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Properties

Label 2790.2.a.y.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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -3.00000 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -3.00000 q^{7} +1.00000 q^{8} +1.00000 q^{10} -3.00000 q^{11} -2.00000 q^{13} -3.00000 q^{14} +1.00000 q^{16} +4.00000 q^{17} -3.00000 q^{19} +1.00000 q^{20} -3.00000 q^{22} -5.00000 q^{23} +1.00000 q^{25} -2.00000 q^{26} -3.00000 q^{28} -4.00000 q^{29} +1.00000 q^{31} +1.00000 q^{32} +4.00000 q^{34} -3.00000 q^{35} -3.00000 q^{38} +1.00000 q^{40} -4.00000 q^{41} +1.00000 q^{43} -3.00000 q^{44} -5.00000 q^{46} -10.0000 q^{47} +2.00000 q^{49} +1.00000 q^{50} -2.00000 q^{52} -3.00000 q^{53} -3.00000 q^{55} -3.00000 q^{56} -4.00000 q^{58} -6.00000 q^{59} -2.00000 q^{61} +1.00000 q^{62} +1.00000 q^{64} -2.00000 q^{65} +2.00000 q^{67} +4.00000 q^{68} -3.00000 q^{70} -7.00000 q^{71} +5.00000 q^{73} -3.00000 q^{76} +9.00000 q^{77} -1.00000 q^{79} +1.00000 q^{80} -4.00000 q^{82} -12.0000 q^{83} +4.00000 q^{85} +1.00000 q^{86} -3.00000 q^{88} -1.00000 q^{89} +6.00000 q^{91} -5.00000 q^{92} -10.0000 q^{94} -3.00000 q^{95} -10.0000 q^{97} +2.00000 q^{98} +O(q^{100})\)

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 \(-0.691875\pi\)
−0.566947 + 0.823754i \(0.691875\pi\)
\(8\) 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
\(17\) 4.00000 0.970143 0.485071 − 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) 0 0
\(19\) −3.00000 −0.688247 −0.344124 − 0.938924i \(-0.611824\pi\)
−0.344124 + 0.938924i \(0.611824\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) −3.00000 −0.639602
\(23\) −5.00000 −1.04257 −0.521286 − 0.853382i \(-0.674548\pi\)
−0.521286 + 0.853382i \(0.674548\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −2.00000 −0.392232
\(27\) 0 0
\(28\) −3.00000 −0.566947
\(29\) −4.00000 −0.742781 −0.371391 − 0.928477i \(-0.621119\pi\)
−0.371391 + 0.928477i \(0.621119\pi\)
\(30\) 0 0
\(31\) 1.00000 0.179605
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 4.00000 0.685994
\(35\) −3.00000 −0.507093
\(36\) 0 0
\(37\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) −3.00000 −0.486664
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) −4.00000 −0.624695 −0.312348 − 0.949968i \(-0.601115\pi\)
−0.312348 + 0.949968i \(0.601115\pi\)
\(42\) 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
\(46\) −5.00000 −0.737210
\(47\) −10.0000 −1.45865 −0.729325 − 0.684167i \(-0.760166\pi\)
−0.729325 + 0.684167i \(0.760166\pi\)
\(48\) 0 0
\(49\) 2.00000 0.285714
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) −2.00000 −0.277350
\(53\) −3.00000 −0.412082 −0.206041 − 0.978543i \(-0.566058\pi\)
−0.206041 + 0.978543i \(0.566058\pi\)
\(54\) 0 0
\(55\) −3.00000 −0.404520
\(56\) −3.00000 −0.400892
\(57\) 0 0
\(58\) −4.00000 −0.525226
\(59\) −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 \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 1.00000 0.127000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −2.00000 −0.248069
\(66\) 0 0
\(67\) 2.00000 0.244339 0.122169 − 0.992509i \(-0.461015\pi\)
0.122169 + 0.992509i \(0.461015\pi\)
\(68\) 4.00000 0.485071
\(69\) 0 0
\(70\) −3.00000 −0.358569
\(71\) −7.00000 −0.830747 −0.415374 − 0.909651i \(-0.636349\pi\)
−0.415374 + 0.909651i \(0.636349\pi\)
\(72\) 0 0
\(73\) 5.00000 0.585206 0.292603 − 0.956234i \(-0.405479\pi\)
0.292603 + 0.956234i \(0.405479\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) −3.00000 −0.344124
\(77\) 9.00000 1.02565
\(78\) 0 0
\(79\) −1.00000 −0.112509 −0.0562544 − 0.998416i \(-0.517916\pi\)
−0.0562544 + 0.998416i \(0.517916\pi\)
\(80\) 1.00000 0.111803
\(81\) 0 0
\(82\) −4.00000 −0.441726
\(83\) −12.0000 −1.31717 −0.658586 − 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 0 0
\(85\) 4.00000 0.433861
\(86\) 1.00000 0.107833
\(87\) 0 0
\(88\) −3.00000 −0.319801
\(89\) −1.00000 −0.106000 −0.0529999 − 0.998595i \(-0.516878\pi\)
−0.0529999 + 0.998595i \(0.516878\pi\)
\(90\) 0 0
\(91\) 6.00000 0.628971
\(92\) −5.00000 −0.521286
\(93\) 0 0
\(94\) −10.0000 −1.03142
\(95\) −3.00000 −0.307794
\(96\) 0 0
\(97\) −10.0000 −1.01535 −0.507673 − 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) 2.00000 0.202031
