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Properties

Label 2790.2.a.p.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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -2.00000 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -2.00000 q^{7} +1.00000 q^{8} -1.00000 q^{10} +4.00000 q^{11} -4.00000 q^{13} -2.00000 q^{14} +1.00000 q^{16} -2.00000 q^{17} -8.00000 q^{19} -1.00000 q^{20} +4.00000 q^{22} +8.00000 q^{23} +1.00000 q^{25} -4.00000 q^{26} -2.00000 q^{28} -4.00000 q^{29} -1.00000 q^{31} +1.00000 q^{32} -2.00000 q^{34} +2.00000 q^{35} -12.0000 q^{37} -8.00000 q^{38} -1.00000 q^{40} -10.0000 q^{41} +8.00000 q^{43} +4.00000 q^{44} +8.00000 q^{46} +4.00000 q^{47} -3.00000 q^{49} +1.00000 q^{50} -4.00000 q^{52} -6.00000 q^{53} -4.00000 q^{55} -2.00000 q^{56} -4.00000 q^{58} -2.00000 q^{59} +10.0000 q^{61} -1.00000 q^{62} +1.00000 q^{64} +4.00000 q^{65} -6.00000 q^{67} -2.00000 q^{68} +2.00000 q^{70} -6.00000 q^{71} -4.00000 q^{73} -12.0000 q^{74} -8.00000 q^{76} -8.00000 q^{77} -8.00000 q^{79} -1.00000 q^{80} -10.0000 q^{82} -4.00000 q^{83} +2.00000 q^{85} +8.00000 q^{86} +4.00000 q^{88} +8.00000 q^{91} +8.00000 q^{92} +4.00000 q^{94} +8.00000 q^{95} -18.0000 q^{97} -3.00000 q^{98} +O(q^{100})\)

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
\(7\) −2.00000 −0.755929 −0.377964 − 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) −1.00000 −0.316228
\(11\) 4.00000 1.20605 0.603023 − 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 0 0
\(13\) −4.00000 −1.10940 −0.554700 − 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) −2.00000 −0.534522
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −2.00000 −0.485071 −0.242536 − 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0 0
\(19\) −8.00000 −1.83533 −0.917663 − 0.397360i \(-0.869927\pi\)
−0.917663 + 0.397360i \(0.869927\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) 4.00000 0.852803
\(23\) 8.00000 1.66812 0.834058 − 0.551677i \(-0.186012\pi\)
0.834058 + 0.551677i \(0.186012\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −4.00000 −0.784465
\(27\) 0 0
\(28\) −2.00000 −0.377964
\(29\) −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
\(34\) −2.00000 −0.342997
\(35\) 2.00000 0.338062
\(36\) 0 0
\(37\) −12.0000 −1.97279 −0.986394 − 0.164399i \(-0.947432\pi\)
−0.986394 + 0.164399i \(0.947432\pi\)
\(38\) −8.00000 −1.29777
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) −10.0000 −1.56174 −0.780869 − 0.624695i \(-0.785223\pi\)
−0.780869 + 0.624695i \(0.785223\pi\)
\(42\) 0 0
\(43\) 8.00000 1.21999 0.609994 − 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) 4.00000 0.603023
\(45\) 0 0
\(46\) 8.00000 1.17954
\(47\) 4.00000 0.583460 0.291730 − 0.956501i \(-0.405769\pi\)
0.291730 + 0.956501i \(0.405769\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) −4.00000 −0.554700
\(53\) −6.00000 −0.824163 −0.412082 − 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) −4.00000 −0.539360
\(56\) −2.00000 −0.267261
\(57\) 0 0
\(58\) −4.00000 −0.525226
\(59\) −2.00000 −0.260378 −0.130189 − 0.991489i \(-0.541558\pi\)
−0.130189 + 0.991489i \(0.541558\pi\)
\(60\) 0 0
\(61\) 10.0000 1.28037 0.640184 − 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) −1.00000 −0.127000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 4.00000 0.496139
\(66\) 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
\(71\) −6.00000 −0.712069 −0.356034 − 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) 0 0
\(73\) −4.00000 −0.468165 −0.234082 − 0.972217i \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) −12.0000 −1.39497
\(75\) 0 0
\(76\) −8.00000 −0.917663
\(77\) −8.00000 −0.911685
\(78\) 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
\(82\) −10.0000 −1.10432
\(83\) −4.00000 −0.439057 −0.219529 − 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) 2.00000 0.216930
\(86\) 8.00000 0.862662
\(87\) 0 0
\(88\) 4.00000 0.426401
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 8.00000 0.838628
\(92\) 8.00000 0.834058
\(93\) 0 0
\(94\) 4.00000 0.412568
\(95\) 8.00000 0.820783
\(96\) 0 0
\(97\) −18.0000 −1.82762 −0.913812 − 0.406138i \(-0.866875\pi\)
−0.913812 + 0.406138i \(0.866875\pi\)
\(98\) −3.00000 −0.303046
