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Properties

Label 2790.2.a.u.1.1
Level $2790$
Weight $2$
Character 2790.1
Self dual yes
Analytic conductor $22.278$
Analytic rank $1$
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] = [2790,2,Mod(1,2790)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2790, 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("2790.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2790 = 2 \cdot 3^{2} \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2790.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: \(22.2782621639\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 930)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2790.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +3.00000 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +3.00000 q^{7} +1.00000 q^{8} -1.00000 q^{10} -3.00000 q^{11} -2.00000 q^{13} +3.00000 q^{14} +1.00000 q^{16} -8.00000 q^{17} -7.00000 q^{19} -1.00000 q^{20} -3.00000 q^{22} -7.00000 q^{23} +1.00000 q^{25} -2.00000 q^{26} +3.00000 q^{28} +8.00000 q^{29} -1.00000 q^{31} +1.00000 q^{32} -8.00000 q^{34} -3.00000 q^{35} -4.00000 q^{37} -7.00000 q^{38} -1.00000 q^{40} +1.00000 q^{43} -3.00000 q^{44} -7.00000 q^{46} -6.00000 q^{47} +2.00000 q^{49} +1.00000 q^{50} -2.00000 q^{52} -5.00000 q^{53} +3.00000 q^{55} +3.00000 q^{56} +8.00000 q^{58} -6.00000 q^{59} +2.00000 q^{61} -1.00000 q^{62} +1.00000 q^{64} +2.00000 q^{65} +10.0000 q^{67} -8.00000 q^{68} -3.00000 q^{70} -9.00000 q^{71} +1.00000 q^{73} -4.00000 q^{74} -7.00000 q^{76} -9.00000 q^{77} +13.0000 q^{79} -1.00000 q^{80} +16.0000 q^{83} +8.00000 q^{85} +1.00000 q^{86} -3.00000 q^{88} +3.00000 q^{89} -6.00000 q^{91} -7.00000 q^{92} -6.00000 q^{94} +7.00000 q^{95} +6.00000 q^{97} +2.00000 q^{98} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 3.00000 1.13389 0.566947 − 0.823754i \(-0.308125\pi\)
0.566947 + 0.823754i \(0.308125\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) −1.00000 −0.316228
\(11\) −3.00000 −0.904534 −0.452267 − 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) 0 0
\(13\) −2.00000 −0.554700 −0.277350 − 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 3.00000 0.801784
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −8.00000 −1.94029 −0.970143 − 0.242536i \(-0.922021\pi\)
−0.970143 + 0.242536i \(0.922021\pi\)
\(18\) 0 0
\(19\) −7.00000 −1.60591 −0.802955 − 0.596040i \(-0.796740\pi\)
−0.802955 + 0.596040i \(0.796740\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) −3.00000 −0.639602
\(23\) −7.00000 −1.45960 −0.729800 − 0.683660i \(-0.760387\pi\)
−0.729800 + 0.683660i \(0.760387\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −2.00000 −0.392232
\(27\) 0 0
\(28\) 3.00000 0.566947
\(29\) 8.00000 1.48556 0.742781 − 0.669534i \(-0.233506\pi\)
0.742781 + 0.669534i \(0.233506\pi\)
\(30\) 0 0
\(31\) −1.00000 −0.179605
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −8.00000 −1.37199
\(35\) −3.00000 −0.507093
\(36\) 0 0
\(37\) −4.00000 −0.657596 −0.328798 − 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) −7.00000 −1.13555
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 1.00000 0.152499 0.0762493 − 0.997089i \(-0.475706\pi\)
0.0762493 + 0.997089i \(0.475706\pi\)
\(44\) −3.00000 −0.452267
\(45\) 0 0
\(46\) −7.00000 −1.03209
\(47\) −6.00000 −0.875190 −0.437595 − 0.899172i \(-0.644170\pi\)
−0.437595 + 0.899172i \(0.644170\pi\)
\(48\) 0 0
\(49\) 2.00000 0.285714
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) −2.00000 −0.277350
\(53\) −5.00000 −0.686803 −0.343401 − 0.939189i \(-0.611579\pi\)
−0.343401 + 0.939189i \(0.611579\pi\)
\(54\) 0 0
\(55\) 3.00000 0.404520
\(56\) 3.00000 0.400892
\(57\) 0 0
\(58\) 8.00000 1.05045
\(59\) −6.00000 −0.781133 −0.390567 − 0.920575i \(-0.627721\pi\)
−0.390567 + 0.920575i \(0.627721\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 − 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) −1.00000 −0.127000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 2.00000 0.248069
\(66\) 0 0
\(67\) 10.0000 1.22169 0.610847 − 0.791748i \(-0.290829\pi\)
0.610847 + 0.791748i \(0.290829\pi\)
\(68\) −8.00000 −0.970143
\(69\) 0 0
\(70\) −3.00000 −0.358569
\(71\) −9.00000 −1.06810 −0.534052 − 0.845452i \(-0.679331\pi\)
−0.534052 + 0.845452i \(0.679331\pi\)
\(72\) 0 0
\(73\) 1.00000 0.117041 0.0585206 − 0.998286i \(-0.481362\pi\)
0.0585206 + 0.998286i \(0.481362\pi\)
\(74\) −4.00000 −0.464991
\(75\) 0 0