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 1.00000 0.0995037 0.0497519 − 0.998762i \(-0.484157\pi\)
0.0497519 + 0.998762i \(0.484157\pi\)
\(102\) 0 0
\(103\) 16.0000 1.57653 0.788263 − 0.615338i \(-0.210980\pi\)
0.788263 + 0.615338i \(0.210980\pi\)
\(104\) −2.00000 −0.196116
\(105\) 0 0
\(106\) −3.00000 −0.291386
\(107\) −9.00000 −0.870063 −0.435031 − 0.900415i \(-0.643263\pi\)
−0.435031 + 0.900415i \(0.643263\pi\)
\(108\) 0 0
\(109\) 20.0000 1.91565 0.957826 − 0.287348i \(-0.0927736\pi\)
0.957826 + 0.287348i \(0.0927736\pi\)
\(110\) −3.00000 −0.286039
\(111\) 0 0
\(112\) −3.00000 −0.283473
\(113\) −9.00000 −0.846649 −0.423324 − 0.905978i \(-0.639137\pi\)
−0.423324 + 0.905978i \(0.639137\pi\)
\(114\) 0 0
\(115\) −5.00000 −0.466252
\(116\) −4.00000 −0.371391
\(117\) 0 0
\(118\) −6.00000 −0.552345
\(119\) −12.0000 −1.10004
\(120\) 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
\(127\) 8.00000 0.709885 0.354943 − 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) 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
\(133\) 9.00000 0.780399
\(134\) 2.00000 0.172774
\(135\) 0 0
\(136\) 4.00000 0.342997
\(137\) 10.0000 0.854358 0.427179 − 0.904167i \(-0.359507\pi\)
0.427179 + 0.904167i \(0.359507\pi\)
\(138\) 0 0
\(139\) −14.0000 −1.18746 −0.593732 − 0.804663i \(-0.702346\pi\)
−0.593732 + 0.804663i \(0.702346\pi\)
\(140\) −3.00000 −0.253546
\(141\) 0 0
\(142\) −7.00000 −0.587427
\(143\) 6.00000 0.501745
\(144\) 0 0
\(145\) −4.00000 −0.332182
\(146\) 5.00000 0.413803
\(147\) 0 0
\(148\) 0 0
\(149\) 11.0000 0.901155 0.450578 − 0.892737i \(-0.351218\pi\)
0.450578 + 0.892737i \(0.351218\pi\)
\(150\) 0 0
\(151\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) −3.00000 −0.243332
\(153\) 0 0
\(154\) 9.00000 0.725241
\(155\) 1.00000 0.0803219
\(156\) 0 0
\(157\) −5.00000 −0.399043 −0.199522 − 0.979893i \(-0.563939\pi\)
−0.199522 + 0.979893i \(0.563939\pi\)
\(158\) −1.00000 −0.0795557
\(159\) 0 0
\(160\) 1.00000 0.0790569
\(161\) 15.0000 1.18217
\(162\) 0 0
\(163\) 14.0000 1.09656 0.548282 − 0.836293i \(-0.315282\pi\)
0.548282 + 0.836293i \(0.315282\pi\)
\(164\) −4.00000 −0.312348
\(165\) 0 0
\(166\) −12.0000 −0.931381
\(167\) 19.0000 1.47026 0.735132 − 0.677924i \(-0.237120\pi\)
0.735132 + 0.677924i \(0.237120\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 4.00000 0.306786
\(171\) 0 0
\(172\) 1.00000 0.0762493
\(173\) 22.0000 1.67263 0.836315 − 0.548250i \(-0.184706\pi\)
0.836315 + 0.548250i \(0.184706\pi\)
\(174\) 0 0
\(175\) −3.00000 −0.226779
\(176\) −3.00000 −0.226134
\(177\) 0 0
\(178\) −1.00000 −0.0749532
\(179\) −4.00000 −0.298974 −0.149487 − 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) 0 0
\(181\) 5.00000 0.371647 0.185824 − 0.982583i \(-0.440505\pi\)
0.185824 + 0.982583i \(0.440505\pi\)
\(182\) 6.00000 0.444750
\(183\) 0 0
\(184\) −5.00000 −0.368605
\(185\) 0 0
\(186\) 0 0
\(187\) −12.0000 −0.877527
\(188\) −10.0000 −0.729325
\(189\) 0 0
\(190\) −3.00000 −0.217643
\(191\) −16.0000 −1.15772 −0.578860 − 0.815427i \(-0.696502\pi\)
−0.578860 + 0.815427i \(0.696502\pi\)
\(192\) 0 0
\(193\) −6.00000 −0.431889 −0.215945 − 0.976406i \(-0.569283\pi\)
−0.215945 + 0.976406i \(0.569283\pi\)
\(194\) −10.0000 −0.717958
\(195\) 0 0
\(196\) 2.00000 0.142857
\(197\) 6.00000 0.427482 0.213741 − 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) 0 0
\(199\) 7.00000 0.496217 0.248108 − 0.968732i \(-0.420191\pi\)
0.248108 + 0.968732i \(0.420191\pi\)
\(200\) 1.00000 0.0707107
\(201\) 0 0
\(202\) 1.00000 0.0703598
\(203\) 12.0000 0.842235
\(204\) 0 0
\(205\) −4.00000 −0.279372
\(206\) 16.0000 1.11477
\(207\) 0 0
\(208\) −2.00000 −0.138675
\(209\) 9.00000 0.622543
\(210\) 0 0
\(211\) −19.0000 −1.30801 −0.654007 − 0.756489i \(-0.726913\pi\)
−0.654007 + 0.756489i \(0.726913\pi\)
\(212\) −3.00000 −0.206041
\(213\) 0 0
\(214\) −9.00000 −0.615227
\(215\) 1.00000 0.0681994
\(216\) 0 0
\(217\) −3.00000 −0.203653
\(218\) 20.0000 1.35457
\(219\) 0 0
\(220\) −3.00000 −0.202260
\(221\) −8.00000 −0.538138
\(222\) 0 0
\(223\) −6.00000 −0.401790 −0.200895 − 0.979613i \(-0.564385\pi\)
−0.200895 + 0.979613i \(0.564385\pi\)
\(224\) −3.00000 −0.200446
\(225\) 0 0
\(226\) −9.00000 −0.598671