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 18.0000 1.79107 0.895533 − 0.444994i \(-0.146794\pi\)
0.895533 + 0.444994i \(0.146794\pi\)
\(102\) 0 0
\(103\) −14.0000 −1.37946 −0.689730 − 0.724066i \(-0.742271\pi\)
−0.689730 + 0.724066i \(0.742271\pi\)
\(104\) −4.00000 −0.392232
\(105\) 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
\(109\) 18.0000 1.72409 0.862044 − 0.506834i \(-0.169184\pi\)
0.862044 + 0.506834i \(0.169184\pi\)
\(110\) −4.00000 −0.381385
\(111\) 0 0
\(112\) −2.00000 −0.188982
\(113\) 6.00000 0.564433 0.282216 − 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 0 0
\(115\) −8.00000 −0.746004
\(116\) −4.00000 −0.371391
\(117\) 0 0
\(118\) −2.00000 −0.184115
\(119\) 4.00000 0.366679
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 10.0000 0.905357
\(123\) 0 0
\(124\) −1.00000 −0.0898027
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 4.00000 0.354943 0.177471 − 0.984126i \(-0.443208\pi\)
0.177471 + 0.984126i \(0.443208\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 4.00000 0.350823
\(131\) −10.0000 −0.873704 −0.436852 − 0.899533i \(-0.643907\pi\)
−0.436852 + 0.899533i \(0.643907\pi\)
\(132\) 0 0
\(133\) 16.0000 1.38738
\(134\) −6.00000 −0.518321
\(135\) 0 0
\(136\) −2.00000 −0.171499
\(137\) −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
\(142\) −6.00000 −0.503509
\(143\) −16.0000 −1.33799
\(144\) 0 0
\(145\) 4.00000 0.332182
\(146\) −4.00000 −0.331042
\(147\) 0 0
\(148\) −12.0000 −0.986394
\(149\) 18.0000 1.47462 0.737309 − 0.675556i \(-0.236096\pi\)
0.737309 + 0.675556i \(0.236096\pi\)
\(150\) 0 0
\(151\) −16.0000 −1.30206 −0.651031 − 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) −8.00000 −0.648886
\(153\) 0 0
\(154\) −8.00000 −0.644658
\(155\) 1.00000 0.0803219
\(156\) 0 0
\(157\) −22.0000 −1.75579 −0.877896 − 0.478852i \(-0.841053\pi\)
−0.877896 + 0.478852i \(0.841053\pi\)
\(158\) −8.00000 −0.636446
\(159\) 0 0
\(160\) −1.00000 −0.0790569
\(161\) −16.0000 −1.26098
\(162\) 0 0
\(163\) −6.00000 −0.469956 −0.234978 − 0.972001i \(-0.575502\pi\)
−0.234978 + 0.972001i \(0.575502\pi\)
\(164\) −10.0000 −0.780869
\(165\) 0 0
\(166\) −4.00000 −0.310460
\(167\) 8.00000 0.619059 0.309529 − 0.950890i \(-0.399829\pi\)
0.309529 + 0.950890i \(0.399829\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) 2.00000 0.153393
\(171\) 0 0
\(172\) 8.00000 0.609994
\(173\) −22.0000 −1.67263 −0.836315 − 0.548250i \(-0.815294\pi\)
−0.836315 + 0.548250i \(0.815294\pi\)
\(174\) 0 0
\(175\) −2.00000 −0.151186
\(176\) 4.00000 0.301511
\(177\) 0 0
\(178\) 0 0
\(179\) 20.0000 1.49487 0.747435 − 0.664335i \(-0.231285\pi\)
0.747435 + 0.664335i \(0.231285\pi\)
\(180\) 0 0
\(181\) 18.0000 1.33793 0.668965 − 0.743294i \(-0.266738\pi\)
0.668965 + 0.743294i \(0.266738\pi\)
\(182\) 8.00000 0.592999
\(183\) 0 0
\(184\) 8.00000 0.589768
\(185\) 12.0000 0.882258
\(186\) 0 0
\(187\) −8.00000 −0.585018
\(188\) 4.00000 0.291730
\(189\) 0 0
\(190\) 8.00000 0.580381
\(191\) −18.0000 −1.30243 −0.651217 − 0.758891i \(-0.725741\pi\)
−0.651217 + 0.758891i \(0.725741\pi\)
\(192\) 0 0
\(193\) −2.00000 −0.143963 −0.0719816 − 0.997406i \(-0.522932\pi\)
−0.0719816 + 0.997406i \(0.522932\pi\)
\(194\) −18.0000 −1.29232
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) −10.0000 −0.712470 −0.356235 − 0.934396i \(-0.615940\pi\)
−0.356235 + 0.934396i \(0.615940\pi\)
\(198\) 0 0
\(199\) 16.0000 1.13421 0.567105 − 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) 1.00000 0.0707107
\(201\) 0 0
\(202\) 18.0000 1.26648
\(203\) 8.00000 0.561490
\(204\) 0 0
\(205\) 10.0000 0.698430
\(206\) −14.0000 −0.975426
\(207\) 0 0
\(208\) −4.00000 −0.277350
\(209\) −32.0000 −2.21349
\(210\) 0 0
\(211\) 24.0000 1.65223 0.826114 − 0.563503i \(-0.190547\pi\)
0.826114 + 0.563503i \(0.190547\pi\)
\(212\) −6.00000 −0.412082
\(213\) 0 0
\(214\) −8.00000 −0.546869
\(215\) −8.00000 −0.545595
\(216\) 0 0
\(217\) 2.00000 0.135769
\(218\) 18.0000 1.21911
\(219\) 0 0
\(220\) −4.00000 −0.269680
\(221\) 8.00000 0.538138
\(222\) 0 0
\(223\) −8.00000 −0.535720 −0.267860 − 0.963458i \(-0.586316\pi\)
−0.267860 + 0.963458i \(0.586316\pi\)
\(224\) −2.00000 −0.133631
\(225\) 0 0
\(226\) 6.00000 0.399114
\(227\) −4.00000 −0.265489 −0.132745 − 0.991150i \(-0.542379\pi\)
−0.132745 + 0.991150i \(0.542379\pi\)