\(76\) −7.00000 −0.802955
\(77\) −9.00000 −1.02565
\(78\) 0 0
\(79\) 13.0000 1.46261 0.731307 − 0.682048i \(-0.238911\pi\)
0.731307 + 0.682048i \(0.238911\pi\)
\(80\) −1.00000 −0.111803
\(81\) 0 0
\(82\) 0 0
\(83\) 16.0000 1.75623 0.878114 − 0.478451i \(-0.158802\pi\)
0.878114 + 0.478451i \(0.158802\pi\)
\(84\) 0 0
\(85\) 8.00000 0.867722
\(86\) 1.00000 0.107833
\(87\) 0 0
\(88\) −3.00000 −0.319801
\(89\) 3.00000 0.317999 0.159000 − 0.987279i \(-0.449173\pi\)
0.159000 + 0.987279i \(0.449173\pi\)
\(90\) 0 0
\(91\) −6.00000 −0.628971
\(92\) −7.00000 −0.729800
\(93\) 0 0
\(94\) −6.00000 −0.618853
\(95\) 7.00000 0.718185
\(96\) 0 0
\(97\) 6.00000 0.609208 0.304604 − 0.952479i \(-0.401476\pi\)
0.304604 + 0.952479i \(0.401476\pi\)
\(98\) 2.00000 0.202031
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −5.00000 −0.497519 −0.248759 − 0.968565i \(-0.580023\pi\)
−0.248759 + 0.968565i \(0.580023\pi\)
\(102\) 0 0
\(103\) 8.00000 0.788263 0.394132 − 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) −2.00000 −0.196116
\(105\) 0 0
\(106\) −5.00000 −0.485643
\(107\) 3.00000 0.290021 0.145010 − 0.989430i \(-0.453678\pi\)
0.145010 + 0.989430i \(0.453678\pi\)
\(108\) 0 0
\(109\) 4.00000 0.383131 0.191565 − 0.981480i \(-0.438644\pi\)
0.191565 + 0.981480i \(0.438644\pi\)
\(110\) 3.00000 0.286039
\(111\) 0 0
\(112\) 3.00000 0.283473
\(113\) −1.00000 −0.0940721 −0.0470360 − 0.998893i \(-0.514978\pi\)
−0.0470360 + 0.998893i \(0.514978\pi\)
\(114\) 0 0
\(115\) 7.00000 0.652753
\(116\) 8.00000 0.742781
\(117\) 0 0
\(118\) −6.00000 −0.552345
\(119\) −24.0000 −2.20008
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) 2.00000 0.181071
\(123\) 0 0
\(124\) −1.00000 −0.0898027
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −12.0000 −1.06483 −0.532414 − 0.846484i \(-0.678715\pi\)
−0.532414 + 0.846484i \(0.678715\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 2.00000 0.175412
\(131\) −6.00000 −0.524222 −0.262111 − 0.965038i \(-0.584419\pi\)
−0.262111 + 0.965038i \(0.584419\pi\)
\(132\) 0 0
\(133\) −21.0000 −1.82093
\(134\) 10.0000 0.863868
\(135\) 0 0
\(136\) −8.00000 −0.685994
\(137\) −10.0000 −0.854358 −0.427179 − 0.904167i \(-0.640493\pi\)
−0.427179 + 0.904167i \(0.640493\pi\)
\(138\) 0 0
\(139\) −10.0000 −0.848189 −0.424094 − 0.905618i \(-0.639408\pi\)
−0.424094 + 0.905618i \(0.639408\pi\)
\(140\) −3.00000 −0.253546
\(141\) 0 0
\(142\) −9.00000 −0.755263
\(143\) 6.00000 0.501745
\(144\) 0 0
\(145\) −8.00000 −0.664364
\(146\) 1.00000 0.0827606
\(147\) 0 0
\(148\) −4.00000 −0.328798
\(149\) 9.00000 0.737309 0.368654 − 0.929567i \(-0.379819\pi\)
0.368654 + 0.929567i \(0.379819\pi\)
\(150\) 0 0
\(151\) −8.00000 −0.651031 −0.325515 − 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) −7.00000 −0.567775
\(153\) 0 0
\(154\) −9.00000 −0.725241
\(155\) 1.00000 0.0803219
\(156\) 0 0
\(157\) −7.00000 −0.558661 −0.279330 − 0.960195i \(-0.590112\pi\)
−0.279330 + 0.960195i \(0.590112\pi\)
\(158\) 13.0000 1.03422
\(159\) 0 0
\(160\) −1.00000 −0.0790569
\(161\) −21.0000 −1.65503
\(162\) 0 0
\(163\) 18.0000 1.40987 0.704934 − 0.709273i \(-0.250976\pi\)
0.704934 + 0.709273i \(0.250976\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 16.0000 1.24184
\(167\) −15.0000 −1.16073 −0.580367 − 0.814355i \(-0.697091\pi\)
−0.580367 + 0.814355i \(0.697091\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 8.00000 0.613572
\(171\) 0 0
\(172\) 1.00000 0.0762493
\(173\) 6.00000 0.456172 0.228086 − 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 0 0
\(175\) 3.00000 0.226779
\(176\) −3.00000 −0.226134
\(177\) 0 0
\(178\) 3.00000 0.224860
\(179\) −4.00000 −0.298974 −0.149487 − 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) 0 0
\(181\) −21.0000 −1.56092 −0.780459 − 0.625207i \(-0.785014\pi\)
−0.780459 + 0.625207i \(0.785014\pi\)
\(182\) −6.00000 −0.444750
\(183\) 0 0
\(184\) −7.00000 −0.516047
\(185\) 4.00000 0.294086
\(186\) 0 0
\(187\) 24.0000 1.75505
\(188\) −6.00000 −0.437595
\(189\) 0 0
\(190\) 7.00000 0.507833
\(191\) 24.0000 1.73658 0.868290 − 0.496058i \(-0.165220\pi\)
0.868290 + 0.496058i \(0.165220\pi\)
\(192\) 0 0
\(193\) −14.0000 −1.00774 −0.503871 − 0.863779i \(-0.668091\pi\)
−0.503871 + 0.863779i \(0.668091\pi\)
\(194\) 6.00000 0.430775
\(195\) 0 0
\(196\) 2.00000 0.142857
\(197\) −6.00000 −0.427482 −0.213741 − 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) 0 0
\(199\) −3.00000 −0.212664 −0.106332 − 0.994331i \(-0.533911\pi\)
−0.106332 + 0.994331i \(0.533911\pi\)