\(227\) 27.0000 1.79205 0.896026 − 0.444001i \(-0.146441\pi\)
0.896026 + 0.444001i \(0.146441\pi\)
\(228\) 0 0
\(229\) 13.0000 0.859064 0.429532 − 0.903052i \(-0.358679\pi\)
0.429532 + 0.903052i \(0.358679\pi\)
\(230\) −5.00000 −0.329690
\(231\) 0 0
\(232\) −4.00000 −0.262613
\(233\) −15.0000 −0.982683 −0.491341 − 0.870967i \(-0.663493\pi\)
−0.491341 + 0.870967i \(0.663493\pi\)
\(234\) 0 0
\(235\) −10.0000 −0.652328
\(236\) −6.00000 −0.390567
\(237\) 0 0
\(238\) −12.0000 −0.777844
\(239\) −12.0000 −0.776215 −0.388108 − 0.921614i \(-0.626871\pi\)
−0.388108 + 0.921614i \(0.626871\pi\)
\(240\) 0 0
\(241\) −18.0000 −1.15948 −0.579741 − 0.814801i \(-0.696846\pi\)
−0.579741 + 0.814801i \(0.696846\pi\)
\(242\) −2.00000 −0.128565
\(243\) 0 0
\(244\) −2.00000 −0.128037
\(245\) 2.00000 0.127775
\(246\) 0 0
\(247\) 6.00000 0.381771
\(248\) 1.00000 0.0635001
\(249\) 0 0
\(250\) 1.00000 0.0632456
\(251\) 12.0000 0.757433 0.378717 − 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 0 0
\(253\) 15.0000 0.943042
\(254\) 8.00000 0.501965
\(255\) 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
\(259\) 0 0
\(260\) −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
\(265\) −3.00000 −0.184289
\(266\) 9.00000 0.551825
\(267\) 0 0
\(268\) 2.00000 0.122169
\(269\) −2.00000 −0.121942 −0.0609711 − 0.998140i \(-0.519420\pi\)
−0.0609711 + 0.998140i \(0.519420\pi\)
\(270\) 0 0
\(271\) 13.0000 0.789694 0.394847 − 0.918747i \(-0.370798\pi\)
0.394847 + 0.918747i \(0.370798\pi\)
\(272\) 4.00000 0.242536
\(273\) 0 0
\(274\) 10.0000 0.604122
\(275\) −3.00000 −0.180907
\(276\) 0 0
\(277\) 12.0000 0.721010 0.360505 − 0.932757i \(-0.382604\pi\)
0.360505 + 0.932757i \(0.382604\pi\)
\(278\) −14.0000 −0.839664
\(279\) 0 0
\(280\) −3.00000 −0.179284
\(281\) −16.0000 −0.954480 −0.477240 − 0.878773i \(-0.658363\pi\)
−0.477240 + 0.878773i \(0.658363\pi\)
\(282\) 0 0
\(283\) −24.0000 −1.42665 −0.713326 − 0.700832i \(-0.752812\pi\)
−0.713326 + 0.700832i \(0.752812\pi\)
\(284\) −7.00000 −0.415374
\(285\) 0 0
\(286\) 6.00000 0.354787
\(287\) 12.0000 0.708338
\(288\) 0 0
\(289\) −1.00000 −0.0588235
\(290\) −4.00000 −0.234888
\(291\) 0 0
\(292\) 5.00000 0.292603
\(293\) −30.0000 −1.75262 −0.876309 − 0.481749i \(-0.840002\pi\)
−0.876309 + 0.481749i \(0.840002\pi\)
\(294\) 0 0
\(295\) −6.00000 −0.349334
\(296\) 0 0
\(297\) 0 0
\(298\) 11.0000 0.637213
\(299\) 10.0000 0.578315
\(300\) 0 0
\(301\) −3.00000 −0.172917
\(302\) 0 0
\(303\) 0 0
\(304\) −3.00000 −0.172062
\(305\) −2.00000 −0.114520
\(306\) 0 0
\(307\) 16.0000 0.913168 0.456584 − 0.889680i \(-0.349073\pi\)
0.456584 + 0.889680i \(0.349073\pi\)
\(308\) 9.00000 0.512823
\(309\) 0 0
\(310\) 1.00000 0.0567962
\(311\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 30.0000 1.69570 0.847850 − 0.530236i \(-0.177897\pi\)
0.847850 + 0.530236i \(0.177897\pi\)
\(314\) −5.00000 −0.282166
\(315\) 0 0
\(316\) −1.00000 −0.0562544
\(317\) 8.00000 0.449325 0.224662 − 0.974437i \(-0.427872\pi\)
0.224662 + 0.974437i \(0.427872\pi\)
\(318\) 0 0
\(319\) 12.0000 0.671871
\(320\) 1.00000 0.0559017
\(321\) 0 0
\(322\) 15.0000 0.835917
\(323\) −12.0000 −0.667698
\(324\) 0 0
\(325\) −2.00000 −0.110940
\(326\) 14.0000 0.775388
\(327\) 0 0
\(328\) −4.00000 −0.220863
\(329\) 30.0000 1.65395
\(330\) 0 0
\(331\) 12.0000 0.659580 0.329790 − 0.944054i \(-0.393022\pi\)
0.329790 + 0.944054i \(0.393022\pi\)
\(332\) −12.0000 −0.658586
\(333\) 0 0
\(334\) 19.0000 1.03963
\(335\) 2.00000 0.109272
\(336\) 0 0
\(337\) −2.00000 −0.108947 −0.0544735 − 0.998515i \(-0.517348\pi\)
−0.0544735 + 0.998515i \(0.517348\pi\)
\(338\) −9.00000 −0.489535
\(339\) 0 0
\(340\) 4.00000 0.216930
\(341\) −3.00000 −0.162459
\(342\) 0 0
\(343\) 15.0000 0.809924
\(344\) 1.00000 0.0539164
\(345\) 0 0
\(346\) 22.0000 1.18273
\(347\) 32.0000 1.71785 0.858925 − 0.512101i \(-0.171133\pi\)
0.858925 + 0.512101i \(0.171133\pi\)
\(348\) 0 0
\(349\) −24.0000 −1.28469 −0.642345 − 0.766415i \(-0.722038\pi\)
−0.642345 + 0.766415i \(0.722038\pi\)
\(350\) −3.00000 −0.160357
\(351\) 0 0
\(352\) −3.00000 −0.159901
\(353\) −18.0000 −0.958043 −0.479022 − 0.877803i \(-0.659008\pi\)
−0.479022 + 0.877803i \(0.659008\pi\)
\(354\) 0 0
\(355\) −7.00000 −0.371521
\(356\) −1.00000 −0.0529999
\(357\) 0 0