\(228\) 0 0
\(229\) 22.0000 1.45380 0.726900 − 0.686743i \(-0.240960\pi\)
0.726900 + 0.686743i \(0.240960\pi\)
\(230\) −8.00000 −0.527504
\(231\) 0 0
\(232\) −4.00000 −0.262613
\(233\) 10.0000 0.655122 0.327561 − 0.944830i \(-0.393773\pi\)
0.327561 + 0.944830i \(0.393773\pi\)
\(234\) 0 0
\(235\) −4.00000 −0.260931
\(236\) −2.00000 −0.130189
\(237\) 0 0
\(238\) 4.00000 0.259281
\(239\) 4.00000 0.258738 0.129369 − 0.991596i \(-0.458705\pi\)
0.129369 + 0.991596i \(0.458705\pi\)
\(240\) 0 0
\(241\) 10.0000 0.644157 0.322078 − 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) 5.00000 0.321412
\(243\) 0 0
\(244\) 10.0000 0.640184
\(245\) 3.00000 0.191663
\(246\) 0 0
\(247\) 32.0000 2.03611
\(248\) −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
\(253\) 32.0000 2.01182
\(254\) 4.00000 0.250982
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 18.0000 1.12281 0.561405 − 0.827541i \(-0.310261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(258\) 0 0
\(259\) 24.0000 1.49129
\(260\) 4.00000 0.248069
\(261\) 0 0
\(262\) −10.0000 −0.617802
\(263\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) 6.00000 0.368577
\(266\) 16.0000 0.981023
\(267\) 0 0
\(268\) −6.00000 −0.366508
\(269\) 12.0000 0.731653 0.365826 − 0.930683i \(-0.380786\pi\)
0.365826 + 0.930683i \(0.380786\pi\)
\(270\) 0 0
\(271\) 8.00000 0.485965 0.242983 − 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) −2.00000 −0.121268
\(273\) 0 0
\(274\) −6.00000 −0.362473
\(275\) 4.00000 0.241209
\(276\) 0 0
\(277\) 8.00000 0.480673 0.240337 − 0.970690i \(-0.422742\pi\)
0.240337 + 0.970690i \(0.422742\pi\)
\(278\) 4.00000 0.239904
\(279\) 0 0
\(280\) 2.00000 0.119523
\(281\) −6.00000 −0.357930 −0.178965 − 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) 0 0
\(283\) 2.00000 0.118888 0.0594438 − 0.998232i \(-0.481067\pi\)
0.0594438 + 0.998232i \(0.481067\pi\)
\(284\) −6.00000 −0.356034
\(285\) 0 0
\(286\) −16.0000 −0.946100
\(287\) 20.0000 1.18056
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 4.00000 0.234888
\(291\) 0 0
\(292\) −4.00000 −0.234082
\(293\) 6.00000 0.350524 0.175262 − 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) 0 0
\(295\) 2.00000 0.116445
\(296\) −12.0000 −0.697486
\(297\) 0 0
\(298\) 18.0000 1.04271
\(299\) −32.0000 −1.85061
\(300\) 0 0
\(301\) −16.0000 −0.922225
\(302\) −16.0000 −0.920697
\(303\) 0 0
\(304\) −8.00000 −0.458831
\(305\) −10.0000 −0.572598
\(306\) 0 0
\(307\) −2.00000 −0.114146 −0.0570730 − 0.998370i \(-0.518177\pi\)
−0.0570730 + 0.998370i \(0.518177\pi\)
\(308\) −8.00000 −0.455842
\(309\) 0 0
\(310\) 1.00000 0.0567962
\(311\) 30.0000 1.70114 0.850572 − 0.525859i \(-0.176256\pi\)
0.850572 + 0.525859i \(0.176256\pi\)
\(312\) 0 0
\(313\) 20.0000 1.13047 0.565233 − 0.824931i \(-0.308786\pi\)
0.565233 + 0.824931i \(0.308786\pi\)
\(314\) −22.0000 −1.24153
\(315\) 0 0
\(316\) −8.00000 −0.450035
\(317\) 22.0000 1.23564 0.617822 − 0.786318i \(-0.288015\pi\)
0.617822 + 0.786318i \(0.288015\pi\)
\(318\) 0 0
\(319\) −16.0000 −0.895828
\(320\) −1.00000 −0.0559017
\(321\) 0 0
\(322\) −16.0000 −0.891645
\(323\) 16.0000 0.890264
\(324\) 0 0
\(325\) −4.00000 −0.221880
\(326\) −6.00000 −0.332309
\(327\) 0 0
\(328\) −10.0000 −0.552158
\(329\) −8.00000 −0.441054
\(330\) 0 0
\(331\) −12.0000 −0.659580 −0.329790 − 0.944054i \(-0.606978\pi\)
−0.329790 + 0.944054i \(0.606978\pi\)
\(332\) −4.00000 −0.219529
\(333\) 0 0
\(334\) 8.00000 0.437741
\(335\) 6.00000 0.327815
\(336\) 0 0
\(337\) −32.0000 −1.74315 −0.871576 − 0.490261i \(-0.836901\pi\)
−0.871576 + 0.490261i \(0.836901\pi\)
\(338\) 3.00000 0.163178
\(339\) 0 0
\(340\) 2.00000 0.108465
\(341\) −4.00000 −0.216612
\(342\) 0 0
\(343\) 20.0000 1.07990
\(344\) 8.00000 0.431331
\(345\) 0 0
\(346\) −22.0000 −1.18273
\(347\) 4.00000 0.214731 0.107366 − 0.994220i \(-0.465758\pi\)
0.107366 + 0.994220i \(0.465758\pi\)
\(348\) 0 0
\(349\) −30.0000 −1.60586 −0.802932 − 0.596071i \(-0.796728\pi\)
−0.802932 + 0.596071i \(0.796728\pi\)
\(350\) −2.00000 −0.106904
\(351\) 0 0
\(352\) 4.00000 0.213201
\(353\) −34.0000 −1.80964 −0.904819 − 0.425797i \(-0.859994\pi\)
−0.904819 + 0.425797i \(0.859994\pi\)
\(354\) 0 0
\(355\) 6.00000 0.318447
\(356\) 0 0
\(357\) 0 0
\(358\) 20.0000 1.05703
