
\(577\) 4.00000 0.166522 0.0832611 0.996528i \(-0.473466\pi\)
0.0832611 + 0.996528i \(0.473466\pi\)
\(578\) 13.0000 0.540729
\(579\) 0 0
\(580\) −2.00000 −0.0830455
\(581\) −24.0000 −0.995688
\(582\) 0 0
\(583\) 0 0
\(584\) −48.0000 −1.98625
\(585\) 0 0
\(586\) 18.0000 0.743573
\(587\) −28.0000 −1.15568 −0.577842 0.816149i \(-0.696105\pi\)
−0.577842 + 0.816149i \(0.696105\pi\)
\(588\) 0 0
\(589\) −4.00000 −0.164817
\(590\) −12.0000 −0.494032
\(591\) 0 0
\(592\) −8.00000 −0.328798
\(593\) −10.0000 −0.410651 −0.205325 0.978694i \(-0.565825\pi\)
−0.205325 + 0.978694i \(0.565825\pi\)
\(594\) 0 0
\(595\) −4.00000 −0.163984
\(596\) −14.0000 −0.573462
\(597\) 0 0
\(598\) 0 0
\(599\) −16.0000 −0.653742 −0.326871 0.945069i \(-0.605994\pi\)
−0.326871 + 0.945069i \(0.605994\pi\)
\(600\) 0 0
\(601\) 38.0000 1.55005 0.775026 0.631929i \(-0.217737\pi\)
0.775026 + 0.631929i \(0.217737\pi\)
\(602\) −8.00000 −0.326056
\(603\) 0 0
\(604\) 14.0000 0.569652
\(605\) 11.0000 0.447214
\(606\) 0 0
\(607\) −8.00000 −0.324710 −0.162355 0.986732i \(-0.551909\pi\)
−0.162355 + 0.986732i \(0.551909\pi\)
\(608\) −10.0000 −0.405554
\(609\) 0 0
\(610\) 10.0000 0.404888
\(611\) 0 0
\(612\) 0 0
\(613\) 16.0000 0.646234 0.323117 0.946359i \(-0.395269\pi\)
0.323117 + 0.946359i \(0.395269\pi\)
\(614\) 26.0000 1.04927
\(615\) 0 0
\(616\) 0 0
\(617\) 10.0000 0.402585 0.201292 0.979531i \(-0.435486\pi\)
0.201292 + 0.979531i \(0.435486\pi\)
\(618\) 0 0
\(619\) −38.0000 −1.52735 −0.763674 0.645601i \(-0.776607\pi\)
−0.763674 + 0.645601i \(0.776607\pi\)
\(620\) −2.00000 −0.0803219
\(621\) 0 0
\(622\) −4.00000 −0.160385
\(623\) 12.0000 0.480770
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 10.0000 0.399680
\(627\) 0 0
\(628\) −14.0000 −0.558661
\(629\) 16.0000 0.637962
\(630\) 0 0
\(631\) −38.0000 −1.51276 −0.756378 0.654135i \(-0.773033\pi\)
−0.756378 + 0.654135i \(0.773033\pi\)
\(632\) −24.0000 −0.954669
\(633\) 0 0
\(634\) −26.0000 −1.03259
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) −3.00000 −0.118585
\(641\) −18.0000 −0.710957 −0.355479 0.934684i \(-0.615682\pi\)
−0.355479 + 0.934684i \(0.615682\pi\)
\(642\) 0 0
\(643\) −14.0000 −0.552106 −0.276053 0.961142i \(-0.589027\pi\)
−0.276053 + 0.961142i \(0.589027\pi\)
\(644\) 16.0000 0.630488
\(645\) 0 0
\(646\) −4.00000 −0.157378
\(647\) 28.0000 1.10079 0.550397 0.834903i \(-0.314476\pi\)
0.550397 + 0.834903i \(0.314476\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) −18.0000 −0.704934
\(653\) 2.00000 0.0782660 0.0391330 0.999234i \(-0.487540\pi\)
0.0391330 + 0.999234i \(0.487540\pi\)
\(654\) 0 0
\(655\) −12.0000 −0.468879
\(656\) −2.00000 −0.0780869
\(657\) 0 0
\(658\) 8.00000 0.311872
\(659\) −48.0000 −1.86981 −0.934907 0.354892i \(-0.884518\pi\)
−0.934907 + 0.354892i \(0.884518\pi\)
\(660\) 0 0
\(661\) 4.00000 0.155582 0.0777910 0.996970i \(-0.475213\pi\)