\(200\) 1.00000 0.0707107
\(201\) 0 0
\(202\) −5.00000 −0.351799
\(203\) 24.0000 1.68447
\(204\) 0 0
\(205\) 0 0
\(206\) 8.00000 0.557386
\(207\) 0 0
\(208\) −2.00000 −0.138675
\(209\) 21.0000 1.45260
\(210\) 0 0
\(211\) 17.0000 1.17033 0.585164 − 0.810915i \(-0.301030\pi\)
0.585164 + 0.810915i \(0.301030\pi\)
\(212\) −5.00000 −0.343401
\(213\) 0 0
\(214\) 3.00000 0.205076
\(215\) −1.00000 −0.0681994
\(216\) 0 0
\(217\) −3.00000 −0.203653
\(218\) 4.00000 0.270914
\(219\) 0 0
\(220\) 3.00000 0.202260
\(221\) 16.0000 1.07628
\(222\) 0 0
\(223\) −10.0000 −0.669650 −0.334825 − 0.942280i \(-0.608677\pi\)
−0.334825 + 0.942280i \(0.608677\pi\)
\(224\) 3.00000 0.200446
\(225\) 0 0
\(226\) −1.00000 −0.0665190
\(227\) −25.0000 −1.65931 −0.829654 − 0.558278i \(-0.811462\pi\)
−0.829654 + 0.558278i \(0.811462\pi\)
\(228\) 0 0
\(229\) −13.0000 −0.859064 −0.429532 − 0.903052i \(-0.641321\pi\)
−0.429532 + 0.903052i \(0.641321\pi\)
\(230\) 7.00000 0.461566
\(231\) 0 0
\(232\) 8.00000 0.525226
\(233\) 25.0000 1.63780 0.818902 − 0.573933i \(-0.194583\pi\)
0.818902 + 0.573933i \(0.194583\pi\)
\(234\) 0 0
\(235\) 6.00000 0.391397
\(236\) −6.00000 −0.390567
\(237\) 0 0
\(238\) −24.0000 −1.55569
\(239\) −8.00000 −0.517477 −0.258738 − 0.965947i \(-0.583307\pi\)
−0.258738 + 0.965947i \(0.583307\pi\)
\(240\) 0 0
\(241\) −22.0000 −1.41714 −0.708572 − 0.705638i \(-0.750660\pi\)
−0.708572 + 0.705638i \(0.750660\pi\)
\(242\) −2.00000 −0.128565
\(243\) 0 0
\(244\) 2.00000 0.128037
\(245\) −2.00000 −0.127775
\(246\) 0 0
\(247\) 14.0000 0.890799
\(248\) −1.00000 −0.0635001
\(249\) 0 0
\(250\) −1.00000 −0.0632456
\(251\) −20.0000 −1.26239 −0.631194 − 0.775625i \(-0.717435\pi\)
−0.631194 + 0.775625i \(0.717435\pi\)
\(252\) 0 0
\(253\) 21.0000 1.32026
\(254\) −12.0000 −0.752947
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 23.0000 1.43470 0.717350 − 0.696713i \(-0.245355\pi\)
0.717350 + 0.696713i \(0.245355\pi\)
\(258\) 0 0
\(259\) −12.0000 −0.745644
\(260\) 2.00000 0.124035
\(261\) 0 0
\(262\) −6.00000 −0.370681
\(263\) −24.0000 −1.47990 −0.739952 − 0.672660i \(-0.765152\pi\)
−0.739952 + 0.672660i \(0.765152\pi\)
\(264\) 0 0
\(265\) 5.00000 0.307148
\(266\) −21.0000 −1.28759
\(267\) 0 0
\(268\) 10.0000 0.610847
\(269\) 18.0000 1.09748 0.548740 − 0.835993i \(-0.315108\pi\)
0.548740 + 0.835993i \(0.315108\pi\)
\(270\) 0 0
\(271\) 7.00000 0.425220 0.212610 − 0.977137i \(-0.431804\pi\)
0.212610 + 0.977137i \(0.431804\pi\)
\(272\) −8.00000 −0.485071
\(273\) 0 0
\(274\) −10.0000 −0.604122
\(275\) −3.00000 −0.180907
\(276\) 0 0
\(277\) −8.00000 −0.480673 −0.240337 − 0.970690i \(-0.577258\pi\)
−0.240337 + 0.970690i \(0.577258\pi\)
\(278\) −10.0000 −0.599760
\(279\) 0 0
\(280\) −3.00000 −0.179284
\(281\) 28.0000 1.67034 0.835170 − 0.549992i \(-0.185369\pi\)
0.835170 + 0.549992i \(0.185369\pi\)
\(282\) 0 0
\(283\) −16.0000 −0.951101 −0.475551 − 0.879688i \(-0.657751\pi\)
−0.475551 + 0.879688i \(0.657751\pi\)
\(284\) −9.00000 −0.534052
\(285\) 0 0
\(286\) 6.00000 0.354787
\(287\) 0 0
\(288\) 0 0
\(289\) 47.0000 2.76471
\(290\) −8.00000 −0.469776
\(291\) 0 0
\(292\) 1.00000 0.0585206
\(293\) −14.0000 −0.817889 −0.408944 − 0.912559i \(-0.634103\pi\)
−0.408944 + 0.912559i \(0.634103\pi\)
\(294\) 0 0
\(295\) 6.00000 0.349334
\(296\) −4.00000 −0.232495
\(297\) 0 0
\(298\) 9.00000 0.521356
\(299\) 14.0000 0.809641
\(300\) 0 0
\(301\) 3.00000 0.172917
\(302\) −8.00000 −0.460348
\(303\) 0 0
\(304\) −7.00000 −0.401478
\(305\) −2.00000 −0.114520
\(306\) 0 0
\(307\) −12.0000 −0.684876 −0.342438 − 0.939540i \(-0.611253\pi\)
−0.342438 + 0.939540i \(0.611253\pi\)
\(308\) −9.00000 −0.512823
\(309\) 0 0
\(310\) 1.00000 0.0567962
\(311\) 16.0000 0.907277 0.453638 − 0.891186i \(-0.350126\pi\)
0.453638 + 0.891186i \(0.350126\pi\)
\(312\) 0 0
\(313\) −10.0000 −0.565233 −0.282617 − 0.959233i \(-0.591202\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) −7.00000 −0.395033
\(315\) 0 0
\(316\) 13.0000 0.731307
\(317\) −24.0000 −1.34797 −0.673987 − 0.738743i \(-0.735420\pi\)
−0.673987 + 0.738743i \(0.735420\pi\)
\(318\) 0 0
\(319\) −24.0000 −1.34374
\(320\) −1.00000 −0.0559017
\(321\) 0 0
\(322\) −21.0000 −1.17028
\(323\) 56.0000 3.11592
\(324\) 0 0
\(325\) −2.00000 −0.110940
\(326\) 18.0000 0.996928
\(327\) 0 0
\(328\) 0 0
\(329\) −18.0000 −0.992372
\(330\) 0 0
\(331\) 32.0000 1.75888 0.879440 − 0.476011i \(-0.157918\pi\)
0.879440 + 0.476011i \(0.157918\pi\)
\(332\) 16.0000 0.878114
\(333\) 0 0