\(359\) −10.0000 −0.527780 −0.263890 − 0.964553i \(-0.585006\pi\)
−0.263890 + 0.964553i \(0.585006\pi\)
\(360\) 0 0
\(361\) 45.0000 2.36842
\(362\) 18.0000 0.946059
\(363\) 0 0
\(364\) 8.00000 0.419314
\(365\) 4.00000 0.209370
\(366\) 0 0
\(367\) −8.00000 −0.417597 −0.208798 − 0.977959i \(-0.566955\pi\)
−0.208798 + 0.977959i \(0.566955\pi\)
\(368\) 8.00000 0.417029
\(369\) 0 0
\(370\) 12.0000 0.623850
\(371\) 12.0000 0.623009
\(372\) 0 0
\(373\) 34.0000 1.76045 0.880227 − 0.474554i \(-0.157390\pi\)
0.880227 + 0.474554i \(0.157390\pi\)
\(374\) −8.00000 −0.413670
\(375\) 0 0
\(376\) 4.00000 0.206284
\(377\) 16.0000 0.824042
\(378\) 0 0
\(379\) 16.0000 0.821865 0.410932 − 0.911666i \(-0.365203\pi\)
0.410932 + 0.911666i \(0.365203\pi\)
\(380\) 8.00000 0.410391
\(381\) 0 0
\(382\) −18.0000 −0.920960
\(383\) 16.0000 0.817562 0.408781 − 0.912633i \(-0.365954\pi\)
0.408781 + 0.912633i \(0.365954\pi\)
\(384\) 0 0
\(385\) 8.00000 0.407718
\(386\) −2.00000 −0.101797
\(387\) 0 0
\(388\) −18.0000 −0.913812
\(389\) 28.0000 1.41966 0.709828 − 0.704375i \(-0.248773\pi\)
0.709828 + 0.704375i \(0.248773\pi\)
\(390\) 0 0
\(391\) −16.0000 −0.809155
\(392\) −3.00000 −0.151523
\(393\) 0 0
\(394\) −10.0000 −0.503793
\(395\) 8.00000 0.402524
\(396\) 0 0
\(397\) −34.0000 −1.70641 −0.853206 − 0.521575i \(-0.825345\pi\)
−0.853206 + 0.521575i \(0.825345\pi\)
\(398\) 16.0000 0.802008
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −4.00000 −0.199750 −0.0998752 − 0.995000i \(-0.531844\pi\)
−0.0998752 + 0.995000i \(0.531844\pi\)
\(402\) 0 0
\(403\) 4.00000 0.199254
\(404\) 18.0000 0.895533
\(405\) 0 0
\(406\) 8.00000 0.397033
\(407\) −48.0000 −2.37927
\(408\) 0 0
\(409\) 6.00000 0.296681 0.148340 − 0.988936i \(-0.452607\pi\)
0.148340 + 0.988936i \(0.452607\pi\)
\(410\) 10.0000 0.493865
\(411\) 0 0
\(412\) −14.0000 −0.689730
\(413\) 4.00000 0.196827
\(414\) 0 0
\(415\) 4.00000 0.196352
\(416\) −4.00000 −0.196116
\(417\) 0 0
\(418\) −32.0000 −1.56517
\(419\) 10.0000 0.488532 0.244266 − 0.969708i \(-0.421453\pi\)
0.244266 + 0.969708i \(0.421453\pi\)
\(420\) 0 0
\(421\) 2.00000 0.0974740 0.0487370 − 0.998812i \(-0.484480\pi\)
0.0487370 + 0.998812i \(0.484480\pi\)
\(422\) 24.0000 1.16830
\(423\) 0 0
\(424\) −6.00000 −0.291386
\(425\) −2.00000 −0.0970143
\(426\) 0 0
\(427\) −20.0000 −0.967868
\(428\) −8.00000 −0.386695
\(429\) 0 0
\(430\) −8.00000 −0.385794
\(431\) 30.0000 1.44505 0.722525 − 0.691345i \(-0.242982\pi\)
0.722525 + 0.691345i \(0.242982\pi\)
\(432\) 0 0
\(433\) 4.00000 0.192228 0.0961139 − 0.995370i \(-0.469359\pi\)
0.0961139 + 0.995370i \(0.469359\pi\)
\(434\) 2.00000 0.0960031
\(435\) 0 0
\(436\) 18.0000 0.862044
\(437\) −64.0000 −3.06154
\(438\) 0 0
\(439\) −40.0000 −1.90910 −0.954548 − 0.298057i \(-0.903661\pi\)
−0.954548 + 0.298057i \(0.903661\pi\)
\(440\) −4.00000 −0.190693
\(441\) 0 0
\(442\) 8.00000 0.380521
\(443\) −4.00000 −0.190046 −0.0950229 − 0.995475i \(-0.530292\pi\)
−0.0950229 + 0.995475i \(0.530292\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −8.00000 −0.378811
\(447\) 0 0
\(448\) −2.00000 −0.0944911
\(449\) −28.0000 −1.32140 −0.660701 − 0.750649i \(-0.729741\pi\)
−0.660701 + 0.750649i \(0.729741\pi\)
\(450\) 0 0
\(451\) −40.0000 −1.88353
\(452\) 6.00000 0.282216
\(453\) 0 0
\(454\) −4.00000 −0.187729
\(455\) −8.00000 −0.375046
\(456\) 0 0
\(457\) 36.0000 1.68401 0.842004 − 0.539471i \(-0.181376\pi\)
0.842004 + 0.539471i \(0.181376\pi\)
\(458\) 22.0000 1.02799
\(459\) 0 0
\(460\) −8.00000 −0.373002
\(461\) 40.0000 1.86299 0.931493 − 0.363760i \(-0.118507\pi\)
0.931493 + 0.363760i \(0.118507\pi\)
\(462\) 0 0
\(463\) −36.0000 −1.67306 −0.836531 − 0.547920i \(-0.815420\pi\)
−0.836531 + 0.547920i \(0.815420\pi\)
\(464\) −4.00000 −0.185695
\(465\) 0 0
\(466\) 10.0000 0.463241
\(467\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 0 0
\(469\) 12.0000 0.554109
\(470\) −4.00000 −0.184506
\(471\) 0 0
\(472\) −2.00000 −0.0920575
\(473\) 32.0000 1.47136
\(474\) 0 0
\(475\) −8.00000 −0.367065
\(476\) 4.00000 0.183340
\(477\) 0 0
\(478\) 4.00000 0.182956
\(479\) 10.0000 0.456912 0.228456 − 0.973554i \(-0.426632\pi\)
0.228456 + 0.973554i \(0.426632\pi\)
\(480\) 0 0
\(481\) 48.0000 2.18861
\(482\) 10.0000 0.455488
\(483\) 0 0
\(484\) 5.00000 0.227273