\(358\) −4.00000 −0.211407
\(359\) −15.0000 −0.791670 −0.395835 − 0.918322i \(-0.629545\pi\)
−0.395835 + 0.918322i \(0.629545\pi\)
\(360\) 0 0
\(361\) −10.0000 −0.526316
\(362\) 5.00000 0.262794
\(363\) 0 0
\(364\) 6.00000 0.314485
\(365\) 5.00000 0.261712
\(366\) 0 0
\(367\) 2.00000 0.104399 0.0521996 − 0.998637i \(-0.483377\pi\)
0.0521996 + 0.998637i \(0.483377\pi\)
\(368\) −5.00000 −0.260643
\(369\) 0 0
\(370\) 0 0
\(371\) 9.00000 0.467257
\(372\) 0 0
\(373\) −9.00000 −0.466002 −0.233001 − 0.972476i \(-0.574855\pi\)
−0.233001 + 0.972476i \(0.574855\pi\)
\(374\) −12.0000 −0.620505
\(375\) 0 0
\(376\) −10.0000 −0.515711
\(377\) 8.00000 0.412021
\(378\) 0 0
\(379\) 15.0000 0.770498 0.385249 − 0.922813i \(-0.374116\pi\)
0.385249 + 0.922813i \(0.374116\pi\)
\(380\) −3.00000 −0.153897
\(381\) 0 0
\(382\) −16.0000 −0.818631
\(383\) −12.0000 −0.613171 −0.306586 − 0.951843i \(-0.599187\pi\)
−0.306586 + 0.951843i \(0.599187\pi\)
\(384\) 0 0
\(385\) 9.00000 0.458682
\(386\) −6.00000 −0.305392
\(387\) 0 0
\(388\) −10.0000 −0.507673
\(389\) 26.0000 1.31825 0.659126 − 0.752032i \(-0.270926\pi\)
0.659126 + 0.752032i \(0.270926\pi\)
\(390\) 0 0
\(391\) −20.0000 −1.01144
\(392\) 2.00000 0.101015
\(393\) 0 0
\(394\) 6.00000 0.302276
\(395\) −1.00000 −0.0503155
\(396\) 0 0
\(397\) 35.0000 1.75660 0.878300 − 0.478110i \(-0.158678\pi\)
0.878300 + 0.478110i \(0.158678\pi\)
\(398\) 7.00000 0.350878
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −25.0000 −1.24844 −0.624220 − 0.781248i \(-0.714583\pi\)
−0.624220 + 0.781248i \(0.714583\pi\)
\(402\) 0 0
\(403\) −2.00000 −0.0996271
\(404\) 1.00000 0.0497519
\(405\) 0 0
\(406\) 12.0000 0.595550
\(407\) 0 0
\(408\) 0 0
\(409\) 8.00000 0.395575 0.197787 − 0.980245i \(-0.436624\pi\)
0.197787 + 0.980245i \(0.436624\pi\)
\(410\) −4.00000 −0.197546
\(411\) 0 0
\(412\) 16.0000 0.788263
\(413\) 18.0000 0.885722
\(414\) 0 0
\(415\) −12.0000 −0.589057
\(416\) −2.00000 −0.0980581
\(417\) 0 0
\(418\) 9.00000 0.440204
\(419\) −12.0000 −0.586238 −0.293119 − 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 0 0
\(421\) −22.0000 −1.07221 −0.536107 − 0.844150i \(-0.680106\pi\)
−0.536107 + 0.844150i \(0.680106\pi\)
\(422\) −19.0000 −0.924906
\(423\) 0 0
\(424\) −3.00000 −0.145693
\(425\) 4.00000 0.194029
\(426\) 0 0
\(427\) 6.00000 0.290360
\(428\) −9.00000 −0.435031
\(429\) 0 0
\(430\) 1.00000 0.0482243
\(431\) −16.0000 −0.770693 −0.385346 − 0.922772i \(-0.625918\pi\)
−0.385346 + 0.922772i \(0.625918\pi\)
\(432\) 0 0
\(433\) −1.00000 −0.0480569 −0.0240285 − 0.999711i \(-0.507649\pi\)
−0.0240285 + 0.999711i \(0.507649\pi\)
\(434\) −3.00000 −0.144005
\(435\) 0 0
\(436\) 20.0000 0.957826
\(437\) 15.0000 0.717547
\(438\) 0 0
\(439\) 18.0000 0.859093 0.429547 − 0.903045i \(-0.358673\pi\)
0.429547 + 0.903045i \(0.358673\pi\)
\(440\) −3.00000 −0.143019
\(441\) 0 0
\(442\) −8.00000 −0.380521
\(443\) −15.0000 −0.712672 −0.356336 − 0.934358i \(-0.615974\pi\)
−0.356336 + 0.934358i \(0.615974\pi\)
\(444\) 0 0
\(445\) −1.00000 −0.0474045
\(446\) −6.00000 −0.284108
\(447\) 0 0
\(448\) −3.00000 −0.141737
\(449\) −2.00000 −0.0943858 −0.0471929 − 0.998886i \(-0.515028\pi\)
−0.0471929 + 0.998886i \(0.515028\pi\)
\(450\) 0 0
\(451\) 12.0000 0.565058
\(452\) −9.00000 −0.423324
\(453\) 0 0
\(454\) 27.0000 1.26717
\(455\) 6.00000 0.281284
\(456\) 0 0
\(457\) 10.0000 0.467780 0.233890 − 0.972263i \(-0.424854\pi\)
0.233890 + 0.972263i \(0.424854\pi\)
\(458\) 13.0000 0.607450
\(459\) 0 0
\(460\) −5.00000 −0.233126
\(461\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) 4.00000 0.185896 0.0929479 − 0.995671i \(-0.470371\pi\)
0.0929479 + 0.995671i \(0.470371\pi\)
\(464\) −4.00000 −0.185695
\(465\) 0 0
\(466\) −15.0000 −0.694862
\(467\) −24.0000 −1.11059 −0.555294 − 0.831654i \(-0.687394\pi\)
−0.555294 + 0.831654i \(0.687394\pi\)
\(468\) 0 0
\(469\) −6.00000 −0.277054
\(470\) −10.0000 −0.461266
\(471\) 0 0
\(472\) −6.00000 −0.276172
\(473\) −3.00000 −0.137940
\(474\) 0 0
\(475\) −3.00000 −0.137649
\(476\) −12.0000 −0.550019
\(477\) 0 0
\(478\) −12.0000 −0.548867
\(479\) −3.00000 −0.137073 −0.0685367 − 0.997649i \(-0.521833\pi\)
−0.0685367 + 0.997649i \(0.521833\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −18.0000 −0.819878