\(334\) −15.0000 −0.820763
\(335\) −10.0000 −0.546358
\(336\) 0 0
\(337\) −10.0000 −0.544735 −0.272367 − 0.962193i \(-0.587807\pi\)
−0.272367 + 0.962193i \(0.587807\pi\)
\(338\) −9.00000 −0.489535
\(339\) 0 0
\(340\) 8.00000 0.433861
\(341\) 3.00000 0.162459
\(342\) 0 0
\(343\) −15.0000 −0.809924
\(344\) 1.00000 0.0539164
\(345\) 0 0
\(346\) 6.00000 0.322562
\(347\) 12.0000 0.644194 0.322097 − 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) 0 0
\(349\) 20.0000 1.07058 0.535288 − 0.844670i \(-0.320203\pi\)
0.535288 + 0.844670i \(0.320203\pi\)
\(350\) 3.00000 0.160357
\(351\) 0 0
\(352\) −3.00000 −0.159901
\(353\) 18.0000 0.958043 0.479022 − 0.877803i \(-0.340992\pi\)
0.479022 + 0.877803i \(0.340992\pi\)
\(354\) 0 0
\(355\) 9.00000 0.477670
\(356\) 3.00000 0.159000
\(357\) 0 0
\(358\) −4.00000 −0.211407
\(359\) −1.00000 −0.0527780 −0.0263890 − 0.999652i \(-0.508401\pi\)
−0.0263890 + 0.999652i \(0.508401\pi\)
\(360\) 0 0
\(361\) 30.0000 1.57895
\(362\) −21.0000 −1.10374
\(363\) 0 0
\(364\) −6.00000 −0.314485
\(365\) −1.00000 −0.0523424
\(366\) 0 0
\(367\) −18.0000 −0.939592 −0.469796 − 0.882775i \(-0.655673\pi\)
−0.469796 + 0.882775i \(0.655673\pi\)
\(368\) −7.00000 −0.364900
\(369\) 0 0
\(370\) 4.00000 0.207950
\(371\) −15.0000 −0.778761
\(372\) 0 0
\(373\) 13.0000 0.673114 0.336557 − 0.941663i \(-0.390737\pi\)
0.336557 + 0.941663i \(0.390737\pi\)
\(374\) 24.0000 1.24101
\(375\) 0 0
\(376\) −6.00000 −0.309426
\(377\) −16.0000 −0.824042
\(378\) 0 0
\(379\) 19.0000 0.975964 0.487982 − 0.872854i \(-0.337733\pi\)
0.487982 + 0.872854i \(0.337733\pi\)
\(380\) 7.00000 0.359092
\(381\) 0 0
\(382\) 24.0000 1.22795
\(383\) 36.0000 1.83951 0.919757 − 0.392488i \(-0.128386\pi\)
0.919757 + 0.392488i \(0.128386\pi\)
\(384\) 0 0
\(385\) 9.00000 0.458682
\(386\) −14.0000 −0.712581
\(387\) 0 0
\(388\) 6.00000 0.304604
\(389\) 22.0000 1.11544 0.557722 − 0.830028i \(-0.311675\pi\)
0.557722 + 0.830028i \(0.311675\pi\)
\(390\) 0 0
\(391\) 56.0000 2.83204
\(392\) 2.00000 0.101015
\(393\) 0 0
\(394\) −6.00000 −0.302276
\(395\) −13.0000 −0.654101
\(396\) 0 0
\(397\) 25.0000 1.25471 0.627357 − 0.778732i \(-0.284137\pi\)
0.627357 + 0.778732i \(0.284137\pi\)
\(398\) −3.00000 −0.150376
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 35.0000 1.74782 0.873908 − 0.486091i \(-0.161578\pi\)
0.873908 + 0.486091i \(0.161578\pi\)
\(402\) 0 0
\(403\) 2.00000 0.0996271
\(404\) −5.00000 −0.248759
\(405\) 0 0
\(406\) 24.0000 1.19110
\(407\) 12.0000 0.594818
\(408\) 0 0
\(409\) 12.0000 0.593362 0.296681 − 0.954977i \(-0.404120\pi\)
0.296681 + 0.954977i \(0.404120\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 8.00000 0.394132
\(413\) −18.0000 −0.885722
\(414\) 0 0
\(415\) −16.0000 −0.785409
\(416\) −2.00000 −0.0980581
\(417\) 0 0
\(418\) 21.0000 1.02714
\(419\) −16.0000 −0.781651 −0.390826 − 0.920465i \(-0.627810\pi\)
−0.390826 + 0.920465i \(0.627810\pi\)
\(420\) 0 0
\(421\) 34.0000 1.65706 0.828529 − 0.559946i \(-0.189178\pi\)
0.828529 + 0.559946i \(0.189178\pi\)
\(422\) 17.0000 0.827547
\(423\) 0 0
\(424\) −5.00000 −0.242821
\(425\) −8.00000 −0.388057
\(426\) 0 0
\(427\) 6.00000 0.290360
\(428\) 3.00000 0.145010
\(429\) 0 0
\(430\) −1.00000 −0.0482243
\(431\) −32.0000 −1.54139 −0.770693 − 0.637207i \(-0.780090\pi\)
−0.770693 + 0.637207i \(0.780090\pi\)
\(432\) 0 0
\(433\) 19.0000 0.913082 0.456541 − 0.889702i \(-0.349088\pi\)
0.456541 + 0.889702i \(0.349088\pi\)
\(434\) −3.00000 −0.144005
\(435\) 0 0
\(436\) 4.00000 0.191565
\(437\) 49.0000 2.34399
\(438\) 0 0
\(439\) −26.0000 −1.24091 −0.620456 − 0.784241i \(-0.713053\pi\)
−0.620456 + 0.784241i \(0.713053\pi\)
\(440\) 3.00000 0.143019
\(441\) 0 0
\(442\) 16.0000 0.761042
\(443\) −19.0000 −0.902717 −0.451359 − 0.892343i \(-0.649060\pi\)
−0.451359 + 0.892343i \(0.649060\pi\)
\(444\) 0 0
\(445\) −3.00000 −0.142214
\(446\) −10.0000 −0.473514
\(447\) 0 0
\(448\) 3.00000 0.141737
\(449\) −34.0000 −1.60456 −0.802280 − 0.596948i \(-0.796380\pi\)
−0.802280 + 0.596948i \(0.796380\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −1.00000 −0.0470360
\(453\) 0 0
\(454\) −25.0000 −1.17331
\(455\) 6.00000 0.281284
\(456\) 0 0
\(457\) −38.0000 −1.77757 −0.888783 − 0.458329i \(-0.848448\pi\)
−0.888783 + 0.458329i \(0.848448\pi\)
\(458\) −13.0000 −0.607450
\(459\) 0 0
\(460\) 7.00000 0.326377
\(461\) 20.0000 0.931493 0.465746 − 0.884918i \(-0.345786\pi\)
0.465746 + 0.884918i \(0.345786\pi\)
\(462\) 0 0
\(463\) −40.0000 −1.85896 −0.929479 − 0.368875i \(-0.879743\pi\)