0.0777910 + 0.996970i \(0.475213\pi\)
\(662\) 26.0000 1.01052
\(663\) 0 0
\(664\) −36.0000 −1.39707
\(665\) −4.00000 −0.155113
\(666\) 0 0
\(667\) 16.0000 0.619522
\(668\) 20.0000 0.773823
\(669\) 0 0
\(670\) −6.00000 −0.231800
\(671\) 0 0
\(672\) 0 0
\(673\) 14.0000 0.539660 0.269830 0.962908i \(-0.413032\pi\)
0.269830 + 0.962908i \(0.413032\pi\)
\(674\) 6.00000 0.231111
\(675\) 0 0
\(676\) 0 0
\(677\) 6.00000 0.230599 0.115299 0.993331i \(-0.463217\pi\)
0.115299 + 0.993331i \(0.463217\pi\)
\(678\) 0 0
\(679\) 32.0000 1.22805
\(680\) −6.00000 −0.230089
\(681\) 0 0
\(682\) 0 0
\(683\) −16.0000 −0.612223 −0.306111 0.951996i \(-0.599028\pi\)
−0.306111 + 0.951996i \(0.599028\pi\)
\(684\) 0 0
\(685\) 6.00000 0.229248
\(686\) 20.0000 0.763604
\(687\) 0 0
\(688\) −4.00000 −0.152499
\(689\) 0 0
\(690\) 0 0
\(691\) −10.0000 −0.380418 −0.190209 0.981744i \(-0.560917\pi\)
−0.190209 + 0.981744i \(0.560917\pi\)
\(692\) −10.0000 −0.380143
\(693\) 0 0
\(694\) −20.0000 −0.759190
\(695\) −4.00000 −0.151729
\(696\) 0 0
\(697\) 4.00000 0.151511
\(698\) −20.0000 −0.757011
\(699\) 0 0
\(700\) −2.00000 −0.0755929
\(701\) 18.0000 0.679851 0.339925 0.940452i \(-0.389598\pi\)
0.339925 + 0.940452i \(0.389598\pi\)
\(702\) 0 0
\(703\) 16.0000 0.603451
\(704\) 0 0
\(705\) 0 0
\(706\) 30.0000 1.12906
\(707\) −4.00000 −0.150435
\(708\) 0 0
\(709\) 32.0000 1.20179 0.600893 0.799330i \(-0.294812\pi\)
0.600893 + 0.799330i \(0.294812\pi\)
\(710\) −8.00000 −0.300235
\(711\) 0 0
\(712\) 18.0000 0.674579
\(713\) 16.0000 0.599205
\(714\) 0 0
\(715\) 0 0
\(716\) −12.0000 −0.448461
\(717\) 0 0
\(718\) −4.00000 −0.149279
\(719\) −36.0000 −1.34257 −0.671287 0.741198i \(-0.734258\pi\)
−0.671287 + 0.741198i \(0.734258\pi\)
\(720\) 0 0
\(721\) −32.0000 −1.19174
\(722\) 15.0000 0.558242
\(723\) 0 0
\(724\) 6.00000 0.222988
\(725\) −2.00000 −0.0742781
\(726\) 0 0
\(727\) 40.0000 1.48352 0.741759 0.670667i \(-0.233992\pi\)
0.741759 + 0.670667i \(0.233992\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −16.0000 −0.592187
\(731\) 8.00000 0.295891
\(732\) 0 0
\(733\) 12.0000 0.443230 0.221615 0.975134i \(-0.428867\pi\)
0.221615 + 0.975134i \(0.428867\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 40.0000 1.47442
\(737\) 0 0
\(738\) 0 0
\(739\) −6.00000 −0.220714 −0.110357 0.993892i \(-0.535199\pi\)
−0.110357 + 0.993892i \(0.535199\pi\)
\(740\) 8.00000 0.294086
\(741\) 0 0
\(742\) −12.0000 −0.440534
\(743\) 8.00000 0.293492 0.146746 0.989174i \(-0.453120\pi\)
0.146746 + 0.989174i \(0.453120\pi\)
\(744\) 0 0
\(745\) −14.0000 −0.512920
\(746\) 38.0000 1.39128
\(747\) 0 0
\(748\) 0 0
\(749\) −16.0000 −0.584627
\(750\) 0 0
\(751\) 32.0000 1.16770 0.583848 0.811863i \(-0.301546\pi\)
0.583848 + 0.811863i \(0.301546\pi\)
\(752\) 4.00000 0.145865
\(753\) 0 0
\(754\) 0 0
\(755\) 14.0000 0.509512
\(756\) 0 0