\(483\) 0 0
\(484\) −2.00000 −0.0909091
\(485\) −10.0000 −0.454077
\(486\) 0 0
\(487\) 32.0000 1.45006 0.725029 − 0.688718i \(-0.241826\pi\)
0.725029 + 0.688718i \(0.241826\pi\)
\(488\) −2.00000 −0.0905357
\(489\) 0 0
\(490\) 2.00000 0.0903508
\(491\) 37.0000 1.66979 0.834893 − 0.550412i \(-0.185529\pi\)
0.834893 + 0.550412i \(0.185529\pi\)
\(492\) 0 0
\(493\) −16.0000 −0.720604
\(494\) 6.00000 0.269953
\(495\) 0 0
\(496\) 1.00000 0.0449013
\(497\) 21.0000 0.941979
\(498\) 0 0
\(499\) 4.00000 0.179065 0.0895323 − 0.995984i \(-0.471463\pi\)
0.0895323 + 0.995984i \(0.471463\pi\)
\(500\) 1.00000 0.0447214
\(501\) 0 0
\(502\) 12.0000 0.535586
\(503\) −20.0000 −0.891756 −0.445878 − 0.895094i \(-0.647108\pi\)
−0.445878 + 0.895094i \(0.647108\pi\)
\(504\) 0 0
\(505\) 1.00000 0.0444994
\(506\) 15.0000 0.666831
\(507\) 0 0
\(508\) 8.00000 0.354943
\(509\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) −15.0000 −0.663561
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 23.0000 1.01449
\(515\) 16.0000 0.705044
\(516\) 0 0
\(517\) 30.0000 1.31940
\(518\) 0 0
\(519\) 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
\(523\) −43.0000 −1.88026 −0.940129 − 0.340818i \(-0.889296\pi\)
−0.940129 + 0.340818i \(0.889296\pi\)
\(524\) −6.00000 −0.262111
\(525\) 0 0
\(526\) −24.0000 −1.04645
\(527\) 4.00000 0.174243
\(528\) 0 0
\(529\) 2.00000 0.0869565
\(530\) −3.00000 −0.130312
\(531\) 0 0
\(532\) 9.00000 0.390199
\(533\) 8.00000 0.346518
\(534\) 0 0
\(535\) −9.00000 −0.389104
\(536\) 2.00000 0.0863868
\(537\) 0 0
\(538\) −2.00000 −0.0862261
\(539\) −6.00000 −0.258438
\(540\) 0 0
\(541\) 22.0000 0.945854 0.472927 − 0.881102i \(-0.343197\pi\)
0.472927 + 0.881102i \(0.343197\pi\)
\(542\) 13.0000 0.558398
\(543\) 0 0
\(544\) 4.00000 0.171499
\(545\) 20.0000 0.856706
\(546\) 0 0
\(547\) 22.0000 0.940652 0.470326 − 0.882493i \(-0.344136\pi\)
0.470326 + 0.882493i \(0.344136\pi\)
\(548\) 10.0000 0.427179
\(549\) 0 0
\(550\) −3.00000 −0.127920
\(551\) 12.0000 0.511217
\(552\) 0 0
\(553\) 3.00000 0.127573
\(554\) 12.0000 0.509831
\(555\) 0 0
\(556\) −14.0000 −0.593732
\(557\) 1.00000 0.0423714 0.0211857 − 0.999776i \(-0.493256\pi\)
0.0211857 + 0.999776i \(0.493256\pi\)
\(558\) 0 0
\(559\) −2.00000 −0.0845910
\(560\) −3.00000 −0.126773
\(561\) 0 0
\(562\) −16.0000 −0.674919
\(563\) 4.00000 0.168580 0.0842900 − 0.996441i \(-0.473138\pi\)
0.0842900 + 0.996441i \(0.473138\pi\)
\(564\) 0 0
\(565\) −9.00000 −0.378633
\(566\) −24.0000 −1.00880
\(567\) 0 0
\(568\) −7.00000 −0.293713
\(569\) 23.0000 0.964210 0.482105 − 0.876113i \(-0.339872\pi\)
0.482105 + 0.876113i \(0.339872\pi\)
\(570\) 0 0
\(571\) 22.0000 0.920671 0.460336 − 0.887745i \(-0.347729\pi\)
0.460336 + 0.887745i \(0.347729\pi\)
\(572\) 6.00000 0.250873
\(573\) 0 0
\(574\) 12.0000 0.500870
\(575\) −5.00000 −0.208514
\(576\) 0 0
\(577\) 4.00000 0.166522 0.0832611 − 0.996528i \(-0.473466\pi\)
0.0832611 + 0.996528i \(0.473466\pi\)
\(578\) −1.00000 −0.0415945
\(579\) 0 0
\(580\) −4.00000 −0.166091
\(581\) 36.0000 1.49353
\(582\) 0 0
\(583\) 9.00000 0.372742
\(584\) 5.00000 0.206901
\(585\) 0 0
\(586\) −30.0000 −1.23929
\(587\) −44.0000 −1.81607 −0.908037 − 0.418890i \(-0.862419\pi\)
−0.908037 + 0.418890i \(0.862419\pi\)
\(588\) 0 0
\(589\) −3.00000 −0.123613
\(590\) −6.00000 −0.247016
\(591\) 0 0
\(592\) 0 0
\(593\) −22.0000 −0.903432 −0.451716 − 0.892162i \(-0.649188\pi\)
−0.451716 + 0.892162i \(0.649188\pi\)
\(594\) 0 0
\(595\) −12.0000 −0.491952
\(596\) 11.0000 0.450578
\(597\) 0 0
\(598\) 10.0000 0.408930
\(599\) 27.0000 1.10319 0.551595 − 0.834112i \(-0.314019\pi\)
0.551595 + 0.834112i \(0.314019\pi\)
\(600\) 0 0
\(601\) 42.0000 1.71322 0.856608 − 0.515968i \(-0.172568\pi\)
0.856608 + 0.515968i \(0.172568\pi\)
\(602\) −3.00000 −0.122271
\(603\) 0 0
\(604\) 0 0
\(605\) −2.00000 −0.0813116
\(606\) 0 0
\(607\) 29.0000 1.17707 0.588537 − 0.808470i \(-0.299704\pi\)
0.588537 + 0.808470i \(0.299704\pi\)
\(608\) −3.00000 −0.121666
\(609\) 0 0
\(610\) −2.00000 −0.0809776
\(611\) 20.0000 0.809113
\(612\) 0 0
\(613\) −22.0000 −0.888572 −0.444286 − 0.895885i \(-0.646543\pi\)
−0.444286 + 0.895885i \(0.646543\pi\)
\(614\) 16.0000 0.645707
\(615\) 0 0
\(616\) 9.00000 0.362620