−0.929479 + 0.368875i \(0.879743\pi\)
\(464\) 8.00000 0.371391
\(465\) 0 0
\(466\) 25.0000 1.15810
\(467\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 0 0
\(469\) 30.0000 1.38527
\(470\) 6.00000 0.276759
\(471\) 0 0
\(472\) −6.00000 −0.276172
\(473\) −3.00000 −0.137940
\(474\) 0 0
\(475\) −7.00000 −0.321182
\(476\) −24.0000 −1.10004
\(477\) 0 0
\(478\) −8.00000 −0.365911
\(479\) 27.0000 1.23366 0.616831 − 0.787096i \(-0.288416\pi\)
0.616831 + 0.787096i \(0.288416\pi\)
\(480\) 0 0
\(481\) 8.00000 0.364769
\(482\) −22.0000 −1.00207
\(483\) 0 0
\(484\) −2.00000 −0.0909091
\(485\) −6.00000 −0.272446
\(486\) 0 0
\(487\) −20.0000 −0.906287 −0.453143 − 0.891438i \(-0.649697\pi\)
−0.453143 + 0.891438i \(0.649697\pi\)
\(488\) 2.00000 0.0905357
\(489\) 0 0
\(490\) −2.00000 −0.0903508
\(491\) 29.0000 1.30875 0.654376 − 0.756169i \(-0.272931\pi\)
0.654376 + 0.756169i \(0.272931\pi\)
\(492\) 0 0
\(493\) −64.0000 −2.88242
\(494\) 14.0000 0.629890
\(495\) 0 0
\(496\) −1.00000 −0.0449013
\(497\) −27.0000 −1.21112
\(498\) 0 0
\(499\) −20.0000 −0.895323 −0.447661 − 0.894203i \(-0.647743\pi\)
−0.447661 + 0.894203i \(0.647743\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 0 0
\(502\) −20.0000 −0.892644
\(503\) −32.0000 −1.42681 −0.713405 − 0.700752i \(-0.752848\pi\)
−0.713405 + 0.700752i \(0.752848\pi\)
\(504\) 0 0
\(505\) 5.00000 0.222497
\(506\) 21.0000 0.933564
\(507\) 0 0
\(508\) −12.0000 −0.532414
\(509\) 24.0000 1.06378 0.531891 − 0.846813i \(-0.321482\pi\)
0.531891 + 0.846813i \(0.321482\pi\)
\(510\) 0 0
\(511\) 3.00000 0.132712
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 23.0000 1.01449
\(515\) −8.00000 −0.352522
\(516\) 0 0
\(517\) 18.0000 0.791639
\(518\) −12.0000 −0.527250
\(519\) 0 0
\(520\) 2.00000 0.0877058
\(521\) 18.0000 0.788594 0.394297 − 0.918983i \(-0.370988\pi\)
0.394297 + 0.918983i \(0.370988\pi\)
\(522\) 0 0
\(523\) 29.0000 1.26808 0.634041 − 0.773300i \(-0.281395\pi\)
0.634041 + 0.773300i \(0.281395\pi\)
\(524\) −6.00000 −0.262111
\(525\) 0 0
\(526\) −24.0000 −1.04645
\(527\) 8.00000 0.348485
\(528\) 0 0
\(529\) 26.0000 1.13043
\(530\) 5.00000 0.217186
\(531\) 0 0
\(532\) −21.0000 −0.910465
\(533\) 0 0
\(534\) 0 0
\(535\) −3.00000 −0.129701
\(536\) 10.0000 0.431934
\(537\) 0 0
\(538\) 18.0000 0.776035
\(539\) −6.00000 −0.258438
\(540\) 0 0
\(541\) 14.0000 0.601907 0.300954 − 0.953639i \(-0.402695\pi\)
0.300954 + 0.953639i \(0.402695\pi\)
\(542\) 7.00000 0.300676
\(543\) 0 0
\(544\) −8.00000 −0.342997
\(545\) −4.00000 −0.171341
\(546\) 0 0
\(547\) −14.0000 −0.598597 −0.299298 − 0.954160i \(-0.596753\pi\)
−0.299298 + 0.954160i \(0.596753\pi\)
\(548\) −10.0000 −0.427179
\(549\) 0 0
\(550\) −3.00000 −0.127920
\(551\) −56.0000 −2.38568
\(552\) 0 0
\(553\) 39.0000 1.65845
\(554\) −8.00000 −0.339887
\(555\) 0 0
\(556\) −10.0000 −0.424094
\(557\) −17.0000 −0.720313 −0.360157 − 0.932892i \(-0.617277\pi\)
−0.360157 + 0.932892i \(0.617277\pi\)
\(558\) 0 0
\(559\) −2.00000 −0.0845910
\(560\) −3.00000 −0.126773
\(561\) 0 0
\(562\) 28.0000 1.18111
\(563\) 36.0000 1.51722 0.758610 − 0.651546i \(-0.225879\pi\)
0.758610 + 0.651546i \(0.225879\pi\)
\(564\) 0 0
\(565\) 1.00000 0.0420703
\(566\) −16.0000 −0.672530
\(567\) 0 0
\(568\) −9.00000 −0.377632
\(569\) −37.0000 −1.55112 −0.775560 − 0.631273i \(-0.782533\pi\)
−0.775560 + 0.631273i \(0.782533\pi\)
\(570\) 0 0
\(571\) −22.0000 −0.920671 −0.460336 − 0.887745i \(-0.652271\pi\)
−0.460336 + 0.887745i \(0.652271\pi\)
\(572\) 6.00000 0.250873
\(573\) 0 0
\(574\) 0 0
\(575\) −7.00000 −0.291920
\(576\) 0 0
\(577\) −8.00000 −0.333044 −0.166522 − 0.986038i \(-0.553254\pi\)
−0.166522 + 0.986038i \(0.553254\pi\)
\(578\) 47.0000 1.95494
\(579\) 0 0
\(580\) −8.00000 −0.332182
\(581\) 48.0000 1.99138
\(582\) 0 0
\(583\) 15.0000 0.621237
\(584\) 1.00000 0.0413803
\(585\) 0 0
\(586\) −14.0000 −0.578335
\(587\) −16.0000 −0.660391 −0.330195 − 0.943913i \(-0.607115\pi\)
−0.330195 + 0.943913i \(0.607115\pi\)
\(588\) 0 0
\(589\) 7.00000 0.288430
\(590\) 6.00000 0.247016
\(591\) 0 0
\(592\) −4.00000 −0.164399
\(593\) −30.0000 −1.23195 −0.615976 − 0.787765i \(-0.711238\pi\)
−0.615976 + 0.787765i \(0.711238\pi\)
\(594\) 0 0
\(595\) 24.0000 0.983904
\(596\) 9.00000 0.368654
\(597\) 0 0
\(598\) 14.0000 0.572503
\(599\) −35.0000 −1.43006 −0.715031 − 0.699093i \(-0.753587\pi\)
−0.715031 + 0.699093i \(0.753587\pi\)
\(600\) 0 0
\(601\) −30.0000 −1.22373 −0.611863 − 0.790964i \(-0.709580\pi\)