\(485\) 18.0000 0.817338
\(486\) 0 0
\(487\) −12.0000 −0.543772 −0.271886 − 0.962329i \(-0.587647\pi\)
−0.271886 + 0.962329i \(0.587647\pi\)
\(488\) 10.0000 0.452679
\(489\) 0 0
\(490\) 3.00000 0.135526
\(491\) 8.00000 0.361035 0.180517 − 0.983572i \(-0.442223\pi\)
0.180517 + 0.983572i \(0.442223\pi\)
\(492\) 0 0
\(493\) 8.00000 0.360302
\(494\) 32.0000 1.43975
\(495\) 0 0
\(496\) −1.00000 −0.0449013
\(497\) 12.0000 0.538274
\(498\) 0 0
\(499\) −36.0000 −1.61158 −0.805791 − 0.592200i \(-0.798259\pi\)
−0.805791 + 0.592200i \(0.798259\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 0 0
\(502\) −20.0000 −0.892644
\(503\) −12.0000 −0.535054 −0.267527 − 0.963550i \(-0.586206\pi\)
−0.267527 + 0.963550i \(0.586206\pi\)
\(504\) 0 0
\(505\) −18.0000 −0.800989
\(506\) 32.0000 1.42257
\(507\) 0 0
\(508\) 4.00000 0.177471
\(509\) −16.0000 −0.709188 −0.354594 − 0.935020i \(-0.615381\pi\)
−0.354594 + 0.935020i \(0.615381\pi\)
\(510\) 0 0
\(511\) 8.00000 0.353899
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 18.0000 0.793946
\(515\) 14.0000 0.616914
\(516\) 0 0
\(517\) 16.0000 0.703679
\(518\) 24.0000 1.05450
\(519\) 0 0
\(520\) 4.00000 0.175412
\(521\) −18.0000 −0.788594 −0.394297 − 0.918983i \(-0.629012\pi\)
−0.394297 + 0.918983i \(0.629012\pi\)
\(522\) 0 0
\(523\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(524\) −10.0000 −0.436852
\(525\) 0 0
\(526\) 0 0
\(527\) 2.00000 0.0871214
\(528\) 0 0
\(529\) 41.0000 1.78261
\(530\) 6.00000 0.260623
\(531\) 0 0
\(532\) 16.0000 0.693688
\(533\) 40.0000 1.73259
\(534\) 0 0
\(535\) 8.00000 0.345870
\(536\) −6.00000 −0.259161
\(537\) 0 0
\(538\) 12.0000 0.517357
\(539\) −12.0000 −0.516877
\(540\) 0 0
\(541\) 2.00000 0.0859867 0.0429934 − 0.999075i \(-0.486311\pi\)
0.0429934 + 0.999075i \(0.486311\pi\)
\(542\) 8.00000 0.343629
\(543\) 0 0
\(544\) −2.00000 −0.0857493
\(545\) −18.0000 −0.771035
\(546\) 0 0
\(547\) 2.00000 0.0855138 0.0427569 − 0.999086i \(-0.486386\pi\)
0.0427569 + 0.999086i \(0.486386\pi\)
\(548\) −6.00000 −0.256307
\(549\) 0 0
\(550\) 4.00000 0.170561
\(551\) 32.0000 1.36325
\(552\) 0 0
\(553\) 16.0000 0.680389
\(554\) 8.00000 0.339887
\(555\) 0 0
\(556\) 4.00000 0.169638
\(557\) −6.00000 −0.254228 −0.127114 − 0.991888i \(-0.540571\pi\)
−0.127114 + 0.991888i \(0.540571\pi\)
\(558\) 0 0
\(559\) −32.0000 −1.35346
\(560\) 2.00000 0.0845154
\(561\) 0 0
\(562\) −6.00000 −0.253095
\(563\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) 0 0
\(565\) −6.00000 −0.252422
\(566\) 2.00000 0.0840663
\(567\) 0 0
\(568\) −6.00000 −0.251754
\(569\) 28.0000 1.17382 0.586911 − 0.809652i \(-0.300344\pi\)
0.586911 + 0.809652i \(0.300344\pi\)
\(570\) 0 0
\(571\) −12.0000 −0.502184 −0.251092 − 0.967963i \(-0.580790\pi\)
−0.251092 + 0.967963i \(0.580790\pi\)
\(572\) −16.0000 −0.668994
\(573\) 0 0
\(574\) 20.0000 0.834784
\(575\) 8.00000 0.333623
\(576\) 0 0
\(577\) −6.00000 −0.249783 −0.124892 − 0.992170i \(-0.539858\pi\)
−0.124892 + 0.992170i \(0.539858\pi\)
\(578\) −13.0000 −0.540729
\(579\) 0 0
\(580\) 4.00000 0.166091
\(581\) 8.00000 0.331896
\(582\) 0 0
\(583\) −24.0000 −0.993978
\(584\) −4.00000 −0.165521
\(585\) 0 0
\(586\) 6.00000 0.247858
\(587\) −28.0000 −1.15568 −0.577842 − 0.816149i \(-0.696105\pi\)
−0.577842 + 0.816149i \(0.696105\pi\)
\(588\) 0 0
\(589\) 8.00000 0.329634
\(590\) 2.00000 0.0823387
\(591\) 0 0
\(592\) −12.0000 −0.493197
\(593\) 30.0000 1.23195 0.615976 − 0.787765i \(-0.288762\pi\)
0.615976 + 0.787765i \(0.288762\pi\)
\(594\) 0 0
\(595\) −4.00000 −0.163984
\(596\) 18.0000 0.737309
\(597\) 0 0
\(598\) −32.0000 −1.30858
\(599\) 30.0000 1.22577 0.612883 − 0.790173i \(-0.290010\pi\)
0.612883 + 0.790173i \(0.290010\pi\)
\(600\) 0 0
\(601\) −26.0000 −1.06056 −0.530281 − 0.847822i \(-0.677914\pi\)
−0.530281 + 0.847822i \(0.677914\pi\)
\(602\) −16.0000 −0.652111
\(603\) 0 0
\(604\) −16.0000 −0.651031
\(605\) −5.00000 −0.203279
\(606\) 0 0
\(607\) −14.0000 −0.568242 −0.284121 − 0.958788i \(-0.591702\pi\)
−0.284121 + 0.958788i \(0.591702\pi\)
\(608\) −8.00000 −0.324443
\(609\) 0 0
\(610\) −10.0000 −0.404888
\(611\) −16.0000 −0.647291
\(612\) 0 0
\(613\) 8.00000 0.323117 0.161558 − 0.986863i \(-0.448348\pi\)
0.161558 + 0.986863i \(0.448348\pi\)
\(614\) −2.00000 −0.0807134
\(615\) 0 0