\(757\) −22.0000 −0.799604 −0.399802 0.916602i \(-0.630921\pi\)
−0.399802 + 0.916602i \(0.630921\pi\)
\(758\) 30.0000 1.08965
\(759\) 0 0
\(760\) −6.00000 −0.217643
\(761\) −38.0000 −1.37750 −0.688749 0.724999i \(-0.741840\pi\)
−0.688749 + 0.724999i \(0.741840\pi\)
\(762\) 0 0
\(763\) 32.0000 1.15848
\(764\) −20.0000 −0.723575
\(765\) 0 0
\(766\) 4.00000 0.144526
\(767\) 0 0
\(768\) 0 0
\(769\) 40.0000 1.44244 0.721218 0.692708i \(-0.243582\pi\)
0.721218 + 0.692708i \(0.243582\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 4.00000 0.143963
\(773\) 54.0000 1.94225 0.971123 0.238581i \(-0.0766824\pi\)
0.971123 + 0.238581i \(0.0766824\pi\)
\(774\) 0 0
\(775\) −2.00000 −0.0718421
\(776\) 48.0000 1.72310
\(777\) 0 0
\(778\) 38.0000 1.36237
\(779\) 4.00000 0.143315
\(780\) 0 0
\(781\) 0 0
\(782\) 16.0000 0.572159
\(783\) 0 0
\(784\) 3.00000 0.107143
\(785\) −14.0000 −0.499681
\(786\) 0 0
\(787\) 38.0000 1.35455 0.677277 0.735728i \(-0.263160\pi\)
0.677277 + 0.735728i \(0.263160\pi\)
\(788\) 22.0000 0.783718
\(789\) 0 0
\(790\) −8.00000 −0.284627
\(791\) 28.0000 0.995565
\(792\) 0 0
\(793\) 0 0
\(794\) 8.00000 0.283909
\(795\) 0 0
\(796\) −24.0000 −0.850657
\(797\) −50.0000 −1.77109 −0.885545 0.464553i \(-0.846215\pi\)
−0.885545 + 0.464553i \(0.846215\pi\)
\(798\) 0 0
\(799\) −8.00000 −0.283020
\(800\) −5.00000 −0.176777
\(801\) 0 0
\(802\) 10.0000 0.353112
\(803\) 0 0
\(804\) 0 0
\(805\) 16.0000 0.563926
\(806\) 0 0
\(807\) 0 0
\(808\) −6.00000 −0.211079
\(809\) 18.0000 0.632846 0.316423 0.948618i \(-0.397518\pi\)
0.316423 + 0.948618i \(0.397518\pi\)
\(810\) 0 0
\(811\) −14.0000 −0.491606 −0.245803 0.969320i \(-0.579052\pi\)
−0.245803 + 0.969320i \(0.579052\pi\)
\(812\) 4.00000 0.140372
\(813\) 0 0
\(814\) 0 0
\(815\) −18.0000 −0.630512
\(816\) 0 0
\(817\) 8.00000 0.279885
\(818\) 12.0000 0.419570
\(819\) 0 0
\(820\) 2.00000 0.0698430
\(821\) −18.0000 −0.628204 −0.314102 0.949389i \(-0.601703\pi\)
−0.314102 + 0.949389i \(0.601703\pi\)
\(822\) 0 0
\(823\) −24.0000 −0.836587 −0.418294 0.908312i \(-0.637372\pi\)
−0.418294 + 0.908312i \(0.637372\pi\)
\(824\) −48.0000 −1.67216
\(825\) 0 0
\(826\) 24.0000 0.835067
\(827\) 8.00000 0.278187 0.139094 0.990279i \(-0.455581\pi\)
0.139094 + 0.990279i \(0.455581\pi\)
\(828\) 0 0
\(829\) −34.0000 −1.18087 −0.590434 0.807086i \(-0.701044\pi\)
−0.590434 + 0.807086i \(0.701044\pi\)
\(830\) −12.0000 −0.416526
\(831\) 0 0
\(832\) 0 0
\(833\) −6.00000 −0.207888
\(834\) 0 0
\(835\) 20.0000 0.692129
\(836\) 0 0
\(837\) 0 0
\(838\) −12.0000 −0.414533
\(839\) 24.0000 0.828572 0.414286 0.910147i \(-0.364031\pi\)
0.414286 + 0.910147i \(0.364031\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 40.0000 1.37849
\(843\) 0 0
\(844\) 12.0000 0.413057
\(845\) 0 0
\(846\) 0 0
\(847\) −22.0000 −0.755929
\(848\) −6.00000 −0.206041
\(849\) 0 0
\(850\) −2.00000 −0.0685994