−0.611863 + 0.790964i \(0.709580\pi\)
\(602\) 3.00000 0.122271
\(603\) 0 0
\(604\) −8.00000 −0.325515
\(605\) 2.00000 0.0813116
\(606\) 0 0
\(607\) −13.0000 −0.527654 −0.263827 − 0.964570i \(-0.584985\pi\)
−0.263827 + 0.964570i \(0.584985\pi\)
\(608\) −7.00000 −0.283887
\(609\) 0 0
\(610\) −2.00000 −0.0809776
\(611\) 12.0000 0.485468
\(612\) 0 0
\(613\) 26.0000 1.05013 0.525065 − 0.851062i \(-0.324041\pi\)
0.525065 + 0.851062i \(0.324041\pi\)
\(614\) −12.0000 −0.484281
\(615\) 0 0
\(616\) −9.00000 −0.362620
\(617\) 3.00000 0.120775 0.0603877 − 0.998175i \(-0.480766\pi\)
0.0603877 + 0.998175i \(0.480766\pi\)
\(618\) 0 0
\(619\) −18.0000 −0.723481 −0.361741 − 0.932279i \(-0.617817\pi\)
−0.361741 + 0.932279i \(0.617817\pi\)
\(620\) 1.00000 0.0401610
\(621\) 0 0
\(622\) 16.0000 0.641542
\(623\) 9.00000 0.360577
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −10.0000 −0.399680
\(627\) 0 0
\(628\) −7.00000 −0.279330
\(629\) 32.0000 1.27592
\(630\) 0 0
\(631\) −5.00000 −0.199047 −0.0995234 − 0.995035i \(-0.531732\pi\)
−0.0995234 + 0.995035i \(0.531732\pi\)
\(632\) 13.0000 0.517112
\(633\) 0 0
\(634\) −24.0000 −0.953162
\(635\) 12.0000 0.476205
\(636\) 0 0
\(637\) −4.00000 −0.158486
\(638\) −24.0000 −0.950169
\(639\) 0 0
\(640\) −1.00000 −0.0395285
\(641\) 30.0000 1.18493 0.592464 − 0.805597i \(-0.298155\pi\)
0.592464 + 0.805597i \(0.298155\pi\)
\(642\) 0 0
\(643\) 19.0000 0.749287 0.374643 − 0.927169i \(-0.377765\pi\)
0.374643 + 0.927169i \(0.377765\pi\)
\(644\) −21.0000 −0.827516
\(645\) 0 0
\(646\) 56.0000 2.20329
\(647\) −49.0000 −1.92639 −0.963194 − 0.268806i \(-0.913371\pi\)
−0.963194 + 0.268806i \(0.913371\pi\)
\(648\) 0 0
\(649\) 18.0000 0.706562
\(650\) −2.00000 −0.0784465
\(651\) 0 0
\(652\) 18.0000 0.704934
\(653\) 6.00000 0.234798 0.117399 − 0.993085i \(-0.462544\pi\)
0.117399 + 0.993085i \(0.462544\pi\)
\(654\) 0 0
\(655\) 6.00000 0.234439
\(656\) 0 0
\(657\) 0 0
\(658\) −18.0000 −0.701713
\(659\) 32.0000 1.24654 0.623272 − 0.782006i \(-0.285803\pi\)
0.623272 + 0.782006i \(0.285803\pi\)
\(660\) 0 0
\(661\) −46.0000 −1.78919 −0.894596 − 0.446875i \(-0.852537\pi\)
−0.894596 + 0.446875i \(0.852537\pi\)
\(662\) 32.0000 1.24372
\(663\) 0 0
\(664\) 16.0000 0.620920
\(665\) 21.0000 0.814345
\(666\) 0 0
\(667\) −56.0000 −2.16833
\(668\) −15.0000 −0.580367
\(669\) 0 0
\(670\) −10.0000 −0.386334
\(671\) −6.00000 −0.231627
\(672\) 0 0
\(673\) −46.0000 −1.77317 −0.886585 − 0.462566i \(-0.846929\pi\)
−0.886585 + 0.462566i \(0.846929\pi\)
\(674\) −10.0000 −0.385186
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) 43.0000 1.65262 0.826312 − 0.563212i \(-0.190435\pi\)
0.826312 + 0.563212i \(0.190435\pi\)
\(678\) 0 0
\(679\) 18.0000 0.690777
\(680\) 8.00000 0.306786
\(681\) 0 0
\(682\) 3.00000 0.114876
\(683\) 15.0000 0.573959 0.286980 − 0.957937i \(-0.407349\pi\)
0.286980 + 0.957937i \(0.407349\pi\)
\(684\) 0 0
\(685\) 10.0000 0.382080
\(686\) −15.0000 −0.572703
\(687\) 0 0
\(688\) 1.00000 0.0381246
\(689\) 10.0000 0.380970
\(690\) 0 0
\(691\) 5.00000 0.190209 0.0951045 − 0.995467i \(-0.469681\pi\)
0.0951045 + 0.995467i \(0.469681\pi\)
\(692\) 6.00000 0.228086
\(693\) 0 0
\(694\) 12.0000 0.455514
\(695\) 10.0000 0.379322
\(696\) 0 0
\(697\) 0 0
\(698\) 20.0000 0.757011
\(699\) 0 0
\(700\) 3.00000 0.113389
\(701\) −33.0000 −1.24639 −0.623196 − 0.782065i \(-0.714166\pi\)
−0.623196 + 0.782065i \(0.714166\pi\)
\(702\) 0 0
\(703\) 28.0000 1.05604
\(704\) −3.00000 −0.113067
\(705\) 0 0
\(706\) 18.0000 0.677439
\(707\) −15.0000 −0.564133
\(708\) 0 0
\(709\) −15.0000 −0.563337 −0.281668 − 0.959512i \(-0.590888\pi\)
−0.281668 + 0.959512i \(0.590888\pi\)
\(710\) 9.00000 0.337764
\(711\) 0 0
\(712\) 3.00000 0.112430
\(713\) 7.00000 0.262152
\(714\) 0 0
\(715\) −6.00000 −0.224387
\(716\) −4.00000 −0.149487
\(717\) 0 0
\(718\) −1.00000 −0.0373197
\(719\) 30.0000 1.11881 0.559406 − 0.828894i \(-0.311029\pi\)
0.559406 + 0.828894i \(0.311029\pi\)
\(720\) 0 0
\(721\) 24.0000 0.893807
\(722\) 30.0000 1.11648
\(723\) 0 0
\(724\) −21.0000 −0.780459
\(725\) 8.00000 0.297113
\(726\) 0 0
\(727\) −29.0000 −1.07555 −0.537775 − 0.843088i \(-0.680735\pi\)
−0.537775 + 0.843088i \(0.680735\pi\)
\(728\) −6.00000 −0.222375
\(729\) 0 0
\(730\) −1.00000 −0.0370117
\(731\) −8.00000 −0.295891
\(732\) 0 0
\(733\) 2.00000 0.0738717 0.0369358 − 0.999318i \(-0.488240\pi\)
0.0369358 + 0.999318i \(0.488240\pi\)
\(734\) −18.0000 −0.664392
\(735\) 0 0
\(736\) −7.00000 −0.258023