\(617\) 27.0000 1.08698 0.543490 − 0.839416i \(-0.317103\pi\)
0.543490 + 0.839416i \(0.317103\pi\)
\(618\) 0 0
\(619\) −38.0000 −1.52735 −0.763674 − 0.645601i \(-0.776607\pi\)
−0.763674 + 0.645601i \(0.776607\pi\)
\(620\) 1.00000 0.0401610
\(621\) 0 0
\(622\) 0 0
\(623\) 3.00000 0.120192
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 30.0000 1.19904
\(627\) 0 0
\(628\) −5.00000 −0.199522
\(629\) 0 0
\(630\) 0 0
\(631\) 17.0000 0.676759 0.338380 − 0.941010i \(-0.390121\pi\)
0.338380 + 0.941010i \(0.390121\pi\)
\(632\) −1.00000 −0.0397779
\(633\) 0 0
\(634\) 8.00000 0.317721
\(635\) 8.00000 0.317470
\(636\) 0 0
\(637\) −4.00000 −0.158486
\(638\) 12.0000 0.475085
\(639\) 0 0
\(640\) 1.00000 0.0395285
\(641\) −26.0000 −1.02694 −0.513469 − 0.858108i \(-0.671640\pi\)
−0.513469 + 0.858108i \(0.671640\pi\)
\(642\) 0 0
\(643\) 19.0000 0.749287 0.374643 − 0.927169i \(-0.377765\pi\)
0.374643 + 0.927169i \(0.377765\pi\)
\(644\) 15.0000 0.591083
\(645\) 0 0
\(646\) −12.0000 −0.472134
\(647\) −3.00000 −0.117942 −0.0589711 − 0.998260i \(-0.518782\pi\)
−0.0589711 + 0.998260i \(0.518782\pi\)
\(648\) 0 0
\(649\) 18.0000 0.706562
\(650\) −2.00000 −0.0784465
\(651\) 0 0
\(652\) 14.0000 0.548282
\(653\) 26.0000 1.01746 0.508729 − 0.860927i \(-0.330115\pi\)
0.508729 + 0.860927i \(0.330115\pi\)
\(654\) 0 0
\(655\) −6.00000 −0.234439
\(656\) −4.00000 −0.156174
\(657\) 0 0
\(658\) 30.0000 1.16952
\(659\) 4.00000 0.155818 0.0779089 − 0.996960i \(-0.475176\pi\)
0.0779089 + 0.996960i \(0.475176\pi\)
\(660\) 0 0
\(661\) −14.0000 −0.544537 −0.272268 − 0.962221i \(-0.587774\pi\)
−0.272268 + 0.962221i \(0.587774\pi\)
\(662\) 12.0000 0.466393
\(663\) 0 0
\(664\) −12.0000 −0.465690
\(665\) 9.00000 0.349005
\(666\) 0 0
\(667\) 20.0000 0.774403
\(668\) 19.0000 0.735132
\(669\) 0 0
\(670\) 2.00000 0.0772667
\(671\) 6.00000 0.231627
\(672\) 0 0
\(673\) 26.0000 1.00223 0.501113 − 0.865382i \(-0.332924\pi\)
0.501113 + 0.865382i \(0.332924\pi\)
\(674\) −2.00000 −0.0770371
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) −27.0000 −1.03769 −0.518847 − 0.854867i \(-0.673639\pi\)
−0.518847 + 0.854867i \(0.673639\pi\)
\(678\) 0 0
\(679\) 30.0000 1.15129
\(680\) 4.00000 0.153393
\(681\) 0 0
\(682\) −3.00000 −0.114876
\(683\) 19.0000 0.727015 0.363507 − 0.931591i \(-0.381579\pi\)
0.363507 + 0.931591i \(0.381579\pi\)
\(684\) 0 0
\(685\) 10.0000 0.382080
\(686\) 15.0000 0.572703
\(687\) 0 0
\(688\) 1.00000 0.0381246
\(689\) 6.00000 0.228582
\(690\) 0 0
\(691\) −31.0000 −1.17930 −0.589648 − 0.807661i \(-0.700733\pi\)
−0.589648 + 0.807661i \(0.700733\pi\)
\(692\) 22.0000 0.836315
\(693\) 0 0
\(694\) 32.0000 1.21470
\(695\) −14.0000 −0.531050
\(696\) 0 0
\(697\) −16.0000 −0.606043
\(698\) −24.0000 −0.908413
\(699\) 0 0
\(700\) −3.00000 −0.113389
\(701\) 21.0000 0.793159 0.396580 − 0.918000i \(-0.370197\pi\)
0.396580 + 0.918000i \(0.370197\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −3.00000 −0.113067
\(705\) 0 0
\(706\) −18.0000 −0.677439
\(707\) −3.00000 −0.112827
\(708\) 0 0
\(709\) −25.0000 −0.938895 −0.469447 − 0.882960i \(-0.655547\pi\)
−0.469447 + 0.882960i \(0.655547\pi\)
\(710\) −7.00000 −0.262705
\(711\) 0 0
\(712\) −1.00000 −0.0374766
\(713\) −5.00000 −0.187251
\(714\) 0 0
\(715\) 6.00000 0.224387
\(716\) −4.00000 −0.149487
\(717\) 0 0
\(718\) −15.0000 −0.559795
\(719\) −26.0000 −0.969636 −0.484818 − 0.874615i \(-0.661114\pi\)
−0.484818 + 0.874615i \(0.661114\pi\)
\(720\) 0 0
\(721\) −48.0000 −1.78761
\(722\) −10.0000 −0.372161
\(723\) 0 0
\(724\) 5.00000 0.185824
\(725\) −4.00000 −0.148556
\(726\) 0 0
\(727\) 45.0000 1.66896 0.834479 − 0.551040i \(-0.185769\pi\)
0.834479 + 0.551040i \(0.185769\pi\)
\(728\) 6.00000 0.222375
\(729\) 0 0
\(730\) 5.00000 0.185058
\(731\) 4.00000 0.147945
\(732\) 0 0
\(733\) −18.0000 −0.664845 −0.332423 − 0.943131i \(-0.607866\pi\)
−0.332423 + 0.943131i \(0.607866\pi\)
\(734\) 2.00000 0.0738213
\(735\) 0 0
\(736\) −5.00000 −0.184302
\(737\) −6.00000 −0.221013
\(738\) 0 0
\(739\) −20.0000 −0.735712 −0.367856 − 0.929883i \(-0.619908\pi\)
−0.367856 + 0.929883i \(0.619908\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 9.00000 0.330400
\(743\) 37.0000 1.35740 0.678699 − 0.734416i \(-0.262544\pi\)
0.678699 + 0.734416i \(0.262544\pi\)
\(744\) 0 0