\(616\) −8.00000 −0.322329
\(617\) −42.0000 −1.69086 −0.845428 − 0.534089i \(-0.820655\pi\)
−0.845428 + 0.534089i \(0.820655\pi\)
\(618\) 0 0
\(619\) −20.0000 −0.803868 −0.401934 − 0.915669i \(-0.631662\pi\)
−0.401934 + 0.915669i \(0.631662\pi\)
\(620\) 1.00000 0.0401610
\(621\) 0 0
\(622\) 30.0000 1.20289
\(623\) 0 0
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 20.0000 0.799361
\(627\) 0 0
\(628\) −22.0000 −0.877896
\(629\) 24.0000 0.956943
\(630\) 0 0
\(631\) 8.00000 0.318475 0.159237 − 0.987240i \(-0.449096\pi\)
0.159237 + 0.987240i \(0.449096\pi\)
\(632\) −8.00000 −0.318223
\(633\) 0 0
\(634\) 22.0000 0.873732
\(635\) −4.00000 −0.158735
\(636\) 0 0
\(637\) 12.0000 0.475457
\(638\) −16.0000 −0.633446
\(639\) 0 0
\(640\) −1.00000 −0.0395285
\(641\) 4.00000 0.157991 0.0789953 − 0.996875i \(-0.474829\pi\)
0.0789953 + 0.996875i \(0.474829\pi\)
\(642\) 0 0
\(643\) −8.00000 −0.315489 −0.157745 − 0.987480i \(-0.550422\pi\)
−0.157745 + 0.987480i \(0.550422\pi\)
\(644\) −16.0000 −0.630488
\(645\) 0 0
\(646\) 16.0000 0.629512
\(647\) −8.00000 −0.314512 −0.157256 − 0.987558i \(-0.550265\pi\)
−0.157256 + 0.987558i \(0.550265\pi\)
\(648\) 0 0
\(649\) −8.00000 −0.314027
\(650\) −4.00000 −0.156893
\(651\) 0 0
\(652\) −6.00000 −0.234978
\(653\) 6.00000 0.234798 0.117399 − 0.993085i \(-0.462544\pi\)
0.117399 + 0.993085i \(0.462544\pi\)
\(654\) 0 0
\(655\) 10.0000 0.390732
\(656\) −10.0000 −0.390434
\(657\) 0 0
\(658\) −8.00000 −0.311872
\(659\) 2.00000 0.0779089 0.0389545 − 0.999241i \(-0.487597\pi\)
0.0389545 + 0.999241i \(0.487597\pi\)
\(660\) 0 0
\(661\) 46.0000 1.78919 0.894596 − 0.446875i \(-0.147463\pi\)
0.894596 + 0.446875i \(0.147463\pi\)
\(662\) −12.0000 −0.466393
\(663\) 0 0
\(664\) −4.00000 −0.155230
\(665\) −16.0000 −0.620453
\(666\) 0 0
\(667\) −32.0000 −1.23904
\(668\) 8.00000 0.309529
\(669\) 0 0
\(670\) 6.00000 0.231800
\(671\) 40.0000 1.54418
\(672\) 0 0
\(673\) 8.00000 0.308377 0.154189 − 0.988041i \(-0.450724\pi\)
0.154189 + 0.988041i \(0.450724\pi\)
\(674\) −32.0000 −1.23259
\(675\) 0 0
\(676\) 3.00000 0.115385
\(677\) −6.00000 −0.230599 −0.115299 − 0.993331i \(-0.536783\pi\)
−0.115299 + 0.993331i \(0.536783\pi\)
\(678\) 0 0
\(679\) 36.0000 1.38155
\(680\) 2.00000 0.0766965
\(681\) 0 0
\(682\) −4.00000 −0.153168
\(683\) −4.00000 −0.153056 −0.0765279 − 0.997067i \(-0.524383\pi\)
−0.0765279 + 0.997067i \(0.524383\pi\)
\(684\) 0 0
\(685\) 6.00000 0.229248
\(686\) 20.0000 0.763604
\(687\) 0 0
\(688\) 8.00000 0.304997
\(689\) 24.0000 0.914327
\(690\) 0 0
\(691\) 28.0000 1.06517 0.532585 − 0.846376i \(-0.321221\pi\)
0.532585 + 0.846376i \(0.321221\pi\)
\(692\) −22.0000 −0.836315
\(693\) 0 0
\(694\) 4.00000 0.151838
\(695\) −4.00000 −0.151729
\(696\) 0 0
\(697\) 20.0000 0.757554
\(698\) −30.0000 −1.13552
\(699\) 0 0
\(700\) −2.00000 −0.0755929
\(701\) −2.00000 −0.0755390 −0.0377695 − 0.999286i \(-0.512025\pi\)
−0.0377695 + 0.999286i \(0.512025\pi\)
\(702\) 0 0
\(703\) 96.0000 3.62071
\(704\) 4.00000 0.150756
\(705\) 0 0
\(706\) −34.0000 −1.27961
\(707\) −36.0000 −1.35392
\(708\) 0 0
\(709\) −42.0000 −1.57734 −0.788672 − 0.614815i \(-0.789231\pi\)
−0.788672 + 0.614815i \(0.789231\pi\)
\(710\) 6.00000 0.225176
\(711\) 0 0
\(712\) 0 0
\(713\) −8.00000 −0.299602
\(714\) 0 0
\(715\) 16.0000 0.598366
\(716\) 20.0000 0.747435
\(717\) 0 0
\(718\) −10.0000 −0.373197
\(719\) 24.0000 0.895049 0.447524 − 0.894272i \(-0.352306\pi\)
0.447524 + 0.894272i \(0.352306\pi\)
\(720\) 0 0
\(721\) 28.0000 1.04277
\(722\) 45.0000 1.67473
\(723\) 0 0
\(724\) 18.0000 0.668965
\(725\) −4.00000 −0.148556
\(726\) 0 0
\(727\) −2.00000 −0.0741759 −0.0370879 − 0.999312i \(-0.511808\pi\)
−0.0370879 + 0.999312i \(0.511808\pi\)
\(728\) 8.00000 0.296500
\(729\) 0 0
\(730\) 4.00000 0.148047
\(731\) −16.0000 −0.591781
\(732\) 0 0
\(733\) 6.00000 0.221615 0.110808 − 0.993842i \(-0.464656\pi\)
0.110808 + 0.993842i \(0.464656\pi\)
\(734\) −8.00000 −0.295285
\(735\) 0 0
\(736\) 8.00000 0.294884
\(737\) −24.0000 −0.884051
\(738\) 0 0
\(739\) 20.0000 0.735712 0.367856 − 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(740\) 12.0000 0.441129
\(741\) 0 0
\(742\) 12.0000 0.440534
\(743\) 16.0000 0.586983 0.293492 − 0.955962i \(-0.405183\pi\)
0.293492 + 0.955962i \(0.405183\pi\)