\(737\) −30.0000 −1.10506
\(738\) 0 0
\(739\) −40.0000 −1.47142 −0.735712 − 0.677295i \(-0.763152\pi\)
−0.735712 + 0.677295i \(0.763152\pi\)
\(740\) 4.00000 0.147043
\(741\) 0 0
\(742\) −15.0000 −0.550667
\(743\) −1.00000 −0.0366864 −0.0183432 − 0.999832i \(-0.505839\pi\)
−0.0183432 + 0.999832i \(0.505839\pi\)
\(744\) 0 0
\(745\) −9.00000 −0.329734
\(746\) 13.0000 0.475964
\(747\) 0 0
\(748\) 24.0000 0.877527
\(749\) 9.00000 0.328853
\(750\) 0 0
\(751\) 6.00000 0.218943 0.109472 − 0.993990i \(-0.465084\pi\)
0.109472 + 0.993990i \(0.465084\pi\)
\(752\) −6.00000 −0.218797
\(753\) 0 0
\(754\) −16.0000 −0.582686
\(755\) 8.00000 0.291150
\(756\) 0 0
\(757\) −2.00000 −0.0726912 −0.0363456 − 0.999339i \(-0.511572\pi\)
−0.0363456 + 0.999339i \(0.511572\pi\)
\(758\) 19.0000 0.690111
\(759\) 0 0
\(760\) 7.00000 0.253917
\(761\) −49.0000 −1.77625 −0.888124 − 0.459603i \(-0.847992\pi\)
−0.888124 + 0.459603i \(0.847992\pi\)
\(762\) 0 0
\(763\) 12.0000 0.434429
\(764\) 24.0000 0.868290
\(765\) 0 0
\(766\) 36.0000 1.30073
\(767\) 12.0000 0.433295
\(768\) 0 0
\(769\) 9.00000 0.324548 0.162274 − 0.986746i \(-0.448117\pi\)
0.162274 + 0.986746i \(0.448117\pi\)
\(770\) 9.00000 0.324337
\(771\) 0 0
\(772\) −14.0000 −0.503871
\(773\) 1.00000 0.0359675 0.0179838 − 0.999838i \(-0.494275\pi\)
0.0179838 + 0.999838i \(0.494275\pi\)
\(774\) 0 0
\(775\) −1.00000 −0.0359211
\(776\) 6.00000 0.215387
\(777\) 0 0
\(778\) 22.0000 0.788738
\(779\) 0 0
\(780\) 0 0
\(781\) 27.0000 0.966136
\(782\) 56.0000 2.00256
\(783\) 0 0
\(784\) 2.00000 0.0714286
\(785\) 7.00000 0.249841
\(786\) 0 0
\(787\) −27.0000 −0.962446 −0.481223 − 0.876598i \(-0.659807\pi\)
−0.481223 + 0.876598i \(0.659807\pi\)
\(788\) −6.00000 −0.213741
\(789\) 0 0
\(790\) −13.0000 −0.462519
\(791\) −3.00000 −0.106668
\(792\) 0 0
\(793\) −4.00000 −0.142044
\(794\) 25.0000 0.887217
\(795\) 0 0
\(796\) −3.00000 −0.106332
\(797\) 2.00000 0.0708436 0.0354218 − 0.999372i \(-0.488723\pi\)
0.0354218 + 0.999372i \(0.488723\pi\)
\(798\) 0 0
\(799\) 48.0000 1.69812
\(800\) 1.00000 0.0353553
\(801\) 0 0
\(802\) 35.0000 1.23589
\(803\) −3.00000 −0.105868
\(804\) 0 0
\(805\) 21.0000 0.740153
\(806\) 2.00000 0.0704470
\(807\) 0 0
\(808\) −5.00000 −0.175899
\(809\) −13.0000 −0.457056 −0.228528 − 0.973537i \(-0.573391\pi\)
−0.228528 + 0.973537i \(0.573391\pi\)
\(810\) 0 0
\(811\) −23.0000 −0.807639 −0.403820 − 0.914839i \(-0.632318\pi\)
−0.403820 + 0.914839i \(0.632318\pi\)
\(812\) 24.0000 0.842235
\(813\) 0 0
\(814\) 12.0000 0.420600
\(815\) −18.0000 −0.630512
\(816\) 0 0
\(817\) −7.00000 −0.244899
\(818\) 12.0000 0.419570
\(819\) 0 0
\(820\) 0 0
\(821\) 24.0000 0.837606 0.418803 − 0.908077i \(-0.362450\pi\)
0.418803 + 0.908077i \(0.362450\pi\)
\(822\) 0 0
\(823\) 14.0000 0.488009 0.244005 − 0.969774i \(-0.421539\pi\)
0.244005 + 0.969774i \(0.421539\pi\)
\(824\) 8.00000 0.278693
\(825\) 0 0
\(826\) −18.0000 −0.626300
\(827\) 44.0000 1.53003 0.765015 − 0.644013i \(-0.222732\pi\)
0.765015 + 0.644013i \(0.222732\pi\)
\(828\) 0 0
\(829\) −21.0000 −0.729360 −0.364680 − 0.931133i \(-0.618822\pi\)
−0.364680 + 0.931133i \(0.618822\pi\)
\(830\) −16.0000 −0.555368
\(831\) 0 0
\(832\) −2.00000 −0.0693375
\(833\) −16.0000 −0.554367
\(834\) 0 0
\(835\) 15.0000 0.519096
\(836\) 21.0000 0.726300
\(837\) 0 0
\(838\) −16.0000 −0.552711
\(839\) 27.0000 0.932144 0.466072 − 0.884747i \(-0.345669\pi\)
0.466072 + 0.884747i \(0.345669\pi\)
\(840\) 0 0
\(841\) 35.0000 1.20690
\(842\) 34.0000 1.17172
\(843\) 0 0
\(844\) 17.0000 0.585164
\(845\) 9.00000 0.309609
\(846\) 0 0
\(847\) −6.00000 −0.206162
\(848\) −5.00000 −0.171701
\(849\) 0 0
\(850\) −8.00000 −0.274398
\(851\) 28.0000 0.959828
\(852\) 0 0
\(853\) 45.0000 1.54077 0.770385 − 0.637579i \(-0.220064\pi\)
0.770385 + 0.637579i \(0.220064\pi\)
\(854\) 6.00000 0.205316
\(855\) 0 0
\(856\) 3.00000 0.102538
\(857\) 22.0000 0.751506 0.375753 − 0.926720i \(-0.377384\pi\)
0.375753 + 0.926720i \(0.377384\pi\)
\(858\) 0 0
\(859\) −24.0000 −0.818869 −0.409435 − 0.912339i \(-0.634274\pi\)
−0.409435 + 0.912339i \(0.634274\pi\)
\(860\) −1.00000 −0.0340997
\(861\) 0 0
\(862\) −32.0000 −1.08992
\(863\) 31.0000 1.05525 0.527626 − 0.849477i \(-0.323082\pi\)
0.527626 + 0.849477i \(0.323082\pi\)
\(864\) 0 0
\(865\) −6.00000 −0.204006
\(866\) 19.0000 0.645646
\(867\) 0 0
\(868\) −3.00000 −0.101827
\(869\) −39.0000 −1.32298
\(870\) 0 0
\(871\) −20.0000 −0.677674
\(872\) 4.00000 0.135457
\(873\) 0 0
\(874\) 49.0000 1.65745
\(875\) −3.00000 −0.101419