\(745\) 11.0000 0.403009
\(746\) −9.00000 −0.329513
\(747\) 0 0
\(748\) −12.0000 −0.438763
\(749\) 27.0000 0.986559
\(750\) 0 0
\(751\) −50.0000 −1.82453 −0.912263 − 0.409605i \(-0.865667\pi\)
−0.912263 + 0.409605i \(0.865667\pi\)
\(752\) −10.0000 −0.364662
\(753\) 0 0
\(754\) 8.00000 0.291343
\(755\) 0 0
\(756\) 0 0
\(757\) 50.0000 1.81728 0.908640 − 0.417579i \(-0.137121\pi\)
0.908640 + 0.417579i \(0.137121\pi\)
\(758\) 15.0000 0.544825
\(759\) 0 0
\(760\) −3.00000 −0.108821
\(761\) −45.0000 −1.63125 −0.815624 − 0.578582i \(-0.803606\pi\)
−0.815624 + 0.578582i \(0.803606\pi\)
\(762\) 0 0
\(763\) −60.0000 −2.17215
\(764\) −16.0000 −0.578860
\(765\) 0 0
\(766\) −12.0000 −0.433578
\(767\) 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
\(772\) −6.00000 −0.215945
\(773\) 31.0000 1.11499 0.557496 − 0.830179i \(-0.311762\pi\)
0.557496 + 0.830179i \(0.311762\pi\)
\(774\) 0 0
\(775\) 1.00000 0.0359211
\(776\) −10.0000 −0.358979
\(777\) 0 0
\(778\) 26.0000 0.932145
\(779\) 12.0000 0.429945
\(780\) 0 0
\(781\) 21.0000 0.751439
\(782\) −20.0000 −0.715199
\(783\) 0 0
\(784\) 2.00000 0.0714286
\(785\) −5.00000 −0.178458
\(786\) 0 0
\(787\) −35.0000 −1.24762 −0.623808 − 0.781578i \(-0.714415\pi\)
−0.623808 + 0.781578i \(0.714415\pi\)
\(788\) 6.00000 0.213741
\(789\) 0 0
\(790\) −1.00000 −0.0355784
\(791\) 27.0000 0.960009
\(792\) 0 0
\(793\) 4.00000 0.142044
\(794\) 35.0000 1.24210
\(795\) 0 0
\(796\) 7.00000 0.248108
\(797\) −2.00000 −0.0708436 −0.0354218 − 0.999372i \(-0.511277\pi\)
−0.0354218 + 0.999372i \(0.511277\pi\)
\(798\) 0 0
\(799\) −40.0000 −1.41510
\(800\) 1.00000 0.0353553
\(801\) 0 0
\(802\) −25.0000 −0.882781
\(803\) −15.0000 −0.529339
\(804\) 0 0
\(805\) 15.0000 0.528681
\(806\) −2.00000 −0.0704470
\(807\) 0 0
\(808\) 1.00000 0.0351799
\(809\) −17.0000 −0.597688 −0.298844 − 0.954302i \(-0.596601\pi\)
−0.298844 + 0.954302i \(0.596601\pi\)
\(810\) 0 0
\(811\) −27.0000 −0.948098 −0.474049 − 0.880498i \(-0.657208\pi\)
−0.474049 + 0.880498i \(0.657208\pi\)
\(812\) 12.0000 0.421117
\(813\) 0 0
\(814\) 0 0
\(815\) 14.0000 0.490399
\(816\) 0 0
\(817\) −3.00000 −0.104957
\(818\) 8.00000 0.279713
\(819\) 0 0
\(820\) −4.00000 −0.139686
\(821\) −40.0000 −1.39601 −0.698005 − 0.716093i \(-0.745929\pi\)
−0.698005 + 0.716093i \(0.745929\pi\)
\(822\) 0 0
\(823\) −18.0000 −0.627441 −0.313720 − 0.949515i \(-0.601575\pi\)
−0.313720 + 0.949515i \(0.601575\pi\)
\(824\) 16.0000 0.557386
\(825\) 0 0
\(826\) 18.0000 0.626300
\(827\) −20.0000 −0.695468 −0.347734 − 0.937593i \(-0.613049\pi\)
−0.347734 + 0.937593i \(0.613049\pi\)
\(828\) 0 0
\(829\) −35.0000 −1.21560 −0.607800 − 0.794090i \(-0.707948\pi\)
−0.607800 + 0.794090i \(0.707948\pi\)
\(830\) −12.0000 −0.416526
\(831\) 0 0
\(832\) −2.00000 −0.0693375
\(833\) 8.00000 0.277184
\(834\) 0 0
\(835\) 19.0000 0.657522
\(836\) 9.00000 0.311272
\(837\) 0 0
\(838\) −12.0000 −0.414533
\(839\) −51.0000 −1.76072 −0.880358 − 0.474310i \(-0.842698\pi\)
−0.880358 + 0.474310i \(0.842698\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) −22.0000 −0.758170
\(843\) 0 0
\(844\) −19.0000 −0.654007
\(845\) −9.00000 −0.309609
\(846\) 0 0
\(847\) 6.00000 0.206162
\(848\) −3.00000 −0.103020
\(849\) 0 0
\(850\) 4.00000 0.137199
\(851\) 0 0
\(852\) 0 0
\(853\) −41.0000 −1.40381 −0.701907 − 0.712269i \(-0.747668\pi\)
−0.701907 + 0.712269i \(0.747668\pi\)
\(854\) 6.00000 0.205316
\(855\) 0 0
\(856\) −9.00000 −0.307614
\(857\) −42.0000 −1.43469 −0.717346 − 0.696717i \(-0.754643\pi\)
−0.717346 + 0.696717i \(0.754643\pi\)
\(858\) 0 0
\(859\) 32.0000 1.09183 0.545913 − 0.837842i \(-0.316183\pi\)
0.545913 + 0.837842i \(0.316183\pi\)
\(860\) 1.00000 0.0340997
\(861\) 0 0
\(862\) −16.0000 −0.544962
\(863\) −43.0000 −1.46374 −0.731869 − 0.681446i \(-0.761351\pi\)
−0.731869 + 0.681446i \(0.761351\pi\)
\(864\) 0 0
\(865\) 22.0000 0.748022
\(866\) −1.00000 −0.0339814
\(867\) 0 0
\(868\) −3.00000 −0.101827
\(869\) 3.00000 0.101768
\(870\) 0 0
\(871\) −4.00000 −0.135535
\(872\) 20.0000 0.677285
\(873\) 0 0
\(874\) 15.0000 0.507383
\(875\) −3.00000 −0.101419
\(876\) 0 0
\(877\) 46.0000 1.55331 0.776655 − 0.629926i \(-0.216915\pi\)
0.776655 + 0.629926i \(0.216915\pi\)
\(878\) 18.0000 0.607471
\(879\) 0 0
\(880\) −3.00000 −0.101130