\(744\) 0 0
\(745\) −18.0000 −0.659469
\(746\) 34.0000 1.24483
\(747\) 0 0
\(748\) −8.00000 −0.292509
\(749\) 16.0000 0.584627
\(750\) 0 0
\(751\) 16.0000 0.583848 0.291924 − 0.956441i \(-0.405705\pi\)
0.291924 + 0.956441i \(0.405705\pi\)
\(752\) 4.00000 0.145865
\(753\) 0 0
\(754\) 16.0000 0.582686
\(755\) 16.0000 0.582300
\(756\) 0 0
\(757\) 36.0000 1.30844 0.654221 − 0.756303i \(-0.272997\pi\)
0.654221 + 0.756303i \(0.272997\pi\)
\(758\) 16.0000 0.581146
\(759\) 0 0
\(760\) 8.00000 0.290191
\(761\) 20.0000 0.724999 0.362500 − 0.931984i \(-0.381923\pi\)
0.362500 + 0.931984i \(0.381923\pi\)
\(762\) 0 0
\(763\) −36.0000 −1.30329
\(764\) −18.0000 −0.651217
\(765\) 0 0
\(766\) 16.0000 0.578103
\(767\) 8.00000 0.288863
\(768\) 0 0
\(769\) −34.0000 −1.22607 −0.613036 − 0.790055i \(-0.710052\pi\)
−0.613036 + 0.790055i \(0.710052\pi\)
\(770\) 8.00000 0.288300
\(771\) 0 0
\(772\) −2.00000 −0.0719816
\(773\) −14.0000 −0.503545 −0.251773 − 0.967786i \(-0.581013\pi\)
−0.251773 + 0.967786i \(0.581013\pi\)
\(774\) 0 0
\(775\) −1.00000 −0.0359211
\(776\) −18.0000 −0.646162
\(777\) 0 0
\(778\) 28.0000 1.00385
\(779\) 80.0000 2.86630
\(780\) 0 0
\(781\) −24.0000 −0.858788
\(782\) −16.0000 −0.572159
\(783\) 0 0
\(784\) −3.00000 −0.107143
\(785\) 22.0000 0.785214
\(786\) 0 0
\(787\) 28.0000 0.998092 0.499046 − 0.866575i \(-0.333684\pi\)
0.499046 + 0.866575i \(0.333684\pi\)
\(788\) −10.0000 −0.356235
\(789\) 0 0
\(790\) 8.00000 0.284627
\(791\) −12.0000 −0.426671
\(792\) 0 0
\(793\) −40.0000 −1.42044
\(794\) −34.0000 −1.20661
\(795\) 0 0
\(796\) 16.0000 0.567105
\(797\) 30.0000 1.06265 0.531327 − 0.847167i \(-0.321693\pi\)
0.531327 + 0.847167i \(0.321693\pi\)
\(798\) 0 0
\(799\) −8.00000 −0.283020
\(800\) 1.00000 0.0353553
\(801\) 0 0
\(802\) −4.00000 −0.141245
\(803\) −16.0000 −0.564628
\(804\) 0 0
\(805\) 16.0000 0.563926
\(806\) 4.00000 0.140894
\(807\) 0 0
\(808\) 18.0000 0.633238
\(809\) −28.0000 −0.984428 −0.492214 − 0.870474i \(-0.663812\pi\)
−0.492214 + 0.870474i \(0.663812\pi\)
\(810\) 0 0
\(811\) −16.0000 −0.561836 −0.280918 − 0.959732i \(-0.590639\pi\)
−0.280918 + 0.959732i \(0.590639\pi\)
\(812\) 8.00000 0.280745
\(813\) 0 0
\(814\) −48.0000 −1.68240
\(815\) 6.00000 0.210171
\(816\) 0 0
\(817\) −64.0000 −2.23908
\(818\) 6.00000 0.209785
\(819\) 0 0
\(820\) 10.0000 0.349215
\(821\) 4.00000 0.139601 0.0698005 − 0.997561i \(-0.477764\pi\)
0.0698005 + 0.997561i \(0.477764\pi\)
\(822\) 0 0
\(823\) 28.0000 0.976019 0.488009 − 0.872838i \(-0.337723\pi\)
0.488009 + 0.872838i \(0.337723\pi\)
\(824\) −14.0000 −0.487713
\(825\) 0 0
\(826\) 4.00000 0.139178
\(827\) −36.0000 −1.25184 −0.625921 − 0.779886i \(-0.715277\pi\)
−0.625921 + 0.779886i \(0.715277\pi\)
\(828\) 0 0
\(829\) −46.0000 −1.59765 −0.798823 − 0.601566i \(-0.794544\pi\)
−0.798823 + 0.601566i \(0.794544\pi\)
\(830\) 4.00000 0.138842
\(831\) 0 0
\(832\) −4.00000 −0.138675
\(833\) 6.00000 0.207888
\(834\) 0 0
\(835\) −8.00000 −0.276851
\(836\) −32.0000 −1.10674
\(837\) 0 0
\(838\) 10.0000 0.345444
\(839\) −42.0000 −1.45000 −0.725001 − 0.688748i \(-0.758161\pi\)
−0.725001 + 0.688748i \(0.758161\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) 2.00000 0.0689246
\(843\) 0 0
\(844\) 24.0000 0.826114
\(845\) −3.00000 −0.103203
\(846\) 0 0
\(847\) −10.0000 −0.343604
\(848\) −6.00000 −0.206041
\(849\) 0 0
\(850\) −2.00000 −0.0685994
\(851\) −96.0000 −3.29084
\(852\) 0 0
\(853\) 14.0000 0.479351 0.239675 − 0.970853i \(-0.422959\pi\)
0.239675 + 0.970853i \(0.422959\pi\)
\(854\) −20.0000 −0.684386
\(855\) 0 0
\(856\) −8.00000 −0.273434
\(857\) −10.0000 −0.341593 −0.170797 − 0.985306i \(-0.554634\pi\)
−0.170797 + 0.985306i \(0.554634\pi\)
\(858\) 0 0
\(859\) −20.0000 −0.682391 −0.341196 − 0.939992i \(-0.610832\pi\)
−0.341196 + 0.939992i \(0.610832\pi\)
\(860\) −8.00000 −0.272798
\(861\) 0 0
\(862\) 30.0000 1.02180
\(863\) −16.0000 −0.544646 −0.272323 − 0.962206i \(-0.587792\pi\)
−0.272323 + 0.962206i \(0.587792\pi\)
\(864\) 0 0
\(865\) 22.0000 0.748022
\(866\) 4.00000 0.135926
\(867\) 0 0
\(868\) 2.00000 0.0678844
\(869\) −32.0000 −1.08553
\(870\) 0 0
\(871\) 24.0000 0.813209
\(872\) 18.0000 0.609557
\(873\) 0 0
\(874\) −64.0000 −2.16483
\(875\) 2.00000 0.0676123
\(876\) 0 0
\(877\) −2.00000 −0.0675352 −0.0337676 − 0.999430i \(-0.510751\pi\)