\(876\) 0 0
\(877\) 2.00000 0.0675352 0.0337676 − 0.999430i \(-0.489249\pi\)
0.0337676 + 0.999430i \(0.489249\pi\)
\(878\) −26.0000 −0.877457
\(879\) 0 0
\(880\) 3.00000 0.101130
\(881\) −18.0000 −0.606435 −0.303218 − 0.952921i \(-0.598061\pi\)
−0.303218 + 0.952921i \(0.598061\pi\)
\(882\) 0 0
\(883\) −13.0000 −0.437485 −0.218742 − 0.975783i \(-0.570195\pi\)
−0.218742 + 0.975783i \(0.570195\pi\)
\(884\) 16.0000 0.538138
\(885\) 0 0
\(886\) −19.0000 −0.638317
\(887\) −22.0000 −0.738688 −0.369344 − 0.929293i \(-0.620418\pi\)
−0.369344 + 0.929293i \(0.620418\pi\)
\(888\) 0 0
\(889\) −36.0000 −1.20740
\(890\) −3.00000 −0.100560
\(891\) 0 0
\(892\) −10.0000 −0.334825
\(893\) 42.0000 1.40548
\(894\) 0 0
\(895\) 4.00000 0.133705
\(896\) 3.00000 0.100223
\(897\) 0 0
\(898\) −34.0000 −1.13459
\(899\) −8.00000 −0.266815
\(900\) 0 0
\(901\) 40.0000 1.33259
\(902\) 0 0
\(903\) 0 0
\(904\) −1.00000 −0.0332595
\(905\) 21.0000 0.698064
\(906\) 0 0
\(907\) −40.0000 −1.32818 −0.664089 − 0.747653i \(-0.731180\pi\)
−0.664089 + 0.747653i \(0.731180\pi\)
\(908\) −25.0000 −0.829654
\(909\) 0 0
\(910\) 6.00000 0.198898
\(911\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 0 0
\(913\) −48.0000 −1.58857
\(914\) −38.0000 −1.25693
\(915\) 0 0
\(916\) −13.0000 −0.429532
\(917\) −18.0000 −0.594412
\(918\) 0 0
\(919\) −10.0000 −0.329870 −0.164935 − 0.986304i \(-0.552741\pi\)
−0.164935 + 0.986304i \(0.552741\pi\)
\(920\) 7.00000 0.230783
\(921\) 0 0
\(922\) 20.0000 0.658665
\(923\) 18.0000 0.592477
\(924\) 0 0
\(925\) −4.00000 −0.131519
\(926\) −40.0000 −1.31448
\(927\) 0 0
\(928\) 8.00000 0.262613
\(929\) −3.00000 −0.0984268 −0.0492134 − 0.998788i \(-0.515671\pi\)
−0.0492134 + 0.998788i \(0.515671\pi\)
\(930\) 0 0
\(931\) −14.0000 −0.458831
\(932\) 25.0000 0.818902
\(933\) 0 0
\(934\) 0 0
\(935\) −24.0000 −0.784884
\(936\) 0 0
\(937\) −16.0000 −0.522697 −0.261349 − 0.965244i \(-0.584167\pi\)
−0.261349 + 0.965244i \(0.584167\pi\)
\(938\) 30.0000 0.979535
\(939\) 0 0
\(940\) 6.00000 0.195698
\(941\) 28.0000 0.912774 0.456387 − 0.889781i \(-0.349143\pi\)
0.456387 + 0.889781i \(0.349143\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) −6.00000 −0.195283
\(945\) 0 0
\(946\) −3.00000 −0.0975384
\(947\) −38.0000 −1.23483 −0.617417 − 0.786636i \(-0.711821\pi\)
−0.617417 + 0.786636i \(0.711821\pi\)
\(948\) 0 0
\(949\) −2.00000 −0.0649227
\(950\) −7.00000 −0.227110
\(951\) 0 0
\(952\) −24.0000 −0.777844
\(953\) −12.0000 −0.388718 −0.194359 − 0.980930i \(-0.562263\pi\)
−0.194359 + 0.980930i \(0.562263\pi\)
\(954\) 0 0
\(955\) −24.0000 −0.776622
\(956\) −8.00000 −0.258738
\(957\) 0 0
\(958\) 27.0000 0.872330
\(959\) −30.0000 −0.968751
\(960\) 0 0
\(961\) 1.00000 0.0322581
\(962\) 8.00000 0.257930
\(963\) 0 0
\(964\) −22.0000 −0.708572
\(965\) 14.0000 0.450676
\(966\) 0 0
\(967\) −32.0000 −1.02905 −0.514525 − 0.857475i \(-0.672032\pi\)
−0.514525 + 0.857475i \(0.672032\pi\)
\(968\) −2.00000 −0.0642824
\(969\) 0 0
\(970\) −6.00000 −0.192648
\(971\) 54.0000 1.73294 0.866471 − 0.499227i \(-0.166383\pi\)
0.866471 + 0.499227i \(0.166383\pi\)
\(972\) 0 0
\(973\) −30.0000 −0.961756
\(974\) −20.0000 −0.640841
\(975\) 0 0
\(976\) 2.00000 0.0640184
\(977\) −46.0000 −1.47167 −0.735835 − 0.677161i \(-0.763210\pi\)
−0.735835 + 0.677161i \(0.763210\pi\)
\(978\) 0 0
\(979\) −9.00000 −0.287641
\(980\) −2.00000 −0.0638877
\(981\) 0 0
\(982\) 29.0000 0.925427
\(983\) −48.0000 −1.53096 −0.765481 − 0.643458i \(-0.777499\pi\)
−0.765481 + 0.643458i \(0.777499\pi\)
\(984\) 0 0
\(985\) 6.00000 0.191176
\(986\) −64.0000 −2.03818
\(987\) 0 0
\(988\) 14.0000 0.445399
\(989\) −7.00000 −0.222587
\(990\) 0 0
\(991\) 5.00000 0.158830 0.0794151 − 0.996842i \(-0.474695\pi\)
0.0794151 + 0.996842i \(0.474695\pi\)
\(992\) −1.00000 −0.0317500
\(993\) 0 0
\(994\) −27.0000 −0.856388
\(995\) 3.00000 0.0951064
\(996\) 0 0
\(997\) −6.00000 −0.190022 −0.0950110 − 0.995476i \(-0.530289\pi\)
−0.0950110 + 0.995476i \(0.530289\pi\)
\(998\) −20.0000 −0.633089
\(999\) 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 2790.2.a.u.1.1 1
3.2 odd 2 930.2.a.e.1.1 ✓ 1
12.11 even 2 7440.2.a.x.1.1 1
15.2 even 4 4650.2.d.ba.3349.1 2
15.8 even 4 4650.2.d.ba.3349.2 2
15.14 odd 2 4650.2.a.bk.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
930.2.a.e.1.1 ✓ 1 3.2 odd 2
2790.2.a.u.1.1 1 1.1 even 1 trivial
4650.2.a.bk.1.1 1 15.14 odd 2
4650.2.d.ba.3349.1 2 15.2 even 4
4650.2.d.ba.3349.2 2 15.8 even 4
7440.2.a.x.1.1 1 12.11 even 2