\(851\) −64.0000 −2.19389
\(852\) 0 0
\(853\) −24.0000 −0.821744 −0.410872 0.911693i \(-0.634776\pi\)
−0.410872 + 0.911693i \(0.634776\pi\)
\(854\) −20.0000 −0.684386
\(855\) 0 0
\(856\) −24.0000 −0.820303
\(857\) 38.0000 1.29806 0.649028 0.760765i \(-0.275176\pi\)
0.649028 + 0.760765i \(0.275176\pi\)
\(858\) 0 0
\(859\) −28.0000 −0.955348 −0.477674 0.878537i \(-0.658520\pi\)
−0.477674 + 0.878537i \(0.658520\pi\)
\(860\) 4.00000 0.136399
\(861\) 0 0
\(862\) 12.0000 0.408722
\(863\) 36.0000 1.22545 0.612727 0.790295i \(-0.290072\pi\)
0.612727 + 0.790295i \(0.290072\pi\)
\(864\) 0 0
\(865\) −10.0000 −0.340010
\(866\) 14.0000 0.475739
\(867\) 0 0
\(868\) 4.00000 0.135769
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 48.0000 1.62549
\(873\) 0 0
\(874\) 16.0000 0.541208
\(875\) −2.00000 −0.0676123
\(876\) 0 0
\(877\) −16.0000 −0.540282 −0.270141 0.962821i \(-0.587070\pi\)
−0.270141 + 0.962821i \(0.587070\pi\)
\(878\) 16.0000 0.539974
\(879\) 0 0
\(880\) 0 0
\(881\) −30.0000 −1.01073 −0.505363 0.862907i \(-0.668641\pi\)
−0.505363 + 0.862907i \(0.668641\pi\)
\(882\) 0 0
\(883\) −28.0000 −0.942275 −0.471138 0.882060i \(-0.656156\pi\)
−0.471138 + 0.882060i \(0.656156\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 24.0000 0.806296
\(887\) 16.0000 0.537227 0.268614 0.963248i \(-0.413434\pi\)
0.268614 + 0.963248i \(0.413434\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 6.00000 0.201120
\(891\) 0 0
\(892\) 22.0000 0.736614
\(893\) −8.00000 −0.267710
\(894\) 0 0
\(895\) −12.0000 −0.401116
\(896\) 6.00000 0.200446
\(897\) 0 0
\(898\) −6.00000 −0.200223
\(899\) 4.00000 0.133407
\(900\) 0 0
\(901\) 12.0000 0.399778
\(902\) 0 0
\(903\) 0 0
\(904\) 42.0000 1.39690
\(905\) 6.00000 0.199447
\(906\) 0 0
\(907\) 44.0000 1.46100 0.730498 0.682915i \(-0.239288\pi\)
0.730498 + 0.682915i \(0.239288\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −20.0000 −0.662630 −0.331315 0.943520i \(-0.607492\pi\)
−0.331315 + 0.943520i \(0.607492\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −8.00000 −0.264616
\(915\) 0 0
\(916\) 8.00000 0.264327
\(917\) 24.0000 0.792550
\(918\) 0 0
\(919\) −32.0000 −1.05558 −0.527791 0.849374i \(-0.676980\pi\)
−0.527791 + 0.849374i \(0.676980\pi\)
\(920\) 24.0000 0.791257
\(921\) 0 0
\(922\) −34.0000 −1.11973
\(923\) 0 0
\(924\) 0 0
\(925\) 8.00000 0.263038
\(926\) 14.0000 0.460069
\(927\) 0 0
\(928\) 10.0000 0.328266
\(929\) −22.0000 −0.721797 −0.360898 0.932605i \(-0.617530\pi\)
−0.360898 + 0.932605i \(0.617530\pi\)
\(930\) 0 0
\(931\) −6.00000 −0.196642
\(932\) −14.0000 −0.458585
\(933\) 0 0
\(934\) −36.0000 −1.17796
\(935\) 0 0
\(936\) 0 0
\(937\) 38.0000 1.24141 0.620703 0.784046i \(-0.286847\pi\)
0.620703 + 0.784046i \(0.286847\pi\)
\(938\) 12.0000 0.391814
\(939\) 0 0
\(940\) −4.00000 −0.130466
\(941\) −50.0000 −1.62995 −0.814977 0.579494i \(-0.803250\pi\)