\(881\) −26.0000 −0.875962 −0.437981 − 0.898984i \(-0.644306\pi\)
−0.437981 + 0.898984i \(0.644306\pi\)
\(882\) 0 0
\(883\) 11.0000 0.370179 0.185090 − 0.982722i \(-0.440742\pi\)
0.185090 + 0.982722i \(0.440742\pi\)
\(884\) −8.00000 −0.269069
\(885\) 0 0
\(886\) −15.0000 −0.503935
\(887\) 46.0000 1.54453 0.772264 − 0.635301i \(-0.219124\pi\)
0.772264 + 0.635301i \(0.219124\pi\)
\(888\) 0 0
\(889\) −24.0000 −0.804934
\(890\) −1.00000 −0.0335201
\(891\) 0 0
\(892\) −6.00000 −0.200895
\(893\) 30.0000 1.00391
\(894\) 0 0
\(895\) −4.00000 −0.133705
\(896\) −3.00000 −0.100223
\(897\) 0 0
\(898\) −2.00000 −0.0667409
\(899\) −4.00000 −0.133407
\(900\) 0 0
\(901\) −12.0000 −0.399778
\(902\) 12.0000 0.399556
\(903\) 0 0
\(904\) −9.00000 −0.299336
\(905\) 5.00000 0.166206
\(906\) 0 0
\(907\) −52.0000 −1.72663 −0.863316 − 0.504664i \(-0.831616\pi\)
−0.863316 + 0.504664i \(0.831616\pi\)
\(908\) 27.0000 0.896026
\(909\) 0 0
\(910\) 6.00000 0.198898
\(911\) −20.0000 −0.662630 −0.331315 − 0.943520i \(-0.607492\pi\)
−0.331315 + 0.943520i \(0.607492\pi\)
\(912\) 0 0
\(913\) 36.0000 1.19143
\(914\) 10.0000 0.330771
\(915\) 0 0
\(916\) 13.0000 0.429532
\(917\) 18.0000 0.594412
\(918\) 0 0
\(919\) −38.0000 −1.25350 −0.626752 − 0.779219i \(-0.715616\pi\)
−0.626752 + 0.779219i \(0.715616\pi\)
\(920\) −5.00000 −0.164845
\(921\) 0 0
\(922\) 0 0
\(923\) 14.0000 0.460816
\(924\) 0 0
\(925\) 0 0
\(926\) 4.00000 0.131448
\(927\) 0 0
\(928\) −4.00000 −0.131306
\(929\) 41.0000 1.34517 0.672583 − 0.740022i \(-0.265185\pi\)
0.672583 + 0.740022i \(0.265185\pi\)
\(930\) 0 0
\(931\) −6.00000 −0.196642
\(932\) −15.0000 −0.491341
\(933\) 0 0
\(934\) −24.0000 −0.785304
\(935\) −12.0000 −0.392442
\(936\) 0 0
\(937\) −52.0000 −1.69877 −0.849383 − 0.527777i \(-0.823026\pi\)
−0.849383 + 0.527777i \(0.823026\pi\)
\(938\) −6.00000 −0.195907
\(939\) 0 0
\(940\) −10.0000 −0.326164
\(941\) −40.0000 −1.30396 −0.651981 − 0.758235i \(-0.726062\pi\)
−0.651981 + 0.758235i \(0.726062\pi\)
\(942\) 0 0
\(943\) 20.0000 0.651290
\(944\) −6.00000 −0.195283
\(945\) 0 0
\(946\) −3.00000 −0.0975384
\(947\) −30.0000 −0.974869 −0.487435 − 0.873160i \(-0.662067\pi\)
−0.487435 + 0.873160i \(0.662067\pi\)
\(948\) 0 0
\(949\) −10.0000 −0.324614
\(950\) −3.00000 −0.0973329
\(951\) 0 0
\(952\) −12.0000 −0.388922
\(953\) 52.0000 1.68445 0.842223 − 0.539130i \(-0.181247\pi\)
0.842223 + 0.539130i \(0.181247\pi\)
\(954\) 0 0
\(955\) −16.0000 −0.517748
\(956\) −12.0000 −0.388108
\(957\) 0 0
\(958\) −3.00000 −0.0969256
\(959\) −30.0000 −0.968751
\(960\) 0 0
\(961\) 1.00000 0.0322581
\(962\) 0 0
\(963\) 0 0
\(964\) −18.0000 −0.579741
\(965\) −6.00000 −0.193147
\(966\) 0 0
\(967\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(968\) −2.00000 −0.0642824
\(969\) 0 0
\(970\) −10.0000 −0.321081
\(971\) 54.0000 1.73294 0.866471 − 0.499227i \(-0.166383\pi\)
0.866471 + 0.499227i \(0.166383\pi\)
\(972\) 0 0
\(973\) 42.0000 1.34646
\(974\) 32.0000 1.02535
\(975\) 0 0
\(976\) −2.00000 −0.0640184
\(977\) −38.0000 −1.21573 −0.607864 − 0.794041i \(-0.707973\pi\)
−0.607864 + 0.794041i \(0.707973\pi\)
\(978\) 0 0
\(979\) 3.00000 0.0958804
\(980\) 2.00000 0.0638877
\(981\) 0 0
\(982\) 37.0000 1.18072
\(983\) 56.0000 1.78612 0.893061 − 0.449935i \(-0.148553\pi\)
0.893061 + 0.449935i \(0.148553\pi\)
\(984\) 0 0
\(985\) 6.00000 0.191176
\(986\) −16.0000 −0.509544
\(987\) 0 0
\(988\) 6.00000 0.190885
\(989\) −5.00000 −0.158991
\(990\) 0 0
\(991\) −25.0000 −0.794151 −0.397076 − 0.917786i \(-0.629975\pi\)
−0.397076 + 0.917786i \(0.629975\pi\)
\(992\) 1.00000 0.0317500
\(993\) 0 0
\(994\) 21.0000 0.666080
\(995\) 7.00000 0.221915
\(996\) 0 0
\(997\) −42.0000 −1.33015 −0.665077 − 0.746775i \(-0.731601\pi\)
−0.665077 + 0.746775i \(0.731601\pi\)
\(998\) 4.00000 0.126618
\(999\) 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.y.1.1 1
3.2 odd 2 930.2.a.a.1.1 ✓ 1
12.11 even 2 7440.2.a.v.1.1 1
15.2 even 4 4650.2.d.y.3349.1 2
15.8 even 4 4650.2.d.y.3349.2 2
15.14 odd 2 4650.2.a.bv.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
930.2.a.a.1.1 ✓ 1 3.2 odd 2
2790.2.a.y.1.1 1 1.1 even 1 trivial
4650.2.a.bv.1.1 1 15.14 odd 2
4650.2.d.y.3349.1 2 15.2 even 4
4650.2.d.y.3349.2 2 15.8 even 4
7440.2.a.v.1.1 1 12.11 even 2