−0.0337676 + 0.999430i \(0.510751\pi\)
\(878\) −40.0000 −1.34993
\(879\) 0 0
\(880\) −4.00000 −0.134840
\(881\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) 20.0000 0.673054 0.336527 − 0.941674i \(-0.390748\pi\)
0.336527 + 0.941674i \(0.390748\pi\)
\(884\) 8.00000 0.269069
\(885\) 0 0
\(886\) −4.00000 −0.134383
\(887\) 56.0000 1.88030 0.940148 − 0.340766i \(-0.110687\pi\)
0.940148 + 0.340766i \(0.110687\pi\)
\(888\) 0 0
\(889\) −8.00000 −0.268311
\(890\) 0 0
\(891\) 0 0
\(892\) −8.00000 −0.267860
\(893\) −32.0000 −1.07084
\(894\) 0 0
\(895\) −20.0000 −0.668526
\(896\) −2.00000 −0.0668153
\(897\) 0 0
\(898\) −28.0000 −0.934372
\(899\) 4.00000 0.133407
\(900\) 0 0
\(901\) 12.0000 0.399778
\(902\) −40.0000 −1.33185
\(903\) 0 0
\(904\) 6.00000 0.199557
\(905\) −18.0000 −0.598340
\(906\) 0 0
\(907\) −10.0000 −0.332045 −0.166022 − 0.986122i \(-0.553092\pi\)
−0.166022 + 0.986122i \(0.553092\pi\)
\(908\) −4.00000 −0.132745
\(909\) 0 0
\(910\) −8.00000 −0.265197
\(911\) −24.0000 −0.795155 −0.397578 − 0.917568i \(-0.630149\pi\)
−0.397578 + 0.917568i \(0.630149\pi\)
\(912\) 0 0
\(913\) −16.0000 −0.529523
\(914\) 36.0000 1.19077
\(915\) 0 0
\(916\) 22.0000 0.726900
\(917\) 20.0000 0.660458
\(918\) 0 0
\(919\) 4.00000 0.131948 0.0659739 − 0.997821i \(-0.478985\pi\)
0.0659739 + 0.997821i \(0.478985\pi\)
\(920\) −8.00000 −0.263752
\(921\) 0 0
\(922\) 40.0000 1.31733
\(923\) 24.0000 0.789970
\(924\) 0 0
\(925\) −12.0000 −0.394558
\(926\) −36.0000 −1.18303
\(927\) 0 0
\(928\) −4.00000 −0.131306
\(929\) −32.0000 −1.04989 −0.524943 − 0.851137i \(-0.675913\pi\)
−0.524943 + 0.851137i \(0.675913\pi\)
\(930\) 0 0
\(931\) 24.0000 0.786568
\(932\) 10.0000 0.327561
\(933\) 0 0
\(934\) 0 0
\(935\) 8.00000 0.261628
\(936\) 0 0
\(937\) −2.00000 −0.0653372 −0.0326686 − 0.999466i \(-0.510401\pi\)
−0.0326686 + 0.999466i \(0.510401\pi\)
\(938\) 12.0000 0.391814
\(939\) 0 0
\(940\) −4.00000 −0.130466
\(941\) −12.0000 −0.391189 −0.195594 − 0.980685i \(-0.562664\pi\)
−0.195594 + 0.980685i \(0.562664\pi\)
\(942\) 0 0
\(943\) −80.0000 −2.60516
\(944\) −2.00000 −0.0650945
\(945\) 0 0
\(946\) 32.0000 1.04041
\(947\) −28.0000 −0.909878 −0.454939 − 0.890523i \(-0.650339\pi\)
−0.454939 + 0.890523i \(0.650339\pi\)
\(948\) 0 0
\(949\) 16.0000 0.519382
\(950\) −8.00000 −0.259554
\(951\) 0 0
\(952\) 4.00000 0.129641
\(953\) −22.0000 −0.712650 −0.356325 − 0.934362i \(-0.615970\pi\)
−0.356325 + 0.934362i \(0.615970\pi\)
\(954\) 0 0
\(955\) 18.0000 0.582466
\(956\) 4.00000 0.129369
\(957\) 0 0
\(958\) 10.0000 0.323085
\(959\) 12.0000 0.387500
\(960\) 0 0
\(961\) 1.00000 0.0322581
\(962\) 48.0000 1.54758
\(963\) 0 0
\(964\) 10.0000 0.322078
\(965\) 2.00000 0.0643823
\(966\) 0 0
\(967\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(968\) 5.00000 0.160706
\(969\) 0 0
\(970\) 18.0000 0.577945
\(971\) 2.00000 0.0641831 0.0320915 − 0.999485i \(-0.489783\pi\)
0.0320915 + 0.999485i \(0.489783\pi\)
\(972\) 0 0
\(973\) −8.00000 −0.256468
\(974\) −12.0000 −0.384505
\(975\) 0 0
\(976\) 10.0000 0.320092
\(977\) 46.0000 1.47167 0.735835 − 0.677161i \(-0.236790\pi\)
0.735835 + 0.677161i \(0.236790\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 3.00000 0.0958315
\(981\) 0 0
\(982\) 8.00000 0.255290
\(983\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) 0 0
\(985\) 10.0000 0.318626
\(986\) 8.00000 0.254772
\(987\) 0 0
\(988\) 32.0000 1.01806
\(989\) 64.0000 2.03508
\(990\) 0 0
\(991\) −56.0000 −1.77890 −0.889449 − 0.457034i \(-0.848912\pi\)
−0.889449 + 0.457034i \(0.848912\pi\)
\(992\) −1.00000 −0.0317500
\(993\) 0 0
\(994\) 12.0000 0.380617
\(995\) −16.0000 −0.507234
\(996\) 0 0
\(997\) 18.0000 0.570066 0.285033 − 0.958518i \(-0.407995\pi\)
0.285033 + 0.958518i \(0.407995\pi\)
\(998\) −36.0000 −1.13956
\(999\) 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.p.1.1 1
3.2 odd 2 930.2.a.h.1.1 ✓ 1
12.11 even 2 7440.2.a.m.1.1 1
15.2 even 4 4650.2.d.b.3349.1 2
15.8 even 4 4650.2.d.b.3349.2 2
15.14 odd 2 4650.2.a.bg.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
930.2.a.h.1.1 ✓ 1 3.2 odd 2
2790.2.a.p.1.1 1 1.1 even 1 trivial
4650.2.a.bg.1.1 1 15.14 odd 2
4650.2.d.b.3349.1 2 15.2 even 4
4650.2.d.b.3349.2 2 15.8 even 4
7440.2.a.m.1.1 1 12.11 even 2