−0.814977 + 0.579494i \(0.803250\pi\)
\(942\) 0 0
\(943\) −16.0000 −0.521032
\(944\) 12.0000 0.390567
\(945\) 0 0
\(946\) 0 0
\(947\) 12.0000 0.389948 0.194974 0.980808i \(-0.437538\pi\)
0.194974 + 0.980808i \(0.437538\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) −2.00000 −0.0648886
\(951\) 0 0
\(952\) 12.0000 0.388922
\(953\) 30.0000 0.971795 0.485898 0.874016i \(-0.338493\pi\)
0.485898 + 0.874016i \(0.338493\pi\)
\(954\) 0 0
\(955\) −20.0000 −0.647185
\(956\) 8.00000 0.258738
\(957\) 0 0
\(958\) 0 0
\(959\) −12.0000 −0.387500
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 0 0
\(963\) 0 0
\(964\) 12.0000 0.386494
\(965\) 4.00000 0.128765
\(966\) 0 0
\(967\) 58.0000 1.86515 0.932577 0.360971i \(-0.117555\pi\)
0.932577 + 0.360971i \(0.117555\pi\)
\(968\) −33.0000 −1.06066
\(969\) 0 0
\(970\) 16.0000 0.513729
\(971\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 0 0
\(973\) 8.00000 0.256468
\(974\) −18.0000 −0.576757
\(975\) 0 0
\(976\) −10.0000 −0.320092
\(977\) 38.0000 1.21573 0.607864 0.794041i \(-0.292027\pi\)
0.607864 + 0.794041i \(0.292027\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −3.00000 −0.0958315
\(981\) 0 0
\(982\) 40.0000 1.27645
\(983\) 32.0000 1.02064 0.510321 0.859984i \(-0.329527\pi\)
0.510321 + 0.859984i \(0.329527\pi\)
\(984\) 0 0
\(985\) 22.0000 0.700978
\(986\) 4.00000 0.127386
\(987\) 0 0
\(988\) 0 0
\(989\) −32.0000 −1.01754
\(990\) 0 0
\(991\) 16.0000 0.508257 0.254128 0.967170i \(-0.418211\pi\)
0.254128 + 0.967170i \(0.418211\pi\)
\(992\) 10.0000 0.317500
\(993\) 0 0
\(994\) 16.0000 0.507489
\(995\) −24.0000 −0.760851
\(996\) 0 0
\(997\) −42.0000 −1.33015 −0.665077 0.746775i \(-0.731601\pi\)
−0.665077 + 0.746775i \(0.731601\pi\)
\(998\) 10.0000 0.316544
\(999\) 0 0
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 7605.2.a.d.1.1 1
3.2 odd 2 2535.2.a.l.1.1 1
13.5 odd 4 585.2.b.a.181.2 2
13.8 odd 4 585.2.b.a.181.1 2
13.12 even 2 7605.2.a.p.1.1 1
39.5 even 4 195.2.b.b.181.1 2
39.8 even 4 195.2.b.b.181.2 yes 2
39.38 odd 2 2535.2.a.e.1.1 1
156.47 odd 4 3120.2.g.a.961.2 2
156.83 odd 4 3120.2.g.a.961.1 2
195.8 odd 4 975.2.h.d.649.2 2
195.44 even 4 975.2.b.b.376.2 2
195.47 odd 4 975.2.h.a.649.1 2
195.83 odd 4 975.2.h.a.649.2 2
195.122 odd 4 975.2.h.d.649.1 2
195.164 even 4 975.2.b.b.376.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
195.2.b.b.181.1 2 39.5 even 4
195.2.b.b.181.2 yes 2 39.8 even 4
585.2.b.a.181.1 2 13.8 odd 4
585.2.b.a.181.2 2 13.5 odd 4
975.2.b.b.376.1 2 195.164 even 4
975.2.b.b.376.2 2 195.44 even 4
975.2.h.a.649.1 2 195.47 odd 4
975.2.h.a.649.2 2 195.83 odd 4
975.2.h.d.649.1 2 195.122 odd 4
975.2.h.d.649.2 2 195.8 odd 4
2535.2.a.e.1.1 1 39.38 odd 2
2535.2.a.l.1.1 1 3.2 odd 2
3120.2.g.a.961.1 2 156.83 odd 4
3120.2.g.a.961.2 2 156.47 odd 4
7605.2.a.d.1.1 1 1.1 even 1 trivial
7605.2.a.p.1.1 1 13.12 even 2